mathoid-texvcjs

{ "00001453ad6771567b7ec0e7404a1e79": "P=3B_0\\left(\\frac{1-\\eta}{\\eta^2}\\right)e^{\\frac{3}{2}(B_0'-1)(1-\\eta)}", "0000239ab0143b8cd72151bf852d7af7": "\\beth_{d-1}(|\\alpha+\\omega|^{2^{\\aleph_0}})", "00004cd84d4d46d0e37b841cd7509c2c": "\\mathrm{REC}(N)", "00009654348eebd7ab85d8599c25aace": "W(2, k) > 2^k/k^\\varepsilon", "0000a3595ace35143948315a2841b307": "h=-1", "000138f6a2210ff1f9bb5eb7bc25ab6c": "(X,\\Sigma)", "0001e5b7a90547e1d7bc8be8a5c1e161": "1-\\left[\\frac{15}{16}\\right]^{16} \\,=\\, 64.39%", "00021ba3771fa2c4684c5639fecea94e": "\\tan\\frac{3\\pi}{20}=\\tan 27^\\circ=\\sqrt5-1-\\sqrt{5-2\\sqrt5}\\,", "00023b9224ac900169410ee72115cea4": "\n\\chi(T) = T^{2g} + a_1T^{2g-1} + \\cdots + a_gT^g + \\cdots + a_1q^{g-1}T + q^g,\n", "00029fdbca88b454dc6e742a8f404ca2": "(p-1)!^n", "0002a4a9343567b9a285b034b9a38ecb": "p = { E \\over c } = { hf \\over c } = { h \\over \\lambda }. ", "0002a95f3d21c0d5e5f455a832f2c17d": "\\psi\\to e^{i\\gamma_{d+1}\\alpha(x)}\\psi\\,", "0002cea3a95fae1835af3910d7ca6930": "e \\Delta\\rho \\simeq \\epsilon_0 k_0^2 \\Delta\\phi", "000303c8a822cfee55c1bd97c1d4cc4a": "f_c(z) = z^2 + c", "000334cb9f0bccc26284c6ef02725e06": "H \\rightarrow G/N \\times G'/N'", "0003cda16b8f0d055034e3c54846c175": "m_{\\text{o}}", "0003d3bfc208df07075efc742b3af376": "\\mathbf{J_2}", "0003ff41496b4d8a9a60cf3e03db80f2": "\\{ (p \\to q), (p \\to \\neg q) \\} \\vdash \\neg p", "00040a566d6ca57745bff5a2514f424c": "A_\\mu(x_i)", "00040baa2353e06d351f6c9dac889ece": " ds^2 = g_{00} \\, dt^2 + g_{jk} \\, dx^j \\, dx^k,\\;\\; j,\\; k \\in \\{1, 2, 3\\} ", "00047597e6585d2a8d77e2c4bb610401": "\n\\bar{h}(s,i;L)=\\prod_{c=1}^i\\sum_{k_c=2+k_{c-1}}^{L-1-2(i-c)}\\bar{f}_{k_c}(s)\n", "0004c246ad141d5412a457dc81323857": "H_1(\\mathrm{A}_3)\\cong H_1(\\mathrm{A}_4) \\cong \\mathrm{C}_3,", "000592a04b7c6c5cc9a9429a048b2757": "\n \\mu = 2C_1~\\sum_{i=1}^5 i\\,\\alpha_i~\\beta^{i-1}~I_1^{i-1} \\,.\n ", "0005a6b0b0b3be71744f935c4a5eeb3a": "f:\\mathcal{H}_g \\rightarrow V", "0005eff4a121d51b65af0ee36bc65e70": "q(\\mathbf{\\pi}) \\prod_{k=1}^K q(\\mathbf{\\mu}_k,\\mathbf{\\Lambda}_k)", "000643b3754284c8b2aeb53d4394f021": "(\\forall i\\in I) f[V_i]\\subseteq V_i", "0006b557602a072b21da57443b92f449": "254 = 2^8 - 2", "000723a6105c190f41462d560ad7458a": "R(X_1, \\ldots, X_{n})", "000736cda6b8807641f5244f27742f56": "\nP_{ij}(f)=\\frac{ A_{ij}(f)}\n{\\sqrt{\\mathbf{a}^{*}_j(f)\\mathbf{a}_j(f)}}\n", "00073e38a79657d8dfb58930122512ce": "A(x, y)\\,dx + B(x, y)\\,dy", "00078c12a085f724c262a7295f8d70b0": "\\frac{\\$\\text{40m}}{\\$\\text{30m}} = 1 \\frac{1}{3} \\approx 1.33", "00079c0fe89f86a710a201e0689b2172": "\\int u \\, dv=uv-\\int v \\, du.\\!", "0008510cb7881764a542e8502fc95b28": "\\Psi(w,v)=w^\\alpha \\cdot v = \\sum_{i=1}^n w_i^q v_i", "0008c41df7229f6c3753f8c45db87f04": "{f_x}(m)", "0008d640a21f52b6b7067d7b03547108": "v_i = \\frac{\\partial \\Phi}{\\partial x_i}", "000904ee9bee58b7b339bfe4b842e49a": "\\forall x \\, \\forall y \\, P(x,y) \\Leftrightarrow \\forall y \\, \\forall x \\, P(x,y)", "000931b2d65a0f6ce57156ed9e2f457e": "\\mathrm{resultant}(p, T)=0", "000945530b96364391c181a406d4fa29": "P(X_i=a)", "0009d412dbeb47c56fe78c99cfd4dc08": "p = c \\cdot u \\cdot \\rho", "0009d7ff4e372f215e5fc71b37a42038": "\\;^+R_{\\alpha \\beta} - {1 \\over 2} g_{\\alpha \\beta} \\;^+R = 0.", "000a91452ffe8335b67f0e5ff2c0a767": "\\textstyle P ( A \\Delta f^{-1}(B) ) = 0. ", "000ab33a85842800e48143f212ac5fc0": "p = 1\\; \\text{GeV}/c = \\frac{(1 \\times 10^{9}) \\cdot (1.60217646 \\times 10^{-19} \\; \\text{C}) \\cdot \\text{V}}{(2.99792458 \\times 10^{8}\\; \\text{m}/\\text{s})} = 5.344286 \\times 10^{-19}\\; \\text{kg}{\\cdot}\\text{m}/\\text{s}.", "000ad1eb8a2c2182ff048350cc9eb0e8": "\\alpha(x)", "000ae84c0190bb851b585c79e3b8449f": "\\,2", "000af2fae5bdfcd63e6dc3e5bce0dea3": "f^*(x^*) = \\sup_{x\\in X}(\\langle x^*,x\\rangle-f(x)),\\quad x^*\\in X^*", "000b1d2bea2949b83a2325c116ed0f04": "\\nabla T = \\omega\\otimes T. \\, ", "000b37155b94f927910c738a2cb82536": "f(\\lambda x + (1 - \\lambda)y)>\\min\\big(f(x),f(y)\\big)", "000b55413dd8e51c6a5331d756bb35cd": "r_{k} = \\frac{B_{0} - B_{k}}{B^{*} - B_{0}}", "000b60e64695a061524870992c804694": "\\mathfrak{H} =\n\\begin{pmatrix}\nZ_\\infty & - \\gamma_1 \\gamma_2 \\\\\n1 & - z_\\infty\n\\end{pmatrix}, \\;\\;\nZ_\\infty = \\gamma_1 + \\gamma_2 - z_\\infty.\n", "000bdb583c44e7082a31ebb9e6d3270e": "Y_{8}^{6}(\\theta,\\varphi)={1\\over 128}\\sqrt{7293\\over \\pi}\\cdot e^{6i\\varphi}\\cdot\\sin^{6}\\theta\\cdot(15\\cos^{2}\\theta-1)", "000c0ecd3b1cdd0c543c83fb72777e40": "\\|u\\|=\\sqrt{(u|u)}.", "000c247a72b758a4a7b58c94ef5c0143": " C_T',", "000c2d05999df03021184202a05ed589": "\\frac{\\Box p}p", "000c2fdc9d5f7e0d8645da414718e55b": "(a+bi) (c+di) = (ac-bd) + (bc+ad)i.\\ ", "000c509e2ba315d93d74f4358779d6db": "V=5 (Y/19.77)^{0.426}=1.4 Y^{0.426}", "000ccb0783ce670a6c05781e17c96ac4": "H=H_e + H_h +V(r_e -r_h)", "000dd16a691352805a456b763a587df9": "E \\cup F", "000dd846c45c943c8bc9924ef48d1f0d": "e^{i\\mathbf{k \\cdot r_{12}}}", "000de4afc6a32a049d59aeacdb9ef318": "f(x) = x^2 - x + 2", "000dfe97e8b66bd454b3cee3f7fdd708": "e^{c(\\ln n)^\\alpha(\\ln\\ln n)^{1-\\alpha}}", "000e03d98da2c9a1864a463164762254": "\\frac{1}{\\ln p}", "000e18741a314511f1bc6557ae754035": " \\mbox{E} =\\frac{\\sqrt{1.64 \\cdot N} \\cdot \\sqrt{ 120\\cdot \\pi}}{2\\cdot \\sqrt{\\pi}\\cdot d} \n\n \\approx 7\\cdot\\frac{ \\sqrt{N}}{d}", "000e540b8ebc9ff725e5bb41d49be814": " \\text{Spec }B ", "000e5c1739ea28760d66f6d05f0e18d1": "\nJ_{\\alpha} = \n\\int_{0}^{\\infty} \\frac{dx}{\\left( x + b^{2} \\right) \\sqrt{\\left( x + a^{2} \\right)^{3}}}\n", "000ec8a8686baebba2fe12442b863020": "U_{11} - U_{21}", "000f32a1b8f6232759a658d470fe72c5": "y = p(x)", "000f743b3f56fd60b28545a4a844b238": "|{\\Psi}\\rangle=\\sum_{i_1,i_2,\\alpha_1,\\alpha_2}\\Gamma^{[1]i_1}_{\\alpha_1}\\lambda^{[1]}_{\\alpha_1}\\Gamma^{[2]i_2}_{\\alpha_1\\alpha_2}\\lambda^{[2]}_{{\\alpha}_2}|{i_1i_2}\\rangle|{\\Phi^{[3..N]}_{\\alpha_2}}\\rangle", "000f9bd1ad9b3b09c9aa4c60c45692fc": "e = O( n^{2/3} m^{2/3} + n + m )", "000febfeef5745a752e85b94b75cf713": "(t_2,t_1,F_{t_1,t_0}(p)) \\in D(X)", "000ff44c1346a4a8419c634aa6792a6b": "\\scriptstyle (m\\mid k)", "0010ce961820b14519f4edb042677035": "\\vec{b} \\equiv \\vec{B}/B", "0010d521b3b9b45b628e76ac7a7e0477": "\\mathit{MPC} = \\frac{\\Delta C}{\\Delta Y}", "00114d741d2031bf778fd8e43ac0cbeb": "(r,\\theta_r,\\phi_r)", "00114eb3ada60483709d9dc80af6eb9e": "\nL_\\mathrm{dB} = 10 \\log_{10} \\bigg(\\frac{P_1}{P_0}\\bigg) \\,\n", "0011faa0f320ff9b7bc5a9e9ec93bd19": "\\sqrt{\\det g}\\mathcal{D}\\Sigma.", "001222b8821d1da420dbe52f697b6ceb": "(x',y') = (x,y) A + b\\,", "00123391b9f305cfe97c99078735ae00": "\\tilde{k}\\,", "00124f922ab1a17e5e2a9a6c50b17a11": "\\displaystyle{AB=-BA,\\,\\,\\,\\,A^2-B^2 =I.