{ "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 }={\\sqrt {5}}-1-{\\sqrt {5-2{\\sqrt {5}}}}\\,", "00023b9224ac900169410ee72115cea4": "\\chi (T)=T^{2g}+a_{1}T^{2g-1}+\\cdots +a_{g}T^{g}+\\cdots +a_{1}q^{g-1}T+q^{g},", "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": "{\\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)", "0004c246ad141d5412a457dc81323857": "H_{1}(\\mathrm {A} _{3})\\cong H_{1}(\\mathrm {A} _{4})\\cong \\mathrm {C} _{3},", "000592a04b7c6c5cc9a9429a048b2757": "\\mu =2C_{1}~\\sum _{i=1}^{5}i\\,\\alpha _{i}~\\beta ^{i-1}~I_{1}^{i-1}\\,.", "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": "P_{ij}(f)={\\frac {A_{ij}(f)}{\\sqrt {\\mathbf {a} _{j}^{*}(f)\\mathbf {a} _{j}(f)}}}", "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}}={\\begin{pmatrix}Z_{\\infty }&-\\gamma _{1}\\gamma _{2}\\\\1&-z_{\\infty }\\end{pmatrix}},\\;\\;Z_{\\infty }=\\gamma _{1}+\\gamma _{2}-z_{\\infty }.", "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}=1.4Y^{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}}\\approx 7\\cdot {\\frac {\\sqrt {N}}{d}}", "000e540b8ebc9ff725e5bb41d49be814": "{\\text{Spec }}B", "000e5c1739ea28760d66f6d05f0e18d1": "J_{\\alpha }=\\int _{0}^{\\infty }{\\frac {dx}{\\left(x+b^{2}\\right){\\sqrt {\\left(x+a^{2}\\right)^{3}}}}}", "000ec8a8686baebba2fe12442b863020": "U_{11}-U_{21}", "000f32a1b8f6232759a658d470fe72c5": "y=p(x)", "000f743b3f56fd60b28545a4a844b238": "|{\\Psi }\\rangle =\\sum _{i_{1},i_{2},\\alpha _{1},\\alpha _{2}}\\Gamma _{\\alpha _{1}}^{[1]i_{1}}\\lambda _{\\alpha _{1}}^{[1]}\\Gamma _{\\alpha _{1}\\alpha _{2}}^{[2]i_{2}}\\lambda _{{\\alpha }_{2}}^{[2]}|{i_{1}i_{2}}\\rangle |{\\Phi _{\\alpha _{2}}^{[3..N]}}\\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": "L_{\\mathrm {dB} }=10\\log _{10}{\\bigg (}{\\frac {P_{1}}{P_{0}}}{\\bigg )}\\,", "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{aligned}S_{1}&=&a_{1}&&&\\\\S_{2}&=&a_{1}&{}+a_{2}&&\\\\S_{3}&=&a_{1}&{}+a_{2}&{}+a_{3}&\\\\\\vdots &&\\vdots &&&\\\\S_{N}&=&a_{1}&{}+a_{2}&{}+a_{3}&{}+\\cdots \\\\\\vdots &&\\vdots &&&\\end{aligned}}", "001384455f0b171fd018da65ca08ae9a": "V\\otimes V/(v_{1}\\otimes v_{2}+v_{2}\\otimes v_{1}{\\text{ for all }}v_{1},v_{2}\\in V).", "0013ada8dc886f1e875984bee5fdea27": "\\rho _{x^{n}\\left(m\\right)}=\\rho _{x_{1}\\left(m\\right)}\\otimes \\cdots \\otimes \\rho _{x_{n}\\left(m\\right)}.", "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 })0,c\\neq 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": "{\\frac {1}{(p+1)\\left(b^{2}-4a\\,c\\right)\\left(c\\,d^{2}-b\\,d\\,e+a\\,e^{2}\\right)}}\\,\\cdot ", "00178f10a40e91f76517d52061ef2a42": "(n+1)!", "00179f58dfc9cf36493673f0dacf255e": "s_{V}({\\mathcal {R}})", "0017f09b2d0eb84ef7d74112761e5ca2": "{\\begin{aligned}\\Phi _{1}&=\\Phi _{2}\\equiv \\Phi (x_{\\perp })\\\\&=-2p_{1}\\cdot A_{1}+A_{1}^{2}+2m_{1}S_{1}+S_{1}^{2}\\\\&=-2p_{2}\\cdot A_{2}+A_{2}^{2}+2m_{2}S_{2}+S_{2}^{2}\\\\&=2\\varepsilon _{w}A-A^{2}+2m_{w}S+S^{2},\\end{aligned}}", "0017fa64796d63c8af98928a15b3662c": "-F\\mathbf {e} _{y}", "001803962c3d9e04abb4057c65fa219a": "d_{\\phi }=1", "00180c42d14cacb3f499b74661393fb8": "|f(s)g(s)|\\leq {\\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}=qV-(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\\leq 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}}}}}={\\tfrac {1}{2}}+{\\tfrac {1}{\\pi }}\\arcsin {\\sqrt {0.75}}={\\tfrac {5}{6}}.", "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(kz-\\omega t)})", "001c03cd18548eff08e44a1c6a40460b": "{\\begin{bmatrix}0&1&0&1&0&0&0&0&0\\\\0&0&1&0&0&0&0&0&0\\\\0&0&0&0&0&0&0&0&0\\\\0&0&0&0&1&0&1&0&1\\\\0&0&0&0&0&1&0&0&0\\\\0&0&0&0&0&0&0&0&0\\\\0&0&0&0&0&0&0&1&0\\\\0&0&0&0&0&0&0&0&0\\\\0&0&0&0&0&0&0&0&0\\end{bmatrix}}", "001c1c698265214507f5814c8c9bbe62": "f(x)={\\begin{cases}{\\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;\\\\{\\frac {\\Gamma ({\\frac {\\nu +1}{2}})}{{\\sqrt {\\pi \\nu }}\\Gamma ({\\frac {\\nu }{2}})}}\\exp \\left(-{\\frac {\\mu ^{2}}{2}}\\right),&{\\mbox{if }}x=0.\\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": "={\\begin{bmatrix}W_{11}&W_{12}&&\\\\&W_{22}&W_{23}&\\\\&&W_{33}&W_{34}\\\\&&&W_{44}\\\\\\end{bmatrix}}", "001f090921d4950e090223a9db6fb0be": "\\mu _{k}(A-A_{k})<\\epsilon ,~\\forall k\\geq 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": "V={\\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})}}", "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}}=\\lim _{s\\to 0}{\\frac {\\langle E(s)\\rangle }{A}}=-{\\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 _{n}^{0},", "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}-i&i\\\\0&i\\end{pmatrix}}", "00290f11d9ba0677c1614e97a3e1f097": "v(t)=\\int _{t_{0}}^{t}i(\\tau )d\\tau .\\,", "002917cdd4458fc6214ed9aaf24cd803": "{\\frac {v^{2}}{2c^{2}}}\\approx 10^{-10}", "0029190f5afee4bdfbdd64cd63bc229b": "\\delta _{0}^{\\prime }\\Omega _{0}^{\\prime }=\\left(\\delta _{0}^{-1}+k^{2}+kx-1\\right)\\delta _{0}\\Omega _{0}.", "002938e91e1d12948fb82e55131c99e7": "\\|Df\\|_{\\infty ,U}\\leq 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": "\\mathbf {E} =\\xi \\exp[i(kx-\\omega t)]\\mathbf {\\hat {x}} ", "002e06e607e26e75da249f7016a07881": "(-m_{i}\\partial _{tt}+\\gamma _{i}T_{i}\\nabla ^{2})n_{i1}=Z_{i}en_{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{aligned}{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{aligned}}", "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}\\cdot 5\\cdot 17\\cdot 3719", "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.\\,zMN", "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 11)(\\lambda \\lambda \\lambda 1(\\lambda \\lambda 1)((\\lambda 441((\\lambda 11)(\\lambda 2(11))))(\\lambda \\lambda \\lambda \\lambda 13(2(64)))))(\\lambda \\lambda \\lambda 4(13)))))(\\lambda \\lambda 1(\\lambda \\lambda 2)2)", "003206d6d973a25d27d7badeae180f6a": "{\\begin{aligned}Area&{}={\\frac {1}{2}}*base*height\\\\&{}={\\frac {1}{2}}*2\\pi r*r\\\\&{}=\\pi r^{2}\\end{aligned}}", "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}}\\geq A_{\\alpha +}^{\\sigma (i_{0})}", "00345c04233a175efdd1e2494c42a238": "\\phi _{1}=-30^{\\circ }...+30^{\\circ }", "0034991f8f6e6f84b95247f345004bb4": "{\\binom {S}{k}}\\,", "0034befe82b7a681848dd6ebb6634a0e": "{\\begin{cases}1&(e^{-p}){\\mbox{ no disaster}}\\\\1-b&(1-e^{-p}){\\mbox{ disaster}}\\\\\\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": "\\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_{2}P_{2}^{*}f_{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_{G}^{*}X,", "003a5820c464d82eca6633352a4c42b9": "r_{m}=r_{c}(1-t)\\,", "003a5ac3c6316db47dde21e454be0a6c": "S=-k_{B}\\,\\sum _{i}p_{i}\\ln \\,p_{i},", "003a70ac099d1c13e037072a7f78ca76": "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}+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]\\right\\}{\\text{d}}x{\\text{d}}y", "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": "{\\begin{array}{l}s_{0}=1\\qquad s_{1}=0\\\\t_{0}=0\\qquad t_{1}=1\\\\\\ldots \\\\s_{i+1}=s_{i-1}-q_{i}s_{i}\\\\t_{i+1}=t_{i-1}-q_{i}t_{i}\\\\\\ldots \\end{array}}", "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": "S_{ij}:=r_{ij}^{(t)}+g_{ij}^{(t)}+b_{ij}^{(t)}", "003dc09bb55482b2f72537dd1850d588": "\\sigma _{N}^{2}={\\frac {(N-1)\\,\\sigma _{N-1}^{2}+(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": "ds^{2}=d\\tau ^{2}-{\\frac {r_{g}}{r}}d\\rho ^{2}-r^{2}(d\\theta ^{2}+\\sin ^{2}\\theta d\\phi ^{2})", "003f38a83670c4350403298b1f4364b6": "e_{ij}=\\mathbf {e} _{i}\\cdot \\mathbf {e} _{j}.", "003f38e45eec556ade8244f8870ae85e": "{S_{3} \\over S_{2}}={{16 \\over 15}\\div {135 \\over 128}}", "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}.", "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": "\\left({\\frac {dr}{d\\tau }}\\right)^{2}={\\frac {E^{2}}{m^{2}c^{2}}}-c^{2}+{\\frac {r_{s}c^{2}}{r}}-{\\frac {h^{2}}{r^{2}}}+{\\frac {r_{s}h^{2}}{r^{3}}}", "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 {mL^{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": "D=O^{T}AO={\\begin{bmatrix}\\lambda _{-}&0\\\\0&\\lambda _{+}\\end{bmatrix}}", "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}{\\frac {\\partial L_{2}}{\\partial w}}=0\\quad \\to \\quad w=\\sum \\limits _{i=1}^{N}\\alpha _{i}\\phi (x_{i}),\\\\{\\frac {\\partial L_{2}}{\\partial b}}=0\\quad \\to \\quad \\sum \\limits _{i=1}^{N}\\alpha _{i}=0,\\\\{\\frac {\\partial L_{2}}{\\partial e_{i}}}=0\\quad \\to \\quad \\alpha _{i}=\\gamma e_{i},\\;i=1,\\ldots ,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.\\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)\\leq \\left({\\sqrt {3/2}}+\\varepsilon \\right)^{d}", "00489f32547332d509d28f64be77a6c3": "{\\begin{cases}N_{j}\\left(U^{\\left(n\\right)}\\right)=\\Gamma _{jk}U_{k}^{\\left(n\\right)}-U_{j}^{\\left(n\\right)}\\\\M_{j}\\left(U^{\\left(n\\right)}\\right)=p_{i}~a_{ijkl}{\\frac {\\partial U_{k}^{\\left(n\\right)}}{\\partial x_{l}}}+\\rho ^{-1}{\\frac {\\partial }{\\partial x_{i}}}\\left(\\rho ~a_{ijkl}~p_{l}U_{k}^{\\left(n\\right)}\\right)\\\\L_{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)\\end{cases}}", "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 {3c^{4}}}\\left({A \\over 2}T_{\\alpha \\beta }+{B \\over 2}T\\eta _{\\alpha \\beta }\\right)\\left(\\delta _{\\nu }^{\\mu }+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 \\ M'}{\\displaystyle M\\|N\\ \\rightarrow \\ M'\\|N}};\\qquad \\qquad {\\it {(Comp2)}}\\quad {\\frac {\\displaystyle M\\ \\rightarrow \\ M'\\qquad \\displaystyle N\\ \\rightarrow \\ N'}{\\displaystyle M\\|N\\ \\rightarrow \\ M'\\|N'}}", "004a9f231095f3c08e2f82e54dd4643f": "\\exp \\left(\\sum _{n=1}^{\\infty }{a_{n} \\over n!}x^{n}\\right)=\\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 yv_{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}),", "004ba7069754fed522854714a8660e16": "{\\overline {z}}=z\\!\\ ", "004bc28bf353a7a7dae3f540aa4c86a5": "I_{c}", "004c00048d155c6aaeee77859a8b45a8": "\\,A\\mapsto M\\alpha (A)M^{-1},", "004c04db969c835339fb23593190d46f": "E{\\bar {X}}_{A}=\\mu _{HA}{\\frac {p_{HA}}{p_{HA}+p_{LA}}}+\\mu _{LA}{\\frac {p_{LA}}{p_{HA}+p_{LA}}},", "004c69ff4b40f7cceab9e42b8f7370fa": "{d^{2}{\\bar {h}}^{i} \\over ds^{2}}+2\\Gamma _{j}^{i}{d{\\bar {h}}^{i} \\over ds}+{d\\Gamma _{j}^{i} \\over ds}{\\bar {h}}^{j}+\\Gamma _{j}^{i}\\Gamma _{k}^{j}{\\bar {h}}^{k}+{\\bar {R}}_{j}^{i}{\\bar {h}}^{j}=0", "004c72301f64855e456aa920a32a1d7c": "{\\tbinom {2}{4}}", "004cc0101dda11ac74e94adc07c9aae2": "det(A)\\neq 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 \\mathbb {N} }\\times (b_{n})_{n\\in \\mathbb {N} }=\\left(\\sum _{k=0}^{n}a_{k}b_{n-k}\\right)_{n\\in \\mathbb {N} }.", "004dadc66378395b6a21b73bdbab86e3": "C=\\{C_{k}^{i}\\}", "004dbfe6dc52810c3e2192e98e8edac0": "M(X)=\\left({\\begin{array}{*{20}c}\\mu \\\\\\Sigma \\\\\\end{array}}\\right)", "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": "(0,653,1854,4063)\\rightarrow (653,1201,2209,4063)\\rightarrow (548,1008,1854,3410)\\rightarrow ", "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": "cr^{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 _{2}n_{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 \\mathbb {R} ^{n}", "0053bd74249ba2edd4ff39532c528ca8": "c_{2}=2.04901523,\\,\\!", "00546b61d4996074c0643b1be8cf5802": "\\{|\\phi _{i}\\rangle \\}", "0054cb6e5b751157081556d7e575ca24": "L(w)", "0054e06028ca38fa0a1cc337ae69ed98": "\\mathrm {core} _{2}", "005503b59bc42d27c5c1ba90c5099d82": "a={\\frac {a^{4}+b^{4}+c^{4}+a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}}{\\left(a^{2}+b^{2}+c^{2}\\right)^{2}}}\\Delta ", "0055139ef653b9bfbedea5d4c316a3d4": "{\\mathcal {E}}(\\exp )=\\{0\\}", "00552124bea53f3a68f87e28129a5903": "e_{i}^{(1)}=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}}{2m_{0}}}+{\\frac {\\hbar ^{2}}{m_{0}^{2}}}\\sum _{\\gamma }^{B}{\\frac {p_{x\\gamma }^{y}p_{\\gamma x}^{y}}{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}+IX", "00572f45e35e977389316f0eef29c429": "\\psi _{0}|0\\rangle +\\int _{x}\\psi _{1}(x)|1;x\\rangle +\\int _{x_{1}x_{2}}\\psi _{2}(x_{1},x_{2})|2;x_{1}x_{2}\\rangle +\\ldots \\,", "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 '}_{\\alpha _{l}}^{[l]})^{2}=\\sum \\limits _{\\alpha _{l}=1}^{\\chi _{c}}(c_{\\alpha _{l-1}\\alpha _{l}})^{2}{\\frac {(\\lambda _{\\alpha _{l}}^{[l]})^{2}}{R}}={\\frac {S_{1}}{R}}", "00576e1590136e3c819062a933b43d7c": "\\mu (A)={\\begin{cases}1&{\\mbox{ if }}0\\in A\\\\0&{\\mbox{ if }}0\\notin A.\\end{cases}}", "00578b5ebbc08a904cf34a0c1a0819ea": "\\theta =90^{\\circ }", "0057a1113ace7fce93043cd1f12d3d08": "J:X\\to (X'_{\\beta })'_{\\beta }.", "0057baf398e7cfd6f637c36ce0d9990a": "\\ell _{({M},\\varphi )}({\\bar {x}},{\\bar {y}})=\\sum _{p=(x,y) \\atop x\\leq {\\bar {x}},y>{\\bar {y}}}\\mu {\\big (}p{\\big )}+\\sum _{r:x=k \\atop k\\leq {\\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": "g(s)=\\int _{0}^{\\infty }(st)^{-k-1/2}\\,e^{-st/2}\\,W_{k+1/2,\\,m}(st)\\,f(t)\\;dt,", "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": "\\mathrm {B} ;\\ G;\\ \\Upsilon ", "005aff1ab64bae2fbd389e08eedceaee": "g\\in [(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{aligned}E_{f_{1}+f_{2}}&=kE_{f_{1}}E_{f_{2}}\\\\E_{f_{1}-f_{2}}&=kE_{f_{1}}E_{f_{2}}\\end{aligned}}", "005b76ddf58418b5840fbcd038a55157": "\\nabla _{\\mathbf {u}}{\\mathbf {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_{x}M", "005cf2bd315336ccfc51a82fbc1d011b": "D[p]=[q,\\_,p]::[x,\\_,f]::\\_", "005cfe08ac4514176ec9114ed86f5227": "(y+[y/4]+5(c\\mod 4)-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{aligned}{\\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{aligned}}", "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+2ac+2ad+2bc+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": "\\|u\\|_{L^{p}}\\leq C\\|u\\|_{L^{q}}^{\\alpha }\\|u\\|_{H_{0}^{s}}^{1-\\alpha },", "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\\leq 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": "\\inf _{g}\\sup _{f}\\iint K\\,df\\,dg={\\frac {3}{7}}.", "0062c755efea0b9be6ef3dd55ccc30c6": "{\\overline {I}}={\\overline {\\overline {I}}}", "0062d94d1a6a6962840096804a79eb6f": "{\\mathcal {L}}\\{f''\\}=s^{2}{\\mathcal {L}}\\{f\\}-sf(0)-f'(0)", "0062df2399c2fbd55c34251620e6f357": "{\\begin{aligned}{\\boldsymbol {F_{12}}}&=m_{1}{\\boldsymbol {a_{1}}},\\\\{\\boldsymbol {F_{21}}}&=m_{2}{\\boldsymbol {a_{2}}},\\end{aligned}}", "0062f69c43f50b5e581711b6f431a0af": "\\textstyle 3+\\log _{2}(n)", "0063113efc28a4d2117081f92b8a8e22": "{\\begin{bmatrix}a_{11}&a_{12}&a_{13}\\\\0&a_{22}&a_{23}\\\\0&0&a_{33}\\end{bmatrix}}", "00634867b24389e3680d995d91df3a9e": "0\\rightarrow B\\rightarrow A\\oplus B\\rightarrow A\\rightarrow 0.", "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}1&a&c\\\\0&1&b\\\\0&0&1\\\\\\end{pmatrix}}", "0063a9838403b9181103f102ed4f2286": "{\\begin{aligned}N(x)&=[{y}_{k}]+[{y}_{k},{y}_{k-1}]sh+\\cdots +[{y}_{k},\\ldots ,{y}_{0}]s(s+1)\\cdots (s+k-1){h}^{k}\\\\&=\\sum _{i=0}^{k}{(-1)}^{i}{-s \\choose i}i!{h}^{i}[{y}_{k},\\ldots ,{y}_{k-i}]\\end{aligned}}", "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\\times 10^{-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_{1}V_{1}=A_{2}V_{2}", "00697121901844d211b29641023e5ffe": "Rev_{t}", "0069a29184ac94f333c07b1dea9e3f8c": "C_{2}\\leq 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": "z^{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)}}", "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 {6G}{2-K}}-{\\frac {2G(K+4)e^{4\\phi _{0}}}{(2-K)^{2}}}-1", "0077c400b0161a221aa7adb882c272d7": "{o.p.d.}=\\Delta \\,n\\cdot t", "0077e6527194ccd11d6c32c045f506f0": "\\tau ={\\sqrt {|\\mathbf {t} |^{2}-\\sigma ^{2}}}", "0077ee7d7b6ea8618a0ade235c73ef68": "\\{p_{1},r_{1}\\}", "0078761465bd6fd6b3215e5c47313b31": "s^{2}={\\frac {w_{1}}{(1-w_{2})^{2}}}", "00787af9608e8ef51e433a7c51d94e00": "\\omega ^{2}r", "00790b5b8c2899d32e6f8362444877cc": "k_{f}^{water}=400s^{-1},k_{u}^{water}=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 )}=\\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": "\\int _{0}^{\\infty }x^{2l+2}e^{-x}\\left[L_{n-l-1}^{(2l+1)}(x)\\right]^{2}dx={\\frac {2n(n+l)!}{(n-l-1)!}}.", "007a632787fbac1c7b731d3853db5170": "y=b\\,", "007a9a1e8463ef195d0a9b1dc88e057e": "{\\begin{bmatrix}V_{1}\\\\V_{0}\\end{bmatrix}}={\\begin{bmatrix}z(j\\omega )_{11}&z(j\\omega )_{12}\\\\z(j\\omega )_{21}&z(j\\omega )_{22}\\end{bmatrix}}{\\begin{bmatrix}I_{1}\\\\I_{0}\\end{bmatrix}}", "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{aligned}\\iint _{R_{C}}s(x,t)dxdt&=-\\int _{x_{i}-ct_{i}}^{x_{i}+ct_{i}}g(x)dx+cu(x_{i},t_{i})-cf(x_{i}+ct_{i})+cu(x_{i},t_{i})-cf(x_{i}-ct_{i})\\\\&=2cu(x_{i},t_{i})-cf(x_{i}+ct_{i})-cf(x_{i}-ct_{i})-\\int _{x_{i}-ct_{i}}^{x_{i}+ct_{i}}g(x)dx\\end{aligned}}", "007f2e086cc219e6031d2b739e28790a": "\\Delta y=\\Delta X*{\\frac {1}{(1-b_{C})(1-b_{T})+b_{M}}}", "007f3d5eec88b2997787156a0da80d1b": "a^{k}=(a_{i}^{k})_{i\\in I}", "007f6d48228b41dbfec441fdb60f208d": "{\\begin{aligned}\\Gamma (z)&=\\int _{0}^{\\infty }d\\lambda e^{-\\lambda }\\lambda ^{z-1}\\\\&=-\\int _{0}^{\\infty }d\\left(e^{-\\lambda }\\right)\\lambda ^{z-1}\\\\&=-\\left[e^{-\\lambda }\\lambda ^{z-1}\\right]_{0}^{\\infty }+\\int _{0}^{\\infty }d\\left(\\lambda ^{z-1}\\right)e^{-\\lambda }\\\\&=0+\\int _{0}^{\\infty }d\\lambda \\left(z-1\\right)\\lambda ^{z-2}e^{-\\lambda }\\\\&=(z-1)\\Gamma (z-1)\\\\\\end{aligned}}", "007f7cfb6c836265f0ee259f9795c82e": "0\\rightarrow G/\\ker \\,f\\rightarrow H\\rightarrow \\operatorname {coker} \\,f\\rightarrow 0", "007f7db03b1721f021e315ea7df8efac": "\\Omega \\,\\!", "007f7f641efda0619b3f766fb9789e1d": "{\\Bigl [}{\\begin{smallmatrix}\\mathrm {*} &\\mathrm {T} &\\mathrm {*} \\\\\\mathrm {*} &\\mathrm {*} &\\mathrm {*} \\\\\\mathrm {*} &\\mathrm {*} &\\mathrm {*} \\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\\geq 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}}_{k}s[n-k]+(\\sum _{k=0}^{N-1}{\\hat {h}}_{k}s[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^{j}b'(x)\\mod (x^{2t-1}-1)", "00839570c7cb93cd4611c23bd52bbef1": "B_{1}+B_{2}a=C_{r}e^{iak_{0}}+C_{l}e^{+ak_{0}}", "0083b07c7fb9fba73f101e2b1eecfba3": "\\{C:K_{X}\\cdot C=0\\}", "0083c4e87edab8507e96fdde5c911ab3": "q(\\mu ,\\tau )=q(\\mu )q(\\tau )", "0083dcac1f5eaa37fd0eb3503722e9b2": "\\Theta \\wedge (d\\Theta )^{n}\\neq 0", "0084209ec3306ab04a193d13223f53d3": "H(p,q)={\\mathcal {F}}\\left\\{h(x,y)\\right\\}", "00843f9d223ff4c5c126d001c62f48c3": "{\\mathcal {P}}_{2}(-p_{2})=a_{20}(-p_{2})^{2}+a_{11}(-p_{2})+a_{02}=0", "00844945aabd62ba8956c429106513d1": "\\sum \\limits _{i=0or1}^{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\\approx 0", "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))\\leq \\sum _{k=0}^{n-1}{\\frac {{\\bigl (}\\mu (I_{s,t}){\\bigr )}^{k}}{k!}}\\leq \\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": "\\{kx: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}nRr^{2}=\\left(\\pi r^{2}\\right)\\left(2\\pi nR\\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 {d} E_{\\text{i}}(\\omega _{\\text{i}})", "008a30ccbc263d1d59165530204391c4": "\\mathrm {[Cr]=[CrO_{4}^{2-}]+[HCrO_{4}^{-}]+2[Cr_{2}O_{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{aligned}x:\\;\\;\\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)&=-{\\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}\\\\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)&=-{\\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}\\\\z:\\;\\;\\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)&=-{\\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}.\\end{aligned}}", "008b1a26a2217fe7dfd19fbfb8bab404": "\\scriptstyle x_{n+1}\\;=\\;{\\frac {x_{n}}{2}}\\,+\\,{\\frac {1}{x_{n}}}", "008b977cbc0bd622e23da22e303e3107": "I_{v}=\\log _{2}", "008bc43ba83b29504bc182aa7b9357b9": "x^{(2)}={\\begin{bmatrix}0.000&-0.1875\\\\0.000&-0.1193\\end{bmatrix}}\\times {\\begin{bmatrix}0.5000\\\\-0.8636\\end{bmatrix}}+{\\begin{bmatrix}0.6875\\\\-0.7443\\end{bmatrix}}={\\begin{bmatrix}0.8494\\\\-0.6413\\end{bmatrix}}.", "008bcc3802c42a9731f0425bd63421c4": "\\int x^{m}\\left(a+b\\,x^{n}\\right)^{p}dx={\\frac {x^{m-n+1}\\left(a+b\\,x^{n}\\right)^{p+1}}{b(m+n\\,p+1)}}\\,-\\,{\\frac {a(m-n+1)}{b(m+n\\,p+1)}}\\int x^{m-n}\\left(a+b\\,x^{n}\\right)^{p}dx", "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}\\left[x_{i},p_{j}\\right]&=&i\\hbar \\delta _{i,j}\\\\\\left[x_{i},x_{j}\\right]&=&0\\\\\\left[p_{i},p_{j}\\right]&=&0\\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": "\\psi \\to \\psi '=U\\psi ", "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}1&x&z\\\\0&1&y\\\\0&0&1\\\\\\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": "\\psi ^{\\dagger }\\gamma ^{0}\\psi =2\\langle {\\bar {\\Psi }}\\Psi \\rangle _{SR}", "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": "{\\begin{aligned}C_{+}&=+{\\frac {1}{2}}C_{0}\\cos {\\left(\\theta -{\\frac {\\pi }{4}}\\right)},\\\\C_{-}&=-{\\frac {1}{2}}C_{0}\\sin {\\left(\\theta -{\\frac {\\pi }{4}}\\right)}.\\end{aligned}}", "00931e7be1a5c92a5c4f77c90428b2ab": "{\\begin{aligned}s&=g(x,u)+\\omega _{s}\\\\{\\dot {x}}&=f(x,u)+\\omega _{x}\\end{aligned}}", "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": "{\\begin{aligned}u=\\operatorname {prox} _{R}(x)\\iff &0\\in \\partial \\left(R(u)+{\\frac {1}{2}}\\|u-x\\|_{2}^{2}\\right)\\\\\\iff &0\\in \\partial R(u)+u-x\\\\\\iff &x-u\\in \\partial R(u).\\end{aligned}}", "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}]\\geq 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\\leq seqs\\leq 6", "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 _{\\eta }^{\\xi }-g^{\\xi \\nu }\\eta _{\\eta }^{\\nu }-g^{\\xi \\nu }\\eta _{\\eta }^{\\mu })\\Omega _{\\xi }^{\\eta }\\;", "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": "\\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)", "009c16baf1e28508b213dca7d341e659": "\\int _{S}fg\\,\\mathrm {d} \\mu =\\|f\\|_{p}\\,.", "009c405b8a6248a30dcbdf70d58f2757": "{\\frac {\\partial z}{\\partial x}}=2x+y", "009cd0a8f8dc2ab7263d4ef99ca9715f": "s_{p-2}\\equiv 0{\\pmod {M_{p}}}.", "009d31cbcf40a1cc2bab52af464ddb35": "C_{D,{\\text{induced}}}=\\pi A\\!R\\sum _{n=1}^{\\infty }nA_{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\\geq 0\\\\-x^{2},&{\\text{if }}x\\leq 0.\\end{cases}}", "009f4fb1e0486cae6830706cbe42128d": "X_{k}=\\sum _{n=0}^{N-1}x_{n}e^{-{\\frac {2\\pi i}{N}}nk}\\qquad k=0,\\dots ,N-1.", "009fbee309e9784672526a00264c93ef": "F=\\{(x,y):x\\in {\\mathcal {R}}^{b},\\,y\\in {\\mathcal {R}}^{n},\\;x=y\\}.", "009fd4b226543561f01553326ecbfee8": "T''(t)=Kc^{2}T(t)\\,", "00a00703dcf8dd553d1384468debf2f3": "N={\\binom {n+d}{d}}-1", "00a020df6280a23ac9daf6baf98439e9": "\\omega _{i},1\\leq i\\leq n", "00a02689dfc44a622ba4f7906e0469f6": "h_{\\text{out}}(G)\\leq \\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^{2}p^{*}\\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 qpq^{*}\\,\\!", "00a2dc92c377a5fcf3f907cf42ef0962": "\\left({\\frac {K_{0}+a}{1+a}}\\right)^{\\gamma }={\\frac {K_{0}}{\\phi }}.", "00a34cdb4626d2fd31087d976797e802": "V_{i}(\\omega _{k})\\rightarrow V_{ik}", "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": "F=A*E^{2}\\propto L^{2}", "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}\\,dA_{ij},", "00a554bca784b6cfea952ee3e5f75cb7": "\\operatorname {nil} \\equiv \\operatorname {false} ", "00a59b76ebf9abfb3d9fe3eefeb9e3f6": "L^{q_{\\theta }}", "00a59dc981df6c75b3538a4ba059633f": "{\\partial {L} \\over \\partial q_{i}}={\\mathrm {d} \\over \\mathrm {d} t}{\\partial {L} \\over \\partial {\\dot {q_{i}}}}.", "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{aligned}\\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]\\\\\\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]\\end{aligned}}", "00a8c8452e84811dd3222f97d0c094e0": "D_{j}={\\frac {y_{U}-y_{L}}{r-1}}\\quad (j=i+1,\\ldots ,i+r-1).", "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^{T}VX,", "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": "{\\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-{\\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": "{\\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)", "00abd9a9738fa0ab1bd4fe864640ac5f": "n\\log ^{O(\\log k)}n", "00ac8c8a2346f2a39cc30536fc519d74": "{\\frac {T_{A}}{T}}={\\bigg (}{\\frac {P_{A}}{P}}{\\bigg )}^{(k-1)/k}", "00acac7e9220e49e733584596a5f11e7": "\\{x,p_{x}\\}_{DB}=\\{y,p_{y}\\}_{DB}={\\frac {1}{2}}", "00acc68c74cb8ce7b35958b2a46e1f5d": "\\epsilon _{S}", "00acc9ac69fc3f2d366801f96e53c565": "({\\hat {k}},{\\hat {l}})", "00ace3be08aece29574b1c573b12f1f0": "{\\text{GF}}(2)^{n}", "00ad2f6e3b361d991d10c82a582bcf5a": "a_{2}={\\frac {-b_{1}+{\\sqrt {b_{1}^{2}-4b_{2}b_{0}}}}{2b_{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 vZ_{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}}{\\mbox{ and }}[z^{n}]{\\frac {1}{1-z}}\\sum _{k=1}^{m}{\\frac {z^{k}}{k}}=H_{m}{\\mbox{ for }}n\\geq m", "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": "{\\mathcal {L}}_{s}=-{\\frac {1}{2}}\\left[\\sigma ^{2}h^{\\alpha \\beta }\\partial _{\\alpha }\\phi \\partial _{\\beta }\\phi +{\\frac {1}{2}}{\\frac {G}{l^{2}}}\\sigma ^{4}F(kG\\sigma ^{2})\\right]{\\sqrt {-g}},", "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\\leq \\xi ", "00b463dbda2a23f566e8f81d9c0824ae": "T(s,x)=s(x)", "00b4662abd3732893063c7d52118bff1": "Lu=-(pu')'+qu=-(pu''+p'u')+qu=-pu''-p'u'+qu=(-p)D^{2}u+(-p')Du+(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": "\\sum _{\\stackrel {1\\leq k\\leq n}{\\gcd(k,n)=1}}f(\\gcd(k-1,n))=\\varphi (n)\\sum _{d\\mid n}{\\frac {(\\mu *f)(d)}{\\varphi (d)}},", "00b56b0fba4e86b56fa04b2abdc00d76": "\\operatorname {Var} [s^{2}]=\\operatorname {Var} \\left({\\frac {\\sigma ^{2}}{n-1}}\\chi _{n-1}^{2}\\right)={\\frac {\\sigma ^{4}}{(n-1)^{2}}}\\operatorname {Var} \\left(\\chi _{n-1}^{2}\\right)={\\frac {2\\sigma ^{4}}{n-1}}.", "00b5d8a91cc4d17d1f997fbba2dddff8": "(D\\nabla ^{2}\\psi -{\\mathbf {u}}({\\mathbf {u}}\\cdot D\\nabla ^{2}\\psi ))", "00b6157255eee3ea721509333534bcf1": "{\\mathcal {L}}_{X}Y=[X,Y]", "00b61ea310c446de9872ca46c979294d": "Volume={\\frac {\\pi }{6}}\\times L_{1}\\times L_{2}^{2}", "00b679c1724ef634f99d7959237a9ee6": "G'+*m", "00b6d2f480c9b40fe618f9917868f9b5": "\\left({\\begin{smallmatrix}\\;\\;\\;1&0&0\\\\\\;\\;\\;1&0&0\\\\-1&1&1\\end{smallmatrix}}\\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 {N}{i}}", "00b7f809d353a2d63350999bc4ad696d": "~~~~~U,V,\\{N_{i}\\}\\,", "00b7ffe43d793c9f4a697c6f2434bdb9": "z(\\infty )={\\frac {a(1-Q)-b}{aQ}}", "00b812e8b8737c6c5e614149e14930c7": "\\,d(X_{t}Y_{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}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}^{}}}\\;.", "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 }}_{2n+1}^{1}", "00c0c1609947a2417500d95a5d8ccd32": "f(x_{0}),f(f(x_{0}))", "00c142954a1c191f60016455013875bc": "G(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_{2}t_{3}{\\bigr )}^{3n}\\,=\\,(-1)^{n}{\\binom {3n}{n,n,n}},", "00c1d4a23bb74b1715414e4c510aede7": "\\{x\\geq 1,y\\geq 1\\}", "00c202de02da2a860d73448b4a381129": "\\mathbf {J^{T}W\\ \\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}}\\left[{\\frac {r_{cf_{1}}+r_{cf_{2}}+\\cdots +r_{cf_{k}}}{\\sqrt {k+2(r_{f_{1}f_{2}}+\\cdots +r_{f_{i}f_{j}}+\\cdots +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": "U(I|J)={\\frac {H(I)-H(I|J)}{H(I)}},", "00c46f60f691d54e4c15ca82cef20abd": "{\\text{refresh overhead}}={\\frac {0.246\\,{\\text{ms}}}{64\\,{\\text{ms}}}}=.0038\\,", "00c48d25333e2da00ac708770f86c606": "u(\\lambda ,T)\\partial \\lambda ={8\\pi hc \\over \\lambda ^{5}}{1 \\over e^{hc/\\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_{a}h_{b}h_{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({\\mathbf {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": "{\\begin{aligned}&{}\\quad L\\left(x_{1},x_{2},\\ldots ,x_{N},\\lambda _{1},\\lambda _{2},\\ldots ,\\lambda _{M}\\right)\\\\&=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)}.