{ "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 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": " = 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_2,\\ldots))", "0063b0581ca767e70c55c38053505d09": "hom_D(d_1, d_2) = hom_C(d_1, d_2)", "0063be012d6f0372fbc5275df643e0e2": " \\sum_{n \\in \\mathbb{Z}^d} |\\psi(t,n)|^2 |n| \\leq C ", "0063c4f869877e207c7899c6524d6be8": "\\{y_1, \\dots, y_n \\}", "0063d7d97893cc32e29093238de98deb": "\\begin{pmatrix} 1 & x & z \\\\ & 1 & y \\\\ & & 1\\end{pmatrix}\\Gamma", "006431705901f4b0c40c087dddfbbe25": "\\int_{t_1}^{t_2} \\sqrt{\\left(\\frac{dr}{dt}\\right)^2 + r^2 \\left(\\frac{d\\theta}{dt}\\right)^2 + \\left(\\frac{dz}{dt}\\right)^2}", "00646d01d11376afbd540912f57493e0": "v \\ll c_{a}", "0064a66eb067fc3deb0891fe68173932": "\\mu\\!\\left(X\\right) = 1", "0064ef0ce826596fc2c66bc568d1cfaf": " p = {\\frac{-x\\pm\\sqrt{x^2-4(\\frac{-gx^2}{2v^2})(\\frac{-gx^2}{2v^2}-y)}}{2(\\frac{-gx^2}{2v^2}) }}", "0064f09258ef604746b88546e170dbad": "Z(k,z)=\\cosh(kz)\\,\\,\\,\\,\\,\\,\\mathrm{or}\\,\\,\\,\\,\\,\\,\\sinh(kz)\\,", "0065753065aa05f26494ba26ae99b06a": "E_s[n]", "00658707cdcacc18f896c09e3708968e": "u_{c,i}=\\frac{10.872+0.404 (c_r/c_t) c_{t,i} - 4 (d_r/d_t) d_{t,i}}{16.518+1.481 (c_r/c_t) c_{t,i} - (d_r/d_t) d_{t,i}}", "0065971788f31a3645db9df9fa09b8e8": "2\\leq l", "0066b0150dd9d84ad2d7a66b9f64f64f": " H(\\omega) ~ ", "0066d1eaa4a6602f51da84c5573afd00": "A=\\operatorname{E} (\\Gamma).", "0067045a28deee4d1cc3e1100034e3b4": "\\! J = 2", "006787201e51940f0e2132a7e8c36236": "g(x)=ax^2\\,\\!", "006790d57484d7d46cec4fb2bc2f83e0": "M_{n,k} = \\{ c : P_k(c) = P_{k+n}(c) \\}\\,", "0067c21f52d6fe72e6cf2bd2fd547157": "\\alpha \\in \\Gamma^*", "0067d840510bf6084a7c967d2c0fd5ad": "\\mathrm{K_a = 10^{-4.19} = 6.46\\times10^{-5}}", "006817227c30a11f53ee96def5bcbd71": "V_A = C_A \\cdot \\exp\\!\\left[{-z\\over\\lambda_A}\\right]", "0068402f045ff74b8daa0abfa498dbb4": "\\left\\{x,y\\right\\} \\overset{\\mathrm{def.}}{=} \\left\\{z : z=x \\vee z = y\\right\\}", "0068434645ac8d5310e51e8c2277158d": "\\frac{5 \\sqrt{3\\pi}}{16}", "00684a778b4930e2e20f2bc5f0c50eb1": "0 \\div 0 = 0", "0068b8e2e9d348cbb8a0ada31556ef9e": "\\left\\{ z \\in H: \\left| z \\right| > 1,\\, \\left| \\,\\mbox{Re}(z) \\,\\right| < \\frac{1}{2} \\right\\}", "00691626fff7a61da09dd5f51a1a4414": "\\neg p \\wedge q", "006950912b8eb67b89c69baec75894f5": "A_1V_1 = A_2V_2", "00697121901844d211b29641023e5ffe": "Rev_t", "0069a29184ac94f333c07b1dea9e3f8c": "C_2 \\le Y_2 + S_1(1 + r)", "0069cf3e398f8b96544ad051c1f41085": "dq = \\lambda_q dl", "0069eb02ccf993aec658878fb31857c6": "K^2 = {C_N^2 \\over {p_{N_2}}}", "0069fd5cf07098f5022e7b98d242e05b": "T-\\lambda I", "006a1e610fe46c7d6abaaca8a311fc11": "\\int_V e^{-\\pi\\langle \\phi,S\\phi\\rangle}\\, \\mathcal D\\phi", "006a682b5c5c4619cf07219e28a451aa": "\\frac{G^\\mathrm{ig}-G}{RT} = \\int_V^\\infty (1-Z) \\frac{\\mathrm{d}V}{V} + \\ln Z + 1 - Z", "006a6e8bcc60e65733b803f7a1f098c0": "m = p^{\\alpha}", "006a8cae222813804405593109e83c2b": "L\\to\\infty\\,\\!", "006a9988bd5ec6cc57698b026e107a6c": "\\exists a\\in A(x,G)\\colon d(x,z)0. ", "00777b09aa9274526b5613b72b177737": "\nz^{2} = \\frac{\\left( c^{2} + \\lambda \\right) \\left( c^{2} + \\mu \\right) \\left( c^{2} + \\nu \\right)}{\\left( c^{2} - b^{2} \\right) \\left( c^{2} - a^{2} \\right)}\n", "0077897d7efba5094e15cff44f8922aa": "\\scriptstyle\\varphi:T \\mapsto \\mathbb{R}", "00779058ba3e2e1281a3bec1701ddf0b": " d \\approx 1.3 ", "0077929132c8c5223d2f96f5e3e43972": "\\sqrt{\\frac{3}{8}}\\!\\,", "00779355fc7d27f81ccd426981e0b1ec": "w\\bar{y}z", "00779d89488f552b532b9648fd849d5a": " (\\partial U)_S=-(\\partial S)_U=\\frac{PC_P}{T}\\left(\\frac{\\partial V}{\\partial P}\\right)_T+P\\left(\\frac{\\partial V}{\\partial T}\\right)_P^2", "0077bac0533450e9c240f9c0b1d9c223": "F^{\\alpha \\beta} = g^{\\alpha \\gamma} F_{\\gamma \\delta} g^{\\delta \\beta} \\,.", "0077c1ecca87764343e8bd1108d65919": " \\alpha_2 = \\frac{6 G}{2 - K} - \\frac{2 G (K + 4) e^{4 \\phi_0}}{(2 - K)^2} - 1", "0077c400b0161a221aa7adb882c272d7": "{o.p.d.} =\\Delta\\,n \\cdot t", "0077e6527194ccd11d6c32c045f506f0": "\n \\tau = \\sqrt{|\\mathbf{t}|^2 - \\sigma^2}\n ", "0077ee7d7b6ea8618a0ade235c73ef68": "\\{ p_1, r_1\\}", "0078761465bd6fd6b3215e5c47313b31": " s^2 = \\frac{ w_1 } { ( 1 - w_2 )^2 } ", "00787af9608e8ef51e433a7c51d94e00": "\\omega^2 r", "00790b5b8c2899d32e6f8362444877cc": "k^{water}_{f}=400 s^{-1}, k^{water}_{u}=2*10^{-5} s^{-1}, m^{}_{f}", "00791df9a93c86a6fde17616aaf15160": "\\{u_1,...,u_n\\}", "00792c8717b528bf0a5fd2e6a5431e47": "g(p_1,p_2,\\ldots,p_n)=\\sum_{j=1}^n p_j.", "007942efad833175d711142ea4ea22ae": "A\\mapsto (B \\Rightarrow A).", "007955300525c5dcbf90db9082725d8a": " \\frac{\\partial}{\\partial x} \\Bigl( \\frac{1}{\\phi}\\frac{\\partial\\phi}{\\partial t}\\Bigr)\n= \\nu \\frac{\\partial}{\\partial x} \\Bigl( \\frac{1}{\\phi}\\frac{\\partial^2\\phi}{\\partial x^2}\\Bigr) ", "0079557046a685511fac69d35552fb03": "p + 2b^2", "00799b185302341d53d975633fa34d9e": "\\theta \\approx 0", "0079bdea239515bb75d307ab7896cfd9": "[(i\\hbar)^{2j}\\gamma^{\\mu_1 \\mu_2 \\cdots \\mu_{2j}} \\partial_{\\mu_1}\\partial_{\\mu_2}\\cdots\\partial_{\\mu_{2j}} + (mc)^{2j}]\\Psi = 0 ", "0079e61773dda1b7e7372d850ec820d3": "\n\\int_0^\\infty x^{2l+2} e^{-x} \\left[ L^{(2l+1)}_{n-l-1}(x)\\right]^2 dx =\n\\frac{2n (n+l)!}{(n-l-1)!} .\n", "007a632787fbac1c7b731d3853db5170": "y = b\\,", "007a9a1e8463ef195d0a9b1dc88e057e": "\n\\begin{bmatrix}\n V_1 \\\\\n V_0\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n z(j\\omega)_{11} & z(j\\omega)_{12} \\\\\n z(j\\omega)_{21} & z(j\\omega)_{22}\n\\end{bmatrix}\n\\begin{bmatrix}\n I_1 \\\\\n I_0\n\\end{bmatrix}\n", "007ab0be21f37e943739ddcfc116f94c": "(a+b)\\cdot c", "007aba174663400614c30e668f8d31a0": " \\text{DWF} = \\exp\\left( -\\langle [\\mathbf{q}\\cdot \\mathbf{u}]^2 \\rangle \\right)", "007ae2204727cb1c044fd7212c2a5481": "C = C_0 \\dots C_n", "007c1c9966b9f4e95a018fb4cdd39a1f": " \\phi_{hc}(r) = \\frac{ 1.5 \\left(r+\\left| r \\right| \\right)}{ \\left(r+2 \\right)} ; \\quad \\lim_{r \\rightarrow \\infty}\\phi_{hc}(r) = 3", "007c8b779fa5b2fbfa9e5808d7d7d932": "T_i = t_i \\cdot \\pi \\left[ \\alpha_i^K \\cdot \\frac{K_{i}}{K} + \\alpha_i^L \\cdot \\frac{L_{i}}{L} + \\alpha_i^S \\cdot \\frac{S_{i}}{S} \\right].", "007d19037e264909bb548db2771d0311": "\\frac{ \\partial f}{ \\partial x} = f_x = \\partial_x f.", "007d588147947102f485cf41305639e8": "\\Omega = \\Sigma_{X|Y} \\Sigma_{XX}^{-1} = I - \\Sigma_{XY} \\Sigma_{YY}^{-1} \\Sigma_{XY}^T \\Sigma_{XX}^{-1}.\\,", "007d5999b8d9d820537b24078de96cc1": "k=\\frac{f_o^2-f_e^2}{f_o^2+f_e^2}.", "007d8191fccdd53f9153ce227ad75b6a": "\\exp(-c)", "007daa94b35faa31165f05da0bd78f8b": "S=\\theta (X_H)", "007e0ecfa25a72cc2f383f39b86540d9": "N = 7", "007e27d57fcd1d2c45476345d34bab59": "SU(3)_L \\times SU(3)_R", "007ebd2662d0cb69047e6bf0843a8ad2": " \\zeta(s,a)=\\sum_{n=0}^\\infty \\frac{1}{(n+a)^s} \\!", "007ede1f44d6d865a3eea50077444c9e": "O_{9}", "007efd2017dc9af726a9fd0111631f45": "H_{p - 1} \\equiv 0 \\pmod{p^3}\\, ,", "007f1e60ec26cf5a7bfdd270125f45ba": "T(\\Delta V) \\approx \\sum_{n = 0}^N a_n (\\Delta V)^n", "007f217f136e1b043c9093734d532e13": "\\begin{align}\n\\iint_{R_C} s(x,t) dx dt &= - \\int^{x_i + c t_i}_{x_i - c t_i} g(x) dx + c u(x_i,t_i) - c f(x_i + c t_i) + c u(x_i,t_i) - c f(x_i - c t_i) \\\\\n&= 2 c u(x_i,t_i) - c f(x_i + c t_i) - c f(x_i - c t_i) - \\int^{x_i + c t_i}_{x_i - c t_i} g(x) dx\n\\end{align}", "007f2e086cc219e6031d2b739e28790a": "\\Delta y = \\Delta X * \\frac{1}{(1 - b_C)(1 - b_T) + b_M}", "007f3d5eec88b2997787156a0da80d1b": " a^{k} = (a^{k}_{i})_{i \\in I} ", "007f6d48228b41dbfec441fdb60f208d": "\\begin{align}\n\\Gamma(z) & = \\int_0^\\infty d\\lambda e^{-\\lambda} \\lambda^{z-1} \\\\\n & = - \\int_0^\\infty d\\left(e^{-\\lambda}\\right) \\lambda^{z-1} \\\\\n & = - \\left[e^{-\\lambda}\\lambda^{z-1}\\right]_0^\\infty + \\int_0^\\infty d\\left(\\lambda^{z-1}\\right) e^{-\\lambda} \\\\\n & = 0 + \\int_0^\\infty d\\lambda\\left(z-1\\right) \\lambda^{z-2} e^{-\\lambda} \\\\\n & = (z-1)\\Gamma(z-1) \\\\\n\\end{align} ", "007f7cfb6c836265f0ee259f9795c82e": "0 \\rightarrow G / \\ker\\, f \\rightarrow H \\rightarrow \\operatorname{coker}\\, f \\rightarrow 0", "007f7db03b1721f021e315ea7df8efac": "\\Omega\\,\\!", "007f7f641efda0619b3f766fb9789e1d": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{*}&\\mathrm{T}&\\mathrm{*}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "007fa54549c55de57cdbbc180eb5dbc3": " P_x = P - \\{ a\\mid a \\geq x\\} ", "00803d249818b788bcdef1e281e2fc83": "dx = udt", "008046f09b8004ec023907a58e377465": " D_1 \\psi = \\frac{A}{\\lambda-\\alpha} \\psi ", "008047cb51b2857a9421b19533c9180f": "\\left( \\tfrac{a}{n}\\right)", "00805d6ff79c98d6900575db1265bf54": "G(A)", "008068aab035eff3a79d9645d5fcaef3": "\\left[1 + \\frac{x}{\\sigma}\\right]^{-\\alpha}", "008080e78f109a140688c229fa3545d6": "\\eta = \\frac{ work\\ done } {heat\\ absorbed} = \\frac{ Q1-Q2 }{ Q1}", "0081356314aa1829716309fc76c3ea7f": "df = {\\partial f \\over \\partial x}dx + {\\partial f \\over \\partial y}dy = pdx + vdy", "008194e7ea2ac22286d9a9c3d4abd909": "h_{r,s}", "0081ad49d57a81653cef2dff3b7f1640": "r \\ge a^{1/4}", "0081c4012db924a05de5e2a64aaf3683": "\\varphi: G \\to G^{op}", "0081dec84e8f982234193c1af00fe0f4": "L=\\frac 1 {N(N-1)}\\sum_{i=1}^{N-1}Q_i", "008200c589d4f31f1b4dd239daae3427": "s_\\lambda= \\sum_\\mu K_{\\lambda\\mu}m_\\mu.\\ ", "00821b05a1ce106b2ceefb3f2331880b": "Z_{F \\circ G}(x_1, x_2, x_3, \\dots) = Z_F( Z_G(x_1, x_2, x_3, \\dots), Z_G(x_2, x_4, x_6, \\dots), Z_G(x_3, x_6, x_9, \\dots), \\dots )", "008222f28187648a32637e1e52306723": "\\frac{\\partial E}{\\partial \\hat{h}_i} = \\frac{\\partial}{\\partial \\hat{h}_i} \\sum_{n=-\\infty}^{\\infty}[x[n]^2 - 2x[n]\\sum_{k=0}^{N-1}\\hat{h}_ks[n-k] + (\\sum_{k=0}^{N-1}\\hat{h}_ks[n-k])^2 ]", "0082381366eb7e186fe3e2b7d31b2cd4": "J_z \\to 0", "0082ab2d2297a4ea938a0d25d6dd5c9a": "h[f] = \\lim_{\\Delta \\to 0} \\left(H^{\\Delta} + \\log \\Delta\\right) = -\\int_{-\\infty}^{\\infty} f(x) \\log f(x)\\,dx,", "0082d037b3e5c48137de5c9b8591c500": " K^M_*(k) := T^*(k^\\times)/(a\\otimes (1-a)) ", "0082f7dbe06d887ba8c2fd1c7252fe18": "(C*(1-A)+G)", "008382f5c4fb4614a13b561e58ecfa66": "b(x) = x^jb'(x) \\mod (x^{2t-1}-1)", "00839570c7cb93cd4611c23bd52bbef1": "B_1+B_2a=C_re^{iak_0}+C_le^{+ak_0}", "0083b07c7fb9fba73f101e2b1eecfba3": "\\{C : K_X \\cdot C = 0\\}", "0083c4e87edab8507e96fdde5c911ab3": "q(\\mu,\\tau) = q(\\mu)q(\\tau)", "0083dcac1f5eaa37fd0eb3503722e9b2": "\\Theta \\wedge\n(d\\Theta)^n \\neq 0", "0084209ec3306ab04a193d13223f53d3": "H(p,q) = \\mathcal{F} \\left\\{ h(x,y) \\right\\} ", "00843f9d223ff4c5c126d001c62f48c3": " \n\\mathcal{P}_2(-p_2) = a_{20}(- p_2)^2 +a_{11}(- p_2) +a_{02} = 0\n", "00844945aabd62ba8956c429106513d1": "\\sum\\limits_{i=0 or 1}^{n}P_n (i) W_n (i)", "00844d9977810a20cb96afff0ba5e562": "P(x) = \\sum_{n=0}^\\infty p_n x^n", "008451b474538e1acb9f7d5d1403b167": "D\\left(\\rho u_i\\right)/Dt\\approx0", "00848e2a02240ee7911a90ba2b2495be": " \\|\\alpha^\\prime \\| = 1", "0084c491a7482112d248bc4acefe66ef": "-[OH^-]_{0^{ }}10^{b_0}", "00862d911b12b7cbe90d7a220cf173ec": "f^{-1}\\colon P(Y)\\to P(X)", "00867b570a40821310cbfddda66378f2": "n_{ij}=\\left|U_i\\cap V_j\\right|", "00870c0e8d811a41fc05bb405771d12e": "H(x+y) = H(x) + H(y)", "0087371d07e71fac449e36f68f88dc18": "10.1) \\ \\mbox{Potential adopters}\\ -= \\mbox{Valve New adopters} ", "00875f86af8c866407a4d164d5cbf7db": "z^2 + c", "00877c9ea9a300fe50856e46eb628dde": " \\mathbf{B}. ", "00879a95cd49c2d5871a2f360db7450d": "\\mathcal{M} = (r,\\mathbf{b},\\mathbf{\\delta},\\mathbf{\\sigma}, A,\\mathbf{S}(0)) ", "0087b1f1983a2b9f594fbccc653b4472": " z_{t} = \\lambda_{1}z_{t-1} + \\varepsilon_{t} ", "0087b3df9b66ced1b6c44e67e0e3ba6b": "(u^2+v^2)^n = u^n+v^n.\\,", "0087f8f28b29f87e843973201011c49b": "\\sum_{k=0}^{n-1} \\mu^{\\otimes k}(A_k(s,t))\n\\le\\sum_{k=0}^{n-1} \\frac{\\bigl(\\mu(I_{s,t})\\bigr)^k}{k!}\n\\le\\exp\\bigl(\\mu(I_{s,t})\\bigr)", "00884c5e389a26ffde2fb1e712dac2e2": "k.", "00887fabd495a45f79d1e7c9cb7c02ee": "f(R) = a_0 + a_1 R + a_2 R^2 + \\ldots", "0088aea01f674fa148b588b5b7f441a7": " \\zeta(x,y,t) ", "0088e106641908cc6bfff060e2e61501": "\\{ k x : k \\in K \\}", "00890a623786dd585b07fa38923f0392": " G_x = \\{g \\in G: g \\cdot x = x \\} ", "0089102d73f673ad70c3a48c34bfe2ec": "f\\left(r\\right) = \\frac{\\left(1 - r^2\\right)^{\\frac{n - 4}{2}}}{\\mathbf{B}\\left(\\frac{1}{2}, \\frac{n - 2}{2}\\right)},", "0089200d6d75460d55a8abd9087b580c": "V = 2\\pi^2 n R r^2 = \\left( \\pi r^2 \\right) \\left( 2\\pi n R \\right). \\,", "008929a9fceab6c14956bee05f48132b": "l_2(\\theta) = \\theta + \\alpha/2", "008953b32e8473e4f9c6e11f36a6aab8": "504 = 2^3 \\cdot 3^2 \\cdot 7", "008962134e77f17fc6b7daee74c10f90": "p = \\operatorname{char}(F)", "0089ee1cbf646ce073b7ab871f9804c2": "{-1 \\choose n} = (-1)^n", "0089fb36fd68801cf2d544380aef3c24": "((\\mathbf{a} - \\mathbf{p}) \\cdot \\mathbf{n})\\mathbf{n}", "008a0dcf2169a1219c1c35ece550f609": "\\operatorname dE_{\\text{i}}(\\omega_{\\text{i}})", "008a30ccbc263d1d59165530204391c4": "\\mathrm{[Cr] = [CrO_4^{2-}] + [HCrO_4^-]+2[Cr_2O_7^{2-}]; pCr=-log_{10}[Cr] }", "008a4740f682e2dc462f3a52807b2bdc": "\\bar{\\partial} : \\Omega^{(p,q)} \\rightarrow \\Omega^{(p,q+1)}", "008a57162fb19a48692b5111234e6b2f": "p_0>0", "008a667e4624af2f89b2a3b153b2b1af": " 2\\mu(K)=\\mu(K+v)+\\mu(K)<\\mu(U)\\,", "008a9a3318419402e1f2b47fba0f5e81": "\\begin{align}\nx:\\;\\; \\rho \\left(\\frac{\\partial u_x}{\\partial t} + u_x \\frac{\\partial u_x}{\\partial x} + u_y \\frac{\\partial u_x}{\\partial y} + u_z \\frac{\\partial u_x}{\\partial z}\\right)\n &= -\\frac{\\partial P}{\\partial x} + \\frac{\\partial \\tau_{xx}}{\\partial x} + \\frac{\\partial \\tau_{xy}}{\\partial y} + \\frac{\\partial \\tau_{xz}}{\\partial z} + \\rho g_x \n\\\\\n y:\\;\\; \\rho \\left(\\frac{\\partial u_y}{\\partial t} + u_x \\frac{\\partial u_y}{\\partial x} + u_y \\frac{\\partial u_y}{\\partial y}+ u_z \\frac{\\partial u_y}{\\partial z}\\right)\n &= -\\frac{\\partial P}{\\partial y} + \\frac{\\partial \\tau_{yx}}{\\partial x} + \\frac{\\partial \\tau_{yy}}{\\partial y} + \\frac{\\partial \\tau_{yz}}{\\partial z} + \\rho g_y\n \\\\\nz:\\;\\; \\rho \\left(\\frac{\\partial u_z}{\\partial t} + u_x \\frac{\\partial u_z}{\\partial x} + u_y \\frac{\\partial u_z}{\\partial y}+ u_z \\frac{\\partial u_z}{\\partial z}\\right)\n &= -\\frac{\\partial P}{\\partial z} + \\frac{\\partial \\tau_{zx}}{\\partial x} + \\frac{\\partial \\tau_{zy}}{\\partial y} + \\frac{\\partial \\tau_{zz}}{\\partial z} + \\rho g_z.\n\\end{align}\n", "008b1a26a2217fe7dfd19fbfb8bab404": "\\scriptstyle x_{n+1} \\;=\\; \\frac{x_n}{2} \\,+\\, \\frac{1}{x_n}", "008b977cbc0bd622e23da22e303e3107": "I_v = \\log_2", "008bc43ba83b29504bc182aa7b9357b9": " x^{(2)} =\n \\begin{bmatrix}\n 0.000 & -0.1875 \\\\\n 0.000 & -0.1193\n \\end{bmatrix}\n \\times\n \\begin{bmatrix}\n 0.5000 \\\\\n -0.8636\n \\end{bmatrix}\n +\n \\begin{bmatrix}\n 0.6875 \\\\\n -0.7443\n \\end{bmatrix} \n =\n \\begin{bmatrix}\n 0.8494 \\\\\n -0.6413\n \\end{bmatrix}. ", "008bcc3802c42a9731f0425bd63421c4": "\n\\int x^m \\left(a+b\\,x^n\\right)^p dx = \n \\frac{x^{m-n+1} \\left(a+b\\,x^n\\right)^{p+1}}{b (m+n\\,p+1)}\\,-\\,\n \\frac{a (m-n+1)}{b (m+n\\,p+1)}\\int x^{m-n}\\left(a+b\\,x^n\\right)^pdx\n", "008bd2981aaddb22cd636b66bdbdb486": "t_{A/B} = {\\int_A}^B \\frac{M}{EI} \\bar{x} \\;dx", "008bda504aea93cf0967b4159571e8ed": "\\mathbf{\\mathit{F}}", "008befef6f37a943603ef39d3d673039": "M(1):=\\lbrace 1,\\dots,d\\rbrace", "008c04f9682f741da40cae14aec5d4ee": "c(x) = \\frac{1-\\sqrt{1-4x}}{2x}=\\frac{2}{1+\\sqrt{1-4x}}", "008c0b0b97f245c3871f09555aba25ef": "T ( w[t] ) \\in \\Sigma", "008c2a31b08704c913be54aa60532f1d": "\\ \\Delta H_{vH}(T)= -R\\frac{dlnK}{dT^{-1}}", "008c2d3a1ce1fe187a52d241c017876f": "\\exp y = 1 + y + {1 \\over 2!} y^2 + {1 \\over 3!} y^3 + \\dots = \\lim_{N\\to \\infty} \\sum_{r=0}^N {N! \\over r! (N-r)!} ({y \\over N})^r = \\lim_{N\\to \\infty} (1 + {y \\over N})^N.", "008c9070a3e7cec8fd2d50ac94c09e66": "\\begin{matrix}\n\\left[x_i , p_j \\right] &=& i\\hbar\\delta_{i,j} \\\\\n\\left[x_i , x_j \\right] &=& 0 \\\\\n\\left[p_i , p_j \\right] &=& 0\n\\end{matrix}", "008c9eebf46eca9e41c9013fc59cc320": "1+i = (1+r) (1+\\pi^e)", "008cb707ff5b45a885fdf6804228837d": "F_{\\alpha \\beta} = \\partial_{\\alpha} A_{\\beta} - \\partial_{\\beta} A_{\\alpha} \\,", "008cdd207cd628ca5ec8c88a91047345": "\\phi \\colon S^{2n-1} \\to S^n", "008cf3a04a0fe2cacd8cf9d6065ad0d8": "\n\\psi \\to \\psi '=U\\psi \n", "008d0eb7c4baa772b60d8c3c01730196": "y_i=\\sum_l p_l r_{li},", "008db6e1aca3b942f9ccddabf20703b9": "\\scriptstyle{| d^\\prime \\rangle , \\ | s^\\prime \\rangle}", "008dd0838341b126a7fc9f80b8e1039f": "b_3", "008e0fd066ddf76297e8ce6ca41eea1d": "C_{}^{}", "008e306ad46f9f8427da6cbaae1c04ae": "\\left\\{ \\begin{pmatrix}\n 1 & x & z\\\\\n 0 & 1 & y\\\\\n 0 & 0 & 1\\\\\n\\end{pmatrix},\\ x,y,z \\in \\mathbb{Z}\\right\\} ", "008ec35c8e46ac1830429906dd7800c7": "\\{S_k\\} \\subset G", "008edcb19e3d7caf12c7925d31018929": "L = \\frac{1}{d}[194.4 - 0.162t]", "008f42d8b190bb3f0e4986f32541c4bf": "weight_i", "008f693046e7ccc149361a57cb40596c": "| L \\rangle ", "008fd8644838f0babbadd5f639875791": "\\phi(z_i,z_{i+1})", "009071732a22f66ece9a28dc61b02e59": "\\mbox{tr} X = \\sum_{\\{i\\}} \\lambda_i e^*_i(e_i). \\,", "0090ab8183aa1428deeff6d4923b5924": "\\upsilon_r\\,", "0090f3e8899492024afb44d709081ec4": "\\mathrm{str}(X) = \\mathrm{tr}(X_{00}) - (-1)^{|X|}\\mathrm{tr}(X_{11})\\,", "009198f5d7b698ceb14b44a81b9499fa": "\n\\psi^\\dagger \\gamma^0 \\psi = 2 \\langle \\bar{\\Psi}\\Psi \\rangle_{S R} \n", "0092851ab1475289854441df38ef5638": " \\text{sara}= r\\cdot \\frac{s^2}{(2^2 - 2)r^2} - \\Big[ r\\cdot \\frac{ s^2}{(2^2-2)r^2}\\cdot \\frac{s^2}{(4^2-4)r^2} -\\Big[ r\\cdot \\frac{ s^2}{(2^2-2)r^2}\\cdot \\frac{s^2}{(4^2-4)r^2}\\cdot \\frac{s^2}{(6^2-6)r^2}-\\cdots\\Big]\\Big] ", "00928bda52f7935d12b019dcacda9fd6": "\\left\\Vert r(T) \\right\\Vert \\leq \\left\\Vert r \\right\\Vert_{X} = \\sup \\left\\{\\left\\vert r(x) \\right\\vert : x\\in X \\right\\}", "0092907fb658877fa806da38ad96fcea": "-\\lambda_{n+1}e_{n+1}", "009296bab85172292e4b1a2391aa2ca8": "\\mathfrak{I} \\vDash \\Phi ", "0092f65c01eeef8c51053657ecf2e9d4": "\n \\begin{align}\n C_{+} &= + \\frac{1}{2} C_0 \\cos{\\left(\\theta - \\frac{\\pi}{4}\\right)},\n \\\\\n C_{-} &= - \\frac{1}{2} C_0 \\sin{\\left(\\theta - \\frac{\\pi}{4}\\right)}.\n \\end{align}\n", "00931e7be1a5c92a5c4f77c90428b2ab": "\\begin{align}\n s & = g(x,u)+\\omega_s \\\\\n \\dot{{x}} & = f(x,u)+\\omega_x\n\\end{align}", "00932f79cb15cc290228b09d758bb627": "w(n,j) = g(y_n|X_P(t^{}_n,j),t^{}_n,\\theta(t_{n-1},j))", "0093a55f284a1beb41832ac953e88581": "A' \\leq B'", "0093c10abe7bc5272a6abdeb3340bf76": "L = 0, F = 0.", "009417c186afcedf81f53493793177d1": " E[Y|X] = \\Pr(Y=1|X) =x'\\beta,", "009425bc73ae2c42a73998a3bcb964d2": "\\displaystyle{{1\\over \\pi}\\left |\\int_a^b {\\sin t \\over t}\\, dt\\right|}", "0094382b75b38eb14a29da862cd12754": "\\boldsymbol\\Sigma_{22}^{-1}", "00948e47613cadeb9121a131170a6474": " \\Delta W = \\int_{V_1}^{V_2} p \\mathrm{d}V \\,\\!", "00949bada66ee0d6400e1c1436d199b1": "\n\\begin{align}\nu = \\operatorname{prox}_R(x) \\iff & 0\\in \\partial \\left(R(u)+\\frac{1}{2}\\|u-x\\|_2^2\\right)\\\\\n\\iff & 0\\in \\partial R(u) + u-x\\\\\n\\iff & x-u\\in \\partial R(u).\n\\end{align}\n", "0094a6c6ea2cc89a880d95abe9d57da5": "\\operatorname{Aff}(A) = V \\rtimes \\operatorname{GL}(V)", "009507a7597f0473f041bb1fbf6f7922": "\\dim f(Z) > n", "0095576b64d0617187cbe451a644022a": "\\mathit{W}_{1-2}+\\mathit{Q}_{2-3}+\\mathit{W}_{3-4}+\\mathit{Q}_{4-1} = 0", "009573de099f0176195cc34a1d74cd00": "\\Phi_{00}=\\frac{M(u)_{\\,,\\,u}}{r^2}", "009585f132416dd4fd5b0ecc4862da1e": "\\mbox{dim } A = n\\,", "0096064837a09b4ff9f4bf35bc16334f": " n > 0 ", "00960935edc55c1cfe049dfb9068697f": "\\tilde{E}_i^a = \\sqrt{det (q)} E_i^a", "00961e8857a00528f8585aecefe6b93a": "\\! (x_1, y_1), \\ldots, (x_m, y_m)", "00966ec70506d450552970ef0e81c794": "\\mathcal{L}=\\frac{1}{2}(\\partial_t\\phi)^2 -\\frac{1}{2}\\delta^{ij}\\partial_i\\phi\\partial_j\\phi - \\frac{1}{2}m^2\\phi^2-\\frac{g}{4!}\\phi^4.", "00968933d49233c3d97f1d0e30a6c2b1": "\\aleph_0 + 4 \\cdot \\aleph_0 = \\aleph_0 \\,.", "0097292bb6d0270b952befc3a0a95249": "\\theta\\in\\Theta\\,\\!", "0097f84e4d935108709946d992c18cd9": "{}E[X_{n+1}|X_1,\\ldots,X_n] \\ge X_n.", "009831967ffc0b0f7f70e153342ce2dc": "C_p = \\frac{(P_m - P_s)L}{AE}", "00983828282a5f9747d1c65fda904761": " \\Theta_{\\pi} ", "0098457f5b516ebc2a71dd02a6d33b57": " t_1 = t_3 = 0, \\; t_4 = t_2^2/4 ", "0098929f79427f1159a6da9916fa1347": "\\frac{d}{dt} \\log_e t = \\frac{1}{t}.", "0098ab864f39da989fe54e22e6b63380": "h_i : X \\to \\{-1,+1\\}", "0098cd3b3fba6cb3178d7b737d7f7b34": "1-\\varepsilon", "0099427a668b698dd94196acde87e495": "2\\le seqs \\le6", "0099d25546919981aca4a5225481b56a": "c_{n} = \\sum_{\\mathbf{i}\\in \\mathcal{C}_{n}} a_{k} b_{i_{1}} b_{i_{2}} \\cdots b_{i_{k}}, ", "0099f5592168eaed97eee19da22a06b1": "d_j \\,", "009a16bae546036e134016ec108f2f5e": "T^{\\mu\\nu}={1\\over 16\\pi G} (g^{\\mu\\nu}\\eta^\\xi_\\eta-g^{\\xi\\nu}\\eta^\\nu_\\eta-g^{\\xi\\nu}\\eta^\\mu_\\eta)\\Omega^\\eta_\\xi\\;", "009a19f7f4737812d772e4192a306a03": "\\varepsilon_m(\\boldsymbol{k}) = E_m - N\\ |b (0)|^2 \\left(\\beta_m + \\sum_{\\boldsymbol{R_n}\\neq 0}\\sum_l \\gamma_{m,l}(\\boldsymbol{R_n}) e^{i \\boldsymbol{k} \\cdot \\boldsymbol{R_n}}\\right) \\ ,", "009a22483ac6a78d97d501ad686a99b8": "\\beth_{0} = \\aleph_0,", "009a879ec10bd307867c8213dd430802": "Q'_{lid} = kA_{lid} \\left ( \\frac{T_b - T_{surr}}{\\Delta x} \\right ) + hA_{lid} \\left (T_b - T_{surr} \\right ) + A_{lid} \\epsilon _{p.p.} \\sigma \\left [ \\left (T_c + \\frac {T_{surr} \\Delta S_{p.p.}}{c_p^{p.p.}} \\right )^4 - T_{surr}^4 \\right ]", "009ac1e61fa36dd671c3e2ec3f322313": "A_{\\nu ; \\rho \\sigma} - A_{\\nu ; \\sigma \\rho} = A_{\\beta} R^{\\beta}{}_{\\nu \\rho \\sigma} \\,,", "009ac90597ad34875b81bebee3c5d62b": "\\left(\\frac{7}{\\sqrt{10}},\\ \\frac{-1}{\\sqrt{6}},\\ \\frac{-4}{\\sqrt{3}},\\ 0\\right)", "009aeb6f91ab0b013d6042a748040d73": "\\left( {{{\\partial s} \\over {\\partial T}}} \\right)_P = {{c_P } \\over T}, \\left( {{{\\partial s} \\over {\\partial P}}} \\right)_T = - \\left( {{{\\partial v} \\over {\\partial T}}} \\right)_P ", "009b45b6731f6596d60061cd9b8138d5": "\n\\Delta \\hat{z}\\ =\\ 2\\pi\\ \\frac{J_3}{\\mu\\ p^3}\\ \\frac{3}{2}\\ \\cos i\\ \\left(\\ e_h\\ (1-\\frac{15}{4}\\ \\sin^2 i)\\ \\hat{g}\\ -\\ e_g\\ (1-\\frac{5}{4}\\ \\sin^2 i)\\ \\hat{h}\\right)\n", "009c16baf1e28508b213dca7d341e659": "\\int_S f g \\, \\mathrm{d}\\mu = \\|f\\|_p\\,.", "009c405b8a6248a30dcbdf70d58f2757": "\\frac{\\partial z}{\\partial x} = 2x+y", "009cd0a8f8dc2ab7263d4ef99ca9715f": "s_{p-2}\\equiv0\\pmod{M_p}.", "009d31cbcf40a1cc2bab52af464ddb35": " C_{D,\\text{induced}} = \\pi A\\!R \\sum_{n=1}^{\\infty} n A_n^2 ", "009de0b1f57210af8d7b220db90ac5cc": "(\\sigma_i)", "009df7d50ec2590cb99c99b08f87a5d0": "\\tau(\\mathcal{H})\\leq 2\\nu(\\mathcal{H})", "009e5791fcbaffcdf91cac58bbf57761": "\\Pr(3;3,6,1)=\\Pr(3;1,3,6)=\\Pr(3;3,1,6)", "009efa14b5e6f3dcef2aa3d81abfe4cc": "\\rho^{\\mathrm{ent}}(X) = \\frac{1}{\\theta}\\log\\left(\\mathbb{E}[e^{-\\theta X}]\\right) = \\sup_{Q \\in \\mathcal{M}_1} \\left\\{E^Q[-X] -\\frac{1}{\\theta}H(Q|P)\\right\\} \\,", "009f00dd75b22d7d959618d5339ed742": "f(x) = \\begin{cases} +x^2, & \\text{if }x\\ge 0 \\\\ -x^2, & \\text{if }x \\le 0.\\end{cases}", "009f4fb1e0486cae6830706cbe42128d": "X_k = \\sum_{n=0}^{N-1} x_n e^{-\\frac{2\\pi i}{N} nk }\n\\qquad\nk = 0,\\dots,N-1. ", "009fbee309e9784672526a00264c93ef": " F = \\{ (x,y) : x \\in \\mathcal{R}^b,\\, y \\in \\mathcal{R}^n,\\; x=y \\}.", "009fd4b226543561f01553326ecbfee8": "T''(t) = Kc^2T(t) \\,", "00a00703dcf8dd553d1384468debf2f3": "N=\\binom{n+d}{d} -1", "00a020df6280a23ac9daf6baf98439e9": "\\omega_i , 1 \\le i \\le n", "00a02689dfc44a622ba4f7906e0469f6": "h_{\\text{out}}(G)\\le \\left(\\sqrt{4 (d-\\lambda_2)} + 1\\right)^2 -1", "00a05ca86bbbff83a90eb6ecb485762e": "(10)_{10} ", "00a10b2b00b93021fbcc49cae4bc0e7e": "(a+b\\sqrt{p^*})(a-b\\sqrt{p^*}) = a^2 - b^2p^* \\in \\beta \\cap \\mathbf Z = (q),", "00a125e0ad76f163105b9a1a97acbafe": "f^{-1}(t)", "00a149b25c8cff2d15e3bdeaef87a6bf": "n_s(\\vec r)\\ \\stackrel{\\mathrm{def}}{=}\\ n(\\vec r)", "00a16782186c3588c2313c1b11a1bcf6": "f(a\\vec{v}) = af(\\vec{v})", "00a18ba5575387e4e4edf85672346d6d": "b=2(\\mathbf{l}\\cdot(\\mathbf{o}-\\mathbf{c}))", "00a1c4642424356e90a07d4bdeb3a369": "\\delta V", "00a1de149855ddb3fd113f4c7bb2e8fb": "\\hat{\\alpha}= - \\frac{1}{\\frac{1}{N}\\sum_{i=1}^N \\ln X_i}= - \\frac{1}{ \\ln \\hat{G}_X} ", "00a225db74c83931acde710cabd1020f": "\\psi(x) =\\frac{d}{dx} \\ln{\\Gamma(x)}= \\frac{\\Gamma'(x)}{\\Gamma(x)}.", "00a2658138798c6630a6c6b75896a8ea": " p \\mapsto q p q^* \\,\\!", "00a2dc92c377a5fcf3f907cf42ef0962": "\\left(\\frac{K_0+a}{1+a}\\right)^\\gamma=\\frac{K_0}{\\phi}.", "00a34cdb4626d2fd31087d976797e802": "\nV_i(\\omega_k) \\rightarrow V_{ik}\n", "00a3567e070481826aedde5b194fd120": "\\Rightarrow^*_{amb}", "00a3b8c17922464b3763885a0a072622": "\\pi_i p_{ij} =\\pi_j p_{ji}, \\,", "00a3e681e7f16483324136c5f343c197": "\\vartheta(x)", "00a3e7c4907d298e04c2705b5217de48": " \\hat{p} = \\frac{n_1}{n}", "00a4060ce409de2a13ba982d9b63055d": "G = \\frac {4 \\pi A_{eff} } { \\lambda^2 } = \\frac {4 \\pi A_{phys} e_a } { \\lambda^2 } \\,", "00a41b5522f83aa7f1dc471f9ba0051a": "c_3=c_1+c_2=|c_1|\\cdot\\left(\\alpha_1+\\alpha_2\\tfrac{|c_2|}{|c_1|}\\right)", "00a467eb4fe6caeadcab17fd68b6d169": "xz\\leq yz", "00a48783273589c885aa79c58705f779": "\nF = A * E^2 \\propto L^2\n", "00a4cfe1a4c0720a2a520713f425e0c4": "D-1", "00a5129b37e70c31ec37a9f1f3b012fe": "\\sum_{n=0}^{\\infty}(n+1)x^n={1\\over(1-x)^2},", "00a51502fb27f23962367cc2d17ce18c": "X_i(\\omega)=\\omega_i", "00a552471b51f76a41fcbc95b4938fd2": "d(\\det(A)) = \\sum_i \\sum_j \\mathrm{adj}^{\\rm T}(A)_{ij} \\,d A_{ij},", "00a554bca784b6cfea952ee3e5f75cb7": "\\operatorname{nil} \\equiv \\operatorname{false} ", "00a59b76ebf9abfb3d9fe3eefeb9e3f6": "L^{q_\\theta}", "00a59dc981df6c75b3538a4ba059633f": "\n{\\partial{L}\\over \\partial q_i} = {\\mathrm{d} \\over \\mathrm{d}t}{\\partial{L}\\over \\partial{\\dot{q_i}}}.\n", "00a5ae7ab7a84d3ba9306ecc2364d6a8": "\\hat{\\xi}^{i} \\rightarrow \\acute{\\hat{\\xi}^{i}}=\\hat{U}^{+}\\hat{\\xi}^{i}\\hat{U}.", "00a67155ff3cd8fab09e943bfe257614": "x_7", "00a67a8d2d4bf0bf959743c81b7aa446": "\\sum_{n=0}^{\\infty} {\\left( \\frac{(-1)^{n}}{2n+1} \\right) }^3 = \\frac{1}{1^3} - \\frac{1}{3^3} + \\frac{1}{5^3} - \\frac{1}{7^3} + \\cdots = \\frac{\\pi^3}{32}\\!", "00a6d384f5362987e87b4ce8b1320bfa": "x*", "00a6dc4d3f87b23761b272ea6b80ce2d": "X_n\\,", "00a78f6c69d27a486ccb1f1d4d2bf147": "|\\psi\\rangle ", "00a793998ab632a05917678ea364f76b": "\\,dN", "00a7b393dbd294b592f68c62459fec49": "A_r \\left ( {\\rm X} \\right ) = \\frac{\\langle m \\left ( {\\rm X} \\right ) \\rangle }{m \\left ( ^{12}{\\rm C} \\right ) / 12} ", "00a7d1ba4a6e1afcbafad38a541a341e": "\\frac{n}{12}", "00a80cce08868ef13e04f34b7f3043fd": "R = Ef/(Ts + Th) ", "00a870853110df52ed102384d3708385": "\\begin{align}\n\\Delta S_F &= \\frac{s - s_i}{c_p} = ln\\left[\\left(\\frac{M}{M_i}\\right)^\\frac{\\gamma - 1}{\\gamma}\\left(\\frac{1 + \\frac{\\gamma - 1}{2}M_i^2}{1 + \\frac{\\gamma - 1}{2}M^2}\\right)^\\frac{\\gamma + 1}{2\\gamma}\\right] \\\\\n\\Delta S_R &= \\frac{s - s_i}{c_p} = ln\\left[\\left(\\frac{M}{M_i}\\right)^2\\left(\\frac{1 + \\gamma M_i^2}{1 + \\gamma M^2}\\right)^\\frac{\\gamma + 1}{\\gamma}\\right]\n\\end{align} ", "00a8c8452e84811dd3222f97d0c094e0": "\n D_j = \\frac{y_U-y_L}{r-1} \\quad (j=i+1,\\ldots,i+r-1).\n ", "00a90587036019f4279b0ec99206f3a7": "\\Pi_n", "00a9bbdd8b6b224a61ab201c9b39ed06": " \\lambda u.x ", "00a9c8c8443a68289eb90415df7d306a": "\\alpha^2,", "00a9cdb637559fcf0fa52b15f0d24067": "p\\sum_{i=1}^n \\left ( \\frac{Y_i - \\hat\\mu \\left (x_i \\right )}{\\delta_i} \\right )^2+\\left ( 1-p \\right )\\int \\left ( \\hat\\mu^{\\left (m \\right )}\\left ( x \\right ) \\right )^2 \\, dx", "00a9cfc6f644a9f8f3258b8864da1c9b": "\\sigma^2 = X^TVX,", "00aa056f604bfb7a08392d451f0a3cf6": "\\varphi_1, \\varphi_2, \\varphi_3, ...", "00aa46e6ebcd45c14cef047bb689f248": "\\gamma= 0.95 (95\\%)", "00aa7d063f46dae0935f3b140e61941d": "\\int_{\\mathbb{R}^n}f\\,dx = \\int_0^\\infty\\left\\{\\int_{\\partial B(x_0;r)} f\\,dS\\right\\}\\,dr.", "00aa8d463550a1ee7942e5dd3330f818": "f(\\gamma,u)", "00ab0b9d2bb48616a1ee5225eecd77df": "\\max(A_1(x_1, \\dots, x_{r-1}), \\dots, A_{n_A}(x_1, \\dots, x_{r-1})) \\leq \\min(B_1(x_1, \\dots, x_{r-1}), \\dots, B_{n_B}(x_1, \\dots, x_{r-1})) \\wedge \\phi", "00ab11b2c84e14c5bc0372acf71d3baf": "(a - b)(a + b) = b(a - b) \\,", "00ab188055de2af9da6158e79db624ad": "{}_{\\ 86}^{220}\\mathrm{Rn} \\xrightarrow{\\ \\alpha\\ } {}_{\\ 84}^{216}\\mathrm{Po}\\ \\mathrm{(55\\ s,\\ 0.54\\ MeV)}", "00ab347375308522c9fc211d16779712": "\\mathbf{N} \\equiv \\mathbf{n}_0", "00ab4f6d5ab07f3403bb7b46b92fbeac": "du = -3u {da\\over a}", "00ab97e57c2e4a4589b34dfa9b6bc551": "\n{\\nabla^2 u -\\dfrac 1{c_0^2}\\frac{\\partial^2 u}{\\partial t^2} + \\tau_\\sigma^\\alpha \\dfrac{\\partial^\\alpha}{\\partial t^\\alpha}\\nabla^2 u\t- \\dfrac {\\tau_\\epsilon^\\beta}{c_0^2} \\dfrac{\\partial^{\\beta+2} u}{\\partial t^{\\beta+2}} = 0.} ", "00aba73ffc448c45f8d1122ee9b3c9d6": "\\{r_1, r_2, r_3,r_4\\}", "00abc855b0107a8e4b9c4a38af54aed6": "\n\\frac{d^{2}\\eta}{d\\tau^{2}} = \\frac{dt}{d\\tau} \\frac{d}{dt} \\left( \\frac{d\\eta}{d\\tau} \\right) = - y^{2} \\ddot{y} = -\\frac{y^{3}}{mr} F(r)\n", "00abd9a9738fa0ab1bd4fe864640ac5f": "n\\log^{O(\\log k)}n ", "00ac8c8a2346f2a39cc30536fc519d74": "\\frac{T_A}{T} = \\bigg(\\frac{P_A}{P}\\bigg)^{(k-1)/k}", "00acac7e9220e49e733584596a5f11e7": "\n\\{x, p_x\\}_{DB} = \\{y, p_y\\}_{DB} = \\frac{1}{2}\n", "00acc68c74cb8ce7b35958b2a46e1f5d": "\\epsilon_S", "00acc9ac69fc3f2d366801f96e53c565": " (\\hat{k} , \\hat{l})", "00ace3be08aece29574b1c573b12f1f0": "\\text{GF}(2)^n", "00ad2f6e3b361d991d10c82a582bcf5a": "a_2 = \\frac{-b_1 + \\sqrt{b_1^2 - 4 b_2 b_0}}{2 b_2}, \\!", "00ad646ae19cb465bf7119d513412743": " (1-X) \\sim \\textrm{Kumaraswamy}(a, 1)\\,", "00ad734308e565a05b76573ee16fce9d": " A_{i_1 i_2 \\cdots } + B_{i_1 i_2 \\cdots} = C_{i_1 i_2 \\cdots} ", "00ad89837a9fd8ba452937e8cb62cb70": "\\Phi_{\\text{P}} (x)=\\frac{ m}{4 \\omega^2} \\left[g (x)\\right]^2", "00adcb82c4f67853e8c543504656cd0c": "\\frac{\\lambda_c}{d} = \\pi v Z_0 C", "00ae016ab9b477f5e9eceaa787a7be83": " [\\phi, L_z] = i \\hbar \\ \\psi(\\phi) \\quad (8) ", "00ae3c1c548819e0a5af11b628c731d7": "\\exp X = e^X = \\sum_{n=0}^\\infty {\\frac{X^n}{n!}}.", "00ae4809938cb083caa9c3b61e1fcde4": "\\tilde{\\mathit{A}}\\subseteq\\mathbb{R}", "00ae48d6eac642900416e0978697565d": "j(i) = 1728", "00ae6724da7c06588a062b10129e7c4a": "\\sum_{k=1}^n k! S_2(n,k),", "00ae73e221ae8438c7e9050b0321f9fb": "G_{k, \\sigma} (y)= 1-(1+ky/\\sigma)^{-1/k} ", "00aee84d876ee6e51ad144b53e456586": " |L| \\cdot {2^j \\choose 2} \\leq {n \\choose 2}", "00af7f6512d73e19bf172e3b9a8b875d": "g = \\frac{V}{P}", "00b02d842e499f5d430d91c9fb0e6d25": "a = \\frac {x} {d}", "00b02ebac24fbe8e7858b4e7f5cd2e98": " \\frac{1}{1-z} \\sum_{k=1}^m \\frac{z^k}{k}\n\\mbox{ and }\n[z^n] \\frac{1}{1-z} \\sum_{k=1}^m \\frac{z^k}{k} = H_m\n\\mbox{ for }\nn \\ge m\n", "00b0b9b3a532cbcdad77535a337d5005": "n_c \\sim A + B (p - p_c) + C (p - p_c)^2 + D_\\pm |p - p_c|^{2 - \\alpha}", "00b0e6bd9379de899a741e524e1efac3": "\\textstyle \\{C_{i}\\}", "00b0e846b6f072fabff3bb11adb32af5": "A'(x)u_1(x)+B'(x)u_2(x)=0.\\,", "00b13d3f8df02a71e391fce9b198d45f": "\n{\\mathcal L}_s=-\\frac{1}{2}\\left[\\sigma^2h^{\\alpha\\beta}\\partial_\\alpha\\phi\\partial_\\beta\\phi+\\frac{1}{2}\\frac{G}{l^2}\\sigma^4F(kG\\sigma^2)\\right]\\sqrt{-g},\n", "00b173f71cdad4f4e5401621a19f24cc": "g(x_i|D)", "00b1f489539a947438d556bfbc27f889": "\\Sigma^T \\Sigma", "00b1f6a425ccc8c636dda4b95ae7e6a7": "n>e^{3100}\\approx 2 \\times 10^{1346}", "00b212863f999c8af73aa32e38ae23e4": " K ", "00b281f46653d754535354d0947ebd62": "\\Psi_1 = C_\\text{Ion}\\Phi_\\text{Ion} + C_\\text{Cov}\\Phi_\\text{Cov},", "00b285739a3b02cf66484aa107d8f5da": "x_3= \\sin i \\cdot \\sin \\omega", "00b2cbab416e71fc8fef9b1d69d40f3e": "P_n' = C_{n-1}' \\oplus E_K^{-1}(C_n)", "00b3285c8751e46d6815fe231be45f26": "L(H_B) \\otimes C(X)", "00b38b79c51077f93a85760f804d9b6b": "\\frac {G(x)}{F(x)^n}", "00b3aeceaed7d552c07adacf9cf0e201": "\\forall n\\in\\mathbb N\\colon n\\cdot 1\\le\\xi", "00b463dbda2a23f566e8f81d9c0824ae": "T(s, x) = s(x)", "00b4662abd3732893063c7d52118bff1": "Lu = -(pu')'+qu=-(pu''+p'u')+qu=-pu''-p'u'+qu=(-p) D^2 u +(-p') D u + (q)u.\\;\\!", "00b4e09b9649761dc59a6883e8136a7c": "\\Phi_{abc} = x_a \\otimes x_b \\otimes x_c", "00b501613bc59cc20f9c60a2996c41c1": "\\alpha:H\\rightarrow G", "00b514f874a9bea96964c9df44eafa61": "\\mbox{affinity} = \\alpha[A][B]\\!", "00b51803048eb8b9edf4d0405bdbf331": "\n\\sum_{\\stackrel{1\\le k\\le n}{ \\gcd(k,n)=1}} f(\\gcd(k-1,n))\n=\\varphi(n)\\sum_{d\\mid n}\\frac{(\\mu*f)(d)}{\\varphi(d)},\n", "00b56b0fba4e86b56fa04b2abdc00d76": "\n \\operatorname{Var}[s^2] =\\operatorname{Var}\\left(\\frac{\\sigma^2}{n-1} \\chi^2_{n-1}\\right)=\\frac{\\sigma^4}{(n-1)^2}\\operatorname{Var}\\left( \\chi^2_{n-1}\\right)=\\frac{2\\sigma^4 }{n-1}.\n ", "00b5d8a91cc4d17d1f997fbba2dddff8": "\n(D\\nabla^2\\psi-{\\bold u}({\\bold u}\\cdot D\\nabla^2\\psi))\n", "00b6157255eee3ea721509333534bcf1": "\\mathcal{L}_X Y = [X,Y]", "00b61ea310c446de9872ca46c979294d": "Volume = \\frac{\\pi}{6} \\times L_1 \\times L_2^2", "00b679c1724ef634f99d7959237a9ee6": "G'+*m", "00b6d2f480c9b40fe618f9917868f9b5": "\\left( \n\\begin{smallmatrix}\n\\;\\;\\;1 & 0 & 0 \\\\\n\\;\\;\\;1 & 0 & 0 \\\\\n-1 & 1 & 1 \n\\end{smallmatrix}\n\\right)", "00b6d8509e28d8c213b6f79878b1c687": "\\,L \\preceq M\\,", "00b71e8d40251307824661f54fe74704": "A_m=U^\\dagger \\partial_m U.", "00b7603bca787fe483f240835e48118f": "\\alpha^p \\smile \\beta^q = (-1)^{pq}(\\beta^q \\smile \\alpha^p)", "00b77e7e542a1b87d284e0c82f74b268": "\\pi_i = 2^{-N} \\tbinom Ni", "00b7f809d353a2d63350999bc4ad696d": "~~~~~U,V,\\{N_i\\}\\,", "00b7ffe43d793c9f4a697c6f2434bdb9": " z(\\infty) = \\frac{a(1-Q)-b}{aQ} ", "00b812e8b8737c6c5e614149e14930c7": "\\,d(X_tY_t)=X_{t-}\\,dY_t + Y_{t-}\\,dX_t+\\,dX_t \\,dY_t,", "00b83964f5f4e4cc74ce5e79d08753eb": "t_0^{\\frac{n}{n+1}}=({x_1 \\cdots x_n})^{\\frac{1}{n+1}},", "00b85b6e01c4bc53e0ea8cedc1b1ba71": "I \\stackrel{\\sim}\\to A_5 < S_5", "00b8a470bc756c7d053a4711b9a7b6ee": " v = \\frac{V_{\\max}[S]}{K_{m} + [S]}", "00b8c2978829c03f88bb7e05abb4da07": "s^2=\\frac{\\varepsilon^\\prime\\varepsilon}{n - p}.", "00b8dc5c0c1b1d0e68f0937ee77cc768": "I_3=\\frac{1}{2}[(n_u-n_\\bar{u})-(n_d-n_\\bar{d})],", "00b8fc4327416d1d8dfb9ead450412fb": "{}^{\\rm T}", "00b96c916f45f7efc4cf7ef334160561": "E^{\\star}", "00b9ff0a65b3004a3824631ea96c4312": "w(X)\\triangleq\\,", "00ba8bf73fc24298f04c5ced60e538d8": "\n\\beta_{k} = \\omega_{k}^{2} + \\frac{1}{4}\n", "00ba98ef451a5b3614d4b1cd2be24e33": "\\forall x\\,(x \\neq \\varnothing \\rightarrow \\exist y \\in x\\,(y \\cap x = \\varnothing))", "00bb4fbbf623dfb74a06868dd7e77789": "\nx = a \\ \\cosh \\mu \\ \\cos \\nu \\ \\cos \\phi\n", "00bb5fcd96c00665ed7216ecfd87c760": "(\\sqrt{p_1}, \\cdots ,\\sqrt{p_n})", "00bb83d631620faae8005e751c1b09e4": "2Pt(+O_2)\\rightleftharpoons 2Pt; \\;\\; PtO (+CO)\\rightleftharpoons Pt(+CO_2\\uparrow).", "00bc06d20b176a7590ab85e4c4109ecb": "a_n = \\frac{g_{n + 1}^2}{g_n}", "00bc150a7555c2c920dabcb58aae9719": "\\begin{align}\\binom{-4}{6} &= \\frac\n{-10\\cdot-9\\cdot-8\\cdot-7\\cdot-6\\cdot-5\\cdot-4}\n{1\\cdot2\\cdot3\\cdot4\\cdot5\\cdot6\\cdot7},\\\\\n&=(-1)^7\\;\\frac{4\\cdot5\\cdot6\\cdot7\\cdot8\\cdot9\\cdot10}\n{1\\cdot2\\cdot3\\cdot4\\cdot5\\cdot6\\cdot7},\\\\\n&=\\left(\\!\\!\\binom{-7}{7}\\!\\!\\right)\\left(\\!\\!\\binom{4}{7}\\!\\!\\right)=\\binom{-1}{7}\\binom{10}{7};\\end{align}", "00bc6441475813a9eb844a7642eb35d2": "\\frac {H^2}{\\mu}", "00bceadfb2a61f92952f04849109c872": " = \\frac{32e^4}{4(k+p)^4} \\left( (k \\cdot k') (p \\cdot p') + (k' \\cdot p) (k \\cdot p') \\right) \\,", "00bd1daee8deee2a0aa8025a6249447d": "H_{DR}^{n+i}(M)", "00bd6540355981104f96f819fcf7f3cf": "\\mathrm{sys\\pi}_1 \\leq 6\\; \\mathrm{FillRad}(M),", "00bdbff791648cce14b0647d6f68e215": "\\Beta", "00be62a0bdb8f1ff449ffe4204e32266": "a_i=\\begin{cases} \na \\mbox{ if } a\\in \\Sigma_k \\\\\n\\varepsilon \\mbox { otherwise }.\n\\end{cases}", "00beb505483ea27413e7cff5c1ceac84": "\\rho(u)\\sim\\frac{1}{\\xi\\sqrt{2\\pi u}}\\cdot\\exp(-u\\xi+\\operatorname{Ei}(\\xi))", "00bf412dae60cc909904f011967d6c0f": "\\boldsymbol{s}", "00bf432558a83b0fff96f92cc78e5903": "[-1, 1] \\times [-1, 1]", "00bf4feecbbec752edf2af3e77445d7f": "K > 0.\\!", "00bf7884e9bb0be697e4d75d83c636e3": "a_i = \\sqrt{\\sum \\limits_{j = 1}^{3}\\left (\\frac{\\partial x_j}{\\partial u_i}\\right )^2}", "00bf8fca49478b06c2393e07bd1d6351": "\\left( {} - \\frac{1}{2} \\nabla^2 + V \\right) \\psi = E \\psi \\qquad \\mbox{with} \\qquad V = {} - \\frac{1}{r_a^{}} - \\frac{1}{r_b^{}} \\; .\n", "00bf93081482d3b84ae03460925087b5": "c' \\ll \\bar{c}", "00bfd7f2d95c69816361b32ce6b642c5": " \\phi_{r}\\,= \\phi_{N} ", "00bfe9f1e05b83d7d1d4fbde9234c054": " \\Omega_{n}=\\frac{2\\pi^{n/2}}{\\Gamma \\left (\\frac{n}{2} \\right )} \\,", "00bfea28379eb8b01462876e924f558a": "N_\\beta \\beta + N_r\\frac{d\\mu}{dt} + N_p p = 0", "00c0358623f65485416f7facdc3f0e29": "\\scriptstyle\\mathbf{X}", "00c06f5f6d455dd42012fba388a8f492": "\\pi = \\frac{4}{1.25} = 3.2", "00c0a905ca111dd2c1be3c5e7a47e645": "r=\\frac{{\\rm ln} X_2 - {\\rm ln} X_1}{\\Delta t}", "00c0bc9e189771f3338f271b094ecf1f": "\\frac{1}{\\varepsilon_0 c^2}\\frac{\\partial^2}{\\partial t^2}\\mathbf{P}^{NL},", "00c0bef6100867ab83e2f807fe4e3f77": "\\boldsymbol{\\Pi}^1_{2n+1}", "00c0c1609947a2417500d95a5d8ccd32": "f(x_0), f(f(x_0))", "00c142954a1c191f60016455013875bc": "\nG(2n,2n,2n) \\, = \\, \\bigl[t_1^{2n}t_2^{2n}t_3^{2n}\\bigl] (-1)^{3n} \\bigl(t_1 t_2 + t_1 t_3 +t_2t_3\\bigr)^{3n} \\, = \\, (-1)^{n} \\binom{3n}{n,n,n},\n", "00c1d4a23bb74b1715414e4c510aede7": "\\{x \\geq 1, y \\geq 1\\} ", "00c202de02da2a860d73448b4a381129": "\\mathbf{J^TW\\ \\Delta y}", "00c2b8f91dee216f6d966d2325f5977b": "E_{5} = \\Delta x \\Delta y \\Delta z \\Delta p^{2}", "00c2c62104ec723473cccd30d67bb175": "K_R=\\frac{\\sin(\\gamma)}{\\sin(\\beta)}", "00c2caa4b42b5416f74a0ad24214444a": " \\lambda m,p,q.(\\lambda g.\\lambda n.(n\\ (g\\ m\\ n)\\ (g\\ q\\ n)))\\ \\lambda x.\\lambda y.p\\ x\\ y ", "00c3092d6cd6589202586d7237b3461e": "\\bar V^*", "00c344caab36d9481e834e98b8323acc": "{{{\\hat{\\vec{I}}}}_{\\mathit{i}}}", "00c35a4b55d8067671553d5466c7adaf": "\\mathrm{CFS} = \\max_{S_k}\n\\left[\\frac{r_{c f_1}+r_{c f_2}+\\cdots+r_{c f_k}}\n{\\sqrt{k+2(r_{f_1 f_2}+\\cdots+r_{f_i f_j}+ \\cdots\n+ r_{f_k f_1 })}}\\right].", "00c363318e77d4e6e8e7c5dc4639ded4": " \\max_w R(w)= \\max_w \\frac{w^{T}Aw}{w^{T}Bw} ", "00c36eb0977c445260330806fc4eb747": "1 - 1/e", "00c3ca8d01abf5f923081a409096570e": "\\mathcal{N}(\\theta,\\sigma_\\theta)", "00c3e4693330fbba3b5583d98916c007": "\\operatorname{Var}(Y|X=x) = \\operatorname{E}((Y - \\operatorname{E}(Y\\mid X=x))^{2}\\mid X=x),", "00c437fbd49f1bffaeb29cfee7c74828": "\nU(I|J) = \\frac{H(I)-H(I|J)}{H(I)} ,\n", "00c46f60f691d54e4c15ca82cef20abd": "\\text{refresh overhead} = \\frac {0.246\\,\\text{ms}}{64\\, \\text{ms}} =.0038 \\,", "00c48d25333e2da00ac708770f86c606": "u(\\lambda,T)\\partial \\lambda = {8\\pi h c\\over \\lambda^5}{1\\over e^{h c/\\lambda kT}-1}\\partial \\lambda.", "00c4b86ffa5024242c69ab93cf3ffd77": "\\;\\frac{(n+\\delta-1)(n+\\delta-2)\\cdots n}{(\\delta-1)!}\\;", "00c50e95caad16094592549fb9f8173b": "\\epsilon_r, \\mu_r", "00c5368656dba9dbd0a8b29cd5175cde": "L_{\\text{o}}\\,\\!", "00c5664aadb43e0c76cbddcbaeab354d": "(\\boldsymbol{\\sigma}\\cdot \\mathbf{a})(\\boldsymbol{\\sigma}\\cdot \\mathbf{b}) = \\mathbf{a}\\cdot\\mathbf{b} + i\\boldsymbol{\\sigma}\\cdot \\left(\\mathbf{a} \\times \\mathbf{b}\\right)", "00c5f2e03ecffb3bb9a4d0e23bb04433": "T = \\frac{1}{2}[abch_ah_bh_c]^{1/3},", "00c6284781367cce9c24eca48ddc6b4d": "L_{\\alpha} = \\bigcup_{\\beta < \\alpha} \\operatorname{Def} (L_{\\beta}) \\! ", "00c6591c0602abb03d5832073d15ecfd": " T_{r} ", "00c6995d19447eda6861d53156af9b8e": "y(x)=a\\,\\operatorname{cosh}(x/a)", "00c6a639415adf84772a637ad27aac19": "J^n", "00c6bc6ad287f1eceb8ee7a7159c6ad4": "\\sum_f P(h_m^y | f, m, a_1) = \\sum_f P(h_m^y | f, m, a_2).", "00c6ceeac7b79177efb24f261dc5d36f": "{\\operatorname{d}\\Gamma_{(y)}\\over\\operatorname{d}y}", "00c704c7014de82863b96e129bb84f17": "Q= \\begin{pmatrix} {*} & {\\kappa\\pi_C} & {\\pi_A} & {\\pi_G} \\\\ {\\kappa\\pi_T} & {*} & {\\pi_A} & {\\pi_G} \\\\ {\\pi_T} & {\\pi_C} & {*} & {\\kappa\\pi_G} \\\\ {\\pi_T} & {\\pi_C} & {\\kappa\\pi_A} & {*} \\end{pmatrix}", "00c726e5ea52a40b4734ac16674a1fec": "A_{22}^{-1}", "00c74ad991c8ba7cba0a93b9e3a6e7a7": " W_{cu} = W_{S} - W_{c}", "00c758eb717145e027408cbd9a7204ed": "\\min f(\\bold x) = x_1^2+x_2^4 ", "00c77f1e6530daa5e26b5f1e8707ee58": "y_k[n]", "00c7a3396464a0f586e8f19d21426030": "\\nabla\\cdot\\vec{V}=0", "00c7b1f225ffee0b32c619833a234f8c": "J(x_t,u_t)", "00c7fda347c5973fe115f30555bbce33": " \\mathbf{Y} ", "00c8046617f5ed936f0bcb8cd79a21c8": "W_C = \\frac{e^2}{2C}. \\ ", "00c83e3709ab5c1370983c0e0a4f4028": " f(\\varepsilon) ", "00c84b4621cbf4da3aeac00e1374ad7e": "F(Tr(g),\\ X)", "00c893e55fea61a9e204c59337813468": "\n\\begin{align}\n& {} \\quad L\\left( x_1, x_2, \\ldots , x_N, \\lambda_1, \\lambda_2, \\ldots, \\lambda _M \\right) \\\\\n& = f\\left( x_1, x_2, \\ldots, x_N \\right) - \\sum\\limits_{k=1}^M {\\lambda_k g_k\\left( x_1, x_2, \\ldots , x_N \\right)}.\n\\end{align}\n", "00c8b4f2f2a97f1bd9eb9a94b8ea4421": "p(n) = p(0) + K\\sum_{i=0}^{n-1}\\sin(x(i))", "00c8c65b70d5576a583742e1530223c2": " \\| \\mathbf{v} \\times \\mathbf{u} \\| \\leq \\|v\\| \\cdot \\|u\\|.\\, ", "00c90641cf5ae67fd080516a748cddae": " N_D", "00c91d4e3ddd4a273fe8af6a44db4c1b": " N_i", "00c93451009bebf6d145c17b33b0b61d": "T_a = e^\\mu_a T_\\mu \\,", "00c9633c6fc31447b561ff0cec0e8c50": "\\phi=\\tfrac12(1+\\sqrt5)", "00c974b6be3cc1121a014e27602a281e": "r = \\lim_{n\\rightarrow\\infty} \\left| \\frac{c_n}{c_{n+1}} \\right|.", "00c982a871aa5fb28aa4186582d05810": "\\frac {F_{out}}{F_{in}} = \\eta \\frac {d_{in}}{d_{out}} \\, ", "00c9c03d069959e21c69983cf6238113": " \\|v-Pv\\|\\leq (1+\\|P\\|)\\inf_{u\\in U}\\|v-u\\|.", "00ca4220c56859dd1ca71a62e2fc97c9": "\\widehat T", "00ca467dc56cecb5da4be603d7f9582f": "T_{b_1} (T_{b_2} f) = (T_{b_1} \\circ T_{b_2}) f = T_{b_1+b_2} f.", "00ca46c0aa0147bfe7d01dcf3f4657a7": "\\oint \\mathbf{F}\\cdot d\\mathbf{l}=0", "00cab8bb09fe0d5af3f8d9e2b363d1f8": "\\ S_c ", "00cb24fe95a03f1afda4697049b5d046": "Opex_t", "00cc0486308f7424f7540183f4032c16": "S_\\ell=e^{2i\\delta_\\ell}", "00cc312cb3d81d4822bf81d7f2cad8e5": "\\begin{pmatrix} 2&-2 \\\\ -2&2 \\end{pmatrix}", "00cc32f85b209a9b142e09a9274ae106": "\ny\\rightarrow y^5-10 y^3 x^2 + 5 y x^4 + y_0", "00cc3e1c11be4342a07b3de6c9960cf8": "n < 1000_b", "00cc8ba3289130d190f3412b29a48685": "\n\\begin{align}\n B_0 &= \\quad a\\left(1-n+\\frac{5}{4}n^2-\\frac{5}{4}n^3+\\frac{81}{64}n^4-\\frac{81}{64}n^5+\\cdots \\right),\\\\[8pt]\n B_2 &= - \\frac{3}{2}a\\left(n-n^2+\\frac{7}{8}n^3-\\frac{7}{8}n^4+\\frac{55}{64}n^5-\\cdots \\right),\\\\[8pt]\n B_4 &= \\quad \\frac{15}{16} a\\left(n^2-n^3+\\frac{3}{4}n^4-\\frac{3}{4}n^5+\\cdots \\right),\\\\[8pt]\n B_6 &= - \\frac{35}{48} a\\left(n^3-n^4+\\frac{11}{16}n^5-\\cdots \\right),\\\\[8pt]\n B_8 &= \\quad \\frac{315}{512} a\\left(n^4-n^5+\\cdots \\right).\n\\end{align}\n", "00cc99d8927d9d46bf01ea3b4b9b3c77": "\\Omega(n^{k/4})", "00cd6312034e4528828ad17f5cb244a4": "{{y_1} ^2 \\over 2}+{q^2 \\over g{y_1}} = {{y_2} ^2 \\over 2}+{q^2 \\over g{y_2}}", "00cd895fbdebe9bdaa2d2e00777b0fda": "Y\\,\\! ", "00cd8ac8d03943325f2d48850aae3516": "(\\mathrm{det} (q)) q^{ab} = \\sum_{i=1}^{3} \\tilde{E}_i^a \\tilde{E}_i^b,", "00cd8f3fc4f954f02edf2e9b38fc64ad": "\\Re \\left (\\langle T y - m y, z \\rangle \\right) = 0.", "00cdba28214b48a1f791d20ff3774516": " \\omega_a = \\frac{2}{T} \\tan \\left( \\omega \\frac{T}{2} \\right) \\ ", "00cdfd3eba9e2b2ca90c08411366466c": "i = 1, \\ldots, p", "00ce12eb39455e0d4e6192d551e2aa16": "\\,P_1, \\ldots,P_4\\,", "00ce6ee441322cd8fb8e36106653af4f": "\\delta\\geq \\Big(1-R-\\epsilon\\Big) H^{-1}_2\\big(\\frac{1}{2}-\\epsilon\\big) \\sim \\frac{1}{2}(1-R-\\epsilon)", "00cea663cc9f3f4477ee32a282088a0b": "((n+2^{i-1})", "00cedb9e857cecf13657fc572c4abc3d": "a_{\\mathrm{in}}", "00cf4b7ac745cb3f5f1688e17f916e9b": "\\dot{\\mathbf{x}}(t) = A \\mathbf{x}(t) - B K \\mathbf{y}(t) + B \\mathbf{r}(t)", "00cf63659905603862c27f4a1a0af03c": "\\ln f =\\ln (u\\cdot v)=\\ln u + \\ln v.\\, ", "00cf95436c7c77d27e82bc13d8c6aabc": " \\mathbf{g}_{hk\\ell} = h \\mathbf{b}_1 + k \\mathbf{b}_2 + \\ell \\mathbf{b}_3 .", "00cfb43e97ff9b34c9c9e3b7f377b854": "(\\forall x \\ \\neg \\phi(x)) \\leftrightarrow \\neg (\\exists x \\ \\phi(x))", "00cfd502e07da068aa1251041be305ad": "(x\\le y \\and y\\le x) \\rightarrow x = y.", "00cfea03d60df13c7b510407aa538de4": "\\hat x' = R \\hat x R^\\dagger = e^{-i\\hat v \\frac{\\theta}{2}} \\hat x e^{i \\hat v \\frac{\\theta}{2}} = \\hat x \\cos^2 \\frac{\\theta}{2} + i (\\hat x \\hat v - \\hat v \\hat x) \\cos \\frac{\\theta}{2} \\sin \\frac{\\theta}{2} + \\hat v \\hat x \\hat v \\sin^2 \\frac{\\theta}{2}", "00cfec326228bb38450262d954608ea5": "a,b>0", "00cff248b36cd708630d75a0f8d5578d": "\\langle ax_1+bx_2, y\\rangle = a\\langle x_1, y\\rangle + b\\langle x_2, y\\rangle.", "00d01ce332cd24bfb260d7405c784721": "f(x) = \\frac{2}{2^{k/2} \\Gamma(k/2)} x^{k-1} \\exp\\left(-\\frac{x^2}{2}\\right)", "00d02be0050fe6b53904e4a4b469d708": "\\mathrm{adj}(\\mathbf{A})_{ij} = \\mathbf{C}_{ji} \\,", "00d03569e01b8be4b186f40df949ae2d": "F(\\nu)=\\frac{8\\pi h\\nu^3}{c^3}", "00d068fab91da8db80e20baf8367ae5f": "p_i'=\\rho_i c D\\Psi_i,\\qquad i=L,G.\\,", "00d0b5c77159a2b0473eb45c80c6446f": "\\vec{v} = P - R", "00d0ce68fc33da33c1ce0e4f1d9a5066": "0 = - \\rho [\\vec{x},t] + \\epsilon_0 \\nabla \\cdot \\vec{E} [\\vec{x},t] ", "00d157cc401f53c2a7fbfb07bda65556": "\\begin{bmatrix}1 & 1 \\\\0 & 1\\end{bmatrix}", "00d17f8035a0d96ed31b6c7d4f68d407": "\\mu_R", "00d1816c30a2064d8a33fb3b72968a7b": "\\frac{\\partial N}{\\partial t} + \\nabla\\cdot\\vec J = 0", "00d183d55d2b196abb82932ba311b65f": "a\\int_{-\\infty}^\\infty e^{-y^2/c^2}\\,dy,", "00d18957cb2173f2ed89a8c17c18c6d5": "y(t) = \\int_{t_0}^{t} f(\\tau) d\\tau\\,", "00d19c4426d87e5afe94809d4244e5fb": "r(t) \\in L_1[0,T] ", "00d1b7a7930501dc59f8789c05987ea3": "\\{X_{1},\\ldots,X_{n}\\}", "00d1ca048ffe0d52e58241c23cab4edc": "\\begin{align}\nx &= r\\sin\\theta\\cos\\phi \\\\\ny &= r\\sin\\theta\\sin\\phi \\\\\nz &= r\\cos\\theta \\end{align}", "00d24f2a938028537e5ec1e402fb025e": " r = L\\cos^2\\lambda ", "00d2996c35870a082c4257b025d1e05c": "\\left\\langle Q[F]\\right\\rangle =0.", "00d2c33a3d9573d1640ecac2b9b4840e": "f^{64}(4) = G;\\, ", "00d309d510caebc30ceba5f8950bbbd3": "f= \\left( 0.79 \\ln \\left(\\mathrm{Re}_D \\right)-1.64 \\right)^{-2}", "00d325a2fdf76f62cae935baca1795c1": " x(N)={1\\over N+1}\\sum_{n=0}^N T^n(x). ", "00d336848fa1cadb1f0bc947ef5fe26f": "\\left(-4\\right), \\left(-1\\right), 1, 1, 3", "00d37c0bdd7265bef0e6c59d9ed57c7e": "\\boldsymbol{\\mu}_\\text{I} = g_\\text{I}\\mu_\\text{N}\\mathbf{I}", "00d39b47b89dbd09d391dfaf690ff54d": "g(\\lambda z,\\overline{\\lambda}\\bar{z}) = \\overline{\\lambda}^{2s} g(z,\\bar{z}).", "00d416245b926fa94db6707e1bfa26f3": "\\Lambda_n = \\Lambda \\cap \\mathrm{QSym}_n", "00d45880eeb858a9c271cdc1ee503b18": " \\mathbf{f}+\\operatorname{div}\\,\\sigma=0", "00d45e14f600ae168770d540cd1ba279": "\n\\sigma_t \\equiv \\frac{8\\pi}{3}r_e^2\n", "00d4698687efb283b8b2efc7d4eadbd7": "\\mathrm{GF}(q).", "00d4789a3dec5360bb488b32283ae6e5": "0100km s^{-1}", "00d8f4690ab8ba747fbef705e87f85ea": "{RSF} = \\frac {D_{V0.9} - D_{V0.1}} {D_{V0.5}}", "00d91888083257bc9da64df8b5b77495": "\\mathfrak M=\\langle P, G,\\textrm I\\rangle", "00d91d0dbc2f9fe1df573c3630e695da": "\\mathrm{Inv}\\langle X | T\\rangle", "00d91e801972be464fa4a166f9632c82": "\\alpha=1\\,", "00d943bb09a594302694dbd086a23e67": "CH_4 + e^- \\to CH_4^+ + 2e^-", "00d94a624b0292143baac796a7a2c061": "A = \\frac{1}{4}\\frac{N}{V} v_{avg} = \\frac{n}{4} \\sqrt{\\frac{8 k_{B} T}{\\pi m}} . \\,", "00d955c498045606e5500803af522135": "\\mathrm{Taxicab}(5, 2, 2) > 1,024,000,000,000,000,000 = 1.024 * 10^{18}.", "00da16cf18f3f2b19a5dda51c87224f1": "{ \\partial^2 \\psi \\over \\partial t^2 } = c^2 \\nabla^2\\psi ", "00da453affacc526f052e4e8e298f098": " \\delta_X(t) \\ge c \\, t^q, \\quad t \\in [0, 2].", "00da99ec5c19e6d0e85396ae7a00cbd0": "\\tfrac{n}{m}\\,", "00dacbdfd9de8e8a8f1de82579834b1a": "A = \\lbrace q : q^* = -q \\rbrace \\!", "00db0fb33c36c75487183306752b416d": "\\nabla({\\boldsymbol\\mu}\\cdot{\\boldsymbol B})", "00db1fa81e4e01b636cd8d68cab8af6b": "ab+bc+ca=s^2+(4R+r)r,", "00dba71d6100580a7bcebbaf8cbe77c5": "C = 15 d^2", "00dbc826534ab999725ea212f1c69ead": "Y^{\\mu}(\\tau)", "00dbd349ac88a050015b40f536c37b37": "V = \\frac{4}{3}\\pi r^{3}", "00dbe5b634a4e98c045d14c8e50b29a0": "{\\tilde{D}}_{5}", "00dc099636c10a19826ff7617ad552d9": " \\mathcal{F} = \\frac{\\Delta\\lambda}{\\delta\\lambda}=\\frac{\\pi}{2 \\arcsin(1/\\sqrt F)},", "00dc240282b8eb8a8da6e88a060ae253": "x \\in L_{n+1} (\\pi_1 (X))", "00dc888cd757386d5ca7fec6f428fd8f": "P(x_1,x_2) = \\frac{p_1^3-p_1p_2}{2} + \\frac{p_1^2-p_2}{2} \\,,", "00dcdbff0ef7631903745ed151e888eb": "\nH(2^1) = \\begin{bmatrix}\n1 & 1 \\\\\n1 & -1 \\end{bmatrix},\n", "00dd261574c58b34290bf82201117286": "\\cos^{-1}\\langle v_{i}, v_{j}\\rangle", "00dd2adbf272ed1c0c561673c17b0abb": "M_2(\\tau+1) = e^{-2\\times 25\\pi i/168} M_2(\\tau)", "00dd34e39b176f5b5af123e9c219d851": "(14)\\qquad \\theta_{(n)}=\\hat{h}^{ba}\\nabla_a n_b=\\bar m^b m^a\\nabla_a n_b+m^b\\bar m^a\\nabla_a n_b=\\bar m^b \\delta n_b+m^b\\bar \\delta n_b=\\mu+\\bar\\mu\\,.", "00dd434f3b19ea165df7db7617d6b649": "a d^2 + b d + c = 0", "00dd43e22370a716f4aa72e3780e1383": "M _{BC} ^f = - \\frac{qL^2}{12} = - \\frac{1 \\times 10^2}{12} = - 8.333 \\mathrm{\\,kN \\,m}", "00dd441ee2e71bf8cc375cf8676fb415": "g(f(k)) + O(|x|^c)", "00dd5e4951f7aed71b8408ed927f31d4": " y_{ij} = \\mu + \\tau_i + \\epsilon_{ij} ", "00ddcb1d5007fb9bd4f82cacfee3e2f7": "C_1, C_2, C_3, C_4", "00dde2f7a53805b6a926341e3ffe11fe": "\\exp(i\\varphi) = \\cos(\\varphi) + i\\sin(\\varphi) \\,", "00ddfe1c0682e4afa3cdfa3764c60765": "E_{1,1}= 510,260 * \\frac {260}{510,260} * \\frac {10,060}{510,260}", "00de10b46d39cabce52c002b4a33ecc9": "\\dot{\\textbf{x}}=f(\\textbf{x},u) ", "00deaa3867a2ab2e7e90ea94042ebe23": "\\{\\hat{1}, \\hat{5}\\}", "00deb5e44ecc1d9f4d0eec4311dd44e6": "\\pi_k(O)=\\pi_{k+8}(O) \\,\\!", "00debd5d6cdedd0fd8d32f39cb8c00d8": "\\frac{a^x\\Gamma(\\frac{ax+b}{a})}{\\Gamma(\\frac{a+b}{a})}\\,", "00ded4313ff02634b6674dd079500b24": "\\sqrt{|\\Delta_K|}", "00deea9376f926019407f400638e861d": "\\ln (n+1) = \\ln(n) + 2\\sum_{k=0}^\\infty\\frac{1}{2k+1}\\left(\\frac{1}{2 n+1}\\right)^{2k+1}.", "00df09b96a36904ebb578eb1f05f77a4": "cm \\cdot \\sqrt{Hz}/ W", "00df2a8a4c44eb54e671d77699afa8ef": "\nF_1(a,b_1,b_2,c; x,y) = \\frac{\\Gamma(c)} {\\Gamma(a) \\Gamma(c-a)} \n\\int_0^1 t^{a-1} (1-t)^{c-a-1} (1-xt)^{-b_1} (1-yt)^{-b_2} \\,\\mathrm{d}t, \n\\quad \\real \\,c > \\real \\,a > 0 ~.\n", "00df3631d22d38ff63d952305dfdcbf4": "\\{\\alpha_{j1}, \\ldots ,\\alpha_{jm}\\} \\subseteq \\{\\alpha_1, \\ldots ,\\alpha_n\\}", "00df8c39fad899a9c54e7bf525399a9b": "\\oint_C \\left({1 \\over z^5}+{z \\over z^5}+{z^2 \\over 2!\\;z^5} + {z^3\\over 3!\\;z^5} + {z^4 \\over 4!\\;z^5} + {z^5 \\over 5!\\;z^5} + {z^6 \\over 6!\\;z^5} + \\cdots\\right)\\,dz ", "00df972d2c271a82d92810d7c5896ebf": "d(x,y) = \\| f_x - f_y \\|,", "00dfb406b411d6b4f4747a589f08a0bd": "B_O", "00dfd04fcd66ecfaa75cbd6216f8ecfa": " \\varphi(\\mathbf{r},t) = \\frac{1}{4\\pi \\varepsilon_0} \\int\\frac{\\mathbf{\\rho}(\\mathbf{r'},t)}{R}d^3r'", "00dfd3b1d5aa76c34b086e7bd80ad512": "x_P,y_P,a", "00e00e01f453611770fe6e93d8e3a976": " \\ddot{t} + \\frac{2}{x} \\, \\dot{x} \\, \\dot{t} = 0, \\; \\ddot{x} + x \\, \\dot{t}^2 = 0, \\; \\ddot{y} = 0, \\; \\ddot{z} = 0", "00e0135b44d128f41d10a54cfa1582d7": " \\textbf{P} = [T(\\phi, \\mathbf{d})]\\textbf{p} = \\begin{bmatrix} \\cos\\phi & -\\sin\\phi & d_x \\\\ \\sin\\phi & \\cos\\phi & d_y \\\\ 0 & 0 & 1\\end{bmatrix}\\begin{Bmatrix}x\\\\y\\\\1\\end{Bmatrix}.", "00e040d159567545fcc73346bcede176": " \\mathcal{S}", "00e05f27e4728ed01881d0110e63112e": " \\mu^+(E) = \\mu(P\\cap E)", "00e07493b2a973570f63aef3d235fa02": "\\Delta\\lambda_B", "00e078273a56777927d4d1ebad370dd0": "\\bigcup_{k\\in\\mathbb{N}} \\mbox{DSPACE}(2^{n^k})", "00e0a9b8a3df6878b80a59ae9f99da2d": "\n\\int \\exp\\left[ \\int d^4x \\left ( -\\frac 1 2 \\varphi \\hat A \\varphi + i J \\varphi \\right) \\right ] D\\varphi \\; \\propto \\;\n\\exp \\left( - { 1\\over 2} \\int d^4x \\; d^4y J\\left ( x \\right ) D\\left ( x - y \\right ) J\\left( y \\right ) \\right)\n", "00e0bc6b6fa01b4434f090b3b0dc6335": "f\\colon (x,h)\\to(x',h')", "00e0dd01c0d7c832bd2d85ed799213eb": " \\frac{d}{dx}\\arccos(x)= -\\frac{1}{\\sqrt{1-x^2}}, -1 1.96) = 0.025, \\,", "00e9ff58fd26233d196727decbb8299e": "\\psi(n) = H_{n-1}-\\gamma\\!", "00ea04e63b470b5a388a603743ca5e0c": "F(X, Y)", "00ea34016973645f9300ad306688a80c": "\\lambda=1/3^n", "00ea34d26b099e9a8fcb9c46e0c53f85": "\\lambda \\in \\Lambda", "00eaea6b6d04912ef0e1d19dec0c8de6": "\\Lambda(x,\\lambda,\\nu) = f_0(x) + \\sum_{i=1}^m \\lambda_i f_i(x) + \\sum_{i=1}^p \\nu_i h_i(x).", "00eb1b2042bf13d3cd835d1322eeaf6f": "\\sqrt{s_{NN}}=200", "00eb20cabf12f793a27c2a5efc5c83e3": "F^{-1}(p;n,1)", "00eb39716d8b7640272128c3d1efcb5a": "f_\\mathbb{H}(\\alpha)=\\omega^\\alpha.", "00eb86947f7d681c7e38a469d78c4e10": "(h*g)^* = h^* * g^*", "00eb8ddc3e102a880f8830fa40184bdf": "x_n \\to 0", "00eb9bb834af2565c19f18328604c050": "a\\quad", "00ec6670f291a54bd603a01ed1b5d802": "C_{P, el} = \\gamma T = \\frac{\\pi^2}{2}\\frac{k_B}{\\epsilon_F}nk_BT", "00eca1b27a7f6fcdb1c102ad67cfa641": "p_i(s) \\ne p_j(s)", "00ecba9a4dd7bd3d2981a76e7464ea45": " \n\\nu_{k}(\\mathbf{J}) = \\frac{1}{T}\n", "00ecf52d65fb00be76ea52bbc333dd67": "y_4=y_3+h(\\tfrac14k_1 + \\tfrac34k_2)=\\underline{1.335079087}.", "00ed278ec09422df6c1b6c7544693a3a": "\n\\Delta E=\\frac{1}{2}\\alpha_0\\left(T-T_0\\right)P_x^2+\\frac{1}{4}\\alpha_{11}P_x^4+\\frac{1}{6}\\alpha_{111}P_x^6\n", "00ed3794f143bbcf0aea4a78715c707a": " \\theta(\\xi)=\\sum\\limits_{n=0}^\\infty a_n \\xi^n ", "00eda8772cea2311b2a365f89fdfcb9b": "\\mathbf{F}_{\\mathrm{net}} = m\\mathbf{a}_{\\mathrm{cm}}", "00ee06cf2adda9c1fea6cbdeb588ea2f": "\\delta(\\varnothing)=\\varnothing", "00ee2e53e92542458ff31715b7a81ebf": "lim^*", "00ee31b0657b8616be40541c4d326199": "\n\\tan \\theta = \\sin \\lambda \\tan(15^{\\circ} \\times t)\n", "00ee8205d9738aee1e2ee3086ae05f53": " y \\ge 0.398", "00ee92b891492c30771ce8b238d0e5be": "\\left(\\sqrt{\\frac{2}{5}},\\ -\\sqrt{\\frac{2}{3}},\\ \\frac{-5}{\\sqrt{3}},\\ \\pm1\\right)", "00ee9a89b1bf53def17c6ec0901ef41d": "pf = {P_a + P_b + P_c \\over |P_a + P_b + P_c + j(Q_a + Q_b + Q_c)|}", "00eeedc7b69405e57deac906e57c5f19": "j = 2, 3, \\ldots, m\\ ", "00eefb2b6b06be1004f91ffa8db3dce5": "\\tan \\gamma = \\frac {d} {R} \\,;", "00ef41d18ca5f3e9deaf55d719272b28": " W := (W_1, \\dots, W_d)", "00ef434594abd949d326cfe092280abc": "v_i : A \\longrightarrow R_+", "00ef776b74b13504b900b0e68fca544c": "\\frac{\\partial c}{\\partial x} = \\frac{\\partial c}{\\partial \\xi} \\frac{\\partial \\xi}{\\partial x} = \\frac{1}{2 \\sqrt{t}} \\frac{\\partial c}{\\partial \\xi}", "00ef987a5388b0b127138d0aef79b6f1": "\\mathcal R=(<_1,\\dots,<_t)", "00efa6a77deaafdb2502b9c077cde286": "L_{g}L_{f}^{i}h(x)", "00efd6280759fc6e3b506689467d003a": "\\chi={C \\over T}", "00f01f2e549a95f7050c54482197c866": " P(R_{NP} \\cap R_A^c, \\theta_1)= \\int_{R_{NP}\\cap R_A^c} L(\\theta_{1}|x)\\,dx \\geq \\frac{1}{\\eta} \\int_{R_{NP}\\cap R_A^c} L(\\theta_0|x)\\,dx = \\frac{1}{\\eta}P(R_{NP} \\cap R_A^c, \\theta_0)", "00f039d45804b9bcb48cda188a6dc085": "g_i(0) = \\left. \\frac{\\partial f(z)}{\\partial z_i} \\right|_{z=0}", "00f0746da2f28aa1374e48ae048cb4b5": "\\begin{align}\\binom{-r}{k} &= \\frac{-r\\cdot-(r+1)\\dots-(r+k-2)\\cdot-(r+k-1)}{1\\cdot2\\cdot3\\cdot4\\cdot5\\cdots k}\\\\\n&=(-1)^k\\;\\frac{r\\cdot(r+1)\\cdot(r+2)\\cdots(f-2)\\cdot(f-1)\\cdot f}{1\\cdot2\\cdot3\\cdot4\\cdot5\\cdots k}\\\\\n&=(-1)^k\\binom{f}{k}\\\\\n&=(-1)^k\\left(\\!\\!\\binom{f-k+1}{k}\\!\\!\\right)\\\\\n&=(-1)^k\\left(\\!\\!\\binom{r}{k}\\!\\!\\right)\\;,\\end{align}", "00f0ae08d8aa3c5c08a83a108da9c688": "x_{i+1},\\ldots,x_n", "00f16a47475ad1385451f8781b66a7e3": "r_i,s_i\\in \\mathbb{R}", "00f1935351a51f42498a297e61a5cacd": "\\ C-\\text{vertex}= 1 : -1 : -1 ", "00f1a523058441ae4e449e8959edc01b": "\\phi (t) \\to (\\exists x \\ \\phi (x))", "00f1c82d17358dd9b5dfc14705f26f50": " C \\subseteq \\{0,1\\}^t, |C| = n ", "00f1d2f8c59c696529d591a3d697d1e2": " \\lambda^{2p} c_H ( \\lambda^p t, \\lambda^q H) = \\lambda^d c_H(t, H) \\, ", "00f20d86ef06cc0932330c692d8027bb": "\\gamma(i_j)=\\gamma(n)", "00f21c1aafe1f46bf3844636e73bc995": "\\epsilon=0\\,\\!.", "00f27297d54b3aeba08e7ce05172c51e": "P(\\mathbf{s})", "00f2806a43b3c8c594f16bd6c54f139e": "\\tilde{\\kappa}_{tr}=\\scriptstyle -0.4\\pm0.9\\times10^{-10}", "00f2a7fb18ef9f999f11d41d5d06f6cc": "p^2-p+1", "00f2ac1cfefd7f10d8f0f8602e8ada08": "\\mathrm df_x(X) = \\langle (\\mathrm d Y)_x(X),x\\rangle + \\langle Y_x, X_x\\rangle = 0.\\,", "00f2b472121ef098a7da40fcc25bb3e0": "\\theta_{\\text{hr}} = \\frac{1}{2}M_\\Sigma = \\frac{1}{2}(60H + M)", "00f2bef40423a891f0b44fa7b5ef62be": "\\delta_{\\theta}", "00f2d62661d2ba1bfeb24b5a69831f7c": "\\pi^{-n}|F(z)|^2 \\exp(-|z|^2)", "00f2f5f0f7f040bd0228ea0b965dd0f8": "V(y) = \\sup_{\\tau \\le \\tau_\\mathcal{S}} J^\\tau (y) = \\sup_{\\tau \\le \\tau_\\mathcal{S}} \\mathbb{E}_y \\left[ M(Y_\\tau) + \\int_0^\\tau L(Y_t) dt \\right]. ", "00f2f6810ac3900653117fb397b4bcec": "F : X \\to X ", "00f2f97d990f02788d955ded67325c25": "\\displaystyle{Q_y(a)=Q(a)Q(y),\\,\\,\\, R_y(a,b)=R(a,Q(y)b).}", "00f312a0444a815e3379b768a36f9a82": "\\alpha x_i + (1 - \\alpha)x'_i >_i x_i^*", "00f322b619703b467e6a25a969fb3e69": "\\sec(M_i)\\ge -1", "00f35a9b6f60fec19b77496b2355a1a0": "(S\\otimes T)^{i_1\\ldots i_l i_{l+1}\\ldots i_{l+n}}_{j_1\\ldots j_k j_{k+1}\\ldots j_{k+m}} =\nS^{i_1\\ldots i_l}_{j_1\\ldots j_k} T^{i_{l+1}\\ldots i_{l+n}}_{j_{k+1}\\ldots j_{k+m}},", "00f3743aa47d5bf6a020ca4a31e90398": "L = D - W", "00f38015779ac8f08efec2b41add8a5b": "{y^k}'(0)", "00f39c473af512d02fb6bd50fe4f6256": "d_x(p) := d(x,p)\\,", "00f3b6143499cc3b862de3e62062daf5": "s = 2^{0} + 2^{1} + 2^{2} + \\cdots + 2^{63}.", "00f3c9966987607a99730c76bc433930": "\\Delta\\sigma", "00f4bd49d1a7004d90ea380d36c41546": "f_0,\\dots,f_m", "00f5193589c35c3beceb543b25ad3032": "k = \\log_{b} w = \\log_{b} b^k", "00f5739e4f39eed7cbbac7fac1a6117f": "F(x,y)=0\\,\\!", "00f59200f79c84fea9991cbd3819b621": " L(P, t) = \\frac{7}{4}t^2 + \\frac{5}{2}t + \\frac{7 + (-1)^t}{8}. ", "00f5a703c61aa0fe9d1d810367643f36": "x' = x_1 = v/2a ,\\ \\ y' = y_1 + v^2/4a \\ ", "00f65f89c91d577837233107e1c43638": "\\Phi : A \\rightarrow B(H),", "00f6824b92f276a2a322ca8918ac7d0c": "= (\\lambda f.(\\lambda x.f\\ (x\\ x))\\ (\\lambda x.f\\ (x\\ x)))\\ g", "00f6920c4ab9951d3e65397852efa61a": "d \\Phi = d S - \\frac {T d U - U d T} {T^2}", "00f69a8f51d74253c95d4bc78917bfdf": "T \\geq T_0", "00f70e54a98dc30bff28031d0471efd8": "\\{\\phi_i\\}_{i=1}^{\\infty}=\\{\\alpha_i\\}_{i=1}^{\\infty}\\cup\\{\\beta_i\\}_{i=1}^{\\infty} ", "00f74621e6765a1bfdf213ef5caca455": "C_p = C_{p~max}", "00f7604203a5423216dc67057ce0215a": "{\\Gamma}^{*}_n", "00f7788fa5d413bf26f99d916e262801": "\nr_{1}^{2} r_{2}^{2} \\left( \\frac{d\\theta_{1}}{dt} \\right) \\left( \\frac{d\\theta_{2}}{dt} \\right) - \n2a \\left[ \\mu_{1} \\cos \\theta_{1} + \\mu_{2} \\cos \\theta_{2} \\right],\n", "00f7871570c9fbe85f3d77ce2a47ed28": "(r, \\theta, \\phi)", "00f7afd7395deaf6c4c7b1225d1be196": "\\scriptstyle M^{-T}", "00f7c047a2bb558b2d9cacf653e904f9": "g(x)\\partial_x", "00f7c9af6fe1a26a5273fd624549bd78": "\nP_j^n = \\left( \\begin{array}{l} n \\\\ j \\end{array} \\right) j! = \\frac{n!}{(n-j)!}.\n", "00f8110b1646fdf7e83e71ec60699c1c": "H_S = H_{0,S} + H_{1, S} ~.", "00f82de0b4c0784560759a470ba1e2db": "v_n\\in V_n, a(u_n,v_n) = f(v_n)", "00f8b941960594da446f06dcb43c24d5": "\\ T\\Delta G_S^\\circ = T\\Delta H_A^\\circ + T\\Delta H_B^\\circ - T\\Delta S_{AB}^\\circ ", "00f8e2c516640e6fcd650b00d542df09": "\nC_{k} = \\left( \\frac{1}{k} \\right) \n\\int d\\theta^{\\prime}\n\\int d\\rho^{\\prime} \\left(\\rho^{\\prime}\\right)^{k+1} \n\\lambda(\\rho^{\\prime}, \\theta^{\\prime}) \\cos k\\theta^{\\prime}\n", "00f90abe1ab45bcd354b79173a50be07": "D(d) \\wedge \\underline{\\neg D(f(d)) \\wedge D(f(d))} \\wedge \\neg D(f(f(d)))", "00f922907920a1c5bf1ffab1976c3ab4": "M - 1", "00f9741740f00e3a15167a9eabc1141e": "U = - m \\sum G \\frac{ M}{r} ", "00f97a8df6a3e6b2656c97f895be7cea": " \\gamma \\ \\stackrel{\\mathrm{def}}{=}\\ \\partial u_x /\\partial y ", "00f9bfef84d607575d466d8e2cf206be": "\\mbox{then} \\quad U B_1 g = B_2 U g = \\sum_{i=0}^n (B_2^*)^i A h_i.", "00f9f64d586edc538f07598e75bd7e6a": "\\{1, 5, 9, 13\\}.", "00f9f8af4014b9c9ba89a00e688d61a8": "L \\approx 4\\pi R^2\\sigma T_I^4\\frac{l}{R}\\approx \\frac{(4\\pi)^2}{3^5}\\frac{\\sigma}{k^4}G^4\\bar m^4 \\langle \\rho \\rangle l M^3", "00fa1b09d5593180d106bf84f3aeb25e": "\\boldsymbol{\\varepsilon} = \\boldsymbol{0}", "00fa91012c19f237403f36589a916e06": "n(x,y)", "00faf620268a7727621272df0cb5d004": "s_1=\\sum_{i=1}^m \\log x_i", "00fafea58bdae9fde99ae911df2dc687": "y_c = \\left({q^2 \\over g} \\right)^{1 \\over 3}", "00fb22b24186d4bec2293e66f62c28ec": "(A_1A_2)^2-r_1^2-r_2^2 \\, ", "00fb626b41cdaeed91618e2c143511ec": "\n\\begin{align}\n\\Delta \\hat{e}\\ & =\\ \\frac{P}{2\\pi}\\ \\frac {1}{V_0}\\ \\int\\limits_{0}^{2\\pi}\\left( (-\\sin(u)\\ \\hat{k}\\ +\\ \\cos(u)\\ \\hat{l}) \\ F\\ \\cos(u)\\ + \\ 2\\ (\\cos(u)\\ \\hat{k}\\ +\\ \\sin(u)\\ \\hat{l})\\ F\\ \\sin(u)\\right)\\ du \\\\\n& = P\\ \\frac{3}{2}\\ \\frac {1}{V_0}\\ \\ F\\ \\hat{l} \n\\end{align}\n", "00fc431ddc28efbd388ad723f0f0ee25": "\\psi(0,x)", "00fc606c713c687da931b916520aa0ab": "\nV(x_1 ... ,x_N) = V_{1,2}(x_1,x_2) + V_{1,3}(x_2,x_3) + V_{2,3}(x_1,x_2)\n\\,", "00fc95fa70207762082f4c24704e320d": "x^2 - Ny^2 = 1", "00fcd7684e4b7476132e4898a5e1ef1e": "a_{11} x_1", "00fcff732898300c9f752e2a5e1f933d": "2^1 \\times 0.1000_2 - 2^1 \\times 0.0111_2", "00fd1da21e8b4ef31d987665dc575099": "3/2", "00fd89f696a1863b6ca202e1cc674619": "(k+l)", "00fdabb96d5bc35cf466e45d8c0e7ea3": "1-\\left(1-\\frac{1}{d}\\right)\\left(1-\\frac{2}{d}\\right)\\cdots\\left(1-\\frac{n-1}{d}\\right)\\geq \\frac{1}{2}.", "00fe5914b3da55ad956f46423c2e2db6": " M_t = (M_{1,t}M_{2,t}\\dots M_{\\bar{k},t}) \\in R_+^\\bar{k}. ", "00fe6a8d6543b053688d56904a800884": "\\displaystyle e^{ 2\\pi iax} f(x)\\,", "00fecd587aaaf41ac1ae0de228e72700": " \\mathbf{B} = \\mathbf{v} \\times \\frac{1}{c^2} \\mathbf{E} ", "00fed98a386431f51ecf1b300fc572f9": " \\frac{\\partial^2 f}{\\partial x^i \\, \\partial x^j} = \\frac{\\partial^2 f}{\\partial x^j \\, \\partial x^i}", "00fee35d098e7ede04688e054b0bcd95": "\\int_\\gamma \\rho\\,|dz|", "00ff8b525150181f600d4d6469d72e48": "\\varphi = \\begin{bmatrix} \\varphi_{stator} \\\\ \\varphi_{rotor} \\end{bmatrix}", "00ffe4e1b0b3c2080a17caf8b4dd5ec2": "Y_\\mathrm {i \\Pi} = \\sqrt {Y^2 + \\frac{Y}{Z}}", "00fff65d34e1aa4cec757836ae3802fb": "\\mathbf{p^{n+1}=p^n +\\delta p}", "01001680de1dcb97337713b5e92dbbae": "\\neg p \\or q", "01001b39914230da09b6548877a4cb99": "135 = (1 + 3 + 5)(1 \\times 3 \\times 5)", "01002661415b311f875cbb1b0149cabf": "x\\in\\mathbb R", "010056e8dd4c8176092bfd7c448d3ef3": "\\ell_2=r'+a'", "01009cc723b713a37f31197e765611ac": "\n \\lim_{n\\to\\infty} \\frac{\\log |W_n|}{n^2} = h > 0.\n", "0100c57389c7ef9cbf33292dc5557d3f": "\\mathcal{M}_{ij} = \\begin{cases} 1 /L(p_j) , & \\mbox{if }j\\mbox{ links to }i\\ \\\\ 0, & \\mbox{otherwise} \\end{cases}\n", "0100feb2d04bb42c8d668cb8c1f745de": "\\left[ \\begin{alignat}{6}\n1 && 0 && -3 && 0 && 2 && 0 \\\\\n0 && 1 && 5 && 0 && -1 && 4 \\\\\n0 && 0 && 0 && 1 && 7 && -9 \\\\\n0 && \\;\\;\\;\\;\\;0 && \\;\\;\\;\\;\\;0 && \\;\\;\\;\\;\\;0 && \\;\\;\\;\\;\\;0 && \\;\\;\\;\\;\\;0 \\end{alignat} \\,\\right] ", "01013feccf496a2036355f101f8262c0": "\\scriptstyle{a=6.1121\\ \\mathrm{millibar};\\quad\\;b= 17.368;\\quad\\;c= 238.88^\\circ \\mathrm{C}:\\quad\\quad\\! 0^\\circ \\mathrm{C}\\le T\\le +50^\\circ \\mathrm{C}\\;\\;(\\le0.05%)}", "0101b1db64c8b0252ec743708a73f160": "t\\mapsto (t,f(t)).", "0101d89bbbc91b5bcb24aefc9c85d788": "\\psi_l", "01026a06c53da688c72cb0a160dfbfa9": "\n\\mu(x,y) = \\begin{cases}\n{}\\qquad 1 & \\textrm{if}\\quad x = y\\\\[6pt]\n\\displaystyle -\\sum_{z : x\\leq z -1", "010312fc903173511c916ac83e307399": "(A,B;C,D) = \\frac {AC}{AD}.\\frac {BD}{BC} = -1. \\, ", "01033f6e1fef6a26df8d24ae68b5ea94": "{{I}_{OUT}}\\approx \\frac{{{V}_{CC}}-1.4}{R1}", "01034c22987fb23fe2470a98cac59a6c": "\n\\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\ 4 & 2 & 7 & 6 & 5 & 8 & 1 & 3 \\end{pmatrix} =\n\\begin{pmatrix} 1 & 4 & 6 & 8 & 3 & 7 & 2 & 5 \\\\ 4 & 6 & 8 & 3 & 7 & 1 & 2 & 5 \\end{pmatrix} =\n(146837)(2)(5)", "0103d9083b6297bd3abb5e70f74e36fd": "CM\\,", "0103f34b09470ebfb13324efd2ea958a": "\\scriptstyle \\dot m_{01} \\,0\\, p_{21} \\,", "010445a7575314b56e76038a7323011e": "V=U", "01044947b534a2326edc87845aaf5e73": "\\left [\n\\begin{smallmatrix}\n 2 & -1 & 0 & 0 \\\\\n-1 & 2 & -1 & -1 \\\\\n 0 & -1 & 2 & 0\\\\\n 0 & -1 & 0 & 2 \n\\end{smallmatrix}\\right ]", "01045123e83db59cdcae28d0568aefb7": " E'' = \\frac{E \\tau_0 \\omega}{\\tau_0^2 \\omega^2 + 1} ,", "01045a9d6f8880f1083a52e42c4fa3a2": "\n \\underline{\\underline{\\boldsymbol{A}_1}} = \\begin{bmatrix}-1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix} ~;~~\n \\underline{\\underline{\\boldsymbol{A}_2}} = \\begin{bmatrix}1 & 0 & 0 \\\\ 0 & -1 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix} ~;~~\n \\underline{\\underline{\\boldsymbol{A}_3}} = \\begin{bmatrix}1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & -1 \\end{bmatrix}\n ", "010486e2b1be8fe0ef4030d3d106dc74": "\\Gamma^\\alpha_{\\beta\\gamma} = \\frac{1}{2} g^{\\alpha\\epsilon}(g_{\\beta\\epsilon,\\gamma} + g_{\\gamma\\epsilon,\\beta} - g_{\\beta\\gamma,\\epsilon}).", "0104c0311e39f9860b10d55583ae02ea": " A^i {}_{k;\\ell} = A^i {}_{k,\\ell} + A^{m} {}_k \\Gamma^i{}_{m\\ell} - A^i {}_m \\Gamma^m{}_{k\\ell}, \\ ", "0104f8d1b787cbc747f4e3be26a2f983": " W^{1,1}(\\Omega)", "01050f5a6c40e55ce8a661e9e261001c": " \\psi_i^m(0) = 0, \\quad\\qquad \\frac{d\\psi_i^m}{dx}(0) = 1. ", "01055417f16cfe6c9cdafe71d15a7601": "\n\\begin{pmatrix}\n0 & i \\bar{\\partial}\\\\\ni \\partial & 0\n\\end{pmatrix} \n\n\\begin{pmatrix}\n \\bar{\\Psi}^\\dagger P_3 \\\\ \\Psi P_3\n\\end{pmatrix}\n\n= m\n\\begin{pmatrix}\n \\bar{\\Psi}^\\dagger P_3 \\\\ \\Psi P_3\n\\end{pmatrix}\n", "01055c25e62d0efce371faeb74de6790": "p_c = cS_w^{-a}", "010573deb29a21cce0b460e782579ca7": "Z_X = \\int_0^\\infty exp\\left\\{ -\\frac{1}{2} \\left(\\Delta X^T\\left( \\frac{k_B T}{\\gamma} \\Gamma^{-1} \\right)^{-1} \\Delta X \\right) \\right\\}d\\Delta X", "0105c98648eac04e84c78046ebe79281": "\\aleph_{0} = \\omega", "01064cc0b83abd93a55e276f959997f1": "P_{4n+1}", "01066775fb82cd31f4d24ad9f105eb72": "|\\Phi\\rangle_\\nu=|\\Phi_0\\rangle_\\nu \\oplus |\\Phi_1\\rangle_\\nu \\oplus |\\Phi_2\\rangle_\\nu \\oplus \\ldots = b_0 |0\\rangle \\oplus |\\phi_1\\rangle \\oplus \\sum_{ij} b_{ij}|\\phi_{2i}, \\phi_{2j} \\rangle_\\nu \\oplus \\ldots", "01069784c44be3f6a432ae18ad52500a": "b > a", "010699bca1525939ffb3d3afc84724c7": " p <_\\mathcal{O} r ", "01072c6ff185236a9e28ab3740190dba": "h \\circ in = f \\circ Fh", "0107992ec5fe58000025a2b4678726bb": " \\langle y^2 \\rangle = \\frac{1}{P} \\int{I(x,y) (y - \\langle y \\rangle )^2 dx dy}, ", "01083b716768aff86c8863df3ec483c5": "\\frac{| \\text{median} - \\text{mode} |}{\\text{standard deviation}} \\leq \\sqrt{3}", "01087980a48f55dc43c8509c3340e6c7": "\n\\begin{align}\nL &= T-U = \\frac {1}{2} M \\dot{\\mathbf{R}}^2 + \\left( \\frac {1}{2} \\mu \\dot{\\mathbf{r}}^2 - U(r) \\right) \\\\\n &= L_{\\mathrm{cm}} + L_{\\mathrm{rel}}\n\\end{align}", "0108997daf7ca086f0286c453fbd686a": "\n\\left\\langle \\int \\phi( \\boldsymbol{x}, t ) \\, d \\boldsymbol{x} \\, dt \\right\\rangle = \\int \\langle \\phi(\\boldsymbol{x},t) \\rangle \\, d \\boldsymbol{x} \\, dt.\n", "010902b462092577c279b155d9b6c730": "\n\\sqrt{\\frac{1}{N}\\sum_{i=1}^N(x_i-\\overline{x})^2} = \\sqrt{\\frac{1}{N} \\left(\\sum_{i=1}^N x_i^2\\right) - \\overline{x}^2} = \\sqrt{\\left(\\frac{1}{N} \\sum_{i=1}^N x_i^2\\right) - \\left(\\frac{1}{N} \\sum_{i=1}^{N} x_i\\right)^2}.\n", "01092b385f6bf4c8c66a4fe0eb43fce3": "\\nabla\\times\\nabla\\times", "0109adca58e0b5448c672b496c42d700": "2I", "0109b8038d6a1ce251f1b33fc594c43b": "t = \\frac{1}{s}", "0109ff5e08bee125b08f8871f5faf5ef": "\\frac{\\mathrm{d} \\det(A)}{\\mathrm{d} \\alpha} = \\det(A) \\operatorname{tr}\\left(A^{-1} \\frac{\\mathrm{d} A}{\\mathrm{d} \\alpha}\\right).", "010a5867de53b91da45a532bba2c19f1": "\\sigma(E)", "010a602110241800fb96b131799ae444": "\\ V_c", "010a6ae3278e36a894ba2dd26eff1d38": " \\mathbf{a_{\\mathrm{Cfgl}}}", "010a783383fdb44f6c116b76d54dcac5": "\\Rightarrow P_0-M_aTe^{-rT}=0", "010a8d6811366852e1099de8bd2a17e5": "m\\left( x^\\mu \\right) =\\Omega \\tilde{m}_0,", "010aa76873e9d7e8d8f046f780325dce": " \\sum_{i \\neq j} \\pi_i q_{ij} = \\sum_{i \\neq j} \\pi_j q'_{ji} = \\pi_j \\sum_{i \\neq j} q_{ji} = -\\pi_j q_{jj}", "010ac74caa3412b1b118d4fdf7845578": "\\rho(T) = \\rho_0[1+\\alpha (T - T_0)]", "010adf4c6ced9a728df5d15df83737a9": "~A \\triangle B \\triangle C", "010b89e573d00053cdb94543806beef2": "K_\\mu-K^{(0)}_\\mu\\,", "010b9b7813c77c13706e107bc6ed3970": "g(a)", "010ba4b68d115c03803566f5fb23aa33": "\\text{and}", "010bc0d1c798e3c3ffe66e58fd8b9aa1": "({x}_{1}, {x}_{2}, {y}_{1}, {y}_{2}, z)", "010bcb271a01cbc1992ae84a01c933cd": "\\forall f,\\ \\langle \\pi_1 \\circ f,\\pi_2 \\circ f \\rangle = f", "010bd9525b51288f53aa1b96f9df78ba": "\\sum_{n=-\\infty}^{\\infty} x[n] \\cdot \\delta(t-nT) = \\underbrace{\\sum_{k=-\\infty}^{\\infty} X[k]\\cdot e^{i 2 \\pi \\frac{k}{NT}t}}_{\\text{Fourier series}} \\quad\\stackrel{\\mathcal{F}}{\\Longleftrightarrow}\\quad \\underbrace{\\sum_{k=-\\infty}^{\\infty} X[k]\\ \\cdot\\ \\delta\\left(f-\\frac{k}{NT}\\right)}_{\\text{DTFT of a periodic sequence}},", "010ce63d12e72ccf4c6b7734c013ac74": "f(x_0, ..., x_n) = 0", "010d0031e0378397227e26ac79fdbb22": " P V^{\\gamma} = \\operatorname{constant} = 100,000 \\operatorname{pa} * 1000^{7/5} = 100 \\times 10^3 * 15.8 \\times 10^3 = 1.58 \\times 10^9 ", "010d11347ba394e5de251b56ee5cffc5": "S(t) = 1 - e^{- \\rho t} \\ \\frac { \\sin \\left( \\mu t + \\phi \\right)}{ \\sin( \\phi )}\\ ", "010d198ed3e886b2bd899031be35afc8": "I = \\frac{\\pi}{2\\sqrt{2}} \\left(17 - 5^{\\frac{3}{4}} 2^{\\frac{9}{4}} \\right) = \\frac{\\pi}{2\\sqrt{2}} \\left(17 - 40^{\\frac{3}{4}} \\right).", "010d2d61606dea3f3c9ac92797b33cde": "(1,0,0)\\,", "010d67b17db62d4120254fa78329f430": "m\\frac{d^{2}\\mathbf{x}}{dt^{2}}=-\\lambda \\frac{d\\mathbf{x}}{dt}+\\boldsymbol{\\eta}\\left( t\\right).", "010d82cce5da096194db036398fa6268": " \\geq 3 ", "010da5ca94a3a09d473eede273468b57": "y = R \\sqrt{1 - {x^2 \\over L^2}}", "010ded9ac15b567d0d703ee999cda567": " (\\text{Total COE Quota})_{qy} = g.(\\text{Motor vehicle population})_{y-1} + (\\text{Projected de-registrations})_{y} + (\\text{Unallocated quota})_{qy-1} ", "010dfad868e4db1f46382a085599dcf1": "C(f)", "010dfcb5c3f2da6b3324559ac8c4a947": "v = kT + T - \\tau", "010e015cee9b35816b245769a1312f5a": " (1 2)(3 4),\\;(1 3)(2 4),\\; (1 4)(2 3)", "010e1df78a41ec6f33dc926c7e788f53": "a = d \\sin\\alpha \\text{ and }b = d \\sin\\beta. \\, ", "010e22805899e839e8ad0357d6291459": "\\begin{align}\\mathrm{d}^kX &= \\left(\\mathrm{d}x^{i_1} e_{i_1}\\right) \\wedge \\left(\\mathrm{d}x^{i_2}e_{i_2}\\right) \\wedge\\cdots\\wedge \\left(\\mathrm{d}x^{i_k}e_{i_k}\\right) \\\\\n&= \\left( e_{i_1}\\wedge e_{i_2}\\wedge\\cdots\\wedge e_{i_k} \\right) \\mathrm{d}x^{i_1} \\mathrm{d}x^{i_2} \\cdots \\mathrm{d}x^{i_k}\\end{align}", "010e2eac6374591a1bd1915c8aad321b": "\\overline{A_i(\\alpha_1, \\ldots, \\alpha_{dim(A_i)})}", "010e406df2463597c58286a93f8b3160": "5959", "010e6246a2bf3a7312443a891f0d6807": "{\\mathrm{d}H\\over \\mathrm{d}\\theta}=v^2 2\\cos(\\theta)\\sin(\\theta) /(2g)", "010ed64a18f5a752fb8dc04b4cbb15c7": "", "010ee67f9b45e754482ee25dc169e448": "\n\\begin{matrix}\n& & 0\\\\\n &0 & \\\\\n0& &B_{i-2,3}\\\\\n &B_{i-1,2}& \\\\\n1& &B_{i-1,3}\\\\\n &B_{i,2}& \\\\\n0& &B_{i,3}\\\\\n &0& \\\\\n& & 0\\\\\n\\end{matrix}\n", "010f0cc465fa1897532a16c9a7bebccf": "\n K(\\overline{\\alpha}, \\alpha' ) = \\langle \\alpha| \\alpha'\\rangle =\n \\left[{\\mathcal N}(\\vert \\alpha\\vert^2) {\\mathcal N}(\\vert \\alpha'\\vert^2)\\right]^{-\\frac 12}\n \\sum_{n=0}^\\infty \\frac {(\\overline{\\alpha} \\alpha')^n}{\\varepsilon_n!}\\; .\n", "010f1dc08f5b3205173de9b3ef97f8d5": "\\mathbf{a} \\cdot \\mathbf{b} = \\frac{1}{2}(\\mathbf{ab} + \\mathbf{ba}).", "010f45b224c66649fd24a2d41cca9077": "A_3, BC_3,", "010f64129d9fe13f5403409b74e435de": "1+k", "010f7fafaef8d2512449da2d87f661f7": " {d^2 X^\\mu \\over ds^2} = {q \\over m} {F^{\\mu \\beta}} {d X^\\alpha \\over ds}{\\eta_{\\alpha \\beta}}.", "010facc8491c6cc8f90b1b691e331eec": " \\nabla \\times \\mathbf{E} = \\nabla \\times \\left( - \\nabla \\phi - \\frac { \\partial \\mathbf{A} } { \\partial t } \\right) = - \\frac { \\partial } { \\partial t } (\\nabla \\times \\mathbf{A}) = - \\frac { \\partial \\mathbf{B} } { \\partial t }. ", "010fbe9ba54ae3ce64ecc869a1d1f16b": "A(x)=\\sum_{n=0}^{\\infty}A_{n}\\frac{x^{n}}{n!}.", "010fddfcd902a3a23f8062b501729920": "G(S,T) = \\Pi_{i=0}^n (a_iS -b_iT)", "010ff055cb8498e38fd1928cdb931835": "z^{2M}-1 = (z^M - 1) (z^M + 1) \\,", "0110381eee9e40ad90f85de1fd4b4c11": "\\scriptstyle{E}", "01104c023b0e663624f2860e3a834417": "\\mathbf{C} \\otimes \\mathbf{C} \\to \\mathbf{C}", "01108ee28e7d36d435864892ef5d7472": "P_t (f) = f \\cdot \\Omega_t .", "0110bce9efd49901c1280eb57432d9d4": "z=\\zeta^{-2}.", "01110fd744e06804b1349f3028504fb4": "\n \\boldsymbol{u}^{(0)} = \\boldsymbol{x} - \\boldsymbol{X}, \\qquad \\boldsymbol{u}^{(1)}=\\boldsymbol{x'} - \\boldsymbol{x}\n ", "011136a856e1b55439f93cddb217cd15": "Q=\\left(1+\\frac{r^2}{d^2}\\right)^{1/2} ", "011181d1c8b4d961d70145417a40cad4": "\\Phi_V(G,k)=\\min_{S\\subseteq V} \\left\\{|\\Gamma(S)\\setminus S| : |S|=k \\right\\}", "0111dc8658ba9e9ea247f960fc04d49c": "\\{\\dot{x}_1, . . . , \\dot{x}_n\\}", "011207bf80cb24795234a1ac1028d7bd": " \\frac{n! \\cdot e^{-\\tau s}}{(s+\\alpha)^{n+1}} ", "01121327b29599ef36ed6dd2721c5249": "e_1>e_2>e_3", "011216940992ec86880f2fbb4775e8a3": "\\alpha\\in\\mathbb{C}", "01122e120967df8acc9cefaa8e670083": "X \\leftarrow Y \\rightarrow Z,", "011234a9a5e2e0dee096ea7d2e3583f5": "{AE}_{6}", "01123baec994e153a1a611a7722dfd43": "\nr^m \\sin^m\\theta \\sin m\\varphi = \\frac{1}{2i} \\left[ (r \\sin\\theta e^{i\\varphi})^m \n- (r \\sin\\theta e^{-i\\varphi})^m \\right] =\n\\frac{1}{2i} \\left[ (x+iy)^m - (x-iy)^m \\right].\n", "01125bda091746a740325341056ffbb5": "\\frac{1-2p}{\\sqrt{np(1-p)}}", "01132bb6f4147773832a0f398d70b353": "\\sum_{n=1}^\\infty S_n(s)x^n = {sx(1+x) \\over (1-x)^3 + 4sx(1-x)}.", "011334d3fb3f0590045702635200c3b2": "\\alpha=1/N", "011393892db3c5f70775f612f769abe3": "W_nW_{n+1}", "0113bc090b7893e7fe1785bd13a56f66": "\\bold{g}=-\\bold{\\nabla}\\phi_g\\,\\quad \\bold{E}=-\\bold{\\nabla}\\phi_e\\quad", "0113c2181543e683a6e08f0de1b2d2c2": " and \\; E' = \\frac{E}{y_c}", "0113eea8a622904bea55f29f3b0d8b5f": " \\mathbf{M}_{\\rm orb}=\\frac{e}{2\\hbar}\\sum_{n}\\int_{\\rm BZ}\\frac{d^{3}k}{(2\\pi)^{3}}\\,f_{n\\mathbf{k}}\\;{\\rm Im}\\;\\langle\n\\frac{\\partial u_{n\\mathbf{k}}}{\\partial{\\mathbf{k}}}|\\times(H_{\\mathbf{k}}+E_{n\\mathbf{k}}-2\\mu)|\\frac{\\partial u_{n\\mathbf{k}}}{\\partial{\\mathbf{k}}} \\rangle, ", "0115006b38e647df4fd59a12e8ca5ec7": "Pj_{\\mu\\nu} = \\delta_{\\mu\\nu}\\delta_{\\mu m}", "01157d38ff33bacb82305caaf0563185": " \\mathbf{\\hat{n}} \\,\\!", "011587386377fee6fa116ed1e0a7632f": "\\ T_c", "01158c0052a10dbede5392256528da42": "\\,\\! x=x^+-x^-", "0115a1827a27116f17a185a58c8bf45d": "S = 2160 \\text{ miles}", "0115a3cd741a626fa1ccdab6e49377bf": "\\hat x = x_0 - x_1", "0115b2bcf65b76e3a6dc869dbb461f40": " \\mathcal F ", "0115b6d0e853baa84d1d57bfc6cb34d5": "[n]_q x^{n - 1}", "0115ef6c5ad3b514a2a841b68d55fb29": ".\\qquad NP/N,\\; N/N,\\; N,\\; \\underbrace{(NP\\backslash S)/NP, \\quad NP}", "011676bac198587c1ee2747ff140304e": "g(x, y, t) = g(x, t) \\, g(y, t)", "01170a7d6571521be3cca093412de98d": "= \\arctan \\frac{120}{119} + \\arctan \\frac{-1}{1}", "0117523b217d98d3af216a5eeee428bb": "\\{0, 1/(p-1), ... , 1-1/(p-1), 1\\}", "0117813d7915d44dc57392c29a517cf2": "\\begin{align}\nV_1(\\mathbb R^n) &= S^{n-1}\\\\\nV_1(\\mathbb C^n) &= S^{2n-1}\\\\\nV_1(\\mathbb H^n) &= S^{4n-1}\n\\end{align}", "0117bd8f3283f282e12383f128066e0d": "\\textstyle v^2 = \\mathbf{v} \\cdot \\mathbf{v}", "0117e5251f4d4c8d5069db88662ea843": "\\zeta_n\\in \\mathcal{O}_k ", "01184af66a83cfabcec15e5008b7b908": "v \\mapsto \\overline v", "0118c7b56b11e08311e39ddd217b13e4": " \\frac{1}{2T}\\int_{-T}^{T}\\,F(a+it)G(b-it)\\,dt= \\sum_{n=1}^{\\infty} f(n)g(n)n^{-a-b} \\text{ as }T \\sim \\infty. ", "01192796a31d5ddef12c5932427015be": "Z = \\sqrt{{R + j \\omega L} \\over {G + j \\omega C}}", "01194ee3ef2be78544698c591b41cc29": "\\mathcal O(E)", "01195b5c3c65a2e936bbc59624736582": "\\varphi(r) = {\\sin (\\ell r)\\over \\ell \\sinh r}", "0119c33834388b477ea829d9ecdd5f5b": "\\frac{1}{T(s)}\\cdot\\frac{dT(s)}{d\\varphi}=-\\frac{t}{n}.", "0119fba08ad14ba30732514039d870fa": " p_i = q_i \\,", "011a0c4f97e9e5bdb6f22186853bb8b0": "\\{v_1, v_2, \\ldots,v_k,v_{k+1},\\ldots,v_n\\}", "011a670a56fa85d571b453901af53cc2": "n_s = (1 - \\frac {\\beta}{\\beta_0}) \\frac{n_i}{n_0}", "011a673f27b86385a3a6d173aa0a72ee": "p^2=\\mu^2\\sqrt{\\frac{\\lambda}{2}}.", "011a6a6252b5fd1cda01edea029e39b5": "q = \\frac{\\pi}{4} T\\, v(\\theta)\\, \\cos^4\\theta", "011a6dbbbf4c9061f8112708331f0778": "\\ v = k[A][B]", "011a7f737228f34b4db13701be8561fb": "s(h,k)\\,", "011abb8ca80eebdf6873f48e7569541e": "(\\forall F\\subseteq U_p)(QUA(F) \\iff (\\forall x,y)(F(x)\\wedge F(y) \\Rightarrow \\neg x<_p y))", "011aff52d9f198a5ad6e9adbf8309dc2": "\n\\begin{bmatrix}\n x & 1 \\\\\n 1 & x \\\\\n \\end{bmatrix} \\times\n \n\\begin{bmatrix}\n c_1 \\\\\n c_2 \\\\\n \\end{bmatrix}= 0\n", "011b0fa13253a12989641a4f775d6a93": "L(\\hat{y}, y)", "011b6e3f128e0de494b3cf0dbecebdb4": "u_j = | \\langle r A_j \\rangle | ", "011b76b2657900f43aeb2eb6f00f3078": "1 + z = \\left(1 + \\frac{v}{c}\\right) \\gamma.", "011ba3f0db4cd8865f15adc08b9b1e4a": " c_2 = 0.988622465, \\,\\!", "011c329b23a7ef28a2ac2e3acf831905": " \\lim_{z\\to 0} \\frac1{z}\\left\\{\\frac1{\\Gamma(1+z)} - \\frac1{\\Gamma(1-z)} \\right\\} = 2\\gamma", "011c45f9300361dab2a3178eb0de4fc1": "{\\varphi}", "011cc0e22684bf7c68fafa96e57bfea9": "\n\\Pr \\left \\{ \\lambda_\\max \\left ( \\sum_k \\mathbf{X}_k \\right ) \\geq t \\right \\}\n\\leq \\inf_{\\theta > 0} \\left \\{ e^{-\\theta t} \\operatorname{tr} e^{\\sum_k \\log \\mathbf{M}_{\\mathbf{X}_k} (\\theta) } \\right \\}\n", "011cfaea9b775115c2ed7cd4e365c19a": "\\vec{v}_p = \\frac{m}{qB^2}\\frac{d\\vec{E}}{dt}", "011d1eb205a58a64270d1f8db8d71496": "(X-\\alpha)\\cdot H)=C\\cdot P(X)", "011d265e68fcf4fcd5c8bfeadff3d883": "\\Gamma \\vdash \\psi", "011d7055b12ac9c6011b288ea4369e4c": "F(x) = \\sum^{\\infty}_{n=0}f_nx^n", "011d85cc1dadb7c594c567b1bf84ed15": "\\Delta \\chi", "011d91f0f55bdcbfab3374d21a45f206": " m_1 e^{s_1}+m_2 e^{s_2}=m_1 e^{s_3}+m_2 e^{s_4} ", "011d945eff010dfb86e59178d558599d": "O(\\epsilon)", "011db249cbb421ddbd4646f0427b875a": "\\mathrm{d}U = T \\mathrm{d}S - P \\mathrm{d}V.", "011dd4023c135f144b52f7281f0a9283": "\\partial{C}.", "011e597046539907efaf6c364d599b7d": "\\begin{align}\n\\frac{\\partial \\mathbf{u}}{\\partial t}+\\left(\\mathbf{u}\\cdot \\nabla \\right)\\mathbf{u} &= -\\frac{\\nabla p}{\\rho} + v \\nabla^2 \\mathbf{u}\\\\\n\\nabla\\cdot \\mathbf{u} &= 0\\\\\n\\mathbf{u}_\\text{bd} &=\\mathbf{u}_\\text{s}.\n\\end{align} ", "011e6b034711ad7c2533ec10a802a236": "R_a=\\sqrt{MN}=\\frac{a^2b}{(a\\cos\\varphi)^2+(b\\sin\\varphi)^2}\\,\\!", "011f09b611e8dfdf2839e129107b57cb": "\\displaystyle I_M(\\gamma,f)", "011f9f40084dbe619093c6799fd364ca": "p_t", "011fbf27c05a51fd558715cb15ca9e6c": "(a_n X'_n + b_n) \\,", "011fe1abc0dc78ffe7389e8e075b346c": "\\varphi(m, n, p) = m\\uparrow^{p - 1}n.\\,\\!", "012042451fb61a8bc8a16fc2d9496d7a": "\\begin{bmatrix}3 & 1\\\\7 & 5\\end{bmatrix}\n\\rightarrow\n\\begin{bmatrix}0.393919 & -0.919145\\\\0.919145 & 0.393919\\end{bmatrix}", "01207a4ae4426161f9a15ba082019284": "C_N", "01208a2f1c00f274d657da007e07bcad": "\\mathbf{J}^2\\Psi = \\hbar^2{j(j+1)}\\Psi", "0120bb85314e516e67fd9e122b322d02": "w=e^\\phi\\in A_{p}", "0120c11249c5dbe88939b4d3a428bfdd": "H_k^{l,p}=Z_k^l/(B_k^{l+p}\\cap Z_k^l)", "0121170b7b0a6ca554e7f22887a4bbbd": "\\prod_{i=0}^{k-1} (x - z_i)", "01218e3452eea40edd9d230ab0057bd8": "\\gamma_{\\|}", "0121be62d098e0058b48c4f32cc2e579": " \\mu=\\Lambda ", "0121cd1b8f6435a7f637b39c96f742b6": "g=\\frac{4\\pi\\hbar^2 a_s}{m}", "01224cb59366d304002144491499e8c1": " A\\ ", "0122a035f0874d830f4198e2804ccd16": "\\omega_1+\\omega_2", "0122e6feaedd1975ebdea673a294b23d": "\\csc\\left(\\frac{\\pi}{2} - A\\right) = \\sec(A)", "0123454bed8d8b55e908efad5eeae92c": "\\Omega_{\\lambda} = .0001\\ldots_2", "012388ec0c34cb5ea3af47429243ba62": "\\mathrm{V}_4 = \\langle a,b \\mid a^2 = b^2 = (ab)^2 = 1 \\rangle.", "0123a92e3e442417076106d28f7ae281": "\\lim_{q \\rightarrow 1} {}^q\\!D = \\exp\\left(-\\sum_{i=1}^S p_i \\ln p_i\\right)", "01243d3114be219db97be76d0831b7f3": " b_n ", "0124683f164f8d31d6b54164cf7dba14": "R=U\\Sigma'V^*,\\,\\!", "01246900c35b6d82eb37621d9094a5e9": "\\scriptstyle\\bar{x} = \\frac{1}{n}\\sum_{i=1}^n x_i", "01247e727fdc6aca334e4996d78b0ec6": "\\check{f} ", "01249b96b456dc3c29cf0a71502a489c": "\\liminf_{x\\to x_0} f(x)\\ge f(x_0)", "0124aa6c23fdb3fc1f3d174333d49c6a": " \\int_0^2 \\! \\int_{0}^{\\pi/2} \\! \\int_0^2 \\! \\bar{f}(r,t,h) r \\, dh \\, dt \\, dr = 16 + 10 \\pi", "0124b193fbc8b25177f41093f23080f9": "{\\color{Blue}x^2}+{\\color{YellowOrange}2x}-{\\color{OliveGreen}1}", "0124bcf3a2001bc3da170761ee0a4ba5": "w(X,\\tau')\\leq nw(X,\\tau)\\,", "01251a850a18fe5ef4a9a02076099e5e": "J^{\\star}", "01256288155bfb1804d71b253962c5e3": "df/f=dn/(n-1)=1/n", "012571aa32cea51f459f1af58b7ba349": "N \\cdot m^{-2}\\!", "01257cc3117225db04024ce9155f2ca3": "\\sin \\theta", "0125adcbb2ba01b3e0093cea861e567d": "P_\\beta(\\sigma) ={e^{-\\beta H(\\sigma)} \\over Z_\\beta},", "01260f820ff769acbea7ee0dd2d678d1": "(p \\leftrightarrow q) \\vdash ((p \\to q) \\land (q \\to p))", "012678e9bb0cf8d9740d1be60944d8cb": "\nT_T = \\sum_{i=1}^m s_i T_{T_i} + \\sum_{i=1}^m s_i \\log{\\frac{\\overline{x}_i}{\\overline{x}}}\n", "0126a60313e72eefaf6c46737d9b41a1": "d = s_1u_1 + s_2 u_2 +s_3(v_1+v_2+h)", "0126edb486b8c0b0b88b24f0440672ba": "W_{1-i} = W''_{1-i} \\cup B", "012750d4fc9e49702ad721133305438e": "c.", "012763afcb19637d2ec85a93fc8ebcc1": "10^{-12}", "0127bc801b5fc9a97fa76be519913071": "\\operatorname{Aut}_X(X_j) \\to \\operatorname{Aut}_X(X_i)", "012809c2e71817addfcf8ab58d7d62e3": " \\tilde{P}(X_1, \\ldots, X_{n-1})=\\tilde{Q}(\\sigma_{1,n-1}, \\ldots, \\sigma_{n-1,n-1})", "01283759cb5b7d72323d613004d5c6cb": "\\operatorname{pf}\\begin{bmatrix} 0 & a & b & c \\\\ -a & 0 & d & e \\\\ -b & -d & 0& f \\\\-c & -e & -f & 0 \\end{bmatrix}=af-be+dc.", "0129236b0bf87eadf6e0c48815ec29fc": "D=A \\cdot B-C \\neq 0", "0129a9ee48ce2de0728ccc23b5d32fd2": "0 \\le \\delta < 1 - \\frac{1}{q}", "012a6bf5f2d5689d4b61f63efb7d36e9": "x_3=0", "012af98c41fa64353b10d071979f4ae5": "\n \\cfrac{\\partial g}{\\partial g_{ij}} = 2~J~\\cfrac{\\partial J}{\\partial g_{ij}} = g~g^{ij}\n ", "012afeab512cdc3b69024644abf16bff": "\\nabla_{\\mathrm{X}_i} \\mathrm{X}_j = 0 \\, ,", "012b05f6f7bec834265a393fcdb608b7": "|B^*|", "012b08e1dab01b8b9706c324265ad777": "b + c", "012b29917c1c6a0e2d2171090701d548": " Tr(K) \\,\\!", "012b2b76378399778cccd4cad4146838": "d\\omega^j = \\sum_{i=1}^r \\psi_i^j \\wedge \\omega^i", "012baadc023e1e82d21fb22b1aecf7b5": "|\\psi(x,t_1)|^2 = |\\psi(x,t_0)|^2\\quad", "012be5e7056d1da507286f526e4b3bc5": "\\|x'\\| = \\sup_{x \\in X,, \\|x\\| = 1} | \\langle x', x \\rangle |", "012c71509a2548925edcec9c39967a8a": "t \\in \\{0,1,\\dots,T\\}", "012c8cccd5e31063edc5ff7db706695a": " \\mathsf{ZFC} ", "012c91f015fe9872e2612e2fb0c33f03": "[0:1:0]", "012d01b09de6abd503712ac7ab36595d": " f(x)= \\begin{cases} x^3, & \\text{if } x\\in \\mathbb Q \\\\ \n \\arctan{x} ,& \\text{if } x\\in \\mathbb R\\backslash \\mathbb Q \\\\\n \\end{cases} ", "012d35d00b383e446f3f084fa0cff8fa": "K_{-0} \\ \\stackrel{\\mathrm{def}}{=}\\ K_{--} \\cup K_{0}", "012dfd4f0d3c6100c8810ad0b61389c8": "t=D\\,T", "012e25daf4b340530125e7655d29e5b2": "(p_1, p_2, \\dots, p_n)", "012e30acfe0f610448dce473af2107a9": "x_{n+1} = \\frac{x_n}{8} \\cdot (15 - y_n \\cdot (10 - 3 \\cdot y_n)).", "012e71358bcdf91b0dd0cdeb1e887aad": " V = \\sum_i \\left. v^i \\frac{\\partial}{\\partial v^i} \\right|_{(x,v)}.", "012e794869b8318a9c5c7bc810a12fbe": "\\mathit{H}\\mathit{H}^*", "012ea3637d253a7387d80d824b8b5876": "i = 0,1", "012ea4761337a8a050b97a456aebd691": "\\scriptstyle\\boldsymbol{f}(\\boldsymbol{x}) = \\left( f_1(\\boldsymbol{x}), f_2(\\boldsymbol{x}), f_3(\\boldsymbol{x}) \\right)", "012eaa0ffeb592014ddd33f1f0a8466a": "\\displaystyle{[(a_1,T_1,b_1),(a_2,T_2,b_2)]=(T_1a_2-T_2a_1,[T_1,T_2]+L(a_1,b_2)-L(a_2,b_1),T_2^*b_1-T_1^*b_2)}", "012eb411e6f12e33648440ca8b078a34": "z_o= \\frac{{\\frac{F_o} {m}}}{\\sqrt{(\\omega_n^2 - \\omega^2) + (\\frac{\\omega_n \\omega} {Q})^2}}, \\; \\theta=\\arctan\\left [\\frac{\\omega_n \\omega} {Q(\\omega_n^2 - \\omega^2)} \\right ]\\,\\!", "012eb63873c1483f3d0c45fabeaa5392": "m_t \\; = \\; M(u_t,v_t) \\; = \\; \\mu u_t^a v_t^b", "012f8a1247eb79e8f0a2dbdf34ac7285": "T(n)\\in O(n^2) \\, ", "012f8be8085d9a15d7e98ad5095835fb": "\\mathbf{B} = \\mathbf{A}_q", "012fe8748ee1f4ab919629265a10db9a": "a^{n-1} \\equiv 1 \\pmod{n}", "01301819a754ae52e9cb29cd2f99f39f": "y = \\int_0^L \\sin s^2 ds", "0130481a486fff641d732f80c081debb": "\\ \\mathbf{A}^3 - \\mathrm{I}_A \\mathbf{A}^2 +\\mathrm{II}_A \\mathbf{A} -\\mathrm{III}_A \\mathbf{E}= 0", "01306a128b6e5bf1c6818d9e6db26151": " r=1-p, A=\\rho", "01307bec59a2a8c59ea2dee9e62884d7": "\\mathcal{M}_{fg}", "01308b69a6af75f2703b8530739d1aad": "\\scriptstyle 1 = \\sum_{i=1}^{r}S_i Q_i", "0130abb5ce2d09836b11370a1f0b9675": "P A - (P + \\text{d}P) A - (\\rho A \\text{d}h) g_0 = 0 \\,", "0130b9feeffff34774c6552e694f8dd2": "d\\geq d_c=4\\,", "0130d4b66578b7cb583e18ffbf58e966": "\\,l_{x + 1} = l_x \\cdot (1-q_x) = l_x \\cdot p_x", "0130f7556a53fd628ce6c7711a7b6741": "y(t_0)", "013184e4ae039b6ec28d676a46c91160": "t(t-1)(t-2)(t^7-12t^6+67t^5-230t^4+529t^3-814t^2+775t-352)", "013196c1528820c696c05fdd019f2bc1": "(g,1)(h,0)=(gh^{-1},1)", "0131b645b7fb3092f2c6185c5e574abb": "k \\approx aF^b(\\rho T_{2lm})^c", "0132354d2539ebfd5df65b84a86c147c": "\\frac{1}{2}L_1 \\rightarrow L_1 ", "01323bc7d0450c490d6e7fe0e6d834c3": " \\gamma\\dot{x}(t) = - k( x(t) - x_0 ) + \\xi(t) ", "0132501af7f43013a2238ba00589a8ea": "L(a_1,\\ldots,a_n)", "01327f8d65f79d07ca14f8009102022e": "conc(\\langle a \\rangle, conc(\\langle b \\rangle, S, \\langle b \\rangle), \\langle a \\rangle)", "013281a45bcd3f3b0be61a2925d85467": "\n \\hat\\theta = \\operatorname{arg}\\min_{\\theta\\in\\Theta} \\bigg(\\frac{1}{T}\\sum_{t=1}^T g(Y_t,\\theta)\\bigg)' \\bigg(\\frac{1}{T}\\sum_{t=1}^T g(Y_t,\\theta)g(Y_t,\\theta)'\\bigg)^{\\!-1} \\bigg(\\frac{1}{T}\\sum_{t=1}^T g(Y_t,\\theta)\\bigg)\n ", "01328eae0ef136dadbc4e8035cf57e95": "z \\mapsto \\frac{az+b}{cz+d}\\;\\;\\;\\;\\mbox{ (where }a,b,c,d\\in\\mathbf{R}\\mbox{)}.", "0132c942f2b5ca4b5cf2451a37f81760": "\\Phi(v_i,z)", "0132d9f378a8d8f7ecb7c048653c4f0c": "\\mathrm{not}~s", "0132dec062ea905a7a546d908745115e": "[h_i,f_j]=-c_{ij}f_j\\ ", "01335a55c757948d19b802db16cbf961": "\\cot \\theta \\,\\! .", "01336f72f56be6d66c128e12b1710ada": "D_j, j=1,\\cdots,N", "0133fbb1b33d299c11fd161f2dca2193": "\\left(\\frac{a}{n}\\right) = \n\\begin{cases}\n\\;\\;\\,0\\mbox{ if } \\gcd(a,n) \\ne 1\n\n\\\\\\pm1\\mbox{ if } \\gcd(a,n) = 1\\end{cases}\n", "013408c14b63d227243d789a3e82deb2": "s_0(t)=\\frac{\\alpha\\,e^{\\beta t}\n-\\beta\\,e^{\\alpha t}}{\\alpha-\\beta},\\quad\ns_1(t)=\\frac{e^{\\alpha t}-e^{\\beta t}}{\\alpha-\\beta}\\quad", "013430fa683e50e86ae691586e6b6348": " |x_\\theta\\rangle ", "0134475bc0c73018a4d06bb200daf95a": "\\begin{align}\n e^{ix} &{}= 1 + ix + \\frac{(ix)^2}{2!} + \\frac{(ix)^3}{3!} + \\frac{(ix)^4}{4!} + \\frac{(ix)^5}{5!} + \\frac{(ix)^6}{6!} + \\frac{(ix)^7}{7!} + \\frac{(ix)^8}{8!} + \\cdots \\\\[8pt]\n &{}= 1 + ix - \\frac{x^2}{2!} - \\frac{ix^3}{3!} + \\frac{x^4}{4!} + \\frac{ix^5}{5!} - \\frac{x^6}{6!} - \\frac{ix^7}{7!} + \\frac{x^8}{8!} + \\cdots \\\\[8pt]\n &{}= \\left( 1 - \\frac{x^2}{2!} + \\frac{x^4}{4!} - \\frac{x^6}{6!} + \\frac{x^8}{8!} - \\cdots \\right) + i\\left( x - \\frac{x^3}{3!} + \\frac{x^5}{5!} - \\frac{x^7}{7!} + \\cdots \\right) \\\\[8pt]\n &{}= \\cos x + i\\sin x \\ .\n\\end{align}", "01344dfb9ae3295888fc7757943505b8": "k \\equiv (k \\hbox{ mod } 2^n) + \\lfloor k/2^n \\rfloor \\pmod{2^n - 1}.", "01345567169ac6c885df21b57c5d1b39": "\\displaystyle \\operatorname{Tr}(R(f)) = \\sum_{\\pi} m(\\pi)\\operatorname{Tr}(R(f)|\\pi)", "013464259bd9a3f765d987a56677237c": "(sa)\\div b = \\begin{cases} \ns & \\mbox{if } a=b \\\\\n(s\\div b)a & \\mbox{if } a \\ne b\n\\end{cases}", "0134ce711aba1a5d3734b9e36f77ba51": " \\mathcal{L}^* = \\{ \\mathbf{v} \\in V \\quad | \\quad \\langle \\mathbf{v},\\mathbf{v}_i \\rangle \\in R \\}.", "0134d1a09490d2d081f8ff1c72ed5668": "\\Gamma,x\\!:\\!\\sigma \\vdash t\\!:\\!\\tau", "0134f5dca9c6d943b80f334ba20d441d": "y(t)=y_0 \\left( x - \\frac{1}{5} x^2 - \\frac{3}{175}x^3 \n - \\frac{23}{7875}x^4 - \\frac{1894}{3931875}x^5 - \\frac{3293}{21896875}x^6 - \\frac{2418092}{62077640625}x^7 - \\cdots \\right) \\ \n", "0134fe896f2db72d72ee8faad50ead66": "|(a,b,c)|^2", "013555e4d53232dc5e312301b6b684f1": "\\displaystyle{e_\\alpha(z)={z^\\alpha\\over \\sqrt{\\alpha!}}}", "01356f495dd2fd66b165b161ea7acc6c": "t=t_0", "013572c04c0d30a2f5bb460305929605": "X \\mapsto \\mathcal{P}^\\perp_{n_\\infty \\wedge n_o}\\left( \\frac{X}{- X \\cdot n_\\infty}\\right)", "0135e0d854ad3f435241fd00e79366c6": "\ny = b\\ \\sinh\\ \\mu\n", "0135efc53a1ef0d8b71ebd8bd463323c": "mol Fe_2O_3 = \\frac{20.0 g}{159.7 g/mol} = 0.125 mol\\,", "0135f990c3ed5081e26a1dc50109e6b9": "\n2 \\left\\langle T \\right\\rangle_\\tau = -\\sum_{k=1}^N \\left\\langle \\mathbf{F}_k \\cdot \\mathbf{r}_k \\right\\rangle_\\tau.\n", "0136048f886a13a8dee3dbc967d039e2": "T^i_j", "013624a40fe42347cbab24b181f961d9": "U_{\\mathbf{Q}_p}^{(n_p)} \\subseteq N_{L^\\chi_\\mathfrak{p}/\\mathbf{Q}_p}(U_{L^\\chi_\\mathfrak{p}})", "0136443ddd0a3e0e1ca5e28f7e915067": "f_{xy}(a,b)=f_{yx}(a,b)=\\frac{e^x}{1+y}\\bigg|_{(x,y)=(0,0)}=1.", "0136abaee416482ed746a2d07d3381a8": "\\frac{1}{2}(b^2+c^2-a^2) = \\frac{1}{2}[d^2+c^2-(c-d)^2] = cd.", "0136deb84efa9af07881f5f481ee2151": "\\mathbf{\\theta} = \\begin{bmatrix} \\theta_1 \\\\ \\theta_2 \\\\ \\vdots \\\\ \\theta_M \\end{bmatrix},", "0136e23e1209d9507f0afe19336223e7": "T = \\sum_{j\\in J} T_j.", "0137032f8601ff0bd8f8a9c5de8c1f00": " P_{ij}|\\sigma_i \\sigma_j\\rangle = |\\sigma_j \\sigma_i\\rangle \\,. ", "01372bd994cf6aee276abce370612dda": "\\frac{9}{8}", "01373bf85c08b0b1eef230d11547c93d": "{\\tilde{I}}_{1}", "01373f77e95fc864e269d6387936e36a": "\n\\int_1^M\\frac1{x^{1+\\varepsilon}}\\,dx\n=-\\frac1{\\varepsilon x^\\varepsilon}\\biggr|_1^M=\n\\frac1\\varepsilon\\Bigl(1-\\frac1{M^\\varepsilon}\\Bigr)\n\\le\\frac1\\varepsilon<\\infty\n\\quad\\text{for all }M\\ge1.\n", "01379ffd7c9f52bc373e34fc543c4b1c": " \\frac { \\sum x } { n } > 3 ", "0137d5bdba35d4616da310282828d112": "\\|xy\\| \\leq K \\|x\\| \\cdot \\|y\\|", "0137da2584fce8c4ab6fe12b68f12778": "I(\\mathcal{B})", "013846c9405f2d183f7979fd321a1928": "\\mathrm{Mode}[X] = e^{\\mu - \\sigma^2}.", "01387e617044b4cd37c33fa98a537db7": " \\int d^2\\theta\\; \\lambda_1\\; U^c D^c D^c ", "0138f8ca532ca5d01b5a4eb3c962bdd6": "\\left[\\frac{\\hbar^2(k+K)^2}{2m}-E_k\\right]\\cdot\\tilde{u}_k(K)+\\frac{A}{a}\\sum_{K'}\\tilde{u}_k(K')=0", "013903b9bbe6818a50bc7b29512189a4": " = \\frac{1}{2} \\left[ \\int_{0}^L \\frac{x^2}{L^2} \\rho(x) \\,dx \\right] v^2", "013922ccbb127dbe27c9a978177138bd": "\\nu_2:P^n \\to P^{n^2+2n}.\\ ", "013993d94d704fb68b0ba51eb11a18b7": "U-normalized", "0139a23ca80a4549e4a2f73e25c9302a": " [-\\nabla^4] \\Phi(\\mathbf{x},\\mathbf{x}') = \\delta(\\mathbf{x}-\\mathbf{x}')", "0139c39e61d8a08307dfe610c7467571": "d\\Omega^0(S^1)", "0139d3a2f779afa6965569049f7bd6dd": "I(F)=\\theta(F)\\mathbf{Z}[G_F]\\cap\\mathbf{Z}[G_F].", "0139ed53e26900cdc7a372ed7d81be32": " \\mathbf{u}_2 ", "013a806c5b18a7014d4325dd7fc8e4dd": " c \\in \\mathbb{R} ", "013a8eb52f67ec903e6752c45adbfb33": "F_v(t)=\\frac{M_a}{r}(e^{rt}-1).", "013ac752899599fe44ebf2b906d5a864": "\\mathbf{x}_k=\\mathbf{x}(t_k)", "013b4eb45a78130f797a2dea2b68b27d": "\\nabla_{x,y} f = - \\lambda \\nabla_{x,y} g", "013b56b61e6a2523649feee13d0abe00": "N \\leftarrow pq", "013b595732673993d8f6a29fcedc9499": "\\frac{s}{H_{N,s}}\\sum_{k=1}^N\\frac{\\ln(k)}{k^s}\n+\\ln(H_{N,s})", "013b5c53cbe17e6b04407a139ef7622e": "\\mathfrak{B}(V_+)=k[x]\\qquad \\mathfrak{B}(V_-)=k[x]/(x^2)", "013b5c7f3983c6cf2eb3c287ddd40c76": "h^{-1}\\left( {d \\over dx}\\right) p_n(x) = n p_{n-1}(x).", "013b729bdc2d69ac52cbf745a00e2bb2": "\\begin{bmatrix}\n1 & 1.25 \\\\\n0 & 1 \\end{bmatrix}", "013b9a12d32bc22945f202bbd856a308": "DPO = \\dfrac{ending~A/P}{COGS/day}", "013ba350bad36a45381a4c3468c365ad": "Y_{n-1}", "013c59dc8b95b3395c38812433707626": "D \\approx \\frac{32400}{\\Theta_{1d}\\Theta_{2d}}", "013c7b0046a59ef24d47814c8160f180": "\\mathbf{\\hat{X}}_{k - 1}\n = \\mathbf{X} -\n \\sum_{s = 1}^{k - 1}\n \\mathbf{X} \\mathbf{w}_{(s)} \\mathbf{w}_{(s)}^{\\rm T} ", "013ca4d3d0ca9e09faa9a4a2f9c6ffd8": "E_2^{p,q}", "013d64b225b6fad0f99cbc59325a03c3": "\\sim, \\nsim, \\backsim, \\thicksim, \\simeq, \\backsimeq, \\eqsim, \\cong, \\ncong \\!", "013d75c88956c6d4e232c94bd3e11fcc": "\\theta_{ij}=c_1(y_{ij}^1+y_{ij}^2)", "013d7d5fdc6bc9b4d9b40de734b01a46": "f(x) = \\frac{x^3-2x}{2(x^2-5)}", "013db70019ec75bc0be1f6adebed5348": "S: Y\\to X", "013dd7a65cf9bfed91be0dbeb88c422c": "\\Theta_{\\Gamma_8}(\\tau) = 1 + 240\\,q^2 + 2160\\,q^4 + 6720\\,q^6 + 17520\\,q^8 + 30240\\, q^{10} + 60480\\,q^{12} + O(q^{14}).", "013e372dd4bf309a78c78ec451c3628e": "\\frac{d P}{d T} = \\frac {s_{\\beta} - s_{\\alpha}}{v_{\\beta} - v_{\\alpha}} = \\frac {\\Delta s}{\\Delta v}.", "013e7bc41b539a8fc5c60ec8472b7c8e": "\\theta=\\frac{s}{r}", "013e95f41d324472d1342dff611d9b64": " F_{r}-F_{l}\\,=0", "013edb7a7f459107e469450b6eda9fff": "\\mathbf{q} - \\mathbf{p} = (q_1-p_1, q_2-p_2, \\cdots, q_n-p_n)", "013eeafbc28cf4bcadc09bb91b4a7d51": "n \\quad,", "013f16caff7ad5915666e826c746b9cd": "\\operatorname{wnchypg}(x;n,m_1,m_2,\\omega) = ", "013f1769aaa2cf3168d91e6b066995f0": "R'(W) > 0", "013f5496efefa9c604dfa4a23b0f1f1a": "\\partial(\\sigma \\frown \\psi) = (-1)^q(\\partial \\sigma \\frown \\psi - \\sigma \\frown \\delta \\psi). ", "01404b28718df1e227d2530317ab93dd": "i = 0\\,\\!", "01408c57bdb6e3c5fd797ea9a1b13946": "\\mathrm{[A]}(t) = \\mathrm{[A]}_{0} \\cdot e^{-k\\cdot t}. ", "01410ef47323fdf3853b5d3786197b0f": "d(x_m, x)<\\varepsilon/2", "01410f71538f21258f58a8932aa10cb2": "\\ln B=\\ln \\big(\\lambda(I+K)\\big)=\\ln (\\lambda I) +\\ln (I+K)= (\\ln \\lambda) I + K-\\frac{K^2}{2}+\\frac{K^3}{3}-\\frac{K^4}{4}+\\cdots", "0141643b8d5556400f163c6049a0741e": "u(w_0 + WTA , 1) = u(w_0 , 0).", "01416a661c3418153eb0c0f922b5653d": "\ng_n = \\binom{N+n-1}{n}\n", "0141f319f9169fa31c63fc24d4dfeded": "\\int x^m\\arccsc(a\\,x)\\,dx=\n \\frac{x^{m+1}\\arccsc(a\\,x)}{m+1}\\,+\\,\n \\frac{1}{a\\,(m+1)}\\int \\frac{x^{m-1}}{\\sqrt{1-\\frac{1}{a^2\\,x^2}}}\\,dx\\quad(m\\ne-1)", "01423856a0862b51452642523a8e6997": "D(u,v) = \\frac{\\sum_{i=0}^{72}D_i}{\\sum_{i=0}^{72}w_i}", "01424af9614de9aa07f3932f4576e5e6": "\n{dP_x \\over dt } = c(P_{x+1} - 2P_{x} + P_{x-1})\n\\,", "01424d0db12f64e329b4234095f9ac26": "y_j=\\beta_0+\\beta_1 x_{1j}+\\beta_2 x_{2j}+\\cdots+\\eta_j \\, ", "014264151437df888613e0559ae86350": "\n\\begin{array}{ll}\nd\\in D & \\mbox{the decision being made, chosen from space } D \n\\\\\nx\\in X & \\mbox{an uncertain state, with true value in space } X\n\\\\\nz \\in Z & \\mbox{an observed sample composed of } n \\mbox{ observations } \\langle z_1,z_2,..,z_n \\rangle\n\\\\\nU(d,x) & \\mbox{the utility of selecting decision } d \\mbox{ from } x\n\\\\\np(x) & \\mbox{your prior subjective probability distribution (density function) on } x\n\\\\\np(z|x) & \\mbox{the conditional prior probability of observing the sample } z\n\\end{array}\n", "0142f80ddbc8ec29aba02c8582b99ee3": "\\text{STr}", "014314fe17876564f0e241b4c4a11b77": " H = \\frac{h}{l_c}", "014345ae6ac2bde9bfec0158c4e850e7": " \n\\begin{align}\nR^J_{pq} & \\equiv R_{pq}(\\theta_2)\\, R_{pq}(\\theta_1),\\text{ with} \\\\[8pt]\n\\theta_1 & \\equiv \\frac{\\pi - 2\\phi_1}{4} \\text{ and } \\theta_2 \\equiv \\frac{\\phi_2}{2},\n\\end{align}\n", "01434663733c3165cd88685687e87f8c": " {\\rm full\\;red\\;circle}=\\left\\{X\\mid \\; \\left( (XC),(XD)\\right) \\; = \\theta+k\\pi\\right\\}", "0143754b30f0af816b24a5427d1e7956": "\\mathbb{S}^\\lambda E", "01438fd76ecd96ff355b59de43f5e3ec": "\n\\left\\{ D_{i}, D_{j}\\right\\} = -\\sum_{s=1}^{3} \\epsilon_{ijs} L_{s} ~.\n", "0143aaa26015fff3cefa48a7fd7fd569": "\\ell(\\theta|X,Y) = \\log L(\\theta|X,Y) = \\sum_{i=1}^m \\left( y_i \\theta' x_i - e^{\\theta' x_i} - \\log(y_i!)\\right)", "0143c5e4040a58ba580bc87c042d165d": "b=\\sum_{i=1}^{N}b_{i}<+\\infty", "014400a930b949e0295bcdedf4489bbe": "\\ln w_r^+-\\ln w_r^-", "0144332ce95ebfe5905078ab8fe7596c": "p = \\tfrac{1}{2}", "0144689cf3b51353d42d0924a5dfbd53": "[x:=x+1]x \\ge 4\\,\\!", "0144fff1444f6169eb0a57fde0a7ce17": " e_n = O(h^p) ", "0145132faf44644d666f66456a528e6e": "c,\\!\\ c_{n-1}, c_n, c_{n+1}", "01457426424450f533996762e5f70dd6": "r \\geq -1 /(K-1)", "0145b12d7f30173a17c26272f9e647f5": " m \\geq - n+1", "0145e9ad21bdf0e4020a665891253d82": "\\omega \\in W", "014630b5a2a36b9eba221efd748a5ef7": "2^{65-1}", "01464ae3746d71824e581b72b8f8d7ef": "Loves(g(x),x)", "0146716079826b80a3d251aa9c8a3a7a": "d=-3,-4,-7,-8,-11,-19,-43,-67,-163.