}", "0012c829b2e3bbb683c9a17381e15b4e": "\\frac{\\mathbf{T}(s+\\Delta{s})-\\mathbf{T}(s)}{\\Delta{s}}=-\\mathbf{q}(s). ", "0013269ea11adb76b0e5c55c5d2da6e3": "34^2", "0013271afabc2f00efdeafe99dabfc9c": "\\; P(s_i)", "0013383b9f26d293e8432ded6c3e5520": "\\begin{align} S_1 &=& a_1& & &\\\\\nS_2 &=& a_1& {}+ a_2& &\\\\\nS_3 &= &a_1& {}+ a_2& {}+ a_3&\\\\\n\\vdots & &\\vdots & & &\\\\\nS_N &=& a_1& {}+ a_2& {}+ a_3& {}+ \\cdots \\\\\n\\vdots & &\\vdots & & &\\end{align}", "001384455f0b171fd018da65ca08ae9a": "V \\otimes V / (v_1 \\otimes v_2 + v_2 \\otimes v_1 \\text{ for all } v_1, v_2 \\in V).", "0013ada8dc886f1e875984bee5fdea27": "\n\\rho_{x^{n}\\left( m\\right) }=\\rho_{x_{1}\\left( m\\right) }\\otimes\n\\cdots\\otimes\\rho_{x_{n}\\left( m\\right) }.\n", "0013b318ce7c8b8ca29b706aaa5ec54d": " \\mathbf{A}\\mathbf{B} = \\mathbf{A} \\cdot \\mathbf{B} + \\mathbf{A} \\times \\mathbf{B} + \\mathbf{A} \\wedge \\mathbf{B}. ", "00141348cd6cabc06166525b88bb1493": "\\lim \\sup _{\\alpha} (n_{\\alpha}/m_{\\alpha}) < r", "00143ba3149a2dfac0bbad577d553b6c": " \\vec{A} = \\frac{B}{2}(x\\hat{y} - y\\hat{x})", "001462c07545b4ba9084efef2a96cf16": "\n\\begin{align}\nq &= q \\left(p + 2 q + r\\right)\\\\\n&= q p + 2 q^2 + q r\\\\\n&= q^2 + q (p + r) + q^2\\\\\n&= q^2 + q (p + r) + p r\\\\\n&= \\left(p + q\\right) \\left(q + r\\right)\\\\\n&= q_1\n\\end{align}\n", "00147bc5b79f2b9ed52b22af8d073758": "z*x\\le y", "00148c7652375ca3d73b9b13e86e6c09": "\\psi(\\hat{\\alpha}) - \\psi(\\hat{\\alpha} + \\hat{\\beta})= \\ln \\hat{G}_X", "0014c0cbfae8735b260b1d36141ba2fb": "\\lim_\\alpha \\gamma := \\bigcap_{n\\in \\mathbb{R}}\\overline{\\{\\varphi(x,t):t<n\\}}.", "0014ef9d03a61d379058ee81b8306ba2": "a_{12}\\,dx\\wedge dy + a_{13}\\,dx\\wedge dz + a_{23}\\,dy\\wedge dz;", "0014f6558469ba4380401518bc112eab": "\\rho(-X)", "0014fd219fc32a104f17c85feed0ec75": " \\frac{d}{dx}\\left( \\log_c x\\right) = {1 \\over x \\ln c} , \\qquad c > 0, c \\ne 1", "001526024fa254f09f605fe336f1efb9": "\\textstyle x+C_{i}", "0015764e9f5498369d691b91d3e231a0": "{f_{xy}\\;=\\;f_{yx}}", "0015c94baa30e618e20880703cd9574e": "\\kappa( \\cdot, \\cdot)", "00160f32f654a73bc70209c66ba07704": " K = \\mathbb{Q} ", "001664050cbc76569028d6ac26295a53": "\\theta = n \\times 137.508^\\circ,", "0016dac7c84a2f7a9a5b064c68d1af56": "B^\\prime=-(n_b-n_\\bar{b})", "0017516c449d71df2d3f9b14a22cab76": "RD = \\min\\left(\\sqrt{{RD_0}^2 + c^2 t},350\\right)", "001758801bb0a24a60d89d6ed42620aa": "\\displaystyle{g^\\prime=\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix},}", "00178a6c0a72a69875dabaf4d5ccc192": "\n \\frac{1}{(p+1)\\left(b^2-4 a\\,c\\right) \\left(c\\,d^2-b\\,d\\,e+a\\,e^2\\right)}\\,\\cdot\n", "00178f10a40e91f76517d52061ef2a42": "(n+1)!", "00179f58dfc9cf36493673f0dacf255e": "s_V(\\mathcal{R})", "0017f09b2d0eb84ef7d74112761e5ca2": "\\begin{align}\n\\Phi _{1} & =\\Phi _{2}\\equiv \\Phi (x_{\\perp }) \\\\\n& =-2p_{1}\\cdot A_{1}+A_{1}^{2}+2m_{1}S_{1}+S_{1}^{2} \\\\\n& =-2p_{2}\\cdot A_{2}+A_{2}^{2}+2m_{2}S_{2}+S_{2}^{2} \\\\\n& =2\\varepsilon _{w}A-A^{2}+2m_{w}S+S^{2}, \n\\end{align}", "0017fa64796d63c8af98928a15b3662c": "-F\\mathbf{e}_y", "001803962c3d9e04abb4057c65fa219a": "d_{\\phi}=1", "00180c42d14cacb3f499b74661393fb8": "|f(s)g(s)| \\le \\frac{|f(s)|^p}p + \\frac{|g(s)|^q}q,\\qquad s\\in S.", "001848ad365fbadd5ad138e8c017229c": "c_{\\rm s}", "0018ea864cfdaca5dd616457e5376705": "X,Y,Z", "001906e750dc40c74b91cf7d58e53031": "S^k \\,", "001914d9d31353c1e3f3a0cc4f5d1b26": "\\mathbf{a}_{\\mathrm{average}} = \\frac{\\Delta\\mathbf{v}}{\\Delta t} ", "0019535400d4fd1cc406673a5c837318": " \\sum_i {}^\\phi{V}_i= q V - (q-1)\\sum_i V_i \\,", "0019561cf8dcc36cdbaef1e31544dba0": "WL", "00195c93942fa87df4fc3cc6475b99f9": "h = \\frac{1}{4} kd \\theta^2", "0019c83f9d0e4f79dbb27fa6520759ef": "\\ell(m)", "001a3615880485d99edbd2bcfd14bbd6": "id_\\tau", "001a607e35251386d2e1be0dfd149e51": " \\mathbf{L} = \\mathbf{r} \\times \\mathbf{p} = \\mathbf{I} \\cdot \\boldsymbol{\\omega} ", "001ab4e8bcdb353a5c9bd1db301c1b29": "x+n+a = \\sqrt{ax+(n+a)^2 +x\\sqrt{a(x+n)+(n+a)^2+(x+n) \\sqrt{\\cdots}}}", "001ac223727c30afb98538642f53b42f": "\\left( \\frac{2}{3} \\right) ^3 \\times 2^2", "001ad3e03ed6e69c3304e438fa6e082b": "\\mathbb{P} (Y \\le 0.75|X=0.5) = \\int_{-\\infty}^{0.75} f_{Y|X=0.5}(y) \\, \\mathrm{d}y = \\int_{-\\sqrt{0.75}}^{0.75} \\frac{\\mathrm{d}y}{\\pi \\sqrt{0.75-y^2} } = \\tfrac12 + \\tfrac1{\\pi} \\arcsin \\sqrt{0.75} = \\tfrac56.", "001b05b435b5ca1ad78f35000decd950": "{\\log}\\circ g: x\\mapsto \\log x^2 = 2 \\log |x|", "001bae4d7ab52c8a0edd0a57e8d85701": "\\mathrm{Poi}\\left(\\frac{C(23, 2)}{365}\\right) =\\mathrm{Poi}\\left(\\frac{253}{365}\\right) \\approx \\mathrm{Poi}(0.6932)", "001bde6f639fbdb6285b504b829d3dce": "bx-x^2", "001c03be5066415d5004e2ad5cd961da": " \\mathbf{E}(z,t) = e^{-z / \\delta_{skin} } \\mathrm{Re} (\\mathbf{E}_0 e^{i(k z - \\omega t)})", "001c03cd18548eff08e44a1c6a40460b": "\n\\begin{bmatrix}\n0&1&0&1&0&0&0&0&0\\\\\n0&0&1&0&0&0&0&0&0\\\\\n0&0&0&0&0&0&0&0&0\\\\\n0&0&0&0&1&0&1&0&1\\\\\n0&0&0&0&0&1&0&0&0\\\\\n0&0&0&0&0&0&0&0&0\\\\\n0&0&0&0&0&0&0&1&0\\\\\n0&0&0&0&0&0&0&0&0\\\\\n0&0&0&0&0&0&0&0&0\n\\end{bmatrix}\n", "001c1c698265214507f5814c8c9bbe62": "f(x)= \\begin{cases} \n\\frac{\\nu}{x} \\left \\{ F_{\\nu+2,\\mu} \\left (x\\sqrt{1+\\frac{2}{\\nu}} \\right ) - F_{\\nu,\\mu}(x)\\right \\}, &\\mbox{if } x\\neq 0; \\\\\n\\frac{\\Gamma(\\frac{\\nu+1}{2})}{\\sqrt{\\pi\\nu} \\Gamma(\\frac{\\nu}{2})} \\exp\\left (-\\frac{\\mu^2}{2}\\right), &\\mbox{if } x=0.