{ "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": "{\\hat {a_{1}}}=0.0135", "0852986d0ab05f20afeb2fa9d9aefe46": "f_{i}(r_{1},\\dots ,r_{i-1},0)", "0852b97b4adc36613e04de12789470ae": "(f*\\Delta )(x)=\\sum _{n=-\\infty }^{\\infty }f(x-n).", "0852f6aa94a4da23a34ea914a9cf154e": "L\\psi _{n}(x)=\\omega _{n}\\psi _{n}(x)", "08530922c2c40dd06bb8e686fa483d8a": "\\mathrm {Eq} (f,g):=\\{x\\in X\\mid f(x)=g(x)\\}{\\mbox{.}}\\!", "08533b111de373fc0db9daf66054c188": "z_{1}=x_{1}y_{1}-x_{2}y_{2}-x_{3}y_{3}-x_{4}y_{4}+u_{1}y_{5}-u_{2}y_{6}-u_{3}y_{7}-u_{4}y_{8}", "08535a977c369045347032867917dc94": "P^{-1}{\\mathcal {F}}P", "08542bb73cc183c2d33c10db7cd7cabf": "P={\\begin{bmatrix}{\\frac {2}{right-left}}&0&0&-{\\frac {right+left}{right-left}}\\\\0&{\\frac {2}{top-bottom}}&0&-{\\frac {top+bottom}{top-bottom}}\\\\0&0&{\\frac {-2}{far-near}}&-{\\frac {far+near}{far-near}}\\\\0&0&0&1\\end{bmatrix}}", "0854472019d887246601ca81de5b0db2": "M(bx_{1},\\ldots ,bx_{n})=bM(x_{1},\\ldots ,x_{n})", "085457224bf55d74398d25780ba8fd76": "\\int _{a}^{b}{\\sqrt {1+{\\bigg (}{\\frac {dy}{dx}}{\\bigg )}^{2}}}\\,dx,", "08547d225b1df1e24d9be337f9e8af3e": "xy^{-1}", "0854dfc743e3e4a31e7518f7d497e948": "i=2i+2\\left(-{\\frac {1}{2}}i\\right)=10.2_{2i}", "0855377f16232c112cc7334456e21ad1": "x_{i}=r_{i}^{-\\beta }", "08555285a6e8334c4942b72b27d88873": "\\ f*g", "085555ba4b282a05cc52051f8edcd94e": "G/\\tau =K_{I}", "085611d3a8521b914adcd17e88f6a519": "EW=0", "0856444a7eabf3601113168835553a3e": "d\\Omega _{k^{\\prime }}", "08570ba4434de2bfe6476ed45505337d": "{\\mathcal {H}}\\subset {\\mathcal {K}}", "08571639e1475d1bb81ae83fd725b12b": "N\\in {\\mathcal {F}}", "08577d8512a05cc521fbf111658ff6c0": "\\!g_{i}", "0857bf5d4e53bc8479a34538b4bdc91e": "\\!c", "0857c3604f27a1e80c6f56addbaa2d84": "UltOsc=100\\times {4\\times avg_{7}+2\\times avg_{14}+avg_{28} \\over 4+2+1}", "0857c70c656c7aaf930e3277a1139c94": "\\scriptstyle dp_{B}(R,t)", "0857e7053d4575070702ecd46cf668f7": "\\lVert x-y\\rVert ", "085827177b928f9c0eff1964846cb77b": "0<\\alpha \\ <2^{160}\\,", "0858850a595e42d35f5eeb6b1c650e70": "\\operatorname {Hdg} ^{*}(X)=\\sum _{k}\\operatorname {Hdg} ^{k}(X)\\,", "0858a3974acfeb642f2364ee49fb05c2": "\\Pr[c\\in B(y,pn)]=\\Pr[y\\in B(c,pn)]=\\mathrm {Vol} (y,pn)/q^{n}\\geq q^{-n(1-H_{q}(p))-o(n)}\\,", "0858c12dc07329bf7e1c4bad5c47cd67": "{s}", "0858c8578fe0a24f05bb5502453ed3c0": "A_{c,b}", "0858eea75fa6bc1ea46d85bbe8f3c291": "-\\omega _{n}^{2}f(t)q", "085967d7b3ee450c85364a7461e22302": "u_{2}(z)", "085a08ff09618180f9c43a171281093c": "-{\\tilde {J}}=M\\nabla (\\mu _{a}-\\mu _{b})", "085a49f2fa9d72fbe2d5d4073bfa1677": "{\\tilde {S}}_{2}=M_{2}-m_{2}+{\\frac {i}{2}}G\\gamma _{1}\\cdot {\\partial }{\\mathcal {L}}{.}", "085a66174db412c03f8c7e11e844c21e": "\\lim _{t\\to \\infty }{\\frac {1}{t}}g(t)={\\frac {\\mathbb {E} [W_{1}]}{\\mathbb {E} [S_{1}]}}.", "085abc1a4ea4269aa59a73c8ecb59330": "B_{MX}^{\\phi }=\\beta _{MX}^{(0)}+\\beta _{MX}^{(1)}e^{-\\alpha {\\sqrt {I}}}.", "085b40c8e85992b9b35eb73c29a5798e": "\\oint _{K}\\kappa \\,ds>4\\pi .\\,", "085b91511496a5a05a5ac52d40c9489e": "DL_{j}", "085bee4f4ccd429ad60e9e96744839c3": "|\\Psi _{m}^{p}\\rangle ={\\mathcal {A}}(\\phi _{1}(\\mathbf {r} _{1}\\sigma _{1})\\phi _{2}(\\mathbf {r} _{2}\\sigma _{2})\\cdots \\phi _{p}(\\mathbf {r} _{m}\\sigma _{m})\\phi _{n}(\\mathbf {r} _{n}\\sigma _{n})\\cdots \\phi _{N}(\\mathbf {r} _{N}\\sigma _{N})),", "085c19cf432cbcb959b468a6924bcb61": "L(p;q_{1},\\ldots q_{n})", "085c57e6606d29c7e177ef0385d05a40": "\\Psi _{A}(x)=C_{A}\\Psi _{0}(x-x_{A})", "085c9b94c5877f6967fe47e0253394c7": "\\{e_{3}\\equiv z_{yy}+{\\frac {1}{y^{2}}}(xy^{3}-x^{2}-y)z_{y}-{\\frac {1}{y}}(x^{3}-x+y)z=0,e_{2}=z_{x}+{\\frac {1}{y}}z_{y}+xz=0\\}.", "085c9f0df11642cf704f40ffdf753055": "s'", "085ca404311ed2fa47059761173a876d": "x,y\\in \\Sigma ^{*}", "085cfa49aa86db198f20d193d5080c92": "\\phi (a\\mathbf {x} +b\\mathbf {y} ,c\\mathbf {x} +d\\mathbf {y} )={\\frac {1}{|ad-bc|}}\\phi (\\mathbf {x} ,\\mathbf {y} ),", "085d1f59b60f6655415fb3f3b13a4d44": "\\gamma \\in \\{2^{-15},2^{-13},\\dots ,2^{1},2^{3}\\}", "085d63c9f7b2e59353b8432c9b054d04": "P\\to (f(U)=f(V))\\,", "085d6e33ee4c55f4e65be030cd507c17": "o(w)", "085da8820e7465711a2a2a5d30250753": "\\Delta ({\\tilde {w}},w')\\leq t", "085dafbb0398cf2b6ab06d53c6987444": "e(n)=d(n)-{\\hat {d}}(n)", "085de9be16c6e59a32235966c7b6cbe3": "F_{L}=\\textstyle {\\frac {1}{5}}k^{4}\\left(5L_{A}\\right)+\\textstyle {\\frac {1}{10}}{(1-k^{4})}^{2}{\\left(5L_{A}\\right)}^{1/3}", "085df8dcd78197333723d2f5b9024b43": "x=R\\lambda /{\\sqrt {2}}", "085e2dcea21c77cf0eef6e999639ee33": "q<1", "085e6f6c82be4cd865387cd12742ea9b": "\\alpha _{i}=2\\cdot \\pi {\\frac {iK}{N}}", "085f012b4860c10d49cea9e685eda831": "BD", "085f471060f928587ef77d613de7a6ab": "E_{CFG}=\\{\\langle G\\rangle \\mid G{\\text{ is a CFG and }}L\\left(G\\right)=\\emptyset \\}", "086052c76ef59b2fb7311dcf92821286": "h_{crit}=\\left({\\frac {9B^{2}}{4}}\\,{\\frac {EI}{\\rho g\\pi r^{2}}}\\right)^{1/3}", "0860a2c408a30ca4bbdd158c6798a132": "A={\\frac {W}{(L)(U_{w})}}", "0860b4bbf2e14b710c80e7a4e35aada9": "{\\dot {Q}}^{\\mathrm {T} }Q+Q^{\\mathrm {T} }{\\dot {Q}}=0", "08610ec20d070997513de6c2ac0f5571": "y=b", "08618cc058c79edd11827393563f4a4f": "P_{i}'", "086192d60aa50b35adfeb8d6794eb4b5": "R({\\vec {x}})\\approx \\tanh p\\,", "0861a74c12b96b48f283c77d29514cda": "W=A^{-1}", "0861adc3d3e17d9c558f0b8939514aee": "{\\bar {n}}_{i}={\\frac {\\displaystyle \\sum _{n_{i}=0}^{1}n_{i}\\ e^{-\\beta (n_{i}\\epsilon _{i})}\\quad \\sideset {}{^{(i)}}\\sum _{n_{1},n_{2},\\dots }e^{-\\beta (n_{1}\\epsilon _{1}+n_{2}\\epsilon _{2}+\\cdots )}}{\\displaystyle \\sum _{n_{i}=0}^{1}e^{-\\beta (n_{i}\\epsilon _{i})}\\qquad \\sideset {}{^{(i)}}\\sum _{n_{1},n_{2},\\dots }e^{-\\beta (n_{1}\\epsilon _{1}+n_{2}\\epsilon _{2}+\\cdots )}}}", "0861e8ef3451e1ec1608f8d18768dc74": "p\\to q", "08622ab1ba41cb55303f381dc2f38327": "y(x^{2}+y^{2})=b(x^{2}-y^{2})+2cxy", "08625098ce1218d596df0cb7a2ff1f49": "S_{0}(p)={\\begin{bmatrix}(I_{x}(p))^{2}&I_{x}(p)I_{y}(p)&I_{x}(p)I_{z}(p)\\\\[10pt]I_{x}(p)I_{y}(p)&(I_{y}(p))^{2}&I_{y}(p)I_{z}(p)\\\\[10pt]I_{x}(p)I_{z}(p)&I_{y}(p)I_{z}(p)&(I_{z}(p))^{2}\\end{bmatrix}}", "0862609ee90f693442fc43e0c9f758bb": "Z=V", "0862629191318ef2d28418cacf18bcd6": "5F_{6}^{2}=320\\equiv -5{\\pmod {13}}\\;\\;{\\text{ and }}\\;\\;5F_{7}^{2}=845\\equiv 0{\\pmod {13}}", "08627f02774fb21464ae91032340eba4": "(\\nabla _{X}Z+(I-\\Delta S)X)+(h(X,Z)+d_{X}\\Delta ){\\mathbf {A}}=0,", "0862c73df2fef82eab278377b6859017": "\\eta _{B}", "08630453b7b0223407e9bd890d7ae35a": "H_{n}=\\sum \\limits _{l=1}^{N}K_{1}^{[l]}+\\sum \\limits _{l=1}^{N}K_{2}^{[l,l+1]}.", "0863360a323a329bc6470dbba564fa56": "K\\;", "086337e68e2d113e81e0e9db7655844e": "{\\frac {\\pi }{\\sqrt {12}}}\\approx 0.9069.", "08634f6ade2e4ad977a9dd7fe3172646": "\\Delta S=\\alpha k_{B}\\ln N\\,", "08639b03744654d4b36f1f47b913755a": "{\\mathbf {r}}", "08639c77b9ec5ddb0c6e94cb46ee0eec": "\\scriptstyle A_{\\parallel }", "0863e4a2343c010d5ccfdde4d7a8c0f1": "\\rho _{A}=\\operatorname {tr} _{B}\\rho _{AB}", "0863f741d7093a2bec6ece488bdae3b0": "y\\cup \\{y\\}", "086403e136c6d79c35384e36319d7181": "c^{2}=ac\\cos \\beta +bc\\cos \\alpha .\\,", "08641454a6623d6aaaf7653303d81903": "\\operatorname {Re} (\\epsilon (\\mathbf {r} ,\\omega ))", "08643b522426f00d876b6175c5125f5e": "S\\in {\\mathcal {A}}(G)", "08644c3901ca0610e482e615b7c82f5e": "\\sum _{a_{i}\\in A}\\phi (a_{i})=0", "08647d7f90875053446ea8763b27d956": "{\\boldsymbol {\\beta }}", "08648e92de68de9b39c67d08bb3a28a3": "*_{N}", "0864a3f53947fcdb99026514b527fb90": "H\\left(f\\right)*{\\frac {1}{T_{s}}}\\sum _{k=-\\infty }^{+\\infty }\\delta \\left(f-{\\frac {k}{T_{s}}}\\right)=1", "0864b70f0056f87446ef403e849a2014": "T_{0},", "0864eb673fda276a4f3a8813aa282976": "\\pi _{\\omega }(x)\\xi _{\\omega }\\mapsto \\pi _{\\phi }(x)\\xi _{\\phi }\\oplus \\pi _{\\psi }(x)\\xi _{\\psi }.", "08650fba58a14f3bc4c15c1bff29986f": "\\int \\rho {\\big (}\\left\\Vert \\mathbf {x} -\\mathbf {c} _{i}\\right\\Vert {\\big )}\\,d^{n}\\mathbf {x} =1", "0865a01dd577912c2e853f4fa232e797": "{\\mathcal {M}}_{1}", "0865a90eb4b59b80ad40594550d43e52": "d\\approx 3.57{\\sqrt {h}}\\,,", "0865ac8f14c29e628449d4808fc6a2e4": "{\\begin{matrix}{\\frac {1}{2}}\\end{matrix}}mv^{2}=gmr.", "0865b7b5d73f8d2d267b2b4d48b4d47b": "X=-\\left\\langle {\\frac {dE_{r}}{dx}}\\right\\rangle \\,", "0865d6c4efb551567c3c2cab22e94bd7": "\\langle introd(j,b,x),s\\rangle ", "0865ee2ee41e748e0ab653b785ea92c3": "\\mathrm {high} =R_{1}C\\cdot \\ln \\left({\\frac {2V_{\\textrm {cc}}-3V_{\\textrm {diode}}}{V_{\\textrm {cc}}-3V_{\\textrm {diode}}}}\\right)", "0865fa8ab44cc8cbe3f68273fd2c03c4": "|G_{a}(z)|<(M+1)\\epsilon \\!", "086675e64252d1197dee3eebd34dec1e": "{\\hat {H}}", "0866827b2b471138b8f46081602e57ab": "E_{1}=q_{1}^{2}-p_{1}r_{1}=0.00000\\,", "0866b23e49b4cf45803e32e679ebed22": "l={\\frac {\\hbar }{mc}}", "0866b33bb22a761df4df09be6ddacbae": "{4\\pi \\over c}j^{\\beta }=\\partial _{\\alpha }F^{\\alpha \\beta }+{\\Gamma ^{\\alpha }}_{\\mu \\alpha }F^{\\mu \\beta }+{\\Gamma ^{\\beta }}_{\\mu \\alpha }F^{\\alpha \\mu }\\ {\\stackrel {\\mathrm {def} }{=}}\\ \\nabla _{\\alpha }F^{\\alpha \\beta }\\ {\\stackrel {\\mathrm {def} }{=}}\\ {F^{\\alpha \\beta }}_{;\\alpha }\\,\\!", "0866ee36a47474c5ca2fb1811dffef82": "GF(p^{t})", "08670fd819285e6e4639e2541972b049": "\\quad \\beta _{n}\\sim f(\\beta _{n}|\\theta )", "086730665882e9b8832ad63899709bf7": "D(\\xi _{a})D(\\xi _{b})=D(\\xi _{a}\\xi _{b}).", "086747ee9945206e6ea6d9ab022b125f": "F=W_{1}\\cdot W_{3}=\\sum _{j=1,3}W_{i}", "08681c27ad524a8e515a19b4d3de2807": "\\int {\\frac {x+A}{\\sqrt {x^{4}+ax^{3}+bx^{2}+cx+d}}}\\,dx", "086834a5ae253549ee9026fec8d279eb": "x^{\\lambda }=e/x\\qquad x^{\\lambda }x=e", "0868420816995522683147f384d56ac3": "{\\begin{aligned}&{\\hat {\\beta }}_{1}={\\frac {s_{yy}-\\delta s_{xx}+{\\sqrt {(s_{yy}-\\delta s_{xx})^{2}+4\\delta s_{xy}^{2}}}}{2s_{xy}}}\\\\&{\\hat {\\nu }}_{1}={\\frac {-1}{{\\hat {\\beta }}_{1}}}={\\frac {-2\\delta s_{xy}}{s_{yy}-\\delta s_{xx}-{\\sqrt {(s_{yy}-\\delta s_{xx})^{2}+4\\delta s_{xy}^{2}}}}},\\\\&{\\hat {\\beta }}_{0}={\\overline {y}}-{\\hat {\\beta }}_{1}{\\overline {x}},\\\\&{\\hat {x}}_{i}^{*}=x_{i}+{\\frac {{\\hat {\\beta }}_{1}}{{\\hat {\\beta }}_{1}^{2}+\\delta }}(y_{i}-{\\hat {\\beta }}_{0}-{\\hat {\\beta }}_{1}x_{i}).\\end{aligned}}", "08684be4f5f5c73459c888f3b47527ac": "p_{11}/(p_{11}+p_{10})", "086886c7642c35706d22fd59d30d2689": ":{\\hat {b}}_{1}^{\\dagger }\\,{\\hat {b}}_{2}\\,{\\hat {b}}_{3}:\\,={\\hat {b}}_{1}^{\\dagger }\\,{\\hat {b}}_{2}\\,{\\hat {b}}_{3}", "0868a10a419e0f3f0833aa59fe8330a5": "\\mu =G(m_{1}+m_{2})", "0868e6113e0afe135438eef879d719e6": "{\\mathit {l_{j}}}\\,", "0868fa8e93f62f819441deb81c05a85f": "x'_{i}\\rightarrow x_{i}", "0869167f15b8c06f68939754920c6817": "p-p_{0}=c_{0}^{2}(\\rho -\\rho _{0})", "0869348ba87ca5f6cefc2e30ddb36314": "F(z,m)=\\sum _{k=0}^{\\infty }f(kT+m)z^{-k}", "086992e911ad666940c60fac809643a8": "V_{g}f\\in L_{m}^{p,q}(\\mathbb {R} ^{2d})", "0869965a2e1546a7606aff31a1f767f4": "y={\\frac {1}{1+e^{-x}}}", "086a625a452572a6878a544d30624348": "P(x)=\\sum _{j=0}^{n-1}u_{j}x^{j}", "086ad3563a5e862713aaf1cd8fd6f466": "(M\\ /\\ s)\\ /\\ t=M\\ /\\ (ts)", "086adc074fc09817139a0e2cd17879c9": "p({\\vec {\\theta }})\\propto {\\sqrt {\\det I({\\vec {\\theta }})}}\\,", "086b8289e29be5115bbfd329f2433486": "{\\text{precision}}={\\frac {\\text{number of true positives}}{{\\text{number of true positives}}+{\\text{false positives}}}}", "086beb6a6c8a029942238364e5a8beab": "(X,d)", "086c08f4887854fbb8ac4b1dd3d58ee1": "x=R\\cos(\\theta )", "086c1c983eef5f4ec1e9183b80dea13e": "{\\tilde {x}}", "086c62f0fe0c5804428eb0133152fd95": "{\\frac {\\theta }{(I)_{J}}}{J \\choose n}\\int _{0}^{1}(Ix)_{n}(I(1-x))_{J-n}{\\frac {(1-x)^{\\theta -1}}{x}}\\,dx", "086cebacd2b29692f317fd1d9adc9b91": "\\alpha ^{\\vee }={2 \\over (\\alpha ,\\alpha )}\\,\\alpha .", "086d4c3a2db4cd707b5e03137be03b8e": "y^{G}=f(k^{G})", "086e143ecfedac00e31a515d138d3a3c": "\\sum \\limits _{i=1}^{\\infty }\\mathrm {P} (X\\geq i)=\\sum \\limits _{i=1}^{\\infty }\\sum \\limits _{j=i}^{\\infty }P(X=j).", "086e1709ae075fde91365e73006c78d5": "\\left({\\frac {F/\\mathbf {Q} }{(n)}}\\right)=\\left({\\frac {L/\\mathbf {Q} }{(n)}}\\right){\\text{ (mod }}H).", "086e1e913727da2fa28e31c854b859f4": "U(x,\\omega )=P(x,\\omega )A(x,\\omega )", "086e77c05b32e57ee30727de66af5caf": "\\mathrm {ERB} (f)=6.23\\cdot f^{2}+93.39\\cdot f+28.52", "086ee2d9b611075135d32fe2a9b5bf61": "a+b'x_{i1}+c'x_{i2},", "086efc867d6a52e34eb88795dbb61efc": "\\gamma _{\\mathrm {SG} }", "086f2a64fea3650776476f320fc982ec": "T(n)=2^{n}T\\left({\\frac {n}{2}}\\right)+n^{n}", "086f5e4bee2482de92c8baefce510f22": "{\\mathcal {F}}(t_{m})", "086f5ebd6937c1a196ff9cde7fbace81": "E>mgl", "086f75fe0a018304ff2817a784eba8ed": "\\log(\\gamma )", "086fce0618cbc52b2baecb5059e0db1f": "\\Diamond \\varphi ", "086fe0b2a0abd7dd3c433deb189569f9": "{\\begin{aligned}{\\text{N}}_{\\text{s}}\\omega _{\\text{s}}+{\\text{N}}_{\\text{p}}\\omega _{\\text{p}}-({\\text{N}}_{\\text{s}}+{\\text{N}}_{\\text{p}})\\omega _{\\text{c}}&=0\\\\{\\text{N}}_{\\text{a}}\\omega _{\\text{a}}-{\\text{N}}_{\\text{p}}\\omega _{\\text{p}}-({\\text{N}}_{\\text{a}}-{\\text{N}}_{\\text{p}})\\omega _{\\text{c}}&=0\\end{aligned}}", "086fe4af32eaa0d8e320267ad0ffefa7": "T:{\\mathcal {F}}\\to {\\mathcal {F}}", "087017cefe10d64b6f95a15506032319": "m_{1}=[12.3,7.6]+[1.697,0]=[13.997,7.6]", "08703596bd26613a0f2d79d0a2c03428": "f(z)={\\frac {e^{tz}-Q_{t}(z)}{P(z)}}", "0870b9f545ddc94563bcca5270b07e1e": "a_{5}=\\lfloor 6^{\\frac {1}{2}}\\rfloor =\\lfloor 2.449\\dots \\rfloor =2,", "0870f159f171f243deba0d2ee71655e7": "\\tan {\\frac {\\theta }{2}}", "08718e42cb5b04f1489f7b5d86544ed3": "x/\\epsilon c", "087195bbcff743d55cb9fc7d061ef781": "{\\widehat {\\varepsilon }}_{i}=X_{i}-{\\overline {X}}.", "0871c793427749c626b5771230f271a8": "\\left\\{{\\begin{array}{ll}1&n=1\\\\2&{\\text{otherwise}}\\end{array}}\\right.", "0871f51ba0068ce4fdbe0e593444563c": "X_{n}\\ \\xrightarrow {\\mathcal {D}} \\ X,", "087249e982bffce2426824d3de6de8e5": "\\mathrm {conv} ", "08727052444f8cdf3e308d5863a722a9": "I_{m,n}=\\int \\sin ^{m}{ax}\\cos ^{n}{ax}dx\\,\\!", "08728ea65350ce111f5ae89cda8832b8": "f\\left(x_{0}\\right)=px_{0}+b", "0872b027f615858ceefa69208cceab47": "N_{2}+8H^{+}+8e^{-}\\to 2NH_{3}+H_{2}", "0872bf33180c1f89aa2fb8a0bfb5afc9": "{EF={\\frac {Q_{e}}{Q_{e}+Q_{h}}}={\\frac {1}{1+B}}}", "0872cfce743c4aa442423f20f85c2437": "\\langle P\\rangle \\,\\!", "0872d8a6ea05decf0ebfefd6837d7da0": "x^{2}+bx+c\\,=\\,\\left(x+{\\tfrac {1}{2}}b\\right)^{2}+k,", "0872fbbc55e4fdf5b6840c958be53aa5": "F_{2}(r)=F_{1}(r)+{\\frac {L_{1}^{2}}{mr^{3}}}\\left(1-k^{2}\\right)", "087319626a720e82ab10f4322bb92895": "c_{v}={\\frac {R}{\\gamma -1}}", "0873225a6aae94dfc0be630f38c83ec3": "y^{5}+y^{4}-24y^{3}-17y^{2}+41y-13", "087322ef246341ad8463c6eb80264c15": "x_{crit}", "08732b8ff0ad81ffc9a3105e18d3e5d3": "\\sigma _{A,B}^{\\dagger }=\\sigma _{A,B}^{-1}:B\\otimes A\\rightarrow A\\otimes B", "087368f6ea9528420df49d2316f9b888": "\\scriptstyle \\land ", "08736f4045e96abf886c45562b8e98a1": "\\scriptstyle {\\epsilon ={\\sqrt {1-v^{2}/c^{2}}}}", "087396a1975686bdeec8258553c79390": "x=a_{0}+{\\frac {1\\mid }{\\mid a_{1}}}+{\\frac {1\\mid }{\\mid a_{2}}}+{\\frac {1\\mid }{\\mid a_{3}}},", "0873dec567d24ed7d6c14976d4a0adb4": "f:V\\to V", "0874064e73f2352985b5c8fc7331051d": "m=0,1,\\dots ,", "0874289dabdd17e6c3fdce45ccbd973f": "\\Lambda ^{n}A:\\Lambda ^{n}V\\rightarrow \\Lambda ^{n}V", "0875135b6dfecc84cfb62efcdd77c8fc": "\\scriptstyle a=\\partial u/\\partial x=\\partial v/\\partial y", "0875ba40ea78e27c50780ff19d1ceb10": "z=(z_{1},z_{2},\\ldots ,z_{n})", "08760f01ea0d9b37e4e4e6f8686ebe01": "\\scriptstyle {\\frac {\\sqrt {3}}{9}}\\,\\sum \\limits _{n=0}^{\\infty }{\\frac {(-1)^{n}}{27^{n}}}\\,\\left\\{\\!{\\frac {18}{(6n+1)^{2}}}-{\\frac {18}{(6n+2)^{2}}}-{\\frac {24}{(6n+3)^{2}}}-{\\frac {6}{(6n+4)^{2}}}+{\\frac {2}{(6n+5)^{2}}}\\!\\right\\}", "0877834805793b7a7f0c4bfb9cb88b1f": "{\\frac {1}{iz}}\\,dz=dt.", "0877e3a4e881d97aeb3bba90f55bae18": "\\langle U(x+y)-(Ux+Uy),U(x+y)-(Ux+Uy)\\rangle =0", "0877ed54238aefb676d353b0f3eb6615": "x_{i}\\,", "087847d30cf204fc8cdd76973b6dcc15": "k_{xo}=k_{o}{\\sqrt {1-({\\frac {m\\pi }{k_{o}a}})^{2}-({\\frac {k_{z}}{k_{o}}})^{2}}}\\ \\ \\ \\ \\ \\ \\ (26)", "08788987bf75976c364309a002529686": "d\\tau ={\\sqrt {\\left(1-1.3908\\times 10^{-9}\\right)dt^{2}-2.4069\\times 10^{-12}\\,dt^{2}}}=\\left(1-6.9660\\times 10^{-10}\\right)\\,dt.", "08789c1c789d3a0d70d6883cb62aa942": "\\pi _{jt}", "0878be8a1c8c0a60e498a63a19d7aed5": "\\displaystyle 5^{2}+12^{2}=13^{2}\\,.", "087932e8c5faf7ec468c1c2f3a058aaf": "{3 \\over 2}\\cdot {3 \\over 2}\\cdot {3 \\over 2}\\cdot {3 \\over 2}\\cdot {8 \\over 5}\\cdot {1 \\over 2}\\cdot {1 \\over 2}\\cdot {1 \\over 2}={81 \\over 80}", "0879894f2225d8ae7eb9b1b1a3d76aaf": "d_{1}>d_{2}", "087a00dfeebaed1d79009912a028e5bc": "={\\frac {1}{331}}", "087a6d79325f7cc88bdaa3ca5e84acc2": "\\ell ^{\\infty }(\\mathbb {R} )", "087ae63322dcc247674a04de75f515a8": "\\iota ^{*}:H^{*}(E)\\longrightarrow H^{*}(F)", "087b4c104aff1c97fb230dc27131f9f8": "a_{0}={\\frac {\\hbar ^{2}}{m_{e}e^{2}}}", "087b7ad7a6693503cc710a0bbd54e040": "\\epsilon _{H}", "087b8782dd88a72e599d2867ec8e4ecc": "\\mu _{\\max }\\leq 1-\\varepsilon ", "087b9c21b7033fa2e1a50634d25f400a": "N\\in \\mathbb {Z} ^{+}", "087bc030c1ad70f4a5b42c443dd2a3fe": "S_{0}=\\left|{\\mathcal {F}}\\left[{\\frac {dW(t)}{dt}}\\right](\\omega )\\right|^{2}={\\text{const}}", "087bd5f8d197b0f1b129664107c086c2": "g_{\\mathrm {e} }=2.0023", "087bd6c601163c61803bff1872872a6b": "\\ c_{1}=c_{01}(1-y_{d})+c_{11}y_{d}", "087c213852df24f142c921a142fa2f59": "M,N", "087c3e00f68962d086837cec05e6b1fb": "H_{0}^{2}(\\Omega )", "087c5d667ebfa8eb0281e46f055f2764": "\\partial _{\\mu }A^{\\mu }\\equiv A^{\\mu }{}_{,\\mu }=0\\!", "087cea17dd3a808fc89b1a1de530a1f4": "2\\pi \\left(1+\\cos {\\theta }\\right)", "087cf94aefdf5f470e33d919960c43b5": "a=15-15i", "087d07c94a3cce22b1dd9f377e5b0315": "k,~k_{e}", "087d27d1a0b418fadacfdf9521391cee": "{\\text{Passer Rating}}_{\\text{NCAA}}={(8.4\\times {\\text{YDS}})+(330\\times {\\text{TD}})+(100\\times {\\text{COMP}})-(200\\times {\\text{INT}}) \\over {\\text{ATT}}}", "087d28b05c8f87a4bb020c3ccf563f23": "Y_{i}\\sim \\mathrm {Pois} (\\lambda \\cdot p_{i}),\\rho (Y_{i},Y_{j})=0", "087d30f0f21740efad9b744a16ef9885": "{\\begin{cases}{\\text{Mesh 1, 2: }}-V_{s}+R_{1}I_{1}+R_{2}I_{2}=0\\\\{\\text{Current source: }}I_{s}=I_{2}-I_{1}\\end{cases}}\\,", "087df3bfe54732c537c1397765620510": "Lz\\equiv l_{2}l_{1}z={\\Big (}\\partial _{x}+\\partial _{y}+{\\frac {2}{x+y}}{\\Big )}{\\Big (}\\partial _{x}-\\partial _{y}+{\\frac {2}{x+y}}{\\Big )}z=0", "087e3e311cf84d5eb4c0daefc55ab1db": "{\\dot {y}}", "087eebb0f949fb53c7e4be08ecf843d3": "{\\frac {30,000\\ \\mathrm {N} }{(111\\ \\mathrm {kg} )(9.807\\ \\mathrm {m/s^{2}} )}}=27.6", "087efa7ff3198fd6fa7b7bae1aeb9d8b": "\\theta \\mapsto p(x\\mid \\theta )\\!", "087f30a536338c2546ba6bc229296d24": "T_{p}^{\\mathrm {SW} }(x,y)={\\begin{cases}T_{\\mathrm {D} }(x,y)&{\\text{if }}p=-1\\\\\\max \\left(0,{\\frac {x+y-1+pxy}{1+p}}\\right)&{\\text{if }}-10", "0889f771f64a54f2f4e00c59eab935ff": "n=(n_{l}n_{l-1}\\dots n_{0})_{2}", "0889fb824147f396d1a1d815553e4fe4": "H={\\frac {1}{\\sqrt {2}}}a", "088a3dcfc3dd3697b2fad6ea74d45e92": "\\ltimes \\!\\,", "088a4edb5ffe3aa76b2ff5406265a8f2": "T_{s}=303~\\mathrm {K} =30~\\mathrm {C} ", "088aa0453d2418193fb066dd406ff6bd": "h=h(-,-).\\,", "088aa21ca6d89bd3e67eb5ec3d720324": "{\\begin{array}{lll}\\eta &=&1-{\\frac {trace(W_{1}^{T}AW_{1})}{trace(D_{B}^{-1/2}P^{T}APD_{B}^{-1/2})}}\\\\&=&1-{\\frac {trace({\\hat {D_{B}}}^{-1/2}{\\hat {P}}^{T}A{\\hat {P}}{\\hat {D_{B}}}^{-1/2})}{trace(D_{B}^{-1/2}P^{T}APD_{B}^{-1/2})}}\\end{array}}", "088aa33c190e59659b39778bd19b5986": "\\lambda <0", "088aaa73541213fd91084b37fd0422bf": "y=C(w(t))x(t)+D(w(t))u(t)", "088b09b0ab517ad1f368845c9818c722": "\\det(E+(n-i)\\delta _{ij})=0", "088b147aae80c99347aefe2dc1fefa09": "G={\\frac {2e^{2}}{h}}MT", "088bd0a0a6ad8aa37cb69bee357297a1": "U[\\mu _{1},\\mu _{2}]=U-\\mu _{1}N_{1}-\\mu _{2}N_{2}", "088bdf436f5d9d1dea35097f67b43c12": "\\sum _{v(p)=0}{\\frac {P(A_{p})}{\\det A_{p}}}=\\int _{M}P(i\\Theta /2\\pi )", "088c3b4690d85e237c04bc75b9ace31a": "\\mathbf {R} =n_{1}\\mathbf {a} _{1}+n_{2}\\mathbf {a} _{2}+n_{3}\\mathbf {a} _{3},", "088c4ffb0bb4687563ca3bf314176dba": "{\\begin{aligned}\\sin(A)&=RQ\\\\&={\\text{length of arc }}PS\\\\&=\\angle POS{\\text{ in radians}}\\\\&={\\frac {\\pi }{180\\times 60}}\\left(m+{\\frac {s}{60}}+{\\frac {t}{60\\times 60}}\\right).\\end{aligned}}", "088c77d6e29bdb1d671b2df67aca9373": "x\\cdot p+y\\cdot q=0", "088c8b835560feac3300998ec1332a92": "|\\psi \\rangle ", "088d03f1074199581e9cd748695d5521": "g(z)=z+b_{0}+b_{1}z^{-1}+b_{2}z^{-2}+\\cdots ", "088d15a0bdfecaccd1c62383f94b8cbc": "-\\varepsilon _{i}.", "088d531507cfb1a6302c6226049e80dc": "M^{\\mu \\nu }\\,", "088d68d8a312039e05c361d2341ac71c": "E_{X}=61.5\\ \\mathrm {mV} \\log {\\left({\\frac {[X^{+}]_{\\mathrm {out} }}{[X^{+}]_{\\mathrm {in} }}}\\right)}=-61.5\\ \\mathrm {mV} \\log {\\left({\\frac {[X^{-}]_{\\mathrm {out} }}{[X^{-}]_{\\mathrm {in} }}}\\right)}", "088dbb6eed854331cdab6c34e5e8852e": "\\left|\\Gamma \\left(\\omega \\right)\\right|^{2}", "088dc58a11fd066e189ee7f8a7725546": "\\kappa (M)\\in H^{4}(M;\\mathbb {Z} /2\\mathbb {Z} )", "088de5149c77b89b1f21d59d3df5e987": "\\mathbf {S} ={\\frac {\\epsilon _{0}}{2i\\omega }}\\int \\left(\\mathbf {E} ^{\\ast }\\times \\mathbf {E} \\right)d^{3}\\mathbf {r} .", "088e53945d396a436984c8403162be33": "K_{\\mathrm {rot} }={\\tfrac {1}{2}}I\\omega ^{2},", "088f2b5f22c21c3051502aa50d7597ec": "i:=1;\\qquad S:=\\emptyset ,\\qquad f^{*}:=f;", "088f6df761802b49580d5bb103e7a428": "L_{i}=C_{i}-d_{i}", "088fc910583005a88db3211c8be16637": "Q={\\big (}\\alpha K^{\\lambda }+(1-\\alpha )N^{\\lambda }{\\big )}^{1/\\lambda },\\,", "088ff137b55f0dce3cce12801b5c9125": "{\\bar {V^{E}}}_{i}=RT{\\frac {\\partial (\\ln(\\gamma _{i}))}{\\partial P}}", "08900515bff2bad7ba26a0b4ae1b93a5": "48+32+1+64+33+17+16+49=260", "089017e414db79a49991a6af70d24a61": "\\scriptstyle f_{0}\\,", "08915b56dbc2dc98df111e43d0cdcbe3": "\\int \\ln(x^{2}+a^{2})\\;dx=x\\ln(x^{2}+a^{2})-2x+2a\\tan ^{-1}{\\frac {x}{a}}", "08917d48d77824f06da1c6c0ca89d811": "2^{2}=4,\\,2^{3}=8,\\,2^{5}=32,\\,2^{7}=128,\\,2^{11}=2048,\\,2^{13}=8192,\\,2^{17}=131072,\\,2^{19}=524288,\\,\\dots ", "0891adbe9e3ea1673fe06c84c4b13fa8": "{\\frac {S(\\omega )}{H_{1/3}^{2}T_{1}}}={\\frac {0.11}{2\\pi }}\\left({\\frac {\\omega T_{1}}{2\\pi }}\\right)^{-5}\\mathrm {exp} \\left[-0.44\\left({\\frac {\\omega T_{1}}{2\\pi }}\\right)^{-4}\\right]", "0891c3a975872f033667b0eddf7c0ce3": "V={\\frac {\\partial }{\\partial \\theta }}\\ln f(X;\\theta )", "0891c88603858c7033d48290fe995b5a": "(AB)^{+}=B^{+}A^{+}\\,\\!", "0891d9a9c062411339abf08c6b6d2179": "H_{L}", "0892355b40f608ef28e5f27f63f91144": "p_{k+1}\\leftarrow M^{-1}r_{k+1}+\\beta _{k}\\cdot p_{k}\\,", "089255d1a3230d76d837329d738f9e48": "c_{1}'(t)={\\dfrac {\\mu _{01}}{i\\hbar }}\\epsilon (t)\\exp \\left(-i{\\dfrac {E_{0}-E_{1}}{\\hbar }}t\\right)\\Rightarrow c_{1}(t')={\\dfrac {\\mu _{01}}{i\\hbar }}\\int _{0}^{t'}\\epsilon (t)\\exp \\left(-i{\\dfrac {E_{0}-E_{1}}{\\hbar }}t\\right)\\mathrm {d} t", "089291bee7f9ab0573749b9336958005": "A_{2N}\\alpha ", "08950ec45604b6b600898b4ae5f44608": "\\;C_{\\Phi }=\\sum _{i=1}^{nm}v_{i}v_{i}^{*}.", "089517dc1e251ebe3f3d53ece382600e": "r={2\\mu \\over v_{\\infty }^{2}}", "0895194b16feb437e209b4f12bb9d537": "{\\begin{aligned}&\\nabla \\cdot \\mathbf {j} (\\mathbf {r} ,t)=-{\\frac {\\partial \\rho (\\mathbf {r} ,t)}{\\partial t}}+\\sigma (\\mathbf {r} ,t),\\\\&\\nabla \\cdot \\mathbf {j} +{\\frac {\\partial \\rho }{\\partial t}}=\\sigma \\quad \\rightleftharpoons \\quad \\nabla \\cdot (\\rho \\mathbf {v} )+{\\frac {\\partial \\rho }{\\partial t}}=\\sigma .\\\\\\end{aligned}}", "08953dcf0a515e7cff3d6c7d8683977c": "h(t)={\\mathcal {L}}^{-1}\\left\\{K(p)\\right\\}=\\delta (t)-\\omega _{0}e^{-\\omega _{0}t}=\\delta (t)-{\\frac {1}{\\tau }}e^{-{\\frac {t}{\\tau }}}", "0895880b7b819b766849d690ad7ecf98": "{\\mbox{Skew}}[Y]={\\mbox{Skew}}[X]/{\\sqrt {n}}", "0895c40bab16acf26444c072230a909e": "{\\mathit {bar}}", "0896910db3486c36b4e9ebde666350a3": "ZFC\\vdash \\operatorname {Con} (ZFC)\\leftrightarrow \\operatorname {Con} (ZFL)", "08973f6125605e2de0399c5626706e37": "M^{2}dM=-{\\frac {K_{\\operatorname {ev} }}{c^{2}}}dt\\;", "08975895667eb87dd2a534ba79385954": "\\mathbf {Z} _{4}=\\{{\\overline {0}},{\\overline {1}},{\\overline {2}},{\\overline {3}}\\}", "089777aeed0359e947eaf33e96ad1248": "{\\frac {dp_{\\alpha }}{d\\tau }}\\,=q\\,F_{\\alpha \\beta }\\,u^{\\beta }", "0897b1e4c0c539268fd4ba6abbe20a05": "F(\\lambda )=e^{\\lambda M^{*}}Te^{-\\lambda N^{*}}.", "0897da084ee46227777f2e99f0a8c598": "{\\begin{array}{c|cccc}0&0&0&0&0\\\\1/3&1/3&0&0&0\\\\2/3&-1/3&1&0&0\\\\1&1&-1&1&0\\\\\\hline &1/8&3/8&3/8&1/8\\\\\\end{array}}", "0897e667e0116679b830496fa73eca25": "\\varphi \\,\\!", "08980a580ec39c643d6ab2a2e5e8abbd": "i\\hbar {\\frac {\\partial }{\\partial t}}\\psi =-\\sum _{k=1}^{N}{\\frac {\\hbar ^{2}}{2m_{k}}}\\nabla _{k}^{2}\\psi +V\\psi ", "08980e54192c9ed6458c9ff7f566bb95": "G=eD", "089817b894737febdd472b2c9af430fe": "\\mathrm {D} _{t}", "089843f7731a27d67155ee011d14ce3c": "\\phi ={\\frac {V_{\\mathrm {V} }}{V_{\\mathrm {T} }}}", "089887b5053c9d8239fd1c3dc583ed86": "F_{in}>{\\frac {1}{2}}", "0898cff846b040b8dffbf037986688b4": "\\int _{\\gamma }f(z)=\\int _{a}^{b}f(z(t))z'(t)\\,dt.", "0898efa852b099482ca09860d347f545": "{|\\mathbf {F} |+n-1 \\choose n}\\geq {\\frac {|\\mathbf {F} |^{n}}{n!}}.", "0898efbeab821569bdc5ae5d3c6f3ab5": "\\sum _{j=1}^{N}c_{j}\\left(H_{kj}-\\varepsilon S_{kj}\\right)=0\\quad {\\text{for}}\\quad k=1,2,\\dots ,N.", "0898fa79b5b86eefb8fc18aad25f635f": "\\lim _{t\\to 1^{-}}G_{a}(tz)=\\sum _{k=0}^{\\infty }a_{k}z^{k}\\!", "08991589b018456998d95c6ae785f81f": "\\mathbb {C} P^{1}", "0899241e18e47bf3b78e9b2a0d632872": "g\\colon \\mathbb {R} ^{n}\\to \\mathbb {R} ^{n}", "08993f6c65b816ce607a024c300a1f84": "\\textstyle 2^{n}/(n2^{l-1}+1)", "08995f987e57f81e4b5ad4c0aaee5866": "{\\frac {b^{2}}{a}}", "08996b0dbb09a185cbc0a964a70e4ea5": "\\sigma _{j,n-1}", "0899c037f2676e8ef14b5f15d3b840b6": "\\omega _{k}=ck", "089a4b8acabe5054582d53f836e07ddb": "(n,{\\tilde {m}},l,\\epsilon )", "089a7ceda77cb567e8854fe8de175b53": "\\|X_{(1)}-x\\|\\leq \\dots \\leq \\|X_{(n)}-x\\|", "089a91d37a3c9cfe64804a32a3ed8cfe": "\\Sigma \\subset {\\mathcal {M}}^{3}", "089b2872354cf4599b6a22e2a3f020be": "x^{*}\\in \\{x_{1},\\ldots ,x_{N}\\}", "089b34600dc598cce585d42cc679d061": "p(z)=a(z-\\alpha _{1})(z-\\alpha _{2})\\cdots (z-\\alpha _{n})", "089b5f8ea190475360f8bb85d181e265": "\\omega (j)", "089b7716be9fe6981ca3351fb15479e8": "{\\boldsymbol {v}}={\\frac {{\\text{d}}{\\boldsymbol {s}}}{{\\text{d}}t}}", "089b9ac86c5f461f59c9f8f674002eba": "2^{-k}\\exp \\left(-(1+o(1)){\\frac {\\log x\\,\\log \\log \\log x}{\\log \\log x}}\\right)", "089ba3f87c5ecee1db40e8302f0a9667": "|s\\rangle ", "089bd4f09c60d3da74d1d7b67bf8a345": "\\textstyle \\delta _{\\nu }=0", "089bf6da03747ee690cfed1a43415203": "e_{k}(X_{1},\\ldots ,X_{n})", "089c078bf9a7ea8ae5092ccede071e22": "{\\hat {\\theta }}={\\underset {\\theta \\in \\Theta }{\\operatorname {arg\\,max} }}\\;S_{n}(\\theta ),\\quad {\\text{where }}\\ S_{n}(\\theta )=\\ln \\!\\!{\\sqrt[{n+1}]{D_{1}D_{2}\\cdots D_{n+1}}}={\\frac {1}{n+1}}\\sum _{i=1}^{n+1}\\ln {D_{i}}(\\theta ).", "089c39ac42361671a5b39e01d28be16d": "\\Phi _{M}\\,", "089c50d0baecda42b714997ca31d8343": "r\\ ", "089c649017c10f59ef86716ad1835192": "T_{0}^{2}(q)={\\dfrac {\\sqrt {6}}{2}}q_{zz}", "089cf08b631ba1f5c463828ee196439b": "Z_{0,0}^{M}=1", "089d079ccc7bcd387140a47ce4f24e24": "\\nu '=\\gamma \\nu -\\gamma \\beta \\nu \\cos \\theta =\\gamma \\nu \\left(1-\\beta \\cos \\theta \\right)", "089d52bdbe45a24dffc733b3cfe769a1": "F(a_{1},\\dots ,a_{n})=\\sum _{i=1}^{n}F(a_{1},\\dots ,a_{i}-1,\\dots ,a_{n}).", "089d6aa9edd04167c42d45f2dae6e667": "C=S(e^{rT}-1)\\,", "089de14241a817bc8b4b39d127f00b72": "\\sigma _{\\mathrm {n} }=\\sigma _{1}n_{1}^{2}+\\sigma _{2}n_{2}^{2}+\\sigma _{3}n_{3}^{2}.", "089de5f0be1afcf0ec62ae9a704ad4bb": "c=110001111", "089e1624882f007bba90fb810b57fe1c": "{\\begin{aligned}C(S,t)&=N(d_{1})S-N(d_{2})Ke^{-r(T-t)}\\\\d_{1}&={\\frac {1}{\\sigma {\\sqrt {T-t}}}}\\left[\\ln \\left({\\frac {S}{K}}\\right)+\\left(r+{\\frac {\\sigma ^{2}}{2}}\\right)(T-t)\\right]\\\\d_{2}&={\\frac {1}{\\sigma {\\sqrt {T-t}}}}\\left[\\ln \\left({\\frac {S}{K}}\\right)+\\left(r-{\\frac {\\sigma ^{2}}{2}}\\right)(T-t)\\right]\\\\&=d_{1}-\\sigma {\\sqrt {T-t}}\\end{aligned}}", "089e690bafe32a984251d428be1be4f2": "\\rho _{Q}={\\sqrt {\\frac {L_{QA}}{C_{QA}}}}={\\sqrt {\\frac {\\phi _{0}^{2}}{e^{2}}}}={\\frac {h}{e^{2}}}=R_{H}\\ ", "089e932f69f266f26e918d19d6217b09": "b=\\lambda ^{-k}", "089eb6e35ca6130e89725199e9bf4c00": "\\phi ^{-}(a_{i})\\geq \\phi ^{-}(a_{j})", "089ed104ae0893c5ab1a8ee6a5797241": "\\sigma _{\\text{mean}}={\\frac {1}{\\sqrt {N}}}\\sigma ", "089ed62808b2967c53cea2f8804edbf7": "x\\,y''+(\\alpha +1-x)\\,y'+{\\lambda }\\,y=0{\\text{ with }}\\lambda =n.\\,", "089eeacfc3eb9e564575aa6d835c3673": "\\log _{10}K_{\\textrm {a}}=\\log _{10}[{\\textrm {H}}^{+}]+\\log _{10}\\left({\\frac {[{\\textrm {A}}^{-}]}{[{\\textrm {HA}}]}}\\right)", "089ef266a52b9ca4e23dce607a48b556": "\\{n_{\\alpha }\\},\\{m_{\\alpha }\\}\\subset \\mathbb {N} ", "089f12531b1c4a0d3560fc99de22290f": "\\left|{|gA\\cap F_{i}| \\over |F_{i}|}-{|A\\cap F_{i}| \\over |F_{i}|}\\right|=\\left|{|A\\cap g^{-1}F_{i}| \\over |F_{i}|}-{|A\\cap F_{i}| \\over |F_{i}|}\\right|", "089fb2bbc2d7478893a96420ff1623f0": "i=1,2,...,s", "089fef7d2053a7b26771146ccfcf5513": "H(\\alpha )=\\max\\{|x|,|y|,|z|\\}.\\,", "089ff39a17b4ec2d40f4f00851804b5b": "a_{5}", "089ffe1694bd5f2c3ad2d15976da5ce7": "sp(x:=x-5,x>15)\\ =\\ \\exists y,x=y-5\\wedge y>15\\ \\Leftrightarrow \\ x>10", "08a0097b172b5cd5c4556d2930dfa993": "P^{2}=I", "08a06702edc84571e50f183d0af6d160": "{\\bar {\\Gamma }}(\\tau )=\\forall \\ {\\hat {\\alpha }}\\ .\\ \\tau \\quad \\quad {\\hat {\\alpha }}={\\textrm {free}}(\\tau )-{\\textrm {free}}(\\Gamma )", "08a0ba2451427d21e13c4775c6e3dd30": "d\\mathbf {S} ", "08a0bc384275a2e1e7e17f6b8ed21138": "{\\vec {p}}_{2}=\\partial _{r}", "08a0ff71551639cc72531876a9c21125": "{\\frac {163}{\\ln 163}}\\approx 2^{5}", "08a0ffadbb96a515913099edbd30c490": "J={\\begin{pmatrix}0&I\\\\-I&0\\end{pmatrix}}", "08a104e9574a90a74209a609d3daacc7": "T\\times \\gamma ", "08a136f9796e1034de4d5b33d1dc2bb4": "m(x,y)", "08a13947c8a56d0118b31a4ffe572d7a": "\\nabla \\cdot \\mathbf {u} =0.", "08a13fa295805c91afc1d69f8976fab5": "\\mathbf {y} _{k}\\sim \\mathbf {\\bar {C}} _{k}\\,\\mathbf {\\bar {x}} =\\mathbf {C} _{k}\\,\\mathbf {x} ", "08a175f5bfe368e1ee60c7fec13a057e": "{\\frac {\\Delta y}{\\Delta x}}=f'(x)+\\varepsilon ,", "08a1811bd8c272fdc022c364e723563f": "q_{r,\\nu }=w/s.\\,", "08a20142f9fd84a6f1b44094576e4e2a": "\\sum _{i=0}^{n}{i{\\binom {n}{i}}^{2}}={\\frac {n}{2}}{\\binom {2n}{n}}", "08a22e50f7516840508a72a6481646b7": "\\sin(\\pi z)=\\pi z\\prod _{n=1}^{\\infty }\\left(1-{\\frac {z^{2}}{n^{2}}}\\right)", "08a29dd11a36e96ebf5f676b6bcd793a": "f_{t,a,b}", "08a31d631f1f0e371e354b024649cfea": "color\\ ", "08a32bbc01161ef7587f6fc60892f9b8": "S(t)=\\sum _{i=1}^{\\infty }S_{i}\\Phi _{i}(t)", "08a35e1c70e126b9d1ea1a35ccc73bc0": "{\\mathbf {P} }(t)", "08a3669763ee42ad3197e9bd45e004ef": "L=K", "08a37568bdfc4282dadce1dce6f48d27": "f_{(\\xi ,\\mu ,\\sigma )}(x)={\\frac {1}{\\sigma }}\\left(1+{\\frac {\\xi (x-\\mu )}{\\sigma }}\\right)^{\\left(-{\\frac {1}{\\xi }}-1\\right)},", "08a386134202a72add2ca015fc6201c6": "\\theta =2\\pi \\nu \\,", "08a398a26cb80312d9c83cbe83379c5a": "\\tan {\\frac {\\delta '}{2}}={\\sqrt {\\frac {1-\\beta }{1+\\beta }}}\\cdot \\tan {\\frac {\\delta }{2}}", "08a39da9b0aa34d4e84a4d67294811db": "G:C\\to D", "08a3fff08bec0bb773f00db8d38ec5ea": "z^{2}=r_{1}^{2}-x^{2}-y^{2}", "08a422c3ea7ffca07f2210324d12db75": "{\\vec {k}}\\|{\\vec {B}}", "08a4377d7ed260cbc3b26e05f2493110": "A^{*}A+\\Gamma ^{T}\\Gamma ", "08a493b23d014a745d4b9daf67519802": "\\mathrm {Id} _{A}(a,b)", "08a4d85fdf5424e20249248d2c9cd5f9": "\\epsilon =ArcCot(\\pm AR)", "08a4eecf61f158c09769acd2e87b8591": "L={\\frac {1}{2}}\\left(M+m\\right){\\dot {x}}^{2}-m\\ell {\\dot {x}}{\\dot {\\theta }}\\cos \\theta +{\\frac {1}{2}}m\\ell ^{2}{\\dot {\\theta }}^{2}-mg\\ell \\cos \\theta ", "08a554081bf4c684c8f4cb0cf90f78cb": "\\mathrm {n} +{}_{\\ 90}^{232}\\mathrm {Th} \\rightarrow {}_{\\ 90}^{233}\\mathrm {Th} \\xrightarrow {\\beta ^{-}} {}_{\\ 91}^{233}\\mathrm {Pa} \\xrightarrow {\\beta ^{-}} {}_{\\ 92}^{233}\\mathrm {U} ", "08a5c71d6fd63e1955df451525935d1c": "a_{P}T_{P}=a_{w}{T_{w}}^{0}+a_{e}{T_{e}}^{0}+[{a_{P}}^{0}-(a_{w}+a_{e}-S_{P})]{T_{P}}^{0}+S_{u}", "08a5d5c8d39219b8b3b0a6c2f7d4a143": "P(x_{1},...,x_{n})=0", "08a5e5edf4fe6f511952a8e8f27c0f65": "\\alpha =e^{\\frac {-2\\pi i}{n}},", "08a625db3223bd84256d71f8110857d1": "u_{11}=4", "08a62bf6fcb9536ca79feb932e81dcd2": "\\int \\exp \\left[-\\theta ^{T}M\\theta \\right]\\,d\\theta ={\\begin{cases}2^{n \\over 2}{\\sqrt {\\det M}},&n{\\mbox{ even}}\\\\0,&n{\\mbox{ odd}}\\end{cases}}", "08a637a7533dcdaf3e830671dc549003": "\\varepsilon _{1}\\Psi ={\\frac {-P^{2}+m_{1}^{2}-m_{2}^{2}}{2{\\sqrt {-P^{2}}}}}\\Psi ", "08a64a2cc12e61fefbaab0023cadfca3": "f:(-2,+\\infty )\\rightarrow \\mathbb {R} ", "08a6f3b40d66c7170552d2b2e87fd2c0": "|\\psi (t)\\rangle =\\sum _{n}a_{n}(t)|n\\rangle ", "08a75a112849152419b42feee123eb44": "x=x_{1}(1,\\lambda ,\\lambda ^{2},\\dots ),", "08a792918893b79247dec464219b5d9f": "f^{abc}", "08a7a7ab25e5448135a705ebc67c67dc": "r_{t+1}", "08a7c0c5e953a77c63ff8527b4d8bd5b": "\\gamma W\\cap W\\,", "08a7d8607501784ff17a3c4e5eeb6e92": "\\mathrm {R_{B\\beta }} ", "08a85f4d2b6708c8a604cb7710a29ee6": "(Sv)(ds)=\\lim _{\\epsilon \\to 0,\\epsilon >0}{\\frac {1}{\\epsilon }}\\int _{0}^{1-\\epsilon }(f(t+\\epsilon \\mu (ds))-f(t))dt", "08a877f3a35b15505a8eb7b18d5fdc73": "\\sum _{v=0}^{r-1}{r-v}", "08a88658e7ed556dd1eecc98587f7199": "H={\\frac {1}{2m}}\\left(\\mathbf {p} -{\\frac {e}{c}}\\mathbf {A} \\right)^{2}+e\\phi -{\\frac {e\\hbar }{2mc}}{\\boldsymbol {\\sigma }}\\cdot \\mathbf {B} .", "08a8c01d82983e9bdcf0e6741f5c1e75": "{\\text{min:}}\\operatorname {Tr} (\\sigma _{B})", "08a97bcb5749c8f469c55ce3dcc470d9": "R_{max}=v^{2}/g\\,", "08a97ca308307132125ee096959ee1df": "\\operatorname {Li} _{2}(0)=0", "08a9d74163be02ea90ab7b7a0351b5f3": "G=SU(N_{c})", "08aa2420736e7ec5fd436f4229f48ad4": "n_{2}", "08aa8c63acbec691aec44ddacf455fc6": "{\\Phi }", "08aab844ad8b5788a4d2e94e6124ca22": "{\\frac {d}{dz}}\\left[z^{1-a_{1}}\\;G_{p,q}^{\\,m,n}\\!\\left(\\left.{\\begin{matrix}\\mathbf {a_{p}} \\\\\\mathbf {b_{q}} \\end{matrix}}\\;\\right|\\,z\\right)\\right]=z^{-a_{1}}\\;G_{p,q}^{\\,m,n}\\!\\left(\\left.{\\begin{matrix}a_{1}-1,a_{2},\\dots ,a_{p}\\\\\\mathbf {b_{q}} \\end{matrix}}\\;\\right|\\,z\\right),\\quad n\\geq 1,", "08aade75fb531a156e4329ffcfb87ceb": "t=(n+1)\\Delta t\\,", "08aaecd3a878f9134a06a38b6ac7aad9": "R_{M}", "08ab13e7e260a41c4e1131ccf103e57c": "A,B\\in K", "08ab14743db555f395c558fd89ee49fb": "\\alpha |f|^{p}=\\beta |g|^{q}\\,", "08ab3226355569dd4cc1e814528a2795": "(x_{1})^{n}=\\sum _{k_{1}=n}{n \\choose k_{1}}x_{1}^{k_{1}};\\ \\ k_{1},n\\in \\mathbb {N} _{0}", "08ab50acbda2fd422f9f2bc299e3858b": "RSI=100-{100 \\over {1+RS}}", "08ab809371ce6537c1a7b6fa1e535dc6": "A\\otimes B", "08ab8a1a461f7f1f2aaf5e560df56c7b": "\\textstyle \\{v_{1},\\dots ,v_{m}\\}", "08abbb203e035f5e7cd178c8dc9e9bb7": "R(f)", "08ac18c283b358ab1356c98861464d28": "{\\frac {250}{50}}=5", "08ac2718fba3838072d91d1aacb83b41": "r=kC_{S}\\theta _{A}C_{B}", "08acb54f0d911db77c96528a6caa7189": "{\\begin{bmatrix}1&0&0\\\\0&4&0\\\\0&0&-3\\\\0&0&0\\\\\\end{bmatrix}}", "08acd20803da1f3613a020ebb5e7ed8d": "(f*g)(t)={\\begin{cases}f(2t)&0\\leq t\\leq {\\tfrac {1}{2}}\\\\g(2t-1)&{\\tfrac {1}{2}}\\leq t\\leq 1\\end{cases}}", "08ace0ea6db5bc67a6dbbc71ccee47e0": "\\prod _{x\\in \\Sigma }\\delta ({\\hat {H}}(x))", "08acee831506e12f7d064a1081242f06": "\\theta _{t}(t^{\\prime })=\\delta _{tt^{\\prime }},", "08ad08f6491037714d09263a79bebfba": "MT", "08ad36651c8fd2c5dc250c5cc8da1c11": "6.02\\approx 20\\log _{10}2", "08ad4c042fe94b11e3c1f61a141fcae3": "\\sigma _{p}", "08ad53c0fc97bb056df5311a351bfdb4": "S_{2}S_{1}S_{2}", "08ad58beca7a2d9538f3ab070d072807": "\\ln \\Gamma (\\alpha )+\\ln \\Gamma (\\beta )-\\ln \\Gamma (\\alpha +\\beta )", "08ad5a353879b2190b43380c06834129": "\\forall x\\in a\\exists y\\,\\phi (x,y)\\rightarrow \\exists b\\forall x\\in a\\exists y\\in b\\,\\phi (x,y)", "08ad7e857232ce296c5b63f8c7fc5b3b": "\\Delta H-T\\Delta S_{int}\\leq 0\\,", "08ae28c054879dfbcac0417af765b298": "\\delta \\;", "08ae6dea47bd74fd7cfd3e24ee234038": "\\sum _{n=0}^{\\infty }{\\left({\\frac {(-1)^{n}}{2n+1}}\\right)}^{5}={\\frac {1}{1^{5}}}-{\\frac {1}{3^{5}}}+{\\frac {1}{5^{5}}}-{\\frac {1}{7^{5}}}+\\cdots ={\\frac {5\\pi ^{5}}{1536}}\\!", "08aea994c648e6990d27839bd3c335a6": "\\mathbf {e'} =\\mathbf {J} \\mathbf {e} .", "08aecc8a1ce3b7392dbba000ac74f843": "\\log {\\mathcal {L}}=-12492.9", "08aee2f8fd1c4ebe83333b4910549142": "{\\mathbf {q} _{u}}({\\mathbf {r} },t)", "08aee5b6c0c5621b81882cb6639cbdc8": "{\\begin{pmatrix}0&-1\\\\1&0\\end{pmatrix}}", "08af80b34b54c2e45bd728257f84fb40": "b={\\hat {\\beta }}", "08af862e381d5a6ed11787a203c54175": "v_{m}={\\frac {2D_{e}-h\\nu _{0}}{h\\nu _{0}}}", "08afa9f77f75b8d63c304f964c0539e5": "\\Psi (x)=C_{A}{\\textrm {Ai}}\\left({\\sqrt[{3}]{U_{1}}}\\cdot (x-x_{1})\\right)+C_{B}{\\textrm {Bi}}\\left({\\sqrt[{3}]{U_{1}}}\\cdot (x-x_{1})\\right).", "08aff5439448f31759bc5ef7e1b0a164": "Z_{n+1}", "08b0104e514f16d489cc743b6f66d906": "P_{i}", "08b046e112ef09823fefab0abc701f96": "S\\rightarrow ba", "08b0c7ac7fa7a06b33ef89840a3ec0ed": "-X_{1}^{2}-X_{2}^{2}+\\sum _{i=3}^{n+1}X_{i}^{2}=-\\alpha ^{2}", "08b1a784c63207cc2f2c69e089160748": "\\langle x,Ax\\rangle =0", "08b24ad19f2e115acbb706c256b9271e": "M_{3,X}=M_{3,A}+M_{3,B}+\\delta ^{3}{\\frac {n_{A}n_{B}(n_{A}-n_{B})}{n_{X}^{2}}}+3\\delta {\\frac {n_{A}M_{2,B}-n_{B}M_{2,A}}{n_{X}}}", "08b29601b034f0a05b294662bbb971cb": "T{\\underline {A}}", "08b29e6aff4e6ec1e064d0855d652a99": "x,z,w_{1},\\ldots ,w_{n}\\!", "08b2c2214887981149ff0c4db51e48fb": "{\\overline {\\overline {V}}}\\to V", "08b2fd3b424c21a551d770c23e0ca994": "\\omega _{k}", "08b327813e8ad4772779f2e81d6144f7": "m=\\sum _{i=1}^{n}a_{i}", "08b3da0bf6532d03c95780f4e504eebe": "\\mathbf {P} \\left[\\sup _{0\\leq t\\leq T}B_{t}\\geq C\\right]\\leq \\exp \\left(-{\\frac {C^{2}}{2T}}\\right).", "08b41e0e3b3d28607a0109f92fc960a8": "P_{ij}=\\mu +\\alpha _{i}+\\alpha _{j}+d_{ij}", "08b4271e7a9111b5ad74901134a20477": "\\Delta (x)=\\int _{0}^{\\infty }d\\tau e^{-m^{2}\\tau }{1 \\over ({4\\pi \\tau })^{d/2}}e^{-x^{2} \\over 4\\tau }", "08b43a28f93233cde0befa6872957246": "b_{14}+a_{15}-c_{13}", "08b44f9bcb2425a5c61732bd5671ce38": "{\\begin{vmatrix}0&{\\frac {\\lambda _{2}-\\lambda _{1}}{(\\lambda _{2}-\\lambda _{1})}}&{\\frac {\\lambda _{2}^{2}-\\lambda _{1}^{2}}{(\\lambda _{2}-\\lambda _{1})}}&\\cdots &{\\frac {\\lambda _{2}^{n-1}-\\lambda _{1}^{n-1}}{(\\lambda _{2}-\\lambda _{1})}}\\end{vmatrix}}\\quad {\\begin{vmatrix}{\\frac {\\lambda _{2}^{n+m}-\\lambda _{1}^{n+m}}{(\\lambda _{2}-\\lambda _{1})}}\\end{vmatrix}}", "08b48c475b79d192ef03ea5415b09f89": "{\\begin{matrix}{4 \\choose 2}{3 \\choose 1}^{2}{36 \\choose 2}\\end{matrix}}", "08b4c57626841de9aed16f68df25fd2c": "z_{v}", "08b511fff209c4e37fbffee768045446": "S_{i',j'}^{t}=1", "08b57638e0309a5476c09e84d8846ae3": "\\mathbb {V} \\left[\\log {\\frac {{\\hat {p}}_{x-1}}{{\\hat {p}}_{x}}}\\right]\\approx {\\frac {1}{np_{x}}}+{\\frac {1}{np_{x-1}}}", "08b5804db7683df220712df0f72937f1": "2m+2.2{\\sqrt {m}}", "08b5805912e3751068405dacd06fb6ab": "\\Pi _{n}^{1}", "08b5d77da7e35ed91a7e17de5c5a4613": "{\\text{Level 5:}}\\ \\ 266=2\\uparrow \\uparrow \\uparrow 2\\uparrow \\uparrow 2+2\\uparrow \\uparrow \\uparrow 2\\times 2+2", "08b5fb3e2d65e1c5a8d30fa4cd40cfef": "\\beta =[\\beta _{0},\\beta _{1}]^{T}", "08b61afeeb4fede6cf7ff8b0cd85612a": "\\exp \\left({\\frac {1}{2}}\\log {\\frac {1+z}{1-z}}\\right)\\prod _{m\\geq 1}\\cosh {\\frac {z^{2m}}{2m}}={\\sqrt {\\frac {1+z}{1-z}}}\\prod _{m\\geq 1}\\cosh {\\frac {z^{2m}}{2m}}.", "08b6237946830fd085f7d87a758ace10": "M(L)\\sim L^{d_{\\text{f}}}\\,\\!", "08b62a2628cff624d81f97b92e16892a": "\\int _{0}^{\\infty }J_{\\lambda }d\\lambda ", "08b64845c8b030cd2a35885708c08f7d": "x=0", "08b69591f93ee237dabcb49caf282947": "c_{\\Lambda }(\\eta ,\\xi )>0", "08b743423bf9352c267d4fc151dc45ab": "\\Delta f=-{\\frac {2f_{0}^{2}}{A{\\sqrt {\\rho _{q}\\mu _{q}}}}}\\Delta m", "08b747a01be91249f4d1fcd29f666490": "(2^{4}/7!!)\\pi ^{3}=(16/105)\\pi ^{3}", "08b750e879638f8dbfbaf8b02aa9b69e": "y_{3}={\\frac {(x_{1}^{2}+y_{1}^{2}-2(x_{1})^{2}}{-4(y_{1}^{2}-1)y_{1}^{2}+(x_{1}^{2}-y_{1}^{2})^{2}}}y_{1}=0", "08b79fc9a0c43abce9cbddf55408dc95": "{\\begin{aligned}{\\boldsymbol {M}}\\colon {\\vec {x_{i}}}(t_{i})&\\to {\\vec {x}}_{i+1}(t_{i+1}=t_{i}+T,{\\vec {x_{i}}}).\\end{aligned}}", "08b8018af644fa58cea50b82ecffd271": "\\mathbb {N} \\cup \\{\\infty \\}", "08b8176cc3ff6af94f409722f33e2822": "\\left(1-{\\frac {2GM}{rc^{2}}}\\right)c^{2}dt^{2}=r^{2}{\\textrm {sin}}^{2}\\theta d\\phi ^{2}", "08b83bc14afc91e039b47d5ab69d29d9": "{\\frac {\\mathrm {d} }{\\mathrm {d} t}}(\\varphi (\\mathbf {x} ,t))={\\frac {\\partial \\varphi }{\\partial t}}+\\nabla \\varphi \\cdot {\\frac {\\mathrm {d} \\mathbf {x} }{\\mathrm {d} t}}.", "08b840d813799099f64d75175669e025": "{s\\triangleleft s}", "08b85058ed80237cdd212f6c41e0ce44": "\\mathrm {DOF} \\approx {\\frac {2Nc\\left(1+m/P\\right)}{m^{2}}}\\,,", "08b8810dc87a9e8dbebbfd61d2df5071": "v={\\frac {\\omega }{\\beta }}.", "08b8d7d03f84ebb2675bbe9b3ce74d65": "ee_{product}=ee_{max}ee_{catalyst}", "08b8d96d5bc3fb56e1f646098021ec43": "f\\colon X\\times [0,1]\\to B\\,", "08b8e29f1416bafafa36c6ad1adbae80": "\\scriptstyle A\\;\\not \\Rightarrow \\;B", "08b91b01ae001d95cdf87ddf9e6c94ce": "C_{n}^{knn}", "08b93bb791922b09b2463dc9b2c84400": "{\\widehat {\\mathbf {C} }}:=\\mathbf {C} \\cup \\{\\infty \\}", "08b949d0ab5a03d90069b893b62e2d0c": "{\\begin{aligned}U(x,z)&=aW{\\frac {\\sin \\left[{\\frac {\\pi Wx}{\\lambda z}}\\right]}{\\frac {\\pi Wx}{\\lambda z}}}\\\\&=aW~\\mathrm {sinc} {\\frac {\\pi Wx}{\\lambda z}}\\end{aligned}}", "08b98c197628f1e05eb912d81617d4e8": "OPEN_{d}", "08b9cef8bfb2a152bacf7fded6d24b7d": "\\kappa =(3-\\nu )/(1+\\nu )", "08b9e5eccad5ff4a01eb737ac1606e62": "m\\geq 2", "08b9f71543f531ab69671c06d811fa72": "{\\bar {x}}", "08ba0b6c5a1063fb3b632c78558aa40a": "\\rho \\epsilon =\\sum _{i}g_{i}\\,\\!", "08ba7ea174f14c34b7ae89cdce8b8d4b": "\\operatorname {F} (-)", "08ba969995304a7a289e3514b1cedc76": "\\prod _{x}a^{x}=Ca^{{\\frac {x}{2}}(x-1)}\\,", "08bac121d68536b08a2a3ecc08df2ebd": "d_{A},d_{B}", "08bad8a366dc9f2a7e981fb27f26779b": "M^{2}f=-\\triangle _{n}P(f)+{\\frac {n-2}{x_{n}}}{\\frac {\\partial P(f)}{\\partial x_{n}}}-\\left(\\triangle _{n}Q(f)-{\\frac {n-2}{x_{n}}}{\\frac {\\partial Q(f)}{\\partial x_{n}}}+{\\frac {n-2}{x_{n}^{2}}}Q(f)\\right)e_{n}", "08bbdf0546471a25454ba6b44f887bdb": "U(\\alpha ,y)", "08bbecb3b1bc01e463e1288ba630f556": "b\\ {\\pmod {\\Phi _{n}(q)}}", "08bbf6749490cb5d38cf9f389f415159": "{\\tilde {\\mu }}", "08bbfeef3cc2675c8bbf079058085fe0": "2^{*}=-4,8,-8{\\text{ according as }}m\\equiv 3{\\pmod {4}},2{\\pmod {8}},-2{\\pmod {8}}.", "08bc601fee971bf4823d1b75d5e289d4": "p_{n}x^{n}+p_{n-1}x^{n-1}+\\cdots p_{0}", "08bce46625100c66af1b4437a3afd1c5": "V_{\\text{max}}=\\pm {\\frac {3}{4(\\kappa _{0}+\\kappa )}}[e_{\\text{33}}-2(1+\\nu )e_{\\text{15}}-2\\nu e_{\\text{31}}]{\\frac {a^{3}}{l^{3}}}\\nu _{\\text{max}}", "08bd716722d5176704191cec94def68e": "g(r,r^{\\prime })\\,", "08be0a16219fcd5e1108755dfd1acf40": "\\beta >-0.5", "08be3901b3bc9236982b3ce3999c8905": "tan({\\frac {a}{D}})", "08be6b931af173ebda4a85fc8d65ce68": "x\\mapsto x", "08be8677937e8336887409cfc509fc9f": "b_{i}=\\gamma ^{2}+\\mathrm {sin} ^{2}({\\frac {i\\pi }{n}}),(i=1,2,...n),", "08bf08d8952c48c9e8bb08ddf5626cbf": "T=300\\,K", "08bf5c38d31fc03b9e3bfebb088c5bbf": "K={\\frac {\\Delta Y}{\\Delta I}})", "08bf620f629deefe2442203f052415f0": "\\mathbf {z} =[x_{1}\\ q_{1}\\ x_{2}\\ x_{1}\\ x_{1}\\ 0\\ 1\\ \\sin x_{1}]^{T}", "08bf9621bd9962a098052301bac589b4": "H_{n}^{-1}(\\beta )", "08bf9852415b519ab9b50a3817f9e035": "R_{\\infty }", "08bfc2eff7da80e42dc3fd78a966d298": "z=z_{t}\\,", "08c06dc493a3191a3fdb7a101581bf8a": "|\\psi _{\\alpha }\\rangle ", "08c08149c2d5d6f9d6f8b93f717c489e": "{\\begin{aligned}\\mathbf {F} &=q\\left(\\mathbf {E_{1}} \\left(x,y,z\\right)-\\mathbf {E_{2}} \\left(x,y,z\\right)+{\\frac {d(\\mathbf {x} _{1}-\\mathbf {x} _{2})}{dt}}\\times \\mathbf {B} \\right)\\\\&=q\\left(\\mathbf {E_{1}} \\left(x,y,z\\right)+\\left((\\mathbf {x} _{1}-\\mathbf {x} _{2})\\cdot \\nabla \\right)\\mathbf {E} -\\mathbf {E_{1}} \\left(x,y,z\\right)+{\\frac {d(\\mathbf {x} _{1}-\\mathbf {x} _{2})}{dt}}\\times \\mathbf {B} \\right).\\\\\\end{aligned}}", "08c096871a3243c00055371f939c4e14": "<{\\overline {16}}_{H}>16_{f}\\phi ", "08c11a24017ed7c2e3444f1d521216b0": "\\mu ={\\frac {D}{k_{B}T}}.", "08c18179c206656b472a78c324d00953": "{O}(M{\\cdot }{\\chi }^{2})", "08c196210fec44d9878c8f9f67db8a21": "g={\\begin{array}{c}x\\\\\\longrightarrow \\\\\\left[{\\begin{array}{rrrrrrrr}-76&-73&-67&-62&-58&-67&-64&-55\\\\-65&-69&-73&-38&-19&-43&-59&-56\\\\-66&-69&-60&-15&16&-24&-62&-55\\\\-65&-70&-57&-6&26&-22&-58&-59\\\\-61&-67&-60&-24&-2&-40&-60&-58\\\\-49&-63&-68&-58&-51&-60&-70&-53\\\\-43&-57&-64&-69&-73&-67&-63&-45\\\\-41&-49&-59&-60&-63&-52&-50&-34\\end{array}}\\right]\\end{array}}{\\Bigg \\downarrow }y.", "08c1ce8a801d8c0ea8f51f30f03e88fd": "\\operatorname {cov} ({\\textbf {X}},{\\textbf {Y}})=\\mathrm {E} \\left[({\\textbf {X}}-\\mathrm {E} [{\\textbf {X}}])({\\textbf {Y}}-\\mathrm {E} [{\\textbf {Y}}])^{\\rm {T}}\\right].", "08c20f58365d71b8909771c1dd1f9a23": "S_{\\text{RST}}=-{\\frac {\\kappa }{8\\pi }}\\int d^{2}x\\,{\\sqrt {-g}}\\left[R{\\frac {1}{\\nabla ^{2}}}R-2\\phi R\\right]", "08c2423c8ee336bbd5124362873b2a74": "{\\begin{aligned}p_{s,0}(z)&=p(z)\\mod \\left(z^{2^{n-s}}-1\\right)&\\quad &{\\text{and}}\\\\p_{s,m}(z)&=p(z)\\mod \\left(z^{2^{n-s}}-2\\cos \\left({\\tfrac {m}{2^{s}}}\\pi \\right)z^{2^{n-1-s}}+1\\right)&m&=1,2,\\dots ,2^{s}-1\\end{aligned}}", "08c2626783e1a71ff9462110eef97fac": "v\\in T_{p}M\\,", "08c2749476c6a68ec930bf745ad93f29": "\\delta _{y}", "08c28d1701d799446b540cbf8c38b7cd": "s(t)=w(t)*R(t)\\!", "08c2de929cd11b682d6aef555dab2402": "F_{4}(a,b)=(x\\to x^{x})^{\\log _{2}(b)}(a)", "08c33b31860c198279fe65d253985a3b": "k\\in Z_{q}", "08c367501a71e8bb4bd723e02823601e": "W(c)=W_{+}(c)+W_{-}(c)", "08c39ddf5b344b4efa8b0134b4c34dce": "a_{i}\\cong {\\frac {\\rho _{it}}{\\sqrt {1-\\rho _{it}^{2}}}}", "08c3ef68c8815c26f82d2cf3f6b1655a": "{\\frac {1}{T}}\\int _{0}^{T}Z(t)^{4}dt\\sim {\\frac {1}{2\\pi ^{2}}}(\\log T)^{4}", "08c422e29bee5503b52e7b047d3e86a2": "C_{1},C_{2},\\ldots C_{n}", "08c48687763aba301ed30d1c33d0ea91": "F={\\frac {\\alpha }{r^{2}}}=\\alpha u^{2}", "08c48857f2f798fbf60b3a200ffff49f": "VT^{{\\hat {c}}_{v}}/N", "08c48ac0fe367d30b4adb3f9132fa1f2": "B_{n}\\cap B_{n+1}\\,=\\,\\emptyset \\,\\forall n", "08c4daf476bf72c27460411bcf8eedba": "\\psi =(x^{\\mu },x^{a})\\,", "08c4fb03f7aa53e8693a5e9f6e3097bc": "\\alpha _{i}", "08c675bff461a17050f942a866b071d0": "X\\to \\mathbb {C} ", "08c756222d35d5fd3aa7bbba4cb2b8ee": "{\\begin{array}{ccc}{\\text{Classical Maximin Format}}&&{\\text{MP Maximin Format}}\\\\\\displaystyle \\max _{d\\in D}\\ \\min _{s\\in S(d)}\\ g(d,s)&=&\\displaystyle \\max _{d\\in D,\\alpha \\in \\mathbb {R} }\\{\\alpha :\\alpha \\leq \\min _{s\\in S(d)}g(d,s)\\}\\end{array}}", "08c7863bcb6fd69fb923572118e873bb": "c_{\\mu }=1", "08c796188adbfd960080a41ac99cde0e": "m=2j-3", "08c797c3299a67b74dfa852ed16fb9bf": "\\scriptstyle a^{2}+b^{2}+c^{2}\\,=\\,1", "08c7b3665204557f7addf4503e5880d6": "L_{M}=kL_{P}", "08c7eb5ed8da2aaf4bdbc9d79ceab260": "|AB|={\\frac {|AC||FE|}{|FC|}}", "08c7eca5e4fa5d220a38a438ea19f3c1": "h_{\\phi }=a\\cosh \\mu \\ \\cos \\nu ", "08c8037f3dcab378895a4256bf65c30a": "{\\begin{cases}\\mu +\\sigma {\\frac {\\Gamma (1-\\xi )-1}{\\xi }}&{\\text{if}}\\ \\xi \\neq 0,\\xi <1,\\\\\\mu +\\sigma \\,\\gamma &{\\text{if}}\\ \\xi =0,\\\\\\infty &{\\text{if}}\\ \\xi \\geq 1,\\end{cases}}", "08c8253419d3e5353ad1b1ed4764f590": "\\Omega ({\\text{ }})\\,\\!", "08c82a904741b3547cd07bd0133a4b1a": "p'=(R'_{1},\\ldots ,R'_{n})", "08c8527a545b434e1ef1a2bca515f75b": "P^{2}{\\begin{pmatrix}x\\\\y\\\\z\\end{pmatrix}}=P{\\begin{pmatrix}x\\\\y\\\\0\\end{pmatrix}}={\\begin{pmatrix}x\\\\y\\\\0\\end{pmatrix}}=P{\\begin{pmatrix}x\\\\y\\\\z\\end{pmatrix}}.", "08c856d8f409003cdee6c9222fa5e399": "(b\\times b)+b+b+a", "08c87a3a9844f1d117a5ebb4730602c5": "{\\overrightarrow {F}}_{Hp}=+q({\\overrightarrow {v}}_{p}\\times {\\overrightarrow {B}}_{z})", "08c885160457626604ed9c52b2a8f0a4": "(1.00,0.00);", "08c88ebb97d05adc97468f81c1fb528b": "\\mathrm {D} _{H}F\\;", "08c8a49d2aa38647e4353bb9efccd38e": "{\\frac {d^{\\alpha }}{dx^{\\alpha }}}x^{\\beta }={\\begin{cases}{\\frac {\\beta !}{(\\beta -\\alpha )!}}x^{\\beta -\\alpha }&{\\hbox{if}}\\,\\,\\alpha \\leq \\beta ,\\\\0&{\\hbox{otherwise.}}\\end{cases}}\\qquad (1)", "08c8d8c4a9434d723579ed18502c4e7c": "{\\mathfrak {der}}(A)\\oplus {\\mathfrak {der}}(B)", "08c91383a07e35c6243eaea2c0ca0f56": "{\\frac {\\langle {y,Ly}\\rangle }{\\langle {y,y}\\rangle }}={\\frac {\\int _{a}^{b}{y(x)\\left(-{\\frac {d}{dx}}\\left[p(x)y'(x)\\right]\\right)}dx+\\int _{a}^{b}{q(x)y(x)^{2}}\\,dx}{\\int _{a}^{b}{w(x)y(x)^{2}}\\,dx}}", "08c9515739b61737ec2867c854042cde": "var(e)=var((I-H)Y)=(I-H)var(Y)(I-H)'=\\sigma ^{2}(I-H)^{2}=\\sigma ^{2}(I-H)", "08c954bd6b08eeeeaed0e08fc6821089": "P\\land Q\\land x\\leq P", "08c9ac1bf5d350fa6e3d49cd32fc3fac": "{\\mathcal {L}}_{xy}^{0}", "08c9c52ff003e23c56c3c4d27f19baa3": "\\{x\\in B\\mid \\phi \\}", "08c9e1162747a6a960f27c76d067aea1": "\\scriptstyle (p^{2}-q^{2},\\,2pq,\\,p^{2}+q^{2})", "08c9fd72ae28c76aed5805c3549437b1": "\\phi _{<}=-{\\frac {3}{\\kappa +2}}E_{\\infty }r\\cos \\theta \\ ,", "08ca46b884a7d9c3da24a279e3a7d043": "f(n)\\sim g(n)\\!", "08ca9adbc56c52fa20b365d678e052b7": "E=\\sum _{n=-\\infty }^{\\infty }e[n]^{2}", "08cabaffce0531f5356640d2eb4235ec": "\\sin(\\delta _{1}+\\delta _{2}+\\delta _{3}\\,)", "08cb4fb03016be77d395704a252db69b": "[x_{1},\\;x_{2},\\;x_{3},\\;x_{1}^{2},\\;x_{1}x_{2},\\;x_{1}x_{3},\\;x_{2}^{2},\\;x_{2}x_{3},\\;x_{3}^{2}]", "08cb64a670be8e46fa99d9df94b9802a": "\\Omega _{B}", "08cb7a1538984812ad8215c44d24e4e9": "{\\frac {\\pi }{4}}\\ (45^{\\circ })", "08ccbd2b322d70ceb83bea7412b75e9c": "\\sum _{k=0}^{\\infty }T^{k}", "08ccc306d3620da015dc0d127b388cc1": "\\rho (S({\\hat {D}}),S(D))\\leq \\epsilon ", "08ccd2f0ad5b24a408c81fced4e598a4": "{(x+y)/2+z \\over 2}\\neq {x+(y+z)/2 \\over 2}\\qquad {\\mbox{for all }}x,y,z\\in \\mathbb {R} {\\mbox{ with }}x\\neq z.", "08ccd5825593984505f863b96864bc35": "u(1)_{A}\\oplus u(1)_{B}\\oplus u(1)_{C}", "08ccefdecc07c994e16cac8520b58554": "{\\frac {1}{2}}\\hbar \\omega _{q}", "08cdcc437830fe62c1399d3c5072678f": "logK_{a}+logK_{b}=logK_{w}", "08ce00ce1306d33f0f93b7a0060e8e85": "H_{k+1}=H_{k}+{\\frac {(\\Delta x_{k}-H_{k}y_{k})(\\Delta x_{k}-H_{k}y_{k})^{T}}{(\\Delta x_{k}-H_{k}y_{k})^{T}y_{k}}}", "08ce088f83959c520215c592a78f02ec": "\\int _{0}^{\\infty }{\\frac {\\sin ^{2}px}{x^{2}}}\\ dx={\\frac {\\pi p}{2}}", "08ce26bd38e03dfedf2ed49be057bd69": "A,B\\in {\\mathcal {C}}", "08ceb334207d72356e9e6909e1c14dc4": "S_{x_{B-l}}=\\mathbb {I} \\otimes S_{x_{l}}", "08cf1b851c60ce9b51bd7b140373baae": "{\\begin{matrix}{2 \\choose 1}{2 \\choose 1}{3 \\choose 2}{40 \\choose 1}\\end{matrix}}", "08cf3af0c3887289761a35076df9adf3": "{PV \\over {\\operatorname {constant} }}=T={{2.50\\times 10^{6}*100} \\over {3.33\\times 10^{5}}}=751", "08d0526ee3dfd44324b874909c5f655f": "I(t):={\\overline {g}}(0,y,0)", "08d05900d6e0b782ea2e61a4d8eff2e1": "\\{H(N),H(M)\\}=C(K)", "08d07705270c0dbe8b96f70df98a89b6": "1\\leq p\\leq 2", "08d095f730c930b731cb4773868b7355": "h=T^{1/\\alpha }", "08d0c59c593458184f0119269d48af3f": "Z_{n^{s+1}}^{*}", "08d0df5077e5dfb76cf50b5aecc1d6a3": "x_{T}=\\sum _{i\\in T}x_{\\{i\\}},", "08d0f85eefa9adb6af147f8109fe85a8": "c_{o}={\\frac {1}{\\alpha _{o}}}\\left[C_{a}\\alpha _{a}+C_{b}\\alpha _{b}\\left(1-\\alpha _{a}\\right)\\right]", "08d17855acddd72fa221f533508acae2": "\\phi =LI\\ ", "08d1afc218ec4f314312137ef8a8193c": "{\\mathcal {E}}", "08d23c49ab9b985b0c3672033948c724": "k_{0},k_{1},k_{2}", "08d2ea5371ea545a6007314da073a4a9": "\\left(S(0),E(0),I(0),R(0)\\right)\\in \\left\\{(S,E,I,R)\\in [0,N]^{4}:S\\geq 0,E\\geq 0,I\\geq 0,R\\geq 0,S+E+I+R=N\\right\\}", "08d2f62b50b798eb9750f76c060727de": "{\\tfrac {{\\vec {x}}(t_{n+1})-{\\vec {x}}(t_{n})}{\\Delta t}}", "08d335a1bfafde67d9640c6b4561ba3b": "V_{S}^{n}", "08d352e23960b64a603f36ea901d63a3": "m(X)={\\text{max}}(X)", "08d42eb1eec575bbc358c513432853bd": "h_{x}", "08d4418ed0d2874a95b923b8d6685fe6": "A_{\\lambda }:=\\bigcup _{\\alpha <\\lambda }A_{\\alpha }", "08d457e8ccd5d50d4647ef64ee79694a": "Z={\\frac {\\zeta ^{N}}{N!}}.", "08d47bee5cd96f3c0a4b7f1c50ccb441": "\\!-\\!\\left(1\\!+\\!{\\frac {\\nu }{2}}\\right)\\psi \\!\\left({\\frac {\\nu }{2}}\\right)", "08d4bac20c0b44fd65b7146822121760": "\\Gamma ({\\tfrac {1}{4}})", "08d4c589a98368dfe97e2102f0c7ebdc": "A_{fcc}={\\frac {4J_{ex}S^{2}}{a}}", "08d4e8fa01149867c4a3b1fc8ef0519e": "\\mathrm {Nu} =2+0.6\\,\\mathrm {Re} ^{\\frac {1}{2}}\\,\\mathrm {Pr} ^{\\frac {1}{3}},~0\\leq ~\\mathrm {Re} <200,~0\\leq \\mathrm {Pr} <250", "08d50ae005bfbbe2b027c14c1780a45b": "{\\dot {r}}={\\frac {dr}{d\\theta }}\\cdot {\\dot {\\theta }}", "08d581da776c2a7cb178e6cdc514a4b4": "\\alpha _{1}=1-(1-\\alpha )^{1/n}.", "08d58d7d444e14ae1eb15fd4569e3256": "{\\vec {\\chi }}", "08d5934ad37a87e6335ee6438c52487a": "{\\frac {\\delta Z}{\\delta N}}=0=\\int \\left.{\\frac {\\delta I[g_{\\mu \\nu },\\phi ]}{\\delta N}}\\right|_{\\Sigma }\\exp \\left(-I[g_{\\mu \\nu },\\phi ]\\right)\\,{\\mathcal {D}}{\\mathbf {g}}\\,{\\mathcal {D}}\\phi ", "08d620536156da2a2bfe95a74d5ef516": "{\\frac {\\partial }{\\partial s}}p(x,s)=-{\\frac {\\partial }{\\partial x}}[\\mu (x,s)p(x,s)]+{\\frac {1}{2}}{\\frac {\\partial ^{2}}{\\partial x^{2}}}[\\sigma ^{2}(x,s)p(x,s)]", "08d62a4ee85116546a8eecf9f4ecf747": "f'(x)=\\lim _{h\\to 0}{f(x+h)-f(x) \\over h}.", "08d682768c9c10f0c3f69c4a2c91a6e6": "K_{Y}(t)={\\mbox{ln}}E[e^{tY}]={\\mbox{ln}}E[E[e^{tY}|N]]={\\mbox{ln}}E[e^{NK_{X}(t)}]=K_{N}(K_{X}(t)).\\,", "08d69c15c33bd9c730364b5f6518971b": "L_{1}=T_{1}-{\\frac {1}{2}}\\left((V_{R})_{1}^{2}+(V_{A})_{1}^{2}\\right)", "08d6bb270ed44924f2dc62751f4baa42": "\\{n1,n2\\}", "08d6c0c272a07fe80bbdbf7df87c50ab": "E_{0}=\\omega A_{0}", "08d6ce9d113abdaa391d55c2f666861a": "(\\ln x,x),", "08d6d8834ad9ec87b1dc7ec8148e7a1f": "PQ", "08d6f8414f9de6c65eeb2fc47ac3e0f8": "p^{*}:{\\check {H}}^{*}(P_{\\delta },\\partial P_{\\delta })\\to {\\check {H}}^{*}(E_{\\delta },\\partial E_{\\delta })", "08d7697676392eb85a784c4a70455b1c": "\\sum _{n}\\mathbb {P} \\left(|X_{n}-X|>\\varepsilon \\right)<\\infty ,", "08d7e6cbce5d42754894ff52cbdb394a": "{\\overset {\\circ }{\\boldsymbol {\\sigma }}}={\\boldsymbol {R}}\\cdot \\left[{\\cfrac {d}{dt}}\\left({\\boldsymbol {R}}^{-1}\\cdot {\\boldsymbol {\\sigma }}\\cdot {\\boldsymbol {R}}^{-T}\\right)\\right]\\cdot {\\boldsymbol {R}}^{T}={\\boldsymbol {R}}\\cdot \\left[{\\cfrac {d}{dt}}\\left({\\boldsymbol {R}}^{T}\\cdot {\\boldsymbol {\\sigma }}\\cdot {\\boldsymbol {R}}\\right)\\right]\\cdot {\\boldsymbol {R}}^{T}", "08d802e12ff6931b47bc2378f02c10ff": "K\\leq {\\tfrac {1}{2}}{\\sqrt {(a^{2}+c^{2})(b^{2}+d^{2})}}", "08d8044fea0d480e139acaf126e1773a": "V_{C}\\ =\\ G_{C}V_{in}e^{j\\phi _{C}}", "08d82c4f51a5000f45b30c9daf559333": "A\\;=\\;\\int _{-r}^{r}2{\\sqrt {r^{2}-x^{2}}}\\,dx\\;=\\;\\pi r^{2}", "08d83ed8382ea113854c1f7d51ce90ee": "{\\big .}{\\frac {\\Delta Q}{\\Delta t}}={\\frac {A\\,(-\\Delta T)}{{\\frac {\\Delta x_{1}}{k_{1}}}+{\\frac {\\Delta x_{2}}{k_{2}}}+{\\frac {\\Delta x_{3}}{k_{3}}}+\\cdots }}.", "08d84a38b4e91bc582967a44cf358ede": "d(1)=0", "08d901ef964a996c92dafaea0b1e35f6": "n(0.5;H)\\approx 1.1774{\\sqrt {H}}.\\,", "08d94cb53e35bca6273ede095c2609a1": "F\\in S", "08d971745912cb478423a7848b89a938": "Y=Z^{-1}={\\frac {1}{R+jX}}=\\left({\\frac {1}{R^{2}+X^{2}}}\\right)\\left(R-jX\\right)", "08d9f577d9d8c84e7b863e54ce8070d6": "X_{n}={\\frac {(1+r)}{(1+r-2rp)}}", "08da3c6c4e61f529ce1690ff8bba7e3a": "x=\\sum _{i=0}^{n}10^{n-i}a_{i}/10^{n}=\\sum _{i=0}^{n}{\\frac {a_{i}}{10^{i}}}", "08da9695cd4a3c791da5473a5b4a5a53": "\\psi (\\Omega )^{\\psi (\\Omega )^{\\psi (\\Omega )}}", "08dab8b0af8f3513821d06101bc32c88": "\\bigwedge L=a_{1}\\land \\cdots \\land a_{n}", "08daf53dede6ad2e156adddcc66f51cd": "\\sum _{k=0}^{\\infty }{\\frac {1}{16^{k}}}\\left({\\frac {4}{8k+1}}-{\\frac {2}{8k+4}}-{\\frac {1}{8k+5}}-{\\frac {1}{8k+6}}\\right)=\\pi \\!", "08db150fdf3a2ba97c9aa41f12fe4e9b": "f(u)=0", "08db3e889f5b99601143bdab5ad9c09a": "\\gamma ={\\frac {\\left|P_{1}-P_{2}\\right|^{2}\\left(P_{3}-P_{1}\\right)\\cdot \\left(P_{3}-P_{2}\\right)}{2\\left|\\left(P_{1}-P_{2}\\right)\\times \\left(P_{2}-P_{3}\\right)\\right|^{2}}}", "08db52dff2b122f64ac22b1bf8455ca9": "{\\binom {p^{2}}{i}}", "08db6ba19dc5c27b3903d4c237b662f0": "0\\leq \\lambda \\perp x_{1}\\geq 0", "08dbbb1365a9f1fd8616be57dd6b4d46": "{\\frac {\\partial \\xi }{\\partial t}}+V\\cdot \\nabla \\eta -f{\\frac {\\partial \\omega }{\\partial p}}=\\left(\\xi {\\frac {\\partial \\omega }{\\partial p}}-\\omega {\\frac {\\partial \\xi }{\\partial p}}\\right)+k\\cdot \\nabla \\omega \\times {\\frac {\\partial V}{\\partial p}}", "08dbf67ca14ddc19fc2dc46bdf70bf3e": "=rV_{a}+\\theta V_{d},", "08dc1a0308ff0b2ed55b865f9ab9a288": "\\partial _{\\mu }\\partial _{\\nu }E_{n}", "08dc2c67c879d6bc197b16bd888ae776": "P_{K}(p^{0},p^{1},u)={\\frac {C(u,p^{1})}{C(u,p^{0})}}", "08dc4994bf9b90f91834c1122e5d1ee8": "U(\\theta ,\\phi )", "08dd266487113c297ab1a73113b8b7ba": "-{\\frac {\\partial v(p^{0},w^{0})/(\\partial p_{i})}{\\partial v(p^{0},w^{0})/\\partial w}}=x_{i}(p^{0},w^{0}),i=1,\\dots ,n.", "08dd6653d00fcf3035261c77857a5574": "{}_{S}", "08dda92627e6174380bae79bf9e78876": "t{\\begin{Bmatrix}r,q,p\\end{Bmatrix}}", "08ddeffa622d13c5c5cb4502ab3444bf": "\\mathbf {ABCD} =((\\mathbf {AB} )\\mathbf {C} )\\mathbf {D} =(\\mathbf {A} (\\mathbf {BC} ))\\mathbf {D} =\\mathbf {A} ((\\mathbf {BC} )\\mathbf {D} )=\\mathbf {A} (\\mathbf {B} (\\mathbf {CD} ))=(\\mathbf {AB} )(\\mathbf {CD} )", "08de2608d958ae73328adbf4d7149456": "f:X\\rightarrow \\mathbb {R} ", "08de4748517e2bf16fb6f251f2e255f3": "\\mathrm {NA} =n\\sin \\theta ,\\;", "08df396a34293d78ce62abd5a8d01d25": "\\varphi (n)={\\mathcal {F}}\\left\\{\\mathbf {x} \\right\\}[1]=\\sum \\limits _{k=1}^{n}\\gcd(k,n)e^{{-2\\pi i}{\\tfrac {k}{n}}}.", "08df6c446109617e11f8af593a09a12b": "\\scriptstyle \\psi ^{\\dagger }", "08dfe7bed8bef41a32d8495aa7a0ddef": "(A^{4}+A^{3}+A^{2}+A+I)b4+63", "08dfea5322f1c1491fcdbd43ee5b762e": "w(C_{1}\\mid C_{2})\\geq \\min\\{2w(C_{1}),w(C_{2})\\}", "08dfebdcd4a178d5446dfcc32049f5c4": "S_{D}(\\lambda )=S_{0}(\\lambda )+M_{1}S_{1}(\\lambda )+M_{2}S_{2}(\\lambda )", "08e004525d0a192ccee7fa267bc3b937": "B=-V_{XY}V_{YY}^{-1}.", "08e03148e101fdabf93fbe0c12664581": "{\\tilde {g}}:=\\lambda g_{0}+(1-\\lambda )g_{1},\\qquad \\lambda \\in [0,1],", "08e0cf27ef94307f45431d40de5505fc": "D^{2}=\\{z:|z|<1\\}", "08e0fa8ae305e4b7d100a092cc27378b": "dV=\\left({\\frac {\\partial V}{\\partial T}}\\right)_{P}dT+\\left({\\frac {\\partial V}{\\partial P}}\\right)_{T}dP\\,", "08e12ee7ce42bda0ba715ded03878d99": "q\\times p", "08e13b1079166a55617771f7b334c6cd": "{\\begin{bmatrix}\\cos \\delta \\cos h\\\\\\cos \\delta \\sin h\\\\\\sin \\delta \\end{bmatrix}}={\\begin{bmatrix}\\sin \\phi _{o}&0&\\cos \\phi _{o}\\\\0&1&0\\\\-\\cos \\phi _{o}&0&\\sin \\phi _{o}\\end{bmatrix}}{\\begin{bmatrix}\\cos a\\cos A\\\\\\cos a\\sin A\\\\\\sin a\\end{bmatrix}}", "08e19bf3ab3ba9d80091f6f4c84d1ce4": "x=\\lambda \\sinh \\left({\\frac {1}{\\sigma }}\\Phi ^{-1}(U)-\\gamma \\right)+\\chi ", "08e1edd9d4a32a6033d51f585d081b02": "\\omega =180^{\\circ }", "08e1f9820e9a919ef61eeaea7cc71eca": "F^{\\times }/F^{\\times 2}", "08e23c9c42a120c5b625a0453617f930": "\\beta ={b+a \\over 2}", "08e27ef958c6435f3ed4d3ef20f8cb80": "\\beta (x)={\\frac {1}{16\\pi }}\\int _{1}^{\\infty }u^{-3/2}e^{-xu}du", "08e2b4fddba3995e9853afd85c02088d": "\\int \\ln(ax+b)\\;dx={\\frac {(ax+b)\\ln(ax+b)-ax}{a}}", "08e2c000fe3961fe1d9c8e52c87fdd99": "\\gamma (n)", "08e2dbe2f7a3e5f55706d9f3598f786c": "\\sin ^{2}\\theta \\!", "08e2e7391e60bde55864396663d8b388": "\\phi ^{0}=A\\left({\\begin{array}{c}\\Psi _{1}^{0}\\\\\\Psi _{2}^{0}\\end{array}}\\right)", "08e3447e2a35226511816695287a9376": "u^{2}", "08e364f63399e5ab2d1aeef34e61a3b3": "{\\mathbb {M}}", "08e383fdce48b725492a61dc8b59a0e9": "\\neg (A\\lor B)\\to \\neg A\\land \\neg B", "08e3952227151a571c6047d62029cea6": "{\\begin{aligned}\\pi _{1}&=E/As\\\\\\pi _{2}&=\\ell /A.\\end{aligned}}", "08e3baeb8ba48e236aaacd9da759f946": "f(\\phi )", "08e41fb36dc83be86be1dd34396f9c27": "\\cot {\\frac {3\\pi }{20}}=\\cot 27^{\\circ }={\\sqrt {5}}-1+{\\sqrt {5-2{\\sqrt {5}}}}\\,", "08e484af580944a18f3169b488733888": "|\\mu |\\,", "08e513f7ded212e7c5fcaf7f9d44fe5f": "L:\\{0,1\\}^{n}\\rightarrow \\{0,1\\}", "08e514d2648a9a22a96ebd8241799f70": "\\mu ^{*}=\\mu ", "08e5a2b68e1046fae8bebff11af3aacb": "\\mathbf {B} '=\\mathbf {B} -{\\frac {1}{{c_{0}}^{2}}}\\mathbf {v} \\times \\mathbf {E} ,", "08e60a0397275b61f281a6eaa3d8a744": "-\\pi \\leq \\arccos(u)\\leq \\pi ", "08e60b1b7f79153557b39e185fbcf162": "f(x)=e^{x}", "08e6197d77729f2c31ddf3127eda43ab": "C_{\\beta J}^{\\;\\;\\;K}e_{I}^{\\alpha }e_{K}^{\\beta }", "08e64f7a0a4d807bfac9e2f1158dd22c": "\\lambda _{\\alpha }\\,\\!", "08e65e29a9040aa76d13e56a74406807": "\\mathrm {Distance} =\\mathrm {Speed} \\cdot \\mathrm {Time} ", "08e67c9ac8c32eb0b951448037073274": "[\\![x_{1}]\\!]\\in [\\![{\\mathsf {T}}_{1}]\\!],~[\\![x_{2}]\\!]\\in [\\![{\\mathsf {T}}_{2}]\\!],~\\ldots ,~[\\![x_{n}]\\!]\\in [\\![{\\mathsf {T}}_{n}]\\!]", "08e6a0743b311f3036b4b17558812e21": "A=\\int \\lambda \\,d\\operatorname {E} _{A}(\\lambda ),", "08e6f27bf45594a08739a6c832914f49": "{\\begin{aligned}F&={\\frac {{\\text{lack-of-fit sum of squares}}/{\\text{degrees of freedom}}}{{\\text{pure-error sum of squares}}/{\\text{degrees of freedom}}}}\\\\[8pt]&={\\frac {\\left.\\sum _{i=1}^{n}n_{i}\\left({\\overline {Y}}_{i\\bullet }-{\\widehat {Y}}_{i}\\right)^{2}\\right/(n-p)}{\\left.\\sum _{i=1}^{n}\\sum _{j=1}^{n_{i}}\\left(Y_{ij}-{\\overline {Y}}_{i\\bullet }\\right)^{2}\\right/(N-n)}}\\end{aligned}}", "08e745a2e1847d10cb104681ed0ff38f": "T=R_{01}R_{02}\\cdots R_{0N}", "08e757e7a55156f0010fadd95bbf1781": "{\\tfrac {1}{12}}\\pi ^{2}-{\\tfrac {1}{2}}\\ln ^{2}2\\,", "08e76895ab552c92a9cd75fb45b9c754": "\\nabla ^{2}\\phi =0", "08e76d529ee95a53711040601988d621": "Z=\\sum _{n=0}^{\\infty }{\\frac {(644n+41)\\left({\\frac {1}{2}}\\right)_{n}\\left({\\frac {1}{4}}\\right)_{n}\\left({\\frac {3}{4}}\\right)_{n}}{(n!)^{3}5^{n}{72}^{2n+1}}}\\!", "08e7930b39961cdad59c78701ed88402": "V=\\left\\{(g_{i},h_{j}):1\\leq i\\leq n,1\\leq j\\leq m\\right\\}", "08e7f6feafd6d6c18b0ff35c21d29897": "\\psi \\varphi \\neq \\emptyset ", "08e8057498083c0f8b73ec308f18e43a": "{\\frac {\\mathrm {d} {\\boldsymbol {H^{\\prime }}}}{\\mathrm {d} t}}=-\\nabla {\\boldsymbol {H^{\\prime }}}\\cdot \\nabla {\\vec {v}}\\qquad {\\boldsymbol {H^{\\prime }}}(t=0)={\\boldsymbol {I}}", "08e80def44c6cdec7c177f66635a0525": "s_{1}=\\left[0,{\\sqrt {2}}\\right]^{T},\\quad s_{2}=\\left[-{\\sqrt {3 \\over 2}},-{\\sqrt {1 \\over 2}}\\right]^{T},\\quad s_{3}=\\left[{\\sqrt {3 \\over 2}},-{\\sqrt {1 \\over 2}}\\right]^{T}", "08e8326c8e0191eb1b8bf451aa73d72d": "\\mathbf {I} =\\sum _{i=1}^{3}\\sum _{j=1}^{3}I_{ij}\\mathbf {e} _{i}\\otimes \\mathbf {e} _{j}.", "08e8441a508c8088981c7dfc9e15d5a8": "I_{D}\\propto V_{GS}", "08e884b06934b8b0bffe39483a339a86": "{\\partial \\rho \\over \\partial t}=-\\nabla \\cdot \\mathbf {J} .", "08e8a9929b064bc2e3ac205901d12970": "r_{i+31}=y_{64+2i+1}", "08e8b5adf68cce4d461bb42cd893fc94": "\\ \\psi (x,t)", "08e8b5e2173db295038b46d2c914b352": "x^{2}\\cdot g(x),\\quad (x^{2}+1)\\cdot g(x),\\quad (x^{2}+x)\\cdot g(x),\\quad (x^{2}+x+1)\\cdot g(x).", "08e8c77df8d74a752f8c1c5ebf78ab88": "B_{E}:X_{E}\\to X_{E}^{*}", "08e90359c2a70d26e2a5a0f8caa48260": "X\\Rightarrow Y", "08e91d1ed178c767ef40109a1da1d511": "n_{g}={\\frac {c}{v_{g}}}", "08e937fa71b77a8dc61e97d3b8abbad8": "{\\mathbf {\\sigma } }=(\\sigma _{x},\\sigma _{y},\\sigma _{z})=\\left[{\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix}},{\\begin{pmatrix}0&-{\\rm {i}}\\\\{\\rm {i}}&0\\end{pmatrix}},{\\begin{pmatrix}1&0\\\\0&-1\\end{pmatrix}}\\right]", "08e9eea18e247c6df9837a00b859452e": "DPA_{n}", "08ea2a5d8434785c5d721f12956f2c15": "\\omega _{c}=qeB/2\\pi m", "08ea6f30ade95e39c2d19cff292b14fa": "f(a)\\cdot h\\cdot (1/2)", "08eafe38f780b3fdc0cd19d06299b793": "E_{0}=E_{k}", "08eb318d1ddf22c353da60249b9bc9f0": "g(\\psi )", "08eb4315e76231f7296b0834191aa7b5": "p(\\theta ,\\psi )", "08eb9f2a5d8489a805a9c51efcd74352": "{\\sqrt {T}}{\\Big (}{\\tfrac {1}{T}}\\textstyle \\sum _{t=1}^{T}z_{t}-\\operatorname {E} [z_{t}]{\\Big )}\\ \\xrightarrow {d} \\ {\\mathcal {CN}}(0,\\,\\Gamma ,\\,C),", "08ebe2b53b1ea788e252b02e21e2c786": "\\left(p(e_{0}),\\ldots ,p(e_{n+1})\\right)", "08ec3075a527c2b93364d1db50f6aab0": "\\Phi _{Y,X}(f)=g:Y\\to GX", "08ecbf4e98447b699f181a3c9a5357ce": "{\\frac {U}{t}}\\sim \\nu {\\frac {U}{y^{2}}}", "08ecc0e3d5b5840e9d211abb2ff652f3": "H(Y,p,s)\\geq H(X,p,r)+\\lambda ||X-Y||", "08ecc318fdf6cc03e3cb7b7e1870479b": "P_{j},", "08ece0b3ba7be4d6f630cc745e32d610": "{\\frac {-{\\sqrt {b}}+ae^{\\tfrac {(-1+a)^{2}}{4b}}{\\sqrt {\\pi }}\\operatorname {erfc} \\left({\\tfrac {-1+a}{2{\\sqrt {b}}}}\\right)}{a^{2}{\\sqrt {b}}}}", "08ecfc1ed8d5360233d57617c89c6442": "P(T|H)", "08ecfdd539947e2df450f0c1226bf8cf": "\\Phi _{n}((-1)^{\\frac {n-1}{2}}z)=C_{n}^{2}(z)-nzD_{n}^{2}(z).", "08ecff33bb1c40c3d72cfe1fc7da929b": "(p,\\gamma )", "08ed0c5ef3a91bdb75903bc3b7ec2189": "X={\\begin{bmatrix}0&X_{12}&X_{13}&\\cdots &X_{1,r-1}&X_{1r}\\\\&0&X_{23}&\\cdots &X_{2,r-1}&X_{2r}\\\\&&&\\ddots &\\\\&&&\\cdots &0&X_{r-1,r}\\\\&&&&&0\\end{bmatrix}}", "08ed3f9b2550f875b183b4c1765e2b85": "{x^{2}+x-1=0}", "08eda79b1e3a8f2682f6c3063fab9638": "p\\rightarrow \\Box \\Diamond p", "08edd9e5c20acddea49f699e8b0970f8": "\\xi _{0}(z)=\\pi ^{-1}(1+|z|^{2})^{-2}", "08edecbef79ec2cdca39e715813389ec": "B'\\subset k", "08ee50ad97f51e6782fdd8f3593453e1": "\\langle A(a)B(b)\\rangle =\\langle A(a')B(b)\\rangle =\\langle A(a')B(b')\\rangle ={\\frac {1}{\\sqrt {2}}}", "08ef56c3ed484b26a592570b29a8f1d8": "x={\\frac {p}{2}}+{\\sqrt {\\left({\\frac {p}{2}}\\right)^{2}-q}}", "08ef5a82b09cbeaea8d09b50aa9d2076": "^{\\;}H(\\xi )=H(c(\\xi ,\\tau ))", "08ef9e22b9761a2996927ae79ea7cc1c": "I_{1},J_{2},J_{3}", "08efcbca510dcc7213342e34cdb3ba1d": "0\\leq y\\leq \\pi \\,", "08f01b23c07ab3c3d3375bb54a7fb5b1": "A+B\\leftrightharpoons AB;K_{AB}={\\frac {[AB]}{[A][B]}}", "08f04976d008faca6991d0f8c07dd154": "O(N_{k,n})", "08f056072a1eec09f51282bcdf204126": "U_{-}=\\bigcap _{n\\geq 0}\\alpha ^{-n}(U)", "08f05db3265da9ffc7132056f15a212f": "{\\overline {(z+w)}}={\\overline {z}}+{\\overline {w}}\\!\\ ", "08f0663a778138bd3d254e0ac4b8d901": "{\\frac {(P\\leftrightarrow Q)}{\\therefore (Q\\to P)}}", "08f0aa58402bc9ed27aa92b910bf91e5": "\\partial \\Phi /\\partial {y}", "08f0cd3e07ea8522b812b23e197673a3": "X=1", "08f0e891b1fe8a269be49d376002fa33": "{\\mathcal {H}}=\\operatorname {span} \\{\\phi _{i}\\}", "08f11a635e5d15f16491f8e112cfa934": "{\\frac {\\partial }{\\partial t}}(\\nabla ^{2}\\psi )+(\\nabla \\times {\\vec {\\psi }})\\cdot \\nabla (\\nabla ^{2}\\psi )=\\nu \\nabla ^{4}\\psi ", "08f1245e8b0e04c1c0b8dcaea0f98a30": "f(x,y)=\\left(1.5-x+xy\\right)^{2}+\\left(2.25-x+xy^{2}\\right)^{2}", "08f1447d656403acec8fa6758b80ce07": "(\\vee ,0)", "08f1503414f0c9303c9f7a17d0bf956e": "N(a+b{\\sqrt {-5}})=a^{2}+5b^{2}", "08f16ff29dca37546da4e3616abfcf14": "\\gamma _{x}^{+}:=\\{\\Phi (t,x):t\\in (0,t_{x}^{+})\\}", "08f17254282d7207a708c0966a83803a": "dl^{2}=e^{-\\lambda (r)}{dr^{2}}+r^{2}d\\theta ^{2}+r^{2}\\sin ^{2}\\theta d\\phi ^{2}\\,", "08f1ae61806daf3ff87a5bfaf55ee0c4": "{\\frac {\\partial ^{2}F}{\\partial x^{2}}}=\\lim _{\\epsilon \\rightarrow 0}{\\frac {[F(x+\\epsilon )-F(x)]+[F(x-\\epsilon )-F(x)]}{\\epsilon ^{2}}}.", "08f2053209f10b4d7478efc6d7e86fbd": "r_{upb}={\\frac {M_{1}-M_{0}-1}{\\sqrt {{\\frac {n^{2}s_{n}^{2}}{n_{1}n_{0}}}-2(M_{1}-M_{0})+1}}}.", "08f2266fab746e856bb9d7cf8bb426eb": "f(z)=\\sum _{k=1}^{\\infty }a_{k}z^{\\lambda _{k}}=\\sum _{n=1}^{\\infty }b_{n}z^{n}\\,", "08f28ccc584b23779d02a87c7dd66bae": "{\\hat {S}}={\\hat {\\sigma }}_{+}+{\\hat {\\sigma }}_{-}", "08f28e7f4d4560325c62e65007df8c02": "{\\begin{aligned}&z\\left({x_{1}\\,\\,x_{2}}\\right)\\,\\,\\,\\,\\approx \\,\\,\\,z\\left({{\\bar {x}}_{1}\\,\\,{\\bar {x}}_{2}}\\right)\\,\\,\\,+\\,\\,\\,\\,{{\\partial z} \\over {\\partial x_{1}}}\\left({x_{1}-\\,\\,{\\bar {x}}_{1}}\\right)\\,\\,\\,+\\,\\,\\,{{\\partial z} \\over {\\partial x_{2}}}\\left({x_{2}-\\,\\,{\\bar {x}}_{2}}\\right)\\,\\,\\,\\\\&\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+\\,\\,\\,{1 \\over 2}{{\\partial ^{2}z} \\over {\\partial x_{1}\\partial x_{2}}}\\left({x_{1}-\\,\\,{\\bar {x}}_{1}}\\right)\\left({x_{2}-\\,\\,{\\bar {x}}_{2}}\\right)\\,\\,\\,+\\,\\,\\,{1 \\over 2}{{\\partial ^{2}z} \\over {\\partial x_{2}\\partial x_{1}}}\\left({x_{2}-\\,\\,{\\bar {x}}_{2}}\\right)\\left({x_{1}-\\,\\,{\\bar {x}}_{1}}\\right)\\\\&\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+\\,\\,\\,{1 \\over 2}{{\\partial ^{2}z} \\over {\\partial x_{1}\\partial x_{1}}}\\left({x_{1}-\\,\\,{\\bar {x}}_{1}}\\right)\\left({x_{1}-\\,\\,{\\bar {x}}_{1}}\\right)\\,\\,\\,+\\,\\,\\,{1 \\over 2}{{\\partial ^{2}z} \\over {\\partial x_{2}\\partial x_{2}}}\\left({x_{2}-\\,\\,{\\bar {x}}_{2}}\\right)\\left({x_{2}-\\,\\,{\\bar {x}}_{2}}\\right)\\end{aligned}}", "08f2b9e70911eb2d15fa8370c13dfc3f": "P^{S}=P^{D}=P>0", "08f2dfea71c4a47fc51caaedb6300a23": "\\Delta \\lambda \\approx \\Delta \\theta \\left({\\partial \\Theta \\over \\partial \\lambda }\\right)^{-1}", "08f330a24acd9b122e68adc7c6387ef5": "z\\mapsto z^{d}+c.\\ ", "08f334f6cae066f8b601b73dc24fea15": "{\\tilde {f}}(\\lambda )=\\int _{0}^{\\infty }F({a^{2}+a^{-2} \\over 2})a^{-i\\lambda }\\,da=\\int _{0}^{\\infty }F(\\cosh t)e^{-it\\lambda }\\,dt.", "08f3643f0984f22493e11d0f8c464f13": "{\\frac {(p_{02})_{actual}}{p_{01}}}=(1+{\\frac {\\eta _{stage}\\delta (T_{0})_{isentropic}}{T_{01}}})^{\\frac {\\gamma }{\\gamma -1}}\\,", "08f41e2b56730d87f1232d525303ba14": "vp", "08f4283df511534554d8ce82cb0d0941": "\\pi '\\,\\!", "08f429d0b28f1b961a3851966f2b6fc0": "t\\in [0,T]", "08f445ca40c14a51e65bad8c914d838b": "{\\frac {\\mathrm {d} f}{\\mathrm {d} x}}=2x\\neq {\\frac {\\partial f}{\\partial x}}=y=x", "08f46744671674fe1bf17fdd58382347": "[0,255]", "08f4769d05723f8c9116f1368697368b": "{\\bar {z}}=a-bi", "08f4a2c8e617f89fc58744e072773f2a": "u\\Vdash A[e]", "08f4ad80a2083e76ed11bdbd090de249": "E_{a}", "08f4b86a3b9ffd5604ee625a45321733": "{\\overline {C_{2}P_{n}}}", "08f514122fefcde5aa6d8972c196a49d": "v_{j,k}", "08f583fc3516e68625a5f2cdeb7fb21a": "BP={\\begin{pmatrix}I_{r}&G\\\\0&0\\end{pmatrix}}", "08f5aa8ca129fa89172b53a52f8df83a": "\\forall \\beta .\\beta \\rightarrow \\ {\\mathit {int}}", "08f5af0d3874b9d9388108ed70fa0102": "E_{k}", "08f5c03968fa3794fa9f9c09a43bf37f": "U^{n}=[-1,+1]^{n}", "08f5cb9da1875f9fc5d801ff4bc687d5": "A_{xx}", "08f5cf5212a6cc3d28fa40a8c49a7be9": "\\displaystyle \\zeta (t)=g_{t}(\\gamma (t))", "08f5f837cd0abcc5b245418c1c56fdba": "\\left\\{{\\begin{matrix}\\ln \\ \\gamma _{1}=x_{2}^{2}\\left[\\tau _{21}\\left({\\frac {G_{21}}{x_{1}+x_{2}G_{21}}}\\right)^{2}+{\\frac {\\tau _{12}G_{12}}{(x_{2}+x_{1}G_{12})^{2}}}\\right]\\\\\\\\\\ln \\ \\gamma _{2}=x_{1}^{2}\\left[\\tau _{12}\\left({\\frac {G_{12}}{x_{2}+x_{1}G_{12}}}\\right)^{2}+{\\frac {\\tau _{21}G_{21}}{(x_{1}+x_{2}G_{21})^{2}}}\\right]\\end{matrix}}\\right.", "08f6708c744a72a9ee3275f804894fb5": "a_{5}+b_{5}+c_{5}=2a_{1}", "08f6b46a4d42bcd73f38312383ad0604": "\\varphi =2\\cos(\\pi /5)=2\\cos 36^{\\circ }\\,", "08f6beca8f2a7c87668da3283c059dec": "\\textstyle N^{x}", "08f70ccac8039a54202c8369fd67c020": "-\\lambda -e^{-\\lambda }=0.\\,", "08f722f992cb9d089321e018b1a0dba1": "G_{\\infty }\\simeq 20G_{N}", "08f73e2aec7961927cffbd6e4f119531": "\\blacksquare \\,", "08f7c2c2243e386a631d31ffbc332451": "c_{1},c_{2}\\in \\mathbf {C} ", "08f7c7586e0cd494d7649321d99680af": "\\lim _{Z,Y\\to 0}Z_{\\mathrm {i} }=k", "08f8407d9559edaedba4edf49fcf5a87": "Z+(1+Z)^{2}t+(1+Z)^{3}t^{2}\\ .", "08f856dbd85b3ced877a379cb806013b": "G_{vv}=C_{xy}G_{yy}", "08f88b18a8abf049883523c51807ff25": "s=r_{0}\\int _{t_{0}}^{t}v/r\\sin \\theta \\,dt", "08f8e3f34408a8d9d0b3670ad56f3d22": "\\gamma z_{j};", "08f940ddfad6c62491b79a3b697f48cc": "(k-1,k)", "08f96b806f2515d0673fb8819ede102c": "\\operatorname {tr} (T_{a})=0\\,", "08f9c9d30750bf03ac5ff958956449c6": "H[i]", "08fa46a40b282211210ef4f3f2e38f69": "\\textstyle P({N}(b(o,r))=1\\mid o)", "08fa75039cafb2eeafa027d6384114e9": "{\\frac {\\tau _{b}}{(\\rho _{s}-\\rho )(g)(D)}}=f\\left(Re_{p}*\\right)", "08fac7bcdbacfa0e6696833e3192e2a2": "x^{i},\\;i=0,1,\\dots ", "08fad96e90651b7c9df05a4cb197ca09": "n_{\\Sigma }^{2}\\cos \\theta _{\\Sigma }\\left(\\sin \\theta _{\\Sigma }d\\theta _{\\Sigma }d\\varphi \\right)=n_{S}^{2}\\cos \\theta _{S}\\left(\\sin \\theta _{S}d\\theta _{S}d\\varphi \\right)", "08fadeb8b0203c165c724d1fbbeb89e5": "S={\\sqrt {(1+(p_{1\\cdot }+p_{\\cdot 1})(R-1))^{2}+4R(1-R)p_{1\\cdot }p_{\\cdot 1}}}.", "08fb08286daf9c5c7d2e9e1203294e52": "\\mathbf {F} =\\mathrm {d} \\mathbf {A} \\,.", "08fb1d4a3481d9f764119ebdf36fe7a4": "x-x\\wedge y\\wedge x", "08fb2f24befe571d4e697653bc136e1d": "f(z)={\\frac {1}{\\pi }}\\iint _{|\\zeta |<1}F(\\zeta ){\\frac {dS}{(1-{\\overline {\\zeta }}z)^{2}}},\\quad |z|<1.", "08fb4ce0b18fafadf98deb10acb9a10e": "{\\begin{alignedat}{8}\\zeta (s)&{}={}&1^{-s}+2^{-s}&&{}+3^{-s}+4^{-s}&&{}+5^{-s}+6^{-s}+\\cdots &\\\\2\\cdot 2^{-s}\\zeta (s)&{}={}&2\\cdot 2^{-s}&&{}+2\\cdot 4^{-s}&&{}+2\\cdot 6^{-s}+\\cdots &\\\\\\left(1-2^{1-s}\\right)\\zeta (s)&{}={}&1^{-s}-2^{-s}&&{}+3^{-s}-4^{-s}&&{}+5^{-s}-6^{-s}+\\cdots &=\\eta (s)\\\\\\end{alignedat}}", "08fbc478214d30cab2edfa3dd2a0c524": "A=\\bigoplus _{e\\in \\min A}Ae", "08fbd3614d3a3f94edc844a85fe5540b": "{\\sqrt {3+2{\\sqrt {2}}}}=1+{\\sqrt {2}}\\,,", "08fc3223f172dcf9af891a46d5d2beab": "f''(x)=12x^{2}\\geq 0", "08fc9b2c8399229b288bfba89e8f3a79": "-{\\frac {1}{\\rho _{L}}}{\\frac {\\partial P}{\\partial r}}={\\frac {\\partial u}{\\partial t}}+u{\\frac {\\partial u}{\\partial r}}-\\nu _{L}\\left[{\\frac {1}{r^{2}}}{\\frac {\\partial }{\\partial r}}\\left(r^{2}{\\frac {\\partial u}{\\partial r}}\\right)-{\\frac {2u}{r^{2}}}\\right]", "08fce8bf68b68b998c7c84655bf396aa": "{\\mathbf {t} }_{1},\\dots ,{\\mathbf {t} }_{N-1}", "08fd046194d03be4fc66f6a81cda3a37": "\\mathrm {B} \\,\\beta \\,", "08fd2ca023e23a0884c73452c55c35ba": "\\mathbf {R} (\\alpha ,\\beta ,\\gamma ){\\begin{pmatrix}0\\\\0\\\\r\\\\\\end{pmatrix}}={\\begin{pmatrix}r\\cos \\alpha \\sin \\beta \\\\r\\sin \\alpha \\sin \\beta \\\\r\\cos \\beta \\\\\\end{pmatrix}},", "08fd5269678a63899dce46c344cbaf76": "\\sum _{n=1}^{\\infty }{\\frac {1}{n^{2}}}={\\frac {1}{4\\pi }}\\int _{-\\pi }^{\\pi }x^{2}\\,dx={\\frac {\\pi ^{2}}{6}}", "08fdb06222897cd3f6cd1a53ba125c71": "C\\cap C=C", "08fdc98f7dfc53b6b266be475e4a8c2d": "f:V\\to W", "08fde725d0dd6e5ac5189e097f881655": "\\varphi _{T_{\\mathrm {High} }}=90^{\\circ }-\\tan ^{-1}(f/f_{1}).\\ ", "08fe9ad4134c034fa936c90e9aaad316": "{\\dot {s}}(\\theta )=({\\dot {s}}_{1}(\\theta ),\\,\\ldots ,\\,{\\dot {s}}_{k}(\\theta ))", "08fed0441b8e84e35c6c3e169378c427": "{\\frac {N_{n}(E)}{n}}\\to p{\\text{ as }}n\\to \\infty .\\,", "08fedbd24dea0f2476982f017b2d3363": "(a^{b})^{c}", "08ff3f541a4e8c6dd994b18253fbbe9e": "M\\times S^{k}", "08ff51325c7d2bf3bcc866b81c9f6614": "\\varphi _{s}=\\sum C_{s,t}\\psi _{t}.", "08ff9c52ed95f4faf0e6908e48b82f82": "\\scriptstyle \\langle F\\mid e^{-itH}\\mid I\\rangle ", "08ffb90ef746c7d6e1d3e9284604fb1a": "-2-2\\gamma ", "08ffc90bc5212727b8ea94c5788ce752": "|\\cdot \\cdot \\cdot \\cdot \\cdot |\\cdot \\cdot \\cdot \\cdot \\cdot |\\,", "08fff5fac6a59d92d6fad7b5fcffd2f5": "\\rho ^{A}", "090024b5d5d5259e89f900dfcc29345a": "\\partial _{\\mu }\\rightarrow D_{\\mu }=\\partial _{\\mu }-ieA_{\\mu }", "09004f6119e485d220772bcdcd301c00": "{\\dot {\\hat {P}}}(t)=1/2\\left(\\tau (t)\\Psi _{1}(t)+\\Psi _{1}(t)\\tau '(t)\\right),{\\hat {P}}(0)=E({\\mathbf {x} }(0))E({\\mathbf {x} }(0))',rank({\\hat {P}}(t))=n_{r}", "09010d3a3d1916d44af57c1166000e39": "y^{k}", "090118cc5b7d861869e8dc2d79654cc7": "{\\overline {D}}=\\{(x,y)\\in {\\mathbb {R} ^{2}}:(x-a)^{2}+(y-b)^{2}\\leq R^{2}\\}.", "09013dbb6539a971962b368da54a79fa": "\\Omega _{d}=(d\\theta /dt)_{drift}", "0902222f3889cc1f9ab4aef8c8004530": "y(z)=\\sum _{k=0}^{\\infty }y_{k}z^{k}", "09022c5adf858547281f956e790976ed": "{\\frac {1}{2k-1}}-{\\frac {1}{2(2k-1)}}-{\\frac {1}{4k}},\\quad k=1,2,\\dots .", "09024465bb6c4592c6155672c67e68f8": "{\\begin{aligned}q(x)&=\\lim _{d\\to 0}{\\Big (}F\\delta (x)-F\\delta (x-d){\\Big )}\\\\&=\\lim _{d\\to 0}\\left({\\frac {M}{d}}\\delta (x)-{\\frac {M}{d}}\\delta (x-d)\\right)\\\\&=M\\lim _{d\\to 0}{\\frac {\\delta (x)-\\delta (x-d)}{d}}\\\\&=M\\delta '(x).\\end{aligned}}", "090256db8431378a5ffdffff00d5f23b": "h_{1}=|\\mathbf {h} _{1}|;\\;h_{2}=|\\mathbf {h} _{2}|;\\;h_{3}=|\\mathbf {h} _{3}|", "09027347334d1e8527fe6d519f159a80": "(x_{i}^{},x_{i+1})", "0902a406c971b15df5ed00b74a416955": "\\lambda =\\kappa ", "0902c2b382996a827924175c9b51493f": "\\nabla ^{2}\\omega _{1}+{\\frac {f^{2}}{\\sigma }}{\\frac {\\partial ^{2}\\omega _{1}}{\\partial p^{2}}}={\\frac {1}{\\sigma }}\\left[{\\frac {\\partial }{\\partial p}}J(\\phi ,\\eta )+{\\frac {1}{f}}\\nabla ^{2}J\\left(\\phi ,-{\\frac {\\partial \\phi }{\\partial p}}\\right)\\right]-{\\frac {f}{\\sigma }}{\\frac {\\partial }{\\partial p}}\\left({\\frac {\\partial \\omega }{\\partial y}}\\cdot {\\frac {\\partial u}{\\partial p}}-{\\frac {\\partial \\omega }{\\partial x}}\\cdot {\\frac {\\partial v}{\\partial p}}\\right)-{\\frac {f}{\\sigma }}{\\frac {\\partial }{\\partial p}}\\left(\\xi {\\frac {\\partial \\omega }{\\partial p}}-\\omega {\\frac {\\partial \\xi }{\\partial p}}\\right)", "0903077b7ebae3ba4019324b3c4f876d": "\\!{\\mathcal {A}}", "090311abe55c69d4df0815b84955e08d": "{\\widehat {a}}(\\phi )=\\phi (a)", "09033e60847ab2388b63e15d975ff401": "\\approx 4m^{3}", "09034cb45fbf22cf89e5849562318705": "S's", "09037fa5ae70155b5532a471adbef010": "2+{\\sqrt {5}}", "090397a5c4875be3c748c6e5a7c5a68d": "p-p_{0}-(\\rho _{2}-\\rho _{1})gz=\\gamma \\left({\\frac {1}{R_{3}}}+{\\frac {1}{R_{4}}}\\right)\\!", "09039a7ae7fc996a69186b52d5d211e4": "w=f(z)+{\\bar {f}}\\left({\\frac {a^{2}}{\\bar {z}}}\\right)", "0903dc8ea9fb88e72c09961ea7c5232b": "\\Delta Y/Y=k-c\\Delta u\\,", "0903e6516263dd169dc47e9d63cb2443": "X_{1},\\ldots ,X_{n}", "0903f68a216a8d137794b64f9820702f": "{\\mathfrak {P}}^{30}", "0903fd81b795dd4b63fe5fbfa874b2ff": "\\operatorname {lambda-free} [\\lambda F.X]=\\operatorname {false} ", "0904322ab23e920093715fee2d6978da": "{\\frac {1}{2}}\\,\\left(1\\,+\\,k\\,h\\,{\\frac {1\\,-\\,\\tanh ^{2}\\,(k\\,h)}{\\tanh \\,(k\\,h)}}\\right)", "090438b81c4ea40671144d64a867cbc6": "{\\frac {P'(t)}{P(t)}}=r{\\frac {(1-c)}{p(t)}},", "09044d7a546675c0a0e799fe0a4017f4": "\\Gamma _{12}(u,v,\\tau )=\\lim _{T\\to \\infty }{\\frac {1}{2T}}\\int _{-T}^{T}E_{1}(t)E_{2}^{*}(t-\\tau )dt", "0904593e6b7980b27a67975abb84fbfd": "x^{2}-1", "090481e5a9874bfb89e693fdfbd419d8": "~g=\\int G(x(a),y(a),z(a))~{\\rm {d}}a~", "09049a6b76e1d64f1b39d338c3697d93": "A=\\varepsilon \\ell c=\\alpha \\ell \\,", "0904a74176c0b12a21f36e1f5c37c382": "(W;M_{0},M_{1})", "0905131e539fb1ce3326f0c848a4cc51": "A=A_{o}\\cos[2\\pi (x/\\lambda -ft)+\\varphi ]\\,", "090560794c199c84aec3a0335aeb82bf": "\\log \\left(P_{r}\\right)=\\log \\left[\\left\\langle \\exp \\left(-\\beta H-\\log \\left(Z\\right)\\right)\\right\\rangle _{r}\\right]\\geq \\left\\langle -\\beta H-\\log \\left(Z\\right)\\right\\rangle _{r}\\,", "090604235cde82e36ec6a33d4acd3881": "{\\begin{matrix}Q^{2}+U^{2}+V^{2}=I_{p}^{2},\\end{matrix}}", "090614c64a0b7bb53db9ff6a62933770": "S_{x}", "090631c41da3627ed4277dc052a6fae0": "z^{-}", "09068042156731af491a5c34fb003f9a": "-r\\ln(1-p)", "09070160b71b284bb5d1db079a05202b": "y=c_{1}y_{1}+c_{2}y_{2}", "0907149553ed8aca1069f30fb615f6c8": "C(u,v)w+C(v,w)u+C(w,u)v=0_{}^{}.", "0907236e045a531fb1788589b18f13a8": "\\scriptstyle \\leq 1\\times 10^{-32}", "09074aa69d9199d45e04b6dbcaed26a8": "SO\\to PSO", "09078b651072f973326ca32b8250f097": "K\\cup L", "0907ba999a01b3d8370a02def4746c33": "\\Gamma ={\\frac {z_{T}-1}{z_{T}+1}}\\,", "0907e0e9c988dfb54802ca60ce8ccdda": "S=\\sum _{i=1}^{n}X_{i}\\sim IG\\left(\\mu _{0}\\sum w_{i},\\lambda _{0}\\left(\\sum w_{i}\\right)^{2}\\right).", "09080b3dcb9b69f8ad794eb2f0a7c206": "dF/dt\\geq 0", "09082d688d90d15b3eb6c90b7c4ee670": "{\\frac {3\\alpha a-1}{4}}", "0908412f11abec075891cc1356ce38bc": "\\scriptstyle Q\\,\\sim \\,{\\mathcal {N}}(\\mu _{2},\\sigma _{2}^{2})", "09085992776ef7750477f9ab1a811a19": "\\Gamma =C([0,1],X)", "0908874e8bc0d69cca3cbec7552687d7": "\\sum _{i\\in V}f_{iv}\\leq c(v)\\qquad \\forall v\\in V\\backslash \\{s,t\\}", "0908877bc430c2539894ff49b8501430": "0\\leq y\\leq 5", "0908c024103c8b53923511f5ed209cb7": "{\\hat {\\mathbf {a} }}", "090903469044bbd8468f7540393ad95d": "(M\\circ ({\\mbox{Id}}\\times M))(x,y,z)=M(x,M(y,z))", "090934bdc2ecd593440d431a42e88d78": "Y=\\textstyle \\sum _{j=1}^{n}X_{j}", "09094947d14c8f93f371c4496bd94e83": "V={\\tfrac {1}{\\sqrt {K_{0}}}}", "090953475374dfc6660beacb417939f5": "\\mathbf {x} _{k}", "0909714bd7f63294d0ec10692e39b6da": "\\langle b_{1}|", "09097d012954ffa36ea5438198f5c662": ":\\quad T_{c}\\ {\\sqrt {1-V^{2}/c^{2}}}", "0909ce96ee22ccd9cfb5db0777f041a7": "(c/n)\\sin(\\psi )", "0909e4cab35224cfe599d8dad0b026f2": "{\\vec {x}}_{n+1}=2{\\vec {x}}_{n}-{\\vec {x}}_{n-1}+{\\vec {a}}_{n}\\,\\Delta t^{2},\\qquad {\\vec {a}}_{n}=A({\\vec {x}}_{n}).", "0909fbc7d9231ebf9bfee3980c866651": "\\ N=s_{0}=\\sum _{k=1}^{N}{x_{k}^{0}}.", "090a081743cd21b4e4e3b8b25b278b7a": "k_{1}=\\gamma /J_{1}", "090a47f13e1d78b36997329e4f9a8641": "\\operatorname {diag} (a)", "090a4e2c887e51b16ce80db7d33a5b9d": "p\\sim n^{\\gamma }", "090a60633f94c20645c0ec8efda7b3f7": "\\lambda h.h\\ (\\operatorname {const} \\ f)=\\lambda h.h\\ x", "090b0c112a40ab3f181030ffc737f841": "\\mathrm {KDF} ", "090b1bcf5f5afc37b39f1aa5f1447051": "-{q^{2} \\over g{y_{1}}}+{q^{2} \\over g{y_{2}}}={{y_{1}}^{2} \\over 2}-{{y_{2}}^{2} \\over 2}", "090b4380400e0e79224039a8d9dbeaea": "{\\frac {R_{1}+R_{2}}{R_{1}-R_{2}}}={\\frac {2\\left(n^{2}-1\\right)}{n+2}}\\left({\\frac {i+o}{i-o}}\\right)", "090b44bb4cd2b003e253f9e3d7a76d44": "f_{p}^{\\mathrm {Y} }(x)=(1-x)^{p}.", "090b4d45fceb164c69b20ef0c313a082": "\\theta (f)\\propto f", "090b612dac015543b25b65297408332e": "{\\mathit {nil}}", "090b7fa32ce18867bcb864421db27c83": "\\{x\\mid M,x\\Vdash p\\}", "090b93a1b8d8fa7697e4827cdd3b876c": "\\textstyle P_{N}=C_{P}+I+D_{g}+E_{e}-S_{w}\\,\\ ", "090b9a46571289134180e192d4f85a72": "g(x)=\\lambda _{x}\\,", "090c045279e0dcc8da18608d10ec145f": "|\\mathbf {a} _{1}|", "090c3aba4b18c527d8c73c5dffd82059": "q_{m+n}=p_{m}q_{n}+p_{n}q_{m}\\,\\!", "090cb61ede0c514c9e2ac7619337231e": "{\\text{rk}}(T)+{\\text{nul}}(T)={\\text{dim}}(V).", "090d3b42b2e06b1d30e7f0ff5574f5c0": "\\lnot \\forall x\\phi ", "090d65554639d85bbc53c179be37aeeb": "{\\frac {Y}{X}}={\\frac {s+z}{s+p}}", "090d6975eb8c8400f516a685b200c533": "{\\overline {X}}={\\overline {X}}_{n}=(X_{1}+\\cdots +X_{n})/n", "090d7079eaeb7ebd8aedc2d16e75a58b": "E_{A}=E_{A}(q_{0,A})+3f_{A}(\\Delta q_{A})^{2}", "090d73ded95ff5578163137441626833": "g^{ij}S_{ij}=0", "090de40fbb68d3836049ec5fdf143db7": "\\mathbf {C} ", "090f096b160b3afe3ad09588ae776d94": "N={\\frac {f}{d}}\\,;", "090f0bc1d14ee038fe2e60b2cf32a1f3": "{\\frac {M_{1}}{(R-r)^{2}}}={\\frac {M_{2}}{r^{2}}}+\\left({\\frac {M_{1}}{M_{1}+M_{2}}}R-r\\right){\\frac {M_{1}+M_{2}}{R^{3}}}", "090f0cebfe76afb1815d64cbc08dd432": "2^{n+1}-1", "090f239dad336f0b2c793cc91c7d9402": "(\\gamma ,\\alpha )\\neq 0\\,\\forall \\gamma \\in \\Phi ", "090f31c0a7baeb58d233964ec2bc3dda": "F_{s}=2r\\pi \\gamma \\!", "090f5532e658323e653f0ab86d0f7d22": "\\psi ^{R}", "090f6678031c36a9c6914db4b15a10f5": "i:\\lambda ", "090f89368c5c383e8dbf356200c5f379": "{\\mathfrak {P}}^{26}", "090f92439b671c9f0666f3d1d13dd30c": "\\Delta v\\ ", "090fc3942a2606112ed6fcad2ac30187": "\\langle x,x\\rangle ^{1/2}", "090ff1d7b103de4e53e2bb4e3c6fef65": "S(T)={\\frac {C}{\\exp \\left({\\frac {c_{2}}{\\lambda _{x}T}}\\right)-1}}", "09106688ed486be9ada25ca35b5094c4": "{\\begin{aligned}&{\\begin{cases}{\\dot {\\hat {\\mathbf {x} }}}(t)=f{\\bigl (}{\\hat {\\mathbf {x} }}(t),\\mathbf {u} (t){\\bigr )},\\\\{\\dot {\\mathbf {P} }}(t)=\\mathbf {F} (t)\\mathbf {P} (t)+\\mathbf {P} (t)\\mathbf {F} (t)^{\\top }+\\mathbf {Q} (t),\\end{cases}}\\qquad {\\text{with }}{\\begin{cases}{\\hat {\\mathbf {x} }}(t_{k-1})={\\hat {\\mathbf {x} }}_{k-1|k-1},\\\\\\mathbf {P} (t_{k-1})=\\mathbf {P} _{k-1|k-1},\\end{cases}}\\\\\\Rightarrow &{\\begin{cases}{\\hat {\\mathbf {x} }}_{k|k-1}={\\hat {\\mathbf {x} }}(t_{k})\\\\\\mathbf {P} _{k|k-1}=\\mathbf {P} (t_{k})\\end{cases}}\\end{aligned}}", "0910b0c05db4ed89df134d7ecb559bd4": "\\left\\langle {\\Delta S_{x}}^{2}\\right\\rangle \\left\\langle {\\Delta S_{z}}^{2}\\right\\rangle \\geq {\\frac {1}{4}}\\left|\\left\\langle \\left[S_{x},S_{z}\\right]\\right\\rangle \\right|^{2}", "0910b8087f0078b46339c4f5de5f5c08": "[{\\hat {x}},{\\hat {p}}]", "0910d95ce2e6351b3a3d9beac5675583": "\\gamma _{m}={\\underset {\\gamma }{\\operatorname {arg\\,min} }}\\sum _{i=1}^{n}L\\left(y_{i},F_{m-1}(x_{i})-\\gamma {\\frac {\\partial L(y_{i},F_{m-1}(x_{i}))}{\\partial f(x_{i})}}\\right).", "0910e2480ee5b08d924fb9c9b47c5ad4": "\\sum _{i=1}^{n}\\|x_{i}\\|^{2}=\\left\\|\\sum _{i=1}^{n}x_{i}\\right\\|^{2}.", "09111a979677cea23448e84408371b87": "\\pi _{0}\\,{\\text{Diff}}^{+}(D^{n})\\to \\pi _{0}\\,{\\text{Diff}}^{+}(S^{n-1})\\to \\Gamma _{n}\\to 0.\\,\\!", "0911296bb845dcfedcab7c5b305b663c": "B(x;r)=B(y;r)", "09115053006b9b3efef572492e9b0345": "p\\neq 2", "0911a8435ba395fac34959ec4936a6e4": "(\\nu x)({\\overline {x}}\\langle z\\rangle .0|x(y).{\\overline {y}}\\langle x\\rangle .x(y).0)|z(v).{\\overline {v}}\\langle v\\rangle .0", "0911cad5408a1512f27757e7a6a4c49c": "\\deg(a_{k+1})+\\deg(b_{k+1})<\\deg(a_{k})+\\deg(b_{k})\\ ,", "0912ac95ff597192a02e15e971c03024": "v_{3}=v_{1}-v_{2}", "0912d3522828b666040b654d97187312": "p_{w}(\\theta )=\\sum _{k=-\\infty }^{\\infty }{p(\\theta +2\\pi k)}.", "0913221ab6181cbea810f1d7729df659": "f(x;\\lambda )=\\lambda e^{-\\lambda x},\\;x\\geq 0.", "091337f4025e3c5d7c06a2aed1f14e97": "M=4", "09136155a08c6f0f3e6a78119ac6cdc8": "{\\bar {\\nu }}=\\omega _{0}+(B^{\\prime }+B^{\\prime \\prime })m+(B^{\\prime }-B^{\\prime \\prime })m^{2}-2(D^{\\prime }+D^{\\prime \\prime })m^{3},\\quad \\omega _{0}=\\omega _{e}(1-2\\chi _{e})\\quad m=\\pm 1,\\pm 2\\ etc.", "091388cbf96e1ff77f238bf8676ddd42": "\\alpha _{\\tau }=\\Lambda p_{1},\\ \\beta _{\\tau }=\\Lambda p_{2},\\ \\gamma _{\\tau }=\\Lambda p_{3}\\ \\mathrm {at} \\ \\tau \\to \\infty ", "091391060d54f3163293f348303b488e": "f(z+1)=W(e^{z+1})-1\\,", "091392c0aaecc4e4a88d9eed5c910092": "F_{p-\\left({\\frac {p}{5}}\\right)}\\equiv 0{\\pmod {p}},\\;\\;\\;F_{p}\\equiv \\left({\\frac {p}{5}}\\right){\\pmod {p}}.", "0913c4bdcd8eb434e8ec3c664c41e77a": "{\\tfrac {1}{2}}A_{0}+\\sum _{n=1}^{\\infty }\\left(A_{n}\\cos nx+B_{n}\\sin nx\\right).", "0913eaf074752c0fe0b7c90f77674845": "\\,r", "0913f73fe0f9119fe826894691ee11b3": "S=I+A+A^{2}+\\cdots +A^{n}", "091442621ed7ba03dc4f1e190e229ff6": "\\langle f^{*}\\star f\\rangle =\\int (f^{*}\\star f)\\,W(x,p)\\,dxdp\\geq 0.", "09147841158d447a0f07d8cba2958953": "i\\omega V_{c}+{\\frac {1}{RC}}V_{c}={\\frac {1}{RC}}V_{s}", "091482325c8967bc9aa92530f5c483d5": "\\exists b", "09149a6119fb94a9381cce68ea4883e0": "+{\\frac {C_{1}R_{1}^{2}+C_{3}R_{1}R_{3}}{C_{2}R_{2}}}+{\\frac {C_{2}R_{1}R_{3}+C_{1}R_{1}^{2}}{C_{3}R_{2}}}+{\\frac {C_{1}R_{1}^{2}+C_{1}R_{1}R_{2}+C_{2}R_{1}R_{2}}{C_{3}R_{3}}}", "0914c56bbd7b1d4950a1244946007dfa": "\\alpha \\,", "0914cdacfd497034622eeb281289ba2a": "k_{0}=1", "09150cc48eb71ff741d74ff0cf7123e0": "S_{\\eta }", "091526342c696504959a405c529a1e3c": "\\{f_{j}^{\\dagger },f_{j}\\}=1", "0915415c5fb1271071d91ec64e89f332": "N(x|\\mu ,v)", "0915564dbdf6347f9e7ce597d022defe": "C(N)={\\frac {N}{1+\\alpha ((N-1)+\\beta N(N-1))}}", "09158ac493eb982be2f92b1ac044362b": "p(x)\\,", "0916b28226473d5c571ec2c9d6df70c9": "\\left\\vert \\left({\\frac {{\\text{Doppler Frequency}}\\times C}{2\\times {\\text{Transmit Frequency}}}}\\right)\\right\\vert >{\\text{Velocity Threshold}}", "0916c7a201125add754533bf2d2d2753": "\\rho (\\xi )=X_{H_{\\xi }}", "091700d80f6fd1d6c66c3c1c5ae506e5": "\\psi ^{(m)}(z)=(-1)^{m+1}\\;m!\\;\\sum _{k=0}^{\\infty }{\\frac {1}{(z+k)^{m+1}}}", "09174c4395232cb6f09a34a36c96387e": "{\\mathcal {O}}_{X}|_{U}", "09177b8a56fc7aa328c67b8dc9d9b9ce": "T={\\frac {1}{4}}{\\sqrt {4a^{2}b^{2}-(a^{2}+b^{2}-c^{2})^{2}}}", "09179fb350fcf2c4ad4d4a13604e2fbb": "x^{q^{i}}-x\\mod f^{*},", "0917f3f82c1c7b416ffa17d770347370": "G(x,y)={\\frac {1}{(2\\pi )^{4}}}\\int d^{4}p\\,{\\frac {e^{-ip(x-y)}}{p^{2}-m^{2}\\pm i\\epsilon }}", "09180b2796aa3fb7217cfe4aa6b6701e": "C=\\varepsilon \\cdot {{A} \\over {d}}", "09183c0edb89036c69e82ba20a0f78a1": "P^{4}", "09183d46744c20de019525df2bfbaaed": "\\mathbf {V} ^{\\rm {T}}=\\mathbf {V} .", "0918647082ead4f66cb5e373b72c094d": "\\scriptstyle \\Pi \\,\\subset \\,\\mathbb {R} ^{n}", "09186b7f4437732f403f59e0e14094d5": "{\\frac {2}{mn}}={\\frac {1}{m}}{\\frac {1}{k}}+{\\frac {1}{n}}{\\frac {1}{k}}", "09186f6c25ca52f6cd3aaf5691ac67aa": "GVD(\\lambda _{0})\\equiv \\left({\\frac {1}{\\nu _{g}(\\lambda _{0}-\\delta \\lambda /2)}}-{\\frac {1}{\\nu _{g}(\\lambda _{0}+\\delta \\lambda /2)}}\\right)", "0918afec57b0c192615d6f0f2b725b82": "0\\leq a_{n}+|a_{n}|\\leq 2|a_{n}|", "0918b6a685fb23ad7b6df7b5e00fa538": "ax^{3}+bx^{2}+cx+d", "0918ba105278e2b9fc166e0c5fa155bb": "L^{p}(d\\nu )", "0919356f7a53777bd2fcd1ce122eef41": "+1.737\\times \\log _{10}[Bilirubin(\\mu mol/L)]-1.184\\times [ApoA1(g/L)]+0.301\\times Sex(female=0,male=1)-5.54", "091947a8ee1bac70789700c52856045d": "z_{3}=-5", "0919631fb604e60147b9fd3ac28f72c5": "\\gamma \\ >0", "0919d464d820465b51cf0c4536d1c64b": "a^{1}b^{1}c^{1}", "091a2f4d9bcd3ca7daddb5db1288606c": "\\kappa (u_{ij})", "091a36c336b9f86ac488b9e8ac0e0ffa": "X^{(0)}", "091a513a46acbcd729c737600655f9eb": "{\\frac {8}{3}}m^{3}", "091a64999eda86b4be1c5164f85fb338": "{\\mathbb {C}}\\times H/\\Gamma ", "091aab6f44ab6c366341e673c52d735e": "M_{\\phi }:\\mathbb {R} ^{S}\\rightarrow \\mathbb {R} ", "091ad9dff9f1baf25af71de76ec09794": "{\\frac {d{\\vec {\\omega }}}{dt}}=({\\vec {\\omega }}\\cdot {\\vec {\\nabla }}){\\vec {v}}", "091b1fd63ce5a18e61a0518154295f19": "\\det \\left(1-z{\\mathcal {L}}\\right)=\\prod _{i}\\left(1-\\rho _{i}z\\right)", "091b3f5e20d3d55b356e0a885bb80303": "\\displaystyle {D(\\varphi )=D(\\varphi )|_{\\Omega }\\oplus D(\\varphi )|_{\\Omega ^{c}},\\,\\,\\,\\,S(\\psi )=S(\\psi )|_{\\Omega }\\oplus S(\\psi )_{\\Omega ^{c}}.}", "091b6c4afc95f4f6d9963fc09eb9ea66": "y=\\sin(t)(R+r\\cos(u)),", "091b852783c17814fe9f2137779de4a8": "\\langle \\mathbf {p} \\rangle =\\left\\langle \\psi \\left|-i\\hbar \\nabla \\right|\\psi \\right\\rangle =\\int _{\\mathrm {all\\,space} }\\psi ^{*}(\\mathbf {r} ,t)(-i\\hbar \\nabla )\\psi (\\mathbf {r} ,t)d^{3}\\mathbf {r} =\\hbar \\mathbf {k} ", "091beef37747159cbda93c3e0947c753": "{\\frac {\\langle E(s)\\rangle }{A}}=\\hbar \\int {\\frac {dk_{x}dk_{y}}{(2\\pi )^{2}}}\\sum _{n=1}^{\\infty }\\omega _{n}\\vert \\omega _{n}\\vert ^{-s}.", "091c17327bb2a9dccebb1bd7ebd1faa5": "\\theta \\in \\mathbb {C} ", "091c33b9617c06f3c8f9211fca7a951f": "\\geqslant ", "091c54bf7ab3f2927aa6af06024c6071": "{\\hat {\\theta }}_{i}", "091c70939a397005815ced426a1069a6": "q\\geq cp", "091cce78f0094f4cd21f6e9e7d4ddf9d": "{\\color {Blue}~2.23}", "091ccf7c41a5b2ff7f123fb420dd9187": "\\epsilon _{i}(b)=\\max\\{n\\geq 0:{\\tilde {e}}_{i}^{n}b\\neq 0\\}", "091d9c0c19b64e2ca4fc411517aa216b": "{\\boldsymbol {\\mathcal {P}}}={1 \\over 4\\pi c}\\mathbf {E} (\\mathbf {r} ,t)\\times \\mathbf {B} (\\mathbf {r} ,t)", "091dacedc989f940358e42b35434c737": "\\scriptstyle \\sum _{n=1}^{\\infty }{\\frac {x^{n}}{n^{s}}}", "091dc0e3964e60041a23eea09883ba4f": "{\\sqrt {8}}(3\\rho ^{3}-2\\rho )\\sin \\theta ", "091dfabee698d3d25988e357afccdbbd": "d/n", "091e57d91fb91625042c6c78cda2cdf5": "{\\begin{aligned}q_{\\text{n}}^{H}(k):&{}\\quad {\\frac {1}{\\tau }}+(d_{u}^{2}+{\\frac {1}{\\tau }}d_{v}^{2})k^{2}&=f^{\\prime }(u_{h}),\\\\[6pt]q_{\\text{n}}^{T}(k):&{}\\quad {\\frac {\\kappa }{1+d_{v}^{2}k^{2}}}+d_{u}^{2}k^{2}&=f^{\\prime }(u_{h}).\\end{aligned}}", "091e7e16ecc5c72b86b33fda6cceb6d1": "{\\frac {1}{d}}={\\frac {1}{r'}}\\left[1-2\\cos(\\theta '-\\theta ){\\frac {r}{r'}}+\\left({\\frac {r}{r'}}\\right)^{2}\\right]^{-{\\tfrac {1}{2}}}.", "091e9a7fb6b7b0e6e04327d6cd8c0776": "C(x,x'')=\\int _{x'}A(x,x')B(x',x'')\\,,", "091ebbc1e1c0749c4df6b535fac39f05": "P_{2}\\uparrow S(X,Y,{\\mathfrak {E}},H)", "091edb42262c877ba740f874405bd33d": "\\alpha ^{-1}(n)", "091eecf569963d2f229b88abf5675a62": "R(5,5)", "091f2f2d60bc57704a3577cac23da4a2": "{\\tfrac {dI}{dT}}=\\beta SI-\\gamma I-\\mu I", "091f5aee3c7167080396ed1c86bd3740": "y_{n}\\to 0", "091f5b5f08472066423a2645778c3222": "{\\boldsymbol {C}}={\\boldsymbol {F}}^{T}\\cdot {\\boldsymbol {F}}=(\\mathbf {G} ^{i}\\otimes \\mathbf {g} _{i})\\cdot (\\mathbf {g} _{i}\\otimes \\mathbf {G} ^{i})=(\\mathbf {g} _{i}\\cdot \\mathbf {g} _{j})(\\mathbf {G} ^{i}\\otimes \\mathbf {G} ^{j})", "091f6b5d6f09937e0570f516690cfa54": "\\ {\\overline {u'w'}}={\\overline {\\xi '^{2}}}\\left|{\\frac {\\partial {\\overline {w}}}{\\partial z}}\\right|{\\frac {\\partial {\\overline {u}}}{\\partial z}}", "091fd227c6b83e7aa1b9d0e9b3fd2d44": "2^{3/12}={\\sqrt[{4}]{2}}", "09200a73966b5c9559e1495e92dcc7ce": "{\\hat {x}}\\in C(\\theta )", "0920682753de2d80a47d6fcd3c8c195a": "T_{n}=(1-{\\frac {n}{N}}){\\frac {a+1}{D^{2}-(1-{\\frac {n}{N}}){\\frac {b-1}{n}}}}", "0920856d25e3fec73110d603dd5f56cb": "{\\begin{aligned}a&=0.14285933+0.06404502i,\\\\b&=0.14362386+0.06461542i,{\\text{ and}}\\\\c&=0.18242894+0.81957139i,\\end{aligned}}", "09209777121192f0d82359664658b0b9": "{\\begin{cases}-{\\frac {b}{a}}{\\frac {\\Gamma \\left(-{\\tfrac {1}{a}}\\right)\\Gamma \\left({\\tfrac {1}{a}}+p\\right)}{\\Gamma (p)}}&{\\text{if}}\\ a>1\\\\{\\text{Indeterminate}}&{\\text{otherwise}}\\ \\end{cases}}", "0920bf711f19ec9e01fbe7f82a2bb21e": "D_{1}\\left(E\\right)={\\frac {1}{\\sqrt {c_{k}(E-E_{0})}}}\\ .", "0920f444b3780881c9612b5674c99063": "g(x)=g_{n}x^{n}+g_{n-1}x^{n-1}+\\ldots +g_{0},\\,", "092144927dc432cb394abb16777fe138": "\\tau =\\mu {\\frac {\\partial u}{\\partial y}},", "09215af2256fcc860a2cd7c773c224ed": "\\psi ", "09219171febf5fe82c2ce71e4ff3a25b": "\\mathbf {A} (\\mathbf {r} ,t)=\\nabla \\times \\int {\\frac {\\mathbf {B} (\\mathbf {r'} ,t)}{4\\pi R}}d^{3}r'", "0921c67f29ec8472f8258719e160047b": "{\\frac {1}{2\\pi i}}\\int _{-i\\infty }^{i\\infty }{\\frac {\\Gamma (a+s)\\Gamma (b+s)\\Gamma (c+s)\\Gamma (1-d-s)\\Gamma (-s)}{\\Gamma (e+s)}}ds", "0921c77ddde769fdfaaa757b3aee9d78": "a_{z}\\,dx\\wedge dy+a_{y}\\,dz\\wedge dx+a_{x}\\,dy\\wedge dz.", "0921c9b64221e4678a9a050a4a5c0191": "\\partial _{1}(c_{1})=0", "09221170da1edff943267d36b77d0915": "(x^{3}+x)-(x^{3}+x^{2})=-x^{2}+x", "0922219ad86894a35df46db958dc34ab": "C\\left(a_{n}(q),q,x\\right)={\\frac {CE(n,q,x)}{CE(n,q,0)}}", "09224714f54be3f67c05ec6c983114ea": "{\\vec {b^{i}}}=(b_{1}^{i},...,b_{m}^{i})", "09228e6d1aa18c62dfcdad14cc61a415": "\\phi :F\\to B", "0922b63a680dc5bd8e75bf150cb2fe4f": "\\Phi :A\\rightarrow \\mathbb {R} ", "0922c9b5c15dac27efc38f77731b83d6": "g(E)=g(F)", "0922eb3f7b10e8d6bc4c559e4af7d6de": "e\\in E,", "09233cbe5f03d5b8a1b4daae8d5a5422": "|\\quad |_{p}:{\\textbf {Q}}\\to {\\textbf {R}}", "0923d20800931dbd64476c69de90eee0": "g\\colon S^{2}\\to g(S^{2})", "0923f3540c0efb59c522fdc507bb3443": "\\eta \\colon K\\to A", "09240fc1caa97137cb5135e7ce2eff8a": "{\\frac {c}{{\\sqrt {2}}\\chi }}{\\sqrt {(\\chi ^{2}-2p-1)+{\\sqrt {\\chi ^{2}(\\chi ^{2}-4p+2)+(1+2p)^{2}}}}}", "09248813a49c15ac9e7dd76614b04168": "1200\\,", "0924922346e7a3d5709e819c477745db": "ROIC={\\frac {{\\textrm {NetOperatingProfit}}-{\\textrm {AdjustedTaxes}}}{\\textrm {InvestedCapital}}}", "09249b0be3fab2baad2a6f48d8ea31c9": "\\|R_{i}\\|_{0}\\leq q", "09249e3cc2e9cab5df7a3dd31364e664": "p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{2}x^{2}+a_{1}x+a_{0},\\,", "0924f87e189924b1b5d9fdc82330cdc6": "\\nabla ^{2}\\left({\\frac {1}{r}}\\right)=-4\\pi \\delta ({\\vec {r}})", "09254c1405dd35d9599cefc5e2936364": "\\mathrm {[A_{p}B_{q}H_{r}]=\\beta _{pqr}[A]^{p}[B]^{q}[H]^{r}} ", "0925546441796437a496ee868b2c93c4": "{\\frac {d}{|\\mathbf {v} |}}={\\frac {d}{\\sqrt {a^{2}+b^{2}+c^{2}}}}", "0925e8fbbe7807223499ff6ef7beb9aa": "{\\frac {R}{2}}{\\sqrt {2+{\\sqrt {2}}}}={\\frac {a}{2}}\\left({\\sqrt {2}}+1\\right)\\!\\,", "092623bc48b6f3f0ec15dc1a57904e85": "{\\widetilde {t}}", "092628c2a346f6d8880320c5a437e015": "H=U+pV,\\,", "09265569b156bfb21b7ee605767c94d3": "\\lambda <\\omega _{1}^{CK}", "092673dc2898d724d94e6019fd391c01": "x\\in \\Lambda ", "09269acbde301d27b2091d4edc20134c": "\\varepsilon _{i}", "092779faa466d9c56c44d640bb4f9ba3": "_{N}C_{2}", "0927c033adbec8f6ac2024ab9790ef2d": "\\,v_{i}", "0927fa3288b43c389822c7dc835fb146": "M_{n}=M_{1}^{n}", "09280fa4ae67a9e7d4df69369d8a4d74": "F(x)=\\sum _{n=0}^{\\infty }f_{n}(x),\\qquad x\\in \\mathbb {R} ,", "09281df7c03d16d683bb2d8b7745fa14": "y=ax^{2}+bx+c\\,", "09287cfee425829c734357bf7cb658bd": "M_{[ab]}={\\frac {1}{2!}}(M_{ab}-M_{ba}),", "092895644979bb03db1e0a4729d41175": "{\\text{Percentage change}}={\\frac {\\Delta V}{V_{1}}}={\\frac {V_{2}-V_{1}}{V_{1}}}\\times 100.", "0928a80b08bcd2fb57a34b7d710dd541": "M_{0}^{2}", "09296204606853b14d1a5a6b4413469a": "P_{c}\\sim 10^{5}\\,\\mathrm {Mbar} .", "092962f1ddafe24557e8dd0c44b38815": "{\\frac {d\\theta }{ds}}={\\frac {s}{R_{c}s_{o}}}", "092993b80f35b2720f5bb39d6f4161d7": "\\Pi _{i\\in I}A_{i}/U", "0929a96a4d662615fd83073af51b6220": "{\\frac {2\\alpha ^{2^{n}}+2\\alpha ^{-2^{n}}-b}{2a}}", "0929ab6e77c5fbfcf609f5889f9b5eff": "S''=0", "092a6856d4af9220c0686d5209ecc781": "{\\frac {1}{p}}_{\\theta }={\\frac {\\theta }{p_{1}}}+{\\frac {1-\\theta }{p_{0}}}.", "092a9e583eb2e54916933fcb25b728f0": "{\\begin{aligned}N_{\\alpha \\beta ,\\alpha }&=0\\quad \\quad N_{\\alpha \\beta }:=\\int _{-h}^{h}\\sigma _{\\alpha \\beta }~dx_{3}\\\\M_{\\alpha \\beta ,\\alpha \\beta }-q&=0\\quad \\quad M_{\\alpha \\beta }:=\\int _{-h}^{h}x_{3}~\\sigma _{\\alpha \\beta }~dx_{3}\\end{aligned}}", "092a9e9394d963738b1f267d01aa3cbe": "x^{5}+ax^{2}+b", "092a9f1c921d20c4f8037bb84eb3fb05": "\\phi (Q)=c|L|", "092ad01e706529b98dedece42bf96d20": "\\mathbf {E} \\,\\!", "092adfb87a939e758ac84c119b2248ee": "N=n\\ell A", "092af1486cf4cdfa3abe7c3efa85f299": "\\varphi ^{n+1}=\\varphi ^{n}+\\varphi ^{n-1}.", "092b1a820808e13c82521cb113834bf4": "((24\\,{\\bmod {\\,}}5)(3^{-1}\\,{\\bmod {\\,}}5))=4\\cdot 2=8\\neq 3", "092b1c41202287b7e2a8a4ef87815c68": "a_{1}+2a_{2}+3a_{3}+\\cdots +na_{n}=n.\\,", "092b210e890b7ea0cacda2de9f0455cc": "W_{0}^{1,p}(X)\\subseteq L^{\\varphi }(X)", "092b6b8745ddea3da8a98a16f31e4711": "\\left\\langle {\\frac {1}{2}}k\\left({\\hat {x}}-x_{0}\\right)^{2}\\right\\rangle \\left\\langle {\\frac {1}{2m}}{\\hat {p}}^{2}\\right\\rangle \\geq \\left({\\frac {\\hbar }{4}}\\right)^{2}{\\frac {k}{m}}.", "092b77229c3b8582317d0e224fbb17a6": "-{1 \\over 4\\pi }\\nabla ^{2}\\left({1 \\over r}\\right)=\\delta \\left(\\mathbf {r} \\right)", "092b88f48f20045681113a8e99c25345": "\\lambda _{n}=\\log n", "092be609dae29cc54c062bf0377d77be": "\\mathbf {L} =C_{L}qA_{ref}", "092c0630a9e58b8d7ad12263d3f937aa": "(12),\\;(13),\\;(14),\\;(23),\\;(24),\\;(34)", "092c366a0f3e517c2ed8a49e9e8e1d17": "x_{/_{\\cong _{{\\mathcal {B}},\\epsilon }}}", "092c4239fce753aaffbc04bcfe87810f": "=C\\int _{-{\\frac {a}{2}}}^{\\frac {a}{2}}e^{\\frac {ikxx^{\\prime }}{z}}\\,dx^{\\prime }", "092c7ba21f6ef3e3f77a03a36e21cb3d": "\\mathbf {1} _{A}", "092c91d017df5d7ca930fb161e6657c4": "\\eta (\\xi )", "092c9f8f119bd88b4e72f4e1d4f1c3a2": "\\sin y=x\\ \\Leftrightarrow \\ y=(-1)^{k}\\arcsin x+k\\pi ", "092caf48acbcfe27d7c3e4134d645bfe": "I\\in [4.769A,5.417A]", "092cbee0a46deeba73e1b0d8036cd6b6": "\\omega '_{0}={\\sqrt {\\frac {1}{LC}}}", "092cd680cd1d538539d41f0b9a191245": "q(uv)=au^{2}", "092d21f798b50357be84affffebe5aa4": "\\Delta _{2}^{\\prime }F(J)^{calculated}=2B^{\\prime \\prime }\\left(2J+1\\right)", "092d3c21515cf56d7f46acaeacebe7f4": "\\displaystyle \\det(\\partial _{ij}\\phi )=", "092d684280262b67670001706a335165": "k_{1}=15.957", "092d6a697df463c3beaaeafbff30f771": "{\\frac {6}{\\pi ^{2}}}e^{\\gamma }\\leq \\limsup _{t\\rightarrow +\\infty }{\\frac {1/|\\zeta (1+it)|}{\\log \\log t}}\\leq {\\frac {12}{\\pi ^{2}}}e^{\\gamma }", "092d73a7de8efeaa2cb788c40d954aa1": "r={\\text{min}}\\left(1,{\\frac {w(\\mathbf {y} _{1},\\mathbf {x} )+\\ldots +w(\\mathbf {y} _{k},\\mathbf {x} )}{w(\\mathbf {x} _{1},\\mathbf {y} )+\\ldots +w(\\mathbf {x} _{k},\\mathbf {y} )}}\\right)", "092d8cf0e1b1919e20eb9b5b670cb7ea": "0\\leq t<\\infty ", "092d8ff63435567921b7b2139a485d70": "\\exp(\\pm kt)e_{i}(t)", "092da07cb35a7e571d676643fbf67db8": "Y_{(j+{\\frac {1}{2}},{\\frac {1}{2}})jm}=\\left({\\begin{array}{c}-{\\sqrt {\\frac {j-m+1}{2j+2}}}Y_{j+{\\frac {1}{2}},m-{\\frac {1}{2}}}\\\\{\\sqrt {\\frac {j+m+1}{2j+2}}}Y_{j+{\\frac {1}{2}},m+{\\frac {1}{2}}}\\end{array}}\\right)", "092dd4a09cd11dcb6305bcd0a92d0884": "d(v_{1},v_{2})=0", "092e66f7933af9712ed95ad2cb76aa3c": "\\{H,L^{2},L_{z}\\}", "092e7e21e24e3f1eda6c1ca932af40cd": "(M+m){\\ddot {x}}+m\\ell {\\ddot {\\theta }}\\cos \\theta -m\\ell {\\dot {\\theta }}^{2}\\sin \\theta =0", "092e86b698513c5298c66b2520ea446c": "D(f)\\leq R_{1}(f)R_{2}(f)", "092ee1e3b9ac117fc99cc2c31ad726d0": "G=S_{m}", "092f4e2c9a879152faab7be3496c54be": "{\\begin{aligned}\\sin x&=\\sum _{n=0}^{\\infty }{\\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1},\\\\\\cos x&=\\sum _{n=0}^{\\infty }{\\frac {(-1)^{n}}{(2n)!}}x^{2n}.\\end{aligned}}", "092f71002698fa24a1010c8459148153": "2^{7\\times 8}+2^{8}=72057594037928192", "092fe1dcd5865e2de38bd7f431441b44": "N((ab)\\sigma )=N(\\sigma )\\pm 1", "09304a387a9b9611762eb244c17a91db": "q<3", "0930bc94c84a1bf6d78bc9c94903e7ea": "E_{n}^{(2)}={\\frac {ma^{2}}{2\\hbar ^{2}}}\\sum _{k}\\left|\\left(\\delta _{n,1-k}+\\delta _{n,-1-k}\\right)\\right|^{2}/(n^{2}-k^{2})", "09311b6a1e840b295537603664157e2c": "\\left(Y,\\Sigma _{Y}\\right)", "0931214e038fcf39e2141121f5bcbce4": "A:=\\{x_{1},x_{2},\\dots ,x_{k}\\}", "093138125585eb86a5b026d0d4184efd": "f(x)\\leq f(y)", "09316b9d3832684c56a8ba24d2a8f404": "G\\times 1", "0932420622ff1696520914e8552efa05": "{\\mbox{MAC}}={\\frac {S}{b}},", "09325b10204d4297bf53de310497577a": "{\\frac {1+{\\scriptstyle {\\frac {4}{5}}}z+{\\scriptstyle {\\frac {3}{10}}}z^{2}+{\\scriptstyle {\\frac {1}{15}}}z^{3}+{\\scriptstyle {\\frac {1}{120}}}z^{4}}{1-{\\scriptstyle {\\frac {1}{5}}}z}}", "09325c95b25d932a3b059ba5d686d067": "\\ A_{\\mu }=\\sum _{a}A_{\\mu }^{a}T^{a}", "093279bb243792c430c94675ef8cb8a0": "(2+x)", "0932ae8f457aa22eaca13a3ce516ccf0": "\\mathbf {p} =\\gamma (v)m\\mathbf {v} ", "0932baa9d80dca487c221eb90f1f6382": "c(X^{*},X)", "0932bda7173e52c1ef331b9cf15d7017": "y=f(x_{1},x_{2},\\dots ,x_{n})", "093352bd5f5cf7937624857f6d5aaff9": "P_{i}(x)", "09338ab83351bc720ba02756829157dd": "P(E)=\\int _{\\omega \\in E}\\mu _{F}(d\\omega )\\,", "09338e33293992563d2b8b7428030b9f": "W_{X}(t,f)=W_{x}(-t,-f)\\,", "0933b9de7a6c3848ed3bb40a5f4e9c9b": "\\pi \\sim _{st}\\pi '", "0933d7dbf9b56f8c222620eedf008dd5": "p={{\\tbinom {10}{1}}{\\tbinom {14}{11}}}/{\\tbinom {24}{12}}={\\tfrac {10!~14!~12!~12!}{1!~9!~11!~3!~24!}}\\approx ", "09343b9e135a7633d2bbd941140e27af": "v={\\frac {c}{n(\\lambda _{0})}},", "09345faacdf4f9e55a9c523272b8efbc": "\\pi ^{ij}", "093465bedcb6b80aae243c0049ba49a8": "\\displaystyle u_{t}=6(u+\\epsilon ^{2}u^{2})u_{x}+u_{xxx}", "0934869ba047b11cf4094c25e60740fe": "x\\in \\mathbb {R} ^{d}", "09348b0b972dea30099e6d228d47c246": "H|n\\rangle =E_{n}|n\\rangle ", "0934df980baf56854b50ce790276ce00": "{\\frac {\\partial \\Delta E}{\\partial P_{x}}}=\\alpha _{0}\\left(T-T_{0}\\right)P_{x}+\\alpha _{11}P_{x}^{3}+\\alpha _{111}P_{x}^{5}-E_{x}=0", "0934f7e6dbac2dd708e040808412f724": "\\ {L}", "09352585aa32af7e20ce9dd32c1d2467": "u=u_{0}+u_{1}+u_{2}+u_{3}+\\cdots ", "093555e79df91ae6b7e73db054f0e17c": "\\mathbf {S} (p)={\\begin{bmatrix}s_{11}&s_{12}\\\\s_{21}&s_{22}\\end{bmatrix}}", "09355d08965267aa05b5e651b748e3bd": "\\mathbb {C} ^{n}", "093565b44e4b02d268e5552c2c892a55": "Z={\\frac {\\nu }{\\omega }}\\ll 1", "09356d00e4cc296c78e526b0e06d708e": "(p',u)", "0935b9ae317c4f6355ae238e936416c8": "\\{l_{a},l_{b}\\}=\\epsilon _{abc}l_{c},\\ \\{l_{a},n_{b}\\}=\\epsilon _{abc}n_{c},\\ \\{n_{a},n_{b}\\}=0", "0935d32289a4b25680d1ef04b4d5d0ba": "\\{\\Delta \\ X\\}", "0935d653167a2f43f6b60532caedc09c": "w^{\\perp }/w", "093603c12920c42248564b69bfa16890": "f_{4}(\\omega )\\,", "09360629420960e3f6cc8b343e048fb2": "O_{4}", "0936080d51060e42d81f404e26dbb610": "|\\pm \\rangle \\equiv |m_{J}=\\pm 1/2,m_{I}=m_{F}\\mp 1/2\\rangle ", "09360b5582197fecc1a7c4eb7531f2c8": "\\textstyle (x_{0},x_{1},\\ldots )", "09362f0f818c3fe663de23d2d787a9cd": "x_{i}=x_{j}", "093674ccfc9d481cfc042f76831c3f56": "{\\begin{aligned}Y_{i}^{1\\ast }&={\\boldsymbol {\\beta }}_{0}\\cdot \\mathbf {X} _{i}+\\varepsilon _{1}\\,\\\\Y_{i}^{2\\ast }&={\\boldsymbol {\\beta }}_{1}\\cdot \\mathbf {X} _{i}+\\varepsilon _{2}\\,\\\\\\ldots &\\ldots \\\\Y_{i}^{m\\ast }&={\\boldsymbol {\\beta }}_{m}\\cdot \\mathbf {X} _{i}+\\varepsilon _{m}\\,\\\\\\end{aligned}}", "0936a4bba5aa2ce62d737887a7883b52": "W^{T}=C_{Y}^{-1}C_{YX}.", "09372411cfdc2b47abc230c2e60059b2": "m=log_{e}{\\frac {p_{1}}{p_{2}}}", "093738e3f6e59e278474a5500092d00d": "k_{y}^{i}={\\frac {k_{y}}{M}}", "09373d79c9a95008a565ecb19efd58d9": "\\langle {\\bar {\\psi }}(k_{1}){\\bar {\\psi }}(k_{2})...{\\bar {\\psi }}(k_{n})\\psi (k'_{1})...\\psi (k_{n})\\rangle =\\sum _{\\mathrm {pairings} }(-1)^{S}\\prod _{\\mathrm {pairs} \\;i,j}\\delta (k_{i}-k_{j}){1 \\over \\gamma \\cdot k_{i}-m}", "0937619de23cd643fbc344b16ab749d2": "h_{n}=|f_{n}-f|^{p}", "093777afbabf9bd67c9197dfea796ea2": "|v_{0}v_{1}|=|v_{1}v_{2}|=\\cdots =|v_{d-1}v_{d}|", "09377a48da75477893c34af1acbb27cf": "F_{p-\\left({\\frac {p}{5}}\\right)}\\equiv 0{\\pmod {p^{2}}}.", "09378bff675841379c42ee11de4a9b6b": "f(0)\\oplus f(1)=0", "0937ebf35ea28cec4b23b3108ab7e6f6": "r^{2}R''+rR'+r^{2}k^{2}R-n^{2}R=0.\\,", "093827ded29f910ac93f96adc251ef57": "L[y]=y\\circ (t+1)-y\\circ t=\\Delta y", "0938313f7cc2962dd463e47bd554c671": "\\Delta t\\rightarrow 0", "093846933be57dbaa33db94d3041c666": "\\textstyle {\\sum _{i}p_{i}=1}", "09387dd5f37856997143c6a1b56a0e3f": "\\int _{G}|f(x)|^{2}\\ d\\mu (x)=\\int _{\\widehat {G}}|{\\widehat {f}}(\\chi )|^{2}\\ d\\nu (\\chi )", "093898651fcefd9850fd8f3f661ebbd3": "E_{1,0}=510,260*{\\frac {260}{510,260}}*{\\frac {500,200}{510,260}}", "0938f06a54a0e2cc6636ddf65cabfaba": "(|1\\rangle -|0\\rangle )/{\\sqrt {2}}", "09398cf2fd5a8aa9788f4701480a00ef": "{\\mathcal {F}}^{2}(\\mathbb {C} ^{n})=\\{f\\colon \\mathbb {C} ^{n}\\to \\mathbb {C} \\mid \\Vert f\\Vert _{{\\mathcal {F}}^{2}(\\mathbb {C} ^{n})}<\\infty \\}", "093990db1805c5a4b0fdc77165490776": "\\,p(x)", "09399c2fa2b785004547456b40180b29": "V=\\{v:[0,1]\\rightarrow {\\mathbb {R}}\\;:v{\\mbox{ is continuous, }}v|_{[x_{k},x_{k+1}]}{\\mbox{ is linear for }}k=0,\\dots ,n{\\mbox{, and }}v(0)=v(1)=0\\}", "0939ba99c8703268d6a7034a4b018596": "y_{2i}={\\begin{cases}y_{2i}^{*}&{\\textrm {if}}\\;y_{1i}^{*}>0\\\\0&{\\textrm {if}}\\;y_{1i}^{*}\\leq 0.\\end{cases}}", "093a254955308018b03188e94a6e6936": "{\\frac {dg(p)}{dp}}={\\frac {1}{f''(g(p))}}", "093a5e576b1d3ad7036b1fa685356f09": "{\\bar {J}}=\\Gamma (\\omega )*c/4\\pi ", "093a6deacfacb4edd1fc263f67ce8c10": "a\\mapsto a.x-x.a", "093a74151e91c98a425681676eda6248": "\\lim _{x\\to 0^{-}}{x^{-1}}=-\\infty ", "093a9ced6065e190272528e9f3dddae2": "\\lim _{n\\to \\infty }{\\widehat {S}}(n)=0", "093b3a3752fce3871c5b49f0d88303b0": "v^{*}=-\\infty ", "093b5e2d3cc4d86e5bd2bc9048c22958": "{\\begin{array}{rcl}s&=&\\displaystyle 1+2+4+8+\\cdots \\\\[1em]&=&\\displaystyle 1+2(1+2+4+8+\\cdots )\\\\[1em]&=&\\displaystyle 1+2s\\end{array}}", "093bb55322de6df7782d6ad0deb16272": "[u,v]_{p,q}=\\sum _{i=1}^{n}\\left({\\frac {\\partial q_{i}}{\\partial u}}{\\frac {\\partial p_{i}}{\\partial v}}-{\\frac {\\partial p_{i}}{\\partial u}}{\\frac {\\partial q_{i}}{\\partial v}}\\right).", "093bd73b2bf47508a3b37ef266bc141e": "\\phi (\\exp(x))=\\exp(\\phi _{*}(x)).\\,", "093bebfe1d89bb89e7421efe818289e0": "{\\mathcal {L}}=\\left\\{L=L':~x^{\\{m\\}'}Lx^{\\{m\\}}=0\\right\\}.", "093c14b44ec3998a6609d5725d2b3b9f": "D_{2}\\left(E\\right)={\\frac {2\\pi }{c_{k}^{2}}}\\left(E-E_{0}\\right)", "093c3d0431033c8751d76e763e7c5ea0": "{^{(4)}}\\Gamma _{ij}^{0}", "093c3ed8582f4044c020b3bad49c3892": "z_{i}=(1,z_{2i},\\dots ,z_{pi})", "093c6c2ad58bb8156d94c5d102f3dc2d": "{\\begin{aligned}x&=r\\sin \\theta \\cos \\varphi \\\\y&=r\\sin \\theta \\sin \\varphi \\\\z&=r\\cos \\theta \\end{aligned}}", "093c8e205679042f3dafbae84a800cc0": "h=\\sum _{i}dy_{i}\\;dy_{i}", "093cafe29f6d7452dec1e5eb9e01e9ed": "O(n^{d+1-\\delta })", "093cfbd44b73b914c0b5564c925438c3": "a\\equiv ^{\\ast }\\!b\\,(\\mathrm {mod} \\,\\mathbf {p} ^{\\nu })\\Leftrightarrow \\mathrm {ord} _{\\mathbf {p} }\\left({\\frac {a}{b}}-1\\right)\\geq \\nu ", "093d2b0e27fbaa55989e56c93c71fd18": "\\left(\\pm {1 \\over 2},\\pm {1 \\over 2},\\pm {1 \\over 2},\\pm {1 \\over 2},\\pm {1 \\over 2},\\pm {{\\sqrt {3}} \\over 2}\\right).", "093d2b892a55ce67f922c5ce1eb4e32b": "(x,y,y',y'',\\dots ,y^{(k)}).", "093dd868c534d861bd8b03ff5e583ced": "{\\mathcal {L}}_{I}=-e{\\bar {\\psi }}\\gamma _{\\mu }A^{\\mu }\\psi \\,-\\,(Z_{1}-1)e{\\bar {\\psi }}\\gamma _{\\mu }A^{\\mu }\\psi ", "093e377d554e2d8aec8ff3269c3a0356": "SU_{\\mu }(2)=(C(SU_{\\mu }(2),w)", "093e3b158570c405f18d08369be44de9": "\\scriptstyle d", "093e3be7518e3afcccefe9181c4542e6": "\\mathbf {x} _{1}=(\\mathbf {P} _{B}^{\\perp }\\mathbf {A} )^{+}\\,\\mathbf {d} ,\\qquad \\mathbf {x} _{2}=(\\mathbf {P} _{A}^{\\perp }\\mathbf {B} )^{+}\\,\\mathbf {d} .", "093e91ccce3d59664b5cc8bd7d533b87": "{\\mathcal {A}}={\\mathfrak {M}}\\{{\\mathcal {B}}\\}", "093eb46b95faa8734786a59215412808": "U_{t}\\geq X_{t}", "093f686f70f93af3ca0b28e12be67463": "||\\phi (a)\\land \\psi (b,c)||=||\\phi (a)||\\ \\land \\ ||\\psi (b,c)||", "093f8a967400f274ba09d084a57ef1af": "{\\frac {c}{a}}", "093fa97e24e55eeb98f14d3f51154ad4": "{F_{y}}'=qE'=-q\\gamma vB.", "094035c27ab8d07bed11709557d69220": "u_{12}=3", "09411d4d4235225bb422cacee3826d77": "\\epsilon ^{abc}", "094128623422276a599624300c3908f7": "S_{40:1}=107", "09415d667c9259d5a873c275204c2c85": "\\left\\lbrace x_{i}\\right\\rbrace ", "09418474ca66bee017b81a3563512aca": "x^{2}=0.\\,", "0941bedf48472baa899d2e8ea1564b90": "\\epsilon _{c}=A\\ln(r_{s})+B+r_{s}(C\\ln(r_{s})+D)\\ ,", "0941fb30d6a8f8d602b9b3dc5a4bdff1": "\\mathbf {v} =\\nabla \\phi \\qquad \\qquad (1)", "094214a889d7620abdcca4fa84418868": "X_{1}...X_{n}", "094227dfb2405ce5b48863d9f5819939": "A_{n,k}=f^{-1}(I_{n,k})\\,", "0942735dea8794d4f2e3deaa83ae7997": "\\scriptstyle >4\\times 10^{16}", "0942a5ac181bc56a8d018034c0a78251": "{\\bar {X}}_{n}={\\frac {1+r}{1+r-2rp}}", "0942b63c4f0f7af563b78c11bc220ee3": "\\lambda '_{k}", "0942c78c287263bcca177acf87db85da": "{\\vec {e}}_{1}={\\frac {1}{\\sqrt {2}}}\\left(-\\partial _{u}+\\partial _{v}\\right)", "0942e92262a38e93d87c6efc63ec10fc": "M\\times \\mathbf {x} =\\mathbf {r} ", "09430f293c53fa1d2b50300a4c97c432": "{n-1 \\choose k}", "0943b817f7702c7884cb9a84e1f1f76d": "\\Delta l^{a}-Dn^{a}=(\\gamma +{\\bar {\\gamma }})l^{a}+(\\varepsilon +{\\bar {\\varepsilon }})n^{a}-({\\bar {\\tau }}+\\pi )m^{a}-(\\tau +{\\bar {\\pi }}){\\bar {m}}^{a}\\,,", "0943ce1ac70bfe14432454af0974699a": "f'(c-):=\\lim _{h\\to 0^{-}}{\\frac {f(c+h)-f(c)}{h}}\\geq 0,", "09443db0f31f1eda9eb405c7bb3b2506": "{R_{0}}^{6}={\\frac {9\\,Q_{0}\\,(\\ln 10)\\kappa ^{2}\\,J}{128\\,\\pi ^{5}\\,n^{4}\\,N_{A}}}", "094465bf966548d186abe0852603823b": "f^{\\#}:\\Gamma (U,{\\mathcal {O}}_{\\mathfrak {Y}})\\to \\Gamma (f^{-1}(U),{\\mathcal {O}}_{\\mathfrak {X}})", "094483aeb9acc83cc696e281ad54e71f": "Y^{I}", "094485f24b16909fa0ad4595e0af8cb0": "\\ K_{m}{\\frac {\\partial {\\overline {u}}}{\\partial z}}=u_{*}^{2}", "0944896e940927c2c8cddf4e98ba9e6e": "\\kappa ={\\cfrac {6(1+\\nu )}{7+6\\nu }}", "0944ba12c6859ff459b96bb3df8e4c13": "\\textstyle k=0,\\ldots ,m,", "0944dccc974c8ecd2eb635e9e07249df": "x_{1}=g(x_{2})", "0945350e808ebfc8a74d3721ffa12fe6": "\\mathrm {K} \\,", "0945cf4b6889981499e131660b7a201b": "b={\\frac {\\sqrt {nt_{2}-t_{1}}}{n}}={\\frac {\\sqrt {na_{1}^{2}+a_{1}-2na_{2}}}{n}}", "09461d4a47d5f335d7a099083bdddde8": "f''(x)>0", "094622e462581ca208963660990c46c5": "{1 \\over {\\sqrt {5}}}", "09466659cb844a81439d1ed468f4e8d0": "|(j_{1}j_{2})JM\\rangle =\\sum _{m_{1}=-j_{1}}^{j_{1}}\\sum _{m_{2}=-j_{2}}^{j_{2}}|j_{1}m_{1}\\rangle |j_{2}m_{2}\\rangle \\langle j_{1}m_{1}j_{2}m_{2}|JM\\rangle ,", "09467e327f947a9328702c0c798537c1": "\\mathrm {d} \\,U=\\delta \\,Q-\\delta \\,W", "0946ff2e62a99a1118d5b2ca3914f9a0": "b={\\frac {L^{2}}{12A}}", "094718e343088ac76a79ade7d407761c": "L_{d}", "09472afa4e28f372f4e468dd1ba70728": "G^{\\circ }:=\\{x'\\in X^{*}:\\sup _{x\\in G}|\\langle x',x\\rangle |\\leq 1\\}", "09472d594f6078f0940ddcb4784fe3a4": "\\vert G\\vert \\leq 1.", "09473a49c3ab4d762f8532d1d99901bc": "\\mathrm {Sh} =2+0.552\\,\\mathrm {Re} ^{\\frac {1}{2}}\\,\\mathrm {Sc} ^{\\frac {1}{3}}", "094753ba0fc36eec5e9a3c66ff174f1c": "r_{n}=\\sum _{i}^{I}x_{ni}", "0947580233afe28450870ca2a4865c7d": "\\neg r", "094764b8dd5d98ad5d48fd84ed303fd8": "\\sum _{i=1}^{n}F_{i}^{Q}d_{i}^{P}=\\int _{\\Omega }\\sigma _{ij}^{Q}\\epsilon _{ij}^{P}\\,d\\Omega ", "09478deae56637f9624b72d6a3896bd7": "{\\tilde {E}}_{n-1}", "09478ff6b31f01fd19507cab34433c16": "\\phi _{1},\\dots ,\\phi _{k-1}\\in H_{0}^{1}(\\Omega )", "09479a199fdef109da421b3a4d5f020f": "\\lambda _{D}=\\left(4\\pi \\,\\lambda _{B}\\,\\sum _{j=1}^{N}n_{j}^{0}\\,z_{j}^{2}\\right)^{-1/2}", "0947f85161b05919d96940f3de14852e": "B", "094831ae988bb10fad007385038a78c2": "A\\land B", "0948394bec59903800ad2992fabb0b30": "\\scriptstyle 0\\,<\\,D\\,<\\,1", "0948a636d65bdc47b8770c1b99e8f236": "\\zeta (s)\\sim \\sum _{n=1}^{N-1}n^{-s}+{\\frac {N^{1-s}}{s-1}}+N^{-s}\\sum _{m=1}^{\\infty }{\\frac {B_{2m}s^{\\overline {2m-1}}}{(2m)!N^{2m-1}}}", "0948c8c44d9f535faf8e2e0dbc712c8e": "k+\\lfloor (d-1-k)/2\\rfloor .", "094902a95b24144c9df29e7578cb9601": "c^{d}\\;mod\\;n^{s+1}", "09496e5f9f581bb18f582f397efba651": "a_{0}={\\sqrt {\\frac {K/\\rho }{(1+V/a)[1+(K/E)(D/t)c]}}}", "09497279b3ce7b167a7bdecaeba65651": "|Re(\\lambda _{t})|", "094984d07f40c2bd9f5bf69963c4fe10": "\\nabla \\cdot {\\mathbf {u}}=0", "0949e1a2b5df281c5408ad24e3da6931": "\\delta (P,Q)\\leq \\epsilon ", "094a23869a432b4fc6713f5fc4a63f1a": "x(S_{j})", "094a4b7f64178507193890bac0caf4c8": "\\mu >\\mu _{0}", "094a507b0d1fcf973868aa32292bf876": "F_{a,P}", "094a56977f70cfb482781d28e7e569c7": "{\\frac {}{}}V=-\\cos \\phi ", "094a758d2221c5331523ad29ab9bc6c3": "t\\mapsto (x(t),y(t)),", "094a7a5a07adf16ec5826539c2846cac": "c(t,s)=0", "094aa8306996bf985a4ec6eebc2a7559": "Fx=g\\ ", "094ae3d247fb35321e48fcb479f8b285": "~~~\\land ~~~", "094b49264ee8427729dfcb00c8d27505": "L(\\lambda )=\\lambda {\\begin{bmatrix}M&0\\\\0&I_{n}\\end{bmatrix}}+{\\begin{bmatrix}C&K\\\\-I_{n}&0\\end{bmatrix}},", "094b52849e774704a0f97670237272ac": "\\rho _{\\rm {total}}({\\mathbf {r}}_{0})=\\rho ({\\mathbf {r}}_{0})-{\\mathbf {\\nabla _{\\mathbf {r_{0}}}\\cdot }}{\\mathbf {p}}({\\mathbf {r}}_{0})\\ ,", "094b80c4c580155f7ca62ea8367d09f4": "k(x,y)\\,", "094bd2fee5adc790afce9d8a740ac813": "\\Pi (T(u),(x)T(u'(x)))", "094c00021bc538a30547d55d7eafedea": "S_{2}=S_{1}-k\\ln {\\frac {c_{2}}{c_{1}}}.", "094c47d866750d6fdc716e0852e437ef": "\\Delta u=0", "094c67fbae9c46095a4a35113b0b6e81": "W={\\frac {1}{2}}\\left[q+c\\left(e^{Q}-1\\right)\\right]", "094c875557a127d33aeb67998d9c3a21": "e=H(x)-H({\\hat {x}})", "094c8c2a19feab2ae05565f68406784f": "\\lim _{t\\to 0}f(t)=\\lim _{s\\to \\infty }{sF(s)}.\\,", "094c98b7a30d55d18d41864d4d83220d": "H_{1}=-{\\vec {d}}\\cdot {\\vec {E}}", "094c9a72eb8a8b5945dadfed823e2029": "{\\mathit {a}}_{\\mathrm {InPAs} }={\\mathit {x}}{\\mathit {a}}_{\\mathrm {InP} }+(1-{\\mathit {x}}){\\mathit {a}}_{\\mathrm {InAs} }", "094d18158607da9878b4c45058a581ca": "g(x)=x^{3}+b_{2}x^{2}+b_{1}x+b_{0}\\,", "094d592d14f557ddfd0ba2559d45dd63": "u_{\\eta }=\\left(\\nu \\epsilon \\right)^{1/4}", "094db3efa7426e81b63abdbefd035742": "E=Q{\\sqrt {\\frac {2U(\\varphi _{0}/{\\sqrt {2}})}{\\phi _{0}^{2}}}}.\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(2)", "094de9a81e0e7af9e7a1956c2e1c5ad8": "W(x,0)\\approx W_{0}(x)(1-\\beta f_{0}x),", "094e9576987031cbd4bcba4500b740df": "\\{(a\\vee \\neg b)\\wedge b,\\neg a\\}", "094ecc18ebbedb0f9fd0a15c6bb5a067": "\\operatorname {Tor} _{1}^{R}(M,R/{\\mathfrak {m}}_{R})=0.", "094f269e7efe7e22e584f9ef7a476fba": "F_{r}=m{\\ddot {r}}-mr{\\dot {\\varphi }}^{2}\\ ", "094f2e693aff125779d7580dfd75e9bc": "k^{*}/k^{*\\ell }\\cong H^{1}(k,\\mu _{\\ell })", "094f314060e898c0178a790629c3da3c": "dQ\\equiv \\left({\\frac {\\partial Q}{\\partial x}}\\right)_{y,z}dx+\\left({\\frac {\\partial Q}{\\partial y}}\\right)_{z,x}dy+\\left({\\frac {\\partial Q}{\\partial z}}\\right)_{x,y}dz,", "094f317341c60034004f2dd57ee56726": "\\mathrm {Re} ={{{\\mathbf {\\mathrm {U} }}L} \\over {\\nu }}", "094f690fc31adc857e41175cbe31e63a": "\\delta <1", "0950734193e735d40fa7b541922bd3cc": "X={x_{i}}\\,", "0950954fe0e0b45837c415da31e1b313": "n_{f}=2{\\sqrt {n_{\\mathrm {osc} }/\\omega }}pcos(\\theta )", "09509bb217a36c28fc8091bf80905aa8": "\\mathbf {x} \\cdot \\nabla f(\\mathbf {x} )=kf(\\mathbf {x} ).", "09510045c7bdf1dc74d9878ff851a5fd": "\\xi \\mapsto h(\\psi (\\xi ))", "095105337e8db3125f8cae0140074f61": "S_{1},S_{2}", "09511782eb08c632e717fe4deee18516": "\\ln {a}", "09511e38861116d97d796c6a9f3bce24": "u_{a}", "0951fbe5c8cbca415187b32a26cfbbe8": "\\int |gh|\\leq \\left(\\int |g|^{p}\\right)^{1 \\over p}\\cdot \\left(\\int |h|^{q}\\right)^{1 \\over q},", "09521a5c8a46e66f356125b92d2d5a45": "K_{j}", "095233c697a7178fcbfdfe24ed52456b": "\\mathrm {RFN} _{T}", "09523a1cd616ab8d1bafb32ee2bd2974": "P(x,\\partial )u(x)=0{\\mbox{ for all }}x\\in W", "0952519696eb3154aac392fec57baf21": "E_{h}=E_{0}+{\\frac {0.05916}{n}}\\log \\left({\\frac {\\{A\\}^{a}\\{B\\}^{b}}{\\{C\\}^{c}\\{D\\}^{d}}}\\right)-{\\frac {0.05916h}{n}}{\\text{pH}}", "09525cb1fa2ba08fe77fcb33734de90a": "\\langle s_{\\lambda /\\mu },s_{\\nu }\\rangle =\\langle s_{\\lambda },s_{\\mu }s_{\\nu }\\rangle .", "095290f62fd852c7ed7b3478abd0817c": "a_{0}(-\\infty )^{n}\\,", "0952c66c88bf229d857571d4c0b4d059": "{\\begin{Bmatrix}p\\ \\ \\ \\ \\ \\\\q,r,s\\end{Bmatrix}}", "0952c9ffb538736990bcb5dae0528dec": "\\mathbf {d^{2}F} =-{\\frac {kII'}{2r^{2}}}\\left[(3-k){\\hat {\\mathbf {r_{1}} }}(\\mathbf {dsds'} )-3(1-k){\\hat {\\mathbf {r_{1}} }}(\\mathbf {{\\hat {r_{1}}}ds} )(\\mathbf {{\\hat {r_{1}}}ds'} )-(1+k)\\mathbf {ds} (\\mathbf {{\\hat {r_{1}}}ds'} )-(1+k)\\mathbf {d's} (\\mathbf {{\\hat {r_{1}}}ds} )\\right]", "0952d38192ddf1f35cbe57e2dbcf270c": "{\\frac {\\eta (c_{\\eta }(0,\\xi _{1})-c_{\\eta }(0,\\xi _{2}))}{\\xi _{1}^{2}-\\xi _{2}^{2}}}", "0952d4d5548366cfbb0508ddef98ac7f": "X=-{\\frac {1}{\\Omega \\left(E\\right)}}\\sum _{Y}Y\\Omega _{Y}\\left(E\\right)\\,", "09536cf4e7246ac68ed5cbcdacc496c7": "\\{B_{i}\\}", "095370a99280d227a2dfcf51c7db6377": "P(E_{1})=P(E_{2})", "0953df157b0e3bdea30ca43e8f93c589": "{G_{1}}={\\frac {1}{4}}", "095403dfb91e5f1a03b3921da626a07e": "\\mathbb {E} (A)=\\operatorname {Tr} (AS)=\\operatorname {Tr} (SA).", "095456c652d061a8dcc43a3dd76190b1": "q=A\\wedge B\\wedge C", "0954a08a2c8cf0f4e7bd803270720932": "x_{k+3}", "0954ad8f140b293c7db51dd3a944f980": "\\mathbf {x} _{i}^{\\rm {T}}", "0954d50e959a8d0bea34b757cfb81f8a": "87^{2}", "0954eaab43d03a708e1aea773b0503fe": "\\mathbf {x} ={\\boldsymbol {\\varphi }}(\\mathbf {X} ,t)~;\\qquad \\implies \\qquad {\\boldsymbol {F}}(\\mathbf {X} ,t)={\\boldsymbol {\\nabla }}_{\\circ }{\\boldsymbol {\\varphi }}~.", "0954f477e7638d610dc3ce6944a8cd82": "H(\\omega )", "0955997ef9a58a6e7c3c4c89f19d9eb8": "p(x)=\\sum _{i=0}^{20}d_{i}\\ell _{i}(x).", "09559aac3e7165615ee2854c47ea561c": "{\\tfrac {\\hbox{Volume before the event}}{\\hbox{Volume after the event}}}", "0955b0c09211920de4e8edcfa209da3a": "v(p,t)", "0955b39526c10a18f246e8a2d04c8f2d": "\\scriptstyle {\\left({\\frac {dk}{dt}}\\rightarrow 0\\right)}", "09562e84f3e57fe020a11623c906039a": "\\chi (T)", "09567e4092aa67f23d128e2ae835c66c": "P_{3}=(P_{0}(1+r)^{2}-c(1+r)-c)(1+r)-c", "0956aa0dc3f908f6ca6850a27799b75d": "{\\begin{aligned}\\min &\\left(f_{1}(x),f_{2}(x),\\ldots ,f_{k}(x)\\right)\\\\{\\text{s.t. }}&x\\in X,\\end{aligned}}", "09573235d73083444dcdace7a99d51ff": "P(S^{t}\\mid S^{t-1})", "095742bb8f0481559e2d3fe85c2f71b0": "\\deg ^{+}(v)=0", "095779904ee10259ed98c005f322a8b4": "\\partial (x_{1}\\wedge \\cdots \\wedge x_{p+1})={\\frac {1}{p+1}}\\sum _{j<\\ell }(-1)^{j+\\ell +1}[x_{j},x_{\\ell }]\\wedge x_{1}\\wedge \\cdots \\wedge {\\hat {x}}_{j}\\wedge \\cdots \\wedge {\\hat {x}}_{\\ell }\\wedge \\cdots \\wedge x_{p+1}.", "0957cd0bd5d645ae98f6d045b075e2ed": "0\\leq i\\leq n", "0957f3abe0246a2d5ae3691f3285252c": "|a-b|={\\sqrt {(a-b)^{2}}}.", "095854a35f357108bdd889fb3fa9c180": "{\\mathcal {H}}=\\hbar \\omega [a_{1}^{2}(t)+a_{2}^{2}(t)]=\\hbar \\omega |a|^{2}\\ ", "095884dd846abad1353040287b637f0f": "\\tan \\theta \\approx \\sin \\theta ={\\frac {h_{a}+h_{b}}{G}}", "095898131dafd77eeb3371eeb43a1d4b": "p(x)=a_{2}x^{2}+a_{1}x+a_{0}\\,", "0958bca8472d89437654500252cca8f2": "E\\psi =-{\\frac {\\hbar ^{2}}{2L_{1}}}{\\frac {d^{2}\\psi }{dQ_{1}^{2}}}-{\\frac {\\hbar ^{2}}{2L_{2}}}{\\frac {d^{2}\\psi }{dQ_{2}^{2}}}-{\\frac {\\hbar ^{2}}{m}}{\\frac {d^{2}\\psi }{dQ_{1}dQ_{2}}}+{\\frac {1}{2}}L_{1}\\omega ^{2}Q_{1}^{2}\\psi +{\\frac {1}{2}}L_{2}\\omega ^{2}Q_{2}^{2}\\psi ", "0958ff175cfcf08f7dd21940e9ad3e48": "N_{rep}", "095909d830f978137dbdd05dd4c799c8": "X_{j}\\sim \\Gamma (r_{j},\\lambda _{j})\\!", "09590f214e0a7a442685461e7fef3c13": "Ae^{i\\omega t}.", "095927ee22e00b24e9abbe3f2919526f": "{\\mathcal {X}}\\times {\\mathcal {U}}\\rightarrow \\mathbb {R} ", "09593c7ef06dd83105a50da58b009551": "F\\circ j_{1}=\\mathrm {id} ,\\;F\\circ j_{0}=Q.", "0959426bf646ca3acbb3a35e1008f06c": "\\mu _{z}=-g_{S}\\mu _{\\mathrm {B} }m_{s}", "09596554c8152b84ed61c789abf5f508": "c(t)=A_{c}\\sin \\left(\\omega _{\\mathrm {c} }t+\\phi _{\\mathrm {c} }\\right).", "0959b686c09969aacd1c134f708d3edd": "\\phi =(s+d)/2", "0959d68c51153e63e94f119cf882e31a": "{\\hat {\\mathbf {e} }}_{z}", "0959e4a345fb6e190468837bc88f1617": "{\\text{if }}A\\subseteq B{\\text{ then }}A\\times C\\subseteq B\\times C,", "095a5c3d1f428a09a5d5698d62870b04": "\\varnothing =X_{-1}\\subset X_{0}\\subset \\cdots X_{n}=P.", "095abe81cfbe7bd18c9d4d2db25c704e": "a_{i}^{j}x^{i}=y^{j}", "095ad04ab63221de2e90e52c139ecdc3": "\\mathrm {C_{A}=[A]+\\Sigma p\\beta _{pqr}[A]^{p}[B]^{q}[H]^{r}} ", "095afb28a327e39c1b8aa65dad1aa047": "p=\\rho _{f}gz.\\,", "095b351e35806fa18cf49a8f9f876d25": "\\varphi ^{2}=\\varphi +1", "095b376e98c5ae57cf7f7db6b68cb33d": "{\\tan \\delta }", "095b571b26ca92845328976c579ecbef": "Y\\sim \\Gamma (n,{\\tfrac {1}{\\lambda }})", "095bd0c7e0196c6060c03bf7f85677b5": "T:z\\mapsto z^{-1},", "095c73b948a40367964d73e43c5d0034": "1+\\varepsilon ", "095c9baecd446ee5d348d90a78a618ac": "\\mathbf {Z} =\\mathbf {Q} {\\boldsymbol {\\beta }}+\\mathbf {f} ,", "095ca029065b3107721a232b440137f0": "{\\begin{aligned}(a\\cdot \\phi )(x)&=a\\cdot \\phi (x)\\\\(\\phi \\cdot a)(x)&=\\phi (a\\cdot x).\\end{aligned}}", "095cafcc7a3a21ca5b06eed0089db8f8": "\\pm 17.8455995405\\ldots ", "095cb6dd9d2b7d3ccf7e5fa7c5c1c26c": "\\xi _{\\lambda }^{(k)}=DE(\\lambda )\\eta ^{(k)},", "095cc73e369d7a4543580ecb7be622fc": "\\ H_{i}^{BM}(\\mathbb {R} ^{n})", "095cfd7f7873d712a4219a2b816c7d2e": "\\varepsilon _{0}+1,\\qquad \\omega ^{\\varepsilon _{0}+1}=\\varepsilon _{0}\\cdot \\omega ,\\qquad \\omega ^{\\omega ^{\\varepsilon _{0}+1}}=(\\varepsilon _{0})^{\\omega },\\qquad {\\text{etc.}}", "095d2252d6f62e164bbabe2e7134c830": "w=d+[2.6m-2.2]+5R(y,4)+3R(y,7)+6R(c,7)\\mod 7", "095d2469b2b30ffee0188cd05f74140d": "D(f\\circ g)=\\left(Df\\circ g\\right)\\cdot Dg,", "095d5e9d33719f85e7a88caa2a8380a1": "y=2/3E", "095d62f182c280893f56fa8c5bd0beb4": "\\nabla ={\\hat {e}}_{i}\\partial _{i}", "095de5c44408794f1b2bdbd9cc73cfbf": "{\\frac {P_{1}}{T_{1}}}={\\frac {P_{2}}{T_{2}}}\\,", "095e0b4f6b0d16656cdec03341422ee9": "f''(x_{n})={\\frac {1}{n}}", "095e0c08dfbc23525edd438905f70751": "\\lim _{x\\to c}ax+b=ac+b", "095e5eb9ad0d34dd1394619992bb9638": "y\\in H_{q}(M)", "095e67779d8ca7e28c9da82d178f032f": "X\\perp \\!\\!\\!\\perp Y\\,|\\,W", "095e9e9f272d6de146b05cff6bcf29ec": "\\beta <\\delta ", "095efc2c0cd01c7483233734c6026f31": "x\\neq {\\tfrac {\\pi }{2k}}", "095f79d6947db27bd085bd08c7fb74fc": "\\left(m_{\\mathrm {p} }+{\\frac {\\rho _{\\mathrm {c} }V_{\\mathrm {p} }}{2}}\\right){\\frac {\\mathrm {d} \\mathbf {v} _{\\mathrm {p} }}{\\mathrm {d} t}}=\\sum \\mathbf {F} +{\\frac {\\rho _{\\mathrm {c} }V_{\\mathrm {p} }}{2}}{\\frac {\\mathrm {D} \\mathbf {u} }{\\mathrm {D} t}},", "095feae595d9a84c43b3fec5c50c4456": "{\\cfrac {\\partial {W}}{\\partial \\lambda _{1}}}=2C_{1}\\lambda _{1}~;~~{\\cfrac {\\partial {W}}{\\partial \\lambda _{2}}}=2C_{1}\\lambda _{2}~;~~{\\cfrac {\\partial {W}}{\\partial \\lambda _{3}}}=2C_{1}\\lambda _{3}", "0960050b9e095c506fc2a530b5b9f57e": "\\Sigma _{(W)\\tau }", "0960565040e50308c60d3f31261850bf": "\\scriptstyle Q({\\sqrt {d}})", "0960744036893f503a758250ed66ae44": "\\mathbf {a} ={d\\mathbf {u} \\over dt}", "09607f1e65b76e690a8c8ccd5f19c9b1": "d^{4}x", "0960f0fa6f0edf1c5617bed02d35d400": "{\\overleftarrow {Y}}=-jY_{\\varepsilon }cot(k_{x\\varepsilon }w)", "096132a030180689ee7dec2c0cee9b0d": "x^{\\alpha }{}_{,\\gamma }=\\delta ^{\\alpha }{}_{\\gamma }", "0961b7929edcd1adc1b80168efe1f1e9": "P\\times P/G", "09621f8d9b25f524f08c1c3be42f74ff": "{\\mathcal {K}}_{X}", "096249664ae745478eaf33d3ccf96078": "\\cap \\!\\,", "096254c7552111f593bb632a91205f32": "g(t)", "09626e60fb4ec8af0338e7c2d8ced428": "\\kappa _{3}", "0962b4b8c4ba6c674eba0d8a139f1847": "s_{j}=High[j]-High[j-2]+Low[j]-Low[j-2],\\!", "0962b784783898155442e71bcc2c61d0": "S:X\\times U\\times \\Omega \\to \\mathbb {R} ", "0962c4a09af7ff74899dd9c28456f237": "|S|=\\gamma n\\,", "0962cbab3034401a572b6225c8adce34": "\\cos(t\\sin(x))=J_{0}(t)+2\\sum _{k=1}^{\\infty }J_{2k}(t)\\cos(2kx)", "0962d64102fe6eb19848f88d2f216ee6": "g'_{\\mu 4}=g_{\\mu 4}", "096341dbe0702c7f67fef27239d01d6b": "\\rho _{d}", "0963a5ceac41578b77c25c7a6bf9ff3c": "\\gamma _{n}\\sim {\\frac {B}{\\sqrt {n}}}e^{nA}\\cos(an+b)", "0963af90edef723ecf56942cfea46cc8": "R_{s}={\\frac {2v^{2}\\cos \\theta \\sin \\theta }{g}}\\left(1-{\\frac {\\sin \\theta }{\\cos \\theta }}\\tan \\alpha \\right)\\sec \\alpha ", "0963beafb48c2673a3205af9c373448f": "bb^{-1}=b^{-1}b=1", "0963cfbb90c638cd43ca6a43115c2a2b": "\\beta _{31}", "0963fdc248538691a611f3dde51ee512": "\\Delta f=\\sum _{i=1}^{n}{\\frac {\\partial ^{2}f}{\\partial x_{i}^{2}}}", "0964764f7e9a82f9a61e983b5981755e": "T_{5}(x)=16x^{5}-20x^{3}+5x\\,", "0964ed758f2f62337e158772917ea184": "P([\\omega _{1},\\omega _{2},\\cdots ,\\omega _{n}])=p^{k}(1-p)^{n-k}", "09650b07280b0293af1c51f1b32e2714": "I_{1}=[\\alpha _{1},b],\\ I_{2}=[\\alpha _{2},\\alpha _{1}],\\ldots ,\\ I_{n}=[\\alpha _{n},\\alpha _{n-1}],\\ldots ", "09650ffec4f07c5d7550e4d9ecff7216": "K_{\\text{ww}}(s)={\\frac {1}{2\\pi }}\\int _{-\\infty }^{+\\infty }c_{\\text{ww}}(k)\\,{\\text{e}}^{iks}\\,{\\text{d}}k.", "096544be8a676d2c106621014410cb9d": "{\\bar {y}}", "0965765e2fb0e390110bf394067143ec": "s\\approx 9.017.", "0965ac349d461503070235f7f7f72a67": "a+b=101", "0965c09c0aec92659986192f7e01b563": "\\theta _{C}\\,", "0965d99537bdd25a6f0e283cc2c8ae31": "V(x)\\ =\\ V_{max}(e^{-x/\\lambda })", "09678f6206c24aaec93304fa2dd73fad": "p(x+0)=(x+0)^{3}-7(x+0)+7\\Rightarrow p(x+0)=x^{3}-7x+7,v_{0}=2", "0967b0a71f7dfdef1aae03b702abe6af": "g_{L}", "0967bfbb79279f6a37470eb9702c8c9a": "[a,a,[b,b,x]]=[b,b,[a,a,x]]", "0967cd071190719cbd546020d108a4d9": "\\Phi _{m}=\\int \\!\\!\\!\\!\\int _{S}\\mathbf {B} \\cdot \\operatorname {d} \\mathbf {S} ", "0967f23ea1b49959e7044d2dca5a3939": "\\epsilon =h\\nu .", "096803b2bed1cbe0fd72476a0ce77066": "AQHI=({\\frac {1000}{10.4}})\\times [(e^{0.000537\\times O_{3}}-1)+(e^{0.000871\\times NO_{2}}-1)+(e^{0.000487\\times PM_{2.5}}-1)]", "09683c59a82119f510512351a6894820": "R_{2k+1}(V)=\\left({\\frac {(2k+1)!!V}{2^{k+1}\\pi ^{k}}}\\right)^{1/(2k+1)}.", "096878ac00613b27f36f59926121e828": "|\\psi '_{r}\\rangle =\\alpha _{0}|100\\rangle +\\alpha _{1}|011\\rangle ", "0968b3b2971c0609830e848ab80d8ae7": "{\\boldsymbol {\\Psi }}+\\mathbf {C} +{\\frac {\\kappa _{0}n}{\\kappa _{0}+n}}(\\mathbf {\\bar {x}} -{\\boldsymbol {\\mu }}_{0})(\\mathbf {\\bar {x}} -{\\boldsymbol {\\mu }}_{0})^{T}", "0968c8e554f07e3dbeff90d4f29c633c": "e=[2;1,2,1,1,4,1,1,6,1,1,8,1,1,\\ldots ,2n,1,1,\\ldots ].\\,", "0968fca9807941e80550ee99ce0c2240": "6*8=48", "096904478ccd486df77f3feb9047b943": "x+t", "09691935278a5c625d71d8ea76d5e437": "\\left[{\\begin{matrix}1&0&0&0\\\\0&\\cos(\\theta )&0&\\sin(\\theta )\\\\0&0&1&0\\\\0&-\\sin(\\theta )&0&\\cos(\\theta )\\end{matrix}}\\right].", "0969414f4d05136d2436d4134a68cb99": "F(x,y)=x+y+xy.\\ ", "09694f8475d80b14df6bbc83c1fb57af": "a+d+b=180", "096971d45197ed1d9a75ce0cd47dae24": "\\{1,i_{1},i_{2},i_{3}\\}", "0969898997e5522aca9617e7de94e68c": "~{\\hat {b}}{\\hat {b}}^{\\dagger }-{\\hat {b}}^{\\dagger }{\\hat {b}}=1~", "0969b386f6f36c93005270a70dbcd718": "(a,b)\\cup (c,d)", "0969d5ec1b40b50173b662915d2969ec": "g=dz^{2}+dx_{1}^{2}+dx_{2}^{2}+\\cdots +dx_{n}^{2}\\,", "096a17a4aca0892fc0c5f93fc6424d76": "\\omega ^{\\left(\\omega ^{\\left(\\omega ^{7}\\cdot 6+\\omega +42\\right)}\\cdot 1729+\\omega ^{9}+88\\right)}\\cdot 3+\\omega ^{\\left(\\omega ^{\\omega }\\right)}\\cdot 5+65537", "096a465d7390b3d6ad85475dc16dfa5b": "m{\\ddot {\\mathbf {r}}}\\cdot \\delta {\\mathbf {r}}=m\\sum _{j}\\left[\\sum _{i}{\\ddot {r_{i}}}{\\partial r_{i} \\over \\partial q_{j}}\\right]\\delta q_{j}", "096aa3f81d7e58d9779a0cbe73ec51f0": "\\epsilon _{\\mu \\nu \\lambda \\sigma }\\,", "096ace34b111266e5369566ddc2de1d6": "H(t)=\\int _{0}^{\\infty }f(E,t)\\left[\\log \\left({\\frac {f(E,t)}{\\sqrt {E}}}\\right)-1\\right]\\,dE", "096afb8c4a105c41a11e79dbb2d0a86a": "\\lim _{x\\to \\infty }f(x)=\\infty ,\\lim _{x\\to a^{+}}f(x)=-\\infty .\\,", "096b4eceb1c9aa32f30a59eba18b9cd6": "\\scriptstyle {\\tan \\beta ={\\frac {b}{a}}\\tan \\phi }\\,\\!", "096b57dec2d3fdbb30eacfd5138c09bc": "\\Gamma =\\sum _{i=1}^{N}s_{i}\\Delta x_{i}", "096b5dbdd0efe5fdefa4e357e9a7225c": "E.", "096b9d223a695b5f6bee9412cc117441": "\\mathbf {P} (X\\geq \\mu +a)\\leq e^{\\frac {-2a^{2}}{n}},\\qquad a>0", "096c1dd7a54c26d0f0b35cef8f9ed3ea": "\\mathrm {\\frac {M}{Lt}} ", "096c5d547901e3a6fc893504e5f2db06": "Q_{\\text{s}}=325+P-30P_{\\text{rg}}", "096c98367a59dffb36f8c2bbe5fb86d4": "T[L]\\subset S\\rightarrow S/R", "096ca414092552ff722add6950d3e4e3": "\\nu (d)=2d+1", "096cb9d94c653880be7ef740c2fb0dbd": "h_{2}", "096ccebe745cf5215e225d1d69097027": "\\left|\\sum _{k=1}^{n}a_{k}b_{k}\\right|\\leq \\operatorname {max} _{k=1,\\dots ,n}|B_{k}|(|a_{n}|-a_{n}+a_{1}),", "096d1141a089ba9d5257120eab9c6937": "\\sigma =\\sum _{j}L_{ij}{\\frac {\\partial F_{i}}{\\partial x_{i}}}{\\frac {\\partial F_{j}}{\\partial x_{j}}}", "096d60169c19cc448abda87fcde082ab": "y_{n}\\to y", "096d6299450f63a8c1fd81f17f031baa": "C_{\\bullet }:{\\mathbf {Top}}^{2}\\to {\\mathcal {CC}}", "096d66d696d9b0171337485d3c0ad5ac": "(\\neg A)\\wedge (\\neg B)", "096d8aa5f4c64cad3813a5d52e1d6b9f": "\\displaystyle {f(e^{i\\theta })=\\sum _{m=1}^{N}a_{m}e^{im\\theta }+a_{-m}e^{-im\\theta },\\,\\,\\,\\,\\,\\,\\,a_{-m}={\\overline {a_{m}}}.}", "096d9ba631683970176f5af9f654e4aa": "{d \\over dx}\\left[(1-x^{2}){d \\over dx}f\\right]+n(n+1)f=0.", "096dea68262a84f3056c534dcef6f6aa": "{\\mathfrak {g}}_{i}=(0)", "096e0e6fde47c8b3321f49f7dc713146": "u", "096e4febe6746dcbe9598b5460874688": "\\rho _{g}(X)=\\rho _{g}(Y)", "096e85756c63167153b43c5c4e7b8be3": "\\eta _{v}\\approx 90\\%", "096ea0cb824615090a2302797944e5c0": "v_{\\infty }=\\,\\!", "096ea70a8a79650e3bf8e72465a2c2a0": "{\\begin{aligned}\\mathbf {E} '&\\approx \\mathbf {E} +\\mathbf {v} \\times \\mathbf {B} \\\\\\mathbf {B} '&\\approx \\mathbf {B} -{\\frac {1}{c^{2}}}\\mathbf {v} \\times \\mathbf {E} \\\\\\mathbf {j} '&\\approx \\mathbf {j} -\\rho \\mathbf {v} \\\\\\rho '&\\approx \\left(\\rho -{\\frac {1}{c^{2}}}\\mathbf {j} \\cdot \\mathbf {v} \\right)\\end{aligned}}", "096ecc68b832602e6dc1a3286457a994": "v'={\\frac {dx'}{dt'}}={\\frac {dx^{\\beta }}{dt^{\\alpha }}}\\,,\\quad \\alpha ,\\beta >0", "096ed71a2569492eb8aaf1db3cb4dc24": "n\\leq r", "096f040c920b4733618df7324e1675cd": "f\\left({\\frac {az+b}{cz+d}}\\right)=(cz+d)^{k}f(z)", "096f05198f447797874c0e5c54fcf0b7": "{\\begin{aligned}R_{\\frac {\\lambda }{2}}&=60\\operatorname {Cin} (2\\pi )=60\\left[\\ln(2\\pi \\gamma )-\\operatorname {Ci} (2\\pi )\\right]=120\\int _{0}^{\\frac {\\pi }{2}}{\\frac {\\cos \\left({\\frac {\\pi }{2}}\\cos \\theta \\right)^{2}}{\\sin \\theta }}d\\theta ,\\\\&=15\\left[2\\pi ^{2}-{\\frac {1}{3}}\\pi ^{4}+{\\frac {4}{135}}\\pi ^{6}-{\\frac {1}{630}}\\pi ^{8}+{\\frac {4}{70875}}\\pi ^{10}\\ldots -(-1)^{n}{\\frac {(2\\pi )^{2n}}{n(2n)!}}\\right],\\\\&=73.1296\\ldots \\;\\Omega ;\\end{aligned}}\\,\\!", "096f2e30968f0d3a41237fe94c9d134d": "T={\\text{PA}}\\cdot {\\text{PB}}=(s-a)(s-b).", "096f459f6961efcf35ad08b2a9150215": "{\\frac {1}{m}}\\sum _{j=1}^{m}1-{\\mbox{erf}}({\\sqrt {c}})=1-{\\mbox{erf}}({\\sqrt {c}})", "096f98947922bba34654053c41d3e754": "{\\mathsf {cap}}(\\mathbb {Z} )\\approx 1-0.5{\\mathsf {H}}(p)\\,", "096fc7fbca9f40181b5e487c4f5ad897": "{\\bar {x}}_{i}={\\frac {1}{N}}\\sum _{j=1}^{N}x_{j}", "097009ed75ef0bb863cc82b7f8a132de": "f^{\\star }:{\\mathcal {P}}(Y)\\rightarrow {\\mathcal {P}}(X)", "09700b05417a749d1d44aca99c89f6e1": "\\S ", "097073df6da15a0976e270a725533f9c": "[(D_{w}+F_{w}/2)+(D_{e}-F_{e}/2)+(F_{e}-F_{w})]\\phi _{P}=(D_{w}+F_{w}/2)\\phi _{W}+(D_{e}-F_{e}/2)\\phi _{E}", "097089d7c14702d3841de37b9e73f981": "\\rho (p,q)", "0970993acbcca31a8ad1daf4cde9bd04": "\\log _{2}(N)+1", "0970a84585c8793159a6ed20a9d3ec28": "\\infty \\infty ", "0970f1ef8543c935a4485f71f7e71f3d": "\\beta ={\\frac {1}{v}}\\left({\\frac {\\partial v}{\\partial T}}\\right)_{p}={\\frac {-1}{\\rho }}\\left({\\frac {\\partial \\rho }{\\partial T}}\\right)_{p}", "0971043f6845805b93760346b10539d0": "I=\\epsilon \\sigma T^{4}\\,", "09712e1bc31a3cbca8b59ee62161c421": "\\,dE", "09720b310739f42f8f43f75de39d76a2": "\\operatorname {true} ", "09722107a1fc3e97695db13a2c8920f5": "={\\sqrt {1^{2}+\\sinh ^{2}(t)}}", "09727656860d33f2e92652fad35ceebf": "{\\frac {\\partial C}{\\partial a}}", "0972a2790e564eb2c668120dd341a753": "C={\\frac {\\mu _{0}\\mu _{B}^{2}}{3k_{B}}}Ng^{2}J(J+1)", "0972b0a00bae0824536f35954e43094d": "\\rho _{\\$}", "0973689429469b272bee6ad9b5c6ba0a": "z^{2}+(r^{2}-R^{2}+a^{2})=2ax.", "097391bd3e1e7dcc273a48e568291d20": "\\|\\mathbf {v} +\\mathbf {u} \\|\\leq \\|\\mathbf {v} \\|+\\|\\mathbf {u} \\|", "0973d0605145d4dba13f07b809abbab8": "\\|\\sum _{k=1}^{n}\\mathbf {v} _{k}\\|^{2}=\\sum _{k=1}^{n}\\|\\mathbf {v} _{k}\\|^{2}.", "09741544dadae6628807fe3d1bf7c5c2": "{d^{n} \\over dx^{n}}f(g(x))=\\sum _{k=1}^{n}f^{(k)}(g(x))B_{n,k}\\left(g'(x),g''(x),\\dots ,g^{(n-k+1)}(x)\\right).", "09741c5ffc900a64f1974cf18780e5bb": "\\theta _{n}^{*}", "0974695ef8a05c316d5ac876ac20d024": "z+n=\\prod _{p_{i}\\in P}p_{i}^{b_{i}}", "09752d7b87a2851f9fc95ecea062f4ea": "M_{int}", "0975621693e34cf3b4ad2d38cd1967ac": "\\textstyle P={\\frac {1}{\\mathcal {Z}}}e^{(\\mu N-E)/(kT)}", "0975646846c2a4df7b4a157872a6eea3": "{\\begin{aligned}u_{1}&={\\frac {F_{1}(\\kappa +1)\\ln |x_{1}|}{4\\pi \\mu }}+{\\frac {F_{2}(\\kappa +1){\\text{sign}}(x_{1})}{8\\mu }}\\\\u_{2}&={\\frac {F_{2}(\\kappa +1)\\ln |x_{1}|}{4\\pi \\mu }}+{\\frac {F_{1}(\\kappa +1){\\text{sign}}(x_{1})}{8\\mu }}\\end{aligned}}", "09756f541757a26b30c46350ebcaecac": "N_{t+1}=\\lambda \\ N_{t}\\ [1-f(N_{t},P_{t})]", "09759381df78e67a0bb93494641ed110": "E[m]\\cong (\\mathbb {Z} /m\\mathbb {Z} )*(\\mathbb {Z} /m\\mathbb {Z} )", "0975ce570b45bf71988686e6cf9c31ad": "gate5", "0975d2363205bf5e1fd7faf7c0f8aeee": "m1=1", "0975dbd2002cc29494d43c86f625b851": "{\\text{s.t.}}={\\begin{cases}g_{1}\\left(x,y\\right)&=6.5-{\\frac {x}{6}}-y\\geq 0\\\\g_{2}\\left(x,y\\right)&=7.5-0.5x-y\\geq 0\\\\g_{3}\\left(x,y\\right)&=30-5x-y\\geq 0\\\\\\end{cases}}", "0976007f6874da15abe6fa496b78f514": "h\\circ f", "097633138af488186f3186b7b891d704": "\\sigma (M)=\\langle L_{n}(p_{1}(M),\\dots ,p_{n}(M)),[M]\\rangle .", "09763da86a7cd69f749f8167611472a9": "w^{2}+x^{2}+y^{n+1}=0", "0976495e501d07840e173eed15d0c8b1": "\\theta _{ab}", "09775492fcfb9eb777ac4cb1abb6321b": "C_{0}\\;", "0977848f1d538d445f8ca37db3fbacaf": "\\rho _{XY\\cdot \\mathbf {Z} }={\\frac {\\rho _{XY\\cdot \\mathbf {Z} \\setminus \\{Z_{0}\\}}-\\rho _{XZ_{0}\\cdot \\mathbf {Z} \\setminus \\{Z_{0}\\}}\\rho _{Z_{0}Y\\cdot \\mathbf {Z} \\setminus \\{Z_{0}\\}}}{{\\sqrt {1-\\rho _{XZ_{0}\\cdot \\mathbf {Z} \\setminus \\{Z_{0}\\}}^{2}}}{\\sqrt {1-\\rho _{Z_{0}Y\\cdot \\mathbf {Z} \\setminus \\{Z_{0}\\}}^{2}}}}}.", "0978441c65781ef8858885e566c51159": "T_{f}", "09784872ea44d9b0dc26740990266bf0": "\\scriptstyle \\rho (r)", "0978927b31f441753e587f876ddb9834": "\\rho ", "0978c8d19fe08434e9e1f689b148d94f": "H^{n-1}", "0978fe6ad5995cfd6c1e14c4f6dc78ce": "{\\frac {dm}{dt}}=\\alpha _{m}(1-m)-\\beta _{m}m", "0979d887e36a5e1e07b517a7fee02c95": "\\gcd(z,n)", "0979f533e9b621efce4a1efc0ee32001": "\\scriptstyle \\blacksquare ", "097a3aff7cb38c8ea745a2906532297d": "\\nabla \\cdot \\mathbf {E} =4\\pi \\rho \\ ", "097a50be807bb22aed485bdee4e92d8b": "\\iiint _{D}f(x,y,z)\\,dx\\,dy\\,dz=\\iiint _{T}f(\\rho \\cos \\phi ,\\rho \\sin \\phi ,z)\\rho \\,d\\rho \\,d\\phi \\,dz.", "097a559ec433955b587d0a6ce5197b55": "-\\log {\\text{det}}(X)", "097a84ad0f37c7cab16c83a005271e8b": "v_{n}(\\xi \\otimes e_{\\alpha })=(v_{n}\\xi )\\otimes e_{\\alpha }.", "097a98dbbca91e3b2b0e85b8d9d9523b": "\\lambda _{1},\\lambda _{2},\\ldots ,\\lambda _{r}", "097a9b36222e3b07ed9c68a582c9c8d9": "\\mathrm {SML} :E(R_{i})-R_{f}=\\beta _{i}(E(R_{M})-R_{f}).