\\ ", "014697f828a07bbea10e47ea5765e8b3": "\\circlearrowright", "0146ccb228cab83f16fd6d6c3924d625": "\\mathit{dr}(n)=0 \\Leftrightarrow n=0.", "0146d3d7054dd057cbad9fd5bf13022b": "p=1/2,", "0146d9113ce3837a4b6112b4ae1f6fc0": "b_N,", "0146f443aa1723821a8c6b5e62aef18e": "\\mathbf{k}:=2", "01474699320c261cb8187c2a811c377f": "3\\times 1 + 3\\times 2 = 9.", "0147bbbbab72e5087a9aba9244149c4d": "\\mathbb{P}(X_1 >0, X_2<0)", "0147bbd757bfe2c961495720cec9049e": "|S|\\geq\\sum_{i=0}^{r} {n\\choose i}", "0147f8c7194007e2af9895e02f6014bf": "c_\\alpha", "014827672ea8d9165a63dd11f8ff0710": "\\mathrm{wt}_y(c)=5", "0148385d6a69af88889c1eae177d300f": "m=-1", "01484cb4279fed5d58f6aa2afcdb856a": "k \\leq n", "01488e3b6f08bd71f5366a9724e63920": "(b, a) = \\{x | x \\in \\mathbb{R}, b < x\\} \\cup \\{\\infty\\} \\cup \\{x | x \\in \\mathbb{R}, x < a\\}", "01489ec50aaac6e2d55551d0f5d416cc": "\\bar{X}_n\\to\\frac{1+r}{1-r}", "0148c7ca12630d2b0eb824ea90794844": "= 21 + 15P(3,1) + 21P(2,2) + 6P(1,3) ", "0148d2ff685f599bfb7796f1b58c0c11": " r'=\\sqrt{x'^2+y'^2}", "0149071d3784d921cb7a919fc4f4005c": " \\ C_{D_i} = \\frac{{C_L}^2}{\\pi \\text{AR} e} ", "01496b07a67abfd9d5f70961c1ea489f": "\\begin{align}\n \\mathcal{L}(\\beta,\\sigma^2|X) \n &= \\ln\\bigg( \\frac{1}{(2\\pi)^{n/2}(\\sigma^2)^{n/2}}e^{ -\\frac{1}{2}(y-X\\beta)'(\\sigma^2I)^{-1}(y-X\\beta) } \\bigg) \\\\\n &= -\\frac{n}{2}\\ln 2\\pi - \\frac{n}{2}\\ln\\sigma^2 - \\frac{1}{2\\sigma^2}(y-X\\beta)'(y-X\\beta)\n \\end{align}", "0149875825e15879e6cbe7e57e49f5d6": "\\rho=\\rho(p)", "0149902c0da462b82d0ff292ba3c7172": "\\beta_\\pm", "014a2dc2cfd0a12dc810658b9101af62": "\\qquad\\qquad R = a + bv + cv^2 ", "014a6a73d02ef7e9279c17b3ce8d17d0": "\\scriptstyle {BD = {1 \\over 2}BC}", "014a78b4ae24d42b26c604ef368dc19f": "( X \\in X ) \\to Y ", "014a8e239917a19e72eef967fd050eb3": "x=\\lim_{n\\to\\infty} x_n ", "014ad654d6fdbdf7914c210f3d490eda": "E=kmc^2", "014aec3bd0d1aaa2f1ba56e8d9a928f6": "F (E/As, \\ell/A) = 0, \\!", "014bead22ed0f89588c13d5080403591": " g_1 = 1 - \\frac{L}{R_1} ,\\qquad g_2 = 1 - \\frac{L}{R_2}", "014c497b06e9c0a1db88ca51c81148a0": "w+\\lambda v", "014c5b5767dd0135cf5882331b2b7f51": "(v, u)", "014c963ac0879ae7bfdf132b685494b5": "\\mathbf{P}^1 \\times \\mathbf{P}^1", "014c9cdd3a043e564c9aec42cd9bccd0": "\\int_0^{\\theta}\\log(\\sin x)\\,dx=-\\tfrac{1}{2}\\text{Cl}_2(2\\theta)-\\theta\\log 2", "014cbcdc18b0ba7be280c53c590ab8f5": " F_\\text{n} =\\frac{l_\\text{n}}{l_\\text{m} + l_\\text{n}} (W-L) + \\frac{h_\\text{cg}}{l_\\text{m} + l_\\text{n}} \\left(\\frac{a_\\text{x}}{g} W - D + T\\right).", "014cf89bc136d25dac6c59e89aa34bd2": "a_nX^n+\\dotsb+a_1X+a_0", "014d0143c6144f9a538fa9a8b34b70e5": "\\overline{\\Delta}", "014d82f7288863e6f215c9e6b4f8d53d": "\nC_w = \\frac {A_w}{L_{pp} \\cdot B}\n", "014e14b6b5d49acd76469e6fbc7a82fd": " \\text{error} = -\\frac{(b-a)^2}{12N^2} \\big[ f'(b)-f'(a) \\big] + O(N^{-3}). ", "014e23762e9d85d7f79dc5e98920b90b": " edg=k.(p-1)(q-1) + g", "014e7d2183b5947d954bc5262a310518": "g(x)=\\left\\lbrace\n\\begin{array}{lr}\n0 & 0\\leq x\\leq \\frac{1}{\\alpha} \\\\ \n+\\infty & \\text{otherwise}, \n\\end{array} \n\\right.\n\\,", "014e864142722fcc9b02b202e48ef62f": " L_{ni} (\\beta) = {exp(\\beta z_{ni}) \\over {\\sum_{j=1}^J exp(\\beta z_{nj})}}", "014e9666a4157b13ffa22c5f9120d61a": "\\begin{pmatrix} E_{0x} e^{i\\phi_x} & E_{0y} e^{i\\phi_y} \\end{pmatrix}^\\top ", "014ef7a82d4d84a5e5340f4085b234ac": "\\gamma_\\mu a\\!\\!\\!/ \\gamma^\\mu = -2 a\\!\\!\\!/ ", "014f13dc68a1ebef4081614936b7feea": "C^* = \\left \\{y\\in X^*: \\langle y , x \\rangle \\geq 0 \\quad \\forall x\\in C \\right \\},", "014f703deb391191425c853b0250a8b1": "{\\rm cov}(I)=\\min\\{|{\\mathcal A}|:{\\mathcal A}\\subseteq I \\wedge\\bigcup{\\mathcal A}=X\\big\\}", "01502d95785ebee778cd52c94b60fce8": "a = \\sqrt{c^2 - b^2}. \\,", "0150cf714657426a7ddac6d88b8350aa": "c<1/2", "0150da4a6df78bd2c47cdb860657017b": "R_{abcd} \\, {{}^\\star \\!R}^{abcd}", "01511265bf0fcc2e6daed57b80e0e61a": "\\dot\\theta", "0151880dd88fd5d6acf8c7246832402d": "\\frac{dW}{d\\omega}=\\frac{2e^2 \\gamma}{9 \\varepsilon_0c}S(y)\\qquad (11)", "01521b06b9713104aa72e655f28da19e": "\\varphi(p^n)", "01523d5515466eff29876c2a65a92242": "\\frac{(P_{high} * T_{high})\\, + (P_{low} * T_{low})} {T_{high} + T_{low}}", "01524f01f8f58579baef14bad535a9be": "Q_i \\subset Q_{i+1}", "01526e53e2e365f14881d561f7410482": "\\sigma: V \\to V\\,", "015272f0d625851e17fb5cec814ef476": "\nz=\\frac{x+i\\gamma}{\\sigma\\sqrt{2}}.\n", "0152b5f04f0a6a89537d2ebdefb64886": "{\\rm as}_5(5,3,4,1,2) = 5, ", "0152d4f07238ad06773c417c3029fb33": "U_{12}", "0152f4a919fc7e636a2127c374c6a820": "\\textstyle \\mathcal{F}; ", "01531ef83653cd13932e4eb31d4293a9": "\\, y(k) = Cx(k) + Du(k)", "01534343fafe188300ea31bb26178936": "\\exists C\\,\\exists M\\,\\forall n\\,\\forall m\\dots", "015344b96bbd0ec4fc811ca619d41bcd": "(\\mathbf{x-v})^\\mathrm{T} A (\\mathbf{x-v}) = 1,", "0153668e4fc7c3978d2019d3c8e0def3": "\\Phi + a = \\Phi \\ ", "0153fdca964ff1733b038341e15052a9": "\nZ_\\mathrm{in} (l)=-j Z_0 \\cot(\\beta l). \\,\n", "01544c6a478e896df7db7c6dfd37d03f": "\\lambda(t|\\theta)", "0154fd706580c907e37d92635916f59c": "E(v,h)", "01551c0ff28abafe81eea0b994e71cb0": "y_{n+1}", "01558774c0f2617ae915ab0f149b9bf6": "E[\\xi]=\\int_0^1\\Phi^{-1}(\\alpha)d\\alpha", "01559153a5f303bf5e242cfeb258376a": "MN-1-s = -s \\mod (MN - 1)", "0155b7d3ab4ed3e60406a3806351083e": "6H_2O \\rightarrow 4H_3O^+ + O_2(g) + 4e^- ", "0155c14a223bc3169ca9d52b033fe1f3": " (k[V]_P)_0", "01564d7f7c423595f5790218a0a7e506": "\\delta W = -(m_1+m_2)gL_1\\sin\\theta_1\\delta\\theta_1 - m_2gL_2\\sin\\theta_2\\delta\\theta_2,", "0156971872bbc66b173c86eaec924f90": "(X, <)", "01569b69c8445b7f7c3037a09f4f81df": " {s_{\\overline{X}_1 - \\overline{X}_2}}^2 ", "01571dc3494b356ca4a3220d0e3ae28b": "\\text{Semiperimeter}=mn(m+n) \\, ", "0157cdacfd6b753d07183d1fe676230e": " [AB,C] = A[B,C] + [A,C]B", "0157d1a94cadd34686afe0c111678caa": "-2x-2x^2+2y-2y^2+y^3+4xy-2xy^2+xy^2", "0157d210fb38c21d6d8986daffcae92d": " A_s = 1 - \\left [ \\frac {sP \\left (2\\theta_i - 2\\theta_k \\right )^2}{\\tan \\theta_k} \\right ] ", "0158382c22c0d624719e46eeeca3ba67": "\\dot m_k", "0158893c0ec80b2f8844b1d1e53143b2": "(T^*T)^p \\ge (TT^*)^p", "0158990fc044957d92d255d9848c471d": "f(w) = \\langle w, v \\rangle_W", "0158a2b95ba1b791ea183477dca28334": "466/885\\cdot 2^n - 1/3 + o(1)", "0159457a3702492f06b73e3e917c3bf6": "\\sum_{n=0}^{\\infty} \\| f_n\\|_U < \\infty", "015969d3f5f4b1046ee951f3509fa327": "\\rho(z)|dz|=\\sigma(f(z))|f^{\\prime}(z)||dz|. \\, ", "01597e9f2530e86683f34ade31503009": "BV(]0,1[)", "015a01db76489cb43c8003036548e316": "\\mathfrak a = (a)", "015a2ad28d0a278bd960b1b0eefdcc00": "\\ S_e = \\alpha_e \\times S_c ", "015a3b39628d270dd345abba52744268": "A = \\dots \\to 0 \\to A^0 \\to 0 \\to \\cdots,", "015a7ef685aafdee58899abccc2e2609": "\\alpha_0 = a/D", "015ae660b57165728429854982d05b05": "D_B>0", "015b39d72d4a0fd8e22283fd59a03971": "\\dot S_i = 0 ", "015b6ed3d17e4320673a7fbb4f08a890": "G(0) = a", "015bbc9bfb8a96e6ab5b0997090b8855": "p\\cdot2^p\\le W(2,p+1)", "015bcbdeb6916e3d184715b7d8f76ddb": "1 + \\cfrac{1}{2 + \\cfrac{1}{2 + \\cfrac{1}{2 + \\cfrac{1}{2 + \\ddots}}}}", "015bd78022d8197e1f89f26566ed30cb": "\n\\begin{align}\n f&=\\frac{a-b}{a}, \\qquad e^2=2f-f^2, \\qquad e'^2=\\frac{e^2}{1-e^2}\\\\\nb&=a(1-f)=a(1-e^2)^{1/2},\\qquad n=\\frac{a-b}{a+b}.\n\\end{align}\n", "015c8a7bf6cdc8659ba3da587412882f": " L=\\{p^*_1,...,p^*_n\\} ", "015e2542aa509f972b2d9cac61e0da8b": "\\begin{cases}\n f(x) \\ge 0 \\\\\n g(x) > 0\\\\\n f(x) < \\left [ g(x) \\right ]^2 \\quad\n\\end{cases}", "015e37f07fb14db21431e6f806b6f914": "\n\\rho_i ' \\simeq \\rho_j ' \\quad \\mbox{and} \\quad \\sigma_i ' \\simeq \\sigma_j ' \\quad \\mbox{for all} \\quad i,j \\;. \n", "015e537a57c7a99528d0b4be0dbff505": " f(x;\\mu,\\sigma_1,\\sigma_2)= A \\exp (- \\frac {(x-\\mu)^2}{2 \\sigma_2^2}) \\quad \\text{otherwise}\n", "015eb7acbdb731c32e8526d3b999986c": "G= \\int_0^\\infty I(\\lambda)\\,\\overline{g}(\\lambda)\\,d\\lambda", "015ec6b712f877cf4b399641b0afcfa4": "x(0) = 1,\\,", "015ed9def054d00e7c577035a29b1c6f": "c^{-2}", "015eeb1f8a5a6a7adc4a4c42bd49a653": " ~ \\bold J(x,y,z) ~ = ~ \\sum_j ~ J_j ~ \\bold J_j(x,y,z) ~~~~~~~~~~~~~~~~~(3.4) ", "015f185f6d3620e612d1d6106a17db4b": "0 \\leq n < N", "015f511f97b854214a366171a4880d0d": " \\gamma =(\\pi Q_r)^{-1} ", "015f7b01c6c52d90a663ea2cc944f8f1": "r_b", "015f801465c03bb660e00a97dbbf9996": "\\log\\left(\\frac{m_{\\rm closed}(t)}{m_{\\rm closed}(0)}\\right) \\sim -\\overline{n^2}t,", "016005918866f2507543201a199b5a9e": "K_{\\mathrm{max}} = \\frac {1} {2} m v^2_{\\mathrm{max}}", "01606d154021cfd719f7faccfbb3519b": "P(A^c) = 1 - P(A).", "0160828f5223b7e57f403a13ef5bef77": "\\displaystyle{\\mathfrak{t}_{\\mathbf{C}} \\oplus \\bigoplus_{\\alpha\\in \\Delta_1} \\mathfrak{g}_\\alpha}", "0160f522c2aab4f57bd959ae587a2b26": "\\alpha_t^j", "0161266e84b0ac7f3091917255670e9b": " \\|\\lambda\\alpha^n\\| \\to 0, \\quad n\\to\\infty. ", "0161387d2f38b3bee6e710f7bc05e728": "P_i = {\\bold 1}'\\otimes\\dots\\otimes{\\bold 1}'\\otimes P \\otimes {\\bold 1}\\otimes\\dots\\otimes{\\bold 1}", "01614247f736bfe380d93aeb04fffe8b": "a^k \\equiv 1 \\pmod p. \\,\\!", "01616afd77e7b2f30b075c048e56828d": "\\frac{V_o}{V_i}=-\\frac{V_i\\, D^2 \\, T}{2L\\, I_o}", "0161a35f5d52b351bcad5e296026ec19": "x(t)=\\frac{Q_t-Q_0}{P}.e^{ -A.t}-\\frac{R_t-R_0}{P}.e^{ -B.t}", "0161ac05b03c2fbc4c46e5942706c976": "F_{n}=\\sum_{k=0}^{\\lfloor\\frac{n-1}{2}\\rfloor} \\tbinom {n-k-1} k.", "0161b0e1647a6aae6ad875c24dce391b": "F_2(x)", "0161b664a9d7ab02df4940a7e0e8f5ea": "V_\\mu", "0161b76065b0cf6479281020f5ac4109": "y \\cdot S", "01625deea2b34a280cf53462d5cd8e89": "\\mathfrak{sl}_2(\\mathbb{C})", "0162f6fc9cba0bc4e038bb69b2fca6ba": "\n\\langle p \\rangle_{IV} = \\langle p \\rangle_{VI} \\equiv \\langle \\langle p \\rangle_I \\rangle_V\n", "01631e49d5e996818ad1e2bc639b30eb": " \\varphi: X \\rightarrow F ", "016330ad775f855967988d01493ad559": "A=\\epsilon c l", "0163630e147945512a566c81d04f5a9e": "SH_k(X) \\cong H_k(X)", "01637dc15f870d2ade39e7cccf90ceff": " F_D ", "016385857bd33bf936a2805d8224e9eb": " \\mbox{E}i(x)=-\\int_{-x}^{\\infty} \\frac{e^{-t}}{t}\\, dt", "0164f6f7287067a014f8969590008cb7": "\\frac {d}{dt} \\iint_{\\Sigma (t)} \\mathbf{F} (\\mathbf{r}, t) \\cdot d \\mathbf{A} = \\iint_{\\Sigma (t)}\\left(\\mathbf{F}_t (\\mathbf{r}, t) + \\left[\\mathrm{\\nabla} \\cdot \\mathbf{F} (\\mathbf{r}, t) \\right] \\mathbf{v} \\right) \\cdot d \\mathbf{A} -\\oint_{\\partial \\Sigma (t)} \\left[ \\mathbf{v} \\times \\mathbf{F} ( \\mathbf{r}, t) \\right] \\cdot d \\mathbf{s} ", "01650aac5453836cb8bc89e6ab9487a0": "p_G(z) = \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{ -\\frac{(z-\\mu)^2}{2\\sigma^2} }", "016557a36376f71d54e07fcb6d1b06a4": "\\partial (ax+by) = a\\,\\partial x + b\\,\\partial y.", "0166611b358712a14531de7bd37b3a92": "\n 0 \\subset T^o \\subset S^o \\subset V^*.\n ", "01666a9de09edfa09d29f736d0e88e6b": "\\frac{2}{1} = 2.0,\\quad \\frac{9}{4} = 2.25,\\quad \\frac{161}{72} = 2.23611\\dots,\\quad \\frac{51841}{23184} = 2.2360679779 \\ldots", "0167263fe51c584357b10f31948924a8": "k = \\begin{cases}\n-j-\\tfrac 1 2 & \\text{if }j=\\ell+\\tfrac 1 2 \\\\\nj+\\tfrac 1 2 & \\text{if }j=\\ell-\\tfrac 1 2\n\\end{cases}", "0167e58e36e9adcdc2dcfe53ad02ceed": "\n \\frac{\\partial \\boldsymbol{A}^T}{\\partial \\boldsymbol{A}}:\\boldsymbol{T} = \\boldsymbol{\\mathsf{I}}^T:\\boldsymbol{T} = \\boldsymbol{T}^T\n", "01683b5e9a02503019a64e3f0e356dd4": "l_a m^a = l_a \\bar{m}^a = n_a m^a = n_a \\bar{m}^a=0\\,.", "016847c3d5354736012b76dde02f985c": "x = x_1 A_1 x_2 ... x_n A_n x_{n+1}, y = x_1 w_1 x_2 ... x_n w_n x_{n+1}", "016918640d9fe4358ca6eecdaf2eb227": "\n(\\mathbf{\\hat{f}_{0:t}})^T = c^{-1}\\mathbf{O_t}(\\mathbf{T})^T(\\mathbf{\\hat{f}_{0:t-1}})^T\n", "016951effe27b2c913c0e4a30f5e3a67": " \\frac{d}{d t} \\left(p_1+ p_2\\right)= 0. ", "01695aacbcb4d3555cf4cd3ebac79d36": " f_X(x) = F_X'(x) = \\frac{1}{2\\pi}\\int_{\\mathbf{R}} e^{-itx}\\varphi_X(t)dt,", "01699f7f623d2f8633068b70b0104fe9": "(x, t)", "016a34677eebc38601dcce71b906a414": "\\min_{w\\in\\mathbb{R}^d} \\frac{1}{n}\\sum_{i=1}^n (y_i- \\langle w,x_i\\rangle)^2+ \\lambda \\|w\\|_1, ", "016a6eae4c21c9122a74c4734080dfc6": "\\frac{G^{ex}}{W_wRT} = f(I) +\\sum_i \\sum_j m_im_j\\lambda_{ij}(I)+\\sum_i \\sum_j \\sum_km_im_jm_k\\mu_{ijk}+\\cdots", "016a7a4366f6d2ba6d02ba1689f6dac4": "\\;\\delta\\;=90^\\circ\\;", "016ac14c46a8e10d3bd0959c48a55a7e": "\\int\\cosh^n ax\\,dx = \\frac{1}{an}\\sinh ax\\cosh^{n-1} ax + \\frac{n-1}{n}\\int\\cosh^{n-2} ax\\,dx \\qquad\\mbox{(for }n>0\\mbox{)}\\,", "016aff365e8041c20b749e28419cb1c5": "R_C(f)= {12200^2\\cdot f^2\\over (f^2+20.6^2)\\quad(f^2+12200^2)}\\ ,", "016b05d112234eafb35fc0882c90a117": "1\\to\\Gamma(N)\\to\\Gamma\\to\\mbox{PSL}(2,\\mathbf{Z}/n\\mathbf{Z})\\to 1", "016b0d87cb90b7ccb8c076c1d8ac6cbe": " [\\varnothing]_p = \\varnothing \\! ", "016b26921e4163616d2ab88325528257": "u(x,t) =X(x) T(t) ", "016b8314d6c42adcdaabe7e963b832af": " \\delta W = \\left(\\mathbf{M}\\cdot \\frac{\\partial\\vec{\\omega}}{\\partial\\dot{\\phi}}\\right) \\delta\\phi.", "016bc796e0261b4c5d6f436c79ceb3da": "\\mathrm{Pr} \\gg 1", "016c089a7c3a63f75a2c314fb02c6b24": " \\hat{H}_{\\mathrm{el}}=\\sum_{i=1}^N\\frac{p_{i}^2}{2m}+\\sum_{i 1 ", "016ee1c2de4d8637ac7a334f5a041990": "\\mu(x,G):= B(x,1/2n(x,G))", "016f0e72e89380b5e989d40148b2cbf4": "\\mathcal{F}=\\{F\\subset E\\vert G[F]\\hbox{ has property }\\mathcal{P}\\}", "016f1ceed5059915d5942a945d89825d": "\\frac{\\partial^2\\rho}{\\partial t^2}-c^2_0\\nabla^2\\rho = \\nabla\\cdot\\left[\\nabla\\cdot(\\rho\\mathbf{v}\\otimes\\mathbf{v})-\\nabla\\cdot\\sigma +\\nabla p-c^2_0\\nabla\\rho\\right],", "016f73e62aa171292b5559ef227c2799": "B(x;r) = \\{y \\in M : d(x,y) < r\\}.", "016f8ae636ec76129cbd95f8c08d0233": " \\frac{p_e}{\\rho g} + \\frac{V_e^2}{2 g} + z_e = \\frac{p_{0}}{\\rho g} + z_{0} + h_f", "016f8b07340bbc06290df981d225ae51": "\\frac{1}{n}\\mathbb{E}(f -\\mathbb{E} f)^2", "016faeaa7ce8d1af117195f3ff1e8d5c": "+ 7 \\cdot 9^{(7 \\cdot 9^7 + 7 \\cdot 9^6 + 7 \\cdot 9^5 + 7 \\cdot 9^4 + 7 \\cdot 9^3 + 7 \\cdot 9^2 + 7 \\cdot 9 + 6)} + \\cdots", "016fc51b812746e6f6bec0ceebd4024d": "F = {1 \\over 4 N_e u + 1}", "016fcb192b7b9cd0a9d3349d1adeda0d": "\n\\begin{align}\n\\left(\\begin{matrix}r_0 \\\\ r_1 \\\\ r_2 \\\\ r_3 \\\\ r_4\\end{matrix}\\right) & {} =\n\\left(\\begin{matrix}\n1 & 0 & 0 & 0 & 0 \\\\\n1 & 1 & 1 & 1 & 1 \\\\\n1 & -1 & 1 & -1 & 1 \\\\\n1 & -2 & 4 & -8 & 16 \\\\\n0 & 0 & 0 & 0 & 1\n\\end{matrix}\\right)^{-1}\n\\left(\\begin{matrix}r(0) \\\\ r(1) \\\\ r(-1) \\\\ r(-2) \\\\ r(\\infty)\\end{matrix}\\right) \\\\\n& {} =\n\\left(\\begin{matrix}\n 1 & 0 & 0 & 0 & 0 \\\\\n 1/2 & 1/3 & -1 & 1/6 & -2 \\\\\n -1 & 1/2 & 1/2 & 0 & -1 \\\\\n-1/2 & 1/6 & 1/2 & -1/6 & 2 \\\\\n 0 & 0 & 0 & 0 & 1\n\\end{matrix}\\right)\n\\left(\\begin{matrix}r(0) \\\\ r(1) \\\\ r(-1) \\\\ r(-2) \\\\ r(\\infty)\\end{matrix}\\right).\n\\end{align}\n", "017109036d98967b18753204f3ba2212": "\\Psi_A(1,2,\\dots,N_A) \\Psi_B(N_A+1,N_A+2,\\dots,N_A+N_B)", "01710b6db82c22ba1e3978b63169aa6f": "S_3 = I p \\sin 2\\chi\\,", "01711dd7f632f52b2769dcb1287d01fc": "\\|G_n-B_n\\|_\\infty", "01716f028414939d0760ec49f73781db": "\\tfrac{1}{2j}\\left[X(z)-X^*(z^*) \\right]", "0171b613b3d21c820963ba706362f057": " e^{\\tfrac{a(1-x^b)}{b}} x^{-2+b} (1-b+a x^b) ", "0171b9d3ea18f0244b52e9734214a6e4": "\\mathbf{F} = q (\\mathbf{v} \\times \\mathbf{B})", "0171ba21e34fac98573041370c9d7a1d": "d(x, m) = d(y, m) = d(x, y)/2", "0171c40165c03cc43ac0db36b95a2fe5": "\\gamma(t) = \n\\left(\\begin{matrix}a&b\\\\ c&d\\\\ \\end{matrix}\\right) \\left(\\begin{matrix}e^{t/2}&0\\\\ \n 0&e^{-t/2}\\\\ \\end{matrix}\\right) \\cdot i\n = \\frac {aie^t +b} {cie^t +d}. ", "0171f1bbb919ffcbf791d5ad55acbbd0": "b, ab, aab, aaab, \\dots", "01723ab614766914630de2e30e351ee5": "X0", "017e2034450b47b6670afdb40731e1b0": "\\; L (H_A)", "017e328293c91381e0341ae5c4e34e90": "a_n\\equiv\\frac{\\omega_nq_n+ip_n}{\\sqrt{2\\hbar\\omega_n}}\\,.", "017e9fec61107957ffe9121078b1f7df": "\\int_0^t H\\,dX =\\int_0^t H_s\\sigma_s\\,dB_s + \\int_0^t H_s\\mu_s\\,ds.", "017f12fc82880d3915c18beb5536cfcd": "\\,2646798 = 2^1+6^2+4^3+6^4+7^5+9^6+8^7", "017f1770e5cf555aaa36edf411633b6e": "\\forall z \\exists y \\forall x [x \\in y \\leftrightarrow ( x \\in z \\land \\phi(x))].", "017f58c5378216f7df65cba52f62c15a": "I_1 - i \\, I_2 = 16 \\, \\left( 3 \\Psi_2^2 + \\Psi_0 \\, \\Psi_4 - 4 \\, \\Psi_1 \\Psi_3 \\right)", "017fd493213155b36bce0bb5acd4b4b0": "1+x(-3+x(4+x(0+x(-12+x\\cdot 2))))=1-3x+4x^2-12x^4+2x^5", "017fd7e93eb4c2c900f221d7bf7b01e2": "b^2 + c^2 = 2m^2 + 2d^2\\,", "0180116fd314296f5bce2923f3534f80": "\\scriptstyle\\hat\\theta_{(i)}", "018031bc1c840403b6fc3312c1055a50": "\\left|F_n(x) - \\Phi(x)\\right| \\le {C \\rho \\over \\sigma^3\\,\\sqrt{n}}.\\ \\ \\ \\ (1)", "01808b41247d4647bbf4ef4b1ffc3e32": "\\or~(\\neg x_1 \\and ... \\and \\neg x_n)", "0180e774f2926393f199367f3ce20eb5": "p(\\mathbf{x}) = \\prod_{u \\in U} f_u (\\mathbf{x}_u)", "0180f298c26994c189a1fc6dc264955b": "w=D_L[F(K,L)]\\,", "01811d6565a5b93b98f52c00c4d45e0d": "\\frac{c}{c_0}=\\frac{t}{t_0}=e^{-\\frac{1}{8}\\left (\\xi_0^2-\\xi^2 \\right )}.", "018147f062e207970e698fac48499e9b": " 1 \\le j \\le n, 1 \\le i \\le m ", "018149d8f8a32fa92ace0794088c0b4d": "\n\\begin{align}\n& {} \\qquad D(X_1,\\ldots,X_n) \\\\[10pt]\n& \\equiv \\left[ \\sum_{i=1}^n H(X_1, \\ldots, X_{i-1}, X_{i+1}, \\ldots, X_n ) \\right] - (n-1) \\; H(X_1, \\ldots, X_n) \\\\\n& = \\left[ \\sum_{i=1}^n H(X_1, \\ldots, X_{i-1}, X_{i+1}, \\ldots, X_n ) \\right] + (1-n) \\; H(X_1, \\ldots, X_n) \\\\\n& = H(X_1, \\ldots, X_n) + \\left[ \\sum_{i=1}^n H(X_1, \\ldots, X_{i-1}, X_{i+1}, \\ldots, X_n ) - H(X_1, \\ldots, X_n) \\right] \\\\\n& = H\\left( X_1, \\ldots, X_n \\right) - \\sum_{i=1}^n H\\left( X_i | X_1, \\ldots, X_{i-1}, X_{i+1}, \\ldots, X_n \\right)\\; .\n\\end{align}\n", "01817fcfbf955c5fd03383d2d5346629": "J^k_0\\rho:J^k_0({\\mathbb R}^n,{\\mathbb R}^n)\\rightarrow J^k_0({\\mathbb R}^n,{\\mathbb R}^n)", "0181e6d5fbcfc15a4ad8b8859441d6f4": " {u_z}_{max}=\\frac{R^2}{4\\mu} \\left(-\\frac{\\partial p}{\\partial z}\\right). ", "01825790c8778cf3f0f332dd06e9125e": " y_t==i ", "0182ae4649da29eb355c50ee5ce8454a": "\\|Ax-b\\|_P^2 + \\|x-x_0\\|_Q^2\\,", "0182cf033974d31a4d153589c68124ec": "B_m(0)=\\sum_{k=0}^m \\frac {(-1)^k k!}{k+1}\n\\left\\{\\begin{matrix} m \\\\ k \\end{matrix}\\right\\}. ", "018360106cc490d81523fe4ec165a677": "\\!\\ c_\\mathrm w", "01837fa8d184e94e252faf806d15565e": " H ", "0183a5cbf260eb9a551564bb32d7aecb": "E_a^2", "0183d8f984ac194cefa846fbd594b5ad": "\\sigma= (\\mu/\\rho) m_a/N_A", "0183fa90419e0bc4ec7bfea03d866cbf": "a \\rightarrow \\sqrt{6}", "01849d49cfc83c0f4a80b66876178a5b": "E=\\sum_{x}\\left [F(x+h)-G(x)\\right ]^{2}.", "0184a2180ba67dc66ef0098b88df3ae6": "\\ S_{\\sigma,\\varepsilon} \\geq N_{\\varepsilon}+1", "0184a6b9877bb3fad2d6650d5e11a8d0": "\\frac{[\\Gamma(\\tfrac13)]^6\\sqrt{10}}{12\\pi^4}=\\sum_{k = 0}^{\\infty} \\frac{(6k)!(-1)^k}{(k!)^{3}(3k)! 3^{k}160^{3k}}", "0184e54b0baba3e4c86ea92a1c3d43c0": "\\delta_1", "0184e6891a43a7356a48dd6188722dc6": "O(N^{3})", "0184fa9b8ce42508657f2c6b37e58170": "L=L_1L_2", "0184ff16fdf7203d30e05788cb0e8678": "S_0(t)", "0185085d739e30630a5c731f0b2e8fb6": " \\hat {\\textbf{Q}}(t)", "0185086322cce19b502df2c6748868a1": "\\scriptstyle{Rt=g(Y,X)}", "01852a43328cfaa6fbf421f7dde01d4e": "\\scriptstyle A_{33} \\;=\\; 0", "0185ab7d0afe257f39839e75beffabe1": "\\left( e^{ix} \\right)^n = e^{inx} .", "0185c175db308f0e18002f2108d38515": "a+b\\omega", "0186254595b6cf79ca29f60c731e597b": "f(A) = \\begin{cases} \\frac1{\\det (A)}, & \\det (A) > 0; \\\\ + \\infty, & \\det (A) \\leq 0; \\end{cases}", "0186cef4e734d3086ddd8e1d98c96217": "\nL = \\frac{1}{2} \\langle F F \\rangle_S - \\langle A \\bar{j} \\rangle_S\\,,\n", "0186d0570a2ecc636534c55241780f3e": "x = 0.", "01870684c1cf92509c6d2448a3ce7c04": "\\frac {\\mu_m}{\\mu_f} ,", "018709b6fc0fe0bf0f220fbffbbf1772": "O(\\theta^n)", "01871299a7fe7070ddc52d1944caab4e": "\n\\int_{B} \\! p_{X,A,B}(x,a,b) = \\int_{B} \\! p_X(x) p_{A,B}(a,b)\n", "01871a14188995b8fe6571db67cc270c": "\\mathcal{F} = \\oint \\mathbf{H} \\cdot \\operatorname{d}\\mathbf{l}", "0187489c33857c111d84ec1dc319fd28": " z^{1-c}\\; {}_2F_1(1+a-c,1+b-c;2-c;z),", "0187756e545d7544471db750ed81ed68": "U(t_k)", "0187e2567f8db40b594ff55be6a7c5f5": " L\\left(s,\\dfrac{x}{p}\\right).\\, ", "01883cd10fb57b382f0043bb0fa82da3": "(y, z)_{x} = \\frac1{2} \\big( d(x, y) + d(x, z) - d(y, z) \\big).", "01883db4def9f5811143f99b22b6e85b": "\n [A_{\\bold{x}}, A_{\\bold{y}}] = A_{\\bold{z}}, \\quad\n [A_{\\bold{z}}, A_{\\bold{x}}] = A_{\\bold{y}}, \\quad\n [A_{\\bold{y}}, A_{\\bold{z}}] = A_{\\bold{x}}.\n", "01883f5cab3c6b493515462df4feb4e3": "\\hat{H} = \\hat{T}^{\\mathrm {translational}} + \\hat{T}^{\\mathrm {rotational}}+ \\hat{V}", "0188847e219e48a741f4df4a3976163b": "\n\\begin{array}{|rcccl|}\n\\hline\n\\color{MidnightBlue}{\\mbox{eval left}}&&(11+9)\\times(2+4)&&\\color{MidnightBlue}{\\mbox{eval right}}\\\\\n&\\color{MidnightBlue}{\\swarrow}&&\\color{MidnightBlue}{\\searrow}&\\\\\n20\\times(2+4)&&&&(11+9)\\times 6\\\\\n&\\color{MidnightBlue}{\\searrow}&&\\color{MidnightBlue}{\\swarrow}&\\\\\n\\color{MidnightBlue}{\\mbox{eval right}}&&20 \\times 6&&\\color{MidnightBlue}{\\mbox{eval left}}\\\\\n&&\\color{MidnightBlue}{\\downarrow}&&\\\\\n&&120&&\\\\\n\\hline\n\\end{array}\n", "0188a6b7a7247688f2e91ba5b50ae1ea": "P_{\\text{ph}}^{2}=0", "0188beea0e3e38c1805d75a62e67f5b2": "[X; \\mathbf{P}^\\infty(\\mathbf{R})] = H^1(X; \\mathbf{Z}/2\\mathbf{Z})", "0188fac9faa5803a2d5be6739bbdb18c": "\\vec{v} \\times \\vec{v} = V_b^2 ", "0189762ab4d0514e0168562d37157d08": "F\\Big(L_-(x),L_0(x),L_+(x),x\\Big)=0", "0189fb8f36c001bc2835b994411aa362": " V \\otimes V ", "018a00a33f83a32a23e2e7738411dc5a": " \\operatorname{Ber}_{+-} J_{\\alpha\\beta} = \\operatorname{sgn}\\, \\operatorname{det} A\\, \\operatorname{Ber} J_{\\alpha\\beta}.", "018a170ccf5243a9d91ea9ac6dda4b8a": "T \\times A", "018a4e8598e4a263216b9e2972506c3e": "\\gamma:S\\ddot\\to x", "018a51db0c95700dedde07b803d7a4e7": "\\left\\{e^{\\frac{2 \\pi i}{6}},e^{-\\frac{2 \\pi i}{6}}\\right\\}=\\left\\{ \\frac{1 + i \\sqrt{3}}{2}, \\frac{1 - i \\sqrt{3}}{2} \\right\\}.", "018a580e2e81afcf158eebfc5ca43427": "\\begin{cases}\n\\Phi(x) - \\left[ \\varphi(0) - \\varphi(x) \\right] / x & x \\ne 0 \\\\\n1 / 2 & x = 0 \\\\\n\\end{cases}", "018aadf04c43b931fc8e2a9d169fbb1b": "C(x_j,x_k) = \\left.\n\\frac{\\partial}{\\partial J_j}\n\\frac{\\partial}{\\partial J_k}\n\\log Z(\\beta,J)\\right|_{J=0}\n", "018abe6cf12c05c131fe7ecb89c3378e": "{13 \\choose 5}{4 \\choose 1} - {10 \\choose 1}{4 \\choose 1}", "018ae3eec6dac7cb676d047a674eb383": "C_{3v}", "018b01cef997a99bb8f9bbf216a8bb84": "({\\mathbf P},F_{\\mathbf P})", "018b443a60f53655f4c367a819fc1d33": "(1+\\sqrt{2})^n", "018b6427d2e0647e427fd1de26c4c7c2": "|R|=p", "018b72eff79fe262ed869e07f933ee9e": "\n\\overline{\\mathrm{Var}(z)}=1-\\overline{R}\\,\n", "018c362a6c80b84a0b7e3ed096b2947f": "\\frac{dx(t)}{dt}=a*(y(t)-x(t))", "018c7a969d34311bed9a89e7f6187eb8": "c_{\\sigma}", "018ca55ecf6e69d4925ffa6c737d4d35": "t \\ ", "018cc819d8e1e37f057e83c9ec40173e": "H_1(z)", "018cdc0ed50078c0b68acdc099e531e7": "m_i \\in \\mathcal{M}", "018cdd89d9f5a4e5292b17b162d577c3": "f_0=0", "018cf3b6c662f14fb1d227939807b1a6": "f \\cdot (g*h)=(f \\cdot g)*(f \\cdot h)", "018d226118c2140faf0afe31a03002ce": "\\Phi(x) = -\\int_c \\vec{F} \\cdot \\mathrm{d}\\vec{r}.", "018d41f1d089523907b3ec9607588ef2": "\\textstyle\\mathbf{IPC}+\\bigvee_{i=0}^n\\bigl(\\bigwedge_{j\\ne i}p_j\\to p_i\\bigr)", "018d600acb6ef28c9e56e616cd9030ae": "\\Delta_n(\\mathcal{C}, x_1, \\ldots, x_n)", "018d86e2178b32a5b9a72537a8070bf7": "^\\bullet", "018dabb09a219a855a8236d1f5c20b33": " h(f_s(z))", "018dabebcae8c7f38c369d0781e1894c": "\\phi(\\beta)=\\frac{3}{4\\beta^{2}}\\left(\\frac{1+\\beta^{2}}{2\\beta}\\lg\\frac{1+\\beta}{1-\\beta}-1\\right), ", "018dbb3fc4985ed840a2e6a8fae944fe": "P_1(u,v)=\\left\\langle \\mathbf{F}(\\psi(u,v)) \\bigg| \\frac{\\partial \\psi}{\\partial u} \\right\\rangle, \\qquad P_2(u,v)=\\left\\langle \\mathbf{F}(\\psi(u,v)) \\bigg| \\frac{\\partial \\psi}{\\partial v} \\right\\rangle ", "018dd87f3449089eb9b8d0831f337a4f": "\\mathfrak{sp}_6(\\mathbf R)", "018ddd56a4074767957790db9d01a62e": " R = \\Sigma \\, \\Phi ", "018de11eca10ed1f4438540470dbb080": "\\nabla^2 = \\partial_{\\rho\\rho}+\\frac{1}{\\rho}\\,\\partial_\\rho +\\partial_{zz}", "018de2aa4efc1422d9375ecc1c106d94": "b^j", "018e15c37302c06def03be4efe13a3fb": "|\\phi\\rangle_A \\otimes |\\psi\\rangle_B", "018e17bcd3719db8c3a20d97cf78c8c7": "\\mathrm{V}", "018e5bf2411bdca849ce5d9cbd6594be": " \\sum_{n=1}^\\infty \\frac{1}{a_n x_n} ", "018e74d99d2d488fc1a3842be6a115f9": "k^{\\prime}\\,", "018e788359518637f7fdb08b607b8193": " \n\\ln y(r_{12})=\\rho \\int \\left[h(r_{13}) - \\ln g(r_{13}) - \\frac{u(r_{13})}{k_{B}T}\\right] [g(r_{23})-1]\\, d \\mathbf{r_{3}}. \\, ", "018e94b6ffd568f76dc90f08af295dbd": "0 \\le i \\le n", "018eb0ac4a321ccaf301048b102f6286": "\\mbox{Debtor days} = \\frac {\\mbox{Year end trade debtors}} {\\mbox{Sales}} \\times {\\mbox{Number of days in financial year}}", "018ebc97d2b46fa45b3b70520e587f57": "\\|y_{n+1}-z_{n+1}\\|\\leq\\|y_{n}-z_{n}\\|", "018ee86a3d311275b87c0a5933942455": "P(s,n) = (s-2)T_{n-1} + n = (s-3)T_{n-1} + T_n\\, .", "018f06e87c6e53a96a3defa23a69a1ad": "f : X_1 \\rightarrow X_2 \\, ", "018f10d214276c7cffc662cc1da2de5f": "|\\psi\\rangle\\in \\mathbb {C}P^N", "018f1b5a032989588141548a05459a83": " q_c = 1 - \\frac {1} {R_0} ", "018f250265d9d5ddae18ce4282d77ff2": "Y_\\mathrm{sun} = 0.25 ", "018f365b18a97f02a2c5e8924fd8540b": "\\zeta \\in F^\\ast", "018feb5d830b583433f1194bc19cf790": " \\operatorname{E}[|X|] = \\operatorname{E}[X] = \\int_0^\\infty \\lbrace 1-F(t) \\rbrace \\, \\mathrm{d}t,", "018fed5a51edfe03ee3443fdc213d0d3": " \\sum_{j=0}^k \\tbinom n j", "01901f6fd51e5c28dd5dfa2e1870d592": " dY_n(t) = S_n(t)\\left[b_n(t)dt + dA(t) + \\sum_{d=1}^D \\sigma_{n,d}(t)dW_d(t) + \\delta_n(t)\\right] , \\quad \\forall 0\\leq t \\leq T, \\quad n = 1 \\ldots N. ", "019049186c3201e21d9c6f8acc6f4762": "0 = L(\\varphi_t,\\nabla_x\\varphi) = (\\varphi_t)^2 - c(x)^2(\\nabla_x \\varphi)^2.", "0190e84d88093784116c0cf414c1f684": "V = \\left(k \\right)\\sqrt\\frac{\\rho_L - \\rho_V}{\\rho_V}", "0190ea3bade1ab5ef3fb72a836b5ae92": "\\mathbf{m}=(m_1,\\ldots,m_c) \\in \\mathbb{N}^c", "01910c3f0e4afd9ab6d71da6a7559ebf": "\\mathfrak J^k(a)_n=b_n=\\sum_{i=0}^n(-k)^{n-i}\\binom{n}{i}a_i.", "0191546962f47fcb2feea1480f82d70d": "(b_0,\\dots, b_{M-1})", "0191c5dbe0a0bed1e8ea409b3ea9449b": " \\sqrt{R^2 - \\left(\\frac{h}{2}\\right)^2},\\qquad\\qquad(1) ", "0191df0133e7e83cbc8b65b67e29cd36": "\\alpha_i(1)=\\pi_i b_i(y_1)", "01922f9fafe5c08cea75ae7237b5ac8f": "\\alpha = \\frac{1}{V} \\biggl(\\frac{\\delta V}{\\delta T} \\biggr)_{P}\\ ", "019236eee89a1bcd87153e945caff4f0": "2(2j+1)", "0192481c1a5fcd1e7f4cce09def4bcb2": "\\!\\mu \\in X", "0193d3271468b5f68bdced7c336ecffb": "\\scriptstyle\\{e^{(a)} = e^{(a)}_{\\mu} dx^\\mu\\}_{a=1\\dots4}", "0193d4d3f614be7ffb688f4a5e71a62d": "|G|", "0193deabfbc61eba0387e52afe5500f0": "unroll : \\mu\\alpha.T \\to T[\\mu\\alpha.T/\\alpha]", "0193ee11c894b0d747dcc9513cbca04c": "Em = 1 \\tfrac{2}{3} 3 + 1 \\tfrac{1}{3} 1 + 0 \\tfrac{1}{3} 2 + 0 \\tfrac{2}{3} 0", "019426145271b9b66f466862efb452a1": "(U, \\phi)", "0194420c9cd297af834bc0fc68b0d0f0": "f(h,k) - f(h,0) - f(0,k) + f(0,0)", "0194949fdd2683fca054957d9a3631f8": " Fr < 1 ", "0194f157695e76edad5de7a928aa3f27": "\\{\\xi_k\\}", "0194fc10bbd26154d932af9c338fb3e8": "y \\in \\mathbb{R}^q", "019503e25e037825852e80e771d92dda": "(n - 1)! = 1 \\times 2 \\times 3 \\times \\cdots \\times (n - 1)", "0195049235f6c32595e6551efc2c4c1b": " F(X)=\\inf_{S}\\sup_{I}\\frac{|I|}{|S|}, ", "01951ec559cd6c4cdc5e189332a65175": "F_3", "019522c5b32a9528c88582d494a9bef5": "\\{(x,t): t < f(x)\\}", "01952cbf349006bc6a12c6661316b4cc": " S = \\frac{U}{T} + N * ~ S = \\frac{U}{T} + N k_B \\ln Z - N k \\ln N + Nk ~", "01952e78f364ba78bdd3844b9917fcf9": "S\\cup\\{x\\}", "01957e8e751f1d74c88c70cc8d9610d8": "f_n(x) = \\sum(x_i - \\bar x_n)^2/(n-1)", "0195a95d13d16542bc61d5cdcf36994f": "A_1=A", "0195cf3c7a5b7625e528f831c3f063ed": "{d\\alpha_j \\over dt} = -{\\alpha_j^{19}\\over \\prod_{k\\ne j}(\\alpha_j-\\alpha_k)} = -\\prod_{k\\ne j}{\\alpha_j\\over \\alpha_j-\\alpha_k} . \\,\\!", "0196026af4a0d9cef3563be5ca49e199": "P( A_K ) = \\prod_{j=1}^n ( a_{Kj} ) ^{w_j}, \\text{ for } K = 1, 2, 3, \\dots , m. ", "01960518dbc691be905340184be10534": "d\\;", "01962c356f57eec60b239226921067db": " \\frac{\\partial}{\\partial u} g(z, u) \\Bigg|_{u=1} = \n\\frac{1}{1-z} \\sum_{k\\ge 1} b(k) \\frac{z^k}{k} =\n\\frac{1}{1-z} \\frac{z^m}{m}", "01965566c95014beea4169411276fafe": "c_p = \\frac{\\gamma R}{\\gamma - 1}", "01969bfd511865852ab937396eededfb": "p_idq^i-H(t,q^i,p_i)dt", "0196b2c7974f456231a64af2c1d2d18d": "I = \\frac{m\\left(W^2D^2+L^2D^2+L^2W^2\\right)}{6\\left(L^2+W^2+D^2\\right)}", "0196dab427cb3a38ec0befeb00f22cff": "J^{\\mu} \\, = \\, \\partial_\\nu \\mathcal{D}^{\\mu \\nu} \\,", "0196f3dfcad95bb6f7a13918cb04874a": "\\sigma_N=\\frac{\\nu_L \\sigma_L - \\nu_S \\sigma_S}{\\nu_L - \\nu_S}\\,", "01973b25656d8aaeca3411bb822c6c6c": "\\mu \\ne 4", "01973d13d4ecf598fbdc26a4efd0ac17": "\ng(z) = \\sum_{n=1}^\\infty z^n \\,\n", "01974885b9d6f1a92fea39de85c21f25": "p+\\delta p", "01975388b40bd8aafd509042d627c2ca": "\\tfrac{26}{11}.", "01975e71b69de760ff53071819946a93": "\\scriptstyle W[k] = (-1)^k\\cdot W_0[k].", "019777033a1f7b80d44ed3e8c1e8fe32": "\n\\psi(x)=\\sum_{n\\le x}\\Lambda(n). \\;\n", "0197a1981e16dbe4aca1cd37b7b8207b": "x_1 = c \\times10^{b_0}", "0197ac0c144881e2e604b20b3b77cd9a": "c_i/f_\\mathrm{eq}", "0197cf1222e54259f28ec57ef981c0b0": "\\frac{1}{T}\\int_0^T \\mu_{\\max}(t)\\, dt \\leq 1 - \\varepsilon", "0197ee0201e14a0f4ce5849df7285c34": "s \\ne t", "01983a10224cc4ef657692c9bc7ee5db": " \\exists^p L := \\left\\{ x \\in \\{0,1\\}^* \\ \\left| \\ \\left( \\exists w \\in \\{0,1\\}^{\\leq p(|x|)} \\right) \\langle x,w \\rangle \\in L \\right. \\right\\}, ", "0198791ecd1dcf1db07cac8e68982320": "\\operatorname{Categorical}(\\boldsymbol\\phi_{x_{t-1}})", "01988c67f402ddd3b5b82a07e8eaa2db": "e=3", "01989ba6f1aee5e66d1a613a1965baa3": "\\Delta E_{UVW}", "0198d3a65976defb7f5c155ce7d12591": "\\bigg(\\frac{a x_a+b x_b+c x_c}{P},\\frac{a y_a+b y_b+c y_c}{P}\\bigg) = \\frac{a(x_a,y_a)+b(x_b,y_b)+c(x_c,y_c)}{P}", "0198dff3f918245c5bddc7308da0886d": " \\begin{bmatrix} 2 & 4 \\\\ 3 & -8 \\\\ 1 & 2 \\\\ 2 & 4 \\end{bmatrix}", "01991eec76ddfec566e5d7ec6dc3e9d6": "\\dot{v}(t)", "01997a8f18de1bfbddb9cfc0c93010af": "L^{3-}+3H^+\\leftrightharpoons LH_3:[LH_3]=\\beta_{13}[L^{3-}][H^+]^3", "01997e740836ece90f7b7fc509abedd2": "0 = t_{0} < t_{1} < \\dots < t_{k} = T", "0199832e9fae5d0586f0bdd9a6282656": "\\sigma_{A}(R\\cup P)=\\sigma_{A}(R)\\cup\\sigma_{A}(P)", "0199c39c779f03bea969358615a0a035": "H^2(\\mathbb{D},\\mathbb{C})", "0199c7407ecd3009ea90841e94a08477": "\\oint_{R} B ds \\cos{\\theta} = \\mu_0 I_{enc}", "019a46fcbcf20b8f001fd25085727497": "\\int_{\\mathbf{R}^n} f(\\mathbf{x})\\delta\\{d\\mathbf{x}\\} = f(\\mathbf{0})", "019a54c4cefc7364b20ea8984379b3bb": "\n\\begin{bmatrix}\n\tZ'_{11} & Z'_{12} \\\\\n\tZ'_{21} & Z'_{22}\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\tZ_{11} & Z_{12} \\\\\n\tZ_{21} & Z_{22}\n\\end{bmatrix}^{-1}\n", "019a5e4654b34048ad6c2144b93e07e6": " A = LDU, \\, ", "019a98cbd788b7cd3fb93a24dd151521": "12^2+35^2=37^2", "019ac375a68a1ba2d3dfaceecb04ba36": " \\pi_2(x) =O\\left(\\frac {x(\\log\\log x)^2}{(\\log x)^2} \\right).", "019ae5e37093a2386ddbdbbd1511ac0b": "\\, TA(t_1)A(t_2)=A(t_1)A(t_2)\\!", "019b26b86ba081c79cd0e6bea83697fe": "GE(\\alpha) =\n\t\t\\frac{1}{N}\n\t\t\\sum_{i=1}^N\n\t\t\\ln\n\t\t\\left(\n\t\t\t\\frac{\\overline{y}}{y_i}\n\t\t\\right),\n\t\t\t\\quad\n\t\t\t\\quad\\quad\n\t\t\t\\text{ for } \\alpha = 0,\n", "019b681173c537822faed84e02198141": "[\\mathtt{Gen}]", "019c168d94e358f1fb02f88a964deaba": " {x^2+y^2} - L^2=0, ", "019c3374d2c4d9109cf304dccaffe1e0": "\\,\\int_0^{\\frac{\\pi}{2}}\\,\\frac{\\sin^2\\,x\\;\\mathrm{d}x}{\\left(a\\,\\cos^2\\,x+b\\,\\sin^2\\,x\\right)^2}\\;=\\;\\frac{\\pi}{4\\sqrt{ab^3}}.", "019c5f010f647f3c10973259318d5764": " E_i E_j=E_j E_i, \\mathrm{if} \\left\\vert i-j \\right\\vert \\geqslant 2,", "019c7ab6604aaf7125613d86788a2b3e": "S^s_0(t)", "019c877780717c1350fd15a60edef362": "((A \\to B) \\land (B \\to C)) \\to (A \\to C)", "019c8cf9139fb7a1a644f74c07140b2a": "Q_A = qL", "019ca4f32ed99b42a77e7c9915a2e76c": "b=0.2", "019d18f1da08be3c865ae8bafbd72d8a": "(x + 5)^{2/3} = 4,\\,", "019d3a959349b8026a9f20a8ae55d8d6": "\n\\begin{array}{lcl}\nr_j & = & FI(KI_{i,j}, l_{j-1} \\oplus KO_{i,j}) \\oplus r_{j-1} \\\\\nl_j & = & r_{j-1}\n\\end{array}\n", "019df08eb3c50b2bcd1c650bf32042fe": " \\phi_{l}\\,= \\phi_{L} ", "019e218615c919ebfbe64ec3eedf30db": " =\\frac{E}{m|\\vec{q}|} \\left(\\frac{|\\vec{q}|^2}{E}, q_x, q_y, q_z \\right) \\,", "019e3232ce1861ab115fe64ee345355b": " |\\psi(t)\\rang = \\sum_n c_n(t) e^{- i E_n t / \\hbar} |n\\rang ", "019e5a89b9b9ee6e978db1badbc7386d": "L = \\int{ \\mathcal{L} \\, d x d y d z}", "019eb126c193432d1a2a2642c3f3f09b": "P(t|M_d)", "019eb4af7f580200e254f4c75953d7e7": "R = \\frac{1}{2}\\rho \\frac{f L S}{A^3} \\equiv \\frac{1}{2}\\rho\\frac{f L}{R_{h} A^2} \\equiv \\frac{1}{2}\\rho\\frac{4 f L}{D_{h} A^2} \\equiv \\frac{1}{2}\\rho\\frac{\\lambda L}{D_{h} A^2}", "019eb91545f1d3444c54206c8d8c5e16": "\\begin{align}\n ~x_k &= (2d_k - 1) + a_k\\\\\n ~y_k &= 2( Y_k - 1) + b_k\n\\end{align}", "019f15b6e3e80ff2e19818c8ea597924": "f^b(t_i, w)", "019f2057772144bbe5ec352cbb1608ff": "RC = \\frac{(H+BB-CS) \\times (TB+(.55 \\times SB))}{AB+BB}", "019fb747da2029da6d18a634727f0dc8": " R = \\Delta T/\\dot Q_A", "019ff996753dcc37e7955f547c2c5fc4": "d_{i,j} = 0 \\mbox{ if } i \\ne j\\ \\forall i,j \\in \\{1, 2, \\ldots, n\\}", "01a015c92ea04385819cf8a2965e22d1": "i,j\\in E.", "01a02c3662ecb476540620d065822d3a": "2(n-1)", "01a07b0b22723e6d6fe75e73e815ecdd": "a\\geq 0", "01a0a57943be3ebbb0a780d7b36eb6bb": "= \\hat{a}_j^\\dagger \\hat{a}_l^\\dagger\\, \\hat{a}_i \\hat{a}_k + \\delta_{il} \\hat{a}_j^\\dagger \\, \\hat{a}_k + \\delta_{kl}\\hat{a}_j^\\dagger \\,\\hat{a}_i + \\delta_{ij}\\hat{a}_l^\\dagger\\hat{a}_k + \\delta_{ij} \\delta_{kl} ", "01a0a8dfe48ae3444a60e0371df84d8e": "\\nu>0", "01a0bc19715f6f2a983ee153b0470c9b": "i^2=j^2=\\eta", "01a0ddeeb3da341949a04413c40519cf": "p(c_j | x_i) \\,", "01a0ef71ac5c04bdabf03022f1a6834f": "G^p=(V,E^p)", "01a0f00f7a6a71f0b50f6450179ac3b7": "|\\mathbf x|_p := \\left(\\sum_i |x_i|^p \\right)^{1/p}", "01a0f25168bedbc4a6f70dee4398308c": "f=\\sum_{k=0}^\\infty A_kz^k.", "01a0fdfb3cd1f76eb8002eba9c586f57": "g_{bf}", "01a11a1c84ff2134d636dbb5a8c4d861": "\n f_b\\left( \\frac{F_D}{\\frac12\\, \\rho\\, A\\, u^2},\\, \\frac{u\\, \\sqrt{A}}{\\nu} \\right)\\, =\\, 0.\n", "01a120c756b1f6cf4f08e0fca0cfa6fe": "dl", "01a1def7f98541e100920165c8ab315a": " x \\rightarrow \\infty ", "01a2003b637de11f1584eddec16efd69": "\\psi(\\lambda)\\,", "01a2090d57c865bb7b277857d0659e3e": "1.\\overline{36}", "01a22ae8fbd128bb63fd9f0304c7d584": "L=\\frac{\\Theta}{2 \\pi} \\cdot 2 \\pi R \\, \\Rightarrow \\, \\Theta = \\frac{L}{R}", "01a2bd187bcc97465946cda426857db6": "H(f)(x) = \\frac{1}{i}(F_+(x) + F_-(x))", "01a2d354eeb4299748e097f987ad06a1": "k = 0,\\ldots,N", "01a303aa74d54caa7d7fd469294630a4": "\\mathbf{f}(\\mathbf{x})\\neq1", "01a35d410aebfade90b90ef175faa85d": " \\begin{bmatrix} a_{1 1}(x) & a_{1 2}(x) & \\cdots & a_{1 n}(x) \\\\ a_{2 1}(x) & a_{2 2} (x) & \\cdots & a_{2 n}(x) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{n 1}(x) & a_{n 2}(x) & \\cdots & a_{n n}(x) \\end{bmatrix} ", "01a395a6f409a18d75821974a1afefbc": "3-", "01a3a511f91e8722706af52507970b22": " \\overline{u'^2} = \\overline{v'^2} = \\overline{w'^2}. ", "01a3c2cb482cb206e9e4d2776bf35a5d": " \\operatorname{let} x : (x\\ x = \\lambda f.f\\ (x\\ x\\ f)) \\operatorname{in} x\\ x ", "01a462b57cdce353e860cb43e567bd43": "K_1=\\frac{{[NO]} {[NO_3]}} {{[NO_2]}^2}", "01a4a59350f82cc6452b8fbfdc645a84": "\\Delta p = 2410 \\left( {m \\over V} \\right)^{0.72}", "01a4b0901e7094539c040c5659f2e2eb": "\n \\Delta^2 w := \\frac{\\partial^2}{\\partial x_\\alpha \\partial x_\\alpha}\\left[\\frac{\\partial^2 w}{\\partial x_\\beta \\partial x_\\beta}\\right]\n = \\frac{\\partial^4 w}{\\partial x_1^4} + \\frac{\\partial^4 w}{\\partial x_2^4} + 2\\frac{\\partial^4 w}{\\partial x_1^2 \\partial x_2^2} \\,.\n ", "01a4e7db800567fb110001e8d958a2f3": "\n\\begin{align}\n\\sum_{i=1}^6 \\tfrac{1}{6}(i - 3.5)^2 = \\tfrac{1}{6}\\sum_{i=1}^6 (i - 3.5)^2 & = \\tfrac{1}{6}\\left((-2.5)^2{+}(-1.5)^2{+}(-0.5)^2{+}0.5^2{+}1.5^2{+}2.5^2\\right) \\\\\n& = \\tfrac{1}{6} \\cdot 17.50 = \\tfrac{35}{12} \\approx 2.92.\n\\end{align}\n", "01a4f56afd9079f8140cbc858c20bcf7": " \\frac{\\delta F[\\varphi(x)]}{\\delta \\varphi(y)} = g(y) F[\\varphi(x)]. ", "01a4fc6069dbeeb5bdf837895affc245": "x \\neq y", "01a54313673b4d77705210d217a7ef37": " \\mathbf{[Z]}=\\begin{bmatrix} Z_{11} & Z_{12} & Z_{13} \\\\ Z_{21} & Z_{22} & Z_{23} \\\\ Z_{31} & Z_{32} & Z_{33} \\end{bmatrix}", "01a550404fe927b314049e8d33de9fa7": " Q^{m}u=\\int_B T_y^mu\\left( x\\right) \\psi\\left( y\\right) \\, dx, ", "01a5708bc021cd7d1eb44d684951de2f": "\\begin{matrix}(14 - x) \\times 4\\end{matrix}", "01a5bc2d7acd353afebdaa9633dffec1": " f(x) = f(a) + f'(a)(x - a) + R_2\\ ", "01a5be688e3a97ad77ab71e95f396757": "\\ R = \\sqrt{(X_{12} - X_{11})^2 + (X_{22} - X_{21})^2}", "01a5be88fb56693e02fbee27521b5063": "{\\mathcal C}_n(z) = \\sum_{k=0}^\\infty \\pi(k+n) \\frac{z^k}{k!}", "01a5e298f2604e78351a4f9efa94aeab": "~k_a", "01a634527dee50e7bd73d69a8a63110d": "\\hat{\\mathcal{O}}", "01a6819a07cedf575f0f299dc4badf1c": "\\displaystyle \\frac{\\mathrm{d}^2 x^i}{\\mathrm{d}t^2} = -\\frac{c^2}{2} \\varepsilon \\gamma_{00|i}", "01a6aba88971cca0b2f59fab085fbe80": "10000=10^4", "01a6c084f5e59595c64196d929743f4d": "\\lambda_4 = \\sqrt{2},", "01a6e061b2e927945bb4fa00e7e344a1": "=\\!\\!(t_1 \\ldots t_n)", "01a72d27fa295850a617bf49fe186a27": "v_\\mathrm{p} = \\frac{\\lambda}{T}.", "01a764cee7384d7c873165dd7c7dd066": "\\textstyle \\leq c", "01a76635894af1be7c454818e15e864d": "g(2^n,2) = 3", "01a78c2c81fa65870adab1526aa3dd6c": "\\left\\{{5'\\atop3}\\right\\}", "01a7ec08e4a4d2e1052b46850941e4e9": "0, 1, \\ldots, n-1", "01a873e523d00e4ab7d05e3b47213d08": "F_{eq}", "01a87ab17ca903f95241c866f531ba64": "(-b-h(a))^2 + h(a)(-b-h(a))", "01a8bf77078848236bfd2d223e761215": "\n \\epsilon=a\\tan\\theta\n ", "01a8c485b579cc073a33b76977543ee8": "\n\\operatorname{Li}_s(-z) + \\operatorname{Li}_s(z) = 2^{1-s} \\,\\operatorname{Li}_s(z^2) \\,.\n", "01a8f4b4ea69d5f48e5aec150b9a938a": "(\\mathbf{J}_2, \\mathbf{E}_2)", "01a92ffdaad37599e891789ee4dc6daa": "{{f}_{M}}", "01a94aa32af850c75db975e05b64e709": "T^{-1/p} + T^{-1/p^2} + T^{-1/p^3} + \\cdots ; \\, ", "01a94f41297bd40bf5881c4b69ad38c7": "\\operatorname{sqsum}(x, y) = x \\times x + y \\times y", "01a95497fd188fe421728b66ae3e94fc": "|\\mathbf{r}|", "01a979edcb34ad6b9c69310e1ba3f01d": " \\varphi_a (g) = a g a^{-1} ", "01a9f31ab16bb54eac94bffcc7fcf7e8": "s_1,s_2 \\in S, r \\in R", "01a9fe25a65a2201a9bb94d9bb9d1c98": "\n f\\;a\\;b\\;c\\;:\\;1 \\to D\n", "01aa6b95d98a04759a36335b2c7b96b0": "\\forall\\beta.\\beta\\rightarrow\\alpha", "01aa89836a9d9b1f38be001498955085": "\\begin{matrix}{4 \\choose 4}\\end{matrix}", "01aac14134bf2df7be4dc632323e4a46": "k_{1}, k_{2}\\in K \\subset A\\,", "01aadeeff6a395a7087f2ba67c85afe6": "e_4", "01aafa82db291da77997f6b1c472899f": " RSS = y^T y - y^T X(X^T X)^{-1} X^T y = y^T [I - X(X^T X)^{-1} X^T] y = y^T [I - H] y", "01abb031e803ea01a54831fbd2ac7af4": "\\sum_{n=1}^\\infty \\frac {z^n}{n!} H_n = -e^z \\sum_{k=1}^\\infty \\frac{1}{k} \\frac {(-z)^k}{k!} = e^z \\mbox {Ein}(z)", "01ac0d10469c3dfa1296b1d1bb690511": " U_n^{(a)}(x;q) = (-a)^nq^{n(n-1)/2}{}_2\\phi_1(q^{-n}, x^{-1};0;q,qx/a)", "01ac191dd7e8a53ee4c24bdda542fec2": "f_c(k,r)\\approx f_0(E,E_{Fn},T_n)", "01ac54386a9d909da3b638139ce7966e": "\\frac{1}{(i\\omega)^2-\\xi^2}", "01acc0905f397bf2ffad7857cb5f3384": "\\|f\\|_p = \\left (\\int |f|^p\\,d\\mu\\right)^{\\frac{1}{p}}", "01ace1d9d151a6069bf22973a57eca16": "\\int_{0}^{\\infty }\\frac{f(ax)-f(bx)}{x}\\ dx=[{f(0)-f(\\infty)}]\\ln \\frac{b}{a}", "01acecf10001f0540a51caf58766b224": "g(x) = f(x + a)", "01acfbf706dd22836aba8f79192bf009": " Y^\\ast = X'\\beta + \\varepsilon, \\, ", "01ad76a59829a51dcb3b63290c1efe8c": "\\Phi(M)", "01ad985307e177d5f92f0cc6a075051d": "a_i = f^i(n)", "01ada4c6e7ff46e9d9c6fa5fb36a69cd": " \\tilde{f}", "01adb584cb3be702a413e84135e5f0df": "h_f = r \\cdot Q^{n}", "01addcd0e7e699e500b24ddb246983b9": "\\gamma(\\mathbf{h})=\\frac{1}{2N(\\mathbf{h})}\\sum^{N(\\mathbf{h})}_{i=1}\\left(Z(x_i)-Z(x_i+\\mathbf{h})\\right)^2", "01adfa4dbebcb0ee3a196cbe0b5adde0": "B = Y_2ZZ_1Z_1 = \\sqrt{3}", "01ae55de5e9698c5db36423be6c05224": "\\omega_e = \\mathrm{id} : T_eG\\rightarrow {\\mathfrak g},\\text{ and}", "01ae6d84773d96ef563f2be9dacf5e9e": "\\frac{\\mathrm{d}N_B}{\\mathrm{d}t} = -\\lambda_B N_B + \\lambda_A N_A.", "01ae83345a7d932357d44a263ec78119": " F_{Y}(y) = P(Y \\leq y) = P(\\mathrm{log}(1 + e^{-X}) \\leq y) = P(X > -\\mathrm{log}(e^{y} - 1)).\\,", "01aee35ed3a5b074de86299a81ccaa03": "\\text{Moeb}(\\mathbf{S}^1)\\subset \\text{Diff}(\\mathbf{S}^1) \\subset \\text{QS}(\\mathbf{S}^1)", "01aee9f729a432e09c332da539eeb8d3": "mn=\\mathrm{N} \\mathfrak{p} -1 ", "01aefae77efb224ae0167b114ce3556b": "\\alpha_{\\text{object}}", "01af897a5f8a1e5cd17232f87c20c21a": "r/2", "01afdbc2d4543f51eeea1f8df91ee9de": "Y = A * F(K,L) \\,", "01aff9b8c6c4296ae629c6fed72f30c7": "\\int\\frac{dx}{ax^2+bx+c}", "01b0782a0a4f89160fd5022c5284e501": "T_{T}=\\frac{2L_{T}}{\\sqrt{c^{2}-v^{2}}}=\\frac{2L_{T}}{c}\\frac{1}{\\sqrt{1-\\frac{v^{2}}{c^{2}}}}", "01b0e951629ccbc18721289f1cd8cdf3": "P\\left(k\\right)\\sim ck^{-\\gamma} \\, ", "01b0f693fc41e3fa99a9a2e13c88dbcf": " S_u = \\frac{\\hbar}{2}(u_x\\sigma_x + u_y\\sigma_y + u_z\\sigma_z)", "01b11daa5ac39cdda7f6b4aa6b489ce4": "\\mod 2^n-1", "01b1abef0f3a7d80becdbbfdccc7763d": "f^*(\\omega + \\eta) = f^*\\omega + f^*\\eta,", "01b1c662c9569eaadaeb573607ea8644": "W_{\\text{Yuk}}", "01b20c4e008e677c5b4594db36fbc925": "H_p(B(S^{-1}S)^0) = \\varinjlim H_p(BS_n) = \\varinjlim H_p(BGL_n(R)) = H_p(BGL(R)), \\quad p \\ge 0", "01b20e5cf9941b9e9034764b678beacb": "U\\,\\!", "01b265fee6a0869c2e4f9adafa319138": "\n\\frac{\\mathrm{d}^3 x}{\\mathrm{d} t^3}+A\\frac{\\mathrm{d}^2 x}{\\mathrm{d} t^2}+\\frac{\\mathrm{d} x}{\\mathrm{d} t}-|x|+1=0.\n", "01b28d0dae8226bfb1d154b5662decc5": "\\frac{P \\or Q, \\neg P}{\\therefore Q}", "01b2c8fc03e1f4bfc3606d60022ac277": "[Q^\\dagger,b^\\dagger\\}=0", "01b2cafcfc2480ecbe9b33cb21ccf6fa": "\\pi_k^n(x_1,\\dots,x_n)=x_k,", "01b2cf201aee34804cd87795fbaf6d24": "\\frac{\\omega-\\omega_o}{\\omega_o}=\\frac{\\Delta \\omega}{\\omega_o}=\\frac{\\mu HH_k}{2kL_e^2(H+H_k)},", "01b2dfca9f0240a5b8f0859525d2e570": "N\\sin \\theta ={mv^2\\over r}", "01b2f3fed2a492efd95302ad3b7b0165": "\\rho\\;\\!", "01b33ba9f800285f0859adf08818f1e7": "x = a\\ \\operatorname{arcsinh}(s/a) + \\alpha.\\,", "01b3fe9e9446fa564c2e05d03313e91c": "v_0(\\xi\\otimes e_\\alpha)=(v,\\alpha) \\xi \\otimes e_\\alpha.", "01b415a0ef6a764eb78f82f7169c051f": "\n\\int_{-1}^{+1}\\frac{T_m(x)\\log(1+x)}{\\sqrt{1-x^2}}dx = \\sum_{n=0}^{\\infty}a_n\\int_{-1}^{+1}\\frac{T_m(x)T_n(x)}{\\sqrt{1-x^2}}dx,\n", "01b449b4ef96135c708dd6a66a52ee28": " \\varphi_\\lambda(e^t i)={1\\over 2\\pi}\\int_0^{2\\pi} (\\cosh t - \\sinh t \\cos \\theta)^{-1-i\\lambda} \\, d\\theta.", "01b4a3c7c3d1f0e5303c0740eda30fd1": " \\frac{n(n-1)}{2} ", "01b59247bdf3ed5106ba8e6ac3cceef3": "\\sum_{g \\in G} r_g g", "01b6af627963b834d56274feae25b317": "\\frac{\\exp(-\\beta \\varepsilon(\\mbox{state}))}{\\mathbb{Z}}", "01b6d4d188517123c07fa5aecdad31ec": " (\\delta f)(x) = {{ f(x+h) - f(x) } \\over h }", "01b706e17abb0a059a02379fa29619c6": "\nA =\n\\begin{bmatrix}\n ~4 & -1 & ~0 & -1 & ~0 & ~0 & ~0 & ~0 & ~0 \\\\\n -1 & ~4 & -1 & ~0 & -1 & ~0 & ~0 & ~0 & ~0 \\\\\n ~0 & -1 & ~4 & ~0 & ~0 & -1 & ~0 & ~0 & ~0 \\\\\n -1 & ~0 & ~0 & ~4 & -1 & ~0 & -1 & ~0 & ~0 \\\\\n ~0 & -1 & ~0 & -1 & ~4 & -1 & ~0 & -1 & ~0 \\\\\n ~0 & ~0 & -1 & ~0 & -1 & ~4 & ~0 & ~0 & -1 \\\\\n ~0 & ~0 & ~0 & -1 & ~0 & ~0 & ~4 & -1 & ~0 \\\\\n ~0 & ~0 & ~0 & ~0 & -1 & ~0 & -1 & ~4 & -1 \\\\\n ~0 & ~0 & ~0 & ~0 & ~0 & -1 & ~0 & -1 & ~4\n\\end{bmatrix}\n", "01b7477bcbbd2d0769a6b4d3bf074f71": "a(k)\\left |0\\right\\rangle = 0.", "01b77ce6c8f81c618bb9968aa25d7455": "\\Delta(z)", "01b7e243a05b2f042a5bc115256a7477": "y = \\frac{1}{2} \\ln \\left(\\frac{E+p_\\text{L}}{E-p_\\text{L}}\\right)", "01b810f080f6385798780e7fc3463c97": "\\pi_1:U\\times G \\to U,\\quad \\pi_2 : U\\times G \\to G", "01b859364dda742772c2f949845f4e52": "F \\times \\mathsf{S}(a) = \\mathcal{P}_B^{\\perp} (a \\cdot \\partial F),", "01b85a530f6ff948dd2ad38b65ec707e": "p_\\mathrm{int} = p_1+s_\\mathrm{int}\\cdot\\mathbf{u}", "01b881814da8e0aaa3c96b9b650e95fd": "\\dot m_{out}=K \\cdot C \\qquad (4)", "01b894dc22ae1082677a08d6b924e48d": "\\scriptstyle v ", "01b8c391a9e2849d70e4175f47a596d5": " \\Phi(y) ", "01b8d973d292f2a29aea13ee5ef47880": " t \\mapsto \\bold{X}(\\bold{u}_0) + t \\bold{A}(\\bold{u}_0). ", "01b95a3d7abea8628080371744d90d22": "\n\\left[ \\left.\n\\begin{array}\n[c]{cccc}\n1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0\\\\\n0 & 1 & 0 & 0\\\\\n0 & 0 & 0 & 1\n\\end{array}\n\\right\\vert\n\\begin{array}\n[c]{cccc}\n0 & 0 & 0 & 0\\\\\n1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0\\\\\n0 & 0 & 1 & 0\n\\end{array}\n\\right] .\n", "01b978a6a034de8a736775b90de615dc": "S_0 = 0", "01b97ce8ecaa783b96225eadb98e51b2": "\\operatorname{d}=\\tau", "01b997311f718112df1bbbe9a5accb6d": "H(A: B |\\Lambda)=0", "01b9b96e98ff2e57ef3171d562d79a55": "r = \\sqrt{3 / \\Lambda}", "01ba19784c735ed8b3a29614ac1c98c4": "C_\\pm(j,m)", "01ba6a18d617927034af344cc636f91f": "\\Delta u \\ge 0,", "01ba77110113019916a9054319ae7c05": "f(0)", "01baae0577e052f0eb6660747516f26f": " [n_{PQ}]_{PQ \\sim QQ} ", "01bae0ecde1960fec732c7c0a51fff82": "\\widehat{U}(t - t_0) \\equiv U(t, t_0)", "01bb12a4f0524a838974075b1126f307": " \\{ fg,h \\} = f \\{ g,h \\} + g \\{ f,h \\} ", "01bbb262477c343e2c65ed5e8a4ad417": "K_- = span\\{ \\phi_- = a \\cdot e^{-x} \\}", "01bbb6d958ab9dc5347b3e281037fa00": "F(x)=E(1_{X_1\\le x})", "01bbc47541ee02be2bda7635619e5043": "\nH(\\mathbf{Y})=-\\frac{1}{N}\\sum_{t=1}^N \\ln p_\\mathbf{s}(\\mathbf{y}^t)+\\ln|\\mathbf{W}|-H(\\mathbf{x})\n", "01bbd03668954499b3f7781400b2da2f": "w, \\mathcal O_L, \\mathfrak p", "01bc44e1b633cb8d01df71ae6569036e": "\\langle \\Sigma \\rangle = \\mbox{diag}( i \\sigma_2 f_3, i\\sigma_2 f_3, i\\sigma_2 f_3, i\\sigma_2 f_2, i \\sigma_2 f_2)", "01bc48c212bc834f5225cc8dba7ee47f": "\\mathbf{\\hat f}", "01bc4ad5ea6ec57f062a5766c6bb6f3b": "\nZ=\\int e^{-\\frac{F(r)}{kT}}dr\n", "01bc6c772c8830bc450cfa7414f52319": "\\scriptstyle v_2", "01bd0aea7abc570c68a7636e14c2650c": "P = \\sum_{n=0}^p A(n)x^n = A(0)+A(1)x+A(2)x^2+ \\cdots +A(p)x^p", "01bd5c65c6572b9a691551059117c64a": "y_{1}^\\star", "01bd5fea77e2aaf26d5127ea526462f9": "\\sum_{k = 0}^{\\infty} \\frac{16^{n-k}}{8k+1} = \\sum_{k = 0}^{n} \\frac{16^{n-k}}{8k+1} + \\sum_{k = n + 1}^{\\infty} \\frac{16^{n-k}}{8k+1}. \\!", "01bd73abf03dde76e3597ba1a3373a6e": "f(x_1,x_2,x_3)", "01bdbed18e06abe6fcd6d4204c9fde4b": "\\chi \\equiv \\{\\operatorname{Tr}\\big(\\Gamma(R)\\big)\\;|\\; R \\in G\\}", "01bde7da2af3bb5b592b0bd89c8a1a84": "n = u", "01bdfb42401098f22e4e65082a25782b": "\\alpha = \\frac{\\lambda - n}{p}", "01be15b45c8c746eeab570c7f9afb5fc": "\\scriptstyle{\\pi^*}", "01be1daa1ef16435bbd120ce445acd8f": "O(1,n)/(O(1) \\times O(n)).", "01befff5e286d6de097c46c7deb5d0e1": "\\omega_L = \\frac{1}{2}\\left[ -\\omega_c + (\\omega_c^2+4\\omega_p^2)^{1/2} \\right]", "01bf691f9de147784b9aa33fc2671716": " \\cot \\theta =\\!", "01bf7c92ab2b2bc69503bac0f5f03dc4": "\\int_0^\\infty |f(t)e^{-st}|\\,dt", "01bf930c851369a27e34ba27a127a9d1": "\\operatorname{span}(\\mathbf{v})", "01bfc575b9da2f84e9e45e0538a0d95f": "T_b=\\frac{I_{\\nu}c^2}{2k\\nu^2}", "01bfd3f7a9d4b122649fac52f46f33da": "[\\omega]^{\\omega}", "01bff782684ce9a8b67e0c4858691369": " \\phi : \\mathbb{R}^4 \\rightarrow \\{ 0 \\} ", "01c047fad210fd39854ed9a0de836647": " T(n) ", "01c06a44570541f591261037bce6aebb": " \\tau_{xy} = \\frac {\\mu b} {2 \\pi (1-\\nu)} \\frac {x(x^2 -y^2)} {(x^2 +y^2)^2}", "01c07da14ee5b66ac914af46f54c98b4": "r_n = b", "01c0a6c761628d72dbdf978bda335e81": "\nL = \\frac{qB}{2c}(x\\dot{y} - y\\dot{x}) - V(x, y)~,\n", "01c0c8b1e311e98d09a5188569dfad2f": "\\frac{(f'(\\theta),\\ f(\\theta))}{|f'(\\theta),\\ f(\\theta)|} = (\\cos \\psi,\\ \\sin \\psi)", "01c0e78b5baed0feef5041fe7545dfef": "\\begin{align}\n& {} \\quad \\langle\\phi(x_1)\\phi(x_2)\\phi(x_3)\\rangle\\\\\n&=\\langle\\phi(x_1)\\phi(x_2)\\phi(x_3)\\rangle_\\text{con}\n +\\langle\\phi(x_1)\\phi(x_2)\\rangle_\\text{con}\\langle\\phi(x_3)\\rangle_\\text{con}\n +\\langle\\phi(x_1)\\phi(x_3)\\rangle_\\text{con}\\langle\\phi(x_2)\\rangle_\\text{con} \\\\\n&+\\langle\\phi(x_1)\\rangle_\\text{con}\\langle\\phi(x_2)\\phi(x_3)\\rangle_\\text{con}\n +\\langle\\phi(x_1)\\rangle_{con}\\langle\\phi(x_2)\\rangle_\\text{con}\\langle\\phi(x_3)\\rangle_\\text{con}\n\\end{align}", "01c11575ea95126fcd60f809f8da5bcf": "E_\\mathrm{v} = 10^{(-14.18-M_\\mathrm{v})/2.5}", "01c154aeb7c0087556908ee407b5d53d": "\\displaystyle\\frac{d^ns}{dt^n}", "01c160807c5af832724af0c6fc6c2ff9": "n \\log_2 n - \\frac{n} {\\ln 2}", "01c1ebc232309ad5d6535a37d3390e4d": "C\\ell_{i,j}", "01c21154df41f632afa61480f9b835f4": "k\\times n", "01c23d55901e9c54a2271c6f35213c45": "\\tilde H,", "01c23ece393899dd12ed251f005a308a": "i^2=id_A", "01c2795090f1eece4dd7433d6ba002ff": "\\theta_A=\\frac{k_1C_A\\theta_E}{k_{-1}+kC_S\\theta_B}", "01c2da53089a42414a8f92ccc46ee9a8": " t_2 = \\sum x_i^2 ", "01c3104f950e4db29466791bda1d743f": "\n d_{(ij)k} = \\alpha_i d_{ik} + \\alpha_j d_{jk} + \\beta d_{ij} + \\gamma |d_{ik} - d_{jk}|, ", "01c3ae91742e02bc43e53d948351f27b": "{\\left(\\frac fg\\right)}' = \\frac{f'g - fg'}{g^2}.", "01c3e9c1e55d5895c44543209649d809": "\n\\begin{array}\n[c]{cccccccc}\nI & X & I & Y & Z & I & I & \\cdots\n\\end{array}\n,\n", "01c3f37234550aae5c346d656a2cbe48": "\\omega = \\frac{1}{\\sqrt{LC}}", "01c3f71b1579582146a7326c7765985e": "q\\begin{Bmatrix} p \\\\ q \\end{Bmatrix}", "01c3fde8626debd09db6b81b0ad7d2d3": "\\forall A\\exists b\\exists c\\exists d\\; bA\\and cA\\and dA\\and \\lnot b=c\\and \\lnot b=d\\and \\lnot c=d ", "01c48f2a351834b3e827311bbde7137d": "\\tilde{\\rho}:K[G]\\rightarrow \\mbox{End} (V).", "01c490edb230e5ce38488ed375fc43de": "r_{j-1}0", "01cd4bb547aef88a022ade3aa2751492": " B_k(j) \\!", "01cd4e3f01a20a9194cf0e90f97cb556": "\\mathbb{P}(V)", "01cdbc2e9e7b9e782f1f84c0125c7150": "\n\\Delta g\\ =\\ \\int\\limits_{0}^{2\\pi}\\bar{V} \\bar{h}\\frac{r^2}{\\sqrt{\\mu p}}d\\theta\n", "01ce38e7e92f345f1d8657e6c1167623": "\n\\begin{align}\nI =& B_{1_1} + B_{2_1} \\\\\nA =& 3/4 B_{1_1} + B_{1_2} + B_{2_1} + B_{2_2} \\\\\nA^2 =& (3/4)^2 B_{1_1} + (3/2) B_{1_2} + B_{2_1} + 2 B_{2_2} \\\\\nA^3 =& (3/4)^3 B_{1_1} + (27/16) B_{1_2} + B_{2_1} + 3 B_{2_2}\n\\end{align} ", "01ce5bc8ee09686540fea99c45d34c3e": "\\left.\\frac{{\\rm d}W}{{\\rm d}z}\\right|_{z=0}=1.", "01ce72ac07ffca2b84ce8f610856d4cd": "\\Delta \\theta = 2 \\pi \\frac{R_W} {D} \\frac{T_1-T_2} {T_R}", "01ce7f5c6112c876528db18ce012f72e": "\nf(s) =\\liminf_{n\\to\\infty} f_n(s),\\qquad s\\in S.\n", "01ceb417b02f63b4e5d46e62973cf371": "\\Delta t = R_N^{-1} ", "01cfb4708aefc4c4a31ab61902490d5a": "y_L = F(x-\\delta, \\hat\\theta)", "01d0242e64a2c042c8683e7c24984b6c": "z:=x+iy\\in\\Omega.", "01d05171f7824558056e284722f832ec": "\\bar{e}_{x}^{ch}=\\,", "01d0587525d2f3cc498963f7b7f882aa": "X \\sim \\chi^2 \\left( 2 \\right)", "01d075d91893ddcb311ee9cd943239eb": "t_2 = \\gamma \\frac{1}{f^\\prime}", "01d09dc5f46ce25eb6baf35afb266fb2": "F=\\left\\lceil\\frac{\\ln(\\epsilon^2/4)}{\\ln(1-\\epsilon/2)}\\right\\rceil", "01d0beba1ca746f01eaa89df05659ab6": "M(a,0,0) \\to S_{ah}", "01d0db18593a536cfb2695353995a6ab": "{dN \\over dt} = aN^2 - bN", "01d1026a9bcf4926e9c62684289f26b0": " \\int X \\mathrm{d}^n x \\equiv \\int X \\mathrm{d} V_n \\equiv \\int \\cdots \\int \\int X \\mathrm{d} x_1 \\mathrm{d} x_2 \\cdots \\mathrm{d} x_n \\,\\!", "01d115d57b9fecdfa8787bc3f7558428": "\\vec J_\\sigma = \\frac{ND\\Omega}{kT}\\nabla H", "01d154d178cf32c6a6cec4a660bd644f": "\\displaystyle 2\\pi f(-\\nu)\\,", "01d1664a7b946d902bed06c864bfb264": "|\\psi\\rangle \\in \\mathcal{H}", "01d1680b2dc1fd8d7d098eb724977024": "b - f(x_0)\\,", "01d17effc3caa2eb0769b9c887809b2b": "G(\\theta|\\alpha)", "01d1a9cb178333516fb523774c0365c2": "y \\,\\!", "01d1b955c6b4390f2d079bce20c322a9": "x - 1 = 1 + \\frac{1}{2} + \\frac{1}{4} + \\frac{1}{5} + \\frac{1}{6} + \\frac{1}{9} + \\cdots", "01d1cc0172190c64a713cea6a1b206ca": "C_{123}=2C", "01d20d6ae7c979e280cb6fcb05563978": "IMA = \\frac {F_{out}} {F_{in}} = \\frac {V_{in}} {V_{out}}.", "01d246be21a9a22158d722e1dda3a217": "A_1, \\ldots, A_n \\vdash B_1, \\ldots, B_k", "01d278bc6a56ef0773914beb858779ce": "[(i-1)w, i w)", "01d27ce00111b7f339e9646c360e5a8f": "1/\\ell", "01d2b09e94363278c3ac681bba860bd8": "G(\\vec r,t) = \\frac{1}{4\\pi r} \\Theta(t) \\delta\\left(t - \\frac{r}{c}\\right)", "01d2b84cce753408277d414ef7185571": "Wins = 52 + fWAR", "01d2c43c8adf5d085baf21b62fa2a944": "(c,\\varepsilon)", "01d2dd56f011f08a8d2bae92b224777c": "\n<\\mid F_{in}\\cdot e_{ex}\\mid^{2}> \\propto \\int \\sin \\theta_{1}^{in}d\\theta_{1}^{in}A_{in}(\\theta_{1}^{in}) \\times \\int \\sin \\theta_{ex}d\\theta_{ex}O(\\theta_{ex})U_{ex}(\\lambda_{in},\\theta_{1}^{in}.\\theta_{ex})", "01d34ea7454b3823348e4f8abe9c5b77": "= X \\oplus N',", "01d35b53d4425d9850ede5b316e98ba2": " \\Phi = \\frac{ { k}_{ f} }{ \\sum_{i}{ k}_{i } } ", "01d37fd202fcf8fa4aba57bb5e0e69f7": "\\; P_i \\pi(a) P_i = a", "01d380085acc4d17c2e69127c713199b": "O(2^n) \\bigcup O(n^2)", "01d3a872cf541b472ef41f84273d36e8": " D_H = \\frac{4 \\cdot 0.25 \\pi (D_o^2 - D_i^2)} {\\pi (D_o + D_i)} = D_o - D_i ", "01d415b15fdb845cc85c3ed324f1fbde": "{\\rm d}N {\\rm d} x", "01d42f036bdb98bef530250193b25fa7": "\n \\begin{matrix}\n ^{^b{b}} \\bar a = & \\underbrace{a_{}^{a^{{}^{.\\,^{.\\,^{.\\,^a}}}}}} & \n\\\\ \n & {{^b} \\bar a}\\mbox{ copies of }a\n \\end{matrix}\n ", "01d43272586b34117a0e3de96023a955": "n_1 + \\ldots + n_r = n\\,\\!", "01d44451a24dd18410cb1ed7c2ba5fce": "\\,q_x = d_x / l_x", "01d47a70793565e13c6acbc537e08978": "\\frac{\\partial}{\\partial g_i}(u^{-1}) = -u^{-1}\\frac{\\partial}{\\partial g_i}(u)", "01d4a01c24e9a19b5520a3836a691600": "\\lambda g", "01d4be5fb686cc741ae27340fe0e1539": " P(X_1,\\ldots,X_n)= P_{\\mbox{lacunary}} (X_1,\\ldots,X_n) + X_1 \\cdots X_n \\cdot Q(X_1,\\ldots,X_n). ", "01d4be8c48d4c3f4c375779c2ae1fc92": "E^{\\prime}", "01d58c08ebdb5b0e80ab88f8d72caf12": " p(a,d) \\leq (1+o(1))\\varphi(d)^2 \\ln^2 d \\; ,", "01d58e290042cd241210e2f4f8bef268": "w_r^-", "01d5ec850531d49cb9513324ec9935db": "\\,x_0\\leq x\\leq x_1\\,\\,", "01d66037af654b16d04c660764651244": "\\det S''_{zz}(z^0) = 0", "01d6ff79c4bd0e0ac0d1b6dbd6680846": "n\\# = \\prod_{i=1}^{\\pi(n)} p_i = p_{\\pi(n)}\\# ", "01d71ed39474d8e2ccecd373f1808342": "A\\mathbf{x} = \\begin{bmatrix} \\mathbf{a}_1 \\cdot \\mathbf{x} \\\\ \\mathbf{a}_2 \\cdot \\mathbf{x} \\\\ \\vdots \\\\ \\mathbf{a}_m \\cdot \\mathbf{x} \\end{bmatrix}.", "01d7389dd8daac6fa380ea48d18da2e3": "\\pi_{10}", "01d776eea2c34f8eec530b7f7a7ef049": " A_o = 0.999 \\approx 8 \\ hours \\ down \\ time \\ per \\ year", "01d779db54d10909296a3e0e20fc6c3a": "V = 2\\pi^2 r^3", "01d7eff18535ee23b9a228919c186e21": "I(s) = \\frac{V_{in}(s)}{R + Ls}", "01d830dbc637ebc6eef10832e456861a": " P(X=5) = f(5;50,5,10) = {{{5 \\choose 5} {{45} \\choose {5}}}\\over {50 \\choose 10}} = {1\\cdot 1221759\n\\over 10272278170} = 0.0001189375\\dots, ", "01d84c0b06afc9c27c5264692ec2ee41": " a, b, c \\in N ", "01d85502cacd9acc332bacd50f367f00": "\\left( -\\nabla^{2}_{\\mathbf{u}}+\\frac{1}{4}ku^{2} +\\frac{1}{u}\\right)\\Phi(\\mathbf{u}) = E_{\\mathbf{u}}\\Phi(\\mathbf{u}).", "01d882ec38abc94b1064ee49b0256d5b": "\\mathbf{P}^n", "01d922986dd7527cf78bf949673bfb1f": "A,B,X,C", "01d931498a3d7b6d7e1bc6f0ed6a4a06": " \\begin{pmatrix} x & y \\\\ -y & x \\end{pmatrix}.", "01d941f4013dec7eb7aba37d2dc11780": "\\mathcal{E}(\\rho) = \\sum_{m,n} \\chi_{mn} E_{m} \\rho E_n^\\dagger", "01d9c67c97e8047bf2bfdaa2ad5c8808": "M = \\frac{-f_2}{f_1},", "01d9db4c2459ffc051305ad74e2f4256": "I(\\bold{x}, t;\\bold{\\hat{n}},\\nu)", "01d9fc38090a5436fadf1b8b06471409": "\\frac{\\mathrm{m}/\\mathrm{s}^2}{\\mathrm{Pa}}", "01da8f763bcd3533e23d82c937942e20": "\nz = \n\\frac{1}{2} \\left( A + B - \\lambda - \\mu -\\nu \\right)\n", "01daa0079732e4a1f48600a4a3251a53": "y_n(x)=(2n\\!-\\!1)x\\,y_{n-1}(x)+y_{n-2}(x)\\,", "01dab2020cf38b41842d6c211501b787": "\ns e^{i \\Delta k \\Lambda}=e^{i \\Delta k \\Lambda} -e^{i 2 \\Delta k \\Lambda n}+e^{i 3 \\Delta k \\Lambda}+...+(-1)^N e^{i \\Delta k \\Lambda (N-1)}-(-1)^N e^{i \\Delta k \\Lambda N}.\n", "01dae64584b988a11f4f653b3359640e": "ds^2 = d\\chi^2 + \\sin^2(\\chi/\\alpha) ds_{dS,\\alpha,n-1}^2,", "01daebc5b411677123fc9f4734fa8fed": "\nC_1^+(\\beta) = \\frac{\\alpha}{2} \\log \\left( 1 + (c_{31}^2 + c_{21}^2) P_1^{(1)} \\right)\n + \\frac{1-\\alpha}{2} \\log \\left( 1 + (1-\\beta) c_{31}^2 P_1^{(2)} \\right)\n", "01db16108e95588e314e7db20af284b5": "z(m,n;s,t) < (s-1)^{1/t} (n-t+1) m^{1-1/t} + (t-1)m.", "01db34fef6aa29ed0a4092f1812ca6d3": " M = \\left\\{ (a,b); a=b; a \\in A; b \\in B \\right\\} ", "01db3f2f5f32c0e5476bacd9e378b24d": "F(\\overrightarrow{x},s)", "01db8fd1c607c20f073a9e4e01267aed": " \\nabla^2 f(x) ", "01dba1731ae06e01d5e4cb38b470dbcc": "\\frac {f{(x)}-f{(-x)}}{2}", "01dbd8419c18df8b5400d24cd60ab691": "q=\\frac{\\sqrt{Fb}}{b}", "01dc19e3571d9dfc66ab0771f91f5180": "(\\alpha_0,\\beta_0,id)", "01dc1d552c5547bade52f5f9c8d22afb": " Q=Q(p) ", "01dc276d84de2a77b12d92dcd2d354b2": "2\\log k", "01dc4239e20dd0a7c6cccfd8ddf4e7f0": "[J_{ij},Q_a] = \\frac 14 (\\gamma_i\\gamma_j-\\gamma_j\\gamma_i)_{ab} Q_b,", "01dc43fb7bd88609eb84d081d609513f": "{}_1F_1(0;b;z)=1", "01dc5007081749b7a310feccf1354232": "\\lambda_1^k,\\lambda_2^k,\\dots,\\lambda_n^k", "01dc58ec3ac830294a6a937ae668cff7": "\\hat e(\\mathbf{s}_0)", "01dc619881fa5961b4ecdd8bcfe256b5": "(k_{f_1},k_{b_1},k_{f_2},k_{b_2})", "01dc735e3025852bf1c8ab7517a735d7": "r_2,\\ p_1", "01dc774e9c2c01320bd7e31b53d233f7": " \\mathbb{Q}[Y_1,\\ldots, Y_s]. \\, ", "01dc82dd0f686daf69ba2dfbc1edd95c": "\\,\\langle P_W\\rangle", "01dccf6774f6bbc2a9ae99375f9b7a91": "\\phi:\\mathcal{G}\\to\\mathcal{N}", "01dd07ab53a078a180fd9b599836ded6": "t_r", "01dd6d12c7f43ffb157c0c0a7b3ad810": "(z_1, z_2; z_4, z_3) = {1\\over\\lambda}", "01dd7550747902bc9e4a872463a3fd20": "\\lfloor p/m \\rfloor", "01ddd304f3cc045b31ac39874c845209": "C=\\frac\\sqrt{\\mu^2c^4-E^2}{\\hbar c}", "01de27534bbeea30c00ebfa9d73e5366": "so(3, 1)", "01de33337a5e025911424042a1359e86": " C = -\\frac{dC_v(K,\\sigma(K))}{dK} = -\\frac{\\partial C_v}{\\partial K} - \\frac{\\partial C_v}{\\partial \\sigma} \\frac{\\partial \\sigma}{\\partial K}", "01de5d01b9bb6c5ea26b690f212d9b04": "n(\\mu) \\propto e^{\\mu/k_B T}", "01def30326acf780125644d83affad21": "E_{yz,3z^2-r^2} = \\sqrt{3} \\left[ m n (n^2 - (l^2 + m^2) / 2) V_{dd\\sigma} +\nm n (l^2 + m^2 - n^2) V_{dd\\pi} - m n (l^2 + m^2) / 2 V_{dd\\delta} \\right]", "01df00b61d5692071c8cbfb211a02dfa": " \\sin iy = i \\sinh y. \\,", "01df03d7ac1229038aa710b8743b3fbe": " \\sin\\left( \\pi/2-\\theta\\right) = \\cos \\theta", "01df2292ddf37ed672196cd88db9568c": "\\mathrm{gain}_{\\mathit{TE}}=a_{\\mathrm{form}} \\cdot (1 - a_{\\mathit{vf}})", "01df2eeb9b53add0e1612df83eefdc35": "X_\\alpha, and X_\\delta", "01df4445f0f3bae5903887db6e0805b0": "\\partial_t i(t,a) + \\partial_a i(t,a) = s(a,t)\\int_{0}^{a_M}{k(a,a_1;t)i(a_1,t)da_1} -\\mu(a) i(a,t) - \\nu(a)i(a,t) ", "01dfd632e0c8ed78c80b807404bd8a38": " L(t) = \\mathbf{R}^* + t\\vec{k}.", "01dff6f37fbd572a4603f7672037ad3a": "[F_{\\lambda}]", "01dffc9159b2ed3efc44c711463dc491": " \\text{HSIC}(X, Y) = \\left| \\left| \\mathcal{C}_{XY} - \\mu_X \\otimes \\mu_Y \\right| \\right|_{\\mathcal{H} \\otimes \\mathcal{H}}^2 ", "01e010f88dff08d008c6d62171d214ca": "U_\\beta = \\left(U_0, U_1, U_2, U_3 \\right) = \\gamma \\left(-c, u_x, u_y, u_z \\right) \\, ,", "01e022402b9df1bc43a30582c69795f7": "p(4063467631k+30064597)\\equiv 0\\pmod{31}.", "01e0258e23148f019a0b12b93d87c9d1": "I=I_S\\left(e^{V_D/(nV_T)}-1\\right)", "01e0415e98dbe03a58400cd4f881e666": "\\boldsymbol{m}\\cdot\\boldsymbol{N}=0", "01e0b28d6603dd6d5ca5fc5502075ec9": " \\frac{a L}{R}", "01e0e8dabc4ce2ac3887ce67a33f1296": " \\Box_2 P ", "01e108d0a2ec9beb42187c4278af1be4": "P_{D-}", "01e1586a49ee5e8cb6148aaade4882f2": "\\zeta(s)=\\frac{1}{s-1}+\\sum_{n=0}^\\infty \\frac{(-1)^n}{n!} \\gamma_n \\; (s-1)^n.", "01e19fdfa95d98838f56f6a0b6f84126": "{\\tilde{BC}}_2", "01e1f9190d77917fa124a3d4ead6a8c4": "Q_{in:friction} = C_d \\rho |\\mathbf{u}|^3", "01e21538af1d452befee558b81565532": "{}^3_2\\mathrm{He} + {}^2_1\\mathrm H \\to {}^4_2\\mathrm{He} + \\mathrm p", "01e26383848cb410a14cb8b3d2b92239": " A = (a_{i,j})_{1\\leq i,j\\leq d} ", "01e2a690c2a1b53a21a63bf4493e6cc6": "[(a,b)] - [(c,d)] := [(a+d,b+c)].\\,", "01e2db5f559085fb07c80e964a30ef0e": "P_3 = (X_3,Y_3,Z_3,ZZ_3) = (48-8\\sqrt{39},296\\sqrt{3}-144\\sqrt{13},2,4)", "01e323cb5b79d946495eab0fd0f6a9c9": "\\begin{bmatrix}\n \\cos\\beta\\cos\\lambda\\\\\n \\cos\\beta\\sin\\lambda\\\\\n \\sin\\beta\n\\end{bmatrix} = \\begin{bmatrix}\n 1 & 0 & 0 \\\\\n 0 & \\cos\\epsilon & \\sin\\epsilon\\\\\n 0 & -\\sin\\epsilon & \\cos\\epsilon\n\\end{bmatrix}\\begin{bmatrix}\n \\cos\\delta\\cos\\alpha\\\\\n \\cos\\delta\\sin\\alpha\\\\\n \\sin\\delta\n\\end{bmatrix}", "01e3cacd2b9c2b395a3126c85a799f03": "\n \\boldsymbol{\\tau} = \\varphi_{*}[\\boldsymbol{S}] = \\boldsymbol{F}\\cdot\\boldsymbol{S}\\cdot\\boldsymbol{F}^T~.\n", "01e44b0dc54b8b019e635f7283b75df2": "\\mu^{\\otimes 0}(A_0(s,t)):=1.", "01e46395993aa5169a4f46c994e057c2": "\\textstyle(x\\pm1, y, z\\mp1)", "01e4a15095bab293c07843429213637e": " \\ddot{q} = M^{-1}Q + M^{-1/2}\\left(AM^{-1/2}\\right)^+(b-AM^{-1}Q). ", "01e4b0c7be863d94cb74865e74285978": "%B=\\frac{f_H-f_L}{f_c}=2 \\frac{f_H-f_L}{f_H+f_L} ", "01e4b9416f3b7a700735850d73bbd049": "\\lim_{n\\to\\infty} a_n = L.", "01e51ae055d2edd7e4320fe80ffe1073": "F_0(x) = \\underset{\\gamma}{\\operatorname{arg\\,min}} \\sum_{i=1}^n L(y_i, \\gamma),", "01e53e5c0b2839cf0c169069276f73e5": " | \\rho | = \\frac \\mathrm {VSWR-1} \\mathrm {VSWR+1}", "01e640b0a6ced27eeac99f6f1da9bb05": " X \\sim N(\\mu, \\sigma^{2}) \\!", "01e649334a2c1ed88b47ade97a8c785f": "\\Omega\\equiv\\frac{d^3\\varphi}{dz^3}+i\\alpha Re\\left[\\left(c-U\\left(z_2=1\\right)\\right)\\frac{d\\varphi}{dz}+\\varphi\\right]-i\\alpha Re\\left(\\frac{1}{Fr}+\\frac{\\alpha^2}{We}\\right)\\frac{\\varphi}{c-U\\left(z_2=1\\right)}=0,", "01e671dbd13bce2f51b07af455e57608": "x_{i,m+j}\\geq 0", "01e6ccfc99e178a8c5bc8f927841d736": "u(x,t)=\\frac{\\lambda}{4}\\int_{E_\\lambda}u(x-y,t-s)\\frac{|y|^2}{s^2}ds\\,dy,", "01e74f04804f931cc50fbfa868d0eaf6": "\\,K_{1B},\\ K_{2B}", "01e74f89e4d6421b5c028282fe6fbf4e": "[e]=\\{f\\in E|f\\leq e\\}", "01e77ba3199f76d686f03552d12c79b2": "\\vec{v}(t + \\Delta t)", "01e78043796bc55062f208abf997af9e": " \\lambda x.\\operatorname{drop-formal}[D, \\lambda o.\\lambda y.o\\ x\\ y, F] ", "01e78c601610f6c7b2a224a6cfb15dd2": "b_{\\nu, n}(x)", "01e8066e145a375d8f6910bb91bc45ec": "(s-1)\\zeta(s) = \\int_{-\\infty}^\\infty \\frac{(1/2 + i t)^{1-s}}{(e^{\\pi t}+e^{-\\pi t})^2} \\, dt.\n", "01e8153ecd79d5daf0df9fc8579edd9e": "\\mathfrak{H}_b", "01e86ced95c51596f778d74df8c8bf96": "V=1096.7 \\sqrt{H/d}", "01e9b4e5ba85de9ac8931c518c75329d": "\\scriptstyle s_{\\infty}(x)", "01ea358477bd18b369f5831702e6e4a7": "\nF_{hkl} = \\begin{cases} 4f, & h,k,l \\ \\ \\mbox{all even or all odd}\\\\\n 0, & h,k,l \\ \\ \\mbox{mixed parity} \\end{cases}\n", "01eaaa17d9dce7e235b677bc79046182": "\\sigma^2=\\lambda^{-1}", "01eadb9d7afc7f715e95d21f4ade3bb0": "(p,\\, t) = (i,\\, 2j+i)", "01eb0cdc32ef16bfba610e677a4823ca": "\\textstyle\\sqrt{e}", "01eb240e2bfb2732e6941810498adfb2": "((p_x q_w-q_x p_w)^2+(p_y q_w-q_y p_w)^2,\\mathrm{sign}(p_w q_w)(p_w q_w)^2)\\,.", "01eb3a530c08e2193c36adc9fab5107d": "y_2 = \\left. \\frac{\\partial y}{\\partial c}\\right |_{c=\\alpha} = a_0 s^{\\alpha }\\sum_{r=0}^{\\infty } \\frac{(\\alpha )_{r}(\\alpha +1-\\gamma )_{r}}{(1)_r (1)_r}\\left( \\ln(s) +\\sum_{k=0}^{r-1} \\left( \\frac{1}{\\alpha +k}+\\frac{1}{\\alpha +1-\\gamma +k}-\\frac{2}{1+k} \\right) \\right)s^r", "01ecb5cec1a178baac07c1d3161bbe12": "B_\\lambda(T) = \\frac{2 c^2}{\\lambda^5}~\\frac{h}{e^\\frac{hc}{\\lambda kT}-1} \\approx \\frac{2c kT}{\\lambda^4}", "01eccb8e17d972949e03580c41d08994": "(X,Z)", "01ecefd8e8946da30610fe9a89d437e0": "c_{T-2}(k) \\, = \\, \\frac{Ak^a}{1+ab+a^2b^2}", "01ecf76e7b919e8f093d393b99d25b96": " |x_1|=|x_2|=\\cdots|x_n|=1", "01ecfad7082922f85b35330787b6a893": " I = I_{cont} \\cdot \\frac{1 + K_n}{1 + 1.71 K_n + 1.33 {K_n}^2}", "01ed60cad5987fe9b72dfafdb6998db4": "\\mathit{ARA}(w) =-\\frac{u''(w)}{u'(w)}", "01ed7a9778d320559052bb613ab06943": "\\varphi_{X+Y}(t)=\\varphi_X(t)\\varphi_Y(t)=(1 - \\theta\\,i\\,t)^{-k_1}(1 - \\theta\\,i\\,t)^{-k_2}=\\left(1 - \\theta\\,i\\,t\\right)^{-(k_1+k_2)}.", "01edb4f49ec6aa80e62fa89946994808": " p \\times 1 ", "01edc57c51203044a554ae8a187fc31e": "X \\sim {\\rm Beta}(\\alpha, \\beta)", "01edc5ac6e7a583842e808f0ac05b1f3": "\\sum_{i \\in I} a_i X^i ", "01ee071d0ac5779eb2dd04415cac4812": "\\max_{x \\in S_{k-1}^{\\perp}, \\|x\\| = 1} (Ax, x) \\ge \\lambda_k", "01eea2c97b7016f8b1d32cec91e85538": "I,J", "01eec55e6318535a8351f82099461fc9": "H + 1 , H + 2 , H + 3 , H + 4 , ... , H + k", "01eecc08088d2dd3a1f402ce7f92772b": "\\eta =\\dfrac {\\pi Ze^{2}m^{1/2}\\ln \\Lambda } {\\left( 4\\pi \\varepsilon_{0}\\right) ^{2}\\left( k_{B}T\\right)^{3/2}}", "01eed3297ad06ed2478e3279d7c7ae69": " AH = t\\ \\text{Crd}\\ 10^\\circ \\approx t\\ \\frac{600}{3438}", "01eee35e8a902584c0b63d1d8bb80ebc": "\\sum_{n=-\\infty}^\\infty |c_n|^2 < \\infty", "01ef7a7dc58a56553149a519ca69a021": "N = \\frac{1}{\\sigma (C + D)}", "01efcc04cd663bb90911383a56399190": "X_R", "01efea36ed99f50ede86d8fdabd95ab9": "F(x): \\mathbb{R}^n \\to \\mathbb{R}^n", "01eff3a47e7e5d237fbf738a52537ca9": "R_e = \\frac{\\max\\left\\{ \\left| \\boldsymbol{U}_p - \\boldsymbol{U}_f \\right| \\right\\}\\, d_p}{\\mu/\\rho_f}", "01f01f007d2da6d2e11e1a1078602332": "D'= [P] + [R] - 2[O]", "01f0a3e33029e37179c066622a70be96": " \\int_E |f| d\\mu < \\epsilon ", "01f166c2df9b362185cbfb587b145efb": "\\frac{1}{q}=-\\exp(\\pi \\sqrt{163})", "01f19e23d7a338320ccc53e6f461c601": "f_{\\text{Aeolian}}\\ =\\ \\frac{\\alpha v}{d}", "01f1aa9773a2bc7c9abd38f608c57ae7": "xx^T - \\Delta \\in S_+", "01f1b5844156ea62392e3fe67819686a": "I(X, Y)", "01f1c233a51e9d045b83d50e5426de86": "a_{k+1},\\dots,a_n", "01f1f0ba6ee5f9907d32c0a36befefe2": " P(\\partial_t)G = 0, \\; \\partial^j_t G(0) = 0, \\quad 0\\leq j \\leq m-2, \\; \\partial_t^{m-1} G(0) = 1/a_m. ", "01f1ffe110c901fcfaefbb12c9e9960f": "\\scriptstyle \\leq1.9\\times10^{-33}", "01f266d4782c987e450bbaa0c56f9353": "1{\\to}\\tau", "01f3a391a61df4f8bf52765c05d92877": "a_{ii}", "01f3c699a2735a0d9a7311d672fd676c": "n_p", "01f414ce69bc416ef26e3b1aa09a3efc": "\\forall x, y, z\\,\\left(xFy \\wedge xFz \\to y = z\\right)", "01f41e5176fe1b6de7af480700737b0f": "E_{\\rm barrier} = W_{\\rm e}", "01f45976384f297b8e2d9f5229576785": "\\Delta = \\{\\alpha_1 \\ldots \\alpha_n\\}", "01f46efd1c4daed220ee2b124342dffa": "\\Delta g_{AB}=O_{B}^{crys}g_{B}(O_{A}^{crys}g_{A})^{-1}", "01f481a88cc19ffe6d6db95ccaa8dd92": "\\tilde{N} < N", "01f48294df6f44567d8f296da1a45a43": "F\\,= \\rho u A", "01f4e1ed87059e734917a9565f6ddf94": "\\tilde{H}_u", "01f4ea6adcaca1e0780939c0de27db0f": "\n\\begin{align}\nW' &= y_1' y_2' + y_1 y_2'' - y_1'' y_2 - y_1' y_2' \\\\\n& = y_1 y_2'' - y_1'' y_2.\n\\end{align}\n", "01f4ef7625842d787dd646758c3b1cf4": "\\operatorname{SO}(\\mathfrak{g})", "01f50df09f313f99b927b433e5677de5": "S(\\phi,\\mathbf A) = \\int {1\\over 4g^2} \\mathop{\\textrm{tr}}(F^{\\mu\\nu}F_{\\mu\\nu}) + |D\\phi|^2 + V(|\\phi|)", "01f524f17e4b4fcb00157c2698aca042": "u_{xy}=u_{yx}.", "01f53789ea66e1ffe67f4ac57dba6499": "y = \\frac {\\pm \\pi \\left(P Q - A \\sqrt{\\left(A^2 + 1\\right)\\left(P^2 + A^2\\right) - Q^2} \\right)} {P^2 + A^2}", "01f5710b2f73e55be4aadfba476a45c8": "\\Delta (x)", "01f5eaffb4b61b9fd64f74b251ece7db": "\\scriptstyle u,v\\in BV(\\Omega)", "01f628b27e8860217ef5fc754e8d60a6": "f_2(1) = 1 \\quad\\text{and for}\\quad d|k,\\; d>1,\\quad \\sum_{m|d} f_2(m) = 0.", "01f63496dff248313e3d9395692dbf61": "f_\\ell^m", "01f64d98287d4a6cfeaf14b94c993ba1": "\\partial_{-}C", "01f65fc413190c418d946b3c95119447": " u^2 - dv^2 = \\pm 4 \\, ", "01f6b15a5434d848c8b6899052b997b7": "\\scriptstyle x \\;\\in\\; W", "01f70036cfc9760ed393feb3b4fd8ad6": "\\scriptstyle \\cos \\theta_c = \\frac{c}{nv}", "01f708ec8a33bf3b68b15d3462a5fc8b": "a = \\left ({\\text{COMP} \\over \\text{ATT}} - .3 \\right ) \\times 5", "01f709eea689f82ea1ea61ca3c385613": "\\beta^{a} \\beta^{b} \\beta^{c} + \\beta^{c} \\beta^{b} \\beta^{a} = \\beta^{a} \\eta^{b c} + \\beta^{c} \\eta^{b a}", "01f70a960eb91ed4f3aadeab35b6deb4": "\\dot{z} = -2z (\\alpha + xy), \\, ", "01f788399c97985044f2437a18aab69e": "|\\Phi^-\\rangle ", "01f78be6f7cad02658508fe4616098a9": "550", "01f7c18c56f6d93726f78c234d1868da": " ((P \\or Q) \\and \\neg P) \\to Q", "01f824346fd27a8e5ae32409c29ab9e0": " a_n \\ne 0 ", "01f864dc442db64bf93663760fa8dae7": "\\begin{align}\n\\Vert\\vec a\\Vert^2 & = \\Vert\\vec b - \\vec c\\Vert^2 \\\\\n& = (\\vec b - \\vec c)\\cdot(\\vec b - \\vec c) \\\\\n& = \\Vert\\vec b \\Vert^2 + \\Vert\\vec c \\Vert^2 - 2 \\vec b\\cdot\\vec c.\n\\end{align}", "01f8b80e36b662229cbd834a93134c87": "\\textstyle u\\in W_p^k(\\Omega)", "01f8cede02e588da726936d313dcaa9b": "\nP(\\vec x|\\vec y) = \\frac {1} {(2 \\pi)^{m n/2} | \\boldsymbol {S_x} |}\n\t\\exp \\left [ -\\frac{1}{2} (\\vec{x} - \\widehat{x}) ^T\n\t\\boldsymbol {S_x}^{-1} (\\vec{x} - \\widehat{x}) \\right ]\n", "01f8f7e003bf6961951efb20b8a6959e": "\\gcd(a_1, a_2) = 1", "01f93ef84b3860edd2c0508453d523ee": " \\Delta_rG^{\\ominus} = -RT \\ln K_{eq} ", "01f94a2e8b3a86a1eac37f3a307d74ef": "\\left( \\lambda_i \\right)", "01f9b8831d5ce67ce115b33c7d1e9478": "Q = f(X_1,X_2,X_3,\\dotsc,X_n)", "01fa15b00eab23e5d544b290e9299048": "550 P_e = \\frac{\\eta_c H h J}{3600},", "01fa5ded58e5d08e631aba5bd2b0feb1": "\\{x\\}_{1}\\equiv \\min(x,1)", "01faaf3be3d2ed3aa7ecd4f6850926b9": "a\\frac{\\partial \\mathbf{U}}{\\partial x}", "01fae99ca641d883ac858c905d86728e": "c \\equiv z^Q \\pmod p", "01faf716f16570e46fec6b9b0d42144b": "f(x, y) = x^2+y^2 - L^2=0,", "01fb2beb7ef70ed58c2ce56badc91b74": "{\\mathfrak\ng}", "01fb56ab71a1da87b572193a63a2feba": " -\\dot{\\hat{S}}(t)=1/2 \\left(\\tau'(t)\\Psi_2(t)+\\Psi_2(t)\\tau(t) \\right),\\hat{S}(T)=0, rank(\\hat{S}(t))=n_r", "01fb78309dc15b8c8b7bf1bc935d2ee1": "\\begin{smallmatrix}M_v\\ =\\ m + 5 (\\log_{10}{\\pi} + 1)\\ =\\ 0.03 + 5 (\\log_{10}{0.12893} + 1)\\ =\\ 0.58.\\end{smallmatrix}", "01fb9a99551dc0d48536ac23ef87c14e": "\\sum_{j = 1}^n x_{ij}\\leq W_i \\text{ for }i = 1, \\ldots, m, \\, ", "01fc58c8b0da0e07d6945f090fb567a1": "P(d)=\\log_{b}(d+1)-\\log_{b}(d)=\\log_{b} \\left(1+\\frac{1}{d}\\right).", "01fcc590495900b89daf89ded70ece09": "\\frac{d}{dt}(x^2+y^2)=\\frac{d}{dt}(h^2)", "01fced4faaa49a4d66f16eb26a0f1e8c": "\\langle f,g \\rangle = \\int_0^\\infty f(x) g(x) e^{-x}\\,dx.", "01fd1ad8daecc094e7dadd6a86273241": "\\tbinom42", "01fd4990d79a022c9f0f6ddb6c474e72": "\\geq _i", "01fdc5c5a4963039312de9a5909dae41": "\\mathbb{R}^d ", "01fde0360ee4e92ea642bfb8db1c042a": " t = t_4 = 2 ", "01fde5258ca4a48d85b73df2431b1c83": "L= \\frac{\\Pr(1)}{\\Pr(-1)} = 1", "01fdf8295daff5a8c956e998c84a1ab0": "e_{(1)}=\\frac{1}{\\sqrt{4+2(x^3)^2}}\\left[ \\left(x^3-\\sqrt{2+(x^3)^2}\\right)\\partial_0+\\left(1+(x^3)^2-x^3\\sqrt{2+(x^3)^2}\\right)\\partial_1+\\partial_2\\right]", "01fe027d59aa17835a0670a9d11d416a": " M_1 = f, N_1 = q\\ q ", "01fe37d9e5cac4cfc89965f899710fa9": " |\\boldsymbol{\\Omega} | = \\frac {d \\theta }{dt} = \\omega (t), ", "01fe48e9996766b42771f70a1bddd9df": "x_1=X_1/Z_1", "01fe558ce89cef29447b50d1c9a2454d": "\\scriptstyle\\tau_s\\,\\sim\\,10^{-6}", "01fe9cac15c05ddb569271027aa28cdf": "C_{3}", "01feeca3ca3b39eaf174f3e80a0bfb08": "O_i(v)", "01ff9831f25527e34621442ec94c296f": "E = hf.", "01ffcf4a001f4377b9230f06043102af": "\\left({\\mathit{He}}_n^{[\\alpha]}\\circ {\\mathit{He}}^{[\\beta]}\\right)(x)=\\sum_{k=0}^n h^{[\\alpha]}_{n,k}\\,{\\mathit{He}}_k^{[\\beta]}(x)\\,\\!", "020018fbc60643a41b9e6556782676f7": "H=\\begin{pmatrix}0 & -i\\\\ i & 0\\end{pmatrix}", "0200643b433a73480343668a47e713b3": "2\\pi i \\xi", "0200653e29381832b95d44a03206abe1": "\\Omega(\\alpha^{-i_k}).", "02008f14e8257624a6629c3fcf01da8f": " y=\\frac{\\int xe^{-x}}{e^{-x}}", "0200bc9485f667875f6505fff4142a32": " \\alpha=\\left(\\frac{D}{R}\\right)\\left (\\frac{\\partial f}{\\partial y}\\right)", "0200cf69d44dc36712c52a3e3981910a": " \\mathbb{R}^m ", "0200dac127ea6040113c5129053902bb": "\\alpha = 2: \\quad \\operatorname{E}\\left [- \\frac{1}{N}\\frac{\\part^2\\ln \\mathcal{L} (\\alpha, \\beta, a, c|Y)}{\\partial \\alpha \\partial a}\\right ] = {\\mathcal{I}}_{\\alpha, a} ", "02013e1085d9c40ceb24d4dcfe30ea95": "[P_\\mu, P_\\nu] = 0\\,", "020158f273ee5b33f137179c93aaeb98": " \\frac{ i \\Omega }{ 2\\pi} ", "02017cb282b7b8578298acc062ceb4e3": "\\Delta \\omega_2\\ =\\ -\\cos i\\ \\Delta\\Omega \\ =\\ 2\\pi\\ \\frac {J_2}{\\mu\\ p^2}\\ \\frac{3}{2}\\ \\cos^2 i\\,", "0201cd4a7d2672f6df21747ba08cc2db": " \\alpha = 1,2\\,,\\dot{\\alpha} = \\dot{1},\\dot{2}", "0201e01c5002bfe328e7411a47d24dfa": "b_{MP}", "0201e8827a113f4f24e40b69706103df": "\n \\dot{\\mathbf{f}}(\\mathbf{x},t) = \n \\frac{\\partial \\mathbf{f}(\\mathbf{x},t)}{\\partial t} + [\\boldsymbol{\\nabla} \\mathbf{f}(\\mathbf{x},t)]\\cdot\\mathbf{v}(\\mathbf{x},t) ~.\n", "02025a1490050e9d1a58211869ac18ad": "0 = \\tau_{0} < \\tau_{1} < \\cdots < \\tau_{N} = T \\mbox{ and } \\Delta t = T/N;", "020289bdba9a5fc746fab9a3dc637da0": "-\\frac{1}{2}\\left[N(x + \\Delta x, t) - N(x, t)\\right]", "02034ec46591073018d6dbdcf4b653c3": "[\\Sigma Z,X]", "0203ad3d019cbfbc650562b9c791af13": "S(\\widehat{g})=\\int_P R(\\widehat{g}) \\;\\mbox{vol}(\\widehat{g})\\,", "0203c4d906c07569b9177cf884cf4601": " (\\mathbf{y}')^{T} \\, \\mathbf{E} \\, \\mathbf{y} = 0", "02049ecc75727af40b1a127c3547ecad": "A= \\begin{bmatrix} 1 & 2 & 0\\\\0 & 2 & 0\\\\ 0 & 0 & 3\\end{bmatrix},", "0204a441c70e3b74dda69b9dfbe5531c": "\\kappa' = \\frac{h\\nu}{4\\pi}~(n_1 B_{12}-n_2 B_{21}) \\,", "0204bcb90858077f463000cb8d1caa7f": "1 \\times 10^{-9}", "0204fcae90c7db37cec6e71af85f4ae2": " {1\\over D_0 ... D_n } = n! \\int_{\\mathrm{simplex}} {1\\over (v_0 D_0 +v_1 D_1 ... + v_n D_n)^{n+1}} dv_1 dv_2 ... dv_n ", "0205592edc62f25eb27d8c8e385d75c1": " \\tau = 50 + 0.6\\sigma _n", "0205d7a21e1ba59606ed6215d1ba84ca": "\\Sigma X = (X\\times I)/(X\\times\\{0\\}\\cup X\\times\\{1\\}\\cup \\{x_0\\}\\times I)", "02061f46096bfadaf285ce34044c0bb6": "y \\succeq z ~\\forall z \\in B'", "02063a9756469712d13d8db5ef2b90af": "\\begin{matrix} \\frac{8}{5} \\end{matrix}", "02066b25f031c16743e7183b4f47aa32": "x^2y''+xy'+\\left (x^2-\\nu^2 \\right )y=0", "0206d8fd533aeb1efbae23598b7752c5": "\n u(x) - u_\\epsilon(x) = O(\\epsilon^2), \\quad 0 < x < 1\n", "0206f3c7893217c7e70b760392a3580a": "\\Delta\\mu = \\Delta T - e \\Delta \\phi = 0", "02076d8f964e19ea76719f7b7fa4c54b": "\\| \\cdot \\|_{L^{p} (\\Omega)}", "02079b7cc8d61d1266ecc3968c9e0b67": "h(x):=\\lim_{\\delta\\to 0} \\frac{1}{\\delta}\\Pr[x < X \\leq x + \\delta | X > x]", "0207d4c83228fd7956a87bc94fb66bc2": "B(\\rho,\\tilde{p})", "02081576a3bc4a07bd86dcbeff6dc169": "p_A = p_B", "0208ceecd9a3efb97ebc79813aa56e3f": " \\mbox{CNOT} = \\begin{bmatrix}1&0&0&0\\\\0&1&0&0\\\\0&0&0&1\\\\0&0&1&0\\end{bmatrix} ", "0209231423580dbddef190641b0dbb33": "\n\\begin{align}\n(Tf)(x) = {e^D - 1 \\over D}f(x) & {} = \\sum_{n=0}^\\infty {D^n \\over (n+1)!}f(x) \\\\\n& {} = f(x) + {f'(x) \\over 2} + {f''(x) \\over 6} + {f'''(x) \\over 24} + \\cdots ~.\n\\end{align}\n", "02096cceea00a25d136b7df9be53e74b": "E-E_{eq} = a - b \\log(i)", "0209a79418e85433101d56bf370871e8": "P(s^2+st)\\cdot P(t^2)=P(t^2+st)\\cdot P(s^2)", "0209c5dba895a295b64d5cd10e412979": "^b", "020a13ff8c9833908347dc24fcb38981": "\\sigma(K)", "020a3fee1e9ad37dd2d9f5e874cce0dd": "r=\\frac{4\\pi\\hbar^{2}n^{2}\\varepsilon \\varepsilon _{r}}{q^{2}m^{*}} \\; \\; (4)", "020a65bc08570b1375de0229ebd438c9": "\n{\\partial \\mathbf{x} \\over \\partial q^i}{\\partial q^i \\over \\partial s} = \\sum_k \\left(\\sum_{i=1}^3 h_{ki}~{\\partial q^i \\over \\partial s}\\right) \\mathbf{e}_k ~;~~ \n{\\partial \\mathbf{x} \\over \\partial q^j}{\\partial q^j \\over \\partial t} = \\sum_m \\left(\\sum_{j=1}^3 h_{mj}~{\\partial q^j \\over \\partial t}\\right) \\mathbf{e}_{m}\n", "020ab50253f4b02b502d59ed210fdfa5": "\\sigma \\left(e^{\\frac{-\\alpha+\\sqrt{\\alpha^2+\\beta\\log{16}}}{2\\beta}}\\right)", "020ab726931c19275865811cf4641d23": "AM^{-1/2}", "020b818d3824f4c951d41124d1faf01f": "F_c\\,", "020bcccbcc330eba43647a35337c3b4b": " A_{kl}[\\nabla]=\\frac{1}{\\rho} \\, \\partial_i \\, C_{iklj} \\, \\partial_j\\,\\!", "020c2838568f25652b3a81cff1c9af84": "\n\\begin{align}\nH_0&=1+\\frac{n^2}{4}+\\frac{n^4}{64}+\\cdots\\qquad\\qquad\\qquad &\nH_6&=\\frac{35}{48}n^3+\\cdots\\\\[8pt]\nH_2&=\\frac{3}{2}\\left(n-\\frac{n^3}{8}+\\cdots\\right) &\nH_8&=\\frac{315}{512}n^4+\\cdots\\\\[8pt]\nH_4&=\\frac{15}{16}\\left(n^2-\\frac{n^4}{4}+\\cdots\\right)\n\\end{align}\n", "020ce2605d01f04976dde1bf02898e01": "\\eta_h = \\frac{\\pi}{2\\sqrt{3}} \\approx 0.9069.", "020d6bb14fd92378223068b95a273811": " P_c=\\gamma\\left (\\frac{1}{R_1}+\\frac{1}{R_2}\\right)\\!", "020d7e9916dbc07f176b19924f410686": "1+r=(1+i)/(1+\\pi) \\approx (1+i)(1-\\pi) \\approx 1+i-\\pi", "020d7ea57aafa51aeb9616ae25a4deef": "z_i\\neq z_j\\quad", "020e2809fcd895b31e4d5f9942b900d1": "\\sigma_{r t} = \\frac{1}{\\mu_0} B_r B_t - \\frac{1}{2 \\mu_0} B^2 \\delta_{r t} \\,.", "020e37332fc37169cb038657a020ff1b": " a^2 = \\frac {T}{\\rho_{solution} g}", "020e7ba90d2fd623965329b56e0f6a6a": "t_o = \\frac{t}{\\gamma},", "020e8f4ceae6ff5b053183b573b5f9fa": "D = b^2-4ac", "020ea27d91a8ee4ec996c7d823adfbc3": "C:a+bx \\rightarrow a + \\omega bx\\,", "020eb6226721a0d3ded3968d8ad8165a": " \\mbox{tr}^2\\,\\mathfrak{H}= \\mbox{tr}^2\\,\\mathfrak{H}'.", "020f083998200b2c752bf46fe39dce27": "\\frac{\\Gamma \\vdash \\alpha \\rightarrow \\beta \\qquad \\Gamma \\vdash \\alpha}{\\Gamma \\vdash \\beta}\\quad\\text{Modus Ponens}", "020f335054558fdff4f387056e345abb": "\n D\\,\\left(\\frac{\\partial^4 w}{\\partial x^4} + 2\\frac{\\partial^4 w}{\\partial x^2\\partial y^2} + \\frac{\\partial^4 w}{\\partial y^4}\\right) = -q(x, y, t) - 2\\rho h\\, \\frac{\\partial^2 w}{\\partial t^2} \\,.\n ", "020f5b0b54d45f3933659d20b4b8901d": "n_1=0\\,\\!", "020fa72193e9f80f8e86a914a89ede7a": " dQ_c = T_cdS_c ", "020fce0965fdbd8eacd65d8c5e7f735f": "\\text{Qv}", "021064cb27a88a2fae850a9fe57034df": "J=1", "0210b7fe254ee8a721aaf0c418a6199b": " \\langle \\mathrm{d} f,X_{g} \\rangle = {X_{g}}(f) = \\{ f,g \\} = - {X_{f}}(g) = - \\langle \\mathrm{d} g,X_{f} \\rangle ", "0210db18cab5ffed55e6f049b2fa4f3d": "x^2-2x-2", "0211181d10cf569cc3d19a52820c511c": "I_2\\to I_1", "02115da4b2995df7446571d92f05311d": "[D] \\cdot C \\geq 0", "0211867b79048bdcee8fc6a90f152e6a": "^{*}\\!H", "0211b2e435b909cc70950f4fcc598b49": "\\displaystyle S_{PQR} = S_{ABC} - S_{ARC} - S_{BPA} - S_{CQB} ", "0211c922a052804e564f1efc1e2421c9": "S(\\boldsymbol{\\beta}) = \\sum_{i=1}^{m}\\bigl| y_i - \\sum_{j=1}^{n} X_{ij}\\beta_j\\bigr|^2 = \\bigl\\|\\mathbf y - \\mathbf X \\boldsymbol \\beta \\bigr\\|^2.", "0211e14fbe450ba44f2fb225d7d00b04": "\\sqrt{\\lambda_1}", "0211eb04eab4ff94e9660c0fd989a0a2": " x \\in S | x\\leq a ", "02121de6d4ac8dfb9e1f7f93345e0368": "\\frac{V_\\mathrm{out}}{V_\\mathrm{in}}=\\frac{10}{1}=10\\ \\mathrm{V/V}.", "021243b9d14264da9db22721350ba73b": " \\mathbf{x}_\\text{p}(t) ", "02126e65a1ef1c21549d2c40cea26d1d": " \\frac{100}{2+2} = 25 ", "02128bd13bcbf456f93f4482b09b34ea": "\\mathbf{Z}(p^\\infty)=\\{\\exp(2\\pi i m/p^n) \\mid m\\in \\mathbf{Z}^+,\\,n\\in \\mathbf{Z}^+\\}.\\;", "02129029f8a82c8440d0197aa5c9f513": "(3+2\\sqrt{2})/6 \\approx 0.971.", "02129bb861061d1a052c592e2dc6b383": "X", "0212a55821995d1dc111723616ae41d0": "V_\\mathrm{th}", "0212c3b4e7f92cca974995c579ceb1c3": " \\lim_{p \\to \\infty} \\; cr(K_p) \\; 64/p^4 = 1. ", "02133fd6a4bdbc80080ccffe4488b883": "\\chi_{mn} = \\sum_{i} a_{mi} a_{ni}^*", "0213c767132c12afbd3114964a9b195b": "e^{i(2h-1)\\theta}", "02145e0b2385830a1d7937a47f81bc6f": "\\mathrm{Ad}_{\\exp X} = \\exp(\\mathrm{ad}_X).\\,", "02146cf17db8911d232615c5935aaea8": "= 1 + 7 + 8 + 2", "0214c7818340dcf25159250a5275c7c5": "y=\\pm \\sqrt{1-x^2}.\\,", "0214e802e89a2a43a1c326b8677eecb5": "\\Delta \\boldsymbol \\beta\\,", "0214fc667b35c1eb4d80bed3631873ee": "A_n,", "02150b6afdf71a00d7a4426c11a03137": "\\lim_{t \\rightarrow \\infty} \\phi(t,i)", "021531540e1f9a1767dc972aba2ce46d": "\\mathrm{Pe}_L = \\frac{L U}{\\alpha} = \\mathrm{Re}_L \\, \\mathrm{Pr}.", "021597b33041ab03bb7d57420dbd92bb": "Q^-(5,q)", "0215f6acde9bc1c91b8536d77d2359b2": "V=w^3 \\left (h/ \\left (\\pi w \\right ) -0.142 \\left (1-10^ \\left (-h/w \\right ) \\right ) \\right ),", "0216b10bb914f682c31527a6dfa29c5a": "\\mathcal{D}\\phi e^{i\\mathcal{S}[\\phi]}", "0216c138751070dfbabb96ef5d1eb18e": "\\tfrac{1}{k}", "0216d9f13fca6900a5faa75a2641597c": "C_2 = 'la'", "02172bc6af05615d441828bb86303fe2": "\\displaystyle{K(x,y)=\\int a (t, {x+y\\over 2})e^{i(x-y)t}\\, dt.}", "0217368dd47e4d3ae870d33145d5fbea": "\\frac{u_{i}^{n + 1} - u_{i}^{n}}{\\Delta t} = \\frac{\\alpha}{\\Delta x^2} \\left(u_{i + 1}^{n} - 2 u_{i}^{n} + u_{i - 1}^{n}\n\\right)", "0217664181f68eccfdfa54bd94f38295": "\\hat{\\mathrm{Td}}^R(E)", "021776ba3e03f3a12b76cfb6038d460f": "\\,\\!d(x,x) = 0", "0217e727821f8b6d0f4ba70aaa0b9289": "L_\\Phi", "0217f1a1daa60d4eec6e1b17556a7691": "\\gamma p", "0218284b131eb117257a718bf33f02f1": "\\boldsymbol\\Omega \\times \\mathbf{u}_{\\theta} =-\\omega \\mathbf{u}_R \\ ,", "02186e91c74c1347bf9dea47ea4d51b3": "e^{\\frac{{\\delta}}{2}F}=\\prod_{odd \\ \\ l}e^{\\frac{{\\delta}}{2}F^{[l]}}", "021879fd8c747c0eec644ff0731fdcd6": " \\frac{VK}{Y} ", "02187fbad579b9a45c66d0ddeef4dcd4": "\\ \\displaystyle \\min \\ ", "0218892c31c600419c38902a989c1080": "\\{\\varphi_m ; m=1,2, \\cdots ,p \\}", "0218956e3e9799b38ec2e73ccb0c29c3": "[0, -\\infty)", "0218ae0a0d2cfb36098a911162226efd": " v(\\sigma) ", "0218aecfd99bbe3201441c46846f8e1b": "L^\\infty (U)", "0218d7b007a1854a503622ac667d4ead": " H=\\frac{\\phi^2}{2L} + \\frac{1}{2} L \\omega ^2 Q^2", "0218ed240c075274c8bfc76ea63844dd": "a(z)", "0218f809672e55a317f05c582cb8c1f5": "S_3 \\to S_2", "021924c0f6a483b67a498c027ad1a005": "150^\\circ", "0219415e6b09c2b7b94b95529d8d248e": "s = 2^{1/12}", "02199f601cbf0f16a3bd2030f8f6732b": "1-e^{-4\\lambda}. \\, ", "0219b34b096b2e436803a6f11c17626e": "a(v) = b(v) = d(v) = 1, \\text{ and } e(v) = 0 \\,.", "0219e915c2ebd3302c323a485855264e": "\\operatorname{Li}_n(z)=\\sum_{k=1}^{\\infty} \\frac{z^k}{k^n}\\,\\!", "021a383ade6882a9507adb8eef538985": "Eq.6", "021a5393fce02c4f57c3adce8e5a8ffe": "2^{w-1} - 1 + {n \\over w}", "021a6af6071cb77c364718edc0ca959b": "A\\oplus B", "021a90bff98f6e9cd1ef938f9968fffc": "\\left \\langle v \\right \\rangle = \\sqrt{\\frac{8 k_b T}{\\pi m}}", "021ad144f1e0aae4df5d8e05c210feed": "\\mathbb P^n_{\\mathbf k}", "021ae1d076393de740cd55333757daa7": "\\pi : \\tilde{\\mathbf{C}^n} \\to \\mathbf{C}^n.", "021b2cae67f9ff7e602432fe2c468f12": "\\begin{matrix}\\mathrm{Cabtaxi}(4)&=&2741256&=&108^3 + 114^3 \\\\&&&=&140^3 - 14^3 \\\\&&&=&168^3 - 126^3 \\\\&&&=&207^3 - 183^3\\end{matrix}", "021b7f98fa40c4921966ab2f3a10c847": " x^5+320x^2-1000x+4288", "021bdc824da4b0d0db8a7001d988daef": "|W_\\alpha(x)-W_\\alpha(y)|\\le C|x-y|^\\alpha", "021c165cdf6f1229bf98835b81614e1b": "\\frac{a}{p}=0.\\overline{a_1a_2a_3\\dots a_na_{n+1}\\dots a_{2n}}", "021c34847126ffcff029c3109c6a2c94": "\\frac{1-e^{-k}}{1+e^{-k}}\\!", "021c5216fdc8ef5520e350ba1b4d04ab": "w_m(x) = w_m(\\pm r^j A) = w_m(A)", "021c663214ddb1c48b2f4caa55d303f9": "\\oint_\\gamma (u\\,dx-v\\,dy) = \\iint_D \\left( -\\frac{\\partial v}{\\partial x} -\\frac{\\partial u}{\\partial y} \\right )\\,dx\\,dy ", "021c760eb4da2c1574bae8d8224eb616": "\\bold{j}_{{\\rm m}, \\, i} = \\rho \\left ( \\mathbf{u}_i - \\langle \\mathbf{u} \\rangle \\right ) ", "021c7d1154a7ba92517fd48bf5cdfb5d": " -\\smile \\ \\mathrm{or} \\ \\smile\\smile\\smile \\ \\mathrm{or} \\ -- \\ \\mathrm{or} \\ \\smile\\smile- \\ \\ ", "021cd5b20499445d7adc8e55e46dcd37": "(x^n - \\lambda_1) \\cdots (x^n - \\lambda_k)", "021d5907c132d4a5a77d11607b940299": "\\sqrt{\\log t}", "021d5bb84628145baa4d65616d42d6d6": "o=f(d)", "021d8d9fca3bd619f7dd60d32c8fbfa3": "F_{out}", "021d90aaf328fde1c5143da6819944a3": " \\varepsilon_t = 0.5 \\left( ( + {\\Delta p}_D + \\overline {\\Delta q} ) - {\\Delta x}_{t-1} \\right) \\,,", "021dcc12da0e15851dc65ba76ab03998": "-1 < \\lambda \\le -0.75", "021dcceba82bdc9cb593fcc99c34d32b": "\\displaystyle \\Re(u)(u_{rr}+u_r/r+u_{zz}) = (u_r)^2+(u_z)^2", "021e2a185b50c03a079d3e0c3e4cb494": "C_\\nu(x) = \\mbox{Re} \\chi_\\nu (e^{ix})", "021e2af83661cfa2eeeff8fc5786363c": "\\mathbf X = (x_{i, j})", "021e2b3f189905b173b82d764385f3d0": "\\bar{\\omega}^{\\frac{M_p+1}{4}}", "021e31c56481b62335929e55ee5cef17": "{\\color{Blue}~5.1}", "021e366c5269ccb6488fd92a2cb8d8d2": " S_r = \\frac{dQ/dT}{Q} .", "021e73b795f4ac022970b23ccbba839b": "H=\\frac{N}{N-1}(1- \\sum_{i}x_i^2)", "021eef71ae47ec077aa3a8094ad10b03": "x \\in \\{-1, 0, 1\\}", "021f05368040315edf8116f146d414ba": " w = \\frac{I_SR}{nV_T} \\left(\\frac {I}{I_S}+1\\right ) ", "021f0d1a78e8ff3d2af1c85f679c945e": "e^{e^{e^{e^{7.705}}}}<10^{10^{10^{963}}}.", "021f10c51ad1c40dd6e0d68ed8e1c041": "\\sum_{n=-\\infty}^{\\infty} x[n]\\cdot \\delta(t-n T) = \\mathcal{F}^{-1}\\left \\{X_{1/T}(f)\\right\\} \\ \\stackrel{\\mathrm{def}}{=} \\int_{-\\infty}^\\infty X_{1/T}(f)\\cdot e^{i 2 \\pi f t} df.", "021f33e28fcf3162445b4cd6c4e6db06": "L_{\\triangle}.", "021f4c71cdce422705204798c756df5b": "(x, y) \\mapsto x", "021f565e2917eb04dc9820f81ac24fe1": "\\varphi:X\\to X", "021f637f4fd183a6797d40bfbc226244": "G^o=\\sum_{i \\in S}{p_i\\log_2{(er_i)}}+(1-\\sum_{i \\in S}{p_i})\\log_2{(R(S^o))} ,", "021fcb0e87fc1b892001c1010be7b9f4": "P=P(X).", "022022f289db140169cd9514f74ee648": "[a, b]", "0220807ccee2f8fefd14155f7ac80aaa": "X_k = \\sum_{n=0}^{N-1} x[n]\\cdot e^{-i 2\\pi \\frac{kn}{N}}.", "022087273905a69a92023e3722643f9a": "f(\\mathbf{r}) = \\frac{1}{\\left(2\\pi\\right)^{3}} \\int F(\\mathbf{q}) e^{\\mathrm{i}\\mathbf{q}\\cdot\\mathbf{r}} \\mathrm{d}\\mathbf{q}", "022132bb3ebcec11d7f81d3f504e9ee6": "y_P-y_0=R_{12} (X-X_0)+ R_{22}(Y-Y_0) + R_{32} (Z-Z_0)", "02213f99cdbec26b01922ac7c2c6a735": "\\mathrm{Ass}_R(M')\\subseteq\\mathrm{Ass}_R(M)\\,", "022174fdae6a4922a7b170c1ee094787": "n_{\\rm e}T\\tau_{\\rm E}", "02219a66af946058fd7efd21b3ee5036": " \\oint_{\\partial \\Sigma(t)}\\mathrm{d} \\boldsymbol{\\ell} \\cdot \\mathbf{E}(\\mathbf{r},\\ t) = - \\ \\iint_{\\Sigma(t)} \\mathrm{d} \\mathbf {A} \\cdot {{\\mathrm{d} \\,\\mathbf {B}(\\mathbf{r},\\ t)} \\over \\mathrm{d}t } ", "02219e95bb4d29afb2dbd06a72de57d7": "y^2 = x^3 + x^2", "0221d4398bfb14e28b879e50c313d424": "O( |E| |V|^{1 / 2} )", "02220173c31977d9839303516a09da5b": "\n{dL\\over dt} = i[H,L] = 0\n\\, ,", "022217a91d9b643de752294096d7f6aa": "p_4, p_1", "0222491b800049563d888f2664f4a8a6": "\n\\sigma(t) = \\frac { 1 }{ b }* log {\\frac{10^{\\alpha}(t-t_n)+1}{10^{\\alpha}(t-t_n)-1}} ", "02224ce925b278fca46db66a1da98c3e": "\n\\Sigma(A\\mathbf{x}) = A\\, \\Sigma(\\mathbf{x})\\, A^\\mathrm{T}", "022307e1bd54450e4783926cdb153408": "V_v = V_r", "02230e656b591d8f31a1b7eb03dfdaab": "\\{a_1 , a_2 , a_3 , a_4 \\}", "02234033881254ba9f33e1b63e381585": "\\text{Holant}(G, f_u T^{\\otimes (\\deg u)}, (T^{-1})^{\\otimes (\\deg v)} f_v).", "022399746d452f7fe708c5414a3ab4dd": "Ac^2\\alpha\\left(-\\rho_G-\\rho_L\\right)=Ag\\left(\\rho_G-\\rho_L\\right)-\\sigma\\alpha^2A.\\,", "0223f2bdbda18a7154bf1f35126ea943": "\\bar{\\Gamma}^{\\beta}_{\\alpha \\gamma} \\, = \\, \n\\frac{\\partial \\bar{x}^{\\beta}}{\\partial x^{\\epsilon}} \\,\n\\frac{\\partial x^{\\delta}}{\\partial \\bar{x}^{\\alpha}} \\,\n\\frac{\\partial x^{\\zeta}}{\\partial \\bar{x}^{\\gamma}} \\,\n\\Gamma^{\\epsilon}_{\\delta \\zeta} \\,\n+ \n\\frac{\\partial \\bar{x}^{\\beta}}{\\partial x^{\\eta}}\\, \n\\frac{\\partial^2 x^{\\eta}}{\\partial \\bar{x}^{\\alpha} \\partial \\bar{x}^{\\gamma}} \\,", "0223fb7c8a6750e68f52034474fcc627": " c_{t+1} = (1-R^{-1}) \\left[A_{t+1} + \\sum_{j=0}^{\\infty}\\left( \\frac{1}{R}\\right) ^{j} E_{t+1} y_{t+j+1} \\right] \n", "02246878093cc4bb4582527127390aba": "\\operatorname{dist}", "02246d3ddf4a376189129511f7aed444": "x = \\left(\\lambda - \\lambda_0\\right) \\cos \\varphi", "0224bf3a2802504318677efcf183c5d8": "(192, 20, 64)", "02251a6d64eac16e4975615fa1729053": "\nR_{\\mathrm{g}}^{2} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{1}{2N^{2}} \\sum_{i,j} \n\\left( \\mathbf{r}_{i} - \\mathbf{r}_{j} \\right)^{2}.\n", "02252b9c9ecfb5255c48bc40e9468ec5": "q^1 =~q", "0225837728c76ed9a4151ba7478ef822": "sw_G= a_G -r_G+1 \\, ", "02258fb9a54bb12a8fd0d91ae705c352": "T_{\\alpha=0}", "0225cf1ee782416bebc60b86c20f7391": "A \\cap B = \\emptyset", "022601f8e00084d1493d9936c5ec4e53": "\\hat{\\lambda}_x", "02260b5ffe7e69417fdffae16ddfdf4c": "\\sigma: F \\subset \\mathbf C", "02267005721eca4c8753e098ebdbea87": "(1)-(3)", "02267731f45f5958fda3e43298fa70f7": "u=(u_n)\\in \\mathbb{R}^{\\mathbb{N}}", "02268fc40c7bbedc4d1267c6e227803f": " v_{0x} = v_0\\cos\\theta", "0226a80fb3896b26afb862b440b47b44": "H(X|Y)\\leq H(P(e))+P(e)\\log(|\\mathcal{X}|-1),", "02273fbbef6b8ed7f587354c0c979f7b": "g^{(2)}(\\tau) \\leq g^{(2)}(0)", "022740cb79459ef196f8b90f51e7c189": "\\bigcap A", "022767b288e7e3aa5058ce3415b9782c": "|Q_0| = |Q_L| = \\tfrac{P}{2}", "02277c0892b59bb77a84b6acc8da10da": " dA= r^{-2}\\, dx\\,dr", "02279e280508ce5ad88446b2647ccf9b": "A = \\begin{bmatrix} 3 & 1\\\\1 & 3 \\end{bmatrix},", "0227d59d472519da01fc1193ec83f83d": " \\frac{3}{2} (n - s_3 (n)) - 2 e_3 (n) - e_3 (n-1) ", "02283262a7b9c92bc0bfe063321d535d": "p(q\\in Q)", "0228336631a10f396ac503f882dcd26a": " P_{1}{v_{1}^{\\,n}} = P_{2}v_{2}^{\\,n}= ... = C", "0228599d96fca4db83d812af38236b09": "\\ \\det(\\mathbf A) = a_{i1}C_{i1} + a_{i2}C_{i2} + a_{i3}C_{i3} + ... + a_{in}C_{in} = \\sum_{j=1}^{n} a_{ij} C_{ij} ", "02285ca77f2b48eb0afa7341dfaf9276": " \\mathbf{y}_{2} = \\mathbf{y}'_{2} ", "02288438fd1d4c7bffb4fb864c115a70": "\\Delta S^\\circ", "0228edc841b87a34088290c1a53b4356": "\\pi=\\frac{72}{Z} \\!", "022938fe967f9e5cda854d269d72d2dc": "m = \\frac{\\sqrt{1-4c}}{2}", "02294e55210a4b616cafd39611b8fc96": "\\mathbf{A}'=\\boldsymbol{\\Lambda}\\mathbf{A}\\,\\!", "02296e14035c2116e1904e948325e16c": "\n \\begin{align}\n \\boldsymbol{\\nabla}\\cdot\\boldsymbol{S} & = \\left[\\cfrac{\\partial S_{ij}}{\\partial q^k} - \\Gamma^l_{ki}~S_{lj} - \\Gamma^l_{kj}~S_{il}\\right]~g^{ik}~\\mathbf{b}^j \\\\[8pt]\n & = \\left[\\cfrac{\\partial S^{ij}}{\\partial q^i} + \\Gamma^i_{il}~S^{lj} + \\Gamma^j_{il}~S^{il}\\right]~\\mathbf{b}_j \\\\[8pt]\n & = \\left[\\cfrac{\\partial S^i_{~j}}{\\partial q^i} + \\Gamma^i_{il}~S^l_{~j} - \\Gamma^l_{ij}~S^i_{~l}\\right]~\\mathbf{b}^j \\\\[8pt]\n & = \\left[\\cfrac{\\partial S_i^{~j}}{\\partial q^k} - \\Gamma^l_{ik}~S_l^{~j} + \\Gamma^j_{kl}~S_i^{~l}\\right]~g^{ik}~\\mathbf{b}_j\n \\end{align}\n ", "0229715bcd0b8ee6e85eb1137020a050": " (\\beta,\\gamma) ", "0229964a1c9475bb8e607e5b9c838930": " \\lor, \\land", "0229f6d302ed458cdbc9d3bfd86ab90c": "\\varphi,\\psi\\ ", "022a32b622291f9215bb9f3e62cbe044": "k>2", "022a6d034a5abd59a24248cbb3b0941b": "\\ A = \\frac{\\partial v}{\\partial x} + \\frac{\\partial u}{\\partial y}", "022a74c2b1d6e9d7052170bc67377d01": "~A \\cap B \\cap C", "022a90134e784ec490f2f2b6d7282f9c": " \n \\nabla^2 \\varphi - {1 \\over c^2} {\\partial^2 \\varphi \\over \\partial t^2} = - {4 \\pi \\rho } ", "022ab9646a0ab3afe4b5defbe5ccfbb8": "V = an", "022b198209b9c2837ed81d53cd974382": "1 + z=\\frac{1}{\\sqrt{1-\\frac{v^2}{c^2}}}", "022b40c01061319faca833a52952fb3a": "J^{\\prime\\prime}{\\leftarrow}J^{\\prime}", "022b84826ecca1d4efb4a7a396d11302": "\\mathbf{V}_i=\\mathbf{V}+\\frac{d\\mathcal{R}}{dt}\\mathbf{r}_{io}", "022b951b5041e6dad209818e3e896f84": " H_p(X,X-x, G)", "022bd032b7f0ee32c7730b7644c6240e": "|\\langle x, y\\rangle| \\le \\|x\\|\\,\\|y\\|", "022bdeb8bfa3ba5a77935025118e9e2c": "\\scriptstyle{1/2}", "022c1333b68909412b5b0041396caaee": "2^{l + 1} - 1", "022c20fd59376ce997de8331ffaedbd3": "A_t = \\{x\\in X\\mid f(x)\\ge t\\}", "022c32018a8e85ae989512fd7ecec25e": "V_n(r)", "022c3b80bef2c0f17f57ed150c1f4652": " 1 + \\frac14 + \\frac19 + \\frac1{16} + \\frac1{25} + \\cdots = \\sum_{n=1}^\\infty \\frac{1}{n^2} ", "022cbd378cab471ae5be73488db3b604": "t=\\tfrac{x-x_1}{x_2-x_1}", "022cde90c52840683f79ce7a7e627c22": "d = 2\\pi / |\\mathbf{g}_{h k \\ell}|", "022d283fc823640c77ed0a4b510ed33b": "-\\frac{\\partial}{\\partial t}p(x,t)=\\mu(x,t)\\frac{\\partial}{\\partial x}p(x,t) + \\frac{1}{2}\\sigma^2(x,t)\\frac{\\partial^2}{\\partial x^{2}}p(x,t)", "022d434b912cb7fa1b0b4644e8b4e2ae": "Y\\ \\sim\\ \\mathrm{Herm}(a_1,a_2)\\,", "022d8aa2bcbc12f4324820915872f900": "\\mathbf{e}_2 \\times \\mathbf{e}_3 = \\mathbf{e}_5, \\quad \\mathbf{e}_3 \\times \\mathbf{e}_5 = \\mathbf{e}_2, \\quad \\mathbf{e}_5 \\times \\mathbf{e}_2 = \\mathbf{e}_3,", "022daeb34db6dd6d51b0de65cf250648": "\\max_{d\\in D}\\min_{s\\in S} dist(d,s)", "022dbecbb7fa5d325462bd7a0ce699d5": "\\alpha^*F:=\\{H'\\le H|\\alpha(H)\\in F\\}", "022dcce091d8dc74031d8dbf34662dab": "n = ax^2 + 2bxy + cy^2", "022e3414e6427b3cc27c5a5911fd9588": "D G(x, s)=0", "022ed48ce122fb6d02b20ffd57a86105": "\\sin (2 \\theta) = 2 \\sin \\theta \\cos \\theta\\,", "022f7d80be231d713945ca4d7beed1cf": "\\left(\n\\frac{\\pi}{6}\n\\right)^{\\frac{1}{3}} \\approx 0.806", "022f9ef548f37cc6101d5e59875cc945": "\n \\alpha_{\\rm{THz}}(\\omega) = \\mathrm{Im}\\left[ \\frac{\\sum_{\\nu, \\lambda} S^{\\nu, \\lambda} (\\omega) \\Delta N_{\\nu,\\lambda} - \\left[ S^{\\nu, \\lambda}(-\\omega) \\Delta N_{\\nu,\\lambda}\\right]^{\\star} }{ \\omega (\\hbar \\omega + \\mathrm{i} \\gamma(\\omega))} \\right]\\;.\n", "022fb3dab2be5bff82479c16cc1780ef": " a \\otimes b \\mapsto (-1)^{|a||b|} b \\otimes a ", "022fb3e873d2b56001a689daec1b9e7d": "\\lim_{x\\to c}{|f(x)|} = \\lim_{x\\to c}{|g(x)|} = \\infty,", "02301b578da6ac04d27ae1fefb9a9133": "\nX^{\\{q\\}}=\\lambda^{-1}([m-q,m])\n", "0230363ab1c553703171c76386773875": "\\psi_1=\\psi_1\\big(\\vec{\\sigma},\\vec{\\rho}\\big)=\\Big({\\textstyle\\sum\\limits_{i=1}^n\\sigma_i^2}\\Big)^{-1/2}\\cdot\\max_{1\\le\ni\\le n}\\frac{\\rho_i}{\\sigma_i^2}.", "023068de560204c0cf3f00e2e4568840": "\\begin{matrix}\\mathrm{Cabtaxi}(1)&=&1&=&1^3 \\pm 0^3\\end{matrix}", "0230ba0ac3a6fc775e42d81c10dfbbea": "0+1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+\\cdots", "02311b615f1556b2414572ce7f4561f0": "(F,m)", "023124f437c87fccd318c8d546e8e40a": "\\textit{VERB} \\; \\textit{NOUNPHRASE}", "02312e4fc33d871628eeb7617f6beebe": "\\displaystyle{\\widehat{P_y}(t)=e^{-y|t|},}", "023142ea5abc2a69bc90e8ed3bfb1ecf": "\\tau\\,\\sim R/c_s,", "0231aa9e0f86af7105e43b77bc14b1c5": " (X+Y)_i", "0231c298d893d2f32ee58bc2edde3c2d": "\\mathbf{F}_k", "0231c2e6053495105d4154730d981449": "n_{c1}(\\mathbf{k})", "0231d206e2f7860d2ba70ffa7f9a4391": "\\langle \\psi|\\psi \\rangle = \\sum_i |c_i|^2 = 1.", "0231f3829641a85262e0bdfaf24857ed": "K^{\\ominus} = \\mathrm{\\frac{[A^-] [H_3O^+]}{[HA] [H_2O]}\\times \\frac{\\gamma_{A^-} \\ \\gamma_{H_3O^+}}{\\gamma_{HA} \\ \\gamma_{H_2O}} =\\mathrm{\\frac{[A^-] [H_3O^+]}{[HA] [H_2O]}}\\times\\Gamma}", "02322c86ce10049b1cac9d06a539264f": "\\text{subject to } \\dot{x_t} = f(x_t,u_t)", "02323b856adbbedeca994dac706eece8": "\\tfrac{863}{60480}", "0232502a9b8aa410be0731dfefa96d89": "\\left(\\mu, \\frac{\\alpha - \\frac12}{\\beta}\\right)", "023278c95dccd52f1c1ede88e3a9bbaa": "\\psi = \\sqrt{\\rho} \\; \\exp \\left( \\frac{i \\, S}{\\hbar} \\right) ", "0232941ed731510409a6f815ba885bd8": "P_3=(0,-72,2\\sqrt{3},12)", "0232b079f9617a0dd2bf92e0501a6baf": "v = Z \\alpha c", "0232e0a4d7ff7211cc29b99bfcd79c60": "\n\\begin{align}\n&{\\partial\\rho\\over\\partial t}+\n\\nabla\\cdot(\\rho\\bold u)=0\\\\[1.2ex]\n&{\\partial(\\rho{\\bold u})\\over\\partial t}+\n\\nabla\\cdot(\\bold u\\otimes(\\rho \\bold u))+\\nabla p=\\bold{0}\\\\[1.2ex]\n&{\\partial E\\over\\partial t}+\n\\nabla\\cdot(\\bold u(E+p))=0,\n\\end{align}\n", "0232f27be40b2b647f260050dd308eb8": "D'", "0232f592c77a40287056489966672f9a": "|W|^2", "023330e0f448e77e7f36d5b64003a4af": "S_L \\, \\dot= \\, 1 - \\mbox{Tr}(\\rho^2) \\,", "023332f3c3c330f5e090368eb88239de": "\\phi:S^p\\to M", "02339a5ca7d0e8ebcf600e7a71af43ac": "\\ker(\\partial_n)=Z_n(X) ", "0233a635281a006b5ef593fd13c442bb": "\\tilde{f} : A \\to B", "0233e398d6cef3db1cb3373918134e2d": "x^2 + 2ay = 0 \\,", "0234461cdd7e945e51351ff44168c87c": "\\displaystyle{G_0=K\\cdot \\exp i\\mathfrak{p} =K\\cdot P_0 = P_0\\cdot K}", "0234566f1fbd03b0d70fb63760de4af9": "F_g = m g \\, ", "0234997dc624d9faffaabdc308aaf0bf": "\\left.\\left(\\frac{d}{dt}\\exp(tY)\\cdot v\\right)\\right|_{t=0}=Y\\cdot v.", "0234d553f8a114a57c79e5a07d1b5f30": " ({\\mathcal F}_a f)(t,y) = (2\\pi)^{-n / 2} \\int_{{\\mathbf R}^n}f(x)e^{-a |x-y|^2/2} e^{-ix \\cdot t}\\, dx.", "0234f8b37d074721fc182323d786a3b8": "\\mathcal{D}_{T*}", "02359fe87f7bece7408ee4c2fb05309d": "\\frac{\\partial \\mathbf m}{\\partial t} = - |\\gamma| \\mathbf{m} \\times \\mathbf{H}_\\mathrm{eff} + \\alpha \\mathbf{m}\\times\\frac{\\partial \\mathbf{m}} {\\partial t}", "0236397aa3334d97ef48265bd70cc65c": "\\mathrm{[HA]} = C_a - \\Delta", "0236b26572404fdd74e9b216b80ec598": "\n\\sup_p h_p(x) \\ge 1\n", "0236c4fd43865dc027e02a12932a3d38": " \\mathbf{\\xi} = \\nabla \\times \\mathbf{h} \\,\\!", "0237339d7ab322085ac4d6fe016b9180": "~ G_0=\\frac{ND}{\\sigma_{\\rm ap}+\\sigma_{\\rm ep}}~", "02373c1f5bede1d3a97be96e5bc98fa2": "30~\\mathrm{dB}", "0237559e0dd6b6ad12596f53e2b0b576": "H(X) = -\\sum_{i=1}^n {p(x_i) \\log p(x_i)}.", "02375cdd0732a6d66d1e458fa1b50b80": "\\mathbf{Q}(t)", "0237a5f569f0044a5bbb8f2192c986ad": "c \\gamma^2 - (a - d) \\gamma - b = 0 \\ ,", "0237f18a6d321a7442c3fee447abeb1d": "{1\\over 2} \\sqrt{2}", "0237f6329ee550690931c6833531edfe": "\\ell_j(x) = \\frac{\\ell(x)}{x-x_j} \\frac{1}{\\prod_{i=0,i \\neq j}^k(x_j-x_i)}", "02386c655ee72999c62ae715ab5d7292": "f(\\pi , \\pi) = -1", "023873b1bddb202be30b5afdcd5749df": "T_{AMB} = 70 \\ ^{\\circ}\\mbox{C}", "02389fda3095cddda9021cb2d21e3cd2": "\n |\\mu|(\\partial B) = 0\\,.\n ", "0238bce249da3358e4f5ed91094a93e7": "ODF(\\boldsymbol{g})=\\frac{1}{V} \\frac{dV(\\boldsymbol{g})}{d g}.", "0238c5283423c18589620888e3e89f6f": "x^2 + 6x + 5 = 0,\\,\\!", "02392d528baa8b5145109fb192d3b1d8": "\\frac{\\frac{L_1}{2l}}{\\frac{L_2}{2l}}\\approx 4 {\\left ( \\frac {L_2}{L_1} \\right ) }^2 \\Longrightarrow \\,\\!", "02393ef35f0969894f61ddf410d7f06d": "x,y \\in \\mathbb{R}^\\times_{>0}", "02398fd5e498663131fd5316fe7ee86e": " E\\left[ \\Lambda(n+1) \\right] = \\Lambda(n) + E\\left[ \\left( \\frac{\\mu\\, \\left( v(n)-r(n) \\right) \\mathbf{x}(n)}{\\mathbf{x}^H(n)\\mathbf{x}(n)} \\right)^H \\left( \\frac{\\mu\\, \\left( v(n)-r(n) \\right) \\mathbf{x}(n)}{\\mathbf{x}^H(n)\\mathbf{x}(n)} \\right) \\right] - 2 E\\left[\\frac{\\mu|r(n)|^2}{\\mathbf{x}^H(n)\\mathbf{x}(n)}\\right]", "02399bb49108a9455c6827292a30f6ea": "\\begin{align}\n r &{}= \\sqrt{6^2 + 5^2} = 7.8102 \\\\\n c &{}= 6 / r = 0.7682\\\\\n s &{}= -5 / r = -0.6402\n\\end{align}\n", "023a62bc62f64d7623a58945e76525ed": " O_k = x_1x_5x_9 \\cdots x_{2N-3}+ x_3x_7x_{11} \\cdots x_{2N-1}", "023a6f9688af4f89a59c4ba647f93d89": "\\nabla^2\\psi=0.\\,", "023a91025f1455379b2c7b284e046e79": "\\mathrm{Hom}^{\\bullet}(\\Gamma_c(X;I^{\\bullet}_X),k)= \\cdots \\to \\Gamma_c(X;I^2_X)^{\\vee}\\to \\Gamma_c(X;I^1_X)^{\\vee}\\to \\Gamma_c(X;I^0_X)^{\\vee}\\to 0", "023adfe845a552b23bef1cb0b61328c7": "\\{x\\in F;\\,x\\Vdash p\\}\\in V", "023b183c77a3cdbe50fe1f990a63a6de": "2^T-s", "023b3db14b66f7d0ee89fdb89c64e57d": "(K,\\, \\nu)", "023b800a1806490ff857cf9d69a260df": "\\neg \\!\\,", "023bdb642d2bb73325b663deba16c00e": " Q_N \\equiv \\frac{1}{N} \\sum_{i=1}^N \\frac{f(\\overline{\\mathbf{x}}_i)}{p(\\overline{\\mathbf{x}}_i)}", "023bf722272578ea8a889efa19070288": "\n w(x,y) = \\frac{q_0}{\\pi^4 D}\\,\\left(\\frac{1}{a^2}+\\frac{1}{b^2}\\right)^{-2}\\,\\sin\\frac{\\pi x}{a}\\sin\\frac{\\pi y}{b} \\,.\n", "023c00c41b5fca2d9265161353de9776": "\\color{Magenta}\\text{Magenta}", "023c2f48e975917d540465a883af89f3": "A_n = A + \\alpha q^n, \\,", "023c91a928cb3ea433ca767d460dcbe2": "{\\rm d}A = {\\rm d}U - (T{\\rm d}S + S{\\rm d}T)\\,", "023cc810f0ec386a71e5846889b5d75e": "k(i)\\geq 1", "023ccd363bceb5d8ec574167a06d1242": "\\delta=0,w(x_1,x_2)=\\mathbb{I}(x_1+x_2 0", "024d0aedab64caf6b9088f1ea5817f0a": " H_\\xi = \\xi^i\\frac{\\partial}{\\partial x^i}\\Big|_{(x,\\xi)} - 2G^i(x,\\xi)\\frac{\\partial}{\\partial \\xi^i}\\Big|_{(x,\\xi)}. ", "024d90f2a1a1613e65739f9c2e526069": "\\mathcal{L}_{QP}=\\mathbf{P} \\cdot \\dot{\\mathbf{Q}} - K(\\mathbf{Q}, \\mathbf{P}, t)", "024d9dbb7ceb42faf9928f6391d3aef8": "\\ddot{V}", "024e0406ca30841312cba458db27f8b9": "A_{i}\\to A_{i}", "024e5399b10635924f8ea3f5619c63da": "f(x, \\Phi_j(x)) \\leq \\Phi_i(x) \\, ", "024e9f2ae45357083043d5794ad82d19": "X_1(t)", "024ea2eb3be1b0fc776b67d7a4e6de18": " \\frac{Y}{L}=A.