\n\\end{cases}", "001c5d215d3b2e814fd7cd1aa4ff25d9": "\\Sigma \\chi(n)\\,", "001c5d9c01ea2876ea70689bc638e282": "\\omega_{k}", "001c9503cb4f65ca231b9ff284672084": "\\mathbf{m}_1", "001ce3f609a62621c609e14916adfe6d": "s_2 = r_2 - cx_2 (\\mathrm{mod}\\,q) ", "001d17159eebbaefe304508512f197cc": "(-3n,5+5n)", "001d433c42ed4314705b2e49be9be3c5": " \\operatorname{Weight}(\\sigma) = \\prod_{i=1}^n a_{i,\\sigma(i)}.", "001da83ce80e2772b581b06641d3ca0c": "\\hat{U}^{\\dagger}\\hat{U} = I,", "001de956296095739ae9e0dc253c9269": "C\\ell(E) = F(E) \\times_\\rho C\\ell_n\\mathbb R", "001df96de10d73eb37ced28a37eed908": "\\theta=\\zeta_n^{a_{g,n}}", "001e2e0eb8437d7fafe16bdea61c10f3": "A/4\\ell_\\text{P}^2", "001e37a6336dbdddd5ac30dfc8964b0d": "r_{ij}", "001e7337ad903328d8889cc1ede11dc1": "h_{\\bar{a}}(\\bar{x})^{\\mathrm{strong}} = (a_0 + \\sum_{i=0}^{k} a_{i+1} x_{i} \\bmod ~ 2^{2w} ) \\div 2^w ", "001ea95cf12dc19b9749fa4c5600c6ed": " =\n\\begin{bmatrix}\nW_{11} & W_{12} & & \\\\\n & W_{22} & W_{23} & \\\\\n & & W_{33} & W_{34} \\\\\n & & & W_{44} \\\\\n\\end{bmatrix}\n", "001f090921d4950e090223a9db6fb0be": " \n\\mu_k(A-A_k)<\\epsilon,~\\forall k\\geq N.\n", "001f1531e895160d2f69783938a8d931": "\\Leftrightarrow P(B|A) \\ = \\ P(B)", "001f223d90ce21bb776d2afe729bfeac": "\\mathcal{C} = \\{ \\mathbf{q} \\in \\mathbb{R}^N \\}\\,,", "001f504393a856e45d22e00796231c32": "\\vec r (t)", "001f53b99bd91a14b91c2e4d6d62757a": " Z = \\sum_{j} g_j \\cdot \\mathrm{e}^{- \\beta E_j}", "001fb78130e343f9c200bd3aa484a3f7": "\\tau = \\int_{E_{th}}^{E'} dE'' \\frac{1}{E''} \\frac{D(E'')}{\\overline{\\xi} \\left[ D(E'') {B_g}^2 + \\Sigma_t(E') \\right]}", "001fdd3fb9e94017c83e467233ef49ec": "\\displaystyle{H=f-P(f_{\\overline{z}})}", "001fdfda5cdd7974a1f1e9f94673914b": "\nV = \\frac{w_{1}(q_{1}) + w_{2}(q_{2}) + \\cdots + w_{s}(q_{s}) }{u_{1}(q_{1}) + u_{2}(q_{2}) + \\cdots + u_{s}(q_{s}) }\n", "00201b4361e4f3f5e5e6700e906ab77e": "f_1,\\dots,f_{2^n} : \\{0,1\\}^k\\to \\{0,1\\}", "002094dbb4ecaa0e1203ad652f1688dc": "\\theta_{k}-\\theta_{k-1}", "00213d222a8d87df7a615d7276c5a6cc": "s_0(1-s_0)", "0021503bde14e7a6b4016da9424dcf7d": "\\frac{e^x}{x^x}\\,", "002155c7baeb5176edda09dbdefab697": "\\frac{\\langle E \\rangle}{A} = \n\\lim_{s\\to 0} \\frac{\\langle E(s) \\rangle}{A} = \n-\\frac {\\hbar c \\pi^{2}}{6a^{3}} \\zeta (-3).", "0021c015403002b9cd758587bb4b6964": "q_2 = 1+\\frac{k+1}{6N}+\\frac{k^2}{6N^2}. ", "00222862eb12394ac0c8c08e36208b90": "R = R_{\\alpha \\beta}^{\\;\\;\\;\\; IJ} e_I^\\alpha e_J^\\beta.", "00223afcebe050cdafb431b459794ef3": "<k> = pN(N-1)", "00225356a24bd1ec942aeca27c1a547a": " {v} \\,", "0022573b4553c3cd0fcebdfc5e357e55": " \\langle 0 | R\\phi(x)\\phi(y) + \\phi(y)R\\phi(x)|0\\rangle = 0 \\, ", "00226656ea0692401f9834fe6994da11": "S'", "0022669f61dc6da750ad3b0b6cd0ab48": "\\text{Ker} (k_* - l_*) \\cong \\text{Im} (i_*, j_*).", "0022f6407bd7dc02538291c1ffe49744": "x=\\frac{X-X_0}{\\lambda}", "00231e43bf02e01b0e106fc44adb74e5": "Y_1,Y_2,Y_3", "002326506700d44c9abb37d147e43b5b": "2v_c \\sin(\\alpha + \\beta) = c (\\cos(\\alpha - \\beta) - \\cos (\\alpha + \\beta)).\\,", "002366902dffd8673e5f838a29448df7": " e(\\mathbf{p},u)", "0023c250d7374bd8d6cec3b306e3c490": "p_1 = p_2", "002506aecf8a8eca0bddf976a3e83647": "x_r(\\theta_r(t))", "0025775d9f14d8821126387b6fa5c846": "D(G,H) = \\sum_{i=1}^{29} | F_i(G) - F_i(H) |", "0025b36cbda8365c09737acc9159df57": "\\gamma-", "0025cd57f9b2bd585ee2e2b8a93ef1ad": "P(X_1, \\ldots, X_N) = \\frac{e^{-\\frac{E}{k_{\\rm B} T}}}{\\int dX_1\\,dX_2 \\ldots dX_N e^{-\\frac{E}{k_{\\rm B} T}}}", "0025e1301274e14414e139894060dc23": "C(x_j,x_k)", "0025e75d1ffda9c4bff6b3de9560fe9d": "(gu)h = (gh^{-1})u", "00262cd78d796a5bb0baa8fd774728fd": "\\Delta^0_n,", "00267af4bf244fb88fc329938fac577c": "rK=D_K[F(K,L)]*K\\,", "00269b430e579348929cba8ca3c9990c": "p \\mid m_i", "00269e3bc1fc99fff7bc6d83b0d70bd0": "\\! t", "0026a625f7d3fd336acca8ae2bfcc06e": "\\! E_\\mathrm{h} / a_0 ", "0026b62d6355a23f08830d835b366f02": "2\\omega", "00279c44b6f5f02d0d5a761218b91ce4": " E_\\text{k} = E_t + E_\\text{r} \\, ", "0027acfd0c7490167b612c4b8b787509": "\\mathrm{ber}(x) / e^{x/\\sqrt{2}}", "0027e0646c279e8a69c9579dbef60613": "((-g)(T^{\\mu \\nu} + t_{LL}^{\\mu \\nu}))_{,\\mu} = 0 ", "002825bde096fa03b809c2b7fa66fe47": " \\sum_{g \\in G} f(g) g", "00287e7aa89ea392e3ecb9cb2837eeb9": "\\tilde{\\boldsymbol{\\Sigma}}", "002884828b36c8d042d8a853f57e5eec": "P(X > x) = Q(x) = \\frac{1}{\\sqrt{2\\Pi}} \\int_{x}^{+\\infty} e^-\\tfrac{X^2}{2}", "0028c604c387c78bc42c47b30010b464": "\\begin{pmatrix}\n-i & i\\\\\n0 & i\n\\end{pmatrix}", "00290f11d9ba0677c1614e97a3e1f097": "v(t) = \\int_{t_0}^{t} i(\\tau) d\\tau.\\,", "002917cdd4458fc6214ed9aaf24cd803": "\\frac{v^{2}}{2c^{2}}\\approx 10 ^{-10}", "0029190f5afee4bdfbdd64cd63bc229b": "\\delta^\\prime_0 \\Omega^\\prime_0 = \\left ( \\delta_0^{-1} + k^2 + kx - 1 \\right ) \\delta_0 \\Omega_0.", "002938e91e1d12948fb82e55131c99e7": "\\|Df\\|_{\\infty,U}\\le K", "00293e3339b4ec9cb5f75b6d8ad16918": "(z_0,\\dots,z_n)", "002978af538e0cb31098f49ab472ca41": "n! [z^n] Q(z).", "0029b0f2bac08e3532a265b95a74cde9": "\\lambda(L(B)) \\leq d", "0029c61e83cd7d4546a128f79bd99822": "A,A^2, A^4,...,A^{2^L}", "002a1bd731bf132e2f5b74a55b6f5c19": "R_A=R/A=5R/3", "002a358521632ae5e656e6a8b93ab594": "\\left(\\frac{\\partial \\mathbf{u}}{\\partial x}\\right)^{\\rm T}", "002ad7526d493f4eff5ee031f9462971": "PFB = \\frac{(3200)(FC)}{(FW)(MC)}", "002aeef2f67a7ab68b15f786fe0b673c": "L\\left(C\\right) \\leq L\\left(T\\right)", "002aef6e85c21276cf6521320260f5a6": " P^{\\, a} {}_{\\, ;\\tau} = (q/m)\\,F^{\\,ab}P_b", "002af1a2280bc443756033b1f386b056": "v = \\frac{c}{n}", "002b0f6cbb93d8febf576f9419105ab4": "\\eta =1-\\frac{\\mathit{u}_1 - \\mathit{u}_4 }{ \\left(\\mathit{u}_2 - \\mathit{u}_3\\right)} = 1-\\frac{(1-4)}{ (5-9)} = 0.25 ", "002b6847b0190969eb52946cc76f76ea": "\\left\\{\\begin{matrix}ax+by&={\\color{red}e}\\\\ cx + dy&= {\\color{red}f}\\end{matrix}\\right.