~", "097b15b5b599c8cef7289a97dd545f49": "{\\mbox{estimated median}}={\\widehat {\\kappa }}\\cdot 2^{1/{\\widehat {\\theta }}},\\,", "097bacc79ab8726dc04e7647f8d32742": "\\Psi (\\rho )={\\begin{bmatrix}\\langle F_{1},\\rho \\rangle \\\\\\vdots \\\\\\langle F_{n},\\rho \\rangle \\end{bmatrix}}", "097bf9d105523a0a559358845998755b": "{\\begin{aligned}2\\cdot R_{*}&={\\frac {(160\\cdot 4.24\\cdot 10^{-3})\\ {\\text{AU}}}{0.0046491\\ {\\text{AU}}/R_{\\bigodot }}}\\\\&\\approx 146\\cdot R_{\\bigodot }\\end{aligned}}", "097bfb21de52e2233346207b45995bbd": "\\sigma (f,g):=\\inf _{\\lambda \\in \\Lambda }\\max\\{\\|\\lambda -I\\|,\\|f-g\\circ \\lambda \\|\\},", "097c01f668e1b9181575da3b69b94777": "E_{n,k}=\\bigcup _{m\\geq n}\\left\\{x\\in A\\,{\\Big |}\\,|f_{m}(x)-f(x)|\\geq {\\frac {1}{k}}\\right\\}.", "097c0fa7da7dfe4ae638d32e1a2f7af7": "b=u^{2}+v^{2},", "097c595ad1fdd1f412534364752b3eb6": "\\mathbb {E} [X_{n}]>0", "097c7b6870e070b68a6127fa5aafc6d5": "{\\mathcal {G}}_{2}", "097ce48eb7ba9f62a32222253205111f": "T((\\lambda {\\dot {q}}_{i})^{2},(\\lambda {\\dot {q}}_{j}\\lambda {\\dot {q}}_{k}),\\mathbf {q} )=\\lambda ^{2}T({\\dot {q}}_{i},{\\dot {q}}_{j}{\\dot {q}}_{k},\\mathbf {q} )\\,,\\quad L(\\mathbf {q} ,\\mathbf {\\dot {q}} )\\,,", "097cf59b7d9ce35264b75bf4efddd0a1": "{\\begin{matrix}{\\frac {\\pi }{2}}\\end{matrix}}", "097d48183f5023757b1e0aa89c426b67": "{\\zeta _{g}}={{\\partial v_{g} \\over \\partial x}-{\\partial u_{g} \\over \\partial y}={1 \\over f_{o}}({{\\partial ^{2}\\Phi \\over \\partial x^{2}}+{\\partial ^{2}\\Phi \\over \\partial y^{2}}})={1 \\over f_{o}}{\\nabla ^{2}\\Phi }}", "097d51db028cbbe5ad2bd557eb51310c": "[\\mathbf {A} ,\\mathbf {B} ],\\qquad \\mathbf {A} \\in \\mathbb {R} ^{n\\times m},\\qquad \\mathbf {B} \\in \\mathbb {R} ^{p\\times m},\\qquad m\\geq n+p.", "097d55f668e4274885302f68024762d8": "{\\frac {1}{5-{\\frac {1}{z}}}}=-z\\left(1+5z+5^{2}z^{2}+5^{3}z^{3}+\\cdots \\right)", "097d8759686fb29d878843530d39965b": "\\nu _{0}'{\\sigma _{0}^{2}}'", "097da62f4a63ee6af7d9e705f624067f": "(H\\ or\\ E)", "097ded293601685da87fcd695852c05f": "z_{4}={\\frac {z_{1}k_{1}+z_{2}k_{2}+z_{3}k_{3}\\pm 2{\\sqrt {k_{1}k_{2}z_{1}z_{2}+k_{2}k_{3}z_{2}z_{3}+k_{1}k_{3}z_{1}z_{3}}}}{k_{4}}}.", "097e4dc3186bba142989fce838e56df2": "R={\\frac {1}{r}},\\ \\Theta =\\theta ,", "097ee796eed6c815586d9373311afaef": "pA+(1-p)C", "097f0656cb205e1963956c521fb4f408": "[1]\\,\\!", "097f20080dce1b9b0584fa371b948475": "\\ell ={\\frac {d}{\\tan \\alpha }}+{\\frac {d}{\\tan \\beta }}", "097f2e4dfb5c66ef949c438dc540d1c4": "H_{5}=0\\,", "097f8961516236e8c71a5eaf265098ee": "N(t)=N_{0}e^{-t/\\tau }", "097fcb4b95369f3515606dfdc7c39acc": "{\\hat {p}}=-j\\hbar {\\frac {\\partial }{\\partial q}}", "09800103d9cdd149c7fa40ada0bd230a": "s_{\\hat {\\beta }}={\\sqrt {\\frac {{\\tfrac {1}{n-2}}\\sum _{i=1}^{n}{\\hat {\\varepsilon }}_{i}^{\\,2}}{\\sum _{i=1}^{n}(x_{i}-{\\bar {x}})^{2}}}}", "098009d0faadd3d7b3fd754d84183ed2": "M_{CB}={\\frac {2EI}{L}}\\left(2\\theta _{B}+4\\theta _{C}\\right)=0.4EI\\theta _{B}+0.8EI\\theta _{C}", "0980296df0730a80f387b01bafb64e9d": "t\\colon A\\rightarrow \\ V", "09805d958fc7ab3d6c57c02f71afdc39": "(1)\\,", "09807a84ef1a3f642a4f2033a4854a8f": "P=\\tau \\omega .\\!", "0980aca54f873f12790fe3506eb3d61f": "\\Omega =2\\ F", "0980cf9e91aeb7ea0ddc6fddbad535ee": "-f(x)=f(-x),\\,", "0980fbfa5727b6f2ca6a7c1b17b48674": "\\exists x_{1}\\exists x_{2}(x_{1}\\not =x_{2})", "0980feeeaa6ae8fef931eedafc1c55b4": "A\\in \\mathbb {R} ^{2\\times 2},{\\textbf {b}}\\in \\mathbb {R} ^{2\\times 1},c\\in \\mathbb {R} ", "09810d9ecc81282741b9dc8f9c503895": "{\\begin{matrix}\\mathrm {ELEMENTARY} &=&\\mathrm {EXP} \\cup \\mathrm {2EXP} \\cup \\mathrm {3EXP} \\cup \\cdots \\\\&=&\\mathrm {DTIME} (2^{n})\\cup \\mathrm {DTIME} (2^{2^{n}})\\cup \\mathrm {DTIME} (2^{2^{2^{n}}})\\cup \\cdots \\end{matrix}}", "09815b3a7e8e43b07cd97cadb9a90e18": "x\\mapsto (\\tau _{j}\\circ \\tau _{i}^{-1})_{x}", "09819451d727e92e946d4b286b90e8f2": "\\eta =1-\\left({\\frac {1}{{r}^{(\\gamma -1)}}}\\right)", "0981db6047065ebfd3318d3ccbe75e29": "\\langle x\\rangle =\\int _{-\\infty }^{+\\infty }x|\\psi |^{2}dx=\\int _{-\\infty }^{+\\infty }\\psi ^{*}x\\psi dx", "09820eb4bcdd600f9e532ab865563fcd": "c\\,\\delta {\\frac {d\\tau }{dq}}=-{\\frac {r^{2}}{c}}{\\frac {d\\varphi }{d\\tau }}\\delta {\\frac {d\\varphi }{dq}}=-{\\frac {r^{2}}{c}}{\\frac {d\\varphi }{d\\tau }}{\\frac {d\\delta \\varphi }{dq}}\\,.", "098242e80f673a13cffbd1503594e264": "\\Lambda :\\mathbb {R} ^{n}\\times \\mathbb {R} ^{m}\\times \\mathbb {R} ^{p}\\to \\mathbb {R} ", "0982a90d01632f7f9530ba11ea06b118": "N={\\{x_{1},x_{2},x_{3},...,x_{N}\\}}", "0982dadeae8a228dd4b9f70dcb434e2c": "A={\\tfrac {1}{2}}{\\begin{pmatrix}1&1\\\\1&-1\\end{pmatrix}}\\ :", "0982e34a1f03c5af8411265f9f7d7154": "\\cong F", "09830b5cfdd205dad87790289c7e9759": "h_{ab}=\\mathrm {atan2} (b^{*},a^{*})\\;", "09831a0e4fa9247ca945da792ebcb688": "\\lambda _{CWL}(M_{1}\\#M_{2})=\\left\\vert H_{1}(M_{2})\\right\\vert \\lambda _{CWL}(M_{1})+\\left\\vert H_{1}(M_{1})\\right\\vert \\lambda _{CWL}(M_{2})", "0983244c70964fc8799fa40690b30455": "2x^{2}+x^{3}=0", "09835a6d51b9fdffd8d39d0c73cdec77": "{\\begin{aligned}s_{t}&=\\alpha x_{t}+(1-\\alpha )(s_{t-1}+b_{t-1})\\\\b_{t}&=\\beta (s_{t}-s_{t-1})+(1-\\beta )b_{t-1}\\\\\\end{aligned}}", "0983c99e7e7a64f926891c02ae09517f": "\\%dMU={\\frac {\\frac {d^{2}u}{dc^{2}}}{\\frac {du}{dc}}}=-{\\frac {\\theta }{c}}", "0983ca34c5e32b69c4927eb0d82f0016": "1+f(z)+\\ln(1+f(z))=z.\\,", "0983f2ee5a9870a798b2e318275cfb01": "A_{q}(n,d)\\times m=A_{q}(n,d)\\times {\\begin{matrix}\\sum _{k=0}^{t}{\\binom {n}{k}}(q-1)^{k}\\end{matrix}}\\leq q^{n}.", "0984a5058c519be102dee9ec5d428e1b": "\\leq e^{-{\\sqrt {d+1}}}+{\\frac {1}{\\sqrt {d+1}}}.", "09852f1866f0b516b83d14195bfab1ce": "f_{i}x_{i}+\\sum _{j=1}^{n}{b_{ij}t_{j}}\\geq h_{i}", "09858f28dd06ae24d4e450112c317d06": "1x=x1=x", "0985c6389e59e6d9bc65f468ba01dced": "\\int _{\\gamma }|z|^{-1}\\,ds\\geq \\int _{\\gamma }|z|^{-1}\\,d|z|=\\int _{\\gamma }d\\log |z|=\\log(r_{2}/r_{1}).", "0985ea2ce5a835ddd8ca34e92602ee75": "X=(X_{t}:t\\geq 0)", "09861baa3c8470520f191421c0e07035": "{d \\over dx}\\cot y={d \\over dx}x", "0986362c046227cbb01c6ab5a48cb4c3": "{\\frac {\\operatorname {Vol} (B_{\\varepsilon }(p)\\subset M)}{\\operatorname {Vol} (B_{\\varepsilon }(0)\\subset {\\mathbb {R} }^{n})}}=1-{\\frac {S}{6(n+2)}}\\varepsilon ^{2}+O(\\varepsilon ^{4}).", "09866ffbc8cb073dbec02b910afdd2c1": "{\\text{post-money valuation}}={\\text{new investment}}\\,\\cdot \\,{\\frac {\\text{total post investment shares outstanding}}{\\text{shares issued for new investment}}}", "0986cdefd6ee502366e89f598a303154": "J={\\begin{bmatrix}0&-I_{V}\\\\I_{V}&0\\end{bmatrix}}.", "0986d7afc7049e26fe594ed1dbc5e7f5": "\\left|\\alpha -{\\frac {p}{q}}\\right|>{\\frac {C(\\alpha ,\\epsilon )}{q^{2+\\epsilon }}}", "0986ea696bbd39ed9ca6bb0590272f87": "(1-R-\\epsilon )", "0987157267dae2f3bf8698474206f699": "\\cos(C)=0\\,", "098721a2282b5c4c233baf5582a53d36": "f'(x)=f(x)\\times {\\Bigg \\{}h'(x)\\ln(g(x))+h(x){\\frac {g'(x)}{g(x)}}{\\Bigg \\}}=g(x)^{h(x)}\\times {\\Bigg \\{}h'(x)\\ln(g(x))+h(x){\\frac {g'(x)}{g(x)}}{\\Bigg \\}}.", "09877b064fa5471c8194cddb57d64020": "{\\overline {T}}", "0987c1a0688089dca4c4e0fc012d44ae": "y'e^{\\int _{s_{0}}^{x}P(s)ds}+P(x)ye^{\\int _{s_{0}}^{x}P(s)ds}=Q(x)e^{\\int _{s_{0}}^{x}P(s)ds}", "0987e45a4deecff31f9af6654c280af1": "p^{k+1}=p^{k}+{\\text{urf}}\\cdot p^{'}", "0987ee8197421cb30cc7bc847561ba6d": "{\\frac {P_{i}}{\\varepsilon _{0}}}=\\sum _{j}\\chi _{ij}^{(1)}E_{j}+\\sum _{jk}\\chi _{ijk}^{(2)}E_{j}E_{k}+\\sum _{jk\\ell }\\chi _{ijk\\ell }^{(3)}E_{j}E_{k}E_{\\ell }+\\cdots .\\!", "098818059bfd2dde17efbbde43cf8edf": "y(t)={\\frac {v(t)-v_{n}}{v_{n}}}={\\frac {v(t)}{v_{n}}}-1", "098829e32a19cef921503374218122d4": "N=pq\\,", "0988378c122ad2df86bd1b52f42438e2": "\\mu (n)=(-1)^{2x+1}=-1.\\,", "098842703beb9d0a23797626421f2871": "\\Delta I_{L_{Off}}=\\int _{DT}^{T}{\\frac {\\left(V_{i}-V_{o}\\right)dt}{L}}={\\frac {\\left(V_{i}-V_{o}\\right)\\left(1-D\\right)T}{L}}", "098858c2e462b2a4074f0f59317fd2b2": "p_{4}(x)=-512x+192x^{2}-24x^{3}+x^{4};", "0988883d6d31b6e2be361660d5a75515": "A(x)=\\Omega (x^{1/3})", "0988a06cb7c934ce557c3a2e87cee7bd": "O{\\Big (}\\sum _{i0", "0995841f0f62a95f7c242f4ec4930b53": "C\\in M(r,n;\\mathbb {K} )\\,\\!", "099593cda6e698b53cfbe594b61a8174": "\\alpha _{SID}=1-(1-\\alpha )^{\\frac {1}{m}}", "099621954608cb6bf333415eb4507a34": "R_{\\mu \\nu }\\cong {\\frac {1}{2}}\\left[\\ln(-g)\\right]_{|\\mu |\\nu |}-[\\mu \\nu ,\\beta ]_{|\\beta }", "09962f051b570d1498d494053eaf8026": "f_{\\alpha }-f_{\\beta }", "09965877ff74a5cb88cf265b134c7dc6": "H(\\omega )={\\frac {1}{\\sqrt {2}}}\\sum _{k\\in Z}h_{k}e^{j\\omega k}", "09967d670c98fb8f3b9be63df7a81c6a": "-{\\frac {\\hbar ^{2}}{2m}}{\\frac {d^{2}}{dx^{2}}}\\psi (x)+V(x)\\psi (x)=E\\psi (x)", "0996b06f4d715a0f9551e5c219fdff2f": "x_{1}\\in X_{1}", "0997199f820cb74dd1a3a1e6541c4748": "f,g:M\\to N", "099752f006bd3a40c926dc076fee8930": "\\rho _{1}", "0997a652bf2570678009f56f910281bd": "{\\frac {\\Delta P}{\\Delta T}}", "0997d68bf07ac02c9a14ef557a6dde99": "E_{0}({\\textrm {volts}})=-{\\frac {\\Delta G^{\\ominus }}{nF}}", "0997e7d9040a21ee00bb261f76367d1b": "R^{-1}\\nabla ^{2}R\\rightarrow \\infty .", "099834bdfc700dd0dd86dc64749ebc5b": "{\\frac {1}{m^{*}}}={\\frac {1}{\\hbar ^{2}}}{\\frac {\\partial ^{2}E(k)}{\\partial k^{2}}}", "0998676d55300fb29951cd4e7100e2f8": "f(p,q,p_{c})=\\left[{\\frac {q}{M}}\\right]^{2}+p\\,(p-p_{c})\\leq 0", "09989a650103e6c656dd117d1001b026": "\\lim _{y\\to q}{\\big (}\\lim _{x\\to p}f(x,y){\\big )}.\\,", "0998aca5b089d13d466f93c28245dc7e": "[-1,1]^{2}\\,", "0998afbc0acded704c1318a2ceb36e90": "\\scriptstyle f_{n}(x)\\leq f_{n+1}(x)", "0998d79e7094eb9a79037c47426c8aae": "d\\mathbf {x} ^{2}=d\\mathbf {X} \\cdot \\mathbf {C} d\\mathbf {X} \\,\\!", "0999538dbc8306d9f6ca28d74d26a279": "N\\geq {\\frac {1}{n}}\\left(q^{n}-\\sum _{p|n,\\;p{\\text{ prime }}}q^{\\frac {n}{p}}\\right).", "09997759dc23f2fe1bae1dbbba258a9e": "T\\colon \\Omega _{c}^{m}(M)\\to \\mathbb {R} ", "09999eba6c9c8f12c42e47a54c4979e7": "g(z)={\\frac {1}{f(z)-b}}", "099aa13b8bb2583dcacc7fe55f54c919": "a_{k}=C/k", "099ab665b59216cde0d59e51ae4cbd4a": "\\,\\mu ", "099b34a42fd8c504e6ccc1ee1a5d30e1": "{\\text{CIRC}}(T;P,Z)=\\{M~|~M\\models T{\\text{ and }}\\not \\exists N{\\text{ such that }}N\\models T,~N\\cap P\\subset M\\cap P{\\text{ and }}N\\cap Z=M\\cap Z\\}", "099b564e6be3850fa2c3b1c16a5a47a1": "ind(P)+\\gamma ^{5}/20", "099b5d1a5253a577bdabdd87b17e70d0": "\\operatorname {O} (M)\\times _{\\sigma _{-}}\\{-1,+1\\}.", "099b6c1a56664593d1b6baf5c240bd7e": "\\sup _{x\\in E}N_{r}(B_{R}(x)\\cap E)\\leq C\\left({\\frac {R}{r}}\\right)^{\\alpha }.", "099b90d035d3ee7c61f2e1a1b38ae971": "\\|x\\|_{C(X)}=\\max \\nolimits _{x\\in X}|f(x)|", "099be9a554513f5cd5028350bc13b07b": "P(x)\\propto \\exp {\\left(-{\\frac {{\\frac {1}{2}}kx^{2}}{K_{B}T}}\\right)}", "099c0558d1e4d2c531680a7b25cb24de": "f^{*}(x)=\\lim _{h\\to 0}{\\left({f(x+h) \\over {f(x)}}\\right)^{1 \\over {h}}}", "099c27c38ffdb1e023a27c1360e68671": "{\\frac {\\partial ^{2}f}{\\partial \\sigma ^{2}}}<0", "099c5c1404941de8b00a095cbfd3d27b": "\\Pr(X-\\mu \\geq k\\sigma )\\leq {\\frac {1}{1+k^{2}}}.", "099c75fe0488e6ca9fa7616ea6ff72dc": "\\mathrm {auth} (p)=\\displaystyle \\sum _{i=1}^{n}\\mathrm {hub} (i)", "099cdfabb805c4d513bdf867d3de30f3": "X_{2}={\\frac {(t+1)x^{2}-1}{(t-1)x^{2}+1}},\\qquad X_{4}={\\frac {(t_{2}+1)X_{2}^{2}-1}{(t_{2}-1)X_{2}^{2}+1}},\\qquad X_{8}={\\frac {(t_{4}+1)X_{4}^{2}-1}{(t_{4}-1)X_{4}^{2}+1}}.", "099d2e1000b227a383ce43097c4f36fa": "d\\mathbf {r} ", "099d3447462e93a4f340dd930752e239": "A\\to C,(A\\to B)\\to C\\vdash C", "099d7af978b6fb5ce727211c86f5639a": "{\\frac {1}{\\operatorname {Tr} ((I-E)S)}}(I-E)S(I-E).\\,", "099db936cba2b046dc8c979ad61933a5": "T_{j',j}", "099dbeb744724fbdd14aa9c20458e166": "\\sigma _{w}=0.1W_{20}", "099e47733b317dcafe1c45cea8a29dfe": "\\Phi :F\\dashv G", "099ee94a7a9d6a6809f685c2464c98be": "\\lambda _{11}=3.83171", "099efdd7be2c05f33125555513005dc0": "\\mathbf {\\epsilon } ^{\\beta }\\ ", "099f1fa68187b4d345e532bbda55e66f": "((x,g_{U})\\in U\\times G)\\sim ((x,g_{V})\\in V\\times G)\\iff {\\mathbf {e} }_{V}={\\mathbf {e} }_{U}\\cdot h_{UV}{\\text{ and }}g_{U}=h_{UV}^{-1}(x)g_{V}.", "099f92f26f75f6d919b945ce7c852b2b": "2\\cdot 3\\equiv 0{\\pmod {6}}", "099fa53577482b5d4ff29eec93cda21d": "R\\ )", "099fc11abb2842272fd9eb824470481a": "{\\begin{aligned}{\\frac {\\theta \\Gamma (\\theta +n+\\alpha )\\Gamma (\\theta +1)}{\\alpha \\Gamma (\\theta +n)\\Gamma (\\theta +\\alpha )}}-{\\frac {\\theta }{\\alpha }}.\\end{aligned}}", "099fc59c3efaacedf8a1e069446342bc": "(-\\alpha ,-\\alpha )", "09a038f959bd2e8ba81e1fe6eeca2853": "{\\frac {[A]_{f}}{[A]_{i}}}=e^{-kt}", "09a0467830b0cb52cae6bef87d434df9": "\\mathbf {v} _{1}", "09a09296c8c2628cd63ec2555bc1d848": "g^{2}(q;\\tau )", "09a0bc9555fb8df32a66cf04f113a318": "{\\begin{array}{c}{\\text{area of}}\\\\{\\text{rectangles}}\\end{array}}=1\\,+\\,{\\frac {1}{2}}\\,+\\,{\\frac {1}{3}}\\,+\\,{\\frac {1}{4}}\\,+\\,{\\frac {1}{5}}\\,+\\,\\cdots .", "09a0f8ccbe3337a8e95dd36497891ca7": "g_{kl}=2\\chi \\Delta n{\\overline {n}}\\,", "09a11106d1a04300591ebc86734b9a08": "\\Psi (z)={\\frac {\\cos 2\\pi (z^{2}-z-1/16)}{\\cos 2\\pi z}}", "09a12cb751fe5626d7c138d2ef3a928b": "X_{t}=\\mu +X_{t-1}+\\epsilon _{t}\\,", "09a14ed70f3bdce0e9288d7e5cd5994b": "abs(\\lambda )=1\\,", "09a194a0a61c475d6260611b5255a5a2": "\\sigma {\\sqrt {2}}{\\mbox{erf}}^{-1}(1/2)", "09a1c11135b445f9a9f9e8893d08cfe7": "x_{i}\\geq x_{\\min }", "09a1cb4663947e85224ba06569cebebf": "{\\frac {95}{48}}", "09a1fd285dd3dcad58e135b2a8acc0eb": "{\\mathit {C}}F=cashflow\\,", "09a20bfbac6c09b7be92573d4379ed97": "i\\in \\lbrace 1,\\ldots k\\rbrace ", "09a26fa2581b375c5c1770a395187c29": "\\lambda _{2}^{2}=\\lambda _{3}", "09a29cc014fcfc407ba0b0533f1a91f8": "a_{n}={\\begin{cases}1&{\\mbox{if }}n=2k-1,\\\\0&{\\mbox{if }}n=2k\\end{cases}}", "09a3516831e4dc4bd70986f5a2ee84fb": "\\ \\|f\\|_{p}\\leq \\mu (S)^{{\\frac {1}{p}}-{\\frac {1}{q}}}\\|f\\|_{q}", "09a35bb3db23ca623e8eb2cdf7e1a960": "SH_{0}({\\text{point}})\\cong \\mathbb {Z} ", "09a3d6a4d9e07bb432d49075c87b1097": "\\scriptstyle p_{\\theta }={\\dot {\\theta }}", "09a3eab23fe0510255994528a4d7299e": "\\Delta p_{x}\\approx 2{\\frac {h}{\\lambda }}\\sin \\varepsilon /2.", "09a452b6295ddeadbf1ad24de8f304c9": "\\mathbf {w} (n)=\\mathbf {w} (n-1)+\\,\\alpha (n)\\mathbf {g} (n)", "09a539a89442cc879f1cb4fd7a1b09e4": "x\\in U,y\\in V", "09a55ca392d1b3a31643fc87a6fb6c4d": "0.886\\langle v\\rangle =v_{p}<\\langle v\\rangle <{\\sqrt {\\langle v^{2}\\rangle }}=1.085\\langle v\\rangle .", "09a5631b96f33c7cd08a5f7ac94507a5": "Z_{2}\\times Z_{2}\\times Z_{2}", "09a62612c4e4f4043c87ec6d3790429d": "P_{i}(S_{x})", "09a715fce75d188d2129ab1ebece5336": "\\delta _{\\epsilon }\\Omega ^{(d+2)}=d\\delta _{\\epsilon }\\Omega ^{(d+1)}=d^{2}\\Omega ^{(d)}(\\epsilon )=0", "09a736721e0a5081dcbbccd2c01c71ff": "p\\land q\\Rightarrow r", "09a7422672629ad14d62321da76d93ec": "ln(N)", "09a76ccc2b2efa52ac4a8976ec5792cb": "a_{7}'=a_{3}\\oplus a_{4}\\oplus a_{5}\\oplus a_{6}\\oplus a_{7}\\oplus 0=1\\oplus 0\\oplus 0\\oplus 1\\oplus 1\\oplus 0=1.", "09a78a9ee276650e6fd9475985427662": "a+dx=a", "09a82fea9a85ce8517c992c36fc868f2": "f^{(t)}=[f_{ij}^{(t)}]_{i=1...N,j=1...M}", "09a85bd3efa5f39ea790dec70d3732f3": "\\left(b_{k}\\right)", "09a88b73190a3ef2ce7c051c9e867a8c": "{\\frac {1}{2}}\\sum _{i=1}^{n}\\sum _{j=1}^{n}\\left(x_{i}y_{j}-x_{j}y_{i}\\right)^{2}=\\sum _{i=1}^{n}x_{i}^{2}\\sum _{i=1}^{n}y_{i}^{2}-\\left(\\sum _{i=1}^{n}x_{i}y_{i}\\right)^{2}.", "09a89796812d795a2deaedc5b3f57284": "\\scriptstyle {T}", "09a8ebdc269ac1f2f9b2f53ee3fc46e3": "{\\begin{matrix}{9 \\choose 4}{4 \\choose 1}^{4}\\end{matrix}}", "09a9a9b63f858809e165da927547a4fa": "pH=6.1+\\log \\left({\\frac {[HCO_{3}^{-}]}{0.03\\times pCO_{2}}}\\right)", "09aa028c019b5d2d4f9a42656c40af79": "p/q", "09aa47885632645c9ac93fd74c1dfc9b": "\\alpha \\,\\rightarrow \\,\\phi _{\\alpha }(0)", "09aa64ff4add7aa3790267246c0d1d48": "G=(V,\\,\\Sigma ,\\,::=,\\,S)", "09aad069203af52f17d6432471a8d93f": "{\\rm {E}}(n,x)", "09aadd014986908dd9072062ccbc3bd0": "\\scriptstyle OA/OB=8.27", "09ab2495cb1a5b4be76ce6032d51060a": "B_{\\lambda }=B\\cap (L_{\\lambda }/qL_{\\lambda }),", "09ab69a2e511b7912d3386f5f4c60569": "\\rho ={\\frac {r_{12}v_{1}v_{2}-r_{14}v_{1}v_{4}-r_{23}v_{2}v_{3}+r_{24}v_{2}v_{4}}{{\\sqrt {v_{1}^{2}+v_{3}^{2}-2r_{13}v_{1}v_{3}}}{\\sqrt {v_{2}^{2}+v_{4}^{2}-2r_{24}v_{2}v_{4}}}}}", "09ab8743a652f3ef0ea9c02772bfe183": "{\\frac {16}{9}}", "09abfd122217ce3f94278c0512f49774": "u\\neq v", "09ac9bbecc7d24533986b71f8fc53f0d": "q_{i}=\\left\\lfloor {\\frac {n}{5^{i}}}\\right\\rfloor ,\\,", "09ac9d81af463f83ae29bbc63ff265b1": "{\\tfrac {mg}{L}}", "09acc9b155de86f6781ab94ab5a81efe": "\\scriptstyle E_{K}", "09ad38db86a3ffdf2b92d66d95906625": "w^{R}", "09ad40c39c6ed860401d70432632ae82": "\\varphi :{\\vec {x}}\\rightarrow {\\vec {x}}-2{\\frac {f({\\vec {p}},{\\vec {x}})}{f({\\vec {p}},{\\vec {p}})}}{\\vec {p}}", "09ad72048b4e4932f84109f09cffb5ec": "H_{n}=H_{n-1}+{\\frac {1}{n}}.", "09adee3111d09346fb6da046f1bd266f": "R_{\\alpha }:S^{1}\\rightarrow S^{1}", "09adfd442038c1cbc7686167ee7d00f5": "B(S,i,j)=\\max _{S'\\subset X_{i} \\atop S=S'\\cap X_{j}}A(S',i)", "09ae008ae4ceab83db8969dbbdaa426a": "\\Pi _{5}=", "09ae3c49e6634a769a653b51b1283ea3": "\\psi (\\zeta _{0})=\\zeta _{0}", "09ae87bf709ae355f1af53b3e08176f2": "{\\begin{bmatrix}\\epsilon _{\\rm {xx}}\\\\\\epsilon _{\\rm {yy}}\\\\\\epsilon _{\\rm {zz}}\\\\2\\epsilon _{\\rm {yz}}\\\\2\\epsilon _{\\rm {zx}}\\\\2\\epsilon _{\\rm {xy}}\\end{bmatrix}}={\\begin{bmatrix}{\\tfrac {1}{E_{\\rm {x}}}}&-{\\tfrac {\\nu _{\\rm {yx}}}{E_{\\rm {y}}}}&-{\\tfrac {\\nu _{\\rm {zx}}}{E_{\\rm {z}}}}&0&0&0\\\\-{\\tfrac {\\nu _{\\rm {xy}}}{E_{\\rm {x}}}}&{\\tfrac {1}{E_{\\rm {y}}}}&-{\\tfrac {\\nu _{\\rm {zy}}}{E_{\\rm {z}}}}&0&0&0\\\\-{\\tfrac {\\nu _{\\rm {xz}}}{E_{\\rm {x}}}}&-{\\tfrac {\\nu _{\\rm {yz}}}{E_{\\rm {y}}}}&{\\tfrac {1}{E_{\\rm {z}}}}&0&0&0\\\\0&0&0&{\\tfrac {1}{G_{\\rm {yz}}}}&0&0\\\\0&0&0&0&{\\tfrac {1}{G_{\\rm {zx}}}}&0\\\\0&0&0&0&0&{\\tfrac {1}{G_{\\rm {xy}}}}\\\\\\end{bmatrix}}{\\begin{bmatrix}\\sigma _{\\rm {xx}}\\\\\\sigma _{\\rm {yy}}\\\\\\sigma _{\\rm {zz}}\\\\\\sigma _{\\rm {yz}}\\\\\\sigma _{\\rm {zx}}\\\\\\sigma _{\\rm {xy}}\\end{bmatrix}}", "09aec5d2afe56504020f7bed3154173b": "{\\begin{bmatrix}\\sigma _{x}^{2}&\\sigma _{xy}^{2}&\\sigma _{xz}^{2}&\\sigma _{xt}^{2}\\\\\\sigma _{xy}^{2}&\\sigma _{y}^{2}&\\sigma _{yz}^{2}&\\sigma _{yt}^{2}\\\\\\sigma _{xz}^{2}&\\sigma _{yz}^{2}&\\sigma _{z}^{2}&\\sigma _{zt}^{2}\\\\\\sigma _{xt}^{2}&\\sigma _{yt}^{2}&\\sigma _{zt}^{2}&\\sigma _{t}^{2}\\end{bmatrix}}=\\sigma _{R}^{2}{\\begin{bmatrix}d_{x}^{2}&d_{xy}^{2}&d_{xz}^{2}&d_{xt}^{2}\\\\d_{xy}^{2}&d_{y}^{2}&d_{yz}^{2}&d_{yt}^{2}\\\\d_{xz}^{2}&d_{yz}^{2}&d_{z}^{2}&d_{zt}^{2}\\\\d_{xt}^{2}&d_{yt}^{2}&d_{zt}^{2}&d_{t}^{2}\\end{bmatrix}}\\ (7)", "09af09bba361f9ad8f05f4af1c5d5fb4": "F_{\\Lambda ,K}(\\lambda ,k)=F_{\\Lambda |k}(\\lambda |k)F_{K}(k)=F_{K|\\lambda }(k|\\lambda )F_{\\Lambda }(\\lambda )", "09af2ef1d44ba3233c9e9854410dc129": "{\\frac {1}{(1-p)^{2}}}=2kt[COOH]^{2}+1=X_{n}^{2}", "09af4b0852cb85adfa02fd9486d130cd": "S_{y}(f)", "09af7cb8e7a8338187ad899971cee128": "\\mathrm {DR} =\\mathrm {SNR} =20\\log _{10}{\\left(2^{n}{\\sqrt {\\tfrac {3}{2}}}\\right)}\\approx 6.0206\\cdot n+1.761", "09af8a8d97b96b0a6df07384bb8a76f9": "N(\\mathbf {z} )=\\#\\{k:z_{k}=x_{k}\\}", "09af8f5c75957906420936806b7e36a0": "p={\\frac {\\rho _{0}C_{0}^{2}\\chi \\left[1-{\\frac {\\Gamma _{0}}{2}}\\,\\chi \\right]}{\\left(1-s\\chi \\right)^{2}}}+\\Gamma _{0}E;\\quad \\chi :=1-{\\cfrac {\\rho _{0}}{\\rho }}", "09af910d326766be00f054e0074b3a65": "\\operatorname {E} (X_{t})", "09af935a5b2855356e62bf3ce934f9f2": "\\pi r^{2}+\\pi rl", "09afa0c0079b3c7288e66c8fcb2f1dca": "{\\frac {\\partial y}{\\partial \\mathbf {x} }}={\\begin{bmatrix}{\\frac {\\partial y}{\\partial x_{1}}}\\\\{\\frac {\\partial y}{\\partial x_{2}}}\\\\\\vdots \\\\{\\frac {\\partial y}{\\partial x_{n}}}\\\\\\end{bmatrix}}.", "09afc5cab7cb9dafea474218a304f4d5": "g_{ij}=E[\\partial _{i}\\ell \\partial _{j}\\ell ]=E[(F_{i}-\\eta _{i})(F_{j}-\\eta _{j})]=V[\\eta ]", "09b058f7ce46d3d531434b039686fcb6": "k_{R}={\\frac {k_{r}[UA]}{[AB][RA]}}", "09b078b00331fc7eb842cbea01642da1": "\\log _{2}\\ln n+\\theta (m/n)", "09b08150ec41bac8fdfe2446e8ce3fa3": "E_{1}=y_{1}+{\\frac {q^{2}}{2gy_{1}^{2}}}=5.0ft+{\\frac {\\left(10{\\frac {ft^{2}}{s}}\\right)^{2}}{2\\left(32.2{\\frac {ft}{s^{2}}}\\right)(5.0ft)^{2}}}=5.06ft", "09b14be5d3b3a683cb9074db4a29911f": "reward=-1", "09b1722405c47b120f5070bbb0ed7884": "D=2^{\\frac {1}{2}}{\\frac {\\left|\\mu _{1}-\\mu _{2}\\right|}{\\sqrt {(\\sigma _{1}^{2}+\\sigma _{2}^{2})}}}", "09b2466d6c0a7409856798bb5fe09d32": "\\alpha _{ij}=\\alpha _{ij_{0}}+\\alpha _{ij_{1}}T", "09b26d835893491a6f700db1e668edfb": "S_{a}", "09b2761e082a80d565646fa5f8487cc4": "L_{n},{\\bar {L}}_{n},-\\infty 1\\\\{\\frac {d}{dx}}\\operatorname {arccsc} x&{}={\\frac {-1}{|x|\\,{\\sqrt {x^{2}-1}}}};\\qquad |x|>1\\end{aligned}}", "09cb61347bcd01f92e78c57f9604363e": "s=(\\ldots ,(s_{i},t_{si},t_{ei}),\\ldots )", "09cb76df6751573c0507d3fda343c080": "s={\\frac {a+b+c}{2}}\\,", "09cbca1050980a56a89eb976b41c899d": "f(x)={\\begin{cases}\\exp(-1/x)&{\\text{if }}x>0,\\\\0&{\\text{if }}x\\leq 0,\\end{cases}}", "09cc24bc90b0b2cf900ae6f618af13d5": "\\Psi (A)=\\sum _{i=1}^{N}K_{i}AK_{i}^{*}", "09cc573a2780647ee7322462ab1b2582": "U_{n}(T)=\\sup\\{U_{n}(p):p\\in S(T)\\}", "09cce1c4ae66cffc2e2c66d6ddad4335": "{\\vec {r}}(s,0)", "09ccf4c384153f2029f149bb2980ea81": "T_{HL}=\\tau \\cdot \\mathrm {ln} \\,2.", "09cd9cd43514bcc2f18fa416dab5903b": "L_{d}\\left(t_{0},t_{1},q_{0},q_{1}\\right)=\\int _{t_{0}}^{t_{1}}dt\\,L(t,q(t),v(t))+{\\mathcal {O}}\\left(t_{1}-t_{0}\\right)^{3}", "09cdce8b229a87d3685ae8365da6e31b": "T(n,t)=e^{-t}I_{n}(t)", "09cddb6125ab410065ddc37956d54785": "\\sum _{i=1}^{m}w(P_{i})\\geq w(C).", "09ce11e63fc7ee44fe80193b9334c43f": "\\Box A_{1};\\ldots ;\\Box A_{n};\\neg \\Box B_{1};\\neg \\Box B_{2}", "09ce1a7980b265974b030377b97186de": "V", "09ce5ea37c0278ea35cdf4eff7ea6d57": "P_{c}^{n}(z)", "09ceaba2dec9f15b202d7c002a975d75": "\\psi (x,t)=A\\cos(2\\pi (kx-\\omega t)+\\varphi )", "09ceabf864f361caaece7be6782e33f3": "\\mathbf {j} (\\mathbf {r} ,t)={\\frac {D_{\\alpha }\\hbar }{i}}\\left(\\psi ^{*}(\\mathbf {r} ,t)(-\\hbar ^{2}\\Delta )^{\\alpha /2-1}\\mathbf {\\nabla } \\psi (\\mathbf {r} ,t)-\\psi (\\mathbf {r} ,t)(-\\hbar ^{2}\\Delta )^{\\alpha /2-1}\\mathbf {\\nabla } \\psi ^{*}(\\mathbf {r} ,t)\\right),", "09ceb85e5d5d470bf97e527eacb8d3e6": "\\delta _{0}={\\frac {a_{P}}{A^{1/2}}}.", "09cfa02d6b4b9b1cec97243b8b7607f9": "\\operatorname {build-param-lists} [g\\ q\\ p,D,V,T_{6}]\\land \\operatorname {build-param-lists} [n,D,V,K_{6}]", "09cfb0416e8bdbcd65c08db1d0c865c0": "\\operatorname {get-lambda} [x,x=\\lambda q.f\\ (q\\ q)]", "09cfc4fa3a6c540362efd6cbb50072ca": "\\;=-(\\sum _{x}\\;(\\sum _{y'}p(x,y')\\log \\sum _{y'}p(x,y'))+\\sum _{y}(\\sum _{x'}p(x',y)\\log \\sum _{x'}p(x',y)))", "09cfc8d62cd7b33a08908c896b3d6a53": "v=\\sum \\lambda _{i}b_{i}'", "09d021ad89f00335a8a8cc4877f2fe98": "f_{i}=\\sum _{j\\neq i}(F_{ij}^{C}+F_{ij}^{D}+F_{ij}^{R})", "09d0ac15b9c43792ddb3847c13e5ad66": "{\\sqrt {\\sigma _{S}^{P}}}", "09d0b6bad23d4e4ef211a55c136d5c3a": "{\\frac {[A_{ad}]}{p_{A}\\,[S]}}=K_{eq}^{A}", "09d0b7ddacbd62f19ed54e802b583cb5": "\\mathrm {Re} (s)=1/2,3/2,\\dots ,", "09d0bb0d0b379ba0f2776269f09d885b": "N_{0}=0", "09d0f4571592140e2de18d53edf41944": "{d\\sigma \\over d\\Omega }.", "09d1069b692f87841955de4c63585fb9": "{\\begin{aligned}\\eta (x,t)&=\\eta _{2}+H\\,\\operatorname {cn} ^{2}\\left({\\begin{array}{c|c}\\displaystyle {\\frac {x-c\\,t}{\\Delta }}&m\\end{array}}\\right),\\\\\\eta _{2}&={\\frac {H}{m}}\\,\\left(1-m-{\\frac {E(m)}{K(m)}}\\right),\\\\\\Delta &=h\\,{\\sqrt {{\\frac {4}{3}}\\,{\\frac {m\\,h}{H}}\\,{\\frac {c}{\\sqrt {g\\,h}}}}}&&={\\frac {\\lambda }{2\\,K(m)}},\\\\\\lambda &=h\\,{\\sqrt {{\\frac {16}{3}}\\,{\\frac {m\\,h}{H}}\\,{\\frac {c}{\\sqrt {gh}}}}}\\;K(m),\\\\c&={\\sqrt {gh}}\\,\\left[1+{\\frac {H}{m\\,h}}\\,\\left(1-{\\frac {1}{2}}\\,m-{\\frac {3}{2}}\\,{\\frac {E(m)}{K(m)}}\\right)\\right]&&{\\text{and}}\\\\\\tau &={\\frac {\\lambda }{c}}.\\end{aligned}}", "09d16272a5eb774a03bd54ef17f4acec": "N(xy)=N(x)N(y)\\,", "09d16dbe76356199420f0cc04c2a7c9f": "\\rho ^{*}", "09d1a180ff7daea0a42c505c86deac69": "C_{V}=\\partial Q/\\partial T\\,\\!", "09d1b8589ae3d19902c3d5494a993711": "dH\\left(S,p,N_{i}\\right)=TdS+Vdp+\\sum _{i}\\mu _{i}dN_{i}", "09d22ed6274ce33f310c6e4de82b9d9e": "\\Gamma _{\\beta \\gamma }^{\\alpha }+T_{~\\beta \\gamma }^{\\alpha }+S_{~\\beta \\gamma }^{\\alpha }.", "09d26432709dc3c5b05ae907c4324311": "LC_{50}(mixture)\\leq 5000{\\tfrac {mL}{m^{3}}}", "09d2788103c2c371e6fb606ad3021a9e": "\\alpha =\\arccos \\left({\\frac {\\cos a-\\cos b\\ \\cos c}{\\sin b\\ \\sin c}}\\right),", "09d288b9607b2ab6024845c6279c7c28": "\\int {\\frac {1}{2}}\\xi _{d}^{2}{\\frac {d}{d\\zeta }}\\left(\\left({\\frac {d\\theta }{d\\zeta }}\\right)^{2}\\right)+{\\frac {1}{2}}{\\frac {d}{d\\zeta }}\\left(\\sin ^{2}{\\theta }\\right)\\,d\\zeta \\,=0", "09d2b138724407ae8ca5490e1b6f5839": "\\sup _{y^{*}\\in Y^{*}}-F^{*}(0,y^{*})\\leq \\inf _{x\\in X}F(x,0),", "09d2c1055863bdad1eda764739bfc1ea": "{\\begin{aligned}{\\mathcal {A}}^{AB}\\Psi _{A}(1,2,\\dots ,N_{A})&\\Psi _{B}(N_{A}+1,N_{A}+2,\\dots ,N_{A}+N_{B})\\\\&={\\tilde {\\mathcal {A}}}^{AB}\\Psi _{A}(1,2,\\dots ,N_{A})\\Psi _{B}(N_{A}+1,N_{A}+2,\\dots ,N_{A}+N_{B}),\\end{aligned}}", "09d33e642b0ce4ce2b732db7468e0452": "EAP=BP=EC(I_{r},G)", "09d34d017b4611c13ec1a5818df7f248": "\\kappa (R)", "09d396adb41a8af8f97997736a8e073c": "{\\frac {1}{b}}", "09d3c623910a65e31a8428ced4dc5c9c": "\\scriptstyle \\{u_{n}\\}_{n\\in \\mathbb {R} }", "09d3c9bcaa224a234df83194834f139c": "{\\begin{aligned}\\int _{0}^{\\delta }t^{\\lambda +n}e^{-xt}\\,dt&=\\int _{0}^{\\infty }t^{\\lambda +n}e^{-xt}\\,dt-\\int _{\\delta }^{\\infty }t^{\\lambda +n}e^{-xt}\\,dt\\\\&={\\frac {\\Gamma (\\lambda +n+1)}{x^{\\lambda +n+1}}}-\\int _{\\delta }^{\\infty }t^{\\lambda +n}e^{-xt}\\,dt,\\end{aligned}}", "09d411ccbbb420e73808542f7e4e4f10": "y_{n}\\approx y(t_{n})", "09d434d65384c517c8cff943c17f0986": "U_{i}\\subset X_{i}", "09d43b8d3ed520382e7ee4129065cd32": "\\delta m_{0}={\\frac {1}{4\\pi G}}\\left[{\\frac {1}{\\rho _{0}c^{2}}}{\\frac {\\partial P}{\\partial t}}-\\left({\\frac {1}{\\rho _{0}c^{2}}}\\right)^{2}{\\frac {P^{2}}{V}}\\right]", "09d4de471943d3408acc76800051dd20": "F_{q^{s}}", "09d4ee4f722c7c32ddd590a8049431f9": "\\displaystyle {\\|\\delta _{h}u\\|_{(k+1)}\\leq C\\|\\Delta _{1}\\delta _{h}u\\|_{(k-1)}+C\\|\\delta _{h}u\\|_{(k)}\\leq C\\|\\delta _{h}\\Delta _{1}u\\|_{(k-1)}+C\\|[\\delta _{h},\\Delta _{1}]u\\|_{(k-1)}+C\\|\\delta _{h}u\\|_{(k)}\\leq C\\|\\Delta _{1}u\\|_{(k)}+C^{\\prime }\\|u\\|_{(k+1)}.}", "09d548d1045c2e0019d7008c386dcead": "MA={\\frac {F_{out}}{F_{in}}}={\\frac {v_{in}}{v_{out}}}\\,", "09d56ff0e3c7334978dc9e7d89cdd35d": "\\{l_{1},\\ldots ,l_{n}\\}", "09d5c372ed012b465e11a6e80e90ebdf": "{\\hat {T}}_{i_{n+1}\\dots i_{m}}^{i_{1}\\dots i_{n}}({\\bar {x}}_{1},\\ldots ,{\\bar {x}}_{k})={\\frac {\\partial {\\bar {x}}^{i_{1}}}{\\partial x^{j_{1}}}}\\cdots {\\frac {\\partial {\\bar {x}}^{i_{n}}}{\\partial x^{j_{n}}}}{\\frac {\\partial x^{j_{n+1}}}{\\partial {\\bar {x}}^{i_{n+1}}}}\\cdots {\\frac {\\partial x^{j_{m}}}{\\partial {\\bar {x}}^{i_{m}}}}T_{j_{n+1}\\dots j_{m}}^{j_{1}\\dots j_{n}}(x_{1},\\ldots ,x_{k}).", "09d5d7aadb6a1368aa5739312a06d2f9": "n{\\stackrel {\\text{def}}{=}}\\lim _{x\\rightarrow a}(f(x)-mx)", "09d64b1f2d0a2c2b477cdd20332f1572": "\\left|{zf^{\\prime }(z) \\over f(z)}\\right|\\leq {1+|z| \\over 1-|z|}", "09d66dde72d13e8a4f7660348a02e6fb": "1000^{2}+3^{2}", "09d6749d3ef8e02f3078ba7b8e384be8": "{\\overline {X}}_{n}\\,\\xrightarrow {P} \\,\\mu \\qquad {\\text{for}}\\qquad n\\to \\infty .", "09d69efcb4b1aaadda75c329f393629b": "f(x)-af'(x)+{a^{2} \\over 2!}f''(x)-{a^{3} \\over 3!}f'''(x)+\\cdots ", "09d6bd0c879b8e358c7a02853763fa54": "P(x=v|c)", "09d6d15e9836b8296c0c911144b9df2a": "2^{2N}", "09d6eb9f4f6ab048899f1aa9ed5f458c": "A=A_{0}\\left(e^{-k\\Delta t_{p}}\\right)^{n}", "09d6f72812aa2c9ee6f331c7a0a3528a": "\\pi \\ :\\prod _{i\\in \\mathrm {N} }\\Sigma \\ ^{i}\\to \\mathbb {R} ^{\\mathrm {N} }", "09d71b61e67754dcce6f3b2515454830": "F_{N}(J)", "09d763c29f4504840551407303e802de": "x=\\int _{0}^{L}\\cos s^{2}ds", "09d794cd66c560bb1571a71998ec043f": "\\psi ^{(m)}(1)", "09d7b5f7a7237a855b70da62770d96c5": "0<\\left|aq_{n}-bp_{n}\\right|<1\\,", "09d7c32dcbd5b25eb8bea40446e42f71": "Q=I_{3}+{\\frac {1}{2}}(B+S+C+B^{\\prime }+T)", "09d83f476c0bc42331604b8aaede7e39": "A=U^{T}U", "09d88a548409e850ede3c3c77fbbfe52": "i=1,...,N", "09d8bdf28f56b26306e94b1c71d38385": "{\\mathcal {L}}=\\sum _{i=1}^{n}\\pi _{i}+\\mu (1-\\sum _{i=1}^{n}\\pi _{i})-n\\tau '\\sum _{i=1}^{n}\\pi _{i}h(y_{i};\\theta )", "09d8f09c410d180586acdf584f613921": "p={\\frac {nRT}{V}}.", "09d96d9708822e3f5dd4ae9a5411c5fd": "_{r=0}\\!", "09d9741f2d433a9fa7b26f06f645e665": "P(\\neg A|B)=c\\cdot P(\\neg A)\\cdot P(B|\\neg A)\\cdot ", "09d9be2b53bc6bc928726e444e6ce4e9": "{\\frac {m}{s}}+{\\frac {n}{t}}:={\\frac {tm+sn}{st}}", "09da455b257c7a7e3ef915407da20770": "\\mathbf {\\mu _{z}} =\\pm {\\frac {1}{2}}g{\\mu _{B}}", "09da71c37a9d91596ce304eb275b66bc": "\\alpha ^{\\dagger B}", "09daae4fda46071d452272bef8583ea2": "r\\,", "09dacfe7401a6cca9a31a65c156803a4": "I_{fl}=0", "09dadba2958f69b94b39163b664cd5cf": "G=1/R\\,\\!", "09daec7e238ac4770fccd55dd2d1294a": "{\\boldsymbol {B}}^{-1}={\\begin{bmatrix}1&-\\gamma &0\\\\-\\gamma &1+\\gamma ^{2}&0\\\\0&0&1\\end{bmatrix}}", "09dafcb3f81c6eccf2f3a989ae3bbed9": "{\\mathcal {F}}^{3}={\\mathcal {F}}^{-1}={\\mathcal {P}}\\circ {\\mathcal {F}}={\\mathcal {F}}\\circ {\\mathcal {P}}", "09db9abdce7ea350cea097b1d3a5ab78": "\\sum _{n=1}^{\\infty }{\\frac {|a_{n}|+|b_{n}|}{n^{2}}}<\\infty ,\\,", "09dbf08604b4dc478635f025988f1a26": "{\\begin{aligned}&[{\\hat {L}}_{a},{\\hat {L}}^{2}]=0\\\\&[{\\hat {S}}_{a},{\\hat {S}}^{2}]=0\\\\&[{\\hat {J}}_{a},{\\hat {J}}^{2}]=0\\\\\\end{aligned}}", "09dbf9e83adffdef3971ef9418272ff8": "M(x)=e^{\\int P(x)\\,dx}", "09dc0725384db62ec77b846d1c7ec865": "|\\phi (t)|0,", "09dcd14058ff41bad89ff5aa3ace012a": "\\Theta ({\\tfrac {t^{2}}{\\log t}})", "09dce38b81405727a859db4802fd7e6f": "\\mathbf {\\tau } =\\lim _{\\Delta S\\to 0}{\\frac {\\Delta F_{\\mathrm {s} }}{\\Delta S}}={\\frac {dF_{\\mathrm {s} }}{dS}},", "09dd14c6fe7f59c5602f210e3142b8db": "{\\check {~}}", "09dd853dcb749c8d580c791b7fab2648": "T_{c0}", "09dd8cb5f9087b6119dc83c5958dbc4f": "\\mu _{\\infty }\\subset \\mathbb {C} ^{\\times }", "09ddcbc4a9e8848bf252ee0736084673": "w=e^{\\frac {2\\pi i}{c}}", "09de29557c6bd203a40b515cfa078e88": "t_{a}^{2}+mn=bc", "09de51c51c98cad7c39f6bc4977539e5": "\\scriptstyle P(c_{t}|s_{t})", "09ded49103a530234c52a1504e2554c7": "\\mu (B)^{-1/q}\\|u-u_{B}\\|_{L^{q}(B)}\\leq C{\\text{rad}}(B)\\mu (B)^{-1/p}\\|\\nabla u\\|_{L^{p}(\\lambda B)}.", "09dedad0ec263786be85542ea1a1345b": "x\\mapsto ax", "09def1137ecd849efa023974f3df766f": "u_{03}", "09df226ba36c784607bcd794757b54c5": "f(2j+1)=2f(j)+1\\;.", "09df569742d4fc5d2ce2489d780f277c": "a_{2}=-0.23\\,", "09df6b640027b6b670c9f51b1b1a192e": "T_{1,n}(z)=g_{1,n}(z)", "09dfb5998f333da54a19f4aeea972ccb": "{\\frac {1}{D}}", "09dfc9ac5bd61e3d3016e0f9ecbc0055": "p(t)=c_{0}+c_{1}t+\\cdots +c_{n-1}t^{n-1}+t^{n}~,", "09dfd2695f0cc23acfc6c2f0a81aadbe": "N(t)=N(0)\\ e^{rt}", "09e02c6ffd1343d0573ac5a1d8831bc0": "2\\cos \\lambda =1+(l-l^{-1})/(\\lambda \\sin \\mu )", "09e07484cbb22259750f534b4bdaeec9": "\\mathbf {U} \\mathbf {V} ^{*}", "09e0a8f5ba23ebc0a9a77636e6abe5c8": "(0,t)", "09e0df0a1f9dddadd41884ef1fb7722e": "u=\\cos \\theta _{n}\\,", "09e163edc3362baed4f46442bfdc008d": "{\\begin{smallmatrix}m=M_{v}+5\\cdot ((\\log _{10}3.64)-1)=2.6\\end{smallmatrix}}", "09e1e5312b29562b52c657bb9a24fd62": "dF\\ =-PdV\\ +\\gamma dA", "09e2578e5a0e9d9de4491045ed909c79": "\\Phi ^{2}", "09e25e57528d466cbc406b1671ada025": "S={\\frac {s}{\\ell }}L\\approx {\\frac {12+1/3}{1/3}}(67+1/3)=2491+1/3\\approx 2490", "09e2781a05bfb8eb8ea17d9fb0e81f02": "I\\ ", "09e286fd78ece730d55049353bdbadd9": "p=p^{0}\\mathbf {e} _{0}+p^{1}\\mathbf {e} _{1}+p^{2}\\mathbf {e} _{2}+p^{3}\\mathbf {e} _{3}", "09e2a004a2228b5fb7494a36d5b4150a": "\\theta \\log {\\tan \\theta }-{\\frac {1}{2}}\\int _{0}^{2\\theta }\\log \\left(\\tan {\\frac {x}{2}}\\right)\\,dx=", "09e2ae91a9cfaf52ab6c4b3a9571b387": "{\\frac {\\Gamma _{w}}{\\delta x_{WP}}}A_{w}", "09e2e4bf1bdf175b0238e47d27fa775f": "\\lim _{n\\to \\infty }\\int _{X}|f_{n}-f|d\\mu =0", "09e32a2e3d5e1edba5c9aa402a5e29d9": "g^{(n)}(\\mathbf {r} _{1},t_{1};\\mathbf {r} _{2},t_{2};\\dots ;\\mathbf {r} _{n},t_{n})={\\frac {\\left\\langle E^{*}(\\mathbf {r} _{1},t_{1})E^{*}(\\mathbf {r} _{2},t_{2})\\cdots E^{*}(\\mathbf {r} _{n},t_{n})E(\\mathbf {r} _{1},t_{1})E(\\mathbf {r} _{2},t_{2})\\cdots E(\\mathbf {r} _{n},t_{n})\\right\\rangle }{\\left\\langle \\left|E(\\mathbf {r} _{1},t_{1})\\right|^{2}\\right\\rangle \\left\\langle \\left|E(\\mathbf {r} _{2},t_{2})\\right|^{2}\\right\\rangle \\cdots \\left\\langle \\left|E(\\mathbf {r} _{n},t_{n})\\right|^{2}\\right\\rangle }}", "09e33a58644a519ce7428cd3b9313ed1": "\\boxplus _{c}", "09e344fbe56d259a8d1a743061bb2f72": "\\beta =d_{2}/d_{1}", "09e38052b56f3b0bc2d4b39a676236db": "\\Gamma \\models \\psi ", "09e391eb85c13e8a661bf3fed45bac3e": "\\int _{x_{i}-ct_{i}}^{x_{i}+ct_{i}}-u_{t}(x,0)dx=-\\int _{x_{i}-ct_{i}}^{x_{i}+ct_{i}}g(x)dx.", "09e3adb135c2ea395bafc12726012c30": "x_{1}\\geq \\dots \\geq x_{n}", "09e3ca144517bfff7b8ea1f9f50c9f33": "\\scriptstyle {\\hat {H}}", "09e445ea5e0c0ebe60cc96f14fa16029": "D={Y_{1}}^{2}", "09e46311281e6bd1b35dab523bc9087d": "0A,X=x)+{\\frac {1}{2}}\\cdot E({\\frac {A}{2}}|BA,X=x)+{\\frac {1}{4}}\\cdot E(A|B0)", "0a01d7147859b6fe005bc76ef8aa25d2": "\\mathbb {Q} (\\zeta _{p^{\\alpha }}+\\zeta _{p^{\\alpha }}^{-1})", "0a01eb6aedc5dab91c3cc43749bcac23": "\\{x\\}", "0a022142514b3c1e104fa5b1cdf3fc55": "\\mathbf {H} (x)=\\nabla \\times \\mathbf {A} (x).\\quad \\quad \\quad \\quad (6)", "0a024764aab75341d90009565c6d4cb9": "{\\begin{matrix}{48 \\choose 4}=194,580\\end{matrix}}", "0a024a743f9863ed7550c3cafbbce117": "d\\Omega ^{2}=d\\theta ^{2}+\\sin ^{2}\\theta d\\phi ^{2}\\ ", "0a028cc625a51dfc59809735c518368d": "{\\tfrac {3(M-K)}{4}}", "0a02f7be7d6ac20b97182a389777935e": "l,j,m_{\\text{l}},m_{s},m_{j}", "0a0320933ebeb7d6ab4df72acfad0336": "\\delta '_{2}(n)={\\frac {\\pi }{4}}\\left({\\frac {c_{1}(4n+1)}{1}}-{\\frac {c_{3}(4n+1)}{3}}+{\\frac {c_{5}(4n+1)}{5}}-{\\frac {c_{7}(4n+1)}{7}}+\\dots \\right).", "0a0393ec57d613f07fc734dd9f9a0b7b": "K^{n}", "0a040cfb4e98304d43e3ccd84b052be0": "Z_{\\mathbf {x} }^{(\\ell )}(\\mathbf {y} )", "0a04315fff14859d66e75bebbaaa6990": "M_{1}", "0a049896dfb5b04d3d36f2b0ad0d3940": "y_{n}=x_{n+1}", "0a0499322e54c5f529336329e75e18ec": "x-c", "0a04d013f6688b54e521e8ee50bb15d4": "\\lambda \\leftarrow \\ln(2|{\\mathcal {U}}|)", "0a04da4e4e9362ea7bc5dee4b63d0712": "\\phi _{x}(D|m_{x})", "0a04ec73d991624208cd624968c34bac": "\\mathbf {\\chi } _{\\mu }^{A}(1)\\mathbf {\\chi } _{\\nu }^{B}(1)", "0a0530a98021395cb4224cca9fdc8bc1": "=\\int _{N}\\left[{\\frac {\\partial {\\mathcal {L}}}{\\partial \\phi }}-\\partial _{\\mu }{\\frac {\\partial {\\mathcal {L}}}{\\partial (\\partial _{\\mu }\\phi )}}\\right]Q[\\phi ]\\,\\mathrm {d} ^{n}x+\\int _{\\partial N}{\\frac {\\partial {\\mathcal {L}}}{\\partial (\\partial _{\\mu }\\phi )}}Q[\\phi ]\\,\\mathrm {d} s_{\\mu }", "0a053819ad3af1dda9071956e694622a": "v_{0}", "0a0562897ee1360e2dd957aa1bec6d53": "\\psi (\\alpha )", "0a0573881713e11008d46bd9b5d9727a": "\\scriptstyle p_{2}=(x_{2},y_{2})", "0a058a5d190a4921c7c6af726ba77944": "x\\sqsubseteq d", "0a0591ded9dcf0fcf4ff693ecb4d86de": "{\\frac {\\partial E}{\\partial {\\hat {h}}_{i}}}=\\sum _{n=-\\infty }^{\\infty }[-2x[n]s[n-i]+2(\\sum _{k=0}^{N-1}{\\hat {h}}_{k}s[n-k])s[n-i]]", "0a0596a02eb219bd6336b93543a68c06": "m>0", "0a0605b51c34900c74441839273b47e3": "A[C]=\\int _{t=t_{0}}^{t_{1}}P\\cdot {\\dot {X}}\\,dt.\\,", "0a061ac6197ca0594b9287722fbba36c": "{{\\varepsilon }_{particle}}+2{{\\varepsilon }_{medium}}\\approx 0", "0a070c5763067f38ff8ee3f2dcfdf7cb": "Y_{x}={\\frac {P_{1}+P_{2}+\\cdots +P_{x}}{x}}.", "0a07202355a4e1313a4b71f1fb1d61a4": "g=e^{\\epsilon \\theta }", "0a07aac793879300f27a7a20f159f4ba": "w,w'", "0a07bc667ce4bcae7d2a5d54b5c08e55": "{\\mbox{Span}}\\,(S)\\leq {\\mbox{Ann}}({\\mbox{Ann}}\\,(S))", "0a07e2591063bc1d36dc01c4097998c3": "or\\,\\!", "0a07f13c7b9a840b338d1961c92c8807": "\\gamma ^{\\mu }\\rightarrow S(\\Lambda )\\gamma ^{\\nu }S(\\Lambda )^{-1}={{({\\Lambda }^{-1})}^{\\mu }}_{\\nu }\\gamma ^{\\nu }:={\\Lambda _{\\nu }}^{\\mu }\\gamma ^{\\nu },", "0a081241e7e619f0622c185799e99544": "{\\dot {x}}^{2}+{\\dot {y}}^{2}+{\\dot {z}}^{2}=2U-C_{J}", "0a0812d1b1e6512f0159ddf948309e05": "k\\mapsto {\\begin{pmatrix}0&i\\\\i&0\\end{pmatrix}}", "0a082adf97abe109ec396bf328091144": "\\pi \\int _{c}^{d}[R(y)]^{2}\\ \\mathrm {d} y", "0a0837a73ffb8cb5135a9646a6dd351a": "\\|y_{n}-y_{m}\\|^{2}=2\\|y_{n}-x\\|^{2}+2\\|y_{m}-x\\|^{2}-4\\|{\\frac {y_{n}+y_{m}}{2}}-x\\|^{2}", "0a083b07c1608f61022a3131aba076d6": "A_{t}(t,T)-(1+B_{t}(t,T))r-\\mu (t,r)B(t,T)+{\\frac {1}{2}}\\sigma ^{2}(t,r)B^{2}(t,T)=0", "0a087ad70673831a93762e4721b53524": "{\\frac {\\omega _{2}}{\\omega _{1}}}={\\sin \\alpha \\over \\sin \\beta }", "0a089b6f769b3241b7359f896b64894f": "z=.656747-.129015i", "0a08a2ce6ea4ef079afd59c51f31dacf": "\\sigma _{k}^{*}(n)=(-1)^{n}\\sum _{d|n}(-1)^{d}d^{k}={\\begin{cases}\\sum _{d\\,|\\,n}d^{k}=\\sigma _{k}(n)&{\\mbox{if }}n{\\mbox{ is odd }}\\\\\\sum _{\\stackrel {d\\,|\\,n}{2\\,\\mid \\,d}}d^{k}-\\sum _{\\stackrel {d\\,|\\,n}{2\\,\\nmid \\,d}}d^{k}&{\\mbox{if }}n{\\mbox{ is even}}.\\end{cases}}", "0a08ff81588ada550c35ae81a497d6ed": "\\csc \\theta \\!", "0a09182e06899a88785977c1907c9d4c": "\\mathbf {a} _{2}=\\mathbf {a} -\\mathbf {a} _{1}.", "0a09e28a82dcbd975816d85cac710a67": "\\textstyle \\alpha >0", "0a0aa7d0c51fa6a5fdf68f855694cf73": "\\mathbf {S} ={\\frac {1}{24}}{\\begin{bmatrix}2&1&1\\\\1&2&1\\\\1&1&2\\\\\\end{bmatrix}}", "0a0acc2973119b91a1e057dc53d5ff08": "Q(\\alpha _{i})=P(\\alpha _{i})E(\\alpha _{i})=y_{i}E(\\alpha _{i})=0", "0a0acec7322df5bc74afe0a063cbd7d9": "c={\\sqrt {{\\frac {g\\lambda }{2\\pi }}\\tanh \\left({\\frac {2\\pi d}{\\lambda }}\\right)}}", "0a0ae5b96c69de86b336cf26626b835f": "T_{c}={\\frac {Jz}{k_{B}}}", "0a0b30ecf732c9d4c7f2319172b15ea5": "L_{e+r}", "0a0b69766627682f4fe79abd0d9f599e": "\\prod _{x}x^{x}=C\\,e^{\\zeta ^{\\prime }(-1,x)-\\zeta ^{\\prime }(-1)}=C\\,e^{\\psi ^{(-2)}(z)+{\\frac {z^{2}-z}{2}}-{\\frac {z}{2}}\\ln(2\\pi )}=C\\,\\operatorname {K} (x)\\,", "0a0b803d9f6fe0824910872584f104bf": "b'=b^{2},c'={\\sqrt {c}},", "0a0bc17d6e6c88654d22f312cd8c44e7": "A-A(h)=a_{0}h^{k_{0}}+a_{1}h^{k_{1}}+a_{2}h^{k_{2}}+\\cdots ", "0a0bdc23ec0e4077adbe7d7f1b13d244": "\\chi ^{2}(1)", "0a0c026ed94f063e7f1a0f7a13104f96": "E[S_{j}^{2+\\delta }]", "0a0c05f9888b0c45d99e1a03cc13d41a": "\\beta ={\\frac {d-q}{p}}\\equiv {\\frac {\\nu }{2}}(d-2+\\eta ).", "0a0c40bc46312c39bda2d9b43c967a92": "a_{m}", "0a0c5f0ff0933ac16229d97240b8cce4": "\\int _{S}\\omega =\\int _{T}d\\omega ", "0a0c6eec7f717e267b12606b755f1bbd": "\\mathbf {B} _{\\mathbf {P} _{0}}(t)=\\mathbf {P} _{0}{\\text{, and}}", "0a0c793df5df8098779330436e984ef4": "{\\sqrt {2}}", "0a0cb81b6cd87583715b2825975836ca": "3.\\quad f\\star 1=1\\star f=f", "0a0cdd2bebf7825cfa46895160aa822e": "a,b,c,...", "0a0ceaa7cdc694b21f25e591b9cb0401": "\\Delta _{2}^{1}", "0a0cfa7a34002d212ce11c71179c9e52": "\\forall i,j\\,\\!", "0a0d174daa91c84e87a3f8c5fa8407cd": "-j1.49={\\frac {-j}{\\omega L_{1}Y_{0}}}\\,", "0a0d1b1b07712a9f7f301f4050740790": "[\\nabla ^{2}+E]G({\\mathbf {r}},{\\mathbf {r}}')=\\delta ({\\mathbf {r}}-{\\mathbf {r}}')", "0a0d2cb7add652380463f1bbb320d9f4": "\\beta =45^{\\circ }", "0a0d925b0b4c4f0b84ff04bfbb84178e": "\\int p(x\\mid I)f_{k}(x)dx=F_{k}\\qquad k=1,\\dotsc ,m.", "0a0da101911f41c75d61aec64817f5ce": "\\lfloor \\log _{2}(n)\\rfloor ", "0a0db1022a14e3de3f0fc480cded0fd8": "1-d=\\left(1-{\\frac {d^{(p)}}{p}}\\right)^{p}", "0a0deb2f3354b5de893085ec672f2273": "a_{i}^{-1}", "0a0e2f98a5da8fa805e23efcebe41404": "2Z_{oc}", "0a0e33f9866ac37568007ecf61d9f1d3": "\\sigma ={\\frac {F}{A}}+{\\frac {M}{\\rho A}}+{\\frac {M}{{I_{x}}'}}y{\\frac {\\rho }{\\rho +y}}", "0a0e3d6a910f97b9338039d9bfe659d9": "R_{1}={\\frac {R_{b}R_{c}}{R_{T}}}", "0a0e8ada664ee9af47ca00b2f64a276b": "2p\\sigma _{\\mu }", "0a0eeb59095b3c37820ae82916392683": "G^{I}", "0a0f00b9cbc0d133ff747d28924f3720": "W^{1,p}(\\Omega )", "0a0fd6f3b9e47e223f1ef377c143f1c8": "{\\Gamma }_{\\beta \\mu \\nu }={\\frac {1}{2}}\\left({{\\partial {g}}_{\\beta \\nu } \\over {\\partial x^{\\mu }}}+{{\\partial {g}}_{\\beta \\mu } \\over {\\partial x^{\\nu }}}-{{\\partial {g}}_{\\mu \\nu } \\over {\\partial x^{\\beta }}}\\right)", "0a0ffdf01e297cca4f2365ba8d56b69e": "f(re^{i\\theta })\\rightarrow f_{1}(e^{i\\theta })", "0a1055edc119afe3aebd434d1467df28": "x\\succ y", "0a109b075fa56d2ab98435c19d91c57b": "\\Phi (x)+\\Phi (-x)=100\\%", "0a10a3832eece68774016f1c564bb546": "C_{scat}={\\frac {1}{6\\pi }}\\left({\\frac {2\\pi }{\\lambda }}\\right)^{4}|\\alpha |^{2}", "0a10b0421b04b87997dc7360586db48b": "{\\color {Blue}~6.12}", "0a10d3013e3245e74a1a9eb214a85ca1": "H=H_{0}+V(t)\\,", "0a10fd2acbe4c093cb9003d62fdfd7f8": "\\,{\\mathcal {M}}(n+1,n)<{\\mathcal {M}}(n+x,n)\\,", "0a1112aeea57dc846577716461c6ea59": "rate={k_{p}}\\left({\\frac {fk_{d}}{k_{t}}}\\right)^{1/2}[I]^{1/2}[M]", "0a1163b70d0b6dcc052ad623f808082f": "SO_{2}", "0a1174d4999b5f870fb05ffc243b88b2": "Z_{in}=Z_{11}-{\\frac {Z_{12}Z_{21}}{Z_{22}+Z_{L}}}", "0a11841004e2dc9d292fa0e359e65b03": "{\\begin{aligned}D_{1}&=\\operatorname {diag} \\{\\lambda _{1},\\dots ,\\lambda _{m}\\}\\\\D_{0}&=R-D_{1}.\\end{aligned}}", "0a1187e2367d522a94e57abca9ab8c75": "\\kappa (A)=\\left|{\\frac {\\lambda _{\\max }(A)}{\\lambda _{\\min }(A)}}\\right|,", "0a11de642fc16acc02b7678fc438b50c": "A=P^{-1}BQ", "0a11de6665c74759c18c9e823843d45c": "{\\frac {d}{dx}}\\ \\operatorname {csch} \\,x=-\\coth x\\ \\operatorname {csch} \\,x\\,", "0a11e0c53e246a6c14dd080cfbe47964": "q_{T}", "0a11eb0ab11f5cfbc4163f7315bdbb74": "J(y):=\\int _{a}^{\\infty }dx\\ f(x,\\ y)", "0a12651908668b936d408556cc9f74db": "b_{i,j}", "0a12fde9c8c7181e4fc7e8f03e00ed0a": "f(x|ai}{\\frac {2q_{i}}{r_{ij}^{3}}}\\mathbf {r} _{ij}\\times \\mathbf {\\hat {p}} _{j}\\right]", "0a138410f3f81e11efb5a7dcf9d0f33d": "\\left\\|\\mathbf {q} \\right\\|", "0a13c40d2f3d9ded6249c64dc128e3ff": "\\mathbf {S} (\\mathbf {p} (t))={\\mathcal {S}}_{z}\\boxtimes _{n=1}^{N}\\mathbf {w} _{z,n}(p_{n}(t)),", "0a13d3b7887910a10c259f09a2646fa1": "\\sigma _{c}=1", "0a13ec02a5dcb29dc8099db4d6eacef9": "A^{j_{1}}=A^{j_{2}}", "0a140c9a1e20994313e50835ee7e8d06": "S=\\{\\,a\\mid \\exists x_{1},\\ldots ,x_{k}[p(a,n_{0},x_{1},\\ldots ,x_{k})=0]\\,\\}.", "0a1422b05d8add93e61974d7c7e7031c": "5(x-1)\\left(x^{2}+x+1\\right)", "0a1445becbb9bc326fce80739e4b6327": "\\ \\Gamma _{a}", "0a1465333ef8e2ab6117a5633b95fdf1": "\\displaystyle x^{n}\\,", "0a14abb19a0c4fbaf70a5c3a1907fd29": "{\\boldsymbol {P}}_{k|k}=({\\boldsymbol {I}}-{\\boldsymbol {K}}_{k}{\\color {Red}{\\boldsymbol {H}}_{k}}){\\boldsymbol {P}}_{k|k-1}", "0a150b071d92bb038c3bfffea4083996": "\\pi _{\\nu ,k}{\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}}^{-1}f(z)=|cz+d|^{-2-i\\nu }\\left({cz+d \\over |cz+d|}\\right)^{-k}f\\left({az+b \\over cz+d}\\right).", "0a15207b66dc7a2e5de49762410f0b3c": "\\sum _{j=0}^{m}(-1)^{j}\\dim \\wedge ^{2j}\\mathbb {C} ^{2m+1}=(-1)^{{\\frac {1}{2}}m(m+1)}2^{m}=(-1)^{{\\frac {1}{2}}m(m+1)}(\\dim \\mathrm {S} ^{2}S-\\dim \\wedge ^{2}S)", "0a153fa2bc343e91250d2347191e7aaf": "v_{2}={\\frac {u_{2}(m_{2}-m_{1})+2m_{1}u_{1}}{m_{1}+m_{2}}}", "0a155fc145e3d52afd5016ec51491704": "W(V):=T(V)/(\\!(v\\otimes u-u\\otimes v-\\omega (v,u),{\\text{ for }}v,u\\in V)\\!),", "0a156b115d741a8534a7fcb88c366938": "|a_{n}|\\geq |b_{n}|", "0a156fe394221ad15eba61e8c1344458": "[M_{i},M_{j}]=i\\epsilon _{ijk}M_{k}", "0a1574d202cb4d29e08def85b56ae6f3": "\\lambda y.e", "0a159364b389c3a6eb66c6b8b87268cb": "{\\widetilde {K}}^{n+2}(X)={\\widetilde {K}}^{n}(X).", "0a1629d84ec3de449a6de90b2afb5662": "t={\\frac {{\\overline {x}}-\\mu _{0}}{(s/{\\sqrt {n}})}},", "0a165087348425a6196424c0cc77cf88": "K_{\\mathit {row}}={\\frac {(1-S_{\\mathit {wn}})^{L_{o}}}{{(1-S_{\\mathit {wn}})^{L_{o}}}+{E_{\\mathit {o}}}{S_{\\mathit {wn}}}^{T_{\\mathit {o}}}}}", "0a165b8fb0a0e9907882c37f44669bcb": "\\alpha _{g}", "0a167e932da019919268f9c58921ded3": "\\lambda _{i}=(\\lambda _{1,i},\\dots ,\\lambda _{n,i})", "0a16a3f8f8f5fd3299ccfd0d5c70e487": "A={\\begin{bmatrix}{\\frac {(x_{1}-x)}{R_{1}}}&{\\frac {(y_{1}-y)}{R_{1}}}&{\\frac {(z_{1}-z)}{R_{1}}}&-1\\\\{\\frac {(x_{2}-x)}{R_{2}}}&{\\frac {(y_{2}-y)}{R_{2}}}&{\\frac {(z_{2}-z)}{R_{2}}}&-1\\\\{\\frac {(x_{3}-x)}{R_{3}}}&{\\frac {(y_{3}-y)}{R_{3}}}&{\\frac {(z_{3}-z)}{R_{3}}}&-1\\\\{\\frac {(x_{4}-x)}{R_{4}}}&{\\frac {(y_{4}-y)}{R_{4}}}&{\\frac {(z_{4}-z)}{R_{4}}}&-1\\end{bmatrix}}", "0a16ac818e03048c86de65e128453433": "p_{m}", "0a1729a453ae53351145b0216e15ac77": "B={\\frac {0.5a+1.5}{5.5a+b+1.5}}", "0a1733bd93771f3262048eb223b5034b": "u_{i}(\\overbrace {\\mathbf {x} ,z_{1},z_{2},\\dots ,z_{i}} ^{\\triangleq \\,\\mathbf {x} _{i}})={\\frac {1}{g_{i}(\\mathbf {x} _{i})}}\\left(\\overbrace {-{\\frac {\\partial V_{i-1}}{\\partial \\mathbf {x} _{i-1}}}g_{i-1}(\\mathbf {x} _{i-1})\\,-\\,k_{i}\\left(z_{i}\\,-\\,u_{i-1}(\\mathbf {x} _{i-1})\\right)\\,+\\,{\\frac {\\partial u_{i-1}}{\\partial \\mathbf {x} _{i-1}}}(f_{i-1}(\\mathbf {x} _{i-1})\\,+\\,g_{i-1}(\\mathbf {x} _{i-1})z_{i})} ^{{\\text{Single-integrator stabilizing control }}u_{a\\;\\!i}(\\mathbf {x} _{i})}\\,-\\,f_{i}(\\mathbf {x} _{i-1})\\right)", "0a17939d6c58d4440be5331771c23f79": "(e,e')\\mapsto g(e)=g'(e')", "0a17f6cc447fa91f2569507065e93f1d": "W^{4}\\,", "0a183ed5142c1166275da8fb1cbbd43f": "\\rightarrow ", "0a1862a6c53d19b2232845e9cd763465": "2J(1-M^{2})=kTg'(H/2JM)", "0a186446102ecd71a5ebea951e4efbfd": "\\sum _{n=0}^{\\infty }f(n)-\\int _{0}^{\\infty }f(x)\\,dx=f(0)/2+i\\int _{0}^{\\infty }{\\frac {f(it)-f(-it)}{e^{2\\pi t}-1}}\\,dt", "0a1892480841c9ee6c99ebfde012563c": "\\int \\limits _{\\Omega }f\\partial _{x_{i}}\\mathbf {\\varphi } _{i}=-\\int \\limits _{\\Omega }\\mathbf {\\varphi } _{i}\\partial _{x_{i}}f", "0a18ab5ec27e2fd3b14e4b8b4d8dfe40": "E(v)={\\frac {1}{2}}hv+{\\frac {hv}{e^{hv/kT}-1}}", "0a18fb53e9264af4daca7f7c683b97af": "X_{n}=\\sum _{v\\in T_{n}}1_{v\\in K}.", "0a1913fe386d80c7e18cc21f0b6d2a23": "{{dv} \\over {dt}}=\\epsilon (\\beta u-v).", "0a193563f17826c797e4fc1dcffc3f78": "|f(x)g(z-x)e^{-2\\pi iz\\cdot \\nu }|=|f(x)g(z-x)|", "0a19359a53b67d41c5bec2ab8020de4e": "\\forall m_{\\bullet }\\forall X_{\\bullet }((\\varphi (0)\\land \\forall n(\\varphi (n)\\rightarrow \\varphi (Sn))\\rightarrow \\forall n\\varphi (n))", "0a1967fa696523c5094b76eb31a2299b": "P(G,k)=P(G+uv,k)+P(G/uv,k)", "0a19a174c8c305734ee08cd9c59b8173": "\\textstyle \\rho (m)", "0a19cb88a54ddc8c0ab871ea2d899da4": "\\|x\\|_{bv_{0}}=TV(x)=\\sum _{i=1}^{\\infty }|x_{i+1}-x_{i}|.", "0a19e88be978af42cd1c72fcc7c43272": "\\tau =F\\left({\\frac {\\partial u}{\\partial y}}\\right)", "0a19e8a90704fee00143624d2d601811": "(zI-A)^{-1}", "0a19fc74f4f19b6b29af362f2dd933df": "y^{1}=r\\cos(x/r)", "0a1a18cb25e26836240de2d3b4427a35": "y\\tan \\varphi =y{\\frac {dy}{dx}}.", "0a1a1fba115965b654ff0834746c6b98": "{\\mathfrak {P}}^{92}", "0a1a579c6cca0ed87742348865b344b5": "N\\rtimes F", "0a1a596db1fce034b1ad4ac503c0f4b6": "EAS=TAS\\times {\\sqrt {\\frac {\\rho }{\\rho _{0}}}}", "0a1a88e81e4cf6ea8911ef0aa78e0652": "a\\ {\\pmod {\\Phi _{n}(q)}}", "0a1ace4f9fb9fd22d914c7769070d597": "EIF_{i}", "0a1af8df9006ec8d2e60e181da9209a1": "\\psi d\\mathbf {s} =\\iiint _{V}\\nabla \\psi \\,dV", "0a1b186df888dd0885b9dfafa85a27e1": "P\\left[A|H\\right]{\\mbox{ }}={\\mbox{ }}{\\frac {P\\left[A\\right]*P\\left[H|A\\right]}{P\\left(H\\right)}}{\\mbox{ }}={\\mbox{ }}{\\frac {P\\left[A\\right]*P\\left(H|A\\right)}{[P\\left(A\\right)*P\\left(H|A\\right)+P\\left(R\\right)*P\\left(H|R\\right)]}}{\\mbox{ }}", "0a1b4cb7586f33a09e22ccf7b42d3e44": "\\mathbf {J} _{n,{\\text{diffusion}}}/(-q)=-D_{n}\\nabla n,\\qquad \\mathbf {J} _{p,{\\text{diffusion}}}/q=-D_{p}\\nabla p.", "0a1b5bdfd6e0fe5f49dcfe3e4ef3dd4f": "\\eta _{e}\\,\\!", "0a1bb414a4b7febbfc581c32f2a7252e": "\\lambda x.x^{2}+2", "0a1bc7b13d508efee843b56c89707c37": "m({}^{12}{\\rm {C}})={\\frac {12M_{\\rm {u}}}{N_{\\rm {A}}}}", "0a1cad5fc1dc53ca85e162eec66e3907": "\\Psi (x,t)=\\psi (x)e^{-iEt/\\hbar }\\,.", "0a1cc9e93748b91ebcd91d264e6f8073": "L={\\rm {st}}(x_{H})\\,", "0a1da683b8497c3da0c344c52c5135cb": "f(x)=6x^{4}-2x^{3}+5", "0a1daa1ce4d23fac8b1d3626e6adb740": "\\gamma ^{k}={\\begin{pmatrix}0&\\sigma ^{k}\\\\-\\sigma ^{k}&0\\end{pmatrix}}", "0a1dc1fdc410edc85d748b690474c267": "Y(s)={\\mathcal {L}}\\left\\{y(t)\\right\\}\\ {\\stackrel {\\mathrm {def} }{=}}\\ \\int _{-\\infty }^{\\infty }y(t)e^{-st}\\,dt", "0a1e05d68b37bc1cd4778f6c5bf5a2cc": "{\\mathcal {F}}(\\mathbf {x} )={\\mathcal {S}}\\boxtimes _{n=1}^{N}\\mathbf {w} _{n}(x_{n}).", "0a1e1094647b377be14178c4cd356895": "r_{1}={\\frac {\\delta }{\\sigma }}", "0a1e2df73a299b0402f03d7076ac60d5": "{\\sqrt {xy}}={\\sqrt {x}}{\\sqrt {y}}", "0a1eb5132685a7e2077db51c722cfbe7": "-{\\frac {1}{2}}+{\\frac {1}{2}}i{\\sqrt {3}}", "0a1f3d8dbc79d1c9275aeb3ea50c129a": "c_{0}\\equiv a_{0}+b_{0}\\mod p", "0a1f3d9a63905b83635d9d5f43a0c508": "\\mathrm {^{244}_{\\ 96}Cm\\ {\\xrightarrow[{18.11\\ yr}]{\\alpha }}\\ _{\\ 94}^{240}Pu} ", "0a1f64ce55f7f835e2510cc481fcb532": "r,n(i)\\in \\mathbb {Z} ", "0a1fa36c48ca2abec461e7b9b57875b3": "{\\frac {1}{500\\log(1/\\varepsilon )}}\\log n=\\int \\Delta X\\cdot \\Delta X^{T}p(\\Delta X)d\\Delta X={\\frac {k_{B}T}{\\gamma }}\\Gamma ^{-1}", "0a3d4a2c3fbb7071c2288f7542dc6452": "{\\big [}\\partial _{x}+\\partial _{y}+{\\tfrac {1}{2}}(y-x){\\big ]}\\,{\\big [}...{\\big ]}", "0a3d655931e50806941c7f9edc142324": "i{\\frac {\\partial }{\\partial t}}u(x,t)=-{\\frac {\\partial ^{2}}{\\partial x^{2}}}u(x,t)+g(|u(x,t)|^{2})u(x,t),", "0a3da209c870dd96b8746e853cb6575c": "\\varepsilon _{n}=-{\\frac {\\partial z_{n}}{\\partial p_{n}}}{\\frac {p_{n}}{z_{n}}}", "0a3da9973aa4bba2479d8110395a0c96": "H(s)={\\frac {G_{0}}{B_{n}(a)}}", "0a3db6a040700fa21b51e59ceb52ea75": "\\langle e_{n},e_{n}\\rangle =1.", "0a3dd035bbc04a67ea2221a3eef66713": "L(x,y)", "0a3e16c1412b8f0d57c11d8e63862f25": "{\\dot {m}}=CA{\\sqrt {k\\rho _{0}P_{0}\\left({\\frac {2}{k+1}}\\right)^{\\frac {k+1}{k-1}}}}", "0a3ed5734d668253b37a5fdab6f7ae16": "\\operatorname {Var} _{X_{i}}\\left(E_{{\\textbf {X}}_{\\sim i}}\\left(Y\\mid X_{i}\\right)\\right)", "0a3ef145e2b4c50f43092d30caf5cbc9": "\\mathrm {d} S={\\frac {C_{P}}{T}}\\mathrm {d} T-\\alpha _{V}V\\mathrm {d} P.", "0a3f14a03e1179ff5fcf85f56d22aefe": "\\mathrm {SNR} ={\\frac {|{(R_{v}^{1/2}h)}^{\\mathrm {H} }(R_{v}^{-1/2}s)|^{2}}{{(R_{v}^{1/2}h)}^{\\mathrm {H} }(R_{v}^{1/2}h)}}\\leq s^{\\mathrm {H} }R_{v}^{-1}s.", "0a3f18d6b2eb22610e573e77a96d8e2f": "X[p\\Delta _{F}]\\,", "0a3f1fdbf8d4999a6721385856cb28ba": "E_{1j}={\\frac {O_{j}}{N_{j}}}N_{1j}", "0a3f6e6fcf2b2547bf675fa98d064c79": "g(t|x)={\\frac {f(x|t)g(t)}{\\int _{x}^{\\infty }f(x|t)g(t)dt}}.", "0a4002d3471a20dab34fba559688fc25": "{\\begin{smallmatrix}\\left[{\\frac {Fe}{H}}\\right]\\ =\\ +0.10\\ \\pm \\ 0.03\\end{smallmatrix}}", "0a404084903fdc45863ca72a4c05f66a": "E[R_{a}-R_{b}]", "0a4043df7ad41149fd439dce8bc75afb": "j_{m}=\\lim \\limits _{A\\rightarrow 0}{\\frac {I_{m}}{A}}", "0a406eb6da79e917940143234380ba81": "N(x)", "0a40ac116550b45e7c09756fa6356f95": "\\sin \\theta ={\\frac {2x\\sin \\theta '}{(x+y)+(x-y)\\sin ^{2}\\theta '}},", "0a4173ea4d4535bd3068215590c761da": "T_{M}", "0a41b44f2382f76cb9bc852a21ac42e8": "f\\ ", "0a41e223b52e58bf4824d6d9189b5d97": "{\\mathfrak {P}}^{71}", "0a41f2dd923676871312de0e840d302e": "TP", "0a4240e180e6f0d9807b58cea49d1b05": "\\tan \\phi ={\\frac {X_{L}-X_{C}}{R}}\\,\\!", "0a42a92fa31f71446817e999772362d6": "H(e^{j\\omega })", "0a42b6293e7172cea22786b67425f43b": "\\left|\\Theta \\right\\rangle ", "0a42cb1feaed1096d983d4406f88cbfa": "s_{n}=\\left\\lfloor E^{2^{n+1}}+{\\frac {1}{2}}\\right\\rfloor ", "0a42fe7084bd3a52d2c7e5f77e7ccd3d": "bWAR=(P_{runs}-A_{runs})+(A_{runs}-R_{runs})", "0a4342572c283f55ea626fe939845e4e": "h_{i}^{t}=e^{t}\\times {E_{i}^{t} \\over E^{t}}", "0a43470746f696689b3bb4673bc2c7e6": "{\\textbf {P}}_{k\\mid n}={\\textbf {P}}_{k\\mid k}+{\\textbf {C}}_{k}({\\textbf {P}}_{k+1\\mid n}-{\\textbf {P}}_{k+1\\mid k}){\\textbf {C}}_{k}^{T}", "0a434fc2469bd6166b56190158c24c75": "P=(u-v)\\cdot (s-t)+(u\\cdot t)(v\\cdot s)-(u\\cdot s)(v\\cdot t),\\,", "0a4361107b0ec4674e31bc3ea176d76a": "\\ell _{j}(\\xi )=\\prod _{i=0,\\,i\\neq j}^{k}{\\frac {\\xi -x_{i}}{x_{j}-x_{i}}},", "0a4388db865396276769880ba66c6d92": "u={\\begin{bmatrix}1\\\\3\\\\4\\end{bmatrix}}\\quad \\quad \\quad ", "0a43cabef5cdf4b5a97a4dcb7268f2cb": "A[Y_{1},\\ldots ,Y_{n}]", "0a43f21155e59f2ed6206ba470a89fac": "{n \\choose r}2^{1-{r \\choose 2}}.", "0a4421c54bb9409c499165e5910c54b9": "g_{L}=1", "0a44261e59194cea7e82f93dc22d8a9f": "{\\frac {\\mathrm {d} }{\\mathrm {d} t}}", "0a443b86d7bc3f2681fd175ebc1aae68": "v_{m}={\\sqrt {2gD}}", "0a44b2fc5e822be617ca2cb296916d9a": "c=C/N_{A}", "0a45051524bb6da37aa3a1f3b99d1184": "\\operatorname {Pr} \\{y\\}=\\operatorname {Tr} (SU^{*}\\operatorname {E} _{y}U)", "0a4528c48a099c5cffb227173fb2c503": "{\\sqrt {\\frac {7}{3}}}", "0a455a9a2a862bf914721d43176f8c01": "E_{\\mathrm {force} }=\\int {F\\mathrm {d} x=FL(1-\\cos \\theta )}", "0a45887b6fd3b1f83832a6dcf9039b18": "\\gamma _{*}:C_{*}\\to C_{*+1}", "0a4589e7895335316611a906d22ef9ba": "\\ {\\bar {T}}", "0a45a76ee727bf5b49651de72ccada7e": "R(X,Y)=\\nabla _{X}\\nabla _{Y}-\\nabla _{Y}\\nabla _{X}-\\nabla _{[X,Y]}", "0a45f5a5faba1511b18e77455330af6b": "\\Delta =\\sum _{k}e_{kk}={\\text{div}}\\;\\mathbf {v} ,", "0a46638600a6198ef3916077c2d91306": "\\operatorname {core} ", "0a468ad0b29472abe84e8132d50032c4": "L'\\to L={\\frac {\\omega _{c}'Q}{\\omega _{0}}}L'\\,,\\,C={\\frac {1}{\\omega _{0}\\omega _{c}'Q}}{\\frac {1}{L'}}", "0a473bb4fe983d5d609760478314c4d8": "v=b^{*}+{\\frac {\\Pr(b^{*}\\ {\\textrm {wins}})}{\\partial \\Pr(b^{*}\\ {\\textrm {wins}})/\\partial b}}", "0a47984f4692cab6153f38cadaf2f462": "{\\mathfrak {a}}+{\\mathfrak {b}}", "0a479924b84b40e86d990664c0afed85": "B\\subset \\mathbb {R} .", "0a47ad8c1e7ebde115be82a55e91b587": "\\textstyle V_{\\theta }^{2}", "0a47c8f4848e78ac5d2f0c4d5bb2fe47": "n\\geq {\\frac {1}{(p-{\\frac {1}{2}})^{2}}}\\ln {\\frac {1}{\\sqrt {\\varepsilon }}}.", "0a47db5f4074a82206c4cc01f1a60d67": "C_{p}(p,T)-C_{V}(V,T)=\\left[p(V,T)\\,+\\,\\left.{\\frac {\\partial U}{\\partial V}}\\right|_{(V,T)}\\right]\\,\\left.{\\frac {\\partial V}{\\partial T}}\\right|_{(p,T)}", "0a483b2fb471e4d19089462c552fed26": "(a\\cdot d+b\\cdot c)\\cdot 10", "0a491e43f5231fc38462b12ea2309d18": "g'N'", "0a4934396f709f76cf3007f2c0437159": "S(n_{1},\\ldots ,n_{l})=\\exists m_{1}\\cdots \\exists m_{k}R(n_{1},\\ldots ,n_{l},m_{1},\\ldots ,m_{k})", "0a496937c8a48ab07f4313bfb4dd6f49": "\\sum _{n=-\\infty }^{\\infty }a^{n(n+1)/2}\\;b^{n(n-1)/2}=(-a;ab)_{\\infty }\\;(-b;ab)_{\\infty }\\;(ab;ab)_{\\infty }.", "0a498ea7a459d8602f53de446b0d5e67": "KE={\\frac {1}{2}}I\\omega _{max}^{2}=\\left(10^{-3}{\\frac {\\ell ^{2}}{3}}\\right)\\left({\\frac {254}{\\ell /2}}\\right)^{2}=43{\\text{ erg}}", "0a499d901a3f2af99c4d726590ac2006": "(A\\oplus B)\\subseteq C", "0a49dd19c59685ed54a4a3abb5cac0af": "{\\widehat {\\mathcal {M}}}", "0a49ffa5288576b9a809b7c539eb25b2": "\\operatorname {Alph} (s)", "0a4a1ae98e9b80e839929846248e45cf": "p(n)\\approx 1-e^{-n^{2}/(2\\times 365)},\\,", "0a4a1da19ac2b7e365c805b2d2bbcb04": "3^{\\frac {2}{13}}", "0a4a56947caf6fdcf237a5c3a5db941a": "(E,{\\mathcal {E}})", "0a4aa2cee55c184ce52c4a2094ac6319": "\\tau _{g}=-{\\frac {d}{d\\omega }}\\arg(H(j\\omega ))", "0a4aafda9ad959024478e6df5a1c2af2": "U_{i}(N),i=1,2", "0a4ae321700a998ee69c8d71b7f9b202": "f':X\\to S'", "0a4b018f4a1382228c85c71a987d566f": "k=n", "0a4b247283f01625b902164464fc5acd": "{\\frac {dt}{d\\tau }}", "0a4b7f7e00aac7ea3381e0a89d874f12": "Z_{\\epsilon }", "0a4bc9d99594802468034e0960f9acb7": "L=\\mathrm {DE} =L'_{0}/\\gamma =18\\ \\mathrm {cm} .", "0a4bd081c8a5481e1c8b320466b3ba96": "|\\cdot |_{p}", "0a4c006fca86431acd1924cdce0b3448": "P\\to Q\\vdash P\\to (P\\land Q)", "0a4c09917c25decc00d66a6ddb424731": "\\mathrm {Re} ({\\tilde {n}})={\\frac {ck}{\\omega }}", "0a4c316a8a007da3bb7d27aea35d6e37": "\\epsilon =(1+\\alpha ){{\\sigma _{0}}/{E}}\\,", "0a4cf0cfd79a52b60ab6b7a20017bc94": "\\lambda _{1}=\\lambda ,\\lambda _{2}=0", "0a4d1af755ff4e6d0e38b74b93c35e36": "{\\mathcal {L}}_{X}g=\\lambda g", "0a4d3cd1b183bc23051aef1c2b51d315": "{\\frac {\\overline {P}}{A}}<\\sigma _{0}", "0a4d7ad4b2e31fbaafcb3d69a750184f": "T_{1}=\\sum _{F}{\\text{(link to the root)}}", "0a4d8ebe42988bac58dc92b4e849b2d5": "\\arccos \\left({23 \\over 27}\\right)", "0a4dc552dd3b32a12e72ed6211f03279": "\\mathbf {f_{0:1}} =\\mathbf {\\pi } \\mathbf {T} \\mathbf {O_{1}} ", "0a4e17d97de673d6f7034c10c7a917d8": "c_{F}(a,b)\\equiv c_{-}(a,b)", "0a4e2ef1e300d5eb178e96272ea61f7a": "23=10", "0a4e4f84dadbf64c5fe90b04c82326d8": "b\\geq 5", "0a4e5aa782a10295367e7ba65948e6b0": "V_{\\mathbb {R} }\\subset V", "0a4ea28d6b34a3d8aee23c8e2a9a77c6": "{\\textbf {P}}_{0\\mid 0}={\\begin{bmatrix}0&0\\\\0&0\\end{bmatrix}}", "0a4eb72f94439f4c3a6f4d0f75572a29": "E_{x\\in _{R}D_{n}}[{\\frac {t_{A}(x)^{\\epsilon }}{n}}]\\leq C", "0a4f65c643313b1349de59af0e7c9c01": "v_{\\mathrm {x} }", "0a4f773ab833d7c432c8fb9c2a336426": "[J_{i},K_{j}]=i\\epsilon _{ijk}K_{k},\\,\\!", "0a4f9cfa2118d339103529d669a0593d": "{\\vec {\\omega }}={\\frac {|\\mathrm {\\mathbf {v} } |\\sin(\\theta )}{|\\mathrm {\\mathbf {r} } |}}\\,{\\vec {u}}", "0a4faaa65d0949db17f80824f26d180f": "21=3\\times 7,231=3\\times 7\\times 11,744=24\\times 31\\,", "0a4fb033c3d6707ba1ec464b9355f69c": "={\\begin{vmatrix}{\\boldsymbol {i}}&{\\boldsymbol {j}}&{\\boldsymbol {k}}\\\\0&0&\\omega \\\\-\\omega tv\\sin \\alpha &\\omega tv\\cos \\alpha &0\\end{vmatrix}}\\ \\ ,", "0a5000fe8b6b5570dd5a1ce00b828ef6": "\\theta \\,\\!", "0a50292b96b71152e9a1c7ccdeecd176": "\\left[T_{\\mathrm {n} }+E_{k}(\\mathbf {R} )+{\\mathcal {T}}_{\\mathrm {k} }(\\mathbf {R} )\\right]\\;\\phi _{k}(\\mathbf {R} )=E\\phi _{k}(\\mathbf {R} )\\quad \\mathrm {for} \\quad k=1,\\ldots ,K,", "0a503598148b46a9c32f9283624e9c14": "{\\hat {A}}_{m_{j}}={\\begin{cases}0&m_{j}{\\text{ odd}}\\\\{\\frac {1}{\\pi }}\\int _{-\\pi /2}^{\\pi /2}f{\\bigl (}\\mathbf {X} (s){\\bigr )}\\cos \\left(m_{j}\\omega _{j}s\\right)ds&m_{j}{\\text{ even}}\\end{cases}}", "0a5044fd8223c03d094e80122b5dbc4d": "h^{1,1}=h_{+}^{1,1}+h_{-}^{1,1}", "0a50606f7c0591dc0dc6bbe8dd555eed": "s+\\Delta s", "0a5096e79e49cf6788475bb19ca34c68": "\\langle X^{N}\\rangle X", "0a50d4e09a2c9dff3e302b50f78abbc3": "x=x^{0}+x^{1}\\mathbf {e} _{1}+x^{2}\\mathbf {e} _{2}+x^{3}\\mathbf {e} _{3},", "0a514266c43d80e53ee002673b749ffa": "\\!p=-\\rho ", "0a51463a7c4566600ed863d7e1c35f40": "{\\it {\\nu }}", "0a515a686edfb83439cc39e028394a01": "I=T_{1}\\cap \\dots \\cap T_{n}", "0a51ae5658d0346929f307121a89cb69": "V_{\\mathrm {T} }=V_{\\mathrm {iL} }[(1+{\\mathit {\\Gamma }})\\cosh(\\gamma x)+(1-{\\mathit {\\Gamma }})\\sinh(\\gamma x)]\\,\\!", "0a523d3edcc9ca50f6ba3b2d1952ab21": "{\\frac {\\sqrt {D}}{D}}={\\frac {1}{\\sqrt {D}}}={\\sqrt {\\frac {T}{\\tau }}}", "0a52b4c489ad207a5f3d40ef8247686e": "D_{x}^{\\alpha }f(x)={\\begin{cases}{\\frac {d^{\\lceil \\alpha \\rceil }}{dx^{\\lceil \\alpha \\rceil }}}I^{\\lceil \\alpha \\rceil -\\alpha }f(x)&\\alpha >0\\\\f(x)&\\alpha =0\\\\I^{-\\alpha }f(x)&\\alpha <0.\\end{cases}}", "0a52c71202d682d910f2b3c756221706": "\\otimes _{o}", "0a52dda0fa627cd533f5065c3aec9b31": "\\lambda \\subset {\\begin{pmatrix}A&B\\\\C&-A^{T}\\end{pmatrix}}", "0a53440524558b8f5866430d1661b32e": "[[M]_{a\\;\\|\\;u}\\;\\|\\;N]_{{\\overline {a}}\\;\\|\\;v}\\rightarrow M\\;\\|\\;[N]_{u\\;\\|\\;v\\;\\|\\;x}", "0a5349c263db987e473ec20e0593428f": "\\forall x,y\\in I:x-y\\in I", "0a5380be71df066e64f061c2cd91c47a": "{\\tilde {\\nu }}_{J^{\\prime }\\leftrightarrow J^{\\prime \\prime }}=2{\\tilde {B}}\\left(J^{\\prime \\prime }+1\\right)-4{\\tilde {D}}\\left(J^{\\prime \\prime }+1\\right)^{3}\\qquad J^{\\prime \\prime }=0,1,2,...", "0a53a6e8d3e302eef4d3b3519cd3414a": "|s-t|\\to 0.", "0a53d5d631516bd9558f88a99c227a89": "\\left[{\\begin{smallmatrix}\\cos 2\\pi /p&\\sin 2\\pi /p\\\\-\\sin 2\\pi /p&\\cos 2\\pi /p\\\\\\end{smallmatrix}}\\right]", "0a53df76736bc93a648b9f5df26a187a": "S=\\sum _{j=h}^{n}C_{n,j}p^{j}(1-p)^{n-j},", "0a53eb045ed05e22443717ebeeabac23": "D(p+x\\|p)", "0a547477d57298d8feac2fea4ccfff3d": "T_{\\mathrm {total} }=", "0a5508caaf3523cd69a2ffdd24a60c6d": "\\int {\\frac {dx}{xR}}=-{\\frac {2{\\sqrt {a}}}{b}}\\int du=-{\\frac {2{\\sqrt {a}}}{b}}u", "0a554cb3e5efaabd40bfc7a0d9eabacc": "\\scriptstyle \\leq 5.9\\times 10^{-35}", "0a559e47dc1acc4c8e570a52d4288ee3": "Molecule~1+Molecule~2\\xrightarrow {} macromolecule", "0a55ec379e8b3cb43d7b6861be6ab73d": "\\lambda _{A}\\colon I\\otimes A\\cong A", "0a562b077f9747f6cc22f2eb308b09d3": "m_{\\ell }\\in \\{0\\cdots n-1\\}\\,\\!", "0a563403cc1e7ce64647faa684876a38": "{\\mbox{If }}a\\in X\\in Con{\\mbox{ then }}X\\vdash a", "0a56463af1f51954a097e806eed96109": "\\|e^{X+Y}-e^{X}\\|\\leq \\|Y\\|e^{\\|X\\|}e^{\\|Y\\|},", "0a56d591f6a4e9f6c320727988a55428": "\\varphi _{n}=\\arctan \\left({\\frac {1}{\\sqrt {n}}}\\right).", "0a56e6a85c9b3a1d47edf7c33b457352": "u(t)^{2}/x(t)", "0a57063a19daee162bcad8e9e289f77b": "\\sum _{1\\leq i_{1}{\\bar {u}}&{\\text{if }}X_{t}<{\\bar {x}},\\\\{}={\\bar {u}}&{\\text{if }}X_{t}={\\bar {x}},\\\\{}<{\\bar {u}}&{\\text{if }}X_{t}>{\\bar {x}},\\end{cases}}", "0a5fd752a9644d8825cd5b7a8fcce732": "q_{2}=y_{2}\\cdot {\\sqrt {2g{\\bigl (}E_{1}-y_{2}{\\bigr )}}}=q_{g}=y_{g}\\cdot {\\sqrt {2g\\cdot h}}", "0a5ffabb24f666e8972f7ed2ed0f4853": "CH_{4}+CH_{3}^{+}\\to C_{2}H_{5}^{+}+H_{2}", "0a60891d81e36a701c40fbb0823bae2e": "n-x", "0a60cd6434fd1f01723c45aa7576eacb": "m={\\frac {\\Delta y}{\\Delta x}}={\\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}={\\frac {8-2}{13-1}}={\\frac {6}{12}}={\\frac {1}{2}}", "0a60d72cea2f6396ba352177aece4b53": "Z'_{w}", "0a6101ec72d4317b4ace994ce83f8bfa": "Y_{t}\\,", "0a610230cbd463bfe6aa18ab3266993a": "B={\\frac {2D_{ox}C_{s}}{N_{i}}}", "0a613492a0d0d42d4dc849b0d83418ab": "\\Delta _{K_{n}}=(-1)^{\\varphi (n)/2}{\\frac {n^{\\varphi (n)}}{\\displaystyle \\prod _{p|n}p^{\\varphi (n)/(p-1)}}}", "0a61a403929176a6efd85d77d069ba64": "\\{(i,x,t)\\in \\mathbb {N} ^{3}|\\Phi _{i}(x)=t\\}", "0a61a7fefcc21f13fad4e7d38909cc7d": "2p_{2}(E\\oplus F)=2p_{2}(E)+2p_{1}(E)\\smile p_{1}(F)+2p_{2}(F)", "0a61cda98eef76a4da74b652bd387729": "R\\bowtie S=S\\bowtie R\\,", "0a622ad3bd0ad03e8e14a9753aa5b251": "1+\\ln(2b)\\,", "0a628ba9b1845b6acce4b4ff2c9ec575": "\\sigma ^{2}", "0a62a30803055f49cbec5088a0686d08": "\\mu \\left[\\left(1+{\\frac {9\\mu ^{2}}{4\\lambda ^{2}}}\\right)^{\\frac {1}{2}}-{\\frac {3\\mu }{2\\lambda }}\\right]", "0a63028aad661f43712b1b5e9db36109": "{\\text{(Eq. 5)}}\\qquad \\sum _{c=1}^{N}\\mu _{ab}^{(c)}(t)\\leq \\mu _{ab}(t)\\qquad \\forall (a,b),\\forall t", "0a63136247dd6edd0140d3aea242fbbb": "\\displaystyle A^{(2)}", "0a631a194dea59eee5b644ff7abb6b93": ")\\to ", "0a6389f8d48024b70b0842cd26d51e06": "\\Delta S_{i=1}=S_{i=2}-S_{i-1}={\\big (}2000.0{\\text{ ft}}\\cdot 1{\\text{ ft}}\\cdot 3.63{\\text{ ft}}{\\big )}-7130.5{\\text{ ft}}^{3}=123.2{\\text{ ft}}^{3}", "0a63cec8a33bf055104bf41af88f757f": "\\langle \\mu \\rangle ={\\frac {\\int \\mu dP}{\\int dP}},", "0a63d57241466bad0086dc935d81cc9b": "||f||_{\\mathcal {H}}^{2}=\\langle f,f\\rangle _{\\mathcal {H}}=\\sum _{i=1}^{n}\\sum _{j=1}^{n}c_{i}c_{j}K(x_{i},x_{j})=c^{T}\\mathbf {K} c", "0a640a1e098c7880f9a1f6bb988cf283": "x+\\infty =\\infty ", "0a645fc515fe6128f91758f996ebe307": "x_{i}\\,", "0a647fdd7aa9d6bbb8633f8f034bb50e": "\\xi \\,", "0a65623d37b26c0f768d4cdc8d638a5c": "u^{a}=\\delta _{0}^{a}=(1,0,0,0)", "0a656d0d581c435e643a9b0934173723": "\\mathrm {E} (e_{t})=0\\,", "0a6595104e3b1b4e61b4568e450766af": "S=\\sum \\limits _{i=\\lfloor {\\frac {n}{2}}\\rfloor +1}^{n}{\\binom {n}{i}}p^{i}(1-p)^{n-i}.", "0a65aaed8624538799b7d70b3995a053": "\\sin \\theta ={\\frac {f}{J}}\\,.", "0a65bf2d85a1f17f162770ba6033347d": "distance={\\frac {c\\delta t}{2}}", "0a65fcf09bca7b5c622bdcda56c71a3d": "R_{\\text{vertical}}={\\frac {R_{12,34}+R_{34,12}}{2}}", "0a6618e95d0397e28cda00fbb663a84f": "-R_{3}i_{3}-\\epsilon _{2}-\\epsilon _{1}+R_{2}i_{2}=0", "0a66218865919d546ff8b3ae261d7c65": "A(0,u_{k},v_{k})", "0a663d246ff9370ba9ee4fec7ab4019d": "{\\frac {1}{N_{f}}}={\\frac {1}{N_{f}^{fatigue}}}+{\\frac {1}{N_{f}^{oxidation}}}+{\\frac {1}{N_{f}^{creep}}}", "0a6652d80a542bbc53855f89bd1bb37f": "b,d,u,a\\in [0,1]\\,\\!", "0a665387bb4989331d083413c8cb4000": "T(z)=z+c\\int _{0}^{1}tdt.", "0a672740a279b44d366aea32423472ea": "\\Omega =[0,1]", "0a6737eb40ad5cc13a887df1d1a4ece7": "\\{\\gamma ^{0},\\gamma ^{1},\\gamma ^{2},\\gamma ^{3}\\}", "0a67f1ca9f0df9c3b6ee44a53db67626": "u_{t}+au_{x}=0\\,", "0a680d21adfaeb3eb50a4526856097f4": "M({\\text{H}}_{2}{\\text{O}})\\rightleftharpoons M({\\text{OH}})+H:[M({\\text{OH}})]=\\beta ^{*}[M][{\\text{H}}]^{-1}", "0a681bee5f23b54137db80c1da9962eb": "u(t,r)={\\frac {1}{r}}F(r-ct)+{\\frac {1}{r}}G(r+ct),\\,", "0a6857a4920c79ba8f989d4b829f4975": "{\\frac {\\operatorname {d} I_{\\text{L}}}{\\operatorname {d} t}}={\\frac {V_{i}}{L}}", "0a685ddd0c07efb9d44fa2b2b2be41f1": "\\omega _{x\\lor y}^{A}=\\omega _{x}^{A}\\sqcup \\omega _{y}^{A}\\,\\!", "0a689066f9815ca92d65d9ca05109454": "P/{\\rm {hp}}={\\frac {\\tau /{\\rm {(lbf\\cdot ft)}}\\times 2\\pi \\times \\omega /{\\rm {rpm}}}{33,000}}.", "0a689befe02045c71043d3986ad85df3": "{\\partial x_{i}(\\mathbf {p} ,w) \\over \\partial p_{j}}={\\partial h_{i}(\\mathbf {p} ,u) \\over \\partial p_{j}}-{\\partial x_{i}(\\mathbf {p} ,w) \\over \\partial w}x_{j}(\\mathbf {p} ,w),\\,", "0a68a7675f3f06a491925480020a14a1": "p_{1}=0,\\ldots ,p_{k}=0\\,", "0a6920a9dfa5c6654196b50cb217b7af": "b_{T}", "0a692cc37fe6d13991a12630a7192bb2": "\\scriptstyle a\\,=\\,\\infty ", "0a69534ad90cd440a667583706dfc338": "2y=-{\\frac {x^{2}}{\\tau ^{2}}}+\\tau ^{2}", "0a698956847f166a1be859c688ee56a4": "\\left\\langle \\left(\\sum _{n=0}^{\\infty }u^{n}\\langle f,\\psi _{n}\\rangle \\psi _{n}\\right),g\\right\\rangle =\\int \\int E(x,y;u)f(x){\\overline {g(y)}}\\,\\mathrm {d} x\\,\\mathrm {d} y\\rightarrow \\int f(x){\\overline {g(x)}}\\,\\mathrm {d} x=\\langle f,g\\rangle ,", "0a69c137345c1e1f6d6db9420926f630": "V_{min}(z)=\\sum _{k=0}^{p}v_{min}(k)z^{-k}", "0a6a87863e4d4926f2a3142f39674144": "\\omega _{x}^{A{\\underline {\\diamond }}B}=\\omega _{x}^{A}\\;{\\underline {\\oplus }}\\;\\omega _{x}^{B}\\,\\!", "0a6a8d4c7720abc9dbb38ad6f23501e4": "f(x+h)=f(x)+f'(x)h+\\psi _{1}(h)\\qquad \\qquad g(x+h)=g(x)+g'(x)h+\\psi _{2}(h)", "0a6a9d1e426c22b7531b48623a60a061": "F_{Y}", "0a6ad0e3d13e1c665f3e5ae978804976": "0\\leq p\\leq 1", "0a6adf905fcb2028fad6ca232aba0356": "S_{\\theta }", "0a6ae37a6d4d363bb1a99e91c1eb3606": "{\\text{value}}=1.25\\times 2^{-3}=0.15625", "0a6af32f750ef00668109cb8793b91ec": "\\sigma _{y'}", "0a6bda3dcba2df8b8b56579d2cfbe7b6": "\\phi _{e}(z)={\\frac {1}{2\\pi i}}{\\mathcal {P}}\\int _{C}{\\frac {\\phi (\\zeta )d\\zeta }{\\zeta -z}}-{\\frac {1}{2}}\\phi (z).\\,", "0a6c11d1ae791d08aa93c8a8d69c4bfa": "d(p,A)=0", "0a6c22504db2d969c9646fb2a01d125a": "S(x,t)", "0a6c4a0d788571abfe1997ac6a5036b7": "Z_{2}^{9}", "0a6c8bc1bdbc0f9d233ef23d4a5d9096": "74^{2}", "0a6c98807229adaf2dd97660ed557168": "d+2c=180", "0a6d5c4ac7b6bdc0e2aab86e76db1c3a": "t=\\lceil 1+n/{\\sqrt {2}}\\rceil ", "0a6d626a9902cd0fb7d84a8b4143a54e": "L(h)\\leq {\\frac {1}{8}}h^{2}={\\frac {1}{8}}\\lambda ^{2}(b-a)^{2}", "0a6d7ddd16fab563c4184450807d5d38": "1\\to A\\to B\\to C\\to 1\\,", "0a6db390f9b1e646c441fb088a8e955a": "U\\left(x_{1},x_{2},\\dots ,x_{n}\\right)", "0a6dd1ec55a668d7e84633f49faf281e": "y\\in V", "0a6e6c60dcb0072ca15787ef4b64a5b7": "\\mathrm {SU} (2n)\\,", "0a6ec12ebd600cd251837bbe54b5d774": "\\varepsilon _{ijk}\\varepsilon _{imn}=\\delta _{jm}\\delta _{kn}-\\delta _{jn}\\delta _{km}\\,.", "0a6f0332d0ad817314fb1aa5ec002c31": "\\alpha \\varphi ={d\\varphi \\over dz}=0", "0a6f4e954b60efed010eea2f3fdfadad": "t,\\lim _{n\\rightarrow \\infty }n_{t}/n=c_{t}>0", "0a6f8c0f86fc463dd96cedfc3da0fe82": "\\sin(x)=\\sum _{n=0}^{\\infty }{\\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}=x-{\\frac {x^{3}}{3!}}+{\\frac {x^{5}}{5!}}-{\\frac {x^{7}}{7!}}+\\cdots ,", "0a6feeb4b14b09fd29a2f22c6b6823c1": "M=|M_{1}|e^{i\\theta _{1}}e^{i\\phi _{1}}+|M_{2}|e^{i\\theta _{2}}e^{i\\phi _{2}}", "0a6ffa5b309e21885b7586e8add2d152": "I_{1}+I_{2}+I_{3}=0", "0a70033be217e33353951677643105bc": "\\cdots \\to H_{n+1}(X_{n+1},X_{n})\\to H_{n}(X_{n},X_{n-1})\\to H_{n-1}(X_{n-1},X_{n-2})\\to \\cdots .", "0a701f7d5819334492641c129208568b": "{1+{\\underbrace {1+1+1+\\cdots +1} _{b}}}", "0a70e1e06ec023f11f8060faf1806c32": "\\textstyle 2m+1", "0a7101b12e80841b388888740faa7a24": "H_{i}(S,T)={\\frac {G(S,T)}{(a_{i}S-b_{i}T)}}", "0a7148eabe2f26dae5d42f1c2d2c41ae": "\\{0^{n}1^{n}\\mid n\\geq 0\\}", "0a715bd905c3dc34931ecad5f34e6903": "\\mathbf {\\omega } _{i}", "0a716816b16d082398377809ec003f3d": "V:=\\bigcup _{\\alpha }V_{\\alpha }.", "0a71b1e3b2981ffa4ec126d033b3a84b": "F(x;k)=P(k/2,x^{2}/2)\\,", "0a71bb9f26976994082eb22e40d21ed1": "E_{sig}(\\omega ,\\tau )", "0a71cb5e502f901c74ba4fb99204f28f": "\\lambda _{s}(2)=s+1\\,", "0a71d3285d990312ab1850bffa37c53b": "X\\to \\mathbb {R} ", "0a72e76e86c46a3c2a5f9fdb35203737": "(1-\\epsilon )\\log n", "0a72f9801e1a0f4a03e56e2297e0c848": "x_{0}\\in \\Omega ", "0a730050106636e0e0bf8b35516e87eb": "U_{q}({\\mathfrak {g}})", "0a7369114d303ba522a8ec9862141869": "|n|_{\\ast }=\\Pi _{i0", "0a804b1ed1ebf0230eb8783baeffb821": "G_{2}:\\{0,1\\}^{\\ell }\\to \\{0,1\\}^{h}", "0a80dcfba6f4ce38eb766846c8ec3dfb": "f(x)={\\frac {A(x+h)-A(x)}{h}}-{\\frac {(RedExcess)}{h}}", "0a810140f0ba4bdac1592a894508ba5b": "x-1=\\log _{2}3\\,", "0a81cc5c3a9950fc082db4fae414dabe": "W=\\int _{0}^{\\epsilon }{\\boldsymbol {\\sigma }}:d{\\boldsymbol {\\epsilon }}~;~~{\\boldsymbol {\\epsilon }}={\\tfrac {1}{2}}\\left[{\\boldsymbol {\\nabla }}\\mathbf {u} +({\\boldsymbol {\\nabla }}\\mathbf {u} )^{T}\\right]~.", "0a81e71240394f1c60598f3331850cae": "\\mathrm {Le} ={\\frac {\\alpha }{D}}={\\frac {\\mathrm {Sc} }{\\mathrm {Pr} }}", "0a821d19fda56d7f9d685aefc4c05b9e": "X\\sim {\\mbox{Inv-Gamma}}(\\alpha ,{\\tfrac {1}{2}})", "0a8224e787df13e07e706dd41a254db0": "H_{i}^{BM}(X)=Ker(\\partial :C_{i}((X))\\to C_{i-1}((X)))/Im(\\partial :C_{i+1}((X))\\to C_{i}((X))).", "0a833a08c0c5b875621902cc4823c0ed": "O_{4}^{(\\alpha )}(t)={\\frac {(1+\\alpha )(4+\\alpha )}{2t}}+4{\\frac {(1+\\alpha )(2+\\alpha )(4+\\alpha )}{t^{3}}}+16{\\frac {(1+\\alpha )(2+\\alpha )(3+\\alpha )(4+\\alpha )}{t^{5}}}.", "0a835d61f03bea60a6b5e1427c71e7db": "(p,q\\in X)", "0a836a2e911068a17983cb3e917e0bbf": "p({\\boldsymbol {\\theta }}|{\\rm {{data})}}", "0a83dbb275ef74fc4b5f58024079d61a": "{\\begin{aligned}H(z)&=Z\\{h[n]\\}\\\\&=\\sum _{n=-\\infty }^{\\infty }h[n]z^{-n}\\\\&=\\sum _{n=0}^{N}b_{n}\\,z^{-n}\\end{aligned}}", "0a83fc11a7fdf57b8f0f3b74916a39c2": "a=mx\\pm c,\\qquad b=nx\\pm d", "0a841365cc7530bcafc5f2f5772c3bdf": "NS_{i}=\\sum _{k=t+1}^{t+n}\\left[e_{i}^{k-1}\\left(G^{k}\\right)\\right]", "0a849de52079d2df084c26a570f3d692": "U_{iy}\\subset \\mathbb {R} ^{p}", "0a84bea043146c2c23559f53f5bca690": "Pr[\\sigma (x)\\in R]\\leq t\\cdot max\\left(Pr[|{\\frac {1}{m}}\\left(\\sum _{i}w_{\\sigma _{i}}^{j}-\\sum _{i}w_{\\sigma _{m+i}}^{j}\\right)|\\geq {\\frac {\\epsilon }{2}}]\\right)", "0a84fa4e8c926eeee85ac67e7778455d": "i\\rightarrow j", "0a8522d7a711bc4e6a0322233052639e": "F_{f}", "0a854e78a3801634aafa3b8e9250806f": "N\\,l", "0a856823f9689d20db459d9393a0fb46": "\\rho _{E}={\\frac {4\\cdot {\\pi }\\cdot a\\cdot R_{W}}{1+{\\frac {2\\cdot a}{\\sqrt {a^{2}+4\\cdot b^{2}}}}-{\\frac {a}{\\sqrt {a^{2}+b^{2}}}}}}\\,", "0a8610a6ac60a5cf79607be23b516f0b": "F={\\frac {{\\vec {G}}\\cdot ({\\vec {e}}-{\\vec {u}})}{RT}}f^{eq}", "0a86bea1739ad845794a217a35b78548": "C_{n^{*}l^{*}}", "0a86e709eb8c2d0751d7e827a50e6c6d": "S_{3}(x,y)=QM(x,y,GM(x,y))", "0a874c119e759077832d65598daf9116": "W=\\int _{V_{i}}^{V_{f}}\\,P\\,dV", "0a874c974a315930ff823fd0dda9d886": "=(ac)'+(bc)'=(ac+bc)',", "0a8768b319a9f85e712dd926f3b06c08": "\\mathrm {B} _{\\mathrm {2} }", "0a876cb41049e2533d9bc9e878ba712f": "\\ \\rho _{water}", "0a886d8d7f69aab15bc76ccaf23ce109": "x_{1},", "0a88fa202330bdab11ca18b855a91c38": "x_{1},x_{2},x_{3},...,x_{n}", "0a89563d09558842989a6d6d48cce69d": "(\\mu )", "0a8990794d6f0ef65c82e7f6e973b1b9": "\\prec _{K}", "0a899eced66c9882eb4ae08e037d962c": "e=1", "0a8a30cc75ba663acfb56efaf4c9d6eb": "{\\tilde {\\mathbf {y} }}", "0a8ad3b9896c6c195de0243fb722fe9d": "~x~", "0a8b270d88f2be91095f4a78b91e0043": "r=e^{i\\Phi }", "0a8b5a8efe1c2d7a9bc12fe9bbdbf2c6": "{\\begin{matrix}{2 \\choose 1}^{2}{11 \\choose 1}{4 \\choose 3}\\end{matrix}}", "0a8ba492b4583452d259471de722b717": "\\mu _{\\rm {N}}=\\mu _{\\rm {B}}{\\frac {A_{\\rm {r}}({\\rm {e}})}{A_{\\rm {r}}({\\rm {p}})}}", "0a8bad72209b87d0405e59af1b0f4af9": "f*g=fg+{\\mathcal {O}}(\\hbar ),", "0a8bb3670a5fb39e668d1263facaf0be": "\\sigma _{zz}-\\sigma _{yz}+\\sigma _{xz}", "0a8ca3202af084bc79fd7067012d19e3": "{\\sqrt {p}}", "0a8cc1b87b4343c9034b76390099e82a": "\\phi \\wedge \\phi '=(-1)^{|\\phi ||\\phi '|+[\\phi ][\\phi ']}\\phi '\\wedge \\phi ", "0a8cdf337307f461204f9eebeb8dd35b": "1/\\gamma ", "0a8ce59fb0d11e553c6c47fae50b59e1": "f_{1}=3x", "0a8d583f9b3016c6649985d43255e601": "f\\colon (X,\\Sigma )\\rightarrow (Y,T)", "0a8d74f7b0ba39c4d917f53a5d900a0a": "\\nabla \\cdot {\\mathbf {A} }+{\\frac {1}{c}}{\\frac {\\partial \\varphi }{\\partial t}}=0.", "0a8d87f95ec4e258f7cc7202d7017a14": "\\langle x|\\psi _{1}\\rangle ={\\sqrt {\\frac {2}{L}}}~{\\rm {sin}}\\left({\\frac {\\pi x}{L}}\\right)", "0a8e77d0114866c191519d9660f13c76": "\\textstyle \\mathbb {R} ^{\\mathbb {R} }", "0a8e8cf8fb77f8e1282adc97cf88ee36": "{\\bar {A}}", "0a8e97e89ac8c137a33840d748dc678d": "\\left({\\frac {p}{q}}\\right)={\\begin{cases}\\left({\\frac {q}{p}}\\right)\\;\\;{\\text{ if }}q\\equiv 1{\\pmod {4}}\\\\\\left({\\frac {-q}{p}}\\right){\\text{ if }}q\\equiv 3{\\pmod {4}}\\end{cases}}", "0a8e9b29b44548aeb1cc70a8bfc48d07": "B=Q(nI_{p})Q^{-1}=nI_{p}", "0a8f453e32cc455eeb7ed1a2bdd2bfaa": "y\\in E", "0a8f59f8f520dceede68cf8327ddec6f": "\\prod _{s}\\left(1+ieA_{\\mu }{dx^{\\mu } \\over ds}ds\\right)=\\exp \\left(ie\\int A\\cdot dx\\right).", "0a8f733556ca682c414d403f00a4af25": "{J}", "0a8f7ce71cfbc4945b21943b018926f0": "\\omega _{r}={\\frac {2{\\pi }n_{s}}{60}}={\\frac {4{\\pi }f_{s}}{p}}", "0a8f94f3432a5152ce3bc7abf6dd44cd": "e^{{\\boldsymbol {\\beta }}_{0}\\cdot \\mathbf {X} _{i}}=e^{\\mathbf {0} \\cdot \\mathbf {X} _{i}}=1", "0a8fd40516398a9469b6c80084f07a84": "g:S\\to X", "0a90190b6f26b7c751e4da864b3ddd18": "T_{1L}", "0a902501c935ad0c486e42b88cfd1379": "Q\\neq d", "0a9037f807f4d3bc91c92d3ea20b2cdd": "x_{i}\\cup y_{i}", "0a904396c0d81e52e04289476e38ddd2": "x_{1}^{2}+x_{2}^{2}+\\cdots +x_{n}^{2}=1", "0a90819df25db81e5acbf6c20e691e15": "75^{2}", "0a914118779a075496338053c599939d": "{\\textrm {E}}(X(X-1)\\cdots (X-k+1))", "0a9151c4b0c3161781098770c1ea6f4a": "{r_{\\rm {c}}}", "0a9165a6c7ecdbc9b81e07c105ff534e": "c\\subset X", "0a916e42ffa1a41216e11adfaa299931": "J_{1}^{s}=J_{1}^{m}", "0a918b99a6447f03255f3f92d306f9f4": "\\gamma =\\epsilon ", "0a918f8974a600eb7fe0bfa01a70bcc1": "\\sum _{n=-\\infty }^{\\infty }x[n]\\ z^{-nL}=\\sum _{n=-\\infty }^{\\infty }x[n]\\ e^{-i\\omega Ln}={\\frac {1}{T}}\\sum _{k=-\\infty }^{\\infty }\\underbrace {X\\left({\\tfrac {\\omega L}{2\\pi T}}-{\\tfrac {k}{T}}\\right)} _{X\\left({\\frac {\\omega -2\\pi k/L}{2\\pi T/L}}\\right)},", "0a920178557d322b2a517b518b2eb40d": "\\theta =(\\mu ,\\sigma ^{2})", "0a924a15542ccd8be300610b9183dad2": "\\rho (t)=\\cos(6\\pi t)[\\cos(2\\pi t)+i\\sin(2\\pi t)],0\\leq t\\leq 1", "0a926558a06e972d65972e320dca0fbf": "r\\ =\\ (1+j)^{12}-1", "0a928ec05dba4fa8bfd87ad1e689fb35": "B_{2}\\cong C_{2},", "0a92bd8b9549d23c78ff57834870be04": "M_{p}^{z_{+}}X_{q}^{z_{-}}", "0a92c9a80b141a4e630b8d64d8b3fbc9": "a_{B}=\\gamma _{BY}\\cdot \\lambda _{0},\\ ", "0a92f9148f59a4f3201a012a2ea2bd83": "f:M\\to M", "0a9309c3100f5a94bbe74ca1c147a4e3": "P(x)\\downarrow (\\exists {y}{\\in }\\mathbf {Y} \\,Q(y))\\equiv \\ \\forall {y}{\\in }\\mathbf {Y} \\,(P(x)\\downarrow Q(y)),~\\mathrm {provided~that} ~\\mathbf {Y} \\neq \\emptyset ", "0a9312a549a978ed84c7064d0d223961": "-2F{\\dot {H}}=\\rho _{\\rm {m}}+{\\frac {4}{3}}\\rho _{\\rm {rad}}+{\\ddot {F}}-H{\\dot {F}},", "0a93584edf76c7fe398aae935f870169": "dx\\;dy\\;dz=\\det {\\frac {\\partial (x,y,z)}{\\partial (r,\\theta ,h)}}dr\\;d\\theta \\;dh={r}\\;dr\\;d\\theta \\;dh\\;", "0a93c280d6342c9f31eb89c18318cf48": "{\\widehat {\\widehat {\\sigma _{e}^{2}}}}", "0a93d5cb5658a362c2dd2959f37882ac": "{\\frac {d}{dt}}e^{tX}=Xe^{tX}=e^{tX}X.\\qquad (1)", "0a942588666e9613660d4008fb645435": "\\forall {f}{\\in }L_{2}(X)\\ f=\\sum _{i=1}^{\\infty }\\phi _{i}", "0a94586053aea0803c6c6568b297697b": "H(x)={\\tfrac {1}{2}}(1+\\operatorname {sgn}(x)).", "0a947e57051daae9c0686c36b9d8fad6": "\\theta \\operatorname {E} [Z]", "0a94a3e84aa8777c3777b07079dac68f": "{\\frac {287}{50}}=5.74", "0a94d0769182456bcbfc9adfaf254297": "0\\leq \\theta \\leq \\pi ", "0a94d21fc48e5b3dc98b40d792072125": "t(m-s+1)>{\\dfrac {N(m-s+1)+s(k-1)}{s+1}}", "0a94dcd9092e2d2fa316c8a63fa30572": "d_{VE}={{\\sqrt {6}} \\over 2}a={\\sqrt {3 \\over 2}}\\,a\\,", "0a95000e1f68f5f6cc9fbaccb19d5b1f": "e^{1/e}\\approx 1.444667", "0a952f495a10afbc4457fa7f8de85758": "a_{i}", "0a953d48bc6b465b4928c6660ec0589b": "g(\\epsilon )", "0a954ab13484b2e5d99c48734627c736": "3x+6=16-x", "0a95710a9af9fa8aeabdbeb961539268": "a\\cdot x^{3}+y^{3}+1=d\\cdot x\\cdot y", "0a95e114dfef3b0d87cb5cb7dc0e734f": "\\scriptstyle \\alpha _{\\Lambda }", "0a960d7a39e0e682207dad4f5d56ef5f": "(I\\ nat\\ 3\\ 4)\\to \\bot ", "0a96d8c7f8ed21ef9129691da22a7869": "T={\\frac {\\tan \\alpha }{4}}(b^{2}+c^{2}-a^{2})", "0a96f5ac8902687fc56cb291773c7f3c": "\\,\\sin(x+y)=\\sin x\\cos y+\\cos x\\sin y,\\,", "0a9735be087434f174ef639f54d07043": "B^{2}-4AC=0", "0a97a89f8c0697a75bd39cf54c005a92": "p_{2n}=p_{n}^{2}+S\\cdot q_{n}^{2}\\,\\!", "0a97a8c27855254be6692a23af223973": "[{\\dot {S}}]={\\begin{bmatrix}{\\dot {\\Omega }}&-{\\dot {\\Omega }}{\\textbf {d}}-\\Omega {\\dot {\\textbf {d}}}+{\\ddot {\\textbf {d}}}\\\\0&0\\end{bmatrix}}={\\begin{bmatrix}{\\dot {\\Omega }}&-{\\dot {\\Omega }}{\\textbf {d}}-\\Omega {\\textbf {V}}_{O}+{\\textbf {A}}_{O}\\\\0&0\\end{bmatrix}}", "0a97b1b4e725c505b64ed5d6f9b345e0": "_{s.1.left=s.8.right\\,}\\!", "0a97d621b275da59155513a87e101c48": "\\sigma _{i}=\\sigma _{0}", "0a981d7056c1b87c83ee1465ac52ac87": "(I\\rightarrow \\neg R)", "0a986c890e5f73e088e630438f5e69d9": "u(x,t)=\\sum _{n=1}^{\\infty }u_{n}(t)\\sin {\\frac {n\\pi x}{L}},", "0a987546c57d6264c4d1a3478e950009": "m_{n}\\ {\\stackrel {\\mathrm {def} }{=}}\\ {\\sqrt {\\frac {c}{2\\pi }}}\\int _{0}^{\\infty }{\\frac {e^{-c/2x}\\,x^{n}}{x^{3/2}}}\\,dx", "0a98acf7be52f18fb4ff6d5fe1dee96f": "V_{out2}", "0a98d6b4652616a19b039e886f937347": "{\\frac {\\Delta _{h}}{h}}~(x)_{n}=n~(x)_{n-1}~,", "0a98dfc5128dbef73fb03082c7cae8b7": "S^{\\prime }", "0a990cf78e97fe6d5421e6c1b3bb6199": "L[u]=\\det D^{2}u-f(\\mathbf {x} ,u,Du)=0\\qquad \\qquad (1)", "0a9926ab592a35a9227434228c772204": "{\\mathcal {D}}=\\{(x,y)\\in \\mathbb {R} ^{2}:y=x^{3}\\}\\cup \\{(x,y)\\in \\mathbb {R} ^{2}:y=0\\}\\ .", "0a99f33438b68b81e10f51f2c4399ade": "e_{1}^{2}=e_{2}^{2}=-1,\\ e_{1}e_{2}=-e_{2}e_{1}", "0a9a2322e8e851c5dfeb2ce16a171d54": "(z_{1},z_{2},z_{3})\\approx e^{i\\phi }(z_{1},z_{2},z_{3}).\\,\\!", "0a9a5bf39687a3dacfe506b4a8800934": "\\langle x|\\cdots |y\\rangle ", "0a9b1d8367a5eb48b97a66146f14798c": "\\omega _{\\text{res}}(\\theta )={\\frac {2\\pi c}{\\lambda _{\\text{res}}(\\theta )}}", "0a9b4e176f5fe418787c0b804e21bc35": "{\\mathfrak {g}}^{*}", "0a9b55a587051fd1b02048d5839f8b93": "C_{H}=[H]+[HA]", "0a9b98f4433f94fccad33fcc9d8d0cd0": "|i\\rangle |\\epsilon \\rangle ", "0a9bb6dce4060d8e63e8f6c92a03b820": "\\delta (\\mathbf {x} )", "0a9bf7c54060c195d2b9dcbdd8e24ade": "a_{1},a_{2},\\dots ", "0a9c7304273c4adc9c26d081641510a4": "v_{F}({\\mathfrak {D}}_{F/K})={1 \\over e_{L/F}}\\sum _{s\\not \\in H}i_{G}(s).", "0a9c7ef7e5f10053d6d922098854889c": "{\\frac {d}{dt}}\\left({\\frac {\\mathbf {r} }{\\Vert \\mathbf {r} \\Vert }}\\right)={\\frac {1}{{\\Vert \\mathbf {r} \\Vert }^{3}}}\\mathbf {r} \\left(\\mathbf {r} \\wedge {\\frac {d\\mathbf {r} }{dt}}\\right)={\\frac {1}{\\mathbf {r} }}\\left({\\hat {\\mathbf {r} }}\\wedge {\\frac {d\\mathbf {r} }{dt}}\\right)", "0a9c83410c31325de7d857efc927021a": "NEP={\\frac {S_{n}}{\\mathfrak {R}}}", "0a9c88a25944fa0301997e445fadcdb4": "e^{i\\omega t{\\boldsymbol {\\sigma }}\\cdot \\mathbf {\\hat {n}} }={\\begin{pmatrix}e^{i\\omega t}&0\\\\0&e^{-i\\omega t}\\end{pmatrix}}.", "0a9cb88f9137d01c15d27bba942872ab": "\\mathbf {Y} (s)=C\\mathbf {X} (s)+D\\mathbf {U} (s),", "0a9cfe58a4d927dce6c7c9de47ace4e9": "\\alpha _{0}", "0a9d1fce409fbef1d2c676ddcdcc1cf2": "c_{i},c_{j}", "0a9d5bfd22b632e8af15c30be5d1bf6e": "\\beta =\\,", "0a9d617b99bfd42b68fd099ebdd7d633": "M=(d\\colon H\\longrightarrow G)\\!", "0a9e1b7a4a58bcdae6480e3e812658e8": "d(f,u)={\\frac {1}{2}}d(f,g)+{\\frac {1}{2(r-2)}}\\left[\\sum _{k=1}^{r}d(f,k)-\\sum _{k=1}^{r}d(g,k)\\right]\\quad ", "0a9e26cd6e65f7315777ce26502fdfae": "E={\\frac {1}{2}}LI_{L}^{2}", "0a9e4c6d789326ff9579297a1c7a91c8": "\\aleph _{1}\\rightarrow (\\aleph _{1})_{2}^{\\aleph _{1}}", "0a9ea40761d743bbcf3a7fbc858d6c7e": "\\int (d+e\\,x)^{m}\\left(a+b\\,x+c\\,x^{2}\\right)^{p}dx={\\frac {(d+e\\,x)^{m+1}\\left(a+b\\,x+c\\,x^{2}\\right)^{p}}{e(m+1)}}\\,-\\,{\\frac {p(d+e\\,x)^{m+2}(b+2c\\,x)\\left(a+b\\,x+c\\,x^{2}\\right)^{p-1}}{e^{2}(m+1)(m+2p+1)}}\\,+\\,{\\frac {p(2p-1)(2c\\,d-b\\,e)}{e^{2}(m+1)(m+2p+1)}}\\int (d+e\\,x)^{m+1}\\left(a+b\\,x+c\\,x^{2}\\right)^{p-1}dx", "0a9edd6479d648f2c47364595502d970": "X_{v}=", "0a9f5fdc7c75be3ff3aea0cffaa8ef0f": "c^{2}=8.98755\\times 10^{16}\\,m^{2}sec^{-2}", "0a9f9049fd85001ece99c93c1e74f7b1": "2n_{\\rm {film}}d\\cos {\\big (}\\theta _{2})=m\\lambda ", "0a9f9dd57729ca20f6e3fded066ccc54": "{\\rho }_{f}\\propto a^{-3(1+w_{f})}\\,.", "0a9faac9096f735c3f42c9c14414aaac": "s\\;", "0a9fc72e1e67cda37754ef8bb00c1be5": "{\\frac {1}{\\sqrt {f}}}=1.1364\\ldots +2\\log _{10}(D_{\\mathrm {h} }/\\varepsilon )-2\\log _{10}\\left(1+{\\frac {9.287}{\\mathrm {Re} (\\varepsilon /D_{\\mathrm {h} }){\\sqrt {f}}}}\\right)", "0aa002e9418974bbe6ec275d784dbf8b": "b=(k-1)/3", "0aa02de685fc954cffa62e48f1dc7341": "\\,g_{ab}", "0aa0515be1376978b735ce4fae64f0be": "2^{1+4}", "0aa0882275dfdd9ce70b984d3f043f99": "p^{2}=A(r^{2}-a^{2})", "0aa0b66092a88ce2f3b0df720cbac3ae": "1\\;2\\;+", "0aa0f8474f5a966e62a5cb2e9dd42366": "\\sigma =E\\varepsilon ", "0aa10f27130cebfeb62737802a79fdff": "{\\begin{aligned}Q_{\\mathbf {u}}(\\theta )&{}={\\begin{bmatrix}0&-z&y\\\\z&0&-x\\\\-y&x&0\\end{bmatrix}}\\sin \\theta +(I-{\\mathbf {u}}{\\mathbf {u}}^{T})\\cos \\theta +{\\mathbf {u}}{\\mathbf {u}}^{T}\\\\&{}={\\begin{bmatrix}(1-x^{2})c_{\\theta }+x^{2}&-zs_{\\theta }-xyc_{\\theta }+xy&ys_{\\theta }-xzc_{\\theta }+xz\\\\zs_{\\theta }-xyc_{\\theta }+xy&(1-y^{2})c_{\\theta }+y^{2}&-xs_{\\theta }-yzc_{\\theta }+yz\\\\-ys_{\\theta }-xzc_{\\theta }+xz&xs_{\\theta }-yzc_{\\theta }+yz&(1-z^{2})c_{\\theta }+z^{2}\\end{bmatrix}}\\\\&{}={\\begin{bmatrix}x^{2}(1-c_{\\theta })+c_{\\theta }&xy(1-c_{\\theta })-zs_{\\theta }&xz(1-c_{\\theta })+ys_{\\theta }\\\\xy(1-c_{\\theta })+zs_{\\theta }&y^{2}(1-c_{\\theta })+c_{\\theta }&yz(1-c_{\\theta })-xs_{\\theta }\\\\xz(1-c_{\\theta })-ys_{\\theta }&yz(1-c_{\\theta })+xs_{\\theta }&z^{2}(1-c_{\\theta })+c_{\\theta }\\end{bmatrix}},\\end{aligned}}", "0aa126687e2a7320ba7e1f54a3e79a03": "\\mathbf {A} +(-1)\\mathbf {B} =(A^{0},A^{1},A^{2},A^{3})+(-1)(B^{0},B^{1},B^{2},B^{3})=(A^{0}-B^{0},A^{1}-B^{1},A^{2}-B^{2},A^{3}-B^{3})", "0aa13b137923a11b7abe08075780a0c1": "K_{M}^{N+2}", "0aa1d09f0db66f42459ec42124f727c8": "\\epsilon \\circ \\nabla =\\nabla _{0}\\circ \\epsilon _{2}:(B\\otimes B)\\to K", "0aa1eabb9b373d56901ceefbfcdbf7cd": "\\int _{a}^{b}{\\frac {d}{dx}}\\left(u(x)v(x)\\right)\\,dx=\\int _{a}^{b}u'(x)v(x)\\,dx+\\int _{a}^{b}u(x)v'(x)\\,dx", "0aa23d614ac0c1ffdf58b658286dd862": "j^{\\star }=\\varepsilon \\sigma T^{4}", "0aa261c140856613b91c3b65582470a8": "a/b.\\,", "0aa267ba142027a329ab28c424c75c06": "2^{n-2}\\equiv 1{\\pmod {n}}\\,\\!", "0aa270e347c65525faa1bb7772a8383e": "{\\frac {A{\\hbox{ prop}}\\qquad B{\\hbox{ prop}}}{A\\vee B{\\hbox{ prop}}}}\\ \\vee _{F}\\qquad {\\frac {A{\\hbox{ prop}}\\qquad B{\\hbox{ prop}}}{A\\supset B{\\hbox{ prop}}}}\\ \\supset _{F}\\qquad {\\frac {\\hbox{ }}{\\top {\\hbox{ prop}}}}\\ \\top _{F}\\qquad {\\frac {\\hbox{ }}{\\bot {\\hbox{ prop}}}}\\ \\bot _{F}", "0aa27d946056cdd21388d7cd18a9f969": "\\sum _{n=1}^{\\infty }{\\frac {1}{2n+1}}[\\zeta (n)-1]=1-\\gamma -{\\frac {1}{2}}\\ln 2.", "0aa283b20ca498410b8cd10e5fbcd42a": "D_{4}", "0aa2ebfdc74934ffe8b784eb6b5df847": "\\langle v_{i}\\rangle ", "0aa318c99e04acabc52e39ef2f1700f2": "\\textstyle R_{k+qm}=R_{k}", "0aa36281330671a817e747f6271edbd1": "\\{\\langle X_{1},X_{2},....,X_{n}\\rangle \\mid p(\\langle X_{1},X_{2},....,X_{n}\\rangle )\\}", "0aa3a9d67938be9eb0691442c674b5e0": "\\,r=\\sin(k\\theta )", "0aa41bd74e4bd77649484a2c86037926": "{\\mathcal {E}}=\\oint _{C}{\\boldsymbol {\\left[E+v\\times B\\right]\\cdot }}d{\\boldsymbol {\\ell }}\\ ", "0aa43d06971528b73dd777b39d508cd7": "\\delta _{x}", "0aa460b5aef85265cc24d65df0458c2a": "x_{k}(\\zeta )=\\Re \\left\\{\\int _{0}^{\\zeta }\\varphi _{k}(z)\\,dz\\right\\}+c_{k},\\qquad k=1,2,3", "0aa48d0248dcbaf032405853d1ebf0ac": "e_{x}=E_{x}-B", "0aa4b777716a125e6154c748ceb06760": "R_{W}={\\frac {\\rho _{E}}{2\\cdot \\pi \\cdot a_{W}}}\\,", "0aa505623e91b2f462841c46926f5cfe": "\\lfloor k\\rfloor ", "0aa505ea75841dbbecff1a050bc871ab": "f:\\mathbb {R} ^{N}\\rightarrow \\mathbb {R} ", "0aa56aaa257aeb04de384bf8dfb1950f": "{\\begin{aligned}F_{\\vec {r}}(p)dp&={\\frac {4\\pi p^{2}dp}{{\\frac {4}{3}}\\pi p_{f}^{3}({\\vec {r}})}}\\qquad \\qquad p\\leq p_{f}({\\vec {r}})\\\\&=0\\qquad \\qquad \\qquad \\quad {\\text{otherwise}}\\\\\\end{aligned}}", "0aa56d842fa681d0a6781ed3d64113e8": "{\\underline {\\lnot \\lnot \\varphi }}\\,\\!", "0aa572144bed5ebd56399e1885058a0c": "w_{it}", "0aa5a728821daf1bb1b22e48ea6dd74b": "Q(\\xi ,\\eta )=\\int \\omega _{b}^{n-k}\\wedge \\xi \\wedge \\eta .", "0aa5b58fc191f2a686da3948ce651e8b": "T(r,t)", "0aa5d573ac848216e8757642c45b3d06": "i\\in \\{1,...,N\\}", "0aa618eb29ee0d363381f618364a87db": "\\lambda _{1}=1\\,\\!", "0aa64b4d34c62565dedba0a3cc00ad56": "\\langle \\rangle ", "0aa66bb3c56357d702290d7492977fdc": "\\textstyle {\\frac {\\partial C_{2}}{\\partial t}}", "0aa673048938f656942ba7966bd4eb7e": "Usage(time)/Lifetime(time)", "0aa6a6089cef1d17c9169364a245f6a8": "bRc", "0aa6d88d57434b58f1ec59a9b300b593": "\\sum _{n=-\\infty }^{\\infty }s(nT)\\cdot \\delta (t-nT)=\\underbrace {\\int _{-\\infty }^{\\infty }{\\tfrac {1}{T}}\\ S_{1/T}(f)\\cdot e^{i2\\pi ft}\\,df} _{\\text{inverse Fourier transform}}\\,", "0aa6e9efdaef42043ddd1cfd7a7570ad": "(E-mc^{2})\\psi _{+}={\\frac {1}{2m}}\\left[{\\boldsymbol {\\sigma }}\\cdot \\left(\\mathbf {p} -{\\frac {e}{c}}\\mathbf {A} \\right)\\right]^{2}\\psi _{+}+e\\phi \\psi _{+}", "0aa6fb1141586b1c9c383b08eea3cb19": "R\\simeq {\\frac {R_{1}}{1-A}}", "0aa75fd6e89c9fff43198d93dd0ce521": "x[v_{1}\\otimes v_{2}]=x[v_{1}]\\otimes v_{2}+v_{1}\\otimes x[v_{2}].", "0aa76511f8ea47f6099088bb89f41350": "{{\\frac {\\Delta S}{R}}={\\frac {\\gamma }{\\gamma -1}}\\ln \\left({\\frac {2}{\\left(\\gamma +1\\right)M_{x}^{2}}}+{\\frac {\\gamma -1}{\\gamma +1}}\\right)+{\\frac {1}{\\gamma -1}}\\ln \\left({\\frac {2\\gamma }{\\gamma +1}}M_{x}^{2}-{\\frac {\\gamma -1}{\\gamma +1}}\\right)}", "0aa781ca2b9ef42658866b6f4202e8e6": "\\int _{S}f\\,dS=\\iint _{T}f(\\mathbf {x} (s,t))\\left|{\\partial \\mathbf {x} \\over \\partial s}\\times {\\partial \\mathbf {x} \\over \\partial t}\\right|\\,ds\\,dt", "0aa78426c5b6a66a12015e8c45a199e6": "{\\text{Tr}}\\left\\{\\Lambda \\rho \\right\\}\\leq {\\text{Tr}}\\left\\{\\Lambda \\sigma \\right\\}+\\left\\Vert \\rho -\\sigma \\right\\Vert _{1}.", "0aa7c74d2c060853eb0c6fa511f1795d": "M\\mapsto M_{\\mathfrak {g}}:=M/{\\mathfrak {g}}M.", "0aa7d7c2b205c78e1885a02371db2f90": "{\\hat {C}}", "0aa7dae70ca29f652d842d18b360e397": "A\\Delta B", "0aa82b70834da9cbe82c3cb48547a35f": "B_{n}", "0aa8725509013ca63cd2925b4642cae1": "d_{yz}", "0aa8d34dc55a888709ee80973c870293": "\\ F_{forward}=drag", "0aa8f5a09588e383fd454d26a61841c1": "E=1-{F\\,'_{\\rm {D}}}/{F_{\\rm {D}}}\\!", "0aa91177aa774b550375479c883d5055": "L_{g}=N\\Lambda \\,", "0aa914f6f9f119c85099c1b830a85470": "{\\begin{pmatrix}x\\\\y\\end{pmatrix}}\\mapsto {\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}}{\\begin{pmatrix}x\\\\y\\end{pmatrix}}={\\begin{pmatrix}ax+by\\\\cx+dy\\end{pmatrix}}.", "0aa91cfe48cb3947e4ea10ed12d22c87": "e^{2}=[7;2,1,1,3,18,5,1,1,6,30,8,1,1,9,42,11,1,1,12,54,14,1,1\\dots ,3k,12k+6,3k+2,1,1\\dots ]\\,\\!.", "0aa92911dc8219c92909933f6c71482e": "\\scriptstyle n\\geq 2", "0aa9cb87647412672a8dd65c5ced873b": "Z_{\\mathrm {sun} }=0.02", "0aa9e64de9dc0e93ca03f21a48f6ee05": "(\\leftarrow /)\\quad {Z\\leftarrow \\Delta Y\\Delta '\\qquad X\\leftarrow \\Gamma \\over Z\\leftarrow \\Delta (Y/X)\\Gamma \\Delta '}", "0aaa10ceadf13196a2cfebd2223b1468": "00\\,", "0ab17fc73bca257d46f02f7379e36f7d": "Q_{Actual}=Q_{Fan}*{\\sqrt {\\rho _{Ref} \\over \\rho _{Actual}}}\\,\\!", "0ab19a1c58a36156bbb4c0300ad21cf9": "W(C;1,0)=A_{n}=1{\\mbox{ iff }}(1,\\ldots ,1)\\in C\\ {\\mbox{ and }}0{\\mbox{ otherwise.}}", "0ab1cbb034fc7fdede7b3a8856b87e52": "{\\frac {dX_{i}}{dt}}=\\sum _{j}\\mu _{ij}\\cdot \\gamma _{j}\\prod _{k}X_{k}^{f_{jk}}\\,", "0ab22dfd8e88a3a1b5d793bdfee0e7d4": "{\\frac {d}{dt}}\\langle \\mathbf {p} (t)\\rangle =q\\mathbf {E} -{\\frac {\\langle \\mathbf {p} (t)\\rangle }{\\tau }},", "0ab23d4889c810546d523bdf5d8248f0": "X_{C}={1 \\over j\\omega C}={1 \\over j2\\pi fC}\\ ;\\ X_{L}=j\\omega L=j2\\pi fL", "0ab24aaa993c96392f6ee45a11e4f28d": "A_{v}|\\psi \\rangle =|\\psi \\rangle ,\\,\\,\\forall v,\\,\\,B_{p}|\\psi \\rangle =|\\psi \\rangle ,\\,\\,\\forall p,", "0ab24ca116a09a877f37060009993050": "\\varepsilon _{t}", "0ab268dacd2dbfdb07ae780787be79bb": "\\mathbb {D} ^{q}\\mathbb {D} ^{-q}=\\mathbb {I} ", "0ab277d305236c1f5274035ba1309cba": "\\because \\!\\,", "0ab2b8020f296fb27c395434c8a47cc1": "x={\\sqrt {y}}", "0ab2c1f8b698090755718ab622e6d685": "\\pi _{i}(S^{n})\\otimes \\mathbb {Q} ", "0ab2c4719593d2b5fb026bbe28c3e916": "\\eta :S\\to T", "0ab2c4cd9fceb89ac2ffd80b5b39e1e1": "e_{0}^{2}=e_{1}^{2}=1", "0ab2fabfc07ab3a98e7dc3c7ab92926c": "\\kappa {\\Big .}", "0ab31a5fed53d96bc37cd22f8061a09a": "f_{1}({\\text{AA}})=p^{2}=f_{0}({\\text{A}})^{2}", "0ab31be6a0962a3aed1868ef918a2cab": "\\sum _{k=0}{\\frac {B_{k}(x)}{z^{k}}}{\\frac {1-s \\choose k}{s-1}}=z^{s-1}\\zeta (s,x+z),", "0ab3ca7f3234bec5859ba72752227c05": "\\int f(x)d_{h}x", "0ab3f6c138cd8032ac51d26763b2bac8": "p(x|y)", "0ab3f82999513e36de6f4b0dbf5e81e0": "\\textstyle W(y|x,s)", "0ab4652fd102e74f7e19bc45fba4a79a": "d\\left({{\\vec {x}};{\\vec {y}}}\\right)=r\\left({{\\vec {x}}-{\\vec {y}};q}\\right)", "0ab492f6cbc2ecfa64374d88f7755a66": "{\\textrm {coversin}}(\\theta ):={\\textrm {versin}}\\!\\left({\\frac {\\pi }{2}}-\\theta \\right)=1-\\sin(\\theta )\\,", "0ab4d99d9556c4727ac5b865641d1144": "\\sum _{n=0}^{\\infty }H_{n}^{(r)}z^{n}=-{\\frac {\\ln(1-z)}{(1-z)^{r}}}.", "0ab5276ab725610bcf839e332f305313": "\\ f_{2,M}=exp(Ax_{1}^{2})\\,", "0ab52d27d1ea9bfd8c482872fb504761": "=1-\\alpha x\\gamma +(1/2)(\\alpha x)^{2}(\\gamma +1)\\gamma -...\\,", "0ab5516223f24f0779e99a66212e9bf9": "h={\\frac {3.5{\\hat {\\sigma }}}{n^{1/3}}},", "0ab55a10a44a3f243d05b5d6a7a9a3fb": "n=1,", "0ab6334bf1d68abf09a3756cbac5f4ae": "\\log V^{2}=2\\log V", "0ab652abbab16f55bb79c87e548bd85a": "{\\boldsymbol {\\sigma }}={\\begin{bmatrix}\\sigma _{1}&\\sigma _{2}&\\sigma _{3}&\\sigma _{4}&\\sigma _{5}&\\sigma _{6}\\end{bmatrix}}^{T}\\equiv {\\begin{bmatrix}\\sigma _{11}&\\sigma _{22}&\\sigma _{33}&\\sigma _{23}&\\sigma _{31}&\\sigma _{12}\\end{bmatrix}}^{T}.\\,\\!", "0ab654034920736609f88f614b9ce9f1": "\\Omega (n^{1/3})", "0ab65726c4090804a96edeaf6e93fa78": "f_{k}=\\sum _{j=0}^{n-1}v_{j}e^{{\\frac {-2\\pi i}{n}}jk}.", "0ab66fe892087d10e4b4ee1b1d98ec10": "f(n_{i})=\\ln(W)+\\alpha (N-\\sum n_{i})+\\beta (E-\\sum n_{i}\\varepsilon _{i})", "0ab6a58d85c4548fe3c88935db2395af": "\\scriptstyle {V_{c}}", "0ab6a98da3418135a067dad824ddc9ae": "\\mathbb {E} [R_{P}]=\\sum _{i=1}^{n}x_{i}\\mathbb {E} [R_{i}]", "0ab78ad8cc490342fe04b15966c38f54": "1093=1111111_{3}=3^{6}+3^{5}+3^{4}+3^{3}+3^{2}+3^{1}+3^{0}\\,.", "0ab7fe497d59ec59b94eb2ea4205a9d1": "{\\mathcal {L}}={\\tfrac {1}{2}}\\ \\partial ^{\\mu }{\\hat {n}}\\cdot \\partial _{\\mu }{\\hat {n}}", "0ab81192bbc2616a6f8007e05cb82572": "\\nu =\\rho \\neq \\mu ", "0ab8520530f4ccebdb0a0f095263d5f7": "\\mathbf {\\bar {y}} =\\mathbf {T} \\,\\mathbf {y} ", "0ab8b46d861c347adde3545bd0ffd8c1": "t'\\equiv 1{\\pmod {p}}", "0ab928ed87d8bdde337ecba2a7b2a5cd": "\\tau =T-t", "0ab979249bba64a40d6f296b462c09d3": "z_{i}=e^{i\\theta _{i}}=\\cos(\\theta _{i})+i\\sin(\\theta _{i})", "0ab98bc4f818e112bb3e48b331513a81": "{\\begin{aligned}Q(AC)&\\equiv (C_{x}-A_{x})^{2}+(C_{y}-A_{y})^{2}\\\\&=((b\\lambda \\ +A_{x})-A_{x})^{2}+((-a\\lambda \\ +A_{y})-A_{y})^{2}\\\\&=(b\\lambda \\ +A_{x}-A_{x})^{2}+(-a\\lambda \\ +A_{y}-A_{y})^{2}\\\\&=(b\\lambda )^{2}+(-a\\lambda )^{2}\\\\&=b^{2}\\lambda ^{2}+(-a)^{2}\\lambda ^{2}\\\\&=b^{2}\\lambda ^{2}+a^{2}\\lambda ^{2}\\\\&=(a^{2}+b^{2})\\lambda ^{2}\\end{aligned}}", "0ab9dc6cc662d678f58b6cb2d3410560": "H=\\sum _{k}{{\\dot {q}}_{k}}p_{k}-L", "0ab9fe1ed9e288a89499d4b770c0b083": "\\left|\\langle A|\\Psi (t)\\rangle \\right|^{2}", "0aba042963dddbddc3ccc114ab9f6f13": "\\lambda _{i}\\neq 0\\;\\forall \\,i", "0aba1ae1cc808cfd1b2b910fcd97e4b8": "{\\overset {\\circ }{\\boldsymbol {\\sigma }}}=J^{-1}~{\\dot {J}}~{\\boldsymbol {\\sigma }}+{\\boldsymbol {F}}\\cdot {\\dot {{\\boldsymbol {F}}^{-1}}}\\cdot {\\boldsymbol {\\sigma }}+{\\dot {\\boldsymbol {\\sigma }}}+{\\boldsymbol {\\sigma }}\\cdot {\\dot {{\\boldsymbol {F}}^{-T}}}\\cdot {\\boldsymbol {F}}^{T}", "0aba2480809eb0eb04bf4e87f2bbbd3b": "{\\frac {G}{N}}={\\frac {G}{N}}^{\\circ }+kT\\ln {\\frac {p}{p^{\\circ }}}", "0aba3e92d95491df32aff9dc2ca23577": "\\log _{2}k", "0aba57f2b2ada99e67ec0704ee05340f": "~\\Phi _{4}(x)=x^{2}+1", "0aba5b1ef550dba0157bd45062077bca": "u(x)=\\int \\limits _{\\partial B(x,r)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-\\,u(y)\\,\\mathrm {d} S(y).", "0abaf6d8fcb1c5a5d0705d44f0b61c9c": "y_{2},\\dots ,y_{k}", "0abb5cf9d9de43ae7b97d69b767281b5": "H_{L}=10\\times \\log _{10}(A_{510}/A_{610})", "0abbd0b00b2ce865975fece655a362bf": "L^{\\infty }([0,1]\\cup \\mathbf {N} ).", "0abbff2aef73d6ac0227476b9763fa0b": "X^{\\beta }:=\\{x\\in X\\mid \\sum _{i=1}^{\\infty }x_{i}y_{i}<\\infty \\quad \\forall y\\in X\\}.", "0abc0c9c6c047030a0f21bad8ac9c59e": "z_{0}^{n}{\\overline {p(1/{\\bar {z_{0}}})}}=z_{0}^{n}{\\overline {p(z_{0})}}=z_{0}^{n}{\\bar {0}}=0", "0abc436f9f35683683660cf5ddeafa26": "f_{X}(x)={\\begin{cases}{\\frac {1}{24}}x^{4}&0\\leq x\\leq 1\\\\{\\frac {1}{24}}\\left(-4x^{4}+20x^{3}-30x^{2}+20x-5\\right)&1\\leq x\\leq 2\\\\{\\frac {1}{24}}\\left(6x^{4}-60x^{3}+210x^{2}-300x+155\\right)&2\\leq x\\leq 3\\\\{\\frac {1}{24}}\\left(-4x^{4}+60x^{3}-330x^{2}+780x-655\\right)&3\\leq x\\leq 4\\\\{\\frac {1}{24}}\\left(x^{4}-20x^{3}+150x^{2}-500x+625\\right)&4\\leq x\\leq 5\\end{cases}}", "0abc5149e2921fb41c2ffc3e22041930": "{\\dfrac {d^{\\frac {1}{2}}}{dx^{\\frac {1}{2}}}}2\\pi ^{-{\\frac {1}{2}}}x^{\\frac {1}{2}}=2\\pi ^{-{\\frac {1}{2}}}{\\dfrac {\\Gamma (1+{\\frac {1}{2}})}{\\Gamma ({\\frac {1}{2}}-{\\frac {1}{2}}+1)}}x^{{\\frac {1}{2}}-{\\frac {1}{2}}}=2\\pi ^{-{\\frac {1}{2}}}{\\dfrac {\\Gamma ({\\frac {3}{2}})}{\\Gamma (1)}}x^{0}={\\dfrac {2{\\sqrt {\\pi }}x^{0}}{2{\\sqrt {\\pi }}0!}}=1,", "0abc701af9cfd1e2ba8ab6f1e06bcb83": "f\\in (0,1]", "0abc8da009882f9bc53c674c9b3700fd": "\\,A", "0abc8f40d3557a10146995de647b4e92": "CG=V_{rt}/Q_{rt}", "0abcc20f29c5824d20220cba6b4124cf": "\\mathbf {v} =v'^{a}\\mathbf {e} '_{a}=v^{b}(R^{-1})_{b}^{a}R_{a}^{c}\\mathbf {e} _{c}=v^{b}\\mathbf {e} _{b}.", "0abcc9598ea4cfb8a14666566cef04c4": "\\sigma (A)", "0abccdeb306ac58bca84a20aae3b6314": "\\operatorname {NWScore} ", "0abccdf17d169d9ce64ab640a90a8ecb": "t=m_{1}^{2}+m_{3}^{2}-2p_{1}\\cdot p_{3}\\,", "0abcd7271f7e8889e06d8476b46ddab1": "A=\\pi r(2h+a)\\;.", "0abd863cfdd959caa83b6068b6b000fc": "\\{x_{i}-b_{i}\\}", "0abde827b4af407fe765bd03cfe196f5": "X^{L}", "0abdfa3e821850c47c87bf08ab4cac16": "R=\\sigma ^{2}I", "0abe4605e6740e35d7daeccc1e4f00c2": "\\mu (X\\,\\triangle \\,Y)=0", "0abe7ba39b1d683718b7c084530e24cb": "A(x)>c{\\sqrt[{3}]{x}}", "0abe80eceafe12537afc61f6b61330dd": "{\\breve {\\theta }}_{j,i}", "0abe9ad771462462c459bec7c0671601": "L(k_{2})=32\\left\\lceil l/16\\right\\rceil +40", "0abee1df9e40d481a91c2e9d80897141": "{\\vec {v}}(t+{\\frac {dt}{2}})={\\vec {v}}(t-{\\frac {dt}{2}})+{\\vec {F}}(t)\\,dt", "0abeec726e06d5d11b456075fac1ffeb": "R_{1}+R_{3}={\\frac {R_{b}(R_{a}+R_{c})}{R_{T}}}.", "0abf02f1817477d9f300bc0176df38fe": "{\\begin{bmatrix}0&1\\\\0&-1\\end{bmatrix}}:\\mathbf {b} ", "0abf23ff097df4c2775f0e5d365d0867": "-2\\ln(LR)=\\textstyle -2\\sum _{i=1}^{k}x_{i}\\ln(\\pi _{i}/p_{i}).", "0abf379a9400e41ff5f59cf92afc12fc": "{\\vec {F_{a}}}\\cdot {\\vec {x}}=F_{a}\\,x.", "0abf7646f0446cd16f1f40e0315399f4": "S={V^{1.85}A^{1.85} \\over k^{1.85}\\,C^{1.85}\\,R^{1.17}\\,A^{1.85}}={Q^{1.85} \\over k^{1.85}\\,C^{1.85}\\,R^{1.17}\\,A^{1.85}}", "0abf86882f5fb08f2528e412a3038d47": "f:{\\mathcal {P}}_{=n}(\\kappa )\\to \\{0,1\\}", "0abfa74cbcb7d42716d0250cc31ed341": "C=B\\log _{2}\\left(1+{\\frac {S}{N}}\\right)\\ ", "0abfd5a1bd50bac038306ec372049bb1": "p_{1}=-p_{2}", "0abfdc7c810161cb10ca3eeb0ab4ea1d": "b={\\frac {n_{solute}}{m_{solvent}}}", "0abfe9cf38ade7c5ce88b1eaebf4616f": "\\operatorname {dim} R\\geq \\delta (R).", "0ac0475c8557d20fcb473088f789b720": "x^{p^{n}}\\in F", "0ac0521677fdeac70934d8e03fa19a64": "{\\begin{aligned}P_{t+\\Delta t}&=P_{t}+(rP_{t}-M_{N})\\Delta t\\\\[12pt]{\\dfrac {P_{t+\\Delta t}-P_{t}}{\\Delta t}}&=rP_{t}-M_{N}\\end{aligned}}", "0ac0a1e3154c59f80839598aca5acbd7": "H(S,P)\\,", "0ac11eba32233953849b5da16a58f8e5": "p\\leq 2", "0ac14e7ca95955736a136a4830f2781c": "\\mathbf {p} _{k}=-B_{k}^{-1}\\nabla f(\\mathbf {x} _{k})", "0ac1a7ccf1ccf6a88d07e9e6866da2c6": "R\\bowtie S:=\\{f\\mid f\\quad (x\\cup y){\\hbox{-tuple}},\\quad f[x]\\in R,\\;f[y]\\in S\\}.\\,", "0ac2253736858e5bfb08f67a1f7bd58b": "\\nabla \\!\\,", "0ac2662c42f9ae476827ec285805a274": "K^{\\prime }(x)=e^{-2u}(K(x)-\\Delta u),", "0ac2740199bc183cf310066f7c28cd71": "\\mathrm {l} ", "0ac2a46e7711261101eea324418915ff": "n=8", "0ac2a50c48210c0fa7da24740f88b0bd": "C=\\int \\Phi _{a}(r_{1})^{2}\\left({\\frac {1}{R_{ab}}}+{\\frac {1}{r_{12}}}-{\\frac {1}{r_{a1}}}-{\\frac {1}{r_{b2}}}\\right)\\Phi _{b}(r_{2})^{2}\\,dr_{1}\\,dr_{2}", "0ac2adcf642727c46385a5ba43c4e9d9": "g(e)=(g_{1}(e),\\ldots ,g_{k}(e))", "0ac2c4bc51028124710747b9983fbaa3": "[v,v']\\in \\Gamma (T)", "0ac340c58b8c0cd2b02f256f5faedc90": "\\max(a)={\\frac {1}{A}}\\int _{A}\\left\\lbrace \\max _{j}P\\left[j|{\\vec {y}}({\\vec {r}})\\right]\\right\\rbrace \\,d{\\vec {r}}", "0ac3c341e931a390385440690571c8e2": "\\operatorname {ad} _{\\mathfrak {g}}x", "0ac3c4ca3ed07532181422d15886b71b": "{\\frac {\\partial \\psi }{\\partial t}}+{\\frac {\\partial ^{3}\\psi }{\\partial x^{3}}}-3(u-\\lambda ){\\frac {\\partial \\psi }{\\partial x}}=C\\psi +D\\psi \\int {\\frac {dx}{\\psi ^{2}}}", "0ac44265bdd3bf7252067cceed598b1f": "x^{2}-ny^{2}=-1.\\,", "0ac452807ab0097a402dc73d594f0ec3": "\\sum _{n=1}^{\\infty }n^{-p},", "0ac473fc014fcd1891811192855b759c": "f_{\\sim }(x)=1-x,", "0ac55db71f8b7290a038d3c72fa574d3": "f(x)={\\frac {{\\sqrt {x}}+{\\sqrt {\\frac {1}{x}}}}{2\\gamma x}}\\phi \\left({\\frac {{\\sqrt {x}}-{\\sqrt {\\frac {1}{x}}}}{\\gamma }}\\right)\\quad x>0;\\gamma >0", "0ac56a500561c30cac86bc43d460dd0c": "He(2^{3}S)+H_{2}O\\to H_{2}O^{+\\bullet }+He(1^{1}S)+e^{-}", "0ac573a76cef06d901ac68f961535fff": "u(0)=0,", "0ac592f604247c87b2029467e777cf78": "R={\\sqrt {x^{2}+y^{2}+z^{2}}}", "0ac59dee686571fd9ee649da40104aa7": "f(\\pi (X))=\\pi (f(X))=f(X)\\ {\\bmod {\\ }}f(X)=0", "0ac5ab8967850137b6c5158beca7bada": "P(y)\\ ", "0ac5fd87a03f69428cbdf790c11d7a80": "\\textstyle \\varphi ={\\frac {1+{\\sqrt {5}}}{2}}", "0ac672a1322969bd6147fc2fd816893c": "n\\$=(n!)\\uparrow \\uparrow (n!).\\,", "0ac6b5246d4da4e79135e76b5eaf96e5": "v'=-h-v\\mod u'", "0ac6ef0d8fa553cb3a0fcb9ec32a23d3": "\\{1,\\alpha ,\\cdots ,{\\alpha }^{n-1}\\}", "0ac6f910c5b3ea8272dad86670481ce5": "R_{1}=R/f_{1}R", "0ac727eafb8be039ae8da9a8a77eed76": "V\\in \\mathbb {C} ^{n,n}", "0ac73ece2591b7ddc5158def8b45137e": "v(s)={\\frac {\\mathrm {d} s}{\\mathrm {d} t}}\\ .", "0ac75f2375edd1cdde6c04189d334bf9": "{\\frac {R}{2}}={\\frac {a}{6}}{\\sqrt {3}}\\!\\,", "0ac764a152cd8addee84feecdfd53b50": "{\\frac {\\partial {(\\rho T)}}{\\partial t}}+{div\\,(\\rho uT)}={div\\,(k\\,grad\\,T)}", "0ac7a9bea3b54a075f8b578330a6e6db": "\\mathbf {S} ={\\begin{bmatrix}0&\\tau &\\kappa &0\\\\\\tau &0&0&\\kappa \\\\\\kappa &0&0&\\tau \\\\0&\\kappa &\\tau &0\\end{bmatrix}}", "0ac7c564084ad99b4ee431f67f2af6c6": "f_{\\psi (\\varepsilon _{\\Omega +1})}(n)", "0ac7f3d784bd054e6971b30ac1e1251a": "AT=TJ", "0ac8143003d388fd0ef1e531da66b31d": "\\lambda _{z,n}", "0ac83a05f1717c5f689e22d9b96476b2": "{\\bar {x}}={\\frac {1}{n}}\\sum _{i=1}^{n}x_{i}={\\frac {1}{n}}(x_{1}+\\cdots +x_{n})", "0ac8d8837a8bd760f91fed392f9dc9c6": "(\\alpha \\neq \\beta )", "0ac928f63faf51766bad45cbd4984295": "\\pi (x)-\\pi \\left({\\sqrt {x}}\\right)+1", "0ac9f1b0c58e3860c196d17efa806bcc": "\\exp \\left(-(x-2\\theta )^{2}/4\\right).", "0aca355f8b843e21cb17e7c768edab18": "\\scriptstyle 6.283\\,185\\,307\\,179\\,586\\,476", "0aca76669b779dd4a3db42245be82b30": "a_{i,n}=\\left[{\\frac {n-\\sum _{j=0}^{i-1}k_{j}+[L/2]-1-i}{[L/2]-i}}\\right];i>0,", "0aca9c0a69b59c541a40a4e48e52077c": "n<1\\,", "0ad0e36e3613657408c4d6b739302693": "\\mathrm {vec} (A)=[a_{1,1},...,a_{m,1},a_{1,2},...,a_{m,2},...,a_{1,n},...,a_{m,n}]^{T}", "0ad13033291025cbcb017455d940a878": "H_{\\omega }(1)-1", "0ad193a4119b84ddf0274386f762faa6": "\\Delta =\\nabla \\cdot \\mathbf {v} ", "0ad1a0ea11728a00f81a93d254d466c4": "p_{j}\\,", "0ad246dae821be375536d8ecc4fd3b30": "E_{rot}={\\frac {l(l+1)\\hbar ^{2}}{2\\mu r_{0}^{2}}}\\ \\ \\ \\ \\ l=0,1,2,...\\,", "0ad2f3cf3bc80559d3b96d9151d7d481": "V^{\\pi }(s)=E[R|s,\\pi ],", "0ad2f6d09ade22978c25bb174f3122a9": "\\operatorname {Trace} (\\rho _{f}(\\operatorname {Frob} _{p}))=a_{p}\\ ", "0ad37bfccd3b49545be5ed21871119c8": "n_{\\mathrm {i} }", "0ad3b74694389e4dc6b9faf0477d359d": "|n|_{p}=p^{-v_{p}(n)}", "0ad3fc8da420819e2173a3de1a335668": "R^{1/2}\\,", "0ad4548531b9ee1545b5eb7827313825": "{\\dot {x}}=b-x/\\tau +{\\sqrt {Dx}}\\xi (t)", "0ad461bd015a1320ef9c9b53ad032490": "\\,x=\\theta /2\\pi \\,", "0ad48f3dc6536fc08529e4289fd28941": "T={\\tfrac {1}{2}}ab", "0ad492cef6b429ca87eb038e54617396": "{\\begin{aligned}\\left|\\int _{C}f(x)e^{\\lambda S(x)}dx\\right|&\\leqslant \\int _{C}\\left|f(x)\\right|\\left|e^{\\lambda S(x)}dx\\right|\\equiv \\int _{C}\\left|f(x)\\right|e^{\\lambda M}\\left|e^{\\lambda _{0}[S(x)-M]}e^{(\\lambda -\\lambda _{0})(S(x)-M)}dx\\right|\\\\&\\leqslant \\int _{C}\\left|f(x)\\right|e^{\\lambda M}\\left|e^{\\lambda _{0}[S(x)-M]}dx\\right|=\\underbrace {e^{-\\lambda _{0}M}\\int _{C}\\left|f(x)e^{\\lambda _{0}S(x)}dx\\right|} _{\\text{const}}\\cdot e^{\\lambda M}.\\end{aligned}}", "0ad4966f96e4977e245ce2fe7e0dff81": "\\tau :\\;V\\otimes V\\longrightarrow V\\otimes V\\,", "0ad4d55cda7cac30484bb00deacd7253": "i^{i}=\\left(e^{i(\\pi /2+2k\\pi )}\\right)^{i}=e^{i^{2}(\\pi /2+2k\\pi )}=e^{-(\\pi /2+2k\\pi )}", "0ad5110e61d07cd15d0e7ff5c8e20b50": "m:I\\to F^{*}J", "0ad549abfc6ed3f68e2b890d96c0fc58": "ax^{2}+2bxy+cy^{2}", "0ad54a6de1d35b7484afaf01d8c1ebfa": "{\\frac {\\partial ^{3}\\varphi _{1}}{\\partial x_{1}^{3}}}+{\\frac {\\partial ^{3}\\varphi _{1}}{\\partial x_{1}\\partial x_{2}^{2}}}+{\\frac {\\partial ^{3}\\varphi _{2}}{\\partial x_{1}^{2}\\partial x_{2}}}+{\\frac {\\partial ^{3}\\varphi _{2}}{\\partial x_{2}^{3}}}=-{\\frac {q}{D}}\\,.", "0ad574c599c118a4d1a3aea85c14c627": "\\mathrm {Re} _{m}=\\mathrm {Re} \\;", "0ad5898a1dc24678b6d0646be3695306": "{\\hat {\\mathbf {x} }}_{i}'", "0ad5a5795233954acfdfed9a5275e070": "q=-k{\\frac {\\partial T}{\\partial r}}", "0ad5ac4713db38888edd4381d2cd724d": "0\\neq 1", "0ad5d68eca854d52a82faf85fc094ddc": "\\left|{\\overline {z}}\\right|=\\left|z\\right|", "0ad6195ac2881486c287c4b96340561a": "r_{1},\\;r_{2}\\geq 0", "0ad6223bdd76dffe58cf64b43882d742": "A\\to A", "0ad66224f1d204b6bb1a6f9dcfe8c91b": "Var(A)=f(bb)a_{bb}^{2}+f(Bb)a_{Bb}^{2}+f(BB)a_{BB}^{2},", "0ad663559a6e673a15b7ee91d4451f29": "F_{c}=-3\\gamma \\pi R\\,", "0ad666ceea810cbb9b46e5e602c1b7f3": "\\left(\\lambda +1\\right)/n", "0ad6836c5749718d50ac958a6b474df4": "F=1-{\\frac {\\operatorname {O} (f(\\mathbf {Aa} ))}{\\operatorname {E} (f(\\mathbf {Aa} ))}}=1-{\\frac {\\operatorname {ObservedNumber} (\\mathbf {Aa} )}{n\\operatorname {E} (f(\\mathbf {Aa} ))}},\\!", "0ad68add8ede6dd61b7eec9e07ce231d": "x+y=2,\\,\\,\\,\\,\\,2x+2y=4", "0ad6e304efa22426be7a8bde52422264": "{\\mathcal {L}}=-{1 \\over 2\\kappa ^{2}}ee_{I}^{\\mu }e_{J}^{\\nu }\\Omega _{\\mu \\nu }^{\\;\\;\\;\\;IJ}[\\omega ]+e{\\overline {\\psi }}(i\\gamma ^{\\mu }\\nabla _{\\mu }-m)\\psi ", "0ad6fbcabc0a265328c16db7c814e3ed": "TT={\\frac {\\sum _{v=v_{\\min }}^{N}vP(v)}{\\sum _{v=v_{\\min }}^{N}P(v)}}", "0ad701e82310fb9b57bd31b4046dacb2": "r={\\frac {y'(t)}{y(t)}}", "0ad75fdf00a4056733b425273446c041": "\\mathbb {E} \\left[{\\mbox{ Charles }}|{\\mbox{ calling }}\\right]={\\frac {4}{42}}\\cdot (P+2)-{\\frac {38}{42}}\\cdot 1", "0ad80da9c5d99359d7f02677f40c4412": "\\sin {\\frac {\\gamma _{s}}{2}}={\\frac {a_{s}}{2r_{u}}},", "0ad81ebc1a4a0d6b0f64bfb84e0e934c": "-{\\tfrac {3}{2}}", "0ad8206218e987db6f1cf49e5b20b103": "{\\frac {\\hbar \\omega }{2}}\\left(-{\\frac {d^{2}}{d\\chi ^{2}}}+\\chi ^{2}\\right)\\psi (\\chi )=E\\psi (\\chi ).", "0ad82926eed098f07871a7b712353e0e": "{\\frac {\\operatorname {d} }{\\operatorname {d} \\!\\theta }}\\,\\tan \\theta =\\lim _{\\delta \\to 0}{\\frac {\\tan \\delta }{\\delta }}\\times \\lim _{\\delta \\to 0}\\left({\\frac {1+\\tan ^{2}\\theta }{1-\\tan \\theta \\tan \\delta }}\\right).", "0ad894a67715cf79e1c6e8ca60c4bd93": "B\\rightarrow A", "0ad8b7348b790908c11239b7d0f46f6d": "t_{ff}={\\frac {1}{4}}{\\sqrt {\\frac {3\\pi }{2G\\rho }}}\\simeq 0.5427{\\frac {1}{\\sqrt {G\\rho }}}\\simeq 66430\\,{\\rm {s}}{\\frac {1}{\\sqrt {\\rho }}}", "0ad932359c1bcae7aff5609deeeb46a9": "{\\mathfrak {a}},{\\mathfrak {b}}", "0ad939a6a380786dbbb2202b3e62f5fa": "K=ab\\cdot \\sin {A}.", "0ad93ff7bfb00f57273d1262d31ac61c": "\\eta _{therm}", "0ad9676d5d1302e127cac77bba485049": "\\psi _{L}\\ {\\stackrel {\\mathrm {def} }{=}}\\ \\left({\\cos \\theta +i\\sin \\theta \\over {\\sqrt {2}}}\\right)\\exp \\left(i\\alpha _{x}\\right)=\\left({\\exp(i\\theta ) \\over {\\sqrt {2}}}\\right)\\exp \\left(i\\alpha _{x}\\right)", "0ad989cf548d5faf9dcd88c4f0f3fdac": "\\int x^{n}Sdx={\\frac {2}{a(2n+3)}}\\left(x^{n}S^{3}-nb\\int x^{n-1}Sdx\\right)", "0ad9b0cbf0774ca90fc4d215a7050772": "E_{i,j}={\\frac {\\left(\\sum _{n_{c}=1}^{c}O_{i,n_{c}}\\right)\\cdot \\left(\\sum _{n_{r}=1}^{r}O_{n_{r},j}\\right)}{N}}\\,,", "0ad9efe4940c9b7a57eefc4ba4c7b105": "Z_{L}", "0ada268d42742fc19970ed1d4411cfe3": "d=|E|/|V|^{2}", "0ada4206c2d86ceb2abc5c74d52ae4dc": "y(t)=e^{-{\\frac {t}{5}}}\\sin(t)", "0ada9e98ac83205ab173c7a958814284": "p_{1}={\\frac {0+0+0+0+2+7+3+2+6+0}{140}}=0.143", "0adaabea1e27f22df013859217cc49b4": "r={\\frac {a}{\\theta }}", "0adace6e1f771a8d412057231ae9e11b": "\\Gamma (z)\\approx {\\sqrt {\\frac {2\\pi }{z}}}\\left({\\frac {1}{e}}\\left(z+{\\frac {1}{12z-{\\frac {1}{10z}}}}\\right)\\right)^{z},", "0adb263e90b7c77dc1ee0a994fe6ccd2": "\\displaystyle \\Psi _{\\gamma \\alpha }(u)=\\Psi _{\\gamma \\beta }(u)\\Psi _{\\beta \\alpha }(u)", "0adb3fac1ad5ad0517a3c2c32cc01424": "Q_{flow}=k{\\frac {D}{\\tau }}", "0adb7dafdc41b9a4760f617066612a6c": "[[A,B],C]=[A,[B,C]]-[B,[A,C]]\\,", "0adba1de183aafddd4146813f294fa2e": "{\\begin{array}{rcl}\\operatorname {var} (PX)&=&\\mathbb {E} [PX~(PX)^{\\dagger }]\\\\&=&\\mathbb {E} [PX~X^{\\dagger }P^{\\dagger }]\\\\&=&P~\\mathbb {E} [XX^{\\dagger }]P^{\\dagger }\\\\&=&P~\\operatorname {var} (X)P^{-1}\\\\\\end{array}}", "0adbd3ca4cb163c59d16e3e875fe4e8f": "p_{1}=5", "0adbdcbb5329dd46c2d3bf5317afcc81": "\\mu (A)=U{\\textrm {-}}\\lim {|A\\cap F_{i}| \\over |F_{i}|}.", "0adc1270c320cab6abf8308fc8d3687e": "\\scriptstyle \\log _{e}({\\frac {760}{101.325}})-5.381564\\log _{e}(T+273.15)-{\\frac {2626.728}{T+273.15}}+1.601858\\times 10^{-05}(T+273.15)^{2}", "0adc1c00f3e59bea21899491bf14fdd3": "V=\\{v_{1}\\ldots v_{n}\\}", "0adc3d60e9744ac3ed6621168c52222c": "{\\frac {{}_{(1)1}\\partial x^{-1}}{\\partial x}}=x^{-1}\\,\\!", "0adc4805cb939b46ae33577d014985ff": "|\\Psi \\rangle _{\\nu }", "0adcaf6f179bdbc47234308c993cf28a": "{C_{L}}", "0adcb3f03a670941fddd67415651a8bb": "\\Sigma _{i}\\cdot \\partial {\\mathcal {G}}\\Sigma _{i}", "0adcc9ab86b40f6eeb1da2fd4f986813": "\\alpha \\approx 0.85", "0adcd34bcfa72a9acb4b3a5e6e50944c": "\\sigma _{n}={\\frac {1}{2}}", "0adce572f800120bb12403ce77380bb3": "<\\alpha ,\\beta >", "0adceb1a4e17e1f6982bcda5746b5f75": "{\\hat {B}}(\\xi )=\\int {\\frac {d^{2n}\\eta }{(2\\pi \\hbar )^{n}}}\\exp(-{\\frac {i}{\\hbar }}\\eta _{k}(\\xi -{\\hat {\\xi }})^{k})\\in \\mathbb {V} .", "0add083d28e9974a35eaa9d9bbc32728": "\\tan \\theta _{1}=\\tan \\eta \\cos \\chi \\,", "0add35947d40601f06f88757edb8cb34": "y_{1}^{n}", "0add49170ca83ceb0e481d9b65b9c4b2": "(a_{1},a_{2},...,a_{10}),\\qquad a_{k}=k^{2}.", "0adda38e472ea3f2b5b6abfca71b0326": "A_{n}(R)={\\frac {d}{dR}}V_{n+1}(R).", "0addbbfa78e302f758e311020b8bebe3": "\\mu _{\\operatorname {eff} }({\\dot {\\gamma }})", "0addcf96e7f409d886f8d1e649907116": "t\\rightarrow P_{t}f(x)", "0ade0c77b850b0197fb9f8bcbd9b016d": "\\|Ax_{n}-b\\|=\\|{\\tilde {H}}_{n}y_{n}-\\beta e_{1}\\|,\\,", "0ade1bfcbe41fcfd15d15576a4587922": "A_{jk}=f_{j}(g_{k})", "0ade45986bd634701e3ef6b1f42187ef": "\\Phi =e^{\\beta (\\epsilon _{i}-\\mu )}-1", "0aded1a42b92b3af244a984feab5f5d6": "ev=0", "0adee9c4df3e9f23d4d45d934b2595fc": "J_{ij}={\\frac {\\partial \\theta _{i}}{\\partial \\xi _{j}}}.", "0adf243c7df85a9fde4f478ed121ba95": "{\\text{Minimize}}={\\begin{cases}f_{1}\\left(x,y\\right)&=\\left[1+\\left(A_{1}-B_{1}\\left(x,y\\right)\\right)^{2}+\\left(A_{2}-B_{2}\\left(x,y\\right)\\right)^{2}\\right]\\\\f_{2}\\left(x,y\\right)&=\\left(x+3\\right)^{2}+\\left(y+1\\right)^{2}\\\\\\end{cases}}", "0adfd0bb7d5a814c2f7bb3116146c882": "f((\\mathbf {v} ,\\mathbf {u} ))=\\sin {(\\mathbf {v} ,\\mathbf {u} )}", "0adfe03b2e4c67613fa5b5e6975fc42f": "s={\\frac {2\\pi r}{z}}", "0ae00baece84cb7a8f2d1c42e71fa7c9": "{\\begin{pmatrix}A&B\\\\B^{*}&\\end{pmatrix}}{\\begin{pmatrix}x_{1}\\\\x_{2}\\end{pmatrix}}={\\begin{pmatrix}b_{1}\\\\b_{2}\\end{pmatrix}},", "0ae00d7feea9f0b0f8b0154779ef7ec8": "\\lim _{x\\rightarrow x_{0}}\\operatorname {ap} \\ f(x).", "0ae01416e90b9cd36a31269816379f88": "S={\\frac {e^{-\\beta H}}{\\operatorname {Tr} (e^{-\\beta H})}}.", "0ae0353f26d6d3c71136114942f1ff14": "\\mu _{i1}=\\mu _{i2}\\,", "0ae0f9542148acfda68f4177c534a18a": "{\\begin{aligned}\\mathbf {T} ^{6}={\\begin{pmatrix}{\\frac {1}{4096}}&{\\frac {3}{1024}}\\\\[4pt]0&{\\frac {1}{4096}}\\end{pmatrix}},\\end{aligned}}", "0ae1285ce5610001567ddb53236e50fe": "M.", "0ae14b7377bc33c387c2661a0b65ad91": "k_{x},k_{y}", "0ae160e4c7d3842ee9eaed697676bb3f": "\\Omega _{n}(E)={\\frac {c_{n}}{c_{k}^{n/p}}}\\left(E-E_{0}\\right)^{n/p}\\ ,", "0ae185a9a0d7167749b407efcc6a1ced": "G({\\vec {r}},t)=\\langle {\\frac {1}{N}}\\int \\rho ({\\vec {r}}'+{\\vec {r}},t)\\rho ({\\vec {r}}',0)d{\\vec {r}}'\\rangle ", "0ae185fb3e4dc62970b6e11005be8c1d": "\\mathbf {R} ={\\frac {1}{M}}\\int _{V}\\rho (\\mathbf {r} )\\mathbf {r} dV,", "0ae1f248a8ba4353c0934d07dbca3423": "k|x_{t}|<1", "0ae21678a1f069a78dab02b4c6ea17d3": "k=A+\\ N(0,s^{2}/y_{k})", "0ae2353e5d59b84f52906b7025b8e628": "{\\displaystyle }P_{2}=(x_{2},y_{2})=(\\sin {\\alpha _{2}},\\cos {\\alpha _{2}})", "0ae241997f4e8defc892e1b015cb1e7f": "A_{(m+r)(n+r)}-A_{mn}\\approx r\\;\\left({dA \\over dJ}\\right)_{mn}\\,,", "0ae25898d277b9e1445a22ab0e83b67e": "{\\begin{aligned}(\\gamma a_{1})(\\gamma a_{2})\\dots (\\gamma a_{m})&\\equiv {\\zeta _{n}^{b(1)}a_{\\pi (1)}}{\\zeta _{n}^{b(2)}a_{\\pi (2)}}\\dots {\\zeta _{n}^{b(m)}a_{\\pi (m)}}\\\\&\\equiv \\zeta _{n}^{b(1)+b(2)+\\dots +b(m)}a_{\\pi (1)}a_{\\pi (2)}\\dots a_{\\pi (m)}\\\\&\\equiv \\zeta _{n}^{b(1)+b(2)+\\dots +b(m)}a_{1}a_{2}\\dots a_{m}{\\pmod {\\mathfrak {p}}},\\end{aligned}}", "0ae25e97958e0619d97709cbe4d8e6e1": "g(x)=f_{n}^{(0)}(x)+{\\mathcal {O}}(\\epsilon )", "0ae260b632c6ffcbd3e92c01ef262348": "t_{r}=t_{2}-t_{1}=2t", "0ae271f7bf529e2058de7094e9cf7978": "x_{0}=r^{2}~mod~N", "0ae2890e26891a12d8612233d1b5b97b": "R({\\vec {x}})=P(2|{\\vec {x}})-P(1|{\\vec {x}})={\\frac {P(2,{\\vec {x}})-P(1,{\\vec {x}})}{P(1,{\\vec {x}})+P(2,{\\vec {x}})}}", "0ae2c9e1494b3b6e3d7e073751f8e08d": "{\\mathcal {O}}(1/k)", "0ae2d2635748aba7216b75e0e6c9e9f3": "S=W(q_{1},q_{2}\\cdots q_{N})-Et", "0ae349e680e10cb11344b7f7fd67849d": "K_{\\text{GN}}(x,x')=\\sigma ^{2}\\delta _{x,x'}", "0ae387cd1eeb766689f643ba1231b705": "\\textstyle \\sim \\left\\langle {\\sin }^{2}x\\right\\rangle ={\\frac {1}{2}}", "0ae3f521fce4775f8cc83a4023c20bdc": "\\mathbf {U} ={\\frac {d\\mathbf {x} }{d\\tau }}", "0ae49173b1ab7f770fc3927f1bd65c8e": "e_{0}=1,\\ e_{1}=i,\\ e_{2}=j,\\ e_{3}=k,\\!", "0ae4ff12c49634470ce0cebe2cb78ea5": "\\langle L(t)\\rangle \\equiv \\int _{-\\infty }^{\\infty }L(x,t)P(x,t)dx,", "0ae537600491b69d6f8857b3ddbc9368": "1=\\sum _{n=2}^{\\infty }(\\zeta (n)-1).", "0ae53c4958cb19e1efe9e2638ebb1d4f": "\\scriptstyle V\\,{\\overset {\\sim }{\\to }}\\,V^{*}", "0ae57e7fd2baa3e23e6fa5d341168439": "i:X\\to \\mathbf {P} _{A}^{r}", "0ae59705a0e645366c129299debb00ef": "(3^{n+1}(3^{n+1}-1)/2,3^{n}(3^{n+1}+1)/2,3^{n}(3^{n}+1)/2)", "0ae5cdaf55de16ec636ac97b974aca5a": "\\int _{1}^{\\infty }{\\frac {\\ln \\ln x}{x^{3}}}\\,dx=-{\\frac {1}{2}}(\\gamma +\\ln 2).", "0ae5d18287a4c831fb0e8eb0bf97a046": "f(x,y,z)=x^{2}+y^{2}\\longrightarrow \\rho ^{2}\\sin ^{2}\\theta \\cos ^{2}\\phi +\\rho ^{2}\\sin ^{2}\\theta \\sin ^{2}\\phi =\\rho ^{2}\\sin ^{2}\\theta ", "0ae61f6339e973f2a71e43f843e6bd35": "\\operatorname {sgn}(x)=[x>0]-[x<0]\\,", "0ae65f4bc72b6d04c26f108f4cd35b3b": "x(\\theta )=(R+r)\\cos \\theta -d\\cos \\left({R+r \\over r}\\theta \\right),\\,", "0ae65f853c825e3107071d966d62514f": "D^{2}({\\mathcal {F}})\\cong {\\mathcal {F}}", "0ae6c26938eb6f75d3dd165425ff0ce7": "{\\hat {H}}_{0}=\\sum _{i}{\\frac {{\\hat {p}}_{i}^{2}}{2m_{i}}}+V", "0ae6e25fafe20bb259b6da2b4c12ea48": "\\,w=w_{i}e^{i}={\\begin{bmatrix}w_{1}&w_{2}&\\cdots &w_{n}\\end{bmatrix}}{\\begin{bmatrix}e^{1}\\\\e^{2}\\\\\\vdots \\\\e^{n}\\end{bmatrix}}", "0ae6fb7e963837b2070b887c63c117ee": "\\mathbf {Q} _{p}", "0ae74f20e2e511df0f36adcfd193acc4": "(\\exists y)\\mu yR(y)=\\{{\\mbox{the least (natural number)}}\\ y\\ {\\mbox{such that}}\\ R(y)\\}", "0ae7b35ba67c9740b964635410608cbf": "W_{ijkl}", "0ae7ea772114e874d075b0beb258e48f": "{\\rm {J={}{\\rm {{\\frac {kg\\cdot m^{2}}{s^{2}}}=N\\cdot m={\\rm {Pa\\cdot m^{3}={}{\\rm {W\\cdot s=C\\cdot V}}}}}}}}", "0ae7f4a7afa0b92f7e129a66cc208d6b": "c_{d}={\\frac {1}{\\pi \\omega _{d-1}}}={\\frac {\\Gamma [(d+1)/2]}{\\pi ^{(d+1)/2}}}.", "0ae83b9a4460c4d9ac182eb506deddb3": "G'_{k}(u)=|F(u)|e^{i\\phi _{k}(u)}", "0ae8b117e9f0a44cfda7c8d7cf8eaf2d": "\\sum _{1\\leq k\\leq n \\atop (k,m)=1}1=n{\\frac {\\varphi (m)}{m}}+{\\mathcal {O}}\\left(2^{\\omega (m)}\\right),", "0ae914d4537770229b7846cfdcd84292": "114{\\frac {1}{2}}", "0ae92c14c0e9def62b2c99369d444c10": "B\\to x", "0ae9357ebdd3d570acab982976a61da9": "\\left\\{{\\frac {x_{1}+(1+x_{2})x_{3}}{x_{1}x_{2}}},{\\frac {x_{1}+x_{3}}{x_{2}}},{\\frac {(1+x_{2})x_{1}+(1+x_{2})x_{3}}{x_{1}x_{2}x_{3}}}\\right\\},", "0ae946a990b8fb199c7f60f2bc27706b": "x\\geq 0", "0ae98f0799e3f806a4a35c66a5ccfd40": "(\\Omega ,{\\mathcal {F}},P)", "0ae9b0da0724bc884adf9942c3c6f074": "{\\begin{bmatrix}a&b\\\\c&d\\\\\\end{bmatrix}}^{-1}{\\begin{bmatrix}1&0\\\\1&3\\\\\\end{bmatrix}}{\\begin{bmatrix}a&b\\\\c&d\\\\\\end{bmatrix}}={\\begin{bmatrix}x&0\\\\0&y\\\\\\end{bmatrix}}", "0ae9d13aad0812ba6f090b3e7baf1383": "B[v']=AB[u']", "0aea0f1ee68a135cbcff00702c976d7e": "q^{n}", "0aea24903360bfd846680a41c1291623": "O(\\log ^{2}p)", "0aea252426836658d80f6c6681ecd4a0": "M^{*}\\,", "0aea403a758eb15358c65d6a99385e2e": "\\mathbb {R} ^{n}\\ni x\\mapsto \\Psi _{r}(x)=f(r^{2}-\\|x\\|^{2})", "0aea552e9914182ab1da8e050e89082a": "\\int _{-\\infty }^{\\infty }x\\Phi (a+bx)\\phi (x)\\,dx={\\tfrac {b}{t}}\\phi ({\\tfrac {a}{t}}),\\qquad t={\\sqrt {1+b^{2}}}", "0aea63c51c29649339627194050c41cb": "A={\\begin{pmatrix}2&1\\\\-1&0\\end{pmatrix}}.", "0aeab43d70097555ea9c86e0daa800fe": "{x^{2} \\over a^{2}}+{y^{2} \\over b^{2}}=1\\,", "0aeaba0177223bf6907edb6aa8811e88": "\\phi _{1},...,\\phi _{m}", "0aeaf52485cd4a50b7ab767a3361a6cd": "w_{i}\\,", "0aec3c266f29dda82b652734669009af": "\\phi _{\\mu }", "0aec811717d3b2e514ec9f3639182d90": "l^{a}=g^{ab}l_{b}", "0aecdaad710b30cd6b6be0996cffe248": "a_{n}=a_{n-1}+a_{n-2}", "0aed50a25843e0b74b7a4b43ad0074f9": "\\operatorname {tr} [\\rho F(A)]\\,,", "0aed50d0ba7fda9096b4b8a37447313e": "(\\lambda -k_{1})(\\lambda -k_{2})", "0aed8dbb14390f58a2145583d3230522": "E_{n}=E_{n}^{0}+\\langle \\varphi _{n}^{0}\\vert V\\vert \\varphi _{n}^{0}\\rangle ", "0aee4f081c8ef3ab812df7e4bac5e97b": "i_{2}", "0aeec20960bed943019da0d278e35151": "p(11^{3}\\cdot 13\\cdot k+237)\\equiv 0{\\pmod {13}}.", "0aef083e866296e552fdfaab6985870b": "P\\land (Q\\lor R)=(P\\land Q)\\lor (P\\land R)", "0aef321f8dbf730edc1a9efd62d604ae": "C_{i}=\\sum _{j}u_{ij}B_{j}.\\quad ", "0aef5e0e685b879acde1a40eca9a0064": "n={\\frac {1}{\\xi }}(\\ln E_{0}-\\ln E)", "0aef8a8a93b67e85ecfdb7aa60b632ab": "E_{c}^{\\rm {GGA}}", "0aefe5e1e1d54d587fbd314141cc0804": "\\cos ^{-1}\\left(-{\\sqrt {{\\frac {1}{15}}\\left(5+2{\\sqrt {5}}\\right)}}\\right)", "0af013a1d081a6e99a7721592d6b5555": "\\scriptstyle Z_{\\theta }\\,\\sim \\,{\\mathcal {N}}(0,\\,I_{q(\\theta )}^{-1})", "0af04b45f774ed65b2a68fed6a17daec": "{\\vec {R}}\\cdot {\\vec {s}}=0", "0af0e907e0626bac05dbaeb759954376": "=2\\pi {\\text{ rad}}", "0af149732181b7aa0644109d228e7585": "r\\neq 1", "0af164866f37679d9b4ac8ae84fbb54f": "\\mathrm {Cu_{(aq)}^{2+}+2e^{-}\\rightarrow Cu_{(s)}\\ } ,", "0af1b25de9436c937f4fb90c3f969eb7": "M\\leftarrow M\\oplus (P_{1}\\cup P_{2}\\cup \\dots \\cup P_{k})", "0af28a4718bf5b5f51f3f5243cb0be8a": "r_{m}={\\frac {a}{2}}=0.5\\cdot a", "0af2a3d0977d08a9a23b3c9103aadb2c": "HP_{S}(n)=HF_{S}(n)", "0af2c6231b40809e0630d39f985e4b33": "f:\\mathbb {R} \\to \\mathbb {R} ;\\qquad f(x)={\\begin{cases}e^{-{\\frac {1}{x^{2}}}}&x>0,\\\\0&x\\leq 0.\\end{cases}}", "0af32ee352608a6f3ad7fc3fe606415d": "F_{\\nu ,\\mu }(x)={\\begin{cases}{\\tilde {F}}_{\\nu ,\\mu }(x),&{\\mbox{if }}x\\geq 0;\\\\1-{\\tilde {F}}_{\\nu ,-\\mu }(-x),&{\\mbox{if }}x<0,\\end{cases}}", "0af331690bd48325646f240e9e06b554": "V_{R}\\ =\\ G_{R}V_{in}e^{j\\phi _{R}}", "0af35b25271436e00741d88a7b6a6d0f": "\\nabla \\cdot \\mathbf {E} =\\rho /\\epsilon _{0}", "0af410b858f1e8cb088109e76b93195a": "FV=PV(1+i)^{n}\\,", "0af4187e03f4bf1693e268465b811f3d": "1-(37/38)^{35}", "0af521c5bbd8e2d5cc34cffd2e1b0a5a": "t=u+vj\\ ", "0af522d7c2423d04e4e821cf815d7136": "{\\text{EVaR}}_{1-\\alpha }(X):=\\inf _{z>0}\\{z^{-1}\\ln(M_{X}(z)/\\alpha )\\}.\\,", "0af52bad2bda865a62d72acc25d7d15e": "{\\bar {f}}", "0af576cb9c17bf59fcd123781de39882": "A,S1(K_{x}(S0,response)", "0af5ddc24e0b7c7c747e11337ae5106e": "A^{\\prime }\\rightarrow \\epsilon \\,|\\,\\alpha _{1}A^{\\prime }\\,|\\,\\ldots \\,|\\,\\alpha _{n}A^{\\prime }", "0af5f6ade3b2bbc1d9af090a1880c854": "\\textstyle c\\in {\\mathcal {C}}", "0af61c2cd218420fc8773ff9ed193ead": "{(S\\ll 1)}", "0af61f6ca8fa904eacc0f2e5952c51b5": "\\land (S_{6}\\implies (\\operatorname {equate} [A_{6},n]\\land V[F_{6}]=n))\\land D[F_{6}]=D[n])", "0af67b699bf07611f74c7577c40159e6": "{\\begin{matrix}{r \\choose 1}{4 \\choose 2}{r-1 \\choose 2}{4 \\choose 1}^{2}\\end{matrix}}", "0af682c0c49d58279b7f2208094cbde6": "\\sum _{n=1}^{x}{\\tfrac {1}{n}}", "0af6b6d6fb20ccf078ac3c4efaa9aafb": "\\varepsilon ={v^{2} \\over {2}}-{\\mu \\over \\left|\\mathbf {r} \\right|}", "0af6f160224749ce767c375359dd8875": "({\\text{Gal}}(K^{sep}/K),\\varinjlim C_{L})", "0af74242b153e40a79fe7c46c0e8915e": "\\bigcup A\\subseteq A", "0af791063649141160e8541600194230": "G(x,Q)=\\sum _{k\\varepsilon +\\sum _{s\\in S_{-p}}u_{p}(j's)x_{s}\\Longrightarrow x_{j'}^{p}=0.", "0afd2c56d8ebe40e1e5bee5960694016": "{\\frac {E\\varepsilon (t)}{\\sigma _{0}}}", "0afd5b870674d4efd365affbe8d23052": "\\sigma _{c}=\\limsup _{n\\to \\infty }{\\frac {\\log |a_{1}+a_{2}+\\cdots +a_{n}|}{\\lambda _{n}}}.", "0afde54cc308eaa5f496587f74b638ec": "n=x_{1}^{k}+\\cdots +x_{\\ell }^{k}.\\,", "0afe2bff60b793ff51934fd0aa5acbb6": "\\delta \\mathbf {Z} _{0}", "0afe7b9c1786b0de754b74542b65f408": "\\max _{0\\leq i\\leq m+n}{(|u_{i}|,|v_{i}|)}\\leq 2b^{9(m+n)}.", "0afe9b2a7b30304eb14249802957f8aa": "r'_{8}(n)={\\frac {({\\frac {1}{2}}\\pi )^{4}}{6}}(n+1)^{3}\\left({\\frac {c_{1}(n+1)}{1}}+{\\frac {c_{3}(n+1)}{81}}+{\\frac {c_{5}(n+1)}{625}}+\\dots \\right).", "0aff0e21abfc190bf229abb142b4a011": "Lu=au_{xx}+bu_{yy}", "0aff737e794c23be86ae194d542a64be": "{\\begin{aligned}t'&=t\\\\x'&=x-vt\\end{aligned}}", "0aff9a174a6ab7bfc3c979c9764b5874": "{\\sqrt {\\hbar G/c^{5}}}", "0affa9fd0694b61247359ab5acf58a21": "\\sum _{n=1}^{\\infty }q^{n}\\sigma _{0}(n)=\\sum _{n=1}^{\\infty }{\\frac {q^{n}}{1-q^{n}}}", "0affd5716e67407b2620c73849705fee": "(n,k,2t+1)_{\\mathcal {F}}", "0affe29f92bcca5fe28a947d53d7e709": "v(p+r,t)\\approx v(p,t)+E(p,t)(r)+R(p,t)(r),", "0b00abade0dfec7157808f254f284f9a": "\\ v_{1}", "0b00ea88e6a73f75f8751591843d912f": "O(n^{c}),\\;00", "0b0b4bf40868b9a659f9d4417631dce1": "{\\rm {ad}}(x){\\rm {ad}}(x)(y)=[x,[x,y]\\,]", "0b0b4c7241b24789623828c7de52aff7": "F=F_{2}(q,P,t)-QP\\,\\!", "0b0b52d7a2c57333f60176e9b1891e61": "20\\div 4=5", "0b0b556d31bcc75a0e259af55920a0ea": "K_{s}(A)", "0b0bb06e3244e6beeb99703fe680e1af": "\\textstyle \\rho _{L}", "0b0bce7f66cd1f7fad91a9fc34772308": "\\,\\zeta (3/2)\\approx 2.6124.", "0b0c03a2f9d4447d282d026b4b11fcd4": "\\mathbf {F} _{\\text{c}}=\\mathbf {F} _{\\text{g}}", "0b0c07872c021789774d17e3a6ef13b0": "g(\\zeta )=f(\\zeta ^{-2})^{-{1 \\over 2}}", "0b0c1f24e1b8461dd05a0c2025a7b479": "z={\\frac {1}{2}}\\cdot ((a^{2}+1)\\cdot (a^{2}+b^{2})^{2}+4\\cdot b^{2})", "0b0c4ac9eb4f84101e896551bd342c40": "\\beta (b)", "0b0cb65efd353d5ee4b9f4db0f5829aa": "\\mathbf {p} (t)=m\\mathbf {v} (t)", "0b0d098dc2ae0d225a4d67eef246897e": "{\\Gamma (c=K^{-1})}=\\Gamma _{max}{\\frac {KK^{-1}}{1+KK^{-1}}}={\\frac {\\Gamma _{max}}{2}}", "0b0d73136bacf275838a54965c8b5aeb": "x=-a/\\omega ^{2}=-{\\frac {eE}{m\\omega ^{2}}}\\,\\exp(-i\\omega t)=-{\\frac {e}{m\\omega ^{2}}}{\\sqrt {\\frac {2I_{0}}{c\\epsilon _{0}}}}\\,\\exp(-i\\omega t)", "0b0d77792b0117723c77f1950e4e3487": "S={\\frac {1}{2}}f(m)+f\\left(m+1\\right)+\\cdots +f\\left(n-1\\right)+{\\frac {1}{2}}f(n)", "0b0dae69b9ae2c69ce7ed51ebc2cc966": "C(\\alpha ,0)", "0b0db187ca92b5f64873942717d227fe": "(S-K)^{+}", "0b0dd242bf350e6ef06f569229a0278a": "\\scriptstyle {\\sqrt {\\det({\\mathcal {I}}(\\alpha ,\\beta ))}}={\\sqrt {\\psi _{1}(\\alpha )\\psi _{1}(\\beta )-(\\psi _{1}(\\alpha )+\\psi _{1}(\\beta ))\\psi _{1}(\\alpha +\\beta )}}", "0b0de107f276831321196bb7588887a1": "[fg](x)=f(x)\\cdot g(x)", "0b0e1bfa525f11f7e5d1e67eb3da2751": "\\textstyle e\\left(P,Q\\right)\\neq 1", "0b0ec3f1721a875d4742be25f204fd26": "\\mathbb {R} \\times \\mathbb {R} ^{d}", "0b0f1e7540e2fdb8b8cd059e7e733ec0": "{\\dot {\\hat {z}}}=A(u(t)){\\hat {z}}+\\phi (y,u(t))-L(t)\\left(C{\\hat {z}}-y\\right)", "0b0f8b3863d5d24a9a310a1bac13aa94": "Obs\\,", "0b1189ce7bd5f85b616515d10c845404": "f={\\frac {1}{2L}}{\\sqrt {\\frac {F}{\\mu }}},", "0b11a19533849ad711d522ec0e7d02a2": "\\lim _{k\\rightarrow \\infty }\\,\\mu (T^{-k}A\\cap B)=\\mu (A)\\cdot \\mu (B)", "0b11d212fd09a23a6854e90476d3479b": "U_{F}(x,1)\\mapsto U_{A}(x,1),\\quad U_{F}(1,0)\\mapsto U_{A}(1,0).", "0b11d32f960aba9accc04cfa0000514c": "\\gamma ={\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}}", "0b120962493753d604c18b6afbf30ace": "T={\\frac {1}{2}}\\left|\\det {\\begin{pmatrix}x_{B}&x_{C}\\\\y_{B}&y_{C}\\end{pmatrix}}\\right|={\\frac {1}{2}}|x_{B}y_{C}-x_{C}y_{B}|.", "0b1250212d4a214f1d3666745265e7d0": "\\exists x_{r}~S", "0b129a4834611f48173672ce51bb8811": "F=RT\\sum _{i}N_{i}\\left(\\ln \\left({\\frac {N_{i}}{V}}\\right)-1+{\\frac {\\mu _{i}^{\\ominus }(T)}{RT}}\\right)", "0b12bf82691ca5c6fe95e7720e56b7d0": "\\chi _{N}=e\\left(\\sum _{i=1}^{t}\\mu ^{(g_{i})}\\right),", "0b12d021397567943d56d0b0ae311eb8": "e=", "0b1345f9433f6a6fd7a57ba8b2f3d241": "r{\\dot {r}}=-{\\frac {\\kappa }{2}}r^{2}", "0b135c45909648708d01224c30b7e96b": "a\\uparrow ^{4}b=a\\uparrow \\uparrow \\uparrow \\uparrow b", "0b1387add57864bff81a792d3d924760": "\\int _{\\Gamma }\\phi (g)=0.", "0b13a818b2887173fbf96623d279f660": "\\aleph _{\\alpha }^{\\aleph _{\\beta }}", "0b1403b7ab144745401b0c4244ddcc6d": "\\phi (z,t_{1},\\dots ,t_{n})", "0b14c77c94a8cc9dc95491b15e6c7bfd": "m=r", "0b14fa01ab6aa848696cce164e50621a": "\\scriptstyle 6\\times 6\\times 6", "0b156bc724c5cd9d72272527688af806": "{\\mathbf {D}}=\\epsilon _{0}{\\mathbf {E}}+{\\mathbf {P}}\\,", "0b158224c3b168f9e94c1f9e67a5f673": "Z={\\frac {P\\,V_{m}}{R\\,T}}={\\frac {1}{1-h}}\\ -{\\frac {A^{2}}{B}}{\\frac {h}{1+h}}", "0b15aedcf31f894e268005e908009745": "0\\leq 2\\beta <\\kappa ", "0b15b424c248ea8ef83e0ff920366072": "{\\bar {u^{\\prime }}}=0", "0b15c970ea8fba5c0d08b137adcd09a2": "\\Lambda ^{2}\\mathbb {C} ^{m}", "0b169eaa5bdca18c3e25237c486a9f26": "\\varphi _{tt}=a(\\varphi )[a(\\varphi )\\varphi _{x}]_{x}.", "0b16e46cbcd7a6cb6953cfdfd0e7cc35": "v.", "0b17104790bd48e99631fa516b1b6645": "(A_{1}\\land \\cdots \\land A_{n})", "0b17149f3dd48b885dad4c30a7fadfec": "\\tau _{\\text{wind}}=\\rho _{\\text{air}}C_{D}U_{h}^{2},", "0b176ad622b6f8b2b6183620a35df7a8": "x_{i}(t)\\neq 0", "0b177e9bdbcda67f6f25d3d599cfbfe1": "x={\\sqrt {c}}", "0b1794d4203b483781426926429e00d8": "{\\frac {P}{F}}={\\frac {{\\frac {1}{2}}{{\\dot {m}}v^{2}}}{{\\dot {m}}v}}={\\frac {1}{2}}v", "0b17b85ebf2a90b7dc0fde77abd8812f": "F'(\\mathbf {x} _{0})", "0b17d47a86849b9a3f651fdbbb7ca717": "\\mathbf {d} _{i}^{[0]}=\\mathbf {d} _{i}", "0b1807c6bd9a0a1112d19fe419539758": "e^{br\\epsilon }e^{-ar},", "0b1847e036f5a594981d76b99c6a1789": "\\langle P,Q\\rangle ={\\frac {1}{2}}{\\bigl (}{\\hat {h}}(P+Q)-{\\hat {h}}(P)-{\\hat {h}}(Q){\\bigr )}.", "0b184877c6e0899328336741280a9e90": "{\\begin{array}{lll}&\\exp \\left(\\left[{\\begin{smallmatrix}.&.&.&.&.&.&.&.&.&.&.&.\\\\-5&.&.&.&.&.&.&.&.&.&.&.\\\\.&-4&.&.&.&.&.&.&.&.&.&.\\\\.&.&-3&.&.&.&.&.&.&.&.&.\\\\.&.&.&-2&.&.&.&.&.&.&.&.\\\\.&.&.&.&-1&.&.&.&.&.&.&.\\\\.&.&.&.&.&0&.&.&.&.&.&.\\\\.&.&.&.&.&.&1&.&.&.&.&.\\\\.&.&.&.&.&.&.&2&.&.&.&.\\\\.&.&.&.&.&.&.&.&3&.&.&.\\\\.&.&.&.&.&.&.&.&.&4&.&.\\\\.&.&.&.&.&.&.&.&.&.&5&.\\end{smallmatrix}}\\right]\\right)=\\left[{\\begin{smallmatrix}1&.&.&.&.&.&.&.&.&.&.&.\\\\-5&1&.&.&.&.&.&.&.&.&.&.\\\\10&-4&1&.&.&.&.&.&.&.&.&.\\\\-10&6&-3&1&.&.&.&.&.&.&.&.\\\\5&-4&3&-2&1&.&.&.&.&.&.&.\\\\-1&1&-1&1&-1&1&.&.&.&.&.&.\\\\.&.&.&.&.&0&1&.&.&.&.&.\\\\.&.&.&.&.&.&1&1&.&.&.&.\\\\.&.&.&.&.&.&1&2&1&.&.&.\\\\.&.&.&.&.&.&1&3&3&1&.&.\\\\.&.&.&.&.&.&1&4&6&4&1&.\\\\.&.&.&.&.&.&1&5&10&10&5&1\\end{smallmatrix}}\\right].\\end{array}}", "0b186e7690e82c74c7fcc8446c67a5e4": "a=u_{0}s\\geq 0.", "0b1e12b1503353093abcfb4df37e6647": "\\|{\\mathcal {F}}\\|_{q,p}=\\left(p^{1/p}/q^{1/q}\\right)^{n/2}.", "0b1e1e805333aeee29d50936831b35ac": "{\\rm {cov}}(V,T)={\\rm {E}}\\left(T\\cdot {\\frac {\\partial }{\\partial \\theta }}\\ln f(X;\\theta )\\right)", "0b1eaced739d5edca2d5694bdd8111f2": "k=1,2,3,\\dots ,N-1\\,", "0b1f03115d54972b55cbf0f14c09b660": "G\\,", "0b1f72bba20f2d5396ec27233856c8b4": "v_{1}\\wedge \\cdots \\wedge v_{k}", "0b1f90cb65a1a1a9f2207c586e00c39e": "\\int _{0}^{x}{\\frac {\\sin(\\theta )}{\\theta }}\\,d\\theta =\\mathrm {Si} (x)\\,\\!", "0b1fba5cfa06e468385c1fd682070237": "x_{1}\\in X", "0b2030c799283bb8285d536dba50d7e5": "x_{i}y_{j}=x_{j}y_{i}", "0b2047f99518b5b6010d0665158c817d": "|C|", "0b2052cab85752cdcbe29fbaaf097cad": "{\\overline {X}}_{1}-{\\overline {X}}_{2}=0.095.", "0b20a20785af243b973b8af20c3f80e3": "R=R_{s}", "0b20a45a98a411d873cc917db0fe4d43": "2.2RC", "0b20c0fb32195402e02f40a20f33a1be": "F_{2}={\\frac {F_{load}}{\\left[{\\frac {Sin(\\beta )Cos(\\alpha )}{Sin(\\alpha )}}+{\\frac {Cos(\\beta )Sin(\\alpha )}{Sin(\\alpha )}}\\right]}}\\,", "0b214a069e513fbc215742275989029d": "A=D+R\\qquad {\\text{where}}\\qquad D={\\begin{bmatrix}a_{11}&0&\\cdots &0\\\\0&a_{22}&\\cdots &0\\\\\\vdots &\\vdots &\\ddots &\\vdots \\\\0&0&\\cdots &a_{nn}\\end{bmatrix}}{\\text{ and }}R={\\begin{bmatrix}0&a_{12}&\\cdots &a_{1n}\\\\a_{21}&0&\\cdots &a_{2n}\\\\\\vdots &\\vdots &\\ddots &\\vdots \\\\a_{n1}&a_{n2}&\\cdots &0\\end{bmatrix}}.", "0b2165ab0fac44657e0035f107ffcc9f": "\\mathrm {Nu} _{D}=3.66", "0b21870c30fc7ebe105fd4fc5a5f80c1": "\\sigma :\\mathbb {C} \\rightarrow \\mathbb {C} ", "0b219b5321ab8079aa13ea0f9349b0e9": "Q_{3},R_{3},\\ldots ,Q_{r-2},R_{r-2},Q_{r-1}", "0b21a666a81629962ade8afd967826ed": "x_{0}", "0b21b88be8cbda85d6f623b3486a48e8": "\\sum _{j=1}^{J}\\mu _{j}Q_{j}", "0b21e10c346c5a26e9cb6bf5c028a58c": "k(k-1)(k-2)", "0b21efde9fbdc6c709090d137e2bd8c3": "{\\frac {Z}{R_{0}}}={\\frac {R_{0}}{Z'}}", "0b22047efa7cb78fd4d53cf973b83ba3": "\\,X_{t}=LX_{t+1}", "0b2209dade114962405c2de07e75bcbd": "f(x)={\\frac {1}{\\pi (1+x^{2})}}.", "0b2220be4d2cf621d428c7190f82354c": "l_{1},l_{2}{\\text{ and }}m", "0b226e2265605852cb52401ab8a65069": "I_{o_{lim}}={\\frac {V_{i}-V_{o}}{2L}}DT", "0b22ce22af97047e180f675fea7c5657": "\\upharpoonright ", "0b22db96440b0d85f4002b499b43f8ac": "max_{i=1}^{p}(w_{i})+max_{i=1}^{p}(h_{i}g)+l", "0b230d76c19fa0d8afe6350e90b27891": "\\vdash \\Box P\\rightarrow P", "0b2355e380add9df4b5d69b2575f16e7": "-3*y^{2}+8*y-8*x+8*z=0", "0b235f7701bde6967516add87539a715": "N\\wedge M:=N\\cap M", "0b23b1a155aaf147c0ab8ec166c7e9a3": "f_{0}({\\vec {x}})=-\\sum _{i=1}^{n}x_{i}\\log x_{i}", "0b241a844bbaee5f5bb205f2cf859842": "\\sigma _{y}^{2}(n\\tau _{0},N)={\\text{AVAR}}(n\\tau _{0},N)={\\frac {1}{2n^{2}\\tau _{0}^{2}({\\frac {N-1}{n}}-1)}}\\sum _{i=0}^{{\\frac {N-1}{n}}-2}(x_{ni+2n}-2x_{ni+n}+x_{ni})^{2}", "0b244b15ac660518a8f3bdca42320595": "{\\biggl (}\\int _{S}|g|^{q}\\,\\mathrm {d} \\mu {\\biggr )}^{1/q}.", "0b249e8fe5495ceb98df510cae5d402e": "f(q)=q^{-1}", "0b2529d7b282db3baa35e50b0cfee33d": "S'\\subseteq S", "0b253b148901d21647b9e9890fd4ca9c": "\\sum _{0\\leq k\\leq K}{{K \\choose k}{N-K \\choose n-k} \\over {N \\choose n}}=1", "0b256f92caebeb2c8a38e5fadefbfa9a": "P={\\frac {\\omega ^{4}}{12\\pi \\varepsilon _{0}c^{3}}}|\\mathbf {p} |^{2}.", "0b25be1699beddd6778cc201df0fba41": "a{\\sqrt {D}}", "0b264cf7209fa033b90cf8606430b55b": "a_{1},\\ldots ,a_{n}\\in A^{n},", "0b268f2c5085ff9b8ed74480420e7055": "\\cot A={\\cos A \\over \\sin A}", "0b26e34f2880a5c5ef298bcac7b6c449": "\\operatorname {Ad} (g)", "0b2733a7de366b183dbe1e433a8ca467": "\\langle T\\varphi ,\\psi \\rangle =\\left\\langle {\\frac {\\partial \\varphi }{\\partial x_{k}}},\\psi \\right\\rangle =-\\left\\langle \\varphi ,{\\frac {\\partial \\psi }{\\partial x_{k}}}\\right\\rangle ", "0b27ae7f4d381c49d68ead1b5f719039": "E(r)\\neq o\\left(r^{1/2}(\\log r)^{1/4}\\right),", "0b27aefb6806e8e8edc032737e607a7d": "({v_{0}+v_{i}})10^{pH_{i}}{\\text{ vs. }}v_{i^{}}", "0b27bcdeadef678b4b0487d10460c0b1": "j\\ ", "0b27e1a137a4ec2b3549293e1ab5398f": "=\\left(R_{\\mathrm {S} }+r_{\\pi }\\right)\\left(1+{\\frac {R_{\\mathrm {E} }}{R}}\\right)\\ ", "0b28276780ee2d33b2c731fbe038004b": "[H_{\\lambda },H_{\\mu }]=0{\\text{ for all }}\\lambda ,\\mu \\in \\Delta ", "0b28d0b65a19b37b3298372f6f6d4dc3": "\\int _{a}^{b}{\\sqrt {E\\,u'(t)^{2}+2F\\,u'(t)v'(t)+G\\,v'(t)^{2}}}\\,dt.", "0b28df2945cc8dc8a74150a229542793": "{\\dot {m}}=\\lim \\limits _{\\Delta t\\rightarrow 0}{\\frac {\\Delta m}{\\Delta t}}={\\frac {{\\rm {d}}m}{{\\rm {d}}t}}", "0b28ebdcf58d96ce257dc32270881447": "f={\\frac {Gr^{2}}{r^{4}+a^{2}z^{2}}}\\left[2Mr-Q^{2}\\right]", "0b2911d97f2720d8b916b1c7255dbae2": "{\\hat {a}}_{i}\\leq a_{i}+\\epsilon |a|", "0b29236672ecf8ef0f1fefba7fdbcf03": "\\sigma _{D}^{(k)}", "0b2928186ff178ba04dd8784c4ed1ad5": "p(a,b)\\ ", "0b29372c93042a9778d4c1869d945604": "\\pi \\,\\mathrm {sr} ", "0b2941f34bde6aca723264dc6c64094b": "\\mathbb {R} ^{3}\\times (0,T)", "0b294e8cb1d79a1323265f5c34b4cff3": "D(af+bg)=a\\cdot Df+b\\cdot Dg", "0b2a80fa78d0d070c8411f415b431033": "h\\in C_{0}^{\\infty }(\\Omega ),", "0b2ac02010ae83184d1a37e5929689f4": "\\int \\cosh(ax+b)\\sin(cx+d)\\,dx={\\frac {a}{a^{2}+c^{2}}}\\sinh(ax+b)\\sin(cx+d)-{\\frac {c}{a^{2}+c^{2}}}\\cosh(ax+b)\\cos(cx+d)+C\\,", "0b2acc9ad0136806ac97d3cc12bfaef7": "\\mathbf {F} _{q^{f}}", "0b2b49e9804f7b52c60587afb300572c": "\\Phi (z)=1-{\\mbox{erf}}(z/{\\sqrt {c}})", "0b2b5ea98816f56e406c3ff216cd0e6a": "{a_{0}\\gg 1}", "0b2ba2852a16ef75fbc23710d711e74f": "I_{L_{Max}}+{\\frac {\\left(V_{i}-V_{o}\\right)\\delta T}{L}}=0", "0b2c131ed63c888e310641b056a64224": "OOO\\ldots \\Diamond \\varphi ", "0b2c15c15aa09b9d946fbf35557fc25c": "i=n", "0b2c241d365373d87c151ed6b1c50a51": "|\\nu |\\geq q^{(1-\\varepsilon )k}", "0b2c4725c762c6fc48c6edfffe3567fc": "k_{1}0", "0b315ab77a0fbabccfe3baad902f518b": "{\\Delta p}_{D}\\,", "0b319be72b5e417ebb4efd9ea4a20fc6": "x_{1}=N_{1}/N=p_{1}\\;\\;and\\;\\;x_{2}=N_{2}/N=p_{2}\\,", "0b31d52116763ba9c239b8a0d733d4cd": "\\langle \\alpha ,\\beta ,\\gamma ,\\delta \\rangle ", "0b32123d093442bad32b4f36773bbf47": "\\mu _{x}+i\\mu _{y}", "0b322569cc973a3780b69dc16b13af02": "\\alpha ,{\\bar {\\alpha }}", "0b323adbc99127564984a808b4fed4ed": "1.5\\mu m", "0b3246ff754cb8eb28a5b9a60e3cd487": "\\alpha (\\omega )=\\mathrm {Im} \\left[\\sum _{\\lambda }{\\frac {F_{\\lambda }}{E_{\\lambda }-\\hbar \\omega -\\mathrm {i} \\gamma _{\\lambda }(\\omega )}}\\right]", "0b3248d59bd55b92e5e5ca559f1913da": "\\left[t_{i}-{\\tfrac {\\delta }{2}},x_{j}\\right],\\quad \\left[x_{j},t_{i}+{\\tfrac {\\delta }{2}}\\right].", "0b327f127472e0d81eb45dfd30ab028b": "\\phi ^{*}\\phi ", "0b32ae4617c6b52a3827afa3b23dc49c": "{\\frac {X^{2}}{2\\lambda a}}+{\\frac {Y^{2}}{2\\lambda b}}+{\\frac {Z^{2}}{2\\lambda a}}=0.", "0b32b8738a92bd1d373d79c4b4aec78b": "f(x)={\\sqrt {1-x^{2}}}", "0b32c5e7997c2b0bea66872bb01da630": "K\\partial _{t}^{+}u_{n,i+1/2}+L\\partial _{x}^{+}u_{n+1/2,i}=\\nabla {S}(u_{n+1/2,i+1/2}),", "0b32e239ba54437908d9d6bce9e573f5": "l_{w}", "0b33216ab0930e7b71253505af6a7d24": "f_{1}\\circ g", "0b3343255b4f896c721249e22ba2643e": "x'\\,", "0b338455700e4c0bbef2417dcb5fb529": "\\|u\\|_{C^{0,\\gamma }(\\mathbf {R} ^{n})}\\leq C\\|u\\|_{W^{1,p}(\\mathbf {R} ^{n})}", "0b3392aca95f6d2ad2fdbb923a565f99": "\\pm e_{p}\\pm e_{q}\\pm e_{r}\\pm e_{s}", "0b33ad1a7ecaef3570b7837bac55c923": "\\{z\\in \\mathbb {C} ||\\phi (z)|>\\mathrm {e} ^{z}\\}", "0b33ddd03325ac3d24e36a4c20cb02d6": "\\alpha \\!\\left(\\lambda \\right)={\\frac {\\partial \\log S\\!\\left(\\lambda \\right)}{\\partial \\log \\lambda }}.", "0b33fe96f9ddcea346f1a8a850fff65b": "\\scriptstyle \\mathbf {I} _{3}", "0b34512ea29020e8a0bb065d93a272a4": "{q_{2} \\choose 1}=k{\\begin{pmatrix}A&B\\\\C&D\\end{pmatrix}}{q_{1} \\choose 1}", "0b3485a4e4e29af51f0f664ace9e0527": "m_{ij}:=\\mathbb {E} (x_{ij})", "0b34a4e99b8218212e4549762bf21d70": "\\mu _{\\text{bkg}}", "0b34af9e7e0db9ae77f28c47dfb8807c": "v_{\\mathrm {N} }", "0b34d783ab3b382d1d262ba78ef6ea39": "{\\hat {A}}{\\hat {B}}{\\hat {C}}{\\hat {D}}{\\hat {E}}{\\hat {F}}\\ldots ", "0b351ba74cf928742dc1137471703f6e": "{\\frac {1}{2}}(\\mathbf {ab} -\\mathbf {ba} )", "0b352ac842981f494ea4e2412767cbaa": "i[p_{0},x_{i}]={\\frac {p_{i}}{p^{0}}}=v_{i}", "0b352cad00c6c287f6903f0763ea40b4": "~d\\theta _{n}=n\\lambda ,~n=0,1,2,\\ldots ", "0b35526473b06bf45ca6ff062a5ecc86": "{\\begin{aligned}X^{\\rm {VV}}&=X^{BS}+p_{vanna}X_{vanna}\\Omega _{vanna}+p_{volga}X_{volga}\\Omega _{volga}\\end{aligned}}", "0b355326b4d0cea1c5ee21cac6cbc74f": "\\Lambda ={\\frac {ch\\beta }{2\\,\\pi ^{1/3}}}", "0b355b7f4b9d7286c810baf9a7b152fe": "I=S{\\sqrt {t}}\\ ", "0b3576a0921e1559618e21809f63cfc1": "\\bigcap A_{\\alpha }^{C}=\\left(\\bigcup A_{\\alpha }\\right)^{C}", "0b35b203428ea307180fb6bad619a003": "{\\dot {x}}_{2}(t)\\neq 0", "0b362b7e68df25b49de5540e2be58c23": "Y^{2}\\sim \\lambda \\chi _{1}^{2}", "0b365b79213974e5d4470843ff99c69b": "2^{-32}", "0b3685434c1693d6ad9fbd407b57a6fa": "n({\\vec {r}})", "0b36b2ead10e88ae4f79925bad77ee3a": "\\,\\log 2\\,", "0b36ee693126b34b58f77dba7ed23987": "\\textstyle i", "0b36f11bd33dd8abb296800e2f340e2a": "a^{b^{c}}=a^{(b^{c})}.\\,", "0b3710ff4f482383b3fdfb9c08ff44f1": "\\mu (A)=\\infty \\,\\,", "0b373fc6150d7af6e2b8e3d5677ffefd": "g:S'\\to S", "0b379ea6857d56503227628fe9e5d78b": "p_{n}=P(Y=n)=e^{[-a_{1}+a_{2}]}\\sum _{j=0}^{[n/2]}{\\frac {a_{1}^{n-2j}a_{2}^{j}}{(n-2j)!j!}}", "0b37ca40516473c15b5227f99db7bf3f": "m_{0}={\\frac {E_{0}}{c^{2}}}\\!", "0b38606109a505d6fa951887c28ce859": "e:G_{1}\\times G_{2}\\rightarrow G_{T}", "0b38675dbc370a7e1c0abe8aae2b59a0": "\\mathbf {\\lambda } ", "0b38a9b62f4c408a78614a8169b2bad6": "P={\\frac {2St}{D}}", "0b390514b9b7c4aca74936148d1661fc": "q={\\sqrt {g}}", "0b39aed8dcb38e489628d79d43b2a78d": "k(n)=\\Theta (\\log m'(n))", "0b39af0e243d684bd94af62623260d40": "A^{B}=\\bigcup _{L\\in B}A^{L}", "0b39cba96270c26a1425484611812e54": "[X;Gr_{n}]", "0b3a84d5453bb30accd25ea4fe20d064": "\\triangle ABC\\cong \\triangle DEF\\,", "0b3aed796f2182cc70231e1537b50131": "\\Im e^{i\\omega }=\\sin \\omega ,", "0b3be80ee121f0032102a9aaefaf10d3": "{\\frac {(-1)^{a-1}\\psi ^{(-a-1)}(x)}{\\Gamma (-a)}},\\,a\\in \\mathbb {Z} ^{-}", "0b3c2b8272ca73ae55cef750a165bd89": "I(X_{1};\\cdots ;X_{n})=I(X_{1};\\cdots ;X_{n-1})-I(X_{1};\\cdots ;X_{n-1}|X_{n}),", "0b3c45e4a2d5247ce8e024fcb23a86ba": "\\sum _{n=1}^{\\infty }{\\frac {1}{n}}=\\infty .\\!", "0b3c465dd4c4bded3b4005541f0b24e0": "S_{F}(z)", "0b3c7780ec18ee6ee6e132740571cc25": "{\\begin{aligned}{\\boldsymbol {\\nabla }}\\mathbf {v} &={\\cfrac {\\partial v_{r}}{\\partial r}}~\\mathbf {e} _{r}\\otimes \\mathbf {e} _{r}+{\\cfrac {1}{r}}\\left({\\cfrac {\\partial v_{r}}{\\partial \\theta }}-v_{\\theta }\\right)~\\mathbf {e} _{r}\\otimes \\mathbf {e} _{\\theta }+{\\cfrac {\\partial v_{r}}{\\partial z}}~\\mathbf {e} _{r}\\otimes \\mathbf {e} _{z}\\\\[8pt]&+{\\cfrac {\\partial v_{\\theta }}{\\partial r}}~\\mathbf {e} _{\\theta }\\otimes \\mathbf {e} _{r}+{\\cfrac {1}{r}}\\left({\\cfrac {\\partial v_{\\theta }}{\\partial \\theta }}+v_{r}\\right)~\\mathbf {e} _{\\theta }\\otimes \\mathbf {e} _{\\theta }+{\\cfrac {\\partial v_{\\theta }}{\\partial z}}~\\mathbf {e} _{\\theta }\\otimes \\mathbf {e} _{z}\\\\[8pt]&+{\\cfrac {\\partial v_{z}}{\\partial r}}~\\mathbf {e} _{z}\\otimes \\mathbf {e} _{r}+{\\cfrac {1}{r}}{\\cfrac {\\partial v_{z}}{\\partial \\theta }}~\\mathbf {e} _{z}\\otimes \\mathbf {e} _{\\theta }+{\\cfrac {\\partial v_{z}}{\\partial z}}~\\mathbf {e} _{z}\\otimes \\mathbf {e} _{z}\\end{aligned}}", "0b3d21ed288a486099e963cd716b9268": "|\\psi \\rangle =\\sum _{j=1}^{N}b_{j}|j\\rangle ,\\quad b_{j}\\in \\mathbb {C} .", "0b3d7aae4fb2032a621d138e96a85ff5": "\\pi ^{s}", "0b3d86190245fdcc18dfb6bc9bdbbf90": "1", "0b4370ad219b723cc1117596a6f937e2": "f_{c}(k,r)", "0b437bd59162b6fbe27c269ce648424d": "(\\xi ,\\zeta )", "0b43a0ebe087869e7d6c770de9c77507": "x_{i}^{\\beta _{i}}", "0b43e70dc01855a0d0dffbffb9da3ca5": "Y=1000", "0b448bad3764cfcb21d9270815ef81d3": "\\mathbb {Z} ^{d}", "0b449bc28c66557d4cac6dff2f4ef27d": "\\Delta (r\\#h)=(r^{(1)}\\#r^{(2)}{}_{(-1)}h_{(1)})\\otimes (r^{(2)}{}_{(0)}\\#h_{(2)}),\\quad r\\in R,h\\in H.", "0b4538c7253805be3412ba2226faaf93": "\\Phi _{3\\times 7\\times 31}(x)", "0b455d59c5f3d4aa7111d861ea44780a": "x+y+z=1;\\,\\!", "0b45f53446f36f1b1f94f63de8b50840": "{1 \\over 6}", "0b46807325355c03ae4b3a9cc181a459": "\\scriptstyle f,g\\in C^{1}(\\Omega )", "0b46e4979f9aa9cd935e01ba29bd035c": "-i(r{\\bar {b}}-b{\\bar {r}})/{\\sqrt {2}}", "0b46e7e78079210bcd1e16f7b0936632": "P\\in \\Pi (A),", "0b46f7199cbd1d1daf0700d93bcebcbe": "K(z)=(2\\pi )^{(-z+1)/2}\\exp \\left[{\\begin{pmatrix}z\\\\2\\end{pmatrix}}+\\int _{0}^{z-1}\\ln(t!)\\,dt\\right].", "0b47b135b10f2bc90fa7fcfb3c9fb999": "{\\mathcal {T}}=e^{-\\tau }", "0b47ee3a2a1fb6e2996273ae19a1b3e0": "\\displaystyle {{\\widehat {Rf}}(z)={{\\overline {z}} \\over |z|}{\\widehat {f}}(z),\\,\\,\\,{\\widehat {R^{*}f}}(z)={z \\over |z|}{\\widehat {f}}(z).