\\frac{K}{L} ", "024ecdaa895a112cc1ad509e2a0a27b4": " f_*(a_0 \\otimes \\cdots \\otimes a_n) = (b_0 \\otimes \\cdots \\otimes b_m) ", "024f088ed8ba29d0348d56a5728d486a": "x_j - x_m \\neq 0", "024f75b25db2a5874f7888c41f537693": "\\mathcal{O}(-1)", "024f93a30355166f71c68076fd453c28": "\\mathbb{RFM}_I(D)", "024f9bda6687751643ae724a45f345c5": "u_n = \\sum_{k=0}^n {n\\choose k} a^k (-c)^{n-k} b_k", "024fdc4704a30426ed70030fd55a7e52": "Q = 12.5", "024fe1572700dab589fc3a4eaaee0eee": " y = \\log_{10} {P}^*_i ", "02501709f35cb1e403a42cda6991af2c": "\\pi = \\sqrt{12}\\sum^\\infty_{k=0} \\frac{(-3)^{-k}}{2k+1} = \\sqrt{12}\\sum^\\infty_{k=0} \\frac{(-\\frac{1}{3})^k}{2k+1} = \\sqrt{12}\\left({1\\over 1\\cdot3^0}-{1\\over 3\\cdot3^1}+{1\\over5\\cdot 3^2}-{1\\over7\\cdot 3^3}+\\cdots\\right)", "0250723330e3db998ce955076784f58d": "\\vec w \\propto \\Sigma^{-1} (\\vec \\mu_1 - \\vec \\mu_0)", "02507ba5d8c3288a6d2a2979ffff4f68": "\\hbar \\Omega _m", "0250c111e54fdf6000aec02a0d851bfa": "H(\\omega)\\,", "02510a99289a433f29b5f77146a9836d": "\n \\left(\\bigcup_{i\\in I} A_i\\right)^o = \\bigcap_{i\\in I} A_i^o.\n ", "02513c497f0b6302937c7b0a7851c18d": " \\gamma = \\gamma' = \\frac{2q -d}{p} \\,", "0251597a9057a3470a7a302dcd31b56e": "V_{ion-ion}", "0251880a00c512cf394979313f3766c8": "\\hat{\\Phi}(t) ", "02521927eba8b0cb32a3cc8ff30d4c7f": "\\tau_\\mathrm{s}\\,", "02527c4a4a9931ee779fd7cf66f30eea": "\\hat{x},\\hat{y},\\hat{z}", "0252d21ed53a7d41d3db2caefda95f8b": "S=\\{ s_n \\}_{n\\in\\N},\\,", "02531c9578e100f64befba62e273b529": "20 \\times \\log_{10} \\left(\\frac{5V}{10 \\mu V}\\right) = 20 \\times \\log_{10}(500000) = 20 \\times 5.7 = 114 \\,\\mathrm{dB}", "025329063bb50ed9795e5fe74bd919e9": "\\#(n)=|B_n(G,T)|, ", "02535ae4ac19df62aea3828db87a7817": " X_C= -\\frac{1}{\\omega C}", "0253a63318b1ccb430558dcb2955a281": "A[\\Psi]=\\int\\mathrm{d}t\\ \\langle\\Psi(t)|H-i\\frac{\\partial}{\\partial t}|\\Psi(t)\\rangle.", "0253c84666685857b6ba8cdbe9d6432a": "\\rho \\left(\\frac{\\partial \\mathbf{v}}{\\partial t} + \\mathbf{v} \\cdot \\nabla \\mathbf{v}\\right) = -\\nabla p + \\mu \\nabla^2 \\mathbf{v} + \\frac{\\mu}{3} \\nabla (\\nabla \\cdot \\mathbf{v}) + \\mathbf{f}. ", "0253d2b1cab9f2e800f7a2e06733e33e": "a,b,k", "0254081c26dc9e45ce5c215fee67ed14": "\\langle\\Delta V\\rangle=\\frac{4}{3}\\frac{e^2}{4\\pi\\epsilon_0}\\frac{e^2}{4\\pi\\epsilon_0\\hbar c}\\left(\\frac{\\hbar}{mc}\\right)^2\\frac{1}{8\\pi a_0^3}\\ln\\frac{4\\epsilon_0\\hbar c}{e^2}", "02544ffbb49928005b35b4fc1c66f9c6": "\\mathbb{P}(n \\leq n^* | n_b \\leq n^* , s+b) = \\frac{\\mathbb{P}(n \\leq n^* |s+b)}{\\mathbb{P}(n_b \\leq n^* |s+b)}\n= \\frac{\\mathbb{P}(n \\leq n^* |s+b)}{\\mathbb{P}(n \\leq n^* |b)}.", "025464d3b6a57dde173c670b334b4c7a": "\\mathbf{\\nabla}\\cdot\\mathbf{E}(\\mathbf{x})=-\\frac{i Z_0}{k}\\mathbf{\\nabla}\\cdot\\mathbf{J}(\\mathbf{x})", "0254928e844d7febdfcfccb610b43951": "1_{GX} = G(\\varepsilon_X)\\circ\\eta_{GX}", "0254ab4d45ac475dc19a0f6111a6bee7": "\\mathbf{K}q=\\mathbf{S}\\,q - \\mathbf{V}q", "0254bddbffe3291cb211dc2690d791df": "\\mathcal O_L / \\mathfrak p^{i+1}.", "0254fe457741ff2f8ac65219733d98bc": "\\digamma(\\nu)", "02553bc981384e85483e10a26c47bf1a": "\\mathfrak{R}", "0255ae6678ed9ddf1b37d7fddd7e9cfe": "\\sum_{m=0}^\\infty \\frac{65520}{691}\\left(\\sigma_{11} (m) - \\tau (m) \\right) q^{m} = 1 + 196560q^2 + 16773120q^3 + 398034000q^4 + \\cdots", "0255b016c317e4eae99aeb727b3f3e10": "\\frac{4^n}{\\Gamma(n+1)}.", "0255d35e1cc778d50d639f145ca7a5e7": "\\lfloor", "0256681cebc402c62c9107251b6e62fe": "\\frac{}{\\Gamma_1, \\alpha, \\Gamma_2 \\vdash \\alpha} \\text{Ax}", "02568e22c87a55a649d0b1b61e3529b2": "\\mathbf{e}^i (\\mathbf{e}_j) = \\delta^i_j.", "0256a4b12d15b54af18b148540113e1e": " \\sum S = (x_{1} + x_{2} + x_{3} + ... + w)(p^{0} + p^{1} + ... + p^{k-1}) = \\sigma_{1}(w)(p^{0} + p^{1} + ... + p^{k-1}) ", "02577ce019c0536fca02f2d07889e40a": " kT = \\frac{\\hbar a}{2\\pi c}", "025784302af37d32451f062663ee025c": "\\Rightarrow x=e^{W(\\ln z)}\\, ,", "0257c4faf4027f97471d14f87512c6e1": "nF^{ _{ }}/RT", "0257d237e99f9bd9830e616b6ac54595": " \\delta W = p dV\\;", "0258321027b3e0da182a33942238407b": "Q_{\\alpha \\beta}=\\int d^3\\mathbf{x'}(3 x'_\\alpha x'_\\beta - \\|\\mathbf{x'}\\|_2^2 \\delta_{\\alpha \\beta})", "0258535e986c72130f7f01840532fc24": "\\{ A , V \\}", "025890facebaed2aec288dc7bede99b1": "J_0(kr)", "0258f7634a80d517311163f85c2bc0a9": "(u,v)=(0,0)", "025943f11cd36bf8028cfdba8a40033a": " n(r) = \n \\begin{cases}\n n_1 \\sqrt{1-2\\Delta\\left({r \\over \\alpha}\\right)^g} & r \\le \\alpha\\\\\n n_1 \\sqrt{1-2\\Delta} & r \\ge \\alpha\n \\end{cases}", "02595d47e0006a3ce08238acdaa0fd6b": " \\operatorname{get-lambda}[F, G\\ V = E] = \\operatorname{get-lambda}[F, G = \\lambda V.E] ", "025a04608819638d1b3ffbed85952e1f": "a(t) = ae^{-j\\omega t} \\ ", "025a0946c92f9fba9719cc3328931e9b": "\\begin{align}\n& \\mathbf{(D-\\omega L)^{-1}[(1-\\omega )D+\\omega U]} = \\frac{1}{12} \\begin{pmatrix}\n-1.2 & 4.4 & 6.6 \\\\\n-0.33 & 0.01 & 8.415 \\\\\n-0.8646 & 2.9062 & 5.0073\n\\end{pmatrix},\n\\end{align}", "025a1d6e6a1ae5a9a00bff0dc971b1ed": "\\Delta^4 m_6 = m_6 - 4m_7 + 6m_8 - 4m_9 + m_{10} = \\int x^6 (1-x)^4 d\\mu(x) \\geq 0.", "025a36473308d14aa4c20882682656b8": "\\alpha = (Q\\times F/4)^{1/4}\\,\\!", "025ab80c795d5d2e8499b80ac2b81b60": "X \\sim \\mathrm{Rayleigh}(1)\\,", "025adbddd8ff913fc53236ff7ae8d8ba": "\\frac{8! \\times 3^7}{24}=7! \\times 3^6=3,674,160.", "025b057912045b97ef467c2c2bc9242a": "\\hat{\\bold{H}}_\\operatorname{PI} = \\begin{bmatrix}0.052 & 0.510 \\\\ 0.510 & 8.882\\end{bmatrix}.", "025b2eae2546fafa1fd6b9f756a7700d": "\\alpha_t", "025b36ac0f07709eb91d6fd2e6d704f6": "K / L", "025b3f94d79319f2067156076bf05243": "\\Sigma", "025b580a55042ccea81fbdea600770d5": "\\|u-u_N\\|_{H^1(\\Omega)} \\leqq C \\exp( - \\gamma N )", "025b98f6d511a3d7a32f9e0dcc096d84": " E[X]_{ab} = R_{ambn} \\, X^m \\, X^n", "025bdb4f9244413527859c3df03bd71a": "\nm\\rightarrow m+S~", "025c4256ddf664dffb51d5cd897eb82e": "\\beta/\\alpha", "025c8812189a2392bba31d16f753065d": "r^{n}, r^{n-1}, \\ldots ,r", "025c9146ef1e96410c26a64fdee29d95": "i_{n-2}-i_{n-3}\\,\\!", "025d0c896f43bb3cd40766c406eba75f": "\\ell_i\\,", "025d2e99c2738d5ca731f6a04ed05e1a": " \\begin{bmatrix} y_1\\\\ y_2\\\\ y_3 \\\\ \\vdots \\\\ y_n \\end{bmatrix}= \\begin{bmatrix} 1 & x_1 & x_1^2 & \\dots & x_1^m \\\\ 1 & x_2 & x_2^2 & \\dots & x_2^m \\\\ 1 & x_3 & x_3^2 & \\dots & x_3^m \\\\ \\vdots & \\vdots & \\vdots & & \\vdots \\\\ 1 & x_n & x_n^2 & \\dots & x_n^m \\end{bmatrix} \\begin{bmatrix} a_0\\\\ a_1\\\\ a_2\\\\ \\vdots \\\\ a_m \\end{bmatrix} + \\begin{bmatrix} \\varepsilon_1\\\\ \\varepsilon_2\\\\ \\varepsilon_3 \\\\ \\vdots \\\\ \\varepsilon_n \\end{bmatrix} ", "025dceb6d6fb0f273aa5fae8c6dca7c6": "e^{S}", "025e191de58cbf019d7d91e22fe94bda": "\\frac{1}{\\sqrt{n}} \\sum_{i=1}^{n} \\left [\\mathbf{X_i} - E\\left ( X_i\\right ) \\right ]=\\frac{1}{\\sqrt{n}}\\sum_{i=1}^{n} \\left [ \\mathbf{X_i} - \\mu \\right ]=\\sqrt{n}\\left(\\mathbf{\\overline{X}}_n - \\mu\\right) ", "025e8bf0eb554eb06c314ce8dffbe64a": "\\scriptstyle \\sin \\theta \\approx \\theta\\,", "025e99932b678d1f0120fe0dbe2e13cc": "\\mathbf{P} = m\\mathbf{U} \\,", "025e9e7552edc9d5c6e1ed0eba4f68fb": " \\left| x(t) - x(t + T) \\right| = 0 \\text{ for all } t. \\ ", "025f5f529b0a6dc6d3a158197ebde4cf": "a/bc", "025f6e2d7c040ef7ec04d50fa2fc2108": "(1-2x_0)^{2^{n}}", "025fc04dcc1848a7baf1b9b46fc11fbf": "f\\in \\mathcal{PC}", "026088a2c5ca5cfa2befcb3b43266009": "f(x)=\\sum_\\alpha a_\\alpha x^\\alpha\\text{, where }\\alpha=(i_1,\\dots,i_r)\\in \\mathbb{N}^r \\text{, and } x^\\alpha=x_1^{i_1} \\cdots x_r^{i_r}", "0260ab105a2f8001f01707d2d4465067": "[M]_{v\\;\\|\\;a\\;\\|\\;u} \\rightarrow [[~]_{u\\;\\|\\;x}\\;\\|\\;M]_{v\\;\\|\\;y}", "0260c684c19a0d9dce9a8da81c542162": "V\\otimes V_{II_{1,1}}", "026150509621605b486cae1a27d552c9": "\\mbox{C}_4^6", "0261592341d2501c32a6f3978b802671": "x = t, y = t^2 \\quad \\mathrm{for} -\\infty < t < \\infty.\\,", "0261a3d116001fbd03cf425823a21970": " F(\\omega) ", "026212a7ffb0ab4032a89f1f802c2838": "Z_{F+G} = Z_F + Z_G\\,", "02628a067d4b163832045a47c972c9a5": "|M|^{2}", "0262d312520c54ffaac84863f9eee4ef": "\n\\zeta=\\frac{\\alpha}{\\sigma_0}\\;\\int_K^{F_0}\\frac{dx}{C\\left(x\\right)}\n=\\frac{\\alpha}{\\sigma_0\\left(1-\\beta\\right)}\\;\\left(F_0^{1-\\beta}-K^{1-\\beta}\\right),\n", "02633821424a5649073e041c2c6ebd0f": " \\forall\\text{ internal } f: {^*\\!A}\\rightarrow {^*\\mathbb{R}} \\dots", "02644a9c0510755d4a1390e1cc07f295": " ab \\le f(a) + g(b). \\, ", "02645e252addf19388e4eb4cdf917017": "\\sum_{f_1\\ge f_2\\ge f_N\\ge 0} \\mathrm{Tr}\\Pi_{\\mathbf{f},0}(z_1,z_1^{-1},\\ldots, z_N,z_N^{-1}) \\cdot \\mathrm{Tr}\\pi_{\\mathbf{f}}(t_1,\\ldots,t_N)\n=\\sum_{f_1\\ge f_2\\ge f_N\\ge 0} \\mathrm{Tr}\\sigma_{\\mathbf{f}}(z_1,\\ldots, z_N) \\cdot \\mathrm{Tr}\\pi_{\\mathbf{f}}(t_1,\\ldots,t_N)\\cdot \\prod_{i 1.\\,", "02718a35a1d62e76d3127af4cd4f23cc": "s_\\mathrm{in}\\,", "0271a9f2d735faff963555b6df864814": "r_2 = (A \\to S, \\{r_2\\}, \\{r_1, r_3\\})", "0271cbc3a02561a58d919aecb18029ab": "m_{p}", "0271cfd20d5c7bf792c844373753b4c9": "{\\partial / \\partial r} = -{\\partial /\\partial n}.", "02721aa35b02c75d8d1f5a9d87228d0a": "\\frac{(a+b)h}{2} \\,\\!", "027226d2312eded580526508612ce832": "\\sqrt{2}+\\sqrt{3}\\,", "02724694ac3af41dd73e0fcb69ee2466": "A_0 \\to \\ldots \\to A_{i-1} \\to A_i \\to A_{i+1} \\to \\ldots \\to A_k", "027281910cf4071ee187728510baa84f": "\\sigma_2^2", "0272b29f6e7dd14e7071eb5bf61b57bb": "T=I", "0272c90422f4b23f836598dc016c9d9f": "\\frac1{137}", "0272d268d0534de5245746bcaa96c0e1": " \\sigma^* = G(F^*) ", "02737eddf8250b8f1aaa104754d37249": "\n\\begin{bmatrix}\n 1\n\\end{bmatrix}\n\\quad\n\\begin{bmatrix}\n 1 & 2 \\\\\n 2 & 1\n\\end{bmatrix}\n\\quad\n\\begin{bmatrix}\n 1 & 2 & 3 \\\\\n 2 & 3 & 1 \\\\\n 3 & 1 & 2\n\\end{bmatrix}\n", "0273a173375948ed6cc340447e4a27ed": "\\text{If }\\lim_{x \\to c} f(x) = L_1 \\text{ and }\\lim_{x \\to c} g(x) = L_2 \\text{ then:}", "02742521dd1678400280d212566bfb47": "\\langle\\phi(0,t)\\phi(0,0)\\rangle\\sim \\sum_nA_n\\exp\\left(-\\Delta_nt\\right)", "027441dff48689fb1b7fbd1cc35a5356": "g \\circ f \\colon X \\to \\mathbf{K} \\colon x \\mapsto g(f(x))", "02752b048de7a6e77676f58bb429610f": " t_1", "027543b772146bb664f61c562344bb75": "\\sum_{i=0}^n i^2 = \\frac{n(n+1)(2n+1)}{6} = \\frac{n^3}{3} + \\frac{n^2}{2} + \\frac{n}{6}", "02757c96b2a9eada766a85e99918010d": " L_{\\sigma,\\varepsilon} := \\max\\{ \\sigma(k) | k \\in I_{\\sigma,\\varepsilon} \\}", "0275a8621507190c4edc2ff72a3e4c06": "X^G", "0275ad96d859850a8883d4d869704943": "\\pi(x) \\leq x", "0275b5048a096e7776c9a2a7bf9c39ad": "\\mathbf{x}_{n+1}=\\mathbf{x}_n-\\gamma_n T(A-\\lambda_n I)\\mathbf{x}_n,\\ n \\ge 0.", "0275e7e544c08853c8c58bc04897645b": "\\mathbf{A}\\cdot{\\rm d}\\boldsymbol{\\ell}=-", "0275f7fb66a3fbd19097948981f29d7e": "\\lnot\\ \\forall{x}{\\in}\\mathbf{X}\\, P(x) \\equiv\\ \\exists{x}{\\in}\\mathbf{X}\\, \\lnot P(x)", "02761f43f1ceb181d2090becb35a5739": "\\left | \\mathbf{a} \\right \\vert = \\sqrt{\\mathbf{a} \\cdot \\mathbf{a}} = \\sqrt{{a_1}^2 + {a_2}^2 + {a_3}^2 + {a_4}^2 + {a_5}^2 + {a_6}^2}.", "027623c36bf90a4651c4401fdb3cc270": "(x_i,x_{i+1})\\in E", "027712408326070f9db72d79a34da1c3": "\\int_0^\\infty (1\\wedge x) \\mu(dx) <\\infty.", "027721e0be74c20fbc15d6dff1e61227": "\\mathbb{C}^{2n}", "02773c881227cb8b849971bf0a8b8aa6": " \\mbox{E} =\\frac{\\sqrt{N} \\cdot\\sqrt{R}}{2\\cdot \\sqrt{\\pi}\\cdot d}", "027757997a2330c4386e56b918e88c4f": "\\lim_{h\\to 0}\\frac{f(a+h) - f(a)}{h} = {+\\infty}\\quad\\text{or}\\quad\\lim_{h\\to 0}\\frac{f(a+h) - f(a)}{h} = {-\\infty}.", "027770abe9a99f31d66cb33a30e4494c": "j=1", "0277df1ffe8c7a546f8668e2d023a508": "\\operatorname{Re}(s) ", "0277e41e188b27fc82b47423e62409fe": " [I_1 \\cdots I_r] = [J_1 \\cdots J_s] \\in Cl(R). ", "02783fff904f832fc73014e85e617ff8": "\\frac{\\alpha}{v}\\log{\\left(\\frac{c+vT}{c}\\right)}=1\\,\\!", "02784e94955679b54e8a6d68a96f5c71": "v = H_0 d", "0278850ba059525d5f0e5e514b76f459": "\n\\frac{s}{D} = \\frac{m_b}{k_B T}\n", "0279047637786c035fd0ae1abaabecf0": "s_c = e\\,\\alpha^{c\\,i}", "02790ba6054fae5179e0a8e8a4948088": "m>0\\,", "027939899d5c69c1f82c472e8671fa17": "\\phi(t) = N\\cdot 2\\pi,\\,", "02796b1bee509dbee67ee4b7a0acbeb5": "\\tfrac{1}{X} \\sim \\mathrm{Planck} ", "0279f4c24ed40a329b8ac3dd52cd8ff2": "d(O_{r}, Q)", "027a3dc0a952751d29f62a84c0d48b7a": " \\mathcal{J}_{ij} = \\begin{cases} J & \\mbox{if }i, j\\mbox{ are neighbors} \\\\ 0 & \\mbox{else.}\\end{cases}", "027a4b3e733807da32b0aec4e03387dc": "i_{\\text{1}} = I_{\\text{B}} + i_{\\text{F}}", "027a6a7dc8797392917d232f79c29137": "\\Omega_E=\\binom{N}{(N+j)/2} = \\frac{N!}{\\left(\\frac{N+j}{2}\\right)! \\left(\\frac{N - j}{2}\\right)!}.", "027a8888e2a55823e14377fc154b0f89": "g(\\lambda) = -\\tfrac{1}{2}\\lambda^{T}AQ^{-1}A^{T}\\lambda - \\lambda^{T}b", "027aef1b3ac13ece6cbeab406d386152": "wp(\\textbf{while}\\ E\\ \\textbf{do}\\ S\\ \\textbf{done}, R)", "027b33f37f2eeb78e798fe97e5b02551": " R_N =r_o = \\begin{matrix} \\frac {1/\\lambda + V_{DS}} {I_D} \\end{matrix} = \\begin{matrix} \\frac {V_{E} L + V_{DS}} {I_D} \\end{matrix}\n", "027b3e2314e62461489d1c69ad4dec6c": "\\begin{align}\n&\\deg P_n = n~, \\quad n = 0,1,2,\\ldots\\\\\n&\\int P_m(x) \\, P_n(x) \\, W(x)\\,dx = 0~, \\quad m \\neq n~.\n\\end{align}", "027b580645d6223958e406b837abb816": "\\pm\\sqrt{1 - \\cos^2 \\theta}\\! ", "027b97cb2500a918e169b01e05f1aae4": "-\\nabla\\cdot\\mathbf{g}=\\nabla^2\\Phi=4\\pi G\\rho\\!", "027b9f898690366de9b5d8b3d9e7e41a": " \\nabla^2 f = {1 \\over r^2}{\\partial \\over \\partial r}\\left(r^2 {\\partial f \\over \\partial r}\\right) \n + {1 \\over r^2\\sin\\theta}{\\partial \\over \\partial \\theta}\\left(\\sin\\theta {\\partial f \\over \\partial \\theta}\\right) \n + {1 \\over r^2\\sin^2\\theta}{\\partial^2 f \\over \\partial \\varphi^2} = 0.", "027ba77426858754748114062a46ac88": "p(x|\\overline{y})", "027bb5050684378c588a0384461002dd": "w''=0", "027bce9859bd9d3f00cedb3501833432": "\\big\\updownarrow \\Big\\updownarrow \\bigg\\updownarrow \\Bigg\\updownarrow \\dots \\Bigg\\Updownarrow \\bigg\\Updownarrow \\Big\\Updownarrow \\big\\Updownarrow", "027bfcbbe3242bea7e33988be97c2e88": "G \\to H\\backslash G", "027c3429f98f7c39bab027549e1b9c7b": "a_1", "027cfb67122353f1488768c2823ea7fb": " r_\\mathrm{ corr } = r + \\frac{ 1 }{ n }( 1 - \\frac{ n - 1 }{ N - 1 } ) \\frac{ r s_x^2 - \\rho s_x s_y }{ m_x^2 } ", "027d00a2432091cf782e1dbec39e173f": "\n\\operatorname{Li}_s(z) = {\\Gamma(1 \\!-\\! s) \\over (2\\pi)^{1-s}} \\left[i^{1-s} ~\\zeta \\!\\left(1 \\!-\\! s, ~\\frac{1}{2} + {\\ln(-z) \\over {2\\pi i}} \\right) + i^{s-1} ~\\zeta \\!\\left(1 \\!-\\! s, ~\\frac{1}{2} - {\\ln(-z) \\over {2\\pi i}} \\right) \\right] ,\n", "027d12976af94786c8f656a872dbc10b": "H_{out}\\ =\\ f(H_{in},\\ m)", "027d85e20311d606467f08fa2b3fbad8": "N = M + 1 \\, ", "027e3a0f8b7e284ab68c542a1ae3489e": "V_{GS}=V_{th}", "027e72afb96af576be811f0b0465ed0c": "f:I\\rightarrow \\mathbb{R}^+", "027ea6e711e5f2c509cc7a4e6a5b64a2": "\n\\langle z^m \\rangle = \\oint p_w(z)z^m \\, dz.\n", "027eb3c1e422b4c252a3eebfef6b7432": "\n\\begin{align}\nq_\\mu^*(\\mu) &\\sim \\mathcal{N}(\\mu\\mid\\mu_N,\\lambda_N^{-1}) \\\\\n\\mu_N &= \\frac{\\lambda_0 \\mu_0 + N \\bar{x}}{\\lambda_0 + N} \\\\\n\\lambda_N &= (\\lambda_0 + N) \\operatorname{E}[\\tau] \\\\\n\\bar{x} &= \\frac{1}{N}\\sum_{n=1}^N x_n\n\\end{align}\n", "027ec8425f9c5fa3980d0a78a6024a36": "\\begin{align}\nx&=a\\cosh\\xi\\cos\\eta\\cos\\phi\\\\\ny&=a\\cosh\\xi\\cos\\eta\\sin\\phi\\\\\nz&=a\\sinh\\xi\\sin\\eta\n\\end{align}", "027efd0609b2b1a78ea698c8088fd976": "a_i[\\mathbf{f}] = \\sum_{k=1}^n v^k[\\mathbf{f}]g_{ki}[\\mathbf{f}]", "027fbc8ed2dce4562d06aecc8a04dff8": "R= \\left[\\frac{n_o (n_2)^{2N} - n_s\n(n_1)^{2N}}{n_o (n_2)^{2N} + n_s (n_1)^{2N}}\\right]^2,", "027fe9d27e81b68b9d9ac895264bb6eb": " (\\kappa-n-1)~r^{n+1}~\\cos(n\\theta) \\,", "028016a24cad05e17d89a0634c318ad0": "Y(y)=C_{1}\\cos(k_{y}y)+C_{2}sin(k_{y}y) ", "0280995012f1d43b2acd677acdf88bd1": "\\frac{v_b}{w_b}\\ge\\frac{v_i}{w_i}\\, ", "0280c97d8a46b10a8fcd21c89a15021b": "\\frac{\\partial}{\\partial x_1} f(x_1, x_2, \\ldots, x_n)\\,,\\quad \\frac{\\partial}{\\partial x_2} f(x_1, x_2, \\ldots x_n)\\,,\\ldots, \\frac{\\partial}{\\partial x_n} f(x_1, x_2, \\ldots, x_n) ", "0280e5bde4394d3371051b15d4770877": " \\Phi = \\iint I_\\lambda \\mathrm{d} \\lambda \\mathrm{d} \\Omega", "02813d27899acc3cff6ba6747ec873cc": "v\\in TM", "0281ba5a5aa825aeead474848d07516c": " \\sigma(\\mathfrak{G}^2 \\oplus \\mathfrak{G}^2 \\oplus \\mathfrak{G}^2 \\oplus \\mathfrak{G}^2) = 1", "028246bcf34addfe79858399c1dcfbfb": " I = \\begin{cases}\n 1 &\\text{if } Y \\le 1/3,\\\\\n 0 &\\text{otherwise},\n\\end{cases} ", "02825789173cc14e46546d75a3d6383c": "\\ell(s)\\ge \\ell(t)", "0282b2607138bd84dda06decc05eacd6": "D_{a}", "0283316329c94c014e656bca7c85f6cf": "{1 \\over 168}\\left(n^7 + 21 n^5 + 98 n^3 + 48 n\\right).", "0283648e14bf01495b25d91cf4d0b645": "N^1, N^2", "0283a6960393cb45f987c35f6d59bc40": "8100a + bx ( 180 - x ).\\,", "0283df13c758ef3af6a782345aba0ebd": "\\int_{-\\infty}^0 \\big[t\\inf \\mathrm{supp}X-g'(t)\\big]dt", "0283e8b43a0c3b781521213883348597": "\\rho_w", "0283eca90e0d5c9785555060675c9983": "NL \\left [ u \\right ] (x) = {1 \\over C(x)}\\int_\\Omega e^{-{{(G_a* \\left \\vert v(x+.)-v(y+.)\\right \\vert ^2)(0)}\\over h^2}}v(y)dy.", "02844aa24ce762ec1f7385b1fefac755": " x^0 = ct = c \\gamma \\tau \\, ", "028469922269efffb745f0d802201923": "k_{xo}=\\sqrt{{k_{o}^{2}}-(\\frac{m\\pi }{a})^{2}-\\beta ^{2}}= k_{o}\\sqrt{1-(\\frac{m\\pi }{ak_{o}})^{2}-\\frac{\\beta ^{2}}{k_{o}}}", "02848f7255ed6999eee0a31a8d180d03": "x\\in\\{0,1\\}^n", "0284d692fbed76eaf34b1b7bf1306aa7": "\\int_{\\mathbf{R}} \\delta\\bigl(g(x)\\bigr) f\\bigl(g(x)\\bigr) |g'(x)|\\,dx = \\int_{g(\\mathbf{R})} \\delta(u)f(u)\\, du", "0284f082f8fa0fe0c6a181cf5be904f5": "\n \\operatorname{Div} \\boldsymbol{P}^T = \\rho_0\\ddot{\\boldsymbol{x}'}.\n ", "02850d6a647bc6cdb7f44baeb1f90089": "{}^2", "0285aa7f11df22d43c8f93a2ca31a266": "x^2+y^2,", "028628dc15d1c92860d56ab2ffe88961": "7^2<103", "028653ccb2edb9857e722606c46a7ed0": " g_D=\\frac{dI}{dV}\\Big|_Q = \\frac{I_0}{V_T} e^{V_Q/V_T} \\approx \\frac{I_Q}{V_T} ", "02865d599780a233d3765bd4587aac66": " =Z_{DP}^{2}\\frac{\\hbar \\omega}{8 \\pi ^2 \\hbar\\rho c^{2}} (N_{q}+\\frac{1}{2} \\pm \\frac{1}{2}) g(E \\pm \\hbar \\omega) \\; \\; (18) ", "0287125b21317160ff3a19b3817dfaf5": "\\partial A", "0287249202504fd9925b675320d10892": "\\scriptstyle 2\\,\\frac{7}{12}", "02873b47e4f412bd6cbcf3456b898fc6": "d=701", "0287b3f6ae84d39a46d3b20287f54922": "c_d\\;", "0287b9ac9048d5360e75da1fe4462517": "\\frac{\\delta l}{\\delta t}=\\frac{[P_A+g \\rho (h-l\\sin\\psi)+\\frac{2\\gamma}{r}\\cos\\phi](r^4 +4 \\epsilon r^3)}{8 r^2 \\eta l}", "0287d19d4b2d3fb6122a9bbff178bc63": "f(z_0)", "0287e0b7e48b39993d5c5ddaab509d7f": "M(x) < N(x)", "02881915156de35c74e826ad9e4b4e0c": "\\frac{c^2k^2}{\\omega^2}=1-\\frac{\\omega_p^2}{\\omega^2}\\,\n\\frac{\\omega^2-\\omega_p^2}{\\omega^2-\\omega_h^2}", "02882117772d1d5e3a33c5a7b1d80a07": " \\frac{(\\mu_2-\\mu_1)-(\\bar X_2 - \\bar X_1)}{\\displaystyle\\sqrt{\\frac{S^2_\\mathrm{pooled}}{n_1} + \\frac{S^2_\\mathrm{pooled}}{n_2} }} ", "0288211d1f3ba68261d99ee4081e59ec": " \n\\begin{bmatrix} \n0 & 0 & 0 & \\cdots & 0 & -c_0 \\\\\n1 & 0 & 0 & \\ldots & 0 & -c_1 \\\\\n0 & 1 & 0 & \\ldots & 0 & -c_2 \\\\\n\\vdots & & & & & \\\\\n0 & 0 & 0 & \\ldots & 1 & -c_{n-1} \\\\\n\\end{bmatrix}\n", "028826fbe2034c16c6a0c342139cca8b": "\\frac{v^2}{R} cos(\\theta)", "0288777665dd9e85992878a2ed64b9f3": " \\mathbf{F} = m \\mathbf{g}", "0288bde0c2d593f2b5766f61b826a650": "nu", "0288c1e7f3264a3b7e252e64e194603e": "E_T = t \\cdot C_T = t \\cdot \\sum_{n=1}^N C_n", "0288c82f52338089d23837709db7ef1c": "v(t;\\delta) = [u_0 \\cos \\delta t + v_0 \\sin \\delta t]e^{-t/T} - \\kappa E_0 \\int_0^t dt' \\cos \\delta(t-t')e^{-(t-t')/T}", "0288d538bd34157467f10e06eb170231": "\\sum^{k}_{i=1}1/\\lambda^2_{i}", "0288d57113c941d383db2aabb92482a7": "\\mathcal F(f)", "028954e78ec6d14a116374223c3da8dc": "\\Eta \\, \\eta \\,", "0289624a13094f127d206a236f58b67e": "G = g_{64},\\text{ where }g_1=3\\uparrow\\uparrow\\uparrow\\uparrow 3,\\ g_n = 3\\uparrow^{g_{n-1}}3,", "02896fcc232388e90696f4a0d8854552": "\\underline{P}(A^c)= 1-\\overline{P}(A)", "0289b2445bb4a1bdd64b9408dd9f3c1c": "\\zeta(6,1)+\\zeta(5,2)+\\zeta(4,3)+\\zeta(3,4)+\\zeta(2,5) = \\zeta(7)", "028a0b0b638a3450112e43a6bee6f048": "_{qp+qp'=qp+q'p\\,}\\!", "028a2a14f97cd167204572e12a98662b": "\\Delta p = f \\cdot \\frac{L}{D} \\cdot \\frac{\\rho V^2}{2}", "028a4658f0b06437681dd13bec5f1f4f": "\\operatorname{erf} (-z) = -\\operatorname{erf} (z)", "028abbffb1dd2d77f036b6d6eca2fc8a": "|x^\\mathsf{T} y|\\le\\|x\\|_2\\|y\\|_2.", "028afe481ca2232314d8cb365a1d1036": "\\scriptstyle{|0\\rangle \\equiv |\\psi(t_0)\\rangle}", "028b2eff77bbc39b102bacc22f2647d3": "f\\colon X\\to X", "028b31dd2dc7f3c03703933cc3d7d225": "\\langle r, s \\mid s^2 = 1, srs = r^{-1} \\rangle \\,\\!", "028bb5478e94d5964115b1af37f9c943": "R(\\theta ) = R(0) + G \\sin^2 \\theta ", "028bb9dd25511f8cdf762d3a3ba1b106": "\\ VDOP = \\sqrt{d_{z}^2}", "028c0d0db7547cf5aa11fea485157522": "e > 0", "028cb29c5821522a92e6dea284904c39": "W_n \\propto n,", "028cb3f87cb4f887fc0cefb603c2b051": "c({\\mathbb B})", "028cf3d120efa3656bf48d357ac1db7b": "\\delta(x-\\xi) = \\sum_{n=1}^\\infty \\varphi_n (x) \\varphi_n^*(\\xi). ", "028cf8d5e59f6aba6cf4d037caae5e4b": "\\nu(H) = \\tau(H)", "028d3c56485a12720db1e16c9e5ecc4b": "\\mathcal{L}(\\phi,\\partial\\phi,\\partial\\partial\\phi, ...,x)", "028d70ffc43db3b4f608e733a123472e": "a \\in \\mathcal{U}", "028dffcf9ec5fe1c9242c20d65a37f27": "S(u) = \\mathrm{sinc}^2(u) = \\left( \\frac {\\sin \\pi u}{\\pi u} \\right) ^2 \\ ; ", "028e0e9d0bc8702bdbf96b7d5328a941": "M \\models", "028e6e505cf7e6a9ec95c45c61d40527": "x_1(t)=x_2(s), \\ y_1(t)=y_2(s) \\ .", "028e7c230401be6584e89b2d13f261d6": "P_K=32.1\\,d", "028ec436855047c2bffa0b383f7936ea": "W_2 =2\\gamma(e_{ij})A(e_{ij})", "028ec8468b090fa5c5d1de11a4fbe39c": " \\textstyle \\sigma_A = \\prod_{j \\in A} \\sigma_j ", "028ed1eec4f4304627517d9d1fb582ae": " \\Psi^{(\\operatorname{Sha}) }(w) = \\prod \\left( \\frac {w- 3 \\pi /2} {\\pi}\\right)+\\prod \\left( \\frac {w+ 3 \\pi /2} {\\pi}\\right). ", "028ee608b65c2b27cbb42c981e683264": " z^j p_1^{k_1} p_2^{k_2} \\cdots p_n^{k_n} q_1^{\\ell_1} q_2^{\\ell_2} \\cdots q_n^{\\ell_n} \\, \\mapsto \\, \\partial_{x_1}^{k_1} \\partial_{x_2}^{k_2} \\cdots \\partial_{x_n}^{k_n} x_1^{\\ell_1} x_2^{\\ell_2} \\cdots x_n^{\\ell_n}.", "028f0c8bb26e1c6568d9feab0a9aa322": "L^p(\\mathbb{R}^n)", "028f38e2f0bc026f9a896d05d579d591": "c_{f,g\\circ h} \\cdot c_{g,h}(f^*(x)) = c_{f\\circ g, h}(x)\\cdot h^*(c_{f,g}(x)).", "028f51e014d288ea668ba5bacc32b683": " v_1 ", "028fd9e7f7bd2a7b04fd2c98b58e90b6": "\\cdots \\,\\leq\\, a_3 \\,\\leq\\, a_2 \\,\\leq\\, a_1", "02900c654e9c50288d2d779994a76b8d": "\\displaystyle \\nabla^2\\omega + \\frac{f^2}{\\sigma}\\frac{\\partial^2\\omega}{\\partial p^2} ", "02907a65f48839c3ed37e198eb8c0afd": "\\tau_{zx}=-\\nu \\frac{\\partial \\rho\\upsilon_x }{\\partial z}", "0290924c27e43ac698cc8659e787a33d": "f_1(z)=\\frac{(1-i)z}{2}", "029099ba5237fd8d6213efbcf3af7836": "\\lambda\\ge 0", "0290a332e92b98cd127f2489d929ecf4": "B^\\prime", "0290a88f14fb8b753331e1f64d60cd86": "\\lim_{c\\rightarrow -m}\\frac{{}_2F_1(a,b;c;z)}{\\Gamma(c)}=\\frac{(a)_{m+1}(b)_{m+1}}{(m+1)!}z^{m+1}{}_2F_1(a+m+1,b+m+1;m+2;z)", "0290c93ee0b15b0416d8286de7bce6ed": "dA_1= \\left(\\mathbf{n} \\cdot \\mathbf{e}_1 \\right)dA = n_1 \\; dA,\\,\\!", "02910523462ad6edcc3f5a16357bded9": "{\\mathrm MinN}(L+1,D,n) \\le 2{\\mathrm MinN}(L,n,n)", "02913629d042aa4f6f8a17b3fd183ea9": "\\scriptstyle P\\,\\sim \\,\\rm{Exp}(\\alpha)", "02914341676b562677f2898686ad23a5": "\\Delta(z)=\\sum_{n> 0}\\tau(n)q^n=q\\prod_{n>0}(1-q^n)^{24} = q-24q^2+252q^3+\\cdots", "02914e47be2a3a1357d398d3ea761c04": "\\nu Z.\\phi \\wedge [a]Z", "0291c9aa28e46d45a874837ffcddc44a": "L_{t} = \\lim_{\\varepsilon \\downarrow 0} \\frac1{2 \\varepsilon} | \\{ s \\in [0, t] | B_{s} \\in (- \\varepsilon, + \\varepsilon) \\} |.", "0291f94c8ebd42d6d8c456051f0aa4f0": "\\left [\n\\begin{smallmatrix}\n 2 & -1 & 0 & -1 \\\\\n-1 & 2 & -1 & 0 \\\\\n 0 & -1 & 2 & -1\\\\\n-1 & 0 & -1 & 2 \n\\end{smallmatrix}\\right ]", "02922b6dcee7ce520b4efa977df1ecca": "A + 2B \\rightleftharpoons AB_2; K_\\text{c} = \\frac{[AB_2]}{[A][B]^2} /\\text{M}^{-2}", "0292eaab254e16d4760c6f6bbfcdd495": " \\psi_t = K * \\psi_0 \\, .", "029300e27efb7a2ac2857174169a9d3e": "\\frac{1}{M \\cdot s}", "029303301b802635239a678d54b41738": "\n \\begin{align}\n N_{\\alpha\\beta,\\alpha} & = 0 \\\\\n M_{\\alpha\\beta,\\alpha\\beta} - q & = 0\n \\end{align}\n", "029314defb85735cc6d46b8e19c0e1c9": "\nR=\\frac{-n(n-1)}{\\alpha^2} \n", "02941d9d7a8a4647de2c3487d03cc029": "p_3 = p_1", "02942b66a4d9733e165e86739d9ee08a": "\\begin{align}\n{\\mathbf{S}}_i \\cdot {\\mathbf{S}}_j \n&=& \\sqrt {\\left( {1 - \\sum\\limits_\\alpha {\\sigma ^2 _{i\\alpha } } } \\right)\\left( {1 - \\sum\\limits_\\alpha {\\sigma ^2 _{j\\alpha } } } \\right)} + \\sum\\limits_\\alpha {\\sigma_{i\\alpha } \\sigma _{j\\alpha } } \\\\ \n&=& 1 - \\tfrac{1}{2}\n\\sum\\limits_\\alpha \\left({{\\sigma ^2 _{i\\alpha }} + {\\sigma ^2 _{j\\alpha } } }\\right) + \\sum\\limits_\\alpha {\\sigma _{i\\alpha } \\sigma _{j\\alpha } } + \\mathcal{O}(\\sigma ^4 )\\\\\n&=&\n1 - \\tfrac{1}\n{2}{\\sum\\limits_\\alpha {(\\sigma _{i\\alpha } } - \\sigma _{j\\alpha } )^2 } + \\ldots \n\\end{align}\n", "029455c92e4ec0a699d96e23d763d9d5": "\\hat{R}_n(f) = \\dfrac{1}{n}\\sum_{i = 1}^n \\mathbb{I}(f(X_n) \\neq Y_n)", "0294705005b27d51c6578400f31f9dab": "u_0=1,\\;v_0=0,\\quad u_1=0,\\;v_1=1,\\quad u_{k+1}=u_{k-1}-q_ku_k,\\;v_{k+1}=v_{k-1}-q_kv_k", "02949639dff879b56cef44160bc985c7": "\\mu_G", "0294a2fe08a3f956ae3ebbb08b074ff6": "\\varepsilon \\left[ M \\right]", "0294b227d4b07bc2935e707e4fa80dd3": "V_k(\\mathbf{R}^n)", "0294c46a2098b46bb343e59580803c2e": " \\ v_{1}-v_{2} = u_{2}-u_{1}", "0294f1e2c6a908d9529b5e98da9d3692": "A = \\frac{9}{4}a^2\\cot\\frac{\\pi}{9}\\simeq6.18182\\,a^2.", "0294fa4e2ed32efe774506d31349c49b": "\\pi S \\sin_n \\theta/\\lambda =n \\pi, n = 0, \\pm 1, \\pm 2,..... ", "0294fa66e2860bf2e2edfc6c2b7c3c22": " \n(1-\\epsilon)\\int_{A_k} \\phi \\, d\\mu_k \\geq (1-\\epsilon)\\int_E \\phi \\, d\\mu_k - \\int_{A-A_k} \\phi \\, d\\mu_k.\n", "029504ab8797ac64df7858c45b1e55b7": "R^\\ast", "02950734961dab76e76bf41728978d00": " \n1) \\quad H(\\empty) = 0\n", "029551905c3f89f00943794b4a2472ed": "g(n, m) = g(n,m-1)+X_m g(n-1,m).", "029561ef26841a2f06634549502d4c5c": "n p = \\omega(\\sqrt{n \\log n})", "0295a42715cf1a1df8cbb20cadfc74f8": "m=0.1n", "0295d8022555242acf9d0af9ee886c46": "x \\le \\frac{l}{2}\\sin\\theta.", "02965244c0ce67304c8f1cb2aa6faa6a": "F = DUV^\\top", "0296682e2a43df25157421f698988f49": "A=[0,1],", "0296de2566416c7d1dfeaf80ff6f3e96": "R(\\hat{n},\\phi) \\equiv \\exp\\left(-\\frac{i}{\\hbar}\\phi\\, \\mathbf{J}\\cdot \\hat{\\mathbf{n}}\\right)", "029768646ed4136a5e6baa9fde70eb83": "{m \\choose r}_q = {m \\choose m-r}_q. ", "0297a16e0134ecb464609f6e4d9ff403": "M_{k,j}=\\mathrm{ln}\\;(M_{k,j} / b_k).", "0297d156aad07b49a45ad666f31bbc70": "S(T) = C \\left(1 + \\frac{A}{T}\\right) - B", "029801ec7f67318dff1e0adc221317e4": "H(f) = \\mathrm{rect} \\left( \\frac{f}{2B} \\right)", "02980a825b993aa7c1e70d410418471c": "\nX \\sim \\mathrm{BNB}(n,\\alpha,\\beta).\n", "02983fb8a36cec4b1d87d21cff61e331": " H^G_*(E_{FIN}(G),K^{top}_{l^1})=H^G_*(E_{FIN}(G),K^{top})\\rightarrow H^G_*(\\{\\cdot\\},K^{top})=K_*(C_r(G))", "02985e0a38eeffec4e784d6f82ff6935": "\\operatorname{Cl}_2\\left(\\frac{3\\pi}{4}\\right)=\n2\\pi\\log \\left( \\frac{G\\left(\\frac{5}{8}\\right)}{G\\left(\\frac{3}{8}\\right)} \\right) -2\\pi \n\\log \\Gamma\\left(\\frac{3}{8}\\right)+\\frac{3\\pi}{4}\\log \\left( \\frac{2\\pi}{\\sqrt{2+\\sqrt{2}}} \n\\right)", "0298986fbd7961975bbb5c7b6cc7e7c8": "N_{\\rm A} = \\frac{M_{\\rm u} A_{\\rm r}({\\rm e})}{m_{\\rm e}} = \\frac{M_{\\rm u} A_{\\rm r}({\\rm e})c\\alpha^2}{2R_\\infty h}", "02989b2a62a0a1e102b65c1794ef4d28": "\\lfloor \\frac{d-1}{2} \\rfloor ", "0299254c9469af661203d1a69d80df20": "\n\\bar{\\delta} {\\phi^A}_{,\\sigma} = \n\\bar{\\delta} \\frac{\\partial \\phi^A}{\\partial x^{\\sigma}} = \n\\frac{\\partial}{\\partial x^{\\sigma}} \\left( \\bar{\\delta} \\phi^A \\right)\n\\,.", "0299430ed9ef9635331dcdcbe5ba1cba": "p_{j}", "029945c0ee0ac4a1e09274775c84fb07": "\\|\\hat{f}\\|_{L^q}\\leq p^{1/2p}q^{-1/2q}\\|f\\|_{L^p}", "0299d91026ddf50fefe05f8f092e2b42": "F_{\\nu}(k)\\,", "0299e342cc72a82c1bab9f222b5d88eb": "f''-{1\\over z}f'+{1-z \\over z^2}f=f''-{1\\over z}f'+\\left({1\\over z^2} - {1 \\over z}\\right) f = 0", "029a7fb4d52c5e9553d51cded8ab7924": "\\begin{align}\nL_f \\equiv \\underline{\\int_{a}^{b}} f(x) \\, dx &\\quad U_f \\equiv \\overline{\\int_{a}^{b}} f(x) \\,dx\n\\end{align}", "029b00b7fa24dcb97246d2df373ef28f": "\\frac{59049}{32768}", "029b0be093d8080d1a61e22ee093c57f": "m \\mid p-1", "029b156f8ba178c2301eb71ef498be1c": "y(t) = y_{(1)}(t)+\\frac{y_1-y_{(1)}(t_1)}{y_{(2)}(t_1)}y_{(2)}(t)", "029b363145195144d94b9ec7a854d54c": "dF = -b_\\text{ext} F dx", "029b3cdf1db812cf8147a10c6a08ddce": " \\sum_{m=0}^{p-1}{(-1)^m{p-1\\choose m} m^{2n}}\\equiv\\sum_{m=0}^{p-1}{(-1)^m{p-1\\choose m} m^{2n-\\wp(p-1)}}\\pmod p\\!", "029b46fb564d16ed8e2249b044615d7e": "n \\ge k", "029b596e37b45dede3df04654bec7ad0": "z_j \\mapsto iz_j", "029b72733492d85b84e82a3e01a9f2d2": "\\ge i", "029b93561645fd5d2a54de2c6f1768cd": " S_n = 1,1,\\frac{1}{2},\\frac{1}{3},\\frac{5}{24}, \\frac{2}{15},\\frac{61}{720},\\frac{17}{315},\\frac{277}{8064},\\frac{62}{2835},\\ldots ", "029ba64831d61ee5b6ef200ac8e7d816": " 2ax = -b \\pm \\sqrt{b^2-4ac} ", "029bd0d5c84b6da53e6262aee62b9dd7": "\\Pr(X=k)=F_{\\chi^2}(2\\lambda;2(k+1)) -F_{\\chi^2}(2\\lambda;2k) . \n", "029be1310b8d6075d3d3e51646d05035": "\\; AP = PJ.", "029be45299faa8334bfe288dad23166f": "\\triangle\\delta\\;=\\;\\delta' - \\delta\\; ", "029c34a36a9fe3e058eaadec6db2d0ec": "\\boldsymbol{\\sigma} = -p\\mathbb{I} + \\mathbb{T}", "029c49bad246fd00fb9fe7d17da86435": "\\gamma_2\\,", "029da23ae63f51c12d40401dd23f6d72": "x^5-9x^4-81x^3+729x^2=3888", "029da49e91a3c1e1bf3aa0faa118ad77": "\\frac{d^n\\bigl(f(x)\\bigr)}{dx^n}\\text{ or }\\frac{d^ny}{dx^n}", "029deb3f9f7ca701edbbb85b7090275f": "- \\mathbf{\\hat{n}}", "029e39793c1f7b96788cabcd8b6bf878": "\\varphi_p(x)=\\frac{1-x}{1-px}.", "029e3cfe2abae8da550ed0f34d8e3d4b": "\\sqrt{a^nx^2 + \\frac{a^n - 1}{a - 1}b}", "029e82e73de3aa9bc60f6c5f3f5f69d8": "\\operatorname{E}\\bigl[(X)_r\\bigr] =\\lambda^r.", "029e9fdca2dde780d3f91df94b9e8428": " \\frac{1}{4} |\\langle (\\hat{A} \\hat{B} - \\hat{B} \\hat{A} )x | x \\rangle|^2\\leq \\| \\hat{A} x \\|^2 \\| \\hat{B} x \\|^2.", "029ea052b66d68b4bc63bafdf60f58a9": "w_{i,j} \\,\\sim\\, \\mathrm{Multinomial}( \\phi_{z_{i,j}}) ", "029f0acdea6ba0c6e8d87bba2ca9aeec": " [M]_{C}^{B} = \n\\begin{bmatrix} \\ [b_1]_C & \\cdots & [b_n]_C \\ \\end{bmatrix} \n", "029f1578c56213de6e29eb7279760254": "\\tilde{R}=\\Phi^{2/(d-2)}\\left[ R + \\frac{2d}{d-2}\\frac{\\Box \\Phi}{\\Phi} -\\frac{3(d-1)}{(d-2)}\\left(\\frac{\\nabla\\Phi}{\\Phi}\\right)^2 \\right]", "029f239d5dc25b4312dbcc33e6430b61": "\\sum_i z_i^2 = 1", "029f2a6a6b4614b43ef44e5211ccfeb8": "S = \\operatorname{Spec} A", "029f2b00d7f140a55a20f049cd6819b3": "Z \\sim \\mathrm{Binomial}(2,p) \\,\\! .", "029f47f73eb96e10b6a61db250f1c89d": "\\scriptstyle x \\,-\\, y", "029f58ea2f6c582eff6e3810e56d80e3": "\\exists\\,c>0 \\mbox{ s.t. } \\langle Au-Av , u-v \\rangle\\geq c \\|u-v\\|^2 \\quad \\forall u,v\\in X.", "029f73d4e4e0cb442c83fdf23e0739b3": "\\{b, (o_1,0);(a_1,b_1),\\dots,(a_r,b_r)\\}\\,", "029f7e7ed8abd49bef1f978b22d6d0b7": "\\mathrm{Hol}_p(\\omega) = \\{g \\in G \\mid p \\sim p\\cdot g\\}. \\, ", "029f823c27a63a7007f99583a9699f32": " Z_i= \\left( \\sum_{j=1}^k W_{i,j} \\right) \\pmod {(m_1-1)} ", "029f9d3b498c3d36842b1ef06942b43d": "{}^ap_i = K_a \\cdot H_{ba} \\cdot K_b^{-1} \\cdot {}^bp_i", "029ff1facef6c4a0f4948e8942c3799c": "\\varepsilon_{\\alpha_1 \\dots \\alpha_n} \\,", "02a05bfedf23700420abe2fc04cb2274": "\\bar F(x) = \\sum_{x_i < x}p(x_i) + \\frac 12 p(x)", "02a085c946818133e362fa13db6a2dc9": "r = m - qp_k^{\\sigma_k}", "02a0ef3f90137c9fe63bbd0f51a02934": "\\mathcal{C}_w", "02a12eda9c985f7d14689d0e8727262e": "l = 0,", "02a1a67ae77cb73f7cbcc9268a5cef91": "\\begin{align}\nH_n &= \\int_0^1 \\frac{1 - x^n}{1 - x}\\,dx \\\\\n&=-\\int_1^0\\frac{1-(1-u)^n}{u}\\,du \\\\\n&= \\int_0^1\\frac{1-(1-u)^n}{u}\\,du \\\\\n&= \\int_0^1\\left[\\sum_{k=1}^n(-1)^{k-1}\\binom nk u^{k-1}\\right]\\,du \\\\\n&= \\sum_{k=1}^n (-1)^{k-1}\\binom nk \\int_0^1u^{k-1}\\,du \\\\\n&= \\sum_{k=1}^n(-1)^{k-1}\\frac{1}{k}\\binom nk .\n\\end{align}", "02a25184b4eb25d0888e13e4571c4e73": "v_f = v_i + at\\,\\!", "02a268132f9bea64808285dd7ddfd432": "i_W", "02a27131b649c53f8a3bc5a4220d22b7": "{R_{fu}=-C_R\\rho m_{fu}\\frac{\\varepsilon}{k}}", "02a2740281b8da515d4444d976ed585a": "G^{\\mathrm{A}}(\\omega)", "02a2e0d4b8c6f07c1077ec6933a47ff3": "\\sigma_\\epsilon^2\\,\\!", "02a2f1e9d34b560b2d6578efb66625bb": "\\varepsilon_E \\,", "02a2f9ca16667bfa29f9d5ae7e78bb87": "(x-z_1)", "02a33c6048e63a2364231d9232ecd944": "< \\Psi |", "02a33ecd982937569d6efb469ed94926": "kX \\sim \\beta^{'}(\\alpha,\\beta,p,kq)\\,", "02a36e7fdef57cedde35a70770e6b12d": "\\frac{1}{\\sqrt{0.15625}} = 2.5298221281347", "02a3af029f929b552194e7edc94f90a7": "\\scriptstyle g \\colon X \\to Z^Y ", "02a3ed1c01f6b16819a27864c3db7825": "\n F^{(n)} = F \\circ F \\circ \\cdots \\circ F.\n", "02a496ef738253943cadde022a8c15ce": "\\mathcal{G}(\\omega_n) \\sim 1/|\\omega_n|", "02a4a4d2acef6c919c157f917f61ca0d": "\n\\delta_{\\mathcal{S}_1} = -x \\dfrac{\\partial}{\\partial x}+b\\dfrac{\\partial}{\\partial b},\n\\quad\n\\delta_{\\mathcal{S}_2} = t\\dfrac{\\partial}{\\partial t}-x\\dfrac{\\partial}{\\partial x}-a\\dfrac{\\partial}{\\partial a}\n\\cdot\n", "02a4a53113d4b16223166c1e3fa68968": "h_{i}(x_{k})=h(x_{k})\\,\\!", "02a50e638d6d146745e4e78d25262300": " \\mathrm{d} \\mathcal{A} = \\mathrm{d}^2 \\Sigma \\sqrt{-g} ", "02a52052ff78b9f4ed628a12c5332ab1": "r\\to 0,", "02a565347865103393c99c62bc6a4e30": "\\frac{ dx }{ dt } = y,", "02a62656269e40d2750f95eabc4e5f77": "H = k\\, \\log(1/p),", "02a6394c29f1f96eee407b857c2cc64d": "\n\tE_4 = \\begin{pmatrix}0.792608291163763585\\\\ 0.451923120901599794\\\\ 0.322416398581824992\\\\ 0.252161169688241933\\end{pmatrix}\n", "02a66174fd3b9c026b1e7c5445d8db45": "{\\nu}^{\\lambda}={\\kappa}", "02a66708e288fbc817a0ddf468f38efd": "\\Pr(X_{t_{n+1}} = i_{n+1} | X_{t_0} = i_0 , X_{t_1} = i_1 , \\ldots, X_{t_n} = i_n ) = p_{i_n i_{n+1}}( t_{n+1} - t_n)", "02a6a5c1251f306d41f15f3ef9364d03": "\\prod _x \\cot x \\tan (x+1) = C \\tan x \\,", "02a6d8f854966e30ce61aad85869d6cd": "LC_{50} \\le 3000 \\tfrac{mL}{m^3}", "02a72fcb2a78037d92876aa41ba392c3": " s = j \\omega ", "02a9502832ce7aedad99b585dcb0271d": "u'' \\equiv {{d^2 u} \\over {d z^2}}", "02a953305e87271cb9e0a038acd3fd16": "188461 = 7 \\cdot 13 \\cdot 19 \\cdot 109\\,", "02a97f679d658932e27850dac019213c": "(A\\mid(B\\mid C))\\mid[((D\\mid C)\\mid[(A\\mid D)\\mid(A\\mid D)])\\mid(A\\mid(A\\mid B))]", "02a997eeb22ff6ee536a061e69655fdf": "r + s = t", "02a9d21cec20b2cf4021f7a19c1802b5": "C_1 \\cap C_2, C_1 - C_2 ", "02a9d64ddaff7bb25d1e0c3f412a9f77": "1-P_{m1}=P_a+P_d", "02a9ef7e14fdade7b8f566b5998bc717": "p_K:X \\rightarrow [0, \\infty)", "02aa2c6cb2d9927af06c30ee957aebfa": "sim(q,d)=\\sqrt[p]{\\frac{(1-\\sqrt[p]{(\\frac{(1-w_1)^p+(1-w_2)^p}{2}}))^p+w_3^p}{2}}", "02aa3b183049f14b8b3e5a538dc66ec2": "2^5\\cdot 3^2\\cdot 5\\cdot 7", "02aa9d2075ab9913a0326bc83973377f": "\\hat{e}_{\\nu}", "02ab3cdbc6bd00bdd71ac855c4b01bf6": " S = (\\lambda x.f\\ (x\\ x)) ", "02ab61395ebb9d4dbbb80e6e83da48bd": "A_{\\text{OL}}=\\frac{V_{\\text{out}}}{\\left(V^+-V^-\\right)}", "02aba4f16e4ba0af71aa9b43d3279317": "\\Re(z_n)= -n+1/2", "02ac03c9cdbf7d67a8c49ff8f41b4883": "\\dot{a}>0", "02ad2f550bbcff52fb668d5073b4eebe": "E[F_6].", "02ad9a8cd8bd3d22af44319a20aea411": "SU(N)_L \\times SU(N)_R \\times U(1)_V \\times U(1)_A ~,", "02addf7ddfd0dd5d9906b0daf73bd117": "Rejection Region{{=}} \\frac{{t_{\\alpha/2}}{n-1}}{\\sqrt{n}\\sqrt{n-2+{t_{\\alpha/2}^2}}}\n", "02ae1a416804acd17c7081a0f762dc95": " f:X\\rightarrow Y", "02ae3b92103e815bd84faf7ce6a3bb38": " \\operatorname{sink-tran}[(\\lambda N.B)\\ Y, X] = \\operatorname{sink-test}[(\\lambda N.\\operatorname{sink-tran}[B])\\ \\operatorname{sink-tran}[Y], X] ", "02af5f7fba9c1914973ea7efb35b0492": "\\widehat{R}(\\Delta\\theta,\\hat{\\mathbf{n}})", "02af8eb3eae4a0e30c706a011d84ee57": "W_L = \\frac{L_{QA}I_B^2}{2} = \\frac{\\hbar \\omega_B}{4}. \\ ", "02af94cd045e8619f00bb5bc7f59cae3": "\\tilde{P_n}(x) = P_n(2x-1)", "02b08642ac753e4ada3a05f4844347e0": " \\{ i \\mathbf{e}_{\\rho} \\}", "02b0fd4198f0d70aef64bf4d9bf0494d": "\\sum_{n=1}^N {f(n) \\over g(n)},", "02b1330a8aa507a073ddb972722e0d5a": "n=\\,-1", "02b1609366b6412ab4943bb3f6105417": "S_4 \\times S_2", "02b16947d6295f6f8949e95d0cc21448": "n = \\frac{P}{100}(N+1)", "02b1c9b886a073fe60883860cdefec25": "\\Delta G_{v+\\frac{1}{2}}=G(v+1)-G(v)", "02b1cd66a35997d9835de799aaaf5f86": " \\mathcal{X} ", "02b1ea0129c77d6f313cef123c252948": "\\lim_{V_{m}\\rightarrow0} \\Phi_{S}\\ne0", "02b1f54a98d3481f62b0ce87972d3b66": " (a, b) \\cdot (c, d) = (ac - bd, bc + ad).\\,", "02b219f647d76be16a3327962f87714c": "\\int \\operatorname{Ei}(x) \\, dx = x \\operatorname{Ei}(x) - e^x", "02b22383bb6bbead7f811b992c9a8025": "\\bigvee A", "02b2d0cbe95af83f6677b7aef4714558": "F = F_{\\alpha\\beta}dx^\\alpha\\wedge dx^\\beta", "02b306c557e51c40c3c7a089a87263dd": "\\scriptstyle c=G=1", "02b349a495e3836e460d8df79d664c17": "\\frac{\\pi}{4} = 2 \\arctan\\frac{1}{3} + \\arctan\\frac{1}{7}\\!", "02b37cabaecc0443326393e450028761": "= \\cos(\\phi(t)) + i\\cdot \\sin(\\phi(t)).\\,", "02b39c4bea11d679ef78cad17231b4d8": "a^n", "02b3e5c70d8f5b967b06865e21354dee": " \\scriptstyle E_{\\rm C} - \\mu \\gg kT", "02b40b0b8c70fccbed1c35172fae0ddb": "x_1 = 10^{0.2192318 - 0.2706462} = 0.888353", "02b41d27bd7283f6711f3f642d4eea89": "e = \\sqrt{1+\\frac{2E\\ell^2}{m^3\\gamma^2}}", "02b425e85a51ff18085dd95dbdcd40f7": " r = e^{i\\theta} \\to", "02b43ed6a76fbc6c03fe07e36f93e15b": "\\text{Li}_n(z)=\\sum_{k=1}^{\\infty}\\frac{z^k}{k^n} \\quad \\Rightarrow \\text{Li}_n\\left(e^{i\\theta}\\right)=\\sum_{k=1}^{\\infty}\\frac{\\left(e^{i\\theta}\\right)^k}{k^n}= \\sum_{k=1}^{\\infty}\\frac{e^{ik\\theta}}{k^n}", "02b48d2f289c6fcfcf1390ea7f3c0b78": "g_{n,k}(r)=A\\rho^\\gamma e^{-\\rho/2}\\left(Z\\alpha\\rho L_{n-|k|-1}^{2\\gamma+1}(\\rho)+(\\gamma-k)\\frac{\\gamma\\mu c^2-kE}{\\hbar cC}L_{n-|k|}^{2\\gamma-1}(\\rho)\\right)", "02b4948c18ccacef4be3a4fab3fabefb": "\\angle CAD = \\angle CBD", "02b538208f7a3bfb2142ac071c421c5e": "\\operatorname{Stick}()", "02b54bbc6a2a4c9cc43f338d36eef7e6": "\\omega \\in \\Omega_{Z,[0,t]} ", "02b564d6c4362c7129de39b2869e5277": "\\hat{L} = L(\\hat{x}, \\hat{\\lambda}_x, \\hat{p}, \\hat{\\lambda}_p)", "02b5658332059fc008e0a85535226677": " \\mu_{ij} = \\left\\lbrace\\begin{matrix}\n1 & \\text{if point }m_i\\text{ corresponds to point }s_j\\\\\n0 & \\text{otherwise}\n\\end{matrix}\\right. ", "02b5bb0d9d973a41b97006936e25c039": " \\sum_{n=0}^\\infty \\pi_n x^n = \\prod_{k=1}^p (1-x^{h_k})^{-1} ", "02b5de8a4de2bc034a849e1a42563e30": " \\sum_{n=0}^\\infty z^n,", "02b5ec030f7f9ebd5047150eae4e2b9c": "\\psi: J(E) \\rightarrow E", "02b604b09e79c0129babbb011f6d5661": "\\mu=\\sqrt{2}\\,\\,\\frac{\\Gamma((k+1)/2)}{\\Gamma(k/2)}", "02b62b40df26d691e9ff9341f234e122": "\\ x/y = y/(x/2)", "02b6be5adfc86aa1f46d986bdf1acd2b": "\\delta < f_P", "02b6d397e015e480420f59701c1a1d26": "\\propto \\!\\,", "02b738fcd1ab79e389bb87198fb4b0f0": "xy^2z > x^2z^2 > x^3 > z^2", "02b74a3bdf4e5db66a093867f1ff1eb2": "101011_2", "02b75d49cc982846f5bfcfdcb49bac27": "\\frac{R_{\\text{ac}}}{\\mu L} = a B_{\\text{max}} f + c f + e f^2", "02b78fcd7b8325bc8afd228b00a7e400": "\n\\leqslant \\int_{1}^{\\infty} 2 f(x) \\sqrt{1 + f'(x)^2} \\,\\mathrm{d}x\n", "02b7b8f652b409bddb9defbcd33e9f18": "\\phi^i=-E^{,i} \\, ", "02b7d06a7f926ebbdb471e187c679023": " \\theta\\,\\! ", "02b7f1e422461eb3fd9a2506826d6218": "\\beta_{k} < \\frac{1}{4}", "02b86afb8959f20906caea2f1ee51409": "0\\rightarrow T_xM \\rightarrow T_xP|_M \\rightarrow T_x^\\perp M\\rightarrow 0.", "02b8ac4a93108652a08604e595b2169e": "(a*b)*c = a*(b*c).\\,", "02b8e23f7fb1e05df6f54a70c71e9345": "\\mbox{MS}(a)=\\max_j L_j(a)", "02b8f0fbe6de66899c009ee691ebb11b": "\\mathbb{P} (x \\in X) ", "02b8f697eb1b2083507fcd85e15dc5ca": "f \\in BMO", "02b8ff40bc81a62770626767a7432b0c": "H_{eff}", "02b90b97ee9ab3050bc8933c14dc031c": "\\overline{x} = \\left ( \\alpha_{ij} \\right )^{-1} \\begin{bmatrix} 1 \\\\ 1 \\\\ 1 \\\\ 1 \\end{bmatrix} = \\begin{bmatrix} 0.3013 \\\\ 0.4586 \\\\ 0.1307 \\\\ 0.3557 \\end{bmatrix}.", "02b91f833a4fd18295bd710d3cc01ef8": "\nC^{S_2}_{E_1} = \\varepsilon^{2}_1 / D\n", "02b92d672582f9ece902c8fa66467f62": " ELA\\,\\!", "02b983191ce736331e1184228497076c": "D\\in\\mathcal{D}", "02b98c25514e6be1b3c6a42e6d794aa5": "7x^2y^3 + 4x - 9", "02b99714e6fd241374ae2ac483902911": "1 \\times \\sqrt{7}", "02b9ce5a4ce10b6965558f07c7c900f1": "n_A+n_B=N", "02b9e2009c8050e8d9a6804348fb8695": "(\\mathbf{J}_1, \\mathbf{E}_1)", "02b9ea85f5337414776557bcd19e37d7": "\\cfrac{G+C}{A+T+G+C}\\times\\ 100", "02b9f50cf32fe169808f11b48f682756": "FG = 1", "02ba02bd287c46cfd61b1056c02e4b1c": "\nH(I|J) = -\\sum_{i,~j} P_{I,J}(i,~j) \\log P_{I|J}(i|j) .\n", "02ba0742fe67f11eff16e880b77226b0": "L_0\\,\\!", "02ba5c2d321b212c8e5f1ea179e1785a": " Q_A = \\mathcal{M}.Q", "02ba8cffb6c1b287f7b2e0b801c7e8cc": "g^*(\\tau) = \\left(\\frac{i}{2}\\right)^{k - 1} \\int_{-\\overline\\tau}^{i\\infty} (z+\\tau)^{-k}\\overline{g(-\\overline z)}\\,dz= \\sum_nn^{k - 1}\\overline {b_n}\\beta_k(4ny)q^{-n + 1}", "02ba92ad3d37d5e3f06a828837a17d7a": "(Fa \\or Fb) \\leftrightarrow Fd", "02bb57797ba7e0a76063c2edd9191fbb": " Z=\\frac {a_j}{\\lVert a_j \\rVert} ", "02bc005b15bc011bf2cae4b1c1a79c12": "\n\\begin{pmatrix} y_1 \\\\ y_2 \\end{pmatrix} = \\frac{1}{x_3} \\begin{pmatrix} x_1 \\\\ x_2 \\end{pmatrix} ", "02bc089f3e9a73d15e8b0f8bc64051d7": "\\hat{x} \\in W", "02bc2631e660c2c59fe6f2f762de0e3a": "(1-\\frac{1}{e})", "02bc9a95236d772fe37193c3aa2c41a4": "\\ G, \\ ", "02bd1381b2387efff7922351b7ec5d8e": "\\epsilon_0 ", "02bd3005e4504960ad57347b2cd4a62e": "n > 1/d", "02be125bcc84d55428d554f276f996bb": " m(\\mathbf{f},\\mathbf{g})=\\prod_{i=1}^N (a_i - b_i +1)", "02befb6993a658e5a0ab7db18c9ddd3d": "\\scriptstyle{\\sum_{j =1}^m P_{ij} = 1}", "02bf4aa45a5414e118fc5b8836daabcf": "\\leq \\Pr[B]+\\Pr[A|B^c]", "02bf538b901eaf61cfe4a0d0b78e22a1": "\\max_{j \\neq i} b_j", "02bf98e606c3288280c9519791bc1a48": "m_0,m_3,m_5,m_6", "02c0176317f7c9c389ed000666afd07f": "A + C \\leftrightharpoons AC; K_{AC}=\\frac{[AC]}{[A][C]}", "02c0b388ae9b0a8452bd8b53c3e25707": "3(4x^2y-6y)+7x^2y-3y^2+2(8y-4y^2-4x^2y)\\,\\!", "02c193d45dbd50de7b409c5454f045d6": "\\alpha = 2 - \\frac{\\tau - 1}{\\sigma}\\,\\!", "02c1b2634f3db0d4515eee02d79b0537": "\n\\begin{align}\nE_{2n} &=(-1)^n (2n)!~ \\begin{vmatrix} \\frac{1}{2!}& 1 &~& ~&~\\\\\n \\frac{1}{4!}& \\frac{1}{2!} & 1 &~&~\\\\\n \\vdots & ~ & \\ddots~~ &\\ddots~~ & ~\\\\\n \\frac{1}{(2n-2)!}& \\frac{1}{(2n-4)!}& ~&\\frac{1}{2!} & 1\\\\\n \\frac{1}{(2n)!}&\\frac{1}{(2n-2)!}& \\cdots & \\frac{1}{4!} & \\frac{1}{2!}\\end{vmatrix}.\n\n\\end{align}\n ", "02c200d95543444ec2205ef66b757136": "\\kappa=\\frac{\\omega_r}Q", "02c2022327405565b8cca1b582733df7": "a = d \\ne b = c, \\alpha = \\zeta = 90 ^\\circ, \\beta = \\epsilon \\ne 90 ^\\circ, \\gamma \\ne 90 ^\\circ, \\delta = 180 ^\\circ - \\gamma ", "02c26e1d2ae0c94c2eb478016ccc5442": "\\, \\delta ", "02c2916b1b5886b896d8a537fe8db434": "a = \\sum_{i=0}^{n}d_{i}(-r)^{i}", "02c29cddfbb95e518fee0d87144c595c": "\\frac {\\dot{m}\\sqrt{T_{01}}}{P_{01}}\\,", "02c2a70eab25d2c4784c023a6a316659": "(\\cos\\theta,\\sin\\theta)", "02c2f8de39e5973c727a6d4858107564": "P:=\\{p_{\\vert X} \\mid p \\in P_n \\}", "02c31bde2dae72763bb7030a6836164f": "\\nabla_r", "02c38550b3c579b5cada441aa00fea85": "\n\\frac{E(u+\\tau\\psi) - E(u)}{\\tau} = \\frac{1}{\\tau} \\left( \\int_\\Omega F(u+\\tau\\psi)dx - \\int_\\Omega F(u)dx \\right)\n", "02c40ae85808d86bd7fdd50f8c36d48a": "\n\\gamma(\\mathbf{v}) = \\frac{1}{\\sqrt{1-\\frac{|\\mathbf{v}|^2}{c^2}}}\n", "02c42262fcf5769ee18cf00a44a604ad": " Lower~limit = e^{Log_e (lower~limit)} = e^{1.49} = 4.4", "02c4ceb96e7cd644ace41c8b9f652803": "b^{-(p-1)}/2", "02c52fa215bf128cd71a773caa85464d": "\\Delta u = K^\\prime e^{2u} + K(x).", "02c559a0df7dd3616a610d9033abe4e2": "\\pi_k(O)=\\pi_{k+4}(\\operatorname{Sp}) \\,\\!", "02c58813370ab922c27e5673ce949850": "K=\\frac{e B \\lambda_u}{2 \\pi \\beta m_e c}", "02c59aa8adbc1e1e4182bb76e89b602f": "x\\not =y \\in X", "02c6703935ac8fc407610edf815fa156": "MA = \\frac{F_B}{F_A} = \\frac{V_A}{V_B} = 2.\\!", "02c67906b26d7fe40fdb90adbba3c0cf": "\\scriptstyle \\partial S/\\partial t ", "02c6f0c00b0d1b69cbb4174a5984a5e3": " q=(s,t_e) \\in Q ", "02c6f235d7c1fe631555420d92b2ef2b": "\\pi\\left(10^{10}\\right)", "02c70beac20542de6b37489aaa4d2d45": "\\mathbf{L}=\\int_V dV \\mathbf{r}\\times \\rho(\\mathbf{r}) \\mathbf{v}", "02c73fe4efabc7a908c3768c18d8ffe3": "GI", "02c784ac595f0d4f08e6b274dd7ae877": "1-R-\\varepsilon", "02c78f8d711f84178689990c53db0388": "\\gamma = \\lim_{a\\to1}\\left[ \\zeta(a) - \\frac{1}{a-1}\\right]", "02c791e85339843965044c4e2ed5b0ad": " Q^T Q = I . \\,\\!", "02c7bc501ed2be649a02a9a9fd0a87e8": "(a+b)+c = a+(b+c), (ab)c = a(bc)", "02c7d5176190cc37c48af6a7f2e008b2": "\\langle f,h_k \\rangle=\\frac{a_0}{2}\\langle h_0,h_k \\rangle + \\sum_{n=1}^\\infty \\, [a_n \\langle h_n,h_k\\rangle + b_n \\langle\\ g_n,h_k \\rangle],", "02c7dcf360a6635c00b29a984e96a1b9": "\n\\mathcal{L} =\n\\frac{1}{2} \\left| \\frac{\\partial\\mathbf{n}}{\\partial t} \\right|^2\n- W(\\mathbf{n},\\nabla\\mathbf{n})\n- \\frac{\\lambda}{2} (1-|\\mathbf{n}|^2),\n", "02c810017ce31a2012fef5f3b1634458": "\\alpha = {R \\over 2L} ", "02c886dfae77c4e687a37e9179e15ed2": "\\,\\eta(s)=\\Phi (-1,s,1).", "02c8b72adc1675abd4cf2dc5cf31a7c0": "(GX,\\varepsilon_X)", "02c925889c2b342fcf5ce61a5a4dacff": "t_1^\\prime", "02c965bb60433d59c3a8f2bef9b19469": "\\phi_\\mu", "02c98a141d6feacc270a43be76d6d897": " P_{\\rm wind} = \\begin{matrix} \\frac12 \\end{matrix} \\cdot \\rho \\cdot S \\cdot v_1^3 ", "02c9b6540d7509e2f58bef72d1dc7ec2": " \\prod_{j=1}^n \\left ( \\alpha - \\alpha_j \\right ) = 0 \\,\\!", "02c9d63fab98237220ce40033f33ef78": "\\nabla \\cdot \\vec v", "02c9f55b2e5f5569dea8fdc7eaec7d10": "PK_R", "02ca10636c28dba407e94fc93215d004": "b_M", "02ca156e9359d57db6f6e32cd090d0fb": " a(z)= \\sqrt{\\frac{1}{\\sigma_x^2} z^2 + \\frac{1}{\\sigma_y^2}} ", "02ca38586ab165b0d09038c1e064c730": " \\hat{C}(\\mathbf{k})=\\frac{\\hat{H}(\\mathbf{k})}{1 +\\rho \\hat{H}(\\mathbf{k})} \\,\\,\\,\\,\\,\\,\\, \\hat{H}(\\mathbf{k})=\\frac{\\hat{C}(\\mathbf{k})}{1 -\\rho \\hat{C}(\\mathbf{k})}. \\, ", "02ca5b777b17efadd0adcd3bfbc0f8e9": "P(B)=0", "02ca7e35cb7137050d8d0a7d18caae3f": "\\mathbf{P}(n,r)", "02cac3592c352a9824607d3b18002406": "r^\\ell \\, Y_\\ell^m", "02cb092dd6953f1bc7c0f25ec5d100db": "p(\\theta)", "02cb939e6fb166fb503428c09a364309": "-n_2 + n_3 = 1 \\ ", "02cb9858cb1c4712568d65b854bea41b": "\n\\begin{pmatrix}\nA_1&B_1\\\\\nA_2 & B_2\n\\end{pmatrix}\n\\begin{pmatrix}\nx\\\\y\n\\end{pmatrix} = \n\\begin{pmatrix}\nC_1\\\\\nC_2\n\\end{pmatrix}.", "02cc26a3340d7fe6e3bff902f3ee1e80": " {\\cos \\gamma = \\sin \\theta_{s}\\sin \\theta \\cos \\psi + \\cos \\theta_{s}\\cos \\theta} ", "02cc7184ed2936ba6c062cd2a905f05e": "\\alpha_1, \\ldots, \\alpha_d \\in \\mathbb{R}", "02cc83f99a9b20510ca44489ebd7a36f": "+ \\frac{200}{510,260} log_2\\left(\\frac{200/510,260}{260/510,260 * 500,200/510,260}\\right)", "02cd27c1810fb9f2696e464c60bc37f8": "\\frac{\\mbox{d}}{\\mbox{d} x} ( \\alpha \\cdot f(x) ) + \\frac{\\mbox{d}}{\\mbox{d} x} (\\beta \\cdot g(x))", "02cd346db0f5d0816cdfba9e1655a21a": "P\\cap -P", "02cd503acfc44eeab79685da98bee009": "0 \\le L(M) \\le 2^{64}", "02cdd4dc0f0d9c79c76a14d95b165c76": "\\pi(x)\\sim\\frac{x}{\\ln x}.\\!", "02cdd72d2f898fb9c7e0710bab8fe5d7": "c_{ijk\\ell} = c_{jik\\ell}\\,", "02ce03b273daac91982b3767415710c1": "(n\\mapsto n\\cdot\\log n) \\in O(n\\mapsto n^2)", "02ce325d3513bcf9b951a90e86772e48": "O(n\\text{ }\\log\\text{ }n)", "02ce4923458cb5d5911064299ae41ebd": " C_p = C_p ( \\alpha , M , Re , P) ", "02ce720ac80fe82c41335fbc936122c8": "\\begin{smallmatrix} V = \\sqrt{{V_r}^2 + {V_t}^2} = \\sqrt{11.4^2 + 16.9^2} = 20.4\\, \\end{smallmatrix}", "02ce844bf2a415f3aa1ad996f28825b2": "C=I", "02cea61bce63ab878ef920e395c399a2": " \\mathbf{\\Pi} = \\{\\pi_x, \\ x \\in \\mathcal{E} \\} ", "02cf28ff66964d3f1795f42f84b0291c": "I = I_0\\exp\\left({-\\int\\mu(x,y)\\,ds}\\right)", "02cf86dc7d65bc7133c890866a1bcb66": "f^*L_1 \\cdots f^* L_m \\cdot F = L_1 \\cdots L_m \\cdot f_* F", "02cfad08a7a516a3762139bbcaf6f27c": " r < R = \\frac{1}{\\limsup\\nolimits_{\\ell\\to\\infty} |f_\\ell^m|^{\\frac{1}{\\ell}}}.", "02cff7b0593b80135cd140f4a21e88f8": "P(i) = \\sum_{n=st_i}^{n=fin_i} |S(n)|^2, i = 1, \\dots, 5 ", "02cffdcc45ccae3138c3796cd80198e3": "c^2 {\\gamma^2} - v^2 {\\gamma^2} = c^2", "02d02364ec17f438ee1f0c55e2f0c0c0": "\\textstyle x,y\\in\\mathbb{R}", "02d0505c45ef5338e56863309cb93ddb": "\\ \\displaystyle \\mathcal{Q}\\times \\mathfrak{U} ", "02d080685ff47967c7316e79e7ea9945": "\\nabla \\times \\mathbf{D} = \\varepsilon_{0} \\nabla \\times \\mathbf{E} + \\nabla \\times \\mathbf{P}", "02d082a3eecb5f89fde6a8625ea7efd1": "H\\left(A,C\\right) =\\left\\{00,01,1\\right\\}", "02d08f4ef10416a36b5f5a548095c02a": "\\approx 0.4 m_s R^2", "02d0b1e19d0037c74a070547faf3faa7": "x1\\;", "02d44c58ff40c2ddc94695ab9955e25a": "g:\\textit{George}", "02d48ef1af7c38a5a790aa75c2a922c9": "X_{5}", "02d4b74c78ad4e5700a887e322543640": "F_{A \\rarr A} = 0", "02d4d3b0044ee3603acdcd99de9dcb05": "X = A^{-1}(I-UY)", "02d4e6ea476acf8a700897f537d8731f": "\\frac{\\pi}{2\\sqrt{2}} \\approx 1.11072073", "02d4f4692cd34a7d069bf5afb42fc10b": "\nf[x_0,\\dots,x_n] = \\sum_{j=0}^{n} \\frac{f(x_j)}{q'(x_j)}.\n", "02d53af1d934122f24861e250bb37ad4": " \\max \\{ p,q \\} \\leq p +_\\mathcal{O} q ", "02d58831ed513c0dda9de14234c8d360": "K^M_2 (K) / 2", "02d5abb1b8b92bb55972cc21e78905c1": "u_i = u_{i-1}^2 - 2", "02d5db3bd7b1a41cb5312b6f011488b1": "a_1\\chi_1+a_2\\chi_2 + \\ldots + a_n \\chi_n = 0 ", "02d5ddbbef84b35fb25ef13b66f1bb41": "\\tau_{tt} \\propto N^2", "02d61f902f91f49bb386782b23cbd9bb": "\n\\begin{align}\n |V| e^{j(\\omega t + \\phi_V)} &= |I| e^{j(\\omega t + \\phi_I)} |Z| e^{j\\theta} \\\\\n &= |I| |Z| e^{j(\\omega t + \\phi_I + \\theta)}\n\\end{align}\n", "02d629ac392c3328e14f57fd55b883ff": "H_t - H_{t-1} \\in -K_t \\; P-a.s.", "02d649ffafb5cf15d2638a8d7f8d8551": "q(x)=x^n+b_1x^{n-1}+\\cdots+b_{n-1}x+b_n, \\,", "02d687535053ef9d1ece75b487e31704": "dS = \\left(\\frac{\\partial S}{\\partial E}\\right)_{x}dE+\\left(\\frac{\\partial S}{\\partial x}\\right)_{E}dx = \\frac{dE}{T} + \\frac{X}{T} dx=\\frac{\\delta Q}{T}\\,", "02d6ede6592ceeed003d45034a9dbaf6": "U = a", "02d79611778e638b14936dac9ed4b7d3": "\\gamma = \\alpha+1/2", "02d7c67a0283ebb7d5120a7a349b23d6": "\\frac{H^2}{P^2}=\\frac{2P^2}{P^2}\\, ", "02d84ce998b810b0415fdee2bb7b6b3d": "|C_{\\alpha\\beta}|^2", "02d8e21f2415d7dcdd6b48a2400de0c5": "Ba/Bb= (Pa/Pb)*(Db/Da)square ", "02d91a6161155b99ceb528d3ae00da42": "\\mathrm{DFL} = \\frac{\\mathrm{EBIT}}{\\mathrm{EBIT\\;-\\;Total\\;Interest\\;Expense}}", "02d958ce1d9f78e47423cea80b715c63": "D_k(c) = D_k(E_k(m)) = m\\!", "02d9c758323964d8c3ec18509487a742": "\\,H_s(s=1,\\ldots,S).\\; n_s", "02d9cfbc6b68bba5b015aecfb111b14d": "\\lambda_B \\approx", "02d9d27bf1b8744745ca52cd27d83f8c": "MB(A)", "02da2a736a512660a6018cc00f4993e1": "x_i \\in S_i", "02da3fe99c93e897396595edccdbd632": "Q(e,y)", "02da6b3fb681e3d0a62145c5bc85d032": "(\\log^2 N)/N", "02da8261a5f9e984bc30751c1449a475": "= \\left[\\sum_{i=1}^{N} 2 \\dfrac{x_{i2}-x_{i1}}{2} \\dfrac{x_{i2}-x_{i1}}{2} ' \\right]^{-1} \\left[\\sum_{i=1}^{N} 2 \\dfrac{x_{i2}-x_{i1}}{2} \\dfrac{y_{i2}-y_{i1}}{2} \\right]", "02db2af0d5dd6a4fab25a12e871c8af1": "\\pi_k(\\mathbb{S})", "02db6c5b194fd0a2661890405cf6b1a1": "\\mathbf{R}^+ \\to \\mathbf{R}^+ : x \\mapsto \\sqrt{x}", "02dbbfc4d4d0a67026392826f85fda83": "0 \\leq b \\leq a ", "02dbc2a63dde34b74a8f54f7d0d15603": "\\frac{d\\left(ky\\right)}{dx} = k \\frac{dy}{dx}.", "02dbc6539d3667d420c5fefe0ee0a0f4": "d_k \\isin \\{ -1, 0, 1 \\}", "02dc2a6a3c90eff032f723994a58a5b7": "U(n) = O(2n) \\cap GL(n,\\mathbf{C}) \\cap Sp(2n, \\mathbf{R}).", "02dc55860ca76ed9063448b5ddf5e65a": "\\hat{w},x_1^n(w),y_1^n", "02dc609c4d6fb1101f650b1dbd33c25a": "s=\\sqrt{\\tfrac{m-r_k^2}d}", "02dc86b381521b74d7f7e2ba46110545": "CO = \\frac{VO_2}{C_a - C_v}", "02dc987d3612a3352f010b927d0e0c6c": "S=\\phi^{-1}(\\phi(S))", "02dcc5a30425c86ce04d7f8d70af95d0": "\\textstyle \\overline{a}_{.k}", "02dd00bf549832493910b3af11394659": "\n\\frac{\\partial F^m_{~\\alpha}}{\\partial X^\\beta} = F^m_{~\\mu}\\,_{(X)}\\Gamma^\\mu_{\\alpha\\beta} \\qquad; ~~\n F^i_{~\\alpha} := \\frac{\\partial x^i}{\\partial X^\\alpha}\n", "02dd3f4c7c9b4b389221cd052d2591ad": "y'=f(x,y)", "02dd4a6693bb322acb9a40b57af2174c": "\\ (U,\\ E,\\ N)", "02de23ac02b08d0fd8faba9fc285105c": " z_{xx}>z_{xy}>z_x>z_{yy}>z_y>z", "02de341fdf0e3e72724ec2f1ad15cd77": "\\alpha=\\|g\\|_q^q", "02dea128ced10af253763324a0583252": "\\mathrm{Ran}(A - \\lambda I) \\cap \\mathrm{Ker}(A - \\lambda I) = \\{0\\},", "02dea2dbd4c59b6ded1f65bd0482d579": "\n\\begin{pmatrix}\n{A'}^0 \\\\ {A'}^1 \\\\ {A'}^2 \\\\ {A'}^3\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n1 & 0 & 0 & 0 \\\\\n0 & \\cos\\theta &-\\sin\\theta & 0 \\\\\n0 & \\sin\\theta & \\cos\\theta & 0 \\\\\n0 & 0 & 0 & 1 \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\nA^0 \\\\ A^1 \\\\ A^2 \\\\ A^3\n\\end{pmatrix}\\ .\n", "02df1d67c4f118aa427e34f09b750537": "x_i \\in \\mathbb{C}^m", "02df319e57e85d778ff5b31c78d39e35": "A\\to\\bot", "02df681620026e67b28f02df6909fef1": "\n Pr[C_i=C] \\geq \\left(\\frac{n-2}{n}\\right)\\left(\\frac{n-3}{n-1}\\right)\\left(\\frac{n-4}{n-2}\\right)\\ldots\\left(\\frac{3}{5}\\right)\\left(\\frac{2}{4}\\right)\\left(\\frac{1}{3}\\right).\n", "02df87d44cd6118584fa0d6a0cbe5157": "\\tan\\left(\\frac{\\Phi}{2}\\right)=-\\frac{1+\\sqrt{R}}{1-\\sqrt{R}}\\tan\\left(\\frac{\\delta}{2}\\right) ", "02dfac85105c1461af22205de6bb9430": "F(y,z)\n\n=\\int_{-\\infty}^\\infty f(\\rho,z)\\,dx\n\n=2\\int_y^\\infty \\frac{f(\\rho,z)\\rho\\,d\\rho}{\\sqrt{\\rho^2-y^2}}\n", "02e042d61b93ff101d6b317e2f36930a": " m < n \\to m - n = 0 ", "02e07bca56d1528d156c2808df1d9484": " f(x) = \\sqrt{x}", "02e089b4d472400298ab379e4d000d99": "F = f_1(0), f_2(0), f_1(1), f_2(1), \\dots,", "02e0a100969db76ab338f592fc3d211a": "[w^r]", "02e0faaa7da52b4aabdb0478360c8cd6": "\\frac 1 {b+c} \\ge \\frac 1 {a+c} \\ge \\frac 1 {a+b} ", "02e1313d8de17516c259fd87e4e4cfe6": "v = \\frac{\\lambda}{\\tau} = \\lambda f.", "02e205b6201bc8f4dfbf3b389b9f8dc0": "W_n", "02e21374c1c420aecdfcf29d81c83b3a": "\\sigma_k =\\sqrt{\\lambda}", "02e235deac445d96c2c095d0b1fddc89": "b_m = \\frac{1}{\\pi} \\int_0^{2 \\pi} f(t) \\sin (mt) \\, dt.", "02e25eaa2b39b968aac77563a6a01eda": "\\dot{r}=\\frac{G}{2}\\dot{v}", "02e294d9ded63b8b6a38cf6592d751af": "z^*\\!", "02e2a497116037a708e6b90196648df6": "\\langle F,\\le,V\\rangle", "02e2dfa1293bc1f3b376c5b3d6bc3222": "R(n) = \\frac{r(1)+r(2)+\\cdots+r(n)}{n}. ", "02e359f86df7c9657f8a41f64540e619": "-\\sum_{k=1}^\\infty \\frac{(-x)^k}{(k-1)! \\zeta(2k)}= O\\left(x^{\\frac{1}{4}+\\epsilon}\\right)", "02e3702daaabd058e3a0853e3b051ab6": "\\left (\\partial_t-k\\partial_x^2 \\right )(\\Phi*g)=\\left [\\left (\\partial_t-k\\partial_x^2 \\right )\\Phi \\right ]*g=0.", "02e37bc832b7b258de2f53958c72c84a": "\\color{Black}\\tfrac{n}{2}", "02e388f3ae24f35e4d176cd90608f435": "m_1a=m_1g-T", "02e44e162cc167619a1b9d55a3c57371": " E^{c} ", "02e44f5d8ad6859397e15c9bf505579c": "\\ J = \\frac {500}{1 + \\lambda_i^2}", "02e46e865b0432d517ca68327c3c991b": "\\pi:\\Sigma^*\\to P(A)", "02e4bc881738860c665a51428033be6b": "n_1 , n_2 , n_3,\\ldots ", "02e53a9984840ee52f4ac5b2af181b68": "B^4 = \\{\\mathbf{x}\\in \\mathbb{R}^4 \\mid |\\mathbf{x}|\\leq 1 \\}.", "02e53d2f2041f13a171dee171f905e10": " \\mathit D ", "02e542b155f6cdbecefba13ccf9781c5": " Spin(10,\\mathbb C)", "02e63cd33d8b7b7324750b90cd29360d": "\\sup_{y^* \\in Y^*} -F^*(0,y^*) \\le \\inf_{x \\in X} F(x,0).", "02e658512da0b87b1d9a3154a3ce1de5": "\nr^2 = x^2+y^2", "02e65f4b856e7e66ea3a08999d08fc45": "\\left(\\begin{array}{ccccc} 0\\\\ & 0\\\\ & & \\ddots\\\\ & & & 0\\\\ & & & & 0\\end{array}\\right)", "02e6751b8690ba6ce14a21e47c7b38ca": "\\Delta ^{\\prime }(x)=-\\frac{g(x\\mid x)}{G(x\\mid x)}\\Delta (x)+\\left(W_{2}^{A}(x,x)-W_{2}^{B}(x,x)\\right)", "02e67e02dd0c6f5f1f4e4d979d35bd60": " ^{14}\\text{N}_2\\text{O} \\rightarrow {^{14}}\\text{N}_2, ", "02e6b069dff4f9036ddd12884e55cfa9": "x=0^0", "02e707f46baf83999b1571559404afff": "A_{k}=\\left[\\sum_{i=1}^{N}\\frac{\\partial \\ln Q_i}{\\partial \\beta}(\\beta_{k})\\frac{\\partial \\ln Q_i}{\\partial \\beta}(\\beta_{k})'\\right]^{-1} .", "02e746e1beef65a29426f5d7934c6857": "\\hat{J_z}", "02e747c355fafa7779c45a9f4cabd96f": "\\, T\\!", "02e74f10e0327ad868d138f2b4fdd6f0": "27", "02e76f246d1f9f937fba84f0b5cbaeef": "C \\ge 0", "02e78bee48c90319ceeb0e0576818254": "\\vec{p}=-g\\vec{s}/\\|\\vec{s}\\|\\,\\!", "02e7ba1333997e4e96b6813efa861eb7": "w \\in \\mathbb{\\hat{C}}\\setminus \\overline{\\mathbb{D}}", "02e7e1c29921ca9b44aa0fe0a0c65531": " r_{1}m_{1} = r_{2}m_{2} ", "02e7fde8845e3e7327b77552d992fb00": "\nV \\setminus \\{v\\}", "02e82062942814d6b745ec77ac54faee": "3 \\Rightarrow 2", "02e8631f6a646c77f58ba39cc8219ab7": "J(\\lambda)\\,", "02e88362e58867d505ddad8f3fa4a5db": "\\,e^{+{1\\over 2}i n\\theta}q {e^{-{1\\over 2}i n\\theta}}^{*}", "02e894f585c564d4db6b4f0e06efb3e7": " {\\rm coNP} = \\forall^{\\rm P} {\\rm P} ", "02e8b238d09747bf14aa6e5732af9715": "i_i(U_g)", "02e8bb51124bd7112ebf6627110845ae": "\\frac{nm}{db}", "02e8e818fae5800a439605693f5f6cb5": "\\,p(x)=q(x)=1.", "02e93736ba5f8b4b5688942188265b26": "f(S)=\\max_{i}\\left(\\sum_{x\\in S}a_i(x)\\right)", "02e953a474c2a9818c3d75ae320728fe": " S_{x,i} ", "02e96c03176f660264850cdf7abea153": "\\alpha(k+1)/(2k)", "02e9b302bf1c76b0dfb128e4b0077c7f": "t = \\frac{-1\\pm\\sqrt{1+8(D+Y)}}{2} ", "02e9e0fce35e84f2a148a722f090844f": "L_{ij}\\,\\!", "02ea4194274544c7169a06a68462022d": " e^{-\\alpha t} \\cdot u(t) ", "02ea9721d236a98ee6036f38defef4f2": "\\partial_t s(t,a) + \\partial_a s(t,a) = -\\mu(a) s(a,t) - s(a,t)\\int_{0}^{a_M}{k(a,a_1;t)i(a_1,t)da_1} ", "02eaa3e2db7bb9679cdb0e1e964349af": "y(t) = \\sum_{k=1}^{K} r_k \\cos\\left(2 \\pi k f_0 t + \\phi_k \\right)", "02eaeaa73a579c8c677e8d57b030f7ca": "\\mathbf{F} = m \\mathbf{g}", "02eb0e03ef4fec950b287d600730da63": "R[f^{-1}]", "02eb35dfa3fb4b87ee32d8bae6104661": "\\frac{\\partial^2}{\\partial x_k \\, \\partial x_j} H = \\frac{\\partial^2}{\\partial x_j \\, \\partial x_k} H. ", "02eb3a06f2243826cca0ee012ce70dcf": "\\log^+|f(z)|", "02eb4539112d1f9338820c579a0947ab": " A = \\begin{bmatrix} 1 & 3 \\\\ 4 & 2 \\end{bmatrix}.", "02eb5f614ec7faa2aa8dbe0645e84f11": "B\\triangleleft A", "02eba0344dcb33f69cad205e517233a3": "(B\\to F)\\to F,C\\to F\\vdash(B\\to C)\\to F", "02ec3a241b9dc06b768d2ee479064fcf": "F(t)\n= 1 - \\left(\\prod_{j=1}^a \\lambda_j^{r_j} \\right)\n\\sum_{k=1}^a \\sum_{l=1}^{r_k}\n\\frac{\\Psi_{k,l}(-\\lambda_k) t^{r_k-l} \\exp(-\\lambda_k t)}\n{(r_k-l)!(l-1)!} ,\n", "02ec45c41647c927d5b9699f8e5ee4c3": "0\\to \\mathcal O_{\\mathbb P(V)} \\to \\mathcal O_{\\mathbb P (V)}(1)\\otimes V \\to \\mathcal T_{\\mathbb P (V)} \\to 0 ", "02ec7e2c7d521f73b244175365ebd463": "(12, 35, 37).", "02ecb38af12588bb04b2d6c48e04ed13": "v(t_2)-v(t_1) = \\int_{t_1}^{t_2}{a}\\, dt. ", "02ed06b11041dfa48ccdd36142cf9600": "\\mu : \\mathit{V}_o \\to \\mathit{W}", "02ed0ce07f470a60c3f421e7fb46312d": "\\lnot (G_1 \\lor G_2) \\to (\\lnot G_1) \\land (\\lnot G_2)", "02ed321b1ae36a826de55612083e3d9f": "a_{2,j}={1\\over14}(2y_{j-2} - y_{j-1} -2y_j -y_{j+1} +2y_{j+2})", "02ed9140696aa6bfbec96ccea0c47884": "\\left\\langle M_C\\right\\rangle \\sim l^{d_f}", "02ed986c1ddd4da3b052e8d837651fbb": "J^1_\\Sigma Y", "02edb5f97703fcbcdeff31cf06a5602d": "_{q'p=qp'\\,}\\!", "02edda68d05afa6c789386adba55832f": "C(u_1,\\dots,u_{i-1},0,u_{i+1},\\dots,u_d)=0 ", "02ede70b85ad095f719554400c052a57": "\\prod A_i", "02ee09f9465ada081867c32b7b1b6a4c": "r = \\sin (6 \\varphi) + 2", "02ee209363833e57ee463c2179b6affa": " \\sigma \\, ", "02ee2d78b2edbc04150e6284b67f2099": "S_l\\;", "02eed43b80646dc09ac23605719df1fb": "\\Delta k_{j}^2", "02eee65279bf8969802e50572b86fa23": "1/\\tau = c/L", "02ef45b6a6516432184cea67db307698": " \\lambda_\\mathrm{De} \\cong \\sqrt{ \\frac{ \\varepsilon_0 T_e }{ q n_0 } }.", "02ef8162dc8ba23187a9c2dcebbe8c36": "\\frac{\\pi}{d-1}", "02ef8cfa689777040dd2fd110a88ccc8": "\\pi(z) = \\frac{1}{\\Pi(z)},", "02efc196a68c60bce5943d30386ac376": "\\cdots \\to H^q_{\\mathrm c}(U) \\overset{j_*}{\\longrightarrow} H^q_{\\mathrm c}(X) \\overset{i^*}{\\longrightarrow} H^q_{\\mathrm c}(Z) \\overset{\\delta}{\\longrightarrow} H^{q+1}_{\\mathrm c}(U) \\to \\cdots ", "02efc65c60ed9284d8784b145d836446": "-\\sqrt{\\frac{6}{35}}\\!\\,", "02efcf02b0e9ffa17460ad97a04e1212": "-0.0497", "02efe698dcd85a84e109a8aa7b6f0257": "\nI_{e} = S J_{e} = S n_{e} e \\sqrt{kT_{e}/2 \\pi m_{e}}\n", "02eff460cc45158b329c8a3037fa0aa1": "\nU_{k}(\\beta) - U_{k - 2}(\\beta) = T_k (\\beta). \n\\,\\!", "02f0006b87e5158d1abfc2f7def4fddd": "\\frac{1}{X}\\sum_{i=1}^{X}\\frac{1}{2^{i-1}}=\\text{D}", "02f0127195703a7ac61a91b73419a93f": "A\\mu=\\{ a_i \\zeta_n^j\\;:\\; 1 \\le i \\le m, \\;\\;\\;0 \\le j \\le n-1\\},", "02f02bbbc58e793e57462bb70b1c4462": "P = (x_0,\\ldots,x_n) \\,\\!", "02f03834cea656c9d3eaa15323e5c21f": "z = 1.5", "02f06e1f429788c0928e3fb3a334e3a0": "\\ E = E_\\text{covalent} + E_\\text{noncovalent} \\, ", "02f07196b3f934fa452e95387eddedf4": " P_1", "02f07d2a70e478f7deabb6ec6a8193c5": "\\begin{matrix}4&4&5\\\\6&7\\end{matrix}", "02f0a792fc10f3544da8261e98039cff": "F (q,p)", "02f15f2928387ff24b84aff0f8cc118d": " v = 614.58 g c \\cdot \\frac{p + b}{b i r n}", "02f17525b5aa1c6bfab3c606434ebb2b": " E_{\\textrm i}=E_{\\textrm r} ", "02f17f9e25eac18a4b65a5f79fd128ed": "G=\\mathbf{A}\\cup\\mathbf{B}", "02f19d2e9bd994b96bfae9aac886f424": "c_\\kappa 0=0", "02f24172940929c3c895d6b5a3f88b24": "\\frac{k}{i} i.", "02f25a51be4aaf6eb72064cd2f168aab": "\\phi_T(x)=\\left(\\alpha(x), \\alpha\\left(f(x)\\right), \\dots, \\alpha\\left(f^{k-1}(x)\\right)\\right)", "02f2b48c29610c0f37753082a3853221": "\\forall (a,b)\\ S(a,b) < \\infty", "02f2b66b570e5fe0c8bba0cf56a99d02": "\\sigma^{-1}\\omega", "02f2ba35e3bcf2226b0a61b1a0bdbf55": "S(f) = \\int_{-\\infty}^\\infty R(\\tau) e^{- j 2 \\pi f \\tau} \\, {\\rm d}\\tau.", "02f2ee7a892954eef0166ec5d10ec8d5": "n = 6", "02f3421ca1b6f287a50cf2d6c4a92ff9": "E_5", "02f3933968c77f4bed9e436105dad75b": "\\partial_y\\partial_x f|_{(0,0)} = -1", "02f3d1a8f75e61c94bee74ef609259c8": "\\frac{\\mathrm{d} g_{ij}}{\\mathrm{d} t} + \\frac{1}{2} \\frac{\\partial f^{k}}{\\partial v^{i}} g_{kj} + \\frac{1}{2} \\frac{\\partial f^{k}}{\\partial v^{j}} g_{ki} = 0 \\mbox{ for } 1 \\leq i, j \\leq n, \\quad \\mbox{(H2)}", "02f3eb23895fcb08557def8670db2e7f": "~K = \\sigma/(s t_{\\rm r})~", "02f3ee4e631696fb75edfe70f4191b42": " \\mathbf{r}^\\mathrm{PR} = \\mathbf{r}^\\mathrm{PQ} + \\mathbf{r}^\\mathrm{QR}.", "02f424541a3bf7294037e7d96c538bdc": "Q_{ij}=M(3x_i x_j-\\delta_{ij})\\ ,", "02f45c6c6f3aa7d3178d4268e161b3e8": "G(n,m)", "02f4b99e9280bfcb506bf330e6cf28fb": "\\omega_{\\mathrm{p}}", "02f4d62cb2570c2e4849461111095536": " E^m", "02f4eaec0a182c70f62fc43d4075d7d5": "e^x = \\sum^\\infty_{n=0} {x^n\\over n!} =1 + x + {x^2 \\over 2!} + {x^3 \\over 3!} + {x^4 \\over 4!}+\\cdots\\!", "02f50a6557dcbcacebf468676a1ebb52": "N(r)", "02f59cb82cc3d0082e9493b86c0528c3": "f''(x) \\ge m ", "02f5d6c787071d778ac2357a445f8228": "\\sigma_{i + 1}^{2} \\leq q \\sigma_{i}^{2},", "02f641c12f8d7e031069b09df11f1c27": "\\ h = \\frac{1}{\\sqrt{s^\\mathrm{H} R_v^{-1} s}} R_v^{-1}s.", "02f6b3b7e4a207277d717543b4562e31": "S(10) > \\Sigma(10) > 3 \\uparrow\\uparrow\\uparrow 3", "02f6f42d3d308f65531d018e6142e2c5": "\\{X_1,X_2,\\ldots,X_n\\}", "02f6f7bcbd0ec8e605fe811978b79061": "\\textstyle=min_{a^{*}(\\theta_{k}w_{k}=1)}\\ W_{k}^{*}R_{k}W_{k}", "02f716902388b03e703855a549afbef1": "N = N_1 + xN_2\\,", "02f7217d2be50e1391fd60f3e462c6b9": "x_j(t)", "02f758c3d842ae6a09d7b7a117d46240": "\\kappa_\\nu B_\\nu = j_\\nu\\,", "02f75e457493ec338c3346526cd6847f": "\\left( 1 - \\sum_{i=1}^{p'} \\alpha_i L^i \\right)", "02f7aa4a2ce39f9f29fdc34c01777154": " Lm ", "02f7add3b530415b79a48cecdb94151a": "\\,\\mathrm m", "02f7dac3f892b9379e20661985ade99c": "dt=a(t)d\\tau", "02f7e1cdf7f369ee01bde5b75bcd86c2": "v \\in U(S)\\,", "02f813eccc85fc5db7514773fac25cf6": "g^{\\alpha\\beta}\\frac{\\partial S}{\\partial x^\\alpha}\\frac{\\partial S}{\\partial x^\\beta} + (mc)^2 = 0\\,,", "02f817b9d33bd6cabbc4511bb2bf1b55": "\\sec \\zeta", "02f831a9258e43b2f63c557008338437": "\\left.\\frac{\\partial}{\\partial u} g(z, u)\\right|_{u=1} =\n\\left. \\frac{\\sum_{d\\mid k} z^d}{1-z} \n\\exp\\left(\\sum_{d\\mid k} \n\\left(u^d \\frac{z^d}{d} - \\frac{z^d}{d}\\right)\\right) \\right|_{u=1} =\n\\frac{\\sum_{d\\mid k} z^d}{1-z}.", "02f8a272aa79452db15718cdded95370": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 22.11315\\log_e(T+273.15) - \\frac {13079.73} {T+273.15} + 166.0812 + 1.233275 \\times 10^{-5} (T+273.15)^2", "02f8a69b15624b58629aed898287f86d": "\\Rightarrow x\\ln x = \\ln z\\,", "02f8c68abedd8d8a4a2cbc020788b3bf": " J_\\nu^{(3)}(x;q) = \\frac{x^\\nu(q^{\\nu+1};q)_\\infty}{(q;q)_\\infty} \\sum_{k\\ge 0}\\frac{(-1)^kq^{k(k+1)/2}x^{2k}}{(q^{\\nu+1};q)_k(q;q)_k} ", "02f8e61699aad550bc66075c76b0df14": "{\\mathbb R}^4,S^3\\times {\\mathbb R},M^4\\setminus\\{*\\},...", "02f8eac21681dabf14f7b2d25389d52b": "\\Delta p = \\frac{2 \\gamma}{R}.", "02f90dad6539792ab4cb6b7c9320a5c8": "P = f_3\\!\\left({Q\\over {ND^3}}\\right),\\,", "02f95acfafb779e2ce212284b6520ae2": "h=\\frac{p}{2}", "02f9aa7f5617898aa9489861d7fa465e": "\\cfrac{1}{(\\sigma_3^y)^2}, \\cfrac{1}{(\\sigma_2^y)^2}", "02f9e1a817c1f42f1e788c8549c78857": "\\bar{l}", "02f9ed4db8ebd67aff8c1e92c42405e1": "V=\\{V_1, V_2,\\ldots, V_C\\}", "02fa32878072963ead556d6f86c039f9": " \\coth\\left(x\\right) = \\frac{1+\\exp\\left( -2x\\right)}{1 - \\exp\\left( -2x \\right)}.", "02fa5faadf3bd895faeca53717d8b758": "R_{25} = \\sigma _{call,25} - \\sigma _{put,25} ", "02fa67b478412c4500425d2ef02b635f": " E_0( \\rho) = \\ln \\left( \\sum_{x_i} P(x_i)^{\\frac{1}{1+ \\rho}} \\right)(1+\\rho) \\, ,", "02fa70decbb4316ea8a5a882df882bc4": "\\kappa x{:}1{\\to}\\tau_1\\,.\\,e\\;:\\;\\tau_1\\times\\tau_2\\to\\tau_3", "02faadedb2a4c152940bad34dd95ed89": "\n(A|B) = \n \\left[\\begin{array}{ccc|c}\n 1 & 1 & 2 & 3\\\\\n 1 & 1 & 1 & 1 \\\\\n 2 & 2 & 2 & 5\n \\end{array}\\right].\n", "02fad5ddcac1eb8cc93e98458d70ae2a": " t_r 0", "030be7a21c42dcb01f92e00e7d84e5e2": "Z_G(t_1,t_2,\\ldots,t_n) = \\frac{1}{|G|}\\sum_{g \\in G} t_1^{j_1(g)} t_2^{j_2(g)} \\cdots t_n^{j_n(g)}.", "030c26ccc7f7cb949016a4a8a4f51f59": "\\eta_L \\, = \\, M_w e^{ \\left[ \\sum \\eta_a - 597.82 \\right] / T + \\sum \\eta_b - 11.202 }", "030c484910162ab90dec8f32b48263ef": "d(t) = a(t)d_0,\\,", "030c51923cd08fea140adaad4268787c": "\\text{CV/PS}=0.4\\times i\\times d^2\\times S", "030c6912e2e7c1098fdfe8298120ba85": "\\lambda_1\\approx\\lambda_2\\approx\\lambda_3", "030c8beff7748ff17469998a9a0251c6": "C_P=\\frac{P}{\\frac{1}{2}\\rho A V^3}", "030c9a04d6a2b608cb18ae6cc9d3ce44": " P_0(x) = 1. ", "030cca9b8a4c4a450e7214013b2635ac": " x\\,dx^2=dy^2, \\, ", "030d0afb674a89d5aae8f9d8be13ac35": "2^a", "030d4ac464f2b3ad5f7e54d8beae9c0d": "W^u(f,p) = W^s(f^{-1},p),", "030d4ae8df70d814e136970a8542bcdd": "f(x)=\\sum_{i=0}^l L_i(x^{k_i}), L_i(y) = \\sum_{j=0}^{m_j-1}f_{p^jk_i\\bmod{N}}y^{p^j},", "030d8ac281de329fb8d38a6f021b3d6e": "Q_B", "030e76a732c956a1950899118616099b": " \\iiint_V \\left( {\\partial P\\over \\partial x} + {\\partial Q\\over \\partial y} + {\\partial R\\over \\partial z} \\right) dx \\, dy \\, dz =\n\\iint_\\Sigma \\left( P + Q + R\\right) \\, d\\Sigma ", "030e9548babdfe0825542d893e7297bb": "h\\nu_m/k", "030eb5f5725d2a23d12126c34b1a6e26": "\n~\\epsilon_t = ~\\sigma_t ~\\times z_t\n", "030ec89a5bf9fbdfe59d5d43d1fe8a5a": "\\sqrt{10/[3(5-\\sqrt{5})]}", "030ed4a5d1a603b96002db8c87a864a4": " \\ M ", "030edff1943a3e6d4e8d029853bb2e67": "\\hat{H}_0 \\to \\hat{H}'_0 \\equiv U \\hat{H}_0 U^{-1} = U (\\alpha \\cdot p + \\beta m) U^{-1} = (\\cos \\theta + \\beta \\mathbf{\\alpha} \\cdot \\hat{p} \\sin \\theta ) (\\alpha \\cdot p + \\beta m) (\\cos \\theta - \\beta \\mathbf{\\alpha} \\cdot \\hat{p} \\sin \\theta ) ", "030f24479c38f531d978533a87c0116d": " \\bold{r}(\\theta(t)) =(\\ell\\sin\\theta, -\\ell\\cos\\theta)", "030f75e47654b5eed666144c2a3f774d": "C_\\text{out} = [Q]^K \\rightarrow [Q]^N, C_\\text{in} : [q]^k \\rightarrow [q]^n", "030fa134106f353689682c3cb4373243": "f(x) = \\begin{cases} \\frac{1}{x} & \\mbox{if } x > 0, \\\\ 5 & \\mbox{if } x \\le 0. \\end{cases}", "030fa229d0d6a66562d90a7101fd629c": "\\scriptstyle a \\;=\\; b \\;=\\; q \\;=\\; 1", "030fc21068588c25507efcfcfc01c7f6": "\\cos\\theta \\pm \\mathbf{i}\\sin\\theta", "031010e9f2111e92d564b794ab7b96be": "\\Sigma^k", "0310200af96f75ba3d543491548db12b": " \\dot{V}_A", "0310222002d8373cf997eb7ddee3dc52": "f(x_4)=14.1014", "03103e6579511946d8ca3d542b1440f6": "2^{355.5}", "031062ffa5fbda1b87a49e321295e1bc": "\n\\vec{\\nabla}\\cdot\\left[\\epsilon(\\vec{r})\\vec{\\nabla}\\Psi(\\vec{r})\\right] = -\\rho^{f}(\\vec{r}) - \\sum_{i}c_{i}^{\\infty}z_{i}q\\lambda(\\vec{r})e^{\\frac{-z_{i}q\\Psi(\\vec{r})}{kT}}\n", "031070c74a10b59558c7b83730c6e5f6": "\\mu(\\hat{p},\\mathbf{1},\\hat{p})=1", "0310ce0a6519debf9789f1f3c73a70b3": "\\displaystyle \\Box n = -\\Delta (|u|^2_{})", "031100983d73a2450b5544fed638b7d1": "\\aleph_\\beta", "031157f7409d26f9f5016282da47a89a": "p_2(x)=-4x+x^2;", "0311a31db8f8795194b1dc3d9da5f1e7": "2^{14_{dec}}", "0311b195732b29dc61854381569b0447": "P(o\\mid b,a) = \\sum_{s'\\in S}\\Omega(o\\mid s',a)\\sum_{s\\in S}T(s'\\mid s,a)b(s)", "0311d0522ad7308943910f7fcb6b1eb9": "s > - 2/\\Delta t", "03124b11d599daf3b1fb9b8aaf6b8c82": " F \\subset YX ", "03124d8546e8d1671120757e00607950": "h:\\mathcal A\\rightarrow\\mathcal B", "0312851f79b545a08e8363bb755d9f4f": "\\left( \\frac{8m+61}{3}, \\frac{8+m}{3}, \\frac{m^2-61}{3} \\right).", "031290e1ccd8bcd4fd9c4f42230d7cc5": " MI(row,col)= H(row) + H(col) - H(row,col) \\, ", "0312d8af026d6167c797d49441d84137": "\\Delta f := \\operatorname{div}\\; \\operatorname{grad} f.", "0312da4595e1f000b7366734bb6d8537": "\\displaystyle{f(T)\\xi=\\lim_{r\\rightarrow 1} f_r(T)\\xi.}", "0312dad48a2779999ccfc1fac186c7cd": "h(N) \\leq c(N,P) + h(P)", "0313062e85dc040c41b9a33ef924c201": "a = r, R = 2r", "0314393ec3e1463189a167a4f2c45163": "\\lambda_c\\sin\\theta = n\\lambda\\,\\!", "0314620279358b68099b802277c1ea4c": "OH=3GO.", "031468b74b375cc8ed6d70f63e2e73a6": "i = 1, 2, ..., k", "031499d95612e801ccacb94f8850bd24": "Hx = 0", "031506c1f09d2ec31c224a1f7b427673": "E^+_q", "0315213a8991b040c5c0b50c37c2fd6b": "\\sum_{n=1}^\\infty \\Pi_0(n)x^n = \\sum_{a=2}^\\infty \\frac{x^{a}}{1-x} - \\frac{1}{2}\\sum_{a=2}^\\infty \n\\sum_{b=2}^\\infty \\frac{x^{ab}}{1-x} + \\frac{1}{3}\\sum_{a=2}^\\infty \\sum_{b=2}^\\infty \\sum_{c=2}^\\infty \\frac{x^{abc}}{1\n-x} - \\frac{1}{4}\\sum_{a=2}^\\infty \\sum_{b=2}^\\infty \\sum_{c=2}^\\infty \\sum_{d=2}^\\infty \\frac{x^{abcd}}{1-x} + \n\\cdots ", "03153117637a3d052115e3d9cf307dc0": "{BSA}=0.007184 \\times W^{0.425} \\times H^{0.725} ", "0315513f8afde9dd3363cc1da930c1ba": " \\omega = \\frac{\\operatorname{d}\\theta}{\\operatorname{d}t}", "031551a9e052da5c2ccdb6eab96e49e2": "\\textrm{Labor~Productivity~(output~per~hour)}={\\textrm{Output}\\over\\textrm{Labor ~Inputs}}", "031590f5590f02496da38540db955827": "{{i}_{E3}}=\\frac{\\beta +1}{\\beta }{{i}_{C3}}", "0315de9e1bec426d2fdd18e8cffc516d": "\\tau:X_\\text{reg}\\rightarrow X\\times G_r^n", "0315f119d11928920959df3b5cc610e3": "E' = E/(1-\\nu^2)", "03163673c5da8149d5b745a2d34b58d7": "\\Omega = \\sum_{p \\in P} 2^{-|p|}.", "0316b16e199796da2f21f486e1ae0418": " \\begin{alignat}{4}\nf(x)&=x(\\sqrt{x+1}-\\sqrt{x})\\\\\n & =x(\\sqrt{x+1}-\\sqrt{x})\\frac{(\\sqrt{x+1}+\\sqrt{x})}{(\\sqrt{x+1}+\\sqrt{x})}\\\\\n &=x\\frac{((\\sqrt{x+1})^2-(\\sqrt{x})^2)}{(\\sqrt{x+1}+\\sqrt{x})}\n &=\\frac {x}{(\\sqrt{x+1}+\\sqrt{x})}\n\\end{alignat}", "0316c850d12c519d4a0e6ba29c718df3": "(1,4,2)", "0316e7c928254cd7a3986f7ed83ce256": "\\{x,y,z\\}", "0317100d49f06c725fa4722579232829": "\\textstyle p \\equiv 2 \\mod 3", "0317f7c498e366823c7bad03638baf3d": "\\int_V \\rho(\\mathbf{r})(\\mathbf{r}-\\mathbf{R})dV = 0.", "031801a96c6385da55f551b10027d2f4": "\\operatorname{im}\\, \\kappa", "031882c0e138764b2fd5e51ca2e686d9": "B = {h \\over{8\\pi^2cI_{\\perp}}}", "0318d3e5bc7f0dbfffb433ef97df66b0": "\\varphi_{X+Y}(t)= \\operatorname{E}\\left [e^{it(X+Y)}\\right]= \\operatorname{E}\\left [e^{itX}e^{itY}\\right] = \\operatorname{E}\\left [e^{itX}\\right] E\\left [e^{itY}\\right] =\\varphi_X(t) \\varphi_Y(t)", "03198b53127912f205d924d069a9412d": "B(x_1,y_1) B(x_2,y_2) = B(x_1 x_2 \\pm ty_1 y_2,x_1 y_2 \\pm y_1 x_2).\\,", "0319dd14cc0deff086e99c2188ae997b": "S=\\sum_{i=1}^9 j_i.\n", "0319eff6ca14880cf69584473c382251": "\\hat{z}=\\hat{c}.", "031a2d2a71d63ad0dcd9943fa3a8ad57": "\\phi: \\mathbb{R}^{2} \\rightarrow \\{0\\} ", "031a3a56765a5d292512ddab0cbef40d": "f : M \\to \\mathbb R\\,", "031a472a51c9e9fcbdd21e7dcda00203": "T_{1/2}=\\frac{-0.693\\,N}{\\frac{dN}{dt} }", "031a4a0901cfef55c54f49518f6baf29": "F_{x}", "031a5590aeeb04464126550f43aa0dcf": " \\mathbf{E} \\, \\mathbf{t} = \\mathbf{R} \\, [\\mathbf{t}]_{\\times} \\, \\mathbf{t} = \\mathbf{0} ", "031aa2ffc9b711d2e8568231f22a365d": "J \\colon \\pi_r (\\mathrm{SO}(q)) \\to \\pi_{r+q}(S^q) \\,\\!", "031ad52745d5567e6c39296eea619281": "\\mbox{QMA}(c,s)", "031b4efd8ba7d8c4e90d80a2e199e411": "\nD_\\text{KL}(\\mathcal{N}_0 \\| \\mathcal{N}_1) = { 1 \\over 2 } \\left\\{ \\mathrm{tr} \\left( \\boldsymbol\\Sigma_1^{-1} \\boldsymbol\\Sigma_0 \\right) + \\left( \\boldsymbol\\mu_1 - \\boldsymbol\\mu_0\\right)^{\\rm T} \\boldsymbol\\Sigma_1^{-1} ( \\boldsymbol\\mu_1 - \\boldsymbol\\mu_0 ) - K -\\ln { | \\boldsymbol \\Sigma_0 | \\over | \\boldsymbol\\Sigma_1 | } \\right\\},\n", "031b9bff8fd8e223ac7b3fc4a03cbd51": "C_D = 1.456 \\times 10^5 (\\frac{\\eta P}{\\sigma S V^3})", "031ba9e2f80ace02d96bc0ec234c80a4": "\\dot{x}_i = \\partial H / \\partial p_i", "031c29938cf30deaad28a1e05352c788": " G < f^{64}(4)y) = \\frac{\\int_y^\\infty x g(x) dx}{1 - F(y)} ", "032e291fd1e5592fcd3b7abe2ea18bef": "\\vec a", "032e9e63fc63d330dfec39bd6bd9e61d": "h_x(\\alpha), \\beta_x(\\alpha), \\lambda_x(\\alpha)", "032eae71d2c2662177f5536db706a47a": "G = S_n", "032ef5050a6c2c4507e9fede8c0b9be8": " \\int \\rho(y) e^{i k y} d^n y = \\langle e^{i k y} \\rangle = \\langle \\prod_{i=1}^{n} e^{ih_i y_i}\\rangle \\,", "032f0a403827291bc6d37f54cdbde8a9": "\\mathcal{L}()", "032f27fb65b3aa322bd965750957a372": " CPP = MAP - ICP ", "032f3d72715da0c48307515af3ed66e1": "number_{(base)} = \\sum_{i=0}^n {digits_i \\times base^i}", "032f3e25fb024bf84e86c774d20682a2": " \\quad 1", "032fc87d7eaf33a24c0c566729002bc8": "\nH = \\frac{1}{a^2} + \\frac{n-1}{b^2}.\n", "032fcbe05b92f61b4fd08f0722ec1cc7": "\\Gamma^i {}_{j k}", "032ff6a1dbdbb6918b10c7310cfe2be5": "(m, P)", "03302b9afd10bdb58e56b2c229a96f77": "(\\forall n\\in\\mathbb{Z}_+):A_{n+1}(x)=\\int_0^xyA_n(y)\\,dy;", "03302f88497644ad4db5e6052d763ba3": "(2^m - m - 1)/(2^m-1)", "033041bab2326817f03a5c4fcb30a589": "\\mathbf{E} [x^m] = \\frac {\\Gamma (Nk - m)} {\\Gamma(Nk)} y^m", "03304451b4f871ed5fa8b77ff7e5a355": "r^2~\\ln r", "03307643d8dd24479f0fb0d50726c9aa": "+\\hbar k_{max}/m", "0330786fa12c9466393d0c36958a5e26": "\\scriptstyle \\theta/(2 \\pi)", "03307c2a355cb08ccf414ce55fe1dd46": "= u(\\sigma(p)) = u(\\sigma(x)) = \\sigma(x) \\,", "033088a3e831ea9c495aa021f0d91f99": "\\Delta\\Delta G_{i, j}^{stat} = \\sqrt{\\sum_x (ln P_{i|\\delta j}^x - ln P_i^x)^2}", "0330e6322712b9e884f75ba8908f4bf3": "(S,S) = 0 . ", "03315ba0f07a3c0afb394d187b635e5a": "\\mathbf{r}_1-\\mathbf{r}_3", "03315f3fc919ff9c467f51369cdb0525": "2 - 1", "0331640d8f7864d0de31445dc0a005e4": "H_n(X;A)\\simeq A", "0331c2749aa13f4e61217ec85f967f29": "E_{x,z} = l n V_{pp\\sigma} - l n V_{pp\\pi}", "0331e3a1d454230fe07a41190807ce88": "\\sqrt{1+2\\sqrt{1+3 \\sqrt{1+\\cdots}}}.", "03323da5fe872cf192f04e86b4b6d097": "x = R_H", "0332a3a0fb4aa99a9f4daa8b9b306250": "\\omega_{0}=1/\\sqrt{R_2 R_4 C_1 C_2}", "0332a6e175c2ea9fab3b0e7dc3287806": " {1 \\over \\lambda} = R \\left( {1 \\over (n^\\prime)^2} - {1 \\over n^2} \\right) \\qquad \\left( R = 1.097373 \\times 10^7 \\ \\mathrm{m}^{-1} \\right)", "0332b621d562f1bc5526f4b1879c3a50": "\\partial \\omega / \\partial x = 0", "0332ed61d1dd26753da1ea5b26a23387": "\n\\nabla:TM\\times\\Gamma(E)\\to\\Gamma(E) \\quad ; \\quad \\nabla_Xv := \\kappa(v_*X)\n", "033314f8c2ab97c4d497d7ae5b0889d0": "x_{t=0}(x,p) = x", "03334a7d3910583154d57b92ff5e90ee": " \\Delta z_{\\rm{bias}}", "0333a113cad1852120b10f74af753df0": "\nP_s\\left(T \\right) = 6.1121 \\exp \\left(\\left( 18.678 - \\frac{T} {234.5}\\right)\\left( \\frac{T} {257.14 + T} \\right)\\right)\n", "0333eb5744fa9eeb4a90a746c738ac85": "f(x) \\geq f(y) + f'(y)(x-y)", "03343105f7e2274b02c3185b0bb305b0": "\\prod _x ax^2+bx = C\\,a^x \\Gamma (x) \\Gamma \\left(x+\\frac{b}{a}\\right) \\,", "03343acd809ae93effe3a7985482a132": "(r_i-i)^2", "03345fcc32c81a8fc9e3697dcac7a670": "\\chi_{0}(\\mathbf{q} | \\Gamma)", "03347a6365e48bfd261160967f23fa18": "A,B,C \\in \\mathcal{C}", "0334c669d93e04082a201fb3aa1afc4f": "f(a)-f(x)\\quad", "0334cb1648ac9ff1ee1d24f11ada8c2f": " dt = \\gamma(\\mathbf{u})d\\tau \\,.", "0334cb73be3efe3f0d830347d285179b": "f_i^{(k\\ell)}", "033546280bfbef560db2e14aca08fce8": "\\overline{\\{0,\\dots,n\\}}", "03355c9959a615b999c58afc2c9c177f": "D(X, Y) = \\sum_{i} \\sum_{j} |x_i-y_j|P(X=x_i)P(Y=y_j).\\,", "0335bb6966e9f95fffbbbdd15844f939": "\\delta(x-z)", "0335f44e12ff05d522834047bc1d8611": "E\\bigl(g(X)(X-\\mu)\\bigr)=\\sigma^2 E\\bigl(g'(X)\\bigr).", "033612701cd03227f71fd00b13d902e7": "\\!\\mathcal A \\models_Z^+ \\psi", "0336bde25c6dbb9bacfa998e2b5016c0": "(b_{14}-a_{14})+(b_{15}-a_{15})=77", "0336c6ab921432effb4a4fda380f55e4": "a^{\\dagger}a", "0336d7f982fdbf9208732a1e267584e1": "K = \\frac{1-\\left|S_{11}\\right|^2-\\left|S_{22}\\right|^2+\\left|\\Delta\\right|^2}{2\\left|S_{12}S_{21}\\right|}\\,", "0336e62688f0d883c689d6ef13e65119": "P(X_i=a \\cap X_{i+1}=b)", "03371dc87912c24e81c3d0c1a453b4b3": "P\\in\\operatorname{Hom}(H,H)^{\\times}", "0337c65c3af4b8bb8626311abef3e21a": "\\scriptstyle \\sigma \\,=\\, 0.5", "0338003e027771272ed23c1e6a62c522": "\\rho_{XY} =E[(X-E[X])(Y-E[Y])]/(\\sigma_X \\sigma_Y) \\;", "033827b39b706414493b6dfede94ee51": "{\\left ( \\frac{\\partial z}{\\partial x} \\right )}_y {\\left ( \\frac{\\partial x}{\\partial z} \\right )}_y = 1.", "0338c024c047094ddaeb91ab7612cdfb": "y_0=y(t_0), \\ \\ y_1=y(t_0+h), \\ \\ y_2=y(t_0+2h), \\ \\dots", "03392cc816f562c05c7cddb75f43d0ab": "d x = {\\left ( \\frac{\\partial x}{\\partial y} \\right )}_z \\, d y + {\\left ( \\frac{\\partial x}{\\partial z} \\right )}_y \\,dz", "033946767fb687a8acafe00a11dc39ef": "X_1\\times\\cdots\\times X_n = \\{(x_1, \\ldots, x_n) : x_i \\in X_i \\}.", "03395260509a989d7c0a6ed0871e76b3": "\\mathrm{ext}[K(X)]", "033a2debd7ace8a1f02030a6418b95c7": "A = \\textstyle 2a^2 \\cot \\frac{\\pi}{8} = 2(1+\\sqrt{2})a^2 \\simeq 4.828427 a^2.", "033a59ab09479bd41cff2a9bb8a2be03": "|A\\rangle", "033a5c232f479b005190cd88d26bd326": "\\mathbf{NTIME}(f(n)) \\subsetneq \\mathbf{NTIME}(g(n))", "033a5e7c5a9565b911c8496e0ba88b23": "{\\mathcal L} = -\\frac{1}{4} F_{\\mu\\nu}F^{\\mu\\nu}-\\frac{n_f g^2\\theta}{32\\pi^2}\nF_{\\mu\\nu}\\tilde F^{\\mu\\nu}+\\bar \\psi(i\\gamma^\\mu D_\\mu - m\ne^{i\\theta'\\gamma_5})\\psi", "033abc09c3acf249153f99047185c031": "\n\\omega^{-1} = \\sqrt{2} + 1.\\,\n", "033ad9c60d57150d5fc5b9c0b201b554": "\n\\mathbf{N} = \\begin{bmatrix}\n -1 & 0 & 0 & 0 \\\\\n 1 & 1 & 0 & 0 \\\\ \n 0 & -1 & -1 & 0 \\\\\n 0 & 0 & 1 & -1 \\\\\n 0 & 0 & 0 & 1 \\\\\n\\end{bmatrix}\n", "033aeb8a43250203f680dcd041b14ea1": "6n - 1", "033b4d0d76a0a2b53b3a977e22c00e6e": "f_C = \\frac{1}{2 \\pi n\\tau_T} \\ , ", "033b571c237d78ae1c9908427fdf52ce": "\\frac{a}{b}", "033b794d576f4bbfe4f3bbee3044741b": "X_1, X_2, \\ldots, X_N", "033b7d86b0dfb4ace148e2d293734efa": " k = 2, \\ldots , r", "033bb10b01e31642bebef44240a150f9": "x(u, v) = \\left(R + r\\cos{v}\\right)\\cos{u}, ", "033bd7948909fec272d4cde4fc3a9d59": "X(0)= \\eta", "033c331ac596852538de39eb0b3f3b96": "\\int_0^1 \\left(S_n(s) - {1 \\over 2}\\right) \\left(S_m(s) - {1 \\over 2}\\right){ds \\over \\sqrt{s(1-s)}}=0.", "033c94564ba398681d4405e630ff5379": "\\scriptstyle {2\\cos{\\tfrac{2\\pi}{7}} \\approx 1.247}", "033cf64947ed82dfdd4acedc393e56b4": "\\delta Q\\ =C^{(p)}_T(p,T)\\, \\delta p\\,+\\,C^{(T)}_p(p,T)\\,\\delta T", "033d24ff17ac6a0c27b2a6ee5db80984": "p+dp", "033d252565b13f3e612b0fa5abfc05ce": "i \\hbar \\frac{d\\psi}{dt} = - \\frac{\\hbar^2}{2 L} \\nabla^2 \\psi+\\frac{Q^2}{2C} \\psi ", "033d2c3acd83e1a970dfd82cd9be15fa": "x^0 + \\Delta x^0 = x^0 + \\tfrac{1}{2} \\left ( dx^{0(2)} + dx^{0(1)} \\right ).", "033d30d82100677736d9d12f77f0de17": "\\Re\\left(\\sum b_k - \\sum a_j\\right)>0", "033d3a80af90a171e50a9601b34ea1fc": "C_\\nu", "033d7fc1e2ae2eac0742a1097f858515": "t^2 = 2\\frac{d-d_i - v_it}{a} = 2\\frac{\\Delta d - v_it}{a}", "033dd2348284fbd3b6479e6b9599faae": "\\scriptstyle \\int r\\,d\\theta = 2\\pi r", "033dff11140d43b95e2e7ae5d19c42d2": " Y_i", "033f61cdfc761a03f151199e89a1d96a": "\\Pi^1_0", "033f7f4c3eb918d9336804647277a218": "10^{8}", "033f9e7b053ff7d55c8790005fc6fdde": " \\tbinom nk ", "033fe8d62666e7954e987a7e43d72f53": "U: x\\mapsto ax\\pmod p", "034024b1ae850a9f144606083c40a03b": "\\cos\\left(\\tfrac{\\pi}{2}\\,(2k+1)\\right)=0", "0340253c8a6f812b5baaffc88152e24c": "\\ \\mathcal{L}_\\mathrm{gf} = - \\frac{1}{2} \\operatorname{Tr}(F^{\\mu \\nu} F_{\\mu \\nu}) ", "034028a4b04ca029b2eff7d3062092b5": "\\displaystyle p_n(x;a|q) = {}_2\\phi_1(q^{-n},0;aq;q,qx) = \\frac{1}{(a^{-1}q^{-n};q)_n}{}_2\\phi_0(q^{-n},x^{-1};;q,x/a) ", "03406fcfd1a0a53351ef741c5988d692": "(R_1^T)^{-1} b", "0340aa31c15462805b477ff46b680260": "F_n(\\lambda)/F_{n-1}(\\lambda)\\cong A(\\mu)", "0340c06debcea43df493c2c8130f3ef1": "Tr(t^at^b) = \\frac{1}{2}\\delta^{ab}.", "0340eae31c3aa0d609eb49e60bc1a4d9": " \\lambda_{CW}(M) = 2 \\lambda(M) ", "0340f80858ff9393d898c1c06d2bb192": "f \\colon U \\times Q \\to V", "034173eabcc634d08a0d3abe459e0c5a": "T\\in \\operatorname{Hom}\\left(\\wedge^2 TM, TM\\right).", "0341ae6f0ad16522997f8ac07e7b6b06": "\\frac{k_B T}{\\gamma} \\Gamma^{-1}", "0341af483b2c59f352de9ff7be013758": "{3\\pi\\over 5}\\ {\\pi\\over 3}\\ {\\pi\\over 2}", "03421840ed5f91b4bf9ed2c83642c61e": "B1-B2", "03421cef1aba2eedcd955e387f7abea5": "\\mathbf{P}(X > (1+\\delta)\\mu) < \\frac{\\prod_{i=1}^n\\exp(p_i(e^t-1))}{\\exp(t(1+\\delta)\\mu)} = \\frac{\\exp\\left((e^t-1)\\sum_{i=1}^n p_i\\right)}{\\exp(t(1+\\delta)\\mu)} = \\frac{\\exp((e^t-1)\\mu)}{\\exp(t(1+\\delta)\\mu)}.", "03422e6c61867719daf2bd867bbc22da": "\n \\ln\\mathcal{L}(\\mu,\\sigma^2)\n = \\sum_{i=1}^n \\ln f(x_i;\\,\\mu,\\sigma^2)\n = -\\frac{n}{2}\\ln(2\\pi) - \\frac{n}{2}\\ln\\sigma^2 - \\frac{1}{2\\sigma^2}\\sum_{i=1}^n (x_i-\\mu)^2.\n ", "0342381b29fc3888b071b4bc6fac9d5c": "P\\simeq \\frac{1}{3} \\epsilon=0.52\\times 10^{31}\\, \\mbox{bar}.\n", "034240c73f7a3435503c27a7cfb0e88e": "U_\\mathrm{in}(t)", "0342852b3b6d1dc4bd117a60e9c54334": "f(\\xi,\\rho,\\theta) = 0 \\,", "0342b646bdcb5436267848732910280c": "C = X_1-A = 1", "0343305d22cfe9adb435704d04f30c9a": "1. \\; \\; \\mathrm{NO}_2 \\; \\xrightarrow{h \\nu } \\; \\mathrm{NO + O}", "03435ed85217ab8779bbcd4f312fc645": "a_{T}^{\\pm} \\rightarrow \\gamma W_L^\\pm", "0343a6b2e82f9d4cdd49eb0d78dcd015": "A(D)=D^2+k^2", "0343ce9c1be67fe20e0d5cedb5044a6b": "\\rho_{\\text{e}}", "03441a463fb12d0679dcc77740797155": "\\tfrac{(M-\\lambda)(M+2\\lambda)}{M+\\lambda}", "034460feb7080db6f557cfd7a3780154": "\\Lambda(n)", "03449dc8a842ae7ae49e08f0af9205c2": "4a^2-4ab+4b^2=(2a-b)^2+3b^2, \\,\\!", "0344b0d9ea0118da3c511f5ccca766d4": "d\\mathbf x'=\\mathbf R \\,d\\mathbf X\\,\\!", "0344b1178a956f175a86eb6b03fd2032": "D_F^q(p, q)", "0344f847b22a6e355a58990d65b75ce2": "\nS = k N \\ln\n\\left[ \\left(\\frac VN\\right) \\left(\\frac UN \\right)^{\\frac 32}\\right]+\n{\\frac 32}kN\\left( {\\frac 53}+ \\ln\\frac{4\\pi m}{3h^2}\\right)\n", "03452f9a2b862e540da40d6e9858ab40": "U = -0.147 \\times R - 0.289 \\times G + 0.436 \\times B", "03456c32718e6804c8a92a89315cddc7": "(\\det\\Phi)'=\\sum_{i=1}^n\\det\\begin{pmatrix}\n\\Phi_{1,1}&\\Phi_{1,2}&\\cdots&\\Phi_{1,n}\\\\\n\\vdots&\\vdots&&\\vdots\\\\\n\\Phi'_{i,1}&\\Phi'_{i,2}&\\cdots&\\Phi'_{i,n}\\\\\n\\vdots&\\vdots&&\\vdots\\\\\n\\Phi_{n,1}&\\Phi_{n,2}&\\cdots&\\Phi_{n,n}\n\\end{pmatrix}.", "0345716ff0625c5efcca01c286036654": "k = e'\\cos\\alpha_0,", "03458f33713d60710c6aca5db16d5bfd": "\\frac D R", "0345e2efcbef367d84fc1770dbe49332": "\nH(2^2) = \\begin{bmatrix}\n1 & 1 & 1 & 1\\\\\n1 & -1 & 1 & -1\\\\\n1 & 1 & -1 & -1\\\\\n1 & -1 & -1 & 1\\\\\n\\end{bmatrix},\n", "03463cd817eaa6de0f808b641f59a7d6": "u = x + \\frac{b}{2c} ", "03471bcbc90fb56290b25b87fffce665": " (ae - bf - cg - dh) + (af + be + ch - dg) \\mathbf{i} + (ag - bh + ce + df) \\mathbf{j} + (ah + bg - cf + de) \\mathbf{k}", "03479877a489a72aeb0d78d3bfef37c1": "\\epsilon^{-d}", "0347b4898c96cbc2a86d3e8e2dc3f2b6": "\\mbox{Tr} \\mathcal {L} = \\sum_n \\langle \\psi_n , \\mathcal{L} \\psi_n \\rangle", "0347fffa7ae2123c9fd659b954b7ae7f": "[A + m(t)]\\cdot \\sin(\\omega_c t),\\,", "03480067b9a23e49fef8dec4890d1591": "f(\\sum\\nolimits_i a_i \\sigma_i) = \\sum\\nolimits_i a_i f(\\sigma_i)", "03483c0575c70ca078b4d06c463cfd93": " (x y)^* = y^* x^*", "0348948eec49e40cc114d0052df04810": "T_f\\;", "0348b58f302d593b58c1d3bb37944b25": "0<\\alpha^{\\,}<1", "03490af427ab865371d9aa274292dd84": "\\sim 1 nm", "034912b6711e851d4b4b85c5db46db55": "\\neg P\\,", "03491dcee43720d276368a4879f55105": "M_Z=\\frac{v\\sqrt{g^2+{g'}^2}}2,", "03498f1d79bdc5f60c4a3890d104e871": "H_{m n}^{\\text{eff}}\\left(x^{\\mu }\\right)=\\langle m|H|n\\rangle +\\langle m|\\partial _{\\mu }H|n\\rangle x^{\\mu }+\\frac{1}{2!}\\sum _{l\\in\\mathcal{H}_H} \\left(\\frac{\\langle m|\\partial _{\\mu }H|l\\rangle \\langle l|\\partial _{\\nu }H|n\\rangle }{E_m-E_l}+\\frac{\\langle m|\\partial _{\\nu }H|l\\rangle \\langle l|\\partial _{\\mu }H|n\\rangle }{E_n-E_l}\\right)x^{\\mu }x^{\\nu }+\\cdots.", "034a110f06a0fb2676560eeb40e30aaa": "\\boldsymbol{u}_e=\\boldsymbol{u}-\\boldsymbol{u}_g.", "034b20078cf947f201ab4f8238c147a8": " \\Delta\\mathbf{r}_i^\\perp = (\\mathbf{r}_i-\\mathbf{R}) - (\\mathbf{S}\\cdot(\\mathbf{r}_i-\\mathbf{R}))\\mathbf{S} = [[I]-[\\mathbf{S}\\mathbf{S}^T]](\\Delta\\mathbf{r}_i),", "034b471f7c8f16bdaa36d9c8403de256": "|C_v|", "034b6293a338c9fd04e49062fcf5946e": "k\\cdot 2^{-j}", "034bcd264c3b7361e9fd371edbde15bc": "T_1[i,j]=\\max_{k}{(T_1[k,j-1]\\cdot A_{ki}\\cdot B_{iy_j})} ", "034bd8c7af8d5b7c12b76495fdadf2d9": "\\frac{\\partial u}{\\partial t} - \\alpha \\nabla^2 u=0 ", "034c0b19b15dece85c48f9d46635d5d4": "\nV = \\int_{1}^{\\infty} f(x) \\cdot \\pi f(x) \\,\\mathrm{d}x\n\\leqslant \\int_{1}^{\\infty} {M \\over 2} \\cdot 2 \\pi f(x) \\,\\mathrm{d}x\n\\leqslant {M \\over 2} \\cdot \\int_{1}^{\\infty} 2 \\pi f(x) \\sqrt{1 + f'(x)^2} \\,\\mathrm{d}x\n", "034c2f092a5f4253243eeb08480739e7": "E_A = \\frac{Q_A}{Q_A + Q_B}", "034c3f4b6980de5052e02e3aad6dcf93": "{\\color{Periwinkle}f'}(x_0) = \\frac{1}{4}", "034c48eb7f86a75a7e074b19200b2b87": "r\\approx\\frac{\\ell c}{2 \\pi f}", "034d36cd80de7dc8c6c57f505eff084d": "{2+\\|\\mu-\\nu\\|_{TV} \\over 4}", "034d4b15ad7de43ef240fdc74360fdd5": " Z_q(V_o,T)=\\int_0^{\\infty}\\sigma(E)[1+(q-1)\\beta E]^{-\\frac{q}{(q-1)}} dE\\,,", "034d4db04be85fef0334b6527626d63c": "O(M+N)", "034d7e5fee2fec89ba717192a586abe2": " \\eta_1 = \\eta_2 ", "034d8d44230f5e1974123a2fed5a38a8": "\\lim_{r \\to 0} f_r(x)", "034d93e729f59291d0aad712e0c96eb5": "(1-x^2)^{1/2}=1-\\frac{x^2}2-\\frac{x^4}8-\\frac{x^6}{16}\\cdots", "034db7fc980bc9d70e5a32c6423d0d5a": "R = \n\\begin{bmatrix}\n\\cos \\theta & -\\sin \\theta \\\\\n\\sin \\theta & \\cos \\theta \\\\\n\\end{bmatrix}\n", "034e098033cdd65d48996c47865c7ad6": "\\ I \\cdot I^{-1} = R", "034e32ca4e515d9ff4c8ea9faebbdc81": "\\frac{a}{2^b}-\\frac{c}{2^d}=\\frac{2^{d-b}a-c}{2^d} \\quad (d\\ge b)", "034e72f956cd45baa15c8832ec645b25": "\\operatorname{Hom}_R\\biggl( \\bigoplus_{i \\in I} M_i,L\\biggr) \\cong \\prod_{i \\in I}\\operatorname{Hom}_R\\left(M_i,L\\right).", "034f01c8ffbcfb03f06bd4919ddbeb87": "T_A^1~|~T_A^2", "034f29f30627882fc04fcfcfd26c65bf": "d_i, v_f =0", "034f48302f81deed68d4491eb032f8fb": " {(0,1,1)}", "034f59279c8bfa23635cdd456fa3e8a9": "\\frac{\\pi}{\\sin \\pi z}", "03501a92ee6948b2bb347b444193e40a": "\\hat\\beta = (X'X)^{-1}X'y = (X'X)^{-1}X'(X\\beta+\\varepsilon) = \\beta + (X'X)^{-1}X'\\mathcal{N}(0,\\sigma^2I)", "0350479dcfff9ffbf51732a10e5adfab": " b^2 c = 4(a-e)e = 4ae - 4e^2.", "03505471828bba790b78b5d6ae1426e5": "\\ell = \\frac{k_{\\rm B}T}{\\sqrt 2 \\pi d^2 P}\\, , \\;\\;\\; v_T=\\sqrt{\\frac{8k_{\\rm B}T}{\\pi m}}\\, .", "03510f0873a68eddaf477e68a9191052": "f(a) = \\mu \\log_2(a)\\,", "0351223e5532e427abfc10e9e4c8770a": "p \\cdot (\\Sigma _i x'_i) \\geq r", "03517dc5bbed7c241b29b04aafd77b11": " y_s(x) = -(1/2)x^2 + (-(1/2)x)^2 = -(1/4) \\cdot x^2. \\,\\!", "0351993ccc8ddb7723ec12b0d61bf7c2": "\\upsilon_D = \\frac{Vq}{2\\pi} \\qquad(3)\n", "0351e4eef5a980a9675866d564e970c6": "X^n/G", "0352187c748afe8507513f0d16b9d224": " |\\phi (t+dt) \\rangle - |\\phi (t) \\rangle = -i\\hat{H} dt |\\phi (t)\\rangle ", "03526787395d9c9cfeeb852f1489558e": "\\pi(A)=[A]", "0352924c1493364e408c14b645a3e297": "\\scriptstyle <10^{-12}", "035295d112bec33185dba2624c9d50c6": "\\phi (\\omega)\\triangleq\\arg K(j \\omega)=\\arctan \\frac{\\omega_0}{\\omega},", "035338c2498de04a3a6f3d3dc9c456cc": "\\sum_i a_i\\sigma_i\\,", "035355567ff818400354891fead4ca0e": " \\lambda= \\frac{D_x\\Delta t}{2 \\Delta x^2}", "03536bcbf2e10f360201eccb72dd34b0": "(x)_{n+1} = \\sum_{k=0}^n\n\\frac{n+1}{k+1}\n\\left[ \\begin{matrix} n \\\\ k \\end{matrix} \\right]\n\\left(B_{k+1}(x) - B_{k+1} \\right) ", "03544dddcb8d13bd5a09e7e442e394cb": "\n H^*_{\\overline{p}}x (n) = \\prod^{\\overline{p}}_{j=1}{x(n - \\tau_{j})} \n", "035451646fcceec539999e4521091551": "da(t)=\\delta_{t}a(t)\\,dt\\,", "0354cb942e62f3909d73c66a52072437": "\\mathbb{C}^{N/2}", "0354d96c50c762db44e348bf0fe7f48b": "3\\times3", "0354f3238103bfb970d5fea51b94adeb": "\\begin{cases}\n\\text{always} \n\\begin{cases}\n\\text{always } 0 \\\\\n\\text{if } y , +1\n\\end{cases}\n\\\\\n\\text{if } x , +2\n\\end{cases}", "03551109fb26fcb802cf4443e4d0a1bc": "\\tau_{\\beta,\\alpha}", "03551e591f616e8f74eec8a006ee40fc": "\nu_\\varepsilon \\left( \\xi ,\\eta ,z\\right) = \\frac{w_{0}}{w\\left(\nz\\right) }\\mathrm{C}_{p}^{m}\\left( i\\xi ,\\varepsilon \\right) \\mathrm{C}\n_{p}^{m}\\left( \\eta ,\\varepsilon \\right) \\exp \\left[ -ik\\frac{r^{2}}{\n2q\\left( z\\right) }-\\left( p+1\\right) \\psi _{GS}\\left( z\\right) \\right] ,\n", "03555a99a410f13b428f1ae7f0b65966": "\\operatorname{MSPE}(L)=g'(I-L)'(I-L)g+\\sigma^2\\operatorname{tr}\\left[L'L\\right].", "035571d0c387810ba8c29b16f26d4873": "s(t)=\\sum_{m=-\\infty}^\\infty \\sum_{n=-\\infty}^\\infty C_{m,n}h(t-mT)e^{jnt\\Omega}", "03557f6220dd37ac9bd22a4d3c605a20": "\\sqrt{I_2} = \\lambda \\sqrt{I_1}", "0355849f21ea7d1b1ac082a874360fbd": "\n \\boldsymbol{\\sigma}_r = \\boldsymbol{Q}\\cdot\\boldsymbol{\\sigma}\\cdot\\boldsymbol{Q}^T ~;~~ \\boldsymbol{Q}\\cdot\\boldsymbol{Q}^T = \\boldsymbol{\\mathit{1}}\n", "0355c3d493eb27ac190768aae8309697": "\n (5) \\qquad \\cfrac{\\partial^3\\varphi}{\\partial x^3} = -\\cfrac{m}{\\kappa AG}~\\cfrac{\\partial^4 w}{\\partial x^2\\partial t^2} + \\cfrac{\\partial^4 w}{\\partial x^4} + \\cfrac{1}{\\kappa AG}~\\frac{\\partial^2 q}{\\partial x^2}\n", "0355cf52d0d548ef1797bde3e95c8626": "\\begin{array}{rcl}\n C\\dfrac{d V}{d t} &=& -I_\\mathrm{ion}(V,w) + I \\\\ \\\\\n \\dfrac{d w}{d t} &=& \\phi \\cdot \\dfrac{w_{\\infty} - w}{\\tau_{w}}\n\\end{array}", "0355fdad639ee1a519a51a03d9577982": "\n1/\\eta_f = q_4 S\n", "0356051658dbbe05dfe8095bd591a425": "\n \\mathcal{I}_j = \\frac 2{\\sqrt{-\\mu_j}\\sqrt{\\lambda}} \\int_0^{\\infty} e^{-\\xi^2/2} d\\xi = \\sqrt{ \\frac{2\\pi}{\\lambda}} (-\\mu_j)^{-1/2}.\n", "0356c7833ecb6be4248c48f846b39891": " B_r =0 , \\quad B_{\\theta} =0 , \\quad B_z=a r^k ~f(\\psi)", "0356e5d88c047cd5055748098f28e8f8": " (u^2 + dv^2)^2 - d(2uv)^2 = 4. \\, ", "035721a27302ab4cb4c360e442ba1412": "H^2 = \\frac{8 \\pi G}{3} \\rho - \\frac{kc^2}{a^2}", "0357a9fb1ab694c9ed122a29c5441768": " \\nu_{\\mathrm{F}} ", "0357f7f863cd4e8ac4242f071798b6a7": "x \\mapsto x'=f(x)", "035872fd1b17cfe817547feef6286761": "s_{0} = \\sigma_{0}+iT", "03591a93124aad4e699d57c084dc5bb0": "b_n\\,", "0359270cc899a5f343ea43b879a7c757": "\\sigma_{ij} = \n\\begin{bmatrix}\n\\sigma_{11} & \\sigma_{12} \\\\\n\\sigma_{21} & \\sigma_{22}\n\\end{bmatrix} \n\\equiv \n\\begin{bmatrix}\n\\sigma_{x} & \\tau_{xy} \\\\\n\\tau_{yx} & \\sigma_{y}\n\\end{bmatrix}", "035937e14a7259dddea132a7dc81610f": "(A \\lor \\lnot A)", "03594f7a5482079c0f1f6cb2e7ba42cd": "=1/2+2\\epsilon_1\\epsilon_2\\ ", "035955e25306ff79019df1214e7e7780": "K_m=\\tfrac{1}{2} (k_1 + k_2).", "03599cb3a1147c64f7995421d197c7ea": "T(*)=B", "035a1895933f9ad2344ba70e8b3ce4a0": " \\mathbf{A} \\circ \\mathbf{B} = \\begin{pmatrix} A_{11} & A_{12} & \\cdots & A_{1m} \\\\\n A_{21} & A_{22} & \\cdots & A_{2m} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n A_{n1} & A_{n2} & \\cdots & A_{nm} \\\\\n\\end{pmatrix}\\circ\\begin{pmatrix}\n B_{11} & B_{12} & \\cdots & B_{1m} \\\\\n B_{21} & B_{22} & \\cdots & B_{2m} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n B_{n1} & B_{n2} & \\cdots & B_{nm} \\\\\n\\end{pmatrix} =\\begin{pmatrix}\n A_{11}B_{11} & A_{12}B_{12} & \\cdots & A_{1m}B_{1m} \\\\\n A_{21}B_{21} & A_{22}B_{22} & \\cdots & A_{2m}B_{2m} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n A_{n1}B_{n1} & A_{n2}B_{n2} & \\cdots & A_{nm}B_{nm} \\\\\n\\end{pmatrix}", "035a36827f08ba9bcb795d03f1caf852": "H = \\ln(2\\pi I_0(\\kappa))-\\kappa\\phi_1 = \\ln(2\\pi I_0(\\kappa))-\\kappa\\frac{I_1(\\kappa)}{I_0(\\kappa)}", "035aa272cdc59be3d2059ed9d259fef2": "{V_s}", "035b0c810fe8694a8e962c8b902e9be5": "m(x,\\beta)", "035b5644665e0eff926385976add04d4": "\\psi(\\Omega^\\Omega)", "035b7dde426417915d7434b285015c83": "y = \\varphi-\\varphi_0 + \\cot(\\varphi) (1 - \\cos((\\lambda - \\lambda_0)\\sin(\\varphi)))\\,", "035b830fb147793943518050a9f77f23": "{dt}", "035ca07e19340efe54cf6714b3900a19": "m = 3.4~m_e", "035d005b37b1e448ca08dcff67204ba2": " I_2", "035d0437c9ab039649490a1f5da46923": " \\widehat X^\\mathrm{T} = P_Z X", "035d0f09bc271c400235212bc27f4300": " \\frac {\\tau_1} {\\tau_2} \\approx A_v \\frac {R_i} {R_i+R_A}\\sdot \\frac {R_L} {R_L+R_o} \\ , ", "035d1e2a7c93db50eb315ea047c8eb33": "T_a f(x)", "035d37a27f6a2f1387b1af89f4252aa2": "i\\hbar\\frac{\\partial}{\\partial t} \\Psi_\\alpha(\\mathbf{r},\\,t) = \\hat H \\Psi = \\left(-\\frac{\\hbar^2}{2m}\\nabla^2 + V(\\mathbf{r})\\right)\\Psi_\\alpha(\\mathbf{r},\\,t) = -\\frac{\\hbar^2}{2m}\\nabla^2\\Psi_\\alpha(\\mathbf{r},\\,t) + V(\\mathbf{r})\\Psi_\\alpha(\\mathbf{r},\\,t)", "035dbe2b93958b4c5753c67fb434e32b": "n!!!", "035de2830abae7a5890b234834d1e68a": "y(t) = (x * h )(t) = \\int_{a}^{b} x(\\tau) h(t - \\tau)\\, d\\tau", "035e2654176d12b3ebd686c286e749bc": "E_K^{-1}(C) := D_K(C) = D(K,C): \\{0,1\\}^k \\times \\{0,1\\}^n \\rightarrow \\{0,1\\}^n,", "035e6b8d56d3d2fcb2895b95238775c6": "L(x, y, t)", "035e923baf7cc63504d6e27f6afc8d55": "k \\in K", "035ebe0ca0b5b38dcb30fbbfd813e632": "K \\equiv \\prod_{\\omega \\,\\in\\, \\Omega} A_\\omega", "035f95f8636704951c2795c1d3deb25c": "L_0-R_0 = L_{n+1}-R_{n+1}", "036007e5c3f89086b75d6326fb344d74": "\\begin{array}{rcl}\n \\int_0^1 x^{1-t} y^t\\ \\mathrm{d}t\n&=& \\int_0^1 \\left(\\frac{y}{x}\\right)^t x\\ \\mathrm{d}t \\\\\n&=& x \\int_0^1 \\left(\\frac{y}{x}\\right)^t \\mathrm{d}t \\\\\n&=& \\frac{x}{\\ln \\frac{y}{x}} \\left(\\frac{y}{x}\\right)^t|_{t=0}^{1}\\\\\n&=& \\frac{x}{\\ln \\frac{y}{x}} \\left(\\frac{y}{x}-1\\right)\\\\\n&=& \\frac{y-x}{\\ln y - \\ln x}\n\n\\end{array}", "036017945dd7313c43d579e4f8ce2826": "\\varepsilon\\gamma_{\\mu\\nu}", "0360351f5986320b5342bfd2430991ad": "\np(t)\n =\\left(\\tfrac12L\\|F'(\\mathbf x_0)^{-1}\\|^{-1}\\right)t^2\n -t+\\|\\mathbf h_0\\|\n", "03603730b2b3b89b93daebe682ecc09c": "\\scriptstyle\\tbinom{-1}0=\\frac{(-1)^{\\underline0}}{0!}=1", "036045c0ca5b6b4db6970a8bfcabea54": "4 \\cdot m\\ ", "03607b43f29d20703ab22323966ca076": " w < -1", "0360936b2c68a3610f00a35a333887ec": " L_\\text{DC} = L_\\text{cen} + L_\\text{shd} + L_\\text{ext}\\, ", "0360d98007184ca17561e52001606f3c": "x[n/k] \\!", "0360ed882d872b7b815000f26aea9b5b": "\\textrm{PPPrate}_{X,i}=\\frac{\\textrm{PPPrate}_{X,b}\\cdot \\frac{\\textrm{GDPdef}_{X,i}}{\\textrm{GDPdef}_{X,b}}}{\\textrm{PPPrate}_{U,b}\\cdot \\frac{\\textrm{GDPdef}_{U,i}}{\\textrm{GDPdef}_{U,b}}}", "036128b1342fd074969bb61e2f64717e": "\\exp(\\gamma\\,t)\\;, \\qquad\\text{with}\\quad \\gamma={\\sqrt{\\mathcal{A}g\\alpha}} \\quad\\text{and}\\quad \\mathcal{A}=\\frac{\\rho_{\\text{heavy}}-\\rho_{\\text{light}}}{\\rho_{\\text{heavy}}+\\rho_{\\text{light}}},\\,", "036163a369759a544e290723ded6ac64": "-\\frac{b}{2a},", "03616a7b205bb8bae29cb8f2dd3c3672": "x^8=\\left(\\left(x^2\\right)^2\\right)^2.", "0361859e21297ddefe083b152eca8a59": "\\lfloor R^n / n \\rfloor", "036193184a8b39ee07604218efc190ca": "\\scriptstyle 1/2(1-x^2)", "0361f56d756082b809fc65c43892e692": "u: \\mathbb{R}^l \\rightarrow[0, \\infty)", "0362281c967583ca8fe3c72e5117067c": " \\{x_1,...,x_n\\}", "03627a7c6b959f06762436e5cebce5c5": "H= 13 + 6{,}93 \\cdot D", "0362846a7f7340b70c29116354f2812b": " \\lambda_2 ", "0362b84dc2a94087be5aec918183dabc": "\\boldsymbol{H}^\\prime", "0362ba7c7b21c002a805627676c67aa7": "t = \\int\\frac{dy}{iy+F}", "0362bb4e96567e31dff05cb30ebb0da5": " L(n,k+1) = \\frac{n-k}{k(k+1)} L(n,k).", "0362c8290d2dfc0d92533cea11259e76": "Loves(", "0362fd4cc3c69a4b89f60252cbec028d": "\\displaystyle \\alpha_k", "03637d758e05cd5e09ab92f25aed6305": "Cl(p,q)", "03637e55edc44c456709a0bbfe6ad999": "|\\alpha_i\\rangle, \\; a_i", "0364531c758ef179874228d39d030061": "l\\alpha_1,\\dots,l\\alpha_n", "0364911541927bcc589331b468da4b85": "\\frac{1}{24} + \\frac{1}{48} = \\frac{1}{16}", "03650ea866ab2b553590f75acb9f9163": " a = -\\log ( 1 - w_2 ) ", "0365431cc30d2b64a93712155623ba23": "dx^2 - adx + b^2c = 0", "0365ad89f3cba76103d2c14e92a69b7f": "\\frac{\\partial (\\mathbf{U} \\otimes \\mathbf{V})}{\\partial x} =", "0365dcab177cfb0b250d751763492687": "\\langle 0|\\Phi(x)\\Phi^\\dagger(y)|0\\rangle=\\sum_n\\langle 0|\\Phi(x)|n\\rangle\\langle n|\\Phi^\\dagger(y)|0\\rangle.", "03662063cfa5a0174240ca60eb9c35a3": "H,", "03663e2f2fba6abde1a8d8248e274cdd": "\\left(\\mathbb{Q},+\\right)", "0366a9adf0b7ed517a545111a592bb2b": "[ax+by,z] = a[x,z] + b[y,z]", "0366e9804d0813d197a05778bff33376": "t + C_2 = \\pm \\int \\frac{dx}{\\sqrt{2 \\int f(x) dx + C_1}}", "0366fd7f9a3a4c4b41add2c894a60ecf": "\\textstyle\\left \\lfloor {{d-1} \\over 2}\\right \\rfloor", "036761f0aa2fa0464cac43f69851e0d7": "\\,y = \\sin^2(t)", "036775ea0d50fa662df6070e56949133": "\\frac{dS}{dt} = - \\beta SI + \\mu (N - S) + f R ", "0367f3665bc4a38c20508951978243e6": "\\mu_2 = \\mu'_2 - \\mu^2\\,", "0367f40a08359f679931b045a08b1682": "\\varphi\\left(\\mathbb{E}\\left[X\\right]\\right) \\leq \\mathbb{E}\\left[\\varphi(X)\\right].", "03680e271932f7839cee3b91aad8c099": "q = q_s + \\vec{q}_v,", "03688d6eb8297d6f8e0b141fb14bc775": " \\left(\\frac{\\partial U}{\\partial V}\\right)_{T} = T \\left(\\frac{\\partial S}{\\partial V}\\right)_{T} - p = T \\left(\\frac{\\partial p}{\\partial T}\\right)_{V} - p ", "0368a97cbd357e2b4f7789335b837e3e": "S\\subseteq \\cup_{j=1}^t T_{i_j}", "03695c25ced5019184f11784b37187e4": "x = 1 + 5u", "03696c698681b2078c7ad19a3cd30f42": " \\langle\\phi_1\\otimes\\phi_2,\\psi_1\\otimes\\psi_2\\rangle = \\langle\\phi_1,\\psi_1\\rangle_1 \\, \\langle\\phi_2,\\psi_2\\rangle_2 \\quad \\mbox{for all } \\phi_1,\\psi_1 \\in H_1 \\mbox{ and } \\phi_2,\\psi_2 \\in H_2 ", "03697d33bfdc54be41ff20d2c4b87847": "\\partial_\\mu j^\\mu = 0 \\!", "0369d9d9cb78d36ed02ef92d31d3160e": "B_1 = b_1", "0369e06b57f43e284c7cd55476a1c41f": "\\int {\\frac{1}{\\sigma\\sqrt{2\\pi}} e^{ -\\frac{1}{2}\\left(\\frac{x-\\mu}{\\sigma}\\right)^2 }}\\; \\mathrm{d}x= \\frac{1}{2} \\left(\\operatorname{erf}\\,\\frac{x-\\mu}{\\sigma \\sqrt{2}}\\right)", "0369fb0bda2ddf27ae799bc1f11673a2": "M_\\mu", "036a38eb6d71058ff09ac9faa8013704": "\\lambda = (1/15,2/15,3/15,4/15,5/15)", "036a67b72bc85b3ad1e8c257a2c27189": "\\tfrac{1}{\\sqrt{2}} (1 - \\sigma_1 \\sigma_2) \\, \\{a_1+a_2\\sigma_1\\sigma_2\\}=\n\\frac{a_1+a_2}{\\sqrt{2}} + \\frac{-a_1+a_2}{\\sqrt{2}}\\sigma_1\\sigma_2", "036ac1c2364d728da1e49043b7b8106f": "\\mathfrak{P}^{78}", "036ad5b0da4302c0b0d5a78c05547737": " \\mathbf{U} \\cdot \\mathbf{V} = U_0 V_0 - U_1 V_1 - U_2 V_2 - U_3 V_3\\,. ", "036aec19e4bdc4d7637c0b4e0d4b9fb9": " Z_{i} ", "036af450785fc8345206720f53793be1": "\\eta_G = \\eta_C \\, (-1)^I", "036af815806a0089ffc36b2a2156e548": "\\mathrm{erfc}", "036b87bb6514b45d3b3a8150e05190a6": "F=\\rho_{air} \\Gamma (V_{\\infty}+V_{induced}) l\n", "036bab4918e8d35cf702026d645809b4": "\\begin{bmatrix} 0 & \\cdots & 0 \\\\ \\vdots\n& \\ddots & \\vdots \\\\ 0 & \\cdots &\n0\\end{bmatrix} ", "036bc3149dd61d35c0393c32613911fd": "f_k(v_1, \\cdots, v_k) = \\frac{1}{k!