\\ ", "002b89f0fa3e9036b33e69d614b18060": "= [P^{(\\pm)} F, G]^{IJ} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; Eq.8", "002b94338d3ad1e2adc60862582ccff2": "\\text{bind}\\colon A^{?} \\to (A \\to B^{?}) \\to B^{?} = a \\mapsto f \\mapsto \\begin{cases} \\text{Nothing} & \\text{if} \\ a = \\text{Nothing}\\\\ f \\, a' & \\text{if} \\ a = \\text{Just} \\, a' \\end{cases}", "002b9647d9a7aacbaaf44a4c005c7f54": "\\Delta \\tau = \\sqrt{\\frac{\\Delta s^2}{c^2}},\\, \\Delta s^2 > 0", "002ba4169a0d47f5c24244d1f9a82cfd": " f^{*} = \\frac{bp - q}{b} = \\frac{p(b + 1) - 1}{b}, \\! ", "002c115aa5aba4aac873a44e7ec65ae1": "\\alpha_{\\tau\\tau}-\\beta_{\\tau\\tau}=e^{4\\beta}-e^{4\\alpha},\\,", "002c1f766558995e2b1166f45a9eb1b0": "\\scriptstyle w[n]", "002c5051d7053790557612d8d2ef2019": "h=C{{\\left[ \\frac{k_{v}^{3}{{\\rho }_{v}}g\\left( {{\\rho }_{L}}-{{\\rho }_{v}} \\right)\\left( {{h}_{fg}}+0.4{{c}_{pv}}\\left( {{T}_{s}}-{{T}_{sat}} \\right) \\right)}{{{D}_{o}}{{\\mu }_{v}}\\left( {{T}_{s}}-{{T}_{sat}} \\right)} \\right]}^{{}^{1}\\!\\!\\diagup\\!\\!{}_{4}\\;}}", "002c8bca1a57ee65188cb4adb14e632c": "f:M \\mapsto N", "002c92314c9a6e81956f72dbe61c39b2": "F=\\overline{(A \\wedge B) \\vee (C \\wedge D)}", "002cbb90309335ef7183f232ac4bf55d": "a^2+c^2=b^2.\\quad", "002ccee36eec167b5d69bb76524b75fd": "SU_{\\mu}(2) = (C(SU_{\\mu}(2),u)", "002cddc16ea0c92f40c38202e128497f": "\\mathrm{2\\ Squares\\ of\\ Land} =(\\frac{\\mathrm{77\\ acres}}{\\mathrm{3\\ Squares\\ of\\ Land}}) \\cdot 2\\ Squares\\ of\\ Land\\ = 50.82\\ acres ", "002d84f8f9870a8115b7866dae7d6d31": "\\sigma_y^2(\\tau) = \\frac{2\\pi^2\\tau}{3}h_{-2}", "002d94ef85bc9ea2c41a550659eb05eb": "\n \\mathbf{E} = \\xi \\exp[i(kx - \\omega t)] \\mathbf{\\hat{x}}\n", "002e06e607e26e75da249f7016a07881": "(-m_i\\partial_{tt}+\\gamma_iT_i\\nabla^2)n_{i1} = Z_ien_{i0}\\nabla\\cdot\\vec E ", "002e08819822cb8016bf5d8593615452": "\\varphi = 2\\cos{\\pi\\over 5} = \\frac{1+\\sqrt 5}{2}\\qquad\\xi = 2\\sin{\\pi\\over 5} = \\sqrt{\\frac{5-\\sqrt 5}{2}} = 5^{1/4}\\varphi^{-1/2}.", "002e5e677339873ae56de031260218b0": "N / \\Gamma ", "002e83cd7e3a8308c5836320f9ac437c": "\\langle x, y \\rangle\\ M\\ N = M\\ x\\ y\\ N", "002ec7f385d551b2c31aedcf1fce7f32": "f_{k,i}", "002f374ea2a9a5316d9dc2de5ba0db82": "\\begin{align} {z \\choose k} = \\frac{1}{k!}\\sum_{i=0}^k z^i s_{k,i}&=\\sum_{i=0}^k (z- z_0)^i \\sum_{j=i}^k {z_0 \\choose j-i} \\frac{s_{k+i-j,i}}{(k+i-j)!} \\\\ &=\\sum_{i=0}^k (z-z_0)^i \\sum_{j=i}^k z_0^{j-i} {j \\choose i} \\frac{s_{k,j}}{k!}.\\end{align}", "002f4e6f409b9611d103847696ce30dd": "C_j^n", "002f4eb7b268cb63dbf1116acb66ed23": " \\Psi(x,t) = \\sum_n a_n \\Psi_n(x,t) = a_1 \\Psi_1(x,t) + a_2 \\Psi_2(x,t) + \\cdots ", "002f8af8f796e82cb12d524429901412": "\\rho: S \\times X \\rightarrow \\{0,1\\}", "002fa9e5e3ba534bf208264e185bab38": "u\\equiv\\frac{r}{\\alpha^2}", "002fde17fb2df61903a3cb830c71241b": "(a_1,\\ b_1,\\ c_1,\\ d_1) + (a_2,\\ b_2,\\ c_2,\\ d_2) = (a_1 + a_2,\\ b_1 + b_2,\\ c_1 + c_2,\\ d_1 + d_2).", "0030177130f83768f8c7205d73fdfadc": "P(y)\\,dy + Q(x)\\,dx =0\\,\\!", "00306d74825ca4c699ac02b1aa3caa18": "=2^2\\cdot5\\cdot17\\cdot3719", "00308fa277a754af480d4ed68cce2a56": "A=\\frac{2}{3}bh", "0030dd6def07c2a872c23491e5c9ac7d": "\\displaystyle{K=\\begin{pmatrix} I & 0 \\\\ 0 & -I \\end{pmatrix}.}", "0030ee1373b5795f95a2d5c2a66b49e5": "\\Delta_{\\mathrm{adv}}(x-y)", "003101e85d556302192b466977a60a8d": "\\langle M, N \\rangle = \\lambda z.\\, z M N", "00310fe1c22c34624ec5fd12b34213a3": " R_{s\\ normal} = \\sqrt{ \\frac{\\omega \\mu_0} {2 \\sigma} }", "003144652c05f21650272d2e79242048": "s, h' \\models P", "003163f025c02255900f7c4225a576b1": " ([\\mathbf{t}]_{\\times})^{T} = \\mathbf{V} \\, (\\mathbf{W} \\, \\mathbf{\\Sigma})^{T} \\, \\mathbf{V}^{T} = - \\mathbf{V} \\, \\mathbf{W} \\, \\mathbf{\\Sigma} \\, \\mathbf{V}^{T} = - [\\mathbf{t}]_{\\times} ", "0031b1a0e5881f6b0ca5ce52f4ab1b04": "f(x)=(x + 1)^{2}(x - 1), \\,", "0031b38c8e97ea03a011524a0ea2b77f": "\\lambda (\\lambda 1 (1 ((\\lambda 1 1) (\\lambda \\lambda \\lambda 1 (\\lambda \\lambda 1) ((\\lambda 4 4 1 ((\\lambda 1 1) (\\lambda 2 (1 1)))) (\\lambda \\lambda \\lambda \\lambda 1 3 (2 (6 4))))) (\\lambda \\lambda \\lambda 4 (1 3))))) (\\lambda \\lambda 1 (\\lambda \\lambda 2) 2)", "003206d6d973a25d27d7badeae180f6a": "\\begin{align}\n Area &{}= \\frac{1}{2} * base * height \\\\\n &{}= \\frac{1}{2} * 2 \\pi r * r \\\\\n &{}= \\pi r^2\n\\end{align}", "003222c0d800ed511b981e1590fd5579": " 0.0000182\\dots,\\, ", "003248f7ade6dc2990d6ae7a805628a8": "\\frac{1,310,000\\ \\mathrm{N}}{(2,430\\ \\mathrm{kg})(9.807\\ \\mathrm{m/s^2})}=54.97", "003275472d45cd9706e6d88486831729": "\\phi_1,\\phi_2,\\phi_3 ", "0032b9d5134fe210abc9011e684a4d23": " a _{i}", "0032f418f93bbaab612a5213f21b9122": "T_r = {T \\over T_c}", "0033322e706f0c7b7dbae50459e4e1a2": "\\Pi\\,", "00338841eb1ca80fef553f18dd02d7db": "\\forall x \\Big(\\forall y (y \\in x \\rightarrow P[y]) \\rightarrow P[x]\\Big) \\rightarrow \\forall x \\, P[x]", "003395de5184f994ecb8f96a60890b6e": "\\chi_G(\\lambda) = (-1)^{|V|-k(G)} \\lambda^{k(G)} T_G(1-\\lambda,0),", "0033aa54194929b25fd3cf4bb6c7d369": "z^p\\overline{z}^q.", "0033ccc8d80038ec44629c31966dfe06": "v_{(G; c)}(\\{1,3\\})=23", "003411f88f779a77e67b7eccd9c6d41a": "\\rho _{\\alpha +} ^{i_0 } \\ge A_{\\alpha + }^{\\sigma (i_0 )} ", "00345c04233a175efdd1e2494c42a238": " \\phi_1 = -30^\\circ...+30^\\circ", "0034991f8f6e6f84b95247f345004bb4": "\\binom Sk\\,", "0034befe82b7a681848dd6ebb6634a0e": "\\begin{cases} \n 1 & (e^{-p})\\mbox{ no disaster} \\\\\n 1-b & (1-e^{-p})\\mbox{ disaster} \\\\\n\\end{cases}", "003529eda35d403c850d8aed6ca10aef": "y = \\psi^{-1}(x)", "003532a7886018f1e650314b310a3290": "x^{q^{2}}\\neq x_{\\bar{q}}", "00354ed1ef1395977fc43f8e6c9aed64": "G_{\\delta\\sigma\\delta}", "0035522d0c7bcb717f215070b1eeef30": "\\log_2 (1-p) + 1-R", "0035587f66355cdac3b284b1fd4645dd": "\\displaystyle{R(Q(b)a,a)Q(b)=Q(b)R(Q(a)b,b)=R(b,Q(a)b)Q(b),}", "00355b116feb4556455199c0b3622e04": "\\gamma_I", "00357df66075bc66d2f4339108604c92": "T \\rightarrow \\infty", "00359027c15ea5ebdf1e499d7c8bec3a": "\n\\langle \\varphi, \\varphi_j \\rangle = \\int_\\mathcal{T} \\varphi(t)\\varphi_j(t) dt, \\text{ for } j = 1, \\dots, k-1. ", "0035cdf76a30ed71e027ee0cc502d979": "1928 = [43, 36]_{44}", "0035ff7f60718d7d705c9d61c4ab5431": "\\ \\beta = \\pi - tan^{-1}(\\frac{1}{10}) - tan^{-1}(L/D) ", "003625928997e0a4a1b8483667736ec6": " \\vec X(n) = \\{ X_d(n) \\}, d = 1..D.", "003656b0a5cdfdf2326d037c9864a835": "dU=TdS-PdV + \\sum_i \\mu_i dN_i.