}", "0b480b680753e6b32893cee15782b019": "\\left(\\left(B\\rightarrow \\lnot A\\right)\\land \\lnot C\\right),\\left(B\\lor C\\right)\\vdash \\lnot A", "0b4820c7770e2712070011be456b5b48": "||{\\Phi _{\\alpha _{l-1}}^{[l-1..N]}}||", "0b483c7ddef11a9b4906fcc0b718739a": "s,t\\in B", "0b48592d55ea07026adfa4153d8ff58b": "f'(x)\\neq 0.", "0b485f9e9d945234b168961c69e322a8": "e^{i\\omega t}u,", "0b48cba4d29474c277c9654b1d414cda": "(Y_{i})", "0b48d9c3258d89b22772e52a7b450263": "[\\eta ,\\eta ]", "0b490cb3b069a46b5b1806fab6e72858": "1=-1", "0b497c52e8285d21db629e8a36475d36": "Ra=\\{ra\\mid r\\in R\\}\\,", "0b49aaba7dbc4888a9714a28ff623044": "R_{\\kappa \\mu \\nu }^{\\lambda }=2\\gamma ^{\\lambda \\sigma }U_{,\\sigma [\\mu }\\Psi _{\\nu ]}\\Psi _{\\kappa }", "0b49dbb29aad13ea115b8958b56d2fdd": "{\\begin{pmatrix}y_{1}\\\\y_{2}\\\\\\vdots \\\\y_{m}\\end{pmatrix}}={\\begin{pmatrix}X_{1}&0&\\ldots &0\\\\0&X_{2}&\\ldots &0\\\\\\vdots &\\vdots &\\ddots &\\vdots \\\\0&0&\\ldots &X_{m}\\end{pmatrix}}{\\begin{pmatrix}\\beta _{1}\\\\\\beta _{2}\\\\\\vdots \\\\\\beta _{m}\\end{pmatrix}}+{\\begin{pmatrix}\\varepsilon _{1}\\\\\\varepsilon _{2}\\\\\\vdots \\\\\\varepsilon _{m}\\end{pmatrix}}=X\\beta +\\varepsilon \\,.", "0b49dd9c4d7c313032638ed763dbfefc": "m_{\\rm {e}}={\\frac {2R_{\\infty }h}{c_{0}\\alpha ^{2}}}", "0b49ef655fe7a8625dd45c3536dd2611": "Q_{n}(z)\\left(c_{0}+c_{1}z+c_{2}z^{2}+\\cdots +c_{m+n}z^{m+n}\\right)=P_{m}(z)", "0b4a58e47dc473931b0f945171c3ae5e": "{\\begin{aligned}f_{X_{1}^{n}}(x_{1}^{n})&=\\prod _{i=1}^{n}{\\tfrac {1}{\\sqrt {2\\pi \\sigma ^{2}}}}\\,e^{-(x_{i}-\\theta )^{2}/(2\\sigma ^{2})}=(2\\pi \\sigma ^{2})^{-n/2}\\,e^{-\\sum _{i=1}^{n}(x_{i}-\\theta )^{2}/(2\\sigma ^{2})}\\\\&=(2\\pi \\sigma ^{2})^{-n/2}\\,e^{-\\sum _{i=1}^{n}((x_{i}-{\\overline {x}})-(\\theta -{\\overline {x}}))^{2}/(2\\sigma ^{2})}\\\\&=(2\\pi \\sigma ^{2})^{-n/2}\\,\\exp \\left({-1 \\over 2\\sigma ^{2}}\\left(\\sum _{i=1}^{n}(x_{i}-{\\overline {x}})^{2}+\\sum _{i=1}^{n}(\\theta -{\\overline {x}})^{2}-2\\sum _{i=1}^{n}(x_{i}-{\\overline {x}})(\\theta -{\\overline {x}})\\right)\\right).\\end{aligned}}", "0b4a6de4fb58a851b6b85c1830e882c1": "KE={\\frac {3}{5}}E_{F}={\\frac {3}{5}}{\\frac {\\hbar ^{2}k_{F}^{2}}{2m_{e}}}={\\frac {2.21}{r_{s}^{2}}}{\\textrm {Ryd}}", "0b4a767cc89a2f6babf4be5bf11bd46d": "{\\tilde {E}}_{i}^{a}\\mapsto {\\tilde {E}}_{i}^{a}/\\beta ", "0b4aa42c8dbf765ee551b156555a8e05": "\\mathbb {A} _{Y}^{n}", "0b4b4f642cc158eb20ae3e76522b9580": "3\\omega ", "0b4b56414ff00d2b87d0fa16dab2d83b": "{\\hat {k}}\\cdot \\nabla \\omega \\times {\\frac {\\partial V}{\\partial p}}={\\frac {\\partial \\omega }{\\partial y}}{\\frac {\\partial u}{\\partial p}}-{\\frac {\\partial \\omega }{\\partial x}}{\\frac {\\partial v}{\\partial p}}", "0b4b8247106f143dbad733e6aa383a07": "z\\mapsto bz", "0b4b879fc55f9b929c29ccb03b53bbbf": "j_{r}={\\frac {-(1+e)\\mathbf {v} _{r}\\cdot \\mathbf {\\hat {n}} }{{m_{1}}^{-1}+{m_{2}}^{-1}+({\\mathbf {I} _{1}}^{-1}(\\mathbf {r} _{1}\\times \\mathbf {\\hat {n}} )\\times \\mathbf {r} _{1}+{\\mathbf {I} _{2}}^{-1}(\\mathbf {r} _{2}\\times \\mathbf {\\hat {n}} )\\times \\mathbf {r} _{2})\\cdot \\mathbf {\\hat {n}} }}", "0b4bb8d3371748c78de7498a9b6b87be": "z'_{k+1}=f'(z_{k})z'_{k}", "0b4be7198491383c310e2a553098acfa": "\\langle j_{1}m_{1}j_{2}m_{2}|JM\\rangle ", "0b4c6e3346811cd92440cbb7bf202fbf": "T=L_{0}/v", "0b4c9bdb81382bb918cf6d666f9d9ddb": "\\beta _{2}(T_{e})\\approx 2\\times 10^{-16}T_{e}^{-3/4}\\ \\mathrm {[m^{3}s^{-1}]} ", "0b4d44d9509a20bd4952af13a21ed7d6": "F={\\frac {1}{4\\pi \\varepsilon _{r}\\varepsilon _{0}}}{\\frac {q_{1}q_{2}}{r^{2}}}", "0b4d4705a0400a9f7b0d4b0dcd7e9faf": "{\\begin{bmatrix}A&U\\\\V&C\\end{bmatrix}}={\\begin{bmatrix}I&0\\\\VA^{-1}&I\\end{bmatrix}}{\\begin{bmatrix}A&0\\\\0&C-VA^{-1}U\\end{bmatrix}}{\\begin{bmatrix}I&A^{-1}U\\\\0&I\\end{bmatrix}}", "0b4d5ac429d333fac141e0fdddaaa11b": "Ar\\rightarrow \\exists xAx", "0b4d9331ae0dcc86b4d23eeb633c3fda": "v\\in T^{0}\\left(V\\right)", "0b4de37b4e42b7d92be622a3b145446e": "{\\begin{bmatrix}R\\\\G\\\\B\\end{bmatrix}}={\\begin{bmatrix}1.0498110175&0.0000000000&-0.0000974845\\\\-0.4959030231&1.3733130458&0.0982400361\\\\0.0000000000&0.0000000000&0.9912520182\\end{bmatrix}}{\\begin{bmatrix}X\\\\Y\\\\Z\\end{bmatrix}}", "0b4e3a5cc248dd732bee56ba460c0b6b": "\\vert \\omega _{n}\\vert ^{-s}", "0b4e6282f174e5939e96d35d7fc1befb": "\\varepsilon _{total}=(-1)^{n+1}\\prod _{i=1}^{n}\\varepsilon _{i},", "0b4e73f935b9bd42529b784cdcf99784": "\\lambda _{k}\\alpha _{k}\\alpha _{k}'", "0b4e77b610ea5f8d13a1eb8730f8caa3": "P^{2m}\\,\\!", "0b4ed3dabb89aec97ad9fc9e37d7c145": "\\delta \\ ", "0b4f31fb64207229e6341ff4a9d08a6c": "m-M=5\\log _{10}d_{L}", "0b4f4100756a7a65689d964352244d61": "e^{{-E_{a}}/{(RT)}}\\ \\ \\ ", "0b4f9454cb92e7210ef2beabfbe33d26": "[micelle]", "0b4fcca6163464bf22afde8686b7c117": "\\omega ,", "0b502b678c8955fca4be0e828fd25089": "m=K_{1}+K_{2},\\ b={\\frac {1}{2}}{\\frac {\\sum \\lambda _{i}}{\\sum \\lambda _{i}^{2}}}", "0b5048058ec1d207710ccf28bec890e0": "\\displaystyle {P(p,q)={\\begin{pmatrix}A_{pp}&B_{pq}\\\\0&D_{qq}\\end{pmatrix}}}", "0b50f6f218d322b03688752cddd7a06e": "\\Delta w_{k}", "0b5102bc014ec7e93cff02cc06d3b726": "p_{0}=\\left(v\\rho \\omega \\right)s_{0}\\,\\!", "0b511a3eab3f963551086c615faf80e2": "f(i)M^{k+1}(i,j)=\\sum _{n=0}^{N}M^{k}(n,i)(f(n)M(n,j))", "0b513d869d9c68fb6d1a500d9fc2c2dc": "V_{wd}", "0b51776edcd81842ab89bd8f0f1ebe9e": "X\\cdot \\alpha ={\\begin{bmatrix}X_{00}\\,\\alpha &X_{01}\\,{\\hat {\\alpha }}\\\\X_{10}\\,\\alpha &X_{11}\\,{\\hat {\\alpha }}\\end{bmatrix}}.", "0b51878fe9cfe8092c1aa6a72bfe5ce4": "D_{2}\\psi ={\\frac {B}{\\lambda +\\alpha }}\\psi ", "0b5198913caacbfdc98b89688e427ab5": "C\\equiv C_{i}\\,{\\bmod {\\,}}N_{i}", "0b520835d59a4f7e03e4cdead48b4699": "x_{i+1}=y_{i+1}\\oplus y_{i}", "0b520e27e385332015d1e92702212db3": "\\lambda _{k}=\\sum _{j=0}^{N-1}c_{j}\\gamma ^{kj}", "0b52664134c34dd9e3b7ae3525aa06c4": "L_{M}\\;", "0b52a7be011ff9fb54334bb6523f31fd": "d\\rho ^{2}=\\left(1+h^{\\prime }(r)^{2}\\right)\\,dr^{2}+r^{2}\\,d\\phi ^{2},\\;r_{1}", "0b58596743f39389da0f13f74d2ccdfe": "e^{\\bar {\\alpha }}=L^{\\bar {\\alpha }}{}_{\\beta }e^{\\beta }", "0b58bc6627165700f64e10e69c59e764": "{\\frac {\\left(x+{\\sqrt {a^{2}+b^{2}}}\\right)^{2}}{a^{2}}}-{\\frac {y^{2}}{b^{2}}}=1", "0b594a8e6ab09a0103551d0aaf155aac": "\\det(cA)=c^{n}\\det(A)", "0b59500cb2c74858986609d125f13794": "(P,J,E)", "0b598f0913ad093d7637585a7c89e143": "\\Pi _{i}={\\frac {\\eta _{i}k_{i}}{\\sum _{j}\\eta _{j}k_{j}}}.", "0b5aa703832eb3248df7472b3d7199a2": "\\nabla _{{\\vec {e}}_{0}}{\\vec {e}}_{1},\\;\\nabla _{{\\vec {e}}_{0}}{\\vec {e}}_{2},\\;\\nabla _{{\\vec {e}}_{0}}{\\vec {e}}_{3}", "0b5ab034c9c5ed41d722e7065cda0c8f": "11", "0b5c5e49946b8ec71fee24dabe7239ec": "G_{T}\\approx G_{a}+{\\bar {b}}^{2}G_{C}+{\\bar {C}}^{2}G_{b}", "0b5c82e45f633a4ca6b49dc55febcd9f": "B(x)=-\\log x+xB(1/x)", "0b5c84fc3e26501a42c5da7fde17b735": "-f(x)", "0b5ca8b5c58140508cdd62446744ed59": "X_{a,b}=Y_{1,b}^{1/a},", "0b5cb9bb91a9eb32198f4467839649c1": "10^{21}-10^{24}", "0b5d1b8e0d8708d71a9d9899dd7d42d3": "T(t=0)=T_{1}\\cdot \\Theta (-z)", "0b5d5663844e90972c92593a9a98eba1": "{\\mathcal {S}}_{E}", "0b5d676e57a419922b786b5a4ddeec2e": "\\operatorname {codim} (N)=\\dim(M)-\\dim(N).", "0b5dbc95b984bf150d5815fec585407a": "4\\pi G=c=\\hbar =\\varepsilon _{0}=1", "0b5dc9a79b2441c4e04cc15ee2a3e2bc": "a_{n}(q),\\,b_{n}(q)={\\frac {2mEl^{2}}{\\hbar ^{2}}}-{\\frac {2m^{2}gl^{3}}{\\hbar ^{2}}}", "0b5dd45570bf4d63b247b96df1293589": "O_{r}^{*}=\\min _{O_{r}\\in N_{in}}d(O_{r},O_{n})-r(O_{r})", "0b5df7868e30c503f4b88730220b67cb": "k\\cdot 2^{n}+1", "0b5df7f323131fea5e143e6b778c4aa3": "{\\hat {\\beta }}(q)", "0b5e023a7f4801d1a76947c84f319d42": "\\gamma \\to -\\infty ", "0b5e14f4dac6e7a27ef9f5888c1ee8c3": "{\\tilde {e}}", "0b5e57a269146a42c483657ea3ecfec9": "C_{Di}={\\frac {C_{L}^{2}}{\\pi eAR}}", "0b5eb6c22c5d5483a0696f1dcd6b8a35": "e=2^{8}+1", "0b5efb365d0830d1c5030b78ed56cb21": "p^{H}(r)=\\left({\\cfrac {3F^{H}}{2\\pi a^{2}}}\\right)\\left(1-{\\cfrac {r^{2}}{a^{2}}}\\right)^{1/2}", "0b5f022d687df82e544eeca553fffd04": "\\sum _{0}^{\\infty }(-x)^{n}", "0b5f71fef68b47c962faaa14d15d9262": "D^{1/2}", "0b5ff567a1734093359d384a92bed018": "(l_{1}+l_{2})^{\\theta }f(x)=f((l_{1}+l_{2})x)=f(l_{1}x)+f(l_{2}x)=(l_{1}^{\\theta }+l_{2}^{\\theta })f(x)", "0b6042563f051dfb18da92766c7d5f58": "x_{0}={\\frac {\\hbar k_{y}}{m\\omega _{c}}}", "0b6050078de4499eb0912b8ee7027a27": "B[x/a]", "0b605fdfebe77225086319d595c41a96": "v\\wedge w", "0b60878e9116c231af283a4395d0ca96": "Q=q^{N}/N!", "0b60ad9f2ad7854a48c85e3d36560585": "g=f\\circ \\psi ", "0b60ccd69f728800c74993348d55560f": "\\,x\\geq 2.\\,", "0b615f2d18fce9a50f34e032b96d5273": "\\mathbf {u} =\\sum _{k}\\langle \\mathbf {v} \\mid \\mathbf {e} _{k}\\rangle {\\tilde {\\mathbf {e} }}_{k}", "0b616b5b31e08d8ab7214ba854ff03a5": "2\\cdot m\\ ", "0b61715d93ea950011b950efbbd3d865": "\\left[{n \\atop j}\\right]", "0b61c05a0a6fc25a93b91be417c22acf": "\\sigma \\times \\sigma ", "0b61de5eb16898d82b661c53c8830b13": "\\Delta ^{2}=0", "0b61fb4685f42449df062ba690b2a00b": "y\\in [0,1]", "0b621240763da1148bf3bd77fe1675f6": "f(t)=\\int _{a}^{t}K(t,s)\\,x(s)\\,ds", "0b6214965799cc6f65709439e4ae45bf": "\\textstyle y_{i+1}=y_{i}+h{\\text{Slope}}_{\\text{ideal}}", "0b6261c1922d72efc5ad25e8f4ffc50b": "\\left({\\frac {-b}{2a}},{\\frac {-D}{4a}}\\right)", "0b6264b77360383344976e727640d7a6": "H(z)={\\frac {(z+1)^{2}}{(z-{\\frac {1}{2}})(z+{\\frac {3}{4}})}}", "0b62b3d09050e46eb8811411db2e7747": "GCD(n,b)=1", "0b62f4565ab2ee04fad0c8c7bbc9bf06": "\\left(a,z\\right)=\\left(c,x\\right)", "0b632de5206d36d5c867d33a9e8935b9": "M([\\Gamma (1)])=\\mathrm {Spec} (R[j])\\,", "0b635bacdb4dd7d15eb8b395c9f46fab": "X={\\begin{cases}1&{\\text{with probability }}p,\\\\0&{\\text{with probability }}1-p,\\end{cases}}", "0b63b73a88c26b47925a9e302be22e1c": "N=", "0b64043437f8b6719eca36c49922cb52": "F_{Y}(y)", "0b641d1199526ae18b344c1d59ef5f81": "B_{r}^{p,q}=d_{0}^{p,q}(Z_{r-1}^{p-r+1,q+r-2}).", "0b64c093edf1d116179e53ae0f1bad3e": "{\\tilde {D}}_{6}", "0b64de3b3a7b9c3e1f7637cff37191db": "E=E_{o}\\;E_{u}", "0b64e5e33d57439d698c5f8e627f6d9c": "L^{*}G(x,s)=\\delta (x-s),", "0b64fc5c88797971e95f2a1c4db33015": "{\\frac {1}{i}}[M_{\\mu \\nu },M_{\\rho \\sigma }]=\\eta _{\\mu \\rho }M_{\\nu \\sigma }-\\eta _{\\mu \\sigma }M_{\\nu \\rho }-\\eta _{\\nu \\rho }M_{\\mu \\sigma }+\\eta _{\\nu \\sigma }M_{\\mu \\rho }\\,", "0b654724b02f2476bb1732f98b89fc54": "Y\\subseteq X\\subseteq H", "0b65a75973388be6d2a7c13d69503df9": "\\liminf _{x\\to +\\infty }x^{2}q(x)>{\\tfrac {1}{4}}.", "0b65d521b64b955ddb3c1db6ab4be245": "f(x)=f(p)+f'(p)(x-p)+o(x-p).", "0b65ea8ba9dcbae4c186be4ae3ebcc78": "(a_{1}+ib_{1})+(a_{2}+ib_{2})=(a_{1}+a_{2})+i(b_{1}+b_{2}),", "0b667a3a47daed6d7d1b1ecb6899d19b": "\\mu ^{+}(E)=\\sup _{B\\in \\Sigma ,B\\subset E}\\mu (B)", "0b668d48204bd73fe63c81584ee5c552": "\\langle n_{i}\\rangle ={\\frac {1}{e^{(\\epsilon _{i}-\\mu )/k_{B}T}+1}},", "0b66b723daa0490205546121416b5c06": "\\Pi _{H}(2m)\\,\\!", "0b66c2051c7a0333fb28047b33eeb19e": "\\mu (A)>\\mu (B)>0.\\,", "0b6717c5b70b6a8b60a6434579dd268a": "TM_{mnp}", "0b673d1dadfee24055406c633d5980b9": "n_{ij}=|X_{i}\\cap Y_{j}|", "0b674220ba0c38156ebcc2198df97600": "4\\sin ^{-1}\\left({1 \\over 3}\\right)", "0b675c6942f1cdda7c3ce4cedf3e60da": "\\mathbf {m} ^{\\phi }", "0b67d686b322c47dbe48350b9102df4d": "{\\frac {1}{16}}\\int _{0}^{1}{\\frac {x^{12}(1-x)^{12}}{1+x^{2}}}\\,dx={\\frac {431\\,302\\,721}{137\\,287\\,920}}-\\pi ", "0b67d78e715d104395a4550d2565f36a": "C=\\cosh ^{-1}{\\frac {Q_{b}}{Q_{t}}}=3.0022", "0b67ed251272f8b32816542f9b4551bc": "\\ \\exists x\\in S", "0b6808e17f615a59c53ee277db89279d": "\\mu (t)=e^{\\int _{a}^{t}{\\Gamma _{\\gamma }(s)}ds}", "0b680cf4e5d86fa9fd7c424f84056884": "\\Omega ^{p,q}=\\Omega ^{1,0}\\wedge \\dotsb \\wedge \\Omega ^{1,0}\\wedge \\Omega ^{0,1}\\wedge \\dotsb \\wedge \\Omega ^{0,1}", "0b6813a6b7451fa84910d4886bdec9f4": "P'Q'", "0b681556ac0e4af173508210691dc21e": "{\\frac {\\partial \\phi }{\\partial t}}=\\nu {\\frac {\\partial ^{2}\\phi }{\\partial x^{2}}}+f(t)\\phi ", "0b688f26c5b321af582b842fc74dbdd5": "ds^{2}=-q\\,\\sin(\\omega u)^{2}\\,du^{2}-2\\,du\\,dv+dx^{2}+dy^{2},", "0b68917c1a3c3f18b30876c87b711db2": "D(P_{\\mathit {max}})=0.", "0b689cfe83964e9a104a3235c7eb192c": "E=n_{1}\\epsilon _{1}+n_{2}\\epsilon _{2}+\\ldots +n_{6}\\epsilon _{6}", "0b693616955a1f920aa3d090850b3017": "\\nabla ^{4}A+\\nabla ^{4}B+\\nabla ^{4}C=3\\left({\\frac {\\partial ^{2}A}{\\partial x^{2}}}+{\\frac {\\partial ^{2}B}{\\partial y^{2}}}+{\\frac {\\partial ^{2}C}{\\partial z^{2}}}\\right)/(2-\\nu ),", "0b69e718c59eedc900655d007d0ccd13": "{f_{n}}(10)", "0b69ed85215d84927bc3d7f0c70a025e": "(S,\\times )", "0b6a19a2be7f050a5185b1c0b3235742": "{\\hat {\\mu }}_{MAP}={\\frac {n\\sigma _{m}^{2}}{n\\sigma _{m}^{2}+\\sigma _{v}^{2}}}\\left({\\frac {1}{n}}\\sum _{j=1}^{n}x_{j}\\right)+{\\frac {\\sigma _{v}^{2}}{n\\sigma _{m}^{2}+\\sigma _{v}^{2}}}\\mu _{0}.", "0b6a3da19681846c5ace240a86199720": "\\beta (2k+1)={{({-1})^{k}}{E_{2k}}{\\pi ^{2k+1}} \\over {4^{k+1}}(2k)!},", "0b6a5514e51091b8fe805aa502d5282e": "\\Lambda _{E}", "0b6adf7a97ec05a201250417232edfdf": "7^{7^{7}}", "0b6b37065a9be1e67e3d7927eb156a68": "a_{k}^{0}={\\frac {1}{2\\pi i}}\\int _{|z|=c}f(z)O_{k}(z)\\,dz,\\!", "0b6b46cb74b06b90456888e1a8451871": "g(x)={\\sqrt[{3}]{x^{2}}}", "0b6baecfc75c8101ee3c9f0caeb2ff00": "{\\mathcal {M}}_{g;k_{1},\\dots ,k_{n}}", "0b6bb2e6140acec75bc33da8062d4470": "S(\\lambda )=S_{0}(\\lambda )+M_{1}S_{1}(\\lambda )+M_{2}S_{2}(\\lambda )", "0b6bc1ef048fb3dfd69aefab41c05ded": "\\operatorname {V} _{\\mathbb {P} ^{n}}.", "0b6bde769d746e35d252dc89cac92bb6": "\\left({\\frac {\\partial f}{\\partial r}},{\\frac {1}{r}}{\\frac {\\partial f}{\\partial \\theta }}\\right)\\cdot \\left({\\frac {\\partial }{\\partial r}}(r-R),{\\frac {1}{r}}{\\frac {\\partial }{\\partial \\theta }}(r-R)\\right)=0", "0b6c88979589e6a7ec2f6d64dd2b8339": "F_{N}(j)\\approx N/N\\approx 1", "0b6caf25d1ee05ee016a35872b511036": "\\limsup _{n\\to \\infty }x_{n}:=\\inf _{n\\geq 0}\\,\\sup _{m\\geq n}x_{m}=\\inf\\{\\,\\sup\\{\\,x_{m}:m\\geq n\\,\\}:n\\geq 0\\,\\}.", "0b6ccccae14c323f947c9bfd908804a2": "{\\textit {on}}(1)", "0b6d0394794d86b24d024166d43159fa": "\\mathrm {slog} _{a}x>-2", "0b6d2e45ad5d4e6e032c3c4182efada7": "[P'_{i},P'_{j}]=0\\,\\!", "0b6d652bb1312a01a2433aaeee8cf45b": "i=j=k=l,\\alpha =\\beta =\\gamma =\\delta =0", "0b6dbb40ed7be2440d57913a546bd27f": "y=c\\cosh {\\frac {v}{c}}\\sin u", "0b6df40e3a9a5560b3b91fd653531c4f": "\\sum _{i=1}^{n}w_{i}h(x_{i})=\\sum _{i=1}^{n}w_{i}r(x_{i})", "0b6df66735fe57143f9af3d20fe74b0a": "\\Gamma ^{[{D]}}", "0b6e1acfb23d9f0326d87794384872df": "\\scriptstyle {\\langle \\psi _{m}|{\\dot {\\psi _{n}}}\\rangle }", "0b6e1ff740e4d5e5de339a7270ee2e5a": "\\displaystyle {R^{2}>1+2\\|H\\|_{2^{n}}R}", "0b6e4aed685ec6ab72efe70617e9f2d2": "q_{0}>0", "0b6e6884a8c9955bf578c90ac95c8094": "\\psi _{2,5}=1", "0b6e6cf649347bfd91ddcd676f030ab4": "\\alpha +\\beta :=(\\alpha ,\\beta )\\in A^{*}(X\\times X)\\oplus A^{*}(Y\\times Y)\\hookrightarrow A^{*}((X\\coprod Y)\\times (X\\coprod Y))", "0b6e80017f7bf943c4eb0cb0463aa820": "y'(t)=f(t,y(t)),\\ \\ y(t_{0})=y_{0},", "0b6e8bb0bcbe5c5fd93a2b945bb71c99": "Q(q)=-{\\frac {\\hbar ^{2}}{2m}}{\\frac {\\Delta R}{R}}", "0b6f1c18d140564c9a02b66c0079751a": "S_{\\rho }={\\frac {1}{2}}\\varepsilon _{\\lambda \\mu \\nu \\rho }U^{\\lambda }J^{\\mu \\nu }", "0b6f6c0f23cf3b29f3652c7315c456aa": "\\textstyle V", "0b6f706c387e5fc4de34329b85cda7bd": "b_{i}^{*}(k)={\\frac {\\sum _{t=1}^{T}1_{y_{t}=v_{k}}\\gamma _{i}(t)}{\\sum _{t=1}^{T}\\gamma _{i}(t)}}", "0b6fff4df0982015d7628a4ad461a31b": "d\\mathbf {x} =d\\mathbf {X} +d\\mathbf {u} \\,\\!", "0b7006de07931805fce1f3b177cfde1c": "\\operatorname {PSL} (2,\\mathbb {R} )", "0b700c6fee8029dc0635590c37e78470": "\\delta W=M\\delta \\phi ,\\!", "0b7014a25bef20e32e159157ef1847c8": "S^{k}(U)=\\{f\\in C^{\\infty }(M,N):(j^{k}f)(M)\\subseteq U\\}.", "0b70554efd6d108f1e593b3d4debf190": "2:nat", "0b705b005b9a77aaac8462a1a3379c92": "X^{n-1}\\cup e^{n}", "0b7069acf3cda4e07c03dba565a9c3ba": "x_{\\text{upper}}", "0b7098a64c3819d855cc94ec732d79eb": "t_{b}=\\cos i\\ \\cos u\\,", "0b712cbfd644933a8a95708263367dc2": "\\exp {\\mathfrak {m}}_{+}", "0b7197879e05bde29b9e1ba8b41a70c9": "\\int _{-\\infty }^{\\infty }f_{X}(x)L_{X\\mid Y=y}(x)\\,dx", "0b71bbe9511a2e0f3f6b8a00ec9ea7f4": "\\lambda =1/\\tau .", "0b723599354b5f36b6100668d08d606d": "L_{n}={\\begin{cases}2&{\\mbox{if }}n=0;\\\\1&{\\mbox{if }}n=1;\\\\L_{n-1}+2L_{n-2}&{\\mbox{if }}n>1.\\\\\\end{cases}}", "0b725c6bb5d13b891fdec55fad18b172": "=\\lambda ^{-1}\\mathbf {P} (n-1)\\mathbf {x} (n)-\\mathbf {g} (n)\\mathbf {x} ^{T}(n)\\lambda ^{-1}\\mathbf {P} (n-1)\\mathbf {x} (n)", "0b72709f27ee0cd5d5051072ad206763": "\\langle a_{i},x_{k}\\rangle =b_{i}", "0b7293253c6ac4331548948b2ab33645": "B=(b_{1},b_{2},\\dots ,b_{n})", "0b72a3031b3ddc7646abde00a44ee8a9": "\\sum _{i\\mathop {=} m}^{n}a_{i}=a_{m}+a_{m+1}+a_{m+2}+\\cdots +a_{n-1}+a_{n}.", "0b72b5bb2041cc2cc46cff117dfdf333": "g=(A\\to B,B^{*}A^{+})", "0b72f70632ea534fd5aa33b8b6e87dda": "3(2P)=4P\\boxplus 2P", "0b731d2ebd143daed5111a68cffef1b3": "m_{s}", "0b7355a770d945d1a7487c6267c36192": "x=x_{1}x_{2}\\cdots x_{p}", "0b7360f59db9ca086f599d3cdf738fd5": "{\\mathit {\\Gamma }}={\\frac {jX_{\\mathrm {L} }-R_{\\mathrm {0} }}{jX_{\\mathrm {L} }+R_{\\mathrm {0} }}}", "0b738bfd1c9f8e654b72d49014ae3ee0": "R_{S}=R_{H}\\left(1-{\\frac {\\sin(\\alpha )\\sin(\\theta -\\alpha )}{\\cos(\\alpha )\\cos(\\theta -\\alpha )}}\\right)\\sec(\\alpha )\\,", "0b739b38d8cd7055c8303f36736548e6": "E^{\\perp }=(\\operatorname {Sp} (E))^{\\perp }=({\\overline {\\operatorname {Sp} (E)}})^{\\perp }.", "0b73e3d244b0fe91f14699654c42f621": "{\\begin{aligned}dA_{\\text{ellipse}}&=\\det {\\begin{pmatrix}{\\frac {\\partial {\\mathbf {T}}}{\\partial r}}&{\\frac {\\partial {\\mathbf {T}}}{\\partial \\theta }}\\\\\\end{pmatrix}}\\,dr\\,d\\theta \\\\&=\\det {\\begin{pmatrix}a\\cos \\theta &-ra\\sin \\theta \\\\b\\sin \\theta &rb\\cos \\theta \\end{pmatrix}}\\,dr\\,d\\theta \\\\&=abr\\,dr\\,d\\theta .\\end{aligned}}", "0b742933011d816b807ddc248f5defc4": "x(\\theta )", "0b748af62424599c585ce90450e45bc7": "\\mathbf {\\Sigma } =\\lambda _{1}\\alpha _{1}\\alpha _{1}'+\\lambda _{2}\\alpha _{2}\\alpha _{2}'+...+\\lambda _{p}\\alpha _{p}\\alpha _{p}'", "0b74b6b66713946fddaf3eebd0507ab5": "\\langle f^{*}\\star f\\rangle ={\\begin{bmatrix}a^{*}&b^{*}&c^{*}\\end{bmatrix}}{\\begin{bmatrix}1&\\langle x\\rangle &\\langle p\\rangle \\\\\\langle x\\rangle &\\langle x\\star x\\rangle &\\langle x\\star p\\rangle \\\\\\langle p\\rangle &\\langle p\\star x\\rangle &\\langle p\\star p\\rangle \\end{bmatrix}}{\\begin{bmatrix}a\\\\b\\\\c\\end{bmatrix}}\\geq 0.", "0b74ea3628ef1e7160808180f41be5ca": "1{\\text{ rad}}=1\\cdot {\\frac {180^{\\circ }}{\\pi }}\\approx 57.2958^{\\circ }", "0b74f2b2b4353057526a60a67b01c84c": "\\sinh(\\mathrm {arsinh} (1/\\varepsilon )/n)", "0b7509c4a2f2085cd2ee34085f371766": "t_{1/2e}={\\frac {t_{1/2p}\\times t_{1/2b}}{t_{1/2p}+t_{1/2b}}}", "0b754285f5d11bc7d1e3e08682f96f9e": "{\\begin{aligned}y_{11}\\,&{\\stackrel {\\text{def}}{=}}\\,\\left.{\\frac {I_{1}}{V_{1}}}\\right|_{V_{2}=0}\\qquad y_{12}\\,{\\stackrel {\\text{def}}{=}}\\,\\left.{\\frac {I_{1}}{V_{2}}}\\right|_{V_{1}=0}\\\\y_{21}\\,&{\\stackrel {\\text{def}}{=}}\\,\\left.{\\frac {I_{2}}{V_{1}}}\\right|_{V_{2}=0}\\qquad y_{22}\\,{\\stackrel {\\text{def}}{=}}\\,\\left.{\\frac {I_{2}}{V_{2}}}\\right|_{V_{1}=0}\\end{aligned}}", "0b756e98d772b24e85f9f7ecfc502035": "T_{i}^{(n)}=\\sigma _{ji}n_{j}\\,\\!", "0b7587cb6f3b2d1d5ec17aa0b2be1880": "{\\tfrac {3}{4}}{\\tbinom {6}{3}}", "0b75b1c42c8e3ffbda0ee14d1b055a72": "\\gamma =1/\\lambda ", "0b75c1483fe9060ba5169500b2554896": "(z^{k})^{n}=z^{kn}=(z^{n})^{k}=1^{k}=1.", "0b75fbfb66a0bee29dd40f2e567fdd0e": "[f(x_{1})\\dots f(x_{l+u})]", "0b769abfd7b91b02f417ef457003f14f": "C_{ij}={\\frac {\\partial Q_{i}}{\\partial V_{j}}}", "0b76b99e897d365e388fc443ee3d0cf4": "\\wp '(z)", "0b76baf1bddbd75b6771ed733ad611e5": "{\\hat {Y}}(X_{0})=\\alpha (X_{0})+\\sum \\limits _{j=1}^{d}{\\beta _{j}(X_{0})X_{0}^{j}}", "0b76be8e813b7522591d66b7ffcf6e5c": "u_{k}=M^{-1}r_{k},\\,", "0b76c38abc9a752466ce2f1749b19229": "{\\frac {f(n+1)}{f(n)}}={\\frac {1}{2}}\\,", "0b7762b81b67ed61e897562de92407e2": "(x_{1},x_{2},x_{3},x_{4})=(x,y,z,ict)", "0b777b7178383d7c06b8516b53eaae5d": "R=\\left|{\\frac {\\left(1+y'^{\\,2}\\right)^{3/2}}{y''}}\\right|,\\qquad {\\mbox{where}}\\quad y'={\\frac {dy}{dx}},\\quad y''={\\frac {d^{2}y}{dx^{2}}},", "0b779dfe034244f21565f934f8e5bc13": "H_{\\varepsilon }^{h}f(\\zeta )={1 \\over \\pi i}\\int _{|H(z)-H(\\zeta )|\\geq \\varepsilon }{\\frac {f(z)}{z-\\zeta }}dz={1 \\over \\pi i}\\int _{|H(z)-H(\\zeta )|\\geq \\varepsilon }{f(z)-f(\\zeta ) \\over z-\\zeta }\\,dz+{\\frac {f(\\zeta )}{\\pi i}}\\int _{|H(z)-H(\\zeta )|\\geq \\varepsilon }{dz \\over z-\\zeta }.", "0b77a0c57e4d83ea934d4ef032e10c08": "[0,c]", "0b77d91370c91e27e44676e02c7678f7": "{\\hat {\\sigma }}^{2}=0.", "0b77df392dd262ffe4f22b543de6bd5e": "\\mathbf {a} \\cdot \\mathbf {b} \\equiv a_{i}b_{i}", "0b780ab920f0fce6bd1baaac1e639265": "\\vert \\pi ^{0}\\rangle ={\\frac {1}{\\sqrt {2}}}\\left(\\vert u{\\overline {u}}\\rangle -\\vert d{\\overline {d}}\\rangle \\right)", "0b783ec3ce75c6e1e4f7984140e672fb": "\\rho ={2r \\over {na_{0}}}", "0b78429649cb061c397b9a4325f68362": "C_{1}:=\\{z:|z|=r_{1}\\}", "0b785832743f30eccf333e693a61e1d5": "{\\hat {F}}[\\{\\phi _{j}\\}](1)", "0b785bc8ff8d426af7a1c5f01a0937b8": "\\delta ", "0b785f8d039c5349d836f063bfa934e7": "I_{P}={\\frac {V_{P}}{|Z|}}", "0b78a36f0e1ff678ddef51be448e179f": "\\pi \\models \\phi ", "0b78e62c7228e3fc96d7a3fcef9c4e5e": "{\\begin{aligned}f(x;\\mu ,\\sigma )&={\\frac {1}{x\\pi \\sigma \\left[1+\\left({\\frac {\\ln x-\\mu }{\\sigma }}\\right)^{2}\\right]}},\\ \\ x>0\\\\&={1 \\over x\\pi }\\left[{\\sigma \\over (\\ln x-\\mu )^{2}+\\sigma ^{2}}\\right],\\ \\ x>0\\end{aligned}}", "0b78f77ae85a50f0ced942e1148f80ae": "PG(2,q)", "0b7982431e7175ff473e7c4d291ee43f": "F(x)=Z_{G}(f(x),f(x^{2}),f(x^{3}),\\ldots ,f(x^{n}))", "0b7993280f4518243cd5d52453c7a69d": "r(f,D)=\\inf _{g\\in C}\\{\\|g\\|:f+g\\notin S(D)\\},", "0b79aed814b1fbd7a838366fb80481fb": "\\theta ={\\frac {\\pi }{2}}-\\phi ;\\,\\!", "0b79c106592d3d9bba9529ec85298e70": "\\tan(7\\pi /16)", "0b7a6f1d3ae1e25d27ed71bb50718eff": "[f_{j}^{\\dagger },f_{k}]=0", "0b7a7742cb4d88af0ca594abda1becad": "V_{n}(P,Q)\\,", "0b7a957ae7ea7b4094d82ccf74915758": "F={\\frac {1}{2}}\\times \\rho \\times S\\times C\\times V^{2}", "0b7ab652df8f24992ff5bf2958e35191": "{\\frac {1}{m(n-1)}}E(SSW)=\\sigma ^{2}", "0b7aec5c0e36c799bc6480a697db16fc": "EL(\\Gamma _{2})=0", "0b7b4ee416bb8bde0d91fa0a499bec10": "\\displaystyle {\\Delta V_{bat}}", "0b7b6ee857e6eebe95609fb5aeff1347": "{\\frac {-\\partial I}{\\partial \\mathrm {z} }}=2\\alpha I=Q", "0b7bc128112f0449f6454d03f3870350": "\\,n\\times n", "0b7c0e06c0b65ed9f714989a493c1276": "P_{5}<0.00009", "0b7c3549265ec90d580049f054047889": "Rl=(3,14)nl\\!", "0b7c3b28f45eb152b42b115ba89875c0": "l_{\\mathrm {tot~core~run} }", "0b7c4d88068200f067fe8a4df30139d2": "\\{y_{i}\\}_{i=1}^{n}", "0b7cb79dd1db776ff58adf439972f641": "U\\circ {\\text{hom}}(-,-)\\simeq {\\text{Hom}}(-,-)", "0b7cc55e9b054c615ea097a5b07a1f5b": "\\tan {\\frac {A}{2}}\\tan {\\frac {C}{2}}=\\tan {\\frac {B}{2}}\\tan {\\frac {D}{2}}=1.", "0b7cc6cabd3ea8f34e93e77c6a5a44e6": "|V_{n}(i)|={\\binom {n}{i}}", "0b7d3ac30d3a62125e3d564a55b96f75": "J_{k}={\\frac {\\lambda }{k}}{\\frac {\\sin k\\theta ^{\\prime }}{\\left(\\rho ^{\\prime }\\right)^{k}}}.", "0b7dafc26b990d2ded1472b3337dc000": "\\scriptstyle \\cup _{i=1}^{n}A_{i}", "0b7dbfa54fa2d188a0f1dfe6968ecbf8": "\\alpha _{\\xi }=\\alpha _{i}(x,\\xi )dx^{i}|_{x}\\in T_{x}^{*}M", "0b7e1bac8263678a8aa543b985365a34": "a_{R}\\approx -4.5\\times 10^{-6}(1+r)A/m", "0b7ecd901143f6121e34a8323f58b464": "{S^{a}}_{b}\\,{S^{b}}_{a}", "0b7f14da27fad5a8049557711a714987": "(E_{a})^{2}=((m_{k}c^{2})^{2}+(m_{t}c^{2})^{2}+2(m_{t}c^{2})(m_{k}c^{2}))", "0b7f5aed32134c95ca4ec4859df2b923": "\\gamma _{zx}^{\\mathrm {core} }={\\tfrac {2h+f}{2h}}~\\gamma _{zx}^{\\mathrm {beam} }", "0b7fc2ac7312ef72e7549d8a0bc98e61": "\\beta =1-{\\tfrac {1}{2}}\\gamma ^{-2}-{\\tfrac {1}{8}}\\gamma ^{-4}-{\\tfrac {1}{16}}\\gamma ^{-6}-{\\tfrac {5}{128}}\\gamma ^{-8}+\\cdots ", "0b7fcfec239e74d9c3829893de187094": "(x_{c},y_{c})", "0b7ff5942359e394540ad9acc83a0a0a": "\\mathrm {sinc} (0)=1", "0b806bcecbe657d1d1cc139c80bd037b": "n_{e}{\\tfrac {6}{29}}\\\\3\\left({\\tfrac {6}{29}}\\right)^{2}\\left(t-{\\tfrac {4}{29}}\\right)&{\\text{otherwise}}\\end{cases}}", "0b9ddea1481307ad13403f4b5a212a12": "f(\\alpha )=\\alpha +\\delta ,\\delta \\neq 0", "0b9dfd5131e36b288d1a52a5c0d689fd": "\\nu +d\\nu \\,", "0b9e1db56abf9af7ff37786e6623c613": "xy\\leq 0", "0b9e478282e5034680c2ea7c97ac48cf": "n^{\\text{left}}", "0b9e62db7302fcf1cd21af484cb62778": "\\oplus _{0}^{k}M_{j}", "0b9e7a0d9b5e57d18dfd0a16ed277ffe": "Se^{-q\\tau }\\phi (d_{1}){\\sqrt {\\tau }}=Ke^{-r\\tau }\\phi (d_{2}){\\sqrt {\\tau }}\\,", "0b9e96409092b9369d0e24e8fedbc3b4": "BF=5", "0b9ef3122465e76b15b0db02c07bb898": "\\ R_{*}=R/M_{\\mathrm {air} }", "0b9f2991087ddb13a722a3319a0bbe2e": "T_{n}", "0b9f7998ac41c70208a48a3acee941b5": "Df", "0ba0102b18fa1fe0edb6bc8981743036": "f\\left(x\\right)=1234+166x+94x^{2}\\,\\!", "0ba067726be762b158e3644944a702b6": "\\ N_{j}", "0ba11ebdc4068356183c1d861d1c0bd8": "O(h)", "0ba183661fb2d56720d7a83f9359ee95": "\\langle \\mathbf {M} ,\\mathbf {N} \\rangle =\\operatorname {trace} (\\mathbf {N} ^{*}\\mathbf {M} )", "0ba193407c682efade2e58f0f5661b44": "L_{n}[\\alpha ,c]=L_{n}[1,c]=e^{(c+o(1))\\ln n}=n^{c+o(1)}\\,", "0ba1a0d58d15ffa72426b99d7b710a15": "\\sum _{\\omega \\neq 0}{\\frac {1}{\\left|\\omega \\right|^{3}}}", "0ba1d143245d8c31a437c8d56cbc814c": "P(\\nabla )", "0ba20a4199f33137fb4c08b3360e1ea8": "A=D", "0ba20e4e3cf6179712b7ac48b0876593": "\\log _{2}(n)/2", "0ba23bba106c02feed2787ae4c29b6fa": "A,B\\subseteq S", "0ba26bb837fefb2b28a751cda5a80607": "L^{3}", "0ba2a3fab3d792c693f6d61c99755b86": "\\,\\!V_{-}", "0ba2b0949f5422099a2e43e25c1d76d9": "{}-{\\frac {d{\\boldsymbol {\\Omega }}}{dt}}\\times \\mathbf {r} _{B}\\ ,", "0ba30127b3f99aa18a8691a08bb716c8": "TC=5000", "0ba323d6320471121122b0a1b38454a2": "|0\\rangle ", "0ba35cc02fc7944cd344bda793fbabab": "L=\\rho M", "0ba3d2cb9da846198feac5a045253dd9": "C_{4,n}=1+4\\,T_{n-1},\\,", "0ba3dbb9afc91fad9c30897769ac7b59": "|P|=(st+1)(s+1)", "0ba4a4576a6dec5580ab283d2fd10ef7": "\\textstyle {\\bar {M}}_{\\mathrm {e} }R{\\bar {M}}_{\\mathrm {e} }^{-1}=T^{-1}", "0ba4f57789ecd10b0f9bb54c80d3e5a9": "\\ \\Delta =r^{2}-r_{s}r+\\alpha ^{2}+r_{Q}^{2}\\,,", "0ba50b63d8d29baee511b4f6c80e91a7": "\\|v(t)\\|\\leq \\|u(0)\\|e^{(-2\\alpha +\\lambda )t}", "0ba574dcaccfc86993ea40105a40130d": "{\\hat {Y}}(X_{0})=\\left(1,X_{0}\\right)\\left(B^{T}W(X_{0})B\\right)^{-1}B^{T}W(X_{0})y", "0ba59a1224d9fc2b5b158b2b2d6e3b56": "c(r,z)={\\bar {c}}(z)+c'(r,z)", "0ba5d06350e48e10f43e830cdef2b249": "\\eta \\colon K\\to A", "0ba5e8b33627efd46f38aea8e8599218": "0\\leq i\\mu (U).\\,", "0bb3c2f0f0ea75b0bad599c233bc2b0e": "\\{(s_{0}s_{1}\\cdots s_{N},s_{1}s_{2}\\cdots s_{N}s_{N+1})|\\;s_{i}\\in A\\;s_{i}\\neq s_{i+1}\\}\\,", "0bb4588fefe4ed4886d2aeed785adada": "({\\tfrac {1}{2}}(a+b),b)", "0bb46ea81527e6aaf89636a3181cb791": "T(S_{liquid}-S_{solid})>H_{liquid}-H_{solid}", "0bb488c36ba76bb7cd57243ba5fd84da": "S={\\underset {i\\in D}{\\times }}Q_{i}", "0bb4c734060fc454d0a1e9caa1990e07": "{\\begin{aligned}\\operatorname {E} [\\ln(X)]&=\\psi (\\alpha )-\\psi (\\alpha +\\beta )=-\\operatorname {E} \\left[\\ln \\left({\\frac {1}{X}}\\right)\\right],\\\\\\operatorname {E} [\\ln(1-X)]&=\\psi (\\beta )-\\psi (\\alpha +\\beta )=-\\operatorname {E} \\left[\\ln \\left({\\frac {1}{1-X}}\\right)\\right].\\end{aligned}}", "0bb4d1bfd576055d1e7004c6af6b0e23": "\\mathbf {q} =\\mathbf {q} (t)\\,\\!", "0bb4fd171cdcbdae0167f08e3ae3aae9": "{\\text{engine}}\\;{\\overset {\\textstyle \\tau }{\\underset {\\textstyle \\omega }{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\\rightharpoondown }}}\\;{\\text{wheel}}", "0bb51c7bd27f97d725be1e502378c921": "\\otimes ", "0bb58f5bc5165ff923fa6cb33b5c7134": "{\\text{Aortic valve area}}\\approx {\\frac {3.5}{\\sqrt {50}}}\\approx 0.5\\ {\\text{cm}}^{2}", "0bb5b6d91e2278fc25029b9a99d25108": "\\,\\sim \\,", "0bb622ab2f73b87f543937a23cccd7fe": "r(0)=r(\\pi )", "0bb62a16ccfe325cf239669304dc28f8": "{\\begin{aligned}\\operatorname {E} (X)&=\\mu \\quad \\quad \\quad {\\text{for }}\\,\\nu >1,\\\\{\\text{var}}(X)&=\\sigma ^{2}{\\frac {\\nu }{\\nu -2}}\\,\\quad {\\text{for }}\\,\\nu >2,\\\\{\\text{mode}}(X)&=\\mu .\\end{aligned}}", "0bb66270cf300e2423596cf871666074": "G=\\langle X|S\\rangle ", "0bb68ac9c6290a78fcae1b7818f29d5d": "\\sum _{k=0}^{\\infty }\\|\\mathbf {x} _{k}\\|<\\infty .", "0bb6bbed431441aa372baf8034bd7218": "\\mathrm {LCM} (T_{p^{k}},T_{q^{\\ell }},T_{r^{m}},\\ldots )", "0bb75646b7cd8e490515c98a08c31c2e": "\\ k=k_{1}/k_{2}", "0bb76dfe7af80a332d7997e59f8af29a": "\\sum _{n=1}^{\\infty }{\\frac {1}{n(4n^{2}-1)}}=2\\ln 2-1.", "0bb771ba9cf53a2fc6e6b8f05f77c732": "0=-\\mu [{\\vec {x}},t]\\delta \\zeta [{\\vec {x}},t]-{2 \\over 8\\pi G}(\\nabla \\zeta [{\\vec {x}},t])\\cdot (\\nabla \\delta \\zeta [{\\vec {x}},t]).", "0bb782aa8f9e133e487458690d3fbfb8": "g:rcl(R)\\to A", "0bb78497083c20789a86269b118db177": "{\\hat {\\mathbf {a} }}=\\mathbf {e} _{i}{\\widehat {a}}_{i}\\,,\\quad {\\hat {\\mathbf {b} }}=\\mathbf {e} _{j}{\\widehat {b}}_{j}", "0bb7f7258bbbe786bdfa37a71d9852e1": "x_{3}=(a+3b)-(a^{2}+3b^{2})^{2},x_{4}=(a^{2}+3b^{2})^{2}-(a-3b)", "0bb8215c3b5e83e3d5790537bbfe5a46": "(b\\rightarrow c))", "0bb84a7003dc3d6f17a64868475384fc": "{\\frac {en}{e}}", "0bb85a393354d3218aeb6b9dea16d825": "\\left[x^{(1)}\\right]\\supset \\left[x^{(2)}\\right]\\supset \\cdots \\supset \\left[x^{(k)}\\right]", "0bb886f3be44195026180987e61f4940": "\\alpha _{k}=\\gamma _{k}/\\lVert g^{(k)}\\rVert _{2}", "0bb8ecebb3198a020ba7803ab55bc4d7": "T_{\\text{f}}", "0bb96e96279951e5c687efc5cce6d14c": "t\\rightarrow \\infty ", "0bb975e1c6680c7dbfc3d5e37e730a90": "R_{abcd}\\,R^{abcd}=12\\omega ^{4},\\;R_{abcd}{{}^{\\star }R}^{abcd}=0.", "0bb97e67fb3e13e62c23ab7a73cd118a": "G(x)=F(x+h)=\\sin(x+h).", "0bb98e9f6d33ed4940a6369aa13a984d": "{d \\over a+b{\\sqrt {c}}}={ad-bd{\\sqrt {c}} \\over a^{2}-b^{2}c}.\\,", "0bb9adf98182c82718f846c0a2cf49d4": "0+j(\\omega -r)", "0bb9c23f489429bcb78a03199065b77c": "{\\tfrac {1}{X}}\\sim {\\mbox{Inv-}}\\chi ^{2}(\\nu )\\,", "0bba2efaf800616d232acb7ce5c5e60d": "\\textstyle \\mathbf {a} ={\\frac {d\\mathbf {v} }{dt}}", "0bba49c57a1b37832d4476dc22691b08": "{\\begin{aligned}\\nabla ^{2}\\Phi ={\\frac {\\left(\\cosh \\tau -\\cos \\sigma \\right)^{3}}{a^{2}\\sin \\sigma }}&\\left[{\\frac {\\partial }{\\partial \\sigma }}\\left({\\frac {\\sin \\sigma }{\\cosh \\tau -\\cos \\sigma }}{\\frac {\\partial \\Phi }{\\partial \\sigma }}\\right)\\right.\\\\[8pt]&{}\\quad +\\left.\\sin \\sigma {\\frac {\\partial }{\\partial \\tau }}\\left({\\frac {1}{\\cosh \\tau -\\cos \\sigma }}{\\frac {\\partial \\Phi }{\\partial \\tau }}\\right)+{\\frac {1}{\\sin \\sigma \\left(\\cosh \\tau -\\cos \\sigma \\right)}}{\\frac {\\partial ^{2}\\Phi }{\\partial \\phi ^{2}}}\\right]\\end{aligned}}", "0bba835a648bebcea1aad3108949ee32": "H_{n}(j\\omega )", "0bba885ffd7144990c41cc925a2ee79c": "a_{n}=C\\lambda _{1}^{n}+D\\lambda _{2}^{n}", "0bba99ffd3d7db3f2e823531e79f9254": "{\\hat {X}}={\\frac {\\alpha -1}{\\beta +1}}", "0bbb7c3aee026872dea6bde392956454": "\\land ", "0bbb7c9fd6468deb705a5b480b8128a9": "r(N)=\\sum _{k_{1}+k_{2}+k_{3}=N}\\Lambda (k_{1})\\Lambda (k_{2})\\Lambda (k_{3}),", "0bbb8bb1a9761d2cd8e7992ad3ea197e": "\\omega _{c}", "0bbb91e5e3b2e3285082422166cfa4b3": "H^{n}(R,S;M)", "0bbb956a418945e7e6d7f8a019942c20": "{\\frac {d}{d\\theta }}\\operatorname {Cl} _{2m+1}(\\theta )={\\frac {d}{d\\theta }}\\sum _{k=1}^{\\infty }{\\frac {\\cos k\\theta }{k^{2m+1}}}=-\\sum _{k=1}^{\\infty }{\\frac {\\sin k\\theta }{k^{2m}}}=-\\operatorname {Cl} _{2m}(\\theta )", "0bbbd1765bfca7ee728a0396b8d96f09": "\\ {\\textbf {f}}\\cdot {\\textbf {f}}_{q}=1{\\pmod {q}}", "0bbc52963f22710610ce13d699d91841": "H^{i}(C(f))=0{\\text{ for }}i\\neq -1,0.\\ ", "0bbc534b447e14a0810504b8e33c5a61": "Y_{i}=C+\\alpha \\sin(\\omega T_{i}+\\phi )+E_{i}", "0bbc85f829e11c2be0ceaa95bd95d95c": "\\gamma '(t)\\cdot \\gamma ''(t)", "0bbc8626b11849e7019772fdd114ea02": "\\lambda _{1}=-a,\\lambda _{2}=a", "0bbcf6d6ffcb14d427333af4905ee9bd": "A_{xy}", "0bbeb807fb5ad085cbaf6d6bb3705ee0": "\\beta =n^{o(1/\\log {\\log {n}})}", "0bbed5fe35b5362d05c53b182782155e": "i:N_{0}\\rightarrow S", "0bbef43d9a302c1d65634f240e1236c0": "\\mathbf {M} _{3}:=\\mathbf {A} _{1,1}(\\mathbf {B} _{1,2}-\\mathbf {B} _{2,2})", "0bbf043db89bfd3995b143df42391734": "\\{a^{n}b^{n}c^{n}:n\\geq 0\\}", "0bbf314db1695862c71b47b15d6b8ef6": "SU(2n)\\supset Sp(2n).", "0bbfa2d9e91f9be7c28d2455eb0ff26a": "\\displaystyle \\mu _{c}=[B_{1},B_{2}]+IJ.", "0bbfac647b7dcd97dc59e6d426d7b79e": "E\\subseteq \\mathbb {R} ", "0bc07dcc6357cbd9ebf56650693d0c9a": "\\textstyle M_{\\mathrm {f} }=\\left({\\begin{array}{cc}3&1\\\\2&0\\end{array}}\\right)", "0bc07fa5b3a1f1c797979b4f1208e20e": "{\\frac {\\partial f}{\\partial t}}+r\\cdot \\nabla _{r}f-{\\frac {1}{\\hbar }}\\nabla _{r}V\\cdot \\nabla _{k}f+\\sum _{\\alpha =1}^{\\infty }{\\frac {(-1)^{\\alpha +1}}{\\hbar 4^{\\alpha }(2\\alpha +1)!}}\\times (\\nabla _{r}\\nabla _{k})^{2\\alpha +1}Vf=\\left({\\frac {\\partial f}{\\partial t}}\\right)_{c}", "0bc099db2499122f4ab36efc41ac5454": "xAB\\rightarrow A-(B-A)_{x-1}-B", "0bc0e415644021b6a85649eaa903726a": "\\scriptstyle \\rho ", "0bc0e5973e45def3494453fd0af29455": "J=\\det({\\boldsymbol {F}})", "0bc15763ad9f28237b6db86df863162b": "{\\frac {Z_{q}}{mU}}", "0bc17f83938237ac1e12af3b13ea7d4f": "{\\mathbb {P} }^{2k+1}{\\mathbb {C} }\\,", "0bc26165fe3c36961425216e7a080afb": "(\\delta ^{\\dagger }\\otimes 1_{A})\\circ (1_{A}\\otimes \\delta )=\\delta \\circ \\delta ^{\\dagger }", "0bc2b0dbabdfc0a993ba856984d937fa": "x=c\\cosh {\\frac {v}{c}}\\cos u", "0bc2b3518a89b0b9834c2f36baebfcd5": "Q(x)={\\frac {1}{\\sqrt {2\\pi }}}\\int _{x}^{\\infty }\\exp \\left(-{\\frac {u^{2}}{2}}\\right)\\,du.", "0bc2e48be05a5a2797a081045a86ac86": "=f(n-1)-n^{2}+3n-2+\\sum _{i=1}^{n-1}\\left(ni-i^{2}\\right).", "0bc3b26061a337cdb5e9f583194ae53d": "\\Phi _{G}", "0bc3f11ccb324d1d244c87353c3e47d5": "\\scriptstyle {\\vec {L}}={\\vec {J}}-{\\vec {S}}", "0bc463cb43711b684210ce06466a3b04": "\\displaystyle \\zeta =(m_{0}-m_{f})/m_{0}=m_{p}/(m_{p}+m_{f})=1-m_{f}/m_{0}", "0bc4a76f4cc040f13e4bba0cf4f4f1f5": "\\sigma (m_{1}+1)<\\ldots <\\sigma (n_{1})=q.", "0bc4e7d532f726745d65abccd61add65": "\\scriptstyle {R_{m}^{0}}", "0bc5273141e5f3d29344c50b379125d7": "A={\\mathcal {O}}_{K},B={\\mathcal {O}}_{L}", "0bc53c1d437f2532a357082f46f603e7": "{\\hat {\\theta }}={\\hat {z}}", "0bc57f651e8658f14da371561fc429a9": "{\\begin{aligned}(a+b\\,\\mathrm {i} )\\cdot (c+d\\,\\mathrm {i} )&=a\\cdot c+a\\cdot d\\,\\mathrm {i} +b\\cdot c\\,\\mathrm {i} +b\\cdot d\\cdot \\mathrm {i} ^{2}\\\\&=(a\\cdot c-b\\cdot d)+(a\\cdot d+b\\cdot c)\\,\\mathrm {i} \\end{aligned}}", "0bc583482b159ca5300f3e65f8b8be6a": "\\lambda ={\\frac {2\\pi }{k}}", "0bc5f5f1569c4e6b56d9dd7f77c838c5": "s_{0}\\left(t\\right)=s\\left(t\\right)", "0bc661c30d2fefc01718ba2ad71a86a6": "B_{C}\\approx {\\frac {1}{T_{M}}}", "0bc671de69a93127085e028316c98721": "{\\dot {X}}={\\frac {\\partial X}{\\partial \\tau }}", "0bc698bc701562e4aaea7119cdb75255": "y_{U}=F(x+\\delta ,{\\hat {\\theta }})", "0bc70e047b0bbb1bc42a49652812fd3e": "A_{1}\\supset A_{2}\\supset A_{3}\\supset \\cdots .", "0bc74c2667a19a3265e2df805e24eeb9": "|[A,B;C,P]|=1.\\ ", "0bc752a29ae295f2c2c82f6c208e2d46": "s(x)=\\mathop {\\mathrm {ess\\,sup} } _{y\\in \\mathbb {R} ^{n}}f\\left({\\frac {x-y}{1-\\lambda }}\\right)^{1-\\lambda }g\\left({\\frac {y}{\\lambda }}\\right)^{\\lambda }.", "0bc7f379a1459ae5bd43b131b1a7a6c9": "u=dx/dt", "0bc811a05f19125ea669f88d7579b575": "|x^{\\mathsf {T}}y|\\leq \\|x\\|_{p}\\|y\\|_{q}\\qquad {\\frac {1}{p}}+{\\frac {1}{q}}=1.", "0bc8465c05ddf597c1c12d240735938a": "N\\in M_{X}", "0bc8dff343096783dbeba168596b1630": "d_{2}\\,", "0bc91df3b6cb92c5438e0975d295bdfd": "\\sum _{m\\neq 0}|m|\\left|\\sum _{n\\neq 0}c_{mn}\\lambda _{n}\\right|^{2}\\leq \\sum _{m\\neq 0}{1 \\over |m|}|\\lambda _{m}|^{2}", "0bc9207f6cedc6b8e8623d6344be2c83": "\\tan \\left({\\frac {1}{4}}\\Omega \\right)={\\sqrt {\\tan \\left({\\frac {\\theta _{s}}{2}}\\right)\\tan \\left({\\frac {\\theta _{s}-\\theta _{a}}{2}}\\right)\\tan \\left({\\frac {\\theta _{s}-\\theta _{b}}{2}}\\right)\\tan \\left({\\frac {\\theta _{s}-\\theta _{c}}{2}}\\right)}}", "0bc938f703277381c26db47d6032d529": "{\\boldsymbol {k}}\\cdot {\\boldsymbol {p}}", "0bc95db8cd7650ad9b1972255c6c136f": "z_{1},\\ldots ,z_{4}", "0bc99c248f639ab8797e8c61fe0efc6c": "x\\in A^{c}\\cup B^{c}", "0bc99da241ced5fb23e6cdc615f2b260": "\\langle x,y\\rangle :=\\sum _{i=1}^{\\infty }x_{i}y_{i}\\quad x\\in E,y\\in E^{\\beta }", "0bc9bba6c22030e1d4dd1dbe6415ff4f": "\\mu _{k,i}", "0bca490eb344191fa2a7a379be377d10": "s=\\sigma +ti", "0bca595265c852b0a4d3b070983bef00": "\\Omega _{ij}={\\frac {1}{2}}(\\partial u_{i}/\\partial x_{j}-\\partial u_{j}/\\partial x_{i})", "0bcab909c1e3e7e18a1c2c898d813ff7": "p:=-2D_{1}~J(J-1)~;~~\\mathrm {dev} ({\\bar {\\boldsymbol {B}}})={\\bar {\\boldsymbol {B}}}-{\\tfrac {1}{3}}{\\bar {I}}_{1}{\\boldsymbol {\\mathit {1}}}~;~~{\\bar {\\boldsymbol {B}}}=J^{-2/3}{\\boldsymbol {B}}~.", "0bcac5527a2d22d1e6c4dd5517c100c1": "A\\models R(a_{1},\\ldots ,a_{n})", "0bcadf78891f225454549428315cd9fd": "Tg=f*g.", "0bcb57347d875cf79605789c9c8aad68": "I\\cap K[Y].", "0bcb8572d9754ce8a6a9cebc01b50402": "-c_{1}\\,", "0bcb965d93d54dded0e1b55d743a9fb6": "\\pm 1,\\pm \\omega ,\\pm \\omega ^{2}", "0bcbc672badf11f08d13d1be16644d04": "\\psi _{ij}(t)", "0bcc36d2da085f786426a6e93c420afe": "P(\\phi )=P_{\\mu }(\\phi )=P_{\\Omega ,\\mu }(\\phi )={\\frac {\\mu (\\{w\\mid \\phi (w){\\text{ and }}w\\in \\Omega \\})}{\\mu (\\{w\\mid w\\in \\Omega \\})}}", "0bcc6cea8bbade2794131c2853c7cdff": "\\textstyle {\\frac {p}{q}}", "0bcc6eb07326241ec09fb86e20e4f2b1": "y(x)", "0bcc7e60f7edb910aac88601954736c3": "P_{n,\\theta }", "0bcca8722a6e60adc53ffb74024df3f4": "{\\mathsf {Q}}=(\\mathbf {Q} -\\mathbf {P} ,\\mathbf {P} \\times \\mathbf {Q} )=([A](\\mathbf {q} -\\mathbf {p} ),[A](\\mathbf {p} \\times \\mathbf {q} )+\\mathbf {d} \\times [A](\\mathbf {q} -\\mathbf {p} ))", "0bccc911690f253ff87914a518c2ae41": "(r_{1})^{2}", "0bccd12acbaf7f0423a2d20ef346f355": "N_{1}\\subseteq N_{2}", "0bccdd2f481faa1e28a724569e730b0f": "B_{m}(x)=\\sum _{n=0}^{m}{\\frac {(-1)^{n}}{n+1}}\\Delta ^{n}x^{m}.", "0bcd48a71b2a676420645b1e0c0c137e": "({\\tfrac {p}{b}})=({\\tfrac {p}{B}})=-1,", "0bcd55df2b8c038c0cd464d3f06e4947": "ds^{2}=e^{2\\Phi }\\left(dt-w_{i}\\,dx^{i}\\right)^{2}-k_{ij}\\,dx^{i}\\,dx^{j}.", "0bcd584e0b1b1d9061b6b700ac8237ea": "[a_{1}\\otimes \\cdots \\otimes a_{n},x]=a_{1}\\otimes \\cdots a_{n}\\otimes x\\quad {\\text{for }}a_{1},\\ldots ,a_{n},x\\in V.", "0bcd8bfe7251e94179d7a24236b10374": "m_{1}u_{1}^{2}+m_{2}u_{2}^{2}=m_{1}v_{1}^{2}+m_{2}v_{2}^{2}\\,\\!", "0bcd91fb6432f991d5b4cbb079e562c7": "z_{t}", "0bcdcc38d1d9395793a01a0dc52c2093": "{\\frac {1}{((i\\omega )^{2}-\\xi ^{2})^{2}}}", "0bce8c6df4159efc00ea47dce2236a38": "f(a_{1})=a_{2}", "0bcebb29fb0eee8f8be42be4436a4823": "D=2(Y_{1}-B)=2({\\sqrt {13}}-{\\sqrt {3}})", "0bcf4c6c79db1875135dc475e94e12b8": "V_{1}\\times V_{2}\\times V_{3}\\to \\mathbb {F} ,", "0bcf4da023d9e69b2c555d7e5188b7be": "a_{n+1,k}=-2na_{n-1,k}\\ \\ k=0", "0bcf7aa0874b301cba529c8c62e67501": "\\operatorname {tr.deg} _{k}k(p)+\\dim D\\leq \\operatorname {tr.deg} _{k}K", "0bcffc346310869894990a469979e6ee": "{\\frac {}{}}\\Delta ", "0bd010bc6889f4a443f788d8c6e055e1": "2\\times 3\\times 5=6\\times 5=30", "0bd084b7b4a41909762a1bbf1afaec13": "\\beta ={\\frac {R_{1}}{R_{1}+R_{2}}}", "0bd08649070cfe010ad66e771bd2faea": "\\tau _{0}={\\frac {n}{c}}\\cdot {\\frac {l}{1-R+X}}", "0bd0ce81fe8a365ec05d91496ed88df4": "=\\operatorname {st} \\left({\\frac {u\\cdot \\mathrm {d} v+(v+\\mathrm {d} v)\\cdot \\mathrm {d} u}{\\mathrm {d} x}}\\right)", "0bd10836058ca3bee3d54e3f313bf5ab": "^{\\;}W(\\xi ,\\tau )=W(\\star q(\\xi ,-\\tau ),0)", "0bd1511db1dea8758b4ebbc48d8109a1": "{\\frac {1}{Z_{\\text{eq}}}}={\\frac {1}{Z_{1}}}+{\\frac {1}{Z_{2}}}+\\cdots +{\\frac {1}{Z_{n}}}", "0bd160c9fcc2db715029d73211989a2d": "y(x)=A\\cdot (Ax^{2}+2Bx+C)", "0bd1a732c88b7ea0de3ba2c93d80a57f": "{\\begin{matrix}{\\frac {6}{5}}\\end{matrix}}", "0bd1aa585ad016db886eaad1aeec7307": "\\oint _{\\mathbf {X} _{A}}^{\\mathbf {X} _{B}}({\\boldsymbol {\\nabla }}\\times {\\boldsymbol {\\epsilon }})\\cdot d\\mathbf {X} ={\\boldsymbol {0}}", "0bd1ecc6359ce275e3a626020b904c8c": "c_{1}", "0bd22ab446a4297363bc1fa5fbaeddd0": "d\\theta _{1}=\\omega \\wedge \\theta _{2},\\,\\,d\\theta _{2}=-\\omega \\wedge \\theta _{1}", "0bd257cdae1a39ee054c760b6bae9db1": "SO^{+}(1,3)", "0bd2778564ec3915ca420cd6ee4382fc": "\\Pr _{y\\in \\{0,1\\}^{k}}{\\big [}({\\text{Had}}(x))_{y}=1{\\big ]}=\\Pr _{y\\in \\{0,1\\}^{k}}{\\big [}\\langle x,y\\rangle =1{\\big ]}\\,.", "0bd29a46079c8cd3b3522e581fef4356": "(y-y_{0})(x_{1}-x_{0})=(y_{1}-y_{0})(x-x_{0})", "0bd29f0b3414278302206922a703b271": "\\left\\|u\\right\\|^{2}=\\left|{\\frac {\\langle u,v\\rangle }{\\langle v,v\\rangle }}\\right|^{2}\\left\\|v\\right\\|^{2}+\\left\\|z\\right\\|^{2}={\\frac {|\\langle u,v\\rangle |^{2}}{\\left\\|v\\right\\|^{2}}}+\\left\\|z\\right\\|^{2}\\geq {\\frac {|\\langle u,v\\rangle |^{2}}{\\left\\|v\\right\\|^{2}}},", "0bd2a420fa1d46354b133aa3817af6f7": "\\Theta =\\sum _{ij}g_{ij}{\\dot {q}}^{i}dq^{j}", "0bd2a702207692aa5fbf30d6fd1a4f82": "\\langle x|x'\\rangle =\\delta ^{3}(x-x')", "0bd2aee67df207805b6e1cdc246e9e82": "f_{1}(n)=f_{0}^{n}(n)", "0bd2bf1364551fb50300e90528343264": "f:\\mathbb {H} \\rightarrow R", "0bd2c988cdd871a62c6286e6bc3e11e3": "Powerloss=\\Delta p_{LS}\\cdot Q_{tot}", "0bd2d9b1b8b40f6ec5e4340e9d48ca20": "{\\widehat {\\neg \\alpha }}:=\\mathbb {I} -{\\hat {\\alpha }}", "0bd32d2ab93c4c68fc4e9f0c9d778d37": "\\langle 0|\\varphi _{\\mathrm {in} }(x)|p\\rangle ={\\frac {e^{-ip\\cdot x}}{(2\\pi )^{3/2}}}", "0bd3704899f112b1e2dd72dca836ed8f": "A_{\\mu }^{a}", "0bd3956bd1cae5bb15558010eed3bf43": "p\\rightarrow 0", "0bd3fedc5f9682ccb2726345b23d4b7a": "\\mathbb {E} _{X}\\left\\{\\left\\Vert {\\sqrt {\\Lambda }}\\rho _{X}{\\sqrt {\\Lambda }}-\\rho _{X}\\right\\Vert _{1}\\right\\}\\leq 2{\\sqrt {\\epsilon }}.", "0bd41f80d2a8fab5a5d8ce9317d612cc": "\\mathbf {W} \\in \\mathbb {R} ^{C\\times N}", "0bd4482e10ee6708f45f070244a7bd50": "{\\begin{aligned}&m=L-\\textstyle {\\frac {1}{2}}C\\\\&(R,G,B)=(R_{1}+m,G_{1}+m,B_{1}+m)\\end{aligned}}", "0bd4558eaeb1eeb35fd9d6c1097c0aad": "\\forall A:A\\times \\varnothing =\\varnothing \\,.", "0bd47873c1c6dd17c6e21690f3fa043a": "H(e^{j\\omega })=H_{c}(j\\omega /T)\\,", "0bd4b1c75e866ea36627f4d7ffe665bc": "\\sum _{A=1}^{N}M_{A}\\,{\\big (}\\delta _{ij}|\\mathbf {R} _{A}^{0}|^{2}-R_{Ai}^{0}R_{Aj}^{0}{\\big )}=\\lambda _{i}^{0}\\delta _{ij}\\quad \\mathrm {and} \\quad \\sum _{A=1}^{N}M_{A}\\mathbf {R} _{A}^{0}=\\mathbf {0} .", "0bd544bc9dfe2625b4c6fce15e8b2979": "J_{l+1/2}(\\rho )", "0bd54c5c230b4f3ccbbfb56d362fba68": "y_{\\mathrm {low} }[n]=\\sum \\limits _{k=-\\infty }^{\\infty }{x[k]g[2n-k]}", "0bd5973cce44b58ebb67f6111158b43c": "{6 \\choose 2}{43 \\choose 4}", "0bd5f8494fe5d199df6ce2a800f0cdf1": "{\\widehat {f}}", "0bd5f9ccbc354415ff0c32ca61c332d0": "|\\{d\\in D:t\\in d\\}|", "0bd60fb8b717bc81c038596f85907717": "{\\tilde {\\mathbf {M} }}=\\mathbf {U} _{t}{\\boldsymbol {\\Sigma }}_{t}\\mathbf {V} _{t}^{*}", "0bd610c28dec19473137174ad39ba64a": "d_{1}-d_{2}", "0bd617f99d6f3fcf268a2368b340bb90": "f_{T}=S_{T}-K", "0bd6c3b263df94127bec3beb1c23e495": "[a]={1 \\over n!}\\sum _{\\sigma }x_{\\sigma _{1}}^{a_{1}}\\cdots x_{\\sigma _{n}}^{a_{n}},", "0bd6df5eec3eae69e41fdf2b4c262670": "m=O(n)", "0bd7637a4b50d8043cebfa0734bf51f4": "{\\frac {\\partial \\phi (\\mathbf {r} ,t)}{\\partial t}}=\\nabla \\cdot \\left[D(\\phi ,\\mathbf {r} )\\right]\\nabla \\phi (\\mathbf {r} ,t)+{\\rm {tr}}{\\Big [}D(\\phi ,\\mathbf {r} ){\\big (}\\nabla \\nabla ^{T}\\phi (\\mathbf {r} ,t){\\big )}{\\Big ]}", "0bd7654b37bf9c6525fb12802c525268": "\\rho (g)\\mapsto {\\tilde {\\rho }}(e_{g})", "0bd7c641c6d170178c725d4f5e10b9c0": "|k_{0}|=|k_{i}|", "0bd7d76313f69fd4b7e08d257ad02239": "(1+x)^{r}\\geq 1+rx\\!", "0bd890413bed9e9302718c8ccc74cda7": "E_{g}", "0bd8c9689b286a31ca75037b750a3599": "\\nabla ^{2}\\mathbf {v} =\\nabla (\\nabla \\cdot \\mathbf {v} )-\\nabla \\times \\nabla \\times \\mathbf {v} ", "0bd8f39c14d9d0684cb0def61f6e679e": "E={\\begin{cases}\\displaystyle \\sum _{n=1}^{\\infty }{\\frac {M^{\\frac {n}{3}}}{n!}}\\lim _{\\theta \\to 0}\\left({\\frac {\\mathrm {d} ^{\\,n-1}}{\\mathrm {d} \\theta ^{\\,n-1}}}\\left({\\frac {\\theta }{\\sqrt[{3}]{\\theta -\\sin(\\theta )}}}^{n}\\right)\\right),&\\epsilon =1\\\\\\displaystyle \\sum _{n=1}^{\\infty }{\\frac {M^{n}}{n!}}\\lim _{\\theta \\to 0}\\left({\\frac {\\mathrm {d} ^{\\,n-1}}{\\mathrm {d} \\theta ^{\\,n-1}}}\\left({\\frac {\\theta }{\\theta -\\epsilon \\cdot \\sin(\\theta )}}^{n}\\right)\\right),&\\epsilon \\neq 1\\end{cases}}", "0bd8fd5d9ebbc10592b8b7ee70663d3f": "{\\rm {E}}(V)", "0bd9126c47b8120c51d3fc38383d1f32": "a,b\\in K", "0bd9522f573aabff1cef7779d290d074": "y\\in Y\\,", "0bd97e0d2983e0467b978524a19f345e": "f_{cr}\\equiv {\\frac {\\pi ^{2}E_{T}}{({\\frac {KL}{r}})^{2}}}\\qquad (2)", "0bd9b26f982e5198437202595f401b30": "[a+(n-1)d]r^{n-1}", "0bda1b7ad85d8fdf990c212c8104d548": "(T,\\mu ^{T},\\eta ^{T})", "0bda24b10d7355305438917b41c4401c": "{\\underline {\\mathrm {Hom} }}(Y,\\Omega ^{-1}(X))", "0bda386678ba6aadb3b0f8f3fd6e35cb": "0\\,", "0bda879984a5765e34987f75b2b0a3a6": "\\mathbf {y} _{k}\\in \\mathbb {R} ^{q}", "0bdad16f8453e5f5bf2dff2e17efd057": "\\mathbf {P} =\\hbar \\mathbf {K} \\,.", "0bdb0f615b65ba86faffbe528a0292e5": "\\mathbf {t} ={\\boldsymbol {\\sigma }}\\cdot \\mathbf {n} ", "0bdb2fe3cd993978c4fd09ae6eff2a6f": "\\sum _{k=0}^{n}{\\tbinom {n}{k}}=2^{n}", "0bdb4aab659219ef245b752bb69a7359": "\\gamma \\equiv g{\\frac {q}{2m}}", "0bdb4b388170a062e3c8a2a944af6943": "\\varepsilon ^{-1/2}", "0bdb4e12b1ab01e6471e3e0ab10057c5": "{\\tfrac {25}{8}}", "0bdb5788606155ecae30d988287f8693": "{\\mathit {C}}_{\\mathit {G}}", "0bdb762ff68ee041fbf58148884827cc": "{\\frac {dm}{dt}}={k_{a}}C_{i}", "0bdba9dd5fcca535a640dae7916c9265": "\\operatorname {E} _{GB1}(Y^{h})={\\frac {b^{h}B(p+h/a,q)}{B(p,q)}}.", "0bdbc2acf29d6ee4517a0e34b96df1c5": "P_{ni}\\equiv Prob({\\text{Person }}n{\\text{ chooses alternative }}i)=G(x_{ni},\\;x_{nj},\\;j\\neq i,\\;s_{n},\\;\\beta ),", "0bdca44d0c3ec8d151498f4708c286e3": "f(\\varphi )=\\int _{0}^{2\\pi }e^{\\varphi \\cos \\theta }\\cos(\\varphi \\sin \\theta )\\;\\mathrm {d} \\theta .", "0bdcb8231d1fc88e3b054956ee8e3c15": "a_{2}=\\lfloor 5^{\\frac {3}{2}}\\rfloor =\\lfloor 11.180\\dots \\rfloor =11,", "0bdce0e3d67fc40224ae9c2b9aea501f": "[C_{i},H]=iP_{i}\\,\\!", "0bdcfaa5eac1af4d3bb4e58026235984": "\\bigstar \\bigstar \\bigstar \\bigstar \\bigstar ", "0bdd69d4f0d5820d0dca2a2545303644": "i{\\text{ such that }}\\sum _{j=1}^{i-1}p_{j}\\leq 0.5{\\text{ and }}\\sum _{j=1}^{i}p_{j}\\geq 0.5", "0bdd823c8102ecf09ed8510d48c0142c": "-i\\epsilon ^{\\sigma 123}\\gamma _{\\sigma }\\gamma ^{5}=-i\\epsilon ^{0123}(\\gamma ^{0})(i\\gamma ^{0}\\gamma ^{1}\\gamma ^{2}\\gamma ^{3})=\\epsilon ^{0123}\\gamma ^{1}\\gamma ^{2}\\gamma ^{3}", "0bdda2c1bf4fcb1c06401a4e7e8e53ba": "M\\leftarrow \\lambda _{B},i\\leftarrow 0", "0bddb2181901c5b50efce95fbe7aa4fb": "\\mathbb {Z} _{3}", "0bde3675f7237bb4bc2e814796a7c5a5": "E[K]={\\boldsymbol {\\tau }}(I+T+T^{2}+\\cdots ){\\boldsymbol {1}}={\\boldsymbol {\\tau }}(I-T)^{-1}{\\boldsymbol {1}}=4.5.", "0bded11b2f244789bbc090b3e5f2ebed": "q\\succ _{P}r", "0bdf10aa4228053856428648ff7e387c": "\\Phi _{12}(r)", "0bdf4f5af065b8936d26688308d59ac0": "M_{Y}\\ ", "0bdf7af0093ab06fee4ef824e72e6adb": "\\phi (a)(h)\\in G", "0bdfb7b5e97c75fcf87685ae72dd0ae3": "{\\mathcal {M}}_{p}\\,", "0be059b215b74db9e94d5b9ef4455b70": "(\\mathbf {A} ^{T}\\mathbf {P} _{B}^{\\perp }\\mathbf {A} )^{-1}", "0be072b811505937e7ccfd4318fd5753": "\\sum _{k=0}^{n}\\mu _{i}q_{k}(i)q_{\\ell }(i)=v\\mu _{k}\\delta _{k\\ell }.\\quad (9)", "0be094014704082336d1d05a451b00e6": "\\Lambda _{n}(a)=\\sum _{x\\mod p}{\\binom {D_{n+1}(x,a)}{p}}", "0be0a0a107ade58d198608b9411f90bb": "Qp_{n}(x)=np_{n-1}(x)\\,.", "0be0b5c7ba61aa2ebce298f0caf055b2": "=<(A-a)\\psi |(A-a)\\psi >=0", "0be0c173c0ac9e0a3ce4a4c649668980": "\\chi (t-t')\\!", "0be25044b639fc8cff948ea87e363d33": "{\\ddot {S}}-(m^{2}\\sigma ^{2}+n^{2})S=0", "0be250a091dcb46adbed4d45a369cc3f": "a^{2}+b^{2}\\quad ", "0be2827c27c5e28e4740fcad1427fc3d": "\\langle \\Psi _{n}|\\Psi _{n}\\rangle _{\\nu }=\\lim _{M\\to \\infty }\\sum _{i_{1},\\ldots i_{n},j_{1},\\ldots j_{n}0", "0c09f7cf43477384a29c85b212d4011b": "\\lambda <\\kappa ", "0c0a0efb31a051e3494731d99ef48198": "F_{rad}={\\frac {dp}{dt}}={\\frac {1}{c}}{\\frac {dE}{dt}}={\\frac {1}{c}}\\sigma _{t}{\\frac {L}{4\\pi r^{2}}}", "0c0a8efa62790179582465b52d21ebb8": "u_{i}^{s}", "0c0a964dc284211be87e0f8e07921516": "x^{5}-5x^{4}+30x^{3}-50x^{2}+55x-21=0\\,,", "0c0a9f9a225af75271a5858c14e5151b": "{\\begin{matrix}e^{{\\frac {-2\\pi i}{N}}(k+N/2)}&=&e^{{\\frac {-2\\pi ik}{N}}-{\\pi i}}\\\\&=&e^{-\\pi i}e^{\\frac {-2\\pi ik}{N}}\\\\&=&-e^{\\frac {-2\\pi ik}{N}}\\end{matrix}}", "0c0aae11e0e021ef3794eb3fbd032516": "t_{0}0", "0c12dbdd0dec0d13fb71ead181d2a7c4": "s_{y}=0", "0c131335e1615c37f1d231c6e17b4327": "\\lbrace w_{1},w_{2},\\dots ,w_{k}\\rbrace ", "0c1355394c2c2e76394c663a71c0f0d9": "c={\\frac {1}{\\sqrt {\\mu _{0}\\varepsilon _{0}}}}\\ .", "0c13642ada76dc2d40ef3ac5e7da2912": "\\langle \\alpha ,\\beta \\rangle ", "0c13aa1b6f9f53919ab6009029121371": "\\varepsilon _{\\gamma }", "0c13c702dfc8ccdce6f30103b5f8beba": "\\precsim ,\\precnsim ,\\precapprox ,\\precnapprox \\,", "0c13f9b96bd652367361c120ce52317a": "{\\tfrac {3}{2}}q", "0c14546d5fd1458589f4f31478067969": "\\hbar =0", "0c14d40b843d92a11b43eb71747e4fa7": "A[n]\\times A[n]\\longrightarrow \\mu _{n}.", "0c157b34bbc0875a3e53e16edb8de05e": "{\\frac {\\partial \\langle H\\rangle }{\\partial a_{n'}^{*}}}=i\\hbar {\\frac {\\partial a_{n'}}{\\partial t}}", "0c1619879060b38ba844e510c8e4430c": "H_{0}:\\theta _{1}=\\theta _{2}=\\cdots =\\theta _{k}", "0c1632efe6d8dc956be0e68daef11fce": "A'(z)=A(z/\\gamma )", "0c16c24482ddd3b621a2213b6a2407cc": "{\\hat {m}}_{ij}=a_{i}b_{j}", "0c16e66a0b815eb7c3781ac7a77033e3": "{\\frac {1}{d}}\\sum _{\\alpha }\\Pi _{\\alpha }=I", "0c16eb8e1e1c2393545fe127a7550705": "{\\frac {\\partial L}{\\partial t}}=0", "0c16f3856d50cb76bc98107d3c03e3ec": "{\\mathbf {p}}\\rightarrow R({\\mathbf {\\hat {n}}},\\theta ){\\mathbf {p}}", "0c171df1127b2ac1e31f4580ba3021f2": "|\\nu _{i}(L)\\rangle =e^{-im_{i}^{2}L/2E}|\\nu _{i}(0)\\rangle .", "0c176f787421c87d7155eacd810d561b": "L_{\\nu ,p}(R^{+})", "0c17b53846ed571eb2c65aa430c050aa": "{\\bar {v}}_{N}:={\\frac {1}{N}}\\sum _{n=1}^{N}v_{n}", "0c18075a4e9e42ee0f96fe15c675b171": "\\kappa E_{0}", "0c1897114b3e3243f27526700c6f1f00": "(a\\cdot b)\\cdot c=(a\\cdot (b\\cdot c))\\cdot (a,b,c)", "0c189b999666c77000c1f09ae29cb7de": "y_{1}\\geq y_{2}\\geq \\cdots \\geq y_{n},", "0c18b4cb842730428d3b92fba3042f6d": "\\alpha =f\\,dx_{i_{1}}\\wedge \\dots \\wedge dx_{i_{k}}+g\\,dt\\wedge dx_{j_{1}}\\wedge \\dots \\wedge dx_{j_{k-1}}.", "0c190c34e6cb454f22ec65fa338ca766": "\\omega \\times \\omega ", "0c191749537a5ff60f927f96b504f2e0": "{\\sqrt {det(q)}}H", "0c195e94426b178d1e295fdf862998c2": "M=N_{1}\\#N_{2},", "0c197f3d26123c6fe4dd354f798891eb": "{\\frac {1.4388}{1.438}}", "0c19e4da62f6dd962a132438c9a7f5f6": "2g+1", "0c1a219e0036fa19beaca1c41a0c5fdc": "N=-z{\\frac {\\partial \\Omega }{\\partial z}}\\approx {\\frac {{\\textrm {Li}}_{\\alpha }(z)}{(\\beta E_{c})^{\\alpha }}}", "0c1a5d0bc626112747fb63c296c65b0e": "{n \\choose 4}\\times 6.", "0c1a725b546068462a7dbad787e6e0ac": "\\exists W(x\\in W);", "0c1a8a8b0416b2799486361bd34b37d2": "f_{n+2}-f_{n+1}=\\sum _{k=0}^{2^{n}-1}{\\bigl (}f(x_{n,k})-f_{n+1}(x_{n,k}){\\bigr )}s_{n,k}=\\sum _{k=0}^{2^{n}-1}a_{n,k}s_{n,k}", "0c1b346494d63e806e5ffe4f4526887c": "H=H_{\\rm {entropic}}+H_{\\rm {external}}={\\frac {1}{2}}k_{B}T\\int _{0}^{L_{0}}P\\cdot \\left({\\frac {\\partial ^{2}{\\vec {r}}(s)}{\\partial s^{2}}}\\right)^{2}ds-xF", "0c1b3dd11f5e83d97f632130dbca773a": "e^{\\log(z)}=z", "0c1b5d1678bea6cf742a08b99d0946bc": "am\\equiv 1{\\pmod {b}}", "0c1ba3d8616d00752aba9843186681bd": "P_{c}(t)={\\frac {1}{2N_{e}}}e^{-{\\frac {t-1}{2N_{e}}}}.", "0c1baa806043f51f22726a9a7768d5af": "\\cos nx=\\sum _{k=0}^{n}{\\binom {n}{k}}\\cos ^{k}x\\,\\sin ^{n-k}x\\,\\cos \\left({\\frac {1}{2}}(n-k)\\pi \\right).", "0c1bc22e629b9d940664b19c6525adf4": "\\alpha \\,\\!", "0c1c3b736d4f580275f93960c47792c2": "t_{r}'", "0c1ce8988c239efcb2d8cc865a4c8116": "-a\\,", "0c1cf7ea98e097d69d6bf7acb1d6a01d": "x={\\frac {r_{1}^{2}-r_{2}^{2}}{4a}}", "0c1d01b7ce7a8d098177395f4b31b977": "S=\\{s|s\\;is\\;a\\;source\\}", "0c1d1f0ee4fffd269dc2f032b8aa4b2c": "h(\\mathrm {B} (\\alpha ,\\beta ))=h(\\mathrm {B} (\\beta ,\\alpha ))", "0c1d56af4b6f9632b0c85c630b629921": "(x-3)(x^{15}-22x^{13}+x^{12}+184x^{11}-26x^{10}-731x^{9}+199x^{8}+1383x^{7}-576x^{6}-1061x^{5}+561x^{4}+233x^{3}-151x^{2}+4x+4)^{2}", "0c1d9062bf6be15690bdffb6543d888f": "{\\frac {{\\dot {m}}_{0}}{{\\dot {m}}_{01}}}={\\sqrt {\\frac {T_{01}}{T_{0}}}}{\\sqrt {\\frac {\\epsilon _{0}^{2}(1-\\epsilon _{c})^{2}-(\\epsilon _{2}-\\epsilon _{c}\\epsilon _{0})^{2}}{\\epsilon _{01}^{2}(1-\\epsilon _{c})^{2}-(\\epsilon _{21}-\\epsilon _{c}\\epsilon _{01})^{2}}}}", "0c1da82d929af9e3fef154ae5ca7fa67": "\\!\\mu _{1}", "0c1db10a37337f20040d4edeae94a163": "\\alpha _{2}<4\\times 10^{-7}\\,", "0c1dc327f2873cc0b82566eab4ad1ade": "x\\geq \\theta ", "0c1dc69a469ddd00e3c6b018d304cea6": "\\mu ={\\sqrt {\\lambda ^{2}-k^{2}}}", "0c1e1043fbe7bb2bf09a5ef1bfb46969": "-\\int _{a}^{b}\\mathbf {E} \\cdot \\mathrm {d} {\\boldsymbol {\\ell }}\\neq V_{(b)}-V_{(a)},\\,", "0c1e32860509b3d417a0130e5fe1c423": "{\\frac {du_{i}}{dt}}+{\\frac {1}{\\Delta x_{i}}}\\left[F\\left(u_{i+{\\frac {1}{2}}}\\right)-F\\left(u_{i-{\\frac {1}{2}}}\\right)\\right]=0,", "0c1e60e464562eb6bd5f2c698ac02edf": "E[X]=E[Y]=0", "0c1e6f7f4e55639c4927413bf8adab48": "\\lambda (x)=C(t-x)^{2}\\;", "0c1e7bab84b67daaddea150ea0bf1f66": "N_{\\mu }^{\\perp }=\\{\\sigma \\in ba(\\Sigma ):\\mu (A)=0\\Rightarrow \\sigma (A)=0{\\text{ for any }}A\\in \\Sigma \\},", "0c1eede8238255b8365fcc1adb7bfd04": "\\Pr(R\\cap B\\mid Y)=\\Pr(R\\mid Y)\\Pr(B\\mid Y),\\,", "0c1ef12c1e2845808d48b347e95f12cf": "y,z", "0c1f0fcf7fde4b7ed268de595f92dfcd": "F_{eq}=F_{1}=F_{2}", "0c1f13a0859aaf0f480228f6abff028d": "F=\\{f_{i}\\},\\ i=1\\ldots n", "0c1fa56e9836372889e4687fb9815fd5": "{\\mathbf {B}}", "0c1fc80974b467f49ca07ae0ea6d96be": "c_{V}\\,", "0c1fd9ed7287bf152bc0ce6850184446": "I_{i}=0^{+}\\rightarrow I_{f}=1^{+}\\Rightarrow \\Delta I=1", "0c1fed1ee0ea5c982b1aea22e5413c10": "x^{4}=x^{2}y-y^{3}.\\,", "0c1fff627dcee3b3df85c770986d9163": "\\scriptstyle {\\bar {\\theta }}", "0c20063a215893c46ff0dbc158869941": "A_{(\\alpha \\beta )\\gamma \\cdots }={\\dfrac {1}{2!}}\\left(A_{\\alpha \\beta \\gamma \\cdots }+A_{\\beta \\alpha \\gamma \\cdots }\\right)", "0c203fdedbfe4c27483d4dfbc1ddc306": "5394826801=7\\cdot 13\\cdot 17\\cdot 23\\cdot 31\\cdot 67\\cdot 73\\,", "0c20a0a5b6db7c7e1912f8f8d321f00d": "A\\rightarrow S:\\left.A,B\\right.", "0c20c5eb68af7b3ffaf0ed46c392f788": "\\langle ", "0c20fe4791299aca97aaebe841e23bcb": "\\mu _{1}\\dots \\mu _{j}", "0c21007f78283a9873ead0e94f6a8abb": "\\|s\\|_{n}=\\sum _{j=0}^{n}\\sup _{x\\in M}|D^{j}s|", "0c21b31adf16d37874174fb589b0b454": "{\\mathfrak {f}}_{(p)}(\\chi )", "0c233d9a76d35bb0c0655f5d0d36e552": "\\exp(ahr)\\exp(bhr)=\\exp((a+b)hr).\\!", "0c2373886b2b0cf01f37b7681a519837": "x*(y+z)={\\overline {x(y+z)}}={\\overline {xy+xz}}={\\overline {xy}}+{\\overline {xz}}=x*y+x*z,", "0c2391a2b9ae4fbc9bd219126028702c": "f^{n+1}~{\\stackrel {\\mathrm {def} }{=}}~f\\circ f^{n},\\,", "0c23bc8a42ed5c2babc4197a8e0de622": "\\scriptstyle \\epsilon _{c}\\,", "0c23ceae2a6f54e4907dd361a3494e7a": "F(x)=\\mu (-\\infty ,x]=\\int _{-\\infty }^{x}f(t)\\,dt.", "0c23d1ca571cf2d600bea055f89a751b": "R^{(p)}=A[X_{1},\\ldots ,X_{n}]/(f_{1}^{(p)},\\ldots ,f_{m}^{(p)}).", "0c23f865dcb1eb12861e54b0c505ca4d": "\\mathrm {O} _{2}", "0c24169b81a093490bc0585b22aca710": "\\Gamma ,\\Lambda ", "0c24362f3df681547cd3e39e4d503fae": "y_{l}", "0c24653cc31cd12e2c68cff3f7889b5c": "\\nabla \\times \\mathbf {H} ={\\frac {4\\pi }{c}}\\mathbf {J} _{\\text{f}}+{\\frac {1}{c}}{\\frac {\\partial \\mathbf {D} }{\\partial t}}", "0c24b2132650458e6b7be43680d4a0e4": "Wp", "0c24f9e1d3a1d8bd98a052d6bd20c2e8": "{N}^{(n)}(B_{1}\\times ,\\dots ,\\times B_{n})=\\sum _{(x_{1}\\neq ,\\dots ,\\neq x_{n})\\in {N}}\\prod _{i=1}^{n}\\mathbf {1} _{B_{i}}(x_{i})", "0c24fe0412722140a94debe3cf15a57c": "M=\\sup \\limits _{x\\in C}\\Re [S(x)]<\\infty \\quad ", "0c2577954896e67091440e5e07485860": "\\forall x\\forall y\\exists z\\forall w[w\\in z\\leftrightarrow (w=x\\lor w=y)].", "0c25c6ac9bc84c799b0647e64fead93b": "d_{w}=0.767+(2*0.1(1-0.767))=0.813", "0c26082951aba3022da93a06b6a1965f": "\\gamma (u)=\\gamma u\\gamma ^{*}=\\gamma ^{2}u\\,", "0c26b457ba9d8a105597dfd7a18616f1": "\\alpha \\colon S\\times S\\to A", "0c26ef5902519ca467f9a1880c167862": "1+{\\frac {a_{1}\\dots a_{p}}{b_{1}\\dots b_{q}}}{\\frac {z}{1!}}+{\\frac {a_{1}(a_{1}+1)\\dots a_{p}(a_{p}+1)}{b_{1}(b_{1}+1)\\dots b_{q}(b_{q}+1)}}{\\frac {z^{2}}{2!}}+\\dots ", "0c26f9d81df7481339ccdb89f436a653": "A,U,B{\\text{and}}V", "0c272339c176ecd5eb12f16a953fcb87": "\\operatorname {E} [X_{t+1}-X_{t}\\vert X_{1},\\dots ,X_{t}]=0\\,,", "0c272fee72d2b94860c3c747bf23b02f": "f(\\zeta )={\\frac {1}{2\\pi i}}\\int _{\\partial D}{\\frac {f(z)dz}{z-\\zeta }}+{\\frac {1}{2\\pi i}}\\iint _{D}{\\frac {\\partial f}{\\partial {\\bar {z}}}}(z){\\frac {dz\\wedge d{\\bar {z}}}{z-\\zeta }}.", "0c273a92c5148554044d961a49029fd6": "{\\mathbb {P} }^{2k+1}{\\mathbb {C} }\\,,", "0c27533d5dd29798f72ffb7ac1ffc57d": "{\\text{with }}{_{2}{\\text{F}}_{1}}(a,b;c;z)=\\sum _{k=0}^{\\infty }[(a)_{k}(b)_{k}/(c)_{k}]z^{k}/k!", "0c275399c73ee1b70950b836943c1995": "m_{i}\\leq q_{i}\\rho ,i=1,2,...,N_{sd}", "0c27d6ba59972ab85de78ca644ffe9cb": "y_{\\text{ave}}", "0c2880c0faaadc90a6e329ffd33ce241": "h(k2)", "0c28aa9ccb33a2d9e8302c51f12f8131": "\\tau ", "0c28b81f87c79a4272b324a221f99848": "b_{1}=V_{1}^{-}", "0c28f1f7661894e6619a2fea912dd938": "{\\hat {\\boldsymbol {\\beta }}}={\\underset {\\boldsymbol {\\beta }}{\\operatorname {arg\\,min} }}\\,S({\\boldsymbol {\\beta }}),", "0c28f2e47422c88c275ea7ffeeee7e41": "P_{\\alpha }^{A}", "0c28f8287babf9fac9272dc8d1cee8ce": "\\chi \\rightarrow \\{-1,+1\\}", "0c28fd74f84a23796ccb3084e1928034": "WBGT=0.7T_{w}+0.3T_{d}", "0c29162a8daac1a3781c97a656db5da0": "\\scriptstyle g(U)\\,\\cap \\,U\\;\\neq \\;\\emptyset ", "0c2929e1f6890b113452d9bc5c896609": "f(z)={\\frac {z+2}{(z-5)^{2}(z+7)^{3}}}", "0c29d9a0a7f5fb012b48025e83af4c99": "D\\to {\\frac {k_{B}T}{8\\eta _{m}ha}}{\\frac {L_{sd}}{a}}", "0c29ec88e10b2d0655aed9f334c998c4": "q_{0}\\,\\in Q\\,", "0c29f1ca956ebc631b02139b4503ee14": "v,v'\\in V\\,", "0c2a24393c01f34e1ec0c3f6a9bf3391": "E_{x}^{\\rm {PBE,SR}}(\\omega )", "0c2b7be478cb3c91c3c7f50457471d8c": "\\sum _{n=1}^{\\infty }(-1)^{n}a_{n}=-a_{1}+a_{2}-a_{3}+\\cdots \\!", "0c2b81a4d22a30289a24ed0cf4ea8e22": "\\forall y\\exists x(x^{2}=y)", "0c2b95541a022c82ebbf0248ee90dca2": "\\sin 87^{\\circ }=\\cos 3^{\\circ }={\\frac {{\\sqrt {60+12{\\sqrt {5}}}}+{\\sqrt {20+4{\\sqrt {5}}}}+{\\sqrt {30}}+{\\sqrt {2}}-{\\sqrt {6}}-{\\sqrt {10}}}{16}}", "0c2b9ba7364d8db8614b0d602ee6de2f": "\\mathrm {Efficiency} ={\\frac {\\mathrm {Useful\\ power\\ output} }{\\mathrm {Total\\ power\\ input} }}", "0c2b9c94aa69394d0652550c8f20f64a": "(f_{1}+f_{2})-(f_{3}+f_{4})", "0c2bb2aebb69d4ace76e27deb93f2643": "{\\hat {\\lambda }}\\,", "0c2bbff22a71fef5b363acb30f0dc041": "{\\frac {x-\\lambda -a_{1}}{a_{2}-a_{1}}}", "0c2be49ff2494b14ce9713e5af78ea81": "~\\epsilon _{t-1}^{-}=~\\epsilon _{t-1}", "0c2c06c343d5d73164aba01fd3fef344": "V^{*},", "0c2c747f9ef05a0fffb04221c89403bd": "i=\\mathbf {e} _{2}\\mathbf {e} _{3},j=\\mathbf {e} _{3}\\mathbf {e} _{1},k=\\mathbf {e} _{1}\\mathbf {e} _{2},\\,\\,\\varepsilon =\\mathbf {e} _{1}\\mathbf {e} _{2}\\mathbf {e} _{3}\\mathbf {e} _{4}.\\!", "0c2cb0029c3418660112059ac23cc16f": "x\\in (1/4,1/2]", "0c2cb856cefb469efe1610c9d08a2485": "X_{0}=N", "0c2cc1d3937d818248ebe7a3c8858e3a": "{\\widehat {K_{\\alpha }}}(\\xi )=|2\\pi \\xi |^{-\\alpha }", "0c2d020c8454a43077e27d19dfa9f909": "\\det T_{g*h}=\\det T_{g}\\cdot \\det T_{h}\\cdot \\mathrm {res} (g_{-},h)", "0c2d1663b1ccfe8134cb66c248945b06": "A={\\text{ constant}}", "0c2d251e761eefb82204185f07130afa": "{\\textbf {r}}=-1+X^{2}+X^{3}+X^{4}-X^{5}-X^{7}", "0c2d954320884adb96cc0b8ee65596ab": "(1-q)(1-q+q^{2})p^{2}-q(1-q)(1+2q)p-q^{3}>0\\,", "0c2d9b32387dd0c133969c924f6a9f20": "\\scriptstyle n_{i}", "0c2e12b1333082e00715b3b4f93d0278": "{\\mbox{Free}}(\\phi \\vee \\psi )={\\mbox{Free}}(\\phi )\\cup {\\mbox{Free}}(\\psi )", "0c2e423c9aecb851b82593f9b4b37709": "\\omega =y_{n}", "0c2e55a5f834d6666a6c66f1606127de": "\\left.{}-33\\left(\\theta _{2}(0,q)^{4}+\\theta _{3}(0,q)^{4}\\right)\\cdot \\theta _{2}(0,q)^{4}\\theta _{3}(0,q)^{4}\\right]", "0c2e6c44260a4a2281013648a888fa43": "x\\circ y={\\frac {1}{2}}(x\\cdot y+y\\cdot x),", "0c2e7b202ad7b0f4f1d92eab5ebde3f0": "A_{k}\\subseteq A_{k+1}", "0c2e8467d8bfa408220d11ac6afbc324": "|P|=|S|\\cos \\varphi ", "0c2f20b2bd4e6fcce67554bcdad32753": "\\max \\left[F_{l},\\left(D_{l}+{\\frac {F_{l}}{2}}\\right),0\\right]", "0c2f24503b43c4d42ad2c7d55e3d6acf": "x|\\sigma ^{2},\\mu ,\\lambda \\sim \\mathrm {N} (\\mu ,\\sigma ^{2}/\\lambda )\\,\\!", "0c2f8cc05147d963bf195e3335014324": "\\left[{\\hat {x}},{\\hat {p}}\\right]={\\hat {x}}{\\hat {p}}-{\\hat {p}}{\\hat {x}}=i\\hbar .", "0c2fbc79e19643b0dd3d5cc0f4479d54": "\\sigma (t)={\\frac {\\sigma _{0}}{1-[1-(t/t*)(1^{1-n})]}}", "0c2fd170e5e47a8e27ece0f7b6739959": "\\sigma \\in \\Delta ", "0c301a02221077fa56c1922a93606f56": "\\ GL_{n}(\\mathbb {Z} _{p})", "0c30282696cb5839ad1ecf78035363a5": "K^{\\ddagger '}=e^{\\frac {-\\Delta G^{\\ddagger }}{RT}}", "0c316a80e3e9ab49a2585086b231ec6a": "I\\subseteq R_{+}", "0c31e09df9adb9361574e270ccac3457": "\\tau _{c}*=0.03", "0c31f2c011d37fc982efa83760ff76b7": "\\tau _{ri}", "0c32205b35444bd5bac9b1478702b651": "return:fail", "0c323b0d06b6ea86c6963fec7cd0505b": "{\\frac {x}{\\cos \\gamma }}+{\\frac {y}{\\sin \\gamma }}=1,", "0c328a9e72a1840913bfc6da64ebee16": "\\{U_{i}\\}_{i=1}^{k}", "0c328b77399b3a1de4a25ff8cdccb10e": "F_{Y}(y)=P(Y\\leq y)", "0c32eae5ea0138451eca1d877243b730": "f_{B}", "0c32f76e19392dbf43da275a97e5fc13": "P(B\\mid A)={\\frac {P(A\\cap B)}{P(A)}}.", "0c332bde051526f0124f664b00d90eba": "\\textstyle i>m", "0c338bd49d71c8383995854260581ed4": "{\\mathcal {H}}_{Heis}={\\frac {1}{2}}(-2J\\sum _{i,j}{\\vec {S}}_{i}\\cdot {\\vec {S}}_{j}\\quad )=-\\sum _{i,j}J{\\vec {S}}_{i}\\cdot {\\vec {S}}_{j}", "0c33ab1e5d620c71ecbdc529f20ca4ec": "[t,t+h]", "0c342ec47723a3a13b624bc44c98d94e": "\\nabla f={\\frac {{\\frac {\\partial f}{\\partial x}}-\\sin(\\phi ){\\frac {\\partial f}{\\partial z}}}{\\cos(\\phi )^{2}}}\\mathbf {e} _{1}+{\\frac {\\partial f}{\\partial y}}\\mathbf {e} _{2}+{\\frac {-\\sin(\\phi ){\\frac {\\partial f}{\\partial x}}+{\\frac {\\partial f}{\\partial z}}}{\\cos(\\phi )^{2}}}\\mathbf {e} _{3}.", "0c344f8ccee98f30cff89cbda229bf4d": "\\tau _{\\mathrm {delay} }={\\underset {t}{\\operatorname {arg\\,max} }}((f\\star g)(t))", "0c3451207604f71dde486c37745fad7e": "_{5}^{6}", "0c3455048afb4a63fde93c0bc96c8b95": "\\scriptstyle {\\frac {1}{2}}m^{2}A_{\\mu }A^{\\mu }", "0c3470f889fbae5a424685f7fcef7143": "W=\\Delta F", "0c34afb31a801725cb54bf74e16e5b83": "(r,r-1)", "0c34d7a043bd5ea7924c45aaec774e98": "x+1/x=-1", "0c351e3856c4c03a080caae1cf5bca22": "E_{n}^{(1)}=\\langle \\psi ^{0}\\vert H'\\vert \\psi ^{0}\\rangle =-{\\frac {1}{8m^{3}c^{2}}}\\langle \\psi ^{0}\\vert p^{4}\\vert \\psi ^{0}\\rangle =-{\\frac {1}{8m^{3}c^{2}}}\\langle \\psi ^{0}\\vert p^{2}p^{2}\\vert \\psi ^{0}\\rangle ", "0c353cae3830e4aa1ac6fca418796f60": "\\cos(\\arcsin x)={\\sqrt {1-x^{2}}}", "0c355f0ebd208bc310f3494422a23908": "\\langle y|R\\rangle \\langle R|x\\rangle ,", "0c361efc0d92fc39bd5527f847f69f71": "[x]={\\textrm {cl}}\\{x\\}=\\bigcap {\\mathcal {N}}_{x}.", "0c362270aed94e591d1c374764a5a327": "f(\\varepsilon )\\sim \\varepsilon ^{\\gamma }\\,", "0c364d87aabab19b046e5f5d7149d4c0": "k_{1},\\ldots ,k_{m}", "0c364fc7967fd3dd4ba9c17d9be44536": "\\langle p_{\\mathbf {R} ,i}|{\\tilde {\\phi }}_{\\mathbf {R} ,j}\\rangle _{rT_{setup}+T_{ko}+T_{skew}", "0c3d701f6160b320f39c480588c6ffa4": "T={\\begin{bmatrix}T_{11}&T_{12}\\\\0&T_{22}\\end{bmatrix}}", "0c3d72395d7576ab13b9e9389f865960": "P(X)", "0c3e08e83a5937916dcb653ad1cb799a": "a_{i},1\\leq i\\leq n", "0c3ea8f4cdbf7b626bb9b2f09477b632": "M=M_{ii}={\\frac {\\mu _{0}}{4\\pi }}\\left(\\oint _{C}\\oint _{C'}{\\frac {\\mathbf {ds} \\cdot \\mathbf {ds} '}{|\\mathbf {R} _{ss^{\\prime }}|}}\\right)_{|\\mathbf {R} |>a/2}+{\\frac {\\mu _{0}}{2\\pi }}lY+O\\left(\\mu _{0}a\\right).", "0c3ec16ce2495229ab5b269d764562b4": "\\textstyle \\exp(rT)=R_{0}", "0c3ee0b09d3c337e066308ae140704c1": "{\\frac {d\\phi (t,t_{0})}{dt}}=A(t)\\phi (t,t_{0})", "0c3ef3f7b93ab6b21572c7134390378a": "1-(\\lambda +\\mu )\\Delta t", "0c3f0e3a4608bac89f6cebab322f2d22": "b\\neq 0", "0c3f7f0c6f72effbc12058ea4a3ba10e": "M=-\\int _{\\mathcal {X}}\\left({\\frac {\\partial \\psi (x,\\theta )}{\\partial \\theta }}\\right)_{T(F)}dF(x)", "0c3f8fc2259e29eee543ec9029e53603": "Y\\sim N_{n}(0,\\sigma ^{2}I_{n})", "0c3f980d93ca95ace0071700f6088ce3": "d\\tau ^{2}=dt_{r}^{2}\\left[1-{\\frac {2M}{r}}+2{\\sqrt {\\frac {2M}{r}}}{\\sqrt {\\frac {2M}{r}}}-{\\sqrt {\\frac {2M}{r}}}^{2}\\right]=dt_{r}^{2}", "0c3fbe9776491ed80ea80723fc6503e2": "z^{2}+w^{3}=0", "0c3ff8a5bbfaeafb675fca98d0741432": "\\tau ^{\\alpha \\beta }\\,", "0c4017f6a2401c4b5bd1c02adebe33da": "\\gamma ^{2}=\\gamma ", "0c4024bf6c05fb544400bdd856c58b91": "{\\mathcal {L}}^{-1}(x)={\\begin{cases}1.31\\tan(1.59x)+0.91x&\\quad \\mathrm {for} ~|x|<0.841\\\\{\\tfrac {1}{\\operatorname {sgn}(x)-x}}&\\quad \\mathrm {for} ~0.841\\leq |x|<1\\end{cases}}", "0c402d854ca6f159ecfcd2dddc05e23b": "A\\subseteq {\\rm {pcf}}(A)", "0c404174baa6c416a814b2486a5841d0": "f''", "0c40462f440862674116eec19a92ecb2": "(x-x_{1})(x-x_{2})\\cdots (x-x_{n}),", "0c4056d0f08a2d26e7374c773b39d165": "\\nabla \\times \\nabla \\times \\mathbf {E} =-\\mu _{o}{\\frac {\\partial }{\\partial t}}\\nabla \\times \\mathbf {H} =-\\mu _{o}\\varepsilon _{o}{\\frac {\\partial ^{2}\\mathbf {E} }{\\partial t^{2}}}", "0c4059314ecc30ff4ebc41f6fc6138ed": "Q_{1}^{({\\text{green}})}(t)-Q_{2}^{({\\text{green}})}(t)=2", "0c408f36576e603377026d3e20ce57dd": "f(x)={\\begin{cases}0&{\\mbox{if Goldbach's conjecture is false}}\\\\1&{\\mbox{if Goldbach's conjecture is true}}\\end{cases}}", "0c40923b462b71b5b7a9ccc35962430a": "\\sum \\limits _{a_{i}\\in S}a_{i}=|S|a_{j}", "0c40a0a2d82dd414c5ac5deb9b589647": "0.{\\overline {571428}}", "0c4134fd43aada42849770ac26005c8c": "{\\frac {r}{m}}-\\ln(m)-1.24=0.", "0c4147b63c68ff83b76223526c582702": "|0\\rangle ^{\\otimes n}|1\\rangle ", "0c41d355882df8f3b078b2ce48100f31": "a,b,\\dots ,k", "0c41e040e70e48def3f230eddd9001a0": "4+\\sum _{i=1}^{n}i\\leq 4+\\sum _{i=1}^{n}n=4+n^{2}\\leq 5n^{2}", "0c4209a8faecbcee233e5df1e1751e24": "{\\frac {6}{5}}", "0c42406640583a31bf9fd0948db1b66f": "\\delta _{a}:Q\\rightarrow Q", "0c4266cc913627ce579cfed1d48652ed": "P_{0}\\cdot C_{0}", "0c4281bde96d7f1b2681313f3538a989": "{\\begin{alignedat}{2}ax_{1}x_{2}+bx_{1}y_{2}+cx_{2}y_{1}+dy_{1}y_{2}&=&\\alpha \\\\ex_{1}x_{2}+fx_{1}y_{2}+gx_{2}y_{1}+hy_{1}y_{2}&=&\\beta \\end{alignedat}}", "0c42896a9b4e598aa2b9fab46328985a": "\\partial _{a}={\\frac {\\partial }{\\partial x^{a}}}", "0c42918eb31efe0ba6711ac312f23003": "t_{l}=2w", "0c42a7a275ba8863cd1732e2a3be4b66": "e_{q}(\\ln _{q}(x))=x", "0c42b89ac347eca8baa6f35f36ec6fc1": "\\mathbb {E} [1_{D(y)\\neq m}]", "0c42e7346e7a3f464d1648dfcf625ea1": "P(78\\leq X\\leq 88)=P\\left({\\frac {78-80}{5}}\\leq Z\\leq {\\frac {88-80}{5}}\\right)=P(-0.40\\leq Z\\leq 1.60)=P(Z\\leq 1.60)-P(Z\\leq -0.40)=0.9452-0.3446=0.6006", "0c42fdfcc5b5e6601e398d7c71216db3": "\\operatorname {E} [S_{N}]=\\operatorname {E} [N]\\,\\operatorname {E} [X_{1}].", "0c431b73a43e17926497f3204abccdc4": "{\\frac {\\partial f}{\\partial t}}+r\\cdot \\nabla _{r}f-{\\frac {1}{\\hbar }}\\nabla _{r}V\\cdot \\nabla _{k}f=\\left({\\frac {\\partial f}{\\partial t}}\\right)_{c}", "0c432282fcdfca30f53df2a2479edc8a": "|\\lambda |<1", "0c436009c4714ade991c78a204cc8a50": "poly(\\lambda )", "0c438a31479ea3ed8039f29013e82599": "A_{n}={\\frac {P}{\\sqrt {n(n+1)}}}", "0c43b5bc69ca67f4d78f82cead712168": "\\sigma _{fail}=\\sigma _{Y}", "0c447ddede8f365b49ffc8c963bda7d1": "\\{x_{1},x_{2},\\dots ,x_{n}\\},", "0c45507111319c8b237201dca0fd2c4a": "dT_{V}=pd\\Theta ", "0c455c9e4175f2093f9c3b61a97e40e3": "\\mu _{r}=[B_{1},B_{1}^{\\dagger }]+[B_{2},B_{2}^{\\dagger }]+II^{\\dagger }-J^{\\dagger }J,", "0c45ac606062178381e49d38b9a979b1": "a-x_{n}={\\frac {-2f(x_{n})f'(x_{n})}{2[f'(x_{n})]^{2}-f(x_{n})f''(x_{n})}}-{\\frac {2f'(x_{n})f'''(\\xi )-3f''(x_{n})f''(\\eta )}{6(2[f'(x_{n})]^{2}-f(x_{n})f''(x_{n}))}}(a-x_{n})^{3}.", "0c45da996841cbcd3fe0838e9687efd6": "ds^{2}=-\\left(1-{2V(r) \\over c^{2}}\\right)c^{2}dt^{2}+dx^{2}+dy^{2}+dz^{2}", "0c461183e2e4008c3661a14cc34a8865": "\\sqcap X", "0c46240353e703ab4ab54295d7bc9850": "(x-1)^{3}x", "0c4633ddfd76a62570b420c5a7856370": "{\\begin{aligned}\\varphi (g)\\colon V&\\to V\\\\v&\\mapsto \\Phi (g,v)\\end{aligned}}", "0c4676a2a5438e2ca500a1b4cfa25964": "\\mathbf {b} :[0,T]\\times \\mathbb {R} ^{N}\\rightarrow \\mathbb {R} \\in L_{2}[0,T]", "0c467ede3869016ea059e85bc3cfb6e1": "m^{2}=\\eta _{0}^{2}(p_{\\mu },\\eta )-\\eta _{i}^{2}(p_{\\mu },\\eta )", "0c4692635510aa9eeba461cf1f217118": "V^{*}=\\bigcup _{i\\in \\mathbb {N} }V_{i}=\\{\\varepsilon \\}\\cup V\\cup V_{2}\\cup V_{3}\\cup V_{4}\\cup \\ldots .", "0c46bfc49d48f31f1cdb0d6336e67ddb": "f_{n}=X_{n}+X_{n+1}+X_{n+2}+\\ldots ", "0c46ccd184562b893e8633b518624511": "X_{k_{1},k_{2},\\dots ,k_{d}}=X_{N_{1}-k_{1},N_{2}-k_{2},\\dots ,N_{d}-k_{d}}^{*},", "0c46cdb789fd6cab181a3d2b342dfcae": "P\\ =\\ 2\\pi \\ a\\ {\\sqrt {\\frac {a}{\\mu }}}\\,", "0c473235a11c4f3d289fe2eed7c5ebe6": "{{\\mathit {l}} \\over n}={1 \\over 2}\\pm {1 \\over 2n}", "0c473afc7ba9849eae3939c0013625b4": "r_{li}", "0c4748f3be6412c3e9b547e11d0a0131": "R_{nl}(r)=r^{l}f_{nl}(r)\\exp \\left(-{\\frac {Zr}{n}}\\right),", "0c474e9e82aa677acf9b6a4014372a97": "\\mathrm {^{241}_{\\ 95}Am\\ \\xrightarrow {(n,\\gamma )} \\ _{\\ 95}^{242}Am} ", "0c477408ace07085a18b6c66198492a0": "Y=\\log(X)", "0c4787b8aac88ac5488ee69523a05381": "K={\\frac {(1-e^{2}\\sin ^{2}\\phi )^{2}}{b^{2}}}={\\frac {b^{2}}{a^{4}(1-e^{2}\\cos ^{2}\\beta )^{2}}}.", "0c4795612ee01990b45d17463059ab7b": "\\textstyle {\\frac {a}{b}}", "0c4809ff81769d7213ad56e521c117f5": "w\\mapsto R(u,v)w", "0c480c04e486fb18c3d016dd168a5798": "Q_{3}=(\\sigma \\,z_{3})+X", "0c482cabe693c6a06288dbfac3527be2": "\\textstyle q^{n-r}", "0c48abe0126552b7382e70c3801b8bc2": "{\\begin{bmatrix}u_{11}&u_{12}&.&u_{1n}\\\\0&u_{22}&.&u_{2n}\\\\.&.&.&.\\\\0&0&.&u_{nn}\\end{bmatrix}}=LDU={\\begin{bmatrix}1&0&.&0\\\\l_{12}&1&.&0\\\\.&.&.&.\\\\l_{1n}&l_{2n}&.&1\\end{bmatrix}}", "0c4927294c93f32340903c76ccd6210d": "{\\mathcal {A}}_{\\mu }=t_{a}{\\mathcal {A}}_{\\mu }^{a}", "0c4937f25a689e866c45c67435cabb4c": "W(dx)=dx/(\\pi x^{2})", "0c493e43973bd9cc8905d8d7c3fbacd9": "\\Gamma (z)={\\frac {1}{z}}-\\gamma +{\\frac {1}{6}}\\left(3\\gamma ^{2}+{\\frac {\\pi ^{2}}{2}}\\right)z+O(z^{2})", "0c4949e23cea11495f9a6e94c2077a54": "\\theta =\\arctan \\left[{\\sqrt {{\\frac {8c^{2}(r_{1}^{2}+r_{2}^{2}-2c^{2})}{r_{1}^{2}-r_{2}^{2}}}-1}}\\right]", "0c4953b32792bf8316d92b514c4e3f48": "C\\subseteq B\\,\\!", "0c49546ecb73718a1a091ff88d5ab885": "n=0,1,2,...", "0c4987875329472336262f519e093915": "h\\sim {2\\times 10^{-20}/{\\sqrt {\\mathrm {Hz} }}}", "0c49af1109a97c7ae982d0e122b1b72d": "\\operatorname {Dom} (n)=\\left(\\bigcap _{p\\in {\\text{preds}}(n)}^{}\\operatorname {Dom} (p)\\right)\\bigcup ^{}\\left\\{n\\right\\}", "0c49d9c162f8bb9bcab5897a56dbc73b": "A(xy)=A(x)A(y).\\,", "0c49e677829106468cb9c7eee716a544": "M_{0}\\to _{G,t_{i_{1}}}M_{1}\\wedge \\ldots \\wedge M_{n-1}\\to _{G,t_{i_{n}}}M_{n}", "0c4a3445efd5488a8469e7b10f68d4b1": "\\gamma ^{0}={\\begin{pmatrix}0&\\sigma ^{2}\\\\\\sigma ^{2}&0\\end{pmatrix}},\\quad \\gamma ^{1}={\\begin{pmatrix}i\\sigma ^{3}&0\\\\0&i\\sigma ^{3}\\end{pmatrix}}", "0c4aee9384242c43a99883a6b9c9190c": "x^{16}+x^{10}+x^{8}+x^{7}+x^{3}+1", "0c4b0c56ecf0f027224babb73cec7044": "h(\\psi (h(\\psi (0))))", "0c4b316ecf27f5494e1f5fbd74dcd75b": "Y=G_{i}\\exp \\left(M^{(i)}\\log(x_{i})+\\sum _{j=1}^{r_{i}}{\\frac {T_{j}^{(i)}}{x_{i}^{j}}}\\right).", "0c4b3df32e5674357d7508ae3d5e706c": "M_{-}(z)={\\overline {M_{+}(z)}},\\quad z\\in \\Sigma .", "0c4b3f7c6c486c4f940706dbe2a77f00": "{\\mathcal {P}}={\\big \\{}P_{\\theta }\\ {\\big |}\\ \\theta \\in \\Theta {\\big \\}}.", "0c4b7cd6210e693a094d76bf2c06b1d1": "{\\vec {c}}_{2}(v)", "0c4ba1c2bd327fea8113992ca1f3c77a": "f(P,w)=\\left\\{{\\begin{matrix}0&{\\mbox{if}}\\ w\\neq 1\\ {\\mbox{in}}\\ H\\\\{\\mbox{undefined/does not halt}}\\ &{\\mbox{if}}\\ w=1\\ {\\mbox{in}}\\ H.\\end{matrix}}\\right.", "0c4bb2e2df8049bdf0a178b17cba4c7b": "V_{0}=RC{\\frac {dV(t)}{dt}}+V(t).", "0c4bc85518833bc2d14944c53eb6b518": "\\left(\\mathrm {PL} _{j}\\right)_{i}={\\frac {\\left(X_{j\\bullet [j]}\\right)_{i}^{2}}{\\sum _{k=1}^{n}\\left(X_{j\\bullet [j]}\\right)_{k}^{2}}}", "0c4c72c4a523f41c517b4185683a153f": "Q=A_{1}{\\sqrt {{\\frac {2}{\\rho }}\\cdot {\\frac {\\left(p_{1}-p_{2}\\right)}{\\left({\\frac {A_{1}}{A_{2}}}\\right)^{2}-1}}}}=A_{2}{\\sqrt {{\\frac {2}{\\rho }}\\cdot {\\frac {\\left(p_{1}-p_{2}\\right)}{1-\\left({\\frac {A_{2}}{A_{1}}}\\right)^{2}}}}}", "0c4c7926af504700991938b35978d4cb": "|\\mathbf {J} |=\\hbar {\\sqrt {j(j+1)}}\\,\\!", "0c4c9575ee93c2265f2886ffbc32e601": "E_{1},E_{2},E_{3},\\ldots ,E_{N}", "0c4ca2473c7cda2aa5d6f89535728326": "\\psi _{L}(0)=\\psi _{C}(0)", "0c4d11f509846e3a2211ca4236fc6aab": "{\\begin{aligned}\\int {\\frac {du}{\\sqrt {a^{2}+u^{2}}}}&=\\sinh ^{-1}\\left({\\frac {u}{a}}\\right)+C\\\\\\int {\\frac {du}{\\sqrt {u^{2}-a^{2}}}}&=\\cosh ^{-1}\\left({\\frac {u}{a}}\\right)+C\\\\\\int {\\frac {du}{a^{2}-u^{2}}}&=a^{-1}\\tanh ^{-1}\\left({\\frac {u}{a}}\\right)+C;u^{2}a^{2}\\\\\\int {\\frac {du}{u{\\sqrt {a^{2}-u^{2}}}}}&=-a^{-1}\\operatorname {sech} ^{-1}\\left({\\frac {u}{a}}\\right)+C\\\\\\int {\\frac {du}{u{\\sqrt {a^{2}+u^{2}}}}}&=-a^{-1}\\operatorname {csch} ^{-1}\\left|{\\frac {u}{a}}\\right|+C\\end{aligned}}", "0c4d45a1ce65df9fb01caac3918a3fbc": "A_{a}", "0c4d91873d2a7a8dd1d2c0ba7f8f271e": "u_{i}(x)", "0c4db32718b8c53b3209df130c005d2d": "\\forall k\\ \\|\\mathbf {e} _{k}\\|=c", "0c4e1df2a7e19274eea5df7c4d27ec9a": "\\left.+\\left|F(\\mathbf {p} /2-\\mathbf {k} )\\right|^{2}(n_{c1}(\\mathbf {k} )+n_{a2}(\\mathbf {k} )-n_{c2}(\\mathbf {k} )-n_{a1}(\\mathbf {k} ))\\right]", "0c4e660a83e8684cf55369080717c873": "A={\\frac {a}{|r_{0}^{2}-a^{2}|}}", "0c4ea554c815215628b2c79e48df6405": "\\sigma _{ij}", "0c4f20bc198645607ac566df18b423ba": "0<-K_{X}\\cdot C_{i}\\leq \\operatorname {dim} X+1", "0c4f58f68d2ca2e6273cd323c66a3aef": "OPEN_{d}'", "0c4f7da8b40888d0d73de1e5e4b9927b": "{\\frac {r_{E}}{r_{O}}}\\ll 1", "0c4f98d7758dd7569b4e2cff204f06e3": "f:x\\mapsto h(Rx)", "0c4f9ea17b32cab6493f89bf2e3978f3": "\\langle x\\rangle ={\\frac {1}{P}}\\int {I(x,y)xdxdy},", "0c4fa624c3da1689fefc711f07124f8a": "\\gamma ^{\\mu \\nu }", "0c5010870ba0629086e10e3db1038559": "{\\frac {1}{FG}}={\\frac {1}{2}}\\left({\\frac {1}{AB}}+{\\frac {1}{DC}}\\right).", "0c502ee79f3b09999ba9d57212bceb67": "\\pm e_{i}", "0c503b5cb68185209fe963aaf64a5938": "\\left(\\mathbf {P} ^{T}\\mathbf {A} \\mathbf {P} \\right)\\left(\\mathbf {P} ^{T}\\mathbf {x} \\right)=\\mathbf {P} ^{T}\\mathbf {b} .", "0c506595cbd5a3ca706f8795eed679ea": "\\left[{\\begin{array}{ccc|c}2&3&5&0\\\\-4&2&3&0\\end{array}}\\right].", "0c50995c83b30a81e4e4a5f09ece0cb7": "H|\\alpha \\rangle =E|\\alpha \\rangle ", "0c51d1715f3246dff44a573ac21b7ca3": "x=y=0", "0c51dde9586da2e526cb82809b896e6c": "\\mathbf {B} =\\mathbf {DN} \\qquad \\qquad \\qquad \\qquad \\mathrm {(8)} ", "0c51df2af65ec969dd356261620760d4": "V_{i}^{*}=P_{i}V^{*}", "0c51f684199167e97a4ae798262d0e68": "(1,2,\\cdots ,k)", "0c51f9e080ca217ae2d6a581f142e12d": "R_{m}(R_{n}(\\xi ,\\xi ),R_{n}(\\xi ,x))=R_{m\\cdot n}(\\xi ,x)\\,", "0c5207fdccbbe3b48a269af808f3fc78": "\\phi _{n}(x)=\\phi (x-n)", "0c5212ae373f51c67b6e1a83f65e0c60": "\\approx 10^{7.8\\times 10^{41}}", "0c5263856259f30ccd1df1fb3146055c": "F(k,x)\\leftarrow Dec(sk,c)", "0c5312947e9f4146d389419eedbc5088": "H(X)=\\operatorname {E} \\left[\\log _{b}\\left({\\frac {1}{p(X)}}\\right)\\right]\\leq \\log _{b}\\left(\\operatorname {E} \\left[{\\frac {1}{p(X)}}\\right]\\right)=\\log _{b}(n)", "0c531c88463688ae6b25af1b44a9ed3b": "\\mu =\\operatorname {(} m(\\vartheta ))", "0c534f8590e3e5cbcdc5b1a9a52798df": "i{d \\over dt}U(t,t_{0})\\Psi (t_{0})=V(t)U(t,t_{0})\\Psi (t_{0}).", "0c53694d9ea0dfd6af455ed2c85df9e3": "I_{d}=\\oint _{S}\\mathbf {J} _{\\rm {d}}\\cdot {\\rm {d}}\\mathbf {A} ,\\,\\!", "0c5464c819c71febf835909a603f1d9a": "a=\\iota x(\\pi (x,b)\\land x=a)", "0c5470706c3157b1ce3605d76d00e10b": "D^{*}={\\frac {q\\lambda \\eta }{hc}}\\left[{\\frac {4kT}{R_{0}A}}+2q^{2}\\eta \\Phi _{b}\\right]^{-1/2}", "0c54cc5a72c233e96ccad4105727a9b1": "\\omega =0^{\\circ }", "0c550cab08f6617ca3c13a930c4191e1": "\\mathbb {F} _{q}^{k}", "0c553bfe89889ede1798457cda24fa37": "\\Lambda ^{r}U=\\bigoplus _{p+q=r}(\\Lambda ^{p}S)\\otimes (\\Lambda ^{q}T).", "0c55c3c8730eaa2047bfadb9620b9a0f": "\\partial ^{i}\\phi =\\theta ^{i}", "0c56a2bc23d94cfab41587837e1c431b": "\\mathbf {H} \\mathbf {C} =\\mathbf {M} \\mathbf {C} {\\boldsymbol {\\Phi }},", "0c56e17183d1902018b37da28c58d631": "m\\in \\mathbb {F} _{q}^{k}\\backslash \\{0\\}", "0c574dde467b9c4e59307b3663b4d5ae": "f(x)-f(u)\\geqq g(x,u)\\cdot \\nabla f(u),\\,", "0c575a1fa28128a5a078b3757f9f1d73": "V_{\\mathrm {int} }=-\\mathbf {F} \\cdot {\\boldsymbol {\\mu }}.", "0c57a8b7a6166a2162d1670278e64e97": "s_{\\infty }(z)=z^{\\nu }(1+{\\mathcal {O}}({\\tfrac {1}{z}})).", "0c57e67e9d7593f78c74880b9d98cef6": "H(u)(t)\\approx A\\cdot \\sin(\\omega t+\\phi _{m}(t))", "0c57f3025fc06db223af09ef43f26afe": "f(x)=x^{5}+x^{4}+x^{2}+x+2", "0c588743b8102cfe7602fcbc999d55da": "M+C_{2}H_{5}^{+}\\to [M+C_{2}H_{5}]^{+}", "0c58a1e9ef4203c464eb36b7c3d95a10": "a\\otimes b=b\\otimes a", "0c58b9f000aeade0a46a9e77889fa65f": "{\\boldsymbol {\\Delta }}_{1}^{1}={\\boldsymbol {\\Sigma }}_{1}^{1}\\cap {\\boldsymbol {\\Pi }}_{1}^{1}", "0c58ed46b137c32253dbde5909e691a6": "R={\\frac {A_{1},\\dots ,A_{n}}{B}}", "0c595e423e56612a862efd5dee8d72a4": "M^{\\!\\!\\!\\!\\!{}^{\\beta }}", "0c596f5b75ceb6f44c51a8c09ee83bab": "\\mathbf {x} _{B}=R\\mathbf {u} _{R}\\ ,", "0c5987d824c0cdb4c19a2437393ae9f4": "\\mathbf {A} =\\mathbf {P} =r\\mathbf {\\hat {r}} +z\\mathbf {\\hat {z}} ", "0c59de0fa75c1baa1c024aabfa43b2e3": "\\textstyle n", "0c59e3f0198f16be9ac1e499453576c9": "k={\\frac {J_{12}}{\\sqrt {b_{1}b_{2}}}}", "0c59f8825ec535fc8bce1ee7249dbf24": "\\textstyle {\\mathcal {N}}(m_{k},\\sigma _{k}^{2}C_{k})", "0c5a44a1b1a0d8803827cd2704539596": "p(x_{n+1})\\ =\\ p_{1}(x_{n+1})-p_{2}(x_{n+1})\\!\\cdot \\!E\\ =\\ f(x_{n+1})-(-1)^{n+1}E", "0c5a4f80f1cbf88f90b7c2fc33bbe0b1": "\\gamma _{tx}", "0c5a5ad9b4f40ffed32a6a66af599367": "{\\tilde {S}}(q,\\omega )", "0c5a7af66122a274bf6d463e213d4f04": "-(0.75)^{n}u[-n-1]\\ ", "0c5a91c0f09ecd88b2afb669cabf0f85": "\\int _{-a}^{a}f(z)\\,dz=\\oint _{C}f(z)\\,dz-\\int _{\\text{Arc}}f(z)\\,dz", "0c5ab7bb5d9ecadc029bb27f422269a5": "\\{y_{1},y_{2},y_{3},y_{4}\\}=\\{e^{(2+i)x},e^{(2-i)x},e^{ikx},e^{-ikx}\\}.", "0c5ac033023f12586ef43a2662a663ff": "f\\approx g", "0c5ac03ccf3b49620b84d3caebc56686": "c=\\left\\lfloor {\\frac {y-1}{100}}\\right\\rfloor \\quad {\\text{and}}\\quad g=y-1-100c.", "0c5af2f63a912f189c8e67a4c8bd90e7": "(N,y)\\to (N,y).", "0c5b1102f4c09c6281c7a01967d3afe4": "{\\hat {q}}=2^{-1/2}({\\hat {a}}^{\\dagger }+{\\hat {a}})", "0c5b5d9bcbb922ae90f87cf90e7d8b44": "n\\left[{n \\atop k}\\right]", "0c5bee0a4c62c3501013220b5ff97c0d": "{\\frac {MacD}{(1+y_{k}/k)}}=-{\\frac {1}{V(y_{k})}}\\cdot {\\frac {\\partial V}{\\partial y_{k}}}\\equiv ModD", "0c5c092e63829f6fd5825f6b291800ca": "P_{J}[f](x)=P_{k}[f](x)+D_{k}[f](x)+\\dots +D_{J-1}[f](x)", "0c5c13dd202360245712aadba935119f": "{\\ddot {T}}-(m^{2}\\tau ^{2}-n^{2})T=0", "0c5c5826298887ada15b9c8e18dd48a5": "[S_{x},S_{y}]=i\\hbar S_{z}", "0c5d030d5cc98694da183cd68f7d4d48": "\\epsilon =1/3", "0c5d1bf05f146172ec5834b147a81bba": "\\mathrm {P} (A=0,B=0)=P\\{1\\}={\\frac {1}{6}},\\;\\mathrm {P} (A=1,B=0)=P\\{4,6\\}={\\frac {2}{6}},", "0c5d54c3e4515867c393721f9565fd9c": "n={\\frac {p}{2\\delta p}}", "0c5df5806926312c23820e442b872b86": "Y\\in W", "0c5e844118728ac06c998f442ff26e1a": "\\pi _{1}(U(1))=\\mathbf {Z} ,", "0c5ea4b2197e5749474800f0f622986f": "S=\\left({\\begin{array}{cc}i&0\\\\0&-i\\end{array}}\\right),V=\\left({\\begin{array}{cc}0&i\\\\i&0\\end{array}}\\right),U={\\frac {1}{\\sqrt {2}}}\\left({\\begin{array}{cc}\\epsilon &\\epsilon ^{3}\\\\\\epsilon &\\epsilon ^{7}\\end{array}}\\right),", "0c5f6f09098a49b8143693323b2c1d58": "k=N", "0c5fad867a03ba7f8cf94149d93bb08d": "{\\frac {\\mathrm {B} (\\alpha +n,\\beta +k)}{k\\mathrm {B} (\\alpha ,\\beta )\\mathrm {B} (n,k)}}", "0c611f83ef132f8728def8bb96fbd214": "X_{1}\\sim \\mathrm {Pois} (\\lambda _{1})\\,", "0c61907468e5c3d49d261330fd536e63": "IR\\subseteq I", "0c61b41995f9d2da561b12294e455c6c": "{\\boldsymbol {\\beta }}=(\\beta _{1},\\beta _{2},\\dots ,\\beta _{n}),", "0c61c6dddb3585d926995b40476f94d8": "{\\begin{aligned}{\\frac {(\\mu _{2}-\\mu _{1})-({\\bar {X}}_{2}-{\\bar {X}}_{1})}{\\displaystyle {\\sqrt {{\\frac {S_{1}^{2}}{n_{1}}}+{\\frac {S_{2}^{2}}{n_{2}}}}}}}&={\\frac {\\mu _{2}-{\\bar {X}}_{2}}{\\displaystyle {\\sqrt {{\\frac {S_{1}^{2}}{n_{1}}}+{\\frac {S_{2}^{2}}{n_{2}}}}}}}-{\\frac {\\mu _{1}-{\\bar {X}}_{1}}{\\displaystyle {\\sqrt {{\\frac {S_{1}^{2}}{n_{1}}}+{\\frac {S_{2}^{2}}{n_{2}}}}}}}\\\\[10pt]&=\\underbrace {\\frac {\\mu _{2}-{\\bar {X}}_{2}}{S_{2}/{\\sqrt {n_{2}}}}} _{{\\text{This is }}T_{2}}\\cdot \\underbrace {\\left({\\frac {S_{2}/{\\sqrt {n_{2}}}}{\\displaystyle {\\sqrt {{\\frac {S_{1}^{2}}{n_{1}}}+{\\frac {S_{2}^{2}}{n_{2}}}}}}}\\right)} _{{\\text{This is }}\\cos \\theta }-\\underbrace {\\frac {\\mu _{1}-{\\bar {X}}_{1}}{S_{1}/{\\sqrt {n_{1}}}}} _{{\\text{This is }}T_{1}}\\cdot \\underbrace {\\left({\\frac {S_{1}/{\\sqrt {n_{1}}}}{\\displaystyle {\\sqrt {{\\frac {S_{1}^{2}}{n_{1}}}+{\\frac {S_{2}^{2}}{n_{2}}}}}}}\\right)} _{{\\text{This is }}\\sin \\theta }.\\qquad \\qquad \\qquad (1)\\end{aligned}}", "0c61f2d07cff0aaf902ed995f465c49f": "l_{G}={\\frac {1}{n\\cdot (n-1)}}\\cdot \\sum _{i\\neq j}d(v_{i},v_{j})", "0c6218811da6eaf8e9b3e7b74b3335a8": "J'_{k}", "0c62330322871b7e83daaea9322208d1": "\\xi \\simeq {\\frac {2}{A+1}}", "0c623332b8cb236713967189dce7dec1": "144^{5}=27^{5}+84^{5}+110^{5}+133^{5}", "0c6304421d16c88be1d3f8548a96e55e": "H_{n}", "0c632da34337fd5fca09226f7400aedb": "{\\frac {1}{2}}\\,|x_{1}y_{2}-x_{2}y_{1}|.\\,", "0c633d028b6f9368c1bc0bb6aedde1ea": "Sg=\\{s.g\\,:\\,s\\in S\\}.", "0c637ad9003e2dc63ea25dee54540c4f": "i_{G}(tst^{-1})=i_{G}(s).", "0c6388b0be02ea5716b00a9918746c08": "{\\begin{bmatrix}0&1\\\\0&0\\end{bmatrix}}.", "0c640324c5f7765196f2d64755d89d27": "d={\\sqrt {\\ell ^{2}+w^{2}}}", "0c6491a608f3bae7bc2cd5bc636c4e32": "\\lnot (P\\Rightarrow Q)\\equiv \\lnot (\\lnot P\\vee Q)\\equiv P\\wedge \\lnot Q", "0c64e0d6b9e20a0b9e155939330804ea": "-r^{-1}\\,", "0c64f49edcfa74c8d1a99056480ef113": "V_{\\text{out}}\\approx {\\frac {V_{\\text{in}}}{\\beta }}={\\frac {V_{\\text{in}}}{\\frac {R_{\\text{1}}}{R_{\\text{1}}+R_{\\text{2}}}}}=V_{\\text{in}}\\left(1+{\\frac {R_{2}}{R_{1}}}\\right)", "0c64f5b682abac8f6828e4376296d35b": "MS_{2}+{\\frac {1}{2}}O_{2}+H_{2}SO_{4}\\rightarrow MSO_{4}+S^{0}+H_{2}O", "0c655ef77d1f0c23c345ce4ccbd4b066": "H^{n}(K(G,n);G)=\\mathrm {Hom} (H_{n}(K(G,n);\\mathbf {Z} ),G)=\\mathrm {Hom} (\\pi _{n}(K(G,n)),G)=\\mathrm {Hom} (G,G),", "0c659059c211ded790ba512c461c543f": "X^{T}{\\hat {e}}=X^{T}[I-X(X^{T}X)^{-1}X^{T}]y=0", "0c65fbc500c305f2e6c16a441d7866fd": "a_{1}a_{2}+a_{1}b_{2}i+a_{1}c_{2}j+a_{1}d_{2}k", "0c666d1d54aa989cf74523eb22c998e5": "X_{\\sigma (X,Y)}", "0c667959eeceea9c31f3f104d8ad7704": "\\nabla _{{\\dot {\\gamma }}(t)}X=0{\\text{ for }}t\\in I.\\,", "0c67327855d5eb7ae6a279f60afd2793": "f:X\\rightarrow Y", "0c67ba5b3caaab0958a0605ff7077b37": "C^{*}\\cong C^{*}\\otimes 1\\rightarrow C^{*}\\otimes (C\\otimes C^{*})\\cong (C^{*}\\otimes C)\\otimes C^{*}\\rightarrow 1\\otimes C^{*}\\cong C^{*}", "0c67f72120fe279fb4c87c4b82e06858": "{\\bar {\\delta }}=1-{\\sqrt {2}}", "0c6851e4dde516a9d8a786657323a2bd": "Probability[Parameter|Data]={\\frac {Probability[Data|Parameter]XProbability[Parameter]}{Pr[Data]}}", "0c68620ee2ea4f1286fcd672a47ea080": "t\\,", "0c68644751a7cf4d050e06a67d3aa3b7": "\\mathbf {I} ", "0c687e4644cf41b98867b88224312874": "L_{xx}(x,y)=L(x-1,y)-2L(x,y)+L(x+1,y).\\,", "0c68c2ef0818a2c3943ec4050d941aef": "PdV", "0c69000328ccb9755609833ca7d5f184": "a^{2}+nb^{2}", "0c6906a2a9c5fcc2d4875a48421dcdaa": "p_{1}p_{2}p_{3}(1-p_{4})\\,", "0c6950ddc28d1b6dbfcfc93ab4352d8c": "\\omega \\rightarrow 0", "0c69bfa2c09b9a9ffea0ebf0c0b09acd": "l_{i}A_{ij}=m_{j}T'", "0c69e77866ab83d257e967e1f50f68f1": "{\\tilde {\\Omega }}(n)", "0c6a4b8eb9b002fc0b4aa236d725245a": "(Df)(a,b)=\\left[{\\frac {\\partial f}{\\partial x}}(a,b)\\ \\ {\\frac {\\partial f}{\\partial y}}(a,b)\\right]=[2a\\ \\ 2b]", "0c6a5c48af63d4600ea68025bb66a745": "\\liminf _{\\varepsilon \\downarrow 0}\\varepsilon \\log \\mathbf {W} _{\\varepsilon }(G)\\geq -\\inf _{\\omega \\in G}I(\\omega ).", "0c6a5e1b1fb691057c255e5cc342c54a": "\\langle V^{2}\\rangle ={\\frac {e^{4}}{(l+1/2)n^{3}a_{0}^{2}}}", "0c6a885b287c52cda1cc8bf19a3a6ae0": "f_{xy}(x,y)\\approx {\\frac {f(x+h,y+k)-f(x+h,y-k)-f(x-h,y+k)+f(x-h,y-k)}{4hk}}~.", "0c6aaee0c6d12e877d64f310b51e80c5": "\\,=A(I-B)A',", "0c6af210a17e2dd1d3054491801f8488": "A_{i}\\neq A_{j}\\quad \\Rightarrow \\quad \\left|A_{i}\\cap A_{j}\\right|<\\infty .", "0c6af5c6ed61a69097374b7c77240481": "L(x,\\xi )={\\tfrac {1}{2}}F^{2}(x,\\xi ),", "0c6b37c282d00505f9af3038f67c3d3d": "\\,p_{n}\\,", "0c6b38f61bd6ff96449417fa4245c83c": "~x=x(t)~", "0c6b6f0cd4d1493335b02bdf0b382686": "D(\\omega )=1-{\\frac {\\omega ^{6}}{225}}+{\\frac {\\omega ^{8}}{1125}}+\\cdots .", "0c6b758e919b4d692ab5bef515db4dc2": "15/16", "0c6b9ad1f6be150d79f233ae7bad21d2": "\\mathbf {t} (s)=\\mathbf {u} (s)\\times \\mathbf {T} (s),", "0c6ba098deaec26a35b1b9aaad036eea": "\\mathrm {^{238}_{\\ 92}U\\ \\xrightarrow {+\\ 15n,7\\beta ^{-}} \\ _{\\ 99}^{253}Es} ", "0c6bd65c04a5bdf49cfffff7bf6a7996": "t_{1}^{2}+13u_{1}^{2}", "0c6bddefda0643f606a063cf6233c15e": "\\!-1/3", "0c6c04dc73a1cd71cbb7388a21c7a3b7": "\\nu :A\\rightarrow \\mathbb {Z} \\cup \\{\\infty \\}", "0c6c44fb9c7765160ae80e13f9442708": "G=(\\{S\\},\\{a,b\\},P,S)", "0c6c67f5a95594342cfc1d3acecaaa93": "\\textstyle \\theta ", "0c6cec00fc3dea60cf65cc0454dca06b": "\\cos {\\frac {\\pi }{3}}=\\cos 60^{\\circ }={\\tfrac {1}{2}}\\,", "0c6d7b7e6c7bb12759ac88371d0bc337": "\\mathrm {d} F=-p\\,\\mathrm {d} V-S\\mathrm {d} T+\\sum _{i}\\mu _{i}\\,\\mathrm {d} N_{i}\\,", "0c6d9ea002ab80da90453d21d27c1014": "y(x_{0}-h)=y(x_{0})-hy'(x_{0})+{\\frac {h^{2}}{2!}}y''(x_{0})-{\\frac {h^{3}}{3!}}y'''(x_{0})+{\\frac {h^{4}}{4!}}y''''(x_{0})-{\\frac {h^{5}}{5!}}y'''''(x_{0})+{\\mathcal {O}}(h^{6})", "0c6dbfd4aaab56d84af722c986e34f19": "a^{3}{\\sqrt {1-3\\cos ^{2}\\alpha +2\\cos ^{3}\\alpha }}", "0c6e333e3f3ce894d6ca6cc9c7bcefbe": "\\scriptstyle \\mathrm {E} (V_{k}-U_{k})=\\infty ", "0c6e92b7fe51d53249264573b5c285d9": "\\mathbf {A} _{k}^{1}", "0c6eb9ecabdbc668407d83e6968c9b08": "A_{d}\\cong Sym^{d}V^{*}.", "0c6f16b943707092fa2868d6a73dca92": "\\,(r+e)\\cdot (r+e)\\leq x", "0c6f65f0f2000c072241774b499d8868": "\\int \\sec ^{3}x\\,dx=\\int {\\frac {dx}{\\cos ^{3}x}}=\\int {\\frac {\\cos x\\,dx}{\\cos ^{4}x}}=\\int {\\frac {\\cos x\\,dx}{(1-\\sin ^{2}x)^{2}}}=\\int {\\frac {du}{(1-u^{2})^{2}}}", "0c6fd96b3b6b3a5c6ffb5d7cc4efecf6": "\\ {\\frac {\\partial p'}{\\partial \\varepsilon _{s}}}={\\frac {\\partial q}{\\partial \\varepsilon _{s}}}={\\frac {\\partial \\nu }{\\partial \\varepsilon _{s}}}=0", "0c700189aeb81935ad53bef18a822a75": "vT_{3}", "0c70e652b8789f2f8ccf05ae2b0f8967": "h_{J}\\leftarrow {\\hat {h}}", "0c7166ce0310c4509cea3864d0bc554e": "S_{1},S_{2},...,S_{n}", "0c71948cb63010f5c951019c2c274f48": "G_{q}(x)={\\begin{cases}0&{\\text{if }}x<-\\nu \\\\[12pt]\\displaystyle {\\frac {1}{c(q)}}\\int _{-\\nu }^{x}E_{q^{2}}^{-q^{2}t^{2}/[2]}\\,d_{q}t&{\\text{if }}-\\nu \\leq x\\leq \\nu \\\\[12pt]1&{\\text{if }}x>\\nu \\end{cases}}", "0c71d1ceea31a94b3011202b936ffbc0": "\\,e_{x}=\\sum _{t=1}^{\\infty }\\ _{t}p_{x}", "0c722fb940f4d14d9d61f35bbf43a622": "x^{2}=4.", "0c728d08a0652654851a9636e13da825": "\\{1,2,3,4,5,6\\}", "0c72974f2266ce7a3fe004260404465e": "\\delta W=\\sum _{i}\\mathbf {F} _{i}\\cdot \\delta \\mathbf {r} _{i}+\\sum _{i}\\mathbf {C} _{i}\\cdot \\delta \\mathbf {r} _{i}-\\sum _{i}m_{i}\\mathbf {a} _{i}\\cdot \\delta \\mathbf {r} _{i}=0.", "0c72b92f7a69ca1c84cf1291884de4a0": "1-s={\\frac {(Q_{1}+Q_{2}-Q_{3})^{2}}{4Q_{1}Q_{2}}}.\\,", "0c7315b75069674e5ebb404e10137252": "V(\\phi )=0", "0c7377fe042986b13af5e3587962e26c": "\\{s,1,4,t\\}", "0c73ad357d888e8dfe85892afffe64ee": "{\\frac {d}{dx}}\\int _{0}^{x}t^{3}\\,dt", "0c73af3efe0bc8a0720b7c8b0f961ce8": "u\\in -\\mathrm {int} K_{M}\\Rightarrow u1\\not \\in A", "0c73ca9ff414bc142a59798e3b5a2713": "T_{G}(-2,0)", "0c743cab686e268b848cf2533435165f": "\\mathbf {v} _{\\mathrm {rot} }=\\mathbf {v} \\cos \\theta +(\\mathbf {k} \\times \\mathbf {v} )\\sin \\theta +\\mathbf {k} (\\mathbf {k} \\cdot \\mathbf {v} )(1-\\cos \\theta ).", "0c7481f8b50f5151a8d3a173e8a473b5": "E_{\\omega }", "0c753ac7e424d2dcd09c2aaa3e3255db": "{\\boldsymbol {B}}^{-1}={\\boldsymbol {B}}\\cdot {\\boldsymbol {B}}-I_{1}~{\\boldsymbol {B}}+I_{2}~{\\boldsymbol {\\mathit {1}}}", "0c755fb3b184017793cbc66859500e13": "Y[\\mathrm {100} ]=c_{11}+c_{12}-2\\left({\\frac {c_{12}^{2}}{c_{11}}}\\right)", "0c757b96131f1e02288ab9af6504c115": "x_{0},y_{0},x_{1},y_{1},\\dots .", "0c75b2ec7bf52d7b5c1fbd70f4effcac": "x^{4}+2y^{2}=8\\,", "0c75b76d77e44aece8df88c26f5defb3": "f(z)=z+\\sum _{n\\geq 2}a_{n}z^{n}", "0c76642fe963a290459752d4cab37ab5": "(1+j)(1-j)=0", "0c7669a058fb48d21246b85bb82d8cbb": "\\arccos \\left({-1 \\over 3}\\right)=2\\arctan({\\sqrt {2}})\\,", "0c76e5bd96c465b86e08ee50f6225631": "\\mathbf {p} =m\\mathbf {v} ", "0c7702a1bb5256ebdbaddd1f0dca2bdf": "ds={\\frac {ds}{dy}}dy", "0c770b4608a4e2798039e68f1fb4d85c": "S_{P}^{2}={\\frac {(n_{1}-1)S_{1}^{2}+(n_{2}-1)S_{2}^{2}+\\cdots +(n_{k}-1)S_{k}^{2}}{(n_{1}-1)+(n_{2}-1)+\\cdots +(n_{k}-1)}}", "0c77513ee982ab821316a1161b2557d6": "D:={\\cfrac {2h^{3}E}{3(1-\\nu ^{2})}}\\,.", "0c77993efa79a80ed8d4e135778926d3": "{\\frac {A}{P}}+{\\frac {B}{Q}}", "0c77ce52be17ea1aec2f74045a15f964": "\\sin \\theta _{3}+\\sin \\theta _{2}\\cos(\\theta _{2}+\\theta _{3})=\\sin(\\theta _{3}+\\theta _{2})\\cos \\theta _{2}\\,", "0c77e14fa839780c36e6dddb6ee0d5d2": "\\Omega _{n,\\mu \\nu }=\\epsilon _{\\mu \\nu \\xi }\\,\\mathbf {\\Omega } _{n,\\xi }", "0c781e57035ef9349d599e89b8e864e9": "\\mathbf {x} _{0i}", "0c78565a768d5385b45c136f034a8bfa": "\\lambda _{1}\\cdot \\lambda _{2}=\\alpha ^{2}+\\beta ^{2}=-B,", "0c786c118b9dc1f2f7c4356e6fb087f9": "\\alpha _{2}={{6\\alpha _{0}+1\\alpha _{1}} \\over 7}", "0c7889d0b29cad50525d0d86e73e9a88": "E_{v}=A_{v}+T_{v}", "0c7892b2bc0ab81f603b60e1fd8778f1": "\\mathbf {C} (\\mathbf {q} ,{\\dot {\\mathbf {q} }})=0", "0c7895ac374a5356f2b8c75e45233bf8": "2^{-L(x)}\\leq {\\frac {1}{2}}p(x)", "0c78a907f9fc3a8b3b3ee3f9431efa88": "\\{\\alpha ,-\\alpha \\}", "0c78b1cd825b78669a9de88c4d45d8da": "H_{q/p,m}=\\zeta (m)-p^{m}\\sum _{k=1}^{\\infty }{\\frac {1}{(q+pk)^{m}}}", "0c79519cd692d16c4c0652cf742b8a43": "a_{1},a_{2},\\ldots ,a_{n}\\in H", "0c7968fb20078e88450de13719977de8": "r{\\frac {\\partial \\log R}{\\partial r}}={\\frac {\\partial \\Phi }{\\partial \\theta }},\\ \\ \\ \\ \\ \\ {\\frac {\\partial \\log R}{\\partial \\theta }}=-r{\\frac {\\partial \\Phi }{\\partial r}},", "0c7a0cbd944ab0d940d60dea9f4e2420": "R'G'B'", "0c7a131c1e0d807dcf1efb41634a6ef0": "E_{1},E_{2}\\in {\\mathcal {E}}", "0c7a2a7f55339301ed38a9be0738ec2e": "R_{12}", "0c7a2fb98cb07936bb6cba7eb8830dd1": "\\phi ^{\\mathrm {even} }(x)", "0c7aa40e42354f8320f9b4413c9b101e": "f=P", "0c7bca0498049e4fc0c97f5a8847aebf": "\\beth _{n}^{+}\\rightarrow (\\aleph _{1})_{\\aleph _{0}}^{n+1}", "0c7be2564c6c563f40d7dda8a8178931": "\\psi \\in T", "0c7c0005c08a0ec74e0bac1f2eb831c8": "\\omega _{2}=1-a/c", "0c7c561b89a43c2dd995973579beb435": "|\\psi _{\\text{tot}}(t)\\rangle =e^{-i{\\hat {H}}_{\\text{JC}}t/\\hbar }|\\psi _{\\text{tot}}(0)\\rangle =\\sum _{n}C_{n}\\left[\\cos \\left({\\frac {\\alpha _{n}}{2}}\\right)|n,+\\rangle e^{-iE_{+}(n)t/\\hbar }-\\sin \\left({\\frac {\\alpha _{n}}{2}}\\right)|n,-\\rangle e^{-iE_{-}(n)t/\\hbar }\\right].", "0c7c768ef7173df92d2246ddd211daec": "n\\times n", "0c7c8fa329d91de0bb7e4d6aaa64ad23": "\\omega ^{(2)}=\\sum _{i1", "0ca1d1ade23787214f38b27d585f12e8": "\\circ L\\circ ", "0ca1d2f6c11a8727d6355f2341ec9fbc": "D\\cdot r=0", "0ca1f156cbfd5262babc97c9d8929bc1": "{\\begin{aligned}\\left[\\sigma _{a},\\sigma _{b}\\right]+\\{\\sigma _{a},\\sigma _{b}\\}&=(\\sigma _{a}\\sigma _{b}-\\sigma _{b}\\sigma _{a})+(\\sigma _{a}\\sigma _{b}+\\sigma _{b}\\sigma _{a})\\\\2i\\sum _{c}\\varepsilon _{abc}\\,\\sigma _{c}+2\\delta _{ab}I&=2\\sigma _{a}\\sigma _{b}\\end{aligned}}", "0ca203894b5e392d3561749af721cec2": "L_{k}:E[k-n/2]\\to E[-k-n/2].", "0ca2345e55c97b9511c232eb2a8c4365": "C=ND(N\\otimes N)", "0ca283c39942c267245f6dbf895f3851": "\\scriptstyle \\oint _{C}", "0ca332a0f65e064b6f8bb8e214b481cc": "\\Omega ^{1}(P,{\\mathfrak {g}})\\cong C^{\\infty }(P,T^{*}P)\\otimes {\\mathfrak {g}}", "0ca334f046ebafdcda716447c2cee856": "{\\mathcal {C}}|\\psi \\rangle ", "0ca33564341edc71a9bfba55950587fb": "\\gamma _{il}\\langle \\Xi _{l}(t)x_{k}\\rangle =\\gamma _{kl}\\langle x_{i}\\Xi _{l}(t)\\rangle ", "0ca3844350590799d77b082ea2b4282f": "\\operatorname {Perf} (f,r)=\\sum _{x\\in X_{n}}f(x)r(x)D_{n}(x).", "0ca3b782b8b2ecc6581936592d248f04": "r={\\cfrac {1}{{\\cfrac {i\\hbar ^{2}k}{m\\lambda }}-1}}\\,\\!", "0ca3cca0f40831f88fc93d5936a8e286": "j=k-p\\ ", "0ca3d493bb99dbbcdd3144cdde7a208e": "\\exp(At)={\\frac {1}{2\\pi i}}\\int _{\\gamma }e^{\\lambda t}(\\lambda \\mathrm {id} -A)^{-1}\\,\\mathrm {d} \\lambda ,", "0ca3ff159b485b7f01d7530242aa6995": "{\\mathcal {A}}=\\Theta {\\mathcal {B}}", "0ca405309a63a1ab5ee4975dae4d29bd": "\\!{\\frac {e^{itb}-e^{ita}}{it(b-a)}}", "0ca40f1a44c808e58818344026fe76ff": "\\phi :X\\to \\mathbf {P} _{A}^{n}", "0ca4689faf88795afdd993e05da65e1d": "(2j_{1}+1)(2j_{2}+1)", "0ca47d9a481af371d1210a620c1945db": "O(\\log n)", "0ca4dcc971e167bd9e798e711888e872": "=\\sum _{p=1}^{n}{(-1)^{(p-1)}v_{p}(\\mathrm {d} x^{1}\\wedge \\cdots \\wedge \\mathrm {d} x^{p-1}\\wedge {\\widehat {\\mathrm {d} x^{p}}}\\wedge \\mathrm {d} x^{p+1}\\wedge \\cdots \\wedge \\mathrm {d} x^{n})}.", "0ca52a2b0cfe5ec11dacd71abb2968cf": "r_{B1}={{\\sqrt {4\\pi }}m_{1}v_{1} \\over a_{1}B}={\\sqrt {2\\hbar \\over m_{1}\\omega _{c}}}", "0ca57439bc11892ba17547b9c9f4d8cf": "x'=x_{0}", "0ca5767775207095948bfc1d523d7a78": "\\|X\\|_{\\Psi _{p}}=c\\rightarrow \\lim _{x\\rightarrow \\infty }f_{X}(x)\\exp(|x/c|^{p})=0,", "0ca595220b95e897840383f2625e2618": "\\zeta (m)", "0ca5abfa72e14bc5752540760326d97c": "{\\widehat {g}}", "0ca5e4e27ca8dc4c891d2c8598529ed1": "O(mn)", "0ca63f5565c65f0d7ec78e881ff06ac4": "\\ p_{1}=p_{x}\\quad ;\\quad p_{2}=p_{y}\\quad ;\\quad T_{1}=T_{x}\\quad ;\\quad T_{2}=T_{y},", "0ca6547a21c90740045c4b22142ed8e5": "{\\overline {\\phi }}", "0ca683cdf5b19a16f7ff74a7b9ce8775": "\\color {Cerulean}{\\text{Cerulean}}", "0ca6b6b74e2fb1dfe2ebbbf1a9c5d38a": "a={\\sqrt {2Rd}}", "0ca779e844ce9b79a91b286c685f6d6d": "dr_{t}=(\\theta -\\alpha r_{t})\\,dt+{\\sqrt {r_{t}}}\\,\\sigma \\,dW_{t}", "0ca77c052f06eda44b2a175baeeaec70": "T=P^{-1}", "0ca786ff0bb6cb577da29861e3fcce79": "~D=\\mathbb {C} \\backslash \\{-1,1\\}.", "0ca7c67385baae31054df97d472b8133": "(n\\times n)", "0ca8943e703b79f50f8d275ae3690c4c": "\\varphi (x)=2\\left(\\ln(x)-\\int _{0}^{\\infty }e^{-t}\\ln |x-t|dt\\right).", "0ca89893c6eca17dd72d3de085e4f120": "\\sum _{j=1}^{n_{S}}\\sum _{b_{j}=0}^{a_{j}}\\sum _{\\beta _{j}}x_{b_{j}}\\ {_{a_{j}}^{b_{j}}}{\\text{S}}_{j}^{\\beta _{j}}\\rightarrow \\sum _{h=1}^{n_{P}}\\sum _{d_{h}=0}^{c_{h}}\\sum _{\\gamma _{h}}u_{\\gamma _{h}}\\ y_{d_{h}}\\ {_{c_{h}}^{d_{h}}}{\\text{P}}_{h}^{\\gamma _{h}}.\\qquad \\qquad (1)", "0ca8a4bfcbd003fd6789a211b32778fc": "\\scriptstyle (p^{2}-q^{2},2pq,p^{2}+q^{2})", "0ca8fbd1011176d3dfce3280504e3058": "\\mathrm {0.{\\overline {3}}} ", "0ca91cc0a1de75d66d963e7ef676c315": "{\\text{Equivalent force, }}F_{eq}=\\int {F{\\bar {u}}}dx", "0ca9259bf34924ebd68bc06113bb1aa9": "{\\frac {p\\lor q}{p,q}}.", "0ca94001a900e3664c8236280aa588b8": "\\sum _{p|n}{\\frac {1}{p}}-\\prod _{p|n}{\\frac {1}{p}}=1.", "0ca97bc136074a054953da4e5beb7e32": "(\\cdot )_{+}", "0ca99ce67c9024c1fdd644423937c30d": "\\mathbb {E} [X_{i}^{?}]=\\Pr[X_{i}^{?}=1]={2\\omega _{i} \\over d}", "0ca9ae4fc6daec1f60a3abfb510d3772": "\\Psi _{0}", "0ca9b50c4f630610a4851c789b270a9a": "C_{x}", "0caa07bd1550b64e2cf05e3f667704e6": "(2ab)^{2}+(2cd)^{2}=(a^{2}+b^{2}-c^{2}-d^{2})^{2}", "0caa2cedb72f96156dd1a4e18260e138": "\\nabla ^{2}A+k^{2}A=0", "0caa58bc12643f0869d115cb897c3d48": "\\epsilon _{i}^{\\mu }", "0caa5a4a28bdb7391c0f6ca6cfa20af3": "\\mathbf {z} '=\\mathbf {X} ", "0caa6bd42800ea3d156d6ea968dc65fa": "z_{T}={\\frac {Z_{T}}{Z_{0}}}\\,", "0caa7a5e7d7c217df561cc4130aff48b": "\\sigma _{1}=\\sin \\psi \\,d\\theta -\\cos \\psi \\sin \\theta \\,d\\phi ", "0caac215c7b2944e3421c0ff167d24d7": "c_{n}=\\langle f,\\phi _{n}\\rangle ", "0cab651f73376762d67453c4f9e81596": "\\mathbf {x} \\in {\\mathcal {D}}", "0cabebec8c6ee5374a14880b88361279": "{\\frac {\\mathrm {opposite} ^{2}+\\mathrm {adjacent} ^{2}}{\\mathrm {hypotenuse} ^{2}}}", "0cabf5dc9b6072e9af1ba86b0b7c73e3": "\\forall x,y\\in U,~x\\neq y", "0cabfee9eab46860c3475c180e8d2535": "{\\begin{aligned}R&={\\sqrt {(X_{12}-X_{11})^{2}+(X_{22}-X_{21})^{2}+(X_{32}-X_{31})^{2}}}\\\\L&={\\sqrt {(x_{12}-x_{11})^{2}+(x_{22}-x_{21})^{2}}}\\end{aligned}}", "0cac58c3600f6525663fb73a0eed9711": "(z_{1},z_{4};z_{2},z_{3})={{\\lambda -1} \\over \\lambda }", "0cac661a1b56040be72a916df99cae87": "\\zeta _{1}=e_{1}+ie_{2},\\zeta _{2}=e_{3}+ie_{4},\\zeta _{3}=e_{5}+ie_{6}", "0cacd044f46d4fac2da878dabb62670f": "M:(u,v)\\mapsto (u',v')\\,", "0cacf149a2873f48cefc9be8d08f9001": "n\\mapsto \\dim _{k}M_{n}", "0cad27b02b1f2e63c2183cbef9398000": "\\beta _{j}^{-}(r_{m}^{-}-r_{f})_{t}", "0cad4207285dc7e8c337e5e356b766d3": "L(\\sigma )=(L(\\sigma )_{1},\\ldots ,L(\\sigma )_{n})\\quad {\\text{where}}\\quad L(\\sigma )_{i}=\\#\\{j>i:\\sigma _{j}<\\sigma _{i}\\},", "0cad5cfd12c48bf9a0830159b7226234": "\\sin {5x}=16\\sin ^{5}x-20\\sin ^{3}x+5\\sin x\\,", "0cadec12065f027759a15c0ccbc217fc": "=\\left.{\\frac {\\partial \\sigma }{\\partial x}}\\right|_{p}=\\sigma '(x)", "0caeb8cec6aa4b6d9d92cb6db9fd39d3": "p\\left(\\mathbf {\\hat {x}} \\right)", "0caebd4279121a0e06a1d74578b4cbd9": "E_{3}\\subseteq {\\mathcal {D}}", "0caef1bc2e2b2924340168899c5f797c": "{\\text{Let }}q^{*}=(-1)^{\\frac {q-1}{2}}q\\;\\;{\\text{ (in other words }}|q^{*}|=|q|{\\text{ and }}q^{*}\\equiv 1{\\pmod {4}}{\\text{).}}\\;", "0caf21d663b985d202ec4044cf359c5d": "x=-{\\frac {a}{1+p}}+{\\frac {a(1+p)}{1-p+p^{2}}}={\\frac {3ap}{1+p^{3}}},\\ y=px", "0caf2ecff3d7c68b2c5f76daa64f1570": "\\eta _{\\mu \\nu }\\Lambda _{\\alpha }^{\\mu }\\Lambda _{\\beta }^{\\nu }=\\eta _{\\alpha \\beta }\\ .", "0caf4169891bbb020dd34842a374df1f": "p=49+12\\cdot \\log _{2}{(f/440)}\\,", "0caf5a81c305a4be64739ac76f8e9ac9": "p_{s}(C)=H_{n}(C)=\\int _{C}dH(\\theta ).", "0cafb2d94c6dc611e7d2092ce1600cea": "c^{2}=gH", "0cafb96a5935cc47dc3f03032d76b151": "[P^{+}F,P^{-}G]^{IJ}", "0cafc962e0f1285d801261fa846ff972": "I\\Delta _{0}+\\exp ", "0cb00aea1f952f9060869f1d6c3f7932": "{\\Psi \\in R^{\\it {N\\times N}}}", "0cb0109eec8a399e0deeadf6780ef0bc": "PV^{\\gamma }={\\text{constant}}", "0cb04864e158c2a33a42532e661d7330": "M_{x}=\\left\\{f\\in A(\\mathbb {T} )\\,\\mid \\,f(x)=0\\right\\},\\quad x\\in \\mathbb {T} .\\,", "0cb08fd8ee924890ec98dd856fbe6064": "y\\in (0,h)", "0cb0f7a63ff9aca39bdaeb1c4a31813b": "F[y]=f\\circ y", "0cb1390497ccb1f298960b08b8cc421a": "K(t,x,y)=\\sum _{n=0}^{\\infty }e^{-\\lambda _{n}t}\\phi _{n}(x)\\phi _{n}(y).", "0cb17ce8729e421ae5624ba19b67d262": "j_{1}=|j_{2}-j_{3}|,\\ldots ,j_{2}+j_{3}", "0cb1deb06987b513858298d18dc8d293": "{\\underline {\\underline {{\\mathsf {A}}_{\\varepsilon }}}}={\\begin{bmatrix}1&0&0&0&0&0\\\\0&1&0&0&0&0\\\\0&0&1&0&0&0\\\\0&0&0&-1&0&0\\\\0&0&0&0&-1&0\\\\0&0&0&0&0&1\\end{bmatrix}}", "0cb1e8758d07263b720b83ea48ec5749": "\\forall a\\in A,\\;0+a=a", "0cb1fc058c4d8c109deb2f17c35ebe21": "d(G\\circ F)(u;x)=dG(F(u);dF(u;x))", "0cb23a098de6be1634d94054278db86f": "M\\times 1", "0cb284fdeb9d28eceb9fa3a80e2840da": "t>{{D+k-1} \\over {m-s+1}}", "0cb2f2aefdd662e02c0509cbfb363c60": "{\\frac {1}{(2\\pi )^{n/2}}}\\int \\cdots \\int \\left|\\prod _{r}{\\frac {2(x,r)}{(r,r)}}\\right|^{\\gamma }e^{-(x_{1}^{2}+\\cdots +x_{n}^{2})/2}dx_{1}\\cdots dx_{n}=\\prod _{j=1}^{n}{\\frac {\\Gamma (1+d_{j}\\gamma )}{\\Gamma (1+\\gamma )}}", "0cb2fb86788e2104843dcc5908255f01": "{{S}_{1}=(\\varphi +2)(1+\\varphi +1)=(\\varphi +2)^{2}=4+4\\varphi +\\varphi ^{2}=5+3\\varphi }", "0cb325cdc56e2fb74b40b3c0f2813302": "{\\mathcal {O}}_{L}", "0cb34c2b882f2f237022f9b3d9f9df04": "(-1)^{3}+7(-1)^{2}+8(-1)+2.", "0cb36a503efa898db5a73b1121c97dab": "\\rho _{In}\\,\\!", "0cb36cdb888c9cb34281f4041ef42be4": "2n_{\\rm {oil}}d\\cos {\\big (}\\theta _{2})=m\\lambda ", "0cb36da122e8269ae0fb02da9afcb9d4": "h_{\\mathrm {FOH} }(t)\\,={\\frac {1}{T}}\\mathrm {tri} \\left({\\frac {t}{T}}\\right)={\\begin{cases}{\\frac {1}{T}}\\left(1-{\\frac {|t|}{T}}\\right)&{\\mbox{if }}|t|0\\\\1,&{\\text{otherwise}}.\\end{cases}}", "0cb5a2a6e179aa01c6c98da94ac1b785": "{\\frac {\\mbox{Net Income}}{\\mbox{Total Assets}}}", "0cb635e0ec9033a24849eee750ec13b8": "{\\mbox{N}}\\,{\\mbox{m}}\\,{\\mbox{radian}}^{-1}\\,", "0cb63dcf644aeda4419850c0de81b363": "\\mu (x,G)\\cap \\mu (y,H)", "0cb683a8dc4215bc8de951d6fe521675": "\\psi ={\\begin{pmatrix}\\psi _{1}\\\\\\psi _{2}\\\\\\psi _{3}\\end{pmatrix}}", "0cb6bc99c7df8da0c95d7e046c33215d": "I(\\rho ^{AB})=S(\\rho ^{AB}\\|\\rho ^{A}\\otimes \\rho ^{B})", "0cb6f583427a8a958b28c43000ab816b": "\\lim _{x\\to \\infty }f(x)=L,", "0cb710e2a38f16363db88ff24efd7ce6": "\\mathbf {x} _{2}", "0cb72d3496c8cfdf612b8bae9b833ffa": "I_{R}={\\frac {V_{in}}{R}}\\,", "0cb7636c2f52a60a30187b70e623b90f": "m^{\\star }=m+c\\left(t-\\tau \\right)\\,", "0cb796817cd329dda950cf2f4a004851": "A\\subseteq \\operatorname {cl} (A)\\!", "0cb7b130fdbd6c06e8c0bfc432871c26": "d_{1}\\times d_{2}\\times \\cdots \\times d_{n}", "0cb7b46ddeafb4808766f896b00ee896": "z\\mapsto \\alpha z+\\beta .\\,", "0cb7d7d7221392e6761d2aee98f1f715": "\\int _{0}^{1}{\\frac {\\ln x}{1-x}}\\,dx=-{\\frac {\\pi ^{2}}{6}}", "0cb84591243d40b0a2c5725012098c1f": "f(x_{i},{\\boldsymbol {\\beta }})", "0cb86af24539d8df8294ca3ad0cf958b": "N=2^{k}-1", "0cb86f3698858595a8778d13f3ef8c36": "\\nabla ^{2}\\left({\\frac {\\partial \\varphi _{1}}{\\partial x_{2}}}-{\\frac {\\partial \\varphi _{2}}{\\partial x_{1}}}\\right)=-{\\frac {2\\kappa Gh}{D(1-\\nu )}}\\left({\\frac {\\partial \\varphi _{1}}{\\partial x_{2}}}-{\\frac {\\partial \\varphi _{2}}{\\partial x_{1}}}\\right)\\,.", "0cb92a0b33596118301fd5b999fccae1": "L=I_{1}\\supseteq I_{2}\\supseteq I_{3}\\supseteq \\cdots \\supseteq I_{n}=\\{0\\}", "0cb92d71d5382feb078d6a2a4fd9b5ac": "\\varphi _{t}:X\\to X", "0cb931a2f1d81ceddc8fbd878d788c0d": "R_{j}f:=p.v.c_{n}\\int _{\\mathbb {R} ^{n}}{\\frac {f(y)(x_{j}-y_{j})}{|x-y|^{n}}}dy", "0cb93f94b03c365daf039170f339b844": "\\sum _{n}(m_{n}c^{2})^{2}-(M_{0}c^{2})^{2}=2\\sum _{n0{\\text{ for }}-10", "0cd57b73cfe488974d599ff76802fe3f": "U=Y^{D}-D(L^{S})", "0cd57c396b4caaae14406507c3771af2": "D_{cl}", "0cd591fbeeab306f3c45fb3b4f31e4c8": "{\\dot {V}}", "0cd599e76be9d96a8bc688e1b916d9dc": "{\\gamma }_{\\mathit {i}}", "0cd5c7a8ec567b2d124e40d976fc5065": "x=X/T", "0cd648fb0680d7bf9feeb3c075531437": "\\Delta _{k}^{EXP}=\\mathrm {EXP} ^{\\Sigma _{k-1}^{P}}", "0cd6664dc7852f9081f95fdaf7afa4d4": "\\mathrm {HA\\rightleftharpoons A^{-}+H^{+}} ", "0cd6688af29da404b8939f7a9229fa8a": "\\mathbf {p} _{1}-\\mathbf {p} _{0}", "0cd68cd7fd8e4023dfc000331849f121": "A/I", "0cd719ae39e1e4d693299105bc56424a": "\\coth \\tau ", "0cd79a334f9f261e390e5c72d439c5da": "p_{t}:=1000(SG-1)\\,", "0cd7fc20752e73dccc5907f9a8c553d9": "Na_{2}O(SiO_{2})+CO_{2}\\rightleftharpoons Na_{2}CO_{3}+2SiO_{2}+Heat", "0cd8ae3c93c68f708ab076c7141ec22f": "\\pi _{Y\\Sigma }:Y\\to \\Sigma ,\\qquad \\pi _{\\Sigma X}:\\Sigma \\to X.", "0cd8aeec5bcd05e759e3b662cf927127": "\\delta W\\leq -\\mathrm {d} A,", "0cd8f6c142f63b07396f872a873c85bf": "Ax-b=0.", "0cd9c6affb70b313199258f50acd512f": "\\nabla \\otimes \\nabla \\circ \\Delta _{2}=\\Delta \\circ \\nabla :(B\\otimes B)\\to (B\\otimes B),", "0cda1788b34a1fa926a45c93da40c496": "\\textstyle b>\\textstyle \\delta ", "0cda4520f331b5b849a10bb7ee5fc926": "f({\\textbf {x}}_{r})>1", "0cdc061630a95c435664cc7d4cdb8918": "M_{a}=\\lim _{N\\to \\infty }N\\cdot x(N)={\\frac {P_{T}\\cdot r}{e^{rT}-1}}", "0cdc140d3e335fdc677f589d200c9542": "DF=D\\cdot F+D\\wedge F", "0cdc647ef9420e43910611bbaf335a5e": "=\\det(\\Lambda -\\mu I)\\det[(\\Lambda -\\mu I)^{-1}V^{-1}\\delta AV+I]", "0cdccaa518c7cb0c6acf322db6574f0e": "V_{an}=V_{cn}={\\frac {V_{ac}}{2}}=120V", "0cdce0dc6368a9432e47e5745eb2724d": "M_{2}:=M_{0}+M_{1}", "0cdd76c5a6cd41df11b6b0b2095f95d3": "S={\\cfrac {bh^{2}}{6}}", "0cdd95ec8842546fdc16096c18cb7689": "{\\frac {9}{5}}+{\\sqrt {\\frac {9}{5}}}=3.1416^{+}", "0cde0a2c46558755e521eb7ccc2be233": "\\forall x\\in L,z\\in \\{0,1\\}^{*},{\\text{View}}_{\\hat {V}}[P(x)\\leftrightarrow {\\hat {V}}(x,z)]=S(x,z)", "0cde38afd98441cc0f0a4a638c7d8b0b": "=\\sum _{a^{n}\\in T_{\\delta }^{\\mathbf {p} ^{n}}}\\Pr \\left\\{E_{a^{n}}\\right\\}\\Pr _{\\mathcal {S}}\\left\\{\\exists E_{b^{n}}:b^{n}\\in T_{\\delta }^{\\mathbf {p} ^{n}},\\ b^{n}\\neq a^{n},\\ E_{a^{n}}^{\\dagger }E_{b^{n}}\\in N\\left({\\mathcal {S}}\\right)\\backslash {\\mathcal {S}}\\right\\}", "0cdea8de9e449fc10716ff2b9a5c1990": "\\mathbf {R} '_{i}=\\mathbf {R} _{i}+\\mathbf {t} \\;\\;{\\textrm {(translation)\\;\\;and}}\\;\\;\\mathbf {R} '_{i}=\\mathbf {R} _{i}+{\\frac {\\Delta \\phi }{|\\mathbf {s} |}}\\;(\\mathbf {s} \\times \\mathbf {R} _{i})\\;\\;{\\textrm {(infinitesimal\\;\\;rotation)}},", "0cdec0f568c75a0993780c7d0c6251e7": "C\\ell (T^{*}M)\\otimes S(M)\\to S(M)", "0cdef127b3d44f3491d62761b9a87c63": "x-y-z=(x-y)-z\\qquad {\\mbox{for all }}x,y,z\\in \\mathbb {R} ;", "0cdf3be7f329c4e9f54870549d480181": "{\\mathcal {F}}^{-1}f", "0cdf4e90d139f3e620251bad3ceeb2da": "q^{*}(\\mathbf {\\pi } )\\sim \\operatorname {Dir} (\\mathbf {\\alpha } )\\,", "0cdf7b47c0b409953f652a0c00b22be5": "\\rho _{c}^{*}=0.3009-0.00785\\mu ^{*2}-0.00198\\mu ^{*4}", "0cdf83a5640a96a3a65ba0e457f5c638": "\\gg \\!\\,", "0cdfb0d135e620979fd70e5da8f8c37d": "V={\\frac {n}{6}}hs^{2}\\cot {\\frac {\\pi }{n}}.", "0cdfb578894c146850e012b1c4a4150f": "\\therefore \\neg B", "0cdfc32e0e580878cd949792c1eda664": "\\displaystyle d", "0cdffe9a56d14cabe6c9be1bf4d7f534": "\\mathbf {C} ^{0}={\\frac {1}{24}}{\\begin{bmatrix}2&1&0\\\\1&2&0\\\\0&0&0\\\\\\end{bmatrix}}={\\frac {1}{48}}{\\begin{bmatrix}1\\\\-1\\\\0\\end{bmatrix}}{\\begin{bmatrix}1&-1&0\\end{bmatrix}}^{\\mathrm {T} }+{\\frac {1}{16}}{\\begin{bmatrix}1\\\\1\\\\0\\end{bmatrix}}{\\begin{bmatrix}1&1&0\\end{bmatrix}}^{\\mathrm {T} }", "0ce154ab57ea6dd5e6d09024c780da6b": "f:X\\otimes I\\to Y\\otimes I", "0ce15b40c001f98bd7f3c5b14eca646d": "=2(1/2+\\epsilon _{1})(1/2+\\epsilon _{2})-(1/2+\\epsilon _{1})-(1/2+\\epsilon _{2})+1\\ ", "0ce16a4e0ae23b16c4236156542655a1": "r_{e}^{2}", "0ce16c1d11eebd4e1827918769385e60": "C=(zs^{-1}+rd_{A}s^{-1})\\times G", "0ce1acfc79636daed6568606e31e9c2c": "{\\omega }_{c}", "0ce1ae819f2406c66e42c8033d03fbc8": "\\ \\mathbb {R} \\ ", "0ce1e54d072e8ec1e740bfd0ff8769fa": "{\\frac {\\alpha _{s}}{\\alpha _{n}}}={\\frac {2}{\\hbar \\omega }}\\int _{\\Delta }^{\\infty }{{\\frac {\\left|{{\\text{E(E + }}\\hbar \\omega {\\text{)}}+\\Delta ^{2}}\\right|[f(E)-f(E+\\hbar \\omega )]}{(E^{2}-\\Delta ^{2})^{1/2}[(E+\\hbar \\omega )^{2}-\\Delta ^{2}]^{1/2}}}dE}{\\text{ + }}{\\frac {1}{\\hbar \\omega }}\\int _{\\Delta -\\hbar \\omega }^{-\\Delta }{{\\frac {\\left|{{\\text{E(E + }}\\hbar \\omega {\\text{)}}+\\Delta ^{2}}\\right|[1-2f(E+\\hbar \\omega )]}{(E^{2}-\\Delta ^{2})^{1/2}[(E+\\hbar \\omega )^{2}-\\Delta ^{2}]^{1/2}}}dE}", "0ce1fbcdf3430039873847c8dc739a74": "a={\\frac {1}{A}}\\int _{A}P\\left[c({\\vec {r}})|{\\vec {y}}({\\vec {r}})\\right]\\,d{\\vec {r}}", "0ce1fee993f8af0c45258cd70db19018": "\\epsilon =\\Delta D/D\\,\\!", "0ce25cebede817fe94972017de2e945f": "(a,b,k)", "0ce281f67b9fb7931114920a3c3fe235": "{}_{m}^{n}\\{x\\}", "0ce292d442ca33bdbeacbc4cf5cb0f08": "\\dim _{\\mathbb {R} }V=2\\dim _{\\mathbb {C} }V", "0ce339d3047a6ceaab1d86abc421f026": "{\\frac {\\rho }{\\rho -1}}\\,", "0ce3989150a739016aa9b8deb6e327ac": "{\\frac {1}{u}}+{\\frac {1}{v}}={\\frac {1}{f}}\\,;", "0ce3d837d6a6f9b76b9e6be9f50ebabb": "{\\frac {dy}{dx}}\\,\\cdot \\,{\\frac {dx}{dy}}=e^{x}\\cdot {\\frac {1}{y}}={\\frac {e^{x}}{e^{x}}}=1", "0ce4ab360eff33a4488c3538ce2b8203": "I/I_{0}=e^{-m\\tau }.\\,", "0ce4b90297d0c9f6ba4a167f3fead9b7": "e_{b}(k,i+1)=e_{b}(k-1,i)-\\kappa _{b}(k,i)e_{f}(k,i)\\,\\!", "0ce4d53d0d87d88c9eb190ca4871a44f": "r^{J^{2}}=\\varepsilon r\\varepsilon ^{-1}", "0ce50bb8ed733a9418321398bdb9b6da": "\\int {d^{3}k \\over \\left(2\\pi \\right)^{3}}\\;\\left[\\mathbf {1} -\\mathbf {\\hat {k}} \\mathbf {\\hat {k}} \\right]\\;{\\exp \\left(i\\mathbf {k} \\cdot \\mathbf {r} \\right) \\over k^{2}+m^{2}}={1 \\over 2}{e^{-mr} \\over 4\\pi r}\\left\\{{2 \\over \\left(mr\\right)^{2}}\\left(e^{mr}-1\\right)-{2 \\over mr}\\right\\}\\left[\\mathbf {1} +\\mathbf {\\hat {r}} \\mathbf {\\hat {r}} \\right]", "0ce52e93d53fab11fb3c7e2c50502cf1": "\\beta _{T}\\equiv -{\\frac {1}{V}}\\left({\\frac {\\partial V}{\\partial p}}\\right)_{T}\\,", "0ce530b7aa2791ceb89ce1ccc835f863": "\\omega _{2}=-{\\frac {1}{2}}+{\\frac {\\sqrt {3}}{2}}i", "0ce55abeed1f6cf4d229dc679eb8f4e1": "\\scriptstyle {\\frac {c}{2a}}", "0ce55bdd51579c55ec4bd9f0a5afce82": "d\\epsilon _{i,j}^{p}=s_{i,j}d{\\lambda }", "0ce562bf3369700559bcca78ac4f22a0": "S={\\frac {1}{3}}{\\frac {(\\alpha -2\\beta )(\\alpha +\\beta +1)^{1/2}}{(\\alpha +\\beta -2/3)(\\alpha \\beta )^{1/2}}}", "0ce5af08f20265b14618a82387ad2689": "\\log \\ a+\\log \\ b", "0ce6761745867b58d92a4dee05b15e3d": "x_{0}\\in {\\mathcal {O}}", "0ce67eaa3167b284cbff224f801002d7": "\\|x'\\|_{\\theta ,p;J}\\simeq \\inf {\\Bigl \\{}{\\Bigl (}\\sum _{n\\in \\mathbf {Z} }{\\bigl (}2^{\\theta n}\\max(\\|x'_{n}\\|_{X'_{0}},2^{-n}\\|x'_{n}\\|_{X'_{1}}){\\bigr )}^{p}{\\Bigr )}^{1/p}\\!:\\,x'=\\sum _{n\\in \\mathbf {Z} }x'_{n}{\\Bigr \\}}.", "0ce69b13de14bf7732b2d5b583711949": "{\\begin{pmatrix}\\varphi A_{L}\\\\\\varphi A_{S}\\end{pmatrix}}={\\begin{pmatrix}1&1\\\\1&0\\end{pmatrix}}{\\begin{pmatrix}B_{L}\\\\B_{S}\\end{pmatrix}}={\\begin{pmatrix}2&1\\\\1&1\\end{pmatrix}}{\\begin{pmatrix}A_{L}\\\\A_{S}\\end{pmatrix}}\\,,", "0ce6dc8c08c27a4d34b981e23f9bab7e": "P_{\\mathrm {dBm} }=-174+10\\ \\log _{10}(\\Delta f)", "0ce6ee2bc39c562268d7d73a02024d10": "RRA=-{\\frac {d(u'(c_{t}))}{d(c_{t})}}{\\frac {c_{t}}{u'(c_{t})}}=-u''(c_{t}){\\frac {c_{t}}{u'(c_{t})}}", "0ce7b01724be7da5ee4f097d9ab42e0b": "\\lambda _{1}=\\lambda _{2}=k", "0ce7e12ff3f27732ff211de93411c2ae": "\\,k=4", "0ce82642b763feeb9fede2b2e912a2f8": "\\left\\{\\omega _{k}\\right\\}", "0ce8e38ac84d7004f0a6a0ddfb68b284": "1+z={\\sqrt {\\frac {1-{\\frac {2GM}{c^{2}r_{\\text{receiver}}}}}{1-{\\frac {2GM}{c^{2}r_{\\text{source}}}}}}}", "0ce8e5baec8c4f25858c9a333a0dc930": "T_{s}\\approx 2\\rightarrow q_{s}*=5.7\\left(\\tau *-0.047\\right)^{3/2}", "0ce91cce3204b6098c3bada646eb360a": "\\mathbf {r} _{k}=\\mathbf {r} _{k-1}-\\alpha _{k-1}A\\mathbf {p} _{k-1}", "0ce91e3f5fb94d4cac8d73e3647221c8": "((c+n)^{2}+c-n)/2", "0ce936e0cbdb88b5b7c2e807585a3169": "\\omega =1", "0ce95c69019bdbc1dafb6a47238fcbcb": "d_{4}", "0ce970ed47a1bd5f07826b3aa1d46755": "{\\mathfrak {d}}({\\mathbb {P} })=\\min {\\big \\{}|Y|:Y\\subseteq {\\mathbb {P} }\\ \\wedge \\ (\\forall x\\in {\\mathbb {P} })(\\exists y\\in Y)(x\\sqsubseteq y){\\big \\}}", "0ce98a3bec553daa19d96480e71b775b": "w_{t}=w_{t+1}", "0ce9ce2807d9033e86f21c083c29917d": "x^{16}+x^{14}+x^{13}+x^{11}+1.\\,", "0ce9d4245034107569e3757f4b412795": "{\\frac {Ncm}{f}}", "0cea0d63ff59608cf5af41b2abc35e2b": "x_{step}", "0ceb25dfa3c84228ae91c44b67b648af": "C_{4}=G_{3}+G_{2}\\cdot P_{3}+G_{1}\\cdot P_{2}\\cdot P_{3}+G_{0}\\cdot P_{1}\\cdot P_{2}\\cdot P_{3}+C_{0}\\cdot P_{0}\\cdot P_{1}\\cdot P_{2}\\cdot P_{3}", "0cebec6260b66a20615ea8fc50d5107f": "I_{n}=-{\\frac {-e^{ax}}{(n-1)x^{n-1}}}-{\\frac {a}{n-1}}I_{n-1}\\,\\!", "0cec1df1676582f4eb7961182f63f874": "e-\\ ", "0cec677abaabe1dad7dcb899ca2a1f33": "{\\begin{cases}u_{t}=-\\Delta u&{\\textrm {on}}\\ \\ \\Omega \\times (0,T),\\\\u=0&{\\textrm {on}}\\ \\ \\partial \\Omega \\times (0,T),\\\\u=f&{\\textrm {on}}\\ \\ \\Omega \\times \\left\\{0\\right\\}.\\end{cases}}", "0cec7498c15902a77a7883ea5ab3e6b1": "\\Phi (x)=\\int _{-\\infty }^{x}\\phi (t)\\ dt={\\frac {1}{2}}\\left[1+\\operatorname {erf} \\left({\\frac {x}{\\sqrt {2}}}\\right)\\right]", "0cec91bb2affb75cdadcc0a4a33d7e9f": "E(p_{i},r-R_{i})={\\frac {3(p\\cdot {\\hat {r}}){\\hat {r}}-p}{r^{3}}}", "0ceca3ceff4858eada85c4fcc3033858": "I_{n}={\\frac {2x^{n}{\\sqrt {ax+b}}}{a(2n+1)}}-{\\frac {2nb}{a(2n+1)}}I_{n-1}\\,\\!", "0ced22ddbe3e56715933e90d766c055c": "\\sigma (Y)\\subset \\sigma (X)", "0ced288748230e00030fa95fc07a2f59": "={\\sqrt {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}}}.", "0ced5b2cea4dc6450d621f9e497da64a": "{1 \\over k_{1}}\\times \\int \\limits _{[A_{1}]^{0}}^{[A_{1}]}{d[A_{1}'] \\over [A_{1}']}={1 \\over k_{2}}\\times \\int \\limits _{[A_{2}]^{0}}^{[A_{2}]}{d[A_{2}'] \\over [A_{2}']}", "0ced9780c9b676853846b24ceea83642": "{\\begin{bmatrix}\\mathbf {A} ^{T}\\mathbf {A} &\\mathbf {A} ^{T}\\mathbf {B} \\\\\\mathbf {B} ^{T}\\mathbf {A} &\\mathbf {B} ^{T}\\mathbf {B} \\end{bmatrix}}^{-1}={\\begin{bmatrix}(\\mathbf {A} ^{T}\\mathbf {A} -\\mathbf {A} ^{T}\\mathbf {B} (\\mathbf {B} ^{T}\\mathbf {B} )^{-1}\\mathbf {B} ^{T}\\mathbf {A} )^{-1}&-(\\mathbf {A} ^{T}\\mathbf {A} )^{-1}\\mathbf {A} ^{T}\\mathbf {B} (\\mathbf {B} ^{T}\\mathbf {B} -\\mathbf {B} ^{T}\\mathbf {A} (\\mathbf {A} ^{T}\\mathbf {A} )^{-1}\\mathbf {A} ^{T}\\mathbf {B} )^{-1}\\\\-(\\mathbf {B} ^{T}\\mathbf {B} )^{-1}\\mathbf {B} ^{T}\\mathbf {A} (\\mathbf {A} ^{T}\\mathbf {A} -\\mathbf {A} ^{T}\\mathbf {B} (\\mathbf {B} ^{T}\\mathbf {B} )^{-1}\\mathbf {B} ^{T}\\mathbf {A} )^{-1}&(\\mathbf {B} ^{T}\\mathbf {B} -\\mathbf {B} ^{T}\\mathbf {A} (\\mathbf {A} ^{T}\\mathbf {A} )^{-1}\\mathbf {A} ^{T}\\mathbf {B} )^{-1}\\end{bmatrix}}", "0cedb27cb5ec58f010b6760c0ecd2b5a": "{\\mathfrak {g}}^{\\mathrm {red} }", "0cee8a32ff09a6e7aab1926b4eee7a25": "SVI={SV \\over BSA}={(CO/HR) \\over BSA}={CO \\over {HR\\times BSA}}", "0ceea09946cdce0ccecaed08641ab99d": "P(H|I,N)=P(T|I,N)=P(S|I,N)=1/3", "0ceee9bc38e54fbfc96e4fbd215f5a3e": "F_{0}=0", "0cef01f449f7fa2c81c6a9d3fe1afcd9": "\\Pr(Y1,~{\\text{ Fib}}(n):={\\text{Fib}}(n-1)+{\\text{Fib}}(n-2).", "0cf2e3538c045cb29bf99a649413e339": "\\mathrm {Fr} ={\\frac {u}{\\sqrt {g'h}}}", "0cf34a6b848c5a4d7a4387a603372c9b": "{\\bar {\\epsilon }}_{sh}\\propto {\\sqrt {t-t_{0}}}", "0cf35cbac313fbd2806a836d9ecfc880": "p({\\textbf {x}}_{k}\\mid {\\textbf {Z}}_{k})={\\frac {p({\\textbf {z}}_{k}\\mid {\\textbf {x}}_{k})p({\\textbf {x}}_{k}\\mid {\\textbf {Z}}_{k-1})}{p({\\textbf {z}}_{k}\\mid {\\textbf {Z}}_{k-1})}}", "0cf3d3aa91772f039ba61463720bd429": "\\left|r_{i}{\\frac {\\partial ^{2}r_{i}}{\\partial \\beta _{j}\\partial \\beta _{k}}}\\right|\\ll \\left|{\\frac {\\partial r_{i}}{\\partial \\beta _{j}}}{\\frac {\\partial r_{i}}{\\partial \\beta _{k}}}\\right|", "0cf42fbad4728db36a804a252cf52dbe": "I_{1z}(t)=2\\sigma _{12}tI_{2z}^{0}+I_{1z}^{0}", "0cf435010a92d67f399c3db28c306076": "G^{*}+M\\to M^{+\\bullet }+e^{-}+G", "0cf4692446b962403224fc038effa58a": "\\epsilon (a)=1", "0cf47fa4cfb90d33829a9a1663e3cb8c": "n((I+J-1+\\left\\lfloor {\\frac {n}{2}}\\right\\rfloor )\\,{\\bmod {\\,}}n)+((I+2J-2)\\,{\\bmod {\\,}}n)+1", "0cf4a55dc8534c02b15eff8870c88c80": "C^{*}\\,", "0cf526df5ccfcccc04b4091962b7fcbc": "\\{b_{i}\\}", "0cf54197a06d7d168cf35bc8aa380c62": "E_{g}(f;N)", "0cf5419a0ebaf6a44e771da8e279d25d": "confidence_{i}", "0cf66929e0a12e33f474775dcc2fcfab": "\\tau =Fr\\sin \\alpha \\,", "0cf6fbb22bfbf6ef5ff74545f2062061": "\\textstyle \\lim _{p\\to -\\infty }M_{p}=M_{-\\infty }", "0cf7553c3b2935fb33a46f4aad4b3bdb": "O_{1},O_{2},O_{3},...", "0cf78cf58cceb47cbc9ba2694b5afbdb": "\\scriptstyle p_{\\phi }=\\sin(\\theta )^{2}{\\dot {\\phi }}", "0cf790fab9f788c5a45bda912bf82b8c": "{\\tilde {u}}({\\vec {e}}_{j})=\\sum _{i}u_{i}({\\tilde {\\omega }}^{i}({\\vec {e}}_{j}))=\\sum _{i}u_{i}\\delta ^{i}{}_{j}=u_{j}", "0cf7acb4860a686b81d9fd804a72a986": "(d\\mathbf {X} )^{\\rm {H}}", "0cf81f9038402e85910cfad17d0051b3": "1s", "0cf82af57dac0e5a2406c265ddbc0da5": "\\chi _{n}(z)=2^{-n}z\\,\\Phi (z^{2},n,1/2).\\,", "0cf8332524f1d7df0458ccaed391105d": "x^{(k+1)}=x^{(k)}+\\omega \\left(b-Ax^{(k)}\\right),", "0cf86e6161c4982781a626dde8bac485": "X\\not \\Vdash A", "0cf890331835ee06782b8ec782a7e634": "H=T^{a+\\varepsilon }", "0cf8c44b1a780162367cb2338bbd2d35": "-\\textstyle {\\frac {1}{3}}", "0cf8caeefdfb84063b53a7a4f0bf7363": "{\\begin{aligned}p(\\sigma ^{2}|D,I)\\;\\propto \\;&{\\frac {1}{\\sigma ^{n+2}}}\\;\\exp \\left[-{\\frac {\\sum _{i}^{n}(x_{i}-{\\bar {x}})^{2}}{2\\sigma ^{2}}}\\right]\\;\\int _{-\\infty }^{\\infty }\\exp \\left[-{\\frac {\\sum _{i}^{n}(\\mu -{\\bar {x}})^{2}}{2\\sigma ^{2}}}\\right]d\\mu \\\\=\\;&{\\frac {1}{\\sigma ^{n+2}}}\\;\\exp \\left[-{\\frac {\\sum _{i}^{n}(x_{i}-{\\bar {x}})^{2}}{2\\sigma ^{2}}}\\right]\\;{\\sqrt {2\\pi \\sigma ^{2}/n}}\\\\\\propto \\;&(\\sigma ^{2})^{-(n+1)/2}\\;\\exp \\left[-{\\frac {(n-1)s^{2}}{2\\sigma ^{2}}}\\right]\\end{aligned}}", "0cf8da118fda3210904625dd5d34d53d": "I'_{R}=I_{R}x10^{-at}", "0cf9d66a30fbfe388026e95dd8f03f97": "c_{ad}", "0cfaa0a6b0d363502e91dc2fe6c8f1f2": "{\\frac {\\mbox{Enterprise Value}}{\\mbox{EBITDA}}}", "0cfab02aabd092dc431c0715cb7d0f42": "\\,c", "0cfac76573d2b2389bf793fd1f3527f8": "\\Gamma (V)=\\{\\varphi :G\\rightarrow \\mathbb {V} \\;:\\;\\varphi (gh)=\\rho (h^{-1})\\varphi (g)\\;\\forall \\;g\\in G,\\;h\\in H\\}.", "0cfaf9758400c6a695279edb495f892b": "\\mathbf {P} _{A}^{n}=\\operatorname {Proj} A[x_{0},\\ldots ,x_{n}].", "0cfb720281d8649ae3698766c3d85b4e": "{\\mathcal {N}}(X)\\cong [X,G/O].", "0cfbbe316fab78bf851858700fd65e1f": "\\alpha \\in \\mathbb {F} _{q^{k}}", "0cfbf719aa6482f331d4479b8f666634": "\\left({\\frac {-1}{\\sqrt {10}}},\\ {\\frac {-1}{\\sqrt {6}}},\\ {\\frac {2}{\\sqrt {3}}},\\ \\pm 2\\right)", "0cfc0c41bb3ac291ba83fa1b1db6fc4d": "\\scriptstyle {h}", "0cfc0cf414540af98a0d77c6e49489ec": "ds=-{\\frac {g}{k^{2}}}\\cos \\theta \\,d\\theta \\,", "0cfc3b7f5e5f43fa82a98ab3547e9850": "f\\in M^{\\ast }=\\operatorname {Hom} _{R}(M,R),\\quad f(m)\\neq 0.", "0cfc40cdcdcc532008908d73ece9c049": "\\exp _{10}^{3}(5.84259)", "0cfc7038f239995e93b5a2cb40a3d82d": "((1+{\\sqrt {5}})/2)^{n+m}=O(1.6180^{n+m})", "0cfc81ac400b9f2ba5839a38d0ee3823": "T\\to T_{c}", "0cfcbb8db350f1eac94263aea8278a04": "\\int _{c}\\mathrm {Hom} _{\\mathbf {X} }(F(c),G(c))=\\mathrm {Nat} (F,G)", "0cfce0a8b040eec0d5cecc2dffc37276": "\\,l_{x+t}=(1-t)l_{x}+tl_{x+1}", "0cfce3299466e89630a4d3ea1788e86c": "R{\\mathcal {F}}f", "0cfcfb0b72b5ba2a9d31a84bbad9468d": "x\\Leftrightarrow y\\equiv (x\\Rightarrow y)\\ast (y\\Rightarrow x)", "0cfd8cfdca3ab6eb1f90b6ae91a79ad9": "A,B,C,D\\in z", "0cfda2fa771f7205d2711104b4483032": "y(\\mathbf {x} )=\\sum _{i=1}^{N}w_{i}\\,\\phi (\\|\\mathbf {x} -\\mathbf {x} _{i}\\|),", "0cfdbdab6a5cf7327cdd7e783ad9f857": "P'=P+RQ", "0cfdcfd651bc8a9ea7b34e170fcc0cd6": "Z[J]", "0cfdf5b3acd19a5cbfdd9d92d87c5b77": "2^{2}+13^{2}", "0cfe589be715a942147af4a8464b82c0": "\\mathbf {x} (i),\\mathbf {x} (j),1\\leq i\\leq 49", "0cfe77ddc931028b21983ba34507dfb9": "\\langle v_{1},...,v_{g},q_{1},...q_{r},h|v_{i}h=h^{\\epsilon _{i}}v_{i},q_{i}h=hq_{i},q_{j}^{a_{j}}h^{b_{j}}=1,q_{1}...q_{r}v_{1}^{2}...v_{g}^{2}=h^{b}\\rangle ", "0cfe817f1f47a4e716b2e97325fd08be": "\\kappa ={\\frac {\\sqrt {(z''y'-y''z')^{2}+(x''z'-z''x')^{2}+(y''x'-x''y')^{2}}}{(x'^{2}+y'^{2}+z'^{2})^{3/2}}}.", "0cfea7c0f28d70f0d2380bb277ba38f5": "P_{n}", "0cff0a6b89e5b68de16a9177f78b9eab": "K=\\mathbb {Q} ({\\sqrt {7}})", "0cff164bc14d4ef25a639dc45b19b891": "f\\colon V\\to W\\,", "0cff42aaf3f77897c0ffd11286abedaf": "ln\\left(s\\right)+2=ln\\left(\\sigma \\right)+1.404576+,", "0cff9eed0b0b0c2e271ca076bc9ca638": "\\varphi (x)=({\\sqrt {(}}\\sigma _{i})\\phi _{i}(x))_{i}", "0cffb8a51f191cda38c44cbc688a24d9": "{\\cfrac {{\\cfrac {A\\wedge B{\\hbox{ true}}}{B{\\hbox{ true}}}}\\ \\wedge _{E2}\\qquad {\\cfrac {A\\wedge B{\\hbox{ true}}}{A{\\hbox{ true}}}}\\ \\wedge _{E1}}{B\\wedge A{\\hbox{ true}}}}\\ \\wedge _{I}", "0d00278997afc38630f338a5f3f8eeec": "j", "0d00bad7feceeb55c817d0b581c389b4": "e^{-x}\\,", "0d00c8234d77c4bb38d0fa90c64d19c9": "|\\psi \\rangle _{A}|0\\rangle _{B}\\rightarrow \\sum _{i}M_{i}|\\psi \\rangle _{A}|i\\rangle _{B},", "0d00d293426942203a2b25819fde31cf": "\\psi =\\sin x\\sin yF(t)\\,{\\hat {\\mathbf {z} }}.", "0d011e142283619e55af90cdb93aa7f6": "{\\begin{bmatrix}\\mathbf {p} _{1}&\\cdots &\\mathbf {p} _{i}\\end{bmatrix}}", "0d017a5760ded5225e2030e8e93a8777": "T_{M}(d)=T_{MB}(1-{\\frac {4\\sigma \\,_{sl}}{H_{f}\\rho \\,_{s}d}})", "0d01867a48fee511b6f2b96c53c0c619": "F(\\cosh t)={2 \\over \\pi }\\int _{0}^{\\infty }{\\tilde {f}}(i\\lambda )\\cos(\\lambda t)\\,d\\lambda .", "0d024a80597c3095196968fdc8587699": "2\\pi ^{2}r^{3}\\,", "0d027f4001f1491f2cc8f28fb2f1b588": "(\\kappa _{x},\\kappa _{y}),\\,", "0d028976429128aa4dcd80b80b13ab5e": "\\left[{\\begin{smallmatrix}2&-\\phi \\\\-\\phi &2\\end{smallmatrix}}\\right]", "0d02c112f2688f2563c5c0dabcca2ac9": "Z_{1},\\cdots ,Z_{k-1}", "0d038d7795f9e1102a7fe09b09f9c9fe": "1\\leftrightarrow 1,2\\leftrightarrow 4,3\\leftrightarrow 9,4\\leftrightarrow 16,5\\leftrightarrow 25,\\ldots ", "0d0392b985a1e23b0e73072bbc4a58fd": "z_{3}(x,y)={\\displaystyle \\int }xe^{-x}H{\\big (}{\\bar {y}}+{\\frac {1}{2}}x^{2}{\\big )}dx{\\Big |}_{{\\bar {y}}=y-{\\frac {1}{2}}x^{2}}", "0d03c8f189c2a0d5aa0341b4c08dd1c5": "{\\frac {\\psi ^{(m)}(n)}{(-1)^{m+1}\\,m!}}=\\zeta (1+m)-\\sum _{k=1}^{n-1}{\\frac {1}{k^{m+1}}}=\\sum _{k=n}^{\\infty }{\\frac {1}{k^{m+1}}}\\qquad m\\geq 1", "0d03d469c53743948ce6851a4c02415b": "1{\\tfrac {1}{3}}{\\text{ }}\\xrightarrow {\\text{yields}} {\\text{ }}[3,4,5],{\\text{ 2}}{\\tfrac {2}{5}}{\\text{ }}\\xrightarrow {\\text{yields}} {\\text{ }}[5,12,13],{\\text{ 3}}{\\tfrac {3}{7}}{\\text{ }}\\xrightarrow {\\text{yields}} {\\text{ }}[7,24,25],{\\text{ 4}}{\\tfrac {4}{9}}{\\text{ }}\\xrightarrow {\\text{yields}} {\\text{ }}[9,40,41],{\\text{ }}\\ldots ", "0d051830c81900841fa766716f05eb06": "PA=FO+{\\text{pos}}\\,TC", "0d05f91d31fef9c0a01768860bcc1962": "f\\colon X\\to Y", "0d060b66360f765db24937577f11fc97": "{\\frac {1+\\sin(\\theta )}{2}}", "0d0639a1d88e1de88c3a6a8145709609": "{4 \\over 1}\\cdot {5 \\over 4}=5.", "0d06c13cecd572f6e3dcc12802bef855": "\\displaystyle {K_{Z}(x,y)=e^{-(Ax^{2}+2Bxy+Dy^{2})}.}", "0d06e9adeb7d619dd6a583bd39e4ba33": "n=3", "0d06f68062b9b478a9ff91e424340b44": "\\wedge ,", "0d077d0efba3f9d45d4c783944fa2036": "M=\\varepsilon \\sinh H-H", "0d078d604e795cdd84a59f3faa798805": "b_{S}", "0d07e3ff354d6e7a18ebc1d38772e053": "\\in _{r}", "0d0806fdc14c8cc7338470ddd5c29849": "(n-1)(n-2)", "0d089509cd64838816edec60b14fcc1e": "\\Phi _{0}(w)\\equiv 1.", "0d08be02f8cfd0b9bbc6a00e8e4049a6": "GF(m)", "0d093ae321c2640045f76be4f1a8c47f": "{\\mathcal {FSL}},or{\\mathcal {FSRI}}", "0d0964133fb3658ac1eae8e99649f30f": "h(X)=-\\int _{X}f(x)\\log f(x)\\,dx.", "0d097894ffcc23dd17ee0faceed4b963": "f(x)=\\sum _{i=0}^{n}a_{i}x^{i};g(x)=\\sum _{i=0}^{m}b_{i}x^{i}", "0d09ed5e9bc899f392fb4410db3f5db1": "P:=\\sum _{\\chi (k)=1}|\\omega _{k}\\rangle \\langle \\omega _{k}|", "0d09ffd312d11ca8ddeca2b81765e2ff": "v={\\sqrt {2\\mu \\over {r}}}", "0d0a240622524a7ad459b367a0f333d7": "{\\rm {li}}(x)=O\\left({x \\over \\ln x}\\right)\\;.", "0d0a31a6c0cf03d2b7b95e81c7d73bc7": "\\displaystyle {(f_{W_{1}},f_{W_{2}})=\\det(1-W_{1}{\\overline {W_{2}}})^{-1/2}.}", "0d0a33da47501a1a7132b47f9b89c78f": "n/w", "0d0a368cb1d733b9643faead9188e41f": "\\Gamma (t)=x^{t}\\sum _{n=0}^{\\infty }{\\frac {L_{n}^{(t)}(x)}{t+n}},", "0d0a84f60b80bd1e8cf76cd687bcaed5": "{\\text{Re}}[Y_{\\ell m}]", "0d0ad5c08cc077cd5406656ce138aa07": "(R:I)=\\{x\\in K:xI\\subseteq R\\}", "0d0ae0f1cc569f2f867bd62b25b84438": "R\\leq 1-{1 \\over n}\\cdot \\log _{q}\\cdot \\left[\\sum _{i=0}^{\\lfloor {{\\delta \\cdot n-1} \\over 2}\\rfloor }{\\binom {n}{i}}(q-1)^{i}\\right]", "0d0afcdef60073f474eb7c479e606c41": "{\\rm {Tr}}(B^{1/2}AB^{1/2})^{rq}\\leq {\\rm {Tr}}(B^{r/2}A^{r}B^{r/2})^{q},", "0d0afce1725c6280c62552ed9281eb5e": "r_{N}=\\left[\\left(N-{1 \\over 2}\\right)\\lambda R\\right]^{1/2},", "0d0b46920e711135fbeac93621d0597c": "\\Delta t={\\frac {n}{2\\omega }}", "0d0b4e3b3ded537ba0e471c90904e19e": "Hy", "0d0b6b0172cfddb91c55a6d03eae6124": "{\\hat {e}}_{\\mu }", "0d0bb9d5a1002bcb284d4f7a6a0c4ba0": "N=90581", "0d0bbaf42806d64474d5e606ee0007cc": "f_{illusions.hu}(a,b)=a^{(2^{2(0.5-b)})}", "0d0c32d836fec88a92a66b1e092e2587": "\\neg ,\\lor ,\\land ,\\Rightarrow ", "0d0c64cb51e10c8933bd814c2ac752ba": "\\mathrm {Be} ={\\frac {\\Delta PL^{2}}{\\mu \\nu }}", "0d0c7355b9b6868b9755378210f53e1f": "\\mathrm {M{\\xrightarrow[{}]{h\\nu \\ (5\\ {\\text{eV}})}}M^{*}{\\xrightarrow[{}]{h\\nu \\ (5\\ {\\text{eV}})}}M^{+\\cdot }+e^{-}} ", "0d0cb760819acadc5040fbecaed81ca8": "M_{x}=\\left.\\left({\\frac {-x^{4}}{2}}+{\\frac {4x^{3}}{3}}-20x^{2}+80x\\right)\\right|_{0}^{2}", "0d0d30b28ad72c5338a5fcce70e06b37": "{\\tilde {\\kappa }}_{o+}=\\scriptstyle 0.7\\pm 1\\times 10^{-14}", "0d0d525d41ae34cd02b4707000dd7a26": "{n(3n-1)(3n-2) \\over 2}", "0d0d62c2fdffa2e5cedc35e099329266": "[(\\gamma _{1})_{\\mu }(p_{1}-{\\tilde {A}}_{1})^{\\mu }+m_{1}c+{\\tilde {S}}_{1}]\\Psi =0,", "0d0d6e27773a4d8daaa2ccd06fae3849": "\\qquad \\qquad \\mathbf {j} _{e}=-{\\frac {e_{c}}{\\hbar ^{3}}}\\sum _{p}\\mathbf {u} _{e}f_{e}^{\\prime }=-{\\frac {e_{c}}{\\hbar ^{3}k_{\\mathrm {B} }T}}\\sum _{p}\\mathbf {u} _{e}\\tau _{e}(-{\\frac {\\partial f_{e}^{\\mathrm {o} }}{\\partial E_{e}}})(\\mathbf {u} _{e}\\cdot \\mathbf {F} _{te}),", "0d0d7fee17b1432ea07a56ec95579f96": "F={\\overline {A\\vee (B\\wedge C)}}", "0d0db343dafbe7e23849033e3346288a": "\\{x:p_{\\alpha _{1}}(x)<\\epsilon ,\\cdots ,p_{\\alpha _{n}}(x)<\\epsilon \\}", "0d0db7d76c9052c8ba42d17c449d117c": "m_{solvent}", "0d0dde6d755b96b85d59014e3168f700": "{\\frac {x}{\\ln x+2}}<\\pi (x)<{\\frac {x}{\\ln x-4}}", "0d0ddeb64a1d1e92f4adfccee1c285d8": "\\delta t=-6.5\\pm 7\\ (\\mathrm {stat.} )\\pm 6\\ (\\mathrm {sys.} )", "0d0e02cb783b4d37bfd5da3ecd0a506f": "\\left\\Vert \\mathbf {x} -\\mathbf {c} \\right\\Vert ^{2}=r^{2}", "0d0e0485539af8381bf0e653bc083810": "z_{m}", "0d0e7b94e4fd1fbb404e609f2d6427a9": "pV", "0d0e9e99c8639eaf90300e449a52d109": "\\mu (\\sigma )\\geq \\mu (\\tau )", "0d0ec4dfa71d81b6c06a605bf9a415a1": "\\displaystyle \\int ", "0d0ee2c2ad53de6180b1ab5bc1addeda": "E=\\int _{S}\\rho e\\left(\\rho \\right)\\,dS.", "0d0ee4482f7e3d9029649a80c4cdd6f1": "k\\geq l-1", "0d0f0b11b41a7b6522da4bca1630b24a": "{\\frac {d^{2}y}{dx^{2}}}={\\begin{cases}{\\mbox{unbounded}}&{\\mbox{if }}y=0{\\mbox{ and }}x\\neq 0\\\\0&{\\mbox{if }}y=0{\\mbox{ and }}x=0\\\\{\\frac {3a^{6}(y^{2}-x^{2})}{y^{3}(a^{2}+2x^{2}+2y^{2})^{3}}}&{\\mbox{if }}y\\neq 0\\end{cases}}", "0d0f13ed6bdce1de9b4b961c513509a7": "{T}_{\\mathrm {m} }", "0d0f7304637ecc05e6032ed806668242": "H(X_{1})\\leq H(X_{1},X_{2})", "0d103e597a557acac8e624df87c26cca": "2-\\eta ={\\frac {\\gamma }{\\nu }}=d{\\frac {\\delta -1}{\\delta +1}}", "0d104191c889282aa464b7fb36d6bed4": "{\\bar {v}}={\\frac {\\bar {q}}{\\bar {k}}}", "0d107ece98c0895e7a3803954ed8074a": "m|\\lambda (n)", "0d115e6b11b58cad4046c41d4bd0471e": "1,2,\\ldots ,n", "0d11771ee8b0b71ee5e2679d9323f860": "\\left({\\vec {\\mu }}_{0},\\Sigma _{y=0}\\right)", "0d118be65949bca98b49c59f8b401130": "\\mathbf {M} _{\\rm {orb}}={\\frac {-e}{2m_{e}}}\\sum _{n}\\int _{\\rm {BZ}}{\\frac {d^{3}k}{(2\\pi )^{3}}}\\,\\langle \\psi _{n\\mathbf {k} }\\vert \\mathbf {r} \\times \\mathbf {p} \\vert \\psi _{n\\mathbf {k} }\\rangle \\,,", "0d11a9233dfb621f73612b720b9ae770": "A_{xx}x^{2}+2A_{xy}xy+A_{yy}y^{2}+2B_{x}x+2B_{y}y+C=0\\,", "0d11c9f5cffe121efbb4a96857a4107a": "~~~=C+\\tau +2{\\sqrt {2GM\\tau }}+4GM\\ln \\left({\\sqrt {\\frac {\\tau }{2GM}}}-1\\right)", "0d11f291eb75b1dc6cb4a8541de39433": "N_{J}=0{\\mbox{ and }}d\\omega =0\\,", "0d11fda126df5d8404401c618ebb59ce": "x\\ \\rightarrow \\ -x", "0d12237ccc6035b81781f5621d4cadb2": "\\sum _{j=1}^{3}\\partial _{j}\\tau _{ij}+\\rho g_{i}-\\partial _{i}p=Q,\\,", "0d127a74dba109a97b99c424ab1b89eb": "g_{n+1}={\\sqrt {g_{n}\\cdot a_{n}}}\\geqslant {\\sqrt {g_{n}\\cdot g_{n}}}=g_{n}", "0d12c259ace57e4c18cfb55619336a62": "Sq\\,60\\times 9", "0d12e1d934c94beaf6f3803f2c236b11": "KC(x_{i},x_{j})=\\int K(x,x_{i})\\cdot K(x,x_{j})dx", "0d1321efd35b20d2a2391609a9b59dc9": "r,s\\in T", "0d1379966d5dd80e86a3044371ec5898": "W_{\\theta }((z_{1},z_{2}))=(e^{i\\theta /2}{\\overline {z_{2}}},e^{-i\\theta /2}{\\overline {z_{1}}}).\\,", "0d13a1b68829647af5b6b0fbc1c03da7": "V\\otimes V/(v\\otimes v{\\text{ for all }}v\\in V).", "0d13aa952054b8a8bf34017a75aa4f03": "f(z)=z+a_{2}z^{2}+\\cdots .", "0d13b875c8183e4e96486212feab69b3": "log(n),log{\\frac {R}{S}}(n)", "0d13b9c0b860a192d858d90f2002d01c": "E=-\\sum _{i}h_{i}S_{i}", "0d13cdba6cf1defa57ddb9df7ccca439": "{\\frac {4bh}{3}}", "0d13f7d127bd0965408c48ff337af703": "\\scriptstyle \\varphi _{AB}", "0d1401b9be38833c006a4fa098717f44": "\\int _{\\mbox{arc}}{e^{itz} \\over z^{2}+1}\\,dz\\rightarrow 0\\ {\\mbox{as}}\\ a\\rightarrow \\infty .", "0d14467f35cab959457c4b565a1471da": "C^{*}(\\theta )\\subset C(\\theta )", "0d149b90e7394297301c90191ae775f0": "it", "0d14aed5be8c8756e4a52158149750de": "v_{x}={\\frac {dx}{dt}}", "0d14ef0f5c907a1deb056a660280790f": "{\\rm {tr}}(\\mathbf {A} ^{\\rm {T}}\\mathbf {B} )=\\operatorname {vec} (\\mathbf {A} )\\cdot \\operatorname {vec} (\\mathbf {B} ),", "0d1578a15ef9db477c7e63c5c62e699f": "a={\\frac {p_{1}+1}{2}}", "0d1586512455d6ef9241f6874e15a601": "\\mathbf {f} \\mapsto \\mathbf {f} '=\\left(\\sum _{k}X_{k}a_{k1},\\dots ,\\sum _{k}X_{k}a_{kn}\\right)=\\mathbf {f} A", "0d1591192899a5b38c1894780efdd6cb": "\\phi _{\\Gamma }(\\omega )", "0d159fded7bade094c53802c76e57d9f": "q=23", "0d168fa842421ee66899e756bbb4b7c5": "{\\frac {\\Delta R}{R}}=GF\\varepsilon +\\alpha \\theta ", "0d16a6cb2ca34d65da949608a7bc01d1": "0.02", "0d16b120ce6eb62b8466601b5259fbaf": "{\\begin{aligned}\\mathbf {u} &=\\sum _{n=-\\infty ,n\\neq 1}^{n=\\infty }\\left[{\\frac {(n+3)r^{2}\\nabla p_{n}}{2\\mu (n+1)(2n+3)}}-{\\frac {n\\mathbf {x} p_{n}}{\\mu (n+1)(2n+3)}}\\right]+\\sum _{n=-\\infty }^{n=\\infty }[\\nabla \\Phi _{n}+\\nabla \\times (\\mathbf {x} \\chi _{n})]\\\\p&=\\sum _{n=-\\infty }^{n=\\infty }p_{n}\\end{aligned}}", "0d16f72296bd681931be5741bc1e2964": "\\kappa \\,.", "0d1713700b056ccc81c30672d64323af": "\\operatorname {cov} (X,Y)", "0d171d6a16ab48052105c6921f8e78d4": "y^{(n)}", "0d179d6c5cb2567c8d1f809017799eb5": "\\delta _{X}(X)", "0d17ae5473412be33cb013507f078adb": "b_{2}\\approx \\left[{\\begin{matrix}0.64676\\\\0.40422\\\\0.64676\\\\\\end{matrix}}\\right],~\\mu _{2}\\approx 5.2418", "0d17c96f7f4592cf6458686db32b9676": "f:\\mathbb {R} \\to \\mathbb {R} ;x\\mapsto x^{2}", "0d17ed49c7a259bba8b0ab4134cdaf84": "\\displaystyle {R_{j}}", "0d17f15ddb059d52d289b254bcf74f4b": "\\,O_{i}", "0d17f8dc895d6b2f90775b63f6db0163": "\\langle P,\\leq \\rangle ", "0d18110c6be9d42370e9d8ee2feab8d2": "R^{1/2}\\ ,", "0d1820932a6e23cb1a23a6602c4d5851": "\\psi _{f}", "0d186644a6b0048d2f4760e14fa16196": "V=V_{1}=V_{2}=\\ldots =V_{n}", "0d1878dfddb55a2ebeb3124eb58be1d9": "\\vert :A\\times A\\rightarrow A", "0d188a9c7c08a4806555b1201fefa9d8": "x(0)=\\phi (-t_{0})x(t_{0})-\\phi (-t_{0})\\int _{0}^{t_{0}}\\phi (t_{0}-\\tau )[Bu(\\tau )+Ew(\\tau )]d\\tau ", "0d188eed59acc58f3f2da960f2a158b1": "f(\\cdot ;\\mathbf {v} ^{(t-1)})", "0d18e073bc56712e22cc4599b12ff989": "=a_{1}e_{1}+a_{2}e_{2}+a_{3}e_{3}.\\,", "0d18f3d858056d92fbca83a7741fed77": "\\sigma _{S}", "0d1901a17a5c3234ff79910c818f4531": "\\left\\{\\rho _{x^{n}\\left(m\\right)}\\right\\}", "0d192fb144e165a94ce0ad3d999e284f": "{\\rm {d}}(ab-ba)=0,\\ \\forall a,b\\in A", "0d194afea9be631172525ea9db6f15bb": "-{\\frac {\\hbar ^{2}}{2m}}\\nabla ^{2}\\phi =i\\hbar {\\frac {\\partial }{\\partial t}}\\phi .", "0d1a5a2b554fab5b3076c0152981e6e3": "u=(u_{1},...,u_{n})", "0d1a73530f7a59d7fc84b3a5094d09e1": "\\Delta \\lambda ", "0d1a76ef259a793f7a800ae90b0870de": "D_{**}^{(p)}(\\mathbf {X} ,\\mathbf {Y} )=\\left({\\sum _{i}{D_{**}(X_{i},Y_{i})}^{p}}\\right)^{\\frac {1}{p}}", "0d1ab40684001e761f4fb4f2256bde93": "=\\gamma {\\frac {\\omega _{\\mathrm {obs} }}{c}}-\\beta \\gamma k_{\\mathrm {obs} }^{1}\\,", "0d1b52bc46246e7df5acc42385979615": "a\\times a", "0d1b62cac42bf3fbd446acf638ce1978": "{\\begin{smallmatrix}\\left[{\\frac {Fe}{H}}\\right]\\ =\\ 0.5\\end{smallmatrix}}", "0d1b80ad19cafad14503d2f2060bd38c": "(V_{x},V_{y})", "0d1bc873fc7a89d2a5fecab6aa8a3265": "{\\tfrac {dy}{dt}}", "0d1c137f5055f97e670d1393065b97f9": "5\\times 7=35\\,", "0d1c62636094364b4c5ca07192d0fd48": "f(\\xi )\\star g(\\xi )=Tr[{\\hat {B}}(\\xi ){\\hat {f}}{\\hat {g}}]", "0d1cfc2b565cc8ef6c591b870149fdee": "GL(R)", "0d1d0541ed5b1a49323e3e5b1a426d06": "{\\mathbf {} }P(t),S(t),{\\hat {P}}(t),{\\hat {S}}(t)", "0d1d353a531d0c830d2ee514f36b3e6d": "\\operatorname {dim} [D(\\cdot )]=q\\times p", "0d1d512c5a77deaf46d335aa42d873aa": "P(Y\\in S)=\\int _{\\phi ^{-1}(S)}p_{x}(x)~dx=\\int _{S}p_{x}(\\phi ^{-1}(y))~\\left|{\\frac {d\\phi ^{-1}}{dy}}\\right|~dy.", "0d1d5de64a1eca2f5186aaf6f5529bb2": "(D-\\lambda )\\varphi _{\\lambda }=0,\\quad (D-\\lambda )\\theta _{\\lambda }=0", "0d1da73a7e2cdd65c45bd490652776a3": "\\Delta L/Lo", "0d1dce0001d87680ea382492c73dedf7": "Y=mX+b", "0d1f051084914b4857972354998479d6": "{\\text{Si}}(z)=\\int _{0}^{z}{\\frac {\\sin \\zeta }{\\zeta }}\\,d\\zeta ", "0d1f538321149df9161ae9bcb8321589": "\\Delta Y=V\\Delta K+W\\Delta L+\\Delta Y'", "0d1fb4cc5a4abdb666d33e0ceb2bb04c": "A_{n}=A_{n-1}-r_{avg,n}\\Delta t_{p}", "0d1fe771863fbaaf863d6b545e5b7b1c": "(g(T)\\xi ,\\eta )=\\int _{0}^{1}g(\\lambda )\\,d(E(\\lambda )\\xi ,\\eta )=\\int _{0}^{1}g(\\lambda )\\,d\\rho _{\\xi ,\\eta }(\\lambda ).", "0d20024c7340f05df7f7abdbd9858389": "n\\geq (m+1)", "0d2008537bbce6671711b431aa5bfc1e": "t_{n}\\leftarrow t_{l}+ta(s);", "0d202ce153f5df5cb6fe44b9461a20b5": "H\\left(X\\right)\\equiv -\\sum _{x}p_{X}\\left(x\\right)\\log p_{X}\\left(x\\right).", "0d207726d1b7ee91d77a749a0fe04bc8": "\\chi (R)", "0d20bff51398891724de0e5ebe4ee297": "\\ln \\left({\\frac {1}{p}}-1\\right)=\\sum _{i=1}^{N}\\left[\\ln(1-p_{i})-\\ln p_{i}\\right]", "0d20c1dbe3e3a3ad0d4a224004f63af5": "\\log =\\exp ^{-1}", "0d20ca8c039a274aa997eff14b25f11c": "r=a", "0d20cd48b4784d29c208353fb3e79545": "\\sigma _{a}^{2}>0", "0d20df5445563118b38f0a894c4f7e98": "X\\subset \\mathbb {R} ^{n}", "0d20fcd3e97607f522c75657445e2ab3": "{D(R)}_{m'm}^{(j)}=\\langle j,m'|U(R)|j,m\\rangle ", "0d211795903196a13bc257387de9b858": "\\scriptstyle ratio={\\frac {PV(B_{2})-PV(B_{1})}{PV(C_{2})-PV(C_{1})}}", "0d217b2d4e34f78f6c6fd2e02695c591": "k={\\frac {\\omega n_{1}}{c}}\\sin(\\theta _{I})", "0d2183bca51700162b8418f4239f35d1": "E_{\\rm {g}}", "0d2194541ab3767828573818fcaad8d3": "f_{k}(x)=e^{kx}", "0d21ed18a98150ae571d46a2d4ff254d": "\\bigoplus _{p=1}^{n}S^{2}(\\Omega ^{p}M),", "0d21edba5c892ac3ec8de9f8d602eb34": "{\\dot {q_{i}}}", "0d2205031b1c6b2e8d118218cf68ee51": "ed\\equiv 1{\\pmod {(p-1)(q-1)}}", "0d22066efc96d7e7d4b6c2a428d76646": "\\exp(X)=\\sum _{k=0}^{\\infty }{\\frac {X^{k}}{k!}}=I+X+{\\frac {1}{2}}X^{2}+{\\frac {1}{6}}X^{3}+\\cdots ", "0d2272aed78c76153375fa478fdb5b77": "C_{D,i}={\\frac {C_{L}^{2}}{\\pi A\\epsilon }}", "0d22b0574833ed0702cb4cc05c077fc6": "\\scriptstyle a\\equiv b{\\pmod {m}}", "0d22be7fc4410551eecdd682be98569f": "p\\sim \\mathrm {Beta} \\left(\\alpha +n,\\ \\beta +\\sum _{i=1}^{n}k_{i}\\right).\\!", "0d22d4f5aa39ec827e00911e251a5ae0": "{\\vec {D}}=|{\\vec {C}}.{\\vec {X_{p}}}(t)-{\\vec {X}}(t)|", "0d23d33bf55d973e8dc20d2434e2ebc5": "\\mathbf {V} ^{-1}\\mathbf {C} \\mathbf {V} =\\mathbf {D} ", "0d23f09a073cdb873cead2349fd0748a": "\\int _{U_{\\alpha }}f=\\int _{\\phi _{\\alpha }(U_{\\alpha })}t_{\\alpha }\\circ f\\circ \\phi _{\\alpha }^{-1}d\\mu ", "0d24044f81ee0e3875e213e76122b1ca": "\\mathrm {Inv} ^{1}\\langle X|T\\rangle =(X\\cup X^{-1})^{*}/(T\\cup \\rho _{X})^{\\mathrm {c} }.", "0d249fd47fefa390fa28d1ae2c141d9a": "{\\mathbf {y} }-{\\boldsymbol {\\nu }}", "0d24c605000ecb1ba6dc1183f36dd153": "{\\rm {Vol}}(M)<\\varepsilon (n)", "0d250ccf9b29327fc9cd0be9f2d1a8b3": "(9)", "0d2522f2702995f340f189e0f0b81559": "{\\sqrt {e^{\\sigma ^{2}}\\!\\!-1}}", "0d25530d2daf54334b54c7ddd7dd2a81": "\\kappa (A,Q_{1})\\neq 0\\neq \\kappa (K_{M},A')", "0d255fc40699018dfbc407a92e64e25d": "s_{j}(x)=s_{i}(x)\\cdot t_{ij}(x).", "0d25727aed4204122915f50b57f45e0a": "\\upsilon _{p}=\\upsilon _{o}=\\pm (\\upsilon _{D}-|\\upsilon _{M}|)\\qquad (8)", "0d257d22b5be1adb99c63cf8207a7818": "c(x,\\eta )=\\left\\{{\\begin{array}{l}1\\quad {\\text{if}}\\quad |\\{y\\in x+{\\mathcal {N}}:\\eta (y)\\neq \\eta (x)\\}|\\geq T\\\\0\\quad {\\text{otherwise}}\\\\\\end{array}}\\right.", "0d261e8e4a3dd10b67d214d2b0d8d4ea": "D_{k-1}\\wedge ", "0d2623e10c0aea105f257c1188978a19": "B_{0}=-V\\left({\\frac {\\partial P}{\\partial V}}\\right)_{P=0}", "0d265bb29d3ea13319bfc6c65d006e71": "t+k\\,\\Delta t", "0d266404d59506165f02b7f9330257af": "O(n)\\subset O(n+1)", "0d267342c9e958b2d9571c87d97604c9": "\\int (A+B\\,x)(a+b\\,x)^{m}(c+d\\,x)^{n}(e+f\\,x)^{p}dx={\\frac {(A\\,b-a\\,B)(a+b\\,x)^{m+1}(c+d\\,x)^{n+1}(e+f\\,x)^{p+1}}{(m+1)(a\\,d-b\\,c)(a\\,f-b\\,e)}}\\,+\\,{\\frac {1}{(m+1)(a\\,d-b\\,c)(a\\,f-b\\,e)}}\\,\\cdot ", "0d2686a84e52ad3ea4d6d16f8d2c5336": "I=\\bigcap _{i=1}^{k}P_{i}", "0d26eb023d1e859d4e55319835f5dd70": "T_{\\rm {h}}", "0d27c8fab6667d760378094752161e20": "c=\\mathbf {st} (x_{i_{0}})", "0d27d9bd6674d75a69331e42c1ac0b45": "\\mathbf {A} \\mathbf {x} =\\mathbf {b} ,\\,x_{i}\\geq 0", "0d27de988caad8fbd3c1dfd807e09332": "\\mathrm {Eu} _{m}=\\mathrm {Eu} \\;\\quad \\quad {\\mbox{i.e.