}\\sum_{\\sigma\\in S_k}{\\rm sgn}(\\sigma)\\, v_{\\sigma(1)}\\cdots v_{\\sigma(k)}", "036bd5766fa637f165b47eec5ef654e9": "\\hat{H}_{\\text{JC}}=\\Omega_+ \\hat{A_+}^{\\dagger}\\hat{A_+}+\\Omega_- \\hat{A_-}^{\\dagger}\\hat{A_-}+C", "036c4d51804f4cfc1f7b6be7da534417": "x\\in\\left[0,2\\pi\\right)", "036c9e68aa344d939f39628025dcabb2": " \\{y\\in\\mathbb{R}^n: y\\cdot x \\le h_A(x) \\}", "036cbde5c4a7803acf92c88221f7a5a9": "\\varphi(\\mathbf{x},t) \\triangleq [ \\varphi_1(\\mathbf{x},t), \\varphi_2(\\mathbf{x},t), \\ldots, \\varphi_n(\\mathbf{x},t) ]^{\\operatorname{T}} : \\mathbb{R}^{n+1} \\mapsto \\mathbb{R}^n", "036ce8d6961cc156ff87ea219a16e141": "h \\circ f = k \\circ g", "036d0a1262bc79e04bf6920ed13153f9": "\\mathbf{v}=\\nabla v", "036d3e0797d404a40093360cdd543cfe": "\\mathbf{C}_{ij} = (-1)^{i+j} \\mathbf{A}_{ij} \\,", "036d75d36a3508b66e2a675c91442f09": "{\\overline P}X = \\mathbb{U}", "036db09816e8def403de9d47dea610c2": "\\lim_{y \\to \\infty} t(y) = 1,", "036e030db648c47d62ddb0d4e10323b8": "\\omega = \\frac {W}{L}", "036e266a7f7a81431af068f2315d04b7": "\\kappa z + \\lambda = \\nabla \\cdot \\mathbf{\\hat{n}}", "036e345124cff1e03283e850d1de41be": "l_{11} \\cdot u_{11} + 0 \\cdot 0 = 4", "036e4d68fbe40841581387b1d11a3814": "d\\theta^i=-\\frac12 \\sum_{jk} c_{jk}^i\\theta^j\\wedge\\theta^k", "036e5d0a20bd67842cd5ee1038993706": "A_{k1},A_{k2},\\dots,A_{kn}, (k=1\\dots m)", "036e72e42c2a752b4fcdf09fd6ae1906": "x = \\sqrt[m]{a^n}", "036e8acc23240ab856207bb391337d76": "\\left( x = y \\right) \\to \\left( \\phi[z:=x] \\to \\phi[z:=y] \\right)", "036ea9d74180c5d430ffbe4a5eb6aa73": "2*10^8", "036f131b92b23cf3a3f92416421aa04d": "n=2,4,\\dots", "036f37927080836aeaa0729bd8f1f6b3": " \\operatorname{int}(A \\cap B) = \\operatorname{int}(A) \\cap \\operatorname{int}(B) \\! ", "036f3aded6a9f3a4f82302cdfb4adc9e": "\\pi\\varepsilon", "036f5baac14db26e5b399f531e73aafe": "Z(S_0) = 1", "036fb716ae56dc260376c97fdd78067d": " \\begin{align} \n \\bold{r}(t) \n & \\equiv \\bold{r}\\left(x,y,z\\right) \\equiv x(t)\\bold{\\hat{e}}_x + y(t)\\bold{\\hat{e}}_y + z(t)\\bold{\\hat{e}}_z \\\\\n & \\equiv \\bold{r}\\left(r,\\theta,\\phi\\right) \\equiv r(t)\\bold{\\hat{e}}_r(\\theta(t), \\phi(t)) \\\\\n & \\equiv \\bold{r}\\left(r,\\theta,z\\right) \\equiv r(t)\\bold{\\hat{e}}_r(\\theta(t)) + z(t)\\bold{\\hat{e}}_z \\\\\n & \\,\\!\\cdots \\\\\n\\end{align}", "036fc1649ceb56182bc4b4a7e2bd80d9": "\\vartheta^{\\perp}", "03702fc3ab6229d5ae9f464d2eec9807": "3 * \\frac{\\sin{\\pi} - 2}{e}", "037054e3d3f2ced9c1c7d049305029a2": "\\mathcal{F}_{L^1}:L^1(\\mathbb{R}^d) \\to L^\\infty(\\mathbb{R}^d)", "037056ef8b9a61d9b133d95776ab0cc9": "H^{(n,h)}_R", "03706c64687ca7cbfa954244ae7e7743": "\\pi_i F", "0370934d456d35275de618335162f5d1": " s_\\lambda(x_1,\\ldots,x_n) = \\sum_T w(T), ", "0370cd983fecaef435601e4a2e5e87b1": "E = E + \\Delta E", "03714d539da1be0f8f3208e8df276539": "\\mathcal{F}_{T}=\\mathcal{F}_{0}+\\mathcal{F}_{d}", "03717e6f404cf96b76df449185e9e5b2": "\n\\lim_{\\lambda\\to 0}W_\\lambda\\chi_E(x)=\\chi_E(x)\n", "037184bd8627aad6f55db0c18e2bdfcc": "K/9IP = 9 \\cdot \\frac{K}{IP}", "03722d47184d8f1325b819f8f8d1c13e": "{{\\left\\{ {{\\phi }_{\\gamma }} \\right\\}}_{\\gamma \\in \\Gamma }}", "03723a5ff6915f3c97d53616e7e69f14": "\\delta \\psi = u \\delta y\\,", "03726358bae9d9db14fb16f36d2406c0": "\\sigma_{ij}= s_{ij} + \\pi\\delta_{ij},\\,", "0372baedaa6d0c831c5186a5ea6a194b": " \\sum_{m=1}^{n} m k_m = n", "0372c1cbf6c34ba77fbe60249129bc39": " h \\ \\bmod q^n-1 ", "03731de8db9bfa07b88bd460a4e69725": "\\frac{e^{\\mu z}\\gamma K_1(\\delta \\sqrt{ (\\alpha^2 -(\\beta +z)^2)})}{\\sqrt{(\\alpha^2 -(\\beta +z)^2)}K_1 (\\delta \\gamma)} ", "037342e71c300f6ba4b66ba429a1906e": "\\{EG,AF,AU\\}", "037368dcfb66641f9adc3d8cf0d1449e": "\\sigma(\\varphi)", "037369c5d056ef947d47b1438cfdd880": "\\theta_s = \\arccos(v \\cos\\theta/v_s)\\,", "03737d5105885405c3250ef71d679334": "H_{nop}", "0373863add1d0e45ebdfc2d4906e2978": "\\frac{36}{p}", "037390b143773f6fb1522a11c7b75522": "{\\det}_p", "037396c55d8e4398ea0187e334dd8bd8": "\n\\xi(\\alpha) \\approx \\sqrt{1 - \\alpha^3}\t\t\t\t\t\t\n", "0373c933a885a2f6a1231a8cb5412b68": "L(q, \\dot{q}, t)", "0373d856fc469552a17c78297e391d35": "\n\\begin{align}\nF(A^1, \\dots, cA^j, \\dots) & = \\sum_{\\sigma \\in S_n} \\sgn(\\sigma) ca_{\\sigma(j)}^j\\prod_{i = 1, i \\neq j}^n a_{\\sigma(i)}^i\\\\\n& = c \\sum_{\\sigma \\in S_n} \\sgn(\\sigma) a_{\\sigma(j)}^j\\prod_{i = 1, i \\neq j}^n a_{\\sigma(i)}^i\\\\\n&=c F(A^1, \\dots, A^j, \\dots)\\\\\n\\\\\nF(A^1, \\dots, b+A^j, \\dots) & = \\sum_{\\sigma \\in S_n} \\sgn(\\sigma)\\left(b_{\\sigma(j)} + a_{\\sigma(j)}^j\\right)\\prod_{i = 1, i \\neq j}^n a_{\\sigma(i)}^i\\\\\n& = \\sum_{\\sigma \\in S_n} \\sgn(\\sigma)\n\\left( \\left(b_{\\sigma(j)}\\prod_{i = 1, i \\neq j}^n a_{\\sigma(i)}^i\\right) + \\left(a_{\\sigma(j)}^j\\prod_{i = 1, i \\neq j}^n a_{\\sigma(i)}^i\\right)\\right)\\\\\n& = \\left(\\sum_{\\sigma \\in S_n} \\sgn(\\sigma) b_{\\sigma(j)}\\prod_{i = 1, i \\neq j}^n a_{\\sigma(i)}^i\\right) \n + \\left(\\sum_{\\sigma \\in S_n} \\sgn(\\sigma) \\prod_{i = 1}^n a_{\\sigma(i)}^i\\right)\\\\\n&= F(A^1, \\dots, b, \\dots) + F(A^1, \\dots, A^j, \\dots)\\\\\n\\\\\n\\end{align}\n", "0373e738ee42ca54133f6dfabc5843f5": "b\\eta e^{bx}e^{\\eta}\\exp\\left(-\\eta e^{bx} \\right)", "0373eae7aead6e972464419409a60233": "\\text{(***)}\\qquad \\left\\|\\int_a^b v(t)\\,dt\\right\\|\\leq \\int_a^b \\|v(t)\\|\\,dt.", "0373ee19f5a13f88cece7ae2f9882d67": "A_0(h) = \\frac{f(x+h) - f(x)}{h}", "0373f990d02a2e5a460fe5a515bc5ae8": "\\frac{\\mathrm{d}^{2}u}{\\mathrm{d}\\theta ^{2}}+u=Cu^3", "0373febfed9fdcbc4a6eaf82c5c55b90": "\\Lambda_n[V_n]/\\langle V_n^2-\\Delta\\rangle", "0374055d88f1f7d950a5d60b0583acfc": "h:x\\to e", "03746ec1664988d3d138cf8b140d356f": "\\mathrm{Tr}_3\\big((|000\\rangle + |111\\rangle)(\\langle 000|+\\langle 111|) \\big) = |00\\rangle \\langle 00| + |11\\rangle \\langle 11|", "037488506c0517b9873957758730ac8b": " \\begin{align}\n \\lambda_1 - 3 \\lambda_2 &{}= 0 , \\\\\n \\lambda_1 + 2 \\lambda_2 &{}= 0 .\n\\end{align} ", "0374b768cce8b8c2881225f6d35574a5": "d \\geq 1", "0374c643357c023c9eef0d52be33a5ed": "W' \\subseteq \\mathcal{B}'", "0374d3d4a26403ce7b7a67f21d47ce70": "\\operatorname{int}A = \\operatorname{core}A", "0375bbdf779e79128ac563157984445d": "\\left\\{\n\\begin{array}{ll}\n\\Delta \\phi + \\lambda \\phi = 0 & \\mathrm{in\\ }\\ U\\\\\n\\phi=0 & \\mathrm{on\\ }\\ \\partial U.\n\\end{array}\\right.\n", "0375c9020d5e36fb357fe60906fbc20c": "Re(K(\\omega))", "0375d6841aac6c9721a1c86a11259576": "t=\\tan^{1/3}\\theta. \\,", "0375eedd14436a3708e34f1a115e69f5": " \\tau = \\frac{\\theta}{2\\pi}+\\frac{4\\pi i}{g^2}.", "0376020ce4e93edc22fe4f0c3b0dcb24": "{D \\zeta \\over Dt} + \\beta v = f_0 {\\partial w \\over \\partial z}", "0376445b7709ebfaf078aeaa6257bde0": "\\overrightarrow{T} = \\frac{3Gm}{r^3}(C-A)\\sin\\delta\\cos\\delta\\begin{pmatrix}\\sin\\alpha\\\\-\\cos\\alpha\\\\0\\end{pmatrix}", "037647636bf1c631caf827d1421458f2": "{{\\varepsilon }_{medium}}", "037654bdad0176286f44c6244d63d6f3": " f(z) = (z - r_1)\\cdots (z-r_n),\\qquad (n\\ge 2) ", "0376892fd567d547374e1c2e56c625da": "e^{{\\rm{i}} \\theta}", "037699b6e9751320acbe2421c3267bb4": "\\sigma_y^2(\\tau) = \\frac{1}{2}\\langle(\\bar{y}_{n+1}-\\bar{y}_n)^2\\rangle = \\frac{1}{2\\tau^2}\\langle(x_{n+2}-2x_{n+1}+x_n)^2\\rangle", "0376bf22b868a6ad8a73538d359b46f7": "r_e", "0376cab0078bf0d02682cbcd698e56ca": "~T(\\gamma)=\n4\\gamma^{1/2}\n\\left(1+O(1/\\gamma)\\right) ~", "0376f2805d5898fcf697d33df97d0262": "|\\mathbf{x} \\times \\mathbf{y}| = |\\mathbf{x}| |\\mathbf{y}|~\\mbox{if} \\ \\left( \\mathbf{x} \\cdot \\mathbf{y} \\right)= 0.", "0377064bd785a7da5f8d035e6b2c1961": "\\operatorname {dn}\\; u = \\sqrt {1-m\\sin^2 \\phi}.\\,", "037714735ed79521f7c696480eaf9693": "(1+z)^u = e^{u\\log(1+z)} = \\sum_{k = 0}^\\infty (\\log(1 + z))^k \\frac{u^k}{k!},", "037744af11534dbacf73329e86342442": "q^{2}-1", "0377566f0297e84c0cdac00b4475134e": " \\nu_{\\rm yx}", "0377a708aa8dcc7522b35aabf8036027": "F(\\mu)=\\frac{m + \\delta}{n}", "03780996512787f65644298a2159d74c": "\\frac{|v-c|}{c}<2\\times10^{-9}", "037896f8e77eff0f9551b4591460d25c": "\\begin{align}\n \\cosh (2x) &= \\sinh^2{x} + \\cosh^2{x} = 2\\sinh^2 x + 1 = 2\\cosh^2 x - 1\\\\\n \\sinh (2x) &= 2\\sinh x \\cosh x\n\\end{align}", "0378c940cadd4e337b0acae7655a2d9b": "P(X^2+1) = (X^2+1)^2-1 = X^4+2X^2", "0378d5b8c35130a441bbee1e68bac4f3": "\\vec{x}_1", "0378da57d283b686b2341fde12e3dd55": "\\eta=1/P(o\\mid b,a)", "0378eab415625ea5efd4ea711c1e59ce": "y' = re^{rx} \\, ", "0378fa6ba5361846976d3dea1cc64c50": " \\alpha \\in \\mathbb{R}", "037935127383c734dcb152dc775276b2": " f(\\boldsymbol{x}) + f(\\boldsymbol{y}) \\ge f(\\boldsymbol{x} \\wedge \\boldsymbol{y}) + f(\\boldsymbol{x} \\vee \\boldsymbol{y}) ", "037997c657ca62c38f560b041de1a550": "\\frac{33}{32}", "0379bbec0e9b0605c52b2127dd734599": "\\frac{(2n)!}{2^n\\,(n!)^2}\\,", "037a233b676370a840d2f51971fd3ebb": "G_{r}^{n}", "037a38293205b83b7920a20def8d7126": "F_X(x) = \\operatorname{E} \\left [\\mathbf{1}_{\\{X\\leq x\\}} \\right],", "037a412ff67eef88af781bd5728f0689": "J_{\\mathrm{eff}}", "037a4753e17bea1ed40d64798a88cef7": "\\{\\textit{SENTENCE}, \\textit{NOUNPHRASE}, \\textit{VERBPHRASE}, \\textit{NOUN}, \\textit{VERB}, \\textit{ADJ} \\}", "037a66437e12145951fdac16d22f8374": "\\pm\\left(\\sqrt{\\frac{5}{2}},\\ \\frac{-7}{\\sqrt{6}},\\ \\frac{-1}{\\sqrt{3}},\\ \\pm3\\right)", "037aa4171e65cc137237f2797823fe38": "\\mathrm{NPV}(R(t)) = \\langle w, R\\rangle = \\int_{t=0}^\\infty \\frac{R(t)}{(1+i)^{t}}\\,dt.", "037ab61010b38491adbf983275260958": "\nx_i = \\bigvee_{j=1}^n (g_{ij}\\wedge y_j), i = 1, 2, \\ldots, m,\n", "037afe33ba5d543202f3a709f46394c1": "\\left(\\frac{\\partial S}{\\partial T}\\right)_{P}", "037b43965d2d07927fafe6ab4fa8f84a": "|K(x-y)-K(x)| \\leq C \\frac{|y|^{\\gamma}}{|x|^{n+\\gamma}},", "037b4b461271363e5902aeffe6c25a00": " \\textbf{h} = p\\textbf{f}_q \\cdot \\textbf{g} \\pmod q. ", "037b89b5bbe84570515c008c3674fd09": "v_{\\rm e} = g_0 I_{\\rm sp} \\,", "037bc51126f8b95d163083b50e45e317": "\\{\\to,\\land,\\lor,\\bot\\}", "037c0702f308e62ee6b430717aa5007c": "\\mathcal{H}_n = ([n], \\{E \\subseteq [n] \\mid | E \\cap [2k]| = | E \\setminus [2k]|\\})", "037c0a2568e7e1416ebc8f304b58dc04": "(i\\omega-\\xi)^{-1}", "037c7474cf061e8fe282c1bef172ad40": "\\{ w \\in \\Sigma_1^* | \\exists q \\in F . (q_0,w,\\epsilon) \\vdash^* (q,\\epsilon,\\epsilon)\\}", "037c7ff9d16428cd312a69859f16b8c8": " \\varphi(\\mathbf{r},t) = \\int\\frac{\\nabla'\\cdot{\\mathbf E}(\\mathbf{r'},t)}{4\\pi R}d^3r'-\\frac{\\partial{\\psi(\\mathbf{r},t)}}{\\partial t}", "037c812791cba4937537a02529d1ac95": "\\bar{V}_i\\otimes V_j", "037d2b07b54fd4e8670c12ecdabcd7f3": "v=v_0", "037d75c55ba6dd3a38f3ef9fcd478337": "K ", "037da86913143f31402a12f5f79f2000": "\\xi _i ", "037e1daa38c30fd321c1d5fa53b5a86d": "\\mathrm{Hom}_{D(A)}(X, Y) = \\mathrm{Hom}_{K(A)}(X, Y).", "037e2e99d8c00d57092b7c7eaf086180": "G:=(V,E)", "037e3dad8c52cb42410e614ed79453aa": "\n\\left|{\\partial \\mathbf{x} \\over \\partial s}\\times {\\partial \\mathbf{x} \\over \\partial t}\\right| = \\left|\\left(\\sum_i {\\partial \\mathbf{x} \\over \\partial q^i}{\\partial q^i \\over \\partial s}\\right) \\times \\left(\\sum_j {\\partial \\mathbf{x} \\over \\partial q^j}{\\partial q^j \\over \\partial t}\\right)\\right|\n", "037e3f00eac3133acd34a2d34c5d8521": "S_6,", "037e711358b5150f62f4b528adaaebed": " \\Sigma = N \\, \\sigma ", "037efc824cbe954dbd5aa581e0156503": "\\,\\hat{m}_1", "037f2c9ddd738e43d71a260275ad5049": "1-g", "037f2d988c51e5390be2a0ecc20ba321": "\\ v_o = A_v v_i \\ . ", "037f54d1963f8292451829bd2a72c387": "\nD_{t}(x_i,x_j)^2 =\\sum_y (p(y,t|x_i)-p(y,t|x_j))^2 w(y)\n", "037fa690256a7bb1884c165d56a87cfb": "u = \\int \\frac{du}{dx} \\,dx", "03803c2f702255059aa8704d1a8da64d": "\\Diamond_i P", "0380957958d369064832e39c069858f0": "t_2^\\prime = 1/f^\\prime", "03809891f820600376128c1d84da3ef0": "0 < b < 1", "03809e73c75758bd32204599b06f608d": "q^{\\eta \\sum_j t_{\\lambda_j} \\otimes t_{\\mu_j}}", "0381a7f5503b5d8e3cc2998f04027717": "\\displaystyle{T_sf(x) ={1\\over 2\\pi}\\int_{\\mathbf{R}^2} {s f(x)\\over (|x-t|^2 + s^2)^{3/2}}\\, dt.}", "0381aa4bb63ef5cbc074eb74c1cb31e3": "\\sigma(V_{\\mathbb{R}})\\subset V_{\\mathbb{R}}\\,", "0381d613bf0e2d5cb45fc42bd256b6c2": "\nk=\\frac 1\\hbar (\\frac E{D_\\alpha })^{1/\\alpha },\\qquad 1<\\alpha \\leq 2.\n", "0381fa2dcaf53009ebe5ffb0d2d872c0": " \\mathrm{Ric}^{}_{}(X_p) = f(p) X_p", "038210fc308436633142e5b8e5d10f0b": "H(e^{i\\theta})=e^{ih(\\theta)},\\,\\,\\, h(\\theta+2\\pi)=h(\\theta)+2\\pi,", "038215c7e1001f5f2fb42c3e577451d0": "\\text{s.t.} =\n\\begin{cases}\n g_{1}\\left(x,y\\right) & = y + 9x \\geq 6 \\\\\n g_{1}\\left(x,y\\right) & = -y + 9x \\geq 1 \\\\\n\\end{cases}\n", "038282908e59d4d3896567c5ba9ce4be": " (\\boldsymbol{\\sigma} \\cdot \\mathbf{p})(\\boldsymbol{\\sigma} \\cdot \\mathbf{q}) = \\mathbf{p} \\cdot \\mathbf{q} + i(\\mathbf{p}\\times\\mathbf{q})\\cdot \\boldsymbol{\\sigma} ", "0382cfd42a408734bc1b8068a658c72c": "Z = 1 + \\frac{2}{t} - \\sqrt{\\left (1 + \\frac{2}{t} \\right)^2 - 1}", "0382f49a76f265907c281437afcb4abb": "\\frac{x_1+x_2+\\cdots+x_n}{n} \\ge \\sqrt[n]{x_1x_2 \\cdots x_n}", "0382f611dfb3c7f92e157811043231f8": "\\mathrm{Be} = \\frac{\\Delta P L^2} {\\mu \\alpha}", "03834bf648aa1cc9df49b37db483c193": " U^{A}_{jj'} = H_{jj'} + \\sum^{B}_{\\gamma \\neq j,j'} \\frac{H_{j\\gamma}H_{\\gamma j'}}{E_0-E_{\\gamma}} = H_{jj'} + \\sum^{B}_{\\gamma \\neq j,j'} \\frac{H^{'}_{j\\gamma}H^{'}_{\\gamma j'}}{E_0-E_{\\gamma}} ", "0383533bb2f23d79a9552e71e8605b77": "f : X \\to Y\\,", "03835cb8c1117fd6b5484f1178c92773": "f\\left(x,u\\right) = -x + u,", "038364fa2b3d85cc9ac202f9fc9af2e5": "C_k=X\\setminus U_k=\\varnothing", "038365d99efa684daea1919d085eeeca": "K=\\mathbb{C}(x)", "0383667c0334bc7e23f0a0b76e426761": "\n\\tilde{p}(\\bar{b}) = \\frac{1}{I} \\sum_{i=1}^I E[\\delta_{b,\\bar{b}}]\n", "0383bd103c8fe4f7aaba585a82a6de6b": "V_\\Sigma^*Y", "0383d0389f143a22feb1c6753e05ace8": "\\scriptstyle \\lambda_j", "0383d6783b0d47a01a151169487f49d9": "\\left \\|\\Gamma^m \\varphi_1 - \\Gamma^m\\varphi_2 \\right \\| \\leq \\frac{L^m\\alpha^m}{m!}\\left \\|\\varphi_1-\\varphi_2\\right \\|", "03840f00a94261bd5ad1dfdd0fff9208": "{dx_1 \\over dt} = r_1x_1\\left({K_1-x_1-\\alpha_{12}x_2 \\over K_1}\\right)", "03840f4a67085a30dee8e4227ede1797": "D_{\\mathbf{v}}{f}(\\mathbf{x}) = \\lim_{h \\rightarrow 0}{\\frac{f(\\mathbf{x} + h\\mathbf{v}) - f(\\mathbf{x})}{h}}.", "038436a5a91c995fcffc56af94062aed": "\\mu(\\xi)", "03846b88d87d5b51da1bb22ca3018143": "\\mathbf{C} = (C_x,C_y)", "0384894ad2cad3e86d932dbdb7202511": "\\psi_n(x)", "0384a5f21aee7a0fd4f40bf0148fa01a": "p\\equiv l \\pmod{q}", "0384c9b28853e0bc446fd4cc1c30e479": "N_t = N_d \\quad \\frac{\\ln 10}{2 \\ln \\frac{b_n}{a_n}}", "03854265fc1eaaf28071f1456a906318": "T_{f\\cdot g} x = T_f x \\cdot T_g x", "038577fa9ec9e7ab8ded2bf564099f4b": " F(\\mathbf{x}^{(0)})=58.456 ", "0385a9fbf3fd60aef19d34c99a222e51": "\\lambda_{0i}", "0385bbcf6ef717868c25f848630bcc47": "\\mathcal{} f", "0385d388b31cb723c1c665715d753b65": "H_{\\frac{1}{a}} = \\frac{1}{a}\\left(\\zeta(2)-\\frac{1}{a}\\zeta(3)+\\frac{1}{a^2}\\zeta(4)-\\frac{1}{a^3}\\zeta(5)+\\cdots\\right)", "0385f072bb5b43c5ba07181eb9c1a71f": "\\begin{align}(a+b+c+d)(x+y+z+w)=&\\,((a+b)+(c+d))((x+y)+(z+w)) \\\\ =&\\,(a+b)(x+y)+(a+b)(z+w) \\\\ &\\,{}+(c+d)(x+y)+(c+d)(z+w) \\\\ =&\\,ax+ay+bx+by+az+aw+bz+bw \\\\ &\\,{}+cx+cy+dx+dy+cz+cw+dz+dw. \\end{align}", "03864907a9123f61c411aa972d7242d8": "density_s", "03864965bdf3283ac3eeab59ae88c7c4": "\n\\int_{\\boldsymbol{\\theta}} \\prod_{j=1}^M P(\\theta_j;\\alpha) \\prod_{t=1}^N P(Z_{j,t}|\\theta_j) d\\boldsymbol{\\theta} = \\prod_{j=1}^M \\int_{\\theta_j} P(\\theta_j;\\alpha) \\prod_{t=1}^N\nP(Z_{j,t}|\\theta_j) \\, d\\theta_j .\n", "03864fca9d766129e5a058b5d8a57fa2": "x+y+xy", "03872bc0e3bb7b2465a7fe1660aaf724": "\\{l\\ |\\ l \\in L\\}", "038747d1464dcbcf21ba14def68dad8e": "\\Omega\\subseteq\\Omega_1", "03879ded5b5a9b27f4bb081cdcff6e8e": "\\mathcal{J}^{\\alpha\\beta\\gamma} = (X^\\alpha - Y^\\alpha )T^{\\beta\\gamma} - (X^\\beta - Y^\\beta )T^{\\alpha\\gamma} ", "0387f562901f1262ed64c811ee661c7a": "\\nu = \\frac{\\mu}{\\rho}", "038830c168c0bf1e26b9e4cca663e4e5": "\\mathcal{E}^{(0)} = \\left \\{z \\in \\mathbb{R}^n : (z - x_0)^T P_{(0)}^{-1} (z-x_0) \\leq 1 \\right \\}", "03884f8e09487d85d849360a2a492a45": "\\frac{1+2x}{(1-x)(1-4x)}", "03885d64afc508cc84f91e010ef53d4c": "d_{\\text{eff}}", "0389297e08112ab3dd87454d65e510ab": " \\frac{Q_1}{T_1}-\\frac{Q_2}{T_2}=0 .", "038935d89deb3eedd1ff79a044a67303": "J_{n\\,(\\text{base})} = \\frac{q D_n n_{bo}}{W} e^{\\frac{V_{\\text{EB}}}{V_{\\text{T}}}}", "0389392b62cc02dc8df5ebfcd3b9629e": "x_n=1", "0389c4f1ff10f51b0b157c6495a1f36f": "\\lambda_{ex}", "0389c86b454b10b0ece9a9aa04d5e305": " \\sum_{i=1}^d{x_i^k},k \\ge 1 ", "0389cc5d3438522a7bbb1dee11cb4bb8": "\\beta(z) = \\beta^* + \\dfrac{z^2}{\\beta^*}", "038a2fc2c166bd4d273bb926a4e69f5b": "(a_d,b_d,c_d)", "038a3710e4ba601e5d1e85105805dd44": "n(n-1) + \\frac{1}{4} - a = \\left(n-\\frac{1}{2}\\right)^2 - a = 0", "038a8f14beb2f599d17425044bf88299": "[2^m - 1, 2^m-m-1, 3]", "038ac5d15adf387f4e3972f82f2ef051": " \\dot{m} ", "038ad5ab428e374131b0791fc13a461e": "\\left ( \\int_{-\\infty}^\\infty e^{-x^2}\\,dx \\right )^2=\\pi,", "038ada52f52d90583a3f20612b16e70f": "1 + z = \\sqrt{\\frac{g_{tt}(\\text{receiver})}{g_{tt}(\\text{source})}}", "038b327459f71c65a49e22ae54589573": "-\\frac{\\hbar^2}{2m} \\nabla^{2}\\psi_{2} + (\\tilde{u}_{2}- E)\\psi_{2} + \\frac{\\hbar^2}{2m} [2\\mathbf{\\tau}_{12}\\nabla + \\nabla \\mathbf{\\tau}_{12}]\\psi_{1} = 0 ", "038b6ccee470797d7128ab49a3e7d9d4": "\\gamma = \\frac{1}{A} (E_{1} - E_{0})", "038b85dc1de6025556249750059aab2c": "\\int \\! L \\, \\mathrm{d} \\theta = \\frac{T}{2 \\pi} ( - ( \\cos{\\theta_1}) \\cdot R - r \\cdot \\theta_1) - \\frac{T}{2 \\pi} ( - ( \\cos { \\theta_2}) \\cdot R - r \\cdot \\theta_2 ))", "038bde9b44dcff2cf0283077ef222b24": "\\textstyle\\text{rate(propagation)} = k_p[\\text{M}^+] [\\text{M}]", "038bdf3c2dedfcdb4211b26a3f6e84b4": "\\langle A_n \\rangle", "038c5c7732d26f91aa976779587da51c": "\\scriptstyle \\dot{}", "038c70349fa67c5870b3aedd6f57ab15": "C = \\frac{\\varepsilon A}{d}", "038ca19ab913b6bcbce87bac5dc2bbc4": "ba^{RC} \\notin \\mathcal{O}", "038cc55a61f269484f124b61d5b6464d": " \\mathcal{U} \\equiv \\frac{H\\, \\lambda^2}{h^3}.", "038cd962d0b2d6a5694382619cb14319": "E_N(\\rho) \\equiv \\log_2 ||\\rho^{\\Gamma_A}||_1", "038cf6c6d831d275d680a6e60d66265e": "dh = C_pdT", "038d04516677e9fb54be37b4707517f8": "nk\\log k", "038dd71f1b87d3876200c0f8a51df38e": "\\frac{\\mbox{d}A_1 \\ \\, \\mbox{d}A_2 \\ \\cos{\\theta_1} \\ \\cos{\\theta_2}}{r^2}", "038e2c07c3e25f2d4a199d3f8c7a6665": "T = \\sqrt[4]{ \\frac{(1-a)S}{4 \\epsilon \\sigma}}", "038e51f8a130284e07e8d1d57a5a9d1d": " P_\\mu = \\left(E/c,\\vec p \\right)", "038e93d36bfcdca83a124aeb85dac36f": "=\\int_{\\mathbb{R}^n} f(x)e^{-2\\pi i x\\cdot \\nu} \\left( \\int_{\\mathbb{R}^n} g(y) e^{-2 \\pi i y\\cdot\\nu}\\,dy \\right) \\,dx", "038eeb291fdda2c431db906966293543": "\\langle \\psi |\\phi\\rangle = \\langle \\phi |\\psi\\rangle^*", "038ef3077cd8f59b36164684c2b427d5": "\\mbox{Golden rule savings rate: } s^G=\\frac{mpk^G}{apk^G}", "038efeea3253bb25ad6c3f04702bec33": "\\omega_1\\to(\\alpha)^2_k", "038f0a9c16ed5bc7b6b8d5a91a5379aa": "Y = \\{ Y_1, Y_2, \\ldots , Y_s \\}", "038f257a4df11011d255f6987b88b940": "(S)\\,", "038f2cc623f044cd4cdbafbe77e5ea11": "F^* \\tilde g = e^{2\\varphi} g", "038f2d4e909f2f51fc3d89a8def54df5": "\\alpha \\smile \\alpha", "038f81da29e7124964889dd8a1dbdaca": "\\frac{\\sqrt{a^2-b^2}}{a}", "03900ab4df72f17a1c2261a18ba70aae": "f(x, \\boldsymbol \\beta)=\\beta_1 +\\beta_2 x", "039030e4c23c03efd88de8e91f221d3f": "\\langle W,R,\\{D_i\\}_{i\\in I},\\Vdash\\rangle", "03905a878126f449a38a2d1da24ea191": "69^{7} \\approx 2^{43}", "0390a850ed66d32e5fdb51566ec563ff": "a|\\alpha\\rangle=\\alpha|\\alpha\\rangle", "0390fbf11b84d5d0a243a933d468d354": " E=E^0+\\frac{RT}{nF}\\ln{A}", "03913b430d130efa8df572544967cd4c": "0 < r < R \\leq \\rho,", "03914a417c6d4d150c99af7d069a7a44": "{ }P_x=\\frac{A_{x}}{\\ddot{a}_{x}}", "0391aa9bf208f16bf05af1ac2ded30db": "E_z", "0391e09ef9dc27ac9ad3b0df4a59be9f": "\\int_{\\Lambda^{m\\mid n}}f\\left( x,\\theta\\right) \\mathrm{d}\\theta\\mathrm{d}x=\\int_{\\Lambda^{m\\mid n}}f\\left( x\\left( y,\\xi\\right)\n,\\theta\\left( y,\\xi\\right) \\right) \\varepsilon\\mathrm{Ber~J~d}\\xi\\mathrm{d}y", "0391e1608ca99e405c9a44258dec721b": "\nw=\\sqrt{KW-UV}\n", "039200a02ae33a09f916fd490682d171": " A = {\\rm adj}(B) ", "03927152c0cd08b3bf1cfe62e032f5b8": "P(y,x)", "03931c8ee9b99a12d3351d369f078bd2": "x^2 + x + 1", "03934e7e0730754682575bf7f01b9daf": "\\frac{\\alpha}{\\beta}.\\beta = \\alpha\\beta^{-1}.\\beta=\\alpha", "03936b068bd5908d018878c57e10ad28": "T = \\frac{Rh_bh_c}{a}", "0393f0e43221b5545069d03e804595fb": "\n \\boldsymbol{S} = \\cfrac{\\partial W}{\\partial \\boldsymbol{E}} ~.\n ", "039476b65dd78b8c2638531f2b81a989": "\\Sigma_k \\hat{\\textbf{d}}_j ", "0394815b45d7615e683c44728aa19e9c": "\\forall x. A", "03949a21221acca1f2d9e53c68dea93b": "(n^2)", "0394b5c9805f0ec750e4c1d4aa92e611": "(\\mathbb{Z}/n\\mathbb{Z})^*.", "0394d2f46758455667a8e7827f20e15b": "\n\\begin{align} \n& A=-\\frac{1}{2}R_{0}\\frac{d\\Omega}{dr}|_{R_{0}} \\\\\n& B=-\\frac{1}{2}R_{0}\\frac{d\\Omega}{dr}|_{R_{0}}-\\Omega \\\\\n\\end{align}\n", "03952d34eb58a0d6ee46a795d07f43f9": "s={1\\over{2}}(3n^2-3n+1)(3n^2-3n+2)={9n^4-18n^3+18n^2-9n+2\\over{2}}", "03952d6280fd93ee0b7fee4d8f356b8b": "|f(\\xi)| \\le M", "03953be2782e96e30064637cec023351": "~t_0~", "03955481cb22251600ac99715ce44954": "\\cos 2\\theta_{\\mathrm{eq}}=1", "03955a7f978fb28a1a5858c31b7ffdeb": "l\\ ", "039570c1315ecd56c74d5bce803cb764": "\n\\left\\langle {J(t;F_e )} \\right\\rangle = \\left\\langle {J(0)\\exp [ - \\beta V\\int_0^t {J( - s)F_e \\;ds]} } \\right\\rangle _{F_e }. \n\\,", "0395886a7bcb04205448e0c0ff07f46f": "z_3 = 0", "0395a1f3d0bccf84440ef0c88fd6f116": " C(t) - P(t) = S(t) - K \\cdot B(t,T)\\ - D(t), ", "0395e6460de9f9ccd696d7688c4e79a3": "nT\\,", "03962b2def67fd1b6e569f33ceb7e43b": "\\gamma_{00}", "03962ffa1b76426a2e565d2343eb3e39": "\\bar{H}", "03967700d8392344b52204a6b5d4a916": " \\sqrt{n} ", "03968360cd326aa15e41060ff247e95c": " xP = x(1 - x - y) = x - x^2 - xy ", "039695a9ac4d4131c8d6f032ce0e9942": "\\beta = \\alpha / \\gamma\\,", "0396ccb4fd35747b42a23349aca95c36": "A_{sn} = A_s \\left ( \\frac {\\left (1 - \\frac {25}{1000} \\right )}{\\left (1 + \\frac {\\delta^{13}C}{1000} \\right )} \\right )^2", "0396d685f66076c856145763149ca6c6": "\\Omega_\\mu =\\frac{1}{4}\\partial_\\mu\\omega^{ij} (\\gamma_i\\gamma_j-\\gamma_j\\gamma_i)", "03975ee84ae58a2154ad5571b39dfce8": " = \\frac{G_{wr}}{G_w}", "039761dd6b0195e2cc69e3b8a0ebf0c3": "\\frac{x_j-x_m}{x_j-x_m} = 1", "03980a4b987a89d69667e8ca894e25ae": "\\{ 1,~i,~\\varepsilon ,~i_0 \\}", "039860856ae56d90e8ba163a0eaaffbc": " \\forall x \\in \\mathbb{Q}", "0398646ed7de77d19d2f3390030229fa": "Coupon yield = \\frac{C}{F}", "039882931e9bc6bdeac618f686046310": "z_2=-\\sin i \\cdot \\cos \\Omega", "0398b9a6b5e23d80d8ebabc77a019add": "t_1^\\prime = t_1 + \\frac{D_L+v\\delta t \\cos\\theta}{c}", "03998219323c607ac1d77af4c8000da7": "P=NM\\bar{v^2}", "03998cf235e1a1cf413937ef0499a229": "c_{\\omega}", "03999f856bdffa949cc76f8934546e7f": " W \\approx \\left[ \\frac{2\\epsilon_r\\epsilon_0}{q} \\left(\\frac{N_A + N_D}{N_A N_D}\\right) \\left(V_{bi} - V\\right)\\right]^\\frac{1}{2} ", "0399a8b21fd225ba9f533525b72406a4": "1\\cdot 10^5", "0399b9f388741e610b515ede8934c7ad": "f(x;\\mu,\\sigma^2) = \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{ -\\frac{1}{2}\\left(\\frac{x-\\mu}{\\sigma}\\right)^2 }\n", "0399d6bdc879a8da017d06cefc8394c8": " \\int_\\R f(x)g(x) \\,d\\mu(x) \\geq \\int_\\R f(x)\\,d\\mu(x) \\, \\int_\\R g(x)\\,d\\mu(x),", "0399e7b324d2fb169e3d5e32d60bf175": "(\\det A)^2\\det S,", "039a097480359dd46e06126cb958bd49": "\nF^{*}_{A} \\subseteq F^{+}\n", "039a7636d144be29d8a6ed15256a28a6": "Z_k=\\int_{[a,b]} X_t e_k(t) \\,dt", "039a76ecfb559a21482c32588caec25a": "\\lim_{a\\to -\\infty}\\int_a^cx\\,\\mathrm{d}x + \\lim_{b\\to\\infty} \\int_c^b x\\,\\mathrm{d}x", "039a9658e762bf76358e34540d2c836c": " g = \\frac{ \\mathrm{d}\\nu }{ \\mathrm{d}\\mu }, ", "039aa2ecb77465fea6001b3b7814e161": "C = X_1-A", "039af6f5a9f7f409d6e6313f7fd010bf": "Z(\\beta) = \\langle\\exp(-\\beta E)\\rangle,\\,", "039b1ac63174801fbffe57cfb3d978a3": " { \\partial^2 u \\over \\partial x ^2 } - {1 \\over c^2} { \\partial^2 u \\over \\partial t ^2 } = 0 ", "039b450257d5575e8a3bf64eea27daf4": "\\int_{-\\infty }^{\\infty } \\frac{1}{x^2+1} \\, dx = \\pi", "039b496636f8ab3c33af3cbed3658ec8": "x^{-1} \\in \\mathfrak{m}_R", "039ba6dcd91075aa703fe285ffd1a726": "\\mathbb{R}\\times L,", "039bad22b928082e52bea4f704b01452": "\\bigcup U_{\\alpha}", "039bdcdd943188d7b26552ea22218ee8": " \\mathrm{d}^2 \\sigma \\rightarrow \\mathrm{d}^2 \\tilde{\\sigma} = \\mathrm{J} \\mathrm{d}^2 \\sigma \\, ", "039bfefb3498857197e9fcc21bdc9632": "\n p(s) = w \\frac{1}{N} + (1-w) \\sum_{i=1}^M \\frac{1}{M} p(s|i)\n", "039c27b50473008fa98fe6152e7f2bfe": "\\gamma=\\frac{1}{2}\\frac{F}{L}", "039c3c087ca086aee04ff3db89b92004": "f'(a) := \\lim_{x\\rightarrow a} \\frac{f(x) - f(a)}{x - a} = \\lim_{x\\rightarrow a} \\frac{h(x)}{g(x)} = \\lim_{x\\rightarrow a}f'(x)", "039c7db777be9bbb8e6c6410ecbd3418": "s \\not\\in \\{0, 2^T/2, 2^T\\}", "039cf06d449c20ec023daf70c02b8097": "[a,b) \\subset \\mathbb{R}", "039cff40c15dd448289a4b84b6604d78": "\\displaystyle \\frac{1}{\\sqrt{2 \\alpha}}\\cdot e^{-\\frac{\\omega^2}{4 \\alpha}}", "039da3dbb17327ebc7e8cf893adb3f5a": "\n\\Lambda = \\frac{\\prod_{n} \\prod_{i} \\exp(x_{ni}(\\beta_n-\\delta_i))}{\\prod_{n} \\prod_{i}(1+\\exp(\\beta_n-\\delta_i))}.\n", "039decec58684bcda79db11267bd7847": " N_j = 4 \\cdot 2^{\\left \\lceil \\frac{j}{2} \\right \\rceil}", "039df402bca763321a9a067703bceba9": "\\textstyle t_1", "039e0153bfb965b25d937758aefa2524": "\\Delta f = { -\\ f_0^{3/2} ( \\eta_l \\rho_l / \\pi \\rho_q \\mu_q )^{1/2} } ", "039e09a38557a8615a87956c7fb1e5da": " \\Delta G_{\\rm em} = 3{\\gamma V\\over\\ R_{\\rm f}} ", "039e2712925b844ceba50e31574d1b49": "A, P, Q \\in \\mathbb{R}^{n \\times n}", "039ede700c93bb84c1ac5772a0b42af4": "\n\\Omega^{2}(t) = \\omega_{n}^{2} \\left[1 + f(t) \\right],\n", "039f3623ac3651c6f16a28470c652940": " SG_\\text{true} = \\frac {\\rho_\\text{sample}}{\\rho_{\\rm H_2O}} = \\frac {(m_\\text{sample}/V)}{(m_{\\rm H_2O}/V)} = \\frac {m_\\text{sample}}{m_{\\rm H_2O}} \\frac{g}{g} = \\frac {W_{V_\\text{sample}}}{W_{V_{\\rm H_2O}}} ", "039f9c9de1ed3462a1896906001c3d8f": "\n {d \\over dx}\\tan y\n = {d \\over dx}\\frac{\\sin y}{\\cos y}\n = \\frac{{dy \\over dx} \\cos^2 y + \\sin^2 y {dy \\over dx}}{\\cos^2 y}\n = {dy \\over dx} \\left (1 + \\tan^2 y \\right)\n", "039fbdf64d87e5e749f9be62f03c1ac7": "\\frac{|A(x)|}{|R|} > 1 - \\frac{1}{2^{|x|}}", "039fbf45379f8551e9ef60aed04178e4": " e_d ", "039fe118d348a89d7b553966bb4e3a92": "\\mu_i = \\left( \\frac{\\partial U}{\\partial N_i} \\right)_{S,V, N_{j \\ne i}}", "03a0000b65d4ac9a8010e24b859031a3": "A_{m,n} = A_{m,n-2}+A_{m,n-1}", "03a0241631bcde8a890989d3fe6c657e": "\\sum_{p|n} f(p)\\;", "03a04fe6fd3748e89a61b4bc79624682": "\n\\sigma_1 = \\sigma_x =\n\\begin{pmatrix}\n0&1\\\\\n1&0\n\\end{pmatrix}\n\\,,\\quad \\sigma_2 = \\sigma_y =\n\\begin{pmatrix}\n0&-i\\\\\ni&0\n\\end{pmatrix}\n\\,,\\quad \\sigma_3 = \\sigma_z =\n\\begin{pmatrix}\n1&0\\\\\n0&-1\n\\end{pmatrix}\n", "03a05c1da417a41dae0da916caedc5c2": "\n\\int (d+e\\,x)^m\\left(a+b\\,x+c\\,x^2\\right)^pdx=\n -\\frac{(d+e\\,x)^{m+1} (b+2 c\\,x)\\left(a+b\\,x+c\\,x^2\\right)^p}{(m+1)(2 c\\,d-b\\,e)}\\,+\\,\n \\frac{2c (m+2p+2)}{(m+1)(2 c\\,d-b\\,e)} \\int (d+e\\,x)^{m+1} \\left(a+b\\,x+c\\,x^2\\right)^pdx\n", "03a08b718087ae8e3fa114b826d96305": "|\\mathcal{U}|=9", "03a0cc58ab774f8680e9fd94d5caf7b5": "\\mathbb{F}_p", "03a0eec6c55e041c5145403b6be3b9cb": "0.33PC+0.55U+0.12EG=0.37SW+0.63BK", "03a10d5c52c6fec06b9bf6ec97a5b6b2": "\n\\begin{matrix}\nX X^T &=& (U \\Sigma V^T) (U \\Sigma V^T)^T = (U \\Sigma V^T) (V^{T^T} \\Sigma^T U^T) = U \\Sigma V^T V \\Sigma^T U^T = U \\Sigma \\Sigma^T U^T \\\\\nX^T X &=& (U \\Sigma V^T)^T (U \\Sigma V^T) = (V^{T^T} \\Sigma^T U^T) (U \\Sigma V^T) = V \\Sigma^T U^T U \\Sigma V^T = V \\Sigma^T \\Sigma V^T\n\\end{matrix}\n", "03a19315749fee66e45a008739366d39": "the : NP/N \\qquad dog : N \\qquad John : NP \\qquad bit : (S\\backslash NP)/NP", "03a22045e75b911170c35acf9d050fd7": "\\frac {d\\ln K} {d(1/T)} = -\\frac{{\\Delta H_m}^{\\Theta}} {R}", "03a235cfdf80ee75657323bbebf0e2ca": "- m c^2 \\frac{d \\tau[t]}{d t} = - m c^2 \\sqrt {1 - \\frac{v^2 [t]}{c^2}} = -m c^2 + {1 \\over 2} m v^2 [t] + {1 \\over 8} m \\frac{v^4 [t]}{c^2} + \\dots ", "03a2d642bba3f4875961243979e8c601": "1-\\frac12-\\frac14+\\frac18-\\frac{1}{16}+\\cdots=\\frac13.", "03a2feb0dba6eba8510cddeb66e8ef1f": " r' = r \\frac{1}{1- pq} > r. ", "03a3560e6753571ad048af88264c0bb9": "\\frac{a*(b+1)}{1*(2*3)}", "03a3c39aa9852a7f991d31078a07cc97": "2\\pi R", "03a3ccf388449808794c9ddaee624540": "B(t,T) = \\mathbb{E}[(1 + r(t,t+1))^{-1} \\cdots (1 + r(T-1,T))^{-1} \\mid \\mathcal{F}_t] = \\frac{1}{1 + r(t,t+1)} \\mathbb{E}[B(t+1,T) \\mid \\mathcal{F}_t]", "03a424c9a0f9fce55418280301f6553b": "a=2.1.", "03a4330f5af1bae4248d69142fb7b656": "\\mathbf{u}_x\\mathbf{v}_x\\mathbf{w}_x", "03a48bf579647edafb8fd0d2d0d6f96f": "j = l \\pm 1/2", "03a4a751eac357b8f7b952d73f1e376b": "\\int_X g\\,d\\mu=\\sup_{f\\in F}\\int_X f\\,d\\mu.", "03a4c56878ff40b6b9dfbe1cb171a96d": "\nw = f(z) = \\frac{a}{c + dz},\\,\n", "03a4db2002c4dffb4de9aabebe5bca27": " X \\times I \\to Y", "03a53ab33c76e6a15fe0dde332242c69": "2^{<\\omega}", "03a541b53b52ea0653764c1ac51f4c8a": "Q[(\\text{d} R / \\text{d} Q) (1+\\mu ) - \\mu(\\text{d} C / \\text{d} Q)] = 0,", "03a6280a5c40bd162b6a2d6d6fc8a03a": "R_n^{(l)}(\\rho) = \\sqrt{2n+D}\\sum_{s=0}^{(n-l)/2}\n(-1)^s{(n-l)/2 \\choose s}{n-s-1+D/2 \\choose (n-l)/2}\\rho^{n-2s}", "03a6467de429b3a11edbde8b6b8fbcc7": "\\frac{1}{2} \\,+\\, \\frac{1}{4} \\,+\\, \\frac{1}{8} \\,+\\, \\frac{1}{16} \\,+\\, \\cdots \\;=\\; 1", "03a66803bd302876b9298ec05ac0e0a6": "\\left(1 - \\frac{it}{\\lambda}\\right)^{-1}\\,\\exp \\{ i\\mu t - \\frac{1}{2}\\sigma^2 t^2 \\}", "03a6a91018a0e7992fcf2af5d2a48bc8": "\\alpha(f_n(x))=\\alpha(x)+n ~.", "03a6afa2815a8b4207ffe936b29c7421": "N^{-3}", "03a6bdcc17724a68d4ff72ae74c17cec": "a \\leq 0", "03a6e7cdccb7aeca2e1f9049927c896e": "=\\max_{\\lambda\\in\\sigma(A)} \\frac{1}{|\\lambda-\\tilde{\\lambda}|}=\\frac{1}{\\min_{\\lambda\\in\\sigma(A)}|\\lambda-\\tilde{\\lambda}|}", "03a73aaac912a3068b0525b8b5ee69b9": "k \\gg 1", "03a7baed00a6193186899c8a0c823b90": "CH_3OH", "03a7f97055201ba94c7471d1764ed4b5": "\nm(\\varphi) = b\\int_0^\\beta\n\\sqrt{1 + e'^2\\sin^2\\beta}\\,d\\beta,\n", "03a8087bd765f80deab2bb94bb5e8c53": "\\langle T x, y \\rangle = \\langle x, T y \\rangle, \\quad x, y \\in H.", "03a8296383cfef6a36b5fb4cf4b14313": "Q_A \\,", "03a87f3d2b231e4aa09ed311b752792f": "x=x", "03a8ecac6e0e640d7f5e79f9103413a2": "N_i = \\frac{g_i}{\\Phi}", "03a91572f241ff32ab94abcc18edefd1": "\\operatorname{ker}(f) \\triangleq \\{(x,x') \\mid f(x) = f(x')\\}", "03a9172e91e7a329897548b920e4b3b1": "x^2-2y^2 = -1", "03a99382839fa0dec99c9d6655bdd747": "\\psi_m(x) = \\sqrt{\\frac{2}{L}} \\sin{\\left(\\frac{m \\pi x}{L} \\right)}, \\,", "03a9c9913020201c1c16cc4806153f2f": "P_B(\\lambda_B)", "03aa4c947fa35f3863419a879ae94189": " \\psi = 0 ", "03aa8a7e9f93b174c1ac6a2ce0776774": "\\scriptstyle t_B \\;=\\; 1", "03aab29df61fc300ace2a4ae56e8b9b1": "|\\phi_{m}^{'}\\rangle", "03aadaf02ca0751fe6a467e10803e850": "\\operatorname{sh}\\,k, \\operatorname{ch}\\,l, \\operatorname{th}\\,m, \\operatorname{coth}\\,n \\!", "03aae9f1bd007e50cdb222061e3e230e": " g(\\mu_m) = \\eta_m = \\beta_{m,0} + X_1 \\beta_{m,1} + \\ldots + X_p \\beta_{m,p} \\,", "03ab00c3face1903468063ad259fa551": "\\mathbb Q/\\mathbb Z", "03ab662b71a6e9ecc0a51e8938a9f26b": "q\\in Q", "03ab79c8d0ebe3b954ef4ae63d73bfbf": "\\phi_i(x)=x^{**}(\\phi_i)", "03ab79d3d14652b6742807f5f5225cb7": "b_i(x)^m=0", "03ac0d241a6fd9fc66a77b2e7ce6db2c": "\n\\begin{align}\np(t) &= (\\cos(2t), \\sin(2t), 0)\\\\\nr(t) &= ( \\cos t \\cos 2 t , \\cos t \\sin 2 t, \\sin t )\n\\end{align}\n", "03ac1ea5f08f206e2c6999b331b58c7d": "\\theta=2\\pi ft\\,\\!", "03ac21b328dbeb50b8d8ae916394f9ef": " \\approx 2.6 \\times 10^{36,305} ", "03ace338f1e2fc15e58f72228a56d525": "A=\\frac{\\sqrt{3}}{4} a^2", "03ad1c3de00115c51e9da7a17ed99162": "\\Phi^{-1}=({\\mathrm d\\varphi_x})^{-1} \\in GL(T_{\\varphi(x)}N, T_xM).", "03ad41ed4ad52afeed2ba2bb320eb8f9": "R'(x) = H(x)\\ \\mathrm{if}\\ x \\ne 0", "03ad4e8445553bf6abad05b0c9eb8c6c": " \\mathfrak{-a} = \\mathfrak{a} \\iff \\mathfrak{a} = \\mathfrak{0}\\qquad \\forall \\mathfrak{a} \\in \\mathfrak{G} ", "03ad7a39f4cae58812e5257e3fb50b3c": "t(tx-2at)+x=0,\\ x(t^2+1)=2at^2,\\ x=\\frac{2at^2}{t^2+1}", "03ad8c5766037005c87fa2d541860ea8": "\\forall s_{-i}\\in S_{-i}\\left[u_i(s^*,s_{-i})\\geq u_i(s^\\prime,s_{-i})\\right]", "03adc5e31d061a36824fd2d2df985b11": "\\operatorname{E}(X^n)=\\mathrm{e}^{n\\mu+\\frac{n^2\\sigma^2}{2}}", "03ae56bc99fbf11c5cbdb0123aac6830": "e^{(\\theta/2)(e_i \\wedge e_j)}= \\cos(\\theta/2)+ \\sin(\\theta/2) e_i \\wedge e_j", "03aef4a1e68615e165e412c64d14399b": "(k_1+k_2+k_3+k_4)^2=2\\,(k_1^2+k_2^2+k_3^2+k_4^2).", "03af1fa0e3d985e14ab133c0d5dfcc3f": "F_{\\theta}", "03af2ad37614e9c2dde9b231e47efac0": "\\gamma \\rightarrow 0 ", "03af3c2704860a125cb8a7cb179f62ef": "\n\\begin{cases}\n\\mathrm{out}_A = 1 \\\\\n\\mathrm{out}_{RGB} = \\mathrm{src}_{RGB} \\mathrm{src}_A + \\mathrm{dst}_{RGB} (1 - \\mathrm{src}_A)\n\\end{cases}\n", "03af47296bd992a62d24d037a7cc1c63": "\\Delta U\\;", "03af5b7be9fc5d6ae89a338a73585a98": "Z = \\frac{1}{V}\\int_\\Omega e^{-\\beta H(\\boldsymbol{r})} \\, d\\boldsymbol{r}.", "03af8f387cbac79f063aeed5e31002f0": "\\vec{a} = (1/\\lambda_a)A^TA\\vec{a}", "03aff3b2154d6187c80d748b16746e7e": "\\sqrt{a ^2+ r} \\approx a + \\frac{r}{2 \\cdot a}", "03b002da7c63cabcb42234272136bc6d": "v(n) \\ne 0", "03b062a002773b4b2e3b3d46fc3a32e3": " s_1 - s_2 = 2A \\quad (4)", "03b07069253fff670c3f652b869afb60": "| \\alpha/\\sqrt{2} \\rangle", "03b0b51d46fcb7fc4289d6674ffd59a0": "\\Sigma_u \\left \\lfloor qu/p \\right \\rfloor", "03b0d432c78ae131fdc3d3ee81f1cb40": "\\mathrm{OTF}(0)=\\mathrm{MTF}(0)", "03b0e351027bad181abe45ab499a1679": "7.72\\approx\\frac{5\\pi}{2}", "03b0ec9ffa0cc8774c2ec4894a2b06c2": "n_{adatom}=n_0 e^\\frac{-\\Delta G_{adatom}}{k_BT} \\qquad (4)", "03b0f61f6a67f7c1157f3ba6976c6f3e": " \\mathbf{b} \\prec^w \\mathbf{a} ", "03b10549cef7810b87a7c29c34ad2309": "x = (x_1, x_2, \\ldots, x_n) \\in \\mathbb{R}^n", "03b174fb05c3ec6889c14cfb9f469d03": " 0< \\delta <1 (e.g. \\delta=0.97) ", "03b183e24368a6a72f056e417204c0d9": "f(k) = -\\frac{1}{2k} + \\frac{\\pi}{2}\\coth\\left(\\pi k\\right)\n", "03b196e9ce523c30d49f5d2d7bda1ed5": "0 + 0 = 0.\\,", "03b1b44a6955cf0b191b94301740d63a": "\\Gamma_+(M)", "03b1e442a01f8c4b877e9526471fd3ea": "\\mathfrak{sl}_4 \\cong \\mathfrak{so}_6", "03b1ecb60736d072099b4bea3dbbf11e": "\nK_H(x') = x' - x_0(T),\n", "03b25e2f947ca0c07d48c53d93f617fc": "N = (P, T, F)", "03b28f6abb6be057c8e59d765aaf78c4": "\\sum_{i=1}^n \\mathrm{Bernoulli}(p) \\sim \\mathrm{Binomial}(n,p) \\qquad 0s} \\, ds", "03cc1382a913a0468b9f71a6736e27c8": "\n H^{(\\lambda)}(X)\n =P_1(X)\n +O\\left(\\prod_{\\kappa=0}^{\\lambda-1}\n \\left|\\frac{\\alpha_1-s_\\kappa}{\\alpha_2-s_\\kappa}\\right|\n \\right)\n", "03cc3ae00c3c4b556427df9ecacebce0": "i_s=i_1\\sin(\\Delta\\varphi_a^*)+i_1\\sin(\\Delta\\varphi_b^*).", "03cc43f844df88ceccd395979a438084": "m_1, m_2\\,", "03cc89a6cd7e58ab528970c70bff387f": "\\ E_{+/-} = E_{(0)} + \\frac{C \\pm J_{ex}}{1 \\pm B^2}", "03ccbc87d41cf99a35e38b27fa7c32b6": "\\sigma_{ij} = \\epsilon_{0}E_{i}E_{j} + \\frac{1}{\\mu_{0}}B_{i}B_{j} - \n\\left(\\frac12\\epsilon_{0}E^2 + \\frac{1}{2\\mu_{0}}B^2\\right)\\delta_{ij} \\,.", "03ccc140b40080330afee07a5170b9d0": "M=[1.44 0; 0 2.89]", "03ccea9e9891a786a0e5f7d8bec31bd3": "A = \\bigoplus_{n\\in \\mathbb N}A_n", "03cd9af73a09b7d07621d3f80c839abc": "(10)", "03cdeac2c9e5cf309c622bb36903465e": "\\chi_6", "03ce19dab62e4b5bf1ac3e2be81be05a": "\\frac{\\gamma}{2\\alpha\\delta K_1(\\delta \\gamma)} \\; e^{-\\alpha\\sqrt{\\delta^2 + (x - \\mu)^2}+ \\beta (x - \\mu)}", "03ce6166bf66aafeb930e5b683790242": "x_D = \\begin{cases}\n0.244063 + 0.09911 \\frac{10^3}{T} + 2.9678 \\frac{10^6}{T^2} - 4.6070 \\frac{10^9}{T^3} & 4000K \\leq T \\leq 7000K \\\\\n0.237040 + 0.24748 \\frac{10^3}{T} + 1.9018 \\frac{10^6}{T^2} - 2.0064 \\frac{10^9}{T^3} & 7000K < T \\leq 25000K\n\\end{cases}", "03cecb10712a6deafefb7b008f96ee7f": " \\lambda = \\sqrt{\\frac{m}{4 \\mu_0 e^2 \\psi_0^2}}, ", "03ced8b82b8d4ff11cc137b7b22188f4": "B\\left(\\frac{\\alpha}{2}; x, n - x + 1\\right) < \\theta < B\\left(1 - \\frac{\\alpha}{2}; x + 1, n - x\\right)", "03cee26504e3be30e6a4799eaa1ebb0a": "\\frac{\\sqrt{\\frac{(d_1\\,x)^{d_1}\\,\\,d_2^{d_2}}\n{(d_1\\,x+d_2)^{d_1+d_2}}}}\n{x\\,\\mathrm{B}\\!\\left(\\frac{d_1}{2},\\frac{d_2}{2}\\right)}\\!", "03cf1d3306a36f956af672193dc515a5": "y=f(x)\\,\\!", "03cf69b8e1926e9304e27bf70f136390": " kT \\gg \\varepsilon_i-\\mu ", "03cf6c16a00da319754020f534d0f73e": "x(n)=\\sum_{k=0}^q b_n(k) d(n-k)+v(n)", "03cf83e7d9741f6bf162b1fbfe0860ac": "-\\ell k\\,", "03cf8f9ecf69055531c30a1f5c0af149": "\\mathbf{p}_{\\mathrm{1}} = m\\mathbf{v} + \\mathbf{u}\\mathrm{d}m", "03cfa63594a40ccb8e679d26ff261748": "F(0)= \\sum\\nolimits_{n\\in \\mathbf{Z}} c_n", "03cfb11ec181418826af113a24111e44": "A = -kT\\log\\left(Z\\right)\\,", "03cfbeda8a8727cdc5c429ce1debc64d": "10 | 560", "03cfdf3d21a85ad10328dba37ba17761": "m\\times n\\!", "03d024b9f26631e6a453e7841f822ddb": "y = \\prod_{\\sigma} \\sigma(x)", "03d0a2ad9a3042e12a7f671666711f1b": "\\sum_{jN+1}^{(j+1)N} X_k", "03d0e2a2646d1ced604bea3690da5cc0": " \\sigma_m = 1/R ", "03d1a28c36a9ecd807f616236d2f152d": "x_t = 1/(\\eta + w_t)", "03d25453bed03733343b2e4bb36cb56f": "q=", "03d25e22170a4308bb4e1cb54a983b33": "\n \\boldsymbol{\\sigma} = \\lambda~\\mathrm{tr}(\\boldsymbol{\\varepsilon})~\\boldsymbol{\\mathit{1}} + 2\\mu\\boldsymbol{\\varepsilon}\n ", "03d2db8193aa097c9a76ecbe7056dc98": "\\delta (\\hat{M_E}) = \\lim_{T \\rightarrow \\infty} \\int_{-T}^T dt e^{i t \\hat{M}_E}", "03d2e5f5855c3331821ae802f99355c6": "g^{(N+1)}", "03d340d167c24ad78bc190e1ab0f0988": "\\vec{e}\\!", "03d35c5129ead46219fd4b22f81f5d04": "2\\pi/3", "03d3ca3fa2226c9a550d3f4cef0a1dd5": "d_1", "03d3fc8d7471cbd34af7dcc0e716e956": " E\\left[ \\Lambda(n+1) \\right] = E\\left[ \\left| \\hat{\\mathbf{h}}(n) + \\frac{\\mu\\,e^{*}(n)\\mathbf{x}(n)}{\\mathbf{x}^H(n)\\mathbf{x}(n)} - \\mathbf{h}(n) \\right|^2 \\right]", "03d41346ade9e0e7d16038913451be15": " \\text{d} C / \\text{d} Q", "03d424e371484e73cbf8027cb75f646d": "[0,1]_{\\ast},", "03d48597117c2bd97aab24e7075b6b41": "FWER = P_{any}(V \\ge 1) ", "03d4ad33274debc1fa4faa5abb7715c0": "Q_{T-1}(W_{T-1})", "03d4c263ed0a4ab7c10fe02f319d64f1": "L \\, ", "03d4f5f0b9f03f5c76dc44e448eaf6d6": "\\ r ", "03d5004c82b6f47b105ca61a9e650cf6": "D_i(u,v) = w_i \\times \\frac{|log(u_i + 1) - log(v_i + 1)|}{log(max\\{u_i, v_i\\} + 2)}", "03d549124b4658a4d2ed56f1aaef6eed": "\ndW = \\sum_{r=1}^{D} Q_{r} dq_{r} = \\sum_{k=1}^{N} \\mathbf{F}_{k} \\cdot d\\mathbf{r}_{k} = \\sum_{k=1}^{N} m_{k} \\mathbf{a}_{k} \\cdot d\\mathbf{r}_{k}\n", "03d5be0871da01d01a8b0c64b5638ae3": " [ d/dz - E_{11}/z , - E_{21}/z] = [ d/dz , - E_{21}/z] + [ - E_{11}/z , - E_{21}/z] = E_{21}/z^2 -E_{21}/z^2 = 0 ", "03d5db1d5b264656b1f664ca2656a118": "\\cot\\frac{\\pi}{12}=\\cot 15^\\circ=2+\\sqrt3\\,", "03d5f2201251112296c9be71d22bcf97": "\\csc \\left(\\frac {\\pi} {z}\\right)", "03d60f1de7d97a09a377389404300b6d": "b \\equiv a \\,\\bmod{f'(a)\\mathfrak m}.", "03d63acd89770450159bbe51452ac3ae": " v_{i1} ", "03d66cdf7525281b402a60c4ef007aab": "z^1 = 0.499997032420304 - (1.221880225696050\\times10^{-6})i{\\;}{\\;}{\\mathrm {(red)}},", "03d69e6e0442b1f7218f02ce91363f32": "P_A = l_A", "03d6a6d3b41fa8d2e621a564b04822c3": " \\scriptstyle \\langle ", "03d6cb574e9b5c08083092a6e6119912": "e_\\infty = \\frac{m}{(m, \\deg(f))}", "03d6d59efc67a54e5f9d03e115ab40ea": " C_\\pm (j,m) = \\sqrt{ j(j+1) - m(m\\pm 1) } ", "03d73b3a369ef15a8cc02d20249f80fc": " g\\cdot z=\\frac{az+b}{cz+d}", "03d78a18d747ce1a9b217ba0c0fb59f3": "\\scriptstyle |\\psi_n\\rang ", "03d7f263292f2c210cf729306da936ea": "M = \\max_{1 \\le i \\ne j \\le m} \\left| a_i^H a_j \\right|.", "03d81762c9fa45c6edf4a7bd0586d464": "\\sqrt{m_\\mathrm{i}/m_\\mathrm{e}}", "03d8a890f439d09045c78c4a62d5a45f": "\n\\left.\n\\begin{matrix}\nx*y*z=(x*y)*z\\qquad\\qquad\\quad\\,\n\\\\\nw*x*y*z=((w*x)*y)*z\\quad\n\\\\\n\\mbox{etc.}\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\ \\ \\,\n\\end{matrix}\n\\right\\}\n\\mbox{for all }w,x,y,z\\in S\n", "03d8d2ba7d77a6b9d8e40cbe310326db": "u_G''(x)=A(x)u_1''(x)+B(x)u_2''(x)+A'(x)u_1'(x)+B'(x)u_2'(x).\\,", "03d934ff2e808e9b0ffcdbdb1df1d6d9": "Z_0 = \\sqrt{\\frac{L}{C}}", "03d994023612fce11d588b24854c5050": "A > B > C", "03d9c9a1ed9d2003fc9725ed0d858420": "\\beta(2)", "03d9da20066f41452bbd390bc1981167": "= (1 + i 2\\pi fT) e^{-i 2\\pi fT} \\mathrm{sinc}^2(fT)) \\ ", "03d9e27b16b8626d6f715803f24950ed": "\\sqrt{x^2}+\\sqrt{y^2}=|x|+|y|", "03da8aa755310b93007325e732dc509b": "x\\ne 1", "03daab1d4024edcc09938b1d5acab536": "F = k X,", "03dab7746e3ecc9822922b5a75b18e61": "\\lim_{z\\rightarrow 1^-} G_a(z) = \\sum_{k=0}^{\\infty} a_k,\\qquad (*)\\!", "03dadf7919b83d68f7682da0245ec658": "\\left[ B \\right]=\\left\\{ \\begin{array}{*{35}l}\n \\left[ A \\right]_{0}\\frac{k_{1}}{k_{2}-k_{1}}\\left( e^{-k_{1}t}-e^{-k_{2}t} \\right) & k_{1}\\ne k_{2} \\\\\n \\left[ A \\right]_{0}k_{1}te^{-k_{1}t}+\\left[ B \\right]_{0}e^{-k_{1}t} & \\text{otherwise} \\\\\n\\end{array} \\right.", "03db060902394497adbf217f4ae97338": "{\\delta W}=P\\mathrm{d}V.", "03db63a398f2b45c18059782c4f36dfe": "f(x) = a e^{- { \\frac{(x-b)^2 }{ 2 c^2} } }+d", "03db64206cf3a86a497c1121e7ac5d19": "M^{\\mathbf r}=\\left(I-H \\right) M \\left(I-H \\right)^{\\rm T}.", "03dbbab2600cebe8b540da30ace6353f": "| B \\rang", "03dbf0197647d1d16db44f5235b2b043": " \\frac { \\text {density of object}} { \\text{density of fluid} } = \\frac { \\text{weight}} { \\text{weight} - \\text{apparent immersed weight}}\\,", "03dbf3e1d43b8a823c32f0f134cd5d5f": "\n f = \\sum_{i = 1}^n \\alpha_i \\varphi(x_i) + v,\n", "03dc38392b2f7a44b03e4a8f381276ec": "P= \\sum_{i,j} u_i \\left( \\langle\\mathbf{v}, \\mathbf{u}\\rangle^{-1}\\right)_{j,i} \\otimes v_j", "03dc396c1aca49f0bf4b6f361ed5c365": "p(F_a=v_i)", "03dc5d1fd972f00dcd03a56ec82313f6": "\\tan A = \\frac{3}{7},", "03dc896e15bfb09b05f1f2cb0aa67fac": "\\lambda\\colon kG \\otimes F(X) \\to F(X) ", "03dc93a786dba8c41080ea2b97c34df5": "\\ln\\left(\\left(\\frac {1}{2}\\right)^{t/t_{1/2}}\\right) = \\ln(e^{-t/\\tau}) = \\ln(e^{-\\lambda t})", "03dd2d723a4862c986838c5169342b94": "PCER = E \\left[ {\\frac{V}{m}} \\right] ", "03dd3245478c9e35f4cf9bcf89c4a226": "\\psi( \\dots, x_j=0, \\dots ) =\\psi(\\dots, x_j=L,\\dots )", "03dd4b4beb2d8bf6ce7ff194f46c1ef7": "Z(X_0, F_0, t) = \\prod_{i}\\det(1-F^* t|H^i_c(F))^{(-1)^{i+1}}", "03dd505cbfabb8a184ff4d2fca82b1ac": " c_j | N_1, N_2, \\dots, N_j = 1, \\dots \\rangle = (-1)^{(N_1 + \\cdots + N_{j-1})} | N_1, N_2, \\dots, N_j = 0, \\dots \\rangle ", "03dd6d27f3d6f4c01e7b2dec3b9ae715": "\\tfrac{4}{5} \\pi", "03ddbc0a475c1bd43b4672cafa104e47": "S \\ni x", "03ddc9b055532c322b2a9620740d6136": "a\\left(\\frac{1}{q}-\\frac{1}{r}+1\\right)=\\frac{1}{q}-\\frac{1}{p}.", "03dde93cc610c1307a6fc777f6e2b044": "CF_i", "03de0b280f757ebe94424def39e130fc": " h_k (X_1, X_2, \\dots,X_n) = \\sum_{1 \\leq i_1 \\leq i_2 \\leq \\cdots \\leq i_k \\leq n} X_{i_1} X_{i_2} \\cdots X_{i_k}.", "03de7cfc6e7b321c4c17410f62e445f9": "\n c^2 = \\frac{2\\kappa G h}{D(1-\\nu)} \\,.\n", "03dea73658b8ca48d2bab910746fed4f": "\\,F_v", "03dedeca4f11b2b3094602be33a3fd12": "\\min_x \\|x\\|_1", "03df32a80ca426f52e6e058a7fe0e3f1": "M_p(n)", "03df43b24d79f21ac2f7483bfe649a78": "\\displaystyle a^2=cd", "03df5d2a6370f99e14f97f689511ecd9": "\\Phi^a", "03df888fc1e5a7faf8843d69f7c2ff0b": "\n\\frac{I_f^0}{I_f} = 1+k_q\\tau_0\\cdot[\\mathrm{Q}]\n", "03df91cb9676e0aefdd35bbb3285ba96": "\\omega_1'", "03df96fbb957bad6c44aab5fd6cd474a": "(\\phi \\lor \\exists x \\psi) \\leftrightarrow \\exists x (\\phi \\lor \\psi)", "03dfa0f901b84523f5dbce2cb9985583": " f(x) \\approx f(a)+f'(a)(x-a). ", "03dfeab6a13f6249ccaf3f30922fdb8d": "J(u) = \\int_D |\\nabla u|^2 \\mathrm{d}x.", "03e005d925a144ea2e6f8860daa4fdca": "\\sigma(q,t)", "03e048b5b3bddc7362e6620e6dcd4d75": "p(f_{ik})\\ ", "03e074dff06fc87c2f5feb819e612bd0": "\\chi_3\\left(z\\right)=\\frac{1}{\\alpha}\\int_\\infty^z\\sinh\\left[\\alpha\\left(z-\\xi\\right)\\right]Ai\\left[e^{i\\pi/6}\\left(\\alpha Re\\right)^{1/3}\\left(\\xi-c-\\frac{i\\alpha}{Re}\\right)\\right]d\\xi,", "03e161b98da6a296b962d010a04970f6": "H_{inv}(s) = \\frac{D(s)}{A(s)}", "03e16b8c66c7a6ca01f976ae30c9586b": "\\mathbf{x}_{k+1} = \\mathbf{f}(k,\\mathbf{x}_k)", "03e18d83841f9780acb30eeb0ec7b5d7": "a_{k,\\ell}", "03e1aca4c970b3b37ad7f63c6ef6fbb7": "\n |j_1-j_2| \\leq J \\leq j_1+j_2.\n", "03e1b478601c13078e20acf0aae90f75": "I_x=E_xL+M_x", "03e1d1c59a1694c2260609f969a05aee": "S_\\mathit{wir}", "03e1d82e99b158334f241aacf764b61c": "(n-1)^2", "03e22ec25b5ec0e94a5589c25909b951": "W_{m} = diag\\{w_{m}\\}", "03e2462350c41d7ec1a5ed27576b9572": "\\gamma \\delta \\gamma^{-1} \\delta^{-1} = \\epsilon", "03e248075c04b3c9e2f9121851c92f42": " I_i \\, ", "03e256589a2c8c87cf5cc0de0c0c70a6": "\\hat{F}_\\mathrm{inconcl.}= 1-\\hat{F}_{\\psi}-\\hat{F}_{\\phi},", "03e3015e5ade7bfebee4372443308fc7": "A_m(1,2) = 1,2,3,4,5,6,7,8,9,10,\\ldots", "03e31b4413745326637d7ee75b266a25": "\\alpha\\div\\beta", "03e344c5d678f065203d644e6cd8f6a0": "q = 2", "03e350a66d4f39798189dac57cffc007": "\\forall x\\in W\\,(x\\Vdash A)", "03e35c79fbb25874863c3a9ea4f6c69a": "\\dbinom{n}{k}", "03e3810cee17a572d4ebe76a0aac1c97": "L^{1,w}", "03e3c8c207091356616e7205551325f0": " \\frac{1}{2} k_BT ", "03e42f7f0cba50e05a4ef28ef7a119ce": "\n\\frac{dT(s)}{ds}/\\frac{T(s)}{r}=-\\frac{t}{n}", "03e44f74cd340ba2739c352b535d868a": "n=\\frac{T}{\\delta}", "03e4a025a0c7424b008ba3875a2c4e8f": " Z[J] = \\sum_{x \\in \\mathcal{X}} \\exp \\left(\\sum_{k} w_k^{\\top} f_k(x_{ \\{ k \\} }) + \\sum_v J_v x_v\\right)", "03e5091cb5cd32369b9252594e4113a4": "5^6", "03e52b5f14c4a6c618eb416e99f7772b": "\\varepsilon_{\\phi_2(0)+1}", "03e55e8dc3eaea1688a82b38f2b00412": "\\ \\|y(t)\\|_{\\infty} < \\infty", "03e58c894b4026627a6c2f57dc122d9f": " f(x)=1/x^2", "03e5aff75dead8f836733b8199d63c49": "\n\\Phi(z,s,a)=\n\\frac{1}{2a^s}+\n\\int_0^\\infty \\frac{z^t}{(a+t)^s}\\,dt+\n\\frac{2}{a^{s-1}}\n\\int_0^\\infty\n\\frac{\\sin(s\\arctan(t)-ta\\log(z))}{(1+t^2)^{s/2}(e^{2\\pi at}-1)}\\,dt\n", "03e5c4ff8dac745730829e5dc3d136da": "\n\\Delta E = E_{n+1} - E_n = {dE \\over dJ}(J_{n+1} - J_n) = {1 \\over T} \\,\\Delta J \n", "03e5cb9a4a8ab0a96d48912ee44ccb82": "\\frac{\\omega_s}{c} \\,", "03e5d5a49cc05c533f2fc8b4fabf1032": "2^b=N", "03e61b72f90197524cecd7aabf0c3b7f": "\\frac{1}{1+a} = 1 - a + a^2 - a^3 + \\cdots \\pm a^n \\mp \\frac{a^{n+1}}{1+a},", "03e647e6672941060cc02ee23aaafacb": "k^{-1}\\bmod\\,q", "03e66241d1f7bff74afcde3a56427d83": "CABED", "03e69d68bb4b8e69ad734ffe3d1595b8": "\\equiv_D", "03e70a4e0f5c37e711494be10964acba": " \\left|\\,{x\\over a}\\,\\right|^n + \\left|\\,{y\\over b}\\,\\right|^n =1 ", "03e72bde921ee249d8d5f0fcec11ac43": "{SU(3)_C\\times SU(2)_L \\times SU(2)_R \\times U(1)_{B-L}\\over \\mathbb{Z}_6} \\rtimes \\mathbb{Z}_2.", "03e74c0242bfd5c0fa3088820daea46c": "\\overline{op_1}'", "03e7620ee41838088ae281c1602cab97": " {s_1 / \\sqrt{n_1} \\over \\sqrt{s_1^2/n_1 + s_2^2/n_2}}. ", "03e79d8069f4e38f91e49006e4259284": "\\bold{u}_1^\\prime = \\bold{u}_1 - \\bold{V} , \\quad \\bold{u}_2^\\prime = \\bold{u}_2 - \\bold{V} ", "03e7b061f9b7de024ec507077863eb49": "\\langle \\sigma_A\\rangle", "03e7e6908f866c4dab2769c4c2b87175": "a=b>c", "03e862a39055364b665a144d012f1465": "\\scriptstyle L_2", "03e86768053aae06fb07c3ec55402e83": "\\gcd(p,q)=\\gcd(p,kq)", "03e8bc5f2d83295cd14c9d24945b18ab": "A= \\dfrac{n}{n_{e}}", "03e90fb862fd9c3b022d97e17a824bc6": "x_2 = 1.000000000000000 .", "03e9615aba27e5f307db8ba3ba2107ca": "\n \\begin{bmatrix}\n 1 & 31 & 12& -3 \\\\\n 7 & 2 \\\\\n 1 & 2 & 2\n \\end{bmatrix}\n", "03e96c424485be59990cd22a79cffc64": "\\Re {zh^\\prime(z)\\over h(z)} \\ge 0", "03e97b92772cadb788968043edbab486": "\\frac{b-a}{2}", "03e98c90ab94388ee0a3fd11220908ef": "D_{ij}={\\delta_{ij}\\over (r_i,r_i)}", "03e9d588acff06de769e9d810c61c133": "x = \\cos\\theta", "03ea481a6377c6c94f3f6293db8773dd": "Y_{MIN}", "03ea4c2ef4e4da1779a45b12f5a23f64": " P = P_e + \\frac{Y-Y_n}{a} ", "03ea501e22ef0596ae87b533bdfca027": "\\Omega, \\Omega_+, \\Omega_-", "03ea9f29d543d258a40f84e07d044afd": " \\{ |e_n\\rang\\} ", "03eac388ebfcefa1384716da5ac392d9": "V_{1} (K, L) = \\lim_{\\varepsilon \\downarrow 0} \\frac{V (K + \\varepsilon L) - V(K)}{\\varepsilon},", "03eb2bb2e0599d81d80cfaa9b03c4c7b": " x : I \\mapsto X ", "03eb6d5cb381b4a0f04113069b1e2a61": " \\frac{N}{4\\cdot \\pi \\cdot d^2}= \\frac{E^2}{R}", "03eb8a3cb7a391c69fecac33667fe4eb": "\\mathbf{x}^{(n)}", "03ebd69aa4068e875588677f031bef5a": " L = d\\cos\\alpha_{crit}\\,\\!", "03ec0562facaeea4f5cc5b21b991f65e": "A=\\frac{1}{2}(20+\\sqrt{5(145+58\\sqrt{5}+2\\sqrt{30(65+29\\sqrt{5})})})a^2\\approx32.3472...a^2", "03ec4481d5dd16bd36562db5929d3a11": "\n\\begin{bmatrix}\n\\varepsilon_1\\\\\n\\varepsilon_2\n\\end{bmatrix}\n\\mid X\n\\sim \\mathcal{N}\n\\left(\n\\begin{bmatrix}\n0\\\\\n0\n\\end{bmatrix},\n\\begin{bmatrix}\n1&\\rho\\\\\n\\rho&1\n\\end{bmatrix}\n\\right)\n", "03ec8730b2ff44a8a89494b272f75d86": "\\eta = \\frac{-dW}{-dQ_h} = \\frac{-dQ_h - dQ_c}{-dQ_h} = 1 - \\frac{dQ_c}{-dQ_h}", "03ed06b5d14dff9fea8727cd4f53e63f": "\\scriptstyle {g_{\\mu\\nu}}", "03ed20965c7afae03a27561dbd28d372": "[V, W] (x) = \\mathrm{D} V(x) W(x) - \\mathrm{D} W(x) V(x),", "03ed351400e6de29ff95107a28d66c09": "x_{n+1} = x_n - \\frac{f(x_n)}{f'(x_n)} = x_n - \\frac{1/x_n - b}{-1/x_n^2} = 2x_n - bx_n^2 = x_n(2 - bx_n).", "03ed8335cb4bf1e669605934b01240f1": "\\mathbf{I}^{(1)}\\cdot\\mathbf{J}^{(1)} = \\sum_{n=-1}^{+1}(-1)^nI_{n}^{(1)}J_{-n}^{(1)} = I_0^{(1)}J_0^{(1)} - I_{-1}^{(1)}J_{+1}^{(1)} - I_{+1}^{(1)}J_{-1}^{(1)},", "03ed9ef12bf7cee79361a68fea5cb8cd": "f:M\\to S^1", "03edb2b6a493e3ea759399c7c4acd4cd": "GL_2( \\Bbb{R})^m.", "03edcaedb348abb3e7085289903a7564": " P=DEC_{k_1}(DEC_{k_2}(...(DEC_{k_n}(C))...))", "03ede58af7d86c129b4edb6af3e9cb0f": "\\rho^{\\text{induced}}(\\mathbf{r})", "03edfa39384656ea5b57eea7c49f7532": "S_\\text{baker-unfolded}(x,y)=\n\\left(2x-\\left\\lfloor 2x\\right\\rfloor \\,,\\,\\frac{y+\\left\\lfloor 2x\\right\\rfloor }{2}\\right).", "03ee304aa69c486d234a44f3378b09da": "\\Delta \\mu _{H+} = -F \\Delta \\psi + 2.3RT \\Delta pH", "03ee86361185b580eb773f753586ddb7": "H(x, v)", "03ee92d0d65558637cb6d34b82d39509": "\\,\\{\\Upsilon_j \\} = \\{\\ddagger\\sigma_m,\\ddagger\\sigma_{m-1}, \\ldots, \\ddagger\\sigma_1 \\} \\in (\\ddagger\\Gamma^+)^*", "03eea484ca5b44485dd3f8d3c741cb3c": "1,x,x^2/2,x^3/3!,\\dots,x^n/n!", "03eec948e92e06a8398b5a7fdec62758": " \\operatorname{V_r}(\\theta)= Constant", "03ef420905afef95aa9d5571cd418501": "\n\\psi_T(y) = \\int_{x} \\psi_0(x) K(x,y;T) dx = \\int^{x(T)=y} \\psi_0(x(0)) e^{i S[x]} Dx\n\\,", "03ef7f06681f7649eaf5dd13c9d53f77": "\\dot{V} - \\frac{U_{osm}}{P_{osm}}\\dot{V}", "03ef81b127bfbe48fe215949105d7e28": "Z_{ij} \\, ", "03efb900cbe0906009ca8cdf2f28ee12": "x_1x_2", "03efbb3a1702295d54d47558026f336a": "N_{MSY}", "03f0d2c858daddfd2cd839e35fdd09c7": " h_A(x+y)\\le h_A(x)+ h_A(y), \\qquad x,y\\in \\mathbb{R}^n.", "03f106c3f162e380f505214595a8b110": "\\begin{align}\nR &={1\\over 2\\pi} \\int_0^{2\\pi} e^{-i\\theta} U_\\theta H^{(1)} U_\\theta^* \\, d\\theta,\\\\\nR_\\varepsilon &={1\\over 2\\pi} \\int_0^{2\\pi} e^{-i\\theta} U_\\theta H^{(1)}_\\varepsilon U_\\theta^* \\, d\\theta.\n\\end{align}", "03f110bd9e7ea18ea3d59dd66e63a23c": "\\bar r_2\\ ", "03f1267db64f5b147017f41c868a4d94": "\\begin{bmatrix} \\dfrac{1}{y_{11}} & \\dfrac{-y_{12}}{Y_{11}} \\\\ \\dfrac{y_{21}}{y_{11}} & \\dfrac{\\Delta \\mathbf{[y]}}{y_{11}} \\end{bmatrix}", "03f190eb9234c0b19deba4f7e0bb8b4c": "4(\\pi)", "03f1e37a6367be8da35b90171b743001": "F\\,.", "03f20cc4a24a90939ad2151268aa2dcd": " d = at^2\\,", "03f23727275ecf230d0235edfff68fbd": "d = c_1d_1 + c_2(v_1+v_2+ h)", "03f23f225a2c154236dfaae5a0bf7e51": "E\\left[u(w(y(e))) - c(e)\\right] \\geq \\bar{u}", "03f275c725ccf747e0b18d0e917cf240": " (e, h, f) ", "03f2803a1e4332c25240375dae0cd931": "x = r \\cos \\phi", "03f2946ba41fedfea05608b274a24e3c": "\\tau=u^\\lambda\\partial_\\lambda", "03f2aee5882c28bc87275d8661fe382b": "f=:\\sum a_{j,k}e^{i(jx+ky)}", "03f2b21268ff5b4cfb3e212a7a352e5e": " x(\\lambda) ", "03f2ee18ea6e349e91f45e9c6d4bf77a": "E(Particle_{i,j}) = k_{s}E_{s,i,j} + k_{b}E_{b,i,j} + k_{g}E_{g,i,j}", "03f33cc02be56b7c09cc5cf7442a7ea9": "\\theta = \\operatorname{atan2} \\left( \\frac{\\partial f}{\\partial y} , \\frac{\\partial f}{\\partial x}\\right)", "03f3657c7cfeab1f4c34e813583841ed": "\\begin{align}L & = \\{uvwxy : u,y \\in \\{0,1,2,3\\}^*; v,w,x \\in \\{0,1,2,3\\} \\and (v=w \\or v=x \\or x=w)\\} \\\\ & \\cup \\{w : w \\in \\{0,1,2,3\\}^*\\and \\text {precisely 1/7 of the characters in }w \\text{ are 3's}\\}\\end{align}", "03f37a2889d1ff304acb68428ed6045b": "p_\\sigma=0", "03f3bf8fecca7e1e602a83a9b7562a11": "a - b", "03f3ca9db6a166009561d00518b1049e": "\\vartheta_{01}, \\vartheta_{10}, \\vartheta_{11}", "03f3ccfc0b3e2d7093afb0146ecb3a23": "\\displaystyle{K_p= \\|z^{-1}(z-1)^{-1}\\|_q/\\pi.}", "03f53547f0d309456588e2688b239aac": "\\begin{matrix}{52 \\choose 4} = 270,725\\end{matrix}", "03f5edbac70ba21f4f43a8ed3c68c926": "\\Lambda = {{8\\pi G} \\over {3c^2}} \\rho\\!", "03f5f86eac108f38f088b7bada9f37ad": "0\\leq \\beta < 1", "03f60de2e8eec9a071f2f23d0c648367": "h_{11}(t)", "03f65e0eeb6bf535749354fd92b970dc": "v_{g} = - \\frac{1}{\\rho_{max} \\tau_{del, jam}^{(a)}}\\qquad\\qquad(1)", "03f6c7272f9a77e0c06f5fb7290a470d": "MPK=R/P", "03f6f0f1d77b4bc5af4704cac07c9681": "x'=V(x)", "03f7e107a2d26b135be2c430d2f00f20": "\\epsilon^2 \\cdot n", "03f855a103cbcabbcbdc053b2a42274a": "\\mathfrak{t}\\ominus \\mathfrak{s}", "03f90abaf79f4744b8b7b766c6df2326": "885.7\\pm0.8~s", "03f94c2d32a2e3e9dedb87522e89d573": "\\pi a^2", "03f960a96507df5ea172c666631d9f7d": "\\left.g\\right.", "03f9745b3fb68caf25bae38a9047b451": "_SM", "03f98599e7e2a6894f748aeb548e6af0": "t \\sigma_1 \\equiv t_1", "03f98cb374db9d443f57a6b3871e2aad": " \\mathbf{A} = \n\\left[\n\\begin{array} {c | c}\n\\mathbf{A}_{11} & \\mathbf{A}_{12} \\\\\n\\hline\n\\mathbf{A}_{21} & \\mathbf{A}_{22}\n\\end{array}\n\\right]\n= \n\\left[\n\\begin{array} {c c | c}\n1 & 2 & 3 \\\\\n4 & 5 & 6 \\\\\n\\hline\n7 & 8 & 9 \n\\end{array}\n\\right]\n,\\quad\n\\mathbf{B} = \n\\left[\n\\begin{array} {c | c}\n\\mathbf{B}_{11} & \\mathbf{B}_{12} \\\\\n\\hline\n\\mathbf{B}_{21} & \\mathbf{B}_{22}\n\\end{array}\n\\right]\n= \n\\left[\n\\begin{array} {c | c c}\n1 & 4 & 7 \\\\\n\\hline\n2 & 5 & 8 \\\\\n3 & 6 & 9 \n\\end{array}\n\\right]\n,\n", "03fa0933c441dc2ac015801a807d2693": "x_{11} = p_{1} q_{1}", "03fa0bc00e305c5fcfd2959a9cce90da": "\\widehat{\\sigma_e^2} = \\frac{1}{n} \\sum_{i=1}^n (x_i-\\hat{x_i})^2.", "03fa132f865cb806ec697d4984b69b1a": "Q_r=\\frac{\\prod_j a_{j(t)}^{\\nu_j}}{\\prod_i a_{i(t)}^{\\nu_i}}", "03fa28067b7f7c8257cc8700d0957e88": "\\begin{align}\n\\alpha & = \\cos a = \\frac{{\\mathbf v} \\cdot \\mathbf{e}_\\text{x} }{ \\left | {\\mathbf v} \\right | } & = \\frac{v_\\text{x}}{\\sqrt{v_\\text{x}^2 + v_\\text{y}^2 + v_\\text{z}^2}} ,\\\\\n\\beta & = \\cos b = \\frac{{\\mathbf v} \\cdot \\mathbf{e}_\\text{y} }{ \\left | {\\mathbf v} \\right | } & = \\frac{v_\\text{y}}{\\sqrt{v_\\text{x}^2 + v_\\text{y}^2 + v_\\text{z}^2}} ,\\\\\n\\gamma &= \\cos c = \\frac{{\\mathbf v} \\cdot \\mathbf{e}_\\text{z} }{ \\left | {\\mathbf v} \\right | } & = \\frac{v_\\text{z}}{\\sqrt{v_\\text{x}^2 + v_\\text{y}^2 + v_\\text{z}^2}}.\n\\end{align}\n", "03fa496e35fda74947e0ecf357c79f5a": "C_o", "03fa5627e5525e969a05f15229892021": "x^3-x-1", "03fa815b0b6dd461c3d05fcb636eeea8": "8x^3 - 4x^2 - 4x + 1 = 0", "03fb606e136573b6a73d962b643adf6b": "\\mathcal{F}_{\\tau} := \\left\\{A\\in\\mathcal{F}:A\\cap\\{\\tau \\leq t\\}\\in\\mathcal{F}_t, \\ \\forall t\\geq 0\\right\\} ", "03fb82180093b4b3ddca81ddebf24ac1": "\\oint_{\\Gamma} \\mathbf{F}\\, d\\Gamma = \\iint_S \\nabla\\times\\mathbf{F}\\, dS ", "03fbd188089fa3e308aa0da3890b0c54": "-b^{-1}", "03fbde77646393d7fc1446b1f79e2bfc": "\\displaystyle{\\nabla D(\\varphi)=D(\\dot{\\varphi}\\mathbf{t}) + S(\\partial_t(\\dot{\\varphi}\\mathbf{n})),}", "03fbe811e8cf5e8eb9ef932cbe6cd17a": "ROC = \\left\\{ z : \\left|\\sum_{n=-\\infty}^{\\infty}x[n]z^{-n}\\right| < \\infty \\right\\} ", "03fbf6a0135f8a1716848e343c2ab8b3": " P_y = P_{y0}(2e^{-\\frac{\\pi|\\epsilon|^2}{2\\alpha_0}}-1)", "03fc0cf8bec9b1e6b4ea35e95f590044": " P = AMB \\bmod d ", "03fcd006e9c861273d6a04e143a20d8b": "r\\arctan(\\frac{y}{x}) = \\frac{1}{1}\\cdot\\frac{ry}{x} -\\frac{1}{3}\\cdot\\frac{ry^3}{x^3} + \\frac{1}{5}\\cdot\\frac{ry^5}{x^5} - \\cdots , ", "03fce92d16e9587b8788dfff21a7abcc": " O_{fg} ", "03fd678e6a278e851da7a244a5956614": "\\varepsilon_1\\varepsilon_2", "03fd8b322be7d8e81f0420f02fe0a57d": "\n4\\pi\\varepsilon_0 V(\\mathbf{R}) \\equiv \\sum_{i=1}^N q_i v(\\mathbf{r}_i-\\mathbf{R})\n", "03fda4629b973f8ce23b7f20635dc7a7": "x_k = - \\frac{1}{3a}\\left(b\\ +\\ u_k C\\ +\\ \\frac{\\Delta_0}{u_kC}\\right)\\ , \\qquad k \\in \\{1,2, 3\\}", "03fe3cb0e67115aaaf2269c58319360d": "\\left \\{ \\sqrt[3]{x} : x \\mbox{ is constructible} \\right \\}", "03fe618e2fdb93cae336d2862b07a167": "h\\otimes v\\in V_h", "03feabf32b5ec498b9917df7f5cdb691": "c(V) = c_0(V) + c_1(V) + c_2(V) + \\cdots .", "03fec2e47d5c99405d591f252239312d": " \\left(\\frac{a}{-1}\\right) = \\begin{cases} -1 & \\mbox{if }a < 0, \\\\ 1 & \\mbox{if } a \\ge 0. \\end{cases} ", "03ff0f1aa2432df4f947e0570f58f967": "h(a)=h_0+\\sum_{i=1}^n h_ia_i\\,", "03ff61c1d4b3054b2fea1f017bf9a0f8": " \\psi^\\dagger \\sigma_j \\frac{\\partial\\psi}{\\partial t} + \\frac{\\partial\\psi^\\dagger}{\\partial t} \\sigma_j \\psi = \\frac{\\partial \\left( \\psi^\\dagger \\sigma_j \\psi\\right)}{\\partial t} ", "03ff64736f09ee66889b1e12aa6ab45a": "\n\\begin{align}\n\\frac{d E_{\\lambda}}{d\\lambda} &= \\frac{d}{d\\lambda}\\langle\\psi(\\lambda)|\\hat{H}_{\\lambda}|\\psi(\\lambda)\\rangle \\\\\n&=\\bigg\\langle\\frac{d\\psi(\\lambda)}{d\\lambda}\\bigg|\\hat{H}_{\\lambda}\\bigg|\\psi(\\lambda)\\bigg\\rangle + \\bigg\\langle\\psi(\\lambda)\\bigg|\\hat{H}_{\\lambda}\\bigg|\\frac{d\\psi(\\lambda)}{d\\lambda}\\bigg\\rangle + \\bigg\\langle\\psi(\\lambda)\\bigg|\\frac{d\\hat{H}_{\\lambda}}{d\\lambda}\\bigg|\\psi(\\lambda)\\bigg\\rangle \\\\\n&=E_{\\lambda}\\bigg\\langle\\frac{d\\psi(\\lambda)}{d\\lambda}\\bigg|\\psi(\\lambda)\\bigg\\rangle + E_{\\lambda}\\bigg\\langle\\psi(\\lambda)\\bigg|\\frac{d\\psi(\\lambda)}{d\\lambda}\\bigg\\rangle + \\bigg\\langle\\psi(\\lambda)\\bigg|\\frac{d\\hat{H}_{\\lambda}}{d\\lambda}\\bigg|\\psi(\\lambda)\\bigg\\rangle \\\\\n&=E_{\\lambda}\\frac{d}{d\\lambda}\\bigg\\langle\\psi(\\lambda)\\bigg|\\psi(\\lambda)\\bigg\\rangle + \\bigg\\langle\\psi(\\lambda)\\bigg|\\frac{d\\hat{H}_{\\lambda}}{d\\lambda}\\bigg|\\psi(\\lambda)\\bigg\\rangle \\\\\n&=\\bigg\\langle\\psi(\\lambda)\\bigg|\\frac{d\\hat{H}_{\\lambda}}{d\\lambda}\\bigg|\\psi(\\lambda)\\bigg\\rangle.\n\\end{align}\n", "03ff7f007de20f47914ea971fd576bb4": " f_1(x_1,\\ldots,x_n), \\ldots, f_k(x_1,\\ldots,x_n).", "03ff922d126da125152f09f9cbabcbd1": "\n\\begin{matrix}\nx + y &=& y + x\\\\\n(x+y)+z&=& x+(y+z)\\\\\nx+x&=&x\\\\\n(x+y)\\cdot z &=& (x\\cdot z) + (y\\cdot z)\\\\\n(x \\cdot y)\\cdot z &=& x \\cdot (y \\cdot z)\n\\end{matrix}\n", "03ffbe62d3a73b362ddbd6ad63e02e40": "t_A = t_B", "04000d383194855a059ae7cba74fa374": "(m)_n = m(m-1)(m-2) \\cdots (m-n+1).", "0400c0a41bae7e3c544019662d174a8c": "\\mathbb{Q}^+,\\cdot", "0400c798906749e6e1e973746d3f0d55": "\\phi ( \\bold{r} ) = \\frac {1}{4 \\pi \\varepsilon_0}\\int \\frac { \\rho ( \\bold{ r}_0 )-\\bold{\\nabla_{\\bold {r_0}}\\cdot} \\bold{p} ( \\bold{ r}_0 )} {| \\bold{ r}- \\bold{r}_0 | } d^3 \\bold{ r}_0 \\ , ", "04012bc7cc9262fca293ae5fe12e8f71": "(f_k)", "04015b657c225a30dfba8934d61e4b23": "\\chi_1(\\omega) = {1 \\over \\pi} \\mathcal{P}\\!\\!\\!\\int \\limits_{-\\infty}^\\infty {\\chi_2(\\omega') \\over \\omega' - \\omega}\\,d\\omega'", "0401a19094ed2243a12b57cb2d91899c": "F/k", "0401a41c89b868b700cb99bb29813b49": "\\lim_{N\\to\\infty}\\left(1+\\frac{r}{N}\\right)^{Nt}=e^{rt}", "0402308150bcd0917bfccf90bf221222": " R_{in} = \\frac {v_x} {i_x} = r_{\\pi} + (\\beta + 1) ({r_O} || {R_L}) ", "0402326c7b73ae0fc89190b47b957bf1": " \\gamma (t) = 4\\pi t + i\\cos(4\\pi t) 0 \\leq t \\leq 1", "04026d001c412a65713544da11c6caf6": "x \\wedge \\left( y \\vee z \\right)\n= \\left( x \\wedge y \\right) \\vee \\left( x \\wedge z \\right) ", "0402b088614dbd675146aa12c9226915": "x\\ne 1\\ ", "0402e0626e7835f8c4b12e5778648846": "\n\\nabla \\cdot \\mathbf F =\n\\frac{1}{H}\\frac{\\partial}{\\partial q^k} \\left(\\frac{H}{h_k} F_k\\right)\n", "0402e9bced3d440d72c3e362204a1255": "\\left(Ax\\right)_i", "040317e39ab6225b2f64a7b2c7012b4f": "2a_k \\ge a_{k+1} \\, \\forall \\, k \\ge 1", "040320f7a3acf4ea621f9cdab62dc440": "N=\\frac{g_0z}{1-z}+\\frac{f}{(\\hbar\\omega\\beta)^3}~\\textrm{Li}_3(z)", "04036a75e479ca9ff8489c2ae2510683": "= A_1 \\mathbf{e_1} (\\mathbf{e_2 e_3})^2 +A_2 \\mathbf{e_2} (\\mathbf{e_3 e_1})^2 +A_3 \\mathbf{e_3}(\\mathbf{e_1 e_2})^2 \\ ", "04037abd8428e254cae323da3f211bac": "VCA(64x^3-112x+56,(0,2)) \\cup VCA(64x^3+192x^2+80x+8,(2,4))", "0403f53cae7b1c1e3791cc34264bddba": " X_i(s)=x_0 + s\\sum_{j=1}^m a_{ij} f(X_j(s)),\\,\\,\\, x(s)=x_0 + s \\sum_{j=1}^m b_jf(X_j(s))", "0403f58796bae7a024a8a63dfc6cff48": "T_6( n^2 + n ) + T_5( n^2 + 3n ) + (n + 1)T_4 + T_1 + T_2 + T_3 + T_7 \\le k( n^2 + n ) + k( n^2 + 3n ) + kn + 5k", "0404085b4df5835395033d5218ff0967": " \\pi_4 = L^q \\mu^r k^s \\beta^t g^u h", "040409df3b8501385ad3738fc2580981": "\\beta<\\alpha", "04047ee4aafa6ea65dbc529a47c97f69": "\\mathbf{3}\\otimes\\mathbf{3}\\otimes\\mathbf{3}=\\mathbf{10}_S\\oplus\\mathbf{8}_M\\oplus\\mathbf{8}_M\\oplus\\mathbf{1}_A", "0404ab2b2d5eae0e14317530984cd375": "\\beta(g) \\propto g^\\alpha", "0404d3f8a99190f20fca883f8fca0385": "\\mathbf{u}_k=\\left[u_0, u_1, \\dots, u_{k-1} \\right],", "040515eac86f681bafb3b7c9852a4d58": "\n\\bar{\\mu}_{\\text{min}} = \\lambda_{\\text{min}}\\left( \\frac{1}{n} \\sum_{k=1}^n \\mathbb{E}\\, \\mathbf{X}_k \\right) \\quad \\text{and} \\quad\n\\bar{\\mu}_{\\text{max}} = \\lambda_{\\text{max}}\\left( \\frac{1}{n} \\sum_{k=1}^n \\mathbb{E}\\, \\mathbf{X}_k \\right).\n", "040548e2562d68d8aba49c12072fbbff": "v_3(t) = \\int_{t_0}^{t} (K_1i_1(\\tau)+K_2i_2(\\tau)) d\\tau.", "0405c9f9d2147a9e6088cbc4a30a8707": "B=B(b, \\lambda)", "0406546e4269ae5098ed91c8999bfa5e": "\\{x \\in V \\colon x = a + n, n \\in W\\}", "0406baf1245fc32c7fcf9e6f50931e91": "\nG = G[\\tilde{S}(\\omega)] = \\int_{-\\infty}^\\infty \\eta(\\omega) \\tilde{S}(\\omega) \\, d\\omega \n", "0406fb29ebe211df5b5b03aeec27b35d": "Re = Re_c", "0407aa1318c41683bf20fd50ff5172e1": " 0.082 H_s^2 ", "0407c900cd036ac4b5e1a43acc9cab35": "\\delta^{(k)}[\\varphi] = (-1)^k \\varphi^{(k)}(0).", "0407f208210c245681a6f4ba985097f2": "y_c = {2 \\over 3} E_{lake} \\,\\!", "040892129b35344eedc8972773e4c4f4": "-(-h)", "0408a2aa720367d75c62a7526d968221": "N \\leq \\frac{Br}{r+1}", "0408e3851c8cdd649c5cee4d7cd7a0c5": "\\alpha \\in[0,1]", "04093a271f00d21635b22f616853e6d3": " A+0=A", "040947dcf5fdde48307e915d313b0839": "\\pi^{-1}\\mathcal{I}\\cdot\\mathcal{O}_{\\operatorname{Bl}_\\mathcal{I} X}", "04099cf0c261f27bc39c95cba442e0c0": " \\forall x \\in A, \\ \\exists y \\in 2^B, \\ x \\in y ", "0409bae34a658d6a4b0c560f9aafb3ac": "\n\\{a_{11}, a_{12}, a_{13}, a_{22}, a_{23}, a_{33}\\}\n", "040a0fe1af61c8a25e80b82326132bc1": "\\left\\{\n\\bar{Z}_{1},\\ldots,\\bar{Z}_{s+c},\\bar{X}_{s+1},\\ldots,\\bar{X}_{s+c}\\right\\} ", "040a906dae13f008ae8164b64adc2eec": "\\frac{3b}{4}", "040ad4c564a4398f895a2bfa60d1e23e": "\n\\begin{align}\n2 \\int \\sec^3 x \\, dx &{}= \\sec x \\tan x + \\int \\sec x\\,dx \\\\\n&{}= \\sec x \\tan x + \\ln|\\sec x + \\tan x| + C.\n\\end{align}\n", "040adb5020d648afa0b1fae88ab194d6": "\\bar{R}^ 2", "040addb211b23f8dec306ce628709283": "\\varepsilon_1''' = -\\frac{\\nu}{E}\\sigma_3", "040aea4c66b6894f163c22953d213a86": "A \\cap B = \\overline{\\overline{A} \\cup \\overline{B}} ", "040b6c2244036a6c9bc837b62ea230b5": "x \\in \\mathbb{R}^L_+ \\ .", "040c0704d1d15a4b3fba31918f2a21b7": "D(\\alpha)", "040c11cf6b8898bb86eaf8d66253d425": "e^\\cdots", "040c19fd5867974a7cea1e053feb6984": "\\{ u', u \\} \\in E \\setminus M", "040c39e51f49c201f5780618028af2ac": " {DB} \\equiv \\frac{1}{N}\\displaystyle\\sum_{i=1}^N D_i", "040c3bbfc6598ecb26e80f76230f92b1": "\\epsilon^1: \\quad 2S_0'S_1' + S_0'' = 0.", "040c456e46507d5bcb155bfcc94d261a": " I_{KAR} = (\\frac {2 Z^2}{n^2 F r})^n ", "040cabec1114ed4c6f505e979e430e5d": " a + (180", "040cdd5b0489fa26d9225262e0eb498c": "P_n = {\\bold 1}'\\otimes\\dots\\otimes{\\bold 1}'", "040cf9b47973c6fc123715d3e59a55da": "\\frac{1}{G_\\mathrm{total}} = \\frac{1}{G_1} + \\frac{1}{G_2} + \\cdots + \\frac{1}{G_n}", "040d2d4d9d9a6775698afb13b0929807": "\\Delta \\lambda", "040d391cdb42c491cc9e569cb39f6860": "{\\mathcal C}_n(z) = \\frac{1}{2 \\pi i} \\oint_C \\frac{\\exp(z+z/t)}{t^{n+1}}\\, dt = \\frac{1}{2 \\pi}\\int_0^{2 \\pi} \\exp(z(1+\\exp(-i\\theta))-ni\\theta))\\,d\\theta.", "040d65a49095e3ca05abbfe6aea6bc68": "\n N_{\\alpha\\beta} := \\int_{-h}^h \\sigma_{\\alpha\\beta}~dx_3 ~;~~\n M_{\\alpha\\beta} := \\int_{-h}^h x_3~\\sigma_{\\alpha\\beta}~dx_3~.\n", "040d891bf42b3af1a37a77b06fdf60b9": "\\oint \\mathbf{B} \\cdot d\\boldsymbol{\\ell} = \\mu_0 I_{\\mathrm{enc}},", "040e3118a4a6e49bffe502dd69465b8e": "\\ v_i = \\sqrt {2gd}\\ ", "040e60d5d63c56e5c5c0203a79d41b50": " I=(a,b)", "040e7a524dcfb640f0ad6571cb348051": "v_{\\text{in}}", "040ebb3e39938fa7bdf7d1275aabb189": "M(E)", "040ef16ee427a4f5b8955fe1d0653ce8": "QE_\\lambda=\\eta =\\frac{N_e}{N_\\nu}", "040f4e6aad36a049d12ca18e6df07c24": "\\tanh", "040f8b1063d9fe4ac7f5d765a4f561a7": "\\hat{C} = \\sum_{i=1}^r c_i\\bar{Y}_i", "040f915801fa8603100ca166fbcec507": " U(\\mathfrak{g})/I", "040ff8a72b1f900e7b36fee6bc0cf2ed": "\nE[\\Delta(t)] \\leq B - \\epsilon \\sum_{i=1}^N E[Q_i(t)]\n", "04103810029df237b1be42a58f7fda1b": "2 \\uparrow \\uparrow \\uparrow 4 = 2 \\uparrow \\uparrow 2\\uparrow \\uparrow 2 \\uparrow \\uparrow 2=2 \\uparrow \\uparrow 2 \\uparrow \\uparrow 2 \\uparrow 2=2\\uparrow \\uparrow 2 \\uparrow \\uparrow 4=2 \\uparrow \\uparrow 2 \\uparrow 2 \\uparrow 2 \\uparrow 2 = 2 \\uparrow \\uparrow 65536", "04104fe57542b5399441f651a80081c4": " = -I I' ds ds' \\left[cos(xds)cos(rds)+cos(rds)cos(xds')\\right] ", "041061f5b7aa1fa7a7a0725b9bb244a3": "1 p_{j}\\\\\n \\end{array}\n \\right.\n\\end{array}\n", "042223f2344fc81a7c09aa69b55a73cf": "X = g_{1}^{x_1}g_{2}^{x_2}", "042306651af18bcacca1f43ab885ce08": "(\\mathbf{D_1} - \\mathbf{D_2})\\cdot \\hat{\\mathbf{n}} = D_{1,\\perp} - D_{2,\\perp} = \\sigma_\\text{f} ", "042311da4bf0cfeb58499992324c9656": "\\frac{Y(z)}{z}", "0423372acc78e5e1965fadc7052d2e63": "E(-)\\,", "04235fbcb43527845cca755f3c862950": "j = H,T", "0423631118dc235bc1c532da9069e111": "f(x) = 3 + 2 x + 1 x^2 + 0 x^3 + 0 x^4 + \\cdots \\,", "04236b0dbc6277364b244d7deb26a24c": "t \\in S", "0423a27c892d4b106a01e930565cfe7e": " A = QR\\,\\!", "0423a45525cec11e3fc7df3731d804e4": " P_{\\rm fwd}, \\, P_{\\rm bwd}", "0423c9cf2fc5bae11fe3c51366abf6cf": "\\scriptstyle S ", "0423e9f4497d84a49a61aad4d9a28793": "\\Delta := \\min \\{c(i,j)-y(i)-y(j): i \\in Z \\cap S, j \\in T \\setminus Z\\}", "04244cd38e478f660ecaab328a1b0191": "|\\{(x,y)\\;:\\; \\operatorname{lcm}(x,y)=D\\}|= 3^{\\omega(D)},\\;", "0424739beee9f4d56c88daa503a7daaf": " \\left( T(n) \\right)_{n = 1}^{\\infty} ", "0424c8a3c1bc4e3b7d8d0ff7d0f61a85": "\\Delta g_{i,\\mathrm{mix}}=RT\\ln x_i", "0424d05bf07a4693eeff7999232c683f": "\\delta W = -mg\\delta y = -mgL\\sin\\theta\\delta\\theta.", "04250f98f961b75fab11084a07494a65": " \\sum_{k=0}^\\infty a_k z^k = A(z) < \\infty \\quad \\Rightarrow \\quad {\\textstyle \\sum} a_kz^k = A(z) \\,\\, (\\boldsymbol{B},\\,\\boldsymbol{wB}). ", "0425a405b5515fb35e3cffb968a7883b": "B \\supseteq \\{c\\}", "0425a6596203e91bbf992827d5b4f628": "\\mathbf v = v_1 \\mathbf e_1 + v_2 \\mathbf e_2 + v_3 \\mathbf e_3", "0425ec80bf7831d3ae52f578c64e1ae2": "\\ \\gamma \\, ", "04262cba3e5105195da110567fadb84a": "f^{-1}\\mathcal{G}", "0426798c7976774172f3b693c5f04192": " \\frac{\\mathrm{d}}{\\mathrm{d} x} \\int_{\\Omega} \\, f(x, \\omega) \\mathrm{d} \\omega = \\int_{\\Omega} \\, f_x ( x, \\omega) \\mathrm{d} \\omega ", "0426819fccb67b54198a009965df4775": "s_{ln} \\,", "04272fe09e6c1a08802e4b3cf35b7411": "10\\uparrow\\uparrow 10\\uparrow\\uparrow (10\\uparrow)^{497}(9.73\\times 10^{32})=(10\\uparrow\\uparrow)^{2} (10\\uparrow)^{497}(9.73\\times 10^{32})", "04274f736adbd0c9342ce19544b22c48": "\\begin{bmatrix} R \\\\ G \\\\ B \\end{bmatrix} = \\begin{bmatrix} 3.1956 & 2.4478 & -0.1434 \\\\ -2.5455 & 7.0492 & 0.9963 \\\\ 0.0000 & 0.0000 & 1.0000 \\end{bmatrix} \\begin{bmatrix} X \\\\ Y \\\\ Z \\end{bmatrix}", "042799d05b97293e7376791b08298fc4": "f_1 \\Leftrightarrow f", "0428292809fdc49a2fa94bb50d7afab4": "\\Pi_\\beta\\, ", "04282b9625be9da1a5f988133a7f400f": "\\int P\\left(A,\\tilde{A}\\right)dAd\\tilde{A} = N\\int exp\\left(L\\left(A,\\tilde{A}\\right)\\right)dAd\\tilde{A},", "042833ea03a8a157fa009a9183156145": "N\\Delta F", "04284904414567d9d27199ed98b105d9": "V(\\rho,\\varphi,z)=\\sum_n \\int dk\\,\\, A_n(k) P_n(k,\\rho) \\Phi_n(\\varphi) Z(k,z)\\,", "04286d274644a21dfaa0c7eb4dd2b3ed": "\\gamma \\in \\mathbb R", "04289e638f16b4cb648cef93380133f1": "\\Delta v \\ll v_\\text{e}", "0428ff8815ad7c4958f8ccb8fa0451ea": "F'(R:BL < Q < BU:AL < P < AU)=\\sum_{T\\!B=1}^{U\\!B=\\infty}\\left(\\sum_{T\\!A=1}^{U\\!A=\\infty}\\frac{F'(R:Q_{(tb)}:P_{(ta)})}{U\\!A}\\right)\\frac{1}{U\\!B}.\\,\\!", "042917fde562bb8fa3e30d31f8a8a6e6": " | /2 ", "042924d24de5e4ace957814fc8e65a87": "A(x, \\vec{y}", "04295addffc8ca7a550c45469b49a0f8": "(\\pm P_i,\\pm P_i)", "04298604d0fbd5d2a4f1402f79f7dacc": "\\delta_{ij}=0", "042986dd20834643d3315851ee8ef56b": "H'=1/M\\Delta\\tau", "04298c6a5f765a2ae1bcb0b60873cd6d": " R = \\frac {4 n \\rho V D \\left( 1 - a X - b X^2 - cX^3 \\right)} {\\mu_{Wall} \\left(3 n + 1 \\right) } ", "0429a04217d4df18daa15804db0d5d25": "R_i \\leftrightarrow R_j", "0429a3a73913d53ed1fa712b8687ad85": "\nw_{jj}\\sim\\sigma_{jj}\\chi^2_m", "0429f41635a9ddf84e37da9ed6d4a610": "dS_{l,t} = \\theta (S_{t}-S_{l,t})dt + \\sigma S_{l,t} \\left(\\rho dW_t + \\sqrt{1-\\rho^2}dW_t^\\bot\\right)", "0429f4e3edd6ac67209d78a7dd3d922a": "\\left(\\Lambda^k(V)\\right)\\wedge\\left(\\Lambda^p(V)\\right)\\sub \\Lambda^{k+p}(V).", "042a038f750e6fa4c18ce6a96253daaf": "I_x \\xrightarrow{\\omega \\tau I_z} \\cos (\\omega \\tau)I_x - \\sin (\\omega \\tau)I_y ", "042a8292539f0499c2cdf28a1aad6163": " Ehr_P(z) = \\frac{\\sum_{j=0}^d h_j^\\ast z^j}{(1 - z)^{n + 1}}, ", "042a8a01b244f575db31a30e7b5e388d": "M(x)=e^{\\int \\frac{-2}{x}\\,dx} = e^{-2 \\ln x} = {(e^{\\ln x})}^{-2} = x^{-2} ", "042ab9d4e16857e079311cb3883c1e59": " \\zeta_Q(s) = \\sum_{(m,n)\\ne (0,0)} {1\\over Q(m,n)^s}.\\ ", "042acfc4e101f725e2ce84bb5b669702": "\\Delta_B =\\frac{2\\textrm{arctanh}\\sqrt{(1-\\omega^{2})(1-v_\\text{K}^2)}}{\\sqrt{1-\\omega^{2}}}", "042afd40606d18eb257b24816c76e3ad": "\\Sigma_{a \\in H} W(a) \\leq k \\cdot \\Sigma_{j \\in S} p_j", "042b0dcf8f6e22d59f523af8081a2c29": "u : E \\to F", "042b362b178d92bb939071045c77d8c3": "f(x) \\ge f(x+y) - f(y)", "042b4d725d323ea7cfa7c77ee4f036c6": "\\langle \\operatorname{exp}(aX)\\rangle \\ \\stackrel{\\mathrm{def}}{=} \\ \\sum_{i=0}^\\infty\\frac{a^i}{i!} X^{'i}", "042b7d5dd05ae9c5d3171737bfa67079": " L_c^-(f) = \\left\\{ (x_1, \\cdots, x_n) \\, \\mid \\, f(x_1, \\cdots, x_n) \\leq c \\right\\} ", "042c3f68e3584df2fe195249113afa9c": "UIRP: {\\Delta}E_t(S_{t + k}) = E_t(S_{t + k}) - S_t = i_\\$ - i_c", "042cb15498fff8dd0cd98aba671b0e0c": " \\tfrac{256}{81} ", "042d04c8dfc93ba83eef36b401af7875": " \\overset{d}{=}", "042d7fe5a77bd8e1275f7167c5b284e4": "M _{CB} ^f = \\frac{qL^2}{12} = \\frac{1 \\times 10^2}{12} = 8.333 \\mathrm{\\,kN \\,m}", "042dd3fd944d5f0a44723261e66bd217": "\\chi(x,X)\\triangleq\\,", "042e6c2d87f56c5d1217c6f0a2db6b06": "x \\, = \\, \\zeta_{L}\\exp \\left( \\frac {\\mu}{k_{B}T} \\right)", "042e85197d4730fb77d56abfe20074c3": "V,f", "042efe365991582b13e0ece6de5b3a44": " e_{ij} = \\begin{cases} a_i, & \\mbox{if } i \\in [1,N] \\\\ b_{ij}, & \\mbox{if }i \\in [N+1,2N] \\end{cases} ", "042f143e616aa57ea4785ce3899aeb42": "L = k\\;S\\;V^2\\;C_L", "042f14746f4d46a704fa744eaf24c57f": "\\Phi(x) =\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^x e^\\tfrac{-t^2}{2}\\,\\mathrm dt = \\frac{1}{2}\\left[1+\\operatorname{erf}\\left(\\frac{x}{\\sqrt{2}}\\right)\\right]=\\frac{1}{2}\\,\\operatorname{erfc}\\left(-\\frac{x}{\\sqrt{2}}\\right)", "042f162c686ff22bb8e0c1c671b60b78": "H_2(z)", "042f26c0a172bfee32ccf9f4c212ad6a": "\nv = B \\sin ax \\cos by \\sin cz,\n", "042f2bf0ad28bf02656cedf3586e2125": "\\Rightarrow 0=\\frac{x^2}{2}+\\frac{x}{2}-F r_1^2\\qquad a=1/2,\\;b=1/2,\\;c=F r_1^2", "042f4855a013920daafd24d2b58267f4": "\n\\mathcal{N} \\int e^{-\\beta H(p, q)} d\\Gamma = 1,\n", "042f9220e89cca453bf5b4997f7392a3": "sig=(sig'||auth_0||auth_1||...||auth_{n-1})", "042fa1acd04ab993476705b196a351a6": "{\\Lambda^\\mu}_\\nu", "043007c6e9c962c3726d72e9b02baa57": "180^\\circ", "043013e71f91933c14ea6dd85fed8c5a": "m_{k}", "0430197ca6f8a78c3702a93ac5086129": " \\rho(a, c) -\\rho(b, a) - \\rho(b, c) \\le 1,", "043067e7c691cff1f7f7ef1221121ff0": "\n w(r) = -\\frac{q}{64 D} (a^2 -r^2)^2 \\quad \\text{and} \\quad \n \\phi(r) = \\frac{qr}{16 D}(a^2-r^2) \\,.\n", "04306fafcc30958126e5d8faf2bce637": "\\eta=(| n_\\Phi-n_{\\overline\\Phi}|)/ (n_\\Phi+n_{\\overline\\Phi})", "043099855a1d6c61e5614168914a5ca1": " h \\nu = \\Delta E ", "0430c7501f1c2d1615938f64fae168c1": "\\mathrm{(A/m^2)}", "0430fbb3b8721c34af98118178317474": "\\sigma(\\hat x) = \\{ \\chi(x) : \\chi \\in \\Delta(A) \\}", "04310407c8322c3c57b1a7224d2bdb8c": "\\begin{align}\n\\partial_1 c\n&= \\partial_1(t_1 + t_2 + t_3)\\\\\n&= \\partial_1(t_1) + \\partial_1(t_2) + \\partial_1(t_3)\\\\\n&= \\partial_1([v_1, v_2]) + \\partial_1([v_2, v_3]) + \\partial_1([v_3, v_4]) \\\\\n&= ([v_2]-[v_1]) + ([v_3]-[v_2]) + ([v_4]-[v_3]) \\\\\n&= [v_4]-[v_1].\n\\end{align}\n", "0431155a0b5d633b35f5b34cd4191f8d": "\nw(r) = \\int \\frac{\\mathrm{d}r}{\\sqrt{\\frac{r}{r_{s}}-1}} = 2 r_{s} \\sqrt{\\frac{r}{r_{s}}- 1} + \\mbox{constant}\n", "0431b0eef983a554d7d7afa238d5895d": " T_{ij}^{(3)}=s_{ik}s_{kj}-s_{mk}s_{km}\\frac{1}{3} \\delta_{ij} ", "0431bb4100c94e9cff839832e9afbf02": "(cdt)^2=dx^2+dy^2+dz^2+(cd\\tau)^2", "0431c3a638a1ed25d525799cf534562c": "\\theta_i = \\frac{x_i}{x_{max}} 2 \\pi ", "0431de28c694cf6c6b13a4b3a20ccf37": "\\begin{align}\np(\\mu|\\mathbf{X}) &\\propto p(\\mathbf{X}|\\mu) p(\\mu) \\\\\n& = \\left(\\frac{\\tau}{2\\pi}\\right)^{\\frac{n}{2}} \\exp\\left[-\\frac{1}{2}\\tau \\left(\\sum_{i=1}^n(x_i-\\bar{x})^2 + n(\\bar{x} -\\mu)^2\\right)\\right] \\sqrt{\\frac{\\tau_0}{2\\pi}} \\exp\\left(-\\frac{1}{2}\\tau_0(\\mu-\\mu_0)^2\\right) \\\\\n&\\propto \\exp\\left(-\\frac{1}{2}\\left(\\tau\\left(\\sum_{i=1}^n(x_i-\\bar{x})^2 + n(\\bar{x} -\\mu)^2\\right) + \\tau_0(\\mu-\\mu_0)^2\\right)\\right) \\\\\n&\\propto \\exp\\left(-\\frac{1}{2} \\left(n\\tau(\\bar{x}-\\mu)^2 + \\tau_0(\\mu-\\mu_0)^2 \\right)\\right) \\\\\n&= \\exp\\left(-\\frac{1}{2}(n\\tau + \\tau_0)\\left(\\mu - \\dfrac{n\\tau \\bar{x} + \\tau_0\\mu_0}{n\\tau + \\tau_0}\\right)^2 + \\frac{n\\tau\\tau_0}{n\\tau+\\tau_0}(\\bar{x} - \\mu_0)^2\\right) \\\\\n&\\propto \\exp\\left(-\\frac{1}{2}(n\\tau + \\tau_0)\\left(\\mu - \\dfrac{n\\tau \\bar{x} + \\tau_0\\mu_0}{n\\tau + \\tau_0}\\right)^2\\right)\n\\end{align}", "0431ea012994d9769b8b4f3c65c2b705": "\\tfrac{4}{256}", "0432161c17ddb5c40865fbf0e8330d4f": " S \\approx \\frac{ \\beta }{ 3 \\alpha + 0.2 } ", "04322155a0969e53dd2af27801c6f824": " \\lbrace \\bold{x}^{ \\left( k \\right) } \\rbrace _{k = 0}^{\\infty} ", "04324364800ac799e323e692305adc0c": "\\{e^{-a_0t}\\}", "043266aabb099e64fc8c537494a7aca8": "z=c \\sinh v", "0432bd149aa64859b8197adf82771c1e": "T_h", "04337624b6e95c7568e5c14798f4015d": "A, \\ ", "043398bf57dc97792d6b2447ed511c2c": " (y_1, y_2) ", "04339e3757686e989e74a5b743cf957a": "F=m(\\ddot{r}-r\\dot{\\theta }^2)=-m\\left(h^{2}u^{2}\\frac{\\mathrm{d}^{2}u}{\\mathrm{d}\\theta ^{2}}+h^{2}u^{3}\\right)=-mh^{2}u^{2}\\left(\\frac{\\mathrm{d}^{2}u}{\\mathrm{d}\\theta ^{2}}+u\\right)", "0433a0b4cdc6725c6ef740321fe194b1": " \\frac{di}{dt} + \\frac{di}{da} = \\delta(a) \\lambda ( S + \\sigma R ) - \\gamma(a) i - \\mu(a) i - m_i(a) i, ", "04345179e5470e70f5f2e79e5a741a11": "n^2-n > 2\\times365\\ln 2 \\,\\! .", "0434626709a496ba269fb912232668c5": "\\# X (\\mathbf{F}_q) = q^{\\operatorname{dim}X} \\sum_{i \\ge 0} (-1)^i \\operatorname{tr}(f; H^i(X(\\mathbf{F}_q), \\mathbb{Q}_l)),", "0434ad71ea4dbab5a8fabc778c60ce13": "\\Delta t' = \\frac{\\Delta t}{\\sqrt{1-\\frac{v^2}{c^2}}}", "0434bb534c7fc17969197e1f0600f80d": "P^{a}P^{b} = \\sum_i (-1)^{a+i}{(p-1)(b-i)-1 \\choose a-pi} P^{a+b-i}P^i", "0434f618df958d57676fc580c89c7c54": "a\\, \\frac{\\sinh\\, \\bigl( k\\, (z+h) \\bigr)}{\\sinh\\, (k\\, h)}\\, \\cos\\, \\theta\\,", "04352aa9441c869cf4ba1cc540a1b71a": "\\sum_m (-1)^{j-m}\n\\begin{pmatrix}\n j & j & J\\\\\n m & -m & 0\n\\end{pmatrix} = \\sqrt{2j+1}~ \\delta_{J0}\n", "0435531cb50ee00cd4e3167505c10d6e": "\\dot{\\mathbf{x}},\\dots,\\mathbf{x}^{(n)}", "04358bea9d231b9a487564055a5ab70e": "\\gamma(k,i)\\,\\!", "0435d2bf45e2d8102d75cbcfd5f25301": " \\mathbf{s} = \\mathbf{B} \\mathbf{d}. ", "043615f47e6c4c88de50220114b1a304": "\\, K = 2n\\pi/a", "0436c6c8b55041676fb391e7ee0214ae": "r_1,r_2,\\cdots,r_a", "043747fa54887321886921e4ceef8ba3": "R_{ix}(t)=M_{i}A_{ix}(t)\\frac{}{}", "0437674141b352e9e6a80b329e9dfa93": "A_\\lambda>0", "0437d63f527b355a2f93abafb5739d1b": "i.", "04384da8fde85931f668ea7ab2435340": "HA_i", "04388a49ab38977d0ec391e4c0510877": " \\lim_{k \\to \\infty} \\| \\bold T^k \\| = 0, ", "04389dc3e787e23ef2e5982982017cfc": "T = 50 + 10 {x- \\mu \\over \\sigma}", "0438a0e8326b4167818569ed6f179378": "f : V^k \\to K\\ ", "0439479af4192d42884cc58105facddf": "\\sum_{j=1}^{n_S} \\sum_{b_j=0}^{a_j} \\sum_{ \\beta_j } x_{b_j} \\ b_j = \\sum_{h=1}^{n_P} \\sum_{ d_h=0 }^{c_h} \\sum_{ \\gamma_h } u_{\\gamma_h} \\ y_{d_h} \\ d_{h}. ", "04397443b09ae04010032ff6bbcce1c5": "f \\mapsto \\mathbb{P}_n f", "04399fe68406275419e18c0e85eab335": "\\frac{a}{x}=o(S_0')\\,", "043a0a32bc14f031f8299bcd330a0e9b": "\\hat{f}(t) = f(t) \\, ", "043a0b9537ea34a66dd44536ef1635cf": "{{\\mathbf{k}}}[\\mathbf{x}]", "043a1613656191ec43c873898661e76e": "\\mathbb{E} \\log(S_t)=\\log(S_0)+(\\mu-\\sigma^2/2)t", "043a46836b4b629ac65945ceda7d90d4": "Y'=YM_i.", "043a49f81e88957db2da952cc274bca9": "f = {1 \\over 2 \\pi \\sqrt {LC}}", "043a4ba8841199b14d188dc969115fdb": "BA = \\frac{\\pi \\times (DBH/2)^2}{144}", "043a93f86a9f805fabace17b1c6aff92": "(X^*_{b}, Y^*_{b}; Z)", "043ada99412e7bc11f2cd700d32c0917": "\\rho\\mu = 1_Y", "043b27c3e4f7d051bb8ef7131fcbc79e": "F^\\dagger", "043b3b9bfb851fabf350c5784ec38c2f": "r e^{a j}, - r e^ {a j}, r j e^{a j}, - r j e^{a j}, \\quad r > 0", "043b43b22560464bcd85b27ca7e9bffb": "x_p^2\\equiv\\frac{2\\xi^2\\sqrt{G}}{\\sqrt{8\\xi^2(\\xi^2\\!+\\!1)+12G\\xi^2-G^3}-\\sqrt{G^3}}", "043b8526035e9453eaf9471988c9bb5c": " R_{01} = \\frac {W_{cu}} {3{I_{S}}^{2}}", "043ba4b2180cec84c17497942ebfad63": "C_\\max, L_\\max, E_\\max, T_\\max, \\sum C_i, \\sum L_i, \\sum E_i, \\sum T_i", "043bdd448fd560f75d1648edf7a1a4b1": " dN_i = \\sum_k \\nu_{ik} d\\xi_k. \\,", "043c10c6bba91fd5ba82f14b1aea724f": "\\min_{\\alpha,\\,\\beta}Q(\\alpha,\\beta)", "043c183552a5d1a083989d2e2c340959": " \\scriptstyle C_c^1(\\Omega,\\mathbb{R}^n)", "043c1ee76d36717817e06c05c9e1087e": "e(S)", "043c5dd74964ad33ceff323d809cdc8b": "\\frac{d\\alpha}{dt}=q+\\frac{Z}{mU}", "043c6bc108326a3fb8ac1410de54d183": "O(\\sqrt{V})", "043c6f9f45b13e3a1395d5a31c341bad": " \\mathcal{L} \\, = \\, \\mathcal{L}_{\\mathrm{field}} + \\mathcal{L}_{\\mathrm{int}} = - \\frac{1}{4 \\mu_0} F^{\\alpha \\beta} F_{\\alpha \\beta} - A_{\\alpha} J^{\\alpha} \\,.", "043c772f6e1bdcee98556418393b3ad3": "\\displaystyle{\\mathfrak{g}=\\oplus_{i=1}^N \\mathfrak{g}_i,}", "043cb308a9d20de572bd4c1e19cc7699": "S^{n-1} \\to G", "043cedd5ff1ac6df55cef007bad07ce7": "\\varphi = \\frac11+\\frac12+\\frac19+\\frac1{145}+\\frac1{37986}+\\cdots", "043d4aa08c6b9f69d15db48f9992471a": "w\\,R\\,u\\land w\\,R\\,v\\Rightarrow u\\,R\\,v", "043d75ee748f4359e858a79b5c6a705a": "\\cos \\theta = \\frac{e^{i\\theta} + e^{-i\\theta}}{2} \\,", "043d986c307ab908b2420d8c88cad08f": "\\textstyle [x] = [\\mathbf{v}_1, \\ldots, \\mathbf{v}_m]", "043dd1f3ff7f0961963cf74d666128f5": " a_{i \\pm \\frac{1}{2}} \\ ", "043dfa0dce4e1eaa000c4ab46ff93863": "\nL_x=\n\\begin{pmatrix}\n0& 0& 0\\\\\n0& 0& -1\\\\\n0& 1& 0\n\\end{pmatrix},\nL_y=\n\\begin{pmatrix}\n0& 0& 1\\\\\n0& 0& 0\\\\\n-1& 0& 0\n\\end{pmatrix},\nL_z=\n\\begin{pmatrix}\n0& -1& 0\\\\\n1& 0& 0\\\\\n0& 0& 0\n\\end{pmatrix}.\n", "043dfb1219b6cab3960f60c45853999d": " U_\\theta =\\begin{bmatrix} e^{i \\theta} & 0 \\\\ 0 & 1 \\end{bmatrix}, ", "043dfb9dbd2157ad42c6fe313393ef24": "mv^2", "043e0e32ada017cc478a144049396d2c": " n_{\\nu_j}", "043e63402705f9ad750f879c5e552c00": "x = x_A", "043e9e440597410021257b5f9afa39d2": "|A|=|A\\times A|", "043eb6de5f4b5487d88efaa861518ec4": "f\\left(E\\right)", "043f027b254081d100d667508ddcd4b6": "\\kappa_b(k,i)\\,\\!", "043f334c7f494be53a0fd5e6e0af9bca": "ogd", "043f7fd770592fb93fc45041bfd6ba33": "f(n)=O\\left(n^n\\right)", "043fc37a324e096884d731e132cbab12": "\\frac{f(n)}{n^{\\log_b a}} = \\frac{\\frac{n}{\\log n}}{n^{log_2 2}} = \\frac{n}{n \\log n} = \\frac{1}{\\log n}", "04400f3aa13f4ee2969b1ee5599e8570": "I:f^\\infty= \\{g\\in R| (\\exists k\\in \\mathbb N) f^ng\\in I\\}", "04404eb69c936453785be20232e1d157": "N_s={120\\times{50}\\over{6}}", "04408afa08486fceff20014e0af5c106": "k=0,1,2,... ", "0440cfa6cdd7d42bd092724ef8503f2c": "m_{em}=E_{em}/c^2", "0440f47714e39f5332168d41b2abdc51": "C=\\frac{\\pi }{24}=0.131", "0440f7ae83b9eb81c046f1fb8da9960e": "{\\rm 1~Rayl = 1~\\frac{dyn \\cdot s}{cm^3}}", "04415ad3f122fd85386427796ab790c3": "{M_{2^\\infty}(\\mathbb{C})}", "0441a46f2b935dfc0b70c5760e6755a2": "z\\mapsto\\pm z", "0441cee755e1999da93fa506f511b548": "\\pi_{n-1}(Ff)", "04422f2574ca15e544d9de9538e45e3e": "\\lim_{n \\to \\infty} |{\\frac{a_{n+1}}{a_n}}| = r.", "044288e8ffc49792d28cd86657921099": " \\mathbf{B} = \\boldsymbol{\\nabla} \\times \\mathbf{A}.", "0443c0ef188080b775ff95a0103cf0d0": "L_x(x, y)=-1/2\\cdot L(x-1, y) + 0 \\cdot L(x, y) + 1/2 \\cdot L(x+1, y)\\,", "044408da84990ea6593e36887b3579c4": "f(x)=2x\\,", "044425ed6cbeb742f7e87d152a4edf4f": " g(X,\\theta) ", "04447f59c1bf03ba62ca2bbed7933c06": "K/2,", "04459080d01e3b6016b4a1f5c038ed0f": "C_n=2^{n\\choose 2} - \\frac{1}{n}\\sum_{k=1}^{n-1} k{n\\choose k} 2^{n-k\\choose 2} C_k.", "04459395e8049f50c0de25d4afa6dec3": "(\\sqrt[5]{100})^{5--1.47}\\approx 387", "0445b220b0fa7b1df97191bf5c256d76": "\\mathbf{B}(t)=(1-t)^3\\mathbf{P}_0+3(1-t)^2t\\mathbf{P}_1+3(1-t)t^2\\mathbf{P}_2+t^3\\mathbf{P}_3 \\mbox{ , } t \\in [0,1].", "0445d20b06e8f0006930d71db82bed73": "\\left(\\frac{C}{h}\\right) = {2 \\pi \\epsilon \\over \\ln(D/d)}= {2 \\pi \\epsilon_0 \\epsilon_r \\over \\ln(D/d)}", "0445eb70e4d914478364efbaacf737c7": "c_V", "04460a1d550c894c5fed25ac9ca64815": "\\int_{\\mathbb{R}^{n}} f(x) \\, \\mathrm{d} x < + \\infty.", "0446322a5f408e8fd1f22d8f5700ecd4": "\\operatorname{relint}(S)", "0446751577e6b290779b469d5dbe7331": "v^2=Q(v)", "04468d1922634dfcdc37a7c70f64af9e": "C_n^{(\\alpha)}(x) = \\frac{(-2)^n}{n!}\\frac{\\Gamma(n+\\alpha)\\Gamma(n+2\\alpha)}{\\Gamma(\\alpha)\\Gamma(2n+2\\alpha)}(1-x^2)^{-\\alpha+1/2}\\frac{d^n}{dx^n}\\left[(1-x^2)^{n+\\alpha-1/2}\\right].", "04469c5ecd43e9b9dfd4fc24d43dde7d": "f_{\\alpha} = F_{\\alpha\\beta}J^{\\beta} .\\!", "0446d05a6479f6c947639821b6c5f13a": " \\sum_{i=1}^d S_i + \\sum_{iB)=A_3 \\cdot \\overline{B}_3+x_3 A_2 \\overline{B}_2+x_3 x_2 A_1 \\overline{B}_1+x_3x_2x_1 A_0 \\overline{B}_0", "046ca8782936938ebb7b5935d7d0c664": "f\\in C^\\alpha(\\Omega)", "046cb06e29f4c1e90331985640ad776a": " \\iint_D \\ f(x,y ) \\ dx \\,dy , ", "046cfdd94af44ab54b498ffcbd636e5b": " \\epsilon_0 = E_0 - m_0 c^2 ", "046d5aa2546f969b1fb0ece5691050d1": "\\textstyle n \\le 2^{r-b+1} -1, ", "046d857e166f77713c3c68ecdbdb9a34": "\\{ \\, (1, 111)\\}", "046d9e9007d432a078332c178710a516": "\\beth_{k+1} = 2^{\\beth_k}", "046db2abf4d0adf4240409c783152fcb": "A \\rightarrow \\varepsilon", "046e9ae403c12efe619ba669e1955a2f": "\\scriptstyle\\bar\\eta", "046ea3ef22af403d11c828ec72d711a0": "m+S", "046ebb1b48895f3d72525897d595788c": "L(p;q_1)", "046ec622fe5bd72e7deacff1d2482bf4": "\n\\begin{align}\n\\varepsilon_0 & \\sim \\operatorname{EV}_1(0,1) \\\\\n\\varepsilon_1 & \\sim \\operatorname{EV}_1(0,1)\n\\end{align}\n", "046f886b34977dca56c25e836e34862e": "X=(X_1,\\ldots,X_n)", "046fb317cb80756569c408df6d76c37e": "\\int\\frac{\\sin^n ax\\;\\mathrm{d}x}{\\cos^m ax} = \\frac{\\sin^{n+1} ax}{a(m-1)\\cos^{m-1} ax}-\\frac{n-m+2}{m-1}\\int\\frac{\\sin^n ax\\;\\mathrm{d}x}{\\cos^{m-2} ax} \\qquad\\mbox{(for }m\\neq 1\\mbox{)}\\,\\!", "046fce63a6a4f7895b14e73e2f1fac79": "A+uv^T=A\\left( I+wv^T \\right)", "047018d7a66d0aefa7616a72267b0557": "\nm(\\varphi) = B_0\\varphi + B_2\\sin 2\\varphi + B_4\\sin4\\varphi + B_6\\sin6\\varphi + B_8\\sin8\\varphi + \\cdots,\n", "04703982a8c13f3e647afa36dc258a3c": "(I_n \\mid S)", "0470a15db5621100067ced7c9ad71923": "F(f):F(X) \\rightarrow F(Y) \\in D", "0470d0befff72541c46222414a829fe5": "\\operatorname{P} (Z_i=2) = \\tau_2 = 1-\\tau_1", "04715c5a2b4e62e7fb226a438528c1cb": "\\begin{align}\n(\\pi_{m,n}(J_i))_{a'b' , ab} &= \\delta_{b'b}(J_i^{(m)})_{a'a} + \\delta_{a'a}(J_i^{(n)})_{b'b},\\\\\n(\\pi_{m,n}(K_i))_{a'b' , ab} &= i(\\delta_{a'a}(J_i^{(n)})_{b'b} - \\delta_{b'b}(J_i^{(m)})_{a'a}),\n\\end{align}", "0471615797404a49fc735c65e449a7aa": "\\mathbf{C}^\\alpha\\ ", "047174dc95a12d05b955f620f3b80798": "E_{kin}=mc^2\\left(\\frac1{\\sqrt{1-\\frac{v^2} {c^2}}}-1\\right)", "04717d25637c0f10e2095645d8f35dcb": "a \\in \\mathbb{N}", "04718f70581064c7db5652ac8bacfa5f": "(c_i = C_\\text{in}(y_i'))", "0471a6b996bf88ea837c16b82f80f25e": "v=y", "047229ccf2f40a3743e0af8092077297": "r=r_c", "047238b589c5452e86b56a33d8210972": "N \\cdot N^r \\cdot S \\cdot N^l \\cdot N ~\\leq~ S \\cdot N^l \\cdot N ~\\leq~ S", "04724bbd90d9b3ede19fc46685c32688": "C_{Hb}", "047276de01eb8ede93eb68722af37dec": "deg(p)", "04728917a32ca84813f26b5ce295bb62": "\\ R_j", "0472acfadb86c949f89853252fe915a4": "F_T + A_T \\Leftrightarrow TC", "0472b8c476010287bff5fd05acec7b2a": "\\omega_0=\\gamma|\\mathbf{B}_{\\|}|", "0472fc4df375a85af48212c820aef7ba": "\\frac{(n-2)}{2n}", "04733873f4f4988b008fa55ca9dcdef5": " (\\Gamma(V,L) \\setminus \\{0\\})/k^{\\ast}, ", "04734da7ad96610d9cd72413217e28e4": "{1\\over2}\\hbar\\omega , \\quad {3\\over2}\\hbar\\omega ,\\quad {5\\over2}\\hbar\\omega \\quad ......", "047352019ed5c2f4b607eac7ba16c621": "(n+1)\\,P_{n+1}(x) = (2n+1)x\\,P_n(x) - n\\,P_{n-1}(x).\\,", "0473831900bfa6b0690f73d1d600aa94": "\\textstyle \\sum c_n = (1,1+2,1+2+3,1+2+3+4,\\dots)", "04740a16dd5a12c6c8d3dcb1388d3a11": "\\ln(2)\\,", "04742d300ace362c7f609b6e2bf98aee": "X(x)=C_{3}e^{-jk_{x}x}+C_{4}e^{jk_{x}x}", "047454141f7d3762203d9c9c0fe94068": "\n{P}=\\left[\\begin{matrix}{T}&\\mathbf{T}^0\\\\\\mathbf{0}&1\\end{matrix}\\right],\n", "047483c44de8b6bf46e64800ab13386f": "\n\\theta\\in[-U,U]\n", "047495f722547a6cabc2b7cf66b3a722": "\\sum_{p^k|n} f(p^k)\\;", "0474b45700b2a1a17ad723d4a260200f": "\n = (1-\\frac{2}{2^s})\\zeta_{2n}(s) + \\frac{2}{2^s}(\\frac{1}{{(n+1)}^s}+\\ldots+\\frac{1}{{(2n)}^s})\n = (1-\\frac{2}{2^s})\\zeta_{2n}(s) + \\frac{2n}{{(2n)}^s}\\,\\frac{1}{n}\\,(\\frac{1}{{(1+1/n)}^s}+\n \\ldots + \\frac{1}{{(1+n/n)}^s}).\n", "04751be39631e2fb1959ca2ffee461d7": "(1-x)^\\alpha(1+x)^\\beta\\,", "047537d879c0fdf93eb53abdba46c5be": "\\Lambda:=\\lbrace 1 \\rbrace", "047542a251ee3d53b0c9912009e84238": "p = r \\Bigg[ \\frac{(1+r)^{n} B_{0}-B_{n}}{(1+r)^{n}-1} \\Bigg]", "047623dba90c6f4d24876c5193f0b4bb": "\\tau(y_i;\\lambda, \\alpha) = \\begin{cases} \\dfrac{(y_i + \\alpha)^\\lambda - 1}{\\lambda (\\operatorname{GM}(y))^{\\lambda - 1}} & \\text{if } \\lambda\\neq 0, \\\\ \\\\\n\\operatorname{GM}(y)\\ln(y_i + \\alpha)& \\text{if } \\lambda=0,\\end{cases}", "04763654620554c15ae64b5aca942bc7": "\\mathbf{A}^0 = \\mathbf{I}", "047790787c56fe7b6abfc4b0aec99d0d": "s_{i_1}s_{i_2}\\dots s_{i_m}", "0477e1ecbf939462595f6bba903295c6": "\\Phi_X(f) = (Ff)u.\\,", "047805207e4e77a99e33063ff9f5ad16": "\\mu\\left(1-\\sigma\\mathrm{log}{\\tfrac{X}{\\sigma}}\\right) \\sim \\textrm{GEV}(\\mu,\\,\\sigma,\\,0)", "04784816d74868a93174f87c0236fe76": "dS_w\\,", "04785525962c6de79ea6eaacbb289d00": "\\mathit{gl}_n \\to \\mathit{gl}_n", "04787fb006bf601b807a0d6d88daf948": " d \\theta = d\\theta_1\\cdots \\, d\\theta_n ", "047893115b0628c644d180c0034540fb": "M_\\oplus", "0478c680906e9ec42d6d9ec19c2f9a68": "\\frac{1}{2^s}\\zeta(s) = \\frac{1}{2^s}+\\frac{1}{4^s}+\\frac{1}{6^s}+\\frac{1}{8^s}+\\frac{1}{10^s}+ \\ldots ", "0478f4e13391a5b6d468b2db291a878f": "dqo", "047913a157084d7cad54db010c56d85a": "\\mathbf{X}=\\{X_1,X_2,\\ldots,X_n\\}", "0479683ca61d38fd063ec79a30e86707": " U_0(r) \\approx a(r) e^{-ikr} ", "04798ef3c700d5ba9bb0a92b0498e9b0": "\\ln K= \\sum_k \\ln {a_k}^{m_k}-\\sum_j \\ln {a_j}^{n_j};\nK=\\frac{\\prod_k {a_k}^{m_k}}{\\prod_j {a_j}^{n_j}} \\equiv \\frac{{\\{R\\}} ^\\rho {\\{S\\}}^\\sigma ... } {{\\{A\\}}^\\alpha {\\{B\\}}^\\beta ...}\n", "04798f7b3ef3f02402eeb94577aa85dc": "\\ \\mathbb{D} _X ", "0479913b6a7d32d643336fb4840b0f06": " F_{ST} = \\frac{ \\pi_\\text{Between} - \\pi_\\text{Within} } { \\pi_\\text{Between} } ", "0479b6d1d786d3ad2be4b2ed143f74be": " (A-\\lambda I)v_2 = v_1. ", "0479bbf4aba93492b525c780cddee25f": "\\overline{\\left ( \\tau_s - \\bar{\\tau}_s \\right )^2} = \\sum_{p,q=0}^{s-1} \\left ( \\overline{\\xi_p\\xi}_q - \\bar{\\xi}_p\\bar{\\xi}_q \\right ) = s \\sum_{p=0}^{s-1} \\left ( \\overline{\\xi_0\\xi}_p - \\bar{\\xi}^2 \\right ).", "047a4e1101708bfb9fd8dc21bdbf43ce": "\\mathcal S(\\gamma) := \\int_a^b L(\\gamma(t),\\dot\\gamma(t))dt", "047acde79363c5f1670a147074d84ff3": "\\{x \\mid \\phi \\}", "047afbff91f0af5c13696532a6c2c8a0": "m\\leq O(n^{(16/15)-\\epsilon})", "047b35daba9c1d2d66362745051dc5f1": "\\partial_t u=\\delta_v H(u,v)", "047bf2ef5af416c7c7b36b6a2f66edc0": "T(V)= \\bigoplus_{k=0}^\\infty T^kV = K\\oplus V \\oplus (V\\otimes V) \\oplus (V\\otimes V\\otimes V) \\oplus \\cdots.", "047c507abd502bb88a4f60732c851832": "\\operatorname{plus} \\equiv \\lambda m.\\lambda n.\\lambda f.\\lambda x. m\\ f\\ (n\\ f\\ x)", "047c508038bcdb0a82a908d184bc2002": "=2\\pi \\varepsilon a\\left\\{ 1+\\frac{1}{2D}+\\frac{1}{4D^2}+\\frac{1}{8D^{3}}+\\frac{1}{8D^{4}}+\\frac{3}{32D^{5}}+O\\left( \\frac{1}{D^{6}}\\right) \\right\\}", "047c65a315d2c3664f293e07153b2b41": " v = \\frac{(m_\\textrm{b}+m_\\textrm{p}) \\cdot \\sqrt{2\\cdot g\\cdot h}}{m_\\textrm{b}}", "047ce8f8e02e71b5b46b73258eebddf6": " \\mathbf{V} \\cdot \\mathbf{W} = \\| \\mathbf{V} \\|\\| \\mathbf{W} \\| \\cos a . ", "047e367f8518f5559adf2909b6e264e6": "(x\\pm i0)^{-k}=x_+^{-k} + (-1)^kx_-^{-k}\\pm\\pi i \\frac{\\delta^{(k-1)}}{(k-1)!},", "047e9d8fda718ccca99693995e9444cc": "L_{p,\\mathrm{loc}}(\\Omega),", "047f60eee20519278eb4e46c31c436f1": "\n \\rho~\\dot{\\eta} \\ge - \\boldsymbol{\\nabla} \\cdot \\left(\\cfrac{\\mathbf{q}}{T}\\right) \n + \\cfrac{\\rho~s}{T} \n ", "047f70ea396f58388c9fa6da42fbc7fb": "\\dot\\omega", "047f9a646fcaed8d8a620b5208eb6c1b": " \\ b = \\frac12 \\times \\rho_{water} \\times S_b \\times C ", "047fd02e5f0ffeb0eea6d81c7bda7d05": "\n\\begin{bmatrix} R \\\\ G \\\\ B \\end{bmatrix}\n=\n\\begin{bmatrix}\n 1 & 0 & 1.28033 \\\\\n 1 & -0.21482 & -0.38059 \\\\\n 1 & 2.12798 & 0\n\\end{bmatrix}\n\\begin{bmatrix} Y' \\\\ U \\\\ V \\end{bmatrix}\n", "047fd1729016dd23ae1d2a19ffd9337c": "\\phi.", "047fde3816d96e562e3871ac2f50059d": "B_1,\\dots, B_k", "047fded37529a2cf6747b2cf845182b7": "\\vdash A \\to B.", "048044ce9dd5c1f3e267135d99f723a9": "b(t)=\\frac{1}{M}\\sum_{i=0}^{i=M-1}{w_i r_i(t-t_i)}", "0480a86160daf12d942f899757a33974": "y \\succ z", "0480c7ca01d301a310b5963cdcaef5e3": "f \\in S", "0480db02c29fbaec48531cb9d43929fe": "mn \\times mn", "04810e2033e49bc7641e329bfe04ea6c": "f \\in \\overline{K}(C)^{*}", "04811704feb2abd5e747e199718b3dab": "\\int_{\\tau_1}^{\\tau_2} \\mathbf{F}_\\mathrm{rad} \\cdot \\mathbf{v} dt = -\\frac{\\mu_0 q^2 r^2}{24 \\pi c^3} \\frac{d \\mathbf{a}}{dt} \\cdot \\mathbf{a} \\bigg|_{\\tau_1}^{\\tau_2} + \\int_{\\tau_1}^{\\tau_2} \\frac{\\mu_0 q^2 r^2}{24 \\pi c^3} \\frac{d^2 \\mathbf{a}}{dt^2} \\cdot \\mathbf{a} dt = -0 + \\int_{\\tau_1}^{\\tau_2} \\frac{\\mu_0 q^2 r^2}{24 \\pi c^3} \\mathbf{\\ddot{a}} \\cdot \\mathbf{a} dt", "0481771e0d238c6608d2f2acaa3ea5ea": "\\sigma = \\pi R^2 P / 2 \\pi R h = R P / 2 h", "048182459491fe2e9c939465e1c541d0": " D(a,s) \\cdot D(b,s) = \\sum_{n=1}^\\infty (a*b)(n) n^{-s} \\ ", "0481d3dfd06bfbe944d6dd475fbb60cc": " ((a + u^2(u^2 - a))^2 - 1)(n + 4dy)^2 + 1 - (x + cu)^2 ", "0481fc89cc8b5cd263273622583380b5": "\\frac{B_1}{h_1^2}=\\frac{B_2}{h_2^2}=\\frac{\\sqrt{B_1 B_2}}{h_1 h_2} =", "0482428fffffe7022ae2cc636c2236fe": "(S)^H\\,", "0482c2d36892eb4589b30cb08c1a360d": "f(n)=\\sum_{d\\,\\mid\\, n}\\mu(d)g(n/d)\\quad\\text{for every integer }n\\ge 1", "048316cdd8ee2f8fe08bfdf69e9b8146": " U \\subset M", "0483319d6300833ac825096bfed9e32e": "1000\\sqrt{\\ell/g }", "048342b8a951b3064014559c5611e2fd": "{\\eta_N}", "048350c2d6b47a176f5d038af2465484": " 7 ", "048365e39b6afdfb2ff84dfd585e9fa1": "q_\\max = \\sqrt{{(E_u^3)(g)\\over (1.5)^3}}=\\sqrt{{(3.04^3)(32.2)\\over (1.5)^3}}=16.4\\text{ ft}^2/s", "0483e16137c69e5676d9801cdd79875b": "\\textstyle\\left(\\frac{{p}}{{5}}\\right)", "048405977db606e46a43b4816b84f43a": "\\frac{R_1}{c}", "048429d0d991b94250f92b125a63c173": "W_{1}^{A}(x,z)", "0484565f18e2eab70b9bbd55ccde7fda": "m_b = m'_b - k", "0484c29ed7efcc6b03fc1c0b6f725a19": "x_2 \\ge 0 ", "0484d7d451687f3e79f67ec3bde75b6e": " \n\\begin{align}\n&u^{0} &=& \\alpha + \\beta x + \\gamma x^{2}/2 \\\\\n&u^{1} &=& -\\frac{1}{2} L^{-1}(u^{0}u^{0''}) &=& -L^{-1} A_{0} \\\\\n&u^{2} &=& -\\frac{1}{2} L^{-1}(u^{1}u^{0''}+u^{0}u^{1''}) &=& -L^{-1} A_{1} \\\\\n&u^{3} &=& -\\frac{1}{2} L^{-1}(u^{2}u^{0''}+u^{1}u^{1''}+u^{0}u^{2''}) &=& -L^{-1} A_{2} \\\\\n&&\\cdots&\n\\end{align}\n", "04850234a56406c23418f463a67eb060": "n \\geqslant 2", "048505e7c44acdca06cbd3d5acdd7df1": "\\theta=\\theta_i", "048538144c496ea0741a737a736eb874": "\\dot{\\bold{r}}\\times\\bold{H}=\\mu\\bold{u} + \\bold{c}", "048549fa6f951b01bd4dcd6e53002584": "\\{\\{i,j\\}: a_{i,j}\\neq 0, 1\\leq i 0", "048a1002418410b27fd943f343cb1d41": "G(\\tau)=\\frac{\\langle\\delta I(t)\\delta I(t+\\tau)\\rangle}{\\langle I(t) \\rangle^2}=\\frac{\\langle I(t)I(t+\\tau) \\rangle}{\\langle I(t)\\rangle^2}-1", "048a5cf6de0c07f4751738e85a0121a9": "\\langle Tx, y\\rangle = \\int_{\\mathbb{R}} \\lambda\\,d\\langle E_\\lambda x,y\\rangle.", "048a9c3303a1553a6aafaf48914ef2be": "2^4\\cdot 3^2\\cdot 5^2\\cdot 7", "048ad459af88f782594d6a04498110ba": "91^2", "048ae06427bbb90c8c794e33b7a6a94b": "\n2\\pi\\gamma RB^{5/2}\\Sigma^2K_1\\left(\\frac{\\ell}{L_c}\\right)\n", "048af646d5b23f889b63067b9014b488": "\\textbf{t}_i", "048b3e9d2796c31a9580d700f5ca6e28": "(\\alpha A)^+ = \\alpha^{-1} A^+\\,\\!", "048b4edc73ba6a55e2f377a459bdeabd": "\\chi = \\frac{\\mathbf{M}}{\\mathbf{H}}", "048b6a58da82ee0994e07c3f235cb954": " z=r e^{\\varphi i} \\text{ with } -\\pi < \\varphi \\le \\pi, ", "048c3096809e88057149e93d08871f7d": " z_k(s) \\leftarrow x_j(s) ", "048cc2757d099299037aca88706d9e7f": "P(y, x_1, \\ldots x_n)=P(y, x_i)P(x_1, \\ldots x_n\\mid y, x_i).", "048cf7531c4567ad53512c73a9f1f870": "l = G'^{-1}(w)", "048cfececa0469f517c2806522571044": "P\\in z", "048d497b67f361d97a7a3c42fe008e19": "n^{(1)},...,n^{(q)}", "048d7a28a426531c29096ab8086f1ab0": "p(x) = 0", "048db65b0805ba9bc7b142f961d8507b": "I(s) = \\frac{1}{ R + Ls + \\frac{1}{Cs} } V(s) ", "048dc7809459e8186f3ea67285bd3140": " H_g(P,Q) ", "048e01282e67d104e634022373d1e75d": "k_2=\\sqrt{2m (E-V_0)/\\hbar^2}", "048e3e85be0499b018b06704c9e3fdf6": " U(0,1) ", "048e48f2ecf712f857fb33ca50e51e3b": " \\ \\phi(x)", "048e52e077417008ca12b5667a8836d1": " k_{GT}", "048e6eeee89b275e038da2b31b481b6f": "\\alpha(T_r) = T_r^{N \\left( M-1 \\right)} exp \\left( L \\left( 1- T_r^{ M N }\\right) \\right)", "048e9b1d644a6d551990258826f47c94": "C(d) = \\sigma^2 \\Bigg(1 + \\frac{ \\sqrt{5}d }{\\rho} +\\frac{ 5d^2}{3 \\rho^2 } \\Bigg) \\exp \\Bigg(-\\frac{\\sqrt{5}d}{\\rho} \\Bigg) \\quad \\quad \\nu= \\tfrac{5}{2}.