\\end{aligned}}", "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_{a}^{\\mu }T_{\\mu }\\,", "00c9633c6fc31447b561ff0cec0e8c50": "\\phi ={\\tfrac {1}{2}}(1+{\\sqrt {5}})", "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": "y\\rightarrow y^{5}-10y^{3}x^{2}+5yx^{4}+y_{0}", "00cc3e1c11be4342a07b3de6c9960cf8": "n<1000_{b}", "00cc8ba3289130d190f3412b29a48685": "{\\begin{aligned}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]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]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]B_{6}&=-{\\frac {35}{48}}a\\left(n^{3}-n^{4}+{\\frac {11}{16}}n^{5}-\\cdots \\right),\\\\[8pt]B_{8}&=\\quad {\\frac {315}{512}}a\\left(n^{4}-n^{5}+\\cdots \\right).\\end{aligned}}", "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 Ty-my,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_{2}^{-1}{\\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)-BK\\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\\leq y\\land y\\leq 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}cD\\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{aligned}x&=r\\sin \\theta \\cos \\phi \\\\y&=r\\sin \\theta \\sin \\phi \\\\z&=r\\cos \\theta \\end{aligned}}", "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": "\\sigma _{t}\\equiv {\\frac {8\\pi }{3}}r_{e}^{2}", "00d4698687efb283b8b2efc7d4eadbd7": "\\mathrm {GF} (q).", "00d4789a3dec5360bb488b32283ae6e5": "0100kms^{-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 {8k_{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)\\geq 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=15d^{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_{1}p_{2}}{2}}+{\\frac {p_{1}^{2}-p_{2}}{2}}\\,,", "00dcdbff0ef7631903745ed151e888eb": "H(2^{1})={\\begin{bmatrix}1&1\\\\1&-1\\end{bmatrix}},", "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": "ad^{2}+bd+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}{2n+1}}\\right)^{2k+1}.", "00df09b96a36904ebb578eb1f05f77a4": "cm\\cdot {\\sqrt {Hz}}/W", "00df2a8a4c44eb54e671d77699afa8ef": "F_{1}(a,b_{1},b_{2},c;x,y)={\\frac {\\Gamma (c)}{\\Gamma (a)\\Gamma (c-a)}}\\int _{0}^{1}t^{a-1}(1-t)^{c-a-1}(1-xt)^{-b_{1}}(1-yt)^{-b_{2}}\\,\\mathrm {d} t,\\quad \\Re \\,c>\\Re \\,a>0~.", "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^{3}r'", "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": "\\int \\exp \\left[\\int d^{4}x\\left(-{\\frac {1}{2}}\\varphi {\\hat {A}}\\varphi +iJ\\varphi \\right)\\right]D\\varphi \\;\\propto \\;\\exp \\left(-{1 \\over 2}\\int d^{4}x\\;d^{4}yJ\\left(x\\right)D\\left(x-y\\right)J\\left(y\\right)\\right)", "00e0bc6b6fa01b4434f090b3b0dc6335": "f\\colon (x,h)\\to (x',h')", "00e0dd01c0d7c832bd2d85ed799213eb": "{\\frac {d}{dx}}\\arccos(x)=-{\\frac {1}{\\sqrt {1-x^{2}}}},-11.96)=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_{B}T", "00eca1b27a7f6fcdb1c102ad67cfa641": "p_{i}(s)\\neq p_{j}(s)", "00ecba9a4dd7bd3d2981a76e7464ea45": "\\nu _{k}(\\mathbf {J} )={\\frac {1}{T}}", "00ecf52d65fb00be76ea52bbc333dd67": "y_{4}=y_{3}+h({\\tfrac {1}{4}}k_{1}+{\\tfrac {3}{4}}k_{2})={\\underline {1.335079087}}.", "00ed278ec09422df6c1b6c7544693a3a": "\\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}", "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": "\\tan \\theta =\\sin \\lambda \\tan(15^{\\circ }\\times t)", "00ee8205d9738aee1e2ee3086ae05f53": "y\\geq 0.398", "00ee92b891492c30771ce8b238d0e5be": "\\left({\\sqrt {\\frac {2}{5}}},\\ -{\\sqrt {\\frac {2}{3}}},\\ {\\frac {-5}{\\sqrt {3}}},\\ \\pm 1\\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{aligned}{\\binom {-r}{k}}&={\\frac {-r\\cdot -(r+1)\\dots -(r+k-2)\\cdot -(r+k-1)}{1\\cdot 2\\cdot 3\\cdot 4\\cdot 5\\cdots k}}\\\\&=(-1)^{k}\\;{\\frac {r\\cdot (r+1)\\cdot (r+2)\\cdots (f-2)\\cdot (f-1)\\cdot f}{1\\cdot 2\\cdot 3\\cdot 4\\cdot 5\\cdots k}}\\\\&=(-1)^{k}{\\binom {f}{k}}\\\\&=(-1)^{k}\\left(\\!\\!{\\binom {f-k+1}{k}}\\!\\!\\right)\\\\&=(-1)^{k}\\left(\\!\\!{\\binom {r}{k}}\\!\\!\\right)\\;,\\end{aligned}}", "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\\pm 0.9\\times 10^{-10}", "00f2a7fb18ef9f999f11d41d5d06f6cc": "p^{2}-p+1", "00f2ac1cfefd7f10d8f0f8602e8ada08": "\\mathrm {d} f_{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 \\leq \\tau _{\\mathcal {S}}}J^{\\tau }(y)=\\sup _{\\tau \\leq \\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})\\geq -1", "00f35a9b6f60fec19b77496b2355a1a0": "(S\\otimes T)_{j_{1}\\ldots j_{k}j_{k+1}\\ldots j_{k+m}}^{i_{1}\\ldots i_{l}i_{l+1}\\ldots i_{l+n}}=S_{j_{1}\\ldots j_{k}}^{i_{1}\\ldots i_{l}}T_{j_{k+1}\\ldots j_{k+m}}^{i_{l+1}\\ldots i_{l+n}},", "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 =dS-{\\frac {TdU-UdT}{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": "r_{1}^{2}r_{2}^{2}\\left({\\frac {d\\theta _{1}}{dt}}\\right)\\left({\\frac {d\\theta _{2}}{dt}}\\right)-2a\\left[\\mu _{1}\\cos \\theta _{1}+\\mu _{2}\\cos \\theta _{2}\\right],", "00f7871570c9fbe85f3d77ce2a47ed28": "(r,\\theta ,\\phi )", "00f7afd7395deaf6c4c7b1225d1be196": "\\scriptstyle M^{-T}", "00f7c047a2bb558b2d9cacf653e904f9": "g(x)\\partial _{x}", "00f7c9af6fe1a26a5273fd624549bd78": "P_{j}^{n}=\\left({\\begin{array}{l}n\\\\j\\end{array}}\\right)j!={\\frac {n!}{(n-j)!}}.", "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": "C_{k}=\\left({\\frac {1}{k}}\\right)\\int d\\theta ^{\\prime }\\int d\\rho ^{\\prime }\\left(\\rho ^{\\prime }\\right)^{k+1}\\lambda (\\rho ^{\\prime },\\theta ^{\\prime })\\cos k\\theta ^{\\prime }", "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 UB_{1}g=B_{2}Ug=\\sum _{i=0}^{n}(B_{2}^{*})^{i}Ah_{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 lM^{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_{1}A_{2})^{2}-r_{1}^{2}-r_{2}^{2}\\,", "00fb626b41cdaeed91618e2c143511ec": "{\\begin{aligned}\\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\\\\&=P\\ {\\frac {3}{2}}\\ {\\frac {1}{V_{0}}}\\ \\ F\\ {\\hat {l}}\\end{aligned}}", "00fc431ddc28efbd388ad723f0f0ee25": "\\psi (0,x)", "00fc606c713c687da931b916520aa0ab": "V(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})\\,", "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\\lor q", "01001b39914230da09b6548877a4cb99": "135=(1+3+5)(1\\times 3\\times 5)", "01002661415b311f875cbb1b0149cabf": "x\\in \\mathbb {R} ", "010056e8dd4c8176092bfd7c448d3ef3": "\\ell _{2}=r'+a'", "01009cc723b713a37f31197e765611ac": "\\lim _{n\\to \\infty }{\\frac {\\log |W_{n}|}{n^{2}}}=h>0.", "0100c57389c7ef9cbf33292dc5557d3f": "{\\mathcal {M}}_{ij}={\\begin{cases}1/L(p_{j}),&{\\mbox{if }}j{\\mbox{ links to }}i\\ \\\\0,&{\\mbox{otherwise}}\\end{cases}}", "0100feb2d04bb42c8d668cb8c1f745de": "\\left[{\\begin{alignedat}{6}1&&0&&-3&&0&&2&&0\\\\0&&1&&5&&0&&-1&&4\\\\0&&0&&0&&1&&7&&-9\\\\0&&\\;\\;\\;\\;\\;0&&\\;\\;\\;\\;\\;0&&\\;\\;\\;\\;\\;0&&\\;\\;\\;\\;\\;0&&\\;\\;\\;\\;\\;0\\end{alignedat}}\\,\\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} \\leq T\\leq +50^{\\circ }\\mathrm {C} \\;\\;(\\leq 0.05\\%)}", "0101b1db64c8b0252ec743708a73f160": "t\\mapsto (t,f(t)).", "0101d89bbbc91b5bcb24aefc9c85d788": "\\psi _{l}", "01026a06c53da688c72cb0a160dfbfa9": "\\mu (x,y)={\\begin{cases}{}\\qquad 1&{\\textrm {if}}\\quad x=y\\\\[6pt]\\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": "{\\begin{pmatrix}1&2&3&4&5&6&7&8\\\\4&2&7&6&5&8&1&3\\end{pmatrix}}={\\begin{pmatrix}1&4&6&8&3&7&2&5\\\\4&6&8&3&7&1&2&5\\end{pmatrix}}=(146837)(2)(5)", "0103d9083b6297bd3abb5e70f74e36fd": "CM\\,", "0103f34b09470ebfb13324efd2ea958a": "\\scriptstyle {\\dot {m}}_{01}\\,0\\,p_{21}\\,", "010445a7575314b56e76038a7323011e": "V=U", "01044947b534a2326edc87845aaf5e73": "\\left[{\\begin{smallmatrix}2&-1&0&0\\\\-1&2&-1&-1\\\\0&-1&2&0\\\\0&-1&0&2\\end{smallmatrix}}\\right]", "01045123e83db59cdcae28d0568aefb7": "E''={\\frac {E\\tau _{0}\\omega }{\\tau _{0}^{2}\\omega ^{2}+1}},", "01045a9d6f8880f1083a52e42c4fa3a2": "{\\underline {\\underline {{\\boldsymbol {A}}_{1}}}}={\\begin{bmatrix}-1&0&0\\\\0&1&0\\\\0&0&1\\end{bmatrix}}~;~~{\\underline {\\underline {{\\boldsymbol {A}}_{2}}}}={\\begin{bmatrix}1&0&0\\\\0&-1&0\\\\0&0&1\\end{bmatrix}}~;~~{\\underline {\\underline {{\\boldsymbol {A}}_{3}}}}={\\begin{bmatrix}1&0&0\\\\0&1&0\\\\0&0&-1\\end{bmatrix}}", "010486e2b1be8fe0ef4030d3d106dc74": "\\Gamma _{\\beta \\gamma }^{\\alpha }={\\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": "{\\begin{pmatrix}0&i{\\bar {\\partial }}\\\\i\\partial &0\\end{pmatrix}}{\\begin{pmatrix}{\\bar {\\Psi }}^{\\dagger }P_{3}\\\\\\Psi P_{3}\\end{pmatrix}}=m{\\begin{pmatrix}{\\bar {\\Psi }}^{\\dagger }P_{3}\\\\\\Psi P_{3}\\end{pmatrix}}", "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}dxdy},", "01083b716768aff86c8863df3ec483c5": "{\\frac {|{\\text{median}}-{\\text{mode}}|}{\\text{standard deviation}}}\\leq {\\sqrt {3}}", "01087980a48f55dc43c8509c3340e6c7": "{\\begin{aligned}L&=T-U={\\frac {1}{2}}M{\\dot {\\mathbf {R} }}^{2}+\\left({\\frac {1}{2}}\\mu {\\dot {\\mathbf {r} }}^{2}-U(r)\\right)\\\\&=L_{\\mathrm {cm} }+L_{\\mathrm {rel} }\\end{aligned}}", "0108997daf7ca086f0286c453fbd686a": "\\left\\langle \\int \\phi ({\\boldsymbol {x}},t)\\,d{\\boldsymbol {x}}\\,dt\\right\\rangle =\\int \\langle \\phi ({\\boldsymbol {x}},t)\\rangle \\,d{\\boldsymbol {x}}\\,dt.", "010902b462092577c279b155d9b6c730": "{\\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}}}.", "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_{a}Te^{-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_{\\mu }^{(0)}\\,", "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^{i2\\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": "PV^{\\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": "(12)(34),\\;(13)(24),\\;(14)(23)", "010e1df78a41ec6f33dc926c7e788f53": "a=d\\sin \\alpha {\\text{ and }}b=d\\sin \\beta .\\,", "010e22805899e839e8ad0357d6291459": "{\\begin{aligned}\\mathrm {d} ^{k}X&=\\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)\\\\&=\\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{aligned}}", "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": "{\\begin{matrix}&&0\\\\&0&\\\\0&&B_{i-2,3}\\\\&B_{i-1,2}&\\\\1&&B_{i-1,3}\\\\&B_{i,2}&\\\\0&&B_{i,3}\\\\&0&\\\\&&0\\\\\\end{matrix}}", "010f0cc465fa1897532a16c9a7bebccf": "K({\\overline {\\alpha }},\\alpha ')=\\langle \\alpha |\\alpha '\\rangle =\\left[{\\mathcal {N}}(\\vert \\alpha \\vert ^{2}){\\mathcal {N}}(\\vert \\alpha '\\vert ^{2})\\right]^{-{\\frac {1}{2}}}\\sum _{n=0}^{\\infty }{\\frac {({\\overline {\\alpha }}\\alpha ')^{n}}{\\varepsilon _{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 }}{dX^{\\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_{i}S-b_{i}T)", "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": "{\\boldsymbol {u}}^{(0)}={\\boldsymbol {x}}-{\\boldsymbol {X}},\\qquad {\\boldsymbol {u}}^{(1)}={\\boldsymbol {x'}}-{\\boldsymbol {x}}", "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": "r^{m}\\sin ^{m}\\theta \\sin m\\varphi ={\\frac {1}{2i}}\\left[(r\\sin \\theta e^{i\\varphi })^{m}-(r\\sin \\theta e^{-i\\varphi })^{m}\\right]={\\frac {1}{2i}}\\left[(x+iy)^{m}-(x-iy)^{m}\\right].", "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_{n}W_{n+1}", "0113bc090b7893e7fe1785bd13a56f66": "{\\mathbf {g}}=-{\\mathbf {\\nabla }}\\phi _{g}\\,\\quad {\\mathbf {E}}=-{\\mathbf {\\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 {\\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{aligned}V_{1}(\\mathbb {R} ^{n})&=S^{n-1}\\\\V_{1}(\\mathbb {C} ^{n})&=S^{2n-1}\\\\V_{1}(\\mathbb {H} ^{n})&=S^{4n-1}\\end{aligned}}", "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": "{\\begin{bmatrix}x&1\\\\1&x\\\\\\end{bmatrix}}\\times {\\begin{bmatrix}c_{1}\\\\c_{2}\\\\\\end{bmatrix}}=0", "011b0fa13253a12989641a4f775d6a93": "L({\\hat {y}},y)", "011b6e3f128e0de494b3cf0dbecebdb4": "u_{j}=|\\langle rA_{j}\\rangle |", "011b76b2657900f43aeb2eb6f00f3078": "1+z=\\left(1+{\\frac {v}{c}}\\right)\\gamma .", "011ba3f0db4cd8865f15adc08b9b1e4a": "c_{2}=0.988622465,\\,\\!", "011c329b23a7ef28a2ac2e3acf831905": "\\lim _{z\\to 0}{\\frac {1}{z}}\\left\\{{\\frac {1}{\\Gamma (1+z)}}-{\\frac {1}{\\Gamma (1-z)}}\\right\\}=2\\gamma ", "011c45f9300361dab2a3178eb0de4fc1": "{\\varphi }", "011cc0e22684bf7c68fafa96e57bfea9": "\\Pr \\left\\{\\lambda _{\\max }\\left(\\sum _{k}\\mathbf {X} _{k}\\right)\\geq t\\right\\}\\leq \\inf _{\\theta >0}\\left\\{e^{-\\theta t}\\operatorname {tr} e^{\\sum _{k}\\log \\mathbf {M} _{\\mathbf {X} _{k}}(\\theta )}\\right\\}", "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 _{n=0}^{\\infty }f_{n}x^{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{aligned}{\\frac {\\partial \\mathbf {u} }{\\partial t}}+\\left(\\mathbf {u} \\cdot \\nabla \\right)\\mathbf {u} &=-{\\frac {\\nabla p}{\\rho }}+v\\nabla ^{2}\\mathbf {u} \\\\\\nabla \\cdot \\mathbf {u} &=0\\\\\\mathbf {u} _{\\text{bd}}&=\\mathbf {u} _{\\text{s}}.\\end{aligned}}", "011e6b034711ad7c2533ec10a802a236": "R_{a}={\\sqrt {MN}}={\\frac {a^{2}b}{(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}}\\rightarrow {\\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)\\geq 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": "T_{T}=\\sum _{i=1}^{m}s_{i}T_{T_{i}}+\\sum _{i=1}^{m}s_{i}\\log {\\frac {{\\overline {x}}_{i}}{\\overline {x}}}", "0126a60313e72eefaf6c46737d9b41a1": "d=s_{1}u_{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\\leq \\delta <1-{\\frac {1}{q}}", "012a6bf5f2d5689d4b61f63efb7d36e9": "x_{3}=0", "012af98c41fa64353b10d071979f4ae5": "{\\cfrac {\\partial g}{\\partial g_{ij}}}=2~J~{\\cfrac {\\partial J}{\\partial g_{ij}}}=g~g^{ij}", "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} \\\\\\arctan {x},&{\\text{if }}x\\in \\mathbb {R} \\backslash \\mathbb {Q} \\\\\\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_{1}a_{2}-T_{2}a_{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": "PA-(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": "{\\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 )}", "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)={\\begin{cases}\\;\\;\\,0{\\mbox{ if }}\\gcd(a,n)\\neq 1\\\\\\pm 1{\\mbox{ if }}\\gcd(a,n)=1\\end{cases}}", "013408c14b63d227243d789a3e82deb2": "s_{0}(t)={\\frac {\\alpha \\,e^{\\beta t}-\\beta \\,e^{\\alpha t}}{\\alpha -\\beta }},\\quad s_{1}(t)={\\frac {e^{\\alpha t}-e^{\\beta t}}{\\alpha -\\beta }}\\quad ", "013430fa683e50e86ae691586e6b6348": "|x_{\\theta }\\rangle ", "0134475bc0c73018a4d06bb200daf95a": "{\\begin{aligned}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]&{}=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]&{}=\\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]&{}=\\cos x+i\\sin x\\ .\\end{aligned}}", "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}s&{\\mbox{if }}a=b\\\\(s\\div b)a&{\\mbox{if }}a\\neq b\\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}-{\\frac {23}{7875}}x^{4}-{\\frac {1894}{3931875}}x^{5}-{\\frac {3293}{21896875}}x^{6}-{\\frac {2418092}{62077640625}}x^{7}-\\cdots \\right)\\ ", "0134fe896f2db72d72ee8faad50ead66": "|(a,b,c)|^{2}", "013555e4d53232dc5e312301b6b684f1": "\\displaystyle {e_{\\alpha }(z)={z^{\\alpha } \\over {\\sqrt {\\alpha !}}}}", "01356f495dd2fd66b165b161ea7acc6c": "t=t_{0}", "013572c04c0d30a2f5bb460305929605": "X\\mapsto {\\mathcal {P}}_{n_{\\infty }\\wedge n_{o}}^{\\perp }\\left({\\frac {X}{-X\\cdot n_{\\infty }}}\\right)", "0135e0d854ad3f435241fd00e79366c6": "y=b\\ \\sinh \\ \\mu ", "0135efc53a1ef0d8b71ebd8bd463323c": "molFe_{2}O_{3}={\\frac {20.0g}{159.7g/mol}}=0.125mol\\,", "0135f990c3ed5081e26a1dc50109e6b9": "2\\left\\langle T\\right\\rangle _{\\tau }=-\\sum _{k=1}^{N}\\left\\langle \\mathbf {F} _{k}\\cdot \\mathbf {r} _{k}\\right\\rangle _{\\tau }.", "0136048f886a13a8dee3dbc967d039e2": "T_{j}^{i}", "013624a40fe42347cbab24b181f961d9": "U_{\\mathbf {Q} _{p}}^{(n_{p})}\\subseteq N_{L_{\\mathfrak {p}}^{\\chi }/\\mathbf {Q} _{p}}(U_{L_{\\mathfrak {p}}^{\\chi }})", "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": "\\int _{1}^{M}{\\frac {1}{x^{1+\\varepsilon }}}\\,dx=-{\\frac {1}{\\varepsilon x^{\\varepsilon }}}{\\biggr |}_{1}^{M}={\\frac {1}{\\varepsilon }}{\\Bigl (}1-{\\frac {1}{M^{\\varepsilon }}}{\\Bigr )}\\leq {\\frac {1}{\\varepsilon }}<\\infty \\quad {\\text{for all }}M\\geq 1.", "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}}}+\\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)=np_{n-1}(x).", "013b729bdc2d69ac52cbf745a00e2bb2": "{\\begin{bmatrix}1&1.25\\\\0&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}=\\mathbf {X} -\\sum _{s=1}^{k-1}\\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 {dP}{dT}}={\\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": "g_{n}={\\binom {N+n-1}{n}}", "0141f319f9169fa31c63fc24d4dfeded": "\\int x^{m}\\operatorname {arccsc}(a\\,x)\\,dx={\\frac {x^{m+1}\\operatorname {arccsc}(a\\,x)}{m+1}}\\,+\\,{\\frac {1}{a\\,(m+1)}}\\int {\\frac {x^{m-1}}{\\sqrt {1-{\\frac {1}{a^{2}\\,x^{2}}}}}}\\,dx\\quad (m\\neq -1)", "01423856a0862b51452642523a8e6997": "D(u,v)={\\frac {\\sum _{i=0}^{72}D_{i}}{\\sum _{i=0}^{72}w_{i}}}", "01424af9614de9aa07f3932f4576e5e6": "{dP_{x} \\over dt}=c(P_{x+1}-2P_{x}+P_{x-1})\\,", "01424d0db12f64e329b4234095f9ac26": "y_{j}=\\beta _{0}+\\beta _{1}x_{1j}+\\beta _{2}x_{2j}+\\cdots +\\eta _{j}\\,", "014264151437df888613e0559ae86350": "{\\begin{array}{ll}d\\in D&{\\mbox{the decision being made, chosen from space }}D\\\\x\\in X&{\\mbox{an uncertain state, with true value in space }}X\\\\z\\in Z&{\\mbox{an observed sample composed of }}n{\\mbox{ observations }}\\langle z_{1},z_{2},..,z_{n}\\rangle \\\\U(d,x)&{\\mbox{the utility of selecting decision }}d{\\mbox{ from }}x\\\\p(x)&{\\mbox{your prior subjective probability distribution (density function) on }}x\\\\p(z|x)&{\\mbox{the conditional prior probability of observing the sample }}z\\end{array}}", "0142f80ddbc8ec29aba02c8582b99ee3": "{\\text{STr}}", "014314fe17876564f0e241b4c4a11b77": "H={\\frac {h}{l_{c}}}", "014345ae6ac2bde9bfec0158c4e850e7": "{\\begin{aligned}R_{pq}^{J}&\\equiv R_{pq}(\\theta _{2})\\,R_{pq}(\\theta _{1}),{\\text{ with}}\\\\[8pt]\\theta _{1}&\\equiv {\\frac {\\pi -2\\phi _{1}}{4}}{\\text{ and }}\\theta _{2}\\equiv {\\frac {\\phi _{2}}{2}},\\end{aligned}}", "01434663733c3165cd88685687e87f8c": "{\\rm {full\\;red\\;circle}}=\\left\\{X\\mid \\;\\left((XC),(XD)\\right)\\;=\\theta +k\\pi \\right\\}", "0143754b30f0af816b24a5427d1e7956": "\\mathbb {S} ^{\\lambda }E", "01438fd76ecd96ff355b59de43f5e3ec": "\\left\\{D_{i},D_{j}\\right\\}=-\\sum _{s=1}^{3}\\epsilon _{ijs}L_{s}~.", "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\\geq 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} ,b0", "015b39d72d4a0fd8e22283fd59a03971": "{\\dot {S}}_{i}=0", "015b6ed3d17e4320673a7fbb4f08a890": "G(0)=a", "015bbc9bfb8a96e6ab5b0997090b8855": "p\\cdot 2^{p}\\leq W(2,p+1)", "015bcbdeb6916e3d184715b7d8f76ddb": "1+{\\cfrac {1}{2+{\\cfrac {1}{2+{\\cfrac {1}{2+{\\cfrac {1}{2+\\ddots }}}}}}}}", "015bd78022d8197e1f89f26566ed30cb": "{\\begin{aligned}f&={\\frac {a-b}{a}},\\qquad e^{2}=2f-f^{2},\\qquad e'^{2}={\\frac {e^{2}}{1-e^{2}}}\\\\b&=a(1-f)=a(1-e^{2})^{1/2},\\qquad n={\\frac {a-b}{a+b}}.\\end{aligned}}", "015c8a7bf6cdc8659ba3da587412882f": "L=\\{p_{1}^{*},...,p_{n}^{*}\\}", "015e2542aa509f972b2d9cac61e0da8b": "{\\begin{cases}f(x)\\geq 0\\\\g(x)>0\\\\f(x)<\\left[g(x)\\right]^{2}\\quad \\end{cases}}", "015e37f07fb14db21431e6f806b6f914": "\\rho _{i}'\\simeq \\rho _{j}'\\quad {\\mbox{and}}\\quad \\sigma _{i}'\\simeq \\sigma _{j}'\\quad {\\mbox{for all}}\\quad i,j\\;.", "015e537a57c7a99528d0b4be0dbff505": "f(x;\\mu ,\\sigma _{1},\\sigma _{2})=A\\exp(-{\\frac {(x-\\mu )^{2}}{2\\sigma _{2}^{2}}})\\quad {\\text{otherwise}}", "015eb7acbdb731c32e8526d3b999986c": "G=\\int _{0}^{\\infty }I(\\lambda )\\,{\\overline {g}}(\\lambda )\\,d\\lambda ", "015ec6b712f877cf4b399641b0afcfa4": "x(0)=1,\\,", "015ed9def054d00e7c577035a29b1c6f": "c^{-2}", "015eeb1f8a5a6a7adc4a4c42bd49a653": "~{\\mathbf {J}}(x,y,z)~=~\\sum _{j}~J_{j}~{\\mathbf {J}}_{j}(x,y,z)~~~~~~~~~~~~~~~~~(3.4)", "015f185f6d3620e612d1d6106a17db4b": "0\\leq n0{\\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 _{i1", "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_{0}^{2}\\nabla ^{2}\\rho =\\nabla \\cdot \\left[\\nabla \\cdot (\\rho \\mathbf {v} \\otimes \\mathbf {v} )-\\nabla \\cdot \\sigma +\\nabla p-c_{0}^{2}\\nabla \\rho \\right],", "016f73e62aa171292b5559ef227c2799": "B(x;r)=\\{y\\in M:d(x,y)0", "017e2034450b47b6670afdb40731e1b0": "\\;L(H_{A})", "017e328293c91381e0341ae5c4e34e90": "a_{n}\\equiv {\\frac {\\omega _{n}q_{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|\\leq {C\\rho \\over \\sigma ^{3}\\,{\\sqrt {n}}}.\\ \\ \\ \\ (1)", "01808b41247d4647bbf4ef4b1ffc3e32": "\\lor ~(\\neg x_{1}\\land ...\\land \\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\\leq j\\leq n,1\\leq i\\leq m", "018149d8f8a32fa92ace0794088c0b4d": "{\\begin{aligned}&{}\\qquad D(X_{1},\\ldots ,X_{n})\\\\[10pt]&\\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})\\\\&=\\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})\\\\&=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]\\\\&=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)\\;.\\end{aligned}}", "01817fcfbf955c5fd03383d2d5346629": "J_{0}^{k}\\rho :J_{0}^{k}({\\mathbb {R} }^{n},{\\mathbb {R} }^{n})\\rightarrow J_{0}^{k}({\\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}}\\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 ({\\tfrac {1}{3}})]^{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_{1}L_{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}{\\frac {1}{\\det(A)}},&\\det(A)>0;\\\\+\\infty ,&\\det(A)\\leq 0;\\end{cases}}", "0186cef4e734d3086ddd8e1d98c96217": "L={\\frac {1}{2}}\\langle FF\\rangle _{S}-\\langle A{\\bar {j}}\\rangle _{S}\\,,", "0186d0570a2ecc636534c55241780f3e": "x=0.", "01870684c1cf92509c6d2448a3ce7c04": "{\\frac {\\mu _{m}}{\\mu _{f}}},", "018709b6fc0fe0bf0f220fbffbbf1772": "O(\\theta ^{n})", "01871299a7fe7070ddc52d1944caab4e": "\\int _{B}\\!p_{X,A,B}(x,a,b)=\\int _{B}\\!p_{X}(x)p_{A,B}(a,b)", "01871a14188995b8fe6571db67cc270c": "{\\mathcal {F}}=\\oint \\mathbf {H} \\cdot \\operatorname {d} \\mathbf {l} ", "0187489c33857c111d84ec1dc319fd28": "z^{1-c}\\;{}_{2}F_{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}={\\frac {1}{2}}{\\big (}d(x,y)+d(x,z)-d(y,z){\\big )}.", "01883db4def9f5811143f99b22b6e85b": "[A_{\\mathbf {x}},A_{\\mathbf {y}}]=A_{\\mathbf {z}},\\quad [A_{\\mathbf {z}},A_{\\mathbf {x}}]=A_{\\mathbf {y}},\\quad [A_{\\mathbf {y}},A_{\\mathbf {z}}]=A_{\\mathbf {x}}.", "01883f5cab3c6b493515462df4feb4e3": "{\\hat {H}}={\\hat {T}}^{\\mathrm {translational} }+{\\hat {T}}^{\\mathrm {rotational} }+{\\hat {V}}", "0188847e219e48a741f4df4a3976163b": "{\\begin{array}{|rcccl|}\\hline \\color {MidnightBlue}{\\mbox{eval left}}&&(11+9)\\times (2+4)&&\\color {MidnightBlue}{\\mbox{eval right}}\\\\&\\color {MidnightBlue}{\\swarrow }&&\\color {MidnightBlue}{\\searrow }&\\\\20\\times (2+4)&&&&(11+9)\\times 6\\\\&\\color {MidnightBlue}{\\searrow }&&\\color {MidnightBlue}{\\swarrow }&\\\\\\color {MidnightBlue}{\\mbox{eval right}}&&20\\times 6&&\\color {MidnightBlue}{\\mbox{eval left}}\\\\&&\\color {MidnightBlue}{\\downarrow }&&\\\\&&120&&\\\\\\hline \\end{array}}", "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}\\Phi (x)-\\left[\\varphi (0)-\\varphi (x)\\right]/x&x\\neq 0\\\\1/2&x=0\\\\\\end{cases}}", "018aadf04c43b931fc8e2a9d169fbb1b": "C(x_{j},x_{k})=\\left.{\\frac {\\partial }{\\partial J_{j}}}{\\frac {\\partial }{\\partial J_{k}}}\\log Z(\\beta ,J)\\right|_{J=0}", "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": "{\\overline {\\mathrm {Var} (z)}}=1-{\\overline {R}}\\,", "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\\neq 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": "\\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\\leq i\\leq 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_{\\mu }^{(a)}dx^{\\mu }\\}_{a=1\\dots 4}", "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):t0", "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_{k}z^{k}.", "01a0fdfb3cd1f76eb8002eba9c586f57": "g_{bf}", "01a11a1c84ff2134d636dbb5a8c4d861": "f_{b}\\left({\\frac {F_{D}}{{\\frac {1}{2}}\\,\\rho \\,A\\,u^{2}}},\\,{\\frac {u\\,{\\sqrt {A}}}{\\nu }}\\right)\\,=\\,0.", "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} )\\neq 1", "01a35d410aebfade90b90ef175faa85d": "{\\begin{bmatrix}a_{11}(x)&a_{12}(x)&\\cdots &a_{1n}(x)\\\\a_{21}(x)&a_{22}(x)&\\cdots &a_{2n}(x)\\\\\\vdots &\\vdots &\\ddots &\\vdots \\\\a_{n1}(x)&a_{n2}(x)&\\cdots &a_{nn}(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": "\\Delta ^{2}w:={\\frac {\\partial ^{2}}{\\partial x_{\\alpha }\\partial x_{\\alpha }}}\\left[{\\frac {\\partial ^{2}w}{\\partial x_{\\beta }\\partial x_{\\beta }}}\\right]={\\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}}}\\,.", "01a4e7db800567fb110001e8d958a2f3": "{\\begin{aligned}\\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)\\\\&={\\tfrac {1}{6}}\\cdot 17.50={\\tfrac {35}{12}}\\approx 2.92.