}}\\quad {p_{m} \\over \\rho _{m}{v_{m}}^{2}}={p \\over \\rho v^{2}}\\;,", "0d281732e55b5b0446f4cb26f620ee3a": "V_{0}=2,V_{1}=A,V_{j}=AV_{j-1}-V_{j-2}", "0d28187719daa8439ef00bdce60d0cdc": "\\{y_{k}\\}_{k=1}^{M}", "0d2876b89cf5452c755b6685369b8591": "x>a", "0d289ca0712b0e45db926c51f7e77153": "rv_{\\theta }=\\Gamma /(2\\pi )", "0d28b4516f695f007e7f87f8f95529d4": "x=as+x_{0}=at+x_{0}\\,", "0d28d8ceee019945cdd265ac580aec67": "\\gamma \\in E", "0d2901ef433b8e8b8a1fe785330e4edd": "\\|{\\boldsymbol {x}}\\|:={\\sqrt {x_{1}^{2}+\\cdots +x_{n}^{2}}}.", "0d2941a258bdb4c6652738256c19daf2": "MU*(MU)", "0d2952936deadd6f35f0ce351fa6710f": "|{\\mathcal {Z}}|=n(n^{2}+1).", "0d2990b73c38ab89d3eb9287b0432b24": "T(X_{1}^{n})=\\left(\\prod _{i=1}^{n}{x_{i}},\\sum _{i=1}^{n}x_{i}\\right)\\,", "0d299763a1317c3b9bf559c527627b48": "6v-2e=6\\sum _{i=1}^{D}v_{i}-\\sum _{i=1}^{D}iv_{i}=\\sum _{i=1}^{D}(6-i)v_{i}=12.", "0d29a3a7d93ff82adf7d58cea12cc7d1": "{\\frac {dK_{i}}{dt}}=1\\,\\forall \\,i\\in S", "0d29c6241f4e8753818cf25871124385": "b=\\sum _{i}x_{i}\\;b_{i}", "0d29d0111b21f34ce720c6c4a665cdea": "{\\frac {\\lambda \\Delta \\Theta }{4\\pi \\Delta t}}", "0d2aec71e040b90af40a5c82f1cd40f4": "\\alpha \\varphi ={d\\varphi \\over dx}=0", "0d2af1bb9bdb0e828e0b1643ce5dfbe9": "\\scriptstyle E_{0}(x)={\\frac {x}{e{\\sqrt {\\pi }}}}", "0d2af255ed8deb7614e9b97cc3e82df5": "{\\mathcal {M}}_{XY}(s)={\\mathcal {M}}_{X}(s){\\mathcal {M}}_{Y}(s)", "0d2b3128c7ae59366163f66691bab00b": "\\nabla _{\\boldsymbol {r}}H({\\boldsymbol {r}})={\\begin{bmatrix}\\partial _{\\boldsymbol {q}}H({\\boldsymbol {q}},{\\boldsymbol {p}})\\\\\\partial _{\\boldsymbol {p}}H({\\boldsymbol {q}},{\\boldsymbol {p}})\\\\\\end{bmatrix}}", "0d2b4014b1181599e39765af3670387d": "Q=I^{2}\\cdot R\\cdot t", "0d2b4ac581f80000a4abcdc1e2115f27": "\\lambda =\\lambda _{0}\\,", "0d2b65af50b37263fc83b5d435647433": "\\displaystyle g", "0d2b922fc35916c85df042d4e0bb7ddc": "\\textstyle {\\tilde {u}}", "0d2ba1be115838af1b49c1a6139674d4": "1-\\delta /2", "0d2bbcb2cff894f24437683c56beb41c": "B/I", "0d2c25d79d691c6d36b745715404093f": "A\\subseteq _{\\omega }G", "0d2c5875cb4535292638d4a381417bb6": "p=-\\left({\\frac {\\partial A}{\\partial V}}\\right)_{T}={\\frac {NkT}{V-Nb'}}-{\\frac {a'N^{2}}{V^{2}}}.", "0d2cab7545531536ea36a769019f921b": "{\\text{PoP}}=C\\times A", "0d2d275575f90b900f8fe693c3f1e79b": "ZY=U+iV\\,\\!", "0d2d5aac0c0d0ec3ffb080cc6ed30ab2": "{\\begin{pmatrix}A'(x)\\\\B'(x)\\end{pmatrix}}={\\begin{pmatrix}u_{1}(x)&u_{2}(x)\\\\u_{1}'(x)&u_{2}'(x)\\end{pmatrix}}^{-1}{\\begin{pmatrix}0\\\\f\\end{pmatrix}}", "0d2d8db530bbee3a8caf21c607110a87": "ALG(\\sigma )-c.OPT(\\sigma )\\leq \\alpha ", "0d2d998328a6fe85713bedef922fa466": "\\pi (x_{k}|x_{0:k-1},y_{0:k})\\,", "0d2da4e5aa8e8c7d8aaf1020ebb5fd55": "\\{\\cdot ,\\cdot \\}_{\\eta }", "0d2dc97221a8c1d67c2cc5a89f0c4812": "g\\cdot \\left(x,y,{\\frac {dy}{dx}}\\right){\\stackrel {\\text{def}}{=}}\\left({\\overline {x}},{\\overline {y}},{\\frac {d{\\overline {y}}}{d{\\overline {x}}}}\\right).", "0d2e17ef5417e82f59050aaf118ab302": "p=vi\\,", "0d2e751ea16d8df35051ef5c09af15b4": "\\mathrm {H} ^{n}(G;M)=\\mathrm {Ext} _{kG}^{n}(k,M)", "0d2e80e7b52d3ae4c59192eec337947b": "\\{U_{\\lambda }\\vert \\lambda \\ \\in \\ \\Lambda \\}", "0d2e823943aec1ab24408beedcc9116c": "\\sigma _{P}", "0d2e858bd7f89eed5461e5637d6e0a50": "\\log n", "0d2e89ded8b8c63f4ebde42591795bcb": "p^{ij}=p\\left(Q_{b}^{(i)}|Q_{c}^{(j)}\\right)", "0d2f44a7b4269b89b6b5610eabdcec76": "|p||q|", "0d2f45f312359620d82962b2ec6be309": "\\tau _{0}", "0d2f6250f9472057738f1e2f6a783f20": "a:=J/M", "0d2fc0ce4b2672c236ee1e7727be2a63": "\\pm {\\frac {z_{1-\\alpha /2}}{\\sqrt {N}}}", "0d2fc3dce992c1a20056f3ca9d16f03b": "{\\frac {\\partial ^{2}y}{\\partial x_{i}\\partial x_{j}}}=\\sum _{k}{\\frac {\\partial y}{\\partial u_{k}}}{\\frac {\\partial ^{2}u_{k}}{\\partial x_{i}\\partial x_{j}}}+\\sum _{k,\\ell }{\\frac {\\partial ^{2}y}{\\partial u_{k}\\partial u_{\\ell }}}{\\frac {\\partial u_{k}}{\\partial x_{i}}}{\\frac {\\partial u_{\\ell }}{\\partial x_{j}}}.", "0d300d8bb92e7abcb5f1fc0d3637425e": "\\mathbb {RFM} _{I}(D)\\,", "0d306f1bd424d62905a90745404a17f7": "a_{n}\\approx {\\sqrt {\\frac {\\rho +\\rho ^{2}B'(\\rho ^{2})}{2\\pi }}}{\\frac {\\rho ^{-n}}{n^{3/2}}},", "0d309d7aef3404621e0719ccdbb900ea": "f_{r}(\\theta )=F(re^{i\\theta }),", "0d30a10e004417673985e972bf8fd97f": "\\oint _{\\gamma }f(z)\\,dz=0", "0d30a83d5cbeb54b7220236e608f1e90": "j=1\\ldots p", "0d319acc1a9015aab6b189e45c77f7b5": "\\sigma (n)\\leq H_{n}+\\ln(H_{n})e^{H_{n}}", "0d31a97e764a72427abaed22f8483aba": "H_{1}:\\theta =0", "0d31e16b4e6149c32102911d79cf4e0f": "{\\dot {M}}", "0d31f258914ebf9e21fbd3cd1b55801e": "(x,\\,y,\\,z)=(1,\\,2,\\,3)", "0d32149b4ee4096ea0a8bf9a2d5d41ed": "{\\mathcal {G}}\\times {\\mathcal {H}}", "0d322a6883cd9363afefdb344596517f": "\\Psi :G\\to \\mathrm {Aut} (G)\\,", "0d3244f649381020e137893daa357eee": "{{\\frac {|AB|}{|BD|}}\\sin \\angle \\ BAD={\\frac {|AC|}{|DC|}}\\sin \\angle \\ DAC}", "0d32c7b080b9c0ab6b422b59eac02eef": "5/4E_{\\mathrm {h} }\\,", "0d331c82cd6dfbe6ef081a4c5facc021": "L_{c}\\ll \\lambda ", "0d3376240ee4aa12f272f6bd4ce0a4ef": "\\Omega =2\\pi \\nu ", "0d337c802e7e5c198e21a6ab6eb17ec8": "E(\\mathbb {F} _{p})\\cong \\mathbb {Z} _{2^{k}n}", "0d3381bc741b5c2ae9f71b2ad21f4058": "{\\begin{aligned}F_{x}&=-q\\left({\\frac {\\partial \\phi }{\\partial x}}+{\\frac {\\partial A_{x}}{\\partial t}}\\right)+q\\left[{\\dot {y}}\\left({\\frac {\\partial A_{y}}{\\partial x}}-{\\frac {\\partial A_{x}}{\\partial y}}\\right)+{\\dot {z}}\\left({\\frac {\\partial A_{z}}{\\partial x}}-{\\frac {\\partial A_{x}}{\\partial z}}\\right)\\right]\\\\&=qE_{x}+q[{\\dot {y}}(\\nabla \\times \\mathbf {A} )_{z}-{\\dot {z}}(\\nabla \\times \\mathbf {A} )_{y}]\\\\&=qE_{x}+q[\\mathbf {\\dot {r}} \\times (\\nabla \\times \\mathbf {A} )]_{x}\\\\&=qE_{x}+q(\\mathbf {\\dot {r}} \\times \\mathbf {B} )_{x}\\end{aligned}}", "0d3385c0bc07dd05265d2bfeb61ff11d": "i\\in N.", "0d3387f183b071349623c1d18272f415": "z^{*}:=f(x^{*})\\in \\mathbb {R} ^{k}", "0d33e9f5b106df3293cdf9a9ba1e4c77": "E(z,s)={1 \\over 2}\\sum _{(m,n)=1}{y^{s} \\over |mz+n|^{2s}}", "0d3412cfa681f1b4049bcd69b873b892": "{\\overline {f}}\\colon {\\overline {V}}\\to {\\overline {W}}", "0d342dc5b57d999d0e51e4e8e1dba385": "\\Sigma =1", "0d3438af5eafe64e58b63a68d9d73f4c": "\\operatorname {Ind} _{H}^{G}\\pi =\\{f:G\\rightarrow V|f(hg)=\\Delta _{G}^{-1/2}(h)\\Delta _{H}^{1/2}(h)\\pi (h)f(g){\\text{ and }}f\\in L^{2}(G)\\}.", "0d348c4753b1623b79d6f1c8420766ef": "a_{0}x^{n}+a_{1}x^{n-1}+\\cdots +a_{n-1}x+a_{n}\\,", "0d34c45e8cc424127988be70470886b2": "\\displaystyle ={\\frac {f}{\\sigma }}{\\frac {\\partial }{\\partial p}}\\mathbf {V} _{g}\\cdot \\nabla _{p}(\\zeta _{g}+f)+{\\frac {R}{\\sigma p}}\\nabla _{p}^{2}(\\mathbf {V} _{g}\\cdot \\nabla _{p}T)", "0d34c87c64f32752b59c9fc56b15ad09": "x^{2}+y^{2}-1=0", "0d34ce5114059b25027f3669291236cf": "\\nabla _{{\\mathbf {e} }_{j}}{\\mathbf {u} }=\\nabla _{j}{\\mathbf {u} }=\\left({\\frac {\\partial u^{i}}{\\partial x^{j}}}+u^{k}\\Gamma ^{i}{}_{jk}\\right){\\mathbf {e} }_{i}", "0d34f3f304db97277d1b5aa730a452df": "\\rho (\\mathbf {r^{\\prime }} )", "0d3521d9f3df8bd3fdd04f19b3e2d9f6": "\\sigma '_{p}", "0d3562e83fb6cb804b5f73faaa1a47be": "K'_{0}", "0d35715a228671a7b6bf190a507c97b8": "T={\\frac {1}{2}}{\\boldsymbol {\\omega }}\\cdot \\mathbf {I} \\cdot {\\boldsymbol {\\omega }}={\\frac {1}{2}}I_{1}\\omega _{1}^{2}+{\\frac {1}{2}}I_{2}\\omega _{2}^{2}+{\\frac {1}{2}}I_{3}\\omega _{3}^{2}", "0d3596c58bb98272bcdb7e2061d0e69f": "\\mathrm {supp} \\,X=\\mathrm {supp} \\,X_{*}{\\mathfrak {P}}.", "0d35ec8c42f9191581d0b11cb1d1af9f": "d=\\partial +{\\bar {\\partial }},\\ \\ \\ \\ d^{*}=\\partial ^{*}+{\\bar {\\partial }}^{*}", "0d35fc1f504d5a3ca234e17aed31b978": "\\scriptstyle {\\sqrt {1-x^{2}}}", "0d3600f23d218a8d6ef9599d23cab885": "m^{2}=0", "0d3626a345c75f44137598236ab85e8c": "U(P)=-{\\frac {i}{2\\lambda }}{\\frac {ae^{ikr_{0}}}{r_{0}}}\\int _{S}{\\frac {e^{iks}}{s}}(1+\\cos \\chi )\\,dS", "0d362bd3c9965386899a8e1bdc8cd27b": "X\\leftarrow Y,\\;Y\\backslash X", "0d362e64d8ac6558fa534d852165aaa3": "y\\,=\\,r_{1}(\\theta )\\sin(\\theta )r_{2}(\\phi )\\cos(\\phi )", "0d368de2a395c090b46e83e71f24ed4f": "y\\equiv 0", "0d36a44f95642058433c588e2c564d8f": "(c_{i}\\neq C_{\\text{in}}(y_{i}'))", "0d36b40da67336f180fd1667993048f2": "{\\begin{aligned}\\rho \\left({\\frac {\\partial u}{\\partial t}}+u{\\frac {\\partial u}{\\partial x}}+v{\\frac {\\partial u}{\\partial y}}+w{\\frac {\\partial u}{\\partial z}}\\right)&=-{\\frac {\\partial p}{\\partial x}}+\\mu \\left({\\frac {\\partial ^{2}u}{\\partial x^{2}}}+{\\frac {\\partial ^{2}u}{\\partial y^{2}}}+{\\frac {\\partial ^{2}u}{\\partial z^{2}}}\\right)+\\rho g_{x}\\\\\\rho \\left({\\frac {\\partial v}{\\partial t}}+u{\\frac {\\partial v}{\\partial x}}+v{\\frac {\\partial v}{\\partial y}}+w{\\frac {\\partial v}{\\partial z}}\\right)&=-{\\frac {\\partial p}{\\partial y}}+\\mu \\left({\\frac {\\partial ^{2}v}{\\partial x^{2}}}+{\\frac {\\partial ^{2}v}{\\partial y^{2}}}+{\\frac {\\partial ^{2}v}{\\partial z^{2}}}\\right)+\\rho g_{y}\\\\\\rho \\left({\\frac {\\partial w}{\\partial t}}+u{\\frac {\\partial w}{\\partial x}}+v{\\frac {\\partial w}{\\partial y}}+w{\\frac {\\partial w}{\\partial z}}\\right)&=-{\\frac {\\partial p}{\\partial z}}+\\mu \\left({\\frac {\\partial ^{2}w}{\\partial x^{2}}}+{\\frac {\\partial ^{2}w}{\\partial y^{2}}}+{\\frac {\\partial ^{2}w}{\\partial z^{2}}}\\right)+\\rho g_{z}.\\end{aligned}}", "0d36b70987f5e50a7c1e4c2ba5b17c85": "\\displaystyle {X=Y^{*},\\,\\,\\,XY-YX=2I.}", "0d370d005e07668b18ad75cdefc288ab": "P_{\\lambda }", "0d37622c2f3e173c0b258e5eb164842e": "b_{c}=2\\left(\\Delta c\\right)\\tan \\phi ", "0d3785ba869328d58bb93b1d2d9b1f91": "M=", "0d37d5cb4cc927f3d9371a5b3196d83c": "{\\mathcal {O}}={\\begin{bmatrix}C\\\\CA\\\\CA^{2}\\\\\\vdots \\\\CA^{n-1}\\end{bmatrix}}", "0d37d7a6a31e214764d679d2550860a5": "{\\dot {m}}={\\frac {d}{dt}}m=\\,", "0d37f08551c16c1b3df722f9a230f833": "{\\begin{aligned}c_{2}&=(a_{1}+a_{2}+c_{1}-1)/2-K\\\\c_{3}&=(3a_{1}+3a_{2}-c_{1}-3)/2-K\\\\c_{4}&=2a_{1}+2a_{2}-c_{1}-2\\\\c_{5}&=(3a_{1}+3a_{2}-c_{1}-3)/2+K\\\\c_{6}&=(a_{1}+a_{2}+c_{1}-1)/2+K.\\end{aligned}}", "0d3819fcc545ed0bdf8d3fdb1362a12f": "\\rho {\\vec {\\nabla }}\\times {\\vec {v}}", "0d384ef21957d905c670977d9b67945e": "\\bot \\rightarrow \\psi ", "0d3854a0747b1e2b2e2dc4aaee53058e": "{\\begin{aligned}{\\dot {e}}&={\\frac {\\operatorname {d} }{\\operatorname {d} t}}H(x)-{\\frac {\\operatorname {d} }{\\operatorname {d} t}}H({\\hat {x}})\\\\&={\\frac {\\operatorname {d} }{\\operatorname {d} t}}H(x)-M({\\hat {x}})\\,\\operatorname {sgn} (V(t)-H({\\hat {x}}(t))),\\end{aligned}}", "0d386210025f2537646ea1bd9f49b2bb": "\\kappa _{2n}={\\frac {2^{2n-1}(2^{2n}-1)B_{2n}}{n\\,(3^{2n}-1)}},\\,\\!", "0d386aef9fd3aa282a8e0f0d56b661c4": "{={{V\\times I} \\over {m}}}", "0d3898ab45954a4419634dad94d3185c": "(X,p)", "0d38bd1865594124901028a3e55f08a6": "\\mathbf {R} _{B/A}=\\mathbf {P} _{B}-\\mathbf {P} _{A}=(x_{B}-x_{A},y_{B}-y_{A},z_{B}-z_{A}).", "0d390802d6e6490916db3adb36e5513a": "\\langle f_{thm}(s)f_{thm}^{T}(t)\\rangle =-\\left(2k_{B}{T}\\right)\\left(\\mu \\Delta -\\Lambda \\Upsilon \\Gamma \\right)\\delta (t-s).", "0d390d989d38d2710c82f29a2b01fe00": "{\\bar {x}}={\\frac {1}{3N}}\\sum _{n=1}^{N}(x_{n,1}+x_{n,2}+x_{n,3})", "0d392315c70d1c7a09e50ddddb0ee8d1": "2\\otimes 2", "0d393a9bd11da911fdba0436fc08b613": "\\arccos {\\frac {\\langle \\mathbf {r} (\\theta ),\\mathbf {r} '(\\theta )\\rangle }{\\|\\mathbf {r} (\\theta )\\|\\|\\mathbf {r} '(\\theta )\\|}}=\\arctan {\\frac {1}{b}}=\\phi .", "0d39b1b7b912ed75ff9e9aa383b6e7e6": "{\\frac {(\\gamma /\\delta )^{\\lambda }}{{\\sqrt {2\\pi }}K_{\\lambda }(\\delta \\gamma )}}\\;e^{\\beta (x-\\mu )}\\!", "0d39bd7aba5ab8f666ff14b2b638cd86": "\\gamma _{k}(G)\\leq G^{p^{k-1}}", "0d39ccf34f19b82c6e63a147dcad4a38": "\\alpha _{1}=4{\\frac {G}{K}}((2K-1)e^{-4\\phi _{0}}-e^{4\\phi _{0}}+8)-8", "0d39efc22d906cf481dcfcbfece93069": "a_{11}x_{1}+a_{12}x_{2}\\leq b_{1}", "0d3a377330c597627c46f63d35f04031": "{\\frac {1-p}{p^{2}}}\\!", "0d3a3aacf4c3bd97b785057b39abd0de": "\\mathbf {p} \\rightleftharpoons \\mathbf {q} ,\\quad H\\rightarrow -H.", "0d3a74d610de42789cd0ca20b2a3c320": "{\\begin{matrix}2&2&4\\\\3&5\\end{matrix}}", "0d3abdc1ea0c2ebffbd6b221553a0504": "\\sum _{R\\in G}^{|G|}\\;\\Gamma ^{(\\mu )}(R)_{nm}=0", "0d3b0c01e429b0a310dc32dfb61f6c59": "{\\frac {\\partial \\rho {\\vec {u}}}{\\partial t}}+\\nabla \\cdot \\Pi =0", "0d3b265f26ca4093a6af09e954cba72a": "f(z)=\\int _{D}f(\\zeta )K(\\zeta ,z)\\,d\\mu (\\zeta ).", "0d3b3b0a92b0da46f8e512429229e033": "\\scriptstyle \\chi _{o}(G)", "0d3bda1e8f0c93cb21cdb05311977fb1": "\\mathbf {K} ({\\hat {n}},\\nu )={\\vec {a}}(\\nu )\\mathbf {I} +\\int _{4\\pi }\\mathbf {Z} ({\\hat {n}}^{\\prime },{\\hat {n}},\\nu )\\mathrm {d} {\\hat {n}}^{\\prime }", "0d3c7749fc39bfa9d6d4dfba34da5891": "1\\leq k\\operatorname {Li} (x)+{\\frac {1}{3}}{\\frac {\\sqrt {x}}{\\log x}}\\log \\log \\log x,", "0d3ea2ad17ad549d8e81ad9150607b53": "\\mathrm {v} =(A^{T}WA)^{-1}A^{T}Wb", "0d3ee1c744c2568cfdd269e202457839": "\\operatorname {var_{GX}} =e^{\\operatorname {var} [\\ln X]}", "0d3ee75d810bc2dfb163d0e221524bce": "{\\widehat {\\varepsilon }}={\\frac {D_{0}}{E_{0}}}=|\\varepsilon |e^{i\\delta }.", "0d3f04f1fa0ed58189e3e643518d2147": "V_{7}={\\frac {16\\pi ^{3}r^{7}}{105}}", "0d3f18f580556b9f285a21347527e61f": "\\mathbf {u} _{k}^{(i-1)}", "0d3f197ee292ba0cca526cc015bac497": "y\\preceq x", "0d3f1a2930fca5e62b5babb4769895f6": "{\\begin{aligned}h_{1}&=1\\\\h_{2}^{2}&={\\frac {\\lambda ^{2}(\\mu ^{2}-\\nu ^{2})}{(\\mu ^{2}-a^{2})(b^{2}-\\mu ^{2})}}\\\\h_{3}^{2}&={\\frac {\\lambda ^{2}(\\mu ^{2}-\\nu ^{2})}{(\\nu ^{2}-a^{2})(\\nu ^{2}-b^{2})}}\\end{aligned}}", "0d3f57f00e9d8fe9e759be900b06465f": "s\\cdot s^{2},\\qquad s\\cdot s^{2}\\cdot s^{2},\\qquad s\\cdot s^{2}\\cdot s^{2}\\cdot s^{2},\\cdot ", "0d3f9f9ac9ecc98c3f41eccea575b445": "w(abcd;ef)\\equiv \\sum _{z}{\\frac {(-1)^{z+\\beta _{1}}(z+1)!}{(z-\\alpha _{1})!(z-\\alpha _{2})!(z-\\alpha _{3})!(z-\\alpha _{4})!(\\beta _{1}-z)!(\\beta _{2}-z)!(\\beta _{3}-z)!}}", "0d3fae5c51f1845e1bc1521dc69da04f": "\\epsilon _{1}=\\epsilon _{2}=0,\\epsilon _{3}=\\epsilon _{4}=\\epsilon _{5}=\\epsilon _{6}>0", "0d3fd5ea850d4397ac9a15e17dcaa56d": "\\mu \\Psi (\\mathbf {r} )=\\left(-{\\frac {\\hbar ^{2}}{2m}}\\nabla ^{2}+V(\\mathbf {r} )+g\\vert \\Psi (\\mathbf {r} )\\vert ^{2}\\right)\\Psi (\\mathbf {r} )", "0d401919d1a7b58fa1a7027637587d9e": "\\mathbb {F} =GF(2)", "0d4026d4486d710582f7f63dd7d8d285": "J_{A}=q_{\\mathrm {A} }j_{\\mathrm {A} }", "0d405d76bbb1bdd28be41db327144066": "h\\ll R", "0d4083d1b03dc87a52689b8524ca6c98": "{\\mathcal {S}}=\\int {{\\mathcal {L}}\\mathrm {d} ^{4}x}.", "0d410447eb3b039f627f6256109cb740": "{\\frac {p_{01}}{p_{1}}}=(1+{\\frac {\\gamma -1}{2}}M_{1}^{2})^{\\frac {\\gamma }{\\gamma -1}}", "0d418bc6ac2910e8f42bb722a1fc219d": "y=r\\sin \\theta ", "0d41a092f385d4e00d361683eddc0b23": "\\textstyle x_{1}", "0d41ab17705002a473b26a8376762b2c": "y'=(2-y)y", "0d423e580708b1c1f686d142a635f618": "S_{\\max }=1+\\exp \\left(-\\pi {\\frac {\\rho }{\\mu }}\\right).", "0d424c34c2273d08727fac1d6bfc4b15": "{d,e,n}", "0d426f79201ec92973381653f2d165f0": "\\scriptstyle \\pi ^{\\!*}F=dA.", "0d432fa7630523417b09114e6ebcc7c0": "r(x)=r(y)+1", "0d435509b55b9239d154e1f311994b40": "\\Phi (Y_{i})=", "0d436a7d5a6abc8b5a66f2a87d0b943a": "\\varepsilon _{\\text{Total}}=\\varepsilon _{D}=\\varepsilon _{S}.", "0d437439f04c80a6c9fd2019664de555": "{\\tilde {A}}={h \\over {8\\pi ^{2}cI_{C}}}", "0d43981b3f452ea51c95a8906d662d7b": "\\alpha _{0}\\leq \\alpha _{1}", "0d43dd2ee0e2ae30c5b0d08a89df4a0d": "f^{(n)}(x)=a^{2^{n}-1}(x-x_{0})^{2^{n}}+x_{0}\\,\\!", "0d445ec375ae7f77c87bdc321004baa3": "00", "0d547bafcbe7b374ba8c3413a8b12d7e": "f(z)={\\frac {1}{\\sin z}}.", "0d5486f654b3c2ec9836b6aaaf9453ca": "q={\\frac {\\alpha }{\\beta }}", "0d5512b584084204e6230c673b27345c": "l_{i}\\left(A_{ij}{\\frac {\\partial u_{j}}{\\partial t}}+a_{ij}{\\frac {\\partial u_{j}}{\\partial x}}\\right)+l_{j}b_{j}=0", "0d5513ddae1a1b8e60cacbb09d4fb18a": "\\langle \\ ,\\ \\rangle \\!\\,", "0d559c91acaa09c111d864b27ded63c6": "\\sigma =1-{\\frac {1.98}{z}}", "0d55b3badf33bb084c8e840debf460a1": "L_{n}(y)\\equiv \\left[\\,D^{n}+A_{1}(t)D^{n-1}+\\cdots +A_{n-1}(t)D+A_{n}(t)\\right]y", "0d55d80d215a530ba36225030f843d3e": "\\operatorname {Ti} _{2}(\\tan \\theta )=\\theta \\log {\\tan \\theta }+{\\frac {1}{2}}\\operatorname {Cl} _{2}(2\\theta )+{\\frac {1}{2}}\\operatorname {Cl} _{2}(\\pi -2\\theta )\\,.\\,\\Box ", "0d55df8967de2ccce1d9838c4d87389e": "(a_{i,j})_{i,j\\in \\mathbb {N} }", "0d57694264eb7f5424f7feff938d0c87": "\\mathrm {OPT} -c", "0d58cedbd2b300987098452ecd4fa49c": "(\\partial \\langle N\\rangle /\\partial V)_{\\mu ,T}=N/V", "0d58da4f428867347c6d83af51455530": "b^{2}(-P^{2},m_{1}^{2},m_{2}^{2})=\\varepsilon _{1}^{2}-m_{1}^{2}=\\varepsilon _{2}^{2}-m_{2}^{2}\\ =-{\\frac {1}{4P^{2}}}(P^{4}+2P^{2}(m_{1}^{2}+m_{2}^{2})+(m_{1}^{2}-m_{2}^{2})^{2})\\,.", "0d591c1b9d3cec28690e2e3a6fbce794": "Z_{L}=V/I=j\\omega L\\,", "0d59ab2b1aa5dfc093b5c68bc4f60eff": "p+q-1<3{\\sqrt {N}}", "0d59b4710476ea5bc11b2e04498d8e9c": "Z=\\int e^{-S}D\\phi =e^{-HT}=e^{-\\rho V}", "0d59e1ac2fa6f3ba813b32b95c8b0fe6": "Z_{AC}=Z_{ref}+c{dC_{l} \\over dC_{y}}+c{dC_{m} \\over dC_{x}}", "0d5a2e64d007b197c34c58e151fb7aec": "\\mathrm {Ec} ={\\frac {V^{2}}{c_{p}\\Delta T}}={\\frac {\\mbox{Kinetic Energy}}{\\mbox{Enthalpy}}}", "0d5a8ad7bb045db82195a3f5e99ea7a5": "pI={{pKa}+{pKb} \\over 2}", "0d5a9c236678da2c6adcaaf02997bfcc": "e_{n}={\\frac {1}{\\ln {2}}}\\times {\\frac {s-1}{2n}}", "0d5ac1b33e4733486bd3025bbbae4587": "(1+\\lambda _{2})", "0d5adcb163927d4b17d3826018322070": "R_{1}={\\tfrac {1}{2}}(\\sigma _{2}-\\sigma _{3})", "0d5ae2347ec769236abb2bb54822f72a": "M\\times \\{0\\}\\cup \\partial M\\times [0,1]", "0d5b015f7055943c22d58f1610233b1e": "L_{L}", "0d5b3fd549d306dcd5d8f29622e7c419": "{\\begin{matrix}{4 \\choose 1}{3 \\choose 1}^{2}\\end{matrix}}", "0d5b43518d23bc0bf26e946517649a3c": "p\\land \\neg q\\rightarrow s.", "0d5b509e6b353c7cc601dbb637434e13": "T_{\\mu \\nu }\\,", "0d5b7a43c4bb1dd3dcdb03bd6ab47db8": "{\\kappa }_{I}=C\\cdot d^{2}", "0d5b7f3aa14fbd23e85918400bcff9dc": "{\\dot {Q}}={\\frac {T_{1}-T_{2}}{\\left({\\frac {L}{kA}}\\right)}}", "0d5c19f4325680412cb9b347bdbecdcd": "FV(A_{1})\\not \\subset \\{p,q,m\\}", "0d5c5da5ef7534b852fc07d37d1b2aa9": "{\\begin{array}{ccc}{\\frac {0.693}{r-r^{2}/2}}&=&{\\frac {69.3}{R-R^{2}/200}}\\\\&&\\\\&=&{\\frac {69.3}{R}}{\\frac {1}{1-R/200}}\\\\&&\\\\&\\approx &{\\frac {69.3(1+R/200)}{R}}\\\\&&\\\\&=&{\\frac {69.3}{R}}+{\\frac {69.3}{200}}\\\\&&\\\\&=&{\\frac {69.3}{R}}+0.34\\end{array}}", "0d5cda0df025be1855baffdca92a4a10": "g(6)=73", "0d5d6e921dcfa41128697ffa75f360d0": "{\\frac {\\partial }{\\partial z_{i}}}\\left(f\\cdot g\\right)={\\frac {\\partial f}{\\partial z_{i}}}\\cdot g+f\\cdot {\\frac {\\partial g}{\\partial z_{i}}},\\quad {\\frac {\\partial }{\\partial {\\bar {z}}_{i}}}\\left(f\\cdot g\\right)={\\frac {\\partial f}{\\partial {\\bar {z}}_{i}}}\\cdot g+f\\cdot {\\frac {\\partial g}{\\partial {\\bar {z}}_{i}}}", "0d5da1fefa03f21983ee7b5a7fb61f7b": "|\\psi (t)\\rangle =c_{a}exp(-i\\omega _{a}t)|a\\rangle +c_{b}exp(-i\\omega _{b}t)|b\\rangle +c_{c}exp(-i\\omega _{c}t)|c\\rangle ", "0d5df42c8e9316a4178e7862647973c0": "L=\\{X+\\xi \\in (\\mathbf {T} \\oplus \\mathbf {T} ^{*})\\otimes \\mathbb {C} \\ :\\ {\\mathcal {J}}(X+\\xi )={\\sqrt {-1}}(X+\\xi )\\}", "0d5df489e2f1a84135cd1b87e837f796": "\\alpha ={\\frac {[X]}{K_{d}^{R}}}", "0d5e6a0fb08469455a5cc24e05d38bf5": "t_{\\textrm {lock}}\\approx {\\frac {wa^{6}IQ}{3Gm_{p}^{2}k_{2}R^{5}}}", "0d5e80f0ab15deab09d277f5298b5c2d": "(Y\\cap Z)\\cup \\{x\\}\\subseteq X", "0d5ea8d130e95ceaa130e27156d59746": "(\\alpha -1)E_{0}/\\hbar c", "0d5ee01bc7d547ebe25fd08f615ebbeb": "U_{\\theta }", "0d5f09252973a93666c74fa009a36445": "S+I\\xrightarrow {\\alpha } 2I", "0d5f1644d03c7307a32fd1fca52f200b": "Z\\sim \\mathrm {Normal} (\\mu ,\\sigma ^{2})\\implies \\mathbb {P} (Z\\in A)=\\gamma _{\\mu ,\\sigma ^{2}}^{n}(A).", "0d5fa3f335333b23d4aaf795d1336587": "X_{1}", "0d5fad88ed0b9085be3722b4f1e9a95f": "\\beta _{m}>kn_{1}", "0d5fcac139ae50e313382ffb3afae15f": "h=A*a_{i}", "0d5fff6174419b8a0e3c044e7e01e991": "\\lambda _{p}(x)={\\frac {1}{1+x}}.", "0d6004e23db3d54429858eda9a46c71c": "J(x,t;X_{0},X_{1})=\\max(\\|x\\|_{X_{0}},t\\|x\\|_{X_{1}}).", "0d603094ff1076c7047ce8c93cdd6638": "\\scriptstyle \\Omega ((n/\\log n)^{1/3})", "0d60d123c0dc96377db47739457728ad": "m_{n}=\\int _{0}^{1}x^{n}\\,d\\mu (x)\\,", "0d61138809433c4493e1b1c6c16b7fcc": "={\\frac {Rise}{Run}}*100\\%", "0d61152f9ff3c169afdbeb34ca366412": "c:=\\lfloor -\\log _{10}x\\rfloor ", "0d612e27b8e64dd5d3aa4b4d5046113b": "\\mathbf {e} _{123}", "0d617cf30a4f2c270fa6b0958c1e2699": "\\{T^{-1}w_{j}\\}", "0d619e148f335c2d2fe38adeb5b9ff9e": "\\operatorname {ad} _{x}(y)=[x,y]\\ ", "0d61f8370cad1d412f80b84d143e1257": "C", "0d622acc0d5a62ee206f37f487dcd945": "f(x)={\\frac {x^{\\alpha -1}(1-x)^{\\beta -1}}{B(\\alpha ,\\beta )}}", "0d6283de008c2ce94def5fafb1ca337a": "t>{\\sqrt {\\frac {2(n+1)d^{2}}{d}}}-{\\frac {d}{2}}-1", "0d62eef61909a38edc064cc0294a6d2f": "H(u)[n]=\\scriptstyle {DTFT}^{-1}\\displaystyle \\{U(\\omega )\\cdot \\sigma _{H}(\\omega )\\}", "0d62f082a0c9aec576d53956a9d4bccc": "{\\dot {m}}=C\\;A_{2}\\;{\\sqrt {2\\;\\rho _{1}\\;{\\bigg (}{\\frac {k}{k-1}}{\\bigg )}{\\bigg [}{\\frac {(P_{2}/P_{1})^{2/k}-(P_{2}/P_{1})^{(k+1)/k}}{(P_{1}-P_{2})/P_{1}}}{\\bigg ]}(P_{1}-P_{2})}}", "0d62f184812c807e551b7f617a4f23b3": "G\\backslash X", "0d6378c66cc570ece9b6f9dc50044d57": "{\\mathfrak {se}}(d)", "0d638921e13e353de35287c0a2745e94": "e^{j2\\pi f_{0}t}x(t)\\rightarrow S_{x}(t,f-f_{0})", "0d6395a2d0abcc5cc6aa489130bd95e1": "{\\frac {\\partial }{\\partial \\tau }}q^{i}(\\xi ,\\tau )=\\{\\zeta ^{i},H(\\zeta )\\}|_{\\zeta =\\star q(\\xi ,\\tau )}", "0d63e93bad860e148833242986589815": "U\\subseteq V", "0d63f2112e7cc4c2255f14e46301a253": "W^{(\\beta )}(t)={\\tilde {W}}(t)+\\beta t", "0d64470b28db4216d30496c046fb0e98": "\\mathbf {p} \\psi (p)=p\\psi (p)\\,,", "0d644badd96f4ef5a1681f3d94a531c6": "\\ Z_{\\mathrm {in} }={\\frac {R^{2}}{Z_{\\mathrm {load} }}}", "0d6488b7edb024f2ed321a3300623550": "{\\frac {1+3x}{(1-x^{2})(1-4x)}}", "0d648f393e6994136ec46e0c2baf9a9a": "M_{m}={\\frac {D}{D_{ep}}}={\\frac {254}{7}}\\approx 36", "0d649f87214df1393d310af177218e94": "f\\colon X\\rightarrow S,\\,", "0d64bccf62bc5c087ff4b2017f9c98dd": "A=N+Z", "0d64bd52634bdf799ee46236a0beaf29": "U>U*", "0d64fd7602820f08b4e8d7908846253c": "\\epsilon ,\\delta >0", "0d65347d215124be539bde62d863dfbe": "\\mathbb {DICKENS} ", "0d654f6b8093c0cf7124cfe990f7e2f4": "(p,q,t)\\cdot (p',q',t')=\\left(p+p',q+q',t+t'+{\\frac {1}{2}}(pq'-p'q)\\right).", "0d6572662db30822bbed759906067c5b": "\\gamma \\lambda (L)", "0d6578e80287ec2dda7958e9436486e9": "PR(A)=PR(B)+PR(C)+PR(D).\\,", "0d65a596011d0a7f904751aaccd84258": "I=S{\\sqrt {t}}\\ +A_{1}t", "0d65abb818f0229c460007c7df628c47": "L(\\theta ,a)=L({\\bar {g}}(\\theta ),a^{*})", "0d65abbe010477c25bc6300b8a81ef91": "x',y'", "0d65e90e51759966dd4bfd0255008e53": "p(x_{i})", "0d661ed63a08ce573e30035b0f5f3c8a": "L(\\chi _{2},s)=\\beta (s).\\,", "0d66747437616002e0f421e3d3075642": "g'({\\tilde {\\theta }})\\,\\xrightarrow {P} \\,g'(\\theta ),", "0d66ba5ff5ba9c4d3a15a0d24e9d6da8": "\\mu _{n}(X+Y)=\\mu _{n}(X)+\\mu _{n}(Y){\\text{ provided }}1\\leq n\\leq 3.\\,", "0d66e857aa9252d6a8b315535559e204": "j=1,\\dots ,m", "0d66ee15f1123cdb5ef6f0e7d17fdfd4": "{n \\choose k}p^{k}(1-p)^{n-k}", "0d66f2288c0073410c0cb98ef61c400e": "\\det :M_{n}(\\mathbb {K} )\\rightarrow \\mathbb {K} ", "0d675882c51ec3a94a97eef709cd620c": "{\\binom {C}{A}}", "0d6761e5304f1c6981198514c5e02b3e": "L_{\\rm {star}}", "0d67cf596e4689d0f119be28c1c8daab": "\\prod _{\\eta \\leq p\\leq \\xi }\\left(1-{\\frac {w(p)}{p}}\\right)^{-1}<\\left({\\frac {\\ln \\xi }{\\ln \\eta }}\\right)^{\\kappa }\\left(1+{\\frac {C}{\\ln \\eta }}\\right).", "0d67e91be40d4d654130b8b74855f0d3": "\\Phi =e^{B+i\\omega }\\Omega ", "0d680ccdea0752dc4b99ccd27b37edf2": "(15)\\quad Z^{c}\\nabla _{c}\\sigma _{ab}=-{\\frac {2}{3}}\\theta \\sigma _{ab}-\\sigma _{ac}\\sigma _{\\;b}^{c}-\\omega _{ac}\\omega _{\\;b}^{c}+{\\frac {1}{3}}h_{ab}\\,(\\sigma _{cd}\\sigma ^{cd}-\\omega _{cd}\\omega ^{cd})+C_{cbad}Z^{c}Z^{d}+{\\frac {1}{2}}{\\tilde {R}}_{ab}\\,.", "0d686788eeefdcd1f956dcab761da7b4": "\\langle \\phi |A|\\phi \\rangle ", "0d6880c00d2e3c4f03e192217b0dbe10": "y(x)=A\\sin(x)+B\\cos(x).\\,", "0d6900f7ca5e6af1058b9c74f9040852": "V_{D}\\approx 0.18V", "0d69048cce41d29392b99b246139214e": "{\\tfrac {347897}{7558272}}", "0d69073fca9788ab650eea1ebbf8d4af": "y=Cx,", "0d693922b7d69a86092944da8979294c": "R_{xx}=R_{yy}=-\\left(p_{xx}+p_{yy}\\right).", "0d6940f87d5c0140cb7db39082082bfb": "|x-y|<\\delta \\,", "0d694f6b87eedd8efdc17b4ca0fea10f": "Y[x,y]=y+x'{\\frac {x'^{2}+y'^{2}}{y''x'-x''y'}}", "0d69682f10dc5e27b194fc7ce0420293": "b", "0d6bff90cbf5767cb046830946cbfd62": "f\\in L(G-D)", "0d6c999ff34fa251ea9b190207e9cf58": "\\phi _{\\mathbf {R} ,i}", "0d6d5a256719e4e1b64b484e5a313b0a": "\\scriptstyle {{\\sqrt {2}}/2}", "0d6d7560650c020295aec4b16601399a": "f(x_{2})=0", "0d6da45ef0794e291028c7801c723fa3": "\\sin(x)={\\frac {2t}{1+t^{2}}}{\\text{ and }}\\cos(x)={\\frac {1-t^{2}}{1+t^{2}}}{\\text{ and }}e^{ix}={\\frac {1+it}{1-it}}", "0d6dcb926e0374df3cfa539818cca85b": "\\operatorname {ev} _{p}:C(X)\\to \\mathbf {R} ", "0d6e1af3345dd778a3b40359b4e06579": "\\ln \\left(-\\ln \\left(1-F)\\right)\\right)", "0d6e3de138f515b1b2012f563c73f741": "\\nabla \\cdot \\mathbf {g} =4\\pi G\\rho \\,\\!", "0d6e9a066cd61dda70aad1dc7d8300fd": "{\\bar {H}}_{n}^{(c)}=-1+{\\frac {1}{2^{c}}}-{\\frac {1}{3^{c}}}+\\cdots ", "0d6e9a28d2ce026d1366909a0e0d103f": "{\\begin{aligned}x_{n+1}&=x_{n}-{\\frac {f(x_{n})}{f'(x_{n})}}\\\\&=x_{n}-{\\frac {a-{\\frac {1}{x_{n}}}}{\\frac {1}{x_{n}^{2}}}}\\\\&=x_{n}\\,(2-ax_{n})\\end{aligned}}", "0d6e9a4fb698f6ee4d436b455b22af40": "d_{12}-d_{13}", "0d6ee871ca9996cf47c5891afcb05cce": "\\scriptstyle {\\dot {e}}", "0d6f46c188dfe8d51d42dd1409bc0fae": "H=(V(G),E(G))", "0d6f80b2d562827628b98a29989f5a2e": "16=1+3+5+7\\;\\;\\Longrightarrow \\;\\;n=4\\;{\\hbox{contains}}\\;s\\oplus p\\oplus d\\oplus f.", "0d6fd7e4ae56eaba3eafa60955bc5e24": "r=\\min ", "0d701637da3c44a8d61c8121a65e5736": "N_{A}", "0d70a494ea5b172de048f91098b76e9b": "B=(I-\\rho F)^{-1}E\\;", "0d70a66a036eb5d07505356a7ee4c7f4": "E_{t}c_{t+1}=c_{t}", "0d70d6897f2205642d7847aeb43dca66": "{m^{+}}_{1}=[15.927,0.497]", "0d70f38109b310382048347d736b8a3b": "{\\mathcal {P}}=\\{{\\text{all distributions}}\\}", "0d70f9806aaf7d48b1823eba6d92ed96": "C_{N}^{2}(h)", "0d711edb02bbc7e66e7f61722cc00e4b": "\\ c=c_{0}(1-z_{d})+c_{1}z_{d}.", "0d712dcf00bf69151f4fca8326ec1efa": "L_{e}\\propto \\langle I\\rangle _{e}\\langle I\\rangle _{e}^{-1.66}", "0d7177e2bb8b7ad65c67b20d79923cbc": "T^{2}=s(s-a)(s-b)(s-c)", "0d717ba8ab3b783601f1cf4c17aefde3": "C(\\langle 1,\\omega \\rangle ),\\ C(\\langle 1,\\omega ^{\\omega }\\rangle ),\\ C(\\langle 1,\\omega ^{\\omega ^{2}}\\rangle ),\\ C(\\langle 1,\\omega ^{\\omega ^{3}}\\rangle ),\\ldots ,C(\\langle 1,\\omega ^{\\omega ^{\\omega }}\\rangle ),\\ldots ", "0d71d145ab8905cced508a3fade4dde2": "\\xi ={\\frac {x}{\\lambda _{\\mathrm {D} }}}", "0d71e2ae8a51fc92fec3e976c84d8d4d": "I(X,Y)=\\sum _{x\\in X}{\\sum _{y\\in Y}{p(x,y)log_{2}\\left({\\frac {p(x,y)}{p_{1}(x)p_{2}(y)}}\\right)}}", "0d7262b93d906c0a0b3a1f73d66d566e": "\\Delta _{uv}=\\pm 0.05", "0d726b1a94abbfa016ae08054fd2297c": "0\\rightarrow L\\rightarrow M\\rightarrow N\\rightarrow 0", "0d728c5911111cfbbf9cc5a8ba354140": "{\\mathit {m}}", "0d72ba127726c94c6e544ca5d8add32b": "{1 \\over 2}K\\rho c^{4}=4\\pi G\\rho \\,", "0d73026876bc64c53169dd10eaf06a4f": "2\\zeta \\ {\\stackrel {\\mathrm {def} }{=}}\\ {\\frac {b}{\\sqrt {ac}}}.", "0d7337197d8106c809e4b387a8adc055": "{\\begin{aligned}\\mathbf {u} \\times \\mathbf {v} =&(u_{1}\\mathbf {i} +u_{2}\\mathbf {j} +u_{3}\\mathbf {k} )\\times (v_{1}\\mathbf {i} +v_{2}\\mathbf {j} +v_{3}\\mathbf {k} )\\\\=&u_{1}v_{1}(\\mathbf {i} \\times \\mathbf {i} )+u_{1}v_{2}(\\mathbf {i} \\times \\mathbf {j} )+u_{1}v_{3}(\\mathbf {i} \\times \\mathbf {k} )\\\\&+u_{2}v_{1}(\\mathbf {j} \\times \\mathbf {i} )+u_{2}v_{2}(\\mathbf {j} \\times \\mathbf {j} )+u_{2}v_{3}(\\mathbf {j} \\times \\mathbf {k} )\\\\&+u_{3}v_{1}(\\mathbf {k} \\times \\mathbf {i} )+u_{3}v_{2}(\\mathbf {k} \\times \\mathbf {j} )+u_{3}v_{3}(\\mathbf {k} \\times \\mathbf {k} )\\\\\\end{aligned}}", "0d733b8d7b7c6f032507207dccaec2b8": "C_{2}\\,\\!", "0d738eb8fef2f1ceaeb9f87d11e69fc2": "\\omega _{e}=\\omega _{r}-\\omega _{c}", "0d7397e218aaac0ca67d4013be769177": "n_{solute}", "0d73b9977315c46391da27462b3f8f8c": "{\\tilde {\\chi }}(\\omega )", "0d73fcd2920035cb8250286d82e7c5c8": "\\dim Y_{r}=\\dim Z_{r}", "0d7416c56cf275658c3ff63a1f17c03c": "A\\subset X\\times Y", "0d7442b0b30a469fe600882e9dca3675": "C\\subseteq A\\cap B\\,\\!", "0d745c26ae530b582affb288201aca65": "\\mu (B(x,2r))\\leq C\\mu (B(x,r))\\,", "0d74b29d914f387cef90b8b18c01db4e": "\\Theta =\\sum _{i}e_{i}\\Theta ^{i}(\\mathbf {e} ).", "0d74ee22e926e3475f2f694b36898c51": "\\sigma _{x'}=-10{\\textrm {MPa}}", "0d750e13123b81d84808b6866686767d": "V_{1}=", "0d755e78dfe816f5a522133c47e6ec9d": "E=\\alpha -x\\beta ", "0d757efa484ce21608162f2e81d28415": "\\Delta u^{j}=0\\,", "0d75959d44373b17d51c95b0ba9368d5": "\\{1,2,..,n\\}", "0d75fe0afc9367ef2832fdb4fa534a71": "x(t)=e^{-\\zeta \\omega _{0}t}(A\\cos \\,(\\omega _{\\mathrm {d} }\\,t)+B\\sin \\,(\\omega _{\\mathrm {d} }\\,t))\\,", "0d7600fc9e8e0eca288e77a5d3367566": "{\\begin{aligned}\\Psi _{1}(t)&={\\sqrt {2}}\\pi ^{-1/4}te^{(-t^{2}/2)}\\\\\\Psi _{2}(t)&={\\frac {2}{3}}{\\sqrt {3}}\\pi ^{-1/4}(1-t^{2})e^{(-t^{2}/2)}\\\\\\Psi _{3}(t)&={\\frac {2}{15}}{\\sqrt {30}}\\pi ^{-1/4}(t^{3}-3t)e^{(-t^{2}/2)}\\end{aligned}}", "0d7610ee830324f4a14318811be5dd14": "\\{\\{a,b\\},\\{a\\}\\}=(a,b)\\neq (b,a)", "0d761d5130edcc610e160ac935bd9c38": "\\alpha =0\\,", "0d764f666a2a99db1a46a7457c315d01": "x\\to -\\infty ", "0d766c9b7c2f9fe10bc8f34c5556cf8a": "(a\\cdot a_{i})^{(n-1)/2}\\not \\equiv \\left({\\frac {a\\cdot a_{i}}{n}}\\right){\\pmod {n}}.", "0d76b01c4468af710e193c26939b8f87": "n^{O(1)}", "0d76ce2a875072de1dd8b901ad2499e7": "\\Delta \\!\\,", "0d76d7f33686631db7c764ea5b0f425c": "{\\bigg [}\\ F{\\frac {\\quad }{\\quad }}H\\ {}^{-}\\!F\\quad \\longleftrightarrow \\quad F^{-}\\ {}\\!H{\\frac {\\quad }{\\quad }}F\\ {\\bigg ]}", "0d772cf39de3a6a5aa23d5f89ec9f7c9": "\\partial _{\\mu }\\partial ^{\\mu }A^{\\nu }=0", "0d77387cf99d9bddbd3075eae2479286": "=\\left(n\\cos \\alpha _{1},n\\cos \\alpha _{2},n\\cos \\alpha _{3}\\right)=n\\mathbf {\\hat {e}} ", "0d775cdaa064177df3f8d024c5de76d7": "\\nu _{i}~", "0d7797d3bdb64e5c98546b183e815141": "g_{jk}", "0d781ab61c6ca4cb383356a2692a0956": "a(t)={\\frac {F(t)}{m}}", "0d7857e88614b1893fe9bc955830c0ee": "l(0)=N/2K\\gg 1", "0d78924a9147e4baf0753ba46c3ad2c3": "f^{(n)}(a)={\\frac {n!}{2\\pi i}}\\oint _{\\gamma }{\\frac {f(z)}{(z-a)^{n+1}}}\\,dz.", "0d78d69b27b43a0a7d4ccab265e59736": "C={\\frac {Ng^{2}S(S+1)\\mu _{B}^{2}}{3k}}", "0d798f87d5b8422df5a1f3624eb3b9c6": "\\left[J_{a},K_{b}\\right]=i\\varepsilon _{abc}K_{c}", "0d79d11581d802295d7629c3e4203a84": "r={\\frac {a_{2}}{\\cos(\\theta -\\alpha _{2})}}", "0d79f1095ebff93e416c9f3347cf71b4": "(p_{x},p_{y})", "0d7a060b526554544b47c643330b80e6": "a(t)=\\sum _{i=1}^{N}K(i)(I(i,t-1))^{2}", "0d7a08596bda9d186ec59382ec8697f2": "\\scriptstyle {\\bar {\\eta }}", "0d7a08acc06406fdfb8011754580022a": "H^{\\prime }", "0d7a613622106410ab383f63f294a9ef": "|\\langle f|g\\rangle |^{2}={\\bigg (}{\\frac {\\langle f|g\\rangle +\\langle g|f\\rangle }{2}}{\\bigg )}^{2}+{\\bigg (}{\\frac {\\langle f|g\\rangle -\\langle g|f\\rangle }{2i}}{\\bigg )}^{2}", "0d7a78ab723beebe175d722b94d53190": "X(fg)=(Xf)g+f(Xg).", "0d7aa8a6bcbfa03ede53f63b3483efb9": "{\\mu }={\\mu }_{0}{\\frac {T_{0}+C}{T+C}}\\left({\\frac {T}{T_{0}}}\\right)^{3/2}.", "0d7ade4f83b26fb7a6ae6e9824e15c45": "f^{m_{k}-1}", "0d7ae37e75939904a44f4858f63b1437": "[x_{i},x_{j}]=0,\\quad [x_{i},t]=i\\lambda x_{i}", "0d7b48fb530cfc179e59ced04d320a34": "Q(p;k,\\lambda )=\\lambda {(-ln(1-p))}^{1/k}", "0d7b5a5cc84765fb28c091e7ca468560": "\\{{\\mathbf {x}}|Ax0", "0d828a89660289c7cf1e6dc5c0c9a299": "S,X_{1},X_{2},\\ldots ,X_{n}\\subseteq \\Omega ", "0d82f533ed1b5a41a1ee6ce63a53eb8e": "{\\frac {1+7x+12x^{2}-4x^{3}}{(1-x^{2})(1-16x^{2})}}", "0d836d4d1c166228e3faafc03be46f50": "O_{K}", "0d8373d159ea431f8f000e8f4d948ed8": "{\\hat {A}}", "0d8402aed8d581ac2572b976aa51bd66": "{\\frac {\\partial ^{2}u}{\\partial x_{i}\\partial x_{j}}}=-R_{i}R_{j}\\Delta u+P_{ij}(x)", "0d843f33ade67aa1d186596d6ff4c82a": "{\\vec {x}}'={\\vec {x}}+2a({\\vec {\\omega }}\\times {\\vec {x}})+2({\\vec {\\omega }}\\times ({\\vec {\\omega }}\\times {\\vec {x}})).", "0d846812ddc803b47558631d214b9714": "\\psi ,{\\bar {\\psi }}", "0d847e21f5a2049d9348a17fd7892cce": "\\Phi ", "0d849d387fc670057ed5a185f063ba97": "B^{\\mu }\\,", "0d84d24d7e9ca33cdf9b6334d3b8fd98": "\\omega _{\\alpha I}^{\\;\\;\\;\\;J}", "0d851c4baee08c08a36545df2c12303f": "\\gamma _{jk}=\\int _{-\\infty }^{\\infty }x(t){\\frac {1}{\\sqrt {2^{j}}}}\\psi \\left({\\frac {t-k2^{j}}{2^{j}}}\\right)dt", "0d8544831f20e8809b2a8dd0c1031191": "{\\tilde {C}}(\\alpha ,\\beta )", "0d85553795fe9cda12a72a5b288431af": "{\\frac {{\\text{d}}[{^{d_{h}}_{c_{h}}}P_{h}^{\\gamma _{h}}]}{{\\text{d}}t}}=\\sum _{i}u_{\\gamma _{hi}}y_{d_{hi}}{\\text{k}}_{3(i)}C_{i}\\qquad \\qquad (8c)", "0d856f7538042b8cd88ffb9e4f13f5aa": "V_{n}(R)={\\frac {R^{n}}{n}}\\textstyle B({\\frac {n-1}{2}},{\\frac {1}{2}})B({\\frac {n-2}{2}},{\\frac {1}{2}})\\cdots B({\\frac {2}{2}},{\\frac {1}{2}})\\cdot 2B({\\frac {1}{2}},{\\frac {1}{2}}).", "0d8687e0f639768283dc7fcdc66fa162": "\\rho _{SE}(0)\\,", "0d86d4127a593a07e5fcdba5055c879b": "{\\ddot {y}}+y-\\varepsilon \\left({\\frac {{\\dot {y}}^{3}}{3}}-{\\dot {y}}\\right)=0.", "0d86d76f19612258365c96121add2094": "I=\\varprojlim I_{n}", "0d87337780b0396d92db0f255a5684c5": "g(u,v)=\\exp[(\\theta _{1}-\\theta _{12})(u-1)+(\\theta _{2}-\\theta _{12})(v-1)+\\theta _{12}(uv-1)]", "0d873d3ee3bd152e069d0a006a489baa": "\\tau (n)\\equiv n\\sigma _{9}(n)\\ {\\bmod {\\ }}7^{2}{\\text{ for }}n\\equiv 3,5,6\\ {\\bmod {\\ }}7", "0d8790697dc884fde9b378eeb376e128": "I:T\\to P", "0d879b526f3f4a257d912a3b9a4f7a64": "s_{n}=\\sum _{k=0}^{n}{\\frac {x^{k}}{k!}},\\ t_{n}=\\left(1+{\\frac {x}{n}}\\right)^{n}.", "0d87f4c1968c24d8491b3940378dd1e7": "\\sigma _{x}={\\frac {x_{0}}{\\sqrt {2}}}{\\sqrt {1+\\omega _{0}^{2}t^{2}}}", "0d880c5cc27fba976bdc442c95a140e9": "\\operatorname {vol} (B)=\\prod _{i=1}^{n}(b_{i}-a_{i})\\,.", "0d883be6eccff1b524bc7e5f629238c2": "r=2{\\sqrt {\\tfrac {Z_{1}Z_{2}e^{2}L}{E_{0}}}}", "0d8877599db5915e1e60b80d202a2b2a": "|\\alpha ^{k}|\\leq q^{kd/2+1/2}", "0d89646470c473a5e74e47613c2716f5": "F=(1+E\\cdot N)^{\\lambda }.", "0d8a501a499c359df4a6bfcb3da1a508": "\\gamma \\,", "0d8a781134042c17b3787a21725e1f42": "-{\\frac {\\partial T_{o}}{\\partial x}}^{TM}-{\\frac {k_{z}}{\\omega \\mu }}{\\frac {\\partial T_{o}}{\\partial y}}^{TE}=-{\\frac {\\partial T_{\\varepsilon }}{\\partial x}}^{TM}-{\\frac {k_{z}}{\\omega \\mu }}{\\frac {\\partial T_{\\varepsilon }}{\\partial y}}^{TE}", "0d8a9a86f426f2dfb3611526ce1b4ddc": "dim\\;f_{i}(X_{i})=k", "0d8ac044e13e73bba35fc58eec938f5e": "\\scriptstyle Z_{L}", "0d8b4abfbca5ae40fbab2837e8284d88": "\\sigma _{H}(\\omega )={\\begin{cases}i=e^{+{\\frac {i\\pi }{2}}},&{\\mbox{for }}\\omega <0\\\\0,&{\\mbox{for }}\\omega =0\\\\-i=e^{-{\\frac {i\\pi }{2}}},&{\\mbox{for }}\\omega >0\\end{cases}}", "0d8b83a7c24f34d046f5477ff841c48c": "2\\omega _{1}", "0d8bcb2482102b6c838150e3a33e583f": "\\mathbb {P} .", "0d8bd35ed636bb503b7a7101de12f349": "\\sigma _{N+1}=\\sigma _{1}", "0d8bd3be4aad812cd2d4d7f5c982cd3d": "\\sum _{k=1}^{\\infty }u_{k}", "0d8bf374888f310d4b9577d46d3be73e": "xR+yR\\equiv zR{\\pmod {N}}", "0d8bfc86536b04eae0696bd8e74ee1fc": "X\\subseteq \\Omega ", "0d8c70cd344458e4dfc23a334f968371": "C_{12}", "0d8c8b3fc79b5a7feed7a8d247a7152e": "Squarenumbersendon7", "0d8c960b49353f0cad057a8a3c53d373": "V_{t}\\approx pq\\left(1-\\exp \\left\\{-{\\frac {t}{2N_{e}}}\\right\\}\\right).", "0d8cca4b3596f47e1064473dc6f6bce1": "E(x^{2})=2\\sigma ^{2},E(\\ln(x))={\\frac {\\ln(2\\sigma ^{2})-\\gamma _{E}}{2}}\\,", "0d8d3750e579ba4f5f2f90e828b5cf02": "Z=\\int d\\mu (\\sigma )e^{-H(\\sigma )}", "0d8d7338a1a1eb83be267ee8b9071951": "\\Delta P=\\rho Vv\\qquad {\\text{(ignoring }}{\\frac {\\rho }{2}}v^{2}),\\,", "0d8d8d31eebf4c6049d640e5d7f03530": "=_{E}", "0d8e349f39c7fe8106368f0566c8c6bb": "\\eta ^{2}(2n+1)\\ll 1,", "0d8e3642269dde4efec6d45fad7005e5": "{\\begin{array}{c|cc}0\\\\1&1\\\\\\hline &1/2&1/2\\\\&1&0\\end{array}}", "0d8e893d539d243b9021c3756fee06f0": "\\langle i_{1}i_{2}\\rangle =\\lim _{T\\rightarrow \\infty }{\\frac {E^{4}}{T}}\\int _{0}^{T}\\sin ^{2}(\\omega t)(\\sin(\\omega t)\\cos(\\phi )+\\sin(\\phi )\\cos(\\omega t))^{2}\\,dt", "0d8eb21b869efd85672f2a6910284b9a": "E(R_{i})=R_{f}+\\beta _{ip}[E(R_{p})-R_{f}]\\,", "0d8ec496a199e12042a3566cef8e68ec": "\\pi (a_{i},a_{j})+\\pi (a_{j},a_{i})\\leq 1", "0d8ec776616a87f5f6843c8a7c72755e": "[x_{2}]=[x_{2}]\\ \\cap \\ \\ [a]-[x_{1}]", "0d8edf7ff8f86d0f702dd560ed4e0422": "{\\dot {C}}", "0d8edfe0089b4be6b1844bafd54e3ec9": "\\operatorname {GrpRng} \\colon \\mathbf {\\operatorname {Grp} } \\to R\\mathbf {\\operatorname {-Alg} } ", "0d8ef5afb6abcfc1be71d1d0d9dd9350": "N=n_{+}+n_{-}=2n_{-}+1\\,", "0d8f078bb545686c5037c64461fa17c3": "{\\vec {\\gamma }}=(\\gamma _{1},\\ldots ,\\gamma _{N})", "0d8f12173d7de1e804cb17084ab236de": "{\\text{EVaR}}_{1-\\alpha }(X)={\\text{EVaR}}_{1-\\alpha }(Y)", "0d8f4b55462e0c90c52e71210df9e662": "{\\frac {p(\\xi )}{q(\\xi )}}=\\left(t\\to {\\frac {p(t)}{\\xi -t}}\\right)[x_{1},\\dots ,x_{n}].", "0d8f6af4ec47b5b34b45fb130bda8473": "\\lim _{r\\to +\\infty }z(r)={\\frac {1}{\\sqrt {1-{\\frac {r_{s}}{R^{*}}}}}}-1", "0d902d1c547b5626d16f7ded98fdec10": "banana\\$", "0d9036db3e0411fac61e9a0b5e3fbad1": "x^{8}+u^{3}\\,", "0d90c489fa71b6a650bff5c4a564cf48": "\\lambda ={\\frac {2}{n}}\\mathrm {div} X", "0d91608af95f1972da1faa6ebf4f0670": "\\int \\mathbb {I} [\\mathbb {E} (r,a,x,\\theta )=\\max _{a'}\\mathbb {E} (r,a',x,\\theta )]P(\\theta |{\\mathcal {D}})\\,d\\theta ,", "0d917891a29af05b2e25833fd4e8868d": "{GOR}", "0d91790a096807684aa124eb24935e0e": "r\\geq 2", "0d91b663ee1745060d67671bce7efba4": "{\\bar {n}}\\in S\\iff (\\exists {\\bar {m}}\\in \\mathbb {N} ^{k})(P({\\bar {n}},{\\bar {m}})=0)", "0d91b6a83546f93c2a3e82aa5b318a2f": "N\\to \\infty ,\\quad {\\text{Problem }}B^{\\lambda N}\\to {\\text{Problem }}B^{\\lambda }", "0d91c55b235fd2e0c72da0f8537f7e6a": "D=K^{2}\\sum C(n)", "0d921dabab09e7874da27eb1e490b62d": "D=D_{ij}", "0d92c778b34d0ee3a946196b10ab8647": "\\nu =\\log _{2}N", "0d92f68ad626b36b2ddcdc3a2d7adb67": "\\omega ={\\frac {\\rho gQS}{b}}", "0d931ee352169206ea51ab112d6037af": "{\\sqrt {(1+v)(1-u) \\over (1-v)(1+u)}}={\\sqrt {1+(v-u)/(1-vu) \\over 1-(v-u)/(1-vu)}}\\,,", "0d932117fe50aa894b84cb4606584588": "-{\\frac {2}{r}}\\rightarrow {\\frac {1}{2}}kr^{2}.", "0d9390cf6a960f7966b005c57cbb4a67": "L(p,q)", "0d93a1f2b62aad5e244f8572a8a1e15e": "{\\mathfrak {F}}\\{f\\cdot g\\}={\\mathfrak {F}}\\{f\\}*{\\mathfrak {F}}\\{g\\}", "0d93daab7d03c6f24c72c9f6c75b32a0": "R(x,\\xi )=e^{-{\\langle x,\\xi \\rangle }}R(e^{\\langle x,\\xi \\rangle }).", "0d940266b39d150a39730577c04989e9": "h_{v}", "0d940bfebf3a61b5532643e2ae2b495d": "m:=\\rho A", "0d944e10715625c7b47eacc48f058fec": "\\langle z^{n}\\rangle =\\int _{\\Gamma }e^{in\\theta }\\,f_{WE}(\\theta ;\\lambda )\\,d\\theta ={\\frac {1}{1-in/\\lambda }},", "0d951cbc3b600f618691ecc971c377cd": "k_{1},k_{-1}", "0d9534c0577572c7aff812faf75fb86b": "A\\subseteq {\\widehat {\\mathbb {R} }}", "0d958bd7b6f2a10ef8f5f555facfbb9a": "y_{n+1}^{(i)}\\equiv a_{i}(y_{n}^{(i)})^{p_{i}-2}+b_{i}{\\pmod {p_{i}}}\\;{\\text{, }}n\\geqslant 0\\;{\\text{where}}\\;y_{0}\\equiv m_{i}(y_{0}^{(i)}){\\pmod {p_{i}}}\\;{\\text{is assumed. }}", "0d95c8ca14996a01b0c3b9e22524cc06": "{3 \\over 12}", "0d9624f2c813b89586817e034d242a22": "T(n,k)=a(\\gcd(n,k))", "0d9626a634d22ccdd7bcda4c3d1a8b21": "\\tan 2\\theta _{n}=+2{\\frac {\\sqrt {ac}}{b}},", "0d96559fc4b37fc17b7c7fdac02404ba": "d=2e-1", "0d96bdce1d0ec71638f06b38d425b9e7": "E_{eq}=E_{1}+E_{2}", "0d979240d6a15d146b37bded6712df21": "U(\\{c_{t}\\}_{t=0}^{\\infty })=\\sum _{t=0}^{\\infty }{f(t)u(c_{t})}", "0d9799861c2ecca1b8430be02ca949f5": "E=E^{\\ominus }+{\\frac {RT}{nF}}\\ln {\\frac {{\\{S\\}}^{\\sigma }{\\{T\\}}^{\\tau }...}{{\\{A\\}}^{\\alpha }{\\{B\\}}^{\\beta }...}}", "0d97dcfff837a7366366cadd81930638": "-5^{\\frac {3}{4}}\\exp \\left({\\tfrac {\\log(2)+\\pi i}{4}}\\right)=-\\exp({\\tfrac {\\pi i}{4}})5^{\\frac {3}{4}}2^{\\frac {1}{4}}.", "0d980f2d8d94f8ba2c3267fc8bd120aa": "=z^{2}-i^{2}\\cdot 5", "0d9860954c3dc996600bc4fa045b6ccc": "\\int {\\frac {x\\,dx}{r}}=r", "0d98cce60df8b24994abae1c5706ff9b": "S_{r}(m)", "0d9935671c83cac6a1430aa071841a8a": "E_{m}", "0d99ebbe2fb15f25b413cf711e284518": "{1 \\over 10^{5}}", "0d9a35d1367f1de3730ff26dfdaa6790": "\\,\\zeta ", "0d9a88ecae09acdc8a8d8d0a461052d8": "F+110={\\begin{bmatrix}1&1&0\\\\0&0&0\\\\0&1&1\\end{bmatrix}}=F^{(1,2)_{R}(1,2)_{C}}=F^{(1,2,3)_{R}(1,2,3)_{C}}.", "0d9ac7439d83b6630ffd611888cba21e": "~z=L^{*}", "0d9acdb3906955c73c796ee195ce9b71": "\\eta ^{\\rho \\sigma }{\\Lambda ^{\\mu }}_{\\rho }{\\Lambda ^{\\nu }}_{\\sigma }=\\eta ^{\\mu \\nu }~,", "0d9afbdd7a18b8bec173de537a3ae6c6": "{\\begin{aligned}a_{0}&=6-4{\\sqrt {2}}\\\\y_{0}&={\\sqrt {2}}-1\\end{aligned}}", "0d9b495148c73d958a19ee689fcd9657": "v=au+x", "0d9b538e2dec4884ac972fb1a729373b": "(X_{c},Y_{c})", "0d9b9cee83854ae41b3597477bb6d472": "{\\frac {\\pi }{4}}=\\arctan {\\frac {1}{2}}+\\arctan {\\frac {1}{3}}\\!", "0d9b9de2e406366685d40fb4b36bdd39": "PE={\\frac {N}{2}}4\\pi \\rho \\int _{0}^{\\infty }r^{2}u(r)g(r)dr", "0d9ba6c9ac7634112345957eaf08f934": "{\\tilde {I}}=I", "0d9bcf457d3598211209dcca52aff02b": "{\\text{SE}}={\\frac {1}{\\sqrt {n-3}}}.", "0d9bfbe01f32d7330c825d66adfa4afd": "\\Delta {\\bar {V}}=\\sigma {\\bar {V}}_{S}+\\tau {\\bar {V}}_{T}-\\alpha {\\bar {V}}_{A}-\\beta {\\bar {V}}_{B}", "0d9c2f943fb2edeff2d9fff3502cd455": "V(p)", "0d9c360b262efb1f99c82429864648c3": "{\\hat {\\theta }}=f(m_{1}(x_{1}),\\cdot ,m_{N}(x_{N}))", "0d9cf0523988611bc3c0fed5302b1321": "\\tau <0", "0d9d12abf76751ce8d6607a11f22093b": "X(s)=\\int _{0^{-}}^{\\infty }x(t)e^{-st}\\,dt", "0d9d1ee87c3898fb1a6a179d4f55c147": "-(1+\\log(-x^{*}))", "0d9d6440e765032275216c2fab5c5a51": "\\emptyset \\notin {\\mathcal {C}}", "0d9d7e71af402d4fd82b1276cf93cadb": "P(t)=e^{Qt}=e^{U^{-1}(\\Lambda t)U}=U^{-1}e^{\\Lambda t}\\,U\\,,", "0d9da8392ccebd5adff2be98fc763838": "f=\\Omega (g)\\,\\!", "0d9db82e77ec6716d3d058b95c371704": "e^{-V}", "0d9df1edc26e29d206e7d47df559bd86": "a\\neq 0", "0d9df69f83bf20d54eb9891850fe15c2": "{\\tbinom {0}{0}}={\\tfrac {0!}{0!0!}}=1", "0d9e036e7c1664d2dccf2aff96e018ab": "[xz,y]=[x,y]^{z}\\cdot [z,y].", "0d9e13d8b0ced0e69c66b6c622f547bb": "\\Lambda _{00}=1", "0d9e2fc952809dbff3f9c576eef26bc1": "\\mid P(x)-f(x)\\mid ", "0d9e4b03d57dd5a9c596ef5f2a258a0d": "\\left({\\frac {1}{2j}}\\sum _{r=1}^{2j}\\gamma _{r}^{\\mu }{\\hat {P}}_{\\mu }-mc\\right)\\Psi =0", "0d9eb00907c40a8c20b5762a2ec7db33": "\\beta =v/c\\,\\!", "0d9ef4b60f7b39147365039ae7b5123d": "E(\\alpha )=k(S^{max}-S(\\alpha ))", "0d9efec78b6da7f0c74dfc6da051ca3a": "\\mathrm {lim} :{\\mathcal {C}}^{\\mathcal {J}}\\to {\\mathcal {C}}", "0d9f1f87399a79ef4abb5a536581b702": "\\Delta _{\\infty }", "0d9f75f3fbec0025079eb21e40eef01e": "p(\\theta _{1},\\cdots ,\\theta _{n})={\\frac {1}{Z_{n,\\beta }}}\\prod _{1\\leq kt_{1}\\qquad \\qquad (5)", "0db4b0393eae168851671b28e5f03c5f": "b(x)", "0db5255d6802d17c55fb65e18cb1cc59": "{\\varphi ^{3}=\\varphi -\\varphi ^{2}}", "0db531324552bb9ad528b76dc082521b": "C_{A}={\\frac {C_{\\bar {v}}}{\\lambda _{\\text{b:air}}+{\\dot {V}}_{A}/{\\dot {Q}}_{c}}},", "0db55d39e76fa07a82d16668d0109203": "\\mathrm {response} =\\mathrm {MD5} {\\Big (}\\mathrm {HA1} :\\mathrm {nonce} :\\mathrm {HA2} {\\Big )}", "0db5a1b30e14d296f63ec74b883b725e": "I^{+}(S)", "0db5ffcddf7c21d712643920d0ea4b74": "f(t,n)=\\sum _{i=0}^{n}{\\binom {t}{i}}", "0db60e579b241e1b6a426329cc8315cd": "w(\\mathbf {x} ,\\mathbf {y} )=\\pi (\\mathbf {x} )Q(\\mathbf {x} ,\\mathbf {y} )\\lambda (\\mathbf {x} ,\\mathbf {y} )", "0db69cac78f8256f871a23aaf8695bf1": "\\mathrm {i} \\hbar {\\frac {\\partial }{\\partial t}}\\Delta \\langle {\\hat {N}}\\rangle =\\mathrm {T} \\left[\\Delta \\langle {\\hat {N}}\\rangle \\right]+\\mathrm {NL} \\left[\\langle {\\hat {1}}\\rangle ,\\Delta \\langle {\\hat {2}}\\rangle ,\\cdots ,\\Delta \\langle {\\hat {N}}\\rangle \\right]+\\mathrm {Hi} \\left[\\Delta \\langle {\\hat {N}}+1\\rangle \\right]\\,,", "0db6a18b8be8e566091e4aeb7e81b3b4": "1,2,3,...", "0db6bed8e27c8514a63c0f8ce62da993": "r(A^{n})\\leq r(A)^{n}", "0db6e19be8496f3f75a41c559c4fa2a6": "f:S^{G}\\to S^{G}", "0db6e7cea6e6a56ce0842aec8b1f8210": "{\\begin{bmatrix}1\\\\0\\end{bmatrix}}", "0db73c5556e237fe31c84e83c328080a": "\\phi \\in {\\mathcal {A}}", "0db7571130969b3cbc9e6814cef41d68": "Happens", "0db7729bbe6ad3641c16988b7bada910": "\\langle \\cdot ,\\cdot \\rangle \\colon \\Phi \\times \\Phi \\to \\mathbb {Z} ", "0db79af945b3192786e9d27f1fdcb1bb": "x=\\left(1+{\\frac {\\text{VAT}}{100\\%}}\\right)\\cdot EC_{\\text{rate}}\\cdot Ex_{\\text{rate}}", "0db7dc2f9e7ae017e1281632cd1bfc99": "{\\dfrac {dW(t)}{dt}}=\\sigma _{t}ln(1-P)W(t)\\,", "0db7ff929037b95c239359017351afaf": "\\displaystyle {\\mathbf {v} (t)-\\mathbf {u} =-\\lambda \\mathbf {n} (0)+t\\mathbf {t} (0)+{t^{2} \\over 2}\\kappa (0)\\mathbf {n} (0)+O(t^{3}).}", "0db8237cedf539d942605907e9075cb1": "[J_{x},J_{y}]=iJ_{z},\\quad [J_{z},J_{x}]=iJ_{y},\\quad [J_{y},J_{z}]=iJ_{x},", "0db83033b346e32af275913ef5dbbff3": "\\mathbf {x} (\\mathbf {X} ,t)={\\boldsymbol {F}}(t)\\cdot \\mathbf {X} +\\mathbf {c} (t)", "0db86e2288a2c79e909c4b9c1190217f": "I_{\\parallel }", "0db8f6db68f2cba2652164a7062bfd62": "{\\frac {p}{r}}=1+\\varepsilon \\cos \\theta .", "0db9428b706928c8ac951a858402fc4f": "\\ \\mu _{\\delta }", "0db99f8334d54ad40c6fd90a0c72d59a": "L=T-V", "0dba18841141c773b1aee7c9b4f86426": "(d_{1}e_{1},\\ldots ,d_{k}e_{k})", "0dba2f26a19a1e55673e2f42abcb3886": "f:X_{n}\\to S_{n}", "0dbaa5b6a288f6484b4f1e8ef8f35130": "\\partial _{t}g_{ij}=-2R_{ij}+{\\frac {2}{n}}R_{\\mathrm {avg} }g_{ij}", "0dbadaf02264f2c4bb453cc22ff82177": "(\\phi \\leftrightarrow \\chi )\\to (\\chi \\to \\phi )", "0dbafe9cc6ef43dcd8b1fda7b4e93ff4": "{\\mathfrak {g}}/I", "0dbb54b92e0af3184d4fb02d2273585c": "r_{1}=-p,", "0dbb78e7ff9f4f06af326d3d9aad0387": "\\textstyle l(x)", "0dbba5fecbae3204a7eda8c3d9ac3527": "\\psi (\\alpha )^{\\psi (\\alpha )}", "0dbbb6b623e7af334b6b83ffe0edf541": "\\sum _{n=1}^{\\infty }\\left({\\frac {f(n)}{n}}\\right)^{1+\\epsilon }\\varphi (n)=\\infty ", "0dbbd48f77affc660112daa8ca762f09": "H_{\\mathrm {Darwinian} }={\\frac {2n}{m_{e}c^{2}}}\\,E_{n}^{2}", "0dbc2f898dd53548ed2546ca8bf2cd83": "c=Am{|S_{2}-S_{1}| \\over S_{2}}\\,.", "0dbc5caee18a77db082a28b97e862f0d": "f''(x_{0})<0", "0dbc6f19102b5fc99c196ae2ef31cad0": "{\\boldsymbol {u}}_{x}({\\boldsymbol {x}},z,t)\\,", "0dbcab05347576af95b05f84e9bee5a8": "\\sigma _{X}", "0dbd1d200bec7fe91f697b75038cad3f": "3\\times 6\\times 9\\times 12\\times 15\\times 18\\equiv 3\\times 6\\times 2\\times 5\\times 1\\times 4\\equiv 1\\times 2\\times 3\\times 4\\times 5\\times 6{\\pmod {7}};\\,\\!", "0dbd2a7d28db235e9a024cd1a49a0605": "R^{n}\\varprojlim :C^{I}\\rightarrow C.", "0dbda1de1761908c58141ef18cae70b2": "SubCipher_{1}=DEC_{b_{1}}(k_{b_{1}},s_{1})", "0dbdee735176e5313bc4632874ba03ae": "00", "0dcbf4aa02c6fae39b1843a1db82624f": "u(c,l)={\\frac {1}{1-\\gamma }}\\left(c-\\psi {\\frac {l^{1+\\theta }}{1+\\theta }}\\right)^{1-\\gamma }", "0dcc320cc0864c35eb52f74cdc09c5bd": "{\\begin{aligned}\\mathbf {P} (\\tau ,\\mu |\\mathbf {X} )&\\propto \\mathbf {L} (\\mathbf {X} |\\tau ,\\mu )\\pi (\\tau ,\\mu )\\\\&\\propto \\tau ^{n/2}\\exp[{\\frac {-\\tau }{2}}\\left(ns+n({\\bar {x}}-\\mu )^{2}\\right)]\\tau ^{\\alpha _{0}-{\\frac {1}{2}}}\\,\\exp[{-\\beta _{0}\\tau }]\\,\\exp[{-{\\frac {\\lambda _{0}\\tau (\\mu -\\mu _{0})^{2}}{2}}}]\\\\&\\propto \\tau ^{{\\frac {n}{2}}+\\alpha _{0}-{\\frac {1}{2}}}\\exp[-\\tau \\left({\\frac {1}{2}}ns+\\beta _{0}\\right)]\\exp \\left[-{\\frac {\\tau }{2}}\\left(\\lambda _{0}(\\mu -\\mu _{0})^{2}+n({\\bar {x}}-\\mu )^{2}\\right)\\right]\\\\\\end{aligned}}", "0dcc375d5558262f89c8d543fed92970": "Q=2-{\\frac {4}{p}}\\sin {p}+{\\frac {4}{p^{2}}}(1-\\cos {p}),", "0dcc56e97688059028e18955bdd0df7e": "-J\\sum _{n,\\alpha }\\left({\\hat {A}}_{n,\\alpha }^{\\dagger }{\\hat {A}}_{n+1,\\alpha }+{\\hat {A}}_{n,\\alpha }^{\\dagger }{\\hat {A}}_{n-1,\\alpha }\\right)", "0dcc9f9668d172a5523d09e35169c65e": "er_{1}", "0dcca3f1738b8292859653984c2c4e9a": "\\pi \\circ \\sigma =\\pi \\circ p_{2}", "0dcca5ab1eeb43a675b41ef35f3e5e91": "\\cdots \\rightarrow H^{n}(X)\\,\\xrightarrow {\\rho } \\,H^{n}(U)\\oplus H^{n}(V)\\,\\xrightarrow {\\Delta } \\,H^{n}(U\\cap V)\\,\\xrightarrow {d^{*}} \\,H^{n+1}(X)\\rightarrow \\cdots ", "0dccc4ec9f4ac7222bbbb51cdcd6910b": "\\operatorname {cr} (G)\\geq c_{r}{\\frac {e^{r+2}}{n^{r+1}}}.\\,", "0dcce12a8089392e5d31682faf41c9fd": "\\Sigma ^{x}", "0dccebb44d4e0544dabd7d15a26ecedf": "{\\textrm {mes}}E_{\\lambda }\\leq {\\frac {{\\textrm {mes}}E}{2}}", "0dcd3fec7c3c8eccf74a69d3e5e9d986": "5x^{3}-5", "0dcd53ac2db434537c97ea9d1a131290": "z\\cdot x", "0dcd93864516050fea9fa81976fd799f": "\\forall a\\forall b\\exists c\\;a\\leq c\\wedge b\\leq c\\wedge \\forall d\\;a\\leq d\\wedge b\\leq d\\rightarrow c\\leq d", "0dcdc726d1f8b0176f4dccda0d157652": "\\omega _{\\mathrm {c} }={\\frac {1}{\\alpha }}", "0dcddc8d5a88c7a270f0bb57892d00e5": "R\\propto {\\frac {\\int {p(X,A|\\theta ,O_{fg})p(\\theta |X_{t},A_{t},O_{fg})}d\\theta }{\\int {p(X,A|\\theta _{bg},O_{bg})p(\\theta _{bg}|X_{t},A_{t},O_{bg})}d\\theta _{bg}}}={\\frac {\\int {p(X,A|\\theta )p(\\theta |X_{t},A_{t},O_{fg})}d\\theta }{\\int {p(X,A|\\theta _{bg})p(\\theta _{bg}|X_{t},A_{t},O_{bg})}\\,d\\theta _{bg}}}", "0dcde92fd09c5978c5c2ff064d405219": "{\\overline {(C\\wedge (A\\vee B))}}\\wedge (C\\vee A\\vee B))", "0dce3f25b599c43e8a365cecf4c14bfd": "D_{A}^{*}D_{A}\\phi =\\nabla _{A}^{*}\\nabla _{A}\\phi +{\\frac {1}{4}}R\\phi +{\\frac {1}{2}}\\langle F_{A}^{+},\\phi \\rangle .", "0dce40d4e6911723f6409624cd67707b": "F_{c}", "0dce4e864791fb762c24fa203327c0b7": "x_{n}={\\frac {1}{k_{n+1}+{\\frac {1}{k_{n+2}+\\cdots }}}}~;", "0dce8ed5a0fc3585f99eeb61dde05f46": "_{w}={\\frac {\\sum _{i=1}^{N}w_{i}}{(\\sum _{i=1}^{N}w_{i})^{2}-\\sum _{i=1}^{N}w_{i}^{2}}}\\ .\\ \\sum _{i=1}^{N}{\\frac {w_{i}.(x_{i}-{\\overline {x}}^{\\,*})^{2}}{(\\sigma _{x_{i}})^{2}}}", "0dce98c6a29a1f14df7e23241a7f528a": "\\sum _{i=1}^{k}p_{i}=1.", "0dcef858935daa507d82b40efc97fa09": "\\displaystyle 2\\Delta =(ab)^{2}", "0dcf0594b30defd35403d9e4cf7263fe": "(p_{x},p_{y},p_{z})", "0dcf2a6a16718eb52daa1e014a2e3469": "\\langle \\psi _{V}(t)|E_{1}^{(-)}(t)E_{2}^{(+)}(t)|\\psi _{V}(t)\\rangle ", "0dcf345068bbe391fa3fdcf24aee3d9d": "\\mathbf {A} ={\\frac {\\mathbf {J} +i\\mathbf {K} }{2}}\\,,\\quad \\mathbf {B} ={\\frac {\\mathbf {J} -i\\mathbf {K} }{2}}\\,,", "0dcfa2b582e1560b89a1378e290d4adf": "\\mathbb {Z} ", "0dcfafc998e729c39c6f4f858e3374c0": "\\mathrm {CNR_{dB}} =10\\log _{10}\\left({\\frac {C}{N}}\\right)=C_{dBm}-N_{dBm}", "0dcfdf4d0d176ca8a1b055efe367970b": "\\mathbf {x} _{k}^{T}\\,\\mathbf {H} \\,\\mathbf {x} _{k}=0,", "0dcffc4f63f6391d6b93de949071487e": "K_{2}F\\rightarrow \\oplus _{\\mathbf {p} }K_{1}A/{\\mathbf {p} }\\rightarrow K_{1}A\\rightarrow K_{1}F\\rightarrow \\oplus _{\\mathbf {p} }K_{0}A/{\\mathbf {p} }\\rightarrow K_{0}A\\rightarrow K_{0}F\\rightarrow 0\\ ", "0dd031b4a27219b06f3c3f7032bc423c": "x\\leq y", "0dd068bb6a336601f11f835bed96dc88": "{\\begin{aligned}\\langle \\Phi _{0}|({\\hat {H}}-{\\hat {F}})|\\Phi _{0}\\rangle &\\neq 0&&\\Longleftrightarrow &E_{\\text{HF}}&\\neq 2\\sum _{i=1}^{N/2}\\varepsilon _{i}.\\end{aligned}}", "0dd068d5ccbfb3690ce73266713a56ca": "L_{\\mathrm {z} }Y_{lm}=\\hbar mY_{lm},", "0dd07ef5eda99a8c90abad049a4b43a5": "v(s)=\\exp {\\biggl (}{-}\\int _{a}^{s}\\beta (r)\\,\\mathrm {d} r{\\biggr )}\\int _{a}^{s}\\beta (r)u(r)\\,\\mathrm {d} r,\\qquad s\\in I.", "0dd0c01b3941ca0423b208d457c83a04": "\\displaystyle {f^{\\sim }(T)=f(T^{*})^{*}.}", "0dd0c5146a112cdefb9d9032d39c94d4": "\\tau _{U}", "0dd123404451caf8bcbae6021c8ed4a1": "({\\tfrac {a}{m}})=1", "0dd129b645807e94f3d1c1758bddf07c": "1+\\epsilon \\sum _{\\mu =1}^{\\mu =M}P_{\\mu }", "0dd17927c0f85a707fcef426b6d9f627": "\\mathbf {y} _{1}^{*}:=\\min\\{f_{1}(\\mathbf {x} )\\mid \\mathbf {x} \\in X\\}", "0dd1fb01acd07995cd9a25e3f0b6a44b": "2^{-H_{\\min }(X|B)}=\\max _{x}P_{X}(x)~.", "0dd21e13c78dfd27099fa3774ec84179": "f\\propto {\\frac {1}{l}}", "0dd2737d5f487aa384eb6b3a8610f213": "f_{d}=f_{r}-f_{t}=2v{\\frac {f_{t}}{(c'-v)}}", "0dd2ca036accb223f7e3c603b8c21932": "D_{J}", "0dd2e3c20f161ca67c45bf59d6ff7abe": "P=(a,b)", "0dd2fa9ccc77ba058673950ac4170883": "\\sin(n\\theta )\\,{\\hbox{or}}\\,\\cos(n\\theta ),{\\text{ and }}J_{n}(k_{m,n}r).", "0dd3165978b723be3b1e4b0759660f13": "Q=Q_{1}^{T}Q_{2}^{T}\\cdots Q_{t}^{T},", "0dd360338af997f4b181d323e119aeea": "{\\mathfrak {F}}'", "0dd3626c452207ee77bc3b375f9cd3f4": "f(k,0)=k!-\\sum _{i=1}^{k}(-1)^{i+1}{k \\choose i}(k-i)!", "0dd3aaeac1183872c4aaba300d6e948c": "0.622={{MM_{H_{2}O}} \\over {MM_{dry\\,air}}}", "0dd3fcf36f75087415064bbda0c3bee6": "b_{k}=\\dim H^{k}(V)=\\sum _{p+q=k}h^{p,q},\\,", "0dd40a0ed1b4273dedd041e1414d9499": "X_{i}\\leq E(X_{i})+a_{i}+M", "0dd431587e81bc100334b3d78d532a54": "L(A)", "0dd46cd1d97279852df5f200f0b4ddc6": "\\neg ({\\overline {\\beta _{1}}}\\wedge {\\overline {\\beta _{2}}})", "0dd4959bdd4ec461917a6fb760733cea": "\\operatorname {E} [|V^{S}-V^{B}|]\\approx \\alpha \\mu \\;.", "0dd49b69061ee7d4332577b638a3d799": "\\Delta g=g-\\gamma .\\,", "0dd4ba47f064411e18ebd674c2a68418": "y(i)+y(j)=c(i,j)", "0dd4e75028a101ddb9d374d10280738e": "\\ MAE={\\frac {\\sum _{t=1}^{N}|E_{t}|}{N}}", "0dd4f39b473a17635819421778abdd49": "d\\mathbf {X} _{1}\\,\\!", "0dd51a155ce9e119a738e8358ecb6bb2": "I_{m,k}={\\frac {\\partial {\\boldsymbol {\\mu }}^{T}}{\\partial \\theta _{m}}}{\\boldsymbol {C}}^{-1}{\\frac {\\partial {\\boldsymbol {\\mu }}}{\\partial \\theta _{k}}}+{\\frac {1}{2}}\\mathrm {tr} \\left({\\boldsymbol {C}}^{-1}{\\frac {\\partial {\\boldsymbol {C}}}{\\partial \\theta _{m}}}{\\boldsymbol {C}}^{-1}{\\frac {\\partial {\\boldsymbol {C}}}{\\partial \\theta _{k}}}\\right)", "0dd51db8af6f06e461cd2b2796e6e165": "D(x,y)={\\frac {1}{2}}", "0dd52f6d1dd6a0aa557cc1dac3101c17": "k=A\\exp \\left[-\\left({\\frac {E_{a}}{RT}}\\right)^{\\beta }\\right]", "0dd57315448852ac5e85d1d7f4227461": "{\\frac {d\\omega _{r}}{dt}}={\\frac {1}{J(T_{e}-B_{m}\\omega _{r}-T_{L})}}", "0dd5e36179a728854337e7914b0acfab": "\\forall \\ x,{\\overline {Bx}}\\ \\rightarrow \\ {\\overline {Rx}}", "0dd70ae73c03aef43b8738b7e30e4365": "w^{2}x^{2}y^{2}+w^{2}x^{2}z^{2}+w^{2}y^{2}z^{2}+x^{2}y^{2}z^{2}+wxyzQ(w,x,y,z)=0", "0dd77d1443b8864daa663a4dc1ac777d": "Interest=Principal\\left[\\left({\\frac {APY}{100}}+1\\right)^{Days~in~term/365}-1\\right]", "0dd7a2e8644f61a9cf6f8be6f07c8aab": "{M}=0.88128485{\\sqrt {\\left[\\left({\\frac {p_{t}}{p}}+1\\right)\\left(1-{\\frac {1}{[7M^{2}]}}\\right)^{\\frac {5}{2}}\\right]}}", "0dd7a863ab4bf0f72c2d31922c4bdb08": "Z={\\frac {\\xi ^{N}}{N!}}.", "0dd7cd6051c76f98b1f7b7ee038198dc": "\\scriptstyle c^{\\underline {k-1}}", "0dd83b06966e0638ae985fb96d5cb7f3": "Q(x)={\\frac {6x}{\\pi ^{2}}}+O\\left(x^{1/2}\\exp \\left(-c{\\frac {(\\log x)^{3/5}}{(\\log \\log x)^{1/5}}}\\right)\\right).", "0dd85a63d2c5a98bedd56fd3f174c67a": "J(x)(f)=f(x),\\qquad f\\in X',", "0dd864c88ec238950bdf95793fd8fc64": "F_{\\nu ,max}", "0dd86a3bd9445a746e6f4255e61909ba": "EL(\\Gamma ):=\\sup _{\\rho }{\\frac {L_{\\rho }(\\Gamma )^{2}}{A(\\rho )}}\\,,", "0dd86b4fc234be303d4b954116484140": "\\Pr[y_{i}^{\\prime \\prime }=?]=\\Pr[\\theta \\in [0,{2\\omega _{i} \\over d}]]={2\\omega _{i} \\over d}.", "0dd87427da2fabf8b593c8de198fa91d": "{\\frac {1}{Z_{\\beta ,n}}}\\prod _{k=1}^{n}e^{-{\\frac {\\beta n}{4}}\\lambda _{k}^{2}}\\prod _{i0\\!", "0ddf5b362207534fd9ec8c845018b5a6": "\\sum _{x\\in A}f_{X}(x)=1", "0ddfc81bf6e635063312fc353ab81bb5": "\\Delta _{m}", "0ddfdbadc8663e0638229c2c60be28c9": "x=[{\\overline {a_{0};a_{1},a_{2},\\dots ,a_{m}}}],", "0de08414c38de494106254e16823f05a": "{\\text{Maximize}}\\,\\,pQ-wL(w)-rK\\,\\,{\\text{with respect to}}\\,\\,Q,\\,w,\\,{\\text{and}}\\,K", "0de0ca0514b0197d643ca5b3290932a5": "\\sin \\theta ={\\frac {y'(s)}{\\sqrt {x'(s)^{2}+y'(s)^{2}}}}=y'(s)\\ ;", "0de0d49ed704dbf3bce7b6e63ce33876": "J={\\frac {1-\\psi ^{'}}{\\phi ^{'}}}\\,", "0de0dc6ba8b5f2802aea992a9a7782d8": "\\mu _{\\operatorname {eff} }({\\dot {\\gamma }})=\\mu _{\\operatorname {\\inf } }+(\\mu _{0}-\\mu _{\\operatorname {\\inf } })\\left(1+\\left(\\lambda {\\dot {\\gamma }}\\right)^{2}\\right)^{\\frac {n-1}{2}}", "0de14e0aed75a8dbb5d64ea6a20a721e": "{\\frac {d}{2}}", "0de23b9e67fe517252937091644dd937": "E[G|H]=\\int _{0}^{T}k(t)E[x(t)|H]dt=0", "0de31b19055091711fccd433999c3e62": "\\log K_{0}=\\log 2+{\\frac {1}{\\log 2}}\\left[{\\mbox{Li}}_{2}\\left({\\frac {-1}{2}}\\right)+{\\frac {1}{2}}\\sum _{k=2}^{\\infty }(-1)^{k}{\\mbox{Li}}_{2}\\left({\\frac {4}{k^{2}}}\\right)\\right].", "0de32fab8a1927bb9864cfa1dc42a480": "{\\frac {dX(t)}{dt}}=u[X(t),t]", "0de35b0a6b8fc32350e8c500c52089af": "\\land T_{7}=[F_{7},S_{7},A_{7}]::[F_{6},S_{6},A_{6}]::K_{5}", "0de3738a81e5265ef899cc00f7d5edc2": "\\sigma _{1}^{2}=\\sigma _{3}^{2}=I,\\;\\sigma _{1}\\sigma _{3}=-\\sigma _{3}\\sigma _{1}=e^{\\pi i}\\sigma _{3}\\sigma _{1}.", "0de3886d300b5c7dfc14329ba1983246": "\\scriptstyle x\\;\\in \\;[0,\\,\\infty )\\!", "0de39f4449f39bc0eeae178121bdf307": "\\mu _{n}=\\{1,\\zeta _{n},\\zeta _{n}^{2},\\dots ,\\zeta _{n}^{n-1}\\}", "0de3a8a2ea897397f77ac970ce3a4173": "{\\hat {H}}(\\mathbf {k} )", "0de3f48e92d7e79380da650f41a56e23": "L_{\\Sigma }=10\\,\\cdot \\,{\\rm {log}}_{10}\\left({\\frac {{p_{1}}^{2}+{p_{2}}^{2}+\\cdots +{p_{n}}^{2}}{{p_{\\mathrm {ref} }}^{2}}}\\right)=10\\,\\cdot \\,{\\rm {log}}_{10}\\left(\\left({\\frac {p_{1}}{p_{\\mathrm {ref} }}}\\right)^{2}+\\left({\\frac {p_{2}}{p_{\\mathrm {ref} }}}\\right)^{2}+\\cdots +\\left({\\frac {p_{n}}{p_{\\mathrm {ref} }}}\\right)^{2}\\right)", "0de47177820aa228efeb43a86c53fc5c": "L^{\\prime }", "0de4ef4752110c23926d7fc377784f36": "{\\mbox{Diameter in millimetres}}=2\\times T\\times A/100+W\\times 25.4", "0de4ef9255a21226302c6def1ec1cc7b": "{\\mathcal {J}}_{p+2}", "0de5580bc106089722df88ed8fa1762f": "u={\\frac {\\omega }{2\\pi c}}(x_{1}-x_{2})", "0de56f5d8c7016548339ec76bca76f3d": "f={\\frac {V_{a}+V_{w}}{V_{s}+V_{a}+V_{w}}}", "0de6b31f36ae71429eaf0e8e367eadb5": "{{K-B \\over N-K}{K \\choose B}{N-K \\choose K-B} \\over {N \\choose K}}", "0de7047912a29c5ae7004b971d26148f": "p(\\chi )={\\frac {2}{3}}\\left(\\left(1-{\\frac {\\chi }{\\pi }}\\right)\\cos {\\chi }+{\\frac {1}{\\pi }}\\sin {\\chi }\\right).\\!\\,", "0de70fe771d6e5d7bd1c7af1d6d5c9f9": "\\mid ", "0de73479fb335937ebcb9a84715982ad": "{\\mbox{Exp}}_{certificateofdeposit}=0.5\\times 500+0.3\\times 500+0.2\\times 500=500", "0de74ceae7ca99234f0e9aae44a1af97": "Vt={\\frac {Vk}{Re}}Rw", "0de77c0791da7b4cf334d4dad8d82808": "D_{o}=\\sum _{u=1}^{N}{\\frac {m_{u}}{n}}D_{u}={\\frac {1}{n}}\\sum _{u=1}^{N}{\\frac {1}{m_{u}-1}}\\sum _{i=1,j=1}^{m}{_{metric}}\\delta _{c_{iu}k_{ju}}^{2}", "0de7a5b26bb28e5a76e8b60a4150636f": "b=90^{\\mathrm {o} }-\\lambda _{\\mathrm {A} }\\,", "0de7b6a61a70688b26e6eeb3113531a3": "dz", "0de9401c1235d8595cbf5e197c2b57dd": "{}E[X_{t}|\\{X_{\\tau }:\\tau \\leq s\\}]\\leq X_{s}\\quad \\forall s\\leq t.", "0de9a79981ec28b4d341491681803722": "{\\textbf {m}}=c{\\textbf {h}}+c-q{\\textbf {f}}_{p}\\cdot K{\\pmod {p}}", "0de9a97807dd2fd5bcbdc5ad60d34fc6": "\\!\\ {1 \\over {{\\sqrt {2}}-1}}={\\sqrt {2}}+1", "0de9dc0f5ff52aee4ee7c9f99f1c39d2": "{\\overrightarrow {Y}}=Y_{o}\\ \\ \\ ,\\ \\ \\ {\\overleftarrow {Y}}=jY_{\\varepsilon }tan(k_{x\\varepsilon }w)", "0de9f248831c58969286349a3d852f91": "\\eta =a\\,\\cos \\,(kx-\\omega t)=a\\,\\cos \\,\\theta ,", "0dea0736fdbd2b398c7ed50f5844d075": "\\displaystyle {0.58\\,W}", "0dea22f489272c79924fe58f2cd48103": "f_{P}(f_{P}(X))=X{\\text{ for all }}X\\in [\\ell ]", "0dea598bd161f2566f652dcbb5e9feb4": "\\Xi \\,\\xi \\,", "0dea8e7d4176801176a52c0236b6c808": "\\left(x^{2}+y^{2}\\right)+\\left(z-a\\coth \\tau \\right)^{2}={\\frac {a^{2}}{\\sinh ^{2}\\tau }}", "0deac00ba48cb30e568820886a46dfae": "=C_{H_{2}O}", "0deae5992a4b076ce59a9a31792603ee": "\\partial ^{4}B/\\partial x^{4}", "0deaea42c9ef1c67ec9775c7f3b92938": "\\left(3{\\sqrt {\\frac {2}{5}}},\\ \\pm {\\sqrt {6}},\\ 0,\\ \\pm 2\\right)", "0deafa7525eb2625f8d7dcaaa6f3490b": "{\\overline {W}}", "0deb0294ad9faaaf5c9024339b47581a": "x_{1}\\in {\\mathfrak {g}}_{\\lambda _{1}}", "0deb57a0dc714b0f8c52bcde78e8718f": "z_{match}\\,", "0debd733e70395cb0e53177ac6ac8a0c": "{\\tfrac {3KE}{9K-E}}", "0dec13788f1388692a2432f552b1fdfc": "c_{jk}", "0dec58ac609485be2477b46e58a7e162": "f\\circ \\Phi (x,y)=\\textstyle {\\frac {1}{2}}x^{T}Bx+h(y).", "0dec6a3e3d3f26835acad2233af715df": "\\omega _{mn}={\\sqrt {\\frac {D\\pi ^{4}}{2\\rho h}}}\\left({\\frac {m^{2}}{a^{2}}}+{\\frac {n^{2}}{b^{2}}}\\right)\\,.", "0decf9b8b8f8301b6b36efbee0f182aa": "g\\!", "0dee0fd3d3e936f88ca9e7fd45906b5e": "\\displaystyle {\\varphi _{s,t}(z)=f_{t}^{-1}\\circ f_{s}(z)=e^{s-t}(z+a_{2}(s,t)z^{2}+a_{3}(s,t)z^{3}+\\cdots )}", "0def02d4e7a3051a5c1de75372cae213": "{\\overline {A}}", "0def164a7fa0a232f15c93cfb75e2818": "\\lim _{w\\to 0^{+}}w\\log _{2}w=0", "0def4b1affe6183b9f9dd3dcf095c235": "\\lim _{x\\to p}f(x)=L", "0def4d70826d93e077b226405da4b6a2": "{\\frac {\\mu ^{3}}{\\lambda }}", "0def5bcd31fd6c0c87ed2adda37bddc4": "\\pi _{1}f_{1}", "0defb44b8ac8c676aa39aaa503de5106": "Q_{i}^{j}=9.6{\\text{ m}}^{3}/{\\text{s}}", "0defb66af90a0ea6b261aab5f6cc54f9": "{\\frac {x}{2}}\\coth {\\frac {x}{2}}=\\sum _{n=0}^{\\infty }c_{n}x^{n}=1+{\\frac {1}{12}}x^{2}-{\\frac {1}{720}}x^{4}+\\cdots ", "0deff6ac005223f150c6e9935a4024ad": "\\omega _{m}\\ll \\omega _{c}\\,", "0deffa862b19684c17dc2fd4fdfdcf44": "\\ln \\left(e^{A}e^{B}e^{-A}e^{-B}\\right)=[A,B]+{\\frac {1}{2!}}[(A+B),[A,B]]+{\\frac {1}{3!}}\\left([A,[B,[B,A]]]/2+[(A+B),[(A+B),[A,B]]]\\right)+\\cdots .", "0df02c8889f4670a3dba1d2ed8ffb4de": "\\displaystyle {\\|f\\|_{(k)}^{2}=\\sum _{j=0}^{k}{k \\choose j}\\|\\partial _{x}^{j}\\partial _{y}^{k-j}f\\|^{2}.}", "0df0695845e62978683766baf2d7c2fc": "n={\\frac {4\\,(z_{\\alpha }+z_{\\beta })^{2}}{d\\log ^{2}{\\lambda }}}", "0df0b267e87f9bb70729600a8b527488": "\\eta \\,\\!", "0df0f9fca8a0c95d27e0b2f4a560a310": "\\displaystyle {f(z)={\\overline {f({\\overline {z}})}},}", "0df14837305561d74a847771eb5f106d": "{\\bar {HP}}_{3}", "0df18f1eec7c67d65b06a79355d7e49e": "[r_{i}-S]\\mathbf {y} =(\\mathbf {r} _{i}-\\mathbf {S} )\\times \\mathbf {y} ,", "0df1ba97c4e027a135a467f26561f612": "q{\\overset {\\alpha }{\\rightarrow }}q'", "0df1f51621a8e294c813056976211d1c": "X_{\\tau (X^{*},X^{**})}^{*}", "0df20f79abb6b93039da4e0cb1e6d172": "\\operatorname {Pref} (L)=\\bigcup _{s\\in L}\\operatorname {Pref} _{L}(s)=\\left\\{t\\vert s=tu;s\\in L;t,u\\in \\operatorname {Alph} (L)^{*}\\right\\}", "0df2440ac89ecfa63a7e317ed22acba7": "J=J_{1}(1)\\oplus J_{1}(2)\\oplus J_{2}(4)={\\begin{bmatrix}1&0&0&0\\\\0&2&0&0\\\\0&0&4&1\\\\0&0&0&4\\end{bmatrix}}.", "0df26f62149f5814c6372a101993654a": "GL(\\mathbb {R} ^{7})", "0df2dfc600298b2706ad4ab774c858c3": "L_{r}=n_{e}n_{H}P(T)~~{W~m^{-3}}", "0df308f3c3554c5ef08358bb562130c3": "J_{gas}=h_{g}(C_{g}-C_{s})", "0df32534ae1f73ea43b99cddf272ca05": "\\lVert z\\rVert =zz^{*}=z^{*}z=x^{2}-y^{2}.", "0df336d237764d8d6747521fa443fece": "e^{-sT}{\\dot {=}}{\\frac {(sT)^{2}-6sT+12}{(sT)^{2}+6sT+12}}", "0df338892151c4df85941524eb3a1d37": "\\neg A\\to (A\\to B)", "0df3dd737ed6ef805d378925bc123797": "V={\\frac {h_{1}B_{1}-h_{2}B_{2}}{3}}", "0df4015a869eb9ec434a9bc9d33a9856": "a_{\\mu }", "0df41b034c4ba373a6f62d6410b4874c": "a=b^{c}", "0df45099cc2928ae59be9c703fe1aa97": "a_{1}=320", "0df4b8c24f1b02af924df7d79c9dac91": "\\mathbb {Z} [{\\sqrt {-5}}].", "0df4d2f9436603c67a4edcf5d4d54ff6": "{\\boldsymbol {\\sigma }}*\\in \\mathbf {b} ({\\boldsymbol {\\sigma }}*)", "0df4dc59f99a775d94e31df99368e1c5": "\\vert n\\vert \\leq N", "0df510feb88385bdc04683b4975d2fa9": "D(x,y)", "0df5d6a0542f397b75c9db6ef0087b0c": "\\,\\,{\\boldsymbol {\\sigma }}=2\\mu \\,{\\boldsymbol {\\varepsilon }}+\\lambda \\,{\\text{tr}}({\\boldsymbol {\\varepsilon }})\\,{\\boldsymbol {I}}\\,\\,", "0df601c6accb491001c07df9c6c1c4e4": "dU=C_{v}dT", "0df62b1e73d01d8a354c785d6684383b": "J_{w}=A\\left(\\Delta \\pi -\\Delta P\\right)", "0df63f38b0a445aae34f1f5bb42f5cb7": "F_{n}(x)=U_{n}(x,-1),\\,", "0df696ed178fe22a9068eaf4ceb586c9": "\\,\\prod _{n}(z-c_{n})", "0df6bd159e216d9baa128178f0cae69e": "z\\in \\mathbb {C} ", "0df6f101fe227cb36d54334e794301d2": "Ker(df_{x})", "0df719d0291aa256b61a04397d63e754": "\\left[{\\mathrm {d} \\over \\mathrm {d} t}{\\partial {T} \\over \\partial {\\dot {q_{i}}}}-{\\partial {(T-V)} \\over \\partial q_{i}}\\right]=0", "0df73d835f83efa414b6762e69e2fa98": "K=\\mathbb {C} ", "0df757a3729a50b14003c69b4089f5b4": "r^{2}-R^{2}", "0df77b9d7105fdc39bae30bdc23addfb": "p(A)=1", "0df8444ac0c9455e206e607ab5cdff73": "{\\vec {r}}_{v}.", "0df859cc5c058f03a493b68070433539": "X|\\mu ", "0df872997025404baedfb272a051f1c8": "V=k_{2}T\\qquad (2)", "0df92300e9acc53840cdb5e971134346": "{\\frac {dC_{AS}}{dt}}=0=k_{1}C_{A}C_{S}(1-\\theta )-k_{2}\\theta C_{S}-k_{-1}\\theta C_{S}", "0df9cc6763484db7df8a7418bff18a65": "w_{k}", "0df9dd2dfec0c0ded5aa8e16e0b69b5a": "e^{\\lambda (r)}-1={\\frac {r_{s}}{r-r_{s}}}\\;", "0dfa1c3d9778bf0bd5ed348cac016a6f": "\\left(-{\\frac {n}{2}}f_{\\mathrm {s} },-{\\frac {n-1}{2}}f_{\\mathrm {s} }\\right)\\cup \\left({\\frac {n-1}{2}}f_{\\mathrm {s} },{\\frac {n}{2}}f_{\\mathrm {s} }\\right)", "0dfa7d19db7627045b6292f0fb99629c": "0=N_{0}\\subsetneq N_{1}\\subsetneq \\cdots \\subsetneq N_{n}=M", "0dfa9bd2538905ec9d491e230d88dd17": "\\Lambda _{0}^{3}\\mathbb {C} ^{6}", "0dfaca0f8aad55df388e07eeb190a086": "{\\boldsymbol {K}}", "0dfad386bed71d50d2656d4320dd0a61": "Y=f(X)\\ ", "0dfbf929067de6d44384294102818d6a": "\\pi ={\\frac {3528}{Z}}\\!", "0dfbfd3cab6588265e46ad52deb0063b": "s\\in \\{1,2,3\\}", "0dfc402e042b5d814aa39f24bbdd96d9": "1,0,-1", "0dfc516dec7cec7d54092a206af5c494": "\\scriptstyle {\\mathbf {L} }", "0dfc76213a3f46666199ce2419e22a79": "f(x)={\\frac {1}{2b}}\\exp \\left(-{\\frac {|x-\\mu |}{b}}\\right)", "0dfc846aa4eaf09afee3fcb0340fe877": "A\\otimes _{R}A^{\\circ }", "0dfc97cb3615a7fac8eb365474728340": "(z-z_{0})^{-2}-{\\frac {z_{0}}{10}}(z-z_{0})^{2}-{\\frac {1}{6}}(z-z_{0})^{3}+h(z-z_{0})^{4}+{\\frac {z_{0}^{2}}{300}}(z-z_{0})^{6}+\\cdots ", "0dfc9c4427c33d64c997bd801ac82c4b": "W_{X}=\\left({\\frac {x/x_{n}}{y/y_{n}}}-1\\right)V_{J}", "0dfcfec2a00afd662748a25bc26cae64": "r\\neq 0", "0dfcfecd43bec0005964b5f47653c306": "x\\neq 1", "0dfd02c657f5b499d64e0cb88f03e6a5": "g^{ab}\\theta _{ab}=\\theta ", "0dfd21de8f1d0744a94d21dc2daf5f18": "L_{3}=1/3", "0dfd5f87577526aa7c6dd5e77595f035": "\\displaystyle {\\pi _{ij}(xy)=\\pi _{ij}(x)\\pi _{ij}(y),\\,\\,\\,\\pi _{ij}(1)=I.}", "0dfd8560df76e8c860748d9efcd07607": "R_{5,4}=11r^{5}-10r^{4}", "0dfdb95b77a645c11063543994f025a9": "\\Gamma _{N}(w|a_{1},...,a_{N})=\\Gamma _{N-1}(w|a_{1},...,a_{N-1})\\Gamma _{N}(w+a_{N}|a_{1},...,a_{N})", "0dfde751679220aa2dfabe52aa975e41": "{\\frac {p}{q}}", "0dfe1bcb396d9fef2614363d96980238": "d>12", "0dfe776fedcc377da1f838a5e43d5a11": "p_{i_{1},i_{2},\\ldots ,i_{N}}\\approx \\sum _{t}^{T}p_{t}\\,\\prod _{n}^{N}p_{i_{n},t}^{n},", "0dfec86dabeaf3d3d7624b7348b75506": "y_{n}(x)={\\sqrt {\\frac {\\pi }{2x}}}Y_{n+{\\frac {1}{2}}}(x)=(-1)^{n+1}{\\sqrt {\\frac {\\pi }{2x}}}J_{-n-{\\frac {1}{2}}}(x).", "0dfec9ab94481562873a30208a5b5c54": "\\lfloor n^{2}/4\\rfloor ", "0dff4b4e798fcf9b6fcbf2a08879653c": "{\\mathcal {M}}={\\mathcal {M}}^{K}+{\\frac {\\mathcal {B}}{1+\\nu }}\\,q+D\\nabla ^{2}\\Phi ", "0dff5d5251f5163826306c2785f8de0a": "{\\begin{aligned}\\tan x&{}=\\sum _{n=0}^{\\infty }{\\frac {U_{2n+1}x^{2n+1}}{(2n+1)!}}\\\\&{}=\\sum _{n=1}^{\\infty }{\\frac {(-1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}}\\\\&{}=x+{\\frac {1}{3}}x^{3}+{\\frac {2}{15}}x^{5}+{\\frac {17}{315}}x^{7}+\\cdots ,\\qquad {\\text{for }}|x|<{\\frac {\\pi }{2}}.\\end{aligned}}", "0dff77b567fa34bfe97953fa0708bcae": "X=X_{0},X_{1},\\ldots ,X_{n-1},X_{n}=Y", "0dff9220b8a143e3bb43279d9bd8f62e": "y=x^{3/2}", "0e0039c3a43d982732917b456a3ff4e9": "R/I^{n}", "0e00f7b53d80743af1c5cc0c6eb3ba67": "U=-G\\int _{0}^{R}{\\frac {(4\\pi r^{2}\\rho )({\\tfrac {4}{3}}\\pi r^{3}\\rho )}{r}}dr=-G{\\frac {16}{15}}{\\pi }^{2}{\\rho }^{2}R^{5}", "0e0163e4ab8ec7b5c915ce81276976c5": "\\int \\limits _{0}^{2\\pi }t_{g}\\ \\left({\\frac {p}{r}}\\right)^{2}\\ {\\frac {3}{2}}\\ du\\ =\\ -{\\frac {3}{2}}\\ \\int \\limits _{0}^{2\\pi }\\ \\left(1\\ +\\ e_{g}\\ \\cos u\\ +\\ e_{h}\\ \\sin u\\right)^{2}\\ \\ \\sin u\\ du\\ =\\ -3\\ e_{h}\\ \\int \\limits _{0}^{2\\pi }\\sin ^{2}u\\ du\\ =\\ -2\\pi {\\frac {3}{2}}\\ e_{h}", "0e017fcf2d4accdc16cc6af46f76263a": "W(k)\\approx {\\frac {1}{k^{2}\\ln 2}}.", "0e01b96d4340e500cc02588adac68db8": "t(1/n,\\epsilon )\\,", "0e01e49a06e77b96edf6d0d45aa44cbe": "s^{2}={\\frac {N\\sum _{i=1}^{N}n_{i}({\\hat {\\theta _{i}}}-{\\hat {\\mu }})^{2}}{(N-1)\\sum _{i=1}^{N}n_{i}}}.", "0e021dc2488637367e3ce0a851ef89b1": "ax^{3}+bx^{2}+cx+d=0\\qquad (1)", "0e024938216add9a87cdc7ae7570ea4a": "\\omega ={\\sqrt {k/m}}", "0e0275c21f40fecb1512deda66339ddc": "\\mathbf {\\gamma _{0}} ", "0e02a725bd42081ec64219e4fec4152b": "\\tau _{l}={\\frac {1}{D_{\\mathrm {rot} }l(l+1)}}", "0e02ef4a6916cb54430c8d30a044d106": "d_{2,1}^{2}=-{\\frac {1}{2}}\\sin \\theta \\left(1+\\cos \\theta \\right)", "0e03245f9c26dbaf07c5f8d56e92cdec": "\\underbrace {A_{i}\\land \\dots \\land A_{i}} _{f(i)}", "0e036688ebce66792099c72a2ce03045": "L_{\\mathrm {loc} }^{1}\\;", "0e0375cf7a64599001ce1371ede02acd": "\\sigma _{x}^{2}/n", "0e037e4350ff36f937860a924c30d1b7": "Q\\cap I", "0e03cf7c5a5cf6ed1c8823318a60aab7": "d=\\min {\\Delta (c_{1},c_{2})}", "0e03fa752807abea7e0adc56c5775c85": "F=-{\\frac {GMm}{r^{2}}}-{\\frac {\\Lambda Mmr}{3}}", "0e0416f6936830bfe3d2f8ffae19975e": "L=3.4~\\mathrm {a.u.} ", "0e0457cd7c7a4011c0db5c2063d1dd61": "Lf(x)=\\Delta f(x)-x\\cdot \\nabla f(x).", "0e045c92b526e900b3b124c807340acd": "i", "0e0480e97af34e0ffe9687ad2e7c700a": "X_{1},...,X_{n}", "0e04da75e1bed4e55dacfc73b528f8ab": "E_{0}=\\left({\\frac {Ze^{2}}{\\alpha D_{\\alpha }^{1/\\alpha }\\hbar }}\\right)^{\\alpha /(\\alpha -1)}.", "0e04e08cca7a0724785106f1f25f9c21": "\\scriptstyle b^{4}", "0e04e8cf0eed70b7effc626e02d36d60": "\\mathrm {deg} \\,C>{\\sum _{i=1}^{r}m_{i} \\over {\\sqrt {r}}}.", "0e04f8b48af9158b690fda9111dcb813": "n^{2}=1-{\\frac {X\\left(1-X\\right)}{1-X-{{\\frac {1}{2}}Y^{2}\\sin ^{2}\\theta }\\pm \\left(\\left({\\frac {1}{2}}Y^{2}\\sin ^{3}\\theta \\right)^{2}+\\left(1-X\\right)^{2}Y^{2}\\cos ^{2}\\theta \\right)^{1/2}}}", "0e0519181090c0e90fcdb129d7ff1695": "\\sum _{i,j}r_{i,j}(x,y)\\partial _{x}^{i}\\partial _{y}^{j}", "0e053935a6d965bc16ca903289028186": "y=A_{y}\\cos \\left(\\omega _{0}t+\\phi _{y}\\right)", "0e0543aa32a4db1e5482305372fa7197": "h=\\theta _{G}-\\lambda _{o}-\\alpha ", "0e054b1618f28ebe35d61832126a510c": "{\\lVert a_{i}\\rVert ^{2}}", "0e0558ac202bf4f6d81ad71b6ee92c11": "L(x_{1},x_{2},x_{3})=(2x_{1}+5x_{2}-3x_{3},\\;4x_{1}+2x_{2}+7x_{3})", "0e05b9ce1187fd800c4aaed120def3c2": "S^{2n+1}", "0e05bed100c5460dd523f83322b3e4d7": "f,", "0e05d4047132bf390f2a8690c09c34a9": "g_{2}(\\tau )={\\frac {\\langle I(t)I(t+\\tau )\\rangle _{t}}{\\langle I(t)\\rangle _{t}^{2}}}", "0e05e60e2f4ff79d43dea8d46f0aeb13": "y(t_{0})=y_{0}.\\,", "0e06c59c460ecd57bee11ecaacaf1751": "L=\\omega ^{l}\\,\\cap \\,(\\omega ^{l})^{-1}", "0e06d9824fe4426d1a65b6fd6300a927": "\\Delta \\lambda _{0}={\\frac {4\\lambda _{o}}{\\pi }}\\arcsin \\left({\\frac {n_{2}-n_{1}}{n_{2}+n_{1}}}\\right),", "0e0709113ba0fd364d521802e28c023c": "-2k\\cdot p'\\approx \\,", "0e0738c84dfb6eec9f26a53301899b4f": "{\\frac {1}{2}}-1.", "0e2ce498ed457deb087daa280b4b5d63": "H(u)(t)=-{\\frac {1}{\\pi }}\\lim _{\\epsilon \\downarrow 0}\\int _{\\epsilon }^{\\infty }{\\frac {u(t+\\tau )-u(t-\\tau )}{\\tau }}\\,d\\tau ", "0e2d27af092b5d56d79c38a5e64372b6": "(x^{2}+y^{2})^{2}=2a^{2}(x^{2}-y^{2})", "0e2d27e61835a094469f4b27f1acdbbc": "f(t)={\\frac {1}{{\\sqrt {2\\pi }}\\,i}}\\int _{c-i\\infty }^{c+i\\infty }(ts)^{1/2}\\,I_{\\nu }(ts)\\,g(s)\\;ds,", "0e2d5fdb037e188e3e8c3fd9a96730c6": "{\\frac {\\pi }{8}}\\ (22.5^{\\circ })", "0e2e241c1e9f5ed14c32f64521d6c661": "Pr[|{\\frac {1}{m}}\\left(\\sum _{i}\\left(w_{\\sigma _{i}}^{j}-w_{\\sigma _{m+i}}^{j}\\right)\\right)|\\geq {\\frac {\\epsilon }{2}}]=Pr[|{\\frac {1}{m}}\\left(\\sum _{i}|w_{i}^{j}-w_{m+i}^{j}|\\beta _{i}\\right)|\\geq {\\frac {\\epsilon }{2}}]\\,\\!", "0e2e83b675862caa6e233408abf829fa": "V:{\\mbox{Var}}\\rightarrow 2^{S}", "0e2e9deb9c13e2c424fa1dc16fc36854": "{\\text{(2)}}\\qquad W=\\int _{V_{1}}^{V_{2}}P\\,dV", "0e2ef5b993a4cfa66e4620c76c86d0e1": "=\\mathbf {w} _{n-1}+\\mathbf {g} (n)\\alpha (n)", "0e2efece425172fce21d95a7f4685c80": "|c_{1}(t)|^{2}", "0e2f5778d73f8c2ad524d627c03eef79": "f={\\frac {Nv}{2d}}\\qquad \\qquad N\\in \\{1,2,3,\\dots \\}", "0e2f8bccc90ecb3f8e0390074a54e3c0": "{\\begin{aligned}\\mu _{X}&={\\frac {\\sum _{i}N_{X_{i}}\\mu _{X_{i}}}{\\sum _{i}N_{X_{i}}}}\\\\\\sigma _{X}&={\\sqrt {{\\frac {\\sum _{i}N_{X_{i}}(\\sigma _{X_{i}}^{2}+\\mu _{X_{i}}^{2})}{\\sum _{i}N_{X_{i}}}}-\\mu _{X}^{2}}}={\\sqrt {{\\frac {\\sum _{i}N_{X_{i}}\\sigma _{X_{i}}^{2}}{\\sum _{i}N_{X_{i}}}}+{\\frac {\\sum _{i{\\frac {4}{3}}\\log g", "0e3508e72485ea6a0da30226d62f390e": "w^{(2)}(r)", "0e358e80cc416bcf774c7f622b02b660": "f(x;\\alpha ,\\beta )={\\frac {\\beta ^{\\alpha }}{\\Gamma (\\alpha )}}x^{-\\alpha -1}\\exp \\left(-{\\frac {\\beta }{x}}\\right)", "0e3593936f8a03ab46ed68a4a730fa01": "I={\\frac {V}{R}}\\quad {\\text{or}}\\quad V=IR\\quad {\\text{or}}\\quad R={\\frac {V}{I}}.", "0e359fff73cf5d682013ae71ecd6bced": "Z_{P}={\\cfrac {bh^{2}}{4}}", "0e35cdac20c814a3bf7f9fca634e08b9": "k_{0}{\\frac {\\partial E_{0}}{\\partial z}}+\\omega _{0}\\,\\mu _{0}\\,\\varepsilon _{0}\\,{\\frac {\\partial E_{0}}{\\partial t}}-{\\tfrac {1}{2}}\\,i\\,\\Delta _{\\perp }E_{0}=0.", "0e35de83aca3aa66631bb7fa668e51fb": "\\Pr \\left(-c\\leq T\\leq c\\right)=0.95\\,", "0e360cc8d6e90ceb6a2f9aec2772df0a": "{\\frac {de}{dt}}=f(1-e)-b(e)e=f(1-e)-e^{2}", "0e36616592497c1ef341680f03266599": "k>p", "0e36b291b88dc4997979ea9336896bb3": "\\gtrdot ", "0e36b3f6a21b235088364364e1d1079e": "X_{v}\\,)", "0e36e690d00f1e4f555c93794e4ec0ae": "(25)\\quad ds^{2}=-{\\Big (}1-{\\frac {M}{r}}{\\Big )}^{2}dt^{2}+{\\Big (}1-{\\frac {M}{r}}{\\Big )}^{-2}dr^{2}+r^{2}d\\theta ^{2}+r^{2}\\sin ^{2}\\theta \\,d\\phi ^{2}\\,.", "0e3750ec580b2ea02b090bd212b6dbe8": "\\omega _{R}", "0e3762719d56d05eb3314097c7994591": "{\\begin{aligned}\\partial _{t}u&=d_{u}^{2}\\,\\nabla ^{2}u+f(u)-\\sigma v,\\\\\\tau \\partial _{t}v&=d_{v}^{2}\\,\\nabla ^{2}v+u-v\\end{aligned}}", "0e3817aa8f75717c7cd02ccfe60c2ecf": "x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0.", "0e382264f8eaf08579b73f95e3883e36": "\\scriptstyle \\lambda _{B}", "0e383bb4f702745557b66ca9f69d3515": "b_{i}^{\\dagger }", "0e383f2c5d9881e1c8417c257c0af981": "\\mathbb {Z} ^{x}=\\{n\\in \\mathbb {Z} :X_{n}(x)=1\\}\\,", "0e3882068f0e733a2e6689f2eb8c7f91": "map(-,-)", "0e389fad6a525d62196c5af0f4e708d9": "n=p_{1}^{n_{1}}\\ p_{2}^{n_{2}}\\ \\cdots \\ p_{k}^{n_{k}}", "0e38d211b77423742da51f7beaa897b7": "{\\boldsymbol {\\omega }}_{2}", "0e3964e09b7a3d311d3bd7bd682592a1": "{\\rm {\\ 4VF_{5}+5SiO_{2}\\rightarrow 2V_{2}O_{5}+5SiF_{4}}}", "0e39cf1e9da4fea5ef94f061c420afab": "H-{\\frac {B}{8}}-{\\frac {B}{4}}---{\\frac {B}{2}}-------B", "0e39f0a2b76a2c43e4801ff5518d9367": "\\operatorname {H} ^{*}(\\mathbb {R} P^{\\infty };\\mathbb {F} _{2})=\\mathbb {F} _{2}[\\alpha ]", "0e3a846357326022783cc0ae44d238ac": "k_{y}=k~\\sin \\theta ~\\sin \\phi ", "0e3a9c695b4b12d38866e352d6147a08": "x={\\frac {a\\beta -\\alpha t^{2}}{a-t^{2}}}", "0e3ad1a187abcabcc794360b5a2c1be7": "|\\psi \\rangle _{A}|0\\rangle _{B}|A'\\rangle _{C}=(\\alpha |0\\rangle _{A}|0\\rangle _{B}+\\beta |1\\rangle _{A}|0\\rangle _{B})|A'\\rangle _{C}", "0e3af61ba058fc81d64432a47685efc4": "{2n \\choose n}={\\frac {(2n)!}{(n!)^{2}}}{\\text{ for all }}n\\geq 0.", "0e3b25193a349e31e044b072cde1e2fd": "(x,y)=(x_{0},y_{0})+t(x_{1}-x_{0},y_{1}-y_{0}),\\,", "0e3b5eb3b41180b0bc5bbc763c6fcbe9": "a^{2}+b^{2}+c^{2}+d^{2}=8R^{2}.", "0e3ba4c66cb27e6098a25bdd55659bd5": "tan\\delta ={\\frac {\\epsilon _{r}''}{\\epsilon _{r}'}}\\,", "0e3bcbbb056672b489498e98b5c4d234": "(x,y)\\,", "0e3c19f27936f55c4b0ce89b9a8bde47": "0.3W", "0e3cb4d9543a5190c02f0c39a2762414": "\\displaystyle P(I|c)=\\prod _{i}\\sum _{j}P(x_{i}|z_{j},c)P(z_{j}|c)", "0e3d703d747485a13ed85f57fda2dc8f": "X\\sim \\chi ^{2}(\\nu )", "0e3d95c373b75507711c5d58fbe36298": "\\mathbf {F} _{p^{n}}", "0e3df9f6b096b5005468115858a46e52": "=2E/N=p(N-1)", "0e3dfe5af34c3d3a23ba821596ef7b49": "\\Sigma _{n}a_{n}", "0e3e011c1151754d74a4289616db9f02": "e^{i\\theta }|1\\rangle ", "0e3e572ebbd47a5e09b62ce00c97b193": "M_{e,\\lambda }(\\lambda ,T)={\\frac {c_{1}\\lambda ^{-5}}{\\exp \\left({\\frac {c_{2}}{\\lambda T}}\\right)-1}}", "0e3ec46a8137f4bb7a0a7214dbf48d02": "{\\begin{aligned}t'&=\\gamma \\left(t-{\\frac {\\mathbf {r} \\cdot \\mathbf {v} }{c^{2}}}\\right)\\\\\\mathbf {r'} &=\\mathbf {r} _{\\perp }+\\gamma (\\mathbf {r} _{\\|}-\\mathbf {v} t)\\end{aligned}}", "0e3eccbc5241d6041d95e68f746abe10": "\\mathrm {P} (A\\cap B)=\\mathrm {P} (A)\\mathrm {P} (B)", "0e3f05e6d7fe07ac545afd9b0595006f": "\\mathbf {P} \\cdot \\left(\\nabla \\times \\nabla \\times \\mathbf {Q} \\right)-\\mathbf {Q} \\cdot \\left(\\nabla \\times \\nabla \\times \\mathbf {P} \\right)=\\nabla \\cdot \\left(\\mathbf {Q} \\times \\nabla \\times \\mathbf {P} -\\mathbf {P} \\times \\nabla \\times \\mathbf {Q} \\right)", "0e3f15091d753046f2573f961351f5f0": "ax^{2}+bx+c=0.", "0e3f3d83e9682e71ad49c5a4dff2b19c": "U_{E}(r)=-\\int _{\\infty }^{r}q\\mathbf {E} \\cdot \\mathrm {d} \\mathbf {s} ", "0e3f492334b43049635142320789ffa3": "\\textstyle \\rho ", "0e3f4b8fdb3385f44d93fc6ec0935ef6": "{G_{\\mathrm {total} }}={G_{1}}+{G_{2}}+\\cdots +{G_{n}}", "0e3f7cc2ce3ee08ad40d7955f74bb0e5": "a_{30}=p_{1}p_{4},", "0e407a544a82af598b71cdeb271c1bc8": "F=-dP\\cdot dA=\\rho \\cdot dA\\cdot dz\\cdot a", "0e408ec7923b18a8b2c4921bb6439b08": "{\\begin{smallmatrix}10^{5.96}\\approx 912,000\\end{smallmatrix}}", "0e40baa4a01c82a0c1646f71f22ae51a": "I(\\alpha h+\\beta k)=\\alpha Ih+\\beta Ik", "0e40ea9522ffef1cb516cc5cf5e3b951": "{\\frac {\\partial p_{1}}{\\partial t}}=D_{p}{\\frac {\\partial ^{2}p_{1}}{\\partial x^{2}}}-\\mu _{p}p{\\frac {\\partial E}{\\partial x}}-\\mu _{p}E{\\frac {\\partial p_{1}}{\\partial x}}-{\\frac {p_{1}}{\\tau _{p}}}", "0e4100e47300f75a7397a0950cce7d81": "y_{j}^{\\star }(\\mathbf {w} \\cdot \\mathbf {x_{j}^{\\star }} -b)\\geq 1,", "0e41624ffe7b1f9bcfdf84c51d2b3e58": "{\\bar {R}}=(x\\,{\\bmod {\\,}}2^{L})+2^{L}", "0e417785a33f28e2e3be5a0a74b1d333": "\\displaystyle {{\\mathfrak {h}}={\\mathfrak {k}}\\oplus {\\mathfrak {m}},}", "0e417e81ec79c8916b95f7e5ff02ad89": "\\int _{a}^{b}e^{nf(x)}\\,dx", "0e419894a4c35c539999ee44e8da0101": "l=({\\frac {A}{\\lambda g}})^{\\frac {1}{3}}", "0e41eeec38c96ed65649bd137a2e396c": "\\psi _{n}^{*}", "0e42174cf10d612474caa6118843c46c": "f(15x)=\\ln 15+f(x)", "0e42ebed04c01a43fa5951d60a5532bf": "\\mathbf {u\\times v} ={\\begin{vmatrix}\\mathbf {i} &\\mathbf {j} &\\mathbf {k} \\\\u_{1}&u_{2}&u_{3}\\\\v_{1}&v_{2}&v_{3}\\\\\\end{vmatrix}}", "0e431002c06f00e69a9ab8a234c93fd8": "\\beta _{t,k}|\\beta _{t-1,k}\\sim N(\\beta _{t-1,k},\\sigma ^{2}I)", "0e43383374dc49cc0556dab28af259e7": "f(t)=u(t)+iv(t).", "0e437f4c8bb87de63c2d99d1405245a3": "{\\dot {X}}_{a}\\,X_{b}+X_{a;b}=\\theta _{ab}+\\omega _{ab}", "0e43bfb90514c9073b22f978c80bbd6b": "\\scriptstyle {\\mathrm {R} }", "0e43f48d0084b9f951109beb09eeded8": "(d_{i})\\neq R", "0e43f92345417252175108d320557ba7": "\\zeta ^{\\prime }(-4)={\\frac {3}{4\\pi ^{4}}}\\zeta (5)", "0e44d5bc246f8a491a9f7fb02a36da4b": "|\\Omega \\rangle =\\sum _{i=1}^{n}|e_{i}\\rangle \\otimes |e_{i}\\rangle ", "0e4501d8227353c9bcfcec37baf96fbb": "\\beta ={\\frac {(1-m)(\\alpha +1)}{m}}.", "0e453023bccb249dcc1ad5c5ead0c658": "T_{d}^{u}", "0e455bd8e5345a4569d3dd8b6f70de3a": "{\\boldsymbol {x}}|_{t=0}={\\boldsymbol {x}}_{0}", "0e459281a00a90d4a7ab45569dbaf153": "K_{d}=k_{r}/k_{f}", "0e4625c586d92a93fb161c1ecfd57b39": "\\textstyle D(x)", "0e4652af07e520df72d81a6c65f28a10": "E_{x}={\\frac {k_{o}^{2}\\varepsilon _{r}-k_{z}^{2}}{k_{o}^{2}-k_{z}^{2}}}[{\\frac {jk_{xo}k_{z}}{\\omega \\varepsilon _{o}\\varepsilon _{r}}}(A\\ e^{-jk_{x\\varepsilon }w}+B\\ e^{jk_{x\\varepsilon }w})+{\\frac {m\\pi }{a}}(C\\ e^{-jk_{x\\varepsilon }w}+D\\ e^{jk_{x\\varepsilon }w})]e^{-jk_{xo}(x-w)}sin({\\frac {m\\pi }{a}}y)e^{-jk_{z}z}\\ \\ \\ \\ \\ \\ \\ \\ (39)", "0e46659fd87e5c6e6b0591ebded4ffbe": "\\nu _{x}", "0e472905fd2ff578258fb384fba0e9f9": "\\rho _{2k}(y):=\\rho \\left({\\frac {y-x_{2k}}{\\delta }}\\right)", "0e47dfeb284dc257cafaa427ee4568c6": "{\\hat {h}}^{ab}={\\hat {h}}^{ba}=m^{b}{\\bar {m}}^{a}+{\\bar {m}}^{b}m^{a}", "0e481442918de87548cd0cfc660f8c1c": "\\int k{\\frac {dy}{dx}}dx=k\\int {\\frac {dy}{dx}}dx.\\quad ", "0e483c1ecb634e62fd5cf16ce8ea27f5": "f({\\boldsymbol {\\mu }},{\\boldsymbol {\\Sigma }}|{\\boldsymbol {\\mu }}_{0},\\lambda ,{\\boldsymbol {\\Psi }},\\nu )={\\mathcal {N}}\\left({\\boldsymbol {\\mu }}{\\Big |}{\\boldsymbol {\\mu }}_{0},{\\frac {1}{\\lambda }}{\\boldsymbol {\\Sigma }}\\right){\\mathcal {W}}^{-1}({\\boldsymbol {\\Sigma }}|{\\boldsymbol {\\Psi }},\\nu )", "0e486dd0adf1b09ff606d1bc43e8ff87": "{\\begin{aligned}x_{ji}|G_{j}&\\sim \\sum _{k=1}^{\\infty }\\pi _{jk}F(\\theta _{k}^{*})\\end{aligned}}", "0e487858411e8e712068596b3e346cb3": "{n \\choose i,j,k}={\\frac {n!}{i!\\,j!\\,k!}}\\,.", "0e48d1d898a137f8f0af6d701499c1b5": "\\lambda _{1}+\\lambda _{2}+\\lambda _{3}=1\\,", "0e492d0d4471ed40d44a2335a40ec3bd": "P=k_{3}T\\qquad (3)", "0e49b63bf7dc3972b1fb7057cbe8ddc8": "\\rho _{xx'}={\\frac {\\sigma _{T}^{2}}{\\sigma _{X}^{2}}}=1-{\\frac {\\sigma _{E}^{2}}{\\sigma _{X}^{2}}}", "0e4a42cd12bffa76aab9d472302e3af9": "{\\frac {kT}{e}}0\\\\0&{\\mbox{ otherwise.}}\\end{cases}}", "0e54eb737ec974bb7c0251abd16b1197": "{\\mathfrak {S}}_{n},", "0e54f0b9318da4fa67fd7079b1ac7fcb": "p_{c}\\gets \\underbrace {(1-c_{c})} _{\\!\\!\\!\\!\\!{\\text{discount factor}}\\!\\!\\!\\!\\!}\\,p_{c}+\\underbrace {\\mathbf {1} _{[0,\\alpha {\\sqrt {n}}]}(\\|p_{\\sigma }\\|)} _{\\text{indicator function}}\\overbrace {\\sqrt {1-(1-c_{c})^{2}}} ^{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\\text{complements for discounted variance}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!}\\underbrace {{\\sqrt {\\mu _{w}}}\\,{\\frac {m_{k+1}-m_{k}}{\\sigma _{k}}}} _{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\\text{distributed as}}\\;{\\mathcal {N}}(0,C_{k})\\;{\\text{under neutral selection}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!}", "0e552a46a02bc20a57cf11a164ab7255": "{\\begin{aligned}\\mathbb {P} {\\biggl (}\\bigcup _{i=1}^{n}A_{i}{\\biggr )}&{}=\\sum _{i=1}^{n}\\mathbb {P} (A_{i})-\\sum _{i0})^{2}\\right]=E[X_{n}^{2}]\\,P(X_{n}>0).", "0e57387985e70b02d850ab34c2c20620": "\\left({\\frac {27379}{8658}}\\right)^{2}=10+{\\frac {1}{8658^{2}}}.", "0e574efea80a3f6ab02c82627206f751": "B\\mapsto +A-BB--B-A++A+B", "0e57800c01e7644dc1ceefedbdfb37f7": "10^{9}", "0e57bea2518b074117a36ecce418afcd": "w_{i}(x)\\ :=\\ {\\frac {u_{i}(x)-m_{i}}{M_{i}-m_{i}}}", "0e582ecca47b8f41349d57134732cfab": "s_{\\lambda }", "0e58a0b8b84bdf25dedf3d49c07a2072": "~A~", "0e58c7ec919aa81794e4b8cebd90e3bb": "{\\mathcal {C}}=", "0e58d6499420076c9ca91c45cc092c61": "\\mathrm {~_{92}^{238}U} \\rightarrow \\mathrm {~_{90}^{234}Th} +\\mathrm {~_{2}^{4}He} ", "0e58ed1250d654c85c529cb6fd71fdf4": "v^{2}=t^{3}+\\left({\\frac {3-A^{2}}{3B^{2}}}\\right)t+\\left({\\frac {2A^{3}-9A}{27B^{3}}}\\right)", "0e590a0b7ac9fd4d5ef629caa808f11b": "\\displaystyle {\\mu _{F\\circ h^{-1}}=0,}", "0e59586c4297b74a76f8da1396ddf153": "f(t)={\\begin{cases}t^{1/3}&{\\text{if }}t>({\\frac {6}{29}})^{3}\\\\{\\frac {1}{3}}\\left({\\frac {29}{6}}\\right)^{2}t+{\\frac {4}{29}}&{\\text{otherwise}}\\end{cases}}", "0e596f42c8962f8ac39913dd61ef8ab8": "F_{j}(a,b)=a\\cdot b", "0e59a0d23041499b470c639cfd48f6ab": "{\\widehat {R}}_{x},{\\widehat {R}}_{y},{\\widehat {R}}_{z}", "0e59d15dbc38599ddbf2fe1296c53fef": "\\int _{-\\infty }^{\\infty }xf(x)\\,dx.\\qquad \\qquad (1)\\!", "0e59daf62a798bcc308af5087b40681c": "3^{4}=81", "0e5a34fbce6d65c2e1fa0d7fa1aaecc7": "\\Omega (n)=\\Omega {\\sqrt {n+1}}", "0e5a52b2f19124fbec2b75a99c2a8edf": "M\\in \\left[0,l-1\\right]", "0e5aa17cc9306c5f67d6ed823a2471aa": "T=Rtan({\\frac {\\Delta }{2}})", "0e5afc54aaa1828ea8d4c20428e411ed": "E_{u}(r,\\theta ,\\phi )~=~2\\pi j~(k~\\cos \\theta )~{\\frac {e^{-jkr}}{r}}~E_{u}(k~\\sin \\theta ~\\cos \\phi ,k~\\sin \\theta ~\\sin \\phi )~~~~~~~~~~~~(2.2)", "0e5b2e44cbb7455368a30b018c758a42": "e^{4\\kappa ^{2}}+2e^{3\\kappa ^{2}}+3e^{2\\kappa ^{2}}-6", "0e5b5133b4e91f22b9257469755ce2db": "Av{\\hat {a}}r({\\hat {\\beta }}_{FD})={\\hat {\\sigma }}_{u}^{2}(\\Delta X'\\Delta X)^{-1},", "0e5b5938e813ae78ce4e9f591c85f3af": "\\scriptstyle {\\frac {1}{\\lambda }}{\\sqrt {\\sigma /(\\rho g)}}", "0e5b98132ffabb056a20b633d72f4a4b": "|E(S,T)-{\\frac {d\\cdot |S|\\cdot |T|}{n}}|\\leq d\\lambda {\\sqrt {|S|\\cdot |T|}}", "0e5b9d285e57ab81b1406ea35c1b37da": "\\,\\Sigma _{xx}=Cov(X)", "0e5bdcf11353a8b33887cb0ab7c1c3f0": "\\scriptstyle {R_{a}^{0}}", "0e5c4a7842cdfee21f996a6590c615bc": "\\scriptstyle {\\mathcal {F}}_{x}", "0e5c592f6179d77efcfac5a23e61e91b": "{\\frac {\\partial k_{i}}{\\partial t}}={\\frac {m_{}}{m_{0}+t}}{\\frac {k_{i}}{\\sum _{j\\in Local}k_{i}^{}}}", "0e5c8d944ef669d1162dc08a4ff901a3": "Y(u,v)=r\\,v\\,\\sin 2u,", "0e5cc2a53fc6f8fe7d3953a8b1e3d35a": "J_{f}:=\\left\\langle {\\frac {\\partial f}{\\partial z_{i}}}:1\\leq i\\leq n\\right\\rangle .", "0e5ce1029c3413777fc598d3a940c156": "\\Gamma _{8}'=\\left\\{(x_{i})\\in \\mathbb {Z} ^{8}:{{\\textstyle \\sum _{i}}x_{i}}\\equiv 0({\\mbox{mod }}2)\\right\\}\\cup \\left\\{(x_{i})\\in (\\mathbb {Z} +{\\tfrac {1}{2}})^{8}:{{\\textstyle \\sum _{i}}x_{i}}\\equiv 1({\\mbox{mod }}2)\\right\\}.", "0e5d00366864a19737b2d9e211fe8321": "C_{p}^{\\ominus }=A+BT+CT^{-2}", "0e5d01486e4dad226ca2dd44bd9307dd": "{dy \\over dt}=-\\alpha xy", "0e5d11d44a887140cf6871818dcca9a0": "\\Delta w_{ij}~\\propto ~\\langle x_{i}y_{j}\\rangle -\\epsilon \\left\\langle \\left(c_{\\mathrm {pre} }*\\sum _{k}w_{ik}y_{k}\\right)\\cdot \\left(c_{\\mathrm {post} }*y_{j}\\right)\\right\\rangle ,", "0e5d5cf88dcf28c29f7e6ef458c10383": "x_{k+1}=x_{k}-a", "0e5e53628cd495c704f319535cd6040b": "xs=0", "0e5ec19a3f022ce99b71fcbcf81a59ba": "\\underbrace {(A^{T}A)^{-1}A^{T}} _{A_{\\text{left}}^{-1}}A=I_{n}", "0e5ec778b4c37d0f3d59a1f8dd690877": "\\lambda _{1},\\,\\lambda _{2}", "0e5eea747ad8dc040b571d59796b4eb3": "IMM_{i-1}(S_{x,{i-1}},a)", "0e5f7f4c06ba1ac30c3de64f06de38b6": "(x_{0},y_{0},z_{0}).", "0e5f8aba2621357e79b7955122bd9aad": "{\\boldsymbol {R}}={\\frac {1}{m_{0}}}\\sum _{k=1}^{N}\\ m_{k}{\\boldsymbol {x_{k}}}\\ ;", "0e5fc02a6555377d76c347a2315fe2f5": "n_{k}^{(-n)}", "0e5fc78b51ca8c03b44eb8de293a0117": "C_{yy}(y_{i})", "0e5fcba9d2fe11b4e36f4f127d5afd9c": "{\\mathfrak {P}}^{27}", "0e5fdd452a2fe2dec8673b2a9f9490ee": "\\Gamma '(m+1)=m!\\left(-\\gamma +\\sum _{k=1}^{m}{\\frac {1}{k}}\\right)\\,.", "0e5febdf21ee53bc0903071fe3a80923": "~\\cos ^{2}(x)~", "0e5fef39fa2d8f23849aedab2700298e": "(C_{e})_{m+1}", "0e6000cb1205b89cb03091572d394bde": "m=p^{k}", "0e6057401cf26e7c5635402da5d7d045": "k=(-3\\pm 13)\\times 10^{-5}", "0e6105f1f5ac555d624847439d38804a": "G=\\langle R_{G}\\mid S_{G}\\rangle ", "0e610aaf7e1a947d4ad1a2fd92c4807c": "{\\mathit {KP}}({\\mathit {PRIMES}})\\leq 176", "0e612125b2af2057bd855505125e99c2": "f(m-h)=f(m)-f'(m)\\cdot h+f''(m)\\cdot {\\frac {h^{2}}{2!}}-f'''(m)\\cdot {\\frac {h^{3}}{3!}}+\\dots ", "0e61495531347f2ec0a4fc519312339e": "{\\mathfrak {gl}}(n)\\,", "0e61b502087a025beb3fb74eb31af787": ".\\qquad \\underbrace {NP/N,\\;\\quad N} ,\\;\\qquad (NP\\backslash S)", "0e621e0e8531edc16cf1c11d314468ec": "\\tau \\,", "0e627122246550af59bad2416c5f60a6": "{\\overline {P}}=(a,-b-h(a))", "0e6286f13c24ac8baf48afb5b28bfe6d": "{\\frac {1}{\\rho _{max}}}", "0e6359bd8d2685435913e7413a1a001c": "0\\geq \\alpha \\geq 1", "0e63aaca16fff0c9bca8aa3a3ce88538": "\\{e\\}", "0e63d1d741a847802d6dffc6c1744d53": "F_{\\mu \\nu \\rho }=\\partial _{\\mu }A_{\\nu \\rho }+\\partial _{\\rho }A_{\\mu \\nu }", "0e63d3069a2a8b90b96877f9b4264fe1": "\\omega =2\\pi f\\,", "0e63e3f173ba95881dab44b02456e785": "d=d_{1}+d_{2}+\\cdots +d_{n},\\;\\;d_{1}\\geq d_{2}\\geq \\cdots \\geq d_{n}", "0e63eb64a1ab439e81a2505d60a95589": "m_{00}^{S}", "0e64090fbaaf05e3ed96bebfbd1f12eb": "{\\begin{aligned}d(x,y)&\\leq d(x,T(x))+d(T(x),T(y))+d(T(y),y)\\\\&\\leq d(x,T(x))+qd(x,y)+d(T(y),y)\\end{aligned}}", "0e640f7338892b86b49a775df12a99d4": "dS^{2}=\\left(1-{\\frac {\\ell _{P}^{2}}{r^{2}}}\\right)c^{2}dt^{2}-{\\frac {dr^{2}}{1-{\\ell _{P}^{2}}/{r^{2}}}}-r^{2}(d\\Omega ^{2}+\\sin ^{2}\\Omega d\\varphi ^{2})", "0e6426d647a469ef35981f6791b83613": "S_{2m+1}={\\frac {\\left(1+{\\sqrt {2}}\\right)^{2m+1}+\\left(1-{\\sqrt {2}}\\right)^{2m+1}}{2}}.", "0e642f75f7debf5032bd936f63b0134b": "\\left(A-sI\\right)^{-1}", "0e6460d58708bdc32db82dd88100f86a": "K_{i+1}", "0e6466f3433f7363e327045917bbe102": "\\Pi _{P}=\\{z\\in \\mathbb {C} \\,\\colon \\,Oz\\cap L_{P}=\\varnothing \\},", "0e64a72837afa7c8e4d25339d070e959": "S[k]={\\frac {1}{P}}\\int _{P}s_{P}(t)\\cdot e^{-i2\\pi {\\frac {k}{P}}t}\\,dt", "0e64ab01a3fb91403ec8e6c0e54dc736": "n=0", "0e64d03e6dff82ca747c2e8e269fe52b": "\\scriptstyle i\\;=\\;1,\\,2,\\,3,\\,4,\\,..,\\,n", "0e651ccb9422e0fbce88e3ba1440907a": "{\\hat {\\boldsymbol {\\beta _{1}}}}=\\mathbf {X_{1}} ^{+}(\\mathbf {y} -\\mathbf {X_{2}} {\\boldsymbol {\\beta _{2}}})", "0e661bfdf4ef0af694145d65d68de5fe": "P_{1}=(1,{\\sqrt {2}})", "0e664cd80eb5c9005092ef58d05238ca": "s\\in \\{0,0.5,1,1.5,\\ldots \\}", "0e66565dabe8def123b5db70d6675171": "f(x_{1},\\ldots ,x_{d})\\ ", "0e667ad4f5fd2ea7b2031ae6108d39e8": "{\\begin{aligned}s(t)&=\\Re \\left\\{\\left[I(t)+iQ(t)\\right]e^{i2\\pi f_{0}t}\\right\\}\\\\&=I(t)\\cos(2\\pi f_{0}t)-Q(t)\\sin(2\\pi f_{0}t)\\end{aligned}}", "0e66cc1365bf91848c84919f8cb72993": "X_{\\mathrm {L} }=-X_{\\mathrm {S} }.\\,", "0e66d3571abdb50c9039b3fb633f0cd5": "{\\underline {p}}", "0e66da71e3f8b765d70cc119d9ae2b1b": "\\operatorname {E} [A_{t}]=\\operatorname {E} [S_{t}]+{\\frac {\\mu \\alpha _{t}(1-\\delta _{t})}{\\epsilon +\\mu \\alpha _{t}(1-\\delta _{t})}}(S_{G}-\\operatorname {E} [S_{t}])\\;.", "0e671582e02ba55e3e8bf04849361941": "Z_{i}(N)/Z_{i}(N-1)=e^{-\\mu /kT}\\,", "0e673b3ec895573f88973739eb34edd2": "G_{n}(x)={\\sqrt {n}}(F_{n}(x)-F(x))\\,", "0e67c4bb9bb313f3abe247da26fc8773": "\\Phi (\\rho ,\\theta )={\\frac {-\\lambda }{4\\pi \\epsilon }}\\left\\{2\\ln \\rho +\\ln \\left(1-{\\frac {\\rho ^{\\prime }}{\\rho }}e^{i\\left(\\theta -\\theta ^{\\prime }\\right)}\\right)\\left(1-{\\frac {\\rho ^{\\prime }}{\\rho }}e^{-i\\left(\\theta -\\theta ^{\\prime }\\right)}\\right)\\right\\}", "0e6858173e8c6f41ac6a1554586d75dd": "[A,B]", "0e6868f4117768e71d1fbf50900987ed": "n>1.", "0e695b4e85dfd6b87c76d4c0e79a48fe": "\\lambda _{1}=\\lambda _{2}=\\lambda \\,", "0e6975c9fac4c9bbfd30c785b9db0d92": "1-2\\times 10^{8}", "0e69c1004b3b608015fff068369e54c6": "(g^{a},g^{b},g^{ab})", "0e6a17a54034d388482311b96343dbd1": "l{\\sqrt {3}}>{\\tfrac {265}{153}}\\,.", "0e77eab22bef534de433ebdef7de8f33": "dY", "0e780e22992a79b7fa9ab719b1192869": "c_{1}(V)\\not =0.", "0e7847bd0023db3a7fd5cfe1bfd2eaae": "X={\\mathcal {F}}^{n}", "0e78d84b4c7794c8842203b24e3b128e": "{\\hat {y_{j}}}=h(x_{j})\\approx y_{j}", "0e78ecfa41b1af1c7c1aff112932c801": "\\nabla \\cdot \\mathbf {E} =4\\pi \\rho _{\\mathrm {e} }", "0e791b0d44ebe091f4f06082320a15b6": "T=a+b\\log _{2}{\\frac {D}{W}}.", "0e7943219b1a626638d7aae7881e1b72": "{\\text{MI}}={\\frac {\\text{PNP}}{\\sqrt {F_{c}}}},", "0e7955975eb9e08c82eb9230bd25926a": "X\\mathbf {\\operatorname {s} } Y", "0e7a2013cddaf77ef0d962b6c9f445f4": "\\Gamma =\\partial \\Omega ", "0e7a22f2ba9cc6801a437bbac55a4b7c": "-302\\pm 2.8\\%", "0e7a2d7fe661a952bebfc4cfbfd30477": "y_{3}=-3", "0e7a6d507eeee6158d5811e47583f2bd": "\\;{}_{2}F_{1}(a+1,b;c;z)-\\,{}_{2}F_{1}(a,b;c;z)={\\frac {bz}{c}}\\;{}_{2}F_{1}(a+1,b+1;c+1;z)", "0e7a846b2b3af579cc2a173e24fadb45": "x_{1}^{2},x_{2}^{2},\\ldots ,x_{n}^{2}", "0e7a85794228b8f302e066b928731cc8": "G_{\\text{III}}=biL_{\\text{p}}=G_{\\text{III}}^{\\circ }e^{{-U^{*}/k(T-T_{\\text{0}})}-(K_{\\text{g}}/T\\Delta T)}", "0e7a921229386a13d9aea1c8bec8f21d": "(g\\circ f)^{-1}=f^{-1}\\circ g^{-1}", "0e7a963235ccd9827b5eb34131ab7d1b": "I(f,g,h)\\leq I(f^{*},g^{*},h^{*})", "0e7ad70b43873a81aa4a9d782b748489": "{\\begin{matrix}x=\\lambda _{1}x_{1}+\\lambda _{2}x_{2}+(1-\\lambda _{1}-\\lambda _{2})x_{3}\\\\y=\\lambda _{1}y_{1}+\\lambda _{2}y_{2}+(1-\\lambda _{1}-\\lambda _{2})y_{3}\\\\\\end{matrix}}\\,", "0e7ae4b871c5bb3a045b883dca7ff8e2": "\\sigma _{i}^{(n)}", "0e7aee4ad8d55cb6c6b95068735ef07e": "f_{1}=-f", "0e7af9e2cf8c217ad49efa8a35643328": "{\\frac {V_{\\mathrm {i} }}{V_{xn}}}=\\left(1+{\\frac {\\delta Z}{Z_{0}}}+\\delta Z\\delta Y\\right)^{n}", "0e7b3dea352b4a1671800721d4eb2dcb": "I_{L1}-I_{L2}*0.5-I_{L3}*0.5+j*{\\frac {\\sqrt {3}}{2}}*\\left(I_{L2}-I_{L3}\\right)", "0e7b511e89a9a8d01b831fece25c94d8": "\\beth _{\\alpha }(\\kappa )", "0e7b85202267a353810a9053cf6e6eba": "\\omega \\notin \\mathbb {D} ", "0e7bcfce42627811c1d4d16d7cd4d2d5": "\\phi (-x)\\phi (x)\\,", "0e7be073a8345e1c52de04c0948ffe10": "b_{3}", "0e7cc0170848276a8b767d2bea2b97a4": "\\displaystyle f'(x_{0})=0.", "0e7cda6cad88d6351c1c73727a869a2e": "{\\frac {H_{N,q,s-1}}{H_{N,q,s}}}-q", "0e7d2878d4316ea7b314af92acb0e03a": "K_{2}(R)", "0e7d5a107407ddd2bf8389cb0b95d5c1": "h_{t}(x,y)", "0e7dacb33a49a3af1c1ec6dd2fe6409b": "\\gamma \\in {\\hat {\\Gamma }}", "0e7dccda8d9698d24d86cdaf9bf8b1aa": "\\scriptstyle {\\mathcal {G}}\\,\\subset \\,{\\mathcal {F}}", "0e7e4bf766655feac26fe2ddc47fb2dd": "\\mathbb {R} \\,", "0e7e4e8f4bed0fbebbb09c42c872b224": "A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}A_{2}B_{2}A_{2}^{-1}B_{2}^{-1}\\cdots A_{n}B_{n}A_{n}^{-1}B_{n}^{-1}=1", "0e7e57320b041af765bf24ac7a92a362": "P_{D}", "0e7e968f153c9d186c0303c4e770d4e9": "V_{Th}=I_{No}R_{No}\\!", "0e7eaf8393b157790c06d73ff40ce0c1": "\\mod \\,n", "0e7fc4ef11a3ec0a05fc14e422899621": "\\chi =-30", "0e7fc75092c22ffcaa1d4b1a6470bead": "\\chi ^{(\\rho )}\\left({\\mathcal {P}}\\left\\{e^{\\int _{\\gamma }A}\\right\\}\\right)", "0e80372b33db5f1db461b8829dea34f4": "2\\cdot 6^{2}+2\\cdot 6-1=2\\cdot 6^{2}+6+5", "0e8046f4d4248b53fb47cb9d4821a613": "\\mathbb {Z} _{12}", "0e80a87a6dfe47dd63066c421a5bef1e": "E(t,T)=E(a_{\\rm {T}}\\,t,T_{0})\\,.", "0e80b4379d0c15b15a6ca966eeb69fcd": "T_{ij}^{(2)}=s_{ik}w_{kj}-w_{ik}s{kj}", "0e80e9d18de2044aad15a99744d2ac5b": "(x_{1},y_{1}),\\ldots ,(x_{m},y_{m})", "0e815a3e8a7a4ede8e1273e3d3b732e1": "\\sigma _{a}:=\\sigma _{g}+{\\frac {k_{h}}{\\sqrt {l}}}+K\\varepsilon _{\\rm {p}}^{n},", "0e81a4aa5171a2240b1f29856b8cd00e": "c_{o}", "0e81cec3a7a1611d7a0ef655e542a3ae": "M_{refl}=\\left({\\begin{matrix}r_{p}e^{j\\chi _{p}}&0\\\\0&-r_{s}e^{j\\chi _{s}}\\\\\\end{matrix}}\\right)", "0e81e19fc5410f6fb1e99fff9e5bf0df": "A\\Rightarrow _{amb}^{*}B", "0e821792f27eb7013db23cc809852e55": "F(x)=\\sum _{x}f(x)\\,", "0e828e00b5680078364a6646db15b076": "\\tau ^{2}", "0e82d5bfb5e62cfcc905bcabf32524f4": "\\neg A\\phi \\equiv E\\neg \\phi ", "0e836d236d498955e959318bcbdb7319": "\\lim _{x\\to \\infty }{\\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}=\\lim _{x\\to \\infty }{\\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}=\\lim _{x\\to \\infty }{\\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}=\\dots .", "0e838abef7a32fb45ee727b9dd3a6995": "\\partial ={\\frac {\\partial }{\\partial z}}={\\frac {1}{2}}\\left({\\frac {\\partial }{\\partial x}}-i{\\frac {\\partial }{\\partial y}}\\right)", "0e8398a1ab8ab23b4ee460c8161887a9": "{\\mathcal {O}}_{\\lambda +\\rho }", "0e83d570202919199cd6861e3a4e9f82": "\\mathbf {r} _{3}\\,", "0e83e694ccff9283de93438758996837": "\\left(x_{1},y_{1}\\right)\\ =\\left({\\frac {-b_{1}m}{m^{2}+1}},{\\frac {b_{1}}{m^{2}+1}}\\right)\\,", "0e83ed0ae798208d3a4015f6d02c41d9": "\\det(V(\\lambda _{2},\\ldots ,\\lambda _{n}))=\\prod _{3\\leq j\\leq n}(\\lambda _{j}-\\lambda _{2})^{2}\\prod _{3\\leq i0", "0ea40f3876f15cbbd7948bf38a0767cc": "\\sum _{s}\\langle X_{s}(z)\\Phi (z_{1},v_{1})\\cdots \\Phi (X_{s}v_{i},z_{i})\\cdots \\Phi (v_{n},z_{n})\\rangle =\\sum _{j}\\sum _{s}\\langle \\cdots \\Phi (X_{s}v_{j},z_{j})\\cdots \\Phi (X_{s}v_{i},z_{i})\\cdots \\rangle (z-z_{j})^{-1}.", "0ea41fdeff58dccbf0cbc3363d7ed4cb": "{\\mathcal {Z}}(\\mu )\\,=\\,\\sum _{N=0}^{N_{S}}\\,\\exp \\left(\\,{\\frac {N\\mu }{k_{B}T}}\\right){\\frac {\\zeta _{L}^{N}}{N!}}\\,{\\frac {N_{S}!}{(N_{S}-N)!}}\\,", "0ea4545a40750c2254fee418264394e5": "i=\\{1,\\ldots ,n\\}", "0ea499b797fa117da837bda69406d8c3": "s_{y}(t)=-{\\frac {mg}{k}}t+{\\frac {m}{k}}(v_{yo}+{\\frac {mg}{k}})(1-e^{-{\\frac {k}{m}}t})", "0ea58128218c9e6e3382c31b1b386426": "p'_{n}(x_{i})", "0ea6088dd979025a8a95bd13fb5649bb": "\\sigma _{{\\text{x}},{\\text{y}}}", "0ea66280e98f872e896d80254f66649f": "~{\\hat {\\Theta }}~", "0ea6e928a1a396d44cfadaa79f77aa24": "d=-\\left\\lfloor {\\frac {m}{2}}\\right\\rfloor ", "0ea71c8e2f6e47d54943ab5bf85c0593": "N(t)=N_{0}\\left({\\frac {1}{2}}\\right)^{t/t_{1/2}}=N_{0}2^{-t/t_{1/2}}=N_{0}e^{-t\\ln(2)/t_{1/2}}", "0ea72b4ffc1411af2cad90045be51d22": "{\\begin{aligned}\\nabla \\cdot \\mathbf {E} &=0\\quad &\\nabla \\times \\mathbf {E} =\\ -&{\\frac {\\partial \\mathbf {B} }{\\partial t}},\\\\\\nabla \\cdot \\mathbf {B} &=0\\quad &\\nabla \\times \\mathbf {B} ={\\frac {1}{c^{2}}}&{\\frac {\\partial \\mathbf {E} }{\\partial t}}.\\end{aligned}}", "0ea74eeb53d853501f568548900bb6a5": "V_{\\text{out}}=V_{\\text{in}}\\left(1+{\\frac {R_{2}}{R_{1}}}\\right)\\,", "0ea7641bf655d3747f7b93ee1f162888": "\\varepsilon _{FY}", "0ea77b901652b89fda0e00a6deaa5100": "ds^{2}=-{\\frac {4ma\\,\\log(r)}{\\pi \\,(1+a^{2}u^{2})}}\\,du^{2}-2du\\,dv+dr^{2}+r^{2}\\,d\\theta ^{2},", "0ea77d56ce7fe8633de9e82863bf023a": "y(x)=g(f^{-1}(x))", "0ea7af9a703d10b174cec7d427ac8fd8": "{\\mathfrak {P}}^{1}", "0ea8222a199c773ccac25000c065b29f": "a_{1}\\neq 0", "0ea8370254d92d397cac00e9a24b0d5d": "f={\\frac {h}{L}}", "0ea89006232ee5198a67fe066f4a4fcb": "\\sigma (q)=\\sum _{n\\geq 0}{q^{(n+1)(n+2)/2}(-q;q)_{n} \\over (q;q^{2})_{n+1}}", "0ea8a45bfadd623a462a0ad2fa2398b4": "\\phi ={\\sqrt {\\frac {\\chi ^{2}}{N}}}", "0ea8ee7e7edc006d1506e9c91978a29b": "F_{\\overline {z}}", "0ea904611993a9c83e48f06369decb56": "\\mathbf {\\ddot {r}} ", "0ea91bab06b6e13b1ba12c54d465881c": "\\ |\\psi \\rangle ", "0ea9255f2f01bb39e43b5a9c22985569": "p\\rightarrow p_{\\infty }", "0ea9436afe559a8cf0358f5cd0a4b745": "(a_{n+N})", "0ea98df10970c673ab6ddd8172b4b008": "\\mathbf {y} ^{1}", "0ea9ba1994230367d549f17d25a18fbd": "A(<\\lambda )", "0eaa1b9d4512f660f8e1eb9624672fc8": "|k_{f}|=|k_{i}|", "0eaa6c551efd2d5ff66244802addc954": "\\Delta f_{\\mathrm {actual} }", "0eab853da730fe3600c61987eee3002d": "d={\\frac {i}{(1+i)}}\\,", "0eac433369f47c36e5875937ec7c17be": "x=6\\in Z_{11}", "0eac5ec7886d3a7d662ff0307f6cf978": "H_{\\mathrm {grav} }=-\\int _{0}^{R}{\\frac {4\\pi r^{2}G}{r}}M(r)\\,\\rho (r)\\,dr,", "0eac7603efc07fcae78802f010c80224": "\\langle p\\rangle _{S}={\\frac {1}{2}}(p+{\\overline {p}}),", "0eacb2fece59b0e087e845f1f55c7067": "C'(B,\\succeq )~=~\\left\\{x\\in B:x\\succeq y~\\forall y\\in B\\right\\}", "0eacc199b1fc9db6afab38ccc227d72e": "X:\\mathbb {C} ^{n}\\rightarrow \\mathbb {C} ^{n},\\,", "0eaccda6096d719f3cb13763d10c297e": "d_{3}\\times b^{3}+d_{2}\\times b^{2}+d_{1}\\times b+d_{0}", "0eace2000256d1de6702664352976e11": "F=\\pi r\\lambda \\sin(2\\theta )\\,\\!", "0ead594b15aece8e5aadeb93d13b78ad": "\\theta (0)", "0ead7d3650792a3226b74adc4aae6644": "F(u)=-mh^{2}u^{2}\\left({\\frac {\\mathrm {d} ^{2}u}{\\mathrm {d} \\theta ^{2}}}+u\\right).", "0ead9b4466b2633025cd99e9faedcb07": "I=\\sum _{j}I_{j}", "0eadd2bea2d38de75b20ebaeff25fad4": "D_{\\mathbf {a} }(f\\circ g)=D_{g(\\mathbf {a} )}f\\circ D_{\\mathbf {a} }g,", "0eadda835ed7183e777155529bbed555": "H_{Moon}\\!\\,", "0eade14ee80fed4210e48965652baf76": "h={\\frac {X^{2}}{a^{2}}}+{\\frac {Y^{2}}{b^{2}}}+{\\frac {Z^{2}}{c^{2}}}=1,", "0eade55b64f11d713e935351b9434aaa": "{}^{\\mathrm {N} }\\mathbf {v} ^{\\mathrm {P} }={\\frac {{}^{\\mathrm {N} }\\mathrm {d} }{\\mathrm {d} t}}(\\mathbf {r} ^{\\mathrm {OP} })", "0eadfb15d26ff037d56ec5650fedc41c": "\\delta :Q\\times \\Gamma \\rightarrow {\\mathcal {P}}(Q\\times \\Gamma \\times \\{L,R\\})", "0eae10467cf4176e3ac165c77fdce3ad": "\\scriptstyle 1/f", "0eae39a6c136aa55e51cbc81a413ce0d": "t_{th}", "0eae599a5797c4f24b9d8bde86ef434b": "q=\\infty ", "0eae5cdc121f487d0428983593133c63": "\\Gamma =0", "0eaf57088b3bfe3d6038300067ed2f51": "A=b\\cdot h.", "0eaf9b08cbadf1e158810847cba8f84e": "{\\overline {M}}=0", "0eafc8452643a846ba7b67a4f7570af6": "{\\begin{aligned}\\operatorname {ev} _{A}{\\bigl (}p(t)I_{n}{\\bigr )}&=\\operatorname {ev} _{A}((tI_{n}-A)\\cdot B)\\\\p(A)&=\\operatorname {ev} _{A}(tI_{n}-A)\\cdot \\operatorname {ev} _{A}(B)\\\\p(A)&=(A\\cdot I_{n}-A)\\cdot \\operatorname {ev} _{A}(B)=0\\cdot \\operatorname {ev} _{A}(B)=0.\\end{aligned}}", "0eafcfd59844a23aeb530fbcf1f44346": "{\\frac {N^{2}+N}{2}}", "0eafde8ccfb1918122abaefe9eb3e305": "\\Delta _{6}", "0eafee684a7fd2ca9f99533b805ba871": "{\\frac {x}{y}}=\\sum {\\frac {1}{2a_{i}+1}},", "0eb01a1012ba4254e5648ff757c8a8bd": "4N-1", "0eb039e9fa57bc62a2ffd651e4e06d2f": "0={\\frac {1}{2}}+\\cos x+\\cos 2x+\\cos 3x+\\cdots .", "0eb0423bffb0ce0a63efafbeab654932": "u_{1}(\\mathbf {q} )", "0eb0b8ed9242ea5a66e657a41508339c": "x(x^{2}+y^{2})=cx^{2}+dy^{2}.\\,", "0eb0d988746ad9b46b3a64ffbf3af50e": "\\scriptstyle {\\hbar }", "0eb0e5797824b9acc1b6c778994a2f06": "{{\\Delta }E_{t}(S_{t+k})}/{S_{t}}", "0eb115e04e172ce70a740f80d7c89222": "\\not \\rightarrow ", "0eb13082fcc7fcc7941c377a4c7a8b51": "Z_{F'}(x_{1},x_{2},\\dots )=\\left({\\frac {\\partial }{\\partial x_{1}}}Z_{F}\\right)(x_{1},x_{2},\\dots )", "0eb1701a503a9c9abefb212f4c384a2b": "\\mathbf {TK} q=\\mathbf {T} q", "0eb1b3584d3fb2bf38b4efe1eb497c2d": "\\{z:\\lVert z\\rVert =a^{2}\\}", "0eb1d13b31aff26e8224411572059dba": "(x)_{n}=x(x-1)(x-2)\\cdots (x-n+1)\\,", "0eb1e2be5ee93cd7950a7aafdf6b2a89": "{\\frac {1}{1-z}}", "0eb1fbfd143fc6d52de70f0efd612cdc": "F''+F=f.\\,", "0eb21bb38a1b60e5754d211e8d13b0b3": "\\textstyle p(c_{j})\\sum _{f_{i}\\in F}\\sum _{k=1}^{m}p(f_{ik}|c_{j})^{2}", "0eb2980d4f992c8c1729fcdbc256f284": "\\{A_{x}\\}_{x\\in [0,1]}", "0eb2c84f5eca0bc90216498b830257c5": "S^{2n}", "0eb2d613657ce34a03da3119a30a594c": "U\\setminus A=A^{C}\\,\\!", "0eb2de6ace0d4e2a9238f4ad175420da": "\\theta ([x])=t(x)", "0eb2fa3888181ea09b14ad17ec1d9c49": "\\mu =c+\\varphi \\mu +0,", "0eb30a3d05c82c4f7fbd37fe2648841b": "m=1/R,", "0eb32500513c88a447c2338109f2aaa4": "\\scriptstyle f(x)=K\\left(\\sum _{i}w_{i}g_{i}(x)\\right)", "0eb329a32751aee64c2f7bc8ff03c566": "w(x_{1},x_{2})", "0eb398dd11a17572e69fa184063bcea0": "1/[y_{1},y_{2}]=[-\\infty ,\\infty ]", "0eb3d001233bf2fc4d1a5540f8f5325a": "1/\\Gamma (z)=\\lim _{n\\to \\infty }{\\frac {1}{n!}}(1-(\\ln n)z/n)^{n}\\prod _{m=0}^{n}(z+m).", "0eb41e837b0d3ad5ebf45624d37d5460": "x_{n}/\\|x_{n}\\|", "0eb43228eee2e4a595fad4870fccd8ca": "\\int _{\\Omega }\\mathrm {d} \\omega =\\int _{\\partial \\Omega }\\omega \\ \\ \\left(=\\oint _{\\partial \\Omega }\\omega \\right).", "0eb4d83fb3d823251fbbac5272ad5b57": "\\mu _{1},\\dots ,\\mu _{K}", "0eb4de4b5c3827fb484f61d84319dfd2": "a=a_{\\mathrm {now} }", "0eb4f14757345f3cd4beb47aafe6dfa4": "|\\mathbf {a} \\times \\mathbf {b} |^{2}=\\sum _{1\\leq it)=\\Phi \\left({\\frac {y-\\mu t}{\\sigma t^{1/2}}}\\right)-e^{2\\mu y/\\sigma ^{2}}\\Phi \\left({\\frac {-y-\\mu t}{\\sigma t^{1/2}}}\\right).", "0ec1287eff4c1d7c0a941a838addc21a": "_{p}", "0ec12bf327a3585fd5f64657b9fcb4c5": "\\lim _{x\\to \\infty }N^{-x}=\\lim _{x\\to \\infty }1/N^{x}=0{\\text{ for any }}N>1", "0ec186827925aba30f82debbc7af6d47": "D=C\\oplus \\nabla ", "0ec1c590b5337e876a71ec9125484f0f": "Ob", "0ec1f9a10ddacd21beb1b90b1f78197d": "\\int xdy=xy-\\int ydx", "0ec1fd2fe41393140048a18c255a70c1": "\\sum _{k=0}^{\\infty }kn_{k}=n", "0ec26e0b3d41f0b9abeda4cd852351a7": "s_{2}\\leq t_{2}", "0ec2aa080ebb37b1eaa058f9bab09b66": "{\\frac {p_{1}\\cdot V_{1}}{T_{1}\\cdot n_{1}}}={\\frac {p_{2}\\cdot V_{2}}{T_{2}\\cdot n_{2}}}=constant", "0ec2bf8227c09c739045105578da1301": "a\\,P(X\\in [a;a+da])", "0ec2cc1749e80ae632dadacef9aca98c": "{u_{ij}^{n+1}-u_{ij}^{n+1/2} \\over \\Delta t/2}=\\left(\\delta _{x}^{2}u_{ij}^{n+1/2}+\\delta _{y}^{2}u_{ij}^{n+1}\\right).", "0ec2cfd176b6c3aad4aace67330c0d43": "{\\mathfrak {k}}^{n},", "0ec30d3a9169df113a8f2070b84ba303": "M(M-1)d\\leq {\\frac {1}{2}}nM^{2}", "0ec32ff534e449bd0b1a54a264512207": "\\mathbf {MTF_{sensor}} (\\xi ,\\eta )", "0ec35c6714dffb0bc052982d0d6342a2": "\\land S_{7}\\implies A_{7}=p", "0ec36571f70b0345be34c1d77173346c": "J_{1k}", "0ec3bdb2d0ab8fc23fa6af566ae046ab": "E(g_{i}^{-1}g_{j})=\\delta _{ij}1", "0ec3ec0e1b8132f23d046f7acd4e5ad7": "\\partial _{t}(f_{t}(z))h^{\\prime }(f_{t}(z))=\\mu e^{\\mu t}h(z)=\\mu h(f_{t}(z)),", "0ec3fe2fd44add93e95a70a0e8ea4329": "{}=-4\\cdot (-4)-(-8)\\cdot (-8)+(-12)\\cdot (-4)+(-4)\\cdot (-12)-(-8)\\cdot (-8)+(-4)\\cdot (-4)", "0ec418f9c266219ec2fc6892197e8c4f": "e(p_{0},v(p_{0},w+EV)=e(p_{0},u_{1})", "0ec455fcaeb0560387593e685c6e4118": "\\,\\,\\sigma _{ij}=2\\mu \\varepsilon _{ij}+\\lambda \\varepsilon _{kk}\\delta _{ij}+\\lambda '\\,\\varepsilon _{ik}\\,\\varepsilon _{kj}", "0ec4b8a5b91e0ed523fce5e73b5f096b": "{\\frac {A_{n-1}}{B_{n-1}}}\\approx {\\frac {A_{n}}{B_{n}}}\\quad \\Rightarrow \\quad {\\frac {A_{n-1}}{A_{n}}}\\approx {\\frac {B_{n-1}}{B_{n}}}=k\\,", "0ec4ee2d22e6e65b90e4fa0208c90190": "{\\begin{aligned}\\sum _{k=1}^{\\infty }{\\frac {1}{k^{2}}}\\left(1+{\\frac {1}{2}}+{\\frac {1}{3}}+\\cdots +{\\frac {1}{k}}\\right)^{2}={\\frac {17\\pi ^{4}}{360}}.\\end{aligned}}", "0ec5bbdf6a503d1e92ff7962eaaac0e9": "\\iota _{[X,Y]}={\\mathcal {L}}_{X}\\iota _{Y}-\\iota _{Y}{\\mathcal {L}}_{X}.", "0ec6d2ba1223048e8b289d76aa74d0d7": "\\scriptstyle (x_{0},x_{1})", "0ec7704560dcaa2dec9da4c63ed5ad92": "AI_{T}=100\\times {\\frac {d}{n}}", "0ec7829f6bb626aa266bc8b6c752d366": "{(3/2)^{4} \\over 2^{2}}={81/16 \\over 4}={81 \\over 64}\\approx {5 \\over 4}={5\\cdot 16 \\over 4\\cdot 16}={80 \\over 64}.", "0ec788e46e02fa5b65722bb563f2162f": "\\psi \\left({\\frac {1}{6}}\\right)=-{\\frac {\\pi }{2}}{\\sqrt {3}}-2\\ln {2}-{\\frac {3}{2}}\\ln(3)-\\gamma ", "0ec7a9b5256fbee66a791ec70c88ee6c": "\\mathbf {S} _{j}", "0ec7d815383c3810635d8782a201009c": "V(n)=(1-P)^{n}\\,", "0ec82a0a81033b59075155f9d0fd00c9": "{\\mathbf {v} }_{n}({\\mathbf {k} })={\\frac {1}{\\hbar }}\\nabla _{\\mathbf {k} }E_{n}({\\mathbf {k} }).", "0ec8382bc2e59b199f8d5fe125b5da64": "{\\bar {\\boldsymbol {\\omega }}}", "0ec84b534799c3dff2eb8ab8874aa716": "{\\sqrt {m_{f}c^{2}/V}}", "0ec8900ebf65161f341ae7bb10febbff": "S(\\omega )=E(\\omega )W(\\omega )\\,", "0ec8f2c925ac1636f04c351cb5290308": "(g^{c})^{a}=g^{ca}", "0ec9aeabff7320cea63e64c3d9797f59": "\\operatorname {erf} ^{-1}(x)", "0ec9d46d25e751c96b5b3beef5f1ac45": "f={\\frac {n_{1}}{n_{0}}}={\\frac {2(1-a)}{a}}", "0ec9e6875e4c6e6702e1b81813a0b70d": "B:1", "0eca1aecaf6c0328ef50d8ab6b579376": "2+15{\\sqrt {\\frac {2}{17}}}", "0eca4aad9f6de63434e73c6a583cdb45": "O(n^{2+\\epsilon })", "0ecaa4dc9275d64ce26ff1ab4cfb2b4d": "R={(n-1) \\over \\phi },", "0ecae8568f75b2269e53630285776138": "X\\Rightarrow Z.", "0ecb2700dabebda5fccf228cb4fbf171": "\\eta _{GX}", "0ecb4b6317baf46cb7a2240545f28107": "x={\\hat {x}}x_{\\text{scale}}+x_{\\text{shift}}", "0ecbd394fa38e6c1512f8e017d83e280": "t=t_{\\mathrm {now} }+\\lambda _{\\mathrm {now} }/c\\,.", "0ecbead79a995072ed0c664a53cef2ff": "\\left(E_{n}^{(0)}-E_{k}^{(0)}\\right)\\langle k^{(0)}|n^{(1)}\\rangle =\\langle k^{(0)}|V|n^{(0)}\\rangle ", "0ecc3fda4595563dda26f7e36c9789f4": "(S_{a}f)(z)=f(z+a)=\\exp(a\\partial _{z})~f(z)", "0ecc72384da5d90e5e16bd5149acbdd1": "\\lim _{x\\to \\infty }{\\frac {\\pi (x,a,q)\\phi (q)}{x/\\log x}}=1.", "0ecc934f5a973ff364f77fe46cac2ca3": "k_{4}(s)=u_{0}+l_{1}s^{1}+l_{2}s^{2}+u_{3}s^{3}+u_{4}s^{4}+l_{5}s^{5}+\\cdots \\,", "0ecc9d58ea7385c1bd00a6446a707e01": "\\,\\!f={\\frac {1}{2\\ln(3)RC}}", "0ecca81206d9341099b254a8ec0b69bf": "A+1=1", "0ecce93dedf82b925357742dedd3c189": "\\varepsilon _{xx}^{\\mathrm {topface} }=-z~{\\cfrac {\\mathrm {d} ^{2}w_{b}}{\\mathrm {d} x^{2}}}-\\left(z-h-{\\tfrac {f}{2}}\\right)~{\\cfrac {\\mathrm {d} ^{2}w_{s}}{\\mathrm {d} x^{2}}}~;~~\\varepsilon _{xx}^{\\mathrm {botface} }=-z~{\\cfrac {\\mathrm {d} ^{2}w_{b}}{\\mathrm {d} x^{2}}}-\\left(z+h+{\\tfrac {f}{2}}\\right)~{\\cfrac {\\mathrm {d} ^{2}w_{s}}{\\mathrm {d} x^{2}}}", "0ecd0835f3f532af0ce7a90a18c6fd6e": "\\displaystyle {|\\beta |={b \\over |\\alpha +\\alpha ^{-1}|}={|\\alpha |^{-1}-|\\alpha | \\over 1-\\delta }>|\\alpha |^{-1}-|\\alpha |.}", "0ecda0976ce67796e1989cd9a3eb6f9f": "y\\ =\\ {\\log(x) \\over \\log(b)}", "0ecdf23207e45eaa19859ecf6b5c49e0": "m=\\mu +\\nu ", "0ecdf30b118b10d69b3e85c36dd8e323": "1+c_{n+1}=(1+c_{n})(1-c_{n}/2)^{2}\\,\\!", "0ece1664733cb0255dffbdd4d06d9654": "\\sum \\limits _{m}\\left[p_{m}\\nabla ^{2}q_{m}-q_{m}\\nabla ^{2}p_{m}\\right]=\\sum \\limits _{m}\\left[\\nabla \\cdot \\left(p_{m}\\nabla q_{m}-q_{m}\\nabla p_{m}\\right)\\right].", "0ece4b923fe95e8a309279f49649d168": "{n(5n^{2}-5n+2) \\over 2}", "0ece67573f5fc0400f2c65818b169579": "5.6\\%", "0ece9c02190ce10d3600e89a796e2fd5": "\\Omega _{\\perp }\\tau =\\pi ", "0ecea947b5c57ab62a754ff9286d7849": "(\\sigma >0)", "0ecf32e5d1127f2804d30f10c050eecf": "{\\frac {b-a}{24}}(11f_{1}+f_{2}+f_{3}+11f_{4})", "0ecfb6c09c20db0d526e72fe7c24ad2a": "{\\hat {\\varepsilon }}(\\omega )=\\varepsilon _{\\infty }+{\\frac {\\Delta \\varepsilon }{1+i\\omega \\tau }},", "0ecfd6e83303dd5b0cf3fc4c5db989e0": "\\theta _{1}+\\theta _{2}=\\theta _{3}+\\theta _{4}=90^{\\circ }\\;", "0ed0442ea254842e1dc623f2dbb706cb": "X(t)=\\sum _{n=-\\infty }^{\\infty }e^{2\\pi int/T}X_{n}", "0ed0868d9ae61951946a25ec61a58a16": "n^{\\Omega (1)}", "0ed0bebdc5b5d5c768912b4165659c61": "{\\frac {\\partial }{\\partial \\tau }}y^{k}{\\frac {\\partial }{\\partial {\\overline {\\tau }}}}", "0ed142e2a90bf0065034c8d2f8ac63cc": "\\lambda _{+}", "0ed18fe290d9a1af0845e5870a9f2d7b": "G^{\\alpha \\beta }", "0ed1aa40537c18854f570601252c351f": "\\rho _{s}\\,", "0ed1bb07211cc10f4a50ab8c1742bd6e": "{\\text{max}}_{i}{\\text{ min}}\\{d(x_{i})+1,i\\}", "0ed26e610e062b54a93ef2723b4dee36": "\\{\\mathbf {x_{i}} \\leftrightarrow \\mathbf {x_{i}} '\\}", "0ed26fc4d8fb4f32a72dcf9ae8f58ec4": "f:X\\to \\mathbb {R} ", "0ed2c2a9542b29bf43054e6df3ac4a40": "{\\bar {\\Phi }}\\left[\\mathbf {r} \\right]={\\frac {1}{2}}\\int d\\mathbf {r} \\int d\\mathbf {r} '{\\hat {\\rho }}(\\mathbf {r} ){\\bar {\\Phi }}(\\left|\\mathbf {r} -\\mathbf {r} '\\right|){\\hat {\\rho }}(\\mathbf {r} ')-{\\frac {1}{2}}nN{\\bar {\\Phi }}(0).\\qquad (3)", "0ed2efea5afd7380a81da0052837e3cf": "r=1,2", "0ed37d3e2811fc9224aac747ccf2b731": "|{\\mathit {before}}\\rangle ", "0ed3abcc3853fb85bcab9873a6aef42e": "c_{K}=1.151c_{B}(d_{BU})^{-0.5}\\,", "0ed3f1170c3833b5f4837d47f586d80b": "\\int \\mathbf {f} =\\left(\\int f_{1},\\,\\dots ,\\int f_{n}\\right).", "0ed4237587c3da5dabdf15501e0e4e0f": "\\omega (x)=1/{\\sqrt {1-x^{2}}}\\,", "0ed45c17a2c820cd87fc9ba6dd4ccc3c": "y={\\frac {{\\sqrt {2}}\\sin(\\phi )}{\\sqrt {1+\\cos(\\phi )\\cos \\left({\\frac {\\lambda }{2}}\\right)}}}", "0ed4fab98433d204d4710264cddd9a68": "\\partial _{t}\\rho =-wd\\rho _{0}/dz", "0ed547de0c8db395d6065e7fe6f99851": "\\nu (A)=\\sum _{n=1}^{\\infty }w_{n}{\\frac {\\mu (A\\cap V_{n})}{\\mu (V_{n})}}", "0ed5aea834e1a3f60b154dfa9f512882": "A_{o}=\\mathrm {min} _{x}(\\mathrm {max} _{S_{o}}(U(S_{o},x)-T_{o}S_{o}))=\\mathrm {min} _{x}(A_{o}(T_{o},x))", "0ed5b2ae271063d16122ea149f76a85c": "{\\mathcal {M}}(P(x))=1.176280818\\dots \\ .", "0ed5d7bcf12581d86bf0c6ff6011f0fb": "co(ci,x,y)=M_{0}M_{1}M_{2}M_{4}", "0ed646b67fe7cd6c54e32e199acdded1": "m{\\frac {du^{\\tau }}{ds}}=eF^{\\tau \\sigma }u_{\\sigma },", "0ed646cdd4eaecf57a0080659f51ad98": "\\mathbf {Wv} ", "0ed663e2b5710eefd700a198c0752a8d": "N=M^{k}", "0ed6705acdd00af7708b96fe7f52d7b9": "C=A^{0}B^{0}-A^{1}B^{1}-A^{2}B^{2}-A^{3}B^{3}={A'}^{0}{B'}^{0}-{A'}^{1}{B'}^{1}-{A'}^{2}{B'}^{2}-{A'}^{3}{B'}^{3}", "0ed748968b473c78a269ab07e5d09bd2": "E_{\\mathrm {t} c}", "0ed76d1800949b99e254688920ce92e7": "L={mg \\over {\\cos \\theta }}", "0ed775167e5c7887ef5b798190d3ed1b": "y=-1", "0ed781e2fe28c2a05577d4fcb080472f": "molAl={\\frac {8.00g}{26.98g/mol}}=0.297mol\\,", "0ed7b37cbfa9b2f866c4a6df84dd1fcd": "r={\\sqrt {(at+b)^{2}+c^{2}}}", "0ed814303885052c91b1ab408b9f27ff": "\\sum _{k=1}^{n}{n-1 \\choose k-1}=2^{n-1}.", "0ed83380783162190fbcd90fb7aa8892": "xy-yx\\neq \\mathbf {1} ", "0ed8f065ac012b25a8085814f03e2ae9": "\\gamma =\\lim _{n\\rightarrow \\infty }-{\\frac {\\ln N(T)}{T}}", "0ed96ca5d9cae4cb98875cb4e4b704e2": "({\\boldsymbol {\\sigma }}\\times {\\boldsymbol {k}})", "0ed9f4b9d22f8074b83ad3331d54b57b": "{\\frac {\\sqrt {3}}{6^{5}}}\\sum _{k=0}^{\\infty }{\\frac {((4k)!)^{2}(6k)!}{9^{k+1}(12k)!(2k)!}}\\left({\\frac {127169}{12k+1}}-{\\frac {1070}{12k+5}}-{\\frac {131}{12k+7}}+{\\frac {2}{12k+11}}\\right)=\\pi \\!", "0eda0d7540824ac98a394fc759a932f4": "f(x)\\in B", "0eda365b536bec1bd78a7337561918e5": "r_{2}=\\$36.67*3.9=\\$143", "0eda70d6900607af8b18e75b47106df4": "!n=\\left\\lfloor (e+e^{-1})n!\\right\\rfloor -\\lfloor en!\\rfloor ,\\quad n\\geq 2,", "0edac5e3cfa9812a2fb87852d30e76ed": "X=(X_{t},\\ t\\in I)", "0edae8dec406acd8a0fc1b63a10810e6": "{\\frac {\\Delta E_{x}}{\\Delta y}}-{\\frac {\\Delta E_{y}}{\\Delta x}}=2L'{\\frac {\\partial H_{z}}{\\partial t}}", "0edaf460f3bc9297904bb0d4ed85aa2b": "\\varepsilon =\\mu ^{-1}\\sigma ^{2}=\\kappa _{1}^{-1}\\kappa _{2},\\,", "0edb12d8bef2ed0d0bab91c4ef569df4": "f({\\boldsymbol {\\sigma }},{\\boldsymbol {\\varepsilon }}_{p})=0", "0edb76fe6c9280fff4b4557075f55fda": "\\gamma =(t_{ox}/\\epsilon _{ox}){\\sqrt {2q\\epsilon _{si}N_{A}}}", "0edba0d61ce70a0fd0d34a2050a5043e": "\\textstyle s>0", "0edbd92c7d47dd8bc471f3a61b2f8dff": "2^{S}", "0edc278bf60452487b33872bd8e32291": "{\\frac {d\\mathbf {r} }{dt}}=\\mathbf {v} ,\\qquad {\\frac {d\\mathbf {v} }{dt}}=\\mathbf {F} (\\mathbf {r} ,\\mathbf {v} ,t),", "0edc54b87a77d6d1ee703bf8c2272213": "G={\\begin{bmatrix}1&\\dots &1&0&\\dots &0\\\\\\ast &\\ast &\\ast &&G'&\\\\\\end{bmatrix}}", "0edc68bc802e369920d69cb090a9ea4b": "\\scriptstyle p_{c}\\,", "0edccb953a137f27a52a69fd72a3d0a6": "\\mathbb {H} ^{\\bullet }", "0edd0e9ac1e5cd291807a66a81016e7a": "\\mathbf {x} (t)\\triangleq {\\begin{bmatrix}x_{1}(t)\\\\x_{2}(t)\\\\\\vdots \\\\x_{n-1}(t)\\\\x_{n}(t)\\end{bmatrix}}\\in \\mathbb {R} ^{n}", "0edd18309ba3f2b38f6fa8ce6be11eb2": "\\operatorname {Spec} A_{i}", "0eddb1c5ac349c6b0e4105604acb4905": "\\mathbf {S} \\cdot d\\mathbf {A} +\\int _{V}\\mathbf {J} \\cdot \\mathbf {E} dV", "0eddb5d6c96d704e54155ab661e7e308": "\\mathbf {N(s)X(s)} +\\mathbf {M(s)Y(s)} =\\mathbf {1} ", "0eddefd18f4e88079ea756e371965dcd": "S(p/q)={\\frac {I_{p/q}(2/q)}{I_{1+p/q}(2/q)}},", "0eddf6cb023b65a580f569bb8cdf1001": "{\\begin{aligned}S_{B}&=n({\\overline {Y}}_{1}-{\\overline {Y}})^{2}+n({\\overline {Y}}_{2}-{\\overline {Y}})^{2}+n({\\overline {Y}}_{3}-{\\overline {Y}})^{2}\\\\[8pt]&=6(5-8)^{2}+6(9-8)^{2}+6(10-8)^{2}=84\\end{aligned}}", "0ede64db6d7538cae16dcf2d5b59a591": "h(-r)=h(r),\\ \\vert h(r)\\vert \\leq M\\left(1+\\vert \\Re (r)\\vert ^{-2-\\delta }\\right),", "0ede7929d658edab751c7667b22f158f": "ax=b\\,", "0edeab0f57af0df5f6c2ca0f014701fd": "{\\frac {dr}{dt}}=\\alpha r", "0edef6f74c6fedc43a450cd88b398866": "S_{n}=X_{1}+\\cdots +X_{n}.", "0edf461ed9d813d53b6d0a1e1efc7b32": "m{\\dot {x}}={\\frac {\\partial H}{\\partial v}}", "0edf97e42e0295df0ec452351d3c1c90": "f_{i}(p)\\in [y_{i}]", "0edfbaf65e24d4db501fcf3a8b2e99d6": "{\\tilde {a}}", "0ee03f592af0190f77a1274ebc5431be": "\\beta _{0}+\\beta _{1}x", "0ee0872605988295f8a5ece95a7e27b6": "\\langle A|B\\rangle =\\left(\\,\\langle A|\\,\\right)\\,\\,\\left(\\,|B\\rangle \\,\\right)", "0ee0c2a4abf7bdb1935bb2bfa867050a": "U_{0}\\cap \\cdots \\cap U_{k}\\neq \\emptyset \\,\\!", "0ee0eee0acf84ea91e59ab0803351baa": "+\\varepsilon ", "0ee11ea708afd96b74b79cd2fd744853": "\\gamma \\in \\Gamma _{g}(N),", "0ee13b5dbfaa24408d659cbe134d5bcc": "S(y)", "0ee15d5fa9f07c1d646885626fbbd38a": "{\\frac {\\overrightarrow {DC}}{\\overrightarrow {DB}}}\\times {\\frac {\\overrightarrow {EA}}{\\overrightarrow {EC}}}\\times {\\frac {\\overrightarrow {FB}}{\\overrightarrow {FA}}}=1,", "0ee1bdb71f255a6a8281017c43451f98": "{\\vec {f}}_{0}=\\partial _{T},\\;{\\vec {f}}_{1}=\\partial _{X},\\;{\\vec {f}}_{2}=\\partial _{Y},\\;{\\vec {f}}_{3}=\\partial _{Z}", "0ee2320b3516e72e3a91ab12d0b3a122": "P\\lor (\\exists xQ(x))", "0ee26194cbb26ae2beb47079f365fba3": "\\{1,2,\\ldots ,n\\}.", "0ee28892ec70ca17ef0e8f81f4ff294f": "e_{p}(x)=0", "0ee2c3781ce63ae5c02ef7eb5cdc4873": "||\\Phi (t)P\\Phi ^{-1}(s)||\\leq Ke^{-\\alpha (t-s)}{\\mbox{ for }}s\\leq t<\\infty ", "0ee2ff3e3c0e579a843b7a06e5de2544": "s\\models _{K}{\\mathcal {P}}_{\\sim \\lambda }(f_{1}{\\mathcal {U}}f_{2})", "0ee32875a74d8115b278f69ea964add1": "{\\hat {X}}(z)={\\hat {X}}_{Bayes}\\left(\\Pi ^{-\\top }\\mathbf {P} _{Z}\\odot \\pi _{z}\\right)", "0ee368f975a90e2cf638b1deb6837c6f": "\\Theta (\\theta )", "0ee38e81a17473b55bc575f2d550cd9a": "\\operatorname {let} x:\\operatorname {get-lambda} [x,x=\\lambda f.f\\ (x\\ f)][x:=x\\ x]\\operatorname {in} x\\ x", "0ee446231316d6a668ea1014cb043e67": "x_{i,1}", "0ee4646802c4ab236a205caccde1f0ca": "i(e,n)", "0ee47777604e68e889857383ebc0e8fa": "\\mathbf {b_{1}} =2\\pi {\\frac {\\mathbf {a_{2}} \\times \\mathbf {a_{3}} }{\\mathbf {a_{1}} \\cdot (\\mathbf {a_{2}} \\times \\mathbf {a_{3}} )}}", "0ee4b1217318e94d4e6ee3dcc1653336": "{\\frac {A}{H}}={\\frac {\\frac {k_{c}}{m+m_{A}}}{\\omega ^{2}-{\\frac {k_{c}}{m+m_{A}}}}}", "0ee4c8cdbf49efbb065a11e64045dddb": "{\\frac {L}{L_{\\odot }}}\\approx {\\left({\\frac {M}{M_{\\odot }}}\\right)}^{3.9}", "0ee4ea2d2548299e7d4d951d6e2a049b": "{\\begin{array}{rl}{\\text{Ax. 1.}}&\\left\\{P(\\varphi )\\wedge \\Box \\;\\forall x[\\varphi (x)\\to \\psi (x)]\\right\\}\\to P(\\psi )\\\\{\\text{Ax. 2.}}&P(\\neg \\varphi )\\leftrightarrow \\neg P(\\varphi )\\\\{\\text{Th. 1.}}&P(\\varphi )\\to \\Diamond \\;\\exists x[\\varphi (x)]\\\\{\\text{Df. 1.}}&G(x)\\iff \\forall \\varphi [P(\\varphi )\\to \\varphi (x)]\\\\{\\text{Ax. 3.}}&P(G)\\\\{\\text{Th. 2.}}&\\Diamond \\;\\exists x\\;G(x)\\\\{\\text{Df. 2.}}&\\varphi {\\text{ ess }}x\\iff \\varphi (x)\\wedge \\forall \\psi \\left\\{\\psi (x)\\to \\Box \\;\\forall y[\\varphi (y)\\to \\psi (y)]\\right\\}\\\\{\\text{Ax. 4.}}&P(\\varphi )\\to \\Box \\;P(\\varphi )\\\\{\\text{Th. 3.}}&G(x)\\to G{\\text{ ess }}x\\\\{\\text{Df. 3.}}&E(x)\\iff \\forall \\varphi [\\varphi {\\text{ ess }}x\\to \\Box \\;\\exists y\\;\\varphi (y)]\\\\{\\text{Ax. 5.}}&P(E)\\\\{\\text{Th. 4.}}&\\Box \\;\\exists x\\;G(x)\\end{array}}", "0ee55d794f17e382f8980ebd1adcf211": "\\Delta (t).", "0ee599876ea529b25749b16ad57e86c2": "({\\ddot {r}}-r{\\dot {\\theta }}^{2}){\\hat {\\mathbf {r} }}+(r{\\ddot {\\theta }}+2{\\dot {r}}{\\dot {\\theta }}){\\hat {\\boldsymbol {\\theta }}}=\\left(-{\\frac {\\mu }{r^{2}}}\\right){\\hat {\\mathbf {r} }}+(0){\\hat {\\boldsymbol {\\theta }}}", "0ee5e5a0affa7b765858efea96611f62": "V_{n}(x_{1},\\ldots ,x_{n})=\\sum _{i\\exp(1/\\mathrm {e} )", "0eebbae6d53eec48e94abbd5f1051ceb": "T_{\\text{s}}", "0eebc6779815507fbde89d4852e65337": "P(i|s_{j})={\\frac {P(i)p(s_{j}|i)}{p(s_{j})}}", "0eebef778df80f071321658c5c428db1": "\\textstyle v", "0eec1197ecf816c5bab696c64594491c": "x,y\\in M", "0eec2cc5ecada971045e34620070f7dd": "{\\check {H}}^{*}(X;A)", "0eede95a493fc729cc19afceefbdf4b1": "p_{B}", "0eedf80c1dae3ef6cc4996e1d2da41e0": "A_{\\gamma -norm}", "0eee2fc28c40b7b24b5514b60c6a3fa3": "a_{0}=-1", "0eee4b1b73a7073779fc70dfd2febf68": "(a,0)\\circ (b,1)=(a+b,1)", "0eeed4f4a72ac653419603e778a37f8f": "ds^{2}=\\rho (du^{2}+dv^{2})", "0eef586a2d9ef3503cd41c7f4338b7f9": "\\rho _{\\text{perm}}(r)", "0eeff2c7b49ef6ec59883c9686d3b3fa": "{\\bar {X}}_{P},{\\bar {X}}_{N}", "0eeff349dbe28511462ce4daa7904b4e": "\\sin x=x-{\\frac {x^{3}}{3!}}+{\\frac {x^{5}}{5!}}-{\\frac {x^{7}}{7!}}+\\cdots ", "0ef07d555b1f79a0fd62ea633b584a8c": "F_{\\mathrm {Anchor} }={\\frac {\\mathrm {Weight} }{2\\cos({\\frac {1}{2}}{\\theta _{\\mathrm {Bottom} }})}}\\approx \\mathrm {Weight} \\times 0.5+O({\\theta _{\\mathrm {Bottom} }}^{2})", "0ef07d7c7e409b1e69ccdde2c9df1042": "c/R", "0ef0bf7c8ea22fb54e995ff882972e7c": "{\\begin{aligned}\\left(\\int _{R}+\\int _{M}+\\int _{N}+\\int _{r}\\right)f(z)\\,dz&=2\\pi i\\left(\\mathrm {Res} _{z=i}f(z)+\\mathrm {Res} _{z=-i}f(z)\\right)\\\\&=2\\pi i\\left(-{\\frac {\\pi }{4}}+{\\frac {1}{16}}i\\pi ^{2}-{\\frac {\\pi }{4}}-{\\frac {1}{16}}i\\pi ^{2}\\right)\\\\&=-i\\pi ^{2}.\\end{aligned}}", "0ef0de55df063c5d53bea8fffbcc95ba": "x\\propto {\\sqrt {t}}", "0ef0e13322d1d789c2396e89df1ab7c3": "\\mathbf {r} =\\left(r+R_{0}\\cos \\theta \\right)\\mathbf {e} _{r}-R_{0}\\sin \\theta \\mathbf {e} _{\\theta }", "0ef10eefc4d96e2911d20a2263a3180e": "a_{i}>0", "0ef134486d3a3012f2eb99f526d1cd25": "a^{-1}+b^{-1}=c^{-1}", "0ef13611c7bb7573ea79b51dfd3cc540": "{\\frac {20\\varphi ^{2}}{3}}", "0ef14bfaeb767ee868194c783d99099f": "X_{n}\\ \\xrightarrow {p} \\ Y", "0ef178116dd59cfc7c4a667fcd96e583": "{\\breve {\\theta }}_{j}=h^{-1}(s,{\\breve {z}}_{1}^{j},\\ldots ,{\\breve {z}}_{m}^{j})", "0ef1cdb14e471d556fdeba8feb761e71": "{\\it {K\\ll N}}", "0ef2101f275ca4f95f311320a6ed7d83": "L_{\\text{r}}(\\omega _{\\text{r}})", "0ef232ece43e3aea83e7acb3a20ee377": "x^{n}(x^{4}-x^{3}-1)=-(x^{4}+x-1)", "0ef2cb651f6d31b5d32f89c05207eb23": "\\scriptstyle {\\overline {\\mathbb {Q} }}", "0ef302b82db7d48f8072bca5738db8cd": "f(1)=-{\\tfrac {1}{2}}(1-1-1-1)=1,f(i)=-{\\tfrac {1}{2}}(i-i+i+i)=-i,f(j)=-j,f(k)=-k", "0ef329f38c93680ff747a65b33bc993d": "a=2\\arctan \\left\\{\\tan \\left({\\frac {1}{2}}(b-c)\\right){\\frac {\\sin \\left({\\frac {1}{2}}(\\beta +\\gamma )\\right)}{\\sin \\left({\\frac {1}{2}}(\\beta -\\gamma )\\right)}}\\right\\},", "0ef3b61f17457ab10cd1d299bc448d13": "(x,y,z)=(4\\cos \\vartheta ,+3+5\\sin \\vartheta ,3\\cos \\vartheta )\\,\\!", "0ef3c9f2756d0f1d870db34b925037f9": "h(k_{2})", "0ef488c2c0a63879ff015daf0b260196": "(\\lambda x.t)s", "0ef4ab58edc89f1ecd9a4241e4dc3d87": "{\\mathbf {v} }_{n}({\\mathbf {k} })={\\frac {1}{\\hbar }}\\nabla _{\\mathbf {k} }E_{n}({\\mathbf {k} }),", "0ef4b0799abb7cbaadffb54ae5d6479f": "{\\begin{aligned}\\qquad {\\frac {x}{84}}&={\\frac {3}{2}},\\\\{84}\\cdot {\\frac {x}{84}}&={\\frac {3}{2}}\\cdot {84},\\\\\\qquad {x}&={\\frac {3\\cdot 84}{2}},\\\\\\qquad {x}&={126}\\\\\\end{aligned}}", "0ef50e7d25b4821a71f4bfb2f5b9606e": "n^{2}=n_{1}^{2}+n_{2}^{2}+n_{3}^{2}.", "0ef55c818acd22814738d42716125193": "p=kT{\\frac {\\partial \\ln Q}{\\partial V}}", "0ef565c20297a1b2eb143db7936cc63d": "(x-3)(x-2)^{9}(x-1)^{18}x^{10}(x+1)^{18}(x+2)^{9}(x+3)(x^{2}-6)^{12}", "0ef5bcbd2e8dc44fee945fa8e9dbf0a0": "{\\hat {x}}_{\\mathrm {fl} }(t_{j})=-{\\frac {{\\hat {\\phi }}_{j}}{2k_{p}}}", "0ef648e26b0e2aa05230dba5cb097a02": "-x_{1}x_{5}x_{6}x_{7}", "0ef672b64161cba04d542b30995895e0": "x^{2}-2y^{2}=0.\\,", "0ef68273cb4b7933668abaab86701d1a": "\\lambda x_{s}^{*}", "0ef6d776c2a05cd278364a75f349e622": "x+S(x)=(1,1,1,\\ldots )", "0ef6f67a7e5f705a94bc71f101921250": "h=\\{h_{n}\\}_{n\\in \\mathbb {Z} }", "0ef73008b57d4b71ddeab2efa3b39cd5": "(\\mathbb {Z} S,\\partial )", "0ef75bc3c2227172bb22d931991ed806": "U_{e}^{2}=0", "0ef8079040fa6b9ab7a0d248b2ef3dd9": "q(n,t)", "0ef8ae60eea281dcdb4b6016b2045016": "P=P_{1}{\\begin{bmatrix}I&0\\\\0&P_{2}\\end{bmatrix}}{\\begin{bmatrix}I&0\\\\0&P_{3}\\end{bmatrix}}\\cdots {\\begin{bmatrix}I&0\\\\0&P_{r}\\end{bmatrix}}", "0ef8f309cd59de106dce18bff6cf3998": "u=m(S\\otimes {\\text{id}})({\\mathcal {R}}_{21})", "0ef933b93605a459e87fe63f3abf33af": "e^{ix}=\\cos x+i\\sin x\\ ", "0ef98851438e1c3b9983222f4f479fb6": "\\pi =3.1459...", "0ef98868107285f53fc0131b51c3d232": "s_{i}=ln(r)", "0ef9abca75a25203ad92dc53e6a1b8c8": "{\\frac {\\partial {\\boldsymbol {F}}}{\\partial {\\boldsymbol {S}}}}:{\\boldsymbol {T}}={\\frac {\\partial {\\boldsymbol {F}}_{1}}{\\partial {\\boldsymbol {F}}_{2}}}:\\left({\\frac {\\partial {\\boldsymbol {F}}_{2}}{\\partial {\\boldsymbol {S}}}}:{\\boldsymbol {T}}\\right)", "0ef9b6a8231cc0f7586f7ece8e4b4b3c": "\\Delta W=P\\Delta V", "0ef9cd6d614e159e77c82faa9e1e7793": "\\scriptstyle b(x+e)-(x+e)^{2}=bx+be-x^{2}-2xe-e^{2}\\;\\sim \\;bx-x^{2}.", "0ef9e3eb60826d8df291b2d958435ffd": "\\Psi ^{k}(\\rho )=N_{k}(\\Lambda ^{1}\\rho ,\\Lambda ^{2}\\rho ,\\ldots ,\\Lambda ^{d}\\rho )\\ ", "0ef9f5d21a51b5f2d4de1a9552522a3c": "\\scriptstyle \\{V_{k}-U_{k},k\\geq 1\\}", "0efa0e81552b8709489c7abc4b7e8c30": "\\sigma =1.661687949633594121296\\dots \\;", "0efa34689d3c4d4c40187804e30fa63c": "(*)\\qquad a_{1}x_{1}^{r}+\\cdots +a_{n}x_{n}^{r}=0,\\quad a_{i}\\in K,\\quad i=1,\\ldots ,n", "0efa76cd29585aa1d733d872fadb52e4": "7/8", "0efa813aeb3b56b6d97cc534d229e351": "d(x)=[[x]](x)+1", "0efac8bccb03a57a4e822317e2498f47": "\\psi _{1}=\\phi _{1}+e_{B}-e_{C}", "0efaff8c84eb1a9dce594c56ffa0a1af": "\\Delta v=v_{\\mathrm {rel} }\\ln {\\frac {m_{0}}{m_{1}}}", "0efb61223cf94255be128c750a839e58": "D_{4}^{-}", "0efb66636f1cdbe8813790a141ca2c77": "CM+\\alpha G", "0efbde7501dec5cbf1cb358503cbe134": "\\Delta _{d}", "0efbeaf74629bc34826bae9a002ad92b": "S:H\\to H", "0efbf0885f3e6692ef3edce7f49b7cd1": "{\\begin{aligned}&{\\mathit {CV}}_{i,j,k-{\\tfrac {1}{2}}}h_{i,j,k-1}^{m}+{\\mathit {CC}}_{i-{\\tfrac {1}{2}},j,k}h_{i-1,j,k}^{m}+{\\mathit {CR}}_{i,j-{\\tfrac {1}{2}},k}h_{i,j-1,k}^{m}\\\\&+\\left(-{\\mathit {CV}}_{i,j,k-{\\tfrac {1}{2}}}-{\\mathit {CC}}_{i-{\\tfrac {1}{2}},j,k}-{\\mathit {CR}}_{i,j-{\\tfrac {1}{2}},k}-{\\mathit {CR}}_{i,j+{\\tfrac {1}{2}},k}-{\\mathit {CC}}_{i+{\\tfrac {1}{2}},j,k}-{\\mathit {CV}}_{i,j,k+{\\tfrac {1}{2}}}+{\\mathit {HCOF}}_{i,j,k}\\right)h_{i,j,k}^{m}\\\\&+{\\mathit {CR}}_{i,j+{\\tfrac {1}{2}},k}h_{i,j+1,k}^{m}+{\\mathit {CC}}_{i+{\\tfrac {1}{2}},j,k}h_{i+1,j,k}^{m}+{\\mathit {CV}}_{i,j,k+{\\tfrac {1}{2}}}h_{i,j,k+1}^{m}={\\mathit {RHS}}_{i,j,k}\\end{aligned}}", "0efc06fcab5210a83d659bae0f38abd6": "K_{il(m+1)}=1", "0efcaa4ee592761dfbaac353df2b7e36": "[0,1).", "0efd20a8c567fb28d43428ee20e2bbcb": "\\phi _{Y}", "0efd61bc41390992a77f80a5a06c47bf": "x\\in {0,1}", "0efd788024567518a5b4293e0931ae9b": "n=2k+1", "0efdc919754730e8a1d775867dd9a765": "EL(\\Gamma )=EL(\\Gamma _{0}\\cup {\\hat {\\Gamma }})\\geq EL(\\Gamma _{0})", "0efdd35ff55faf6c57660f8f2fabe06a": "{1 \\over 1}", "0efddb396a169aae735d4c6338bbd573": "{\\tilde {X}}\\to X", "0efddecf476bfe08bf5614cecb579e81": "D_{\\odot }=8\\cdot 10^{22}Tm^{3}", "0efe3aa0265316cfd541fa2d8ddec772": "X_{\\gamma }^{*}", "0efe603d75586124a0c9063f2efa1ee9": "\\operatorname {succ} (C,\\langle \\alpha ,\\beta ,\\gamma ,\\delta \\rangle )", "0efe7438c4655ff1db613dc403c879c8": "P_{CJ}", "0eff006a60fd5cc965f8cb0c1deb0abf": "G(z,v)=\\left.{\\frac {d}{dv}}(Q_{1}(z,v)-Q_{2}(z,v))\\right|_{v=1}", "0eff7f098e5d3f54f62d64451e7d73f2": "{s_{4}}", "0efff695b0b58001158ce4db0a4b2bcc": "\\psi _{\\alpha }(z)=\\left|f(z)\\right|^{\\alpha }", "0f00b3c10d0376443564d5f3caee0d94": "\\mathbf {p} _{1}", "0f00bc2c7720fa6ad525f36dcdeb1e12": "{\\tbinom {n}{r}}p^{r}q^{n}\\!", "0f00c9c3df8cec18099ead98b0fe798f": "R_{\\odot }", "0f00f1ac31b2a114d7604981bf0b4506": "(\\Theta +w_{1})\\cap (\\Theta +w_{2})=\\{w_{1}-w_{2},0\\}", "0f016364aa5f908606195967b5720016": "(12)\\quad \\psi _{SS}={\\frac {1}{2}}\\ln {\\frac {L-M}{L+M}}\\,,\\quad \\gamma _{SS}={\\frac {1}{2}}\\ln {\\frac {L^{2}-M^{2}}{l_{+}l_{-}}}\\,,", "0f0220afee2fda153a57d4917ccc9c73": "P_{C}(t)=\\sum _{j=0}^{n}\\dim(C^{j})t^{j}.", "0f02415893f392ae3a49b7f094582894": "\\ A\\oplus B", "0f027be980543d6a3190da0644af525e": "\\theta =\\pi /2", "0f028a9b2ff22fc19f614142846f2b60": "p={\\dot {f}}\\left(x_{0}\\right)", "0f02d2af8d4cb3398befb330521499aa": "T_{\\text{H}}=\\hbar g/(2\\pi ck_{\\text{B}})", "0f032e6603cd3acec91e101e4151608a": "p\\left(x\\right)=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}", "0f039bc7f9fbb9d104edfd36ff8e1a8f": "y_{2c}=\\left({\\frac {q_{2}^{2}}{g}}\\right)^{\\frac {1}{3}}=\\left({\\frac {50.0^{2}}{32.2}}\\right)^{\\frac {1}{3}}=4.27{\\text{ ft}}", "0f040012f32ccda8df006d37de843dca": "vx/c^{2}", "0f04410c5d06eb16acee9bd6dba56da1": "\\nabla ^{2}{\\tilde {E}}+n^{2}(\\omega )k_{0}^{2}{\\tilde {E}}=0", "0f044fd89a02f0faaa33f601cd4456c6": "(x_{1},x_{1}^{2}),\\dots ,(x_{n},x_{n}^{2})", "0f04760baf5558d4bbe75cf61baf6c3e": "\\,\\!E(p_{1},p_{2},m):=p_{1}x_{1}^{*}(p_{1},p_{2},m)+p_{2}x_{2}^{*}(p_{1},p_{2},m)", "0f0479dde21b538e20613c236524e8bd": "\\alpha _{m_{1},\\ldots ,m_{a},k}", "0f04830c34e7cc214636ea1943b847ff": "|\\psi _{1}\\rangle ", "0f04f756ef2ffd214dfa965190df0ea5": "1,4,13,40,121,\\ldots ", "0f0531115e3e77f256c4793f6b5b7cba": "\\mathbf {p} _{\\rm {i}}=m_{i}\\mathbf {v} _{\\rm {i}}\\,\\!", "0f0548e3edbe197a085783f43c886fd7": "\\Delta f\\Delta t\\geq 1", "0f05ed8a80db39a44f2a36e3add4ce6d": "F^{g}=\\mathbb {I} +[\\lambda ^{g}-1]f_{0}\\otimes f_{0}\\,", "0f065de05a3af4d3b9d35512c5b8fb1b": "\\epsilon =-1,0,1", "0f068c855c612be613d98074d8104729": "n_{H}", "0f068e12955a8bb3921ff41539198f29": "L=20\\ \\log _{10}\\left({\\frac {4\\cdot \\pi \\cdot d\\cdot f}{c}}\\right)", "0f0692c6a127aeb0927f9560b5d0957c": "t^{+}(X)", "0f0732ee7778c809bc99b171b67bb183": "\\psi \\,{\\sim }\\,\\psi +2\\pi ", "0f073743b3bcca827a11bfdbbff36525": "L_{1}\\pitchfork L_{2}", "0f076a371caa07eec622be5168e5063f": "[u_{1}(+1,-1)+u_{1}(-1,+1)]-[u_{1}(+1,+1)+u_{1}(-1,-1)]=[u_{2}(+1,-1)+u_{2}(-1,+1)]-[u_{2}(+1,+1)+u_{2}(-1,-1)]", "0f076a79e69ae714cb4f6e082f1452ff": "MA={\\frac {T_{B}}{T_{A}}}={\\frac {r_{B}}{r_{A}}}.", "0f076f8b744e97a21de9fe1a95a8472b": "43^{8}+96222^{3}=30042907^{2}\\;", "0f07873018bf1a703d5afadfd6970ea4": "d(t)", "0f07c7495a7216ac745bb0e93b9042ef": "R={\\begin{bmatrix}1/2&-1/{\\sqrt {2}}\\\\-1/{\\sqrt {2}}&2\\end{bmatrix}},", "0f07d5fcda7e3b7c2421c5abf2e3f009": "V_{f}", "0f081a488e5950f3db1dff195e784251": "{\\frac {\\partial }{\\partial z_{1}}},...,{\\frac {\\partial }{\\partial z_{n}}}", "0f092a85c54621235644012feb10b93e": "I(2\\omega ,l)=I(\\omega ,0)\\tanh ^{2}{\\left({\\frac {E_{0}\\omega d_{\\text{eff}}l}{n_{\\omega }c}}\\right)}", "0f095189b6b24dfc311fb7cb95426510": "{\\begin{aligned}\\int x\\cos(x)\\,dx&=\\int u\\,dv\\\\&=uv-\\int v\\,du\\\\&=x\\sin(x)-\\int \\sin(x)\\,dx\\\\&=x\\sin(x)+\\cos(x)+C,\\end{aligned}}\\!", "0f0977808653871ae4a6844920bad475": "\\langle 0|\\Phi ^{\\dagger }(x)\\Phi (y)|0\\rangle ", "0f09f0a472a0128d7648ba178a3ef497": "g(x,y,t)\\geq 0", "0f09f7f67aaff7917507b729ca4475fb": "{\\sqrt {\\scriptstyle {R(R-2r)}}},", "0f0a198fa4bfaca64e402dec5c7432c6": "(\\mathbf {A} +\\mathbf {A} ^{\\rm {T}})\\mathbf {X} ", "0f0a6fff8b02a5f891baedc44d75db3a": "{\\hat {G}}_{j}|\\psi \\rangle =0", "0f0a803810eee22c12ffe5fda15c68fb": "M_{t}", "0f0ad94c353a22cb807dec55d5804ccf": "ax^{3}+bx^{2}+cx+d=0.\\,", "0f0ae744d0a1c066d4e7909bf74b8047": "\\scriptstyle {p_{i}-1}\\,>\\,{p_{i-1}}", "0f0b2f09c81ae61048bb5dcbfd62f374": "\\,\\!G_{D}=G_{m}A_{D}\\tanh(L_{D})/L_{D}", "0f0b647a6d76b42ee1f3c9318356cd0a": "\\ Z\\in \\,\\Gamma ", "0f0ba9e9598750f90d669b0e93f473ae": "x(\\alpha -\\beta y)=0\\,", "0f0bb9f9d04a17a54202e85cb5e2e518": "s_{1}=s_{2}=1", "0f0bc587de127b293209cbed48446924": "h=e-f", "0f0bc80015e3f6b8ec2013685387aa07": "k_{B}TC", "0f0c270d43cf35e63758f8e587e5d8e2": "a_{1}\\equiv a_{2}{\\pmod {n}}", "0f0c65c2fea532dde8a1fd0c56c6ccf7": "Z_{e}^{0}", "0f0c8fadf0bc978fc4b57422dfd06eae": "\\kappa =\\rho _{0}c_{0}^{2}~.", "0f0d089095143c98b63de7fa90778416": "{\\mathcal {A}}_{i_{n}=i}", "0f0d472568fd30411f161b15937203c8": "\\Gamma ={\\frac {V_{R}}{V_{F}}}\\,", "0f0d877106d519c364414428ed6c8343": "\\textstyle x!\\{{n \\atop x}\\}\\,", "0f0d999e448e9d8ad5620a349374e05d": "\\sum _{k=0}^{\\infty }(-1)^{k}k!=\\int _{0}^{\\infty }\\left[\\sum _{k=0}^{\\infty }(-x)^{k}\\right]\\exp(-x)\\,dx", "0f0dc63e1f0389e7906126b17bd3d363": "B(a,b)", "0f0dd321a08735ebfd9622515597fc3e": "R\\times R{\\stackrel {m}{\\to }}R", "0f0dfad96415df06399d5889f925e884": "(x,z)\\in R_{i}", "0f0eef7027405e6f329f95badada1bea": "C-P=D(F-K)=S-DK", "0f0f088f38f4ff29f23b1485d1037801": "B_{\\lambda }(T)\\neq B_{\\nu }(T)", "0f0f2b960c4a5542f476c18e6ce1cf72": "\\left\\{Q_{\\alpha i},{\\overline {Q}}_{\\dot {\\beta }}^{j}\\right\\}=2\\delta _{i}^{j}\\sigma _{\\alpha {\\dot {\\beta }}}^{\\mu }P_{\\mu }", "0f0f408101834f1e9fe619a9b93fa289": "\\int _{1}^{\\infty }\\left(\\int _{1}^{\\infty }{\\frac {x^{2}-y^{2}}{\\left(x^{2}+y^{2}\\right)^{2}}}\\ dx\\right)\\ dy={\\frac {\\pi }{4}}\\ .", "0f0fa540d7e1cb650a0556a76b2d6edd": "d(y,z)+d(x,y)\\geq d(x,z)", "0f102b13d5901c3496616db74fc186e3": "\\langle 1/n_{i+1}\\rangle \\subseteq \\langle 1/n_{i}\\rangle ", "0f102ea0f824fa658f2cdb21c083c06f": "{\\bar {u}}(T)={\\frac {0.860117757+1.54118254\\times 10^{-4}T+1.28641212\\times 10^{-7}T^{2}}{1+8.42420235\\times 10^{-4}T+7.08145163\\times 10^{-7}T^{2}}}", "0f1054ee3143dafa62f0ef3bd496790a": "\\scriptstyle \\varphi (\\alpha )\\,", "0f1064238c29a38c14fd70abd3efeb4f": "\\omega (\\lambda )=W(\\Phi _{\\lambda },\\mathrm {X} _{\\lambda }),", "0f106db4509c14c554251e3dbd751c4a": "{}_{p}F_{q}(a_{1},\\dots ,a_{p};c_{1},\\dots ,c_{q};z)=\\sum _{n=0}^{\\infty }{\\frac {(a_{1})_{n}\\cdots (a_{p})_{n}}{(c_{1})_{n}\\cdots (c_{q})_{n}}}{\\frac {z^{n}}{n!}}", "0f10a24f958a56e8a29ea76f8d3c9d62": "\\displaystyle {L(a,b)=2([L(a),L(b)]+L(ab)).}", "0f10c55dd6b23de94ce63ac8620f6a93": "a_{r}={\\frac {(c)_{r}(c+1-\\gamma )_{r}}{(c+1-\\alpha )_{r}(c+1-\\beta )_{r}}}a_{0}\\quad \\forall r\\geq 0", "0f11322364697effa291c09cbb4b329b": "T^{0}", "0f113d2a51cfa347752c80b86ccdaf4c": "{\\text{The particle horizon }}H_{p}{\\text{ exists if and only if }}N>2", "0f11a79b80fc384f5fc8bf0c1b982015": "-{\\frac {c^{4}}{8\\pi G}}{\\frac {v_{s}^{2}(y^{2}+z^{2})}{4g^{2}r_{s}^{2}}}\\left({\\frac {df}{dr_{s}}}\\right)^{2},", "0f11ac423daf13c71340554654c1f774": "{1 \\over 2}{\\begin{pmatrix}1&1&0&0\\\\1&1&0&0\\\\0&0&0&0\\\\0&0&0&0\\end{pmatrix}}\\quad ", "0f11e991ac9a2e24be89fd5b6fc73522": "0<|a|<1", "0f120720fca7dc520b10b29c78724048": "e^{2.5}", "0f12112fad912306098877356a13460e": "R\\,exp(iS/\\hbar )", "0f1211b4dca7a3046a352d3ca0fc7922": "{\\frac {1}{4}}{\\begin{bmatrix}1+c&a-ib&\\pm (1+c)&\\pm (a-ib)\\\\a+ib&1-c&\\pm (a+ib)&\\pm (1-c)\\\\\\pm (1+c)&\\pm (a-ib)&1+c&a-ib\\\\\\pm (a+ib)&\\pm (1-c)&a+ib&1-c\\end{bmatrix}}", "0f12224ab3e145715557d1856b6e7d27": "4a^{2}x^{2}+4abx+4ac=0", "0f126900012f02725a504d249bd78a10": "(\\mathbb {Z} _{6},+)", "0f1270a0b146db3c08cb6bbd545bb785": "D{\\overline {D}}A=-J", "0f12cd3827b8b5e1ad8ade89ead4e378": "\\langle C_{\\xi }:\\xi <\\alpha \\rangle \\,", "0f13444fc91bddebf3da47ff668a633b": "f^{(i)}(0)", "0f135e01a8d4d50018a46c4051f79c3e": "h(\\cdot )", "0f136d4a5ddfc2d79894a45600d12934": "{\\begin{matrix}\\beta _{0}&=\\beta _{0}^{(0)}&&&\\\\&&\\beta _{0}^{(1)}&&\\\\\\beta _{1}&=\\beta _{1}^{(0)}&&&\\\\&&&\\ddots &\\\\\\vdots &&\\vdots &&\\beta _{0}^{(n)}\\\\&&&&\\\\\\beta _{n-1}&=\\beta _{n-1}^{(0)}&&&\\\\&&\\beta _{n-1}^{(1)}&&\\\\\\beta _{n}&=\\beta _{n}^{(0)}&&&\\\\\\end{matrix}}", "0f136f00ead8866abe9a7ea038843828": "{\\sqrt {\\tfrac {32}{5}}}={\\sqrt {\\tfrac {16\\cdot 2}{5}}}=4{\\sqrt {\\tfrac {2}{5}}}", "0f1371682cd4c8a8e9a1be893361676a": "\\sum _{j=1}^{D-1}p_{j}^{2}=1.", "0f13f3c39690d785a4a1d04c50b0f1f8": "(x,y,z)", "0f141a44688b80e310d7959ec6ae93ac": "{\\frac {1}{3}}", "0f1438efb3841f09489d9ab8bd19e004": "\\gamma =1+{\\frac {M_{\\mathrm {orb} }}{(M_{\\mathrm {spin} }+M_{\\mathrm {orb} })}}", "0f1484c7ec1f75de907fdf1d3d5a73d8": "\\Delta _{KN}\\,:=\\,r^{2}-2Mr+a^{2}+Q^{2}\\,,\\;\\;\\rho _{KN}\\,:=\\,r^{2}+a^{2}\\cos ^{2}\\!\\theta \\,,\\;\\;\\Sigma ^{2}\\,:=\\,(r^{2}+a^{2})^{2}-\\Delta _{KN}a^{2}\\sin ^{2}\\theta \\,.", "0f1495edf029c94f6e2d799433c86b01": "k\\to k[X_{1},\\ldots ,X_{n}].", "0f150e200abc8fd48aa0cf3adac855bc": "m\\leftarrow N", "0f1529f45f27a0a59079bba2adc07d8f": "F_{\\mathrm {e} }={\\frac {\\varepsilon _{0}\\varepsilon _{r}AV^{2}}{2d^{2}}}", "0f158ae8922e991854741f120ada1f98": "(l_{k},l_{k},m_{k})\\,", "0f15cf873eeb280fdc5d799347af871f": "\\gamma _{\\xi }=-{\\frac {1}{4}}\\xi +{\\frac {1}{8}}\\xi \\left(2\\chi _{\\xi }^{2}+\\mathrm {ch} 2\\chi +1\\right).", "0f15eb788b44e710faa9b3e94ad71e85": "U_{L,0}={\\mathcal {O}}_{L}^{\\times },U_{L,i}=1+{\\mathfrak {p}}^{i}", "0f1632cbfe9047c43071f9922adcc684": "{\\frac {1}{2}}C\\omega ^{2}=-{\\frac {1}{2}}CU^{2}\\left({\\frac {2k}{rd}}\\right)\\theta ^{2}", "0f1667fcf4945b65ed8e562246210463": "Y=\\{-(1+x):x\\in X\\}", "0f1670cd146f6acc74d478f311d74b0d": "n\\mathbin {:} {\\mathbb {N} }", "0f16aabf3789de8261bbaa408b02037c": "n_{e}=N_{\\rm {C}}(T)\\exp((E_{\\rm {F}}-E_{\\rm {C}})/kT),\\quad n_{h}=N_{\\rm {V}}(T)\\exp((E_{\\rm {V}}-E_{\\rm {F}})/kT),", "0f16ac37fade9b3e37b91246bac75b19": "\\rho _{w}", "0f174fa569af15a5c309f7a6ab4ae9ce": "S+E+I+R=N", "0f177369a3b71275d25ab1b44db9f95f": "SG", "0f177b64ba415650092993ad7ffd08bf": "m_{n}=\\int _{I}t^{n}\\,\\rho (t)\\,dt,", "0f17909efaa6c4600b27b53e1464afb8": "\\angle DOC=\\angle EOC-\\angle DOE.", "0f180b6b6420caeb7a3ac04d7673d097": "\\forall i\\exists j:\\;j\\geq i\\quad \\rho (w_{j})\\in F.", "0f18b0719a8a25c84045838eed640d5d": "\\operatorname {drop-param} [(g\\ q\\ p\\ n),D,V,\\_]", "0f18d177552f76439714df1a070557ef": "n_{R}", "0f18e1e5b5e7572e9ae0771f74f505de": "\\psi ={\\begin{pmatrix}\\psi _{1}\\\\\\psi _{2}\\\\\\psi _{3}\\end{pmatrix}},{\\overline {\\psi }}={\\begin{pmatrix}{\\overline {\\psi }}_{1}^{*}\\\\{\\overline {\\psi }}_{2}^{*}\\\\{\\overline {\\psi }}_{3}^{*}\\end{pmatrix}}", "0f1931a673d4a539da6ab4ebe1a89c12": "{\\tilde {g}}_{33}=r_{0}^{2}\\sin ^{2}\\theta ", "0f195c56b2ea0e2d3dbbdd06985642a1": "z^{n}-y^{n}", "0f19c7a357993d0b24f33526aae81e6c": "Z_{in}=Z_{C}\\,", "0f1a0bf001f82a76496bc5b40d6efaf7": "C^{*}", "0f1a0f019de9f84c8da32d01bdae9be1": "\\rightarrow D_{n}", "0f1af1f75945c10f599368811e2d8a64": "{\\frac {1}{8}}", "0f1af5f00bf02e3b287ede795dd97000": "x_{i}y_{j}=x_{j}y_{i}", "0f1b0622732b9a85b94f0276c68bcbe2": "T_{\\mathbf {v} }^{-1}=T_{-\\mathbf {v} }.\\!", "0f1b46270f6500d1db212c0b97fcf23d": "{\\mathcal {O}}_{X}\\to k(x)\\xrightarrow {F} k(x)", "0f1b81d9663ffffd7db0fc133954caff": "{\\frac {Y}{X}}={\\frac {(s+z_{1})(s+z_{2})}{(s+p_{1})(s+p_{2})}}.", "0f1c584c74c18e33b7af55168a49c417": "I_{o}={\\frac {\\left(V_{i}-V_{o}\\right)DT\\left(D+\\delta \\right)}{2L}}", "0f1c5de67149c89cfb36bdce0db33fdb": "L_{\\text{o}}(\\mathbf {x} ,\\,\\omega _{\\text{o}},\\,\\lambda ,\\,t)\\,=\\,L_{e}(\\mathbf {x} ,\\,\\omega _{\\text{o}},\\,\\lambda ,\\,t)\\ +\\,\\int _{\\Omega }f_{r}(\\mathbf {x} ,\\,\\omega _{\\text{i}},\\,\\omega _{\\text{o}},\\,\\lambda ,\\,t)\\,L_{\\text{i}}(\\mathbf {x} ,\\,\\omega _{\\text{i}},\\,\\lambda ,\\,t)\\,(\\omega _{\\text{i}}\\,\\cdot \\,\\mathbf {n} )\\,\\operatorname {d} \\omega _{\\text{i}}", "0f1c5ecfd263f374d42b3ed098cdbdbc": "N\\triangleleft {\\text{mess}}", "0f1cd825b8ef9dfdc6adac0a716ce7df": "c_{k}={\\frac {{\\frac {1}{2}}f(b_{k})a_{k}-f(a_{k})b_{k}}{{\\frac {1}{2}}f(b_{k})-f(a_{k})}}", "0f1d1c919beaad5e662854164f3547ee": "v_{k+1}=U_{k}(\\alpha )\\,\\!", "0f1d3168bb177f371b7ae01ff6e619da": "(V_{r},V_{s})", "0f1d457ed91a664e4a8558572a6ca2e6": "m>\\lfloor n/2\\rfloor ", "0f1d50fd14bb478046961dbf4a645642": "\\ln(2)", "0f1d6bb4ba23b6f35e3491360c82f119": "\\sigma _{N}", "0f1d8f39f3821707300f85643412a8d6": "\\omega ={\\frac {(6+\\sigma )^{2}}{24}}", "0f1dc919c903c93b11a16d99c3cb2971": "F:A\\rightarrow B", "0f1e1872b95b8bbc4bf0d8a0eb159198": "\\cos x+\\cos 2x+\\cos 3x+\\cdots =\\sum _{k=1}^{\\infty }\\cos(kx).", "0f1e44ebca5015f56450922eee10efde": "V={\\frac {\\partial }{\\partial \\sigma ^{2}}}\\log L(\\sigma ^{2},X)", "0f1e914f26d9342611b4ede510773172": "\\sigma _{f}^{2}=a^{2}\\sigma _{A}^{2}", "0f1f1d95b54202965f0eba7e56ee7fa8": "\\mathbf {a} \\times (\\mathbf {b} \\times \\mathbf {c} )=(\\mathbf {a} \\cdot \\mathbf {c} )\\mathbf {b} -(\\mathbf {a} \\cdot \\mathbf {b} )\\mathbf {c} ", "0f1fbd76ac5335872f5e662dc522ed9d": "\\left(P_{2}-P_{1}\\right)", "0f1fcc336ac5b6f2dd73e6eb152031cb": "d(x,y)\\geq 0", "0f20b9bf9ac0229f73b3cafb15befa2a": "\\rho =(U\\otimes U)\\rho (U^{\\dagger }\\otimes U^{\\dagger })", "0f20ed920f547b473a9490757c6cd0e7": "{\\tilde {C}}", "0f22779771cafdad017cfaa8e4555d89": "{\\vec {H}}", "0f229b490c383cf89a339be0147ed7a8": "V_{\\text{out}}={\\begin{cases}V_{{\\text{S}}+}&{\\text{if }}V_{1}>V_{2},\\\\V_{{\\text{S}}-}&{\\text{if }}V_{1}1.8\\times 10^{15}", "0f48087903db5651c7f25b0d254ff050": "\\mathrm {vis} ", "0f482605492c028a78db0ba355d1cafb": "([g_{1},h_{1}]\\cdots [g_{n},h_{n}])^{s}=[g_{1}^{s},h_{1}^{s}]\\cdots [g_{n}^{s},h_{n}^{s}]", "0f48716ddd3296ba459b851fcd483f27": "={M \\over 2}\\cdot A.", "0f4871dd0ff9551e39ca7a2e73b37305": "\\Upsilon \\,", "0f487c94170acb324233fad4d4d18fac": "ZFC+\\lnot \\operatorname {Con} (ZFC+H)\\vdash \\lnot \\operatorname {Con} (ZFC)", "0f48a335a7c9e4528be77f00cad07d57": "Y(s)=Z(s)G(s)\\Rightarrow Z(s)={\\dfrac {Y(s)}{G(s)}}", "0f48aeabf7edfb0b62703187e367cd8c": "x^{2}{\\frac {d^{2}y}{dx^{2}}}+x{\\frac {dy}{dx}}+(x^{2}-\\alpha ^{2})y={\\frac {4{(x/2)}^{\\alpha +1}}{{\\sqrt {\\pi }}\\Gamma (\\alpha +{\\frac {1}{2}})}}", "0f48ba77a92cfc6534ac0cdaefa06495": "a_{-1}=1", "0f48da724906e6cd83a23bfc1a5e7229": "z(r,\\theta )=\\Re (2i(-\\ln(1-r^{2}e^{2i\\theta })+\\ln(1+r^{2}e^{2i\\theta }))=2\\tan ^{-1}\\left({\\frac {2r^{2}\\sin 2\\theta }{r^{4}-1}}\\right)", "0f48f62c0f7893f9fb7fccc7817ee5ed": "s^{\\mathfrak {n}}(t)", "0f48fff9a3b17d030f4b22a90b332a89": "d^{-1}", "0f490342247f08cfa2930b46aaaa92af": "x=ka\\sin \\theta ={\\frac {2\\pi a}{\\lambda }}{\\frac {q}{R}}={\\frac {\\pi q}{\\lambda N}}", "0f492b6e31bca36580398bb1d0be3685": "\\displaystyle {a^{-1}=(a-b)^{-1}+(a-Q(a)b^{-1})^{-1}=(a-b)^{-1}+Q(a)^{-1}(a^{-1}-b^{-1})^{-1}.}", "0f4956319d3e1831c8d3a2a341fd0323": "\\sigma (\\mathbf {x} )=\\mathbf {0} ", "0f49ab8db7d4552fe5370831bd1ffd3d": "A\\succeq B\\succeq C", "0f4a36a71d180e1c06d76d770ce919cc": "\\Gamma (D(P),{\\mathcal {O}}_{\\mathbb {P} (V)})", "0f4a720d82fb0ece5e4582e3d497e773": "g\\in {\\mathcal {H}}", "0f4acbf419a68b1e231f5a17f1dfe74a": "\\eta _{0}\\propto {\\begin{cases}const.