\\,", "003695b09b8e5ddc7fcca8ee1aed316c": " S \\subseteq [n] ", "0036ac1e1ae00ff6a59a729ecdb0ca91": "T_c", "00372ba6f6a4645a32d220eb15577468": "\\mathbb{CFM}_I(R)", "0037ecfd65cf97652c38001750960741": "t^{\\mbox{th}}", "003920cd429ea833122f2971b7944ce1": "\\ P_2= x_2P^*_2f_{2,M}\\,", "00392327200f6a4d35e9c33e723c7e26": "m = n \\sqrt{2}", "003935cf7152b790d696b09642eeea6b": "r_n = (1/2) - x_n h_n", "003941bb8340136488f449dfee574111": "dn_1", "003987dd42d31ffec69d55619deb3d97": "P_1(X)=P(X)/(X-\\alpha_1)", "0039cbae10746ef0b5c1afe4589e9a3e": "(S; \\wedge, \\vee)", "0039f36e9885ebeb4de300eb0f22ebe4": "H^*_GX,", "003a5820c464d82eca6633352a4c42b9": "r_m = r_c ( 1 - t ) \\, ", "003a5ac3c6316db47dde21e454be0a6c": "S = -k_B\\,\\sum_i p_i \\ln \\,p_i,", "003a70ac099d1c13e037072a7f78ca76": "\n U = \\frac{1}{2} \\int_0^a \\int_{-b/2}^{b/2}D\\left\\{\\left(\\frac{\\partial^2 w}{\\partial x^2} + \\frac{\\partial^2 w}{\\partial y^2}\\right)^2 +\n 2(1-\\nu)\\left[\\left(\\frac{\\partial^2 w}{\\partial x \\partial y}\\right)^2 - \\frac{\\partial^2 w}{\\partial x^2}\\frac{\\partial^2 w}{\\partial y^2}\\right]\n \\right\\}\\text{d}x\\text{d}y\n", "003ab5cf816a2d6306acef92162bd5e5": "n < \\lambda \\leq n+p", "003af996ea8f154c29fdcff0f9762f62": "\\theta_k(z) = \\sum_{\\gamma\\in\\Gamma^*} (cz+d)^{-2k}H\\left(\\frac{az+b}{cz+d}\\right)", "003b112cec5f2a74b4eaafc0d1627242": "\\tfrac{\\vec x_{n+1}-\\vec x_n}{\\Delta t}", "003b125ee6a3d44d4f40c957f2611b54": "\\phi _1 , \\phi _2 , \\dots , \\phi _{n-1} \\,", "003b2ceba9c9fca8743b7ada1a22e559": "V_0 = 0,1 ", "003b435dc6f1352fe48d6ab32e5dfd2a": "\\int_{-\\infty}^0 f(x)\\,\\mathrm{d}x=\\pm\\infty", "003b627e9de797d9a9ce175fb6392235": "\\frac{d^2}{dx^2}X=-\\frac{\\omega^2}{c^2}X\\quad\\quad\\quad", "003be3626a91a1ff64ddfc5dbd4edb48": "\\|f_{\\theta}-f_{\\theta'}\\|_{L_1}\\geq \\alpha,\\,", "003c39c6732e6fff7f2947459f7fa5df": "\n\\begin{array}{l}\ns_0=1\\qquad s_1=0\\\\\nt_0=0\\qquad t_1=1\\\\\n\\ldots\\\\\ns_{i+1}=s_{i-1}-q_i s_i\\\\\nt_{i+1}=t_{i-1}-q_i t_i\\\\\n\\ldots\n\\end{array}\n", "003c664848c04c53bedfd7853a47516d": "(-\\mu_j)^{-1/2}", "003c67ab880e13638d98d028457ce502": " V_1 = k_1 [E_{1T}], ", "003ccc5b040e4941beaf0e1c7b71604c": "n \\geq n_0 ", "003d17dfe0f53c5ec3bb56ba64d54d39": "\\{a_n\\} \\subset G", "003d1b455ffe1cfd3d52390be60afabc": "\\|f\\|_{L^{p,\\infty}(X,\\mu)}^p = \\sup_{t>0}\\left(t^p\\mu\\left\\{x\\mid |f(x)|>t\\right\\}\\right).", "003d5dbcdaf031030dca9e8aeb0b7e5d": "= \\frac{k}{n}.", "003d667ac140e61d45eb1c0148ce6885": " {\\alpha \\choose k} = \\frac{(-1)^k} {\\Gamma(-\\alpha)k^ {1+\\alpha} } \\,(1+o(1)), \\quad\\text{as }k\\to\\infty. \\qquad\\qquad(4)", "003d9844a3d178796ad777fa6e22e467": "\nS_{ij} := r_{ij}^{(t)} + g_{ij}^{(t)} + b_{ij}^{(t)}\n", "003dc09bb55482b2f72537dd1850d588": "\\sigma^2_N = \\frac{(N-1) \\, \\sigma^2_{N-1} + (x_N - \\bar x_{N-1})(x_N - \\bar x_{N})}{N}.", "003dd9b388c28533104e73e1b5429c89": "(\\psi'(\\theta))^2/I(\\theta)", "003de6af834956a356ade65eef50d280": "\\Delta\\ W_{ij}(n) = \\gamma\\ \\Delta\\ W_{ij}(n-1) \\Delta\\ R(n) + r_i(n) ", "003e239d39f2c653d6e74c9ddf2f4fe4": "\\kappa = v \\frac{\\mu \\Delta x}{\\Delta P}", "003e40578e8a8611e92faedeebe7f2b8": "x_i(\\mathbf{w}, y) = \\frac{\\partial c (\\mathbf{w}, y)}{ \\partial w_i}", "003e4578d0879dbf7092d45082daf55e": "d^* = \\sup_{y^* \\in Y^*} \\{-f^*(A^*y^*) - g^*(-y^*)\\}", "003e570691573cf65b75f9d7f3d399c1": "\\alpha_c : S(c,c)\\to T(c,c)", "003e75b4ed582eaf7e6001a024932ecf": "n = \\prod_{i=1}^r p_i^{a_i}", "003eae0fd1605ab2c3d9cb22c0e610ac": "H(j \\omega) = \\mathcal{F}\\{h(t)\\}", "003ec252d81828cf0f19388f49018e57": "X_3", "003f2cd1d7c8d8357deec5a359889df5": "\nds^{2} = d\\tau^{2} - \\frac{r_{g}}{r} d\\rho^{2}\n- r^{2}(d\\theta^{2} +\\sin^{2}\\theta\nd\\phi^{2})\n", "003f38a83670c4350403298b1f4364b6": "e_{ij} = \\mathbf{e}_i\\cdot\\mathbf{e}_j.", "003f38e45eec556ade8244f8870ae85e": " {S_3 \\over S_2} = {{16\\over15} \\div {135\\over128}} ", "003f7619ae0c1da19bd1ae62e01dcd2d": "\\pi/4", "003fa3ffdad3e57a239d9a8ce9ff8556": "N=O(n)", "003fcba6cfeca74b28e6a63de15178d5": "(S^0, S^1,\\dots)", "003ffcbad12d7b85054a98ad396622b9": "A = 2\\left(6+6\\sqrt{2}+\\sqrt{3}\\right)a^2 \\approx 32.4346644a^2", "004004a61e6f526c6c2bf255a5010811": "\\mathfrak M (K)", "00400e43c571b943e3788f989b6e4f4d": "\\scriptstyle(\\lnot u)\\Rightarrow v", "00404e17a85b5f39a7eb42f087f3c3ff": "(x+y)^n = \\sum_{k=0}^n {n \\choose k}x^{n-k}y^k = \\sum_{k=0}^n {n \\choose k}x^{k}y^{n-k}.\n", "004079a9e10ff7052646221da1745005": "\\,Q", "00409987890d39631dfb17ba290a11db": "t_a = t+\\frac{|\\mathbf r - \\mathbf r'|}{c}", "0040a8d09dc53fcd583183a7b90c38eb": "\\operatorname{Ext}_R^i(M,\\overline\\Omega) = \\operatorname{Hom}_R(H_m^{d-i}(M),E(k))", "0040bc7d53402e15e76efd567502219f": " D_x = \\frac{1}{i} \\frac{\\partial}{\\partial x}. \\,", "0040ddcb1ff90a92a8701bef0dc2e6f7": "\n\\left( \\frac{dr}{d\\tau} \\right)^{2} = \n\\frac{E^2}{m^2 c^2} - c^{2} + \\frac{ r_{s} c^2}{r} - \n\\frac{h^2}{ r^2 } + \\frac{ r_{s} h^2 }{ r^3 }\n", "00410f0f22d52a5b186f73d0c721e3b2": "\\varphi = \\frac{1 - \\sqrt{5}}{2} = -0.6180\\,339887\\dots", "00415718523d2088141fa516e7cb17cb": "T_\\mathrm{W}[\\rho] = \\frac{1}{8} \\int \\frac{\\nabla\\rho(\\mathbf{r}) \\cdot \\nabla\\rho(\\mathbf{r})}{ \\rho(\\mathbf{r}) } d\\mathbf{r} = \\int t_\\mathrm{W} \\ d\\mathbf{r} \\, ,", "00417172fd9a1d80f3d7ce0d1bdbefa7": "I_{\\mathrm{center}} = \\frac{m L^2}{12} \\,\\!", "00418dc4838b3092afa6d069011fefd0": "Y_\\alpha(z)\\sim-i\\frac{\\exp\\left( i\\left(z-\\frac{\\alpha\\pi}{2}-\\frac{\\pi}{4}\\right)\\right)}{\\sqrt{2\\pi z}}\\text{ for }-\\pi<\\arg z<0", "00423a7a5fd53953495fb4aed95bc108": " h(-,Z) = d\\Delta", "00424861f5673267a2705f68bf870be6": " \\displaystyle