\\end{aligned}}", "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}^{m}u\\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' \\atop 3}\\right\\}", "01a7ec08e4a4d2e1052b46850941e4e9": "0,1,\\ldots ,n-1", "01a873e523d00e4ab7d05e3b47213d08": "F_{eq}", "01a87ab17ca903f95241c866f531ba64": "(-b-h(a))^{2}+h(a)(-b-h(a))", "01a8bf77078848236bfd2d223e761215": "\\epsilon =a\\tan \\theta ", "01a8c485b579cc073a33b76977543ee8": "\\operatorname {Li} _{s}(-z)+\\operatorname {Li} _{s}(z)=2^{1-s}\\,\\operatorname {Li} _{s}(z^{2})\\,.", "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)=aga^{-1}", "01a9f31ab16bb54eac94bffcc7fcf7e8": "s_{1},s_{2}\\in S,r\\in R", "01a9fe25a65a2201a9bb94d9bb9d1c98": "f\\;a\\;b\\;c\\;:\\;1\\to D", "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)^{n}q^{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 _{i=1}^{N(\\mathbf {h} )}\\left(Z(x_{i})-Z(x_{i}+\\mathbf {h} )\\right)^{2}", "01adfa4dbebcb0ee3a196cbe0b5adde0": "B=Y_{2}ZZ_{1}Z_{1}={\\sqrt {3}}", "01ae55de5e9698c5db36423be6c05224": "\\omega _{e}=\\mathrm {id} :T_{e}G\\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\\geq 0", "01b20e5cf9941b9e9034764b678beacb": "U\\,\\!", "01b265fee6a0869c2e4f9adafa319138": "{\\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.", "01b28d0dae8226bfb1d154b5662decc5": "{\\frac {P\\lor 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": "\\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,", "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": "A={\\begin{bmatrix}~4&-1&~0&-1&~0&~0&~0&~0&~0\\\\-1&~4&-1&~0&-1&~0&~0&~0&~0\\\\~0&-1&~4&~0&~0&-1&~0&~0&~0\\\\-1&~0&~0&~4&-1&~0&-1&~0&~0\\\\~0&-1&~0&-1&~4&-1&~0&-1&~0\\\\~0&~0&-1&~0&-1&~4&~0&~0&-1\\\\~0&~0&~0&-1&~0&~0&~4&-1&~0\\\\~0&~0&~0&~0&-1&~0&-1&~4&-1\\\\~0&~0&~0&~0&~0&-1&~0&-1&~4\\end{bmatrix}}", "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 {\\mathbf {X}}({\\mathbf {u}}_{0})+t{\\mathbf {A}}({\\mathbf {u}}_{0}).", "01b95a3d7abea8628080371744d90d22": "\\left[\\left.{\\begin{array}{cccc}1&0&0&0\\\\0&0&0&0\\\\0&1&0&0\\\\0&0&0&1\\end{array}}\\right\\vert {\\begin{array}{cccc}0&0&0&0\\\\1&0&0&0\\\\0&0&0&0\\\\0&0&1&0\\end{array}}\\right].", "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\\geq 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}\\leq x})", "01bbc47541ee02be2bda7635619e5043": "H(\\mathbf {Y} )=-{\\frac {1}{N}}\\sum _{t=1}^{N}\\ln p_{\\mathbf {s} }(\\mathbf {y} ^{t})+\\ln |\\mathbf {W} |-H(\\mathbf {x} )", "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": "Z=\\int e^{-{\\frac {F(r)}{kT}}}dr", "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": "L={\\frac {qB}{2c}}(x{\\dot {y}}-y{\\dot {x}})-V(x,y)~,", "01c0c8b1e311e98d09a5188569dfad2f": "{\\frac {(f'(\\theta ),\\ f(\\theta ))}{|f'(\\theta ),\\ f(\\theta )|}}=(\\cos \\psi ,\\ \\sin \\psi )", "01c0e78b5baed0feef5041fe7545dfef": "{\\begin{aligned}&{}\\quad \\langle \\phi (x_{1})\\phi (x_{2})\\phi (x_{3})\\rangle \\\\&=\\langle \\phi (x_{1})\\phi (x_{2})\\phi (x_{3})\\rangle _{\\text{con}}+\\langle \\phi (x_{1})\\phi (x_{2})\\rangle _{\\text{con}}\\langle \\phi (x_{3})\\rangle _{\\text{con}}+\\langle \\phi (x_{1})\\phi (x_{3})\\rangle _{\\text{con}}\\langle \\phi (x_{2})\\rangle _{\\text{con}}\\\\&+\\langle \\phi (x_{1})\\rangle _{\\text{con}}\\langle \\phi (x_{2})\\phi (x_{3})\\rangle _{\\text{con}}+\\langle \\phi (x_{1})\\rangle _{con}\\langle \\phi (x_{2})\\rangle _{\\text{con}}\\langle \\phi (x_{3})\\rangle _{\\text{con}}\\end{aligned}}", "01c11575ea95126fcd60f809f8da5bcf": "E_{\\mathrm {v} }=10^{(-14.18-M_{\\mathrm {v} })/2.5}", "01c154aeb7c0087556908ee407b5d53d": "\\displaystyle {\\frac {d^{n}s}{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_{1}C_{A}\\theta _{E}}{k_{-1}+kC_{S}\\theta _{B}}}", "01c2da53089a42414a8f92ccc46ee9a8": "t_{2}=\\sum x_{i}^{2}", "01c3104f950e4db29466791bda1d743f": "d_{(ij)k}=\\alpha _{i}d_{ik}+\\alpha _{j}d_{jk}+\\beta d_{ij}+\\gamma |d_{ik}-d_{jk}|,", "01c3ae91742e02bc43e53d948351f27b": "{\\left({\\frac {f}{g}}\\right)}'={\\frac {f'g-fg'}{g^{2}}}.", "01c3e9c1e55d5895c44543209649d809": "{\\begin{array}{cccccccc}I&X&I&Y&Z&I&I&\\cdots \\end{array}},", "01c3f37234550aae5c346d656a2cbe48": "\\omega ={\\frac {1}{\\sqrt {LC}}}", "01c3f71b1579582146a7326c7765985e": "q{\\begin{Bmatrix}p\\\\q\\end{Bmatrix}}", "01c3fde8626debd09db6b81b0ad7d2d3": "\\forall A\\exists b\\exists c\\exists d\\;bA\\land cA\\land dA\\land \\lnot b=c\\land \\lnot b=d\\land \\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": "\\Delta g\\ =\\ \\int \\limits _{0}^{2\\pi }{\\bar {V}}{\\bar {h}}{\\frac {r^{2}}{\\sqrt {\\mu p}}}d\\theta ", "01ce38e7e92f345f1d8657e6c1167623": "{\\begin{aligned}I=&B_{1_{1}}+B_{2_{1}}\\\\A=&3/4B_{1_{1}}+B_{1_{2}}+B_{2_{1}}+B_{2_{2}}\\\\A^{2}=&(3/4)^{2}B_{1_{1}}+(3/2)B_{1_{2}}+B_{2_{1}}+2B_{2_{2}}\\\\A^{3}=&(3/4)^{3}B_{1_{1}}+(27/16)B_{1_{2}}+B_{2_{1}}+3B_{2_{2}}\\end{aligned}}", "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": "f(s)=\\liminf _{n\\to \\infty }f_{n}(s),\\qquad s\\in S.", "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,iw)", "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": "<\\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": "{\\begin{matrix}^{^{b}{b}}{\\bar {a}}=&\\underbrace {a_{}^{a^{{}^{.\\,^{.\\,^{.\\,^{a}}}}}}} &\\\\&{{^{b}}{\\bar {a}}}{\\mbox{ copies of }}a\\end{matrix}}", "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 \\over 10272278170}=0.0001189375\\dots ,", "01d84c0b06afc9c27c5264692ec2ee41": "a,b,c\\in N", "01d85502cacd9acc332bacd50f367f00": "\\left(-\\nabla _{\\mathbf {u} }^{2}+{\\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({\\mathbf {x}},t;{\\mathbf {\\hat {n}}},\\nu )", "01d9fc38090a5436fadf1b8b06471409": "{\\frac {\\mathrm {m} /\\mathrm {s} ^{2}}{\\mathrm {Pa} }}", "01da8f763bcd3533e23d82c937942e20": "z={\\frac {1}{2}}\\left(A+B-\\lambda -\\mu -\\nu \\right)", "01daa0079732e4a1f48600a4a3251a53": "y_{n}(x)=(2n\\!-\\!1)x\\,y_{n-1}(x)+y_{n-2}(x)\\,", "01dab2020cf38b41842d6c211501b787": "se^{i\\Delta k\\Lambda }=e^{i\\Delta k\\Lambda }-e^{i2\\Delta k\\Lambda n}+e^{i3\\Delta k\\Lambda }+...+(-1)^{N}e^{i\\Delta k\\Lambda (N-1)}-(-1)^{N}e^{i\\Delta k\\Lambda N}.", "01dae64584b988a11f4f653b3359640e": "ds^{2}=d\\chi ^{2}+\\sin ^{2}(\\chi /\\alpha )ds_{dS,\\alpha ,n-1}^{2},", "01daebc5b411677123fc9f4734fa8fed": "C_{1}^{+}(\\beta )={\\frac {\\alpha }{2}}\\log \\left(1+(c_{31}^{2}+c_{21}^{2})P_{1}^{(1)}\\right)+{\\frac {1-\\alpha }{2}}\\log \\left(1+(1-\\beta )c_{31}^{2}P_{1}^{(2)}\\right)", "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 {1}{4}}(\\gamma _{i}\\gamma _{j}-\\gamma _{j}\\gamma _{i})_{ab}Q_{b},", "01dc43fb7bd88609eb84d081d609513f": "{}_{1}F_{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 ^{2}c^{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[mn(n^{2}-(l^{2}+m^{2})/2)V_{dd\\sigma }+mn(l^{2}+m^{2}-n^{2})V_{dd\\pi }-mn(l^{2}+m^{2})/2V_{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 },andX_{\\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 {aL}{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": "{}_{2}^{3}\\mathrm {He} +{}_{1}^{2}\\mathrm {H} \\to {}_{2}^{4}\\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}\\cos \\beta \\cos \\lambda \\\\\\cos \\beta \\sin \\lambda \\\\\\sin \\beta \\end{bmatrix}}={\\begin{bmatrix}1&0&0\\\\0&\\cos \\epsilon &\\sin \\epsilon \\\\0&-\\sin \\epsilon &\\cos \\epsilon \\end{bmatrix}}{\\begin{bmatrix}\\cos \\delta \\cos \\alpha \\\\\\cos \\delta \\sin \\alpha \\\\\\sin \\delta \\end{bmatrix}}", "01e3cacd2b9c2b395a3126c85a799f03": "{\\boldsymbol {\\tau }}=\\varphi _{*}[{\\boldsymbol {S}}]={\\boldsymbol {F}}\\cdot {\\boldsymbol {S}}\\cdot {\\boldsymbol {F}}^{T}~.", "01e44b0dc54b8b019e635f7283b75df2": "\\mu ^{\\otimes 0}(A_{0}(s,t)):=1.", "01e46395993aa5169a4f46c994e057c2": "\\textstyle (x\\pm 1,y,z\\mp 1)", "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+it)^{1-s}}{(e^{\\pi t}+e^{-\\pi t})^{2}}}\\,dt.", "01e8153ecd79d5daf0df9fc8579edd9e": "{\\mathfrak {H}}_{b}", "01e86ced95c51596f778d74df8c8bf96": "V=1096.7{\\sqrt {H/d}}", "01e9b4e5ba85de9ac8931c518c75329d": "\\scriptstyle s_{\\infty }(x)", "01ea358477bd18b369f5831702e6e4a7": "F_{hkl}={\\begin{cases}4f,&h,k,l\\ \\ {\\mbox{all even or all odd}}\\\\0,&h,k,l\\ \\ {\\mbox{mixed parity}}\\end{cases}}", "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 {2c^{2}}{\\lambda ^{5}}}~{\\frac {h}{e^{\\frac {hc}{\\lambda kT}}-1}}\\approx {\\frac {2ckT}{\\lambda ^{4}}}", "01eccb8e17d972949e03580c41d08994": "(X,Z)", "01ecefd8e8946da30610fe9a89d437e0": "c_{T-2}(k)\\,=\\,{\\frac {Ak^{a}}{1+ab+a^{2}b^{2}}}", "01ecf76e7b919e8f093d393b99d25b96": "|x_{1}|=|x_{2}|=\\cdots |x_{n}|=1", "01ecfad7082922f85b35330787b6a893": "I=I_{cont}\\cdot {\\frac {1+K_{n}}{1+1.71K_{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)\\geq \\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 _{t}^{j}G(0)=0,\\quad 0\\leq j\\leq m-2,\\;\\partial _{t}^{m-1}G(0)=1/a_{m}.", "01f1ffe110c901fcfaefbb12c9e9960f": "\\scriptstyle \\leq 1.9\\times 10^{-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}}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 ^{bc}+\\beta ^{c}\\eta ^{ba}", "01f70a960eb91ed4f3aadeab35b6deb4": "{\\dot {z}}=-2z(\\alpha +xy),\\,", "01f788399c97985044f2437a18aab69e": "|\\Phi ^{-}\\rangle ", "01f78be6f7cad02658508fe4616098a9": "550", "01f7c18c56f6d93726f78c234d1868da": "((P\\lor Q)\\land \\neg P)\\to Q", "01f824346fd27a8e5ae32409c29ab9e0": "a_{n}\\neq 0", "01f864dc442db64bf93663760fa8dae7": "{\\begin{aligned}\\Vert {\\vec {a}}\\Vert ^{2}&=\\Vert {\\vec {b}}-{\\vec {c}}\\Vert ^{2}\\\\&=({\\vec {b}}-{\\vec {c}})\\cdot ({\\vec {b}}-{\\vec {c}})\\\\&=\\Vert {\\vec {b}}\\Vert ^{2}+\\Vert {\\vec {c}}\\Vert ^{2}-2{\\vec {b}}\\cdot {\\vec {c}}.\\end{aligned}}", "01f8b80e36b662229cbd834a93134c87": "\\textstyle u\\in W_{p}^{k}(\\Omega )", "01f8cede02e588da726936d313dcaa9b": "P({\\vec {x}}|{\\vec {y}})={\\frac {1}{(2\\pi )^{mn/2}|{\\boldsymbol {S_{x}}}|}}\\exp \\left[-{\\frac {1}{2}}({\\vec {x}}-{\\widehat {x}})^{T}{\\boldsymbol {S_{x}}}^{-1}({\\vec {x}}-{\\widehat {x}})\\right]", "01f8f7e003bf6961951efb20b8a6959e": "\\gcd(a_{1},a_{2})=1", "01f93ef84b3860edd2c0508453d523ee": "\\Delta _{r}G^{\\ominus }=-RT\\ln K_{eq}", "01f94a2e8b3a86a1eac37f3a307d74ef": "\\left(\\lambda _{i}\\right)", "01f9b8831d5ce67ce115b33c7d1e9478": "Q=f(X_{1},X_{2},X_{3},\\dotsc ,X_{n})", "01fa15b00eab23e5d544b290e9299048": "550P_{e}={\\frac {\\eta _{c}HhJ}{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 {g}}", "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": "{\\tbinom {4}{2}}", "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_{n,k}^{[\\alpha ]}\\,{\\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 {\\partial ^{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": "{\\dot {\\mathbf {f} }}(\\mathbf {x} ,t)={\\frac {\\partial \\mathbf {f} (\\mathbf {x} ,t)}{\\partial t}}+[{\\boldsymbol {\\nabla }}\\mathbf {f} (\\mathbf {x} ,t)]\\cdot \\mathbf {v} (\\mathbf {x} ,t)~.", "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^{2}y''+xy'+\\left(x^{2}-\\nu ^{2}\\right)y=0", "0206d8fd533aeb1efbae23598b7752c5": "u(x)-u_{\\epsilon }(x)=O(\\epsilon ^{2}),\\quad 0x]", "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": "{\\begin{aligned}(Tf)(x)={e^{D}-1 \\over D}f(x)&{}=\\sum _{n=0}^{\\infty }{D^{n} \\over (n+1)!}f(x)\\\\&{}=f(x)+{f'(x) \\over 2}+{f''(x) \\over 6}+{f'''(x) \\over 24}+\\cdots ~.\\end{aligned}}", "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": "{\\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}~;~~{\\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}", "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": "{\\begin{aligned}H_{0}&=1+{\\frac {n^{2}}{4}}+{\\frac {n^{4}}{64}}+\\cdots \\qquad \\qquad \\qquad &H_{6}&={\\frac {35}{48}}n^{3}+\\cdots \\\\[8pt]H_{2}&={\\frac {3}{2}}\\left(n-{\\frac {n^{3}}{8}}+\\cdots \\right)&H_{8}&={\\frac {315}{512}}n^{4}+\\cdots \\\\[8pt]H_{4}&={\\frac {15}{16}}\\left(n^{2}-{\\frac {n^{4}}{4}}+\\cdots \\right)\\end{aligned}}", "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 _{rt}={\\frac {1}{\\mu _{0}}}B_{r}B_{t}-{\\frac {1}{2\\mu _{0}}}B^{2}\\delta _{rt}\\,.", "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": "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}}}\\,.", "020f5b0b54d45f3933659d20b4b8901d": "n_{1}=0\\,\\!", "020fa72193e9f80f8e86a914a89ede7a": "dQ_{c}=T_{c}dS_{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 im/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 {LU}{\\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}-2u_{i}^{n}+u_{i-1}^{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 {8k_{b}T}{\\pi m}}}", "021ad144f1e0aae4df5d8e05c210feed": "\\mathbb {P} _{\\mathbf {k} }^{n}", "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)|\\leq C|x-y|^{\\alpha }", "021c165cdf6f1229bf98835b81614e1b": "{\\frac {a}{p}}=0.{\\overline {a_{1}a_{2}a_{3}\\dots a_{n}a_{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": "{\\mathbf {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 \\leq -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_{S}R}{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-nT)={\\mathcal {F}}^{-1}\\left\\{X_{1/T}(f)\\right\\}\\ {\\stackrel {\\mathrm {def} }{=}}\\int _{-\\infty }^{\\infty }X_{1/T}(f)\\cdot e^{i2\\pi ft}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^{-i2\\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": "{dL \\over dt}=i[H,L]=0\\,,", "022217a91d9b643de752294096d7f6aa": "p_{4},p_{1}", "0222491b800049563d888f2664f4a8a6": "\\sigma (t)={\\frac {1}{b}}*log{\\frac {10^{\\alpha }(t-t_{n})+1}{10^{\\alpha }(t-t_{n})-1}}", "02224ce925b278fca46db66a1da98c3e": "\\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 ^{2}A.\\,", "0223f2bdbda18a7154bf1f35126ea943": "{\\bar {\\Gamma }}_{\\alpha \\gamma }^{\\beta }\\,=\\,{\\frac {\\partial {\\bar {x}}^{\\beta }}{\\partial x^{\\epsilon }}}\\,{\\frac {\\partial x^{\\delta }}{\\partial {\\bar {x}}^{\\alpha }}}\\,{\\frac {\\partial x^{\\zeta }}{\\partial {\\bar {x}}^{\\gamma }}}\\,\\Gamma _{\\delta \\zeta }^{\\epsilon }\\,+{\\frac {\\partial {\\bar {x}}^{\\beta }}{\\partial x^{\\eta }}}\\,{\\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]", "02246878093cc4bb4582527127390aba": "\\operatorname {dist} ", "02246d3ddf4a376189129511f7aed444": "x=\\left(\\lambda -\\lambda _{0}\\right)\\cos \\varphi ", "0224bf3a2802504318677efcf183c5d8": "(192,20,64)", "02251a6d64eac16e4975615fa1729053": "R_{\\mathrm {g} }^{2}\\ {\\stackrel {\\mathrm {def} }{=}}\\ {\\frac {1}{2N^{2}}}\\sum _{i,j}\\left(\\mathbf {r} _{i}-\\mathbf {r} _{j}\\right)^{2}.", "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))-2e_{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": "{\\begin{aligned}{\\boldsymbol {\\nabla }}\\cdot {\\boldsymbol {S}}&=\\left[{\\cfrac {\\partial S_{ij}}{\\partial q^{k}}}-\\Gamma _{ki}^{l}~S_{lj}-\\Gamma _{kj}^{l}~S_{il}\\right]~g^{ik}~\\mathbf {b} ^{j}\\\\[8pt]&=\\left[{\\cfrac {\\partial S^{ij}}{\\partial q^{i}}}+\\Gamma _{il}^{i}~S^{lj}+\\Gamma _{il}^{j}~S^{il}\\right]~\\mathbf {b} _{j}\\\\[8pt]&=\\left[{\\cfrac {\\partial S_{~j}^{i}}{\\partial q^{i}}}+\\Gamma _{il}^{i}~S_{~j}^{l}-\\Gamma _{ij}^{l}~S_{~l}^{i}\\right]~\\mathbf {b} ^{j}\\\\[8pt]&=\\left[{\\cfrac {\\partial S_{i}^{~j}}{\\partial q^{k}}}-\\Gamma _{ik}^{l}~S_{l}^{~j}+\\Gamma _{kl}^{j}~S_{i}^{~l}\\right]~g^{ik}~\\mathbf {b} _{j}\\end{aligned}}", "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": "\\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 |\\leq \\|x\\|\\,\\|y\\|", "022bdeb8bfa3ba5a77935025118e9e2c": "\\scriptstyle {1/2}", "022c1333b68909412b5b0041396caaee": "2^{l+1}-1", "022c20fd59376ce997de8331ffaedbd3": "A_{t}=\\{x\\in X\\mid f(x)\\geq t\\}", "022c32018a8e85ae989512fd7ecec25e": "V_{n}(r)", "022c3b80bef2c0f17f57ed150c1f4652": "1+{\\frac {1}{4}}+{\\frac {1}{9}}+{\\frac {1}{16}}+{\\frac {1}{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} _{hk\\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'\\leq H|\\alpha (H)\\in F\\}", "022dcce091d8dc74031d8dbf34662dab": "n=ax^{2}+2bxy+cy^{2}", "022e3414e6427b3cc27c5a5911fd9588": "DG(x,s)=0", "022ed48ce122fb6d02b20ffd57a86105": "\\sin(2\\theta )=2\\sin \\theta \\cos \\theta \\,", "022f7d80be231d713945ca4d7beed1cf": "\\left({\\frac {\\pi }{6}}\\right)^{\\frac {1}{3}}\\approx 0.806", "022f9ef548f37cc6101d5e59875cc945": "\\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]\\;.", "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": "X^{\\{q\\}}=\\lambda ^{-1}([m-q,m])", "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\\leq i\\leq 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_{3}O^{+}]}{[HA][H_{2}O]}}\\times {\\frac {\\gamma _{A^{-}}\\ \\gamma _{H_{3}O^{+}}}{\\gamma _{HA}\\ \\gamma _{H_{2}O}}}=\\mathrm {\\frac {[A^{-}][H_{3}O^{+}]}{[HA][H_{2}O]}} \\times \\Gamma } ", "02322c86ce10049b1cac9d06a539264f": "{\\text{subject to }}{\\dot {x_{t}}}=f(x_{t},u_{t})", "02323b856adbbedeca994dac706eece8": "{\\tfrac {863}{60480}}", "0232502a9b8aa410be0731dfefa96d89": "\\left(\\mu ,{\\frac {\\alpha -{\\frac {1}{2}}}{\\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": "{\\begin{aligned}&{\\partial \\rho \\over \\partial t}+\\nabla \\cdot (\\rho {\\mathbf {u}})=0\\\\[1.2ex]&{\\partial (\\rho {\\mathbf {u}}) \\over \\partial t}+\\nabla \\cdot ({\\mathbf {u}}\\otimes (\\rho {\\mathbf {u}}))+\\nabla p={\\mathbf {0}}\\\\[1.2ex]&{\\partial E \\over \\partial t}+\\nabla \\cdot ({\\mathbf {u}}(E+p))=0,\\end{aligned}}", "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}=mg\\,", "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": "\\sup _{p}h_{p}(x)\\geq 1", "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": "|\\mu |(\\partial B)=0\\,.", "0238bce249da3358e4f5ed91094a93e7": "ODF({\\boldsymbol {g}})={\\frac {1}{V}}{\\frac {dV({\\boldsymbol {g}})}{dg}}.", "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} _{>0}^{\\times }", "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]-2E\\left[{\\frac {\\mu |r(n)|^{2}}{\\mathbf {x} ^{H}(n)\\mathbf {x} (n)}}\\right]", "02399bb49108a9455c6827292a30f6ea": "{\\begin{aligned}r&{}={\\sqrt {6^{2}+5^{2}}}=7.8102\\\\c&{}=6/r=0.7682\\\\s&{}=-5/r=-0.6402\\end{aligned}}", "023a62bc62f64d7623a58945e76525ed": "O_{k}=x_{1}x_{5}x_{9}\\cdots x_{2N-3}+x_{3}x_{7}x_{11}\\cdots x_{2N-1}", "023a6f9688af4f89a59c4ba647f93d89": "\\nabla ^{2}\\psi =0.\\,", "023a91025f1455379b2c7b284e046e79": "\\mathrm {Hom} ^{\\bullet }(\\Gamma _{c}(X;I_{X}^{\\bullet }),k)=\\cdots \\to \\Gamma _{c}(X;I_{X}^{2})^{\\vee }\\to \\Gamma _{c}(X;I_{X}^{1})^{\\vee }\\to \\Gamma _{c}(X;I_{X}^{0})^{\\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": "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}}\\,.", "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 _{k=0}^{\\infty }{\\frac {(-3)^{-k}}{2k+1}}={\\sqrt {12}}\\sum _{k=0}^{\\infty }{\\frac {(-{\\frac {1}{3}})^{k}}{2k+1}}={\\sqrt {12}}\\left({1 \\over 1\\cdot 3^{0}}-{1 \\over 3\\cdot 3^{1}}+{1 \\over 5\\cdot 3^{2}}-{1 \\over 7\\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": "\\left(\\bigcup _{i\\in I}A_{i}\\right)^{o}=\\bigcap _{i\\in I}A_{i}^{o}.", "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 \\mathbb {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)}}={\\frac {\\mathbb {P} (n\\leq n^{*}|s+b)}{\\mathbb {P} (n\\leq n^{*}|b)}}.", "025464d3b6a57dde173c670b334b4c7a": "\\mathbf {\\nabla } \\cdot \\mathbf {E} (\\mathbf {x} )=-{\\frac {iZ_{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 _{j}^{i}.", "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=pdV\\;", "0258321027b3e0da182a33942238407b": "Q_{\\alpha \\beta }=\\int d^{3}\\mathbf {x'} (3x'_{\\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)={\\begin{cases}n_{1}{\\sqrt {1-2\\Delta \\left({r \\over \\alpha }\\right)^{g}}}&r\\leq \\alpha \\\\n_{1}{\\sqrt {1-2\\Delta }}&r\\geq \\alpha \\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{aligned}&\\mathbf {(D-\\omega L)^{-1}[(1-\\omega )D+\\omega U]} ={\\frac {1}{12}}{\\begin{pmatrix}-1.2&4.4&6.6\\\\-0.33&0.01&8.415\\\\-0.8646&2.9062&5.0073\\end{pmatrix}},\\end{aligned}}", "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 {\\mathbf {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": "m\\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 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": "{\\frac {1}{137}}", "0272d268d0534de5245746bcaa96c0e1": "\\sigma ^{*}=G(F^{*})", "02737eddf8250b8f1aaa104754d37249": "{\\begin{bmatrix}1\\end{bmatrix}}\\quad {\\begin{bmatrix}1&2\\\\2&1\\end{bmatrix}}\\quad {\\begin{bmatrix}1&2&3\\\\2&3&1\\\\3&1&2\\end{bmatrix}}", "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 _{n}A_{n}\\exp \\left(-\\Delta _{n}t\\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\\geq 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": "{\\frac {s}{D}}={\\frac {m_{b}}{k_{B}T}}", "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}}", "027b3e2314e62461489d1c69ad4dec6c": "{\\begin{aligned}&\\deg P_{n}=n~,\\quad n=0,1,2,\\ldots \\\\&\\int P_{m}(x)\\,P_{n}(x)\\,W(x)\\,dx=0~,\\quad m\\neq n~.\\end{aligned}}", "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)+{1 \\over r^{2}\\sin \\theta }{\\partial \\over \\partial \\theta }\\left(\\sin \\theta {\\partial f \\over \\partial \\theta }\\right)+{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 {rs_{x}^{2}-\\rho s_{x}s_{y}}{m_{x}^{2}}}", "027d00a2432091cf782e1dbec39e173f": "\\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],", "027d12976af94786c8f656a872dbc10b": "H_{out}\\ =\\ f(H_{in},\\ m)", "027d85e20311d606467f08fa2b3fbad8": "N=M+1\\,", "027e3a0f8b7e284ab68c542a1ae3489e": "V_{GS}=V_{th}", "027e72afb96af576be811f0b0465ed0c": "f:I\\rightarrow \\mathbb {R} ^{+}", "027ea6e711e5f2c509cc7a4e6a5b64a2": "\\langle z^{m}\\rangle =\\oint p_{w}(z)z^{m}\\,dz.", "027eb3c1e422b4c252a3eebfef6b7432": "{\\begin{aligned}q_{\\mu }^{*}(\\mu )&\\sim {\\mathcal {N}}(\\mu \\mid \\mu _{N},\\lambda _{N}^{-1})\\\\\\mu _{N}&={\\frac {\\lambda _{0}\\mu _{0}+N{\\bar {x}}}{\\lambda _{0}+N}}\\\\\\lambda _{N}&=(\\lambda _{0}+N)\\operatorname {E} [\\tau ]\\\\{\\bar {x}}&={\\frac {1}{N}}\\sum _{n=1}^{N}x_{n}\\end{aligned}}", "027ec8425f9c5fa3980d0a78a6024a36": "{\\begin{aligned}x&=a\\cosh \\xi \\cos \\eta \\cos \\phi \\\\y&=a\\cosh \\xi \\cos \\eta \\sin \\phi \\\\z&=a\\sinh \\xi \\sin \\eta \\end{aligned}}", "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_{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}}}\\geq {\\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}1&{\\text{if }}Y\\leq 1/3,\\\\0&{\\text{otherwise}},\\end{cases}}", "02825789173cc14e46546d75a3d6383c": "\\ell (s)\\geq \\ell (t)", "0282b2607138bd84dda06decc05eacd6": "D_{a}", "0283316329c94c014e656bca7c85f6cf": "{1 \\over 168}\\left(n^{7}+21n^{5}+98n^{3}+48n\\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": "\\operatorname {Div} {\\boldsymbol {P}}^{T}=\\rho _{0}{\\ddot {{\\boldsymbol {x}}'}}.", "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})}{8r^{2}\\eta l}}", "0287d19d4b2d3fb6122a9bbff178bc63": "f(z_{0})", "0287e0b7e48b39993d5c5ddaab509d7f": "M(x)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 \\geq 0", "0290a332e92b98cd127f2489d929ecf4": "B^{\\prime }", "0290a88f14fb8b753331e1f64d60cd86": "\\lim _{c\\rightarrow -m}{\\frac {{}_{2}F_{1}(a,b;c;z)}{\\Gamma (c)}}={\\frac {(a)_{m+1}(b)_{m+1}}{(m+1)!