&{\\text{, }}a<{\\sqrt {a_{m}}}\\\\\\left|t\\right|^{1/4}&{\\text{, }}a={\\sqrt {a_{m}}}\\\\\\left|t\\right|^{1/2}&{\\text{, }}a>{\\sqrt {a_{m}}}\\end{cases}}", "0f4ad3bfb282249852d4a44f2da783d9": "q_{\\text{opt}}=F^{-1}\\left({\\frac {7-5}{7}}\\right)=F^{-1}\\left(0.285\\right)=D_{\\min }+(D_{\\max }-D_{\\min })\\cdot 0.285=58.55\\approx 59.", "0f4b5c2053084288ba37032ffc17b004": "i=f(I\\cdot t^{p})", "0f4b5cb0d3f20255da0a4f9aa377ba6d": "s=a\\tan \\varphi ", "0f4baa38d51338570ac03f2cde849699": "\\operatorname {Hom} _{\\rm {Schemes}}(X,\\operatorname {Spec} (A))\\cong \\operatorname {Hom} _{\\rm {CRing}}(A,{\\mathcal {O}}_{X}(X)).", "0f4bf550421793761fe8501bd8dcad19": "y_{i}\\in \\{0,1\\}^{k}", "0f4c1baf6f3b4b49b635036b02520aa0": "x=\\left({\\frac {1}{k}}\\right)y,", "0f4c4ce0863d100a12c90c114fd9abeb": "{\\vec {b}}", "0f4c8bcae3d428804c468ce4c2e374c4": "K_{m,n},", "0f4cd392fdcd029636175fbee752d895": "X\\setminus Y", "0f4cd952c45670d1a7b9c0f59f1319b6": "y_{3}=\\sin i\\cdot \\cos \\omega ", "0f4d44a6d1ab415a985235815794ffd8": "n\\approx 2.5{\\sqrt {\\frac {c}{2j+1}}}", "0f4d4d48f82159f4e29325b88c84a4a8": "{\\frac {\\cos(x)x^{2}-\\sin(x)2x}{x^{4}}}", "0f4d6327a617f88e3d585b30e6eab420": "Z>m_{N}", "0f4d948653fa27cc90071f85d5b6d8d8": "{\\frac {\\partial c}{\\partial t}}+\\nabla \\cdot {\\vec {j}}=R,", "0f4dab6356d1fd9de79ec5678d07fa83": "\\tau _{\\varepsilon }", "0f4e0ab41c9ac56bd3ec84bc8d93671f": "\\langle \\Phi (\\rho \\otimes \\omega ),I\\otimes O\\rangle =\\langle \\rho ,O\\rangle ", "0f4e76b575249460870a83c151a53d67": "{\\begin{array}{lcll}I_{2}(d^{2}\\theta /dt^{2})&=&-K-M(d\\theta /dt)&\\cdots (Eq.7)\\\\(d^{2}\\theta /dt^{2})+M/I_{2}(d\\theta /dt)&=&0&\\cdots (Eq.8)\\end{array}}", "0f4ec8f1bf97041b050e900f1aebf684": "{XX'' \\over YY'}={DX \\over BY},", "0f4f7ffc015aa89b86f99b7d40446152": "\\{\\mu _{i}\\}_{i=1\\dots \\infty }", "0f4fbe4fb9f670136bd4cabbddb71a4e": "\\nabla ^{2}G=\\delta (x).", "0f4fdbf640f71457a4f8156b659621b6": "p=\\hbar k\\,\\!", "0f500804d115099c77b8aa3ac2fd901f": "b\\in I^{j}/I^{j+1}", "0f5010640e0352579f10c06ffc92375d": "\\{x\\}\\mapsto [x]_{R}", "0f50309a698653a330504a19be7b3bfb": "{\\begin{bmatrix}[x=1]\\\\\\vdots \\\\{[x=k]}\\end{bmatrix}}", "0f503ea579eaee3098543fbfcc343b03": "\\rho _{2}=\\rho _{3}=z^{-2}+9z^{-3}+80z^{-4}+965z^{-5}-\\cdots .", "0f5073c04b8fcde03f80f3e1324933e5": "Lx", "0f508098e84c6b780ca835ec5862a647": "\\scriptstyle {Z_{21}}", "0f50c9baa20cdd71cf493e06a21c5bee": "\\{\\ X^{/}\\}", "0f50ce1725284da939e3907c2521e787": "\\eta _{v_{e}}", "0f515746a540b5b8f8e2096d33a55c75": "S=\\int Tr\\partial _{\\mu }A_{\\nu }\\partial ^{\\mu }A^{\\nu }+f_{jk}^{i}\\partial ^{\\nu }A_{i}^{\\mu }A_{\\mu }^{j}A_{\\nu }^{k}+f_{jr}^{i}f_{kl}^{r}A_{i}A_{j}A^{k}A^{l}+Tr\\partial _{\\mu }{\\bar {\\eta }}\\partial ^{\\mu }\\eta +{\\bar {\\eta }}A_{j}\\eta \\,", "0f52708aa0d2f74a63230012b40d1ca8": "k_{q}", "0f52d112f4addae265628151fa31b328": "\\mathbf {m} _{i}={\\frac {1}{l_{i}}}\\sum _{n=1}^{l_{i}}\\mathbf {x} _{n}^{i},", "0f52d94344143bc8b1fd698cae16e83d": "\\psi (\\beta )<\\delta ", "0f52dec5c9d961f937211f6db1952f92": "{\\frac {|\\triangle SCA|}{|\\triangle SDA|}}={\\frac {|\\triangle SCA|}{|\\triangle SCB|}}", "0f53350338713b8a22fe41905d7749c5": "\\kappa _{xy}(f)={\\frac {A_{xy}^{2}}{\\Gamma _{xx}(f)\\Gamma _{yy}(f)}},", "0f537a85a20e520b18abcce2df9eb893": "{\\mathcal {P}}={\\Big \\{}\\ f_{\\theta }(x)={\\tfrac {1}{{\\sqrt {2\\pi }}\\sigma }}e^{-{\\frac {1}{2\\sigma ^{2}}}(x-\\mu )^{2}}\\ {\\Big |}\\ \\theta =(\\mu ,\\sigma ):\\mu \\in \\mathbb {R} ,\\,\\sigma \\!>0\\ {\\Big \\}}.", "0f53e98758ba9b2b118b728482471323": "(a^{n}-1)^{2n}+(a^{n}-1)^{2n+1}=[a(a^{n}-1)^{2}]^{n};", "0f5478b96f3b8b3c582269a8dd1f6f4c": "[x_{0},x_{1},\\ldots ,x_{n}]", "0f549e0a2668dcee3144d997fee0ae7a": "J_{0}^{1}(\\varphi \\circ f)(t)=tv^{i}{\\frac {\\partial f}{\\partial x^{i}}}(p)", "0f54c07dc29cc38240e9bd81c391cd8e": "Y=A\\times K^{\\alpha }\\times L^{\\beta }", "0f54ee04ab1446600e4220c0d4232b67": "\\chi _{\\lambda }(e^{X})={\\rm {Tr}}\\,\\pi _{\\lambda }(e^{X}),(X\\in {\\mathfrak {t}})),\\,\\,\\,d(\\lambda )={\\rm {dim}}\\,\\pi _{\\lambda }.", "0f54f750550d134b74f42ef611710b7c": "B_{\\lambda }(\\lambda ,\\ T)\\ =\\ -\\ {\\frac {\\mathrm {d} \\nu }{\\mathrm {d} \\lambda }}B_{\\nu }(\\nu (\\lambda ),\\ T).", "0f554ea390eeb2d4997d482777249fcd": "{\\text{grant}}(j)", "0f555f8a6116983db9ff85a0ac6bd491": "{\\mathcal {M}}=(r,\\mathbf {b} ,\\mathbf {\\delta } ,\\mathbf {\\sigma } ,A,\\mathbf {S} (0))", "0f55612e676ceca86078c208ebbb02d8": "\\alpha =1/{\\sqrt {\\beta _{c}}}", "0f55c3cdf8180af20d86a06eb31954f6": "M\\sqcup (-N)", "0f55cab694c029be42a8f8a8c2e0a99e": "\\int _{\\Omega }vL[u]\\ d\\Omega =\\int _{\\Omega }uL^{*}[v]\\ d\\Omega +\\int _{S}{\\boldsymbol {M\\cdot n}}\\,dS,", "0f55e29d9f506eb65b6a9dbcf1ebc719": "H(\\omega )=\\left\\{{\\mathcal {H}}f\\right\\}(\\omega )={\\frac {1}{\\sqrt {2\\pi }}}\\int _{-\\infty }^{\\infty }f(t)\\,{\\mbox{cas}}(\\omega t)\\mathrm {d} t,", "0f560a2e916b1cb8e819012fae3491ae": "\\ln(x)=\\sum _{n=1}^{\\infty }{\\frac {(-1)^{n+1}}{n}}(x-1)^{n}", "0f560d3528ac0cf5b4bf0da4eaf64f80": "\\psi (\\Omega \\omega )=\\varepsilon _{\\omega }=\\phi _{1}(\\omega )", "0f5647baff8f89c8ba99759ee35a03bd": "K=2\\pi -\\alpha -\\beta -C.", "0f5670c10524c3f9ea5d7d47f6fe4ab8": "{\\mathcal {B}}[f](\\alpha u+\\beta v,\\dots )=\\alpha {\\mathcal {B}}[f](u,\\dots )+\\beta {\\mathcal {B}}[f](v,\\dots ),{\\text{ when }}\\alpha +\\beta =1.\\,", "0f571e9eadc7f8e6c36922c6ad8157dc": "(p\\land (q\\land r))\\vdash ((p\\land q)\\land r)", "0f572d482ec9fe1a0feb546afe74cec7": "q'_{\\text{P}}={\\sqrt {\\epsilon _{0}\\hbar c}}={\\frac {e}{\\sqrt {4\\pi \\alpha }}}=5.291\\times 10^{-19}", "0f572e78d729f39d24acc591924ebc00": "1\\,{\\text{pdl}}=1\\,{\\text{lb}}_{m}\\cdot 1\\,{\\frac {\\text{ft}}{{\\text{s}}^{2}}}", "0f5745a39590195f97cefda202c1a00f": "{\\it {M}}=2", "0f574cafb461928fde1420211e049020": "{\\textbf {F}}={\\begin{bmatrix}1&\\Delta t\\\\0&1\\end{bmatrix}}", "0f57680523558dcf650fc1aa21528949": "F_{\\alpha \\beta }\\,=\\,\\partial _{\\alpha }A_{\\beta }\\,-\\,\\partial _{\\beta }A_{\\alpha }\\,.", "0f57acdd48e0530d2e2fda2d824788bc": "f\\geq 0", "0f57f1313a8e92722688d3aa000f2f14": "e=\\lim _{n\\to \\infty }\\left(1+{\\frac {1}{n}}\\right)^{n}.", "0f581636dd5a8183f5a0484d96d7cb19": "D_{t}=", "0f581f31e63cdce64f590ceb83d7f824": "\\operatorname {Cl} _{2}(2\\pi z)=2\\pi \\log \\left({\\frac {G(1-z)}{G(1+z)}}\\right)-2\\pi \\log \\left({\\frac {\\sin \\pi z}{\\pi }}\\right)", "0f583d100fec13c47aa4f911b268158c": "\\epsilon (\\mathbf {q} )=1+{\\frac {k_{0}^{2}}{q^{2}}}", "0f5853b0f8b5c20679b2c5bd78c91fef": "{\\tfrac {h_{i}}{k_{i}}}", "0f586042fa46296b3eb8b940f9b3a0f8": "\\Box \\mathbf {A} ={\\frac {4\\pi }{c}}\\mathbf {j} ", "0f5883fcd146319d282838deb4c174ec": "f(x)=x^{T}Ax", "0f58de596713bb42ec73c8b379d46c59": "LC_{50}(mixture)\\leq 200{\\tfrac {mL}{m^{3}}}", "0f58fac7c62960d176682b1d6404ea02": "e=2\\sum _{k=0}^{\\infty }{\\frac {k+1}{(2k+1)!}}", "0f5906ce29c97f3d473437f693f8addf": "\\oint _{K}\\kappa \\,ds\\leq 4\\pi ,", "0f59120e4ede42e765b23cd5e35d32b8": "\\Lambda =\\sum \\limits _{i=1}^{\\infty }\\Lambda _{i}.", "0f596901845f805cb6305b7b365cf5c4": "z<=-0.25", "0f596ef6330dafa2e50880d1b5ec22b5": "\\theta =\\arccos \\left({\\frac {\\mathrm {trace} (R)-1}{2}}\\right)", "0f597eba7730a8d5cce3b1b23cf733e8": "A=\\mathrm {d} ^{2}/\\mathrm {d} x^{2}", "0f598644b790d37b8f90de2586a0c90d": "E_{a'b'}=\\mathrm {diag} (\\lambda _{1},\\lambda _{2},\\lambda _{3})", "0f5994a9965ffe3f1965ecf6e89abe25": "U_{1}(T)=1", "0f59952ffe586f13939805ca01426332": "h(n)", "0f59a737f68757fbd97c1987919dcb9b": "{\\frac {X_{\\tau _{\\nu }}}{\\sqrt {\\nu }}}", "0f5a077a904db6bf28b28ea5957b55a1": "3(b^{2}-c^{2})(c^{2}-a^{2})(a^{2}-b^{2})(a^{2}yz+b^{2}zx+c^{2}xy)\\,\\,+", "0f5a6bacf345b81f7666359b3b79e6fe": "k=37", "0f5a911c58332baa07c6d99dadd954bf": "\\{\\{64x^{3}+384x^{2}-1024x+512,(1,{\\tfrac {3}{2}})\\},\\{64x^{3}+576x^{2}-64x-64,({\\tfrac {3}{2}},2)\\},\\{64x^{3}+192x^{2}+80x+8,(2,4)\\}\\}", "0f5aa0a8e99463f2a36455aba15bb5b5": "\\sigma _{\\mathrm {oct} }\\,\\!", "0f5aa3c68338c7fbb692df971a15955e": "\\ell _{i}\\in \\mathbb {R} \\cup \\{-\\infty \\}", "0f5b0dbee721c33ca7fe31936a32a42c": "\\sigma _{\\bar {x}}=\\sigma /{\\sqrt {n}}", "0f5b11f8a843355b4382ead6e182c57f": "{\\begin{aligned}R_{1}&={\\frac {R_{b}R_{c}}{R_{a}+R_{b}+R_{c}}}\\\\R_{2}&={\\frac {R_{a}R_{c}}{R_{a}+R_{b}+R_{c}}}\\\\R_{3}&={\\frac {R_{a}R_{b}}{R_{a}+R_{b}+R_{c}}}\\end{aligned}}", "0f5bfa684083cdd5d5a17fb3ebe51d8b": "P=\\{0,\\dots ,76\\}", "0f5bfc82b754e0f20c1aec792c462dfa": "\\operatorname {tr} (\\gamma ^{5})=\\operatorname {tr} (\\gamma ^{\\mu }\\gamma ^{\\nu }\\gamma ^{5})=0", "0f5bff0227f7d270d110b0fd2671a345": "SU(3)_{C}\\times SU(3)_{L}\\times U(1)_{X}", "0f5c02812f1eaadfdbcab8608a5e9889": "\\rho \\left(1\\right)={\\begin{bmatrix}1&0\\\\0&1\\\\\\end{bmatrix}}\\qquad \\rho \\left(u\\right)={\\begin{bmatrix}1&0\\\\0&u\\\\\\end{bmatrix}}\\qquad \\rho \\left(u^{2}\\right)={\\begin{bmatrix}1&0\\\\0&u^{2}\\\\\\end{bmatrix}}.", "0f5ca855c830bbecb5eb874a90957d52": "{\\vec {G}}_{m}", "0f5cc5a41884b677bd2718fd26a7b78b": "\\{re^{2\\pi i\\theta }:0\\leq r\\leq 1,\\theta \\in \\mathbb {Q} \\}\\subset \\mathbb {R} ^{2}", "0f5cd19852fbfa265cf774e761adb7e2": "P_{0}={\\frac {M_{a}}{r}}(1-e^{-rT})", "0f5cd37268db1469d362d94329c65859": "k\\leq n-\\log(\\sum _{i=0}^{t}{n \\choose i})", "0f5d33b506f67ed0ab8624a34197ccba": "O\\left[m(1+\\log n)+n\\log n\\right]", "0f5d6266c8b9e375b15b230a70c77838": "x'=x\\cos \\theta +y\\sin \\theta ", "0f5e0853f82372e76e84d712f57cc02c": "s\\ {\\stackrel {\\mathrm {def} }{=}}\\ {\\frac {\\mathrm {entropy} }{\\mathrm {volume} }}={\\frac {p+\\rho }{T}}={\\frac {2\\pi ^{2}}{45}}g_{*}(T)T^{3}", "0f5ec36d93ad2d1052db24fc39f55c23": "*Prob(D", "0f5f44d5b4a90b72616c4852072a34c4": "p(x)=x^{3}+2x-3", "0f5f69e7e68e63a5d666bbed7a37c9da": "(1-p+pe^{it})^{n}\\!", "0f5f8024248470026e785696ce48e843": "\\ k={\\frac {k_{B}T}{h}}e^{-{\\frac {\\Delta G^{\\ddagger }}{RT}}}", "0f5ff46bdd9b1ef57c896c74a5d5367c": "\\mathrm {Var} [X]=K''_{X}(0)=A''(\\theta )\\,.", "0f5ff4c85303ec0f41d1c03a6d049943": "\\mathbf {r} =r{\\hat {\\mathbf {r} }}", "0f603579577c12956a74c1ce88f48220": "P_{+}={1 \\over 5}", "0f60666f0d48d92bfe0a891a311d1db6": "\\Phi \\left(\\eta ,\\tau \\right)=1", "0f6073904491b22c04a145532cd6b1e4": "J_{\\kappa }^{(\\alpha )}(x_{1},x_{2},\\ldots ,x_{m})=\\sum _{\\mu }J_{\\mu }^{(\\alpha )}(x_{1},x_{2},\\ldots ,x_{m-1})x_{m}^{|\\kappa /\\mu |}\\beta _{\\kappa \\mu },", "0f607506cdeea84e8447574da8a17a96": "\\scriptstyle {(\\lambda _{i}^{+},\\lambda _{i}^{-})}", "0f609244a8c2f9b549f734e7387bd13d": "\\,{\\frac {(1-p)^{r}}{(1-pe^{t})^{r}}}", "0f60967e91d8f401f5e796b14ee0d126": "P_{c}\\sim 10^{5}\\,\\mathrm {Mbar} ", "0f60a09ff0df09fb54ca7b12045f573d": "\\sigma (t)=\\left({\\frac {1}{1-t}},0\\right)", "0f60f8ba7733b53856f3f587c096ccfd": "\\textstyle P\\in G_{1},Q\\in G_{2}", "0f6189ff0cded3aab0ba434d8c325982": "\\mathrm {C_{n}H_{m}+{\\frac {n}{2}}\\ O_{2}\\rightarrow n\\ CO+{\\frac {m}{2}}\\ H_{2}} ", "0f61cd7a7973003a62651054f6f9fbf5": "c_{s}=(\\gamma ZkT_{e}/m_{i})^{1/2}=9.79\\times 10^{5}\\,(\\gamma ZT_{e}/\\mu )^{1/2}\\,{\\mbox{cm/s}}", "0f624bdf2befce40d2eb69d434373f3e": "gfg^{-1}", "0f6286fcc6f4dc3786c2c54159d9c24f": "n\\geq \\ N", "0f62ce61480470c586f2c6c152d17100": "{1/b^{2}}", "0f6307077598fc956455fbf831228f16": "\\Delta (h)=h_{(1)}\\otimes h_{(2)}", "0f631e166d309cad911b55ed9fab0c71": "E_{image}=w_{line}E_{line}+w_{edge}E_{edge}+w_{term}E_{term}", "0f63b3432883b8eaf1d7c54f60be6110": "p_{\\textrm {b}}=p_{\\textrm {(p+b)}}\\,", "0f6422ea0ca94a25194a53c75b3f4096": "j(\\tau ')", "0f64303c04289769db0982cf1bd7f148": "{\\boldsymbol {\\alpha }}=(\\alpha _{1},\\dots ,\\alpha _{\\widetilde {n}})^{T}", "0f6430ec8a66e7789db8f0ad21b72bde": "f_{2}^{*}(g)={\\overline {f_{2}(g^{-1})}}", "0f648da14647174ee9df6f22e8216164": "(0,0,1)", "0f64b4f9251d46ad162ce7bb2d92d687": "2T_{G}(3,3)", "0f651da3e03453030d123f907e8e7116": "\\eta _{so}={\\frac {8-4{\\sqrt {2}}-\\ln {2}}{2{\\sqrt {2}}-1}}\\approx 0.902414\\,.", "0f655b911040e025b2df4941f848658e": "COD={\\frac {8000(b-s)n}{sample\\ volume}}", "0f6561dff7b4d3b35589805aab6b9f1a": "1+{1 \\over 2}+{1 \\over 3}+{1 \\over 4}+{1 \\over 5}+\\cdots =\\sum _{n=1}^{\\infty }{1 \\over n}.", "0f65b1418dfc6bf4395ba18b17ad596c": "f_{a}=e^{3.158x10^{-5}*h}", "0f65b521fb825c6ac584c26609b41e25": "\\{\\{64x^{3}-112x+56,(0,2)\\},\\{64x^{3}+192x^{2}+80x+8,(2,4)\\}\\}", "0f65dbbd5678818c72f1a6085ab17ff4": "{\\frac {h\\nu '}{c}}=\\gamma {\\frac {h\\nu }{c}}-\\gamma \\beta \\left\\Vert p\\right\\|\\cos \\theta =\\gamma {\\frac {h\\nu }{c}}-\\gamma \\beta {\\frac {h\\nu }{c}}\\cos \\theta ", "0f65ec39a1a40a045a876f743f3db79f": "{\\bar {X}}\\xrightarrow {n\\to \\infty } N(k,2\\cdot k/n)=N(2,4/3)", "0f66112133d2bb49d036732670cb14d4": "{\\text{ Recovery}}=\\left({\\frac {\\text{Mass of protein in the foam}}{\\text{Mass of protein in the initial feed}}}\\right)*100\\%", "0f6619145e3b27578a526a55ca52c3e6": "{}_{t}p_{x}", "0f663749d5f854395e25f43b77e222ee": "J={\\begin{bmatrix}J_{m_{1}}(\\lambda _{1})&0&0&\\cdots &0\\\\0&J_{m_{2}}(\\lambda _{2})&0&\\cdots &0\\\\\\vdots &\\cdots &\\ddots &\\cdots &\\vdots \\\\0&\\cdots &0&J_{m_{s-1}}(\\lambda _{s-1})&0\\\\0&\\cdots &\\cdots &0&J_{m_{s}}(\\lambda _{s})\\end{bmatrix}}", "0f665891419cf6212ac6e273c7a526be": "F^{+}", "0f66bb703c825146400acc9251364cf5": "p_{k}^{(i)}", "0f66db32cdff104091dc8334eda4ec5d": "\\mathrm {For} \\quad 00", "0f68f62a556484635461c1ebc134ce7f": "2^{a}3^{b}", "0f69fc38b4d429f823d2b6f35352aaad": "\\max _{w}{w^{T}Aw}", "0f6a2127a9175b5ddf193f0850dd3f52": "{\\mathcal {O}}_{\\mathfrak {X}}", "0f6a4684a0e94788263f4648b08ad591": "-\\nabla \\Phi =a", "0f6a516c14b744036ed6bdd9522de8bb": "N_{0}=", "0f6aaa23b4c4abafb8af57a96f00db52": "\\scriptstyle 1,\\dots ,m", "0f6accea29c9b0cd88f02c0bb8ad2f5c": "D(x,y,z)", "0f6af10bb3ba2496ca65735fea21bf55": "\\mathrm {CV(RMSD)} ={\\frac {\\mathrm {RMSD} }{\\bar {x}}}.", "0f6b1ff44038c4dceccde97fde4ec400": "T_{p}(S)", "0f6b371db1cc3b0fee9e2f437b95479b": "\\textstyle \\oint _{C}", "0f6b8dae6e45b5d8e94d2eab32ccd094": "e^{-iHt}\\vert J,\\gamma \\rangle =\\vert J,\\gamma +\\omega t\\rangle \\;,", "0f6b8e39c7c8a53464afccb74eb48c66": "{\\sqrt {a^{2}+b^{2}}}", "0f6b959e66ff686f45bc9c74333ab4c7": "\\xi _{1}=2/3u_{1},\\xi _{2}=2/3u_{2},\\xi _{3}=-2/3(u_{1}+u_{2})", "0f6b9aa74b7b873b1242a04c381c15d1": "n(n+1)(n+2)(n(4n+1+1/2)+(4n+1/2))", "0f6bc7f53815aac4f372dd7aeaacc00b": "\\infty ", "0f6bd8a1491c8d8d8217074717ec5f2b": "\\approx \\int _{\\partial N}f^{\\mu }\\,\\mathrm {d} s_{\\mu }.", "0f6c1667564081fa0f06039091c3fb8e": "(z-i{\\sqrt {5}})", "0f6c9d456bbd7638819aea3447715a81": "{\\big \\{}v{\\Big (}{\\textstyle \\sum _{k=1}^{n}}\\mathbf {1} {\\Big )}:n\\in \\mathbb {N} {\\big \\}}", "0f6cd4a6cd050026e41db79e658db0d5": "Rec(w',SS(w))=w", "0f6ce229c71b3b00f3e58391bf4daec7": "U^{c}=\\emptyset .", "0f6d58e9376213cef83c595ce9d7a057": "{\\frac {dr}{dt}}=Ar", "0f6d68c8eb550bb4361f03d69ebacefe": "\\langle F|\\exp \\left({-{i \\over \\hbar }{\\hat {H}}T}\\right)|0\\rangle =\\left({-im \\over 2\\pi \\delta t\\hbar }\\right)^{N \\over 2}\\left(\\prod _{j=1}^{N-1}\\int dq_{j}\\right)\\exp \\left[{i \\over \\hbar }\\sum _{j=0}^{N-1}\\delta t\\left({1 \\over 2}m\\left({q_{j+1}-q_{j} \\over \\delta t}\\right)^{2}-V\\left(q_{j}\\right)\\right)\\right]", "0f6d87b5c4f3eb0c263bd87de4894096": "\\epsilon _{F}", "0f6dd6b7ce3aa84e81ec8ec2eb0eacd6": "t=2{\\frac {d}{V}}", "0f6df9808fdd3fa5f2a8f534c49ed0a5": "{\\mathcal {O}}^{\\times }", "0f6e0da8f27afccc12baa42a54545e11": "w[k]", "0f6e67b0b39bda7131c549f02ccc2654": "\\operatorname {dim} _{\\mathrm {Haus} }(X\\times Y)\\geq \\operatorname {dim} _{\\mathrm {Haus} }(X)+\\operatorname {dim} _{\\mathrm {Haus} }(Y).", "0f6e9f089a23b00f0e39ab4adce1e5f0": "TP+FP+FN+TN=n(n-1)/2", "0f6eda019c8a968ce0a901771ab35492": "a{\\tfrac {b}{c}}=a\\times {\\tfrac {b}{c}}", "0f6f045bc2bc0d0ad36fd8d538290b36": "K(u)={\\frac {1}{\\sqrt {2\\pi }}}e^{-{\\frac {1}{2}}u^{2}}", "0f6f26ea2b255696850c427bce93fb29": "\\Delta f=r^{1-n}{\\frac {\\partial }{\\partial r}}\\left(r^{n-1}{\\frac {\\partial f}{\\partial r}}\\right)+r^{-2}\\Delta _{S^{n-1}}f.", "0f6f3c72940b1f4e44a9c456f5f7feb7": "I=c\\epsilon _{0}E_{a}^{2}/2", "0f6f41df58abee1a5911cc2abe770c6e": "p_{n}(t^{2})=(a+b)_{n}(a+c)_{n}(a+d)_{n}\\;{}_{4}F_{3}\\left({\\begin{matrix}-n&a+b+c+d+n-1&a-t&a+t\\\\a+b&a+c&a+d\\end{matrix}};1\\right).", "0f6f4e577681817576b132a249576597": "\\gamma \\cdot z=(\\sigma _{1}(\\gamma )z_{1},\\dots ,\\sigma _{m}(\\gamma )z_{m})", "0f6f79b4e7f27028a46817ea08796b80": "\\int \\!\\!\\!\\!\\!\\int \\!\\!\\!\\!\\!\\int _{\\partial V}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\;\\;\\;\\subset \\!\\supset \\mathbf {D} \\cdot \\mathrm {d} \\mathbf {A} ", "0f6f8b0c71033aca81444d3c33bf5fdc": "E^{\\alpha \\beta \\gamma \\delta }=-g^{\\alpha \\zeta }g^{\\beta \\eta }g^{\\gamma \\theta }g^{\\delta \\iota }E_{\\zeta \\eta \\theta \\iota }\\,.", "0f700bb9c48e768a657326fdcf5d01ba": "50i+10", "0f703e8a6f24bff70706f1c825e5a18f": "[I_{C}]=-\\sum _{i=1}^{n}m_{i}[\\Delta r_{i}]^{2},", "0f708423ecaa127142ef36159287b73c": "E[h^{2}(X_{1},X_{2})]<\\infty ,\\,E|h(X_{1},X_{1})|<\\infty ,", "0f70c4a9ee079ab888365a33cc36ffe3": "A_{21}g_{2}(e^{h\\nu /kT}-1)+B_{21}g_{2}F(\\nu )=B_{12}g_{1}e^{h\\nu /kT}F(\\nu )\\,", "0f70db458164242f6b547fe5d7664fb8": "\\langle A\\rangle _{\\rho }", "0f711b414fb6ed6b3be70c83c9bbf63a": "h_{j}\\,", "0f71445e144d4763dc511b5a6d9a75f9": "\\operatorname {Div} (X)", "0f717f78bace9a6cfea28c74e156e770": "\\langle x,y\\rangle +\\langle y,x\\rangle \\leq 2|\\langle x,y\\rangle |", "0f718175813b99b38b10d8ed1b77239f": "X^{(3)}", "0f71e50e24e47df7de4c47cb3b30cc47": "NL=1000*({ELA \\over A_{Floor}})*({H \\over H_{Ref}})^{0.3}\\,\\!", "0f71eb7a24845d771b41131af87b288d": "b_{i}", "0f72416ba535f2e5ee53e26ba25388fd": "\\Psi (r)\\propto {\\frac {e^{ikr}}{4\\pi r}}\\int \\!\\!\\!\\int _{\\mathrm {aperture} }E_{inc}(x',y')e^{-i(k_{x}x'+k_{y}y')}\\,dx'\\,dy',", "0f727abd001d9a3a2d6b720d4e8a2e15": "z=\\ z", "0f72937f28cee25bd6d096a3414d1c35": "K_{b}(message')=K_{b}(message)", "0f72d22546951413f8170379faf822d4": "\\int _{\\mathbf {R} ^{n}}f\\,dV=\\int _{0}^{\\infty }\\int _{S^{n-1}(r)}\\exp \\left(-r^{2}/2\\right)\\,dA\\,dr,", "0f72fd197f769468040706159452ea25": "\\operatorname {Pic} (X)=H^{1}(X,{\\mathcal {O}}_{X}^{*})", "0f7324a963a0272333f1d3710b76dacc": "R_{i}A_{j}\\subseteq A_{i+j}", "0f7354834f8744aeba4a32c639224701": "2z={\\frac {x^{2}+y^{2}}{\\sigma ^{2}}}-\\sigma ^{2}", "0f73b5f9fb185c4b07daace297b590aa": "f_{0}={1 \\over 2\\pi R_{0}}{\\sqrt {3\\gamma p_{0} \\over \\rho }}", "0f73e18171d0366d89444bb39f0192c3": "\\exists k>0\\;\\exists n_{0}\\;\\forall n>n_{0}\\;f(n)\\leq g(n)\\cdot k", "0f742253a17b7057d3fb9914245a04bd": "p\\times k", "0f74665df199b3055a82560d54109e71": "(x_{1},\\ldots ,x_{k})\\preceq (y_{1},\\ldots ,y_{k})\\ ", "0f75579333df0e185a4dbb96c76a535a": "\\pi (x)=\\Pi (x)+O({\\sqrt {x}}).", "0f75a3dad716a836e5a8a14392ec7132": "\\sin {\\frac {\\pi }{5}}=\\sin 36^{\\circ }={\\tfrac {1}{4}}[{\\sqrt {2(5-{\\sqrt {5}})}}]\\,", "0f75ace06351b0d194a690efb2c87b45": "(x_{1}-x_{2})=-(x_{2}-x_{1})", "0f75be97bbec5955c71233e79d221a17": "F_{142}x-F_{141}=0", "0f75d3ab56044b4d125acf01586561fb": "Z\\cap S", "0f763abdee89b55d8004ae55ea25a795": "|\\phi \\rangle ", "0f768ac5d5dea8d93716a27da05871de": "a_{2}", "0f77141e51439de02bfaea54740504b8": "x(p,w)", "0f773e1a0284e3af2e9fef701a31b379": "n=\\int _{E_{0}}^{E_{0}+E_{F}}g(E)dE", "0f77e84fdaf785dac996824c9d4f7916": "m\\left(x^{\\mu }\\right)", "0f780cb28944a42fe461040d31cd2447": "{\\frac {\\partial L}{\\partial {\\dot {x}}}}", "0f789187e3943ccab592d1192aa153b1": "\\psi (\\xi )", "0f78f2358bab8308cede6a7c7c16ffa3": "-\\infty 4", "0f7f03c76044a68faf851319543a29ac": "I_{r,r-1}", "0f7f1890a81292060210bbf8dcd4218a": "\\delta ^{n}[f](x+h/2)", "0f7f23ae06b9d3ef30259d0b033e8234": "\\mathrm {Vol} \\,B(p,r)\\leq \\mathrm {Vol} \\,B(p_{k},r).", "0f7f40d5077a28acd9e794cd42d7d78e": "r_{\\mathrm {corr} }=r{\\frac {1+\\theta c_{xy}}{1+\\theta c_{x}^{2}}}", "0f7f7ab585fb1fca9e53c736a03fad97": "E\\not \\in \\operatorname {FV} [G]\\land E\\in \\operatorname {FV} [H]\\to \\operatorname {sink} [(\\lambda E.G\\ H)\\ Y,X]=\\operatorname {sink-test} [G\\ \\operatorname {sink-test} [(\\lambda E.H)\\ Y,X]]", "0f7f867274a4ad48d09c66ea6498656a": "xy=x", "0f7fd84b7c85c66f952364e4d10bab0f": "y_{n}\\approx y(t_{n})", "0f7ff6789afae668ff60803160b67208": "{\\begin{smallmatrix}\\left[{\\frac {Fe}{H}}\\right]\\ =\\ -0.04\\end{smallmatrix}}", "0f80b58328ab87252779eb2851a8ceef": "\\textstyle 2^{5}-1=31", "0f80cb13f63ec51875946b4ccbbeb7d2": "N=m(g+a)", "0f80d4c54b7c1098bfd0e274d5f5850d": "{\\bar {x}}=\\left(\\prod _{i=1}^{n}{x_{i}}\\right)^{\\tfrac {1}{n}}", "0f80d78e8efcab9a201d22706043965b": "a=x_{0}x", "0f81dc166bb16a659e0955c88a53b1eb": "V_{\\rm {as}}=\\rho \\cdot c^{2}\\cdot S_{\\rm {d}}^{2}\\cdot C_{\\rm {ms}}", "0f81fcaf86f636038761640ab3705284": "|x\\rangle =U(g(x))|\\psi \\rangle \\in {\\mathfrak {H}}.", "0f82555be6cd4b6d885314d1df76ac3f": "R_{\\text{D}}\\,", "0f82c0ed2cb6cbffade2408f25fd7b34": "\\operatorname {sgn}(x)={\\begin{cases}-1&{\\text{if }}x<0,\\\\~~\\,0&{\\text{if }}x=0,\\\\~~\\,1&{\\text{if }}x>0.\\end{cases}}", "0f82d548df24685f230e386d952d3332": "U=\\langle E\\rangle =\\langle \\sum _{i=1}^{N}E_{i}\\rangle =\\sum _{i=1}^{N}\\langle E_{i}\\rangle ", "0f82f41bd79dafae16c7d83b3176f326": "d_{1},", "0f834a186f2e5eee334482f0cb70ecf9": "D\\chi _{E}", "0f836e48ba01cc613d9a133379003dec": "\\Delta \\cup \\{A\\}\\vdash B", "0f837b7ae1e5c94134c017416474bcb8": "\\langle f,f\\rangle =1\\,", "0f837d9e15f43ad43049981331acd0e7": "\\wp '(\\omega _{i}/2)^{2}=\\wp '(\\omega _{i}/2)=0", "0f83cabb59b8731a0b5df7a0af120654": "i(e)=e", "0f83e2f37c8d0867fa853a78a32c4c8e": "A={\\begin{pmatrix}a_{1}&\\dots &a_{n}\\\\b_{1}&\\dots &b_{n}\\end{pmatrix}},\\quad B={\\begin{pmatrix}c_{1}&d_{1}\\\\\\vdots &\\vdots \\\\c_{n}&d_{n}\\end{pmatrix}}.", "0f83e95a31c99346fb57a6bf364b8a37": "j\\omega \\ ", "0f83f2488a3704ac5e4e42c33a31833e": "n_{s}={\\frac {V_{s}}{V_{v}}}", "0f84b7c96c4d278db55fd56096d4fa41": "\\mathrm {stsys} _{1}{}^{n}\\leq \\gamma _{n}\\mathrm {vol} _{n}(M),", "0f8530c37c5027d2a2ea974460bf65fd": "{t=t_{2}}\\ ", "0f85394623f7bf6d1c1aaf8da61cda12": "\\mathbf {x_{2}} ", "0f85856a4e282968d7f20a340070a3c2": "x_{k}^{1}", "0f859fba06e1f0ab16df9ef2f523fea8": "\\mathbf {j} _{3}", "0f85a70fb1740380f25ecd468dbd6093": "g(x,b)", "0f85b89606bc6a4d440b60da136140a4": "\\langle X^{N1}\\rangle ", "0f85bc3596e7cf85b5f4101792d05b6f": "\\mathrm {Volume} =\\int _{0}^{2\\pi }d\\phi \\int _{0}^{R}h\\rho \\ d\\rho =h2\\pi \\left[{\\frac {\\rho ^{2}}{2}}\\right]_{0}^{R}=\\pi R^{2}h", "0f86346404e78b9a0e706e0e4f573f08": "tan^{2}\\theta ", "0f8639ee60c9a331a1d10d1eeea27dd4": "\\textstyle (x_{n}-{\\bar {x}}_{n})={\\frac {n-1}{n}}(x_{n}-{\\bar {x}}_{n-1})", "0f86434a4ebb51e2fd16540da2ae1193": "\\langle {\\mathbb {N} },S\\rangle ", "0f8648406e1a728158ba22f115699c3a": "k={{k_{0}+k_{\\inf }{\\gamma ^{0.5}}_{r}} \\over {1+{\\gamma ^{0.5}}_{r}}}", "0f869d71054fa3024f7693f4afb9c542": "{\\begin{aligned}p(z_{i}\\in \\mathbf {Q} |\\mathbf {z} _{1,\\dots ,i-1},\\alpha )&=\\lim _{K\\to \\infty }\\sum _{u\\in \\mathbf {Q} }{\\frac {\\alpha /K}{i-1+\\alpha }}\\\\&={\\frac {\\alpha }{i-1+\\alpha }}\\lim _{K\\to \\infty }{\\frac {K-L}{K}}\\\\&={\\frac {\\alpha }{i-1+\\alpha }}\\end{aligned}}", "0f86d0b48ede27b4e8d8ee6cd8400ce4": "{\\frac {e^{ik_{0}r}}{r}}=i\\int \\limits _{0}^{\\infty }{dk_{\\rho }{\\frac {k_{\\rho }}{k_{z}}}J_{0}(k_{\\rho }\\rho )e^{ik_{z}\\left|z\\right|}}", "0f86d2258150f7079f2660a8e48262f4": "\\beta >1", "0f86d5ddc9a449255d2b09de9f58cb96": "E'_{y}={E_{y} \\over {\\sqrt {1-v^{2}/c^{2}}}}", "0f86ef08ff272986f2b2f002a9e23d7a": "\\sum _{n=i}^{\\infty }{\\frac {\\left[{n \\atop i}\\right]}{n(n!)}}=\\zeta (i+1)", "0f876286cee3b5efed5e0fa42ff90a73": "i_{C}", "0f8768c7908fb5025bd1ccb70d730119": "c(t)={\\frac {1}{2\\pi }}{\\frac {d^{2}\\phi (t)}{dt^{2}}}", "0f8775338487ed50957f07771d8a8e0a": "y=Dx", "0f87a4ca5fbfa50fdc9f583068ad054b": "T_{p}M\\to T_{p}M", "0f87c81c859fa514e7097b84ea63a58f": "\\theta -\\alpha ", "0f87d593cb2ab07cb1a091232e007dbb": "(b^{z})^{u}=b^{zu}", "0f87e9db119b585ffa88f2d11d1e814a": "-r^{-2}\\,", "0f880e3e23ef56452a2e4a4ff19a20c1": "b=\\infty ", "0f888189578e25ed99e12c83011fc3f0": "\\dim \\,\\operatorname {Sym} ^{k}(V)={N+k-1 \\choose k}.", "0f888640ea5fdc8569e9a5876a0134cb": "\\operatorname {de-let} [f\\ (x\\ x)]", "0f893f2edc2b22d08c3f91ac13e77e4e": "\\mu \\left(f^{n}(U)\\cap U\\right)=0.\\,", "0f8957fab6484c33b5001d50cc3130e9": "c_{I}", "0f897d1e8594a75f1f48d424b4641c88": "\\phi ~{\\mathcal {U}}~\\psi ", "0f89a646bc78da6ff833e9f337d1ffb2": "{\\frac {e^{iat}-e^{i(b+1)t}}{n(1-e^{it})}}", "0f89d0a22ae8af89860425fe8b060dc3": "dG\\left(T,p,n_{i}\\right)=-SdT+Vdp+\\sum _{i}\\mu _{i}dN_{i}", "0f89fc758d1ebc6683f7bf0227582da7": "\\int \\!\\!\\!\\!\\int \\!\\!\\!\\!\\int _{V}\\nabla \\times \\mathbf {F} \\,dV=", "0f8a2408502495b1b7ad16253d2962e9": "S(x_{1},x_{2},\\ldots ,x_{n})=(x_{2},\\ldots ,x_{n},0).", "0f8a3165ed272b22750e56b9a30cdde9": "q=m/n", "0f8b06f3561bb6f6955827c926766a86": "N.k+M_{e}", "0f8b1d784f2fb7550f60fc4cf5d35427": "R(q)=P(q)\\cdot q", "0f8b57e8522423166fce1d4e0fc5a59b": "{\\mathcal {F}}_{n}^{+}", "0f8b7217482e5c28b4cd673605bed1c3": "\\Gamma _{a\\;\\;i}^{\\;\\;j}", "0f8b7c3dc93cfd7502480ed615925d62": "\\rho =N/L", "0f8b9b1f5e5f90f00ba08ba8e341c92b": "(v_{P})_{\\mathcal {P}}", "0f8bcc58dfd18b199fd474c5120d0c46": "{\\cfrac {\\partial ^{2}\\mathbf {v} }{\\partial t^{2}}}-c_{0}^{2}~\\nabla ^{2}\\mathbf {v} =0\\qquad {\\text{or}}\\qquad {\\cfrac {\\partial ^{2}p}{\\partial t^{2}}}-c_{0}^{2}~\\nabla ^{2}p=0,", "0f8bf99dd27c2e921f76fc479848b221": "\\vdash A,\\lnot A", "0f8ca084a5043d103e4ef5e85549c8c7": "H(x)\\triangleq {\\begin{bmatrix}h_{1}(x)\\\\h_{2}(x)\\\\h_{3}(x)\\\\\\vdots \\\\h_{n}(x)\\end{bmatrix}}\\triangleq {\\begin{bmatrix}h(x)\\\\L_{f}h(x)\\\\L_{f}^{2}h(x)\\\\\\vdots \\\\L_{f}^{n-1}h(x)\\end{bmatrix}}", "0f8ca64a950171d84952b80f660ccf24": "|x|<1", "0f8cde1d90db67b5e9e84fddd1783dc1": "\\mathbf {S} ={\\begin{pmatrix}1&d\\\\0&1\\end{pmatrix}}", "0f8ce94b375579bd280ecbe491064831": "v\\in \\mathbb {R} ", "0f8d2b097d78573fb144352f861e978c": "{\\hat {G}}({\\boldsymbol {k}},\\omega ):", "0f8d6f8da96e5d4e7eae1971269e2d38": "s=(\\ldots ,(s_{i},t_{ei}),\\ldots )", "0f8ddd36e329e9106facd25b79876b52": "|z_{k}|<1,k=1,\\ldots ,n", "0f8e4372551617be928b5729e09a3725": "\\{a,b,c,d\\}\\in \\mathbb {R} ", "0f8e959c393de05b2faafb5e92cd9edb": "F:M\\times N\\to \\mathbb {R} .", "0f8eb2799178e2e3df91d90fafb61084": "p=\\omega \\cdot \\varepsilon _{r}''\\cdot \\varepsilon _{0}\\cdot E^{2},", "0f8ece4ba3b54c75fc3f090ddae8f901": "^{\\;}q^{i}(\\xi ,\\tau )", "0f8fc9f8362111886d55e05f39729230": "\\beta =\\partial _{t}F_{t}|_{t=0}.", "0f901e21172331d03ccfe8e3553e99bb": "\\Delta =L/16", "0f90453c26a5fc908a5345e189a52440": "\\textstyle W_{p}^{m}(\\Omega )", "0f909cb7dfab5fa7b870ca9f386c910c": "\\mathrm {D} \\,\\mathbf {F} \\left(\\mathbf {X} \\right)={\\frac {\\partial \\,\\mathrm {vec} \\ \\mathbf {F} \\left(\\mathbf {X} \\right)}{\\partial \\left(\\mathrm {vec} \\ \\mathbf {X} \\right)^{\\prime }}}.", "0f90bc5155ea84c4352bc03ba37e0517": "\\alpha >\\beta ", "0f90be5d1b4c19c754d912e0729add74": "\\mathbf {\\hat {e}} _{\\bot }\\,\\!", "0f90c57425a10c0c6b841d1d82f634bb": "L_{\\frac {1}{2}}", "0f911c00e08e9f7cfeff6527ed9ef486": "c_{\\text{deep}}", "0f911c2e59c279f47901d4467276b92d": "{\\frac {1}{H_{N,s}}}\\sum _{n=1}^{N}{\\frac {e^{nt}}{n^{s}}}", "0f913aa03754914c7c1c442943c5b5ce": "(1,0),(0,1)", "0f916a9cab3d3315eb76b88ee8ddd16c": "\\nabla \\cdot \\mathbf {v} ={\\frac {\\partial {\\dot {x}}_{1}}{\\partial x_{1}}}+{\\frac {\\partial {\\dot {x}}_{2}}{\\partial x_{2}}}+{\\frac {\\partial {\\dot {p}}_{1}}{\\partial p_{1}}}+{\\frac {\\partial {\\dot {p}}_{2}}{\\partial p_{2}}}={\\frac {\\partial }{\\partial x_{1}}}{\\frac {\\partial H}{\\partial p_{1}}}+{\\frac {\\partial }{\\partial x_{2}}}{\\frac {\\partial H}{\\partial p_{2}}}-{\\frac {\\partial }{\\partial p_{1}}}{\\frac {\\partial H}{\\partial x_{1}}}-{\\frac {\\partial }{\\partial p_{2}}}{\\frac {\\partial H}{\\partial x_{2}}}=0", "0f916aebbb8b12e8f96bf781fc854e26": "(x,y,z)\\rightarrow (f(x,y,z)+x_{0},g(x,y,z)+y_{0},h(x,y,z)+z_{0})", "0f91836fce3a27a5dfe0e56fc658e00d": "\\Delta u=0", "0f91906a35c86ed83e3f85281673751f": "Q_{2}", "0f9190e956ee7347cfd61a1f0488793f": "D=\\Gamma /\\delta x", "0f91a78f4699260331be744450bc6e06": "f:X\\to \\mathbb {R} ^{d}", "0f920ab28129df0e212b4c561554520a": "S^{-1}S", "0f921b0605129a2e67da2b634d296c43": "C_{ML}={\\frac {1}{n}}S.", "0f92446c6a38c0bc35d6003f43809059": "V_{C}={\\frac {F_{v}\\cdot a}{a+b+c}}", "0f926a776da9725cf76c44c715bc70aa": "\\varphi (x)=\\mathbf {E} ^{x}\\left[f(X_{\\tau _{H}})\\right],", "0f927dc9c2eb688d1821f31d6234c43a": "{\\frac {\\Gamma _{1},A,\\Gamma _{2},B,\\Gamma _{3}\\vdash \\Sigma }{\\Gamma _{1},B,\\Gamma _{2},A,\\Gamma _{3}\\vdash \\Sigma }}", "0f929cb64eaa6ae1dcf1267c0da63e01": "\\varphi (t)=\\int _{-\\infty }^{\\infty }F(x)e^{itx}\\,dx", "0f933a81fadd3cfb58a83144f1d1f784": "F_{o}=\\int (f_{c}+\\lambda )dA+\\gamma \\oint _{C}ds", "0f938ae0c3b91676bd112eb548fb6263": "{\\frac {V_{i}-V_{o}}{L}}t_{\\mathit {on}}-{\\frac {V_{o}}{L}}t_{\\mathit {off}}=0", "0f93b9917f7d148c9156acd04c4a4768": "M\\rightarrow [~]_{n^{\\searrow }\\;\\|\\;{\\overline {n}}^{\\nwarrow }}", "0f93e4769d2b006339748a5dc99f8568": "H_{*}(C_{*}(X\\times Y))\\cong H_{*}(C_{*}(X)\\otimes C_{*}(Y)).", "0f9425f5e2b2df044511faced9303ef4": "\\displaystyle {a^{-1}\\leq {|f(z_{1})-f(z_{2})| \\over |f(z_{1})-f(z_{3})|}\\leq a.}", "0f943fe9d35f55afd59f64c531070806": "{\\tilde {r}}", "0f944c42f590fb7f5995b075e389e60d": "{\\bar {\\lambda }}_{e}\\equiv {\\tfrac {\\lambda _{e}}{2\\pi }}\\simeq 386~{\\textrm {fm}}", "0f94ba5a0860b67d84f738a44150adf4": "\\{\\gamma ^{\\mu },\\gamma ^{\\nu }\\}=2g^{\\mu \\nu }\\,", "0f94f7b73affbce3a4f189203c6db4dd": "\\alpha _{n}=(1-aq^{2n})\\sum _{j=0}^{n}{\\frac {(aq;q)_{n+j-1}(-1)^{n-j}q^{n-j \\choose 2}\\beta _{j}}{(q;q)_{n-j}}}.", "0f953f13ef3001c1c8049820623f1b68": "{\\hat {R}}_{m}=2({\\hat {L}}_{m}\\cdot {\\hat {N}}){\\hat {N}}-{\\hat {L}}_{m}", "0f95b1c1ead7bb1fe5a5bf09ddad8a00": "i,a,b\\in \\mathbb {Z} ", "0f95d2f2265095c8032bb38dd0790e74": "\\lambda =1", "0f95fdb4329c089877c5ddfb6bfb4c82": "\\phi (t)=\\omega _{0}t-kx=\\omega _{0}t-{\\frac {2\\pi }{\\lambda _{0}}}\\cdot n(I)L", "0f9604a9c8698a2fb16fdaf0012a69c1": "H_{n}(X)={\\tilde {H}}_{n}(X)", "0f9611cb17be237d2866355fcaaac3bb": "n^{4/3};", "0f967f36c6c9fa31619c810c7f7cacf9": "\\partial _{\\mu }{\\hat {A}}^{\\mu }", "0f969be3fd4a88a56a91e846d23a7aea": "u_{i}(x,t,\\theta _{i})=\\theta _{i}x_{i}+t_{i}.", "0f96abfc454fc36ff095577f9c27f967": "\\langle \\phi (k_{1})\\phi (k_{2})\\phi (k_{3})\\phi (k_{4})\\rangle ={i \\over k_{1}^{2}}{i \\over k_{2}^{2}}{i \\over k_{3}^{2}}{i \\over k_{4}^{2}}i\\lambda \\,", "0f96c8d82fcfd9adf03bd9ed16ba6496": "\\tau _{E}={\\frac {W}{P_{\\rm {loss}}}}", "0f97884bd76cfbb8588c95f6faa27905": "ds^{2}=-g^{2}x^{2}dt^{2}+dx^{2}+dy^{2}+dz^{2},\\;\\forall x>0,\\forall t,y,z", "0f97aa07d7c5242faf011e9f4bfd86cd": "k/s\\leq j/r", "0f982eb4e607b6bd0e95c5c17aa061bd": "-{\\frac {d^{2}}{dq^{2}}}+q^{2}=\\left(-{\\frac {d}{dq}}+q\\right)\\left({\\frac {d}{dq}}+q\\right)+1", "0f988f6267e703b3e704be5c92e5928f": "\\beta *\\,", "0f98ecbc950c82f6766d05a0de81302d": "{\\dot {a}}=-{\\frac {i}{\\hbar }}[a,H_{\\mathrm {sys} }]-[a,c^{\\dagger }]\\left({\\frac {\\gamma }{2}}c+{\\sqrt {\\gamma }}b_{\\mathrm {in} }(t)\\right)+\\left({\\frac {\\gamma }{2}}c^{\\dagger }+{\\sqrt {\\gamma }}b_{\\mathrm {in} }^{\\dagger }(t)\\right)[a,c]\\,.", "0f9925457a8362dfbd541adaef71e405": "\\hbar \\omega ", "0f9960105518def1fcd4dc394d1e608a": "\\int _{0}^{T}R_{N}(t,s)k(s)ds=\\sum _{i=1}^{\\infty }\\lambda _{i}S_{i}\\int _{0}^{T}R_{N}(t,s)\\Phi _{i}(s)ds=\\sum _{i=1}^{\\infty }S_{i}\\Phi _{i}(t)=S(t)", "0f999d9a8625b402e2d9ff7f1054a20a": "w_{j}\\leftarrow Av_{j}\\,", "0f99d8fa133089747bcf97f037598cea": "\\pi _{T}=\\left({\\frac {\\partial U}{\\partial V}}\\right)_{T}", "0f99ff7ba20d49484a5b9c59e2cc92b1": "{{x{:}1{\\to }\\tau \\;\\in \\;\\Gamma } \\over {\\Gamma \\vdash x:1{\\to }\\tau }}", "0f9a473c826fd7b5b9dda2be2e16ace5": "\\Delta :R\\to R\\otimes R", "0f9ac29b5a9afd7b205ae9659595546a": "\\log \\left|R-R_{+}\\right|-\\log \\left|R-R_{-}\\right|=-2Dk_{1}t+\\phi _{0}", "0f9b351a277d51977639c72a8f9ebd3f": "Q={\\frac {20,000}{10,000}}\\times {\\frac {20}{50}}=0.8", "0f9b4c013307df46756dc4df84c3299a": "I=\\{0,1,\\ldots n-1\\}", "0f9b5d101e34509491b9c7bcb93e2d91": "j=1,2,\\dots ,N", "0f9ba803bfaff87945d4e4b62681181c": "\\alpha ={\\epsilon _{0}k_{\\rm {B}}T \\over p}\\chi _{\\rm {e}}", "0f9bb27f4749a30ccde83856e055831c": "M_{2}={\\frac {1}{\\sin(\\beta -\\theta )}}{\\sqrt {\\frac {1+{\\frac {\\gamma -1}{2}}M_{1}^{2}\\sin ^{2}\\beta }{\\gamma M_{1}^{2}\\sin ^{2}\\beta -{\\frac {\\gamma -1}{2}}}}}.", "0f9bc1d2168f177a75610f41d437bccf": "\\varphi (U^{0},t)={\\mbox{e}}^{t{\\mathcal {A}}}U^{0}", "0f9bdddd9de1feabce131880c2803018": "\\scriptstyle f_{s}>2B\\!", "0f9be017bd8dfde699a6a7839273faa5": "y(N)=\\sum _{k=0}^{N}x(k)e^{-2\\pi i{\\frac {kK}{N}}}", "0f9c5e421c4ba26dc4a5b39ebf8f89f7": "c_{f}=\\prod _{p\\ prime}x_{i}\\left(1-{\\frac {\\omega \\,\\!_{f}(p)}{p^{2+q_{p}}}}\\right)", "0f9c742170552489d91dc8aa4fa58199": "PFB", "0f9c815e60b0f4a8eb2fd850c4cbf730": "F\\gtrsim 1", "0f9c960b45e724c54078fc607fe18bfd": "\\beta _{m}=-\\int \\varphi _{m}^{*}({\\boldsymbol {r}})\\Delta U({\\boldsymbol {r}})\\varphi _{m}({\\boldsymbol {r}})\\,d^{3}r\\ ", "0f9cdfcc8eb270ce86bfa80ca12449ee": "\\gamma ^{*}\\,", "0f9d27408478b3ecc0c718eb7c9d3028": "{\\sqrt {I}}\\cdot S^{-1}R={\\sqrt {I\\cdot S^{-1}R}}", "0f9d9792bf9bd583f1e954ce17dced17": "{\\sqrt {2}}\\times {\\sqrt {3}}", "0f9d9ba652e6e902ad2e6b231b09711d": "d(X,Y)=\\inf \\!{\\big \\{}\\varepsilon >0:\\ \\Pr {\\big (}|X-Y|>\\varepsilon {\\big )}\\leq \\varepsilon {\\big \\}}", "0f9dda58fd559ce047deddfae4f53712": "\\sigma _{X}^{2}=\\left({\\frac {\\partial f}{\\partial A}}\\sigma _{A}\\right)^{2}+\\left({\\frac {\\partial f}{\\partial B}}\\sigma _{B}\\right)^{2}+\\left({\\frac {\\partial f}{\\partial C}}\\sigma _{C}\\right)^{2}+\\cdots ", "0f9e59f0c9a528be1fc5cce7b139fe14": "\\varepsilon ={\\frac {T-T_{c}}{T_{c}}}", "0f9e6cee356d9b261c5547e57347689c": "\\sigma _{log}={\\sqrt {\\ln \\!\\left(1+\\!\\left({\\frac {0.42}{5.33}}\\right)^{2}\\right)}}=0.079", "0f9ec194e7c7f5d384944c8feca146d1": "W_{n-1}", "0f9ecbfac90ad2cf197f5f62472b49e0": "\\omega :=*L^{*}", "0f9eee8d9d1e1546df7290a7a181a8a3": "r_{d,s}=h_{d,s}x_{s}+n_{d,s}\\quad ", "0f9f16fbacf4e36950560330937206d0": "37-57=-20", "0f9f729d515983aa8a73f4718e7aee70": "a=1+i,p=z^{2}-1", "0f9f9cb5d8adf0a1a4e2aba7f7168da9": "y'(x)={\\frac {T_{y}-y}{T_{x}-x}}", "0f9f9d2d5ae4771fed1989ac89ba1645": "1-{\\sqrt {2R}}", "0f9fc49fd9bb017e81cdd79c8d3580ef": "A_{q}(n,d)=\\max _{C\\in C_{q}(n,d)}|C|.", "0f9ff5d924d853a7a6800ab77927f905": "S(\\mathbf {r} ,i)", "0fa01fa203eb55df5ee784dfb5fa9351": "\\min(x,y)\\leq H(x,y)\\leq HG(x,y)\\leq G(x,y)\\leq GA(x,y)\\leq A(x,y)\\leq \\max(x,y)", "0fa10cd5e24994a12e059ce244e612c6": "Q_{1}\\bot Q_{2}\\equiv Q_{1}\\supset \\neg Q_{2}", "0fa180793647215fc680ba3a3edcd004": "O(\\cdot )", "0fa1d9dfdeacb5e65637b90b99097ad2": "\\displaystyle {f_{z}(x)=e^{izx^{2}/2}}", "0fa2055bb25c164ee502717117e52848": "\\scriptstyle G=\\langle H,t|t^{-1}Kt=L\\rangle ", "0fa2c6c326f1a09438e512daf732a44a": "\\lim _{a\\to 0;\\,a^{3}\\cdot Q\\to {\\rm {const.}}}", "0fa2ebc31b4d8afc07009d1e871a6864": "e_{n}\\sim {\\mathcal {N}}(0,1),", "0fa330514cbf4c99e71d00c822ee69f7": "{\\frac {dP_{\\text{Magnetic dipole}}(\\mathbf {x} )}{d\\Omega }}={\\frac {Z_{0}}{32\\pi ^{2}}}k^{4}\\|\\mathbf {m} \\|_{2}^{2}\\sin ^{2}\\theta ", "0fa37b1f811139e9bf7e1a5476784fc3": "x^{5}+ax+b=0\\,", "0fa3a2a5fd0bf9b1850bced816f95052": "\\Delta {\\mathcal {F}}\\propto \\Delta {\\mathcal {O}}^{-1}", "0fa41df8aa07b065276c83b4afd6e7ae": "E(X^{-n})", "0fa42ca38f870da9bd73afc47e8bbc9a": "\\alpha \\ominus \\beta =\\alpha -\\beta ", "0fa47e711bb72a7351dfd0c1cf8ad6df": "01}v_{i}Constant\\times Side\\ Lobe} \\end{cases}}", "0fb7f6309aabf56c13d63ddf30256eb8": "C_{11}^{\\lambda }:=x^{\\lambda }", "0fb7fe4f3a90cfb5c3e852ae216f38c2": "\\,{\\mathfrak {g}}", "0fb83a14032c05ee52464f257e53eaed": "n_{AB}={\\frac {n_{A}}{n_{B}}}\\,", "0fb869ec3121a34c20cd3fe1bde0178d": "v_{OLS}[{\\hat {\\beta }}_{OLS}]=s^{2}(\\mathbb {X} '\\mathbb {X} )^{-1},s^{2}={\\frac {\\sum _{i}{\\hat {u}}_{i}^{2}}{n-k}}", "0fb8a446a1dda88e8eeb853f0c7936a6": "={\\frac {e^{4}}{(k-k')^{4}}}{\\Big (}({\\bar {v}}_{k}\\gamma ^{\\mu }v_{k'})^{*}({\\bar {u}}_{p'}\\gamma _{\\mu }u_{p})^{*}{\\Big )}{\\Big (}({\\bar {v}}_{k}\\gamma ^{\\nu }v_{k'})({\\bar {u}}_{p'}\\gamma _{\\nu }u_{p}){\\Big )}\\,", "0fb8cae157e7181ee7643085bda5172c": "\\Pi _{n}^{0,Y}", "0fb8d05cb55fe538e6c907ff89fbb3f9": "\\gamma _{1}={\\frac {C'\\left(F_{\\text{mid}}\\right)}{C\\left(F_{\\text{mid}}\\right)}}={\\frac {\\beta }{F_{\\text{mid}}}}\\;,", "0fb8d7cbbc10479a5fa1b84e17912de8": "\\Delta d={\\frac {\\lambda }{\\pi \\cdot n\\cdot {\\sqrt {\\frac {I}{Isat}}}}}", "0fb932c91ae74c269eacedb368b1c915": "{\\mathit {g^{(1)}}}", "0fb942c8435592280811d88cce7ba74e": "V\\sim {\\chi '}_{k}^{2}(\\lambda )", "0fb966248096f610895496745e95e368": "_{p}F_{q}^{(\\alpha )}(a_{1},\\ldots ,a_{p};b_{1},\\ldots ,b_{q};X,Y)=\\sum _{k=0}^{\\infty }\\sum _{\\kappa \\vdash k}{\\frac {1}{k!}}\\cdot {\\frac {(a_{1})_{\\kappa }^{(\\alpha )}\\cdots (a_{p})_{\\kappa }^{(\\alpha )}}{(b_{1})_{\\kappa }^{(\\alpha )}\\cdots (b_{q})_{\\kappa }^{(\\alpha )}}}\\cdot {\\frac {C_{\\kappa }^{(\\alpha )}(X)C_{\\kappa }^{(\\alpha )}(Y)}{C_{\\kappa }^{(\\alpha )}(I)}},", "0fb99c9bdd971cbc7ac97e8bf3ca38e2": "\\ Z=R+jX", "0fb99eb483202bc8ceff4467a6a78101": "\\zeta =\\prod _{i}\\zeta ^{i}=\\zeta ^{trans}\\zeta ^{rot}\\zeta ^{vib}\\zeta ^{e}.", "0fb9b328b82db6997fb4a2b824955052": "R(1)={\\frac {1}{2\\pi }}\\int _{-\\pi }^{\\pi }\\delta (\\omega -\\omega _{0})e^{i\\,\\omega }d\\omega ", "0fb9c2648be84b10fd2194ddb4e77174": "X^{t}=X", "0fb9ed11385a262ae353289145e773d6": "\\operatorname {Var} _{Y\\mid X}(Y|x).", "0fba181ef88c54352735b493592944b0": "W(\\alpha ,\\alpha ^{*})={\\frac {2}{\\pi }}\\int P(\\beta ,\\beta ^{*})e^{-2|\\alpha -\\beta |^{2}}\\,d^{2}\\beta ", "0fba4dc11d5071cba86fe42327c07ffb": "PCIOPS(mean)=1/[0.5/SSDIOPS(Iwrite)]+[0.5/SSDIOPS(Iread)]", "0fba620c0859484228589f5a45227fce": "\\alpha ={\\frac {(y_{LK}/x_{LK})}{(y_{HK}/x_{HK})}}=K_{LK}/K_{HK}", "0fbaa9e3951f9c47557bd14c53631adb": "z\\cdot {\\overline {r}}", "0fbabc341d97ce5bdfe1cef255fc5655": "\\mu _{2}={\\frac {k_{0}p_{2}}{p_{0}}}={\\frac {10\\times 0.1}{0.5}}=2.0", "0fbafa8a78e5b8e4b060b74a5a439ccd": "\\nabla ^{2}\\equiv \\operatorname {div} \\ \\operatorname {grad} \\equiv \\nabla \\cdot \\nabla ", "0fbb921c724f9f860cb5c1f1bfb90cf9": "2^{10,000}", "0fbbaf0cc0e84424bca4f843cdd4f702": "a^{2}=\\left[{\\frac {\\partial p}{\\partial \\rho }}\\right]_{S}.", "0fbbc606b21134ea0c9f01a1e834043e": "b_{1}^{*}:=b_{1},B_{1}:=\\langle b_{1}^{*},b_{1}^{*}\\rangle ", "0fbbef1e1ce114dc34416faf8eab0ec8": "u_{k}(\\mathbf {r} )\\equiv \\left.{\\frac {\\partial ^{k}}{\\partial t^{k}}}{\\big (}v(\\mathbf {r} ,t)-v'(\\mathbf {r} ,t){\\big )}\\right|_{t=t_{0}}", "0fbc597b405aece846d7df0d3dfc94df": "{\\begin{aligned}\\sum &{\\Big (}Y_{1}Y_{2}\\ln \\Phi (X_{1}\\beta _{1},X_{2}\\beta _{2},\\rho )\\\\[4pt]&{}\\quad {}+(1-Y_{1})Y_{2}\\ln \\Phi (-X_{1}\\beta _{1},X_{2}\\beta _{2},-\\rho )\\\\[4pt]&{}\\quad {}+Y_{1}(1-Y_{2})\\ln \\Phi (X_{1}\\beta _{1},-X_{2}\\beta _{2},-\\rho )\\\\[4pt]&{}\\quad {}+(1-Y_{1})(1-Y_{2})\\ln \\Phi (-X_{1}\\beta _{1},-X_{2}\\beta _{2},\\rho ){\\Big )}.\\end{aligned}}", "0fbc611a81d11b3acd10004138f4184d": "{\\text{volume}}=nAr/3", "0fbc683f4b9203db0fd009ec23827f47": "{1 \\over 1}+{1 \\over 3}+{1 \\over 6}+{1 \\over 10}+{1 \\over 15}+{1 \\over 21}+\\cdots =2.", "0fbcad5fadb912e8afa6d113a75c83e4": "a\\!b", "0fbce2fc7995c733099c64b7b03c9437": "x\\geq 0\\land y\\geq 0", "0fbcf41bc04edf4d4cfc4fcb369628df": "\\;q_{c}=q\\left(1+{\\frac {M^{2}}{4}}+{\\frac {M^{4}}{40}}+{\\frac {M^{6}}{1600}}...\\right)\\;", "0fbd1776e1ad22c59a7080d35c7fd4db": "]", "0fbd7fe649111cedf3b16e0a383d02ac": "{\\mathcal {N}}(X)", "0fbdae42cf54b2dc02cff2eab55462e5": "{\\hat {\\mathbf {e} }}_{1}", "0fbdb60d9f64ab2f0173a41becc7cbc4": "\\phi \\lor \\neg \\phi ", "0fbdddf2f1ccac4c2fade79aa78af39f": "\\mathrm {diag} \\left(J_{\\alpha ,l},J_{\\beta ,m},J_{\\gamma ,n}\\right)", "0fbe0ea246a7ee254d0be8503ab0b3ec": "{\\mathcal {H}}={\\mathcal {H}}^{1}\\otimes {\\mathcal {H}}^{2}\\otimes {\\mathcal {H}}^{3}.", "0fbe536b0ecd3ec51540e2bb9b1b9557": "x\\in {\\widehat {\\mathbb {R} }},A\\subseteq {\\widehat {\\mathbb {R} }}", "0fbe9e972640b6ebe63d973f40b73d75": "R_{B}(f)={12200^{2}\\cdot f^{3} \\over (f^{2}+20.6^{2})\\quad {\\sqrt {(f^{2}+158.5^{2})}}\\quad (f^{2}+12200^{2})}\\ ,", "0fbedf9f1798f041a491fe19bd77c5d0": "{\\frac {4+\\delta +\\delta ^{2}}{2-\\delta -\\delta ^{2}}}", "0fbef3b2b762736dc45de3374d94ffac": "{\\mathcal {F}}(X)", "0fbf0728a057b8b1d9ea65f6b97477ca": "\\!a", "0fbf16349e67cb24b0634960bae7cc28": "{\\begin{aligned}|V|&=|I||Z|\\\\\\phi _{V}&=\\phi _{I}+\\theta \\end{aligned}}", "0fbf5183bc4af7336a65ef477a06732c": "E_{n}(mx)={\\frac {-2}{n+1}}m^{n}\\sum _{k=0}^{m-1}(-1)^{k}B_{n+1}\\left(x+{\\frac {k}{m}}\\right)\\quad {\\mbox{ for }}m=2,4,\\dots ", "0fbf629057576a0f948c3b41dd6c730f": "(r{\\bar {g}}+g{\\bar {r}})/{\\sqrt {2}}", "0fbfba2d66eb7c1b0571c7ff7d1c09ae": "c=2\\sin {\\tfrac {\\pi }{5}}={\\sqrt {\\tfrac {5-{\\sqrt {5}}}{2}}}", "0fbfe4d8e7731db2599020587eaa92b0": "t=w-{\\frac {p}{3w}}", "0fc047ec6ce679fea54586089980d199": "\\omega (x_{i},y_{j})=-\\omega (y_{j},x_{i})=\\delta _{ij}\\,", "0fc063d19f47aaa28eafe8145b73a9a8": "{\\tilde {f}}(\\lambda )=\\int _{G}f(g)\\varphi _{-\\lambda }(g)\\,dg.", "0fc0747d110cbc84c925e08d75c6c948": "{\\tilde {g}}_{lk}{\\overline {n}}=g_{kl}", "0fc07624dee8cd68b111a92599adb39d": "n(I)=n+n_{2}I", "0fc084aaa92db3b493c21915b46d3994": "\\ x=s+v,\\,", "0fc0931f7407f09477b2f5b809be7165": "e_{1}=2585.25381092892231", "0fc099675c6d8e7ec4c70b8fcbd9e2e2": "\\beta =i", "0fc09a771f874f23e0384316ca8e424a": "\\rho ^{\\prime }\\equiv \\rho +{\\frac {\\Lambda }{8\\pi G}}", "0fc0b8a7f599213481452408a8675c6f": "D_{\\mathrm {KL} }(P\\|Q)\\equiv \\sum _{i=1}^{n}p_{i}\\log _{2}{\\frac {p_{i}}{q_{i}}}\\geq 0.", "0fc0cfa73492e389aa75471ad6707b30": "{\\frac {\\partial C_{1}}{\\partial t}}={\\frac {\\partial }{\\partial x}}[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_{2}}{\\partial x}}]]", "0fc10db026bd13e5a0c606b5cdaa2765": "\\max _{\\alpha }\\sum _{i=1}^{n}\\alpha _{i}-{\\frac {1}{2}}\\sum _{i=1}^{n}\\sum _{j=1}^{n}y_{i}y_{j}K(x_{i},x_{j})\\alpha _{i}\\alpha _{j},", "0fc14e67d1fbfb874e033bf12360b3cc": "\\mathbf {T} _{U,i}", "0fc22bf62c1ffe928ed175dcb02070c8": "G[\\mu _{j}]=U+pV-TS-\\mu _{j}N_{j}\\,", "0fc294aa66aba0ffb41529bbf232188f": "u{\\bar {u}}=1", "0fc2adfe6beba9cd4bf68bd5b9f9b89b": "S={\\frac {\\sigma _{\\text{between}}^{2}}{\\sigma _{\\text{within}}^{2}}}={\\frac {({\\vec {w}}\\cdot {\\vec {\\mu }}_{y=1}-{\\vec {w}}\\cdot {\\vec {\\mu }}_{y=0})^{2}}{{\\vec {w}}^{T}\\Sigma _{y=1}{\\vec {w}}+{\\vec {w}}^{T}\\Sigma _{y=0}{\\vec {w}}}}={\\frac {({\\vec {w}}\\cdot ({\\vec {\\mu }}_{y=1}-{\\vec {\\mu }}_{y=0}))^{2}}{{\\vec {w}}^{T}(\\Sigma _{y=0}+\\Sigma _{y=1}){\\vec {w}}}}", "0fc322eb39e7f8cfe71f7b568b4fba61": "\\phi (x)", "0fc38e493c1285b0ee01777c163bcc49": "\\varphi (t,x)/\\varepsilon ", "0fc3a467b257571cbedb9d01bf706fd0": "\\mathbf {Ax} ={\\lambda }\\mathbf {x} ", "0fc3b37acac582ac8327a49b0a9b544d": "90^{\\circ }/m", "0fc3d8831995c4401b8e744f8d9c42e8": "-e_{2}", "0fc3dfa9b5b8b7f307e246f6859bb17c": "h_{\\text{fe}}", "0fc420468409c0648c1b0a06cac24384": "\\sin ^{2}\\theta ", "0fc4266b6f939b070809ed195d978557": "{\\text{angle}}=\\arctan \\left({\\frac {\\text{slope}}{100\\%}}\\right)", "0fc4397d3642ba456c3416794c063e53": "v_{L}", "0fc47a5ff0f41a4c2d286a2f66d63d8b": "|z|=|a+\\sum {b_{n}i_{n}}+\\sum {c_{n}\\varepsilon _{n}}+d|:={\\sqrt[{4}]{(a^{2}+b_{n}^{2}-c_{n}^{2}-d^{2})^{2}+4(ad-b_{n}c_{n})^{2}}}", "0fc4a16b0f6ba85aac16d1a0bd34d732": "3.096262735\\times 10^{78}", "0fc4db473d51d022ae8c7527ca72f385": "\\left\\langle hk\\ell \\right\\rangle ", "0fc54c0fbc206bc462232612ba3d73d2": "{\\frac {\\Omega ^{(s)}}{\\Omega ^{(0)}}}=\\exp \\left(\\sum _{p=0}^{s-1}\\xi _{p}\\right).", "0fc588ce3c156f858cf5b0e4781a78bf": "{\\tilde {K}}_{0}\\left(A\\right)=\\bigcap \\limits _{{\\mathfrak {p}}{\\text{ prime ideal of }}A}\\mathrm {Ker} \\dim _{\\mathfrak {p}},", "0fc64d72d017c4e6de9ef38be23fe050": "{\\frac {J_{2k}(n)}{J_{k}(n)}}", "0fc691491f464ec04fefa7aa584ed639": "x_{2}\\rightarrow x_{s}(t)", "0fc6afa86487fa6984a3d272536df644": "={\\frac {1}{Z_{0}\\cdot \\left(\\pi \\epsilon _{0}\\epsilon _{r}\\right)\\cdot \\left[\\ln \\left({2D/d}\\right)\\right]}}", "0fc72a1f05c9f8309e64772f7958e09c": "a^{2}\\,", "0fc7313c481025d4585c3667f306dbff": "itH", "0fc74c75e4205a0bfbf953d3d5f05dec": "U(\\lambda )", "0fc7777459eb9dbe16ecaa0511f94c10": "M_{t}=p(W_{t},t)-\\int _{0}^{t}a(W_{s},s)\\,\\mathrm {d} s,", "0fc7926c975ed551728c1035c0ee12b0": "{\\hat {F}}(i)={\\hat {h}}(i)+\\sum _{j=1}^{n/2}[2{\\hat {J}}_{j}(i)-{\\hat {K}}_{j}(i)]", "0fc7aaa887faafddeaeeab150764302e": "(a_{1}b_{7}+a_{2}b_{8}+a_{3}b_{5}-a_{4}b_{6}-a_{5}b_{3}+a_{6}b_{4}+a_{7}b_{1}-a_{8}b_{2})^{2}+\\,", "0fc7d5ea1b013420e10444a51403375c": "\\mathrm {DQE} ={\\frac {\\mathrm {NEQ} }{q}}", "0fc7f31f553e020baf4ccc6deaf31e4c": "T=u^{\\alpha }\\gamma _{\\mu }^{\\alpha \\beta }(x-x')_{\\mu }{\\overline {u}}^{\\beta }.", "0fc81ae8fa1e7f4f359aee819e173686": "\\scriptstyle \\{U_{n}\\}", "0fc859fbb8f85ff0f76f2e63963578ca": "{\\vec {e}}_{i}=c\\times {\\begin{cases}(0,0,0)&i=0\\\\(\\pm 1,0,0),(0,\\pm 1,0),(0,0,\\pm 1)&i=1,2,...,5,6\\\\(\\pm 1,\\pm 1,0),(\\pm 1,0,\\pm 1),(0,\\pm 1,\\pm 1)&i=7,8,...,17,18\\\\\\end{cases}}", "0fc87d0ee76aaa7e2c012abba01dbf28": "\\mathbf {F} =m\\mathbf {a} \\quad \\to \\quad \\mathbf {a} =\\mathbf {F} /m", "0fc931801fbf2cc5230b5d966debc290": "disc({\\mathcal {H}})=O({\\sqrt {n}}).", "0fc94bc17ba6ef87f29aaebfc83454c2": "a\\uparrow ^{n}b=a\\uparrow ^{n-1}\\left(a\\uparrow ^{n-1}(...(a\\uparrow ^{n-1}a))...)\\right)", "0fc95ba5988654853f65ca319d9dd63f": "\\delta \\sim {\\frac {1}{T}}\\sim {\\frac {v}{L}}", "0fc9a0a4bbba13ec06c952fa2c29344f": "Z'=Y/X,Y'=Z/X\\ ", "0fc9b9f524180e5d78fce0a906acc595": "r=x\\cos \\theta +y\\sin \\theta ", "0fc9d065d91cefbcddf512dd10310cdc": "L_{\\mathrm {p} }=L_{\\mathrm {W} }+10\\,\\log _{10}\\left({\\frac {S_{0}}{4\\pi r^{2}}}\\right)\\,", "0fc9f41c7244f732ea2ed08bc9b58edd": "x^{1}\\cdot 2^{0}=x", "0fca3f83324d05eae9454da929175411": "p0,\\beta \\neq 1", "0fd06dc95a40a81847c6258a87fda96e": "\\mathbf {X} V\\Sigma ^{+}U^{\\rm {T}}=U\\Sigma V^{\\rm {T}}V\\Sigma ^{+}U^{\\rm {T}}=UPU^{\\rm {T}},", "0fd08afa70f7ac95d28fc1330d2ed79e": "\\scriptstyle \\leq 1.6\\times 10^{-14}", "0fd0ad1ca1deb2e3f3ede003ede23703": "\\mathrm {hub} (p)=1", "0fd12e9afec1f7098461e3da25d98001": "ds^{2}=-f(t,r)^{2}\\,dt^{2}+g(t,r)^{2}\\,dr^{2}+r^{2}\\left(d\\theta ^{2}+\\sin ^{2}\\theta \\,d\\phi ^{2}\\right),", "0fd13dfb64dbf5362eacc93bac0445f8": "E(\\mathbf {\\hat {f}} +i\\mathbf {\\hat {s}} )\\mathrm {e} ^{i(kz-\\omega t)},", "0fd19ab2182c2be5eee79f4941a5f5a3": "R/I,", "0fd1da4a72751e2d8d60e1794c22495a": "\\alpha '=\\alpha /(4\\pi \\varepsilon _{0})", "0fd1ec1b238c8d6315503d97d4825e4a": "{\\sigma _{S}^{D}}=18.0~\\mathrm {mN/m} ", "0fd24248803990e22edde58e4b456f76": "c_{i}=p_{i}\\oplus F(x_{1i},x_{2i},x_{3i})", "0fd2c3d3bbac540097dbc4ebc1e04bb2": "{\\mathcal {H}}=\\left({\\begin{array}{rr}1&-1\\\\-1&1\\end{array}}\\right)", "0fd2edfc8daa121e2356ebdd9e09e5d5": "{\\sqrt {\\frac {g}{k}}}\\,=\\,{\\frac {g}{\\sigma }}\\,", "0fd335ac99693b80d5f6ac83fe0bb513": "L\\rho _{L}\\,", "0fd752a060af3224aba5bad07de07ded": "2\\pi f", "0fd75ec1092b8a849eb9b6400f1f2041": "0\\leq r\\leq 1", "0fd7a5887b765d548197900f109f293e": "\\nabla \\left(\\alpha f+\\beta g\\right)(a)=\\alpha \\nabla f(a)+\\beta \\nabla g(a).", "0fd7d07f37158be28be1f04a2f435de8": "f(x)=a(x-x_{0})^{2}+x_{0}=h^{(-1)}(g(h(x)))\\,\\!", "0fd7f98c0bb1731fd618055871c5f973": "\\sup _{n}\\{a_{n}\\}.", "0fd85b89337c24dc207587957b8624e3": "\\langle k\\rangle *\\alpha ", "0fd8ba339a14c61da1a517401157e7de": "{\\frac {\\partial F}{\\partial n_{1}}}=n_{1}\\sigma _{1}^{2}-2n_{1}\\sigma _{1}\\left(\\sigma _{1}n_{1}^{2}+\\sigma _{2}n_{2}^{2}+\\sigma _{3}n_{3}^{2}\\right)+\\lambda n_{1}=0\\,\\!", "0fd96b818247d78b9375742cd6586e04": "M={\\frac {Q^{2}}{gA}}+{\\overline {y}}A", "0fd99ff654fb7b30bfa43221653a32f9": "2^{3n/2}", "0fd9a0ad48af1c3aad45a6d8ea90101e": "dp={\\frac {\\partial p}{\\partial t}}dt+{\\frac {\\partial p}{\\partial x}}dx+{\\frac {\\partial p}{\\partial y}}dy+{\\frac {\\partial p}{\\partial z}}dz", "0fd9c63d545d81c4da8caa0e0d2ce436": "p_{e}=p_{amb}", "0fd9e1aafc2e37fdba1f20cec8266816": "\\mathrm {1\\,sb=10^{4}\\,nit=10^{7}\\,millinit} ", "0fda17baf45cbf6af5b6b69232abee04": "\\prod _{n=1}^{\\infty }(1-x^{n})=\\sum _{n=0}^{\\infty }a_{n}x^{n}", "0fda22d202dc2502ce40892fe16b45e5": "{\\tilde {\\textbf {P}}}^{2n}(t)=[H^{2n}(t)]{\\textbf {P}},", "0fda8cca8ce1020383e61c7b2eb140a3": "(U;z^{A},c^{a})", "0fda905a8eaa285d890e74b373493ca7": "H_{j}", "0fdacd20ce254f34fbc464cc2ec3b740": "f(x)=s", "0fdae7d74b475dc1c455c0efe3590ddf": "m_{0}=0\\,\\!", "0fdb21e4f9bce557a6fb4cdfa4292f3d": "{\\dot {\\lambda }}\\geq 0~,~~f\\leq 0~,~~{\\dot {\\lambda }}\\,f=0\\,.", "0fdb8e258a6856772851a8f961e8d9a7": "{\\frac {\\partial \\varphi }{\\partial t}}=v|\\nabla \\varphi |.", "0fdbe2baf71976743a48d73949d22d55": "{\\text{Sl}}_{2m-1}(\\theta )={\\frac {(-1)^{m}(2\\pi )^{2m-1}}{2(2m-1)!}}B_{2m-1}\\left({\\frac {\\theta }{2\\pi }}\\right)", "0fdbe4ba43f5904b8e7cb8cdd0b78439": "\\mathbf {x} \\times \\mathbf {y} =T_{\\mathbf {x} }(\\mathbf {y} ).", "0fdc25637dbf5d907528731ab295504b": "\\Omega =\\Lambda +\\Sigma ", "0fdc804330bcf315883436ed70a62126": "{\\sqrt {n}}({\\hat {\\sigma }}^{2}-\\sigma ^{2})\\simeq {\\sqrt {n}}(s^{2}-\\sigma ^{2})\\ \\xrightarrow {d} \\ {\\mathcal {N}}(0,\\,2\\sigma ^{4}).", "0fdd8448f4767b2e1b1b875faf0e5f49": "\\ C_{x}", "0fde09bee2e3e11bb85c024acf0a2b10": "T^{n}\\times P^{n}", "0fdeb3d0d02691eee0f388da567410e0": "C_{p}\\left(T_{2}-T_{1}\\right)\\;", "0fdedd1216fd5c16a7f10583779b40e5": "{\\frac {1}{2}}|\\mathbf {AB} \\times \\mathbf {AC} |.", "0fdef2041c00867622e5cfb18e44590a": "P\\land \\neg P", "0fdf7a0f59f999169e75e4f41c435188": "{\\rm {Li}}(x)=\\int _{2}^{x}{\\frac {dt}{\\ln t}}\\,", "0fdf89089449682d95e380f9a23dfe74": "{\\begin{array}{cc}\\mathbf {I} ={\\begin{pmatrix}1&0&0\\\\0&1&0\\\\0&0&1\\end{pmatrix}};&\\mathbf {I} ^{-1}={\\begin{pmatrix}1&0&0\\\\0&1&0\\\\0&0&1\\end{pmatrix}}\\\\\\\\\\mathbf {R} ={\\begin{pmatrix}1&0&0\\\\0&0&1\\\\0&1&0\\end{pmatrix}};&\\mathbf {R} ^{-1}={\\begin{pmatrix}1&0&0\\\\0&0&1\\\\0&1&0\\end{pmatrix}}\\\\\\\\\\mathbf {S} ={\\begin{pmatrix}+1&0&0\\\\0&-1&0\\\\0&0&-1\\end{pmatrix}};&\\mathbf {S} ^{-1}={\\begin{pmatrix}+1&0&0\\\\0&-1&0\\\\0&0&-1\\end{pmatrix}}\\\\\\end{array}}", "0fdfe61eb9713f3d86d27e9c2e7aba41": "{\\frac {d}{dt}}\\langle A\\rangle ={\\frac {d}{dt}}\\int \\Phi ^{*}A\\Phi ~dx^{3}=\\int \\left({\\frac {\\partial \\Phi ^{*}}{\\partial t}}\\right)A\\Phi ~dx^{3}+\\int \\Phi ^{*}\\left({\\frac {\\partial A}{\\partial t}}\\right)\\Phi ~dx^{3}+\\int \\Phi ^{*}A\\left({\\frac {\\partial \\Phi }{\\partial t}}\\right)~dx^{3}", "0fe04ccd07305d4426cb2d0fe1e06b24": "24\\times 1,312,000", "0fe06a7178cf7287a9a4870318cd5060": "r_{SOI}=a\\left({\\frac {m}{M}}\\right)^{2/5}", "0fe082d15fa88a3ae09e057d205a551e": "\\sin _{k}^{2}(i)\\equiv (2^{-1}{\\bmod {p}})\\cdot (1-\\cos _{k}(2i).", "0fe0a2514a9cf2ba888ecca28eb1b71d": "\\forall M'", "0fe1540a9034b396e8c0ca9a8d41ca84": "\\mathbf {m} =\\mathbf {m} _{\\rm {orb}}+\\mathbf {m} _{\\rm {spin}}", "0fe20a4f5323ed611f6a14ba590dd252": "I_{2}=-{Z_{21} \\over Z_{22}}\\,I_{1}", "0fe22f29aac3cd821c4aa3c2c5a999b0": "Au_{xx}+2Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+F=0\\,", "0fe34beb493986d579b09cf842e4a7ef": "\\sum _{ij;i\\neq j}\\psi _{i}^{*}\\psi _{j}\\phi _{j}^{*}\\phi _{i}", "0fe34f3258a18e913bba77be0d89243c": "m_{0}D+m_{1}\\sigma (D)+\\cdots +m_{n-1}\\sigma ^{n-1}(D),~~~~{\\text{where }}m_{i}=O(\\ell ^{1/(n-1)})=O(q^{g})", "0fe394cf6df0566fd8e4cddae78758b5": "{\\tilde {\\lambda }}_{i}", "0fe398c270ae0c7f308943145950d12d": "{\\boldsymbol {u}}_{e}=-{\\hat {\\boldsymbol {z}}}\\times {\\Big (}{\\frac {1}{f}}{\\frac {\\partial {\\boldsymbol {\\tau }}}{\\partial z}}{\\Big )}.", "0fe3a286ae4e08b0d994222bd306afaf": "{\\frac {|\\Delta x|}{|x|}}", "0fe3d81d2b000c5f8d99b84dd98b32ab": "\\displaystyle {Q(Q(a)b)=Q(a)Q(b)Q(a),\\,\\,\\,Q(a^{m})=Q(a)^{m}\\,\\,(m\\geq 0).}", "0fe3d9c6b4bcc5525435458790c04fec": "b_{max}", "0fe42c26de1d22fba3d54abb76741191": "\\chi '(G)\\geq \\Delta (G).\\,", "0fe4669801f948c6d9240747889d269b": "\\ln(t)\\cdot u(t)", "0fe522862f36b8bca8b846197b66e331": "\\Pi _{f}", "0fe58892c79e94268a59a7078a7814f4": "-\\int _{a}^{b}\\mathbf {F} \\cdot d\\mathbf {l} =P(\\mathbf {b} )-P(\\mathbf {a} )", "0fe5a9b80f6d0f0b299d28c2da30fa63": "\\lambda (x,y)=(\\lambda x,y)=(x,\\lambda y)\\!", "0fe5d36b3fec3f7d074868a303a99434": "\\partial F={\\mathcal {P}}_{B}(\\nabla )F", "0fe6683a06cd0338b7ab4cd63633b78a": "{\\textbf {P}}_{k|k}=(I-{\\textbf {K}}_{k}{\\textbf {H}}_{k}){\\textbf {P}}_{k|k-1}", "0fe6b96de49ee5f6d313c5e7b3136835": "c_{T-j}", "0fe6e35b6e6e36652c8774aae0628915": "\\scriptstyle \\Gamma _{(z_{0},r)}=\\partial D(z_{0},r)", "0fe6edbc6bd9c973761ea25268521545": "P\\left(t,T\\right)=e^{-\\int _{t}^{T}f\\left(t,s\\right)\\,ds}.", "0fe74d16c564008427727959c6e71222": "{n \\choose 5}{5 \\choose 3}\\times 2", "0fe74d4e0208bc631753e803f0353928": "36\\operatorname {Li} _{2}\\left({\\frac {1}{2}}\\right)-36\\operatorname {Li} _{2}\\left({\\frac {1}{4}}\\right)-12\\operatorname {Li} _{2}\\left({\\frac {1}{8}}\\right)+6\\operatorname {Li} _{2}\\left({\\frac {1}{64}}\\right)={\\pi }^{2}", "0fe75a5189c2ea3f123621d098ddd03e": "PR", "0fe7c0b5e58d24e3fba107270bb953e8": "{\\mathcal {E}}(s)", "0fe7ddc565ebe9d16eb3444a392f8df6": "\\lbrace q\\mid q<_{\\mathcal {O}}p\\rbrace ", "0fe86cc2b6a3198af1cc93df5d4d6193": "D_{y}", "0fe8a89af2c204db83ef653388ae6fbc": "c_{0}(\\log |\\Delta |)^{2}", "0fe8f38e1d2c3868701be659beaa5848": "C{\\ddot {\\phi }}+G{\\dot {\\phi }}+\\phi /L=i\\,", "0fe960ea7d4104ef9d09cbbd41307d6e": "E(\\nu )", "0fe9686935e2be0226b24959f6713536": "a\\uparrow b\\uparrow c=a\\uparrow (b\\uparrow c)", "0fe97f260eae33f063386d2443ca4e2d": "\\gamma _{K}={\\dot {k}}/k=s.f(k)/k-(n+\\delta )\\ ,", "0fe992a391e39803222f9713e2142be1": "r=2a\\cos {\\theta \\over 3}", "0fe9e27a4bf412e50275321f5c5d431c": "\\lim _{n\\rightarrow \\infty }\\left|{\\frac {a_{n+1}}{a_{n}}}\\right|=1", "0fe9e816422c723d1c76b691e1d676bd": "x\\in A", "0fea552abd4c2f09dd0ecb41dcbda411": "c_{pqrs}=l_{pi}~l_{qj}~l_{rk}~l_{s\\ell }~c_{ijk\\ell }", "0fea78208e465b4e57cb2fa209f49f99": "E_{min}=\\alpha E", "0fea8f9347ee90ad9d31bc31d443fd0e": "\\alpha +h_{x}(\\alpha )=K(x)+O(1)", "0feab75901667616678429cfbe6114a3": "C_{I}", "0feabcc18804c47f0abde563618a85f4": "B_{t}=\\sum _{k=1}^{\\infty }Z_{k}{\\frac {{\\sqrt {2}}\\sin(k\\pi t)}{k\\pi }}", "0feac4661c7f76a4e0842ef697f4eb98": "\\operatorname {E} [\\cdot ]", "0feb33d8c32fa19df125704507c8d2b5": "q_{1}=x_{11}+x_{21}", "0feb34f24556a69af98fb532876080b1": "C_{H}^{d}(S):=\\inf {\\Bigl \\{}\\sum _{i}r_{i}^{d}:{\\text{ there is a cover of }}S{\\text{ by balls with radii }}r_{i}>0{\\Bigr \\}}.", "0feb44d857d3510ab18975469799b2d1": "|j,m,l,1/2\\rangle ", "0feb4f87e6273c51fb878ac983e01629": "{\\bar {t}}", "0feb6c48073bf9ee45aeebdc468d612e": "R_{j}^{i}", "0febe3443dc70c82815095ddf6a334ae": "K_{\\lambda \\mu }=K_{\\lambda \\mu }(1)=K_{\\lambda \\mu }(0,1).\\ ", "0fec6acbfbb750f86b1469612f8d0327": "{\\frac {c^{2}-s^{2}}{sc}}={\\frac {a_{\\ell \\ell }-a_{kk}}{a_{k\\ell }}}.", "0fecb1d76491880ab53dea1bdedd5b66": "f^{n}(x)\\in U", "0fecbc6612b90bff28ec553d603dba8c": "hf_{ij}={\\text{estimated frequency of haplotype }}ij=gf_{i}\\;gf_{j}=0.0215\\!", "0fecf021df2f33adeea6a5c8501bff95": "{\\vec {\\mu }}_{s}", "0fecfcebc29fc9f890ec12e787727a09": "\\vartheta (x)=\\sum _{p\\leq x}\\log p=\\log \\prod _{p\\leq x}p=\\log(x\\#).", "0fed1bb26cfec02d08fc67fe7d1d6bb8": "\\theta (F)={\\frac {1}{m}}{\\underset {(a,m)=1}{\\sum _{a=1}^{m}}}a\\cdot \\mathrm {res} _{m}\\sigma _{a}^{-1}\\in \\mathbf {Q} [G_{F}].", "0fed219a5c9ca170788a66534e8c7700": "P=\\exp {\\left(A+{\\frac {B}{C+T}}+D\\cdot T+E\\cdot T^{2}+F\\cdot \\ln \\left(T\\right)\\right)}", "0fed3cdb09247f9d104ba6015df01a75": "C_{T}^{(p)}(p,T)\\ ", "0fed9e0fd38ec4502d9b5a064ec52c0b": "{\\frac {1}{1-w}}=\\sum _{n=0}^{\\infty }w^{n}.", "0fee49d4621b4f2d14111406a999e287": "a'_{kk}=a_{kk}-ta_{k\\ell }\\,\\!", "0feea555e91f76eba0d3ac6a94022f67": "S(t)=P_{2}(t){\\mbox{ , }}t_{1}\\leq t=X", "0ff30fbff074d55c7f69d5b00c010ea6": "(1+x)^{p}\\approxeq 1+px{\\mbox{ when }}x<<1", "0ff32026b7bc78e4c1ed622ae5891bde": "V(\\zeta )=\\int _{0}^{\\infty }G(|\\varphi (t,\\zeta )|)dt", "0ff33750ad45820ec1319aa926e4d788": "(\\alpha ,\\beta )=(\\zeta ,1).", "0ff35e7a721728fd06d7ff81dcfead3f": "x={\\frac {p}{2^{n}5^{m}}}={\\frac {2^{m}5^{n}p}{2^{n+m}5^{n+m}}}={\\frac {2^{m}5^{n}p}{10^{n+m}}}", "0ff36d579dbadedcab2d801a1fe9fe45": "P_{t}", "0ff38944aefc9f54e7f6c2089ae858f0": "\\forall j\\in \\{1,\\ldots ,n\\}", "0ff39a3b850ebc88cd39c4618de3e411": "\\delta \\propto k^{-2}", "0ff3d3ff61190d6d56da7390e8cb1313": "t_{4}=1213121121312", "0ff4178ce42f74534ee213b6b691c76e": "k^{n-k}", "0ff428303f7c5ab220c0a4c0b97b57fb": "I_{\\text{s}}", "0ff45e9ec87a1bb80ddff335180180ec": "O(n\\log k)", "0ff4dee6e989771a41e8975cccd3be25": "\\rho ({\\mathbf {D}})", "0ff4efd46bc3379a65250abc6d288333": "C\\subseteq X", "0ff5464870c6c1e0a8fe6ed07d3f0666": "[x^{2}+(\\cot \\varphi _{1}-y)^{2}]^{1/2}", "0ff548518306c8ede8f90e492317ebe1": "T_{Out}\\,\\!", "0ff54d80e3d4fc62feaa8d2c376722da": "W(\\mathbf {C} )={\\frac {\\mu }{2}}(I_{1}^{C}-3)", "0ff592606b9f2dfa45548d2415fd8e9d": "\\mathbf {Gr} (r,{\\mathcal {E}})_{s}", "0ff5e71bf4aab2580c3788cd6c9b8f39": "\\Omega \\!", "0ff60dcae60fe1793331843f23602c21": "a_{{\\mathbf {k} }_{l}}^{\\dagger }|n_{{\\mathbf {k} }_{1}},n_{{\\mathbf {k} }_{2}},n_{{\\mathbf {k} }_{3}}...n_{{\\mathbf {k} }_{l}},...\\rangle ={\\sqrt {n_{{\\mathbf {k} }_{l}}+1}}|n_{{\\mathbf {k} }_{1}},n_{{\\mathbf {k} }_{2}},n_{{\\mathbf {k} }_{3}}...n_{{\\mathbf {k} }_{l}}+1,...\\rangle ", "0ff695b719f5589ce68ae5fad5f9b970": "f(\\sigma _{\\bar {C}},\\cdot )-{\\bar {C}}=0\\,", "0ff6a5ca4fe06952d6997c564515c254": "3\\in X", "0ff760323df54c472bbe3df83eea3368": "X\\to Z", "0ff7650e1dcef535f34c1e7d9b49a55e": "A=-k_{\\mathrm {B} }T\\ln {\\bigg (}{\\frac {V_{\\mathrm {A} }^{N}}{N!\\Lambda ^{3N}}}{\\bigg )}", "0ff790ce5aa9be38a3af2a6c3212a452": "V_{o}=V_{i}", "0ff7e8099b9fba40e069c21722c9ac77": "\\|f\\|_{B}=\\sup _{x\\in X}\\left|f(x)\\right|", "0ff81db00ea358a73d443a7b10ac962e": "E[p]", "0ff86ade0c00234d6a86395e7e9e7fc0": "2^{2}\\cdot 3\\cdot 5", "0ff870d34fce892ecf9019b59bc51c1b": "{\\frac {1}{F(p)}}=\\sum _{n=0}^{\\infty }a_{n}p^{-n}", "0ff886fcf5436652cb6c0f6c82ac0500": "x\\!\\!\\sim y{\\text{ iff}}{\\begin{cases}x=y,{\\mbox{ or }}\\\\x=\\gamma (t){\\text{ and }}y=g(t){\\text{ for some }}t\\in [0,1]{\\mbox{ or }}\\\\x=g(t){\\text{ and }}y=\\gamma (t){\\text{ for some }}t\\in [0,1]\\end{cases}}", "0ff895cc17d6587aaa31e2b5b9c43334": "x_{2},\\ldots ,x_{K-1}", "0ff930fda5dd413e5a5848cb2806a992": "\\pi (\\theta |x)={\\frac {p(x|\\theta )\\pi (\\theta )}{p(x)}}={\\frac {f(x-\\theta )}{p(x)}}", "0ff9557af1eab7c66e0fa804660ea21c": "\\textstyle {1}", "0ff95c81b92a9909c7e15bae42b2b919": "[A\\cup B]_{\\text{seq}}=[A]_{\\text{seq}}\\cup [B]_{\\text{seq}}", "0ff96b58177c94df21db350bc3720e21": "k(\\tau )=\\int \\limits _{-\\infty }^{\\infty }S(\\nu )\\exp \\left(-i2\\pi \\nu \\tau \\right)d\\nu =\\exp \\left(-\\pi ^{2}\\tau ^{2}\\Delta \\nu ^{2}\\right)\\exp \\left(-i2\\pi \\nu _{0}\\tau \\right)", "0ff99fa99eda51bf18c8ceaa737f5b00": "Q_{n}(i)", "0ff9b1f42708a1392c719e29425d8322": "a_{n}=t_{n}-{\\frac {1}{2(n+1)}}", "0ff9dfefd9a847bab174b2b0386ee229": "\\scriptstyle \\emptyset \\in {\\mathcal {F}}", "0ff9e4b796741eb00f0178d7c994a8e4": "\\forall x_{1}\\ldots \\forall x_{n}\\;\\exists y\\;P(y)", "0ffa1c61f3af78f7a579dd62f6f7e741": "\\chi (3,7)=q_{1}+q_{1}q_{2}-q_{1}", "0ffac5cd4249e1754a8e1bac9497ad67": "M_{1}\\prec _{K}M_{2}", "0ffaf658e74aee96b2410fc4f1ab9266": "\\sum _{n\\in \\mathbb {Z} }\\left|x[n]\\right|^{p}<\\infty .", "0ffb7e5fc4f9fc7ea4d0c4ac9738e8b5": "\\iiint _{D}(x^{2}+y^{2}+z)\\,dx\\,dy\\,dz=\\iiint _{T}(\\rho ^{2}+z)\\rho \\,d\\rho \\,d\\phi \\,dz;", "0ffbdef80663d0b5f7be54455a3f3736": "{\\hat {f}}(n)=c_{n}", "0ffbf8e663951350614606c52ea3f31c": "\\left({\\frac {\\lambda \\ }{2a}}\\right)^{2}={\\frac {\\sin ^{2}\\theta \\ }{h^{2}+k^{2}+l^{2}}}.", "0ffc2fb0831add8c4d1b889f6360805a": "\\Delta v_{r}=\\int {\\frac {qE_{r}(r,z)}{m_{0}v_{z}}}dz", "0ffc49f6b711b28603c43491fcbc9639": "A\\equiv _{T}B\\,", "0ffd7c709a323baf97fe8b678f3b0114": "P_{r}\\approx P_{t}({\\frac {\\lambda G}{4\\pi d}})^{2}\\times ({\\frac {4\\pi h_{t}h_{r}}{\\lambda d}})^{2}", "0ffd89bc3eaa39ebd01f8b349e0c8d39": "\\mu (S)={\\frac {1}{2\\pi }}m\\left(f^{-1}(S)\\right),", "0ffdf10da47650638f0b5d716a7c9407": "i{\\overline {\\psi }}(\\partial _{\\mu }+A_{\\mu })\\Gamma ^{\\mu }\\psi ", "0ffdff24d8460133a3425569b27c1f5d": "P_{i}={\\frac {a_{i}^{\\text{o}}}{a_{i}^{\\text{w}}}}=\\exp \\left[{\\frac {z_{i}F}{RT}}(\\Delta _{\\text{o}}^{\\text{w}}\\phi -\\Delta _{\\text{o}}^{\\text{w}}\\phi _{i}^{\\ominus })\\right]=P_{i}^{\\ominus }\\exp \\left[{\\frac {z_{i}F}{RT}}\\Delta _{\\text{o}}^{\\text{w}}\\phi \\right]", "0ffe48bc7bbbfbd49b3b2ec98c888f77": "\\mathbf {Q} ^{m}", "0fffa731659443521b209273e179ca6d": "E=a\\cdot E_{0}+B", "0fffc4e9ea6664cdbc8a33802ccb863e": "{\\hat {F}}", "0fffe46f0dea0cc08391a462dc86f6a6": "\\{|f\\rangle =|u\\rangle \\otimes |t\\rangle \\otimes |s\\rangle \\otimes |r\\rangle \\}", "10001d1b13648c4fde049b96dcc46879": "dQ=TdS", "10004ee38a47bc9fa1c843c56c73436a": "Y(s)={\\mathcal {L}}\\left\\{y(t)\\right\\}", "10006d0675a431acba482acde502d102": "F={\\begin{smallmatrix}{\\frac {1}{{\\sqrt {2\\pi }}\\sigma }}P^{2}\\exp(-(P-P_{0})^{2}/\\sigma ^{2})\\end{smallmatrix}}", "100089280ba9a89fc86b84be968c610c": "\\mathbb {E} {\\big [}\\nabla f\\cdot v{\\big ]}=\\mathbb {E} {\\big [}f\\delta v{\\big ]}.", "1001858f21202f87a4dbc2ef0ecc0e33": "x\\not =y", "10024912e355b3ef15847978acc8c900": "\\log \\zeta (s)=-\\sum _{p}\\log(1-p^{-s})=\\sum _{p,n}p^{-ns}/n.", "100263ddff847c180682c3b41ae33f60": "{\\frac {d\\mathbf {\\rho } }{dt}}=\\mathbf {\\Omega } \\times \\mathbf {\\rho } ", "100283dea92f90aa425d28ec59528000": "K=H-\\mu N", "1002b06e12aafa8dcf8b88d347f28951": "{\\begin{aligned}{\\frac {\\mathrm {d} F_{\\varepsilon }}{\\mathrm {d} \\varepsilon }}&={\\frac {\\mathrm {d} x}{\\mathrm {d} \\varepsilon }}{\\frac {\\partial F_{\\varepsilon }}{\\partial x}}+{\\frac {\\mathrm {d} g_{\\varepsilon }}{\\mathrm {d} \\varepsilon }}{\\frac {\\partial F_{\\varepsilon }}{\\partial g_{\\varepsilon }}}+{\\frac {\\mathrm {d} g_{\\varepsilon }'}{\\mathrm {d} \\varepsilon }}{\\frac {\\partial F_{\\varepsilon }}{\\partial g_{\\varepsilon }'}}\\\\&={\\frac {\\mathrm {d} g_{\\varepsilon }}{\\mathrm {d} \\varepsilon }}{\\frac {\\partial F_{\\varepsilon }}{\\partial g_{\\varepsilon }}}+{\\frac {\\mathrm {d} g'_{\\varepsilon }}{\\mathrm {d} \\varepsilon }}{\\frac {\\partial F_{\\varepsilon }}{\\partial g'_{\\varepsilon }}}\\\\&=\\eta (x){\\frac {\\partial F_{\\varepsilon }}{\\partial g_{\\varepsilon }}}+\\eta '(x){\\frac {\\partial F_{\\varepsilon }}{\\partial g_{\\varepsilon }'}}\\ .\\\\\\end{aligned}}", "1002cd7181a8c9442bdc0a372e458272": "\\sum _{i=1}^{k}\\ln \\Gamma (\\alpha _{i})-\\ln \\Gamma \\left(\\sum _{i=1}^{k}\\alpha _{i}\\right)", "1002f925a2f1bc9dcda108dc38692100": "c_{t}=y_{t}+(1+r)b_{t}", "100356104d78ff40589ac8dfeb5f32a0": "{\\overrightarrow {\\Gamma }}={\\overrightarrow {m}}\\times {\\overrightarrow {B}}", "1003664af1b58fe33f27838a7802c2c8": "\\mathrm {B} (k,n+1-k)", "1003b738a3a1ab33f9aeb768e36fb2db": "E(A)", "1003effa0a74afd23c46e2ac1e1864c3": "\\scriptstyle \\zeta [{\\vec {x}},t]", "10040034a9fa378aa45741429194af62": "x=l_{0}\\|r_{0}", "100484d176dca301883760832989c297": "\\xi _{d}=d^{-1}{\\sqrt {\\frac {K_{2}}{\\epsilon _{0}\\Delta \\chi _{e}E^{2}}}}", "10048d2e8b8a081a92405a9c2daa3203": "{\\frac {d}{dx}}\\left(\\prod _{i=1}^{n}u_{i}(x)\\right)=\\sum _{j=1}^{n}\\prod _{i\\neq j}^{n}u_{i}(x){\\frac {du_{j}(x)}{dx}},", "10048e1db55647a2c536090802a2013c": "ds^{2}=-c^{2}\\left(1-{\\frac {2GM}{rc^{2}}}\\right)dt^{2}+{\\frac {dr^{2}}{1-{\\frac {2GM}{rc^{2}}}}}+r^{2}(d\\theta ^{2}+\\sin ^{2}\\theta \\,d\\phi ^{2}).", "1004bdae5ef831a50bfee125f1342fa4": "d(x^{2}(x^{2}+y^{2}))-d(a^{2}y^{2})=0", "1004dc886a9953c947eabc29e72e22d3": "I=(i_{1},i_{2},\\cdots ,i_{k})\\in \\Im ", "100521c77d873735e9b77843868ce73b": "H_{ij}={\\begin{bmatrix}{\\partial ^{2}V_{ij} \\over \\partial x_{i}\\partial x_{j}}&{\\partial ^{2}V_{ij} \\over \\partial x_{i}\\partial y_{j}}&{\\partial ^{2}V_{ij} \\over \\partial x_{i}\\partial z_{j}}\\\\{\\partial ^{2}V_{ij} \\over \\partial y_{i}\\partial x_{j}}&{\\partial ^{2}V_{ij} \\over \\partial y_{i}\\partial y_{j}}&{\\partial ^{2}V_{ij} \\over \\partial y_{i}\\partial z_{j}}\\\\{\\partial ^{2}V_{ij} \\over \\partial z_{i}\\partial x_{j}}&{\\partial ^{2}V_{ij} \\over \\partial z_{i}\\partial y_{j}}&{\\partial ^{2}V_{ij} \\over \\partial z_{i}\\partial z_{j}}\\end{bmatrix}}", "100540e73a7630cf24c3e6030fd4573b": "\\forall {\\mathbf {X} ,\\mathbf {Y} }\\ D_{**}^{(p)}(\\mathbf {X} ,\\mathbf {Y} )=0\\ \\nLeftrightarrow \\ \\mathbf {X} =\\mathbf {Y} \\,", "100543673f111aba9912aaaa01cf5bf5": "C_{p}^{(T)}(p,T)=\\left.{\\frac {\\partial U}{\\partial T}}\\right|_{(p,T)}\\,+\\,p\\left.{\\frac {\\partial V}{\\partial T}}\\right|_{(p,T)}\\ ", "1005484cbee0886fa3105c50d997e297": "\\sum d_{i}^{2}=194", "1005893aad2aa15fc21e483bfa2a404e": "f=f_{y}={\\frac {R_{M}}{2}}\\cdot {\\frac {1}{\\cos \\theta }}", "10059ba0b3d5b51f386cfd1993ef5e9a": "\\Sigma _{XY}=\\operatorname {cov} (X,Y)", "1005bab0ce24fe12b04c23f4dbf0c9f0": "\\mathrm {not} ~p", "1005d7f37f874a7268b92222a8476894": "H_{{\\frac {3}{4}},3}={({\\tfrac {4}{3}})}^{3}-27\\zeta (3)+\\pi ^{3}", "1006079f9d88cb3b4a13fd7b8d54c8d8": "S(\\rho ^{123}||\\rho ^{1}\\otimes \\rho ^{23})-S(\\rho ^{12}||\\rho ^{1}\\otimes \\rho ^{2})=S(\\rho ^{12})+S(\\rho ^{23})-S(\\rho ^{123})-S(\\rho ^{2})\\geq 0,", "1006649e21624f24f9fe35836a56c387": "\\gamma ^{\\mu }\\gamma ^{\\nu }\\gamma _{\\mu }=-2\\gamma ^{\\nu }.\\,", "1006cc2b33e4c62fdba3b0dc068d08e9": "\\left|\\Gamma (\\omega )\\right|^{2}=\\sin ^{2}\\phi _{\\Gamma }(\\omega )=1-\\sin ^{2}\\phi _{\\tau }(\\omega )", "10071bcd9da4a9cbc26be5144494ff55": "n=f(x)=A(x,x)", "10073179d2cd088b9054d5a3068f446a": "v_{\\lambda }", "1007377099f7b8b812f04c87a775f8de": "{\\sqrt {n}}{mn \\choose n}\\geq {\\frac {m^{m(n-1)+1}}{(m-1)^{(m-1)(n-1)}}}", "10074c8f698696461ea3375b3c9a88d7": "dz=\\left({\\frac {\\partial z}{\\partial x}}\\right)_{y}dx+\\left({\\frac {\\partial z}{\\partial y}}\\right)_{x}dy", "1007e5b4d0e8e4801c245326fe2d911d": "{\\begin{cases}\\varphi _{X}\\!:\\mathbf {R} \\to \\mathbf {C} \\\\\\varphi _{X}(t)=\\operatorname {E} \\left[e^{itX}\\right]=\\int _{\\mathbf {R} }e^{itx}\\,dF_{X}(x)=\\int _{\\mathbf {R} }e^{itx}f_{X}(x)\\,dx=\\int _{0}^{1}e^{itQ_{X}(p)}\\,dp\\end{cases}}", "1007f2eb6a35e3f978c8552d0899df30": "(x_{1},y_{1})+(x_{2},y_{2})=\\left({\\frac {x_{1}y_{2}+y_{1}x_{2}}{1+dx_{1}x_{2}y_{1}y_{2}}},{\\frac {y_{1}y_{2}-ax_{1}x_{2}}{1-dx_{1}x_{2}y_{1}y_{2}}}\\right)", "10082936419735934dab86df00c4b1ab": "t_{r}\\!", "10084ae97f86c9321131e6c3e31a9a9a": "\\displaystyle {\\frac {1}{2\\pi }}\\iint f(x,y)e^{-i(\\omega _{x}x+\\omega _{y}y)}\\,dx\\,dy", "1008ac9834bd5cefffc465a4b26a7ac7": "{\\frac {1}{r_{a}}}+{\\frac {1}{r_{c}}}={\\frac {1}{r_{b}}}+{\\frac {1}{r_{d}}}.", "1008bc3b2ccaa1ce32a5da86f8899ccf": "A\\times P", "10090dedbdfa037f2f0860222fdcbaa7": "\\left\\langle \\sum _{i}a_{i}\\otimes \\lambda _{i},\\sum _{j}b_{j}\\otimes \\mu _{j}\\right\\rangle =\\sum _{i,j}(\\lambda _{i})_{0}(\\mu _{j})_{0}\\int _{X}a_{i}\\smile b_{j}.", "1009208bf66795bbc26f94eca59122f8": "c_{k}=c(x_{k},u_{k})", "10092d92ea7b2e67ea8cb963bd134450": "X_{i1}=1", "1009ac27f02d37e5db7f3462885a1acd": "E_{1}=\\Delta x+\\Delta y+\\Delta z+2\\Delta p=0", "1009dde431503a7d94d31b1ef4516cf8": "R=mg\\pm {\\frac {mv^{2}}{r}}", "100a1c0d2f14ad7111e3e6ea5d887c3a": "h=H(m)", "100a6918357a83bfcf174046ccd5f870": "d\\Phi =\\omega \\wedge \\Phi ", "100a7d1ed937ebe8e0db1ca02ab75f4e": "m_{\\rm {u}}={\\frac {N_{\\rm {A}}}{M_{\\rm {u}}}}={\\frac {A_{\\rm {r}}({\\rm {e}})}{m_{\\rm {e}}}}={\\frac {A_{\\rm {r}}({\\rm {e}})c\\alpha ^{2}}{2R_{\\infty }h}}", "100ac5c6c6d0b755e826bd0096f0347b": "{I_{c}}=", "100ae91942a961d32bebd12bbe368146": "\\eta _{A}:A\\to T(A)", "100b483f3fe1552228dc3b6f46f699c8": "{\\overline {\\mathcal {M}}}_{g,n}(X,A)", "100b54fa9bb67b6b3e2f1032dd430fff": "L(p;q)", "100b581e900eef1a0a5515e3a5d23761": "\\sim 10^{2075}\\,\\!", "100b8e815cab90d99a5855816c38ce0f": "h{(x)}", "100b908b432230b727445c2b4f09310a": "KG(n,2)", "100bb23db4f39d534c682204f8a80108": "m(m-1)\\cdots (m-N+1)+a_{N-1}m(m-1)\\cdots (m-N+2)+\\cdots +a_{1}m+a_{0}=0.", "100bfde3965eb604a6606f2b4035ea94": "\\nu ^{2}0,s=1,\\beta \\neq 1", "10270df5d24fcbb9dd6a00cd96610039": "1/100000th", "102758538c3a776dbb7789b560578eb8": "f_{A}(x)={\\begin{cases}{\\frac {1}{n}}&{\\text{if }}x{\\text{ is rational and }}n{\\text{ is minimal so that }}x\\in F_{n}\\\\-{\\frac {1}{n}}&{\\text{if }}x{\\text{ is irrational and }}n{\\text{ is minimal so that }}x\\in F_{n}\\\\0&{\\text{if }}x\\notin A\\end{cases}}", "1027d252747926648c413216c056ffbd": "f(p)={\\frac {1}{4\\pi m^{3}c^{3}\\theta K_{2}(1/\\theta )}}\\exp \\left(-{\\frac {\\gamma (p)}{\\theta }}\\right)", "1027d37c74892a9be58ce55dd3a68e5f": "Math\\geq medium", "1027d569c08705a1d176ca26904144ab": "f_{\\theta }(x)", "1027d86971238e2c8890888ea6757a37": "{\\text{storage efficiency}}={\\frac {{\\text{effective capacity}}+{\\text{free capacity}}}{\\text{raw capacity}}}.", "10283d17d27eaf286f6a21bd089b06bd": "sim(d_{j,q})={\\frac {P(R|{\\vec {d}}_{j})}{P({\\bar {R}}|{\\vec {d}}_{j})}}", "102855b21b719ae1b495b15c22051259": "{\\tbinom {n+k-1}{n-1}}={\\tbinom {n+k-1}{k}}", "10285e0066647eeb9add855d58741547": "q+r\\leq 1", "10285f558ab4b167191f884c9d5a06ab": "4^{5}=1024", "1028a7c1188bd3d81ab75fe764f5f036": "|\\psi '\\rangle =\\alpha _{0}|000\\rangle +\\alpha _{1}|111\\rangle .", "1028b5307b619b51cb78c0796b3ef506": "y_{1}=1.066869388", "1028f2df4211d4057f8892585c2b98b8": "{\\hat {x_{i}}}", "1029b6f472a96d3b952f1f823448c5c0": "20^{2}=400\\equiv 15", "1029cd07e0ea07c469dcc29d97aba969": "\\left(x,t\\right)", "1029cd3f41bd4c684ea220434bb119dd": "{\\boldsymbol {\\nabla }}\\cdot (\\varphi ~\\mathbf {v} )=\\varphi ~{\\boldsymbol {\\nabla }}\\cdot \\mathbf {v} +\\mathbf {v} \\cdot {\\boldsymbol {\\nabla }}\\varphi ", "102a10cb398bd3e335fe8a0668bfd8f1": "v={\\sqrt {\\frac {T}{\\mu }}},\\,", "102a28c1c081668e97ec05cbab03a3ed": "{\\begin{bmatrix}a&b/2\\\\b/2&c\\end{bmatrix}}.", "102a6408ae9e7cbd8ad32c664bd9de81": "{\\vec {\\tau }}=I{\\vec {\\alpha }}", "102ab1c6749dce0c13fddf3990dfdfe7": "y_{2}", "102b67b7bb51826fc86b2470c1a9b531": "\\langle -\\alpha |{\\hat {\\rho }}|\\alpha \\rangle =\\langle -\\alpha |\\psi \\rangle \\langle \\psi |\\alpha \\rangle ", "102b7e8d3f76a9c950d08bfdf98830ef": "C_{n}{\\mathbb {R}}^{2}", "102bae106140aeaf227180024a6ae2b2": "{4 \\over 5}", "102bb348fd2c1410e444140ec8889158": "X_{(a,b,c,d)}(u)={\\sqrt {-i}}\\cdot e^{i\\pi {\\frac {d}{b}}u^{2}}\\int _{-\\infty }^{\\infty }e^{-i2\\pi {\\frac {1}{b}}ut}e^{i\\pi {\\frac {a}{b}}t^{2}}x(t)\\;dt\\,,", "102bc530ef1253fd4431edb7880f4466": "b=1;", "102bd4c258a00714d7dd11ce3e7075e0": "(1+year+[year/4]+[(year-1600)/400]-[(year-1600)/100])\\mod 7", "102bdc15f139a867d8b0f3396a968e9d": "J=\\Psi \\Psi ^{\\dagger },", "102c2b378276eb5b124af69720498b7a": "{\\begin{pmatrix}j_{1}&j_{2}&j_{3}\\\\m_{1}&m_{2}&m_{3}\\end{pmatrix}}=(-1)^{j_{1}+j_{2}+j_{3}}{\\begin{pmatrix}j_{2}&j_{1}&j_{3}\\\\m_{2}&m_{1}&m_{3}\\end{pmatrix}}=(-1)^{j_{1}+j_{2}+j_{3}}{\\begin{pmatrix}j_{1}&j_{3}&j_{2}\\\\m_{1}&m_{3}&m_{2}\\end{pmatrix}}.", "102c32b26bbc907b501556281aa53841": "f(x)=x^{3}-12x^{2}-42\\,", "102cab400d8f3c240d4e50d658da24f3": "{\\mathfrak {sl}}(n,\\mathbb {C} )", "102d2bf2ff3751ac8226713a73d8ff96": "M=\\mathrm {diag} [m_{1}I_{n_{1}},m_{2}I_{n_{2}},\\cdots ,m_{N}I_{n_{N}}]", "102d6049a3b8e8b67a70919709a316ed": "{\\frac {1}{\\alpha }}\\tanh(\\alpha gt)=v", "102d6202c25478272229f898a5365033": "R_{J}=\\int _{0}^{\\pi /2}3\\sin \\theta (\\cos \\theta )^{2}R_{F}(\\cos \\theta )d\\theta ", "102da3191257df7b8c5bc3369b4b631f": "\\int \\cos(x)\\,dx=\\sin(x)+C.", "102dc7004515c2ac64327380faa80844": "L\\,\\!", "102dd3e45c84a4142f0b2b0bf905e243": "=\\left.{\\frac {\\partial \\sigma }{\\partial x}}\\right|_{p}={\\dot {\\sigma }}(x)\\,", "102de6bc0991c781305934d2c92a8d9f": "f_{X}(x)={\\begin{cases}{\\frac {1}{6}}x^{3}&0\\leq x\\leq 1\\\\{\\frac {1}{6}}\\left(-3x^{3}+12x^{2}-12x+4\\right)&1\\leq x\\leq 2\\\\{\\frac {1}{6}}\\left(3x^{3}-24x^{2}+60x-44\\right)&2\\leq x\\leq 3\\\\{\\frac {1}{6}}\\left(-x^{3}+12x^{2}-48x+64\\right)&3\\leq x\\leq 4\\end{cases}}", "102e0c46a0c48c3d19943bae38ca1c77": "\\ln \\gamma _{i}=\\ln \\gamma _{i}^{c}+\\ln \\gamma _{i}^{r}", "102e39fb67fe113e0d04e5bbc5e040ce": "I\\approx {\\frac {c\\,n\\,\\epsilon _{0}}{2}}|E|^{2},", "102e6cd8bf051901c3c2144d287d75a8": "\\varphi \\to \\varphi e^{k\\omega }.", "102e7a07844dffac65a4e4000cc2394a": "A=(a_{1},\\dots ,a_{n})\\subseteq V", "102ea74c2046dc9560869cce40aec725": "P=\\mathbf {v} \\cdot \\mathbf {F} ", "102ec24202515e1cd4c4556cea5da73b": "V_{z}", "102ed868d63f74136287b8fe0cc5d661": "V_{n}={\\frac {2(2\\pi )^{(n-1)/2}}{n!!}}R^{n}.", "102edd41a0f0268c29d257e10ca467b9": "T_{p}^{\\mathrm {H} }(x,y)={\\begin{cases}T_{\\mathrm {D} }(x,y)&{\\text{if }}p=+\\infty \\\\0&{\\text{if }}p=x=y=0\\\\{\\frac {xy}{p+(1-p)(x+y-xy)}}&{\\text{otherwise.}}\\end{cases}}", "102f7748c227f8ef79cd8e77c89ff989": "{\\mathcal {R}}_{1/2}^{K}", "102f7b26809a107e81be9e0371929a6e": "{\\frac {\\partial \\theta }{\\partial t}}=\\nabla \\cdot D(\\theta )\\nabla \\theta ", "102f8896841ad94d3cbd761d904c422f": "\\nabla ^{2}\\phi =\\rho (x,y,z)\\;.", "102fba291d5242c58260fa7a6ec6a869": "n=2\\,", "102fca3929563807011e10100f19722a": "\\int \\sigma _{x}dA=0", "102fe0e8f33a162633812a71731ca902": "\\prod _{t\\in T}(1-x^{t})^{-1}.", "10302ca9e8f4048ee0a24fc4d9705e81": "d_{G}(p,q)=d_{G'}(f(p),f(q))", "1030448c29ab3533e26c8ab0dfd0c0a1": "x^{k+1}=x^{k}+{\\frac {b_{i}-\\langle a_{i},x^{k}\\rangle }{\\lVert a_{i}\\rVert ^{2}}}a_{i}", "10304638e0cede49b46e5ec345077525": "Coenergy=area~OACO=W'_{stored}=\\int _{0}^{i}\\lambda (i)~di\\;", "10308439b407adff8dd4ce208f49302b": "1/2B", "103084ce820b78321322b7981badf2bd": "T={\\frac {m}{2}}\\mathbf {\\dot {r}} \\cdot \\mathbf {\\dot {r}} ", "1030dad401f7d4683087bf75e1e09de1": "a=0.5", "1030dec6085b4ce36d0ea5d25132b0e3": "S=R-\\{0\\}", "1030e9eaa4fcba3835fb2b77d219599c": "g_{m}(z)={\\frac {1}{|S_{m}|}}\\left(\\log {\\frac {1}{1-z}}\\right)^{m}={\\frac {1}{m!}}\\left(\\log {\\frac {1}{1-z}}\\right)^{m}.", "103196e771d34f5b93e7f5390638f8fd": "\\Delta I=0\\Rightarrow ", "1031b74ac2bf56c60578f00f3506b94c": "\\zeta (-k)=-{\\frac {B_{k+1}}{k+1}}", "1031e9537fd9a53d196fde746ac8572f": "f:E\\mapsto R", "103214e1a717b7c1a5a33e55e5757c11": "\\arcsin x=2\\arctan {\\frac {x}{1+{\\sqrt {1-x^{2}}}}}", "10326a7f44a2aed29848023d9e0ed0d1": "{\\frac {m_{2}u_{2}-m_{2}u_{1}+m_{1}u_{1}+m_{2}u_{2}}{m_{1}+m_{2}}}={\\frac {u_{1}(m_{1}-m_{2})+2m_{2}u_{2}}{m_{1}+m_{2}}}", "1032735dcc0ca5203b269cf9a4e448bf": "(\\coth \\alpha -1)\\left(e^{2\\alpha }-1\\right)=2", "1032779a9946de78d71667c4eccf0941": "F^{ab}=\\partial ^{b}A^{a}-\\partial ^{a}A^{b}\\,\\!", "1033162fd547cdebb5f1f70c765f4286": "Tf(\\psi )=f(T^{*}\\psi ).\\,", "103353cc9b8470d53fdcecdfd235bef1": "\\Sigma _{1}(L_{\\alpha }[B])", "10337fe2b560f392b4a56867f5c5698a": "O(c^{12}\\log {n})", "1033a62855a3339dba801a62c385cd0c": "\\displaystyle u_{xt}=\\sinh u", "1033cf9540529a545f88b8b29f6dda13": "A\\subset B\\subset C\\quad {\\textrm {and}}\\quad |A|=|C|\\qquad \\Rightarrow \\qquad |A|=|B|=|C|", "1033d1b50c9529f52307e1d9b8f83550": "T|\\alpha ,\\beta \\sim \\mathrm {Gamma} (\\alpha ,\\beta )\\!,", "1033ec26434eeb17994f6b4f38bd4b9b": "{\\frac {V_{1}}{T_{1}}}={\\frac {V_{2}}{T_{2}}}\\qquad \\mathrm {or} \\qquad {\\frac {V_{2}}{V_{1}}}={\\frac {T_{2}}{T_{1}}}\\qquad \\mathrm {or} \\qquad V_{1}T_{2}=V_{2}T_{1}.", "103408b48618d71c7e1baf910b144e75": "P_{n}=n{\\binom {n+1}{2}}-{\\binom {n+1}{3}}.", "103418c721caddb55f16da1b7e733e91": "G_{\\mu \\nu }^{a}\\,", "10341cab940e4f0d0dfb4e8d0c197db0": "P\\propto 1/K^{\\beta }", "10342149cd04422c7be79506cb544ac5": "x>y", "10343a0ffa3897b059bf833c27346d2c": "\\omega ^{*}={\\underset {\\omega \\in \\Omega }{\\textrm {argmax}}}\\sum _{n=1}^{\\ell }y_{n}h({\\boldsymbol {x}}_{n};\\omega )\\lambda _{n}.", "10345a62ca611efe33e873521938b1d4": "\\forall xD(x)", "1034bb957d8d9ac84e4fdf700718c97b": "\\zeta ^{\\prime }(-8)={\\frac {315}{4\\pi ^{8}}}\\zeta (9).", "1034e605ec7bc7833f2217ed2bb6f683": "{\\begin{aligned}A&{}={\\begin{bmatrix}1&2\\\\0&1\\end{bmatrix}},&&{\\mbox{SNF}}(xI-A)={\\begin{bmatrix}1&0\\\\0&(x-1)^{2}\\end{bmatrix}}\\\\B&{}={\\begin{bmatrix}3&-4\\\\1&-1\\end{bmatrix}},&&{\\mbox{SNF}}(xI-B)={\\begin{bmatrix}1&0\\\\0&(x-1)^{2}\\end{bmatrix}}\\\\C&{}={\\begin{bmatrix}1&0\\\\1&2\\end{bmatrix}},&&{\\mbox{SNF}}(xI-C)={\\begin{bmatrix}1&0\\\\0&(x-1)(x-2)\\end{bmatrix}}.\\end{aligned}}", "103558daf835ebcf075ebdad1bf46046": "\\delta ^{(s+1)}=1-{\\frac {\\left(k^{(s)}/x^{(s)}+1\\right)\\delta ^{(s)}}{1+\\delta ^{(s)}k^{(s)}\\left(1+x^{(s)}+1/x^{(s)}\\right)}},", "10357eca6dae03ef9444d6959d569d68": "(\\mathbb {Z} /p^{k}\\mathbb {Z} )^{\\times }\\cong \\mathrm {C} _{p^{k-1}(p-1)}\\cong \\mathrm {C} _{\\varphi (p^{k})}.", "103581bf307c91c65d905f3713cd4c3a": "a^{\\dagger }|\\alpha \\rangle =\\left({\\partial \\over \\partial \\alpha }+{\\alpha ^{*} \\over 2}\\right)|\\alpha \\rangle ", "1035f984542a0935bb1b52a2107c2b55": "{\\mathcal {D}}\\phi ", "1036107b80b3c20e8a4b5d0f3d435f3b": "\\psi (y_{1},y_{2})=A\\!\\int _{-\\infty }^{\\infty }dpe^{-p^{2}/4\\sigma ^{2}}e^{-ipy_{2}/\\hbar }e^{ipy_{1}/\\hbar }\\exp[-{(y_{1}+y_{2})^{2}/16\\Omega ^{2}}]", "1036c54502c98d1a6544ef96cbb3931d": "N^{c}H^{k}(X,\\mathbf {Z} )=H^{k}(X,\\mathbf {Z} )\\cap (H^{k-c,c}(X)\\oplus \\cdots \\oplus H^{c,k-c}(X)).", "10372075e5d94832483f4b3ba8003d64": "n=\\sum _{i}n_{i}", "10373f982f06c0fde45572daabc28069": "\\textstyle 1/e", "10375bb6626b2dbfa8d09354e45c8f00": "(0,1/4)", "103765e308cd130adf02b875f87339fa": "V\\otimes V\\otimes V^{\\otimes N}", "1037a5eaa16f867b3ec4d2e3719c6335": "\\Delta \\left(\\omega /2\\right)={\\frac {8}{3\\rho _{\\mathrm {F} }Z_{q}}}f_{f}^{\\,4}m_{\\mathrm {F} }^{3}n^{3}\\pi ^{2}J^{\\prime \\prime }", "1037b761864b6ad19d60a53dc6e183ff": "{\\frac {1}{q}}={\\frac {1}{R}}-j{\\frac {\\lambda }{\\pi w^{2}}}", "1037d8a52b9d26cb6583d4814eee6a22": "{n \\choose k}={\\frac {n!}{k!(n-k)!}}.", "10383cf58902cd2c31693b239314bb31": "\\displaystyle {v_{1}=L_{-1}^{\\prime }v_{0}.}", "103849b2bf6419f9c09671c422720e16": "f:G\\rightarrow S", "10386b156192707a11884ed2e3a823d2": "x(t)={\\frac {\\phi (t)}{2\\pi v_{n}}}={\\frac {\\Phi (t)}{2\\pi v_{n}}}-t=T(t)-t", "10387d7085857102d3ae38d479874e0e": "{\\alpha _{j} \\over \\alpha _{j}-\\alpha _{k}}", "10388960b06dfe7a831efe3b625f0f6a": "u_{y}(\\mathbf {y} )\\triangleq u_{1}(\\mathbf {x} ,z_{1})", "1038c2d84bf2d2e047578d70cc42f289": "a,r>0", "103900171be17cef5009421ac3d8abb4": "h^{\\alpha \\beta ab(x-y)}=h^{0}=1", "103943d5a8e71e1bddd99215f5bce206": "\\|\\cdot \\|_{\\alpha }", "10395826925c4ee44ccbfd4828f72efe": "\\left(2+{\\sqrt {-6}}\\right)^{6}=\\left(-2+4{\\sqrt {-6}}\\right)\\left(-1-3{\\sqrt {-6}}\\right)=9+2{\\sqrt {-6}}.", "10399bd6f02aa8ee0c26c9fd85d27fcd": "E_{10}", "10399d6467fff318f95995da28a7aee0": "{\\frac {p(y|H2)}{p(y|H1)}}", "1039d4ecfce0a267b94a159b0c756f9f": "z\\in \\mathbb {C} ^{\\times }", "1039d86887ddb36a4b4cbe8332e3f05e": "f=\\sum _{e}c_{e}T^{e}\\,", "103a0423a08cf4835cbdc9ad3990a6bd": "s^{*}", "103a3afea292a4152356bb956807cf5c": "M=i\\lambda \\,", "103a6537463f3fc109fb69744b35a922": "2^{-a}\\theta _{-a}(x/2)", "103a9b591a0c13bf963ee47217364bb9": "A=\\{x\\in X|(\\exists y\\in Y)\\langle x,y\\rangle \\in B\\}.", "103af78d2723cbc7390feca98a2f6627": "S_{np_{i+1}}=F_{p_{i+1}}[S_{n}\\cup S_{n}+n\\cup S_{n}+2n\\cup ...\\cup S_{n}+n(p_{i+1}-1)]", "103b219e350cd93c0481cbe2e34c9702": "\\Delta =-128p^{2}r^{4}+3125s^{4}-72p^{4}qrs+560p^{2}qr^{2}s+16p^{4}r^{3}+256r^{5}+108p^{5}s^{2}", "103b54a6c3552438e651ae24c1e05e15": "{\\frac {dN}{dt}}+\\lambda N=0.", "103bc12bcc604e062151c53eb4c4bfcc": "P>p_{N}", "103be57d2dd0dc101fdc0f1680a43ca9": "{\\bar {H_{i}}}={\\bar {U_{i}}}+P{\\bar {V_{i}}},", "103be638391ff903d89fd4f0cf206271": "h_{n-1}", "103c1fa60eb3dd7a2fd25df702db837e": "S_{\\rm {metal}}={\\frac {\\pi ^{2}k^{2}T}{-3e}}{\\frac {c'(\\mu )}{c(\\mu )}}+O[(kT)^{3}],\\quad \\sigma _{\\rm {metal}}=c(\\mu )+O[(kT)^{2}].", "103c4d1d4b12b42fe38ce7c10f3d1c23": "h_{A}(X)=\\operatorname {Hom} _{\\mathcal {A}}(A,X)", "103c555bb111c3ae1160080c6cbf924b": "\\Leftrightarrow {\\frac {y}{x}}\\cdot {\\frac {2y}{2x+3c}}=-1", "103c8952f01a7876f26a9490bb62ab3b": "\\scriptstyle Z'", "103cbc44576c4035a2df6850bb50ad60": "T({\\text{deg C}})=16.9-4.0\\times \\mathrm {\\delta ^{18}O_{calcite}} -\\mathrm {\\delta ^{18}O_{seawater}} ", "103d195c206db9ddeeee72242a84807b": "{\\boldsymbol {r}}=({\\boldsymbol {q}},{\\boldsymbol {p}})", "103d7a0c324401169a814fe6d5043396": "a_{1}=0.93\\,", "103daec9259d6cc712f853b6c5dde6fa": "^{\\;}f(\\xi )", "103dedc904b031e2459e88d1bd093bcb": "{\\frac {1}{f}}={\\frac {1}{f_{1}}}+{\\frac {1}{f_{2}}}.", "103e0caf979e6731177b723f8edd652c": "-\\infty T_{i-1,j}", "10456541e2e4199768f8dfbbd16ad328": "T([a_{1},a_{2},\\dots ])=[a_{2},a_{3},\\dots ].\\,", "1045703f180d4f164c014b7c8e7ee923": "\\operatorname {aff} (S)", "10466eac46ef87b0c7eaa7a937a50715": "R_{sens}", "104711545dac6c139387e42e0e161892": "|Df(x)|^{2}\\leq K(x)|J(x,f)|\\,", "104721890407035b7824da6b9116dc52": "\\left\\lceil {\\frac {-1}{\\log _{2}(1-p)}}\\right\\rceil \\!-1", "104748d879f2ce4e1e2766b081636757": "t'=C_{n}=t", "10482210c26ef1b1e7ccc8d8c810d803": "b={\\frac {2}{3}}{\\sqrt {-m_{b}^{2}+2m_{a}^{2}+2m_{c}^{2}}}={\\sqrt {2(a^{2}+c^{2})-4m_{b}^{2}}}={\\sqrt {{\\frac {a^{2}}{2}}-c^{2}+2m_{a}^{2}}}={\\sqrt {{\\frac {c^{2}}{2}}-a^{2}+2m_{c}^{2}}},", "1048277d6c56a6c550f44662ab2b4432": "\\arccos \\left(-{\\frac {r}{R}}\\right)", "10483b9c8e8f35edc0f3825631e10fe5": "X_{i}=\\alpha Q_{j}+\\beta Q_{k}+...\\qquad i=1,2,...I\\qquad \\qquad \\mathrm {(4)} ", "10486f564aa00a30bbfbee1f12edb357": "{\\begin{aligned}Q(1)&=Q(2)=1,\\\\Q(n)&=Q(n-Q(n-1))+Q(n-Q(n-2)),\\quad n>2.\\end{aligned}}", "1048bed8822c83d9b10093d3e41076fe": "\\Pi _{\\gamma \\gamma }(q^{2})=q^{2}\\Pi _{\\gamma \\gamma }^{\\prime }(0)+...", "1048c41875dddbcb4527ba359df67eb9": "[z,x]\\subseteq B_{\\delta }([x,y]\\cup [y,z]).", "1048ea90cbe90bc3ad306ba3345fcb8c": "x_{1}(t)\\,", "104907595c4752d247a512e898cd0791": "T=\\sum _{i=1}^{r}\\lambda _{i}\\,v_{i}^{\\otimes k}", "10490d946bbff452c7fac9a474ead806": "W_{i}\\,\\!", "1049776b40f2fa8b36a9d445962591fe": "k_{e}=1/(4\\pi \\varepsilon _{0}\\varepsilon )", "1049e1b7b43dfb935fb8aede1557b12e": "S_{1}(t)=S_{0}(t)^{r}", "104a3743c75e1801116e0675561ec38d": "x={\\frac {1}{a}}\\int _{0}^{L'}\\cos s^{2}\\,ds", "104a45b3f83657b3596ad131b439d49c": "C_{0},\\theta ", "104a7100ab2ed8107e49e48cb74b1d08": "n=i", "104a802daeb358d70acf05982a49eb85": "\\mathbb {F} _{p}", "104a844d8b7debdd9f5c268b8b6c98d9": "=\\int _{0}^{\\infty }G(\\tau )F(t-\\tau )\\,d\\tau ", "104a9c2259ad58975903cd20b822dd18": "g^{\\alpha \\beta }P_{\\alpha }P_{\\beta }+(mc)^{2}=0\\,,", "104af5c0b0bd341f2537166f7f4b64fd": "c_{2}=hc/k\\,", "104b150315cb3f0186f5f7a7890278af": "z=-1", "104b68522344cd182f7e6bf48597f8b4": "U(r)={\\frac {1}{4\\pi }}\\int _{S}\\left[U{\\frac {\\partial }{\\partial n}}\\left({\\frac {e^{iks}}{s}}\\right)-{\\frac {e^{iks}}{s}}{\\frac {\\partial U}{\\partial n}}\\right]dS", "104b6ce5d83caf9db146524664c3bff4": "4:3=12:9", "104b9e820bba415dd2bd69440404dfc1": "Distance>\\left({\\frac {C}{2\\times PRF}}\\right)", "104ba9eca7bf3510ed30e77d6341954a": "\\sum _{\\boldsymbol {y\\in {\\mathcal {N}}(x)}}w_{\\boldsymbol {xy}}=1", "104bfa2b279ec591d6df5e56e43b7fa6": "\\gamma =", "104c7143d90880d3190db00b6bb322c1": "\\neg B\\wedge \\neg C", "104c7ef64fbb8aef5a90561b37c2b0bb": "r=|x_{1}|", "104cd82faa54ffdf7e340acdcf2a122f": "n=N_{D}", "104ce8c59cd154603c1a795b5eb17cc2": "\\mu (\\cdot ,\\omega )", "104cf66a1e9b97dece25ef0fa9192bef": "grE=\\bigoplus _{i}E_{i}/E_{i-1}", "104d462d2f0571247bf1c2f54241b56a": "{S^{k}}_{h}", "104da3274bc566f1a6a79f84204e4100": "s={\\frac {u^{2}}{2g}}.", "104db75fd9c0c1d13834f8a53c765713": "v'=2t", "104e44d1d9ecfb3b95438a9dffe43cec": "{\\mathit {l}}", "104e8538e2be49fb06026350e29bab47": "1\\mathrm {RPM} =\\pi /30", "104e9831686cbd2236305106009ebc0c": "{\\mathcal {O}}_{X}", "104eabba4fcc513e8a7171fe02fa056b": "{\\vec {W}}={\\vec {V}}-{\\vec {U}}", "104eb81fef5e228c6b8a202b62e6f27a": "\\sigma (X)=\\sum _{i", "104f6adbd6ba7b01d4e5871d8b2be687": "X\\in {\\mathcal {A}}\\rightarrow I_{\\mathcal {A}}(X)=1,X\\not \\in {\\mathcal {A}}\\rightarrow I_{\\mathcal {A}}(X)=0", "104f6aea21ae8e217713eaba35450aec": "f(-x)=-(x-1)^{2}(x+1),\\,", "1050049992e5e1da727e42b203f22f5c": "\\left|2,H\\right\\rangle ", "105031f299eba2a0412ef65a31a7bb06": "d(m,n)=n-m", "105034c0fa69d87cd61fe93e6fedfeb7": "r={\\frac {2mcv}{eH}}", "1050657ca01e3ef0856917571ac4bed1": "P=e^{\\frac {\\Omega +\\mu _{1}N_{1}+\\mu _{2}N_{2}+\\ldots +\\mu _{s}N_{s}-E}{kT}},", "10506f3ec86b0c69d40e251d78f4ab6c": "\\cos \\,{\\theta ^{*}}=r\\cos \\,{\\theta }", "1050f432eb116ed842c1036ca0f40bf2": "|F(z)|\\leq 1", "1050fb8ed5cf4421503064a55e9ffbf0": "H=-t\\sum _{\\langle ij\\rangle }c_{i\\sigma }^{\\dagger }c_{j\\sigma }+{\\text{h.c.}}+U\\sum _{i}n_{i\\uparrow }n_{i\\downarrow }", "10510a06c145c567f25257c49e030903": "K(r,r^{\\prime })", "10513679dc5acb961011d3d9a49bacab": "Y_{6}^{-2}(\\theta ,\\varphi )={1 \\over 64}{\\sqrt {1365 \\over \\pi }}\\cdot e^{-2i\\varphi }\\cdot \\sin ^{2}\\theta \\cdot (33\\cos ^{4}\\theta -18\\cos ^{2}\\theta +1)", "10513952894e4f9a943efe7059a3974a": "\\operatorname {E} (X_{i})=np_{i}.\\,", "105156c37f38929fbdef4951b39c6507": "B_{n}(G,T)=\\{x\\in G|x=a_{1}\\cdot a_{2}\\cdots a_{k}{\\mbox{ where }}a_{i}\\in T{\\mbox{ and }}k\\leq n\\}.", "105194c083e82084928df656ac2ae614": "\\operatorname {E} (X)=\\operatorname {E} _{Y}(\\operatorname {E} _{X\\mid Y}(X\\mid Y)),", "1051a9374ef548795bc25032fd225135": "\\lim _{t\\to \\infty }{\\frac {X_{t}}{t}}={\\frac {1}{\\mathbb {E} [S_{1}]}}.", "1051b92c06d017dc0372c525c8f7d8b3": "\\displaystyle {{1 \\over p}+{1 \\over q}=1.}", "1051e8b5104bec04e6c0393852efb4b6": "V=\\pi \\int _{a}^{b}\\vert f^{2}(x)-g^{2}(x)\\vert \\,dx", "1051f0ed74981811fc5f45a5e888de3e": "H_{\\mathrm {dR} }^{\\ast }(X/K)", "1051f481dcda92f5cda1c0ec389f3439": "n_{j}(x):=\\prod _{i=0}^{j-1}(x-x_{i})", "105205252c4ef67036974ff1d4edd43a": "{\\begin{aligned}\\mathbf {E} &={\\frac {1}{2}}\\left(\\mathbf {F} ^{T}\\mathbf {F} -\\mathbf {I} \\right)\\\\&={\\frac {1}{2}}\\left[\\left\\{(\\nabla _{\\mathbf {X} }\\mathbf {u} )^{T}+\\mathbf {I} \\right\\}\\left(\\nabla _{\\mathbf {X} }\\mathbf {u} +\\mathbf {I} \\right)-\\mathbf {I} \\right]\\\\&={\\frac {1}{2}}\\left[(\\nabla _{\\mathbf {X} }\\mathbf {u} )^{T}+\\nabla _{\\mathbf {X} }\\mathbf {u} +(\\nabla _{\\mathbf {X} }\\mathbf {u} )^{T}\\cdot \\nabla _{\\mathbf {X} }\\mathbf {u} \\right]\\\\\\end{aligned}}\\,\\!", "1052174039620e7fde413d924d00add2": "-623\\pm 4.1\\%", "10524aead52f40bf34e46d4acaccfc90": "\\left(\\rho _{t}\\right)_{t=0}^{T}", "10526ecf0692d922c5664a175092df1d": "\\displaystyle {\\|\\partial ^{\\alpha }F_{m}\\|_{\\infty }\\leq 2^{-m}}", "1052dedc4fe612121562cf309599dc8a": "f^{\\prime }\\colon M_{X}\\to N.", "1053297cfec39507c150a031d9807c5c": "d_{Y}(d_{X}\\Delta )=-d_{Y}(h(X,Z)).", "10533ef211fc97c64582c0c197c61ea7": "v_{0}=f(x,y)", "10536c37f23c49d03626478ec55a5176": "\\log(m)=\\log(a)+b\\log(-\\log(p_{0}))", "1053775fa2bb5d21ae8e025d4bedb4c9": "r_{0}\\in E(T_{1}(q_{1},\\epsilon ))\\Rightarrow r_{0}\\in E(T(q_{1},\\epsilon ))", "10539eafc7cde74d6ea0425f0f9d6d93": "z(x,y)=\\operatorname {Re} (x+iy)^{3}.", "1053a176d2a1b179bcf6f9ed52a9d2d5": "F_{n}(a,1)", "1053bcf811bd354caf3d733292e863b5": "d\\theta =7.2923\\times 10^{-5}\\,dt", "10540fc9ca730bb540932da6c5ebcecc": "\\mathbf {k} ={\\frac {4}{3}}({\\sqrt {2}}-1)\\approx 0.5522847498", "105493c7bb9503fc8661a0d6fe1a6568": "H\\,=\\,{\\frac {1}{2}}\\,\\rho \\,\\varphi \\,{\\Bigl [}w\\,\\left(1\\,+\\,\\left|{\\boldsymbol {\\nabla }}\\eta \\right|^{2}\\right)-\\,{\\boldsymbol {\\nabla }}\\eta \\cdot {\\boldsymbol {\\nabla }}\\,\\varphi {\\Bigr ]}\\,+\\,{\\frac {1}{2}}\\,\\rho \\,g\\,\\eta ^{2},", "1054fd36e3154250341c186305517c11": "dU=TdS-PdV+\\sum _{i}\\mu _{i}dn_{i},\\,", "1055954120a6b9502509223a782e0d3f": "L=\\partial Q/\\partial m\\,\\!", "10559a162fc88f61dfdd5eaa334ad70c": "\\lambda :S\\rightarrow Y^{\\phi }", "105603623a8013e2227cd34c0c85f97e": "\\sigma _{ij}=-p\\delta _{ij}.\\,", "10561afac9e2e03fc7e1c71fefd35e3d": "V_{1}\\geq V_{2}\\geq V_{3}>0", "10567a9251d9c6e06845a2d4b72c3db9": "\\lambda ^{2}(x)", "1056c695268c17e80ac95c07c4f66fd6": "m_{2}\\parallel m_{1}", "1056d96fe69e8d0e01edca1be931984a": "RPF={\\frac {\\text{effective RPF}}{\\text{extraction ratio}}}", "1057341a46d972c82ed2b4f710a0de35": "\\sigma ^{\\mu \\nu }=(i/2)[\\gamma ^{\\mu },\\gamma ^{\\nu }]", "10574254a52ea6c81af7e490c6526ab5": "n=N_{c}{\\frac {e^{-(E_{c}-E_{Fn})}}{k_{B}T}}", "1057d99dae7444eca0cdb00c7c919bae": "T_{\\alpha \\beta }{}^{\\lambda }", "1057e69f9d8a63a02d170a18dd07cab0": "{\\frac {1}{2}}m_{\\mathrm {i} }\\,u(x)^{2}={\\frac {1}{2}}m_{\\mathrm {i} }\\,u_{0}^{2}-e\\,\\varphi (x)", "105816d30833811fbe2f31ba7d322893": "H^{2}(z)=H_{0}^{2}\\left(\\Omega _{M}(1+z)^{3}+\\Omega _{k}(1+z)^{2}+\\Omega _{\\Lambda }\\right).", "10587446e71952ff71043b9e01285df8": "EIw_{\\mathrm {max} }={\\cfrac {1}{3}}\\left[{\\dfrac {Pb(L^{2}-b^{2})^{3/2}}{6{\\sqrt {3}}L}}\\right]-{\\cfrac {Pb(L^{2}-b^{2})^{3/2}}{6{\\sqrt {3}}L}}", "10587a0bf20b915f5b40657c0d14c351": "{f_{k}}^{m_{k}}", "1058f207c5827901aacfffbddc133749": "s(mt)=(sm)t\\,\\forall s\\in S,r\\in R,m\\in M", "10590a2a921f3cfb3c78b19c8991567e": "K^{\\ominus }=\\mathrm {{\\frac {[ML]}{[M][L]}}\\times {\\frac {\\gamma _{ML}}{\\gamma _{M}\\gamma _{L}}}=\\mathrm {\\frac {[ML]}{[M][L]}} \\times \\Gamma } ", "10597a859eb8f063b84dcf46a78a340e": "\\pi /4\\approx 79\\%", "1059892d1a83f26026494bad45230314": "\\|\\mu \\|_{ba}=\\sup _{A\\in \\Sigma }|\\mu |(A)", "105a431e5e7208ab9a95885e4c1a6d00": "(-ea,0)", "105a583f856c240861fdbce6a08ef950": "\\mu _{3}=\\mu '_{3}-3\\mu \\mu '_{2}+2\\mu ^{3}\\,", "105a61b297ec79d4df9bf0eda8711ce3": "L(s,E)=\\sum _{n=1}^{\\infty }{\\frac {a_{n}}{n^{s}}}.", "105a93ec9598b442a6dbcac9a9c51afd": "f:A\\otimes C\\to B\\otimes C", "105ac573a342bb2a73b39ce5a67ecfc7": "s={v+u \\over 1+(vu/c^{2})}.", "105ad981a3a494ccfd19f083e3c39181": "s=(p_{1}+p_{2})^{2}=p_{1}^{2}+p_{2}^{2}+2p_{1}\\cdot p_{2}\\approx 2p_{1}\\cdot p_{2}\\,", "105b57eb54d52c16693790affb42b3f1": "a^{2^{\\overset {n}{}}}+1", "105b7dc1803bed57779c3fb7113de4f0": "\\displaystyle 2", "105be3a9948bc0c843cfd71f9cee62c8": "a^{2}+b^{4}", "105bfa8f7a7c8a5c0517811f065d225e": "{\\frac {q_{i}}{p_{i}}}=1", "105c380923405a73fd556a84e71ccc4a": "\\|x\\|^{2}=\\sum _{y\\in A}|\\langle x,y\\rangle |^{2}.", "105c81bdc324722ea6882508c2f161c9": "N=\\sum _{i=0}^{n-1}10^{i}{d_{i}},", "105cd58796c60bffd5c93d411ffec2ee": "dU=\\varepsilon \\sigma dVT^{4}+4\\varepsilon \\sigma VT^{3}dT", "105d095fea9b5dc74027a60ea59480a8": "\\Theta \\subseteq \\mathbb {R} ^{d}", "105d09df013b357a1adaf9cace814542": "\\operatorname {hypg} (x_{c};x_{c-1}+x_{c},m_{c},m_{c-1}+m_{c})\\,,", "105d335879c4dd1c2b8d6804123c75a1": "{\\bigg .}J=-{\\frac {P({\\sqrt {p_{1}}}-{\\sqrt {p_{2}}})}{\\delta }}{\\bigg .}", "105d564a2e246d83b13c6c60da5be00f": "\\lambda _{dB}", "105d791c39f39768722d45fc8de26331": "{\\begin{array}{c||c}{\\textit {Worst-Case\\ Pessimism}}&{\\textit {Best-Case\\ Optimism}}\\\\\\hline Maximin\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ Minimax&Minimin\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ Maximax\\\\\\displaystyle \\max _{d\\in D}\\,\\min _{s\\in S(d)}\\,g(d,s)\\ \\ \\ \\displaystyle \\min _{d\\in D}\\,\\max _{s\\in S(d)}\\,g(d,s)&\\displaystyle \\min _{d\\in D}\\,\\min _{s\\in S(d)}\\,g(d,s)\\ \\ \\ \\displaystyle \\max _{d\\in D}\\,\\max _{s\\in S(d)}\\,g(d,s)\\end{array}}", "105dbffea4bae06bb72aa655a9fd85b9": "\\left[\\mathbf {N} \\right]:\\left(\\mathbb {Z} _{2}\\right)^{2n}\\rightarrow \\left[\\Pi ^{n}\\right]", "105df97cb2b28ce762603e65e7014891": "A(a,\\theta )=0.\\,", "105e0fd711cbf7051fc7c5b074e03dd9": "\\phi \\leftrightarrow \\psi \\in T,", "105e6cc625ee9a47b3de284e18ff0d4d": "|f_{n}(z)|\\leq f_{n}^{\\prime }(0){|z| \\over (1-|z|)^{2}}.", "105e9ffddb82970beeb9758fefa8d67c": "\\mathbf {x} ^{(k+1)}=D^{-1}(\\mathbf {b} -R\\mathbf {x} ^{(k)}).", "105eff5d63f81d987ca1ef3ab59e20e4": "\\alpha >0", "105f04ba9b57f6e5bc4384d8e2c3c60d": "\\sum _{n=1}^{\\infty }\\|v_{n}\\|_{X}<\\infty \\quad {\\text{implies that}}\\quad \\sum _{n=1}^{\\infty }v_{n}\\ \\ {\\text{converges in}}\\ \\ X.", "105f691465c8509b4b0c9ae338874cf8": "u=({\\frac {t}{2\\pi }})^{1/4}", "105fd69314e27aad4330fb666aa8d63e": "\\mathbf {J} _{\\rm {final}}=\\mathbf {S} ,", "1060909feeb5b9fd8aa58acdbc585d98": "\\langle \\Delta {\\hat {a}}_{1}^{2}(t)\\rangle \\langle \\Delta {\\hat {a}}_{2}^{2}(t)\\rangle \\geq {\\frac {1}{16}}\\ ", "10609172afe82afd4f6d2430801db223": "N={\\frac {f}{D}}\\ ", "1060940f9bc195152586c815b695e1c0": "g\\cdot e=q\\in Q", "1060a00607fcbcc7c5e358c396c1dad9": "\\mathbf {K} =l_{1}\\mathbf {g} _{1}+l_{2}\\mathbf {g} _{2}+l_{3}\\mathbf {g} _{3}", "1060a146fb2ebe00e174f0df8ee07b0c": "T_{i_{1}},T_{i_{2}},\\dots ,T_{i_{t}}", "1060a8639a8eaf6a0b95b562be31054e": "\\mathrm {j} _{\\pm }|j\\,m\\rangle =\\hbar C_{\\pm }(j,m)|j\\,m\\pm 1\\rangle ", "1060b3bb974263a2607cc128310f149c": "M(X)=\\left({\\begin{array}{*{20}c}\\mu \\\\0\\\\\\end{array}}\\right)", "10612beb6fc538fc018be844eeb0d072": "a\\uparrow \\uparrow b", "1061c7b1f36aa2444bc0e18691b608a8": "\\operatorname {Vol} (B_{p_{1},\\ldots ,p_{n}})=2^{n}{\\frac {\\Gamma (1+p_{1}^{-1})\\cdots \\Gamma (1+p_{n}^{-1})}{\\Gamma (1+p_{1}^{-1}+\\cdots +p_{n}^{-1})}}.", "1061e17fcbcc8a225fe764cee9e30517": "\\{\\alpha ,\\beta ,[\\alpha ,\\beta ]\\}", "1061e659c9b89c1363a2da84727acae0": "\\mathop {Br} (L/K)\\equiv K^{*}/\\mathop {N} _{L/K}L^{*}\\equiv \\mathop {H} ^{2}(G,L^{*}).", "10620363f64e24c3f8939194ddb2b8ca": "{1 \\over 2}=0.5", "10627075f5bf992d67536050b968ef46": "{\\frac {3\\ \\mathrm {hours} }{90\\ \\mathrm {miles} }}={\\frac {7\\ \\mathrm {hours} }{x\\ \\mathrm {miles} }}\\quad ", "1062845af68f1fc14fa25d718acadd1c": "\\xi =\\alpha ^{-1}>0", "106293382fb634df421dcec22e2df409": "P(t)=\\sum _{i\\geq 0}t^{i}{\\text{Sq}}^{i}", "1062f6a24adfea22683e3b1166ef0138": "{\\overline {2m-n}}\\approx {\\sqrt {\\frac {2n}{\\pi }}},", "1063093cbd65d421dd4883ac9cb5605e": "\\int f(x)d\\alpha (x)", "10635b6b468ac98f48df3506226b35fb": "\\int _{k}^{k+1}f(x)\\,dx=\\int u\\,dv", "10641b531d42332b827f9f555e148999": "A={\\begin{cases}d\\gamma {\\mbox{ , frequency}}<3GHz\\\\R_{f}d\\;+\\;k[1-e^{(R_{f}-R_{i}){\\frac {d}{k}}}]{\\mbox{ , frequency}}>5GHz\\end{cases}}", "106493900bc7a57f4ce726dc7cc3afbd": "\\mathbf {x} ,\\mathbf {u} ", "1064b869b64a3a85c41ceacac520bcd2": "X_{2}=\\{2,4\\}", "1064bf99872d91f6bb96808c3c72d9a9": "n_{1},...,n_{N}", "106507a26fe9e5247042ebc0473ecd96": "R_{n+1}(x)={\\frac {2n+1}{n+1}}\\,{\\frac {x-1}{x+1}}\\,R_{n}(x)-{\\frac {n}{n+1}}\\,R_{n-1}(x)\\quad \\mathrm {for\\,n\\geq 1} ", "10652d4d853204be81405853512bef73": "\\ \\ O", "106559128efc97eaaf5bd9d4e29ca27c": "\\alpha _{k,i}={\\frac {x-u_{i}}{u_{i+n+1-k}-u_{i}}}.", "1065bba4fd73439743384ba9757c756b": "F^{T}(T,r)=1", "1066ab3520d801e5e652b5141851896f": "var(X)", "1067536fbc266714eb70a7abacdbb3bc": "R=\\{(a_{1},\\ldots ,a_{n})\\in M^{n}:{\\mathcal {M}}\\vDash \\phi (a_{1},\\ldots ,a_{n})\\}", "106757ca4d0cefe808860134c91ba89c": "f^{(n)}(x)=h^{(-1)}(g^{(n)}(h(x)))\\,\\!", "10677f4164982825ebd37ae9b9c1fd84": "q_{p}(a)={\\frac {a^{p-1}-1}{p}}.", "1067e4670b029a04916f94026f2b229c": "\\mu _{k}\\equiv E[W^{k}]=g_{k}(\\theta _{1},\\theta _{2},\\dots ,\\theta _{k}).", "106868bb6b6812fb6906433de421aebb": "\\mathbf {d} ^{T}\\mathbf {y} +\\beta t-1=0", "106886c83a135af29eedc43e92d86d97": "\\{{\\mathcal {F}}_{t}|0\\leq t\\leq T\\}", "1068eca0030d9eb179874297e9509db2": "\\int d\\mathbf {r} _{1}d\\mathbf {r} _{2}h(r_{12})e^{i\\mathbf {k\\cdot r_{12}} }=\\int d\\mathbf {r} _{1}d\\mathbf {r} _{2}c(r_{12})e^{i\\mathbf {k\\cdot r_{12}} }+\\rho \\int d\\mathbf {r} _{1}d\\mathbf {r} _{2}d\\mathbf {r} _{3}c(r_{13})e^{i\\mathbf {k\\cdot r_{12}} }h(r_{23}).\\,", "106920849742cae40711297a15e4813d": "\\operatorname {de-let} [\\operatorname {let} p:p\\ f\\ x=f\\ (x\\ x)\\operatorname {in} \\operatorname {let} q:q\\ p\\ f=(p\\ f)\\ (p\\ f)\\operatorname {in} q\\ p]", "106924f892eb6ed989f692796d450f51": "{\\begin{aligned}{\\frac {d}{dx}}\\left[{\\frac {(4x-2)}{x^{2}+1}}\\right]&={\\frac {(4)(x^{2}+1)-(4x-2)(2x)}{(x^{2}+1)^{2}}}\\\\&={\\frac {(4x^{2}+4)-(8x^{2}-4x)}{(x^{2}+1)^{2}}}={\\frac {-4x^{2}+4x+4}{(x^{2}+1)^{2}}}\\end{aligned}}", "10697fb7a3928e4bb592bd7af5615318": "||X^{\\alpha }||^{2}={\\frac {|\\alpha |!}{\\alpha !}},", "10698191eebd9da71b7cdbd8216a1f07": "I[f]=\\displaystyle \\int _{X\\otimes Y}V(f({\\vec {x}}),y)p({\\vec {x}},y)d{\\vec {x}}dy", "10699ae1c07c6fec5ee1ef3213c194f2": "\\mathbf {m} _{i}^{\\phi }", "1069cc47df11b079df3edb6d23e3c4a0": "\\;1", "1069fbe44be8eda06b917423d38613e5": "\\ {\\frac {S}{C}}={\\frac {4^{4}R^{2}}{P_{t}G^{2}\\lambda ^{2}}}(180/\\theta ^{o})^{2}{\\frac {1}{\\sigma ^{o}}}{\\frac {P_{t}G^{2}\\lambda ^{2}}{(4\\pi )^{3}R^{4}}}\\sigma ", "106a039b82e803000fca22f2341e2cf5": "i=0\\ldots i_{\\max },j=0\\ldots j_{\\max },t=0\\ldots t_{\\max }", "106a06f6f1a113339c50ed3982032719": "\\scriptstyle c\\eta =\\Psi \\,", "106a0c7baf3eec15560cea013971df8a": "g^{M}=f^{*}g^{N}\\,,", "106a7b104d047613b80a7434b5a6bd27": "{\\frac {\\partial g}{\\partial x}}\\cdot X+{\\frac {\\partial g}{\\partial y}}\\cdot Y+{\\frac {\\partial g}{\\partial z}}\\cdot Z=ng(X,Y,Z)=0.", "106aebe50c16f51c44fd467a2b68402a": "6n^{2}+O(n)", "106af9cc8a7e2e822f577eae0c33fa42": "\\{c_{i},z_{r}\\}", "106b0afa62ca1964e04a1fa532b14bb9": "{\\mathbf {f}}(t)=\\left({\\begin{array}{c}0\\\\0\\end{array}}\\right),\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\quad (12)", "106b3b5f7073e9ac19dc62fd12cd1493": "si\\,\\ b=0", "106b407530e035bbd26af3ee9bd77f73": "{\\hat {c}}_{P}={\\hat {c}}_{V}+1", "106b582092e6ef60750cf2c00c43cbe5": "Y=y_{j}", "106b58c5304459c0559f99a203384afa": "\\{a_{1},a_{2},\\ldots ,a_{n}\\}", "106b70ebd6d052d831002beef1f7bd15": "e\\ =\\ 0\\,", "106c226ad079a63ed3ab090f239b3634": "newvar", "106c2ccab729dc12e57d0a929c184374": "M\\left(t;\\mu _{1},\\mu _{2}\\right)=G(e^{t};\\mu _{1},\\mu _{2})", "106c3e97042d56cf4370396e9ac1834c": "\\eta ^{\\mu \\nu }=\\left({\\begin{array}{cccc}-1&0&0&0\\\\0&1&0&0\\\\0&0&1&0\\\\0&0&0&1\\end{array}}\\right)~.", "106c69553435b01f5e308a6a2f5ba4d4": "X_{u}", "106c8456a4332abef6463aa131812f54": "{\\begin{aligned}y_{t+h}^{1}&=y_{t}+hf\\left(y_{t},t\\right)\\\\y_{t+h}^{2}&=y_{t}+hf\\left(y_{t+h/2}^{1},t+{\\frac {h}{2}}\\right)\\\\y_{t+h}^{3}&=y_{t}+hf\\left(y_{t+h/2}^{2},t+{\\frac {h}{2}}\\right)\\end{aligned}}", "106cae097fafc14b81d04dc5ddc7e80c": "bulgingfactor=SIF(curved)/SIF(flat)", "106d1549f7650c1526733c7fba356da2": "J^{\\mu }=\\left(c\\rho ,\\mathbf {j} \\right)", "106d77d909081cd0d762dfd51d3e069c": "J_{\\alpha +1}\\cap {\\text{Pow}}(J_{\\alpha })={\\text{Def}}(J_{\\alpha }),", "106dcef9b0ca1bd9a1add24ec797b01e": "\\Omega ^{1}", "106de3f7eb065e89e2dd8b93caa3b18b": "A(\\eta )=A+\\eta B", "106e04faf922d88c3d5b205ed14d2c70": "\\log \\,f=u", "106e17b4c00b964366ca3cc1a8690bba": "\\int _{\\mathbb {R} ^{n}}|\\nabla u^{*}|^{p}\\,d{\\mathcal {H}}^{n}\\leq \\int _{\\mathbb {R} ^{n}}|\\nabla u|^{p}\\,d{\\mathcal {H}}^{n},", "106e4a9917d6719374df3e2eb1a00a81": "\\mu ^{*}", "106e7d1e8966ebd1d549b2934a58ec3d": "r_{\\mathrm {O} }={\\frac {v_{\\mathrm {ds} }}{i_{\\mathrm {d} }}}{\\Bigg |}_{v_{\\mathrm {gs} }=0}", "106ea042a96b6ed470419b87b07ba73d": "O(\\min(m,n))", "106eda0fe8058b452c725b41a207ce86": "f(x)=\\langle A\\varphi (x),\\varphi (x)\\rangle ", "106f13b1ceb9dc898019ee56397ecf33": "\\rho ={\\frac {R_{\\mathrm {thresh} }}{R_{\\mathrm {rms} }}}.", "106f6e95c824ca8a420b97ce357f03bf": "\\nabla \\cdot \\mathbf {D} =\\rho ,\\,", "106f7b2626462a1240e3cc721e7d784c": "A=(T,Con,\\vdash )", "106fbabdc5e1b39ac8dd530312b0b04d": "S[{\\text{Bq/g}}]\\simeq {\\frac {4.17\\times 10^{23}}{T_{1/2}[s]\\times m}}", "106fbbc9a6e399c079525a7f16ef274d": "\\theta _{n}=n\\pi /N", "107058f4765614d00722900ef06d4b58": "A_{\\mu }\\ ", "1070831b69831844c8342c0c7406accd": "\\scriptstyle f:X\\rightarrow Y", "1070e4b7755be48ca68357f6aae76639": "f_{LO}", "1070e805fbcb793a96b1fb2d34d04c67": "{\\frac {\\delta F[\\rho (x)]}{\\delta \\rho (y)}}=\\lim _{\\varepsilon \\to 0}{\\frac {F[\\rho (x)+\\varepsilon \\delta (x-y)]-F[\\rho (x)]}{\\varepsilon }}.", "10710f5febe38ba2da2c436922871706": "FiB_{y}=log((Y_{y}/(TE)^{TL_{y}})/(Y_{0}/(TE)^{TL_{0}}))", "107294afd7a8960ca16629bbfb9daf29": "c_{1}^{2}", "1072971e5b20f51ee61526f67421d422": "\\left(p_{R}+{\\frac {3}{v_{R}^{2}}}\\right)(v_{R}-1/3)=(8/3)T_{R}", "1072e5433390314a628706b6c54d5e03": "D\\varphi _{x},\\;(\\varphi _{*})_{x},\\;\\varphi '(x).", "1072fc17b02aa18abd77f940b9882d69": "{\\frac {1}{4}}+{\\frac {1}{4}}={\\frac {1}{2}}.", "1073454e636af61eb8b1104d87901338": "r=f_{2}(\\theta )-f_{1}(\\theta )", "10735efd8ff2ac607289435c23d52148": "{\\begin{alignedat}{2}\\left(\\cos x+i\\sin x\\right)^{k+1}&=\\left(\\cos x+i\\sin x\\right)^{k}\\left(\\cos x+i\\sin x\\right)\\\\&=\\left[\\cos \\left(kx\\right)+i\\sin \\left(kx\\right)\\right]\\left(\\cos x+i\\sin x\\right)&&\\qquad {\\text{by the induction hypothesis}}\\\\&=\\cos \\left(kx\\right)\\cos x-\\sin \\left(kx\\right)\\sin x+i\\left[\\cos \\left(kx\\right)\\sin x+\\sin \\left(kx\\right)\\cos x\\right]\\\\&=\\cos \\left[\\left(k+1\\right)x\\right]+i\\sin \\left[\\left(k+1\\right)x\\right]&&\\qquad {\\text{by the trigonometric identities}}\\end{alignedat}}", "10737fb13bacd501f750df1ed63ea912": "{\\frac {}{}}-I\\,\\delta ", "1073878c7558aa393950b0f77b35c72c": "\\Re (z)>2", "1073a309599df6c0ae8a0db6caac31dd": "H^{'}|\\psi _{-}\\rangle =E_{-}|\\psi _{-}\\rangle ", "1073e8b593ec12cab16adcf65aed6d12": "\\textstyle Z_{3}", "1073f96f9d0f122ad8eea35c9f401b06": "1-{\\frac {\\ln(1-(1-p)e^{-\\beta x})}{\\ln p}}", "10742a7031c333375f78905b214665c3": "L_{z}(q,p)=p_{i}", "10744650b75b5cc2c34e541becc7a113": "s_{ln}=s_{b}ln(b)\\,", "10748196696737041ef54f908953b664": "\\cdot :A\\times A\\longrightarrow A", "1074b41fe631c4109245966e4475305a": "g=210=2\\cdot 3\\cdot 5\\cdot 7", "10756ac0abc0a953399aa8f5ad73ad20": "P_{1}+{\\frac {1}{2}}\\cdot \\rho \\cdot V_{1}^{2}=P_{2}+{\\frac {1}{2}}\\cdot \\rho \\cdot V_{2}^{2}", "1075b680183f7db57fcb0ba1fec31650": "(p,q){\\overset {\\alpha }{\\rightarrow }}(p',q)", "107626a21fcebdb7d7478578f8243c83": "\\mathbf {{\\hat {f}}_{0:t}} =c^{-1}\\mathbf {{\\hat {f}}_{0:t-1}} \\mathbf {T} \\mathbf {O_{t}} ", "10765a82dac92461a9861c2f2ba7f61d": "\\displaystyle {Q(Q(a)b)=Q(a)Q(b)Q(a),}", "1076761c92b639e2dcfa04d074be11dd": "\\left({\\sqrt {1/45}},\\ 1/6,\\ {\\sqrt {1/28}},\\ {\\sqrt {1/21}},\\ {\\sqrt {1/15}},\\ {\\sqrt {1/10}},\\ {\\sqrt {1/6}},\\ -2{\\sqrt {1/3}},\\ 0\\right)", "10768a2dbab107d673c70a9836cd2e5c": "y_{n+1}^{(0)}+\\tau _{n+1}^{(0)}=y(t+h)", "1076a0378f4783eaad07687b0b2ace00": "J:X\\to (X'_{\\beta })'_{\\beta }", "1076ec62b59f3573ad57ec0d51a70642": "\\sum _{i=1}^{n}{\\widehat {\\varepsilon }}_{i}x_{i}=0.", "1076ecc0eb1f101be0acc6f812dc6e11": "L[t]=-mc{\\sqrt {-g_{\\alpha \\beta }[x[t]]{\\frac {dx^{\\alpha }[t]}{dt}}{\\frac {dx^{\\beta }[t]}{dt}}}}+q{\\frac {dx^{\\gamma }[t]}{dt}}A_{\\gamma }[x[t]].", "10773b949ef4edfc8d8205ac51e06e93": "P\\in {\\mathcal {C}}", "107749f8872102b27f5d67907c906870": "\\sin {5x}=1\\,", "10785587b0bcf02c1b649966d1b4f36f": "\\sigma _{d}", "10785bd2cabc31918e71c6c7f0aad68b": "\\scriptstyle \\times ", "1078af129de448ce026148e2ef6e9781": "x\\geq 0,y\\leq 0", "1078b711da6ad03ecfd409dd0e3bd60e": "\\mathrm {SNR} ={\\frac {|y_{s}|^{2}}{E\\{|y_{v}|^{2}\\}}}.", "1078f7c5c13fc2a607de3f3bf40aa185": "\\mathbf {S} _{k}", "10790bda39fe0fb4ae54f9c5bdb6f8d4": "a'\\omega r", "10790fb041dcdb02a57c3226416a7e62": "\\scriptstyle n\\;", "1079277ea10c07e866cc2077ea4e389d": "\\sigma =\\sigma _{0}e^{-(T_{0}/T)^{1/4}}", "10796f9f3cd25b66c6c7cecdcf37836c": "X=\\operatorname {spec} A,Y=\\operatorname {spec} B", "107982d4f370747346ed1f3a2149643a": "H^{c}(p(x)\\|m(x))=-\\int p(x)\\log {\\frac {p(x)}{m(x)}}\\,dx,", "1079c61dda342bdb1baabd5946ab6c2c": "\\mu _{r}\\approx 1", "1079f23c89fd9a8e69f16d12de519b02": "\\displaystyle {\\langle f_{1},f_{2}\\rangle =\\int f_{1}f_{2}\\,dx.}", "1079f6506d6071b69e7128fd3b43dac1": "|u(x,t)|={\\frac {1}{(1+4t^{2})^{1/4}}}e^{-{\\frac {(x-k_{0}t)^{2}}{1+4t^{2}}}}", "107a1ab538603d47ea3b978e911b5e45": "E_{rel}", "107a22bc2dab2d4f7cfcc1f317c6b9c8": "\\pi \\colon {\\mathcal {M}}\\to M", "107a617402042c7e265e8157bffe7aac": "\\mu (\\mathrm {B} (\\alpha ,\\beta ))=1-\\mu (\\mathrm {B} (\\beta ,\\alpha ))", "107a7214747351a152be8f899c283787": "(p,0011,Z)\\vdash (q,0011,Z)\\vdash (r,0011,Z)", "107a739c20f7dc172b8faafa964dcf19": "{\\widehat {\\beta }}=(X'X)^{-1}X'y", "107a9b12b61ac742b9e22eefd2073de1": "A(n,m)", "107aa4b91a348e14b6d395655526ab5f": "{\\sqrt {\\frac {3}{10}}}\\!\\,", "107b374ab3faeef4b9fe6095835b2bb5": "E^{2}=\\hbar ^{2}\\omega ^{2}\\quad \\mathrm {and} \\quad p^{2}=\\hbar ^{2}k^{2}={\\frac {\\hbar ^{2}\\omega ^{2}}{c^{2}}}", "107b9515e3671f27063de1b517ee8af5": "X_{k}=\\alpha ^{i_{k}},\\ Y_{k}=e_{i_{k}}", "107bdfd4818aea4b03788e63b4c0b1a2": "\\scriptstyle {\\overline {DA}}", "107c09e61217a7d4463696efda8fb5de": "\\Lambda _{CW}\\!", "107c0c78bd971e91000c8e53b24bd2b6": "{\\overline {X}}={\\frac {1}{n}}\\sum _{i=1}^{n}{X_{i}}", "107c630b9895142e62434b8de1d9a31d": "\\varphi (t)=\\int _{0}^{t}h_{s}\\,ds.", "107c69c252a32f9740a7d559b577bbf5": "c_{1},c_{2},\\ldots ", "107c80dc7288f340848c585e41e48729": "{\\vec {x}}(i)\\approx {\\vec {x}}(j),\\,", "107c8fa0b6ea181f94fc2cebdad05c36": "\\scriptstyle {\\tau }", "107cafd46841c433f11b3fdc07a85f43": "W_{C}:=\\mathrm {Tr} \\,(\\,{\\mathcal {P}}\\exp i\\oint _{C}A_{\\mu }dx^{\\mu }\\,)\\,.", "107d3f2106f84c08b1c6329ed7808264": "\\theta =\\tan \\theta -{\\frac {\\tan ^{3}\\theta }{3}}+{\\frac {\\tan ^{5}\\theta }{5}}-{\\frac {\\tan ^{7}\\theta }{7}}+\\quad \\cdots ", "107d5913c5006af1aa9da9ffc953a5f9": "{\\frac {4}{\\pi }}", "107db30076dfd42abad12bcc4f2df54e": "\\ell =\\log \\log n+\\log k+O(\\log(\\epsilon ^{-1}))", "107dd35529337915d96b93c468c5d2db": "{G^{a}}_{b}\\,{G^{b}}_{a}=R^{2}", "107e0df20ebaa373a96eba1d385b3e62": "\\mathbf {U} ,\\mathbf {V} ", "107e493a70ed552c9bf06445170cb933": "m_{4}=[12.3,7.6]-[0,2.404]=[12.3,5.196]", "107e58a61d09bb2662af482f4854c1a6": "K(x,x';t)={\\frac {1}{2\\pi }}\\int _{-\\infty }^{+\\infty }dk\\,e^{ik(x-x')}e^{-i\\hbar k^{2}t/(2m)}=\\left({\\frac {m}{2\\pi i\\hbar t}}\\right)^{1/2}e^{-m(x-x')^{2}/(2i\\hbar t)}", "107eb092479031ee1b070ecb8cbde8c8": "{\\hat {\\beta }}^{(j)}-{\\hat {\\beta }}=-{\\frac {1}{1-h_{j}}}(X'X)^{-1}x'_{j}{\\hat {\\varepsilon }}_{j}\\,,", "107ed2aa825eaf2a59619ac77900e5a1": "(\\rho uA\\phi )_{e}-(\\rho uA\\phi )_{w}=(rA{\\frac {\\partial \\phi }{\\partial x}})_{e}-(rA{\\frac {\\partial \\phi }{\\partial x}})_{w}", "107f1d148daa2d473899b081282f4ee7": "{\\bigg .}J=-D{\\frac {(C_{2}-C_{1})}{\\delta }}{\\bigg .}", "107f55430a19cb9f2d5fe5eaa850d266": "{A_{cross}}\\,\\!", "107f5c8797d7ab2bb7b974cbfc89179e": "\\mathbb {R} ^{+}", "107fa32631456ce154c9be68882e2950": "\\varepsilon _{\\text{eff}}", "1080166158767d99bf842ea8f58eb3b3": "E_{F}^{n}={\\sqrt {(p_{F}^{n})^{2}c^{2}+m_{n}^{2}c^{4}}}\\,", "108033bcad1a58ce4bff250a191c0759": "d((\\varphi -1)b)=bd\\ln \\gamma ", "10808e732947926e55f234f4743fb6c6": "I(t)=I_{ex}\\sin(w_{ex}t)", "10809472c9f904ea5424db169b4e6c8d": "E_{v}=A_{v}+T_{v}=B_{v}+S_{v}\\,,", "1080ae9dfed0a9a6dd056268b45cb8a9": "y=a\\sinh \\xi \\sin \\eta ,", "1080f14dae21503e351cd5bbf7f950d7": "|a|_{\\ast }^{\\log _{a}b}\\leq 1", "1080f9b5a4252bf0eaf82baeefe52701": "0=\\cap Q_{i}\\Leftrightarrow \\emptyset =\\operatorname {Ass} (\\cap Q_{i})=\\cap \\operatorname {Ass} (Q_{i})", "1081056e04121326e85e3e087c15fb68": "{\\rho _{Out} \\over \\rho _{In}}\\,\\!", "10811bb6fe7f685d31276622b2c40bdd": "T_{\\text{C}}", "1081205f9a79d1f491babb9abb6b6323": "(\\mathbf {E} _{1},{\\hat {O}}\\mathbf {E} _{2})=({\\hat {O}}\\mathbf {E} _{1},\\mathbf {E} _{2})", "10812f90b6269226a99b24d864c3efa1": "[ML']=K{\\frac {[ML][L']}{[L]}}=K{\\frac {\\beta _{ML}[M][L][L']}{[L]}}=K\\beta _{ML}[M][L']:\\beta _{ML'}=K\\beta _{ML}", "10816784add7b5da0654f1ca9edd919e": "\\sigma _{f}={\\frac {3FL}{2bd^{2}}}", "1081dae289d8b07256d132eae92792cc": "{}+(a_{1}c_{2}-b_{1}d_{2}+c_{1}a_{2}+d_{1}b_{2})j", "10821b05e6436e9492cf84b0379fba1b": "(n_{i},n_{j})", "10821f8715a1c75669f17b8ffbbaabb1": "\\mathbf {\\bar {x}} ", "10826dbd67d65ada562c2d351963d9b7": "d_{H}(x,y)=k", "1082f2f5dd560b12bf8ce758d58140f2": "{\\mathcal {T}}=e^{-\\alpha \\,x}", "108315147577d0f2627997312d1958e6": "\\operatorname {E} (s^{2})=\\operatorname {E} \\left({\\frac {\\sigma ^{2}}{n-1}}\\chi _{n-1}^{2}\\right)=\\sigma ^{2},", "108324da8fa55a6a317590dd17e235d2": "{\\frac {\\Delta F(P_{0})}{\\Delta P}}={\\frac {F(P_{\\acute {n}})-F(P_{0})}{\\Delta _{\\acute {n}}P}}={\\frac {F(P_{1})-F(P_{0})}{\\Delta _{1}P}}={\\frac {F(P_{1})-F(P_{0})}{P_{1}-P_{0}}}.\\,\\!", "1083a8190456fbd27d9ffb29e2411ce7": "3k-1", "1083d6c1c705a1c7074f6005f4eb38c0": "{\\frac {\\partial f}{\\partial x_{1}}}v_{x_{1}}+{\\frac {\\partial f}{\\partial x_{2}}}v_{x_{2}}+\\,\\,\\,\\cdots \\,\\,\\,+{\\frac {\\partial f}{\\partial x_{N}}}v_{x_{N}}=0", "1084379e31aed4fbb0d59f20eed153b5": "\\zeta \\ll 1.\\,", "1084451459414a3fa16f40378e4378db": "{\\frac {u_{b}-u_{a}}{L}}", "108456c8399350ce4bd12c33f6f1f5b4": "\\ MRS_{xy}=P_{x}/P_{y}", "108458fecbf9807d98c393c840235f11": "{\\partial f \\over \\partial y}", "10846c5e06dc27e8d9336af5928fe805": "f_{U}", "1085b4cad24b8792a98a689c26390907": "\\ h", "1085da3130b1f6574f3a37e8f68f3eee": "{\\tilde {E}}_{7}", "1085ebad3828994a51d84c0387a2f7be": "\\log _{2}(1+m_{y})+e_{y}=-{\\frac {1}{2}}\\log _{2}{(1+m_{x})}-{\\frac {1}{2}}e_{x}", "10862040ad4ab49098ddf7a3314aefc6": "(Y,L)", "10865655faa44f74d5a18a82a5650669": "x=\\left(x_{1},x_{2},\\ldots ,x_{n}\\right)^{\\mathsf {T}}", "1086bd456aeff33774e603f9db50ac07": "W=\\min(Y_{1},Y_{2},\\cdots ,Y_{n})", "1086d86b37eaf879ca8ab9f53a9387a6": "\\lim _{x\\to \\infty }L(x)=b\\in (0,\\infty ),", "1086eda3eff89a0b05819a5ab87c6383": "\\delta W\\,", "108702f3c756c6a444975a31a40b1329": "|w|=1.", "108734a14b5051135ab92742de362841": "x'^{2}+y'^{2}+z'^{2}=c^{2}t'^{2}.", "108739ea7dec5ebd0e5d2fa008f31fa0": "p(y)\\propto \\left[1+{\\frac {y^{2}}{\\alpha ^{2}}}\\right]^{-m}\\exp \\left[-\\nu \\arctan \\left({\\frac {y}{\\alpha }}\\right)\\right]", "1087cd498720f5f1389d5abaa558542c": "{\\begin{aligned}x_{\\mathrm {triangle} }(t)&{}={\\frac {8}{\\pi ^{2}}}\\sum _{k=0}^{\\infty }(-1)^{k}\\,{\\frac {\\sin \\left((2k+1)t\\right)}{(2k+1)^{2}}}\\\\&{}={\\frac {8}{\\pi ^{2}}}\\left(\\sin(t)-{1 \\over 9}\\sin(3t)+{1 \\over 25}\\sin(5t)-\\cdots \\right)\\end{aligned}}", "1087db6db630cca7423ee9106de76bb4": "S\\to aSa|bSb|aa|bb", "1088093dd4df4c0f25bf46f87cc84456": "E_{d}=0", "1088ab7f22bec82db219ebc8815799f3": "\\{X\\mid \\forall n\\exists m\\phi (X,n,m)\\}", "1088d8875db5f61d743db700b3fb1635": "{\\sqrt {n}}({\\hat {\\theta }}_{\\text{mle}}-\\theta _{0})\\ \\ \\xrightarrow {d} \\ \\ {\\mathcal {N}}(0,\\ I^{-1}),", "10891edb4bb2a4636cda37168f5c49fd": "{\\begin{aligned}x_{n+1}-2x_{n}+x_{n-1}&=h^{2}w^{2}x_{n}\\\\\\iff \\quad x_{n+1}-2(1+{\\tfrac {1}{2}}(wh)^{2})x_{n}+x_{n-1}&=0.\\end{aligned}}", "10898ba36924b03a17aad6280082ca26": "\\eta ={\\sqrt {\\frac {6780}{9640}}}=0.8386\\ldots ", "1089f5a290ca9f1743550a7abbd7c9f2": "x^{7}+x^{6}+1", "108a08a709a62f9915e68eb522a08c58": "{v},{\\lambda }\\,", "108a1a65717fa3e21a6a80b49b8e4348": "q>p-1", "108a20afe90bb03e3da6f0919c3915e9": "|Z|_{max}\\approx {RQ_{L}^{2}}", "108a25cab551e4360cf1c6e99aa2798c": "\\displaystyle {T(t)=e^{At},}", "108a3d41363f8f92a05573b228415938": "{\\frac {U}{L^{3}}}=\\int _{0}^{\\infty }u_{\\lambda }(T)\\,d\\lambda ,", "108a4df21b9ff1de4cc8fd13275985b4": "\\mod 0", "108ab83f7305b42e54f202f43ef090fc": "\\displaystyle s={\\frac {n}{{\\text{poly}}(\\epsilon )}}", "108ac618dcb9569f19bea5c82e98c6c5": "c_{1}^{2}\\leq 3c_{2}\\ ", "108af287296db6e7232b21cfa1e1ea02": "dy=f'(x)\\,dx", "108af9c0e086d1fffd162b597a4350d0": "\\sum _{A=1}^{N}M_{A}\\mathbf {d} _{A}=0.", "108afc158333ae823721263268463db7": "W_{LC}=\\hbar \\omega _{LC}={\\frac {\\hbar }{\\sqrt {LC}}}.\\ ", "108b73e74e6ca41dd55657b5a59a9bb3": "\\epsilon ^{T}{\\tilde {\\Lambda }}\\epsilon =\\epsilon ^{T}\\left(\\Lambda +\\Lambda ^{T}\\right)\\epsilon /2", "108ba330334cf046276613d7c159d65a": "T(z)", "108bc405d112e54017ab721bb3ad0db3": "\\beta \\in \\left(0,1\\right)\\;", "108bc63c98914afd3fd36e4091bf1a9c": "EP=({\\tfrac {3}{4}}+{\\tfrac {1}{6}}-1)/(2\\times {\\tfrac {1}{6}}-1)={\\tfrac {1}{8}}", "108c73bc39aec6a533d4ce7efd8a95fe": "B_{k+1}={\\textrm {argmin}}_{B}\\|B-B_{k}\\|_{V}", "108c8126d7205ba92f83795580cbf308": "={\\widehat {D}}(-\\alpha )(\\alpha |\\alpha \\rangle -\\alpha |\\alpha \\rangle )", "108c97a55f5eb2bf236f975bdd437a55": "D^{\\geq 1}\\subset D^{\\geq 0};", "108d00703f36e0f9943ff05832978df4": "G\\mapsto F+G", "108d2c0c1a8150e22483011555779f45": "x>n\\,\\!", "108d3dfa2b81d80b9a374b999af15d79": "c(n)=\\int _{0}^{2\\pi }f(t)e^{-int}\\,dt", "108d5f63d448324e2fb851304313b21a": "\\Delta \\geq {\\frac {n}{2}}", "108d83492dba4cd0d61c903cc5a681cd": "F_{X}(x)=\\int _{-\\infty }^{x}f_{X}(t)\\,dt.", "108dcb2b089e392344da55ed7c23f2d4": "s_{f}={\\sqrt {\\left({\\frac {\\partial f}{\\partial {x}}}\\right)^{2}s_{x}^{2}+\\left({\\frac {\\partial f}{\\partial {y}}}\\right)^{2}s_{y}^{2}+\\left({\\frac {\\partial f}{\\partial {z}}}\\right)^{2}s_{z}^{2}+...}}", "108e0f264777eaf47d084a72ef06a13b": "2n+p+q+z-e", "108e4d17a841743afae7cc214a84312d": "{\\frac {1}{M_{SUSY}^{2}}}", "108e70d04afa261d72eee192a19743f0": "\\pi _{i}(X,A)\\,", "108ebcc631dd4b5414fa9467201dc3f2": "F(K,L)", "108ee72fb98668534397dd40dbf4b38f": "q(x)[\\phi (y)]=\\delta ^{(d)}(X-y)Q[\\phi (y)]\\,", "108f160d64f0f0d892f00b20c5d05d66": "\\textstyle \\prod _{}x{\\hbox{, }}(x{\\hbox{ in }}S)", "108f1cca0b631655b2bacc370d9060a6": "p=\\left[1-{\\frac {{\\rm {Tr}}(\\Omega ^{2})}{8\\pi ^{2}}}+{\\frac {{\\rm {Tr}}(\\Omega ^{2})^{2}-2{\\rm {Tr}}(\\Omega ^{4})}{128\\pi ^{4}}}-{\\frac {{\\rm {Tr}}(\\Omega ^{2})^{3}-6{\\rm {Tr}}(\\Omega ^{2}){\\rm {Tr}}(\\Omega ^{4})+8{\\rm {Tr}}(\\Omega ^{6})}{3072\\pi ^{6}}}+\\cdots \\right]\\in H_{dR}^{*}(M),", "108f3188bced2591245d04cf8fc3576a": "\\ P(X=x\\ {\\mbox{and}}\\ Y=y)=P(X=x)\\cdot P(Y=y)", "108f5465001029d6733711abb3552989": "M\\!\\ '", "108f64135afda75d6e4c2bdc14b1c4a4": "\\sum _{n=1}^{m}n(-1)^{n-1}.", "108fc3080dc8491cc88ec4af36f472f7": "\\tau _{ij}^{r}=L_{ij}+C_{ij}+R_{ij}", "108fc3e6d0b52a362023a1c50501cd3e": "f(k;N,K,n)=f(K-k;N,K,N-n)", "10901d93e9f9697cbe2461384f34c262": "Z_{i}^{'}=f^{'-1}(X_{i})", "10905c67c8b0fbb962072d3a8a7fd279": "\\Pr(Y_{i}=1)={\\frac {e^{{\\boldsymbol {\\beta }}_{1}\\cdot \\mathbf {X} _{i}}}{1+e^{{\\boldsymbol {\\beta }}_{1}\\cdot \\mathbf {X} _{i}}}}={\\frac {1}{1+e^{-{\\boldsymbol {\\beta }}_{1}\\cdot \\mathbf {X} _{i}}}}=p_{i}", "10908f4f03deb67e0ea70b94e4084d60": "(\\pi r^{2})(2\\pi R)=2\\pi ^{2}Rr^{2}", "1090f0cb613db397ada239de8b14f38a": "G^{{\\hat {a}}{\\hat {b}}}=8\\pi \\epsilon \\,\\left[{\\begin{matrix}1&0&0&0\\\\0&1&0&0\\\\0&0&1&0\\\\0&0&0&-1\\end{matrix}}\\right]", "1090fadfb43f68956350e51293aef824": "\\int _{1}^{\\infty }{\\frac {1}{x^{2}}}\\,\\mathrm {d} x", "10917eb4cde3371e74feb7c3521e3c4a": "M^{[2]}", "10919cc68ebafd9d17595d97911b6312": "C_{P}(T)=\\left\\{\\lim _{\\Delta T\\to 0}{\\frac {\\Delta H}{\\Delta T}}\\right\\}=\\left({\\frac {\\partial H}{\\partial T}}\\right)_{p}", "1091c2161de19592006170cddb32fa37": "\\vdash \\phi ", "1091db94750f3580097d97db9b6a62c2": "{\\vec {r}}_{1}", "1092100a3427cacb91c4b831300fa98e": "S={\\frac {1}{4\\pi }}\\int d^{2}x{\\sqrt {g}}(g^{\\mu \\nu }\\partial _{\\mu }\\phi \\partial _{\\nu }\\phi +(b+b^{-1})R\\phi +4\\pi e^{2b\\phi }),", "1092d3acd5d77a0082844280a9a32af5": "{\\begin{aligned}e_{1}&=(1,0,0,\\dots )\\\\e_{2}&=(0,1,0,\\dots )\\\\&\\ \\ \\vdots \\end{aligned}}", "1092fada509c1a2f2071453253ff55b4": "\\theta _{\\mathrm {R} }", "109325b4621a9f7e8fc36c9d59703437": "|{\\sqrt {2}}-a/b|", "109345941ca96584d9d591da970533d4": "{\\frac {256}{243}}", "1093a6ecef8973689f5e3b132269543a": "f:G\\to A", "10940f217b9529cd1671a2f59fef3e1b": "i(t)=C{\\frac {dv}{dt}}", "1094190fcdae75727ef87cbfee568838": "Q_{i}^{j+1}", "109425303bf0d94f257ccd70a86cadc5": "p(t)={\\text{penalty function whose time average must be minimized}}", "109445adb7dbe075dd0140e5705f50dd": "1/(\\exp(ar-br\\epsilon ))=", "10945651c529b0dd04112517c0a02e95": "|S|=O({\\sqrt {N}})\\,", "1094a5d3c18738997a6377d822bc4ebc": "[x:=e](b=c+x)\\equiv b=c+e\\,\\!", "1094a60a924ff4012789a5cd6fd5d7dc": "(\\theta )", "1094e05af58a9a840369c0cbceab87a5": "Y={\\begin{cases}\\xi K,&\\xi >0\\\\Y_{0}{\\frac {L}{L_{0}}}\\left({\\frac {L_{0}}{L}}{\\frac {P}{P_{0}}}\\right)^{\\alpha },&0<\\alpha <1\\end{cases}}", "1094f5e49a3fa41977c0538d88af1670": "{\\overline {\\mathcal {M}}}_{g,n}", "10950e4c8e9403c73a1994e7d5c66cb9": "\\Phi (e,x)=I\\quad ", "1095242412ee0efae54b5add8204474c": "\\rho (\\vartheta )=\\rho _{\\parallel }\\cdot cos^{2}\\vartheta +\\rho _{\\perp }\\cdot sin^{2}\\vartheta ", "10952edcb44b46d9a7e0c67c7384199f": "\\Delta R\\ ", "109554ba5842b0240aa09a03b7c46f70": "p(x+a+1)", "109599a77cd127b7ce94528d1041a521": "CCA=tdUCC", "1095a3448c3c62a66f09fd74ecd18e1d": "A_{k+1}=R_{k}Q_{k}=Q_{k}^{T}Q_{k}R_{k}Q_{k}=Q_{k}^{T}A_{k}Q_{k}=Q_{k}^{-1}A_{k}Q_{k},", "10965da0183ca0c1990c8472aa5fcf3b": "T_{d}={\\frac {\\log(2)}{\\log(1+{\\frac {r}{100}})}}", "1096671ddd8eb7963f45a6de97aaac6b": "\\ \\Delta G(T)=\\Delta H(T_{d}){\\frac {T_{d}-T}{T_{d}}}+\\int _{T_{d}}^{T}\\Delta C_{p}dT-T\\int _{T_{d}}^{T}\\Delta C_{p}dlnT", "1096919679550cac1aab5503a0f8c9cf": "Z(k,z)=e^{kz}\\,\\,\\,\\,\\,\\,\\mathrm {or} \\,\\,\\,\\,\\,\\,e^{-kz}\\,", "1096d230bf1ee03e34f15b5cdd8cf72f": "L'={\\frac {dL}{dl}}\\approx {\\frac {\\mu _{d}}{2\\pi }}\\ln {\\frac {r_{o1}}{r_{i}}}", "1096db202e772b1231e07202c527d6d4": "(\\cdot )_{k}", "1096fccbfe868c49606440b743b7c442": "r_{1}^{2}r_{2}^{2}\\left({\\frac {d\\theta _{1}}{dt}}\\right)\\left({\\frac {d\\theta _{2}}{dt}}\\right)-2c\\left[\\mu _{1}\\cos \\theta _{1}+\\mu _{2}\\cos \\theta _{2}\\right],", "109758ad89b167abc74e919edc865b3e": "{\\bar {Y}}=1", "10976ef65af219f5308e519dacf145b6": "\\kappa _{1}^{F(n)}-\\gamma _{1}=0\\,,", "1097a73ec8b15bcf86aba7fda80ebd7f": "p_{1},\\dots ,p_{r}", "109852bb99a76283a46dbe3ea5d4dff0": "{\\cfrac {\\Gamma \\vdash A,\\Delta }{\\Gamma ,\\lnot A\\vdash \\Delta }}\\quad ({\\lnot }L)", "1098739e1872273f2c88409e32233d44": "{\\frac {1}{r}}{\\frac {\\partial }{\\partial r}}(rE_{r})=qn/\\epsilon _{0},", "10988c3ae1eb27588bf6540cece5b83f": "B_{\\max }k_{\\mathrm {on} }", "10989df6a215e0838610a5a7b58805cc": "(\\xi ,\\eta )", "1098b5481b8f62c4a6db391ff7cfa93d": "p=59", "1098e3736e4bf2238a5c58411942c41d": "\\gamma ^{1,2,3}={\\begin{pmatrix}0&i\\sigma ^{1,2,3}\\\\-i\\sigma ^{1,2,3}&0\\end{pmatrix}},\\quad \\gamma ^{4}={\\begin{pmatrix}0&I_{2}\\\\I_{2}&0\\end{pmatrix}}", "1098f88fbae465e2ceaa6e44e41743a2": "(dX)^{2}\\,\\!", "1098fccd0ee4c85d98ca1b7e9b5f55b1": "\\bigcup _{n=1}^{\\infty }A_{n}", "109922db3cc0e2922b5dc86bf493482f": "P(n)=\\sum _{d\\mid n}d\\sigma (d)\\mu (n/d)", "1099c661d6b9604c64fb9d1b5c56f2d9": "P_{theo,c}=A_{c,t}\\times C_{y}\\times h_{i}", "1099e9ee8b1263137e7da97f986f13a8": "\\bigvee _{i}X_{i}=\\coprod _{i}X_{i}\\;/\\sim ,\\,", "1099f15fee692bf95e4e9786f96cac35": "\\displaystyle X(t)=\\eta +tu(\\eta ,0)", "109a51860c7f47a6a5118614e6c81b25": "F^{\\alpha \\beta }{}_{;\\beta }\\,=\\mu _{0}J^{\\alpha }", "109a60954f638e328c89597bb27bd6d8": "\\Gamma _{ij,k}^{(\\alpha )}=\\Gamma _{ij,k}^{(0)}+\\alpha T_{ijk}", "109a87d8dec44ed62486174a7e074f7b": "\\mu _{s}(l,x_{s},p+\\Pi )=\\mu _{s}^{0}(l,p+\\Pi )+RT\\ln \\gamma _{s}x_{s}", "109a884b34370355ceac60f83ecd5ab6": "n,f=(0,1):(n+1,f\\times (n+1))", "109a99edf06bbf0cafe7955eb6ac0916": "K[T]/T^{2},", "109aaf1c2c923237608aa9f6b7b91a45": "\\delta =-3^{\\circ }", "109ac253e71fcd17f8df1ffbafe1be85": "{\\mathbb {R} }/{\\mathbb {Z} }", "109b1b64deed72bcd74934ef21619fe6": "a,d\\in k\\setminus \\{0\\}", "109b9d05c6bdcbff24d7987dd774eb39": "a-b=0\\;", "109ba257f02a31f82155d48ddfd5eb92": "{\\frac {\\partial \\mathrm {net} }{\\partial w_{i}}}", "109ba6ed595dc1bec4f88e1ed6afc5d6": "\\rightsquigarrow 1", "109c115a8fb6e68a44a03bf67dc6f29e": "A[f]=\\int _{x=x_{0}}^{x_{1}}n(x,f(x)){\\sqrt {1+f'(x)^{2}}}dx,\\,", "109c1fe50f34ceea5a04dbe877e17ab6": "A{\\vec {e}}_{j}=\\sum a_{i,j}{\\vec {e}}_{i}", "109c296a3a88cce2c57abc9cd3a62b1b": "A+L->A^{+}+L->A^{++}", "109d065990a0f0c7cdd3d82178cfe467": "t^{*}=({\\hat {\\theta }}^{*}-{\\hat {\\theta }})/{\\hat {se}}_{{\\hat {\\theta }}^{*}}", "109d14234ca7b15e14d436df4b8a32b3": "{\\begin{aligned}\\mathbf {y} _{p}'(t)&=(e^{tA})'\\mathbf {z} (t)+e^{tA}\\mathbf {z} '(t)\\\\[6pt]&=Ae^{tA}\\mathbf {z} (t)+e^{tA}\\mathbf {z} '(t)\\\\[6pt]&=A\\mathbf {y} _{p}(t)+e^{tA}\\mathbf {z} '(t)~.\\end{aligned}}", "109d26a38e9a65bf49110c621f3e2b43": "x^{\\alpha }", "109d82045882b47372aa6669a8c65a46": "X=\\mathbb {D} .", "109da39c9ef1dc48b2c34face7f1705d": "L(g)=\\{G_{i}^{(g)}\\}", "109eca64ff76194cd2e8e9b1cff8f709": "M={\\frac {r_{screen}}{r_{tip}}}.", "109edb220aff864a2f77f003b4a64608": "\\langle F',\\varphi \\rangle =-\\langle F,\\varphi '\\rangle =-\\int _{-\\infty }^{+\\infty }F\\varphi '=\\langle f,\\varphi \\rangle ", "109eedca4965cc4f1435b5ec4e1bc5a1": "\\rho _{\\text{isBusinessContact / isFriend}}({\\text{addressBook}})", "109f172d0f6309e1793dee513c5c9881": "E=E_{0}+c_{k}k^{p}", "109f310698c67b59267233f30a2dbce8": "{\\mathcal {C}}^{\\mathcal {J}}", "109f6e29e4cdef0b2850bac2d6989db3": "\\sum _{i=0}^{n-1}a^{i}\\otimes _{K}{\\frac {b_{i}}{p'(a)}}", "109f9dba683e7cc9f7bd0f8aa8c16aa3": "{\\begin{aligned}\\left\\|\\mu -m\\right\\|=\\left\\|\\mathrm {E} (X-m)\\right\\|&\\leq \\mathrm {E} \\|X-m\\|\\\\&\\leq \\mathrm {E} (\\left\\|X-\\mu \\right\\|)\\\\&\\leq {\\sqrt {\\mathrm {E} (\\|X-\\mu \\|^{2})}}={\\sqrt {\\mathrm {trace} (\\mathrm {var} (X))}}\\end{aligned}}", "109fa6f56471f8b5c0a9c63236dc50b2": "L_{0}=\\mathrm {AB} =30\\ \\mathrm {cm} ", "109fd3fb75823668d9731ce1f8eef742": "\\beta _{12}=K\\beta _{11}\\,", "109fdeb06203947183fd61e2fe859c86": "r={\\frac {1}{T}}\\left(W(-se^{-s})+s\\right){\\text{ with }}s={\\frac {M_{a}t}{P_{0}}}", "10a08e31a172a3de88d0596827571ff6": "{\\frac {dT_{\\text{core}}}{dt}}=Q_{\\text{cmb}}\\left[A_{\\text{c}}(L+E_{G})\\left({\\frac {R_{i}}{R_{\\text{c}}}}\\right)^{2}\\rho _{i}{\\frac {dR_{i}}{dT_{\\text{cmb}}\\eta _{\\text{c}}}}-{\\frac {R_{\\text{c}}^{3}-R_{i}^{3}}{3R_{\\text{c}}^{3}}}\\rho _{\\text{c}}c_{\\text{c}}\\right]^{-1}", "10a18a252ed439b206e9e09d59c31f35": "i\\in \\{0,1\\}", "10a1a9b21789426655a7cef3eedb323d": "\\eta _{\\mu \\alpha }\\left(\\partial ^{2}+m^{2}\\right)D^{\\alpha \\nu }\\left(x-y\\right)=\\delta _{\\mu }^{\\nu }\\delta ^{4}\\left(x-y\\right).", "10a1c62e7176cf9361da91b78993d8c0": "\\limsup _{n\\to \\infty }(a_{n}+b_{n})\\leq \\limsup _{n\\to \\infty }(a_{n})+\\limsup _{n\\to \\infty }(b_{n}).", "10a22a029ea33e3c0935b81cf178e77f": "w(2+i)=1", "10a234df3a29c89fd2754a22b3e5f55a": "F(\\rho ,\\sigma )=\\lVert {\\sqrt {\\rho }}{\\sqrt {\\sigma }}\\rVert _{\\mathrm {tr} },", "10a25d4db586c0447d3182ac742895f1": "f(x)=\\Omega (g(x))\\Leftrightarrow g(x)=O(f(x))", "10a2b772e79678e1c08cb162664fa34f": "\\gamma _{\\mathrm {se} }", "10a2dab3b8bb6fc3062e4ea3b86d3bfc": "{\\frac {d}{dx}}f(x)={\\frac {{\\frac {dg(x)}{dx}}\\cdot h(x)-{\\frac {dh(x)}{dx}}\\cdot g(x)}{h^{2}(x)}}", "10a301200cbd1ea17a4d727e10e00d9e": "w(v)+w(u)\\geq d", "10a3236fec503e5ee001a7f87b492733": "\\Delta {}E=W+Q+E", "10a35ce791976a5d4ee0442d5817697a": "\\displaystyle {\\|a^{*}\\|=\\|a\\|,\\,\\,\\,\\|\\{a,a^{*},a\\}\\|=\\|a\\|^{3}.}", "10a365729012f70af2556be9cdf5a732": "-{\\sqrt {\\frac {18}{35}}}\\!\\,", "10a3b06a3657e3d1b93c145b064abbcc": "R\\sin x^{\\circ }={\\frac {Rx(180-x)}{10125-{\\frac {1}{4}}x(180-x)}}", "10a3e10030df2b1c63325a75c1820253": "\\mu ^{-}\\,/\\,\\mu ^{+}", "10a3e6b75a65c3e43e9bd70a4aea933f": "{\\tbinom {n}{k}}", "10a40604fd5176f4dc10a23ffb6409b2": "\\scriptstyle E(w)", "10a4442c17e48d9fce1702597cb4750b": "\\left.({\\mathcal {L}}_{\\!X}f)(p)\\triangleq {\\frac {\\operatorname {d} }{\\operatorname {d} t}}f(\\gamma (t))\\right\\vert _{t=0}", "10a460c5cf4a0e9ce6887ccfb89e8ada": "{\\text{at }}z=0:\\quad A{\\frac {\\partial u}{\\partial z}}=\\tau ^{x}\\quad {\\text{and}}\\quad A{\\frac {\\partial v}{\\partial z}}=\\tau ^{y},", "10a491d6e6e5da59dd7a8ade4e7d203a": "{{\\frac {|AB|}{|BD|}}\\sin \\angle \\ BAD=\\sin \\angle BDA}", "10a51d59d8f78ee0d8c665bbc281a995": "f^{*}Y\\to X'", "10a55ad0f78ce6e9dddc9f0d25692522": "(N_{j},N_{k})", "10a55ea60679e457a0ae710cdf5fd184": "Q=A_{1}\\cdot V_{1}=A_{2}\\cdot V_{2}", "10a58fc621da6aa7dcba68639f7ae83a": "(a,b)=\\{\\{a\\},\\{a,b\\}\\}.\\,", "10a5e908edbb2881c5ea62363e7b8e51": "S\\equiv -k_{B}\\sum _{s}P_{s}\\ln P_{s}=k_{B}(\\ln Z+\\beta \\langle E\\rangle )={\\frac {\\partial }{\\partial T}}(k_{B}T\\ln Z)=-{\\frac {\\partial A}{\\partial T}}", "10a5fe428ccc4739924d9299b3d3e925": "\\operatorname {drop-formal} [Z,Y,V]\\equiv Y", "10a6066086cf9605188376749c98653a": "\\angle DOC=\\angle DOE+\\angle EOC.", "10a65dec87bdc5afe4a4c6fc8625d4a2": "10\\uparrow ^{m}n", "10a7360428d7a4ec4d0d5885b227054d": "C_{\\psi }=\\int \\limits _{-\\infty }^{+\\infty }{\\frac {|\\Psi (\\Omega )|^{2}}{|\\Omega |}}\\,d\\Omega <\\infty ", "10a7491db271a208d3436f577f35576a": "{\\vec {r}}_{i}(t+\\delta t_{\\mathrm {MPC} })={\\vec {r}}_{i}(t)+{\\vec {v}}_{i}(t)\\delta t_{\\mathrm {MPC} }", "10a74e7bba9417af59929258c975355e": "M_{Y}\\ ", "10a77078361fbfefd59b647fc1ab06aa": "z={\\frac {({\\overline {x}}_{1}-{\\overline {x}}_{2})-d_{0}}{\\sqrt {{\\frac {\\sigma _{1}^{2}}{n_{1}}}+{\\frac {\\sigma _{2}^{2}}{n_{2}}}}}}", "10a7832cda41bfad9dec2473adf31fc4": "log(t)", "10a7ad705170d5cef996e4b6bde38618": "I{\\underline {A}}", "10a7bcc75aa9d193c304c1937418798d": "Cn_{r}", "10a7cf8ba6f8ee42519b2af39ca0b724": "\\{\\mathrm {X} _{1}(p),\\dots ,\\mathrm {X} _{4}(p)\\}", "10a826b1854b836a0b481dd9d1d251bf": "\\{\\mathbf {Z} _{k}:k=1,2,\\ldots ,n\\}", "10a8ab7ea986641a5c054200fc263648": "\\ y=(d/2){\\sqrt {(\\xi ^{2}-1)(1-\\eta ^{2})}}\\cos \\phi ,", "10a93856574e68c286da272962609390": "x'=x{\\sqrt {1-v^{2}/c^{2}}}", "10a93d56445a9e7abc8455e26825bed7": "\\xi (\\omega )={\\begin{cases}-r,&\\omega {\\text{ is red}}\\\\-r,&\\omega =0\\\\r,&\\omega {\\text{ is black}}\\end{cases}},", "10a9537eed0c904da74fe0fde3041bfc": "\\left\\{{\\begin{bmatrix}0\\\\1\\\\-3\\\\3\\\\-1\\end{bmatrix}}{\\begin{bmatrix}1\\\\-15\\\\30\\\\-1\\\\-45\\end{bmatrix}}\\right\\},\\left\\{{\\begin{bmatrix}0\\\\0\\\\0\\\\0\\\\1\\end{bmatrix}}{\\begin{bmatrix}0\\\\0\\\\0\\\\1\\\\0\\end{bmatrix}}{\\begin{bmatrix}0\\\\0\\\\1\\\\-2\\\\0\\end{bmatrix}}\\right\\}", "10a95d31413b5465b8c83c30d657d6ea": "Q=kd", "10a95df6e59f7a1a40dd176bc5884cb7": "^{2}\\Sigma ^{+}", "10a98d22844a50d53ed669b1a0a5d42f": "\\{(x,z)\\ |\\ x\\,(S\\circ R)\\,z\\wedge \\forall y\\in Y\\ (y\\,S\\,z\\Rightarrow x\\,R\\,y)\\}.", "10aa51128f0a6f16f6ee6253b5b8f2d6": "a(u,v)=\\int _{a}^{b}\\!u'(x)v'(x)\\,dx", "10aa66225d91cadf44874edaee2b0244": "z_{3}={\\frac {z_{1}z_{2}-ax_{1}y_{1}x_{2}y_{2}}{(y_{2}^{2}+(z_{1}x_{2})^{2})}}", "10aa7b8b0b15fd27d28a6612265cba7b": "\\Gamma _{7}", "10aa9887049893fdb1a896fca150a797": "-{\\frac {\\hbar ^{2}}{2m}}\\nabla ^{2}\\delta \\psi +V\\delta \\psi +g(2|\\psi _{0}|^{2}\\delta \\psi +\\psi ^{2}\\delta \\psi ^{*})=i\\hbar {\\frac {\\partial \\delta \\psi }{\\partial t}}", "10aaab75818f6fd1f7490ed8ca10af39": "\\kappa ={\\frac {\\Pr(a)-\\Pr(e)}{1-\\Pr(e)}},\\!", "10aaac2d30cec6245ecdb4c6d83abed2": "w_{i}=\\int _{-s}^{s}{\\frac {1}{y-y_{0}}}d\\Gamma \\qquad (7)", "10ac41137292c9de95b784464a8f33d3": "g_{uc}(\\langle sss\\rangle )=g_{uc}(\\langle {\\text{sß}}\\rangle )=g_{uc}(\\langle {\\text{ßs}}\\rangle )=\\langle SSS\\rangle ", "10ac7828cbfca1c7fa78610954f47c99": "\\mathbf {H} =\\mathbf {F} /q_{m}\\,", "10ac9e1d972523558bc6e58a4d4efd38": "\\nu ^{*}", "10aca448948213b86b8a1aa47d638b33": "\\gamma ^{\\sigma }\\,", "10ad07348572f86c16caa72122080733": "\\eta _{f}={\\frac {\\tanh {mL_{c}}}{mL_{c}}}", "10ad7f98c69273f5be1ccb8fa572f02e": "M_{T}^{2}=(E_{T,1}+E_{T,2})^{2}-({\\overrightarrow {p}}_{T,1}+{\\overrightarrow {p}}_{T,2})^{2}", "10ad8e428e9137aa983c5dd8c252a19b": "f(\\lambda )={\\frac {3}{\\lambda }}\\left[{\\frac {1}{tanh(\\lambda )}}-{\\frac {1}{\\lambda }}\\right]", "10adb068eb6ad18e34a61387770fc0be": "\\Phi :{\\mathcal {H}}\\to {\\mathcal {H}}^{*}", "10adba9a4d1293648444f90c1f98f4af": "<1,2>", "10ae0e1dfeeadf184fab5f11d82e8271": "\\{C_{I},C_{J}\\}=f_{IJ}^{K}C_{K}", "10ae161e3b32f9a344ac89bb428ae2d6": "x_{t}=x_{t}(\\xi _{[t]})", "10ae23b174a87716e3840e53044d1502": "J(u)=\\int _{D}|\\nabla u|^{2}\\mathrm {d} x", "10ae2db675e5043225913ed613ad1697": "{\\begin{aligned}\\mathbf {Y} &={\\frac {d\\mathbf {F} }{dt}}={\\frac {d^{2}\\mathbf {p} }{dt^{2}}}={\\frac {d^{2}(m\\mathbf {v} )}{dt^{2}}}\\\\&=m\\mathbf {j} +\\mathbf {2a} {\\frac {{\\rm {d}}m}{{\\rm {d}}t}}+\\mathbf {v} {\\frac {{\\rm {d^{2}}}m}{{\\rm {d}}t^{2}}}\\\\\\end{aligned}}\\,\\!", "10ae67bf9ad12ccc730ab496b1863a09": "g_{dsF}=g_{ds}+{\\frac {C_{GD}}{C_{T}}}g_{m}", "10af0757ad55e23fa0cea99aafab63d7": "\\lambda x.x+y", "10af4fab96c96f4c326d3a7f0cfc8141": ":{\\hat {b}}^{\\dagger }\\,{\\hat {b}}:\\,={\\hat {b}}^{\\dagger }\\,{\\hat {b}}.", "10af6b7078f7aa9f6d583996bfc30d5a": "s=vt-{\\begin{matrix}{\\frac {1}{2}}\\end{matrix}}at^{2}", "10af71414b9126febdd74ba0d3815105": "y''=-y\\,", "10afc8268f8964502e8824eff7aa42a0": "L_{4k}(\\mathbf {Z} )", "10b001559e5ea111724784ac9136f042": "(x+m+1)\\sum _{i=0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i}=(x-m){\\dbinom {x}{m}}.", "10b01101b78d570970fdd18bc9bca880": "{\\overline {\\lim }}(\\log a_{n})/2^{n}<+\\infty .", "10b04274297b40fe360362f70661531e": "\\quad (7)\\qquad \\qquad {\\frac {d{\\bar {\\rho }}_{i}}{dt}}+{\\frac {1}{\\Delta x_{i}}}\\left[f_{i+{\\frac {1}{2}}}-f_{i-{\\frac {1}{2}}}\\right]=0,", "10b09ad594da0a55c2366a2f2c1e057b": "\\Omega _{i,j}=\\Gamma _{i,i}+\\Gamma _{j,j}-\\Gamma _{i,j}-\\Gamma _{j,i}=K_{i}K_{i}^{T}+K_{j}K_{j}^{T}-K_{i}K_{j}^{T}-K_{j}K_{i}^{T}=(K_{i}-K_{j})^{2}", "10b1127d5f0f8fe6e96b9f49ce875f92": "1\\leq k\\leq |\\mathbf {X} |", "10b112b5372daf1cc5161752bd552176": "n\\in \\{0,1,...,N-1\\}", "10b1161dd4156b738b506af877a96db6": "{\\frac {dy}{dt}}=g(x,y)=-y\\left(\\gamma -\\delta {\\frac {x}{1+x}}\\right)", "10b17aad991be7b34a64cf31c763432b": "\\Lambda (x)=\\sum _{i=1}^{v}\\lambda _{i}x^{i}", "10b1fa405163165d5adb8d287e24a50f": "g={1 \\over 2}h_{\\alpha {\\bar {\\beta }}}\\,(dz^{\\alpha }\\otimes d{\\bar {z}}^{\\beta }+d{\\bar {z}}^{\\beta }\\otimes dz^{\\alpha }).", "10b22174c40bce87be79c006df59ceed": "\\phi _{bc},\\,\\phi _{ac}", "10b23c2b5a687510af844bc0b3fc9ed8": "^{-1}:G\\to G,", "10b25f75c898d917edeb80f0d61a5e5b": "S+C\\supseteq \\{f(x):x\\in M\\}", "10b2666e21341f2b952499d4e22efbef": "{\\hat {r}}={\\frac {\\bar {r}}{|{\\bar {r}}|}}", "10b3bb2bb94ad44f4cca3cc43fe84ff8": "R_{ab}l^{a}l^{b}=R_{ab}l^{a}l^{b}-{\\frac {1}{2}}Rg_{ab}l^{a}l^{b}=8\\pi \\,T_{ab}l^{a}l^{b}", "10b3c7530c0ce99746ed4ad690ac03e3": "\\mathbb {H} ^{p+1}", "10b42f67a9e763181931e1c056d1d04e": "m>\\lambda ", "10b443d7f9559f9b0f14e5f71343bc1a": "{\\begin{aligned}c&{}={\\frac {r(1+r)^{N}}{(1+r)^{N}-1}}P\\\\&{}={\\frac {r}{1-(1+r)^{-N}}}P\\end{aligned}}", "10b4821438b54304dbdc6d8d0fd66a56": "{\\begin{aligned}{\\mbox{D}}({\\mbox{E}}(x))&=a^{-1}({\\mbox{E}}(x)-b)\\mod {m}\\\\&=a^{-1}(((ax+b)\\mod {m})-b)\\mod {m}\\\\&=a^{-1}(ax+b-b)\\mod {m}\\\\&=a^{-1}ax\\mod {m}\\\\&=x\\mod {m}.\\end{aligned}}", "10b4aac1a654918f5814117468e23fd2": "{y\\in R^{\\it {M}}}", "10b4bf357efa33ae062b5b7202d9446a": "\\forall k\\in \\mathbb {N} :\\phi (x,k+1)\\geq \\phi (x,k)", "10b541396eb5a9ffb8b6080f3525595b": "(u,v),\\mathrm {height} (u)\\leq \\mathrm {height} (v)+1", "10b5766bf11dd27b7a7a07a88588bee8": "{\\frac {3}{4}}V_{g}+V_{e}", "10b5ed345a5bbe2bda56fa0def78adcc": "{\\hat {H}}={\\frac {1}{2m}}\\left[\\left(\\mathbf {p} -q\\mathbf {A} \\right)^{2}-\\hbar q{\\boldsymbol {\\sigma }}\\cdot \\mathbf {B} \\right]+q\\phi ", "10b5f67b32006b705c9bd7ff308fe91e": "M\\otimes _{{\\mathbb {Q}}(q)}M'", "10b623ccb527b48de06c362ce120ddc0": "A_{N}=\\int D\\mu \\prod _{0500^{\\circ }C} \\ U+2MgF_{2}} ", "10c7f8a4b7b0a99a6532f5db55d3b665": "\\mathbf {r} \\rightarrow R(\\mathbf {\\hat {n}} ,\\theta )\\mathbf {r} ", "10c81bf8c00e16c19dd22f8096fdd0fb": "S+\\alpha .m_{0}\\lfloor {\\frac {n}{2}}\\rfloor }^{\\infty }{\\frac {z^{k}}{k}}+\\sum _{k=1}^{\\lfloor {\\frac {n}{2}}\\rfloor }{\\frac {z^{k}}{k}}\\right).", "10cff4b7a419264e8c021751dd91a76c": "\\omega _{2}=iR_{2}", "10d00a611c37ed3585b3819fedfc6f0f": "S({\\boldsymbol {\\beta }})=\\mathbf {y} ^{\\rm {T}}\\mathbf {y} -2{\\boldsymbol {\\beta }}^{\\rm {T}}\\mathbf {X} ^{\\rm {T}}\\mathbf {y} +{\\boldsymbol {\\beta }}^{\\rm {T}}\\mathbf {X} ^{\\rm {T}}\\mathbf {X} {\\boldsymbol {\\beta }}.", "10d0392dbbc815dbd45eaa908643f589": "A\\times _{C}B=\\left\\{(a,b)\\in A\\times B\\;{\\big |}\\;\\alpha (a)=\\beta (b)\\right\\}", "10d0599cc949b0f3ae1a61c7eaca8353": "P=\\rho R_{s}T,", "10d0ad92bb3547cb511c2c7beec53a51": "k\\leq 0", "10d112ead69f7b0aeb7bb6823f973f04": "K:G\\sim N(E,{\\frac {N_{0}E}{2}})", "10d17bb6648eec2409b0159ff27356a5": "V_{v}={\\frac {\\hbar }{2}}(2\\omega _{1}+\\omega _{2}-3\\omega )=-{\\hbar e^{2} \\over 16\\pi \\varepsilon _{0}m_{e}\\omega Z^{3}}.", "10d1c53ac47198404da42496e393662e": "E(|X_{n}|,|X_{n}|\\geq K)=1\\ {\\text{ for all }}n\\geq K,", "10d1d029504c5eee87108420925b9733": "\\Phi (x)\\equiv 0", "10d1f58f0d83a8b2481c2943109fd288": "{\\frac {x}{\\sigma {\\sqrt {2}}}}", "10d20370c745ee26c5556cb817b36fd3": "X^{2}=X\\times X", "10d2053af5c2cf7f281292a067ac5380": "f:\\mathbb {Z^{+}} \\rightarrow \\mathbb {R} ", "10d213356e630fc9a88029c082c834b1": "{\\frac {1}{\\sqrt {\\lambda }}}=-2\\log[{\\frac {\\varepsilon }{3.7D}}-{\\frac {5.02}{Re}}\\log({\\frac {\\varepsilon }{3.7D}}+{\\frac {13}{Re}})]", "10d22e184c2215fb046a5d3b2f26304b": "\\!V={\\frac {4}{3}}\\pi r^{3}", "10d241651b61cfdd54169b153ab938e4": "\\varepsilon {\\dot {x}}_{2}=f_{2}(x_{1},x_{2})+\\varepsilon g_{2}(x_{1},x_{2},\\varepsilon ),\\,", "10d27c956fff38e4b59b6486b5538bcb": "u=\\left\\lfloor {\\sqrt {x}}\\right\\rfloor ", "10d2c843b66fe271ad3b5218b499622c": "f_{1}(x),f_{2}(x),\\dots ,f_{m}(x)\\,", "10d301becbebc5d4edbc436250495a7a": "x+z=y", "10d37c0fcf8bfb55880f122ef7476640": "g(x)={\\frac {f(x)}{f(x)+f(1-x)}},\\qquad x\\in \\mathbb {R} ,", "10d4b6c2f87c147e5fa757c20c29ce17": "K=uv[2t(1-uv)-(u+v)(1-t^{2})][2(u+v)t+(1-uv)(1-t^{2})]", "10d5099a2b6b08bf93311d42709413b9": "\\sigma _{3}\\,\\!", "10d54f8ab2aa3bf7b9fbf56532bcdd92": "\\varphi =(1+{\\sqrt {5}})/2,", "10d559a732e1e9b07700a1fdcc1b2077": "f_{1}f_{2}\\rightarrow g", "10d5a142e603155741af2829cef43516": "H_{j-1,j}^{}=H_{j,j+1}^{}\\equiv H_{L}", "10d5a456b521b432b67ab5024ade2409": "\\displaystyle Y_{t}={\\begin{cases}\\delta _{1-t}(W_{t})&{\\text{for }}0\\leq t<1,\\\\0&{\\text{for }}1\\leq t<\\infty ,\\end{cases}}", "10d5c144f59e84fff7a4a9a48c8a40e9": "T=\\tau ", "10d62bf6f1e6566885e584356f173a97": "T_{a}(x)=x+a", "10d634548ef000d636630284051b76eb": "\\sum _{k=n}^{n+17}F_{k}=76F_{n+10}", "10d7016ae8a3134ac187ec27ab164b8e": "=32id_{5}+16d_{4}-8id_{3}-4d_{2}+2id_{1}+d_{0}-{\\frac {i}{2}}d_{-1}-{\\frac {1}{4}}d_{-2}+{\\frac {i}{8}}d_{-3}", "10d748916f1e04efb42fb6ded622eee1": "{\\begin{cases}a\\\\b\\end{cases}}=a_{0}{\\sqrt {\\frac {\\xi }{\\xi _{0}}}}\\left[1\\pm {\\frac {A}{\\sqrt {\\xi }}}\\sin \\left(\\xi -\\xi _{0}\\right)\\right],", "10d785a454a472daa18581b57c13b7a7": "{\\begin{array}{lcl}\\mathbf {A} &=&{\\begin{bmatrix}\\cos \\theta \\cos \\psi &\\cos \\phi \\sin \\psi +\\sin \\phi \\sin \\theta \\cos \\psi &\\sin \\phi \\sin \\psi -\\cos \\phi \\sin \\theta \\cos \\psi \\\\-\\cos \\theta \\sin \\psi &\\cos \\phi \\cos \\psi -\\sin \\phi \\sin \\theta \\sin \\psi &\\sin \\phi \\cos \\psi +\\cos \\phi \\sin \\theta \\sin \\psi \\\\\\sin \\theta &-\\sin \\phi \\cos \\theta &\\cos \\phi \\cos \\theta \\\\\\end{bmatrix}}\\end{array}}", "10d78f524211cb318a4dad4f027664b7": "(\\partial H)_{P}=-(\\partial P)_{H}=C_{P}", "10d7adf5c6cf00828f5abf61c0da4347": "Ux\\cdot \\operatorname {diag} (f(x_{0}),\\dots ,f(x_{n}))=T_{f}x\\cdot Ux", "10d7dcb7b76d35a438b7c571adc509f7": "x(t)=Ae^{\\gamma _{+}t}+Be^{\\gamma _{-}t}\\,,", "10d7ebe08036bf0dca2ea2aebcd6739c": "v={\\frac {\\sum _{}{\\frac {\\pm e_{k}}{R_{k}}}+\\sum _{}\\pm a_{m}}{\\sum _{}{\\frac {1}{R_{k}}}+\\sum _{}{\\frac {1}{R_{i}}}}}", "10d839602e2bd61fd6b89395c2f2b1d8": "\\eta c_{\\eta }(\\xi _{1},\\xi _{2})", "10d867586deb88c9dff1a2bd50fe94e8": "p_{2}={\\frac {m_{1}}{1+m_{1}}}\\ ,", "10d890fc5c36a42fdaec7e589f1c216e": "(2^{\\aleph _{0}})^{\\aleph _{0}}=2^{\\aleph _{0}^{2}}=2^{\\aleph _{0}},\\,", "10d89835595e6e6f8578a5312e942cec": "\\exp(i\\omega t)", "10d8989ebc40b46b347efeb066400891": "\\left({\\frac {p_{2}}{p_{1}}}\\right)^{\\frac {\\gamma -1}{\\gamma }}", "10d8a5f8785fe2bcae31e13760724f71": "\\sum _{k=0}^{\\infty }\\left({\\frac {1}{k!