\n", "048ebaae8fb0e589582b112ccdaf92f4": "S_a(Tr(g^b))=\\left(Tr(g^{(a-1)b}),Tr(g^{ab}),Tr(g^{(a+1)b})\\right)\\in GF(p^2)^3", "048ebefe83e4a8507c500f9bee0f2efe": " [ES] = \\frac{K_i [S][E]_0}{K_m K_i + K_i[S] + K_m[I]}", "048efa823ac43bd64960226b1668c49f": "\\operatorname{sech}\\,x = \\left(\\cosh x\\right)^{-1} = \\frac {2} {e^x + e^{-x}} = \\frac{2e^x} {e^{2x} + 1} = \\frac{2e^{-x}} {1 + e^{-2x}}", "048f1cf76c8a9280aca95ae9a90e3dbf": "z-n", "048ff3eff28beef138d3798e8b153d59": " \n \\begin{align}\n \\hat{\\mu}_1 & = m_1 \\\\\n \\hat{\\mu}_2 & =m_2\n \\end{align}\n", "04902232a7610df3d8e4f38aabc2787a": " x_N \\in X_N", "049029f82b397ae1a5055bf7f706a9ee": "T \\ \\sin \\theta_1 =F_1 \\,\\!", "0490503c6469600795c4219e09b48d4e": "\\{|S,S_z\\rangle\\}\\equiv\\{|1,1\\rangle,|1,0\\rangle,|1,-1\\rangle\\}", "049069e06486046b7174b58402be8888": "\\varphi:\\{E^a\\} \\mapsto \\{\\Phi,E^a,I^a\\}", "0490938bb5edc81e4514ea3ef4bc2f79": "\\mathbb{R}^3,", "04909457bdda7c5c3eb1b12c98278188": "At(room1)", "0491a674d60a00af1b06d832319055c1": "\\mathbf{A} = {}^{*} \\omega_{\\mathbf{A}}= {a}_{1}d{x}_{2} \\wedge d{x}_{3}+{a}_{2}d{x}_{3} \\wedge d{x}_{1}+{a}_{3}d{x}_{1} \\wedge d{x}_{2}", "0491a76b9af525e4dba9daec6c65875b": "\\pi \\gets \\mathrm{Prove}(\\sigma,y,w)", "0491d1a54522592cd19851c5e7e553c1": "W_{t}(n)", "0491f45558966441248f4d2dee9b412d": "P_{\\mathrm{error}\\ 1\\to\\mathrm{any}} \\le M^\\rho \\prod_{i=1}^n \\sum_{y_i} \\left(\\sum_{x_i} Q_i(x_i)[p_i(y_i|x_i)]^{\\frac{1}{1+\\rho}}\\right)^{1+\\rho} ", "0492037f2bf335bbb59e262f8b0da426": "\\gamma:[0, 1]\\to \\mathbb C. ", "04920982c23e776f0cd74b5e114b96c4": "K\\subseteq_s M", "04927783e15da5933265708eacf831b0": " \\prod_{r=1}^4 \\Gamma(\\tfrac{r}{5}) = \\frac{4\\pi^2}{\\sqrt{5}} \\approx 17.6552850814935242483", "04928fe1823546455f9c6b2e93967375": "\\scriptstyle\\tfrac{1}{r}+\\tfrac{1}{s}=1", "0492e77087122537e15019d02c9dc267": "\\cdots \\to \\pi_{i+1} BD \\to \\pi_i B(d \\backslash f) \\to \\pi_i BC \\to \\pi_i BD \\to \\cdots.", "049300f2155adb98dbae6855615508dc": "\\scriptstyle\\leftarrow", "04932c19f04b83ac8733455192b348ed": "A f (x) = b(x) \\cdot \\nabla_{x} f(x) + \\frac1{2} \\big( \\sigma(x) \\sigma(x)^{\\top} \\big) : \\nabla_{x} \\nabla_{x} f(x).", "04937f297c5a4b1df274359eb81322f6": "{50 \\choose 3} = 19,600", "0493b6cc2e6df03a04e667b941a12781": "L_n[1/2, 1] = e^{(1+o(1))(\\ln n)^{1/2}(\\ln \\ln n)^{1/2}}.\\,", "0493c135827d7dd68263ccf670524310": "G(f)", "0493c23f7bffb081dcaf19fae853ceba": " F(x) = f(x) + \\cdots + (-1)^j f^{(2j)}(x) + \\cdots + (-1)^n f^{(2n)}(x),\\quad x\\in\\mathbb{R},\\!", "04941746bc3fbb903fa792b044e2418a": "A\\in \\mathcal{F}", "049441c8ce08c1ec18e16794e80465e6": "d \\vec{\\ell}_2", "04947fc757b781cf6bde09b5e0647d25": "\\varphi=0", "0494e386a35fe24bfb125175572a32a2": "\\rho(\\mathbf r,t)=\\rho[v,\\Psi_0](\\mathbf{r},t)\\leftrightarrow v(\\mathbf r,t)=v[\\rho,\\Psi_0](\\mathbf r,t)", "0494ea13ecdfad57c59dfee70213f05c": " h_2 (X_1, X_2, \\dots,X_n) = \\sum_{1 \\leq j \\leq k \\leq n} X_j X_k,", "049516430ab9768d38217c5b85a4da78": "\\frac{1}{2}(k\\!-\\!\\ln(2)\\!-\\!(k\\!-\\!1)\\psi_0(k/2))", "049526a86f30148986edffdb4168e359": "a_y", "04956b031dfcb109700b760742460d48": "\\displaystyle{\\pi^\\prime_s((g^\\prime)^{-1}) F(x) =|cx+d|^{1-2s} F\\left({ax+b\\over cx +d}\\right).}", "049594571bf34b6300576cefd2297470": " \\omega_{ }^{ } = c k", "0495c0ae17755da6efa9259f9976dc72": "\\int\\frac{x\\;dx}{s^3} = -\\frac{1}{s}", "04960db2bec542bc240a4c537f2bc27c": "\\mathbb{Z}^n_q \\times \\mathbb{Z}_q", "04962fb96f35c2c9217723bc1b531c45": "\\Lambda = \\begin{pmatrix}\n\\lambda_1 & \\ldots & 0 \\\\\n\\vdots & \\ddots & \\vdots \\\\\n0 & \\ldots & \\lambda_4\n\\end{pmatrix}\\,,\n", "04964ec95cb9665e7ba6a188e4300f90": "5\\quad 1\\quad 1\\quad 5\\quad 0\\quad 3\\quad 4\\quad 2", "04973f026f5455096a643b8e6c8e7e6f": "e^{2A} - \\frac{I+A}{I-A}=- \\frac{2}{3} A^3 +\\mathrm{O} (A^4) ~. ", "049769258f04334cccfd306f91a73e38": "e^x\\log(1+y)= y + xy - \\frac{y^2}{2} + \\cdots", "0497d90d14a6754bb11533d7e46cdcff": " = \n[\\textrm{CO}_2]_{eq} \\left(\\frac{[\\textrm{H}^+]_{eq}^2 + K_1[\\textrm{H}^+]_{eq}+K_1K_2}{[\\textrm{H}^+]_{eq}^2}\\right). ", "049817e71b75872219c9769deb9e18d7": "\\scriptstyle \\hbar = \\frac{h}{2 \\pi} \\,", "0498200b37d09b47bbc8d014ad28e86b": "{D}_{8}^{(2)}", "04984b24ee286d6f5dd129c9c1cfa224": " \\Delta h = \\star d \\star d h = \\exp(-2 p) \\, \\left( h_{xx} + h_{yy} \\right)", "049874699cbc9ddacaf4d244d90d3e8d": " 0 \\leq 2 n \\sum_{j=1}^n a_j b_j - 2 \\sum_{j=1}^n a_j \\, \\sum_{k=1}^n b_k,", "0498f50bdce41bf6b06a52a836cbb96f": "E_\\theta={-iI_0\\sin\\theta\\over 4\\varepsilon_0 c r}{L\\over\\lambda}e^{i\\left(\\omega t-kr\\right)}.", "04990f5a51869124035ab5fbdeeaf677": "(p,p^2)", "049956d7e13116db00e8822cdd8244b4": " H = -\\sum\\nolimits_{j=1}^N \\partial^2/\\partial x_j^2 +2c\n\\sum\\nolimits_{1\\leq i< j\\leq N} \\delta(x_i-x_j)\\ , ", "04998b15150c44c8c5bdd75659e311a3": " h \\nu ", "0499dc8d5b79109560ec7ac8ed4ef4d3": "[I-A]", "0499ee1ba90ee7f1581090e6a5dfeda7": "\\scriptstyle \\mathcal{N}_1=\\{-2,-1,0,1,2\\} ", "0499fbc0b0f937d8f3a192a94edcd193": "x\\leq y", "049a40c20859c5b3ecd63a2c347e9b96": "\n\\mathbf{g^{(1)}} = \\begin{bmatrix} \n+5 & +5 & +5 \\\\\n-3 & 0 & -3 \\\\\n-3 & -3 & -3 \n\\end{bmatrix},\\ \n\\mathbf{g^{(2)}} = \\begin{bmatrix} \n+5 & +5 & -3 \\\\\n+5 & 0 & -3 \\\\\n-3 & -3 & -3 \n\\end{bmatrix},\\ \n\\mathbf{g^{(3)}} = \\begin{bmatrix} \n+5 & -3 & -3 \\\\\n+5 & 0 & -3 \\\\\n+5 & -3 & -3 \n\\end{bmatrix},\\ \n\\mathbf{g^{(4)}} = \\begin{bmatrix} \n-3 & -3 & -3 \\\\\n+5 & 0 & -3 \\\\\n+5 & +5 & -3 \n\\end{bmatrix}", "049a4b75a9bb72e3e3a4936fb684bc08": " \\alpha = \\frac{e^2}{(4 \\pi \\varepsilon_0) \\hbar c} = 7.297\\,352\\,5698(24) \\times 10^{-3}.", "049a6650706ebdc9c16d62aae65fa924": "|x+y| \\le |x|+|y|", "049b3498c324a73d0f0c34d9179c2030": " \\ c ", "049b4955f342c15b0dd8f3490f044788": "\\mathrm{C + \\tfrac{1}{2}O_2 \\ \\Rightarrow \\ CO}", "049be4f3f580b85dee4a1872a70e7915": "\\langle \\cdot,\\cdot\\rangle_2", "049cb7ccab6876bb9e9092849353b81b": "p_i = \\frac{1}{1+e^{-(\\beta_0 + \\beta_1 x_{1,i} + \\cdots + \\beta_k x_{k,i})}}. \\, ", "049d1e2f594e1ee91680961985844c5d": "\n W = C_{1} (I_1-3) \\,\n ", "049d39cdb7a32b75ce1cd6ded61882cc": "M\\le w", "049d670c004f59dc4843a509ce3f7127": "\\mathfrak{P}^{84}", "049d9a8da567c5f349c38ba94254918a": "k_{x}=k_{x}'+i k_{x}''=\\left[\\frac{\\omega}{c} \\left( \\frac{\\varepsilon_1' \\varepsilon_2}{\\varepsilon_1' + \\varepsilon_2}\\right)^{1/2}\\right] + i \\left[\\frac{\\omega}{c} \\left( \\frac{\\varepsilon_1' \\varepsilon_2}{\\varepsilon_1' + \\varepsilon_2}\\right)^{3/2} \\frac{\\varepsilon_1''}{2(\\varepsilon_1')^2}\\right].", "049dab7627d4081f8530e7177cad5095": "\n H_x= k_y \\sin k_x x \\cos k_y y \\cos k_z z\n ", "049db5a373f1303f5f652aaa6e00ed88": "\\int^T_0 \\frac{N_0}{2}\\delta(t-s)k(s)ds = S(t) \\Rightarrow k(t) = C S(t), 00),", "04c01f0e4919efa824b9c43529a898c0": " {h_1 \\over h_0} =\\frac{{\\sqrt{1+{{8Fr^2}}} -1}}{2}, ", "04c0375a0567dd3e84683930cb024313": " s_0 = s_n ", "04c05885c12db7ae13199140c4d54225": "\n \\cfrac{\\partial W}{\\partial I_1}\\biggr|_{I_1=3} = \\frac{\\mu}{2} \\,.\n ", "04c0b195b64a5e0327fcdbac013810d0": "p \\oplus q", "04c0d07376defa38f245802bcbd4b3bb": "~\\hat a = X+iP~", "04c110246defb7f6a694db4b679a88ed": "\\left( \\Phi \\cup \\{\\lnot\\phi\\}\\right)", "04c110b5b06c8889aed5b28c383d6e50": "K\\otimes_{\\mathbb Q}K", "04c14940ca11289c43be6206a9c1b646": "\\sigma_{ff}", "04c185182b2f87bbbd72616c17b812da": "p_3(x)=9x^2-3\\,=3(3x^2-1)\\,=3(x\\sqrt{3}-1)(x\\sqrt{3}+1)", "04c1b944cb0a850a29331752ca1bdbd6": "\\mathrm{F=C\\ V^{-1}=A^2kg^{-1}m^{-2}s^4}", "04c1d73c3888fb72cf1de41e95ac8d81": "\\textrm{NM}(k_0,\\,p)", "04c22e12f3c8c6a80f33e0ac3d25fe5b": "\\mbox{EXPSPACE} = \\bigcup_{k\\in\\mathbb{N}} \\mbox{DSPACE}(2^{n^k}) = \\bigcup_{k\\in\\mathbb{N}} \\mbox{NSPACE}(2^{n^k})", "04c2327dc649b2c09a324f7cae1f7d74": "85^2", "04c25020b321831974418d1de8bc2c44": " B_j = (a_j - a_j^*)/(2i) ", "04c26f1786a13ba9848cee465f3fa420": "a_0b_2", "04c314b581e276e14dfef4a0b7e02636": "(cA)_{ij}=cA_{ij}, \\qquad (Ac)_{ij}=A_{ij}c.\\,", "04c318c98c0b1586b6565fbdab350291": "\\Delta a=\\sin^{-1}\\left(\\frac {V_w\\sin(w-d)}{V_a}\\right)", "04c359cfa326b9819aa6afe5bf8c94c3": "m = 6", "04c3c996749f1f84cea72957eb9ad245": "(i_k)_{1 \\le k \\le K}", "04c3ffe149343baf59147dc0804615d0": "\\,^{249}_{97}\\mathrm{Bk} + \\,^{50}_{22}\\mathrm{Ti} \\to \\,^{295}_{119}\\mathrm{Uue} \\,+4\\,^{1}_{0}\\mathrm{n}", "04c41f2b4656b51e364061c051c9b3ec": " \\sum_n \\left( i\\hbar \\frac{\\partial c_n}{\\partial t} - c_n(t) V(t) \\right) e^{- i E_n t /\\hbar} |n\\rang = 0", "04c42d54597e72014a0777dfe9bd9545": " S : K[G] \\to K[G] ~\\text{by}~ S(g) = g^{-1} ~\\text{for all}~ g \\in G_1 ", "04c44949f777e21a9f0581a1a99b6b3e": "\\hat{S}_{i}|\\phi\\rangle = s_{i}|\\phi\\rangle, s_{i}\\in\\mathbb{C}", "04c46410d6f23482e677bb6e5f946e16": "a(b+c)=ab+ac", "04c4669a6f6b89a6f9fc05756117dc52": " v_{Water} = \\sqrt{\\frac{2 \\cdot \\left(p_{Total}-p_{Static}\\right)}{\\rho}} \\,\\!", "04c46f30b4ba78f605ff8c0d3ed1b90a": "\\omega_{p}", "04c47c2b9159dfaec59e2e21bdef8f9f": "a(u_n, e_i) = f(e_i) \\quad i=1,\\ldots,n.", "04c48e87ae666606b70484f5db48f436": "({\\sin \\theta})^2 = -\\frac{(\\mathbf u \\wedge \\mathbf v)^2}{{ \\mathbf u }^2 { \\mathbf v }^2}", "04c49c75b7a3e933536e209d4d8805af": " \\lnot \\lnot x = x,", "04c4c430df526a2f4bb6f83f9539e9d4": " Z_\\text{in} = {v \\over i} = -Z ", "04c4c9a327125dcc9336c608e6c54653": "\\varphi(x)=-\\frac{2}{\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty}te^{-\\frac{t^2}{2}}\\ln|x-t|\\,dt.", "04c5a57d48a350f653f25d4fe36858d2": "r^2 = - \\tan(2\\theta)/2. \\,", "04c5c869a7af06b0d7b63f3085caea1e": "\\scriptstyle (y_1,\\, y_2,\\, \\ldots,\\, y_n)", "04c67acad7c66773a0881af5009e16b3": "\\wedge^{p+1}M_{q-1}\\rightarrow \\wedge^{p}M_{q} \\rightarrow \\wedge^{p-1}M_{q+1}", "04c68062041875d4ffe70413c1372f51": "\\mathsf{(CH_2CH_2)O+H_2\\ \\xrightarrow{Zn\\ +\\ CH_3COOH}\\ CH_2\\!\\!=\\!\\!CH_2+H_2O}", "04c6db9ab3f1ef6e9b80535b5fa6b17b": "2\\eta_{\\mu\\nu}A^\\mu U^\\nu = 0.", "04c7178c91de90bb2a1fea278b1c09b8": " x_{ij} \\in \\{0,1\\}", "04c75fc8c02e137ead6f3efd786e4084": "H_2^{+}", "04c7c45e4c5faecd07a56a00d3d1eaa0": "F_\\mathrm{n}\\,", "04c879a7b484925cf17dc1946509be64": " f_{y}(x,y) \\approx \\frac{f(x,y+k ) - f(x,y-k)}{2k} \\ ", "04c8ed7896b4e9a5fce36f40ce841c1c": "\\frac{1}{2}(l^2-1)", "04c8f846f8901cc910a604daa68910d5": "(r\\bar{b}+b\\bar{r})/\\sqrt{2}.", "04c8fd52917642ff9ec6b7e1ad2b711b": "S_j", "04c95c5523b64e44b5f09bb443214031": " \\tau_n = O(h^{p+1}) ", "04c96de0f1a9b2624fbefac5583c47b5": "V = \\frac{\\pi ^ {\\frac{n-1}{2}}\\, r^{n}}{\\,\\Gamma \\left ( \\frac{n+1}{2} \\right )} \\int\\limits_{0}^{\\arccos\\left(\\frac{r-h}{r}\\right)}\\sin^n (t) \\,\\mathrm{d}t", "04c976c400307140d07071312dd322a7": "\\; \\sum_i\\Omega_i^1\\otimes\\ldots\\otimes\\Omega_i^n", "04c9cbb2ae0a9222fe98f1c128e3567b": "S^{IJ}", "04c9e049ecdf7b697aa92a03cc5dc80b": "conc(\\langle a \\rangle, conc(\\langle b \\rangle, conc(\\langle b \\rangle, conc(\\langle \\epsilon \\rangle, \\langle \\epsilon \\rangle, \\langle \\epsilon \\rangle), \\langle b \\rangle), \\langle b \\rangle), \\langle a \\rangle)", "04ca2451af9f8f56281fed4c9e2566fe": "y(t)=-\\frac{1}{2}gt^2+v_{0}t+y_0", "04ca8c61e468479e7bf1d496316aa78d": "\\mbox{vec}(\\mbox{ad}_A(X)) = (I_n\\otimes A - A^T \\otimes I_n ) \\mbox{vec}(X)", "04caad811d7de70354b943c14d443caf": "U_s U_{\\omega}", "04cafb4a620e52221658357732a348c6": " 2 \\int\\limits_{-\\infty}^\\infty f(t)\\cos\\,{2\\pi \\nu t} \\,dt.", "04cb149fecba1f56811e1d6ff04dcb7d": " a + 2 = b + 2. \\, ", "04cb1ed4b24ae1d0c7051df70770ef69": " (E_{t+1} - E_{t}) y_{t+j+1}", "04cb67c92d7cec7f994a5c5a1f7d4b11": "H \\bmod N \\times 2^L", "04cb7878b651d3c480dfc4e6941d068f": " v' = -\\frac{\\partial \\psi}{\\partial x}", "04cbae5f18b69a1c89403c9af6ae4f65": " [min(r_1, r_2), min(g_1, g_2), min(b_1, b_2)] ", "04cc0e28c90a06698de8ab8ab4269bb9": "L(F, G) := \\inf \\{ \\varepsilon > 0 | F(x - \\varepsilon) - \\varepsilon \\leq G(x) \\leq F(x + \\varepsilon) + \\varepsilon \\mathrm{\\,for\\,all\\,} x \\in \\mathbb{R} \\}.", "04cc38bcb1f8bf6c0b8a797ba4244e11": "x_i \\in \\mathbb{R}^{n+1}, \\, i = 1,...,m", "04cc7eed1a40c64a7be510d9d0d6b51c": " n = \\infty\\!", "04cc91a9b5e3aadb9b97d1921bab8f81": "\\frac{d}{d x}\\left(\\frac{1}{2 - n}\\left(\\frac{d t}{d x}\\right)^{n - 2}\\right) = f(x)", "04ccfac50c13f886fd57d6102c0674c8": "\\sigma_{zz} = \\sigma_{zx} = \\sigma_{yz} = 0", "04cd0e0151f352e7fd414d694a604136": "[1, 2]", "04cd61c128b35877531bd18ad85af8d7": "\\scriptstyle{R_0^0 + R_3^3 = 0}", "04ce4598bd3f73b2b528b57e5e1af6e6": "\\langle 1\\rangle", "04cea56b0b312d7edce09d5dd7596ba9": "\\frac{\\partial u}{\\partial x} + \\frac{\\partial v}{\\partial y} + \\frac{\\partial w}{\\partial z} = 0", "04cec08a0e5858f7e7d7bb8028a0746d": "\\beta \\le -2", "04cf31d6ec3540fee12e8e5ed390d9ba": "A_e = \\frac{3 \\lambda ^2 }{8 \\pi} ", "04cf4ec52b62d7ce63235d8519aa5f88": "\\int_0^\\infty x^{y-1} e^{-x}\\,dx,", "04cf774444cfa3e18887ceddd932d053": " 0\\to C^0 \\stackrel{d_0}{\\longrightarrow} C^1\\stackrel{d_1}{\\longrightarrow} C^2 \\stackrel{d_2}{\\longrightarrow} \\cdots \\stackrel{d_{n-1}}{\\longrightarrow} C^n \\longrightarrow 0. ", "04d06c41023fa9b103747ebb5689f586": "x_{1,t}", "04d12470043ba1d37c0a63948d1c200b": "U=\\sigma_x", "04d15429e1cebd053387fbefbe192dc7": "p^a q^b\\ ", "04d1e7a488ce709acd268317cfb2defe": "X_z(z)=\\frac{-1}{H(1+\\phi(z)\\bar\\phi(z))^2} \\left \\{(1-\\phi(z)^2, i(1+\\phi(z)^2), 2\\phi(z)) \\frac{\\bar{\\partial\\phi}}{\\partial \\bar z}(z) \\right \\}", "04d204bbbcfb647c1628359b7d3f8ec4": "R_h", "04d223813fed198d04db780b0d506017": "A = k[t^2, t^3] \\subset B = k[t]", "04d243e0bc2764202af0a72263fb94e9": "f\\!\\left(x\\right) \\geq f\\!\\left(y\\right)", "04d25560f37662e7a63f9f37757271d2": "\\sum_{n=1}^\\infty \\frac{t^{2n}}{n} \\zeta(2n) = \n\\log \\left(\\frac{\\pi t} {\\sin (\\pi t)}\\right)", "04d265067859ccc4737cd584b0b3c99e": "\n\\hat{\\rho} (\\mathbf{r}) = N \\sum_{j=1}^n \\int_0^1 ds \n\\delta \\left( \\mathbf{r} - \\mathbf{r}_j (s) \\right).\n", "04d29d26a6f00d0951137ace61c9ff20": " |T_j| < \\frac{t}{2d}] \\leq n \\cdot n^{-2d} \\leq n^{-d}", "04d3b323a3ea25db0d1633b89147ece0": "i = 1, \\dots, n", "04d42f232c194ce477ed3d8ef88de683": "A = \\sum_i {x_i\\, A_i}", "04d444c8f2f6c71b8b5785e58eacb9eb": "c_{sound} = \\sqrt{\\left(\\frac{\\partial P}{\\partial \\rho}\\right)_{s}} = \\sqrt{\\frac{\\gamma P}{\\rho}}=\\sqrt{\\frac{\\gamma R T}{M}} ", "04d4721517edfca170ac3802db26813e": "x = 0 \\!", "04d47874f992aef898ec8e9a27bdb7da": " z \\in \\partial \\Pi_A", "04d49a21d2d751d28a93329700556599": "A'(z) = \\sum_{k=0}^{N}a_k\\gamma^kz^{-k}", "04d4a4b969a9937e007085d733918c7f": "k>0\\,", "04d4afd88b88bf73e429f0b39c6abfd3": "C = e^{- \\frac{K \\cdot t}{V} + const } \\qquad(2c)", "04d4f97b34dc23f791fe306b0e995dfc": "\\mathbf{C}\\,\\!", "04d51ab83469a9216904129f03469844": "P_{50} = 5^{50} \\cdot \\frac{\\Gamma \\left(3/5 + 50\\right) }{\\Gamma \\left( 3 / 5 \\right) } \\approx 3.78438 \\times 10^{98}. ", "04d52171a5c6eca9a3d0bbd805b2b536": "VSWR=\\frac{|V|_{max}}{|V|_{min}}=\\frac{1+|\\Gamma|}{1-|\\Gamma|}", "04d59985c001bc8f54707f446ce9fd33": "\\{x'_k:k r", "04deed9912b56be9d7f2882643ef19ca": "e=E-127", "04df27d3472a35811336a4a701d68984": "\\vdash \\dashv, \\vDash, \\Vdash, \\models \\!", "04df55da0404fcec41530fd9c731776f": "\\delta_{x}:S \\times X \\rightarrow S", "04df898e5535afa983878f0186ec6cd9": "\\int_{\\Omega} v_j v_k\\,ds", "04dfaf8dc62e04aadd3b85654d8ea067": "{{documentation}}", "04dfc1faa60e22d2c4b3f89cf549d55a": "|k|/n", "04dfc825159e549dcf4f938211d845fe": "x^n\\in\\mathcal{X}^n", "04dfca30d4aab589307ba2b8d5b82d6e": "a|n\\rangle=\\sqrt{n}|n-1\\rangle", "04dff67803d6445e3af17ecc63827668": " t \\ne t_n ", "04e03460dc9bf154dd788748a06c2472": "\\mathcal{S}^\\prime", "04e056e18f107f3b4f74fcebcb56042c": "S_m(P,T)=S_m(P_0,T_0)+C_P \\ln \\frac {T}{T_0}-R\\ln\\frac{P}{P_0}.", "04e101d162346f2087821eda0a2354fa": "\\lim_{\\delta \\downarrow 0} \\delta \\log \\mu_{\\delta} (S) = - \\inf_{x \\in S} I(x). \\quad \\mbox{(E)}", "04e210fc8bdbcf4c7a865e30d482b05a": "\n\\operatorname{Jacobian}\\left( \\frac{x, y}{A, B} \\right)\n =\\begin{vmatrix}\n -(B^2-4A)^{-\\frac{1}{2}} & \\frac{1+B(B^2-4A)^{-\\frac{1}{2}}}{2} \\\\\n (B^2-4A)^{-\\frac{1}{2}} & \\frac{1-B(B^2-4A)^{-\\frac{1}{2}}}{2} \\\\\n \\end{vmatrix}\n = (B^2-4A)^{-\\frac{1}{2}}\n", "04e22f55c7aa1d710ace6b0dc6be18de": "\\sin^2(\\theta)+\\cos^2(\\theta)=1,", "04e26e6c3597879a21c0cf8662316481": "L((1+n)^x \\mod n^2) \\equiv x \\pmod{n}", "04e2cf6db6627e8b2e12e109e878cf46": "G=\\cfrac{E}{2(1+\\nu)}", "04e2dd42a81fe5e382a2a47dda3af106": "\\lnot \\;\\exists \\;xO(x)", "04e30a138457b99329132428dcf4682c": "\\Omega=2^\\mathbb{N}=\\{H,T\\}^\\mathbb{N}", "04e340f9a578caa3bf7db69949976347": "\\pi_{xy}", "04e346a8813bd90c853d764753d1bc1a": "2 \\rightarrow 1.", "04e3ae971cea84aab401463bc236844d": "\\int_0^{\\theta} \\operatorname{Sl}_{2m+1}(x)\\,dx=\\zeta(2m+2)-\\operatorname{Cl}_{2m+2}(\\theta)", "04e3f5127b45a587cee6af90f1652ebf": "\nc_1(q) = 1, \\;\\;\n c_q(1) = \\mu(q), \\;\n\\mbox{ and }\\; c_q(q) =\n\\phi(q)\n.\n", "04e3f78844e2687a97fa0932a63c94b8": "\\mathbf{e}_i=\\mathbf{e}_{i'} (A^{-1})^{i'}_i,\\,", "04e4a643ec333306aab41015824e77b8": "\\mathcal{P} = \\mathcal{C}\\times\\mathcal{M} = \\{ (\\mathbf{q},\\mathbf{p})\\in\\mathbb{R}^{2N} \\} \\,,", "04e4bc90fc5cb5af671c8ea0303b02b2": " P\\left[ (\\tilde{X}^n,\\tilde{Y}^n) \\in A_{\\varepsilon}^n(X,Y) \\right] \\leqslant 2^{-n (I(X;Y) - 3 \\epsilon)} ", "04e4ea40b54a681cc441a335f195180d": "s_b(z)", "04e50826ed9a1064bb210b8d98d7904e": "\\rho(x_1,x_2)=0", "04e54f1f9c3733f61da4feb2f4b9dd70": " w_{ij}^{\\nu} = w_{ij}^{\\nu-1}\n\t\t +\\frac{1}{n}\\epsilon_{i}^{\\nu} \\epsilon_{j}^{\\nu} \n\t\t -\\frac{1}{n}\\epsilon_{i}^{\\nu} h_{ji}^{\\nu}\n\t\t -\\frac{1}{n}\\epsilon_{j}^{\\nu} h_{ij}^{\\nu}\n\t\t ", "04e5c0b589f343996819a788a67d2ffc": "\\text{R-X}^-\\text{C}^+\\,+\\, \\text{M}^+ \\, \\text{B}^- \\rightleftarrows \\,\\text{R-X}^-\\text{M}^+ \\,+\\, \\text{C}^+ \\,+\\, \\text{B}^-", "04e5c6491f65a8ff500707053264975b": "n=p_1+\\cdots +p_c", "04e60aceefaac27f43d9266b4e898495": "R = R_x(\\gamma) \\, R_y(\\beta) \\, R_z(\\alpha)\\,\\!", "04e6493104c14d65c65a7e3ae307874c": " \\tau=\\frac{t}{|c|} ", "04e6b5ce6f920b15c208e31017181e58": " s^2 = \\frac{ b }{ a ( a + b ) } + \\frac{ d } {c ( c + d ) } ", "04e6e4c84f34a47815aa1c74bddce026": "\\eta(0)=0", "04e6ea5a4cfc7efe45577a4968b32fb4": "H^I_p(H^{II}_q(P_\\bull \\otimes Q_\\bull)) = H^I_p(P_\\bull \\otimes H^{II}_q(Q_\\bull))", "04e7717b13155456972e9ae515c2e5df": "u(x,t)=\\int_{0}^{t} \\frac{x}{\\sqrt{4\\pi k(t-s)^3}} \\exp\\left(-\\frac{x^2}{4k(t-s)}\\right)h(s)\\,ds, \\qquad\\forall x>0", "04e7909cf29056a41c53b565a2ee68c2": "\\mathbf{v} = v(t)\\mathbf{u}_\\mathrm{t}(s) \\ ,", "04e7a342417c7b4a3fd09ac71f00b250": "m_p\\left(r\\right)\\rightarrow m_0", "04e7bce75675f3c6e2bd9cd9de82df4d": "F_\\alpha = \\sum_{\\alpha \\succeq \\beta} M_\\beta, \\, ", "04e7f66067d55d79409ae532dff606d4": "\\operatorname{E}(c) = \\frac{1}{N+1}\\sum_{i=0}^{N} i", "04e7ffbbe1a99fadc9c4cef92ec795e9": "\\overline{O} = O", "04e801a95286ebb4a962bb8f59c4073b": "\\mathcal{D}(A).", "04e83ca06b3a78ee3bda963bd4a2fd56": "\n\\operatorname{Li}_2(u) + \\operatorname{Li}_2(v) - \\operatorname{Li}_2(uv) = \\operatorname{Li}_2 \\left( \\frac{u-uv}{1-uv} \\right) + \\operatorname{Li}_2 \\left( \\frac{v-uv}{1-uv} \\right) + \\ln \\left( \\frac{1-u}{1-uv} \\right) \\ln\\left( \\frac{1-v}{1-uv} \\right),\n", "04e880bbfa917de599eeaae85bc0bc85": "e^2", "04e9155ef246bb508734c8e560f378d9": "\\pi_2 M", "04e96147afb31e6766e43593312db18d": "t=pq^{-1}=\\gamma^r\\gamma^{-s}=\\gamma^{r-s}=h^{\\alpha\\beta(r-s)}", "04e97e24c18920b8bf657dd449790432": "s = a^pb^pc^p", "04e98084ee989d47e1373fa9fddb2d74": "shared(d)", "04e999090e2c17187ef280070a248637": "{d}", "04ea2cd547a555329ca7624d1ecea049": "E^{(+)}(\\mathbf {r}, t) = i\\sum_{i}[\\frac{\\hbar\\omega_{i}}{2}]^{1/2}\\hat{a}_{i}\\mathbf{\\varepsilon}_{i}e^{i(\\mathbf {k}_{i}\\cdot\\mathbf {r} - \\omega_{i}t)}", "04ea6720d76a2f9c83ef10db3f587c23": "y_0 \\in \\{0, 1\\}^m", "04eae6d13d1528605d9dda775789745e": " \\vee: \\mathrm{Con}(\\mathcal{A}) \\times \\mathrm{Con}(\\mathcal{A}) \\to \\mathrm{Con}(\\mathcal{A})", "04eaf227fb05fa613254a4b9ba3713a6": "K(\\!(T_n)\\!)", "04eb06021221b54fb4506e7fd94fb64e": " ||y - A(x_1+x_2+ \\cdots +x_n)|| < \\delta \\, 2^{-n} \\, ; \\quad (2) ", "04eb0f45582acc6ccd08133edaadd7b9": "g(r)=\\exp[-\\beta w(r)]", "04eb7f0e9b851eed2913fd1244b6e9f2": "c \\in \\Sigma^n", "04ebd70849b60dff8a8ef599cd00d654": "v_+ = v_- = v_{\\text{out}}.\\,", "04ec04c064836ad3876f5cbfd3c2ec4f": "|a|(1+a/4)\\pi \\,", "04ec3070cbc012e2cfa4f9fe5f939abd": "v = \\frac{\\omega}{2 \\pi c} (y_1 - y_2)", "04ec42dd7a5255c85e090b33973a8ceb": "s^2 = \\frac{1}{3N} \\left\\{ \\sum_{n=1}^{N} ( x_{n,1} - \\bar{x})^2 + \\sum_{n=1}^{N} ( x_{n,2} - \\bar{x})^2 + \\sum_{n=1}^{N} ( x_{n,3} - \\bar{x})^2\\right\\} ", "04ec6053f6146c2eb3a0bd4e08578401": "(\\tfrac{p}{q})=1", "04ecb34572dfeb5b6f2032e0bfc18806": "y_c = \\frac{2}{3}\\sqrt{M_c} ", "04ecb94754fc963b1045f89f2d595c44": "X\\subseteq V", "04eccba89e407f705f8ef660d7b4d614": "g=G \\frac {m_1}{r^2}=(6.6742 \\times 10^{-11}) \\frac{5.9736 \\times 10^{24}}{(6.37101 \\times 10^6)^2}=9.822 \\mbox{m} \\cdot \\mbox{s}^{-2}", "04ecef256c10ed98b0dcffcef97251c0": "W_T^{(2)}(\\omega)", "04ed090ac7a3e0e380bca8de5f6b41ed": "\\beta \\in \\mathcal{O}_k", "04ed6b29079f24735c5b29745ef0a1b7": "a_{1}+a_{14}", "04eda5539ba8311ed9023276aaf1b885": "\n \\sum_{n=1}^{l_\\lambda} \\; \\Gamma^{(\\lambda)} (R)_{nm}^*\\;\\Gamma^{(\\lambda)} (R)_{nk} = \\delta_{mk} \\quad \\hbox{for all}\\quad R \\in G,\n", "04edb01258a81268e75b640c739649bc": "a(x-y) \\bmod 2^w", "04edf159dde4bfd8233801c022187323": "\\psi(\\alpha+1)=\\psi(\\alpha)=\\delta", "04ee0ff1daec33fb96547c3f6fdfb597": "p_w(\\theta)=\\frac{1}{2\\pi}\\,\\sum_{n=-\\infty}^{\\infty} \\phi(-n)\\,e^{in\\theta} = \\frac{1}{2\\pi}\\,\\sum_{n=-\\infty}^{\\infty} \\phi(-n)\\,z^n ", "04ee3e02987ce86c2a483f4f4cb4dcf0": "\\sigma_{Z_1}^2.", "04ee6592608fcb53fb98eb913894d483": "\nIMD_i = \\left( e_i^t - h_i^t \\right) \\times \\left( G_i - G \\right)\n", "04eed3678d03fd0b3dc1d3f672bdeae1": "e_q(x) = \\exp(x) \\text{ if } q = 1 ", "04eed96382514f7c340f4e53fe09db69": "(hkl)", "04ef1e759d5f3184342d6948487a53d5": "\n\\mathcal{L} [\\varphi (x)] =\n-{1\\over 4} F_{\\mu \\nu} F^{\\mu \\nu} + {1\\over 2} m^2 A_{\\mu} A^{\\mu} + A_{\\mu} J^{\\mu}\n", "04efa9f79c535e637f267063d5460fba": "(F \\cdot G)[A] = \\sum_{A=B+C} F[B] \\times G[C].", "04efe8396f5fc91ac3d7e5b549fcfb7d": "\\begin{matrix}{52 \\choose 5} = 2,598,960\\end{matrix}", "04f052c0bde0bdcb192ed417678a785a": "\\{ www : w \\in \\{a,b\\}^{*}\\}", "04f06ca1499e8a908f20f92cbc1cb863": "g=h^{-1}th", "04f081930149949cf30a1b9b8635c47e": "X \\in \\C", "04f084963d52d685bb83410abe86643e": "x = c_1 c_2 \\ldots p \\ldots p' \\ldots x_n ", "04f0980f6e8fa97d144641ec8b6b8ff4": " (L_0 - \\tilde{L}_0) |\\Psi\\rangle = 0 ", "04f14932ad6780bb4713155a180f0040": "\ny=\\sqrt{a^2-x^2}, \\quad\ny'=\\frac{-x}{\\sqrt{a^2-x^2}}, \\quad\ny''=\\frac{-a^2}{(a^2-x^2)^{3/2}},\\quad\nR=|-a| =a.\n", "04f188c7b8ed64c7fe2137cda960608a": "\n \\mathbf{M}_H\n =\n \\begin{bmatrix}\n \\;\\;\\,0.38971 & 0.68898 & -0.07868 \\\\\n -0.22981 & 1.18340 & \\;\\;\\,0.04641 \\\\\n \\;\\;\\,0.00000 & 0.00000 & \\;\\;\\,1.00000\n \\end{bmatrix}\n", "04f18cafc2ee54e4b6c66b4ecbd09eca": "R_2 - R_1 = R \\sqrt{1 + \\frac{x_2^2}{R^2} + \\frac{y_2^2}{R^2}} - R \\sqrt{1 + \\frac{x_1^2}{R^2} + \\frac{y_1^2}{R^2}}", "04f1a9486e22042a59277d7022778e75": "v_{3}", "04f1dab970ad559b1fe9c0a1a1bd2a38": "1\\le i, j\\le k", "04f1ef249f07bd7a5759fd398eee3f4e": "\\Gamma(s,z) = \\Gamma(s) - \\gamma(s, z)", "04f1ff7f5c9bde7065bb8bfa4ef93d41": "\n\\begin{align}\n\\mathbb{E}\\Bigl[\\liminf_{n\\to\\infty}X_n\\,\\Big|\\,\\mathcal G\\Bigr]\n&=\\mathbb{E}[X|\\mathcal G]\n=\\mathbb{E}\\Bigl[\\lim_{k\\to\\infty}Y_k\\,\\Big|\\,\\mathcal G\\Bigr]\n=\\lim_{k\\to\\infty}\\mathbb{E}[Y_k|\\mathcal G]\\\\\n&\\le\\lim_{k\\to\\infty} \\inf_{n\\ge k}\\mathbb{E}[X_n|\\mathcal G]\n=\\liminf_{n\\to\\infty}\\,\\mathbb{E}[X_n|\\mathcal G].\n\\end{align}\n", "04f236a5d65eb2902a7521e68752fd15": "\\sum _{ v \\neq v0} (q_v - q _{v \\cap w})", "04f23f77a4da740c280d3617cb0c2a1b": "L(G) = \\{ w \\in T^{*} : S \\Rightarrow_{p_1} ... \\Rightarrow_{p_n} w \\}", "04f25fc454fe9c1218a15db30e347a68": "0^\\circ", "04f2e33784b89346f3cb7a773ace6986": " \\scriptstyle p_i = p^{\\star}_i x_i", "04f3137ff098aa5741e25f9e0a30097f": "g_0=1,", "04f37063c459dc5067b3a505eb13254a": "\\scriptstyle \\phi(a)", "04f394df1d823a51f2052efd822ee5ba": "xzy^{-1}xx^{-1}yz^{-1}zz^{-1}yz\\;\\;\\longrightarrow\\;\\;xyz.", "04f4251e7aab69f16e4921ae9c10f3fa": "X_{SC}", "04f42f9c70ae2265168f604d0e77823c": "\\kappa =\\frac{1}{\\rho}=\\frac{C}{R}", "04f45059d9e134e6f04406c34a24902f": "[j]_{TOT} \\,=\\, [j] + \\sum_{i=1}^{N_S}\\, \\nu_{i,j} \\,[i] ", "04f46896df145356b2cfb916ff84bee0": " A^{-1}=\\frac{(-1)^{n-1}}{\\det(A)}(A^{n-1}+c_{n-1}A^{n-2}+\\cdots+c_{1}I_n).", "04f46ef4f610873f0b607299831248f3": " S_m = \\int_0^m {\\left( {x \\over {2\\sqrt{x^2+4}}} + {{m+2} \\over {2m}} \\right)} \\, dx. ", "04f4fbb099ecddf77a8bd49e549a4796": " \\overbrace{\\smile \\smile -\\smile}^{\\mathrm{Foot 9}} | \\overbrace{\\underbrace{-\\smile}_{\\mathrm{Brahma}}}^{\\mathrm{Foot X}} | \\overbrace{\\smile\\smile\\smile-}^{\\mathrm{Foot 11}} || ", "04f5027f7716ebfad5764a4c176a88cf": "Q_{q} = \\frac {1} {\\sqrt {N}} \\sum_{l} u_{l} e^{- i q a l } ", "04f50f7dd4b7b9a5dc57ade5af0e862d": "\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{7}+\\frac{1}{43}+\\cdots=1,", "04f52ca3d98d9b896f04128244d4ddf1": "d\\mathbf x'=\\mathbf U \\,d\\mathbf X\\,\\!", "04f53647bec0920c0d3570f033877ffa": "D_{n,k} = {n \\choose k} \\cdot D_{n-k,0}.", "04f5934929b135782033055aefa70325": "_C^E", "04f59a19dbe6293b61640ab1810b6854": "A=\\frac {4}{3}a^2", "04f5bbaf6b93197b7c2e2061d9751f1e": "\\varphi(z) = \\int (T_z f_z)g_z \\, d\\mu_2", "04f5be0bccdbf812a6640f4b88fd67a0": "a+b+c", "04f5cb345657f4532084a899aec6339b": " \\varphi: \\mathcal F\\rightarrow \\mathcal G ", "04f5f57d53e9bffe95e1969a780328c1": "f^{*}", "04f6027f91c7dee5c61ff63a88813e6d": "P= (X_1:Y_1:Z_1)", "04f60f28c56bcac963753ada77addbb5": "\\mathbf{P} = \\mathrm{d} \\langle \\mathbf{p} \\rangle /\\mathrm{d} V \\,\\!", "04f62906170dc100289eb31b2819479c": " r \\sqrt{4-2\\sqrt{2}} = \\frac{a}{2}\\sqrt{4+2\\sqrt{2}} \\!\\, ", "04f6460714e9c2cc801ea09b76dd543d": "u^2-a_1u+\\frac{a_1^2}{4}=a_0+\\frac{a_1^2}{4} .", "04f6c757ca09e262b8f61e709cd2b567": "\\displaystyle \\partial_t u + \\beta\\, t^n\\, \\partial_x^3 u + \\alpha\\, t^nu\\, \\partial_x u= 0", "04f6ca8294b413fe37a829daee69bee3": "\\Psi_L\\left(0\\right)=\\Psi_G\\left(0\\right)+\\text{H.O.T.},\\,", "04f6ce540b30e0340b87e29e5ece08c5": " m,\\, 0x_0", "05017b16d121273397774b34532bf10b": "k^2-2\\,i\\,k\\,x-1\\,=\\,0", "0501ab330f701f2e5ddaaaa5d8cf2af2": "f(z)=(z-a)^ng(z)\\ \\mbox{and}\\ g(a)\\neq 0.\\,", "0501c25234c86d03e007782268f04893": "x^3=(0,0,1)", "0501ceca7bd0d96b04794c2a514b6f37": "\\mathcal{L} f = -\\partial_t f(t) + r(t) f(t).", "0501eedfb34554c82f3ad105604c242a": "_{\\sim}\\!", "050202b86b163e362266acc78f67be89": "\\Box\\phi", "050244e419735079939749935cfc6c78": "\\begin{align}&(1 + 2\\mu)u_{i,j}^{n+1} - \\frac{\\mu}{2}\\left(u_{i+1,j}^{n+1} + u_{i-1,j}^{n+1} + u_{i,j+1}^{n+1} + u_{i,j-1}^{n+1}\\right) \\\\ & \\quad = (1 - 2\\mu)u_{i,j}^{n} + \\frac{\\mu}{2}\\left(u_{i+1,j}^{n} + u_{i-1,j}^{n} + u_{i,j+1}^{n} + u_{i,j-1}^{n}\\right).\\end{align}", "05027bd684c505bb972c1b177b20c56d": " P( | X | \\ge k ) \\le \\frac{ 4 \\operatorname{ E }( X^2 ) } { 9k^2 } \\quad\\text{if} \\quad k^2 \\ge \\frac{ 4 } { 3 } \\operatorname{E} (X^2), ", "0502b073ec2e7800a308776ab0811922": "\\hat\\psi(\\vec r) = \\sum\\limits_i w_i^\\alpha (\\vec r) b_i^\\alpha", "0502cd530e4f328821d546d4a0944188": "F_n(x) = \\frac{1}{n} \\left(\\frac{\\sin \\frac{n x}{2}}{\\sin \\frac{x}{2}}\\right)^2 =\n\\frac{1}{n} \\frac{1 - \\cos(nx)}{1 - \\cos x} \n", "05035d210f3d1496caf59b529bc1410a": " \\zeta = \\frac{\\delta}{\\sqrt{(2\\pi)^2+\\delta^2}} \\qquad \\text{where} \\qquad \\delta \\triangleq \\ln\\frac{x_1}{x_2}.", "0503d5a4ed130ec62d2bc8a2f654b56a": "Z^\\dagger", "05041b22d390f8f8b61338f65a1724c4": "K_{\\lceil n/r\\rceil, \\lceil n/r\\rceil, \\ldots, \\lfloor n/r\\rfloor, \\lfloor n/r\\rfloor}.", "050456a2a2c341938f221eb3e0b50372": "\\omega = \\frac{-1 + \\sqrt{-3}}{2}=e^\\frac{2\\pi i}{3}", "050458ff63a54df48024c9ebb9932d84": "\\,\\gamma", "050473e9a8a8e6fac1b0dfc8960fb55e": "\\frac{3}{8}\\sqrt{35}\\cos(4\\theta)\\cos^4(\\phi)", "0504c1d23e37f48a62ba1437e9cab3e2": " M_{i,j}", "0505018fc0ed9786c0216099fc3b789c": "a=(v^2-u^2)(v^2+u^2), \\,", "05051b000eeb56e299912b68d5c5e2c0": "y - y_1 = m( x - x_1 ),\\,", "05052479786e4f2b053609801f833d7b": "F(d, k)", "050542d2523a82915c1fdad950acdc5e": "\\mathcal{A} f (x) = 0.", "05055bfc9a5b48f205c595eb622a5fb4": "S({\\Lambda^\\mu}_\\nu) = {(\\Lambda^{-1})^\\mu}_\\nu = {\\Lambda_\\nu}^\\mu \\,", "0505676729e95ec9f4958bceb2658882": "\\bar{\\nu}_e + p \\to e^+ + n", "0505b3b8e4b450288f5985d487fd641c": "\\omega = \\frac{\\lambda \\cdot v}{r}", "05061073560f5cf8ce91f9b49a796c9a": " \\theta_A = \\frac {P}{P+P_0}", "0506af4c0ad7aa17657c8aaf095acc26": "d_\\pm", "050704d18bf227d8d89a90f3209b39bb": "\\displaystyle{u_x=-v_y,\\,\\, u_y=v_x.}", "05070e88dfde3a30bb688c009c8f6bb4": "n_z", "050710f82f53f780d2c7fd7795137c44": "Y_1, Y_2", "05073a04fe1376c3b0c45106273f9187": "a\\sim b", "0507c11a8aee36060834108d45eec574": "\\mathbb R v_1,\\dots,\\mathbb R v_6", "0507ca3317618b35b1e64a4dbc5ad5da": "\\text{MTBF} = \\theta. \\!", "0507d05470ff6520b4965cb227d62218": "10_{123}", "0507e3cf2687b0f76c74a01a26568226": "\\hat{\\lambda}_i", "050802f5a55c0af3f857280e59e25a6d": "S_x(\\omega) = \\hat{x}(\\omega)\\hat{x}^*(\\omega)", "0508352d6beb495b1dffad1f8726fb9e": "\\frac{1}{\\tau} = ar", "0508b61cf5f29dcbc2d668fa5e93fd4f": "\\frac{1}{T_2^*}=\\frac{1}{T_2}+\\frac{1}{T_{inhom}} = \\frac{1}{T_2}+\\gamma \\Delta B_0 ", "05090603b60ddc4d45703252f192d9d6": "\\chi_1^2", "05096bd9c0a26b57faa623e920635e0e": "|U| >1/2,\\ V = W = 0,", "050992bf4515002318edb223863a9ae0": "\\alpha \\in A", "05099cbdcccc7a04282d0f96c127de8a": "R_{abcd} \\, R^{abcd}", "0509b32282371643e6308a79f7d4f5dc": "f_\\ell^m=\\int_{\\Omega} f(\\theta,\\varphi)\\, Y_\\ell^{m*}(\\theta,\\varphi)\\,d\\Omega = \\int_0^{2\\pi}d\\varphi\\int_0^\\pi \\,d\\theta\\,\\sin\\theta f(\\theta,\\varphi)Y_\\ell^{m*} (\\theta,\\varphi).", "0509d73229a2e1c0ce410544d2c0c25d": "(\\partial T)_P=1", "0509f544a02d65ac9b57509058a3a05e": " X_t = c + \\sum_{i=1}^p \\varphi_i X_{t-i}+ \\varepsilon_t .\\,", "050a2bd6fe954b091760195ffaaa0808": "\\ddot x - 2n\\dot y = \\frac{\\delta U}{\\delta x}", "050a4f9d47d3514082e7fa0c2ed2da90": "\\sqrt{2} \\ln(1 + \\sqrt{2})", "050a580104aa0173c165551a3e383357": "Z = \\left(\\overline{X}_n-\\mu\\right)\\frac{\\sqrt{n}}{\\sigma}", "050a90d6a372aebd4a064da88365182c": " \\phi _1(z) = (1-z)/2 \\quad z \\in [0,1].", "050a93457b3b36469a4362c630c68575": "\n\\sum_{n=0}^{\\infin} (-1)^n\n", "050b2d78abf6b855c631c27406f6763f": "(A \\vee B \\vee C) \\wedge (\\overline{A} \\vee \\overline{C}) \\wedge (\\overline{B} \\vee \\overline{C})", "050b377515d021da5001b6ef871978a8": "\\mathcal{F}_i= - \\frac{\\partial \\mathcal{V}}{\\partial q_i}\\, ", "050b5355658ab527c84edb8f00f387d6": "\nH^\\dagger - H = 0\n\\,", "050b57a5f8f2f3a7bf5992a5f74069d3": "c_0 = S-1 \\,\\!", "050b5e0fe4d1ae8a4a9919dc545fa7e7": "e^{ar}", "050b89800d1de9d236fd5a26e225bb5b": " \\csc \\theta\\! ", "050c2a34694f64f4b312fe044bfa151f": "P^{(i)}_{0}", "050c5d1a59538341e67943d438532d5d": "na_0x^n + (n-1)a_1x^{n-1} + \\cdots + 2a_{n-2}x^2 + a_{n-1}x = 0 \\, ", "050c6f71cd07650bd1f7ae739b59ba1d": "F_{1 \\rightarrow 2}", "050c79c03277c6a6ad35246617006d32": "\n \\begin{align}\n p_0 = -\\frac{de_0}{dV} = \\frac{\\rho C_0^2}{2s^4(1-\\chi)} \\Biggl[& \\frac{s}{(1 - s\\chi)^2} \\Bigl (- \\Gamma_0^2(1 - \\chi)(1 -s\\chi) \n + \\Gamma_0 [s \\{4 (\\chi-1) \\chi s-2 \\chi+3\\}-1] \\\\\n & - \\exp(\\Gamma_0\\chi)[\\Gamma_0(\\chi-1) -1](1-s\\chi)^2(\\Gamma_0-3s) + s [3-\\chi s \\{(\\chi-2) s+4\\}]\\Bigr) \\\\\n & - \\exp\\left[-\\tfrac{\\Gamma_0}{s} (1-s\\chi)\\right][\\Gamma_0(\\chi-1) - 1](\\Gamma_0^2 - 4 \\Gamma_0 s + 2 s^2)(\\text{Ei}[\\tfrac{\\Gamma_0}{s} (1-s\\chi )] - \\text{Ei}[\\tfrac{\\Gamma_0}{s}]) \\Biggr] \\,.\n \\end{align}\n ", "050d2253a1b35110e73f5b61e3d64d28": "\\int_{ L_0 + L_1 + L_2 } \\left ( - c^2 u_x(x,t) dt - u_t(x,t) dx \\right ) = \\iint \\limits_{R_C} s(x,t) dx dt. ", "050d2c73ed3c5b1dc2f46f8b057a9a64": "p(\\theta | y, \\xi) = \\frac{p(y | \\theta, \\xi) p(\\theta)}{p(y | \\xi)} \\, ,", "050d3009fd9b0b042952de0e4d937f19": "\\frac{L}{r^{2}} \\frac{d}{d\\theta} \\left( \\frac{L}{mr^{2}} \\frac{dr}{d\\theta} \\right) = -\\frac{{2}L^{2}}{mr^{5}} \\left( \\frac{dr}{d\\theta} \\right)^2 + \\frac{L^{2}}{mr^{4}} \\frac{d^{2}r}{d\\theta^{2}}\n", "050da762cbd9b00e5919fcc071c87259": "\\int_0^\\infty\\frac{\\sin t}{t}\\, dt=\\int_{0}^{\\infty}\\mathcal{L}\\{\\sin t\\}(s)\\; ds=\\int_{0}^{\\infty}\\frac{1}{s^{2}+1}\\, ds=\\arctan s\\bigg|_{0}^{\\infty}=\\frac{\\pi}{2},\n", "050db563e14ba6743e9ce8a9f6a9f9a3": " \\boldsymbol\\beta^{(s+1)} = \\boldsymbol\\beta^{(s)} - \\mathbf H^{-1} \\mathbf g \\, ", "050dce3386e1fefedd89e8bce5018b68": "\\left. {\\color{white}...}\\ \\omega v \\left(\\cos\\alpha + \\omega t \\sin \\alpha \\right) \\right]\\ ", "050e1820f88caa93847c1c1826795b4c": "\\frac{p_k}{p_{k-1}} = a + \\frac{b}{k}, \\qquad k = 1, 2, 3, \\dots", "050e63bdc63bf7ed99a58f1cb20b4610": "\\mathrm{Div}^0(C)", "050ed507c56e133906e661314f467dcf": "\\sup_n\\left|\\sum_{i=1}^n x_i\\right|", "050eede33f602f8ec77ef8203acb103f": "\\mathbf{\\Sigma}^1_1", "050f1cabedeafb9366261993192f1252": " \\ \\alpha_i ", "050f2343beede00d97ce19ffdd84280b": " TE_{01n} ", "050f2abc0b8bc0b355cb908860cf4119": "\\rm \\ SCl_2 + Cl_2 \\xrightarrow{193K} SCl_4", "050f771089c62964d9e54d7ae690bc6a": " \\omega_{\\mu} = e_{\\mu\\nu\\rho\\sigma}\\xi^{\\nu}\\nabla^{\\rho}\\xi^{\\sigma}", "050f89274e7d82bcb8198954f77106ba": "E_c : z \\to e^z + c \\,", "050fd765812551d51d962c54c4a8c8bb": "\\{r,s\\}", "050fdd1960dfdbab718e82719d89afa9": "R_{\\rho z}=8\\pi T_{\\rho z}", "050ffa5ef6f992064ea682bfeae6ac8b": "p^k,", "0510097af5b114125816748fa362a294": "EL(\\Gamma_1\\cup\\Gamma_2)\\ge \\bigl(EL(\\Gamma_1)^{-1}+EL(\\Gamma_2)^{-1}\\bigr)^{-1}.", "05109063960af84cd328819ba140fa94": "[\\cdot,\\cdot]\\colon\n\\mathfrak{g} \\wedge \\mathfrak{g} \\to \\mathfrak{g}", "05109474b54025b7f9e935a25b01b1e1": "exp(-z^2)", "0510e31aee49c3fbd1d39dbd5d5f84f9": "13 = (17-4) \\mod {26}", "05114f16e74e3b815b33172483b79ca2": "\\scriptstyle \\bar{X}_i=\\frac{1}{m}\\sum_{j=1}^{m}X_{ij}", "051188924a0fba93a9c1ecc164215d7d": "\\rho_h", "0511b7daa53ac131c9a6ae3d745ec8db": "\\mathfrak q", "0511fc24827cdcffed5ea19bb6124789": "\\frac{u_{i+1}-u_i}{\\Delta x}\\ f", "05120b211561ca725f9785dfbefe359f": "AX -XB = Y ", "05121e6c9874bc5a4cf7817470a670ed": "\n R_0 = \\cfrac{d\\epsilon^p_2}{d\\epsilon^p_3} = \\cfrac{H}{G} ~;~~\n R_{90} = \\cfrac{d\\epsilon^p_1}{d\\epsilon^p_3} = \\cfrac{H}{F} ~.\n ", "051230c786f41cee9ecd2f4bd8806de0": "s_1 = c_1e_1", "051315e37a1615b3dbaf5ec61fa30952": "\\tau(p^{r + 1}) = \\tau(p)\\tau(p^r) - p^{11}\\tau(p^{r - 1})", "051344f71c00744c96451a881eb6364d": "\\nabla \\times \\mathbf{E} = -\\frac {\\partial \\mathbf{B}}{\\partial t}", "0513a6272599ff46057f412f576460cd": "d(\\gamma A) = Af_{ij}d\\epsilon_{ij}", "0513acacdfeb03bc371c4ebde470299c": "y_{n} = c_{n-1}y_{n-1} + c_{n-2}y_{n-2} + \\cdots +c_{0}y_{0}.", "0514314546f794ec13e571b5c8c4c107": "\\ell = 2 a", "05143c911e7294959a8d8ca0d12c71d7": "D=\\{1,2\\}, P(1)=\\bot, P(2)=\\top, c=1", "05144cc001f66271c26c893017144baf": "\\min E_{T} = \\sum_{i}\\Big[ E_i(r_i) + \\sum_{i\\ne j} E_{ij}(r_i, r_j)\\Big] \\, ", "051452f6a6a5a155a444d89a2ca665bc": "2~\\ln r + 1", "0514845fd4a3e78213e7ab88b9dd492a": "E = \\frac{1}{4} Wkd \\theta^2", "0514afb94e82c61cbaa2a3b503a2fab4": " u(R,t) = \\frac{dR}{dt} = \\frac{F(t)}{R^2} ", "0514c16ec7e9eb98c506535d7438bc92": "\\dot{q_i} \\, ", "05150cfbe7764ff9c0bc04c8544ef7e7": "\\nu_\\mathrm{t}", "05151a93e80308a1e909cf45e63beb65": "K \\otimes_\\Q \\mathbb{R} ", "05152c21814653d312d1a9dc611f3975": "\\Delta\\, G_i \\,\\sim \\Gamma (\\Delta t_i/\\nu, \\nu)", "051535f7bc824e59e73b31aeec32d3b8": "\\mbox{female shoe size (Brannock)} = 3\\times\\mbox{foot length in inches}-21", "05158466407bde46b85a8649ade91ec8": "\\Delta_{\\textrm{B}}", "0515ba4dd3540bee4010b6e2718689a6": "K_{SV}~=~Stern-Volmer~constant~for~oxygen~quenching", "0515e1203a6da3e9b342a993d26bb494": "\\Delta _{\\mathcal{L}}(x_{\\perp }) =-1_{1}1_{2}\\frac{\\mathcal{L}(x_{\\perp })\n}{2}\\mathcal{O}_{1},\\text{scalar}\\mathrm{,} ", "0515ecca071219dfab5ed29f01652c71": " E_\\mathrm{stored} = \\frac{1}{2} C V^2,", "05161917f741c897aba47f69fe891a57": "{T_v(s) = V_{out}(s)/V_{in}(s)}\\,", "05168f730983e424739d63483138d587": "\\mathbf{x}_R = A\\mathbf{x}_L", "0516a583f096aee2d1ef45dbd10159e9": "d(\\sigma) \\geq \\frac12 (d(\\sigma 0)+d(\\sigma 1))", "0516cd87df2bccdd7d83c444138de721": " - \\frac {\\hbar ^2}{2m} \\frac {d ^2 \\psi}{dx^2} = E \\psi.", "05172cdab5fd630a4cb101b18fe4f0f3": " -e_2=<0,-1> ", "05173ca87cf4e63b6588070bdcd42071": "\\displaystyle{W(x)W(y)=e^{-{i\\over 2} \\Im (x,y)} W(x+y).}", "051747d010127b31ff30e257312eecf1": "\nRE_{\\hat g} \\,\\, = \\,\\,{{\\hat\\sigma _g \\,} \\over {\\hat g}}\\,\\,\\, \\approx \\,\\,\\,\\sqrt {\\,\\,\\left( {{{s_L } \\over {n_L \\,\\bar L}}} \\right)^2 \\,\\,\\, + \\,\\,\\,\\,4\\left( {{{s_T } \\over {n_T \\,\\bar T}}} \\right)^2 \\,\\, + \\,\\,\\,\\,\\left( {{{\\bar \\theta } \\over 2}} \\right)^4 \\left( {{{s_\\theta } \\over {n_\\theta \\,\\bar \\theta }}} \\right)^2 \\,}", "05176183bb00310d71e626f5264ff66b": "\\displaystyle{\\frac{1}{2}R(a,b)=L(a)L(b) -L(b)L(a) +L(ab),}", "0517be056a5873b94503d2bd7e5f9cc1": "\\| Mf \\|_{L^{p}} \\leq C_{p} \\| f \\|_{L^{p}}.", "0517d1f36dae4b67ae3986160d121900": "\\int_{-\\infty}^{\\infty} |\\psi (t)|^2 \\, dt <\\infty.", "0517f31b8aacd320eaf6b16b7fa435e1": "\nZ = \\sum_{n=0}^{\\infty} e^{-n\\beta h\\nu} = \\frac{1}{1 - e^{-\\beta h\\nu}}.\n", "0518a46df04592797fb11f5a9d147616": "\\Delta \\mathbf x\\,\\!", "0518a4ea135ccbe3916da92bbe8e8701": "\\Delta \\bar{e}\\ \\,", "0518cc5d7d6d3cdbb5ab9bc1dc3bf0b5": "\nRD = \\frac{W_\\mathrm{air}}{W_\\mathrm{air} - W_\\mathrm{water}}\\,\n", "0518ce225d7219b5b7b398ce8a548f57": "\\liminf_{\\varepsilon \\to +0} \n \\varepsilon^{-1} \\left\\{ \\gamma^n (A_\\varepsilon) - \\gamma^n(A) \\right\\}\n \\geq \\varphi(\\Phi^{-1}(\\gamma^n(A))),", "05195afa5f1b5313ca387bb548c25dc2": "f(n^k) = kf(n).\\,", "05197f4f8923ce9df2ad252bcdfc1343": "\\dot Q(t)\\ =C^{(V)}_T(V,T)\\, \\dot V(t)\\,+\\,C^{(T)}_V(V,T)\\,\\dot T(t)", "05198d0212461cd43f11908164f4213a": "\\Delta(t) = c_0 + c_1 t + \\cdots + c_n t^n + \\cdots + c_0 t^{2n}", "0519bc388c4b70254424e2de54e23721": "\\theta = v/c = \\kappa", "0519cf07a04ebb3ef1e2693196df08e4": "P_{2}^{1}(x)=-3x(1-x^2)^{1/2}", "0519d4dbdec5bcca4c39bcba98058239": "\\gcd(2^a-1, 2^b-1)=2^{\\gcd(a,b)}-1", "051a7eb36d169001282aa8f35dadc66e": "\nV = V(t). \n", "051ae9d0e81bebfd1186c42463742fdf": "n \\geqslant 0 ", "051b2590dc90f6478107992385384d64": "x=y=z=0,\\,s=10,\\,t=15.", "051b2cc28181aacee228bc94d47bc04c": " \\lambda(t_1) = F(t_1) x(t_1)", "051b39b0bcdd0277e6a15d127af4d094": "\\gamma' (1)", "051b65d0bc2ef8fb4cbdcbc778ea00f9": " \\hat{x} ", "051b7e26712f1115cdc466d49d6b3305": "c_m=\\frac{1}{2\\pi}\\int_\\Gamma \\ln(f_w(\\theta))e^{-i m \\theta}\\,d\\theta", "051bad0de5df71fa5a3d047779cc191d": " \\ell(\\gamma)=\\int_\\gamma \\rho(z) \\, |dz|,\\quad A(D)=\\int_D\\rho^2(x+iy) \\, dx \\, dy, \\quad z=x+iy. ", "051bae02c7c2b9c8414016a40fe8e3bf": " \\; \\text{Var}\\left(\\boldsymbol{\\varepsilon}\\right) = \\sigma^2I_{n \\times n} ", "051c2a7ff34934f6fc05c14807b02861": "\n\\begin{bmatrix}\n B_{11} & B_{12} & 0 & \\cdots & \\cdots & 0 \\\\\n B_{21} & B_{22} & B_{23} & \\ddots & \\ddots & \\vdots \\\\\n 0 & B_{32} & B_{33} & B_{34} & \\ddots & \\vdots \\\\\n \\vdots & \\ddots & B_{43} & B_{44} & B_{45} & 0 \\\\\n \\vdots & \\ddots & \\ddots & B_{54} & B_{55} & B_{56} \\\\\n 0 & \\cdots & \\cdots & 0 & B_{65} & B_{66}\n\\end{bmatrix}\n", "051c4c75a8934dd4d7ef677f2918368c": "{N_i}", "051c780fb650536715a3fcf6121dc9e8": "\\displaystyle{\\log z = \\log |z| + i\\arg z}", "051ca4949f88882b6e288b0d5ec6d5fc": "SSR = \\sum_{i=1}^n\\bigg(\\frac{\\varepsilon_i^2}{\\sigma_\\varepsilon^2} + \\frac{\\eta_i^2}{\\sigma_\\eta^2}\\bigg) = \\frac{1}{\\sigma_\\varepsilon^2} \\sum_{i=1}^n\\Big((y_i-\\beta_0-\\beta_1x^*_i)^2 + \\delta(x_i-x^*_i)^2\\Big) \\ \\to\\ \\min_{\\beta_0,\\beta_1,x_1^*,\\ldots,x_n^*} SSR", "051ca55dcaf28074e2cb5a42b1691c17": "y_p(x) = \\sum_{i=1}^{n} c_i(x) y_i(x)\\quad\\quad {\\rm (iii)}", "051d1518eda7defecc640212bc0908df": "\\scriptstyle \\bar\\psi", "051d50d5305afafd7b365b0ed61221a4": "T_i + U_{i-1} \\sqrt{x^2-1} = (x + \\sqrt{x^2-1})^i. \\, ", "051d6224436c5fc199a4c46c1aad0003": " E_{21} =\\frac{d \\ln (c_2/c_1) }{d \\ln (U_{c_1}/U_{c_2})}\n =\\frac{d \\left(-\\ln (c_1/c_2)\\right) }{d \\left(-\\ln (U_{c_2}/U_{c_1})\\right)}\n =\\frac{d \\ln (c_1/c_2) }{d \\ln (U_{c_2}/U_{c_1})}\n = E_{12}\n", "051d6422f391f4a35ceab86263d112f8": "\\int\\frac{dx}{\\sinh^n ax} = -\\frac{\\cosh ax}{a(n-1)\\sinh^{n-1} ax}-\\frac{n-2}{n-1}\\int\\frac{dx}{\\sinh^{n-2} ax} \\qquad\\mbox{(for }n\\neq 1\\mbox{)}\\,", "051db488801812c2b62a83559cddaea0": "W_q=\\frac{\\rho^2+\\lambda^2\\sigma_B^2}{2\\lambda(1-\\rho)}", "051dd97e3eb4556b287a44a9e427a37b": "\\sqrt{\\varphi_1^2 + \\varphi_2^2}", "051e7321f1a7ebbc27505ecd75e6bbe8": "\\mathcal{G}(p,q)", "051e8ca3a671da00d3446e6da1f5ff6e": " \\langle f_1,\\ldots, f_k\\rangle = \\left\\{\\sum_{i=1}^k g_i f_i\\;|\\; g_1,\\ldots, g_k\\in K[x_1,\\ldots,x_n]\\right\\}.", "051ee141bac36d8612e305c8beecf706": "P(t) = \\begin{cases}\n 0 & tt_o+t_p \\\\ \n\\end{cases}\n", "051ee904daa0559210339ff3c6ed52c6": " \\nabla B_z = (d B_z/dA)\\nabla A", "051f0ede9f5842fe5a8e50066845bdc7": "p=p(V,T)\\ ", "051f218870baef81059be4a102dec711": " G\\left(X'_{i}\\beta\\right) ", "051f2871fd7e787c6ec9c8be7702f7f4": "(D V_i)^2 / Z_o = \\eta V_i^2 / Z_i ", "051f58f5abb870ac348fd824566ba1b1": " \\log_{b^n} a = {{\\log_b a} \\over n} ", "051f84bf61e8e26b26ab4cc0cd4d0af6": "\\,\\lambda_i", "051ff4c2be9011cd50b03822e0fef332": "1\\over {\\sqrt 6}", "05208e2e2161e5c9451c0ad985594f0e": "\nE_{a_0} = \\frac{E_S}{4\\pi a_0^2}\n", "0520b5f0f2d99a7a28f5ca9b8ee08bf9": "\\dot{x} \\equiv \\frac{dx}{dt} = \\left(\\begin{array}{c} \\frac{da}{dt} \\\\[6pt] \\frac{db}{dt} \\\\[6pt] \\frac{dc}{dt} \\\\[6pt] \\vdots \\end{array}\\right).", "0520df0adb6ded51ed8afb052e4bded9": "\\frac{2 \\cdot 5}{7}", "0520f68ba263a7a7ec277cc0671d6b23": "V_{out}(t)", "05210b6b9045a1666d5676422477286c": "I_C=I_E-I_B\\,", "05211990618d5f6fdc2ab9065bf70066": "r\\leftarrow p", "05218ec1fe4c2fe5e117e292cb91b5c2": "\\varphi(h(y),s) = h(\\psi(y,t))", "0521b4209637a2fb3ebc86938716bc9b": " = \\int_{-\\infty}^{\\infty} \\left[ \\int_{-\\infty}^{\\infty} x(t) w(t-\\tau) \\, e^{-j \\omega t} \\, dt \\right] \\, d\\tau ", "0521f1cdd4dd30d846d0bd2d196c5b9b": "J(\\mathbf{x}) = (\\mathbf{x}-\\mathbf{x}_{b})^{\\mathrm{T}}\\mathbf{B}^{-1}(\\mathbf{x}-\\mathbf{x}_{b}) + (\\mathbf{y}-\\mathit{H}[\\mathbf{x}])^{\\mathrm{T}}\\mathbf{R}^{-1}(\\mathbf{y}-\\mathit{H}[\\mathbf{x}]),", "052227aa30bd74eafbce3b5cde10ea9b": "\\langle l, r \\rangle_w", "05224930b0615345deb884948267a8ac": " M = 3(N-1-j) + j = 1, \\!", "05225d2f4212a8e279c90b2d9183c6fa": "t=1\\,\\!", "0522718afc8aa16a9af1dc1323991229": "\\psi(x) = C\\,\\exp\\left(-\\frac{(x-x_0)^2}{2 w_0^2} + i p_0 x\\right)", "052274267eaed4f24d4bc546decc403f": " M = \\begin{pmatrix} e(a_1,b_1) & e(a_1,b_2) & \\cdots & e(a_1,b_n) \\\\ e(a_2,b_1) & e(a_2,b_2) & \\cdots & e(a_2,b_n) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ e(a_n,b_1) & e(a_n,b_2) & \\cdots & e(a_n,b_n) \\end{pmatrix} ", "052275e1c5c5fc2132c328a5a1548487": "\\boldsymbol{\\alpha}", "0522ba869064082c64d0e85ec613c34a": " CPI = \\sum_{i=1}^{n} CPI_i*weight_i ", "0522d9bcc2cdf664ad01b74ab937f24e": "\n\\begin{align}\nA_{0} &= u_{0}^{2} \\\\\nA_{1} &= 2 u_{0} u_{1} \\\\\nA_{2} &= u_{1}^{2} + 2 u_{0} u_{2} \\\\\nA_{3} &= 2 u_{1} u_{2} + 2 u_{0} u_{3} \\\\\n & \\cdots\n\\end{align}\n", "05230f95fc3a31bc3b0826c13f8f4a31": " G = \\left\\{(\\Delta ,x):{\\rm{f}}_i (\\Delta ,x) \\le 0,0 \\le i \\le k,\\Delta = xx^T \\right\\} ", "05233dd32f8cc541bd9ced9f0786cda8": "E[G^2|K]=\\int^T_0\\int^T_0k(t)k(s)E[x(t)x(s)]dtds = \\int^T_0\\int^T_0k(t)k(s)(R_N(t,s) +S(t)S(s))dtds = \\rho + \\rho^2", "05234ea4351e32f9183ca278bbf9bac6": "\\mapsto \\!\\,", "0523517f4871af8f21c7335440857928": "D^a \\,", "05235be088fe90ed01afc11dbff739dc": " \\frac {PV}{T} = \\sqrt{k_p k_v k_t} \\,\\!", "0523b7f9b83c5c7489ec4d18839c41a1": "\\mathbf{f} \\,\\colon \\,f_1\\ge f_2\\ge \\cdots \\ge f_N", "0523d5903ad11af2202b4188d345244e": "\n\\| u \\|_{L^4} \\leq C \\| u \\|_{L^2}^{1/4} \\| \\nabla u \\|_{L^2}^{3/4}.\n", "0524ac0139c5674da470ce71e9dc2998": " \\mathbb{Z}_d \\times \\mathbb{Z}_d ", "0525441782af4827a325e2fc2c934ed2": "A = Z+N\\,\\!", "052558595508698e079de150da568929": "\\sqrt 2 \\sinh u,\\,", "05258ad8d57d6ca8ec02a490c078934c": "\nI_{+} = -I_{e} e^{-e V_{+}/(k T_{e} )} + I_{ion}^{sat}\n", "0525b7ccbe9150fc1336beb2b81b5880": "\\eta_2 = \\frac{H}{m}\\, \\left( 1 - m - \\frac{E(m)}{K(m)} \\right),", "0525cccdd7123c339471e6dc1fd332a1": "\\begin{align}\n\t&\\sin(f_c\\cdot t+I\\cdot\\sin(f_m\\cdot t)) \\\\\n\t&\\quad = J_0(I)\\sin(f_c\\cdot t) \n\t+ \\sum_{k=1}^{\\infty} J_k(I)\\left[\\sin(f_c+k f_m)t+(-1)^{k}\\sin(f_c-k f_m)t\\right]\n\\end{align}", "05261178171cbde148f522fd4ce40017": "L \\propto \\log \\log N", "05266488515a4d0c92daba82dd43647f": "x_n=T_1x_n^{(1)}+T_2 x_n^{(2)}+\\dots +T_rx_n^{(r)} \\mod T", "0526732aa1201e9383e0adb4a439229b": "\\left(\\delta_S\\right)", "0526a8bf4fe60ce15b9c93314c984f53": " \\sigma >0", "05274b730a401a9c4ac31d7e4fc653ce": "G=\\langle x_1,x_2 \\mid R \\rangle", "0527514ada9c403ea469ca02ce24f292": "a = \\sqrt{2}\\, , \\quad b = \\log_2 9\\, , \\quad a^b = 3\\, .", "05276a877a3f26d3fed313cc1cadd89a": "\\bold{F}\\;", "05276cf9f4a5efb6dfd91c6b7066883c": " m=0 \\, ", "0527fb7eec99185176dddeecb4105f22": " \\mathbf{A}\\cdot\\mathbf{B} = \\mathbf{A}'\\cdot\\mathbf{B}' ", "0528481eedf437a0564f67864056d139": "\\dot{\\alpha}^*=\\alpha", "0529d8b612fce5de2462245a2978c70c": " X , Y , Z , XX , YY , ZZ , XY , YZ , XZ ", "052a0e1ff447c2e2bec8c6e49313bb2c": "\\mathbf x, \\mathbf y\\in \\mathcal A ", "052a8dde47dfc691fa05737e59c17f16": "\\sum_{i=0}^k\\frac{\\Gamma(\\alpha + i)\\beta^i\\lambda^{k-i}e^{-\\lambda}}{\\Gamma(\\alpha)i!(1+\\beta)^{\\alpha+i}(k-i)!}", "052b48ee967b71c502168486ddb54522": "\\bigl(\\tfrac12,\\tfrac12, \\ldots \\tfrac12\\bigr).", "052b801b8b515aca0898929f40f14ada": "r_c\\,\\!", "052bb990e8596e24d7948c61a3f3a8ed": "\\Delta^\\text{op}", "052be0158cc8b723e885b9b440e1083e": "f(\\phi,\\psi)=0\\,", "052c5652b42a40ec4801dee938109f88": "\nx^{- \\alpha} \\; G_{p,\\,q+1}^{\\,m,\\,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q}, \\alpha \\end{matrix} \\; \\right| \\, \\eta x \\right) = \n\\frac{1}{2 \\pi i} \\int_{c - i \\infty}^{c + i \\infty} e^{\\omega x} \\; \\omega^{\\alpha - 1} \\; G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, \\frac{\\eta}{\\omega} \\right) d\\omega,\n", "052c66e5fa5f447b7f2367c9b102f1d4": "(Y, \\mu, S)", "052c6b13ea3a99fa82c0e51204815d0f": "4r^2\\le e^2+f^2+g^2+h^2 \\le 4(R^2+x^2-r^2)", "052c70b3eed17d912bd74195d3477f66": "R = 45 + 48 = 93", "052ccc833647d5b81b19d27119f7979a": "\\alpha_{\\text{pump on}} (\\omega)", "052cd2ba2e36f8f6f63b1ada442105fc": "h_m(x) = \\sum_{j=1}^J b_{jm} I(x \\in R_{jm}),", "052ce045b2c96c121918eca1de8fc712": "\\ln(n!) - \\tfrac{1}{2}\\ln(n) \\approx \\int_1^n \\ln(x)\\,{\\rm d}x = n \\ln(n) - n + 1,", "052cee2ddaf572eb50e2aeebf52edb60": "a = 2 \\arctan \\left( \\frac {D/2} {f} \\right) = 2 \\arctan \\left( \\frac {D} {2f} \\right)", "052d085a5fe58905b0426ef8b85f0638": " f^+(x) = \\left\\{\\begin{matrix} f(x) & \\text{if } f(x) > 0 \\\\ 0 & \\text{otherwise} \\end{matrix}\\right. ", "052d161539044589cc32eee91b8fda6a": "\\begin{align} P(A |\\text{not }B) &= \\frac{P(\\text{not }B | A) P(A)}{P(\\text{not }B | A)P(A) + P(\\text{not }B |\\text{not }A)P(\\text{not }A)} \\\\ \\\\\n\n &= \\frac{0.01\\times 0.001}{0.01 \\times 0.001 + 0.95\\times 0.999} \\\\ ~\\\\ &\\approx 0.0000105.\\end{align}", "052d776f4e8548cffeb47a2dbd78c129": "\\textstyle \\alpha = d", "052d7db1d302c8e2e7ae04d6a5d0ef2b": "P = \\frac{\\int_0^{\\frac{\\pi}{2}} l\\cos\\theta d\\theta}{\\int_0^{\\frac{\\pi}{2}} t d\\theta} = \\frac{l}{t}\\frac{\\int_0^{\\frac{\\pi}{2}} \\cos\\theta d\\theta}{\\int_0^{\\frac{\\pi}{2}} d\\theta} = \\frac{l}{t}\\frac{1}{\\frac{\\pi}{2}}=\\frac{2l}{t\\pi}", "052df73f7c43029df9b3fcd9c4ad22fa": "\\begin{matrix}2\\end{matrix}", "052e076e9fc04db8b0a520a78c844876": "\\sin 2x = 2\\sin\\frac{x}{2}\\cos\\frac{x}{2}", "052e34d11e812d6bb5902b169db0517f": "W(S)", "052e54d580841636190e637d0333414a": "\\quad W_{2\\,p}=\\frac{2\\,p-1}{2\\,p}\\times\\frac{2\\,p-3}{2\\,p-2}\\times\\cdots\\times\\frac{1}{2}\\,W_0=\\frac{2\\,p}{2\\,p}\\times\\frac{2\\,p-1}{2\\,p}\\times\\frac{2\\,p-2}{2\\,p-2}\\times\\frac{2\\,p-3}{2\\,p-2}\\times\\cdots\\times\\frac{2}{2}\\times\\frac{1}{2}\\,W_0 = \\frac{(2\\,p)!}{2^{2\\,p}\\, (p!)^2} \\frac{\\pi}{2}", "052ee2717d0683b8ef7729b1063002a6": "C = \\text{Tr}_{\\text{CTC}}\\left[ U \\right]", "052f2bc9062738ec52049899cddaa7c0": "\np\\sigma \\xrightarrow\\alpha p'", "052f3e6f6172ebddf8d9a015db13307f": "\n \\overset{\\circ}{\\boldsymbol{\\tau}} = \\dot{\\boldsymbol{\\tau}} - \\boldsymbol{l}\\cdot\\boldsymbol{\\tau} - \\boldsymbol{\\tau}\\cdot\\boldsymbol{l}^T \n", "052f84150425938458bfcda119406ac9": "{{\\overline{P_1P_3}\\cdot \\overline{P_2P_4}}\\over{\\overline{P_1P_4}\\cdot \\overline{P_2P_3}}} =1+{{\\overline{P_1P_2}\\cdot \\overline{P_3P_4}}\\over{\\overline{P_1P_4}\\cdot \\overline{P_2P_3}}}", "052f90c64762cae32b83178a5045cd8d": "\\bold{u}_f\\;", "052f99631cd8d4bfa945c02313d18f40": "U=0", "052fd9b4b90a459ed294c0b9c2d1d4e1": "T \\subseteq [n]", "052fe85ca556dc32e605488ad5560478": "j=1,\\ldots,m\\,\\!", "05304f3e6a805e7506cb7e955b8fa969": "\\ln (1/\\Gamma(z)) \\sim -z \\ln (z) + z + \\tfrac{1}{2} \\ln \\left (\\frac{z}{2\\pi} \\right ) - \\frac{1}{12z} + \\frac{1}{360z^3} -\\frac{1}{1260 z^5}\\qquad \\qquad \\text{for}\\quad |\\arg(z)| < \\pi", "05306690c3b5fd73579ab942e82f1768": "+\\lambda^2\\sum_{m\\neq n}\\sum_{q\\neq n}\\sum_n\\frac{\\langle m|V|n\\rangle\\langle n|V|q\\rangle}{(E_n-E_m)(E_q-E_n)}|m\\rangle\\langle q|+\\ldots", "05306697eb4b61a275ac2cf33c664371": " 4 = \\operatorname{perm} \\left ( \\begin{matrix} 1 & 1 \\\\ 1 & 1 \\end{matrix} \\right )\\operatorname{perm} \\left ( \\begin{matrix} 1 & 1 \\\\ 1 & 1 \\end{matrix} \\right ) \\neq \\operatorname{perm}\\left ( \\left ( \\begin{matrix} 1 & 1 \\\\ 1 & 1 \\end{matrix} \\right ) \\left ( \\begin{matrix} 1 & 1 \\\\ 1 & 1 \\end{matrix} \\right ) \\right ) = \\operatorname{perm} \\left ( \\begin{matrix} 2 & 2 \\\\ 2 & 2 \\end{matrix} \\right )= 8.", "053095516840cd071b5a3b992cd97389": "\\mathcal M_{u}", "0530bed92f633ae1bccc61c5c5d58fdb": " f_x=x/(\\lambda z)=l/\\lambda", "0530cd86ac74e46292792762625a3337": "A(0, b) = 2 b + 1", "0530f18af57b81cb2c913b0d3089f540": "\\displaystyle{\\sum b_n(\\zeta^{-1}) z^{-n} = \\exp \\sum a_m(\\zeta^{-1}) z^{-m} = {g(z)-g(\\zeta)\\over z-\\zeta},}", "0530f4c6b8b0956007dd50dcb0eb0f6c": "\\vec{v} \\vec{w}", "05310eabb8157f12c7f05bf756526726": "F:\\mathcal{P}(S \\times S) \\to \\mathcal{P}(S \\times S)", "053188dd9d2bcfdf6aee570206038125": " E = \\sqrt{\\sum P(n-X)^2} ", "05325fc96d5fe3a9abb9a9ff8ba9465a": "r(t) = q", "0532864d8a133e2306c3965faaf76a2b": "\\delta t=1.7\\pm1.4\\ (\\mathrm{stat.}) \\pm3.2\\ (\\mathrm{sys.})", "0532aa97d0f0b796383aef0266ca31d2": "\\frac{df_{a_1,\\ldots,a_{i-1},a_{i+1},\\ldots,a_n}}{dx_i}(a_i) = \\frac{\\part f}{\\part x_i}(a_1,\\ldots,a_n).", "053326edbd07a01f83831d7f82855e5b": "\\frac{\\sin \\theta}{\\theta \\cos \\theta} > 1\\,", "05332b4197c5bb54ec4d3dbc10d9eec8": "H=\\left(N+\\frac{1}{2}\\right)\\hbar\\omega,", "05337cb337140eb5fcfceb4ab7ba6184": "\\mathit{f}", "0533bcef777c92ce342ea3625c1dfb42": "|c-[c]-\\frac{1}{2}|<\\frac{1}{2}-a", "0533d8bce4d6a4b5f26caa843aec2fdd": "1.05^4-1=21.55%", "053407cd89ffc607ac5304ac11057fd0": "\\begin{align} 2\\cdot R_*\n & = \\frac{(10^{-3}\\cdot 128\\cdot 2.05)\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 56\\cdot R_{\\bigodot}\n\\end{align}", "053456938c3e537cd4b6aad2387c484d": "\nR(\\tau) = \\frac{\\operatorname{E}[(X_t - \\mu)(X_{t+\\tau} - \\mu)]}{\\sigma^2}, \\,\n", "0534e1c6e1c58e65a298449d6def9e2a": " (n,m,l, \\epsilon)", "0534f253fa8bdb5a9a05bfae6b479256": "{{P}_{Disk}}=\\left[ \\varphi ,\\chi \\right]*\\text{ }Vector\\text{ }of\\text{ }Disk\\text{ }performance\\text{ }counter+{{\\lambda }_{constantDisk}}", "0534f91618c21c6115ee879962400bf3": "v = (x_b + k^e)\\mod N", "0535104b327e8cbeacc6b9fcfafd7e10": "\n \\varphi_\\alpha = w^0_{,\\alpha} \\,.\n ", "05355d3274aed8dc2eca2ab604bd85d3": "N_{s}", "0535634f282026214e59aa656824235c": "f(\\mathbf{z}) \\in f(\\mathbf{y}) + [J_f](\\mathbf{[x]}) \\cdot (\\mathbf{z} - \\mathbf{y})", "0535730078adc42f7af6fe8ce72846f3": "t\\geq 0", "053640ae353786b845162394037755d4": "\\mu(X)=\\mu(Z)+\\mu(X\\setminus Z)", "05364d147eb0bc88147e0ba960605f03": " C^*_{(+)}= C_{(+)} ", "0536ad4ce780b9d76a1baa3013ac1918": "\n\\boldsymbol{x}, \\boldsymbol{y} \\sim\\ \\mathcal{N}(\\boldsymbol\\mu_{X,Y}, \\boldsymbol\\Sigma_{X,Y})\n", "0536b9dcbddd3e7ed41903ab2ea8a619": "f_n(x)=x+n", "0536cb091c7b6ab01bbbbcd34f865cfa": "\\displaystyle x_{i}\\rightarrow x_{i}b^{\\left[ x_{i}\\right]}, \\phi _{i}\\rightarrow\n\\phi _{i}b^{\\left[ \\phi _{i}\\right] }. ", "0536ccb9e1940a60e0d1dfb9178ea027": "s = +j \\omega_0\\,", "0536e7e4439f94e77793561079872db8": "\n\\exp \\left\\{ - \\frac{a}{2} x^2 \\right\\} =\n\\sqrt{\\frac{1}{2 \\pi a}} \\; \\int_{-\\infty}^\\infty\n\\exp \\left[ - \\frac{y^2}{2 a} - i x y \\right] \\, dy,\n", "0537054c6c956f5ab503c9fdcd425c06": "- + -", "0537342e84ebcc0997f7ed98ef18c3da": "g(a,a+d,a+2d,\\dots,a+sd)=\\left(\\left\\lfloor\\frac{a-2}{s}\\right\\rfloor+1\\right)a+(d-1)(a-1)-1.", "05374cb757176f04bf864caa67390057": "\\partial W=M \\sqcup N", "05378ca23df01c19a92166951a7a563e": "-S = \\left(\\frac{\\partial F}{\\partial T}\\right)_{V}\\,", "0537a4dc709adae5af1e5ad54a743ec7": "B(\\boldsymbol{u},\\boldsymbol{v}) - F(\\boldsymbol{v}) \\geq 0 \\qquad \\forall \\boldsymbol{v} \\in \\mathcal{U}_\\Sigma ", "0537c34c6bcde95545ce50bb1e94f6d1": "\\mathbf{\\nabla}\\cdot(\\epsilon \\mathbf{\\nabla}\\varphi)= -4 \\pi \\rho_{f}", "0537ca2ca25661ae0d9bbec87714cc55": "e^{-\\beta_{k}\\tau}", "0537d06876f6ff79f8b419fa747a25db": "\\int x e^{c x^2 }\\; \\mathrm{d}x= \\frac{1}{2c} \\; e^{c x^2}", "0538221777a190b91f792b30a8aedad4": " \\sum_{j \\in J} a_j \\mathbf{v}_j = \\mathbf{0} \\,", "0538a1b229fa70423b107e594a783264": "\nh\\,a = \\begin{cases}\n c & \\mbox{if } p\\,a\n \\\\b \\oplus ha' & \\mbox{otherwise}\n \\end{cases}\n ", "0538b394394e22701b79c1ea9a80f9ca": "T = T_i2^{-R/C_V}.", "0539109d4289480e283c341dff4f2491": "\\lambda_p \\approx \\lambda_j", "05391104f231a1aa43ca9f8192d45ab4": "x \\in Y", "05393d10d8fe7779e4cf3c8724c53f01": "r(T) = \\lim_{n \\to \\infty} \\|T^n\\|^{1/n}.", "0539ba5df67017d6394f3669755ba31c": " \\frac{dc}{dl} = G'(l). ", "053a98124485559a12edcc8176574789": "L(\\lambda, \\alpha, s)", "053b39ebd828ace2d4f73180f53ba2a0": "\n\\frac{x^2}{a^2}+\\frac{y^2}{b^2}+\\frac{z^2}{c^2}=m^2\n", "053b9a3cc33be28c94b003ccf8dc0f94": "\n=[E_{12}, E_{11}] E_{22} + E_{11} [E_{12}, E_{22}] -\n[E_{12}, E_{21}] E_{12} - E_{21}[E_{12},E_{12}] +[E_{12},E_{22}]\n ", "053bbf1ecf57ac081e3f0b9156b77ec1": "\\sigma = (x~~1)(x~~2)\\cdots (x~~i)(y~~i+1) (y~~i+2)\\cdots (y~~k)(x~~i+1)(y~~1).", "053bcefa288f5670afb2b08d40a818c1": "\\omega \\in\n\\Omega_{Z,[t_l, t_u]}", "053c104ac907d191d028dc3ce0a0126d": "\\mathbb{J}", "053c115098a7fbf8f6bcdcf83197ae46": "\\scriptstyle f(x,y,z)=w^2", "053c735d694601190d0d4d634d4b9b3c": "\\begin{pmatrix}\n 0 & 2 & 0 \\\\\n 0 & 0 & 3 \\\\ \n 0 & 0 & 0\n\\end{pmatrix}", "053c73b9865caf2fb1ef485da71b9618": "\\ker \\rho = \\left\\{g \\in G \\mid \\rho(g) = \\mathrm{id}\\right\\}.", "053c82ae3b2f75e841f38f565fffbb7b": "\\partial_x", "053ca75a838fa7f797b2d812ae85a03a": "s_{n-1}", "053d1a056e387db09b31665c752971ca": "\\Pi_{(x:A)}B(x)", "053d2eb84321dcdf2526369fc086cfb1": "r < \\operatorname{diam}(\\Omega)", "053d3b5e3c21ef4f364b0b836612264e": "\\sigma_B \\geq 0", "053d472eab7e87d03b517d01f001a2ff": "\\Omega*m", "053e5921874fb15240a8b8be120c1bb0": "{x \\over {a-x}} = D {y \\over {b-y}}", "053e6723664890129ebc3c269c2371d1": "\\sqrt{\\ } \\!\\,", "053e70c93e7e72c8678df6ff21231e17": "\n w(x_1,x_2,0) = \\varphi(x_1,x_2) \\quad \\text{on} \\quad x_1 \\in [0,a] \\quad \\text{and} \\quad\n \\frac{\\partial w}{\\partial t}(x_1,x_2,0) = \\psi(x_1,x_2)\\quad \\text{on} \\quad x_2 \\in [0,b] \n", "053ea84cf5b391a6bc0e2769a337b124": "(v,v-k,v-2k+\\lambda)", "053eb11717e6a67416b0b5b369490f43": "G = \\sum_{nm} G_{nm}", "053eb3c78df7b74a18c177d5fff4640a": " k[\\Delta] = \nk\\oplus\\bigoplus_{0\\leq r\\leq d}\n\\bigoplus_{i_0<\\ldots 0,", "053fb39ecb3a197a84a1e24f7e1036c5": "\\frac{\\Delta \\hat{z}}{P}\\,", "054071638984997309331a922b0939d9": "\\lambda=(gy-u^2-v^2)/L^2", "054079603de534fdc6b53a8ebaf62a52": " h \\equiv \\frac{\\sigma_d}{\\sigma_m} ", "0541088a52783eb8184a0704d885f61a": "\\partial_k:C_k\\to C_{k-1}", "054175e12fea6ba14d68da07557fd856": "q_b = \\iiint \\rho_b dV = -", "05417c800e0b9fa8c72b54a10bf205ad": "x_{(i)}", "05417e69be53514379344dd452419664": "R[t] \\to S, \\quad f \\mapsto \\overline{f}", "05419dda5884bdd874dedbe5304f008a": "\\forall x \\, \\phi (x) \\Leftrightarrow_{\\mathrm{def}} \\forall X \\, ( \\mathrm{set}(X) \\rightarrow \\phi (X) )", "0541c06b519ac3465f57abd96c0aacde": "p^2 \\gg k^2", "0541da45a48c535528249f3115d39b0b": "S_2=52.6 \\text{ mm}", "0541df319595678b2a34001bc11b0a6e": "\\gcd{(A,B)} = 1", "0541f24a007b7b665eeee28fc06f6ea3": "h_i=a_{0,i}", "0542761b5ed427e079c3a5eb0a388bf1": " \\rho \\left( \\frac{\\part \\vec{u}_{x}}{\\part t} + \\nabla_{y}\\cdot\\vec{u}_{x}\\vec{u}_{y}\\right)= -\\nabla_{x}p+\\nu\\nabla_{y}\\cdot\\left(\\nabla_x \\left( \\rho \\vec{u}_y \\right) +\\nabla_y \\left( \\rho \\vec{u}_x \\right)\\right) \\,\\!", "0542f906a77ca3e12b89972bc173b196": "\\Gamma(s) = (s-1)!", "054314d841c8f58c84d5c92bf9af8689": "Q(V,T)\\ ", "0543614a7ded3a57d4f0d0805f7f6818": "r^{-6}", "054395ad5b295a1788d6640f63de9c88": "m_\\mathrm{s}", "0544d0f6b025998039fc986117cb5107": "A\\to\\neg\\neg A", "0544ddcc1d9f04f0b80c59c2a0640cdd": "\\begin{align}dy_{\\text{1}}\\ =\\ I_{\\text{1}}dt\\ +\\ cdW_{\\text{1}}\\ -\\ u(I_{\\text{2}}dt\\ +\\ cdW_{\\text{2}})\\\\\ndy_{\\text{2}}\\ =\\ I_{\\text{2}}dt\\ +\\ cdW_{\\text{2}}\\ -\\ u(I_{\\text{1}}dt\\ +\\ cdW_{\\text{1}})\\end{align},\\quad y_{\\text{1}}(0)\\ =\\ y_{\\text{2}}(0) = 0", "05451ef2f9f21fce1ae0956590d7dc50": "\\ P_{ij\\ldots}=P_{ij\\ldots}(\\mathbf X,t)", "05451fff8d6c48d06fae0418857ee63c": "\\mathbf{x}=(x_1, x_2, \\dots, x_n)", "054521ed7f18b89d0ee32e64fcf995bc": "\n_a I_b^{\\left( D \\right)} 1=\\frac{1}{\\Gamma \\left( {1+\\alpha } \n\\right)}\\int_a^b {\\left( {dt} \\right)^D} ", "05458ad345a3e4f6d52e91a62e829a02": "h_2=0.1935\\times Do-0.455\\times t", "0545afccd5ede14da5822029bb943006": "\\sum_{\\pi\\in S_n} \\frac{\\sigma(\\pi)}{\\nu(\\pi)+1} = \n(-1)^{n+1} \\frac{n}{n+1},", "0545c01d23dca7b361c37f12366c25a2": " f : x = \\{x_n\\} \\in \\ell^1 \\ \\rightarrow \\ \\sum_{n=0}^{\\infty} x_n,", "0545dfdc996460d9837db4932313fb76": "i(x,y)", "054601baeb96b31f4e7eb6fbdc35e3e5": "\\left(\\pm1,\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+2\\sqrt{2}),\\ \\pm(1+2\\sqrt{2})\\right)", "05462613c17fd32778bb06f4e57c8c52": "\\bigl(\\tfrac12,\\tfrac12, \\ldots\\tfrac12, -\\tfrac12\\bigr)", "05462afc37afa847c14b68233c4a6770": " D_q f(x)=\\frac{f(qx)-f(x)}{(q-1)x}.", "0546416dcc8b41b6304edac92d767118": "\\int\\frac{dx}{r} = \\operatorname{arsinh}\\frac{x}{a} = \\ln\\left( \\frac{x+r}{a} \\right)", "05465f7db8e44281594795f0a743bc36": "\\overline{N}_{\\Delta f}(f)", "0546602fbc3752251622c6ad39b77b54": " u(t,x) = T(t) v(x).\\,", "0546fce8761070e7da5fdf6bc0b0bcc1": " v_0 = \\frac{V_{\\max}[\\mbox{S}]}{K_M + [\\mbox{S}]} ", "05473bec95ce3ec988da31f310a63a1d": "\n \\Beta(x,y) =\n \\int_0^\\infty\\dfrac{t^{x-1}}{(1+t)^{x+y}}\\,\\mathrm{d}t,\n \\qquad \\mathrm{Re}(x)>0,\\ \\mathrm{Re}(y)>0\n\\!", "054745ac1b7edb7c864d90535b3feba0": "- \\nabla U(X)", "0547564a05c36f9e94a0163b186dab47": "E_{K_1}(E_{K_2}(P)) = P", "0547870987b7e4e2f8782db9146b430a": "\\left( \\left( x\\ast y\\right) \\ast \\left( x\\ast z\\right)\n\\right) \\ast \\left( z\\ast y\\right) =0", "05478d83f2ad822d957bdf9dba6eff34": "(d_1,e_1) \\cdot (d_2,e_2) = (d_1^{e_2}d_2^{e_1}, e_1e_2) \\ . ", "0547c51ea114c93a980d6dc2b6f904b9": "\\nu = \\alpha c_{\\rm s} H = \\alpha c_s^2/\\Omega = \\alpha p_{\\mathrm{tot}}/(\\rho \\Omega)", "0547e443c12f4232d1d5ea6b222a0525": "\\tilde{T} = \n\\begin{bmatrix}\n0 & \\; & \\cdots & z \\\\\n\\frac{1}{2} & \\ddots & \\ddots & \\; \\\\\n\\; & \\ddots & \\ddots & \\vdots \\\\\n\\; & \\; & \\frac{1}{2} & 0 \n\\end{bmatrix},\n", "054803695498bb95cc220bee5b591ab9": "n_F(\\xi)=\\frac{1}{2}\\left(1-\\mathrm{tanh}\\frac{\\beta\\xi}{2}\\right)", "054826fb48e60800985100a1704a3d58": "\\Gamma = \\Gamma_{ab} + \\Gamma_c", "05487c8bf39af24bd9bca6d3a28aa0cc": "(\\cos(\\theta))", "0548ce06a9515c323b8e4948f12bc697": " h_{x} = -e_{y} / \\eta ", "05494d0a69bc739e85f967733bee4c00": "x_1^2 + \\cdots + x_n^2 + 2a_1x_1 + \\cdots + 2a_nx_n + c = 0", "0549a953f3ba4ef9d116cb0e1132a3bf": " \\vec{s}_a \\cdot \\vec{s}_b ", "0549c34e4484e3cc069290d888bd9e61": "{\\left ( \\frac{\\partial x}{\\partial y} \\right )}_z {\\left ( \\frac{\\partial y}{\\partial z} \\right )}_x {\\left ( \\frac{\\partial z}{\\partial x} \\right )}_y = -1.", "0549c7cd38c25400ab340f4a9e3443db": "\\Delta q = (q-q_0) ", "054a3a4c6ee140a2c53ddaf1b6bb0e96": "|R| < |S|", "054b07d42e4f6e99e7aaac28551aff25": "\\sum_{m=0}^\\infty\\sum_{n=0}^\\infty a_{m-n} \\lambda_m\\overline{\\lambda_n} = 2(1-|z|^2) \\,\\Re\\, f(z).", "054b1b47b17865fe755c44c54b5551a5": "\\beta_{n}^{PR} = \\frac{\\Delta x_n^\\top (\\Delta x_n-\\Delta x_{n-1})}\n{\\Delta x_{n-1}^\\top \\Delta x_{n-1}}\n", "054b6af9ec764b7ae7bdbc715df68090": "\\mathfrak{P}^{36}", "054b947d6a054d09c69bce8840ee4886": "g_N \\left (x_1,\\dots, x_N, t \\right) = G\\left(x_1, t \\right) \\cdots G\\left(x_N, t\\right) ", "054c117040ce2a6676a9e837016aa70d": "z=1\\ldots Z", "054c373fad93a5f8848feda8b6c23c5d": "\\tan a = \\frac{\\sin a}{\\cos a}", "054c8e2980502a6d65a953d78a2f9c4a": " \\rho_1 \\mathbf{v}_1 \\cdot \\mathbf{S}_1 = \\rho_2 \\mathbf{v}_2 \\cdot \\mathbf{S}_2 ", "054c9d24788c0023b22433c604b03c41": "B[u, u] + G \\| u \\|_{L^{2} (\\Omega)}^{2} \\geq C \\| u \\|_{H^{k} (\\Omega)}^{2} \\mbox{ for all } u \\in H_{0}^{k} (\\Omega),", "054cae4d1530dc60449f72b8fe9a5c6f": " \\bar r(t) ", "054cd288d706740e52dddb6f039062f0": "\\mathbf{U} = E[(\\mathbf{X} - \\mathbf{M})(\\mathbf{X} - \\mathbf{M})^{T}]", "054d457a8a66de966ff08d0b72b5b96d": "a^{th}", "054de48d0df9a4458f7703161ba8acd9": "x \\mapsto g(x; 2)", "054dfbc4cbd2f68a06c003ae77f874ea": "\nCIQ_t = \\mathcal{A} \\left ( 1 + \\mathcal{B} \\right )^t\n", "054eb2251868bb2a2cf30c3f43b48e7b": "J_F(x_n) (x_{n+1} - x_n) = -F(x_n)\\,\\!", "054eb94b485d8f79dc65c360a6a539fe": "|\\partial A|\\geq C\\left(\\min \\left( |A| , |G\\setminus A| \\right)\\right)^{(d-1)/d}. \\, ", "054ec7718dc62f34e0fcab7a6bc9e865": "\n\\int_a^{x_0-\\delta} e^{n f(x) } \\, dx + \\int_{x_0 + \\delta}^b e^{n f(x) } \\, dx\n\\le \\int_a^b e^{f(x)}e^{(n-1)(f(x_0) - \\eta)} \\, dx = e^{(n-1)(f(x_0) - \\eta)} \\int_a^b e^{f(x)} \\, dx \n", "054ee465850b8285713271cb44f8f3c9": "M/T", "054f2cc1ac6da37ad4bf572757cf7dd5": "S_{11} = {(1 - Z_0 Y_{11}) (1 + Z_0 Y_{22}) + Z^2_0 Y_{12} Y_{21} \\over \\Delta} \\,", "054f2d87ffd8835b0422c2e3d22d76ae": "\\log (1/\\epsilon)", "054f383837eb3832ac11add3cdd54472": "x_1\\le\\cdots\\le x_n\\quad\\text{and}\\quad y_1\\le\\cdots\\le y_n", "054faf8b16b83a42e23391c90bc802e9": "t=q^{-s}", "055011102cd4a7625e250b7efd8990eb": "x\\in K", "05501f6d0a6bc597511bf940eac005b8": "R_{\\text{vertical}} = \\frac{R_{12,34} + R_{34,12} + R_{21,43} + R_{43,21}}{4}", "055036089d7d464018b5dc9f3d56ed55": " \\rho = ", "055039b61f4f25f8766e98e3f0d1daac": "\\begin{align}\n\\rho_0 \\ddot{\\boldsymbol{x}'} &= \\frac{\\partial^2}{\\partial t^2} (\\boldsymbol{u}^{(0)} + \\boldsymbol{u}^{(1)} + \\boldsymbol{X}) \\\\\n &=\\frac{\\partial^2 \\boldsymbol{u}^{(1)}}{\\partial t^2}\n\\end{align}", "05508790e1f9a13201a1f9fbaabf615d": "J \\ \\stackrel{\\mathrm{def}}{=}\\ P_+ - P_-", "05508e4ff635ae65cf58468bc93febe2": " v'' = 0. \\;", "055202e88e159f6ae4c20e58150084f8": "D_{\\mu}\\tilde{F}^{\\mu\\nu}=0.", "05520c4227b2eabb934ff18323903144": "\\sigma_x \\sigma_p \\ge \\frac{\\hbar}{4}\\sqrt{3+\\frac{1}{2}\\left(\\frac{\\Omega^2}{\\omega^2}+\\frac{\\omega^2}{\\Omega^2}\\right)-\\left(\\frac{1}{2}\\left(\\frac{\\Omega^2}{\\omega^2}+\\frac{\\omega^2}{\\Omega^2}\\right)-1\\right)} = \\frac{\\hbar}{2}.", "055238ed1eb06ba46e9d541b8e74cca3": "k_x^2+k_y^2+k_z^2= k^2", "05524620db947b72a399c99f6a9329a2": "x_k:=x_j+ P_k A^{-1}\\left(b - A x_j \\right),", "0552ae58c274fcd873a6ce452a0c6231": "\\left|\\frac{f(z_1)-f(z_2)}{\\overline{f(z_1)}-f(z_2)}\\right|\\le \\frac{\\left|z_1-z_2\\right|}{\\left|\\overline{z_1}-z_2\\right|}.", "05536b45589d88f483e73d0ef5612dce": "\n\\sum_i n_i(\\bar{Y}_{i\\cdot} - \\bar{Y})^2/(K-1)\n", "05539ed6141a398c93b7065ad4201a6a": "f_1, f_0", "0553c8b8021e37b9e34f9af179254d52": "\\epsilon_c = f\\epsilon_f + \\left(1-f\\right)\\epsilon_m.", "0554246077f1d1b5f2c5f4ffacf646fd": "\\bar x(t)", "055492d3c623cada3d3b79809dded22c": "(|x\\rangle, |\\psi\\rangle)", "0554ac534fbee8422cedc5d3f1fac58f": " f^* = \\min_{x\\in R^n}\\{f(x)\\}", "05551124009a4be6272c004d9493bbc3": "f=f_1 e_1+f_2 e_2+f_3 e_3+\\cdots", "0555229d9f9cefecadb272f844e063bd": "2^{-b}", "055523e005e73c3ae0bd0aa9ce2da1ce": "\\varepsilon_{eff}", "05555364502329a70d4673fa5c6e402b": "M=C+D", "0555554754988fe2b2b12ace1983d53e": "\\hat{ \\theta } =T(\\hat F_n) \\,,", "05556a62b2f29e3ffa699da5ed630595": "\\pm 2x^{7/22}", "0555da2e4f0667910a6d9e9dae93bb32": " e^{M f(x)}. \\,", "0556468dd133208856883806211b55be": "\\Pi^{\\mathbb{Z}^{+}}", "05564f5751a56a8627447ad69654ae77": "\\ln \\frac{\\hat{\\beta} - \\frac{1}{2}}{\\hat{\\alpha} + \\hat{\\beta} - \\frac{1}{2}}\\approx \\ln \\hat{G}_{(1-X)} ", "05566b811fe14c932f8abade2a39dd9b": "\\textstyle b(x)", "05569328ce8674f4e3f0bcddde66e97d": "L_1 = x_2 p_3 - x_3 p_2", "0556d29890d84c63ead4d591cf8fe6c4": " \\ell \\equiv ab\\bmod m ", "055758542eb4d96345b9a2448b5ec284": "\\textstyle e_{\\lambda}(f_{ij}) = \\frac{f_{ij}}{\\lambda}\n\\frac{\\partial \\lambda}{\\partial f_{ij}}", "0557611da6e08448d16d418a19f8c6e9": "\\frac{d^2z}{dt^2} + 2i\\Omega \\frac{dz}{dt} \\sin(\\varphi)+\\omega^2 z=0 \\,.", "05576a866d96093e3b177316997766bd": "G = \\sum_i \\mu_i N_i\\,", "05577e342b9901af7a97641588e704f0": "M(H)", "0557da9106c4a4ca91ffdf200da209ce": "\\Delta = b^2 - 4ac\\,\\!", "0557e4e4424f3b9bb39ec55278ae06ed": "\\,(1-\\varepsilon)L + \\varepsilon N\\, \\prec \\,M \\, \\prec \\,\\varepsilon L + (1-\\varepsilon)N.\\,", "05581e0aec5d7df6fb8e95cfd4e38a9b": "H(Y|X)=0", "0558205a74c10475844dba3327824980": "v\\in M", "05588378ae0dc3377574679e08679a30": "\\Rightarrow \\delta^2 = n^{2\\gamma -1}", "055889aaee38b7c53f994c5e42a40994": "\\Rightarrow", "0558c6b0f1f42f903bd7421dcbe104f1": "\\int_X\\left(\\int_Y f(x,y)\\,\\text{d}y\\right)\\,\\text{d}x=\\int_Y\\left(\\int_X f(x,y)\\,\\text{d}x\\right)\\,\\text{d}y=\\int_{X\\times Y} f(x,y)\\,\\text{d}(x,y).", "0558fd5010fefb8a612245fa9a0f90ff": "\\textrm{var}(X) = \\frac{4 - \\pi}{2} \\sigma^2 \\approx 0.429 \\sigma^2", "055984811ecb3a80e01471dd107feea9": " g(x,X) = \\sqrt{n}\\frac{x - \\overline{X}}{s} ", "0559a0924e84a864de100431a3f3a920": "K_{\\rm d}", "0559ba5a8d8c38f339a9214613f3e418": "\\mbox{Golden rule for capital/labour ratio: } \\frac{ df }{ dk } = (n+d)", "0559e2fa97bfd2c0e077f36b3d65732c": "\\forall\\ x, Rx\\ \\rightarrow\\ Bx", "0559f1beb33bebaeba1551b86a80dfde": "g_2= \\tfrac{1}{\\eta}ij", "0559ff3b5ee5589a99f3d115f5f6ba92": "hmcr", "055a1394374dd31041ec2ee185ca1749": "\n\\overbrace{\\rho \\Big(\n\\underbrace{\\frac{\\partial \\mathbf{v}}{\\partial t}}_{\n\\begin{smallmatrix}\n \\text{Unsteady}\\\\\n \\text{acceleration}\n\\end{smallmatrix}} + \n\\underbrace{\\left(\\mathbf{v} \\cdot \\nabla\\right) \\mathbf{v}}_{\n\\begin{smallmatrix}\n \\text{Convective} \\\\\n \\text{acceleration}\n\\end{smallmatrix}}\\Big)}^{\\text{Inertia}} =\n\\underbrace{-\\nabla p}_{\n\\begin{smallmatrix}\n \\text{Pressure} \\\\\n \\text{gradient}\n\\end{smallmatrix}} + \n\\underbrace{\\mu \\nabla^2 \\mathbf{v}}_{\\text{Viscosity}} + \n\\underbrace{\\mathbf{f}}_{\n\\begin{smallmatrix}\n \\text{Other} \\\\\n \\text{forces}\n\\end{smallmatrix}}\n", "055a2275d6d566469f1e8d55e11773ab": "\\int f^{-1}(y)\\,dy= x f^{-1}(y)-F\\circ f^{-1}(y)+C,", "055a6233561f4f99bbe61895da792be2": "\\scriptstyle \\hat x", "055a6eacf46e89a3e550c074366c46fd": "f(x) = \\int _{-\\infty}^{\\infty} A(\\xi)\\ e^{ i(2\\pi \\xi x +\\varphi (\\xi))}\\,d\\xi,", "055a8649953c17cb8f80b461f1718e5a": " size = 1.22\\frac{\\lambda}{D}distance", "055a9f4f3e3d8618f65d3d30c1089cbf": " a_1b_2-a_2b_1=0", "055b8424d6eeb4f26e4709d51649d526": " \\Rightarrow \\mathbf{u} = \\begin{pmatrix}\n \\mathbf{u}_1 \\\\\n \\mathbf{u}_2 \n\\end{pmatrix} = \\begin{pmatrix}\n -\\mathbf{B} \\\\\n \\mathbf{I}_{n-r} \n\\end{pmatrix}\\mathbf{u}_2 = \\mathbf{X}\\mathbf{u}_2. ", "055c210a7797b4e842635accb13e32a7": "q_{i}", "055c6a0341f1cde9ac1b10222a5ffd1a": "\\frac{D}{\\beta_k}", "055c9f1f08029271f30a3a260d8adc91": "k+d \\le n+1 ", "055cbe8016aa569202dc280991a1cb7c": " W(C;0,1) = A_{0}=1 ", "055cebdfa45752b5a893ad0e84ae382f": "\\Theta\\subseteq\\mathbb{R}^k (k\\geq 1)", "055cf3ed53c85605519082bdafd3564a": "y=k^\\alpha \\,", "055d063e8aa202f11316a89d2481c165": " \\mathbf{V}_g ", "055d47751d6a5128845a7ea5c7646411": "W_0(z)=\\overline{z}\\,", "055d4b1c657a5659f94cd3be29a7eb3c": " E_2 = E_0 \\left (\\frac{4 m_x m_y cos^2 (\\theta_1)}{(m_x + m_y)^2} \\right) ", "055d8efdf967ba0b99fb155e9ba13602": " \\Phi = \\iint L\\left ( \\mathbf{\\hat{e}}_{\\angle} \\cdot \\mathrm{d}\\mathbf{A} \\right ) \\mathrm{d} \\Omega", "055d9ebe3f02e2ecb1aea9b5288f7012": " R = \\frac {\\textrm''{Static \\,\\, pressure \\,\\, rise\\,\\, in \\,\\, rotor}''}{\\textrm''{Total \\,\\, pressure \\,\\, rise \\,\\,in \\,\\,stage}''} ", "055db5bcedf2479d313b23195ccd4dd3": "{\\mathbf D}_a\\leq \\Delta (l,p)", "055e1511ceee6cdf419b958e47302ef5": "A=LDU", "055e525da2ab4585b14653934c601607": "\\lambda_1 = e^\\varphi", "055e6eb5934e1add9077f56cf45af477": "x_{i0}", "055e70d8c3300dfdfb5dad705eec67c5": "S(y)\\equiv \\frac{9\\sqrt{3}}{8\\pi}y\\int_{y}^{\\infty}K_{5/3}(x)dx", "055ee518ef74037299fd85d27199b5e6": "x_i, x_j", "055f16d961b10c799c8d889dfdf4c780": " U \\subseteq \\mathbb{R} ", "055f3ed5b830b80653844071be81430b": "\\aleph_\\alpha=\\beth_\\alpha", "055f95f9c0b1eb17fd7304b613bf47cb": "V(s) = I(s) \\left ( R + Ls + \\frac{1}{Cs} \\right ) ", "055f9b02b3725e514f0d09cf77cf44c5": " \\Delta G < 0 \\,", "055fb5b5fe1af9b954e324a8d3476d91": "i, i = 0, 1, 2, 3", "055fcefc1ef3182b8ed1ae0cf149091b": "b=1.", "055fe14fb7b286dd73fa7f890c73ccd1": "\\lang i|j\\rang = \\delta_{ij}", "05605eb50844d7da0adfa00a79edb154": "\\theta_f\\;", "056071b398767bff60d2fa58ce678cad": "\\exp \\overline{C}", "0560d806ee2ce7a70952aae668a97853": "\\frac{3x^2 + 12x + 11}{(x+1)(x+2)(x+3)} = \\frac{A}{x+1} + \\frac{B}{x+2} + \\frac{C}{x+3}", "0560e4258b913a2bdbd319cac3071786": "x_{k-1}", "0560f8ae7f453aa55d98ef103ee55cf8": "\\mathcal{L}\\left\\{\\frac{df}{dt}\\right\\} = s\\cdot\\mathcal{L} \\left\\{ f(t) \\right\\}-f(0), ", "0560fda2aa9d1b6a5ce3e5ea42e3ecd6": " P_\\ell, ", "0561415c538ec01c092fa3149cb72cfd": "x_{0} := g_{k}(x)", "05615c41a8772058f713dc4a59bc89a4": "k=|S|", "0561871b25f674b5d87401405a71fbe4": "x = \\left \\{ x_1, x_2, \\ldots x_d \\right \\}", "0561cf6257c67b8da7d20e530b4b2854": "\\alpha = \\delta^{\\beta_1}\\gamma_1 + \\cdots + \\delta^{\\beta_k}\\gamma_k", "05621a51b634c7dced18f7261f41c999": "T_1(\\cos(x))=\\cos(x) \\, ,", "05623a342aecb7bfba450a4d6fb10c04": "\\operatorname{Pic}(X) \\to H^2(X, \\mathbb{Z}).", "05627261e44b9fe9c6696c577f1b72a1": "C\\ell_{p,q}(\\mathbf{R}) = C\\ell_{p,q}^{+}(\\mathbf{R})\\oplus C\\ell_{p,q}^{-}(\\mathbf{R})", "0562a1641aab03785af0d3ed968f0e3f": "X(e^{iw})=1-cos(w)", "0562f50892ad4889a964d403daf603e2": "\\frac{d}{ds}{\\mathbf{s}}_u=\\frac{1}{r}\\cdot\\frac{d}{d\\varphi}{\\mathbf{s}}_u=-\\frac{1}{r}\\cdot{\\mathbf{n}}_u.", "0562f79dba54f724a2e46dd90c1848c8": "\\mathbb Z[1/p]/\\mathbb Z", "0562fcb3da51e39c2c9e3680d8dbf12c": "\\implies f_X(x|Y=y) = \\frac{f_Y(y|X=x)\\,f_X(x)}{f_Y(y)}.", "0563119e25413247d5447cf3497ae6bd": "\\sum_{k=1}^\\infty\\frac{1/k^s}{\\zeta(s)}\\log (k^s \\zeta(s)).\\,\\!", "056316892087717f6b1da0054c5b0b71": "\\psi: k\\left[M\\right] \\to \\prod_{i \\in I}k", "05636e876c6535047b56fff578067c07": "D^{k+1}", "0563d3fc61798c4971e0b7150ae040ca": "{T_{cold}}", "05642005d370089d5b8157be5c1f6a19": "t_1 = \\pi \\sqrt{\\frac{a_1^3}{\\mu}} \\quad and \\quad t_2 = \\pi \\sqrt{\\frac{a_2^3}{\\mu}}", "056438d7487f684072ab843845aa6a8b": " I_{ref} = I_{C1} (1 + 1/ { \\beta}_1) \\ ,", "0564620f798e254b5b2933dc44d0b26b": "T_\\alpha^\\pi = F_{\\alpha\\beta} \\mathcal{D}^{\\pi\\beta} - \\frac{1}{4} \\delta_\\alpha^\\pi F_{\\mu\\nu} \\mathcal{D}^{\\mu\\nu}", "05647b627b7a29a511a922dafbca560a": "\\alpha = k/(\\rho c_p)", "0565181079e13a9ab934f370e98d5b6d": "A = 4 \\sin \\frac{\\pi}{4} R^2 = 2\\sqrt{2}R^2 \\simeq 2.828427\\,R^2.", "05657d9ad07e9f51b2f6f3e210e2e2c6": "\\scriptstyle{\\langle L \\rangle \\Phi}", "0565b67cb9aa47f5e9fcf825bb8d8d93": "\\vec{X} f = f_{,a} \\, X^a", "0565be088eea5995b19bf091d936eea7": "\\begin{matrix}\\underbrace{{2^2}^{{\\cdot}^{{\\cdot}^{{\\cdot}^2}}}} - 3 \\\\n\\mbox{ + 3}\\end{matrix}", "0565e48cc9230dbec676919b2d405b4a": "\\displaystyle{z}", "0565f3387aa61808aa3fc267f563fcfe": "\\lambda_m^2 + 2\\lambda_m - J_m - 3 = 0 ", "0565f7962efe7a29de4cf05523effe90": " | \\psi \\rangle = \\int\\limits_R d^3\\mathbf{r} \\, | \\mathbf{r} \\rangle \\langle \\mathbf{r} | \\psi\\rangle = \\int\\limits_R d^3\\mathbf{r} \\, \\psi(\\mathbf{r}) | \\mathbf{r} \\rangle ", "0565f7aacc902330a589569f23bc3777": "\\partial \\alpha = 0", "0566040c991ab961543164e2b6d0add4": "\\psi(b_k) = \\sum_{i+j=k}(b)_{2i}^{j+1}\\otimes b_j", "056680547cf214f9aa06ac445f46ebb1": "\n\\frac{\\partial F_x }{\\partial x} + \\frac{\\partial F_y }{\\partial y} + \\frac{\\partial F_z }{\\partial z} = 0 ", "05668a01779ec8170b9bd5eeb0e7e921": "\\operatorname{U}(n,\\mathbf{C}/\\mathbf{R})(\\mathbf{R}) = \\operatorname{U}(n)", "0566acb6948ed36d10fdd7b86b154624": "n_b", "0566be246667812ef1b9e2d8217c66a1": "t_{ij}=\\sqrt{\\overline{O_iO_j}^2-(R_i-R_j)^2}=\\frac{\\sqrt{R-R_i}\\cdot \\sqrt{R-R_j}\\cdot \\overline{K_iK_j}}{R}", "0566e1a5690a3eb3da63262a66fa0698": "\nL=\\left(\\begin{array}{cc} 1 & x \\\\ 0 & \\partial_x+1+\\frac{1}{x}\\end{array}\\right)\n \\left(\\begin{array}{c}L_1\\\\L_2\\end{array}\\right).\n", "0566e9701b441428077c015ebab72b10": "E(X)=X^q -\\gamma", "0566ebb077e0d89398d4b183b9ffbfe4": "-{1 \\over 4a}((x+c)^2 + y^2 - 4a^2 - (x-c)^2 - y^2) = \\sqrt{(x-c)^2+y^2}", "05676fd044e1b6537d129a1ce35221ac": "\\frac{\\partial u}{\\partial x},\\frac{\\partial u}{\\partial y},\\frac{\\partial v}{\\partial x},\\frac{\\partial v}{\\partial y}", "0567bc11782096059ff91f3b6ecbfe19": "\\, k_n", "0567c4efa9b4404acc969cc3305f88e2": " \\exp(\\psi(x+\\tfrac{1}{2})) = x + \\frac{1}{4!\\cdot x} - \\frac{37}{8\\cdot6!\\cdot x^3} + \\frac{10313}{72\\cdot8!\\cdot x^5} - \\frac{5509121}{384\\cdot10!\\cdot x^7} + O\\left(\\frac{1}{x^9}\\right)\\quad\\mbox{for } x>1\n", "0567ec7054caa1f3022e6ffcbf0f32e3": "(x^3 + x) + (x + 1) = x^3 + 2x + 1 \\equiv x^3 + 1 \\pmod 2", "05680cf08e27cac3ec72e1bf4d4a939e": "a_n \\,\\!", "056830395974567389aa73b5b8e3c465": "b_r / a_{cr}\\,", "05690502ce6f2f155c061072882033a8": "\\{p: f(x) \\neq 0 \\in p\\}", "0569f6de84b11c3e31f8acfd25b439b6": "z \\cdot y", "056a03fd62348998d916bb11cc2be318": "\\mathcal{L}_Y(S\\otimes T)=(\\mathcal{L}_YS)\\otimes T+S\\otimes (\\mathcal{L}_YT).", "056a4fa84dbb17f1133a0fe6af2e2e79": "M_{\\psi}", "056a64d987f4bea4c72ed4877813caf3": "\\bar{f}g", "056a69254949cc31f6cce2b2a84673cf": " A(t) ", "056aa22a39082777d9a918b5e5f781e3": "x_1^2+x_2^2+\\cdots+x_k^2-x_{k+1}^2-\\cdots-x_{k+l}^2,", "056af43822bf2a1b53146e86a0b99a87": "c^{T} x", "056b0564f5f92a6777295b9f1aad72b5": "\\Delta S_m = -k[\\,N_1\\ln\\phi_1 + N_2\\ln\\phi_2\\,]\\,", "056bd278b08b43f49b1036042801de3e": "\\delta = \\left( \\frac{2 \\pi}{\\lambda} \\right) 2 n \\ell \\cos\\theta. ", "056bffe5543d1ee0ce2bc4be836cc566": "A f(x) = r x f'(x) + \\frac1{2} \\alpha^{2} x^{2} f''(x).", "056c0bacc33c7706434191da1d12a4d5": "\\text{left} = 2i", "056c1ebee11842df114fbc54c6c9081f": "m'+\\frac{l^2}{2}", "056c28d5e04ebb0a184ec46f4218dbc6": "dp=-\\rho\\, d\\phi", "056c2ff05baecaa2d9bc281911e67be5": "k \\to \\mathit{gl}_n", "056c3719d885b88534067656768bba41": "A_t = \\{ x \\in \\Omega ~:~ \\rho(A,x) \\le t \\}", "056c6ce531c45bf819f4c2409c94fec0": "\\sum_{k=-\\infty}^{-1} a_k (z-c)^k.", "056cc60fc03db3fd4826b5d6bf8c2a90": "\\langle j||T^k||j'\\rangle", "056d7c9223e14763ef161f68f7a378f1": "f''(x) = \\frac{4}{9}x^{-\\frac{2}{3}} \\!", "056d87295ca84b3e47d233385a121a44": "I({\\mathbf{v}^K})", "056e099b0d247d31a9d840df6faa31f2": "\\frac{T_2}{T_1} =\n \\frac{p_2}{p_1}\\frac{\\rho_1}{\\rho_2}.", "056e4dede838adb3f029756e8b1d4d19": "E = \\int \\vec{F}\\cdot \\vec{dx}", "056ea57ffb8d615466b22c21ec1ec3e9": " \\mathbf{\\bar f} ", "056eb396f5d970c10a1179f85ccad787": "p^f-1", "056ec6e1e7047facb5a711ddc022dd52": "V(S,T)", "056ed43842b510bfef52c7fca7065818": "\\Psi\\;", "056f7e72d793d391b4f94f277da1d068": "\\mathbf{rank}_q", "056fc1a23d9d948fdc2bacf0369c7647": "a_i \\leq b_i", "056fcc85dc2922d5f85c85479988c69d": "\n\\dot{x}=f(x,u), \\quad x(0)=x_0, \\quad u(t) \\in \\mathcal{U}, \\quad t \\in\n[0,T]\n", "056fe0c9c2dcef04b1d833a805918990": "\\omega+\\Omega", "05701db28cca4ac8cf3bb0028784d4a9": "K[T]/(T-1) \\oplus K[T]/(T-1)", "05707f83c6ef547df16bbeae25c9c227": "dx=\\dot { x } dt", "0570a40ecae288f0da3cac967eabfc89": "\\alpha=m \\omega/\\hbar", "0570ed6fb37085a43bc2eada9939c757": "-log_{10}[H^+]_i = b_0 - b_1E_{i^{ }}", "0571057a349615a6d0c7d0eddba6244e": "\n(x,t) \\mapsto (\\epsilon x, \\epsilon t), \\qquad \\epsilon \\to 0.\n", "0571263d18a78ee05fa0bc29cc854b09": "E/n", "05713aa7c6790e4bcf7207ef58e05c91": "\\forall x_1\\dots\\forall x_n(R(x_1,\\dots,x_n)\\leftrightarrow\\phi(x_1,\\dots,x_n))", "0571754f2edf474b173a58110b284e1c": "z = w ", "0571b600ca602cea19fc3dc53d61de9f": " \\int_{-\\infty}^\\infty |f(x)|^2 \\, dx < \\infty, ", "0571fd912bb6c5ca4f7fb043722a808e": "\\dfrac{d}{dx}(u\\cdot v \\cdot w)=\\dfrac{du}{dx} \\cdot v \\cdot w + u \\cdot \\dfrac{dv}{dx} \\cdot w + u\\cdot v\\cdot \\dfrac{dw}{dx}", "0572b30c7c1461bdae9f31e98964ad41": "H^i(K,A)\\times H^{2-i}(K,A^\\prime)\\rightarrow H^2(K,\\mu)=\\mathbf{Q}/\\mathbf{Z}", "0573242c1b0fb2514cab35af5eafc629": " (\\mu^{-1})^*(q)", "05736af293901a39c6de0ddc3e82bc65": " \\tfrac{N(N-1)}{2} ", "05738dc77a464fa1c03491f72dc18291": "\\,\\frac{\\hbar}{2} |c+\\rangle = S_c |c+\\rangle = \\mbox{D}(y, t) S_b \\mbox{D}^{-1}(y, t) |c+\\rangle \\Rightarrow", "0573998e30bb1b067df261bb84e7eaab": "0<=K<=L", "0573a69296711ddc741820ffb78d9b1b": "X\\otimes B_i=X\\setminus (X\\odot B_i)", "0573c586c2ab52ee222cb359c4fec2be": "2 \\pi r = \\pi d", "0573e756682afb04864c599b3d72534a": "\\|x\\|_p = \\left( \\sum_{i=1}^n |x_i|^p \\right) ^{1/p}, ", "0574a27738923dd052ed0b873c176afc": "0.03", "0574bd365c0dd9e5387b993473af7980": "(2t)^{2n}", "0574daa93b94cb4c103ee36aa8b63570": "R_{sd,X}", "05751a6b7a52ceea27491cb8bf2c03ce": "f(p)=p^2", "057527f0300dd9eb9cb5ef4ac291aaac": " \\omega_1=-0.201, \\omega_{2/3}=-0.223 \\pm i 62.768", "057533f317f61ef1df78c0b2dceb5a3a": "\\Rightarrow_{A \\to a}\\ aAAA \\ \\Rightarrow_{A \\to a}\\ aaAA \\ \\Rightarrow_{A \\to a}\\ aaaA \\ \\Rightarrow_{A \\to a}\\ aaaa", "057570839734aa21edc27381769b7236": "\\delta_Y", "0575751be4544d418593d2c63585b1df": "\\pi : (x, v) \\mapsto x,", "05758ba4e5a9443110f6d3250672a985": "\\int x^n\\cos ax\\;\\mathrm{d}x = \\frac{x^n\\sin ax}{a} - \\frac{n}{a}\\int x^{n-1}\\sin ax\\;\\mathrm{d}x\\,= \\sum_{k=0}^{2k+1\\leq n} (-1)^{k} \\frac{x^{n-2k-1}}{a^{2+2k}}\\frac{n!}{(n-2k-1)!} \\cos ax +\\sum_{k=0}^{2k\\leq n}(-1)^{k} \\frac{x^{n-2k}}{a^{1+2k}}\\frac{n!}{(n-2k)!} \\sin ax \\!", "0575ab58409f9aac03cedd5b6338ac3a": "\\mathrm{ADC}(x,y,z)= \\ln [S_2(x,y,z)/S_1(x,y,z)]/(b_1-b_2)", "0575e80830acab1e929cf5c964e0d546": "[n:=n+1]\\,\\!", "05762f5f873ec78b0108f4864bbfd457": "b+ \\lambda b + \\lambda^2 b + ... = b/(1-\\lambda).", "0576553202f580240b1cf104dc47b948": "\ns=\\sqrt{\\ln(1/R^2)} = \\sigma\n", "0576594e182762841595fc8a2491371f": "I_k \\subset I", "0576708a3bf3b6c024636403d7bcc3ef": "x_j \\geq 0", "0576908980c395c2024cbdfd1aafe578": " \\operatorname{cov}(\\mathbf{X}_1 + \\mathbf{X}_2,\\mathbf{Y}) = \\operatorname{cov}(\\mathbf{X}_1,\\mathbf{Y}) + \\operatorname{cov}(\\mathbf{X}_2, \\mathbf{Y})", "05769dcb970800b24eca2cc69b516db5": "\\phi_2(x,z,t) = A e^{kz} \\cos(kx - \\omega t)", "0576c789af7cc797743f3f7cbad5fb80": "(i,j,k)", "05774954cef3a0e293515b97e89be98d": "\\mathrm{tr}(\\varepsilon)", "0577852e37185c6cd0c4ac6777f14a91": "\n\\begin{align}\n& \\int_{\\theta_j} \\frac{\\Gamma\\bigl(\\sum_{i=1}^K \\alpha_i \\bigr)}{\\prod_{i=1}^K \\Gamma(\\alpha_i)} \\prod_{i=1}^K \\theta_{j,i}^{\\alpha_i - 1} \\prod_{i=1}^K \\theta_{j,i}^{n_{j,(\\cdot)}^i} \\, d\\theta_j \\\\\n = & \\int_{\\theta_j} \\frac{\\Gamma\\bigl(\\sum_{i=1}^K \\alpha_i \\bigr)}{\\prod_{i=1}^K \\Gamma(\\alpha_i)} \\prod_{i=1}^K \\theta_{j,i}^{n_{j,(\\cdot)}^i+\\alpha_i - 1} \\, d\\theta_j.\n\\end{align}\n", "057796ea520ace98e007953a19207084": "N_\\text{pop}", "0577c77434821f9f34888a3b9db7a197": "D_{E}/N_{E}", "0577d9f31339603fc68203e27839154c": "\\displaystyle -\\frac{\\sqrt{\\pi / 2}}{\\left| \\omega \\right|} - \\sqrt{2 \\pi} \\gamma \\delta \\left( \\omega \\right) ", "05781736e9c5c16927ec2d12d93f3ed9": "D_X(fY) = X[f]Y + f D_XY, \\qquad \\qquad \\qquad f\\in C^\\infty(M)", "057823af195fce5ca941d996f080e228": "\\frac{4}{3}\\pi\\rho\\left(\\frac{c}{H}\\right)^3", "05782e0451ecf804ff449f8842dbd711": "P_{\\text{ph}} = P_{i} - P_{f}", "0578369344ae0c63685d01cd24cf9e75": "d({\\rm tr}(\\mathbf{X})) =", "05784229b1d380c22ed5bef087564b0f": "x' = - \\log(x) \\in \\mathbb{R}", "057847a3ccfac155db00ca47aa3a8edc": " o ", "0578ba0070874ce131a37d4bf39876ee": "\\{\\phi_n\\}_{n = 0,\\ldots,N}", "0578eb4988c85891fd365ff71c1e66d5": "x_0, x_1", "0578f7bdf6a5a97560ddef0fc8df79da": "\\scriptstyle 0 \\,\\leq\\, k \\,<\\, \\nu_j", "05791934f40a51c096001c8b416d99ee": "m, n", "05792b009c1c76032e4f0d74fc039add": "\\tfrac{n (n-3)}{2}", "057932a6583d43823847b32fbaf5b141": " { \\mathit l^{*} } ", "05794b318d1f4b3639ecf61a6a2f2b90": "\n\\sigma_1^2 = \\sigma_2^2 = \\sigma_3^2 = -i\\sigma_1 \\sigma_2 \\sigma_3 = \\begin{pmatrix} 1&0\\\\0&1\\end{pmatrix} = I", "057973b13e59ab4b88cdff34367b443e": "f'(x) > 0", "057983feab1f42353292fb9bca66f887": "|A \\times B|", "05799aff7960fb4b181ae7028f5574e8": "S(a)M = \\{s(x) | s \\in S(a), x\\in M\\}", "0579e8a56c2c3d0afbdd9af1056865ea": "d(\\lambda)\\delta(e^X)\\Phi_\\lambda(e^X)=\\sum_{\\sigma\\in W} {\\rm sign}(\\sigma) e^{i\\lambda(X)},", "0579fcddb7c1f2aa56be97d20a3a5627": "\nW=\\bigoplus_{i=1}^n x_i V.\n", "057a08003c9b7434a4f4215c423c551e": "p(x) = x^3+6x^2+5x+1", "057a0e33b12f5c7967e15f5832e3385f": "A(U_n)", "057a45f8f29fa8af85ce222327568947": "w_T\\ ", "057a4bc42b0cf828b8296e636cb6a7a4": "U(x,t)+iV(x,t) = \\sqrt \\frac{\\pi}{4t} e^{z^2} \\text{erfc}(z) = \\sqrt \\frac{\\pi}{4t} w(iz)", "057a87890570cca5bd5cef01e20e6ce7": "[x]_1", "057a9b1d97bd44392a456f60a5cbde33": "\\mbox{Vert}_pP \\subset T_pP", "057ab4d73fa0d1a004c4446be1dbd9a1": "c=5^2=25", "057afa46cd59e2df99088a5324cab268": "{{V}_{DS}}", "057b22ee5b69f16d58e5fbbec5bea5ef": " \\cong", "057b88949f199e7e691cbe9ec91c6846": " -\\frac{1+\\xi^2}{2} \\, \\partial_\\xi. ", "057b9e161e98f9412b90e36ef4d481c3": "e = \\frac{a}{d}.", "057ba1a33b85d2a85f1a6270f7910103": "(b_s)_{s\\ge 0}", "057ba3b651a36bc6493c706e135d4ce9": "S_e", "057c11c3e16e3b4182b3d3675dff0386": "\\bar L_n W(z)=0.\\,", "057c438d7d6cce182f0416037a19c28d": " \\mathrm{li}(x)\\;=\\;\\mathrm{li}(x) - \\mathrm{li}(\\mu) ", "057c60af4d800c9e42ae59d4ed84671a": "(a+bi) + (c+di) = (a+c) + (b+d)i.\\ ", "057c85faf8ce31f4f57bd127c79373a8": "\\phi^{-}(a)=\\frac{1}{n-1}\\displaystyle\\sum_{x \\in A}\\pi(x,a)", "057d3e9c71af89337100f6ccd17652b2": "F(x) = \\frac{\\Delta\\,t(i)}{f_s(i)}", "057d7fa74c18228541ece69706f4164f": "\\textstyle h(z)", "057d8c542564872299e0cb0e69aa903f": "\\int\\limits_A \\, n(\\rho u\\phi)\\,dA = \\int\\limits_A \\,n(\\Gamma\\nabla\\phi)+\\int\\limits_{CV}\\,S_\\phi \\,dV", "057de9905acef9693d8927400102d9a4": "j \\neq k \\in [n] ", "057dff4a8daa4545ffc37758c9e8704b": "\\scriptstyle{\\|\\hat{u}\\|_{L^2} = \\|u\\|_{L^2}}", "057e5d99cb244ece8533e316322ba604": "(\\mathcal{L}f)(s) = E\\left[e^{-sX} \\right] \\, ", "057e7a19450e9501183720d33f1b7532": " \\ \\psi_o (\\phi) ", "057eb8b5a5594748fb4a27c6e06ab83a": " 8\\pi^2/105 \\approx 75.2 \\% ", "057ec7fb57573ce682b9938f7dd4bb51": "\\coprod_{X \\in K}{F(X)}", "057f67a43202131848df57b01e4adb2e": "\\Omega = 2 \\pi \\left (1 - \\cos {\\theta} \\right) ", "057f761d37d0308db1e5c5cea71ad24d": " \\begin{bmatrix} V_1 \\\\ V_2 \\end{bmatrix} = \\begin{bmatrix} z_{11} & z_{12} \\\\ z_{21} & z_{22} \\end{bmatrix} \\begin{bmatrix} I_1 \\\\ I_2 \\end{bmatrix} ", "057f89ba663e2b980408c1b4b4cd15c6": "H_{n,m}= H_n^{(m)} = H_m(n).", "057f955857779bb29bd41289dc134374": "\\int_{0}^{1} \\dot{h}_{s} \\cdot \\mathrm{d} x_{s}.", "057fac8904dcbcb9c4e74fc827e80405": "\nS^m_\\ell(x,y,z) = \\left[\\frac{2 (\\ell-m)!}{(\\ell+m)!}\\right]^{1/2} \\Pi^m_{\\ell}(z)\\;B_m(x,y)\n,\\qquad m=1,2,\\ldots,\\ell.\n", "057fec8201937ea7950f7ab6bba5c451": "a \\in U", "057ff7e49c26eaac3acb319b7599dc17": "e_{q}", "05800c2d629fb6726bd3fd05ef8af782": " F_{hkl} = \\sum_{h'k'l'} F_{h'k'l'}F_{h-h',k-k',l-l'} ", "05806b61f30e53a7aa298c5df7e94b19": "_{k+1}V^i_3(x,y)=_kV^r_1(x,y+1)", "0580caa35cb38096c461dc12b333d6da": "\\frac{3}{8}", "0581045b961280329795c8c6a45486b8": "x = f(y) .", "058110797fc814035a19dc84b41ee35f": "\\operatorname{Res}(f,c) = \\frac{g(c)}{h'(c)}.", "058123e87a3a29e28ccc28b96cfbe22c": " \\ell = \\pi \\cdot 2r ", "05813e47f2e6afecad7a27b9b92aedba": " \\sqrt{4\\pi} \\left(\\mathbf{m}, \\mathbf{M}\\right) ", "058172c7f435e28a55decbd97d50a94d": "|\\uparrow_z \\rangle", "0581bf9a4c1c4efe58eb16db28d55ace": "x\\in V(S)", "0582203dbad92451e1ec7a7cbfc1d3e5": "f \\in C^{k+1} (I)", "058310a90451f6f468eed91004066cdb": "D(p||q)\\geq 0", "058316969c3fa24ba9247ba1117d33f1": "= \\frac{600!}{2} \\cdot \\frac{1200!}{2} \\cdot \\frac{720!}{2} \\cdot \\frac{2^{720}}{2} \\cdot \\frac{6^{1200}}{2} \\cdot \\frac{12^{600}}{3}", "05836f96b679b8bd7cdf135bf8242658": "RAC h.p. = (D^2 * n)/2.5 \\,", "05842111e00efaae45238f50d6f79b46": "\\lambda(x,y,z) \\equiv x^2 + y^2 + z^2 - 2xy - 2yz - 2zx", "05842c3d39e2cc3218963be659fd058e": "\\frac{\\theta}{\\theta_b}=e^{-mx}", "05844333cebabb90adb0b1ff0466149e": " \\hat{E} = i\\hbar\\frac{\\partial }{\\partial t} \\,\\!", "05844c6d990659e658f08b35c8afe3b1": " \\operatorname{perm}(A)=\\sum_{\\sigma\\in S_n}\\prod_{i=1}^n a_{i,\\sigma(i)}.", "05845e95a493130bfd283f00883865e6": " \\beta", "05848330faa279f4cb0071cb153a3534": "p(x) = \\frac{\\beta^\\alpha}{\\Gamma(\\alpha)} x^{\\alpha-1}e^{-\\beta x}.", "0584b20d5625be70127643626f43cb71": "f=\\frac{ab}{c}.", "0584ed27d4e794dad1db3f81e7bbbea8": "k^{-s} F(s;kq) = \\sum_{n=0}^{k-1} F\\left(s,q+\\frac{n}{k}\\right).", "058550c85a503e65a1b89ee16888fec4": "(\\varepsilon,\\eta):F\\dashv G", "05857d9d93a6f74ec43cfe51ec11acc6": "p_1 \\equiv \\frac{\\partial}{\\partial q_1} L_d\\left( t_0, t_1, q_0, q_1 \\right)", "058580224fb05a175bdb6d8ddf62a94c": "(a + c) \\mid b", "058611c3621fe41d898d1d9b12e1feb6": "q = \\left\\lfloor {n_1} / {n_0} \\right\\rfloor", "05863ab8b1604fb2e47ac4df8d1bb7dc": "g_y(\\mathbf{y}) \\triangleq \\begin{bmatrix} \\mathbf{0}\\\\ 1 \\end{bmatrix},\\,", "05866caf91be86c7599a6120cfdb5d70": "\\begin{pmatrix} (mc^2 - E + e \\phi) & c\\sigma\\cdot \\left(p - \\frac{e}{c}A\\right) \\\\ -c\\sigma\\cdot \\left(p - \\frac{e}{c}A\\right) & \\left(mc^2 + E - e \\phi\\right) \\end{pmatrix} \\begin{pmatrix} \\psi_+ \\\\ \\psi_- \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 0 \\end{pmatrix}. ", "0586c47757aa46b672cd02d11528b548": "j(\\tau) = N", "05879c918710403f506c5c45a540ed62": "\\scriptstyle \\epsilon/m_{0}\\sim1.76\\times10^{7}", "0587b2d9f6b659f62a0a82a2936f1048": " V \\ne W \\to \\operatorname{let-combine}[\\operatorname{let} V : E \\operatorname{in} \\operatorname{let} W : F \\operatorname{in} G] \\equiv \\operatorname{let} V, W : E \\and F \\operatorname{in} G ", "0587d95a881a0784ec095a16d0720b54": "g(X)=\\frac{dF_1(X)}{dX}", "058801a9a81de4377e2ef6959d0d3d89": "[N_i,P_0]=iP_i\\left(1-\\frac{P_0}{\\eta}\\right)", "058838738338afba3b46fded336a157d": "~V=\\frac{(\\sigma_{\\rm ap}+\\sigma_{\\rm ep})\\sigma_{\\rm as}}{D}~", "058849d454ab21f6fe48b4672fe81b81": "\\displaystyle{Q(a,b)a^{-1} =b.}", "0588607f74debea92ac58e9beda8c0ef": "u \\cdot u_{n}", "0588a554f880dc083ee41f783c9c2cae": "{n^{O(1/\\varepsilon^2)}}", "058912ca192d028678452b1ff1e895df": "x^md(x)", "0589446d07423f1cf786dc2db08901a7": " \\Bigl\\| \\sum_{k=0}^n \\varepsilon_k \\alpha_k b_k \\Bigr\\|_V \\le C \\Bigl\\| \\sum_{k=0}^n \\alpha_k b_k \\Bigr\\|_V ", "0589634ff96a29b5b6027675aa45f6f4": "J_+|j\\,m\\rangle = \\alpha|j\\,m+1\\rangle,\\quad", "05898dbef5eb0036dac5efc7dbc574f1": "P(x) = \\frac{-2}{x}", "0589a243a3b7f31a686ca9321aa64b64": "(|\\text{dead}\\rangle + |\\text{alive}\\rangle)/\\sqrt 2", "0589fe63259c31bc8394d0f1dbfa49b7": " S_i ", "058a0159f6993abb9800a0876f570c53": "L(s,\\pi,r_i)", "058a113e25e870d4154580c91d6ac1c3": "16C,\\;16D,\\;32A,\\;32B,\\;32C,\\;32D,\\;34A,\\;46A,\\;46B\\;", "058a1ce7c2f092541fbafe263690e611": "= \\sgn( \\sin (\\theta+ \\frac{\\pi}{2})) \\frac{\\sqrt{1 - \\sin^2 \\theta}}{\\sin \\theta}", "058a32571bab72f7af24319b1f57d425": "\\tilde \\nu", "058a46442473533fe9c79b81850d8de6": "\\,_2F_1(a,b;c-1;z)-\\,_2F_1(a+1,b;c;z) = \\frac{(a-c+1)bz}{c(c-1)}\\,_2F_1(a+1,b+1;c+1;z)", "058a682eda2aa0b1b205129e1e36c535": "E'", "058a98728120f8e485502ff4c60835c1": "\\,t\\,", "058ad826b638e617036c8e3545c7242f": "I(p_{t_m},p_{t_n},q_{t_m},q_{t_n}) \\le I(p_{t_m},p_{t_r},q_{t_m},q_{t_r})~~\\Leftarrow~~p_{t_n} \\le p_{t_r}", "058af3bca72462c2ad6d47dbfa36aa38": "J_n = -\\frac{\\cos{ax}}{(n-1)x^{n-1}}-\\frac{a}{n-1}\\left [-\\frac{\\sin{ax}}{(n-1)x^{n-1}}+\\frac{a}{n-1}J_{n-2} \\right ]\\,\\!", "058b28fec217060370f1f651de40658b": " P = {2 \\over 3} \\frac{q^2 a^2}{ c^3} \\mbox{ (cgs units)} ", "058b2ca50228deb144d988a9561c1d18": "\\mathrm{R{^{\\cdot}} + O_2 \\ \\xrightarrow {fast} \\ ROO{^{\\cdot}}}", "058b66f546bbae3fb29a7e8259a42364": "\\Delta(a)", "058b77bb9451583a056115e5c62f2dff": "\\exp_{10}^3(2.18726)", "058bb7c4b3cb9d8b1a7bbc860efda23a": "A =\n\\begin{bmatrix}\n 5 & 4 & 2 & 1 \\\\\n 0 & 1 & -1 & -1 \\\\\n-1 & -1 & 3 & 0 \\\\ \n 1 & 1 & -1 & 2\n\\end{bmatrix}", "058bba9129ccab88e489d2730febaa0d": "\n\\left(\\frac{\\partial U}{\\partial T}\\right)_V\n= T\\left(\\frac{\\partial S}{\\partial T}\\right)_V\n- p\\left(\\frac{\\partial V}{\\partial T}\\right)_V ; C_V = \\left(\\frac{\\partial U}{\\partial T}\\right)_V\n", "058c0dbac1a605db3a931a3ad1e62048": "kx-\\omega t = \\left(\\frac{2\\pi}{\\lambda}\\right)(x - vt)", "058c0e5a0bba35fb39f56cb8261396ee": "W(s) = \\sum_{i \\in N} u_i(s),", "058c3cc810faf90fa02b532d44bd93de": "\\Phi(M,x)=n", "058c5b9d6783c45a574f3951275aa144": "\\ell^{(-1)}=\\frac{2}{1-\\alpha}p^{\\frac{1-\\alpha}{2}}=p", "058cae5e7470da022b1a0bd31cf47d57": "(T_h f)(s) = h(s) \\cdot f(s).", "058cbc8415ed139e477dc4d67365153a": "\\sigma_{r} > \\sigma_{f}", "058cde4336e8d993179632cfee6939a4": "\\Omega_\\text{rel} = \\frac {3\\pi G m}{c^2 r}.", "058cdf0a7a1ab69ef3b026214341d476": "\\lim_{t\\rightarrow 0} \\vartheta(x,it)=\\sum_{n=-\\infty}^\\infty \\delta(x-n)", "058ce521659b36dd1c773ed1563dc8a9": "\\quad (A \\cdot B) + (A \\cdot C) = A \\cdot (B + C)", "058cec60659527b3415f49c4d666261a": "p(t)=\\delta (t - \\tau )", "058d38950ec4527f6b9ed00b276195ae": "\\int \\cosh x \\, dx = \\sinh x + C", "058d470bb28dc01348bef8eed55608da": "\\|x\\|_\\infty = \\sup_n |x_n|", "058d7c0d06b525d5cca64ba2414d8579": "=6", "058e043d15d210ad7035a7c62707767c": "\\sum_s P_s = \\frac{1}{Z} \\sum_s \\mathrm{e}^{- \\beta E_s} = \\frac{1}{Z} Z\n= 1. ", "058e20c0187ab310bbfacd83dbe56743": "\\int_{-\\infty}^\\infty H_m(x) H_n(x)\\, \\mathrm{e}^{-x^2}\\, \\mathrm{d}x = \\sqrt{ \\pi} 2^n n! \\delta_{nm}", "058e46087c15f585f9dcec23ceeb8248": " \\left( \\left|x\\right|^{r} + \\left|y\\right|^{r} \\right)^{t/r} + \\left|z\\right|^{t} \\leq 1", "058e5842df9a74c8c65bc58d03e6dfad": "\\sin \\alpha \\cos \\beta = {\\sin(\\alpha - \\beta) \\over 2} + {\\sin(\\alpha + \\beta) \\over 2} \\approx {\\alpha - \\beta \\over 2} + {\\sin(\\alpha + \\beta) \\over 2} ", "058e80a4ab77f55be256e18ba64707c2": "G(s)=K_d \\frac{s^2 + \\frac{K_p}{K_d}s + \\frac{K_i}{K_d}}{s}", "058e8d0e4e13e2ad6f046c0048d08676": "\\langle\\overline{z}\\rangle=e^{i\\mu-\\sigma^2/2}. \\,", "058ea4b0b13ae669b827d4002475a648": "Q_0=m_0 s_b L_{sludge0} ", "058eaefcf0f2e16da2c9741e9cc8f340": "k_{2(3)}\\equiv k_{2(2)}", "058ed80904627b8193cd1fbfd75b502c": "\ne_i^{t+n} - e_i^t = NS_i + IM_i + RS_i + AL_i\n", "058edad9bc884a06ddd1a0290f5d61b1": "\n\\begin{bmatrix}\n Y_1 \\\\\n Y_2 \\\\\n Y_3\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n \\cos\\theta & -\\sin\\theta & 0 \\\\\n \\sin\\theta & \\cos\\theta & 0 \\\\\n 0 & 0 & 1\n\\end{bmatrix}\n\\begin{bmatrix}\n X_1 \\\\\n X_2 \\\\\n X_3\n\\end{bmatrix}\n", "058ee734a301721e209bac7eeae3eeaa": "k \\mod q \\ne 0", "058efe614790dbb05f0134a9ba1f229b": "\\frac{P(R) - P_\\infty}{\\rho_L} = \\frac{P_B - P_\\infty}{\\rho_L} - \\frac{4\\mu_L}{\\rho_LR}\\frac{dR}{dt} - \\frac{2S}{\\rho_LR} = R\\frac{d^2R}{dt^2} + \\frac{3}{2}\\left(\\frac{dR}{dt}\\right)^2", "058f18c4de16ad5e4e948d5a35a5a371": "T' = \n\\begin{bmatrix}\n0 & 0\\\\ T & 0\n\\end{bmatrix}\n\n\\quad \\mbox{and} \\quad\n\nN' = \n\\begin{bmatrix}\nN & 0 \\\\ 0 & M\n\\end{bmatrix}.", "058f1c58404718bfef62fc8469b2451f": "E^\\mathrm{damping}(\\mathbf{x}_j,t)=\\frac{E_j^\\mathrm{ret}(\\mathbf{x}_j,t)-E_j^\\mathrm{adv}(\\mathbf{x}_j,t)}{2}", "058f758c4d6146daa6ee6006adf74bc7": "\\scriptstyle {\\frac{1}{\\sqrt{12}}}\\mathrm{LSB}\\ \\approx\\ 0.289\\,\\mathrm{LSB}", "058f7763c0d8276113bd3071bc73718d": "\\omega(z) = W_{\\big \\lceil \\frac{\\mathrm{Im}(z) - \\pi}{2 \\pi} \\big \\rceil}(e^z).", "058fa67dd1928085ab61e9d09f691a8d": "\n(x+1)(x-1) = 1\\,\n", "058fa75172f003a02c43e23043dc41f7": "O(N^{1.5})", "05903eeb43b8650a76a00736fd97466e": "D_F = k_2 \\cdot\\frac{\\lambda}{{NA}^2}", "0590580c311dff2d7a5d79e10c912e16": "\\displaystyle c_f", "059081225fb1e2be67bb10d0071e1c9d": "\\ \\displaystyle \\{S(d): d\\in D\\}\\ ", "0591382720b9f82853663a2214536734": "H(n,q^2)", "059167c366dbebe947e02a226e082451": "{h_1} + \\frac{V_1^2}{2} = {h_2} + \\frac{V_2^2}{2}", "05919836f7bb584b48725d3190ee2133": "c'=c\\pm kv\\,", "0591e21a312c6022ef7bed37de8def05": "\\Phi_{ij}\\mapsto-\\Phi_{ij}", "0592074890fe569d7e99a0621d8934d2": "\\omega_M, \\omega_N", "059212990734c096c9ecbcbdb51b37d0": "\\rho = \\int^T_0 k(t)S(t)dt = \\int^T_0 S(t)^2 dt = E", "05928b13c02c0dddd7ab38de5a50cdad": "b, c", "0593076a0a8e42ddd486e980d8a7378a": "N(\\mu,1/n),", "05934a6dfdc9db4870b16992573199fa": "u(x) \\lneqq \\max_{y \\in \\partial \\Omega} u(y)", "05935d88c1621e854f158d005cfbbbf1": "\\Omega \\setminus c", "05939125bc21745ade8be1ac850190db": "p_{\\mathrm{c}}=P(\\mathrm{SINR}>t)=1-p_{\\mathrm{out}}", "0593c8f84588c06cde68d0fc8b2a3de3": "(X_n)", "0593ceb5c70d6a8078b25691ac6de147": "(\\widetilde{s}^1, \\dots, \\widetilde{s}^T, \\widetilde{o}^1, \\dots, \\widetilde{o}^T) ", "059400fcdb7b6dbd68f163355e81db6c": "|N - Z|", "0594251298a83049cd9c21646f652c51": "\n\\text{minimize} \\quad \\text{over } \\widehat D \\quad \n\\operatorname{vec}^{\\top}(D - \\widehat D) W \\operatorname{vec}(D - \\widehat D)\n\\quad\\text{subject to}\\quad \\operatorname{rank}(\\widehat D) \\leq r,\n", "05946cf357fed2266088f9437991cf89": "N_{A(i)} = 0\\,", "0594bea4ff32f27fe0c7914658e06984": "\\psi_{2n}/y", "059559a9bd812fc270276feab50a8052": " \\frac{1}{2}[(\\kappa+1) \\theta~\\sin\\theta - \\{1 - (\\kappa-1) \\ln r\\} ~\\cos\\theta] \\,", "0595815d64ecffd529fbc3f684e64c73": " \\Delta G_{\\rm em} ", "0595f7d34ddda3f9763cadecbd9f6547": "\\Gamma^I", "059619a3cebb398372a64b4c73580ada": "m I= \\int_a^b m\\varphi(t)\\,dt \\le \\int^b_aG(t)\\varphi(t) \\, dt \\le \\int_a^b M\\varphi(t)\\,dt = M I,", "05961be985b19697bc6b124e52c48a5a": "I_{n_2, k_2}", "05962269fc7570a5e4b24c53a979a2d6": "Q = \\mathrm{Ran}(A - \\lambda I) \\cap \\mathrm{Ker}(A - \\lambda I) \\neq \\{0\\},", "0596302fde1dc2cc0678f7805b205a15": "\\bar\\psi\\equiv\\psi^\\dagger\\gamma^0", "059663f6390660fe913cf64da7cff186": "u(\\vec{p}, 1) = \\sqrt{E+m} \\begin{bmatrix}\n1\\\\\n0\\\\\n\\frac{p_3}{E+m} \\\\\n\\frac{p_1 + i p_2}{E+m}\n\\end{bmatrix} \\quad \\mathrm{and} \\quad\nu(\\vec{p}, 2) = \\sqrt{E+m} \\begin{bmatrix}\n0\\\\\n1\\\\\n\\frac{p_1 - i p_2}{E+m} \\\\\n\\frac{-p_3}{E+m} \n\\end{bmatrix} ", "059665a79da90a9b27772d692d991814": " \\Delta_1 = 1 \\, ", "0596a6b7d0432b9dcb676aff1041de16": "\\mathrm{ARFCN}=\\frac{f - 300 - 0,0125}{0,025}", "0596bc73eb25f56d08b9afec41e693ae": "\\hat\\theta=\\theta^{(M+1)}", "0596bd1d976b95c82dc2e4113845ffc9": "(2k+1)", "0597078cef136f8ee64dc05937bcc759": " \\Delta = \\det(M) = \\det\\left(\\begin{bmatrix}A_1 & B_1 & B_2\\\\B_1 & A_2 & B_3\\\\B_2&B_3&A_3\\end{bmatrix}\\right) ", "0597920f4622a27a11ef2cf6e0e4a737": "\\rho_0 = F \\cot^{n} (\\frac14 \\pi + \\frac12 \\phi_0)", "0597e2cdcb0557084c936bb7d92f5815": "(2^{2n} - 1) - 2^{n + 1}", "0597ed548c0a81dfa30ae7d7ac201f31": " \\aleph_{1} ", "059806bda3bd68c6db4b7c5ce378a95e": "L_{X} = \\sum_{i} b_{i} \\frac{\\partial}{\\partial x_{i}} + \\frac1{2} \\sum_{i, j} \\big( \\sigma \\sigma^{\\top} \\big)_{i, j} \\frac{\\partial^{2}}{\\partial x_{i} \\, \\partial x_{j}}.", "059820265ec22520c46c7f6f1a6e9e49": "\nA = L_{1}^{-1} L_{1} A^{(0)}\n= L_{1}^{-1} A^{(1)} = L_{1}^{-1} L_{2}^{-1} L_{2} A^{(1)} = \nL_{1}^{-1}L_{2}^{-1} A^{(2)} =\\ldots = L_{1}^{-1} \\ldots L_{N-1}^{-1} A^{(N-1)}.\n", "059899c64f7db0368e50acdb6a707233": " \\mathbf{\\hat T^ \\dagger} (\\varepsilon) \\mathbf{\\hat H} \\mathbf{\\hat T}(\\varepsilon) = \\mathbf{\\hat H} ", "05989d00ccacfb4642b8160a44bc64d9": "\\frac{1}{[A]^{n-1}} = \\frac{1}{{[A]_0}^{n-1}} + (n-1)kt", "0598cbccdd968ba76913621e4f5088b5": "A \\neq B", "05995ce08147baeb5fe26791807d7a84": "\nk_2 \\approx \\frac{1.5}{1+\\frac{19\\mu}{2\\rho g R}},\n", "0599699a20cb859e197a4c564c4c47cd": "xp(x)", "0599b4b872f6708fb7e6347a15a11c95": "u_1=\\mbox{Re}(y_1)=\\tfrac{1}{2} (y_1+y_2) =e^{2x}\\cos(x),", "0599bfc4eb518f0b6f9d122f1c2ff42f": " \n\\Delta m = 0 \\quad\\hbox{and}\\quad \\Delta l = \\pm 1 \n", "0599bfd0ad922eed6e26d767c6ad53e2": "(e, g, e): (A, e) \\rightarrow (A, e)", "059a194fc6285d7e2f2079721ff01fff": "j_g(x)", "059a28895dc6969cc4a70a491c9fdefe": "\\mathcal S = (\\mathcal S^1,\\dots,\\mathcal S^n)", "059a4f4e34c6270dd8579e6185198f2e": "\\gamma(s)=e^{i\\phi}Q^s\n\\prod_{i=1}^k \\Gamma (\\omega_is+\\mu_i)", "059ab98a87e4a57ecf8d4c113c392b7d": "EL(\\Gamma_1)=0", "059af9424ed592bb0476be33188b38f3": "T_\\text{goal} = b \\log_2 \\left( \\frac{A}{W} + 1 \\right)", "059b3b273d826477ff79174fa4e57b02": "y^2=x^3-x,", "059b9b4866ffa54ccf9a56e7517d209e": "x'=ax.\\,", "059bc2d9bc55c7cdbb9e82a0d2023c2b": "|H|= \\sqrt{H_x^2+H_y^2+H_z^2}", "059bebfb939b7c6f14a0c74fc933dea8": "\\text{arcsin} (x) \\approx x", "059c7e548f7ffdb49cfaaba48531baa6": " \\sum_{i=1} \\left(Y_i - g\\left(X'_i \\beta\\right)\\right)^2. ", "059ccf126490e1ffb0d75267846b1ca3": "\\frac{M}{C}", "059cec77641c915e7434c0830ebe5dd9": "1200\\log_2( 3^{1/13} )= 146.3...", "059d13248aa9b3f33a9f03be87389c2d": "P_{TAF}", "059d2e1c5f7e6f4beb099432654f423c": "1\\le q, p < \\infty", "059d8050582684950696d2bf4a0a9c22": "\n\\begin{bmatrix}\n0 & -1 & 0 \\\\\n-1 & 5 & -1 \\\\\n0 & -1 & 0\n\\end{bmatrix}\n", "059dcb7018310a884c8e68f80838958c": "\\,(1 + 9 + 6 + 8 + 3)^3=19{,}683", "059e347bb40b012f97255c18b26df569": "C_{IJK}", "059e75340a274fbea9a34c246670e73f": "z_n", "059e757564c12a958d2ef2d59cfd3bec": "\n\\beta^{0} =\n\\begin{pmatrix}\n0&1&0&0&0\\\\\n1&0&0&0&0\\\\\n0&0&0&0&0\\\\\n0&0&0&0&0\\\\\n0&0&0&0&0\n\\end{pmatrix}\n", "059e7593930844763fce650787af8806": "\n\\mathbf{r} = r(\\hat{ u} \\cos \\theta + \\hat{ v} \\sin \\theta) = r \\hat{ u}(\\cos \\theta + \\hat{ u} \\hat{ v} \\sin \\theta)\n", "059e93c547b1182a9c4aef775da41c5a": "\n f(I_1, J_2, J_3) = 0 \\,\n ", "059fa5813e555d8ad1d205bcd7e7edb1": "\\zeta=+1.", "059fe7b20132b9c261200a7a7bd62966": "k_i = K_i/L_i", "05a03641923964c19f02fab6c874798e": "G(x) = \\sum_{1 \\le n \\le x}F(x/n)\\quad\\mbox{ for all }x\\ge 1", "05a03beaf3b2097c77dcbbabceddbc6a": "\\begin{cases} \\dfrac{\\partial v}{\\partial t}(t, x) = A v (t, x) - q(x) v(t, x), & t > 0, x \\in \\mathbf{R}^{n}; \\\\ v(0, x) = f(x), & x \\in \\mathbf{R}^{n}. \\end{cases}", "05a0f1219d603d2c824bb08383d87c4e": "\\hat{\\mathbf{z}}\\,\\!", "05a10f7b11ab4c4d8367790cf8710ff6": "\\mathbf{E}_{\\mathbf{P}} \\left([Y_t-Y_s]\\chi_F\\right)=0,", "05a14d12d18b07a35c4a3b985bcd8360": "V(S) = \\{x \\in \\mathbb{A}^n \\mid f(x) = 0, \\forall f \\in S\\}", "05a1670a1689d4290c39177e63d72bf0": "\n \\boldsymbol{l} = \\boldsymbol{l}^e + \\boldsymbol{F}^e\\cdot\\boldsymbol{L}^p\\cdot(\\boldsymbol{F}^e)^{-1} \\,.