}}z^{m+1}{}_{2}F_{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 {M} inN}(L+1,D,n)\\leq 2{\\mathrm {M} inN}(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}{\\frac {1}{2\\varepsilon }}|\\{s\\in [0,t]|B_{s}\\in (-\\varepsilon ,+\\varepsilon )\\}|.", "0291f94c8ebd42d6d8c456051f0aa4f0": "\\left[{\\begin{smallmatrix}2&-1&0&-1\\\\-1&2&-1&0\\\\0&-1&2&-1\\\\-1&0&-1&2\\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": "{\\begin{aligned}N_{\\alpha \\beta ,\\alpha }&=0\\\\M_{\\alpha \\beta ,\\alpha \\beta }-q&=0\\end{aligned}}", "029314defb85735cc6d46b8e19c0e1c9": "R={\\frac {-n(n-1)}{\\alpha ^{2}}}", "02941d9d7a8a4647de2c3487d03cc029": "p_{3}=p_{1}", "02942b66a4d9733e165e86739d9ee08a": "{\\begin{aligned}{\\mathbf {S} }_{i}\\cdot {\\mathbf {S} }_{j}&=&{\\sqrt {\\left({1-\\sum \\limits _{\\alpha }{\\sigma _{i\\alpha }^{2}}}\\right)\\left({1-\\sum \\limits _{\\alpha }{\\sigma _{j\\alpha }^{2}}}\\right)}}+\\sum \\limits _{\\alpha }{\\sigma _{i\\alpha }\\sigma _{j\\alpha }}\\\\&=&1-{\\tfrac {1}{2}}\\sum \\limits _{\\alpha }\\left({{\\sigma _{i\\alpha }^{2}}+{\\sigma _{j\\alpha }^{2}}}\\right)+\\sum \\limits _{\\alpha }{\\sigma _{i\\alpha }\\sigma _{j\\alpha }}+{\\mathcal {O}}(\\sigma ^{4})\\\\&=&1-{\\tfrac {1}{2}}{\\sum \\limits _{\\alpha }{(\\sigma _{i\\alpha }}-\\sigma _{j\\alpha })^{2}}+\\ldots \\end{aligned}}", "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_{k}u_{k},\\;v_{k+1}=v_{k-1}-q_{k}v_{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}}\\simeq 6.18182\\,a^{2}.", "0294fa4e2ed32efe774506d31349c49b": "\\pi S\\sin _{n}\\theta /\\lambda =n\\pi ,n=0,\\pm 1,\\pm 2,.....", "0294fa66e2860bf2e2edfc6c2b7c3c22": "(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}.", "029504ab8797ac64df7858c45b1e55b7": "R^{\\ast }", "02950734961dab76e76bf41728978d00": "1)\\quad H(\\emptyset )=0", "029551905c3f89f00943794b4a2472ed": "g(n,m)=g(n,m-1)+X_{m}g(n-1,m).", "029561ef26841a2f06634549502d4c5c": "np=\\omega ({\\sqrt {n\\log n}})", "0295a42715cf1a1df8cbb20cadfc74f8": "m=0.1n", "0295d8022555242acf9d0af9ee886c46": "x\\leq {\\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": "X\\sim \\mathrm {BNB} (n,\\alpha ,\\beta ).", "02983fb8a36cec4b1d87d21cff61e331": "H_{*}^{G}(E_{FIN}(G),K_{l^{1}}^{top})=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)=2\\pi \\log \\left({\\frac {G\\left({\\frac {5}{8}}\\right)}{G\\left({\\frac {3}{8}}\\right)}}\\right)-2\\pi \\log \\Gamma \\left({\\frac {3}{8}}\\right)+{\\frac {3\\pi }{4}}\\log \\left({\\frac {2\\pi }{\\sqrt {2+{\\sqrt {2}}}}}\\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": "{\\bar {\\delta }}{\\phi ^{A}}_{,\\sigma }={\\bar {\\delta }}{\\frac {\\partial \\phi ^{A}}{\\partial x^{\\sigma }}}={\\frac {\\partial }{\\partial x^{\\sigma }}}\\left({\\bar {\\delta }}\\phi ^{A}\\right)\\,.", "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{aligned}L_{f}\\equiv {\\underline {\\int _{a}^{b}}}f(x)\\,dx&\\quad U_{f}\\equiv {\\overline {\\int _{a}^{b}}}f(x)\\,dx\\end{aligned}}", "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}}Fdx", "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\\geq k", "029b596e37b45dede3df04654bec7ad0": "z_{j}\\mapsto iz_{j}", "029b72733492d85b84e82a3e01a9f2d2": "\\geq 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).", "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^{n}y}{dx^{n}}}", "029deb3f9f7ca701edbbb85b7090275f": "-\\mathbf {\\hat {n}} ", "029e39793c1f7b96788cabcd8b6bf878": "\\varphi _{p}(x)={\\frac {1-x}{1-px}}.", "029e3cfe2abae8da550ed0f34d8e3d4b": "{\\sqrt {a^{n}x^{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}={\\begin{bmatrix}\\ [b_{1}]_{C}&\\cdots &[b_{n}]_{C}\\ \\end{bmatrix}}", "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": "{}^{a}p_{i}=K_{a}\\cdot H_{ba}\\cdot K_{b}^{-1}\\cdot {}^{b}p_{i}", "029ff1facef6c4a0f4948e8942c3799c": "\\varepsilon _{\\alpha _{1}\\dots \\alpha _{n}}\\,", "02a05bfedf23700420abe2fc04cb2274": "{\\bar {F}}(x)=\\sum _{x_{i}0", "02ad2f550bbcff52fb668d5073b4eebe": "E[F_{6}].", "02ad9a8cd8bd3d22af44319a20aea411": "SU(N)_{L}\\times SU(N)_{R}\\times U(1)_{V}\\times U(1)_{A}~,", "02addf7ddfd0dd5d9906b0daf73bd117": "RejectionRegion{=}{\\frac {{t_{\\alpha /2}}{n-1}}{{\\sqrt {n}}{\\sqrt {n-2+{t_{\\alpha /2}^{2}}}}}}", "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}\\rightarrow 0}\\Phi _{S}\\neq 0", "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}1&{\\text{if point }}m_{i}{\\text{ corresponds to point }}s_{j}\\\\0&{\\text{otherwise}}\\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 x^{2}z^{2}>x^{3}>z^{2}", "02b74a3bdf4e5db66a093867f1ff1eb2": "101011_{2}", "02b75d49cc982846f5bfcfdcb49bac27": "{\\frac {R_{\\text{ac}}}{\\mu L}}=aB_{\\text{max}}f+cf+ef^{2}", "02b78fcd7b8325bc8afd228b00a7e400": "\\leqslant \\int _{1}^{\\infty }2f(x){\\sqrt {1+f'(x)^{2}}}\\,\\mathrm {d} x", "02b7b8f652b409bddb9defbcd33e9f18": "\\phi ^{i}=-E^{,i}\\,", "02b7d06a7f926ebbdb471e187c679023": "\\theta \\,\\!", "02b7f1e422461eb3fd9a2506826d6218": "\\beta _{k}<{\\frac {1}{4}}", "02b86afb8959f20906caea2f1ee51409": "0\\rightarrow T_{x}M\\rightarrow T_{x}P|_{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": "C_{E_{1}}^{S_{2}}=\\varepsilon _{1}^{2}/D", "02b92d672582f9ece902c8fa66467f62": "ELA\\,\\!", "02b983191ce736331e1184228497076c": "D\\in {\\mathcal {D}}", "02b98c25514e6be1b3c6a42e6d794aa5": "7x^{2}y^{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": "H(I|J)=-\\sum _{i,~j}P_{I,J}(i,~j)\\log P_{I|J}(i|j).", "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 _{n}n^{k-1}{\\overline {b_{n}}}\\beta _{k}(4ny)q^{-n+1}", "02ba92ad3d37d5e3f06a828837a17d7a": "(Fa\\lor Fb)\\leftrightarrow Fd", "02bb57797ba7e0a76063c2edd9191fbb": "Z={\\frac {a_{j}}{\\lVert a_{j}\\rVert }}", "02bc005b15bc011bf2cae4b1c1a79c12": "{\\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^{2}y-6y)+7x^{2}y-3y^{2}+2(8y-4y^{2}-4x^{2}y)\\,\\!", "02c193d45dbd50de7b409c5454f045d6": "\\alpha =2-{\\frac {\\tau -1}{\\sigma }}\\,\\!", "02c1b2634f3db0d4515eee02d79b0537": "{\\begin{aligned}E_{2n}&=(-1)^{n}(2n)!~{\\begin{vmatrix}{\\frac {1}{2!}}&1&~&~&~\\\\{\\frac {1}{4!}}&{\\frac {1}{2!}}&1&~&~\\\\\\vdots &~&\\ddots ~~&\\ddots ~~&~\\\\{\\frac {1}{(2n-2)!}}&{\\frac {1}{(2n-4)!}}&~&{\\frac {1}{2!}}&1\\\\{\\frac {1}{(2n)!}}&{\\frac {1}{(2n-2)!}}&\\cdots &{\\frac {1}{4!}}&{\\frac {1}{2!}}\\end{vmatrix}}.\\end{aligned}}", "02c200d95543444ec2205ef66b757136": "\\kappa ={\\frac {\\omega _{r}}{Q}}", "02c2022327405565b8cca1b582733df7": "a=d\\neq b=c,\\alpha =\\zeta =90^{\\circ },\\beta =\\epsilon \\neq 90^{\\circ },\\gamma \\neq 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": "{\\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)", "02c40ae85808d86bd7fdd50f8c36d48a": "\\gamma (\\mathbf {v} )={\\frac {1}{\\sqrt {1-{\\frac {|\\mathbf {v} |^{2}}{c^{2}}}}}}", "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 {eB\\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\\to 1}\\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": "{\\mathcal {L}}={\\frac {1}{2}}\\left|{\\frac {\\partial \\mathbf {n} }{\\partial t}}\\right|^{2}-W(\\mathbf {n} ,\\nabla \\mathbf {n} )-{\\frac {\\lambda }{2}}(1-|\\mathbf {n} |^{2}),", "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}{\\frac {1}{2}}\\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": "{\\begin{pmatrix}A_{1}&B_{1}\\\\A_{2}&B_{2}\\end{pmatrix}}{\\begin{pmatrix}x\\\\y\\end{pmatrix}}={\\begin{pmatrix}C_{1}\\\\C_{2}\\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\\leq L(M)\\leq 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": "r1\\;", "02d44c58ff40c2ddc94695ab9955e25a": "g:{\\textit {George}}", "02d48ef1af7c38a5a790aa75c2a922c9": "X_{5}", "02d4b74c78ad4e5700a887e322543640": "F_{A\\rightarrow A}=0", "02d4d3b0044ee3603acdcd99de9dcb05": "X=A^{-1}(I-UY)", "02d4e6ea476acf8a700897f537d8731f": "{\\frac {\\pi }{2{\\sqrt {2}}}}\\approx 1.11072073", "02d4f4692cd34a7d069bf5afb42fc10b": "f[x_{0},\\dots ,x_{n}]=\\sum _{j=0}^{n}{\\frac {f(x_{j})}{q'(x_{j})}}.", "02d53af1d934122f24861e250bb37ad4": "\\max\\{p,q\\}\\leq p+_{\\mathcal {O}}q", "02d58831ed513c0dda9de14234c8d360": "K_{2}^{M}(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": "{\\begin{aligned}|V|e^{j(\\omega t+\\phi _{V})}&=|I|e^{j(\\omega t+\\phi _{I})}|Z|e^{j\\theta }\\\\&=|I||Z|e^{j(\\omega t+\\phi _{I}+\\theta )}\\end{aligned}}", "02d629ac392c3328e14f57fd55b883ff": "H_{t}-H_{t-1}\\in -K_{t}\\;P-a.s.", "02d649ffafb5cf15d2638a8d7f8d8551": "q(x)=x^{n}+b_{1}x^{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}\\in \\{-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": "{\\frac {\\partial F_{~\\alpha }^{m}}{\\partial X^{\\beta }}}=F_{~\\mu }^{m}\\,_{(X)}\\Gamma _{\\alpha \\beta }^{\\mu }\\qquad ;~~F_{~\\alpha }^{i}:={\\frac {\\partial x^{i}}{\\partial X^{\\alpha }}}", "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": "{\\begin{pmatrix}{A'}^{0}\\\\{A'}^{1}\\\\{A'}^{2}\\\\{A'}^{3}\\end{pmatrix}}={\\begin{pmatrix}1&0&0&0\\\\0&\\cos \\theta &-\\sin \\theta &0\\\\0&\\sin \\theta &\\cos \\theta &0\\\\0&0&0&1\\\\\\end{pmatrix}}{\\begin{pmatrix}A^{0}\\\\A^{1}\\\\A^{2}\\\\A^{3}\\end{pmatrix}}\\ .", "02df1d67c4f118aa427e34f09b750537": "x_{i}\\in \\mathbb {C} ^{m}", "02df319e57e85d778ff5b31c78d39e35": "A\\to \\bot ", "02df681620026e67b28f02df6909fef1": "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).", "02df87d44cd6118584fa0d6a0cbe5157": "\\tan \\left({\\frac {\\Phi }{2}}\\right)=-{\\frac {1+{\\sqrt {R}}}{1-{\\sqrt {R}}}}\\tan \\left({\\frac {\\delta }{2}}\\right)", "02dfac85105c1461af22205de6bb9430": "F(y,z)=\\int _{-\\infty }^{\\infty }f(\\rho ,z)\\,dx=2\\int _{y}^{\\infty }{\\frac {f(\\rho ,z)\\rho \\,d\\rho }{\\sqrt {\\rho ^{2}-y^{2}}}}", "02e042d61b93ff101d6b317e2f36930a": "m\\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}=\\left.{\\frac {\\sum _{d\\mid k}z^{d}}{1-z}}\\exp \\left(\\sum _{d\\mid k}\\left(u^{d}{\\frac {z^{d}}{d}}-{\\frac {z^{d}}{d}}\\right)\\right)\\right|_{u=1}={\\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\\geq 0}{\\frac {(-1)^{k}q^{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": "(A|B)=\\left[{\\begin{array}{ccc|c}1&1&2&3\\\\1&1&1&1\\\\2&2&2&5\\end{array}}\\right].", "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 AV^{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^{j}k_{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=\\iint _{\\Sigma }\\left(P+Q+R\\right)\\,d\\Sigma ", "030e9548babdfe0825542d893e7297bb": "h\\nu _{m}/k", "030eb5f5725d2a23d12126c34b1a6e26": "~\\epsilon _{t}=~\\sigma _{t}~\\times z_{t}", "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": "{\\mathbf {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\\leq 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": "{\\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}}", "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 }\\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-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}}+\\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{alignedat}{4}f(x)&=x({\\sqrt {x+1}}-{\\sqrt {x}})\\\\&=x({\\sqrt {x+1}}-{\\sqrt {x}}){\\frac {({\\sqrt {x+1}}+{\\sqrt {x}})}{({\\sqrt {x+1}}+{\\sqrt {x}})}}\\\\&=x{\\frac {(({\\sqrt {x+1}})^{2}-({\\sqrt {x}})^{2})}{({\\sqrt {x+1}}+{\\sqrt {x}})}}&={\\frac {x}{({\\sqrt {x+1}}+{\\sqrt {x}})}}\\end{alignedat}}", "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 ^{2}cI_{\\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}.", "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": "D_{\\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\\},", "031b9bff8fd8e223ac7b3fc4a03cbd51": "C_{D}=1.456\\times 10^{5}({\\frac {\\eta P}{\\sigma SV^{3}}})", "031ba9e2f80ace02d96bc0ec234c80a4": "{\\dot {x}}_{i}=\\partial H/\\partial p_{i}", "031c29938cf30deaad28a1e05352c788": "Gy)={\\frac {\\int _{y}^{\\infty }xg(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^{iky}d^{n}y=\\langle e^{iky}\\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": "H={\\frac {1}{a^{2}}}+{\\frac {n-1}{b^{2}}}.", "032fcbe05b92f61b4fd08f0722ec1cc7": "\\Gamma ^{i}{}_{jk}", "032ff6a1dbdbb6918b10c7310cfe2be5": "(m,P)", "03302b9afd10bdb58e56b2c229a96f77": "(\\forall n\\in \\mathbb {Z} _{+}):A_{n+1}(x)=\\int _{0}^{x}yA_{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}(lnP_{i|\\delta j}^{x}-lnP_{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}=lnV_{pp\\sigma }-lnV_{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": "\\nabla :TM\\times \\Gamma (E)\\to \\Gamma (E)\\quad ;\\quad \\nabla _{X}v:=\\kappa (v_{*}X)", "033314f8c2ab97c4d497d7ae5b0889d0": "x_{t=0}(x,p)=x", "03334a7d3910583154d57b92ff5e90ee": "\\Delta z_{\\rm {bias}}", "0333a113cad1852120b10f74af753df0": "P_{s}\\left(T\\right)=6.1121\\exp \\left(\\left(18.678-{\\frac {T}{234.5}}\\right)\\left({\\frac {T}{257.14+T}}\\right)\\right)", "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": "dx={\\left({\\frac {\\partial x}{\\partial y}}\\right)}_{z}\\,dy+{\\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 4.828427a^{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}}}F_{\\mu \\nu }{\\tilde {F}}^{\\mu \\nu }+{\\bar {\\psi }}(i\\gamma ^{\\mu }D_{\\mu }-me^{i\\theta '\\gamma _{5}})\\psi ", "033abc09c3acf249153f99047185c031": "\\omega ^{-1}={\\sqrt {2}}+1.\\,", "033ad9c60d57150d5fc5b9c0b201b554": "\\mathbf {N} ={\\begin{bmatrix}-1&0&0&0\\\\1&1&0&0\\\\0&-1&-1&0\\\\0&0&1&-1\\\\0&0&0&1\\\\\\end{bmatrix}}", "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_{T}^{(p)}(p,T)\\,\\delta p\\,+\\,C_{p}^{(T)}(p,T)\\,\\delta T", "033d24ff17ac6a0c27b2a6ee5db80984": "p+dp", "033d252565b13f3e612b0fa5abfc05ce": "i\\hbar {\\frac {d\\psi }{dt}}=-{\\frac {\\hbar ^{2}}{2L}}\\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_{i}t}{a}}=2{\\frac {\\Delta d-v_{i}t}{a}}", "033dd2348284fbd3b6479e6b9599faae": "\\scriptstyle \\int r\\,d\\theta =2\\pi r", "033dff11140d43b95e2e7ae5d19c42d2": "Y_{i}", "033f61cdfc761a03f151199e89a1d96a": "\\Pi _{0}^{1}", "033f7f4c3eb918d9336804647277a218": "10^{8}", "033f9e7b053ff7d55c8790005fc6fdde": "{\\tbinom {n}{k}}", "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^{a}t^{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": "\\ln {\\mathcal {L}}(\\mu ,\\sigma ^{2})=\\sum _{i=1}^{n}\\ln f(x_{i};\\,\\mu ,\\sigma ^{2})=-{\\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}.", "0342381b29fc3888b071b4bc6fac9d5c": "P\\simeq {\\frac {1}{3}}\\epsilon =0.52\\times 10^{31}\\,{\\mbox{bar}}.", "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": "S=kN\\ln \\left[\\left({\\frac {V}{N}}\\right)\\left({\\frac {U}{N}}\\right)^{\\frac {3}{2}}\\right]+{\\frac {3}{2}}kN\\left({\\frac {5}{3}}+\\ln {\\frac {4\\pi m}{3h^{2}}}\\right)", "03452f9a2b862e540da40d6e9858ab40": "U=-0.147\\times R-0.289\\times G+0.436\\times B", "03456c32718e6804c8a92a89315cddc7": "(\\det \\Phi )'=\\sum _{i=1}^{n}\\det {\\begin{pmatrix}\\Phi _{1,1}&\\Phi _{1,2}&\\cdots &\\Phi _{1,n}\\\\\\vdots &\\vdots &&\\vdots \\\\\\Phi '_{i,1}&\\Phi '_{i,2}&\\cdots &\\Phi '_{i,n}\\\\\\vdots &\\vdots &&\\vdots \\\\\\Phi _{n,1}&\\Phi _{n,2}&\\cdots &\\Phi _{n,n}\\end{pmatrix}}.", "0345716ff0625c5efcca01c286036654": "k=e'\\cos \\alpha _{0},", "03458f33713d60710c6aca5db16d5bfd": "{\\frac {D}{R}}", "0345e2efcbef367d84fc1770dbe49332": "H(2^{2})={\\begin{bmatrix}1&1&1&1\\\\1&-1&1&-1\\\\1&1&-1&-1\\\\1&-1&-1&1\\\\\\end{bmatrix}},", "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": "(xy)^{*}=y^{*}x^{*}", "0348948eec49e40cc114d0052df04810": "T_{f}\\;", "0348b58f302d593b58c1d3bb37944b25": "0<\\alpha ^{\\,}<1", "03490af427ab865371d9aa274292dd84": "\\sim 1nm", "034912b6711e851d4b4b85c5db46db55": "\\neg P\\,", "03491dcee43720d276368a4879f55105": "M_{Z}={\\frac {v{\\sqrt {g^{2}+{g'}^{2}}}}{2}},", "03498f1d79bdc5f60c4a3890d104e871": "H_{mn}^{\\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": "V=\\int _{1}^{\\infty }f(x)\\cdot \\pi f(x)\\,\\mathrm {d} x\\leqslant \\int _{1}^{\\infty }{M \\over 2}\\cdot 2\\pi f(x)\\,\\mathrm {d} x\\leqslant {M \\over 2}\\cdot \\int _{1}^{\\infty }2\\pi f(x){\\sqrt {1+f'(x)^{2}}}\\,\\mathrm {d} x", "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={\\begin{bmatrix}\\cos \\theta &-\\sin \\theta \\\\\\sin \\theta &\\cos \\theta \\\\\\end{bmatrix}}", "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\\geq 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 ^{2}I)", "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)", "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}{\\frac {n+1}{k+1}}\\left[{\\begin{matrix}n\\\\k\\end{matrix}}\\right]\\left(B_{k+1}(x)-B_{k+1}\\right)", "03544dddcb8d13bd5a09e7e442e394cb": "H_{\\overline {p}}^{*}x(n)=\\prod _{j=1}^{\\overline {p}}{x(n-\\tau _{j})}", "035451646fcceec539999e4521091551": "da(t)=\\delta _{t}a(t)\\,dt\\,", "0354cb942e62f3909d73c66a52072437": "\\mathbb {C} ^{N/2}", "0354d96c50c762db44e348bf0fe7f48b": "3\\times 3", "0354f3238103bfb970d5fea51b94adeb": "{\\begin{cases}{\\text{always}}{\\begin{cases}{\\text{always }}0\\\\{\\text{if }}y,+1\\end{cases}}\\\\{\\text{if }}x,+2\\end{cases}}", "03551109fb26fcb802cf4443e4d0a1bc": "\\tau _{\\beta ,\\alpha }", "03551e591f616e8f74eec8a006ee40fc": "u_{\\varepsilon }\\left(\\xi ,\\eta ,z\\right)={\\frac {w_{0}}{w\\left(z\\right)}}\\mathrm {C} _{p}^{m}\\left(i\\xi ,\\varepsilon \\right)\\mathrm {C} _{p}^{m}\\left(\\eta ,\\varepsilon \\right)\\exp \\left[-ik{\\frac {r^{2}}{2q\\left(z\\right)}}-\\left(p+1\\right)\\psi _{GS}\\left(z\\right)\\right],", "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": "{\\boldsymbol {\\sigma }}_{r}={\\boldsymbol {Q}}\\cdot {\\boldsymbol {\\sigma }}\\cdot {\\boldsymbol {Q}}^{T}~;~~{\\boldsymbol {Q}}\\cdot {\\boldsymbol {Q}}^{T}={\\boldsymbol {\\mathit {1}}}", "0355c3d493eb27ac190768aae8309697": "(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}}}", "0355cf52d0d548ef1797bde3e95c8626": "{\\begin{array}{rcl}C{\\dfrac {dV}{dt}}&=&-I_{\\mathrm {ion} }(V,w)+I\\\\\\\\{\\dfrac {dw}{dt}}&=&\\phi \\cdot {\\dfrac {w_{\\infty }-w}{\\tau _{w}}}\\end{array}}", "0355fdad639ee1a519a51a03d9577982": "1/\\eta _{f}=q_{4}S", "0356051658dbbe05dfe8095bd591a425": "{\\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}.", "0356c7833ecb6be4248c48f846b39891": "B_{r}=0,\\quad B_{\\theta }=0,\\quad B_{z}=ar^{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}={\\begin{bmatrix}\\sigma _{11}&\\sigma _{12}\\\\\\sigma _{21}&\\sigma _{22}\\end{bmatrix}}\\equiv {\\begin{bmatrix}\\sigma _{x}&\\tau _{xy}\\\\\\tau _{yx}&\\sigma _{y}\\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}\\\\A_{21}&A_{22}&\\cdots &A_{2m}\\\\\\vdots &\\vdots &\\ddots &\\vdots \\\\A_{n1}&A_{n2}&\\cdots &A_{nm}\\\\\\end{pmatrix}}\\circ {\\begin{pmatrix}B_{11}&B_{12}&\\cdots &B_{1m}\\\\B_{21}&B_{22}&\\cdots &B_{2m}\\\\\\vdots &\\vdots &\\ddots &\\vdots \\\\B_{n1}&B_{n2}&\\cdots &B_{nm}\\\\\\end{pmatrix}}={\\begin{pmatrix}A_{11}B_{11}&A_{12}B_{12}&\\cdots &A_{1m}B_{1m}\\\\A_{21}B_{21}&A_{22}B_{22}&\\cdots &A_{2m}B_{2m}\\\\\\vdots &\\vdots &\\ddots &\\vdots \\\\A_{n1}B_{n1}&A_{n2}B_{n2}&\\cdots &A_{nm}B_{nm}\\\\\\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}}}\\cdot {\\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}\\int _{0}^{1}x^{1-t}y^{t}\\ \\mathrm {d} t&=&\\int _{0}^{1}\\left({\\frac {y}{x}}\\right)^{t}x\\ \\mathrm {d} t\\\\&=&x\\int _{0}^{1}\\left({\\frac {y}{x}}\\right)^{t}\\mathrm {d} t\\\\&=&{\\frac {x}{\\ln {\\frac {y}{x}}}}\\left({\\frac {y}{x}}\\right)^{t}|_{t=0}^{1}\\\\&=&{\\frac {x}{\\ln {\\frac {y}{x}}}}\\left({\\frac {y}{x}}-1\\right)\\\\&=&{\\frac {y-x}{\\ln y-\\ln x}}\\end{array}}", "036017945dd7313c43d579e4f8ce2826": "\\varepsilon \\gamma _{\\mu \\nu }", "0360351f5986320b5342bfd2430991ad": "p(t)=\\left({\\tfrac {1}{2}}L\\|F'(\\mathbf {x} _{0})^{-1}\\|^{-1}\\right)t^{2}-t+\\|\\mathbf {h} _{0}\\|", "03603730b2b3b89b93daebe682ecc09c": "\\scriptstyle {\\tbinom {-1}{0}}={\\frac {(-1)^{\\underline {0}}}{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^{2}c=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)+fR", "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}\\}={\\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", "036bab4918e8d35cf702026d645809b4": "{\\begin{bmatrix}0&\\cdots &0\\\\\\vdots &\\ddots &\\vdots \\\\0&\\cdots &0\\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\\leq 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}=-{\\frac {1}{2}}\\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{aligned}{\\mathbf {r}}(t)&\\equiv {\\mathbf {r}}\\left(x,y,z\\right)\\equiv x(t){\\mathbf {\\hat {e}}}_{x}+y(t){\\mathbf {\\hat {e}}}_{y}+z(t){\\mathbf {\\hat {e}}}_{z}\\\\&\\equiv {\\mathbf {r}}\\left(r,\\theta ,\\phi \\right)\\equiv r(t){\\mathbf {\\hat {e}}}_{r}(\\theta (t),\\phi (t))\\\\&\\equiv {\\mathbf {r}}\\left(r,\\theta ,z\\right)\\equiv r(t){\\mathbf {\\hat {e}}}_{r}(\\theta (t))+z(t){\\mathbf {\\hat {e}}}_{z}\\\\&\\,\\!\\cdots \\\\\\end{aligned}}", "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_{R}^{(n,h)}", "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": "\\lim _{\\lambda \\to 0}W_{\\lambda }\\chi _{E}(x)=\\chi _{E}(x)", "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}mk_{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": "\\xi (\\alpha )\\approx {\\sqrt {1-\\alpha ^{3}}}", "0373c933a885a2f6a1231a8cb5412b68": "L(q,{\\dot {q}},t)", "0373d856fc469552a17c78297e391d35": "{\\begin{aligned}F(A^{1},\\dots ,cA^{j},\\dots )&=\\sum _{\\sigma \\in S_{n}}\\operatorname {sgn}(\\sigma )ca_{\\sigma (j)}^{j}\\prod _{i=1,i\\neq j}^{n}a_{\\sigma (i)}^{i}\\\\&=c\\sum _{\\sigma \\in S_{n}}\\operatorname {sgn}(\\sigma )a_{\\sigma (j)}^{j}\\prod _{i=1,i\\neq j}^{n}a_{\\sigma (i)}^{i}\\\\&=cF(A^{1},\\dots ,A^{j},\\dots )\\\\\\\\F(A^{1},\\dots ,b+A^{j},\\dots )&=\\sum _{\\sigma \\in S_{n}}\\operatorname {sgn}(\\sigma )\\left(b_{\\sigma (j)}+a_{\\sigma (j)}^{j}\\right)\\prod _{i=1,i\\neq j}^{n}a_{\\sigma (i)}^{i}\\\\&=\\sum _{\\sigma \\in S_{n}}\\operatorname {sgn}(\\sigma )\\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)\\\\&=\\left(\\sum _{\\sigma \\in S_{n}}\\operatorname {sgn}(\\sigma )b_{\\sigma (j)}\\prod _{i=1,i\\neq j}^{n}a_{\\sigma (i)}^{i}\\right)+\\left(\\sum _{\\sigma \\in S_{n}}\\operatorname {sgn}(\\sigma )\\prod _{i=1}^{n}a_{\\sigma (i)}^{i}\\right)\\\\&=F(A^{1},\\dots ,b,\\dots )+F(A^{1},\\dots ,A^{j},\\dots )\\\\\\\\\\end{aligned}}", "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{aligned}\\lambda _{1}-3\\lambda _{2}&{}=0,\\\\\\lambda _{1}+2\\lambda _{2}&{}=0.\\end{aligned}}", "0374b768cce8b8c2881225f6d35574a5": "d\\geq 1", "0374c643357c023c9eef0d52be33a5ed": "W'\\subseteq {\\mathcal {B}}'", "0374d3d4a26403ce7b7a67f21d47ce70": "\\operatorname {int} A=\\operatorname {core} A", "0375bbdf779e79128ac563157984445d": "\\left\\{{\\begin{array}{ll}\\Delta \\phi +\\lambda \\phi =0&\\mathrm {in\\ } \\ U\\\\\\phi =0&\\mathrm {on\\ } \\ \\partial U.\\end{array}}\\right.", "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\\geq 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 )=4\\gamma ^{1/2}\\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\\times 10^{-9}", "037896f8e77eff0f9551b4591460d25c": "{\\begin{aligned}\\cosh(2x)&=\\sinh ^{2}{x}+\\cosh ^{2}{x}=2\\sinh ^{2}x+1=2\\cosh ^{2}x-1\\\\\\sinh(2x)&=2\\sinh x\\cosh x\\end{aligned}}", "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}})\\geq 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}}},\\ \\pm 3\\right)", "037aa4171e65cc137237f2797823fe38": "\\mathrm {NPV} (R(t))=\\langle w,R\\rangle =\\int _{t=0}^{\\infty }{\\frac {R(t)}{(1+i)^{t}}}\\,dt.", "037ab61010b38491adbf983275260958": "x_{i}=\\bigvee _{j=1}^{n}(g_{ij}\\wedge y_{j}),i=1,2,\\ldots ,m,", "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^{3}r'-{\\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": "\\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|", "037e3f00eac3133acd34a2d34c5d8521": "S_{6},", "037e711358b5150f62f4b528adaaebed": "\\Sigma =N\\,\\sigma ", "037efc824cbe954dbd5aa581e0156503": "\\,{\\hat {m}}_{1}", "037f2c9ddd738e43d71a260275ad5049": "1-g", "037f2d988c51e5390be2a0ecc20ba321": "\\ v_{o}=A_{v}v_{i}\\ .", "037f54d1963f8292451829bd2a72c387": "D_{t}(x_{i},x_{j})^{2}=\\sum _{y}(p(y,t|x_{i})-p(y,t|x_{j}))^{2}w(y)", "037fa690256a7bb1884c165d56a87cfb": "u=\\int {\\frac {du}{dx}}\\,dx", "03803c2f702255059aa8704d1a8da64d": "\\Diamond _{i}P", "0380957958d369064832e39c069858f0": "t_{2}^{\\prime }=1/f^{\\prime }", "03809891f820600376128c1d84da3ef0": "01-{\\frac {1}{2^{|x|}}}", "039fbf45379f8551e9ef60aed04178e4": "e_{d}", "039fe118d348a89d7b553966bb4e3a92": "\\mu _{i}=\\left({\\frac {\\partial U}{\\partial N_{i}}}\\right)_{S,V,N_{j\\neq i}}", "03a0000b65d4ac9a8010e24b859031a3": "A_{m,n}=A_{m,n-2}+A_{m,n-1}", "03a0241631bcde8a890989d3fe6c657e": "\\sum _{p|n}f(p)\\;", "03a04fe6fd3748e89a61b4bc79624682": "\\sigma _{1}=\\sigma _{x}={\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix}}\\,,\\quad \\sigma _{2}=\\sigma _{y}={\\begin{pmatrix}0&-i\\\\i&0\\end{pmatrix}}\\,,\\quad \\sigma _{3}=\\sigma _{z}={\\begin{pmatrix}1&0\\\\0&-1\\end{pmatrix}}", "03a05c1da417a41dae0da916caedc5c2": "\\int (d+e\\,x)^{m}\\left(a+b\\,x+c\\,x^{2}\\right)^{p}dx=-{\\frac {(d+e\\,x)^{m+1}(b+2c\\,x)\\left(a+b\\,x+c\\,x^{2}\\right)^{p}}{(m+1)(2c\\,d-b\\,e)}}\\,+\\,{\\frac {2c(m+2p+2)}{(m+1)(2c\\,d-b\\,e)}}\\int (d+e\\,x)^{m+1}\\left(a+b\\,x+c\\,x^{2}\\right)^{p}dx", "03a08b718087ae8e3fa114b826d96305": "|{\\mathcal {U}}|=9", "03a0cc58ab774f8680e9fd94d5caf7b5": "\\mathbb {F} _{p}", "03a0eec6c55e041c5145403b6be3b9cb": "0.33PC+0.55U+0.12EG=0.37SW+0.63BK", "03a10d5c52c6fec06b9bf6ec97a5b6b2": "{\\begin{matrix}XX^{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}\\\\X^{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}\\end{matrix}}", "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": "-mc^{2}{\\frac {d\\tau [t]}{dt}}=-mc^{2}{\\sqrt {1-{\\frac {v^{2}[t]}{c^{2}}}}}=-mc^{2}+{1 \\over 2}mv^{2}[t]+{1 \\over 8}m{\\frac {v^{4}[t]}{c^{2}}}+\\dots ", "03a2d642bba3f4875961243979e8c601": "1-{\\frac {1}{2}}-{\\frac {1}{4}}+{\\frac {1}{8}}-{\\frac {1}{16}}+\\cdots ={\\frac {1}{3}}.", "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": "w=f(z)={\\frac {a}{c+dz}},\\,", "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}(-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_{3}OH", "03a7f97055201ba94c7471d1764ed4b5": "m(\\varphi )=b\\int _{0}^{\\beta }{\\sqrt {1+e'^{2}\\sin ^{2}\\beta }}\\,d\\beta ,", "03a8087bd765f80deab2bb94bb5e8c53": "\\langle Tx,y\\rangle =\\langle x,Ty\\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": "{\\begin{aligned}p(t)&=(\\cos(2t),\\sin(2t),0)\\\\r(t)&=(\\cos t\\cos 2t,\\cos t\\sin 2t,\\sin t)\\end{aligned}}", "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_{x}M).", "03ad41ed4ad52afeed2ba2bb320eb8f9": "R'(x)=H(x)\\ \\mathrm {if} \\ x\\neq 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": "{\\begin{cases}\\mathrm {out} _{A}=1\\\\\\mathrm {out} _{RGB}=\\mathrm {src} _{RGB}\\mathrm {src} _{A}+\\mathrm {dst} _{RGB}(1-\\mathrm {src} _{A})\\end{cases}}", "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^{T}A{\\vec {a}}", "03aff3b2154d6187c80d748b16746e7e": "{\\sqrt {a^{2}+r}}\\approx a+{\\frac {r}{2\\cdot a}}", "03b002da7c63cabcb42234272136bc6d": "v(n)\\neq 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_{B}T}}\\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)", "03b196e9ce523c30d49f5d2d7bda1ed5": "0+0=0.