}}\\int _{A^{k}}\\rho _{k}(x_{1},\\ldots ,x_{k})\\,{\\textrm {d}}x_{1}\\cdots {\\textrm {d}}x_{k}\\right)^{-{\\frac {1}{k}}}=\\infty ", "10d8c8cc3c107342f64fb76596c0908b": "0\\to K\\to F{\\overset {u}{\\to }}M\\to 0", "10d8fa9d04d95ac73026b8dc03d7c8c4": "PV^{\\gamma }=\\operatorname {constant} \\qquad ", "10d91283b650cd88188f1de614da1872": "\\approx 10\\times 0.02\\times 0.834", "10d94500d9cce400d297c5f7e179143a": "\\delta n_{k}", "10d9c53b46755279bbb0ec2c8dcacc49": "\\scriptstyle {\\frac {\\sqrt[{12}]{2}}{{\\sqrt[{12}]{2}}-1}}", "10d9ddbd5d2f6453db360b396af9032a": "{1 \\over 2\\pi }\\int _{0}^{2\\pi }|e^{g}|^{2}\\,d\\theta \\leq e^{A},", "10d9ee9be1f790eb1addb53458190cdd": "n-CN(0,I_{N\\times N})", "10da04269de1736c4f92c635f399f7b4": "\\psi _{7,8}=1", "10da7af65b26ed6927b624abdede8994": "x_{1},\\ldots ,x_{m}", "10dab83c45ba3ddfe99ee0a5ec537c8a": "c_{15}", "10dac876187a7e73313fc986fb5cec19": "C(K)=\\lim _{r\\to \\infty }C(\\Sigma ,S_{r}).", "10db389c133e8bc28e9be66025b0064b": "\\gcd(N,du)=d", "10db4ad06b1ae4064c3c12ba4b2bab66": "R={R^{m}}_{m}", "10db7c533c7259766477548ffc284fba": "{\\mathbf {A}}_{z}=\\left[{\\begin{array}{c c c c c}0&0&0&1&0\\\\-uw&w&0&u&0\\\\-vw&0&w&v&0\\\\{\\hat {\\gamma }}H-w^{2}-a^{2}&-{\\hat {\\gamma }}u&-{\\hat {\\gamma }}v&(3-\\gamma )w&{\\hat {\\gamma }}\\\\w[(\\gamma -2)H-a^{2}]&-{\\hat {\\gamma }}uw&-{\\hat {\\gamma }}vw&H-{\\hat {\\gamma }}w^{2}&\\gamma w\\end{array}}\\right].", "10db94130479c95f1e316620e654a436": "\\left[\\int _{0}^{1}(1-U(x))\\,dx,\\int _{0}^{1}(1-L(x))\\,dx\\right].", "10db9ebe9cbcc9a65480d326235caffb": "x^{4}+18x^{3}+23x^{2}+8x+1", "10dc276caea41ecc0e3ecf39fc3aca22": "\\Omega =d\\omega +\\omega \\wedge \\omega =0.", "10dc52167ae97abd7ec6f4d4c89a532c": "z\\mapsto uz.", "10dc75b26ed9777d36a3ba7ed338e8f9": "n=N,\\alpha ,\\,\\beta \\!", "10dca1af11a65ac9a06c0ee2b56b907d": "\\scriptstyle q_{e}^{2}q_{Z}^{2}=q_{e}^{4}\\,", "10dd04ab5dda18d086a6048e29272b7f": "K_{H}=\\int \\limits _{-\\infty }^{x'}{\\big [}k_{H}^{{\\widetilde {u}}(x)}-k_{H}{\\big ]}dx-\\int \\limits _{x'}^{\\infty }{\\big [}1-k_{H}^{{\\widetilde {u}}(x)}{\\big ]}dx", "10dd2476912f3b7ec3a4b86d418944c9": "~~~~~S,p,\\{N_{i\\neq j}\\},\\mu _{j}\\,", "10dd73f7d88ac75c34b6542921b0ae1a": "\\ B\\,", "10dd8fa7ff9196e4b5bd5e200d6d0506": "a_{m,n}+a_{m,n+1}+a_{m+1,n}+a_{m+1,n+1}=34", "10de06fa069a8d1cbcac910f71e8c40c": "[b^{3},b^{4}]", "10de86a15a513180b9deaff8f6145372": "{\\pi }", "10df8ee196666c9e2ebea43c48cc11e3": "\\Phi (r,\\theta ,z;r',\\theta ',z')=a'\\Phi _{\\infty }(r,\\theta ,z;r',\\theta ',z')-a'\\Phi _{\\infty }(r,\\theta ,z;r',\\theta ',-z'-2z_{b})", "10df9a6dcd5de2a5b151f831fca7759c": "\\sum _{l=0}^{N_{j}-1}\\left|2^{\\frac {3j}{4}}{\\hat {\\phi }}_{j,0,0}(r,\\omega -{\\frac {2\\pi l}{N_{j}}})\\right|^{2}=\\left|W(2^{-j}r)\\right|^{2}\\sum _{l=0}^{N_{j}-1}{\\tilde {V}}_{N_{j}}^{2}(\\omega -{\\frac {2\\pi l}{N}})=\\left|W(2^{-j}r)\\right|^{2}", "10dfd3e1f8a43ead38b23b9f32f6948b": "\\mathbb {Z} /p", "10dfdcb4c0a14620873148af4214dc28": "a^{n-1}\\not \\equiv 1{\\pmod {n}}", "10dff05cc4d1d30ebe2e9ec2e4604a2c": "\\rho _{A}(X)=\\inf\\{u\\in \\mathbb {R} :X+u1\\in A\\}", "10e05521d0da50a8c6a59c9bc5dc320d": "V\\rightarrow U", "10e084e983c9507caf9014c40e479cf2": "M_{+}^{1}(A)\\times M_{+}^{1}(A)\\rightarrow [0,\\infty {})", "10e0dd76c46da5fa492f40a634ec23cb": "x_{1},x_{2}\\in X", "10e1042b56900798d95e67427511971d": "\\ln S_{t}", "10e1104ac9b22a46729ada710a651c31": "0\\ f\\ x=x", "10e15af0938930ead144c97eba0e7dec": "\\sigma _{2c}=4.6", "10e16c6a764d367ca5077a54bf156f7e": "\\sigma ^{2}", "10e196a4eee70e2d42ccfd3eeff37969": "\\sum _{c\\,\\in \\,C}(p_{c,t}\\cdot q_{c,t})=\\sum _{c\\,\\in \\,C}[(P_{t}\\cdot p'_{c,t})\\cdot q_{c,t}]=P_{t}\\cdot \\sum _{c\\,\\in \\,C}(p'_{c,t}\\cdot q_{c,t})", "10e1a6140d3569e0b5d723b335277017": "\\epsilon >\\|f(t)-f_{e}\\|", "10e1ce139d2e15d736318ef7e14d3bd9": "d_{X}\\Delta =0", "10e1ef4e195f389a2640d9321a0f6936": "r\\leq d", "10e22a5acb1d5cfe9ce5d3eb73884243": "F(x)=f(x)+ig(x)", "10e31ef2b67b23aed0351f7448f12369": "S^{p,q}", "10e324f2ceef507b29a7581bf01c7b12": "y(0)=0.", "10e343d8eb2ab64dacb8ae345eb765ea": "|\\psi (\\mathbf {r} ,t)|^{2}\\to 0", "10e3b4c8abc4c4b7b6ea1acca1efab41": "M_{\\odot }", "10e400abc162ac9b9e0dabb6b47336f8": "Cv(b)=Cv(a)+\\int _{a}^{b}i(t)dt.", "10e4538433703868de1dd42648521159": "\\mathbf {J} =nq\\langle \\mathbf {v} \\rangle ,", "10e48e317edea309efe28c0c7be85826": "OBV=OBV_{prev}+\\left\\{{\\begin{matrix}volume&\\mathrm {if} \\ close>close_{prev}\\\\0&\\mathrm {if} \\ close=close_{prev}\\\\-volume&\\mathrm {if} \\ close1\\end{aligned}}", "10f3f9c364a3d3ed7a7b6fba42b5e125": "{\\begin{Vmatrix}x&y\\\\z&v\\end{Vmatrix}}", "10f40f397dca533b29193d407258d0da": "e^{2\\pi i/7}", "10f46008ea75bd8724e8b496799d7560": "p_{k,i}^{C}<{\\frac {1}{2}}+{\\frac {1}{Q(k)}}", "10f46706ddd9ad8305035939ad897d92": "d=180", "10f495b4e951124d0018fe8d90a56ba0": "h^{0}(X,L)=1", "10f4a6c0fbf00aaaca912101c7c36c58": "P_{FNL}\\approx \\mathrm {exp} \\left(-{B_{g}}^{2}\\tau _{th}\\right)", "10f4b44378472a2a883fad0f67713fbc": "n_{t+1}(x)=\\int _{\\Omega }k(x,y)\\,f(n_{t}(y))\\,dy,", "10f4c37af9bdc057940bbc18b032c08d": "\\langle \\sigma \\rangle ", "10f4fb3c812f816984bb8e6ad192a818": "\\mathbf {} V(t)", "10f6053ad1dd29614967d5da8ecb36c2": "X[Y/Z]", "10f625ee8d71697fee8df97b3f8dc527": "\\sum _{n=1}^{\\infty }{\\frac {\\sigma _{a}(n)\\sigma _{b}(n)}{n^{s}}}={\\frac {\\zeta (s)\\zeta (s-a)\\zeta (s-b)\\zeta (s-a-b)}{\\zeta (2s-a-b)}}.", "10f6369053f87f9504e22666522480df": "T_{i}^{2}-(x^{2}-1)U_{i-1}^{2}=1.\\,", "10f685669f14a794377659dccc9ed427": "\\textstyle x\\in X", "10f6b407daf02fbdc1b33b14a8fa7751": "\\mathbf {\\hat {p}} |k\\rangle =k|k\\rangle ", "10f6bf04f81a6a4db7ad206d6c8dda0a": "[x_{1}:x_{2}]", "10f6fbd31545a0a9a19989d2bcbad7eb": "{a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}=1", "10f71c9b2dbcab8c31fe6f4d4b98758f": "\\operatorname {ord} (\\phi \\vert _{K})=[O_{K}/\\delta O_{K}:\\mathbf {Z} /q\\mathbf {Z} ].", "10f76616946b0a4e9e33f1a83c4c0955": "{\\begin{aligned}\\Delta &{}={\\frac {\\sqrt {(a+b+c)(a+b-c)(b+c-a)(c+a-b)}}{4}}\\\\&{}={\\frac {1}{4}}{\\sqrt {2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})}}.\\end{aligned}}", "10f7721e867ed84fdd9532ad22f63f53": "\\int _{0}^{\\infty }({\\frac {1}{e^{x}-1}}-{\\frac {e^{-x}}{x}})dx=\\gamma ", "10f78d77368b3170ffbac9c54786d63c": "{\\mathcal {M}}\\cong \\operatorname {PSL} (2,\\mathbf {C} )", "10f7cf89be8ff6206bbca376a30e7930": "(\\omega _{1},0,\\omega _{0}+\\omega _{r}/2)", "10f7db2a7ae08b93d9eefec15cd7a76a": "\\Phi _{1}^{-1}\\circ \\Phi _{2}=\\mathrm {Hom} (\\varphi ,-)", "10f7db55a055f3b0bc669d2c785cf132": "{\\frac {y-Y_{0}}{Y_{1}-Y_{0}}}\\approx {\\frac {x-X_{0}}{X_{1}-X_{0}}}", "10f7fec54220538ced34a65a1a2d28f6": "{\\vec {\\Omega }}={\\frac {{\\vec {V}}_{r}\\times {\\vec {R}}}{{\\vec {R}}\\cdot {\\vec {R}}}}", "10f80fcece2bd8202d21042a418518ed": "D[||\\partial _{j}]=\\partial '_{j}D(p||p)=0", "10f8108e609759e20476df797b027818": "{\\mathcal {L}}_{X}Y:=\\left.{\\frac {1}{2}}{\\frac {\\mathrm {d} ^{2}}{\\mathrm {dt} ^{2}}}\\right|_{t=0}\\Phi _{-t}^{Y}\\circ \\Phi _{-t}^{X}\\circ \\Phi _{t}^{Y}\\circ \\Phi _{t}^{X}=\\left.{\\frac {\\mathrm {d} }{\\mathrm {d} t}}\\right|_{t=0}\\Phi _{-{\\sqrt {t}}}^{Y}\\circ \\Phi _{-{\\sqrt {t}}}^{X}\\circ \\Phi _{\\sqrt {t}}^{Y}\\circ \\Phi _{\\sqrt {t}}^{X}\\,", "10f84c85c8bc5c78a60b666fcd4d6fd9": "H_{q}(x)\\equiv _{def}-x\\cdot \\log _{q}{x \\over {q-1}}-(1-x)\\cdot \\log _{q}{(1-x)}", "10f871805364e3b2edcbeee007c07ede": "\\left({\\frac {N_{1}}{N_{2}}}\\right)_{T}", "10f8b56ea13c46a90c4180bb43498294": "\\Gamma _{\\varphi }\\ =\\ \\int d{q}\\int {\\frac {d\\omega }{2\\pi }}\\,{\\tilde {S}}({q},\\omega )\\,{\\tilde {P}}(-{q},-\\omega )", "10f8cdb2300466e9a4344fb2acd86552": "R(p_{b})={\\frac {C}{1-H_{2}(p_{b})}}.", "10f8fde9f843ff8a9984b509036c49f1": "P,P^{*}", "10f9132e586392a6513eeeb303db630c": "yi", "10f92f7e05286641e8a4baec0d3da30f": "\\pi /2", "10f937112d89ed784552c7af9ac763ec": "[x,y]=xy-yx", "10f95fd0d3e47c5973c7f562260ad572": "y[n]=\\sum _{m=-\\infty }^{\\infty }{h[n,n-m]x[n-m]}", "10fa0278ab54eb34767f6cffab9259b3": "10-3+2\\,", "10fa5ad9536941ac7b45268510f8b815": "\\theta _{s}={\\frac {L_{s}}{2R_{c}}}=L_{s}^{2}", "10facdc361b9034de46cfc789cf18295": "\\mu _{i}=\\mu _{i}^{\\star }+RT\\ln x_{i}\\,", "10faea16cffd1370a2f40d8bb96f6316": "m=(1+\\epsilon )(n+1)\\log {q}", "10fb109c7f10edcbd87bf4b6d1ca7a53": "T_{\\nu \\dots }^{\\mu \\dots }\\,", "10fb2c81feb695801f7f550c18f6361b": "{\\begin{bmatrix}\\alpha -1\\\\-\\beta \\end{bmatrix}}", "10fb2cabe67272af59733c4b404d9a90": "d(i)=\\sum \\limits _{j}w_{ij}", "10fb63698de8e6f0369f75eaebee667f": "e\\cdot B^{-1}", "10fc34ae7d2d44980d9a409eb1938ca6": "A_{0}(x)=0", "10fc4bfdd36f48519636f1529f6abc9e": "b(x)^{m}", "10fc95efe0bf9dd932c53c256d5c3ad6": "{\\frac {V}{l}}={\\frac {I}{a}}\\rho \\qquad {\\text{or}}\\qquad V=I\\rho {\\frac {l}{a}}.", "10fca3f454e370e68e5d4ef5bf3ddf62": "y'(t)=f(t,y(t)),\\quad y(t_{0})=y_{0},", "10fcaa7bf3452453fd00b97960bf9c1a": "{\\begin{aligned}x&=r\\cos \\phi \\\\y&=r\\sin \\phi \\\\z&=z\\end{aligned}}", "10fcf43d858c03dafc054d102f147ce6": "\\bigcap _{i\\in I}C_{i}\\neq \\emptyset ", "10fd22543d21500e6751e6346c984f8c": "G_{i}=1/R_{i}\\,", "10fd6a3702e89c5c360d99db2c7a4805": "\\mathbf {a} _{AB}={\\frac {d^{2}}{dt^{2}}}\\mathbf {X} _{AB}", "10fd9c5893e5e8cb5137151faeb8f3ec": "Ld=1\\,Laborer\\cdot 0.5\\,{\\frac {Ph}{Laborer}}=0.5\\,Ph", "10fdaf2ab79128a041a6403212d1c8f0": "Prob_{pure}=e^{-2G}", "10fe11584f7d5f67aa0aecca0ca95731": "{\\Delta }P", "10fe9a7e35b02412ce6ab517840ff4df": "V_{y}=v\\sin \\theta -{\\frac {gx}{v\\cos \\theta }}", "10fea0b6eda684767263baa4da13daad": "\\nabla ^{2}\\mathbf {E} -\\mu _{0}\\epsilon _{0}{\\frac {\\partial ^{2}\\mathbf {E} }{\\partial t^{2}}}=0.", "10fef7f1ece06914d356929bb2f02d91": "{\\vec {F}}_{12}={\\frac {\\mu _{0}I_{1}I_{2}}{4\\pi }}{\\frac {2}{D}}(0,-1,0)\\int _{L_{1}}dx_{1}", "10ff1a9e235307c647eb79d9503321da": "c=kp\\,", "10ff1b27d7648f47afcbfe2927d442ff": "L^{p}(\\mathbb {R} )", "10ff2ff0656919d5fd52ba752f034ca5": "\\scriptstyle C_{c}^{1}(\\Omega ,\\mathbb {R} ^{n})", "10ff3921d987215a0a44f012c768dc08": "\\langle \\partial _{t}u,e^{ikx}\\rangle =\\langle {\\frac {1}{2}}u^{2}-\\rho \\partial _{x}u,\\partial _{x}e^{ikx}\\rangle +\\langle f,e^{ikx}\\rangle \\quad \\forall k\\in \\left\\{-N/2,\\dots ,N/2-1\\right\\},\\forall t>0.", "10ff3951f3c68b7ef1da98837f2707cb": "k_{2}=f(t_{0}+{\\tfrac {2}{3}}h,y_{0}+{\\tfrac {2}{3}}hk_{1})=2.7139", "10ff4194bb5fbd72798c3a7b057fea0d": "C({\\vec {N}})", "10ffd8f26b66effcadf3af92fc9ad000": "M_{B_{1}}^{i},\\dots ,M_{B_{k}}^{i}", "10ffe7009b277632c4f8045875566bb4": "c_{f}\\approx ", "110022d17790fed7bc43781dfd1b7eb9": "x[0]=\\lim _{z\\to \\infty }X(z).", "110035449b34621bf2da984cd87cffcf": "a^{n}x+{\\frac {a^{n}-1}{a-1}}b", "1100361a91ac2da2bd6a80d0d1ca1636": "\\Omega ^{1}\\to \\Omega ^{2}", "110052477e50e69862dce956ef5e1f89": "(n_{x},n_{y},n_{z})=(3,5,7)", "11008e2549109c23f79fe494bd2b6ed6": "m_{\\text{P}}=c^{n_{1}}G^{n_{2}}\\hbar ^{n_{3}},", "11009eff1b06cea1b8d716c777254188": "n\\,", "1100b76bf84f16689dfc7deb6533146e": "\\delta <\\alpha ", "1100d11beff5e2b62c7f2c8142232f4a": "\\mathrm {S} \\!\\!\\!\\Vert ", "11010df7e53eac5d8da576b6a0ddfbf9": "LR-={\\frac {\\Pr({T-}|D+)}{\\Pr({T-}|D-)}}", "11013a75061a55751f28de2c74ade4d5": "p\\times 1", "110145a930d21967d1147147ea289e66": "\\Rightarrow q_{2}^{*}={\\frac {a-3\\cdot {\\frac {\\partial C_{2}(q_{2})}{\\partial q_{2}}}+2\\cdot {\\frac {\\partial C_{1}(q_{1})}{\\partial q_{1}}}}{4b}}.", "110198bffc5514ac79c683ca681a7600": "f'\\circ f=g'\\circ g\\,", "1101c99afd8028f653cb43f10f4f3ce6": "p_{4}=m_{1}q_{3}(1+m_{2})\\ .", "1101ca65bea66aef754d5bd34892e220": "h(w)=\\sum _{n\\geq 1}{\\lambda _{n} \\over n}\\Phi _{n}(w)+\\sum _{n\\geq 1}{\\lambda _{-n} \\over n}\\Phi _{-n}({a \\over w}).", "1101f7f3d2b4039dea68701f38ea9ed7": "\\operatorname {etr} (A):=\\exp(\\operatorname {tr} (A))", "11020b4f21edd4f09a0ead06b6d63627": "\\Psi =e^{i(kx-\\omega t)}\\,\\!", "1102b844e752112397f05d00e91324a1": "A^{\\alpha }{\\vec {u}}=\\sum _{k=1}^{\\infty }\\lambda _{k}^{\\alpha }u_{k}{\\vec {w_{k}}}", "11030e148c36253940edfc08ede968d1": "h_{\\mu }={\\frac {1}{2}}{\\sqrt {\\frac {\\left(\\nu -\\mu \\right)\\left(\\lambda -\\mu \\right)}{\\left(A-\\mu \\right)\\left(B-\\mu \\right)}}}", "1103137ca878e48a577d7b65f17013ab": "9k\\equiv -1{\\pmod {7}}", "1104572e09c96d0ca09b566665e394ee": "\\{x\\leq 10\\}\\;\\mathbf {while} \\ (x<10)\\ x:=x+1\\;\\{\\lnot (x<10)\\land x\\leq 10\\}", "1104d00b81ae3af412f3d583183c1379": "2\\cdot A_{n}\\to A_{n}.", "1105080e231ecf26cd884aea3ff8b624": "S_{k}=\\sum _{n=1}^{k}(-1)^{n-1}a_{n}\\!", "110555287c78c0f3e148298cda41d666": "\\Lambda _{MS}=217_{-23}^{+25}{\\rm {\\ MeV}}.", "11055c7adb9c8a326c3469790173af18": "\\alpha ={\\frac {\\mu }{a}}", "110574a5bb89681c6bc07de2eb2af8e6": "{\\mathcal {N}}\\int e^{-\\beta H(p,q)}x_{k}{\\frac {\\partial H}{\\partial x_{k}}}\\,d\\Gamma ={\\Bigl \\langle }x_{k}{\\frac {\\partial H}{\\partial x_{k}}}{\\Bigr \\rangle }={\\frac {1}{\\beta }}=k_{B}T.", "11057a773dc90dff2e5c9cd86fe68402": "R(\\theta ,\\delta )=\\operatorname {E} _{F(x\\mid \\theta )}[{L(\\theta ,\\delta (x))]}.\\,\\!", "110585049926feaaa6e156f2e0f2a5e5": "DP_{S}^{T}", "1105a7158f1cba5299aa791c50832c34": "u_{i},~i=1,2,3", "1105b792fa5f2e515c85aa2ea0e94418": "(-1)^{m}\\psi ^{(m)}(1-z)-\\psi ^{(m)}(z)=\\pi {\\frac {d^{m}}{dz^{m}}}\\cot {(\\pi z)}=\\pi ^{m+1}{\\frac {P_{m}(\\cos(\\pi z))}{\\sin ^{m+1}(\\pi z)}}", "1105dc5e3c5a368c1afd2c9721017a8c": "Q_{1}", "1105df8f8e65e5bbf15b59e5c08504bf": "\\displaystyle {\\overline {{\\hat {f}}(-\\nu )}}", "1105e3f63832168d478e6f8ddb440f12": "\\log c", "110629f545ec2d54c06a639d80ec39e1": "\\int f\\,\\mathrm {d} \\mu ", "11063aba125c8de2441aee5feed84ffe": "a_{i}=\\gamma _{xi}x_{i}\\,", "110657a47e7a40ff5221e5bca56ceccd": "2.00356", "11069c05d732517bd08a43546fc02a73": "\\approx {\\frac {1}{2}}{\\begin{bmatrix}1\\\\\\delta \\mathbf {x} \\\\\\delta \\mathbf {u} \\end{bmatrix}}^{\\mathsf {T}}{\\begin{bmatrix}0&Q_{\\mathbf {x} }^{\\mathsf {T}}&Q_{\\mathbf {u} }^{\\mathsf {T}}\\\\Q_{\\mathbf {x} }&Q_{\\mathbf {x} \\mathbf {x} }&Q_{\\mathbf {x} \\mathbf {u} }\\\\Q_{\\mathbf {u} }&Q_{\\mathbf {u} \\mathbf {x} }&Q_{\\mathbf {u} \\mathbf {u} }\\end{bmatrix}}{\\begin{bmatrix}1\\\\\\delta \\mathbf {x} \\\\\\delta \\mathbf {u} \\end{bmatrix}}", "1106c567f4722709c525ecb870fcf51c": "f(x)={\\frac {1}{(x-1)(x^{2}+x+1)}}", "1106f844677df0c805a8cc002fcb1470": "{\\dot {\\varepsilon }}", "11073c2cfeb60b6c04ec95959ab3ca15": "{\\hat {\\alpha }},{\\hat {\\beta }}", "11076c1c9b6b49e95f5f9e377dc42278": "t_{nm}=\\left[\\mathbf {T} \\right]_{nm}", "11076fafbd8872769c4a44689ae803d0": "\\Delta \\epsilon ", "1107be8d11e3f0f354bd0ad697f7562c": "\\alpha ={\\frac {2ax+b\\pm {\\sqrt {(2ax+b)^{2}-16}}}{4}}", "1107f25512e907df730bdb847d8c75d7": "a_{k}=2\\langle \\phi (x),\\phi (2x-k)\\rangle ", "1108517637315bc0360bca1416cfc517": "u\\sim w_{0}\\,", "1108c2522505a49ba4219e2fbe6a5ca9": "\\cos a\\cos b", "110943d95fc5ddcd8a009f6a9b63486b": "|D|>0", "11096792ca8dde823d044cc948f15534": "\\operatorname {Im} ", "11096ba55e57b0ba1b35efb241f87569": "\\epsilon \\neq 0", "110972b39a69695d1bd6b63fcff9b8c9": "{\\mathit {F}}={\\frac {a^{2}}{L\\lambda }}", "11097ba45d6b2fc241b6ffee9e3f07b4": "\\mu =a_{1}+2a_{2}", "11098b06bc5ce941f1283ed5e081ea17": "dW/dt=0", "1109a243eb6e39d5536973d96353cbb4": "\\cdot e^{-{\\frac {\\nu ^{2}}{2a^{2}}}}H_{n}\\left({\\frac {\\nu }{a}}\\right)", "1109e2f82871c8b4af63678e4fae1a52": "\\mathbf {x} _{k}\\propto \\mathbf {A} \\,\\mathbf {y} _{k}", "1109f06297279a41dedd7973cfe667a0": "-D_{1}{\\frac {\\partial C_{1}}{\\partial x}}", "110a38a973ecc3a6d5d642e75c3a6671": "x=x_{i}", "110a3ea61e602565f8caf38c0b2c2c04": "L_{1}\\times L_{2}\\equiv L_{2}\\times L_{3}\\equiv L_{3}\\times L_{1}.", "110a451bff14cf0501f2ebb7c418a581": "u_{0}", "110a8755d8163407ee9c371792d932ef": "C={\\frac {u_{x}\\,\\Delta t}{\\Delta x}}+{\\frac {u_{y}\\,\\Delta t}{\\Delta y}}\\leq C_{max}", "110aca91bcd7ff0e9c4a27cfc5a08ea4": "\\forall x\\forall y\\forall s[(\\langle x,s\\rangle \\in F\\land \\langle y,s\\rangle \\in F)\\rightarrow x=y])].", "110b013cca576480b40669e4d960abd5": "H(A:B)=H(A)+H(B)-H(A,B)\\,", "110b3a66217e97116fbd198a0cb1ba8e": "L(s,\\chi )=\\sum _{n=1}^{\\infty }{\\frac {\\chi (n)}{n^{s}}}", "110b9257f9d7d2d395edec28d272930a": "L_{n}^{(\\alpha )}", "110b9df0a7948a3528fe05ba12a861f5": "\\operatorname {d} L_{\\text{r}}(\\omega _{\\text{r}})", "110ba441223e562aa02c7c32f54f12eb": "1+r={\\frac {M_{1}-C_{1}}{M_{0}}}\\times {\\frac {M_{2}-C_{2}}{M_{1}}}\\times {\\frac {M_{3}-C_{3}}{M_{2}}}\\times ...\\times {\\frac {M_{n-1}-C_{n-1}}{M_{n-2}}}\\times {\\frac {M_{n}-C_{n}}{M_{n-1}}}", "110bda9da3f5e510b3e317f0585d36f7": "y_{1}={\\frac {2y_{g}}{-1+{\\sqrt {1+{\\frac {8gy_{g}^{3}}{q^{2}}}}}}}={\\frac {(2)(0.30)}{-1+{\\sqrt {1+{\\frac {8(9.8)(0.30)^{3}}{3.70^{2}}}}}}}=8.04{\\text{ m}}", "110c2cf45ddb788fbb479bd105c78cd8": "\\delta \\rho -{\\bar {\\delta }}\\sigma =\\rho ({\\bar {\\alpha }}+\\beta )-\\sigma (3\\alpha -{\\bar {\\beta }})+(\\rho -{\\bar {\\rho }})\\tau +(\\mu -{\\bar {\\mu }})\\kappa -\\Psi _{1}+\\Phi _{01}\\,,", "110c8597512e49796437092e71b6c94d": "(1)\\;Fr_{1}={\\frac {v}{\\sqrt {gy}}}={\\frac {10[m/s]}{\\sqrt {9.81[m/s^{2}]*0.5[m]}}}=4.5", "110ccacdf41482aa6a44192990e56bcf": "\\nabla h_{i}(i\\in {\\mathcal {E}})", "110d1ec129ecf4c227b843be04fdfc29": "\\Delta _{*}\\colon H_{\\bullet }(X)\\to H_{\\bullet }(X\\times X)", "110d95a683d386b31d93433baf49db52": "p(x)=p(x|\\theta )", "110d97436114572b3b0ef30429e4fe8b": "(x^{n})'=nx^{n-1},", "110da4245831429b2f43162353ffafa8": "H_{n}(a,b)={\\begin{cases}b+1&{\\text{if }}n=0\\\\a&{\\text{if }}n=1,b=0\\\\0&{\\text{if }}n=2,b=0\\\\1&{\\text{if }}n\\geq 3,b=0\\\\H_{n-1}(a,H_{n}(a,b-1))&{\\text{otherwise}}\\end{cases}}\\,\\!", "110dc2975f1834c40361303b53b676a5": "\\mu (x):\\mathbb {R} {\\rightarrow }\\mathbb {R} ", "110e2bf8ead8b8997c7a14d025ce30a4": "I_{\\mathrm {max} }", "110e3416e03f22194e5e9234d4a1b327": "N_{xx}=\\int _{A}\\sigma _{xx}~\\mathrm {d} A~;~~M_{xx}=\\int _{A}z\\sigma _{xx}~\\mathrm {d} A", "110ebbbe7ffaf067a5dd5dbedb369ac6": "\\operatorname {E} [g(X)]=\\int _{-\\infty }^{\\infty }g(x)f(x)\\,\\mathrm {d} x.", "110eee294603f883d5c15c8179d8baa0": "\\omega _{0}={1 \\over {\\sqrt {LC}}}", "110f0e7b0458f8b860f8d36a2776c659": "{\\textit {ADJ}}\\;{\\textit {NOUNPHRASE}}", "110faf0a710447a4d467cc98d586a0ce": "\\hbar \\omega _{n}=R\\left({\\frac {1}{2^{2}}}-{\\frac {1}{n^{2}}}\\right)\\quad n=3,4,5,...", "110fca3e22a1dbfe1abb234ca3259ea5": "I_{L_{\\text{max}}}={\\frac {V_{i}\\,D\\,T}{L}}", "110fd17ed759bb2d02a8e1a1e1ba93da": "\\int {\\frac {dx}{\\sqrt {x^{2}-1}}}={\\mbox{arccosh}}(x)+C\\quad (x>1)", "110ffbbb53c2696a1252640beb988e3b": "S(q_{1},\\dots ,q_{N},t)=W(q_{1},\\dots ,q_{N})-E\\cdot t", "111019a279c754001bea7f9d5343b55c": "{\\frac {1}{r^{2}}}", "111024e02e58228a7435e46dcdf4bf96": "r(t)=r_{e}\\left[1-\\exp \\left(-\\left({\\frac {2\\gamma _{LG}}{r_{e}^{12}}}+{\\frac {\\rho g}{9r_{e}^{10}}}\\right){\\frac {24\\lambda V^{4}(t+t_{0})}{\\pi ^{2}\\eta }}\\right)\\right]^{\\frac {1}{6}}", "1110544cf450124486e6a6c8a315ff70": "\\int _{-a}^{a}f(z)\\,dz=\\pi e^{-t}-\\int _{\\mathrm {arc} }f(z)\\,dz.", "11107bf7200e8ea8c65157a3ab5eed0e": "\\Gamma \\,", "1110948f63b45eebb689ea45bd20e33f": "n_{t+1}=f'(0)k*n_{t}", "1111091518909c264d8fba533dd169da": "\\triangle {M}=M_{1}-M_{2}=25.5", "11113478595309bdef22cb31bc94b0dc": "\\psi _{i+2}", "111185f76fc35ac247d23bc2ea139aea": "r=4s(1-s)", "111196a2755fc7ac7b4fbf9c739c0422": "0.872", "1111f4cf24ae298dc7a326a55c09c108": "\\ln(I_{z})=-\\sigma Nz+C.\\,", "1111fd11a6fa9dcb7af73680df44de80": "{\\textit {dau}}(m,h)\\leftarrow {\\textit {par}}(h,m)\\land {\\textit {par}}(h,t)\\land {\\textit {par}}(g,m)\\land {\\textit {par}}(t,e)\\land {\\textit {par}}(n,e)\\land {\\textit {fem}}(h)\\land {\\textit {fem}}(m)\\land {\\textit {fem}}(n)\\land {\\textit {fem}}(e)", "1112838385b37079efd51e340b3cbd75": "\\mathbb {C} /L", "111310acf2892086d3e75dd8c5523628": "p({{\\boldsymbol {x}}|{\\rm {label}}})", "11134ff62a3534476fc6cc89687d67bf": "a_{t}", "111355b8fd3c56f629ebb75905df6888": "\\scriptstyle \\delta x", "11136d975f9e0c5a36bbe8c6a6c30c65": "a\\neq d", "11138f63622d9aacf6ddb06b9e57b138": "\\rho ={\\frac {p}{R_{\\rm {specific}}T}}", "1113ddb156abcd31f04d54feb03cda9a": "\\mathrm {gyr} [\\mathbf {u} ,\\mathbf {v} ]\\mathbf {w} =\\ominus (\\mathbf {u} \\oplus \\mathbf {v} )\\oplus (\\mathbf {u} \\oplus (\\mathbf {v} \\oplus \\mathbf {w} ))", "111404248a0360274a22146234c585b4": "s_{(a^{p})}s_{(b^{q})}", "1114799c925afee9bbda93fe79a96633": "P(a_{T+1}|{\\hat {a}}_{1:T},o_{1:T})=\\int _{\\Theta }P(a_{T+1}|\\theta ,{\\hat {a}}_{1:T},o_{1:T})P(\\theta |{\\hat {a}}_{1:T},o_{1:T})\\,d\\theta ", "11147b59ea9e585fca5b2314d6ecbc4a": "ay_{1}v''=0.\\;", "1114c80010a66a42faea3d20cfa4cbd2": "|S|=n", "1114d50f217bc0281228d0c670b26336": "\\delta _{k}", "1114da8d3d28a20e353ebff4ec0da999": "f=-k_{B}TN^{-1}\\log Z.", "1114f50a824355c5a96324d9cc456dd1": "\\int \\!\\!{\\frac {x-a}{(x-a_{1})(x-a_{2})}}\\,\\mathrm {d} x={\\frac {(a_{1}-a)\\ln(x-a_{1})-(a_{2}-a)\\ln(x-a_{2})}{a_{1}-a_{2}}}+C", "11155ed053e766272d7cbd379321ebfb": "{\\frac {1}{2}}mv^{2}", "11156ef4182af303a38779b329afb012": "{\\hat {f}}(5)", "111571612bcd52aea3c64a4121c65d3b": "1/2+1/4+1/8+1/16+1/32+1/64+(5\\ ro)", "111585dd48e959cb11225784a88a06b4": "\\Omega _{p}:\\sigma \\to 2\\pi -\\sigma ", "1115cc55da57b8046f88d722e9c8b8fd": "\\omega |_{\\xi }>0", "1115cc7289d83d9234925bdcfad2b88d": "(x^{i_{1}},\\sigma ^{j_{1}})(x^{i_{2}},\\sigma ^{j_{2}})=(x^{i_{1}+i_{2}2^{^{j_{1}}}},\\sigma ^{j_{1}+j_{2}})", "111652c367c6eff15bf4836e56ecf561": "\\Pr(5{\\text{ heads}})=f(5)=\\Pr(X=5)={6 \\choose 5}0.3^{5}(1-0.3)^{6-5}\\approx 0.0102", "1116bc7b60a1156225cbcf51ea0614f0": "\\scriptstyle x\\;=\\;x_{0}\\,+\\,h,\\;y\\;=\\;0,\\;\\;z\\;=\\;0", "1116d784637b4b4292dd2649b22e6e6b": "x_{i}=\\cos \\left({\\frac {2i-1}{2n}}\\pi \\right){\\mbox{ , }}i=1,\\ldots ,n.", "1116e1424b39c12d01de442e53799d8b": "x+y=y+x", "1116f44a804480205e0fde026b5afa4b": "V_{\\left(p-k\\right)}^{T}{\\boldsymbol {\\beta }}_{*}=\\mathbf {0} ", "11173d648d53d06aca0b1e4024515e2f": "{\\frac {1}{f}}={\\frac {1}{d_{o}}}+{\\frac {1}{d_{i}}}", "11176deba9a0cdb756a12de6db30bd63": "{\\begin{cases}{\\frac {dx}{dt}}=-y-z\\\\{\\frac {dy}{dt}}=x+ay\\\\{\\frac {dz}{dt}}=b+z(x-c)\\end{cases}}", "11178357bc66a35266c727b2f657438c": "\\sigma =\\int _{r_{\\mathrm {obs} }}^{r_{\\mathrm {atm} }}{\\frac {\\rho \\,\\mathrm {d} r}{\\sqrt {1-\\left({\\frac {n_{\\mathrm {obs} }}{1+(n_{\\mathrm {obs} }-1)\\rho /\\rho _{\\mathrm {obs} }}}\\right)^{2}\\left({\\frac {r_{\\mathrm {obs} }}{r}}\\right)^{2}\\sin ^{2}z}}}\\,.", "1117bf4e60f49773ad7e18289e3bb115": "A\\oplus B=\\bigcup _{b\\in B}A_{b}", "1117cdb74845ff280dceb1c070770d44": "p_{2}-p_{1}\\,", "1118433b9ccb65b7c14e48985612938b": "\\int _{0}^{\\infty }{\\frac {\\gamma x+\\log \\Gamma (1+x)}{x^{5/2}}}\\,dx={\\frac {2\\pi }{3}}\\zeta \\left({\\frac {3}{2}}\\right)", "11189aa6b1d37bd8d4e8fe3c79653f03": "a{\\frac {x_{c}}{t_{c}^{2}}}{\\frac {d^{2}\\chi }{d\\tau ^{2}}}+b{\\frac {x_{c}}{t_{c}}}{\\frac {d\\chi }{d\\tau }}+cx_{c}\\chi =Af(\\tau t_{c})=AF(\\tau ).", "1118a7c0f92998eae45123971960361a": "~\\omega _{\\rm {s}}~", "1119725f6a8281d25ce18c40a995f5eb": "Y_{t}=a\\cdot t+b+e_{t}", "1119c8d1911a1511da2c8d3c85c760cd": "f(\\partial X)", "1119e1a2b8d600202c198255528f1946": "B_{\\delta }", "111a2979f2509c2d09f082f039efbb12": "={\\frac {V_{nk_{4}}V_{k_{4}k_{3}}V_{k_{3}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{3}}E_{nk_{4}}}}-E_{n}^{(2)}{\\frac {|V_{nk_{4}}|^{2}}{E_{nk_{4}}^{2}}}-2V_{nn}{\\frac {V_{nk_{4}}V_{k_{4}k_{3}}V_{k_{3}n}}{E_{nk_{3}}^{2}E_{nk_{4}}}}+V_{nn}^{2}{\\frac {|V_{nk_{4}}|^{2}}{E_{nk_{4}}^{3}}}", "111a538025ee0698a7a26c922b5b9a86": "{\\frac {\\partial h}{\\partial t}}+C{\\frac {\\partial h}{\\partial x}}=D{\\frac {\\partial ^{2}h}{\\partial x^{2}}},", "111a63e3fd193e80d2cfd33b187bbc12": "L=F/\\sim ", "111b3cda00313d80e2967f6e045ad149": "r_{O}", "111bb025692e5ba52c23eae1d47b2035": "D=1", "111be2a57cccb50dece2c4598eca667b": "[T]={}^{0}T_{n}=\\prod _{i=1}^{n}{}^{i-1}T_{i}(\\theta _{i}),", "111be2ac0bacdca11c8d0be7c6beae80": "{\\frac {\\left({\\cfrac {q^{2}}{4\\pi \\epsilon _{0}L_{1}^{2}}}\\right)}{\\left({\\cfrac {q^{2}/4}{4\\pi \\epsilon _{0}L_{2}^{2}}}\\right)}}={\\frac {mg\\tan \\theta _{1}}{mg\\tan \\theta _{2}}}\\Longrightarrow 4{\\left({\\frac {L_{2}}{L_{1}}}\\right)}^{2}={\\frac {\\tan \\theta _{1}}{\\tan \\theta _{2}}}", "111c0693607c0ed88a3d6936a4434532": "j_{z}=m_{j}\\,\\hbar ", "111c77400d4672c1936276d2647df165": "m(x)\\leq {\\frac {f(x)-f(y)}{g(x)-g(y)}}={\\frac {{\\frac {f(x)}{g(x)}}-{\\frac {f(y)}{g(x)}}}{1-{\\frac {g(y)}{g(x)}}}}\\leq M(x)", "111cb170603f82a5ee2f584bd4eb838c": "{\\dot {e}}_{1}=0", "111d10ccaa2716291b402bea0680c23d": "\\Psi _{2}:=C_{\\alpha \\beta \\gamma \\delta }l^{\\alpha }m^{\\beta }{\\bar {m}}^{\\gamma }n^{\\delta }\\ ,", "111d25fdd402a4b1eba732d027332bce": "-log_{10}({\\frac {1+2d}{365}})", "111d4300d81fac14bb67ded72ce8222a": "{\\frac {\\xi }{\\xi _{0}}}={\\frac {\\ln {\\tfrac {t}{t_{1}}}}{\\ln {\\tfrac {t_{0}}{t_{1}}}}}.", "111d835b36d8312e993aa099aa72e62a": "(P\\leftrightarrow Q)\\leftrightarrow (Q\\leftrightarrow P)", "111d8940d1b32b4e525396ac09f9bfb5": "(V,E,A-a)", "111deeccc6453c79fbe0fcfce4e14a8f": "\\eta ^{\\mu \\rho }=0", "111e07ca18e0b69d9765d5693c244e7b": "f(x+uR)=(x+u_{1}R,\\ldots ,x+u_{k}R)\\quad {\\mbox{ for every }}x\\in R", "111e454a716b3a61f7ef4de9f14f1ba9": "J(\\mathbf {x} )", "111e5751a8d46d382301a04a61f97ebe": "\\Delta E\\Delta t\\geq {\\frac {\\hbar }{2}}\\ ,", "111e78d1ab40407546acc0c3f2c455e7": "{\\sqrt {ax^{2}+bx+c}}", "111ea26ef5fdbd87adee0b2545a2c421": "f:X\\to Y", "111ecf951fd47af074e7209524576f85": "\\ Z=|Z|e^{j\\arg(Z)}", "111ef680298282cb7ed299f28678acd8": "x\\in \\Sigma ", "111f9a7b168df5ddf62b26391cfc00c6": "\\langle \\psi |{\\mathcal {T}}\\{F\\phi ^{j}\\}|\\psi \\rangle =\\langle \\psi |{\\mathcal {T}}\\{iF_{,i}D^{ij}-FS_{int,i}D^{ij}\\}|\\psi \\rangle .", "111fb75d0fc2d07f03ffec898598a279": "\\langle a_{0},a_{1},a_{2},\\ldots \\rangle \\in A", "111ff7857fbb1e8261592993d61d2e47": "={\\frac {1}{2}}+{\\frac {\\sin(k_{1}x)+\\sin(k_{2}x)}{4}}", "112023b22122f83cef644aabb93fd324": "L=20\\ \\log _{10}\\left({\\frac {4\\cdot \\pi \\cdot d}{\\lambda }}\\right)", "11203164535cb8a09eb060515de2282e": "{\\sqrt {\\theta _{1}^{2}+\\theta _{2}^{2}}}", "11203f393143bd6971260789730f9ed5": "{\\frac {-a_{0}}{a_{1}}}", "11205144e60361dced45e00c2c7d553e": "Q={\\begin{pmatrix}{-(x_{1}+x_{2}+x_{3})}&x_{1}&x_{2}&x_{3}\\\\{\\pi _{1}x_{1} \\over \\pi _{2}}&{-({\\pi _{1}x_{1} \\over \\pi _{2}}+x_{4}+x_{5})}&x_{4}&x_{5}\\\\{\\pi _{1}x_{2} \\over \\pi _{3}}&{\\pi _{2}x_{4} \\over \\pi _{3}}&{-({\\pi _{1}x_{2} \\over \\pi _{3}}+{\\pi _{2}x_{4} \\over \\pi _{3}}+x_{6})}&x_{6}\\\\{\\pi _{1}x_{3} \\over \\pi _{4}}&{\\pi _{2}x_{5} \\over \\pi _{4}}&{\\pi _{3}x_{6} \\over \\pi _{4}}&{-({\\pi _{1}x_{3} \\over \\pi _{4}}+{\\pi _{2}x_{5} \\over \\pi _{4}}+{\\pi _{3}x_{6} \\over \\pi _{4}})}\\end{pmatrix}}", "112096cb5347f695c5271d5d23d3b4bd": "Q_{n}(x;a,b,N)={}_{3}F_{2}(-n,-x,n+a+b+1;a+1,-N+1;1).\\ ", "1120d973c96b8c15fe6d1074cbf1e50a": "R_{O}+R_{B}-F=0\\quad {\\text{and}}\\quad -\\mathbf {r} _{A}\\times \\mathbf {R} _{O}+\\mathbf {r} _{B}\\times \\mathbf {R} _{B}=\\mathbf {0} \\,.", "1120e8d3f2657a841811c6b77f21e996": "t={\\frac {x}{v\\cos \\theta }}", "112123c25cf63310139f9ee076803a17": "x+1\\,\\!", "112140d8a0c2c74b7242625c70b65e83": "-v_{\\mathrm {rel} }{\\frac {\\mathrm {d} m}{\\mathrm {d} t}}=m{\\mathrm {d} v \\over \\mathrm {d} t}", "11217ddea497d67192e898572bd42bb0": "(1+\\epsilon )", "112200292440ad6785be80f1b9b955e6": "z={\\sqrt {x^{2}+y^{2}}}", "112251de2fe1fa944cf29785c2b74638": "RC_{t}=\\ln \\left({\\frac {P_{t}}{P_{t-1}}}\\right).", "112301345da08130e42b524c7d8e7152": "{\\mathcal {E}}=qE+\\mu B+{\\frac {1}{2}}mv_{\\parallel }^{2}+{\\frac {1}{2}}mv_{\\perp }^{2}", "1123039eb00b314adb814e533ee330a2": "\\mathrm {FillRad} (X)\\leq C_{n}\\mathrm {vol} _{n}{}^{\\tfrac {1}{n}}(X),", "11230d45736bd09b3653c926735adf7a": "\\psi ^{n}=\\psi ^{n-1}+\\psi ^{n-2}\\,.", "1123194db6f75bf8855ce75e51a2d454": "\\mathbb {E} [(R_{r}-R_{\\min })_{-}^{2}]", "1123593f39ece8e15eeacf8058003d77": "h(x,y)=f(x)+g(y)", "1123768fcd42c1b6c4cf1d6f5594e57c": "\\mathbf {i} (\\mathbf {i} x)\\leftrightarrow \\mathbf {i} x.", "11238c4a19646952d7282e5e9954a0f9": "u=v", "1123a1fc5fbf885d0ec54e0a5fcdd92b": "(X,Y,Z)=\\left\\{{\\begin{matrix}(0,0,0)&{\\text{with probability}}\\ 1/4,\\\\(0,1,1)&{\\text{with probability}}\\ 1/4,\\\\(1,0,1)&{\\text{with probability}}\\ 1/4,\\\\(1,1,0)&{\\text{with probability}}\\ 1/4.\\end{matrix}}\\right.", "1123bb63b22a9a4309ad4d49afa8b57e": "f(S_{x,i},c)", "1123dc128c4342924f078f79bdc82ef3": "{\\mathcal {J}}({\\hat {T}}):=1+\\exp \\left[-{\\cfrac {1+1/\\zeta }{1+\\zeta /(1-{\\hat {T}})}}\\right]\\quad {\\text{for}}\\quad {\\hat {T}}:={\\frac {T}{T_{m}}}\\in [0,1+\\zeta ],", "1123f425a881d74b45ddb30c264f0e1f": "(H_{i})", "112405e9295c4b9eaf79b5dcafa7427f": "\\forall m\\,\\forall n\\,(\\varphi (n,m)\\iff {\\frac {n}{m}}0\\}.", "113653cfe1e86a56605ac2f73536f6a6": "\\forall V\\in {\\mathcal {T}},f(x)\\in V\\exists U\\in {\\mathcal {T}},x\\in U:U\\subseteq f^{-1}(V)", "113660501ce535ef3f7236adc13f80aa": "\\Sigma _{0}(A)=(A\\!\\ggg \\!2)\\oplus (A\\!\\ggg \\!13)\\oplus (A\\!\\ggg \\!22)", "11366cd4251a48c25e11d34e038655a7": "\\Re .", "1136b03089226940e1d7fb351c702626": "\\left[-{\\hbar ^{2} \\over 2\\mu }\\left({\\frac {1}{r}}{\\partial ^{2} \\over \\partial r^{2}}r-{l(l+1) \\over r^{2}}\\right)+V(r)\\right]R(r)=ER(r)", "113721c395c3e9cd302a4ffb75bf0a6b": "K\\geq 2", "11373d017ce2d9410fb2b9e8f385b849": "K_{p}^{eff}={\\frac {1}{t_{c}}}\\int _{0}^{t}D_{0}exp\\left({\\frac {-Q}{RT(t)}}\\right)dt", "11375ad1dd1e39f1d877d65c107cfd34": "2^{Rb}", "11376cc39053f7daa4e94ce0a7069a68": "\\mu (x,y)", "113780691a78e21f390ac884dd4486e9": "-\\operatorname {Tr} \\rho \\log \\rho .", "1137e2250a6eb176ce798ed682ed2411": "H^{\\rm {T}}+H=2I.\\,", "11387e8f3b98a0ff9fbb0537a2d049f2": "I_{n}(\\rho )=J_{n}(i\\rho )", "11389d2ef487b17c4f8a6dff48b55df0": "{\\frac {\\partial \\theta }{\\partial t}}={\\frac {\\partial }{\\partial z}}\\left[K\\left({\\frac {\\partial \\psi }{\\partial z}}+{\\frac {\\partial z}{\\partial z}}\\right)\\right]={\\frac {\\partial }{\\partial z}}\\left[K\\left({\\frac {\\partial \\psi }{\\partial z}}+1\\right)\\right]", "1138b6443d4c7b2d30095d5741eec4a8": "|I|:=\\sum _{i=1}^{m}I(i)", "1138fb7fb1eafebb618573abbeddb7d5": "ABA^{-1}B^{-1}", "113923a67b215b851c60fc6a9e7dd56b": "{\\frac {F}{\\rho _{f}U^{2}A}}={\\frac {12}{\\mathrm {Re} }}\\left(1+0.15\\mathrm {Re} ^{0.687}\\right).", "113a95b141f7314208c329cc42ab7b52": "\\parallel {\\boldsymbol {\\phi }}_{\\mathcal {B}}(x)-{\\boldsymbol {\\phi }}_{\\mathcal {B}}(y)\\parallel _{2}\\leq \\varepsilon ", "113aa98b9c33bfd73ff331d9a59abb53": "BP^{2}=wB^{2}+Az^{2},", "113aece3fbe3acd17f86223218948956": "n=\\left\\lfloor {\\frac {\\pi }{4\\theta }}\\right\\rfloor \\approx \\left\\lfloor {\\frac {\\pi }{4\\sin(\\theta )}}\\right\\rfloor =\\left\\lfloor {\\frac {\\pi }{4}}{\\sqrt {\\frac {N}{G}}}\\right\\rfloor =O({\\sqrt {N}}).", "113b13f7d253420d7034f48ce979f2b0": "Q={\\frac {2\\pi f_{o}\\,{\\mathcal {E}}}{P}},\\,", "113bb1754905ffd7453260851fe208ba": "M{\\to _{G}}^{*}M'", "113bd144dfec035303bc967572eee272": "{\\begin{aligned}\\varphi \\,=\\,&\\left\\{\\,\\varphi _{b}\\,-\\,{\\frac {1}{2}}\\,(z+h)^{2}\\,{\\frac {\\partial ^{2}\\varphi _{b}}{\\partial x^{2}}}\\,+\\,{\\frac {1}{24}}\\,(z+h)^{4}\\,{\\frac {\\partial ^{4}\\varphi _{b}}{\\partial x^{4}}}\\,+\\,\\cdots \\,\\right\\}\\,\\\\&+\\,\\left\\{\\,(z+h)\\,\\left[{\\frac {\\partial \\varphi }{\\partial z}}\\right]_{z=-h}\\,-\\,{\\frac {1}{6}}\\,(z+h)^{3}\\,{\\frac {\\partial ^{2}}{\\partial x^{2}}}\\left[{\\frac {\\partial \\varphi }{\\partial z}}\\right]_{z=-h}\\,+\\,\\cdots \\,\\right\\}\\\\=\\,&\\left\\{\\,\\varphi _{b}\\,-\\,{\\frac {1}{2}}\\,(z+h)^{2}\\,{\\frac {\\partial ^{2}\\varphi _{b}}{\\partial x^{2}}}\\,+\\,{\\frac {1}{24}}\\,(z+h)^{4}\\,{\\frac {\\partial ^{4}\\varphi _{b}}{\\partial x^{4}}}\\,+\\,\\cdots \\,\\right\\},\\end{aligned}}", "113be821399e699753e79ea62d6bed80": "{\\frac {Q_{hot}}{T_{hot}}}={\\frac {Q_{cold}}{T_{cold}}}", "113c148ae8eb779638a15d7d8103b115": "{\\begin{aligned}Lu&=u''+k^{2}u=f(x)\\\\u(0)&=0,\\quad u\\left({\\tfrac {\\pi }{2k}}\\right)=0.\\end{aligned}}", "113c1f9998f3d206f938020b1356cc53": "-j0.23={\\frac {-j}{\\omega C_{2}Z_{0}}}\\,", "113c604e3acd7c5eccffaab50a14562f": "\\sigma :(u,v)\\mapsto (ru,v/r),\\quad r=e^{b}.", "113c78dec425577876ce0ec66c1e72f0": "P\\equiv ", "113cc4e8feb0006c5e165577550bf7eb": "G/N\\approx G'/N'", "113d0bf3c4ca86be1731bc904755e493": "\\mathbf {B} ^{*}=\\{\\mathbf {b} _{1}^{*},\\mathbf {b} _{2}^{*},\\dots ,\\mathbf {b} _{n}^{*}\\},", "113d1615f9ab61862cd85415d5c99d9a": "{\\begin{aligned}e_{(\\mathbf {I} _{1})}={\\frac {dx_{1}-dX_{1}}{dX_{1}}}&=\\Lambda _{(\\mathbf {I} _{1})}-1\\\\&={\\sqrt {C_{11}}}-1={\\sqrt {\\delta _{11}+2E_{11}}}-1\\\\&={\\sqrt {1+2E_{11}}}-1\\end{aligned}}\\,\\!", "113e06af3cb1014bc4bd20479a8d58fb": "{\\frac {1}{(x-1)(x^{2}+x+1)}}={\\frac {A}{x-1}}+{\\frac {Bx+C}{x^{2}+x+1}}", "113e16ae4724578abc2b1330c8afed5a": "r_{u}=a\\,{\\frac {\\sqrt {3}}{2}}\\varphi ", "113e2755d04d625f296bd3ab8572451a": "\\sigma \\in (0,+\\infty )\\,", "113e2b3a08df2273ee34a171678d95e3": "f(x)=\\Omega _{+}(g(x))", "113e592b280e3c7f6674166a55ebb3e6": "R={\\tfrac {1}{2}}{\\sqrt {a^{2}+c^{2}}}", "113e5d0ee812c73b087d02b4780873b1": "x[n]=\\sum _{k=0}^{N-1}h_{k}s[n-k]+w[n]", "113e8a4a1fd4a91fd6b3664411cbfc5f": "I(\\theta )", "113ec26e14fea1e8b8ef8b9c3955eed6": "(M,m,l,t,\\epsilon )", "113efb35df30996a5b1f383dd52759b9": "x^{3}-2x^{2}+10x-1=5", "113fab14368535575d8feb4a91c7177a": "A_{k}(x)={\\frac {1}{k!}}\\sum _{m=0}^{k}{k \\choose m}x^{m}\\sum _{l=1}^{k-m}{\\frac {(-x)^{l-1}}{l}}", "113fb17b52cc29abd2a8fcd3f0046273": "W_{D}={\\frac {J_{2}}{2G}}\\,\\!", "11404065aad86239f168f1dd0b47a639": "{\\tfrac {5}{8}}", "114059d52cf464dd4c2811cd718c40c8": "B\\leq _{T}A.", "11406a650f6017bb05db0eebd407ccae": "T\\,", "1140ab3ef4ae67e684bb87f119e9b26e": "G_{I}", "1140b8095b48ecf78f052a86c1afe555": "d/dx", "1141633dbd7b6b138d55b000b9cd5284": "Uxy\\rightarrow \\exists z\\forall v[Ovz\\leftrightarrow (Ovx\\lor Ovy)].", "11417f9bd33db4bedc068659fb106d9a": "F(x;a,d,p)={\\frac {\\gamma (d/p,(x/a)^{p})}{\\Gamma (d/p)}},", "1141a31b5508d2972d69626172b1e1fb": "\\oint _{\\partial \\Sigma }\\mathbf {E} \\cdot \\mathrm {d} {\\boldsymbol {\\ell }},\\quad \\oint _{\\partial \\Sigma }\\mathbf {B} \\cdot \\mathrm {d} {\\boldsymbol {\\ell }}\\,,", "1141a52d1df57f90d0a2f75bd56126b0": "V(0)=0", "1141e020d3ce8a28aafc9fcb8ed471c1": "(xy)(xx)=x(y(xx))", "114240fd63fb6e9698ed0036541ba7fa": "B^{n}x+\\alpha =(By+\\beta )^{n}+r'.", "11424a3fb00068cbe8a87391c99a31d6": "T_{s*t}=T_{s}\\circ T_{t}.", "1142794e9db23eddd31936c76c5a947e": "f_{1}\\propto {\\tfrac {1}{L}}.", "1142b5d5391f559645974f438a760192": "{\\frac {k}{d}}=0,{\\frac {1}{5}},{\\frac {29}{146}},{\\frac {117}{589}},{\\frac {146}{735}},{\\frac {555}{2794}},{\\frac {1256}{6323}},{\\frac {5579}{28086}},{\\frac {17993}{90581}}", "1142e3219876d1ba09de22153d8a74a5": "\\chi (G_{K},M)=\\left(\\#R/mR\\right)^{-1},", "1142fa173a39cc5a9ec43a1686cac3cc": "(2n+1)\\times (5n^{2}+5n+1)", "1143071c5dadda6abc647004a79249bc": "r_{i}={\\frac {\\varphi ^{2}a}{2{\\sqrt {3}}}}={\\frac {\\sqrt {3}}{12}}\\left(3+{\\sqrt {5}}\\right)a\\approx 0.7557613141\\cdot a", "11433acd20dbe84bf6bdfa7133f66659": "{\\mathcal {F}}_{\\infty }^{*}=\\sigma \\left\\{X_{s}^{-1}(B):s\\in [0,\\infty ),B\\in {\\mathcal {E}}^{*}\\right\\}.", "11435743bd2a32ab5515f2a21badbf16": "{\\frac {x}{\\sigma ^{2}}}e^{-x^{2}/2\\sigma ^{2}}", "114390c4f3a48c39cc943d255869db29": "F_{\\lambda }", "1143daa3e1751d75996b0a35fc4960f5": "f(x)=f_{n}H_{n}(x)", "1143f59fb20a94b3d4a3ae79ecafa798": "z=(x-a)/(b-a)", "114432e160732831afda796d46442ff1": "2.25>2", "11443ef243378aaddce5d74f152ff6ee": "\\exists {x}\\Box Ax", "11443fad8b35176eeda4fceea98977dc": "-{\\frac {\\hbar ^{2}}{2m}}{\\frac {\\mathrm {d} ^{2}}{\\mathrm {d} x^{2}}}\\Psi (x)+V(x)\\Psi (x)=E\\Psi (x),", "114443ed4006898ab5cc8995cabb2367": "(abc)'=a'b'c'=(-mam^{-1})(-mbm^{-1})(-mcm^{-1})=-ma(m^{-1}m)b(m^{-1}m)cm^{-1}=-mabcm^{-1}\\,", "114446f596abbc4ee7b2d34c6e1fb97c": "s_{n}=(-1)^{n}(\\triangle ^{n}a)_{0}", "11447ab7e8f3a8c95a15c639bdb47f11": "X\\sim N(0,\\sigma ^{2})", "1144b75226347ac9bb7e74e9e135337e": "{\\ f(p,V,T)=0}.", "1145493ee88586261a9948c1369c7cd1": "a^{2}=1-h^{2}\\ ;\\ a={\\frac {1}{2}}{\\sqrt {\\frac {5+{\\sqrt {5}}}{2}}}\\ .", "11455d9cbc4864713ecafb527e30d6c7": "\\Delta Y/Y\\approx \\Delta {\\overline {Y}}/{\\overline {Y}}+c(\\Delta {\\overline {u}}-\\Delta u).", "1145c27a623e14e3abe85706d2794286": "X={5^{7/4}:16},\\ ", "1145c6ca798ae04fe22789746bd5bdbe": "R\\approx _{\\bar {x}}R'", "1145cdcda0c92366117ff24212cc1ecc": "\\alpha ,\\beta \\in \\Phi ", "1145d8ecd66df21777bf5a6d5f6667fa": "\\{y\\in O_{L}:yO_{L}\\subseteq O_{K}[\\theta ]\\};", "1146b5f80e8ee91b4e6d838f6aef84a9": "\\lbrace z=\\sigma je^{aj}:\\sigma \\in R\\rbrace ", "1146e78611867431398a5122a81e2a6d": "g(x)=x", "1146f1015dd4284308212916379113c1": "0\\leq s\\leq t", "1146ff7dd3bc058061f873e6154a42cb": "f_{ab}", "11474654a5e8eda0484361c96e38234c": "{\\frac {\\partial P_{ij}}{\\partial s}}(s;t)=-\\sum _{k}A_{ik}(s)P_{kj}(s;t)", "11477a1a9ef60be0b65b4b2d5c480797": "\\square (O{\\underline {A}}\\to {\\underline {A}})", "114780cc31a9e7b5a0eb16a06a3cf004": "L(z)\\equiv \\sum L_{n}z^{-n-2}={1 \\over 2}\\sum _{i}:v^{(i)}(z)^{2}:", "1147c6f0ef61b104c918a7d8790d496d": "{\\chi }", "1147d0801b73fae5de3c31dba1a8fb09": "{\\frac {e^{-iut}}{\\sqrt {2\\pi }}}", "11480970a23e1290174787419b11539d": "\\phi +\\delta \\phi ", "114809b615bff5a770613797c7171637": "f\\to m,x\\to f", "114893c3454abd2a02cded328ca50c5f": "x^{5}-{\\frac {4}{13}}x+{\\frac {29}{65}}", "1148a2214c517568a6596204429024c8": "Q(p)=\\mu -\\beta \\ln(-\\ln(p)),", "1148bb577e31cb82ddcced154b0d3442": "x*y=\\alpha (x)\\cdot \\beta (y)", "1148bfaab34d174e55512b2a4cd0234a": "\\mathrm {2H} _{2}+\\mathrm {4OH} ^{-}\\longrightarrow \\mathrm {4H} _{2}\\mathrm {O} +\\mathrm {4e} ^{-}", "114943323cca16302ce6e2e45151c030": "z73.2", "115576e1bea0010d2e8420e134e091b2": "{\\frac {\\partial \\langle H\\rangle }{\\partial a_{n}}}=-i\\hbar {\\frac {\\partial a_{n}^{*}}{\\partial t}}", "11558254980c2ddf12c4c292bef600cd": "{\\begin{aligned}r&=ct\\\\r'&=ct'\\end{aligned}}", "1155cf5cac0483c06999787abb3f9c25": "L(s)={\\frac {N(s)}{D(s)}}.", "115604d47d645ff5255cd29680c46b3f": "{\\frac {m_{1}u_{1}+m_{2}u_{1}-m_{1}u_{1}-m_{2}u_{2}}{m_{1}+m_{2}}}={\\frac {m_{2}(u_{1}-u_{2})}{m_{1}+m_{2}}}", "11564ee93d8d0aa8a63309414c54b245": "F={\\frac {2\\cdot \\mathrm {precision} \\cdot \\mathrm {recall} }{(\\mathrm {precision} +\\mathrm {recall} )}}.\\,", "115660db79e1999c639558c9b50e863a": "-\\left(R_{im}{\\frac {\\partial U_{j}}{\\partial x_{m}}}+R_{jm}{\\frac {\\partial U_{i}}{\\partial x_{m}}}\\right)", "11567450d788888c7720845b90bdb00f": "{{\\underline {Z}}(\\mathbf {r} ,\\omega )={\\frac {{\\underline {p}}(\\mathbf {r} ,\\omega )}{{\\underline {v}}(\\mathbf {r} ,\\omega )}}}", "11568c5b8ff966fa1f29789a2f930912": "{\\overline {\\lambda }}=-1000,{\\underline {\\lambda }}=-1", "1156b75f0b2d38b4c20117280e4af3f1": "C_{n}.", "115706f968adfb2f1b1c453fb1f8be04": "{\\begin{array}{cl}\\displaystyle {\\frac {x:\\sigma \\in \\Gamma \\quad \\tau ={\\mathit {inst}}(\\sigma )}{\\Gamma \\vdash x:\\tau }}&[{\\mathtt {Var}}]\\\\\\\\\\displaystyle {\\frac {\\Gamma \\vdash e_{0}:\\tau _{0}\\quad \\Gamma \\vdash e_{1}:\\tau _{1}\\quad \\tau '={\\mathit {newvar}}\\quad {\\mathit {unify}}(\\tau _{0},\\ \\tau _{1}\\rightarrow \\tau ')}{\\Gamma \\vdash e_{0}\\ e_{1}:\\tau '}}&[{\\mathtt {App}}]\\\\\\\\\\displaystyle {\\frac {\\tau ={\\mathit {newvar}}\\quad \\Gamma ,\\;x:\\tau \\vdash e:\\tau '}{\\Gamma \\vdash \\lambda \\ x\\ .\\ e:\\tau \\rightarrow \\tau '}}&[{\\mathtt {Abs}}]\\\\\\\\\\displaystyle {\\frac {\\Gamma \\vdash e_{0}:\\tau \\quad \\quad \\Gamma ,\\,x:{\\bar {\\Gamma }}(\\tau )\\vdash e_{1}:\\tau '}{\\Gamma \\vdash {\\mathtt {let}}\\ x=e_{0}\\ {\\mathtt {in}}\\ e_{1}:\\tau '}}&[{\\mathtt {Let}}]\\end{array}}", "11570cd2d7c0faaf632db1c3ae51fb24": "\\neg (\\neg (\\neg A\\lor B)\\lor (C\\lor (D\\lor E)))\\lor (\\neg (\\neg E\\lor D)\\lor (C\\lor (A\\lor D)))", "115733207da2070a7898aac909f3e2f4": "f(x)\\in o(g(x))", "11578cced88be1dbe51af21d285c9a6e": "{\\widehat {U}}", "1157b715ac82804ec725517a7c91748d": "|m-1|", "1157d4a8ddde804f2f2d454828c82fda": "P_{\\mathbf {k} }", "1157d9753cfd628b4e46d257440ea519": "\\left[{\\widehat {U}},{\\widehat {H}}\\right]=0", "1157ed1eaeceabd9c9b042937c58e410": "{\\mathcal {SHIQ}}^{\\mathcal {(D)}}", "11581a37ec230c38b480deecfe66b9fd": "3^{-1/\\beta }\\alpha ", "11586ca43984fa4adce66fab55c57cf4": "\\langle 2,2,1,0,0\\rangle ", "1158dc93546783a9f8a3a3a987780463": "\\omega ^{i}\\ldots \\omega ^{j}", "11590c5f4a09afd40aa65505af9fc952": "R_{4,0}=5+126r^{4}-280r^{3}+210r^{2}-60r", "1159af01a969666a7affdeefb4a899d0": "9\\cdot 2^{3w}", "1159d1e4729abc812d241c69ffaadf78": "\\epsilon \\approx 23.4^{\\circ }", "115a813b34fe8662f323dcea4ad7d198": "\\chi (\\mathbf {R} )", "115a8e5fba4b0b414f44cc1b4c9d4615": "f(t)=e^{-2\\tau _{D}/t}\\left(I_{0}(2\\tau _{D}/t)+I_{1}(2\\tau _{D}/t)\\right)", "115ab4e4e168872519e917b973ab21f8": "z_{0}={\\frac {\\pi W_{0}^{2}}{\\lambda }}", "115acdbbdb4767f0208d686f8613fef6": "\\beta _{A}=\\beta _{D}\\left({\\frac {D}{V}}\\right)+\\beta _{E}\\left({\\frac {E}{V}}\\right)", "115b12876ce8eb8a4d7ede1d9a802342": "P=\\operatorname {Perm} (A')\\,{\\bmod {\\,}}Q", "115b1b827ac9cfbb5bcb78595d9706f4": "\\partial \\colon {\\mathcal {D}}_{m+1}\\to {\\mathcal {D}}_{m}", "115b757eed221a3b5c2341b1b2222429": "\\alpha =\\theta /2", "115bdc1b9a7cab95337a886c1992c91e": "r^{ed}\\equiv r{\\pmod {N}}", "115c36c53126eb8ada7117af11ee4670": "\\mathbb {R} \\setminus \\mathbb {A} ", "115c469411c5831217307670ea40f92a": "V=d'\\Sigma _{YY}^{-1/2}Y=b'Y", "115c793833709b1514aceb8f8c5e7a20": "c=K_{R}/K_{T}", "115ca2c2ee963546ff5de45343b0d0e5": "{\\begin{bmatrix}\\cosh \\phi &-\\sinh \\phi &0&0\\\\-\\sinh \\phi &\\cosh \\phi &0&0\\\\0&0&1&0\\\\0&0&0&1\\\\\\end{bmatrix}}=\\exp \\left(-\\phi {\\begin{bmatrix}0&1&0&0\\\\1&0&0&0\\\\0&0&0&0\\\\0&0&0&0\\\\\\end{bmatrix}}\\right)\\equiv \\exp(-\\phi K_{x}),", "115ca478164bafb41bb075eedc3bc311": "\\mathbb {C} ^{3}", "115ca536bb0aaaff4b2eccc62af2d3a1": "\\cos {\\theta }_{W}*=r\\cos {\\theta }", "115cab08cf56cd0cbe43382811fe10d5": "{\\frac {1}{\\sec \\theta }}\\!", "115cc42e4c85a83033ad9cf797f9db81": "m(H,t)", "115cdc52306415f28dee695e67870175": "a_{i}^{j}:=(A\\mathbf {u} _{i})^{j}\\in K", "115ce754cd3689beba99006c1070a37f": "\\!\\ \\sum _{n=1}^{N}n^{2}", "115d0271af1b2b2a7467788276944a6f": "{\\overline {x}}", "115d7db3c5060afe2a39c77831ae48b9": "{\\vec {e}}_{1},{\\vec {e}}_{2},\\,{\\vec {e}}_{3}", "115dc1e020322b605c687d29cc08be1c": "{\\hat {g}}\\ ,\\ {\\hat {h}}\\,", "115dfe6472be5fbebe6abd2476687b55": "\\phi {\\mbox{ fixes }}K\\quad \\iff \\quad q{\\mbox{ splits completely in }}K.", "115e1db2d082962d49b2cc7e91d632b2": "x\\in \\mathrm {supp} \\,X.", "115e2c8264d9c247771fc75c882d94cd": "\\left({\\begin{smallmatrix}\\lambda \\\\&\\lambda ^{-1}\\end{smallmatrix}}\\right)\\times \\{\\pm I\\}", "115ef0a4b772bc4d2c8396f2c87b07d8": "m-2,m,m+2", "115f00f446eb59162d4575ac63502376": "F(x;\\mu ,\\sigma ,\\xi )=\\exp \\left\\{-\\left[1+\\xi \\left({\\frac {x-\\mu }{\\sigma }}\\right)\\right]^{-1/\\xi }\\right\\}", "115f5428a916422b09db1bd60a2dd67b": "\\gamma V^{2/3}=k(T_{C}-T)\\,\\!", "115f56d072ef0db3cfb4fddc6f534e13": "x=8", "115f8a166d3bfbee69b62c6f9830a907": "{\\frac {R}{2}}{\\sqrt {3}}={\\frac {a}{2}}{\\sqrt {3}}\\!\\,", "115f8f7a3e1d808db462a94f9ccc5f81": "\\tau _{ph-e}", "115fa04e79b504896a190f6a591439a8": "\\lambda _{2}-\\lambda _{1}", "115fa499c0ef07569c045f29dbe7c40f": "\\tau _{ij}^{\\mu }(t):=\\sum _{\\nu =1}^{n}(\\xi _{i\\nu }(t)-\\xi _{\\nu }^{\\mu }(t))(\\xi _{j\\nu }(t)-\\xi _{\\nu }^{\\mu }(t)),\\qquad \\xi _{i\\nu }(t):=\\sum _{i=1}^{n}\\mu _{i}(t)\\xi _{i\\nu }(t)", "115fc26783cbc21f1da892eb190dbc8d": "{}^{3}i=i^{\\left({}^{2}i\\right)}", "1160a8d1145bcf5538e74df9e491f55a": "c_{V}=(d\\delta _{ij}-n_{ij})_{ij},", "1160daebba71f9f8a58469ac16f75af5": "\\mathbf {R} _{j}", "11610dcae4326433c839040e21f00907": "\\scriptstyle E(z),", "116116f8dff1f18bab37e1bf14d6d8a6": "\\sum _{j=1}^{n}A_{jk}=\\sum _{j=1}^{n}f_{j}(g_{k})=0", "1161217fa260fba5181ae8b30438fd2d": "|\\Psi (t)\\rangle =e^{-iE_{\\Psi }t/\\hbar }|\\Psi (0)\\rangle ", "116164854da736767a9ec29a9c493b33": "(\\Omega ,{\\mathcal {F}},({\\mathcal {F}}_{t})_{t\\geq 0},\\mathbb {P} _{x})", "1161aaaf0a1228651a7ca301949e1b7c": "\\langle \\psi (t)|=\\langle \\psi (t_{0})|U^{\\dagger }(t,t_{0}).", "1161acc9c5893cbbb9980c230f53362b": "L_{2}\\in \\mathbb {R} ^{(n-1)}", "1161f2ab11758784d3b9c3961013965c": "(\\mathrm {proj} _{\\theta }F)\\setminus G_{\\theta }", "1161fc5ddeb59b72a5a49d5c70386aea": "r=L/10", "116202fcd704620524148cc876ca6648": "x\\rightarrow b\\equiv (a_{1},a_{2},\\ldots ,a_{n},b)", "11620499dadd6b0d68a72e06a8bd9bb2": "(256^{\\,\\!256^{257}})^{256^{256^{257}}}=256^{256^{257}\\times 256^{256^{257}}}=256^{256^{257+256^{257}}}", "11628573b0b504edd4ef4ef2d4d290de": "\\sum _{n=1}^{\\infty }a_{n}=1+2+3+4+5+\\cdots ", "11628a882ce9d229db3fd6a37b343edc": "{\\hat {T}}={\\frac {{\\hat {p}}^{2}}{2m}}.", "11629b4d41796922245a01b3463f239c": "h_{k}=\\max \\left\\{\\left\\lfloor 5h_{k-1}/11\\right\\rfloor ,1\\right\\},h_{0}=N", "1162d2547e153449686efdd5fc203b57": "T_{ss}={\\sqrt {2}}\\,T\\approx 1.41\\,T,", "1163720ff25bc474b03273f89160ad5a": "\\alpha _{R}", "1163a4f72e317a52b4c6ee4ca3a98d41": "wp({\\textbf {if}}\\ E\\ {\\textbf {then}}\\ S_{1}\\ {\\textbf {else}}\\ S_{2}\\ {\\textbf {end}},R)\\ =\\ (E\\Rightarrow wp(S_{1},R))\\wedge (\\neg E\\Rightarrow wp(S_{2},R))", "1163ecc17467b15ce8fce2a661471954": "\\phi (5)=4", "11642cfb4226577be605838141fff5b2": "A=f^{-1}(B)", "116436af151c6d132ee4d4c21880be54": "n>-\\delta ", "1164446217f2d40dbed109c14ed03843": "\\left(\\alpha ^{m}\\right)^{-1}=\\alpha ^{-m}", "11644feff7f3d02f1e93c502204d8632": "P_{1},...,P_{k}", "11648838cb046147cc143cdc73f89fcf": "\\left\\{f_{1},\\ f_{2},\\ f_{3},\\dots \\right\\}", "116500e5e8f64c5dacebf52dba6fdb4b": "L(x,y)=\\int _{0}^{1}x^{1-t}y^{t}\\ \\mathrm {d} t", "1165b63f4e07dd7247f469168fbe5e8d": "C_{i}^{n}", "116654a97d39dec49e9ad9310a642740": "\\beta =1/(1-\\pi _{A}^{2}-\\pi _{C}^{2}-\\pi _{G}^{2}-\\pi _{T}^{2})", "11667bdf2486170b6ef0d01dc6429589": "[24,12,8]_{2}", "116682e06468a4e1504bbf1752d36bce": "{\\frac {d}{dx}}e^{x}=e^{x}.", "1166c9cc5504cf193eb09412d20ab295": "=\\sum _{i}p_{i}(\\log p_{i}-\\sum _{j}\\log q_{j}|v_{i}^{*}w_{j}|^{2})", "11676eea716a9c8a71b526204666aa0d": "\\phi \\colon {\\mathfrak {U}}\\to [0,+\\infty )", "1167fd98f4799a8fc909465ca24e5308": "\\mathbf {v} _{T}={\\frac {R}{f}}\\ln \\left[{\\frac {p_{0}}{p_{1}}}\\right]\\mathbf {k} \\times \\nabla _{p}{\\bar {T}}", "11687dc6f6b86e47916885a545f2d942": "\\langle p\\rangle =-{\\frac {\\partial A}{\\partial V}},", "11688718147baa0f26bd72966b0fe42e": "\\lambda _{N}", "1168d34108b07c6d001252fa32bf6607": "L=-{\\tilde {F}}{\\dot {v}}^{2}+2{\\dot {v}}{\\dot {r}}\\,,", "1169116784fe73e9e7f4f08c2071eb6c": "H_{*}(Y)", "11691e618a6a432b47d7a0559b9905bc": "E^{(2)}=\\sum _{l=3}^{\\infty }C_{2l}R^{-2l}", "116948fc0f905f88bc2c8d2e55dd1dab": "P(n)^{2}-P(n+1)P(n-1)=P(-n-7).\\,", "1169726f7c7d63110123a8e83018a8f7": "|\\Psi \\rangle \\sim |\\Psi \\rangle +Q_{B}|\\Lambda \\rangle ", "116988c0814431969204c6aebd934c2f": "\\sum _{1\\leq j\\leq m,x_{j}\\neq y}{\\frac {x_{j}-y}{\\left\\|x_{j}-y\\right\\|}}\\leq \\left|\\{\\,j\\mid 1\\leq j\\leq m,x_{j}=y\\,\\}\\right|.", "116a3bb7832dc0b4a701a0459b9d6039": "\\scriptstyle {(Rc,Rt)}", "116a505ed2fffa33ebb2afe22102bdd8": "{A}_{10}^{(2)}", "116a6bc7babae4e38520eee6a9b23a3b": "{\\mathcal {D}}^{J}", "116ab8beaf22b17c0eb7661719fbd20f": "(\\sigma ,\\tau ,z)", "116aff26524952c0187b4bd82c511e4d": "\\nabla _{\\mathbf {x} _{0}}f\\cdot \\mathbf {e} _{i}=0", "116b1fae0f0afb22dd4fa38693ba8bad": "G(s)={\\frac {Y(s)}{X(s)}}", "116c6f7e9919a523854279e9a38d1ffc": "-\\ln(r)=\\int _{0}^{T_{f}}\\lambda ({\\vec {p}}(t))\\,dt", "116c8d4537aa23b794cdfb220e537561": "\\sin 20^{\\circ }={\\frac {h}{d}}", "116cba1e615fa0c81ef017ca0220696f": "P_{3}=(3/4)-\\epsilon ", "116cf4bfbb0de46e5535f3eaa5925cb0": "(f*g_{T})(t)\\equiv \\int _{t_{0}}^{t_{0}+T}\\left[\\sum _{k=-\\infty }^{\\infty }f(\\tau +kT)\\right]g_{T}(t-\\tau )\\,d\\tau ,", "116cff2d0a695d9b64feb758cfbce710": "{\\frac {\\infty }{\\infty }}", "116d05e0567fd82ed85b9ea1a8209a22": "c=14", "116d23f78bdda731cd508bf1edc2fab0": "C^{*}(\\theta )", "116d24d9bec83e9fa98ba54fca653f20": "C\\ell _{2}(\\mathbb {C} )=\\mathbb {H} \\otimes \\mathbb {C} .", "116e0beaaf40538cb22ff6dd234f210e": "T_{\\epsilon }\\in {\\mathcal {M}}_{\\epsilon }", "116e874e1b0a8d4fe2f6ef411cbc1725": "{\\frac {dG}{dr}}>0", "116e8d0161aa18dcd8697c29107a5444": "\\langle x,\\,y\\rangle ={\\frac {1}{4}}(\\langle x+y,\\,x+y\\rangle -\\langle x-y,\\,x-y\\rangle )", "116f17dbce75f9c59e6d379785145eda": "a(u,v)=L(v)\\,", "116f1a4ea8fa990c13d061391fcdf3aa": "Car\\{f(t)\\}=G_{Car}(p)=p\\int _{0}^{\\infty }e^{-pt}f(t)\\,dt\\qquad (4)", "116f482925f13a16a0f215e52151a684": "{\\frac {V(t)}{V_{0}}}=0.1", "116f721d3da983c298853c95cdc136c3": "(V^{*})^{\\mathbb {C} }=V^{*}\\otimes \\mathbb {C} \\cong \\mathrm {Hom} _{\\mathbb {R} }(V,\\mathbb {C} ).", "116f922091726325efc390df425993bd": "H_{\\alpha }^{(2)}(ze^{im\\pi })", "116fdf0b91652e41f66e1e8dbf32cd86": "e(E)\\cup e(E)", "117036e7c6aab20fab2a2877c4b25ac9": "{\\frac {{\\text{base area}}\\times {\\text{height}}}{3}},", "11706b2094097664eac87a3f06fc7ec3": "\\exp :{\\mathfrak {g}}\\to G", "1170bed0e4fc98b25a8d4eb45e33e368": "=52,900\\pi ", "11710dab78027e9dfd08c7f33cb3549f": "{\\vec {u}}=\\left[{\\begin{array}{c}e^{i2t}s+e^{-i2t}{\\bar {s}}\\\\{\\frac {1-i}{2}}e^{i2t}s+{\\frac {1+i}{2}}e^{-i2t}{\\bar {s}}\\\\-{\\frac {i}{2}}e^{i2t}s+{\\frac {i}{2}}e^{-i2t}{\\bar {s}}\\end{array}}\\right]+{O}(\\alpha +|s|^{2})", "11712808843c7284f8321cfd1737c232": "{\\frac {a^{2}e^{t}}{\\left((a-1)e^{t}+1\\right)^{2}}}", "1171607b0a6433f5c479d07bb2493c08": "v_{g}={g \\over f}{\\partial Z \\over \\partial x}", "1171745fec644de49fc65b3e8decb474": "\\left|\\sum _{m=1}^{N}\\sum _{n=1}^{N}c_{mn}\\lambda _{m}\\lambda _{n}\\right|^{2}\\leq \\sum _{n=1}^{N}{1 \\over n}|\\lambda _{n}|^{2},", "1171a7669353b1c82b746d338c6243fd": "[t_{v}(x),1-f_{v}(x)]", "1171bd68fef4916e94ad457fdc70c348": "S(X)", "1171d7db59f3494bbfedf2a9502651ff": "{\\bar {n}}_{i}", "1171f3c2e687c8f5ff49704b82fd7971": "{\\frac {\\Gamma \\vdash t:\\alpha \\rightarrow \\beta \\qquad \\Gamma \\vdash u:\\alpha }{\\Gamma \\vdash t\\;u:\\beta }}", "1172181a130c2e7d7800c3af251b088a": "\\Psi _{0},\\ldots ,\\Psi _{4}", "117283af27dbb4ca45b5a6eb4b3a1dc7": "\\scriptstyle C\\;=\\;E_{K}(P\\,\\oplus \\,X)\\,\\oplus \\,X", "1172f0828ab9f76ef1abc90365328068": "k\\left\\{{n \\atop k}\\right\\}", "1172f4d0764352ed3247b500c30204c7": "|x|\\geq 2|y|", "1172f77da75dafe6030cbaed4593570a": "i^{2}=+1=j^{2}=k^{2}", "1173055876a7b2905f697d8f9668a637": "x_{11}=x_{22}=x_{33}=1\\ ", "11736f4d29b55d79c10b77ecd975e9fe": "h={\\frac {P}{\\gamma }}", "11739414d73d5a3aaf9d200f04d71b37": "G_{AW}={\\frac {1}{R_{AW}}}", "1173e3ebd1948797d17368626c35f8a9": "F:{\\mathcal {D}}\\rightarrow {\\mathcal {C}}", "117401135dec95dc50ec8608e19b6b25": "{\\frac {\\partial \\mathbf {F} (\\mathbf {X} )}{\\partial \\mathbf {X} }}={\\begin{bmatrix}{\\frac {\\partial f_{1,1}}{\\partial \\mathbf {X} }}&\\cdots &{\\frac {\\partial f_{1,p}}{\\partial \\mathbf {X} }}\\\\\\vdots &\\ddots &\\vdots \\\\{\\frac {\\partial f_{m,1}}{\\partial \\mathbf {X} }}&\\cdots &{\\frac {\\partial f_{m,p}}{\\partial \\mathbf {X} }}\\\\\\end{bmatrix}}", "11746bccbcf8464f21751a70089991a6": "n\\cdot k", "1174993a2831637a3259982405c06635": "{A}_{8}^{(1)}", "1174cb7011fde8918aedd7223b59169b": "{\\frac {n_{2}}{n_{1}}}={\\frac {g_{2}}{g_{1}}}e^{-h\\nu /k_{\\mathrm {B} }T}", "1175a25b4597fb7f0e1943dcc8cb7f76": "1=2-\\phi (2),3=9-\\phi (9)", "1175ba5a589b300ea49f15171dab0b0b": "x{\\frac {dx}{dt}}+y{\\frac {dy}{dt}}=h{\\frac {dh}{dt}}", "1175d3d4ed6e02a31cf9cda3f9c21419": "\\ Ci", "11764209c8d5f4598c920c2508390d3a": "\\mathrm {Re} ={\\frac {Vd}{\\nu }}\\ ", "117647c7157ba9372a6ecf8a3dc0527b": "\\operatorname {Var} [\\,\\varepsilon |X\\,]=\\sigma ^{2}I_{n}", "11765921e09fe6e829549e5dda369257": "{{S}_{5}=1}", "1176c5a284fbbccc28d74d018b05141a": "t^{*}=t-{\\frac {\\epsilon vx^{*}}{c^{2}}}\\cdot ", "1176f078de6b83917830cb6eb5dd04fa": "{\\frac {dQ}{dx}}=-q", "1176f5ea4e4804b2b7a60f5d637c3949": "p_{\\mathbf {y} }(\\mathbf {y} )={\\frac {p_{\\mathbf {x} }(\\mathbf {x} )}{|{\\frac {\\partial \\mathbf {y} }{\\partial \\mathbf {x} }}|}}={\\frac {p_{\\mathbf {x} }(\\mathbf {x} )}{|\\mathbf {W} |}}", "11770404062ba82c7d4c1aa072790ef1": "a_{0}1+a_{1}x+a_{2}x^{2}+\\ldots +a_{n}x^{n}", "11774547ef4cd4c5448862aa2c0a6023": "e_{0},e_{1},\\ldots ,e_{n}", "117776f7745bf3e5c5d51e88f1a23254": "{\\begin{Bmatrix}8\\\\8\\end{Bmatrix}}", "11780b176b58cd6ea28dee0873293bc7": "\\beta >0\\,", "11786ecb9f238b7b8df2d61d3eac99ba": "{\\begin{matrix}g_{m}R_{C}\\end{matrix}}", "11787d2fd817cd9fe9823373c8074a48": "f(t)=\\sum _{k=-\\infty }^{\\infty }f(t_{k}){\\frac {G(t)}{G'(t_{k})(t-t_{k})}},\\qquad \\forall f\\in B_{\\pi }^{2},\\qquad (t\\in \\mathbb {R} ),", "11788ebd5725a052338cd77f6490e97c": "E=0.0100\\,", "117899eaf7ed6ffda5badf88104d63c0": "\\int _{a}^{b}f(x)\\,dx", "11789c067d528fecc5ce45f9bb9c71eb": "\\forall x\\in X\\qquad \\|J(x)\\|''=\\|x\\|,", "11796cdc1086128796382de44b2507bb": "\\min(\\mathbf {x} )\\leq H(\\mathbf {x} )\\leq G(\\mathbf {x} )\\leq L(\\mathbf {x} )\\leq A(\\mathbf {x} )\\leq R(\\mathbf {x} )\\leq C(\\mathbf {x} )\\leq \\max(\\mathbf {x} )", "1179a931d14c6bbfca61f93374717098": "t={\\frac {\\mathrm {arctanh} (\\alpha v)}{\\alpha g}}", "1179b180d2fcc8046e5a1efb81fbf1e3": "K\\supseteq F", "117a094440c2eba0e6e5d0467c49c8db": "\\lambda C_{1}+\\mu C_{2}", "117a2b6f6d6c2fdb27ca47b848e48fee": "H_{2}(p)", "117a45d968b83dcd7914847eb3b00b38": "T_{1}", "117a59245a2de5e897b4261659347e1d": "g(z)=G(z)e^{-z\\tau }", "117a7a6ca88ee132faef24da7ee814cb": "{\\frac {1}{u}}+{\\frac {1}{v}}={\\frac {1}{f}},", "117ab296ecdee1fa53eea4bf18846eb6": "V(a_{0},b_{0},c_{0},d_{0},\\dots )=0\\,", "117b1de65c406a8c921dc07ab9c31d03": "Q=\\mathbf {F} \\cdot {\\frac {\\partial \\mathbf {V} }{\\partial {\\dot {q}}}}+\\mathbf {T} \\cdot {\\frac {\\partial {\\vec {\\omega }}}{\\partial {\\dot {q}}}},", "117b36b2a8dbf6b49b0ec6412754c478": "S={\\frac {A}{L}}.", "117b72513d4f8f6a45e02eaa40705857": "C^{T}B\\leq l_{W\\times L}", "117ba7a7e6c5336d3ac4f55dde406e72": "\\left({\\frac {\\partial f}{\\partial t}}\\right)_{\\mathrm {coll} }=\\iint gI(g,\\Omega )[f(\\mathbf {p'} _{A},t)f(\\mathbf {p'} _{B},t)-f(\\mathbf {p} _{A},t)f(\\mathbf {p} _{B},t)]\\,d\\Omega \\,d^{3}\\mathbf {p} _{A}.", "117c22a09a24b31fa321d92efe8542cc": "{\\begin{aligned}\\phi _{L}&\\to 90^{\\circ }={\\frac {\\pi }{2}}^{c}\\\\\\phi _{R}&\\to 0\\end{aligned}}", "117c24ef265606c75bb90945ed3e13f3": "f^{(m+n)}(x)=f^{m}(f^{n}(x))", "117c2d0addfba1f25d27d36ed2d11d36": "L(e):=L(\\mathbf {Z} [e])=L(\\mathbf {Z} )", "117c33d7e9e2dc477b8404376e5e676c": "\\lambda \\in K", "117c3bf6611d125422aa42d357d4b0fa": "M(\\beta ):=|\\langle \\mathbf {s} _{i}\\rangle |=0", "117c7c5e148e7093e155c0ef153c1a4d": "[A]_{p}\\subseteq [B]_{p}", "117c876c77fcb4838d85de387cbe3891": "S=A[x_{0},\\ldots ,x_{n}]", "117c9316c7e7125ca3655cfd675357c4": "Y(t)", "117cd7625dcd554c0a040e910224f44e": "[A_{n}]", "117ced56c999973b3908400d2b2e3760": "{\\sqrt {\\gamma }}\\,", "117d6831cf06fe1df9285813beea3b0b": "\\{X_{n}\\}_{1\\leq n\\leq \\omega }", "117dcc0fbfbf870f9191d14870821f43": "(x,z)", "117df26a1bbe348410220d5ca5647475": "n={\\sqrt {\\frac {G(M\\!+\\!m)}{a^{3}}}}\\,\\!", "117e1cd49f525ec50ceec22f12c4a6c1": "[a,a,a]=a,\\,[a,a,b]=b,\\,[b,a,a]=b,\\,[b,a,b]=a,", "117e870077ca470d0f60c7078f66dde0": "c(t,s)=d", "117ed0f9be061d102e91686c7ca0235d": "dE_{\\nu }=I_{\\nu }(\\mathbf {r} ,{\\hat {\\mathbf {n} }},t)\\cos \\theta \\ d\\nu \\,da\\,d\\Omega \\,dt", "117ed34abf0ea194a6bbd386056beaa8": "t_{0}=0", "117eeb703fa190e97a3c85a8985b657c": "\\lim _{n\\to \\infty }|f(x_{n})-f(y_{n})|=0.\\,", "117f0d2b15dc619b8808d30c79d0adc0": "\\{\\Omega ,F,Q,\\{F_{t}^{W}\\}\\}", "117f15c8f3faf940d396ef1eeb176ee7": "R_{mn}(r)=\\sum _{s=0}^{\\frac {m-|n|}{2}}(-1)^{s}F(m,n,s,r)", "117f2eada08195a0107d3f68b5cd63dd": "d=\\min d\\left(x_{i},x_{j}\\right)", "117f3f14332af756759f7baf43b54f36": "\\omega _{p}=-{\\frac {3}{2}}{\\frac {R_{E}^{2}}{(a(1-e^{2}))^{2}}}J_{2}\\omega \\cos i", "117f5f3114e0819728f6395981e8c8ad": "\\vert a\\vert ", "117f66712cc5f8d2098910b18a6a5704": "r\\approx {\\frac {d\\,\\Delta \\,E}{e\\,U}}", "118033221b907a59dae49784ba28f73d": "\\langle \\phi (x)\\otimes \\phi (y),\\phi (x')\\otimes \\phi (y')\\rangle =k(x,x')\\otimes k(y,y')", "118058c316ed9495877f9723050eb264": "S=I+NFI\\,\\!", "1180b57a13e84ba8490843f79f8bcce0": "\\tau \\circ i_{1}=i_{2}\\circ \\sigma ,\\ \\nu \\circ \\pi _{1}=\\pi _{2}\\circ \\tau ,\\ \\tau ^{*}\\omega _{2}=\\omega _{1}\\,,", "1180e2e37a24f0571e6bbe5ea1ff681a": "{\\mathcal {H}}_{\\mbox{accept}}", "1180e8b090976498bb302f9a6c780880": "{\\frac {1}{\\nu }}", "1180e9e0ae1b514839448da396de66a7": "\\ h=\\alpha R_{v}^{-1}s.", "118102e7c18a48d8899a119f97769f5e": "\\mathbf {u} (\\mathbf {x} )", "1181773135eee2b565798627a2595adc": "[C]", "1181a08a1f8789c5818aa2684fdc8c47": "i{\\frac {\\partial \\psi }{\\partial \\tau }}+{\\frac {1}{2}}{\\frac {\\partial ^{2}\\psi }{\\partial \\xi ^{2}}}+|\\psi |^{2}\\psi =0", "1181b70ddbc5b4d1bed0efdeb5b624d2": "V(r)={\\frac {k}{r}}", "1181c0a94992c78f2d3fd3e4198a6260": "\\lambda =3", "1181d0815ce412228210b8274de2cbb6": "{\\frac {q}{\\sin(\\theta /2)}}={\\frac {[x\\,y\\,z]^{\\top }}{\\|[x\\,y\\,z]\\|}}", "1181e002c98c3cbccaee0b35ce0e7351": "X_{0}=x_{0}", "1181f6aaa1cea49eac0fef907785b6d0": "\\{c_{r},c_{i},z_{r}\\}", "1182188601478ec0f6faeb247b0de1c8": "Q_{2},Q_{4},Q_{6},\\ldots ", "11829544f8c367926e75bcb8511da71f": "\\lim _{z\\to i}(z-i)f(z)=\\lim _{z\\to i}(z-i){e^{itz} \\over z^{2}+1}=\\lim _{z\\to i}(z-i){e^{itz} \\over (z-i)(z+i)}=\\lim _{z\\to i}{e^{itz} \\over z+i}={e^{-t} \\over 2i}.", "1182c033e5b85ce38cef766da9963bad": "\\forall x,y,z:x0", "11a7eabd1b1c44aa3f4c873b38028ae3": "\\psi _{\\pm }^{(0)}", "11a84a75cf428a917f16ea7ff0945e8b": "EI~{\\cfrac {\\mathrm {d} ^{4}{\\hat {w}}}{\\mathrm {d} x^{4}}}-\\mu \\omega ^{2}{\\hat {w}}=0\\,.", "11a8d86726d302556a29dfa1368e290a": "l_{m}=ct/2", "11a90372ae2b049f9adffee655e2a72a": "K'_{1}", "11a96c38f521725bc19c7ae083c5289b": "\\mathrm {Prog} ", "11a9d0f35ce9d8e4f180c6abe854a211": "{\\begin{aligned}&EI{\\frac {\\mathrm {d} ^{4}w}{\\mathrm {d} x^{4}}}=0\\\\&w|_{x=0}=0\\quad ;\\quad {\\frac {\\mathrm {d} w}{\\mathrm {d} x}}{\\bigg |}_{x=0}=0\\quad ;\\quad {\\frac {\\mathrm {d} ^{2}w}{\\mathrm {d} x^{2}}}{\\bigg |}_{x=L}=0\\quad ;\\quad -EI{\\frac {\\mathrm {d} ^{3}w}{\\mathrm {d} x^{3}}}{\\bigg |}_{x=L}=F\\,\\end{aligned}}", "11aa0643bf16b4b9043967e8de22625f": "i\\leftarrow i+r", "11aa3248f8ca1c728ccaeb6cf8b1add0": "b+d<1\\,\\!", "11aad80233a22332f9547cb445c0b2fb": "G(L,R,E)\\,", "11aae48d8fc93f983099e28586e1a3b3": "d\\colon M\\times M\\rightarrow \\mathbb {R} ", "11aaeb6ceb570fdd697907fbbf7388d5": "1-\\gamma _{k}dt", "11ab2b2ccbfeceba3b5595ca34baca26": "(\\pm 1,0,0,\\dots ,0)", "11ab4b54123ec919b1071b230866710d": "{\\frac {1}{4\\pi }}\\operatorname {Im} {\\Bigl [}\\int d^{4}\\theta {\\frac {dF}{dA}}{\\bar {A}}+\\int d^{2}\\theta {\\frac {1}{2}}{\\frac {d^{2}F}{dA^{2}}}W_{\\alpha }W^{\\alpha }{\\Bigr ]}\\,", "11abcd51751cf261db1b87f46460f1ce": "i_{n}^{2}=-1", "11abd2f806d34ae561ca62929700cf00": "U=\\{v\\mid \\Omega (v)=p\\}", "11abfd083e6b86882c80a3378e2b3d1e": "\\psi _{1}(x)\\sim {\\frac {x^{2}}{2}}.", "11ac146493ae65f4fbf4f05b484e64bc": "p_{0}=\\,p_{1}+\\ {\\tfrac {1}{2}}\\,\\rho \\,v^{2}.", "11ac209ac1341776954956d0f6554cc7": "\\operatorname {P} (Z_{i}=1)=\\tau _{1}\\,", "11ac57ef5edcf4bbe51d6697476aa894": "\\!\\ Re<10^{5}", "11ac6a2f0d3ecf155dca5dd67ec23781": "\\sigma _{v}^{2}", "11acb0d6d50d41bffdecce3e83039b09": "(A^{*}A)^{-1}\\,\\!", "11ad5eb0249d88065004e74b38ee9e66": "(f{\\big |}\\gamma )(\\tau )=(\\rho (C\\tau +D))^{-1}f(\\gamma \\tau ).", "11ae002a0fb31ee6f3fd7596e4b07a4d": "t_{r}={\\frac {4}{\\sigma }}{\\mathrm {erf} ^{-1}(0.8)}\\cong {\\frac {0.3394}{f_{H}}}", "11ae07a6a8c4ff9601203bdead6014bf": "A_{\\sigma }^{\\mu }", "11ae468a3b92b84d2e6b12ae0eebb9b2": "{\\frac {d}{dr}}\\left(p+{\\frac {B_{z}^{2}}{2\\mu _{0}}}\\right)=0", "11ae968c4e2df747c7091ff1e6f8e2b5": "{\\tilde {B}}_{9}", "11aebcef8709da91fea76d93c5545261": "{\\mathfrak {t}}", "11aedd0e432747c2bcd97b82808d24a0": "FR", "11af963ffd06d004bc5daa73379f4e2f": "\\beta ^{a}\\gamma +\\gamma \\beta ^{a}=\\beta ^{a}", "11b054f14693226433206621deab70cf": "{\\hat {i}}", "11b05d3bf74162ae4340a4e8a50ee956": "S_{y}=S_{zx}=\\int _{\\partial {\\mathcal {V}}}[(z-z_{\\text{com}})T^{0x}-(x-x_{\\text{com}})T^{0z}]dxdydz", "11b0f750d5701cde064ead807a85552a": "h_{e}(X)\\,", "11b1198851cf3ab82e536cd11abc14ae": "f(x\\pm h)=f(x)\\pm hf'(x)+{\\frac {h^{2}}{2}}f''(x)\\pm {\\frac {h^{3}}{6}}f^{(3)}(x)+O_{1\\pm }(h^{4}).\\qquad (E_{1\\pm }).", "11b179c60b7c7af416738accb6ef9b91": "\\scriptstyle p\\equiv 1\\;mod\\;4", "11b1aa371e1b4354f6a238494c3d1fd3": "M=k", "11b1adea22b58011cd4f701393cfdc79": "\\scriptstyle V\\left({\\frac {1}{e}}\\right)", "11b1b3bb43582b1f3cb0dc554b4dc7fb": "fg+gf=(f,g),\\,", "11b1c063d3bf3bbac5b0efed84f7001c": "-B_{1}\\left(f(n)+f(m)\\right)={\\frac {1}{2}}\\left(f(n)+f(m)\\right).", "11b228092f47746cbe2158b88f0bceb7": "x_{A}+y", "11b2abf03847e92e346d70d816fba149": "{\\begin{aligned}N_{11}&={\\cfrac {Eh}{2(1-\\nu ^{2})}}\\left[\\left({\\frac {\\partial w}{\\partial x_{1}}}\\right)^{2}+\\nu \\left({\\frac {\\partial w}{\\partial x_{2}}}\\right)^{2}\\right]\\\\N_{22}&={\\cfrac {Eh}{2(1-\\nu ^{2})}}\\left[\\nu \\left({\\frac {\\partial w}{\\partial x_{1}}}\\right)^{2}+\\left({\\frac {\\partial w}{\\partial x_{2}}}\\right)^{2}\\right]\\\\N_{12}&={\\cfrac {Eh}{2(1+\\nu )}}\\,{\\frac {\\partial w}{\\partial x_{1}}}\\,{\\frac {\\partial w}{\\partial x_{2}}}\\end{aligned}}", "11b34e87417616c61bbedffacac1b65f": "xy\\equiv yx\\,.", "11b399b94a25be9b0f801511afabf8e2": "f(r)=(1-r^{2})^{2}", "11b41fef36dd8adbde5ae9673b2b065d": "V(q)={\\begin{cases}0&q\\in \\Omega \\\\\\infty &q\\notin \\Omega \\end{cases}}", "11b4310a65d3510d0c485f8f55f6909d": "f(5)=0.01024\\,", "11b49f590d903ad849cdc607e34e8d16": "Resolution=2^{{group}+{\\frac {element-1}{6}}}", "11b50b8fbbee172b3204f66c98711b23": "c_{i}(k)", "11b5160b179795fe0624b937c8da905c": "(\\mathrm {id} _{V})^{\\mathbb {C} }=\\mathrm {id} _{V^{\\mathbb {C} }}", "11b59e2a38012325167752b7700ebb8a": "\\dim R[T_{1},\\ldots ,T_{n}]=n+\\dim R,\\,", "11b5a76d5751498d4bd31705e9407ae1": "O(n2^{n})", "11b60dff80227709a6cb11540a856c8b": "C'C''", "11b610e497c1c994acebd013e63278ae": "S=g^{ab}(\\Gamma _{ab,c}^{c}-\\Gamma _{ac,b}^{c}+\\Gamma _{ab}^{d}\\Gamma _{cd}^{c}-\\Gamma _{ac}^{d}\\Gamma _{bd}^{c})=2g^{ab}(\\Gamma _{a[b,c]}^{c}+\\Gamma _{a[b}^{d}\\Gamma _{c]d}^{c})", "11b61470da9f0be95bf73a19e9361477": "c={\\sqrt {n}}", "11b617e9bb799b3805e664ad7faea35e": "x_{\\mathrm {i} }={\\frac {p_{\\mathrm {i} }}{p}}={\\frac {n_{\\mathrm {i} }}{n}}", "11b636f081d5f3565c2adb6892000a94": "k/q.", "11b668f53f9a2e32b74f2b5d837664f7": "{\\mathcal {N}}_{R}", "11b6906b668bbd84ebc34fe2f7a3d336": "{\\frac {m_{0}}{m_{1}}}", "11b695a31ec2e50b1f0de0245fdc05fb": "(x'_{i})", "11b73dd6e8ad1c143ca5ab30fa5d6a28": "{\\frac {\\partial y}{\\partial \\mathbf {X} }}.", "11b76c61361bf60df3db38e37da01777": "H(s)={\\frac {s^{2}+\\omega _{z}^{2}}{s^{2}+{\\frac {\\omega _{0}}{Q}}s+\\omega _{0}^{2}}}", "11b78314ceb6e861cca1d51a6f20a9e4": "T=g_{m}\\left(R_{D}\\ ||r_{O}\\right)\\approx g_{m}R_{D}\\ ,", "11b7c96e63bc207eeb4534586c222726": "Impulsiveness", "11b7cda882c050df7c5ec8acbddc3d60": "\\varphi (t)=\\sum _{n=0}^{\\infty }{\\frac {(it)^{n}}{n!}}e^{n\\mu +n^{2}\\sigma ^{2}/2}.", "11b7ebf120fae22e0d7aaa9628d9e3d9": "\\Delta U=NC_{V}\\Delta T\\,\\!", "11b7f71278d18664661fa856b8d7f973": "matrix(3,3)", "11b814048a39d34d64a5f7607b36979d": "mN_{k}-N_{k+1},", "11b8294189f3a23fbe2327e2d3da5885": "{\\sqrt[{3}]{n}}_{s}", "11b8318688ada4ec54f10c83f608bca4": "-6\\pi r_{p}\\mu V_{r}+{\\frac {4}{3}}\\pi r_{p}^{3}{\\frac {V_{t}^{2}}{r}}\\rho _{p}-{\\frac {4}{3}}\\pi r_{p}^{3}{\\frac {V_{t}^{2}}{r}}\\rho _{f}=0", "11b83510388ffdecff441c67f6b7b90f": "(\\mathbf {a} \\otimes \\mathbf {b} )\\cdot \\mathbf {n} =(\\mathbf {b} \\cdot \\mathbf {n} )\\mathbf {a} ", "11b8397342d93d09ee9704377150a7c7": "{\\tilde {\\nu }}={\\frac {\\nu }{c_{\\mathrm {n} }}}", "11b88743ebc05a218dd129c5fb2c7d11": "\\,P(X=1)=p", "11b8dc85f0024f4753b55976d657959d": "B\\subseteq S", "11b8ed25729a8c79af1d22e958effdf1": "{B}_{3}^{(1)}", "11b90f5f8b1c7511f7881725aa078db5": "2^{-7/4}", "11b9547165d679e0d80ed6596d14faa0": "(mR)^{2}+(n/R)^{2}", "11b9732f985f005686a8b77109599314": "\\displaystyle {Q(a)R(b,a)=R(a,b)Q(a)=2Q(Q(a)b,a).}", "11b9ccb302aa3f5f626a860c8b680bf5": "(\\alpha \\to \\beta )", "11b9f85ab74578e12373c1cbfa3ebec5": "{\\frac {du_{i}}{dt}}+{\\frac {1}{\\Delta x_{i}}}\\left[F\\left(u_{i+1/2}^{*}\\right)-F\\left(u_{i-1/2}^{*}\\right)\\right]=0.", "11b9fdfa6a1f1146cd765f616a6740a7": "G_{B}(\\tau =0^{-})={\\frac {1}{\\beta }}\\sum _{i\\omega _{n}}{\\frac {e^{i\\omega _{n}0^{+}}}{i\\omega _{n}-\\xi }}=-n_{B}(\\xi )", "11ba21c9ecec309ae0f871e1232bdb26": "\\mathbf {D} (X)", "11bb45bf03607fa4a8780b2eb967b8f8": "{\\textbf {x}}_{e}={\\textbf {x}}_{o}+\\gamma ({\\textbf {x}}_{o}-{\\textbf {x}}_{n+1})", "11bb48e2b88c2f4b2f122c08b6091665": "\\!Y\\subseteq X", "11bb7801e0ddd1340404d859e0028056": "C_{ijkl}\\Rightarrow C_{\\alpha \\beta }={\\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\\\C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\\\C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\\\C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\\\C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\\\C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\\end{bmatrix}}.\\,\\!", "11bb7fe00bad88246f53d380583586cb": "\\mathbb {E} ^{x}[X_{\\tau -}^{j}X_{\\tau }^{k}]", "11bba3af6b0b6e810be8a9d4b7103a39": "\\pi \\sim (\\ell )^{-1}(t)^{1}(\\ell /t)^{1}\\sim 1", "11bbdf62da879f31eab2cd9947d2b230": "\\delta _{H}(d_{N})=\\delta _{H0}{\\frac {\\exp \\left(-d_{N}/\\lambda _{N}\\right)}{1+d_{N}/d_{0}}},", "11bc1cb1f8cbbec01a2643bbb0c314ac": "L(s_{i_{k}})=L(s_{i_{k}+1})=\\ldots =L(s_{i_{k+1}-1})=L(r_{j_{k}})=L(r_{j_{k}+1})=\\ldots =L(r_{j_{k+1}-1})", "11bc2f7ed78af035a2b47b0adb6a0677": "1/R^{4}", "11bc99ce86e4f6c555e8daecfa5888dd": "R={\\frac {2GM}{c_{s}^{2}}}", "11bcde0d2c833b7c1002feb91b6b4c93": "{\\widehat {C}}_{Y\\mid X}={\\boldsymbol {\\Phi }}(\\mathbf {K} +\\lambda \\mathbf {I} )^{-1}{\\boldsymbol {\\Upsilon }}^{T}", "11bcec96c06854bfe499bc54f4ee0da5": "{\\mathfrak {q}}", "11bd10ddb1f764d6fcbded698dc86707": "\\varphi (-r)=i{\\sqrt {r}}", "11bd2361a6eb807f36b53586e5f80af6": "\\alpha _{H}=a_{1}x_{1}+a_{2}x_{2}+\\cdots ", "11bd9da3b32a4786783d25184a03bd31": "r=\\left(g^{k}{\\bmod {\\,}}p\\right){\\bmod {\\,}}q", "11bdc1e30fd3ed1b9887e8ddeb96c06e": "f(x)=g(x)(h(x))^{-1}", "11bee95572ff1b37d3161ca75fab97c3": "\\delta _{g}={\\frac {\\tau c_{0}}{2cos\\gamma }}", "11befdb7e597dc81e6b7d87cdf7b5e2d": "\\mathrm {SU} (2)=\\left\\{{\\begin{pmatrix}\\alpha &-{\\overline {\\beta }}\\\\\\beta &{\\overline {\\alpha }}\\end{pmatrix}}:\\ \\ \\alpha ,\\beta \\in \\mathbf {C} ,|\\alpha |^{2}+|\\beta |^{2}=1\\right\\}~,", "11bf062195375b554ddaea39d71c36aa": "\\ \\displaystyle \\varphi (q,\\alpha ,u)", "11bfc01102a4c23ece55bfaef15791db": "a_{b}=a", "11bff4ef546eaddaa32849dca0ecbf2f": "\\det(\\varphi I_{n}-A)", "11c0521e5bb89ec0ecce70100746eaf6": "m\\cdot O(1)=O(m)=O(n)", "11c05f4117923f95ac1edf6cd520e3b9": "X_{1},...,X_{n},Y_{1},...,Y_{n}", "11c06d5b5312a1707492fda3188f324d": "s=a\\varphi ", "11c07821cf89ac201e0641174b78632a": "(Eq.9){\\text{ }}{\\text{Subject to: }}\\lim _{t\\rightarrow \\infty }{\\overline {Y}}_{i}(t)\\leq 0{\\text{ }}\\forall i\\in \\{1,...,K\\}", "11c08f093cff73b3d86a5147a5b6cd28": "q^{n}=k^{n}(\\cos n\\theta +\\epsilon \\sin n\\theta )", "11c09e7863d9d3194e9096ac86bae950": "Rec(w',s)=s+dec(w'-s)=w", "11c0cc3b57d3256be2cb8ff9cdaf890b": "s(v_{j},v_{k})=v_{j}\\cdot v_{k}", "11c10ca094f47cfa99c7de7d4dc4eb16": "f_{1}(x;\\nu )={\\frac {2^{-\\nu /2}}{\\Gamma (\\nu /2)}}\\,x^{-\\nu /2-1}e^{-1/(2x)},", "11c114e7209a5960780cfb1ebd9fc9f6": "Q=c_{1}e\\left(\\int _{0}^{1}f_{k}e^{-x}\\,dx\\right)+c_{2}e^{2}\\left(\\int _{0}^{2}f_{k}e^{-x}\\,dx\\right)+\\cdots +c_{n}e^{n}\\left(\\int _{0}^{n}f_{k}e^{-x}\\,dx\\right)", "11c19357ff9addad82843b1fa7701271": "{\\frac {\\partial W}{\\partial t}}=-\\{\\{W,H\\}\\}=-{\\frac {2}{\\hbar }}W\\sin \\left({{\\frac {\\hbar }{2}}({\\stackrel {\\leftarrow }{\\partial }}_{x}{\\stackrel {\\rightarrow }{\\partial }}_{p}-{\\stackrel {\\leftarrow }{\\partial }}_{p}{\\stackrel {\\rightarrow }{\\partial }}_{x})}\\right)\\ H=-\\{W,H\\}+O(\\hbar ^{2}),", "11c19633c6d7b85eec5efafdd655392d": "i+1", "11c1c2288469326fb14435817d4d7d00": "\\textstyle d=4", "11c1ca1e01883a29c81f45fd43419fd3": "\\{x_{n}\\}_{n=1}^{\\infty }", "11c1cfdab01ae7681edfb77fea93eec0": "\\mathbf {x} ^{(0)}={\\begin{bmatrix}1&0\\end{bmatrix}}", "11c1fc7aa258b5e0ae4870e0cc1ba60a": "\\delta \\Gamma _{\\mu \\nu }^{\\lambda }={\\frac {1}{2}}g^{\\lambda a}\\left(\\nabla _{\\mu }\\delta g_{a\\nu }+\\nabla _{\\nu }\\delta g_{a\\mu }-\\nabla _{a}\\delta g_{\\mu \\nu }\\right).", "11c2228ac0051bca1eda480e726b37f9": "{\\vec {p}}_{0}={\\frac {1}{\\sqrt {1-\\omega ^{2}\\,R^{2}}}}\\,\\partial _{T}+{\\frac {\\omega \\,R}{\\sqrt {1-\\omega ^{2}\\,R^{2}}}}\\;{\\frac {1}{R}}\\partial _{\\Phi }", "11c233a746ebb17a0e3a25cfab18831d": "J0}}\\\\0&{\\rm {otherwise}}\\end{cases}}", "11d1c399bc953bd57052df718c97eb28": "D_{\\mathrm {p} }=D_{\\mathrm {maj} }-2\\cdot {\\frac {3}{8}}\\cdot H=D_{\\mathrm {maj} }-{\\frac {3{\\sqrt {3}}}{8}}\\cdot P\\approx D_{\\mathrm {maj} }-0.649519\\cdot P", "11d1fe36981aa8c3eff4a88cc464b32a": "\\scriptstyle a,\\,b", "11d2160cfc9793e262568d9b9bc96b87": "m=(G/2)*u", "11d24fe702c2d1cdd62babbba5708df6": "\\forall x\\in U:Px=x.", "11d25fdc92b7462f27048eced349ab6b": "\\left.\\right.\\left.F(z)\\right.", "11d2bf2ea54935cbfc833e064e593fd4": "u={\\frac {T}{3}}\\left({\\frac {\\partial u}{\\partial T}}\\right)_{V}-{\\frac {u}{3}}", "11d2c9e260cf131c4573d14a6fb40290": "{\\frac {AF}{FB}}\\times {\\frac {BD}{DC}}\\times {\\frac {CE}{EA}}=-1.", "11d2d83d2a8b74903fe01d972b7a3ccf": "F(s-a)\\ ", "11d2fe8f803bb9174e28aab2ad00c0e7": "\\kappa ={\\tfrac {1}{R}}={\\tfrac {\\mathrm {d} \\theta }{\\mathrm {d} t}}=2t", "11d38009cc294822bb57ad979908a7cd": "\\min Z'=\\sum _{k=1}^{u}{r^{k}L^{k}+\\sum _{i=1}^{n}{v_{i}\\left({-N_{i}}\\right)}}", "11d38807371727c1cb969272b2ca024d": "={\\widehat {U}}^{\\dagger }i({\\widehat {a}}^{\\dagger }{\\widehat {a}}{\\widehat {a}}-{\\widehat {a}}{\\widehat {a}}^{\\dagger }{\\widehat {a}}){\\widehat {U}}={\\widehat {U}}^{\\dagger }i[{\\widehat {a}}^{\\dagger },{\\widehat {a}}]{\\widehat {a}}{\\widehat {U}}=-i{\\widehat {U}}^{\\dagger }{\\widehat {a}}{\\widehat {U}}", "11d39ad00e0261287376610f95b99ae0": "\\mathrm {W} (\\theta ,\\Phi )={\\frac {\\mathrm {G} (\\theta ,\\Phi )}{4\\pi r^{2}}}P_{t}", "11d3b633daf16f3fef0913059b75f3db": "f_{c}'(z)={\\frac {d}{dz}}f_{c}(z)=2z", "11d3bad9bd977d95c0bffc7a473760ea": "\\exists \\delta >0\\left[d(x,y)<\\delta \\Rightarrow \\lim _{n\\to \\infty }d\\left(f^{n}(x),f^{n}(y)\\right)=0\\right].", "11d46313ebd320b17294a457b363aa8b": "\\left(x(.),u(.)\\right)", "11d55c055be5f075ce61b2d2ba1f79fa": "\\operatorname {wnchypg} (x-1;n-1,m_{1}-1,m_{2},\\omega ){\\frac {m_{1}\\omega }{m_{1}\\omega +m_{2}}}+", "11d56074081eeffe4c63eb095a38206b": "\\mathrm {NOT} =\\lambda x^{\\mathsf {Boolean}}{.}x\\,{\\mathsf {Boolean}}\\,\\mathbf {F} \\,\\mathbf {T} ", "11d57ab76d6294282af585087ca7334e": "Y_{b}", "11d5866f1abdb1c49cac70dbbbe57902": "\\alpha =c_{1}+2r", "11d5b60ce76bbc641214acb704ab6fe6": "\\left|\\mathbf {E} \\right|={\\sqrt {\\frac {\\epsilon }{\\mu }}}\\left|\\mathbf {H} \\right|\\,\\!", "11d61eaf65925f427f4dc01f1b7e84e7": "{\\hat {M}}=M\\otimes _{R}{\\hat {R}}.", "11d67ea4b9fb6edd5f254b938295e3b5": "{\\frac {dM_{z}(t)}{dt}}=\\gamma ({\\mathbf {M}}(t)\\times {\\mathbf {B}}(t))_{z}", "11d6e47a6720e94b34b288c195ef5b40": "f(x)\\propto x^{\\alpha _{1}}", "11d77f58e03f7ed2001c6f2a0f5b2df1": "\\|\\alpha \\|_{1}=\\sum _{i=1}^{p}|\\alpha _{i}|", "11d78c73e6a2eaeb07f68ceec74c2950": "{\\dot {x}}=f(x)+\\varepsilon ^{2}g(x,y,\\varepsilon )", "11d7c155b970e74f0062d495b8fd4d5a": "n_{-}(V)=\\operatorname {dim} \\ \\operatorname {ran} (V)^{\\perp }", "11d814dce29075fb019d4532066556e0": "L_{+}", "11d901c3f95a6b29f967667d965579b0": "i=0,...,n", "11d91946425bcfec566ce02871ef6028": "V({\\boldsymbol {r}})=0", "11d92354c0639da86952e51c6a7e7b88": "\\lambda ={\\frac {\\lambda _{0}}{n(\\lambda _{0})}}.", "11d939a9a0deb985b2f8665979f5ef30": "{\\begin{array}{rl}\\partial _{t}u&=d_{u}^{2}\\Delta u+\\lambda u-u^{3}-\\sigma v+\\kappa ,\\\\\\tau \\partial _{t}v&=d_{v}^{2}\\Delta v+u-v\\end{array}}", "11d93c03615feab22fe36ba8d244629f": "O(1/i)", "11d93e19e9ff675a9c86b0ffa75cf786": "E=\\{e_{1},e_{2},e_{3},e_{4}\\}=", "11d9767e7ea405a2d815e0cdd080b1fc": "m_{u}^{}", "11d9ebff2514f655e2909ae8eee20ce3": "{\\text{arcsin}}x\\approx x", "11d9fc68eaa08665e5ce85834cf001c1": "\\limsup _{n\\rightarrow \\infty }{\\frac {\\sigma (n)}{n\\,\\log \\log n}}=e^{\\gamma },", "11da1adc767e0c9a0ce7dfa41807a3dc": "{\\frac {1}{3b^{2}}}", "11da63597524890adae32f86363d5f5a": "z\\in \\mathrm {Im} (\\eta )", "11dad195d8e52921a73c86e2e61eb50a": "\\mathbf {C} ^{*}", "11db12869f053838508a4fbbf89f2ddf": "\\mathbf {a} \\succ \\mathbf {b} ~", "11db6ee9d7d579f4ea6b2b98bd1217f4": "\\langle {\\hat {L}}\\rangle =\\sum _{i=1}^{n}\\langle \\phi _{i}\\phi _{i}|{\\hat {L}}|\\phi _{i}\\phi _{i}\\rangle ", "11db89e9fbef3a15e4b6244be8bec816": "(S_{6}\\implies (\\operatorname {equate} [A_{6},x]\\land V[F_{6}]=A_{6}))\\land D[F_{6}]=D[x]", "11dbbb6333e772d15bd6c9ef2a268f90": "x_{i}=u_{i},\\;F_{i}(x)\\leq 0", "11dbcbacd821c1f2fd829823642ba0f2": "[a,b].", "11dbcced51c5e20bdbd0e06e218676c1": "P_{CSWD}={\\sqrt {P_{C}\\cdot P_{HR}}}", "11dbeb470a22725eae723ee964219880": "{\\begin{bmatrix}A_{11}&A_{12}&A_{13}\\\\A_{21}&A_{22}&A_{23}\\\\A_{31}&A_{32}&A_{33}\\end{bmatrix}}^{T}{\\begin{bmatrix}A_{11}&A_{12}&A_{13}\\\\A_{21}&A_{22}&A_{23}\\\\A_{31}&A_{32}&A_{33}\\end{bmatrix}}={\\begin{bmatrix}1&0&0\\\\0&1&0\\\\0&0&1\\end{bmatrix}}", "11dc28755d48bdd248d2e630065d5ca1": "I(X)=H(X)", "11dc32ca6523e30001129dbf8e96f2d5": "a_{i,j}=(-1)^{i+j}\\det(M_{i}^{j})", "11dc7aa165ef6d539a3fe04b99536837": "0\\leq \\operatorname {tr} (AB)^{2}\\leq \\operatorname {tr} (A^{2})\\operatorname {tr} (B^{2})\\leq \\operatorname {tr} (A)^{2}\\operatorname {tr} (B)^{2}", "11dcb4cb1d4066a67678b6bb3e356035": "x_{0}\\in X", "11dcbfc57387afca9e1759de34f5a796": "{\\rm {E}}\\left[{x_{i}-{\\bar {x}}_{i}}\\right]\\,\\,\\,=\\,\\,\\,{\\rm {E}}\\left[{x_{i}}\\right]\\,\\,\\,-\\,\\,\\,{\\rm {E}}\\left[{{\\bar {x}}_{i}}\\right]\\,\\,\\,=\\,\\,\\,\\mu _{i}-\\,\\,\\mu _{i}\\,\\,\\,=\\,\\,\\,0", "11dcecd31e731212639c8d646afb3eb7": "\\scriptstyle p\\;=\\;(xy\\,+\\,1)x", "11dced657f158f0a6f79c137b0a15e5e": "\\Box p\\rightarrow \\Diamond p", "11dceeb5ce1e6f8c5547918b7890fe7d": "(c-\\gamma (s))\\cdot {\\underline {T}}(s)=(c-\\gamma (t))\\cdot {\\underline {T}}(t)=0,", "11dd08fe97945563a17d261788aaacaa": "e^{i(h-(-h))\\theta }=e^{2ih\\theta }", "11dd385761f0a8b2c9ee507971fc7501": "E\\Psi =-{\\frac {\\hbar ^{2}}{2m}}\\nabla ^{2}\\Psi +V\\Psi ", "11ddf3a3bacec1bc9d5b1671849981e8": "T(v)=\\lambda v", "11deb27929c1bd314c59e988aea4da64": "t=2Nc\\left(1+m\\right)\\,.", "11dec9074649889630b840ade80c8205": "{\\hat {g}}(\\xi )", "11df0b7969eec89421afd102b07e604a": "{\\mathcal {N}}(\\mu ,\\sigma ^{2})", "11df4671d951c3d4600067696845f99f": "f\\in W^{*}", "11df48d189a8241e594a1e90b2460b29": "t_{\\Delta }", "11df63dcebb430119b832c602f154893": "\\scriptstyle w(e)", "11dfa998f298ca5e7d31bf2b45f931d8": "{\\gamma }={\\frac {4{\\pi }^{2}\\mathrm {m} {\\mathrm {V} }}{{A}^{2}p{T}^{2}}}", "11dfca20e8ea322b70e7b8ba7c976d5e": "P_{\\mathbf {v} ^{K}}(\\mathbf {x} |spike)=\\langle \\prod _{i=1}^{K}\\delta (x_{i}-\\mathbf {s} \\cdot \\mathbf {v} _{i})|spike\\rangle _{\\mathbf {s} }", "11dfe907f9269ce0adeddafc973d18bd": "\\sum _{A=1}^{N}\\mathbf {s} _{A}^{t}=0\\quad \\mathrm {and} \\quad \\sum _{A=1}^{N}\\mathbf {R} _{A}^{0}\\times \\mathbf {s} _{A}^{t}=0,\\quad t=1,\\ldots ,3N-6.", "11dff278df10b879b16db44ab74f039b": "pH=-\\log\\{H^{+}\\}\\,", "11e018751617e4fabe33aa6c29f27abf": "\\prod _{i=0}^{k}{\\binom {m_{i}}{n_{i}}}{\\pmod {p}}", "11e01ade2d2e023deacc9ad5391caf82": "x_{I}^{*}(\\theta )\\in \\arg \\max _{x\\in X}\\sum _{i\\in I}v(x,\\theta _{i})", "11e02b8cb6226946f595f58d507936cb": "a+rA=\\{a+rx:x\\in A\\}.", "11e04a304f4b6423980f7555bdd4cf41": "{\\frac {64}{16}}={\\frac {\\!\\!\\!\\not 64}{1\\!\\!\\!\\not 6}}={\\frac {4}{1}}=4", "11e07516c9ff1ae1d18f844093bab267": "\\nabla y_{t}=\\delta y_{t-1}+u_{t}\\,", "11e08ce9610b4561dc8a839a3846109b": "{\\begin{bmatrix}c_{3}c_{1}-c_{2}s_{1}s_{3}&-c_{3}s_{1}-c_{1}c_{2}s_{3}&s_{2}s_{3}\\\\c_{2}c_{3}s_{1}+c_{1}s_{3}&c_{3}c_{2}c_{1}-s_{3}s_{1}&-c_{3}s_{2}\\\\s_{1}s_{2}&c_{1}s_{2}&c_{2}\\end{bmatrix}}", "11e0d0c606434a00ace138baef950774": "n=77", "11e0d930026f0dbdb064a96437fbeaf8": "A=A_{0}\\oplus A_{1}", "11e10dde3dff77631a2b72f2505f4a11": "i_{n}i_{m}+i_{m}i_{n}=0", "11e124cc17fe1f2c70ade5b9721b8f60": "\\angle QCB=\\angle QBA=\\angle QAC", "11e17fe92d980c807bdb9d613e8adb15": "Z_{ij}", "11e182d4c29bdf5bad4882f4809ff77e": "2\\pi r", "11e18821fa277ffb37661c872d2073c4": "\\sin(iy)={e^{-y}-e^{y} \\over 2i}=-{e^{y}-e^{-y} \\over 2i}=i\\sinh(y)\\ .", "11e1ad7ee5e0ad4c2904d90355381278": "{\\tilde {\\psi }}", "11e1ed0d090db03164b41f636faa5590": "[-\\nabla ^{2}+k^{2}]\\Phi (\\mathbf {x} ,\\mathbf {x} ')=\\delta (\\mathbf {x} -\\mathbf {x} ')", "11e202dc62949f494c84b7271922bdb7": "T^{2}=\\mathbb {R} ^{2}/\\mathbb {Z} ^{2}", "11e20d73c8440be32494bdbc3d383be1": "{\\sqrt {N}}", "11e24123a5a9a4b549f2d008acc32892": "A=[a_{ij}]", "11e24259e061c7dc8e029ad399fa496a": "\\displaystyle {\\frac {1}{|a|}}\\cdot \\operatorname {tri} \\left({\\frac {\\xi }{a}}\\right)", "11e24faca9d893c790a6fd17042c9a7d": "\\lambda ", "11e265b6f385b00fc20e9a7994802b70": "T(-h,a)=T(h,a)", "11e2660de92e7bf1d36ed791aacf78bc": "{\\vec {x}}(t+\\Delta t)={\\vec {x}}(t)+{\\vec {v}}(t)\\,\\Delta t+{\\tfrac {1}{2}}\\,{\\vec {a}}(t)\\,\\Delta t^{2}", "11e3699d824b4516fb3bc8e0c74dfc39": "{\\frac {\\partial }{\\partial u}}g(z,u){\\Bigg |}_{u=1}={\\frac {1}{1-z}}\\sum _{k\\geq m}{k \\choose m}{\\frac {z^{k}}{k}}={\\frac {1}{1-z}}{\\frac {1}{m}}{\\frac {z^{m}}{(1-z)^{m}}}={\\frac {1}{m}}{\\frac {z^{m}}{(1-z)^{m+1}}}.", "11e3c76bd167154c7dd85234e874b4a5": "\\Omega ({\\sqrt {n\\log n}})", "11e456ec90d6ede89f6ddce301a3ebaa": "(Cut)\\quad {Z\\leftarrow \\Delta X\\Delta '\\qquad X\\leftarrow \\Gamma \\over Z\\leftarrow \\Delta \\Gamma \\Delta '}", "11e4ba83387d0d23a023742c0fc19876": "K(u)={\\frac {70}{81}}(1-{\\left|u\\right|}^{3})^{3}\\,\\mathbf {1} _{\\{|u|\\leq 1\\}}", "11e4c6e9d0b7973f5d0de4dc99316cf8": "{\\hat {h}}(\\xi )={\\overline {{\\hat {f}}(\\xi )}}\\,\\cdot \\,{\\hat {g}}(\\xi ).", "11e4e1075fc71fe7d00a880b73f32068": "\\sum _{n=1}^{\\infty }a(n)n^{-s}", "11e57ab7584e41d57386988166ea22a6": "\\left({\\begin{matrix}x_{3}&x_{1}-x_{2}\\\\x_{1}+x_{2}&-x_{3}\\end{matrix}}\\right)", "11e58c902d658b7346dc294e4d616e39": "\\mathbf {k} ^{e}", "11e59cf7b7aad86b1073be0f4ee186ff": "\\mathbf {P} _{k}^{n}", "11e5aad0c5544602a9cd3731c81e9228": "L_{f,P}=\\sum _{i=1}^{n}(x_{i}-x_{i-1})m_{i}.\\,\\!", "11e5bc6e852aecd259ad9cb97c302d45": "J_{0}=\\left({{\\partial L} \\over {\\partial {\\vec {\\omega }}}},{\\vec {\\omega }}\\right)+\\left({{\\partial L} \\over {\\partial {\\vec {v}}}},{\\vec {v}}\\right)-L,\\quad J_{1}=\\left({{\\partial L} \\over {\\partial {\\vec {\\omega }}}},{{\\partial L} \\over {\\partial {\\vec {v}}}}\\right),\\quad J_{2}=\\left({{\\partial L} \\over {\\partial {\\vec {v}}}},{{\\partial L} \\over {\\partial {\\vec {v}}}}\\right)", "11e5f3e73b68bb531cef3a6b56ca3923": "f_{*}{\\mathcal {O}}_{X}", "11e60ba50ad18c52fd95e243667a62cd": "a\\cdot b", "11e6917b95ef4a0199496878138098f8": "d_{\\odot }=1.58{\\times }10^{-5}\\,{\\text{lyr}}", "11e6af22dbabbf1de1356f75431b9ade": "\\delta =(-1)^{nk+n+1}s\\,{\\star \\mathrm {d} \\star }=(-1)^{k}\\,{\\star ^{-1}\\mathrm {d} \\star }", "11e6df5ff24c641172f4c03ea5556a25": "A={\\frac {V_{o}}{V_{i}}}={\\frac {\\left[{\\frac {M}{C}}+\\left(a{\\frac {M}{C}}\\right)\\left(\\beta ^{2}-1\\right)+{\\frac {\\beta ^{2}+2\\beta +3}{4\\left(\\beta ^{2}+\\beta +1\\right)}}\\right]}{\\left({\\frac {N}{C}}+{\\frac {3\\beta ^{2}+2\\beta +1}{4\\left(\\beta ^{2}+\\beta +1\\right)}}\\right)}}", "11e6f967d89c6728653edad985c83678": "\\theta \\sim p(\\theta |\\alpha )", "11e71825e743d04b43fa2879fb60c35e": "P(E(s),E(s'),T)", "11e7d493ebd781f7727261a9c9627c8d": "\\omega \\in A_{p}", "11e83df9231f795a5ce8b3400e41ef3e": "b\\triangleleft a", "11e8b25fd8e651a8f88303c7a37ca8b7": "4^{k}>600\\gamma ^{2}", "11e8bf6878e9108fbb653ee2f8f3d47e": "{\\frac {\\beta (1-p)}{-p\\ln p}}", "11e8e6dd37899edab9c3442b294a0e31": "x^{23}-1", "11e91e60f0be8349203530e87d1c4afe": "V(\\mathbf {r} )={\\frac {1}{2}}kr^{2}={\\frac {1}{2}}k\\left(x^{2}+y^{2}+z^{2}\\right)", "11e94c7d3fcfc2c32919b430fd1cf845": "0.167,\\ 0.177,\\ 0.181,\\ 0.181,\\ 0.182,\\ 0.183,\\ 0.184,\\ 0.186,\\ 0.187,\\ 0.189\\,", "11e9817dcf2067fd98b2c39842966e07": "x=\\psi (y)", "11e9855eba1e79c22f2029ba4001ea50": "A\\in U", "11e98a93b3bc26cdd4ec517c128c1ea6": "f(\\alpha \\mathbf {v} )=\\alpha f(\\mathbf {v} )", "11e99fdec9c9dd116bf8116a56db548e": "\\operatorname {Cov} [\\mathbf {z} ]=V", "11ea44d24b9497446942458ca8f8d00c": "f_{n}(X_{(n)})=n{\\frac {1}{L}}\\left({\\frac {X_{(n)}}{L}}\\right)^{n-1}=n{\\frac {X_{(n)}^{n-1}}{L^{n}}},0b\\end{cases}}", "11f79cd4cbaf600d68e91e548441bb1e": "\\tau _{D,i}", "11f7c1b5e66c2424cd7260a1cf45b307": "ABAB", "11f7c5713a99d91837c961a1e25dc701": "f(x)\\sim A{(x-\\mu )}^{-\\alpha -1}\\,.", "11f8240fd173bd97d3756ed4b1084e6a": "\\left(1+\\theta t\\right)^{-1/\\theta }", "11f84454ff4380cdb326350928d64e55": "{\\begin{aligned}&{\\begin{bmatrix}0&-z&y\\\\z&0&-x\\\\-y&x&0\\end{bmatrix}}\\mapsto {}\\\\&\\quad {\\frac {1}{1+x^{2}+y^{2}+z^{2}}}{\\begin{bmatrix}1+x^{2}-y^{2}-z^{2}&2xy-2z&2y+2xz\\\\2xy+2z&1-x^{2}+y^{2}-z^{2}&2yz-2x\\\\2xz-2y&2x+2yz&1-x^{2}-y^{2}+z^{2}\\end{bmatrix}}.\\end{aligned}}", "11f873575f831054a4d9f4a5f8cde97a": "{\\hat {H}}(t)={\\hat {H}}_{0}+{\\hat {V}}(t)\\theta (t-t_{0})", "11f8b1b30d0ebbcdd70bfee12d4fd283": "h_{m}(X_{0})=\\left\\|X_{0}-X_{[m]}\\right\\|", "11f9076610e11a6f6e8ff746702402af": "\\Phi _{{\\mathrm {eE} },h}:z_{k}\\mapsto z_{k+1}", "11f92604bf3b962e180703b811d7b000": "\\mathrm {M} \\,\\mu \\,", "11f92a018e196a4387c248253f4d0d19": "m_{k}=\\int t^{k}\\psi (t)\\,dt.", "11f9672216276ff3d800c0693c54ce55": "\\sigma :P(U)\\times P(V)\\to P(U\\otimes V).\\ ", "11f96ce0f1569effd19ad91910fd2107": "{\\frac {(\\alpha -\\beta )\\,\\Gamma (2n+\\alpha +\\beta )}{(n-1)!\\,2^{n}\\,\\Gamma (n+1+\\alpha +\\beta )}}\\,", "11f96d4c7b035493552ad8fd95057d61": "S^{\\prime }(a,q,0)=1", "11f9c97a767b44a71863a0839580ba63": "f(k;r,p)\\equiv \\Pr(X=k)={k+r-1 \\choose k}(1-p)^{k}p^{r}\\quad {\\text{for }}k=0,1,2,\\dots ", "11fa56fcce59e3bcb7031dfaa53c6ef1": "(d+[(m+1)2.6]+y+[y/4]+[c/4]-2c)\\ {\\bmod {\\ }}7-(d+[2.6m-0.2]+y+[y/4]+[c/4]-2c)\\ {\\bmod {\\ }}7", "11fa88866d4e21e2ed0c9428e47e27c4": "{\\frac {d\\lambda }{dt}}=0", "11fa8e5d143041a8e707991fee6ae3f4": "n_{\\mathrm {tot} }", "11fac24dd2b06c4ec14eb363df98b7e0": "\\epsilon _{y}={\\frac {\\partial u_{y}}{\\partial y}}\\,\\!", "11fae8d7d9a95a34400eb6e6af382a1c": "\\cos \\theta ", "11fb3a578987f34f532c6fdcc5ad663e": "{\\mathcal {H}}\\left(q_{j},p_{j},t\\right)=\\sum _{i}{\\dot {q}}_{i}p_{i}-{\\mathcal {L}}(q_{j},{\\dot {q}}_{j},t).", "11fb9ec0afca1866ff70aa8825bc62f7": "+\\sin \\left[2\\pi \\left({\\frac {x}{\\lambda +\\Delta \\lambda }}-(f-\\Delta f)t\\right)\\right]", "11fbf25b430a5ad659dd0ec045b88026": "1-t\\leqslant e^{-t}", "11fc3a0c8fca92ae6bc10ff8241a116e": "Hg", "11fc3d25620edbaebea78565eaee7e97": "y.", "11fcc08dd2049b49044a68ad6227eb70": "m_{1}{\\frac {d^{2}{\\mathbf {r} }_{1}}{dt^{2}}}=-{\\frac {m_{1}m_{2}g({\\mathbf {r} }_{1}-{\\mathbf {r} }_{2})}{|{\\mathbf {r} }_{1}-{\\mathbf {r} }_{2}|^{3}}};\\;m_{2}{\\frac {d^{2}{\\mathbf {r} }_{2}}{dt^{2}}}=-{\\frac {m_{1}m_{2}g({\\mathbf {r} }_{2}-{\\mathbf {r} }_{1})}{|{\\mathbf {r} }_{2}-{\\mathbf {r} }_{1}|^{3}}},", "11fcc0d29b5ca02c22d2743e57f24fdc": "{\\widehat {\\mu }}=1/(2\\pi )^{n}\\mu ", "11fcdff0e4febef6282057017d633eae": "{\\begin{array}{c|ccc}0&1/6&-1/6&0\\\\1/2&1/6&1/3&0\\\\1&1/6&5/6&0\\\\\\hline &1/6&2/3&1/6\\\\\\end{array}}", "11fce5b4c07fc997a961f3fdb3c3b13b": "\\lambda \\mapsto \\int _{X}^{\\oplus }\\ \\lambda _{x}d\\mu (x)", "11fd48dc9847b6765280e4b59eff33d4": "|\\ell -s|\\leq j\\leq \\ell +s", "11fd6d28c2ef34b6bda25ecb312c0464": "{\\frac {5\\pi }{12}}\\ (75^{\\circ })", "11fd73531d2e69e1003cbf0abc35cfe9": "\\forall n0.", "1205d7a46ec30a343a54d574c0a8658c": "r(i,k)\\leftarrow s(i,k)-\\max _{k'\\neq k}\\left\\{a(i,k')+s(i,k')\\right\\}", "120618f829de6dbd77a0a8d1a54bfa36": "f=g+b", "12063f42a48485f4dbbedfd82d00a1f2": "\\langle a^{N}\\rangle \\langle b^{N}\\rangle \\langle a^{N}\\rangle ", "12066669bb72df1468e41090f81e90b3": "{\\frac {\\partial \\,{\\textbf {J}}}{\\partial \\,a}}={\\frac {\\partial }{\\partial \\,a}}\\left({\\frac {\\pi }{2{\\sqrt {ab}}}}\\right)=-{\\frac {\\pi }{4{\\sqrt {a^{3}b}}}}.\\,", "1206766680720eee51aea77c5ab9f6b3": "[t_{a},t_{b}]=if^{abc}t_{c}", "12069ce86e3daea709688687431ab3f2": "x[\\infty ]=\\lim _{z\\to 1}(z-1)X(z).", "120716a31746fb7c6f5a6efec31f2cae": "\\sin \\theta \\approx {\\frac {3.83}{ka}}={\\frac {3.83\\lambda }{2\\pi a}}=1.22{\\frac {\\lambda }{2a}}=1.22{\\frac {\\lambda }{d}}", "12077168e44b7f66b87b686004c403bd": "\\phi _{x}\\in F", "1207df2ea399c1bb2a8a49493445c6cd": "s\\in \\{-1,1\\}", "1207f006c17fce4c5629a09a1800ee1e": "c(p,y,t)=\\sum _{i}b_{ii}\\left(y^{b_{yi}}e^{b_{ti}t}p_{i}+\\sum _{j\\,:\\,j\\neq i}b_{ij}{\\sqrt {p_{i}p_{j}}}y^{b_{y}}e^{b_{t}t}\\right).", "120838cff47154743574e6de33bcf2a5": "\\scriptstyle {X_{L}}", "12087bb53a2f5ea9352922f6c496c879": "P(R_{NP},\\theta _{1})", "1208c67e86ac809b392cd4977bd4a13a": "x_{1}(z)=x_{-1}(z)+a_{0}\\cdot x_{0}(z)", "12093ae7210d96911e186b48a93c55ca": "\\scriptstyle \\sigma _{2}^{2}", "120972ceaf7ac335b5b73c397e86f2ac": "B(x)=\\int {1 \\over e^{-4x}}e^{-2x}\\cosh {x}\\,dx=\\int e^{2x}\\cosh {x}\\,dx={1 \\over 6}e^{x}(3+e^{2x})+C_{2}", "12098aef5d61b130a68dc519f947bf1f": "C_{D,0}", "1209e36df50bd83b87b011f615da329f": "^{\\;}H(\\xi )", "1209ec2da726bed1b2e675521f2c9619": "I(\\mathbf {q} )=\\int _{V}\\gamma (\\mathbf {r} )e^{-i\\mathbf {q} \\cdot \\mathbf {r} }{\\text{d}}\\mathbf {r} ", "1209fc086f15b78a6c45757aa758636e": "\\operatorname {sgn}(\\omega _{n})=\\pm 1", "120a1439c5535d9fa0e985e99b9389d7": "\\mu _{0}=4\\pi \\times 10^{-7}({\\rm {{N/A^{2}})\\approx 1.2566370614\\cdots \\times 10^{-6}({\\rm {{N/A^{2}})}}}}", "120a146a8dfbb5f8af851f3f659c7c59": "V_{n}=2F_{2n-1}-F_{n-1}\\,", "120a29453afe71b97be1434afdc65324": "Q=n\\left(n+2\\right)\\sum _{k=1}^{h}{\\frac {{\\hat {\\rho }}_{k}^{2}}{n-k}}", "120a4b36f3d8bf327b4c84aa8a52678d": "\\epsilon D", "120a527ec2fb6583ec5c438f64ad5c18": "{\\mathcal {O}}(1)", "120a9af57ee747e52cba32da79e7c12c": "\\omega _{1}(t)", "120abe116bf3ca0db9641be23603f06d": "g(\\cdot \\mid x)", "120ac279f23f3c3ae5a6ef9bba6c8a7d": "b_{\\{j,k\\}}={\\begin{vmatrix}a_{1j}&a_{1k}\\\\a_{2j}&a_{2k}\\end{vmatrix}}", "120aef2dd3e826ad00d6f8d361f71d33": "2x^{3}-2x^{2}-3x+2=0", "120af3c5227de4cd6e9a3dd1e45aa9f9": "L_{i_{1}\\alpha _{1}}^{i_{2}\\alpha _{2}}{\\hat {L}}_{i_{2}\\beta _{1}}^{i_{3}\\beta _{2}}S_{\\alpha _{2}\\beta _{2}}^{\\alpha _{3}\\beta _{3}}=S_{\\alpha _{1}\\beta _{1}}^{\\alpha _{2}\\beta _{2}}{\\hat {L}}_{i_{1}\\beta _{2}}^{i_{2}\\beta _{3}}L_{i_{2}\\alpha _{2}}^{i_{3}\\alpha _{3}},\\quad 00.", "1212c7db203d48037731de433b2a6b4f": "G=(V\\,,\\Sigma \\,,R\\,,S\\,)", "1212c9f997adce8def4ba2f8f70c41a9": "\\lim _{n\\rightarrow +\\infty }g_{n}", "121313223ec6d864aab9b8f8e10f2716": "Q_{S}\\left(n\\right)\\,\\!", "1213266e347e6ebc1748790b5d341018": "\\beta _{cr}\\approx (s^{2}+2s)^{\\frac {1}{2}}", "12134aa42c1c5f312d1b532963bc5614": "A\\rightarrow (B\\rightarrow A)", "121352ec22d6a433071dbe726f1e8f42": "\\sum _{i=1}^{N}\\sum _{j=1}^{N}", "12135fa5efea91932025817bf1cdd83d": "\\ln 2={\\frac {1}{1}}-{\\frac {1}{2}}+{\\frac {1}{3}}-{\\frac {1}{4}}+{\\frac {1}{5}}-\\cdots .", "121413ddbd0b4bb71eeefdf13d444b80": "r_{c}=-{\\frac {2\\gamma }{G_{v}}}", "12143c1b992ca08d35b2cd3f97cb394a": "\\alpha \\approx {\\frac {d}{f}}", "12146a84ceb263a0c53f1085faafd27d": "{1-e^{-1}}=0.61", "12147ec9106aee686c4b00da28a45e3c": "\\neg \\Diamond ", "1214ee19f0c86c16a1ceb0746bef34ee": "u=\\sum _{n=0}^{\\infty }f_{i}^{(n)}u_{n}=\\sum _{n=0}^{\\infty }e_{i}^{(n)}v_{n},", "1214efd654949f8bc2a73f106b8d0d8d": "r_{B}(n)>0", "12150c34c7ebacf3ac6d651c9d033d66": "{\\vec {f^{j}}}\\in C^{1}(\\mathbb {R} ^{s},\\mathbb {R} ^{s}),j=1,\\ldots ,d", "12159869635db63b02c0a0ae6176daeb": "\\nabla \\cdot \\mathbf {D} =\\rho _{\\mathrm {free} }", "12159d69e4fc16397dd1cd7ee3bbdc37": "SubCipher_{n+1}=DEC_{b_{n+1}}(k_{b_{n+1}},C)", "1215ae164e4d4120e2e91a83843d6f05": "c_{\\text{fil}}", "12166545159517419d339863a7710ac4": "\\Gamma \\cup \\{\\neg \\varphi \\}", "12170f55dc268f5a2bf48938dc194a16": "\\mathrm {Ad} :G\\to \\mathrm {Aut} ({\\mathfrak {g}})\\subset \\mathrm {GL} ({\\mathfrak {g}})", "1217388d726a03b92ca781521ce0e03a": "\\textstyle c", "1217546f73d17b94510859ff8d59b705": "\\,{\\frac {d\\mathbf {w} }{dt}}~=~\\eta \\,y(t)(\\mathbf {x} (t)-y(t)\\mathbf {w} (t)).", "1217751311083458516371ed22dea7a9": "{\\begin{pmatrix}0&1&1&1&0\\\\1&0&1&0&1\\\\1&1&0&1&1\\\\1&0&1&0&1\\\\0&1&1&1&0\\end{pmatrix}}", "1218482fc44abff9acb5730296209d10": "\\ D_{heel}", "1218a2f546e67a8ae154b8cc595c35b8": "(y_{1},\\dots ,y_{k})\\in [m]^{k}", "1219144e892baf828fb87d05ed2b97d5": "\\gamma ={\\frac {5}{3}}", "12197d5cd38fa1835f1a8f32828923a5": "k=\\int _{a}^{b}g(x)\\,dx.", "1219eb7d81ba826c407356af7b1cc141": "\\phi \\,\\!", "121a319e4238c37b5ab2d03408442f72": "H(X_{i}|...)", "121a466bf97e7e552a66009aa8ddb72f": "\\gamma ={\\frac {1}{3}}", "121a6bda80191ac5d4cdd998566037db": "D_{i}=X_{i}\\oplus Y_{i}\\oplus B_{i}", "121ae3a9ca06e5926057c6b5cce1a6bb": "g(z)=-1/90\\sum _{i=0}^{8}z^{i}", "121b66144d4d915c134836d416e819a4": "{{r}_{O3}}", "121bb53505fa4eb6007832227983b4f8": "|\\psi (t)\\rangle \\equiv {\\begin{pmatrix}\\vdots \\\\\\psi _{j-1}(t)\\\\\\psi _{j}(t)\\\\\\psi _{j+1}(t)\\\\\\vdots \\end{pmatrix}}", "121c602f6f9c6483569216f5e01b8af6": "k\\in \\{1,2,\\dots \\}\\,", "121ca52489106f28aeeb3c6fc0b2da63": "h_{A}(x)=\\sup\\{x\\cdot a:a\\in A\\},", "121cdf43d753c1fe95ac68a7ce679903": "{\\begin{aligned}0&={\\frac {dI'}{d\\epsilon }}[0]=L[\\mathbf {q} [t_{2}],{\\dot {\\mathbf {q} }}[t_{2}],t_{2}]T-L[\\mathbf {q} [t_{1}],{\\dot {\\mathbf {q} }}[t_{1}],t_{1}]T\\\\[6pt]&{}+\\int _{t_{1}}^{t_{2}}{\\frac {\\partial L}{\\partial \\mathbf {q} }}\\left(-{\\frac {\\partial \\phi }{\\partial \\mathbf {q} }}{\\dot {\\mathbf {q} }}T+{\\frac {\\partial \\phi }{\\partial \\epsilon }}\\right)+{\\frac {\\partial L}{\\partial {\\dot {\\mathbf {q} }}}}\\left(-{\\frac {\\partial ^{2}\\phi }{(\\partial \\mathbf {q} )^{2}}}{\\dot {\\mathbf {q} }}^{2}T+{\\frac {\\partial ^{2}\\phi }{\\partial \\epsilon \\partial \\mathbf {q} }}{\\dot {\\mathbf {q} }}-{\\frac {\\partial \\phi }{\\partial \\mathbf {q} }}{\\ddot {\\mathbf {q} }}T\\right)\\,dt.\\end{aligned}}", "121cfdcf4419248f434783a6336936a9": "(1/T)\\,,", "121d499a70c5dc88cf8e65e432afd025": "T=-D^{-1}R", "121d53811f80aaa7b3c66910fe38b54e": "S\\rightarrow A:\\{K_{PB},B\\}_{K_{SS}}", "121d771b8a0e39fbff64b00c95b7a748": "\\rho ^{2}", "121d9aa5f5eddf45d63336459e783c3a": "c_{pd}", "121db019e3a57df3fe993e8dd1fa2fe6": "H_{n}^{(0)}={\\frac {1}{n}}.", "121dc4263a56311cf360403b36a43e0f": "a\\oplus b", "121e2ecb84f8ba698a3a0595d9aeb902": "\\scriptstyle {\\boldsymbol {\\nabla }}V", "121e4918c3a10d79c7cf9d4fe447bb9e": "\\tau _{i,j}=A_{ij}+{\\frac {B_{ij}}{T}}+{\\frac {C_{ij}}{T^{2}}}+D_{ij}\\ln {\\left({T}\\right)}+E_{ij}T^{F_{ij}}", "121e49ddecefa1ac21aa339b189d1d97": "{\\hat {M}}\\approx {\\frac {1}{2}}\\left({\\frac {1}{F_{ST}}}-1\\right)", "121ec24d82aa944c1ef87fa2dd4a4e70": "-100\\pm 10", "121ed2a3ccf49bfb186b15094afafad8": "{\\frac {d}{\\log {d}}}", "121ed985398c676ff487978e8aff933f": "{\\cfrac {\\partial W}{\\partial I_{1}}}=-{\\cfrac {C_{0}}{J_{m}}}", "121eef2a0abaeca6998fed61f1e67dc0": "B_{k+1}=B_{k}+U_{k}+V_{k}\\,\\!", "121effd24b0086552605654b52abfc1b": "p_{1}=x_{1}\\,.", "121f1f1345b88d61e8de929c2fba2aae": "h=-a_{0}", "121f459a3dbf3a52d4e92f9003241485": "\\;{}_{2}F_{1}(a+1,b;c;z)-\\,{}_{2}F_{1}(a,b+1;c;z)={\\frac {(b-a)z}{c}}\\;{}_{2}F_{1}(a+1,b+1;c+1;z)", "121f4a4c60a00cc2780613db122af89e": "G(u)=-{\\frac {2}{mh^{2}}}\\int ^{\\frac {1}{u}}F(r)\\,dr", "121fb37b643d996c919f8e15c44bf226": "L^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}=\\ell (\\ell +1)e.", "121ff5f1d1c30418d5d0376357ae0ae7": "{\\frac {ax^{2}+2bx}{2}}\\,", "121ff81c8e173410c911f6d7461f6413": "{\\boldsymbol {\\omega }}'={\\boldsymbol {\\omega }}+{\\boldsymbol {\\Omega }}\\,\\!", "1220328f561803f9df3a1ea222bf1429": "\\{f,g\\}_{DB}=\\{f,g\\}_{PB}-\\sum _{a,b}\\{f,{\\tilde {\\phi }}_{a}\\}_{PB}M_{ab}^{-1}\\{{\\tilde {\\phi }}_{b},g\\}_{PB}~,", "12204ac9e030bcb065b0d0ca24844451": "10.36\\%", "12204b6f3a627910dc94d797e198cbb4": "\\langle f^{*},w\\rangle =f(w).", "12205bce3fafc529b4e8a2c5b8c4e1a4": "r\\cdot p=p\\cdot r=p", "12211635c7c7c91a4dfed3ecd37652b4": "\\beta (\\phi ,\\psi )=(-1)^{{\\frac {1}{2}}m(m+1)}\\beta (\\psi ,\\phi ),", "1221af30eae8b0aae45f557d7998052e": "{\\textbf {F}}=-m{d^{2}{\\textbf {C}} \\over dt^{2}}.", "1221c109e5e9e5dd2885d90c2771ccb9": "[A]_{p}\\subseteq \\operatorname {cl} (A)", "1221d3d1fd9e0cba179df465b66ef57d": "\\lambda _{j}=c_{0}+c_{n-1}\\omega _{j}+c_{n-2}\\omega _{j}^{2}+\\ldots +c_{1}\\omega _{j}^{n-1},\\qquad j=0\\ldots n-1.", "1221f85a7b80396c80ab7b3944cd2322": "m_{m}\\;", "12221d5e47854ad9debadde1920d1287": "\\int f(r)dxdy=2\\pi \\int f(x)xdx", "12221fe07fa7fe3c15273412710e0444": "\\partial _{t}u=\\Delta _{x}u^{m},\\,\\,m>1,\\,", "1222535e4e93103b092c1eb492d5b777": "e(v)={\\frac {\\int \\limits _{0}^{v}{}yf(y|v)dy}{F(v|v)}}", "122278ab8aa68e7ad3e9eb9bcc0dcf6e": "a(x,t),b(x,t)", "1222b6bc07bdd5ec8fd316dabb48c05c": "{\\mathfrak {so}}_{2n+1}", "1222c45a3bba0883fca335a2bfa4928e": "\\displaystyle D_{q}(f(x)+g(x))=D_{q}f(x)+D_{q}g(x)~.", "1222e75665bd66c2233b8d3ef5669676": "L=T-V", "1223e4fb70d877bba74cf6c4fc19583b": "\\sigma _{ii}>0", "12241245fb0cea72d91bbe8de86b4fca": "R^{r}f_{*}\\to g_{*}\\circ R^{r}f'_{*}\\circ g'^{*}.", "12243e80df90f120933cb7b35c734faa": "f(\\mathbf {y} )", "122477605844386cb335556364107b47": "{\\begin{bmatrix}{\\sqrt {\\dfrac {\\eta _{2}}{\\eta _{1}}}}\\\\[15pt]-2\\eta _{2}\\end{bmatrix}}", "1224b964b62baf5d6570be748c192a0d": "T_{0}=T-{\\frac {1}{n-1}}\\iota (\\operatorname {tr} \\,T)", "1225673626a600669a2579d1ac4d9715": "\\int \\arctan(x)\\,dx=x\\arctan(x)-{\\frac {\\ln \\left(x^{2}+1\\right)}{2}}+C", "12258850e6551184c4a13195460dc147": "\\nabla X", "1225ad822e57b5a9627250100389e6b1": "({\\widetilde {\\rho }},{\\widetilde {V}})", "1225d6d6b564c2317d34f22746a61221": "{\\mbox{d}}T={\\mbox{d}}L\\cos \\varphi -{\\mbox{d}}D\\sin \\varphi ={\\mbox{d}}L(\\cos \\varphi -{\\frac {{\\mbox{d}}D}{{\\mbox{d}}L}}\\sin \\varphi )", "1225e32c37e9a08903e07b59b0688569": "|00\\rangle ,|01\\rangle ,|10\\rangle ,|11\\rangle ", "1225e48212927f828b221d81d9d465e8": "\\mathrm {C} ^{\\alpha }", "1225ecade34057f10b5c5b7093e5c798": "{\\bar {F}}(x)=o(1/x).", "12264b412801e03e9aa31ff07c76bc61": "L_{D}{\\big [}\\rho _{S}(t){\\big ]}=0", "1226af56e9c19ec8e78b7166632fb140": "{\\vec {\\Gamma }}={\\vec {\\mu }}\\times {\\vec {B}}=\\gamma {\\vec {J}}\\times {\\vec {B}}", "12277132c4446ec03bdb8b55891b24d4": "100k", "12279c110c35cf254232fcb75f9c4551": "r_{5}=1267.9", "1227a24589c82a235a53865dd8f95b21": "\\mu _{\\mathrm {N} }={{e\\hbar } \\over {2m_{\\mathrm {p} }c}}", "12280c745065c7236fd62726c50c459f": "\\gamma (x')", "1228f9085b9820aea6ac9da1079d5e4d": "E\\left[\\Lambda (n+1)\\right]=E\\left[\\left(\\mathbf {\\delta } (n)-{\\frac {\\mu \\,\\left(v(n)+r(n)\\right)\\mathbf {x} (n)}{\\mathbf {x} ^{H}(n)\\mathbf {x} (n)}}\\right)^{H}\\left(\\mathbf {\\delta } (n)-{\\frac {\\mu \\,\\left(v(n)+r(n)\\right)\\mathbf {x} (n)}{\\mathbf {x} ^{H}(n)\\mathbf {x} (n)}}\\right)\\right]", "122926e78aad5d17b565114e475b7a28": "L=-{a_{0}^{2} \\over 8\\pi G}f\\left\\lbrack {|\\nabla \\phi |^{2} \\over a_{0}^{2}}\\right\\rbrack -\\rho \\phi \\;", "122963c1d994589b246ccb769ffa284d": "t={\\frac {W(R\\ln p)}{\\ln p}}", "12299fc0f87855188ca4a6c8c4b52689": "h{\\boldsymbol {.}}v=\\epsilon (h)v", "1229d22508f1f40ae348675979810e14": "\\rho _{x}^{A}", "122a602c7504f1711beecbd5be1a1df6": "sim(q_{and},d_{j})=1-{\\sqrt[{p}]{\\frac {(1-w_{1})^{p}+(1-w_{2})^{p}+....+(1-w_{t})^{p}}{t}}}", "122a647e7714db9742c6e5e3351205dc": "-\\nabla ^{2}{\\vec {\\psi }}=\\nabla \\times (\\nabla \\times {\\vec {\\psi }})=\\nabla \\times \\mathbf {v} ", "122a89af348d8fd8f1e20572419fcd87": "\\Sigma _{XX}^{-1}\\Sigma _{XY}\\Sigma _{YY}^{-1}\\Sigma _{YX}", "122a90ef5f11f5ff0f7e0fb42685ba4a": "\\gamma _{n}^{2}=c/(2n)", "122a9d74ef2cd7d8c2f4af8b883440ec": "2nm", "122b02fddeff90c6717eb96df33035ef": "E_{\\mathrm {p,e} }=\\phi q\\,\\!", "122b2c2f74465193a588e41ab16a292e": "d_{r}={\\frac {|x-y|}{\\left({\\frac {|x|+|y|}{2}}\\right)}}\\,.", "122b7298fcd662e04bcd4eb2bc63d32a": "\\mathbf {R} ^{1,3}\\rtimes \\operatorname {O} (1,3)", "122b911c4c74cc684e2610c2b5394a15": "{\\frac {M_{a}}{r}}={\\frac {1}{r}}{\\frac {dP(z)}{dz}}+P(z).", "122bbad4e2259cbc791139a1aba963e4": "(\\tan x)'=\\sec ^{2}x={1 \\over \\cos ^{2}x}=1+\\tan ^{2}x\\,", "122c42ae392a1b8225c6b27954b8c778": "\\nu =\\mu /{\\rho })", "122c5bf75de3bea75e6ac3ae9d9c9ed1": "D_{AB}", "122c8661d375cb55a740549ce54a545f": "(x_{n},v_{n})", "122c9b13feb136f3cb409708edd8d2cc": "(\\mathbf {a} \\cdot \\nabla )=\\left(\\sum _{i}a^{i}e_{i}\\right)\\cdot \\left(\\sum _{i}{\\frac {\\partial }{\\partial q^{i}}}\\mathbf {e} ^{i}\\right)=\\left(\\sum _{i}a^{i}{\\frac {\\partial }{\\partial q^{i}}}\\right)", "122cdedc11e700eb0e2b26bd2302ec44": "\\Re s=1", "122d1605ff98ab654b99fb3ced2b7e29": "(a^{2}+ab+b^{2})^{2}+(c^{2}+cd+d^{2})^{2}=((a+b)^{2}+(a+b)(c+d)+(c+d)^{2})^{2}", "122d7503d3910ab589be38df4994de48": "\\mathbf {q} ^{m}", "122d8a8fb9fb25f50f74a03c5c38eb74": "\\exists x\\,\\phi ", "122db48b9cefab92ca64236b128959ed": "x\\delta '(x)=-\\delta (x).", "122e1f2a7549858b94d561f3a64e9c68": "\\lVert v_{i}\\rVert ^{2}=1", "122e339be0077372b946d2a9e17c87d0": "{\\hat {T}}=T/T_{m}", "122e58646e28a92c19594205fa841430": "r=yieldofoption", "122e92668cca7de2ab7a4336753e2124": "h\\chi ^{2}|\\Phi |^{2}", "122eacaf98b0557f59e75a5c20147b18": "\\mathbf {y} ={\\begin{pmatrix}y_{1}\\\\y_{2}\\\\1\\end{pmatrix}}", "122ecd098eef3aec29493300556f4e23": "\\ K({\\sigma }/{E})^{n}", "122ecfc0c5e1d463b0c835ebf6f02d2c": "\\textstyle S_{1}(r)", "122f35c1eceb0aca505b33e515a1fa0d": "\\mathrm {MA} ={\\frac {f_{O}}{f_{E}}}", "122f3b03fe3a9d6d715b3d87b81f98d1": "{\\hat {t}}(s)\\equiv {\\frac {\\partial {\\vec {r}}(s)}{\\partial s}}", "122fb62a42c73bd5c9505c40ea8fcc19": "{\\frac {347,373,600}{635,013,559,600}}", "122fc242fc2d6bf7d88058e310d2ea39": "1+max(1+MD(p),0)=", "122fcb27b282247b7ffa507dcdd25254": "{\\begin{aligned}\\sin(x+iy)&=\\sin x\\cos iy+\\cos x\\sin iy\\\\&=\\sin x\\cosh y+i\\cos x\\sinh y.\\end{aligned}}", "122fcc4df4015fe4340ce1e50bc3fb53": "{\\mathcal {}}(H_{*}(LM),H_{*}(LM))", "123008b3f7350bfe54c9616559cc267f": "\\mathbf {\\pi } ", "12301fc6d254544e5cb648e15b667997": "\\mathbb {R} ^{N}", "1230293f3b27768bb582604a7a04cd37": "\\scriptstyle \\leq 8\\times 10^{-16}", "12307c23fc78dd0bcb6ec9a9e6d934f0": "(M,g)\\,", "12307fed4e43bdbb9486112b07ec51a2": "\\left\\{{\\begin{matrix}n\\\\k\\end{matrix}}\\right\\}=S(n,k)", "1230dee7c33c22fa088449d30a88f4c5": "{\\frac {1}{2}}0", "1234c59108794113afe9373cdc572041": "\\left(x_{0},y_{0}\\right)=\\left(2,1942\\right);\\left(x_{1},y_{1}\\right)=\\left(4,3402\\right);\\left(x_{2},y_{2}\\right)=\\left(5,4414\\right)\\,\\!", "1234e4cf4710843d42fbb6c9044dab47": "{\\vec {v}}", "1235134fa75f7dac02d1d3d6c16021bb": "\\displaystyle \\mu ={\\sqrt {\\frac {(ab+cd)(ad+bc)}{(a+c)^{2}(ac+bd)}}}={\\sqrt {\\frac {(ab+cd)(ad+bc)}{(b+d)^{2}(ac+bd)}}}", "12356653e3a5d7857d21840f8562edc7": "{\\begin{aligned}L_{c}&={\\Big (}{\\frac {Y_{w}L_{wr}}{Y_{wr}L_{w}}}D+1-D{\\Big )}L\\\\M_{c}&={\\Big (}{\\frac {Y_{w}M_{wr}}{Y_{wr}M_{w}}}D+1-D{\\Big )}M\\\\S_{c}&={\\Big (}{\\frac {Y_{w}S_{wr}}{Y_{wr}S_{w}}}D+1-D{\\Big )}S\\\\\\end{aligned}}", "12357aa804445a44a7a8f7ac3102e32e": "\\mathrm {d} s^{2}=4{\\frac {\\mathrm {d} x^{2}+\\mathrm {d} y^{2}}{(1-(x^{2}+y^{2}))^{2}}}.", "12358cec9dc75a161dbfd5d16ee21c55": "J{\\frac {\\partial }{\\partial z^{\\mu }}}=i{\\frac {\\partial }{\\partial z^{\\mu }}}\\qquad J{\\frac {\\partial }{\\partial {\\bar {z}}^{\\mu }}}=-i{\\frac {\\partial }{\\partial {\\bar {z}}^{\\mu }}}.", "12359b5c5fce40bd85efc8bb603e134b": "\\operatorname {dim} R", "1235d6d9ed53384c24a9be1b59de595e": "(2j+1)\\sum _{m_{1}m_{2}}{\\begin{pmatrix}j_{1}&j_{2}&j\\\\m_{1}&m_{2}&m\\end{pmatrix}}{\\begin{pmatrix}j_{1}&j_{2}&j'\\\\m_{1}&m_{2}&m'\\end{pmatrix}}=\\delta _{jj'}\\delta _{mm'}.", "12360bf6c58ef5cb3fa6137f43ddda52": "x[n]\\cdot y[n]\\!", "12360d64269d04026c458407b7eb77c8": "h,\\,h'", "12360f5d0cd4a103ed69b52d46e4a8b9": "\\|f\\|_{p}={\\biggl \\|}\\prod _{k=1}^{n}f_{k}{\\biggr \\|}_{p}\\leq \\prod _{k=1}^{n}\\|f_{k}\\|_{p_{k}/\\theta _{k}}=\\prod _{k=1}^{n}\\|f\\|_{p_{k}}^{\\theta _{k}}.", "12369323adef2b6ac037ac7509473605": "ds^{2}=\\sum _{k=1}^{d}\\left(h_{k}\\right)^{2}\\left(dq_{k}\\right)^{2}\\ ,", "12369bb81f635b884615d3a06adcd9c1": "M'=M\\Lambda .\\,\\!", "1236b0ee80dab7abc6c4ed5f3fa884d0": "E(t)", "1236ef6d4c88045281fb35fb7233f30b": "{\\frac {{\\tbinom {4}{1}}^{13}}{\\tbinom {52}{13}}}=0.010568\\%=1:9462", "1236f3661b5883b7d2617fa911e48394": "(0,\\pm \\varphi ,\\pm 1)", "123714ae9bbc895bbb005db130c1ea18": "{\\text{Per-unit ohms reactance}}={\\frac {{\\text{ohms reactance * }}{\\text{kva base}}}{kv_{L-L}^{2}*1000}}", "123720fa113ee8a26f845d702bce53b8": "i,j\\neq k", "12374131d5ed6ffcae1aafeda4a5d97e": "c_{L}(s',x)\\theta _{0}.", "12393c86c08d0cbd3233ac67404b5f4c": "g=C*a", "12395d6f992052fda8d9e1a84b2af10a": "Pr=Sc=1", "1239f086e7e91edefaecc2f7e5815f9c": "q(2\\pi r)(2t){\\Bigr |}_{r}-q(2\\pi r)(2t){\\Bigr |}_{r+\\Delta r}-h_{c}(2)2\\pi r\\,\\Delta r\\left(T-T_{e}\\right)=0,", "123a12e616746a8b3425c03b01a48908": "\\wedge ^{3}(\\mathbb {R} ^{6})=20", "123a2a6b5a2242545d496ac741765a0a": "{\\text{Var}}({\\hat {\\theta }})={\\frac {{\\text{Var}}(Y_{1})}{2}}={\\frac {{\\text{Var}}(Y_{2})}{2}}.", "123abc919f58bb86743c32750f05906c": "{\\frac {d}{dt}}\\int _{S}S\\,dS=-\\int _{S}CB_{\\alpha }^{\\alpha }\\,dS", "123aeec2a5af89d9f40e5ecb2228234a": "f\\ ", "123b3a22e1631d6b5ad940988aa3aac9": "\\sum _{a}p_{a}c(a,x)\\leq C", "123b5221e62d1bab9ebad987eec7f585": "{\\frac {d^{2}y}{dw^{2}}}+(a-2q\\cos 2w)y=0.", "123b84ce6042edefb5b79c73453eaa49": "\\varepsilon ={\\begin{pmatrix}\\varepsilon _{xx}'&\\varepsilon _{xy}'+ig_{z}&\\varepsilon _{xz}'-ig_{y}\\\\\\varepsilon _{xy}'-ig_{z}&\\varepsilon _{yy}'&\\varepsilon _{yz}'+ig_{x}\\\\\\varepsilon _{xz}'+ig_{y}&\\varepsilon _{yz}'-ig_{x}&\\varepsilon _{zz}'\\\\\\end{pmatrix}}", "123bafcba93692cc1b377714de1d1fe9": "ACH_{natural}\\,\\!", "123bc18dd032ef9cbbd9c4decf8141fc": "j(-k)=-i\\,", "123bc36c8925dd815c8776ad813f3bc7": "\\ A'_{\\mu }=GA_{\\mu }G^{-1}+{\\frac {i}{g}}(\\partial _{\\mu }G)G^{-1}", "123bdb08585e858c15dcdbe416a84929": "200/\\pi ", "123bdf044cc21fe34e8f3fd079132580": "y={\\begin{cases}a&\\{ft\\}<0.5\\\\-a&\\{ft\\}>0.5\\end{cases}}", "123bf20abd51db581b1c254bff6488cf": "\\ {(D/L)_{\\alpha }}e.", "1241fb664e80c6d150a1ea2f028e9564": "{\\begin{aligned}\\theta _{k}[n]&={\\frac {2\\pi }{f_{\\mathrm {s} }}}\\sum _{i=1}^{n}f_{k}[i]+\\phi _{k}\\\\&=\\theta _{k}[n-1]+{\\frac {2\\pi }{f_{\\mathrm {s} }}}f_{k}[n]\\\\\\end{aligned}}", "124285765ef84e388663269b08fc769e": "\\mathrm {2ROO{^{\\cdot }}\\ \\xrightarrow {} \\ 2RO{^{\\cdot }}+O_{2}\\ \\longrightarrow {}\\ ROH+QO+O_{2}} ", "1242bcaec18bf6267b0f73fe7ba477cc": "x^{2}-y^{2}\\equiv 0{\\pmod {n}}{\\hbox{ , }}(x+y)(x-y)\\equiv 0{\\pmod {n}}", "1242c179234b05ffffd98a49a5ed9539": "x_{1}+x_{2}", "1242de1682fb4474d1201906297e595f": "\\int x\\sin ^{2}{ax}\\;\\mathrm {d} x={\\frac {x^{2}}{4}}-{\\frac {x}{4a}}\\sin 2ax-{\\frac {1}{8a^{2}}}\\cos 2ax+C\\!", "1242e581a0c2efd7027c5d27be1b8661": "A_{5}/\\{e\\}", "1242ecd57e70f6b94b7613dfb7bf013b": "F^{\\alpha \\beta }=\\left({\\begin{matrix}0&{E_{x}}&{E_{y}}&{E_{z}}\\\\-{E_{x}}&0&cB_{z}&-cB_{y}\\\\-{E_{y}}&-cB_{z}&0&cB_{x}\\\\-{E_{z}}&cB_{y}&-cB_{x}&0\\end{matrix}}\\right).", "1243248f7c2490bc40cd0736c008f547": "={\\frac {1}{2}}{\\frac {m}{L}}\\int _{0}^{L}u^{2}\\,dy", "124367e24296d57e22098b3214a95f38": "E_{b\\lambda }(\\lambda ,T)", "1243e8c3af2463f759d6e0936b9a9512": "\\beta _{\\text{max}}\\,", "1244109828a3a22567f537df52ee9fa2": "{x}={\\sqrt {r^{2}+\\alpha ^{2}}}\\sin \\theta \\cos \\phi ", "124425e8319cb8ce9d2c719154aebb33": "\\scriptstyle {f:\\mathbb {R} ^{2}->\\mathbb {R} }", "124429560e9b47daf31f10d235f6e8c4": "{\\begin{aligned}x&=[a_{0};a_{1},a_{2},\\dots ,a_{k},a_{k+1},a_{k+2},\\dots ,a_{k+m},a_{k+1},a_{k+2},\\dots ,a_{k+m},\\dots ]\\\\&=[a_{0};a_{1},a_{2},\\dots ,a_{k},{\\overline {a_{k+1},a_{k+2},\\dots ,a_{k+m}}}]\\end{aligned}}", "124457016256eab931d0737735bdd7ed": "\\phi :U\\times U\\to [0,1]", "1244680ad953edccff29bd4d4d32911e": "f\\colon M\\to N", "1244995069a569a3654e069361470c25": "j(\\tau ),{\\frac {j^{\\prime }(\\tau )}{\\pi }},{\\frac {j^{\\prime \\prime }(\\tau )}{\\pi ^{2}}}", "1244c1bdcebdccb532f8dc75a2f9c090": "\\exp[-x/2]", "1245706d70b0116c2d45bb8cfb88d58c": "x_{j}=x_{m}", "1245717a3d05e6f6fe82e2d3334fdbf9": "\\left|h_{i}(x)\\right|\\in O\\left({\\frac {x}{(\\log x)^{2}}}\\right)", "1245783d4161fda018f51697c88d3b8c": "p(1|0)=0.5\\,\\operatorname {erfc} \\left({\\frac {A+\\lambda }{\\sqrt {N_{o}/T}}}\\right)", "12458d4526d6ebacc01561035773a72e": "{\\boldsymbol {\\omega }}_{\\mathbf {B} }={1 \\over 2}\\mathbf {B} \\times \\mathbf {B'} =-{1 \\over 2}\\tau \\mathbf {B} \\times \\mathbf {N} ={1 \\over 2}\\tau \\mathbf {T} ", "1245cb3de72982a5c8de03e7a500e8cd": "l_{2}=a_{00}-{\\mathcal {L}}(p_{6})+p_{3}p_{6},l_{3}=a_{00}-{\\mathcal {L}}(p_{9})+p_{3}p_{9},l_{31},.....", "1245d412ae9fab2fd7892fb683c6fa61": "O(\\log ^{2}n)", "124664611c787862a0324b191ddaa120": "a_{n+1}=a_{n}+1", "12470230f048f949d059b336e72b3909": "b\\equiv 8^{2^{2-1-1}}\\equiv 8{\\pmod {13}}", "1247271523c63a49a34c35cde5f81c4f": "\\ \\alpha ", "12473f8d7960bda6a3531ff5e50dfe59": "F^{n}", "12474cc97b24718e3063e70fa9666eea": "G=(N,T,P,S)", "12474dbaedded29c0b332c5e45788ade": "r(u,v)>0", "12475580a93d0ee778183ab1d41a8713": "\\sum _{n=-\\infty }^{\\infty }m_{n}{\\hat {f}}(n)e^{int}", "12476384f5a379dd4f040efe08a5f740": "\\int _{a}^{b}f(x)\\,\\mathrm {d} x=F(b)-F(a).", "12476d963b3e2219aae926849ed72a6e": "\\int _{{\\textbf {R}}^{nd}}f(x_{1},\\dots ,x_{n})M^{n}(dx_{1},\\dots ,dx_{n})=E\\left[\\sum _{(x_{1},\\dots ,x_{n})\\in {N}}f(x_{1},\\dots ,x_{n})\\right],", "124793fab7ff1799b76a5f3c32f114a3": "f:A\\to \\mathbb {R} ^{2}", "1247cecf9cbbcb26fe51b3182c261d50": "SO(2k+1)\\cong PSO(2k+1)=PO(2k+1),", "1247dfaf13fa0de630671919222ede1b": "q'_{\\text{P}}", "124817a655ced5c7aac92cde4e35c94d": "EZ=0", "124855233af08ab80a9b816d884a33c4": "(-1.00,0.00);", "12487358679b18e3b307c8c5417d71d7": "{\\frac {D(D+2)}{2(k-1)}}>n{\\begin{pmatrix}r+1\\\\2\\end{pmatrix}}", "1248bb97d27a019202b745b3eb87b54d": "R_{\\mathrm {VARIMAX} }=\\operatorname {arg} \\max _{R}\\left(\\sum _{j=1}^{k}\\sum _{i=1}^{p}(\\Lambda R)_{ij}^{4}-{\\frac {\\gamma }{p}}\\sum _{j=1}^{k}\\left(\\sum _{i=1}^{p}(\\Lambda R)_{ij}^{2}\\right)^{2}\\right).", "1248cc9b9ae89cd84f3a33977f2aa5f5": "\\phi _{l}\\,={\\frac {\\phi _{N}+\\phi _{L}}{2}}", "1248f41cea6c5800827eef338ffe9233": "w=w_{N}+w_{P}", "1249255abc34d9746ea2998bcc4e66a0": "A_{i}\\preceq B_{i}", "124959c70e14f25a350b4433012e93e8": "y=f(x)={\\frac {a+bx}{c+dx}}\\,", "1249881b4e7472f7500ab50d768f1870": "{\\begin{aligned}\\phi _{0}(x)&=1,\\\\\\phi _{k}(x)&=x^{k}=x\\phi _{k-1}(x)\\end{aligned}}", "1249d55a013e09bc608e513a47d10794": "p=A\\to w\\in P", "124a121306fde680279af0804a518d4f": "\\phi =\\cos ^{-1}{\\sqrt {\\frac {2}{3}}}", "124a4f4d11c05d6ff8f576f98bf74766": "U_{0},\\dots ,U_{q-1}", "124a76d25daafbd3b64f44bf0f63b410": "1-4\\pi r^{3}/3V", "124a785e7d95506a9a0a87e1fccf6087": "a_{1}*b_{13}", "124a912c09eef7a570c4c81dd6f021ae": "T_{11}=2\\eta _{0}\\lambda {\\dot {\\gamma }}^{2}\\,", "124aee59b2cc99d23e863b9c8602a69c": "{\\sqrt {r^{2}+h^{2}}}", "124afdb7adcd1ce02e1f749621999b13": "S(p_{x})=-\\int p_{x}(u)\\log p_{x}(u)du", "124b0d7b8a012b551fe318b4e3855f7a": "=\\int {\\Big [}f(x)\\;\\nabla _{\\theta }\\log \\pi (x\\,|\\,\\theta ){\\Big ]}\\;\\pi (x\\,|\\,\\theta )\\;dx", "124b31831039e075ae68ec8b2d471284": "f(x)={\\frac {k}{\\lambda ^{k}}}x^{k-1}\\exp \\left(-{\\frac {x^{k}}{\\lambda ^{k}}}\\right)", "124b40f8730a387b59b373453abc35f4": "G(x,y,", "124b55f9d7f201cc5d9d45857cfe5275": "z(\\mathbf {x} )z(\\mathbf {x'} )\\approx \\varphi (\\mathbf {x} )\\varphi (\\mathbf {x'} )=K(\\mathbf {x} ,\\mathbf {x'} )", "124b80811383990664fbd06048601c7c": "\\sigma _{xy}", "124b94e13e3ab87f0648002b7d014dff": "U=\\int d\\theta \\int \\rho d\\rho \\ \\lambda (\\rho ,\\theta )\\Phi (\\rho ,\\theta )", "124bb64e427a77d713fd3511725a46e4": "y(x,\\ t)=A\\cos \\left(kx-\\omega t\\right)=A\\cos \\left(k(x-vt)\\right)", "124bbed7f1cc8945b9362a028037eb6a": "{\\mathfrak {H}}\\,", "124c37a293ec5fd14a840ed9e6b167a1": "\\nabla ^{2}\\mathbf {B} ={\\frac {1}{c^{2}}}{\\frac {\\partial ^{2}\\mathbf {B} }{\\partial t^{2}}}", "124c3f1456bcae33c34c3705ee9b97b1": "\\scriptstyle {\\mathfrak {C}}({\\mathcal {Z}})", "124c657d230808bb54c9b80b8d8c7951": "I=\\displaystyle s_{w}Y+(s_{c}-s_{w})P_{c}", "124cc99f1a965ccda1f1ebc90e27f0ce": "\\alpha :[0,1]\\rightarrow [0,1]", "124d0d3b5a7d9c9d4eafcab3db0a5e17": "{\\frac {\\mathrm {d} \\sigma }{\\mathrm {d} (\\cos \\theta )}}={\\frac {\\pi \\alpha ^{2}}{s}}\\left(u^{2}\\left({\\frac {1}{s}}+{\\frac {1}{t}}\\right)^{2}+\\left({\\frac {t}{s}}\\right)^{2}+\\left({\\frac {s}{t}}\\right)^{2}\\right)\\,", "124d6e4e12cd3587cfc2e894599ceb83": "C_{rr}={\\frac {F}{N_{f}}}", "124d83121dfc7388b3968154d23df6fb": "(X-X_{\\min })/(X_{\\mathrm {norm} }-X_{\\min })", "124daaa9e105e4908792851293009202": "\\tau (M)", "124ddc7a5c21676c228a1889de2423e5": "=\\sum _{i\\neq m}{\\text{Tr}}\\left\\{\\mathbb {E} _{X^{n}}\\left\\{\\Pi _{\\rho _{X^{n}\\left(i\\right)},\\delta }\\right\\}\\ \\Pi _{\\rho ,\\delta }^{n}\\ \\mathbb {E} _{X^{n}}\\left\\{\\rho _{X^{n}\\left(m\\right)}\\right\\}\\ \\Pi _{\\rho ,\\delta }^{n}\\right\\}", "124df6f2540504dc5ea73945b35b9634": "({v_{0}+v_{i}})K_{w}10^{-b_{1}E_{i}+b_{0}}", "124e0b2d5bbc6c7d1c376e23e5e11b8b": "R={\\sqrt {\\frac {(ab+cd)(ac+bd)(ad+bc)}{(a+b+c-d)(b+c+d-a)(c+d+a-b)(d+a+b-c)}}}.", "124e21988bd76b47a0232d0df89107fa": "i=0,1,..,n", "124e35fd39520da2c58603ffc4c5d101": "f_{pm}", "124e676e83768676656053306aeeffbc": "e^{-At}\\mathbf {y} '-Ae^{-At}\\mathbf {y} =e^{-At}\\mathbf {b} ", "124e8e38590efe9ffe05c3fd7fa802ae": "z=\\infty ", "124ee9170b68be63c66a6508107ec89f": "b+X", "124ef642b189c7c2450cff56c63e29a3": "{\\frac {\\partial u}{\\partial r}}", "124f9472da0fcb9f86874a2b0d7d2bac": "\\alpha ={\\frac {1}{V}}\\left({\\frac {\\partial V}{\\partial T}}\\right)_{P}\\ ={\\frac {1}{V}}\\left({\\frac {V}{T}}\\right)", "124f987ff771391bb00f37927adf8d51": "X_{2}", "124fd5141bc40b5b468edd12592de2a5": "x-y\\in K_{0}", "1250211274ac8dbaea6909a65db43e42": "P,Q,R,S", "12503daaa21a370cca9e4c120ad57057": "\\Delta U(V_{a},V_{b};T^{+};V_{c},V_{d};T^{-})\\ ", "12506f2e450d14afb20ccaab244169bc": "S_{1}\\cdot x_{1}+S_{2}\\cdot x_{2}", "1250b128d30bd4cc75ffced31b5a770a": "\\forall x\\forall X_{\\in A}(Xx\\rightarrow \\exists Y_{\\in B}(Yx\\land \\forall y(Yy\\rightarrow Xy)))", "12510ecc627663872b9a367b7d0a42f9": "({\\overline {C}}\\vee A\\vee B)\\wedge {\\overline {({\\overline {C}}\\wedge (A\\vee B))}}", "125172d7b5e75dcb3bb4cea332268e82": "H_{3}^{*}\\quad \\longrightarrow \\quad H\\ +\\ H\\ +\\ H", "1251ef41f88e98de4d6424fac710c96b": "n_{i}", "1252156f985856912260dec8d312d04e": "A_{x}{\\hat {\\mathbf {x} }}+A_{y}{\\hat {\\mathbf {y} }}+A_{z}{\\hat {\\mathbf {z} }}", "1252205ef76eda7fb2f3243606f87404": "\\mathbf {F} =\\mathbf {m} \\mathbf {a} ", "1252609ae43efbe1758c91376fe5262b": "\\rho \\otimes \\omega .", "12528bd547d254e55632140682b9adec": "t=\\int _{0}^{z_{s}}{ndz \\over c\\cos \\alpha (z)}", "125329315671ce0768bc58235ad51770": "\\left({\\text{P}}-{\\text{V}}\\right)", "12533d0cac5f786c86f7c65d790a9226": "L=T-U.\\quad ", "12538201841371db2c2d38d42eebbcd9": "H(\\phi )\\leq \\log {\\sqrt {2\\pi eV(\\phi )}},", "1253967d5a5a67fb641930ab20fea687": "F_{s,n}=gV(\\rho _{s}-\\rho _{a})", "1253b8522e16c0aef582d10ec00e4fa3": "{\\frac {OA}{OQ}}=\\cos \\alpha \\,", "12542106b65a8ea67158e83abf8b38fc": "X=2/3.", "1254717bd76e6ef0c39fb10d1de24e7a": "Y=g(X)", "125495e1cd74bc8b3c1e181d3be3ec2b": "a_{2}(S,H)={\\frac {\\ln(S/H)+(r-{\\frac {1}{2}}\\sigma ^{2})\\tau }{\\sigma {\\sqrt {\\tau }}}}=a_{1}(S,H)-\\sigma {\\sqrt {\\tau }}", "12550c1f32bd7c369550848520210d4d": "u'(t)={\\frac {d}{dt}}r'(t)={\\frac {d}{dt}}r(t)-v=u(t)-v.", "125559e24a2662e0f31ced8149f1ae6a": "\\operatorname {MTF} _{s}(\\nu )=e^{-3.44\\cdot (\\lambda f\\nu /r_{0})^{5/3}\\cdot [1-b\\cdot (\\lambda f\\nu /D)^{1/3}]}", "1255c60ed83d144951ed7d372a92fd30": "(X,d_{X})", "125600b11d7831dae7398f10239575f4": "(O(loglogn))", "12564d9786930cafbbe0d46ed676a605": "x^{2}-1.786737601482363x+2.054360090947453\\times 10^{-8}=0", "12565974392c9ea7f9aa41160579b8d8": "f\\in [0,1]", "1256e361612e4bf9ff63ba8873202b7b": "c_{k'}^{(1)}=-{\\frac {i}{\\hbar }}\\int _{0}^{t}dt'\\;\\langle k'|H_{1}(t')|k\\rangle \\,e^{-i(E_{k}-E_{k'})t'/\\hbar }", "125706542d7f4c6099a06a4f2910fdb4": "y_{2}(x)=4x-3", "12571fdd653be83248f4c367ebe63526": "\\varepsilon /a=\\varepsilon ", "12572aeb00945406c0aab1a3db8a243d": "-{\\overline {3}}={\\overline {-3}}={\\overline {1}}.", "1257f763703310d4d585c8adfb8c63ce": "\\pm \\sum _{i=-\\infty }^{n}a_{i}p^{i}.", "12580adb261fa0e7dc64dca30ad48490": "O(\\log n/\\log \\log n)", "1258424547d3aa8b9a38c22fd11d64de": "E[F]<1", "125866e83e92694a1d413a7f07ce6cc8": "\\Sigma U_{\\Gamma }=F_{\\Gamma }-A_{\\Gamma 1}A_{11}^{-1}F_{1}-A_{\\Gamma 2}A_{22}^{-1}F_{2},", "125876e06b2da07ede2d22ea19c4cbee": "{1 \\over \\rho }{\\partial \\over \\partial \\rho }\\left(\\rho {\\partial f \\over \\partial \\rho }\\right)+{1 \\over \\rho ^{2}}{\\partial ^{2}f \\over \\partial \\phi ^{2}}+{\\partial ^{2}f \\over \\partial z^{2}}", "12589b831ded820b78312b4a9e184fd6": "\\Psi ", "1258a5e9e73f6d90a2e7f3bbaf6b7858": "1s^{2}\\,", "1258dbf189da09bf418f8f7198e52383": "{\\overline {\\overline {I}}}(h)=\\inf\\{I(f)|f\\in C[a,b],h\\leq f\\}.", "1258dd3bb09ffd5dd023fb9de14d699b": "[H_{k}]", "12590fd370dd0d394f27f2c42e383909": "{\\begin{aligned}T_{A}&=[A]+K[A][B]\\\\T_{B}&=[B]+K[A][B]\\end{aligned}}", "12591df3303c108f4b805aa680b25fdc": "v={\\frac {1}{u}}({u}\\cdot v)+{\\frac {1}{u}}({u}\\wedge v)=({v}\\cdot u){\\frac {1}{u}}+(v\\wedge u){\\frac {1}{u}}", "1259b792279c498370afd3cf199bda9b": "e_{3}(\\tau )={\\tfrac {1}{3}}\\pi ^{2}(\\vartheta _{10}^{4}(0;\\tau )-\\vartheta _{01}^{4}(0;\\tau )).", "1259c9004980830e830c91e7968cdd64": "s\\ {\\stackrel {\\mathrm {def} }{=}}\\ q/f", "1259d55e1521d7150973e4f0c57b19a3": "(\\nabla _{c}T)^{a_{1}\\ldots a_{r}}{}_{b_{1}\\ldots b_{s}}={\\frac {\\partial }{\\partial x^{c}}}T^{a_{1}\\ldots a_{r}}{}_{b_{1}\\ldots b_{s}}+\\,\\Gamma ^{a_{1}}{}_{dc}T^{d\\ldots a_{r}}{}_{b_{1}\\ldots b_{s}}+\\cdots +\\Gamma ^{a_{r}}{}_{dc}T^{a_{1}\\ldots a_{r-1}d}{}_{b_{1}\\ldots b_{s}}", "125a667f96c903122b119b7d55a7679a": "\\tan \\theta ={\\sqrt {{\\mathit {l}} \\over {\\mathit {l}}^{\\prime }}}.", "125ad75d17f3b3cfcdfb0c044c023737": "B_{max}", "125b0511a55b001336a4ac691827b64e": "v\\notin N(T\\setminus S)\\,", "125b3020442c5368e2560aaa34bf6b7d": "\\Pi _{0}(x)", "125ba9c6dfaa60d16409d0823ae0af2d": "m(\\phi )=a(1-e^{2})\\int _{0}^{\\phi }\\left(1-e^{2}\\sin ^{2}\\phi \\right)^{-3/2}d\\phi ,", "125bc3a895f1cace6b98ec45b9843a6f": "(a_{1})+(a_{2}x+a_{2})+(a_{3}x^{2}+a_{3}x+a_{3})=x^{2}-1\\,", "125c1c1e80015d2040a8330045ba5520": "u'''(c)<0", "125c242a0f02072e64f9a0b8ec347ce4": "\\left|{\\frac {f(z_{1})-f(z_{2})}{1-{\\overline {f(z_{1})}}f(z_{2})}}\\right|\\leq \\left|{\\frac {z_{1}-z_{2}}{1-{\\overline {z_{1}}}z_{2}}}\\right|.", "125c44cda569226c190ba8c30f32ecab": "log_{g}", "125c76b791996b305353eb9ef3267bca": "(G\\times SL(n),V\\otimes \\mathbb {F} ^{n})", "125c8a306b5436475841a9ca05531c98": "{\\tau '_{\\rm {pb}}}", "125c916ea28cb0850dbaeae7deea0981": "n=\\mathrm {JD} -2451545.0", "125d3bebec5585c5d1186740cec6c450": "{\\begin{aligned}E_{\\text{out}}&=\\sum _{i=1}^{\\infty }{K_{i}E_{\\text{in}}^{i}\\sin(i\\omega t)}\\\\&=K_{1}E_{\\text{in}}\\sin(\\omega t)+K_{2}E_{\\text{in}}^{2}\\sin(2\\omega t)+K_{3}E_{\\text{in}}^{3}\\sin(3\\omega t)+K_{4}E_{\\text{in}}^{4}\\sin(4\\omega t)+K_{5}E_{\\text{in}}^{5}\\sin(5\\omega t)+\\cdots \\end{aligned}}", "125d3d0b4ab35bf5a521075f21b624f5": "X^{+}3{\\frac {hs}{D}}x^{2}+3\\left({\\frac {hs}{D}}\\right)^{2}x={\\frac {P'}{3}}{\\frac {h^{2}}{D^{2}}}", "125d4958d0e1050ee0240844f52260a7": "\\gcd(a,a)=a", "125d61f85d6153a25aee3145e197dd5c": "\\Omega ={\\left(q+N^{\\prime }-1\\right)! \\over q!(N^{\\prime }-1)!}", "125d70c63d089e961501181575d0109c": "N(\\mu ,I_{k})", "125d83620b063896be9566c9947f4861": "\\alpha _{1}={\\frac {4G}{c^{2}}}{\\frac {M}{b_{1}}}", "125e33aceadefa2ba7f06cc44ab8b2a3": "E\\,=\\,{\\frac {1}{2}}\\,\\rho \\,g\\,a^{2}\\,", "125e7fd7e93fb8bb74b6c13e295e772d": "[ion]_{\\mathrm {out} }", "125eaa7cf8392f6df79524d1a2c79ecc": "y=ax^{3}+bx^{2}+cx+d\\;.", "125ec6280e530d5eb3696c74bdb05b4c": "\\sum _{\\scriptstyle j=1 \\atop \\scriptstyle j\\not =i}^{n}a_{i,j}(\\Phi _{j,1},\\ldots ,\\Phi _{j,n}),", "125eee4fbaf566efa60b078555716f27": "W(T,G):=N_{G}(T)/C_{G}(T).", "125f3838a0d36b6db2bc4839ec1a3313": "v\\in (F_{2})^{d}", "125f70d920c3b077f993f3c8e55fc7a8": "c_{\\mathrm {air} }=331.3\\ \\mathrm {\\frac {m}{s}} (1+{\\frac {\\vartheta ^{\\circ }\\mathrm {C} }{2\\cdot 273.15\\;^{\\circ }\\mathrm {C} }})\\,", "125f86f1bb764f803ffb0ba805dd6e34": "{\\dfrac {dR}{dP}}<0\\!\\ ", "125ff111ae29a09a0d6187c9ad57d47e": "\\Delta _{h}=hD+{\\frac {1}{2}}h^{2}D^{2}+{\\frac {1}{3!}}h^{3}D^{3}+\\cdots =\\mathrm {e} ^{hD}-I~,", "125ff850e79a4937783d86ec5201e79d": "\\mu =\\left({\\frac {\\gamma -1}{2}}\\right)^{2}-{\\frac {1}{4}}\\quad ,\\quad \\gamma =1\\pm {\\sqrt {1+4\\mu }}.", "126181fb770c0f9cc630588ac25faa51": "p={\\tfrac {1}{6}}", "1261e4be5b3b0a32eafdba15427ca7c8": "u(t)=-K(x-e)\\,", "126207c7c197aaac720240a1d6b4e28f": "f(k,i)", "12623305c66087564e25eef11f3a0df6": "z_{1},\\dots ,z_{k-1},z_{k+1},\\dots ,z_{n}", "126248cb7a44893db99179a8b28cb329": "D(f(d))", "1262780e938a441012b3440423e8680f": "(q_{i,1},q_{i,2},q_{i,3},q_{i,4})", "126282e05e46e5803e9f309388d2d38d": "\\epsilon _{ik}={\\frac {\\mathbf {x} _{0k}^{\\top }([\\delta K]-\\lambda _{0i}[\\delta M])\\mathbf {x} _{0i}}{\\lambda _{0i}-\\lambda _{0k}}},\\qquad i\\neq k.", "126288d3a8b385054b6ddac5b34c0a96": "{\\hat {P}}(y\\mid x_{1},\\ldots x_{n})={\\frac {\\sum _{i:1\\leq i\\leq n\\wedge F(x_{i})\\geq m}{\\hat {P}}(y,x_{i})\\prod _{j=1}^{n}{\\hat {P}}(x_{j}\\mid y,x_{i})}{\\sum _{y^{\\prime }\\in Y}\\sum _{i:1\\leq i\\leq n\\wedge F(x_{i})\\geq m}{\\hat {P}}(y^{\\prime },x_{i})\\prod _{j=1}^{n}{\\hat {P}}(x_{j}\\mid y^{\\prime },x_{i})}}", "12629a5d9f1f5f72b4687c383e23d98b": "{\\bar {A}}^{f}e^{i\\phi _{n}^{f}}\\xrightarrow {iFFT} {\\bar {A}}_{n}^{k}e^{i\\phi _{n}^{k}}.", "1262b0faeaad9a7ed7a2bff7d9abba81": "\\mathbb {A} _{k}^{n}", "1262c8e1c18575a059b5cf91f534a970": "\\scriptstyle {\\sqrt {n}}", "12631845e3c8705ce99025dc91efae18": "1-x^{2}", "126322941f06403fe6622d98ef9d16dd": "g_{(a,k)}(u)=k(1-u)^{-{\\frac {1}{a}}},", "126327aabeee288cf4009fa57374087d": "X_{i}\\subseteq M_{i}", "12633add9295ebce5a88194f6045af38": "s(x)=p(x)\\cdot x^{t}-s_{r}(x)\\,.", "1263e48ce658011752e890a21f3cd767": "\\{like\\langle Mary,Sue\\rangle ,like\\langle Mary,Bill\\rangle ,like\\langle Mary,Lisa\\rangle \\}", "1263f69f7776ee47356ed28dface550e": "n_{i}b", "12640f1b86c99509ca20c670e0f8357f": "\\ \\alpha _{j}=A_{j}\\,e^{-i\\Omega \\,t},~~~j=1,2,", "1264377932bb51600a94acc2dc4c931f": "\\Delta _{+}", "12644e32dfd2bae04a4425cc837d092f": "y(t)=Cx(t)+Du(t)\\,", "126452993412054a722fef0d5b76cc79": "S\\subset P", "1264775e19756e6284ae825567ebe908": "|nm_{n}\\rangle ,n=", "1264d76983fee04d9abff194802e7a91": "{\\frac {\\delta J}{J}}=\\varepsilon _{S}^{2}{\\frac {\\delta S}{S}}", "1264e7dd7a4c8d655f80511a1896bdf9": "10\\uparrow \\uparrow \\uparrow 7=(10\\uparrow \\uparrow )^{7}1", "126527e7998eb6e56dca42ba00aa3cdd": "V(z)=\\prod _{p\\in P,pt\\}|=|\\{x:f(x)>t\\}|.", "127ab64bb77b0232593c48234f86d150": "s_{1},y_{1},y_{2}", "127b0d77e28c888c1b5c9333439d1216": "\\forall \\phi ({\\boldsymbol {x}})\\in L(A,{\\boldsymbol {x}})", "127b65b4c2c1cd11aa55916245993a33": "\\displaystyle \\Lambda _{0}(P)=\\min \\sum _{x\\in X,s\\in S}P(x)l(s)", "127b6f24caf5e02a93f9a5d9865dcc45": "{\\mathit {alg}}(A_{1},B_{3})", "127b891ef1ca85b0b16821f10084a492": "|\\mathbf {U} |{\\rm {tr}}\\left(\\mathbf {U} ^{-1}{\\frac {\\partial \\mathbf {U} }{\\partial x}}\\right)", "127ba9d5bb612a2810654de6c4a24fd7": "{\\cfrac {1}{3+{\\cfrac {1}{1+{\\cfrac {1}{1+{\\cfrac {1}{3+{\\cfrac {1}{9+\\ddots }}}}}}}}}}", "127bc10827549d7b09c71fcffb9e83cf": "\\mathbf {e} \\,\\!", "127c423075a3f04249a36367eb845ce9": "f_{a}([n,n+1))=a_{n}", "127c5888c5e3986f030d153f1b7a3f84": "{\\text{stick}}(T(p,q))=2q{\\text{, if }}2\\leq p0", "1282ebdebd93251bba9b2a347a5748bb": "(x_{1},\\ldots ,x_{n})R=A", "12838cc6c9d9f08bd41423b3c18ae865": "{}^{\\mathrm {N} }\\mathbf {a} ^{\\mathrm {R} }={}^{\\mathrm {N} }\\mathbf {a} ^{\\mathrm {Q} }+{}^{\\mathrm {B} }\\mathbf {a} ^{\\mathrm {R} }+2{}^{\\mathrm {N} }{\\boldsymbol {\\omega }}^{\\mathrm {B} }\\times {}^{\\mathrm {B} }\\mathbf {v} ^{\\mathrm {R} }", "1283bca13104450a3d644e1f61cc29b9": "{\\hat {H}}=-J\\sum _{j=1}^{N}\\sigma _{j}\\sigma _{j+1}-h\\sum _{j=1}^{N}\\sigma _{j}", "1283dfdefd9a083ea421b2128b4d41c2": "{\\frac {|\\Psi (X,a)|^{2}}{\\int |\\Psi (X,a)|^{2}\\,dX}}", "1283edbbf0829fb0c5fac0c5378a3754": "{\\overline {X}}_{n}\\pm A{\\frac {S_{n}}{\\sqrt {n}}}.", "12842d1001ac44440b1106c77a6352f7": "z:\\mathbb {R} ^{p}\\rightarrow \\mathbb {R} ", "128446a074f73eeed19baa18c8f10e2c": "g_{i}(x)>0.", "1284501774c40a4e3cd18ff724f171fc": "|{\\text{alive}}\\rangle ", "1284720f4e5ab194021b0ec2614532f6": "\\,_{2}F_{1}", "12848b9e61a8a6e0b78591be42a03c0f": "{\\overline {\\operatorname {Sp} }}(E)={\\overline {\\operatorname {Sp} (E)}}", "1284a37300dff1eef415c4671fc4f550": "\\displaystyle {a(z)={1 \\over |z|^{2}+1}.}", "1284b1184c63d526116f02716eaa0152": "{\\overline {V}}", "1285139fcd6053413cf8aafa4469e2f8": "{\\begin{aligned}C&=LBA\\div (SPT\\times HPC)\\\\H&=(LBA\\div SPT)\\,{\\bmod {\\,}}HPC\\\\S&=(LBA\\,{\\bmod {\\,}}SPT)+1\\end{aligned}}", "128523349a618db89d0a57eb780a0fe4": "{\\begin{bmatrix}u&0\\\\0&u^{-1}\\end{bmatrix}}", "12852673ea50019ba6e2809f54e64b76": "\\operatorname {ad} (x)(y)=[x,y].", "128530b0af27d8a2025932018b18b4d7": "B\\oplus A", "12856e19775c18fd7baba3954d06bbbb": "\\mathrm {Res} _{z=\\infty }{\\frac {f(z)}{5-z}}=\\exp({\\tfrac {\\pi i}{4}})\\left(5-{\\frac {3}{4}}\\right)=\\exp({\\tfrac {\\pi i}{4}}){\\frac {17}{4}}.", "12859eb919f21acda5de15e886748a40": "(T_{s}-T_{o})", "1285bd102cc096dd43afefeb62566669": "O(A_{1}:A_{2}|B)\\triangleq {\\frac {P(A_{1}|B)}{P(A_{2}|B)}}", "1285d4942c89ea2976d708dfa97df851": "X[k]", "1285f0eb201983932c7ca8b8c42480a6": "P(k)\\propto k^{-\\gamma }.", "12860194205a06179d4444609bdc9aaa": "d={\\frac {WL}{4T}}", "12863bb5fc4524cb660f5dab95b580c9": "|\\Psi ^{-}\\rangle ", "128689e74d5fa4c94a5fb7b1c563c756": "\\textstyle (x,y\\pm 1,z\\pm 1)", "12871792bb70b377f281f1645e374975": "P_{e}^{(n)}\\geq 1-{\\frac {1}{nR}}-{\\frac {C}{R}}", "12873aeb286b5c2cca7e9477ec9481c5": "\\left(1+x_{i}\\right)", "128743d2a4c3ecfae83458638d0afc97": "|\\;\\;|", "1287491ec7e30d8610ee89c7fe68f165": "(\\phi \\to \\bot )\\to \\lnot \\phi ", "12878870fa8bca90746deb93abe62bd7": "{\\frac {\\sqrt {2}}{Y}}dt={\\frac {d\\varphi _{1}}{\\sqrt {E\\chi _{1}-\\omega _{1}+\\gamma _{1}}}}={\\frac {d\\varphi _{2}}{\\sqrt {E\\chi _{2}-\\omega _{2}+\\gamma _{2}}}}=\\cdots ={\\frac {d\\varphi _{s}}{\\sqrt {E\\chi _{s}-\\omega _{s}+\\gamma _{s}}}}.", "1287c663689a16554d0912980ce45aed": "e_{s}", "1287e2f653202e49eb141856077e70a1": "a\\oplus b=T(a,1,b)", "12889c5f93db54fa936d8a765d4eae57": "\\diamond ", "1288e7976aeb15fa36c259ee691fa9f6": "\\sigma \\approx 0.45\\lambda N\\ .", "1288f2505e50d59509893aa9ccacaaab": "\\phi -1=\\left(\\sum _{i}m_{i}\\right)^{-1}\\left[If'-f+\\sum _{i}\\sum _{j}\\left(\\lambda _{ij}+I\\lambda '_{ij}\\right)m_{i}m_{j}+2\\sum _{i}\\sum _{j}\\sum _{k}\\mu _{ijk}m_{i}m_{j}m_{k}+\\cdots \\right]", "12893d508020982fb6b2ff7dd7ae1453": "i_{abc}(t)=T^{-1}i_{\\alpha \\beta \\gamma }(t)={\\begin{bmatrix}1&0&1\\\\-{\\frac {1}{2}}&{\\frac {\\sqrt {3}}{2}}&1\\\\-{\\frac {1}{2}}&-{\\frac {\\sqrt {3}}{2}}&1\\end{bmatrix}}{\\begin{bmatrix}i_{\\alpha }(t)\\\\i_{\\beta }(t)\\\\i_{\\gamma }(t)\\end{bmatrix}}.", "1289aae3ff46474aba2e05b4d2280e20": "A_{\\%}(M)", "1289df2c84ee0c3a40a1065037b04e78": "\\mathbf {b} _{i},\\mathbf {b} _{j}", "128a351b89452c4a8d00aaccdb9d768c": "h(x_{1},\\ldots ,x_{m})\\,", "128a3b1f4c29b0e5d8dc4441370d5287": "a_{v}=\\int _{0}^{\\infty }e^{-\\lambda u}{\\frac {(\\lambda u)^{v}}{v!}}{\\text{d}}F(u)~{\\text{ for }}v\\geq 0", "128ac735f20a3719bda08037ea22f57e": "\\scriptstyle X\\subseteq \\mathbb {R} ", "128aeb60dd0b9b835ea64242ee28689f": "E_{n_{1},n_{2}}^{(0)}=E_{n_{1}}+E_{n_{2}}=-{\\frac {Z^{2}}{2}}{\\Bigg [}{\\frac {1}{n_{1}^{2}}}+{\\frac {1}{n_{2}^{2}}}{\\Bigg ]}", "128afe269fe8494d81c53c597f6884ff": "\\omega _{X|Y}", "128b162f3c9bed28b393d74204dea961": "y=x\\cot {\\frac {\\pi x}{2a}}.", "128b2315c119aa4eac8fb764208093fe": "\\Lambda ", "128b2c4604b54fdc17544b8bd6ea1c87": "{\\vec {F}}_{f}=\\int _{f}{\\vec {j}}\\times {\\vec {B}}d^{3}{\\vec {r}}", "128b3fa96868a2126f747c3f359f5c89": "r=a+b\\theta ^{1\\!/\\!x}.", "128b431cae260ba2ecf90420f7788334": "{\\frac {{\\rm {d}}^{2}x}{{\\rm {d}}t^{2}}}=-\\sum _{n}\\omega _{n}^{2}x", "128b4b3d701986d8faf7d987d4fc30d6": "A_{192}=3.1410319509", "128bc044cd07eeaf609bdec5337e028e": "\\lambda _{1}\\lambda _{2}\\cdots \\lambda _{n}vol(K)\\leq 2^{n}vol(R^{n}/\\Gamma ).", "128bed3dc81d6fe9b774969ee1e3f277": "1/e=37\\%", "128c3c0470addee25dc3e3fb07ba16f9": "\\langle cacao\\rangle ", "128c48c1c8125a6befc26888274e95cc": "f'(x)={\\mathrm {d} \\over \\mathrm {d} x}f(x)", "128c5a8ad52331e3b7d7fbbaec8fcd50": "F_{0},F_{1},F_{2}", "128c6a4f7e1a9cb466921f7a7bf54212": "D_{\\mathrm {KL} }(P\\|Q)=\\sum _{i}\\ln \\left({\\frac {P(i)}{Q(i)}}\\right)P(i)\\!", "128c74c8c28c008a33efbd4d8958162e": "Y_{s}(z)", "128c7ad2d429745ab8a80ae6ba90b4ef": "\\mathrm {E} (X)=\\lambda \\Gamma \\left(1+{\\frac {1}{k}}\\right)\\,", "128cc12ac724410d689b09aa3cbf6da3": "{\\begin{bmatrix}e_{x}\\\\e_{y}\\\\e_{z}\\\\e_{t}\\end{bmatrix}}=A^{-1}{\\begin{bmatrix}e_{1}\\\\e_{2}\\\\e_{3}\\\\e_{4}\\end{bmatrix}}\\ (2)", "128d68ecea6a7fe76562024915ca9252": "k=k(n)", "128d7a9bd803506a72862602729ff3ba": "\\ (\\phi ,\\lambda )", "128e0bbe088ffccc96e270fbb234e941": "\\operatorname {dist} _{\\operatorname {robust} }(T({\\mathcal {M}}),{\\mathcal {S}})=\\sum _{m\\in T({\\mathcal {M}})}\\sum _{s\\in {\\mathcal {S}}}g((m-s)^{2})", "128e249456f495173273bd487e4ff69b": "|r\\rangle ", "128e46c906836d75c4f2de51ff9063fc": "P(n)=|\\langle n|\\alpha \\rangle |^{2}=e^{-\\langle n\\rangle }{\\frac {\\langle n\\rangle ^{n}}{n!}}", "128ee115856834e25132858339b49686": "g(s)=2\\gamma \\int _{0}^{\\infty }(st)^{\\gamma +\\rho -1/2}\\;G_{p,\\,q+1}^{\\,q+1,\\,0}\\!\\left(\\left.{\\begin{matrix}\\mathbf {a_{p}} \\\\0,\\mathbf {b_{q}} \\end{matrix}}\\;\\right|\\,(st)^{2\\gamma }\\right)f(t)\\;dt,", "128f81f3c82c6548b1bff4b7275018e0": "u_{2}={\\frac {(x_{1}^{2}+ax_{2}^{2}+x_{3}^{2}+x_{4}^{2})x_{6}-2x_{2}(x_{1}x_{5}+bx_{2}x_{6}+x_{3}x_{7}+x_{4}x_{8})}{c}}", "128f906f18ac639aa2ee0f1422df9bd0": "a+c", "128fa24c558c3507c9276167b0238205": "B\\rightarrow \\neg C", "12902544313a452e8c8346c520e826e8": "=<\\psi |(A-a)^{2}\\psi >", "12902fc62eb1ae422beb9b2dfb7f1928": "n_{1}(\\lambda )", "1290559cd34e18837285c6e1b47342ff": "\\pi :C(X)\\rightarrow L(K)", "12906248bbe21d18d6b8d96391a922a1": "\\forall p:{\\mathcal {BB}}p\\to {\\mathcal {B}}p", "12909992d5d2cbd8a4c871a7228c6621": "\\Gamma (n)=1\\cdot 2\\cdot 3\\cdots (n-1)=(n-1)!\\,", "1290bbb09acfab4c1ea905f753946b06": "\\mathbf {E} ={\\frac {1}{2}}\\left[(\\nabla _{\\mathbf {X} }\\mathbf {u} )^{T}+\\nabla _{\\mathbf {X} }\\mathbf {u} +(\\nabla _{\\mathbf {X} }\\mathbf {u} )^{T}\\cdot \\nabla _{\\mathbf {X} }\\mathbf {u} \\right]\\,\\!", "1290c173f86f2eb00adfa9fd8d6cb59b": "\\varepsilon _{A}^{l}:A^{l}\\otimes A\\to I", "1290eac0ed93251cf8fb17d8d3bdf7c6": "p({\\textbf {z}}_{k}|{\\textbf {z}}_{1:k-1})=\\int p({\\textbf {z}}_{k}|{\\textbf {x}}_{k})p({\\textbf {x}}_{k}|{\\textbf {z}}_{1:k-1})d{\\textbf {x}}_{k}", "1290eeffdf3b1e7b492ce720dd10e08d": "(g^{a})^{b}{\\bmod {p}}", "1290f1ca251579ae051c4cbe263d7a7b": "\\int _{-\\pi }^{\\pi }\\sin((2m-n)x)\\sin ^{n}x\\ dx=\\left\\{{\\begin{array}{cc}(-1)^{m+(n+1)/2}{\\frac {\\pi }{2^{n-1}}}{\\binom {n}{m}}&n{\\text{ odd}}\\\\0&{\\text{otherwise}}\\\\\\end{array}}\\right.", "12910770b73831d398ee1304962983de": "{\\mathcal {Z}}\\left\\{f(t)e^{-a\\,t}\\right\\}=e^{-a\\,m}F(e^{a\\,T}z,m).", "129150c6efea9d3418afcf58346e9b34": "{\\mathcal {L}}(D)", "12919629d5d98028f8dacf03e5acaf66": "\\mathrm {E_{1}} (z)=\\int _{1}^{\\infty }{\\frac {e^{-tz}}{t}}dt", "1291978b40fef6c5d9d4fbba9be0d1c0": "\\mu \\|w\\|_{2}", "12919dc6b2e16aac8bfa19a25c0a2d4b": "VCA(p_{0{\\frac {1}{2}}},(a,m))\\cup \\{[m,m]\\}\\cup VCA(p_{{\\frac {1}{2}}1},(m,b))", "129272275507e29f027b602dc8577020": "{\\frac {\\partial T}{\\partial q_{j}}}=\\sum _{i=1}^{n}m_{i}\\mathbf {\\dot {r}} _{i}\\cdot {\\frac {\\partial \\mathbf {\\dot {r}} _{i}}{\\partial q_{j}}}", "12931a7922d292546b5ca8f85c8af3e1": "\\langle f|\\mathbf {H} _{B}|f'\\rangle \\equiv \\langle u,t,s,r|H_{B}\\otimes \\mathbb {I} \\otimes \\mathbb {I} \\otimes \\mathbb {I} |u',t',s',r'\\rangle ", "129344bb032523706eabf801ce7ee83b": "f(S)=\\sum _{i\\in S}w_{i}", "12935094c05d4fe264f2fc36e73f92c3": "{\\begin{aligned}\\phi _{1}+\\phi _{2}&=\\phi _{\\text{sys}}\\\\{\\frac {\\phi _{1}}{V_{1}}}+{\\frac {\\phi _{2}}{V_{2}}}&=0\\ ,\\end{aligned}}", "12938105a43d5bd52ffd54768a90d4af": "{\\frac {1}{2}}\\rho v^{2}+P+\\rho gh=\\mathrm {const.} ", "1293ec82c0b7832cde09c34855e74c40": "\\perp ,\\angle ,\\sphericalangle ,\\measuredangle ,45^{\\circ }\\!", "1294693eebe7d20084681aada4103956": "{\\vec {F}}\\cdot t=\\Delta m{\\vec {v}}", "129469ac0904fdf0ef810058213e7523": "V_{1}\\otimes V_{2}\\otimes V_{3}", "1294788e2618d4229c3e59cb9570d903": "E_{\\text{S}}=-\\mathbf {d} \\cdot \\mathbf {F} .", "1294c1f68e997ca6d8ecb25082804df9": "\\langle r^{2}\\rangle ={\\frac {6k_{\\rm {B}}T\\tau ^{2}}{m}}\\left(e^{-t/\\tau }-1+{\\frac {t}{\\tau }}\\right).", "1295421e1fcaee2a173e113a9d045194": "P(S\\rightarrow S'|E)", "129543f9083a67fe18059cdb5c2e41a6": "\\Delta W=-\\Delta V\\,\\!", "12959e8725dc55f16e20d57052980217": "{\\cfrac {\\mathrm {d} ^{2}W}{\\mathrm {d} x^{2}}}-\\left({\\cfrac {1+\\alpha }{\\beta }}\\right)~W={\\frac {M}{\\beta D^{\\mathrm {beam} }}}-{\\cfrac {q}{D^{\\mathrm {face} }}}", "1295a859b167ac81f4eb51b1c83b9ed4": "C\\in {\\mathcal {C}}", "1295f7ead87b6432cba321b54eb6a4fe": "phi_{i}", "12962c2f3464d4162ffa46009a480575": "P=K\\rho ^{5/3}", "1296c45635af9dcd2e03f80ad2e7c7e6": "d\\mathbf {X} _{t}={\\boldsymbol {\\mu }}(\\mathbf {X} _{t},t)\\,dt+{\\boldsymbol {\\sigma }}(\\mathbf {X} _{t},t)\\,d\\mathbf {W} _{t},", "129799ef2fac2e0f9e0e28c16c6d636c": "{\\gamma \\times y_{1}^{2} \\over 2}-{\\gamma \\times y_{2}^{2} \\over 2}-F_{d}=\\rho q(v_{2}-v_{1})", "12979e24fa99ffbb7f9dff7add280e9d": "A_{f}(\\infty )", "1297fd52d3a43aef7bb4671d3bdcb0d2": "\\,{\\vec {S}}_{i}\\cdot {\\vec {D}}=\\sum _{A=1}^{N}\\;M_{A}{\\big (}{\\vec {f}}_{i}\\times {\\vec {R}}_{A}^{0}{\\big )}\\cdot {\\vec {d}}^{\\,A}={\\vec {f}}_{i}\\cdot \\sum _{A=1}^{N}M_{A}{\\vec {R}}_{A}^{0}\\times {\\vec {d}}^{A}=\\sum _{A=1}^{N}M_{A}{\\big (}\\mathbf {R} _{A}^{0}\\times \\mathbf {d} _{A}{\\big )}_{i}=0,", "12984b04e8a953c103b5a529a128c99b": "{\\frac {dR}{dt}}=\\gamma I-\\mu R", "1298853a34d859cb701ae96805924528": "{\\vec {F}}\\,({\\vec {q}})\\!", "1298a05a415e6456eb1a36ed80aad200": "\\phi \\to \\bot ", "12992240713fa57dea8beac2f9e6b0d7": "\\sin \\alpha ={\\frac {\\cos U_{1}\\cos U_{2}\\sin \\lambda }{\\sin \\sigma }}\\,", "129930a70d02c994e22afc8b5858fd3f": "*T^{IJ}={1 \\over 2}\\epsilon _{KL}^{\\;\\;\\;\\;\\;\\;IJ}T^{KL}.", "12994c1a1c3de5f91e4b361ce002e0ba": "\\alpha =[\\alpha _{1},\\ldots ,\\alpha _{N}]^{T}", "1299988740ca877169a14d4791f07d2f": "1\\,-\\,{\\frac {1}{3}}\\,+\\,{\\frac {1}{5}}\\,-\\,{\\frac {1}{7}}\\,+\\,{\\frac {1}{9}}\\,-\\,\\cdots \\;=\\;{\\frac {\\pi }{4}}.\\!", "1299f15555e56cf3ad2a405ffaa33c71": "H_{u}:V\\to V\\,", "1299f512529d4c7f8a198b278ba90f3c": "\\forall n(n\\in \\mathbf {N} \\iff ([\\forall k\\in n(\\bot )\\lor \\exists k\\in n(\\forall j\\in k(j\\in n)\\land \\forall j\\in n(j=k\\lor j\\in k))]\\land ", "129a14a0b7f576ef418c269e42a06e0c": "x=\\operatorname {Re} \\,(z)={\\dfrac {z+{\\overline {z}}}{2}}", "129a829c9e91d217377ffc2766a4b4e4": "{\\mathit {E_{g,\\mathrm {InPAs} }}}={\\mathit {x}}{\\mathit {E_{g,\\mathrm {InP} }}}+(1-{\\mathit {x}}){\\mathit {E_{g,\\mathrm {InAs} }}}-{\\mathit {bx}}(1-{\\mathit {x}})", "129a8495f1a7863d379c07b4d6823d93": "wp(\\mathbf {if} \\ E_{1}\\rightarrow S_{1}\\ [\\!]\\ \\ldots \\ [\\!]\\ E_{n}\\rightarrow S_{n}\\ \\mathbf {fi} ,R)\\ ={\\begin{array}{l}(E_{1}\\vee \\ldots \\vee E_{n})\\\\\\wedge \\ (E_{1}\\Rightarrow wp(S_{1},R))\\\\\\ldots \\\\\\wedge \\ (E_{n}\\Rightarrow wp(S_{n},R))\\\\\\end{array}}", "129ab398b7a82f688c39dbc35665aea9": "w(z)=\\left({\\frac {z-a}{z-b}}\\right)^{\\alpha }\\left({\\frac {z-c}{z-b}}\\right)^{\\gamma }\\;_{2}F_{1}\\left(\\alpha +\\beta +\\gamma ,\\alpha +\\beta '+\\gamma ;1+\\alpha -\\alpha ';{\\frac {(z-a)(c-b)}{(z-b)(c-a)}}\\right).", "129b589447eac570919806a4185e0533": "\\Delta _{\\mathbf {Q} ({\\sqrt {d}})}", "129bb404ca829f380029ad271169e967": "(X,X)", "129bc753f57cd83e62b01c65567d3da9": "\\exp(z)", "129c1890bcdbfbf0280c06f64917fbec": "{\\tilde {S}}(\\omega )", "129c62c5d963eb77790ec32a3ba74cee": "P^{*}={\\frac {1}{2}}", "129c9eb5ffb4389eba06a0ef4ad5342a": "(X_{j})_{j\\in S_{i}}", "129cd22e74f096474908e39736962095": "{\\mathcal {F}}^{3}({\\hat {f}})=f.", "129ce768e58bd652764f42e4ef302cfe": "{\\begin{aligned}\\delta K&=-\\int _{0}^{T}\\left\\{\\int _{\\Omega ^{0}}\\left[J_{1}\\left({\\ddot {u}}_{\\alpha }^{0}~\\delta u_{\\alpha }^{0}+{\\ddot {w}}^{0}~\\delta w^{0}\\right)-J_{3}~{\\ddot {w}}_{,\\alpha \\alpha }^{0}~\\delta w^{0}\\right]~\\mathrm {d} A+\\int _{\\Gamma ^{0}}J_{3}~n_{\\alpha }~{\\ddot {w}}_{,\\alpha }^{0}~\\delta w^{0}~\\mathrm {d} s\\right\\}~\\mathrm {d} t\\\\&\\qquad -\\left|\\int _{\\Omega ^{0}}J_{3}~{\\dot {w}}_{,\\alpha \\alpha }^{0}~\\delta w^{0}~\\mathrm {d} A-\\int _{\\Gamma ^{0}}J_{3}~{\\dot {w}}_{,\\alpha }^{0}~\\delta w^{0}~\\mathrm {d} s\\right|_{0}^{T}\\end{aligned}}", "129d83ceec94d5ff850e59267524f9d3": "{\\mbox{E}}={\\frac {\\sqrt {30\\cdot P}}{d}}", "129da5e0f6b8bf1b449e80360ba684ed": "{\\begin{array}{rcl}p&=&{\\frac {\\varphi }{e}}\\\\&=&2{\\frac {1+{\\sqrt {5}}}{\\sqrt {10+2{\\sqrt {5}}}}}\\\\&\\approx &1.70130\\end{array}}", "129db74edf52b763b3c642533a6e2949": "\\sigma _{(ij)}", "129de37aa714507bdda89a1e20644a24": "{\\bar {x}}={\\frac {1}{A}}\\int _{a}^{b}x[f(x)-g(x)]\\;dx", "129e4bd1f992b085d3ec3012930aa1ba": "\\delta (B)={\\frac {\\Pi _{i=1}^{n}||b_{i}||}{\\sqrt {\\det(B^{T}B)}}}={\\frac {\\Pi _{i=1}^{n}||b_{i}||}{d(\\Lambda )}}", "129e73ab6e70f84330cfacb9f01f57dd": "\\mathbf {F} _{12}=-G{m_{1}m_{2} \\over {\\vert \\mathbf {r} _{12}\\vert }^{2}}\\,\\mathbf {\\hat {r}} _{12}", "129f008829c016637ce5cd30dbc975f3": "M={\\mathrm {M} inN}(L,D,n)", "129f61bb06883ef57d30ffda00b7751b": "\\{\\gamma ^{\\mu },\\gamma ^{\\nu }\\}=2g^{\\mu \\nu }", "129fe03e48da442355421bad241d5291": "u\\in W^{1,n}(R^{n})", "12a02d46fb89297274041849b9f651d6": "D=2^{m}-1", "12a03acd2fc3dc8f1083387fa2ee6a76": "\\bowtie \\!\\,", "12a03c1e0b9d556e1538f637dda40611": "C_{3}=-{\\frac {3125}{24EI}}(-1645+4M_{c}+64R_{a})\\quad {\\text{and}}\\quad C_{4}={\\frac {25}{12EI}}\\left(-40325+6M_{c}+120R_{a}\\right)\\,.", "12a07588f9667526eec89d4f256fe18a": "{\\frac {\\partial ^{2}\\eta }{\\partial \\phi ^{2}}}=\\cos \\left(2\\left(\\phi -\\theta \\right)\\right)+h\\cos \\phi >0.\\,", "12a0ad123bfa94caaea75d2ac1def938": "a\\cdot (0.089490\\dots )", "12a0da5171ef3a87a5c61d1c5be421c7": "c={\\frac {w}{n}}", "12a12fc7bc407a230701ed2849ac22c8": "F_{i}=A_{t}uC_{i}\\,", "12a190dea50e93cdec7d9e3969af3b83": "C_{n}\\,\\mathbf {v} =\\mathbf {v} -({\\tfrac {1}{n}}\\mathbf {1} '\\mathbf {v} )\\mathbf {1} ", "12a1961c1fdbc45cfc541d7df4bf4d25": "f_{0}{\\frac {\\partial v_{g}}{\\partial p}}=-{\\frac {R}{p}}{\\frac {\\partial T}{\\partial x}}", "12a1b3daf4f9ca64e9ba471d98626bc0": "T\\delta (t)=\\sum _{n=-\\infty }^{\\infty }m_{n}e^{int}", "12a1bdc5b304c80049c79fab077ab3d0": "B[{\\vec {X}}]_{ab}={{}^{\\star }R}_{ambn}\\,X^{m}\\,X^{n}", "12a20d1603863de6503ddd1777c01212": "\\lVert z\\rVert \\neq 0", "12a221a2818cdec906d60bf80b95f925": "k\\equiv \\left({\\frac {\\partial Z}{\\partial p}}\\right){\\frac {\\partial }{\\partial p}}\\ln \\theta ", "12a2708a6820b1e7b153a8cec8842735": "\\Delta ={\\frac {\\partial ^{2}}{\\partial x_{1}^{2}}}+\\cdots +{\\frac {\\partial ^{2}}{\\partial x_{n}^{2}}}.", "12a321efe5c0ca06395e54fba3545253": "\\|\\beta \\|_{0}", "12a32b6f4b8b0b80a54ec0f9b1da5b0e": "\\Theta =\\Theta _{\\mu }^{a}dx^{\\mu }\\otimes \\vartheta _{a}", "12a369bca305ce8f34204337129c91d5": "P_{a}=P_{0}a^{D/2-1}\\,", "12a3d2b96d5ab8447ee9b9bcdb38af8a": "\\scriptstyle p_{i}", "12a3de3d0a6b9f00973ae83d9f62dfd4": "[A]={\\frac {[A]_{0}+[B]_{0}}{1+{\\frac {[B]_{0}}{[A]_{0}}}e^{([A]_{0}+[B]_{0})kt}}}", "12a42198be200a3a1aee1299f262bc07": "{\\boldsymbol {\\alpha }}=(1,0,\\dots ,0),", "12a43195bb0211d4707eb2df67ce55fe": "{\\rho ~({\\dot {e}}-T~{\\dot {\\eta }})-{\\boldsymbol {\\sigma }}:{\\boldsymbol {\\nabla }}\\mathbf {v} \\leq -{\\cfrac {\\mathbf {q} \\cdot {\\boldsymbol {\\nabla }}T}{T}}.}", "12a470bdb6c72e4187daf15d3a379d99": "{\\frac {t_{r}}{t}}={\\sqrt {1-{\\frac {r_{s}}{r}}}}", "12a47f4e060848cb0c773c81fa537299": "k^{2}E[u_{1}^{2}]+E[x_{2}^{2}]", "12a4b07c5c6410c832dfe6665ec17b24": "t_{m+n}=t_{m}t_{n}+Du_{m}u_{n},\\quad u_{m+n}=t_{m}u_{n}+t_{n}u_{m}\\quad {\\mbox{and}}\\quad t_{n}^{2}-Du_{n}^{2}=N^{n}", "12a51479f6ab7bd981c1142867ef2765": "c^{1}", "12a552b131692d34a36d4127410c4cd9": "\\Omega ^{0}(M,V)\\otimes _{\\Omega ^{0}(M)}\\Omega ^{p}(M)=(V\\otimes _{\\mathbb {R} }\\Omega ^{0}(M))\\otimes _{\\Omega ^{0}(M)}\\Omega ^{p}(M)=V\\otimes _{\\mathbb {R} }(\\Omega ^{0}(M)\\otimes _{\\Omega ^{0}(M)}\\Omega ^{p}(M))=V\\otimes _{\\mathbb {R} }\\Omega ^{p}(M).", "12a5aa756afb5ef3c78826e031a261c6": "{\\mathbf {X}}({\\mathbf {u}})+t{\\mathbf {A}}({\\mathbf {u}})={\\mathbf {X}}({\\mathbf {u}}+d{\\mathbf {u}})+t{\\mathbf {A}}({\\mathbf {u}}+d{\\mathbf {u}})", "12a64ebb1365899db65c0e4c4e1109a1": "\\tau _{H}", "12a6651c5dface30d2dc51df9c9ec7db": "a,b,c", "12a6eb8ec8c9a5eadbc0ff06b945661c": "{\\begin{aligned}(\\gamma a_{1})(\\gamma a_{2})\\dots (\\gamma a_{m})&=\\gamma ^{\\frac {\\mathrm {N} {\\mathfrak {p}}-1}{n}}a_{1}a_{2}\\dots a_{m}\\\\&\\equiv \\left({\\frac {\\gamma }{\\mathfrak {p}}}\\right)_{n}a_{1}a_{2}\\dots a_{m}{\\pmod {\\mathfrak {p}}},\\end{aligned}}", "12a6f951e73b362e1d2821d112d61d4e": "\\tau _{1}\\ ", "12a708077f12a14041703e40d277c12f": "={1 \\over W}{\\begin{pmatrix}u_{2}'(x)&-u_{2}(x)\\\\-u_{1}'(x)&u_{1}(x)\\end{pmatrix}}{\\begin{pmatrix}0\\\\f\\end{pmatrix}},", "12a72d9ca6bf8ca954b8505d2f8698ae": "c_{1}:\\mathrm {Pic} (X)\\to H^{2}(X,\\mathbb {Z} ).", "12a759005c96067536fecbdf90d9eefc": "K({\\sqrt {d}})", "12a816f4089f8b075e6d461f627f11e2": "x=c\\varphi ,\\ s=\\ln \\tan {\\tfrac {1}{4}}(\\pi +2\\varphi ).