M(f) = \\sup_{x\\in D} \\mu(f'(x)).", "00427b119652e0a312fd6a9200137efc": "\\left(\\frac{1 + \\sqrt{1-\\beta^2}}{2}\\right) T", "0042b8b4bd18cd7f590f833a653788ae": "S - S_0 = S - 0 = 0", "0042c1492109c45e812558aac1ee6599": " \nD = O^T A O = \\begin{bmatrix}\n \\lambda_{-}&0\\\\ 0 & \\lambda_{+}\n\\end{bmatrix} \n \n ", "0042d0c90d4c6cc652c0b54ce47f81a1": "f( B_1, B_2, \\ldots, B_m)\\subset B", "0043019f31c2e65deeee14435ed0c2df": " \\nabla \\cdot ( A \\nabla u ) = 0 ", "0043bfae9decf0fe362e422acefcbe4f": "\\hat{ \\textrm{d}}_j", "0043e6787bf9c93b5f9c05ea592c6ef5": " \\operatorname{Var}(X \\mid X>a) = \\sigma^2[1-\\delta(\\alpha)],\\!", "00446ccbf030e3c1559f52147c13d9e7": "(\\tfrac{q^*}{p})=1,", "00448c4852a2cc9d5da56bb6d3a53614": "\\int_{\\mathbf{R}^d}(f*g)(x) \\, dx=\\left(\\int_{\\mathbf{R}^d}f(x) \\, dx\\right)\\left(\\int_{\\mathbf{R}^d}g(x) \\, dx\\right).", "004494b2606a7adaf174db7b6dc17d14": " \\begin{cases}\n \\frac{\\partial L_2 }{\\partial w} = 0\\quad \\to \\quad w = \\sum\\limits_{i = 1}^N \\alpha _i \\phi (x_i ) , \\\\\n \\frac{\\partial L_2 }{\\partial b} = 0\\quad \\to \\quad \\sum\\limits_{i = 1}^N \\alpha _i = 0 ,\\\\\n \\frac{\\partial L_2 }{\\partial e_i } = 0\\quad \\to \\quad \\alpha _i = \\gamma e_i ,\\;i = 1, \\ldots ,N ,\\\\\n \\frac{\\partial L_2 }{\\partial \\alpha _i } = 0\\quad \\to \\quad y_i = w^T \\phi (x_i ) + b + e_i ,\\,i = 1, \\ldots ,N .\n \\end{cases} ", "00449fa9f66ff928b3c0d4f7a0bfd190": "\\Pr\\left\\{E_{a^{n}}\\right\\}", "004573673bb14177fd56ecc3a0259b49": "\\ [A]_t = -kt + [A]_0", "00460704eeb45cb43f638437da0f138c": "T_i = K_i d_i", "00463a2876f07b3e7a8c4ce619c532a5": "\\left\\{\\left(x, y\\right) \\in A \\times B : xRy\\right\\}", "004651c8ecc3cdd380d5ac44723bb634": " [x_t - x^{*}] = A[x_{t-1}-x^{*}]. \\, ", "0046849cd8f4bd8eb09652cf7151a14e": "\\mathbf{aaaaaa}\\,\\xrightarrow[\\;H\\;]{}\\,\\mathrm{281DAF40}\\,\\xrightarrow[\\;R\\;]{}\\,\\mathrm{sgfnyd}\\,\\xrightarrow[\\;H\\;]{}\\,\\mathrm{920ECF10}\\,\\xrightarrow[\\;R\\;]{}\\,\\mathbf{kiebgt}", "0046ab0e7bd8520919d98cc057dbff07": "\\beta_k=\\frac{\\partial S}{\\partial\\alpha_k},\\quad k=1,2 \\cdots N ", "0047362db8e80d2564e21c2adad1ca45": "q^{42}", "004789ef923dbade2d1256e476da60ba": "\\theta_1 < \\theta_2", "0047beba5dbab2fe8e288d1e9b1d5192": "R_{k,l}", "0048528384f5b1b70e8d279c559c5436": "f:I\\rightarrow \\mathbb{R}", "004875f8b2294b19c688df2856489d01": "\\alpha(d) \\le \\left(\\sqrt{3/2} + \\varepsilon\\right)^d", "00489f32547332d509d28f64be77a6c3": " \n\\begin{cases}\nN_j\\left(U^\\left(n\\right)\\right)=\\Gamma_{jk}U_k^\\left(n\\right)-U_j^\\left(n\\right) \\\\\nM_j\\left(U^\\left(n\\right)\\right)=p_i~a_{ijkl}\\frac{\\partial U_k^\\left(n\\right)}{\\partial x_l}+\n\\rho^{-1}\\frac{\\partial }{\\partial x_i}\\left(\\rho~a_{ijkl}~p_l U_k^\\left(n\\right)\\right) \\\\\nL_j\\left(U^\\left(n\\right)\\right)=\\rho^{-1}\\frac{\\partial }{\\partial x_i}\\left(\\rho~a_{ijkl} \\frac{\\partial U_k^\\left(n\\right)} {\\partial x_l} \\right)\n\\end{cases}\n", "00493a8b1b2cb014c676b1c7f2dd1af1": "c = {r \\over {1-(1+r)^{-N}}} P_0", "0049559f98dfaee50543d7d517d24204": "\\mathcal{X}(S(z;u))=\\mathcal{X}(u)+z\\ ", "00495fa4b21e827afa0a14a0556bbb4c": "P_{em} = \\frac{3R_r^{'}I_r^{'2}n_r}{sn_s}", "00496954c373cd5810ba8c18bbaec16c": "\\dot q^\\mathrm{T}", "004984cb0fbd087fc4aa5d6ba33188c2": "dE_\\theta(t+\\textstyle{{r\\over c}})=\\displaystyle{-d\\ell j\\omega \\over 4\\pi\\varepsilon_\\circ c^2} {\\sin\\theta \\over r} e^{j\\omega t}\\,", "0049ea3f4597154927b84fc6183b2ec1": "\\mathfrak{P}^{51}", "004a0f215460cccf77c5be94cd5957a4": "\\gamma=3\\Omega/4\\ ,", "004a0f66dcf0e61c0561ce8c17d34024": " f^{\\mu} = - 8\\pi { G \\over { 3 c^4 } } \\left ( {A \\over 2} T_{\\alpha \\beta} + {B \\over 2} T \\eta_{\\alpha \\beta} \\right ) \\left ( \\delta^{\\mu}_{\\nu} + u^{\\mu} u_{\\nu} \\right ) u^{\\alpha} x^{\\nu} u^{\\beta} ", "004a192738d835e7c80660759807ffb7": "= \\sum_{k=1}^{d} \\left(\\dot v_k \\ + \\sum_{j=1}^{d} \\sum_{i=1}^{d}v_j{\\Gamma^k}_{ij}\\dot q_i \\right)\\boldsymbol{e_k} \\ . ", "004a929cbdcada032006e670aec159ce": "\\qquad{\\it (Comp1)} \\quad \\frac{\\displaystyle M \\ \\rightarrow\n\\ M'} {\\displaystyle M\\|N \\ \\rightarrow \\ M'\\|N}; \\qquad \\qquad {\\it (Comp2)}\n\\quad \\frac{\\displaystyle M \\ \\rightarrow \\ M'\\qquad\\displaystyle N\n\\ \\rightarrow \\ N'} {\\displaystyle M\\|N \\ \\rightarrow \\ M'\\|N'}", "004a9f231095f3c08e2f82e54dd4643f": "\\exp\\left(\\sum_{n=1}^\\infty {a_n \\over n!} x^n \\right)\n= \\sum_{n=0}^\\infty {B_n(a_1,\\dots,a_n) \\over n!} x^n.", "004acfd27331d9504ebbf27a7a9ffcde": "(\\cdot,\\,\\cdot)", "004ad6eb8267d487727c4f2c03c5ceae": "F_0=\\left\\{(1,0,0),(0,1,0),(-1,0,0),(0,-1,0),(2,1,1),(-1,2,-1),(-2,-1,1),(1,-2,-1)\\right\\}", "004b071ceacb7dbbc6505f34eab1216d": " \\frac{D_g u_g}{Dt} - f_{0}v_a - \\beta y v_g = 0 ", "004b15ab050ca1fe6e6092337b1116a3": "(\\alpha_j - \\alpha_i)", "004b1f52d0b2112708389023597f813a": "S \\subset L\\,", "004b8fb50f7aa0ce50232bb773f5f387": "\\operatorname{E} (X_t)=\\operatorname{E} (c)+\\varphi\\operatorname{E} (X_{t-1})+\\operatorname{E}(\\varepsilon_t),\n", "004ba7069754fed522854714a8660e16": "\\overline{z} = z \\!