\\,", "03b1b44a6955cf0b191b94301740d63a": "\\Gamma _{+}(M)", "03b1e442a01f8c4b877e9526471fd3ea": "{\\mathfrak {sl}}_{4}\\cong {\\mathfrak {so}}_{6}", "03b1ecb60736d072099b4bea3dbbf11e": "K_{H}(x')=x'-x_{0}(T),", "03b25e2f947ca0c07d48c53d93f617fc": "N=(P,T,F)", "03b28f6abb6be057c8e59d765aaf78c4": "\\sum _{i=1}^{n}\\mathrm {Bernoulli} (p)\\sim \\mathrm {Binomial} (n,p)\\qquad 0s}\\,ds", "03cc1382a913a0468b9f71a6736e27c8": "H^{(\\lambda )}(X)=P_{1}(X)+O\\left(\\prod _{\\kappa =0}^{\\lambda -1}\\left|{\\frac {\\alpha _{1}-s_{\\kappa }}{\\alpha _{2}-s_{\\kappa }}}\\right|\\right)", "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}-\\left({\\frac {1}{2}}\\epsilon _{0}E^{2}+{\\frac {1}{2\\mu _{0}}}B^{2}\\right)\\delta _{ij}\\,.", "03ccc140b40080330afee07a5170b9d0": "M=[1.440;02.89]", "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}0.244063+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\\\\0.237040+0.24748{\\frac {10^{3}}{T}}+1.9018{\\frac {10^{6}}{T^{2}}}-2.0064{\\frac {10^{9}}{T^{3}}}&7000KB>C", "03d9c9a1ed9d2003fc9725ed0d858420": "\\beta (2)", "03d9da20066f41452bbd390bc1981167": "=(1+i2\\pi fT)e^{-i2\\pi fT}\\mathrm {sinc} ^{2}(fT))\\ ", "03d9e27b16b8626d6f715803f24950ed": "{\\sqrt {x^{2}}}+{\\sqrt {y^{2}}}=|x|+|y|", "03da8aa755310b93007325e732dc509b": "x\\neq 1", "03daab1d4024edcc09938b1d5acab536": "F=kX,", "03dab7746e3ecc9822922b5a75b18e61": "\\lim _{z\\rightarrow 1^{-}}G_{a}(z)=\\sum _{k=0}^{\\infty }a_{k},\\qquad (*)\\!", "03dadf7919b83d68f7682da0245ec658": "\\left[B\\right]=\\left\\{{\\begin{array}{*{35}l}\\left[A\\right]_{0}{\\frac {k_{1}}{k_{2}-k_{1}}}\\left(e^{-k_{1}t}-e^{-k_{2}t}\\right)&k_{1}\\neq k_{2}\\\\\\left[A\\right]_{0}k_{1}te^{-k_{1}t}+\\left[B\\right]_{0}e^{-k_{1}t}&{\\text{otherwise}}\\\\\\end{array}}\\right.", "03db060902394497adbf217f4ae97338": "{\\delta W}=P\\mathrm {d} V.", "03db63a398f2b45c18059782c4f36dfe": "f(x)=ae^{-{\\frac {(x-b)^{2}}{2c^{2}}}}+d", "03db64206cf3a86a497c1121e7ac5d19": "M^{\\mathbf {r} }=\\left(I-H\\right)M\\left(I-H\\right)^{\\rm {T}}.", "03dbbab2600cebe8b540da30ace6353f": "|B\\rangle ", "03dbf0197647d1d16db44f5235b2b043": "{\\frac {\\text{density of object}}{\\text{density of fluid}}}={\\frac {\\text{weight}}{{\\text{weight}}-{\\text{apparent immersed weight}}}}\\,", "03dbf3e1d43b8a823c32f0f134cd5d5f": "f=\\sum _{i=1}^{n}\\alpha _{i}\\varphi (x_{i})+v,", "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_{c}^{i}(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": "c^{2}={\\frac {2\\kappa Gh}{D(1-\\nu )}}\\,.", "03dea73658b8ca48d2bab910746fed4f": "\\,F_{v}", "03dedeca4f11b2b3094602be33a3fd12": "\\min _{x}\\|x\\|_{1}", "03df32a80ca426f52e6e058a7fe0e3f1": "M_{p}(n)", "03df43b24d79f21ac2f7483bfe649a78": "\\displaystyle a^{2}=cd", "03df5d2a6370f99e14f97f689511ecd9": "\\Phi ^{a}", "03df888fc1e5a7faf8843d69f7c2ff0b": "{\\frac {I_{f}^{0}}{I_{f}}}=1+k_{q}\\tau _{0}\\cdot [\\mathrm {Q} ]", "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": "|j_{1}-j_{2}|\\leq J\\leq j_{1}+j_{2}.", "03e1b478601c13078e20acf0aae90f75": "I_{x}=E_{x}L+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_{B}T", "03e42f7f0cba50e05a4ef28ef7a119ce": "{\\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": "\\Phi (z,s,a)={\\frac {1}{2a^{s}}}+\\int _{0}^{\\infty }{\\frac {z^{t}}{(a+t)^{s}}}\\,dt+{\\frac {2}{a^{s-1}}}\\int _{0}^{\\infty }{\\frac {\\sin(s\\arctan(t)-ta\\log(z))}{(1+t^{2})^{s/2}(e^{2\\pi at}-1)}}\\,dt", "03e5c4ff8dac745730829e5dc3d136da": "\\Delta E=E_{n+1}-E_{n}={dE \\over dJ}(J_{n+1}-J_{n})={1 \\over T}\\,\\Delta J", "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": "{\\mathbf {u}}_{1}^{\\prime }={\\mathbf {u}}_{1}-{\\mathbf {V}},\\quad {\\mathbf {u}}_{2}^{\\prime }={\\mathbf {u}}_{2}-{\\mathbf {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": "{\\begin{bmatrix}1&31&12&-3\\\\7&2\\\\1&2&2\\end{bmatrix}}", "03e96c424485be59990cd22a79cffc64": "\\Re {zh^{\\prime }(z) \\over h(z)}\\geq 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}\\rangle \\}", "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}\\approx 32.3472...a^{2}", "03ec4481d5dd16bd36562db5929d3a11": "{\\begin{bmatrix}\\varepsilon _{1}\\\\\\varepsilon _{2}\\end{bmatrix}}\\mid X\\sim {\\mathcal {N}}\\left({\\begin{bmatrix}0\\\\0\\end{bmatrix}},{\\begin{bmatrix}1&\\rho \\\\\\rho &1\\end{bmatrix}}\\right)", "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)^{n}I_{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}({\\mathbb {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)=\\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": "\\psi _{T}(y)=\\int _{x}\\psi _{0}(x)K(x,y;T)dx=\\int ^{x(T)=y}\\psi _{0}(x(0))e^{iS[x]}Dx\\,", "03ef7f06681f7649eaf5dd13c9d53f77": "{\\dot {V}}-{\\frac {U_{osm}}{P_{osm}}}{\\dot {V}}", "03ef81b127bfbe48fe215949105d7e28": "Z_{ij}\\,", "03efb900cbe0906009ca8cdf2f28ee12": "x_{1}x_{2}", "03efbb3a1702295d54d47558026f336a": "N_{MSY}", "03f0d2c858daddfd2cd839e35fdd09c7": "h_{A}(x+y)\\leq h_{A}(x)+h_{A}(y),\\qquad x,y\\in \\mathbb {R} ^{n}.", "03f106c3f162e380f505214595a8b110": "{\\begin{aligned}R&={1 \\over 2\\pi }\\int _{0}^{2\\pi }e^{-i\\theta }U_{\\theta }H^{(1)}U_{\\theta }^{*}\\,d\\theta ,\\\\R_{\\varepsilon }&={1 \\over 2\\pi }\\int _{0}^{2\\pi }e^{-i\\theta }U_{\\theta }H_{\\varepsilon }^{(1)}U_{\\theta }^{*}\\,d\\theta .\\end{aligned}}", "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_{1}d_{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{aligned}L&=\\{uvwxy:u,y\\in \\{0,1,2,3\\}^{*};v,w,x\\in \\{0,1,2,3\\}\\land (v=w\\lor v=x\\lor x=w)\\}\\\\&\\cup \\{w:w\\in \\{0,1,2,3\\}^{*}\\land {\\text{precisely 1/7 of the characters in }}w{\\text{ are 3's}}\\}\\end{aligned}}", "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\\pm 0.8~s", "03f94c2d32a2e3e9dedb87522e89d573": "\\pi a^{2}", "03f960a96507df5ea172c666631d9f7d": "\\left.g\\right.", "03f9745b3fb68caf25bae38a9047b451": "_{S}M", "03f98599e7e2a6894f748aeb548e6af0": "t\\sigma _{1}\\equiv t_{1}", "03f98cb374db9d443f57a6b3871e2aad": "\\mathbf {A} =\\left[{\\begin{array}{c | c}\\mathbf {A} _{11}&\\mathbf {A} _{12}\\\\\\hline \\mathbf {A} _{21}&\\mathbf {A} _{22}\\end{array}}\\right]=\\left[{\\begin{array}{c c | c}1&2&3\\\\4&5&6\\\\\\hline 7&8&9\\end{array}}\\right],\\quad \\mathbf {B} =\\left[{\\begin{array}{c | c}\\mathbf {B} _{11}&\\mathbf {B} _{12}\\\\\\hline \\mathbf {B} _{21}&\\mathbf {B} _{22}\\end{array}}\\right]=\\left[{\\begin{array}{c | c c}1&4&7\\\\\\hline 2&5&8\\\\3&6&9\\end{array}}\\right],", "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{aligned}\\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}}}},\\\\\\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}}}},\\\\\\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}}}}.\\end{aligned}}", "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": "4\\pi \\varepsilon _{0}V(\\mathbf {R} )\\equiv \\sum _{i=1}^{N}q_{i}v(\\mathbf {r} _{i}-\\mathbf {R} )", "03fda4629b973f8ce23b7f20635dc7a7": "x_{k}=-{\\frac {1}{3a}}\\left(b\\ +\\ u_{k}C\\ +\\ {\\frac {\\Delta _{0}}{u_{k}C}}\\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\\geq 0.\\end{cases}}", "03ff0f1aa2432df4f947e0570f58f967": "h(a)=h_{0}+\\sum _{i=1}^{n}h_{i}a_{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": "{\\begin{aligned}{\\frac {dE_{\\lambda }}{d\\lambda }}&={\\frac {d}{d\\lambda }}\\langle \\psi (\\lambda )|{\\hat {H}}_{\\lambda }|\\psi (\\lambda )\\rangle \\\\&={\\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 }\\\\&=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 }\\\\&=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 }\\\\&={\\bigg \\langle }\\psi (\\lambda ){\\bigg |}{\\frac {d{\\hat {H}}_{\\lambda }}{d\\lambda }}{\\bigg |}\\psi (\\lambda ){\\bigg \\rangle }.\\end{aligned}}", "03ff7f007de20f47914ea971fd576bb4": "f_{1}(x_{1},\\ldots ,x_{n}),\\ldots ,f_{k}(x_{1},\\ldots ,x_{n}).", "03ff922d126da125152f09f9cbabcbd1": "{\\begin{matrix}x+y&=&y+x\\\\(x+y)+z&=&x+(y+z)\\\\x+x&=&x\\\\(x+y)\\cdot z&=&(x\\cdot z)+(y\\cdot z)\\\\(x\\cdot y)\\cdot z&=&x\\cdot (y\\cdot z)\\end{matrix}}", "03ffbe62d3a73b362ddbd6ad63e02e40": "t_{A}=t_{B}", "04000d383194855a059ae7cba74fa374": "(m)_{n}=m(m-1)(m-2)\\cdots (m-n+1).", "0400c0a41bae7e3c544019662d174a8c": "\\mathbb {Q} ^{+},\\cdot ", "0400c798906749e6e1e973746d3f0d55": "\\phi ({\\mathbf {r}})={\\frac {1}{4\\pi \\varepsilon _{0}}}\\int {\\frac {\\rho ({\\mathbf {r}}_{0})-{\\mathbf {\\nabla _{\\mathbf {r_{0}}}\\cdot }}{\\mathbf {p}}({\\mathbf {r}}_{0})}{|{\\mathbf {r}}-{\\mathbf {r}}_{0}|}}d^{3}{\\mathbf {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)=\\left(x\\wedge y\\right)\\vee \\left(x\\wedge z\\right)", "0402b088614dbd675146aa12c9226915": "x\\neq 1\\ ", "0402e0626e7835f8c4b12e5778648846": "\\nabla \\cdot \\mathbf {F} ={\\frac {1}{H}}{\\frac {\\partial }{\\partial q^{k}}}\\left({\\frac {H}{h_{k}}}F_{k}\\right)", "0402e9bced3d440d72c3e362204a1255": "\\left(Ax\\right)_{i}", "040317e39ab6225b2f64a7b2c7012b4f": "2a_{k}\\geq a_{k+1}\\,\\forall \\,k\\geq 1", "040320f7a3acf4ea621f9cdab62dc440": "N={\\frac {g_{0}z}{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_{j}f(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}\\leq 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": "{\\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 {\\bar {\\mu }}_{\\text{max}}=\\lambda _{\\text{max}}\\left({\\frac {1}{n}}\\sum _{k=1}^{n}\\mathbb {E} \\,\\mathbf {X} _{k}\\right).", "040548e2562d68d8aba49c12072fbbff": "v_{3}(t)=\\int _{t_{0}}^{t}(K_{1}i_{1}(\\tau )+K_{2}i_{2}(\\tau ))d\\tau .", "0405c9f9d2147a9e6088cbc4a30a8707": "B=B(b,\\lambda )", "0406546e4269ae5098ed91c8999bfa5e": "\\{x\\in V\\colon x=a+n,n\\in W\\}", "0406baf1245fc32c7fcf9e6f50931e91": "G=G[{\\tilde {S}}(\\omega )]=\\int _{-\\infty }^{\\infty }\\eta (\\omega ){\\tilde {S}}(\\omega )\\,d\\omega ", "0406fb29ebe211df5b5b03aeec27b35d": "Re=Re_{c}", "0407aa1318c41683bf20fd50ff5172e1": "0.082H_{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": "\\{a_{11},a_{12},a_{13},a_{22},a_{23},a_{33}\\}", "040a0fe1af61c8a25e80b82326132bc1": "\\left\\{{\\bar {Z}}_{1},\\ldots ,{\\bar {Z}}_{s+c},{\\bar {X}}_{s+1},\\ldots ,{\\bar {X}}_{s+c}\\right\\}", "040a906dae13f008ae8164b64adc2eec": "{\\frac {3b}{4}}", "040ad4c564a4398f895a2bfa60d1e23e": "{\\begin{aligned}2\\int \\sec ^{3}x\\,dx&{}=\\sec x\\tan x+\\int \\sec x\\,dx\\\\&{}=\\sec x\\tan x+\\ln |\\sec x+\\tan x|+C.\\end{aligned}}", "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 {2Z^{2}}{n^{2}Fr}})^{n}", "040cabec1114ed4c6f505e979e430e5d": "a+(180", "040cdd5b0489fa26d9225262e0eb498c": "P_{n}={\\mathbf {1}}'\\otimes \\dots \\otimes {\\mathbf {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_{\\alpha \\beta }:=\\int _{-h}^{h}\\sigma _{\\alpha \\beta }~dx_{3}~;~~M_{\\alpha \\beta }:=\\int _{-h}^{h}x_{3}~\\sigma _{\\alpha \\beta }~dx_{3}~.", "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": "E[\\Delta (t)]\\leq B-\\epsilon \\sum _{i=1}^{N}E[Q_{i}(t)]", "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": "=-II'dsds'\\left[cos(xds)cos(rds)+cos(rds)cos(xds')\\right]", "041061f5b7aa1fa7a7a0725b9bb244a3": "1p_{j}\\\\\\end{array}}\\right.\\end{array}}", "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+2x+1x^{2}+0x^{3}+0x^{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_{k}z^{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:BL2\\times 365\\ln 2\\,\\!.", "0434626709a496ba269fb912232668c5": "\\#X(\\mathbf {F} _{q})=q^{\\operatorname {dim} X}\\sum _{i\\geq 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}{\\begin{pmatrix}j&j&J\\\\m&-m&0\\end{pmatrix}}={\\sqrt {2j+1}}~\\delta _{J0}", "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 }\\|{\\mathbf {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": "re^{aj},-re^{aj},rje^{aj},-rje^{aj},\\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 ={\\frac {1}{1}}+{\\frac {1}{2}}+{\\frac {1}{9}}+{\\frac {1}{145}}+{\\frac {1}{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": "L_{x}={\\begin{pmatrix}0&0&0\\\\0&0&-1\\\\0&1&0\\end{pmatrix}},L_{y}={\\begin{pmatrix}0&0&1\\\\0&0&0\\\\-1&0&0\\end{pmatrix}},L_{z}={\\begin{pmatrix}0&-1&0\\\\1&0&0\\\\0&0&0\\end{pmatrix}}.", "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^{n}g\\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)^{2}t\\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_{3}x_{2}x_{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\\leq 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": "{\\begin{aligned}\\varepsilon _{0}&\\sim \\operatorname {EV} _{1}(0,1)\\\\\\varepsilon _{1}&\\sim \\operatorname {EV} _{1}(0,1)\\end{aligned}}", "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": "m(\\varphi )=B_{0}\\varphi +B_{2}\\sin 2\\varphi +B_{4}\\sin 4\\varphi +B_{6}\\sin 6\\varphi +B_{8}\\sin 8\\varphi +\\cdots ,", "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{aligned}(\\pi _{m,n}(J_{i}))_{a'b',ab}&=\\delta _{b'b}(J_{i}^{(m)})_{a'a}+\\delta _{a'a}(J_{i}^{(n)})_{b'b},\\\\(\\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}),\\end{aligned}}", "0471615797404a49fc735c65e449a7aa": "\\mathbf {C} ^{\\alpha }\\ ", "047174dc95a12d05b955f620f3b80798": "E_{kin}=mc^{2}\\left({\\frac {1}{\\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 \\over 2}\\hbar \\omega ,\\quad {3 \\over 2}\\hbar \\omega ,\\quad {5 \\over 2}\\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": "{P}=\\left[{\\begin{matrix}{T}&\\mathbf {T} ^{0}\\\\\\mathbf {0} &1\\end{matrix}}\\right],", "047483c44de8b6bf46e64800ab13386f": "\\theta \\in [-U,U]", "047495f722547a6cabc2b7cf66b3a722": "\\sum _{p^{k}|n}f(p^{k})\\;", "0474b45700b2a1a17ad723d4a260200f": "=(1-{\\frac {2}{2^{s}}})\\zeta _{2n}(s)+{\\frac {2}{2^{s}}}({\\frac {1}{{(n+1)}^{s}}}+\\ldots +{\\frac {1}{{(2n)}^{s}}})=(1-{\\frac {2}{2^{s}}})\\zeta _{2n}(s)+{\\frac {2n}{{(2n)}^{s}}}\\,{\\frac {1}{n}}\\,({\\frac {1}{{(1+1/n)}^{s}}}+\\ldots +{\\frac {1}{{(1+n/n)}^{s}}}).", "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,\\\\\\\\\\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}};K={\\frac {\\prod _{k}{a_{k}}^{m_{k}}}{\\prod _{j}{a_{j}}^{n_{j}}}}\\equiv {\\frac {{\\{R\\}}^{\\rho }{\\{S\\}}^{\\sigma }...}{{\\{A\\}}^{\\alpha }{\\{B\\}}^{\\beta }...}}", "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^{k}V=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)^{k}x_{-}^{-k}\\pm \\pi i{\\frac {\\delta ^{(k-1)}}{(k-1)!}},", "047e9d8fda718ccca99693995e9444cc": "L_{p,\\mathrm {loc} }(\\Omega ),", "047f60eee20519278eb4e46c31c436f1": "\\rho ~{\\dot {\\eta }}\\geq -{\\boldsymbol {\\nabla }}\\cdot \\left({\\cfrac {\\mathbf {q} }{T}}\\right)+{\\cfrac {\\rho ~s}{T}}", "047f70ea396f58388c9fa6da42fbc7fb": "{\\dot {\\omega }}", "047f9a646fcaed8d8a620b5208eb6c1b": "\\ b={\\frac {1}{2}}\\times \\rho _{water}\\times S_{b}\\times C", "047fd02e5f0ffeb0eea6d81c7bda7d05": "{\\begin{bmatrix}R\\\\G\\\\B\\end{bmatrix}}={\\begin{bmatrix}1&0&1.28033\\\\1&-0.21482&-0.38059\\\\1&2.12798&0\\end{bmatrix}}{\\begin{bmatrix}Y'\\\\U\\\\V\\end{bmatrix}}", "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 Rh=RP/2h", "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\\geq 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}\\geq 0", "0484d7d451687f3e79f67ec3bde75b6e": "{\\begin{aligned}&u^{0}&=&\\alpha +\\beta x+\\gamma x^{2}/2\\\\&u^{1}&=&-{\\frac {1}{2}}L^{-1}(u^{0}u^{0''})&=&-L^{-1}A_{0}\\\\&u^{2}&=&-{\\frac {1}{2}}L^{-1}(u^{1}u^{0''}+u^{0}u^{1''})&=&-L^{-1}A_{1}\\\\&u^{3}&=&-{\\frac {1}{2}}L^{-1}(u^{2}u^{0''}+u^{1}u^{1''}+u^{0}u^{2''})&=&-L^{-1}A_{2}\\\\&&\\cdots &\\end{aligned}}", "04850234a56406c23418f463a67eb060": "n\\geqslant 2", "048505e7c44acdca06cbd3d5acdd7df1": "\\theta =\\theta _{i}", "048538144c496ea0741a737a736eb874": "{\\dot {\\mathbf {r}}}\\times {\\mathbf {H}}=\\mu {\\mathbf {u}}+{\\mathbf {c}}", "048549fa6f951b01bd4dcd6e53002584": "\\{\\{i,j\\}:a_{i,j}\\neq 0,1\\leq i0", "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": "2\\pi \\gamma RB^{5/2}\\Sigma ^{2}K_{1}\\left({\\frac {\\ell }{L_{c}}}\\right)", "048af646d5b23f889b63067b9014b488": "{\\textbf {t}}_{i}", "048b3e9d2796c31a9580d700f5ca6e28": "(\\alpha A)^{+}=\\alpha ^{-1}A^{+}\\,\\!", "048b4edc73ba6a55e2f377a459bdeabd": "\\chi ={\\frac {\\mathbf {M} }{\\mathbf {H} }}", "048b6a58da82ee0994e07c3f235cb954": "z=re^{\\varphi i}{\\text{ with }}-\\pi <\\varphi \\leq \\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}^{MN}\\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}}.", "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": "{\\begin{aligned}{\\hat {\\mu }}_{1}&=m_{1}\\\\{\\hat {\\mu }}_{2}&=m_{2}\\end{aligned}}", "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} }\\leq 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": "Af(x)=b(x)\\cdot \\nabla _{x}f(x)+{\\frac {1}{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 _{s}^{\\prime }((g^{\\prime })^{-1})F(x)=|cx+d|^{1-2s}F\\left({ax+b \\over cx+d}\\right).}", "049594571bf34b6300576cefd2297470": "\\omega _{}^{}=ck", "0495c0ae17755da6efa9259f9976dc72": "\\int {\\frac {x\\;dx}{s^{3}}}=-{\\frac {1}{s}}", "04960db2bec542bc240a4c537f2bc27c": "\\mathbb {Z} _{q}^{n}\\times \\mathbb {Z} _{q}", "04962fb96f35c2c9217723bc1b531c45": "\\Lambda ={\\begin{pmatrix}\\lambda _{1}&\\ldots &0\\\\\\vdots &\\ddots &\\vdots \\\\0&\\ldots &\\lambda _{4}\\end{pmatrix}}\\,,", "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": "=[{\\textrm {CO}}_{2}]_{eq}\\left({\\frac {[{\\textrm {H}}^{+}]_{eq}^{2}+K_{1}[{\\textrm {H}}^{+}]_{eq}+K_{1}K_{2}}{[{\\textrm {H}}^{+}]_{eq}^{2}}}\\right).", "049817e71b75872219c9769deb9e18d7": "\\scriptstyle \\hbar ={\\frac {h}{2\\pi }}\\,", "0498200b37d09b47bbc8d014ad28e86b": "{D}_{8}^{(2)}", "04984b24ee286d6f5dd129c9c1cfa224": "\\Delta h=\\star d\\star dh=\\exp(-2p)\\,\\left(h_{xx}+h_{yy}\\right)", "049874699cbc9ddacaf4d244d90d3e8d": "0\\leq 2n\\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}cr}{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\\sum \\nolimits _{1\\leq i0),", "04c01f0e4919efa824b9c43529a898c0": "{h_{1} \\over h_{0}}={\\frac {{\\sqrt {1+{8Fr^{2}}}}-1}{2}},", "04c0375a0567dd3e84683930cb024313": "s_{0}=s_{n}", "04c05885c12db7ae13199140c4d54225": "{\\cfrac {\\partial W}{\\partial I_{1}}}{\\biggr |}_{I_{1}=3}={\\frac {\\mu }{2}}\\,.", "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^{2}kg^{-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_{0}b_{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\\leq k\\leq K}", "04c3ffe149343baf59147dc0804615d0": "\\,_{97}^{249}\\mathrm {Bk} +\\,_{22}^{50}\\mathrm {Ti} \\to \\,_{119}^{295}\\mathrm {Uue} \\,+4\\,_{0}^{1}\\mathrm {n} ", "04c41f2b4656b51e364061c051c9b3ec": "\\sum _{n}\\left(i\\hbar {\\frac {\\partial c_{n}}{\\partial t}}-c_{n}(t)V(t)\\right)e^{-iE_{n}t/\\hbar }|n\\rangle =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_{2}CH_{2})O+H_{2}\\ \\xrightarrow {Zn\\ +\\ CH_{3}COOH} \\ CH_{2}\\!\\!=\\!\\!CH_{2}+H_{2}O}}", "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}{dx}}\\left({\\frac {1}{2-n}}\\left({\\frac {dt}{dx}}\\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 \\leq -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)=\\log \\left({\\frac {\\pi t}{\\sin(\\pi t)}}\\right)", "04d265067859ccc4737cd584b0b3c99e": "{\\hat {\\rho }}(\\mathbf {r} )=N\\sum _{j=1}^{n}\\int _{0}^{1}ds\\delta \\left(\\mathbf {r} -\\mathbf {r} _{j}(s)\\right).", "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 RT}{M}}}", "04d4721517edfca170ac3802db26813e": "x=0\\!", "04d47874f992aef898ec8e9a27bdb7da": "z\\in \\partial \\Pi _{A}", "04d49a21d2d751d28a93329700556599": "A'(z)=\\sum _{k=0}^{N}a_{k}\\gamma ^{k}z^{-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}:kr", "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\\neq 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": "\\operatorname {Jacobian} \\left({\\frac {x,y}{A,B}}\\right)={\\begin{vmatrix}-(B^{2}-4A)^{-{\\frac {1}{2}}}&{\\frac {1+B(B^{2}-4A)^{-{\\frac {1}{2}}}}{2}}\\\\(B^{2}-4A)^{-{\\frac {1}{2}}}&{\\frac {1-B(B^{2}-4A)^{-{\\frac {1}{2}}}}{2}}\\\\\\end{vmatrix}}=(B^{2}-4A)^{-{\\frac {1}{2}}}", "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": "c_{1}(q)=1,\\;\\;c_{q}(1)=\\mu (q),\\;{\\mbox{ and }}\\;c_{q}(q)=\\phi (q).", "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}+{\\frac {1}{n}}\\epsilon _{i}^{\\nu }\\epsilon _{j}^{\\nu }-{\\frac {1}{n}}\\epsilon _{i}^{\\nu }h_{ji}^{\\nu }-{\\frac {1}{n}}\\epsilon _{j}^{\\nu }h_{ij}^{\\nu }", "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_{p}^{I}(H_{q}^{II}(P_{\\bullet }\\otimes Q_{\\bullet }))=H_{p}^{I}(P_{\\bullet }\\otimes H_{q}^{II}(Q_{\\bullet }))", "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": "\\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),", "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^{p}b^{p}c^{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": "\\sum _{n=1}^{l_{\\lambda }}\\;\\Gamma ^{(\\lambda )}(R)_{nm}^{*}\\;\\Gamma ^{(\\lambda )}(R)_{nk}=\\delta _{mk}\\quad {\\hbox{for all}}\\quad R\\in G,", "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": "IMD_{i}=\\left(e_{i}^{t}-h_{i}^{t}\\right)\\times \\left(G_{i}-G\\right)", "04eed3678d03fd0b3dc1d3f672bdeae1": "e_{q}(x)=\\exp(x){\\text{ if }}q=1", "04eed96382514f7c340f4e53fe09db69": "(hkl)", "04ef1e759d5f3184342d6948487a53d5": "{\\mathcal {L}}[\\varphi (x)]=-{1 \\over 4}F_{\\mu \\nu }F^{\\mu \\nu }+{1 \\over 2}m^{2}A_{\\mu }A^{\\mu }+A_{\\mu }J^{\\mu }", "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 \\mathbb {C} ", "04f084963d52d685bb83410abe86643e": "x=c_{1}c_{2}\\ldots p\\ldots p'\\ldots x_{n}", "04f0980f6e8fa97d144641ec8b6b8ff4": "(L_{0}-{\\tilde {L}}_{0})|\\Psi \\rangle =0", "04f14932ad6780bb4713155a180f0040": "y={\\sqrt {a^{2}-x^{2}}},\\quad y'={\\frac {-x}{\\sqrt {a^{2}-x^{2}}}},\\quad y''={\\frac {-a^{2}}{(a^{2}-x^{2})^{3/2}}},\\quad R=|-a|=a.", "04f188c7b8ed64c7fe2137cda960608a": "\\mathbf {M} _{H}={\\begin{bmatrix}\\;\\;\\,0.38971&0.68898&-0.07868\\\\-0.22981&1.18340&\\;\\;\\,0.04641\\\\\\;\\;\\,0.00000&0.00000&\\;\\;\\,1.00000\\end{bmatrix}}", "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\\leq i,j\\leq k", "04f1ef249f07bd7a5759fd398eee3f4e": "\\Gamma (s,z)=\\Gamma (s)-\\gamma (s,z)", "04f1ff7f5c9bde7065bb8bfa4ef93d41": "{\\begin{aligned}\\mathbb {E} {\\Bigl [}\\liminf _{n\\to \\infty }X_{n}\\,{\\Big |}\\,{\\mathcal {G}}{\\Bigr ]}&=\\mathbb {E} [X|{\\mathcal {G}}]=\\mathbb {E} {\\Bigl [}\\lim _{k\\to \\infty }Y_{k}\\,{\\Big |}\\,{\\mathcal {G}}{\\Bigr ]}=\\lim _{k\\to \\infty }\\mathbb {E} [Y_{k}|{\\mathcal {G}}]\\\\&\\leq \\lim _{k\\to \\infty }\\inf _{n\\geq k}\\mathbb {E} [X_{n}|{\\mathcal {G}}]=\\liminf _{n\\to \\infty }\\,\\mathbb {E} [X_{n}|{\\mathcal {G}}].\\end{aligned}}", "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_{i}^{\\star }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 {Foot9} }|\\overbrace {\\underbrace {-\\smile } _{\\mathrm {Brahma} }} ^{\\mathrm {FootX} }|\\overbrace {\\smile \\smile \\smile -} ^{\\mathrm {Foot11} }||", "04f5027f7716ebfad5764a4c176a88cf": "Q_{q}={\\frac {1}{\\sqrt {N}}}\\sum _{l}u_{l}e^{-iqal}", "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_{1}u+{\\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^{n}u\\,\\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)^{n}g(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{aligned}&(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{aligned}}", "05027bd684c505bb972c1b177b20c56d": "P(|X|\\geq k)\\leq {\\frac {4\\operatorname {E} (X^{2})}{9k^{2}}}\\quad {\\text{if}}\\quad k^{2}\\geq {\\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 {nx}{2}}}{\\sin {\\frac {x}{2}}}}\\right)^{2}={\\frac {1}{n}}{\\frac {1-\\cos(nx)}{1-\\cos x}}", "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": "\\sum _{n=0}^{\\infty }(-1)^{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": "H^{\\dagger }-H=0\\,", "050b57a5f8f2f3a7bf5992a5f74069d3": "c_{0}=S-1\\,\\!", "050b5e0fe4d1ae8a4a9919dc545fa7e7": "e^{ar}", "050b89800d1de9d236fd5a26e225bb5b": "\\csc \\theta \\!", "050c2a34694f64f4b312fe044bfa151f": "P_{0}^{(i)}", "050c5d1a59538341e67943d438532d5d": "na_{0}x^{n}+(n-1)a_{1}x^{n-1}+\\cdots +2a_{n-2}x^{2}+a_{n-1}x=0\\,", "050c6f71cd07650bd1f7ae739b59ba1d": "F_{1\\rightarrow 2}", "050c79c03277c6a6ad35246617006d32": "{\\begin{aligned}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 )+\\Gamma _{0}[s\\{4(\\chi -1)\\chi s-2\\chi +3\\}-1]\\\\&-\\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 )}\\\\&-\\exp \\left[-{\\tfrac {\\Gamma _{0}}{s}}(1-s\\chi )\\right][\\Gamma _{0}(\\chi -1)-1](\\Gamma _{0}^{2}-4\\Gamma _{0}s+2s^{2})({\\text{Ei}}[{\\tfrac {\\Gamma _{0}}{s}}(1-s\\chi )]-{\\text{Ei}}[{\\tfrac {\\Gamma _{0}}{s}}]){\\Biggr ]}\\,.\\end{aligned}}", "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)dxdt.", "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}}}", "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}},", "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})\\geq {\\bigl (}EL(\\Gamma _{1})^{-1}+EL(\\Gamma _{2})^{-1}{\\bigr )}^{-1}.", "05109063960af84cd328819ba140fa94": "[\\cdot ,\\cdot ]\\colon {\\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": "R_{0}={\\cfrac {d\\epsilon _{2}^{p}}{d\\epsilon _{3}^{p}}}={\\cfrac {H}{G}}~;~~R_{90}={\\cfrac {d\\epsilon _{1}^{p}}{d\\epsilon _{3}^{p}}}={\\cfrac {H}{F}}~.", "051230c786f41cee9ecd2f4bd8806de0": "s_{1}=c_{1}e_{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 =2a", "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\\neq 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 _{\\mathbb {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 })}{2}}{\\mathcal {O}}_{1},{\\text{scalar}}\\mathrm {,} ", "0515ecca071219dfab5ed29f01652c71": "E_{\\mathrm {stored} }={\\frac {1}{2}}CV^{2},", "05161917f741c897aba47f69fe891a57": "{T_{v}(s)=V_{out}(s)/V_{in}(s)}\\,", "05168f730983e424739d63483138d587": "\\mathbf {x} _{R}=A\\mathbf {x} _{L}", "0516a583f096aee2d1ef45dbd10159e9": "d(\\sigma )\\geq {\\frac {1}{2}}(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": "RE_{\\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": "Z=\\sum _{n=0}^{\\infty }e^{-n\\beta h\\nu }={\\frac {1}{1-e^{-\\beta h\\nu }}}.", "0518a46df04592797fb11f5a9d147616": "\\Delta \\mathbf {x} \\,\\!", "0518a4ea135ccbe3916da92bbe8e8701": "\\Delta {\\bar {e}}\\ \\,", "0518cc5d7d6d3cdbb5ab9bc1dc3bf0b5": "RD={\\frac {W_{\\mathrm {air} }}{W_{\\mathrm {air} }-W_{\\mathrm {water} }}}\\,", "0518ce225d7219b5b7b398ce8a548f57": "\\liminf _{\\varepsilon \\to +0}\\varepsilon ^{-1}\\left\\{\\gamma ^{n}(A_{\\varepsilon })-\\gamma ^{n}(A)\\right\\}\\geq \\varphi (\\Phi ^{-1}(\\gamma ^{n}(A))),", "05195afa5f1b5313ca387bb548c25dc2": "f(n^{k})=kf(n).