\n ", "05a1b0fe8d72d0636008275c15fd2299": "S(q \\to 0)", "05a1bb8d7daf0ca1440d3671c888141a": " X \\equiv X \\left ( x_1, x_2 \\cdots x_n \\right ) \\,\\!", "05a22200576b61fe2ef5aa3b91e71a2b": "\\phi(x) = \n\\begin{cases} \n1 & \\text{if } x > x_0 \\\\\n0 & \\text{if } x < x_0\n\\end{cases}", "05a23464412cb303e8a83e9594770967": "\n\\begin{align}\nE_1 & = { q_1^2 - p_1 r_1} \\\\\n& = {[(p + q)(q + r)]^2 - (p + q)^2 (q + r)^2 = 0}\n\\end{align}\n", "05a23510438cdd29a1ad83afe9d9f6c2": "(Bxuv \\and Byuz \\and x \\ne u) \\rightarrow \\exists a\\, \\exists b\\,(Bxya \\and Bxzb \\and Bavb).", "05a23c52ac89bc9ebd3c184ca593617c": "\\tilde{d}=\\frac{ncp}{\\sqrt{n}}.", "05a25a16af94aad599d3e12d42ebd8b9": "(\\lambda x.E)y \\equiv E[x:= y]\\,", "05a2631bbe66935c5c813a4fde3b7ab9": "\\Pi(n,m)=\\int_0^{\\pi/2}{\\frac{1}{(1-n \\sin^2 \\theta)\\sqrt{1-m \\sin^2 \\theta }}} d\\theta.", "05a28eddeefb3e7ec9ad3d0d823e3b51": "\\mu:G\\times G\\to G", "05a2adcce0f6ea21fe1d923c764ef033": " \\left \\| \\varphi_{N,x} \\right \\| = \\frac{1}{2 \\pi} \\int_0 ^{2 \\pi} \\left | D_N(x-t) \\right | \\, dt = \\frac{1}{2 \\pi} \\int_0 ^{2 \\pi} \\left | D_N(s) \\right | \\, ds = \\left \\| D_N \\right \\|_{L^1(\\mathbf{T})}.", "05a2bf3560d6f10b0957cd3fba163d77": "\\eta_s", "05a2ec30cdec30ffa6c7a682c010d9a3": "\\langle u_n:n\\in\\mathbb{N} \\rangle", "05a305c8d54caaf41a3043162d35b737": "\\gamma_\\text{LG}\\ ", "05a32025b1c511ac180801de8d64bd1e": "Y_{ij}=1", "05a325ba2832c04eb0a9be1724014c00": "\n\\begin{align}\n -fv &= -\\frac{1}{\\rho_o} \\frac{\\part p}{\\part x}+K_m \\frac{\\part^2 u}{\\part z^2}, \\\\\n fu &= -\\frac{1}{\\rho_o} \\frac{\\part p}{\\part y}+K_m \\frac{\\part^2 v}{\\part z^2}, \\\\\n 0 &= -\\frac{1}{\\rho_o} \\frac{\\part p}{\\part z},\n\\end{align}\n", "05a3898b2b506b889a4b2f5ee0ef71ed": " \\mathrm{Tr}(\\Pi_\\alpha \\Pi_\\beta ) = \\mu^2 \\;", "05a3b69f25b972b48a814ddf8f294836": "\n \\rho_0~\\frac{\\partial\\mathbf{v}}{\\partial t} + \\nabla p = 0 ~.\n ", "05a3dbacc000a46c03ad1d7bf299b5dc": "s_A=\\sum_{i=1}^m u_i a", "05a4c3fe37dd5267d00a2f615e697e44": "f(q)", "05a4eacbd53ee615aa8bce3f4c4b2256": "\\scriptstyle F", "05a502a14e31c468da088859adb01294": " \\left (\\frac{p_2}{p_1} \\right )^\\frac {1}{\\gamma}", "05a50c8d59692e1b21ea86916ad49629": "I(p_{t_m},\\alpha \\cdot p_{t_m},q_{t_m},q_{t_n})=\\alpha \\cdot I(p_{t_m},p_{t_n},q_{t_m},q_{t_n})", "05a52e65d34e8c97528c6325153643e5": "A/A_t", "05a531358c926fc155d2972747886f23": "z^0\\in W", "05a54f7dfbbdffc4e8ed3ecd4d196b54": "\\left\\langle E\\right\\rangle=-\\frac{d\\log\\left(Z\\right)}{d\\beta}= \\frac{\\varepsilon}{2} + \\frac{\\varepsilon}{e^{\\beta\\varepsilon}-1}.", "05a56d47bc7464213948240437d78bb9": "\\cos \\phi_0 = \\frac {\\mathbf{W}} {\\mathbf{V_1} \\mathbf{I_0}} ", "05a576cf1c7ec9ac44b971c70222084d": "P_{\\text{SEN}}", "05a57c17051ccbd74999fe193a96296f": "(M)", "05a5860dee2ffe27c4829192d7b6f653": "\n \\left( {\\frac{\\partial G / T}{\\partial T}} \\right)_P\n =\n \\frac{1}{T}\\left( {\\frac{\\partial G}{\\partial T}} \\right)_P - \\frac{1}{T^{2}}G\n =\n - \\frac{1}{T^{2}}\\left( {G - T\\left({\\frac{\\partial G}{\\partial T}} \\right)_P\n } \\right)\n = - \\frac{H}{T^{2}}\n", "05a5d2f763dd0afef27ee5372b984bef": "v(x) = au(x) + b.", "05a5d998ebc0923def6b40d8ce610619": "\\sqrt[3]{\\left(\\sqrt{2}+ \\sqrt{3}\\right)\\left(5 - \\sqrt{6}\\right) + 3\\left(2\\sqrt{3} + 3\\sqrt{2}\\right)} = \\sqrt{10 - \\frac{13 - 5\\sqrt{6}}{5 + \\sqrt{6}}}. ", "05a5deae321ac8b5499f169726704a3e": "d^*(A)", "05a640d5e5c0b96112adf4645d970cc3": "-\\boldsymbol{ \\nabla \\times}\\left( \\boldsymbol {\\nabla \\times B} \\right ) = \\nabla^2 \\boldsymbol B =\\mu_0 \\epsilon_0 \\frac {\\partial^2}{\\partial t^2} \\boldsymbol {B } = \\frac{1}{c^2} \\frac {\\partial^2}{\\partial t^2} \\boldsymbol {B } \\ , ", "05a69ce951018ee635e7904f95f1b81d": "0 = \\frac{\\mathrm{d}}{\\mathrm{d}t} \\left ( \\frac {\\partial L}{\\partial \\dot{q}_j} \\right ) - \\frac {\\partial L}{\\partial q_j} + \\frac {\\partial D}{\\partial \\dot{q}_j}.", "05a6ab966433656751786a90220bf926": "\\eta_p", "05a6fa309cb2210dce11fff50543885d": "\\bold{D} \\cdot\\bold{\\hat{n}}dS = \\iiint \\rho_f dV ", "05a6fbb6a85505b06737daa61a7804d8": "\\frac{18}{11}", "05a74314523ecaef49f6d096dcfb2db8": "v,w\\in T_pM", "05a755e52977242b6ddc4a0be2586cf0": " \\forall x \\in \\mathbb{R} \\quad x < x+1 ", "05a7925a61f3d86855365871b926a815": "C_{QL} = e^2D_{Gr} = \\frac{e^2m^*}{\\pi\\hbar^2}", "05a7c250a7657563d94fde4d9d389be9": "+0, \\ \\times1, \\uparrow1, \\ \\uparrow\\uparrow1, ", "05a7d68f204dd5981bf9a03c5523d4e3": "\\ 1.39m+2ln(n).", "05a7e0a3c05d0ae38bea6040d5921972": "\\sum_{1\\le k\\le n \\atop \\gcd(k,n)=1}\\!\\!k = \\frac{1}{2}n\\varphi(n)", "05a8582a57ed00fc2b8a616e13d365d4": "\\theta\\sim\\pi\\,\\!", "05a86d6045bfbb3ebe58e44b3e3c6e03": "\\tfrac{\\lambda(1+\\nu)}{3\\nu}", "05a881cca6a168c1289809a58c88862d": " 2 \\times 3 ", "05a89d47da17caf1e29bd6bb8b0679b6": "(-0) + (-0) = (-0) - (+0) = -0\\,\\!", "05a93c269574b71673c27657ba128827": "f(x) = \n\\left\\{\\begin{matrix} \n0 &\\mbox{if}\\ x \\in S \\\\\n\\mbox{undefined/does not halt}\\ &\\mbox{if}\\ x \\notin S\n\\end{matrix}\\right.\n", "05a9a9253de629a22b0de164f7876ec1": " | \\Psi_{E}\\rangle \\langle \\Psi_{E} |", "05a9c4872bef340f03df495e76a13ba9": " P(\\xi) \\, \\hat u (\\xi) = \\hat f(\\xi). ", "05aa01e7c266f58bb67b90354de99c43": "\nA(L(G)) = B(G)^{T}B(G) - 2I_q\\ \n", "05aa172af5cbe36e9633e1c60bc11b8b": "\\mathbf{\\Phi}_{11}= \\sqrt{\\frac{3}{8\\pi}}\\mathrm{e}^{\\mathrm{i}\\varphi}\\left(\\mathrm{i}\\,\\hat{\\mathbf{\\theta}}-\\cos\\theta\\,\\hat{\\mathbf{\\varphi}}\\right)", "05aa1b8cd5d151e87afd734a4c6edf73": "\n\\delta q_{i} = \\epsilon g_{i}(\\mathbf{q}, \\mathbf{\\dot{q}}, t)\n", "05aaf88555241a5c68c772efe40d512c": "B_2\\,", "05ab0872388a5950ce822076bd1199db": "m^*_l , m^*_t ", "05ab6c9f62cdb72589e8b74958e9e61f": " \\frac{P_f}{P}=\\theta ", "05abb692d211f6070acdca75d5371fd6": "\\bar{d}_i ", "05abc0cae09fc2d1037dea041efc2452": "\\mathbf r(s) = \\mathbf r(0) + \\left(s-\\frac{s^3\\kappa^2(0)}{6}\\right)\\mathbf T(0) + \\left(\\frac{s^2\\kappa(0)}{2}+\\frac{s^3\\kappa'(0)}{6}\\right)\\mathbf N(0) + \\left(\\frac{s^3\\kappa(0)\\tau(0)}{6}\\right)\\mathbf B(0) + o(s^3).", "05ac4ab7acee500e0844cf32d45f363f": " \\eta = \\frac{T_H - T_C}{T_H} ", "05ac7a592ac15a18edc4cd1610e87ea8": "\\int", "05ac9133d639722a044b69afebcac8b6": "\\mathbf{g} = \\mathbf{h}\\oplus\\mathbf{p}", "05aca4449050dba074fa25af215863c6": "\n\\langle A \\rangle_\\rho = \\frac{\\int D \\sigma \\; A[\\sigma] \\; \\rho[\\sigma]}{\\int D \\sigma \\; \\rho[\\sigma]} .\n", "05ad9aa7741b8dd9c8282adfbbbb1ee4": " R(r) = \\gamma J_n(\\rho), \\,", "05adeadf32234f91b991e3ffc00d4276": "(p-\\xi)(q-\\xi)=\\frac{k'}{k}(p'+\\xi)(q'+\\xi)", "05ae0c7816ab8c2dba5f57b946a5d3c3": "S=\\cup_{n=1}^\\infty S_n", "05ae0d3dee59e87b4835d0764dd91491": "\\epsilon^0: \\quad S_0'^2 = Q(x),", "05ae382ec6db2593fd16efc16e826e95": "= \\begin{matrix} \\frac{1}{2} \\end{matrix} m_0 v^2 \\ ", "05ae5dfa706b7faa27218940abbb1dd9": "p, a_1, a_2, .., a_n, q", "05ae8ae73c309f38677a320bfe49f71e": "\\alpha(a,b)", "05aeda03cea692eb54855eee884616ca": "G_0/G_1 = H_0/H_1", "05af08f63698bdb185b50d8e1bc418f1": "\\gamma := F_{12}\\,", "05af247981d724e96b1f296624e43462": "F(\\bold{x}, t;\\nu)", "05af80787d3ee70cb19fe32fa4d51a74": "\\displaystyle{z=x+iy=(x,y),}", "05afb5ff344af186af6b7ca460765724": "\\mathrm{Lan}_{Y_D}(\\hat{\\psi})", "05afd513118371378c77886746d05ea1": "\\Psi = (a \\Psi_A + b \\Psi_B)", "05afd9a4b7b92a253abbadc92b701182": "X_{(1)}=\\min\\{\\,X_1,\\ldots,X_n\\,\\}", "05afde080c3c138560fb170da360c2a0": "L^{p_0}", "05b00416881f32ac96954a465e95293b": "Z_{\\text{resistor}} = R", "05b062c7413e26761f3999e1edd9289e": "\\displaystyle{\\psi_r(e^{i\\theta})=1+ {2r\\sin \\theta\\over 1-2r\\cos\\theta +r^2}.}", "05b0da896eeeb72186d5e2fc782bebc3": "R=-8\\pi T=0", "05b0f3e7d663fe1fa32fd698f405ace1": "\\pi_1(u)\\!:\\!\\tau_1", "05b10a093bcb0061605ffe22f10a2566": "C_{out}", "05b1239c4c206422a939bb9e50df3a5b": "f = 2^{12k/12} \\times 440 \\,\\text{Hz} = 2^k \\times 440 \\,\\text{Hz}", "05b1249c37ada36961107370141e5684": "A\\circ B = (AB+BA)/2", "05b142a9273c56306a41b6d17eedee68": "\\tan\\frac{\\pi}{5}=\\tan 36^\\circ=\\sqrt{5-2\\sqrt5}\\,", "05b1bdfd3c31fc9e683a5ac5d58ba23a": " n \\cdot t ", "05b20ea7e11d9daf79426da8d4d13da1": " P_1=(x_1,y_1)", "05b22643726772c8e63a1cc59b6bbb71": "\\psi(k)", "05b2482e9246a5c5644afabf3dad0cc3": "\\mathfrak{P}^{64}", "05b25f3d33c46808e06cbb8b6e831a9a": "\\int_L f(z)\\,dz", "05b269633ce9c9458c44db3519f54255": "\\frac{d_1-2}{d_1}\\;\\frac{d_2}{d_2+2}\\!", "05b29308d9c7c60fb46a662423e01dd4": "\\gcd(p, q, r) = \\gcd(p, \\gcd(q, r)),", "05b29d062cea367973c5af8d196b444d": "Q=2 \\pi \\frac{\\frac{1} {2} k z_o^2} {\\pi D z_o^2 \\omega_n} = \\frac{1} {2\\delta}, \\; \\omega_n=\\sqrt{\\frac{k} {m}}, \\; \\delta=\\frac{D} {2\\sqrt{mk}}\\,\\!", "05b2dec6a25665b36dda44033765c7b7": "\\int g(x) T(f)(x) \\, dx = \\iint g(x) K(x-y) f(y) \\, dy\\,dx.", "05b34b0eb41a6e2a115c9ed8deafe7df": "\\frac{V_1}{V_4}=\\frac{V_1}{V_2}\\cdot\\frac{V_2}{V_3}\\cdot\\frac{V_3}{V_4}=\\sqrt{\\frac{Z_{I1}}{Z_{I4}}}e^{\\gamma_1+\\gamma_2+\\gamma_3}", "05b3be8dbcf0d9a08ea0513410ef1282": "[\\psi^{(i)}(x),\\psi^{(j)}(y)]=0", "05b3c3a2a55dc111c4ee95fcbae49a12": "a\\leq t\\leq b ", "05b435045822a31b2a64190aac7e1bde": "\\{ \\mathbb{P}_N\\}", "05b44446ffc86fb27bcbd7eb7ff3d0f1": "(n,k,d)_{\\mathcal{F}}", "05b4521ead02d78da929929258f36fe3": "\\sum_{n=0}^\\infty nf(n/N)", "05b45fb47724552e5a57963aa09bd75d": "v\\in T^k\\left(V\\right)", "05b4799795585f3c6df304abe1080109": "I_o=\\bar{I_D}=\\frac{I_{L_{max}}}{2}\\delta", "05b4fd04922f80b290ebb1d1a672281a": "{1,...,t}\\,\\!", "05b514c88c9e2123a6c8acc8729ab939": "\\begin{align}\\begin{vmatrix}a&b&c\\\\d&e&f\\\\g&h&i\\end{vmatrix} & = a\\begin{vmatrix}e&f\\\\h&i\\end{vmatrix}-b\\begin{vmatrix}d&f\\\\g&i\\end{vmatrix}+c\\begin{vmatrix}d&e\\\\g&h\\end{vmatrix} \\\\\n& = a(ei-fh)-b(di-fg)+c(dh-eg) \\\\\n& = aei+bfg+cdh-ceg-bdi-afh.\n\\end{align} ", "05b53317814fbfc9799affa3b7f0ca93": "[x:=?]p\\,\\!", "05b54b1d076865bf416da04d9117672e": "H\\left( \\frac{1}{\\sqrt{2}}|0\\rangle-\\frac{1}{\\sqrt{2}}|1\\rangle \\right)= \\frac{1}{2}( |0\\rangle+|1\\rangle) - \\frac{1}{2}( |0\\rangle - |1\\rangle) = |1\\rangle", "05b57be59730cd2e0a174ce9ea691824": "1+2\\cos(x)+2\\cos(2x)+2\\cos(3x)+\\cdots+2\\cos(nx) = \\frac{ \\sin\\left[\\left(n+\\frac{1}{2}\\right)x\\right\\rbrack }{ \\sin\\left(\\frac{x}{2}\\right) }. ", "05b6cbeb7a2a97c6cf4ab673d03b7819": "C_\\mathrm{srgb}", "05b70f0366f4ce67735118a38ae6f4eb": "(x,y,z) = (x \\triangleleft y) \\triangleleft z - x \\triangleleft (y \\triangleleft z)", "05b75f24697420391a4349928ffb07f5": "\\bar{w_m}", "05b7a8ca7581fc1774fdbd1c4816be3b": "\\cup\\,", "05b87076480fc5625941b3a378773516": "X \\xrightarrow{vu} Z \\xrightarrow{m} Y' \\xrightarrow {n} ", "05b8976e246ca198290a3c22bd7b89d8": "F^T(t,r)=e^{A(t,T)-B(t,T)r}", "05b8add3164d2677c9bc109e067cdd30": "\\frac{\\mathrm d\\mathbb P}{\\mathrm d\\mathbb Q}", "05b94098ae4bfb39766992c8b9a9cddc": "\\gamma_{wof}", "05b9b130e331d094677aa7840b3c475e": "q_\\mathrm{net} = \\sum_{i=1}^N q_i \\,\\!", "05b9be3947267b1acbe2173d1ac1afc9": "y_3=Ay_2=AA^{2}y_0=A^{3}y_0,", "05ba28454878dec6cb87da9ff1d82648": "\nP = p + \\frac{1}{2} \\rho r^2 \\Omega^2.\n", "05ba6b36bb720587ff6f7b6e7e49a318": "S_-|s,m\\rangle=\\hbar \\sqrt{s(s+1)-m(m-1)}|s,m-1\\rangle", "05ba6f3563c0b2f7ddc62879c27b5f0e": "{{i}_{IN}}-{{i}_{OUT}}=\\frac{2\\left( \\overline{{{\\beta }_{12}}}-{{\\beta }_{3}} \\right)+2}{\\overline{{{\\beta }_{12}}}{{\\beta }_{3}}+2\\overline{{{\\beta }_{12}}}+2}", "05ba731686e7ea35361ac7af1177ae3e": " \\gamma^2 - \\kappa \\le \\frac{ 186 }{ 125 } ", "05ba874764a8fac7788e7fbe836b9b1e": "\\{0, 1\\}^m", "05ba8e4d35954e38e4dd92b63892e25e": "S_1 \\subseteq S", "05bab9329dd16291f016388c68ce4ae5": " \\tfrac{1+\\text{log}(\\text{tf}_{t,d})}{1+\\text{log}(\\text{ave}_{t \\epsilon d}( \\text{tf}_{t,d}))}", "05bae3b9cf380743cb488d4e5134ef1a": "[[G_{nm}]]", "05bb0da9a9b62787ced0ceedca9849be": "V_{max}\\ \\propto\\ \\sqrt[3]{power/f}", "05bb3d22db6d6ef0482446936fbc4aee": "T \\colon R^3 \\to R \\,", "05bb91e495dcd9f39f806f82422e3b64": " T = (X_1 + X_2 + \\cdots + X_n) ", "05bbbba0f5bc6b36a223e33e2b73afeb": "X \\to p \\to q+1 = g_q^p(1)", "05bbd70b35171806bd5098e27986dcfa": "\\nabla \\cdot (\\mathbf{a} \\mathbf{b}^\\mathrm{T}) = \\mathbf{b}(\\nabla \\cdot \\mathbf{a})+(\\mathbf{a}\\cdot \\nabla) \\mathbf{b} \\ .", "05bbe30c0a898a91ecb1f91c73e5daee": "S_1 (q)", "05bc170a4703117ff03cebcb6f69bb6c": " p(x_n) = y_n, \\quad n=0,\\ldots ,N-1. \\, ", "05bc296af2ad5d1ca4efc5ccc41b0549": " v_0^2 \\left(h_0-{h_0^2 \\over h_1}\\right) + {g \\over 2} (h_0^2 - h_1^2)=0. ", "05bc49ae7d6c96bf37559b7836c96d28": "\\mathfrak{L}", "05bcc0311a05d7aed31aee8615537f89": "DP_{p,c} = \\frac{\\sum_{p,c}(DP)}{count_{p,c}(singular cases)}", "05bd2d1be0d5d9ddb208f535d1145d7c": "s = c + \\frac{2v^2}{d}.", "05bd51e02228180c67c08ace7425da07": "R = \\frac{ \\sum{||F_\\text{obs}| - |F_\\text{calc}|| } }{ \\sum{ |F_\\text{obs}|}}", "05bd6dee39277c57474ce9368b2a6d84": "S=-3N\\langle x-0\\rangle^0\\ +\\ 6Nm^{-1}\\langle x-2m\\rangle^1\\ -\\ 9N\\langle x-4m\\rangle^0\\,", "05bd6e9d7504fc2db9a92e5a207d6ba7": "d\\ln{H}=m d\\ln{\\dot{\\varepsilon_p}}+n d\\ln{h_p}.", "05bdb30cea2ff2f555ea8236e1845287": "0\\leq i\\leq n", "05be2af0da9707467a594f649c32188a": "\n\\gamma = (b^2-c^2)\\cos^2\\beta\\sin^2\\alpha-(a^2-b^2)\\sin^2\\omega\\cos^2\\alpha,\n", "05be5f075a520c2e7d39376d2bc5587f": "a\\rightarrow_W b", "05be88b119ba2a5850f4fffae769e9fc": "L_{k-1}\\wedge\\cdots\\wedge L_{2}\\wedge L_{1}", "05bec6ac2fd604dbdd5da6ed4c98c3ba": " \\lim_i \\epsilon_i = 0", "05befe8285e71329765f393e3519cc18": "X \\cup_{f} Y", "05bf0be6456738fc35ecaacf1f0dc0dc": "\\mathcal{R}\\models\\varphi[a]", "05bf0e6d74499d4ea7bcde463a10aa02": "Z(\\beta,\\mu)\\ \\stackrel{\\mathrm{def}}{=}\\ \\mathrm{Tr}\\left[ e^{-\\beta \\left(H-\\mu N\\right)} \\right]", "05bf188f15a57c458a10ce6ea7d3f30d": " \\eta \\to 3\\gamma, ~~ e^+e^-, ~~ 4e ", "05bf4d1426470438683cc492040d9cda": " i_1=E^2\\sin^2(\\omega t)\\, ", "05bf6956b879bab7b342e627222be8ef": "(N^{1/4}+1)^2", "05bf8bd2f7266b98af0dc87ea7d7e679": "1.43^{-1} + 3.9^{-1} = 0.956", "05bfe51895722cc4d344507815a72c72": "\\mathbf{S}_{i_1,i_2,\\ldots,i_N} ", "05c054bd8daeca25e6dc6b8a1db5736c": "\\| f \\|' := \\sup_{x \\in \\mathbb{R}} \\left| \\int_{- \\infty}^{x} f \\right|.", "05c08881739e0666af07526add9e2c5d": "u(x,t) = \\frac{f(x-ct) + f(x+ct)}{2} + \\frac{1}{2c} \\int_{x-ct}^{x+ct} g(s) ds", "05c0a00d7a781bcea63d9072acfaa2a7": "\n\\frac{\\pi}{2}= \n\\sum_{k=0}^\\infty\\frac{k!}{(2k+1)!!}= \\sum_{k=0}^{\\infty} \\cfrac {2^k k!^2}{(2k + 1)!} =\n1+\\frac{1}{3}\\left(1+\\frac{2}{5}\\left(1+\\frac{3}{7}\\left(1+\\cdots\\right)\\right)\\right)\n", "05c16634a2abdf6e9fb4d0593a176811": "(e^{-x^2/2}u')' + \\lambda e^{-x^2/2}u = 0", "05c1c3d13113e80e93d06955f8a0d3d9": " \\operatorname{E}( \\widehat{\\theta}(X) ) ", "05c1f1d0a83b7ada12034af7ee1d6e5a": " H = X \\left(X^\\top \\Sigma^{-1} X\\right)^{-1} X^\\top \\Sigma^{-1}, \\, ", "05c223937c098ccc7d5fe65df085b311": "P_j(0)=0", "05c2604cd81453f4c92ca92d2ebfd54e": "x^3 - 4x + 7", "05c264d438209c0f6db45dde67a2d801": "P_0^s (S) = \\lim_{\\delta \\downarrow 0} \\sup \\left\\{ \\left. \\sum_{i \\in I} \\mathrm{diam} (B_i)^s \\right| \\begin{matrix} \\{ B_i \\}_{i \\in I} \\text{ is a countable collection} \\\\ \\text{of pairwise disjoint balls with} \\\\ \\text{diameters } \\leq \\delta \\text{ and centres in } S \\end{matrix} \\right\\}.", "05c26c300015aa8a80471774a788670c": "\\sin^5\\theta = \\frac{10 \\sin\\theta - 5 \\sin 3\\theta + \\sin 5\\theta}{16}\\!", "05c287405896816565fa9d9ab0b197dc": "\\langle\\mathbf{v},\\mathbf{w} \\rangle", "05c2a754b08bf56c1e1aa415093b2dae": "G_2=\\langle L,R,F^2,B^2,U^2,D^2\\rangle", "05c331575e79832c7d767e1326882c8e": "\\sum_{i=1}^d{P_i \\cdot \\log_2{\\frac{1}{P_i}}}", "05c34b387fb3657e877769df402b3cda": "\\textstyle \\sqrt {\\frac {1} {5}} \\left ( \\frac {1} {5} - 0 \\right ) + \\sqrt {\\frac {2} {5}} \\left ( \\frac {2} {5} - \\frac {1} {5} \\right ) + \\cdots + \\sqrt {\\frac {5} {5}} \\left ( \\frac {5} {5} - \\frac {4} {5} \\right ) \\approx 0.7497.\\,\\!", "05c381c941317f383b38c16a50d8fcb6": "(t,r,\\theta,\\phi)", "05c390791fd4770b47596dc6994505ab": "\nh_R(t) = \\delta (t) - {1 \\over RC} e^{-t / RC} u(t) = \\delta (t) - { 1 \\over \\tau} e^{-t / \\tau} u(t)\n", "05c391ace5152be60b6040676e6ce83c": "(x+3)^2 = 4.\\,\\!", "05c3a6c8f942d0c0774385ee683c2fcf": "\\mathbf{w}_{(1)}\n = \\underset{\\Vert \\mathbf{w} \\Vert = 1}{\\operatorname{\\arg\\,max}}\\,\\{ \\sum_i \\left(t_1\\right)^2_{(i)} \\}\n = \\underset{\\Vert \\mathbf{w} \\Vert = 1}{\\operatorname{\\arg\\,max}}\\, \\sum_i \\left(\\mathbf{x}_{(i)} \\cdot \\mathbf{w} \\right)^2 ", "05c3e32e9d53ac4884505fa674adcc5b": "{1\\over 3} + {2\\over 9}+{2\\over 27}", "05c43a234b7406db7cf291f9002d5b8d": " Y\\,\\!", "05c440b0bf58018422ebf3802967a159": "\\mathbf {q}_e", "05c46bd969ec4f85c3f245a1ecf1b75f": "{(\\bullet)}_{,j}", "05c4ab41865fa3d449fae5f24c1e8c57": "n(q_i)", "05c4c4f452db3760a7e0c7690c9473d6": "m c^2 \\frac{dt}{d \\tau [t]} = \\frac{m c^2}{\\sqrt {1 - \\frac{v^2 [t]}{c^2}}} = +m c^2 + {1 \\over 2} m v^2 [t] + {3 \\over 8} m \\frac{v^4 [t]}{c^2} + \\dots \\,.", "05c5122c969c2531c1f6764abe5c037a": "M_A = 4 \\frac{EI}{L} \\theta_A + 2 \\frac{EI}{L} \\theta_B = 4 \\frac{EI}{L} \\theta_A", "05c51dcaee2768b6a2b0906e8239e86e": "RPM = {Speed \\over Circumference}={Speed \\over \\pi \\times Diameter}", "05c544046a1e219149406b97ff19a95d": "I[f] = \\frac{1}{4\\pi}\\int \\mathrm{d}\\Omega\\ f(\\Omega) = \\frac{1}{4\\pi}\\int_0^\\pi \\sin(\\theta)\\mathrm{d}\\theta\\int_0^{2\\pi}\\mathrm{d}\\varphi\\ f(\\theta,\\varphi),", "05c5afd9da07bff6fcf6e46abe179aa8": "N_{SV}", "05c5cb92f034600b9f06e9a37b664251": "n\\in\\mathbf{Z}", "05c5d866380a56cb67ce18b5aca4c57a": "\\phi^{\\Rightarrow x} \\leq \\psi^{\\Rightarrow x}\\,", "05c5f32ce2a5ca1c969e85db436088d6": "128^4", "05c609a273b2bea4f5dc3b9537e68b83": "\n \\hat H Y_\\ell^m (\\theta, \\varphi ) = \\frac{\\hbar^2}{2I} \\ell(\\ell+1) Y_\\ell^m (\\theta, \\varphi ). \n", "05c630ec8a6acbca4c93b0ec3c5f857e": "\\int\\frac{\\cot^n ax\\;\\mathrm{d}x}{\\cos^2 ax} = \\frac{1}{a(1-n)}\\tan^{1-n} ax +C\\qquad\\mbox{(for }n\\neq 1\\mbox{)}\\,\\!", "05c671f47458f9ba8a33b361a6bb73d0": "P(n) = \\ln\\left(\\frac{\\Omega}{ n} \\right) \\frac { 1}{ \\ln(\\Omega) c } ", "05c68a5b4405c118dd85a1a78e2a3946": "t = \\frac{2}{2} = 1", "05c6c30055e6f5f1f21a6952ef39bafe": "\\text{pulse dispersion} = \\frac{\\triangle\\ n_1\\ \\ell}{c}\\,\\!", "05c6d02faf6e88de12c96b8ed371ba7f": "F = \\R", "05c71188190df6096f2e8460f93db854": "\\displaystyle{E=\\bigoplus_{i\\le j} E_{ij},}", "05c71f95ec0de948798f5478018b2622": " \\nu_\\mathrm{1}= |\\nu_\\mathrm{n}-a/2|", "05c76f804f1b3dd570c74ecb38db5642": "v_C(t)\\,", "05c7759a446e834f13203a703eceb182": "f^{n+1}\\ x = f (f^n x) ", "05c794638232b29175d15651c34614c6": "\\{2,5\\}\\;", "05c7d9478d9ca050dcb2518e566ee3f2": "G\\setminus A", "05c7f10b3ba002bff3fd3050db03314b": "\\lim_{x \\to c} x^r = c^r \\qquad \\mbox{ if } r \\mbox{ is a positive integer}", "05c7f2cac0b70415e8de7a9802d2e09c": "\\kappa=1/\\sqrt{2}", "05c850ca793caf0aa51e967a2e8e0c39": "n= \\infty ", "05c8649308e29c0869d4693b4139816b": "\\tfrac{1}{2}\\text{social cost}(Z) \\leq E(Z) \\leq \\text{social cost}(Z)", "05c898fc68d9bdc0c397cd800b83a9ca": "(k,\\theta_k,\\phi_k)", "05c8aa6c5f99b293809c083655415e70": "p_{ij}(t)\\ ", "05c8aeef3fd69db58fb38ec0fe1584cb": "\\frac{1}{\\pi} \\int |\\alpha\\rangle\\langle\\alpha| d^2\\alpha = I\n\\qquad d^2\\alpha \\equiv d\\Re(\\alpha) \\, d\\Im(\\alpha)", "05c8b0dcff5822fbde7e0398f05aea00": " x^2 ", "05c8e93d5e664b4fbdd450f7ae0df317": "X=\\sum_{i=1}^N w_i Z(x_i)", "05c916460de60839f9545a943d990edd": "y_c=C_1e^{ \\left ( -b+\\sqrt{b^2 - 4c} \\right )\\frac{x}{2}} + C_2e^{-\\left ( b+\\sqrt{b^2 - 4c} \\right )\\frac{x}{2}}\\,\\!", "05c92b00d6bb7100843d0ca196c4a56a": "A=A_1\\times A_2\\times\\dotsb\\times A_N", "05c93e0130d62c75768900584e84b491": "{{P}_{T}}f(u,\\xi )=\\frac{1}{2\\pi }\\int_{-\\infty }^{\\infty }{\\int_{-\\infty }^{\\infty }{{{P}_{V}}f({{u}^{'}},{{\\xi }^{'}})}}.{{P}_{V}}{{\\phi }_{\\gamma (u,\\xi )}}({{u}^{'}},{{\\xi }^{'}})d{{u}^{'}}.d\\xi '", "05c93f2cfb54e2a5d9a374aacb918e0b": "\\textstyle (\\Omega_2,\\mathcal{F}_2,P_2) ", "05c953543a2f0987bb8f7c87d1f22dfa": "{\\Gamma(\\alpha+\\beta+n)}", "05c9864ab6b670cdb4261c66c183c47a": "r=k \\frac{K_1K_2C_AC_B}{(1+K_1C_A+K_2C_B)^2}. \\, ", "05c9c29453269f69480a108826520535": "\\lambda x . x x x", "05ca8e7728a7f1667453038b1ad6afa3": "K \\in \\mathcal{K}.", "05cab4a49dfc9b7b1327cc15eab681d7": "+1/2\\,", "05cad9232024d1f205509f78fc65df53": "R^h = \\mathbb{H} / \\Gamma", "05caed6c68f525022399020b4bfb94d8": "KL_{i,1}", "05cb3148609df7797d7bcc4141ba84a8": "f \\in \\mathcal{H}_k", "05cb34c6f7a4b2415077be3d9283df20": "(\\phi \\lor \\psi)", "05cbc2cf48741a31cce77a970ff4131d": "J_{ij} < 0 ", "05cc05f25d900129201d72733479ed90": "\\begin{align}\n\\Omega U &\\simeq \\mathbf{Z}\\times BU = \\mathbf{Z}\\times U/(U \\times U)\\\\\n\\Omega(Z\\times BU)& \\simeq U = (U \\times U)/U\n\\end{align}", "05cc3380c308ddf2623f7e43b6a47fc3": "\\,N\\,", "05cc9bb84d986e769d9677d2f30d1c65": "\\det(A) = \\det(X)^{-1} \\det(BX) = \\det(X)^{-1} \\det(B)\\det(X) = \\det(B) \\det(X)^{-1} \\det(X) = \\det(B).\\ ", "05cd66ec0550ba0059b46a660e693e69": ",a,g(", "05cd6b1347352544845e7f45f9b73dcb": "\\mbox{QMA}({2}/{3},1/3)", "05cda3f39c61cc3cd8f9d58bb9b25616": "\\{1, 2,\\dots, k^2\\}", "05cddb1a519e4279f413146bec4ecdb6": " \\, [0, \\theta]. \\, ", "05cdeef4cdeedd41a74966eed95c31e0": "S(c, c) \\to S(c, c')", "05cdf9f7d3e7e9a701a0e383b0d21092": "(-1)^k {z \\choose k}= {-z+k-1 \\choose k} = \\frac{1}{\\Gamma(-z)} \\frac{1}{(k+1)^{z+1}} \\prod_{j=k+1} \\frac{(1+\\frac{1}{j})^{-z-1}}{1-\\frac{z+1}{j}}", "05cdfab85fed1fd5538b08c5b2534f68": "\\hat{X}^n_{k}", "05ce135f532096424b95e65d324e3066": "\\beta(g)=-\\left(11-\\frac{2n_f}{3}\\right)\\frac{g^3}{16\\pi^2}~,", "05ce27299cfa7c645dc43f5900ab7b51": "\\omega_n = \\sqrt{\\frac{K_p K_v}{\\tau_1}}", "05ce58568ba255a9b3ea387369e3b806": "\\succ_{P}", "05ce5e7e1445f7580639409d0b4b229a": "\\theta \\in \\textrm{End}(V)", "05ce62090305d8c81d8e1984a0196b9e": "m - n + k.\\ ", "05ce6b1ede4d11a9c95f3f305238c738": "i\\ne j\\ ", "05ceaa3417df09a995fe929f4e78788f": "\\frac{2^{7-k} \\mod k}{k}\\, .", "05cec527ebdab7d072cc405cf84af1fe": "\\sigma_x,\\!", "05cef6d6ee4d460288d2f29251073318": "\n\\begin{bmatrix}\nE&F\\\\\nF&G\n\\end{bmatrix}\n", "05cf0a41402aa254ea95ef6c4b15c6bb": "A\\cdot (c_1 X + c_2 Y)=c_1 A\\cdot X + c_2 A\\cdot Y\\,", "05cf3bf32200fd37fdf6a1d3a77ffb78": "{0, 1, 6}", "05cf6a211012e62a821e558fc09108e5": "\\varepsilon_{jmn} \\varepsilon^{imn}=2\\delta^i_j ", "05cf867d5245db00b37d986d14c94ea9": "\\dot{\\boldsymbol{e_j}} =\\sum_{k=1}^{d}\\frac {\\partial}{\\partial q_k}\\boldsymbol{e_j}\\dot q_k \\ ", "05cf8fe66345795237ac822f788ef92d": "_2^1\\text{P}", "05cfcfea3e2eb81380e5df3c9c807736": "p=\\frac{\\varepsilon _{2}}{\\sqrt{-P^{2}}}p_{1}-\\frac{\\varepsilon _{1}}{\\sqrt{\n-P^{2}}}p_{2} ", "05cfd4a8c76cc15a15f3c9ac03e73ec2": " \\mathrm{Res}(f, \\infty) = -\\lim_{|z| \\to \\infty} z^2 \\cdot f'(z)", "05d009b38560ef6a0599e43575933115": "[E_{\\beta},E_{\\gamma}]=\\pm(p+1)E_{\\beta+\\gamma}", "05d0330244ce364c66a262afdc74082a": " \\frac{\\partial |\\Psi|^2}{\\partial t}=-\\nabla \\cdot \\mathbf{j} ", "05d05e751a80db7375eae13c25f0ca13": "\\mathbf{C}", "05d07a7c15b65d6b762ec8003c66c0be": "A\\in\\mathbb{R}^{k\\times d}", "05d08383a49f0991b19264fa014a430c": "A|\\psi_n\\rangle=a_n|\\psi_n\\rangle", "05d0e694a8c918c5eb5f137840030045": "\\ i_1^2 = -1, i_2^2 = i_3^2 = +1", "05d0e9032db15f41512942654cf714a2": "\\sigma_{\\mathbf{v}} = \\sqrt{\\frac{3kT}{m}}", "05d0f6b04e586050c1e1a56428ddabee": "|\\theta|", "05d0fabf9ade9a1b534f221d937c152e": "\\binom{m}{n}\\equiv\\prod_{i=0}^k\\binom{m_i}{n_i}\\pmod p,", "05d1398bbc4b0b7e1015d32c12854012": "\\ W=PIt", "05d22b1ac0c732002507d2dedbcad08e": " \\operatorname{build-list}[\\lambda q.\\lambda x.x\\ (q\\ q\\ x), D, D[p]] \\and D[p] = L_1", "05d27529606dcd2415f3490efde4399d": "y_{ij} = x_{ij}'\\beta +\\mu_i +\\epsilon_{ij} \\,", "05d28ad1ff5ee18982d1ab0df406bc51": "\\part_{X_i}", "05d2abd793340a845e9b9d0dcfcd4ca3": "\\ G(\\tau)=G(0)\\sum_i \\frac{\\alpha_i}{(1+(\\tau/\\tau_{D,i}))(1+a^{-2}(\\tau/\\tau_{D,i}))^{1/2}} +G(\\infty)", "05d2caf1e9ce7327126c96ca23ae5520": "\\ \\mathbf a= \\dot{\\mathbf v} = \\ddot{\\mathbf x} =\\frac{d^2\\mathbf x}{dt^2}=\\frac{\\partial^2 \\chi(\\mathbf X,t)}{\\partial t^2} ", "05d2e9dda026d6b2b9d64eb5bd8c5131": "\\left\\lfloor \\frac{n}{2} \\right\\rfloor ", "05d2f836c9edaf00687f79a06ddafa3f": "\\sin^{n} \\alpha", "05d32c428028dd422d7d0393241c095c": "\n \\begin{align}\n N_{\\alpha\\beta,\\beta} & = J_1~\\ddot{u}_\\alpha \\\\\n M_{\\alpha\\beta,\\alpha\\beta} - q(x,t) & = J_1~\\ddot{w} - J_3~\\ddot{w}_{,\\alpha\\alpha}\n \\end{align}\n", "05d3874e7243517143e0d8e86dff6d40": "= (1-2 i \\theta)^{-\\frac{p}{2}}.~~\\blacksquare", "05d39f59c1a1e6ae07acb4cb6c2fa55e": "\\quad = \\gamma \\frac{\\omega_{\\mathrm{obs}}}{c} - \\beta \\gamma \\frac{\\omega_{\\mathrm{obs}}}{c} \\cos \\theta. \\,", "05d3e17d000e32a4bbd754019fdf0c09": "x_{k+2}", "05d3e3a666490ce89fd9a7a7b0651ab5": "V = -\\boldsymbol{\\mu}\\cdot\\bold{B} ", "05d41768c61c9626a407da6464305e0f": "G\\times X\\rightarrow X", "05d4bb3cdffddbd2eb5ee961ebf4967f": "\\hat\\beta = \\big(\\tfrac{1}{n}X'X\\big)^{-1}\\tfrac{1}{n}X'y \n = \\beta + \\big(\\tfrac{1}{n}X'X\\big)^{-1}\\tfrac{1}{n}X'\\varepsilon \n = \\beta\\; + \\;\\bigg(\\frac{1}{n}\\sum_{i=1}^n x_ix'_i\\bigg)^{\\!\\!-1} \\bigg(\\frac{1}{n}\\sum_{i=1}^n x_i\\varepsilon_i\\bigg)", "05d4cc0e17ee5806e4040d34db0e53c5": "[g_{ij}] = \\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & r^2 & 0 \\\\\n0 & 0 & 1 \\\\\n\\end{pmatrix}", "05d4d5a45a7e8e0494c0d39e6d3b1495": "(\\mathcal{L}_Y T)_p=\\left.\\frac{d}{dt}\\right|_{t=0}\\left((\\varphi_{-t})_*T_{\\varphi_{t}(p)}\\right)=\\left.\\frac{d}{dt}\\right|_{t=0}\\left((\\varphi_{t})^*T\\right)_p", "05d4e93dfe94044737fcc6a71dc92ed8": "n(p-1)", "05d52867f9e20a380e34202c6768102d": "\\hat{y}_1 = y_1(1+\\delta_3)", "05d5449c09ffbfbe8b15b17745c5c826": "\\mathbb{H} \\oplus \\mathbb{H}", "05d56f7c3994dedec05a6b75c11b7a18": "F\\, =\\, \\underbrace{\\rho\\, C_m\\, V\\, \\dot{u}}_{F_I} + \\underbrace{\\frac12\\, \\rho\\, C_d\\, A\\, u\\, |u|}_{F_D},", "05d57c5572d005d3c21408594a152be6": " \\mu_{disp} =\\sum_k \\gamma_k A_k", "05d5d3bfc47cb6c934131f2e4aafbe1f": "\\displaystyle \\Phi = \\mu_0\\mu_rNiA/l,", "05d5da12cf320f6b8aa34e8ff21d4eee": "\\mu_t = M_t / m", "05d6052a29659a1972c00d6345dbaf3a": "\n 1 = \\int_{-\\infty}^{\\infty} f(x)\\,dx\n = \\int_{g(-\\infty)}^{g(\\infty)} f(x)\\,dx.\n \\!", "05d65eeb37dfe3eada0857a857820791": "\\tilde{\\lambda}\\notin\\sigma(A)", "05d677a5a308358279bc0ce73a5b66eb": "x^\\ast - \\frac{\\epsilon}{2} \\frac{c}{||c||} \\in P", "05d6ee53814b655a3ffa1748da94df9d": "K_{\\rm J}^2 R_{\\rm K} = K_{\\rm J-90}^2 R_{\\rm K-90}\\frac{mgv}{U_{90}I_{90}}", "05d7133d190e6f49370153a8a3654dff": "\n\\begin{align}\n\\langle d\\Psi/dt | \\hat{x} | \\Psi \\rangle + \\langle \\Psi | \\hat{x} | d\\Psi/dt \\rangle &= \\langle \\Psi | \\hat{p}/m | \\Psi \\rangle, \\\\\n\t\\langle d\\Psi/dt | \\hat{p} | \\Psi \\rangle + \\langle \\Psi | \\hat{p} | d\\Psi/dt \\rangle & = \\langle \\Psi | -U'(\\hat{x}) | \\Psi \\rangle,\n\\end{align}\n", "05d74db688ee9c38cb781c2f1d840ace": "\\mathcal{L}(\\theta \\mid x )= f(x\\mid\\theta). \\!", "05d7cd70b0eb47eca402d4b08e43c92e": "\n\\sum_{t=1}^n \\frac{FCFF_t}{(1+WACC_{g})^t} +\n \\frac{\\left[\\frac{FCFF_{n+1}}{(WACC_{st}-g_n)}\\right]}{(1+WACC_{g})^n}\n", "05d7e8e9e7a7baf15e49422aa7eeea0b": "q(\\xi) = (\\xi-x_0) \\cdots (\\xi-x_n)", "05d87631f336b199f9daae6e57824896": "x_n = z_n \\times z_{n - 1}", "05d893a914644be9aa42f1fa2804e1d3": "\\begin{align}\n\\epsilon( t, \\omega ) \n\t&= \\int x(\\tau) h( t - \\tau ) e^{ -j \\omega \\left[ \\tau - t \\right]} d\\tau \\\\\n\t&= e^{ j \\omega t} \\int x(\\tau) h( t - \\tau ) e^{ -j \\omega \\tau } d\\tau \\\\\n\t&= e^{ j \\omega t} X(t, \\omega) \\\\\n\t&= X_{t}(\\omega) = M_{t}(\\omega) e^{j \\phi_{\\tau}(\\omega)}\n\\end{align}", "05d8d8906414d4466fef2875630a776a": "x \\in \\{ INT\\_MIN , ... , INT\\_MAX \\}", "05d8e55c60e1240c8d9385ad078d8283": "\\frac{dx}{dt}(T) \\cdot (y-Y)=\\frac{dy}{dt}(T) \\cdot (x-X).", "05d90f614706bc6fafc53f97fe45853c": "R_1=mR,\\,", "05d9936b2f9e6a19a257c7444afc3f06": "\\{y_k\\}_{k=1}^M", "05d9d2d1e373b38c4d316bc905925ea9": " w\\left(\\omega\\right)", "05da02550b176b031a52dd445ed13cdd": " D=< \\frac {\\Delta x^2}{\\tau_D}> \\sim \\frac {c^2 \\delta E^2}{B^2k_{\\perp}^2\\,D} \\sim \\frac {c \\delta E}{Bk_{\\perp}} ", "05da2717763ceea3d7f31ca86ae4fb20": "\\alpha M_k-\\beta M_{k+1}=\\left\\{\\begin{array}{cc}-ve, & if\\; \\alpha<0\\\\ ve, & if\\; \\alpha>0 \\end{array}\\right.", "05da351c30b5100133e9819823891df6": "x_{n+1}=x_n^2-c", "05da8d653aade338bf5a13b0ca5f197a": "\\exp_p(z)=\\sum_{n=0}^\\infty\\frac{z^n}{n!}.", "05dab0e648f28f5051ceca435fad3552": "a_0b_3", "05dabdf6797bb34bd0c215e92a3ec706": " f: \\mathbb R_+\\to\\mathbb R; x\\mapsto x^2", "05db644d5820ac7cb87759b8fce150e3": "F^{-1}(p) = a + p (b - a) \\,\\,\\text{ for } 00", "05e53749ca3e0fbdd8ad7b0bb193db2a": "\\Lambda(A)", "05e5912146d2c3ba23e769415892616a": "\\gcd{(a^{(N-1)/p}_p - 1, N)} = \\gcd(7^{2\\cdot 25} - 1, 11351) = 1.", "05e5d4406c44c5bcff6b911a8427d630": "\\nabla\\phi", "05e603a1e451174fdf7ca07065141804": " A = QR, \\, ", "05e61e59147411910cc55ab15f423054": "\\ F = \\frac{1}{4 \\pi \\varepsilon_0} \\frac{q^2}{r^2}. ", "05e666402749e2dafc3acf1a40303ac2": "{{\\partial \\zeta_g \\over \\partial t} = {-\\overrightarrow{V_g} \\cdot \\nabla ({\\zeta_g + f})} + {f_o {\\partial \\omega \\over \\partial p}} }", "05e67768a59320d85e3ebf316725ae67": "\\mathbb{R}\\rarr \\mathbb{R}", "05e6958d35278b9f4186874c8ce2baff": "\\vec{p}^{\\,*}", "05e6a4b2796d446ae06d219936784b31": "k < 2\\times10^{-3}", "05e6aaaf68164c07f41c8b803dc47ea1": " \\Delta P = \\frac{8 \\mu L Q}{ \\pi r^4} ", "05e6b4fcbb73a01861df35fbf63b4a03": " \nW = \\begin{cases}\nmW&\\xi \\leq 1/m \\\\\n0&\\xi > 1/m\n\\end{cases}\n", "05e6e8d1b3ca6f727ff21d16d9f02a8f": "\\Delta\\epsilon\\equiv\\epsilon_\\parallel-\\epsilon_\\perp", "05e71ee635eeaa582f914152270c8d58": "\\overline{\\mathrm{Nu}}=-{{1} \\over {S'}} \\int_{S'}^{} \\mathrm{Nu} \\, \\mathrm{d}S'\\!", "05e730c6e95369f755e64f447b385a85": "f^i \\left (p \\right )", "05e757209bb816fcb90984fa2a4eafda": " 1.57 \\approx \\frac{\\pi}{2} \\leq k_{\\R} \\leq \\mathrm{sinh}(\\frac{\\pi}{2}) \\approx 2.3", "05e7ca63d7e5b2b34f1090ef28bb487b": "\\int r\\cos \\theta dm", "05e7db456cd901f5d80e881bcc27d8e9": "\\tfrac23", "05e82ad825e447fb9a23e8aa7c714fe3": "U\\colon(\\mathbf{Ab},\\otimes_\\mathbf{Z},\\mathbf{Z}) \\rightarrow (\\mathbf{Set},\\times,\\{*\\})", "05e849e1ff7eb44e943f4681b34964c3": "\\textstyle x^jb(x)", "05e8692dbb52599435d0d7f29759f335": "{\\scriptstyle\\frac{1}{120}} (-x^5+25x^4-200x^3+600x^2-600x+120) \\,", "05e88b875c72c95fcaf55ca9bfd22ede": "\\begin{matrix} {9 \\choose 1}{4 \\choose 3} \\end{matrix}", "05e893878365af1b7320f5549d71bc2f": "f(a) \\ne \\varepsilon", "05e8d5fffb9e0eb4ba401fb00c15f755": " \\psi \\ge \\frac{ 3 }{ \\sqrt{ 3 } + 1 } \\quad ( \\approxeq 1.098 ) ", "05e8fea2636da68588744dc377fce281": "f \\colon M \\rightarrow N", "05e960ec1a492cfce14fe3d8072b2b4f": "g_D", "05e9641ef840d8182e4f3c3da469acf1": " y(t) = |H(i \\omega)| \\ a(t - \\tau_g) \\cos \\left( \\omega (t - \\tau_\\phi) + \\theta \\right) \\ ", "05e99b2733f693b9998df767196a54cd": "\\varepsilon^{\\mu_1 \\cdots \\mu_n} = \\delta^{\\mu_1 \\cdots \\mu_n}_{\\,1 \\,\\cdots \\,n} \\,", "05e9ac62f51dbba005e09d300de60664": "\\omega = 2\\pi f", "05ea09c94632221ae3b86541a2ea035c": " = [F_3, S_3, A_3]::[F_2, S_2, A_2]::[F_1, S_1, A_1]::\\_] ", "05ea312c3901219bb261e3ed52010dbc": " \\epsilon = {v^2\\over{2}}-{\\mu\\over{r}} = -{1\\over{2}}{ \\mu^2\\over{h^2}}\\left(1-e^2\\right) = -\\frac{\\mu }{2a} ", "05ea41806d96ec5cb8f44a2da8405f3e": "r_s\\,\\!", "05ea433257df6dc6c44f7152684deb88": "t^3+pt+q=0 ", "05ea673eea6b99802aa0524f719f51ff": "\\frac{d}{dt}\\langle A(t)\\rangle = \\left\\langle\\frac{\\partial A(t)}{\\partial t}\\right\\rangle + \\frac{1}{i \\hbar}\\langle[A(t),H]\\rangle", "05ea7a1d9defbaf990e9eba60e1bcb2b": "r_1,\\ldots,r_k", "05ea91282cb0b72b3b928c2a6ffe9af7": "x^{q^{n_i}}-x \\bmod f", "05eaa64619424b6173e145259a040d6b": "\\boldsymbol\\epsilon_i \\sim N(0, \\boldsymbol\\Sigma_{\\epsilon}^2).", "05eb0089b956a39fe3bc207e4d6a7013": "\\langle\\mu\\mu|\\lambda\\lambda\\rangle", "05eb1dee4e92fb17bf8d6ddfc587a387": "T^*(x_1, x_2, x_3, \\dots) = (0, x_1, x_2, \\dots).", "05eb2a561d1784325ed89cc26246cb9a": "g(x) \\le 0", "05eb726fee1bdb3bf8f6e44507de3cb4": " PV = \\frac {FV} {\\left( 1+i \\right)^n}\\,", "05ebaa7f76fedca83ba62c60d06094ed": "T_{\\Phi} := \\{ \\; \\overline t \\; |\\; t \\in T^S \\} ", "05ebf1eb685f0543a778bf06239aff7f": " |B| \\geq \\binom{n_i}{i-r}+\\binom{n_{i-1}}{i-r-1}+\\ldots+\\binom{n_j}{j-r}. ", "05ec70e3150e60a283d05a974e47b16a": " g_{2m+1} = (2m+1) g_{2m}\\,.", "05ecbc5d16691a7e8d021d4e3e941e5e": "\\mathbb C^m", "05ecfb7f01a85bf497a522d8b6470404": "(X_0, X_1)_{\\theta,1} \\subset X \\subset (X_0, X_1)_{\\theta,\\infty},\\,", "05ed12f72126f770fa209561c79dc1ab": " \\lambda_1 \\simeq \\lambda_2 \\gg \\lambda_3 \n", "05ed131f606dc23ff7455a4c2b68d667": "a_i \\in A_i", "05ed36d37a7191b670f20c048bee54fb": "H(p,m) = H(p) + D_{\\mathrm{KL}}(p\\|m),", "05ed5d9424c97dc464245040474a92cf": " \\cup", "05edcf086e29f22c22810eeaf4c1fff2": " \\operatorname{de-lambda}[x\\ x] = \\operatorname{de-lambda}[f\\ (x\\ x)] ", "05ee121ad3a56550731350e3f6a1b768": "q \\geq p", "05ee3cd596266e4d18eec9b47db9924d": "\n \\boldsymbol{N}^T\\cdot\\mathbf{n}_0~d\\Gamma_0 = d\\mathbf{f} \n", "05ee6c3c79b2396c35dd23c5e78a511c": "A \\in \\mathbb{C}^{n \\times n}", "05ee6e4e0ded01cf48b7400aaf57c2c8": "\n\\lambda'_k = \\begin{cases}\n4 \\lambda_k - 2 m_k, \\, \\text{ if } 0 \\leq k < n \\\\\nL, \\, \\text{ otherwise }\n\\end{cases}\n", "05ee960dda2b244449b33d7306c450fd": "\\zeta = \\sum_{j=1}^N \\sqrt{2 S(\\omega_j) \\Delta \\omega_j}\\; \\sin(\\omega_j t - k_j x \\cos \\Theta_j - k_j y \\sin \\Theta_j + \\epsilon_{j}).", "05eec65bbbf300943c6628e620d68c44": "\\operatorname{Spec} R[x] \\to \\operatorname{Spec} R", "05eef43d884d5cad5a6230f1d8c963c2": "\\forall L\\in \\textrm{PSPACE}, L\\leq_p \\textrm{TQBF}", "05ef3ac55920353bc2600b4f1953f9fa": "\\displaystyle{\\begin{pmatrix} a & b \\\\ c & d\\end{pmatrix},}", "05ef66efbdaf714b32b946c069c9273e": "\\langle(\\Delta E)^2\\rangle=\\langle E\\rangle^2/m", "05ef6956fd1187aa30f900c8668fe0ad": "a,b,c=1\\ldots N^2-1.", "05ef7a0353032e90e6e93136804f9dc9": "e(\\mathbb Z^2)=1.", "05ef9e5e9e92371c226768239fd9905d": "\\begin{align} (Q^T)^T (Q^T) &{}= Q Q^T = I\\\\ \\det Q^T &{}= \\det Q = +1. \\end{align}", "05efc84b2e65c00051a28dc40391de6e": "\\{ x_0, x_1, \\ldots, x_n\\}", "05efc9164c4af919bc6100c7b8c98280": "\\Psi(x) = \\int_0^x (\\Phi')^{-1}(t)\\, dt.", "05f004482ab1df787ffae85b2eda7c27": "h(R)=|a_0|\\,R^{-k}+\\cdots+|a_{k-1}|\\,R^{-1}-|a_k|+|a_{k+1}|\\,R+\\cdots+|a_n|\\,R^{n-k}", "05f0621535b4e916222d563fe82fc59c": "\\Psi\\left(\\mathbf{r},t\\right) \\,\\!", "05f091ec9712d065e1604d2f58048333": "P(\\ell)", "05f0da43780aff7c4fcda9e4381a9b29": "w(\\alpha \\widehat{x} \\beta, \\gamma \\widehat{y} \\delta) = \\alpha x \\gamma \\widehat{y} \\delta \\beta", "05f113a1f78b0dddf389345d9a093844": "j = 0 ", "05f1327cae46de4114ff4ff01c274eec": "C'_{Op}", "05f1400dc41cdacc6e26c688a0573502": "V_t ", "05f146b6cb758883507b70d1df4c8c8f": "\\mathfrak{h}^*", "05f1b458bef42023cc00cfb97f231e71": " = n\\,c_P\\,\\Delta T ", "05f1c3c597476cc48987621dad890333": " n = ( t / D )^2 a m^{( b - 2 )} ", "05f1fc8045e32ac73c7514938f72e5ca": "M_{y} =\\int c_{y}d\\dot{m}", "05f219042d02159d7978ee14b2b9b44f": " L = V - \\{s,t\\} ", "05f2812f1ead7ec30a37ddbc1c0f1f6d": " = 2 + \\frac{2}{3} + \\frac{4}{15} + \\frac{4}{35} + \\frac{16}{315} + \\frac{16}{693}\n+ \\frac{32}{3003} + \\frac{32}{6435} + \\frac{256}{109395} + \\frac{256}{230945} + \\cdots\\! ", "05f297b7a9c0d1251633dc3be8dc3817": "p^{*} = 0.528 p_0", "05f2c01a0a6c2339d4f459ce6354e581": "\n\\begin{align}\n\\boldsymbol\\beta'_1 &= \\boldsymbol\\beta_1 - \\boldsymbol\\beta_K \\\\\n\\cdots & \\cdots \\\\\n\\boldsymbol\\beta'_{K-1} &= \\boldsymbol\\beta_{K-1} - \\boldsymbol\\beta_K \\\\\n\\boldsymbol\\beta'_K &= 0\n\\end{align}\n", "05f2f1f6344d4bdb8370c1ab38b04219": "F(x,y,z)", "05f31527b8adc26005838c7536768283": "\\boldsymbol{F} = \\boldsymbol{F}(\\mathbf{X},t)", "05f315c99f1d4a4add6ed521ed43de20": "{\\Sigma^{\\infty}_{n=1}}(a_{n})", "05f3d6d19f3c15c4eaaa7477b12581ea": "P(H_1|E)", "05f4f243796cfa7f7b0c0c4469571465": "c_1 \\in C_1, c_2 \\in C_2", "05f51a092217d8e2c80ec9b79cf181be": "\\mathrm{d}U = \\delta Q - \\delta W\\,", "05f51a2cdca83e228f598eb61013653c": "(p_1,p_2,\\ldots,p_d),\n\\sum_{i=1}^d p_i=1", "05f53074d1c7f85781e7ffb09990b40a": "l_a=(-1,0,0,0)\\,,\\quad n_a=(-\\frac{F}{2},-1,0,0)\\,,\\quad m_a=\\frac{r}{\\sqrt{2}}(0,0,1,\\sin\\theta)\\,.", "05f55405e2597005963fc687fc0a397a": "H^n(A) = H^0(A[n])", "05f5a83cb1eace02ff37ab6b5ad9d92c": "a(t)=e^{t\\delta}\\,", "05f622803eb90d0598940885b74f7aff": "\n \\hat{y}_j^{(j)} - \\hat{y}_j = x'_j\\hat\\beta^{(j)} - x'_j\\hat\\beta = - \\frac{h_j}{1-h_j}\\,\\hat\\varepsilon_j\n ", "05f6292a398e85d0f6da63e624084153": "m_1 \\ne m_2", "05f6407a1cfade6431a89db24e251d94": "\\ PER = \\left ( uPER \\times \\frac{lgPace}{tmPace} \\right ) \\times \\frac{15}{lguPER} ", "05f66fa6148b8f62b8f3ce1f2639b1a3": "\\left (\\frac{E[V]}{R} \\right )", "05f6fdde4f929e7e7849e061fd649a9e": " \\cos \\theta = \\left(\\frac{\\gamma_s - \\gamma _{ws}^0 +\\frac{CV^2}{2}}{\\gamma_w}\\right) \\,", "05f72aa198a77e03b3e1af085701840b": " \\mathbf{\\bar{x}}=\\frac{1}{N}\\sum_{i=1}^{N}\\mathbf{x}_i. ", "05f74d85a66227de47682bafc2345668": "\n\\Delta \\bar{e}\\ =\\ \\frac {J_2}{\\mu\\ p^2}\\ \\int\\limits_{0}^{2\\pi}\\left(-\\hat{t}\\ \\left(\\frac{p}{r}\\right)^2\\ \\frac{3}{2}\\ \\left(3\\ \\sin^2 i\\ \\sin^2 u\\ -\\ 1\\right)\\ - \\ \\left(2\\ \\hat{r}-\\frac{V_r}{V_t}\\ \\hat{t}\\right)\\ \\left(\\frac{p}{r}\\right)^2\\ 3\\ \\sin^2 i \\cos u\\ \\sin u\\right)du \n", "05f76669a2ca87733a9fd1b530985ebb": "d_2,d_3,\\dotsb,d_m=0", "05f78fe4ec8e018b4d0486c5e98d45d4": "\\int_A f\\,d\\mu = \\int_{[a,b]} f\\,d\\mu", "05f7e498b0add2d4f3a946e530af830f": "\\forall k\\geq N_2 \\Rightarrow \\|A^k\\|^{1/k} > (\\rho(A)-\\epsilon).", "05f7e7c3f82b8ae7fc229dcf117d33ca": " 3 q _{2}q _{3} + 3q _{3}q _{4}+ 3 q _{1}q _{2}+q _{2}q _{4} + q _{1}q _{4} - q _{1} - q _{3} - q _{4}", "05f7ff803727730e147a6364410df1a2": "\\nu(x) = j", "05f846ca0e56a4867fd161ad252d994c": "\\frac{\\sigma^2}{2 a}", "05f86dee230cb1b9bb63d2159ad4449d": "P(B)=0,", "05f875bff224e8a484deca81bd4509fd": "\\frac{|\\text{actual effort} - \\text{estimated effort}|}\\text{actual effort}", "05f8831c4b653ded6674c224df25afb1": "\\left| \\mathbf{q} \\right|", "05f910f9d3bdd8ac40ec33c32909a772": "\\sigma_1(A)", "05f928dfb944e822440d7fe52821a2c3": "\na =x_1+x_2\n", "05f9516e185c7a916bc48bcc2a83f9d8": "S=\\bigcup(S_i\\mid i\\in I)", "05f98e5a81d331404f783b349fbf36f7": "y(r)=e^{\\beta u(r)}g(r)", "05f9c7abda372d91d676d45eae84a70b": " M = \\frac{p\\,(p-1)}{2}. ", "05f9d8f712baaeb6b5630bb6b919c139": "\\mathbb{S}", "05f9ed3a71241f7c04b687316fd91004": "n_s\\,\\!", "05f9f22c2944bba1198040d2a3edc044": "\\lim_{n\\to\\infty} z^{\\pm n}", "05f9f742d73d155fc3e9a8a071c7286f": "\\dot p = -\\frac{\\partial H}{\\partial q} = \\{p,H\\} = -\\{H,p\\} ", "05fa27202aff9c146e3eabdbd86d2ec5": "N = \\rho / (1 - \\rho)", "05fa7db7bce48752a8bfdb32d3b9c2c5": "\\begin{align} \n &\\sum_{r = 0}^\\infty a_r(r + c)(r + c - 1)x^{r + c - 1} -\\sum_{r = 1}^\\infty a_{r - 1}(r + c - 1)(r + c - 2) x^{r + c - 1} +\\gamma \\sum_{r = 0}^\\infty a_r(r + c) x^{r + c - 1} \\\\ \n &\\qquad -(1 + \\alpha + \\beta) \\sum_{r = 1}^\\infty a_{r - 1}(r + c - 1) x^{r + c - 1}-\\alpha \\beta \\sum_{r = 1}^\\infty a_{r - 1} x^{r + c - 1} =0\n\\end{align}", "05fa87902f89e288654d0e2752b0cefa": "M=E-\\varepsilon\\cdot\\sin E.", "05fb05070d1664acb2a478a9111e4853": "\\mathbb{Q} \\cap [0,1]", "05fb116dd77209938c1398a35cd8b116": "\n \\begin{align}\n \\frac{\\partial I_1}{\\partial \\boldsymbol{A}} & = \\boldsymbol{\\mathit{1}} \\\\\n \\frac{\\partial I_2}{\\partial \\boldsymbol{A}} & = I_1~\\boldsymbol{\\mathit{1}} - \\boldsymbol{A}^T \\\\\n \\frac{\\partial I_3}{\\partial \\boldsymbol{A}} & = \\det(\\boldsymbol{A})~[\\boldsymbol{A}^{-1}]^T \n = I_2~\\boldsymbol{\\mathit{1}} - \\boldsymbol{A}^T~(I_1~\\boldsymbol{\\mathit{1}} - \\boldsymbol{A}^T)\n = (\\boldsymbol{A}^2 - I_1~\\boldsymbol{A} + I_2~\\boldsymbol{\\mathit{1}})^T \n \\end{align}\n", "05fb99d59ad9c1d1e6e39dab062a8b33": "m_{1}u_{1} + m_{2}u_{2} = m_{1}v_{1} + m_{2}v_{2} = {0}\\,\\!", "05fbd3a56cf9aeb0671739bab7c850f9": "t \\in [0,1],", "05fbe6c52ef5b1319c97203ef8abc1b5": "\\omega=\\omega_o\\left( 1 +\\frac{\\mu BH_k}{kL_{e}^{2}(B+H_k)}\\right)^{1/2}\\approx \\omega_o\\left( 1 +\\frac{\\mu BH_k}{2kL_{e}^{2}(B+H_k)}+...\\right)\\Rightarrow", "05fc110a8dbad659411d44f326dfbc99": "w : X \\vdash w : X", "05fc67ba192a27ca6b8ae5d30e2978e7": "\\scriptstyle \\partial \\vec{D} / \\partial t", "05fc9804f26c17af7a6f5844a7678d2d": "\\oint_S \\mathbf{B} \\cdot \\mathrm{d}\\mathbf{A} = 0,", "05fce8ddbd2502dc79f9950775edce6f": "v X = \\{ \\lambda \\in X : r ( \\lambda ) = v \\}", "05fd0c333ff9be30a0e6c163a5d092a3": "K(GL(R), 1)", "05fd1c7db07940b7a7afe1e193282045": "{\\mathfrak{m}_B}^s \\subset (y_1, \\dots, y_m) + \\mathfrak{m}_A B", "05fd43ad299946ac38adffa752f99a60": "\\Gamma:={\\Bbb Z}^3\\ltimes{\\Bbb Z}", "05fd66fa5a897319d0fb87bd24af04d3": "\\tfrac{1}{24} \\left ( (\\operatorname{tr}A)^4-6 \\operatorname{tr}(A^2)(\\operatorname{tr}A)^2+3(\\operatorname{tr}(A^2))^2+8\\operatorname{tr}(A^3)\\operatorname{tr}(A) -6\\operatorname{tr}(A^4) \\right )", "05fd7fc4c1d13bec4e3bdd8523ba2fa5": "\\overline{W}_{\\dot{\\alpha}}", "05fd9792691ff82531e230768864e180": "2^{S''}", "05fe086cb3d686ae49d586d8f95414f6": "\\left|\\widehat f(n)\\right|\\le {K \\over |n|}", "05fe12829fff81295b9bef693f5e8779": "\nv(\\mathbf{r}) \\ \\stackrel{\\mathrm{def}}{=}\\ \\int d\\mathbf{r}^{\\prime}\\, \\rho_{uc}(\\mathbf{r}^\\prime) \\ \\varphi_{\\ell r}(\\mathbf{r} - \\mathbf{r}^\\prime)\n", "05fe9c80d66c928b66cdc1e76dd5efbc": "D=\\tfrac{a}{\\sin \\alpha} = \\tfrac{b}{\\sin \\beta} = \\tfrac{c}{\\sin \\gamma}.", "05feaeeb29070ecee88283b395e32236": "\n= \\sum_{k=0}^n k! \\,S(n\\!+\\!1, \\,k\\!+\\!1) \\left({z \\over {1-z}} \\right)^{k+1} \\qquad (n=0,1,2,\\ldots) \\,,\n", "05ffb8526e575ca4f6ccdd8ff33ca71c": "\\bar{f}(s)=\\int_0^\\infty e^{-st}f(t)\\,dt", "05ffbdb4aeadaadb02ee46b499f9ce2d": "\\dot{x}(t)", "06001471dde5949692c7cf2cf7feda6b": "\\begin{matrix} {10 \\choose 1}{4 \\choose 3}{44 \\choose 1} \\end{matrix}", "060038299b950bc5d6c8e81975ed65fe": " \\Rightarrow \\frac{p(y|H2)}{p(y|H1)} \\ge \\frac{\\pi_1}{\\pi_2}", "060064589e48960b70d7488cdb0f6d66": "\\gamma_k =\\frac{1}{y_k^T s_k}.", "0600706e87ee0b62690eaac783c0a96d": "M_{\\pi_T}^2 \\propto \\langle\\bar{T}T \\bar{T}T\\rangle_{M_{ETC}}", "06009ffd2c6b4c9bee322cb86461806e": " \\tfrac5{36} - \\tfrac1{30} \\sqrt{15} ", "0600b0075f1dc8c5beeb7e0c89d1be2e": " K= C_{12}+C_{23}-C_{13} \\le 1", "0600ce9319de00e376d249db90db96eb": "\\mathbf{w}_{n}=\\mathbf{R}_{x}^{-1}(n)\\,\\mathbf{r}_{dx}(n)", "0600eb7f294010969188a9763065934e": "\\left(\\frac{\\Delta Q}{\\Delta t}\\right)_\\mathrm{bar} = \\left(\\frac{\\Delta Q}{\\Delta t}\\right)_\\mathrm{water}", "06010d437e589a532f147b49326d1bb0": "* \\!\\,", "0601d5da1d270f4663f165701f1c9798": "q_1 = 1+\\frac{\\sum_{i=1}^k \\pi_{i}^{-1}-1}{6N(k-1)}. ", "0601ef2dbd4b2eae873ecbaf02ba45cc": "\\delta_t, \\, t \\in G,", "06020d9ff8c01eaaf44943780aa8a89d": "x = jb\\,", "0602535e8203a4f9c0f07088182fe798": " U_s = 2 \\left|s\\right\\rangle \\left\\langle s\\right| - I", "060268a090fed8d9854efb535e06332b": "\\Delta \\vec{F} = \\vec{F}_{n} - \\vec{F}_{n-1} ", "060290f166448ac0480686e89f6a921a": " U \\cap A = T.\\,", "060301f9fe00c278acc161de360ced0d": " A: G\\times M \\to M ", "06031c8d29a41ca293d19d7d397017de": "(11, 5_2,4, 1)", "06031cba9297343eabe2961fa3da37f3": "(NB)/3", "060361fbb611719487b00f78f51cbf9b": "\\int f(x)\\sin(x)\\,dx=F'(x)\\sin(x)-F(x)\\cos(x),", "0603b8974e76582d6b317b0aa99346f7": "\\Pi_{H}(m)\\leq\\left( \\frac{em}{d}\\right)^{d}\\,\\!", "0603ba49d242efbd716fdcc687d4aaf4": "\\mathbf{c}, \\mathbf{b}", "060462a5b69fe5821f4e5c6375706bd6": "\\partial_i\\ell", "0604dfb6d9db52ca41732d4f1c82a753": " \\frac{1}{F_{max}} = \\frac{1}{F_{e}} + \\frac{1}{F_{c}}", "060562d10a1d73260d67b7623181857c": "N = R_{\\ast} \\cdot f_p \\cdot n_e \\cdot f_{\\ell} \\cdot f_i \\cdot f_c \\cdot L", "0605d13e5ecf124707bc65207eb9065c": "\np = {h \\over \\lambda}\n", "0605ec8ff0e079dfb988391330236abf": " w \\cdot ( u \\wedge v) = \\frac{1}{2}( w ( u \\wedge v) - ( u \\wedge v) w)\n", "0606585ddb28d26b541178a2fd750d74": "\\theta = \\begin{cases} \\sin^{-1} \\frac{1}{\\beta}, & \\text{if }\\beta \\ge 1 \\\\ \\pi - \\sin^{-1}{\\beta}, & \\text{if }\\beta\\le 1\\end{cases}", "06066522c496336b0fd736296a1d0d9d": "\\gamma = \\frac{(1+w)G_s\\gamma_w}{1+e}", "0606dd106fbc2416e0716e2252336df9": "\\displaystyle s_\\mu h_r=\\sum_\\lambda s_\\lambda", "0606e81e12923a421971691286f2935c": " x", "0606f21326dd8210d4402885228f181e": "M_0,M_3,M_5,M_6", "0606f740fa19e367728576ebdd03c049": "k_{\\rm{adj}} = k \\left( \\frac{\\mbox{maximum rotor-speed}}{\\mbox{actual rotor-speed}} \\right)", "0607238db7ad004ed43ca8e1dbef539d": "z_1,\\ldots,z_n", "0607262f81fbe5797380178222c0068c": "\\nabla(\\nabla\\cdot\\vec{A}+\\frac{1}{c^2}\\frac{\\partial\\varphi}{\\partial t})=\\mu_0\\vec{J}-\\frac{1}{c^2}\\frac{\\partial^2\\vec{A}}{\\partial t^2}+\\nabla^2\\vec{A} ", "0607374c116257a45022bbd802572b26": " \\gamma_3 = \\sqrt{\\frac{2}{\\pi}}(\\sigma_2-\\sigma_1)\\left[\\left(\\frac{4}{\\pi}-1\\right)(\\sigma_2-\\sigma_1)^2 + \\sigma_1 \\sigma_2\\right]", "060746f5f4519d2e745eaba4708111c1": "E - e\\phi \\approx mc^2", "0607473e9e177ff05e9a6f4d1bf1fd81": "U(s)", "06079a798ccec5f66f8ecb8704f52987": "I=q/t\\,", "0607db2a0900d1cf784c4cd826368deb": "SS_\\text{res}=\\sum_i (y_i - f_i)^2\\,", "0607db8fc32c6ebe9fe571ceec46879d": "T_{11} = \\left(2C_1 + \\frac {2C_2} {\\alpha} \\right) \\left( \\alpha^2 - \\alpha^{-1} \\right)", "060817c208d5981b1485cabd5bdb5139": "pN", "06081fb3e714ed01d831ebd8513f1822": "\\frac{Gross Profit}{Sales}", "060844221f545e5bf6862e60aaec07aa": "\nds^{2} = d\\mathbf{q} \\cdot \\mathbf{M} \\cdot d\\mathbf{q}\n", "06084b087fb41001d770760b25cbe12f": "\\left\\{\\pm\\frac{\\pi}{2}, \\pm\\frac{3\\pi}{2}, \\pm\\frac{5\\pi}{2}, \\ldots \\right\\}\\,.", "0608732f994277f423acfaef18f70d8a": " \\rho \\neq e", "0608e49f58cb46fab57a77087a85d990": "a = 2, \\, b = 2, \\, f(n) = n^2", "0609110318e4878dbb0eeb0ccf3b336e": " \\mathfrak{g} = [\\mathfrak{g}, e] \\oplus \\mathfrak{g}_f ", "06092a49718e3e55aa32259d4a1cbdc0": " \\dim \\mathfrak{d} = \\dim W - 1 ", "0609778b9c9c588b63b6a3732e9fee9d": "K = {k_1,k_2, \\dots ,k_n}", "06099bc35e1dbd78c6c50816a9cd892d": "\\overline{x}\\,", "0609d71ea290e86c7da521ed45f0de14": "\\Delta x'", "0609f218ab24220de50e4a0bca984c61": "\\ln r = x\\ln[A] + \\textrm{constant}", "060a9233be6ac589bd81a3756d5b0a4d": " \\sum{x_i} \\leq k ", "060ab80287a2426f32708c585e447161": "y\\in Q^n", "060ac1614e0969e935138d1e7dd96062": "C_{m}", "060af5819fb36c7d0154761c2b3697c4": "\n \\mathbf{a}\\cdot(\\mathbf{b}\\times \\mathbf{c}) =\n-\\mathbf{a}\\cdot(\\mathbf{c}\\times \\mathbf{b})\n", "060afc5a1b7d77a2e221639a9fe8fee7": "\\left.u_p\\right|_{r= R}=0", "060b84c979ace117697e202f36c77586": "q\\begin{Bmatrix} p, q , r \\end{Bmatrix}", "060b92061c5b5c477d2b4fded0e27d96": " \\vec{e}_2 = \\partial_x ", "060bfd719f6fd57edd4f3521c612dbdb": "\\partial(X,f,\\alpha)=2\\pi \\sqrt{-1}\\sum_i(V_i, f_i, res_{V_i}\\,\\alpha)", "060c301f6ac199cfb7701726bef0dcf4": "\\scriptstyle r", "060c63b9bc19a48246bfbfe3435cbc3a": "A_\\lambda", "060c6d21eec9d49717d5dd5a7c768c0d": "\\mathcal{H} = L^2(\\mathbb{R})", "060cc3fcebf81f5a13d8a8de42b490f2": "(V_i, V_j,)", "060cd7165c91012d5c391e921f7a9930": " \n\\begin{align}\n&y_{0} &=&\\ y(0)+ L^{-1}(-1) &=& -t \\\\\n&y_{1} &=& -L^{-1}(y_{0}^{2}) =-L^{-1}(t^{2}) &=& -t^{3}/3 \\\\\n&y_{2} &=& -L^{-1}(2y_{0}y_{1}) &=& -2t^{5}/15 \\\\\n&y_{3} &=& -L^{-1}(y_{1}^{2}+2y_{0}y_{2}) &=& -17t^{7}/315.\n\\end{align}\n", "060d0527eba6da4161bcb4b833b41c31": "Q(\\boldsymbol{r}) = Q(F_\\boldsymbol{r}), p(\\boldsymbol{r}) = p(F_\\boldsymbol{r}),", "060d13166c74ee3cc0985680289cf42a": "u_{\\max}^{(s+1)}=\\frac{1}{x^{(s)}},\\ k^{(s+1)}=\\left[\\frac{1}{x^{(s)}}\\right].", "060d52f0747e46a84b87ab0515dfdfb1": "\\mu_1, \\mu_2, \\ldots, \\mu_r", "060d68ae440ca0f8f5b87557cefde05b": "\nT=\\frac{V}{A} \\cdot 0.161\\,\\mathrm{s}\n", "060d6ca8599c55633a112da0b64b25bf": "\\int_{-\\infty}^{\\infty} |\\psi (t)|\\, dt <\\infty", "060dc851ace6e3e11ffc450cc603ec99": " \\eta=\\frac{y}{\\delta(x)}=y\\left( \\frac{U}{\\nu x} \\right)^{1/2}", "060e428bbbf6496c2e7d9b8a308ee239": " m", "060ee93d0601609f694cfe42a429e569": "l=d+w", "060f013cc49db63b4af50b03a20996f2": "\\scriptstyle \\mathbf{D}", "060f03bd35e64518bb9744cd7aa00b5a": "R(w) =\\sum_{g=1}^G \\|w_g\\|_2,", "060f40333258faf628efce4f086a01f3": "f: \\mathbb{R} \\rightarrow \\mathbb{R}^+", "060f987de88e7c8d4afad7d4828e3f7b": "\\sup_{\\theta \\in \\Theta} R(\\theta,\\delta^M) = \\inf_\\delta \\sup_{\\theta \\in \\Theta} R(\\theta,\\delta). \\, ", "06108a0b8b6dcaa756c6c3ab6317551d": "p=\\frac{N_0-N}{N_0}", "06109000b497df97e7b4118d2b5f9c41": "\n w = 0 \\,, -D\\frac{\\partial^2 w}{\\partial y^2}\\Bigr|_{y=b/2} = f_1(x) \\,, \n -D\\frac{\\partial^2 w}{\\partial y^2}\\Bigr|_{y=-b/2} = f_2(x) \n", "0610a36cf3ae80b3045fb4b372651650": "\\mathbf{\\nabla}\\times(\\mathbf{\\nabla}\\times\\mathbf{V})=\\mathbf{\\nabla}(\\mathbf{\\nabla}\\cdot\\mathbf{V})-\\mathbf{\\nabla}^2\\mathbf{V}", "061104ac886aef675293663800232f56": " Q^{(1)}, Q^{(2)},\\ldots ", "061107504b5aa7a97959c51cb34e484f": "z\\bar{z}+w\\bar{w}=1.", "06114dd2614cc35393a7c6b2deff8e0a": "Z=\\sum_{n=0}^{\\infty } \\frac{(10n+1) \\left ( \\frac{1}{2} \\right )_n \\left ( \\frac{1}{4} \\right )_n \\left ( \\frac{3}{4} \\right )_n} {(n!)^3{9}^{2n+1}}\\!", "061173042b74c01eb3b2dbbec445897c": "G_{4}(\\mathbf{p}, \\mathbf{P}, t)", "06117fb16c9900d808148064b388381a": "\\bar{n}_i = \\ \\frac {1} {e^{(\\epsilon_i - \\mu)/kT }+1}", "06119872473c06fc42d6f7cf08d6aa41": "E_\\text{K} = \\frac{1}{2}\\sum_{i=1}^n m_i ([\\Delta r_i]\\boldsymbol\\omega)\\cdot([\\Delta r_i]\\boldsymbol\\omega) + \\frac{1}{2}(\\sum_{i=1}^n m_i) \\mathbf{V}_C\\cdot\\mathbf{V}_C.", "0611d5ea94a9498441c4bb70af9d9b60": "\\displaystyle \\frac{1}{|a|}\\cdot \\operatorname{tri} \\left( \\frac{\\nu}{2\\pi a} \\right) ", "06120cc69950c1c1c2a4679a307ac149": "Y_{t} = C_{0} + I_{0} + cY_{t-1} + b (C_{0} + cY_{t-1} - C_{0} - cY_{t-2})", "061299ee08b59ed4968edae3ad322fc8": "S\\in W", "061339dfbd7f3c80d83c9f59490b76fb": " Vol_q(0, \\lfloor {{d-1} \\over 2} \\rfloor) \\le q^{H_q({\\delta \\over 2})n-o(n)} ", "061377df11087841d850ebdd7a81a57c": "M^{0a} = -M^{a0} = K_a \\,,\\quad M^{ab} = \\varepsilon_{abc} J_c \\,.", "0613a14e112170454dd8ee2fac200e33": "b\\cdot a", "06141a1da5d19a810187d649c248c613": "E_c = L^2/(L^2+m^2)", "061453faeff864f7eb127d98843c4c0a": "\\Delta : \\mathcal{C} \\rightarrow \\mathcal{C}^\\mathcal{J}", "0614ad79a0f78028781bb65a4665fcf7": "\\frac{\\pi r}{2}", "0615003c55d5aab471d04225e021cf7a": " Z_{eff} ", "06150743b944ae53760c95d20c1dec95": "q(x,y) = q_0", "061510548cb220ad5348824f657cffca": "k^{-m} E_m(kx)= \\sum_{n=0}^{k-1}\n(-1)^n E_m \\left(x+\\frac{n}{k}\\right)\n\\quad \\mbox{ for } k=1,3,\\dots", "061512d21c171f0f05094bc24900f4ea": "f(x) = x^3-1\\, ,", "06154eca89935c16391249e30b659550": "M_b", "06154fbf5b0d359fdabd084cd66ebc25": "\n\\frac{d}{dx}(x^2)=2x.\n", "061550e9b9bdc85c3f4a8591b42e540b": "\n \\mathbf{J}^{23}\\mathbf{A} = \\left[ \\begin{matrix} 0 & 0& 0 \\\\ a_{31} & a_{32} & a_{33} \\\\ 0 & 0 & 0 \\end{matrix}\\right]\n", "0615609bb804231ecd6e9ea7b59a5ee6": "a_0 + a_1 x_1 + a_2 x_2 + \\cdots + a_n x_n \\leq 0", "06158b7ebe260812220d4e2b7c3ecb90": "0,x_1,\\ldots, x_n\\,", "06159f2da0ee5c095102d190ec683974": "\\ B-\\text{vertex}= 1 : -1 : 1 ", "0615d318431f10aa166cb6d492ff5de2": "d\\colon M \\to M \\colon M_i \\to M_{i+1}", "06161097402a99112a7073c0e6f25328": "H = AF_4 = 0", "06165a207796d90c74c3962037eab3da": " m\\frac{du}{dt}=X_u u -mg\\gamma", "06166b0bfc29b2d32dbc5179ebdab4e7": "[\\nu ]=\\sec ^{-\\mu }", "06168ddfffbb48e0679a58d34ca4e824": "\\pi ab.\\,", "0616964198f654b6b7402626697ec7a4": "V = \\frac {\\pi h^2}{3} (3r-h)", "0616d198ca8080fb18755a5ce61e3e31": " \\partial p/\\partial s=0", "061702b8ec8c978285ef3f1f6486484b": "j_1 ^* \\circ F^* = \\mathrm{id}, \\; j_0^* \\circ F^* = 0.", "061705237c9d8f58e5c9702b0643d447": "Y(s) = \\left( \\frac{P(s)C(s)}{1 + F(s)P(s)C(s)} \\right) R(s) = H(s)R(s).", "06174424b7644856608e3315c3ecadd6": "P_Z ", "06177336f1deb5b1e796c2e24aad5d38": "(\\ g(-\\tau)=g(\\tau)\\ ),", "06181bccf4fdac8791f2b00ede092713": "\\forall x \\in X \\, \\forall y \\in X \\, ( ( x < y \\, \\land \\, \\lnot (y < x) \\, \\land \\, \\lnot( x = y )\\, ) \\lor \\, ( \\lnot(x < y) \\, \\land \\, y < x \\, \\land \\, \\lnot( x = y) \\, ) \\lor \\, ( \\lnot(x < y) \\, \\land \\, \\lnot( y < x) \\, \\land \\, x = y \\, \\, ) ) \\,.", "0618371d023a736616c74fc01e13271b": " {\\rm vec}(\\mathbf{X}^{\\rm T} \\boldsymbol\\Sigma_{\\epsilon}^{-1} \\mathbf{X}(\\mathbf{B} - \\hat{\\mathbf{B}})) = (\\boldsymbol\\Sigma_{\\epsilon}^{-1} \\otimes \\mathbf{X}^{\\rm T}\\mathbf{X} ){\\rm vec}(\\mathbf{B} - \\hat{\\mathbf{B}}), ", "06184cbb2d2852a9c1aa37c1cf47f7a2": "(-\\nabla_x^2 + t) \\langle H(x)H(y) \\rangle = 0 \\rightarrow \\nabla^2 G(x) +t G(x) = 0", "061880dceea6b778b71a416d7c1c4dda": "4. \\; \\; \\mathrm{O} + {\\mathrm{ClO} \\cdot} \\; \\xrightarrow \\; {\\mathrm{Cl} \\cdot} + \\mathrm{O}_2", "0618b0b833d1f4b07fae34fdbd2fe21e": " g(E \\cup F) = g(E) + g(F). ", "0618bec31282a3ef777041db73a0306c": "\\begin{align}\n\\frac{\\partial \\Lambda}{\\partial x} &= 2 x y + 2 \\lambda x &&= 0, \\qquad \\text{(i)} \\\\\n\\frac{\\partial \\Lambda}{\\partial y} &= x^2 + 2 \\lambda y &&= 0, \\qquad \\text{(ii)} \\\\\n\\frac{\\partial \\Lambda}{\\partial \\lambda} &= x^2 + y^2 - 3 &&= 0. \\qquad \\text{(iii)}\n\\end{align}", "0618e26d2e432a00aec6fabf72a09380": "\\Re(s)=1. \\, ", "06193df674d572760c5db06e18741194": " \n{\\eta}", "06195a4fb543bcc3edabb8d8109acf77": "\\ \\frac{S}{C}=\\frac{1}{1300}\\frac{R^3}{P_tG^2\\lambda^2}\\frac{1}{\\tau\\theta\\sec\\psi\\sigma^o}\\frac{P_tG^2\\lambda^2}{(4\\pi)^3R^4}\\sigma", "06198eabc68808757ee3fa9a544a9f9e": "-{T^a}_b \\, Y^b", "0619a61d3d587407b32bb48256faf6bd": "\\color{Black}\\tfrac{4}{m}\\tfrac{2}{m}\\tfrac{2}{m}", "0619cfe36a6a71767c71545a48967dd4": "y=2.870x - 3.000x^2 - 0.275", "061a00186250241da69478b06d200c8f": "F(x) = f(x) - i g(x)", "061a01f63f1d2ef8c5e01e4a5bca087a": " F(x;b,\\eta) = \\left(1 - e^{-bx}\\right)e^{-\\eta e^{-bx}} \\text{ for }x \\geq 0. \\,", "061a43b72d779f9cbd8d90dd97a90679": "i =j", "061a92807a3c76be3ec523cf988cae28": " Sym^k (V) \\to Sym^k (V)", "061a979350a81ea6511a7124316f5899": "\\Delta v = \\sum_{i=0}^{n-1} {Ve}_i \\cdot ln \\frac {Minitial_i} {Mfinal_i} ", "061aa0e33a22e7061ea23b9cc82e4226": "ax + by + c = 0", "061aa8475215a37cad95a2c964bf6c0e": "|x|^2", "061abf8bccd1fe41c7f24818fb10e861": "C_{n+1}(L) = \\pi_n^{-1} ( Z ( L / C_{n}(L) ) )", "061acf5b000d56fe69754ded04693653": "m = v", "061af55e64886354d06554c5b95bbb5c": "y=a\\sin\\left(u\\right)\\sin\\left(v\\right)", "061b477bc57099c87f264be0934ccd6d": "Q_i=\\{(s_i,t_{ei})| s_i \\in S_i, t_{ei} \\in (\\mathbb{T} \\cap [0, ta_i(s_i)])\\} ", "061b8d113b20df75d64f158d0c902cc5": " g'(\\nu) = {1 \\over \\pi } { (\\Gamma / 2) \\over (\\nu - \\nu_0)^2 + (\\Gamma /2 )^2 }", "061b9d3c6eeb7b63db0df009b21a3957": "\\sum \\mathbf{p}", "061be9c6449ff9eb3eb38603c3aafbc8": "\\tau~=~i~C~t ,", "061bf8988a0342a0119609f12317688d": "\\lambda=\\operatorname{E}(X)=\\operatorname{Var}(X).", "061c85ceafcd4d4e2127e7073dbc9b6c": "P_i = wl_i + (1 + r) wl_A a_i", "061c8bf9ac1c45cd53ea36123f57aa4a": "m = ", "061ca2407a364c9ed7be5e86bc31c4cb": "\n\\begin{matrix}\nQ^2+U^2+V^2 \\le I^2.\n\\end{matrix}\n", "061ca39c6c549e7b737974e06028f3f8": "\n\\sum_{k=1}^N \\mathbf{F}_k \\cdot \\mathbf{r}_k = \n\\sum_{k=1}^N \\sum_{j=1}^N \\mathbf{F}_{jk} \\cdot \\mathbf{r}_k.\n", "061cc8162de7c13e3efbbbbba68e7e04": "\\tfrac{m}{n}", "061d4742bddb491b499e54c32cf2b68e": "w_j = \\frac{1}{\\prod_{i=0,i \\neq j}^k(x_j-x_i)}", "061d964ac1f0e7c9bb70b6e25980ecc1": " \\langle \\mathbf{ABC} \\rangle = 0,", "061db0126fb94036224aab33e423811b": "= \\frac{D}{V_\\text{d}}", "061dd8f29ebee8ad5e2369c0fdfac9bb": " p = \\sum_\\alpha p_\\alpha X^\\alpha,\\ ", "061df79c44ad52e26e803baec66b16d6": "d_{X,Y}=1-\\rho_{X,Y}.", "061e4287d2c03e68d2f06a853c9e345c": "\\psi(\\Omega)^3", "061e663fd567a45ac9d046e78a1b3caa": "\\mathsf{I} \\left(\\sum_{i=1}^n r_i \\mathit{1}_{ [x_{i-1},x_i)}\\right) = \\sum_{i=1}^n r_i (x_i-x_{i-1})", "061e68340720962b50a1a1f6b17c14a5": "C_\\text{V}(x) = \\exp(- b_\\text{ext} x)", "061ee6c53cb1c7143dffc932f58467ec": "\\mathbf{q}_{1}", "061fac1ccb610da7c6c849fc1b85d3ed": "E(X~|~X\\le z)\\cdot Pr(X \\Delta r^2\\\\\n s^2 &< 0 \\\\\n\\end{align}", "062034e1b1e77f92fa8d08995925acf7": " s \\in (0,\\infty) ", "062040c65b51cd1374b751276f041f00": "O(n \\cdot \\log n)", "062096cbf7815cd82b26fe175a0f1fb2": "\\begin{align}\n\\mathcal{A}\\left\\{x(t-\\tau)\\right\\}\n&= \\int_{t-a}^{t+a} x(\\lambda-\\tau) \\, \\operatorname{d} \\lambda\\\\\n&= \\int_{(t-\\tau)-a}^{(t-\\tau)+a} x(\\xi) \\, \\operatorname{d} \\xi\\\\\n&= \\mathcal{A}\\{x\\}(t-\\tau),\n\\end{align}", "0620ac251740f66d804ae4a7cae323d4": "\\infty_1 ", "0620e7e777cd8b7e87bc391a420202f3": "6\\cdot 6=36 > 27", "062129255c464930f035184ee28f91d0": " P-P_{-1}\\ \\approx \\pi ", "06213f7b6eb6c9529be8a52b9e59b147": "\\ln \\ln \\frac{\\varepsilon^{(s+1)}}{\\varepsilon^{(s)}} = \\eta_s + \\sum_{p=0}^{s-1} \\xi_p, \\quad \\eta_s = \\ln \\left [ 2\\delta^{(s)} \\left ( k^{(s)} + x^{(s)} - 1 \\right ) \\Omega^{(0)} \\right ] ", "06215cc7ede143df16d1e4a54b219d39": "3n^2", "0621b31ee4fdc1084fa5d458679be123": "\n\\begin{pmatrix}\\mathrm{Cu}\\\\\\mathrm{Ag}\\\\\\mathrm{Au}\\end{pmatrix}\n\\begin{pmatrix}\\mathrm{Al}\\\\\\mathrm{Ga}\\\\\\mathrm{In}\\end{pmatrix}\n\\begin{pmatrix}\\mathrm{S} \\\\\\mathrm{Se}\\\\\\mathrm{Te}\\end{pmatrix}_2\n", "0621d7da3d66fb1c2e19a8e9ba982159": " x^\\alpha = x_1^{\\alpha_1} x_2^{\\alpha_2} \\cdots x_n^{\\alpha_n}. ", "0621fdb9b047e3455b27417e76bc6dbb": "S_k(n,r) \\cong \\mathrm{End}_{\\mathfrak{S}_r} (V^{\\otimes r}).", "0621ffe2cc7912d02595966ce1095472": " p_c ", "06228e7e11688a89df9a6ef09a3684bb": "O(\\frac{n^2}{m^2})", "0622c18f16ae745537349b8d5f629fe5": "|1\\rangle\\leftrightarrow|2\\rangle", "0622d0f2c3a2b81593b8db0108871121": " \\left(\\mathbf{A} - \\lambda_i \\mathbf{I}\\right)\\mathbf{v} = 0. \\!\\ ", "062314ee4515e21c160b657d3f3763b0": "\\langle a \\rangle", "06236523bedf0e5f6a9d963cebdd55b5": "\\!\n\\Bigl\\langle x_{m} \\frac{\\partial H}{\\partial x_{n}} \\Bigr\\rangle = \\delta_{mn} k_{B} T.\n", "0623691cb796dbcb5716c7cc29380dd2": "\\sqrt{8r\\left(\\sqrt{4R^2+r^2}-r\\right)}\\le s \\le \\sqrt{4R^2+r^2}+r", "0623bf85ab6b26f9ea9d75e605791ab6": " \\frac {d \\mathbf{u}_j (t)}{dt} = \\boldsymbol{\\Omega} \\times \\mathbf{u}_j (t), ", "0623cdec0ce139d951d43c42901b2bc7": "= \\left[ {n \\choose 0} \\cot^n x - {n \\choose 2} \\cot^{n-2} x \\pm \\cdots \\right] \\; + \\; i\\left[ {n \\choose 1} \\cot^{n-1} x - {n \\choose 3} \\cot^{n-3} x \\pm \\cdots \\right].", "0623e20e090df47dffc522dba9515e31": "f(q(\\xi ,\\tau))^{\\;}", "062404cdddf8b568d4aec12e5ba37a13": "\\int x^3 r^{2n+1} \\; dx = \\frac{r^{2n+5}}{2n+5} - \\frac{a^2 r^{2n+3}}{2n+3}", "062405359c635dc6fce4eb706b473ab8": "\\mathbf{P} \\left( \\left\\{ \\omega \\in \\Omega \\left| \\lim_{s \\to t} \\big| X_{s} (\\omega) - X_{t} (\\omega) \\big| = 0 \\right. \\right\\} \\right) = 1.", "06240fb43060edd8a406f399def4bbb1": "a\\uparrow^n\\cdots\\uparrow^na\\uparrow^n1", "062429ee30b925bb458e4649dc433a3b": "\n\\tau = \\frac{ f \\rho v^2}{2}\n", "062434cd357e12f6cb0470162bc396d4": "\\scriptstyle A_n = \\{ i \\in I \\,:\\, a_i > 1/n \\}", "062461c7b79714d39684613c8f62ee16": "\\vec{j} = \\vec{j}_{\\text{diffusion}} + \\vec{j}_{\\text{advective}} = -D \\, \\nabla c + \\vec{v} \\, c.", "062536d639888f461dddbae1c1858e50": " F_{ST} ", "06254eaadabdb05aaf423cafa36f26a0": "a b ^{-1}", "0625863f41074bc5bce3370e701b6a31": "B_n(f,f)=0", "0625f9150cc414ca567fc3a4b32adb02": "x^2+ y^2 = -1", "0626359a6d0f2e2c24ba74cc83d6e44d": "\\frac{\\partial \\varphi}{\\partial t} + \\tfrac{1}{2} v^2 + \\frac{p}{\\rho} + gz = f(t),", "06263f482c703d45549ae3fd10f2d143": "d_k = n\\sum_{i=0}^k \\frac{(n+i-1)!4^i}{(n-i)!(2i)!}", "06266b399c5eef9b6d7464698fbefefc": "\\rho_e = \\sqrt{\\frac{L_e}{C_e}} = \\sqrt{\\frac{\\mu_0}{\\epsilon_0}} = \\rho_0 = 2\\alpha R_H. \\ ", "0626bb3ceac07cb0c47c3a18731ff0e2": "L_3 + -4L_2 \\rightarrow L_3", "062762cd1071fc2b58b93de1014f67a8": "S = \\frac{r^2}{4 \\times \\text{focal length}}", "0627b03980bb3b2b488092cd5c1eb4d0": "p(\\mathbf{Z}|\\mathbf{X},\\boldsymbol\\theta^{(t)})", "0627be370a9321cc5711d301bb89d099": "k \\in \\mathbb{C}", "0627c3775170c6cd963fb32678428514": "\\hat{O}' \\Psi [\\gamma] = \\int [dA] (\\hat{O}^\\dagger W_\\gamma [A]) \\Psi [A]", "0627f8d1a01988c1d46a6c3d86d83dd3": "\n\\frac{r_1}{A} = 0.46224\\left(\\frac{M_1}{M_1+M_2}\\right)^{1/3}\n", "06281beef0a11f0bbd026b6a910af3e7": "\\frac{\\pi}{4} = 4 \\arctan\\frac{1}{5} - \\arctan\\frac{1}{239} ", "0628666933060a11dd7308de2080b232": "\\hat{\\mu} \\pm 1.96\\hat{\\sigma}", "0628eaa3ae45bddd90c29a79e6c49e3b": "(A \\vee B \\vee \\overline{C}) \\wedge (\\overline{A} \\vee C) \\wedge (\\overline{B} \\vee C)", "062917f45123be31aae872afa8498ec0": "ds^2=\\frac{dy^2}{y^2}", "06291e2cab443a10fe3b6d9094ca6fe0": "v'=v^2 + R(x)v +S(x),\\!", "062939382c914bd39578f170b79d1b91": "|\\chi(x)| = \\chi(1)", "062984a0416fa2887b6e7445c4cc2563": "\\theta_{\\rho\\sigma}", "0629cb6dcd098c7599fbbd90349c25dc": "R_B", "062a037a163a451ebe08bf1367e6f834": " = \\frac{B_{wr}}{B_w}", "062a199ea7c8a6452693d0cdd2d9d9a3": "\\operatorname{erf}(x)\\approx 1-\\frac{1}{(1+a_1x+a_2x^2+a_3x^3+a_4x^4)^4}", "062a2b397475bd7de2c73915909b205f": "\\{\\mathbf{X}\\}", "062a2ee51454e2cb157488a562fad2bb": "P \\or R", "062a42706cf98fee006f2ad7f10482a0": "\\scriptstyle f\\,^\\prime=g", "062a53f9126f5965ec44bd5b730eed6f": "s(n)^k", "062a9c61db003c41317573201fbc8aa5": " P_k. ", "062ab0db31f98254f062e64eff695414": "A = \\sum_{n=0}^\\infty a_n X^n ", "062acb49d4a0c8289c5db26a369c1145": "\\textstyle C \\cap \\mathbb{Z}^n", "062ad19db449b50e15b179c3c6d31c1f": "c^{2} d\\tau^{2} = \\left( g_{tt} - \\frac{g_{t\\phi}^{2}}{g_{\\phi\\phi}} \\right) dt^{2} \n+ g_{rr} dr^{2} + g_{\\theta\\theta} d\\theta^{2} + g_{\\phi\\phi} \\left( d\\phi + \\frac{g_{t\\phi}}{g_{\\phi\\phi}} dt \\right)^{2}.", "062b8bd326f589ec94b970b64b7f0320": "(X_i)_{i \\in n}", "062ca4ff82afa835f107cbd32ab9c206": "x[m - l]", "062ca8518521c3130bd03e8dea036e0e": "i+j+k+l", "062cb3ee8cef408d72495aefdf4b5ef6": "\\pi_b", "062ccb1bfc2d82ee0d8f0ef0c8ecb329": "(c_1\\mid c_1 + 0 ) \\in C_1\\mid C_2", "062cd57cbca9cbc94a168e813a3bc469": "s_{\\Lambda}=\\sum_{i=1}^m x_i; s_{K}=\\prod_{i=1}^m x_i", "062d16cc5ddb949e53dc2d22874c8cd1": " \\mu_A = \\mu_{A}^{\\ominus} + RT \\ln\\{A\\} \\,", "062d41268eaebed7f369b10257c567ba": "W_k", "062d49b83e583cc240886020adfe2fd9": " E([0,N]) ", "062d769c571f8fdb77f4b1c0dbaf8b4a": "s^m", "062e24315ff4a033f03e116eae78ddb9": " \\cos \\alpha = \\cfrac{|\\mathbf{e}_1|}{|\\mathbf{b}_1|} \\quad \\Rightarrow \\quad |\\mathbf{e}_1| = |\\mathbf{b}_1|\\cos \\alpha", "062e4e19dd4b81651c2d6d1999c0c29f": "C= \\frac{L}{R_0^2}", "062e6e7cc5add0256261b3628cbefa4e": "\\begin{pmatrix}\n1 \\\\\nu \\\\\n v \\end{pmatrix}.", "062f1f11a733f565e3b29f75e34656ec": "O(n^22^{\\sqrt{\\log n\\log\\log n}})", "062f270a7d2ff2caa4245419eb108ebd": "d(E_1,E_2)>0", "062f2bd0056a5b0d04f1547095f638f4": " s = 1 ", "062f38a477980dbb65d6a1f0dde0b0ef": "O(\\textrm{polylog} n)", "062f4baeef5e0644cd7cf013b3bf6d3b": " \\mathcal{G}_\\pi ", "062f64ee5db3f26b9bacfdf42b319493": " [N_2O_2] = K_1 [NO]^2 \\,", "062fd1dbe57ab537da02a03eeb0b1dda": " \\nabla\\times\\left(\\psi\\mathbf{A}\\right)=\\psi\\nabla\\times\\mathbf{A}-\\mathbf{A}\\times\\nabla \\psi ", "0630366d4daa8203300da7fde1b23eab": "\\alpha\\to 0", "063058ce56f748f91c2e666d35d36ff7": "w = x_{i_1} x_{i_2} \\cdots x_{i_n} \\ ", "0630ca48e0da4c93bbb613b1c29eb351": "f(x):\\mathbb{R}{\\rightarrow}\\mathbb{R}", "063153bce09b19630b96a3cafebad55f": "r={{h^2}\\over{\\mu}}{{1}\\over{1+\\cos\\theta}}", "0631b96b08c4bffdfe4d9abc6d7b2970": "\\tbinom63", "0631ce95efa824617f2196451eb7ce04": " c_1 \\sqrt{\\frac{\\log N}{N}}.", "063220b1e41ecb6061d8249c3450ef1f": "P(D_x) \\, e^{i (x - y) \\xi} = e^{i (x - y) \\xi} \\, P(\\xi) ", "063256969a21b46c2fd28cf77583a2b9": " q_K : V\\otimes_L K \\to K ", "06325e5f1e0b49e9844641fd4675cd46": " V_b = V_0 + \\left (\\acute{V}_0 - V_0 \\right)\\hat{T} ", "063291bdce4eb10fd02bc01071a91ffe": "M_{00}", "0632c0a62e5cca50c3fe63e3552f25d8": " \\| r_{k} \\| \\rightarrow 0 ", "0632c3741e6b5a3fa721280b0a00ebe1": "A f (x) = \\tfrac1{2} \\sum_{i, j} \\delta_{ij} \\frac{\\partial^{2} f}{\\partial x_{i} \\, \\partial x_{j}} (x) = \\tfrac1{2} \\sum_{i} \\frac{\\partial^{2} f}{\\partial x_{i}^{2}} (x)", "0633115c0def5ce26a9247e92e2b9c19": "\\vec{F}(\\vec{r})", "06333472307bc5773235531ccafe47ea": "\\textstyle m = v \\oplus H_2\\left(e\\left(d_{ID}, u\\right)\\right)", "0633d94eb9029d87df669cc2abb9bec4": "L^2/mr^3", "0633ff34c1b67102945ea800db03c001": "[x]_P,", "0634484bb068bfd017b502939afa9f4c": "\\nu_{22}\\sigma^2", "063462b0140315805f1159656326c24e": "\\| \\|^2 ", "0634636bdf1467c4314c6ee26d0faee1": "\\mbox{Tor}_p(M,N) \\cong E^\\infty_p = H_p(T(C_{\\bull,\\bull}))", "0634db24f94b09d50d8fd583f7767149": " \\mathbf{A}\\mathbf{x} = \\mathbf{b}", "06352f73ce4d7862f4f367d1125ceec1": "A_{0}+\\displaystyle\\sum_{n=1}^{\\infty}(A_{n} \\cos{nx} + B_{n} \\sin{nx}).", "06353d181a478e98fe7d096dac2832bc": " i =1 - (1 - \\frac{1}{n})\\alpha ", "06357f4aed117fffde206b2b6b99a083": "r_{\\mathit l \\mathit l^{\\prime}} ", "0635983bc182111b1d187d32c25f1aa3": "F \\subseteq F^\\prime", "0635e7adba5b91d7bf56394edf74dee4": "\\sum\\limits_{i=1}^\\infty a_i=A \\in G", "063664973c564954cb2c5df9f51dcb51": "[A,[B,C]_D]_D=[[A,B]_D,C]_D+[B,[A,C]_D]_D.", "06369b8178473296e74994e82b560d19": "0 = \\int_{\\partial N} \\sqrt{-g} \\ \\xi^{\\mu} T_{\\mu}^{\\nu} \\ \\mathrm{d}^3 s_{\\nu} = \\int_{\\partial N} \\xi^{\\mu} \\mathfrak{T}_{\\mu}^{\\nu} \\ \\mathrm{d}^3 s_{\\nu}", "0636e6ee3577c00d82887a3ac9b7d159": "x(0)", "0636fd78a288331071f745ef8dc93b55": "\\left(\\frac3{F_n}\\right)=-1", "06373818e93a19cae644f7ae80a93704": "D_N = A_N - A_{N/2}\\,", "06375828f8aad83b0b94d0bbd47f59dc": "O(m\\sqrt n\\log n)", "06375ff0aa87888a6a812f49c284cc90": " V = n R T / P\\,", "06377dac0f6eced39d3a1e40accfd768": " \\epsilon = 23.439^\\circ - 0.0000004^\\circ n ", "06378f09ca31cdcd679a23af53f7cbbf": "\\ (u, v) \\not \\in E", "0637b03526afa1d3c13e875c1362db58": "I \\to \\infty", "0637da5f81ac21c91ad10834e8f9a76c": "M_X^c + (\\overrightarrow{XG} \\times mg - \\overrightarrow{XG} \\times ma_G - \\dot {H}_G) = 0", "06380e026e2872cb9a9d810c0faca93b": "Re[\\lambda]<0", "06382b959ddcd48f17456b523e2174be": "\\{X\\}", "0638318d29c7a65a7bb96b00246f91f0": "y^{k}", "063839ff99e1701de1eaabd3fbe2a375": "(x^2-y^2)^3 + 8y^4+20x^2y^2-x^4-16y^2=0.", "063841e36aa49cb4034c792b94bdfac0": "\\left(\\mathbf{A} + \\mathbf{UBV}\\right) \\left( \\mathbf{A}^{-1} - \\mathbf{A}^{-1}\\mathbf{UB}\\left(\\mathbf{B} + \\mathbf{BVA}^{-1}\\mathbf{UB}\\right)^{-1}\\mathbf{BVA}^{-1} \\right) ", "0638d22c12fd921a79d40362e1145749": "\\pi_1(X)/p_{*}(\\pi_1(C))", "0638d77c443cb5665c72a4f6c778bfe3": "k\\rightarrow \\infty.\\,", "063910a53398da0628fb1aea3d87e2fd": "\n{\\langle p \\rangle} = \\beta_\\text{max}\\left (1 + \\kappa^2\\right) \\epsilon \n \\left({1 - \\epsilon_B - \\epsilon}\\right)^2 G(\\epsilon) \\left(B_\\text{max}\\right)^2.\n", "0639331d8257eed848c99b4b94feda72": "\\mathbf{P}(t)=\\varepsilon_0 \\int_{-\\infty}^t \\chi(t-t') \\mathbf{E}(t') \\, dt'.", "06394fe2c6de92e648258ceb26e00745": "f_{\\Delta E}", "0639659520fe2a5bf59a5f1a30161ad9": "\\int_X f\\, d\\mu = \\int_X f e ^{i \\theta} \\, d |\\mu|", "0639ad6e78459f1b637d5ab558a8e6b6": "\\tilde{K}_\\pm \\ \\stackrel{\\mathrm{def}}{=}\\ K_\\pm / (K_0 \\cap K_\\pm)", "0639c979079363949c258a4efb310bb2": "\\begin{matrix} {1 \\choose 1}{11 \\choose 4}{4 \\choose 1}^4 \\end{matrix}", "0639f0fbf94d30efe14c4758e1f15aed": "\\omega(y|\\alpha,\\beta) = \n\\begin{cases}|y|^\\alpha\\left[1-i \\beta(\\tan \\tfrac{\\pi\\alpha}{2})\\mathbf{sign}(y)\\right]& \\alpha \\ne 1\\\\\n|y|\\left[1+i \\beta \\tfrac{2}{\\pi} \\mathbf{sign}(y)\\ln |y|\\right] & \\alpha = 1\\end{cases}\n", "063a05f2356036fe0a763ccb577ef79a": "B^x value \\left ( O,t \\right ) = \\left [ indexpartition \\right ]_2 + \\left [ xrep \\right ]_2", "063aa99b954878147d700c12864f8c9e": "\\mu\\ne\\nu", "063ac6c62425d6ffbdfe820371e2bc37": "\\; D_{\\mathrm{REE}} (\\rho) = \\min_{\\sigma} S(\\rho \\| \\sigma)", "063ae6fb13ea090df90cdfb17ec34e4e": "\\Gamma(x)=(\\alpha^8x-1)(\\alpha^{11}x-1).", "063b55bf6e50e78c8cfcbba464256e8b": "I_A = \\frac {\\pi} {2g} \\int_0^{T_d} a (t)^2 dt ", "063b6462ec5e3632c8e2b3df3eff3d60": "\\pi \\colon E \\to B\\, ,", "063bd81d17a4f96befa771de0625676d": "J^\\mu = \\frac{i\\hbar}{2m}(\\psi^*\\partial^\\mu\\psi - \\psi\\partial^\\mu\\psi^*) \\, . ", "063c09eb13b64940701faf1a3dc98c95": "{ \\mathbf{ \\tau } = {d \\mathbf{ L} \\over dt} }", "063c2426e7dbf82691c1d8c2706ae60d": " H = \\int {\\mathbf A}\\cdot{\\mathbf B}\\,d^3{\\mathbf r} ", "063c3a99991296354fd84c499074e27a": "f_p(x)", "063c9a6662fc69d6492027c45d75758d": "\\mathbf{H}_\\mathrm{eff} = \\frac{2A}{\\mu_0 M_s} \\nabla^2 \\mathbf{m} - \\frac{1}{\\mu_0 M_s} \\frac{\\partial \nF_\\text{anis}}{\\partial \\mathbf{m}} + \\mathbf{H}_\\text{a} + \\mathbf{H}_\\text{d}", "063ca26489809732d289ee06eb8552bd": " \\scriptstyle *:A\\times A\\to \\mathfrak{G} ", "063cb646cf29e1d4518894d1cc840635": "\\mathbb{E}(W_i) = \\frac{1+\\rho_i}{2} \\mathbb{E}(C) + \\frac{(1+\\rho_i) \\text{Var}(C_i)}{2 \\mathbb{E}(C)}", "063cc80c42fafab7ea390184547cd686": "\n\\begin{align}\np(\\mathbf{X}\\mid \\mathbf{Z},\\mathbf{\\mu},\\mathbf{\\Lambda}) & = \\prod_{n=1}^N \\prod_{k=1}^K \\mathcal{N}(\\mathbf{x}_n\\mid \\mathbf{\\mu}_k,\\mathbf{\\Lambda}_k^{-1})^{z_{nk}} \\\\\np(\\mathbf{Z}\\mid \\mathbf{\\pi}) & = \\prod_{n=1}^N \\prod_{k=1}^K \\pi_k^{z_{nk}} \\\\\np(\\mathbf{\\pi}) & = \\frac{\\Gamma(K\\alpha_0)}{\\Gamma(\\alpha_0)^K} \\prod_{k=1}^K \\pi_k^{\\alpha_0-1} \\\\\np(\\mathbf{\\mu}\\mid \\mathbf{\\Lambda}) & = \\prod_{k=1}^K \\mathcal{N}(\\mathbf{\\mu}_k\\mid \\mathbf{\\mu}_0,(\\beta_0 \\mathbf{\\Lambda}_k)^{-1}) \\\\\np(\\mathbf{\\Lambda}) & = \\prod_{k=1}^K \\mathcal{W}(\\mathbf{\\Lambda}_k\\mid \\mathbf{W}_0, \\nu_0)\n\\end{align}\n", "063cde69390fa7062c5bda566cfce138": " \\cos E = \\frac{x}{a} = \\frac{ae +r \\cos \\theta}{a} = e+ (1-e \\cos E) \\cos \\theta \\ \\to \\cos E = \\frac{ e + \\cos \\theta }{1 + e \\cos \\theta } ", "063d0555ec921f06621d113cf250fa0c": "\\int_a^b \\frac{d}{dx}\\left(u(x)v(x)\\right)\\,dx = \\left[u(x)v(x)\\right]_a^b", "063d3d08b5833130d9c19cc910b8852c": "[\\hat{x},\\hat{p}]=i\\hbar", "063d582f0b7db6d715605e5d5b186203": "X \\dot = Y", "063d7b9a3cd93212199947067732dcf1": "5. \\; \\; 2\\mathrm{O}_3 \\; \\xrightarrow{h \\nu } \\; 3\\mathrm{O}_2 ", "063d8e899312d0ab03afd72c8f1a16de": "\\mathcal{P} = \\frac{\\mu A}{\\ell}", "063d9c323d1afbafecb1fa625f1d8dbb": "f(x)=L^{-1}(4x-1).\\, ", "063da2738dd86cb86b09b041b907e19a": "\\textstyle x\\in\\left( a,b\\right) ", "063dc8f4d1520bd2822ad410a5085574": "x_i^*\\in[x_{i-1},x_i]", "063df27909b8117eeed224b2b58f2f69": "\\lim_{n \\rightarrow \\infty} \\left( \\max_{a \\leq x \\leq b} | f(x) -P_n(x)| \\right) = 0.", "063e3063e86db87974de028a9dc464d1": "F_{-n} = (-1)^{n+1} F_n. \\, ", "063e806ef9a3b14c27155fefddde363b": " \\boldsymbol{\\beta}_{ut} = \\left(\\beta_{ut}^1, \\dots, \\beta_{ut}^n \\right) ", "063e9a76af25b78ab52ccb69cf835f3c": "(y^m u^r)^{(p-1)(q-1)/r} \\equiv 1 \\mod n", "063e9b555eaa456479aa5abc50feb3e8": " \\psi\\rightarrow \\psi e^{i2ct}", "063f318a4eab2f244bd68ab13d9b64ef": "D_{kn} = \\frac{2}{N} \\cos\\left(\\frac{nk\\pi}{N/2}\\right) \\times \\begin{cases} 1/2 & n=0,N/2 \\\\ 1 & \\mathrm{otherwise} \\end{cases}", "063fa15417cbd7b6b5a5e0087e069e95": "\n p_0 = \\cfrac{2 a E^*}{\\pi R} ~;~~\n p_0' = -\\left(\\cfrac{4\\gamma E^*}{\\pi a}\\right)^{1/2}\n ", "063fa2259f768d9e76c6ded794f42d06": "v_1, v_2, \\cdots, v_m", "063fca31a7350b0628b274d7560a8876": " \\operatorname{get-lambda}[p, p = \\lambda f.\\operatorname{let} x : x\\ q = f\\ (q\\ q) \\operatorname{in} f\\ (x\\ x)] ", "06406f401c9db3d2015a9130b3afc883": "(dx)^2=C_{KL}dX_KdX_L\\,\\!", "0640a6b2b362e804d4520747d2b41256": "u^\\prime = u\\,", "0640b6347878b4e87d5ac0838bc3bcbe": "\\mathfrak{so}_{4} \\cong \\mathfrak{sl}_2 \\oplus \\mathfrak{sl}_2 ", "0640ce2434f184907371d2ac5d917cd5": "x=\\sqrt{t}", "0640d02d6a69d9ca4f41cd8fdb8240b6": "\\mu:\\mathcal{X} \\to \\mathbb{R}", "0640d612b34d116370eafd65b020a831": "G_n^{(1)}", "0640e12048949ecf87c8ccf163d8d404": "\\{ \\tau \\leq t \\}", "0640e5ee3912c73671421e2d89a33b92": "\\theta =180{}^\\circ ", "0640eb22ee0e9aaaa5055a6daed22104": "q_x = - k \\frac{d T}{d x}", "06410e3cc2dfd60b883efabddee06c33": " f(i) = \\beta_0 + \\beta_1 x_{i1} + \\cdots + \\beta_p x_{ip},", "064118dbe891d9f20688a8bf141bd158": "\\left ( \\frac{a}{b} \\right )", "0641205e5b9c3658f86787b4ccf76ca3": "\\langle J,J_z|\\vec \\mu_J|J,J_{{z'}}\\rangle = g_J\\mu_B\\langle J,J_z|\\vec J|J,J_{z'}\\rangle", "064145950df9574945eaa3894624a044": "\\displaystyle \n= \\sum_{\\sigma, \\tau = 1}^n\\left({\\partial^3 F \\over \\partial t^\\alpha t^\\nu t^\\sigma} \\eta^{\\sigma \\tau} {\\partial^3 F \\over \\partial t^\\mu t^\\beta t^\\tau} \\right)\n", "0641727c9e3546b2b9c5b335ea73a3b5": "y_i \\succsim_i x_i", "0641f75e5ef17d434b22e363f6e14dfd": "_{p\\leftarrow q\\,}\\!", "0642209c44c24f23c9172a30286eb802": "\\ 1/x", "06424d3fefb5dbef1d4af971d1e97773": "e^{i \\pi} = (e^{i \\delta})^{\\pi / \\delta},\\,\\!", "0642796128dd4c58f27ed1bcdef2c71d": "\\Delta H^*_{ab}", "06429e91ca50db64fc70570ed66f7380": "q_3", "06429fa86d535a910037f92a580383e4": "(x^2+y^2)^2=2(x^2-y^2)\\,", "0642cc739b2f8c0404acb0c2ac9f5ace": "0,1,\\ldots,n", "064310c16ba9839ac791f793bdd726b7": "\\hbar=c=1", "06431b49bbcb1ac96205e734d1c52fb8": " p(\\theta | I_t, O_{fg}) ", "0643d743ed71e86bd64f547f6e80308a": "\\left(\\frac{d}{dx}\\right)_q f(x)=\\frac{f(qx)-f(x)}{qx-x}.", "06440725ef44eaeb882e81dacf25fb68": "q1=q", "0644122b838b7185dacf29793cade3bd": "\\theta(\\lambda)", "06448fe7815d407ad7ff6bfe9579a7d6": "(f * s)(t)", "0644b871dfafa8cd458b32f664661297": "\\,I^n(t)\\,", "06450841c23388142835699b0ac86913": "\nP_{1D}(x)dx=\\sqrt{\\frac{3}{4{\\pi}Ll_p}}\\exp{\\left(-\\frac{3x^2}{4Ll_p}\\right)}dx \\,\\,\\,;\\,\\,\\,\\,\\,\\,\nP_{2D}(R)dR=2{\\pi}R\\frac{3}{4{\\pi}Ll_p}\\exp{\\left(-\\frac{3R^2}{4Ll_p}\\right)}dR\n", "06450d9fbdf8d1f569a98d268b6d054e": "\\begin{bmatrix}\n a & b \\\\\n c & d\n\\end{bmatrix}\n\\begin{bmatrix}\n x \\\\\n y\n\\end{bmatrix}.\n", "064518314a6112b0e1a062bc7e818e0f": "A^{-1} \\cdot B^{-1}a_{i_1}^{\\varepsilon_1}B\\cdots B^{-1}a_{i_L}^{\\varepsilon_L}B", "064579fefd9a750d9cfbda846d2eb899": "\\mbox{VBN} = 14.534 \\times \\ln\\left[ \\ln(\\nu + 0.8) \\right] + 10.975\\,", "064605eeb7caaf26a5c67dc67b015871": "\\displaystyle{(g,G)\\cdot (h,H)=(gh,K),}", "064642b96331839dec506e878d02cd46": "\\rho_{\\text{Electric dipole}}(\\mathbf{x},t) = \\frac{-i k}{4 \\pi \\epsilon_0}\\frac{e^{i k r - i \\omega t}}{r}\\mathbf{n}\\cdot\\mathbf{p}", "06464d647faa419e1912c0c22bb1f263": "i\\colon A\\hookrightarrow X", "064663015cf54c0606d3673d21187cbc": "-\\overline{v'T'}", "06467448db2258a82dbf67043db3ced2": "S_0''=0=o(S_0')\\,", "064695238557467e60cb5e053aa0fe65": "Q_1 - Q_2 > 0", "0646a890a9669fb4da9d37527102896e": "u = u_{0} + \\Phi(\\pi - \\pi_{t})", "0646dadd49af164112729c1e9e880cf2": "d(O_{r}, O_{n}) <= r(O_{r})", "06471ded130d2e5ce109d3167696c49d": "\\Omega^*(M) = \\bigoplus_{k=0}^\\infty \\Omega^k(M).", "06472b40bd4e5d9619b9b1d15d33bc04": "\\operatorname{ess.inf}", "0647e246b9aa2a45be714732516cd13f": " P(\\partial_t,\\xi)G(t,\\xi) = 0, \\; \\partial^j_t G(0,\\xi) = 0 \\; \\mbox{ for } 0\\leq j \\leq m-2, \\; \\partial_t^{m-1} G(0,\\xi) = 1/a_m. ", "0647f845a1fc3218780e47d701d32dad": "g: Z\\to V", "064832debd50461e7c43ba7fbaab62d1": "dG= \\sum_{j=1}^m \\mu_j\\,dN_j = 0", "0648373fbe646cf8a58abccc71e691a0": "dE = \\delta Q - \\delta W,", "0649273b46cdfccc71cc410bfdd07c09": "\\operatorname{fnchypg}(x;n,m_1,N,\\omega) = \\operatorname{fnchypg}(n-x;n,m_2,N,1/\\omega)\\,.", "0649b1d9ebcba60c8e487b5ff077fd37": "\\mathcal{C} \\times \\mathcal{D}", "064a39ffb4ce76f9f7888c745b194ffb": "f: \\mathcal{X} \\to \\mathcal{Y}", "064a45240a7d91dd83b638733dd88f65": " u(\\theta) = \\frac{ GM }{h^2} + A \\cos(\\theta-\\theta_0)", "064a5481f3b33fe02b5af01dcfce74a3": "\\sum n_P P \\to \\sum n_P P.", "064a7413ce1f710c1ab5175077b85716": "S(t) = \\Pr(T > t)", "064a818b99190e7f13fa62e568d6ad40": "S(P) \\geq S(Q)", "064b035ed9383c2e45edb8795b459dc2": "\\bar{T}_{\\ell_1 \\ell_2 \\cdots \\ell_q}^{k_1 k_2 \\cdots k_p} = \\mathsf{L}_{i_1}{}^{k_1} \\mathsf{L}_{i_2}{}^{k_2} \\cdots \\mathsf{L}_{i_p}{}^{k_p} (\\boldsymbol{\\mathsf{L}}^{-1})_{\\ell_1}{}^{j_1}(\\boldsymbol{\\mathsf{L}}^{-1})_{\\ell_2}{}^{j_2} \\cdots (\\boldsymbol{\\mathsf{L}}^{-1})_{\\ell_q}{}^{j_q} T_{j_1 j_2 \\cdots j_q}^{i_1 i_2 \\cdots i_p}", "064b59a42ba406a97b723db2b4277085": "T(X_1^n)=\\overline{X}=\\frac1n\\sum_{i=1}^nX_i", "064b7f588deebf178356485a12196fb8": "\\sqrt{\\frac{2}{3}}\\!\\,", "064bd950f9ba6d5ad4f577ece6659e46": "\n k = \\cfrac{5 + 5\\nu}{6 + 5\\nu}\n ", "064bd958e3e9578a2676150e5e8bd6d0": "c_{jk} = \\left[W_\\psi f\\right]\\left(2^{-j}, k2^{-j}\\right)", "064c25a4ec47eb9d7119f34745f978dc": " +48(x^2 + y^2)(x^2 - 3y^2)^2 + (x^2 - 3y^2)x[16(x^2 + y^2)^2 - 5544(x^2 + y^2) + 266382] = 720^3.", "064c678d55189edf8539d54cb383f358": "i\\leq n", "064c9727b2531f2a9a150cc5a4a815d7": "\\displaystyle Wg(1^3,d) = \\frac{d^2-2}{d(d^2-1)(d^2-4)}", "064cc9865887da54d41d095b13f33d89": "ay \\bmod 2^w", "064d4bf58e4b377ddc029af6979cdca4": " \\textbf{A} ", "064d566169bd649ee5862d04310e3ff5": " \\mathit{XP} + Y \\longrightarrow \\mathit{XY} + P_i", "064d8a5614ac0a0aac343fd50a644849": "\\widehat{\\sigma}", "064dfa1b3bea950017e1f5042a957127": " \\log_b(xy) = \\log_b(x) + \\log_b(y) \\!\\, ", "064e113d61e97c3b00cd1efd7434bbe5": "x_2=3", "064ed431e31cd627e97ea3addb1493b6": "e^{-\\pi z^{2}}", "064efb9d9fad29c9d848fbb4a42ccec3": "x_{n_1,n_2}", "064f0e3bfda1a64772d3eb4307075b2c": "\\nu \\ll \\omega", "064f3ae713cdc9d4788fa99f1cbee672": "\\pi_1 (\\mathbb{H}/\\Gamma)", "064f80c126d8f90c294a28d0d5205e5b": "\n\n\\sum_{n=0}^{\\infty}\\frac{n!L_{n}^{(\\alpha)}(x)L_{n}^{(\\alpha)}(y)r^{n}}{\\Gamma\\left(1+\\alpha+n\\right)}=\\frac{\\exp\\left(-\\frac{\\left(x+y\\right)r}{1-r}\\right)I_{\\alpha}\\left(\\frac{2\\sqrt{xyr}}{1-r}\\right)}{\\left(xyr\\right)^{\\frac{\\alpha}{2}}\\left(1-r\\right)},\\quad,\\alpha>-1,\\left|r\\right|<1.\n\n", "064fcd42f1e9f514e0fd694aa5c4a2fa": "T_{ij}", "064ff7725e1de916ba94bfc251c551d5": "w_0\\left(t-\\tfrac{(N-1)T}{2}\\right)\\cdot \\operatorname{rect}\\left(\\tfrac{t-(N-1)T/2}{NT}\\right),", "065070c1a970b7bbf09e708029624ccd": "\\text{Liquid} \\xrightarrow[\\text{cooling}]{\\text{eutectic temperature}} \\alpha \\,\\, \\text{solid solution} + \\beta \\,\\, \\text{solid solution}", "065091dc156b96ef2bc9f867ea8d9a90": "\\scriptstyle x \\oplus y = XY", "0650e38c5dfbf20f831af30d7ac69f99": "L_{[\\omega]}^{n-1}: H_{DR}^1(M) \\to H_{DR}^{2n-1}", "0652402621111239f416a1862561d031": "\\ \\mathbf b=0", "0652623115b46f2a9ccf0f54129c1506": "u^{\\alpha} = (1, 0, 0, 0) \\,,", "06529432da242c4cc04f5870bab2cd37": "\\pi_1 (X\\vee Y) \\cong \\pi_1(X) * \\pi_1(Y).", "0653081814d4d647a1bafb2d96b99591": "\\Big( \\pi \\models \\phi_1 \\Rightarrow \\phi_2 \\Big) \\Leftrightarrow \\Big( \\big(\\pi \\not\\models \\phi_1 \\big) \\lor \\big(\\pi \\models \\phi_2 \\big) \\Big)", "06531529788d2b229a9977c605e7607a": "\\frac{\\log_2 N\\,\\log_3 N\\,\\log_5 N}{6}.", "06534b4d5b8d6f8c690344a0f0ef53d3": "\\begin{align}\nL_{x}&\\approx I_{1}(\\dot{\\psi}-\\Omega\\sin\\delta)\\,,\\\\\nI_{2}\\ddot{\\alpha}&\\approx (L_{x}\\Omega\\sin\\delta+I_{2}\\,\\Omega^{2}\\sin^{2}\\delta)\\,\\alpha\\,.\\end{align}", "065375e324898a3a1d67dab3a2452a37": " -\\dot{S}(t) = A^\\mathrm T(t)S(t)+S(t)A(t)-S(t)B(t)R^{-1}(t)B^\\mathrm T(t)S(t)+Q(t),", "0653fd4adefb8c3762a1ce0e81d50d2d": "x=x_{s}(t)", "0654029124451d6c93acee2f2456e142": "C_{QY} = \\frac{\\epsilon_0}{\\lambda_0} = 3.649\\;2417\\;\\mathrm{F/m^2}", "0654865eb641896b269a22f105ba83a3": " \\tau := \\sup \\{ t \\in [0,1] : W_t = 0 \\} ", "0654ced79aa19a4f4cb123a26bef9e4e": "\\sigma:A\\rightarrow \\mathrm{End}(V)", "0654d4bb827f9e0d5e9577c634df1dc2": " p\\in [-1,1]", "06550af4bcc791b2d570e461baefba01": "\\frac{{N-K \\choose n} \\scriptstyle{\\,_2F_1(-n, -K; N - K - n + 1; e^{t}) } }\n {{N \\choose n}} \\,\\!", "06559d4446604dad285c25f86a8b505b": " \\begin{align}\n\\operatorname{E} \\operatorname{tr} e^{\\sum_{k=1}^n \\mathbf{X}_k} & = \\operatorname{E}_0 \\cdots \\operatorname{E}_{n-1} \\operatorname{tr} e^{\\sum_{k=1}^{n-1} \\mathbf{X}_k + \\mathbf{X}_n }\\\\\n&\\leq \\operatorname{E}_0 \\cdots \\operatorname{E}_{n-2} \\operatorname{tr} e^{\\sum_{k=1}^{n-1} \\mathbf{X}_k + \\log(\\operatorname{E}_{n-1} e^{\\mathbf{X}_n} ) }\\\\\n&= \\operatorname{E}_0 \\cdots \\operatorname{E}_{n-2} \\operatorname{tr} e^{\\sum_{k=1}^{n-2} \\mathbf{X}_k + \\mathbf{X}_{n-1} + \\mathbf{\\Xi}_n} \\\\\n& \\vdots\\\\\n& = \\operatorname{tr} e^{\\sum_{k=1}^n \\mathbf{\\Xi}_k}\n \\end{align} ", "0655a9a3d09cb12d562a6a71520bcfc6": "\\mathbf{x}_*", "0655cce591173476dca8441b58faa7c0": " A\\otimes_K K_v \\simeq M_d(K_v). ", "0655cd5b52625898a4522c700969124b": "m(x_i)=\\frac{1}{N(\\mathbf{h})}\\sum^{N(\\mathbf{h})}_{i=1}Z(x_i)", "065646299b64e9db2a42e4a70c2044e4": " \nE[L(t)] - E[L(0)] + V\\sum_{\\tau=0}^{t-1}E[p(\\tau)] \\leq (B + C + Vp^*)t\n", "06566a283d2455c50bf4e6cc2613ffa1": " f_Y(y | \\theta, \\tau) = h(y,\\tau) \\exp{\\left(\\frac{b(\\theta)T(y) - A(\\theta)}{d(\\tau)} \\right)}. \\,\\!", "06567105e3af8240346209deed923e2c": "\\sqrt {S} = \\sqrt{\\frac{\\vert S \\vert + a}{2}} \\, + \\, \\sgn (b) \\sqrt{\\frac{\\vert S \\vert - a}{2}} \\, \\, i \\,.", "065672f2121201154ac873c04e7aaf53": "if\\, (C^{cand} \\neq \\emptyset)", "0656b0158a90a29c81bde47bd93357c0": "\\!\\mu_2(v_3)", "0656eefc86eb14c91fadefc25281bb41": " F(\\epsilon) = \\frac{1}{e^{(\\epsilon-\\mu) / k T} + 1} ", "0656f87fdfc7f7b7c0a52cf06a06bf00": "r(x) = \\sum_{a} r_a \\emptyset ^a (x) = r_a \\emptyset ^a (x) = x", "06571844a2a521364f8605899e26ed06": "H = h_1 h_2 h_3", "0657241774278b7f2381b41ac559b03b": " \\text{PV} = \\text{FV}\\cdot e^{-rt} ", "0657298c5bcea3a7c5842bfc838f74c2": "K=0,", "065729da1dbd83b81b1b168c928169b9": "O(n \\log h)", "06575ab091aca463a8dc616775352dab": "\\sum_{i=1}^n (x_i-\\mu)(x_i-\\mu)^\\mathrm{T} = \\sum_{i=1}^n (x_i-\\bar x)(x_i-\\bar x)^\\mathrm{T} = S", "0657967352cd8a9c703d53b9c13fe4dd": "L_{g}L_{f}^{k}h(x) = 0 \\qquad \\forall x", "0657d4e54d9e5e0d0a393e895b261708": "{\\rm R} + {\\rm L} \\to {\\rm RL}", "065801965a23a5923991a44e5fd950ae": "\\begin{matrix}\np \\oplus q & = & (p \\lor q) \\land \\lnot (p \\land q) \n\\end{matrix}", "065821342a09d802f7b40b0f6b9a88e1": "F(k+1) = f(F(k)) = f(G(k)) = G(k+1).", "065831348fd5839996fb2e09d9e8b681": "k_r^-", "0658bbf1015e2f0d90ca6440473c08c3": "\n\\begin{matrix} X_k= \\underbrace{\\sum \\limits_{m=0}^{N/2-1} x_{2m} e^{-\\frac{2\\pi i}{N/2} mk}}_{\\mathrm{DFT\\;of\\;even-indexed\\;part\\;of\\;} x_m} {} + e^{-\\frac{2\\pi i}{N}k}\n \\underbrace{\\sum \\limits_{m=0}^{N/2-1} x_{2m+1} e^{-\\frac{2\\pi i}{N/2} mk}}_{\\mathrm{DFT\\;of\\;odd-indexed\\;part\\;of\\;} x_m} = E_k + e^{-\\frac{2\\pi i}{N}k} O_k.\n\\end{matrix}\n", "0658bdbe1fcda6797f88e35351e12b9b": "\\begin{array}{l}\nf^1\\big(\\theta^1(t)\\big)=\\cos\\big(\\omega^1 t\\big), \nf^2\\big(\\theta^2(t)\\big)=\\sin\\big(\\omega^2 t\\big) \n\\\\\nf^1\\big(\\theta^1(t)\\big)^2\nf^2\\big(\\theta^2(t)\\big)\nf^2\\big(\\theta^2(t) - \\frac{\\pi}{2}\\big) \n=\n-\\frac{1}{8}\\Big(\n2\\sin(2\\omega^2 t)\n+\\sin(2\\omega^2 t - 2\\omega^1 t)\n+\\sin(2\\omega^2 t + 2\\omega^1 t)\n\\Big)\n\\end{array}\n", "0658d2b10b2036c4126666ff7af50dbb": "a_1,\\ldots, a_n", "065932afb82fb41aaa6f6369d7bdad6b": "DPV= \\int_0^T FV(t) \\, e^{-\\lambda t} dt \\,,", "06594abcd0cdb4baeba7da6a799ec010": "\\pm\\frac{\\tan \\theta}{\\sqrt{1 + \\tan^2 \\theta}}\\! ", "06594f0602e88da2137a52fa52f46333": "M(x) \\cdot x^{n-1} = Q(x) \\cdot K'(x) + R(x)", "0659f3831ebaefaa6fc92860933b3c69": "\\mathcal{V}\\,", "065a26e3c7e0b9576696f8564ceeb46b": " \\displaystyle{(e^\\xi,e^\\eta)=e^{(\\xi,\\eta)}.}", "065a6bc421f03312e07d3cc59ef6a059": "a^2+1", "065af87168ddb3765d35cb956546ff57": "\\sum_{i=1}^k\\sum_{j\\in N_i} w_{ij} x_{ij} \\leq W,", "065b307962ae9ec00ac3c9ed98f9cd29": "E_{inc}", "065b6bee27a66e9bb1bef8853aef3946": "I_\\mathrm{ion}(V,w) = \\bar{g}_\\mathrm{Ca} m_{\\infty}\\cdot(V-V_\\mathrm{Ca}) + \\bar{g}_\\mathrm{K} w\\cdot(V-V_\\mathrm{K}) + \\bar{g}_\\mathrm{L}\\cdot(V-V_\\mathrm{L})", "065b7cd54a67f380c3d5ee0fcb6898fd": "\\omega\\in S,", "065bccda0bb62414e07c0279c1ba8d9c": "q'=h^sg^y", "065bdd18294611a861191a74e16f8502": "\\mu_m = ", "065c472ad60d01cb605c8863aac6ab62": "l_a n^a=-1=l^a n_a\\,,\\quad m_a \\bar{m}^a=1=m^a \\bar{m}_a\\,,", "065c5afdef0432fb8fa2ac1d7b4626a9": "\\,\\!\\theta_{n,k}", "065c7d5a2d7df51c6aaaf7f33e9b137d": " \\Gamma_{ij}{}^k = \\cfrac{\\partial \\mathbf{b}_i}{\\partial q^j}\\cdot\\mathbf{b}^k = -\\mathbf{b}_i\\cdot\\cfrac{\\partial \\mathbf{b}^k}{\\partial q^j} ", "065c7fe485ac4442ec7d169b329c6636": "f\\left(r\\right) = \\frac{\\left(n - 2\\right)\\, \\mathbf{\\Gamma}\\left(n - 1\\right) \\left(1 - \\rho^2\\right)^{\\frac{n - 1}{2}} \\left(1 - r^2\\right)^{\\frac{n - 4}{2}}}{\\sqrt{2\\pi}\\, \\mathbf{\\Gamma}\\left(n - \\frac{1}{2}\\right) \\left(1 - \\rho r\\right)^{n - \\frac{3}{2}}} \\,\\mathbf{_2F_1}\\left(\\frac{1}{2}, \\frac{1}{2}; \\frac{2n - 1}{2}; \\frac{\\rho r + 1}{2}\\right)", "065c8c145cdc659a52d38d79b8ad93ee": "\\scriptstyle f: [0, T]\\longrightarrow X ", "065c90b0effda6c8a4ffcb4ba5e07d3c": "\\kappa ", "065cba6033cd20888b9121ccda1c637b": "(x+1) \\ge 4\\,\\!", "065cd9c9acf9bcd888f6b0880c54b0e9": "m_n\\ :=\\ {0}^{256\\ -\\ \\mathcal{j} m_n \\mathcal{j}} \\mathcal{k} m_n", "065d060554456cdb96fd2f6413e448b0": "f(x)/g(x)", "065d544472d087acac0094a9bc44e9e8": " \\beta x f_0 \\ll 1 ", "065e1647d7aec79e787d42b59bc7ce82": " \\epsilon> 0", "065e40dd8dbb27b2a8088e3687149788": "I_x = I_y = \\frac{m r^2}{2}\\,\\!", "065e6069c6786461750b1077bb083b29": "x = \\sum_{i=1}^{N} S_i", "065ed001d5e92678b71b01fd95fc0623": "\\exists\\lambda\\in\\sigma(A): |\\lambda-\\mu|\\leq\\|\\delta A\\|_2", "065ed0c542bedddf5eecedad6cdbfdbd": "\\{A_1,\\ldots,A_n,\\neg B\\}", "065edee3cddf0c96c2358169e219412a": "m_1=(4x+m)", "065f10721e490b358283f4cda7f9cfad": "\\forall x f\\ x = x", "065f53c5ff9b8139c5562ca4dfa66ee5": "\\forall i: |\\gamma_i|\\leq i", "065f73e8b5eb80210019f2fb94218375": "\n\\operatorname{cov}(X_i, X_j) = \\frac{\\theta_i \\theta_j}{(a-1)^2(a-2)}, \\qquad \\operatorname{cor}(X_i, X_j) = \\frac{1}{a}, \\qquad i \\neq j.\n", "065f9fe034d3e8ef22183ff943ce6d2e": "\\vec w \\cdot \\vec \\mu_{y=0} ", "065fa0367342168be7eea9e42c03a454": "\\theta_1 \\in \\Theta_1", "065fffaa6113a8311f7abb23bdd15688": "z = \\exp(i t)", "066077bc473df482eb54bfbd841d892b": "\\alpha(a, \\, b) \\stackrel{\\mathrm{def}}{=} \\displaystyle\\sum\\limits_{c \\in A} f(a, \\, c, \\, b ) \\cdot \\sum _{d,\\,e \\in A} g(a,\\,d,\\,e) ", "0660be61967e90e49ec26230c2409a36": " {\\boldsymbol{L}_{k-1}} = \\left . \\frac{\\partial f}{\\partial \\boldsymbol{w} } \\right \\vert _{\\hat{\\boldsymbol{x}}_{k-1|k-1},\\boldsymbol{u}_{k-1}} ", "06610b8ddbd684f89a62227c032baf06": "\\mathbf{b_1}", "0661758bfa11e17b001b691146866560": "g: X \\to Y", "06617c3d95af696566773ac2b8a5989e": "\\sup_{\\eta>0} \\int_{-\\infty}^\\infty \\left |f(\\xi+i\\eta) \\right|^2\\,d\\xi = C < \\infty", "0661a980b1208be48c4a350b21ac02b8": "q_x = q_y\\,", "0661d9ddb8cf900ce1d2e896d3d955d0": "L = 20\\log\\left(\\frac{R}{R_0} + 1\\right)\\, \\text{dB}", "0662056719736cf5d0d5999aa4a4ee1d": "\\{ z : e^z = w \\} = \\{ v + 2k\\pi i : k \\in \\mathbb{Z} \\}", "066213b23a5e9281e0eacdc112fdb1cb": " \\|\\cdot\\|_{C^{k, \\alpha}} ", "0662605598c033c0dde3dea8e319a8b2": "{(2n)! \\over (n+1)! (n+1)!}", "0662ae0efdba6803461bd5f402f83232": "\n\\int_{-\\infty}^\\infty \\frac {\\gamma\\left(\\frac s 2, z^2 \\pi \\right)} {(z^2 \\pi)^\\frac s 2} e^{-2 \\pi i k z} \\mathrm d z = \\frac {\\Gamma\\left(\\frac {1-s} 2, k^2 \\pi \\right)} {(k^2 \\pi)^\\frac {1-s} 2}.\n", "0663c6feb5cc4ba436fd441e8858053b": "M:D\\rightarrow C", "06640de893d91e932365dc13a6399d28": "\\scriptstyle \\mathbf{\\nabla}\\cdot\\mathbf{\\sigma} \\,+\\, \\mathbf{F} \\;=\\; {\\mathbf{0}} ", "066418514c51fdc53b0e3419861a1fb5": "\\eta^{\\mu\\mu}", "06642454857236ba2faa1c45d2fb117e": "(\\gamma\\gg 1)", "066464cd43b120eef607ad821fe70b81": "\\vartheta(z|q) = \\prod_{m=1}^\\infty \n\\left( 1 - q^{2m}\\right)\n\\left( 1 + 2 \\cos(2 \\pi z)q^{2m-1}+q^{4m-2}\\right).", "0664735ed61d755a0aec900cc1e7b9f2": "\\operatorname{erf}", "0664b46389c4fcf1ba4d081976dc6cc9": "P(a_i^T(x) \\geq b_i) \\geq p, \\quad i = 1,\\dots,m ", "0664ddd782c0850852945bed0155ef58": "[*:*:0:\\dots:0]", "0665078bf784c47a904a52a59c74f033": "do(move(2,3),S_{0})", "066510e73bd9a053774f50bc7b4d0d6e": "Z_{0} \\approx 376.730\\ 313\\ 461\\ 77 \\ldots \\Omega", "06652f0828e48ad7c33fc94ecc5fcd6b": "\\textstyle \\sigma_k = M_{\\mathrm f} R^k (\\Delta)", "066557ff29a2c4b0590aba3fdd721c60": "\\frac{d}{d x}\\left(\\sin(x^2)\\right) = 2 x \\frac{d}{d u}\\left(\\sin(u)\\right) = 2 x \\cos(x^2)\\,", "066561356344af6c07cdbb6126f8d032": "\\sqrt{31}\\times\\sqrt{31}", "066563f838bfc88c66dfaf2644dd64f6": "\\vec{S}(n) = \\sum_{j=1}^8 b_j K_j^n\\vec{\\xi}_j.", "0665666dbcc9f49f1ccd97b513147dc7": "T_I", "066627838e3b7a9238618ba6dc14be3e": "{\\widehat{HV}}_3", "06666e1bb344f1eedfb6ea7ebda9f844": "Y=2k(\\phi(front)+\\phi(rear))=4k(\\theta-\\psi)+2k\\frac{(b-a)}{V}\\frac{d\\theta}{dt}+2k\\eta", "0666e08d1fd0f044f5a0b7008e449afb": "\\big\\{\\mathbf{F}_{\\alpha}\\big\\}_{\\alpha=1}^{M=N\\times{N}}.", "0666e4b33c71c6516d1b6b295f1b6d55": "\\arcsin( )", "0666fd617b7abb83c7d26d29d45d4fea": "k^{}_{}: ", "06670427b22277cd72dc40510011730d": " u_{tt} - u_{xx} = V'(u) ", "06673df311ebe5b0e2158bf8cda07674": "H=2h", "066758da027cd5480bd8a47a807632c6": "(n - 1)", "0667aa8259991dc1003ae3a7ef3ae3b8": "T= \\sigma_N/A\\sigma_0", "0667bfc2ffc9583d811ab4927f6f7dc0": "p = C^{-1}(-2 \\ln(p_1 p_2 \\cdots p_N), 2N) \\, ", "0667d925c7da0aa40a80f26cb23fcd13": "-x\\sin A + y\\cos A", "0667e4329ba13406226d5cea24b19455": "(\\nabla \\cdot \\mathbf v) f = \\left (\\frac{\\part v_x}{\\part x}+\\frac{\\part v_y}{\\part y}+\\frac{\\part v_z}{\\part z} \\right )f = \\frac{\\part v_x}{\\part x}f+\\frac{\\part v_y}{\\part y}f+\\frac{\\part v_z}{\\part z}f ", "06684da7018ebd3b6f0eae745842b787": "t \\propto x^2", "066850b2749b50590922dcb15469e15d": " {H_1 \\over H_2} = { \\left ( {D_1 \\over D_2} \\right )^2 }", "06685b068927716f1ce3908154ccb395": " \\frac{d^2\\theta}{d\\xi^2}+\\frac{2}{\\xi}\\frac{d\\theta}{d\\xi} + \\theta = 0 ", "066873313a2c794e852cf6adbd160bcf": "n_{1, t+1} = \\lambda n_{1, t}", "06688a9275a9fd584c2ba0fef2bb5a2b": "\\mathbf{x} = [x_1, x_2, \\ldots, x_{N_t}]^T", "066939d1ed61b458e6fcc10842a2d93f": " a^{m*2^k} + b^{m*2^k}.\\!\n", "06695c2683b56718e7253a916e120efc": " \\begin{pmatrix}\nF_\\text{x} \\\\\nF_\\text{y} \\\\\nF_\\text{z} \\\\\n\\end{pmatrix} = q\\begin{pmatrix}\nE_\\text{x} \\\\\nE_\\text{y} \\\\\nE_\\text{z} \\\\\n\\end{pmatrix} - q \\begin{pmatrix}\n0 & - B_\\text{z} & B_\\text{y} \\\\\nB_\\text{z} & 0 & - B_\\text{x} \\\\\n- B_\\text{y} & B_\\text{x} & 0 \\\\\n\\end{pmatrix} \\begin{pmatrix}\nv_\\text{x} \\\\\nv_\\text{y} \\\\\nv_\\text{z} \\\\\n\\end{pmatrix}", "0669b5a06de44528ae887d71ee31ad99": "(\\sin(\\alpha/2))^2\\,", "0669c91f79ea0489e96a0d277743c1bd": " x(t+1) = f \\left [ x(t) \\right ] \\approx \\varphi(t) = \\varphi \\left [ x(t)\\right ] ", "0669f5563d91e61b7916f60874730336": "\\epsilon_\\perp", "066a873d751b6c7c1227ff3ad7a3f235": "\\scriptstyle \\Delta_0", "066ab225e700949fd1fc61bd31b2ca29": "\\Omega(M,\\mathrm{T}M)", "066abd4ee06a1a15551d4d33435e9914": "t_{1} \\leq t_{2} \\implies \\mathcal{F}_{t_{1}} \\subseteq \\mathcal{F}_{t_{2}}.", "066b143b02ca298f94a0e4598f1206e9": " r = \\sqrt{\\frac{1}{2}(\\alpha^{2}-a_{21}\\alpha)} ", "066b3d5450fc5172d18d18e7ce7537f0": " z = x^{1/u} ", "066bb2257a55de86e56d9abaf7981d1d": "\\tan \\psi = -\\cot \\theta,\\, \\psi = \\frac{\\pi}{2} + \\theta, \\alpha = 2 \\theta.", "066c18b4d2049b5dded8990995d51334": "\\beta_{n}(T_e)", "066c1f04ac58a8d7633a539616d0e1a9": "\\tfrac {1}{6} \\pi^2 \\,", "066c26c3fe080fc316eb22ed44c1476a": "E_1 = E_2", "066c2b40daeec388ad222d1c338c3c83": "\\operatorname{E}_k", "066c3292096ad58a643b5c0b1e9ecb3d": " R_{i+1} ", "066ca74ca6eb93c75f218a3ff8ab5a69": "W_{1B}(y)", "066ce4431913d9f04ee5b13c9717e224": "v_d\\gg \\langle v\\rangle", "066d1b059460622cef5ed6cc1dff98ff": "Mod(\\sigma)(M')", "066d3aac7344710c04eaa756f0206458": "a_n=\\prod_{p^k \\mid n} \\frac{1}{k}=\\prod_{p^k \\mid \\mid n} \\frac{1}{k!} ", "066d6876b45d2660cd34cf9c03ee8b03": "a_0 \\leqslant a_1 < b_0", "066d77404b1f613f287dfa835100ea96": " \\displaystyle{f_\\alpha [\\pi] = [\\pi_l]} ", "066e2433c7b2ab71cc8af27d0869768e": "1-(1-\\alpha)^{1/m},1-(1-\\alpha)^{1/(m-1)},...,1-(1-\\alpha)^{1}", "066e6728e97193ce98e6102242dadbb3": "\\mathbf{f_{k}}", "066ea5ee3b091aa219300b229d553f6f": " \\delta z^{\\pm} \\ll B_0 ", "066eb2e8dd3eae537288c8ffa1c59f19": "U(P) = \\frac {1}{4 \\pi} \\int_{S} \\left[ U \\frac {\\partial}{\\partial n} \\left( \\frac {e^{iks}}{s} \\right) - \\frac {e^{iks}}{s} \\frac {\\partial U}{\\partial n} \\right]dS ", "066eb5d6feb36ae6c23ec3f8c70bb3af": "3 \\neq 0", "066ee51fb92bce3c2c2c95cf62538f5b": " \\delta\\left[ n \\right] = H[n] - H[n-1].", "066f30cec05a2c80a69ddcba5330f385": "L^2(\\mathbb R)=\\mbox{closure of }\\bigoplus_{k\\in\\Z}W_k,", "066f366637d8b734cdcb6850a51ed8ea": "f_1(X)f_2(Y)\\le f_3(X\\vee Y)f_4(X\\wedge Y)", "066fb94f6f904ffcd10ec7548b631bad": " \\left(\\frac{P_2}{P_1}\\right)^{-1 \\over \\gamma}=\\frac{V_2}{V_1} ", "066fd05ea8024436d9591d661eb6c36b": "X^2-D", "066ff3eb14fb518035dc413b97261c9f": "\\nu(\\theta)", "06700127a9e442cf05c79c15497d8963": "\\ (u + \\operatorname{d}u, v + \\operatorname{d}v)", "067036e7b5dfa9aad061ff612f962836": "{u \\over t}h \\equiv B_{(p-1)/2} \\pmod{ p}", "06703de5f6099981a749de52b9283778": "\\Box \\varphi", "0670443ef3b1c86a2459506c8098f58c": "wp(S, x=0) \\vee wp(S,x=1)", "067056bff759bf80fa0445c87a986dad": "\\ell_{(M,\\varphi)}(x,y)", "06708913c7fe58477cf8b1551e2394c8": "R^N", "06708de6599880972472ddd95342a36b": " p=u_x, \\quad q=u_y. \\,", "0670c0438d7cb43f7437da0f5bde1d78": "\\frac{1 + {\\scriptstyle\\frac{1}{3}}z}\n{1 - {\\scriptstyle\\frac{2}{3}}z + {\\scriptstyle\\frac{1}{6}}z^2}", "0670ced325ad94af3d97acf4547b6651": "{T^{\\alpha\\beta}}_{,\\beta} + F^{\\alpha\\beta} J_{\\beta} = 0", "0670fd4ad21a740df9f80fbe52d38b28": "N_k", "0670fd9dc081c99fd16be422a65c0366": " e + 2b = 180", "06710bbaf38ff33f645828396e72779e": "\\tau=\\theta", "0671570968abcb7dd61fb0eccc0d7ed7": "\\delta(n) = \\left| \\Pr_{x \\gets D_n}[ A(x) = 1] - \\Pr_{x \\gets E_n}[ A(x) = 1] \\right|.", "0671d7dfca8003422eefad9368fb8abc": "f(x) = \\omega(x) g(x)\\,", "0671dfa1ef2d24b727088aa07c2ade62": "a = x_0 < x_1 < \\cdots < x_n = b . \\,\\!", "06720f5d3d09e97e1aa9247a475d336e": "\\Gamma \\rightarrow ", "06722a5c67c658f46df95ac512149fba": "M \\oplus M^*", "067295cdea96e5c5cd7abdbf3a7443fb": "\\mathbf{w}(\\mathbf{X})", "0672f19f9a46a703e324a2238d5cf2b3": "n^{\\Theta(1)}", "06730233da1d833267e42dc715222711": "\\phi_\\omega(x)\\,", "067372b770afb538bef4527bc27a8d2d": "{\\text{PriorProbability}}(x=p;\\alpha \\text{Prior},\\beta \\text{Prior}) = \\frac{ x^{\\alpha \\text{Prior}-1}(1-x)^{\\beta \\text{Prior}-1}}{\\Beta(\\alpha \\text{Prior},\\beta \\text{Prior})}", "06737bcd735f0d3a2b534a15671e2eff": "\n\\begin{align}\n 1^{2p+1} + 2^{2p+1} &+ 3^{2p+1} + \\cdots + n^{2p+1}\\\\ &= \\frac{1}{2^{2p+2}(2p+2)} \\sum_{q=0}^p \\binom{2p+2}{2q}\n(2-2^{2q})~ B_{2q} ~\\left[(8a+1)^{p+1-q}-1\\right].