\\,", "12a8262222e759f3bcec22b6a05ac3b9": "g(T)v=0", "12a87db75c2d2db107a04c20062ec14b": "s=r\\theta .", "12a8c4f7411c7748b54c230efb166e7b": "\\langle K,\\prec _{K}\\rangle ", "12a8d229ad0ea9f8fdb27d3e5f39c193": "x\\neq \\omega ", "12a9378f2a8d70b0d7d3f185cdd7cf46": "{}_{j\\in {1,2,3}}", "12a966fc3df93e6f1e118da677d74e99": "\\sum _{k=0}^{\\infty }ar^{2k+1}={\\frac {ar}{1-r^{2}}}", "12a9b0a75fd3f74872afddcb626fd436": "\\|X-\\mu \\|_{\\alpha }\\geq k\\sigma _{\\alpha }^{2}", "12a9b5731d3932fdb0a1f12964b2032d": "\\Delta G=\\epsilon _{surfaceatom}-\\epsilon _{adatom}\\qquad (3)", "12aa72cea82cb412ecf50f34283a0c6b": "\\operatorname {cont} _{F}:X\\rightarrow Z", "12aac7e42e7e58d3076235a37300aac7": "(\\gamma =1.4)", "12aaf723354ec1b39f422d06de4c31d1": "(M,L)", "12ab7083e6ae070d8d9438a89631e6eb": "\\sigma ^{2}\\left[1+{\\frac {\\alpha \\phi (\\alpha )-\\beta \\phi (\\beta )}{Z}}-\\left({\\frac {\\phi (\\alpha )-\\phi (\\beta )}{Z}}\\right)^{2}\\right]", "12ab9e63af84b4eb593bb95c514554a6": "{\\frac {\\partial r_{2}}{\\partial t}}+(u-{\\sqrt {\\rho }}){\\frac {\\partial r_{2}}{\\partial x}}=0", "12abc05a54c243e633e3f791fd8d5559": "{\\hat {x}}", "12abe5668eda8e95686ff5a9ff43bcec": "W=i\\theta ^{j}\\mathbf {e} _{j}+\\eta ^{j}\\mathbf {e} _{j},", "12ac0bd4846b6f11fef6826d03b35037": "\\scriptstyle V_{\\mathrm {r} }", "12ac5b96d7e2791264bb4eec4bf2d154": "\\Delta h_{ab}", "12ac71598a569e32691dca97dc431372": "{R_{x}}=1000{\\frac {R^{*}}{M_{x}}}\\,,", "12ac959723e228ae1db6756a33a731be": "\\pi (\\mathrm {E} \\lambda )=\\pi _{\\mathrm {i} }\\pi _{\\mathrm {f} }=(-1)^{\\lambda }\\,", "12aca7d6645830510fc98827099b6395": "f(n)\\leq Cg(Cn+C)+Cn+C", "12acce3b528f43bd40f99793a46b3e06": "C_{lk}", "12ad3df34778052eb6749f0d25faf230": "{\\frac {E}{I}}=Z_{sc}", "12ad55e6e897c11bf1259b6dca16e5f0": "O(n^{2}/(\\log _{2}n)^{\\log _{2}{\\frac {8}{3}}})", "12ad5ff40904a6eb547cee28579be98e": "P=(x^{2}+bx+1)(x^{2}-bx+1).", "12ada8d9797e6fdc1708834cde6139a4": "{\\frac {{\\dot {m}}_{0}}{{\\dot {m}}_{01}}}={\\frac {\\epsilon _{0}}{\\epsilon _{01}}}={\\frac {p_{01}}{p_{0}}}", "12add0ec70859801543f9e1ba97e9840": "\\scriptstyle {\\beta }\\,\\!", "12ae08c812c085c4b53741770231b64c": "G'=\\{*n_{1},*n_{2},\\ldots ,*n_{k}\\}", "12ae0919ab7550811cc121cc4f775965": "H_{1}\\oplus H_{2},", "12ae3b0c6d8a027927a14e470fca74da": "E_{r}(m)=y^{m}u^{r}\\mod n", "12ae637c93e7554e4fa72cb44b296255": "a\\succeq b\\;", "12ae6fa2857922529d49338005d6883e": "-2\\log(\\Lambda )", "12ae78abff2d05e11b92d824bcfaaaa7": "y_{i}=\\beta _{0}+\\beta _{1}x_{i}+\\beta _{2}x_{i}^{2}+\\cdots +\\beta _{p}x_{i}^{p},", "12ae8713f54cd71b3e51e5713792408b": "{\\frac {\\pi }{2\\log(1+{\\sqrt {2}})}}", "12aea7af5c39da7b1b97830eb5f73c80": "-\\hbar ", "12aeca6f151fa63be6365ff4b77c5cef": "s^{-1}", "12af61664176d2f4cc1c32b28f591b53": "\\textstyle {\\frac {1}{2^{n}}}", "12afbb568b98e61f251766e6a00b966f": "k\\in \\{1,..,m\\}", "12afbf166db5e42dba63d6ebeb3d8cb2": "\\Delta K=K\\otimes K", "12aff02f649bf9e309023654620e3f77": "z_{1},\\ldots ,z_{n}", "12b016b79bf0c182438966edb0577263": "I_{1}={\\frac {V_{1}}{|Z_{total}|}}\\angle (-\\theta )", "12b048cefb0fcb2ca8aa36df5734b4fb": "gx=y", "12b05c41f7a1463f8b1eb9c4e0447ba8": "\\delta W=p{\\rm {d}}V", "12b08a615453f593e858f645acfe04ed": "{\\frac {\\mathrm {d} }{\\mathrm {d} t}}\\left({\\frac {\\partial L}{\\partial {\\dot {q}}_{j}}}\\right)={\\frac {\\partial L}{\\partial q_{j}}}", "12b0a80feb6c6d42de9618ccea2424c3": "\\displaystyle {W(x_{1},y_{1})W(x_{2},y_{2})=e^{i(x_{1}\\cdot y_{2}-y_{1}\\cdot x_{2})}W(x_{1}+x_{2},y_{1}+y_{2}),}", "12b0c7d1a029e2fb44cccf9bc0f15207": "(i=1,\\ldots ,d)", "12b0c7e2a4bbc48e21fdd8593ec16f55": "k_{1}=3.796866512", "12b0cafaaeaf87d2d182d4e6dbed36b3": "{\\tilde {I}}(\\omega )={\\frac {K}{2\\pi c}}\\left({\\frac {\\omega _{max}}{\\omega }}-1\\right)", "12b0f5f9deec753718bc0b7afb818521": "I_{v}", "12b13fe271fc463930eab38a2efa88ae": "{\\frac {1}{\\sqrt {\\lambda }}}=-2\\log(({\\frac {\\varepsilon }{3.715D}})+({\\frac {6.943}{Re}})^{0.9}))", "12b1587adf08c65c0511e068c19061ad": "z^{E}=z-\\sum _{i}x_{i}z_{i}^{id}.", "12b1be9bd1b2bbe833dc250bf2f43966": "10{\\sqrt {\\ell /g}}", "12b1dfc0528e77eb301ad31b9bcb08b0": "\\Delta \\nu ", "12b1e0726ac1678cc3029ef8a83b6931": "a_{3,1}x_{1}+a_{3,2}x_{2}=b_{3}", "12b1f4ba736cfa59c1849eda9000f37b": "f^{-1}(U)\\cap X", "12b1f4eb936ea0857b8222e3f347292b": "\\left(\\!\\!{\\ \\choose \\ }\\!\\!\\right)", "12b20aa839be216d65d6c9bce2c581ea": "\\iint _{D}5\\ dx\\,dy", "12b2696c2b28b0068d8c41e62f04176a": "w\\rightarrow {\\overline {w}}^{3}+w_{0}", "12b26e4c1c31b3b76558de6bd66702f9": "s\\mapsto s(\\pi )/\\pi ", "12b30099bd9645905cc065dc7132dcb2": "\\int L(a-\\theta )f(x-\\theta )d\\theta ", "12b36da2f312d7c2499bcf31efc4ebcd": "{\\hat {p}}_{0}\\leftarrow {\\hat {r}}_{0}\\,", "12b3f9b15dec4c28fcdc2bbe3da32007": "S={\\frac {h}{2\\pi }}\\,{\\sqrt {s(s+1)}}={\\frac {h}{4\\pi }}\\,{\\sqrt {n(n+2)}},", "12b41d9247a36a3920fbea7cc586ded0": "\\tan {(\\theta )}\\approx \\mu _{\\mathrm {s} }\\,", "12b451a278920c624e8bfa3974345fa1": "p=x^{2}+y^{2}", "12b45995a97b38006221633f74505dba": "\\Omega \\subseteq \\mathbb {R} ^{n}", "12b4a61685792b17d3859fad5a74e30c": "{\\mbox{Dic}}_{n}<{\\mbox{Pin}}_{-}(2)", "12b4af413cfb7703002f8bbd3b0b5d6a": "{\\begin{bmatrix}\\sigma _{11}\\\\\\sigma _{22}\\\\\\sigma _{33}\\\\\\sigma _{23}\\\\\\sigma _{31}\\\\\\sigma _{12}\\end{bmatrix}}={\\begin{bmatrix}2\\mu +\\lambda &\\lambda &\\lambda &0&0&0\\\\\\lambda &2\\mu +\\lambda &\\lambda &0&0&0\\\\\\lambda &\\lambda &2\\mu +\\lambda &0&0&0\\\\0&0&0&\\mu &0&0\\\\0&0&0&0&\\mu &0\\\\0&0&0&0&0&\\mu \\end{bmatrix}}{\\begin{bmatrix}\\varepsilon _{11}\\\\\\varepsilon _{22}\\\\\\varepsilon _{33}\\\\2\\varepsilon _{23}\\\\2\\varepsilon _{31}\\\\2\\varepsilon _{12}\\end{bmatrix}}", "12b4c2b43e16af79101d4796af888bef": "{\\mathcal {L}}_{t}\\{f(t)\\}(s)=F(s),\\ \\forall s\\in \\mathbb {R} ", "12b504c59476364d7e750ab7d7eebbff": "I_{D}=\\mathbf {L} \\cdot \\mathbf {N} CI_{L}", "12b50e1ac0a5d717e6bb02eafc9cc559": "{\\scriptscriptstyle {\\sqrt[{r}]{a/b}}}", "12b53fb340aa69cb9a729c682d6b202e": "{\\partial f \\over \\partial r}={1 \\over ir}{\\partial f \\over \\partial \\theta }.", "12b5a8ea98b8ab3e6d0dad02316c1b24": "p_{\\textrm {kin}}=p-{\\frac {qA}{c}}\\,\\!", "12b5d858cd06c9470b47c6a3f0dc416e": "\\Phi (s)={\\overline {\\Phi (1-{\\overline {s}})}};", "12b5f7617c346ed49011aa316d494e97": "-{\\boldsymbol {\\nabla }}p=\\rho _{f}{\\frac {{\\text{D}}{\\boldsymbol {u}}_{f}}{{\\text{D}}t}}-\\mu {\\boldsymbol {\\nabla }}\\!\\cdot \\!{\\boldsymbol {\\nabla }}{\\boldsymbol {u}}_{f},", "12b60ac23e59a36f0ee3e9559665ee9b": "\\operatorname {I} (S)=\\{f\\in k[x_{1},x_{2},\\ldots x_{n}]\\ |\\ f(x)=0{\\mbox{ for all }}x\\in S\\}.", "12b63b128dceabbb46222e747274a07d": "L\\!", "12b672be98c714f5bfea96f72e68b3f2": "[x,y]v=x(yv)-y(xv)", "12b6a8e5116dcb1e7bd0773297540bab": "H({\\vec {r}},t)", "12b6c396ed5335c16c9931566e01e01c": "{\\begin{aligned}G_{\\alpha \\beta }&=R_{\\alpha \\beta }-{\\frac {1}{2}}g_{\\alpha \\beta }R\\\\&=R_{\\alpha \\beta }-{\\frac {1}{2}}g_{\\alpha \\beta }g^{\\gamma \\zeta }R_{\\gamma \\zeta }\\\\&=(\\delta _{\\alpha }^{\\gamma }\\delta _{\\beta }^{\\zeta }-{\\frac {1}{2}}g_{\\alpha \\beta }g^{\\gamma \\zeta })R_{\\gamma \\zeta }\\\\&=(\\delta _{\\alpha }^{\\gamma }\\delta _{\\beta }^{\\zeta }-{\\frac {1}{2}}g_{\\alpha \\beta }g^{\\gamma \\zeta })(\\Gamma _{\\gamma \\zeta ,\\epsilon }^{\\epsilon }-\\Gamma _{\\gamma \\epsilon ,\\zeta }^{\\epsilon }+\\Gamma _{\\epsilon \\sigma }^{\\epsilon }\\Gamma _{\\gamma \\zeta }^{\\sigma }-\\Gamma _{\\zeta \\sigma }^{\\epsilon }\\Gamma _{\\epsilon \\gamma }^{\\sigma }),\\end{aligned}}", "12b6e5b1cf0b9416d45705209e7aa455": "(1+s{\\text{Ad}}\\beta )P(t^{p}+t^{p-1}s+\\cdots +ts^{p-1})P(s^{p})", "12b7c38ef6061bf664b5a90fa79b9cee": "F:=\\int d^{D}x\\ \\left(a(T)+r(T)\\psi ^{2}(x)+s(T)\\psi ^{4}(x)\\ +f(T)(\\nabla \\psi (x))^{2}\\ +h(x)\\psi (x)\\ \\ +{\\mathcal {O}}(\\psi ^{6};(\\nabla \\psi )^{4})\\right)", "12b7f2f3eaf91f10c1e1b83df4a34f3f": "W^{\\ast }(s)={\\frac {(1-\\rho )sg(s)}{s-\\lambda (1-g(s))}}", "12b8a17d82f5e38488446cd9ba19ded4": "V_{x},V_{y}", "12b8b41ff058e83ee14c54b1fde37b9a": "\\Lambda (x,\\lambda )=x^{2}+\\lambda (x^{2}-1).", "12b8cb61a9c477d612f909da15d0ae07": "Z'_{k}", "12b8cb811367cf01d77e980ef4b7f0f1": "m+0.5p(q+|q-n|)", "12b8cd801b7f6604560c36433cfd065b": "\\displaystyle E(k,\\phi )=\\int _{0}^{\\phi }{\\sqrt {1-k^{2}\\sin ^{2}\\theta }}d\\theta ,{\\text{ for }}\\left|k\\right|\\leq 1", "12b8cfd5604adf510aef104e8ed08d71": "Pr_{L}", "12b8d73616c0ba0b1111564bb04d97c8": "L_{4k},", "12b8eea5eb1be4eae3e9ef1ad4e1f335": "\\,\\!q(\\tau )=q_{t}(\\tau )+q_{s}(\\tau ).", "12b911ece928743b7fe21ed2d9c9c13e": "\\operatorname {rank} ({\\widehat {D}})\\leq r\\quad \\iff \\quad {\\text{there is full row rank }}R\\in \\mathbb {R} ^{m-r\\times m}{\\text{ such that }}R{\\widehat {D}}=0", "12b91c3bd8e5b26d60407a6ac90a989f": "\\mathbf {E} (Y_{n+1}\\mid X_{1},\\ldots ,X_{n})=Y_{n}.", "12b9448e1273187047ea33cf8d1d27f1": "\\Lambda =0\\;", "12b96dca22ddf063dd5f998ccaa1b46b": "(F,m):({\\mathcal {C}},\\otimes ,I)\\to ({\\mathcal {D}},\\bullet ,J)", "12b9c111bb3ca4ca068a8af30b749dc8": "{\\frac {e}{\\sqrt {1-e^{2}}}}", "12b9cea058dec45e846033fcab119d55": "\\varepsilon _{ijk}\\varepsilon ^{imn}\\equiv \\sum _{i=1,2,3}\\varepsilon _{ijk}\\varepsilon ^{imn}", "12bb10bb09648797e41e349f47192853": "x^{3}-x-1,", "12bb779c25d4905394e26eceeb8181f3": "m^{2}+(a-1)m+b=0.\\,", "12bbb95e280dac1509f437191f1229c1": "Y", "12bbe31ba35d7bd3b225be626a2e8920": "H(z,f)=\\limsup _{r\\to 0}{\\frac {\\max _{|h|=r}|f(z+h)-f(z)|}{\\min _{|h|=r}|f(z+h)-f(z)|}}", "12bbf175cba3329ae6cea2f3b338019a": "(\\land ,\\lor )", "12bc20ee215a9e0e2d31f232d33439a0": "\\pi ={\\begin{pmatrix}0.885&0.071&0.044\\end{pmatrix}}.", "12bc3eb8a56d48ca26deac0ca96a629d": "X^{n}\\left(i\\right)", "12bc96995257c8a723ac0ed50eb06db6": "n\\$=n!^{(4)}n!\\,", "12bca1efb56e3394e1589c9ea2828b7e": "\\mu (gx)=\\mu (x).\\,", "12bcde8f3a659bac809109dd290bce00": "d=d_{\\mathrm {0} }+D\\cos({\\boldsymbol {\\omega }}t+{\\boldsymbol {\\varphi }})\\;", "12bd2ad68ee2f5396d3f4f6d6dbc81b1": "\\omega _{lab}=\\omega _{0}\\left(1\\pm {\\frac {v}{c}}\\right),", "12bd40062aa66bc122b6e84d6784c530": "{\\frac {n}{n+1}}", "12bd9ec91bb43b70654fdd95c0b75228": "A=87.7", "12bdb956bf19b9a7d0bd9024cd2a39f8": "dim_{\\mathbb {C}}M=2", "12bdc80720708ecac900f80afeefb5ec": "\\{(x,1)\\mid x\\in [0,1]\\}", "12bdf1abae9f8c43f5d25ba2c604dfb6": "U-XE", "12be3568f4ded29150c39b4425f69774": "T:\\mathbf {M} \\to \\mathbf {A} ", "12be6c7b1ca2a6f21553dda9c2367f6a": "I(t)=|E(t)|^{2}", "12beacc95a9c786a8e4aedfbdfb10a60": "R''=w^{e}*R'", "12beb2d82a58c794a4d8f92099b94cbc": "{\\mathfrak {P}}^{82}", "12bee37145febaa5d96781f1a7515298": "\\omega _{1}^{2},\\omega _{2}^{2},\\cdots \\omega _{N}^{2}", "12bf91fb53556a4ab09128332f93e500": "\\mathbf {b} _{1},\\mathbf {b} _{2},\\dots ,\\mathbf {b} _{n}", "12bfd56fc9de4bb573aaa882be8a426f": "J_{3}\\,\\!", "12bfd7162403b1d2a02b8587ce79b253": "M(p)=\\exp \\left({\\frac {1}{(2\\pi )^{n}}}\\int _{0}^{2\\pi }\\int _{0}^{2\\pi }\\cdots \\int _{0}^{2\\pi }\\ln {\\Bigl (}{\\bigl |}p(e^{i\\theta _{1}},e^{i\\theta _{2}},\\ldots ,e^{i\\theta _{n}}){\\bigr |}{\\Bigr )}\\,d\\theta _{1}\\,d\\theta _{2}\\cdots d\\theta _{n}\\right).", "12c0b16175a1f25afebfc5d195d366bf": "5(x-1)\\left(x+{\\frac {1+i{\\sqrt {3}}}{2}}\\right)\\left(x+{\\frac {1-i{\\sqrt {3}}}{2}}\\right)", "12c12c6467be80650810898f91a4ae5f": "\\nu (M)\\,", "12c1974296ca8b37c4c5377ecfd88c94": "dW_{t}=0", "12c1c5bc7e48b74a425ca2fcdd28238a": "{\\overline {S}}_{i}", "12c23528a899cb3ac176b76655a1f184": "1+{\\sqrt {2}}", "12c2720d8493b190fa4e41919e992527": "\\{s_{m}-s_{n}:m,n\\in \\mathbb {N} ,n1", "12cba6f59541487695f7d2c06fb7e60a": "(x+5)(x-2)=0.\\,", "12cbb1e0ff00ce62ce8afe2614ae974a": "\\|\\gamma '(t)\\|=1{\\mbox{ }}(t\\in [a,b])", "12cbc1b787322f173fa108dd979cecff": "c=(1,2)", "12cbc33e5829dce2e2b8ef870b446829": "(a+b)^{n+m-1}=\\sum _{i=0}^{n+m-1}{n+m-1 \\choose i}a^{i}b^{n+m-1-i}.", "12cbf63543c8de9ffae334c3d90bf626": "0\\leq ma),", "12cf746417128179be3623825057202f": "\\bigcup _{i\\in I}V_{i}=V", "12cf959e944778eb96642e9aaf58718a": "\\lambda _{\\alpha _{l}}^{[l]}", "12cfe3b552b0704e72cf03bd74b9ad06": "\\ v_{1}", "12d00559154de4707253b83f932533a4": "{\\vec {d}}={\\vec {d}}_{\\text{eg}}|{\\text{e}}\\rangle \\langle {\\text{g}}|+{\\vec {d}}_{\\text{eg}}^{*}|{\\text{g}}\\rangle \\langle {\\text{e}}|", "12d018653b48dffddf983f277f479980": "{\\mathfrak {P}}^{62}", "12d049c9dab4fbdbe631408a4f12c819": "\\Delta q\\,\\!", "12d05fbc8355d50f9d368ca179fee312": "\\left({\\frac {m}{3}}+M\\right){\\ddot {\\bar {x}}}=-k{\\bar {x}}", "12d119262c7d12350b3d5d250c1acace": "P(E)", "12d15fd3f667cf27ad772053b7f9bb61": "\\eta ^{2}+3\\eta +2", "12d1b8d0f7b34da18a98f8719652e363": "{\\overline {S}}", "12d1e9f979bc038f2b1f67c614b7c77d": "{(e_{i})}_{i\\in I}=((\\delta _{ij})_{j\\in I})_{i\\in I}", "12d1f7f456152efbd318e790d068f8c8": "\\scriptstyle dx\\int {dx\\int {Vdx}}", "12d21cc90bf405d6b9283f2b74d112ea": "\\displaystyle R^{+}", "12d30f0e910b7cdd8686f952dffdff9b": "f(x_{1},x_{2},\\ldots ,x_{n}|\\theta )P(\\theta )", "12d31caefcac9be96c3ed4f566c9b38a": "{\\begin{matrix}G(n-1)&=&p^{-1}p^{n}+&q^{-1}q^{n}+&r^{-1}r^{n}\\\\G(n)&=&p^{n}+&q^{n}+&r^{n}\\\\G(n+1)&=&pp^{n}+&qq^{n}+&rr^{n}\\end{matrix}}", "12d35494f5be1c989aa15102988e2ac8": "\\displaystyle T={\\frac {a^{2}+b^{2}+c^{2}}{4{\\sqrt {3}}}}\\quad {\\text{(Weizenbock)}}", "12d3c78e7f224919086c1e2de9b6ad1e": "{f_{qk}}^{m_{qk}}", "12d3cfa1e8fae898e02015336cc7ed51": "\\gamma \\gamma ^{*}=\\gamma ^{*}\\gamma ,", "12d3d52072cefeefb584c02336ab569b": "{\\frac {X_{1}+\\dots +X_{n}}{\\sqrt {n}}}", "12d402d4445f06d63587bdbc2f5b182f": "{\\mathfrak {m}}_{p}^{k+1}", "12d420ddbbed595350db3a7a8006c9ff": "P(\\Delta R)=p(\\Delta X)p(\\Delta Y)p(\\Delta Z)={\\frac {1}{\\sqrt {(2\\pi )^{3N}|{\\frac {k_{B}T}{\\gamma }}\\Gamma ^{-1}|^{3}}}}exp\\left\\{-{\\frac {3}{2}}\\left(\\Delta X^{T}\\left({\\frac {k_{B}T}{\\gamma }}\\Gamma ^{-1}\\right)^{-1}\\Delta X\\right)\\right\\}", "12d4a11b43413deccabd9d8ced763ef0": "U_{2}=R_{2}I", "12d4b4948e84bc8d319e962531ce0802": "{\\textbf {x}}_{2}", "12d4f613bb8743b635ebae050c82dee6": "{\\frac {(p_{1}+q_{1}-a)(p_{2}+q_{2}-c)}{(p_{1}+q_{1}+a)(p_{2}+q_{2}+c)}}={\\frac {(p_{2}+q_{1}-b)(p_{1}+q_{2}-d)}{(p_{2}+q_{1}+b)(p_{1}+q_{2}+d)}}", "12d5224355216b72793cbc747cc10e00": "a=2,\\,b=2,\\,c=1,\\,f(n)=10n", "12d53427a9f8da8e48f0f6fc1e9b20c8": "\\sin a=\\tan(\\pi /2{-}B)\\,\\tan b=\\cos(\\pi /2{-}c)\\,\\cos(\\pi /2{-}A)=\\cot B\\,\\tan b=\\sin c\\,\\sin A.", "12d57689b9da7393795fb7c3bd3fb260": "Y_{4}^{-1}(\\theta ,\\varphi )={3 \\over 8}{\\sqrt {5 \\over \\pi }}\\cdot e^{-i\\varphi }\\cdot \\sin \\theta \\cdot (7\\cos ^{3}\\theta -3\\cos \\theta )={\\frac {3}{8}}{\\sqrt {\\frac {5}{\\pi }}}\\cdot {\\frac {(x-iy)\\cdot z\\cdot (7z^{2}-3r^{2})}{r^{4}}}", "12d58b3c1e3be412397ce706eec58187": "q_{0}\\leq 0", "12d5b8f557eb5e3a965dd3c02f98cbf6": "M(t;s)=E(e^{tX})={\\frac {1}{\\zeta (s)}}\\sum _{k=1}^{\\infty }{\\frac {e^{tk}}{k^{s}}}.", "12d5def95a4093ca6d6dbe53fa67687c": "{\\mathcal {L}}=\\{\\lambda \\in \\mathbb {C} \\,:\\,e^{\\lambda }\\in {\\overline {\\mathbb {Q} }}\\}.", "12d5eef93f7ebd4e4fdfb37121a19705": "\\mathrm {ID} =(n-1)am^{b-1}", "12d61c7987dfe01a11cb402b838666f7": "\\inf \\theta \\leq 7/22.", "12d6c196f8848a8c536eec200f568b7c": "-0.0905\\leq m\\leq 2", "12d6cc2a716bba1f2f74b191f8436a63": "y_{i}\\ell _{i}(x_{i})=y_{i}", "12d6e7bb8f03bbd9223c3de6ea99611c": "\\varphi =\\operatorname {atan2} (y,x)\\quad ", "12d6f502f93f5aa1640f255a5e0fc5d6": "+\\ln \\Gamma _{p}\\left(-{\\Big (}\\eta _{2}+{\\frac {p+1}{2}}{\\Big )}\\right)=", "12d73e9376972e5f053f0e666b9ce193": "f(k;m,r,p)={\\frac {{k+r-1 \\choose k}p^{k}}{(1-p)^{-r}-\\sum _{j=0}^{m-1}{j+r-1 \\choose j}p^{j}}}\\quad {\\text{for }}k\\in {\\mathbb {N} }{\\text{ with }}k\\geq m,", "12d7c74e91f24ee4870480bcf601033a": "\\cup _{\\xi }(\\xi +C_{\\xi })", "12d8065aff3cc649a610c26364257d1b": "7x^{2}y^{3}+4x^{1}y^{0}-9x^{0}y^{0}", "12d815ebe36b7ca49c8a5e1162fa20e6": "-mg(l+a)\\limsup _{z\\to \\zeta ,z\\in U}v(z).", "12dc8dc484ebd95d9dbf99ae2e8d8195": "{\\textstyle \\alpha ^{2}}", "12dc8dee8ac8f2d13df4966e760fb8f8": "q_{1}", "12dcb4179e40d7bf9cda1df436787e88": "\\varphi (t)=|t|^{p}", "12dcb60e61d37830185efb04e20b0cec": "m=3", "12dcccc0c4c40664a7d09962a8d0dedf": "K_{n}x^{n}+K_{n-1}x^{n-1}+\\cdots +K_{1}x+K_{0}\\!", "12dd17ecfd7f0fc3e13e33a6f13e4b97": "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 }}0<\\arg z<\\pi ", "12dd47e1542dbfcbeb676fa9bfdacf40": "\\theta _{E}={\\sqrt {{\\frac {4GM}{c^{2}}}{\\frac {d_{S}-d_{L}}{d_{S}d_{L}}}}}", "12ddbd83f5eb39d90d8006c75801341f": "((a,b)\\downarrow F)", "12de7673992b1735e29cdd211851fa05": "a\\,\\!", "12de9a3db97b28001828569f4376c1d1": "\\operatorname {I} (x)=\\{f\\in \\operatorname {C} (X):f(x)=0\\}.", "12deb437aef5219835c08ea920fd072f": "L\\cap M", "12ded7db537863e87ed65d5764d663ec": "R_{z}R_{y}R_{z}(x)=R_{zyz}(x)", "12defe87bf1ca924857fe2dc72574735": "w_{i}={\\frac {n'_{i}}{n_{i}}}=k-az_{i}+bz", "12df2035c45186af8889f458d0e74b33": "\\lim _{z\\to 1^{-}}G_{a}(z)", "12df2b4ddcfd3c66a95bd3632128b8ad": "\\mathrm {E} (w(t_{1})\\cdot w(t_{2}))", "12df43ff9277a98334d388b0e8d9fec4": "P_{i_{1}}\\oplus P_{i_{2}}\\oplus \\cdots \\oplus C_{j_{1}}\\oplus C_{j_{2}}\\oplus \\cdots =K_{k_{1}}\\oplus K_{k_{2}}\\oplus \\cdots ", "12df555a4a55830f5fa74afdc2b1adaf": "[S(a,c)+S(c,b)-S(a,b)]/15\\,", "12dfaf0593af8628c956faa8f43ce9b6": "\\int _{0}^{1}\\sum _{j,k=1}^{q}|H_{j,k}(t)|dt<\\infty ,", "12dfcd386801054fee0516dcee4afad3": "Z_{21}=R_{c}\\qquad Z_{11}=R_{c}+R_{a}\\qquad Z_{22}=R_{c}+R_{b}\\,", "12e03421a5114dc69b8d1f5dc5481d99": "10^{n}", "12e08c5918486ddebdac638153b174af": "r_{k}(n)=|\\{(a_{1},a_{2},\\dots ,a_{k}):n=a_{1}^{2}+a_{2}^{2}+\\cdots +a_{k}^{2}\\}|", "12e09e44f4c0b1f0ced6c36e48e86687": "H_{0}=1{\\mbox{ and }}P_{0}=0\\,", "12e0ec62de27894b7f6e0b7bfb6f44a3": "h_{0}-h={\\frac {Q}{2\\pi T}}\\ln \\left({\\frac {r}{R}}\\right)", "12e13d157bbadd050990bdc63b3a2cb1": "+w={\\frac {Dz}{Dt}}", "12e16e4a20d9e78115f974d5df9619f5": "\\rho =n_{2}/n_{1}", "12e199b82cc15236dd0fd7c0b3f8230d": "y^{p}-y=f(x),", "12e1c57c5cc2913e702f823692893b6b": "\\eta (\\phi )", "12e208274a580aa422f9b2ed612800e7": "(S,\\sigma )", "12e210132dceee320960d402cabfbade": "H_{p}=\\ker \\alpha _{p}\\subset T_{p}M.", "12e2120355c00c91d80322b211d9aeb9": "Demand_{p,c}+Backlog_{p-1,c}>0", "12e21dc7f9e40df41ebc118bb473d09d": "1-{\\text{e}}^{-2}\\approx 0.8647", "12e2316629a83f4fbcbbbcf04beefcc9": "{\\epsilon _{F}}_{n}", "12e2684254a086311455cbf71bf08a94": "Y^{\\prime }[\\sigma f]\\to Y^{\\prime }[\\sigma ]", "12e2c88e6900f70ae80118e68dc0b4a2": "\\beta (g)=-\\left({\\tfrac {11}{3}}C_{2}(G)-{\\frac {4}{3}}n_{f}C(R)\\right){\\frac {g^{3}}{16\\pi ^{2}}}~,", "12e2d2e74b4da04b62457157494e4794": "w^{f}(f^{*})=\\sum _{e\\in E}f_{e}^{*}(a_{e}\\cdot f_{e}+b_{e})", "12e317dd0f200b6cf3fcd9f1b4426338": "F(\\sigma _{22}-\\sigma _{33})^{2}+G(\\sigma _{33}-\\sigma _{11})^{2}+H(\\sigma _{11}-\\sigma _{22})^{2}+2L\\sigma _{23}^{2}+2M\\sigma _{31}^{2}+2N\\sigma _{12}^{2}+K(\\sigma _{11}+\\sigma _{22}+\\sigma _{33})^{2}=1~.", "12e3701452a420354f5951800b335f1a": "V_{o}", "12e43c0c1847302d6cb841d015c5e7ad": "\\int _{0}^{\\infty }\\cos(2x)\\prod _{n=1}^{\\infty }\\cos \\left({\\frac {x}{n}}\\right)dx\\approx {\\frac {\\pi }{8}}.", "12e4a8c1eea7241a8a6a682eec8b01a5": "\\phi _{va}(r)={\\frac {2r}{1+r^{2}}}\\ ", "12e4f3077c67b562c78b7ad38d95eb65": "V-A=\\{x\\mid x\\not \\in A\\}", "12e50ebc69c490be810bb85a286c598e": "t={\\cfrac {1}{1-{\\cfrac {m\\lambda }{i\\hbar ^{2}k}}}}\\,\\!", "12e5c3f9519de2d2b2515496393afe74": "location_{i}", "12e6547e74dc186eb07a1bc666b6d315": "\\,\\!d(f,g)=|f(x_{0})-g(x_{0})|\\;", "12e6c830f554b4f544d3c71edb1b381a": "w/h>3.3", "12e70da219e570ef638a7cbbdf6fa279": "\\mathbf {A} \\oplus \\mathbf {B} ={\\begin{bmatrix}a_{11}&\\cdots &a_{1n}&0&\\cdots &0\\\\\\vdots &\\cdots &\\vdots &\\vdots &\\cdots &\\vdots \\\\a_{m1}&\\cdots &a_{mn}&0&\\cdots &0\\\\0&\\cdots &0&b_{11}&\\cdots &b_{1q}\\\\\\vdots &\\cdots &\\vdots &\\vdots &\\cdots &\\vdots \\\\0&\\cdots &0&b_{p1}&\\cdots &b_{pq}\\end{bmatrix}}.", "12e74ceffdbabef1dc50642de5997544": "\\varepsilon _{n}\\in \\{-1,+1\\}", "12e78a9a1c063e655f5780d2014e2543": "t\\mapsto S\\,{\\mbox{diag}}(e^{it\\theta _{1}},\\dots ,e^{it\\theta _{n}})\\,S^{-1}.", "12e7f8d87ce2beff61056303f2f2739e": "D(t+1)=|V(x,y,t+1)-V(x,y,t)|", "12e81850af31e2928120bb40138ac4ab": "\\scriptstyle {\\vec {l}}=(l_{x},l_{y})", "12e821270e647e92be6ff93be4ee05a1": "f(n)=\\sum _{i=1}^{k}\\left\\lfloor {\\frac {n}{5^{i}}}\\right\\rfloor =\\left\\lfloor {\\frac {n}{5}}\\right\\rfloor +\\left\\lfloor {\\frac {n}{5^{2}}}\\right\\rfloor +\\left\\lfloor {\\frac {n}{5^{3}}}\\right\\rfloor +\\cdots +\\left\\lfloor {\\frac {n}{5^{k}}}\\right\\rfloor ,\\,", "12e8414a8bccc4c392ddd07cb3783d50": "x'={\\frac {ac(x^{2}+y^{2})+x(ad+bc)+bd}{|cz+d|^{2}}}", "12e84b15e7781c853764804251a11a15": "(M,g)", "12e859b049da9f0f16ce90703552e392": "MRP_{L}=MPP_{L}*P", "12e8816bc42772d56cbf6f39842880ed": "e_{1}(\\tau )={\\tfrac {1}{3}}\\pi ^{2}(\\vartheta ^{4}(0;\\tau )+\\vartheta _{01}^{4}(0;\\tau )),", "12e881b909aaac1d06011d7196fb1b41": "{\\frac {(\\log g)^{2}}{\\pi g}},", "12e8db2338dd0c22bc6e8760d0467124": "g'\\in G", "12e902d3bb1a1e5b89e5ec1e515ab140": "\\int _{y_{0}}^{y_{1}}f(x,y)\\,dy", "12e954346c9e336f877391adc7077202": "94906265.625x^{2}-189812534x+94906268.375", "12e960b12f5df5e10c4c76778194a78c": "h(X)=-\\int _{X}f(x)\\log f(x)\\,d\\mu (x),", "12ea069dc94f7fbae7c58b3519ed2283": "|\\mathrm {card} E(K)-(q+1)|\\leq 2{\\sqrt {q}}", "12ea1f6f35632b89274a1306e1110b63": "1.96x^{2}+19.6x", "12ea4c59e107564a215e0caf5f3098bc": "n\\cdot \\delta p", "12ea9f349f523c8e65466889e361f1f7": "{\\frac {\\pi r^{2}}{4}}", "12eb2bc65d74c0d29efc0c590e233d78": "{\\hat {X}}^{n}(z^{n})=\\left({\\hat {X}}_{1}(z^{n}),\\ldots ,{\\hat {X}}_{n}(z^{n})\\right)", "12eb2ef1e21e6656971ea7034aa87b10": "\\lim _{B\\rightarrow x}{\\frac {1}{|B|}}\\int _{B}f\\,\\mathrm {d} \\lambda ,", "12eb651ede12aaa0b4c6a5d089469d8f": "\\chi _{n}{\\begin{pmatrix}a&b\\\\0&a^{-1}\\end{pmatrix}}=a^{n}.", "12eb8b69f203694b0466180bc2015ca0": "\\textstyle Y", "12eb9bbc4431ae838874afea8be7b85c": "d(x,y)=\\operatorname {arccosh} \\left({\\frac {B(x,y)}{\\sqrt {Q(x)Q(y)}}}\\right).", "12ebb833247183cfac8da5b71f0bc3ec": "{\\text{im}}(A)\\subseteq {\\text{im}}(B).\\,", "12ebe2c770417549f7646e379c12b1dc": "PDOP={\\sqrt {d_{x}^{2}+d_{y}^{2}+d_{z}^{2}}}", "12ebee2e4b45140931d45a2d1b1d90ac": "T(x)=g(x)+\\sum _{i=1}^{k}a_{i}T(b_{i}x+h_{i}(x))\\qquad {\\text{for }}x\\geq x_{0}.", "12ec7ff279101059cf73b8573527e420": "B^{2}-4AC>0", "12ec8e137f4da6e137921159720c0344": "N(X)", "12eccac6ea84d2bf1e281e9339543924": "{\\tilde {D}}_{6}", "12ecf5153e8c8edad5e820479359dfc9": "B=(b_{pq})", "12ed2d0a5bc03c9a56d72b0de4f4af77": "v_{\\it {avg}}", "12ed44a1c273a0fb33aea3ec863853e1": "(x_{1},\\ldots ,x_{k})\\in R_{i}^{A}", "12edae1b896b5cf29c6ec4b0e2bd8789": "1/\\eta _{f}", "12ee191a47ed137dda9783152b9f0da5": "n=31", "12ee24c9d00f630f859812fede24fdae": "{\\{f_{1},f_{2}\\}_{M\\times N}}(x,y)={\\{{f_{1}}(x,\\cdot ),{f_{2}}(x,\\cdot )\\}_{N}}(y)+{\\{{f_{1}}(\\cdot ,y),{f_{2}}(\\cdot ,y)\\}_{M}}(x)", "12ee63ae1f4e1954c6f3b906e1279d6b": "F=a+r\\Psi ^{2}+s\\Psi ^{4}+H\\Psi \\,", "12eef979c0d460492ed2e71958a22f0a": "\\Sigma ^{-1}(C)", "12efe22166c5612553f8ae0be66380ce": "\\scriptstyle p_{0m}\\,", "12efeae7f866427b231a28461c8e890b": "\\vartheta _{4}^{2}(q)=\\sum _{k=-\\infty }^{\\infty }q^{2k^{2}}\\vartheta _{4}\\left({\\frac {k\\ln q}{i}},q\\right).", "12f06183f7b45f8f489d99effe1b2a2a": "\\alpha _{1}=a+b+e;\\quad \\beta _{1}=a+b+c+d;", "12f0a2309c99718f3895b58e8e8c6901": "AX+UY=I", "12f0e11d53301ad87b30c9d353966900": "I=I_{0}\\left[{\\frac {\\sin \\left(\\phi /2\\right)}{\\left(\\phi /2\\right)}}{\\frac {\\sin \\left(N\\delta /2\\right)}{\\sin \\left(\\delta /2\\right)}}\\right]^{2}\\,\\!", "12f0eddd942f836ca3915061b27d62bd": "l(X)", "12f0ee59968eaea8f149568bcbeae97a": "\\qquad \\forall _{i\\neq j}\\,x_{ij}\\geq 0.", "12f11189828a77c8db2ce45b787deba1": "\\left(Z_{i-k},\\ldots ,Z_{i-1},Z_{i+1},\\ldots ,Z_{i+k}\\right)", "12f1734702b5e3c6d8513017e9cf6503": "p_{K}(\\lambda x)=\\inf \\left\\{r>0:\\lambda x\\in rK\\right\\}=\\inf \\left\\{r>0:x\\in {\\frac {r}{|\\lambda |}}K\\right\\}=\\inf \\left\\{|\\lambda |{\\frac {r}{|\\lambda |}}>0:x\\in {\\frac {r}{|\\lambda |}}K\\right\\}=|\\lambda |p_{K}(x).", "12f19ff25bc9100d0900a5aafaea30d7": "\\mathrm {ERH} =a_{w}\\times 100\\%", "12f21fd950c570ec93929f7c86bcdff6": "D:\\Gamma (E\\otimes \\Omega ^{*}M)\\rightarrow \\Gamma (E\\otimes \\Omega ^{*}M)", "12f222dd22cac17760b581cda7dfd714": "{\\overline {\\lambda }}={\\frac {\\lambda }{\\lambda ^{\\text{*}}}}", "12f2526ca72325ccb0e5e5b3bb5287cf": "e^{i\\alpha }=\\left\\langle 0|U(\\infty )|0\\right\\rangle ^{-1}", "12f27527bbcbebfdebc4ffda139ff725": "{\\mathcal {Y}}", "12f2cc8365910da12d20f36f89062d94": "\\Phi _{b}({\\hat {p}})\\Phi _{b}({\\hat {q}})=\\Phi _{b}({\\hat {q}})\\Phi _{b}({\\hat {p}}+{\\hat {q}})\\Phi _{b}({\\hat {p}})", "12f31853fb6eae539a2a6883f2fca0ea": "Q.", "12f384debded95097d5fe47bf193ae62": "0={\\hat {x}}_{1}-x_{1}", "12f388f68ab8e57d3c532d8c8c1e09b6": "\\sin 18^{\\circ }=\\cos 72^{\\circ }={\\frac {{\\sqrt {5}}-1}{4}}", "12f3d8cb2e74ac34a45ec484e937d0f0": "x|U|=y|V|", "12f3e4d62efea37fbd97dcd9c362397a": "\\mathrm {Ir} ={\\frac {\\tan \\alpha }{\\sqrt {H/L_{0}}}}", "12f3f32b7110ffc144c00e99d00ee9e2": "B_{n}^{(k)}", "12f429b33a8c7c4f9a5c96ae1991fe25": "r(x+y)=rx+ry", "12f45c4863bb28d2b9d2de7af6741bbf": "a^{b}=\\left(re^{\\theta i}\\right)^{b}=\\left(e^{\\ln(r)+\\theta i}\\right)^{b}=e^{(\\ln(r)+\\theta i)b}", "12f46d0e829afc435aed9f325365f47b": "(F_{p})^{d}", "12f48fbcc1b971f4e41f52fe2794574d": "\\mathbf {N_{s}} ={\\frac {\\mathbf {120} \\mathbf {f} }{\\mathbf {p} }}", "12f4d86c0f31334e102665a4279361be": "A\\models \\forall x\\phi [x,{\\bar {a}}]\\iff \\phi ^{A,x,{\\bar {a}}}\\in \\forall _{A}", "12f50006a8e143e325aaac5d37b3b744": "kA{\\frac {(T_{m-1}^{i}-T_{m}^{i})}{\\Delta {x}}}+kA{\\frac {(T_{m+1}^{i}-T_{m}^{i})}{\\Delta {x}}}+{e_{m}}A\\Delta {x}=(\\rho c_{p}\\Delta xA){\\frac {(T_{m}^{i+1}-T_{m}^{i})}{\\Delta x}}", "12f517f63b3ddd32c54e8b5232e1033c": "{\\frac {Z(u)u^{1/2}}{\\sqrt {\\pi }}}\\sim \\int _{-\\infty }^{\\infty }e^{i(uV(x)+\\pi /4)}\\,dx", "12f531e442f4d962c2ea9e7aacc89938": "\\gamma ={\\frac {3-\\tau }{\\sigma }}\\,\\!", "12f59d9ef36d087ca186e02ad958f70b": "\\gamma =\\operatorname {atan2} (X_{3},Y_{3}).", "12f5c27e369b035e2caf0bf5bfbd6800": "f(x_{1},x_{2},\\ldots ,x_{n})", "12f6367e88eb5209ac6c4395c08c6cd9": "\\bigcup \\{\\sigma \\,\\colon (\\exists C)(\\langle \\sigma ,C\\rangle \\in G)\\}", "12f664b3e4f2e51e214768edeb12f0c6": "P(x)\\lor (\\exists {y}{\\in }\\mathbf {Y} \\,Q(y))\\equiv \\ \\exists {y}{\\in }\\mathbf {Y} \\,(P(x)\\lor Q(y)),~\\mathrm {provided~that} ~\\mathbf {Y} \\neq \\emptyset ", "12f68af42744fd031ff56476b9c0451e": "\\phi \\vert _{K}\\in \\operatorname {Gal} (K/\\mathbf {Q} )", "12f6ecf7fef507aceae8be0eff761f5c": "P^{n}(x\\smile y)=\\sum _{i+j=n}(P^{i}x)\\smile (P^{j}y)", "12f6f461cd49e0808f7ddee474f1d3b9": "L_{K}=\\left({\\frac {m}{2n_{s}e^{2}}}\\right)\\left({\\frac {l}{A}}\\right)", "12f70b4a9cf4455c58a7aee27bbe05cc": "P_{t}^{}(j,q)", "12f89e127d7459eecc488aa723db1e43": "\\left[{\\begin{matrix}1&0&0&0\\\\0&\\cos(\\theta )&0&\\sin(\\theta )\\\\0&0&1&0\\\\0&-\\sin(\\theta )&0&\\cos(\\theta )\\end{matrix}}\\right]", "12f8aa6419bdd67dfdb3c40ad076b55b": "\\mu (A)>0,", "12f8c70405aeeb9dacf082090dfb54ff": "{\\begin{bmatrix}c_{2}c_{3}&-c_{3}s_{2}c_{1}+s_{3}s_{1}&c_{3}s_{2}s_{1}+s_{3}c_{1}\\\\s_{2}&c_{1}c_{2}&-c_{2}s_{1}\\\\-s_{3}c_{2}&s_{3}s_{2}c_{1}+c_{3}s_{1}&-s_{3}s_{2}s_{1}+c_{3}c_{1}\\end{bmatrix}}", "12f97a3f5226a04def3aa90382c6c340": "\\int _{\\Gamma _{1}}{\\frac {\\mathrm {F} (\\mathrm {X} )}{\\mathrm {X} -\\Xi }}\\mathrm {d} \\mathrm {X} =2\\pi i\\sum _{s=1}^{k}\\mathrm {N} ^{(s)}u^{(s)}\\mathrm {F} (\\Xi )", "12f98ccba91bd634dfd3ea3b577709f0": "W(\\mathrm {T} )", "12f99fe93443d65b450816343da2ca54": "F_{n}=F_{n-1}+F_{n-2},\\,", "12fa76fe3c2380fde076156344a47f42": "\\alpha _{p}", "12fae542fdc986ff628b05c980383420": "v={d \\over dt}(Li)=L{di \\over dt}\\,", "12faf7b241cc52a383b177bbeee59c6a": "\\nabla \\times \\mathbf {E} =-\\mu _{o}{\\frac {\\partial \\mathbf {H} }{\\partial t}}", "12fafa9ac27b00580823b78bf96a95d1": "{\\begin{aligned}\\nabla ^{2}U&=-M_{x}\\left({\\partial ^{2}B_{x} \\over {\\partial x}^{2}}+{\\partial ^{2}B_{x} \\over {\\partial y}^{2}}+{\\partial ^{2}B_{x} \\over {\\partial z}^{2}}\\right)-M_{y}\\left({\\partial ^{2}B_{y} \\over {\\partial x}^{2}}+{\\partial ^{2}B_{y} \\over {\\partial y}^{2}}+{\\partial ^{2}B_{y} \\over {\\partial z}^{2}}\\right)-M_{z}\\left({\\partial ^{2}B_{z} \\over {\\partial x}^{2}}+{\\partial ^{2}B_{z} \\over {\\partial y}^{2}}+{\\partial ^{2}B_{z} \\over {\\partial z}^{2}}\\right)\\\\&=-M_{x}\\nabla ^{2}B_{x}-M_{y}\\nabla ^{2}B_{y}-M_{z}\\nabla ^{2}B_{z}\\end{aligned}}", "12fb282673e43435f7b5ae1853b69187": "G\\rightarrow 0", "12fc6220753b43847d66009e75120f3c": "\\{\\{q_{1}\\},\\{q_{2},q_{3}\\}\\}", "12fc9cdd82579fea64aecf1180555792": "{\\begin{aligned}E(X)&={\\frac {1}{2}}\\\\[6pt]\\operatorname {Var} (X)&={\\frac {1}{24}}\\end{aligned}}", "12fcc337a262b208240745257f841f17": "a^{-2}+b^{-2}=d^{-2}", "12fce0714a5d61d7685c348011b075ff": "y\\cot \\varphi ={\\frac {y}{\\tfrac {dy}{dx}}},", "12fd204b34a3f1c84d262945ddc37d29": "H_{0}^{(i)}(x)=xG_{0}^{(i)}[H_{1}^{(1)}(x),...,H_{1}^{(n)}(x)]", "12fd3b4d703b1f89fc2c84ae5a40a4bb": "ab>1", "12fd3cef765bf5411bfc584ecad7b943": "F_{m}P_{m}=S_{m}F_{N}.\\,", "12fd4f8116f7c244b6d6d3e06cfd0905": "R/p^{n}", "12fd51e8221d40d266a77b12c3648135": "A\\rightarrow X", "12fda988cc40eb4214181d95e00bbac0": "e^{i(\\alpha +\\beta )}=(\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta )+i(\\sin \\alpha \\cos \\beta +\\sin \\beta \\cos \\alpha )", "12fdfe327f07b75055766578c5cc5dc1": "2^{15_{dec}}", "12fe2690b6a38adbb1493e2a6f0173c1": "S:X^{2}\\rightarrow X^{2}", "12fe535347865177bacade254d11ce9c": "\\scriptstyle xa\\equiv 1{\\pmod {p}}", "12fe535c731a0b7fb49eafe24805e3a2": "y_{\\mathrm {atm} }", "12fecd3dd54eb8a0e3aa4ebc5a1cbd9d": "d_{1}", "12fee16699e5bd04103a2932ef58edb1": "{\\boldsymbol {s}}={\\boldsymbol {R_{f}-R_{i}}}=\\Delta {\\boldsymbol {R}}", "12ff71999646c56fc4a6de1db987e194": "\\rightarrow ", "12ff7c40280485def3c4db7c6495791e": "10^{7.6}", "12ffb15026e354690a84a0499fd2f89d": "x_{n+1}={\\frac {x_{n}^{2}+2x_{n}}{x_{n}^{2}+1}}.", "13005a3d0853a1970cbcbfeeda952427": "\\oint _{C}{\\vec {H}}\\cdot \\mathrm {d} {\\vec {l}}=I_{\\mathrm {enc} }", "1300cefb3f126d2f188def5f3998da6d": "1_{x,y}", "1300efb50bfdd3e33f4d776a2a64b038": "\\|T^{2}x\\|\\geq \\|Tx\\|^{2}\\,", "1300f7354e7b132ba06e255f8d22d194": "\\quad {\\tfrac {t^{2}+1}{2a}}", "1301432ccc745f36b88301dc521459b5": "y_{3}\\left(x\\right)=j_{-4}(x)=\\left(-{\\frac {15}{x^{3}}}+{\\frac {6}{x}}\\right){\\frac {\\cos(x)}{x}}-\\left({\\frac {15}{x^{2}}}-1\\right){\\frac {\\sin(x)}{x}}.", "1301973e1d25a624d3f45ec3c45b941c": "_{Mpq}\\!", "13020040a0b735394e11d352cd14ecef": "P(x',t|x,0)", "13021c358dd5db520006c31a59781395": "e_{a}^{\\nu }=e_{a}^{\\mu }e_{\\mu }^{\\nu }\\,", "1302a21c7f0407dde27fce9c3b35317a": "\\Delta _{r}G>0", "1302a34d5d567b58f2e52436000dd9f1": "-n(n+1)~r^{n}~\\cos(n\\theta )\\,", "1302ae880d88b2bdce4868961306bc92": "Z:=Z+L_{i}*w_{i}", "1302b02a9106f64307e0fa4a85386177": "{x^{2}+2x^{3}}", "130357f29fa8718819bdff6d0269f48c": "{\\frac {d}{dt}}{\\mathbf {v}}_{0}(\\varphi _{0}^{-1}(P(t)))=\\left({\\frac {d}{dt}}J_{\\varphi _{01}}(\\varphi _{1}^{-1}(P(t)))\\right)\\cdot {\\mathbf {v}}_{1}(\\varphi _{1}^{-1}(P(t))).", "13035b90ba8b496cfc6d04d439eb8d04": "P\\land \\exists x\\,Q(x)\\Leftrightarrow \\exists x\\,(P\\land Q(x))", "1303613906a404e9888364cc307ce8bf": "\\lambda _{2}\\,", "13037e9c9753cb1f886f429929388f63": "M_{G}", "13039a55a4720d3c2b91c996311a3612": "\\Delta X_{n}^{1}", "1303b0be86ada546ec776915e817c81a": "p_{\\pi }({\\boldsymbol {\\eta }}|{\\boldsymbol {\\chi }},\\nu )=f({\\boldsymbol {\\chi }},\\nu )\\exp \\left({\\boldsymbol {\\eta }}^{\\rm {T}}{\\boldsymbol {\\chi }}-\\nu A({\\boldsymbol {\\eta }})\\right),", "1303c4d55e7579f5bc020153b1461d5a": "{\\dot {\\rho }}={\\frac {\\partial \\rho }{\\partial t}}+{\\boldsymbol {\\nabla }}\\rho \\cdot \\mathbf {v} ~;~~{\\dot {\\eta }}={\\frac {\\partial \\eta }{\\partial t}}+{\\boldsymbol {\\nabla }}\\eta \\cdot \\mathbf {v} .", "1303c5c75d3eae272b9c84920c5a0466": "\\mathbb {C} ^{*}", "13040d378cbb9b9ef31d7bdfa27746ae": "x_{n}=(ax_{n-r}+c_{n-1}){\\bmod {\\,}}b\\,,\\ \\ c_{n}=\\left\\lfloor {\\frac {ax_{n-r}+c_{n-1}}{b}}\\right\\rfloor ,", "1304435c8bc4c0f5b8ecbe7f16727d75": "A^{*}", "13044b70e7e47b34729d39dd200c14ea": "{\\tfrac {t^{2}}{\\log t}}", "13046ae9549d127278b6e26472a79c1b": "\\alpha ={\\frac {\\partial u_{y}}{\\partial x}}\\quad ,\\qquad \\beta ={\\frac {\\partial u_{x}}{\\partial y}}\\,\\!", "1304c9b804161fbf6cd3e1e4f0f3055e": "\\langle {\\hat {L}}\\rangle _{\\textrm {PM}}=\\sum _{A}^{\\textrm {atoms}}\\sum _{i}^{\\textrm {orbitals}}|q_{i}^{A}|^{2}", "1305054881bb4d43e86f723e845e1d43": "m_{k}{\\ddot {\\vec {x}}}_{k}(t)=F_{k}({\\vec {x}}(t))=-\\nabla _{{\\vec {x}}_{k}}V({\\vec {x}}(t))", "130527965708be2b8274e688419cf280": "su(N)", "13058be3d8ceb06273459847a6a3ad30": "{\\boldsymbol {\\mu }}_{X,Y}", "1305cf401f3832d052e786f7c2a81982": "\\textstyle {J}_{\\alpha }", "1305e7910687b7f64827c2808a0ccc91": "6.5\\cdot 10^{4}", "130600787d4889375d2049fd1806ca7d": "R=\\bigcup _{i}A(x)\\oplus t_{i}", "13064eff2904bad4595d093c741db465": "P_{\\text{estimated}}={\\frac {{\\frac {1}{SU}}-1}{{\\frac {1}{NP}}-1}}", "13067925c30c5b7d4111b06cdfac0313": "u_{1}(\\mathbf {q} )", "130693682fe4d9d5612c6bc6f7df878f": "3^{8}", "1306e5d99726c06558d21f4716feed7a": "38\\equiv 14{\\pmod {12}}\\,", "1306eb165f2534c59abd6d11e8367bfb": "a_{1}\\leq a_{2}\\leq \\cdots \\leq a_{n}", "130738856fc9612571b31e7928e2e614": "\\zeta {\\bigl (}{\\tfrac {1}{2}}+it{\\bigr )}", "13073a46df0768c0a6553150f6bf2047": "X_{c}\\leq X_{c}^{0}", "1307530bbad17fe83a008987a06fad15": "\\textstyle \\delta ", "13077de639cf686dc761c7bf9a5c1c1d": "H_{0}={DF}_{T}\\int _{\\omega }H(\\omega )\\,d\\mathbb {P} (\\omega )", "13078955e1101087e2918cfc6f71e594": "g(\\sigma _{1},\\sigma _{2})=\\int _{X}{\\frac {d\\sigma _{1}}{d\\mu }}{\\frac {d\\sigma _{2}}{d\\mu }}d\\mu ", "130810d2c92926f5bb947b917c6fad2a": "D_{T}^{2}z=\\{\\{z,T\\},T\\}=\\{({\\dot {q}},0),T\\}=(0,0)", "13086b5390a51152ae0cc4dd91df65ff": "\\{a^{\\frac {j}{2}}\\psi (a^{j}x-kb)\\}_{j,k\\in Z}", "13091099a7b1fbd221ab526eac2fa699": "P_{c}\\,", "130923d07b3d567bceaf83fe8f7aa71b": "T^{{\\bar {k}}-2}p,\\dots ,Tp,p", "13092adb1b5af45587af133463d706b8": "\\mathrm {pOH} =\\mathrm {pK_{W}} -\\mathrm {pH} ", "1309783b0373ce7dbb64485afa366e47": "\\delta _{m+1}={\\frac {T_{m+1}(0)}{T_{m}(0)}}", "13098eced8194ca1107b909ecaa50cd0": "{\\frac {d}{dt}}{\\begin{bmatrix}\\mathbf {T} \\\\\\mathbf {N} \\\\\\mathbf {B} \\end{bmatrix}}=\\|\\mathbf {r} '(t)\\|{\\begin{bmatrix}0&\\kappa &0\\\\-\\kappa &0&\\tau \\\\0&-\\tau &0\\end{bmatrix}}{\\begin{bmatrix}\\mathbf {T} \\\\\\mathbf {N} \\\\\\mathbf {B} \\end{bmatrix}}.", "1309a3363d95bf456d2f270e7a4f1683": "\\,O,O_{i}", "1309bd6f699c1dd81906c4b087c6478c": "u={\\tfrac {1}{6}}\\left(x^{3}-\\pi ^{2}x\\right),", "1309bd8fe3cb64bf49bd16aeba59ee1b": "{\\text{hom}}(-,X):C^{\\text{op}}\\to C", "1309d31299357ab2306d906f44cce76d": "\\left|\\Gamma _{1}\\right|=\\left|\\Gamma _{2}\\right|", "130a0b7752c8cd8ee8802710b1ed97c2": "Z_{L}=\\infty ", "130a14353b67992ed46ed53b9a666fca": "\\color {Melon}{\\text{Melon}}", "130a3c6d6ca230f74b1e0b2abd20f4b2": "\\nabla f(x^{*})", "130a555bc192bac57517e470a9a35afa": "\\int \\limits _{0}^{1}\\!{\\frac {\\ln \\ln {\\frac {1}{x}}}{1+2x\\cos \\varphi +x^{2}}}\\,dx\\,=\\int \\limits _{1}^{\\infty }\\!{\\frac {\\ln \\ln {x}}{1+2x\\cos \\varphi +x^{2}}}\\,dx={\\frac {\\pi }{2\\sin \\varphi }}\\ln \\left\\{{\\frac {(2\\pi )^{\\frac {\\scriptstyle \\varphi }{\\scriptstyle \\pi }}\\,\\Gamma \\!\\left(\\!\\displaystyle {\\frac {1}{\\,2\\,}}+{\\frac {\\varphi }{\\,2\\pi \\,}}\\!\\right)}{\\Gamma \\!\\left(\\!\\displaystyle {\\frac {1}{\\,2\\,}}-{\\frac {\\varphi }{\\,2\\pi \\,}}\\!\\right)}}\\right\\},\\qquad |\\Re {\\varphi }|<\\pi .", "130a55d0696be58d22e26eca42afb6ee": "t=t_{0}", "130a7ca8d25ee619d31cb1fa29dde94e": "(k+2)(k+1)A_{k+2}+(-2k+1)A_{k}=0\\;\\!", "130a812f388c01a429719b7beefecaf7": "C{\\frac {dV}{dt}}=I-g(V),", "130ab19206799d24bd617e1db01eea1d": "\\Delta \\cong \\operatorname {Hom} (\\operatorname {Gal} (L/K),\\mu _{n})", "130acfcea55210419f8a268e091e918c": "pV=nRT\\,\\!", "130af4e066dee5f97a48b2a5008b2520": "{\\frac {d^{2}u_{\\epsilon }}{dx^{2}}}-c^{\\epsilon }(x)u_{\\epsilon }=-\\phi ^{\\epsilon }(x),\\quad 04\\pi A", "130b67f65c556c8820db1d9554cf439e": "\\vartheta _{00}(z;\\tau )=\\vartheta (z;\\tau )", "130c3b48e8cdf998df86e9f8001569b6": "b\\geq n", "130c4fda13ec679cd4fd8dda55373b41": "{\\frac {1}{\\beta }}-\\log \\left[{\\frac {\\beta }{2\\alpha \\Gamma (1/\\beta )}}\\right]", "130c9c02c2e7e2ffe9eb6fc8efa70c2c": "P::=\\emptyset \\,\\,\\,|\\,\\,\\,a.P_{1}\\,\\,\\,|\\,\\,\\,A\\,\\,\\,|\\,\\,\\,P_{1}+P_{2}\\,\\,\\,|\\,\\,\\,P_{1}|P_{2}\\,\\,\\,|\\,\\,\\,P_{1}[b/a]\\,\\,\\,|\\,\\,\\,P_{1}{\\backslash }a\\,\\,\\,", "130cf98832a0dc4773147358d3e103d1": "2/101=1/101+1/202+1/303+1/606", "130d12788450ada3da40b3d5f89f2e16": "\\operatorname {GL} (V)\\leq \\operatorname {\\Gamma L} (V).", "130d338a47d093b0e5799ab5b99dad49": "{C}_{CF}", "130d60bd2f97bbabe44e82f80080643f": "y=-{g\\sec ^{2}\\theta \\over 2v_{0}^{2}}x^{2}+x\\tan \\theta ", "130db251383a3c49afe77ff591563b28": "{\\frac {1}{\\sigma ^{2}+\\tau ^{2}}}\\left({\\partial ({\\sqrt {\\sigma ^{2}+\\tau ^{2}}}A_{\\sigma }) \\over \\partial \\sigma }+{\\partial ({\\sqrt {\\sigma ^{2}+\\tau ^{2}}}A_{\\tau }) \\over \\partial \\tau }\\right)+{\\partial A_{z} \\over \\partial z}", "130dfb56e1ad751c6a4b5d0f4d3ecb9e": "(Tx,\\mu _{x})", "130e2e1609c1b145132ba01cd3dc31de": "W_{KRA}=\\sum _{n=N}^{n=\\infty }2\\pi \\omega ^{2}p(n-n_{\\mathrm {osc} })^{2}\\int d\\Omega |FT(I_{KAR}\\Psi (\\mathbf {r} ))|^{2}J_{n}^{2}(n_{f},{\\frac {n_{\\mathrm {osc} }}{2}})", "130e3d66e7312d085da14666b55d2338": "2(1+{\\sqrt {2}})s^{2}\\,\\!", "130e5f8c90c6557bae665228a2cf87bb": "u(t)=\\gamma \\sin(\\omega _{0}t+\\varphi _{0})", "130e67d965e74a64298d664f795cf982": "A^{n-1}b", "130ed2b3e1b9d52ab62a058d517715e6": "(X_{n},d_{n},\\mu _{n})", "130f7bcc1a8ab7d37dc972ee3ec3ac4d": "\\neg D\\rightarrow \\neg C", "130fafbaca9391335bd927142d732724": "{\\begin{aligned}\\sum _{i=1}^{n}{\\text{E}}[(S_{i}-S_{i-1})^{2}]&=\\sum _{i=1}^{n}{\\text{E}}[S_{i}^{2}-2S_{i}S_{i-1}+S_{i-1}^{2}]\\\\&=\\sum _{i=1}^{n}{\\text{E}}\\left[S_{i}^{2}-2(S_{i-1}+S_{i}-S_{i-1})S_{i-1}+S_{i-1}^{2}\\right]\\\\&=\\sum _{i=1}^{n}{\\text{E}}\\left[S_{i}^{2}-S_{i-1}^{2}\\right]-2{\\text{E}}\\left[S_{i-1}(S_{i}-S_{i-1})\\right]\\\\&={\\text{E}}[S_{n}^{2}]-{\\text{E}}[S_{0}^{2}]={\\text{E}}[S_{n}^{2}].\\end{aligned}}", "130fe413966ba4e4e3071bb45a910e5a": "v_{T-j}", "1310016cef732a1fac36974f1a2792a9": "{\\frac {\\left\\Vert v_{Target}\\right\\|}{\\sin(\\theta _{Deflection})}}={\\frac {\\left\\Vert v_{Torpedo}\\right\\|}{\\sin(\\theta _{Track}-\\theta _{Deflection})}}", "13102c14d5dfbd8248dae67af3e0660f": "k=|f_{o}-f_{e}|/f_{0},", "131051ea4f96502fce372ee4692631a1": "\\forall x\\,(P(x)\\rightarrow Q(x))\\rightarrow (\\forall x\\,P(x)\\rightarrow \\forall x\\,Q(x))", "1311810f1a89f9ec08638abbe6e4c96e": "dW=Q\\,dV=\\left[\\int _{0}^{V}\\ dV'\\ C(V')\\right]\\ dV\\ .", "1312028ddc981e4e36cd3b91ef665805": "\\{f,{\\mathcal {H}}\\}+{\\frac {\\partial f}{\\partial t}}=0", "131214d4fdf40a43f0102364f4decbf9": "U=-G{\\frac {16}{15}}\\pi ^{2}R^{5}\\left({\\frac {M}{{\\frac {4}{3}}\\pi R^{3}}}\\right)^{2}={\\frac {-3GM^{2}}{5R}}", "131217925a1a78cecae66513e4864e72": "b\\in \\Gamma ", "131220f39668c31132d0c521ec903630": "\\mathbf {P} \\left[\\sup _{0\\leq t\\leq T}{\\big |}{\\sqrt {\\varepsilon }}B_{t}{\\big |}\\geq c\\right]\\leq 4n\\exp \\left(-{\\frac {c^{2}}{2nT\\varepsilon }}\\right).", "13127fa065167f088d0a0877e21b3c91": "|x|=p_{1}^{a_{1}}\\ldots p_{r}^{a_{r}}.", "1312805d1c4c78fbb9dcc5bee3c8a435": "\\alpha ", "1312b06cebbe1e15696198bc426ab9f4": "\\sum _{x}\\psi _{+}(x)|x,\\uparrow \\rangle +\\psi _{-}(x)|x,\\downarrow \\rangle \\,", "1312cd3f180f13ce045e3ad7c920ebe9": "\\|f_{n}\\|\\leq M_{n}", "1312dba4acceffe6ddb18cb7ffa771c0": "t_{\\max }>{\\frac {d}{i_{\\max }}}+{\\frac {d}{j_{\\max }}}+O(1)", "1312eba929a970eba0e90b2571773ede": "{\\begin{aligned}\\delta ({\\bar {\\lambda }})&=2\\delta _{1}({\\bar {\\lambda }})+\\delta _{2}({\\bar {\\lambda }})=2{\\big (}{\\bar {n}}_{1}-1)\\alpha _{1}+{\\big (}{\\bar {n}}_{2}-1)\\alpha _{2}\\ ,\\\\\\Delta &=2{\\frac {\\delta _{1}({\\bar {\\lambda }})}{V_{1}}}+{\\frac {\\delta _{2}({\\bar {\\lambda }})}{V_{2}}}\\ .\\end{aligned}}", "1312efddb47a74d08895428eb0b19cc9": "w(x^{q},y^{q})=(x^{q^{2}},-y^{q^{2}})", "131385d30e2d926d9b2c5ac1e8d3c0e3": "X_{j}=-{\\rm {\\nabla }}{\\frac {\\mu _{j}}{T}}", "1313b87e9846aee17064b2ec8b35d6f6": "ta(s)=\\min\\{t_{si}-t_{ei}|i\\in D\\}.", "1313f245c16223eaee992ec499a5f9e7": "{\\frac {\\partial ^{2}\\Pi }{\\partial x_{i}\\partial x_{j}}}", "1314154d6660eac18e2b083dcf2aa5b7": "f(3)=10.", "131438c56262e9b5f96a3655f3b7919f": "g(z)={\\begin{cases}{\\frac {f(z)}{z}}\\,&{\\mbox{if }}z\\neq 0\\\\f'(0)&{\\mbox{if }}z=0,\\end{cases}}", "131464ff977d56664e7a983f8ded56be": "[Force\\ on\\ Plate]=[Weight\\ of\\ plate]+[Surface\\ Tension\\ Force]-[Buoyant\\ Force]", "131472bdcba05ffd00c336914d3cf8f0": "\\mathbb {C} ^{n}\\otimes \\mathbb {C} ^{n}", "1314763ced2cedfdd8e32b584235fdf4": "{\\frac {2\\pi (x)}{\\frac {x}{\\epsilon ^{ln(lnx)}}}}", "1314c51722f0a48ea12574945e8d223a": "f(-x)=1/f(x)", "1315095d2fedddc2d11f1a8def6c58e7": "\\langle T_{C}\\rangle ={\\frac {1}{\\Delta S}}\\int _{Q_{out}}TdS", "13156036dd3fba86bd5f3354e8e62641": "{\\frac {d^{2}y}{dx^{2}}}=(A+B\\wp (x))y", "1315bd1457554e9533bbc89b53bb90bc": "(T,V)", "1315c1dada771ae80b948d4e2f6d5897": "f(t)={\\frac {1}{2\\pi }}{\\frac {d\\phi (t)}{dt}}", "1315d95a72845a0b5768bc91064530e7": "{\\tilde {f}}(x)={\\frac {1}{2\\pi }}{\\text{ p.v.}}\\int _{0}^{2\\pi }f(t)\\cot \\left({\\frac {x-t}{2}}\\right)\\,dt", "1316a1629f7600e21d93f6d417821aef": "S'=S'''=S'''''", "1316de8e1ab2134b3399cc0d78b45c3b": "\\sigma _{2}\\,\\!", "13173ff34816285ce96238c281b1e72f": "f_{c}\\,,\\,f_{m}\\,", "1317c941b60c67b6081284ff6e30e051": "-8.349\\times 10^{-11}\\times 60\\times 60\\times 24\\times 10^{9}\\approx -7214{\\text{ ns}}", "1318274397f12d88e3920207fb8f460c": "n={\\frac {E}{\\hbar \\cdot \\omega }}-{\\frac {1}{2}}={\\frac {m\\omega A^{2}}{2\\hbar }}-{\\frac {1}{2}}", "13183ff4a9f45b94417d7823bdda0850": "=O\\sum _{n=0}^{\\infty }{\\frac {1}{n!}}(-iHt/\\hbar )^{n}|\\psi _{0}\\rangle ", "13185be3e9220ab954bdedcd05c02e4b": "K(x_{i}-x)=e^{-c||x_{i}-x||^{2}}", "131866f77a89234d6428c42fc06c8d19": "V_{c}=a_{0}{\\sqrt {5{\\Bigg [}{\\bigg (}{\\frac {q_{c}}{P_{0}}}+1{\\bigg )}^{\\frac {2}{7}}-1{\\Bigg ]}}}", "1318c185f5339fc1a60ca1da067d5cac": "T_{m}(k)=V_{1}\\cdot C_{M}(k)+V_{2}\\cdot (\\alpha \\cdot C_{M}(k+1)+(1-\\alpha )\\cdot C_{m}(k+1))+V_{3}\\cdot (\\beta *C_{m}(k-1)+(1-\\beta )\\cdot C_{M}(k-1))", "1318d541272762b8020f290828504e96": "{\\text{length of solar day}}={\\frac {\\text{length of sidereal day}}{1-{\\tfrac {\\text{length of sidereal day}}{\\text{orbital period}}}}}.", "1318dd34e2fea40d32972c9f4002520e": "\\delta =\\arcsin(1/e)\\,", "1318f1898bd8690d1e477b8e6d18e78b": "\\Gamma (s)\\zeta (s)=\\int _{0}^{\\infty }{\\frac {x^{s-1}}{e^{x}-1}}\\,dx,", "1319f5d4d78428de32ec985155bb72fa": "\\sigma \\;\\mid \\;\\tau \\equiv _{b}\\tau \\;\\mid \\;\\sigma ", "131a659fb4e4edd79e2ab569b39b6745": "n(j)", "131a6cc31efd47a5b6bef19568764408": "E=F(\\alpha )", "131a8a5279434571a7c80274988aff10": "\\varphi =2\\pi \\xi {\\frac {}{}}", "131ac7a55fbf77500a953b4eb00d0b84": "G(\\psi )=\\sum _{r=1}^{k_{1}}\\psi (r)e^{2\\pi ir/k_{1}}.", "131ae124a06fd38c77be375a4bcd7b94": "={\\frac {1}{\\sqrt {2\\pi }}}\\sum _{m}e^{im\\theta _{k}}\\int _{0}^{\\infty }r\\operatorname {d} \\!r\\,f_{m}(r)2\\pi i^{m}J_{m}(kr)", "131b25c041e3ea27f4e87ce51a622f71": "H_{n}(M;\\mathbf {Z} )=0", "131b4518a6693db291494c4c8135b1c9": "\\{(U_{\\alpha },\\varphi _{\\alpha })\\}", "131b68643b1ddb1eeb812048affdecd3": "\\,{\\mathcal {R}}\\,", "131b7e143eba2be8fbc97f6084d466e3": "\\ pF^{n}=_{def}\\{x_{1}...x_{n}:F^{n}x_{1}x_{3}...x_{n}x_{2}\\}.", "131b993f4b6b23c36b6d36e8a36496ec": "{\\hat {G}}({\\boldsymbol {k}})=H\\left(k-k_{c}\\right),\\qquad k_{c}={\\frac {\\pi }{\\Delta }}.", "131bd4c2bf2fb2dcd6653f59d2fb8c0c": "{\\frac {P(x)}{Q(x)}}=\\sum _{i=1}^{n}{\\frac {P(\\alpha _{i})}{Q'(\\alpha _{i})}}{\\frac {1}{(x-\\alpha _{i})}}", "131c37e2852da6c21088bbafd9c5de32": "0.d_{1}d_{2}d_{3}\\dots ;\\dots d_{\\infty -1}d_{\\infty }d_{\\infty +1}\\dots ,", "131c40c8d7f7db51a150c10f8c1a4d68": "(\\Sigma ,D,I)", "131c5c32a334854272ed65e40e7b0ca7": "R={\\log _{q}{|C|} \\over n}\\leq 1-H_{q}(J_{q}(\\delta ))+o(1)", "131c7d356a34f25bf2583b2beafd84da": "\\partial f_{i}/\\partial x_{j}", "131c927249ce3b52d8adadd80644be60": "CCl_{4}", "131d04366a99f2e9676b62d2c861a9bf": "u'{\\frac {d}{dx}}{\\frac {\\partial L}{\\partial u'}}={\\frac {d}{dx}}\\left({\\frac {\\partial L}{\\partial u'}}u'\\right)-{\\frac {\\partial L}{\\partial u'}}u''\\,,", "131d22e7bca14bf19d7a0855ebb8adad": "|R(z)|\\leq 1", "131d3742ca3d2b210b7d6d10e4f54b1e": "{\\vec {f}}_{j}", "131d506244289281ea9fb297f4daa846": "P_{\\pi }^{-1}=P_{\\pi ^{-1}}=P_{\\pi }^{T}.", "131d64f3093d2610bdefb597a858c1c3": "P(X_{1},X_{2},\\ldots ,X_{n})", "131d9bdbbb43837df3a5e5b7b9775d7f": "\\underbrace {B_{j}\\land \\dots \\land B_{j}} _{j}.", "131db117593e7d6ffae022b771c6e90b": "y(x)=x\\cdot y'+(y')^{2}.\\,\\!", "131e315d843a666dacdb545dab79d48e": "{\\frac {x}{2}}.", "131ea58e218b38e231f2e0ed9619e5cd": "iS=\\int _{t}\\left[m\\left({\\frac {dy(it)}{dt}}\\right)^{2}+V(y(it))\\right]dt", "131ee96e40a675bd815eded8a6d9c050": "|-4|>|2|+|1|", "131f04ef3dd102f4156afe21bc2f5028": "d'", "131f6304d827bb8c0b31ec47798e3f65": "t\\in {\\mathcal {T}}", "131f9ab1827a2dcbb3e0f39652f2a961": "GlobalCO_{2}Emissions=(GlobalPopulation)\\left({\\frac {GrossWorldProduct}{GlobalPopulation}}\\right)\\left({\\frac {GrossEnergyConsumption}{GrossWorldProduct}}\\right)\\left({\\frac {GlobalCO_{2}Emissions}{GrossEnergyConsumption}}\\right)", "131fcbb7cec215136e10228e0f342aea": "\\mu _{a}\\,", "131fd69e68b644a2b54d36b8d2ae88be": "I_{electrode}^{i}", "131fdc4636b86ac5f9a3e54a8a7ee4f9": "(e,f),(g,h)\\mapsto \\left({\\frac {eh+fg}{1+degfh}},{\\frac {fh-aeg}{1-degfh}}\\right)", "131fecb364a8e0856fd04b557de54bd0": "T>0", "131ff6a36cf10e00931d5734549a1395": "{\\frac {dF}{dx}}={\\frac {e}{\\epsilon \\epsilon _{0}}}\\left(n-n_{a}\\right)", "13200863b1d4d0620055ef7f9ff9a667": "{\\mathcal {L}}_{X}(T(Y_{1},\\ldots ,Y_{n}))=({\\mathcal {L}}_{X}T)(Y_{1},\\ldots ,Y_{n})+T(({\\mathcal {L}}_{X}Y_{1}),\\ldots ,Y_{n})+\\cdots +T(Y_{1},\\ldots ,({\\mathcal {L}}_{X}Y_{n}))", "132010180fdb9b50fc20997fe2f115a8": "P=P_{s}+P_{f}-P_{sf}.", "1320632218b9ecf3cbffdaac3e3ac696": "\\neg (A\\lor A)\\lor A", "132089cacde27ed8b9020f418d357bff": "\\psi (\\Omega ^{\\psi (\\Omega ^{\\psi (0)})})", "1320a2664921eb458cfb03364a61db64": "=\\left(\\sum _{i=1}^{n}\\left|x_{i}-y_{i}\\right|^{2}\\right)^{1/2}", "13211299daae2d9982b9d048f3937784": "{\\bar {x}}_{2}", "1321a455ac4427540303dd5987f5a3c2": "f_{j}^{e}[n]", "1321c5dadf11bd90800751315efe739d": "z=\\Phi (w)", "1321f600924e7d690deccf753b1d4b56": "A[{\\hat {\\mathbf {k} }}]\\,\\!", "13223267f2d1188904b8bf4e5c05efa7": "\\pi \\oplus (\\sigma \\ominus \\tau )\\neq (\\pi \\oplus \\sigma )\\ominus \\tau ", "13230b5bfb612251a19e58baeada5e2b": "0=-\\Delta {\\text{H}}_{\\text{f}}+{\\text{V}}+{\\frac {1}{2}}{\\text{B}}+{\\text{IE}}_{\\text{M}}-{\\text{EA}}_{\\text{X}}+{\\text{U}}_{\\text{L}}", "132390541f97ed12997bc60618bb285b": "{\\mbox{run rate }}={\\frac {\\mbox{total runs scored}}{\\mbox{total overs faced}}}", "1323b5bbb882153c7fa129a3f55e7913": "{\\frac {\\mathbf {B} }{\\left|\\mathbf {B} \\right|}}.", "132487d42bb4f40fcfcc85dd0bea4be9": "a_{\\mathit {wf}}=\\left\\{{\\begin{array}{ll}{\\frac {4\\Delta _{0,50}^{2}}{1+4\\Delta _{0,50}^{2}}},&\\mathrm {for} \\ \\Delta _{0,50}<0\\\\\\\\0,&\\mathrm {for} \\ \\Delta _{0,50}\\geq 0\\end{array}}\\right.", "13249b56109f67473434bff4b692e31d": "{\\frac {A(x\\rightarrow x')}{A(x'\\rightarrow x)}}={\\frac {P(x')}{P(x)}}{\\frac {g(x'\\rightarrow x)}{g(x\\rightarrow x')}}", "1324d399626dfdf63b91c382533ecd09": "S_{\\text{gauge-fixed}}={\\tfrac {1}{2}}\\langle \\Psi |c_{0}L_{0}|\\Psi \\rangle +{\\tfrac {1}{3}}\\langle \\Psi ,\\Psi ,\\Psi \\rangle \\ ,", "132510a47daebc02e46edc2d58b1f513": "\\int _{-\\infty }^{\\infty }f(t)\\,dt=\\lim _{a\\to -\\infty }\\lim _{b\\to \\infty }\\int _{a}^{b}f(t)\\,dt.", "13259605bd89e461ed298c8372a9bbee": "r^{2}\\,d\\phi ^{2}=\\cot(\\phi )^{2}\\,dr^{2}={\\frac {R^{2}}{r^{2}-R^{2}}}\\,dr^{2}", "1325cbab3acd85b62fa10f31c669cd6c": "g_{Q}(x)=R-1\\quad {\\text{if }}R{\\text{ is odd and }}x\\geq a^{2}.", "1325d02cc79ae19ea98ff59226e069de": "\\,x,y\\in M", "132697478da1ebf855774a52fab3b193": "\\chi ^{2}=\\sum ^{k}{\\frac {({\\text{observed}}-{\\text{expected}})^{2}}{\\text{expected}}}", "1326bfa9cab2cc91daed2e2054fa085b": "i_{s}", "132779ec50452e8bd1fe9cf13f9d8834": "=m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}+2\\left(m_{1}^{2}-p_{1}\\cdot p_{1}\\right)\\,", "1327c5a67041c39498c92429828f52bd": "K_{i}^{*}={\\frac {K_{i}k_{4}}{k_{3}+k_{4}}}", "1327c68053a3b44efab0e651016e0c60": "E=V-128", "1327e7eb009322eb5bf8773c2bab7bcb": "g=n+1,\\lambda =\\varphi (n),", "1327fa011c9a108b7e03af4e5e5a6272": "\\displaystyle \\int _{D}f(\\tau ){\\overline {g(\\tau )}}E(\\tau ,s)y^{k-2}dxdy", "1327fcc8c22e8fa5a446fd71df965e37": "{\\frac {dT}{dt}}+{\\frac {1}{\\tau }}T={\\frac {1}{\\tau }}T_{a},", "1328288bf2458c7c38005b885b88f1b2": "\\alpha =f_{1}(d_{a}),\\beta =f_{2}(d_{a})", "13285769c8cd293bfd541406e0621d87": "d=0", "1328f7ac8e44b744072fe5457b1a787a": "\\rho _{I}(t)=\\sum _{n}p_{n}(t)|\\psi _{n,I}(t)\\rangle \\langle \\psi _{n,I}(t)|=\\sum _{n}p_{n}(t)e^{iH_{0,S}t/\\hbar }|\\psi _{n,S}(t)\\rangle \\langle \\psi _{n,S}(t)|e^{-iH_{0,S}t/\\hbar }=e^{iH_{0,S}t/\\hbar }\\rho _{S}(t)e^{-iH_{0,S}t/\\hbar }.", "13291972707bbf15282eb48b3a58b85f": "C\\approx {\\frac {\\pi a(9-{\\sqrt {35}})}{2}}", "13293d96229660b4d58f4d598cb00853": "e^{X}Ye^{-X}=e^{\\operatorname {ad} _{X}}Y=Y+\\left[X,Y\\right]+{\\frac {1}{2!}}[X,[X,Y]]+{\\frac {1}{3!}}[X,[X,[X,Y]]]+\\cdots .", "13294ddcd2ae37f16ca81ab399088ecb": "{1 \\over 3}", "1329561c1be71291edf63fb128558fcf": "\\nabla \\times \\mathbf {H} ={\\frac {4}{c^{2}}}\\left(-4\\pi G\\mathbf {J} +{\\frac {\\partial \\mathbf {g} }{\\partial t}}\\right)\\,\\!", "132956250769ae7e923691734e6e9788": "v_{e}=I_{sp}g_{o}", "132993a1e61dac8fa6cb8da8d59e465a": "\\min(2,3,2,1,3,6)=1", "1329a1ea3cf18fa02d323888a46fb846": "{\\textrm {E}}[\\|{\\textbf {x}}_{k}-{\\hat {\\textbf {x}}}_{k|k}\\|^{2}]", "132a0ec0e90d85720803ceb65ff88b17": "r=c", "132a246a32b3636371660621e977e4ec": "\\therefore ", "132a2b9ff7d611c9947b4fb65921f73e": "R(u,v)w=\\Omega (u\\wedge v)w.", "132a53a090c1a8764a5c3cf45ad356a0": "b\\approx 43AU", "132b39cdff1c47d9b24723eb36d96a4c": "{k_{1} \\over k_{2}}={\\frac {\\ln(1-F_{1})}{\\ln(1-F_{2})}}", "132b45d748a0dc277832d754f6064583": "\\delta _{Y}(\\varepsilon )\\geq c\\,\\varepsilon ^{q},\\quad \\varepsilon \\in [0,2].", "132b7a718a970d6d5c32f5a3954ce494": "\\mathbb {E} \\left[\\left(\\int _{0}^{t}H_{s}\\,dB_{s}\\right)^{2}\\right]=\\mathbb {E} \\left[\\int _{0}^{t}H_{s}^{2}\\,ds\\right]", "132c1d66533a39a087d903f859f1ad4c": "\\phi _{n}(\\kappa )=0.033C_{n}^{2}\\kappa ^{-11/3},\\quad {\\frac {1}{L_{0}}}\\ll \\kappa \\ll {\\frac {1}{l_{0}}}", "132c834c23e12064579e7d5727bd243b": "{2,3,1}\\,\\!", "132c8644bf0affdf6857571014cfe3b9": "f(1)=0", "132ca804c2d37103e4a52c9fc645b444": "{\\partial u \\over \\partial x}={u_{i+1}^{n}-u_{i-1}^{n} \\over 2h}", "132cc50dbfc565f8ad72c623ed36feeb": "\\sum _{k=0}^{\\infty }{\\frac {1}{k+1}}{2k \\choose k}z^{k}={\\frac {1-{\\sqrt {1-4z}}}{2z}},|z|<{\\frac {1}{4}}", "132d340340a15905d4ff1c16aed8f4f2": "2^{n}\\equiv 2{\\pmod {n}}\\,", "132d42e04bdcdb4eac78be6206e7fd6f": "(\\underbrace {-,\\cdots ,-} _{k},\\underbrace {+,\\cdots ,+} _{n})\\,", "132d47002ae5c3ebf5e37764e5e11921": "V^{B}", "132d6ca808cf0abfc68b8b26afc104a4": "\\phi (\\tau ,z+\\lambda \\tau +\\mu )=e^{-2\\pi im(\\lambda ^{2}\\tau +2\\lambda z)}\\phi (\\tau ,z)", "132d86a0486dd349d6c0fbc5e5aec408": "\\,\\theta \\,", "132d9204086f03abfd00f2c0f28f3b0f": "F(x_{1},\\ldots ,x_{n},u,u_{x_{1}},\\ldots u_{x_{n}})=0.\\,", "132da3fb6246e107dcf0c89cf4efb65a": "{\\sqrt {2/(N-1)}}", "132dbb8ebe5d7a782ee5b17a139f28c2": "SP_{x}(t,f)=|ST_{x}(t,f)|^{2}=ST_{x}(t,f)\\,ST_{x}^{*}(t,f)", "132e0c3e2dfafa2973d0c971b80a6187": "f(x)\\in L_{\\omega }^{2}", "132e298c25d7fed88263fa595f83b242": "V[J]=\\int J\\wedge J\\wedge J", "132e2dc6655eefbfee62e01e9b64b571": "\\prod _{i=1}^{r}\\prod _{j=1}^{c}{\\frac {1}{1-x_{i}y_{j}}}.", "132e47b9ee867e241feeb4adb582d9a7": "(S,G,P)", "132e76ecc2f0a55d7627df17687f42f8": "\\left[{\\begin{matrix}\\sigma '_{11}&\\sigma '_{12}&\\sigma '_{13}\\\\\\sigma '_{21}&\\sigma '_{22}&\\sigma '_{23}\\\\\\sigma '_{31}&\\sigma '_{32}&\\sigma '_{33}\\\\\\end{matrix}}\\right]=\\left[{\\begin{matrix}a_{11}&a_{12}&a_{13}\\\\a_{21}&a_{22}&a_{23}\\\\a_{31}&a_{32}&a_{33}\\\\\\end{matrix}}\\right]\\left[{\\begin{matrix}\\sigma _{11}&\\sigma _{12}&\\sigma _{13}\\\\\\sigma _{21}&\\sigma _{22}&\\sigma _{23}\\\\\\sigma _{31}&\\sigma _{32}&\\sigma _{33}\\\\\\end{matrix}}\\right]\\left[{\\begin{matrix}a_{11}&a_{21}&a_{31}\\\\a_{12}&a_{22}&a_{32}\\\\a_{13}&a_{23}&a_{33}\\\\\\end{matrix}}\\right].", "132e89f895e672f7d768c59b730dea85": "1^{2}>0", "132e918e08c9e66aca00c78708edbdb1": "m\\rightarrow \\infty ", "132ed9975c221bd31a82c24d4d298377": "m_{n},\\,1\\leq n\\leq N", "132f2a7909026ad9d99eb3617e5fa5d2": "P={\\frac {\\pi ^{2}f^{2}I^{2}L^{2}}{2c^{2}h\\sigma }}\\,", "132f2e4d607214c35f77a6a8a223976c": "\\langle Tx,y\\rangle =\\int _{\\mathbb {R} }\\lambda \\,dE_{x,y}(\\lambda )", "132f8ba98a13f25af92dd3489c20d3ed": "B_{ij}\\equiv {\\frac {\\partial A_{j}}{\\partial x_{i}}}-{\\frac {\\partial A_{i}}{\\partial x_{j}}}", "132f9970d5ddbe6e4a617c2ee6fa7592": "(L_{n+1},R_{n+1})=(L_{n+1}',R_{n+1}')", "132fbae817d977b2489d77aae8b28b21": "e^{z}=1+{\\cfrac {z}{1-{\\cfrac {{\\frac {1}{2}}z}{1+{\\cfrac {{\\frac {1}{6}}z}{1-{\\cfrac {{\\frac {1}{6}}z}{1+{\\cfrac {{\\frac {1}{10}}z}{1-{\\cfrac {{\\frac {1}{10}}z}{1+-\\ddots }}}}}}}}}}}}.", "132fc9d7ae8570da5cae1c756fe647f3": "3{\\sqrt {N}}", "1330031b2c0b8f743c415044002c5990": "di_{i}", "1330295f7365e6b2525765b8b7c5347a": "\\operatorname {exp} \\equiv \\lambda m.\\lambda n.n\\ m", "13306c2f3619496b7e192f85d1fcad35": "F_{0}|_{A}=f_{0}", "133092a1bf8f87f422416eafe4d02b7f": "{\\mathcal {L}}_{X}\\omega =0.", "1330a9073cead0f3f632d228b833e311": "\\textstyle n^{2}", "1330d7d3a3e04888cb1d564ef967d6ff": "s={\\sqrt {\\frac {(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}-2}}}", "13310bbf243b195ce5d7a1981a908507": "\\lambda _{mod}={\\frac {\\lambda ^{2}}{\\Delta \\lambda }}\\ .", "13310e071799bf88465bbff9b19d8dc9": "\\mathrm {rk} \\,L={\\tfrac {1}{2}}\\mathrm {rk} \\,E", "13311b0d1e75ba697cd320411dade1eb": "{dx_{2} \\over dt}=r_{2}x_{2}\\left({K_{2}-x_{2}-\\alpha _{21}x_{1} \\over K_{2}}\\right).", "13315da8e16ec5944c65ea78df091456": "{\\mbox{eGFR}}={\\frac {{k}\\times {Height}}{Serum\\ Creatinine}}", "13319531c7ebbfaa01fcf4eb07a06132": "u(s)=\\left(K_{P}+K_{I}{\\frac {1}{s}}+K_{D}s\\right)e(s)", "133254c99c18642f9232fd0d29f8935a": "B_{0}=2,\\quad B_{i}=\\min _{0\\leq jx+n]v^{n}=\\,_{n}p_{x}v^{n}", "133706d1605beb1fa881cf4025412167": "\\mathrm {rem} \\left(x_{0},m_{i}\\right)=a_{i}", "13375c22930246d1b824a15c9a2be130": "\\scriptstyle {R_{l}^{0}}", "13379092ec45be1af10deb4c090e3527": "p(x|z)", "1337d9ff34c9a0c6876945f612884af7": "\\vert {\\Psi _{\\mathbf {p} }^{\\circ }}\\rangle ", "1337f98310fd31f9d4ddf2cec78347dd": "e=-d\\phi /dt=-SdB/dt", "13383997fff1a7f77ba27c1e9354ab04": "\\lambda =\\int _{\\,}{\\mathcal {E}}\\,dt", "13383f6e3ebb812b4bc0c0471be38c48": "\\sum _{k=0}^{\\infty }k^{4}{\\frac {z^{k}}{k!}}=(z+7z^{2}+6z^{3}+z^{4})e^{z}\\,\\!", "1338514089e09578d0b1a6700fd6eb29": "U(x,y,z,t)=u(x,y,z)e^{-i(kz-\\omega t)},", "13385cd2626cfc1bfd651830e67e2701": "{\\begin{bmatrix}0.5&0&0&0.5\\\\0&0.5&0&0.5\\\\0&0&0.5&0.5\\\\0&0&0&1\\end{bmatrix}}", "1338939c352a99a4780ebcb080dc2468": "A=B^{T}B=R^{T}Q^{T}QR=R^{T}R", "133919e2f37517dcc77b483f366747b7": "\\left({\\frac {dn_{1}}{dt}}\\right)_{\\mathrm {pos} \\,\\mathrm {absorb} }=-B_{12}n_{1}\\rho (\\nu )", "13397149a94379232e8eeb8a4a8af93b": "C={\\dot {V}}-C_{osm}", "133976d5ba017d7d872b0e6f6e598d7f": "\\,\\zeta <1/{\\sqrt {2}}", "13398f328ba973b9339f9e03fb7bfba4": "\\scriptstyle \\epsilon _{c}=p_{c}/p_{01}\\,", "1339a37da7375f2e51ec474b12db6d12": "{\\hat {H}}_{\\textrm {int}}=\\chi \\sum _{n,\\alpha }\\left[({\\hat {u}}_{n+1,\\alpha }-{\\hat {u}}_{n,\\alpha }){\\hat {A}}_{n,\\alpha }^{\\dagger }{\\hat {A}}_{n,\\alpha }\\right]", "1339d03c679e2fc6efefa28a4ba76646": "{1 \\over 7}", "133a36d041284fc8ce45d46132a4bd0d": "{\\begin{aligned}xy&=\\{X_{L}|X_{R}\\}\\{Y_{L}|Y_{R}\\}\\\\&=\\left\\{X_{L}y+xY_{L}-X_{L}Y_{L},X_{R}y+xY_{R}-X_{R}Y_{R}|X_{L}y+xY_{R}-X_{L}Y_{R},xY_{L}+X_{R}y-X_{R}Y_{L}\\right\\}\\\\\\end{aligned}}", "133a786238a843f2b6f075d701b704ab": "(\\mathbf {b} \\in \\operatorname {Im} (A))", "133a8d8d59795d2ef443dec95ffa2cda": "\\sigma _{\\mathrm {eq} }", "133ae19c283d03c7dcaf8edf823d73ad": "\\lambda ^{2}d\\mu _{h}=d\\mu _{k}\\,", "133ae4baa2da8a7e24135cc4221f547e": "\\rho _{1}\\geq \\cdots \\geq \\rho _{n}\\,", "133b491d22f26dcf8dca63855aefe28b": "G_{1}:\\{0,1\\}^{h}\\to \\{0,1\\}^{n}", "133b4acb325c07849cd7b4b06a813ae2": "X(z)=\\sum _{n=0}^{\\infty }x[n]z^{-n}", "133b7a84b36043b125b9dd0bc5620c69": "\\,{\\hat {C}}=CT", "133bec54d3df720bcda67ce19f16364b": "\\rightarrow (1)", "133c2c358ea4bc9f5734ac6732aea6d5": "{\\frac {f(x_{0}+\\epsilon )-f(x_{0})}{\\epsilon }}>K/2;", "133c5722f16a7c1d197fabe6edf073dd": "height_{midparent}={\\frac {1}{2}}(height_{father}+(1.08\\times height_{mother}))", "133c7f13616bd30d299879c648948055": "e_{1}={\\frac {P\\int K^{2}}{nh^{D}}}", "133c9151ba673d631a9b7e83ff2b57ba": "\\sum _{j=1}^{n}w_{j}x_{j}=W,", "133cb81b5fb186ec165b1ae5a60265a2": "({\\mathbb {N}},\\cdot ,\\uparrow )", "133cc6903f063f92a40f353bcb3091cc": "\\delta X(\\omega )", "133d15a3a6a3956e0a7778ec71d85d39": "\\mathbb {C} ^{n}=\\bigoplus _{i}Y_{i}", "133d5a7e53eb1623544ec86487d4a8fb": "\\scriptstyle {\\text{range}}\\,=\\,\\max _{i}(Y_{i})\\,-\\,\\min _{i}(Y_{i})", "133d5d6776100b4588234f7831489de7": "X=\\coprod _{\\alpha }{X_{\\alpha }}", "133d7f9f31bddd62b6bc8880c8f5d946": "E(\\mathbb {F} _{q})", "133dbb123218e2e771e45cea77bd301a": "cr(K_{m,n})", "133dc26295ba5508bdbd353d10f36ade": "\\mathbf {E} (\\mathbf {r} ,t)=\\mid \\mathbf {E} \\mid \\mathrm {Re} \\left\\{|\\zeta \\rangle \\exp \\left[i\\left(kz-\\omega t\\right)\\right]\\right\\}\\equiv \\mid \\mathbf {E} \\mid \\mathrm {Re} \\left\\{|\\phi \\rangle \\right\\}", "133dd4051053de4acd865b646dc1cfcf": "\\mathbf {M} ^{1},\\mathbf {N} ^{1}", "133dd94b78bcd13c5f8f2aab4623e2c2": "\\lbrace 17,23,29\\rbrace ", "133df2f858fe61455e9d923f2eedb4fe": "f_{e}(x_{e})=\\sin \\left({\\frac {n\\pi x_{e}}{L_{e}}}\\right)", "133e121d08e25c65bdf90665b5ec692d": "p(y,x)=\\sum _{1}^{n}w_{j}p_{j}(y,x),", "133e75244edb014f03c534d36547ad2a": "|f(x)-(S_{N}f)(x)|\\leq K{\\ln N \\over N^{p}}\\omega (2\\pi /N)", "133e76e81033a31efdc0c55fe42c2500": "{\\binom {k}{\\alpha }}={\\frac {k!}{\\alpha _{1}!\\alpha _{2}!\\cdots \\alpha _{n}!}}={\\frac {k!}{\\alpha !}}", "133ebe4e66278209e05d4ef60366950a": "\\lambda f.(\\lambda p.p\\ p)\\ (\\lambda x.f\\ (x\\ x))", "133ec32a11ce5196bf065519d1af434a": "z=x+iy=|z|(\\cos \\phi +i\\sin \\phi )=re^{i\\phi }\\ ", "133f1eb71de5e7e9f33479bf1ffc09cb": "g_{em}\\ =\\ 2\\left({\\theta _{\\mathrm {left} }-\\theta _{\\mathrm {right} } \\over \\theta _{\\mathrm {left} }+\\theta _{\\mathrm {right} }}\\right)", "133f461b9ea2866ad4052fc6f77dde03": "\\mathbf {B} ,", "133f908a0fe7da6392cf707d7532da28": "0\\leq l\\leq r\\leq n", "134047c6f2d1271f1f09a7d57f300050": "{\\begin{aligned}\\sin y=x\\ \\Leftrightarrow \\ &y=\\arcsin x+2k\\pi ,{\\text{ or }}\\\\&y=\\pi -\\arcsin x+2k\\pi \\end{aligned}}", "134072fb981be1fabaa8d3bc8a157fc9": "x^{2}+c=x^{2}-2xt+t^{2}", "1340912de620d2f69c86b7a8f82d9e4f": "\\scriptstyle TE_{mnl}", "13411e15c34c939e042a26cc78feefee": "\\log {g(\\zeta )-g(\\eta ) \\over \\zeta -\\eta }=-\\sum _{m,n\\geq 1}c_{mn}\\zeta ^{-m}\\eta ^{-n}", "1341914dc76b01b5f48b797ada5e2278": "\\mathbf {k} ", "1341a95f233b84a8046b7951e8d8d1bf": "\\left[\\left.{\\begin{array}{cccc}1&0&0&0\\\\0&0&0&0\\\\0&0&1&0\\\\0&0&0&1\\end{array}}\\right\\vert {\\begin{array}{cccc}0&0&0&0\\\\1&0&0&0\\\\0&1&1&1\\\\0&1&1&1\\end{array}}\\right].", "1341b662dcad96b90630ab8d84e9f7c2": "B\\ ", "1341d34474d7c9ae8ae6000ae1621705": "y[n]=(x*g)[n]=\\sum \\limits _{k=-\\infty }^{\\infty }{x[k]g[n-k]}.", "13425d29c65b33d585c51dde0d20e767": "{\\boldsymbol {\\mu }}_{a}", "13425eeed61b9fa166c4f35bd844dfb2": "{\\bar {\\mathsf {\\Omega }}}(a,x)\\mapsto {\\bar {\\mathsf {\\Omega }}}'(a,x)=R{\\bar {\\mathsf {\\Omega }}}(a,x)R^{\\dagger }-2a\\cdot \\nabla RR^{\\dagger }.", "1342da3464211ac0bb6a11410359c008": "O(n^{1/(2k-1)})", "1342de7d0703e1c9abee73fbdd6cb173": "\\langle x(t)\\rangle =\\langle x\\rangle _{0}+\\int \\limits _{-\\infty }^{t}\\!f(\\tau )\\chi (t-\\tau )\\,d\\tau ,", "134352ac75a2b6d3ca5766a3a629b8c5": "k[V]^{G}", "13436c247320284d49c07238c666286d": "c_{3}<0", "1343923278289008b04ad5f121b3ca76": "P(x)", "1343993b14ffa86c32c3c4d15e72073d": "\\{\\{-2,-1\\},\\{1,2\\}\\}", "13439c3307ae1ceba0998aa85991c218": "F_{M}", "1343e2405a6ece8515c1dbc6740430fd": "{\\mathcal {O}}^{X}", "134449c536b1ac37d78b34526fd197d1": "{\\textstyle \\sum }a_{k}z_{0}^{k}=a(z_{0})\\,({\\boldsymbol {wB}})", "1344c33158c30a3d04ede87305193000": "\\scriptstyle \\lfloor {\\frac {i+w}{J}}\\rfloor ", "13451deb165c2bc58d59b2a0673bb9da": "{\\begin{aligned}&\\left\\langle j'_{n},m'_{n},...,j'_{2},m'_{2},j'_{1},m'_{1}|j_{1},m_{1},j_{2},m_{2},...j_{n},m_{n}\\right\\rangle \\\\=&\\langle j'_{n},m'_{n}|...\\langle j'_{2},m'_{2}|\\langle j'_{1},m'_{1}||j_{1},m_{1}\\rangle |j_{2},m_{2}\\rangle ...|j_{n},m_{n}\\rangle \\\\=&\\prod _{k=1}^{n}\\left\\langle j'_{k},m'_{k}|j_{k},m_{k}\\right\\rangle \\end{aligned}}", "13456d2a0b0163e5ed501d708d14f81e": "1922^{12}", "13458606ac0ed75bc3167fb7aa312921": "{\\mathcal {N}}=(1,1)", "1345ceee29863e5fc5152d2cfa6ea016": "(U,\\varphi )\\,", "1345fadf3b0abf77d1f92de0bd6b9dde": "P[X_{1}\\leq u_{1},\\dots ,X_{n}\\leq u_{n}]\\leq P[Y_{1}\\leq u_{1},\\dots ,Y_{n}\\leq u_{n}]", "1346280563307dd9139fd4ce45dd2c57": "{\\begin{aligned}{\\mathcal {L}}_{K}=\\sum _{f}{\\overline {f}}(i\\partial \\!\\!\\!/\\!\\;-m_{f})f-{\\frac {1}{4}}A_{\\mu \\nu }A^{\\mu \\nu }-{\\frac {1}{2}}W_{\\mu \\nu }^{+}W^{-\\mu \\nu }+m_{W}^{2}W_{\\mu }^{+}W^{-\\mu }\\\\\\qquad -{\\frac {1}{4}}Z_{\\mu \\nu }Z^{\\mu \\nu }+{\\frac {1}{2}}m_{Z}^{2}Z_{\\mu }Z^{\\mu }+{\\frac {1}{2}}(\\partial ^{\\mu }H)(\\partial _{\\mu }H)-{\\frac {1}{2}}m_{H}^{2}H^{2}\\end{aligned}}", "134630743b6fb4c5cff9042f53149d72": "\\langle E\\rangle ^{2}=m^{2}c^{4}+\\langle \\mathbf {p} \\rangle ^{2}c^{2}.", "1346582c28b50643bd3a23af930310e4": "{\\begin{aligned}x^{1}&=x\\\\x^{n}&=x^{n-1}x\\quad {\\hbox{for }}n>1\\end{aligned}}", "13465b6d3bf9a84a71ed199ea982c4c8": "\\phi _{a}(\\omega )", "13467c607ceac041aa451742b18fbaa2": "\\beta _{g}=", "1346dc43ec12bd9657742c5ae8d0c0a7": "{\\frac {1}{x}}", "13470969d10dc4a7bb3884354fda0137": "v_{i}", "134766878c4e4b03c0c362b5bfc40f1b": "K_{m}^{\\text{app}}=K_{m}(1+[I]/K_{i})", "13477f050c5019c52feea347bdd90ce8": "\\left[{\\begin{smallmatrix}2&-1&0\\\\-1&2&0\\\\0&0&2\\end{smallmatrix}}\\right]", "134782eba81925e50ae0aff1007edf2c": "F=\\rho SV^{2}\\sin ^{2}(\\theta )", "1347aed7c5f264dbd5a679a2e94ebbda": "V_{-}\\,", "1347f9a8c5c697b7fa4992ba182e9b71": "{\\frac {k}{(k-1)}}", "1347fd20b7d0a229b50d1e432f2a5b68": "\\Lambda ={\\frac {h}{\\sqrt {3mkT}}},", "13480623c74d78a30c24aa310d8bb6cd": "\\nu _{Ti}=(ZeKE/m_{i})^{1/2}=1.69\\times 10^{7}Z^{1/2}K^{1/2}E^{1/2}\\mu ^{-1/2}{\\mbox{s}}^{-1}\\,", "13481239c55b3814795d3944a24f171e": "{\\mbox{Area}}={\\frac {1}{2}}ab\\sin C.", "1348c11881b7d1ba814e82f0b60f2c4a": "[\\omega \\wedge \\eta ]=\\omega \\wedge \\eta -(-1)^{pq}\\eta \\wedge \\omega ,", "1348d4b84f8dd009299afd64b8c1dad5": "\\scriptstyle \\theta ", "1348e80284a9ad55a9b73fde790f76fb": "B=\\{R_{1},\\dots ,R_{K}\\}", "134956319b41a7c6ac16713377bdd9a8": "{\\frac {1}{64}}\\int _{0}^{1}{\\frac {x^{16}(1-x)^{16}}{1+x^{2}}}\\,dx=\\pi -{\\frac {741\\,269\\,838\\,109}{235\\,953\\,517\\,800}}", "13496798b1e2e7bee66667072f59783b": "cr(K_{m,n})=\\lfloor n/2\\rfloor \\lfloor (n-1)/2\\rfloor \\lfloor m/2\\rfloor \\lfloor (m-1)/2\\rfloor .\\ ", "134995a181146891dcb464a453b0c888": "c^{2}d\\tau ^{2}=g_{\\mu \\nu }dx^{\\mu }dx^{\\nu }\\,\\!", "1349965ed2c5baa35fbd46e88db032ed": "S_{n}=\\sum _{i=1}^{n}a_{i}X_{i},", "13499a4e188a19c8e5d1f88037037f17": "\\operatorname {Categorical} ({\\boldsymbol {\\theta }}_{x_{t}})", "1349a8a8d242a4cf4351aa8e8bbe149d": "y=[3,3,5]\\,", "1349aeef7c8ba408bd0c335ef1077f98": "\\displaystyle {(x,y)_{0}=\\varphi (f_{x,y})}", "1349bb664f627dbfaab4c6b3ece4350d": "s^{-1}=S\\,", "1349cd4602162f137d131d73ac9881e7": "\\scriptstyle {A+{\\sqrt {5}}I}", "134a4965cdfb5e02180c5ccb775f1f6d": "\\sum _{n=0}^{N-1}Q_{n}(x)Q_{n}(y)\\pi _{n}={\\frac {1}{\\rho (x)}}\\delta _{x,y}", "134a657a7a8e136d733917d789d955a8": "{\\mbox{House P/E ratio}}={\\frac {\\mbox{House price}}{{\\mbox{Rent}}-{\\mbox{Expenses}}}}", "134a7ce93d7e65ed846fc1208af4aa21": "D\\approx D'+t", "134b086d86dad362441361c409900502": "C_{n-1}'\\not =C_{n-1}", "134b203250c2c1b0ff0c3918a28057b3": "\\langle \\phi (x_{1})...\\phi (x_{n})\\rangle ={\\int e^{-S}\\phi (x_{1})...\\phi (x_{n})D\\phi \\over \\int e^{-S}D\\phi }", "134b31b03360659b2909ce8be973374f": "{\\frac {d\\varphi }{dt}}=\\left(1-{\\frac {r_{s}}{r}}\\right){\\frac {L\\,c^{2}}{E\\,r^{2}}}\\,.", "134ba72d07c49a31137b952b01779e15": "\\operatorname {tr} (x^{k})=0", "134bd8b9d48c083f943399684eba58fb": "k\\propto {\\sqrt {\\frac {T}{M}}}{\\text{ (ideal gas)}}.", "134c0aba8f0f7b4c55a967c6db48ec89": "2^{bh(v)}-1", "134c1f959e7e679d4c31eba0cfd8a5c0": "\\exists x\\,c=x", "134c7737800ed7b1d5738898c7fa5c41": "\\lim _{x\\to \\infty }{\\frac {f(x)}{g(x)}}=0.", "134c7bf1a1dc1364b03b3c6cea450061": "(u^{2}+\\alpha +y)^{2}=(\\alpha +2y)u^{2}-\\beta u+(y^{2}+2y\\alpha +\\alpha ^{2}-\\gamma ),\\qquad \\qquad (3)\\,", "134ca6c9305ac8ec8f1d36a6278183f1": "2\\theta ^{\\circ }", "134cc2f6b5c1d2301ee7a1c89e922efe": "\\not =", "134ce3f5e3b06d8a6d338bd2e55b484b": "{\\begin{matrix}\\operatorname {Ta} (3)&=&87539319&=&167^{3}&+&436^{3}\\\\&&&=&228^{3}&+&423^{3}\\\\&&&=&255^{3}&+&414^{3}\\end{matrix}}", "134cf0be132297c8ce449fba44833d23": "m=(8/3)E/c^{2}", "134d1982658be76a6fe7aac6c740136b": "\\iint \\limits _{S}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\subset \\!\\supset \\mathbf {D} \\cdot \\mathrm {d} \\mathbf {A} ", "134d3af76bbc193f5209e0359b97035a": "{\\frac {1}{10}}+{\\frac {1}{10}}={\\frac {1}{5}}", "134d3c018786324961c4c581d902b4bf": "\\gamma =P(-z", "135de5d2a80d9bdfd581c3fb8abd7ed8": "u,v,", "135df1635b0181e64bb76d735a9dd9ab": "u_{i}^{n+1}={\\frac {u_{i}^{n}+u_{i}^{\\overline {n+1}}}{2}}-a{\\frac {\\Delta t}{2\\Delta x}}\\left(u_{i}^{\\overline {n+1}}-u_{i-1}^{\\overline {n+1}}\\right)", "135e615dad444216df9ffb2d06a290c7": "C_{2\\epsilon }=1.92", "135eaaeed8800c89e945ecb35f714f51": "{\\sqrt {\\frac {1}{56}}}\\!\\,", "135ef7e3a528accf8a7fa14e84c4adb4": "G_{\\mu \\nu }+\\Lambda g_{\\mu \\nu }={\\frac {8\\pi G}{c^{4}}}T_{\\mu \\nu }", "135f0b21da240c2b181561ef360b1890": "\\sigma ^{2}(\\langle A\\rangle )={\\frac {1}{M}}\\sigma ^{2}A\\left[1+2\\sum _{\\mu }\\left(1-{\\frac {\\mu }{M}}\\right)\\phi _{\\mu }\\right],", "135f12dabd027f2b6f8b81a0411945dc": "{\\frac {M}{m}}={\\sqrt {\\pi {\\frac {\\delta }{T}}{\\frac {V^{2}}{2}}}}\\mathrm {e} ^{\\left({\\frac {\\delta }{T}}{\\frac {V^{2}}{2}}\\right)}\\mathrm {erf} \\left({\\sqrt {{\\frac {\\delta }{T}}{\\frac {V^{2}}{2}}}}\\right)", "135f1b3b8adb4ccff6b1fa503cfee73a": "TCR=\\left({\\frac {l_{\\mathrm {sum~of~pieces} }}{l_{\\mathrm {tot~core~run} }}}\\right)\\times 100", "135f2a9e8499913f9280be68e0547196": "\\Psi (x)={\\begin{cases}e^{-{\\frac {1}{1-x^{2}}}}&{\\mbox{ for }}|x|<1\\\\0&{\\mbox{ otherwise}}\\end{cases}}", "135fbfb66efe89f65aba29ec1f57e088": "\\chi (S,{\\mathcal {O}}_{S})={\\frac {1}{12}}(c_{1}(S)^{2}+c_{2}(S))", "135fdeea4e0a32282427ab07523f98a6": "M(t;k,\\lambda )={\\frac {\\exp \\left({\\frac {\\lambda t}{1-2t}}\\right)}{(1-2t)^{k/2}}}.", "136012efaaae697cb42f9977241f0ffb": "(a^{p})", "13601c2d567143550b01a3a015505c6b": "K_{i}={\\frac {y_{i}}{x}}_{i}", "13607753dc207af118a6f5906b12be4f": "E_{r}\\,", "136078e6ebba8718a68add37deb80b44": "L=-\\partial _{x}^{2}+v(x,t)", "1360a23956abea8e55de99cfce0f8bff": "b(v)=E(\\max _{j\\neq i}v_{j}~|~v_{j}\\leq v~\\forall ~j)", "1360fa40d8a9dbb878b994e8bec3916a": "\\mathrm {H_{R}} ={{\\mbox{tube hematocrit}} \\over {\\mbox{feed reservoir hematocrit}}}", "13610ea260bd4212f416eb1fb90d8fc7": "\\ H_{i}^{BM}(X)=H^{-i}(X,\\mathbb {D} _{X}),", "136134b66637e72916e0caf51d42b2a9": "\\mathrm {SL} (2n,\\mathbb {C} )", "136172cde4030f5cdd4c8382afbe12ae": "\\tau _{\\max ,\\min }=\\pm R", "1361add4a3550f1d8d3b2081a3538c95": "{\\mathcal {E}}(\\rho )=\\sum _{i}A_{i}\\rho A_{i}^{\\dagger }", "1361afc324d9da7cdffd4b33b62bc5df": "s=v+u-{vu \\over c}\\,.", "13620a9fb842a4f28288c82d67063545": "\\textstyle \\mathbf {c} _{2}", "136236fbb4a90ceadaf5a46ddabf6222": "X_{1},\\ldots ,X_{n}", "136274abb512b3bbcd47339ef5446eb5": "|\\Psi (t)\\rangle =\\sum _{n}C_{n}(t)|\\Phi _{n}\\rangle ", "136281cbcb35cce6a6c724b1fdc86c7a": "(S\\subseteq X)\\;\\mapsto \\;\\exists _{f}S=\\{\\;y\\in Y\\;\\mid \\;\\exists x\\in f^{-1}\\lbrack \\{y\\}\\rbrack ,x\\in S\\;\\}=f\\lbrack S\\rbrack ", "13629935975995fa4da39df7fc0fcd64": "{\\frac {1}{H}}", "1362a1c16c3cfd959735dd3dd6816fa0": "e^{iz\\cos(\\phi )}=\\sum _{n=-\\infty }^{\\infty }i^{n}J_{n}(z)e^{in\\phi },\\!", "1362b8ef4124561aa318d8102211d65d": "D_{\\mu }=\\partial _{\\mu }\\pm ig_{s}t_{a}{\\mathcal {A}}_{\\mu }^{a}\\,,", "1362ea3e2bcf8011573390f6075eb3b6": "\\psi _{4}(x)=(2{\\sqrt {6}}\\,\\pi ^{1/4})^{-1}\\,(4x^{4}-12x^{2}+3)\\,\\mathrm {e} ^{-{\\frac {1}{2}}x^{2}}", "13630cde0bca1544d12c063c9c44d37a": "R_{1}\\simeq R_{2}\\simeq R", "13636687d2d40633b6914ad156310399": "{\\begin{array}{ccccccc}g_{1}&=&X&Z&Z&X&I\\\\g_{2}&=&I&X&Z&Z&X\\\\g_{3}&=&X&I&X&Z&Z\\\\g_{4}&=&Z&X&I&X&Z\\end{array}}", "136371b757a99c261a0bc00454a8b5dd": "\\lambda _{i}\\neq \\lambda _{j}", "136393b310d4a4d0ce5f4bfab3aed455": "C_{2}=\\prod _{p\\geq 3}{\\frac {p(p-2)}{(p-1)^{2}}}\\approx 0.660161815846869573927812110014\\dots ", "1363958cf6b0f5ca95aa77e05186bd11": "\\sum _{x}a^{x}={\\frac {a^{x}}{a-1}}+C\\,", "1363adc49cab9c2b6c64c76455b4103b": "d\\operatorname {Ad} ", "1364196bb8bcabcdf0c8f12048c92fd3": "W_{o}(t)=\\int \\limits _{0}^{t}e^{A^{T}\\tau }C^{T}Ce^{A\\tau }d\\tau ", "13641c4ed4cd6194596e51428ccac0a5": "\\sigma _{z}^{2}\\approx \\,\\,\\,{\\frac {1}{n}}\\,\\,\\left[{a^{2}\\sigma _{1}^{2}\\,\\,+\\,\\,\\,b^{2}\\sigma _{2}^{2}\\,\\,\\,+\\,\\,\\,2\\,a\\,b\\,\\sigma _{1,2}}\\right]", "13645114b8c545be75a31d2a5b24646e": "P/", "136452cf44c6429d500587445e911df1": "\\|x\\|_{bv_{0}}=\\sum _{i=1}^{\\infty }|x_{i+1}-x_{i}|", "13645b741cdf628074abe7e46f9a79d7": "0\\cdots >T_{j}", "136791245aa2325460447dca2fa71a12": "f_{Y_{[r_{1}:n]},\\cdots ,Y_{[r_{k}:n]}\\mid X_{r_{1}:n}\\cdots X_{r_{k}:n}}(y_{1},\\cdots ,y_{k}|x_{1},\\cdots ,x_{k})=\\prod _{i=1}^{k}f_{Y\\mid X}(y_{i}|x_{i})", "13679fb120b35b229b3e05404f8aa382": "{\\begin{aligned}\\forall w_{1},\\ldots ,w_{n}\\,\\forall A\\,([\\forall x\\in A&\\,\\exists !y\\,\\phi (x,y,w_{1},\\ldots ,w_{n},A)]\\\\&\\Rightarrow \\exists B\\,\\forall y\\,[y\\in B\\Leftrightarrow \\exists x\\in A\\,\\phi (x,y,w_{1},\\ldots ,w_{n},A)])\\end{aligned}}", "1367f6bf32a2668c02fb4a77e76a630c": "\\sigma _{s}={\\frac {n_{e}\\ e^{2}}{m_{e}\\ \\nu }}", "13681016cf9de2173e0ebcdda7581381": "{e \\over n}\\leq {{q-1} \\over q}\\left({1-{\\sqrt {1-{q \\over {q-1}}\\cdot {d \\over n}}}}\\,\\right)=J_{q}({d \\over n})", "13682bd627f2940a3279ea7b0dcdce48": "W_{i-1}", "13683671b6790314712469db219402c9": "d^{2}\\xi ^{\\prime }dz\\rho ({\\vec {\\xi }}^{\\prime },z)", "1368b899de5b7e71900fd806dcb8f14c": "|\\Psi _{C}\\rangle ", "1369100fedcec429e3834ca46d847ea1": "P(\\cdot |\\alpha ,\\beta )", "1369120d47bcb570e5c3465b99073a14": "2^{5}=32>26", "136995b56f396817203f1f5233ce267d": "d_{k}=-H_{k}g_{k}\\,\\!", "1369bb002c76449871bb2b207bdd91ae": "\\scriptstyle {\\vec {J}}={\\vec {L}}+{\\vec {S}}", "1369f8fd43395ddd5e49af69beffed02": "\\displaystyle {\\hat {f}}(-\\xi )={\\overline {{\\hat {f}}(\\xi )}}\\,", "136a1dc73d3479a1482df4e2f293fd4f": "\\xi =0", "136a21801e1ae63fab301f3b19d78b7e": "(4)\\qquad F_{ab}=A_{b\\,;a}-A_{a\\,;b}\\;,", "136a2ea462cf67501e39fe49fb602717": "\\mathrm {tr} (\\mathbf {AB} )=\\sum _{i}\\sum _{k}A_{ik}B_{ki}=\\sum _{k}\\sum _{i}B_{ki}A_{ik}=\\mathrm {tr} (\\mathbf {BA} )", "136a6c2a06fa87a5b1353466bda0788e": "DR_{p,c}={\\frac {\\sum _{p,c}(DR)}{count_{p,c}(singularcases)}}", "136a7d81159f175ef03e659567a7cd63": "g^{r'}~{\\bmod {~}}p", "136aa4cc2deb07514413c06ba58ca9d4": "{\\begin{aligned}\\operatorname {ns} (u)&={\\frac {1}{\\operatorname {sn} (u)}}\\\\[8pt]\\operatorname {nc} (u)&={\\frac {1}{\\operatorname {cn} (u)}}\\\\[8pt]\\operatorname {nd} (u)&={\\frac {1}{\\operatorname {dn} (u)}}\\end{aligned}}", "136ac4cf53968f2310e80756db6601a5": "c(q_{j})", "136acc1e219e7e8465c9bc4dae692e1f": "-(p+1)/r", "136b278e2de250a8682e336712119093": "q\\oplus (a)q", "136b4f8eb8849dc5e87a162e30f1e560": "Q(p)", "136b5d1c2e1a5207a43fa208fac550e5": "\\mathrm {H} (L,R)=(\\sigma (L),R)", "136bce849d912ba119952d760716f63e": "=\\max _{\\beta }\\left[1-{\\frac {1}{3}}\\left[\\exp \\left(-{\\frac {1.10x-20\\beta }{10}}\\right)+\\exp \\left(-{\\frac {1.10x}{10}}\\right)+\\exp \\left(-{\\frac {1.10x+20\\beta }{10}}\\right)\\right]\\right]", "136c740771455e77934c50250eb85b40": "Prob(ranking\\;1,2,\\ldots ,J)={exp(\\beta z_{1}) \\over \\sum _{j=1}^{J}exp(\\beta z_{nj})}{exp(\\beta z_{2}) \\over \\sum _{j=2}^{J}exp(\\beta z_{nj})}\\ldots {exp(\\beta z_{J-1}) \\over \\sum _{j=J-1}^{J}exp(\\beta z_{nj})}", "136cc7c76d8b6b6c3561210b71a26d8c": "A,B,C\\subseteq X\\,", "136ce9d75f387c554d1c7a5716f70dd0": "-{\\frac {d}{dx}}\\left[p(x){\\frac {dy}{dx}}\\right]+q(x)y=\\lambda w(x)y", "136cec92ad00e5750911c5173984995c": "\\varepsilon _{Nd(t)}=\\left[{\\frac {\\left({\\frac {^{143}Nd}{^{144}Nd}}\\right)_{sample(t)}}{\\left({\\frac {^{143}Nd}{^{144}Nd}}\\right)_{CHUR(t)}}}-1\\right]*10000", "136daf68d2fcdf660449579c7684f588": "R=A/nil(A)", "136e057a89522fda3f2160bf4ae29e6e": "\\scriptstyle x_{\\tau }(u)=\\delta (u-\\tau ).", "136e3a7ce0ee1b8181d2b156119b2543": "0=\\sum _{i=1}^{N}w_{i},\\;\\;0=\\sum _{i=1}^{N}w_{i}\\,c_{j,i}\\;\\;\\;(j=1,2,...,nx)", "136e54ce18481154840ff82bf0921877": "n_{i}=\\sum _{\\sigma }{\\hat {a}}_{i\\sigma }^{\\dagger }{\\hat {a}}_{i\\sigma }", "136e7596defa3afe882e06588efceef2": "Z", "136ece3cbc00399cf85b6790a699b61b": "{\\frac {p!}{k!(p-k)!}}", "136f2ef5e86dfc3c3946507a080d32c7": "M^{*}F(t)=QF(t/2).", "136f313de9976a602722f3184c11c17d": "D\\equiv 0{\\pmod {4}}", "136f5279da7978992bbb2f26d8d9233c": "E[M_{t}]=\\int _{0}^{\\infty }mf_{M_{t}}(m)\\,dm=\\int _{0}^{\\infty }m{\\sqrt {\\frac {2}{\\pi t}}}e^{-{\\frac {m^{2}}{2t}}}\\,dm={\\sqrt {\\frac {2t}{\\pi }}}", "136f6025ea6b5f6bcd989ae94f5977e9": "G={\\begin{bmatrix}I_{k}|P\\end{bmatrix}}", "136f8119128bf5713464b09b91b8acbd": "(C^{\\infty })'(M)", "136f8f7b3ff4529f70c24086afff8d8f": "(W-X^{T}1)r_{f}+X^{T}r=\\mu ,", "136f9f7f79f8f3cc054b9b2a650e804f": "W\\theta (0)", "137058ed167a5d2442c69a789769579e": "\\sigma _{11}=-p+2~\\lambda ^{2}~{\\cfrac {\\partial W}{\\partial I_{1}}}~;~~\\sigma _{11}=-p+{\\cfrac {2}{\\lambda ^{2}}}~{\\cfrac {\\partial W}{\\partial I_{1}}}~;~~\\sigma _{33}=-p+2~{\\cfrac {\\partial W}{\\partial I_{1}}}~.", "1370c864eb2f67df67723fa1437f198c": "m\\to 0", "13718bd5f184073cce55400e3c8acc8e": "v_{air}\\,\\!", "1371bdcba6f128e58bca080135f0f683": "\\mathbf {c} _{i}", "1371d4f76f50dcbd7710cbf2bc6d3b1e": "{\\tau }_{1}", "1372369aed4fa7e0b05ac8efac485540": "\\scriptstyle {R_{3}^{3}}", "13727862a20b058e29b86f69bb5b4baf": "500truckloads*.10(ECI)=50meatruckloads", "13728b4491a939a79cbdc12a7141f757": "G(t)=\\sum _{n=0}^{N}\\pi _{n}(t)", "1372a0e75c93f76c8333db2fbd64f7bb": "D_{\\mathrm {r} }", "1372cce5756a6fb83d8d3f2616cdeb1c": "(x_{1},y_{1})+\\ldots +(x_{n},y_{n})=O", "1372eca818fb486c6277ae40b2fa7da4": "\\operatorname {Pr} _{Y}(x,y)=y", "13730889a73e74c8cab459dcc247aa6c": "\\sum _{n=1}^{N}x_{n}", "137321b5e03c14e7c8809c284b484ba4": "R[x]/(x^{N}-1)", "137332dcf48f1aeb4dec20440b3b10a1": "a,b\\in \\mathbb {Z} _{q}", "13737d5cb4230ced7d9f3f722336f34c": "F=\\Delta PA", "1373e19ffbe4761848b6b5155cd321a3": "\\Omega _{1},\\dots ,\\Omega _{n}", "137425aeae8e33d918f4007ec7afe7e3": "s_{k}", "137430895b5795a01ac938af36d78160": "{\\begin{aligned}L&=10\\log _{10}{\\frac {x_{1}^{2}}{x_{2}^{2}}}&\\mathrm {dB} \\\\&=10\\log _{10}{\\left({\\frac {x_{1}}{x_{2}}}\\right)}^{2}&\\mathrm {dB} \\\\&=20\\log _{10}{\\frac {x_{1}}{x_{2}}}&\\mathrm {dB} \\\\&=\\ln {\\frac {x_{1}}{x_{2}}}&\\mathrm {Np} .\\\\\\end{aligned}}", "1374784edd01ebfc4da0acb387524ddd": "W({\\boldsymbol {F}})={\\hat {W}}(I_{1},I_{2},I_{3})={\\bar {W}}({\\bar {I}}_{1},{\\bar {I}}_{2},J)={\\tilde {W}}(\\lambda _{1},\\lambda _{2},\\lambda _{3})", "1374aa7dbf4849a6383e5c0e4e60e53e": "f(n)=\\sum _{d\\mid n}g(d)", "1374d1b45ba405e027bd63dcd276b6fe": "x^{*}", "13752f8569b8fcc49a0bed9f02338545": "{\\frac {\\eta ^{2}}{2}}", "13754d474526b209002526515675f9a2": "x_{1}x_{2}...x_{\\ell }", "1375836cccc27c1845242446967a079c": "10^{r}\\equiv 1{\\pmod {n}}", "13758573b562e906f010844c4001f8b1": "B_{0}\\subset R^{m}", "13759f9e38df443bd0155cff248d883f": "\\int _{0}^{\\infty }{\\frac {f(t)}{t}}\\,dt=\\int _{0}^{\\infty }F(p)\\,dp.", "1375bc8c503cf23cd43dad67cc1e2b8f": "m=k_{B}=1", "137620d6eb133dccc06f294763412a5b": "R_{tot}=1/(1/R_{1}+1/R_{2}+...+1/R_{n})", "137636c30def3f0cdde728451138e0be": "\\lfloor \\log _{2}(n)\\rfloor =19", "1376485059038fd4e7f3969059bf5c6a": "A-B\\geq 0", "1376851cfafa2d01f575745b80c09536": "f^{-1}(y)=\\ln(y)", "137690c4ec2f8ed11ef96b4a882ff7de": "\\phi _{i}(av)=a\\phi _{i}(v)", "137725489a1a2bf3a42dd1c7dc054b03": "C^{\\prime }={\\frac {2\\pi R}{\\sqrt {1-v^{2}/c^{2}}}}", "1377e4580b0c29ac513c81832435a18f": "\\textstyle \\sup _{s\\in [0,t]}\\mathbb {E} |M_{s}|<\\infty ", "1377e4ded193081a4165d3739f0316b8": "v_{x}", "13780c8ce9896bf51fbca91eab265e2f": "\\mathrm {crd} \\ 108^{\\circ }=\\mathrm {crd} (\\angle \\mathrm {ABC} )={\\frac {b}{a}}={\\frac {1+{\\sqrt {5}}}{2}},", "13782ba3c8f54f06459629a04fa764e0": "V={\\frac {nQ}{M}}", "13784c65bea3e22c51e63bc2a76af39d": "\\mu ^{-}", "1378b0ea4f16e22f851bedec7bc2853d": "\\mathbb {Z} /24\\mathbb {Z} ", "1378e529acc7dcee18b444de5acc7a8f": "p\\in [{\\underline {p}},{\\overline {p}}]", "1379135365aff8d4271ab4ddccdc3eb1": "ds^{2}={\\frac {1}{y^{2}}}\\left(dt^{2}-dy^{2}-\\sum _{i}dx_{i}^{2}\\right),", "13793107b866816ca20ccd668553f8ee": "r_{los}^{2}+r_{rg}^{2}", "1379723a3280a93acdde639dff1c2bfe": "\\mathbf {R} _{x}=E[\\mathbf {xx} ^{H}]={\\begin{bmatrix}R_{xx}(0)&R_{xx}^{*}(1)&R_{xx}^{*}(2)&\\cdots &R_{xx}^{*}(N-1)\\\\R_{xx}(1)&R_{xx}(0)&R_{xx}^{*}(1)&\\cdots &R_{xx}^{*}(N-2)\\\\R_{xx}(2)&R_{xx}(1)&R_{xx}(0)&\\cdots &R_{xx}^{*}(N-3)\\\\\\vdots &\\vdots &\\vdots &\\ddots &\\vdots \\\\R_{xx}(N-1)&R_{xx}(N-2)&R_{xx}(N-3)&\\cdots &R_{xx}(0)\\\\\\end{bmatrix}}", "1379bcc3eac33117fe20475bf7794f77": "U_{ji}", "1379ce43cd5c63a5370d841f49fb19e5": "x={\\frac {-b\\pm {\\sqrt {b^{2}-4ac}}}{2a}}={\\frac {1\\pm {\\sqrt {(-1)^{2}-4\\cdot 2\\cdot 4}}}{2\\cdot 2}}={\\frac {1\\pm {\\sqrt {-31}}}{4}}\\,\\!", "1379d37d449fb77abc720e653b2536b6": "RMS_{total}={\\sqrt {{RMS_{1}^{2}}+{RMS_{2}^{2}}+...+{RMS_{n}^{2}}}}", "1379d9368665c4e8a20a437932fbb438": "\\eta =w", "1379e5304064f9b91509ee863b51edf8": "(s+{\\vec {v}})(t+{\\vec {w}})=(st-{\\vec {v}}\\cdot {\\vec {w}})+(s{\\vec {w}}+t{\\vec {v}}+{\\vec {v}}\\times {\\vec {w}})", "137a0dd7db03a19aafa369ca30066012": "C_{k}=C_{i}g_{n_{1}}g_{n_{2}}\\cdots g_{n_{j}}", "137a1de3b187ba130de5cbc87340f310": "{\\hat {H}}={\\frac {1}{2}}\\left[{\\hat {p}}^{2}+\\Omega ^{2}(t){\\hat {q}}^{2}\\right].", "137a50086fe42a48a2e6aa6a40acb69a": "y_{t}=y_{0}+\\sum _{j=1}^{t}\\varepsilon _{j}", "137a5b787c73bd06df0f88940a46538f": "\\theta (u,\\xi )", "137aaccd9f336df140fc5c56ef994637": "W_{C}={\\frac {Q_{C}^{2}}{2C}}=2\\pi \\alpha W_{LC}.\\ ", "137b4d5df5c1fb3094a4c5d427d238f9": "x_{i}={\\frac {n_{i}}{n}}", "137c269981b03f3bf744b25af803f5bd": "\\quad W_{0}={\\frac {\\pi }{2}}\\qquad {\\text{ and }}\\quad W_{1}=1\\,", "137c48e7106e14d2670ac3c64a67245c": "{\\begin{pmatrix}1&0&0&0\\\\2&4&0&0\\\\0&3&3&0\\\\0&0&4&3\\\\\\end{pmatrix}}.", "137c4c8daac011f2092021bf18f91838": "L=\\lim _{n\\to \\infty }x_{n}", "137ca731118c981b9b810850a9f7d762": "p_{k},\\ldots ,p_{n}\\ (kA(i)\\}\\right\\vert ", "13899053ddf98c757bfe45562e2fce84": "n_{1}+n_{2}+\\cdots +n_{r}=n", "13899aa46204b02713a854702142955d": "P_{0}", "13899f65e16e008ce26fa729a050b528": "A_{\\alpha \\beta \\cdots ,\\gamma }=\\partial _{\\gamma }A_{\\alpha \\beta \\cdots }={\\dfrac {\\partial }{\\partial x^{\\gamma }}}A_{\\alpha \\beta \\cdots }", "1389d47d0aa8dc720a497c18ed6481f5": "a\\sin \\left({\\frac {x}{a}}\\right)", "1389f24a8338ce719d16d83142d63331": "(x\\pm i0)^{\\alpha }=x_{+}^{\\alpha }+e^{\\pm i\\pi \\alpha }x_{-}^{\\alpha },", "138a2645a4f744c7655fd5a5e6842431": "(X,{\\mathcal {T}})", "138b41111570a898f0329dc1c4e8cb84": "P(t+\\tau )=P(t)P(\\tau )\\ ", "138b5e1b12b3dc93c1b366b1cecf52fb": "-\\$5.26-3\\cdot \\$9.99", "138b65ddb20677aed17c4ee7be0bab59": "\\left\\{{4 \\atop 4/2}\\right\\}", "138b96d357d1dbf023d63302ca4fa236": "\\sum _{n=1}^{\\infty }{\\frac {1}{n}}=\\prod _{p}{\\frac {1}{1-p^{-1}}}=\\prod _{p}\\left(1+{\\frac {1}{p}}+{\\frac {1}{p^{2}}}+\\cdots \\right)", "138bde2d21ca0ae2352e13c7f4a91a58": "a_{r}={\\frac {(r+c+\\alpha -1)(r+c+\\beta -1)}{(r+c)(r+c+\\gamma -1)}}a_{r-1},{\\text{ for }}r\\geq 1.", "138be8b82adc8730f67c56b92bae0eee": "\\phi ^{*}=-1", "138c6545ca874b5cf3e084cc89a02a63": "\\psi (\\mathbf {x} ,\\tau )=\\mathrm {e} ^{K\\tau }\\psi (\\mathbf {x} )\\mathrm {e} ^{-K\\tau }", "138c8f109e48c641d347f0ca144b0e39": "-U_{A}(\\delta _{B})U_{B}(\\delta _{B})\\leq -U_{A}(\\delta _{A})U_{B}(\\delta _{A})", "138ccf9ccea0f33f6b0c9c6876cb4c16": "\\psi ({\\vec {r}})", "138d127a5421ac7869e068005ccee17e": "\\alpha (t)={\\frac {IR_{P}(t)}{R_{S}(t)}}\\qquad \\qquad (6)", "138d5f82a5c03aa4eee6d1cd17b696e8": "{\\dot {x}}=Ax", "138d640060375a883fbd74f71590abf4": "E\\supset F", "138d6a6fc53d1b77dc76d881dc22a145": "e^{{\\frac {-i\\delta }{\\hbar }}H_{k,k+1}}.", "138d9f0cbc8ae0deb9e8348dec2c4c16": "r(x(t),u(t))dt=dR(x(t),u(t))", "138ddb8f654b4ff218add31482d356d6": "Q={\\sqrt {(R^{2}+a^{2})^{2}-(2az)^{2}}}.", "138eede587eb7cb6233a68550ef6eadb": "H=\\sum _{i}x_{i}{\\partial \\over \\partial x_{i}},", "138ef8b49caa515783e0d7e4b3f394dd": "2(\\nabla _{X}Y,Z)=X\\cdot (Y,Z)+Y\\cdot (X,Z)-Z\\cdot (X,Y)+([X,Y],Z)+([Z,X],Y)+(X,[Z,Y]),", "138f4bf2027bfa7624a45b158b0af615": "M(\\mu ,\\sigma )", "138f7dfe3628974a1ca00535b6634cd0": "d=\\gcd(d_{1},v_{1}+v_{2}+h)=\\gcd(x-1,-6x+22)=1", "138fcbf2237d26bab3c454e45fd87151": "i\\hbar {\\frac {\\partial }{\\partial t}}\\rho ={\\mathcal {H}}[\\rho ]", "138fee2b86d3c0c4604dc123f34bc454": "\\mathbf {b_{2}} =2\\pi {\\frac {\\mathbf {a_{3}} \\times \\mathbf {a_{1}} }{\\mathbf {a_{2}} \\cdot (\\mathbf {a_{3}} \\times \\mathbf {a_{1}} )}}", "139005c289391e4dc52df5e595404b5a": "\\Delta (x)={\\mathcal {O}}\\left(x^{\\theta +\\epsilon }\\right)", "1390442d61c28d944d7a39a240a26530": "(f,{\\hat {O}}g)=({\\hat {O}}f,g)", "1390d4579c5cc0c926cbe7e66b77d54d": "T=C_{1}\\cup C_{2}\\cup \\cdots \\cup C_{m}", "1390ed3f3f8522eaad1f83b338ddee9c": "\\Delta (a)\\in {\\mathcal {C}}^{\\mathcal {J}}", "13912a3ca98677f892f445f5b1300c3d": "\\mu /T", "13913219d68657e24f239b20b865e5e7": "P_{c}^{(k)}(0)=P_{c}^{(k+n)}(0)", "13917f94cefdb0d9ff0f84555477bd01": "2^{2^{\\overset {n}{}}}+1", "139182d52c3c7f28e8ba425ca152c666": "{\\frac {dx}{dt}}=f(x)", "1391f0d9dfdc1bf756433daef0ea0ce0": "P(A\\ {\\mbox{and}}\\ B)=g(P(A),P(B|A))", "1392065f3f8c8367cd26d30dbf021df5": "p_{v,w,k}(G)", "13921f3c95c76b3d2f1646c85060d753": "f(q)\\ =f(w+xi+yj+zk)\\ =wf(1)+xf(i)+yf(j)+zf(k)\\ =w-xi-yj-zk\\ =q^{*}.", "1392370d74da2f451a05f3413430d8c9": "df:T_{x}X\\rightarrow T_{f(x)}Y", "139239d03b317553540c7d23a16223e3": "f(n_{i})=\\sum _{i}(n_{i}+g_{i})\\ln(n_{i}+g_{i})-n_{i}\\ln(n_{i})+\\alpha \\left(N-\\sum n_{i}\\right)+\\beta \\left(E-\\sum n_{i}\\varepsilon _{i}\\right)+K.", "139255cec61f93b2848c497fc367573e": "F_{\\theta }=aF_{A}-bF_{B}=0.\\,\\!", "1392768521f2c4dd9d791ea425772ac7": "P={\\frac {\\rho AV^{3}}{2}}C_{P}", "13930b0ba518cd87f3c7aa5294e62619": "{\\frac {5}{6}}\\rho ", "13937363c58307c2bf9e767cfbc66fde": "\\rho (z_{1},z_{2})=\\log {\\frac {|z_{1}-{\\overline {z_{2}}}|+|z_{1}-z_{2}|}{|z_{1}-{\\overline {z_{2}}}|-|z_{1}-z_{2}|}}", "1393b8b8d829705d728a201b457bfef1": "g_{0es}", "139402e6328f843b622a6bc0069b6942": "\\lambda _{n+1}", "1394092508a282c4515983a71d3cf0da": "{\\frac {\\lambda }{1+\\lambda }}\\nabla \\cdot \\left(\\mathbf {\\Sigma } _{i}\\nabla v\\right)=\\chi \\left(C_{m}{\\frac {\\partial v}{\\partial t}}+I_{\\text{ion}}\\right).", "139436d7ba8aa26f22039d940e898a2f": "h{{\\left[{\\frac {\\mu _{v}^{2}}{g{{\\rho }_{v}}\\left({{\\rho }_{L}}-{{\\rho }_{v}}\\right)k_{v}^{3}}}\\right]}^{{}^{1}\\!\\!\\diagup \\!\\!{}_{3}\\;}}=0.0020{{\\left[{\\frac {4m}{\\pi {{D}_{v}}{{\\mu }_{v}}}}\\right]}^{0.6}}", "139483beaeb74051ce81b3a25e76eca7": "{\\mathfrak {gl}},", "1394cf45e4685fdf36505e6eb6527bc6": "z\\left(x,y\\right)=\\left(x,y+z\\right)", "1394fcc7a60fcc0ad9769c73b423399f": "f\\in \\sigma ", "139580bf570d8ad46ebb7a867d642bd6": "|x-y|\\geq {\\bigg |}|x|-|y|{\\bigg |}.", "13958edccb8b6efb5b844a25ba92126d": "{\\frac {(i\\omega )^{2}+\\xi ^{2}}{((i\\omega )^{2}-\\xi ^{2})^{2}}}", "1395a5b0361707b3c7fb4be218d3690c": "{\\partial \\over {\\partial x^{a}}}\\equiv \\partial _{a}\\equiv {}_{,a}\\equiv (\\partial /\\partial ct,\\nabla )", "1395c2c2e805891838cabda81a9ad18d": "{\\rm {trig}}(M)=(0,2g,g_{3})\\,", "13962c1563475279c2fd0dadbc0f9dcc": "\\cos(2\\sigma _{m})=\\cos \\sigma -{\\frac {2\\sin U_{1}\\sin U_{2}}{\\cos ^{2}\\alpha }}\\,", "139635b4ef68570a2c388dc2959a05c4": "f(x+)=\\lim _{y\\to x^{+}}f(y)", "139650b71b241f871e85a92f3c53e30f": "a\\in \\left\\{0,...,(p-1)(q-1)/4-1\\right\\}", "1396540cbdb53b994df321149be608c6": "\\left\\{{\\frac {(1+x_{2})x_{1}+(1+x_{2})x_{3}}{x_{1}x_{2}x_{3}}},{\\frac {x_{1}+x_{3}}{x_{2}}},{\\frac {(1+x_{2})x_{1}+x_{3}}{x_{2}x_{3}}}\\right\\},", "13970073048c13fa98f8f25354db4073": "{0+1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+\\cdots }", "13975e67033638c94eb919a345ebf8f2": "x=918.082", "1397999bd4e9728137aab993fc4f599a": "f:A\\to Q", "13981791e732963ce7efd1774e65e876": "X[f_{1}\\dotso f_{i}f_{i+1}\\dotso f_{n}]\\to \\alpha Y[f_{1}\\dotso f_{i}]\\beta Z[f_{i+1}\\dotso f_{n}]\\gamma ", "13981c380df46df15370cd0c188cc9f1": "\\operatorname {gr} _{m}O_{X,x}=\\bigoplus _{i\\geq 0}m^{i}/m^{i+1}.", "1398253a5cd2be953826b8be16956b58": "{\\mathbf {} }K(t)=P(t)C^{\\mathrm {T} }(t)W^{-1}(t)", "13982f20c005669cfb6c88f7691bf242": "a_{0}-L", "13983de680535568fb7d4537975b9353": "\\mathbf {y} =n({\\overline {\\mathbf {x} }}-{\\boldsymbol {\\mu }})'{\\mathbf {\\Sigma } }^{-1}({\\overline {\\mathbf {x} }}-{\\boldsymbol {\\mathbf {\\mu } }})", "13991f26a97da351e592720ed88a6f3f": "{25 \\over 2}", "13997636c12724d9f4ce906a479e7b3b": "X_{n}\\ \\xrightarrow {p} \\ X\\quad \\Rightarrow \\quad X_{n}\\ \\xrightarrow {d} \\ X,", "1399834dd4c3f5a18ca05f0ec5187593": "{\\frac {b}{d}}", "1399ab3520ac37ecfe8fffaf77818532": "{\\frac {\\partial \\eta }{\\partial t}}+{\\frac {\\partial \\Phi }{\\partial x}}\\,{\\frac {\\partial \\eta }{\\partial x}}+{\\frac {\\partial \\Phi }{\\partial y}}\\,{\\frac {\\partial \\eta }{\\partial y}}={\\frac {\\partial \\Phi }{\\partial z}}\\qquad {\\text{ at }}z=\\eta (x,y,t).", "1399afeae9068428fb649b2f58e2d837": "\\|A\\|_{F}\\leq \\|A\\|_{*}\\leq {\\sqrt {r}}\\|A\\|_{F}", "1399b73092d3c7cc5930fd0d299319ae": "{\\boldsymbol {\\alpha }}=(\\alpha _{1},\\alpha _{2},\\ldots ,\\alpha _{K})", "1399ee959b9c3e978484e5b38b2de434": "{\\hat {X(t|T)}}=\\sum _{i=1}^{\\infty }{\\hat {X_{i}}}\\Phi _{i}(t)", "1399f16c0e05287ee03576a9a5a594f2": "Qp_{n}(x)=np_{n-1}(x).\\,", "139a5f6c5804614df8d2b96c09d599d4": "{\\mathit {He}}_{7}(x)=x^{7}-21x^{5}+105x^{3}-105x\\,", "139a90c221557c3ed4d82640bb001feb": "1\\!\\!1_{B(k)}", "139aa459357bbc403230f5aeff8c9bd2": "{\\hat {\\varepsilon }}(\\omega )=\\varepsilon '(\\omega )-i\\varepsilon ''(\\omega )", "139aa6db3d5ffb3c73a79d653af4a26d": "u_{8}", "139acf60ce8d35cffa79c9e638bda563": "h_{0}={X_{cross_{1}} \\over 2}{\\sqrt {V_{1}-V_{0} \\over V_{1}+V_{0}}}", "139adac7a4d34459f8200a36670125f4": "{{\\text{f}}_{\\text{P5}} \\over {\\text{f}}_{\\text{T}}}=2^{(7/12)}\\approx 1.49821\\approx {3 \\over 2}", "139b12acfd62c0c9e58ce3074718e8f2": "I=veN/L", "139b19b7ba394648e575d747ce355c5c": "{\\overset {\\triangledown }{\\boldsymbol {\\sigma }}}={\\dot {\\boldsymbol {\\sigma }}}-{\\boldsymbol {l}}\\cdot {\\boldsymbol {\\sigma }}-{\\boldsymbol {\\sigma }}\\cdot {\\boldsymbol {l}}^{T}", "139b2bd3c2a26c61a3ad4f01b039dfae": "f:\\mathbb {N} \\to \\mathbb {N} ", "139b31f52f59a6b0baf38be01ce5c286": "C=\\mathrm {d} q/\\mathrm {d} V\\,\\!", "139b8eb65be2ce7670dc5dd97965ea18": "\\dotso ", "139bf931df0172d67e975268a0da28cf": "\\theta _{0}", "139c1c707c03ecdd16ff500bd5d62795": "{\\hat {\\rho }}=\\exp {\\big (}{\\tfrac {1}{kT}}(A-{\\hat {H}}){\\big )},", "139c7429b09fd9dcb93be6e1b43445e2": "d_{1}={\\begin{pmatrix}1&0&0\\\\1/2&1/2&0\\\\1/2&-1/2&w\\end{pmatrix}}.", "139cd3b40768e63fcc2477a647912f3e": "\\mathbb {R} _{x}^{n}\\times \\mathrm {R} _{\\xi }^{N}", "139ce03bd868c678d141ba0c0694f17e": "\\varphi :{\\mathcal {X}}\\to \\mathbb {R} ", "139ce992f89071078f21c526c73a1129": "{\\tilde {A}}_{2}(R)=2\\pi R", "139d2186190f2751b5744a9c2d2e85c6": "dQ=C_{V}dT+P\\,dV.", "139d31e416990a77ae77f7707e47c283": "R_{n}^{2}(\\xi ,-x)=R_{n}^{2}(\\xi ,x)", "139d5ee757298244ee3fb4a70218cca3": "\\left(\\left({\\frac {b_{n}}{a_{n}}}\\right)^{2}\\right)^{N_{t}}=10^{N_{d}}", "139d7a416ec5e0d694e6f8ccb1897598": "\\mathbf {A} _{i}=\\alpha \\times (\\mathbf {r} _{i}-\\mathbf {R} )+\\omega \\times \\omega \\times (\\mathbf {r} _{i}-\\mathbf {R} )+\\mathbf {A} .", "139d8c95b22903818ae3fc6bdbbaf484": "\\operatorname {arcsec} x", "139da28898a3a8f4a4b540eb4f30197d": "Y=\\beta _{30}+\\beta _{31}X+\\beta _{32}Me+\\varepsilon _{3}", "139dabbd6d49ca7d4173bfc97e309567": "J_{y}\\sim {\\frac {B_{in}}{\\mu _{0}\\delta }},", "139dc70371bc58bd36588eac59273528": "\\Diamond _{i}", "139de60d60eebe461ce944aed52c5ac1": "\\alpha (\\theta )\\,\\,\\equiv \\,\\,\\left[{\\,1\\,\\,\\,+\\,\\,\\,{1 \\over 4}\\sin ^{2}\\left({\\theta \\over 2}\\right)\\,}\\right]^{2}", "139e4170c21d840b942914ca25ec79e2": "q\\neq 0\\in {\\mathbb {C}}", "139e6a0a8ba62b6d91239788708b6c27": "\\Gamma (n+{\\tfrac {1}{4}})=\\Gamma ({\\tfrac {1}{4}}){\\frac {(4n-3)!^{(4)}}{4^{n}}}", "139e9bd0bc65b7788031d1f298ade80f": "{\\mathcal {F}}:L_{\\mu }^{2}(G)\\rightarrow L_{\\nu }^{2}({\\widehat {G}}).", "139f120e78490ad580d52c1c5d7e15cc": "A_{Bq}=nN_{A}{\\frac {\\ln(2)}{t_{1/2}}}", "139f1ab475bef4718960f797cd17c360": "D_{f}", "139f93d9d22a9eb7c5378f539d28c52d": "s_{n,k}(t)=2^{1+n/2}\\int _{0}^{t}\\psi _{n,k}(u)\\,du,\\quad t\\in [0,1],\\ 0\\leq k<2^{n}.", "139f99e3ce496865b4d82434f35be559": "h_{a}(x)=", "139fb3f33193989bbc214dade614669d": "\\textstyle G'", "139fc0c23e71138369bd21fe428e217a": "p_{k}(x_{1},\\ldots ,x_{n})=\\sum \\nolimits _{i=1}^{n}x_{i}^{k}=x_{1}^{k}+\\cdots +x_{n}^{k},", "139ff8801186f3cbe1bb867897c42db8": "\\mathrm {verb\\ form} ={\\mbox{stem}}+{\\mbox{thematic vowel}}+{\\mbox{inflectional suffix}}", "13a004c2b20e756d91a85b04eb5e93a4": "(j,k)", "13a0e3336da790da7f4fe483e4ac4818": "P\\left(y\\right)", "13a13ea0a6972ed94afdfb4256b28b3c": "{\\frac {x}{b}}={\\frac {c}{d}}", "13a14a36b6b8fe3c5ce9c859bb8e008b": "{}'", "13a1a2181217b0f6f4c471f0e1fd1c1c": "F_{2}=\\{e\\}\\cup S(a)\\cup S(a^{-1})\\cup S(b)\\cup S(b^{-1})", "13a1d8c76a4ed9746ec064e1ddd05792": "\\sup _{\\alpha }E(G(|X_{\\alpha }|))<\\infty .", "13a2096381197da1b640b2c00aee9a1b": "f=\\chi _{S}-{\\frac {1}{2}}", "13a248d66e721f2b7dbd3caa0161c593": "E_{0}", "13a25ea40c1a73a51817a2e290ad2661": "(a(x_{0}-m)+b(y_{0}-n))^{2}=a^{2}(x_{0}-m)^{2}+2ab(y_{0}-n)(x_{0}-m)+b^{2}(y_{0}-n)^{2}=(a^{2}+b^{2})((x_{0}-m)^{2}+(y_{0}-n)^{2})", "13a2b22efb5eef810e052d13ccf79e18": "xc\\\\\\end{cases}}", "13a6cd3741dd1e205768fa3c190c623d": "h_{\\text{FE}}", "13a712e2e5e37c4591599e6462a3c539": "P(||X-\\mu ||\\geq k||\\sigma ||)\\leq {\\frac {1}{k^{2}}}.", "13a7980c0175a147ec579c1de932ff7f": "K={\\frac {\\det {\\begin{vmatrix}-{\\frac {1}{2}}E_{vv}+F_{uv}-{\\frac {1}{2}}G_{uu}&{\\frac {1}{2}}E_{u}&F_{u}-{\\frac {1}{2}}E_{v}\\\\F_{v}-{\\frac {1}{2}}G_{u}&E&F\\\\{\\frac {1}{2}}G_{v}&F&G\\end{vmatrix}}-\\det {\\begin{vmatrix}0&{\\frac {1}{2}}E_{v}&{\\frac {1}{2}}G_{u}\\\\{\\frac {1}{2}}E_{v}&E&F\\\\{\\frac {1}{2}}G_{u}&F&G\\end{vmatrix}}}{(EG-F^{2})^{2}}}", "13a7d841fc898c8c02c13f038fe92740": "W=kTN\\ln(V_{2}/V_{1})\\,\\!", "13a806f9b35a643e1fcc46e7004644ca": "{\\frac {F}{\\rho AV^{2}}}=f(R_{n},\\alpha )", "13a819238aabe85e3290a2ff42fabe4c": "{\\frac {d^{2}r}{d\\theta ^{2}}}\\cdot {\\dot {\\theta }}^{2}+{\\frac {dr}{d\\theta }}\\cdot {\\ddot {\\theta }}-r{\\dot {\\theta }}^{2}=-{\\frac {\\mu }{r^{2}}}", "13a81e9b5ec3c7289685fe7ff36093be": "[0,1]\\subseteq \\biguplus _{k}V_{k}\\subseteq [-1,2]", "13a82b51e585f7d1b00306f43451deda": "SK={\\frac {F(1-\\alpha )+F(\\alpha )-2Q_{2}}{Q_{3}-Q_{1}}}", "13a8681d826f0a8ce514209bb3296296": "x^{i}e^{-a|x|^{2}}\\in S(\\mathbf {R} ^{n}).", "13a88c5330b738ed79fd6108e89529ca": "K\\wedge P", "13a8ba603816294cce5461895078cfe0": "{\\nabla }^{2}\\rightarrow {\\delta _{z}}^{2}", "13a8d1c9ab3fdf40db247d818a3923a9": "\\{x|x1", "13b013f5414f688a6d92858b772bd107": "{\\stackrel {\\mathbf {E} _{\\bot }}{}}", "13b056184eb7ad4941965732413b07b3": "p(x)=x^{3}+x^{2}-2x-1", "13b085cc51d035eb76dd0bfa554c839a": "{\\dot {\\gamma }}_{e}", "13b09f6e00a4f6ee5820a7e5dd4041e3": "(\\theta _{(\\alpha )}^{*};\\theta _{(1-\\alpha )}^{*})", "13b17d219f5366cd59f9559444470bba": "I_{D}={\\frac {\\mu _{n}C_{ox}}{2}}{\\frac {W}{L}}(V_{GS}-V_{th})^{2}\\left(1+\\lambda (V_{DS}-V_{DSsat})\\right).", "13b220906096a833240c35ac61bea2f9": "\\Delta v=v_{e}\\ln {\\frac {m_{0}}{m_{1}}}", "13b2605642a0828f17eaf6b14cc39115": "\\mathrm {Distance} (b^{\\mathrm {ideal} },b_{\\mathrm {Inverse~iteration} }^{k})=O\\left(\\left|{\\frac {\\mu -\\lambda _{\\mathrm {closest~to~} \\mu }}{\\mu -\\lambda _{\\mathrm {second~closest~to~} \\mu }}}\\right|^{k}\\right).", "13b2617da4326976297e3fda1be47c60": "S(t|\\theta )=1-F(t|\\theta )", "13b2a29edfa63cb4f376f9f462203cff": "\\lnot a\\Rightarrow b", "13b3083b8f2fb1c7b7167c7eb4dac524": "\\scriptstyle H=h(X)", "13b31244048bb600c8df889e693462e8": "{\\begin{aligned}P&=1{\\text{mW}}\\cdot 10^{\\frac {x}{10}}\\\\P&=1{\\text{W}}\\cdot 10^{\\frac {x-30}{10}}\\end{aligned}}", "13b382773db3b0030a79c8c8879acedf": "\\lambda '-\\lambda ={\\frac {h}{m_{e}c}}(1-\\cos {\\theta }),", "13b3e1dc8890efe0bca8e279e98536c4": "C^{\\prime }(a,q,\\xi )={\\frac {\\partial C(a,q,\\xi )}{\\partial \\xi }}", "13b42d63d4ed41298de411106db9997b": "\\lim _{x\\to 0}x\\ln(x)=0", "13b44a5310ad050fd3659fe683480256": "4p^{3}+27q^{2}\\leq 0\\,,", "13b4a21e630d48396eb5b9858a91b508": "{\\mathcal {M}}=NI", "13b4ac2a4d9f1629bc7d9dbbbdac7953": "f_{0}\\,", "13b4b01f1960159186f3db3a9635dd81": "{\\vec {k}}\\perp {\\vec {B}}_{0}", "13b4dfbf05dcdff8b708363ebea2b131": "\\lim _{i\\to +\\infty }\\lambda _{i}=0.", "13b4fedf6dd767b15eea81b1a8fdba53": "\\mathrm {SINR} (x){=}{\\frac {P}{I+N}}", "13b4ff0f3eb0a5e8c883dea9b78eaa14": "\\left({\\ddot {\\mathbf {r}}}\\right)", "13b51b66631a6b9fd9f3ba5545c49d3a": "d=\\partial +{\\bar {\\partial }}", "13b52edc722dfb4460cd039e9fae414e": "1/(1-z)", "13b56608a775d37fe34ef6935a7d89e5": "\\gamma _{f}(\\theta ))", "13b5dfc6528f8fd69cfe012ac78bff92": "g^{th}", "13b5e00bc189f331925dcde4c4a240f0": "{\\partial u \\over \\partial x}+{\\partial v \\over \\partial y}=0", "13b6f1165f57dec6285dbe3b25ea5fae": "S=kJ^{z}", "13b6fc21d83d6fcaf39403ccef6ff9d6": "L(f,{\\hat {f}})=\\|f-{\\hat {f}}\\|_{2}^{2}\\,,", "13b746f1ef5257fb901be21456ef5001": "V_{Atom}", "13b7b7bb33bf08524eda17f7e13caf9b": "\\scriptstyle -{\\frac {1}{7}}", "13b7c88f95f9fbbfa455dd466f19d3a1": "V_{oc}", "13b7df217eb78c8c450d688982a5c153": "|\\sigma |", "13b80d2c9594a5f63f41835ef889b7e3": "\\mathbb {R} \\supset [,]\\ni t\\to x(t)\\in X", "13b8d00f2d98049f91f702cd5dd5bc82": "\\mathbf {\\gamma _{1}} :I_{1}\\to R^{n}", "13b93faeba2e7c51d1f0a7fe8faa0e3d": "f(x)/f(-x)\\equiv e^{-x}", "13b9926ff965d20a727bfd2bf5e88e50": "\\mathbb {P} (|X-\\mathbb {E} (X)|\\geq a)\\leq {\\frac {{\\textrm {Var}}(X)}{a^{2}}},", "13b998cf24f1b652e64c68bcd1c2a02e": "\\operatorname {de-let} [\\lambda q.f\\ (q\\ q)]", "13b9a1a680edce72d6046544b536a690": "u(x,y,z)", "13b9fe7dd1b095da9d0ae037eca47257": "x'_{3}", "13ba01ba8b7d29e25ee0c6d65ce50b11": "x\\ {\\dot {=}}\\ 0\\rightarrow x\\ {\\dot {=}}\\ 0", "13ba37f62f0b6fe78b1f1fbaff6f951a": "{\\mathcal {L}}\\left(f\\right)=S_{\\phi }(f)/2", "13ba92980b8067fe29e825bc21b479b9": "x\\leq K", "13baac17377b3271ca694e37c98e866e": "\\operatorname {cons} \\equiv \\operatorname {pair} ", "13baad0e3c9fcebffd17a64142690f8a": "\\delta \\in \\Delta ", "13baaf3d45bde9201138e629fb4697dc": "R(S)={\\frac {1-\\sum _{i\\in S}{p_{i}}}{1-\\sum _{i\\in S}{\\frac {\\beta _{i}}{D}}}}", "13bafdd735f14ed4c4203d8f218c39bc": "{\\underline {P}}(Cl_{t}^{\\geq })", "13bb21af2a2261b726e33d0606ec1194": "\\psi ({\\vec {x}})\\,\\!", "13bb4390518aafa7e2c9fbe68e835962": "\\rho _{l}=\\mathrm {density\\ of\\ liquid} ", "13bb511b7b20cd22ef71b7c48dc31ba1": "G_{11}={\\frac {-r\\nu '(r)+e^{\\lambda (r)}-1}{r^{2}}}\\;", "13bbe8151b482046de4ec544a7d63fad": "A\\to {\\widehat {A}}", "13bc894e6f0a405b62a37563e93fb81b": "{\\sqrt {Z}}", "13bcac9a6adf62375319ebc69d857e72": "L_{\\in }", "13bce2e6d2961bc3f71af9fb46195e40": "\\sum r_{i}.", "13bcf04bfe060069699e0b4b92a53f0f": "\\sin 75^{\\circ }=\\cos 15^{\\circ }={\\dfrac {{\\sqrt {6}}+{\\sqrt {2}}}{4}}.\\,\\!", "13bcf14b62b27c7c459cc7dcd9318a10": "id:X\\to X", "13bcfcd70046d935354b1465d626c583": "M_{n}(i,j)", "13bd0bf2ac081b9576ebc5d020a4a006": "\\scriptstyle {\\dot {m}}_{01}\\,", "13bd1be2356eca4452e4648d4acb4195": "\\beta \\ =\\arcsin \\left({\\frac {b}{c}}\\right)\\,", "13bd26cf0060d30f0e487f6a1281d4ab": "a_{ij}=\\left\\{{\\begin{matrix}1&\\mathrm {if} ~v_{i}\\in e_{j}\\\\0&\\mathrm {otherwise} .\\end{matrix}}\\right.", "13bdee301af9bf655137757ae2926528": "i=i_{0}{\\frac {nF}{RT}}\\Delta E", "13bdffe841517fa58bc5bdf3002ef4b7": "\\mathrm {ob} (\\mathrm {Gr} )", "13be0c14eaff4ce2fa73860c7e5931aa": "(n,k)", "13bec184fc576edc414c745fb59404f9": "{\\tbinom {m}{r}}.", "13bef16fb8a1fc44a5e0f6e8cc65becb": "1/2(n-1)", "13bf37566e0f1da297e16fb11ece90a9": "b_{i}=a_{i}-2e_{i}", "13bf6c3f74e23c1279b9d598699e9cfb": "V_{CB}=V_{CE}-V_{BE}", "13bf800328e595b0e3eeae1957d214ed": "^{n}\\mathbf {P} _{r}", "13bfbc7863ef1d8de184392afd3a0baf": "\\displaystyle V_{m}=\\sum \\limits _{n=1}^{K}L_{m,n}{\\frac {di_{n}}{dt}}.", "13bfdb9ea68fb8f938b2e2c2a2319449": "D_{0}<0", "13bff50d081ada63b49056f1bf18df84": "p(x_{k}|y_{0},\\ldots ,y_{k})", "13c01cd3e913665018170fffd3868e66": "{\\dfrac {PR}{RB'}}={\\dfrac {SQ}{B'S}},", "13c0256b854bfa3d7466bf8e465c8554": "(R_{g})_{S,[m]}", "13c04abc55b3bf628f2176f9fe41bf85": "P=\\{(a,b)|a,b\\in R\\}\\cup \\{(a)|a\\in R\\}\\cup \\{(\\infty )\\}", "13c07c8f2638661b4745a601cd650c94": "x{d \\over dx}\\psi =x\\psi '\\neq {d \\over dx}x\\psi =\\psi +x\\psi '", "13c0d15703c030e9aba65821e7cc8c35": "m\\neq 0,1,3", "13c0eba9a3f4908a5af14baad48048b4": "x_{i}:=x(i)\\,", "13c0f073c5afe7200ac95ca6bbc3bd16": "f_{j}^{(1/p)}=\\sum _{\\beta }f_{j\\beta }^{1/p}X^{\\beta },", "13c110c72a21001815d1b9ac6d26c69a": "\\geq 0", "13c1431b6249f455f7e9641bcae2470a": "SU(2)_{R}\\approx S^{3}", "13c15e19e4ded2841ceba7ced1ce5e42": "\\sin(y)=x\\ \\Leftrightarrow \\ y=(-1)^{k}\\arcsin(x)+k\\pi ", "13c1bffe6821c0a73571a98470ee4338": "{\\mathfrak {g}}_{-}", "13c228abf293efad438a0962b1b47859": "f^{-4}", "13c25fb369f84cc7bd181afc909d3b04": "B(x)=B_{0}\\exp \\left(-{\\frac {x}{\\lambda _{L}}}\\right),", "13c2be2d7eed414bc7215717ec722225": "D(\\alpha \\wedge \\beta )=D(\\alpha )\\wedge \\beta +(-1)^{\\ell \\deg(\\alpha )}\\alpha \\wedge D(\\beta ).", "13c2db86d73119fa9729125bb82065fd": "i(t)=Ie^{-{\\frac {R}{L}}t}", "13c37739c4d4334a29f8f209a86e01be": "L_{x}u+L_{y}u+Nu=\\rho (x,y)\\qquad (2)", "13c41e19c4c03b414056ea858fbf6a56": "7+5{\\sqrt {2}}=14.07106\\ldots ", "13c5392d8cff46918314ce12261eda68": "{\\breve {\\rho }}=r\\left[1+{\\frac {1-r^{2}}{2\\left(n-1\\right)}}\\right]", "13c581c0c5cfdde1ab31cb97b88f198d": "C_{M}", "13c5a27dff834d87ddd52aed154e4a14": "A,B\\in {\\mathbb {R} }^{m\\times n}", "13c5cd37b56d25c57f12863cfa8b7fa7": "a\\geq b\\iff a=\\operatorname {lcm} (a,b),\\;", "13c61635b1317c01496fe4efef0662f7": "F_{n}=2+\\prod _{i=0}^{n-1}F_{i}.", "13c69d77429db181b5cfbabe0e49072c": "E\\left(t-D_{F}/c\\right)+E\\left(t-D_{V}\\left(t\\right)/c\\right)", "13c6cafd953eab410e737cb871dfe891": "p(x|\\alpha ,\\beta ,\\theta )={\\begin{cases}{\\frac {\\theta \\Gamma (\\alpha +\\beta )}{\\Gamma (\\alpha )\\Gamma (\\beta )}}{\\frac {(x-a)^{\\alpha -1}(b-x)^{\\beta -1}}{(b-a)^{\\alpha +\\beta +1}}}+{\\frac {1-\\theta }{b-a}}&\\mathrm {for} \\ a\\leq x\\leq b,\\\\[8pt]0&\\mathrm {for} \\ xb\\end{cases}}", "13c6fc34f3023e69ae5eb5458fa132c8": "(x_{j}-x_{i})", "13c7884743a6887fd6d936627abd680f": "R_{X}+R_{Y}\\geq H(X,Y).\\,", "13c79f61af22d8d64c1cc951e0c785bb": "y\\subseteq d(R)\\,", "13c7a6df336f87699953af24ffd581c8": "A={\\frac {A+A^{*}}{2}}+i{\\frac {A-A^{*}}{2i}}", "13c7c3406af6259fef0d1b424749098a": "V_{out}=V_{2}-V_{G}=V_{2}-{\\frac {R_{G}}{R_{G}+R_{1}}}V_{1}.\\,", "13c84c3fcb77ff93263e5c37c22e9417": "{\\begin{aligned}{\\bar {y}}&={\\frac {1}{T}}\\left(\\int _{0}^{DT}y_{max}\\,dt+\\int _{DT}^{T}y_{min}\\,dt\\right)\\\\&={\\frac {D\\cdot T\\cdot y_{max}+T\\left(1-D\\right)y_{min}}{T}}\\\\&=D\\cdot y_{max}+\\left(1-D\\right)y_{min}.\\end{aligned}}", "13c8563ec9ac2e7f9677a569a02c801c": "={\\frac {3}{16^{0}\\cdot 1}}+{\\frac {6}{16^{1}\\cdot 3}}+{\\frac {18}{16^{2}\\cdot 5}}+{\\frac {60}{16^{3}\\cdot 7}}+\\cdots \\!=\\sum _{n=0}^{\\infty }{\\frac {3\\cdot {\\binom {2n}{n}}}{16^{n}(2n+1)}}", "13c88539eb11ea3612be36e0b555b41c": "\\delta _{0}>0", "13c8bbf43b8d850d687d92084164ca6e": "{\\tilde {f}}\\colon X\\times [0,1]\\to E", "13c95bd779f41f236b248d209461863d": "{\\delta }_{2}={\\sqrt {{\\mu }L \\over {\\rho }V}}\\,\\!", "13c966d1a315e3f8cfabd6bf30fce881": "(\\frown \\mu ):H^{k}(C)\\to H_{n-k}(C)", "13c98e4297d8cd8c4cc4924a0c1aeebc": "g_{i}/f_{i}", "13c9bacbd51596eb2e784e95cccda826": "dx=(R+\\chi )\\left(1-cos\\Theta \\right)+\\left(D_{ss}+D_{sp}\\right)U_{x}", "13c9c2059d9486d624292b432f0e8d0e": "\\operatorname {div} \\mathbf {F} =\\nabla \\cdot \\mathbf {F} ={\\frac {\\partial F_{1}}{\\partial x}}+{\\frac {\\partial F_{2}}{\\partial y}}+{\\frac {\\partial F_{3}}{\\partial z}},", "13c9e59705dceb6525ac40e456c4d6cb": "{\\frac {0.088\\ \\mathrm {N} }{(5.3\\ \\mathrm {kg} )(9.807\\ \\mathrm {m/s^{2}} )}}=0.0017", "13c9fcd74294315a00a0b9fa9d0e169a": "\\|u\\|_{E}=(u|u)_{E}^{\\frac {1}{2}}\\,", "13ca99b2c58112befc9682db04a97644": "{\\mathbf {r}}\\rightarrow {\\mathbf {r}}+{\\mathbf {v}}t", "13caa4a3bbb94482cc406fdbfcb150ec": "n=r\\times s^{-1}{\\pmod {m}}", "13cade3cec323560140fe7e7d15babb6": "M=L/\\Delta x", "13caed825f61241d1f0f4ba0433df6c7": "\\theta (L)=1+\\sum _{i=1}^{q}\\theta _{i}L^{i}.\\,", "13cb21eeaa1950c5923069e52e9b557e": "g_{n}=O\\left((\\ln p_{n})^{2}\\right).", "13cb39597596b588fc8abe4809a88fb3": "(0,1/n,0)", "13cb3993ab6dbfa2363407b76a546d3f": "f(x_{1},\\ldots ,x_{n},\\underbrace {0,\\dots ,0} )=(y_{1},\\ldots ,y_{n},\\underbrace {0,\\ldots ,0} )", "13cb8b963522a1dd94ac884ee6be8921": "b=0.1", "13cc0cb81b499dff66d4f0ddd68a2061": "\\tau _{V,W}\\;", "13cc41973e39a7c9f16d908e993272a7": "d(S_{1},S_{2})=0", "13cc462f4a9b73176bc86f7bdec46744": "[P_{i},P_{j}]=0\\,\\!", "13cc48d0937dde00e67a6fd4c71bef29": "m:\\mathbb {Z} _{p}\\rightarrow \\mathbb {Z} _{p}/p\\mathbb {Z} _{p}\\cong \\mathbb {F} _{p}", "13cc4f9d7c953594735c96257da721a6": "\\mu _{i}", "13cc6e45fbff19d28450bc66f843f0fe": "{\\mbox{linking number}}\\,=\\,n_{1}-n_{4}\\,=\\,n_{2}-n_{3}.", "13cc906a4e683aa486dd36cc891784b2": "{\\frac {\\partial v_{i}}{\\partial x_{i}}}-{\\frac {v_{i}\\,v_{j}}{a^{2}}}{\\frac {\\partial v_{i}}{\\partial x_{j}}}=0.", "13cd537318275d481f3fe5fce0df8c4c": "S\\subset V", "13cd6c45ed2ddd2f920cecd0878fb4c2": "C_{i}\\,", "13cd999fb1ceb40d0d23a4305b9b3d69": "(x-a)^{n}\\equiv (x^{n}-a){\\pmod {(n,x^{r}-1)}}\\qquad (2)", "13cdf85ed9179714e663b835a1f9010d": "R_{22}=R_{11}=-\\Delta p.", "13ce1bf9c56de115da720336da04aaae": "{\\begin{aligned}\\tau _{2}&={\\frac {(C_{C}C_{L}+C_{L}C_{i}+C_{i}C_{C})(R_{A}//R_{i})(R_{O}//R_{L})}{(C_{M}+C_{i})(R_{A}//R_{i})+(C_{L}+C_{C})(R_{o}//R_{L})}}\\\\&\\approx {\\frac {C_{C}C_{L}+C_{L}C_{i}+C_{i}C_{C}}{C_{M}}}(R_{O}//R_{L})\\ ,\\\\\\end{aligned}}", "13ce391ffb235b6414114486bbff161f": "1^{\\dagger }=1", "13ce48429d23820700301cea0dc74821": "00\\,", "13dcdf1294abd8124f5076faa766fd87": "H=\\,", "13dd50dcd0266c57673a4897f439ce2b": "{\\tilde {B}}_{3}", "13dd89e569ae31da9a5b4bb8089fe6a8": "D_{r}", "13ddb885bc1daf16b00d254c350777d4": "x\\sim p(x|\\theta )", "13ddf7a74e9f0aea649eef875366d524": "x^{2}=ax+b", "13de3ac790a45f1c0d7de7002d985e75": "{\\sqrt {\\frac {ne^{2}}{m\\epsilon _{0}}}}=", "13de676e2e8099271de972b5df1970c9": "t_{p-r}=1.5[s],\\mu =0.7", "13de6980b7616933c24f210f3bd19b6e": "{L \\over D}={{\\Delta s} \\over {\\Delta h}}={v_{\\text{forward}} \\over v_{\\text{down}}}", "13de8df1c9a03f72d57fb7395a13171b": "D_{\\max }", "13ded1c2fd307ba2bbf16e13259999d3": "i:H\\to E", "13df0d3569f262da1f9f4dd32ea612bf": "\\mathrm {CR} (x_{1},\\dots ,x_{N})=\\sum _{i_{1},\\dots ,i_{N}=-1}^{2}f_{i_{1}\\dots i_{N}}\\prod _{j=1}^{N}b_{i_{j}}(x_{j})", "13df5b3e4f1ccb9953d13f29fa7bd57d": "\\left(\\pm 1,\\ \\pm 1,\\ \\pm (1+{\\sqrt {2}}),\\ \\pm (1+2{\\sqrt {2}}),\\ \\pm (1+2{\\sqrt {2}})\\right)", "13df6c5073effc4e7570c7a98fbb7215": "F[r]=-{\\frac {r^{2}}{r'}}", "13df9d2a5f69fc01626224bacd519663": "\\alpha ,\\ \\ K(\\alpha +1)=K(\\alpha );", "13e049f13df403a560f546f07d83514e": "-j{\\sqrt {\\frac {6}{25}}}", "13e04d93897c4e0a1bd3710a9a5d0fff": "\\Phi (z,s,a)=z^{-a}\\Gamma (1-s)\\sum _{k=-\\infty }^{\\infty }[2k\\pi i-\\log(z)]^{s-1}e^{2k\\pi ai}", "13e05336a7e1dd445acd37d8828a265c": "\\epsilon _{\\alpha \\beta \\gamma \\delta }", "13e06dafb3c3e46440a6610a5df8716d": "m\\leq n\\log b/\\log a+1", "13e07167e2881097861b6091edb70ef6": "t\\left\\{r,{p \\atop q}\\right\\}", "13e076c3da5284fd2ce22a748343c723": "{\\boldsymbol {\\mathsf {E}}}.", "13e095007c180894278b223b29c42aa0": "M_{yy}{\\Bigr |}_{y=-b/2}=M_{yy}{\\Bigr |}_{y=b/2}", "13e09863e41108f3df357d5b22abdaf5": "v=\\infty ", "13e0ca60af320b20ee8888980330fba8": "{\\begin{pmatrix}A&B\\\\C&D\\end{pmatrix}}", "13e0e9033dd0370e857ab9a9dcd316ad": "[\\lambda ]^{\\omega }", "13e1203b77a69ca578b9771243e0878d": "x,y", "13e13c46b4e543e3f349f6b1655edfbe": "\\mathbf {J_{1}} ", "13e141066758b3e71c890019247bc3f3": "Z_{\\mathrm {in} }=Z_{L}=Z_{0}\\,", "13e15946bead62b759bb7390782bbdc1": "\\Phi =HJ-JH^{t}\\,", "13e16d400be638b58fc9139c9c2d12b4": "Md=0.80log10(T)^{2}+1.7log10(T)-0.87", "13e17901066f338d9f6fb1196592b4a2": "w(2,2)", "13e19ef4e4201148f8d3b66b69ed6a80": "c_{p}", "13e21973282a707ec1f05cc88dcda2ed": "\\phi \\left(x\\right)=\\sum _{n=0}^{\\infty }\\lambda ^{n}\\phi _{n}\\left(x\\right)", "13e22d738a6397235374190f6c2a2ea4": "x+p={\\sqrt {(x-p)^{2}+y^{2}}}.", "13e251c3e11c15bf285e0bd12a1b701d": "\\left({\\frac {\\partial S}{\\partial E}}\\right)_{x}={\\frac {1}{T}}\\,", "13e26b7a27c61b9c731e34a12458f944": "{I_{1}}", "13e2e028dbe3df49c5d9c6571eaa9687": "W=\\int _{V_{1}}^{V_{2}}P_{1}\\left({\\frac {V_{1}}{V}}\\right)^{\\gamma }\\,dV", "13e355e1f421a234afc7fd4486449465": "xL_{n}^{(k)\\prime \\prime }(x)+(k+1-x)L_{n}^{(k)\\prime }(x)+(n-k)L_{n}^{(k)}(x)=0,\\,", "13e3784aa841558352b6afbc82efbc88": "{\\mathcal {S}}=\\int _{t_{1}}^{t_{2}}L\\,\\mathrm {d} t.", "13e386ec55fefc5ed41079850023dd15": "\\log _{a}\\lambda ", "13e3a9c7c384fa9920781bc3be101b5f": "\\sigma _{ic},i=1,2,3", "13e3c08c5a53ad234e9848d4a5df09c6": "S\\to C_{K}\\to S^{1}", "13e40115f4f5cf791061425ad093ab2d": "T^{\\bar {k}}p", "13e41043429047515b1a29d607a78c8b": "\\mathbf {n} _{i}=\\mathbf {R} \\mathbf {N} _{i}.\\,\\!", "13e44574cd9c6767e3de212754979d99": "{\\sqrt {4\\pi \\varepsilon _{0}}}\\left(\\mathbf {E} ,\\varphi \\right)", "13e445f9fc74ad0d5cfde729b163c384": "\\gamma (n)=\\prod _{p\\mid n}p", "13e4811f4614d10e5f46131cdd816539": "t\\mapsto (t^{2},t^{3})", "13e490055a59bf24eb6f40b101960256": "D_{i}(\\theta )", "13e490990e435667da0fc7c35e68b700": "T\\colon Q\\times \\Sigma \\to Q.", "13e52c0e33c8d08cae6b94417edfb876": "\\displaystyle T(f)=\\int _{{\\mathfrak {a}}_{+}^{*}}{\\tilde {f}}(\\lambda )|c(\\lambda )|^{-2}\\,d\\lambda ", "13e538ba214c1c43218df090e4c2ab7a": "R=|\\langle z\\rangle |=e^{-\\sigma ^{2}/2}", "13e54a7f943e30f3e5514db13a79323a": "A(z)=1+B(z)+B(z)^{2}+B(z)^{3}+\\cdots ={\\frac {1}{1-B(z)}}.", "13e5832f5571902e48f6615c9aa80eb0": "[HG]", "13e6345a299bf399fff856aa3e3f6606": "y'(t)=-A\\,y+{\\mathcal {N}}(y),\\qquad \\qquad \\qquad (7)", "13e67957a08cec4bbc4721a87204f97e": "\\Delta _{r}G=\\Delta _{r}G^{\\circ }+RT\\ln Q_{r}\\,", "13e6aa219798b128c6b99e97bc0da381": "\\mathbf {v} =v^{r}(r)\\mathbf {Y} _{10}+v^{(1)}(r)\\mathbf {\\Psi } _{10}", "13e6bc6b05b5d9e7e443467900514984": "S_{1},S_{2},S_{3},S_{4}\\,", "13e6de38f569e23c36495c99ad7fbff9": "E[J,z;\\epsilon ]=C\\int d{\\vec {x}}(I({\\vec {x}})-J({\\vec {x}}))^{2}+A\\int d{\\vec {x}}z({\\vec {x}})|{\\vec {\\nabla }}J({\\vec {x}})|^{2}+B\\int d{\\vec {x}}\\{\\epsilon |{\\vec {\\nabla }}\\phi ({\\vec {x}})|^{2}+\\epsilon ^{-1}\\phi ^{2}(z({\\vec {x}}))\\}", "13e6f378f1e8454839eaf17f4eb1bc55": "{\\begin{bmatrix}\\cos(\\theta )&\\sin(\\theta )&0\\\\-\\sin(\\theta )&\\cos(\\theta )&0\\\\0&0&1\\end{bmatrix}}", "13e6fb26737d787325e07ef819c94733": "W^{*}=\\cup _{\\gamma \\in \\Gamma }\\;\\;\\gamma W.", "13e70e36bbdf697429dc57ca79af8d53": "(A,B)\\mapsto Tr(B^{1-p}A^{p})", "13e73281966ecd9181d9b8301e044ec5": "\\pi _{i+}=\\sum _{j=1}^{c}\\pi _{ij}", "13e75326d8bd7b820616c6fd3c1ab3b7": "\\beta >2", "13e75b2d5a262d8e30d67eaa06059361": "[0,0.5]", "13e78439a8c71a3c143a5b3622bfbd9c": "{\\mathit {w}}_{GC}({\\mathit {q}})", "13e7b576b988edc4fae49cac9c78c042": "r_{c}\\leq R(q,u)", "13e7c5d053d41b7be434b7cf382327bd": "Mw=\\langle w,v_{1}\\rangle v_{1}+Tw=(\\langle w,v_{1}\\rangle +\\alpha )v_{1}.", "13e7cb754b1da09c09bebf265b0bddfa": "\\mathbf {d} _{j}=\\nabla X_{j}+(X_{j}-Y_{j})\\nabla (\\ln P)+\\mathbf {g} _{j}\\,;", "13e81abc6e85e7f9d2f9660381161ad4": "\\mathbf {D} =\\mathbf {E} +4\\pi \\mathbf {P} ", "13e8deab1d6ba961f20f865cf76b5d2c": "\\Phi (p)=(\\pi (p),\\varphi (p))", "13e901580cf920b47e6d047b798cce24": "\\sum _{k=0}^{\\infty }{{\\alpha +k-1} \\choose k}z^{k}={\\frac {1}{(1-z)^{\\alpha }}},|z|<1", "13e9553675a4ba9e3a40324a422d8e69": "{\\bar {\\psi }}\\ {\\stackrel {\\mathrm {def} }{=}}\\ \\psi ^{\\dagger }\\gamma ^{0}", "13e9669842c4613654cd545f2fb325ab": "T=C_{F}\\int [n({\\vec {r}})]^{5/3}\\ d^{3}r\\ .", "13e97b6aed3842977e74d66dee531499": "\\scriptstyle {\\boldsymbol {e}}_{i,{\\text{rec,ECEF}}}\\;=\\;-{\\frac {\\partial r({\\boldsymbol {r}}_{i},\\,{\\boldsymbol {r}}_{\\text{rec}})}{\\partial {\\boldsymbol {r}}_{\\text{rec,ECEF}}}}", "13e989548e45c4ec185cd72cc621f1e8": "c_{B}(0,b)=-{\\frac {1}{2b}}\\mathrm {coth} {\\frac {\\beta b}{2}}", "13e9908316bc305c9c6f70118da244ab": "M^{(i)}", "13e9b56294ca43dc5ce78517d8b6ede8": "\\prod _{i=1}^{N}(m_{i}-1)=M\\prod _{i=1}^{N}\\left(1-{\\frac {1}{m_{i}}}\\right)", "13ea8e9ce4ea2782326060f9dad03d2d": "\\mathrm {FWHM} =2{\\sqrt {2\\ln 2}}\\;\\sigma \\approx 2.355\\;\\sigma .", "13eabfc6c56a703fbfa4a181062b8819": "\\ d={\\frac {1}{2}}gt^{2}", "13ead68d11118032f72a8f816e85e31c": "\\scriptstyle \\square ^{2},", "13eb0c37628fb783dd162e4739636211": "D(Rf_{\\ast }({\\mathcal {F}}))\\cong Rf_{!}D({\\mathcal {F}})", "13eb19f99cede15b8a0d52a7759a6189": "{\\frac {dx}{dt}}=x-x^{2}", "13eb38a5c3636f9f57506a3d1e69efec": "g_{N}=0", "13ebc40f0650cc9846452a66b3b535c2": "X\\cdot b_{k}(X)\\mod f(X)=X\\cdot b_{k}(X)-f(X)", "13ebca8a5244c089e4849bae92e46cee": "\\sum _{j=0}^{n}(-1)^{j}{\\tbinom {n}{j}}P(n-j)=n!a_{n}", "13ebcc7113a6d4928b435f7227f33368": "\\left(traces\\left(P\\right),failures\\left(P\\right)\\right)", "13ebdd01b4ceebb2b033052aa19db14c": "Z_{\\alpha }=[Z_{0}:Z_{1}:\\ldots :Z_{n}]", "13ec0d6e556beae382494d2a95f78d63": "F=-kX\\,", "13ecb2c067a3c38537f9f283786e69f2": "\\delta ={\\sqrt {{2\\rho } \\over {\\omega \\mu _{r}\\mu _{0}}}}", "13eda1f72c27b90988812d98c8675b39": "{\\mbox{DG}}={\\frac {a-b}{a}}", "13ede354a74f816880b301bce4aea00e": "\\scriptstyle \\phi '", "13ee4a83bf2887706221c37f10d413e5": "{\\frac {d[W]}{dt}}=-a_{1}[W][E_{1}]+d_{1}[WE_{1}]+k_{2}[W'E_{2}]", "13ee9b2356ceba882c513d31911ab4e9": "r_{1}={\\frac {m_{2}a}{(m_{1}+m_{2})}}", "13ef78a9ffe5c721e87bf22f06afeda8": "\\beta =0.35", "13ef984bab7f31b28f716dec76e994a1": "A_{n}(z)=\\sum _{k=0}^{n}a_{k}z^{k}.", "13ef99fef5af39da92ce9ecc1dc9a80d": "V(x,T)=D(x),\\,", "13efa50c9fc36219f1237a81ed3c1d4b": "\\alpha (m,n)=\\min\\{i\\geq 1:A(i,\\lfloor m/n\\rfloor )\\geq \\log _{2}n\\}.", "13efc496c414effcb2775070a26eb719": "|I|={\\frac {P}{A_{\\mathrm {surf} }}}={\\frac {P}{4\\pi r^{2}}}", "13efd63aac410295eea5fd87df8a1f97": "a(x-\\alpha )(x-\\beta )\\ ", "13f054501b181e44bc8ae6ce4fadf874": "G(\\theta |\\alpha ),", "13f0637e8a2c25741aad7ce49b4afe8b": "(g\\circ f)^{-1}(y)={\\tfrac {1}{3}}(y-5)", "13f0e801c388377344c74c1e979d1003": "N(k,\\epsilon )\\leq \\exp(C(\\epsilon )k).", "13f0f5a748310b9c41469a7a184046f4": "C_{AB}={\\frac {M_{B}}{M_{A}}}={\\frac {1}{2}}", "13f0f7b7f267d9a31e4615ed21bfa332": "{\\sqrt {6^{2}+8^{2}}}=10", "13f15aae861adaae938599523683b922": "ax+by+c=0\\,", "13f1a511f0ff158c02a9c858b6f72652": "\\gamma _{4}\\,", "13f1ba4b116ede3f57030c5fd444ddfe": "{\\frac {W}{m_{\\mathrm {0} }c^{2}}}=\\left(1+{\\frac {\\alpha ^{2}Z^{2}}{(n_{\\mathrm {r} }+{\\sqrt {n_{\\mathrm {\\varphi } }^{2}-\\alpha ^{2}Z^{2}}})^{2}}}\\right)^{-1/2}-1", "13f26e96037bfbcb4d1ee107ef3bd38a": "\\log(*z_{1}/z_{1})=k_{syz}y", "13f29485a675bcd7c648e29cc2b6251f": "q\\psi _{0}+{\\frac {d\\psi _{0}}{dq}}=0", "13f2ca0729b20b2d9017a45be64a3f26": "t_{p}\\equiv t_{1}^{p}\\equiv t_{1},\\quad u_{p}\\equiv u_{1}^{p}D^{(p-1)/2}\\equiv u_{1}", "13f2ce330f6afa6357c7b6fa4fcb7d7b": "{\\begin{smallmatrix}\\left[{\\frac {Fe}{H}}\\right]\\ =\\ -0.25\\end{smallmatrix}}", "13f32c617d706cc01d67728e9df4324d": "J_{q}(n,d,e)", "13f395321385408b8aa03e1460cedaa0": "\\alpha _{k}\\leq {\\frac {3k-4}{4k}}\\quad (4\\leq k\\leq 8)\\ ,", "13f3ce4b40dd26c6080d25a54f5333db": "\\ g_{\\phi }=\\left(9.780327+0.0516323\\sin ^{2}(\\phi )+0.0002269\\sin ^{4}(\\phi )\\right)\\,{\\frac {\\mathrm {m} }{\\mathrm {s} ^{2}}}", "13f3f9e180ad21a7554ace64550a781e": "F',", "13f41fb0aa05b8c4a7953dd9834e0fb2": "\\mathbf {P} ^{2}=-(E_{\\mathrm {rest} }/c)^{2}=-(mc)^{2}.", "13f41fc1fe5e94692d180cf7d6cd121a": "s,e\\in \\mathbb {Z} _{q}", "13f4a66e5e47360fb397d6164f8e08e5": "x_{ij}=Q_{ijkl}\\times P_{k}\\times P_{l}", "13f4ba65c6d3620a34bbaa1d084a7d7c": "r_{i}=\\left\\vert \\left\\{x_{k}:x_{k}\\geqq x_{i}\\right\\}\\right\\vert .", "13f560bcbba95daeb7dc06e9407411f2": "R/\\operatorname {ker} f", "13f5621af050d4974be77b6bf13551a5": "{\\frac {dN}{dt}}=-kN", "13f573d4129ddf7a986e1ec4154936b2": "U(a,b,x)\\sim x^{-a}\\,_{2}F_{0}\\left(a,a-b+1;\\,;-{\\frac {1}{x}}\\right),", "13f5cb1ea0c29245c82805824c46c15f": "g\\left(\\alpha \\,\\right)=1-{\\left(1-{2^{-N}}\\right)^{\\alpha }\\,}:0\\leq \\,\\alpha \\,", "13f627f48d45107a1bc58ebb9253e6c7": "{\\frac {{\\text{d}}[{_{a_{j}}^{b_{j}}}S_{j}^{\\beta _{j}}]}{{\\text{d}}t}}\\simeq -\\sum _{i=1}^{m}{\\frac {x_{b_{ji}}{\\text{k}}_{3(i)}E_{0}{\\overline {S}}_{i}}{{\\overline {S}}_{i}+K_{i}\\left(1+\\displaystyle \\sum _{p\\neq i}{\\dfrac {{\\overline {S}}_{p}}{K_{p}}}\\right)}}\\qquad \\qquad (9a)", "13f62c9aa3411c74f55faa6850ff5fb2": "f(c)", "13f637e4f7207a7983076438ea545fb8": "\\rho /\\sigma ", "13f6618d09bec656658ffdcce15c36c4": "w_{1},\\ldots ,w_{N}", "13f74f1754f3c0e377c3c42c9f890fa6": "\\|x\\|_{\\infty }\\leq \\|x\\|_{1}\\leq n\\|x\\|_{\\infty }.", "13f7557092bfe957f27b4fce6a6220c5": "\\delta _{g}", "13f77b2369f8dac673bb67aca059600e": "\\nabla {\\vec {f}}=0", "13f7961a3fd7a41a92786c7c83efe6a3": "fp_{11}/(p_{11}+p_{10})", "13f7f9af207b597ad46a50a11e29bbf5": "\\mathbf {x} (t)=\\mathbf {r} _{k}e^{\\lambda _{k}t}", "13f7fd9341eebda91119595c3a05f048": "{\\frac {d}{dt}}\\langle {\\dot {\\gamma }}(t),{\\dot {\\gamma }}(t)\\rangle =2\\langle \\nabla _{{\\dot {\\gamma }}(t)}{\\dot {\\gamma }}(t),{\\dot {\\gamma }}(t)\\rangle =0.", "13f80ae362303da193277c011c2fbec9": "D+\\sum _{k=0}^{\\infty }CA^{k}Bz^{k}", "13f812157a5051a7030462246c9401c6": "\\delta ={\\sqrt {(\\Delta H_{v}-RT)/V_{m}}}", "13f8135b589c6a7384811bd0c5cb8337": "\\int x\\arccos(a\\,x)\\,dx={\\frac {x^{2}\\arccos(a\\,x)}{2}}-{\\frac {\\arccos(a\\,x)}{4\\,a^{2}}}-{\\frac {x{\\sqrt {1-a^{2}\\,x^{2}}}}{4\\,a}}+C", "13f89954f5de407793e2443ab28a541a": "\\log p", "13f8ba4cd54290fc83a7a25cc1a3ca4a": "\\mathbf {s} =2\\mathbf {h} /(1+h^{2})", "13f9055fe4e2dcd3cccb57f8999e1b4a": "\\int I(q)q^{2}\\,dx", "13f93390f4b4d864816bee8cb63c98f1": "X=(X(t),t\\geq 0)", "13f95b621b1fe44bc70559490145d485": "n,m\\geq N", "13f962b29fda08256826a204bd3fff0a": "\\omega ^{2}+\\omega ", "13fa132608cee6c9379d9d712e446c54": "{\\color {white}{\\frac {d}{dx}}2^{x}}=2^{x}\\cdot {\\text{constant,}}", "13fa1e63dda9da93f8f44a795341e7d5": "r_{1}", "13fa8829f2a9f276a410c78ddf113e9a": "a_{n}^{2n-2}", "13fad4907a80f051079e9768ab244ea6": "\\forall x,y\\geq 0", "13fb41e8d12af3a134c710e571e7d06f": "y=-{\\frac {3}{7}}x+{\\frac {11}{7}}\\;\\;\\;\\;{\\text{and}}\\;\\;\\;\\;z=-{\\frac {1}{7}}x-{\\frac {1}{7}}{\\text{.}}", "13fb42c883d34195c3d9c45b604fd2e2": "{\\frac {n(q_{1})}{N}}\\cdot {\\frac {n(q_{2})}{N}},", "13fb48d034eafb0ea71b2dab96af0d73": "N(d_{+})", "13fb4a3eab5c5d693731b14e2aa88cb6": "\\scriptstyle t\\rightarrow \\infty ", "13fbebcbae10fb52668c48da27b4c919": "{\\text{d}}", "13fbf2741122c6ab7fda130b3a931527": "SO_{1,n}^{+}\\mathbb {R} ", "13fc7b1a064ccade54f847f4dc938a64": "{x_{1}=z_{1}}", "13fc9de7f44bc19ac212dd2ec9d32d81": "CMUAMA={\\frac {\\left(MUAC-\\left(\\pi \\times {\\frac {TSF}{10}}\\right)\\right)^{2}-6.5}{4\\pi }}", "13fca36f7ef54140559e0d0f6253a895": "{\\mathcal {L}}[\\varphi (x)]", "13fcbed8bcac58d95c0cb989f3aca2de": "{\\sqrt[{n}]{x}}", "13fcdc132c6487bb6bdc892786083b2e": "a=-\\nabla \\Phi ", "13fce668381d93c55c280d0a2f0d3328": "\\operatorname {Var} (X)=\\operatorname {E} [X^{2}]-(\\operatorname {E} [X])^{2}.", "13fd0277843a5b1a1409bfe98ba5da89": "\\int x^{m}\\,\\operatorname {arcoth} (a\\,x)dx={\\frac {x^{m+1}\\operatorname {arcoth} (a\\,x)}{m+1}}+{\\frac {a}{m+1}}\\int {\\frac {x^{m+1}}{a^{2}\\,x^{2}-1}}\\,dx\\quad (m\\neq -1)", "13fd08bf049e64aa2d3bc65965d909c4": "\\mathrm {Re} ={\\frac {\\rho U_{\\infty }L}{\\mu }}.", "13fd1fc00c4c9593b88673c6b494ce6f": "O^{0},\\cdots ,O^{T}", "13fdb2f1a3f178de3b5cbf7da8c6c1ad": "t^{\\prime }=t-vx/c^{2}\\,", "13fe16b5f133e0549dafe9ec7bd2095f": "{\\mathcal {A}}", "13fe38ce74a1252e183b1436189178a1": "d(x_{n},x_{m})\\leq d(x_{n},x)+d(x_{m},x)<\\varepsilon /2+\\varepsilon /2=\\varepsilon ", "13fe6dc777724e1cc5872da88f9acf36": "(X_{1}=1,Y_{1}=0,Z_{1}=0)^{T}", "13fe9f44dbf2af655b47d501b25b9916": "(x-\\alpha _{i})", "13fea8852297c6f282d61e184a0f69d7": "x^{2}-2qy^{2}=1\\ ", "13feb4abaef8f5fbd97e61b4e89deb80": "C_{\\ bc}^{a}=0", "13ff24f35935ad57e15ca5647d54f220": "\\Sigma _{SFR}\\propto (\\Sigma _{gas})^{n}", "13ffcbc1f6b13df1ed8616aac1db77f4": "-P_{1}=(x_{1},-y_{1})", "14004376841054065e5ea453af59c836": "\\lim _{\\nu \\to +\\infty }\\operatorname {P} \\left({\\frac {X_{\\tau _{\\nu }}}{\\sqrt {\\nu }}}\\kappa _{z}\\,", "14026da1fcd875592978e55743abd618": "-1=-A+C", "14029858d3a7ae492219886d2f27ae2f": "\\operatorname {Res} (f,i)=\\lim _{z\\to i}(z-i)f(z)=\\lim _{z\\to i}{\\frac {e^{iz}}{z+i}}={\\frac {e^{-1}}{2i}}", "1402af796cd0c63255a6d85027341273": "f(n)\\sim g(n).\\,", "1402f3316bdc64712a705e5b8e5bb68e": "r^{2}n\\propto r^{-1.5}", "14039829aa64625e99bb06b9c263cdbc": "M=L/R", "1403a9c06353ee7a734082b3b9cb2971": "\\exp \\log {\\frac {1}{1-z}}={\\frac {1}{1-z}}", "1403fa9347c70cfd5b3348a3e0a3fd36": "f(t)=-2t^{3}+3t^{2}", "14041c3532cd6c4878ea09af53e2f83e": "R[t]\\to S,\\quad f\\mapsto f(T)", "1404dd3f4e27c420c5ab03816552ac8c": "[\\mathbb {Z} _{4}^{6}\\times (\\mathbb {Z} _{3}^{7}\\rtimes \\mathrm {S} _{8})\\times (\\mathbb {Z} _{2}^{11}\\rtimes \\mathrm {S} _{12})]^{\\frac {1}{2}}.", "14053ad84391542856ecfd44ca84861d": "1/{\\mbox{trace}}\\mathbf {\\left(J^{T}WJ\\right)^{-1}} ", "1405797ea19ff84d492e5d8bab6d163f": "(a_{1},b_{1},c_{1})", "14057ee278db4784c475107da4a770a1": "\\left\\vert h\\right\\vert <\\pi ", "14059d451ac251e108ddc5c61bd38c1d": "\\operatorname {Li} _{1}(z)=-\\ln(1-z)", "1405e6e9709a6719f1b48c43be5c2ed5": "\\tau _{1}+\\tau _{2}=C_{2}(R_{1}+R_{2})+C_{1}R_{1}\\ ,", "14060f1ac8215b6ec3eaad022f03e484": "r=k^{2}\\cdot {\\frac {1}{d^{2}}}", "14061d083159604d7fcadc603f39ca4a": "\\chi _{e}\\ =\\varepsilon _{r}-1.", "1406f6b20d8beb7d967eb30f17f8c5de": "{{}^{3}R}_{2323}={\\frac {-\\omega ^{2}}{2}}\\,{\\frac {C^{\\prime }\\left({\\frac {q^{2}}{\\omega ^{2}}},{\\frac {q^{2}}{2\\omega ^{2}}},\\omega u\\right)^{2}}{C\\left({\\frac {q^{2}}{\\omega ^{2}}},{\\frac {q^{2}}{2\\omega ^{2}}},\\omega u\\right)^{2}}}", "1406f9934ee0e7fcdc261228e66f3d17": "F(x,\\theta )", "14070ed17a1fa76a2d1328db994351b6": "P(V_{t}\\geq 0)=1{\\text{ and }}P(V_{t}\\neq 0)>0\\,", "1407ab317ac43ec9ecd555dc671cbeb5": "\\omega _{n}", "1407dc401c1bd95b366f793d77da5c7f": "{\\text{cat}}(X\\times Y)\\leq {\\text{cat}}(X)+{\\text{cat}}(Y)", "140835c22e5cd4e2aa12a4238fba7eda": "|{\\psi }\\rangle ", "14084093465d434dbdc4d6ae6e7e8eba": "E_{4}^{2}=E_{8},\\quad E_{4}E_{6}=E_{10},\\quad E_{4}E_{10}=E_{14},\\quad E_{6}E_{8}=E_{14}.", "1408598a7ade84d0aa31dffee9c43ecd": "(\\nabla _{c}\\nabla _{d}-\\nabla _{d}\\nabla _{c})", "14085c4af1c38b5ac993344c422d2316": "\\log _{10}(F(x))=m\\log _{10}(x)+b,", "14088ad333389531f5729ba6a5551b51": "H=max(P1,P2)", "140893524d30a815137e6c0352230082": "K(k)\\,", "1409ea5de7ff862ea96b938784ccec86": "\\lim _{t\\to \\infty }\\mathrm {E} [r_{t}]=b", "140abeefb251599e23e898ebb923f018": "p\\leq N", "140acc06c3f36fcdd9125c47af3bf7ec": "\\scriptstyle i", "140ae05fe8360fc6cf8c02435b06752a": "a_{n+1,k}=-na_{n-1,k}\\ \\ k=0", "140b1bcf66d4ab868b3b5a7c3fa38fb1": "\\pi _{i}A", "140b2d7ed48bf632f77f6a89b68ab71f": "u(x(p,w))", "140b9b19460d154360ca9e032e65d727": "V_{2}=Q/A_{2}", "140be2791373d73b906f152de4ddb9f4": "\\Re (s)>\\sigma _{a}", "140c4da37dec3cb1ae6fda0f71f7a376": "{\\bar {\\Phi }}", "140c56a63228acff004b72df352f8459": "0I_{p}", "141a5d4759e2e45fd42e9e0cf1cd2b24": "|\\Omega \\rangle ", "141a8feeed3aa4546bc622e620217295": "\\mathrm {NH_{4}^{+}>K^{+}>Na^{+}>Li^{+}>Mg^{2+}>Ca^{2+}>guanidinium} ", "141b06aefda2202fd9aa03590738e288": "DR_{*}^{S}", "141b0a5b54a6a4ce72c2d574fcd300ec": "g'=g{\\rho _{1}-\\rho _{2} \\over {\\rho _{1}}}", "141b2459f1671b13ba901e104007b160": "G=H_{1}\\times H_{2}\\times \\cdots \\times H_{l}\\,", "141b6116216b2edc432492a52b138294": "{\\hat {p}}\\psi =-i\\hbar {\\frac {\\partial }{\\partial x}}\\psi ,", "141b79b30a584e36e304c72504c8c32b": "m_{1}\\|m_{2}\\|\\cdots \\|m_{x}", "141b7fbcbf287b5d86e7a7f850071866": "\\ \\sum _{n=-\\infty }^{\\infty }{\\left|h[n]\\right|}=\\|h\\|_{1}<\\infty ", "141bcf4a5cf76ef12ae6f306d336013e": "HK=139.54{\\frac {P}{d^{2}}}", "141bd2f935f1bce44d710ea6c49bc353": "{\\frac {dw/dt}{w}}=g_{w}=\\rho v-\\gamma .", "141c1b932c1a056b00d39687ec9d6d3d": "\\omega ^{4}-(2K+\\kappa ^{2})\\omega ^{2}+K(K+Rd\\Omega ^{2}/dR)=0", "141c8dcd33517cce3525436243c7f6e4": "dx={\\frac {x}{\\sqrt {t^{2}+ab}}}\\,dt", "141c95fd06f83b5427d4d88ca9afa1cb": "\\left(2+{\\sqrt {-6}}\\right)^{2}=4+4{\\sqrt {-6}}-6=-2+4{\\sqrt {-6}}.", "141caf8f69c7bad12c265e37f2adfa54": "0\\to R\\xrightarrow {\\ x\\ } R\\to 0.", "141cf5f371e76a6d4fe45f877c850b11": "(g^{a},g^{b},g^{c})", "141d368a8b802696c217d15316468993": "\\mathbf {J} _{\\mathrm {tot} }=\\mathbf {J} +{\\frac {\\partial \\mathbf {D} }{\\partial t}}", "141d4bcf6119d5d38a60a3beb8cddbdc": "\\{f,\\{g,h\\}\\}+\\{g,\\{h,f\\}\\}+\\{h,\\{f,g\\}\\}=0", "141db70c280e90ffa01d28203151ffb9": "e(v)={\\tfrac {1}{2}}v", "141defc7bb8056ad41ef4076ab548e36": "h^{2}=x^{2}+y^{2}.\\,", "141e9f0203f7ee6056a19ca2042734d2": "m\\geq n", "141eb513e82b0cb4eb21f3e50ca6513d": "{\\begin{cases}\\infty &{\\text{for }}\\alpha \\leq 1\\\\{\\frac {\\alpha \\,x_{\\mathrm {m} }}{\\alpha -1}}&{\\text{for }}\\alpha >1\\end{cases}}", "141eb7cecef1acaf78084308005924e8": "\\alpha _{t_{1}}", "141f10ee8f1c1df43e4455ffb074a3d2": "{\\mathcal {B}}({\\mathcal {H}})", "141f2d7837d1bbc63598518cb5a224eb": "{\\mathcal {S}}[\\mathbf {q} (t)]=\\int _{t_{1}}^{t_{2}}L(\\mathbf {q} (t),\\mathbf {\\dot {q}} (t),t)dt", "141f2f6208c6bf606490e0115ad029f0": "\\lim _{n\\rightarrow \\infty }f_{n}(x)=f(x).", "141f93d115993bc092f008533a014cfa": "\\chi (\\tau ,0)", "141fbad5c237cb390746713b73b9bef9": "x_{2},x_{3},\\ldots ,x_{s}", "141fc3e984bdb86c22325a3962223a0e": "\\mu =C_{1}\\left(1+{\\tfrac {3}{5\\lambda _{m}^{2}}}+{\\tfrac {99}{175\\lambda _{m}^{4}}}+{\\tfrac {513}{875\\lambda _{m}^{6}}}+{\\tfrac {42039}{67375\\lambda _{m}^{8}}}\\right)\\,.", "1420991a473b0e136407785761b536a3": "90<\\theta \\leq 180", "1420ff3150fd64b6b4c9725f96af9200": "{\\partial \\over {\\partial x^{a}}}\\ {\\stackrel {\\mathrm {def} }{=}}\\ \\partial _{a}\\ {\\stackrel {\\mathrm {def} }{=}}\\ {}_{,a}\\ {\\stackrel {\\mathrm {def} }{=}}\\ (\\partial /\\partial ct,\\nabla )", "14210a2d9fa4821b39a58b859826bbe0": "M{\\ddot {x}}+\\gamma {\\dot {x}}+Kx=F\\,", "142172b9546813e11f78ac92518bbdc4": "\\left({\\frac {p^{*}}{q}}\\right)=1.", "1421884db2f413200b9f9272deaebd17": "T:V\\rightarrow V", "1421fc8049dc113ff83bcb794a74bfeb": "{\\hat {f}}(\\xi )={\\mathcal {F}}(f)(\\xi )=\\int _{\\Re ^{n}}f(x)e^{-2\\pi i\\langle x,\\xi \\rangle }\\,dx", "1422462345eebddf87056618ad9d51e5": "V_{R}=C_{R}\\cdot \\exp \\!\\left[{-z \\over \\lambda _{R}}\\right]", "142252dc7029d4b5698971640730a216": "\\tau _{U}=\\inf\\{t\\geq 0\\ :\\ X_{t}\\not \\in U\\}", "142272e402c3424d0c751458f25c7041": "{\\frac {c}{c'}}\\sim 1+\\left(\\beta -\\delta -{\\frac {1}{2}}\\right){\\frac {v^{2}}{c^{2}}}\\sin ^{2}\\theta +(\\alpha -\\beta +1){\\frac {v^{2}}{c^{2}}}", "142351e9a48de4907251a15fbae7462a": "A_{0,1}\\subseteq A_{0,2}\\subseteq A_{0,3}\\subseteq \\dots ", "1423714a046ef79b33d25f61bc7850f6": "\\Delta (G)=n-1", "14237a09b6be33cb266c0a9a467d3627": "i*(TM)", "1423c12947ad7b6a9b8fc0ff70af3691": "\\lim _{x\\to \\infty }{\\frac {f(x)}{g(x)}}=\\lim _{x\\to \\infty }\\left(1+{\\frac {\\sin x}{x}}\\right)=1", "1423f1eca43285573e3e5a981cea044e": "{\\begin{matrix}{\\frac {9}{7}}\\end{matrix}}", "1423fe2a603a0984f3b824f4dd88eeaa": "E[h(y)]=\\int _{-\\infty }^{+\\infty }{\\frac {1}{\\sigma {\\sqrt {2\\pi }}}}\\exp \\left(-{\\frac {(y-\\mu )^{2}}{2\\sigma ^{2}}}\\right)h(y)dy", "1424576de49dc2dcc9ebbe1471441e38": "f(x,y)={\\begin{cases}y&{\\text{if }}y\\neq x^{2}\\\\0&{\\text{if }}y=x^{2}\\end{cases}}", "142480edc25c3f67ac9b6b587d2bebd1": "\\cosh s=\\cosh b\\cdot \\cosh ^{2}l-\\sinh ^{2}l.", "1425709a7e394ffd986d11587dd70b6f": "R_{critical}={\\frac {2\\cdot \\gamma \\cdot V_{Atom}}{k_{B}\\cdot T}}", "142593160f797362160f0238e043b1a5": "|\\psi (x,t_{1})|^{2}\\neq |\\psi (x,t_{0})|^{2}", "1425ce4e1db78ed1c2a3a1efe93d253b": "X_{i}=p_{i}\\geq 0", "1425eaee8efb8e982d94faa96104dc21": "\\mathrm {Bhattacharyya} =\\sum _{i=1}^{n}{\\sqrt {(\\mathbf {\\Sigma a} _{i}\\cdot \\mathbf {\\Sigma b} _{i})}}", "142636266e9be0d0c02aee8ee0eac181": "\\zeta (k)", "142648986c0bb09ee0458b22ebea425f": "\\beta _{1}y(b)+\\beta _{2}y'(b)=0\\qquad \\qquad \\qquad (\\beta _{1}^{2}+\\beta _{2}^{2}>0),", "14265577ff8ee9e24a39b0fb2d437dde": "p\\gamma _{0}=E+\\mathbf {p} ", "14266865cebe9969b708551bfde7e30f": "{\\frac {C_{V}}{Nk}}\\sim 9\\left({T \\over T_{D}}\\right)^{3}\\int _{0}^{\\infty }{x^{4}e^{x} \\over \\left(e^{x}-1\\right)^{2}}\\,dx", "142673c1609416cfa9ceb305bf06a449": "E_{x_{l},p_{k}}={\\frac {\\partial x_{l}(p,w)}{\\partial p_{k}}}\\cdot {\\frac {p_{k}}{x_{l}(p,w)}}={\\frac {\\partial \\log x_{l}(p,w)}{\\partial \\log p_{k}}}", "142683a703877c13109bc6956a3058d2": "(2-\\eta )^{3}=7(\\eta -1)^{2}.", "1426ac4994ec481343a9455c3ab2e8de": "a_{n}=a\\,r^{n-1}.", "1426e8fca3932e4e9613cb0f342743ed": "{\\mathfrak {P}}^{4}", "142767c46cedb3e74f0790a60729f0bb": "a_{p_{j}}^{+}", "142775222b0df67f72be7e0b1a236361": "\\mu -\\sigma {\\sqrt {\\frac {1-q}{q}}}\\leq x_{q}\\leq \\mu +\\sigma {\\sqrt {\\frac {q}{1-q}}}", "14278c3e0b76defbfedf2b6e94d9c822": "d>1/64", "1427d1c3ce6ab5f6510890b718b2bef2": "B_{k}\\mathbf {p} _{k}=-\\nabla f(\\mathbf {x} _{k})", "1428200c45d995f73c2cc15864b00ac7": "\\scriptstyle D'", "14293d58f5914a1382bf8414e2c1e08f": "\\ \\displaystyle u\\ ", "142953a63755044e215e0366ef46e082": "{\\begin{bmatrix}C^{j+1}\\end{bmatrix}}={\\begin{bmatrix}AA^{-1}\\end{bmatrix}}([BB][C^{j}]+[d]).", "142a0904f51a76d54e43a463353867db": "(-\\infty ,a_{n})\\cup (b_{n},\\infty )", "142a31ba36a0c4226833cc6c92cf346f": "\\int _{0}^{1}f(x)v(x)\\,dx=\\int _{0}^{1}u''(x)v(x)\\,dx.", "142a64ccd835b449d00b24f165185036": "{\\overline {\\mathbf {Q} }}[[T]]", "142aeca4468bcc28fa1ad7002b67e517": "\\neg K_{a}\\neg \\varphi ", "142b25dcf1eb58c482b4398f2696275d": "P_{\\mathrm {out} }=K_{p}\\,{e(t)}", "142b37cacd911ba8623fab4dbb59a5bc": "{\\mbox{External virtual work}}=\\int _{V}\\delta {\\boldsymbol {\\epsilon }}^{T}{\\boldsymbol {\\sigma }}\\,dV\\qquad \\mathrm {(1)} ", "142b609233aee1a59ce32a52b7f92cf7": "\\Gamma (W)", "142b7a3a46f09b4078475acffb9656f3": "{\\mathcal {E}}:D\\times D\\to \\mathbb {R} ", "142bdc762b8a4d9528b48131697d07e9": "H_{A}(\\mu )=\\Phi \\left({\\frac {{\\sqrt {n}}(\\mu -{\\bar {X}})}{s}}\\right)", "142c195145bbea2637638446a2f082b1": "\\Omega ^{*}=\\arcsin \\left(n_{1}n_{2}\\right)+\\arcsin \\left(n_{2}n_{3}\\right)+\\arcsin \\left(n_{3}n_{4}\\right)+\\arcsin \\left(n_{4}n_{1}\\right).", "142c59268b5a18704f75a79830173662": "n=1200\\cdot \\log _{2}\\left({\\frac {f_{2}}{f_{1}}}\\right).", "142c796b930c3645a04c12f50136bc68": "f{\\Bigl (}\\sum _{k=1}^{n}A_{k}^{*}X_{k}A_{k}{\\Bigr )}\\leq \\sum _{k=1}^{n}A_{k}^{*}f(X_{k})A_{k},", "142cf40b2574250a374f7e299ebe03b7": "\\mathbb {R} ^{n}", "142d2a0290f5aebfa351affb2596f8ff": "U_{Total}=U_{Repel}+U_{Stretch}+U_{Bend}+U_{Trellis}+U_{Gravity}", "142d3eeaab4eed722c8faf528eb611e9": "\\{X(t)", "142d83e2b8f496be11ec117b6e474b48": "S\\ {\\stackrel {\\mathrm {def} }{=}}\\ M_{pl}^{2}\\int _{M}\\epsilon _{abcd}(e^{a}\\wedge e^{b}\\wedge \\Omega ^{cd})=M_{pl}^{2}\\int _{M}d^{4}x\\epsilon ^{\\mu \\nu \\rho \\sigma }\\epsilon _{abcd}e_{\\mu }^{a}e_{\\nu }^{b}R_{\\rho \\sigma }^{cd}[\\omega ]", "142db804092ae5dfd0153788a41affbc": "c=(p'q'^{-1})^{\\alpha \\beta }=(h^{r}g^{x}h^{-s}g^{-y})^{\\alpha \\beta }=(h^{r-s}g^{x-y})^{\\alpha \\beta }=", "142e106451174957a0f633988242f918": "{\\widehat {D}}^{*}", "142e86f3ae7a030ceab9be75254dbe22": "V_{1}=V", "142ec92f6842e7261b744f2753cb8d13": "\\lim _{n\\rightarrow \\infty }np_{n}=\\lambda ,", "142fbdde23d012ff8976400c3567a8f1": "4.16130x^{2}+9.15933x-11.4207=0", "14311166283b0f182c3234b15228d850": "\\underbrace {\\begin{bmatrix}-1&0&0\\\\0&-1&0\\\\0&0&1\\\\\\end{bmatrix}} _{C_{2}}\\times \\underbrace {\\begin{bmatrix}1&0&0\\\\0&-1&0\\\\0&0&1\\\\\\end{bmatrix}} _{\\sigma _{v}}=\\underbrace {\\begin{bmatrix}-1&0&0\\\\0&1&0\\\\0&0&1\\\\\\end{bmatrix}} _{\\sigma '_{v}}", "14319465a54059760cc6149b457e8e49": "{\\frac {\\partial ^{2}u}{\\partial x^{2}}}+{\\frac {\\partial ^{2}u}{\\partial y^{2}}}=0.", "1431cd84e28cbc734dd7c4e7c464bdff": "f(x)=b", "1431dceac216e994f13b24e4057314e0": "\\forall x\\ \\exists y\\ \\exists z\\ ((x\\lor z)\\land y)", "1432055a1fc723b7b952ed4cf4156263": "{\\mathfrak {D}}_{L/K}", "14324d0a9022731fd83e60bdc012e0c3": "\\mathbf {y} ", "14325b203add24084e67cd76ea6ca2da": "\\scriptstyle k\\leq n+1", "14328c24ee66f319bcafb22b7ed94340": "{\\begin{matrix}{\\frac {1}{a}}\\end{matrix}}", "1432d0964e3b0179409331bd0c357ee4": "\\eta _{th}\\equiv {\\frac {W_{out}}{Q_{in}}}\\equiv {\\frac {\\text{Electrical Power Output + Heat Output + Cooling Output}}{\\text{Total Heat Input}}}", "143330b8f6b24ee34cbc79bcb03855e6": "[I]={\\begin{bmatrix}I_{11}&I_{12}&I_{13}\\\\I_{21}&I_{22}&I_{23}\\\\I_{31}&I_{32}&I_{33}\\end{bmatrix}}={\\begin{bmatrix}I_{xx}&I_{xy}&I_{xz}\\\\I_{xy}&I_{yy}&I_{yz}\\\\I_{xz}&I_{yz}&I_{zz}\\end{bmatrix}}.", "143330e466809978a62d204ac3e40609": "\\!f(x)=1+e^{-ax}", "143368bf77368c8be688ab66460487e1": "u_{n,i}", "1433a38f493f455defec103e3a2dae42": "Z=\\sum _{n=0}^{\\infty }\\left({\\frac {2}{27}}\\right)^{n}{\\frac {(15n+2)\\left({\\frac {1}{2}}\\right)_{n}\\left({\\frac {1}{3}}\\right)_{n}\\left({\\frac {2}{3}}\\right)_{n}}{(n!)^{3}}}\\!", "1433cdacaeee0b8cb98fdde716f69018": "\\xi =\\pm \\,\\Delta \\,\\int _{0}^{\\psi }{\\frac {{\\text{d}}{\\hat {\\psi }}}{\\sqrt {1-m\\,\\sin ^{2}{\\hat {\\psi }}}}}=\\pm \\,\\Delta \\,F(\\psi |m),", "1433d671dffea5fa94244fa7c2c3ab3c": "u_{L}^{C}", "1433d98eac19da5dbbbb05b35277d324": "{\\frac {L}{L_{\\odot }}}={\\left({\\frac {R}{R_{\\odot }}}\\right)}^{2}{\\left({\\frac {T}{T_{\\odot }}}\\right)}^{4}", "1433df7c68d31d87f49f4bf730d2a2b9": "\\sum _{i=0}^{n}\\Delta h_{i}g_{i},", "1433eca3fb6259959ec22064cea8e510": "\\varphi _{\\alpha }(\\beta )<\\delta \\,", "1433fa80a3b91ac2bcce6be91fceb9ea": "p_{i_{1},\\ldots ,i_{n}}(f_{1},\\ldots ,f_{n})", "143405b7914725e1af955bd2894d8f8d": "Z=AR^{b}", "1434101071ef648e8514b2d62094e12f": "\\lbrace \\lambda _{i}\\rbrace ", "1434f4cd0dfeb1fafdb47b468be10cb3": "{\\tfrac {2\\pi }{3}}", "143501ccd3f55f24062586288a1ccc6a": "E\\sum _{j}({\\hat {Y}}_{j}-E(Y_{j}\\mid X_{j}))^{2}/\\sigma ^{2},", "14356a8f663e67fa16797ad9a6256264": "w'=v_{x}-iv_{y}={\\bar {v}},", "14362077aac4af29772f1aa0d584934c": "{\\mathfrak {X}}", "143680e4361eb2a7c43249848743cb0f": "E^{*}(\\mathbf {CP} ^{\\infty })=E^{*}({\\text{point}})[[x]]", "14369e878a206361b693eec2b4075da9": "-[H^{+}]_{0^{}}10^{-b_{0}}/K_{w}", "143757781b5d8489bdd2ac66b5b7ffeb": "v\\in S", "1437711dd636dc151fe1bc7ef51f9f42": "S^{\\circ }=10\\log S+1", "14378b6ea28ddb55ea1762203de351b1": "\\ RED=Ra/R_{0}", "1437f56b00ca3fd0c768bba2e7d21d51": "M(t)=M_{0}\\mathrm {e} ^{-t/T_{2}}", "143854de9c6845037b2f5438102ea02d": "(x-1)/x", "143855010b900c440e8977c1f1b98639": "a=N_{\\mathrm {A} }^{2}a'", "14399ced3698adf9e88f3585b44fb6ea": "\\left(\\!\\!{n \\choose k}\\!\\!\\right)={n+k-1 \\choose k}={\\frac {(n+k-1)!}{k!\\,(n-1)!}}={n(n+1)(n+2)\\cdots (n+k-1) \\over k!},", "1439a448cdcbabcb180f357f369ec0f7": "L(p)\\leq 2^{n}M(p)\\leq 2^{n}L(p);", "1439a845ab6be262395878229ffc01a8": "\\left\\|\\sum _{k=1}^{m}x_{k}-\\sum _{k=1}^{n}x_{k}\\right\\|=\\left\\|\\sum _{k=n+1}^{m}x_{k}\\right\\|\\leq \\sum _{k=n+1}^{m}\\|x_{k}\\|<\\varepsilon ,", "1439cb1b3731deb28c4f986a7c4cc301": "[\\Delta (y,E(m))>(p+\\epsilon )]\\leq 2^{-{\\epsilon ^{2}}n}", "143a0b190ad337896b3083780b42d53f": "S=\\left(\\Omega ,{\\mathcal {A}},\\mu \\right)", "143a75a3c85ba4ff7e34f9dfd9fba0f5": "|{\\mathcal {P}}|<\\infty ", "143a7928d8fb6e86328f39a447ca5cd1": "p\\ll \\rho c^{2}", "143a8a3672c255f0d031064fe50e67ca": "\\left[{n \\atop r}\\right]_{r}", "143b831ae565a37b707c5d0b910e2178": "\\arctan(1/\\lambda )", "143bde353c8bcdd3f40722ab93aa6b21": "={\\frac {1+\\sin(Ax)\\cos(Bx)}{2}}", "143c84eda304ce710729dd74d9280351": "\\phi (\\cdot )", "143cb2c308688f8c25e9d1c0377d53a7": "{100 \\over {\\sqrt {2}}}", "143cc097f631290d61cf4c959a01f98f": "{\\begin{aligned}&\\textstyle {\\frac {2\\log \\left({\\frac {{\\sqrt[{3}]{27-3{\\sqrt {78}}}}+{\\sqrt[{3}]{27+3{\\sqrt {78}}}}}{3}}\\right)}{\\log(2)}},\\\\&^{\\text{or root of}}\\\\&2^{x}-1=2^{(2-x)/2}\\end{aligned}}", "143cc1d4040f5dda5dd38a57fa665d99": "\\sigma '(0)=x.\\,", "143d65ea21768236186b56833b42ae28": "H_{n}(G,M)=H_{n}(F\\otimes _{\\mathbf {Z} [G]}M)", "143d8caf7d4f115d46e8038db9d6f2e0": "\\left[M{\\frac {\\partial }{\\partial M}}+\\beta (g){\\frac {\\partial }{\\partial g}}+n\\gamma \\right]G^{(n)}(x_{1},x_{2},\\ldots ,x_{n};M,g)=0", "143df6b29b8532a06bc3bdaf9df743bb": "\\rho \\simeq \\rho _{0}\\,", "143e11af44f263b2ed499a1b1f5c5f07": "\\ U_{0}=\\,", "143e1593f81c07e8b2b45047a73949c1": "x_{3}=r\\sin \\psi \\sin \\theta \\sin \\phi \\ ", "143e4e4767ae79e5f11f239a4cfc15a5": "\\left[{\\begin{matrix}1+\\alpha ^{2}/2&\\alpha &0&-\\alpha ^{2}/2\\\\\\alpha &1&0&-\\alpha \\\\0&0&1&0\\\\\\alpha ^{2}/2&\\alpha &0&1-\\alpha ^{2}/2\\end{matrix}}\\right]", "143e5c40cf53a2e5cd195bb97f2f676d": "{\\sqrt {\\lambda _{i}}}V_{i}=\\mathbf {X} ^{\\mathrm {T} }U_{i}", "143edc54ae24a633118a04cb763856a7": "E\\{(g_{i}^{*}(y)-x_{i})g_{j}(y)\\}=0,", "143ef5ed8cf7d3292ee5ecaaabc0bd5c": "F_{\\theta }", "143f057790f2444eaac137a2ef01e7f0": "{\\begin{bmatrix}\\ln p_{1}+C\\\\\\vdots \\\\\\ln p_{k}+C\\end{bmatrix}}", "143f72d458ac43bcb0e1f61fe97946eb": "{\\underline {\\mathbf {Z} }}", "14402582008e305fb1d857ccb9991cee": "(M,\\Omega )", "14404e74761d07676dbbfe8a67e02b5e": "\\displaystyle \\iint f(x,y)e^{-2\\pi i(\\xi _{x}x+\\xi _{y}y)}\\,dx\\,dy", "14409d5a6e9d918d7b8e31ee775ea26f": "U\\mapsto \\pi _{U}", "1440c93a7208dc2f18c42ee01ea0e10b": "Y={\\frac {\\omega _{H}}{\\omega }}", "1440da304e956c08becad0e9cda005e7": "\\int \\tan ^{n}ax\\;\\mathrm {d} x={\\frac {1}{a(n-1)}}\\tan ^{n-1}ax-\\int \\tan ^{n-2}ax\\;\\mathrm {d} x\\qquad {\\mbox{(for }}n\\neq 1{\\mbox{)}}\\,\\!", "1440ef44ed08563449987ad3fdb9fd7b": "i_{C}=C{\\frac {d}{dt}}v_{C}\\iff v_{L}=L{\\frac {d}{dt}}i_{L}", "14413d160c862f9212dfe5ca189d99b0": "{\\tfrac {dx}{dt}}", "14414cb709fc8bade32a579c658d9c6f": "{\\begin{aligned}{\\boldsymbol {\\nabla }}\\cdot \\mathbf {v} &={\\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}}\\\\{\\boldsymbol {\\nabla }}\\cdot {\\boldsymbol {S}}&={\\frac {\\partial S_{rr}}{\\partial r}}~\\mathbf {e} _{r}+{\\frac {\\partial S_{r\\theta }}{\\partial r}}~\\mathbf {e} _{\\theta }+{\\frac {\\partial S_{rz}}{\\partial r}}~\\mathbf {e} _{z}\\\\&+{\\cfrac {1}{r}}\\left[{\\frac {\\partial S_{\\theta r}}{\\partial \\theta }}+(S_{rr}-S_{\\theta \\theta })\\right]~\\mathbf {e} _{r}+{\\cfrac {1}{r}}\\left[{\\frac {\\partial S_{\\theta \\theta }}{\\partial \\theta }}+(S_{r\\theta }+S_{\\theta r})\\right]~\\mathbf {e} _{\\theta }+{\\cfrac {1}{r}}\\left[{\\frac {\\partial S_{\\theta z}}{\\partial \\theta }}+S_{rz}\\right]~\\mathbf {e} _{z}\\\\&+{\\frac {\\partial S_{zr}}{\\partial z}}~\\mathbf {e} _{r}+{\\frac {\\partial S_{z\\theta }}{\\partial z}}~\\mathbf {e} _{\\theta }+{\\frac {\\partial S_{zz}}{\\partial z}}~\\mathbf {e} _{z}\\end{aligned}}", "144250497400d99f1ecd90a7602dcd9a": "g(f(k))", "1442b13913e1383c039765fe935eb2c5": "F(b)-F(a)=\\int _{0}^{1}DF(a+(b-a)t)\\cdot (b-a)dt", "1442c814e115d0aca59d389d2ce695c7": "H={50 \\choose 2}{48 \\choose 2}\\div 2!=690,900", "144312c1373c86c3a98d1434a706779e": "\\Delta _{\\Omega }=\\det(I-T_{\\Omega }^{2})=\\prod (1-\\lambda _{n}^{2}).", "14432a7c175f6ba320b4b1acff1461b6": "\\langle \\cdot |\\cdot \\rangle ", "14435595fd595cdabadb60c07124b653": "{\\frac {\\partial W^{*}}{\\partial t^{*}}}+U^{*}{\\frac {\\partial W^{*}}{\\partial X^{*}}}+W^{*}{\\frac {\\partial W^{*}}{\\partial Z^{*}}}\\ =-{\\frac {\\partial p_{d}}{\\partial Z^{*}}}+Pr\\left({\\frac {\\partial ^{2}W^{*}}{\\partial X^{*2}}}+{\\frac {\\partial ^{2}W^{*}}{\\partial Z^{*2}}}\\right)\\ -{Ra_{s}Pr_{s}S}+{Ra_{T}Pr_{T}T}", "14435a5e55ac0326df9b871d3b852e91": "p_{i}=y_{i}^{2}", "14435b40fa72ce72b61ea788a1206d14": "4X^{3}-g_{2}X-g_{3}", "14436843492351317959da20d48ae255": "\\left\\{(x,y)\\in R^{2}:\\sum _{i=1}^{n}{\\sqrt {(x-u_{i})^{2}+(y-v_{i})^{2}}}=d\\right\\}", "144369d646b3419394a676dccc8df430": "F_{1}={x}^{2}+sx+{\\frac {b}{2}}+{\\frac {s^{2}}{2}}-{\\frac {c}{2p}}", "14439ad4ab1ed9260c140b560f76f5d7": "\\alpha _{V}(V,T)\\ ={\\frac {\\left.