\\,", "05197f4f8923ce9df2ad252bcdfc1343": "{\\dot {Q}}(t)\\ =C_{T}^{(V)}(V,T)\\,{\\dot {V}}(t)\\,+\\,C_{V}^{(T)}(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": "V=V(t).", "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^{-im\\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 ^{2}I_{n\\times n}", "051c2a7ff34934f6fc05c14807b02861": "{\\begin{bmatrix}B_{11}&B_{12}&0&\\cdots &\\cdots &0\\\\B_{21}&B_{22}&B_{23}&\\ddots &\\ddots &\\vdots \\\\0&B_{32}&B_{33}&B_{34}&\\ddots &\\vdots \\\\\\vdots &\\ddots &B_{43}&B_{44}&B_{45}&0\\\\\\vdots &\\ddots &\\ddots &B_{54}&B_{55}&B_{56}\\\\0&\\cdots &\\cdots &0&B_{65}&B_{66}\\end{bmatrix}}", "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 _{1}x_{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}})}}={\\frac {d\\left(-\\ln(c_{1}/c_{2})\\right)}{d\\left(-\\ln(U_{c_{2}}/U_{c_{1}})\\right)}}={\\frac {d\\ln(c_{1}/c_{2})}{d\\ln(U_{c_{2}}/U_{c_{1}})}}=E_{12}", "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}0&tt_{o}+t_{p}\\\\\\end{cases}}", "051ee904daa0559210339ff3c6ed52c6": "\\nabla B_{z}=(dB_{z}/dA)\\nabla A", "051f0ede9f5842fe5a8e50066845bdc7": "p=p(V,T)\\ ", "051f218870baef81059be4a102dec711": "G\\left(X'_{i}\\beta \\right)", "051f2871fd7e787c6ec9c8be7702f7f4": "(DV_{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": "E_{a_{0}}={\\frac {E_{S}}{4\\pi a_{0}^{2}}}", "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}}{2w_{0}^{2}}}+ip_{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": "{\\begin{aligned}A_{0}&=u_{0}^{2}\\\\A_{1}&=2u_{0}u_{1}\\\\A_{2}&=u_{1}^{2}+2u_{0}u_{2}\\\\A_{3}&=2u_{1}u_{2}+2u_{0}u_{3}\\\\&\\cdots \\end{aligned}}", "05230f95fc3a31bc3b0826c13f8f4a31": "G=\\left\\{(\\Delta ,x):{\\rm {f}}_{i}(\\Delta ,x)\\leq 0,0\\leq i\\leq k,\\Delta =xx^{T}\\right\\}", "05233dd32f8cc541bd9ced9f0786cda8": "E[G^{2}|K]=\\int _{0}^{T}\\int _{0}^{T}k(t)k(s)E[x(t)x(s)]dtds=\\int _{0}^{T}\\int _{0}^{T}k(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}\\geq f_{2}\\geq \\cdots \\geq f_{N}", "0523d5903ad11af2202b4188d345244e": "\\|u\\|_{L^{4}}\\leq C\\|u\\|_{L^{2}}^{1/4}\\|\\nabla u\\|_{L^{2}}^{3/4}.", "0524ac0139c5674da470ce71e9dc2998": "\\mathbb {Z} _{d}\\times \\mathbb {Z} _{d}", "0525441782af4827a325e2fc2c934ed2": "A=Z+N\\,\\!", "052558595508698e079de150da568929": "{\\sqrt {2}}\\sinh u,\\,", "05258ad8d57d6ca8ec02a490c078934c": "I_{+}=-I_{e}e^{-eV_{+}/(kT_{e})}+I_{ion}^{sat}", "0525b7ccbe9150fc1336beb2b81b5880": "\\eta _{2}={\\frac {H}{m}}\\,\\left(1-m-{\\frac {E(m)}{K(m)}}\\right),", "0525cccdd7123c339471e6dc1fd332a1": "{\\begin{aligned}&\\sin(f_{c}\\cdot t+I\\cdot \\sin(f_{m}\\cdot t))\\\\&\\quad =J_{0}(I)\\sin(f_{c}\\cdot t)+\\sum _{k=1}^{\\infty }J_{k}(I)\\left[\\sin(f_{c}+kf_{m})t+(-1)^{k}\\sin(f_{c}-kf_{m})t\\right]\\end{aligned}}", "05261178171cbde148f522fd4ce40017": "L\\propto \\log \\log N", "05266488515a4d0c92daba82dd43647f": "x_{n}=T_{1}x_{n}^{(1)}+T_{2}x_{n}^{(2)}+\\dots +T_{r}x_{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": "{\\mathbf {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 (}{\\tfrac {1}{2}},{\\tfrac {1}{2}},\\ldots {\\tfrac {1}{2}}{\\bigr )}.", "052b801b8b515aca0898929f40f14ada": "r_{c}\\,\\!", "052bb990e8596e24d7948c61a3f3a8ed": "\\Delta ^{\\text{op}}", "052be0158cc8b723e885b9b440e1083e": "f(\\phi ,\\psi )=0\\,", "052c5652b42a40ec4801dee938109f88": "x^{-\\alpha }\\;G_{p,\\,q+1}^{\\,m,\\,n}\\!\\left(\\left.{\\begin{matrix}\\mathbf {a_{p}} \\\\\\mathbf {b_{q}} ,\\alpha \\end{matrix}}\\;\\right|\\,\\eta x\\right)={\\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 ,", "052c66e5fa5f447b7f2367c9b102f1d4": "(Y,\\mu ,S)", "052c6b13ea3a99fa82c0e51204815d0f": "4r^{2}\\leq e^{2}+f^{2}+g^{2}+h^{2}\\leq 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{aligned}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)}}\\\\\\\\&={\\frac {0.01\\times 0.001}{0.01\\times 0.001+0.95\\times 0.999}}\\\\~\\\\&\\approx 0.0000105.\\end{aligned}}", "052d776f4e8548cffeb47a2dbd78c129": "\\textstyle \\alpha =d", "052d7db1d302c8e2e7ae04d6a5d0ef2b": "P={\\frac {\\int _{0}^{\\frac {\\pi }{2}}l\\cos \\theta d\\theta }{\\int _{0}^{\\frac {\\pi }{2}}td\\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": "p\\sigma \\xrightarrow {\\alpha } p'", "052f3e6f6172ebddf8d9a015db13307f": "{\\overset {\\circ }{\\boldsymbol {\\tau }}}={\\dot {\\boldsymbol {\\tau }}}-{\\boldsymbol {l}}\\cdot {\\boldsymbol {\\tau }}-{\\boldsymbol {\\tau }}\\cdot {\\boldsymbol {l}}^{T}", "052f84150425938458bfcda119406ac9": "{{{\\overline {P_{1}P_{3}}}\\cdot {\\overline {P_{2}P_{4}}}} \\over {{\\overline {P_{1}P_{4}}}\\cdot {\\overline {P_{2}P_{3}}}}}=1+{{{\\overline {P_{1}P_{2}}}\\cdot {\\overline {P_{3}P_{4}}}} \\over {{\\overline {P_{1}P_{4}}}\\cdot {\\overline {P_{2}P_{3}}}}}", "052f90c64762cae32b83178a5045cd8d": "{\\mathbf {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}{1260z^{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)=2b+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\\pm 1.4\\ (\\mathrm {stat.} )\\pm 3.2\\ (\\mathrm {sys.} )", "0532aa97d0f0b796383aef0266ca31d2": "{\\frac {df_{a_{1},\\ldots ,a_{i-1},a_{i+1},\\ldots ,a_{n}}}{dx_{i}}}(a_{i})={\\frac {\\partial f}{\\partial 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{aligned}2\\cdot R_{*}&={\\frac {(10^{-3}\\cdot 128\\cdot 2.05)\\ {\\text{AU}}}{0.0046491\\ {\\text{AU}}/R_{\\bigodot }}}\\\\&\\approx 56\\cdot R_{\\bigodot }\\end{aligned}}", "053456938c3e537cd4b6aad2387c484d": "R(\\tau )={\\frac {\\operatorname {E} [(X_{t}-\\mu )(X_{t+\\tau }-\\mu )]}{\\sigma ^{2}}},\\,", "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": "\\varphi _{\\alpha }=w_{,\\alpha }^{0}\\,.", "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": "{\\boldsymbol {x}},{\\boldsymbol {y}}\\sim \\ {\\mathcal {N}}({\\boldsymbol {\\mu }}_{X,Y},{\\boldsymbol {\\Sigma }}_{X,Y})", "0536b9dcbddd3e7ed41903ab2ea8a619": "f_{n}(x)=x+n", "0536cb091c7b6ab01bbbbcd34f865cfa": "\\displaystyle x_{i}\\rightarrow x_{i}b^{\\left[x_{i}\\right]},\\phi _{i}\\rightarrow \\phi _{i}b^{\\left[\\phi _{i}\\right]}.", "0536ccb9e1940a60e0d1dfb9178ea027": "s=+j\\omega _{0}\\,", "0536e7e4439f94e77793561079872db8": "\\exp \\left\\{-{\\frac {a}{2}}x^{2}\\right\\}={\\sqrt {\\frac {1}{2\\pi a}}}\\;\\int _{-\\infty }^{\\infty }\\exp \\left[-{\\frac {y^{2}}{2a}}-ixy\\right]\\,dy,", "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 xe^{cx^{2}}\\;\\mathrm {d} x={\\frac {1}{2c}}\\;e^{cx^{2}}", "0538221777a190b91f792b30a8aedad4": "\\sum _{j\\in J}a_{j}\\mathbf {v} _{j}=\\mathbf {0} \\,", "0538a1b229fa70423b107e594a783264": "h\\,a={\\begin{cases}c&{\\mbox{if }}p\\,a\\\\b\\oplus ha'&{\\mbox{otherwise}}\\end{cases}}", "0538b394394e22701b79c1ea9a80f9ca": "T=T_{i}2^{-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": "{\\frac {x^{2}}{a^{2}}}+{\\frac {y^{2}}{b^{2}}}+{\\frac {z^{2}}{c^{2}}}=m^{2}", "053b9a3cc33be28c94b003ccf8dc0f94": "=[E_{12},E_{11}]E_{22}+E_{11}[E_{12},E_{22}]-[E_{12},E_{21}]E_{12}-E_{21}[E_{12},E_{12}]+[E_{12},E_{22}]", "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 \\Omega _{Z,[t_{l},t_{u}]}", "053c104ac907d191d028dc3ce0a0126d": "\\mathbb {J} ", "053c115098a7fbf8f6bcdcf83197ae46": "\\scriptstyle f(x,y,z)=w^{2}", "053c735d694601190d0d4d634d4b9b3c": "{\\begin{pmatrix}0&2&0\\\\0&0&3\\\\0&0&0\\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": "w(x_{1},x_{2},0)=\\varphi (x_{1},x_{2})\\quad {\\text{on}}\\quad x_{1}\\in [0,a]\\quad {\\text{and}}\\quad {\\frac {\\partial w}{\\partial t}}(x_{1},x_{2},0)=\\psi (x_{1},x_{2})\\quad {\\text{on}}\\quad x_{2}\\in [0,b]", "053ea84cf5b391a6bc0e2769a337b124": "(v,v-k,v-2k+\\lambda )", "053eb11717e6a67416b0b5b369490f43": "G=\\sum _{nm}G_{nm}", "053eb3c78df7b74a18c177d5fff4640a": "k[\\Delta ]=k\\oplus \\bigoplus _{0\\leq r\\leq d}\\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 {\\partial {\\vec {u}}_{x}}{\\partial 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{aligned}dy_{\\text{1}}\\ =\\ I_{\\text{1}}dt\\ +\\ cdW_{\\text{1}}\\ -\\ u(I_{\\text{2}}dt\\ +\\ cdW_{\\text{2}})\\\\dy_{\\text{2}}\\ =\\ I_{\\text{2}}dt\\ +\\ cdW_{\\text{2}}\\ -\\ u(I_{\\text{1}}dt\\ +\\ cdW_{\\text{1}})\\end{aligned}},\\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": "_{a}I_{b}^{\\left(D\\right)}1={\\frac {1}{\\Gamma \\left({1+\\alpha }\\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}}=(-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(\\pm 1,\\ \\pm (1+{\\sqrt {2}}),\\ \\pm (1+{\\sqrt {2}}),\\ \\pm (1+2{\\sqrt {2}}),\\ \\pm (1+2{\\sqrt {2}})\\right)", "05462613c17fd32778bb06f4e57c8c52": "{\\bigl (}{\\tfrac {1}{2}},{\\tfrac {1}{2}},\\ldots {\\tfrac {1}{2}},-{\\tfrac {1}{2}}{\\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": "\\mathrm {B} (x,y)=\\int _{0}^{\\infty }{\\dfrac {t^{x-1}}{(1+t)^{x+y}}}\\,\\mathrm {d} t,\\qquad \\mathrm {Re} (x)>0,\\ \\mathrm {Re} (y)>0\\!", "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)\\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_{1}e_{2})\\ .", "0547c51ea114c93a980d6dc2b6f904b9": "\\nu =\\alpha c_{\\rm {s}}H=\\alpha c_{s}^{2}/\\Omega =\\alpha p_{\\mathrm {tot} }/(\\rho \\Omega )", "0547e443c12f4232d1d5ea6b222a0525": "{\\tilde {T}}={\\begin{bmatrix}0&\\;&\\cdots &z\\\\{\\frac {1}{2}}&\\ddots &\\ddots &\\;\\\\\\;&\\ddots &\\ddots &\\vdots \\\\\\;&\\;&{\\frac {1}{2}}&0\\end{bmatrix}},", "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_{1}x_{1}+\\cdots +2a_{n}x_{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})}{\\Delta x_{n-1}^{\\top }\\Delta x_{n-1}}}", "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": "CIQ_{t}={\\mathcal {A}}\\left(1+{\\mathcal {B}}\\right)^{t}", "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": "\\int _{a}^{x_{0}-\\delta }e^{nf(x)}\\,dx+\\int _{x_{0}+\\delta }^{b}e^{nf(x)}\\,dx\\leq \\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", "054ee465850b8285713271cb44f8f3c9": "M/T", "054f2cc1ac6da37ad4bf572757cf7dd5": "S_{11}={(1-Z_{0}Y_{11})(1+Z_{0}Y_{22})+Z_{0}^{2}Y_{12}Y_{21} \\over \\Delta }\\,", "054f2d87ffd8835b0422c2e3d22d76ae": "\\log(1/\\epsilon )", "054f383837eb3832ac11add3cdd54472": "x_{1}\\leq \\cdots \\leq x_{n}\\quad {\\text{and}}\\quad y_{1}\\leq \\cdots \\leq 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{aligned}\\rho _{0}{\\ddot {{\\boldsymbol {x}}'}}&={\\frac {\\partial ^{2}}{\\partial t^{2}}}({\\boldsymbol {u}}^{(0)}+{\\boldsymbol {u}}^{(1)}+{\\boldsymbol {X}})\\\\&={\\frac {\\partial ^{2}{\\boldsymbol {u}}^{(1)}}{\\partial t^{2}}}\\end{aligned}}", "05508790e1f9a13201a1f9fbaabf615d": "J\\ {\\stackrel {\\mathrm {def} }{=}}\\ P_{+}-P_{-}", "05508e4ff635ae65cf58468bc93febe2": "v''=0.\\;", "055202e88e159f6ae4c20e58150084f8": "D_{\\mu }{\\tilde {F}}^{\\mu \\nu }=0.", "05520c4227b2eabb934ff18323903144": "\\sigma _{x}\\sigma _{p}\\geq {\\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-Ax_{j}\\right),", "0552ae58c274fcd873a6ce452a0c6231": "\\left|{\\frac {f(z_{1})-f(z_{2})}{{\\overline {f(z_{1})}}-f(z_{2})}}\\right|\\leq {\\frac {\\left|z_{1}-z_{2}\\right|}{\\left|{\\overline {z_{1}}}-z_{2}\\right|}}.", "05536b45589d88f483e73d0ef5612dce": "\\sum _{i}n_{i}({\\bar {Y}}_{i\\cdot }-{\\bar {Y}})^{2}/(K-1)", "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^{Mf(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 }}{\\frac {\\partial \\lambda }{\\partial f_{ij}}}", "0557611da6e08448d16d418a19f8c6e9": "{\\frac {d^{2}z}{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": "\\overbrace {\\rho {\\Big (}\\underbrace {\\frac {\\partial \\mathbf {v} }{\\partial t}} _{\\begin{smallmatrix}{\\text{Unsteady}}\\\\{\\text{acceleration}}\\end{smallmatrix}}+\\underbrace {\\left(\\mathbf {v} \\cdot \\nabla \\right)\\mathbf {v} } _{\\begin{smallmatrix}{\\text{Convective}}\\\\{\\text{acceleration}}\\end{smallmatrix}}{\\Big )}} ^{\\text{Inertia}}=\\underbrace {-\\nabla p} _{\\begin{smallmatrix}{\\text{Pressure}}\\\\{\\text{gradient}}\\end{smallmatrix}}+\\underbrace {\\mu \\nabla ^{2}\\mathbf {v} } _{\\text{Viscosity}}+\\underbrace {\\mathbf {f} } _{\\begin{smallmatrix}{\\text{Other}}\\\\{\\text{forces}}\\end{smallmatrix}}", "055a2275d6d566469f1e8d55e11773ab": "\\int f^{-1}(y)\\,dy=xf^{-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_{1}b_{2}-a_{2}b_{1}=0", "055b8424d6eeb4f26e4709d51649d526": "\\Rightarrow \\mathbf {u} ={\\begin{pmatrix}\\mathbf {u} _{1}\\\\\\mathbf {u} _{2}\\end{pmatrix}}={\\begin{pmatrix}-\\mathbf {B} \\\\\\mathbf {I} _{n-r}\\end{pmatrix}}\\mathbf {u} _{2}=\\mathbf {X} \\mathbf {u} _{2}.", "055c210a7797b4e842635accb13e32a7": "q_{i}", "055c6a0341f1cde9ac1b10222a5ffd1a": "{\\frac {D}{\\beta _{k}}}", "055c9f1f08029271f30a3a260d8adc91": "k+d\\leq 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 {4m_{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": "\\langle i|j\\rangle =\\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": "{\\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_{i}O_{j}}}^{2}-(R_{i}-R_{j})^{2}}}={\\frac {{\\sqrt {R-R_{i}}}\\cdot {\\sqrt {R-R_{j}}}\\cdot {\\overline {K_{i}K_{j}}}}{R}}", "0566e1a5690a3eb3da63262a66fa0698": "L=\\left({\\begin{array}{cc}1&x\\\\0&\\partial _{x}+1+{\\frac {1}{x}}\\end{array}}\\right)\\left({\\begin{array}{c}L_{1}\\\\L_{2}\\end{array}}\\right).", "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\\cdot 6!\\cdot x^{3}}}+{\\frac {10313}{72\\cdot 8!\\cdot x^{5}}}-{\\frac {5509121}{384\\cdot 10!\\cdot x^{7}}}+O\\left({\\frac {1}{x^{9}}}\\right)\\quad {\\mbox{for }}x>1", "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}}_{Y}S)\\otimes T+S\\otimes ({\\mathcal {L}}_{Y}T).", "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)2n\\ell \\cos \\theta .", "056bffe5543d1ee0ce2bc4be836cc566": "Af(x)=rxf'(x)+{\\frac {1}{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)\\leq 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}}}={\\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": "{\\dot {x}}=f(x,u),\\quad x(0)=x_{0},\\quad u(t)\\in {\\mathcal {U}},\\quad t\\in [0,T]", "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_{1}E_{i^{}}", "0571057a349615a6d0c7d0eddba6244e": "(x,t)\\mapsto (\\epsilon x,\\epsilon t),\\qquad \\epsilon \\to 0.", "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 i62.768", "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": "s={\\sqrt {\\ln(1/R^{2})}}=\\sigma ", "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)=Ae^{kz}\\cos(kx-\\omega t)", "0576c789af7cc797743f3f7cbad5fb80": "(i,j,k)", "05774954cef3a0e293515b97e89be98d": "\\mathrm {tr} (\\varepsilon )", "0577852e37185c6cd0c4ac6777f14a91": "{\\begin{aligned}&\\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}\\\\=&\\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}.\\end{aligned}}", "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+fD_{X}Y,\\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": "\\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": "W=\\bigoplus _{i=1}^{n}x_{i}V.", "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}}_{p}P\\subset T_{p}P", "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\\geq 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": "S_{\\ell }^{m}(x,y,z)=\\left[{\\frac {2(\\ell -m)!}{(\\ell +m)!}}\\right]^{1/2}\\Pi _{\\ell }^{m}(z)\\;B_{m}(x,y),\\qquad m=1,2,\\ldots ,\\ell .", "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_{3}^{i}(x,y)=_{k}V_{1}^{r}(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": "RACh.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}\\sim 1.76\\times 10^{7}", "0587b2d9f6b659f62a0a82a2936f1048": "V\\neq W\\to \\operatorname {let-combine} [\\operatorname {let} V:E\\operatorname {in} \\operatorname {let} W:F\\operatorname {in} G]\\equiv \\operatorname {let} V,W:E\\land 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^{m}d(x)", "0589446d07423f1cf786dc2db08901a7": "{\\Bigl \\|}\\sum _{k=0}^{n}\\varepsilon _{k}\\alpha _{k}b_{k}{\\Bigr \\|}_{V}\\leq 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": "=\\operatorname {sgn}(\\sin(\\theta +{\\frac {\\pi }{2}})){\\frac {\\sqrt {1-\\sin ^{2}\\theta }}{\\sin \\theta }}", "058a32571bab72f7af24319b1f57d425": "{\\tilde {\\nu }}", "058a46442473533fe9c79b81850d8de6": "\\,_{2}F_{1}(a,b;c-1;z)-\\,_{2}F_{1}(a+1,b;c;z)={\\frac {(a-c+1)bz}{c(c-1)}}\\,_{2}F_{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}})\\leq I(p_{t_{m}},p_{t_{r}},q_{t_{m}},q_{t_{r}})~~\\Leftarrow ~~p_{t_{n}}\\leq 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={\\begin{bmatrix}5&4&2&1\\\\0&1&-1&-1\\\\-1&-1&3&0\\\\1&1&-1&2\\end{bmatrix}}", "058bba9129ccab88e489d2730febaa0d": "\\left({\\frac {\\partial U}{\\partial T}}\\right)_{V}=T\\left({\\frac {\\partial S}{\\partial T}}\\right)_{V}-p\\left({\\frac {\\partial V}{\\partial T}}\\right)_{V};C_{V}=\\left({\\frac {\\partial U}{\\partial T}}\\right)_{V}", "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 Gm}{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=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": "e_{i}^{t+n}-e_{i}^{t}=NS_{i}+IM_{i}+RS_{i}+AL_{i}", "058edad9bc884a06ddd1a0290f5d61b1": "{\\begin{bmatrix}Y_{1}\\\\Y_{2}\\\\Y_{3}\\end{bmatrix}}={\\begin{bmatrix}\\cos \\theta &-\\sin \\theta &0\\\\\\sin \\theta &\\cos \\theta &0\\\\0&0&1\\end{bmatrix}}{\\begin{bmatrix}X_{1}\\\\X_{2}\\\\X_{3}\\end{bmatrix}}", "058ee734a301721e209bac7eeae3eeaa": "k\\mod q\\neq 0", "058efe614790dbb05f0134a9ba1f229b": "{\\frac {P(R)-P_{\\infty }}{\\rho _{L}}}={\\frac {P_{B}-P_{\\infty }}{\\rho _{L}}}-{\\frac {4\\mu _{L}}{\\rho _{L}R}}{\\frac {dR}{dt}}-{\\frac {2S}{\\rho _{L}R}}=R{\\frac {d^{2}R}{dt^{2}}}+{\\frac {3}{2}}\\left({\\frac {dR}{dt}}\\right)^{2}", "058f18c4de16ad5e4e948d5a35a5a371": "T'={\\begin{bmatrix}0&0\\\\T&0\\end{bmatrix}}\\quad {\\mbox{and}}\\quad N'={\\begin{bmatrix}N&0\\\\0&M\\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": "(x+1)(x-1)=1\\,", "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 _{0}^{T}k(t)S(t)dt=\\int _{0}^{T}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": "{\\text{minimize}}\\quad {\\text{over }}{\\widehat {D}}\\quad \\operatorname {vec} ^{\\top }(D-{\\widehat {D}})W\\operatorname {vec} (D-{\\widehat {D}})\\quad {\\text{subject to}}\\quad \\operatorname {rank} ({\\widehat {D}})\\leq r,", "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": "mI=\\int _{a}^{b}m\\varphi (t)\\,dt\\leq \\int _{a}^{b}G(t)\\varphi (t)\\,dt\\leq \\int _{a}^{b}M\\varphi (t)\\,dt=MI,", "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}1\\\\0\\\\{\\frac {p_{3}}{E+m}}\\\\{\\frac {p_{1}+ip_{2}}{E+m}}\\end{bmatrix}}\\quad \\mathrm {and} \\quad u({\\vec {p}},2)={\\sqrt {E+m}}{\\begin{bmatrix}0\\\\1\\\\{\\frac {p_{1}-ip_{2}}{E+m}}\\\\{\\frac {-p_{3}}{E+m}}\\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}({\\frac {1}{4}}\\pi +{\\frac {1}{2}}\\phi _{0})", "0597e2cdcb0557084c936bb7d92f5815": "(2^{2n}-1)-2^{n+1}", "0597ed548c0a81dfa30ae7d7ac201f31": "\\aleph _{1}", "059806bda3bd68c6db4b7c5ce378a95e": "L_{X}=\\sum _{i}b_{i}{\\frac {\\partial }{\\partial x_{i}}}+{\\frac {1}{2}}\\sum _{i,j}{\\big (}\\sigma \\sigma ^{\\top }{\\big )}_{i,j}{\\frac {\\partial ^{2}}{\\partial x_{i}\\,\\partial x_{j}}}.", "059820265ec22520c46c7f6f1a6e9e49": "A=L_{1}^{-1}L_{1}A^{(0)}=L_{1}^{-1}A^{(1)}=L_{1}^{-1}L_{2}^{-1}L_{2}A^{(1)}=L_{1}^{-1}L_{2}^{-1}A^{(2)}=\\ldots =L_{1}^{-1}\\ldots L_{N-1}^{-1}A^{(N-1)}.", "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": "k_{2}\\approx {\\frac {1.5}{1+{\\frac {19\\mu }{2\\rho gR}}}},", "0599699a20cb859e197a4c564c4c47cd": "xp(x)", "0599b4b872f6708fb7e6347a15a11c95": "u_{1}={\\mbox{Re}}(y_{1})={\\tfrac {1}{2}}(y_{1}+y_{2})=e^{2x}\\cos(x),", "0599bfc4eb518f0b6f9d122f1c2ff42f": "\\Delta m=0\\quad {\\hbox{and}}\\quad \\Delta l=\\pm 1", "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}\\prod _{i=1}^{k}\\Gamma (\\omega _{i}s+\\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\\leq q,p<\\infty ", "059d8050582684950696d2bf4a0a9c22": "{\\begin{bmatrix}0&-1&0\\\\-1&5&-1\\\\0&-1&0\\end{bmatrix}}", "059dcb7018310a884c8e68f80838958c": "\\,(1+9+6+8+3)^{3}=19{,}683", "059e347bb40b012f97255c18b26df569": "C_{IJK}", "059e75340a274fbea9a34c246670e73f": "z_{n}", "059e757564c12a958d2ef2d59cfd3bec": "\\beta ^{0}={\\begin{pmatrix}0&1&0&0&0\\\\1&0&0&0&0\\\\0&0&0&0&0\\\\0&0&0&0&0\\\\0&0&0&0&0\\end{pmatrix}}", "059e7593930844763fce650787af8806": "\\mathbf {r} =r({\\hat {u}}\\cos \\theta +{\\hat {v}}\\sin \\theta )=r{\\hat {u}}(\\cos \\theta +{\\hat {u}}{\\hat {v}}\\sin \\theta )", "059e93c547b1182a9c4aef775da41c5a": "f(I_{1},J_{2},J_{3})=0\\,", "059fa5813e555d8ad1d205bcd7e7edb1": "\\zeta =+1.", "059fe7b20132b9c261200a7a7bd62966": "k_{i}=K_{i}/L_{i}", "05a03641923964c19f02fab6c874798e": "G(x)=\\sum _{1\\leq n\\leq x}F(x/n)\\quad {\\mbox{ for all }}x\\geq 1", "05a03beaf3b2097c77dcbbabceddbc6a": "{\\begin{cases}{\\dfrac {\\partial v}{\\partial t}}(t,x)=Av(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": "{\\boldsymbol {l}}={\\boldsymbol {l}}^{e}+{\\boldsymbol {F}}^{e}\\cdot {\\boldsymbol {L}}^{p}\\cdot ({\\boldsymbol {F}}^{e})^{-1}\\,.", "05a1b0fe8d72d0636008275c15fd2299": "S(q\\to 0)", "05a1bb8d7daf0ca1440d3671c888141a": "X\\equiv X\\left(x_{1},x_{2}\\cdots x_{n}\\right)\\,\\!", "05a22200576b61fe2ef5aa3b91e71a2b": "\\phi (x)={\\begin{cases}1&{\\text{if }}x>x_{0}\\\\0&{\\text{if }}x\\sim {\\frac {c^{2}\\delta E^{2}}{B^{2}k_{\\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_{0}b_{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_{p}^{(N-1)/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} \\rightarrow \\mathbb {R} ", "05e6958d35278b9f4186874c8ce2baff": "{\\vec {p}}^{\\,*}", "05e6a4b2796d446ae06d219936784b31": "k<2\\times 10^{-3}", "05e6aaaf68164c07f41c8b803dc47ea1": "\\Delta P={\\frac {8\\mu LQ}{\\pi r^{4}}}", "05e6b4fcbb73a01861df35fbf63b4a03": "W={\\begin{cases}mW&\\xi \\leq 1/m\\\\0&\\xi >1/m\\end{cases}}", "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_{\\mathbb {R} }\\leq \\mathrm {sinh} ({\\frac {\\pi }{2}})\\approx 2.3", "05e7ca63d7e5b2b34f1090ef28bb487b": "\\int r\\cos \\theta dm", "05e7db456cd901f5d80e881bcc27d8e9": "{\\tfrac {2}{3}}", "05e82ad825e447fb9a23e8aa7c714fe3": "U\\colon (\\mathbf {Ab} ,\\otimes _{\\mathbf {Z} },\\mathbf {Z} )\\rightarrow (\\mathbf {Set} ,\\times ,\\{*\\})", "05e849e1ff7eb44e943f4681b34964c3": "\\textstyle x^{j}b(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)\\neq \\varepsilon ", "05e8d5fffb9e0eb4ba401fb00c15f755": "\\psi \\geq {\\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 _{\\,1\\,\\cdots \\,n}^{\\mu _{1}\\cdots \\mu _{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)\\leq 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}", "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": "{\\boldsymbol {N}}^{T}\\cdot \\mathbf {n} _{0}~d\\Gamma _{0}=d\\mathbf {f} ", "05ee6c3c79b2396c35dd23c5e78a511c": "A\\in \\mathbb {C} ^{n\\times n}", "05ee6e4e0ded01cf48b7400aaf57c2c8": "\\lambda '_{k}={\\begin{cases}4\\lambda _{k}-2m_{k},\\,{\\text{ if }}0\\leq k(\\rho (A)-\\epsilon ).", "05f7e7c3f82b8ae7fc229dcf117d33ca": "3q_{2}q_{3}+3q_{3}q_{4}+3q_{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}}{2a}}", "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": "a=x_{1}+x_{2}", "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{aligned}&\\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}\\\\&\\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\\end{aligned}}", "05fa87902f89e288654d0e2752b0cefa": "M=E-\\varepsilon \\cdot \\sin E.", "05fb05070d1664acb2a478a9111e4853": "\\mathbb {Q} \\cap [0,1]", "05fb116dd77209938c1398a35cd8b116": "{\\begin{aligned}{\\frac {\\partial I_{1}}{\\partial {\\boldsymbol {A}}}}&={\\boldsymbol {\\mathit {1}}}\\\\{\\frac {\\partial I_{2}}{\\partial {\\boldsymbol {A}}}}&=I_{1}~{\\boldsymbol {\\mathit {1}}}-{\\boldsymbol {A}}^{T}\\\\{\\frac {\\partial I_{3}}{\\partial {\\boldsymbol {A}}}}&=\\det({\\boldsymbol {A}})~[{\\boldsymbol {A}}^{-1}]^{T}=I_{2}~{\\boldsymbol {\\mathit {1}}}-{\\boldsymbol {A}}^{T}~(I_{1}~{\\boldsymbol {\\mathit {1}}}-{\\boldsymbol {A}}^{T})=({\\boldsymbol {A}}^{2}-I_{1}~{\\boldsymbol {A}}+I_{2}~{\\boldsymbol {\\mathit {1}}})^{T}\\end{aligned}}", "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": "vX=\\{\\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 :={\\mathbb {Z}}^{3}\\ltimes {\\mathbb {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|\\leq {K \\over |n|}", "05fe12829fff81295b9bef693f5e8779": "v(\\mathbf {r} )\\ {\\stackrel {\\mathrm {def} }{=}}\\ \\int d\\mathbf {r} ^{\\prime }\\,\\rho _{uc}(\\mathbf {r} ^{\\prime })\\ \\varphi _{\\ell r}(\\mathbf {r} -\\mathbf {r} ^{\\prime })", "05fe9c80d66c928b66cdc1e76dd5efbc": "D={\\tfrac {a}{\\sin \\alpha }}={\\tfrac {b}{\\sin \\beta }}={\\tfrac {c}{\\sin \\gamma }}.", "05feaeeb29070ecee88283b395e32236": "=\\sum _{k=0}^{n}k!\\,S(n\\!+\\!1,\\,k\\!+\\!1)\\left({z \\over {1-z}}\\right)^{k+1}\\qquad (n=0,1,2,\\ldots )\\,,", "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)}}\\geq {\\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": "{\\tfrac {5}{36}}-{\\tfrac {1}{30}}{\\sqrt {15}}", "0600b0075f1dc8c5beeb7e0c89d1be2e": "K=C_{12}+C_{23}-C_{13}\\leq 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": "p={h \\over \\lambda }", "0605ec8ff0e079dfb988391330236abf": "w\\cdot (u\\wedge v)={\\frac {1}{2}}(w(u\\wedge v)-(u\\wedge v)w)", "0606585ddb28d26b541178a2fd750d74": "\\theta ={\\begin{cases}\\sin ^{-1}{\\frac {1}{\\beta }},&{\\text{if }}\\beta \\geq 1\\\\\\pi -\\sin ^{-1}{\\beta },&{\\text{if }}\\beta \\leq 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 {GrossProfit}{Sales}}", "060844221f545e5bf6862e60aaec07aa": "ds^{2}=d\\mathbf {q} \\cdot \\mathbf {M} \\cdot d\\mathbf {q} ", "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": "\\mathbf {a} \\cdot (\\mathbf {b} \\times \\mathbf {c} )=-\\mathbf {a} \\cdot (\\mathbf {c} \\times \\mathbf {b} )", "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": "{\\begin{aligned}&y_{0}&=&\\ y(0)+L^{-1}(-1)&=&-t\\\\&y_{1}&=&-L^{-1}(y_{0}^{2})=-L^{-1}(t^{2})&=&-t^{3}/3\\\\&y_{2}&=&-L^{-1}(2y_{0}y_{1})&=&-2t^{5}/15\\\\&y_{3}&=&-L^{-1}(y_{1}^{2}+2y_{0}y_{2})&=&-17t^{7}/315.\\end{aligned}}", "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": "T={\\frac {V}{A}}\\cdot 0.161\\,\\mathrm {s} ", "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": "w=0\\,,-D{\\frac {\\partial ^{2}w}{\\partial y^{2}}}{\\Bigr |}_{y=b/2}=f_{1}(x)\\,,-D{\\frac {\\partial ^{2}w}{\\partial y^{2}}}{\\Bigr |}_{y=-b/2}=f_{2}(x)", "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 )\\leq 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}(-1)^{n}E_{m}\\left(x+{\\frac {n}{k}}\\right)\\quad {\\mbox{ for }}k=1,3,\\dots ", "061512d21c171f0f05094bc24900f4ea": "f(x)=x^{3}-1\\,,", "06154eca89935c16391249e30b659550": "M_{b}", "06154fbf5b0d359fdabd084cd66ebc25": "{\\frac {d}{dx}}(x^{2})=2x.", "061550e9b9bdc85c3f4a8591b42e540b": "\\mathbf {J} ^{23}\\mathbf {A} =\\left[{\\begin{matrix}0&0&0\\\\a_{31}&a_{32}&a_{33}\\\\0&0&0\\end{matrix}}\\right]", "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\\Delta r^{2}\\\\s^{2}&<0\\\\\\end{aligned}}", "062034e1b1e77f92fa8d08995925acf7": "s\\in (0,\\infty )", "062040c65b51cd1374b751276f041f00": "O(n\\cdot \\log n)", "062096cbf7815cd82b26fe175a0f1fb2": "{\\begin{aligned}{\\mathcal {A}}\\left\\{x(t-\\tau )\\right\\}&=\\int _{t-a}^{t+a}x(\\lambda -\\tau )\\,\\operatorname {d} \\lambda \\\\&=\\int _{(t-\\tau )-a}^{(t-\\tau )+a}x(\\xi )\\,\\operatorname {d} \\xi \\\\&={\\mathcal {A}}\\{x\\}(t-\\tau ),\\end{aligned}}", "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": "{\\begin{pmatrix}\\mathrm {Cu} \\\\\\mathrm {Ag} \\\\\\mathrm {Au} \\end{pmatrix}}{\\begin{pmatrix}\\mathrm {Al} \\\\\\mathrm {Ga} \\\\\\mathrm {In} \\end{pmatrix}}{\\begin{pmatrix}\\mathrm {S} \\\\\\mathrm {Se} \\\\\\mathrm {Te} \\end{pmatrix}}_{2}", "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": "\\!