\\ ", "004bc28bf353a7a7dae3f540aa4c86a5": "I_c", "004c00048d155c6aaeee77859a8b45a8": "\\, A \\mapsto M\\alpha(A)M^{-1} ,", "004c04db969c835339fb23593190d46f": "\nE\\bar{X}_A = \\mu_{HA}\\frac{p_{HA}}{p_{HA}+p_{LA}} + \\mu_{LA}\\frac{p_{LA}}{p_{HA}+p_{LA}},\n", "004c69ff4b40f7cceab9e42b8f7370fa": " {d^2 \\bar h^i \\over ds^2} + 2 \\Gamma^i_j {d \\bar h^i \\over ds} + {d \\Gamma^i_j \\over ds} \\bar h^j + \\Gamma^i_j \\Gamma^j_k \\bar h^k + \\bar R^i_j \\bar h^j = 0 ", "004c72301f64855e456aa920a32a1d7c": "\\tbinom24", "004cc0101dda11ac74e94adc07c9aae2": "det(A)\\ne 0", "004cf65ad83a6a03009f6629678c1bde": "i^2= -1", "004d00460322f8ea8cfce85f9084898d": "\\lim_{\\mathbf{h}\\to 0} \\frac{\\lVert f(\\mathbf{a} + \\mathbf{h}) - f(\\mathbf{a}) - f'(\\mathbf{a})\\mathbf{h}\\rVert}{\\lVert\\mathbf{h}\\rVert} = 0.", "004d51a85883bac7a3bd93d24453cd39": "f(x_i) = \\sum_{f=1}^n c_j \\mathbf K_{ij}", "004d61714e5c41d0bc9aff7cb62b7259": "(a_n)_{n\\in\\N} \\times (b_n)_{n\\in\\N} = \\left( \\sum_{k=0}^n a_k b_{n-k} \\right)_{n\\in\\N}.", "004dadc66378395b6a21b73bdbab86e3": "C=\\{C_k^i\\}", "004dbfe6dc52810c3e2192e98e8edac0": "\nM(X) = \\left( {\\begin{array}{*{20}c}\n \\mu \\\\\n \\Sigma \\\\\n\\end{array}} \\right)\n", "004df6f3067e46c45e07b3e9e96f47d3": "\\sigma_\\text{l}", "004e1f9156a736730142d8026957f78e": "\\hat{\\nu}", "004e234f6cdf2e3ff6785774b71b23b2": " \\frac{\\partial F \\left( u \\left( t \\right) \\right)}{ \\partial u}. ", "004e35035b2f412209b351f3df19dbf0": " \\ddot{r} = \\frac{1}{2} \\, \\frac{d}{dr} \\left( (E^2-V) \\, (1+m/r)^4 \\right) ", "004e652b26937bc4fc57cff56c8c45c5": " f,g_1,\\ldots,g_n\\in H", "004ed4a583fb5e14530d8a50c277465f": "\n(0, 653, 1854, 4063) \\rightarrow\n(653, 1201, 2209, 4063) \\rightarrow\n(548, 1008, 1854, 3410) \\rightarrow\n", "004f13ea26fac88c1336de7014e5d86e": " (\\sqrt{2},1); \\quad (-\\sqrt{2},1); \\quad (\\sqrt{2},-1); \\quad (-\\sqrt{2},-1); \\quad (0,\\sqrt{3}); \\quad (0,-\\sqrt{3}). ", "004f36fdc2ad8de69901b2d8334cbdc4": " N_0 k_B", "004f5f4d152754122d438075e243d9fd": "\\frac{b^2}{\\sqrt{a^2-b^2}}", "004f77d74952fece0fe7da9c0e9f362d": "A \\leq_{F} B", "004f97f6e33b7a3b21d1b8ae701da2ef": "u(x,\\dot{x})", "004fb86ed073c6e27d750267bf963bf9": "c r^n \\in I^n", "004fbd61429af6ede34c05cb20415624": "(x-c_2)^2", "004ff877b585feec05fc1619795865b4": "R\\mathcal S(\\mathcal F \\ast \\mathcal G) = R\\mathcal S(\\mathcal F) \\otimes R\\mathcal S(\\mathcal G)", "005011b1c44424b4077226fb6ed12dbd": "p_\\varepsilon (x,t) = 0\\text{ for }x \\in \\partial\\Omega_a", "0050398776b0feb63e2eeb7384b6dcd7": "\\Gamma_{\\infty}", "0050e58f180026f58f4d56eef3a51021": "\\hbar {\\mathbf k'}", "005119eb2768ca72c1837f074d72d0a7": "\\phi(t) = {\\rm Tr}[f(B + tC)]", "0051740ae877c5b18dee89574732c99a": "n_2^2\\sigma_2^2-2\\sigma_2n_2^2\\sigma_\\mathrm{n}+n_2^2\\lambda=0\\,\\!", "0051788326e3478daf0813cdc52388a5": "\\mathrm{SO}(2)", "0051f0b0fff70aba89b8d5352d80722b": "N=g^{\\mu\\nu}K_\\mu K_\\nu\\;", "0052077694b84a2fbc16b07c951977a6": " W= \\frac {1}{iwc_0 Q} (D-R) \\quad (2.6)", "005259dad02c95d61a8dcba7035615ee": "f(b)-f(a)\\geq f(x_n+0)-f(x_1-0)=\\sum_{i=1}^n [f(x_i+0)-f(x_i-0)]+", "005302f209db336a7561fc004e245c6d": " y''(t) = f(t, y(t), y'(t)), \\quad y(t_0) = y_0, \\quad y'(t_0) = a ", "0053479d9005b96a7e238f3c76676ec5": "\\exp(\\lambda (e^{t} - 1))", "00535d682974b6ce2abed6e0d9e65e30": "d^2=4*x*b_{7}*c_{12}^2=", "0053a62968e1874c0e873d21cf4634fa": "\\underline{x} \\in \\R^n", "0053bd74249ba2edd4ff39532c528ca8": "c_2 = 2.04901523, \\,\\!", "00546b61d4996074c0643b1be8cf5802": "\\{| \\phi_i \\rangle\\}", "0054cb6e5b751157081556d7e575ca24": "L(w)", "0054e06028ca38fa0a1cc337ae69ed98": "\\mathrm{core}_2", "005503b59bc42d27c5c1ba90c5099d82": "\na = \\frac{a^4+b^4+c^4+a^2b^2+b^2c^2+c^2a^2}{\\left( a^2+b^2+c^2 \\right)^2} \\Delta\n", "0055139ef653b9bfbedea5d4c316a3d4": "\\mathcal{E}(\\exp)=\\{0\\}", "00552124bea53f3a68f87e28129a5903": "e^{(1)}_i = a_i", "005522a913e457a072a578ef939fb5f3": "\\sigma = 0, \\sigma = 0.2, \\sigma = 0.4, \\sigma = 0.6, \\sigma = 0.8, \\sigma = 1", "00556d8eb6763f7cab142e2c7caf0e95": "D = \\prod_{i=1}^K d_i.", "005589a38037bf9df004958bb97d463c": " I_x(a,b) = \\sum_{j=a}^\\infty \\binom{a+b-1}{j} x^j (1-x)^{a+b-1-j}. ", "0055d263238cda7b7306068f1d676b1f": " B_0 = \\frac{\\hbar^2}{2 m_0} + \\frac{\\hbar^2}{m_0^2} \\sum^{B}_{\\gamma} \\frac{ p^{y}_{x\\gamma}p^{y}_{\\gamma x} }{ E_0-E_{\\gamma} }, ", "0055e644da0728d42924ea03350ea963": "ji=-k", "005629782cc4d869040eb39436ff3edd": "\\sigma_{mk}", "0056b3d282c468d9da43689c4ea780e3": "\\mathcal{O}(x_1,\\ldots,x_n)", "0056b8fd312214ab941b8bb4997b7c96": "\\operatorname{P}(X\\leq m) = \\operatorname{P}(X\\geq m)=\\int_{-\\infty}^m f(x)\\, dx=\\frac{1}{2}.\\,\\!", "0056ed7091c7f8276cbd7eee8c0e5577": "Y = \\beta T_8 + I X", "00572f45e35e977389316f0eef29c429": "\n\\psi_0 |0\\rangle + \\int_x \\psi_1(x) |1;x\\rangle + \\int_{x_1x_2} \\psi_2(x_1,x_2)|2;x_1 x_2\\rangle + \\ldots\n\\,", "005732f2b6be3ee1f925df935f842c6f": "F = GHB", "0057531b8dfcbaf7bf5c9326914adf8d": "k_0 \\in (K_0 \\cap K_\\pm)", "00575feb2a6676e28e72b37df84a3618": "n_{2}=\\sum\\limits_{\\alpha_l=1}^{\\chi_c} (c_{\\alpha_{{{l-1}}}\\alpha_{l}})^2\\cdot({\\lambda'}^{[l]}_{\\alpha_l})^2=\\sum\\limits_{\\alpha_l=1}^{\\chi_c}(c_{\\alpha_{{{l-1}}}\\alpha_{l}})^2\\frac{(\\lambda^{[l]}_{\\alpha_l})^2}{R} = \\frac{S_1}{R}", "00576e1590136e3c819062a933b43d7c": " \\mu (A)= \\begin{cases} 1 & \\mbox{ if } 0 \\in A \\\\ \n 0 & \\mbox{ if } 0 \\notin A.\n\\end{cases}", "00578b5ebbc08a904cf34a0c1a0819ea": "\\theta = 90^\\circ", "0057a1113ace7fce93043cd1f12d3d08": "\nJ:X\\to (X'_\\beta)'_\\beta.\n", "0057baf398e7cfd6f637c36ce0d9990a": "\\ell _{({M},\\varphi )}({\\bar x},{\\bar y})=\\sum _{p=(x,y)\\atop x\\le {\\bar x}, y>\\bar y }\\mu\\big(p\\big)+\\sum _{r:x=k\\atop k\\le {\\bar x} }\\mu\\big(r\\big)", "0057d6a820d541c86b119e50682c74b9": "\\hat{x} = (A^{T}A+ \\Gamma^{T} \\Gamma )^{-1}A^{T}\\mathbf{b}", "0057d78ddfbbb18bd8cb8ff50034d770": "Ax = y.", "0057f7c40c1c3d556269650f184c5d4d": "P(k,k') = \\frac {2 \\pi} {\\hbar} \\mid \\langle k' , q' | H_{el}| \\ k , q \\rangle \\mid ^ {2} \\delta [ \\varepsilon (k') - \\varepsilon (k) \\mp \\hbar \\omega_{q} ]", "005874faf228750704e196df7b32cfb5": "\ng(s) = \\int_0^{\\infty} (st)^{-k-1/2} \\, e^{-st/2} \\, W_{k+1/2,\\,m}(st) \\, f(t) \\; dt,\n", "0058f6dc44d924d18482c23df4fba4c4": "F \\in [0,2]", "0059129c160701104ffc251a2f9a5fd6": "{D}_{4}^{(3)}", "00592dd31623e21f87c674477cadf7b3": "\\lambda_{in}", "0059bd909ff2f47bc4ab8e6cb87b199b": "(A \\vee B) \\wedge C", "0059cfbe87754367ae99f910b2e52325": "~{\\rm slog}_b(z)~", "0059d15cf2bc2d0ef806c8572c4933b4": "\\Omega^8\\operatorname{BSp}\\simeq \\mathbf Z\\times \\operatorname{BSp} ;\\,", "005a21b75723dccee94d965dce65eba8": "rpm_{motor}", "005a491cc79d4933a1bce022a2244fef": "\\frac{\\delta^3}{\\delta J(x_1)\\delta J(x_2) \\delta J(x_3)}Z[J]", "005a5a0f4c8ae71fd658bbf442c91b6a": "1 + 2\\;", "005acbb23e5b52409b16f226c75356f8": "a_{t+1} = (1 + r) (a_t - c_t), \\; c_t \\geq 