\n\\end{align}\n", "06749f2a39f73bebb1d25788272f8214": "0 \\dots r", "0674bc21dbf638814d60e546889f2b8a": "e^{-\\tfrac{1}{2}\\sigma^2}", "0674dd7c8d7f30cc54d56f5ae5560879": "*(R_1,R_2,...R_n)", "0674fd93053ee8fdf28f7aaf69a00aba": "\\langle f,g\\rangle=(\\text{constant term of }f \\overline g \\Delta)/|W|", "0675098fd53d0ed69f56fe72b8740591": "\nS(\\omega) = 155 \\frac{H_{1/3}^2}{T_1^4 \\omega^5} \\mathrm{exp} \\left(\\frac{-944}{T_1^4 \\omega^4}\\right)(3.3)^Y,\n", "067510fba0d995e22350729841c910d3": "314\\,", "0675520d44ff563fc919439cc0dfa866": "i,j)\\in S", "06755e06e02c35fa4a4b189b4ba71d7b": "(S^1)^{\\wedge i} \\wedge (S^1)^{\\wedge j} \\to A", "06757e1103a40a7617c774b23ad18745": "d\\sigma", "067580ccb786f68077eb4c401c4b170b": "\\frac{\\partial g}{\\partial x}(X,Y,Z) \\cdot x+\\frac{\\partial g}{\\partial y}(X,Y,Z) \\cdot y+\\frac{\\partial g}{\\partial z}(X,Y,Z) \\cdot z=0.", "06758409cad5acbccb9b590639b9c987": "s\\in\\mathbb{C}\\setminus\\{1\\}", "0675fc293d7b86277328488400289d70": "\\tau_{ind}\\left(\\omega \\rightarrow 0\\right) = 0", "0676063a2bcdaa36ed8dc22f2d0145c4": "|\\dot{\\sigma}|", "06769d6913d96948a188f6bb10f2a7cb": "E_g=\\gamma\\left(\\frac{2a}{d_t}\\right)", "0676b1b022bc5a569c7083dc99faa8f4": "C_p(p,T)-C_V(V,T)=\\frac{TV\\,\\beta _p^2(T,p)}{\\kappa _T(T,p)}", "0677b553c54d02a66e67001fbbb60e6a": "\\!p", "0677d517e819310328d04a9e9409b1c1": "P_{-\\mu-\\frac12}^{-\\nu-\\frac12}\\biggl(\\frac{z}{\\sqrt{z^2-1}}\\biggr)=\n\\frac{(z^2-1)^{1/4}e^{-i\\mu\\pi} Q_\\nu^\\mu(z)}{(\\pi/2)^{1/2}\\Gamma(\\nu+\\mu+1)}\n", "06780a12fc4e68987e63d6abf66989a4": "\\psi'(g)=\\psi(g)", "067816e3563071c84ee0fcc1c7c96556": "\\alpha_1 \\,\\! ", "0678665b6db497c25d835e29345c2bb8": "f(z) = q - 2q^2 - 3q^3 + 2q^4 - 2q^5 + 6q^6 + \\cdots, \\qquad q = \\exp(2 \\pi i z)", "06788225244c9188ffd21bd2a5b97f14": " U = ", "067883f6a1df461f913f2a9f071d706d": "\\textstyle \\left \\lfloor \\frac{n}{p} \\right \\rfloor", "0678caa04da34220a4e8dc041488b618": "\\hat{\\theta}", "067938f8db0864ed2e506bec3acd6e85": "\\mu < \\mu_0", "0679440b99bfa58585f8a1779401cece": "\\frac{4 \\pi \\epsilon_0 G m_p m_e}{e^2} \\approx 10^{-40}.", "06798435062759cbb501c273d45a8752": "(2/1)^{31}", "0679c47aaa53066c513fade3ce974774": " x = \\mathop{\\rm sign}(x) \\sum_{i\\in\\mathbb Z} a_i\\,10^i", "067a06be194ff19d2c9d7e9e0d8ef1bc": "t = 2 i k_L e^{-i k_R L}\\left[\\frac{M_{11} M_{22} - M_{12} M_{21}}{-M_{21} + k_L k_R M_{12} + i(k_R M_{11} + k_L M_{22})}\\right]", "067a2dd2ec46d3ee5c7057bc20f2603e": "f \\left( z \\right) = \\ln{z}-z ", "067a66737160589adb79c8d2fa2ebf56": "\\mathbb{D}", "067a76ac757e97cb7ba2e8b882d01947": "011011", "067a7959aca16e6dca1750ba02e75af2": "\\mathcal{F}^{-1}g(x):=\\lim_{R\\to\\infty}\\int_{\\mathbb{R}} \\varphi(\\xi/R)\\,e^{2\\pi ix\\xi}\\,g(\\xi)\\,d\\xi,\\qquad\\varphi(\\xi):=e^{-\\xi^2}.", "067ae1930d84e342f43e13831d08d57c": "\\begin{align} 2\\cdot R_*\n & = \\frac{(51.3\\cdot 3.26\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 36\\cdot R_{\\bigodot}\n\\end{align}", "067ae35a0e786d0a745640683ea4e6c3": "U(y)\\!", "067b196a2eb6ed203f74271ad062ffb2": "\\begin{pmatrix}\n1 & 4 & 0 & 0 \\\\\n3 & 4 & 1 & 0 \\\\\n0 & 2 & 3 & 4 \\\\\n0 & 0 & 1 & 3 \\\\\n\\end{pmatrix}.", "067b7c30cc6df3dd0bcdee997c7de9fd": "S(z;u)=\\mathcal{X}^{-1}(z+\\mathcal{X}(u))", "067b926ccfc7b6a30d5dab2d30c3b4df": "Sq^i Sq^j = \\sum_{k=0}^{[i/2]} {j-k-1 \\choose i-2k} Sq^{i+j-k} Sq^k", "067ba60ad838fb71e30ef7c04220dd92": " C_0(x) ", "067bc08281a947a82a191fda0165ece3": "L_z(x)R_z(y) = B_z(xy)", "067beafbfe4c5e47df74c436264c5493": "\\Gamma_k", "067c085e0d9391d7e09b6bdb00ae9e70": "[X,Y] = XY - YX", "067c140cc51ade1b9712233649bd08cc": "r = \\alpha/\\beta", "067c1f736e72bee73cb745c45c4416ba": "\\frac{\\partial \\phi}{\\partial t} = D\\,\\frac{\\partial^2 \\phi}{\\partial x^2}\\,\\!", "067c2c0d217bdec32ae91e410f92dd83": "H_{\\text{bath}} ", "067c371e6c85d102c1598b3d6682ebdf": "h_n^{(2)}", "067c522d835f14123e6db1e083c569e7": "\\mathcal{E} = \\frac{U}{d}", "067cac656ba3bfc52d66aca056d9efe6": "\\mu_n(A) \\to \\mu(A)", "067cdb3665d1900f55d85812a3c2c6a4": "(\\Sigma^*, \\cdot, \\stackrel{*}{\\rightarrow}_R)", "067cf2572cd687b4d0da7249bc451d1a": "\\rho_M = \\Omega_M \\rho_c", "067cf3da45f5c22435e8c1ceb934e29f": "B = \\cup _j B_j", "067cf58f905a00984db1508337e09984": "N_k(A)", "067d01e6ac353c232be79dd25dc35671": "\\frac{\\beta }{k_{0}}= \\frac{c}{v_{ph}}= \\frac{\\lambda _{0}}{\\lambda _{g}}\\simeq sin\\theta _{m}", "067d1ec1784031f26a5dab89f01456ee": "r=\\tfrac12.", "067d4d071ce173a0f3778c9236cf08e4": " = \\frac{\\operatorname{P}[E_1]}{1-(1-\\operatorname{P}[E_1]-\\operatorname{P}[E_2])}\n= \\frac{\\operatorname{P}[E_1]}{\\operatorname{P}[E_1]+\\operatorname{P}[E_2]}\n", "067d5fde8f2c0731bc2f17fb68a7a23b": "A\\neq 0", "067d81383666089aca8b14e4b6ae6ab4": "(n,q,\\pi,G)\\,", "067e11f62a8a4cd7c873c289b749eeb2": " E[\\varepsilon_t]=0 \\, ,", "067e44c2ee9cb7e8ee7b12d675688024": "\\sigma = 1/\\nu d_\\text{f}\\,\\!", "067e7c5fd6c577257323affccb9f2d62": "-(1/3)E/c^2", "067e8173226f0047ad0e439500259b2f": "a=b=c", "067e9262376f55c180f13aa130ca1201": "d J/d x = d A \\rho v/dx = k r \\rho v", "067e99060907090655b8f083897609d8": "0.0\\dot{7}", "067f12a29ae2f078ce3367d6f6ec64f7": "f(\\phi)\\,", "067f4a8a461954dfa4aa5c1661495191": "P^s (S) = \\inf \\left\\{ \\left. \\sum_{j \\in J} P_0^s (S_j) \\right| S \\subseteq \\bigcup_{j \\in J} S_j, J \\text{ countable} \\right\\},", "067f56e80150f458f7f00050e947db33": "x^3 + Ax +B", "067f5b4e31caa7ea54b65c4ad196344e": "w_i = \\{f_i^1, ..., f_i^k\\}.", "068055de35dcdebbc711f4a9afaf13d3": "Pr[\\pi \\gets \\mathrm{Prove}(\\sigma,y,w) : \\mathrm{Verify}(\\sigma,y,\\pi)=\\mathrm{accept}] =1", "068059390899e3d6cf5fd5e76586865e": "Points\\ difference\\ = \\frac{14}{No\\ of\\ teams\\ -\\ 4}", "068099a91211d97da66278ce458cba93": "p = {\\delta(H) \\over l-1}p_c + o(H) p_m", "06809e93fb9570e35c55c9552ebfce10": "\\scriptstyle {L = \\lbrace (a_1,a_2,...a_n)|a_1c_1 + a_2c_2 + ... a_nc_n = d \\rbrace}", "0680a49da435b7058f15bc86ad1d61b4": "\n \\boldsymbol{\\nabla}\\times\\boldsymbol{\\omega} = -\\boldsymbol{\\nabla}\\times\\boldsymbol{\\varepsilon} = - \\boldsymbol{\\nabla}\\mathbf{w}.\n ", "068132e5c9db9a7bf997db2b19e1925d": " = | {v}_1 - {v}_0 |\\;", "068168112d1f7deab9e7f61ac52b7f2c": " \\int \\Phi(a+bx) \\, dx = b^{-1} \\left ((a+bx)\\Phi(a+bx) + \\phi(a+bx)\\right) + C ", "0681892c6549c3efc5cd229822e7e9bc": "Z_{\\rm can} ", "0681b5ac60d6a074cee5e6cf09e4ddda": "\\int d\\mu(\\sigma) \\prod_j \\sigma_j^{n(j)} = 0 ", "0681eb8ecfbc9b5990c22df8c6daa4fe": "\\lambda = \\frac{g}{2\\pi}T^2 \\qquad \\scriptstyle \\text{(deep water).}", "06821a4333616ec92dde102111db818e": "\\alpha_{1}(a,\\, b) \\cdot \\alpha_{2}(a)", "068270d8dcc78b8ada5a0f9fc6f6af11": " \\frac{dV}{dt} = v \\pi R^{2} = \\frac{\\pi R^{4}}{8 \\eta} \\left( \\frac{- \\Delta P}{\\Delta x}\\right) = \\frac{\\pi R^{4}}{8 \\eta} \\frac{ |\\Delta P|}{L}, ", "06829d25041f675aed53bc90d4e4569e": "P_\\mathrm{emp}(x)", "0682f62ed88d2d9e7403ac03e0f06255": "\\displaystyle u \\ ", "0682f8703a7bbe19c459acbbc3312dd5": "\\scriptstyle p=\\tfrac{b}{a}, q=\\tfrac{c}{a}\\!", "0683443785f335e228b1d8fa42899e04": "\\sigma=1/2", "0683afc6d0540771a2bbf8f8bdedf0cb": "K = z/\\sin z", "068423177eff58cdb23c0632588209f6": "\n\\int_{B_{1/2}}\\!\\!\\!\\chi_{B_1}(x-y) \\varphi_{1/2}(y)\\mathrm{d}y= \\int_{B_{1/2}}\\!\\!\\!\n \\varphi_{1/2}(y)\\mathrm{d}y=1\n", "06842a7a139712704a54a8284b5017cd": "m,n,d,c", "06844c045947ee318197c628dcc921ef": "C = -\\ln p_k .", "0684528693d692e4c88e53e6cdc01e39": "~W_{\\rm d}=\\frac{ I_{\\rm p} \\sigma_{\\rm ep}}{ \\hbar \\omega_{\\rm p} }+\\frac{I_{\\rm s}\\sigma_{\\rm es}}{ \\hbar \\omega_{\\rm s} } +\\frac{1}{\\tau}~", "06854d35d6224320ae115a5c6c62d2c5": " {\\lVert x_k-x \\rVert^2} \\leq \\left(1-\\left|\\left\\langle\\frac{x_{k-1}-x}{\\lVert x_{k-1}-x \\rVert},Z_k\\right\\rangle\\right|^2\\right){\\lVert x_{k-1}-x \\rVert^2}. ", "06859cecf6e54eeb8e0f170d13ee9901": "\\mathsf{case}\\ e\\ \\mathsf{of}\\ x \\Rightarrow e_1 | y \\Rightarrow e_2", "0685d737671d4db440ae000f13ddd614": " \\alpha= \\frac{W}{W+k \\cdot (W+M)} \\,\\ ", "0685e6b1212801a886813184201e30b7": "{}^2 E'_{pq} = H_p(Y, H_q(*, \\mathbb{Z})) \\Rightarrow H_{p+q}(Y, \\mathbb{Z}).", "06861e59a653954257336dd2becf45a8": "\\Gamma \\backslash N", "068695c2aa2969123e274e78272c42e1": "\\xi_1,\\dots,\\xi_K", "0686bc15bcef41940b5b94f503be18fc": "a_{\\ell m}^{(M)}=\\frac{-ik^{\\ell+2}}{(2\\ell+1)!!}\\left(\\frac{\\ell+1}{\\ell}\\right)^{1/2}[M_{\\ell m}+M_{\\ell m}']", "0686f68c86e63e8c126077f0c255d319": "\n\\arccsc(z)\n", "068705c208645af032cd5da5e112042d": "q = \\int \\rho \\, dV", "06870b319501adc54583033a50d63092": "a (x-y)(x-z) + b (y-z)(y-x) + c (z-x)(z-y) \\ge 0.", "068722584f10a02209cb3a22738c9e98": "P=2.\\omega_D.i, \\; A=\\xi.\\omega-\\omega_D.i, \\; B=\\xi.\\omega+\\omega_D.i", "0687357451abeef5780b4c3bd9e107a9": "\\ \\mbox{SF} = \\frac{\\mbox{chip rate}}{\\mbox{symbol rate}}", "0687510a024f6831750fafcd94541d38": "\\left | H(j\\omega) \\right | = \\left | \\frac {1}{1+\\alpha j \\omega} \\right | =\\sqrt{ \\frac {1}{1 + \\alpha^2\\omega^2}}", "06876f67e1ffe1319ce5ef2524074a4e": "\\displaystyle \\nabla^2u+\\sinh u=0", "0687a14bf9b9e2b96b257c38fb12ca1d": "\\neg a \\vee c", "0687bc8a5c3584cb26adeaa6ae986916": "U'_a", "0687d164c9a5ac835eeec8b898440f13": " l_A a_D + (1 + \\sigma) (l_A a_B + l_B) l_D ", "0687dc1974f9d312eddda27504a24a8f": "\\mathbb{P}\\left( (Z_1,\\ldots,Z_n) =\n(z_1,\\ldots,z_{i-1},z,z_{i+1},\\ldots,z_n) \\right)\\,.", "06881dc19dc00e4ef1755e9dd4b09094": "\\int_{a}^{b}\\omega(x)f(x)=\\sum_{j=1}^{N}w_{j}f(x_{j}) = w_{i} f(x_{i}).", "06887fa940c07c1481fc61748faaac5a": "(\\phi,\\lambda)", "06889b83613b0fda2a4fdd78dc68d49c": "\\displaystyle{\\Phi(1)=P.}", "0688db2962b45a1527abaf75d43d146b": "\n D = \\cfrac{Eh^3}{12(1-\\nu^2)} \\,.\n", "0689341a386da86cf9563874ad6f75b7": "u(x_1)", "068a7b56b0cdc2e58d9bed0c11515e1f": "\n\\partial_t \\hat{u}_k\n=\n- \\frac{i k}{2} \\sum_{p+q=k} \\hat{u}_p \\hat{u}_q \n- \\rho{}k^2\\hat{u}_k\n+ \\hat{f}_k\n\\quad k\\in\\left\\{ -N/2,\\dots,N/2-1 \\right\\}, \\forall t>0.\n", "068ad15d949ff53e4c05dee669d187ce": "\\mathcal{E} = - \\frac {d \\Phi_B} {dt} = -\\frac {d}{dt}\\iint_{\\Sigma (t)} d \\boldsymbol{A} \\cdot \\mathbf{B} (\\mathbf{r},\\ t) \\ , ", "068af70b4be78ed9b688a40c2d5c8a1d": "\\begin{matrix}\\frac1{128}\\end{matrix} (6435x^8-12012x^6+6930x^4-1260x^2+35)\\,", "068b01dbadf5a8f6fb36d6dd2dda2c34": "k(s) = \\det\\begin{bmatrix}\\beta''(s) & \\beta'''(s) \\end{bmatrix}.", "068b762cc7ed5622a06f8ecbc88396d0": "x^{2} \\equiv 1\\pmod{p} ", "068b85d57c0b9c89af307e6314448557": "\n \\frac{\\partial \\sigma_{xx}}{\\partial x} + \\frac{\\partial \\tau_{xz}}{\\partial z} = 0\n", "068b8ed62a82c0bfedea19a38beda42c": "\\varphi\\circ g", "068bbfb5ca1b6884fd761f3c4dc4d4da": "q_1 q_2\\cdots q_n", "068be23a9d109063b10a313e2f71f7f5": "c_n=h_0^n + h_1^n + h_2^n.", "068c00d7f35e1321fb8899bb83b8db52": " \n \\begin{bmatrix} T_1\\\\T_2 \\\\ T_3 \\end{bmatrix} = \n \\begin{bmatrix}\n \\sigma_{11} & \\sigma_{21} & \\sigma_{31} \\\\\n \\sigma_{12} & \\sigma_{22} & \\sigma_{32} \\\\\n \\sigma_{13} & \\sigma_{23} & \\sigma_{33} \n \\end{bmatrix}\n \\begin{bmatrix} n_1\\\\n_2 \\\\ n_3 \\end{bmatrix}\n", "068c063f5304c4222bac4f60474f6b5d": "\\gamma_{s}", "068c3e650dc389220a48a4be2511f186": "\\mathbb{P}[\\omega = H] = p\\in (0, 1)", "068c4dbb106c87186d1f4c4ed052a676": "A_u", "068cbacf039d96a2e80d3c510ff41f4e": "\\frac{d\\tau}{dt} = 1 - \\frac{U}{c^2} - \\frac{v^2}{2c^2} ", "068d053da819608daa8b38c9cf3118da": "q = q_1 + ... + q_r", "068d06b9ae016c763684a23a994c4c56": "Plato:c-b=1,\\quad \\quad Pythagoras:c-a=2,\\quad \\quad Fermat:\\left| a-b \\right|=1", "068d14bf79c3ee8628bb81520cfb5b27": "g(r) = \n\\begin{cases}\n 0,&r 0", "06a55db1a17d4505e2b15c4aec2d780a": "\\| f \\|_{L^{p, q}} = \\left\\{ \n\\begin{array}{l l} \n\\left( \\int_0^{\\infty} (t^{\\frac{1}{p}} f^{*}(t))^q \\, \\frac{dt}{t} \\right)^{\\frac{1}{q}} & q \\in (0, \\infty),\\\\\n\\displaystyle \\sup_{t > 0} \\, t^{\\frac{1}{p}} f^{*}(t) & q = \\infty.\n\\end{array} \n\\right.", "06a57402a19a50c01966d32ba45a13ca": "\n\\begin{align}\n\\hat{y} = &\\ 25 \\\\\n& + 6.1 \\max(0, x - 13) \\\\\n& - 3.1 \\max(0, 13 - x) \\\\\n\\end{align}\n", "06a577e4bfc6d61013b0a750e883d100": "\\mathit{k} \\in \\mathbb{Z}^+", "06a69789c3a0377cbcfbc5f9b7e25541": "\n{\\mathbb Z}\\backslash \\left(D^n\\times{\\mathbb R}\\right)", "06a69d8fac6c838029f7d39d117759b8": "\\vec{r}\\,'(t)", "06a6cc8549169648b4030dcbf8a34b9b": "\\theta_{min} \\approx \\frac {CD} {AC} = \\frac{\\lambda}{W}", "06a76ed86a8187d5ed2914ac3e56b22c": "\n {d\\vec{\\omega} \\over dt} = (\\vec{\\omega} \\cdot \\nabla) \\vec{v} + \\nu \\nabla^2 \\vec{\\omega}\n", "06a7886775c5a32da669024135dae607": "\\quad\\quad\\int \\arccos(y) \\, dy = y\\arccos(y) - \\sin(\\arccos(y))+C.", "06a7a2204d20af7ec17f478e3bfc0dc5": "Rz = 12.528\\cdot(S^{0.542})/((P^{0.528})\\cdot(V^{0.322}))", "06a7a43679442901409ceea31fd2c63d": "\nP_\\mu (n,t)=\\frac{(\\nu t^\\mu )^n}{n!}\\sum\\limits_{k=0}^\\infty \\frac{(k+n)!}{\nk!}\\frac{(-\\nu t^\\mu )^k}{\\Gamma (\\mu (k+n)+1)},\\qquad 0<\\mu \\leq 1,\n", "06a832e45f0a2df58022d7c2e13998ee": "H = \\sum_{j\\sigma}\\epsilon_f f^{\\dagger}_{j\\sigma}f_{j\\sigma} + \\sum_{\\sigma}t_{jj'}c^{\\dagger}_{j\\sigma}c_{j'\\sigma} + \\sum_{j,\\sigma}(V_j f^{\\dagger}_{\\sigma}c_{j\\sigma} + V_j^* c^{\\dagger}_{\\sigma}f_{j\\sigma}) + U\\sum_{j}f^{\\dagger}_{j\\uparrow}f_{j\\uparrow}f^{\\dagger}_{j\\downarrow}f_{j\\downarrow}", "06a85b19a6802a789494d61c25143645": " \\tau = 0.85 \\sigma _n", "06a896fa8ff8e974397a7c1d4f7e970c": " -\\left(\\eta_2 + \\frac{p + 1}{2}\\right)(p\\ln 2 - \\ln|\\boldsymbol\\Psi|)", "06a8c4a883febf3b3c1e8d98aef4380a": "\\mathbf{w}_n", "06a927df29f1b23a6ddbca364428e099": "r^s\\,", "06a96ec00c9ee7409a4fb60fa521edb5": "U \\subset \\Omega_x", "06a9739646c8387f56abe4303aa9173e": "M=\\begin{bmatrix}\n1 & 0 & 1\\\\\n0 & 1 & -1\n\\end{bmatrix}", "06a994dc739c93a51c1249e678797b3a": "m \\in \\{ 0, \\dots, n-1 \\}", "06a9b7f085a1e129cf9ed70f2fd4cfc7": "P_{\\mathrm{i}}", "06a9c832ac2806ff5170bcd4c8966904": " (s,t_s) \\in S'", "06a9fe91a949298ba79500ecd7ff46ea": "\\mathbf{1}_A(x) = \\begin{cases}\n1 & \\text{ if } x \\in A \\\\\n0 & \\text{ otherwise}\n\\end{cases}", "06aa3ae0f57e9b87bf69f3a38047ebd5": "G \\approx 1/t\\,", "06aa92d94fa8eb721ebd2c5a74e9693d": "N=S-S_0-\\int\\frac{dQ}{T}.", "06ab332b667b5fccd38be74338754cc0": "t\\otimes v\\in V_{tgt^{-1}}", "06ab38ad98e39bcacc7a8082887e64c3": "{*}", "06ab582721055f7507054a8fdd5f9034": "\\scriptstyle W_p", "06aba47b20f4437dddfa1bde395760c3": "g_{i j} = \\mathbf e_i \\cdot \\mathbf e_j", "06abc6a479b86882b9cfa3a247503a29": "a\\frac{\\partial \\mathbf{u}}{\\partial x}", "06abe4267cc458b20d5629ad6ec811d3": " \\ CVI(ESA) = A\\phi\\mu_d\\frac{\\rho_p-\\rho_m}{\\rho_m}", "06abf479832451bfce7629e688b35f86": "(1 - p)^{k-1}\\,p\\!", "06aca9d7637d292ae6f30e90d6492007": " J \\ne R", "06acaebf61246b1dfd3f106e2ab527b3": "E_A(\\log(x))\\geq E_B(\\log(x))", "06acd492d5d9c2d9ad2d2be4be0dbd9b": "d + a\\mathbf{\\hat{i}} + b\\mathbf{\\hat{j}} + c\\mathbf{\\hat{k}}", "06ad0d79511ce778229beb2e7adbbdb8": "\\mathcal{E}_{ijk}", "06adaae10024e7fa1776d1611e210975": "\\phi^2 = \\phi+1", "06add2141d8cd704911cc9766f6b3d74": "\\tilde{\\mathbf{B}}^+ = W(\\tilde{\\mathbf{E}}^+)[1/p]", "06adfd48187e054bb4439d9133640790": "1 \\mathrm{\\ rev} = 360^{\\circ} = 2\\pi \\mathrm{\\ rad}\\mathrm{, and}", "06ae0850263fd1e91f14e535544e034b": "\n\\operatorname{Li}_2 \\left(x \\right) + \\operatorname{Li}_2 \\left(1-x \\right) = \\tfrac{1}{6} \\pi^2 - \\ln(x)\\ln(1-x) \\,,\n", "06aebef211fb825da2eda40aa75499f0": " \\varepsilon_{ni} ", "06af3826fe0062389f5975927a08f573": "\\phi_{sl}=\\frac{\\rho_{s}(\\rho_{sl} - \\rho_{l})}{\\rho_{sl}(\\rho_{s} - \\rho_{l})}", "06af87fb65c8cbd4a2578d1c5a1bf356": "\\beta_n^{ }", "06afbfb6bdc88b7a434fd27845332387": "n\\equiv 1 \\pmod{2^{k}}, \\quad n\\equiv 0 \\pmod{5^{k}}\\, .", "06b048d5acd367ff91c5a6b90010b4ef": "\n\\epsilon_{abc} \\eta_{b\\mu\\nu} \\eta_{c\\rho\\sigma}\n= \\delta_{\\mu\\rho} \\eta_{a\\nu\\sigma}\n+ \\delta_{\\nu\\sigma} \\eta_{a\\mu\\rho}\n- \\delta_{\\mu\\sigma} \\eta_{a\\nu\\rho}\n- \\delta_{\\nu\\rho} \\eta_{a\\mu\\sigma}\n", "06b068f1873379c153e916e9b9211a09": "\\left\\{ S_{\\alpha}^i, \\overline{S}_{\\dot{\\beta}j} \\right\\} = 2 \\delta^i_j \\sigma^{\\mu}_{\\alpha \\dot{\\beta}}K_\\mu", "06b06b1fc241987f0246fb3a4a70fec1": "A \\otimes B", "06b0fd8f01ce256f229860bf283f3e6d": "\n \\sigma_{ij} = \\lambda~\\varepsilon_{kk}~\\delta_{ij} + 2\\mu~\\varepsilon_{ij} = c_{ijk\\ell}~\\varepsilon_{k\\ell} ~;~~ c_{ijk\\ell} = \\lambda~\\delta_{ij}~\\delta_{k\\ell} + \\mu~(\\delta_{ik}~\\delta_{j\\ell} + \\delta_{i\\ell}~\\delta_{jk})\n ", "06b137ca45622b7aec8d85f58f8d164a": "{\\frac{m}{e}}>2.35", "06b16782e68921373cf5edf0493be912": "=\\empty", "06b18d1719a5020a6cdd57551570b346": "a^2+c^2=b^2+d^2", "06b1972b2a0d710bdac4a38f8c8d9db8": "\\forall_1", "06b1a519499bbfb45c46e582db7016bb": "\\sum_{w\\in I_n} f^{1/k}(w) \\mu_n(w) = O(n)", "06b1deed07c8747ed175f7eb5d24496f": "f_{W}/f\\,", "06b1e1d2e04eb5ac021f2575ed7b64a3": "P_{A}=\\frac{|C_{A}|^2}{|C_{A}|^2+|C_{B}|^2}", "06b20283768dfa01a2508eb1553f20d6": "\n\\nabla^{2} \\Phi = \n\\frac{1}{a^{2} \\left( \\sinh^{2}\\mu + \\sin^{2}\\nu \\right)} \n\\left[\n\\frac{1}{\\cosh \\mu} \\frac{\\partial}{\\partial \\mu} \n\\left( \\cosh \\mu \\frac{\\partial \\Phi}{\\partial \\mu} \\right) + \n\\frac{1}{\\cos \\nu} \\frac{\\partial}{\\partial \\nu}\n\\left( \\cos \\nu \\frac{\\partial \\Phi}{\\partial \\nu} \\right)\n\\right] +\n\\frac{1}{a^{2} \\left( \\cosh^{2}\\mu+\\cos^{2}\\nu \\right)}\n\\frac{\\partial^{2} \\Phi}{\\partial \\phi^{2}}\n", "06b21f7325a11c27a747ce38c164237a": "\\mathbf{k}_o", "06b2a00ff6acf41deb8ba139ccae4f6b": " L = m\\hbar", "06b2d3f8e67033ccbdea177b6735505f": "k_i\\sigma", "06b3473cc47c38570634b2fbce24af01": "\\cot A = \\frac {1}{\\tan A} = \\frac {\\textrm{adjacent}} {\\textrm{opposite}} = \\frac {b} {a}. ", "06b37022ce795d7496c4b01ad20c9a12": "\n\\hat{\\mu} \\sim IG \\left(\\mu, \\lambda \\sum_{i=1}^n w_i \\right) \\,\\,\\,\\,\\,\\,\\,\\, \\frac{n}{\\hat{\\lambda}} \\sim \\frac{1}{\\lambda} \\chi^2_{n-1}.\n", "06b398af20108b57457bddd031c674aa": "Z_\\mathrm{in}=Z_L \\,", "06b40e2b1efc279d7974e7b8fddff0b6": " [B]=-\\frac{k_1'}{k_2} ln \\left ( 1 - \\frac{[C]}{[R]_0} \\right )", "06b45a226f0a63a5bb2936785b8bf396": "C \\subseteq X", "06b4633b591c7203cf47c5655d6c763f": "p_{e}", "06b46f7f426d4227c38f05c06e398621": "f_\\text{P1,2} = f_\\text{1,2} \\left( 1- \\vec v \\ast \\frac{\\vec e_\\text{1,2}}{c}\\right)", "06b4726fb98cc908e151656bb50e8540": "(x_1, \\ldots, x_n)", "06b48e3cafb0bdfb0be0e2bf7a6c91bc": "-1 \\le \\rho_{ij}< 1", "06b4979a43155a9f4fbf1a58ee620f44": " u(t,x,y) = tM_{ct}[\\phi] = \\frac{t}{4\\pi} \\iint_S \\phi(x + ct\\alpha,\\, y + ct\\beta) d\\omega,\\,", "06b4c72d5a08353c4adf480325e491e1": "r := 0", "06b510444ae01abaf2cb66a7085a415e": " rN", "06b53cf41884e57bcaf94421e92b2f7e": "\n\\mathcal{L}\\{f(x)\\}=-\\boldsymbol{\\alpha}(sI-\\Theta)^{-1}\\Theta\\boldsymbol{1}\n", "06b56847960f2dbf2669533900541748": "f_\\ast \\colon S_\\ast(X)\\rightarrow S_\\ast(B)", "06b5772a2f9345b1359939e01a53c047": " \\mathcal{P} = \\lbrace p \\mid p <_{\\mathcal{O}} e_d \\rbrace ", "06b58525084d383b44896c04eefd5fd8": "Q \\,", "06b58921a3a45af60b12b2818b208f59": "-j2\\pi /n", "06b5db735ebb6fc09e2c7f37cf8cee27": "T=|t|^2= \\frac{1}{1+\\frac{V_0^2\\sinh^2(k_1 a)}{4E(V_0-E)}}", "06b6181ce1cf9c509c5e45f72fc2af49": " \\exists ! x_n A(x_1, \\ldots , x_n) ", "06b63628beb319968cb673bb6d9aaeb9": "\\textstyle \\oplus_{i=1}^n \\mathbb{C}^m", "06b651a692ef43f1f96580ad0471cace": "\\mu(x,G)=B(x,\\delta/2)", "06b661bd66f0d7407279b496647dfdcf": "\n \\begin{bmatrix}\n 1 & 2 & 3 & 4 & 0 & 0 & 0 \\\\ \n 0 & 3 & 2 & 1 & 1 & 0 & 10 \\\\\n 0 & 2 & 5 & 3 & 0 & 1 & 15\n \\end{bmatrix}\n", "06b6732971a098fefbf1cdcc022a0707": "\\mathcal L \\left\\{J^2f\\right\\}=\\frac1s(\\mathcal L \\left\\{Jf\\right\\} )(s)=\\frac1{s^2}(\\mathcal L\\left\\{f\\right\\})(s)", "06b6942ffb3bd3773ab7dde8bd9d463b": "j < k + m < j + 1 \\,", "06b6b26126dc65a7c0ba235188c2ee93": " Q(\\theta_1,\\theta_2,\\theta_3)= Q_{\\bold{x}}(\\theta_1) Q_{\\bold{y}}(\\theta_2) Q_{\\bold{z}}(\\theta_3) , \\,\\!", "06b6c2f1ae45d6ba5e1a9735ec1140e9": " u'(x) = \\lim_{h \\rightarrow 0} \\frac{u(x + h) - u(x)}{h} \\qquad (2)", "06b72d8385b68498eb3c2025ccafe15b": "C(P)=\\{ \\lambda_1(\\textbf{x}_1, 1) + \\cdots + \\lambda_n(\\textbf{x}_n, 1) \\mid \\textbf{x}_i \\in P,\\ \\lambda_i \\in \\mathbb{R}, \\lambda_i\\geq 0\\}.", "06b742d2a227e5c7633150c9ebaa83bd": "\n\\widehat{\\boldsymbol \\theta}_{JS+} = \n\\left( 1 - \\frac{(m-2) \\sigma^2}{\\|{\\mathbf y} - {\\boldsymbol\\nu}\\|^2} \\right)^+ ({\\mathbf y}-{\\boldsymbol\\nu}) + {\\boldsymbol\\nu}.\n", "06b7536a72d0f7c7c5af9aa188335008": "X^\\alpha _0 = \\{0\\};\\ \\ X^\\alpha_{n+1} = \\bigcup_\\gamma X^{\\beta_{\\gamma+1}}_n\\setminus \\beta_\\gamma", "06b7541704df6ffd80bfe8b062fff09b": "{{i}_{C1}}={{i}_{C2}}\\equiv {{i}_{C}}", "06b77d730793f0bc864492d8ecfeaeb0": "g=\\begin{pmatrix}a & b \\\\ c & d\\end{pmatrix}", "06b7d06f0c2dffd025c2e773acfee9f3": " (f^*g^N)(v,w) = g^N(df(v),df(w))\\,.", "06b8160063b2584d96f2a9f25140cbf1": "x^2 + y^2 + z^2 = c^2t^2.", "06b81f958e3d1c0ee06aa37721af644f": "dT_{V}=pd\\Theta", "06b8622846eb44b8fad8a1bd22c4e77f": "FX/SO(1,3) ", "06b867677f19b7e48c680dda13b99618": "\\max_{x_1,\\ldots, x_n} \\Delta_n(\\mathcal{C}, x_1, \\ldots, x_n)", "06b8bf1b0a58f1bcb74d65a37edcbe2a": "\n \\begin{Bmatrix}\n j_1 & j_2 & j_3\\\\\n j_4 & j_5 & j_6\\\\\n j_7 & j_8 & j_9\n \\end{Bmatrix}\n = \n \\begin{Bmatrix}\n j_1 & j_4 & j_7\\\\\n j_2 & j_5 & j_8\\\\\n j_3 & j_6 & j_9\n \\end{Bmatrix}\n =\n \\begin{Bmatrix}\n j_9 & j_6 & j_3\\\\\n j_8 & j_5 & j_2\\\\\n j_7 & j_4 & j_1\n \\end{Bmatrix}.\n", "06b8c6ff2fc9dc85f76b034d346ebd9f": "\\frac{ \\omega }2 \\, \\partial_\\phi=\\vec{e}_3+\\vec{e}_0", "06b8e92904fdfd7fed1edbec65f6b750": "\\mathcal{T};", "06b9a76f4b9894f3f086afb18cece11f": " \\theta = \\alpha_{0} [\\tan (\\lambda L) \\sin(\\lambda y) + \\cos (\\lambda y) - 1]", "06b9c9ab3e809d880a89970d1e7e78be": "f_1(z) = \\,_1F_1(a+1;b+1;z)", "06b9d9792d77936371978d7765d73abe": "\\scriptstyle(f_n)", "06b9da224eb265a6651cfbe33634ca70": " (-r \\sin \\varphi, r \\cos \\varphi, 0) . \\,\\!", "06b9ddbf8e582b8f76a5cfbe98c05ab5": "\\tau_{\\text{rms}}=\\sqrt{\\frac{\\int_0^\\infty(\\tau-\\overline{\\tau})^2 A_c(\\tau)d\\tau}{\\int_0^\\infty A_c(\\tau)d\\tau}}", "06b9ebc07d19b6e751779c9deec2975a": "(y'_1,y'_2,y'_3,s')", "06bab0275afda592834316418089c3aa": "\\sigma\\text{'s}", "06bae13d55767350862d703a0a443d91": "\\frac{h^4}{180}(b-a) \\max_{\\xi\\in[a,b]} |f^{(4)}(\\xi)|,", "06baefddda163e47d9a815a391f33883": "\\sqrt{9}=3", "06bb25ce4bcf2a16644ef74a68d31886": "t<0 ", "06bb2827a354b8a6a491f8956ffd9c7b": "S=A/4", "06bbe522b15fd4974f2c5dd35df30232": "\n\\begin{matrix}\n7)16^27^66^32^41^60^49^0\n\\end{matrix}\n", "06bc01adbf7210620cc43b392bd15974": "z_k:=z(kh)", "06bc16d78f2bff88c09c264bcab033f5": "P^{-1}=A^{-1},", "06bc28ff8bc6ed7a28751992b6b03fb4": "\\mathcal{F}_{\\tau}", "06bc8f6595dcd604d81c645714a0cc74": "u(x,\\dot{x}) = -(|\\dot{x}|+k+1) \\underbrace{\\operatorname{sgn}(\\overbrace{\\dot{x}+x}^{\\sigma})}_{\\text{(i.e., tests } \\sigma > 0 \\text{)}}", "06bc9999ffd4b7f56d9acb4bc41b7b2b": " \\cfrac{\\Gamma \\vdash A, \\Delta \\qquad \\Sigma \\vdash B, \\Pi}{\\Gamma, \\Sigma \\vdash A \\and B, \\Delta, \\Pi} \\quad ({\\and}R)\n ", "06bcc1b573ae3e3f913f9747f154270e": "\\begin{align}\nS\\left(v\\right)&=\\int_0^T\\frac{d}{dT}E\\left(v\\right)\\frac{dT}{T}\\\\[10pt]\n&=\\frac{E\\left(v\\right)}{T}-k\\log\\left(1-e^{-\\frac{hv}{kT}}\\right)\n\\end{align}\n", "06bcd5269b2a6532a55599dd5187f116": "2^{|V|-1}-1", "06bcf75c2eabd4e09fe93db827cc7deb": "L^1(G//K)\\ni f\\mapsto \\hat{f}", "06bd1846a371cfac4ab8923b7a92d943": "T_{first}", "06bd3be4113d20cac5c1af358416abdf": "x=\\frac{v^2_{bullet}2 \\sin(\\delta\\theta)\\cos(\\delta\\theta)}{g} \\,", "06bda16a2e19377ac3bb0d4d253bc272": "\\langle E(t) \\rangle = \\frac{C}{t^3} + \\textrm{finite}\\,", "06bdaf2d7bde957d9fff2920ba9c8028": " v^{\\mu} \\ \\stackrel{\\mathrm{def}}{=}\\ {dx^{\\mu} \\over d\\tau} = \\left (c {dt \\over d\\tau}, { dt \\over d\\tau}{d\\mathbf{x} \\over dt} \\right ) = \\left ( \\gamma , \\gamma { \\mathbf{v} \\over c } \\right ) ", "06bde2879f214505b9808cc161a2d455": "I_k(\\mathbf{y}, t)", "06be07d35fabd5ce8f0b06d71eee740c": "f(x_1,\\ldots,x_k) \\simeq U(\\mu y\\, T(y,e,x_1,\\ldots,x_k))", "06be37efc4272118aead8209bde71ffa": "VC(C)=VC_0(C)+1.", "06be3857f24e511c1218394307f03b29": "R_{3,3} = r^3", "06be9fa0f9e14a2759fc4fb778ea7ff2": "\\mathrm{C_0 = 0}", "06bf44d7f269895f9d5f46fc5a5955e9": "v_{110}", "06bf55697e3fb2ab1ab2d08cf18a9f2d": "X^n(j)", "06bfaa8bb4bb4f1dd228210ab38fd26b": "\\operatorname{MSE} \\, \\hat{f}(\\bold{x};\\bold{H}) = \\operatorname{Var} \\hat{f}(\\bold{x};\\bold{H}) + [\\operatorname{E} \\hat{f}(\\bold{x};\\bold{H}) - f(\\bold{x})]^2", "06c04e28a21751cb1b8483f2bc2da567": "\\psi_L\\rightarrow e^{i\\theta_L}\\psi_L", "06c0858237a31c8ec7700537002f9227": "J = 4t^2/U", "06c08b6eb8ee26cabbfcd9d4c5a5941b": " |1-z|\\leq M(1-|z|) \\, ", "06c099d6b68186dd2ffad8add13d0141": "V(t) = V_0{H*h}(t) = \\frac{V_0}{\\sqrt{\\pi}}\\int\\limits_{-\\infty}^{\\frac{\\sigma t}{2}}e^{-\\tau^2}d\\tau = \\frac{V_0}{2}\\left[1+\\mathrm{erf}\\left(\\frac{\\sigma t}{2}\\right)\\right]\\Leftrightarrow\\frac{V(t)}{V_0}=\\frac{1}{2}\\left[1+\\mathrm{erf}\\left(\\frac{\\sigma t}{2}\\right)\\right]", "06c0b29db4c8b8b8b21123a934320e3c": "\\{f^*_n\\} \\in A'", "06c107ecfc08a6ee9113a1fc02ca1f06": "\\log(X_i)", "06c1086887a6c7c5155178a1564c7095": "\\mbox{ P1 }:\\begin{cases}\nu''(x)=f(x) \\mbox{ in } (0,1), \\\\\nu(0)=u(1)=0,\n\\end{cases}", "06c12e1824a0f5ee49f8ea651a650f27": "\n\\begin{bmatrix}\nL\\\\M\\\\S\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n0.8951 & 0.2664 & -0.1614 \\\\\n-0.7502 & 1.7135 & 0.0367 \\\\\n0.0389 & -0.0685 & 1.0296\n\\end{bmatrix}\n\\begin{bmatrix}\nX\\\\Y\\\\Z\n\\end{bmatrix}\n", "06c141702f2f0a1c813a676f56b084da": "a=\\theta", "06c1465b20e8f7941c105e39360901d1": "D\\ne 26", "06c15b82a3ad5f71a943c8809fffaeb7": "\\Delta_{n+1} \\equiv \\Omega_{n+1} - \\Omega_n = \\frac{f(u_n)}{u_n} \\delta_n\\Omega_n = \\frac{f(u_n)(1+u_{n-1})}{f(u_{n-1})u_n}\\Delta_n,", "06c18e7e7c0d23c57e7bc656c338b014": "4!/(2!2!) = 6", "06c1c5414cb3043035bfa6eb54717f57": "\\overline{c_i}=c_i", "06c1cf783b14d26057fe32bfa1217003": "\\textstyle Y(\\omega)=y ", "06c1eeb6446fcff690c856056b8a6a02": "\n\\begin{align}\n\\mathrm{ker}_n(x) & = \\frac{1}{2} \\left(\\frac{x}{2}\\right)^{-n} \\sum_{k=0}^{n-1} \\cos\\left[\\left(\\frac{3n}{4} + \\frac{k}{2}\\right)\\pi\\right] \\frac{(n-k-1)!}{k!} \\left(\\frac{x^2}{4}\\right)^k - \\ln\\left(\\frac{x}{2}\\right) \\mathrm{ber}_n(x) + \\frac{\\pi}{4}\\mathrm{bei}_n(x) \\\\\n& {} \\quad + \\frac{1}{2} \\left(\\frac{x}{2}\\right)^n \\sum_{k \\geq 0} \\cos\\left[\\left(\\frac{3n}{4} + \\frac{k}{2}\\right)\\pi\\right] \\frac{\\psi(k+1) + \\psi(n + k + 1)}{k! (n+k)!} \\left(\\frac{x^2}{4}\\right)^k\n\\end{align}\n", "06c2fa5a49cdd7947f43cc504560f878": "\\overline{\\mathbb F}", "06c34d795a2610039740c4ff5e9afd89": "A_{n-1}(1) \\int_0^\\infty \\exp\\left(-r^2/2\\right)\\,r^{n-1}\\,dr.", "06c361310c1d1c2a6d26c628aa50b14f": " \\mathcal{F} = \\frac{\\Delta\\lambda}{\\delta\\lambda}=\\frac{\\pi}{2 \\arcsin(1/\\sqrt F)}", "06c3ce92fd68d8d369f4796a74c8837e": "q_i(F_S)= F_S", "06c4496e824cc08bdb5a8eec610bef16": " \\gamma_s \\,\\!", "06c46b49f0881195c506925d90d158fb": "t_E,t_{E'} 1", "06c98e14c7c3908709f994ff68005384": "(1-R-\\varepsilon)H_q^{-1}(\\frac{1}{2}-\\varepsilon)", "06c9fd9208a35b057752d5172887d84a": "T = \\frac{ \\lambda v w }{v F w} = \\frac{1}{\\sum e_{\\lambda} ( f_{ij} )}", "06ca06eb3872382fbf005c6e44ab7f81": "\nx = R\\lambda, \\qquad\\qquad y = R\\psi,\n", "06ca59b7c7c0502d709e5b1a414fbde0": "X \\sim \\mathrm{GH}(\\lambda, \\alpha, \\beta, 0, \\mu)\\,", "06cab4a31ea57f55055e4d095dc08f6a": "h = \\frac{(v-3)(v-4)}{12}.", "06cafe5de1b67c6a71ea5c1eee766059": "b_0 . b_1 b_2 b_3 b_4 \\ldots = b_0 + b_1\\left({\\tfrac{1}{10}}\\right) + b_2\\left({\\tfrac{1}{10}}\\right)^2 + b_3\\left({\\tfrac{1}{10}}\\right)^3 + b_4\\left({\\tfrac{1}{10}}\\right)^4 + \\cdots .", "06cb240fa85a363b8dfe7dfacce57926": "O_6(2) \\cong S_8.", "06cbc6fe0922a3a82ac909a372c797fd": "(x-3)x^{14}(x+3)(x^2-x-4)^7(x^2-2)^6(x^2+x-4)^7(x^4-6x^2+4)^{14}.\\ ", "06cc6bc06c290863fe9318fabb6cc26f": "f\\colon R^r\\to R", "06cc7b48df48ae4205f45d63023d8274": "^{\\;}\\mathbb{V}", "06cc832179f822ac4714c2853115975f": "d_y", "06cd3ef006ee03dd9dee6be33b34ac95": "\\int_{-\\pi/4}^{\\pi/4} \\ln(\\sin x+\\cos x)\\,dx=-\\frac{\\pi}{4}\\ln 2.", "06cd663ed5bd4d9da3167789c48d0028": "\\begin{align}\n \\frac{\\partial}{\\partial b} \\left (\\int_a^b f(x)\\; \\mathrm{d}x \\right ) &= \\lim_{\\Delta b \\to 0} \\frac{1}{\\Delta b} \\left[ \\int_a^{b+\\Delta b} f(x)\\,\\mathrm{d}x - \\int_a^b f(x)\\,\\mathrm{d}x \\right] \\\\\n &= \\lim_{\\Delta b \\to 0} \\frac{1}{\\Delta b} \\int_b^{b+\\Delta b} f(x)\\,\\mathrm{d}x \\\\\n &= \\lim_{\\Delta b \\to 0} \\frac{1}{\\Delta b} \\left[ f(b) \\Delta b + \\mathcal{O}\\left(\\Delta b^2\\right) \\right] \\\\\n &= f(b) \\\\\n \\frac{\\partial}{\\partial a} \\left (\\int_a^b f(x)\\; \\mathrm{d}x \\right )&= \\lim_{\\Delta a \\to 0} \\frac{1}{\\Delta a} \\left[ \\int_{a+\\Delta a}^b f(x)\\,\\mathrm{d}x - \\int_a^b f(x)\\,\\mathrm{d}x \\right] \\\\\n &= \\lim_{\\Delta a \\to 0} \\frac{1}{\\Delta a} \\int_{a+\\Delta a}^a f(x)\\,\\mathrm{d}x \\\\\n &= \\lim_{\\Delta a \\to 0} \\frac{1}{\\Delta a} \\left[ -f(a)\\, \\Delta a + \\mathcal{O}\\left(\\Delta a^2\\right) \\right]\\\\\n &= -f(a).\n\\end{align}", "06cd70be27adef46544f64c887693177": "J_- = J_x - iJ_y,\\quad", "06cd76d1020ab27735a252602fb177fb": "V(x) = \\dfrac{1}{2}kx^2+e\\epsilon(t)x", "06cd8b34c35f763d4ee1d16e68cf4823": "w(n)= \\frac{1}{2} \\,w_r(n) -\\frac{1}{4} e^{\\mathrm{i}2\\pi \\frac{n}{N-1}} w_r(n) - \\frac{1}{4}e^{-\\mathrm{i}2\\pi \\frac{n}{N-1}} w_r(n)", "06ce256e4f7fcf6035ef0555c52ae624": "\n\\vec{C}=2.\\vec{r_2}\n", "06cea412fb13e3f307acaec972edfdc4": "X_1Y_1Z_1", "06ceef85fc5f1f79b9262f97e16620a2": "\\mathbb{D}^qf(t)=\\mathcal{L}^{-1}\\left\\{s^q\\mathcal{L}[f(t)]\\right\\}.", "06cf26fa7a959c1bc54d9696c5487a15": "q(\\alpha^i)=0", "06cf37c067a62dbcfb0edfef71db7ff9": " x = \\sum_{1\\leq{d}\\leq{D}}{q_d} + \\sum_{D+1\\leq{n}\\leq{N}}{q_n} ", "06cf3f21716ec66fe3b8a0415eca9567": "g=14", "06cf5a60d0ff83a69bf792a9392a470c": "\\sigma_e = \\frac{F}{A_0}", "06cfb7d91d13409686276ea1f8443ac9": "\\frac{4\\% - 3\\%}{3\\%} = 0.333\\ldots = 33 \\frac{1}{3}\\%.", "06cff0a2dfea0ce8968c1f57cacc978a": "\\scriptstyle \\sqrt{3}", "06d02a33a188753e4b675a7fc68c9619": " \\hat{g}(k)+\\hat{f}_{+}(k,0) = \\hat{f}_{-}(k,0)+\\hat{f}_{+}(k,0) = \\hat{f}(k,0) = C(k)F(k,0) ", "06d06445a0db1bdfe59687eb37f15370": "x_{9} \\ ", "06d125ac778c36e3c5a4e4e70a4267ee": "\n\\omega^2 = \\omega_{pe}^2 + \\omega_{ce}^2 + 3 k^2 v_{\\mathrm{e,th}}^2\n", "06d1431d41ebf019f454c85760c3cca8": "C_{70} ", "06d17b63bf91b101ee63e7baab89231f": "Z_n^m(\\rho,\\varphi) = (-1)^m Z_n^m(\\rho,\\varphi+\\pi)", "06d183f092c404a9d7ae381aa654aac0": "\\{e_{i_1}\\wedge e_{i_2}\\wedge\\cdots\\wedge e_{i_k} \\mid 1\\le i_1 < i_2 < \\cdots < i_k \\le n\\}", "06d1b20b6d623a20b32d43a41defd91e": "\\zeta=\\zeta_0 \\exp(-\\frac{\\alpha r^2}{4\\nu}),", "06d1c20fe9248c2e99b70478a991931e": "S(x)=\\sum_{i=0}^{d-2}s_{c+i}x^i.", "06d1eaab950c44221265ebd40c503472": " v_{n+1} = 0 ", "06d1f9b858a17480c87ae3c15577d0ea": " \\text{Var}(\\widehat{\\boldsymbol{\\beta}}_{ols}) - \\text{Var}(\\widehat{\\boldsymbol{\\beta}}_{k}) ", "06d20a4367635fb23cda65269853e538": "\\scriptstyle{ X \\in Y }", "06d221afdb2d5f56d0361f8d769f2fb7": "r_1\\,", "06d22da9e4615a96674abdfaaebce7b8": " \\int \\frac{dx}{a+bx+cx^2} = \\frac1c \\int \\frac{du}{u^2-A^2} = \\frac1c \\int \\frac{du}{(u+A)(u-A)}. ", "06d26585a3a1d268902592d27d27ef6e": "\\frac{\\partial s}{\\partial t}=(1/T)\\frac{\\partial u}{\\partial t}+(-\\mu/T)\\frac{\\partial \\rho}{\\partial t}", "06d2680d2748feb8040aff35f62670ed": "E_{obs|ref2}=E_{obs|ref1} - E_{ref2|ref1}", "06d2a1325df844c2e1f0873af6d6155c": "d \\to -\\infty", "06d2ea6cb2cc72ca2085187314e50d6b": "\n \\delta V_{\\mathrm{ext}} = \\int_{\\Omega^0} q~\\delta w^0~\\mathrm{d}\\Omega\n", "06d321586f58ac57180d3eb4cdae5ad3": "\\widehat{\\mathbf{p}}=\\frac{\\hbar }{i}\n\\frac{\\partial }{\\partial \\mathbf{r}}", "06d355dcadf9485685d2271cbf2020dd": "\\Delta \\rho_{0}=\\rho_{l,0}-\\rho_{v,0}", "06d35a9e9556c323d829d1dff476c29a": "U = \\frac{(3/5)GM^2}{r}", "06d38a545c57e06c2569f0eea1a45fd0": "S \\cdot \\{\\varepsilon\\} = S = \\{\\varepsilon\\} \\cdot S", "06d45da0e7c98c2fb2dfe85c798fc6f4": "s\\in [0, 1],", "06d49a56b1097ffa642fa7fbbb977a91": " r_E \\left( \\approx \\frac {1}{g_m}\\right) ", "06d531222c5f5fd58c59c1bb3e475329": "\\int e^{-c x^2 }\\; \\mathrm{d}x= \\sqrt{\\frac{\\pi}{4c}} \\operatorname{erf}(\\sqrt{c} x)", "06d55f5e2fe1755898c705f016cdea8a": "X \\ \\overset {P}{\\doublebarwedge} \\ Y.", "06d570cbe959835037c081ce647fefae": " \\bigcup_{i\\in\\mathcal{I}}C_i = \\operatorname{cl}(\\bigcup_{i\\in\\mathcal{I}}C_i) ", "06d59f9ae2ed151caaa258fd663de952": " \\operatorname{st}(x y) = \\operatorname{st}(x) \\operatorname{st}(y) ", "06d5e99d7165083157b74072d13eb13b": "P_k= A_k^{-1} A", "06d60c179e16986dd5326985dc670887": "1-\\frac{8}{\\pi^2}", "06d6475d55d6388c67128f65fc7007c3": "(1,1)/(1)", "06d6594a03a1e8d75ce3381314d6b7f1": "\\lim_{x \\to c}{f(x)} = f(c).", "06d677fe7d2a0321ce2495e91150acb4": "\\epsilon(p,t) =\n\\begin{bmatrix}\n\\epsilon_{1 1} & \\epsilon_{1 2} & \\epsilon_{1 3} \\\\\n\\epsilon_{2 1} & \\epsilon_{2 2} & \\epsilon_{2 3} \\\\\n\\epsilon_{3 1} & \\epsilon_{3 2} & \\epsilon_{3 3}\n\\end{bmatrix}\n", "06d67c5b693f72aa5917b99d98d06b20": " \\nabla^2 \\phi = 0 ", "06d6e1872f82cb9a1db5f9c7c64a7571": "\\chi(1)=1,\\quad \\chi(2)=3,\\quad \\chi(3)=5,\\quad \\chi(4)=6\\text{ and } \\chi(k)=6\\text{ for }k > 4.", "06d6e24415fd48d7de21b90bd3179306": "y=4-x", "06d6e2c20abc7aed22f5b1eb55c6199c": "\\sigma^2 = 3.5033e-02", "06d6f807c5685c2cba55d485275b21dd": "x\\sim y \\iff x\\,R\\,y \\land y\\,R\\,x", "06d706413765857ed4231c13dfe495aa": "x<\\mu-s", "06d71cf7be2620b331fae5cf58c948f1": "\\rho(\\mathbf{y}|\\mathbf{X},\\boldsymbol\\beta,\\sigma^{2}) \\propto (\\sigma^{2})^{-n/2} \\exp\\left(-\\frac{1}{2{\\sigma}^{2}}(\\mathbf{y}- \\mathbf{X} \\boldsymbol\\beta)^{\\rm T}(\\mathbf{y}- \\mathbf{X} \\boldsymbol\\beta)\\right).", "06d71d56f0c78aad0ce24077fe8590c9": " (x_1^2+\\cdots+x_r^2) \\cdot (y_1^2+\\cdots+y_s^2) = (z_1^2 + \\cdots + z_n^2) \\ , ", "06d7428d5398711bd4d2ff7e2a122f1b": "\\bar{D}=\\mathbf{E}^\\theta[D(\\theta)]", "06d74648983ecf54bb131566b8c5e418": "\\langle X,D,C \\rangle", "06d800b37ae1c15f719d18fcd511e768": "x=\\tfrac{\\pi}{2k}", "06d80eb0c50b49a509b49f2424e8c805": "dog", "06d8192069321dae13f673e4324cf8f6": "\\prod_{n=1}^{\\infty} \\left( 1+C\\beta_n\\right) =P", "06d843d8a9eee3a075aefeeb8178dd05": "\\Delta E_{max} = (1 - \\alpha) E", "06d8f8215a9dc7088c23faf64a73364d": "\\{x_{(1)},\\ldots,x_{(T)}\\}", "06d9276fc2a30f0b2c971565f06e7347": "\\left(\\tfrac{1}{2}z\\right)^\\nu= \\Gamma(\\nu)\\cdot \\sum_{k=0} I_{\\nu+2k}(z)(\\nu+2k){-\\nu\\choose k} = \\Gamma(\\nu)\\cdot\\sum_{k=0}(-1)^k J_{\\nu+2k}(z)(\\nu+2k){-\\nu \\choose k}", "06d96da660982f1e88498de82cde6f85": "\n\\vec k \\cdot \\vec J = -k_0 J^0 \\rightarrow 0\n,", "06d975f667e17991f33b462cef956c1e": "\\vec\\omega = (b, c, d)", "06d9766f8ef1d304b904154bd5149e18": "F \\longrightarrow E \\ \\xrightarrow{\\, \\ \\pi \\ } \\ B", "06d9ea23ffa6fc9d4f535ef2bcdb1a4e": "g_n(z)=\\frac{z^2}{n^3}", "06da660f3d03e1f9bc27af2962dcc537": "I(t)=\\int_0^{a_M}{i(t,a)da}", "06db2dd3ef340c71b2530fc72f4beff2": "N_s\\,", "06db3756924a7876fa447c44f664476f": " \\int_E w(x)\\ dx,", "06db5db6b567d8497fb8c5750e82c1d7": "p_{eq}", "06dbaccc61a71d13ff91db5c0d2705ca": "r_1 = x_1 i + y_1 j + z_1 k , \\quad r_2 = x_2 i + y_2 j + z_2 k)", "06dbbed2b1179fcff18a3f581ec4a699": "\\scriptstyle \\delta_1 ", "06dbe93abce0797f98a1206ea8edabf1": "\\begin{align}\\text{1 Ci}&=\\frac{3.7\\times 10^{10}}{(\\ln 2)N_{\\rm A}}\\text{ moles}\\times t_{1/2}\\text{ in seconds}\\\\\n&\\approx 8.8639\\times 10^{-14}\\text{ moles}\\times t_{1/2}\\text{ in seconds}\\\\\n&\\approx 5.3183\\times 10^{-12}\\text{ moles}\\times t_{1/2}\\text{ in minutes}\\\\\n&\\approx 3.1910\\times 10^{-10}\\text{ moles}\\times t_{1/2}\\text{ in hours}\\\\\n&\\approx 7.6584\\times 10^{-9}\\text{ moles}\\times t_{1/2}\\text{ in days}\\\\\n&\\approx 2.7972\\times 10^{-6}\\text{ moles}\\times t_{1/2}\\text{ in years}\n\\end{align}", "06dbf7054de09e50f2eb8d9740e39928": "E_{tgu} = 0.5 \\cdot 11.848^2 / 4.54 = \\,", "06dc1f15d5e653961721b66c2f50c546": "Z_t = \\sum_{k=0}^t X_k", "06dc81637103e72fdfa625195ea60e44": "\\beta=(\\beta_1, \\beta_2,\\cdots)", "06dca4ab9922618adfc9155350a5b70a": "1 \\in F,", "06dcebfa58fad42a3a3d8303ca2c014f": "z_0 = \\exp(i\\theta)", "06dd0da5c04f7a6c99e16857f0c29817": "A \\rightarrow A \\wedge A", "06dd7d2c0e5a9dd8e9a8e91452c8590a": "\\alpha = \\pi", "06ddaa5ef23158584ff864431938da9d": "V\\to V^*:v\\mapsto v^*", "06de3fdcff77757723e81468cfb6e1b1": "P_{\\mathrm{error}\\ 1\\to 2} = \\sum_{x_1^n(2)} Q(x_1^n(2))1(p(y_1^n|x_1^n(2))>p(y_1^n|x_1^n(1)))", "06dea3a87ee7de5c6ea467d41933b433": "I_b = -I_x \\frac{R_2}{( R_1 \\parallel r_E ) + r_{\\pi} +R_2} \\ . ", "06debf5df321963c1ff477b0de006c05": "p_i = \\left[ \\max_{a \\in A} \\sum_{j \\neq i} b_j(a) \\right] - \\sum_{j \\neq i} b_j(a^*) ", "06dee034ea49ade3d26fe1e451d96b20": "C^{1}_k", "06def552447a886e0bfa720025cef63f": "\\Rightarrow M_n=\\frac{R^n}{n!}.", "06df73567c0247dd180edd56272d3b69": "F_{12}, F_{13}, F_{23} ", "06dfc3da0a33b852be7fbefed9ef5690": "G_i+G_e=G", "06dfcf3b2c231351f49940fb9396f3dc": "\\lambda_n", "06dfe0fe70749a43c6698ae9fc719087": "\\mathcal D_m(M)", "06dfe6f5484e00e9821e49d2464df754": "K_a = \\frac{[HG]_{eq}}{[H_{eq}][G_{eq}]}", "06dfeb4bb3a0ef570bf0994f83b5ba82": "\\partial_\\mu\\left[\\frac{\\partial\\mathcal{L}}{\\partial(\\partial_\\mu\\phi)}Q[\\phi]-f^\\mu\\right]\\approx 0.", "06e0d2fe4275db5bc0d5005a5e89c591": "p_1, ..., p_d, q\\in\\Z,1\\le q\\leq N", "06e109375fa004e433314744d0521158": "f(x) = \\frac{2 \\beta^{\\frac{\\alpha}{2}}}{\\Gamma(\\frac{\\alpha}{2})} x^{\\alpha - 1} \\exp(-\\beta x^2)", "06e14d1a766e16597ace30c8b513befb": "\\varepsilon ^{\\alpha \\beta }", "06e15c8cb9648247f7cf2d8393f04df6": "\\vert\\hat{f}(\\xi)\\vert \\leq \\int_{\\mathbf{R}^n} \\vert f(x)\\vert \\,dx,", "06e168bb73d2b659d657f44db7c1fc7c": "E^{p,q}_1\n= \\begin{cases}\n0 & \\text{if } p < 0 \\text{ or } p > 1 \\\\\nH^q(C^\\bull) & \\text{if } p = 0 \\\\\nH^{q+1}(A^\\bull) & \\text{if } p = 1 \\end{cases}", "06e18996a9d3c2afb1ce39d09f1e8986": "S_z = m_s \\hbar\\,\\!", "06e18ba5b85397f0770e9943a7b8a808": "f_u \\left(\\begin{pmatrix}\na & b \\\\\n0 & 1 \\end{pmatrix}\\right)=a^u,", "06e1926d5d41cef9bbec354b734e14ec": "\\quad (3) \\qquad \\qquad \\bar{\\rho}_i \\left( t_2 \\right) = \\frac{1}{x_{i+\\frac{1}{2}} - x_{i-\\frac{1}{2}}} \\int_{x_{i-\\frac{1}{2}}}^{x_{i+\\frac{1}{2}}} \\rho \\left(x,t_2 \\right)\\, dx ,", "06e24240b47b74861da0de82940a32fa": "\ne_3 =\n\\begin{pmatrix}\n1&0\\\\\n0&-1\n\\end{pmatrix}\n", "06e2745e0e66a583228563c795212f20": "c=\\pm1", "06e2a3dd682d5d67e6be7c625958c372": " k_{\\mathrm{H,px}} = \\frac{p}{x} ", "06e2b3b868988d6dddb72612c4af5f99": "\\frac{1}{{{D}_{Ae}}}=\\frac{1}{{{D}_{AB}}}+\\frac{1}{{{D}_{KA}}}", "06e2b3f28b474386df1ae3cf6d50cb12": "\\mathbf{i}=\\mathbf{r}_i", "06e2c91cdbf2ba72bfae3686775d5315": " \\mu_{i,j}", "06e327d1d370a4c85d1f1558d7cf4d74": "m = \\gamma m_0 \\,\\!", "06e3623afd16b07c4a3a101e51fdcbad": "\\begin{matrix}\n \\mathrm{if} & p_l=p_{1}(u) & p_m=p_{2}(u) & p_n=p_{3}(u) \\\\\n \\mathrm{then} & p'_l=p_{2}(u-1) & p'_m=p_{1}(u-1) & p'_n=p_{3}(u-1)\n \\end{matrix}", "06e39ddc7614317468f1446eb7cbaafb": "\\cosh c = \\cosh a \\ \\cosh b - \\sinh a \\ \\sinh b \\ \\cos \\gamma \\ , ", "06e3c0415761d467d709f78b6a2f39af": " \\log p_A(n) \\sim C \\sqrt{\\alpha n} ", "06e40264795ae083e71e3d43644b5566": "\\begin{align} \\hat{H} & = \\hat{T} + \\hat{V} \\\\\n & = \\frac{\\bold{\\hat{p}}\\cdot\\bold{\\hat{p}}}{2m}+ V(\\mathbf{r},t) \\\\\n & = -\\frac{\\hbar^2}{2m}\\nabla^2+ V(\\mathbf{r},t)\n\\end{align} ", "06e41b676042e7f6903beb74cfddb357": " BT^{-1}", "06e420fac994e1cab4dce5c4863f2b99": "\\mathit{H}", "06e48758f2485170f5d8a32f64c8e8f4": "\\stackrel{\\vec v}{}", "06e4ff064815e7f80da5f70841d17505": "H(x(t)) = m \\frac{d^2(x(t))}{dt^2} + kx(t)", "06e52dab3c87e4a2d735170d93008ea7": "\\begin{bmatrix}\n0 & 0 & 3 & 0\\\\\n0 & -2 & 0 & 0\\\\\n1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 1\\end{bmatrix}.", "06e5325a33498b0229a4bddf89137d86": " (U_sU_\\omega)^r = M \\begin{pmatrix} \\exp(2rit) & 0 \\\\ 0 & \\exp(-2rit)\\end{pmatrix} M^{-1}", "06e626389f5786c0205c99697e6a294c": "Fi_{22}\\;", "06e62aac6c74d1bc3efe7fe7270c02a7": "\\begin{smallmatrix} \\mu = \\sqrt{ {\\mu_\\delta}^2 + {\\mu_\\alpha}^2 \\cdot \\cos^2 \\delta } = 1907.79\\,\\text{mas/y} \\end{smallmatrix}", "06e640a8860ee0240a0c5237354e40db": "\nP_A \\left( 1 + e^{v_A } \\right) = e^{v_A } \n", "06e680a0049734936819f48f82b575ba": " \\mathfrak{a} \\subset \\mathcal{O}_k", "06e6d4c4500e3f92118c38cc01dc8e4c": "5\\zeta(2)\\zeta(5)+2\\zeta(3)\\zeta(4)-11\\zeta(7)", "06e78e6aa0957b68ca5e1def5adec2db": "[H,\\Pi]=0", "06e7a18e76a7fee5fe83ce36965cf2a1": "\\boldsymbol{L}_{y}\\hat{f}(k,y)-P(k,y)\\hat{f}(k,y)=0,", "06e7a7a01e0bda91c7309fcee8b78a62": "F^\\%(*)\\to F(*)", "06e84b6b0430a6929023040832bbf88e": "{\\partial\\vec{B}\\over\\partial t} = 0.", "06e86219c47ebd4c614b54c0e8b79736": " \\mathit{WER} = \\frac{S+0.5D+0.5I}{N} ", "06e8627e8d832f40ecd387bbc3e69ff4": "\n \\frac{1}{T} \\sum_{t=1}^T \\mathbf{1}_{\\{X_t\\in A\\}} \\ \\xrightarrow{a.s.}\\ \\operatorname{Pr}[X_t\\in A],\n ", "06e89453aad485a4ca9d7eb0e0bd05c2": "Q_B = C_BV_B. \\ ", "06e8a474d11071362f5ff94cf9b4068b": "\\lim_{x\\to1}\\frac{\\ln(x)}{x-1}=1", "06e8e3fcf20954509c3473c4299d2536": "\\int_0^a \\sqrt{a^{2}-x^{2}} \\, dx =\\frac{\\pi a^2}{4} ", "06e96890472fe7ab1384d5fff3917118": "l>0\\,", "06e9c29be8b22b9257e57ec136590683": "f(X)\\,", "06ea2b747cafb6b4903c3acf9f83618a": ".\\qquad \\qquad\\qquad\\quad\\;\\;\\; S", "06ea6eb5ed10b6a90fd6ccf94007ae1b": "\\frac{\\partial}{\\partial t}f(x,t) = -\\frac{\\partial}{\\partial x}\\left[\\mu(x,t)f(x,t)\\right] + \\frac{\\partial^2}{\\partial x^2}\\left[ D(x,t)f(x,t)\\right].", "06eafb7f3c501c5cd3f4da601efda614": "\\langle Hu, v \\rangle \\overset{\\mathrm{def}}{=} \\langle u, -Hv\\rangle", "06eb4f9416bd48082827a6ad5f366fe2": "L\\setminus D", "06eb9a650f16ad424939b9b8dcdd3ceb": "s=(i,j)", "06ebaa2b1ba50cf8a0edb806fb1b6ff8": "D_0(f)D_0(\\hat{f}) \\geq \\frac{1}{16\\pi^2}", "06ebb4aa52a557939af15290156fa983": "s=O(n/\\epsilon^2)", "06ebcb95f371509d486f5e59255afbf4": "\n\\text{Tr}\\left\\{ \\Pi_{\\rho_{X^{n}\\left( m\\right) },\\delta}\\hat{\\Pi}\n_{\\rho_{X^{n}\\left( m-1\\right) },\\delta}\\cdots\\hat{\\Pi}_{\\rho_{X^{n}\\left(\n1\\right) },\\delta}\\ \\Pi_{\\rho,\\delta}^{n}\\ \\rho_{x^{n}\\left( m\\right)\n}\\ \\Pi_{\\rho,\\delta}^{n}\\ \\hat{\\Pi}_{\\rho_{X^{n}\\left( 1\\right) },\\delta\n}\\cdots\\hat{\\Pi}_{\\rho_{X^{n}\\left( m-1\\right) },\\delta}\\Pi_{\\rho\n_{X^{n}\\left( m\\right) },\\delta}\\right\\} ,\n", "06ebe860836e08b5e4a0ad3731cbd535": "y= \\frac{y'}{x'^{g+1}}", "06ec614a0e2ec8c27aab21d24e399139": "\n\\begin{align}\nm\\frac{d}{dt} \\langle \\Psi(t) | \\hat{x} | \\Psi(t) \\rangle &= \\langle \\Psi(t) | \\hat{p} | \\Psi(t) \\rangle, \\\\\n\\frac{d}{dt} \\langle \\Psi(t) | \\hat{p} | \\Psi(t) \\rangle &= \\langle \\Psi(t) | -U'(\\hat{x}) | \\Psi(t) \\rangle.\n\\end{align}\n", "06ec91b0b084c842412cdf066bc7c37c": "f^\\star : X^{*} \\to \\mathbb{R} \\cup \\{ + \\infty \\}", "06ed7e10040e3ee1b3f9a05f97de9c6e": "\\zeta:S\\ddot\\to d", "06edb6b869ad60a20f5feac501131df1": "\\frac{\\partial\\mathcal{L}}{\\partial x_i}=0~~\\forall i", "06ee4a0e90b9a11ae66330843e01977c": "\\hat{f}(x) = \\sum_{i=1}^{k} c_i B_i(x) ", "06ee531b6e99bc6853ac756441b4c77f": "b= 2 a", "06ee789a1ea5e1bc3d9991243b79a4ad": "\\sum_{i}{ q_i \\frac{\\partial f_k}{\\partial k_i} } = -\\sum_{i}{ q_i \\frac{\\partial f_k}{\\partial \\mu} \\frac{\\partial \\epsilon_k}{\\partial k_i} } = -\\sum_{i}{ q_i k_i \\frac{\\hbar^2}{m} \\frac{\\partial f_k}{\\partial \\mu}}\n", "06eec1ccfddb8a755e5938215f7a9657": "V^{-1} (x) \\approx \\sqrt (4\\pi) \\frac{d^{1/2}N(x)}{dx^{1/2}} ", "06eee942ebe1a41550d614af6ad20e90": " i_2=E^2\\sin^2(\\omega t + \\phi)=E^2(\\sin(\\omega t)\\cos(\\phi)+\\sin(\\phi)\\cos(\\omega t))^2\\,", "06ef429c6c40dcb74594090468a61d80": "\\left.\\theta_i\\right.", "06ef9747a150e2ad887581089c78680c": "MV = PT", "06efb9f55f0f8a6b9e0e143007c26d9f": "\\widehat{f^{(k)}}(n) = (in)^k \\hat{f}(n)", "06efc40eaea741de8fa51bbd983437a0": "\\mathrm{D \\cdots H{-}A}", "06efd57083242b84c3eb89bbb1425b40": "2\\,ln\\,\\gamma ", "06efdc8e798ff9ee22c6b30b921daf64": "\\int\\cosh^n ax\\,dx = -\\frac{1}{a(n+1)}\\sinh ax\\cosh^{n+1} ax + \\frac{n+2}{n+1}\\int\\cosh^{n+2}ax\\,dx \\qquad\\mbox{(for }n<0\\mbox{, }n\\neq -1\\mbox{)}\\,", "06efec6d4c0c0da25c6238aaf03fe6a8": "\\sigma(x,x') = \\frac{1}{2} \\eta_{\\alpha \\beta} (x-x')^{\\alpha} (x-x')^{\\beta}", "06f00d95d9d9f3f2c7eb2f30f4736dee": "\\kappa_0 = 0.378893+1.4897153\\,\\omega-0.17131848\\,\\omega^2+0.0196554\\,\\omega^3", "06f01daf487e9d1b5c741b192ce92e64": "-0.75 < \\beta < -0.5", "06f0a515cfb17044e3ca9ec8aa712a6b": "\\overline{p}\\, \\propto \\,V\\,\\frac{\\sigma_1 - \\sigma_e}{\\sigma_1 + 2\\sigma_e}\\,\\overline{E_0}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(4)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,.", "06f0c89efa76c089c159e40cc5eb3609": "xv = v", "06f0c8acd9064fa1d89dfb4a6ed87e16": "q(\\tilde{x},\\tilde{{u}}\\vert \\tilde{\\mu})=\\mathcal{N}(\\tilde{\\mu},C)", "06f0e1edab2cf3baff4208e0f17d4792": " \\tfrac{mg}{Ld}", "06f0f399e2282ba52bfc5a4a3010b68e": "\n \\sum F = 0 ~,~~ \\sum M_{A} = 0 \\,.\n ", "06f14cd36432d7dd4b3e006bbc820201": "\\lambda = \\ell^2.", "06f17ae1b9444097a418c5cdd9f2cdde": "\\hat n=\\textrm{const.}", "06f19e0c3c74019a6fe60e558316752d": "-\\frac{\\partial \\operatorname{cost}}{\\partial \\mu_{ij}}", "06f22bc0a0594b1b47ecd2d686e99cbe": "\\phi \\otimes \\phi^{\\Rightarrow x} = \\phi\\,", "06f234fe042ff4bb0d94cd9463dce0cb": "c[[a,b]]", "06f276fbb135db98d357b0983fd446ed": "\\mu_6=\\kappa_6+15\\kappa_4\\kappa_2+10\\kappa_3^2+15\\kappa_2^3.\\,", "06f27f3f4b6930af17c5a12ad197eaa3": "\\scriptstyle \\mathrm{E}(e^2)", "06f30b25421cacf5df4bcaecc7a8d021": "\\mu_r' = \\bigg({\\frac{1}{2}}\\bigg) \\sum_{k=0}^r \\bigg[{\\frac{r!}{k! (r-k)!}} b^k \\mu^{(r-k)} k! \\{1 + (-1)^k\\}\\bigg]", "06f32ab04e43e768fcdbce800a4054b6": "[\\mathtt{Var}]", "06f3832b60c015244731ae1d6dbc5b20": "\\frac{d}{d\\mu}\np_{\\lambda}-F_{\\lambda}=0.", "06f3c8003382ae94fcf8d0470aadd7f1": "\\forall a \\in A, L(a) = \\mathit{out}", "06f3cd5804a745e27558ff9ca765a6b3": "{\\rm Imm}_\\lambda(A)=\\sum_{\\sigma\\in S_n}\\chi_\\lambda(\\sigma)a_{1\\sigma(1)}a_{2\\sigma(2)}\\cdots a_{n\\sigma(n)}.", "06f3dfda128b2077a1c12feb5ac41d0a": "\\textstyle s^\\alpha+t^\\alpha=1 ", "06f3f5c134185227850a6fb52f7a5cfa": "\\sideset{}{^\\prime}\\sum", "06f465defe4c51ed39be1fdd33c764db": " v(x,\\tau)=\\exp(-\\alpha x-\\beta\\tau) u(x,\\tau).", "06f4797c2b7386626e512d6d2c20c09e": "E_r(r,z)", "06f49d3ac085dde14636ef63d7d311d6": "n(\\vec{r}) ,", "06f4a4eead785d13d8b51f3e7b9290e6": "2x \\in o(x^2) \\,\\!", "06f51ec47cd65ae6ed71d25574b4ade8": "\\mathbf{1}_A (\\omega) = 0.", "06f54460efb0b5c4c906147072b0eef7": "r = 0.0961 = 9.61 \\%", "06f578789605643db73b1890cf52be34": "D = x_{11} - p_1q_1", "06f59c823dd57d3a21d55798c4c302a8": "g_2^2=g_3^3= (g_2g_3)^7=-1,", "06f5d6d42ae6d5f9a9d18907e7392814": " a(x+kv)+ b(y-ku) = ax+by + k(av -bu) =ax+by + k(udv -vdu) =ax + by", "06f6247566f82e01d436b7134b5753d3": "\\scriptstyle f_s.", "06f65750206044de34d3194bf4ff1e0a": "\n \\frac{1}{R} = \\frac{1}{R_1}+\\frac{1}{R_2}\n ", "06f6761ad8398a80686ea3ac861b86c5": "\n\tT_{max} = E {4 M m \\over (m+M)^2 }\n", "06f681e831f53857b7e0edbf9eca5b39": "q_p(1) \\equiv 0 \\pmod{p}", "06f692570e6471fafea645933393cddf": "-2\\Im(\\mathit \\Gamma)=\\tan \\left (\\frac{4\\pi}{\\lambda} x\\right)", "06f695e8d632b1d99c0afb37e1e68a4c": "\\lim_{x\\rightarrow+\\infty}\\arctan(x)=\\pi/2.", "06f6a489209115c5cef3f45036aad3ec": "PA", "06f6c1b6db4342eddb0f52c714b23026": "\\Delta \\mathbf{B} \\in \\mathbf{P}_\\pm(1,0,0)", "06f6df1976c2e03ea84a9f336763f590": "\\overline{X}={X_1 + \\cdots + X_n \\over n}", "06f710cb5ada709d2d6065f0af4f4927": "B_\\infty^{p,q} = \\bigcup_{r=0}^\\infty B_r^{p,q},", "06f745c18b05f95d97cc6f6896de1ff1": "x=s-\\epsilon", "06f765a89dd281c30bd5aa2a4d90f6bc": " \\mathbf{a} = \\sum_{i=1}^N a_i\\mathbf{e}_i = a_1 \\mathbf{e}_1 + a_2 \\mathbf{e}_2 + \\cdots a_N \\mathbf{e}_N", "06f7895cd704b1cb0921cf98aec71926": "\\alpha\\,\\!", "06f7db588b7ed518b4dff3b48f834c1a": "\\int_{X_1\\times X_2} f(x_1,x_2)\\, \\mu(\\mathrm d x_1,\\mathrm d x_2) = \\int_{X_1}\\left( \\int_{X_2} f(x_1,x_2) \\mu(\\mathrm d x_2|x_1) \\right) \\mu\\left( \\pi_1^{-1}(\\mathrm{d} x_{1})\\right)", "06f7fec6a2087c3b4559cc748a44643d": "M(a,b,c) = \\prod_{i=1}^a \\prod_{j=1}^b \\prod_{k=1}^c \\frac{i+j+k-1}{i+j+k-2}.", "06f869a41aa361bf1ad9b85d303467be": "(D_0,\\epsilon)", "06f883a740bbcc55c24333ee8767e954": "M=J", "06f8f61719c9ae54bd872b2f15ac21e8": "{\\mathfrak c} \\leq \\aleph_0 \\cdot 10^{\\aleph_0} \\leq 2^{\\aleph_0} \\cdot {(2^4)}^{\\aleph_0} = 2^{\\aleph_0 + 4 \\cdot \\aleph_0} = 2^{\\aleph_0} ", "06f8f7cc9e0d46723578a08f21b1577e": "u_i = \\overline{u_i} + u_i',\\,", "06f9293d5ce55f612cb7a6ebca367aca": "\\operatorname{tr} (A^* A) = \\sum_j^n |\\lambda_j|^2.", "06f95d0d72cee463dc00300f8b935650": " p_{i} ", "06f95e2140d5bef1d3414796c7d6e0c2": "Z_I\\,\\!", "06f9a75b18c09c0c1a86f9a95630df70": "V(\\varepsilon_i)= \\sigma^2 < \\infty,", "06f9b7b1d3f141742ad1c582b55056ba": "x = \\pm 1", "06f9be585f2e7547a204207eff5fc548": " {R^{\\alpha}}_{\\beta} ", "06fa147a005a6ef2d1e4e2c11a541d97": "\\sum_{i=1}^n(x_i-\\overline{x})(\\theta-\\overline{x})=0", "06fa35c9031e823ee6cfccb5605c4eb6": " x\\mapsto (d_\\lambda f)(x)", "06fa4b907599c8a36554f23497da2208": "C_{\\text{min}, \\text{ss}}", "06fa5385239b7aaf6deb58c60cce8798": "\n\\alpha(u) =\n\\begin{cases}\n \\frac{1}{\\sqrt{2}}, & \\mbox{if }u=0 \\\\\n 1, & \\mbox{otherwise}\n\\end{cases}\n", "06fa62a7df57887836c1e22f862ae08b": "(4~5).", "06faac98935d1cf9b57d0640c6073d4f": "X \\land \\neg X", "06fab9786c1782ba7733c31a17c6c66e": "\\begin{align}a&=6.112\\ \\mathrm{millibar};\\quad\\;b&= 17.67;\\quad\\;c&= 243.5^\\circ \\mathrm{C};\\end{align}", "06fb0756cbb51cef8245388c77460834": "\\int_S F \\, dS", "06fb12324a98e8b31b2819be10b29dca": "\\begin{align}\nx' = \\gamma x - \\frac{\\gamma v}{c}ct & \\Rightarrow & x' = \\gamma(x - vt) \\\\\nct' = -\\frac{\\gamma v}{c} x + \\gamma ct & \\Rightarrow & t' = \\gamma\\left(t-\\frac{vx}{c^2}\\right) \n\\end{align}", "06fbe3c17a36710731842480e1657952": "P=\\frac{T^\\alpha}{R^\\beta}", "06fbf0791485f24f1a0df9ea75544e43": " \n\\begin{align}\ny &= y_{0} + y_{1} + y_{2} + y_{3} + \\cdots \\\\\n & = -\\left[ t + \\frac{1}{3} t^{3} + \\frac{2}{15} t^{5} + \\frac{17}{315} t^{7} + \\cdots \\right] \n\\end{align}\n", "06fbf7e846775b80c4fffad4c0b3055b": "v = \\sum_{i=1}^n v^iX_i,\\quad w = \\sum_{i=1}^n w^iX_i", "06fc0b39a9811f7e78cf9a439d4cef40": "\n\\begin{cases}\n \\frac{dx_1}{dt} = (1-x_2^2)*x_1-x_2+u \\\\\n \\frac{dx_2}{dt} = x_1 \\\\\n \\frac{dx_3}{dt} = x_1^2+x_2^2+u^2 \\\\\n x(t_0) = [0 \\ 1 \\ 0] \\\\\n t_f = 5 \\\\\n -0.3 \\le u \\le 1.0 \\\\\n\\end{cases}\n", "06fc1a78b9aaaee997b0adbfa5992f6c": "(n+1)", "06fc4a6d4d713d72b39ef424e8c7995a": "\\mathrm{Financial\\;leverage}= \\frac{\\mathrm{Total\\;Assets}}{\\mathrm{Shareholders'\\;Equity}}", "06fc5b02a356eaa5b1b33b3f5b7a711f": "\\|u+v\\|^2+\\|u-v\\|^2=2(\\|u\\|^2+\\|v\\|^2).", "06fc5b5f85dbf32589c521ca55e05e10": "\\log (\\operatorname{E}(Y|\\mathbf{x}))=\\mathbf{a}' \\mathbf{x} + b,", "06fcd3f2aa256fd816ec7081a38c30cc": "\n\\operatorname{Var}(X) = \\int_{-\\infty}^\\infty \\frac{(x - \\mu)^2}{\\sqrt{2\\pi \\sigma^2}} e^{ -\\frac{(x-\\mu)^2}{2\\sigma^2} } \\, dx = \\sigma^2.\n", "06fcd5f9f7bf19377c6f7c4560d9ddd3": "\\sqrt{12.746 \\times A_m}", "06fce68ac85e7fd4fe558639c55dff48": "\\operatorname{tanh}(z)", "06fd262059da6c3ef9aebeb89b4eae62": "\\mathbf{e}_1 ", "06fd43a831e994e442b64b77ffb70cfb": "\\begin{bmatrix}\n1 & u_{12}/u_{11} & . & u_{1n}/u_{11}\\\\\n0 & 1 & . & u_{2n}/u_{22} \\\\\n. & . & . & . \\\\\n0 & 0 & . & 1 \\end{bmatrix}", "06fd65a45b7d5147e034e4c037c6bb07": "\\|f\\| = \\max_I |a_I|", "06fd6796cdf75b8da2f096efdd36a09a": "X\\ \\sim\\ BW2(a,b)", "06fd689d7a8096ce961bd4f8a53800d1": "U_B=Q^2\\sin^2(\\omega t + \\phi)/2C\\,\\!", "06fde8f3ea98e2025590255693da5a68": "\\{\\Phi_{ij}\\hat{=}0\\,,\\Lambda_{}\\hat{=}0\\}", "06fe11932a45adb4faff9e1461556ada": "K_\\text{joint}=\\frac{2W}{\\Delta \\theta}", "06fe30b11b4e7f2b5d4ca7eff02fd65b": "\nG(k) = {1 \\over i\\omega - {k^2\\over 2m} }.\n\\,", "06fe3fd50fd4de394e13d4e6c8ca2e2b": " p_m", "06fe9eed3a7ef77fb236b4115bc813df": "L_1(B) \\subseteq V", "06fed6899c66d75d74f56fa57e2e7c97": "k'_L = 0.664{D_{AB} \\over x} Re^{1/2}_L Sc^{1/3}", "06fef6cf9cd6d4b3c27115712d7f9f89": "b\\;", "06ffd1b14a65819e385ba237fcaeeecb": " X^i Y^{n-i}, \\quad 0\\leq i\\leq n ", "06ffeaf4615e304202a27b140949c683": "T=\\{(a,v)\\colon \\|a\\|=1,\\, a\\cdot v=0\\},", "06fff7df730a38b1bce6ec8adf57cd68": "A(\\alpha_1, ..., \\alpha_n)", "06fffc7bee852c3e3a52d94e7637c348": "\\Lambda_{\\mathrm{m}} =\\Lambda_{\\mathrm{m}}^\\circ - K\\sqrt{c}", "07001c08cbfd50263d50d487c27d473f": "\\left[F\\left(-1\\right),F\\left(1\\right)\\right]", "07001e7bd5d796308250f06e997b336f": "\\begin{align}\n& \\{ \\Gamma,\\Gamma \\} =2I && \\{ \\Gamma, Q \\} =0 && \\{ \\Gamma, \\bar{Q} \\} =0\\\\\n&\\{ Q,\\bar{Q} \\}=2Z && \\{ Q, Q \\}=2(H+P) && \\{ \\bar{Q}, \\bar{Q} \\} =2(H-P) \\\\\n& [N,Q]=\\frac{1}{2} Q && [N,\\bar{Q} ]=-\\frac{1}{2} \\bar{Q} && [N-[1-q,\\Gamma]=0 \\\\\n& [N,H+P]=H+P && [N,H-P]=-(H-P) &&\n\\end{align}\n", "07002e45e18227e8552911cf43b3eb74": "ip\\,", "07004971f8da61850b3167f634758095": "\\Delta G^\\circ = -nFE^\\circ \\,", "07009d1fe5a8f3356a54628b0a9a2e2c": "=\\lim_{x\\to\\pm\\infty}\\left[\\left(x-\\frac{1}{x}\\right)-x\\right]", "07015c9bc41543737124da6128a321cf": "|1\\rangle \\otimes |1\\rangle = \\frac{1}{\\sqrt{2}} (|\\Phi^+\\rangle - |\\Phi^-\\rangle).", "07019868d7cf7840cc2569aa632692b5": "\\scriptstyle{E_{2}}", "0701d7e98e5b319a2d6eca4593dbf8ca": "\\Delta E = \\hbar \\omega", "0701e21caf8c27e9c3c3fffaddae03da": "\\gamma=\\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix}", "07023e53e01690646b5b2d31d3a79551": "\\mathbf{AA3} = \\begin{bmatrix}\n-\\beta & 0 & 0 & 0 \\\\\n0 & -\\beta & 0 & 0 \\\\\n0 & 0 & -\\beta & 0 \\\\\n0 & 0 & 0 & -\\beta \\end{bmatrix}", "07024215329a71973769a3641eb82d08": "\\psi^{*}(\\theta|_{W})=0, \\forall \\theta \\in \\Lambda_{C}^{1}\\pi_{r+1,r}.\\,", "07025eb37092f002d5e6a63feb9826fa": "\\AA^{-2}", "0702636caf3faa212e5ca5901d56b7a8": "\n\\begin{align}\nA_j& = \\frac{\\sum_{i=1}^{L} x_{L(j-1)+i}}{L} \\quad \\forall j& = 1,2,\\ldots,N\n\\end{align}\n", "070268441b1ba4ce1b40b9226759b5fd": "M^{\\rm{SN}}(x) = h - eFx -e^2/(16\\pi\\varepsilon_0 x), \\qquad\\qquad (3)", "07029e1595c60a230b6af32248fbbf84": "\nf_Y(y) = f_X \\left( g^{-1}(y) \\right) \\left| \\frac{d}{dy} g^{-1}(y) \\right|\n", "0702e0a272267151de0194ef145a01ed": "\\{X_{\\alpha}\\}_{\\alpha\\in\\Alpha} \\subset L^1(\\mu)", "0702e7e54d88e1deeb932e2f20843bbe": "f: S^1 \\rightarrow \\mathbb R^3.", "0702ec22009626b16d21444a52c6cf46": "\\gamma_{13}", "0702f8fa72ff2c6ff7eb93f0ba58aeee": "\\frac{x-a}{x-c}\\cdot \\frac{b-c}{b-a}", "070322d303658ef53b56abd0278e694f": "((1 \\times 2) \\times 3) \\times 4\\dots", "070333d4041f0bc56f1494be2d8d1ef2": "\\displaystyle \\hat{f}_3(\\omega) \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{1}{(2 \\pi)^{n/2}} \\int_{\\mathbf{R}^n} f(x) \\ e^{-i \\omega\\cdot x}\\, dx = \\frac{1}{(2 \\pi)^{n/2}} \\hat{f}_1\\left(\\frac{\\omega}{2 \\pi} \\right) = \\frac{1}{(2 \\pi)^{n/2}} \\hat{f}_2(\\omega) ", "07034752e26042109fd161506d3571d8": " \\omega_{\\rm orb} = \\frac{L}{r^2} = \\sqrt{m/r^3} ", "07039f7406f216840d06c06d80a1e13b": "Z,", "0703a367605efeb9385d8afc267f2e77": "\\ k_b M = S - \\sum_i( I_i E_i),", "0703f26e8171d3a7864cbfe3e3336935": "G_{ab} + \\Lambda g_{ab} \\, = \\kappa T_{ab}", "0703fb725f82df343b488a1b0d99e7c3": "S = A[x_0,\\ldots,x_n]", "0703fd136dc12b6e3c60af31b2003aed": "I = \\int L(\\mathbf{q}, \\dot{\\mathbf{q}}, t) \\, dt ~,", "070440c136d68b9abc282fd3ef723457": "\\frac{\\partial (\\mathbf{u} + \\mathbf{v})}{\\partial \\mathbf{x}} =", "070458900dad2a691b5356912410f346": " \\sigma_x = \\left( \\begin{matrix} 0 & 1 \\\\ 1 & 0 \\end{matrix}\\right) ", "070465b8297a9c0490a2657287978584": "\n\\|Ax\\|_{\\beta}\\leq \\|A\\|_{\\alpha,\\beta}\\|x\\|_{\\alpha}.\n", "0704911db4e3ec5f12d536fbfd7ed629": "c\\,\\!", "0704b8919da6315a296827d30201318e": "\\mathbf{H}_\\alpha(x) = \n \\frac{2{(x/2)}^{\\alpha}}{\\sqrt{\\pi}\\Gamma(\\alpha+\\frac{1}{2})}\n \\int_{0}^{\\pi/2} \\sin (x \\cos \\tau)\\sin^{2\\alpha}(\\tau) d\\tau.", "0704d96823ccea6a0bdbe0072d4aad24": "H_2^{16}O_{(l)} + H_2^{18}O_{(g)} \\rightleftharpoons H_2^{18}O_{(l)} + H_2^{16}O_{(g)}", "0704dc9bb7fc4caccdf59e000795f364": "\\left[Re(-1/2), Im(0)\\right]", "0704fa08767f443cc1448a563edbbd5d": "r=\\tfrac{1}{2}", "07055850bcbb43dcc9c8609e7cc9e31f": "\\frac{1}{S_1} + \\frac{1}{S_2} = \\frac{1}{f} ", "07059ea6785419c6f38f887b999356f2": "\\rho(A) < 1", "0705a0d1c4faa496c25a0ec3d9162e95": " \\int \\bar\\psi(\\gamma^\\mu \\partial_{\\mu} - m ) \\psi ", "0705d1be3febdcf632f0b687bc4a1e6a": "A\\equiv((B\\equiv C)\\equiv((C\\equiv A)\\equiv B))", "07062a14bdcb7842ff61a0f6e0ea15b9": "u^{\\pm i}", "07064e3e2d782d232254b31c5bbb03d6": "g \\notin F", "07067e13cc0db2b99caced6cca364657": "\\mu_{ab}^{(c)}(t) =0", "0706b9c536eb81f763daa0a36b1eb6fe": "S(E,a_E,a)= \\prod_{u=a_E}^{a-1} \\left[ 1-q(E,a_E,u)\\right]", "0706d7c09de74ed1481735753c2ad5fa": "\nP(k) = {n-1\\choose k} p^k (1 - p)^{n-1-k},\n", "070701aeaccfe5013215c9da112ceed7": "-(1/T)\\nabla\\mu_j", "0707579e62f807dd4a752af8617b0f69": "U(1) \\hookrightarrow S^{2n+1} \\twoheadrightarrow \\mathbf{CP}^n", "0707669836d19443cf6c5cc89ca963e6": "y(t)", "07078c91cc4fdcf0d49cc18bbddc12bd": " \\zeta=\\chi + i \\eta ", "0707afd12d13ec433d645854ca98b125": " I_2 = \\frac{V_2}{|Z_{total}|}\\angle (-120^\\circ-\\theta) ", "0707c48bc143637a3ae1679c42f505f7": "I\\times S^1", "0708149ad8eaaaed1072969025150497": "\\sum F_x = \\Delta (ma_x)", "0708208ffee01d6980a07141fb2ce279": "\\mathbf{A}^{\\mathrm{T}} = -\\mathbf{A} .", "070835c49d3a13f95f614c65665eeebf": "\\frac{1}{\\Gamma(z)} = z e^{\\gamma z} \\prod_{n=1}^{\\infty} \\left(1 + \\frac{z}{n}\\right) e^{-\\frac{z}{n}}", "0708572de1f982adb99029dd6bec9dba": "\\Phi(\\vec{r})=\\frac{1}{4\\pi Dr}\\exp(-\\mu_{eff}r)", "07086da1acd701594ea69101cdaba123": "{kT \\over q}", "07089d218561cbd6cd4ea199a4c78913": " |g\\rangle=|(\\hat{B}-\\langle \\hat{B} \\rangle)\\Psi \\rangle.", "0708a28c7a65508d6f7b18ee71e983dd": "x_1, x_2 \\in I", "0709039bb733667dc30f69865cdf7de2": " \\alpha_{k} = \\frac{\\mathbf{p}_k^\\mathrm{T} \\mathbf{b}}{\\mathbf{p}_k^\\mathrm{T} \\mathbf{A} \\mathbf{p}_k} = \\frac{\\mathbf{p}_k^\\mathrm{T} (\\mathbf{r}_{k-1}+\\mathbf{Ax}_{k-1})}{\\mathbf{p}_{k}^\\mathrm{T} \\mathbf{A} \\mathbf{p}_{k}} = \\frac{\\mathbf{p}_{k}^\\mathrm{T} \\mathbf{r}_{k-1}}{\\mathbf{p}_{k}^\\mathrm{T} \\mathbf{A} \\mathbf{p}_{k}}, ", "070914944ea53f62a72003d0f4842860": "\\bigl\\|\\sum_{k=0}^\\infty u_k \\bigr\\|^2 = \\sum_{k=0}^\\infty \\|u_k\\|^2.", "07091a2d49315b83c62a336e1c6c9dce": "C = \\frac{\\;Q}{u}\\cdot\\frac{\\;f}{\\sigma_y\\sqrt{2\\pi}}\\;\\cdot\\frac{\\;g_1 + g_2 + g_3}{\\sigma_z\\sqrt{2\\pi}}", "0709f02a39c8929c9e10b7e1eb005fd0": "\\left(k,n\\right)", "070b0ab70ce7186a1c9d02a1827f73da": "\\begin{pmatrix}\n\\mathbf{e}_+ \\\\\n\\mathbf{e}_{-} \\\\\n\\mathbf{e}_0\n\\end{pmatrix} = \\mathbf{U}\\begin{pmatrix}\n\\mathbf{e}_x \\\\\n\\mathbf{e}_y \\\\\n\\mathbf{e}_z\n\\end{pmatrix} \\,,\\quad \\mathbf{U} = \\begin{pmatrix}\n- \\frac{1}{\\sqrt{2}} & - \\frac{i}{\\sqrt{2}} & 0 \\\\\n+ \\frac{1}{\\sqrt{2}} & - \\frac{i}{\\sqrt{2}} & 0 \\\\\n0 & 0 & 1\n\\end{pmatrix}\\,,\n", "070b83683a0c810c647073f04a216534": "\\mathbf{i}=(\\mathbf{r}_i,\\Omega_i)", "070b9ebc3aca0b4c9df25a47aa63331c": "\\sum_{q^\\prime}\\left[P_a\\right]_{qq^\\prime}=1", "070bba10304c823c16dffe6457617b82": "\\omega>\\omega_p", "070bbaa9ce926608de688431864bbe8a": "\\tau_\\sigma : V^{\\otimes n} \\to V^{\\otimes n}", "070c05b4f4c6bd8d10fce8d41e488868": "s = \\dfrac {q-1} {1 - a_1} \\bmod{\\ell} ", "070c5b0034631d6d60580faa61c3dd5b": "(x^2 + y^2)^2 = 2a^2 (x^2 - y^2)\\,", "070c5bb99cbfd08e0249b95ecb5d0daa": "I = m(L/2)^2 + m(L/2)^2 = 2m(L/2)^2 = mL^2/2\\,", "070d27d5ae9f59ac9133ce4d69ff2be6": "R_i=\\sum_{j=1}^m r_{i,j} ,", "070d31cec69ed1b7b2bd488cde6138ab": "S^{*} = \\{ (o_{i}, o_{j}) | o_{i}, o_{j} \\in X_{k}, o_{i}, o_{j} \\in Y_{l}\\}", "070d5b305cef4688fddf42beeda3ea45": "\n\\begin{align}\n \\textbf{a}^* &=\\frac{2\\pi\\textbf{b}\\times\\hat{\\textbf{n}}}{|\\textbf{a}\\times\\textbf{b}|}\\\\\n \\textbf{b}^* &=\\frac{2\\pi\\hat{\\textbf{n}}\\times{\\textbf{a}}}{|\\textbf{a}\\times\\textbf{b}|}\n\\end{align}\n", "070d881e0d5e48fdb27cbd9ac84a89f2": " \\cap A_\\alpha ", "070d8dbcdfac6ce17ad9e33703927077": "f'(x)=2x\\sin(1/x)-\\cos(1/x)", "070df022d5055dce70f882c03fa6549d": "x(t)\\in \\mathbb{R}^n", "070e04501111e2ad0b0608cb37c5d2ac": "E_{11}=e_{(\\mathbf I_1)}+\\frac{1}{2}e_{(\\mathbf I_1)}^2\\,\\!", "070e306772d9e79210ef776d8a66a8a7": "\\frac {d} {dx} \\left[x^{n+1}J_{n+1}(x) \\right] =x^{n+1}J_n(x)", "070e80302d8b5797e3cd275e5a2d10fa": " E(Q_t) = \\delta + Q_{t-4} ", "070e9826cccd011b6d5560decbbcc991": " j^{1}\\sigma(p_{1},p_{2}) = (p^{1},p^{2},p^{1}(p^{2})^{2},(p^{2})^{2},2p^{1}p^{2}) \\,", "070eddbbf27e7936e557c6e2ff2bc758": "\\frac{1}{r^4} P^1_3(\\sin\\theta) \\sin\\varphi = \\frac{1}{r^4} \\frac{3}{2}\\ (5\\ \\sin^2\\theta - 1) \\cos\\theta \\sin\\varphi\n", "070edf626867f2e74beb2a9f117e3d17": "Q=\\{(s,t_e)|s \\in S, t_e \\in (\\mathbb{T} \\cap [0, ta(s)])\\}", "070f583f36e31d4401c0aff07df3ece9": " \\mathbf{L} = m r^2 \\boldsymbol{\\omega}", "070fecf848d0313aa08c027f08f73a4f": "\\mathbf{F} = q[- \\nabla \\phi - \\frac{d\\mathbf{A}}{dt} + \\nabla(\\mathbf{A} \\cdot \\mathbf{v})]", "070ff24de65bdcfa9e46b4c5adab778a": "\\ \\alpha_i ", "0710302b40db7631dcd79f68631d2a41": "f\\colon V \\to V", "07104fe9a27abef8ae57d48ed80223a8": "\nJ", "0710afd247c0582195d440cc0e12ba43": "f_1(x)f_2(y)\\le f_3(x\\vee y)f_4(x\\wedge y)", "0710b689e4caa52df41a447f2f810891": "\\operatorname{P}(X\\leq m) \\geq \\frac{1}{2}\\text{ and }\\operatorname{P}(X\\geq m) \\geq \\frac{1}{2}\\,\\!", "0710c12e6cbddc77d94d8a03e078dd27": " \\langle \\varepsilon_q | \\psi_N \\rangle = \\langle \\varepsilon_q | \\frac{1}{\\|\\psi\\|} \\left ( \\sum_{i = 1}^n c_i | \\varepsilon_i \\rangle \\right ) = \\frac{c_q}{\\|\\psi\\|} \\,,", "07110254746dcae91bc441539c119e0e": "\\lambda \\,", "071117439d4d72b051e49c65ff9a4f02": "T_\\text{hold}=T_\\text{load}\\cdot{e}^{-\\mu\\cdot\\phi}\\quad\\text{ or } \\quad T_\\text{load} = T_\\text{hold}\\cdot{e}^{\\mu\\cdot\\phi} ", "0711354093f22350c807383161c718ad": "\\lim_{n\\rightarrow\\infty}\\frac{1}{n}\\sum_{k=0}^{n-1}T^kf", "071148229a4c01ef09ca2c0b77230f2c": "\\cos (\\alpha + \\beta) = OB = OA-BA = OA-RQ = \\cos \\alpha \\cos \\beta\\ - \\sin \\alpha \\sin \\beta\\,", "07119a27e858f421530bbcace27168a0": "\\Omega(G)/G", "07119d5545091b075c4d361f8488abb2": "\\dot{r}_{j} = \\lambda_j r_j + r_{j-1}, j=2,3,\\dots,n", "0711d4567b6e6bc02a1bbb73e87c497b": "\\operatorname{E}\\,\\hat\\sigma^2 = \\frac{n-p}{n} \\sigma^2", "07128930ff48fb3fc74418d68b9f4a23": " \\frac {n!} {(n-k)!k!}. ", "071302c7ae2dc33849d7424158fa7569": "P_A = A (A^\\mathrm{T} A)^{-1} A^\\mathrm{T}. ", "07138f98c839393571d4c74d772b1305": "\t\\begin{array}{rr|rr} \n 1x & \\text{-}13 & 16x & \\text{-}81 \n\\end{array}", "0713a52582a53f4d7ba16c4c6ed27031": "C_1 \\subseteq C_2", "0713a6b3411166cb06f7ab980d9f5ede": "m_{\\mathrm{TNT}}", "071433da6a0b97575672c8502b6da5e8": "01.\n\\end{cases}", "0724b2ff19e6ceca34d0d2c3314d23c3": "\\sum_k \\kappa(u_{ik}) u_{kj} = \\sum_k u_{ik} \\kappa(u_{kj}) = \\delta_{ij} I,", "0724f4e191c17098d6f6b4b92ed70159": "\\{\\max cx \\mid x \\in P\\}", "07250ba09253f459138209af2c9054f7": "\\displaystyle{z_n=re^{2\\pi i n\\over N}}", "07250d8d86eff1000acaf0522dc3ac5f": "\\forall n < t", "07252b644efc2a703f7f681fbfce9b2d": "i\\frac{\\partial}{\\partial t}\\sigma (q,t)=L\\sigma (q,t).", "0725648f358978c7534ec4f1d491abb0": " \\mathcal{O}_X ", "0725a35b3a82eb04f1c74c9d832e9028": "g_{\\mu \\nu}", "0725d5c8eb96d73dd518391b7fd06712": "S^{\\alpha }", "072632ad30a9e26b24bee86e6ce49aae": "\n p := -\\tfrac{1}{3}~\\text{tr}(\\boldsymbol{\\sigma}) ~;~~\n \\boldsymbol{s} := \\boldsymbol{\\sigma} + p~\\boldsymbol{\\mathit{1}}\n", "0726689d2e61a048c115006c9ceea155": " e,", "072693264f50b3f4201bffb3e9a3139b": "\\sigma_{rr}\\, ", "0726a69d0264198a8f1a2a9117893503": "\\frac {\\mathrm{d} \\mathbf{r}}{\\mathrm{d}t} = \\lim_{{\\Delta}t \\to 0} \\frac {\\mathbf{r}(t + {\\Delta}t)-\\mathbf{r}(t)}{{\\Delta}t} = \\frac{\\mathrm{d} \\boldsymbol{\\ell}}{\\mathrm{d}t} \\ .", "07270aab27a420bb7c3951224b597e4b": " = \\frac{1}{n(n - 1)} [(\\sum_{j=1}^k n_{i j}^2) - (n)] ", "072753aa297b0aea0e4624f94d777ff1": "\\langle U_{g} [e], [e] \\rangle_f = f(g)", "0727cde1da04a53168ad6bb916d51f52": "\\mathfrak{sl}_n", "0727ea3e3281289bd156591d3e813ad1": "\\scriptstyle\\boldsymbol{u}(\\boldsymbol{x})=(u_1(\\boldsymbol{x}),\\ldots,u_p(\\boldsymbol{x})) ", "072858900caf8f4db8959233cc538734": " u_{jj} = \\partial_j u_j", "072871070204574fc3d379c09d1b19b9": "\\mathbf{b}=(b_1,\\dots,b_n)^t", "07289bb62b68542477d5e3ecb4cfc7cc": "f(n) = 0", "0728caa5bd49579b7d719edbd2159387": "t_0 = t_f \\sqrt{1 - \\frac{3}{2} \\! \\cdot \\! \\frac{r_0}{r}}\\, .", "0729aaabbe5af8ad7331adac3ad03e01": "\\frac12 a_0-\\frac14\\Delta a_0 +\\frac18\\Delta^2 a_0 -\\cdots = \\frac12-\\frac14.", "0729d51d0f40afdc533043d47fa40214": "10^{15}", "0729ed42759d8aade6e27204f103be48": "\\ S_c", "072a73c94590db2748152d36bc19dccd": "x=2+3t,\\;\\;\\;\\;y=-1+t\\;\\;\\;\\;z=\\frac{3}{2}-4t", "072a8af452e91f60c5eba13af18cf72c": "c_0 \\frac{\\pi}{4} = \\sum_{n=1}^N c_n \\arctan \\frac{a_n}{b_n}", "072adfc230e8952b5cb62880bf69272c": "E_o(\\rho,Q)= -\\ln\\left(\\sum_{y} \\left(\\sum_{x} Q(x)[p(y|x)]^{\\frac{1}{1+\\rho}}\\right)^{1+\\rho}\\right), ", "072af4c81ba2f88175e28f970ce3d1c9": " v_{k+1} = (4 \\beta^2 - 2) v_{k} - v_{k-1}. \\,\\!", "072b01b85e9d24695f60c4ab591dfa6a": "\\frac{d}{dx}\\int_a^x f(t)\\, dt = f(x).", "072b030ba126b2f4b2374f342be9ed44": "60", "072b2f2acfebae9e9efe2b99ea972ccc": "\\displaystyle (\\beta)", "072c54ea4bb453f4eb49b830901b60c8": "Q_i[x^j_\\alpha(t)]=t \\delta^j_i. \\, ", "072cdabf57aff1d612245d5a028b91ee": "\\mathrm{kV_p}", "072ce830e5ecb5328bbda0aad2281265": " 2(\\ell w + \\ell h + wh) \\, ", "072d4b9ebb7d8b49bc729ef951f501ed": " \\mid \\langle \\psi_{x\\pm} \\mid \\psi_{y\\pm} \\rangle \\mid ^ 2 = \\mid \\langle \\psi_{x\\pm} \\mid \\psi_{z\\pm} \\rangle \\mid ^ 2 = \\mid \\langle \\psi_{y\\pm} \\mid \\psi_{z\\pm} \\rangle \\mid ^ 2 = \\frac{1}{2}. ", "072d7b0f3eb30fcd4fccdbe870fa3994": " h_L(t) = \\delta(t) -{R \\over L} e^{-t \\frac{R}{L}} u(t) = \\delta(t) -{1 \\over \\tau} e^{-\\frac{1}{\\tau} t} u(t) ", "072db38fbbae890b4bd5ae5e1ccbb022": " \\lim_{x \\to0^+} 0^x = 0. \\! ~~ (8) ", "072db40c19a10e624b1529e767914484": "U_0=\\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix}", "072e29653f2eef3b350835b84e6b3744": "MD(\\varphi \\rightarrow \\psi) = max(MD(\\varphi), MD(\\psi))", "072e5620f53251f5440ad8fc03bf9736": "v=\\partial/\\partial x_j", "072f40b0d1204d25dab5d17da842c1e0": "\\gamma^0 = \\begin{pmatrix} 0 & I_2 \\\\ I_2 & 0 \\end{pmatrix},\\quad \\gamma^k = \\begin{pmatrix} 0 & \\sigma^k \\\\ -\\sigma^k & 0 \\end{pmatrix},\\quad \\gamma^5 = \\begin{pmatrix} -I_2 & 0 \\\\ 0 & I_2 \\end{pmatrix}.", "072f4ed969ed563ed39bb9a512f7a7cd": "(7.f)\\quad e^\\psi =\\,\\Phi^2-2C\\Phi+1\\,.", "072f4fd15987cf40a5ba20fc14a3c165": "1 + 1 = 2", "072f7417db2400c9952b7145ed281f46": "\\bar{X}+X_\\xi", "072f7c69d1d6211625222dc08a9465c1": "z(n;2)=n^{3/2}(1+o(1)).", "072fa72ea632294b7d71df76773551f8": "\\frac{\\partial \\ln |\\mathbf{X}^{\\rm T}\\mathbf{X}|}{\\partial \\mathbf{X}} =", "072fdbe4e27341c370c721ad551af14f": "1100000 - 1011101 = 11", "072ff341eac911182c5fac7794d611e4": "Solow Residual=g_{(Y/L)}-\\alpha*g_{(K/L)}", "072ffa50d2cf35595e2754cbf0677e95": "Rd\\Omega^2/dR=-3\\Omega^2<0\\ ,", "073016c126332f131da2cb8a9a6bea1a": "a_{N-1} a_{N-2} \\dots a_0", "0730280d3582824e0ec54ddeeab7e33c": "\\alpha = E[R_p-R_b]", "07302b9c5e204ccdad2e0265cf1545a6": "d = revolutions \\times d_f", "07304a7ad2da8514893da22daea73881": "XX = X^2", "0730a3f9a9adfcb75a656c093332d36e": "l_{i}^-\\rightarrow l_{j}^{-}\\nu_{i}\\bar{\\nu_{j}}", "0730b75e96c0453b1b196be7ff4fa194": "vu", "0730f9104d1a5098c4d88fabcdbfc8b9": "q\\, =\\, \\tfrac12\\, \\rho\\, v^2", "073104b4040b9c8ddd9f09b54805cbaf": "\\ln W = \\alpha N+\\beta E\\,", "07311e20ca459e52fef6ff1ed7e74c96": "O_X(U)", "073234664a210a834b8f8d5ff279ec95": "x'=-x.\\,", "07323a39242f586b7f085dc127d795c0": "\\neg X", "0732488d61e60296c33ec87fba7602f0": "R_{ix}(t+ \\Delta t)=P_{ix}(t+ \\Delta t)+\\sum \\frac {T_m(t+ \\Delta t)}{l_m(t+ \\Delta t)} \\times (X_j(t+ \\Delta t)-X_i(t+ \\Delta t))", "0732bcf93456bd5d72097502b87fd541": "\\Delta\\Omega=(\\Omega_{0}-\\Omega_\\mathrm{pole})", "0732c8aad6337e8d8bbd515c4693c2bf": "\\mbox{Certain safety factor} = \\frac{\\mathrm{LD}_{1}}{\\mathrm{ED}_{99}}", "0732ec27993507494f2eb050f361d1fe": "\\textstyle{5 \\div {1 \\over 2} = 5 \\times {2 \\over 1} = 5 \\times 2 = 10}", "0733483cabd437eda80d9da0d3f4018a": "V_y > V_x", "0733b8464937afd4081f25c09be53fa1": "\\begin{bmatrix} c_1 & -s_1 c_3 & -s_1 s_3 \\\\\n s_1 c_2 & c_1 c_2 c_3 - s_2 s_3 e^{i\\delta} & c_1 c_2 s_3 + s_2 c_3 e^{i\\delta}\\\\\n s_1 s_2 & c_1 s_2 c_3 + c_2 s_3 e^{i\\delta} & c_1 s_2 s_3 - c_2 c_3 e^{i\\delta} \\end{bmatrix}. ", "07341b95aeda2633856303d8f9cb497d": "A = \\begin{bmatrix}\\mathbf{a} & \\mathbf{b} & \\mathbf{c}\\end{bmatrix}", "073456157e857bf7ec76ee4ea25d69d0": "-\\sqrt{-r}", "07347921990cb6f18d2e46d3212030e1": "A^D=0.", "073546f534c6e6d0a62a04eefd1aa8bf": "R_{k+1}(a, b) = 1", "07354e7d280d293e90b43918abcfecd4": "S(f) = \\frac{\\sigma_Z^2}{| 1-\\sum_{k=1}^p \\varphi_k e^{-2 \\pi i k f} |^2}.", "073575b3716398185104175a18564140": "p=18", "07357ff8e80f38aa7f7bd72df210f1b2": "\\sqrt{\\exp}", "07358ebcd581a065b84eb48b7362b09c": "D_L \\ = \\ R_0r_1(1+z) = \\frac{c}{H_0q_0^2} \\left[q_0z+(q_0-1)(-1+\\sqrt{1+2q_0z})\\right]", "0735d9aa7920768b9cfe84434d9f18c6": "a^2 k (1-a)^{k-1} \\, ", "0736059add7e8fe4e6ff63b06b623349": "~(x)_n\\equiv (xT_h^{-1})^n=x (x-h) (x-2h) \\cdots (x-(n-1)h)", "073629db8e56188b25f5fc01c858587f": "number = normalized( weight / mean packet size )", "0736a56f3d66e66ec6c1fa27886e637e": "\\varphi(n^{s+1})", "0736ace3b1f283c0190a4fdb6b4451ee": "|df_p(v)\\times df_p(w)|=\\kappa|v\\times w|\\,", "0736b52ab6852acc846c382d0a356ef6": "f(x,y) = 181.617 \\,", "0736be535ab20cdb901f3b10b3f6601c": "\\theta[\\mathbf{f}] = \\begin{bmatrix}\\theta^1[\\mathbf{f}]\\\\\\theta^2[\\mathbf{f}]\\\\\\vdots\\\\\\theta^n[\\mathbf{f}]\\end{bmatrix}.", "0736d932c4f810737387df1b18b79499": "\\begin{pmatrix}\n1 & 1 & 1 & 1 & 0 & 0 & 0 \\\\\n1 & - & 0 & 0 & 1 & 1 & 0 \\\\\n1 & 0 & - & 0 & - & 0 & 1 \\\\\n1 & 0 & 0 & - & 0 & - & - \\\\\n0 & 1 & - & 0 & 0 & 1 & - \\\\\n0 & 1 & 0 & - & 1 & 0 & 1 \\\\\n0 & 0 & 1 & - & - & 1 & 0\n\\end{pmatrix}", "0737179ee66c8461eeafd1b317438d93": "{n \\choose \\lfloor{n/2}\\rfloor} \\ge {n \\choose k}", "0737552727d3f52d5f6ac33e430cccf9": "\\sigma(X', X)", "073883b3807515b371c7103bcd50240f": "\\mathbb{E}\\left [ ((H\\cdot M)_t^*)^p \\right ] \\le C\\mathbb{E}\\left [(H^2\\cdot[M]_t)^{\\frac{p}{2}} \\right ]<\\infty", "07388de6996a4bcd801b6bc90aa9df6c": "(p_n)_n\\,", "0738cf6a34f09bfa105a8f9bb6bfb679": "(u_1,v_1) = (\\cos\\theta\\,w_1-\\sin\\theta\\,z_1,\\,\\sin\\theta\\,w_1+\\cos\\theta\\,z_1)\\,\\!", "0738f11969551fbb00584191dfecd4e5": "(A\\to\\neg B)\\to(B\\to\\neg A)", "07396449ec8b62cc97795b456882987d": "\\delta(g(x)) = \\sum_i \\frac{\\delta(x-x_i)}{|g'(x_i)|}", "0739768b9134483933fc1ee966f3a4cd": "g^{efghcdb}", "07399a25bdc257ad80519b3e10b08e02": "\\prod _x f(x)\\,", "0739a75a70c0fbd83ac74b5789627a2a": "\\alpha\\beta\\gamma\\cdots", "073a512b87277b08f8abadf785cbff48": "J^{\\alpha} = \\, (c \\rho, \\bold{J} ) \\,", "073a52e4766b7792036a3077e4052d23": "{{\\gamma }_{k}}(X)", "073a6593121390a4317e466933e744c6": "s(t) = A\\cdot \\cos(\\omega t + \\theta),\\,", "073a97127b4c8c67e21103fa55663f68": "\\Delta(x) = \\sum_{n=-\\infty}^\\infty \\delta(x-n),", "073aeab6305458edb9db996b527938cf": "F=\\left\\{(x,\\ y):c\\le y \\le d,\\ r(y) \\le x \\le s(y)\\right\\}", "073aef54904465bc5f2c5c88b3fa7d30": "\\{(-,+,+,+)\\,,l^a n_a=-1\\,,m^a \\bar{m}_a=1\\}", "073b23a014cdb561e726fdfd782536de": "\\sin(45 ^\\circ) = \\frac12\\sqrt{2};", "073b301e11bb8a8080bcb487c4d7b7e4": "x^3+bx^2+cx+d=0", "073b63613eef32ad23c021ccc4317e95": "\n \\cfrac{\\partial p}{\\partial t} + \\kappa\\left[\\cfrac{\\partial v_r}{\\partial r} + \\cfrac{1}{r}\\left(\\cfrac{\\partial v_\\theta}{\\partial \\theta} + v_r\\right) + \\cfrac{\\partial v_z}{\\partial z}\\right] = 0 ~.\n ", "073b6f20e1e3a7b116757dcdac5caed2": " x = (x_1, \\ldots, x_n)^\\mathrm{T} ", "073b8c7ae4ce7097c9af53d63637ce0e": "1\\leq i \\leq 2r", "073bcfbe99da78460cf5a9266da799f2": "\\frac{\\pi^2}{12}+\\frac{\\gamma^2}{2}", "073c289961a29f9be578aee613690f6d": "\\boldsymbol{\\alpha} \\leftarrow", "073c3804b1c449d744683e78a8693ab3": "\\ cos\\theta = 1 - \\beta(\\gamma_L - \\gamma_c)\\ ", "073c7b80ea477f9c5b750163479126e6": " \\displaystyle{ We^{-3/2} \\approx 0.22\\, W} ", "073c87a1b4fd09a206f70fe96c79a1cc": "t_{E}", "073cd80184e7f504f8595e8da5cf9a36": " 1 / (\\lambda T) . ", "073d06b35bd22a5c89556e597b8a557d": "s_0 \\approx S", "073d278cb1d43a03237f3a839b0a0826": "f:\\mathbb{N} \\longrightarrow\n\\mathbb{N}", "073d378533393ea8de9b8729e76a7318": " H^{s}(E\\backslash \\bigcup\\Gamma_{i})=0", "073df4e746cfe5c69d95dc4ac562bbe4": "\\Omega_M", "073e27cb1774b505ef111a366414793f": "(1-\\omega)\\phi_i + \\frac{\\omega}{a_{ii}} (b_i - \\sigma)", "073e375add3813aab90e8e4de93e8af7": " 7/4 \\times 5/6 ", "073e61f9fd390745a15dc70c6263c3ce": "X\\times X", "073e72a48c4f261575d477418fa0139e": "{\\rm vec}(\\mathbf{B} - \\hat{\\mathbf{B}})^{\\rm T} (\\boldsymbol\\Sigma_{\\epsilon}^{-1} \\otimes \\mathbf{X}^{\\rm T}\\mathbf{X} ){\\rm vec}(\\mathbf{B} - \\hat{\\mathbf{B}})", "073e894de339048c4adf4abc34b32783": "\\scriptstyle1\\leq j\\leq k", "073e9b97c4bc3878248d4727163b1ae2": "(v-k-1)\\mu = k(k-\\lambda-1).", "073ea3b7cec5eeeeec4164d10c217465": "\\beta^2", "073f18a5623a40477ec466172cba8054": "\\chi:V \\rightarrow \\{-1,1\\}", "073f339583a2f4da6bdad7daaa7f6f11": "F_{electrostatic} = \\frac{1}{2} \\frac{\\partial C}{\\partial z} \\Delta V^2 ", "073f95647088ae7e39f204457c32edef": "\\frac{1}{2}\\int \\frac{d^dp}{(2\\pi)^d} \\tilde{\\phi}^*(p)R_k(p)\\tilde{\\phi}(p)", "073fba3e5f887e0871c7b450e96e0c13": "(AA^*)^{-1}\\,\\!", "073fd12dab3dfe07a12152f9c9671677": "g_{k,n}(z)\\approx z", "073ff1e86bd18f4d9184ddaf913415fd": "x:(S^1)^{\\wedge i} \\to A, \\, \\, y:(S^1)^{\\wedge j} \\to A", "073ffadcdaa43aa78a7732eb3dab54c5": "\\mathbb{Q}(\\sqrt{2},\\sqrt{3})", "074002b9606b4566f3cb61a013bc8a43": "\\frac{dy}{dt} = \\frac{dy}{dx} \\cdot \\frac{dx}{dt} ", "07400b898734e4d35872d9f00e27d8e8": "1 \\cdot x ", "074016936a361c9b3ffa5491eee80e15": " \\Rightarrow 37675 = 34250 + 3425 ", "07402c69c78a7c057efa7c217c98f14e": "0 \\le t \\le 2 \\pi.", "07405e52845785645c3846f46a49323c": " a >0.\\,", "07406589807bb14817217a224a910198": "\\bar{\\theta}(\\mathbf{r},t) = t_n \\theta^n(\\mathbf{r},t)\\,,", "074097ea89225398ceb1128b5405b9fb": "x \\ge 0", "0740ae1e776b13ee7c58dbe7d28a86e0": "\\displaystyle m_1,\\,\\ldots,\\,m_N", "0740b1ad3077f5c9eea8df09f039e468": " [Fu](t,m,n) = [Fp](m,n) \\, \\cos( \\sqrt{m^2+n^2} \\, t ) + \\frac{ [Fq](m,n) \\, \\sin (\\sqrt{m^2+n^2} \\, t) }{\\sqrt{m^2+n^2}} ", "0740d56f5a5831a0fc0323b759552e8a": "f'''(x)\\geq 0", "0740ed673b4645e6673f5176495f3d96": "y = x", "074104ddab3a7e03f350918bfb6aff94": " \\frac {v_{\\ell}} {v_i} = A_v \\frac {R_L} {R_L+R_o}\\,\\!", "0741adc73b5cbce75e96a2c1cb93f96b": "(x_i,\\hat\\mu(x_i))", "0741bc843be2d7c0e6ef88dc85352a6a": "\\Omega(t^2/n^2)", "0741f8b2588f82369bb0dde8c395406c": "\\lambda \\setminus \\mu", "0741fdc506e5886e2d86a0ca9dab339b": "E_x, E_y, E_z", "074202f478d8c34356dce99c61155557": "\\sum_{b \\in B} \\left|x (b)\\right|^2 = \\sup \\sum_{n=1}^N |x(b_n)|^2", "0742460f0c51461ec49dc9faccfe0faa": " f(i) = \\cos(1) + i\\sin(1). \\, ", "0742565886f2dd30b7c53927aa007e51": "h[n] = 0 \\ \\forall n < 0,", "074288624383bc9007623912870acfe8": " V_L = V_S \\frac {T (1 - \\Gamma_S)(1 + \\Gamma_L)} { 2 ( 1 -T^2 \\Gamma_S \\Gamma_L) } \\, ", "07429da45516fb218151c6a0d153cf0b": " s", "0743075233cdfa694552767c9396f30a": "y_{isth} = \\alpha + X_{sith} \\beta + u_{sith}.\\,", "0743243d2af3dd9be8a8cee60adcb8a1": "\\varphi_{i}:M\\supset W_{i}\\rightarrow U_{i}\\subset\\mathbb{R}^{n}", "07432f3e8b4c6dbebf18ef958c46c9a3": "\\, x", "07433d49212843e7076456754ad87639": "\\mu_{z} ((t_{1}, t_{2})) = \\int_{t_{1}}^{t_{2}} 1 + \\| \\dot{z}(t) \\| \\, \\mathrm{d} t", "07434aa38f012db69b7f60d9fa8c0126": " b_0 = -\\infty", "07434fa7b23739fe350bf93e371183f7": "Y_\\beta", "0743bb0be3b046143f0a7657b412ea61": "H \\cong G", "0743bdbf9106df9455b1a340f28b6a88": " d(x) = \\int_a^b g(x,y)\\,m(y)\\,dy ", "07449a04b6a67c2ae7875cbf072c8ccd": "t \\notin \\gamma ", "0744ba02574434f6bcd2be4d203c156c": "x \\ge 3\\,\\!", "0744e0c9460149bf75ae433e78295bbf": "\\sigma \\in G", "0745076424d51f025019d732d61dc90c": "\\frac{1}{\\sqrt{3}}", "07451e54ea5aac17d40e38cad613494d": "Y_0 = K*Y", "07453e6baf8f216467f9b664de795bfc": "g_1(x) = \\sum_{k \\geq 1} \\frac{\\sin(k \\pi / 4)}{k! (8x)^k} \\prod_{l = 1}^k (2l - 1)^2", "074552de1c633701e0fd74715a03d7ba": "b_{i,k}", "07455460393c61f17cba114435aca24a": "[Q^\\dagger,b\\}=\\frac{dx}{dt}+i\\Re\\{W\\}", "0745566b9902ba1c640bcf6d2b22b5c1": "g \\in L^2 (X, \\mu)", "0745579204b7ebffe2fbf2da53f4fc17": "\\lim_{n \\to \\infty}\\frac1{b_n}\\sum_{k=1}^n b_kx_k = 0.", "07456f68b8b64ab4d5f2d065d8800c80": "= \\frac{2L_{L} / \\gamma (v)}{c}\\frac{1}{1-\\frac{v^{2}}{c^{2}}}", "074570409811b2cab66129c35e89ad82": "- T(\\alpha_1, \\alpha_2, \\ldots, \\mathcal{L}_YX_1, X_2, \\ldots) \n- T(\\alpha_1, \\alpha_2, \\ldots, X_1, \\mathcal{L}_YX_2, \\ldots) - \\ldots\n", "0745ba279fe361d42860dd5ee162cc0e": " \\mu \\approx 100 ", "0745dc526a8ba31c9f0b8565be0d8e94": "\n\\operatorname{sgn}(t):=\\left\\{\n\\begin{array}{ll}\n\\frac{t}{|t|},&t\\in \\mathbb{C}\\setminus\\{0\\},\\\\\n0,&t=0.\n\\end{array}\n\\right.\n", "074665b80ba529c5a84b2f98eb137e39": "\n U_{B}=\\{x\\in V:\\quad \\|\\varphi\\|_B<1\\},\\qquad B\\in {\\mathcal B}, \n ", "074713fe645ff55c3503bc5cf350d8af": "\\Theta( L_{a}+\\vert\\mathbb{C}\\vert M_{a})= \\Theta(\\vert\\mathbb{C}\\vert M_{a})", "07472f0af8b30b0ce09edd6d6246708c": "\\Psi : Y \\to (X, \\tau)',\\quad y \\mapsto (x \\mapsto \\langle x, y\\rangle).", "0747420804c56c9a02dc45ab66e3f7c4": "\\frac{2^{4031399}+1}3", "0747c16f20f48e69325d669977d9b8ae": "W=\\textstyle{\\frac{1}{2}}(-X+3Y+Z)", "074835d1d992419a421c82c1fc3f7c3b": "\\left(X, \\Sigma_X\\right)", "0748bb666d7a04bc2763837ecc623db1": "\\mu (U_i) \\leq 2^{-i}", "0748dc9d2f6b852f0060f0b3e68eb8f9": "M_{xy}(t_{SL}) = M_{xy}(0) e^{-t_{SL}/T_{1rho}} \\,", "07491314818d10c7f92ec5a22c7d7e37": "r\\mathbf{a}=(ra_1)\\mathbf{e}_1\n+(ra_2)\\mathbf{e}_2\n+(ra_3)\\mathbf{e}_3.", "07494407d8974fe8c48ef51656d886f4": "\\| (u, v) \\| := \\| u \\|_{L^{1}} + \\| v \\|_{M},", "0749a5a308ccce845eed5c7f350e799f": "{{i}_{E3}}={{i}_{C2}}+{{i}_{B1}}+{{i}_{B2}}={{i}_{C}}+2{{i}_{B}}=\\frac{\\beta +2}{\\beta }{{i}_{C}}", "0749a79fa388599db2e978b658f9081a": "H_0: \\theta=\\theta_0", "074a08f8d3aad5c15ec7217f53b3b60d": "\\Omega^7", "074a2cb57b7a7a4af8af6ceee1a1fa55": "\\begin{cases}\n\\begin{cases}\n\\begin{cases}\n\\begin{cases}\n\\begin{cases}\n\\begin{cases}\n\\begin{cases}\n\\begin{cases}\n\\dot{\\mathbf{x}} = f_x(\\mathbf{x}) + g_x(\\mathbf{x}) z_1 &\\qquad \\text{ ( by Lyapunov function } V_x, \\text{ subsystem stabilized by } u_x(\\textbf{x}) \\text{ )}\\\\\n\\dot{z}_1 = z_2\n\\end{cases}\\\\\n\\dot{z}_2 = z_3\n\\end{cases}\\\\\n\\vdots\n\\end{cases}\\\\\n\\dot{z}_i = z_{i+1}\n\\end{cases}\\\\\n\\vdots\n\\end{cases}\\\\\n\\dot{z}_{k-2} = z_{k-1}\n\\end{cases}\\\\\n\\dot{z}_{k-1} = z_k\n\\end{cases}\\\\\n\\dot{z}_k = u\n\\end{cases}", "074a81a1fb5d7d806f2da321fc83ffb6": " \\, {l_c} ", "074ae6dfaebaa5e5d168b78e98a18de6": "R_r^{'}/s", "074b4a7192acfe468cc1e567ad6e7725": "\\,\\Sigma_{xx}\\beta_k(k=1,\\ldots,K)", "074b4d7cb6805e2618baa131aea71638": " S \\mapsto \\nu (g^{-1} S) \\quad ", "074b87eddded6a895dedc4c361db165c": "\\frac{200-150}{100}=0.50", "074be6c7441cfe773d43ac834a1ee97a": "-E_{act}/R", "074bf1d8d0b30d80372f9918b3845727": "Q = \\left (A^T A \\right )^{-1}\n", "074c14097a4cefef444cc434b45576f1": "3 \\cdot a_n\\ \\mathrm{dB}", "074c5003b6fd3a0d3c02d956852a529c": "\\mu*N", "074c699683f5efad06712c730018baa1": " P_2=(x_2,y_2) ", "074c6a9b9b3d5052dae458c1c1cebef0": "\n\\sum_{i=1}^{n} a_i^k = b^k\n", "074c775b2a19e7ce773d8a43e6e817f0": " \\frac{\\operatorname{d}}{\\operatorname{d}t} \\left( c_1 x_1(t) + c_2 x_2(t) \\right) = c_1 x'_1(t) + c_2 x'_2(t) ", "074c7ca9d8c84795dca480f5ba6fb001": "R_p\\simeq0.3", "074c8c5de4d3a155081a95c480b70a29": "V = D/t", "074c8fa594b5d54226e76ae3b6669fc8": "\\neg\\neg A \\to A", "074cc7670157df51929ce0e6fb025d6d": "9/11 = 0.1\\ 1\\ 3\\ 3\\ 1\\ 0\\ 5\\ 0\\ 8\\ 2_!", "074cfc7052d14a7f29d1d081c6258a63": "\\displaystyle g_i", "074d1e1f2a3c12b8ad152cee342fb779": " m_i \\scriptstyle", "074e4440565d21762c26d1ecc605021f": "x_0 = 0 \\quad (11')", "074e5c7dd44f09cd16d0b75ac973bfa2": "d'_P(P_1,P_2)=(x_1-x_2)(y_1-y_2)", "074eeee3fa471a47ee383d47edca237d": "A=60^\\circ", "074ef0e78e37964f7583f7bd5568218b": "\n\\frac{\\partial S}{\\partial \\alpha_{r}} = \n\\frac{\\partial}{\\partial \\alpha_{r}} \\frac{1}{2} \\sum_{k=1}^{N} m_{k} \\left| \\mathbf{a}_{k} \\right|^{2} = \n\\sum_{k=1}^{N} m_{k} \\mathbf{a}_{k} \\cdot \\left( \\frac{\\partial \\mathbf{a}_{k}}{\\partial \\alpha_{r}} \\right)\n", "074ef76444599eff3f9ae2b65afcca54": " T_{n} = 1,1,1,2,5,16,61,272,1385,7936,50521,353792,\\ldots \\quad (n=1,2,\\ldots) ", "074f0008987d021f1d042094caa89a13": "\\int \\frac{\\delta Q}{T} = 0", "074f5f3bd8ebf2efab25dd2b4933a662": "w\\Vdash p", "074f84aaf0e6e73802b0caf3bbb2b5bb": "r\\gg r_a", "074fb98c1ec909b43c244d8ab7a62170": "i=0 \\ldots n-1 ", "07500d4e5afe4e53a75c4b4c205459d4": "a,b\\in X", "07503ada062ead95b8ee189fa6d93c6e": "\\textstyle{A=\\frac12 log_2(RG) = \\frac12 (log_2(R) + log_2(G))}", "0750524860f0c0d1948a04fb5044a63f": "\\delta Q = \\mathrm{d}U + \\delta W", "0750607522d1f2c4e445f1bb189dd792": "\\operatorname{End}(V) \\times \\operatorname{End}(V) \\to \\operatorname{End}(V)", "07508dd8b529ec382f9c78e0438c4d08": " \\tan\\phi = \\frac{y}{-x} = \\frac{2y}{y^2 - 1} = \\frac{2e^a}{e^{2a} - 1} = \\frac{1}{\\sinh a}. ", "0750fa6c4391138d52861e02cd8f1001": "(x-3)x^{28}(x+3)(x^2-6)^{21}(x^2-2)^{27}.\\ ", "07510ea55969b82b001f63319143f3d7": "\\frac{x}{y}=\\frac{1}{\\lceil y/x\\rceil}+\\frac{(-y)\\bmod x}{y\\lceil y/x\\rceil}", "0751669a6b433328f530f4036618aa53": "\ns_t \\equiv \\vec{S}_t \\cdot \\vec{D} = \\sum_{A=1}^N M_A \\;\\vec{s}^{\\,A}_t \\cdot \\vec{d}^{\\,A} = 0\n\\quad\\mathrm{for}\\quad t=1,\\ldots, 6.\n", "07516a79c61e8deda3a488d2625e3d5c": " E_\\text{cm}", "07517907d40805f1737608a213260a88": "\\ M_{heel} = pressure \\times S \\times A {cos(\\phi)}^n ", "07517cf760c151b716b9d4e70df09cf0": "\\mu \\ = \\mu_r \\,\\mu_0 \\,\\!", "0751a0235b4732f716701e72722cdb33": "= \\int_{0}^{\\infty}C\\, \\operatorname{d}t", "0751ceccee6d9fe57a4aaf339a0e5a35": " p_{4,4}(x) = y_4 \\, ", "075215656b7a0e53567103e62b31ee31": "\n\\begin{align}\n\\overline{Y}_1 & = \\frac{1}{6}\\sum Y_{1i} = \\frac{6 + 8 + 4 + 5 + 3 + 4}{6} = 5 \\\\\n\\overline{Y}_2 & = \\frac{1}{6}\\sum Y_{2i} = \\frac{8 + 12 + 9 + 11 + 6 + 8}{6} = 9 \\\\\n\\overline{Y}_3 & = \\frac{1}{6}\\sum Y_{3i} = \\frac{13 + 9 + 11 + 8 + 7 + 12}{6} = 10\n\\end{align}\n", "0752312c070751129630dd26655e1933": "\n c=f\\lambda\n ", "07527e3ec7a12e1161e53821bc99f95a": "\\mathbf{M} = \\mathbf{F}(\\mathbf{E})", "0752f7fefb594dc5a4937ab6746900b6": " \\theta = 3 \\nu - 2 \\mu ,", "07536b2456b41acd85206334dd863837": "\\frac{q^{r+1}-1}{q-1}", "07537a4b745cadd9926a130320d1488f": "v_0 = s, E_0 = E", "0753b0785e89821ff5957766f811f56e": "N \\geq 2", "0753bc3f9919ba50a4bc8d342b40a6e0": " \\rho_0 ", "0753d9d83d2a2f563f50c4c7ae14d898": "\n\\begin{pmatrix}\n0 & i \\partial_0 + i\\nabla \\\\\ni \\partial_0 - i \\nabla & 0\n\\end{pmatrix} \n\n\\begin{pmatrix}\n \\Psi_L \\\\ \\Psi_R\n\\end{pmatrix}\n\n= m\n\\begin{pmatrix}\n \\Psi_L \\\\ \\Psi_R\n\\end{pmatrix}\n", "075406f6eceea0908aea81e296465c89": "\\psi + \\theta + \\phi = \\tfrac{\\pi}{2}\\, ", "0754dcc9db21c1211d42115b528100d5": "0 U_a) \n= \\Pr( \\alpha P_i + \\beta D_i + \\varepsilon_i > \\alpha P_a + \\beta D_a + \\varepsilon_a ) \\\\ \n& = \\Pr( \\varepsilon_i - \\varepsilon_a > \\alpha P_a + \\beta D_a - \\alpha P_i - \\beta D_i )\n\\end{align}\n", "07629bad05308a4805729a54ee888b3e": "a_6= \\lfloor 2^\\frac{1}{2} \\rfloor = \\lfloor 1.414\\dots \\rfloor = 1. ", "07629de353237b95bd44b92e2b0b5a74": "A[i,j] + A[k,\\ell] \\le A[i,\\ell] + A[k,j].\\,", "0762d4fb3790cfb0107ceddd69321346": "0 \\to \\operatorname{Der}_B(C, M) \\to \\operatorname{Der}_A(C, M) \\to \\operatorname{Der}_A(B, M).", "0762e2065b43aea6df90e731cf029b9b": "(G_n)_{n\\in\\N}", "076323ef7554f221c8070ad5de5c22b8": "[A,C]", "076381b47788f3b58e3c4e8b6804a8c5": "g(r) = h(r) = \\sinh(r)", "0763a4e21bae93349f9f2ae62bb860a9": "1 \\not\\in \\mathfrak{p}R", "0763cdc8f55ed814496308e5f4658f29": "\\Delta \\bar \\nu=0", "0763ce93e8a2656be17560c66cc1dbdb": "p_1(x),p_2(x),\\dots ,p_n(x)", "0763dc4fcdd12467c574a777667a3334": "\\scriptstyle g_k", "0763e9c6e4becc41e920c5126e588bbd": "r_{u,i} = \\frac{1}{N}\\sum\\limits_{u^\\prime \\in U}r_{u^\\prime, i}", "0763edab0eef820e06e1ef0f13de8655": "[X^n]f(X)=\\mathrm{Res}\\left(X^{-n-1}f(X)\\right).\\,", "076402d6722b347aecd7d25f2116f240": "H(X)=0", "0764226bffda183764c6261c01d7d892": "{dQ_h \\over dt} = F_h (C_{art} - {{Q_h} \\over {P_h V_h}})", "076456bd63a74f2491312e4ecf5a9863": "\\tilde{W}_t = W_t + \\frac{\\mu -r}{\\sigma}t,", "0764da0d400071e4a976e5804b264e13": "\\mathbf{P} \\big[ \\| B \\|_{\\infty} > c \\big] \\approx \\exp \\left( - \\frac{c^{2}}{2 T} \\right).", "076504e491bcaa11b7f41e85cdb74513": "14_{11} \\ ", "07654ab947447d79c73ad4a85cd17a9b": "C^{(p)}_T(p,T)=\\frac{C^{(V)}_T(V,T)}{\\left.\\cfrac{\\partial p}{\\partial V}\\right|_{(V,T)}} ", "0765f9cf030a6ef694d7f8e83a51c489": "\\limsup_{n \\to \\infty} \\frac{\\sigma(n)}{n \\ln \\ln n} = e^\\gamma,", "07661b12ffb58e544cdfe5d13653a383": "\\exp(x+y) = (\\exp x)(\\exp y)", "076673014ed0ddbe48a92b2b6617bf58": "\nq_{yy} = \\frac{\\sum (y-\\bar{y})^2 I(x,y)}{\\sum I(x,y)}\n", "0766a65e28d01fdaf213975cdc5bf122": "%C* = %C/6 \\mbox { for } %C \\ge 0.30% ", "0766ff40b4aee25394db43ccc72965d0": "b_1 \\equiv b_2 \\pmod{n}", "0767138437a99c249de200ae67000612": "a; A/\\alpha", "076745b8e5c00ecc8dfe0907506eb923": "\\mathbf{1}_A (\\omega) = 1 ", "076754254670c93760b74ca08d8d532f": " Af = \\lim_{t\\rightarrow 0} \\frac{T_tf - f}{t},", "076771b9e64ea0b1fc4b6722c4321ec3": "F_3(a, b) = a \\uparrow\\uparrow (b + 1)", "0767b602113fbc4116971faab83e9299": "n^a\\partial_a", "0767b7b70887925ed38002857226f4f2": "\\omega_2, \\omega_3, \\ldots, \\omega_\\omega, \\omega_{\\omega + 1}, \\ldots, \\omega_{\\omega_\\omega},\\ldots", "07681e835315bf610815cfbd4d1cc885": " P_{t+h}(S\\rightarrow S' | E) - P_{t+h}(S\\rightarrow S') < P_t(S\\rightarrow S' | E) - P_t(S\\rightarrow S')", "0768ca20f268ea2b57e7e1fcb1668c71": "P_{\\mu}", "076983fb4bd3ca6730ccfef16dd7e0f7": "|\\langle k | \\alpha \\rangle|^2", "076998aa7882ad367aa77901ac1e9418": "\\rho^{f}(\\vec{r})", "0769a79fb3a50f173aa673f653e7492c": "\\Im, \\imath, \\jmath, \\Bbbk, \\ell, \\mho, \\wp, \\Re, \\circledS \\!", "0769ad1292fe7abf0ed65594ac1b4b29": "\\lambda_{1,2}", "076a06c5db36dd4ea8b0b9a39e8f1be0": "T_f(\\varphi)=\\int_{\\mathbf{R}^n}f(x)\\varphi(x)\\,dx", "076a123ca2e849665659a2720ac00d24": "{p_D \\over p_B}", "076a3422506f570b6d2bb3beb560bcbd": "\\textbf{P}_0 \\textbf{X}_0 = \\textbf{P} \\textbf{H}^{-1} \\textbf{H} \\textbf{X} = \\textbf{P} \\textbf{X} = \\mathbf{x}", "076a38cdbb042d14cc1395be90288e60": "\\ V_r = R_0 I[1 + \\pi _L \\sigma _{xx} + \\pi _T (\\sigma _{yy} + \\sigma _{zz} )] ", "076a543427dc3dd3d85167d119e6f96b": "\\psi\\in\\mathrm{End}(A)\\otimes\\mathbb{Q}", "076a5d907fd387870f7cd559d302984b": " \\varphi_1 = \\underset{\\Vert \\mathbf{\\varphi} \\Vert = 1}{\\operatorname{arg\\,max}} \n\\left\\{\\operatorname{Var}(\\int_\\mathcal{T} (X(t) - \\mu(t)) \\varphi(t) dt) \\right\\}, ", "076a7392775093a4109f6d6823f82de7": "Pr(q,r^*)", "076a809092253b8be867c519f4fd165e": " Y = \\frac{1}{2} (c_{11} + 2c_{12}) \\left[ 3 - \\frac{c_{11} + 2c_{12}}{c_{11} + 2(2c_{44} - c_{11} + c_{12}})(l^2m^2 + m^2n^2 + l^2n^2) \\right]", "076a8a4792382ef936eb8d30c0344af3": " \\frac{S'(0)}{S(0)} = 1 ", "076a9536d872162b643b3c12af11462e": "OPT\\,\\!", "076ac46c2af2a73f18609004e868cc25": "\\mathbf{S_z} = \\left| \\mathbf{S} \\right| \\cos \\theta", "076ad1a05ab5a56fba3ab7c501c59247": "r_{U \\subset V} : \\Gamma(V, \\mathcal{F}) \\to \\Gamma(U, \\mathcal{F})", "076ad72025bf258254eabe7ff54d23bc": "\\scriptstyle f(x) = \\sum_{i=1}^n v_i(x)", "076b1020422cb14d6c78aa30d137e4d2": " | \\bar{\\psi} \\rangle = \\sum_{mm'} D^{(j)}_{m'm} | j , m \\rangle \\quad \\Rightarrow \\quad | \\bar{\\psi} \\rangle = D^{(j)} | \\psi \\rangle ", "076b261c5f14add8ce5c20f7d0f8e605": "\\mathrm{Conversion\\ rate} = \\frac{\\mathrm{Number\\ of\\ Goal \\ Achievements}}{\\mathrm{Visits}}", "076b5159c531e835b323a4a37292c8c7": "\\frac{\\alpha-1}{\\alpha+\\beta-2}\\!", "076bac1211cd22f8042a6388a18536da": "G^{ab}= 8 \\pi \\left( \\psi^{;a} \\psi^{;b} - \\frac{1}{2} \n\\psi_{;m} \\psi^{;m} g^{ab} \\right) ", "076bc6f390507ab9471bbdedb7a7a8a6": " t' = \\frac{t - {v\\,x/c^2}}{\\sqrt{1-v^2/c^2}}\\ ,", "076c0cdb2b88594f3c51810670c8fa00": "\\sigma_{L_x} \\sigma_{L_y} \\geq \\frac{\\hbar}{2} \\left| \\langle L_z \\rangle \\right|.", "076c1d84e4c601afa735e9f1981f1481": "\\gamma^{xy}\\text{ given }\\gamma, \\gamma^x\\text{ and }\\gamma^y", "076c5b7ef72f7fe5eef303c071611acf": "s^2=\n{n \\choose 2}^{-1} \\sum_{i < j} \\frac{1}{2}(x_i - x_j)^2 =\n\\frac{1}{n-1} \\sum_{i=1}^n (x_i - \\bar x)^2", "076c60af2c283a925276be8d9ac48463": "u=", "076ccb9743f01c5e8092b493693e4a3b": "\\hat\\gamma", "076cd69391958c8e34d1dec5b79ed219": "g = 2 \\, dx \\, dy.\\, ", "076d689f8e234ddda25328e7eb961ce4": " \\frac{1}{2} \\ddot{h}_{\\hat{\\theta}\\hat{\\phi}} = -R_{\\hat{t}\\hat{\\theta}\\hat{t}\\hat{\\phi}} = -R_{\\hat{r}\\hat{\\theta}\\hat{r}\\hat{\\phi}} = R_{\\hat{t}\\hat{\\theta}\\hat{r}\\hat{\\phi}} = R_{\\hat{r}\\hat{\\theta}\\hat{t}\\hat{\\phi}}\\ ,", "076daa1db6a90ca612866888bb311b54": "z_\\alpha", "076db37d0fffd46b9ed7bfebf9548faa": "\\boldsymbol\\Sigma^+", "076e00399c2caed64fb2f4b94294c74a": "W = w_{i,j} \\in[0, 1],\\; i = 1, . . . , n,\\; j = 1, . . . , c", "076e7d43cb533c3d6200edc7d18bb420": "-\\frac{\\eta}{(n-1)!}\\partial_\\xi^{n-1} n_\\eta(\\xi)", "076eb7575aeb3c05b0f49e0966d8c763": "\\frac{q^2}{g}\\left(\\frac{y_2-y_1}{y_1 y_2}\\right)=\\frac{1}{2}(y_2-y_1)(y_2+y_1).", "076ebbaefa05d01247e9acf77a3ddf30": "m_i(t)", "076eecfb3ad1c653758ea0bb69c7a261": "\\int_0^\\infty \\frac {x^{n-1}}{e^{x}-1}\\ dx=\\Gamma (n)\\zeta (n)", "076f0a34cc9c4868859a1c63fa5e21b3": "\nB_m(x,y) = \\sum_{p=0}^m \\binom{m}{p} x^p y^{m-p} \\sin ((m-p) \\frac{\\pi}{2}),\n", "076f1d8cf906ddd76befeee26faf6378": "x_2=-1\\,\\!", "076f447e151637a04c93d7729fada964": "W_i = W_i - (x_i - \\mathrm{nearHit}_i)^2 + (x_i - \\mathrm{nearMiss}_i)^2", "076f7053aa11ee2e25f239fe56837022": "(M, d)\\,", "076f79eb95affcfa5564c2e7447977c6": " \\neg ", "0770acf09851f96ea1970372b019313b": "\\varphi/n", "0770e4d2141ff0645d384dfa5cba212a": "\\bar B(\\mathbf x_1,\\theta\\|\\mathbf h_0\\|)\\subset\\bar B(\\mathbf x_0,t^*)", "07710b5c43702a8bb7b9104eacc6ba71": "\\Gamma", "0771127c2a542f877110010002618345": "\n \\mathcal{L}(\\theta\\,|\\,x_1,\\ldots,x_n) = f(x_1,x_2,\\ldots,x_n\\;|\\;\\theta) = \\prod_{i=1}^n f(x_i|\\theta).\n ", "077135ee6dd7495ef979732cb77841ad": "k=1,\\dots,K", "07713f465efe7e23092a0ccf5c5e1dfa": "p \\Leftrightarrow q", "0771aa167b47724b08e1cd2007e2521d": "2~\\mu~u_r\\, ", "0771b835887c341bf6a6031dacb3cdfb": "\\eta(\\text{Earth},\\text{Be-Ti})=(0.3 \\pm 1.8)\\times 10^{-13}", "0771f479546178c069c0e216ed1286da": "X= \\left ( - \\infty, \\infty\\right )\\times \\mathbb{R}^{L-1}_+", "0772a4cddee27cf4d09e462427329124": "\\widehat{D}=\\mathcal{A}(D)\\,\\!", "0772f3768a0d5aec0631e82c594a8f1c": "K=\\frac{1}{N} \\sum_{r=0}^N \\left ( Q(t_1)Q(t_2)+Q(t_2)Q(t_3)\n-Q(t_1)Q(t_3) \\right )_r. ", "077375f6e49036ecf60254cb3db53b04": "d \\approx 1.22\\sqrt{h} \\,.", "0773a7f0d6ea3e3de6772d5011308740": "s_1,\\ldots,s_{\\ell}", "0773fb4ec69cb402146dee39c7dfe67c": "\\Phi(\\lambda x) = \\bar{\\lambda} \\Phi(x)", "07746dfcc964e0a44b73ae010adb47b6": "D_3 \\bar R", "0774d16de8a5ea7ec5f8cedcce770371": "f(x,y) = \\frac{xy}{x^2+y^2}.", "0774eb853e768d8440a5decf37352564": "\\Phi_x(t) := \\Phi(t,x)\\,", "0774f8878b7656441abeba8fcead99c1": "p(x)=\\sum_{i=0}^{n}y_i\\cdot\\prod_{0\\leq j\\leq n,j\\neq i}\\frac{x-x_j}{x_i-x_j}.", "0775088d699e8f8e39ed21939e1ea35f": "var(\\epsilon)=\\sigma^2I ", "077517b02a759ac0d88726961e1cbf62": "BGL(R)", "077542d71239fe9a28c0b81e9b943e9c": " RDF = 100 {(p-n)\\over p} ", "077548709960a44c68ae0ec66cd7c4e0": "C \\in \\mathcal{B}_B ", "07755fc3d3fd4a2a1a25d203fc4333b7": "\\forall x. P(x) \\Rightarrow Q(x)", "0775c495f3d2b269d0366b56aa4b64ce": " [\\mathbf{t}]_{\\times} = \\mathbf{V} \\, \\mathbf{W} \\, \\mathbf{\\Sigma} \\, \\mathbf{V}^{T} ", "0775c8c1c41841b9a7e8f6ccc78cc54c": "u_n \\in \\mathrm{ker}(e_i) \\cap M_{\\lambda + n \\alpha_i}", "0775e8e8c04f8675f218725931c50dca": "x_i w_{ji} \\,", "077602586c235d52628faee997c58f2c": "\\pi+2k\\pi=\\theta_1+\\theta_2+\\cdots+\\theta_n-(\\phi_1+\\phi_2+\\cdots+\\phi_m) ", "07766e12ed7514e4f6a97b128fa3d24d": "\\sqrt{2} =\n\\prod_{k=0}^\\infty\n\\frac{(4k+2)^2}{(4k+1)(4k+3)} =\n\\left(\\frac{2 \\cdot 2}{1 \\cdot 3}\\right)\n\\left(\\frac{6 \\cdot 6}{5 \\cdot 7}\\right)\n\\left(\\frac{10 \\cdot 10}{9 \\cdot 11}\\right)\n\\left(\\frac{14 \\cdot 14}{13 \\cdot 15}\\right) \\cdots", "0776dd788bfa2eac06807230d12a51eb": " 2n^2+1 ", "077710d78d9aaf5f53a613b973a43e3f": "\\{|a_i b_j \\rangle\\}", "07776e5b55dbf4f4afc26776e279ba4e": "[S_z,S_x]=i\\hbar S_y", "077771a16d7f24cafdc89a59245a3b16": " p_r = m \\dot r \\ , \\ p_{\\theta} = mr^2 \\dot{\\theta}\\ , ", "077823f1d2b7790b923e2f811f3a50ee": " \\mathbf{W} \\, \\mathbf{\\Sigma} = \\begin{pmatrix} 0 & -s & 0 \\\\ s & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} ", "07782a7e6670123f9540cfe2cb646d3e": " F(a) = \\langle F_a\\mid F\\rangle = \\pi^{-n} \\int_{C^n} \\kappa(a,z)F(z)\\exp(-|z|^2)\\,dz,", "0778bf59ee2c9fae1d1070944026249e": "x:=?\\,\\!", "0778c85c734c50af0af297e0fc1c0d61": " Q_k ", "0778ca1c5f7b9cce86dd69e5f2287181": "u_{mn}(r, \\theta, t) = \\left(A\\cos c\\lambda_{mn} t + B\\sin c\\lambda_{mn} t\\right)J_m\\left(\\lambda_{mn} r\\right)(C\\cos m\\theta + D \\sin m\\theta)", "07790af7823cca418e70823c74f70d21": "(\\hat{b}^\\dagger)", "077932dfe335eb11060da12290cd69e7": "f_c = \\frac{1}{2\\pi RC}", "0779386a9cd0a46f0327a20912ccd06e": " G: L^2(\\mathcal{T}) \\rightarrow L^2(\\mathcal{T}),\\, G(f) = \\int_\\mathcal{T} G(s, t) f(s) ds. ", "0779458e5cb9f831cdfa2b3b7e9553e2": "m,n", "07796dea4d8c0505afef310d2328f6ec": "\\textbf{x}(k+1) = A\\textbf{x}(k) + B\\textbf{u}(k)", "0779870da1d0900176882d27b80a4d7b": "\\textstyle \\lim_{r \\rightarrow \\infty} g(\\mathbf{r}) = 1", "0779e058d2e7c0d16b352bdd33457f3b": "\\begin{align}\n f_{\\theta_1}=f_{\\theta_2}\\ \n &\\Leftrightarrow\\ \\tfrac{1}{\\sqrt{2\\pi}\\sigma_1}e^{ -\\frac{1}{2\\sigma_1^2}(x-\\mu_1)^2 } = \\tfrac{1}{\\sqrt{2\\pi}\\sigma_2}e^{ -\\frac{1}{2\\sigma_2^2}(x-\\mu_2)^2 } \\\\\n &\\Leftrightarrow\\ \\tfrac{1}{\\sigma_1^2}(x-\\mu_1)^2 + \\ln \\sigma_1^2 = \\tfrac{1}{\\sigma_2^2}(x-\\mu_2)^2 + \\ln \\sigma_2^2 \\\\\n &\\Leftrightarrow\\ x^2\\big(\\tfrac{1}{\\sigma_1^2}-\\tfrac{1}{\\sigma_2^2}\\big) - 2x\\big(\\tfrac{\\mu_1}{\\sigma_1^2}-\\tfrac{\\mu_2}{\\sigma_2^2}\\big) + \\big(\\tfrac{\\mu_1^2}{\\sigma_1^2}-\\tfrac{\\mu_2^2}{\\sigma_2^2}+\\ln\\sigma_1^2-\\ln\\sigma_2^2\\big) = 0\n \\end{align}", "077a4a4dd8cee9a21236f43448023670": "\\nabla_{[a} R_{bc]d}^{\\ \\ \\ e} = 0", "077a72a276d6314aafe95c03f531b013": "g(x_1,x_2,x_3,v_1,v_2,v_3)\\,", "077a92483fd1b690cd5582c27fd5e27b": "k \\, \\frac{p_k}{p_{k-1}} = ak + b,", "077ac601e604a3663daf0fef57d18fb7": "\n\\begin{align}\nd^4\\sigma &=\n\\frac{Z^2\\alpha_{fine}^3c^2}{(2\\pi)^2\\hbar}|\\mathbf{p}_+||\\mathbf{p}_-|\n\\frac{dE_+}{\\omega^3}\\frac{d\\Omega_+ d\\Omega_- d\\Phi}{|\\mathbf{q}|^4}\\times \\\\\n&\\times\\left[-\n\\frac{\\mathbf{p}_-^2\\sin^2\\Theta_-}{(E_--c|\\mathbf{p}_-|\\cos\\Theta_-)^2}\\left\n(4E_+^2-c^2\\mathbf{q}^2\\right)\\right.\\\\\n&-\\frac{\\mathbf{p}_+^2\\sin^2\\Theta_+}{(E_+-c|\\mathbf{p}_+|\\cos\\Theta_+)^2}\\left\n(4E_-^2-c^2\\mathbf{q}^2\\right) \\\\\n&+2\\hbar^2\\omega^2\\frac{\\mathbf{p}_+^2\\sin^2\\Theta_++\\mathbf{p}_-^2\\sin^2\\Theta_-}{(E_+-c|\\mathbf{p}_+|\\cos\\Theta_+)(E_--c|\\mathbf{p}_-|\\cos\\Theta_-)} \\\\\n&+2\\left.\\frac{|\\mathbf{p}_+||\\mathbf{p}_-|\\sin\\Theta_+\\sin\\Theta_-\\cos\\Phi}{(E_+-c|\\mathbf{p}_+|\\cos\\Theta_+)(E_--c|\\mathbf{p}_-|\\cos\\Theta_-)}\\left(2E_+^2+2E_-^2-c^2\\mathbf{q}^2\\right)\\right]. \\\\\n\\end{align}\n", "077af31100254b1688d4e5fd515e4f7f": "p(z)=z^3-2z+2", "077b4344e9a2a6f67524501d81bb5f28": "U^2 V^2 + V^2 W^2 + W^2 U^2 - U V W = 0\\,", "077b4d6796ac247752630a652ee7d6a1": "P_2 = 1 - \\frac{\\mathrm{non}\\,\\mathrm{outs}}{\\mathrm{unseen}\\,\\,\\mathrm{cards}} \\times \\frac{\\mathrm{non}\\,\\mathrm{outs} - 1}{\\mathrm{unseen}\\,\\,\\mathrm{cards} - 1}", "077b7b8d72ef0127c406eb30d6fd8ef1": "\\left\\{a_m\\right\\}", "077bd014b6feaae19ad50c0177a1bc11": "\\begin{matrix}\n & & \\text{S} & \\text{E} & \\text{N} & \\text{D} \\\\\n + & & \\text{M} & \\text{O} & \\text{R} & \\text{E} \\\\\n \\hline\n = & \\text{M} & \\text{O} & \\text{N} & \\text{E} & \\text{Y} \\\\\n\\end{matrix}", "077c2aa89f728f5a7458d3ded7baa4da": "\n\\mbox{If } \\left(\\frac{\\alpha}{\\mathfrak{a} }\\right)_n =1\n\\mbox{ then }\\alpha \\mbox{ may or may not be an }n\\mbox{-th power}\\pmod{\\mathfrak{a}}.\n", "077c3744ad2e8cfbdfe217cd4e9d3701": "c_0/c_1-1", "077c62fef3fdef49e8c72e2bc9d15eef": "\\tilde{P}=[p_0(x),p_1(x),...,p_{n-1}(x)]^{T}", "077c65bdfc1bffff22f8b60fdd146757": "f \\cdot u = -g\\frac{\\partial P / \\partial y}{\\partial P / \\partial z} = -g{\\partial Z \\over \\partial y}", "077cb05b34f5d3c3f31ce0fb5c6df809": " F(x) = \\underline{\\int_{a}^{x}} f(x) \\, dx ", "077dcdb31d7ed1a0d801f2a2f0710163": "O(n^2\\ln ^{O(1)} n)", "077dea9a5816ff7eeb169b7d7389645b": "\\rho = \\frac{(1+w)G_s\\rho_w}{1+e}", "077e3451bc1e12c3eca2b923844ba92d": "\\nabla'\\left(\\frac{1}{|\\bold{r}-\\bold{r}'|}\\right) \\equiv \\left(\\bold{e}_x \\frac{\\partial }{\\partial x'} + \\bold{e}_y\\frac{\\partial }{\\partial y'} + \\bold{e}_z\\frac{\\partial }{\\partial z'}\\right)\\left(\\frac{1}{|\\bold{r}-\\bold{r}'|}\\right) = \\frac{\\bold{r}-\\bold{r}'}{|\\bold{r}-\\bold{r}'|^3}", "077e453f76eadfc3ba6d5ee80f19e450": "n = q^k", "077e8355e57bb087ba0e3b627effb8d9": "H(Y|X) = H(Y)", "077e8ade7d473a4697aa61af200f4028": "P^\\alpha g_{\\alpha\\beta}P^\\beta = (m_0 c)^2\\,.", "077ea541cceaa0a121fc034ed98a3c9d": "[\\mathcal{T}_n(f)](x)=ne^{-nx}\\sum_{k=0}^\\infty{\\frac{(nx)^k}{k!}\\int_{k/n}^{(k+1)/n}f(t)\\,dt}", "077ee94ac9eb64530fe8724b2686b402": "2\\sqrt n", "077f0f6b89b5079ea8703b3d7e3cb630": "\\displaystyle{W(x)e^{y} = e^{-\\|x\\|^2/2} e^{-(x,y)} e^{x+y}.}", "077f1307be73c6ab24fb82ff7ebd5c91": "j_Y", "077f143f43b17d11829c03ee32cd3131": " \\operatorname{Ran}(F_1)\\times\\cdots\\times \\operatorname{Ran}(F_d) ", "077f15fc341fcb199dedeafc83810044": "\\frac{\\omega - \\omega_0}{\\omega_0}\\thickapprox \\frac{(\\bar{w_m}-\\bar{w_e})\\cdot\\Delta V}{W}\\,", "077f44a5f5abf839b2b85486f6563533": "\\mathrm{Be} = \\frac{\\Delta P L^2} {\\mu D} ", "077f64801fcff049ad6261e6f58614ec": "\\Delta^k(a_n) = \\Delta^{k-1}(a_{n+1}) - \\Delta^{k-1}(a_n)=\\sum_{t=0}^k \\binom{k}{t} (-1)^t a_{n+k-t}", "077f689e2c97906d8fbb172af3100753": "\\hat{g}_n(u_n)", "077f6952e7a9fe28ddf1404e5a7438fd": " P \\left ( \\mathbf{x} \\right ) = \\int P \\left ( \\mathbf{x} \\land y \\right ) \\, dy = {1 \\over N} \\sum_{i=1}^N \\, \\rho \\big ( \\left \\Vert \\mathbf{x} - \\mathbf{c}_i \\right \\Vert \\big )", "077f998032173465d265b3726ab970b5": "\\mbox{Pin}_+(n) := \\mbox{Pin}(n,0) \\qquad \\mbox{Pin}_-(n) := \\mbox{Pin}(0,n)", "077fc096705bdbf04e4df374dad720c2": "a_c=0.81\\,\\;", "078010d409b4f9155fa3603c1d3468cd": "\\mbox{Intensity} \\ \\propto \\ \\frac{1}{\\mbox{distance}^2} \\, ", "07804a56b1ff9142d02458700ef10b57": " \\vdash A \\and B ", "0780928654b8d68408912cd0ab7411e7": "P-Q \\succeq 0", "0780a1a50544e424461bf141635dcc29": " B_{0} ", "0780a8f12b33e6d16ac035270fbdc88c": "I = \\sum_{n=1}^\\infty \\varphi_n \\varphi_n^\\dagger, ", "0780f3729d0863e7a57fbf560ae42288": "\\vdash \\neg (p \\land \\neg p)", "078103def5c3c8da420e74d0665bfaca": " BC \\to D", "07813d85abecae8553a456cde7fdc47c": "j(\\tau) = {1 \\over q} + 744 + 196884 q + 21493760 q^2 + 864299970 q^3 + 20245856256 q^4 + \\cdots", "078229fb92933bbd30db6b783ddc6550": " d_{\\mathrm H}(X,Y) = \\max\\{\\,\\sup_{x \\in X} \\inf_{y \\in Y} d(x,y),\\, \\sup_{y \\in Y} \\inf_{x \\in X} d(x,y)\\,\\}\\mbox{,} \\! ", "0782c5509849d7802777fd9ae857bdaa": "a(Z)=2^{1-A}(1+Z)^A", "0782fd67789ce951f272d19a88258fa1": " \\delta V = \\delta W_{H,i} + \\delta W_{H,p} - \\delta W_{g,p} + \\delta W_{\\sigma} ", "0782ffe56254b5123a769b1618afe23f": "u_0 = 0 \\, ", "078376930c9985774961ee63c5615a07": "p(X)", "0783c56c8c7d602538ef5970f4d8e594": " L(p) ", "078401532866081dddda6744220b4c75": "EP", "07846d31b4a7947cf5fcaf5185ba4b65": "10^{10^{10^{10^{10^{1.1}}}}}", "07847ca5764ba07964bfa208eee3d90e": "\\operatorname{E}[g(X)] = \\int_a^\\infty g(x) \\, \\mathrm{d} \\mathrm{P}(X \\le x)= \\begin{cases} g(a)+ \\int_a^\\infty g'(x)\\mathrm{P}(X > x) \\, \\mathrm{d} x & \\mathrm{if}\\ \\mathrm{P}(g(X) \\ge g(a))=1 \\\\ g(b) - \\int_{-\\infty}^b g'(x)\\mathrm{P}(X \\le x) \\, \\mathrm{d} x & \\mathrm{if}\\ \\mathrm{P}(g(X) \\le g(b))=1. \\end{cases}", "07849574dc0001eece45c0e0b066167d": "\\int\\frac{x^4\\;dx}{s}\n= \\frac{x^3s}{4}+\\frac{3}{8}a^2xs+\\frac{3}{8}a^4\\ln\\left|\\frac{x+s}{a}\\right| ", "0784bbbafde6513ff77e1888c5fa441a": "\\mathbf{e}_2 ", "0784be20ef0e37eb46ebd36b4a2bf6dd": "\\scriptstyle 1-\\varphi=\\frac12(1-\\sqrt5)", "078526a9ab20c4b04eeb2175fde4a01e": " q^{(t)} = \\operatorname*{arg\\,max}_q \\ F(q,\\theta^{(t)}) ", "0785af11f2b9847cae144679ba6ece53": "\\oint \\frac{\\delta Q}{T} = 0", "0785e11419e998a63a3705b8a7bc84e7": "y=\\operatorname{sign}{f(x)}", "0785e7cdb8333390991176dd4ca77445": "c_{\\pm}=\\frac{c}{1\\pm\\kappa}", "07863730e8e14f03f5853cff18ee08cc": "r^2 - \\frac{2r_0}{r_0^2 - a^2} r\\cos(\\theta-\\theta_0) + \\frac{1}{r_0^2 - a^2} = 0.", "0786b3d3ff8cb066d85837afa952dd2e": "{}^{13}_{7}\\text{N}\\to{}^{13}_{6}\\text{C}", "0787060eb68af7205e261a6d1513aa89": "\\alpha < 1", "078757e10e62f005ec259835c931b771": "\\Delta H_c^\\circ", "07876a8567eb7d684ce5172df8bd487f": "\nd_2 = \\begin{bmatrix}\n-y\\\\\nx\\\\\n\\end{bmatrix}.