{\\Bigl \\langle }x_{m}{\\frac {\\partial H}{\\partial x_{n}}}{\\Bigr \\rangle }=\\delta _{mn}k_{B}T.", "0623691cb796dbcb5716c7cc29380dd2": "{\\sqrt {8r\\left({\\sqrt {4R^{2}+r^{2}}}-r\\right)}}\\leq s\\leq {\\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 ^{n}a\\uparrow ^{n}1", "062429ee30b925bb458e4649dc433a3b": "\\tau ={\\frac {f\\rho v^{2}}{2}}", "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": "ab^{-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": "{\\frac {r_{1}}{A}}=0.46224\\left({\\frac {M_{1}}{M_{1}+M_{2}}}\\right)^{1/3}", "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_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4})^{4}}}", "062a2b397475bd7de2c73915909b205f": "\\{\\mathbf {X} \\}", "062a2ee51454e2cb157488a562fad2bb": "P\\lor 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}+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}1\\\\u\\\\v\\end{pmatrix}}.", "062f1f11a733f565e3b29f75e34656ec": "O(n^{2}2^{\\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_{2}O_{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": "{\\tbinom {6}{3}}", "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": "Af(x)={\\tfrac {1}{2}}\\sum _{i,j}\\delta _{ij}{\\frac {\\partial ^{2}f}{\\partial x_{i}\\,\\partial x_{j}}}(x)={\\tfrac {1}{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_{p}^{\\infty }=H_{p}(T(C_{\\bullet ,\\bullet }))", "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({\\frac {3}{F_{n}}}\\right)=-1", "06373818e93a19cae644f7ae80a93704": "D_{N}=A_{N}-A_{N/2}\\,", "06375828f8aad83b0b94d0bbd47f59dc": "O(m{\\sqrt {n}}\\log n)", "06375ff0aa87888a6a812f49c284cc90": "V=nRT/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^{2}y^{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": "{\\langle p\\rangle }=\\beta _{\\text{max}}\\left(1+\\kappa ^{2}\\right)\\epsilon \\left({1-\\epsilon _{B}-\\epsilon }\\right)^{2}G(\\epsilon )\\left(B_{\\text{max}}\\right)^{2}.", "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}fe^{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 )={\\begin{cases}|y|^{\\alpha }\\left[1-i\\beta (\\tan {\\tfrac {\\pi \\alpha }{2}})\\mathbf {sign} (y)\\right]&\\alpha \\neq 1\\\\|y|\\left[1+i\\beta {\\tfrac {2}{\\pi }}\\mathbf {sign} (y)\\ln |y|\\right]&\\alpha =1\\end{cases}}", "063a05f2356036fe0a763ccb577ef79a": "B^{x}value\\left(O,t\\right)=\\left[indexpartition\\right]_{2}+\\left[xrep\\right]_{2}", "063aa99b954878147d700c12864f8c9e": "\\mu \\neq \\nu ", "063ac6c62425d6ffbdfe820371e2bc37": "\\;D_{\\mathrm {REE} }(\\rho )=\\min _{\\sigma }S(\\rho \\|\\sigma )", "063ae6fb13ea090df90cdfb17ec34e4e": "\\Gamma (x)=(\\alpha ^{8}x-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 F_{\\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": "{\\begin{aligned}p(\\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}}\\\\p(\\mathbf {Z} \\mid \\mathbf {\\pi } )&=\\prod _{n=1}^{N}\\prod _{k=1}^{K}\\pi _{k}^{z_{nk}}\\\\p(\\mathbf {\\pi } )&={\\frac {\\Gamma (K\\alpha _{0})}{\\Gamma (\\alpha _{0})^{K}}}\\prod _{k=1}^{K}\\pi _{k}^{\\alpha _{0}-1}\\\\p(\\mathbf {\\mu } \\mid \\mathbf {\\Lambda } )&=\\prod _{k=1}^{K}{\\mathcal {N}}(\\mathbf {\\mu } _{k}\\mid \\mathbf {\\mu } _{0},(\\beta _{0}\\mathbf {\\Lambda } _{k})^{-1})\\\\p(\\mathbf {\\Lambda } )&=\\prod _{k=1}^{K}{\\mathcal {W}}(\\mathbf {\\Lambda } _{k}\\mid \\mathbf {W} _{0},\\nu _{0})\\end{aligned}}", "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": "p_{0}={\\cfrac {2aE^{*}}{\\pi R}}~;~~p_{0}'=-\\left({\\cfrac {4\\gamma E^{*}}{\\pi a}}\\right)^{1/2}", "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_{K}dX_{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 {dT}{dx}}", "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 =\\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)", "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": "P_{1D}(x)dx={\\sqrt {\\frac {3}{4{\\pi }Ll_{p}}}}\\exp {\\left(-{\\frac {3x^{2}}{4Ll_{p}}}\\right)}dx\\,\\,\\,;\\,\\,\\,\\,\\,\\,P_{2D}(R)dR=2{\\pi }R{\\frac {3}{4{\\pi }Ll_{p}}}\\exp {\\left(-{\\frac {3R^{2}}{4Ll_{p}}}\\right)}dR", "06450d9fbdf8d1f569a98d268b6d054e": "{\\begin{bmatrix}a&b\\\\c&d\\end{bmatrix}}{\\begin{bmatrix}x\\\\y\\end{bmatrix}}.", "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 {-ik}{4\\pi \\epsilon _{0}}}{\\frac {e^{ikr-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 _{t}^{j}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}}={\\frac {1}{n}}\\sum _{i=1}^{n}X_{i}", "064b7f588deebf178356485a12196fb8": "{\\sqrt {\\frac {2}{3}}}\\!\\,", "064bd950f9ba6d5ad4f577ece6659e46": "k={\\cfrac {5+5\\nu }{6+5\\nu }}", "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": "\\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.", "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{aligned}L_{x}&\\approx I_{1}({\\dot {\\psi }}-\\Omega \\sin \\delta )\\,,\\\\I_{2}{\\ddot {\\alpha }}&\\approx (L_{x}\\Omega \\sin \\delta +I_{2}\\,\\Omega ^{2}\\sin ^{2}\\delta )\\,\\alpha \\,.\\end{aligned}}", "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 {\\,_{2}F_{1}(-n,-K;N-K-n+1;e^{t})}}{N \\choose n}}\\,\\!", "06559d4446604dad285c25f86a8b505b": "{\\begin{aligned}\\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}}\\\\&\\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}})}\\\\&=\\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}}\\\\&\\vdots \\\\&=\\operatorname {tr} e^{\\sum _{k=1}^{n}\\mathbf {\\Xi } _{k}}\\end{aligned}}", "0655a9a3d09cb12d562a6a71520bcfc6": "\\mathbf {x} _{*}", "0655cce591173476dca8441b58faa7c0": "A\\otimes _{K}K_{v}\\simeq M_{d}(K_{v}).", "0655cd5b52625898a4522c700969124b": "m(x_{i})={\\frac {1}{N(\\mathbf {h} )}}\\sum _{i=1}^{N(\\mathbf {h} )}Z(x_{i})", "065646299b64e9db2a42e4a70c2044e4": "E[L(t)]-E[L(0)]+V\\sum _{\\tau =0}^{t-1}E[p(\\tau )]\\leq (B+C+Vp^{*})t", "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}}}\\,+\\,\\operatorname {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 )/kT}+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}p\\oplus q&=&(p\\lor q)\\land \\lnot (p\\land q)\\end{matrix}}", "065821342a09d802f7b40b0f6b9a88e1": "F(k+1)=f(F(k))=f(G(k))=G(k+1).", "065831348fd5839996fb2e09d9e8b681": "k_{r}^{-}", "0658bbf1015e2f0d90ca6440473c08c3": "{\\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}\\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}.\\end{matrix}}", "0658bdbe1fcda6797f88e35351e12b9b": "{\\begin{array}{l}f^{1}{\\big (}\\theta ^{1}(t){\\big )}=\\cos {\\big (}\\omega ^{1}t{\\big )},f^{2}{\\big (}\\theta ^{2}(t){\\big )}=\\sin {\\big (}\\omega ^{2}t{\\big )}\\\\f^{1}{\\big (}\\theta ^{1}(t){\\big )}^{2}f^{2}{\\big (}\\theta ^{2}(t){\\big )}f^{2}{\\big (}\\theta ^{2}(t)-{\\frac {\\pi }{2}}{\\big )}=-{\\frac {1}{8}}{\\Big (}2\\sin(2\\omega ^{2}t)+\\sin(2\\omega ^{2}t-2\\omega ^{1}t)+\\sin(2\\omega ^{2}t+2\\omega ^{1}t){\\Big )}\\end{array}}", "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^{s}g^{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 {_{2}F_{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)\\geq 4\\,\\!", "065cd9c9acf9bcd888f6b0880c54b0e9": "m_{n}\\ :=\\ {0}^{256\\ -\\ {\\mathcal {j}}m_{n}{\\mathcal {j}}}{\\mathcal {k}}m_{n}", "065d060554456cdb96fd2f6413e448b0": "f(x)/g(x)", "065d544472d087acac0094a9bc44e9e8": "\\beta xf_{0}\\ll 1", "065e1647d7aec79e787d42b59bc7ce82": "\\epsilon >0", "065e40dd8dbb27b2a8088e3687149788": "I_{x}=I_{y}={\\frac {mr^{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 xf\\ x=x", "065f53c5ff9b8139c5562ca4dfa66ee5": "\\forall i:|\\gamma _{i}|\\leq i", "065f73e8b5eb80210019f2fb94218375": "\\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.", "065f9fe034d3e8ef22183ff943ce6d2e": "{\\vec {w}}\\cdot {\\vec {\\mu }}_{y=0}", "065fa0367342168be7eea9e42c03a454": "\\theta _{1}\\in \\Theta _{1}", "065fffaa6113a8311f7abb23bdd15688": "z=\\exp(it)", "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": "\\int _{-\\infty }^{\\infty }{\\frac {\\gamma \\left({\\frac {s}{2}},z^{2}\\pi \\right)}{(z^{2}\\pi )^{\\frac {s}{2}}}}e^{-2\\pi ikz}\\mathrm {d} z={\\frac {\\Gamma \\left({\\frac {1-s}{2}},k^{2}\\pi \\right)}{(k^{2}\\pi )^{\\frac {1-s}{2}}}}.", "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 }\\left(1-q^{2m}\\right)\\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}{dx}}\\left(\\sin(x^{2})\\right)=2x{\\frac {d}{du}}\\left(\\sin(u)\\right)=2x\\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 {\\partial v_{x}}{\\partial x}}+{\\frac {\\partial v_{y}}{\\partial y}}+{\\frac {\\partial v_{z}}{\\partial z}}\\right)f={\\frac {\\partial v_{x}}{\\partial x}}f+{\\frac {\\partial v_{y}}{\\partial y}}f+{\\frac {\\partial v_{z}}{\\partial 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}}.\\!", "06695c2683b56718e7253a916e120efc": "{\\begin{pmatrix}F_{\\text{x}}\\\\F_{\\text{y}}\\\\F_{\\text{z}}\\\\\\end{pmatrix}}=q{\\begin{pmatrix}E_{\\text{x}}\\\\E_{\\text{y}}\\\\E_{\\text{z}}\\\\\\end{pmatrix}}-q{\\begin{pmatrix}0&-B_{\\text{z}}&B_{\\text{y}}\\\\B_{\\text{z}}&0&-B_{\\text{x}}\\\\-B_{\\text{y}}&B_{\\text{x}}&0\\\\\\end{pmatrix}}{\\begin{pmatrix}v_{\\text{x}}\\\\v_{\\text{y}}\\\\v_{\\text{z}}\\\\\\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}0.", "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}{\\frac {1}{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": "{\\frac {\\partial \\sigma _{xx}}{\\partial x}}+{\\frac {\\partial \\tau _{xz}}{\\partial z}}=0", "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": "{\\begin{bmatrix}T_{1}\\\\T_{2}\\\\T_{3}\\end{bmatrix}}={\\begin{bmatrix}\\sigma _{11}&\\sigma _{21}&\\sigma _{31}\\\\\\sigma _{12}&\\sigma _{22}&\\sigma _{32}\\\\\\sigma _{13}&\\sigma _{23}&\\sigma _{33}\\end{bmatrix}}{\\begin{bmatrix}n_{1}\\\\n_{2}\\\\n_{3}\\end{bmatrix}}", "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)={\\begin{cases}0,&r0", "06a55db1a17d4505e2b15c4aec2d780a": "\\|f\\|_{L^{p,q}}=\\left\\{{\\begin{array}{l l}\\left(\\int _{0}^{\\infty }(t^{\\frac {1}{p}}f^{*}(t))^{q}\\,{\\frac {dt}{t}}\\right)^{\\frac {1}{q}}&q\\in (0,\\infty ),\\\\\\displaystyle \\sup _{t>0}\\,t^{\\frac {1}{p}}f^{*}(t)&q=\\infty .\\end{array}}\\right.", "06a57402a19a50c01966d32ba45a13ca": "{\\begin{aligned}{\\hat {y}}=&\\ 25\\\\&+6.1\\max(0,x-13)\\\\&-3.1\\max(0,13-x)\\\\\\end{aligned}}", "06a577e4bfc6d61013b0a750e883d100": "{\\mathit {k}}\\in \\mathbb {Z} ^{+}", "06a69789c3a0377cbcfbc5f9b7e25541": "{\\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": "{d{\\vec {\\omega }} \\over dt}=({\\vec {\\omega }}\\cdot \\nabla ){\\vec {v}}+\\nu \\nabla ^{2}{\\vec {\\omega }}", "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": "P_{\\mu }(n,t)={\\frac {(\\nu t^{\\mu })^{n}}{n!}}\\sum \\limits _{k=0}^{\\infty }{\\frac {(k+n)!}{k!}}{\\frac {(-\\nu t^{\\mu })^{k}}{\\Gamma (\\mu (k+n)+1)}},\\qquad 0<\\mu \\leq 1,", "06a832e45f0a2df58022d7c2e13998ee": "H=\\sum _{j\\sigma }\\epsilon _{f}f_{j\\sigma }^{\\dagger }f_{j\\sigma }+\\sum _{\\sigma }t_{jj'}c_{j\\sigma }^{\\dagger }c_{j'\\sigma }+\\sum _{j,\\sigma }(V_{j}f_{\\sigma }^{\\dagger }c_{j\\sigma }+V_{j}^{*}c_{\\sigma }^{\\dagger }f_{j\\sigma })+U\\sum _{j}f_{j\\uparrow }^{\\dagger }f_{j\\uparrow }f_{j\\downarrow }^{\\dagger }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}1&0&1\\\\0&1&-1\\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}1&{\\text{ if }}x\\in A\\\\0&{\\text{ otherwise}}\\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_{ij}=\\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\\neq 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": "\\operatorname {Li} _{2}\\left(x\\right)+\\operatorname {Li} _{2}\\left(1-x\\right)={\\tfrac {1}{6}}\\pi ^{2}-\\ln(x)\\ln(1-x)\\,,", "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": "\\epsilon _{abc}\\eta _{b\\mu \\nu }\\eta _{c\\rho \\sigma }=\\delta _{\\mu \\rho }\\eta _{a\\nu \\sigma }+\\delta _{\\nu \\sigma }\\eta _{a\\mu \\rho }-\\delta _{\\mu \\sigma }\\eta _{a\\nu \\rho }-\\delta _{\\nu \\rho }\\eta _{a\\mu \\sigma }", "06b068f1873379c153e916e9b9211a09": "\\left\\{S_{\\alpha }^{i},{\\overline {S}}_{{\\dot {\\beta }}j}\\right\\}=2\\delta _{j}^{i}\\sigma _{\\alpha {\\dot {\\beta }}}^{\\mu }K_{\\mu }", "06b06b1fc241987f0246fb3a4a70fec1": "A\\otimes B", "06b0fd8f01ce256f229860bf283f3e6d": "\\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})", "06b137ca45622b7aec8d85f58f8d164a": "{\\frac {m}{e}}>2.35", "06b16782e68921373cf5edf0493be912": "=\\emptyset ", "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": "\\nabla ^{2}\\Phi ={\\frac {1}{a^{2}\\left(\\sinh ^{2}\\mu +\\sin ^{2}\\nu \\right)}}\\left[{\\frac {1}{\\cosh \\mu }}{\\frac {\\partial }{\\partial \\mu }}\\left(\\cosh \\mu {\\frac {\\partial \\Phi }{\\partial \\mu }}\\right)+{\\frac {1}{\\cos \\nu }}{\\frac {\\partial }{\\partial \\nu }}\\left(\\cos \\nu {\\frac {\\partial \\Phi }{\\partial \\nu }}\\right)\\right]+{\\frac {1}{a^{2}\\left(\\cosh ^{2}\\mu +\\cos ^{2}\\nu \\right)}}{\\frac {\\partial ^{2}\\Phi }{\\partial \\phi ^{2}}}", "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": "{\\hat {\\mu }}\\sim IG\\left(\\mu ,\\lambda \\sum _{i=1}^{n}w_{i}\\right)\\,\\,\\,\\,\\,\\,\\,\\,{\\frac {n}{\\hat {\\lambda }}}\\sim {\\frac {1}{\\lambda }}\\chi _{n-1}^{2}.", "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\\leq \\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": "{\\mathcal {L}}\\{f(x)\\}=-{\\boldsymbol {\\alpha }}(sI-\\Theta )^{-1}\\Theta {\\boldsymbol {1}}", "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": "{\\begin{bmatrix}1&2&3&4&0&0&0\\\\0&3&2&1&1&0&10\\\\0&2&5&3&0&1&15\\end{bmatrix}}", "06b6732971a098fefbf1cdcc022a0707": "{\\mathcal {L}}\\left\\{J^{2}f\\right\\}={\\frac {1}{s}}({\\mathcal {L}}\\left\\{Jf\\right\\})(s)={\\frac {1}{s^{2}}}({\\mathcal {L}}\\left\\{f\\right\\})(s)", "06b6942ffb3bd3773ab7dde8bd9d463b": "j0{\\text{)}}}", "06bc9999ffd4b7f56d9acb4bc41b7b2b": "{\\cfrac {\\Gamma \\vdash A,\\Delta \\qquad \\Sigma \\vdash B,\\Pi }{\\Gamma ,\\Sigma \\vdash A\\land B,\\Delta ,\\Pi }}\\quad ({\\land }R)", "06bcc1b573ae3e3f913f9747f154270e": "{\\begin{aligned}S\\left(v\\right)&=\\int _{0}^{T}{\\frac {d}{dT}}E\\left(v\\right){\\frac {dT}{T}}\\\\[10pt]&={\\frac {E\\left(v\\right)}{T}}-k\\log \\left(1-e^{-{\\frac {hv}{kT}}}\\right)\\end{aligned}}", "06bcd5269b2a6532a55599dd5187f116": "2^{|V|-1}-1", "06bcf75c2eabd4e09fe93db827cc7deb": "L^{1}(G//K)\\ni f\\mapsto {\\hat {f}}", "06bd1846a371cfac4ab8923b7a92d943": "T_{first}", "06bd3be4113d20cac5c1af358416abdf": "x={\\frac {v_{bullet}^{2}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}}({\\mathbf {x}};{\\mathbf {H}})=\\operatorname {Var} {\\hat {f}}({\\mathbf {x}};{\\mathbf {H}})+[\\operatorname {E} {\\hat {f}}({\\mathbf {x}};{\\mathbf {H}})-f({\\mathbf {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}u''(x)=f(x){\\mbox{ in }}(0,1),\\\\u(0)=u(1)=0,\\end{cases}}", "06c12e1824a0f5ee49f8ea651a650f27": "{\\begin{bmatrix}L\\\\M\\\\S\\end{bmatrix}}={\\begin{bmatrix}0.8951&0.2664&-0.1614\\\\-0.7502&1.7135&0.0367\\\\0.0389&-0.0685&1.0296\\end{bmatrix}}{\\begin{bmatrix}X\\\\Y\\\\Z\\end{bmatrix}}", "06c141702f2f0a1c813a676f56b084da": "a=\\theta ", "06c1465b20e8f7941c105e39360901d1": "D\\neq 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": "{\\begin{aligned}\\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)\\\\&{}\\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}\\end{aligned}}", "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 vw}{vFw}}={\\frac {1}{\\sum e_{\\lambda }(f_{ij})}}", "06ca06eb3872382fbf005c6e44ab7f81": "x=R\\lambda ,\\qquad \\qquad y=R\\psi ,", "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{aligned}{\\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]\\\\&=\\lim _{\\Delta b\\to 0}{\\frac {1}{\\Delta b}}\\int _{b}^{b+\\Delta b}f(x)\\,\\mathrm {d} x\\\\&=\\lim _{\\Delta b\\to 0}{\\frac {1}{\\Delta b}}\\left[f(b)\\Delta b+{\\mathcal {O}}\\left(\\Delta b^{2}\\right)\\right]\\\\&=f(b)\\\\{\\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]\\\\&=\\lim _{\\Delta a\\to 0}{\\frac {1}{\\Delta a}}\\int _{a+\\Delta a}^{a}f(x)\\,\\mathrm {d} x\\\\&=\\lim _{\\Delta a\\to 0}{\\frac {1}{\\Delta a}}\\left[-f(a)\\,\\Delta a+{\\mathcal {O}}\\left(\\Delta a^{2}\\right)\\right]\\\\&=-f(a).\\end{aligned}}", "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": "{\\vec {C}}=2.{\\vec {r_{2}}}", "06cea412fb13e3f307acaec972edfdc4": "X_{1}Y_{1}Z_{1}", "06ceef85fc5f1f79b9262f97e16620a2": "\\mathbb {D} ^{q}f(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": "\\omega ^{2}=\\omega _{pe}^{2}+\\omega _{ce}^{2}+3k^{2}v_{\\mathrm {e,th} }^{2}", "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\\leq i_{1}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": "{\\vec {k}}\\cdot {\\vec {J}}=-k_{0}J^{0}\\rightarrow 0,", "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{aligned}{\\text{1 Ci}}&={\\frac {3.7\\times 10^{10}}{(\\ln 2)N_{\\rm {A}}}}{\\text{ moles}}\\times t_{1/2}{\\text{ in seconds}}\\\\&\\approx 8.8639\\times 10^{-14}{\\text{ moles}}\\times t_{1/2}{\\text{ in seconds}}\\\\&\\approx 5.3183\\times 10^{-12}{\\text{ moles}}\\times t_{1/2}{\\text{ in minutes}}\\\\&\\approx 3.1910\\times 10^{-10}{\\text{ moles}}\\times t_{1/2}{\\text{ in hours}}\\\\&\\approx 7.6584\\times 10^{-9}{\\text{ moles}}\\times t_{1/2}{\\text{ in days}}\\\\&\\approx 2.7972\\times 10^{-6}{\\text{ moles}}\\times t_{1/2}{\\text{ in years}}\\end{aligned}}", "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_{k}^{1}", "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 \\mathbb {Z} ,1\\leq 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_{1}^{p,q}={\\begin{cases}0&{\\text{if }}p<0{\\text{ or }}p>1\\\\H^{q}(C^{\\bullet })&{\\text{if }}p=0\\\\H^{q+1}(A^{\\bullet })&{\\text{if }}p=1\\end{cases}}", "06e18996a9d3c2afb1ce39d09f1e8986": "S_{z}=m_{s}\\hbar \\,\\!", "06e18ba5b85397f0770e9943a7b8a808": "f_{u}\\left({\\begin{pmatrix}a&b\\\\0&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": "e_{3}={\\begin{pmatrix}1&0\\\\0&-1\\end{pmatrix}}", "06e2745e0e66a583228563c795212f20": "c=\\pm 1", "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}\\mathrm {if} &p_{l}=p_{1}(u)&p_{m}=p_{2}(u)&p_{n}=p_{3}(u)\\\\\\mathrm {then} &p'_{l}=p_{2}(u-1)&p'_{m}=p_{1}(u-1)&p'_{n}=p_{3}(u-1)\\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{aligned}{\\hat {H}}&={\\hat {T}}+{\\hat {V}}\\\\&={\\frac {{\\mathbf {\\hat {p}}}\\cdot {\\mathbf {\\hat {p}}}}{2m}}+V(\\mathbf {r} ,t)\\\\&=-{\\frac {\\hbar ^{2}}{2m}}\\nabla ^{2}+V(\\mathbf {r} ,t)\\end{aligned}}", "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}0&0&3&0\\\\0&-2&0&0\\\\1&0&0&0\\\\0&0&0&1\\end{bmatrix}}.", "06e5325a33498b0229a4bddf89137d86": "(U_{s}U_{\\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": "P_{A}\\left(1+e^{v_{A}}\\right)=e^{v_{A}}", "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": "{\\frac {1}{T}}\\sum _{t=1}^{T}\\mathbf {1} _{\\{X_{t}\\in A\\}}\\ \\xrightarrow {a.s.} \\ \\operatorname {Pr} [X_{t}\\in A],", "06e89453aad485a4ca9d7eb0e0bd05c2": "Q_{B}=C_{B}V_{B}.\\ ", "06e8a474d11071362f5ff94cf9b4068b": "\\lim _{x\\to 1}{\\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": "{\\text{Tr}}\\left\\{\\Pi _{\\rho _{X^{n}\\left(m\\right)},\\delta }{\\hat {\\Pi }}_{\\rho _{X^{n}\\left(m-1\\right)},\\delta }\\cdots {\\hat {\\Pi }}_{\\rho _{X^{n}\\left(1\\right)},\\delta }\\ \\Pi _{\\rho ,\\delta }^{n}\\ \\rho _{x^{n}\\left(m\\right)}\\ \\Pi _{\\rho ,\\delta }^{n}\\ {\\hat {\\Pi }}_{\\rho _{X^{n}\\left(1\\right)},\\delta }\\cdots {\\hat {\\Pi }}_{\\rho _{X^{n}\\left(m-1\\right)},\\delta }\\Pi _{\\rho _{X^{n}\\left(m\\right)},\\delta }\\right\\},", "06ebe860836e08b5e4a0ad3731cbd535": "y={\\frac {y'}{x'^{g+1}}}", "06ec614a0e2ec8c27aab21d24e399139": "{\\begin{aligned}m{\\frac {d}{dt}}\\langle \\Psi (t)|{\\hat {x}}|\\Psi (t)\\rangle &=\\langle \\Psi (t)|{\\hat {p}}|\\Psi (t)\\rangle ,\\\\{\\frac {d}{dt}}\\langle \\Psi (t)|{\\hat {p}}|\\Psi (t)\\rangle &=\\langle \\Psi (t)|-U'({\\hat {x}})|\\Psi (t)\\rangle .\\end{aligned}}", "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=2a", "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 }}}", "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": "\\sum F=0~,~~\\sum M_{A}=0\\,.", "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 }}p_{\\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_{1}q_{1}", "06f59c823dd57d3a21d55798c4c302a8": "g_{2}^{2}=g_{3}^{3}=(g_{2}g_{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": "{\\frac {1}{R}}={\\frac {1}{R_{1}}}+{\\frac {1}{R_{2}}}", "06f6761ad8398a80686ea3ac861b86c5": "T_{max}=E{4Mm \\over (m+M)^{2}}", "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": "\\alpha (u)={\\begin{cases}{\\frac {1}{\\sqrt {2}}},&{\\mbox{if }}u=0\\\\1,&{\\mbox{otherwise}}\\end{cases}}", "06fa62a7df57887836c1e22f862ae08b": "(4~5).", "06faac98935d1cf9b57d0640c6073d4f": "X\\land \\neg X", "06fab9786c1782ba7733c31a17c6c66e": "{\\begin{aligned}a&=6.112\\ \\mathrm {millibar} ;\\quad \\;b&=17.67;\\quad \\;c&=243.5^{\\circ }\\mathrm {C} ;\\end{aligned}}", "06fb0756cbb51cef8245388c77460834": "\\int _{S}F\\,dS", "06fb12324a98e8b31b2819be10b29dca": "{\\begin{aligned}x'=\\gamma x-{\\frac {\\gamma v}{c}}ct&\\Rightarrow &x'=\\gamma (x-vt)\\\\ct'=-{\\frac {\\gamma v}{c}}x+\\gamma ct&\\Rightarrow &t'=\\gamma \\left(t-{\\frac {vx}{c^{2}}}\\right)\\end{aligned}}", "06fbe3c17a36710731842480e1657952": "P={\\frac {T^{\\alpha }}{R^{\\beta }}}", "06fbf0791485f24f1a0df9ea75544e43": "{\\begin{aligned}y&=y_{0}+y_{1}+y_{2}+y_{3}+\\cdots \\\\&=-\\left[t+{\\frac {1}{3}}t^{3}+{\\frac {2}{15}}t^{5}+{\\frac {17}{315}}t^{7}+\\cdots \\right]\\end{aligned}}", "06fbf7e846775b80c4fffad4c0b3055b": "v=\\sum _{i=1}^{n}v^{i}X_{i},\\quad w=\\sum _{i=1}^{n}w^{i}X_{i}", "06fc0b39a9811f7e78cf9a439d4cef40": "{\\begin{cases}{\\frac {dx_{1}}{dt}}=(1-x_{2}^{2})*x_{1}-x_{2}+u\\\\{\\frac {dx_{2}}{dt}}=x_{1}\\\\{\\frac {dx_{3}}{dt}}=x_{1}^{2}+x_{2}^{2}+u^{2}\\\\x(t_{0})=[0\\ 1\\ 0]\\\\t_{f}=5\\\\-0.3\\leq u\\leq 1.0\\\\\\end{cases}}", "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": "\\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}.", "06fcd5f9f7bf19377c6f7c4560d9ddd3": "{\\sqrt {12.746\\times A_{m}}}", "06fce68ac85e7fd4fe558639c55dff48": "\\operatorname {tanh} (z)", "06fd262059da6c3ef9aebeb89b4eae62": "\\mathbf {e} _{1}", "06fd43a831e994e442b64b77ffb70cfb": "{\\begin{bmatrix}1&u_{12}/u_{11}&.&u_{1n}/u_{11}\\\\0&1&.&u_{2n}/u_{22}\\\\.&.&.&.\\\\0&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": "G(k)={1 \\over i\\omega -{k^{2} \\over 2m}}.\\,", "06fe3fd50fd4de394e13d4e6c8ca2e2b": "p_{m}", "06fe9eed3a7ef77fb236b4115bc813df": "L_{1}(B)\\subseteq V", "06fed6899c66d75d74f56fa57e2e7c97": "k'_{L}=0.664{D_{AB} \\over x}Re_{L}^{1/2}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{aligned}&\\{\\Gamma ,\\Gamma \\}=2I&&\\{\\Gamma ,Q\\}=0&&\\{\\Gamma ,{\\bar {Q}}\\}=0\\\\&\\{Q,{\\bar {Q}}\\}=2Z&&\\{Q,Q\\}=2(H+P)&&\\{{\\bar {Q}},{\\bar {Q}}\\}=2(H-P)\\\\&[N,Q]={\\frac {1}{2}}Q&&[N,{\\bar {Q}}]=-{\\frac {1}{2}}{\\bar {Q}}&&[N-[1-q,\\Gamma ]=0\\\\&[N,H+P]=H+P&&[N,H-P]=-(H-P)&&\\end{aligned}}", "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}-\\beta &0&0&0\\\\0&-\\beta &0&0\\\\0&0&-\\beta &0\\\\0&0&0&-\\beta \\end{bmatrix}}", "07024215329a71973769a3641eb82d08": "\\psi ^{*}(\\theta |_{W})=0,\\forall \\theta \\in \\Lambda _{C}^{1}\\pi _{r+1,r}.\\,", "07025eb37092f002d5e6a63feb9826fa": "\\mbox{\\AA} ^{-2}", "0702636caf3faa212e5ca5901d56b7a8": "{\\begin{aligned}A_{j}&={\\frac {\\sum _{i=1}^{L}x_{L(j-1)+i}}{L}}\\quad \\forall j&=1,2,\\ldots ,N\\end{aligned}}", "070268441b1ba4ce1b40b9226759b5fd": "M^{\\rm {SN}}(x)=h-eFx-e^{2}/(16\\pi \\varepsilon _{0}x),\\qquad \\qquad (3)", "07029e1595c60a230b6af32248fbbf84": "f_{Y}(y)=f_{X}\\left(g^{-1}(y)\\right)\\left|{\\frac {d}{dy}}g^{-1}(y)\\right|", "0702e0a272267151de0194ef145a01ed": "\\{X_{\\alpha }\\}_{\\alpha \\in \\mathrm {A} }\\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": "\\|Ax\\|_{\\beta }\\leq \\|A\\|_{\\alpha ,\\beta }\\|x\\|_{\\alpha }.", "0704911db4e3ec5f12d536fbfd7ed629": "c\\,\\!", "0704b8919da6315a296827d30201318e": "\\mathbf {H} _{\\alpha }(x)={\\frac {2{(x/2)}^{\\alpha }}{{\\sqrt {\\pi }}\\Gamma (\\alpha +{\\frac {1}{2}})}}\\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": "P(k)={n-1 \\choose k}p^{k}(1-p)^{n-1-k},", "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)}}=ze^{\\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}\\mathbf {e} _{+}\\\\\\mathbf {e} _{-}\\\\\\mathbf {e} _{0}\\end{pmatrix}}=\\mathbf {U} {\\begin{pmatrix}\\mathbf {e} _{x}\\\\\\mathbf {e} _{y}\\\\\\mathbf {e} _{z}\\end{pmatrix}}\\,,\\quad \\mathbf {U} ={\\begin{pmatrix}-{\\frac {1}{\\sqrt {2}}}&-{\\frac {i}{\\sqrt {2}}}&0\\\\+{\\frac {1}{\\sqrt {2}}}&-{\\frac {i}{\\sqrt {2}}}&0\\\\0&0&1\\end{pmatrix}}\\,,", "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": "{\\begin{aligned}{\\textbf {a}}^{*}&={\\frac {2\\pi {\\textbf {b}}\\times {\\hat {\\textbf {n}}}}{|{\\textbf {a}}\\times {\\textbf {b}}|}}\\\\{\\textbf {b}}^{*}&={\\frac {2\\pi {\\hat {\\textbf {n}}}\\times {\\textbf {a}}}{|{\\textbf {a}}\\times {\\textbf {b}}|}}\\end{aligned}}", "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_{3}^{1}(\\sin \\theta )\\sin \\varphi ={\\frac {1}{r^{4}}}{\\frac {3}{2}}\\ (5\\ \\sin ^{2}\\theta -1)\\cos \\theta \\sin \\varphi ", "070edf626867f2e74beb2a9f117e3d17": "Q=\\{(s,t_{e})|s\\in S,t_{e}\\in (\\mathbb {T} \\cap [0,ta(s)])\\}", "070f583f36e31d4401c0aff07df3ece9": "\\mathbf {L} =mr^{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": "J", "0710afd247c0582195d440cc0e12ba43": "f_{1}(x)f_{2}(y)\\leq 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^{k}f", "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": "{\\begin{array}{rr|rr}1x&{\\text{-}}13&16x&{\\text{-}}81\\end{array}}", "0713a52582a53f4d7ba16c4c6ed27031": "C_{1}\\subseteq C_{2}", "0713a6b3411166cb06f7ab980d9f5ede": "m_{\\mathrm {TNT} }", "071433da6a0b97575672c8502b6da5e8": "01.