0,", "005ad6c7839bc9f58a588458fb2784be": "\\Beta;\\ G;\\ \\Upsilon", "005aff1ab64bae2fbd389e08eedceaee": "g\\isin [(X\\times Y)\\to Z]", "005b295caf5cffc88b950047571a21b8": "\\underbrace{u_1(\\mathbf{x},z_1)=v_1+\\dot{u}_x}_{\\text{By definition of }v_1}=\\overbrace{-\\frac{\\partial V_x}{\\partial \\mathbf{x}}g_x(\\mathbf{x})-k_1(\\underbrace{z_1-u_x(\\mathbf{x})}_{e_1})}^{v_1} \\, + \\, \\overbrace{\\frac{\\partial u_x}{\\partial \\mathbf{x}}(\\underbrace{f_x(\\mathbf{x})+g_x(\\mathbf{x})z_1}_{\\dot{\\mathbf{x}} \\text{ (i.e., } \\frac{\\operatorname{d}\\mathbf{x}}{\\operatorname{d}t} \\text{)}})}^{\\dot{u}_x \\text{ (i.e., } \\frac{ \\operatorname{d}u_x }{\\operatorname{d}t} \\text{)}}", "005b5ee9184b63d5aae64f486f7762fb": "\\begin{align}\n E_{f_1 + f_2} &= k E_{f_1} E_{f_2}\\\\\n E_{f_1 - f_2} &= k E_{f_1} E_{f_2}\n\\end{align}", "005b76ddf58418b5840fbcd038a55157": "\\nabla_{\\bold u}{\\bold v}(P)", "005b859372ff66ab53af32bd3a95d44c": "\\overline{P}_+:=\\{Q\\in \\mathcal P \\ | \\ Q\\parallel_+ P\\}", "005bee71a96229dc83bdfe3e6a3acd0e": "a + b = 1 + (a + (b - 1)),\\,\\!", "005c84a6de1981ba507fc84f6d002474": "[ES] = \\frac{[E]_0 [S]}{K_m + [S]}", "005cec355090557072bc5242720c1baf": "\\Delta_x \\subset T_xM", "005cf2bd315336ccfc51a82fbc1d011b": " D[p] = [q, \\_, p]::[x, \\_, f]::\\_ ", "005cfe08ac4514176ec9114ed86f5227": " (y + [y/4] + 5(c\\mod4) -1) \\mod 7 ", "005d02c0ccb188f9ce6f80af84add7b2": "E \\left[ \\hat{\\sigma}^2\\right]= \\frac{n-1}{n} \\sigma^2", "005d3c5a843cc4afd4f9459017e79c9b": "v = \\left( \\begin{matrix} \\alpha & \\sqrt{\\mu} \\gamma \\\\ - \\frac{1}{\\sqrt{\\mu}} \\gamma^* & \\alpha^* \\end{matrix} \\right).", "005d4b56062ccf78a1b95d44a904247f": "\\begin{align} \\text{var} (a) &= \\frac{3 \\sigma^2}{2 \\sqrt{\\pi} \\, \\delta_x Q^2 c} \\\\ \\text{var} (b) &= \\frac{2 \\sigma^2 c}{\\delta_x \\sqrt{\\pi} \\, Q^2 a^2} \\\\ \\text{var} (c) &= \\frac{2 \\sigma^2 c}{\\delta_x \\sqrt{\\pi} \\, Q^2 a^2} \\end{align}", "005d5a3817f33dbd656f7b1f926c3ca9": "i/k^2", "005d5be63f060f92e94635636bf5b460": "X_1, X_2, Y_1, Y_2", "005d5f39e6da2cbf9468db66550b1eb5": " r = \\cos^3 \\theta + \\sin^3 \\theta ", "005db61459186328eb26260e77d5c924": "\\mathbb{H}P^2", "005db7c35c2fcc2802e368349fb1dbd2": " \\gamma^\\mu ", "005ddad159bdd4129d68bbf13f9b313c": "{V_{D}} = {V_{P}} + {V_{T}} \\left(\\frac{fu}{fu_{t}}\\right)", "005de217bb2d2c562ddb6ef9b2c6e6af": "a^2 + b^2 + c^2 + d^2 = 2ab + 2 a c + 2 a d + 2 bc+2bd+2cd,\\,", "005e2424c5b287b323d90c18e7d14ebe": "\\begin{cases} y = t^5, \\\\ x = t^3. \\end{cases}", "005e3511011cdc4a24614efd9d0e46eb": "\\mathsf{fv}", "005e882e411a505e927d9403fc95de5a": "\\sum_x \\sum_y I(x,y) \\,\\!", "005ea9a1faaf40201a1fd149fe0df890": "E = R(\\frac {1}{cos(\\frac {\\Delta}{2})}-1)", "005ed603f042c5daf6424e819f284c3c": "charK=2", "005f0f12a2e245b294afb991849fa7e1": "\n\\| u \\|_{L^p} \\leq C \\| u \\|_{L^q}^\\alpha \\| u \\|_{H_0^s}^{1-\\alpha},\n", "005f483aa77c88741fb6a5aca33ab88a": "z = S(r)", "005fa114e9c6b6ee16b3fbe3cd3388d4": "\\langle \\cdot,\\,\\cdot \\rangle \\, ", "005fa1cc2fa20c304d008d28eab9f654": "\\sum_{k=1}^{k=1} \\cos (-2\\pi\\frac{n(k-1)}{1})/1 = 1,1,1,1,1,1,1,1,1...", "005fa74cd2721b0e1f14c33a18a72635": "O(n^2)\\,", "0060137dcd6ebcaf2dd43e3874138898": "\\mathbf{v}=\\mathbf{v}(\\mathbf{x},t)", "00602495b14f9a5268d76e9856935c65": "\\sum_{n=1}^\\infty(\\nu+n)\\sigma_n|a_n|^2", "00602be4ce46f584276cca5f03ce4724": "\\scriptstyle k\\le 3", "006041eaed4c1e105ab451fa672c7eee": "\\boldsymbol{F}_r", "0060430b8c2b4e4aea5fe6f13f242844": "\\mu = ( \\mu_1, \\mu_2, \\mu_3, \\dots , \\mu_N )^T", "00606ffff5f0c9b9833b36681455bd31": "|A|=q", "0060811bf995ea99d0d7af0599037529": "R - R_f = 0.15 ", "0060884b4efc537e5c4e39a03a850a1c": " d=1 ", "0060a9b42c9111cf46baa1f23c60aff3": " \\gamma_1 = \\frac{ 2 \\nu^{ 3 } } { ( \\sigma^2 + \\nu^2 )^{3/2} } ", "0060b049e7e0220cdf2da68756928145": "\\forall x [\\mathrm{Proof}_T(x, \\#\\rho) \\to \\exists z \\leq x \\mathrm{Proof}_T (z,\\mathrm{neg}(\\#\\rho))].", "0060bb0858ef5d84a9930047929fe5b8": "P_{reflect} = \\frac{9.08}{R^2} cos^2 \\alpha ", "0060e120daf207e3782db6738544b75e": "\\text{Average investment} = \\frac{\\text{Book value at beginning of year 1 + Book value at end of useful life}}{\\text{2}}", "00610a4f8b4857300c196650e8badb31": "k_\\mathrm{on}", "006152f03b3939e864f9ac66565b6b58": " \\frac{\\alpha + n}{\\beta + n \\overline{x}}. ", "00617636cc05caa13d75cdc6958d47ce": "K_B", "0062510a5af85f0f1e616f850e5b4e3e": "\n\\inf_g \\sup_f \\iint K\\,df\\,dg=\\frac{3}{7}.\n", "0062c755efea0b9be6ef3dd55ccc30c6": "\\overline{I} = \\overline{\\overline{I}}", "0062d94d1a6a6962840096804a79eb6f": "\\mathcal{L}\\{f''\\}\n = s^2 \\mathcal{L}\\{f\\} - s f(0) - f'(0)", "0062df2399c2fbd55c34251620e6f357": "\n\\begin{align}\n\\boldsymbol{F_{12}} & =m_1\\boldsymbol{a_1},\\\\\n\\boldsymbol{F_{21}} & =m_2\\boldsymbol{a_2},\n\\end{align}", "0062f69c43f50b5e581711b6f431a0af": "\\textstyle 3+\\log_2(n)", "0063113efc28a4d2117081f92b8a8e22": "\n \\begin{bmatrix}\n a_{11} & a_{12} & a_{13} \\\\\n 0 & a_{22} & a_{23} \\\\\n 0 & 0 & a_{33}\n \\end{bmatrix}\n ", "00634867b24389e3680d995d91df3a9e": "0\\rightarrow B\\rightarrow A\\oplus B\\rightarrow A\\rightarrow0.", "0063518e51e9e5ee82646085312dc4ca": "L \\to \\frac{\\omega_c'}{\\omega_c}\\,L", "006352d28b12736b2039ee834b99551c": "r\\;", "00636f68c06830b056c7dc4b296df1b5": "R_T = -2 \\sqrt{\\frac{\\bar{C}'^7}{\\bar{C}'^7+25^7}} \\sin \\left[ 60^\\circ \\cdot \\exp \\left( -\\left[ \\frac{\\bar{H}'-275^\\circ}{25^\\circ} \\right]^2 \\right) \\right]", "006380ed20df9a00246c9f6175355342": "b=3\\,\\!", "0063a4e600bfbf1e870b4704eba7e3c8": "\\begin{pmatrix}\n 1 & a & c\\\\\n 0 & 1 & b\\\\\n 0 & 0 & 1\\\\\n\\end{pmatrix}", "0063a9838403b9181103f102ed4f2286": "\\begin{align}\nN(x) &= [{y}_{k}]+ [{y}_{k}, {y}_{k-1}]sh+\\cdots+[{y}_{k},\\ldots,{y}_{0}]s(s+1)\\cdots(s+k-1){h}^{k} \\\\\n&=\\sum_{i=0}^{k}{(-1)}^{i}{-s \\choose i}i!{h}^{i}[{y}_{k},\\ldots,{y}_{k-i}]\n\\end{align}", "0063afecc3edf643d2ba84bad6572269": "(\\nabla_Y T)(\\alpha_1, \\alpha_2, \\ldots, X_1, X_2, \\ldots) =Y(T(\\alpha_1,\\alpha_2,\\ldots,X_1,X