\n", "07877f1626ec59a1150e271a430efbe4": "M\\,ds\\sqrt{v}", "07878fc6547acfb9bcc8a958e91d7bc3": " c(M,N) = \\left({ \\sum_{k=0}^{N-1} \\binom{M+k}{k} 2^k }\\right)^{-1} \\ . ", "0787dc3b9e4bf3484368a3902c5bbced": "\n \\lambda_{\\mathrm{chain}} = \\sqrt{\\tfrac{I_1}{3}} ~;~~ \\beta = \\mathcal{L}^{-1}\\left(\\cfrac{\\lambda_{\\mathrm{chain}}}{\\sqrt{n}}\\right)\n ", "07885fa4c2e009921cc1f7ebc938cb6a": "c_{A},c_{B} \\in [0,1]", "078869f7e8fdd24930b6b5e77b36dacb": "\\epsilon/\\epsilon_0", "078889072a75e391e732a9144a555c3c": "\\langle x^2 \\rangle =\\int_{-\\infty}^{\\infty}x^2\\frac{1}{\\sqrt{2 \\pi}}e^{-\\frac{x^2}{2}}=1", "0788abe959b8ff1f7c3071845bdc6a6d": "\\Gamma(a,z) \\sim z^{a-1}e^{-z}\\left(1+\\frac{a-1}{z}+\\frac{(a-1)(a-2)}{z^2}\\dots\\right)", "0789a67bdd33013a802f662f3980e22b": "(-a, 0)", "078a398ba2ed0731db1da302aacf0209": " \\zeta(-m, \\beta )-\\frac{\\beta ^{m}}{2}-i\\int_ 0 ^{\\infty}dt \\frac{ (it+\\beta)^{m}-(-it+\\beta)^{m}}{e^{2 \\pi t}-1}=\\int_0^\\infty dp \\, (p+\\beta)^m ", "078a766704afcaa594a3832203bea1cd": "A^{(a-1)/2}\\equiv +1 \\pmod a\\;", "078b07ae6be73b8e120a54b2632b6e41": "H(x^*(t),u^*(t),\\lambda^*(t)) \\equiv \\mathrm{constant}\\,", "078b8bd0aad721cae6a101460fff3766": " \\mathbb P\\big( \\|X - \\mu\\|_\\alpha \\ge k \\sigma_\\alpha \\big) \\le \\frac{ 1 } { k^2 }", "078b98585c52531242818511c6b154bf": "i_1< i_2< \\cdots < i_k", "078baafe2b532e3c9745aa0789297179": "S(\\mathbf{x}) = (\\mathbf{x} \\cdot \\mathbf{d})^2 - (\\mathbf{d} \\cdot \\mathbf{d}) (\\mathbf{x} \\cdot \\mathbf{x}) (\\cos \\theta)^2", "078bb829a1a71cebb8d843d700069253": "\n\\gamma^\\mu p_\\mu \\Psi = 0.\n", "078beef5c9bced4bfe33725d080816f6": "\\vec x= \\vec p + t\\vec r", "078c4da1f294a6312e132f7f0f2c0233": "\\mathbf{X} = \\{\\mathbf{x}_1,\\dots,\\mathbf{x}_n\\} \\sim \\mathcal{N}(\\boldsymbol\\mu,\\boldsymbol\\Sigma)", "078c510cd2d8847c528ea20d6ae1545d": "10^{10}{\\rm cm}", "078c521870ecf21cba86dc7d86a934e4": "\n\\frac{1}{r} = \\frac{1}{b} \\cos\\ (\\theta_1 - \\theta_0)\n", "078c555e9b1b83a63fc1985055b8307c": "b_n = r_n \\sin \\left( \\varphi_n \\right) = - \\frac{2}{T} \\int_{t_0}^{t_0+T} x(t) \\sin(2 \\pi n f_0 t) \\, dt \\ ", "078ca34fedc623cdec4dd1b75dbf9bfc": "\\mathbf{K}\\cdot\\mathbf{R}=2\\pi(k_1 n_1+k_2 n_2+k_3 n_3)", "078cc2efaf037b682afb955a44393851": "g_{s s} = 1 - \\frac{2 \\Phi}{c^2} \\,", "078ce003ac87485b8e59e9ea6fc52c96": "H^i(V, \\mathbb{Z}/\\ell^k\\mathbb{Z})", "078d9fb579d5120de13059b860540f3a": "z_0\\in\\mathbb{C}", "078dcac09b5c6acab007d0b51613421a": "\nL_i = \\int_x (\\psi^\\dagger\\psi)(x_1)(\\psi^\\dagger\\psi)(x_2)\\cdots(\\psi^\\dagger\\psi)(x_n) V(x_1,x_2,\\dots,x_n).\\,", "078dd444551348ea3f6b37f4ec305d38": "Z_{I2} = \\sqrt{\\frac{DB}{CA}}", "078df1ac9e620524e7af4fce16563234": "f_1\\mbox{ }:\\mbox{ }f_2", "078e05545233258bfef3bf484c87874b": "\\frac{\\partial (x,y,z)}{\\partial (\\rho, \\theta, \\phi)} =\n\\begin{vmatrix}\n\\cos \\theta \\sin \\phi & - \\rho \\sin \\theta \\sin \\phi & \\rho \\cos \\theta \\cos \\phi \\\\\n\\sin \\theta \\sin \\phi & \\rho \\cos \\theta \\sin \\phi & \\rho \\sin \\theta \\cos \\phi \\\\\n\\cos \\phi & 0 & - \\rho \\sin \\phi\n\\end{vmatrix} = \\rho^2 \\sin \\phi", "078e342007a7505c9ea9195cd363fe6c": " \\scriptstyle{a=-\\infty} \\ ", "078e3be4ee9973438a1776ec10d0ade5": "(\\neg A\\to B)\\to((B\\to A)\\to A)", "078e3fc4e2dfff81082c2a4a3078b5d7": "\n\\begin{align}\n \\eta &= \\varepsilon\\, \\eta_1 + \\varepsilon^2\\, \\eta_2 + \\varepsilon^3\\, \\eta_3 + \\cdots ,\n \\\\\n \\Phi &= \\varepsilon\\, \\Phi_1 + \\varepsilon^2\\, \\Phi_2 + \\varepsilon^3\\, \\Phi_3 + \\cdots \n \\quad \\text{and}\n \\\\\n \\mathbf{u} &= \\varepsilon\\, \\mathbf{u}_1 + \\varepsilon^2\\, \\mathbf{u}_2 + \\varepsilon^3\\, \\mathbf{u}_3 + \\cdots .\n\\end{align}\n", "078e4b138b54054ddd240d6fd58adba9": " e^{-1/T^{1/2}} ", "078edaa1a5ec22034aefa57ab5f8842c": "I=(a,b],\\ a\\geq -\\infty \\ ", "078f15812c1e340a21f34ec87b774c11": "T\\subseteq\\lambda\\cdot K", "078f15817acfc96b8b6ae79c436756a6": "\n\\begin{align}\n\\omega_r &= {1 \\over r\\sin\\theta}\\left({\\partial \\over \\partial \\theta} \\left( v_\\phi\\sin\\theta \\right) - {\\partial v_\\theta \\over \\partial \\phi}\\right) \\boldsymbol{\\hat r}, \\\\\n\\omega_\\theta &= {1 \\over r}\\left({1 \\over \\sin\\theta}{\\partial v_r \\over \\partial \\phi} - {\\partial \\over \\partial r} \\left( r v_\\phi \\right) \\right) \\boldsymbol{\\hat \\theta}, \\\\\n\\omega_\\phi &= {1 \\over r}\\left({\\partial \\over \\partial r} \\left( r v_\\theta \\right) - {\\partial v_r \\over \\partial \\theta}\\right) \\boldsymbol{\\hat \\phi}.\n\\end{align}\n", "078f414851ca4081636b3f54e7c19cab": "R_n(a_1,a_2,\\cdots,a_l)=(n+1-a_l,n+1-a_{l-1},\\cdots,n+1-a_1)", "078f6ad03d97c8307ecc5627f56b638e": "\\mathbf{g} = \\hat{\\mathbf{e}}\\tan\\left(\\frac{\\theta}{2}\\right)", "078f8b1d87a03db927e913543c17bca6": "\n\\delta = \\ln \\frac{R}{r}\n", "078f902f77268415c8f69289c814149a": " L =\\text{De Jan} \\text{ s}\\ddot{\\mathrm{a}}\\text{it} \\text{ das} \\text{ mer} \\text{ (d'chind)}{}^m \\text{ (em} \\text{ Hans)}{}^n \\text{ es} \\text{ huus} \\text{ h}\\ddot{\\mathrm{a}}\\text{nd} \\text{ wele} \\text{ (laa)}{}^m \\text{ (h}\\ddot{\\mathrm{a}}\\text{lfe)}{}^n \\text{ aastriiche.} ", "078fb69b545281d66be48dbd263cc910": " a = \\frac{p_1 - 1}{2} ", "078fe6201a479ae75ab26179290fcbf0": "\\sigma^2 =a_1+4a_2", "0790293fc4b123eb0f36eb0a6d78ea3c": "\\operatorname{pos}(U \\cup V) = \\max \\left( \\operatorname{pos}(U), \\operatorname{pos}(V) \\right)", "07902f881ffe6a02162aa07d4c1dfa04": "\\mathfrak{m}_1", "07904b4de470b155735daa96f3c8d4d5": "\\,\\Delta\\mathbf{w} ~ = ~ \\eta\\, y(\\mathbf{x}_n) \\mathbf{x}_{n}", "0790767d36e3688ba6d24c1c9fe61aba": "\n [\\boldsymbol{\\nabla}f(\\mathbf{x})]\\cdot\\mathbf{c} = \\cfrac{\\partial f}{\\partial q^i}~c^i = \\left(\\cfrac{\\partial f}{\\partial q^i}~\\mathbf{b}^i\\right)\n \\left(c^i~\\mathbf{b}_i\\right) \\quad \\Rightarrow \\quad \\boldsymbol{\\nabla}f(\\mathbf{x}) = \\cfrac{\\partial f}{\\partial q^i}~\\mathbf{b}^i\n ", "0790a1d764c4d067a8c698fb27272740": "V_i = M_i \\oplus E(M_i,A_i)", "0790e02160593a946b326cbd8a217cd6": "R_{ab}=8\\pi T_{ab}", "07910a4e5d2ec75aaad3b352e1d43829": "2^a3^b\\rho\\ge3", "079176c8e29d1f917b0d00985676d0bc": "M=\\left(\\widetilde{B}\\times F\\right)/\\pi_1B", "0791905f3c905a3bba5b31afbd79ea51": "(-\\infty,\\infty)", "0791a66e82c58f903c9ce378da46b6f3": "\\frac Rr\\ne \\frac{R^*}{r^*}", "0791b56c99a17e19785117b3ec8dac89": "f(x) = \\frac{(a/b)^{p/2}}{2 K_p(\\sqrt{ab})} x^{(p-1)} e^{-(ax + b/x)/2},\\qquad x>0,", "0791d0fd295994201f869fda975930d9": " \\omega \\,", "07924475c362d9ef2d50b8d84ec89d17": "\n p := -\\tfrac{1}{3}\\,\\text{tr}(\\boldsymbol{\\sigma}) = -\\frac{\\partial W}{\\partial J} = -2 D_1 (J-1) \\,.\n ", "0792dac93251d1a2c473f86682ca24a0": "T;Y \\,", "0792f1b63be29965d4194e3a82d304db": " f(x) = x^n+f_n x^{n-1}+...+f_1 \\in \\mathbb{Z}[x] ", "0792fbc0396ce564588008d7ddbac637": "\nQ=\\begin{pmatrix}\n0&0&-1\\\\\n0&J&0\\\\\n-1&0&0\n\\end{pmatrix}\n", "07935152f03e472299580aa15bb39322": " J \\, ", "0793554326d422491f32938dcc782c52": " p\\ K\\ (p\\ K) = K\\ (p\\ K\\ (p\\ K)) ", "07938ce9284d4ca381250ac22878f214": "\\mathbf{D} \\cdot {\\rm d}\\mathbf{A}", "0793f3f0fa1dc48a83b18ef3711fa48f": "\\Gamma(x),", "079407b01126de398d868fa81c01c73f": "\\{b_k\\}", "07942101007b8ecb11dc153c5c8c0da5": "\\Delta\\,T_m(x)=T_{mB}-T_m(x)=T_{mB}\\frac{4\\sigma\\,_{sl}}{H_f\\rho\\,_sx}", "079435779e8a9f09354627bca21b554d": "R(x_1,\\dots,x_n,f(x_1,\\dots,x_n))", "0794e8f6490e2236c2f899d1756f19ad": " u_3 ", "07952402115e4cd280dcd06fa9794ca5": "E[X_i=H\\mbox{ k out of n times}]=\nP(k,n)={n\\choose k} p^k (1-p)^{n-k}", "07953e6ef895cfce3fa72999ffa6d9c3": "(\\mathbf{A} + \\mathbf{A}^{\\rm T})\\mathbf{x}", "0795436aa1f00dc612d66a8ad74c0197": "r'\\,", "079626a29686af62428f258cbba09efe": "\\lambda=+1", "07974ea99dfe454eefeb4bcaa6083fc0": "g_m \\ r_O = \\begin{matrix} \\frac {I_C} {V_T} \\frac {V_A +V_{CE}} {I_C} \\end{matrix} = \\begin{matrix} \\frac {V_A +V_{CE}}{V_T} \\end{matrix} ", "0797d059b1316aa1f391bf60cc948b64": "(r+1)", "0797e4a661c4bb58bf65e11bc7e8fbaf": "X = (x_1, \\dots, x_n)'", "0797eddcd37cbbdbe182b2997e67186e": "f(x,y)=f(x,y+2\\pi)", "07981a091595e82db542568bb13f4064": "\\bar{A}^f_n = \\left[a A^f + (1-a) A_n^f\\right],", "079898aa8d94522b14e48505abb4231e": "v_{(G; c)}(\\{3\\})=7", "0798c843ba97cfbf97fd46dc4183c6b5": "\\overline{f}", "0798e8918be95124f6a68db54fa66f23": "MTTF = Aj^{-n} e^\\left(\\frac{Q}{kT}\\right)", "07994e43fce43f532bfe46704d0a6b30": "5 \\cdot 0 = 5 \\cdot 2 = 5 \\cdot 4 = 5 \\cdot 6 = 5 \\cdot 8 = 0 \\mod 10", "079958cd8a0faf47e193c20057f5d768": "\\{f^{(0)}_n (x)\\}", "079966c92fd1bea34a1e75a0ac35821c": "h[n]={{\\delta[n-1] + 2\\delta[n] - 3h[n-1]} \\over {4}}", "0799a469e665882648f757c5c7d455dd": "0, \\, 1=\\omega^0, \\, \\omega=\\omega^1, \\, \\omega^\\omega, \\, \\omega^{\\omega^\\omega}, \\, \\ldots \\,.", "0799a79811856765395d37a3606f5fad": "p = \\alpha\\overline{\\alpha}", "0799e0fde4ea45bd6f223e49b942fb8d": "\\partial \\subset P_n", "079a7ee3ed6ebc9231495b76ba70762c": "x_\\text{max}=\\frac{X_\\text{max}-X_0}{\\lambda}", "079b3039d1a9af18b3838c740a61d3f1": " \\qquad x_{n+1} = (\\epsilon)[r x_n (1-x_n)]_s + (1-\\epsilon)[r x_n (1-x_n)]_{s-1} ", "079b3b1d62f3e2cebd960448cef8350e": " \\Omega =\\arccos { {n_x} \\over { \\mathbf{\\left |n \\right |}}}\\ \\ (n_y\\ge 0);", "079b548a20f175cb786037c41c5a772e": "\\Delta G_i= \\sum_{j}\\gamma_j O_j~", "079bb003772b9993669167d6f942560e": " P( X < k ) = 0 \\text{ if } E( X ) > k \\text{ and } E( X^2 ) < k E( X ) + M E( X ) - kM ", "079c28d5bb1d3579df78a4ba94b7bc0b": "\\rho(\\vec{r}),\\, r\\, \\epsilon\\, \\reals^3", "079ce6bab2c2c49ca08fc38927b65c14": "\\{|\\psi_i\\rangle\\}_{i = 0,1,2, \\dots}", "079cef4b5a0aee68224afe49dee3806a": "\\int \\mathbf{1}_B \\, |f| \\ d\\mu = \\int_{B} |f(y)| \\, d\\mu(y) > \\lambda \\, \\mu(B).", "079d04bd53369a885a4b28fc72759de9": "f \\in k(x)", "079d0c77d14321410f742b4e6723f265": "\n y = 1.9,\\ 3.7,\\ 5.8,\\ 8.0,\\ 9.6\n ", "079d1298dc1aa97fb05d3e31a34e99ba": "P(x,y)=\\alpha A_{ji}/k_i", "079d1dd1a6bfa359c20a0f0f86b95244": " \\operatorname{let} x : x\\ f = f\\ (x\\ f) \\operatorname{in} x ", "079d3a60adc0db20d1548a37a9f64798": " \\binom{n-1}{n-x}.", "079e0906f54cec5f50d68cef26dcec24": "\n \\mathbf{M}_x = \\int_A \\left(-y\\sigma_{xx}\\mathbf{e}_z + y\\sigma_{xz}\\mathbf{e}_x + z\\sigma_{xx}\\mathbf{e}_y - z\\sigma_{xy}\\mathbf{e}_x\\right)dA =: M_{xx}\\,\\mathbf{e}_x + M_{xy}\\,\\mathbf{e}_y + M_{xz}\\,\\mathbf{e}_z\\,.\n ", "079e65f2e5200ed596fefbc5ca338dab": "x_1,\\,x_2", "079e90fce3aca99e1793748d8cf13797": "=\\frac{\\varepsilon\\cdot(1+\\varepsilon\\cdot\\cos \\theta)+(1-\\varepsilon^2)\\cdot\\cos \\theta}{1+\\varepsilon\\cdot\\cos \\theta}\n", "079ea1bf502c75add05019e423631989": " \\mathbb R^3", "079f3c0fef4432f1916946d862ef1bfc": "a = \\frac{1}{4p}; \\ \\ b = \\frac{-h}{2p}; \\ \\ c = \\frac{h^2}{4p} + k; \\ \\ ", "07a00cd5b0cc3bec1fd9df012b99014e": " \\left| \\int_{C_R} \\frac{f(z)}{5-z} dz \\right| \\le 2 \\pi \\rho \\frac{(3+\\frac{1}{1000})^{\\frac{3}{4}} \\rho^{\\frac{1}{4}}}{2-\\frac{1}{1000}} \\in \\mathcal{O} \\left( \\rho^{\\frac{5}{4}} \\right) \\to 0.", "07a045db2bd1e498a63749f712ea79fb": "\\left(\\tfrac{p}{5}\\right)", "07a067208a5f2d3664c63166c2d42441": "\\partial_\\hat{t} \\phi + 6\\, \\phi\\ \\partial_\\hat{x} \\phi + \\partial_\\hat{x}^3 \\phi =0", "07a0860ff99da4aae32240a53338c565": "[X]:=[X,X] \\, ", "07a0a6c0e56e693b951a065c0e60ece0": "f,g\\colon D^n \\to D^n", "07a13336865f965687fd89cef1847882": "E_5(x)=x^5-\\frac{5}{2}x^4+\\frac{5}{2}x^2-\\frac{1}{2}\\,", "07a1380ab446246937cf802ba6231205": "[a-1,a+1]", "07a13d291ced20db18b7299bcb9ca384": "w^{\\prime\\prime}+\\xi\\sin(2z)w^{\\prime}+(\\eta-p\\xi\\cos(2z))w=0. \\, ", "07a145ff8a030ac01257a1f36db04057": "p=p_0 \\sin(\\omega t \\mp kx)", "07a166ccb950ebdb1898ff46b551a34f": " \\mathcal{F}(t) = \\sigma \\left( \\bigcup_{0\\leq s 3 \\\\ (\\frac{h_M}{3})^2 \\mbox{ if,} h_M \\le 3 \\end{cases}", "07c42fc623708dcff18cf4725c2236de": " I(x) = x \\, \\text{ln} x + (1-x) \\, \\text{ln} (1-x) + \\text{ln}2. ", "07c486fb4b0fe62b6ac0d6812622d104": "\\widehat\\beta_j = c_{1j}y_1+\\cdots+c_{nj}y_n", "07c4b1c417f00bf0185eab23d4c98e0b": "\\tilde{4}", "07c4d198de67c9a2105575ff7ad439a2": "a\\propto t^{\\frac{2}{3(1+w)}},", "07c4e50fb61792c2c6499d99dce0fb86": "rK/Y=D_K[F(K,L)]*K/F(K,L)\\,", "07c517ae18a8634ffd9e648ceebfbb5b": "A \\propto L^2", "07c546846b741996053cf2b6439fa1a0": "\\textbf{V}_O=\\dot{\\textbf{d}},", "07c572ea3a09fb7c69a6343e8e3bf4a8": "\n \\quad (4) \\qquad \\epsilon(x,t) = \\sum_{m=1}^{M} e^{at} e^{ik_m x}\n", "07c5cd354f729bbd65ca75545e335213": "\n\\begin{align}\n\\frac{d \\phi}{d t} & = -k(D-A)\\phi \\\\\n& = -k L \\phi,\n\\end{align}\n", "07c63f19fc3af989a2abc4d944cffb25": "M_{PL}=\\frac{M_{star}V_{star}}{V_{PL}}\\,", "07c66863e9b22c9997ef6cfae0734f87": " BA = q AB", "07c6b9b031c2f39202629eebedcc4fa0": " 1-ee=\\frac{1-c-cee''}{1-c}", "07c6c00f24f9522906d343bad4c19afd": "C_1: f_1(x,y)=0, \\ C_2: f_2(x,y)=0.", "07c6d4483bf3e1a60cdb9705810301bb": " \\lambda_1 = \\lambda_2 = 0 ", "07c7010061587e4178aad0eedb95a1bf": "x, y \\in fRep", "07c72b098a91f2641aa9b6627a9499f1": "y(x,t) = y_0 \\cos \\Bigg( \\omega \\left(t-\\frac{x}{c} \\right) \\Bigg)", "07c72c20b02ad827027b41c6e810155a": " \\vec{r}_1 ", "07c74fccb3a16df1a8a955eb01442dda": "\n\\begin{align}\nP_0^0(\\cos\\theta) & = 1 \\\\[8pt]\nP_1^0(\\cos\\theta) & = \\cos\\theta \\\\[8pt]\nP_1^1(\\cos\\theta) & = -\\sin\\theta \\\\[8pt]\nP_2^0(\\cos\\theta) & = \\tfrac{1}{2} (3\\cos^2\\theta-1) \\\\[8pt]\nP_2^1(\\cos\\theta) & = -3\\cos\\theta\\sin\\theta \\\\[8pt]\nP_2^2(\\cos\\theta) & = 3\\sin^2\\theta \\\\[8pt]\nP_3^0(\\cos\\theta) & = \\tfrac{1}{2} (5\\cos^3\\theta-3\\cos\\theta) \\\\[8pt]\nP_3^1(\\cos\\theta) & = -\\tfrac{3}{2} (5\\cos^2\\theta-1)\\sin\\theta \\\\[8pt]\nP_3^2(\\cos\\theta) & = 15\\cos\\theta\\sin^2\\theta \\\\[8pt]\nP_3^3(\\cos\\theta) & = -15\\sin^3\\theta \\\\[8pt]\nP_4^0(\\cos\\theta) & = \\tfrac{1}{8} (35\\cos^4\\theta-30\\cos^2\\theta+3) \\\\[8pt]\nP_4^1(\\cos\\theta) & = - \\tfrac{5}{2} (7\\cos^3\\theta-3\\cos\\theta)\\sin\\theta \\\\[8pt]\nP_4^2(\\cos\\theta) & = \\tfrac{15}{2} (7\\cos^2\\theta-1)\\sin^2\\theta \\\\[8pt]\nP_4^3(\\cos\\theta) & = -105\\cos\\theta\\sin^3\\theta \\\\[8pt]\nP_4^4(\\cos\\theta) & = 105\\sin^4\\theta\n\\end{align}\n", "07c7d5ba70a439c2672cc9f9ff7fd5c6": " \\eta \\rightarrow 1 ", "07c86837ae050de703b9b4ae927ea74f": " s \\equiv r \\,\\bmod p^k \\Rightarrow f(s) \\equiv f(r) \\,\\bmod p^{k+1}", "07c87896e2f6ba6e78a0aee9cbc12fe9": "a^{n-1} \\equiv 1\\pmod{n}.", "07c8dbeacc8116af36c1b3751d6281b6": "(X,\\mathcal{B},m)", "07c8f060c1725bba7aa487a844c9476b": "M = (M_t)_{t \\ge 0}", "07c9080b749d0c13e4d837ebbbc9e37d": "B_{\\alpha \\beta }", "07c965672f4a5f68c7bd1e6ebcd41757": "\\Psi(x,y)=xu^{O(-u)}", "07c9cc926c59abb1d9faa3929434f9ee": "(n-1)\\times 1", "07c9d9baa051a6669920ce7cfdd6cca9": "Q^{2}_{0}", "07c9df79208d745a5f7d28440089223a": "R = {{\\mathbf{k}}}[x_1, \\ldots, x_n]", "07c9f0f65f673a264a5101683e774507": "\\alpha_{i}", "07ca6bfc1ec481f65b7a5e66ad113a86": "K(-u) = K(u) \\mbox{ for all values of } u\\,.", "07ca7c9ea7468b113c70a30966addd2f": " \\int_{\\theta_j} \\frac{\\Gamma\\bigl(\\sum_{i=1}^K\nn_{j,(\\cdot)}^i+\\alpha_i \\bigr)}{\\prod_{i=1}^K\n\\Gamma(n_{j,(\\cdot)}^i+\\alpha_i)} \\prod_{i=1}^K\n\\theta_{j,i}^{n_{j,(\\cdot)}^i+\\alpha_i - 1} \\, d\\theta_j =1 .", "07ca9562bbb1523000132740402e0821": "\\mathbf{A} = \\left[\\begin{array}{ccc}\n 1 - 2q_2^2 - 2q_3^2 & 2(q_1q_2 - q_3q_4) & 2(q_1q_3 + q_2q_4)\\\\\n 2(q_1q_2 + q_3q_4) & 1 - 2q_1^2- 2 q_3^2 & 2(q_2q_3 - q_1q_4)\\\\\n 2(q_1q_3 - q_2q_4) & 2(q_1q_4 + q_2q_3) & 1 - 2q_1^2 - 2q_2^2\n\\end{array} \\right]", "07caab57203b6fc1892fd63ec88de3b8": "\\textrm{Bl} \\ ([D])", "07cae0c56048358e5028c12ecf5378f9": "E(x,y) + \\lambda V(y).", "07caeb770a1a54b8038e0b7f91471753": "\\{L_i(z)\\}_{i=0, 1, ..., N-1}", "07caf5113a9a1987819f000cef81323a": "c \\leq 0", "07cbc478c48c75e20e5161ce2afe38fe": " (x-1)^{-2n-2} P_{n+1}(x) = \\left( x(1-x)^{-2n-1} P_n(x) \\right)^\\prime ", "07cbcb705ecb00a73efe88560b8111d2": "H:\\mathcal{A}\\to\\mathcal{L}", "07cbd6c155424e110559a84df364be5a": "L_2", "07cbfbc8ecd30c38d7262bd4bb61b1bb": " T_n ", "07cc002e715d0752fa5c15c2b888c436": "x_1, x_2, ..., x_k", "07cc2c497da32faf7daaf07ac443db40": "7^6 = 343^2 \\equiv 5^2 \\equiv 25 \\equiv -1 \\bmod 13.", "07cc3836ebb271c10041263ecfa731fb": "K(x) \\leq K(x,S) +O(1) \\leq K(S)+K(x|S)+O(1) \\leq K(S)+\\log|S|+O(1) \\leq K(x)+O(1)", "07cc5009afa1f350aaebe36f0a3b040f": "R(n_1,\\ldots,n_k) \\Leftrightarrow \\psi(n_1,\\ldots,n_k)", "07cc6948b3ee101934f470bb101d8e0f": "\nV(r) = \\frac{mc^{2}}{2} \\left[ - \\frac{r_{s}}{r} + \\frac{a^{2}}{r^{2}} - \\frac{r_{s} a^{2}}{r^{3}} \\right]\n", "07cc694b9b3fc636710fa08b6922c42b": "time", "07cc72d1e021c27f30df1d6859ad7487": "\\scriptstyle\\star ", "07cc76f54ed9d934037070a5d38936fa": "t_i=0,", "07ccb14a3caf4b8e2a190ad94e61c477": "\\exp \\{ i\\mu t - \\frac{1}{2}\\sigma^2 t^2 \\}", "07ccc8a49ed7e50dae6493dddacb1337": "\\gcd(a, b) =\\gcd( b, a).\\;", "07cd0c9345dc0e317d87b3277fe82d33": "(1)\\Leftrightarrow(2)\\Leftrightarrow(3)\\Leftrightarrow(4)\\Leftrightarrow(5)", "07cd160f356bcd99031846437ffb6778": " R_k(x) = \\frac{f^{(k+1)}(\\xi_C)}{k!}(x-\\xi_C)^k(x-a) ", "07cd864dbf621cda99ed595a7ac398b6": "D(g \\circ f)(x) = Dg(f(x))\\circ Df(x).", "07cde9b882c862e19d4a5eb8681f70e9": "\\overline{K}:=\\{0,1,\\infty\\}", "07cdfd3f2454ba70e05c0cdc4a7854cd": "G(\\chi\\chi^\\prime)=\\chi(N^\\prime)\\chi^\\prime(N)G(\\chi)G(\\chi^\\prime).", "07ce14b7349b70c72fbd8c385c006ca3": "\\psi(x) \\rightarrow D(\\Lambda) \\psi(\\Lambda^{-1}x) ", "07ce389119f86db93f5d510d0b6d587a": "\\limsup_{n\\rightarrow\\infty}\\sqrt[n]{|a_n|},", "07ce6c9e3ddac38a1f039aa7ba3eba7b": "f_{e,\\Gamma, R}=\\sum_{p: \\, e \\in p}{f_p}.", "07cea9830ca9474f6448c247178d5601": "F(X) = \\frac{1}{M}\\sum_{m=1}^M T_m(X) = \\frac{1}{M}\\sum_{m=1}^M\\sum_{i=1}^n W_{im}(X)Y_i = \\sum_{i=1}^n\\left(\\frac{1}{M}\\sum_{m=1}^M W_{im}(X)\\right)Y_i", "07cec40f230f56832c4f520622dbb971": "S(T) = \\frac{1}{\\pi}\\mathop{\\mathrm{Arg}}(\\zeta(1/2+iT)) =O(\\log(T)).", "07cf3bdd7fafae355d8e940d0d0c8ff3": "\\int_X p(x;\\theta)dx =1", "07cf65b648327a23d03aee1d3d01396a": "\n\\begin{align}\ns \n&=p_1 p_2 \\cdots p_m \\\\\n&=q_1 q_2 \\cdots q_n.\n\\end{align}\n", "07cf77dd31bd2f7129a37461b9117b7b": "RSTUV", "07cfb64d1763c263fff4490df998db91": "\\left ( \\phi \\to ( \\psi \\rightarrow \\xi \\right)) \\to \\left( \\left( \\phi \\to \\psi \\right) \\to \\left( \\phi \\to \\xi \\right) \\right)", "07d00fa47dad1fdd6db21a172bf289d0": "\\mathbb E[f(x_n) - f^*] = O(1/n)", "07d041038e9a835f2354401c8e2aac4a": "\\sum_{i} p^{ij} = 1, \\ ", "07d04a0ebd91ae40b0be1239f9b9d28f": "\\frac{dW}{d\\omega}\\approx \\sqrt{\\frac{3\\pi}{2}}\\frac{e^2}{4\\pi\\varepsilon_0 c}\\gamma\\left ( \\frac{\\omega}{\\omega_\\text{c}} \\right )^2 e^{-\\omega/\\omega_\\text{c}}", "07d0bf51630248ffbe90a7052bfa15e5": " Q_B (l_A a_B + l_B) l_B ", "07d148ab82f89959ab34650ead1fe3b6": "\\mathbf{w}=\n\\begin{bmatrix}\n(Q+a-1)&\\frac{1}{3}m&0&0\\\\\nm&Q&\\frac{2}{3}m&0\\\\\n0&\\frac{2}{3}m&Q&m\\\\\n0&0&\\frac{1}{3}m&Q\n\\end{bmatrix}\n", "07d15a9ed668847ae9885c2b04698bf6": "\\tau = rF\\sin\\theta,\\!", "07d1746d1d350c31d3fb0de089483818": "\\mu_{T} = \\left( \\pi_{ST} \\right)_{*} (\\mu_{S})", "07d1810b6e4a730498bf6f95abd7a7bd": "r^0", "07d1b04e8d1599a0a4256c61132b0e27": " \\omega = 2*\\pi /0.1 ", "07d1b8e0a8f21ab94d74bdcc820fac60": "{\\delta} < \\mathrm{error} ; ", "07d1deb679816938dc05177722496beb": " (K \\phi)(x) = \\sum_y K(x,y) \\phi(y) \\,", "07d20f50f5198298e034d36b7a46493d": "2 \\times \\sqrt{3}", "07d22e4f4046963f2eaf5627d0e37d04": "p_{j,t-1}", "07d25ff8ad8b1381e164770c9e90e050": "Z(t)=I(t)+jQ(t)\\,", "07d26a08be43af7cb561d6b6b8eec113": "p^2 + 2pq + 2pr + q^2 + 2qr + r^2 = 1. \\,", "07d2a5bee02b2b0042fc92d05b95818e": "\\boldsymbol{\\omega} = (\\omega_x, \\omega_y, \\omega_z) ", "07d2aa1b053b0001c46c43695eb3655d": "e=C_vT", "07d2bb600c8d9b65679ffedd1bad08bd": "\\mathbf{F}_{\\mathrm{Centripetal}} = \\mathbf{T} + \\mathbf{F}_{\\mathrm{Fict}}\\ , ", "07d3755579f31a45280dfc8ded0e80d7": "e_1,\\ldots,e_m \\in \\mathbb{T}", "07d3936feb19afdacadbe368a18ac88d": "f(xy)=f(x)+f(y), f(1)=0", "07d3a06c3b9f4fdf60055d30a5b2070b": "[\\hat{X},\\hat{P}] = \\hat{X}\\hat{P}-\\hat{P}\\hat{X} = i\\hbar", "07d3c8cf5f9b1d5f12740463fc056102": " \\mathbf{J} = \\mathbf{J_f} + \\nabla\\times\\mathbf{M} + \\frac{\\partial\\mathbf{P}}{\\partial t}", "07d3e0a0783d2d067f6fa1f93664ce1a": "\\nabla \\times \\vec{B} = \\mu_0 \\vec{J}", "07d3e5de2d131680b4ff26c328b4cc6f": "t_{mn} = \\frac{(m+n)(m+n-1)\\cdots(m+1)}{n(n-1)\\cdots 1}.\\ ", "07d41ce9ee6e308e17d75c30e4b6c000": "\\Gamma(n+1/p) = \\Gamma(1/p) \\frac{(pn-(p-1))!^{(p)}}{p^n}", "07d421ec371c7d4d836b60b5a4da084c": "\\frac{\\partial \\rho}{\\partial t} + \\vec \\nabla \\cdot(\\rho \\vec v) = 0 ", "07d422876e555cf72ff10918f1f92485": " H^{-1}(z)", "07d4d6ec5a86f2a9b56a9d012ef281fa": " \nH = \\frac{(l_1)^2}{2I_1}+\\frac{(l_2)^2}{2I_2}+\\frac{(l_3)^2}{2I_3}+ mg (a n_1 + bn_2 + cn_3),\n", "07d52077eaa5865dfc7121020bcf09c1": "\n\\begin{array}\n[c]{cccccc}\ng_{1} & = & Z & X & Z & I\\\\\ng_{2} & = & Z & Z & I & Z\\\\\ng_{3} & = & Y & X & X & Z\\\\\ng_{4} & = & Z & Y & Y & X\n\\end{array}\n", "07d5364be7263d4eaad2c3f82df50154": "K=I\\otimes T", "07d593f0b25ba1a8bf43dac9a1d4d41f": "(x_s, t_s)\\,", "07d5c7099ff999998f0068b6b34ab6d4": "\nCIQ_t = \\mathcal{A} e^{\\mathcal{B} t}\n", "07d63f12586dfdbaebc11e3311a2d36b": "F_+(H) = \\overline{S^*H}", "07d64d6c01234b60032aa525cd2c1f96": "\\mathrm{Rot}_H", "07d68e12866dda148c93268f6bf2ec95": "\\frac{\\partial^2 y}{\\partial x^2}=\\frac{\\mu}{T}\\frac{\\partial^2 y}{\\partial t^2}.", "07d73fc27d368d61cd55cb4d5e1f29e8": "\\Leftrightarrow \\!\\,", "07d7c4352aefd6bda26303c773765454": "a_0 b_n - \\tbinom{n}{1}a_1 b_{n-1} + \\tbinom{n}{2}a_2 b_{n-2} - \\cdots +(-1)^n a_n b_0 = 0", "07d8112f3cf98ff31b7aac846f90cd75": "\\varrho(T_h)", "07d86c31e7078074357f17c2fa997928": "PR(A)= \\frac{PR(B)}{2}+ \\frac{PR(C)}{1}+ \\frac{PR(D)}{3}.\\,", "07d87337f49d692cfd1c1dc4bdc54771": "w = d + m + c + y \\mod 7,", "07d8da455eb16ff3a133f69d7a2964af": "\\zeta(s)=\\frac{\\eta(s)}{1-2^{1-s}},", "07d8fbd2720f2d36a6de65b679b3adea": "\n\\begin{pmatrix} & h& \\\\[-0.9ex] v & & v'\\\\[-0.9ex]& h'& \\end{pmatrix} \n", "07d935680b6501b2e42fe4baea021389": "mk", "07d9577618053507ed710ae0be8a4705": "n-m\\ge 0", "07d9d68a024064595021c95152f318e3": "6 + \\sqrt{3}", "07d9d7cd24111b32653ded6c2e075a8c": "\\sigma = \\pi^2 k^4 / 60 \\hbar^3 c^2 ", "07d9f7a4cfc9c776b7034b04068cce16": "pf_i = C/N = 0.311\\!", "07daf1dddf3b5a8e6724497dfb74d5d6": " [U_h(\\mathrm{M}(a,b,c))]\\psi(x) = e^{i (b \\cdot x + h c)} \\psi(x+h a). ", "07daf43d269c6cb7b45c16ca4062ceb6": "\\mathbf{z} = \\left \\{(x_{i},y_{i}) \\in X \\times Y: i = 1, \\dots, m\\right\\} \\in Z^{m}", "07daff2abb9da1e1697b8a58798985ec": " V = \\pm \\frac{ f R }{2} \\pm \\sqrt{ \\frac{f^2 R^2}{4} - \\frac{R}{\\rho}\\frac{\\partial p}{\\partial n} } ", "07db2bb21ed4bca1aeef150981f8ca83": "\\mathsf C", "07db5288cfa5b7e0cb01a657c5ab31b9": "G_X(t,f) = G_x(-f,t)e^{-j2 \\pi ft} \\, ", "07db94a164b976adbf9fbd45788266b5": "d\\mathbf X\\,\\!", "07dbc28d6621cd56804fd8d4ed5a1205": "\\frac{d^2}{d\\theta^2}\\left(\\frac{1}{\\mathbf{r}}\\right) + \\frac{1}{\\mathbf{r}} = -\\frac{\\mu\\mathbf{r}^2}{\\mathbf{l}^2}\\mathbf{F}(\\mathbf{r})", "07dbfcb7ead62d17fb5e5df064d63b6e": "\\tilde{f}=\\left|\\frac{\\tilde{d}}{2}\\right|", "07dc07bc1536e975103ee20654509c29": "\\frac{\\sqrt{2}}{2} \\left(\\frac{(2m-1)\\Omega}{m}\\right)^{1/2}", "07dc172a833d6915b7c243373714b5dd": "I_{im+}", "07dc6ec99fe20876f73ca2bc44eaf4e6": "A,B \\in E", "07dddf7ed882ed38f02642e10b723f59": "\\mathbf{r}_6 = (a/4)(3\\hat{x} + \\hat{y} + 3\\hat{z})", "07de1a3d19ef9adf4304071b0922a724": "x^* = \\text{null}", "07de7dafd1a757933a70ece3441ce9b7": "\\{\\psi(w) : w\\not=v\\}", "07de97a4f99b2d930a3ab53023301768": "( u \\wedge v) w = - w \\cdot ( u \\wedge v) + w \\wedge u \\wedge v", "07deb2311d8a8b360fbf44fa38230ceb": "A(u) = \\frac{u^2 + 2}{u \\sqrt{u^2 + 4}}", "07df2900bfa96aecf5901be3829a1bdd": "M\\to H_1(M,\\mathbb{R}) /\nH_1(M,\\mathbb{Z})_{\\mathbb{R}}", "07df403dfe51db0194e6c677a582ab10": "\\int d[wx^2] = \\int x^4 dx", "07df5771b077f4a06dd347b292015939": "R(x,y_1,\\dots,y_n)", "07df8dc8930628c9016f6332f6edab8a": "\n\\widetilde{\\theta} = \\frac{\\exp {(- \\beta u)} - \\exp {(- \\beta u_0)}}{1 - \\exp {(- \\beta u_0)}}\n", "07e01a2d436b13f41b9cdf19214307d8": "\\bar{b}^2 G_C", "07e0689fd47aac7cc7b899beb05fabbd": "\\vdash \\in \\Gamma - \\Sigma", "07e1019f0737536293bb710b19de8c60": "\\Gamma^{[k-1]}", "07e10f3656d7cfd24b00d6804a1c41cb": "H_\\Lambda^\\Phi(\\omega) = \\sum_{A\\in\\mathcal{L}, A\\cap\\Lambda\\neq\\emptyset} \\Phi_A(\\omega)", "07e13d322a1dc4341d3d7c3c36993dae": "\\Phi_{1}\\left(\\mathrm{R}_{i}\\right)", "07e1a666e867b50fc7ee58bbfa4544aa": "E_\\lambda", "07e1a8662990a0595e395b6349adbc6c": "\\displaystyle{Tf_n=\\mu_nf_n}", "07e1af018de324ecf10b02348a778236": " \\frac {\\mathrm{DOF}_2} {\\mathrm{DOF}_1} \\approx \\frac {c_2} {c_1} = \\frac {l_2} {l_1} \\,,", "07e200bba2d90c20ed0773de03be3cd9": "\\Omega_{-}", "07e20de5d75966ec1f7ad971c27a9490": "\\phi(v_j,v_k)=\\int_0^1 v_j' v_k'\\,dx", "07e240557ba49686c97a43603c5f1193": "s=-x^3-x^2+x\\ ", "07e25498887751c397b19bb5787ed061": "\\pi \\approx {355 \\over 113}", "07e271ec125747627ea1737274501e63": "\\sqrt{I}=\\{r\\in R|r^n\\in I\\ \\hbox{for some positive integer}\\ n\\}.", "07e2ba2bd104f609d18414d2507428be": "\\{ C (\\vec{N}) , G (\\lambda) \\} = G (\\mathcal{L}_\\vec{N} \\lambda)", "07e2e9364a982bb791f5b5745b9c1d96": "\\vec{N}=\\{0;\\; 1; \\; 0\\}; \\;\\; \\vec{E}=\\{\\frac{\\sqrt{3}}{2}; \\; \\frac{1}{2}; \\; 0\\}; \\;\\; \\vec{L}=\\{-0.6; \\; 0.8; \\; 0\\}; \\;\\; n=3", "07e2ecb3228caaddeed2a9869696a507": "\nm_1\\;\\operatorname{sc}^2(u)+m_1= m_1\\;\\operatorname{nc}^2(u) = \\operatorname{dc}^2(u)-m\n", "07e2f7f391b53640f096df4a27b66ce6": "\\hat{t} = \\operatorname{argmaxminlocal}_{t}(\\nabla^2_{norm} L(\\hat{x}, \\hat{y}; t))", "07e378366ded1264c8b2a4c2fb497a10": "x_1:= x\\,", "07e3c84bd480e3f434119e3fa3b0c84d": "\\nu_x= \\frac{1}{2} \\delta_{-1} + \\frac{1}{2}\\delta_1", "07e41023fdc5086c51ea6aa944023f34": "\\mu_r''=(\\frac{\\lambda_g^2+4a^2}{16a^2})(\\frac{V_c}{V_s})(\\frac{Q_c-Q_s}{Q_cQ_s})\\,", "07e4b3f7df2f71b1c46fa47ce0f29f56": " f(\\mathbf{y}) + [J_f](\\mathbf{[x]}) \\cdot (\\mathbf{z} - \\mathbf{y})=0 ", "07e4f44940d9af1fa9cc5954202a9b9e": "\n \\boldsymbol{s} = 2 K~\\left(\\sqrt{3}\\dot{\\varepsilon}_{\\mathrm{eq}}\\right)^{m-1}~\\dot{\\boldsymbol{\\varepsilon}}_{\\mathrm{vp}}\n ", "07e4fbe766dabd2aeccdf093039dbdad": "A_{i, j, k}", "07e4fc3d3abe469b6ace7dc96abb5e95": "d (f, g) := \\| f-g \\|", "07e51bf43f0a51d6988c5a86b5b9bcc5": "\\omega^2 = \\int_{-\\infty}^{\\infty} [F_n(x)-F^*(x)]^2\\,\\mathrm{d}F^*(x)", "07e557a11281006cc8627851723d1022": "\\Box A_\\mu = \\frac{4 \\pi}{c} J_\\mu", "07e55f58e7dbe0217993dc53faa7b1b2": "f(z) = \\sum_{n=0}^\\infty c_n z^n,", "07e59ea71bf21615c75fc41b46a45d78": "q \\ge 0", "07e5a4a56a57f5c874ebf79bb67a0b18": "\\mathbb R", "07e5b3f26da912ff2b11115cfd81091d": "D_2=kTB_2(1+N_2\\frac{d ln\\gamma_2}{d lnN_2}) ", "07e5b4c482edd88ba7d1c8cfd8d63fb5": "\\frac{d A}{dz} =-i \\gamma_{\\|}|A|^2A", "07e6d08dadd5f7bd8b8b2a7aea06aaf0": "\\phi = \\pm \\pi / 2\\,", "07e7c990ae832000342d7c0251b7e594": "\\delta \\mathbf{Z}_0 \\to 0", "07e83a2180c6cc88a1926d0e7b96f29d": "\\operatorname{E}(\\mathbf{1}_A) = \\operatorname{P}(A). \\; ", "07e84ffdd5db350ab15b792a02f529b4": "\\scriptstyle \\mathfrak{X} (M)", "07e866a596b9a2ab3e7d7da99ebb774b": "\\gamma_2 = \\exp(-\\delta_1-\\delta_2)+\\exp(-\\delta_1-\\delta_3)+\\exp(-\\delta_2-\\delta_3).", "07e866dcdf6518db1b1c1fc125830bd4": "1852 = metres\\ per\\ nautical\\ mile", "07e873918decf42106f6f9d2d99d8188": "x=a(1-\\sin\\psi),\\,y=a\\frac{(1-\\sin\\psi)^2}{\\cos\\psi}.", "07e8fda49b6107fd677f5bf1e507a270": "\\Delta^n_{t\\Delta x} f", "07e922057d45dfb7eb00b9d826750685": "1-n/N", "07e9493b94f772c30c1ab8a66aa96f7b": "\\mathbf{J}(\\mathbf{r}, t) = \\rho(\\mathbf{r},t) \\; \\mathbf{v}_\\text{d} (\\mathbf{r},t) \\,", "07e95931f58a4cf5b9c090be27f0bc6e": "1-F_{Y}(q)", "07e960f2c0ed274e20c6e1c5bb5aa04c": "T_1^{(1)},T_2^{(1)},X_1^{(1)},X_2^{(1)},H^{(1)}", "07e96fc1b25d8051509ab34ac69522e7": " \\mathbf{T} = \\begin{pmatrix} \na_\\text{x} b_\\text{x} & a_\\text{x} b_\\text{y} & a_\\text{x} b_\\text{z} \\\\ \na_\\text{y} b_\\text{x} & a_\\text{y} b_\\text{y} & a_\\text{y} b_\\text{z} \\\\\na_\\text{z} b_\\text{x} & a_\\text{z} b_\\text{y} & a_\\text{z} b_\\text{z}\n\\end{pmatrix}", "07e9990f329c2a9a3d96a09c210f94e8": "v_{1,2} = 5", "07e9b2c777ce42a404f6b03885773b7f": "\n\\tanh(\\alpha + \\beta) = {\\tanh(\\alpha) + \\tanh(\\beta) \\over 1+ \\tanh(\\alpha) \\tanh(\\beta) }\n", "07ea01e1bb346a8adf43797c14bb5e5b": "\n\\frac{1}{2} \\frac{dI}{dt} = \\frac{1}{2} \\frac{d}{dt} \\sum_{k=1}^N m_{k} \\, \\mathbf{r}_k \\cdot \\mathbf{r}_k = \\sum_{k=1}^N m_{k} \\, \\frac{d\\mathbf{r}_k}{dt} \\cdot \\mathbf{r}_k = \\sum_{k=1}^N \\mathbf{p}_k \\cdot \\mathbf{r}_k = G\\,.\n", "07ea67ca0036c2ba5440bb73c375ad1a": " \\Delta\\mathbf{r}_i\\times(\\boldsymbol\\omega\\times(\\boldsymbol\\omega\\times\\Delta\\mathbf{r}_i )) + \\boldsymbol\\omega\\times(\\Delta\\mathbf{r}_i\\times(\\Delta\\mathbf{r}_i\\times\\boldsymbol\\omega))=0,", "07ea97937a2b0e75b07b6a136d022618": "\\Vert f^*_n \\Vert \\le 1", "07ea9eb1f4232484e23c7ec7420df172": "\\frac{1}{a}", "07ebdda21bfd38368e5a089060b7f27b": "\\lbrack\\mathbf z\\rbrack = \\lbrack\\mathbf z\\rbrack_1 + \\lbrack\\mathbf z\\rbrack_2 = 2\\lbrack\\mathbf z\\rbrack_1 = \\begin{bmatrix} 2R_1 + 2R_2 & 2R_2 \\\\ 2R_2 & 2R_2 \\end{bmatrix}", "07ec11425cb4ccb2b0aba3c2ed074fe4": " (\\bullet\\bullet\\bullet)(\\bullet)", "07ec12590399c4f008aeb69aebdfc16c": "\n \\tau_m = \\sigma_m \\sin\\phi + c \\cos\\phi ~.\n ", "07ec3a356588619c88a8fdb0443923da": "i=1...n", "07ec5f49bbac96fc6b295696f31015df": "\\Phi(i)", "07ec7c9d1d7727d8da38fbb903501d01": "dF_\\mathrm n\\,\\!", "07ec8edd29169a1e35f05c1344c8c0ce": "\\chi_a", "07ecf9eacc6696e59015190e4684fcbe": "\\boldsymbol\\omega=\\frac{\\mathbf{r}\\times\\mathbf{v}}{|\\mathrm{\\mathbf{r}}|^2}", "07ed5b28b87ea8650dd99c429d927e28": "R_{0}=\\{(x,x):x\\in X\\}", "07ed62bddcc38436d34bcfdb378e32bc": "\n d\\mathbf{f} = \\boldsymbol{F}\\cdot d\\mathbf{f}_0 = \\boldsymbol{F} \\cdot (\\boldsymbol{S}^T \\cdot \\mathbf{n}_0~d\\Gamma_0)\n", "07ed7b7884737b80357da49facb87ff4": "n \\ge 4.", "07ed9c3f277abe2ae9ca8f83a9b87e83": "M_{2x} = \\dot{m}V_{2x} = - \\rho QV_2 \\quad and \\quad F_{P2x} = \\overline{P}_2A_2", "07ed9cd92782b85be245409f24a9b337": "y (\\theta) = r (k - 1) \\sin \\theta - r \\sin \\left( (k - 1) \\theta \\right). \\,", "07eda6dfa5faa951c2089ae9b256594b": "\n D\\,\\nabla^2\\nabla^2 w = -q(x, y, t) - 2\\rho h \\, \\ddot{w} \\,.\n ", "07ee3609f571e22755490614f22f2f3b": "e^{i \\pi}= -1.", "07ee477f13d903896289636d38728763": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 24.03761 \\log_e(T+273.15) - \\frac {7062.404} {T+273.15} + 166.3861 + 3.368548 \\times 10^{-5} (T+273.15)^2\n", "07ee4e74e6c56bd8d51ed1a555cea2bc": "_k\\mathbf{b}_{l,m,n} = \\mathbf{S}_k\\mathbf{a}_{l,m,n}", "07ee679472c3e77e252a87bcef5a40f7": "\n\\begin{align}\n\\omega_1 &= \\omega - {e^2\\over 32\\pi\\varepsilon_0 m_e\\omega Z^3},\\\\\n\\omega_2 &= \\omega - {e^2\\over 16\\pi\\varepsilon_0 m_e\\omega Z^3}.\n\\end{align}\n", "07ee9b67a0557b8f091293637b1a079b": "(S,\\Sigma)", "07ef19656e5b0e7f28762bfaa1fb9ba8": "[i_{L_1},i_{L_2}]= i_{[L_1,L_2]^\\and}", "07ef275540acce92238e509054c30393": "{\\overline{b}}=(B^{-1}a_1B,\\ldots,B^{-1}a_nB)", "07ef7a9526aa22e94314110e1f000f61": "0,\\ldots,n-1", "07ef82cb261e1d693985694652fda01b": " \\lbrace T \\rbrace ", "07ef8344f0d7f63f29cb988bff684c67": "\\Im z =0", "07efbca572b25c0069d4b524dd94a4a1": "H_{n+1}(x)=2 xH_n(x)-H_n'(x).\\,\\!", "07efc8cc2791419a300e2582688e62f5": "x_1,\\ldots,x_j", "07effcc790d2b70570b1db621da3b832": "\\frac{v_0 [Cl^-]_0-v_i[Ag^+]_0}{v_0+v_i} \\begin{cases} \n\\approx [Cl^-]_i \\text{ or } K_{sp} 10^{-b_1E_i+b_0} & \\text{ when } v_{0^{ }} [Cl^-]_0 > v_i[Ag^+]_0 \\text{ (before equivalence)} \\\\\n= 0 & \\text{ when } v_{0^{ }} [Cl^-]_0 = v_i[Ag^+]_0 \\text{ (equivalence point)} \\\\\n\\approx -[Ag^+]_i \\text{ or } -10^{b_1E_i-b_0} & \\text{ when } v_{0^{ }} [Cl^-]_0 < v_i[Ag^+]_0 \\text{ (after equivalence)} \n\\end{cases} ", "07effe5ef25cfe6080ecea6307b82361": " a_{ab} ", "07f02e761349ad9cc156d0396d6f371f": "U_n=a \\varphi^{n-1} + b \\psi^{n-1} + a \\varphi^{n-2} + b \\psi^{n-2} = U_{n-1} + U_{n-2}.\\,", "07f03b52648fd3107fc19bc5e6f42241": "\\varphi(L)", "07f046a9608ace07fb8896f3f528bf1f": "A = 1", "07f0f62493c98773cd3ff6b8157130d2": "R_0\\ .", "07f1660e1c7fc4ebff5e9335d99c10c4": " \\Delta \\mathbf{X} = \\left(c\\Delta t, \\Delta \\mathbf{r} \\right) ", "07f17b9356c3ca4fa35ea17059be8480": "[u,\\ u+du]", "07f1b37b7ad932e9db4d027659488398": "State(Do(move(box,table,floor), s)) \\circ on(box,table) = State(s) \\circ on(box,floor)", "07f1e15a0cfa370ca2ead6af6abe5d6c": "T^{-3}", "07f2592540d35ea44029a6d5bb864e64": "\\Big(\\frac{R}{M} \\Big)^{i+j}", "07f25ebff1c2f8f67f8b679f9775404e": "|\\Psi\\rangle = c_1(t)|1\\rangle + c_2(t)|2\\rangle.", "07f2e759fb2f9e8e8ed66d8e2b96645a": "X_i, ...,", "07f366f076527f9e875ff995d8ebb87a": "\\forall j \\notin S", "07f36a2e80867725481391901421d6eb": "SCM \\cdot SPE", "07f3daf594fabc1cbdf516a9badce8be": "B_{3/2}=\\{ x: |x| < 3/2 \\}", "07f3ef3bb038afea20cf429f47257089": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{F}&\\mathrm{F}&\\mathrm{*}\\\\\n\\mathrm{F}&\\mathrm{F}&\\mathrm{*}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "07f413b41321d20d192b373c086d8404": "\\mathcal{L}(x)=\\mathcal{L}[\\phi(x), \\partial_\\mu \\phi(x),x].\\ ", "07f434fd4dcd230df79051a964a91db5": "V^\\prime=\\mathrm{Hom}(V,\\mathbf{Q}_p(1))", "07f44f22b588b14ab897df90fd17b90a": "\\exists a \\phi(a) \\rightarrow \\exists a\\, (\\phi(a) \\wedge \\forall x\\in a\\,(\\neg \\phi(x)))", "07f47497b56738455bb427f2da7d5919": "L \\otimes_k \\overline{k}", "07f49c2885a727b0c6d8861e9380bd0b": "\\Phi : \\mathbb{R}^d \\to \\mathbb{R}^N", "07f4a089bd6a73ea154c13cfecd1e8a9": "\\boldsymbol{\\varrho\\varsigma\\vartheta\\varphi} \\!", "07f4aa82ec596a2cf91a6e4b28980c34": "\\forall n:\\int_{a_{n}}^{b_{n}}\\tilde{w}_{n,i}(p_{n})\\tilde{w}_{n,j}(p_{n}) \\, dp_n=\\delta_{i,j},\\quad1\\leq i,j\\leq I_n,", "07f4cbfcb4839a6b6a6e6e1206a25fe9": "\\psi_1(x)-\\frac{x^2}{2} = \\int_0^x \\psi(t)\\,dt - \\frac{x^2}{2} ", "07f58f8add1dfd58f33e288fb79e4f4e": "w^2+x^3+y^4=0 ", "07f5c943b03f215e31fec01edbf147bb": "\\operatorname{erf}^{-1}(x)\\approx \\sgn(x) \\sqrt{\\sqrt{\\left(\\frac{2}{\\pi a}+\\frac{\\ln(1-x^2)}{2}\\right)^2 - \\frac{\\ln(1-x^2)}{a}}\n-\\left(\\frac{2}{\\pi a}+\\frac{\\ln(1-x^2)}{2}\\right)}.", "07f607059536ad6823c8fad0e9aa56b2": "\n \\nabla^2\\nabla^2 w = -\\cfrac{q}{D} ~;~~ D := \\cfrac{2h^3E}{3(1-\\nu^2)} = \\cfrac{H^3E}{12(1-\\nu^2)}\n ", "07f60dbb469e9e9e36d1cf704ee38a45": "d \\Xi = \\frac {U + P V} {T^2} d T - \\frac {V}{T} d P + \\sum_{i=1}^s (- \\frac{\\mu_i}{T}) d N_i", "07f616656d5b4665686f9e2c866e21e1": "V^H", "07f6459108618319abf2c6cad952185a": " L = 2\\bullet10^{-7} \\bullet \\ln \\left ( {D\\bullet e^{1 \\over 4} \\over D_{BE}} \\right )", "07f6572f593a609ba9797612c41a2be9": "X(0)=X_0", "07f6a6057feecafb2624927486cb7df0": "\\hat{F} = \\frac{\\sum_{k=1}^N (\\hat{x}_{k+1}-\\hat{F} \\hat{x}_k)}{\\sum_{k=1}^N \\hat{x}_k^{2}} ", "07f6cb9c30863d5b991fd674fdeb79e9": "{\\rho g h S}=0.06{(\\rho_s-\\rho)(g)(D)}", "07f6cfe0053dea15639b5fe17e345f85": "H = \\int (F_{12}-F_1F_2)^2 \\, dF_{12} \\!", "07f6d1c4a280a629b4b44203827a6a7f": "D(G) = 1 + \\sum_i \\left({p^{e_i} - 1}\\right) \\ . ", "07f6dbd4ebf0d41a81baca3474fe25d4": "\n X_n \\, \\xrightarrow{\\mathrm{a.s.}} \\, X.\n ", "07f71416daa6dea5b7b41941c136e328": "\\rho \\ge 0, \\; \\; \\rho + p \\ge 0 .", "07f71f05062a336104b178ab45f4a546": "H=L^2(G)", "07f723f71af59303c67d2e7acc3f578d": "a(n)=\\sum_{k=0}^{n-1} 2^k \\left\\langle\\begin{matrix} n \\\\ k \\end{matrix}\\right\\rangle=A_n(2),", "07f72d5ba169d5124719806ab8c9b41a": "\\lambda=w\\cdot\\tilde\\lambda,", "07f7326f90f457b8b54604bd0938feea": "\\{P\\in \\R^2 \\ | \\ d_P(P,M)=r\\}", "07f75d0b6a742741fa0c9cf580a93f27": "\n\\langle\\bar{r}^2(s)\\rangle = \\frac{1}{\\bar{k}_\\alpha(s)}\\langle\\bar{r}^2(s/\\bar{k}_\\alpha(s))\\rangle_\\text{nrml}.\n", "07f768b3892b58a5caf49d00c8c9cb10": "N=KM", "07f777a06f68b273ae74156423bf9bbc": "\\sqrt{\\sigma^2 + \\hbar^2/16\\Omega^2}", "07f7c5c34e21d72d3e56aebabd3f4da9": "P \\mathcal{F} P = \\mathcal{F'} ", "07f7e3e2d1c6309fe22aee868b47cb27": "\\textbf I(\\alpha)=\\int_0^{\\frac{\\pi}{2}}\\frac{\\ln\\,(1+\\cos\\alpha\\,\\cos\\,x)}{\\cos\\,x}\\;\\mathrm{d}x, \\qquad 0 < \\alpha < \\pi.", "07f7fe67f245362d980d9ec2797c42a4": "L=\\sum_{e\\in E}L_e", "07f8019cadb67675f5173fc91f7d12fc": " ax^5+bx^4+cx^3+dx^2+ex+f=0,", "07f82936a58503d65eab46a67961a033": "\\alpha(d)", "07f83e3672e2bd67852962e1cbe8f915": "\\theta=0.25", "07f83f87447d449a3fb8f39d020e4b19": "c_n = n^{1/\\alpha} \\,", "07f84114279c3cc0a8fabaf65c411fa7": "{M} = \\left[\\begin{array}{cc} {Q} & -{A}^{T}\\\\ {A} & 0\\end{array}\\right]\\,", "07f86868058a46b043166e507eb716eb": " \\lVert ", "07f8918ac334c18ca1d640e99ad9995a": " f( r_t | M_t = m^i) = \\frac{1} {\\sqrt{2\\pi\\sigma^2(m^i)}}\\exp\\left[-\\frac{(r_t-\\mu)^2}{2\\sigma^2(m^i)}\\right] .", "07f89236a63c2a0eaabc361ef22b62c7": "\\tfrac{\\lambda(1+\\nu)(1-2\\nu)}{\\nu}", "07f89df78e4991b4fdeb361e375d6da0": " q = \\min\\left(a\\ell,\\frac {k} {\\sigma}\\right)", "07f8ed15938f4f4daa46aeb7d5339d09": "\\sum_{g\\in G}f_g g,", "07f92b1474456ac3e8fcf2f80f036cbe": "a \\in A_{n-1}", "07f9b6947a6c88e5e901e4e01b9a9b1e": "\\textbf{P}_{k\\mid k-1} = \\textbf{F}_{k-1} \\textbf{P}_{k-1\\mid k-1} \\textbf{F}_{k-1}^{\\text{T}} + \\textbf{Q}_{k} ", "07f9dd1c4bf69068cebe5bd48eafdc03": "\\nabla^2\\phi=4\\pi\\rho\\;", "07fa4c077b4e8efadcad6b8c50ad86ad": "\\scriptstyle \\Gamma^k_{ji} \\;=\\; \\Gamma^k_{ij}", "07faa54195f92c9fbe1c8ce51e2fcab0": " G_Y(s) = \\sum_{n=0}^\\infty p_n s^n = \\exp(a_1(s-1)+a_2(s^2-1)) ", "07fac7020527711c1f93664a2a8cc2dd": " H = \\{f {\\in}L_2(X)\\mathrel{\\Bigg|} \\sum_{i=1}^\\infty\\frac{^2}{\\sigma_i} < \\infty\\} ", "07fb17c177b0c8b329ab14c6e50c116c": "d - S \\approx \\Delta z / \\cos \\theta - \\Delta z\\theta", "07fb67bf137540829db30aa6a3afa376": "E_{em}=\\frac{1}{2}\\frac{e^{2}}{a},\\qquad m_{em}=\\frac{2}{3}\\frac{e^{2}}{ac^{2}}", "07fbca51be8b36b7db7ad2782684cb2a": "\\psi'(g * h) = \\psi(g * h) = \\psi(h) * \\psi(g) = \\psi(g) \\mathbin{\\ast'} \\psi(h)=\\psi'(g) \\mathbin{\\ast'} \\psi'(h).", "07fc02f658d3b17c2069e849f641c065": "0 < \\delta \\le 1", "07fc397c1492f0ca4e476ff2c7bea004": "a(bc)=(ab)c", "07fc5c45178323ac61380dbd5da6b62f": "\\operatorname{var}(X) = \\operatorname{E}[(X - \\mu)^2] = \\frac{\\alpha \\beta}{(\\alpha + \\beta)^2(\\alpha + \\beta + 1)}", "07fcab2d87b5aa4534599accc14381d1": "{\\mathcal{A}}_{i_{n}=j}", "07fd33ee378880f8d7fc75b7bea8549a": " d = \\lceil \\ln{1/\\delta} \\rceil", "07fd521442a72909417714ce6598665b": " \\mathbf{r}_i = 1 ", "07fd85bb5e9f013abd836a5c4611800f": "\nk^{2} = \\frac{\\mu}{h^2} - 1\n", "07fd9f296aee66d18c6418ef9889831e": "\\frac{\\partial \\rho}{\\partial t} + \\nabla \\cdot (\\rho \\mathbf{v}) = 0", "07fdc7b9d4934d172afa37d71b01ff03": "\\frac{d}{dt}\\langle \\sigma_z \\rangle = -2g\\left(\\langle a^\\dagger \\sigma \\rangle+\\langle a \\sigma^\\dagger \\rangle\\right) -2\\gamma \\langle \\sigma_z\\rangle-2\\gamma ", "07fe21a915b4b6752931a2a04d55b977": "e^{-i\\int H(t) dt_{op}}\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} \\otimes \\begin{bmatrix} 0 \\\\ -1 \\end{bmatrix}=\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} \\otimes \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}", "07fe268cbfb5379831451c4a1454383f": "y' = y + kx", "07fe44b2b7a4bc99918d7ead9b6628d4": "R_n(x)\\ \\stackrel{\\mathrm{def}}{=}\\ T_n\\left(\\frac{x-1}{x+1}\\right)", "07fe896419a35b754e001c99ac31b415": "\\boldsymbol{ \\cdot}\\ ", "07feceb71273d9c3afb7b4411c6a6bcb": "(a + bi,\\ c + di) \\leftrightarrow (a, b, c, d).", "07feef3766c80eca6fa372fdd0d85a74": "{x}_{i}={x}_{k}-(k-i)h", "07ff119b44e0d0b394c9ec0ea60015a5": "\\sin (\\beta) = \\sqrt {1 - Z_3^2}.", "07ff48c90138571dcde03e88b1496a94": "\\omega_1 = 1 \\,", "07ff7187591188d861ab08e40ce7da07": "d\\Omega^2", "07ffd5675f86ea627719a5078abd1233": "F\\triangle G\\in\\mathcal{A}", "07ffe7e828ae69de037252ff612c1296": " h^0(K|_D) - 1 \\le \\frac{1}{2}\\mathrm{deg}_D(K) = \\frac{1}{2}K^2.\\, ", "07ffec902fe52741a043f367f2489075": "(Y_t)_{t\\geq0}", "08001cb417ce6f4521f76272af06aa8a": "1/e.", "080035f725082c1785e2e7fb515ca7c2": "\\sigma = Y \\, \\epsilon \\,", "0800590145a98e0c3db79f9486ba4962": "u^T a u > \\alpha u^T u", "0800cb500a7f35c564a2c2470a235670": "0 \\ (0^\\circ)", "0800f1fd8d8e51a3cfc95338d90f9b9c": "Z = \\left(1 - \\frac {3}{8}n^2\\right)(p + qi)^{2/3}\\qquad\\text{ where }\\; i = \\sqrt{-1}", "0800fc577294c34e0b28ad2839435945": "hash", "08016d6af0dcd8036c15b3241df14c39": "\\lambda f.(p\\ f)\\ (p\\ f) ", "08020db13c98dd0177d79e55fdf35861": "i \\theta = \\ln \\left(ix \\pm \\sqrt{1-x^2}\\right) \\, ", "080221c3cf8912a1f1581d70d3938fea": "\\omega=\\sqrt{|\\det [g_{ij}]|}\\;\\mathrm{d}x^1\\wedge\\cdots\\wedge \\mathrm{d}x^n", "0802233eb3d016cb5bc16c0a2f2e8c83": "y_3 = \\frac{y_2y_1-z_1x_2x_1z_2}{(y_2^2+(z_1x_2)^2)}", "0802c6028987aada44d354c9956377e0": " -13 \\mathbf{e}_1\\wedge \\mathbf{e}_2 -7 \\mathbf{e}_1\\wedge \\mathbf{e}_3 +5 \\mathbf{e}_2\\wedge \\mathbf{e}_3", "0802e3a1e982590022e68ab61f70fe82": "\nS(\\theta) = \n\\begin{bmatrix}\n\\cos \\theta & \\sin \\theta \\\\\n\\sin \\theta & -\\cos \\theta \\\\\n\\end{bmatrix}\n", "0803326ac905a86dc32fd4241ce8ad64": "a + b^2x_{i1} + \\sqrt{c}x_{i2}", "080334dd7dda84677cf51ad5ef4b12b1": "\\frac{1}{j!}\\left(\\begin{matrix}j\\\\ \\alpha\\end{matrix}\\right)=\\frac{1}{\\alpha!}", "08034666a8a0b35592ad928e7a6a6566": "\n[d(\\rho, \\rho+d\\rho)]^2 = \\frac{1}{2}\\mbox{tr}( d \\rho G ),\n", "0803cddddef0f826dc277274439946bf": "M(v)_{,\\,v}>0", "0803d4c8f7ddb5848750e3d993739400": " \nP\\left(C(\\eta)=\\frac{1}{P[\\eta_t(0)\\neq \\eta_t(1)]}\\right)=1.\n ", "0803da122304c1fb30912df9af524179": "x=(x_1,x_2)\\in \\mathbb{R}^2", "0803e4218668589d5c676e448655369f": "\\Delta : \\mathcal C \\to \\mathcal C^{\\mathcal J}", "08042c60a97650d83931631359b0612a": "\\Delta F/2^N", "08046747cf9ae3433d1dc3ad5e362185": "E_k = \\gamma m c^2 - m c^2 \\,", "080496e06f129f12b22f04cc2c63aded": " p = \\frac{m_{A}ng} {A} ", "0804e38d3286e2ba6ec104414c6acf76": "\\lambda=\\operatorname{lcm}(p-1,q-1)", "08051a547149d7059ecdb09c2aced7cb": " 1 \\le \\phi(r) \\le 2, \\left( r > 2 \\right) \\ ", "08051e685083ef235b8272a896fbb30c": "sm = 0", "08052901962833a8403a21b0f8030372": "\\sigma^2=k-\\mu^2\\,", "080582af2aa04b597f3aaa921afb9034": "\\neg\\forall a, b, c: a R b \\wedge b R c \\Rightarrow a R c.", "0805acd495c11ae19f8559768abd03b4": " \\mathbf{F}' = \\mathbf{F} - \\mathbf{F}_\\mathrm{app} ", "0805d97b0541722b463aa4b421226d5c": "\\mathbf{z} = \\mathbf{a} + F(\\mathbf{b}-\\mathbf{c})", "08060285fdc836b29e6ee4d60c078b31": "\n\\lambda_k = \\min \\{ \\max \\{ R_A(x) \\mid x \\in U \\text{ and } x \\neq 0 \\} \\mid \\dim(U)=n-k+1 \\}\n", "0806162ce9bdde35a1a3993fa7952ce1": "|\\Phi^+\\rangle=\\frac{1}{\\sqrt{2}}\\left(|00\\rangle+|11\\rangle\\right)", "080638a7c78d56b009d7b2e6be392450": "f_i \\circ g", "0806690f6db9d9d8969e809422be28ff": "\\mathbf{v} = {\\mathbf{u}\\over\\|\\mathbf{u}\\|},", "0806ef51c9d2a0c9dd910c774ad73949": "\\eta, b > 0\\,\\!", "080716cb5ffc5e8e1b4e6b39a4ac6230": "\\mathcal{Y}=\\mathcal{F}({\\mathbf{x}})", "08071eb8a8034d9ec87257a3c7d59713": "HS_{A^{[d]}}(t)=t^d\\,HS_A(t)\\,.", "080723ca1bf64850a3528333030f5bbd": "(h_1,\\dots ,h_k)\\in Z^k", "0807e932413b4b3ecb21fe4a40041c61": " F( \\mathbf{p}/2-\\mathbf{k}) ", "08081904d0a5eebd7d63b92db97c84a1": "x_{n+1} = x_n Y_{n+1}", "080821376613570566c8bddf3543f70a": "y (\\theta) = (R - r) \\sin \\theta - r \\sin \\left( \\frac{R - r}{r} \\theta \\right),", "08082bd3966f0a2646cd6474adf4051b": "\\scriptstyle |x-a/q|<\\frac{1}{q^c}", "08082f345ef9eb8c4e5bd40840f833ad": "p, q \\in {\\mathcal M}", "0808e87ccbbea546976395761b08b042": "B=\n\\left[\n\\begin{array}{rrrrrrrr}\n-26 & -3 & -6 & 2 & 2 & -1 & 0 & 0 \\\\\n0 & -2 & -4 & 1 & 1 & 0 & 0 & 0 \\\\\n-3 & 1 & 5 & -1 & -1 & 0 & 0 & 0 \\\\\n-3 & 1 & 2 & -1 & 0 & 0 & 0 & 0 \\\\\n1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\n\\end{array}\n\\right].\n", "080996adb536d4f975e1051aae28b43d": "s\\cdot s' = g^{xy}\\cdot g^{y(q-1-x)} = g^{xy}\\cdot g^{(-x)y} =1", "0809ce6efaef359f28799282f3ffcac2": "v_{\\rm e} \\,", "080a383e8f46dc5dbd1a70d492f585f5": "uq \\equiv 1 \\pmod{m_p}", "080a5456e612ac18b948e5b40e6fc28d": "\\frac{\\partial g(\\mathbf{u})}{\\partial \\mathbf{u}} \\cdot \\frac{\\partial \\mathbf{u}}{\\partial x}", "080a54faf6477cb1ec8f81bec6eb4a9a": "Z = {X - \\operatorname{E}[X] \\over \\sigma(X)}", "080a8aa32f1c1671f52da71131a83f0c": "y=\\alpha x + \\gamma_1\\hat{y}^2+...+\\gamma_{k-1}\\hat{y}^k+\\epsilon", "080a8de67d7e3706ec46e3c447d5ce95": "\\forall s \\in S, \\; (T_h f)(s) = \\lambda f(s).", "080a8e28e05a9eccf41684405f8bdf4c": " \\Gamma( \\tfrac{1}{4}) = \\sqrt{ 2G \\sqrt{ 2\\pi^3 } } ", "080ab0960d40c23391570bd3570dd4e5": "\\varphi^{-1}(L)", "080ae704a512940d212a42aba919fd35": "W=360/365.24", "080aea70124a782bc5f04ef69cefe7b8": "{_{metric}} \\delta_{ck}^2", "080b4ec4f1a7c297b2d5f971d101496f": "f() \\,", "080b6a2a645c08f1aeb0d32ed2ddb29f": "T_{j,i}^{(t)} := \\operatorname{P}(Z_i=j | X_i=\\mathbf{x}_i ;\\theta^{(t)}) = \\frac{\\tau_j^{(t)} \\ f(\\mathbf{x}_i;\\boldsymbol{\\mu}_j^{(t)},\\sigma_j^{(t)})}{\\tau_1^{(t)} \\ f(\\mathbf{x}_i;\\boldsymbol{\\mu}_1^{(t)},\\sigma_1^{(t)}) + \\tau_2^{(t)} \\ f(\\mathbf{x}_i;\\boldsymbol{\\mu}_2^{(t)},\\sigma_2^{(t)})} ", "080b810bde9e0944409f5fab33466681": "\n \\operatorname{Var} (x) = \\frac{(b-a)^2(3-2\\theta^2)}{36}.\n", "080bc27fc4bde009fd8d81156bbbee28": "v_g = c \\left( n - \\lambda_0 \\frac{dn}{d\\lambda_0} \\right)^{-1}.", "080be99436600b3a521fc139be91959e": "\\begin{align}\n-i \\pi^2 &= \\left( \\int_R + \\int_M + \\int_N + \\int_r \\right) f(z) \\, dz \\\\\n&= \\left( \\int_M + \\int_N \\right) f(z)\\, dz && \\int_R, \\int_r \\text { vanish} \\\\\n&=-\\int_\\infty^0 \\left (\\frac{\\log(-x + i\\epsilon)}{1+(-x + i\\epsilon)^2} \\right )^2\\, dx - \\int_0^\\infty \\left (\\frac{\\log(-x - i\\epsilon)}{1+(-x - i\\epsilon)^2}\\right)^2 \\, dx \\\\\n&= \\int_0^\\infty \\left (\\frac{\\log(-x + i\\epsilon)}{1+(-x + i\\epsilon)^2} \\right )^2 \\, dx - \\int_0^\\infty \\left (\\frac{\\log(-x - i\\epsilon)}{1+(-x - i\\epsilon)^2} \\right )^2 \\, dx \\\\\n&= \\int_0^\\infty \\left (\\frac{\\log(x) + i\\pi}{1+x^2} \\right )^2 \\, dx - \\int_0^\\infty \\left (\\frac{\\log(x) - i\\pi}{1+x^2} \\right )^2 \\, dx && \\epsilon \\to 0 \\\\\n&= \\int_0^\\infty \\frac{(\\log(x) + i\\pi)^2 - (\\log(x) - i\\pi)^2}{(1+x^2)^2} \\, dx \\\\\n&= \\int_0^\\infty \\frac{4 \\pi i \\log(x)}{(1+x^2)^2} \\, dx \\\\\n&= 4 \\pi i \\int_0^\\infty \\frac{\\log(x)}{(1+x^2)^2} \\, dx\n\\end{align}", "080c1910f3ceddb0b77d33d1677d746f": "N = f_\\textrm{1}", "080c35b77a898fe3f4173f82c095f1e2": "\\lambda \\geqslant 0", "080c67fdb340842524d40951a9e00a01": "\t\n\\sum^n_{ j = 1}\t x_{ij} = 1 (i = 1,2,\\dots, n), \n", "080c684aa7d273801245f9403dfd0d83": "g_S(X_p,Y_p) = [SX_p,Y_p].\\,", "080c8c3cdf354e5e33b36c507a909315": " f(z) = z^2 \\ ", "080cab92cfc99a002d6a4b1dfc9ade56": "S=\\{S,I,R\\}^N", "080cb8b467fe966ceb89c5dd5640a0ec": "Tz = a\\,", "080d24284639989668e6784879144746": "\\theta_E \\,\\!", "080d41d579122ebf1e702b2b1f0ee762": "\\operatorname{GL}(\\infty,A)", "080dbeaf489228439b69b0722dbdae6b": "X(t,\\omega)", "080dc914e350ec23c584090040c21dc6": "A \\cap B\\,\\! = A \\smallsetminus (A \\smallsetminus B) = ((A \\cup B) \\smallsetminus (A \\smallsetminus B)) \\smallsetminus (B \\smallsetminus A)", "080e2919f17a0da7e3728a9c57407470": "\\text{ENTR} = - \\sum_{\\ell=\\ell_\\min}^N p(\\ell) \\ln p(\\ell),", "080e32483ae0e0bb79a46ade8d9e67d3": "\n\\frac{d}{dx}\\ln_{k+1}(x)\n=\\frac{d}{dx}\\ln(\\ln_k(x))\n=\\frac1{\\ln_k(x)}\\frac{d}{dx}\\ln_k(x)\n=\\cdots\n=\\frac1{x\\ln(x)\\cdots\\ln_k(x)},\n", "080e36beb9cfa3069a1b88f5413e3b7c": "\nH = \n\\frac{\\left| \\mathbf{p}_{1} \\right|^{2}}{2m_{1}} + \n\\frac{\\left| \\mathbf{p}_{2} \\right|^{2}}{2m_{2}} + \n\\frac{1}{2} a q^{2},\n", "080e6ed779b2550bb44cfac745578f00": "R=K[V].", "080e7e75310e7da29d363a822d78784b": "\\langle\\mathbf{u},\\mathbf{v}\\rangle = \\cos(\\theta)\\ \\|\\mathbf{u}\\|\\ \\|\\mathbf{v}\\|.", "080e9604620a20dbce9c4f12a20b75a1": "^\\circ", "080ee50acda7a3c58ac74c82aa11d878": " d = S_{k} + C_1 \\ S_{k-1} + \\cdots + C_L \\ S_{k-L}.", "080f4cf957395aaccc1ac5e5ba068128": "\\mathcal{M}_{1,1}\\to\\mathcal{M}_{fg}", "080f8172991adfc7f0e33535de92021c": "k(\\mathbf{x}_i,\\mathbf{x}_j)=\\mathbf{x}_i\\cdot\\mathbf{x}_j", "080fd23ae2ac271d16fda37d8d3cbc36": "GS_f", "080fe291f016a44034864c25ac1eae06": "\\|\\hat{r}\\|^2", "080fe98c239e1d74c1e726857c292e4d": "g_{ij} \\in R[x_1, \\ldots, x_n]", "080fef7c5f9dd8e2f29a0d12f8a53fc0": "\\sqrt{2} = 1.414213562\\ldots", "08108b0366c6bdf6c7e25dc050fafbd2": "v_1\\odot v_2\\odot\\cdots\\odot v_r := \\frac{1}{r!}\\sum_{\\sigma\\in\\mathfrak{S}_r} v_{i_{\\sigma 1}}\\otimes v_{i_{\\sigma 2}}\\otimes\\cdots\\otimes v_{i_{\\sigma r}}.", "0810f4ff59d7f20702a1cc960f24a1e6": " \\pi^{ji} = -(-1)^{(\\left|x^{i}\\right|+1)(\\left|x^{j}\\right|+1)} \\pi^{ij} ", "08116389efc47cea36a58a41961bb6a6": " \\frac{\\partial C_1}{\\partial t}=\\frac{\\partial}{\\partial x}[\\frac{C_1 + C_2}{C} D_1 \\frac{\\partial C_1}{\\partial x} -\\frac{C_1}{C}[D_1 \\frac{\\partial C_1}{\\partial x} - D_2\\frac{\\partial C_1}{\\partial x}]]", "081191eaef5bde95a7d1a30488cfa49d": "p, q, r \\in P", "0811bcd8179acefcb8acd67de7b25dbb": "S(w):=(w''/w')' - (w''/w')^2/2 =f", "081242d676ae2969a930140e1e7274a4": "n=14", "08126368219617f6b2a0d3fcaef58c6f": "\\Sigma _{XX} ^{-1} \\Sigma _{XY} b", "0812a08226756cfa6bfb3f7758aaf11e": "\\frac{0.22}{2.4234}=0.0908", "0812c05386e29f4d1393cc971a38ce2e": " 3x^2 + 4x -5 = 0 \\,", "08131b203dd9cd51e81e0d1480d2acd8": "CAS={EAS\\times[1+\\frac{1}{8}(1-\\delta)M^{2}+\\frac{3}{640}(1-10\\delta+9\\delta^{2})M^{4}]}", "081383cb72feaa3f7812dcdb9c2496eb": "p=p_i(T_p) e^{(E_i-E_{Fp})/(k_BT_p)}", "08138b63a12f3000e86fc0cfa0688955": " \\hat{s} = \\hat{k}z + \\hat{l} ", "0813909e5271885bd5aa895185f9fdcf": " G(s)= \\sum_{n=1}^{\\infty} g(n)n^{-s}. ", "08139df44c0a347f29afd2d37cd80953": "-\\frac{Nc}{4}(\\delta_1 + \\delta_2 + \\delta_3)", "0813cfff53030157c8ddc347189139ab": "\\zeta(1/2) \\approx -1.4603545\\!", "081401fe713bbb02014da353e28b08bb": "\\color{BlueViolet}\\text{BlueViolet}", "081442a592f1940a7dc02beb010e0512": "\\mathrm{Sc} = \\frac{\\nu}{D} = \\frac {\\mu} {\\rho D} = \\frac{ \\mbox{viscous diffusion rate} }{ \\mbox{molecular (mass) diffusion rate} }", "081489eb8d9388a69a4749dc37dafc0e": " \n s=(b-a)/3, \\,", "0814a92bf0ef3dd45cc6d933ad7ef89c": "N(M)=\\{m\\ge1|P_m(M)\\ge1\\},", "0814f52f09d242777fb573267881b8c2": "P = K_1 \\rho^{5\\over 3}", "08151ffc359809b80f90697c49d21a63": "\n\\theta_{eff}=\\cos^{-1}(\\mathbf{\\hat{n}} \\cdot \\mathbf{\\hat{v}}),\n", "08152191c9ff6afb0258f0cca95e8bee": "1.8304 0", "082699a68eb4e1ffa3bda5a8cb741212": "L_n^{\\alpha }(z) = \\frac{z^{-\\alpha }e^z}{n!} \\frac{d^n}{d z^n}\\left(z^{n + \\alpha } e^{-z}\\right)=\\frac{\\Gamma (\\alpha + n + 2)/\\Gamma (\\alpha +2)}{\\Gamma (n+1)} \\, _1F_1(-n,\\alpha +1,z),\n", "0826be3909e4965656ced1974ed6ca41": "\\varphi(s(x)\\cdot g) = g.", "0826d5f1bef545293a9cb5ed76cbda38": "-\\frac{ \\text{polylog}^2(2,1-p)}{\\beta^2\\ln^2 p}", "0826dd36fc273a36a97993c3382d7596": "\\mathbf{\\hat{b}_{t:T}}", "08273b5090556d1402f1105ddf5a4078": "B_{max} = \\tfrac{1}{M}\\cdot\\tfrac{1}{2T},", "082743bdd2a37f8be8077c266c1fbaa5": "N_{E}/N_{NE}\\approx H_{E}/H_{NE}.", "082795f5c36ed57a1b4346e3e867a969": "(x_1 x_2 + N y_1 y_2,", "0827d56851adbf127a6df596e9c23635": "X(t) = \\left( \\frac{\\nu}{\\nu+1} \\right)^{\\nu} K ", "08280de5642c348a3099e37d1398975d": "T(v)", "0828167ec4f6eb3a1077cbb76d4664e2": " \\tfrac{365.242\\ 190\\ 402}{366.242\\ 190\\ 402} ", "0828559d0f38e03671647e62aedded42": "\\text{return} \\colon T \\rarr S \\rarr T \\times S = t \\mapsto s \\mapsto (t, s)", "08288b8cef1b440b233e7b2aa8b74202": "\\texttt{fix}_\\alpha", "0828a0d0be30dd9628148fcafc9c67df": " \n\\sum_{n\\le \\lambda} \\left(1-\\frac{n}{\\lambda}\\right)^\\delta\n= \\frac{1}{2\\pi i} \\int_{c-i\\infty}^{c+i\\infty} \n\\frac{\\Gamma(1+\\delta)\\Gamma(s)}{\\Gamma(1+\\delta+s)} \\zeta(s) \\lambda^s \\, ds\n= \\frac{\\lambda}{1+\\delta} + \\sum_n b_n \\lambda^{-n}.\n", "0828afe50f227250fa69a5f682bc0512": "\n\\int_1^\\infty e^{iax}\\frac{\\ln x}{x} \\, dx = -\\frac{\\pi^2}{24} + \\gamma\\left(\\frac{\\gamma}{2}+\\ln a\\right)+\\frac{\\ln^2 a}{2}-\\frac{\\pi}{2}i(\\gamma+\\ln a) + \\sum_{n\\ge 1}\\frac{(ia)^n}{n!n^2}.\n", "0829042bf44637ca470ca32478ff2b1c": "F(y_1)=F(y_2)=\\cdots=0\\ \\Rightarrow\\ F(y_1+y_2+\\cdots)=0", "082922b6768a542390e435095ceb28ec": " Ly = f", "08292eabfd0980c97be52ef60fc47f6d": "u^T \\nabla f(x)", "0829471378a17c0994cfa3d084c38ad2": "\\pi/2-\\varphi-\\theta_0", "08297f9dcc77d2e8f0a9d461fc8d29a7": "p>0", "082980c9e438c59723e0889fafc1ca87": " \n[0,1]", "0829b30266db4e8501632d3b33671a11": "\\mathrm{SU}(n)", "0829bfcd7e8d1ecee8e9cc2b579d116d": "k(\\mathbf{x}_n,\\mathbf{x}_m^j)", "0829f8fca5e0ab811b2aa5af19d80c80": "u_j^n", "082a31c2eaac4c1aae03bb98e21e5a25": "K(u) = \\frac{15}{16}(1-u^2)^2 \\,\\mathbf{1}_{\\{|u|\\leq1\\}}", "082a5766388b2cb393e8e535301c16a7": " V_0 = \\frac{V_{\\max}[S]}{K_m(1 + \\frac{[I]}{K_i}) + [S]}", "082a82b8b71f83992035b4be4c776ac8": " p(x) = {\\alpha \\over \\lambda} \\left[{1+ {x \\over \\lambda}}\\right]^{-(\\alpha+1)}, \\qquad x \\geq 0,\n", "082b0e41ac5d0f86aa6e51587580f3b7": "\\ln(x)", "082b7febbb152527ae1f05d1bbb8c49b": "\\Delta b_T", "082bd2666489a522185b37cc49581cb8": "\\frac{2}{3} \\times 2", "082be05223beba07f8c61abaf1f9f14b": "\\lambda _j", "082bf236294ff058c05fd953990bafb0": "m=2^k", "082c05cd77606b370a83c09a4a24e33e": "n_\\max", "082c80009e98668e0306f23c4ffac32a": "\n\\mathcal{G}(\\tau - \\beta) = \\zeta \\mathcal{G}(\\tau),\n", "082cc266a32b400880939844673e26b8": "e > 7.5 n,\\,", "082ccfc0d25d2ed24504b86f937e4b22": "N(0)=N_0", "082d71cfb09c97b2c7c4dbc53d80bd8a": "\ndV = \\frac{\\left( \\mu - \\lambda \\right) \\left( \\nu - \\lambda \\right) \\left( \\nu - \\mu\\right)}{8\\sqrt{\\left( A - \\lambda \\right) \\left( B - \\lambda \\right) \\left( A - \\mu \\right) \\left( \\mu - B \\right) \\left( \\nu - A \\right) \\left( \\nu - B \\right) }} \\ d\\lambda d\\mu d\\nu\n", "082d758dbed4791e7613866dcd5ec11a": "\\sgn(x) = \\frac{|x|}{x}.", "082d91fadc58f3d6b7aa476f2597c401": "\\mathbb{E}f^2", "082dccaf0370dae658dfa85c173de3c4": "I_C = C\\frac{dV_C}{dt}", "082ddaa8702fdacb652990136fb6bc1a": "\\mathrm{tr}", "082df00ed1cc1efbbe12198fb5cf2f6d": "C_2= \\left[ \\begin{array}{rrr} \n1 & 0 \\\\ \\\\\n0 & 1 \n\\end{array} \\right] - \\frac{1}{2}\\left[ \\begin{array}{rrr} \n1 & 1 \\\\ \\\\\n1 & 1\n\\end{array} \\right] = \\left[ \\begin{array}{rrr} \n\\frac{1}{2} & -\\frac{1}{2} \\\\ \\\\\n-\\frac{1}{2} & \\frac{1}{2} \n\\end{array} \\right]\n", "082e734e8aa89c91fcc1a922d9d5adca": " -[R][R]= -\\begin{bmatrix} 0 & -z & y \\\\ z & 0 & -x \\\\ -y & x & 0 \\end{bmatrix}^2 = \\begin{bmatrix}\n y^2+z^2 & -xy & -xz \\\\ -y x & x^2+z^2 & -yz \\\\ -zx & -zy & x^2+y^2 \\end{bmatrix}.", "082eb188ca14a4fabe391a68adbed0c0": "m/e", "082eec537b35280f43027e668cbbff39": " \\pi_{(t+1)} ", "082f7bbea8ce98e9c5f929f1f8c2fc5f": "Y[x,y]=y-\\frac{y'\\int_a^t \\sqrt { x'^2 + y'^2 }\\, dt}{\\sqrt { x'^2 + y'^2 }}", "082f847a67de8a9e2091c3751e10723c": "<0.58", "082fb23fc490236c1dfcf8dba5364e34": "I_{\\mathcal Q}(+)\\colon Q\\times Q\\to Q", "08301fef54b97a39b5180cf46bdb7ded": "r_1 = (S \\to AA, \\{r_1\\}, \\{r_2\\})", "083028550121a5b354ece52f326b89be": " \\ x_d = (x - x_0)/(x_1 - x_0)", "08305728f3551363b41853a6bc90f96f": "W=W_1W_2", "08308e352cf2d86d3b78ca048a75d173": "0<\\alpha<1", "083090b0d66349f0269b1f1393604346": "\\displaystyle{(H^\\varepsilon)^*=JUH^\\varepsilon U^*J.}", "08310a10e061182e64df26263c08539c": "\\int_t^ {t+\\Delta t} \\!\\!\\!\\int\\limits_{cv} \\rho c \\frac{\\partial T} {\\partial t}\\,dV\\,dt = \\int_t^ {t+\\Delta t} \\!\\!\\!\\int\\limits_{cv} \\frac{\\partial \\frac{ k \\partial T} {\\partial x}} {\\partial x}\\,dV\\,dt + \\int_t^ {t+\\Delta t} \\!\\!\\!\\int\\limits_{cv} S\\,dV\\,dt", "083151f616e8598aea21074ac234d885": "\\vert\\psi\\rangle=\\vert\\psi_A\\rangle\\otimes\\vert\\psi_B\\rangle", "083174abb898e41376cb3a38fd8b37ce": " z + pl(a - p) + t(2ap - p^2 - 1) - pm ", "0831d1ccb710165c736b75b02f24aa58": "\\varepsilon_i=X_i-\\mu,\\,", "0831f6d6a574c6fe68549253aba4d8e6": " ds^2= \\frac{1}{2\\omega^2} [ -(dt + e^x dz)^2 + dx^2 + dy^2 + \\tfrac{1}{2} e^{2x} dz^2], \\qquad\\qquad -\\infty < t,x,y,z < \\infty,", "0832093a01e09dc2dc09ef5c91b8e566": "\\left\\langle\\sqrt{R},Z_R\\right\\rangle", "083277b08629dc9df83286c5b5b43b18": "\n\\mathbf{N} = \\frac{d\\mathbf{L}}{dt} = \\dot{\\mathbf{r}} \\times \\mu\\dot{\\mathbf{r}} + \\mathbf{r} \\times \\mu\\ddot{\\mathbf{r}} \\ ,\n", "0832cc7679d7455016051857d736b9f9": "K\\subset\\mathbb{P}^3 ", "083319a7938c6e9efe4be8a7e6f2cc5e": "pV^0 = p", "083344515659c94737ef0b7d6a2bf7ff": "(\\hat{c}_P/\\hat{c}_V)", "083355a2c49ec103aa16669eedbb3d32": "\\lambda(s)", "083356749e6c1731ec6a005d841d8f37": "(1,8,1)\\rightarrow(1,3)_0\\oplus(1,2)_{\\frac{1}{2}}\\oplus(1,2)_{-\\frac{1}{2}}\\oplus(1,1)_0", "0833d3565a5f83b2ffa0779c50fc0158": "{\\epsilon} = 1", "083406ed4ca69b03b58821b806ef6a99": "\\displaystyle{Q(a,b)=L(a)L(b)+L(b)L(a)-L(ab).}", "08349d33537fafa342d491d1d322c862": "\n\\ln\\Omega_{E,\\ell} = \n\\ln\\left(\\ell^n n\\pi^{n/2} (2E)^{\\frac{n-1}{2}} \\right) \n- \\ln\\left[ (n/2)! \\right]\n", "0834b3533011a374c3847ad2ab279f68": "\\scriptstyle \\gamma_\\mathrm{sa}", "08354324d53a50f3f2147c9798231c7e": "\n\\sum_{k_1+\\dots +k_y = 0}^x {n\\choose k_1} {n\\choose k_2} {n\\choose k_3} \\cdots {n \\choose x - \\sum_{j = 1}^y k_j } = { \\left( y + 1 \\right) n \\choose x}.\n", "0835593c0e39feb8bbf480625454a569": "\\cos\\theta = \\frac{r^2 + R^2 - s^2}{2rR}.", "083596369fc033fba72e6a4077de3370": "R_{\\mathrm{K-90}} \\,", "0835a821464e35efbf9a2a32606055ec": "M+L \\rightleftharpoons ML:\\log \\beta_{110} =\\log \\left(\\frac{[ML]}{[M][L]} \\right)", "083629f5a933f3d7a97c03d3357eacb2": "\\frac{ d}{ dt}q(t) =~~\\frac{\\partial}{\\partial p}\\mathcal{H}", "08363353e10c2a2c28ee3c648bb8cf95": "d = d_0 + 2d_1 + 2^2d_2 + \\cdots + 2^md_m", "0836499f686226206f227fa15514a074": " D = \\frac{N}{P_d} = \\frac{Np}{\\pi}= \\frac{N}{P_{nd} \\cos\\psi} ", "0836558734369e6ee1e47366bba90ecc": " \\Delta u = \\nabla^2 u = 0\\,", "083666aab531d20c3136aee3399616e8": "\\sinh{(\\chi_{nk}/2)}", "083699762748f512cc92b85e3f9ef4de": "{\\rm Tr} A^{\\dagger}_i A_j \\sim \\delta_{ij} ", "083712f0134574c67ce76b6023adbb11": "l_1 = m_1 = \\frac{1}{\\sqrt{2}}\\sin{y}, \\qquad l_2 = m_2 = \\frac{1}{\\sqrt{2}} ", "08374f72a08d06014d01e9bf8694fba4": "\\mathbf{u}^n", "083755c68cacbe992fce507c1bc2a2ed": "f(x) = x^n", "083788997e7b8122e52fb3f583094fea": "\\phi:M\\to \\prod_{i\\in I}R\\,", "0837a60eda63e2be86c97145714d016b": " \\operatorname{build-param-lists}[q, D, V, T_7] \\and \\operatorname{build-param-lists}[q, D, V, K_7] ", "0838368df43c46fcc7e86b727b6836ef": "-e^2 \\left( \\bar{v}_{k} \\gamma^\\mu v_{k'} \\right) \\frac{1}{(k-k')^2} \\left( \\bar{u}_{p'} \\gamma_\\mu u_p \\right) ", "083882cffb2ef35c88a184362f82f323": "M_k = 0", "0838e5f328b2875d3328a756d49f7cf5": "\\frac{\\mu}{\\mu^3+1} \\to \\frac{\\mu^2}{\\mu^3+1} \\to \\frac{\\mu^3}{\\mu^3+1} \\to \\frac{\\mu}{\\mu^3+1} \\mbox{ appears at } \\mu=\\frac{1+\\sqrt{5}}{2}", "08397d0e51ccf0875df8733befda49ec": " \\Phi ( \\omega) := \\begin{cases}\n\\frac {1}{\\sqrt{2\\pi}} & \\text{if } | \\omega|< 2 \\pi /3, \\\\\n\\frac {1}{\\sqrt{2\\pi}} \\cos\\left(\\frac {\\pi}{2} \\nu \\left(\\frac{3|\\omega|} {2\\pi}-1\\right) \\right) e^{j\\omega/2} & \\text{if } 2\\pi/3<|\\omega|< 4\\pi/3, \\\\\n0 & \\text{otherwise}. \\end{cases}", "0839e30ddc2fefd6f1969bed87662de6": "L_r = \\frac{\\rho}{\\pi}\\cdot \\cos \\theta_i\\cdot L_i", "083a3b265a2be7a470c7a5ed9adbe3ca": "P_\\infin", "083aad334d5cb8d6a6f2849a925a32ca": "\\alpha\\in\\pi_p(M)", "083abb9c2e30db4852bf9496d81dbd29": "\nZ(\\omega)=\\frac{R_{\\text{t}}}{1+R_{\\text{t}}\\,C_{\\text{dl}}\\,\\text{i} \\,\\omega}\n", "083b02e64459f1366c1c661bc62e6bb6": "g_p(X_p,Y_p) = g_p(Y_p,X_p).\\,", "083b179c24b4527a892dc84eea627723": "h_{\\bar{a}}(\\bar{x}) = \\left(\\big( \\sum_{i=0}^{k-1} x_i \\cdot a_i \\big) ~\\bmod ~ 2^{2w} \\right) \\,\\, \\mathrm{div}\\,\\, 2^{2w-M}", "083b51c0c079e0adf3b1cc39aee3f094": " (Bf)(z) = \\int_{R^n} \\exp[-(z \\cdot z - 2 \\sqrt{2} z \\cdot x + x \\cdot x)/2]f(x) \\, dx, ", "083bf6e702235e5aae317792c3039ec5": "\\theta(t_iht_i^{-1})", "083c122be64a93c9170c41190ee88bbb": " T = - \\frac {i_r} {i_t} \\ . ", "083c1e6df0c761e36d8f0b4101147a39": "\\mathbf{U} \\equiv \\{\\mathbf{u}_0,\\mathbf{u}_1\\dots,\\mathbf{u}_{N-1}\\}", "083c297e4f5354265325e6158b31391c": "V(\\rho,\\varphi,z)\n=\\frac{1}{R}\n=\\sum_{n=0}^\\infty \\int_0^\\infty dk\\, A_n(k) J_n(k\\rho)\\cos(n(\\varphi-\\varphi_0))e^{-k|z-z_0|}\n", "083c7e987b886df53d04d5c25bfce4ca": "\\{e^{(2+i)x},e^{(2-i)x}\\}", "083c7ee847c7db9a6eab38b6596f2346": " X = \\mathbb{R}", "083c8120d00c4c74fa7cf4dd43b85702": "(n_x,n_y,n_z)=(3,2,5)", "083cc8cbc7e1b7f714ba0183d816e728": "\\frac{dL}{dt}v+LMv=\\frac{d\\lambda}{dt}v+MLv.", "083d0215515f5e00863f7811058f7fda": "\n\\begin{align}\ng_x (m)&\\overset{\\underset{\\mathrm{def}}{}}{=} m \\cdot x \\mod (2^{32}+15)\\\\\n&=\\textstyle \\sum_{i=1}^k m_i \\cdot x_i \\mod (2^{32}+15)\n\\end{align}\n", "083d108ada4cf78366a41ee889ef5928": "x_1, x_2,\\dots, x_m", "083d1c80c7d3a4a7825bdfe66f59f876": "\n\\begin{matrix}\n\\;\\; x \\;- 10\\\\\n\\quad x^2-2x+1\\overline{) x^3 - 12x^2 + 0x - 42}\\\\\n\\qquad\\qquad \\underline{x^3 - \\;\\;2x^2 + \\;\\;x}\\\\\n\\qquad\\qquad\\qquad\\qquad -10x^2 - \\;x - 42\\\\\n\\qquad\\qquad\\qquad\\;\\;\\; \\underline{-10x^2 + 20x - 10}\\\\\n\\qquad\\qquad\\qquad\\qquad\\qquad\\;\\; -21x - 32\n\\end{matrix}\n", "083e5d4ffdeeedeb3641fd1bbf13c265": "8.314\\,472(15)~\\frac{\\mathrm{J}}{\\mathrm{mol~K}}", "083e7ada1c8323f435fbe16ebb718dd6": "\\frac{\\partial a}{\\partial \\mathbf{x}} =", "083ea38cb2452079b0c6997df773c72c": "\\partial W'", "083ef5b128bb1a4a74b04690aa8e1e2b": "F_x>0", "083f2607d761b21e45382928bd38a900": "145 = 12^2 + 1^2 = 8^2 + 9^2", "083fd908b329b68d7baceb6d862c0884": "\\mathbf F=\\langle F,\\le,V\\rangle", "083ffc82346f253eb19efba5d89fa30f": "D:C\\rightarrow J", "08400f197df75d3490dde4e0fa924b53": "(\\cos \\theta +i\\sin \\theta)^k= \\cos k\\theta +i\\sin k\\theta \\quad \\Rightarrow \\text{Li}_n\\left(e^{i\\theta}\\right)=\\sum_{k=1}^{\\infty}\\frac{\\cos k\\theta}{k^n}+ i \\, \\sum_{k=1}^{\\infty}\\frac{\\sin k\\theta}{k^n}", "0840340ef8c42b4ba4f20e835ea4fefd": "\\mathbf{F}_{\\rm R} = - \\lambda \\mathbf{v} \\, ,", "084074297524e154da07aa0f417397c7": "\\zeta(3)= 14 \n\\sum_{k=1}^\\infty \\frac{1}{k^3 \\sinh(\\pi k)}\n-\\frac{11}{2}\n\\sum_{k=1}^\\infty \\frac{1}{k^3 (e^{2\\pi k} -1)}\n-\\frac{7}{2} \n\\sum_{k=1}^\\infty \\frac{1}{k^3 (e^{2\\pi k} +1)}.\n", "08407cb51853afc254d75629bf04ae2d": "Q = 0", "08408f2e955c657229534b324d6daeac": "t = \\frac{1}{i}\\ln(iy + F) + k", "0840a5e804bda6f0bb5fb19bc26da1b7": " {\\Phi} ", "0840daf69b940521edcb979ce016caac": "d_2(f(x),f(y))=d_1(x,y)\\quad\\mbox{for all}\\quad x,y\\in M_1", "0840f67d544559e630d781f6e7a37260": "\\mathbf{h}P_\\pi\n= \n\\begin{bmatrix} h_1 \\; h_2 \\; \\dots \\; h_n \\end{bmatrix}\n\n\\begin{bmatrix}\n\\mathbf{e}_{\\pi(1)} \\\\\n\\mathbf{e}_{\\pi(2)} \\\\\n\\vdots \\\\\n\\mathbf{e}_{\\pi(n)}\n\\end{bmatrix}\n=\n\\begin{bmatrix} h_{\\pi^{-1}(1)} \\; h_{\\pi^{-1}(2)} \\; \\dots \\; h_{\\pi^{-1}(n)} \\end{bmatrix}\n", "084118391910b6cfbb23e955f4b22d3b": "(A,M)", "08416aaefd3999c7364c585a2356b21e": "\\frac{1}{\\sqrt{2\\pi}}", "08416c42cb79122fe1dc357ff747b62e": "H \\cdot t= a \\cdot b \\cdot ( e \\cdot \\sinh E-E)", "0841816b3be3adc912b9cbf086eafdee": "\\Phi^{(k+1)}(\\omega)= \\frac {1} {\\sqrt 2} H\\left( \\frac {\\omega} {2}\\right) \\Phi^{(k)}\\left(\\frac {\\omega} {2}\\right)", "08423b8a7f05f162d5c5c9a18244439a": " \\int_{E}f\\,d\\mu=\\int_{K}f\\,d\\mu,~~~\\int_{E}f_n\\,d\\mu=\\int_{K}f_n\\,d\\mu ~\\forall n\\in \\N. ", "0842704f9234ba2e15fc47efddceecc5": "t'=t-\\tfrac{vx}{V^{2}}", "0842e280cea535a1d3a1cf35ffe3ab33": "A\\circ B = (A\\ominus B)\\oplus B, \\, ", "0842f9b0000dee93b7b1847d9ee4ff10": "S_s", "084318b834ba16c555d4e360aa779fe3": "\\arccsc (-x) = - \\arccsc x \\!", "084343c957422bb56d98768da6c03fa7": "\\tau^{a}{}_{b}\\,", "084344599a70dded3295c6e46638db85": "J_2\\,", "08438591d46590c6aecfd370bec7d16a": " \\Lambda_{p \\times p} = \\text{diag}\\left[\\lambda_1,...,\\lambda_p\\right] = \\text{diag}\\left[\\delta_1^2,...,\\delta_p^2\\right] = \\Delta^2 ", "0843dae813a7fd5e76f86100feda9ad0": "M_{a}", "084423d98402d5fa10725f9077145a37": "y_{21}-y_{22}", "08443545122b333b15f6ec00f846a254": "y^{\\prime}(s) = \\cos \\frac{s}{\\alpha} \\ ; \\ x^{\\prime}(s) = -\\sin \\frac{s}{\\alpha} \\ , ", "0844d68955e074f574e9d409b6d4d824": "G(\\xi ) = \\frac{3}{{\\xi ^2}}(\\sin \\xi - \\xi \\cos \\xi )", "0845029a236f14f39e20c5ea6b6b684c": "\\sum_{k=0}^\\infty \\frac{\\sin[(2k+1)\\theta]}{2k+1}=\\frac{\\pi}{4}, 0<\\theta<\\pi\\,\\!", "084527d395401ea9842baa5edd84917e": " 0 \\le S \\le 1 - \\log_e( 2 ) ", "08453705d77fef0f311613ac0801a483": "\\nabla^2\\mathbf{B}+\\alpha^2\\mathbf{B}= \\mathbf{B}\\times\\nabla\\alpha ", "0845854e993df19bf2fcf8d8bde95597": "SU(2)_L SU(2)_R", "0845a06ae634f99a58eba196e6e625d3": "H_k(X;A)=A^{r_k}", "0845d29982824ca8a53057468e27ed4b": " X \\leq_{HYP} Y", "0845d6c99d3a90e1ec21ad8c268fba78": "H_{ij} = {- 1 \\over {s_{ij}}^{p+2}} \\begin{bmatrix} {(X_j - X_i)(X_j - X_i)} & {(X_j - X_i)(Y_j - Y_i)} & {(X_j - X_i)(Z_j - Z_i)}\\\\{(Y_j - Y_i)(X_j - X_i)} & {(Y_j - Y_i)(Y_j - Y_i)} & {(Y_j - Y_i)(Z_j - Z_i)}\\\\{(Z_j - Z_i)(X_j - X_i)} & {(Z_j - Z_i)(Y_j - Y_i)} & {(Z_j - Z_i)(Z_j - Z_i)} \\end{bmatrix}", "084604aee805ea5d2248e1c7ea23dd00": "{\\mathit{momentum} \\over N+1} = \\mathit{SMA}_\\mathit{today} - \\mathit{SMA}_\\mathit{yesterday}", "08462e114376476a4ef1bd786c212e13": "(\\cos(\\theta/2) - i \\sigma_3 \\sin(\\theta/2)) \\, \\sigma_3 \\, (\\cos(\\theta/2) + i \\sigma_3 \\sin(\\theta/2))\n= (\\cos^2(\\theta/2) + \\sin^2(\\theta/2)) \\, \\sigma_3 = \\sigma_3.", "084653f60e71e4bdbc21bf91255609bc": " \\sigma_h = K_p \\sigma_v + 2c \\sqrt{K_p} \\ ", "0846b0ed8e72421537d7de82ee54c153": " \\frac{\\partial u_i}{\\partial x_i} = 0 ", "08473ca91ebe8d021888034cd81cb7f4": "\\,^{254}_{99}\\mathrm{Es} + \\,^{48}_{20}\\mathrm{Ca} \\to \\,^{302}_{119}\\mathrm{Uue} ^{*} ", "0847587afea00f4f42715cef67daa019": "\\theta = \\theta^\\prime", "0847c8ffad3aecaedb53f0fa2fd535b3": "\\left [\\begin{smallmatrix}2&-1\\\\-5&2\\end{smallmatrix}\\right ]", "0847d8c57819029175a2455933cfc696": "{R_{abc}}^d+{R_{cab}}^d+{R_{bca}}^d = 0.", "0847df7b9c5fb53a214fc80bc5df2df3": " F(k;n,\\tfrac{1}{2}) \\geq \\frac{1}{15} \\exp\\left(- \\frac{16 (\\frac{n}{2} - k)^2}{n}\\right). \\!", "0847f615f040b9e64a573923df4ad112": "x_\\star", "08488f06ff28d094e8c234b086f25853": " U = 1/(1/h_1 + dx_w /k + 1/h_2) ", "0848995767d6cd9b895abe93ebe53dc5": "v_i(0)", "0848b32d4a785ca97d04be1e69de3936": "\\Gamma_5", "0848d20fb47e7b315af38020c6c07856": "\\lambda(y) = X_1^2(y) + \\cdots + X_k^2(y)", "0849035a0f0432ba8fb8aaeb65740b8f": " \\nabla^2 f(x) - mI", "08493db2571077516e5f8ffcbed059f3": "r_1 > 0", "08494b4722a778ebbcc25dbe73aa019d": "\n \\underline{\\underline{\\mathbf{A}}} = \\begin{bmatrix} A_{11} & A_{12} & A_{13} \\\\ A_{21} & A_{22} & A_{23} \\\\\n A_{31} & A_{32} & A_{33} \\end{bmatrix}~.\n ", "08497086c0bdc9930f7bd56fb588aad2": "S_{mn} = S_{nm}\\,", "0849758f0b601830f420400199146924": "\\Delta p = p_{i,x} - p_{f,x} = p_{i,x} - (-p_{i,x}) = 2 p_{i,x} = 2 m v_x\\,", "0849ede8f3d74e1e41d884bb7b22c900": "I_{L_{Max}}", "084a1377b87cd676874736d2e07744da": "\\frac{E}{m} = K\\left(\\frac{\\sigma}{\\rho}\\right)", "084a36e9759b6be6152a5494fc9f7163": "i_a(t)+i_b(t)+i_c(t)=0", "084a5d14a846449b88da90388d0d1be7": "\\ \\Delta^r(\\alpha_{i,j,k}) = \\alpha_{i+1,j,k} - \\alpha_{i,j,k} ", "084a799f353c0237d5625a23ad626f5d": "[A,B]=0", "084ad8fd849249aa258d43a5aed6914c": " L_k = R(t)e^{\\beta_k} ", "084b34dd6de2eee9941ae886d11b6eef": "(13)\\quad Z^c\\nabla_c B_{ab}=-B^c_{\\;\\;b}B_{ac}+R_{cbad} Z^c Z^d\\;. ", "084bd624309690df3a29b3fcb906d838": "S_k (n, r) = \\mathrm{Hom}_k( A_k (n, r), k)", "084c24cc32297f7667d9742433e36289": "n:=n_0", "084c32ca00e00fb3895b49d744769c4b": "S(t) = \\frac{1}{\\pi}\\arg{\\zeta\\bigl(\\tfrac{1}{2}+it\\bigr)}", "084c6df1514b475c70c295022e8919ad": "\\{P_i, y_i\\}_{i=1}^n ", "084cea1b18a3467fe33db6e9df06f713": "E = \\frac{k\\cdot P\\cdot M}{R\\cdot T_A}", "084d3e56c35a1b9fd8fe110f3c87efba": "P=\\frac{RT}{V_m-b}-\\frac{a(T)}{V_m(V_m+b)+b(Vm-b)}", "084d5ffb9a91a4f0f9b641f773f265e5": "\\max_{s\\in X} U(s)", "084d6dc7cd051708220858e4a210b1ff": "\\sigma^2_c = \\frac{f^2 \\mathcal{L}\\left(f\\right)}{f_{osc}^3}", "084d8b189a67229cdd81f3908435f717": "\\scriptstyle \\mathbb{C}^2\\equiv\\mathbb{R}^4", "084db482325b782cb718e9996b9d11cb": " x \\wedge y = - y \\wedge x. ", "084dbad49b48a53102cc8188e8b6ade0": " \\theta_{p,\\omega}^{A} = { \\mu_{p,\\omega}^{A}, \\Gamma_{p,\\omega}^{A} } ", "084dc1e813d6d2d950d528fce1a6f476": " \\mathcal{C}_{XY}: \\mathcal{H} \\mapsto \\mathcal{H} ", "084de0b9fa9f912d5fd22a4fabe760f0": " B_k r_k ", "084de299e2ce516d42cbc70e8cccdfd0": "D(s,\\mathbf{x})", "084e15e0ce97e3947839329e00df6765": "|z_k - z^*| < \\epsilon", "084e182e56a68d767225d1158ccc4b65": "\\! 1-p+pe^{it}", "084e192fa77eb12c5a06d1f38700f56a": "\n|X|E_{k}=\\sum_{i=0}^n q_k\\left(i\\right)D_i. \\qquad (7)\n", "084e2f1d2e7342f2ac07cc95c74eba59": "[\\alpha]_\\lambda^T = \\frac{ \\alpha}{l \\times c}", "084e6ec023a3270d969c54e4a1962174": "U_\\text{Inner}=U_\\text{Outer} \\, ", "084eaec7015459fe305fcfdc78d71eed": "i_{\\ast}: T_p S \\to T_p M.", "084eb83efe38c09c9af3dc570f63eca3": " \\omega \\in L(\\mathcal{G},t)", "084eddb071091218f302917204e898e3": "\\tau_e = 1/\\dot{\\gamma}_e ", "084ee992ed154852424d0e0d7029d060": "v_{xo}", "084f5c4e013ea2ab9d5e8d1933dcd5ff": "\\mu=\\pi\\left(\\sqrt{m}\\right)-n", "084f86393f8363b1a3fd75a5f11dec3b": "X \\setminus V", "08504aadd8d0c81440a057ac70fd754d": "\\Delta m^2_\\text{atm}\\simeq2.5\\times10^{-3}\\,\\mbox{eV}^2", "08507709a2cd321cc65a78f8ba1ec1ba": "E[\\xi]=\\int_0^{+\\infty}(1-\\Phi(x))dx-\\int_{-\\infty}^0\\Phi(x)dx", "0850f79692ab55eac984ab3f24553961": "\\begin{align}& j = \\ell +s \\\\\n& j \\in \\{|\\ell-s|,|\\ell-s|+1 \\cdots |\\ell+s|-1,|\\ell+s| \\} \\\\\n\\end{align}\\,\\!", "08516cd80144421850e3d9e4a1b94afe": " \\nabla F = \\mu_0 c J ", "085199cb15978a3d6a78c9457b9d493c": "\\frac{Av}{\\|Av\\|}, \\frac{A^2v}{\\|A^2v\\|}, \\frac{A^3v}{\\|A^3v\\|}, \\dots", "0852021ce522cd53d93bd5b14df57407": "\\hat{a_1}=0.0135", "0852986d0ab05f20afeb2fa9d9aefe46": "f_i(r_1,\\dots,r_{i-1},0)", "0852b97b4adc36613e04de12789470ae": "(f*\\Delta)(x) = \\sum_{n=-\\infty}^\\infty f(x-n).", "0852f6aa94a4da23a34ea914a9cf154e": "L\\psi_n(x)=\\omega_n \\psi_n(x)", "08530922c2c40dd06bb8e686fa483d8a": " \\mathrm{Eq}(f,g) := \\{x \\in X \\mid f(x) = g(x)\\}\\mbox{.}\\! ", "08533b111de373fc0db9daf66054c188": "z_1 = x_1 y_1 - x_2 y_2 - x_3 y_3 - x_4 y_4 + u_1 y_5 - u_2 y_6 - u_3 y_7 - u_4 y_8", "08535a977c369045347032867917dc94": "P^{-1}\\mathcal{F}P", "08542bb73cc183c2d33c10db7cd7cabf": "\nP = \n\\begin{bmatrix}\n\\frac{2}{right-left} & 0 & 0 & -\\frac{right+left}{right-left} \\\\\n0 & \\frac{2}{top-bottom} & 0 & -\\frac{top+bottom}{top-bottom} \\\\\n0 & 0 & \\frac{-2}{far-near} & -\\frac{far+near}{far-near} \\\\\n0 & 0 & 0 & 1\n\\end{bmatrix}\n", "0854472019d887246601ca81de5b0db2": " M(bx_1, \\ldots, bx_n) = b M(x_1, \\ldots, x_n) ", "085457224bf55d74398d25780ba8fd76": "\\int_a^b \\sqrt{1+\\bigg(\\frac{dy}{dx}\\bigg)^2}\\,dx,", "08547d225b1df1e24d9be337f9e8af3e": "xy^{-1}", "0854dfc743e3e4a31e7518f7d497e948": "i = 2i + 2\\left(-\\frac{1}{2}i\\right) = 10.2_{2i}", "0855377f16232c112cc7334456e21ad1": "x_i = r_i^{ - \\beta }", "08555285a6e8334c4942b72b27d88873": "\\ f*g ", "085555ba4b282a05cc52051f8edcd94e": "G/\\tau = K_I", "085611d3a8521b914adcd17e88f6a519": "E W = 0", "0856444a7eabf3601113168835553a3e": "d\\Omega_{k^\\prime}", "08570ba4434de2bfe6476ed45505337d": "\\mathcal{H}\\subset \\mathcal{K}", "08571639e1475d1bb81ae83fd725b12b": "N\\in\\mathcal F", "08577d8512a05cc521fbf111658ff6c0": "\\! g_i", "0857bf5d4e53bc8479a34538b4bdc91e": "\\! c", "0857c3604f27a1e80c6f56addbaa2d84": " UltOsc = 100 \\times {4 \\times avg_7 + 2 \\times avg_{14} + avg_{28} \\over 4 + 2 + 1 } ", "0857c70c656c7aaf930e3277a1139c94": "\\scriptstyle dp_B(R,t)", "0857e7053d4575070702ecd46cf668f7": "\\lVert x - y \\rVert", "085827177b928f9c0eff1964846cb77b": "0 < \\alpha\\ < 2^{160}\\,", "0858850a595e42d35f5eeb6b1c650e70": "\\operatorname{Hdg}^*(X) = \\sum_k \\operatorname{Hdg}^k(X)\\,", "0858a3974acfeb642f2364ee49fb05c2": " \\Pr[c \\in B(y, pn)] = \\Pr[y \\in B(c, pn)] = \\mathrm{Vol}(y, pn) / q^n \\ge q^{-n(1-H_q(p)) - o(n)} \\, ", "0858c12dc07329bf7e1c4bad5c47cd67": "{s}", "0858c8578fe0a24f05bb5502453ed3c0": "A_{c,b}", "0858eea75fa6bc1ea46d85bbe8f3c291": "-\\omega_{n}^{2} f(t) q", "085967d7b3ee450c85364a7461e22302": "u_2(z)", "085a08ff09618180f9c43a171281093c": " - \\tilde J = M \\nabla (\\mu_a - \\mu_b) ", "085a49f2fa9d72fbe2d5d4073bfa1677": "\\tilde{S}_{2} =M_{2}-m_{2}+\\frac{i}{2}G\\gamma _{1}\\cdot {\\partial }\\mathcal{\nL}{.} \n", "085a66174db412c03f8c7e11e844c21e": "\\lim_{t \\to \\infty} \\frac{1}{t}g(t) = \\frac{\\mathbb{E}[W_1]}{\\mathbb{E}[S_1]}.", "085abc1a4ea4269aa59a73c8ecb59330": "B^\\phi_{MX}=\\beta^{(0)}_{MX} +\\beta^{(1)}_{MX} e^{-\\alpha \\sqrt I}.", "085b40c8e85992b9b35eb73c29a5798e": "\\oint_K \\kappa\\,ds > 4\\pi.\\,", "085b91511496a5a05a5ac52d40c9489e": " DL_j ", "085bee4f4ccd429ad60e9e96744839c3": "|\\Psi_{m}^{p}\\rangle = \\mathcal{A}(\\phi_{1}(\\mathbf{r}_{1}\\sigma_{1})\\phi_{2}(\\mathbf{r}_{2}\\sigma_{2})\\cdots\\phi_{p}(\\mathbf{r}_{m}\\sigma_{m})\\phi_{n}(\\mathbf{r}_{n}\\sigma_{n})\\cdots\\phi_{N}(\\mathbf{r}_{N}\\sigma_{N})),", "085c19cf432cbcb959b468a6924bcb61": "L(p;q_1,\\ldots q_n)", "085c57e6606d29c7e177ef0385d05a40": "\\Psi_A(x)=C_A \\Psi_{0}(x-x_A)", "085c9b94c5877f6967fe47e0253394c7": " \\{e_3\\equiv z_{yy}+\\frac{1}{y^2}(xy^3-x^2-y)z_y-\\frac{1}{y}(x^3-x+y)z=0, e_2=z_x+\\frac{1}{y}z_y+xz=0\\}.", "085c9f0df11642cf704f40ffdf753055": "s'", "085ca404311ed2fa47059761173a876d": "x,y\\in \\Sigma^*", "085cfa49aa86db198f20d193d5080c92": "\\phi(a\\mathbf{x}+b\\mathbf{y},c\\mathbf{x}+d\\mathbf{y}) = \\frac{1}{|ad-bc|}\\phi(\\mathbf{x},\\mathbf{y}),", "085d1f59b60f6655415fb3f3b13a4d44": "\\gamma \\in \\{ 2^{-15},2^{-13}, \\dots, 2^{1},2^{3} \\}", "085d63c9f7b2e59353b8432c9b054d04": "P \\to (f(U) = f(V))\\,", "085d6e33ee4c55f4e65be030cd507c17": "o(w)", "085da8820e7465711a2a2a5d30250753": "\\Delta(\\tilde{w}, w') \\leq t ", "085dafbb0398cf2b6ab06d53c6987444": "e(n)=d(n)-\\hat{d}(n)", "085de9be16c6e59a32235966c7b6cbe3": "\n F_L = \\textstyle{\\frac{1}{5}} k^4 \\left( 5 L_A \\right) + \\textstyle{\\frac{1}{10}} {(1 - k^4)}^2 {\\left( 5 L_A \\right)}^{1/3}\n", "085df8dcd78197333723d2f5b9024b43": "x = R\\lambda/\\sqrt 2", "085e2dcea21c77cf0eef6e999639ee33": "q<1", "085e6f6c82be4cd865387cd12742ea9b": "\\alpha _i = 2\\cdot \\pi \\frac{iK}{N}", "085f012b4860c10d49cea9e685eda831": " BD", "085f471060f928587ef77d613de7a6ab": "\nE_{CFG} = \\{\\langle G \\rangle \\mid G \\text{ is a CFG and } L \\left( G \\right)= \\empty \\}\n", "086052c76ef59b2fb7311dcf92821286": "h_{crit} = \\left(\\frac{9B^2}{4}\\,\\frac{EI }{\\rho g\\pi r^2}\\right)^{1/3}", "0860a2c408a30ca4bbdd158c6798a132": "A = \\frac {W}{(L)(U_w)}", "0860b4bbf2e14b710c80e7a4e35aada9": "\\dot{Q}^\\mathrm{T} Q + Q^\\mathrm{T} \\dot{Q} = 0", "08610ec20d070997513de6c2ac0f5571": "y = b", "08618cc058c79edd11827393563f4a4f": "P_i'", "086192d60aa50b35adfeb8d6794eb4b5": "\nR(\\vec x) \\approx \\tanh p \\,\n", "0861a74c12b96b48f283c77d29514cda": "W=A^{-1}", "0861adc3d3e17d9c558f0b8939514aee": " \\bar{n}_i = \\frac\n\n{\\displaystyle \\sum_{n_i=0} ^1 n_i \\ e^{-\\beta (n_i\\epsilon_i)} \\quad \\sideset{ }{^{(i)}}\\sum_{n_1,n_2,\\dots} e^{-\\beta (n_1\\epsilon_1+n_2\\epsilon_2+\\cdots)} }\n\n{\\displaystyle \\sum_{n_i=0} ^1 e^{-\\beta (n_i\\epsilon_i)} \\qquad \\sideset{ }{^{(i)}}\\sum_{n_1,n_2,\\dots} e^{-\\beta (n_1\\epsilon_1+n_2\\epsilon_2+\\cdots)} } ", "0861e8ef3451e1ec1608f8d18768dc74": " p \\to q ", "08622ab1ba41cb55303f381dc2f38327": "y(x^2+y^2)=b(x^2-y^2)+2cxy", "08625098ce1218d596df0cb7a2ff1f49": "\nS_0(p) = \n\\begin{bmatrix}\n(I_x(p))^2 & I_x(p)I_y(p) & I_x(p)I_z(p) \\\\[10pt]\nI_x(p)I_y(p) & (I_y(p))^2 & I_y(p)I_z(p) \\\\[10pt]\nI_x(p)I_z(p) & I_y(p)I_z(p) & (I_z(p))^2\n\\end{bmatrix}\n", "0862609ee90f693442fc43e0c9f758bb": "Z=V", "0862629191318ef2d28418cacf18bcd6": "5F_6^2=320\\equiv -5 \\pmod {13} \\;\\;\\text{ and }\\;\\;5F_7^2=845\\equiv 0 \\pmod {13}", "08627f02774fb21464ae91032340eba4": " (\\nabla_XZ + (I - \\Delta S)X) + (h(X,Z) + d_X\\Delta)\\bold{A} = 0 , ", "0862c73df2fef82eab278377b6859017": "\\eta_B", "08630453b7b0223407e9bd890d7ae35a": "H_n=\\sum\\limits_{l=1}^{N}K^{[l]}_1 + \\sum\\limits_{l=1}^{N}K^{[l,l+1]}_2.", "0863360a323a329bc6470dbba564fa56": "K\\;", "086337e68e2d113e81e0e9db7655844e": "\\frac{\\pi}{\\sqrt{12}} \\approx 0.9069.", "08634f6ade2e4ad977a9dd7fe3172646": "\n\\Delta S = \\alpha k_{B} \\ln N \\,\n", "08639b03744654d4b36f1f47b913755a": "\\bold r", "08639c77b9ec5ddb0c6e94cb46ee0eec": "\\scriptstyle A_\\parallel", "0863e4a2343c010d5ccfdde4d7a8c0f1": "\\rho_A=\\operatorname{tr}_B\\rho_{AB}", "0863f741d7093a2bec6ece488bdae3b0": "y \\cup \\{y\\}", "086403e136c6d79c35384e36319d7181": "c^2 = ac\\cos\\beta + bc\\cos\\alpha.\\,", "08641454a6623d6aaaf7653303d81903": " \\operatorname{Re}(\\epsilon (\\mathbf{r}, \\omega)) ", "08643b522426f00d876b6175c5125f5e": "S \\in \\mathcal{A} (G)", "08644c3901ca0610e482e615b7c82f5e": "\\sum_{a_i \\in A} \\phi(a_i)=0", "08647d7f90875053446ea8763b27d956": "{\\boldsymbol{\\beta}}", "08648e92de68de9b39c67d08bb3a28a3": "*_N", "0864a3f53947fcdb99026514b527fb90": "H \\left( f \\right) * \\frac{1}{T_s}\\sum_{k = -\\infty}^{+\\infty} \\delta \\left( f - \\frac{k}{T_s} \\right) = 1", "0864b70f0056f87446ef403e849a2014": "T_0,", "0864eb673fda276a4f3a8813aa282976": "\\pi_{\\omega}(x) \\xi_{\\omega} \\mapsto \\pi_{\\phi}(x) \\xi_{\\phi} \\oplus \\pi_{\\psi}(x) \\xi_{\\psi}.", "08650fba58a14f3bc4c15c1bff29986f": " \\int \\rho \\big ( \\left \\Vert \\mathbf{x} - \\mathbf{c}_i \\right \\Vert \\big ) \\, d^n\\mathbf{x} =1", "0865a01dd577912c2e853f4fa232e797": "\\mathcal{M}_1", "0865a90eb4b59b80ad40594550d43e52": "d \\approx 3.57\\sqrt{h} \\,,", "0865ac8f14c29e628449d4808fc6a2e4": " \\begin{matrix}\\frac12\\end{matrix} mv^2=gmr.", "0865b7b5d73f8d2d267b2b4d48b4d47b": "X = -\\left\\langle\\frac{dE_{r}}{dx}\\right\\rangle\\,", "0865d6c4efb551567c3c2cab22e94bd7": "\\langle introd(j, b, x), s\\rangle", "0865ee2ee41e748e0ab653b785ea92c3": "\\mathrm{high} = R_1 C \\cdot \\ln\\left(\\frac{2V_{\\textrm{cc}}-3V_{\\textrm{diode}}}{V_{\\textrm{cc}}-3V_{\\textrm{diode}}}\\right) ", "0865fa8ab44cc8cbe3f68273fd2c03c4": "|G_a(z)| < (M+1)\\epsilon \\!", "086675e64252d1197dee3eebd34dec1e": "\\hat{H} ", "0866827b2b471138b8f46081602e57ab": "\nE_1 = q_1^2 - p_1 r_1 = 0.00000 \\,\n", "0866b23e49b4cf45803e32e679ebed22": "l=\\frac{\\hbar}{mc}", "0866b33bb22a761df4df09be6ddacbae": " { 4 \\pi \\over c }j^{\\beta} = \\partial_{\\alpha} F^{\\alpha\\beta} + {\\Gamma^{\\alpha}}_{\\mu\\alpha} F^{\\mu\\beta} + {\\Gamma^{\\beta}}_{\\mu\\alpha} F^{\\alpha \\mu} \\ \\stackrel{\\mathrm{def}}{=}\\ \\nabla_{\\alpha} F^{\\alpha\\beta} \\ \\stackrel{\\mathrm{def}}{=}\\ {F^{\\alpha\\beta}}_{;\\alpha} \\, \\!", "0866ee36a47474c5ca2fb1811dffef82": "GF(p^t)", "08670fd819285e6e4639e2541972b049": " \\quad \\beta_n \\sim f(\\beta_n | \\theta) ", "086730665882e9b8832ad63899709bf7": " D(\\xi_a)D(\\xi_b) = D(\\xi_a \\xi_b). ", "086747ee9945206e6ea6d9ab022b125f": " F = W_1 \\cdot W_3 = \\sum_{j=1,3} W_i", "08681c27ad524a8e515a19b4d3de2807": "\\int \\frac{x+A}{\\sqrt{x^4+ax^3+bx^2+cx+d}}\\, dx", "086834a5ae253549ee9026fec8d279eb": "x^{\\lambda} = e/x \\qquad x^{\\lambda}x = e", "0868420816995522683147f384d56ac3": "\\begin{align}\n & \\hat\\beta_1 = \\frac{s_{yy}-\\delta s_{xx} + \\sqrt{(s_{yy}-\\delta s_{xx})^2 + 4\\delta s_{xy}^2}}{2s_{xy}} \\\\\n & \\hat\\nu_1=\\frac{-1}{\\hat\\beta_1} = \\frac {-2 \\delta s_{xy}}{s_{yy}-\\delta s_{xx} - \\sqrt{(s_{yy}-\\delta s_{xx})^2 + 4\\delta s_{xy}^2}}, \\\\\n & \\hat\\beta_0 = \\overline{y} - \\hat\\beta_1\\overline{x}, \\\\\n & \\hat{x}_i^* = x_i + \\frac{\\hat\\beta_1}{\\hat\\beta_1^2+\\delta}(y_i-\\hat\\beta_0-\\hat\\beta_1x_i).\n \\end{align}", "08684be4f5f5c73459c888f3b47527ac": "p_{11}/(p_{11}+p_{10})", "086886c7642c35706d22fd59d30d2689": " : \\hat{b}_1^\\dagger \\,\\hat{b}_2 \\,\\hat{b}_3 : \\,= \\hat{b}_1^\\dagger \\,\\hat{b}_2 \\,\\hat{b}_3", "0868a10a419e0f3f0833aa59fe8330a5": " \\mu = G(m_1 + m_2)", "0868e6113e0afe135438eef879d719e6": "\\mathit{l_{j}}\\,", "0868fa8e93f62f819441deb81c05a85f": "x'_i \\rightarrow x_i", "0869167f15b8c06f68939754920c6817": "p-p_0=c_0^2(\\rho-\\rho_0)", "0869348ba87ca5f6cefc2e30ddb36314": "F(z, m) = \\sum_{k=0}^{\\infty} f(k T + m)z^{-k}", "086992e911ad666940c60fac809643a8": " V_gf\\in L^{p,q}_m(\\mathbb{R}^{2d}) ", "0869965a2e1546a7606aff31a1f767f4": "y = \\frac{1}{1+e^{-x}}", "086a625a452572a6878a544d30624348": " P(x)=\\sum_{j=0}^{n-1} u_j x^j ", "086ad3563a5e862713aaf1cd8fd6f466": "(M \\ / \\ s) \\ / \\ t = M \\ / \\ (ts)", "086adc074fc09817139a0e2cd17879c9": "p(\\vec\\theta) \\propto \\sqrt{\\det I(\\vec\\theta)}\\,", "086b8289e29be5115bbfd329f2433486": "\\text{precision}=\\frac{\\text{number of true positives}}{\\text{number of true positives}+\\text{false positives}}", "086beb6a6c8a029942238364e5a8beab": "(X, d)", "086c08f4887854fbb8ac4b1dd3d58ee1": "x=R\\cos(\\theta)", "086c1c983eef5f4ec1e9183b80dea13e": "\\tilde{x}", "086c62f0fe0c5804428eb0133152fd95": "\n\\frac{\\theta }{(I)_{J}} {J \\choose n}\n\\int_{0}^{1}(Ix)_{n}(I(1-x))_{J-n}\\frac{(1-x)^{\\theta -1}}{x}\\,dx \n", "086cebacd2b29692f317fd1d9adc9b91": "\\alpha^\\vee= {2\\over (\\alpha,\\alpha)}\\, \\alpha.", "086d4c3a2db4cd707b5e03137be03b8e": " y^G = f(k^G) ", "086e143ecfedac00e31a515d138d3a3c": " \\sum\\limits_{i=1}^\\infty \\mathrm{P}(X\\geq i) = \\sum\\limits_{i=1}^\\infty \\sum\\limits_{j=i}^\\infty P(X = j). ", "086e1709ae075fde91365e73006c78d5": "\\left(\\frac{F/\\mathbf{Q}}{(n)}\\right)=\\left(\\frac{L/\\mathbf{Q}}{(n)}\\right)\\text{ (mod }H).", "086e1e913727da2fa28e31c854b859f4": "U(x,\\omega)= P(x,\\omega)A(x,\\omega) ", "086e77c05b32e57ee30727de66af5caf": "\n\\mathrm{ERB}(f) = 6.23 \\cdot f^2 + 93.39 \\cdot f + 28.52\n", "086ee2d9b611075135d32fe2a9b5bf61": "a + b'x_{i1} + c'x_{i2},", "086efc867d6a52e34eb88795dbb61efc": "\\gamma_\\mathrm{SG}", "086f2a64fea3650776476f320fc982ec": "T(n) = 2^nT\\left (\\frac{n}{2}\\right )+n^n", "086f5e4bee2482de92c8baefce510f22": "\\mathcal{F}(t_m) ", "086f5ebd6937c1a196ff9cde7fbace81": "E > mgl", "086f75fe0a018304ff2817a784eba8ed": "\\log(\\gamma)", "086fce0618cbc52b2baecb5059e0db1f": "\\Diamond \\varphi", "086fe0b2a0abd7dd3c433deb189569f9": "\\begin{align}\n \\text{N}_\\text{s} \\omega_\\text{s} + \\text{N}_\\text{p} \\omega_\\text{p} - (\\text{N}_\\text{s} + \\text{N}_\\text{p})\\omega_\\text{c} &= 0 \\\\\n \\text{N}_\\text{a} \\omega_\\text{a} - \\text{N}_\\text{p} \\omega_\\text{p} - (\\text{N}_\\text{a} - \\text{N}_\\text{p})\\omega_\\text{c} &= 0\n\\end{align}", "086fe4af32eaa0d8e320267ad0ffefa7": "T:\\mathcal{F}\\to\\mathcal{F}", "087017cefe10d64b6f95a15506032319": "m_1 = [12.3, 7.6] + [1.697, 0] = [13.997, 7.6]", "08703596bd26613a0f2d79d0a2c03428": "f(z)=\\frac{e^{t z}-Q_t(z)}{P(z)}", "0870b9f545ddc94563bcca5270b07e1e": "a_5= \\lfloor 6^\\frac{1}{2} \\rfloor = \\lfloor 2.449\\dots \\rfloor = 2, ", "0870f159f171f243deba0d2ee71655e7": " \\tan \\frac{\\theta}{2}", "08718e42cb5b04f1489f7b5d86544ed3": "x/\\epsilon c", "087195bbcff743d55cb9fc7d061ef781": "\\widehat{\\varepsilon}_i=X_i-\\overline{X}.", "0871c793427749c626b5771230f271a8": "\\left\\{\\begin{array}{ll}1 & n = 1\\\\ 2 & \\text{otherwise}\\end{array}\\right.", "0871f51ba0068ce4fdbe0e593444563c": "X_n\\ \\xrightarrow{\\mathcal D}\\ X,", "087249e982bffce2426824d3de6de8e5": "\\mathrm{conv}", "08727052444f8cdf3e308d5863a722a9": "I_{m,n} = \\int \\sin^m{ax}\\cos^n{ax}dx\\,\\!", "08728ea65350ce111f5ae89cda8832b8": "f\\left(x_0\\right) = p x_0 + b", "0872b027f615858ceefa69208cceab47": "N_2 + 8H^+ + 8e^- \\to 2NH_3 + H_2", "0872bf33180c1f89aa2fb8a0bfb5afc9": "{EF = \\frac{Q_e}{Q_e + Q_h} = \\frac{1}{1+B}}", "0872cfce743c4aa442423f20f85c2437": " \\langle P \\rangle \\,\\!", "0872d8a6ea05decf0ebfefd6837d7da0": "x^2 + bx + c \\,=\\, \\left(x + \\tfrac{1}{2}b\\right)^2 + k,", "0872fbbc55e4fdf5b6840c958be53aa5": "\nF_{2}(r) = F_{1}(r) + \\frac{L_{1}^{2}}{mr^{3}} \\left( 1 - k^{2} \\right)\n", "087319626a720e82ab10f4322bb92895": "c_v = \\frac{R}{{\\gamma - 1}}", "0873225a6aae94dfc0be630f38c83ec3": "y^5+y^4-24y^3-17y^2+41y-13", "087322ef246341ad8463c6eb80264c15": "x_{crit}", "08732b8ff0ad81ffc9a3105e18d3e5d3": " \\sigma^\\dagger_{A,B}=\\sigma^{-1}_{A,B}:B \\otimes A \\rightarrow A \\otimes B", "087368f6ea9528420df49d2316f9b888": "\\scriptstyle \\land", "08736f4045e96abf886c45562b8e98a1": "\\scriptstyle{\\epsilon = \\sqrt{1 - v^2/c^2}}", "087396a1975686bdeec8258553c79390": "x = a_0 + \\frac{1 \\mid}{\\mid a_1} + \\frac{1 \\mid}{\\mid a_2} + \\frac{1 \\mid}{\\mid a_3},", "0873dec567d24ed7d6c14976d4a0adb4": "f: V \\to V", "0874064e73f2352985b5c8fc7331051d": "m=0, 1, \\dots,", "0874289dabdd17e6c3fdce45ccbd973f": "\\Lambda^n A: \\Lambda^n V \\rightarrow \\Lambda^n V", "0875135b6dfecc84cfb62efcdd77c8fc": "\\scriptstyle a=\\partial u/\\partial x=\\partial v/\\partial y", "0875ba40ea78e27c50780ff19d1ceb10": "z=(z_1,z_2,\\ldots, z_n)", "08760f01ea0d9b37e4e4e6f8686ebe01": "\\scriptstyle\n\\frac{\\sqrt{3}}{{9}}\\, \\sum \\limits_{n=0}^\\infty \n\\frac{(-1)^n}{27^n}\\,\\left\\{\\!\n\\frac{{18}}{(6n+1)^2} - \\frac{{18}}{(6n+2)^2} -\n\\frac{{24}}{(6n+3)^2} -\n\\frac{{6}}{(6n+4)^2} +\n\\frac{{2}}{(6n+5)^2}\\!\\right\\}\n", "0877834805793b7a7f0c4bfb9cb88b1f": " \\frac{1}{iz} \\,dz = dt.", "0877e3a4e881d97aeb3bba90f55bae18": "\\langle U(x+y)-(Ux+Uy), U(x+y)-(Ux+Uy) \\rangle = 0 ", "0877ed54238aefb676d353b0f3eb6615": "x_i \\,", "087847d30cf204fc8cdd76973b6dcc15": "k_{xo}=k_{o}\\sqrt{1-(\\frac{m\\pi }{k_{o}a})^{2}-(\\frac{k_{z}}{k_{o}})^{2}} \\ \\ \\ \\ \\ \\ \\ (26) ", "08788987bf75976c364309a002529686": "d\\tau = \\sqrt{\\left ( 1 - 1.3908 \\times 10^{-9} \\right ) dt^2 - 2.4069 \\times 10^{-12}\\, dt^2} = \\left( 1 - 6.9660 \\times 10^{-10}\\right ) \\, dt.", "08789c1c789d3a0d70d6883cb62aa942": "\\pi _{jt}", "0878be8a1c8c0a60e498a63a19d7aed5": "\\displaystyle 5^2 + 12^2 = 13^2 \\,.", "087932e8c5faf7ec468c1c2f3a058aaf": " {3\\over2} \\cdot {3\\over2} \\cdot {3\\over2} \\cdot {3\\over2} \\cdot {8\\over5} \\cdot {1\\over2} \\cdot {1\\over2} \\cdot {1\\over2} = {81\\over80}", "0879894f2225d8ae7eb9b1b1a3d76aaf": "d_1>d_2", "087a00dfeebaed1d79009912a028e5bc": "= \\frac{1}{331}", "087a6d79325f7cc88bdaa3ca5e84acc2": "\\ell^\\infty(\\mathbb{R})", "087ae63322dcc247674a04de75f515a8": "\\iota^* : H^* (E) \\longrightarrow H^* (F)", "087b4c104aff1c97fb230dc27131f9f8": "a_0=\\frac{\\hbar^2}{m_e e^2}", "087b7ad7a6693503cc710a0bbd54e040": "\\epsilon_H ", "087b8782dd88a72e599d2867ec8e4ecc": "\\mu_{\\max} \\leq 1 - \\varepsilon", "087b9c21b7033fa2e1a50634d25f400a": "N\\in\\mathbb{Z}^+", "087bc030c1ad70f4a5b42c443dd2a3fe": " S_0 = \\left|\\mathcal{F}\\left[\\frac{dW(t)}{dt}\\right](\\omega)\\right|^2 = \\text{const} ", "087bd5f8d197b0f1b129664107c086c2": " g_\\mathrm{e} =2.0023", "087bd6c601163c61803bff1872872a6b": " \\ c_1 = c_{01}(1 - y_d) + c_{11}y_d", "087c213852df24f142c921a142fa2f59": "M, N", "087c3e00f68962d086837cec05e6b1fb": "H^2_0(\\Omega)", "087c5d667ebfa8eb0281e46f055f2764": "\\partial_{\\mu}A^\\mu \\equiv A^\\mu{}_{,\\mu} = 0 \\!", "087cea17dd3a808fc89b1a1de530a1f4": " 2\\pi \\left (1 + \\cos {\\theta} \\right) ", "087cf94aefdf5f470e33d919960c43b5": "a=15-15i", "087d07c94a3cce22b1dd9f377e5b0315": "k,~k_e", "087d27d1a0b418fadacfdf9521391cee": "\\text{Passer Rating}_{\\text{NCAA}} = {(8.4 \\times \\text{YDS}) + (330 \\times \\text{TD}) + (100 \\times \\text{COMP}) - (200 \\times \\text{INT}) \\over \\text{ATT}}", "087d28b05c8f87a4bb020c3ccf563f23": "Y_i \\sim \\mathrm{Pois}(\\lambda \\cdot p_i), \\rho(Y_i, Y_j) = 0", "087d30f0f21740efad9b744a16ef9885": " \\begin{cases}\n\\text{Mesh 1, 2: } -V_s + R_1I_1 + R_2I_2 = 0\\\\\n\\text{Current source: } I_s = I_2 - I_1\n\\end{cases} \\, ", "087df3bfe54732c537c1397765620510": "\nLz\\equiv l_2l_1z=\\Big(\\partial_x+\\partial_y+\\frac{2}{x+y}\\Big)\\Big(\\partial_x-\\partial_y+\\frac{2}{x+y}\\Big)z=0", "087e3e311cf84d5eb4c0daefc55ab1db": " \\dot{y} ", "087eebb0f949fb53c7e4be08ecf843d3": "\\frac{30,000\\ \\mathrm{N}}{(111\\ \\mathrm{kg})(9.807\\ \\mathrm{m/s^2})}=27.6", "087efa7ff3198fd6fa7b7bae1aeb9d8b": "\\theta \\mapsto p(x\\mid\\theta)\\!", "087f30a536338c2546ba6bc229296d24": "T^{\\mathrm{SW}}_p (x,y) = \\begin{cases}\n T_{\\mathrm{D}}(x,y) & \\text{if } p = -1 \\\\\n \\max\\left(0, \\frac{x + y - 1 + pxy}{1 + p}\\right) & \\text{if } -1 < p < +\\infty \\\\\n T_{\\mathrm{prod}}(x,y) & \\text{if } p = +\\infty \n\\end{cases}\n", "087f70bfdc412cff7f905892786e7145": "f : L \\to L", "087f77f0c34b93605f9d6dde51b9278e": "\\Delta x \\left[ f( a + \\Delta x ) + f(a + 2 \\Delta x)+\\cdots+f(b) \\right].", "087f88f33ecfbfaf6b959a0ae04c58e8": "v_C = v_{ \\pi} \\left(1+ \\frac {R_f} {R_1} \\right ) -i_B r_{ \\pi 2} \\ . ", "087fa767d07fdab42a05e99efe06d8b8": "\n \\mathrm{SNR} = \\frac{\\mu}{\\sigma}\n", "087fd50acd38042b6b2178db7a78d50a": "\\frac{d}{dx}g(x)=h(x)\\cdot\\frac{df(x)}{dx}+\\frac{dh(x)}{dx}\\cdot f(x)", "087ff194b49f1dc50600a9316af323c2": "\\Sigma^f\\,", "087ff4c278fe1896c72ec5b417934be7": "\\frac{1}{1260} = \\int_0^1\\frac{x^4 (1-x)^4}{2}\\,dx < \\int_0^1\\frac{x^4 (1-x)^4}{1+x^2}\\,dx < \\int_0^1\\frac{x^4 (1-x)^4}{1}\\,dx = {1 \\over 630}.", "088036c6d173183cf632e89525ce225a": "\\frac{\\pi}{3}", "088052edf469e8126bd37dfc8878ce51": "\\xi\\rightarrow\\infty", "088053bb25455067de4867cd725c7ed1": "\\mathcal{F}_{s}=-\\oint\\frac{1}{2}W(\\mathbf{\\hat{n}}\\cdot\\mathbf{\\hat{\\nu}})^2\\mathrm{d}S", "0880841a6ce8807ae6d6b7e94cbb9c72": " \\hat{g}_{ij}(t,x^k) \\to g_{ij}(t,x^k)", "0880ba6fa331b84320f320c3364180c7": "\\ Y=Min[K,L]", "0880c290c61d2acbaf130f0f8ef83dfb": "\\ \\zeta = \\frac{\\Delta W}{W_1} ", "088144bbe97e45f4e65a31a795dff154": "w=m(\\mathsf{i})", "088165301a2d7e4e5d299fd5b4618682": " V_{j} ", "08819ffe7f6d3e55490596f39495a63c": "\\sum_{j=5}^8 f(j) = 0.17367.", "0881e2a455bfa5b6dae63afe36f581c1": "\\frac{\\partial \\rho}{\\partial t} + \\frac{\\partial \\left( \\rho\\, v_i \\right)}{\\partial x_i} = 0", "0882903c39f87b9eac329bacff4942c8": "f_{clk}/f_o", "08829d8c00150260e3c4e34d1286237c": "p_{p_1}", "0882f7f585c18f32f9227f88d7d90155": "\\displaystyle{ f(a) =(f,E_a)}", "08832020387001bc2e9e82836fb500e4": "| \\Psi \\rangle=\\sum\\limits_{i=1}^{M}c_{i_1i_2..i_N} | {i_1,i_2,..,i_{N-1},i_N} \\rangle", "0883422bed909a1f6799617c4d526225": "{S_i-S_j}", "088364475c1e613e1b64d49dfeee4976": "M_i\\rightarrow M_j", "08838bd3c696e0632c8c31f133af263c": "f(\\tau)=A \\tau^k (1+b\\tau ^{k_1} + \\cdots) \\text{.}", "0883aecb4ef0d44b183e543352f9a6aa": "\\hbar = h/2\\pi", "0883b6c2fe84f73b8a93a07049b7a9d4": " \\frac{\\partial^2}{{\\partial x}^2} \\phi_x(x) = \\kappa^2 N_x e^{\\kappa x} = -\\frac{2m}{\\hbar^2} E_x N_x e^{\\kappa x} ", "0883bd84a2ab66a95fc9e8b50493eb99": "S = 2", "0883c969a7f0461fdc868b48bf89e964": "S^k", "088449fb3a983cc30349e8fe4dda7681": "\\alpha_i(t),", "08846ff3d5a8264373ae22b22d77e05c": "\\mathbf{A} =\n\\begin{bmatrix}\n1 & 2 & 3 \\\\\n4 & 5 & 6 \\\\\n7 & 8 & 9\n\\end{bmatrix}.\n", "088484ecb75d0f052ea2c7e9c13ee443": "\\ell \\gg m, m^\\prime", "0884b7813582ce810d6871a1fecea9c6": "\\frac{1}{T_2} = \\frac{\\rho S}{V}", "0884c1a36661d49a5c2e43ef65956105": "\\mathbf{I} = \\mu_0 \\mathbf{M} \\,", "0884cbace74dc1bfd878c582c5ee2efb": "{1\\over P}{K\\choose B}{N-K\\choose K-B}\\over {N\\choose K}", "08852cee1ca1c59ff4b9f3c157f513d6": "P_1 \\triangleleft C \\triangleright P_2 \\equiv ( C \\land P_1 ) \\lor ( \\lnot C \\land P_2 )", "08852f33943b2f0869e64534ac4f170a": "k_j\\sigma", "08854a2961463c85e318c8ef3fb7ec3c": "P = L\\times R /S", "08854c771fb53ecd0372766c1158f1f4": "b_4=\\frac{r_3,c_5 - b_3\\times a_2 - b_2\\times a_3 - b_1\\times a_4}{a_1}\\mbox{ with remainder }r_4 \\dots", "08857777bdc5a26f6c329c1db3d51570": " {\\pi\\over 4} = 4 \\arctan \\left({1\\over 5}\\right) - \\arctan \\left({1\\over 239}\\right)", "08857781972f452d9e7078a6653b08a1": "L^1(\\R)\\cap L^2(\\R).", "08857a59dc02f2a4358fb4af576e7ba8": "{\\dot u} = -3(p+u)\\frac{\\dot a}{a}", "0885df44ab1609dc6f365b99785154e1": "\\ddot{x_0}=h(x_0)-\\frac{g(x_0)g'(x_0)}{2 \\omega^2}", "08862e2d685ab71496c7eab5cfd9a832": " = 2,800\\,", "08864af48bfe60fd15b59545b5202aa4": "\n \\begin{align}\n x_0 &= \\cos(\\tau) \\\\\n \\text{and }\n x_1 &= \\tfrac{1}{32}\\, \\cos(3\\tau) + \\left( \\omega_1 - \\tfrac{3}{8} \\right)\\, \\tau\\, \\sin(\\tau).\n \\end{align}\n", "0886945705284bb0bdac03c73ee74428": "\n\\bold{Q} = \\boldsymbol\\beta = \n{\\partial S \\over \\partial \\boldsymbol\\alpha}\n", "088699c6fbb45a9f800785271f5b0563": "s:\\Lambda^n_k\\rightarrow X", "0886a4b22a8c9259eb5c296a23c6dd1f": "\\frac{dH}{dt} = \n\\frac{\\partial H}{\\partial \\boldsymbol{p}} \\cdot \\frac{d \\boldsymbol{p}}{dt} + \n\\frac{\\partial H}{\\partial \\boldsymbol{q}} \\cdot \\frac{d \\boldsymbol{q}}{dt} + \n\\frac{\\partial H}{\\partial t}", "0886b0dbeac4c1355c331bb702b288fc": "B^n x < B^n (y+1)^n\\,", "0886caabbc4904e95fc0cae59756e826": "(b_{1}-d)-(a_{1}-d)", "0886d2ea8c2837496c1f6a6c073c381a": "\\operatorname{CG}()", "0886d5b4f1591e426ad21c1b72ae89c0": "[TN] + [T_{M/N}] = [TM]", "08873ca3cc7efb7f3b26fda4b77ccfac": "\\partial_a", "0888292cf8e218d6ececdb638c169df2": " \n\\nabla_{\\hat{\\mathbf{h}}^H} C(n) = \\nabla_{\\hat{\\mathbf{h}}^H} E\\left\\{e(n) \\, e^{*}(n)\\right\\}=2E\\left\\{\\nabla_{\\hat{\\mathbf{h}}^H} ( e(n)) \\, e^{*}(n) \\right\\}\n", "088835ca7e893f42b565ab2bca10e393": "\\mathrm{Res}(fg')=-\\mathrm{Res}(f'g);\\,", "08886096e0572da090f4c40fca33537c": "= {\\begin{matrix} \\frac{1}{10} \\end{matrix} + \\begin{matrix} \\frac{1}{100} \\end{matrix} + \\begin{matrix} \\frac{1}{50} \\end{matrix} + \\begin{matrix} \\frac{1}{80} \\end{matrix} + \\begin{matrix} \\frac{1}{20} \\end{matrix} + \\begin{matrix} \\frac{1}{500} \\end{matrix} \\over 6} ", "0888805f4700a2434c62aacb14080476": "\\int\\frac{\\mathrm{d}x}{\\tan ax - 1} = -\\frac{x}{2} + \\frac{1}{2a}\\ln|\\sin ax - \\cos ax|+C\\,\\!", "08893af1be1bdd9df5eadf537e3bbbe6": "\\pi_1 := \\text{Pr}[P=1,Q=1]=\\sum_{\\omega \\in S_1} \\Psi^2_\\omega", "088967dfe40ea7b71cdc3cec6a28eea7": "\\omega_{\\mu}^{\\ IJ} = e^I_\\nu \\partial_\\mu e^{\\nu J} + e^I_\\nu e^{\\sigma J} \\Gamma^\\nu_{\\sigma\\mu}", "0889ddc3444aae10a93496769c2adb62": " w_1 \\ge w_2 \\ge \\dots \\ge w_\\mu > 0", "0889f771f64a54f2f4e00c59eab935ff": "n=(n_ln_{l-1}\\dots n_0)_2", "0889fb824147f396d1a1d815553e4fe4": "H=\\frac{1}{\\sqrt{2}}a", "088a3dcfc3dd3697b2fad6ea74d45e92": "\\ltimes \\!\\,", "088a4edb5ffe3aa76b2ff5406265a8f2": " T_s = 303 ~\\mathrm{K} = 30 ~\\mathrm{C} ", "088aa0453d2418193fb066dd406ff6bd": "h = h(-,-).\\,", "088aa21ca6d89bd3e67eb5ec3d720324": " \\begin{array}{lll}\n\\eta &=& 1- \\frac{trace(W_1^TAW_1)}{trace(D_B^{-1/2}P^TAPD_B^{-1/2})}\\\\\n &=& 1- \\frac{trace(\\hat{D_B}^{-1/2}\\hat{P}^TA\\hat{P}\\hat{D_B}^{-1/2})}{trace(D_B^{-1/2}P^TAPD_B^{-1/2})}\n \n\\end{array}\n", "088aa33c190e59659b39778bd19b5986": "\\lambda < 0", "088aaa73541213fd91084b37fd0422bf": "y=C(w(t))x(t)+D(w(t))u(t)", "088b09b0ab517ad1f368845c9818c722": "\\det(E+(n-i)\\delta_{ij}) =0 ", "088b147aae80c99347aefe2dc1fefa09": "G = \\frac{2e^2}{h}MT", "088bd0a0a6ad8aa37cb69bee357297a1": "U[\\mu_1,\\mu_2] = U-\\mu_1 N_1-\\mu_2 N_2", "088bdf436f5d9d1dea35097f67b43c12": " \\sum_{v(p)=0}\\frac{P(A_p)}{\\det A_p} = \\int_M P(i\\Theta/2\\pi)", "088c3b4690d85e237c04bc75b9ace31a": "\\mathbf{R} = n_1 \\mathbf{a}_1 + n_2 \\mathbf{a}_2 + n_3 \\mathbf{a}_3,", "088c4ffb0bb4687563ca3bf314176dba": "\n\\begin{align}\n\\sin (A) & = RQ \\\\\n& = \\text{length of arc } PS \\\\\n& = \\angle POS \\text{ in radians}\\\\\n& = \\frac{\\pi}{180\\times 60}\\left( m + \\frac{s}{60}+ \\frac{t}{60\\times 60}\\right).\n\\end{align}\n", "088c77d6e29bdb1d671b2df67aca9373": "x \\cdot p + y \\cdot q = 0", "088c8b835560feac3300998ec1332a92": " |\\psi \\rang", "088d03f1074199581e9cd748695d5521": "g(z)=z+b_0 + b_1z^{-1} + b_2 z^{-2} + \\cdots", "088d15a0bdfecaccd1c62383f94b8cbc": " -\\varepsilon_i.", "088d531507cfb1a6302c6226049e80dc": "M^{\\mu\\nu} \\,", "088d68d8a312039e05c361d2341ac71c": "E_{X} = 61.5 \\ \\mathrm{mV} \\log{ \\left( \\frac{ [X^{+}]_\\mathrm{out}}{ [X^{+}]_\\mathrm{in}} \\right) } = -61.5 \\ \\mathrm{mV} \\log{ \\left( \\frac{ [X^{-}]_\\mathrm{out}}{ [X^{-}]_\\mathrm{in}} \\right) }", "088dbb6eed854331cdab6c34e5e8852e": "\\left|\\Gamma\\left(\\omega\\right)\\right|^2", "088dc58a11fd066e189ee7f8a7725546": "\\kappa(M)\\in H^4(M;\\mathbb{Z}/2\\mathbb{Z})", "088de5149c77b89b1f21d59d3df5e987": "\\mathbf{S}= \\frac{\\epsilon_0}{2i\\omega}\\int \\left(\\mathbf{E}^\\ast\\times\\mathbf{E}\\right)d^{3}\\mathbf{r} .", "088e53945d396a436984c8403162be33": "K_{\\mathrm{rot}} = \\tfrac{1}{2}I\\omega^2,", "088f2b5f22c21c3051502aa50d7597ec": "i:=1;\\qquad S:=\\emptyset,\\qquad f^*:=f;", "088f6df761802b49580d5bb103e7a428": "L_i=C_i-d_i", "088fc910583005a88db3211c8be16637": "Q=\\big(\\alpha K^\\lambda + (1-\\alpha) N^\\lambda\\big)^{1/\\lambda},\\,", "088ff137b55f0dce3cce12801b5c9125": "\\bar{V^E}_i= RT \\frac{\\partial (\\ln(\\gamma_i))}{\\partial P}", "08900515bff2bad7ba26a0b4ae1b93a5": "48+\t32+\t1+\t64+\t33+\t17+\t16+\t49\t=\t\t260", "089017e414db79a49991a6af70d24a61": "\\scriptstyle f_0\\,", "08915b56dbc2dc98df111e43d0cdcbe3": "\\int \\ln(x^2+a^2)\\; dx = x\\ln(x^2+a^2)-2x+2a\\tan^{-1} \\frac{x}{a}", "08917d48d77824f06da1c6c0ca89d811": "2^2=4,\\, 2^3=8,\\, 2^5=32,\\, 2^7=128,\\, 2^{11}=2048,\\, 2^{13}=8192,\\, 2^{17