\\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 in \\over N}}", "07250d8d86eff1000acaf0522dc3ac5f": "\\forall nV_{x}", "0733b8464937afd4081f25c09be53fa1": "{\\begin{bmatrix}c_{1}&-s_{1}c_{3}&-s_{1}s_{3}\\\\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 }\\\\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 ikf}|^{2}}}.", "073575b3716398185104175a18564140": "p=18", "07357ff8e80f38aa7f7bd72df210f1b2": "{\\sqrt {\\exp }}", "07358ebcd581a065b84eb48b7362b09c": "D_{L}\\ =\\ R_{0}r_{1}(1+z)={\\frac {c}{H_{0}q_{0}^{2}}}\\left[q_{0}z+(q_{0}-1)(-1+{\\sqrt {1+2q_{0}z}})\\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/meanpacketsize)", "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}1&1&1&1&0&0&0\\\\1&-&0&0&1&1&0\\\\1&0&-&0&-&0&1\\\\1&0&0&-&0&-&-\\\\0&1&-&0&0&1&-\\\\0&1&0&-&1&0&1\\\\0&0&1&-&-&1&0\\end{pmatrix}}", "0737179ee66c8461eeafd1b317438d93": "{n \\choose \\lfloor {n/2}\\rfloor }\\geq {n \\choose k}", "0737552727d3f52d5f6ac33e430cccf9": "\\sigma (X',X)", "073883b3807515b371c7103bcd50240f": "\\mathbb {E} \\left[((H\\cdot M)_{t}^{*})^{p}\\right]\\leq 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 ,{\\mathbf {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\\leq y\\leq d,\\ r(y)\\leq x\\leq s(y)\\right\\}", "073aef54904465bc5f2c5c88b3fa7d30": "\\{(-,+,+,+)\\,,l^{a}n_{a}=-1\\,,m^{a}{\\bar {m}}_{a}=1\\}", "073b23a014cdb561e726fdfd782536de": "\\sin(45^{\\circ })={\\frac {1}{2}}{\\sqrt {2}};", "073b301e11bb8a8080bcb487c4d7b7e4": "x^{3}+bx^{2}+cx+d=0", "073b63613eef32ad23c021ccc4317e95": "{\\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~.", "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 \\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": "\\scriptstyle 1\\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^{d}p}{(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\\leq t\\leq 2\\pi .", "07405e52845785645c3846f46a49323c": "a>0.\\,", "07406589807bb14817217a224a910198": "{\\bar {\\theta }}(\\mathbf {r} ,t)=t_{n}\\theta ^{n}(\\mathbf {r} ,t)\\,,", "074097ea89225398ceb1128b5405b9fb": "x\\geq 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\\geq 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 }{\\frac {1}{b_{n}}}\\sum _{k=1}^{n}b_{k}x_{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}}_{Y}X_{1},X_{2},\\ldots )-T(\\alpha _{1},\\alpha _{2},\\ldots ,X_{1},{\\mathcal {L}}_{Y}X_{2},\\ldots )-\\ldots ", "0745ba279fe361d42860dd5ee162cc0e": "\\mu \\approx 100", "0745dc526a8ba31c9f0b8565be0d8e94": "\\operatorname {sgn} (t):=\\left\\{{\\begin{array}{ll}{\\frac {t}{|t|}},&t\\in \\mathbb {C} \\setminus \\{0\\},\\\\0,&t=0.\\end{array}}\\right.", "074665b80ba529c5a84b2f98eb137e39": "U_{B}=\\{x\\in V:\\quad \\|\\varphi \\|_{B}<1\\},\\qquad B\\in {\\mathcal {B}},", "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}+(ra_{2})\\mathbf {e} _{2}+(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}{\\begin{cases}{\\begin{cases}{\\begin{cases}{\\begin{cases}{\\begin{cases}{\\begin{cases}{\\begin{cases}{\\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{ )}}\\\\{\\dot {z}}_{1}=z_{2}\\end{cases}}\\\\{\\dot {z}}_{2}=z_{3}\\end{cases}}\\\\\\vdots \\end{cases}}\\\\{\\dot {z}}_{i}=z_{i+1}\\end{cases}}\\\\\\vdots \\end{cases}}\\\\{\\dot {z}}_{k-2}=z_{k-1}\\end{cases}}\\\\{\\dot {z}}_{k-1}=z_{k}\\end{cases}}\\\\{\\dot {z}}_{k}=u\\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}", "074c14097a4cefef444cc434b45576f1": "3\\cdot a_{n}\\ \\mathrm {dB} ", "074c5003b6fd3a0d3c02d956852a529c": "\\mu *N", "074c699683f5efad06712c730018baa1": "P_{2}=(x_{2},y_{2})", "074c6a9b9b3d5052dae458c1c1cebef0": "\\sum _{i=1}^{n}a_{i}^{k}=b^{k}", "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}\\simeq 0.3", "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": "{\\frac {\\partial S}{\\partial \\alpha _{r}}}={\\frac {\\partial }{\\partial \\alpha _{r}}}{\\frac {1}{2}}\\sum _{k=1}^{N}m_{k}\\left|\\mathbf {a} _{k}\\right|^{2}=\\sum _{k=1}^{N}m_{k}\\mathbf {a} _{k}\\cdot \\left({\\frac {\\partial \\mathbf {a} _{k}}{\\partial \\alpha _{r}}}\\right)", "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={\\frac {1}{2}}log_{2}(RG)={\\frac {1}{2}}(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": "s_{t}\\equiv {\\vec {S}}_{t}\\cdot {\\vec {D}}=\\sum _{A=1}^{N}M_{A}\\;{\\vec {s}}_{t}^{\\,A}\\cdot {\\vec {d}}^{\\,A}=0\\quad \\mathrm {for} \\quad t=1,\\ldots ,6.", "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": "{\\begin{aligned}{\\overline {Y}}_{1}&={\\frac {1}{6}}\\sum Y_{1i}={\\frac {6+8+4+5+3+4}{6}}=5\\\\{\\overline {Y}}_{2}&={\\frac {1}{6}}\\sum Y_{2i}={\\frac {8+12+9+11+6+8}{6}}=9\\\\{\\overline {Y}}_{3}&={\\frac {1}{6}}\\sum Y_{3i}={\\frac {13+9+11+8+7+12}{6}}=10\\end{aligned}}", "0752312c070751129630dd26655e1933": "c=f\\lambda ", "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": "{\\begin{pmatrix}0&i\\partial _{0}+i\\nabla \\\\i\\partial _{0}-i\\nabla &0\\end{pmatrix}}{\\begin{pmatrix}\\Psi _{L}\\\\\\Psi _{R}\\end{pmatrix}}=m{\\begin{pmatrix}\\Psi _{L}\\\\\\Psi _{R}\\end{pmatrix}}", "075406f6eceea0908aea81e296465c89": "\\psi +\\theta +\\phi ={\\tfrac {\\pi }{2}}\\,", "0754dcc9db21c1211d42115b528100d5": "0U_{a})=\\Pr(\\alpha P_{i}+\\beta D_{i}+\\varepsilon _{i}>\\alpha P_{a}+\\beta D_{a}+\\varepsilon _{a})\\\\&=\\Pr(\\varepsilon _{i}-\\varepsilon _{a}>\\alpha P_{a}+\\beta D_{a}-\\alpha P_{i}-\\beta D_{i})\\end{aligned}}", "07629bad05308a4805729a54ee888b3e": "a_{6}=\\lfloor 2^{\\frac {1}{2}}\\rfloor =\\lfloor 1.414\\dots \\rfloor =1.", "07629de353237b95bd44b92e2b0b5a74": "A[i,j]+A[k,\\ell ]\\leq 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 \\mathbb {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}}{2T}}\\right).", "076504e491bcaa11b7f41e85cdb74513": "14_{11}\\ ", "07654ab947447d79c73ad4a85cd17a9b": "C_{T}^{(p)}(p,T)={\\frac {C_{T}^{(V)}(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": "q_{yy}={\\frac {\\sum (y-{\\bar {y}})^{2}I(x,y)}{\\sum I(x,y)}}", "0766a65e28d01fdaf213975cdc5bf122": "\\%C*=\\%C/6{\\mbox{ for }}\\%C\\geq 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_{t}f-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')x)\\,\\mathrm {d} x&\\mathrm {if} \\ \\mathrm {P} (g(X)\\geq g(a))=1\\\\g(b)-\\int _{-\\infty }^{b}g'(x)\\mathrm {P} (X\\leq x)\\,\\mathrm {d} x&\\mathrm {if} \\ \\mathrm {P} (g(X)\\leq g(b))=1.\\end{cases}}", "07849574dc0001eece45c0e0b066167d": "\\int {\\frac {x^{4}\\;dx}{s}}={\\frac {x^{3}s}{4}}+{\\frac {3}{8}}a^{2}xs+{\\frac {3}{8}}a^{4}\\ln \\left|{\\frac {x+s}{a}}\\right|", "0784bbbafde6513ff77e1888c5fa441a": "\\mathbf {e} _{2}", "0784be20ef0e37eb46ebd36b4a2bf6dd": "\\scriptstyle 1-\\varphi ={\\frac {1}{2}}(1-{\\sqrt {5}})", "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": "{}_{7}^{13}{\\text{N}}\\to {}_{6}^{13}{\\text{C}}", "0787060eb68af7205e261a6d1513aa89": "\\alpha <1", "078757e10e62f005ec259835c931b771": "\\Delta H_{c}^{\\circ }", "07876a8567eb7d684ce5172df8bd487f": "d_{2}={\\begin{bmatrix}-y\\\\x\\\\\\end{bmatrix}}.", "07877f1626ec59a1150e271a430efbe4": "M\\,ds{\\sqrt {v}}", "07878fc6547acfb9bcc8a958e91d7bc3": "c(M,N)=\\left({\\sum _{k=0}^{N-1}{\\binom {M+k}{k}}2^{k}}\\right)^{-1}\\ .", "0787dc3b9e4bf3484368a3902c5bbced": "\\lambda _{\\mathrm {chain} }={\\sqrt {\\tfrac {I_{1}}{3}}}~;~~\\beta ={\\mathcal {L}}^{-1}\\left({\\cfrac {\\lambda _{\\mathrm {chain} }}{\\sqrt {n}}}\\right)", "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 }\\geq k\\sigma _{\\alpha }{\\big )}\\leq {\\frac {1}{k^{2}}}", "078b98585c52531242818511c6b154bf": "i_{1}0,", "0791d0fd295994201f869fda975930d9": "\\omega \\,", "07924475c362d9ef2d50b8d84ec89d17": "p:=-{\\tfrac {1}{3}}\\,{\\text{tr}}({\\boldsymbol {\\sigma }})=-{\\frac {\\partial W}{\\partial J}}=-2D_{1}(J-1)\\,.", "0792dac93251d1a2c473f86682ca24a0": "T;Y\\,", "0792f1b63be29965d4194e3a82d304db": "f(x)=x^{n}+f_{n}x^{n-1}+...+f_{1}\\in \\mathbb {Z} [x]", "0792fbc0396ce564588008d7ddbac637": "Q={\\begin{pmatrix}0&0&-1\\\\0&J&0\\\\-1&0&0\\end{pmatrix}}", "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 \\,_{s}x}}", "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}}]=P(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}}_{n}^{f}=\\left[aA^{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_{n}^{(0)}(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 )[rx_{n}(1-x_{n})]_{s}+(1-\\epsilon )[rx_{n}(1-x_{n})]_{s-1}", "079b3b1d62f3e2cebd960448cef8350e": "\\Omega =\\arccos {{n_{x}} \\over {\\mathbf {\\left|n\\right|} }}\\ \\ (n_{y}\\geq 0);", "079b548a20f175cb786037c41c5a772e": "\\Delta G_{i}=\\sum _{j}\\gamma _{j}O_{j}~", "079bb003772b9993669167d6f942560e": "P(Xk{\\text{ and }}E(X^{2})\\lambda \\,\\mu (B).", "079d04bd53369a885a4b28fc72759de9": "f\\in k(x)", "079d0c77d14321410f742b4e6723f265": "y=1.9,\\ 3.7,\\ 5.8,\\ 8.0,\\ 9.6", "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": "\\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}\\,.", "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 }}", "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|\\leq 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 s3\\\\({\\frac {h_{M}}{3}})^{2}{\\mbox{ if,}}h_{M}\\leq 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": "\\quad (4)\\qquad \\epsilon (x,t)=\\sum _{m=1}^{M}e^{at}e^{ik_{m}x}", "07c5cd354f729bbd65ca75545e335213": "{\\begin{aligned}{\\frac {d\\phi }{dt}}&=-k(D-A)\\phi \\\\&=-kL\\phi ,\\end{aligned}}", "07c63f19fc3af989a2abc4d944cffb25": "M_{PL}={\\frac {M_{star}V_{star}}{V_{PL}}}\\,", "07c66863e9b22c9997ef6cfae0734f87": "BA=qAB", "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": "{\\begin{aligned}P_{0}^{0}(\\cos \\theta )&=1\\\\[8pt]P_{1}^{0}(\\cos \\theta )&=\\cos \\theta \\\\[8pt]P_{1}^{1}(\\cos \\theta )&=-\\sin \\theta \\\\[8pt]P_{2}^{0}(\\cos \\theta )&={\\tfrac {1}{2}}(3\\cos ^{2}\\theta -1)\\\\[8pt]P_{2}^{1}(\\cos \\theta )&=-3\\cos \\theta \\sin \\theta \\\\[8pt]P_{2}^{2}(\\cos \\theta )&=3\\sin ^{2}\\theta \\\\[8pt]P_{3}^{0}(\\cos \\theta )&={\\tfrac {1}{2}}(5\\cos ^{3}\\theta -3\\cos \\theta )\\\\[8pt]P_{3}^{1}(\\cos \\theta )&=-{\\tfrac {3}{2}}(5\\cos ^{2}\\theta -1)\\sin \\theta \\\\[8pt]P_{3}^{2}(\\cos \\theta )&=15\\cos \\theta \\sin ^{2}\\theta \\\\[8pt]P_{3}^{3}(\\cos \\theta )&=-15\\sin ^{3}\\theta \\\\[8pt]P_{4}^{0}(\\cos \\theta )&={\\tfrac {1}{8}}(35\\cos ^{4}\\theta -30\\cos ^{2}\\theta +3)\\\\[8pt]P_{4}^{1}(\\cos \\theta )&=-{\\tfrac {5}{2}}(7\\cos ^{3}\\theta -3\\cos \\theta )\\sin \\theta \\\\[8pt]P_{4}^{2}(\\cos \\theta )&={\\tfrac {15}{2}}(7\\cos ^{2}\\theta -1)\\sin ^{2}\\theta \\\\[8pt]P_{4}^{3}(\\cos \\theta )&=-105\\cos \\theta \\sin ^{3}\\theta \\\\[8pt]P_{4}^{4}(\\cos \\theta )&=105\\sin ^{4}\\theta \\end{aligned}}", "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\\geq 0}", "07c9080b749d0c13e4d837ebbbc9e37d": "B_{\\alpha \\beta }", "07c965672f4a5f68c7bd1e6ebcd41757": "\\Psi (x,y)=xu^{O(-u)}", "07c9cc926c59abb1d9faa3929434f9ee": "(n-1)\\times 1", "07c9d9baa051a6669920ce7cfdd6cca9": "Q_{0}^{2}", "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}n_{j,(\\cdot )}^{i}+\\alpha _{i}{\\bigr )}}{\\prod _{i=1}^{K}\\Gamma (n_{j,(\\cdot )}^{i}+\\alpha _{i})}}\\prod _{i=1}^{K}\\theta _{j,i}^{n_{j,(\\cdot )}^{i}+\\alpha _{i}-1}\\,d\\theta _{j}=1.", "07ca9562bbb1523000132740402e0821": "\\mathbf {A} =\\left[{\\begin{array}{ccc}1-2q_{2}^{2}-2q_{3}^{2}&2(q_{1}q_{2}-q_{3}q_{4})&2(q_{1}q_{3}+q_{2}q_{4})\\\\2(q_{1}q_{2}+q_{3}q_{4})&1-2q_{1}^{2}-2q_{3}^{2}&2(q_{2}q_{3}-q_{1}q_{4})\\\\2(q_{1}q_{3}-q_{2}q_{4})&2(q_{1}q_{4}+q_{2}q_{3})&1-2q_{1}^{2}-2q_{2}^{2}\\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 {1}}3.", "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": "V(r)={\\frac {mc^{2}}{2}}\\left[-{\\frac {r_{s}}{r}}+{\\frac {a^{2}}{r^{2}}}-{\\frac {r_{s}a^{2}}{r^{3}}}\\right]", "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": "{\\begin{aligned}s&=p_{1}p_{2}\\cdots p_{m}\\\\&=q_{1}q_{2}\\cdots q_{n}.\\end{aligned}}", "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} ={\\begin{bmatrix}(Q+a-1)&{\\frac {1}{3}}m&0&0\\\\m&Q&{\\frac {2}{3}}m&0\\\\0&{\\frac {2}{3}}m&Q&m\\\\0&0&{\\frac {1}{3}}m&Q\\end{bmatrix}}", "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_{v}T", "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": "H={\\frac {(l_{1})^{2}}{2I_{1}}}+{\\frac {(l_{2})^{2}}{2I_{2}}}+{\\frac {(l_{3})^{2}}{2I_{3}}}+mg(an_{1}+bn_{2}+cn_{3}),", "07d52077eaa5865dfc7121020bcf09c1": "{\\begin{array}{cccccc}g_{1}&=&Z&X&Z&I\\\\g_{2}&=&Z&Z&I&Z\\\\g_{3}&=&Y&X&X&Z\\\\g_{4}&=&Z&Y&Y&X\\end{array}}", "07d5364be7263d4eaad2c3f82df50154": "K=I\\otimes T", "07d593f0b25ba1a8bf43dac9a1d4d41f": "(x_{s},t_{s})\\,", "07d5c7099ff999998f0068b6b34ab6d4": "CIQ_{t}={\\mathcal {A}}e^{{\\mathcal {B}}t}", "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": "{\\begin{pmatrix}&h&\\\\[-0.9ex]v&&v'\\\\[-0.9ex]&h'&\\end{pmatrix}}", "07d935680b6501b2e42fe4baea021389": "mk", "07d9577618053507ed710ae0be8a4705": "n-m\\geq 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+hc)}\\psi (x+ha).", "07daf43d269c6cb7b45c16ca4062ceb6": "\\mathbf {z} =\\left\\{(x_{i},y_{i})\\in X\\times Y:i=1,\\dots ,m\\right\\}\\in Z^{m}", "07daff2abb9da1e1697b8a58798985ec": "V=\\pm {\\frac {fR}{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} )/H_{1}(M,\\mathbb {Z} )_{\\mathbb {R} }", "07df403dfe51db0194e6c677a582ab10": "\\int d[wx^{2}]=\\int x^{4}dx", "07df5771b077f4a06dd347b292015939": "R(x,y_{1},\\dots ,y_{n})", "07df8dc8930628c9016f6332f6edab8a": "{\\widetilde {\\theta }}={\\frac {\\exp {(-\\beta u)}-\\exp {(-\\beta u_{0})}}{1-\\exp {(-\\beta u_{0})}}}", "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 _{n}f_{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": "m_{1}\\;\\operatorname {sc} ^{2}(u)+m_{1}=m_{1}\\;\\operatorname {nc} ^{2}(u)=\\operatorname {dc} ^{2}(u)-m", "07e2f7f391b53640f096df4a27b66ce6": "{\\hat {t}}=\\operatorname {argmaxminlocal} _{t}(\\nabla _{norm}^{2}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_{c}Q_{s}}})\\,", "07e4b3f7df2f71b1c46fa47ce0f29f56": "f(\\mathbf {y} )+[J_{f}](\\mathbf {[x]} )\\cdot (\\mathbf {z} -\\mathbf {y} )=0", "07e4f44940d9af1fa9cc5954202a9b9e": "{\\boldsymbol {s}}=2K~\\left({\\sqrt {3}}{\\dot {\\varepsilon }}_{\\mathrm {eq} }\\right)^{m-1}~{\\dot {\\boldsymbol {\\varepsilon }}}_{\\mathrm {vp} }", "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\\geq 0", "07e5a4a56a57f5c874ebf79bb67a0b18": "\\mathbb {R} ", "07e5b3f26da912ff2b11115cfd81091d": "D_{2}=kTB_{2}(1+N_{2}{\\frac {dln\\gamma _{2}}{dlnN_{2}}})", "07e5b4c482edd88ba7d1c8cfd8d63fb5": "{\\frac {dA}{dz}}=-i\\gamma _{\\|}|A|^{2}A", "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 _{t\\Delta x}^{n}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}a_{\\text{x}}b_{\\text{x}}&a_{\\text{x}}b_{\\text{y}}&a_{\\text{x}}b_{\\text{z}}\\\\a_{\\text{y}}b_{\\text{x}}&a_{\\text{y}}b_{\\text{y}}&a_{\\text{y}}b_{\\text{z}}\\\\a_{\\text{z}}b_{\\text{x}}&a_{\\text{z}}b_{\\text{y}}&a_{\\text{z}}b_{\\text{z}}\\end{pmatrix}}", "07e9990f329c2a9a3d96a09c210f94e8": "v_{1,2}=5", "07e9b2c777ce42a404f6b03885773b7f": "\\tanh(\\alpha +\\beta )={\\tanh(\\alpha )+\\tanh(\\beta ) \\over 1+\\tanh(\\alpha )\\tanh(\\beta )}", "07ea01e1bb346a8adf43797c14bb5e5b": "{\\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\\,.", "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 \\leq 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": "\\tau _{m}=\\sigma _{m}\\sin \\phi +c\\cos \\phi ~.", "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": "d\\mathbf {f} ={\\boldsymbol {F}}\\cdot d\\mathbf {f} _{0}={\\boldsymbol {F}}\\cdot ({\\boldsymbol {S}}^{T}\\cdot \\mathbf {n} _{0}~d\\Gamma _{0})", "07ed7b7884737b80357da49facb87ff4": "n\\geq 4.", "07ed9c3f277abe2ae9ca8f83a9b87e83": "M_{2x}={\\dot {m}}V_{2x}=-\\rho QV_{2}\\quad and\\quad F_{P2x}={\\overline {P}}_{2}A_{2}", "07ed9cd92782b85be245409f24a9b337": "y(\\theta )=r(k-1)\\sin \\theta -r\\sin \\left((k-1)\\theta \\right).\\,", "07eda6dfa5faa951c2089ae9b256594b": "D\\,\\nabla ^{2}\\nabla ^{2}w=-q(x,y,t)-2\\rho h\\,{\\ddot {w}}\\,.", "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}", "07ee4e74e6c56bd8d51ed1a555cea2bc": "_{k}\\mathbf {b} _{l,m,n}=\\mathbf {S} _{k}\\mathbf {a} _{l,m,n}", "07ee679472c3e77e252a87bcef5a40f7": "{\\begin{aligned}\\omega _{1}&=\\omega -{e^{2} \\over 32\\pi \\varepsilon _{0}m_{e}\\omega Z^{3}},\\\\\\omega _{2}&=\\omega -{e^{2} \\over 16\\pi \\varepsilon _{0}m_{e}\\omega Z^{3}}.\\end{aligned}}", "07ee9b67a0557b8f091293637b1a079b": "(S,\\Sigma )", "07ef19656e5b0e7f28762bfaa1fb9ba8": "[i_{L_{1}},i_{L_{2}}]=i_{[L_{1},L_{2}]^{\\land }}", "07ef275540acce92238e509054c30393": "{\\overline {b}}=(B^{-1}a_{1}B,\\ldots ,B^{-1}a_{n}B)", "07ef7a9526aa22e94314110e1f000f61": "0,\\ldots ,n-1", "07ef82cb261e1d693985694652fda01b": "\\lbrace T\\rbrace ", "07ef8344f0d7f63f29cb988bff684c67": "\\Im z=0", "07efbca572b25c0069d4b524dd94a4a1": "H_{n+1}(x)=2xH_{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}\\approx [Cl^{-}]_{i}{\\text{ or }}K_{sp}10^{-b_{1}E_{i}+b_{0}}&{\\text{ when }}v_{0^{}}[Cl^{-}]_{0}>v_{i}[Ag^{+}]_{0}{\\text{ (before equivalence)}}\\\\=0&{\\text{ when }}v_{0^{}}[Cl^{-}]_{0}=v_{i}[Ag^{+}]_{0}{\\text{ (equivalence point)}}\\\\\\approx -[Ag^{+}]_{i}{\\text{ or }}-10^{b_{1}E_{i}-b_{0}}&{\\text{ when }}v_{0^{}}[Cl^{-}]_{0}^{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 \\leq 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": "k^{2}={\\frac {\\mu }{h^{2}}}-1", "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\\leq {\\frac {1}{2}}\\mathrm {deg} _{D}(K)={\\frac {1}{2}}K^{2}.\\,", "07ffec902fe52741a043f367f2489075": "(Y_{t})_{t\\geq 0}", "08001cb417ce6f4521f76272af06aa8a": "1/e.", "080035f725082c1785e2e7fb515ca7c2": "\\sigma =Y\\,\\epsilon \\,", "0800590145a98e0c3db79f9486ba4962": "u^{T}au>\\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_{2}y_{1}-z_{1}x_{2}x_{1}z_{2}}{(y_{2}^{2}+(z_{1}x_{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": "S(\\theta )={\\begin{bmatrix}\\cos \\theta &\\sin \\theta \\\\\\sin \\theta &-\\cos \\theta \\\\\\end{bmatrix}}", "0803326ac905a86dc32fd4241ce8ad64": "a+b^{2}x_{i1}+{\\sqrt {c}}x_{i2}", "080334dd7dda84677cf51ad5ef4b12b1": "{\\frac {1}{j!}}\\left({\\begin{matrix}j\\\\\\alpha \\end{matrix}}\\right)={\\frac {1}{\\alpha !}}", "08034666a8a0b35592ad928e7a6a6566": "[d(\\rho ,\\rho +d\\rho )]^{2}={\\frac {1}{2}}{\\mbox{tr}}(d\\rho G),", "0803cddddef0f826dc277274439946bf": "M(v)_{,\\,v}>0", "0803d4c8f7ddb5848750e3d993739400": "P\\left(C(\\eta )={\\frac {1}{P[\\eta _{t}(0)\\neq \\eta _{t}(1)]}}\\right)=1.", "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 mc^{2}-mc^{2}\\,", "080496e06f129f12b22f04cc2c63aded": "p={\\frac {m_{A}ng}{A}}", "0804e38d3286e2ba6ec104414c6acf76": "\\lambda =\\operatorname {lcm} (p-1,q-1)", "08051a547149d7059ecdb09c2aced7cb": "1\\leq \\phi (r)\\leq 2,\\left(r>2\\right)\\ ", "08051e685083ef235b8272a896fbb30c": "sm=0", "08052901962833a8403a21b0f8030372": "\\sigma ^{2}=k-\\mu ^{2}\\,", "080582af2aa04b597f3aaa921afb9034": "\\neg \\forall a,b,c:aRb\\wedge bRc\\Rightarrow aRc.", "0805acd495c11ae19f8559768abd03b4": "\\mathbf {F} '=\\mathbf {F} -\\mathbf {F} _{\\mathrm {app} }", "0805d97b0541722b463aa4b421226d5c": "\\mathbf {z} =\\mathbf {a} +F(\\mathbf {b} -\\mathbf {c} )", "08060285fdc836b29e6ee4d60c078b31": "\\lambda _{k}=\\min\\{\\max\\{R_{A}(x)\\mid x\\in U{\\text{ and }}x\\neq 0\\}\\mid \\dim(U)=n-k+1\\}", "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=\\left[{\\begin{array}{rrrrrrrr}-26&-3&-6&2&2&-1&0&0\\\\0&-2&-4&1&1&0&0&0\\\\-3&1&5&-1&-1&0&0&0\\\\-3&1&2&-1&0&0&0&0\\\\1&0&0&0&0&0&0&0\\\\0&0&0&0&0&0&0&0\\\\0&0&0&0&0&0&0&0\\\\0&0&0&0&0&0&0&0\\end{array}}\\right].", "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": "\\operatorname {Var} (x)={\\frac {(b-a)^{2}(3-2\\theta ^{2})}{36}}.", "080bc27fc4bde009fd8d81156bbbee28": "v_{g}=c\\left(n-\\lambda _{0}{\\frac {dn}{d\\lambda _{0}}}\\right)^{-1}.", "080be99436600b3a521fc139be91959e": "{\\begin{aligned}-i\\pi ^{2}&=\\left(\\int _{R}+\\int _{M}+\\int _{N}+\\int _{r}\\right)f(z)\\,dz\\\\&=\\left(\\int _{M}+\\int _{N}\\right)f(z)\\,dz&&\\int _{R},\\int _{r}{\\text{ vanish}}\\\\&=-\\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\\\\&=\\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\\\\&=\\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\\\\&=\\int _{0}^{\\infty }{\\frac {(\\log(x)+i\\pi )^{2}-(\\log(x)-i\\pi )^{2}}{(1+x^{2})^{2}}}\\,dx\\\\&=\\int _{0}^{\\infty }{\\frac {4\\pi i\\log(x)}{(1+x^{2})^{2}}}\\,dx\\\\&=4\\pi i\\int _{0}^{\\infty }{\\frac {\\log(x)}{(1+x^{2})^{2}}}\\,dx\\end{aligned}}", "080c1910f3ceddb0b77d33d1677d746f": "N=f_{\\textrm {1}}", "080c35b77a898fe3f4173f82c095f1e2": "\\lambda \\geqslant 0", "080c67fdb340842524d40951a9e00a01": "\\sum _{j=1}^{n}x_{ij}=1(i=1,2,\\dots ,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": "{\\frac {d}{dx}}\\ln _{k+1}(x)={\\frac {d}{dx}}\\ln(\\ln _{k}(x))={\\frac {1}{\\ln _{k}(x)}}{\\frac {d}{dx}}\\ln _{k}(x)=\\cdots ={\\frac {1}{x\\ln(x)\\cdots \\ln _{k}(x)}},", "080e36beb9cfa3069a1b88f5413e3b7c": "H={\\frac {\\left|\\mathbf {p} _{1}\\right|^{2}}{2m_{1}}}+{\\frac {\\left|\\mathbf {p} _{2}\\right|^{2}}{2m_{2}}}+{\\frac {1}{2}}aq^{2},", "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_{B}T_{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": "s=(b-a)/3,\\,", "0814a92bf0ef3dd45cc6d933ad7ef89c": "N(M)=\\{m\\geq 1|P_{m}(M)\\geq 1\\},", "0814f52f09d242777fb573267881b8c2": "P=K_{1}\\rho ^{5 \\over 3}", "08151ffc359809b80f90697c49d21a63": "\\theta _{eff}=\\cos ^{-1}(\\mathbf {\\hat {n}} \\cdot \\mathbf {\\hat {v}} ),", "08152191c9ff6afb0258f0cca95e8bee": "1.83040", "082699a68eb4e1ffa3bda5a8cb741212": "L_{n}^{\\alpha }(z)={\\frac {z^{-\\alpha }e^{z}}{n!}}{\\frac {d^{n}}{dz^{n}}}\\left(z^{n+\\alpha }e^{-z}\\right)={\\frac {\\Gamma (\\alpha +n+2)/\\Gamma (\\alpha +2)}{\\Gamma (n+1)}}\\,_{1}F_{1}(-n,\\alpha +1,z),", "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}+Ny_{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\\rightarrow S\\rightarrow T\\times S=t\\mapsto s\\mapsto (t,s)", "08288b8cef1b440b233e7b2aa8b74202": "{\\texttt {fix}}_{\\alpha }", "0828a0d0be30dd9628148fcafc9c67df": "\\sum _{n\\leq \\lambda }\\left(1-{\\frac {n}{\\lambda }}\\right)^{\\delta }={\\frac {1}{2\\pi i}}\\int _{c-i\\infty }^{c+i\\infty }{\\frac {\\Gamma (1+\\delta )\\Gamma (s)}{\\Gamma (1+\\delta +s)}}\\zeta (s)\\lambda ^{s}\\,ds={\\frac {\\lambda }{1+\\delta }}+\\sum _{n}b_{n}\\lambda ^{-n}.", "0828afe50f227250fa69a5f682bc0512": "\\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\\geq 1}{\\frac {(ia)^{n}}{n!n^{2}}}.", "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": "[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|\\leq 1\\}}", "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,", "082b0e41ac5d0f86aa6e51587580f3b7": "\\ln(x)", "082b7febbb152527ae1f05d1bbb8c49b": "\\Delta b_{T}", "082bd2666489a522185b37cc49581cb8": "{\\frac {2}{3}}\\times 2", "082be05223beba07f8c61abaf1f9f14b": "\\lambda _{j}", "082bf236294ff058c05fd953990bafb0": "m=2^{k}", "082c05cd77606b370a83c09a4a24e33e": "n_{\\max }", "082c80009e98668e0306f23c4ffac32a": "{\\mathcal {G}}(\\tau -\\beta )=\\zeta {\\mathcal {G}}(\\tau ),", "082cc266a32b400880939844673e26b8": "e>7.5n,\\,", "082ccfc0d25d2ed24504b86f937e4b22": "N(0)=N_{0}", "082d71cfb09c97b2c7c4dbc53d80bd8a": "dV={\\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 ", "082d758dbed4791e7613866dcd5ec11a": "\\operatorname {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}1&0\\\\\\\\0&1\\end{array}}\\right]-{\\frac {1}{2}}\\left[{\\begin{array}{rrr}1&1\\\\\\\\1&1\\end{array}}\\right]=\\left[{\\begin{array}{rrr}{\\frac {1}{2}}&-{\\frac {1}{2}}\\\\\\\\-{\\frac {1}{2}}&{\\frac {1}{2}}\\end{array}}\\right]", "082e734e8aa89c91fcc1a922d9d5adca": "-[R][R]=-{\\begin{bmatrix}0&-z&y\\\\z&0&-x\\\\-y&x&0\\end{bmatrix}}^{2}={\\begin{bmatrix}y^{2}+z^{2}&-xy&-xz\\\\-yx&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_{1}W_{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 0", "083f2607d761b21e45382928bd38a900": "145=12^{2}+1^{2}=8^{2}+9^{2}", "083fd908b329b68d7baceb6d862c0884": "\\mathbf {F} =\\langle F,\\leq ,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\\sum _{k=1}^{\\infty }{\\frac {1}{k^{3}\\sinh(\\pi k)}}-{\\frac {11}{2}}\\sum _{k=1}^{\\infty }{\\frac {1}{k^{3}(e^{2\\pi k}-1)}}-{\\frac {7}{2}}\\sum _{k=1}^{\\infty }{\\frac {1}{k^{3}(e^{2\\pi k}+1)}}.", "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 }={\\begin{bmatrix}h_{1}\\;h_{2}\\;\\dots \\;h_{n}\\end{bmatrix}}{\\begin{bmatrix}\\mathbf {e} _{\\pi (1)}\\\\\\mathbf {e} _{\\pi (2)}\\\\\\vdots \\\\\\mathbf {e} _{\\pi (n)}\\end{bmatrix}}={\\begin{bmatrix}h_{\\pi ^{-1}(1)}\\;h_{\\pi ^{-1}(2)}\\;\\dots \\;h_{\\pi ^{-1}(n)}\\end{bmatrix}}", "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 \\mathbb {N} .", "0842704f9234ba2e15fc47efddceecc5": "t'=t-{\\tfrac {vx}{V^{2}}}", "0842e280cea535a1d3a1cf35ffe3ab33": "A\\circ B=(A\\ominus B)\\oplus B,\\,", "0842f9b0000dee93b7b1847d9ee4ff10": "S_{s}", "084318b834ba16c555d4e360aa779fe3": "\\operatorname {arccsc}(-x)=-\\operatorname {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\\leq S\\leq 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))=(\\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": "\\,_{99}^{254}\\mathrm {Es} +\\,_{20}^{48}\\mathrm {Ca} \\to \\,_{119}^{302}\\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": "{\\underline {\\underline {\\mathbf {A} }}}={\\begin{bmatrix}A_{11}&A_{12}&A_{13}\\\\A_{21}&A_{22}&A_{23}\\\\A_{31}&A_{32}&A_{33}\\end{bmatrix}}~.", "08497086c0bdc9930f7bd56fb588aad2": "S_{mn}=S_{nm}\\,", "0849758f0b601830f420400199146924": "\\Delta p=p_{i,x}-p_{f,x}=p_{i,x}-(-p_{i,x})=2p_{i,x}=2mv_{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_{\\;\\;b}^{c}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 _{c}^{2}={\\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": "|X|E_{k}=\\sum _{i=0}^{n}q_{k}\\left(i\\right)D_{i}.\\qquad (7)", "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_{\\text{atm}}^{2}\\simeq 2.5\\times 10^{-3}\\,{\\mbox{eV}}^{2}", "08507709a2cd321cc65a78f8ba1ec1ba": "E[\\xi ]=\\int _{0}^{+\\infty }(1-\\Phi (x))dx-\\int _{-\\infty }^{0}\\Phi (x)dx", "0850f79692ab55eac984ab3f24553961": "{\\begin{aligned}&j=\\ell +s\\\\&j\\in \\{|\\ell -s|,|\\ell -s|+1\\cdots |\\ell +s|-1,|\\ell +s|\\}\\\\\\end{aligned}}\\,\\!", "08516cd80144421850e3d9e4a1b94afe": "\\nabla F=\\mu _{0}cJ", "085199cb15978a3d6a78c9457b9d493c": "{\\frac {Av}{\\|Av\\|}},{\\frac {A^{2}v}{\\|A^{2}v\\|}},{\\frac {A^{3}v}{\\|A^{3}v\\|}},\\dots ", "0852021ce522cd53d93bd5b14df57407": "