{\\cfrac {\\partial p}{\\partial V}}\\right|_{(V,T)}}{p(V,T)}}", "1444bcb16f2537b1fbefbc5fcea86b80": "H={p^{2} \\over 2}+{l^{2} \\over 2r^{2}}-{1 \\over r}.", "1444e97b33fb963e49879f931a953b23": "00\\}", "144662c5f1f6ab08e02d5c60fb0bbc45": "S'=S\\cup \\{C\\}", "1446793c9c0ed3142c01fe5ecaef4f8f": "X=[-a,a]\\times [-b,b]\\times [-c,c]=\\{(x,y,z)\\in \\mathbb {R} ^{3}\\,:\\,-a\\leq x\\leq a,-b\\leq y\\leq b,-c\\leq z\\leq c\\}\\,.", "14467cfb1fd04a9ca11c589331e46a25": "a_{H^{+}}=\\exp \\left({\\frac {\\mu _{H^{+}}-\\mu _{H^{+}}^{\\ominus }}{RT}}\\right)", "1446922b8d015f3fc7b6e854a97bee12": "I(\\rho ,{\\mathcal {N}})", "14483780e738c8868f175e46543e763f": "Z(S,H;s)=\\sum _{x\\in S}H(x)^{-s}.", "14484357a61bd495ef4cd9907df395d6": "\\tau =\\pm {\\frac {h}{r^{2}+h^{2}}}.", "14489729ebeff60cdd5ed67bbc0e4ea5": "M'=x^{3}M", "14489dbbe75ed0c92a3a12f2ce1c6c6f": "P=10^{\\frac {-Q}{10}}", "1448a6a0a9de5e5c139f19c6d4832b70": "M_{m}", "1448ca68c4bd5ae9b613ceac0de0e8bd": "\\nabla (fg)=f\\nabla g+g\\nabla f", "1449c17944ea4b2f31c5c417e19e4b6d": "\\displaystyle P(z|d)", "1449d0bd76ba67646e5f8401694f80b0": "a^{\\dagger }(\\phi )\\,", "144a9b252e1102805dd6567a4fe6855f": "a_{31}={\\frac {(\\lambda +2\\mu )(4\\mu +A)+4\\mu B}{4(\\lambda +\\mu )}}", "144acadb0a4d2198041385b5c501a8ae": "\\left({\\frac {1}{z}}+1\\right)\\left(1-\\exp(-z)\\right)={\\frac {1}{z}}+1-\\exp(-z)-{\\frac {1}{z}}\\exp(-z).", "144ad6c6094e76add7517b072c61c080": "f_{(\\xi ,\\mu ,\\sigma )}(x)={\\frac {\\sigma ^{\\frac {1}{\\xi }}}{\\left(\\sigma +\\xi (x-\\mu )\\right)^{{\\frac {1}{\\xi }}+1}}},", "144b0daffd620f02963fa2f758d051d3": "{\\frac {1}{\\tilde {Q_{s}}}}=\\langle {\\frac {1}{Q_{mnp}}}\\rangle _{k\\leq k_{r}\\leq k_{r}+\\Delta k}", "144b5833a81e160f270ebe34acb2ccc2": "{\\frac {e^{E(1)/\\gamma }}{e^{E(1)/\\gamma }+e^{E(2)/\\gamma }}}\\ ", "144b761e4f25f989fbac92f5592a8d10": "{\\begin{aligned}(x_{N}*y)[n]&=\\int _{0}^{1}{\\frac {1}{N}}\\sum _{k=-\\infty }^{\\infty }\\scriptstyle {DFT}\\displaystyle \\{x_{N}\\}[k]\\cdot \\scriptstyle {DFT}\\displaystyle \\{y_{N}\\}[k]\\cdot \\delta \\left(f-k/N\\right)\\cdot e^{i2\\pi fn}df\\\\&={\\frac {1}{N}}\\sum _{k=-\\infty }^{\\infty }\\scriptstyle {DFT}\\displaystyle \\{x_{N}\\}[k]\\cdot \\scriptstyle {DFT}\\displaystyle \\{y_{N}\\}[k]\\cdot \\int _{0}^{1}\\delta \\left(f-k/N\\right)\\cdot e^{i2\\pi fn}df\\\\&={\\frac {1}{N}}\\sum _{k=0}^{N-1}\\scriptstyle {DFT}\\displaystyle \\{x_{N}\\}[k]\\cdot \\scriptstyle {DFT}\\displaystyle \\{y_{N}\\}[k]\\cdot e^{i2\\pi {\\frac {n}{N}}k}\\\\&=\\scriptstyle {DFT}^{-1}\\displaystyle {\\big [}\\scriptstyle {DFT}\\displaystyle \\{x_{N}\\}\\cdot \\scriptstyle {DFT}\\displaystyle \\{y_{N}\\}{\\big ]},\\end{aligned}}", "144b8c7cb82c965e7bc25dfacfcba334": "d\\in N", "144c25411906427be681acc2216a28ca": "D_{\\mu }=\\partial _{\\mu }-(e/\\hbar c)A_{\\mu }", "144cc1a637818aa02c6ad1c2b96b00da": "{\\sqrt[{3}]{2}}", "144ccf1df809efa97b0563822db6dd08": "01", "149f569669d1a44647f4d92bc6c93074": "T_{p}M\\,", "149f7a0280b55b3155e91b98f9b7c3f8": "{\\begin{smallmatrix}T_{\\rm {eff}}\\end{smallmatrix}}", "149f90aa5c8c46265a1028dd3ff372a4": "I_{q}=I_{1}\\sin ^{2}\\left({\\frac {q\\pi g\\sin \\alpha }{\\lambda }}\\right)/\\sin ^{2}\\left({\\frac {\\pi g\\sin \\alpha }{\\lambda }}\\right)\\ ,", "149f97ff8f22132ee28bded5553a3de4": "\\displaystyle {{1 \\over 2\\pi }\\iint |F_{-}(z)|^{2}(1-|z|^{2})^{-1/2}\\,dxdy}", "149fccab5db3016d91d55f7cd34365dc": "e\\approx 1.602\\ 176\\ 565\\times 10^{-19}\\;\\;\\mathrm {C} .", "149ff0831c48cc13d841b48610a256cb": "{\\frac {g({\\boldsymbol {x}}'\\rightarrow {\\boldsymbol {x}})}{g({\\boldsymbol {x}}\\rightarrow {\\boldsymbol {x}}')}}=1", "14a0038e8b9b837517e3c18942ff1318": "\\delta E/\\delta x", "14a04151804b46d0b3a742a46683cc86": "\\Lambda (x,y,0)", "14a04cf4d8af10b63b005597c21c7f49": "p\\oplus _{x}n", "14a086e3774e08338d1ec6da0c9b29c1": "{\\mbox{If }}X\\vdash Y{\\mbox{ and }}Y\\vdash a{\\mbox{, then }}X\\vdash a", "14a0ba194ca4874a1766b95b4077450c": "R_{\\lambda }f=\\int _{0}^{\\infty }e^{-\\lambda t}T_{t}f\\,dt.", "14a105ea781d75edce90d493637386bb": "{1}/{\\sqrt {\\omega }}", "14a12ff2cdba4702c041047ebd8cdf9a": "\\Pi _{k}(z_{1},\\ldots ,z_{n}):=(z_{k}-z_{1})(z_{k}-z_{2})\\cdots (z_{k}-z_{k-1})(z_{k}-z_{k+1})\\cdots (z_{k}-z_{n})\\quad k=1,\\ldots ,n.", "14a1989c28e728ff0a674efbf1fbe751": "2/7", "14a1b756edad99bb5e85197d5826fc65": "p\\not \\ll p", "14a22e69771467be72f32266adf77c29": "W=\\arg \\min _{X}(\\max _{Y}score(Y,X))", "14a3428e846942683433782b55655f3b": "H=p_{a}^{i}\\partial _{i}y^{a}-L", "14a3e8659f24b984dbcdaec48308233b": "c_{t+1}=(1-R^{-1})\\left[A_{t+1}+\\sum _{j=1}^{\\infty }\\left({\\frac {1}{R}}\\right)^{j-1}E_{t+1}y_{t+j}\\right]", "14a451b226111e0db393ca9afcc90d6d": "f(tx+(1-t)y)\\leq tf(x)+(1-t)f(y)-t(1-t)\\phi (\\|x-y\\|),\\,", "14a46dc71d5d7e93bf36cc14931ae283": "P(W_{n}|[Spam=false])={\\frac {1+a_{f}^{n}}{2+a_{f}}}", "14a49175f3782dac4f7d241c68691903": "\\textstyle {\\textbf {R}}^{d}", "14a4a69df046e46988db0621397a6773": "p_{1}:=A^{-1}Bp_{2}", "14a4ae2ab11e8c3a5be1f3d56019b2bd": "{\\mbox{div}}\\,({\\mbox{grad}}\\,f)=\\nabla \\cdot (\\nabla f)=\\nabla ^{2}f=\\Delta f", "14a4b0abbb834a812611bf9959b53831": "i,j,x", "14a4bc40b2af8ab2f195803e5ee2f058": "{\\begin{bmatrix}-2{\\boldsymbol {\\eta }}_{1}\\\\[5pt]-(2\\eta _{2}+p+1)\\end{bmatrix}}", "14a4ebfd49a31e699009ccf08552e124": "B=B_{0}", "14a4ece9e83c329a24d7c343ad313ca7": "h_{\\theta }=R\\,", "14a5beaca6422ab203dd7ba36cc8eea1": "\\Phi _{n}(x)=\\prod _{\\stackrel {1\\leq k\\leq n}{\\gcd(k,n)=1}}(x-e^{2i\\pi {\\frac {k}{n}}})", "14a5e1cbf01b176512513d2a81b048cf": "c_{1}=0", "14a61ba307e6494ae48d290ce6b42aeb": "i<0", "14a6dc93d8b195acc00951d4b36f9be4": "K={\\frac {L^{2}}{t_{D}U}}", "14a6e55d9b5e48cecfb5eb5004754d68": "(M^{2},\\partial M)\\subset S^{3}", "14a74a300d66e724d164da868f36d31b": "w(t,0)=0,", "14a7a87e6d92dc217768e184e793b45f": "dQ=C\\,dV", "14a7fca18e7850a306a2e1a7bd0a3182": "\\mathbf {P} =m\\mathbf {v} +e\\mathbf {A} ", "14a84fa53884614a613c8fc4aa59381b": "\\ \\phi ,\\ \\mu ,\\ \\kappa ", "14a863300a71d05baea19833d441a427": "\\tan \\alpha _{2}={\\frac {1}{\\phi }}-\\tan \\beta _{2}\\,", "14a901f13da44d2c8f324a09fb6b1e44": "\\gamma (a)=x", "14a9cb1785a05fe25a4a6b2a335f9486": "{\\frac {1}{4\\pi c^{2}}}", "14a9e4d8494ee5ecc7c25c58ea341f5e": "{\\text{ESF}}_{i}\\,", "14a9f2fca3e9efd08d709e653b7d10c5": "\\mathbb {X} =(x_{1},\\ldots ,x_{n}),", "14aa2258df61c93d6076dfb8eda363b5": "{\\frac {\\partial C}{\\partial t}}={\\frac {C_{i}^{j+1}-C_{i}^{j}}{\\Delta t}}", "14aa66df6d70aa7bb2ce69faa2b2a55e": "x_{\\perp }", "14aa688ae0424eaa04ede7cff7cddb9b": "\\ell _{P}^{2}={\\frac {\\hbar \\,G}{c^{3}}}", "14ab3eebd427d1772aec2b22274f32cc": "\\sum _{i=0}^{n}P_{i}", "14ab3f17ba9814c384b26da6a881dbff": "\\rho (\\cdot ,\\cdot )", "14ab5490ae43841ffa71a17bce172bde": "\\textstyle (1+X)^{n}=\\sum _{k\\geq 0}{\\binom {n}{k}}X^{k},", "14ac2b02aac8abf2d2ef6748a78db25a": "{\\frac {\\alpha }{\\alpha _{G}}}", "14ac778ed69e05f7a64a2237a643eb09": "{\\begin{array}{l}\\sum \\limits _{n=0}^{\\infty }\\delta \\left(x-\\gamma _{n}\\right)+\\sum \\limits _{n=0}^{\\infty }\\delta \\left(x+\\gamma _{n}\\right)={\\frac {1}{2\\pi }}{\\frac {\\zeta }{\\zeta }}\\left({\\frac {1}{2}}+ix\\right)+{\\frac {1}{2\\pi }}{\\frac {\\zeta '}{\\zeta }}\\left({\\frac {1}{2}}-ix\\right)-{\\frac {\\ln \\pi }{2\\pi }}\\\\[10pt]{}+{\\frac {\\Gamma '}{\\Gamma }}\\left({\\frac {1}{4}}+i{\\frac {x}{2}}\\right){\\frac {1}{4\\pi }}+{\\frac {\\Gamma '}{\\Gamma }}\\left({\\frac {1}{4}}-i{\\frac {x}{2}}\\right){\\frac {1}{4\\pi }}+{\\frac {1}{\\pi }}\\delta \\left(x-{\\frac {i}{2}}\\right)+{\\frac {1}{\\pi }}\\delta \\left(x+{\\frac {i}{2}}\\right)\\end{array}}", "14ad0bfa0595277207a88a0ad161da7e": "f\\circ \\gamma :[0,T]\\to \\mathbb {R} ", "14ad13e0abab5b454966327be399c54f": "\\lim _{p\\to 0}M_{p}(x_{1},\\dots ,x_{n})=\\exp {\\left(\\ln {\\left(\\prod _{i=1}^{n}x_{i}^{w_{i}}\\right)}\\right)}=\\prod _{i=1}^{n}x_{i}^{w_{i}}=M_{0}(x_{1},\\dots ,x_{n})", "14ad464a01c14bac4ba24c17d317efe7": "X={\\frac {Y}{y}}x", "14ad82993c8b8e35ed32cc3d0e10a930": "D_{f}", "14ad8c20e8759cd1e2c26de63cddf745": "{\\tilde {A}}=\\left\\langle H_{0}\\right\\rangle _{0}-TS_{0}+\\left\\langle \\Delta H\\right\\rangle _{0}=\\left\\langle H\\right\\rangle _{0}-TS_{0}\\,", "14add1279f441439f5c5587163ab3090": "\\sin ^{5}\\theta \\cos ^{5}\\theta ={\\frac {10\\sin 2\\theta -5\\sin 6\\theta +\\sin 10\\theta }{512}}\\!", "14addb3443ab67b0b55c7956500c0ec1": "=p(C)\\ p(F_{1},\\dots ,F_{n}\\vert C)", "14adf6a1ecbf5150421904ffa40367b1": "\\mathbf {\\hat {n}} =\\mathbf {\\hat {r}} \\times {\\boldsymbol {\\hat {\\theta }}}\\,\\!", "14ae0cfd4eaffa27e467902d6d622ee1": "\\sigma =\\phi _{1,2}", "14ae2ab8141a09eed79e1ea2b6f24c29": "E={\\frac {1}{2}}mU_{\\mathrm {ion} }^{2}=neV_{1}", "14aea17ff026dc1f45913ef79fcf2fb5": "{\\frac {\\partial \\rho (\\mathbf {r} ,t)}{\\partial t}}+\\nabla \\cdot \\mathbf {j} (\\mathbf {r} ,t)=0,", "14aef51bc2b404fe65a900e4c994a91f": "({\\boldsymbol {\\beta }}-{\\hat {\\boldsymbol {\\beta }}})", "14af1ffd3776c073f97ae89da0508f55": "v\\mapsto e^{t\\log b}v", "14af742eeca25a23943d80ef558da3be": "k_{1}={\\tfrac {(a-c)b}{c(c+1)}},k_{2}={\\tfrac {(b-c-1)(a+1)}{(c+1)(c+2)}},k_{3}={\\tfrac {(a-c-1)(b+1)}{(c+2)(c+3)}},k_{4}={\\tfrac {(b-c-2)(a+2)}{(c+3)(c+4)}}", "14af8245a44e5d43831875ac275efc0c": "{\\begin{cases}p<{\\frac {(d-1)^{d-1}}{d^{d}}}&d>1\\\\p={\\tfrac {1}{2}}&d=1\\end{cases}}", "14aff913ec4987e693202110cad3ce9a": "y=Ae^{t}+2Be^{-5t}.\\,\\!", "14b0191cb071cb958f50f92ddd4753f0": "\\scriptstyle F,", "14b033d9bdc25c33f257c99d09d6845c": "\\lbrack \\mathbf {h} \\rbrack =\\lbrack \\mathbf {h} \\rbrack _{1}+\\lbrack \\mathbf {h} \\rbrack _{2}", "14b05ec3da085dfc96d142f6c04d6409": "\\exp(v)=c(1).\\ ", "14b0d05755a357f0a7a3b7095f0740ee": "O(n+z)", "14b0d0c977d026a668c604c1281c25bb": "p=2\\quad {\\mbox{and}}\\quad d_{K}\\equiv 1{\\pmod {8}},", "14b10b819a86a86e17a8483cffb920bc": "\\rho :\\mathbb {C} ^{n}\\to \\mathbb {R} ", "14b18ed237e269e9f1fb0d0e79aca6a6": "{dx \\over dt}={x(t+\\epsilon )-x(t) \\over \\epsilon }\\,", "14b1c0c9185f9fa4ec15e12b90a65e12": "^{h}p(x,y,z)=z^{\\deg(p)}p({\\tfrac {x}{z}},{\\tfrac {y}{z}})", "14b1d8a94e3282503d370545514f90f5": "[z_{/\\cong _{\\mathcal {B}}}]_{X}=\\{z\\in z_{/\\cong _{\\mathcal {B}}}\\mid z\\in X\\},", "14b1eba3a7dfa2690a41ebf7bb1f6a6a": "\\emptyset ", "14b2214382cfdeac478caa51cc4f72ec": "F(t)=Ks(t)=K((1cm)sin(wt))\\!", "14b2842455aa11b5f613eed22762ed6d": "\\Delta _{1}^{0}", "14b34ae33b0ab6685987aa3692e185b7": "\\delta _{\\nu _{1}\\dots \\nu _{n}}^{\\mu _{1}\\dots \\mu _{n}}=\\varepsilon ^{\\mu _{1}\\dots \\mu _{n}}\\varepsilon _{\\nu _{1}\\dots \\nu _{n}}.", "14b3801428a2fe9857893634334d2207": "q\\to 0", "14b3822faba0d938a0e9abce7f881fe5": "R=|P-Q|", "14b3ac8a919efae53f6dced1d14fb47c": "E_{3}=-q", "14b4184aaaff9c86e38f1c6ed2e5d92a": "T_{K}", "14b4883589c601fdef221d007a845c5e": "{\\frac {{\\text{d}}E}{{\\text{d}}t}}=z{\\frac {{\\text{d}}B}{{\\text{d}}t}}-{\\frac {{\\text{d}}C_{1}}{{\\text{d}}t}}-{\\frac {{\\text{d}}C_{2}}{{\\text{d}}t}}-{\\frac {{\\text{d}}C_{3}}{{\\text{d}}t}}", "14b4a5e4eb6dfd9db07c8a321767272c": "\\mathrm {DF} (t_{\\textrm {tod}},t_{\\textrm {mat}})", "14b4ad51b0f38ab76613a466c7c1b24e": "\\scriptstyle y", "14b4c6ec856b2c53fe76a14bc8660292": "r_{4}(n)=\\pi ^{2}n\\left({\\frac {c_{1}(n)}{1}}-{\\frac {c_{4}(n)}{4}}+{\\frac {c_{3}(n)}{9}}-{\\frac {c_{8}(n)}{16}}+{\\frac {c_{5}(n)}{25}}-{\\frac {c_{12}(n)}{36}}+{\\frac {c_{7}(n)}{49}}-{\\frac {c_{16}(n)}{64}}+\\dots \\right)", "14b4d7614915d3922532fa7b65c00547": "L_{p}=10\\log _{10}\\left({\\frac {{p_{\\mathrm {rms} }}^{2}}{{p_{\\mathrm {ref} }}^{2}}}\\right)", "14b4ffe824a9ea41453b801f7ac7e201": "F=k\\cdot x", "14b54f90bbd7717134c9d927e6965465": "I_{1}(z)", "14b557f0d69c25a059228a4e4e52c4f0": "\\scriptstyle \\pm 2", "14b559d1492835fa6207d56fb7c5abce": "S(\\alpha )=S^{max}{\\sqrt {1-{\\frac {1-{\\sqrt {1-\\Delta ^{2}}}}{2}}}}", "14b57afad6adbd260e99275813c87ff2": "b\\mapsto (\\pi _{*}\\alpha )_{b}", "14b59f15f19074345ff62f75f5a1aabf": "\\tan(\\theta )\\approx \\theta =\\Delta x/R", "14b5b7073193311e38479f57494dc0cd": "F(t,x,y)={\\frac {F(u,x,y)-F(t,x,y)}{u-t}}=0.", "14b60671587932fbd870acd5571406fc": "a=\\omega -1", "14b60dee94ab94cebe3aaffdf0480025": "x\\in V_{\\alpha }\\cap V_{\\beta }", "14b6bcb2240d3eeacabcda0695c0d2ef": "{\\mathfrak {R}}_{k}", "14b6ea586dbb753876056f1bc99821cb": "\\scriptstyle \\leq -2\\times 10^{-21}", "14b6eb9e096de2f56e9ce0b72fef4edb": "(X,Y,\\langle ,\\rangle )", "14b711dbccb916558d43abb0886404da": "\\int _{a}^{b}f(x)\\,dx", "14b742b76633e38af9c74ae51f878201": "y^{5}+py^{3}+qy^{2}+ry+s=0", "14b76ebe44edc9ec2fa73be9de1c55fc": "(M\\geq 0)", "14b78e4112ca310eba1106f1fcc837ce": "\\Delta S_{\\mathrm {overall} }=\\Delta S^{\\prime }+\\Delta S^{\\prime \\prime }-\\Delta S^{\\prime }=\\Delta S^{\\prime \\prime }.", "14b79935be74f7e2ac374a544a4d0128": "(x',t')=(x,t){\\begin{pmatrix}1&0\\\\-v&1\\end{pmatrix}}.", "14b7e9bfa079c71a1c9b01299bdd5380": "{\\frac {x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}}=-{\\frac {\\partial ^{2}}{\\partial x\\partial y}}\\arctan(y/x)", "14b86c80a39556ef49d0ccaef7ffeb0b": "t_{2}\\,\\!", "14b8c68022b2702f0d1788a8acc04fab": "\\theta _{V}=Ad(h^{-1})\\theta _{U}+h^{*}\\omega _{H},\\,", "14b8f97fe567a57911317f7dbbf408aa": "\\scriptstyle p_{21}\\,", "14b931b4b874e8138609a0a5f298e2e0": "A=2\\pi rh.", "14b940f2a549b2c92fc3fa5ccf88cb0b": "{\\boldsymbol {R}}\\cdot {\\boldsymbol {R}}^{T}={\\boldsymbol {\\mathit {1}}}\\quad \\implies \\quad {\\dot {\\boldsymbol {R}}}\\cdot {\\boldsymbol {R}}^{T}=-{\\boldsymbol {R}}\\cdot {\\dot {{\\boldsymbol {R}}^{T}}}", "14b94255fda9b4cbe904a8a8caa9c460": "\\Gamma _{n}^{o}=\\langle \\mathrm {Shannon} _{n}\\rangle ^{+}", "14b95950d75d0bcd4fdb0dec9c5447a2": "\\Delta H=-R*(b+2{\\frac {c}{T}}),", "14b991c77c711171bcad2b04db117190": "v\\in C^{2}({\\bar {\\Omega }})", "14b9b8ea7673699a10967338db764d85": "\\beta (u)=2\\pi \\rho =(4\\pi )u=4\\pi {\\sqrt {2M(r-2M)}};", "14b9bda42976d420e1726c2093e446a8": "\\langle \\Gamma _{i}(t)\\Gamma _{j}(t')\\rangle \\propto \\delta (t,t')", "14b9ebeaf82be5c6d696eec0326f6aef": "i:{\\mathrm {F} }_{\\mathrm {O} }(E)\\to {\\mathrm {F} }_{\\mathrm {GL} }(E)", "14b9ef4df305a651cbf1ed7cee3c9f89": "u_{l,k}", "14ba54fbe57cf3c56f09d5efc509b40a": "\\mathbf {e} ", "14ba5bcc64da4d7995f081772373cb3b": "i\\hbar {\\frac {\\partial }{\\partial t}}\\psi =-{\\frac {\\hbar ^{2}}{2m}}\\nabla ^{2}\\psi +V\\psi ", "14ba5f3d519ded167e29e779009d630e": "(\\alpha ,\\beta ,\\gamma )=(\\alpha (x,y),\\beta (x,y),\\gamma (x,y))=(xy,x-y,x+y)", "14ba6d9c3993c5e1e2f5e62d6553d34b": "{\\frac {\\partial \\Lambda (T_{ij},\\lambda _{i},\\lambda _{j})}{\\partial T_{ij}}}=-\\ln T_{ij}-\\lambda _{i}-\\lambda _{j}-\\beta C_{ij}=0", "14ba9598bb7c26b855e212a4554b8ef7": "\\nabla ^{4}", "14baaef38cb2ea4f613ad36072f37c08": "\\forall s,t:W^{-}[s,t]=W(s,t)", "14bb0cdf38e9a9eb47e7645cd78d39c2": "|w|\\leq \\tau .", "14bb23c2bb070e0086f75a30c3ec8109": "s_{0}(t)=e^{st}\\left(\\cosh(qt)-s{\\frac {\\sinh(qt)}{q}}\\right),\\qquad s_{1}(t)=e^{st}{\\frac {\\sinh(qt)}{q}},", "14bb2fe43021ce9e91a4a20e524d8a17": "{\\dot {\\varepsilon _{\\rm {p}}}}^{*}", "14bb3663551afe19c99efcaab77fca84": "R_{n}^{(n+1)}(t)=f^{(n+1)}(t)", "14bb48486188a76c2d7eed9d5465b9f9": "I_{n,m}=I_{m-2,n-1}-a^{2}I_{m-2,n}\\,\\!", "14bb5a367707365553196bd8d6ff133b": "\\left({\\frac {p_{i}}{p_{\\mathrm {ref} }}}\\right)^{2}=10^{\\frac {L_{i}}{10}},\\qquad i=1,2,\\cdots ,n", "14bbabedcf353aee78305b07b6481765": "{\\tilde {c}}_{i}=c_{i}{\\tilde {b}}_{i-1}\\,", "14bc76512bc1da3166d918135e10e894": "\\forall y\\forall z(P(y)\\land \\exists xQ(x,z))", "14bc8fa16a5059b9115c6e085d02de0a": "E=\\int _{0}^{L}B\\kappa ^{2}(s)ds", "14bc9de520cd7d3dabc2346d66cdeac5": "z_{\\alpha /2}", "14bcaa9506f743b1f8d6bfb3df9fd549": "H+L(\\alpha )\\geq 0", "14bce8d003e9f64192914e43b4f0d383": "{\\frac {1}{\\sqrt {n}}}\\sum _{i=1}^{n}x_{i}\\varepsilon _{i}\\ \\xrightarrow {d} \\ {\\mathcal {N}}{\\big (}0,\\,V{\\big )},", "14bce9a3f18b2d8ae5d231a327a477be": "({\\mbox{STr}})", "14bd08d2e3938c183ff5ec6081e13305": "{\\begin{aligned}&{}\\quad \\left[{\\begin{matrix}A&B\\\\C&D\\end{matrix}}\\right]^{-1}=\\left[{\\begin{matrix}I_{p}&0\\\\-D^{-1}C&I_{q}\\end{matrix}}\\right]\\left[{\\begin{matrix}(A-BD^{-1}C)^{-1}&0\\\\0&D^{-1}\\end{matrix}}\\right]\\left[{\\begin{matrix}I_{p}&-BD^{-1}\\\\0&I_{q}\\end{matrix}}\\right]\\\\[12pt]&=\\left[{\\begin{matrix}\\left(A-BD^{-1}C\\right)^{-1}&-\\left(A-BD^{-1}C\\right)^{-1}BD^{-1}\\\\-D^{-1}C\\left(A-BD^{-1}C\\right)^{-1}&D^{-1}+D^{-1}C\\left(A-BD^{-1}C\\right)^{-1}BD^{-1}\\end{matrix}}\\right].\\end{aligned}}", "14bd28d163393630962ee86e8b212243": "EAC={\\frac {NPV}{A_{t,r}}}", "14bd4970ce01b9e821756add9d1e3e53": "(1+x)^{-2}=1-2x+3x^{2}-4x^{3}+\\cdots \\quad |x|<1", "14bd66ba5aef2092821bf1261b66f766": "BMO", "14bda14f876b8699dcda7eb93a655790": "\\|x-y\\|^{2}+\\|x+y\\|^{2}=2\\|x\\|^{2}+2\\|y\\|^{2}", "14bda6aa361e73c544279855296030e6": "\\varphi _{xx}+\\varphi _{yy}=-\\rho ,", "14bdea8555086393973634251f337792": "r=r_{0}(\\theta )", "14bdf6fd669a50951e3b9f708a744ed7": "\\Phi _{1}(x)=x-1\\,", "14be9791e05dd35e5f535a5720573f58": "W=-{\\frac {\\mu J_{m}}{2}}\\ln \\left(1-\\left({\\frac {\\lambda _{1}^{2}+\\lambda _{2}^{2}+\\lambda _{3}^{2}-3}{J_{m}}}\\right)\\right)", "14bedee5da804bc74223bbcfc5ba89eb": "(A,+,\\cdot ,-,0,1,c_{\\kappa },d_{\\kappa \\lambda })_{\\kappa ,\\lambda <\\alpha }", "14bf287a0a068c1d59dd9e70a602fc86": "-{\\frac {1}{2}}\\lambda dN_{1}=d[(N_{2}D_{1}+N_{1}D_{2}){\\frac {dN_{1}}{d\\lambda }}]", "14bf5ff0df232b4b551de7e07ae01643": "P({\\vec {R}})", "14bf9378dfc74058a4e59e065fccecd2": "\\gamma =\\alpha +j\\beta \\,", "14bf958843dc773e5ad7064be0000af3": "{n \\choose k}_{q}", "14bfd0a90a0114be4ca5aca2f68d8573": "W=C_{1}({\\bar {I}}_{1}-3)+D_{1}(J-1)^{2}=C_{1}\\left[J^{-2/3}(\\lambda _{1}^{2}+\\lambda _{2}^{2}+\\lambda _{3}^{2})-3\\right]+D_{1}(J-1)^{2}", "14bfea5f6966e9a4578f22fe82c9f3d7": "\\scriptstyle \\eta (v,x)", "14c01d5f62dbb18e7b5d370b27bcf7d2": "(\\mathbf {b} +\\mathbf {c} )\\mathbf {a} =\\mathbf {ba} +\\mathbf {ca} ", "14c029a192de512fc04dec22839e1ac0": "I={\\frac {1}{12}}wh^{3}", "14c08a89a28c2265590e3f0cb1d6b809": "T^{00}>0", "14c095a4c2130cb8f51665823633f1b9": "H=\\int \\rho h\\mathrm {d} V,", "14c0c1a69914fe7e8908d13868a014e3": "\\ (U,\\ N,\\ E)", "14c0ff60b4a081e857dbdc419e76dcef": "\\left\\{0\\right\\}=T_{0}\\subseteq T_{1}\\subseteq \\dotsb \\subseteq T_{k}=R", "14c11610e98799a98b74d6973a7f0434": "ROC2=(1-Price/Price(X2))*100;", "14c13e919506c9cd6022c1a2295f40ca": "\\mathrm {P} ", "14c166d8caf03d529673554619ceed96": "{\\frac {\\hat {p}}{\\hat {q}}}", "14c1fbbbbfd081aaee80274d6468fb9a": "\\sigma _{e}\\,", "14c2009e893a1535fb198e51a90e4e2d": "I=I_{o}e^{-Q\\Delta x}=I_{o}e^{-{\\frac {\\Delta x}{\\lambda }}}=I_{o}e^{-\\sigma (\\eta \\Delta x)}=I_{o}e^{-{\\frac {\\rho \\Delta x}{\\tau }}},", "14c2777e467711300832e9e3c1a2a295": "e^{-1/g}", "14c3136e55c1e556b19a70495143cf86": "{\\frac {1}{2}}\\Delta v^{2}", "14c3314ad0051d045b2ff85b02d2dee7": "\\left(z{\\frac {d}{dz}}+a\\right)\\left(z{\\frac {d}{dz}}+b\\right)w=\\left(z{\\frac {d}{dz}}+c\\right){\\frac {dw}{dz}}", "14c3acbe640e1ddbc5220bd84b809171": "|\\mathrm {GHZ} \\rangle ={\\frac {|0\\rangle ^{\\otimes M}+|1\\rangle ^{\\otimes M}}{\\sqrt {2}}},", "14c3b3a6539320ccb18e22aa48e888d2": "\\mathbb {N} ^{\\mathbb {N} }", "14c3b932f35ec23203ada6df588d16b2": "S^{d+1}", "14c50c535c224592c570575125c75374": "J=100\\left(A/A_{w}\\right)^{cz}", "14c517b2613d6d508b61ae507ce2e70e": "f(x)=\\ln x", "14c52509018cc4bbd4348bc3e1b653ab": "\\mathbf {A} \\otimes (\\mathbf {B} +\\mathbf {C} )=\\mathbf {A} \\otimes \\mathbf {B} +\\mathbf {A} \\otimes \\mathbf {C} ,", "14c533e18ef8415bf05d537db2a27732": "x=y+m", "14c561e595ce5709065107017d4d8f09": "\\delta =1/\\beta -1", "14c56683e30b53b498b7174e28c7bd0b": "L(A)\\subseteq L(B)", "14c577760ebc7810fa5601661d622768": "\\operatorname {max} (a,b)={\\frac {a+b+|a-b|}{2}}", "14c600e00c3967278760bfc2053219fe": "\\forall A[A\\not =\\varnothing \\rightarrow \\exists b(b\\in A\\land \\forall c(c\\in b\\rightarrow c\\not \\in A))].", "14c61e1ce270e52467a13ac075504ce2": "{\\frac {(\\mathbf {r} -\\mathbf {p} _{i})}{|\\mathbf {r} -\\mathbf {p} _{i}|^{3}}}=-\\nabla \\left({\\frac {1}{|\\mathbf {r} -\\mathbf {p} _{i}|}}\\right).", "14c628fcecb9a6431f91f65b56a001d8": "x-\\Delta x\\leq \\xi \\leq x+\\Delta x", "14c66ffea255e4ee362236443973cb72": "W(\\varepsilon )", "14c672ab3df14adce6d7abae9757cc6b": "\\partial _{ij}", "14c6873157820ac817c02b4fc3720a89": "\\phi =\\arctan \\left({R \\over L}\\right)", "14c6ca77360582fdb354b9302bfc312d": "{}^{3}x", "14c6e8a587dfdce3609603692f1f9ef9": "1\\leq i\\leq m", "14c70ec8cbb51314c5b6ac584cf2c523": "\\pi P(t)=\\pi ", "14c7452626fa36bf359685372604ae1c": "t_{3}", "14c7a5f699476506c72559b7696a35af": "\\mu :\\mathbb {R} ^{n}\\times [0,T]\\to \\mathbb {R} ^{n};", "14c87522cd363dad9abe585e94f6d1ef": "\\varpi ", "14c8883d2c4b2c396906c46f3a897d9c": "\\varphi _{h(e)}\\simeq \\varphi _{\\varphi _{e}(e)}", "14c88f20b13d670f33fb09f84517d6c5": "g({\\textbf {a}})\\not =0", "14c8ced3a0f6fa9c6d5e8166b3bfcdc9": "{\\sqrt {2}}^{\\sqrt {2}}.", "14c8d739d8b6fd0db889ce221e884b62": "h\\nu =g_{\\mathrm {e} }\\mu _{\\mathrm {B} }B_{\\mathrm {0} }", "14c92c0854fed9dd04da1b2d34fbf96a": "46=6+20+20", "14c92e023182e18fd9d0d8071f441b1b": "q^{*}(\\mathbf {Z} )=\\prod _{n=1}^{N}\\prod _{k=1}^{K}r_{nk}^{z_{nk}}", "14c983b8e5474e64fc30679ed0c6d4b5": "{\\begin{array}{cl}{\\underset {{\\boldsymbol {\\alpha }},{\\boldsymbol {\\xi }},\\rho }{\\min }}&-\\rho +D\\sum _{n=1}^{\\ell }\\xi _{n}\\\\{\\textrm {sb.t.}}&\\sum _{\\omega \\in \\Omega }y_{n}\\alpha _{\\omega }h({\\boldsymbol {x}}_{n};\\omega )+\\xi _{n}\\geq \\rho ,\\qquad n=1,\\dots ,\\ell ,\\\\&\\sum _{\\omega \\in \\Omega }\\alpha _{\\omega }=1,\\\\&\\xi _{n}\\geq 0,\\qquad n=1,\\dots ,\\ell ,\\\\&\\alpha _{\\omega }\\geq 0,\\qquad \\omega \\in \\Omega ,\\\\&\\rho \\in {\\mathbb {R} }.\\end{array}}", "14c9d37cf5a78009bc1a75af48c3b6d5": "\\mu =2", "14c9e1137621b47ab94317b664421b45": "p={\\frac {\\rho }{4}}\\left(\\cos 2x+\\cos 2y\\right)F^{2}(t).", "14ca01b4a2151099cb75d1fbeb9b29bd": "\\quad h(\\phi )\\;=\\;{\\frac {P'K'}{PK}}\\;=\\;{\\frac {\\delta y}{R\\delta \\phi \\,}}.", "14ca3634eb43d7f0b16f867df5ef5ec5": "{\\frac {\\partial \\log {\\mathcal {L}}(\\alpha ,\\beta \\,|\\,x_{1},\\ldots ,x_{n})}{\\partial \\beta }}={\\frac {\\partial \\log {\\mathcal {L}}(\\alpha ,\\beta \\,|\\,x_{1})}{\\partial \\beta }}+\\cdots +{\\frac {\\partial \\log {\\mathcal {L}}(\\alpha ,\\beta \\,|\\,x_{n})}{\\partial \\beta }}={\\frac {n\\alpha }{\\beta }}-\\sum _{i=1}^{n}x_{i}.", "14ca3c9c106d9587d826deee8499c635": "{\\begin{aligned}\\varepsilon _{rr}&={\\frac {\\partial u_{r}}{\\partial r}}\\\\\\varepsilon _{\\theta \\theta }&={\\frac {1}{r}}\\left({\\frac {\\partial u_{\\theta }}{\\partial \\theta }}+u_{r}\\right)\\\\\\varepsilon _{\\phi \\phi }&={\\frac {1}{r\\sin \\theta }}\\left({\\frac {\\partial u_{\\phi }}{\\partial \\phi }}+u_{r}\\sin \\theta +u_{\\theta }\\cos \\theta \\right)\\\\\\varepsilon _{r\\theta }&={\\frac {1}{2}}\\left({\\frac {1}{r}}{\\frac {\\partial u_{r}}{\\partial \\theta }}+{\\frac {\\partial u_{\\theta }}{\\partial r}}-{\\frac {u_{\\theta }}{r}}\\right)\\\\\\varepsilon _{\\theta \\phi }&={\\frac {1}{2r}}\\left[{\\frac {1}{\\sin \\theta }}{\\frac {\\partial u_{\\theta }}{\\partial \\phi }}+\\left({\\frac {\\partial u_{\\phi }}{\\partial \\theta }}-u_{\\phi }\\cot \\theta \\right)\\right]\\\\\\varepsilon _{r\\phi }&={\\frac {1}{2}}\\left({\\frac {1}{r\\sin \\theta }}{\\frac {\\partial u_{r}}{\\partial \\phi }}+{\\frac {\\partial u_{\\phi }}{\\partial r}}-{\\frac {u_{\\phi }}{r}}\\right).\\end{aligned}}", "14ca95091504036f5e660f0cd380b0d0": "00", "14de173a8adf1c85c00d0533d848e0f8": "={\\frac {5-2.34}{5.09}}=.5\\%", "14de304fb8a6abb19c7d5a9fe5ce6057": "C(u\\otimes v)", "14de5f1f13e662c4c6c6562e3549fced": "A(t)=U(-t)AU(t).\\quad ", "14deeddfefcfbacb84366f221cdb02d2": "y={\\frac {B-{\\sqrt {B^{2}-4A}}}{2}}", "14df0ff5c4308223b3e2168e251d4447": "{\\begin{matrix}{\\frac {2}{3}}\\end{matrix}}", "14df3f5d62fa5eaafb76fabe34dbce62": "\\nabla _{X}V=\\lim _{h\\to 0}{\\frac {\\Gamma (\\gamma )_{h}^{0}V_{\\gamma (h)}-V_{\\gamma (0)}}{h}}=\\left.{\\frac {d}{dt}}\\Gamma (\\gamma )_{t}^{0}V_{\\gamma (t)}\\right|_{t=0}.", "14df83ae45ba136ce8958f93b0c695e1": "\\displaystyle {\\|u\\|_{(1)}^{2}=|(\\Delta u,u)|\\leq |(\\Delta _{1}u,u)|+|(Xu,u)|\\leq \\|\\Delta _{1}u\\|_{(-1)}\\|u\\|_{(1)}+C^{\\prime }\\|u\\|_{(1)}\\|u\\|_{(0)}.}", "14df91d39f7a06a8eaca21bc4de6499a": "{\\frac {{\\overline {PA}}\\cdot {\\overline {QA}}}{{\\overline {CA}}\\cdot {\\overline {AB}}}}+{\\frac {{\\overline {PB}}\\cdot {\\overline {QB}}}{{\\overline {AB}}\\cdot {\\overline {BC}}}}+{\\frac {{\\overline {PC}}\\cdot {\\overline {QC}}}{{\\overline {BC}}\\cdot {\\overline {CA}}}}=1.", "14df945f4520d87288c1c406b58fee90": "1+\\lceil p\\log _{10}(2)\\rceil ", "14dfc2a5d3a9a165077201fcfde1b0a9": "M_{\\mathrm {u} }=M({}^{12}\\mathrm {C} )/12\\,", "14dffd39abb6d74064c1393e199dcffd": "D_{med}=E|X-median|=2Cov(X,I_{O})", "14e02c652e90ad7b729ff84ea33b110f": "C={\\frac {3\\times {\\mbox{number of triangles}}}{\\mbox{number of connected triples of vertices}}}={\\frac {\\mbox{number of closed triplets}}{\\mbox{number of connected triples of vertices}}}.", "14e0406efd339db5f4eaa23f638d8005": "v(p+r,t)=v(p,t)+J(p,t)r", "14e0490b83d7555f9832ebb54a7ca51d": "{\\mathbf {p}}={\\frac {\\partial G_{2}}{\\partial {\\mathbf {q}}}}={\\frac {\\partial S}{\\partial {\\mathbf {q}}}}\\ \\rightarrow \\ H({\\mathbf {q}},{\\mathbf {p}},t)+{\\partial G_{2} \\over \\partial t}=0\\ \\rightarrow \\ H\\left({\\mathbf {q}},{\\frac {\\partial S}{\\partial {\\mathbf {q}}}},t\\right)+{\\partial S \\over \\partial t}=0.", "14e04d5363196d308393629af6bcc333": "\\int _{-\\infty }^{\\infty }e^{-ax^{2}+bx+c}\\,dx={\\sqrt {\\frac {\\pi }{a}}}\\,e^{{\\frac {b^{2}}{4a}}+c},", "14e054604a0e03d668baedc5eb23b93e": "\\|f\\|_{1,w}\\leq \\|f\\|_{1}.", "14e05bd95dec768c200f5dbb03a20af4": "{\\frac {\\partial \\mathbf {D} }{\\partial t}}+\\mathbf {J} =\\nabla \\times \\mathbf {H} \\ \\rightarrow \\ \\mathbf {E} \\cdot {\\frac {\\partial \\mathbf {D} }{\\partial t}}+\\mathbf {E} \\cdot \\mathbf {J} =\\mathbf {E} \\cdot \\nabla \\times \\mathbf {H} .", "14e061011b37262c1d6213a4f276bef7": "L_{1}\\cdot L_{2}=\\{w\\cdot z|w\\in L_{1}\\land z\\in L_{2}\\}", "14e08757093344734a670802306cf705": "{\\begin{aligned}e^{ix}&=\\cos x+i\\;\\sin x\\\\e^{-ix}&=\\cos x-i\\;\\sin x\\end{aligned}}", "14e0df45900873845c67edfe0b8d2ae8": "a_{t}", "14e0f56fb1ecd858b267ea86f7e26e86": "\\gamma _{\\text{SG}}\\ ", "14e14a8523502755031185532f5d92b4": "\\forall C[\\lnot \\exists W(C\\in W)\\iff \\exists F(\\forall x[\\exists W(x\\in W)\\Rightarrow \\exists s(s\\in C\\land \\langle s,x\\rangle \\in F)]\\land ", "14e19b344fd2b63f60c7c9d1a8f3a9fd": "U(f)=f(T)\\xi ,\\,C[0,1]\\rightarrow H", "14e1e154857d768e838515e68d36ffc2": "P_{2}=P_{1}(1+r)-c", "14e1fbd7b83da4c8661e1a3cd6eb846a": "T_{o}", "14e2139b9e08d44a4d42c892bc598e13": "\\ x[n]=0.5^{n}u[n]", "14e236e27f2f575b8a87b1b44d881856": "(r,\\phi ,z)\\in [0,\\infty )\\times [0,2\\pi )\\times (-\\infty ,\\infty )", "14e23d2a69b782882a1e6bf5ddbf12d5": "{\\frac {d^{2}x}{dt^{2}}}+\\beta (t){\\frac {dx}{dt}}+\\omega ^{2}(t)x=E(t).", "14e26413a15d7467db850d97b0e6a0e4": "{\\frac {\\mathbf {s(} n\\mathbf {)} \\,\\,{\\mathsf {nat}}}{n\\,\\,{\\mathsf {nat}}}}", "14e2afb8376dc5a31797418eed321b54": "{\\mathcal {L}}.", "14e2ca745d0be4bef09547415c48afc8": "10_{137}", "14e2e0c29c6c6d68e209dd61c72b25fa": "\\operatorname {E} (T)=k\\sigma ^{2}+n\\mu ^{2}+2\\mu \\sum _{i=1}^{k}(n_{i}T_{i})+\\sum _{i=1}^{k}n_{i}(T_{i})^{2}", "14e3390d4539cc1370999e6f31da05fc": "{\\mathcal {P}}_{3}\\,D_{m'm}^{j}(\\alpha ,\\beta ,\\gamma )^{*}=m\\,D_{m'm}^{j}(\\alpha ,\\beta ,\\gamma )^{*},", "14e3d87382feaafd9bd43b4357b87016": "877\\,", "14e3e589f905545fa78bc9df02e436f1": "k_{B}T\\approx 0.026\\,eV", "14e3eeb4d53f990467847d3de906187d": "expr\\subseteq \\{expr_{1},\\dots ,expr_{n}\\}", "14e422dfbefe9a24811dd6951618d87d": "{\\text{gross margin}}={\\frac {0.667}{1+0.667}}=0.4=40\\%", "14e463372eceb1e2a386d27b7f58f296": "e^{-}+H_{2}O\\longrightarrow H_{2}^{+}+O+2e^{-}", "14e4963992aa60057ea937a7979bfca4": "f:\\mathbb {R} _{+}\\to \\mathbb {R} ", "14e498b4f9bd48ea2e533c8f4498402b": "\\mathbb {P} {\\biggl (}\\bigcap _{j\\in J}A_{j}{\\biggr )}={\\frac {(m-|J|)!}{m!}}.", "14e4c98bfe057d46da3da7fb426c0649": "{\\begin{aligned}f^{-(n+1)}(x)&=\\int _{a}^{x}\\int _{a}^{\\sigma _{1}}\\cdots \\int _{a}^{\\sigma _{n}}f(\\sigma _{n+1})\\,\\mathrm {d} \\sigma _{n+1}\\cdots \\,\\mathrm {d} \\sigma _{2}\\,\\mathrm {d} \\sigma _{1}\\\\&={\\frac {1}{(n-1)!}}\\int _{a}^{x}\\int _{a}^{\\sigma _{1}}\\left(\\sigma _{1}-t\\right)^{n-1}f(t)\\,\\mathrm {d} t\\,\\mathrm {d} \\sigma _{1}\\\\&={\\frac {1}{(n-1)!}}\\int _{a}^{x}\\int _{t}^{x}\\left(\\sigma _{1}-t\\right)^{n-1}f(t)\\,\\mathrm {d} \\sigma _{1}\\,\\mathrm {d} t\\\\&={\\frac {1}{n!}}\\int _{a}^{x}\\left(x-t\\right)^{n}f(t)\\,\\mathrm {d} t\\end{aligned}}", "14e4dc358686033decc36698b281bdf2": "{\\dot {p}}(0)=-m\\omega ^{2}x_{0},", "14e4f0aea0e961a68d28dc6b2d387e46": "y=3x-3.", "14e5018d20c891631cae4ae7ba235221": "S(n)=n+1.\\,", "14e51a4c8a314d559a6647d06ab34e47": "A\\cup (B\\cap C)=(A\\cup B)\\cap (A\\cup C)\\,\\!", "14e5500feea0eced4cd7056759255f38": "C=dE/dT=k\\beta ^{2}\\langle E^{2}\\rangle _{c}=k\\beta ^{2}(\\langle E^{2}\\rangle -\\langle E\\rangle ^{2})", "14e56f4ef5e41d392b474a872b31a68c": "\\{\\to ,\\land ,\\lor \\}", "14e5c4069fe2064426aa3e4c95c000ef": "N(u)=1", "14e5dac99630fc6a1bf6df6a3c9bacab": "\\Delta E_{A}", "14e63f3d1ab0e0df716d0062e050ed9f": "\\nabla _{X}Y-\\nabla _{Y}X=[X,Y]\\ ", "14e6981dd7f5eb726461212591524113": "\\gamma ^{r}", "14e6a8efaff1957e45e78449344e0202": "S_{1}=\\alpha ^{i}+\\alpha ^{i'}", "14e6d76fde0d20cf0a85be806816b91d": "i\\neq j", "14e727541ba6c3c44af619177d541452": "\\{\\psi (x),\\psi (y)\\}=\\{\\psi ^{*}(x),\\psi ^{*}(y)\\}=0\\,", "14e728185b3120aff615ef7fc2831b65": "S^{(\\omega )}", "14e7449108e1538f44efcbd44b2eb5f9": "u={\\frac {J_{3}P_{3}^{0}(\\sin \\theta )}{r^{4}}}=J_{3}{\\frac {1}{r^{4}}}{\\frac {1}{2}}\\sin \\theta (5\\sin ^{2}\\theta -3)=J_{3}{\\frac {1}{r^{7}}}{\\frac {1}{2}}z(5z^{2}-3r^{2})", "14e7552457ef87f15eb79aec9f2ff8f8": "{\\mbox{Area}}\\;=\\;T\\,+\\,2\\left({\\frac {T}{8}}\\right)\\,+\\,4\\left({\\frac {T}{8^{2}}}\\right)\\,+\\,8\\left({\\frac {T}{8^{3}}}\\right)\\,+\\,\\cdots .", "14e7a3651122c733dfee2fa075d7036e": "(x_{1},x_{2},\\ldots ,x_{n})", "14e7b6aa827e74a8cb0df9ddc25d580f": "F\\models _{\\text{pref}}G", "14e7d26a80421f01577ac03673b81fdd": "\\beta =0.280169499023\\dots \\,.", "14e7ff9975624bbc495c2d14d82f4717": "\\prod _{k=1}^{4}(k+2)=(1+2)(2+2)(3+2)(4+2)::=3\\times 4\\times 5\\times 6=360", "14e8117c2348050446945d8166ad965e": "c(u,v)-f(u,v)>0", "14e815a1046597f66ca63a4b4133084d": "C^{++}=(C^{+})^{+}", "14e824db17a01698432596793cb25682": "1944=[33,30]_{58}", "14e82900ada7fada0846c3e43a183b0e": "\\Omega _{j}=\\theta _{j}+\\phi _{j}", "14e837d978b467fda5c9915b270fd86d": "m_{c}", "14e84173ffef7c811198aacca371259f": "{v_{C}^{2} \\over 2}-gh_{C}+{P_{\\mathrm {atm} } \\over \\rho }=\\mathrm {constant} ", "14e8421b04c1eda501d764aa0740d9b3": "r=r(s),\\ \\theta =\\theta (s)", "14e853a522a4b902e97bbc63774e5481": "\\deg(cP)=\\deg(P)", "14e8b1f27cf4ec764e59ade3659a8143": "\\phi \\,", "14e8f43a5bae11336e6f8a330756a580": "\\sum _{i}{\\Big (}\\sum _{\\alpha }a_{i\\alpha }X^{\\alpha }{\\Big )}\\otimes b_{i}=\\sum _{i}\\sum _{\\alpha }X^{\\alpha }\\otimes a_{i\\alpha }^{p}b_{i},", "14e927836ae1392b721ac9aa99652fda": "\\int \\mathrm {Det} ({\\partial F \\over \\partial G})e^{iS_{GF}}DA\\,", "14e93652d91ab9b286325e01d678ece0": "{\\vec {W}}=M{\\vec {L}}+{\\vec {P}}\\times {\\vec {C}}", "14e968b38a8cc064505594710aad830d": "\\mu _{2}={\\frac {2}{\\bar {x}}}\\left[1-{\\sqrt {1+2(c^{2}-1)}}\\right]^{-1}", "14e979873497c4bd4af70de5b3b0dee6": "e^{S(x)}\\,", "14e9ec72509a2fedf308bdfd9f2852e8": "2^{803}-2^{402}+1", "14ea01d3a995a921f7ad8b2334a4f1af": "{\\frac {1}{p}}\\!", "14ea08eaf6b4195da9032a3fccad5b73": "S_{xy}", "14eab2048d7664df893a5adbdb453601": "lu=g(x),x\\in \\partial D\\,", "14eac3aadba72699512f712a5324af1e": "\\{ba,abab,aababb,aaababbb,\\dotsc \\}", "14eacef4b222c0df06a463b3a6c7c453": "\\beta =4", "14eafc1151f2f5b3dbee9a115c930330": "{\\frac {d^{2}y}{dx^{2}}}={\\frac {d}{dx}}\\left({\\frac {dy}{dx}}\\right).", "14eb177540f56701669c15f438deac9b": "e^{i\\pi },e^{i\\pi {\\sqrt {2}}},e^{i\\pi {\\sqrt {2}}},e^{2i\\pi }.", "14eba0ad299228f8b098728d0165ba31": "P(x_{2})-f(x_{2})=-\\varepsilon \\,", "14ec0e00142389334d2b900b57ab4409": "P_{u}=uu^{\\mathrm {T} }.\\,", "14ecadbf797b09b7456d26996ad4ffb0": "\\gamma (0)=x", "14ed1849368bff14300b12c5fed3b49d": "A_{j}\\,", "14ed61f971f989af85220c5edc55aae2": "H={\\frac {1}{2m}}\\left(p-{\\frac {e}{c}}A\\right)^{2}+e\\phi -{\\frac {e\\hbar }{2mc}}\\sigma \\cdot B.", "14edb01b881a33a8d514f97e9a7c0d95": "x\\cdot (2+x)", "14ee1a75229267df9f2b4f5c258a1875": "{\\frac {\\Diamond p\\land \\Diamond \\neg p}{\\bot }}", "14ee368469aa2f9e06e2e90f1cc1854a": "\\int _{-\\pi }^{\\pi }f(e^{i\\theta }){\\overline {g(e^{i\\theta })}}\\,d\\mu ", "14ee48626bfe01caf79e6127438e2ff7": "q=\\mathbf {S} (q)+\\mathbf {V} (q)\\,", "14ee7570caf0393954381bb7cf7f249c": "f(a+xh)\\approx f(a)+x\\left({\\frac {\\Delta f(a)+\\Delta f(a-h)}{2}}\\right)+{\\frac {x^{2}\\Delta ^{2}f(a-h)}{2!}}.", "14ee907b38cab40339b0f89490279d33": "FMV\\,=\\,{VBAB*TAB_{factor}}", "14eea6eed865ff34a70d0289907bcf8d": "{\\frac {\\mathrm {d} f}{\\mathrm {d} t}}=ROCOF={\\frac {\\Delta Pf}{2GH}}", "14ef32eece00cf2399260cc8441c2fd8": "{\\frac {P_{2}B}{P_{1}P_{2}}}={\\frac {\\sin \\beta _{1}}{\\sin \\delta }}", "14ef5ea590fe892c51d01b02b899e304": "{\\frac {1}{n!}}\\omega ^{n}=\\operatorname {pf} (A)\\;e^{1}\\wedge e^{2}\\wedge \\cdots \\wedge e^{2n},", "14ef66acbd0d15ac6d0a4d12f4599822": "b_{i}(x)=0", "14ef8444279f153a690863dacd7076dd": "A_{k+1}=A_{k}\\cup \\{\\sigma (k+1)\\}\\,;\\quad S_{k+1}=S_{k}+a_{\\sigma (k+1)}.", "14efd16f77ac0f11999509a4feed1ae2": "{\\bar {r}}_{i\\cdot }={\\frac {\\sum _{j=1}^{n_{i}}{r_{ij}}}{n_{i}}}", "14efd330a815e8144983aff5d97ab918": "p(\\sigma [1]\\sigma [2]\\ldots \\sigma [L]\\gamma [1]\\gamma [2]\\ldots \\gamma [L])=\\prod _{t=1}^{t=L}p(\\sigma [t]\\gamma [t])", "14efdcfb0beb639d80cd0151e0b9a023": "{\\begin{aligned}L'&=\\Delta x'-v\\Delta t'\\\\&=\\gamma L_{0}-\\gamma v^{2}L_{0}/c^{2}\\\\&=L_{0}/\\gamma \\end{aligned}}", "14f088991d1ce8b0abb86ef2ae1c1761": "\\sup _{x\\in \\mathbb {R} }\\left|F_{n}(x)-\\Phi (x)\\right|\\leq {0.33554(\\rho +0.415\\sigma ^{3}) \\over \\sigma ^{3}\\,{\\sqrt {n}}},", "14f0e52b57f95fb4e77f1f09b3c7f990": "H_{n}^{(m)}=\\sum _{k=1}^{n}{\\frac {1}{k^{m}}}", "14f11e2103b92b2f68e484123f74a14d": "W_{4\\to 1}=\\int _{V_{4}}^{V_{1}}P\\,dV,\\,\\,{\\text{zero work if V4 equal V1}}", "14f12a1313d53610d9f8756932be1cc0": "\\omega _{\\Lambda _{1}}\\omega _{\\Lambda _{2}}", "14f1c4eb7f8fdbe942495db69dbc4ad3": "\\scriptstyle {d={\\left({\\frac {{L_{\\ast }}/{L_{\\odot }}}{S_{eff_{\\ast }}}}\\right)^{0.5}AU}}", "14f217fadc773bd00a64218d27e3b66a": "\\delta [f]=|f(0)|q_{0.95}^{\\chi _{k-\\ell }^{2}}", "14f942c48031a21e13e002e587fec84e": "\\underbrace {a\\diamondsuit \\cdots \\diamondsuit a} _{k{\\text{ factors}}}.\\,", "14f9963f71cf4cf7c6ea41a98a2f6264": "A_{n}=(5\\times 92^{\\frac {36-n}{39}})^{2}", "14f9ef1130fcff22dd2d61cb33fab0a0": "y_{j}=\\sum _{i}w_{ij}x_{i}~~{\\textrm {or}}~~{\\textbf {y}}=w{\\textbf {x}}", "14fa875aefc7fc270d389366da600ffb": "\\phi \\left[W_{u}(x)\\cap E_{i}\\right]\\supset W_{u}(\\phi x)\\cap E_{j}", "14fa8b16e3989bcd739bc66b98862309": "{(x_{1}-m')}/{\\sigma },...,", "14faf2b6cef7cba1384bbc9fc72534db": "{\\hat {\\mathbf {R} }}={1 \\over n}\\sum _{i=1}^{n}x_{i}x_{i}^{\\mathrm {T} }.", "14fb1c6a8602e0cbb50d1a78e2fd9f02": "E_{\\ell r}=\\sum _{\\mathbf {k} }{\\tilde {\\Phi }}_{\\ell r}(\\mathbf {k} )\\left|{\\tilde {\\rho }}(\\mathbf {k} )\\right|^{2}", "14fb2328bf4cca2d2c0389f7212d5ddb": "f_{y}(0,0)=p_{y}(0,0)=a_{01}", "14fbc1fca5dd685bed59a70d040cd9c8": "{\\begin{aligned}{\\boldsymbol {\\nabla }}\\mathbf {v} &=\\sum _{i,j=1}^{3}{\\frac {\\partial v_{i}}{\\partial x_{j}}}\\mathbf {e} _{i}\\otimes \\mathbf {e} _{j}=v_{i,j}\\mathbf {e} _{i}\\otimes \\mathbf {e} _{j}\\\\{\\boldsymbol {\\nabla }}\\cdot \\mathbf {v} &=\\sum _{i=1}^{3}{\\frac {\\partial v_{i}}{\\partial x_{i}}}=v_{i,i}\\\\{\\boldsymbol {\\nabla }}\\cdot {\\boldsymbol {S}}&=\\sum _{i,j=1}^{3}{\\frac {\\partial S_{ij}}{\\partial x_{j}}}~\\mathbf {e} _{i}=S_{ij,j}~\\mathbf {e} _{i}~.\\end{aligned}}", "14fc0108eef4e4abd43be4964b3b4205": "\\mu _{v\\to u}(x_{v})", "14fc28c715b45f9fbac4ee662cdcd886": "c_{ii}>0.\\ ", "14fc363aee31b904874a250a5641ce4e": "u>c", "14fc418fd94add531ebafd67f995303c": "{\\frac {\\mathrm {d} (T_{h})_{*}(\\gamma )}{\\mathrm {d} \\gamma }}(x)=\\exp \\left(\\langle h,x\\rangle ^{\\sim }-{\\tfrac {1}{2}}\\|h\\|_{H}^{2}\\right),", "14fc712ae6153c5256ad88904015ee06": "C_{f}={\\frac {0.0583}{Re^{0.2}}},", "14fcd29ee2244be10098df07fc1e5fda": "PDI={\\frac {M_{w}}{M_{n}}}", "14fcfeb03a9b6d1ce7060a7484cf09e9": "f(x,y)=x-2y+2", "14fd27ac01b79e967a2d348e1110449a": "{\\begin{aligned}S_{n+2}&=\\int _{0}^{\\pi /2}S_{1}r.S_{n}R^{n}\\,d\\theta =\\int _{0}^{\\pi /2}S_{1}.S_{n}R^{n}\\cos \\theta \\,d\\theta \\\\&=\\int _{0}^{1}S_{1}.S_{n}R^{n}\\,dR=S_{1}\\int _{0}^{1}S_{n}R^{n}\\,dR\\\\&=2\\pi V_{n+1}\\end{aligned}}", "14fd595278208455417e37733d105cde": "P_{1},...,P_{m}", "14fda63bfc585789c4388e47899edf02": "f=a_{0}+q+\\sum _{i=2}^{\\infty }a_{i}q^{i}", "14fdae055857594646650f6df422d385": "1/r_{12}", "14fdae72d286220a51ca6aab0b568d2c": "a\\equiv \\omega ^{2}LC\\left[\\left({\\frac {R}{\\omega L}}\\right)\\left({\\frac {G}{\\omega C}}\\right)-1\\right]", "14fdd71a84ad2e348e68be392fe131a5": "{\\frac {z-2}{z+2}}=\\left({\\frac {\\zeta -1}{\\zeta +1}}\\right)^{2}.", "14fdd8e49e85cb2b687f53ba28e6ecec": "f*V_{n}=f\\,", "14fde9573cfaf3cfa5b052f0fcb4b08b": "d\\,'=c\\cdot (0-t_{1}')=-c\\,t_{1}'", "14fe3060ebf011d3c394abf02d7792f7": "{}^{n}({}^{1/n}a)=\\underbrace {({}^{1/n}a)^{({}^{1/n}a)^{\\cdot ^{\\cdot ^{\\cdot ^{\\cdot ^{({}^{1/n}a)}}}}}}} _{n}\\neq a", "14fe6905fe9c9dce3fe936ec30de29d5": "\\left({\\frac {\\mathrm {d} \\alpha _{1}}{\\mathrm {d} t}},\\dots ,{\\frac {\\mathrm {d} \\alpha _{n}}{\\mathrm {d} t}}\\right),", "14fe9153f6d0831a07952bf376858324": "{}=p^{03}z_{0}+p^{13}z_{1}+p^{23}z_{2}.\\,\\!", "14fe9938cfd54da7f47f41f9838bf062": "F_{2,n'}", "14ff1c57811659da83f8031d400fc6a7": "|\\langle R_{n},g_{\\gamma _{n}}\\rangle |", "14ff45db1b78c1fdc0dabb7b41c29e98": "c(r)=e^{-\\beta w(r)}-e^{-\\beta [w(r)-u(r)]}.\\,", "14ff730f8df66f8c730b44fd84f4fde4": "b=x-x_{i},\\quad i=2,3,\\ldots ,n", "14ffde9a267388f79d46fbb5af499f56": "{\\frac {b^{2n+1}}{n!}}I_{n}\\left({\\frac {\\pi }{2}}\\right)=P_{n}\\left({\\frac {\\pi }{2}}\\right)a^{2n+1}.", "15004284acb060b9a39420f3c7b2e3b4": "r=R{\\frac {h}{H}}", "150086db11283ee59943128676d31ad8": "b^{\\alpha }", "1500f358bc15e4cdd7f46f7b298fd765": "{\\begin{aligned}2\\cdot R_{*}&={\\frac {(180\\cdot 5.60\\cdot 10^{-3})\\ {\\text{AU}}}{0.0046491\\ {\\text{AU}}/R_{\\bigodot }}}\\\\&\\approx 216.8\\cdot R_{\\bigodot }\\end{aligned}}", "15010fff3925f351e1bbf8957584fcb8": "\\mu (AB+CD)=\\mu (AB)\\mu (CD).\\,", "15011e5d1c43a1d140a287cd0085e337": "\\nabla _{fX}Y=f\\nabla _{X}Y", "15014092a8f15a1121bc8170bbb8f379": "\\scriptstyle {(Rt)}", "15015f837616fa6e4b9bf1e892401f3d": "{\\ddot {x}}_{a}+{\\frac {2}{\\tau _{0}}}{\\dot {x}}_{a}+\\omega _{a}^{2}x_{a}={\\frac {e}{m}}E(t,\\mathbf {r} _{a})", "150182dd3ab31c1af4c7f6cb4518a55b": "TX_{i}=X_{i+1}", "15019f6b74fa698963000a977d7e29fc": "\\sigma _{r\\theta }=0", "1501c9b5f71363c53b3856c0fba84f66": "p_{X}", "1501fed59dfe33c518b7f85d9e4ffc79": "f({\\vec {x}},{\\vec {y}},{\\vec {z}})\\rightarrow g({\\vec {x}},{\\vec {z}})", "1502005c3c9100ff68265f92c5ec021a": "\\left.{\\begin{matrix}L_{1}u\\ {\\stackrel {\\mathrm {def} }{=}}\\ \\sum _{i}f_{1}^{i}(x){\\frac {\\partial u}{\\partial x^{i}}}&=0\\\\L_{2}u\\ {\\stackrel {\\mathrm {def} }{=}}\\ \\sum _{i}f_{2}^{i}(x){\\frac {\\partial u}{\\partial x^{i}}}&=0\\\\\\dots &\\\\L_{r}u\\ {\\stackrel {\\mathrm {def} }{=}}\\ \\sum _{i}f_{r}^{i}(x){\\frac {\\partial u}{\\partial x^{i}}}&=0\\end{matrix}}\\right\\}", "150231d2a099a1286f8acdcc2cc9518c": "(a^{2}+b^{2})(q^{2}+p^{2})=(aq+bp)^{2}+(ap-bq)^{2}\\,", "1502556194c82dc003affc7ed12da9b0": "\\chi _{i}(s)={\\frac {\\langle \\mathbf {e} _{i}'(s),\\mathbf {e} _{i+1}(s)\\rangle }{\\|\\mathbf {r} '(s)\\|}}", "1502b72b6db27630a1994d2983074858": "n\\approx {\\sqrt {1+{\\frac {3Ap}{RT}}}}", "1502edcb8e318e91227d5aa554047da3": "{\\hat {\\alpha }}={\\frac {4GM}{c^{2}b}}", "150316342acccb9435deeb75a212f262": "g_{ij}", "15031fcfb750ea831a00f623b4d9e3af": "q\\prec p", "150327d942f4c55b12c36dffd94535cc": "A={\\begin{bmatrix}0&0&0&-1\\\\1&0&0&0\\\\0&-1&0&0\\\\0&0&-1&0\\end{bmatrix}}.", "1503409db55a18ec1dd9b8dfa7e4e459": "k\\geq 1", "15035a174ce023dce5f584573f1f19cf": "q_{n+1}=2p_{n}q_{n}\\,\\!", "1503775f37a96a5cf4b34f2940cbc146": "\\color {Black}{\\tfrac {8}{m}}{\\tfrac {2}{m}}{\\tfrac {2}{m}}", "15038d33b3f2772ebf6b368b9149b4a6": "P(x)\\to (\\exists {y}{\\in }\\mathbf {Y} \\,Q(y))\\equiv \\ \\exists {y}{\\in }\\mathbf {Y} \\,(P(x)\\to Q(y)),~\\mathrm {provided~that} ~\\mathbf {Y} \\neq \\emptyset ", "1503ccb0af6e39fa198588a0113d9cc6": "p=\\hbar k", "150427b5fb001afd6ec41d30f1bcdce0": "{\\sqrt {-g}}A_{;a}^{a}=({\\sqrt {-g}}A^{a})_{,a}\\;\\mathrm {or} \\;{\\sqrt {-g}}\\nabla _{\\mu }A^{\\mu }=\\partial _{\\mu }\\left({\\sqrt {-g}}A^{\\mu }\\right)", "150462ed90f31805e32b937780144b45": "U(\\zeta )", "15049f9a6e066d06efa33ff8af0fabca": "(p_{1},\\,p_{2},\\dots ,\\,p_{n})", "15053aadb91c9f5a5efc3ce39367f793": "K_{i}={\\frac {P'_{i}}{P}}", "15057f605aa5f1b57a03ff73f2f2e7e9": "g(x)=(x^{2t-1}-1)p(x)", "1505c7ec876ec273313aebb7769a5fdf": "p(r_{1},r_{2})=P-{\\frac {2\\gamma \\,\\rho \\,_{vapor}}{(\\rho \\,_{liquid}-\\rho \\,_{vapor})}}\\left({\\frac {1}{r_{1}}}+{\\frac {1}{r_{2}}}\\right)", "1505f59c1c3d31d3fa713ec9440c6ae2": "f^{\\star }\\left(x^{*}\\right)={\\begin{cases}x^{*}\\ln x^{*}-x^{*},&x^{*}>0\\\\0,&x^{*}=0\\\\\\infty ,&x^{*}<0.\\end{cases}}", "150642ff0a49be7ffd3faffa4fb52762": "t^{3}+pt+q=0,", "15065755f2f20d486095164fef11bb63": "{\\textbf {a}}={\\textbf {f}}\\left(c{\\textbf {h}}+c\\right){\\pmod {q}}", "15067153ac49cb6bc4cf7cf472f28523": "C_{dyn}={\\frac {V_{T}}{PIP-PEEP}}", "15068c177d578c057a0b98d4a8c3b1bd": "\\left(r,s\\right)", "15068c2d533d79587ab59a73fa9d378a": "\\mathbb {E} [2e'+s']\\leq {2 \\over d}\\sum _{i}e_{i}j}^{n}[\\langle \\phi _{i}|{\\vec {r}}|\\phi _{i}\\rangle -\\langle \\phi _{j}|{\\vec {r}}|\\phi _{j}\\rangle ]^{2}", "151a45f165ee02c623c89ae6f4d0e1da": "\\phi '(a)=f'(a)-y>y-y=0", "151a9fff5c179a425b234b5ec3f78b21": "-40<\\Re (t)<40", "151ab59fc4d37d024b9971d280067409": "\\mathrm {D} F(\\sigma )\\;", "151ae48ef2fc97b2d10feea4f5a931e4": "W_{m_{1}}", "151b25977466c263aaea32849e702145": "{\\frac {d[B]}{dt}}=0=k_{1}[A]-k_{2}[B]\\Rightarrow \\;[B]={\\frac {k_{1}}{k_{2}}}[A]", "151b2998e20e1c73380c4a22e2ceaa5f": "[x,y]=(-1)^{\\epsilon ({\\hbox{deg}}\\ x)\\epsilon ({\\hbox{deg}}\\ y)}[y,x]", "151b42e448c23ac59605891968a4cf2b": "{\\mathit {z}}", "151b6b5f1caef6e7becd256d823da774": "M_{n}=\\sum _{k=0}^{n}X_{k},\\quad n\\in \\mathbb {N} _{0},", "151be16dba694615f34300b5671fc282": "f(x_{0})=y_{0}", "151c60a5653d9496d06d107cccff367e": "[{\\textrm {CO}}_{3}^{2-}]_{eq}={\\frac {K_{2}[{\\textrm {HCO}}_{3}^{-}]_{eq}}{[{\\textrm {H}}^{+}]_{eq}}}={\\frac {K_{1}K_{2}[{\\textrm {CO}}_{2}]_{eq}}{[{\\textrm {H}}^{+}]_{eq}^{2}}}.", "151c84c4d0ae26ab2fbdc2e2f98410e0": "{\\Vert r\\Vert }^{2}{\\Vert {u}\\Vert }^{2}=\\sum _{i0", "1523152aead8dabf961fff04942255d4": "\\mathbf {i} ={\\sqrt {-1}}", "152320d41632044685c51990969bccdd": "f(x,y)=U(x,y,z){\\big |}_{z=0}", "15234f112df70078944cac193de16223": "{\\begin{cases}{\\bar {x}}=a_{1}+2a_{2}\\\\\\sigma ^{2}=a_{1}+4a_{2}\\end{cases}}", "1523888e56250d9cb747adbc1486d0cb": "\\omega _{L}={\\sqrt {k_{z}^{2}\\left({\\frac {C_{s}^{2}C_{A}^{2}}{C_{s}^{2}+C_{A}^{2}}}\\right)}}", "1523b4bcdca5d98bfe69baeaf407ed32": "|U(T)|>0\\,", "1523c3fb576603821cbb8ce40a425d5f": "k=\\mathbb {Q} _{p}", "1523e675d9362bbc17a3bf2b0a7b5c9c": "I_{1},I_{2},\\ldots ,I_{n}\\subset C_{1}", "15240e01e99f730acbca49ec179bc689": "{\\sqrt {2}}={\\frac {10}{7}}\\approx 1.429", "1524ca696793b8e9021b13ccce9c7b66": "N_{i}(c)", "1524da5b0d4c26570167caaf1cebcc78": "{\\mathsf {C^{*}\\!-\\!alg}}\\to {\\mathsf {KK}}", "1524e58e20a937c0060ac07255b78d82": "R={\\frac {\\left(z_{2}-z_{1}\\right)}{\\left(z_{2}+z_{1}\\right)}}", "1524ef779a6ad53fa02a94d29b4f9cd8": "C_{1}\\otimes C_{2}\\in {\\mathcal {C}}", "152579ec6455c2e9a01e5f05e6d6ff19": "{\\mathcal {A}}_{2}", "1525ea3b0c78924df135d9692fc11964": "{\\overline {{\\beta }_{12}}}", "15260190b3b39932051d61faa6a803f8": "n^{3}-n^{2}", "15261e886be65a84fe205c3977bdc398": "\\scriptstyle {\\frac {41}{29}}", "15267992b2d9da957e79218e616c9e43": "\\mathbf {F} \\cdot \\Delta \\mathbf {r} =-\\mathbf {\\nabla } E_{\\mathrm {p} }\\cdot \\Delta \\mathbf {r} =-\\Delta E_{\\mathrm {p} }\\Rightarrow -\\Delta E_{\\mathrm {p} }=\\Delta E_{\\mathrm {k} }\\Rightarrow \\Delta (E_{\\mathrm {k} }+E_{\\mathrm {p} })=0\\,.", "1526b87bbbadcc1e771ac725d6bdf16e": "\\left({\\frac {x}{1-y}},0\\right)", "1526e884b546acfdd0d7c75918b37c53": "g(T)", "1526f04a8949c8498db6337f18c56cf3": "\\int _{0}^{\\infty }{\\frac {\\log(x)}{(1+x^{2})^{2}}}\\,dx", "1526f88328a22f432dd5426202393e6f": "N(\\mu ,\\sigma _{v}^{2})", "15270572da93f4c1e0731271ffe34725": "\\|x'\\|>r", "15272a4ba4fd4758d094293266259776": "\\alpha ^{A}", "152761662c1d7fa527c0243bcbce5068": "B(z^{2})", "152773cdcb0db8cbf1b42c4c431be858": "(2n-5)!!={\\frac {(2n-4)!}{(n-2)!2^{n-2}}}.", "1527b4869090aa6a4086722308d38855": "G\\subset U", "152805fc3733c04b94a03a799aef9a71": "U=\\mathbf {d} _{\\rm {e}}\\cdot \\mathbf {E} .", "1528096214598ac2ea2bd31eca0bc8cb": "m=\\rho _{f}V_{\\text{disp}}.\\,", "152834aac46fc61d71ac17144ff80e57": "\\forall x\\,(\\exists y\\,A(x)\\vee B(z))", "15285354e9a8489a4208c92748d0261d": "\\delta (x)={\\frac {1}{2(2\\pi i)^{n-1}}}\\int _{S^{n-1}}\\delta ^{(n-1)}(x\\cdot \\xi )\\,d\\omega _{\\xi }", "1528ea172314511404da128ef88bac03": "L=D(N\\otimes I)", "1528f6665471ce32a32e437bbf1f05c9": "L(G)=\\{a^{*}b^{*}c^{*}\\}", "1528f9498312a3097a58679ee261ca7c": "|x\\rangle |y\\oplus f(x)\\rangle ", "152913f0b6e9eddbd542450d5e506089": "V_{g}=V_{a}", "1529604788d4d5ba54434fa696bdfdef": "E_{s}={\\frac {1}{\\delta }}\\log \\left({\\frac {k_{s}}{k_{\\text{CH3}}}}\\right)", "1529873aacb4caca884e6cb529112111": "xy\\equiv zw\\rightarrow yx\\equiv zw\\,", "152a259377e0c9586aac82af946bfeec": "\\|A^{-1}\\delta A\\|", "152a3c96fc1f453436715a0ba6158581": "f(z)=f(z+\\omega _{1})", "152a4afe03a98f543f34eead9b7c1f91": "{\\mathbf {\\alpha }}", "152aaee4411aac160251a27ba4fbde7e": "(d_{1},e_{1},f_{1})+(d_{2},e_{2},f_{2})=((-1)^{e_{1}e_{2}}d_{1}d_{2},e_{1}+e_{2},[d_{1},d_{2}][-d_{1}d_{2},(-1)^{e_{1}e_{2}}]f_{1}f_{2})", "152aece223b8329f8210d1f20a714d90": "x(v)\\cdot Q", "152b13d725619ed50c7ae73cda0398e4": "X_{2}(z)", "152b1ee0c95d96ab864677eb5e6ffd73": "\\;^{+}T^{IJ}:={1 \\over 2}{\\Big (}T^{IJ}-{i \\over 2}\\epsilon _{KL}^{\\;\\;\\;\\;\\;\\;IJ}T^{KL}{\\Big )}", "152b3ff994a2efd470f838ae16c2771a": "{\\sqrt[{4}]{3^{4}+2^{4}+{\\frac {1}{2+({\\frac {2}{3}})^{2}}}}}={\\sqrt[{4}]{\\frac {2143}{22}}}=3.14159\\ 2652^{+}", "152ba4338d171777433824e1301584a3": "\\gcd(a,b)=1", "152bbb5db0e8b594498189d41f50a517": "W,B,D,", "152bdf8fe8d1f55d0e5e47c9e6bd426b": "Ff\\to X\\to Y", "152c07732064d8afdd0632205ff344ee": "\\Omega (n^{2})", "152c4eb645d0aa2601b5b0e9f96be36e": "R_{t}", "152c979392856a3c360ca78cd801ab30": "\\displaystyle {\\frac {1}{\\sqrt {2a}}}\\cos \\left({\\frac {\\omega ^{2}}{4a}}-{\\frac {\\pi }{4}}\\right)", "152cbe1462fedaaf0758623fea90a8ed": "(x,y)\\ ", "152cf5ce5c2ca62f1e0f9dc665231ec2": "\\Psi _{T}(x(T))={\\frac {\\partial \\Psi (x)}{\\partial T}}|_{x=x(T)}\\,", "152d2f4c5aadab30428a16a832f437f7": "\\operatorname {F} (\\dots \\mid z_{d})", "152d4e0a650c261d3c1c88d99bee1690": "{\\frac {\\partial \\Pi (\\mathbf {\\xi } ,t)}{\\partial t}}=\\sum _{j}\\left(\\sum _{ik}-S_{ij}{\\frac {\\partial f'_{j}}{\\partial \\phi _{k}}}{\\frac {\\partial (\\xi _{k}\\Pi (\\mathbf {\\xi } ,t))}{\\partial \\xi _{i}}}+{\\frac {1}{2}}f'_{j}\\sum _{ik}S_{ij}S_{kj}{\\frac {\\partial ^{2}\\Pi (\\mathbf {\\xi } ,t)}{\\partial \\xi _{i}\\,\\partial \\xi _{k}}}\\right),", "152d4e9fe5b2b1546a6ae0a30fa90f35": "\\sum _{n=1}^{\\infty }a_{\\sigma (n)}=\\infty .", "152dcd4279fa3d4d6ceebbab8b1ee972": "u_{0}(t)=U_{0}\\,\\cos \\left(\\Omega \\,t\\right),\\,", "152ddd99c0309af134f533dce4226747": "P_{(k)}>{\\frac {\\alpha }{m+1-k}}", "152e2e1463822418d92010dfdc8d2687": "J(0)={\\widetilde {J}}(0)=0", "152e4473322f6fa26671e66470368cd3": "m(n)=\\pi (\\phi (e,s(n)))\\,,", "152e71a5a4dd7787348e88741789653b": "\\mathbf {x} =(n_{1},\\dots ,n_{K})", "152e77da2ce3e36646af590bd58bcb15": "I_{t-1}=1", "152e99ca18c7cc35643c38be9bf9515e": "V[c]=c\\left[\\iint _{D}f\\,dx\\,dy+\\int _{C}gds\\right].", "152eb48160422b123e39f6381267e9fd": "y\\in P(x)", "152ece20c43b4f01d6729e2d284791d6": "\\pi _{j}\\ ", "152f34470cd4f7713c1daa9b1b59075b": "p_{D}={\\bar {D}}-D({\\bar {\\theta }})", "152fccda9e719a6a6200efde7ac4aaee": "\\cos \\theta ={\\frac {\\mathrm {adjacent} }{\\mathrm {hypotenuse} }}={\\frac {a}{c}}\\ .", "15301b741ac5d2c56353b66f7b994d62": "GDP=f(M,T)\\,\\!", "15302c4f548df3d14e9b2d09f10cca8d": "Q_{b}", "1530478861498456191bf9fa3865d235": "h(x_{1},\\ldots ,x_{m})=f(g_{1}(x_{1},\\ldots ,x_{m}),\\ldots ,g_{k}(x_{1},\\ldots ,x_{m}))\\,", "15304814be9206143c225fb9d1bce059": "X\\ \\sim \\ {\\text{NB}}(r;p)", "15306861686a7216b23ee92133a80fdc": "C\\in {\\mathcal {B}}_{B}", "1530a731f2f513d9464f84565e45dec7": "0\\leq r_{i+1}<|r_{i}|,", "1530adae49bd73ee1977be18bc2dd09b": "{\\begin{aligned}f*g&=fg+{\\tfrac {i\\hbar }{2}}\\Pi ^{ij}\\partial _{i}f\\,\\partial _{j}g-{\\tfrac {\\hbar ^{2}}{8}}\\Pi ^{i_{1}j_{1}}\\Pi ^{i_{2}j_{2}}\\partial _{i_{1}}\\,\\partial _{i_{2}}f\\partial _{j_{1}}\\,\\partial _{j_{2}}g\\\\&{}-{\\tfrac {\\hbar ^{2}}{12}}\\Pi ^{i_{1}j_{1}}\\partial _{j_{1}}\\Pi ^{i_{2}j_{2}}(\\partial _{i_{1}}\\partial _{i_{2}}f\\,\\partial _{j_{2}}g-\\partial _{i_{2}}f\\,\\partial _{i_{1}}\\partial _{j_{2}}g)+{\\mathcal {O}}(\\hbar ^{3})~.\\end{aligned}}", "1530c55e914bd04ae717ffcb80ff3a91": "k={\\frac {1}{3}}vl{\\frac {C_{V}}{V_{m}}}", "1530edd15ae9a59555a0ef06668ee6af": "{\\frac {2m^{m}}{\\Gamma (m)\\Omega ^{m}}}x^{2m-1}\\exp \\left(-{\\frac {m}{\\Omega }}x^{2}\\right)", "15313d88d0b6e7ff07ff7d0bdf43a278": "3\\uparrow \\uparrow 5=3^{3^{3^{3^{3}}}}=3^{3^{3^{27}}}=3^{3^{7625597484987}}", "15315b5728bfabbb6b18eff436d64eea": "K(\\omega )={\\frac {\\omega }{c}}{\\sqrt {\\frac {\\varepsilon _{1}\\varepsilon _{2}\\mu _{1}\\mu _{2}}{\\varepsilon _{1}\\mu _{1}+\\varepsilon _{2}\\mu _{2}}}}", "1531810abc9870b35577e941803769ab": "\\{\\ \\Delta P_{1}\\}", "1531948bc9853e370bd2cf110472e5a2": "\\ \\mathrm {P} ", "1531c099596cb8b5438e53fbc21550b2": "V_{\\max }", "153253d4a370d4b7ea171bd4153e6da8": "{\\vec {a}}=a_{1}a_{2}a_{3}a_{4}", "15325c052618a7c29ef4f547f1bc70ff": "\\langle G,R\\rangle ", "153266eb50c22e410c33eb0d4fb28514": "\\rho >1", "15333ef0377798fa111a2e02a12dca96": "e^{x}-1\\approx x+{\\frac {x^{2}}{2}}", "1533a14f71e135ab9f0f16e213c5d25f": "\\iota _{X_{H_{\\xi }}}\\omega =dH_{\\xi }.", "1533cd3ace9cc3783309dbb2c7c859b1": "b_{i}<\\max _{j\\neq i}b_{j}8\\end{cases}}\\\\X&=Y\\cdot {\\frac {9u^{\\prime }}{4v^{\\prime }}}\\\\Z&=Y\\cdot {\\frac {12-3u^{\\prime }-20v^{\\prime }}{4v^{\\prime }}}\\\\\\end{aligned}}", "153d86e023b6b7ad827f3b1cb7b3c0c6": "d\\!\\,", "153dae0b85c14068b2bdf5e19521878b": "[0,0.5)", "153e09e49ca01cb423e55a5ed7ac0fbc": "{\\text{Had}}(x)={\\Big (}\\langle x,y\\rangle {\\Big )}_{y\\in \\{0,1\\}^{k}}", "153f3747a2ed0464c1b9e8de5375c2a1": "\\zeta (2)=\\sum _{n=1}^{\\infty }{\\frac {1}{n^{2}}}={\\frac {1}{1^{2}}}+{\\frac {1}{2^{2}}}+{\\frac {1}{3^{2}}}+{\\frac {1}{4^{2}}}+\\cdots ={\\frac {\\pi ^{2}}{6}}\\approx 1.644934.", "153f4f9f9158da7fec07cff141d4192a": "\\left\\{{\\frac {\\partial }{\\partial x^{1}}},\\dotsc ,{\\frac {\\partial }{\\partial x^{n}}}\\right\\}", "153f88290a0056f91e66ecd6d041e92f": "{\\begin{aligned}{\\mathcal {H}}_{1}&:={\\text{Image}}\\;P&=\\operatorname {span} \\{|\\omega _{k}\\rangle \\in B_{\\text{op}}\\;|\\;\\chi (x)=1\\},\\\\{\\mathcal {H}}_{0}&:={\\text{Ker}}\\;P&=\\operatorname {span} \\{|\\omega _{k}\\rangle \\in B_{\\text{op}}\\;|\\;\\chi (x)=0\\}.\\end{aligned}}", "153fc2a5a0a49d52dda62d96ae0a293f": "R\\,", "153ff72c134f0f76d039e76e2cd65430": "{\\frac {\\displaystyle 2\\sum _{i=1}^{n}|X_{i}-\\mu |}{b}}\\sim \\chi ^{2}(2n)\\,", "15402f6b763418e2dd02a339830723eb": "\\mathrm {M} =\\mathbb {R} ^{2}-\\{0\\}", "1540c20ea21399dcc5c4471c6a142cf5": "{\\frac {1}{n}}\\sum _{k=1}^{n}\\mathrm {E} ((M_{k}-M_{k-1})^{2}|M_{1},\\dots ,M_{k-1})\\to 1", "1540c468af640bcaed6b3f89b22a3312": "A_{\\mu }A_{\\mu }+B_{\\mu \\nu }B_{\\mu \\nu }+\\psi _{\\{1}^{\\alpha }\\psi _{2\\}}^{\\alpha }", "1540f458c8e0763b13e9390fab6a3bdb": "t({\\vec {r}})", "154110dd919695283934c98a88e92d6b": "T_{G}(2,0)=(-1)^{|V|}\\chi _{G}(-1)", "15412bf6e129b5dd241dec7b347b5fb0": "\\Omega _{\\mathbf {k} }", "1541946ac8630b7fbe14f4039c91dea2": "(\\theta ,\\psi ,\\phi )", "1541a50027e21ce2fee5a27cf76c6f58": "X_{0}*(M_{1}+M_{2}+\\ldots +M_{n})=X_{1}*M_{1}+X_{2}*M_{2}+\\ldots +X_{n}*M_{n}", "15422b5b15231ad4fd8984a3df13a205": "F=F_{3}(p,Q,t)+qp\\,\\!", "15425dec0e43a568632920f854039cbc": "k\\in \\mathbb {N} ", "154265b2b4d50e678dfb1b46bb1f937c": "C(m,n)", "154275e94de1184fb3838d6971d4ec5d": "2\\rho =2n\\pi ", "15429a42c5db422f056cdf98490f45b0": "\\mu =GM/c^{2}", "1542ba1c3007c0c228c3a0106c590392": "\\{V{\\text{ open }}:(\\exists {U\\in {\\mathcal {O}}}){\\bar {V}}\\subseteq U\\}\\,", "1542c1e2c3eac02cfabefb1d20e09c69": "p(x)={\\frac {\\alpha -1}{x_{\\min }}}\\left({\\frac {x}{x_{\\min }}}\\right)^{-\\alpha },", "1543d6b31be651c36e169d2c8558a387": "R_{n}'", "1543da840d6f786643629782a4ebb352": "\\kappa ={\\sqrt {{\\frac {4\\pi e^{2}}{\\epsilon }}{\\frac {\\partial n}{\\partial \\mu }}}}", "1543dcc83dac32c808bce7474cdea1dc": "{\\begin{matrix}\\left({\\frac {L/K}{\\cdot }}\\right):&I_{K}^{\\Delta }&\\longrightarrow &\\mathrm {Gal} (L/K)\\\\&\\displaystyle {\\prod _{i=1}^{m}{\\mathfrak {p}}_{i}^{n_{i}}}&\\mapsto &\\displaystyle {\\prod _{i=1}^{m}\\left({\\frac {L/K}{{\\mathfrak {p}}_{i}}}\\right)^{n_{i}}.}\\end{matrix}}", "1543f5fcd4b1f5b91bdbafcdd74fe681": "\\mathbb {E} \\left[g(X_{1},\\dots ,X_{d})\\right]=\\int _{\\mathbb {R} ^{d}}g(x_{1},\\dots ,x_{d})\\,dH(x_{1},\\dots ,x_{d}).", "1544123e15697e116718d79b155e96fd": "\\kappa ={\\frac {p}{z}}\\tan \\phi ", "154436025a5a66f99973a4a3466b647a": "(-\\beta )^{2}-4(2y+\\alpha )(y^{2}+2y\\alpha +\\alpha ^{2}-\\gamma )=0.\\,", "15443fa24fc052d2556c913a4cd1ffb2": "{\\dot {V}}\\leq 0", "15444eb27adfc21cf2a51f356dc5fa78": "D(x)", "1544582ed1b77bdef9aa36024e53a754": "\\mathbf {S} _{k}(\\mathbf {p} (t))={\\mathcal {S}}\\boxtimes _{n=1}^{N}\\mathbf {w} _{k,n}(p_{n}(t)),", "154474f6c95de68cca407040716ab320": "\\delta _{r}=c_{0}/2B", "1545ba8fc8492a3ae4431ff622b4cbde": "~E={\\frac {I_{\\rm {s}}G}{I_{\\rm {p}}A}}~", "1545c7f46c4ecfb12b1da8e7f8fb7f93": "I=(a,b)=\\{x\\in \\mathbf {R} \\,|\\,af(x_{n_{j}},\\theta _{n_{j}})", "154a10c4cf4093047ee2a384de5d3d24": "{\\rho _{XT}^{2}}={\\frac {\\sigma _{T}^{2}}{\\sigma _{X}^{2}}}", "154aeb31b7cde4d54a94604d4e5e076a": "\\displaystyle {T(\\beta )={\\begin{pmatrix}1&\\beta \\\\0&1\\end{pmatrix}}.}", "154b0b00834c981f1a541c98e30b00a3": "\\mu _{\\operatorname {eff} }({\\dot {\\gamma }},T)=\\exp \\left(A_{0}+A_{1}\\ln({\\dot {\\gamma }})+A_{2}T\\right)", "154b2edabf59f96e7ae5ca18e0bb2e51": "\\sum _{x}\\cosh ax={\\frac {1}{2}}\\coth \\left({\\frac {a}{2}}\\right)\\sinh ax-{\\frac {1}{2}}\\cosh ax+C\\,", "154b35701e243ec3c8bb863e8396063c": "M'=\\rho \\cdot V^{2}", "154bda1bbf05d3478b67935eced901f0": "g(r,\\theta )", "154be246bdbdb1062d92c5824ca424e4": "{\\frac {1}{2}}\\sum _{i=1}^{K}Q_{i}(t+1)^{2}\\leq {\\frac {1}{2}}\\sum _{i=1}^{K}Q_{i}(t)^{2}+{\\frac {1}{2}}\\sum _{i=1}^{K}y_{i}(t)^{2}+\\sum _{i=1}^{K}Q_{i}(t)y_{i}(t)", "154c149de88269008049e6aa6dd5fd59": "T_{0}={\\frac {abc}{2h}}.", "154c4e3c7c89855d5f79f7f3dbf08b51": "{\\boldsymbol {\\nabla }}\\mathbf {u} ={\\boldsymbol {\\varepsilon }}+{\\boldsymbol {\\omega }}", "154c64630c334177ed9650af5fe09166": "D_{e}", "154ca2557c060d053770a45b78552adc": "\\left.{\\frac {d\\varphi }{dk_{1}}}\\right|_{k_{1}=k}=x-t\\left.{\\frac {d\\omega }{dk_{1}}}\\right|_{k_{1}=k}+\\left.{\\frac {d\\alpha }{dk_{1}}}\\right|_{k_{1}=k}\\ ,", "154caade0d22c60368adb4679ffc6170": "G_{1},~G_{2},", "154cc4a27418889f382f820b8fc07f5b": "I_{5}\\ ={\\frac {1}{5}}\\cos ^{4}x\\sin x+{\\frac {4}{5}}\\left[{\\frac {1}{3}}\\cos ^{2}x\\sin x+{\\frac {2}{3}}\\sin x\\right]+C,\\,", "154ccbfb15b904c565175e8c882f9555": "(x(t_{0}),y(t_{0}),z(t_{0}))", "154d0752c1bac72deba7bc1ecb29457c": "|\\psi \\rangle |\\phi \\rangle \\rightarrow \\sum _{n}c_{n}|\\psi _{n}\\rangle |\\phi _{n}\\rangle \\quad {\\text{(measurement of the first kind),}}", "154d1ce852f85f3403286df390e1ada6": "\\ker \\phi ", "154d3a2080a981deb50b0b18dadad582": "I[{\\vec {g}}]=\\sum _{j=1}^{N}\\sum _{k=1}^{N}f_{j}[{\\vec {g}}]f_{k}[{\\vec {g}}]e^{2\\pi i{\\vec {g}}\\cdot {\\vec {r}}_{jk}}.", "154d6d86c025f9b7795b79991754bb0c": "V_{\\text{Yukawa}}(r)=-g^{2}{\\frac {e^{-kmr}}{r}},", "154d734640776c25790af7c2ccd59718": "N\\to M\\to M''", "154d76c244544d1e944cad50ee55e9a6": "A_{i}\\in N", "154d8b615022832fc79be03f23217b37": "\\phi :{\\mathfrak {g}}_{1}\\to {\\mathfrak {g}}_{2}", "154dcc5feeccfe9765608ea046be4894": "N=\\left(-{^{(4)}g_{00}}\\right)^{-1/2}", "154e62a25c1e6ad0f36057e69326674b": "1/\\alpha \\approx 137", "154e6a4858bb6ab887515a950d028966": "=x\\cdot {\\frac {1}{t}}\\quad {\\mbox{ at }}t=1", "154e76ce6e9778985f7e184fb32c9498": "y\\in \\mathbb {F} _{q^{n}}", "154ea992cb8a2d25f34eec06d2a1df03": "\\displaystyle {gDg^{-1}=D+A}", "154ec6666949cc4e4ab01dc262a72d38": "\\sigma _{0}^{}", "154f0919e57f1e7a8a046c3d58f256f7": "n-p", "154f8988452469bc0d75a0af93ae5772": "Y(1-(b-bt+t)=I", "154f9132441870774ff1345ef56b117e": "0\\equiv z+tf'(r){\\pmod {p^{m}}}.", "154f9e98c064f79f5c0b69bda11f25a6": "R_{sensmin}", "154faf33306e82d5cc078f93d705fcf2": "0\\rightarrow U\\otimes \\Lambda ^{k-1}(W)\\rightarrow \\Lambda ^{k}(V)\\rightarrow \\Lambda ^{k}(W)\\rightarrow 0", "1550108d51092e4907cc997e15dc8253": "a^{d\\cdot 2^{r}}\\equiv -1\\mod n\\quad {\\mbox{ for some }}0\\leq r0}X_{s}(-m)X_{s}(m)\\right],\\,\\,\\,L_{\\pm 1}=(k+h)^{-1}\\sum _{s}\\sum _{m\\geq 0}X_{s}(-m\\pm 1)X_{s}(m).", "155577b3e2cfdd9e2ec28122285ba204": "res_{U_{i}\\cap U_{j},U_{i}}\\colon F(U_{i})\\rightarrow F(U_{i}\\cap U_{j})", "1555a930d1270fe8fe23732784d8466d": "J=J_{\\lambda _{1},m_{1}}\\oplus J_{\\lambda _{2},m_{2}}\\oplus \\ldots \\oplus J_{\\lambda _{N},m_{N}}", "1555b19090de5fcdf8e79162bb6940fe": "x_{1}M_{1}=y-x_{2}M_{2}", "1555d4776408d6c2e59609e4b3b8b632": "\\pi =\\cup _{i=1}^{\\infty }(\\sigma _{i}\\cup \\gamma _{i})", "1555e4119fef0f664e96cb1979e021f7": "dF_{i}=\\sum _{j}\\epsilon _{ij}\\,dA_{j}\\,", "155662295e5f1b6ef807afa96cac6320": "2C_{6}H_{6}+15O_{2}\\rightarrow 12CO_{2}+6H_{2}O", "1556638b996f1898fd9147c1a872003d": "{\\dot {Q}}", "15568c0b18bc33bd3f3ca38b7b9ec153": "f,\\;\\;\\theta _{init}", "15568d5930d7c77f5762eda92702b453": "N", "1556a97e3df85543d499c532ba0c4ba6": "\\{A\\vert A\\subseteq \\kappa \\land \\kappa \\in j(A)\\}\\,.", "1556bfedadd1e72e7c42b82a9496a4e3": "\\left(k_{\\perp }^{2}+{\\frac {1}{\\rho ^{2}}}\\right)G(\\mathbf {k} _{\\perp })=1", "15571b8edc062e222f56cc1eb80e4259": "\\left|A\\right|^{2}=\\left|B\\right|^{2}\\,", "15572737efdba948eb2ca2a6e8de1350": "U+pV-TS", "15576f8043e74f3f26921c77bbdb3db9": "X^{\\star }", "15580c586bb8d5b0cc15743155ce97d0": "|ab|=|a||b|{\\mbox{ for all }}a,b\\in G", "15582d37e233ec3d41b3ae9aa51c97b3": "H=2a+1{\\mbox{ and }}P=c", "1558567a52d52d33add2f0eeb97cf03f": "\\lambda _{0}\\neq 0,", "1558db63955cc27fdf5cf0c1be4c4ffd": "{\\tbinom {4}{7}}", "15592b00063dd40a2051ad58740a5d5d": "A:B:C={\\frac {1}{\\Delta \\Gamma }}:{\\frac {1}{\\Delta E}}:{\\frac {1}{\\Delta kT}}", "155941981b5bb955103afd98cb722a55": "\\beta _{n}", "155945b440ff23143412ebeeb7979dcd": "Y\\sim \\chi ^{2}(\\nu _{2})\\,", "155973fe048353bc11dab589eabd0be5": "B_{i,k,t}(x)={\\frac {x-t_{i}}{h}}k[0,...,k](.-t_{i})_{+}^{k-1}", "155a113803327e5ceba60ca54025602d": "\\scriptstyle \\tau =Fd", "155a2141a0435006a0eea4b8c36234d7": "L_{i}^{X/Y}=0", "155a69be27e79f5d0573f4a388b95e16": "m\\leq H_{\\infty }(X)-2\\log \\left({\\frac {1}{\\varepsilon }}\\right)", "155a738e6a896777be2d95b149cf8c1c": "m={\\frac {\\sqrt {E^{2}-(pc)^{2}}}{c^{2}}}", "155a79ef12c6615b93c9d44ebe9db065": "x(t)w(t-\\tau )={\\frac {1}{2\\pi }}\\int _{-\\infty }^{\\infty }X(\\tau ,\\omega )e^{+j\\omega t}\\,d\\omega .", "155aacb5190b9ddd14f52a060f051df5": "{\\mathcal {}}f'", "155b28a32f4eb488c543ec39d602926e": "\\langle \\cdot [e],[e]\\rangle _{f}", "155b830af7e037c46e586fa71c5d6476": "v(p,w)", "155bd21287ade5e44a71ec14c33d4172": "\\{\\vert n\\rangle \\}_{n\\in \\mathbb {N} _{0}}", "155c1f61a4b66c2ee99f12e0cd82c56c": "f_{W}\\upharpoonright N=0\\,", "155c829097296079097025a5ba8421d9": "E=pc\\,.", "155cbda4ef9aad1c2230d475e8b94181": "g_{\\mu \\nu }=\\,0", "155cf475d37727ff852c380674e4eb4d": "\\delta t=6.5\\pm 7.4\\ (\\mathrm {stat.} ){\\scriptstyle {+8.3 \\atop -8.0}}\\ (\\mathrm {sys.} )", "155d316d634946d7f446ac19deaf9150": "E_{21}={\\frac {d\\ln(c_{2}/c_{1})}{d\\ln(MRS_{12})}}=-{\\frac {d\\ln(c_{2}/c_{1})}{d\\ln(MRS_{21})}}=-{\\frac {d\\ln(c_{2}/c_{1})}{d\\ln(U_{c_{2}}/U_{c_{1}})}}=-{\\frac {\\frac {d(c_{2}/c_{1})}{c_{2}/c_{1}}}{\\frac {d(U_{c_{2}}/U_{c_{1}})}{U_{c_{2}}/U_{c_{1}}}}}=-{\\frac {\\frac {d(c_{2}/c_{1})}{c_{2}/c_{1}}}{\\frac {d(p_{2}/p_{1})}{p_{2}/p_{1}}}}", "155d483107e2ee9b932c880779ce140e": "p=\\min \\left(1,{\\frac {\\exp \\left(-{\\frac {E_{j}}{kT_{i}}}-{\\frac {E_{i}}{kT_{j}}}\\right)}{\\exp \\left(-{\\frac {E_{i}}{kT_{i}}}-{\\frac {E_{j}}{kT_{j}}}\\right)}}\\right)=\\min \\left(1,e^{(E_{i}-E_{j})\\left({\\frac {1}{kT_{i}}}-{\\frac {1}{kT_{j}}}\\right)}\\right),", "155dbc0436ba00db8e4a3629082d121a": "H_{q}(x)=xlog_{q}(q-1)-xlog_{q}x-(1-x)log_{q}(1-x)", "155dcd4f4cf4b9018e087e29a70d3ebb": "\\langle r\\rangle =a\\Gamma ({\\frac {4}{3}})={\\frac {a}{3}}\\Gamma ({\\frac {1}{3}})\\approx 0.893a\\,.", "155e31404d7c8508854066fffd001772": "-21-3\\lambda +7\\lambda +\\lambda ^{2}+16=0\\,\\!", "155e509dc08e54e8bbda792bbc3d255d": "{\\hat {e}}_{j}", "155e6bd548873626fba24ec5f415abc6": "\\ell _{0}", "155e906c6c57b939919e44e46b21ecc0": "\\angle \\theta ", "155e96421d20a886a7a26f2e9d999941": "{\\begin{aligned}f'(x_{2})=\\ell _{0}'(x_{2})f(x_{0})+\\ell _{1}'(x_{2})f(x_{1})+\\ell _{2}'(x_{2})f(x_{2})+\\ell _{3}'(x_{2})f(x_{3})+\\ell _{4}'(x_{2})f(x_{4})+O(h^{4})\\end{aligned}}", "155ea9abf796ec17d07dd994464287a8": "f={\\frac {a-b}{a}}=1-b:a.", "155eb0542d80a49d7303d6ee86b942df": "G=(\\ast A_{v})\\ast F(X)", "155eb8ff9f84f4356ad19ebcdfd92838": "\\mathbb {E} _{X^{n}}\\left\\{{\\frac {1}{M}}\\sum _{m}{\\text{Tr}}\\left\\{\\left(I-\\Pi _{\\rho _{X^{n}\\left(m\\right)},\\delta }\\right)\\Pi _{\\rho ,\\delta }^{n}\\rho _{X^{n}\\left(m\\right)}\\Pi _{\\rho ,\\delta }^{n}\\right\\}\\right\\}", "155ebf3256daa047d3933d06af5fa86a": "R=1", "155ede14a3aec17422ac5779b1545822": "B=\\{x\\}", "155f4cce4dd38781e083ff289f9a2f40": "\\mathrm {Kn} ={\\frac {k_{B}T}{{\\sqrt {2}}\\pi \\sigma ^{2}pL}}", "155f67ecc3a881fb4ecaaedb59877c01": "B=\\{B_{1},...B_{s}\\}", "155f6e225044fe11ce4fabf81552e036": "X_{k}^{i}", "155f894f211e0a351cc118540ed2a44d": "r^{n}=a^{n}\\sin(n\\theta ).\\,", "155fa186ec31446c9962e24d6891d43b": "U_{B}", "155fee16a95f9ad34ab979feed4b7506": "{\\begin{array}{cccc}z^{*}=&{\\stackrel {DM}{\\mathop {Opt} }}&{\\stackrel {Nature}{\\mathop {opt} }}\\quad &g(d,s)\\\\[-0.05in]&d\\in D&s\\in S(d)&\\end{array}}", "156069ac45d280c77ba5cd20960fab8f": "-{\\frac {\\partial p}{\\partial z}}={\\frac {\\Delta p}{L}}", "156081cfc0c000324719b54f01f6af69": "QB=0", "1560b79a9abbb13e102b75f499e893e1": "3=2+1", "1560c9dff2d0d96a151f9b6d2e7a5ddb": "{\\tfrac {\\beta }{2}}", "1560d28f994bdb1c882c28d058b221d7": "{\\frac {\\lambda \\operatorname {E} (S^{2})}{2\\{1-\\lambda \\operatorname {E} (S)\\}}}", "1560d6130a4407023310cb6d8679b3cf": "\\sum _{i=0}^{n}p_{i,j,k}=k", "1560f8483b5e42b9b845ede58dfabfbe": "\\forall \\alpha .\\forall \\beta .\\alpha \\rightarrow (\\beta \\rightarrow \\alpha )", "1561090f4fd4ae85e4fe269d450e0d34": "-1<\\kappa <0", "1561629b26389b872170f53e1be54b6d": "V^{\\alpha }", "1561718106fa14629c5d122cfe614413": "D=\\omega (m_{1}-x_{1})+(m_{2}-x_{2})", "15618ab9efcee346b9dd866ea048d87e": "={\\dfrac {5({\\sqrt {3}}-4)}{3-16}}\\,\\!", "1561be931a03d5c0dea5046b8a6ceda4": "{\\tfrac {v}{c}}", "1561c3d5b80265e5c980916a453b7cc7": "\\Omega [x^{*}-c]", "15620f2bf6b000d170522649a601b2bd": "\\vert {\\Phi _{0}}\\rangle ", "15629e5d956e0d8bd59efb45de72aacf": "T\\in \\mathbb {R} ^{d}", "15630f1d85190e088343788a9b311d78": "\\{y\\mid \\exists x[\\phi (x,y)]\\},", "1563162ecebf94912369dfc0f9af1a05": "T=g_{m}(r_{\\pi }//R_{S})\\ {\\frac {R_{D}//r_{O}}{R_{D}//r_{O}+R_{F}+r_{\\pi }//R_{S}}}\\ .", "1563384a0036c665a7531a88f116d80e": "\\underbrace {\\underbrace {\\mathrm {root+suffix} } _{\\mathrm {stem} }+\\mathrm {ending} } _{\\mathrm {word} }", "156348c36b74f26ca6e7a24235bf3309": "s_{k}t_{k+1}-t_{k}s_{k+1}=(-1)^{k}.", "15636e811bd1b3596a3284eaf3542050": "G^{-1}", "156371fa68f63dc48e4345abd91d613d": "\\nu _{1}\\leq \\nu _{2},k\\leq n,p\\in [0,1]", "1563a72c6993fe81033e39ae5134ed05": "{\\frac {\\partial }{\\partial h}}\\operatorname {AMISE} (h)=-{\\frac {R(K)}{nh^{2}}}+m_{2}(K)^{2}h^{3}R(f'')=0", "1563b8504d179165422dcf11b92fecdf": "c^{2}=\\gamma (p+p^{0})/\\rho ", "156410efaa3f06bcb68056331e7b66a4": "v_{D}\\,", "15641e1f5d1554d4b3fe895154bf1d68": "f^{n}(A)\\cap B\\neq \\varnothing ", "15642b77303878d4edb3ac4771dc5aad": "\\mathrm {e} _{i}", "156460d6b5c9a01f27f8e7e92a789f84": "B_{2}=G^{31}", "1564e81e84079281c809d3c521389fa0": "\\Lambda ^{1}(E)^{K}", "156507a62f3a70e3f759500c9f05d8ad": "({\\frac {x}{z}},{\\frac {y}{z}})", "15650a7541f2a18fd124cb8b7792374d": "h<{\\frac {\\lambda }{8\\cos i}}\\;", "1565b0796797071f4f3823eb26127d0e": "K{\\hbox{--}}{\\bar {K}}", "156605ebad8e67f6a389eeec48c4c19e": "p(X)=X^{2}-5X-2I_{2},", "156615bc3ef86e8f898f45390ccc68bb": "e^{3}", "1566b3941bf0066601d4bcf32286ebf2": "N(x)", "1566db4013b00c30b4a4702d503270c2": "f\\in {\\mathcal {M}}_{g;k_{1},\\dots ,k_{n}}", "1567070f4dcf5fb26ecb608f3c944e75": "{s_{k} \\over {n \\choose \\lfloor {n/2}\\rfloor }}\\leq {s_{k} \\over {n \\choose k}}.", "15676f60f1237fd7dce08cf3db9d5c3f": "D_{12}={\\frac {3}{2n(d_{1}+d_{2})^{2}}}\\left[{\\frac {kT(m_{1}+m_{2})}{2\\pi m_{1}m_{2}}}\\right]^{1/2}", "15677511bae0e800d0667059d477a89e": "\\scriptstyle p(bk\\,+\\,a)\\;\\equiv \\;0{\\pmod {b}}", "15678dc8270a263fb1f8c8f1e6bac9d7": "d^{\\dagger }\\omega =i(\\partial -{\\bar {\\partial }}){\\text{ln}}||\\Omega ||,", "1567c7fb03847cd101a37fa32db4503a": "\\mathbf {a} =\\mathbf {a} _{\\text{x}}+\\mathbf {a} _{\\text{y}}+\\mathbf {a} _{\\text{z}}=a_{\\text{x}}{\\mathbf {i} }+a_{\\text{y}}{\\mathbf {j} }+a_{\\text{z}}{\\mathbf {k} }.", "15688d3238b42b148eecd42a3278cce1": "q_{1}\\ {\\hat {f_{1}}}\\ +\\ q_{2}\\ {\\hat {f_{2}}}\\ +\\ \\ q_{3}\\ {\\hat {f_{1}}}\\ =\\ \\sin {\\frac {\\theta }{2}}\\quad {\\hat {e_{3}}}={\\frac {\\sin {\\frac {\\theta }{2}}}{\\sin \\theta }}\\quad {\\bar {E}}", "1568dd4ff929e70671a51abdf352c665": "O(n^{2}\\log n\\log q)", "1568dd8ceade5b85b2ed2f8156a02290": "1^{2^{k-1}}", "1568eff0ee1d358d6a0fa7b08b12e7c0": "GA(m,\\mathbb {R} )", "1568fd88c570556f7e078a0e481101e7": "-1/2\\mu _{i}\\cdot E_{RF}", "15699da5162c6174d970b58e778138fc": "{\\frac {x}{n}}", "1569b0652f60d5b4eebaaa2de3178f48": "r_{\\mathcal {P}}", "1569b73bb8a869da53435f54c19830d0": "=3+{\\frac {1}{8}}+{\\frac {9}{640}}+{\\frac {15}{7168}}+{\\frac {35}{98304}}+{\\frac {189}{2883584}}+{\\cfrac {693}{54525952}}+{\\frac {429}{167772160}}+\\cdots \\!", "1569ffcfea536239b5328a05e4b4c777": "{\\begin{pmatrix}{\\overline {\\delta }}&-{\\overline {\\gamma }}\\\\-{\\overline {\\beta }}&{\\overline {\\alpha }}\\end{pmatrix}}", "156a349c2cdf2e3ef2ef19e87dc406f0": "T^{ab}{}_{bc}=\\sum _{b}{T^{ab}{}_{bc}}=T^{a1}{}_{1c}+T^{a2}{}_{2c}+\\cdots +T^{an}{}_{nc}=U^{a}{}_{c}.", "156aac4cc6dc985f8be0e74c416f6298": "{\\overline {w_{i}}}", "156ac61a39a7925e5c69ffcc67224523": "\\left|A\\right|_{ij}=a_{ij}-\\sum _{k\\neq i}|A^{kj}|_{il}|A^{ij}|_{kl}^{\\,-1}\\cdot a_{kj}", "156ad0d78e964e2ea2a79aee3ea8227a": "\\mathbf {F} _{\\mathrm {rad} }={2 \\over 3}{\\frac {q^{2}}{c^{3}}}\\mathbf {\\dot {a}} .", "156af2859026537b52e98582ca4d7255": "\\gamma >2", "156af37236f5daa4d889325fe88baedb": "1\\ {\\rm {Np}}=20/\\ln _{}10\\ {\\rm {dB}}\\approx 8{.}685889638\\ {\\rm {dB}}\\,", "156b14c98e5d372d59916e3fea9cb5fa": "dV=\\left({\\frac {\\partial V}{\\partial p}}\\right)_{T}dp+\\left({\\frac {\\partial V}{\\partial T}}\\right)_{p}dT=V\\left(\\alpha dT-\\beta _{T}dp\\right)\\,\\,{\\text{ (2)}}\\,", "156b4321c04913ca7f983483b8a101ba": "|Leader_{i}\\rangle ={\\frac {1}{n^{3/2}}}\\sum _{a=1}^{n^{3}}|a,a,\\ldots ,a\\rangle ", "156b63ebe005a62e831e6d1c301b4be7": "Z_{r}", "156bb9c17a70f18fc1896deb7f987fa5": "I=C(dV/dT)", "156bc7b9bee44723fa628b738e0e8ee2": "c^{2}=5^{2}", "156bca7a24d5769e858a6f11c26c826c": "m_{2}={\\frac {a-a_{2}}{b_{2}(a_{2}-a_{1})}}\\!", "156be0d4e3de061d159b7192f263baf6": "P(x_{1})", "156bf7552c0780e13b206b571067f74e": "{x'}^{j},j=0,1,\\dots ", "156c83cb1e70cb24e5799a30e12c5073": "D(fv)=v\\otimes (df)+fDv", "156c8687dbe59f2546579112fdcc5f61": "N_{j}\\left(U^{\\left(0\\right)}\\right)=\\Gamma _{jk}U_{k}^{\\left(0\\right)}-U_{j}^{\\left(0\\right)}==\\left(\\Gamma _{jk}-\\delta _{jk}\\right)U_{k}^{\\left(0\\right)}==0", "156cae5711114578af2e9398aacf4e86": "r{\\frac {\\partial }{\\partial r}}={\\frac {\\partial }{\\partial \\rho }}", "156cb999e3d3ca190825626a84c48c7b": "x_{k}=-\\cos \\left({\\frac {k\\pi }{N-1}}\\right);\\qquad \\ k=0,1,\\dots ,N-1.", "156cbdbf192274123c4951f88bff2a10": "(z-1)(z+1)=z^{2}-1=(z-j)(z+j)", "156cceab26d40dc7ba5af57cc28d5e6a": "I=I_{0}e^{i\\omega t}.", "156ce9e375e6cb012dc635b488c01c73": "c_{0}={1 \\over {\\sqrt {\\mu _{0}\\varepsilon _{0}}}}.", "156d008bc155eebecf5e8b8e90df1230": "MRF={\\frac {(U-hR_{Fn})(hR_{F1}-L)\\prod _{i=1}^{n-1}(hR_{Fi+1}-hR_{Fi})}{[(U-L)/(n+1)]^{n+1}}}", "156d00a9cb2329569c2b4163cac450af": "{\\operatorname {d} \\Gamma \\over \\operatorname {d} y}", "156d3429cd6916eefe2b3b683ef6721e": "{\\tfrac {1}{2}}(1\\pm j)", "156dd665999efac8c7ef8ded066d5ed3": "w\\,R\\,u\\land w\\,R\\,v\\Rightarrow u=v", "156e05afcdb99b14dbe2671a850c61bc": "P_{ij\\sigma }^{\\lambda }", "156e0d2b1231ce052224f9d4455c7699": "\\delta =\\theta _{0}+\\theta _{2}=\\theta _{0}+{\\text{arcsin}}{\\Big (}n\\,\\sin {\\Big [}\\alpha -{\\text{arcsin}}{\\Big (}{\\frac {1}{n}}\\,\\sin \\theta _{0}{\\Big )}{\\Big ]}{\\Big )}-\\alpha ", "156f1942b102e583cb7bc12c0f88bf26": "|k_{e}|,|k_{h}|", "156f446d2432e8d1f65d1574c09f3f2e": "M\\;[\\mathbf {s} \\;\\mathbf {t} ]\\;=\\;[\\mathbf {s} \\;\\mathbf {t} ]\\Lambda _{\\theta }", "156f828f9f15b6fdd668d56b86504831": "\\mathbf {j} _{r}(t)\\in \\mathbb {R} ^{3}", "156f82c61b9ccc4edaad89686de3d321": "\\delta Q=TdS_{h}\\,", "15704217859c9273fffc6681434d7dae": "{4 \\choose 2}\\left({\\frac {1}{2}}\\right)^{4}=6\\cdot {\\frac {1}{16}}={\\frac {6}{16}}", "1570627dee1e357f70871ee9315ca87e": "V_{tangential}=V_{wall}", "157075c60304563ebf1397dc9d434ad0": "s_{AB}", "1570bee19fe00d903b33e3bab8f1e4b1": "B={\\dfrac {dN}{dt}}=k_{1}M_{T}^{j}(c-c^{*})^{b}", "15712a00763cd12474b0a1ef8ffd3721": "\\{\\lambda _{n}\\}", "157148969e68174643d663fa61499e64": "{\\mbox{SL}}(2,\\mathbf {R} )=\\left\\{\\left({\\begin{matrix}a&b\\\\c&d\\end{matrix}}\\right):a,b,c,d\\in \\mathbf {R} {\\mbox{ and }}ad-bc=1\\right\\}.", "15718ead15aa98daa72cb4f16dd23aff": "H_{\\xi }=\\langle \\mu ,\\xi \\rangle ", "1571a63bd42577b33dd46664cf75b6cb": "{\\begin{aligned}B_{1_{1}}=&{\\begin{pmatrix}0&0&48&-16\\\\0&0&-8&2\\\\0&0&1&0\\\\0&0&0&1\\end{pmatrix}}\\\\B_{1_{2}}=&{\\begin{pmatrix}0&0&4&-2\\\\0&0&-1&1/2\\\\0&0&1/4&-1/8\\\\0&0&1/2&-1/4\\end{pmatrix}}\\\\B_{2_{1}}=&{\\begin{pmatrix}1&0&-48&16\\\\0&1&8&-2\\\\0&0&0&0\\\\0&0&0&0\\end{pmatrix}}\\\\B_{2_{2}}=&{\\begin{pmatrix}0&1&8&-2\\\\0&0&0&0\\\\0&0&0&0\\\\0&0&0&0\\end{pmatrix}}\\end{aligned}}", "157210a3a7133f92876dce0788b068ce": "I:\\{\\mathbb {X} \\subseteq \\mathbb {R} ^{n}\\}\\rightarrow \\{\\mathbb {Y} \\subseteq \\mathbb {R} ^{m}\\}", "1572c4a2036b7316e5687740c9835193": "e\\colon G\\times G\\rightarrow G_{T}", "15731a3600bd3731157459d5d07a3f30": "Z_{P}=b_{1}t_{1}y_{1}+b_{2}t_{2}y_{2}\\,", "15733ef796e744976ebf2e9dc5de8942": "B_{2}", "15737fde927f0c945e4d33784063c612": "(R^{i}f_{*}{\\mathcal {F}})^{\\mathrm {an} }\\cong R^{i}f_{*}^{\\mathrm {an} }{\\mathcal {F}}^{\\mathrm {an} }", "1573f0f7dd2bdd54d3a2e1ae915af0f6": "{\\text{inquire}}(j)", "157443d24eb255d284b70ff30e8146ec": "\\ I(r,d)=\\delta I_{b}R\\exp(\\alpha d)", "157471ee6e4a50a9b18cc321f24d3d6e": "\\mathrm {B} (x,y)={\\dfrac {(x-1)!\\,(y-1)!}{(x+y-1)!}}\\!", "157482be62239900dd14b755b374d327": "W_{n,k}=\\sum _{v=0}^{k}(-1)^{v+k}\\left(v+1\\right)^{n}{\\frac {k!}{v!(k-v)!}}\\ .", "15749dc1f9eaadac752f1c7168a02e27": "(Exa\\leftrightarrow Exb)\\leftrightarrow (a=b),", "1574acf98fadf9844681c5cad9313456": "K_{m}={\\frac {k_{r}+k_{\\mathrm {cat} }}{k_{f}}}", "1574adfffa2f46ff72d8d2aea763ddf0": "6_{0}", "1574b82063b4508684413e5d70e0992c": "{\\mbox{eGFR}}={\\mbox{166}}\\ \\times \\ {\\mbox{(SCr/0.7)}}^{-1.209}\\ \\times \\ {\\mbox{0.993}}^{Age}\\ ", "1574f057e5a4de3b7b53627317967b49": "\\operatorname {Li} _{s}(z)", "15753780721821850ae30ed7309bbdf5": "s,t\\in V", "15756f2fbcbca2dabaeb57e80e52266f": "\\Delta =1", "157644a52f1c4a0566f2a2d2c7afd239": "{\\frac {SS_{AB}}{SS_{E}}}.", "1576aea5a12637288e73fd0a33103ec8": "\\sigma ^{2}=\\operatorname {E} (X^{2})-\\mu ^{2}", "157703d038aa3fa1057bee5a3b799331": "E[X]_{11}=-2m/r^{3},\\;E[X]_{22}=E[X]_{33}=m/r^{3}", "1577506b32c9ed486679b56ca0b2364b": "(p-1)/2\\,", "15776735b8f28a30308748db44fd45cf": "{\\begin{bmatrix}{\\begin{vmatrix}3&-1\\\\-1&-5\\end{vmatrix}}&{\\begin{vmatrix}-1&2\\\\-5&8\\end{vmatrix}}\\\\\\\\{\\begin{vmatrix}-1&-5\\\\1&1\\end{vmatrix}}&{\\begin{vmatrix}-5&8\\\\1&-4\\end{vmatrix}}\\end{bmatrix}}={\\begin{bmatrix}-16&2\\\\4&12\\end{bmatrix}}.", "157859fa19dc8e6f0e199d5c967c471e": "t_{1}[K]=t_{2}[K]", "157861f05e4e565c764b650e9fb6ceb6": "{\\frac {dr}{dt}}={\\frac {d\\mathbf {A} }{dt}}r_{0}={\\frac {d\\mathbf {A} }{dt}}\\mathbf {A} ^{\\mathrm {T} }(t)r(t)", "157863ed597755d3bf01136ff4b3e042": "\\Re ^{2}", "15786f316de05ed69ac95e73b97db10b": "\\prod _{k=0}^{\\infty }\\left(1-{\\frac {1}{q^{2^{k}}}}\\right)=\\left(1-{\\frac {1}{q}}\\right)^{-1}-2.", "1578a045d6f0f59bf351e6fbe3db53c3": "{\\vec {s}}_{b}", "1578b5a9b9e3d9b42b963fa877d903f1": "w_{\\boldsymbol {xy}}", "1578c2aa3dfa7a7786dd148e725ffa9a": "f_{i}(\\alpha )=\\alpha ^{q^{i-1}}", "1578e830b9df7b5f2030526e2690d083": "\\tan \\phi ={\\frac {\\rm {{BC}\\sin \\alpha }}{\\rm {{AC}\\sin \\beta }}}.", "157907749afa659528e72aa2093baec8": "\\displaystyle {|\\partial ^{\\alpha }a(z)|\\leq C_{\\alpha }(1+|z|)^{m-|\\alpha |}}", "15790d55a74f61f69c900e5a63bc414a": "{n \\choose \\nu }", "15790eefb3794d0539ab1399e6510c42": "f(x,y)\\!", "15795f0ad37587786eff313052dbf309": "\\Delta \\mathbf {r} _{i}=\\mathbf {r} _{i}-\\mathbf {R} ", "1579bf75a2ec41a54939df716eed43ff": "{\\begin{bmatrix}1&0&0\\\\0&1&0\\\\0&1&0\\\\0&0&1\\end{bmatrix}}{\\begin{bmatrix}a\\\\b\\\\d\\end{bmatrix}}={\\begin{bmatrix}a\\\\b\\\\b\\\\d\\end{bmatrix}}", "157a024493ed769eac263927c17debd5": "L=\\{a^{n}b^{n}c^{n}:n\\geq 1\\}", "157a40034a71de5c1b9c41a09a0be9d8": "\\{H(N),H(M)\\}=C({\\vec {K}})", "157a539de99045b95023087b009870bc": "QE_{\\lambda }={\\frac {R_{\\lambda }}{\\lambda }}\\times {\\frac {hc}{e}}\\approx {\\frac {R_{\\lambda }}{\\lambda }}{\\times }(1240\\;{\\rm {W}}\\cdot {\\rm {nm/A}})", "157a68adf787734e9db6707a48502e33": "{\\frac {1}{2\\pi i}}\\oint _{C}z^{k}{\\frac {f'(z)}{f(z)}}\\,dz=z_{1}^{k}+z_{2}^{k}+\\dots +z_{p}^{k},", "157a8b4284f5ea75d520c6fb73c366ee": "x\\in \\mu (x,G)\\subseteq G", "157a96185fbdaf3c8c1b01f32bf1f7e7": "sgn(p)", "157b22a03713bdd30c7a0c0425a3d374": "C_{abcd}+C_{acdb}+C_{adbc}^{}=0", "157b565fc02df373b816360feee7ea71": "C={\\frac {Q}{V}}", "157bda1e42dd199c05d9a99ebe441725": "{\\sqrt {x^{2}+b}}", "157bde725ce70e551d064dea3b822188": "v\\in D(\\Delta _{D})", "157be742b8b08b79bda1669bf4ae317a": "A\\left({\\frac {h}{t}}\\right)+{\\frac {A\\left({\\frac {h}{t}}\\right)-A(h)}{t^{k_{0}}-1}}\\approx A\\left({\\frac {h}{s}}\\right)+{\\frac {A\\left({\\frac {h}{s}}\\right)-A(h)}{s^{k_{0}}-1}}", "157bf2f09a472e136bc178b5ae2163ef": "p_{d}=\\left({\\frac {e^{\\sigma {\\sqrt {\\Delta t/2}}}-e^{(r-q)\\Delta t/2}}{e^{\\sigma {\\sqrt {\\Delta t/2}}}-e^{-\\sigma {\\sqrt {\\Delta t/2}}}}}\\right)^{2}\\,", "157c051f2c4b2b53df7a0a2d650e4a75": "{\\boldsymbol {\\nabla }}\\cdot {\\boldsymbol {\\sigma }}=\\mathbf {0} \\qquad \\implies \\qquad {\\cfrac {\\partial \\sigma _{jk}}{\\partial x_{j}}}=0~.", "157cdee1e6c1295915d121a016dddecb": "\\Psi _{id}", "157d51eaa8cea4ad0e0c6bd3f58a4f29": "x(t)=(x_{1}(t),...,x_{N}(t))\\in A", "157d98d2d90cb1539f6c1add75bc4e61": "O(\\log(\\max\\{K(x\\mid y),K(y\\mid x)\\}))", "157da986ab45e59325519a4877db8ca6": "\\mathbf {L} _{i}^{-1}={\\begin{bmatrix}1&&&&&&&0\\\\0&\\ddots &&&&&&\\\\0&\\ddots &1&&&&&\\\\0&\\ddots &0&1&&&&\\\\&&0&-l_{i+1,i}&1&&&\\\\\\vdots &&0&-l_{i+2,i}&0&\\ddots &&\\\\&&\\vdots &\\vdots &\\vdots &\\ddots &1&\\\\0&\\dots &0&-l_{n,i}&0&\\dots &0&1\\\\\\end{bmatrix}},", "157dc79688333227ac047d010705342d": "|2\\Theta _{C}|", "157ddf15c35d49507ec68d0e086e92ef": "_{k}\\mathbf {V} ^{r}=\\mathbf {S} _{k}\\mathbf {V} ^{i}", "157de57f8e3b05f8f67df6e61e14efd7": "\\ I_{A}=A_{11}+A_{22}+\\cdots +A_{nn}=\\mathrm {tr} (\\mathbf {A} )\\,", "157de6eaeb987616e891407eb92aa2e8": "R\\rightarrow R\\otimes _{S}R", "157e26d8bbe8d2ffaca5227bf75c1d08": "{\\frac {2}{n}}={\\frac {1}{n}}+{\\frac {1}{2n}}+{\\frac {1}{3n}}+{\\frac {1}{6n}}", "157e2a2ae28ccabe24227c1625991bd5": "\\omega _{c}={\\frac {eB}{m^{*}}}-\\ ", "157e5bf8bdfc36bcc4639e3254305012": "R_{1}R_{2}+R_{1}R_{3}+R_{2}R_{3}={\\frac {(R_{a}R_{b}R_{c})(R_{a}+R_{b}+R_{c})}{R_{T}^{2}}}", "157ea4320ab3754080350cc7c47ae59f": "f_{k}(x_{1},\\dots ,x_{m})", "157ee5ec7f45bd3399c2fd33315b2d40": "\\scriptstyle p_{1},p_{3}", "157f2752be15df83cb72e60c175f350d": "f\\mapsto \\mathbb {P} _{n}^{0}={\\dfrac {1}{n}}\\sum _{i=1}^{n}\\varepsilon _{i}f(X_{i})", "157f389d05517b0f6162f73c98e1f274": "M(q)=1+240\\sum _{n=1}^{\\infty }{\\frac {n^{3}q^{n}}{1-q^{n}}}=E_{4}(\\tau )", "157f5d953a0086ca508afecb0b131507": "(-1)^{i}E", "157f6e736149f35649909cc817261596": "T_{1}\\neq T_{2}", "157f994f0ccb5bd0438a2f6c6bcc1606": "{\\color {Blue}~6.5}", "157f9e5c73d648c19a5c58341082853f": "hg\\in V(\\Gamma )", "157fa14781f927c02cb756db273ae81c": "\\Delta =-16(4a^{3}+27b^{2})", "157fe76b2767a9665879c415c9260475": "S(r')={\\begin{cases}S_{0},&{\\text{if }}r'\\leq R\\\\\\,0,&{\\text{if }}r'>R\\end{cases}}\\qquad (8)", "1580072f8746210f4608ac8b4efac4c4": "\\ N=f/D", "158043874ca5bd5fcda242b43bcaab2c": "\\sharp (L\\cap \\phi _{H}(L))\\geq \\sum _{k=0}^{n}b_{k}(L;\\mathbb {Z} _{2})", "15805d86b511f3ec49655262b4b6c934": "E_{\\mathrm {v} }", "158060b729a5d6180bdd61ad215c45f1": "\\rho ={\\frac {N_{s}}{N}}", "1580709060bf4c410eb2934cfece4ce0": "u={\\hat {u}}_{0}e^{kt}", "158082a8651cf87b678bc52e481d5af1": "{\\frac {1}{l}}", "15809515d71a8f147e8340023a211ad5": "w\\notin x", "1580ce905b1ec478998b84e85b63c627": "T=2\\pi {\\sqrt {a^{3} \\over \\mu }}", "1580e4269c3ba3dcf55d55ee44d4f13b": "(\\Omega ,{\\mathcal {F}},P)", "1581ddbd53bce67eaba8b152c234a1b0": "k:A\\to X", "1581ee7dc8f306dc9a4cb422b6919ed9": "(I<0)", "158202eb547e8572fecdd9ee36993b11": "i:=i+1;", "158223c871a3023a2b64633a714ca3a2": "\\displaystyle {U^{*}F(e^{it})=2^{3/4-\\sigma /2}\\pi |1-e^{it}|^{1-2\\sigma }F\\left({1+e^{it} \\over 1-e^{it}}\\right).}", "1582719893f12c430ae4c72b5dc0745f": "(x_{\\mu },\\lambda _{\\mu })", "15837d3dc0788df45deea5e53d963801": "{\\Gamma ({\\frac {1}{n}}+1) \\over s^{{\\frac {1}{n}}+1}}", "1583cd31c563db6aba62b068cb3f507a": "{\\underline {a}},{\\underline {b}}\\in U\\Rightarrow {\\underline {a}}+{\\underline {b}}\\in U", "1584194c4b047b0575ede0bdb3a3d614": "Y_{1},Y_{2}\\subset Y", "15844fae553a00461f67e6f1ce892907": "{\\begin{aligned}\\operatorname {cov} (X,Y)&{}=E(XY)-E(X)E(Y)=E(XY)-0=E(E(XY\\mid W))\\\\&{}=E(X^{2})\\Pr(W=1)+E(-X^{2})\\Pr(W=-1)\\\\&{}=1\\cdot {\\frac {1}{2}}+(-1)\\cdot {\\frac {1}{2}}=0.\\end{aligned}}", "1584635f63ff3f4c21d830afafa510e9": "{1 \\over {\\sqrt {\\mathit {f}}}}=-2.0\\log _{10}\\left({\\frac {\\frac {\\epsilon }{d}}{3.7}}+{\\frac {2.51}{Re{\\sqrt {\\mathit {f}}}}}\\right),{\\text{turbulent flow}}.", "15849d08d4a893d77f574d9793e5b9e6": "\\left[{0 \\atop 0}\\right]=1\\quad {\\mbox{and}}\\quad \\left[{n \\atop 0}\\right]=\\left[{0 \\atop n}\\right]=0", "15849f78d47c2c541cd79f62ba0af120": "R(x)=a_{n}{\\prod (x-r)}{\\mbox{ for all }}r\\in R.\\,\\!", "1584a84028095dfb66e7ab623b8533a0": "\\eta _{a\\mu \\rho }\\eta _{b\\mu \\sigma }=\\delta _{ab}\\delta _{\\rho \\sigma }+\\epsilon _{abc}\\eta _{c\\rho \\sigma }\\ ,", "15853bac5f51b81afdf9d055fb34d560": "G\\ ", "158597e702831d0c1e008828c6a5abfd": "D_{\\infty }={\\frac {d+1}{2}}\\infty _{1}+{\\frac {d-1}{2}}\\infty _{2}", "1585b31846140536e76d45d568e4029d": "(1/n)x_{1}+(1/n)x_{2}+...+(1/n)x_{n},", "15862f920d7221720e4040b4fc9be633": "5!=120", "15865584e16de0dc617a9cf70e28f479": "432\\times 204=88,128\\,", "15874906dc3b925104b166ba0993e669": "\\varepsilon (x,C)\\equiv \\exp {(-\\int a(C)dx)}", "158794e5928e3e86899499c6ec4d589c": "\\chi _{M}^{I}:\\mathbb {N} \\rightarrow \\mathbb {N} ", "1587dc4523849612b4667b8f3bdbf322": "{\\frac {\\partial |\\mathbf {U} |}{\\partial x}}=", "1587e777c464476c6bfe19b4f65511b0": "\\left\\{\\rho _{x}\\rho _{x^{\\prime }}\\right\\}=0", "1587eb85745dc11a1a54f97985d06c9b": "\\mathbb {Z} [{\\sqrt {n}}]", "158810ebe4b50630ad20f9e51652ecb0": "|\\Delta E|", "15882d97815327ba452dfbc531651578": "N_{\\varepsilon }:=\\max(\\kappa _{\\varepsilon },\\lambda _{\\varepsilon })", "15882ea00370eb89d1509905e460b940": "m=0", "1588d1b5f367c9ce437155dd0e2182ee": "k\\in \\mathbb {N} ,\\ t_{1},\\dots ,t_{k}\\in \\mathbb {R} ", "1588eb607818ab4c59a45141ce345c72": "f:\\mathbb {C} ^{n}\\to \\mathbb {C} ", "158904e7db3986c7cfe7de001d824370": "\\prime ,\\backprime ,f^{\\prime },f',f'',f^{(3)}\\!,{\\dot {y}},{\\ddot {y}}", "1589102232e24f6a353213849edbdd9e": "{f(x+h)-f(x-h) \\over 2h}.", "15891f39fbba8d83d3976094d9449403": "l\\leq a^{T}x\\leq u", "1589238e395f9f7a82e72405cbacc4bb": "\\exists x(\\forall y((\\pi (y,b)\\land y=a)\\leftrightarrow y=x)\\land a=x)", "158946ad8b0cc12cd9e276f3b8baca2e": "{\\binom {p}{n}}={\\frac {p!}{n!(p-n)!}}.", "15894b92eb08a3a74e300ed34d9a9ca5": "{\\mathbf {A} }\\sim {\\mathcal {W}}({\\mathbf {\\Sigma } },\\nu )", "15895caeec0bde86ecdffa1bf4bef84c": "r_{z}", "1589b31ea13bf4843d464032a72c3523": "k\\geq 0", "1589be0e2979a112c910175cc8ef46ba": "y^{\\prime }=(y_{1}^{\\prime }\\ldots y_{N}^{\\prime })", "1589cfc6c58d5f6b9716a8bd5db181b1": "\\mathbb {C} \\rightarrow \\mathbb {C} ,z\\mapsto z^{2}", "1589daa1872425e0b32116d476beca49": "{\\hat {z}}=z_{1}{\\hat {I}}+z_{2}{\\hat {J}}+z_{3}{\\hat {K}}", "1589fbea7fa2a30d00e6a516bf1c09dc": "B_{\\text{g}}={\\frac {2\\pi Gm}{5rc^{2}T}}", "158a0976973b25e5a3260266a4fa7db0": "(-2)^{n}\\,n!\\,", "158a20d137d6d44b7494019e81734bd5": "\\mathrm {Hom} (N,\\mathrm {lim} F)\\cong \\mathrm {lim} \\,\\mathrm {Hom} (N,F-)", "158a422d836132d8b373e706cf65bb3c": "M_{1}={\\frac {1}{8}}\\,S^{ab}\\,S^{cd}\\,\\left(C_{acdb}+i\\,{{}^{\\star }C}_{acdb}\\right)", "158a598d0471fe0a69e89bf68b6394d8": "\\mathbf {v} _{T}={\\frac {1}{f}}\\mathbf {k} \\times \\nabla _{p}(\\Phi _{1}-\\Phi _{0})", "158ae6940c6897b5dadb98bc99517f44": "V_{\\text{out}}=-{I_{\\text{p}}R_{\\text{f}}}", "158b04bad8e0515e297a059163d7f372": "2^{s+1}", "158b09cb8321dcc604d309a33db556d0": "xAx',xwx'\\in (N\\cup T)^{*}", "158b2caccc1034af7124e705750c1f33": "{\\boldsymbol {\\nabla }}\\times ({\\boldsymbol {\\nabla }}\\times {\\boldsymbol {\\epsilon }})={\\boldsymbol {0}}", "158b62ecfee543c353a075eb9d79f755": "\\lambda =\\exp \\left(\\left|-\\alpha ^{(s)}\\right|\\right)+\\sum _{s=1}^{\\infty }\\sum _{k}{\\frac {\\exp \\left(\\left|-\\alpha ^{(s)}\\right|\\right)}{k^{(1)^{2}}k^{(2)^{2}}\\cdot \\cdot \\cdot k^{(s)^{2}}}}", "158b6aa93f4596f520554719193169a4": "{\\hat {\\mathbf {x} }}_{2}=[-\\cos \\beta \\sin \\gamma _{2}\\,,\\,\\cos \\gamma _{2}\\,,\\,\\sin \\beta \\sin \\gamma _{2}]", "158bc9745d022ce0120157276894023f": "X\\ ", "158bfa6c128a87092838c40fc12d24de": "E(\\mathbf {r} ,t)=E^{(+)}(\\mathbf {r} ,t)+E^{(-)}(\\mathbf {r} ,t)", "158c03d0c0065bfb38fd1da052a406a4": "P(U)", "158c5cf40a5d67c5fca60cdfac2854f7": "I=-I,", "158c63eda0b4a1aab37091cc0fa52725": "[D\\rightarrow D^{'}]", "158cc82496207c65c4094d7b98c860f8": "\\nabla \\cdot \\left(\\mathbf {\\Sigma } _{0}\\nabla v_{0}\\right)=0\\,\\,\\,\\,\\,\\,\\,\\mathbf {x} \\in \\mathbb {T} ,", "158ce2b2b8e794ce3fc61e12f005dd4a": "\\theta _{3},\\ldots ,\\theta _{k}", "158cf43a4455e3772a6ba13f6743e986": "\\ {\\mathcal {L}}_{f},{\\mathcal {L}}_{g},{\\mathcal {L}}_{m}", "158d1bfe08e0474c07dc64238fd950d5": "{\\begin{aligned}\\phi (x)&=ax^{2}-x^{3},\\\\\\psi (x)&=1-bx^{2}.\\end{aligned}}", "158d23aefaecffc74e69e4cc7c9dda48": "{\\mathcal {C}}(S_{\\ast }(X),S_{\\ast }(B))=\\cdots \\xleftarrow {\\delta _{2}} {\\mathcal {C}}_{-2}(S_{\\ast }(X),S_{\\ast }(B))\\xleftarrow {\\delta _{1}} {\\mathcal {C}}_{-1}(S_{\\ast }(X),S\\ast (B))\\xleftarrow {\\delta _{0}} S_{\\ast }(X)\\otimes S_{\\ast }(B),", "158d485d7cad4f8d9f90c7d2d01b72d6": "\\operatorname {Tr} (\\rho ^{2})<1", "158e243a6fd150e10517552ca09a2856": "|f|(x)", "158e312f0c4baacf72801256e4bbde13": "x=y+1", "158e5532c5b58379449c30763b22f680": "{\\boldsymbol {\\sigma }}={\\mathsf {c}}:{\\boldsymbol {\\epsilon }}", "158e71cf7fa0b1517568c40bdb9a5cc9": "k'={\\sqrt {1-k^{2}}}", "158e9a39ade58643c2eb42403702faeb": "\\displaystyle f_{T}(x_{n}+\\Delta x)=f_{T}(x)=f(x_{n})+f'(x_{n})\\Delta x+{\\frac {1}{2}}f''(x_{n})\\Delta x^{2}", "158ebfae026f4352052046f9b9b54aad": "{52.0{\\mbox{ g }}\\mathrm {H_{2}S} }-{28.4{\\mbox{ g }}\\mathrm {H_{2}S} }={23.6{\\mbox{ g }}\\mathrm {H_{2}S} }", "158ed842cd5fa65ff039d48af8b1c928": "\\mathbb {Z} ", "158f1955d5de89a3fdb6152e87cd2eb4": "\\color {WildStrawberry}{\\text{WildStrawberry}}", "158f255aebd5d5af89c72d8fa0268e71": "{\\tilde {C}}_{8}", "158f6b61b11a13b0fa1f1cbecaec80f2": "\\mathbf {C} (A,A)", "15901f103d097397d5f7fdc3313e032f": "(Q\\to P)", "15908194ba88c19c15af20b49a7d1364": "R(t)\\sim (-1)^{N-1}\\left(\\Psi (p)u^{-1}-{\\frac {1}{96\\pi ^{2}}}\\Psi ^{(3)}(p)u^{-3}+\\cdots \\right)", "1590b2a8b109a6280423c57230b295ab": "SLG={\\frac {TB}{AB}}", "1590b4ecf12b76f32e32ee48fad02602": "\\Psi :L(H_{B})\\otimes C(X)\\rightarrow L(H_{A}).", "1590bdd1ac1de3098ae64ae14bb15bc2": "\\lim _{k\\to \\infty }\\left({\\frac {1}{4}}\\right)^{k}=0", "1590bf11662688c51fe74034fac77c5f": "y=4e^{-ln(2)t}=2^{2-t}", "1590cf5200cbfd405fa15a562aeea4dc": "2+2<5", "159108a61907c6a33dca946641595f2a": "\\kappa _{\\nu }", "1591b39ab84d6f8c5eacef5e7f440e1e": "{\\tilde {o}}=(T1'-T1-T2'+T2)/2", "1591e9541c3ccdda4d0adf5e0742a4be": "{\\text{II}}(v,w)=\\langle S(v),w\\rangle ", "159200de3062dfbbea4dac89128f4cf1": "{}^{\\mathrm {N} }\\!{\\boldsymbol {\\omega }}^{\\mathrm {B} }={}^{\\mathrm {N} }\\!{\\boldsymbol {\\omega }}^{\\mathrm {D} }+{}^{\\mathrm {D} }\\!{\\boldsymbol {\\omega }}^{\\mathrm {B} }.", "1592a07a39644dada570b31bb5e35aab": "t_{1/2}={\\frac {[A]_{0}}{2k}}", "1592e428a53437cee99f85c65e741d65": "{\\int }_{S}\\mathbf {F} dS={\\int }_{\\tilde {S}}\\mathbf {F} d{\\tilde {S}}", "1592f21267a349034a3b364022436006": "\\rho (\\theta )=\\rho _{\\perp }+(\\rho _{\\parallel }-\\rho _{\\perp })\\cos ^{2}\\varphi ", "1592f7b158da73e482bdccb713ee0402": "k=0,1,\\cdots ,n", "1592f84858cb4edaf8e71ace005b9aca": "a(f*g)=(af)*g\\,", "159366c42181c8efb408c7139099f3b9": "\\rho _{x^{n}}\\equiv \\rho _{x_{1}}\\otimes \\cdots \\otimes \\rho _{x_{n}}.", "15939189fdbaddb34870f063c2724a5d": "\\sin x\\approx x{\\text{ when }}x\\approx 0.", "1593bbb1848be7f6f0550af1b0ea0567": "\\scriptstyle {{\\hat {U}}(t_{0},t_{0})=1}", "15944619706d6ff55791c24833609ccb": "\\Delta \\left(\\,P-P_{e}\\right)+{\\frac {\\rho }{2}}\\,\\left[{\\frac {\\,f}{\\,D}}+{\\frac {f_{e}}{D_{e}}}\\left({\\frac {F}{F_{e}}}\\right)^{2}\\right]\\,W^{2}\\Delta \\,X+{\\frac {\\rho }{2}}\\left[\\left(\\,2-\\beta \\right)\\,-\\left(\\,2-\\beta _{e}\\right)\\left({\\frac {\\,F}{F_{e}}}\\right)^{2}\\right]\\Delta \\,W^{2}\\,=\\,0", "159464c4c1ffdcd5479f1c8f38f6ee72": "(x_{2},-x_{1},x_{4},-x_{3},\\dots ,x_{2n},-x_{2n-1})", "15946696b88e05d0fdb844f46c31e510": "I=0.16Re^{-1/8}.", "15946b211bc9675a7940db155566ab80": "S(A\\mid BC)\\leq S(A\\mid B)", "159492b89d7997ce974fb61434ca6af8": "\\mathbb {M} _{n}(\\mathbb {C} )", "1594b61cf0b73cbabb74d091025dec41": "nx_{i}R\\ln(V/V_{i})=-nRx_{i}\\ln x_{i}\\,", "1594fb94d8791fc03e5b00694c4129a3": "v_{\\alpha ,0}", "15951ee46361447e80755c8fafb141de": "(v_{0{^{}}}[H^{+}]_{0}-v_{i}[OH^{-}]_{0})/(v_{0}+v_{i})=[H^{+}]_{i}-K_{w}/[H^{+}]_{i}", "1595357dde7d6179ade6e587c127361e": "\\int {\\frac {\\mathrm {d} x}{\\sin ax}}={\\frac {1}{a}}\\ln \\left|\\tan {\\frac {ax}{2}}\\right|+C", "15955b7482fa3f61ddc7b9690e620d97": "h^{\\prime }(x)=e^{x}\\log x+x^{-1}e^{x}=h(x)+f(x)g(x).", "1595ab75534e7e40203ddaf98baeeac1": "s_{ij}^{v}(x)>s_{ji}^{v}(x)", "1595c1a44120bc8bf4cdbbfe228a0de1": "ds^{2}={\\frac {4}{|q|^{2}(\\log |q|^{2})^{2}}}dq\\,d{\\overline {q}}", "1595d365f082f21681ddd4ae39de2430": "h(F_{1})=A", "1595d708e8b3afca37be7a5a8075ce1c": "n^{2}a^{n}u[n]", "15962be4aac1b52d034a3adbb895bd01": "t^{-n}\\int _{\\mathbb {R} ^{n}}f(y)\\varphi (y/t)\\,dy=t^{k}\\int _{\\mathbb {R} ^{n}}f(y)\\varphi (y)\\,dy", "1596a7c62a81030cf307d771fefbacbb": "{\\mathbb {P} }{\\biggl (}\\bigcup _{i=_{1}}^{n+1}A_{i}{\\biggr )}\\leq \\sum _{i=_{1}}^{n}{\\mathbb {P} }(A_{i})+\\mathbb {P} (A_{n+1})=\\sum _{i=_{1}}^{n+1}{\\mathbb {P} }(A_{i})", "1596a98c1da23daee068026373951cde": "\\langle l_{1},r_{1}\\rangle _{w}", "1597122dd9a68542ec0c33d1624f1e2b": "a=\\delta ", "15973cfc8e7ee6c42f9b352459c02c00": "C(h)", "159775a3dd81e203a63c0d55b5c56c2c": "ModVR=1-{\\frac {Kf_{m}-N}{N(K-1)}}={\\frac {K(N-f_{m})}{N(K-1)}}={\\frac {Kv}{K-1}}", "159793a4a769bd8edd528c9b73ed9d95": "w=(1+\\lfloor 2.6\\cdot 11-0.2\\rfloor +(0-1)+\\left\\lfloor {\\frac {0-1}{4}}\\right\\rfloor +\\left\\lfloor {\\frac {20}{4}}\\right\\rfloor -2*20)\\ {\\bmod {\\ }}7", "1597f846f1853c71c44004c21b980359": "-{\\frac {d\\rho }{\\rho }}=-{\\frac {1}{a^{2}}}{\\frac {dp}{\\rho }}=-{\\frac {1}{a^{2}}}{\\frac {-\\rho VdV}{\\rho }}={\\frac {V}{a^{2}}}dV", "15983db6289aae8d3a71fc7e15b67153": "{{i}_{B1}}={{i}_{B2}}", "159871b70efc10fbddd951d127560f6a": "B_{Z}", "15987259d4b76cf46a31c7164f2df634": "y=g(x).", "1598d88f06844008f486c8c51bbcaa14": "\\mathrm {wind} ", "15999f6a0ddec237d55fc56843a97cdd": "z\\mapsto z-R{\\frac {f}{f'}}", "1599a047ffc4a884e936eff8e8572512": "\\{x\\in F:w(x)>0\\}", "159a321fc0413162bbfd5843d689904b": "HRT={\\frac {Volume\\;of\\;aeration\\;tank}{influent\\;flowrate}}", "159a721fa09a49abf0d1fda4f2c4bd5e": "W_{0},W_{1},...,W_{N-1}", "159a92b310043eb6dfb1ec235936a100": "V'_{a}", "159aec9bd0b30b8fa9cc3db72fea629c": "\\Gamma _{u}", "159afee50ea8e265039b582d51a6624f": "0<\\lambda <1", "159b0786e476906d332710b8dc91c53a": "{\\tilde {C}}_{5}", "159b5cd78d1990e1596ce704d559b5fd": "B(x;r)\\subseteq B(y;s)", "159bb2c223b17fb81983b683006e7afa": "\\left\\langle \\rho =\\rho _{2},Z_{2}\\right\\rangle ", "159bd0a488d278e99d15ef00fd7d89fb": "\\lim _{x\\to a}f(x)=L\\,", "159be9da606742ceb0c03faf7b8bf584": "{\\text{gyr}}[\\mathbf {u} ,\\mathbf {v} ]\\mathbf {w} =\\ominus (\\mathbf {u} \\oplus \\mathbf {v} )\\oplus (\\mathbf {u} \\oplus (\\mathbf {v} \\oplus \\mathbf {w} ))", "159c189938462c8730c82eb8831b6333": "\\lambda =(\\lambda _{1},\\lambda _{2},\\cdots ,\\lambda _{k})", "159c9230e219d55346e8def570a2e524": "{\\hat {\\Delta }}=R_{i}-{\\hat {R}}-{\\hat {a}}\\cos {\\theta _{i}}-{\\hat {b}}\\sin {\\theta _{i}}", "159cb9183d9730290ed2f89c463982bd": "{\\begin{aligned}\\varepsilon _{1}(t)&=-{\\frac {1}{2}}{\\sqrt {4a^{2}+(\\hbar \\omega _{0}-2\\mu B(t))^{2}}}\\\\\\varepsilon _{2}(t)&=+{\\frac {1}{2}}{\\sqrt {4a^{2}+(\\hbar \\omega _{0}-2\\mu B(t))^{2}}}.\\\\\\end{aligned}}", "159d02907cc0242abca769b4a6d7a019": "H_{\\nu }(\\omega )", "159d213753d88ac2ba9da6d48f4221ec": "\\mathbf {R} ", "159d32cf7b7e681ee5d5bf88e1057709": "r(t_{1},...,t_{n})", "159d491622b4ec95a20fa5bcfa9d4725": "{\\tfrac {k_{0}}{p_{0}}}\\,p", "159d5f60bba230164e90208becdf12a5": "w=z", "159d817837d8424099a67c86ec84b8b3": "\\mathbf {P} _{1}(R)", "159d8654856a89593e5a6668d67152be": "\\alpha :TM\\rightarrow {\\mathbb {R} },\\quad \\alpha _{x}=\\alpha |_{T_{x}M}:T_{x}M\\rightarrow {\\mathbb {R} }", "159de732021373b213818c1796a9f6fe": "{\\text{for }}t<\\log _{e}(2)", "159e0adc038b248158a210e05fa25bce": "A(i,i)", "159e9457856b41152608d43e897a1de1": "A_{ij}", "159ec4588327ac31c8f201f70e6b20ce": "i\\hbar {\\partial \\psi (x,t) \\over \\partial t}=-{\\frac {\\hbar ^{2}}{2m}}{\\frac {\\partial ^{2}\\psi (x,t)}{\\partial x^{2}}}+U(x)\\psi (x,t)", "159ede73b5141eaf74a84c7714a106d4": "\\scriptstyle J^{\\gamma }", "159f12ce9148954f5f1c7dedd3def78f": "P\\!", "159f1fd015f2c4a16f360de56c3b80d4": "Bq^{\\ast }\\equiv 0{\\bmod {\\ }}(d/z)", "159f461e95b3364702ab257b8a40d8de": "g^{\\mu \\nu }{\\frac {\\partial S}{\\partial x^{\\mu }}}{\\frac {\\partial S}{\\partial x^{\\nu }}}=c^{2}.", "159f5192acff5e8dc7db30f9384240db": "\\varepsilon =0.01", "159f69eb8dee26f9c6e864243fc6b120": "{\\frac {1}{2\\pi i}}\\int _{-i\\infty }^{i\\infty }{\\frac {\\Gamma (a+s)\\Gamma (b+s)\\Gamma (-s)}{\\Gamma (c+s)}}(-z)^{s}\\,ds", "159fac319463a70c7ab8f1f6d02442a3": "E_{F}={\\frac {\\hbar ^{2}}{2m_{p}}}\\left({\\frac {3\\pi ^{2}(6\\times 10^{43})}{1\\ \\mathrm {m} ^{3}}}\\right)^{2/3}\\approx 3\\times 10^{7}\\ \\mathrm {eV} =30\\ \\mathrm {MeV} ", "15a044cd3da8d90f7ad8ae7ade5da46d": "S_{d}[i_{r},i_{z}]={\\frac {S[i_{r},i_{z}]}{N\\Delta V(i_{r})}}", "15a05b43eaf77133cee28c619e0d7f32": "\\{y_{i}\\}", "15a07fdb25797093c39e7b36fa2b6d7c": "N(E)={\\frac {V}{2\\pi ^{2}}}\\left({\\frac {2m}{\\hbar ^{2}}}\\right)^{3/2}{\\sqrt {E-E_{0}}}", "15a0b2b3b65a87583abfc5864a071b9f": "b<2^{w-M}", "15a0ced64f2312cfc5e8a290857a1503": "\\ C_{l_{\\alpha }}", "15a0cff797c349351ab4f600b5f2df0d": "f_{a}(1)=4^{1^{0}2^{0}1^{1}}=4^{1}=4", "15a0eae7f7184d6b692478431ab00098": "\\alpha ={\\frac {\\sqrt {4\\,b_{2}\\,b_{0}-b_{1}^{2}\\,}}{2\\,b_{2}}}.\\!", "15a0edfed7445c8b972b09217adccc65": "d/D\\geq 0.07", "15a0fa890ef5d942c426262efb871dbe": "x^{2}+2xh+h^{2}=(x+h)^{2},", "15a152f6e29e18e521cd0187a5e05b03": "E=\\left({a_{1}\\,a_{2} \\over 2\\pi L_{B}}\\right)\\int _{0}^{\\infty }{{k\\;dk\\;} \\over k^{2}+k_{B}^{2}}{\\mathcal {J}}_{0}\\left(kr_{B1}\\right){\\mathcal {J}}_{0}\\left(kr_{B2}\\right){\\mathcal {J}}_{0}\\left(kr_{12}\\right)=\\left({2e^{2} \\over L_{B}}\\right)\\int _{0}^{\\infty }{{k\\;dk\\;} \\over k^{2}+k_{B}^{2}r_{B}^{2}}{\\mathcal {J}}_{0}^{2}\\left(k\\right){\\mathcal {J}}_{0}\\left(k{r_{12} \\over r_{B}}\\right)", "15a19ff7f8b1ed8508009c382fd1c3f7": "T\\;", "15a1beedfb982e81cc06a2cbe4b8d668": "{\\mathcal {L}}={\\bar {\\psi }}_{B}\\left[i\\gamma _{\\mu }(\\partial ^{\\mu }+ie_{B}A_{B}^{\\mu })-m_{B}\\right]\\psi _{B}-{\\frac {1}{4}}F_{B\\mu \\nu }F_{B}^{\\mu \\nu }", "15a226a912eed749d24be5f0252e7a55": "\\sigma _{y}", "15a2311476b41023ee77af5d711a61ff": "I(t)=M_{a}t-P_{0}+P(t)\\,", "15a2c1783c28f988302c23e8b1d17528": "f(\\mathbf {r} )=\\sum _{\\mathbf {K} }h(\\mathbf {K} )\\cdot e^{i\\mathbf {K} \\cdot \\mathbf {r} },", "15a2cbb963cf3912de9da737cec98841": "1+B_{t}(t,T)+\\alpha (t)B(t,T)-{\\frac {1}{2}}\\gamma (t)B^{2}(t,T)=0", "15a2d6da795620300ad3e583b255098d": "V_{n}({\\sqrt {R}},Q)={\\frac {a^{n}+b^{n}}{a+b}}", "15a35368435047a2bcb47b5ab3c42817": "\\beta (\\alpha _{s})=-\\left(11-{\\frac {2n_{f}}{3}}\\right){\\frac {\\alpha _{s}^{2}}{2\\pi }}~,", "15a3b2d6b390d100390a7dd5598fc4b7": "\\phi =hf_{0}\\,\\!", "15a3e569dbae8b65fe55b2e7be9100ac": "\\wp _{\\tau }(z)", "15a40c10a679907b72ce76117c4535b6": "\\psi _{n_{x},n_{y},n_{z}}={\\sqrt {\\frac {8}{L_{x}L_{y}L_{z}}}}\\sin \\left({\\frac {n_{x}\\pi x}{L_{x}}}\\right)\\sin \\left({\\frac {n_{y}\\pi y}{L_{y}}}\\right)\\sin \\left({\\frac {n_{z}\\pi z}{L_{z}}}\\right)", "15a418f456760c26776fdef7dc372cca": "\\{p,1-p\\}", "15a432f30c01457fcae6b73310e7de3d": "(x-3)(x-1)^{3}(x+1)^{3}(x+3)(x^{2}-3)^{2}.\\ ", "15a49140db2d4d6b0dce2636b730e2d0": "t\\in \\mathbb {R} ", "15a4f2c4f69248dc99cd43b70f28bac4": "\\mathbf {F} =q\\left[-\\nabla \\phi -{\\frac {\\partial \\mathbf {A} }{\\partial t}}+\\nabla (\\mathbf {v} \\cdot \\mathbf {A} )-(\\mathbf {v} \\cdot \\nabla )\\mathbf {A} \\right]", "15a4f479f3cbc8a5f090a936bb9ddffc": "z_{k+1}=z_{k}+hf(q_{k},p_{k+1})\\,", "15a51ca1c3908a562357091e3bb4f285": "\\Box \\phi ", "15a5652489e7465d11da99309009febd": "K=\\mathbb {R} _{+}^{d}", "15a5d314e912346fbcaf2ca8024a10fe": "S=\\sum _{i=1}^{N-1}(c_{i+1}-c_{i})^{2}", "15a5f902bcd0a5374a1b9b80507fa7e9": "\\textstyle n2^{l-1}+1", "15a639ccc4b3e0056a8fc594553642b1": "-{\\frac {dN}{dt}}=\\lambda N", "15a64955780f6ec48902a71956c120e1": "\\omega _{N}=e^{-{\\frac {2\\pi i}{N}}},", "15a668b9bcce8685a0797693dc7e9582": "2\\leq i\\leq m-1", "15a67b8b933a6bd77593e3bd4c779317": "\\alpha <2", "15a764354a3f4ceb778a21690fb952cf": "\\mathbf {M} =\\mathbf {F} ^{\\mathrm {T} }", "15a76ff818d343cc5eef7101f97a63ae": "F_{0}\\colon M\\to M\\times 0", "15a795d5b4688eef60138943925f3111": "r_{a}+r_{b}+r_{c}=4R+r,", "15a7e3687c8dd090497a498de2f1eca6": "X_{n}=\\{x\\in X:\\operatorname {dim} H_{x}=n\\}.", "15a874ae54da1ba7b9d6c6cc8ac42993": "a_{0},\\ldots ,a_{N}>0", "15a8bf428849adbfa7472abb73905ab5": "\\alpha \\times Y", "15a8f2c34eb79dfd32cc8f2a0978e299": "|00...0\\rangle ", "15a900072b20edb02837602480ea2320": "A^{T}A=I,det(A)=+1", "15a96b96777a9f1ad352d52751eaf17c": "\\gamma _{n}(x)", "15a96f27e285951bde18cf32f5d872bb": "\\cdots {\\overset {d_{n+1}}{\\longrightarrow }}E_{n}{\\overset {d_{n}}{\\longrightarrow }}\\cdots {\\overset {d_{3}}{\\longrightarrow }}E_{2}{\\overset {d_{2}}{\\longrightarrow }}E_{1}{\\overset {d_{1}}{\\longrightarrow }}E_{0}{\\overset {\\epsilon }{\\longrightarrow }}M\\longrightarrow 0,", "15a9a8e6cca35e1319518b144ac07fd6": "{\\begin{aligned}\\mathbf {r} _{\\mathrm {cog} }&={\\frac {1}{M\\left|\\mathbf {g} \\left(\\mathbf {r} _{\\mathrm {cog} }\\right)\\right|}}\\int \\left|\\mathbf {g} \\left(\\mathbf {r} \\right)\\right|\\mathrm {d} \\mathbf {m} \\\\&={\\frac {1}{M\\left|\\mathbf {g} \\left(\\mathbf {r} _{\\mathrm {cog} }\\right)\\right|}}\\int \\mathbf {r} \\left|\\mathbf {g} \\left(\\mathbf {r} \\right)\\right|\\mathrm {d} ^{n}m\\\\&={\\frac {1}{M\\left|\\mathbf {g} \\left(\\mathbf {r} _{\\mathrm {cog} }\\right)\\right|}}\\int \\mathbf {r} \\rho _{n}\\left|\\mathbf {g} \\left(\\mathbf {r} \\right)\\right|\\mathrm {d} ^{n}x\\end{aligned}}\\,\\!", "15a9b73c37000fa3e83b737099a299c7": "P\\setminus K", "15a9ccb8060b43d920e39e01ff0156d1": "\\mathrm {} \\ \\Pr(X>40\\mid X>30)=\\Pr(X>10).\\,", "15a9cedbab792101825311615fd74749": "0=\\sum _{m=1}^{d}b_{m,n}u_{m}", "15aa2eaa36aeb7eb649e20eebda1fdf6": "g^{(2)}(\\tau )=1", "15aa3a38fce913805176c7cb29301b64": "F_{0}(a,b)=a+b", "15aa647c827f671b92a1fd276abac7ed": "V(J)=1", "15aa6eaad71b91fa8df72e39fdf6f744": "\\mathbf {p} _{m+1}=\\sum _{i}^{m}c_{i}\\mathbf {p} _{i}.", "15aab1388fcbe47ff5b9f79235412390": "m=101", "15aabc8fd5a35230ac695c5186c508a6": "(v,w){:}", "15aacf1aed839305ac36d38bc6167739": "\\liminf _{n\\to \\infty }\\|x_{n}-x_{0}\\|<\\liminf _{n\\to \\infty }\\|x_{n}-x\\|,", "15aae5fc0a6f417399ed38195bee8892": "P_{1}(y)", "15ab09152ff5b36d264aa5ba472b0c2a": "\\Delta G_{Fe}^{\\ominus }=-nFE_{Fe}^{0};\\Delta G_{Ce}^{\\ominus }=-nFE_{Ce}^{0}", "15ab2d2b0b92c13f328635e5c4bdbe64": "i+1", "15ab8efc379b8627b4b644325a4bb0c0": "\\phi (T\\otimes U)=\\phi (T)\\otimes \\phi (U)", "15ab95def80823e45f1a6dfdb8021b77": "{\\frac {1}{a_{{\\overline {n}}|i}}}-{\\frac {1}{s_{{\\overline {n}}|i}}}=i", "15abeba2ef8c4882259ac119c4ca3ca2": "\\Delta F_{y}={\\frac {1}{2}}C_{L}\\rho w^{2}ldr{\\frac {\\cos(\\beta -\\phi )}{\\cos \\phi }}", "15ac0347a6a9075f425d0f9c1e2bfdf5": "V_{g}f\\in Y", "15ac818c15c8653db90739a08427c531": "l={\\frac {RT}{m_{A}g}}", "15acad78b054830808698dcd07dcba83": "\\tau (w)=w':(w,w')\\in U", "15acd4240aa428eeb50998aaee7835c2": "|H\\rangle ", "15ad03c50e93cfbbc5b1e515a5b5f785": "{\\zeta _{g}}", "15ad92e948ea305c298a2867e0f0c6ba": "\\mathbf {I_{m}} =\\mathbf {I_{0}} \\sin \\phi _{0}", "15adaceddd1a2ac25858b574e8802da7": "\\,{\\hat {\\lambda }}_{i}", "15adbe25adc627c59823907f881c3b1f": "f={\\frac {v}{2l}}", "15adcba3bdb0754dd7aedb4540134925": "x(t)=-{\\frac {qE_{0}}{m\\omega ^{2}}}\\sin(\\omega t)+{\\frac {qE_{0}}{m\\omega }}t-{\\frac {d}{2}}", "15adcbc83e8e5da33cd5fc22108ff0b5": "A=227\\cdot 5^{2}", "15adcf5af2342261722d9cfaf993a7fe": "P(f)={\\frac {1}{f_{\\max }-f_{\\min }}}\\int _{\\Omega }\\delta (f-F({\\boldsymbol {r}}))P({\\boldsymbol {r}})\\,d{\\boldsymbol {r}}={\\frac {1}{f_{\\max }-f_{\\min }}}{\\frac {1}{V}}\\rho (f)\\int _{\\Omega }\\delta (f-F({\\boldsymbol {r}}))\\,d{\\boldsymbol {r}}={\\frac {1}{f_{\\max }-f_{\\min }}}={\\text{constant}}", "15adef1749db58083539ddd73babdc70": "D_{N}({\\mathbf {t} }_{1},\\dots ,{\\mathbf {t} }_{N-1})=1", "15ae52d0a5c11ff8a7cd7bffc075762c": "g*", "15ae64b4a18bf00ce833b6dba8114e8f": "\\forall i\\in C^{*}{\\text{, }}|\\mathbf {c} _{i}|\\geq w_{\\min }", "15ae75960ffd194d3707fee25eda5dea": "e_{j}^{N_{j}}=1=\\omega _{jk}^{N_{j}}=\\omega _{jk}^{N_{k}}\\,", "15ae88fb367511de5388699de99ffa83": "\\delta (q_{1},b,a)=(q_{1},\\varepsilon )", "15ae96c14d1517db2bfa3551aaa2c5aa": "{\\begin{array}{lcl}Precision={\\left\\vert Prelevant\\cap Pretrieved\\right\\vert \\over \\left\\vert Pretrieved\\right\\vert };\\\\\\\\Recall={\\left\\vert Prelevant\\cap Pretrieved\\right\\vert \\over \\left\\vert Prelevant\\right\\vert };\\\\\\\\AveragePrecision=\\int _{0}^{1}{Prec(R_{ecall})}dR_{ecall},\\\\\\end{array}}", "15aeaca53456b1ef193568e90b9f9141": "N_{1}0=32+4+2+1", "15aefc48007673476be6000d0a080ee4": "\\ln {p_{s} \\over p_{0}}={\\frac {2\\sigma M}{RT\\rho \\cdot r_{p}}}", "15af582436a80bdb8c712f7a88315c05": "\\supset ", "15af7211b9e69fa7ac8b2ca415229b02": "e^{x_{1}y_{1}},e^{x_{1}y_{2}},e^{x_{2}y_{1}},e^{x_{2}y_{2}}.", "15af80a32fb35660a7441d811fa4d063": "\\epsilon ^{\\mu }(q,y)\\,", "15af99f4716e26855fed68cbd82b62ce": "F({\\vec {r}},t)=\\varepsilon ae^{i{\\vec {k}}x-\\omega t}+h.c", "15b0021e4b3c087ac029030f5336b0cd": "\\lim _{n\\to \\infty }\\operatorname {Pr} (X_{n}\\in A)=\\operatorname {Pr} (X\\in A)", "15b08699ac7e80090714a81e44781870": "\\lambda :A\\to B", "15b088e42dbe089b5302a75efc6b1687": "b=\\det {\\begin{pmatrix}z_{1}w_{1}&z_{1}&w_{1}\\\\z_{2}w_{2}&z_{2}&w_{2}\\\\z_{3}w_{3}&z_{3}&w_{3}\\end{pmatrix}}\\,", "15b0a8a0c01e5e2f1d901888ed491e00": "n<214928639999", "15b0c6118b4b283ef9b0e174fef1e40e": "\\pm {\\tfrac {1}{2}}+G(x-y)|_{\\mathbf {R} ^{n-1}}", "15b0d170266cc663d85481ecb3ed4fa6": "\\mathbf {F} ^{g}=\\mathbf {I} +[\\vartheta ^{\\perp }-1]\\mathbf {s} _{0}\\otimes \\mathbf {s} _{0}", "15b1165ee52e66c76559f28205268924": "{\\begin{aligned}L'&=L'_{old}+v\\Delta t'={\\frac {L}{\\gamma }}+{\\frac {\\gamma v^{2}L}{c^{2}}}\\\\&=\\gamma L\\end{aligned}}", "15b1359fc78dcd4a855a126c08bb6c32": "\\sup _{\\Lambda }|\\chi (\\lambda )-1|\\leq \\epsilon ,\\quad \\chi \\in {\\hat {G}}.", "15b1cbee5939bc1ea8d8487f639c0691": "{\\rm {pH}}=-\\log _{10}[{\\rm {H^{+}}}]\\,\\!", "15b1d37bb7f5937c66ad4aca512f580e": "\\sum _{\\delta \\mid n}\\Lambda (\\delta )=\\log n.", "15b1e4d6d2feceb255e4acd82cccf6a7": "\\sum _{k=1}^{m}{\\frac {1}{k^{2}}}={\\frac {1}{1^{2}}}+{\\frac {1}{2^{2}}}+\\cdots +{\\frac {1}{m^{2}}}", "15b1e887fd51a27495a10c72d8f14b1e": "\\scriptstyle \\lfloor k\\rfloor \\,", "15b1e88a9c6cc1a7808db4fa2540a280": "\\sum _{j}n_{j}{\\text{Reactant}}_{j}\\rightleftharpoons \\sum _{k}m_{k}{\\text{Product}}_{k}\\equiv \\alpha A+\\beta B...\\rightleftharpoons \\rho R+\\sigma S...", "15b201981b91fa3db08249a57516327b": "{v_{esc}}\\,", "15b2710d28a65222c7942cfad48419b8": "\\sum _{k=2}^{\\infty }(-1)^{k}{\\frac {\\zeta (k)}{k}}=\\gamma ", "15b2d3d77baa02dff47bf537690a3144": "\\mathrm {O_{2}+4\\ e^{-}+4\\ H^{+}\\longrightarrow H_{2}O} ", "15b2da7e7b99c320b0a07dca941a69b5": "z\\in [x,y]", "15b2f1c5744e0329f210a15b1278ecb1": "M(j,j)=1-{\\frac {\\lambda m(j)\\sum _{i=1,i\\neq j}^{20}A(i,j)}{\\sum _{i=1,i\\neq j}^{20}A(i,j)}}", "15b30a02f62fd433a23d4503a4fca3af": "V_{\\mathit {SYNCHSW}}=I_{\\mathit {SYNCHSW}}R_{\\mathit {on}}=(1-D)I_{o}R_{\\mathit {on}}", "15b330d06e8136d00895dd6b480f8420": "H_{3}L\\rightleftharpoons H_{2}L+H:pK_{1}=-\\log \\left({\\frac {[H_{2}L][H]}{[H_{3}L]}}\\right)", "15b39bacaa699958e470c8e2d2cad8b3": "(x_{i}^{1},x_{i}^{2})", "15b3cf12724810c360e9c60e592156da": "P(k,k')={\\frac {2\\pi }{\\hbar }}\\left(D_{ac}\\sum _{q}{\\sqrt {\\frac {\\hbar }{2MN\\omega _{q}}}}|q|{\\sqrt {n_{q}+{\\frac {1}{2}}\\mp {\\frac {1}{2}}}}\\,I(k,k')\\delta _{k',k\\pm q}\\right)^{2}\\delta [\\varepsilon (k')-\\varepsilon (k)\\mp \\hbar \\omega _{q}],", "15b41df8b46ce1c1ff53664445c11f73": "-{\\frac {\\hbar ^{2}}{2M}}\\nabla _{i}\\cdot \\nabla _{j}", "15b47468765e2de05c9e95b5a826d1cb": "\\left({9 \\over 75}\\right)", "15b4a17a9366c9a59085bef8f32de4e5": "x={\\frac {1}{a_{1}}}+{\\frac {1}{a_{1}a_{2}}}+{\\frac {1}{a_{1}a_{2}a_{3}}}+\\cdots .\\;", "15b4ad58baaa08c32f8c67f18e672c3e": "(x_{0},y_{0})", "15b4af05e9582885bed95d18844b9e45": "\\Delta R_{0}\\Delta x_{0}=\\Delta r_{s}\\Delta r\\geq \\ell _{P}^{2}", "15b4f8f00c591228cb92f88164bdc3a3": "(a_{n})", "15b508d2db1619fa3656a5c266274483": "x^{n}(x^{2}-x-1)-(x^{2}-1)\\,", "15b5177220acb64df10fc20902356ff4": "\\displaystyle {Y=\\mathrm {Ad} (X)\\cdot Y=Y+[X,Y]+{1 \\over 2}[X,[X,Y]]\\in {\\mathfrak {m}}_{+}\\oplus {\\mathfrak {k}}_{\\mathbb {C} }\\oplus {\\mathfrak {m}}_{-},}", "15b51ded00bc1cc6fd772d859f151295": "\\mathbb {R} ^{2d}", "15b51e0caff1955020c7bfc6d29e3304": "{\\begin{aligned}&\\nabla ^{2}\\left({\\frac {\\partial \\varphi _{1}}{\\partial x_{1}}}+{\\frac {\\partial \\varphi _{2}}{\\partial x_{2}}}\\right)=-{\\frac {q}{D}}\\\\&\\nabla ^{2}w^{0}-{\\frac {\\partial \\varphi _{1}}{\\partial x_{1}}}-{\\frac {\\partial \\varphi _{2}}{\\partial x_{2}}}=-{\\frac {q}{\\kappa Gh}}\\\\&\\nabla ^{2}\\left({\\frac {\\partial \\varphi _{1}}{\\partial x_{2}}}-{\\frac {\\partial \\varphi _{2}}{\\partial x_{1}}}\\right)=-{\\frac {2\\kappa Gh}{D(1-\\nu )}}\\left({\\frac {\\partial \\varphi _{1}}{\\partial x_{2}}}-{\\frac {\\partial \\varphi _{2}}{\\partial x_{1}}}\\right)\\,.\\end{aligned}}", "15b54adae7e26abe688e1a0bd9dc2da5": "A\\to X", "15b5dc2065bec88d12b000ac4facc905": "{Tonnage}={\\frac {{Length}\\times \\ {Beam}\\times {\\frac {Beam}{2}}\\times {\\frac {3}{5}}\\times {0.62}}{35}}", "15b5dca93fb69d952a3dd9539cd5f20e": "X+{\\mathit {ATP}}\\longrightarrow {\\mathit {XP}}+{\\mathit {ADP}}", "15b698c308009d5c2d3d2cb432d22510": "\\omega .\\!", "15b6e22b44bb28cb88f1df76cafba3f6": "+(\\partial _{b_{1}}X^{c})T^{a_{1}\\ldots a_{r}}{}_{cb_{2}\\ldots b_{s}}+\\ldots +(\\partial _{b_{s}}X^{c})T^{a_{1}\\ldots a_{r}}{}_{b_{1}\\ldots b_{s-1}c}+w(\\partial _{c}X^{c})T^{a_{1}\\ldots a_{r}}{}_{b_{1}\\ldots b_{s}}", "15b70d274ddb172cea22c1998c7f68b4": "{\\sqrt {2\\pi }}{\\Big (}\\varphi (t)+t\\Phi (t)-\\max\\{t,0\\}{\\Big )}", "15b77835caca4e41ac84afea64c86869": "\\sigma ={3}\\varepsilon _{0}{\\frac {\\kappa -1}{\\kappa +2}}E_{\\infty }\\cos \\theta ={\\frac {1}{V}}{\\mathbf {p\\cdot {\\hat {R}}}}\\ .", "15b77a8e21dd857fa3ee2d0edea24650": "k_{i}^{\\beta }", "15b7a68d43b66b052d0184693a12b8b2": "\\sum \\limits _{k=a}^{b-1}f(k)=\\int _{a}^{b}f(x)\\,dx\\ +\\sum \\limits _{k=1}^{m}{\\frac {B_{k}}{k!}}\\left(f^{(k-1)}(b)-f^{(k-1)}(a)\\right)+R_{-}(f,m).", "15b7b1ddb03e6bad0184fcfcd71a8c42": "{\\begin{aligned}V_{\\mathrm {rms} }&={\\sqrt {{\\frac {1}{T}}\\int _{0}^{T}[{V_{pk}\\sin(\\omega t+\\phi )]^{2}dt}}}\\\\&=V_{pk}{\\sqrt {{\\frac {1}{2T}}\\int _{0}^{T}[{1-\\cos(2\\omega t+2\\phi )]dt}}}\\\\&=V_{pk}{\\sqrt {{\\frac {1}{2T}}\\int _{0}^{T}{dt}}}\\\\&={\\frac {V_{pk}}{\\sqrt {2}}}\\end{aligned}}", "15b7b2d36fba832657d78d2045b8f9df": "\\int {\\frac {r^{7}\\;dx}{x}}={\\frac {r^{7}}{7}}+{\\frac {a^{2}r^{5}}{5}}+{\\frac {a^{4}r^{3}}{3}}+a^{6}r-a^{7}\\ln \\left|{\\frac {a+r}{x}}\\right|", "15b7f368a251434ef0f694ee96efd6de": "T(y_{0})=T_{0}\\,", "15b817ca48d2c94356b48bdba4f406e5": "a_{8}\\times (6\\rho ^{4}-6\\rho ^{2}+1)", "15b827ec7905f04f1bf39cb554b08430": "BSC_{p}", "15b83ef1361eb0fb666ef29f9b48c5e2": "f=\\beta y,", "15b841c73f03e6246b7b90dc2167706b": "\\sum _{k=1}^{d+1}H_{B_{k}}\\geq {\\frac {d+1}{2}}\\log({\\frac {d+1}{2}})", "15b858c0d0b1c830e8dc63a4466f6e78": "P={\\frac {P_{max}}{2}}", "15b8730ba75051c58f866cb2ffcc0a45": "\\left(B\\lor C\\right),\\lnot C,\\left(\\left(B\\rightarrow \\lnot A\\right)\\land \\lnot C\\right)\\vdash \\lnot A", "15b88e119ddf26a00567806f64ffee62": "{\\begin{aligned}&\\mathbf {T} ^{2}={\\begin{pmatrix}{\\frac {1}{16}}&{\\frac {1}{4}}\\\\[4pt]0&{\\frac {1}{16}}\\end{pmatrix}},\\quad \\mathbf {T} ^{3}={\\begin{pmatrix}{\\frac {1}{64}}&{\\frac {3}{32}}\\\\[4pt]0&{\\frac {1}{64}}\\end{pmatrix}},\\quad \\mathbf {T} ^{4}={\\begin{pmatrix}{\\frac {1}{256}}&{\\frac {1}{32}}\\\\[4pt]0&{\\frac {1}{256}}\\end{pmatrix}},\\quad \\mathbf {T} ^{5}={\\begin{pmatrix}{\\frac {1}{1024}}&{\\frac {5}{512}}\\\\[4pt]0&{\\frac {1}{1024}}\\end{pmatrix}},\\end{aligned}}", "15b8cce139ab68921eb9889be91d6b10": "p_{y}(x):=\\vert \\langle x,y\\rangle \\vert \\qquad x\\in X", "15b8fe65a153f3d61e3aa92d7d727d28": "mL_{e}^{2}{\\ddot {\\theta }}=-kL_{e}^{2}\\theta -\\mu B\\sin {\\left({\\frac {\\theta H_{k}}{B+H_{k}}}\\right)}\\left({\\frac {H_{k}}{B+H_{k}}}\\right)-2K_{u}V\\sin {\\left({\\frac {\\theta B}{B+H_{k}}}\\right)}\\cos {\\left({\\frac {\\theta B}{B+H_{k}}}\\right)}\\left({\\frac {B}{B+H_{k}}}\\right)\\Rightarrow ", "15b9015edadefaca276c4699b4faa442": "\\{j_{1}j_{2}j_{3}\\}", "15b983900b08826c6039f3daea7be214": "P_{0}=k\\left(V/H\\right)^{n}", "15b9da7a94585faa52f59490cb35a005": "\\lambda m.\\lambda n.n\\operatorname {succ} m", "15ba550d5555a6d2c09df02798070402": "\\sum _{i=N+1}^{\\infty }|a_{i}|=\\sum _{i=1}^{\\infty }|a_{N+i}|<\\sum _{i=1}^{\\infty }r^{i}|a_{N+1}|=|a_{N+1}|\\sum _{i=1}^{\\infty }r^{i}=|a_{N+1}|{\\frac {r}{1-r}}<\\infty .", "15ba5c2c4de89fbb2a2873e771b895ef": "2\\epsilon ", "15ba9457f86ec9a30bc9c1186628cce1": "x\\in A", "15ba9d3b4668ce0a9a087945d9c05734": "y_{tt}=ky_{xxxx}", "15baa5d14f7136e5885fd0a4410b4bce": "{\\frac {\\mathrm {ft^{3}} }{\\mathrm {slug} }}", "15bac40c6a3a27c1ed0594b21fdfa7ca": "y\\left(t\\right)=x(at)", "15bb0105cac825d1941db7ffbd160537": "\\Psi _{N}(f)=E[\\exp(-N(f))]", "15bb0e7dcc81e0af498eaadf86e72a34": "y_{3}={\\frac {y_{1}^{3}-a\\cdot x_{1}^{3}}{a\\cdot y_{1}\\cdot x_{1}^{3}-y_{1}}}", "15bb14daeb013fca7424aea2a459a7c6": "k/q", "15bb233b90fe2d2bfced7d9ea2501886": "(-2c,-c^{2})", "15bb38bb5739e8eceae8bf32af65c15a": "d=1", "15bb9113eb813735527dfe406b41e477": "z_{c}\\leq z\\leq -z_{c}", "15bbc42a514b3cf401204ed7695d63d8": "W_{A}", "15bc21f02f93443df17d3dfbc8880a94": "\\rho =N/V", "15bc2bff3845bdeb9c5a40abdfa8b97c": "x=(RF)a\\lambda ", "15bc4951d5b32e4704d864e13be5decf": "f''-2zf'+\\lambda f=0;\\;\\lambda =1", "15bc4b37ea80d103be9b3611ad98f0af": "\\Psi _{n}={\\sqrt {\\frac {2}{L}}}\\sin \\left({\\frac {n\\pi }{L}}x\\right)", "15bc6df800ec908fe87f4be767eeafa8": "2C_{1}=\\mu \\,", "15bc6e5f4d18ff7409b845d3fbae4544": "T_{1}\\dots T_{n}", "15bc86d481ea9eac6dbd8fa8cbfafedc": "{\\hat {f}}_{i-1/2}^{n}={\\frac {1}{2}}\\left(f_{i-1}+f_{i}\\right)-{\\frac {\\Delta x}{2\\Delta t}}\\left(u_{i}^{n}-u_{i-1}^{n}\\right).", "15bca7119d9e828dead4a9675b32d7d1": "\\psi (\\mathbf {r} ,\\ t=0)=A\\ e^{i\\mathbf {k\\cdot r} }\\ ,", "15bcb9807bc17321a713ffc649618a7a": "\\chi _{\\lambda }(\\exp X)={\\frac {\\sin((2\\lambda +1)X)}{\\sin X/2}}", "15bcbb94f2d649b10cb7ccd09036c5e7": "=\\lim _{x\\to \\pm \\infty }\\left[{\\frac {x^{2}-1}{x}}-x\\right]", "15bcbd06c8084b9a0a4beb424ed4401f": "{\\mathcal {C}}=\\{f\\in V^{*}:\\langle f,e_{s}\\rangle >0\\ \\forall s\\in S\\}", "15bd36046c5714dac49475e884de0db9": "\\pi _{i}\\colon G\\to G_{i}\\quad \\mathrm {by} \\quad \\pi _{i}(g)=g_{i}", "15bdcda88ab6e25112b1d5a2e10de4fe": "{\\frac {e_{c}^{2}|\\langle \\varphi _{v}|e_{c}\\mathbf {x} |\\varphi _{c}\\rangle |^{2}D_{ph-e}[f_{e}^{\\mathrm {o} }(E_{e,v})-f_{e}^{\\mathrm {o} }(E_{e,c})]}{\\epsilon _{\\mathrm {o} }^{2}m_{e,e}^{2}u_{ph}n_{\\omega }\\omega }}", "15be3c2519dc3df50beeab4d9eb20dd8": "p_{1}", "15be438baa231c98073b780e3f12a54f": "\\lambda /d", "15be7062c05163f511f39a3ead28b2fc": "(M,{\\bar {N}})", "15bebe71037e28f744651387ee43421f": "\\displaystyle {|a_{n}|\\leq r^{-n}\\sup _{|z|=r}|f(z)|.}", "15bec5e1a8928d5ef4480b811df43591": "\\sigma _{22}=\\sigma _{33}=0", "15beddf17e3a9b75437d2f2685184871": "U_{2}\\left(x,y\\right)=\\beta ", "15bee9298af680589f79e9b7ec0bd67a": "\\ \\lambda _{i}=\\lambda _{0}[1+K_{i}(\\Delta \\mu /\\mu )],", "15bee9611d495eb5166ba0a51b815b43": "\\sin \\theta ={\\frac {\\text{Rise}}{\\text{Length}}}\\,", "15bf21e981330cd12c2fa37226efbd36": "T_{G}(x,y)=(x-1)^{-k(G)}Q_{G}(x-1,y-1).", "15bf4bfb259c425e6b9389f4817402e5": "{\\hat {N}}=m+{\\frac {m-k}{k}}=m+mk^{-1}-1=m\\left(1+k^{-1}\\right)-1", "15bf62afc7f4ae839219d9deb2e75ba4": "x'=y'^{n}+r'", "15bf8550544059b33960eb174e210629": "{(\\eta _{b})_{max}}={\\cos ^{2}\\alpha _{1}}", "15bfc40c1903572994adff2f388e5277": "({\\hat {c}}-{\\hat {a}})", "15bfd1c1468174a48780ca9ea4480488": "\\Pr(y_{j}=h_{i})=(1-\\pi ){\\frac {\\lambda ^{h_{i}}e^{-\\lambda }}{h_{i}!}},\\qquad h_{i}\\geq 1", "15c09a929654f7dca65a774f4b6fda6b": "{\\frac {n(n-1)}{2}}", "15c0c070e4b5cc0016f225d630343863": "{p \\choose i}\\equiv 0{\\pmod {p}},\\qquad 00\\\\0&{\\text{if}}&\\lambda _{i}=0\\end{array}}\\right\\}", "15cbbcf59b8c3b7a89234a0c521d9679": "3(x+1)=3x+3", "15cbbd53175dd6120a923121ea791e40": "\\mathbb {E} [(R_{r}-R_{\\min })_{+}]", "15cbc670cc40c103cf14ee7fa0188e7b": "P_{y}(t+dt)=P_{y}(t)+dt\\sum _{x}P_{x}R_{x\\rightarrow y}\\,", "15cbe281ef9b7700d442a2e1c5436511": "\\;W\\pi _{1}(a)V_{1}h=\\pi _{2}(a)V_{2}h.", "15cbff19fab422b675636e3aca109199": "D=\\{x^{2}+y^{2}\\leq 9,\\ -5\\leq z\\leq 5\\}", "15cc6fa4e7b4493ea9785dcc0818deb1": "\\left(xzyz^{-1}\\right)\\left(zy^{-1}x^{-1}y\\right)=xzyz^{-1}zy^{-1}x^{-1}y.", "15cc89522a3f7c2dfe6bf3f4279c3cd4": "g_{1}=0,\\ldots ,g_{n}=0", "15ccab65a0d3d8e92c3cd5b53081a59b": "1.\\ \\mathrm {CrO_{4}^{2-}+H^{+}\\rightleftharpoons HCrO_{4}^{-};K_{1}={\\frac {[HCrO_{4}^{-}]}{[CrO_{4}^{2-}][H^{+}]}}} ", "15ccfed3f49be24a67ae0e34109f4939": "f\\circ f^{n}=f^{n}\\circ f=f^{n+1}", "15cd2154da295c744400b94a13df90cb": "+{\\frac {62}{2835}}*x^{9}-{\\frac {1382}{155925}}*x^{11}+{\\frac {21844}{6081075}}*x^{13}+O(x^{15})", "15cd36c81f97890a9ad6f40e9de51c14": "\\|A\\|_{w}=\\sup _{n\\geq 0}(1+n)\\mu (n,A),", "15cd60143bf155c4f74a5051e7bc3f19": "{\\frac {V_{\\mathrm {i} }}{V_{x}}}=\\lim _{\\delta x\\to 0}\\left(1+{\\frac {\\delta Z}{Z_{0}}}\\right)^{\\frac {x}{\\delta x}}", "15cda45f4d5812906a1d6f3eb52a6e46": "b'=c", "15ce175a6c16fb78e557de6749cce437": "\\ L/D>10", "15ce2b3a61533cbd1bc374db738df222": "ta(s,t_{s})=t_{s}.", "15ce48e12ee10e058c8703b7eaf85f39": "\\log K_{0}={\\frac {1}{\\log 2}}\\left[\\sum _{k=3}^{N}\\log \\left({\\frac {k-1}{k}}\\right)\\log \\left({\\frac {k+1}{k}}\\right)+\\sum _{n=1}^{\\infty }{\\frac {\\zeta (2n,N)}{n}}\\sum _{k=1}^{2n-1}{\\frac {(-1)^{k+1}}{k}}\\right]", "15ceb08b14110719751cbc75928ab109": "\\mathbb {} H_{*}", "15cef79d8979bb44a71c1257a028be0e": "{r \\over a}", "15cf356e0be06f5b0b80f69cf474f397": "S(x)={\\frac {P(x)}{R(x)}}\\,\\!", "15cf45b2e38e677f92eec766b59641c8": "\\Psi =\\Phi +b\\;", "15d009971f452f713b4ce9f21694263e": "\\Gamma ({\\tfrac {1}{4}})=(2\\pi )^{3/4}\\prod _{k=1}^{\\infty }\\tanh \\left({\\frac {\\pi k}{2}}\\right)", "15d0227f4335b85f92d67c39ac2346c0": "{\\rm {gcd}}(n,q)=1,", "15d0910d6646a361be7642f8eb24b3a9": "\\mu ={\\frac {mv_{\\perp }^{2}}{2B}}", "15d118905e7b54240a45992e64eaf5af": "\\Lambda \\in \\operatorname {Hom} (L^{\\infty }(G),\\mathbf {R} )\\,", "15d1204794314b34bcc11ae3aae3bdbc": "\\phi ({\\vec {r}},t)=\\phi _{0}({\\vec {r}})e^{i\\omega t}.", "15d12cedba94c857ab59fd9070919861": "z_{1},z_{2},z_{3}", "15d170b7c98f459cbaecc0d663e2fa1e": "1/(NC)", "15d180cdb818fabf450cb5183f8859df": "\\left|{\\frac {x}{a}}\\right|^{r}+\\left|{\\frac {y}{a}}\\right|^{r}\\leq 1", "15d189324d42cce986d3274ac59d4628": "V^{\\mathbb {C} }\\cong V\\oplus iV", "15d1c2bda03172895e59027112c2d5e5": "\\beta =1\\!", "15d1ded340f9aea25761b6325caf4f27": "E[U]=0\\times 1/2+1\\times 1/4+(-1)\\times 1/4=0", "15d23db95ffa01914bf1fa6fc14da8ec": "qq^{*}\\neq 0.", "15d25ba877fd68948884a11c80124302": "\\mu =G(M+m)", "15d29f1fec65b19ceed0e6c5024d06fd": "I_{a}", "15d2cafcaafb889fd0ed095fe22e1857": "C_{p}>0", "15d31fa8dfcae8446857532e9f24ada3": "w(z)=\\left(1-z^{2}\\right)^{\\alpha -{\\frac {1}{2}}}.", "15d34c55bb3a669b1eba7a09b9155036": "\\nu >6\\,", "15d34ca776588afc565a7a294f5d2f64": "G(r)\\propto {\\frac {1}{r^{D-2+\\eta }}},", "15d3c2197ac311e7ffdcaede17dd5728": "\\left[{n \\atop k-1}\\right]", "15d3f77ec905a77ff67bfe4c1bc4a266": "a^{n-1}=38^{220}\\equiv 1{\\pmod {221}}.", "15d44acfccb3e7ebde38cfb234437fee": "D={\\frac {Eh^{3}}{12(1-\\nu )}}.", "15d4f829eada1f04155fdf6c9f57e483": "\\scriptstyle A=\\{x\\in Z^{d},\\eta (x)=1\\}\\subset Z^{d}", "15d5094234d64c66cd150dbbf6b43d67": "T_{lower}={\\frac {Fd_{m}}{2}}\\left({\\frac {\\pi \\mu d_{m}\\sec {\\alpha }-l}{\\pi d_{m}+\\mu l\\sec {\\alpha }}}\\right)={\\frac {Fd_{m}}{2}}\\left({\\frac {\\mu \\sec {\\alpha }-\\tan {\\lambda }}{1+\\mu \\sec {\\alpha }\\tan {\\lambda }}}\\right)", "15d52cf8689750029499a0669b415c94": "H_{0}|n\\rangle =E_{n}|n\\rangle ", "15d54eb537b515e7be0ef1d73f30ecbd": "S(T,V,N)=Nk_{\\rm {B}}\\left[{\\frac {5}{2}}+\\ln \\left({\\frac {n_{\\rm {Q}}}{n}}\\right)\\right]", "15d56205fd6de9645bbdb9734a0a64fb": "\\lim _{n\\to \\infty }{H}_{2n}=\\textstyle \\left({\\frac {1}{2}}\\right)^{\\left({\\frac {1}{3}}\\right)^{\\left({\\frac {1}{4}}\\right)^{\\cdot ^{\\cdot ^{\\left({\\frac {1}{2n}}\\right)}}}}}={2}^{-3^{-4^{\\cdot ^{\\cdot ^{-2n}}}}}", "15d596fbe42e42db84dd109b066e5c37": "{\\frac {\\mbox{Current Assets}}{\\mbox{Current Liabilities}}}", "15d5ac95cdda0dcc3723e8bee0d37015": "\\phi (\\psi )", "15d5b8732f638e752010baaac8090fdf": "\\gamma _{33}=e^{\\psi }\\sim \\xi ^{-(1-s_{1}^{2}-s_{2}^{2})}.\\,", "15d5c12bd2d5c56f0d8b4da4168d6e97": "|C|=A_{q}(n,d).\\,", "15d5c689a52b2ee02abebca3310da138": "{}={\\begin{vmatrix}a^{i}&a^{j}\\\\b^{i}&b^{j}\\end{vmatrix}}", "15d5c6c9d48c5f1887e99d323c2e9bf8": "[x,x]=0", "15d5d14d342c43ff5236d06f987e9539": "H_{m}", "15d64454fa0c07c7d0f79e1c975defb5": "m_{rel}=\\gamma m_{0}\\!", "15d6446efad9c71737042b3dc5b5bb2a": "K^{1}(X)", "15d65a2b0eea0e05a1e334b06c45c6f7": "\\mathbf {X} ={\\begin{bmatrix}X_{11}&X_{12}&\\cdots &X_{1n}\\\\X_{21}&X_{22}&\\cdots &X_{2n}\\\\\\vdots &\\vdots &\\ddots &\\vdots \\\\X_{m1}&X_{m2}&\\cdots &X_{mn}\\end{bmatrix}},\\qquad {\\boldsymbol {\\beta }}={\\begin{bmatrix}\\beta _{1}\\\\\\beta _{2}\\\\\\vdots \\\\\\beta _{n}\\end{bmatrix}},\\qquad \\mathbf {y} ={\\begin{bmatrix}y_{1}\\\\y_{2}\\\\\\vdots \\\\y_{m}\\end{bmatrix}}.", "15d66ec190cc47fe47e714a2d93604b6": "|+\\rangle ={\\frac {1}{\\sqrt {2}}}(|0\\rangle +|1\\rangle )", "15d6bb440f4ed3fe1f5e16fe6077e727": "(A-2I){\\begin{bmatrix}0\\\\0\\\\0\\\\0\\\\1\\end{bmatrix}}={\\begin{bmatrix}-1&0&0&0&0\\\\3&-1&0&0&0\\\\6&3&0&0&0\\\\10&6&3&0&0\\\\15&10&6&3&0\\end{bmatrix}}{\\begin{bmatrix}0\\\\0\\\\0\\\\0\\\\1\\end{bmatrix}}={\\begin{bmatrix}0\\\\0\\\\0\\\\0\\\\0\\end{bmatrix}}", "15d6c26bf4b3159521919775cd0e72bd": "\\langle r,a\\mid r^{3},a^{2},arar\\rangle ", "15d7195739d5145a299530076a2ef2a1": "{\\Delta }_{\\rho }(f):={\\frac {(-1)^{\\left|f\\right|}}{2}}{\\rm {div}}_{\\rho }X_{f}={\\frac {(-1)^{\\left|x^{i}\\right|}}{2\\rho }}\\partial _{i}\\rho \\pi ^{ij}\\partial _{j}f.", "15d728bf354d2df98672f1b31eb40189": "0.{\\overline {5}}{\\overline {4}}", "15d7d36cf87173aa0eb5303f99d4d0cc": "x_{n+1}=f_{\\mu }(x_{n})={\\begin{cases}\\mu x_{n}&\\mathrm {for} ~~x_{n}<{\\frac {1}{2}}\\\\\\\\\\mu (1-x_{n})&\\mathrm {for} ~~{\\frac {1}{2}}\\leq x_{n}\\end{cases}}", "15d7da66d761cf7cd60e43d77c270cc4": "P_{n}(z)", "15d835c33075057ecf09c1184db0a2e3": "{\\textbf {R}}_{K}", "15d8420cf5d547366c0636eed59786d1": "h(X)", "15d8ce27fc7984fac063adc6cbf70b2d": "{\\mathbf {g}}=-G\\sum _{i}{\\frac {M_{i}({\\mathbf {r}}-{\\mathbf {r_{i}}})}{|{\\mathbf {r}}-{\\mathbf {r}}_{i}|^{3}}},", "15d92c476babe876329314876ecf17a8": "PW_{x}(t,f)=\\int _{-\\infty }^{\\infty }w(\\tau /2)w^{*}(-\\tau /2)x(t+\\tau /2)x^{*}(t-\\tau /2)e^{-j2\\pi \\tau \\,f}\\,d\\tau ", "15d93034172d981b98cc8244c3f840b6": "R^{i+1}\\subseteq G", "15d946399729d25977616ee774c81943": "{\\mathcal {L}}\\{f(t)\\}", "15d98038949e111deebe7ff282ee8777": "{\\frac {dR}{dt}}=\\nu I-\\mu R", "15d9bf81da4b5b53c6609c4c8ceb2cdf": "{\\mathcal {A}}=E/\\sigma :", "15d9d8f4f078956d8f9f9d5c8e5539ed": "g(p)=[h_{1}(p),...,h_{k}(p)]", "15da0d15d15ef322370f70871d711615": "a=2\\mu {\\sqrt {\\rho }}", "15da1162a8aa5570ca97019af3768520": "b\\in \\mathbb {Z} ", "15da3864b7ae54434184efc742ca5070": "t0)\\end{array}}", "15e92f6d4a9337d74273af2acf6d83ce": "e={\\sqrt {g(2-g)}}.", "15e955e6c0573aa3ecddefe4f88966f0": "n(n-1)!", "15e97c30e5d70fa1ac640b7ebc6492d7": "i\\hbar {\\frac {\\partial }{\\partial t}}\\Psi ={\\hat {H}}\\Psi ", "15e9c56e6997f3e3ebe05932377de6fc": "\\Gamma (n/24)", "15e9fbcf229f7869d2f5242fc212d595": "{\\rm {MG}}_{p}(\\alpha ,\\beta ,{\\boldsymbol {\\Sigma }})", "15e9fc901189caf056597836e61ffc39": "E_{\\mathrm {p} }=qU\\,", "15ea0d423ee2c460639733a1db3ae844": "\\tan \\psi =\\left({\\frac {f}{v'\\cos \\theta -f}}+1\\right)\\tan \\theta ={\\frac {f+v'\\cos \\theta -f}{v'\\cos \\theta -f}}\\tan \\theta \\,,", "15eacdf189942184cd35d98a8584d7a7": "{\\begin{pmatrix}c&0\\\\0&c\\end{pmatrix}}", "15eb7e2dd76647080b66c72611df326a": "{\\begin{aligned}\\left\\langle \\theta ^{G},\\psi ^{G}\\right\\rangle &=\\left\\langle \\left(\\theta ^{G}\\right)_{K},\\psi \\right\\rangle \\\\&=\\sum _{t\\in T}\\left\\langle \\left([\\theta ^{t}]_{t^{-1}Ht\\cap K}\\right)^{K},\\psi \\right\\rangle \\\\&=\\sum _{t\\in T}\\left\\langle \\left(\\theta ^{t}\\right)_{t^{-1}Ht\\cap K},\\psi _{t^{-1}Ht\\cap K}\\right\\rangle ,\\end{aligned}}", "15ebb45bc4af78c49017a7881eb07d6d": "dz(t)", "15ec0a0315d245215dab2d96b3ee7fda": "\\phi _{2}\\ ", "15ec6bb05ebf0f9d3856768ea8dd0281": "\\pi _{0}:={\\text{Pr}}[P=1,Q=0]=\\sum _{\\omega \\in S_{0}}\\Psi _{\\omega }^{2}.", "15ec79f3ad4f6cbd00c7a78efb5a6e0d": "T^{2}(\\Omega )", "15eca165b219f76d826fe5b4dadb789b": "p", "16035954482dd051630c8c1fcc2c6223": "\\left({\\frac {-1}{\\sqrt {10}}},\\ {\\frac {-1}{\\sqrt {6}}},\\ {\\frac {-4}{\\sqrt {3}}},\\ 0\\right)", "16035c7ec2bd5cd3dbf8d30735874440": "{\\overline {\\mu }}_{j}^{(1)}={\\overline {\\mu }}_{j}^{(2)}", "16037c53e30170ba29e8298dbb695ccd": "X_{1}X_{2}=X_{2}X_{1}", "1603e1b87bad791f5528a1dbd779cb00": "\\mathbf {QP} =\\mathbf {Q} .", "1603f099b1c079b9b99420b9ef432adf": "\\mathbf {x} '\\mathbf {A} \\mathbf {x} =\\sum _{i,j}a_{ij}x_{i}x_{j}", "1604060a637eca70aa8e0de1da9f1434": "\\theta _{2}\\,\\!", "16043eb02289b77bc5fce774490eea50": "{\\begin{bmatrix}a&b\\\\b&d\\end{bmatrix}}", "1604e99e69e844dbb0a3206c7bfe2c94": "\\tau =RC={\\frac {1}{2\\pi f_{H}}}", "1604f4b7b58ce3600b6a9e057acdee1d": "A=UDV^{H}", "16059dbdae38a8d786633b935bb732d3": "M_{\\eta }", "1605ebdd8ada06e88406aab800879f56": "M_{j}(i)=1", "1606168a880f9ceb6218263927db5e96": "x\\cdot y\\leq x^{2}+y^{2}/4", "16066b820736bfc0a2702a4c96742db6": "\\scriptstyle {\\sqrt {b^{2}-4ac}}", "1606742f064c2c2048309ae60b3029a8": "|a(\\tau ,\\zeta )|^{2}", "1606b0a8f39b9678547ff92c5ef862cb": "|r_{1}-r_{2}|.", "1606bf745931de219e701bbd4053b37c": "(\\phi _{1},\\dots ,\\phi _{n})", "160707b27ae191ee560ff49b9395a8ec": "\\ y[n]=\\sum _{k=-\\infty }^{\\infty }h[n-k]x[k].", "16076ef56e279ae6139dfa0f3638975c": "L=\\sum _{i=1}^{n}f_{i}(x){\\frac {d^{i}}{dx^{i}}}\\,;", "1607c9d968bb05dd129cea29a24316e8": "\\omega '={\\frac {\\omega r}{2(R+r)}}", "16083831472ba45fc050280b08e07960": "d_{n}(x,y)=\\max\\{d(f^{i}(x),f^{i}(y)):0\\leq i0),", "161baba6c4c3c626ea666ebfc4105591": "\\quad P(D)u(x)={\\frac {1}{(2\\pi )^{n}}}\\int _{\\mathbb {R} ^{n}}\\int _{\\mathbb {R} ^{n}}e^{i(x-y)\\xi }P(\\xi )u(y)\\,dy\\,d\\xi ", "161bec377d77e25b12d52e69a701829f": "{\\frac {N}{N_{0}}}=e^{-{\\frac {\\Delta E^{\\ddagger }}{kT}}}", "161c51f3d771e4dd423e1ad83114dd24": "\\theta _{1}=90^{\\circ }", "161c6f1c6482c27c11b7e309de6ee14c": "360^{\\circ }\\iff 2\\pi r", "161c72a30bb8e124336571042fec1689": "\\nabla (\\psi +\\phi )=\\nabla \\psi +\\nabla \\phi ", "161d178a63c6ba1197b179f70d1ce602": "J_{\\lambda }", "161d99d74cb4c40acb4155d162d8a0ec": "\\operatorname {U} (n_{1})\\times \\cdots \\times \\operatorname {U} (n_{k}).", "161de52026bee27c27aed967b1f880e7": "{\\bar {g}}", "161e060938bb662de6074c9db3634ba7": "{dy \\over dx}={Y \\over X}", "161e699f709e1ccc5bce07a2a418af13": "\\gamma _{\\mathbf {u} }={\\frac {1}{\\sqrt {1-{\\frac {u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}{c^{2}}}}}}", "161e6b2d1dde57ec6896ce4e858e9fa5": "\\langle s,t\\mid s^{2},t^{3},(st)^{5}\\rangle \\,\\!", "161ea786146feb57cd6abc541686c72e": "f^{-1}(V_{i})=\\bigcup _{j}U_{ij}", "161ee5ee127ecf8e27ffde1b3e26f79f": "T_{\\rm {E}}", "161f4f6af4de76b1aff051e67e3fd1bc": "Q_{n}c_{n}", "161f6f5c4f668245b0918698ae63fbe0": "Fdr(z)", "161fa7b85fdcca5ebd788698081ad8f7": "SUE={\\frac {EPS-Forecast}{\\sigma (EPS-Forecast)}}", "161fddc79b30ddbd8d4d76c1a91b1d63": "n_{e}", "16203be20bacd599b82fea4ec72b194b": "(z,z_{2};z_{3},z)=(z_{1},z;z,z_{4})=\\infty .", "16204ccb11f9d08f3dea8134dfa0005f": "\\Gamma _{tot}=\\Gamma _{rad}+\\Gamma _{nrad}", "16207669b2295a2763f185ae8262a882": "\\,e_{x}", "1620e76d5789119b7f4db797d2c16290": "{\\bar {f}}(x)f(x).", "1620edf3d0bbd27cf2a67e18f5ef26aa": "X_{\\alpha }", "1620fd53d6f66d0fabf51c67d9f74dbf": "(x,y,z)\\mapsto (-x,-y,-z)", "162120f2b19e12af8a16431dd7608fd2": "{\\hat {\\phi }}=Y_{D}\\circ F", "162153d1df4ffa80997fc4f2dd21d36e": "y=\\cos(6\\pi nt)\\sin(2\\pi nt)-\\cos(6\\pi t)\\sin(2\\pi t),0\\leq t\\leq 1", "1621ffb153e69579df73c4e2e4e16ffa": "p_{1}\\,\\!", "162214b565c6fce6fd0a1a966d1e86e6": "\\mathrm {^{244}_{\\ 96}Cm\\ {\\xrightarrow[{}]{(\\alpha ,n)}}\\ _{\\ 98}^{247}Cf\\ {\\xrightarrow[{3.11\\ h}]{\\epsilon }}\\ _{\\ 97}^{247}Bk} ", "1622513279631d306a106afca0a9fb5c": "{\\frac {1}{2}}\\ln {\\frac {1+\\alpha v}{1-\\alpha v}}=\\mathrm {arctanh} (\\alpha v)", "162271f2afbae19d3d56fc1033ab6676": "\\log(a_{\\rm {T}})={\\frac {E_{a}}{R}}\\left({\\frac {1}{T}}-{\\frac {1}{T_{0}}}\\right)", "1622a2eb142201bae06cda9da5993d10": "{\\begin{aligned}&\\left(-{\\frac {d^{2}}{dx^{2}}}+A\\right)\\psi _{1}^{0}=0,\\qquad \\psi _{1}^{0}(0)=0\\quad ,\\qquad {\\frac {d\\psi _{1}^{0}}{dx}}(0)=1,\\\\&-{\\frac {d^{2}}{dx^{2}}}\\psi _{2}^{0}=0,\\qquad \\psi _{2}^{0}(0)=0,\\qquad {\\frac {d\\psi _{2}^{0}}{dx}}(0)=1,\\end{aligned}}", "1622df73b9c5280b05661e8f980bf523": "G_{S_{N}}(z)=G_{N}(G_{X}(z)).", "16231fc31b0ce649586bc7dcdaaebe50": "f_{\\omega +1}(3)-2", "16234783ba2ad8a92d11da2927d33519": "[Z_{i}]={\\begin{bmatrix}\\cos \\theta _{i}&-\\sin \\theta _{i}&0&0\\\\\\sin \\theta _{i}&\\cos \\theta _{i}&0&0\\\\0&0&1&d_{i}\\\\0&0&0&1\\end{bmatrix}},", "16238fc72724d93eefd77414bc68311f": "\\|x\\|^{2}=\\sum _{k\\in B}|\\langle x,e_{k}\\rangle |^{2}.", "162391ddafb66ea6a3b803c04b56b9ff": "||(x,y)||={\\sqrt {x^{2}+y^{2}}},\\,", "1623adeb554952f0ad58fdc5f37b1d41": "\\ln P+n\\ln v=C", "1623b32e64ebd98b825e7b00cdfa1f17": "A_{t,t+1}=A_{t}\\cap L^{p}({\\mathcal {F}}_{t+1})", "1623d62ce069246a0589ab4354c3e7df": "A\\otimes B", "1623f245369f75fbfdbe221248606faf": "B=A+(360/\\pi )\\times 0.0167\\times \\sin(W\\times (D-2))", "1624827224698be508cdef05fa773167": "z\\in D_{R}", "1624aae913f625fb636449be53712b76": "d>4", "1624cf5668bfe6f97ee7ac5dd28caaeb": "d\\Gamma _{0}", "1624dda74d784601344a062ad4043e13": "X=2\\left[{\\begin{matrix}\\xi _{1}\\\\\\xi _{2}\\end{matrix}}\\right]\\left[{\\begin{matrix}-\\xi _{2}&\\xi _{1}\\end{matrix}}\\right].", "162525bf03a300e1303483a4c383f672": "x:S\\to X", "16252d1e952b8d5dbcc652cfa3e1483a": "{\\frac {1}{2}}\\cdot \\delta ", "16252fc0c6a3eb60a27f563e1b7c45f7": "a\\triangleright (b\\triangleright c)=(a\\ \\triangleright b)\\triangleright (a\\ \\triangleright c)", "16253c47c7c3ee7321643f2f8befcc2c": "{\\frac {1+{\\scriptstyle {\\frac {1}{2}}}z}{1-{\\scriptstyle {\\frac {1}{2}}}z}}", "1625957bf68fcc1e04bdaa499cbabd80": "(\\forall x\\phi )\\rightarrow \\psi ", "1625960b01ba1b6aa2b2dd7957139dd7": "\\operatorname {W} (A)(Ax+ix)=Ax-ix\\quad x\\in \\operatorname {dom} (A).", "1626041568028f4a773985cbb8abc73b": "{\\frac {[D_{ad}]}{p_{D_{2}}^{1/2}[S]}}=K_{eq}^{D}", "1626d06ac26e06bdf92327a534afca3e": "\\alpha _{j}(p,q)=P(w_{1(p-1)},N_{pq}^{j},w_{(q+1)m}|G)", "1626f4fa568afb2422c7618b3b319d9c": "ACH_{50}={Q_{50}*60 \\over V_{Building}}\\,\\!", "1627182735a540a231758044b5ba9073": "\\mathbf {x} _{w}^{(k)}={\\underset {\\mathbf {x} _{w}\\in W(\\mathbf {x} _{w}^{(k-1)})}{\\operatorname {argmax} }}\\,\\det(\\mu (\\mathbf {x} _{w},\\sigma _{I}^{k},\\sigma _{D}^{(k)}))-\\alpha \\operatorname {trace} ^{2}(\\mu (\\mathbf {x} _{w},\\sigma _{I}^{k},\\sigma _{D}^{(k)}))", "16276dace2435af3d6698e25188461ab": "g(x)=1", "1627a172c9976b86185ed15f1e74280e": "P_{3}=(4,4{\\sqrt {15}},-8{\\sqrt {3}},16)", "1627e6e02a61a7cbe851b6a5be34cc73": "\\varphi _{\\delta }(\\alpha )", "16280b2575d6aeaf5ab4c103568e641f": "\\theta ={\\pi \\over q}", "16287aa3baeb06275b1c6d54f20027ad": "f(t)={\\frac {\\lambda ^{x}t^{x-1}e^{-\\lambda t}}{(x-1)!}}.", "1628e53f591f0a4ef4d3f846ed33898d": "{\\hat {H}}", "1629501ce2f38bfa07421d63a174bf97": "c=18", "16299b96e4e4d86900f487356828dde9": "M_{y}=\\int _{0}^{2}{\\int _{x}^{4-x}}{}{}x\\,(2x+3y+2)\\,dy\\,dx", "1629bfcf86cacbab43a76db9cd8917fa": "|e_{n}|\\leq {\\frac {\\max _{j}\\tau _{j}}{hL}}\\left(\\mathrm {e} ^{L(t_{n}-t_{0})}-1\\right).", "1629d8150bc0a2e5b4e1cf347cc1707f": "N/{\\sqrt {\\theta }}", "162a6e32f3f45240d9166140d85c14bf": "\\left[{\\frac {q}{p}}\\right]_{3}=1{\\mbox{ if and only if }}{\\begin{cases}q|LM{\\mbox{ or }}\\\\L\\equiv \\pm {\\frac {9r}{2u+1}}M{\\pmod {q}},\\;\\;\\;{\\mbox{ where }}\\\\\\;\\;\\;\\;\\;u\\not \\equiv 0,1,-{\\frac {1}{2}},-{\\frac {1}{3}}{\\pmod {q}}\\;\\;\\;{\\mbox{ and }}\\\\\\;\\;\\;\\;\\;3u+1\\equiv r^{2}(3u-3){\\pmod {q}}\\end{cases}}", "162a78da2f4c128badcaf5d6b8687037": "T_{K}\\,\\!", "162a7a80205f5cac14df3c47e219ba08": "{\\boldsymbol {\\alpha }}+\\sum _{i=1}^{n}\\mathbf {x} _{i}\\!", "162a92b0abe7f4a09cb2760e40c1c3a3": "\\sum _{m,n}\\int d^{d}x_{1}\\cdots d^{d}x_{m}\\,d^{d}y_{1}\\cdots d^{d}y_{n}S_{m+n}(x_{1},\\dots ,x_{m},y_{1},\\dots ,y_{n})f_{m}({\\bar {x}}_{1},\\dots ,{\\bar {x}}_{m})^{*}f_{n}(y_{1},\\dots ,y_{n})\\geq 0", "162acb38e76dc86c88f97b8b0c5464bc": "{\\frac {1}{x-2}}={\\frac {3}{x+2}}-{\\frac {6x}{(x-2)(x+2)}}\\,.", "162afdd94e26c656a21e8cddec79c830": "\\zeta _{i}=dz_{i}", "162b0d976cdeabd7105ebcf9aa9184cd": "b\\leq a", "162b23614f3de15ba9c77440d9b75780": "F_{2}", "162b2d9697b973899bf85b76991cda0c": "\\mathbb {F} _{q^{n}}", "162b505a04f7ea940337b689871f22e0": "{\\hat {\\mathbf {A} }}", "162b6a575da557fe8ed4d3be373390c7": "{\\underline {\\mathbf {e} }}(\\ell )=\\mathbf {F} \\left[\\mathbf {0} _{1xN},e(\\ell N),\\dots ,e(\\ell N-N-1)\\right]^{T}", "162b6ad30dbd1c4133375e92fd5433b5": "\\varphi _{K}(x)={\\chi _{K_{\\delta }}\\ast \\varphi _{\\delta }(x)}=\\int _{\\mathbb {R} ^{n}}\\chi _{K_{\\delta }}(y)\\,\\varphi _{\\delta }(x-y)\\,\\mathrm {d} y,", "162b98f202cb53bd7014b27f4d97a938": "{\\frac {\\partial }{\\partial t}}{\\frac {4(1200t+200)}{t^{2}+4t-2t^{2}}}=4{\\frac {(4t-t^{2})(1200)-(4-2t)(1200t+200)}{(t^{2}+4t-2t^{2})^{2}}},", "162ba20e77c5ec68cd91627573f08ffd": "{\\frac {I}{I_{0}}}={\\frac {1}{2}}\\quad .", "162bd8a46babaeb48e7f6955dd130ca1": "|x+iy|={\\sqrt {x^{2}+y^{2}}}.", "162be0dba6b9a12a356c0133f5d7ba5b": "G(s)={\\frac {1}{s^{2}+s+1}}", "162bf31bc716b1eedc125e11307f7dc7": "\\mathbf {H} \\ \\equiv \\ {\\frac {\\mathbf {B} }{\\mu _{0}}}-\\mathbf {M} ,", "162bf4aecfe4a8271341df1c23e76deb": "P_{k}(x)=1+x+{\\frac {x^{2}}{2!}}+\\cdots +{\\frac {x^{k}}{k!}},\\qquad R_{k}(x)={\\frac {e^{\\xi }}{(k+1)!}}x^{k+1},", "162c565e38457ce882add7bc6486846f": "{\\overline {\\rho }}={\\frac {1}{\\overline {\\sigma }}}", "162c7caa6d566dbf86b5016503631d3a": "(4)\\quad t=v-r-2M\\ln {\\Big (}{\\frac {r}{2M}}-1{\\Big )}\\qquad \\Rightarrow \\quad dt=dv-{\\Big (}1-{\\frac {2M}{r}}{\\Big )}^{-1}dr\\;,", "162cdead5f8571f34875f3ddb11067a1": "g_{p}(Y_{p},aU_{p}+bV_{p})=ag_{p}(Y_{p},U_{p})+bg_{p}(Y_{p},V_{p}).\\,", "162cf91b72aea9a61cc553dfa22cc4d9": "d\\to z", "162d1cef0825ad9dc2eb09ab220a1cba": "\\exp _{10}^{2}(5.84259)", "162d4c413f99ae2763b1ced17ed1a14b": "\\Gamma ", "162d616e67e056a8da8a0ffdcd4f4a91": "T(\\psi (x){\\bar {\\psi }}(y))\\ {\\stackrel {\\mathrm {def} }{=}}\\ \\theta (x^{0}-y^{0})\\psi (x){\\bar {\\psi }}(y)-\\theta (y^{0}-x^{0}){\\bar {\\psi }}(y)\\psi (x).", "162dea7b04a30499ae3eeafd1d424788": "(C)\\int \\,fd\\nu +(C)\\int g\\,d\\nu \\leq (C)\\int (f+g)\\,d\\nu .", "162e1e0ce491b1cdb006392bfaacc15f": "\\rho _{i,T_{0}}", "162e547ee23d422d373ed8b79eb327c1": "\\qquad \\qquad \\mathrm {vibrational} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ E_{f,v,l}=\\hbar \\omega _{f,v}(1+{\\frac {1}{2}})\\ \\ \\mathrm {and} \\ \\ Z_{f,v}\\sum _{j=0}^{\\infty }\\mathrm {exp} [-(l+{\\frac {1}{2}}){\\frac {\\hbar \\omega _{f,v}}{k_{\\mathrm {B} }T}}]={\\frac {\\mathrm {exp} (-T_{f,v}/2T)}{1-\\mathrm {exp} (-T_{f,v}/T)}},", "162ecc8ae7d6b6e88b0df42d01b3ef45": "\\operatorname {Pin} ^{\\pm }(4)\\to \\operatorname {O} (4)", "162eeb5b8cbe5357686bc3d88f203a34": "d\\alpha =\\sin \\phi \\,d\\lambda .", "162efb1eb8f0378c7d837ea8b30286e7": "\\left[1;{\\frac {2t}{t^{2}+1}};{\\frac {t^{2}-1}{t^{2}+1}}\\right].", "162efe6847ed1adcc20122c9f979c1ef": "D_{\\mu }\\phi =(\\partial _{\\mu }\\phi -ieA_{\\mu }\\phi )", "162f2935e6ab84cf7b4d1c8579bfe9ca": "M_{0},M_{1}\\subset N", "162f439ef32e7eb2d173c00579a1527c": "{\\begin{matrix}{11 \\choose 1}{4 \\choose 3}{40 \\choose 2}\\end{matrix}}", "162fbd0fb5cfde94641f4910a529232e": "B_{0}={\\frac {1}{c}}E_{0},", "162fc5abe360a63b7d74988459dbce40": "\\|AT-I\\|_{F},", "162fcede5603c9d7122deaadcad0b4f4": "E_{1}={\\frac {v_{1}^{2}}{2g}}+y_{1}+{\\frac {p_{1}}{\\gamma }}=E_{2}={\\frac {v_{2}^{2}}{2g}}+y_{2}+{\\frac {p_{2}}{\\gamma }}", "162ffde5e606b0343377f5bfb48a82d7": "p(x)\\cdot {d \\over dx}", "1630016f05909cf288d9c8c8c12a22e2": "\\Pi ^{S}+\\Pi ^{I}=1", "16303c2c1bc34fff553826356a07558a": "f_{i}=K_{ij}~d_{j}~.", "1630787bd562dd16135397a0f7e2e84c": "f,g:{\\mathbb {R} }^{n}\\rightarrow {\\mathbb {R} }", "163098679a068ecd801d34f4a150b98d": "\\lim _{n\\to \\infty }{\\frac {\\mathrm {N} (n,S)}{n}}=0.", "1630d5ff7976739ac1a9d3cfd295b54d": "\\mathbf {v} _{k}={\\boldsymbol {\\omega }}\\times \\mathbf {r} _{k}", "163150331dea672eb07db8bde80ed0bb": "{\\begin{pmatrix}p_{n-1}&p_{n}\\\\q_{n-1}&q_{n}\\end{pmatrix}}", "16316bb5c537f4d07cf462dfaf5bfb06": "E=-{\\frac {\\Delta \\phi }{d}}", "16324897458c1d52a5004148bf814b8f": "\\left(f_{1}\\star _{\\inf }\\cdots \\star _{\\inf }f_{m}\\right)^{\\star }=f_{1}^{\\star }+\\cdots +f_{m}^{\\star }.", "16326fbad995488ac0d59a546dd5b5a9": "\\delta ={\\frac {\\rho _{a}p}{k}}", "16327e656d07ae22a9a41e498d631640": "{\\dot {\\gamma }}={\\frac {V}{H}}", "1632ee7fe5d6c343f4479c0968396e31": "\\sigma _{n}", "16333758fad2598cf8c84b7c5df92264": "A+B\\rightleftharpoons |A\\cdots B|^{\\ddagger }\\rightarrow P", "16334ad4eb29342eb8f65930e4d44f2f": "K,N:", "16334eff3966c4d239a05520fc557779": "{\\begin{aligned}P(Presentation~WHOIFPI)=P(Presentation~WHOIFPI~by~condition~1)+\\\\P(Presentation~WHOIFPI~by~condition~2)+\\\\P(Presentation~WHOIFPI~by~condition~3)+etc\\end{aligned}}", "16335d82d31355736892a13801efa0d1": "ku>\\kappa _{z}", "16337592edce67f98b9ad43c98831c29": "X_{w}(a,b)", "163379a708207472f0b48ebd571d58b0": "u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i})>0\\Leftrightarrow \\Phi (a'_{i},a_{-i})-\\Phi (a''_{i},a_{-i})>0", "1633e609f33108f110e1b223a3dd81ee": "{\\begin{aligned}\\mathbf {V} \\mathbf {V} ^{*}&={\\begin{bmatrix}0&0&{\\sqrt {0.2}}&0&-{\\sqrt {0.8}}\\\\1&0&0&0&0\\\\0&1&0&0&0\\\\0&0&0&1&0\\\\0&0&{\\sqrt {0.8}}&0&{\\sqrt {0.2}}\\end{bmatrix}}\\cdot {\\begin{bmatrix}0&1&0&0&0\\\\0&0&1&0&0\\\\{\\sqrt {0.2}}&0&0&0&{\\sqrt {0.8}}\\\\0&0&0&1&0\\\\-{\\sqrt {0.8}}&0&0&0&{\\sqrt {0.2}}\\end{bmatrix}}\\\\&={\\begin{bmatrix}1&0&0&0&0\\\\0&1&0&0&0\\\\0&0&1&0&0\\\\0&0&0&1&0\\\\0&0&0&0&1\\end{bmatrix}}\\equiv \\mathbf {I} _{5}\\end{aligned}}", "163402576a6c172be0e40b729c5c35c0": "{Q_{3}-Q_{1} \\over Q_{3}+Q_{1}}.", "163418a02552d4134c87bde2b66bedb0": "\\alpha ", "163442fc17db463edca4a31791202bcf": "\\vdash B", "1634a736412bd208c0ecf4d7fb7ed6d4": "x=\\pm \\left({\\sqrt[{m}]{a}}\\right)^{n}", "1635515f4a6d6dbe320ce0be8c415af7": "\\csc x=(\\sin x)^{-1}={\\frac {1}{\\sin x}}.", "16355ab7f45478b0519aefe1cc9763c3": "-{\\partial p}/{\\partial s}", "163560912eb173d743579a570e2d6033": "(By+\\beta )^{n}-B^{n}y^{n}", "16357da621cb7d585efb110e0a67808a": "\\Sigma _{3}^{1}", "1635b9b702eb57833b992fbc0e321b97": "y(t)=f(x(t))", "163618078d58100a50990cd8aef08d4b": "h(k)", "16364f65af01370b2b21831899ccbf96": "vblood", "1636a34fb6f146d991358b572651f55d": "e_{j}y_{j}+\\sum _{i=1}^{m}{b_{ij}s_{i}}\\leq d_{j}", "1637153e8e8cd310f1af6b8b9bf085b9": "n_{t}^{j}", "163746cf9c097785b6d2df9d156a8a38": "S^{i}=\\{z_{1},...,\\ z_{i-1},\\ z_{i}',\\ z_{i+1},...,\\ z_{m}\\}", "1637d979a782c3aedb33f29b5d826736": "y_{i}\\left[{w^{T}\\phi (x_{i})+b}\\right]=1-e_{c,i},\\quad i=1,\\ldots ,N.", "1637df3195f37e58a4188ba7d2c17979": "2\\,\\operatorname {Cl} _{2}(\\theta )+2\\int _{\\pi }^{\\pi -\\theta }\\log {\\Bigg |}2\\sin {\\frac {y}{2}}{\\Bigg |}\\,dy=", "163837903b71c6f56e0256b4ff83e00d": "r=f(x)g(y)", "16385244564dfff6bd0a61fc2bf3d7d4": "{\\frac {7}{6}}", "16389d8d2ba844f7e4e27270e357a41a": "\\chi :\\mathbb {Z} /N\\mathbb {Z} \\rightarrow F^{*}\\ ", "1638a8867c7211973ebbeab3f7e28566": "\\omega ={\\frac {\\sigma \\left(k^{2}a^{2}-1\\right)}{2a\\mu _{A}}}{\\frac {1}{k^{2}a^{2}+1-k^{2}a^{2}I_{0}^{2}\\left(ka\\right)/I_{1}^{2}\\left(ka\\right)}}", "1638b77b7d0b364a035923604795f335": "m\\in P", "1638ca3816cd0772e02b3fc1fff85976": "T(R)={\\frac {T(A_{1}),\\dots ,T(A_{n})}{T(B)}}.", "1638e8492e71dee9f64c27cc55719866": "p_{f}>{\\frac {1}{2}}", "1638f2ed0da101523e2880edaef4fc8b": "p(r)\\neq 0", "1639028839d71cccdeaba64bceff9cd4": "\\displaystyle L=\\mu _{0}\\mu _{r}N^{2}A/l.", "163915d96bf1488f866c78c2183e7378": "\\int _{-\\infty }^{\\infty }x\\phi (x)\\Phi (bx)\\,dx=\\int _{-\\infty }^{\\infty }x\\phi (x)\\Phi (bx)^{2}\\,dx={\\frac {b}{\\sqrt {2\\pi (1+b^{2})}}}", "16394c0c93870bc3f6c9298b5f686b93": "r^{\\prime \\prime }=-\\mathbf {k} \\,r\\,\\mathbf {k} .", "16394fca0405a7339ada1becb2298ead": "M=1", "16395f8634039643670729f61a34ab9a": "\\mathbf {P} (X\\leq (1-\\delta )\\mu )\\leq e^{-{\\frac {\\delta ^{2}\\mu }{2}}},\\quad 0<\\delta <1", "1639a20f662391c938749087bebac654": "\\phi (x)=\\sum _{k=-N}^{N}a_{k}\\phi (2x-k).", "163a88682def3da8260f0b7b7977a349": "t\\in [0.7,\\dots ,1.4]", "163a98e1eef1667a79813b694b4cf488": "R\\sim \\mathrm {Rayleigh} (1)", "163af789c6bff5f1a86e66190b323255": "(1-(z-d)^{2}/n)^{n}", "163b3106450e6b8293d55f9ee242b3fc": "\\Box A\\to \\Diamond A", "163b3b34fd9f9e7c51bcb630c6af12d9": "-{\\frac {\\hbar ^{2}}{2m}}\\nabla ^{2}\\psi _{n}(q)=E_{n}\\psi _{n}(q)", "163b7e16ccc676a367d51ddd8354c5a0": "\\sigma =uv=\\prod _{i=1}^{k}p_{i}", "163b97624c5346ea00305d1d176fcc9c": "1,3,7,21,48,112,\\ldots ", "163bdc6c9f31a965fe67eb31f4ef40d3": "(1+z)^{n}=1+z^{n}({\\mbox{mod}}\\;n).", "163be9b6f55ef16b0f1259b7146daacc": "\\forall \\mu ,\\nu ,\\exists \\lambda ", "163bfdbfc2c6ab4f516c0465dc08c8e1": "2^{\\mathrm {intermediate\\ result} }=2^{I+F}=2^{I}\\,2^{F}", "163cde00287e629f33dae509a8414505": "\\phi (x)", "163d42b20ee0642a0e30e9a9bc3a40e9": "SL\\to GL^{+}", "163d513d322c2502434ea06a393e2404": "\\mathbf {S} X:=\\bigcup _{i=0}^{\\infty }\\mathbf {S} _{i}X{\\mbox{.}}\\!", "163dadc589bdbbe1b5e0959baedb8d84": "d\\approx 1.23\\cdot {\\sqrt {h}}", "163dc45aaa57f2225909859887a7b18d": "(V^{*})^{\\mathbb {C} }\\cong (V^{\\mathbb {C} })^{*}.", "163dcd4f1e43a48b97e727112bd591fe": "k\\neq i", "163ddb0e4aa7ad86bb62b4030462e347": "P_{n}(k)", "163dea18c142083900d95b7e1fa82f3f": "({\\bar {4}},1,2)", "163dfce51229b83841c0d51d0fd21651": "\\mu (A)=\\inf\\{\\mu (G)|G\\supseteq A,G{\\mbox{ open and measurable}}\\}", "163e11cbf5ff4357f0bf6207feae7258": "n={\\frac {f}{2-f}},\\quad A={\\frac {a}{1+n}}\\left(1+{\\frac {n^{2}}{4}}+{\\frac {n^{4}}{64}}+\\cdots \\right),", "163e1f321990e6d90185dbd06b0cc11b": "\\partial _{\\omega }(\\omega \\epsilon )", "163e284be69977ee0b7c09456c07f3d1": "l_{2}(\\theta )=(\\vartheta +\\theta )/2", "163e7f8cc14081491672fb252a14a5b5": "i:=i+1;", "163e8ab46fdc03a113690f3b5b860785": "\\operatorname {let} p\\ f\\ x=f\\ (x\\ x)\\land q\\ f=(p\\ f)\\ (p\\ f)\\operatorname {in} q", "163e9ad8bbc63b2b80c6e2b234419c8d": "EL(\\Gamma )\\geq {\\frac {\\log(r_{2}/r_{1})}{2\\pi }}.", "163ea35546c9de550a1851bb7a2cbeaf": "p_{\\mathbf {Y} }(Y)={\\frac {p_{\\mathbf {y} }(\\mathbf {y} )}{|{\\frac {\\partial \\mathbf {Y} }{\\partial \\mathbf {y} }}|}}", "163eb8a639ffa87bb3d5ec3e7a436dd2": "\\delta (x)={\\frac {1}{2\\pi }}\\sum _{n=-\\infty }^{\\infty }e^{inx}", "163f41ddc8645ad40cb50b494ff409f1": "\\sum _{i=0}^{n}\\dim H^{2i}(M,R){\\bmod {2}}", "163f42ac9a96526aa1add67ec35d36b1": "(L_{1}\\times L_{2})\\cdot L_{3}=\\langle L_{1},L_{2},L_{3}\\rangle =0.", "163fbe8cb54f22f59e5cc82b5c729e28": "{\\dot {S}}", "16402685a0e4602f2c951ca8df6ade36": "2\\leq i\\leq m-1", "1640503311e070f7e7bfcf49a037d7b1": "\\Theta _{\\mathrm {div} }\\simeq 2{\\frac {w_{0}}{z_{R}}}.", "1640603d3de2bea8fffaddfe7a143c61": "{\\tfrac {5}{2}}", "1640da851490562e0ca9dae6e8198b64": "(\\mathbf {b_{1}} ,\\mathbf {b_{2}} ,\\mathbf {b_{3}} )", "164126a5c9c38f224248aff32ce6e7ad": "\\varphi \\star \\varphi ^{\\prime }", "164209e30f2287f64347bfdb337dc9fb": "v\\,={\\frac {2\\pi r}{T}}=\\omega r", "1642d7cc623fde3c56a7471079868d7f": "m_{t+1}=m_{t}-{\\overline {\\Delta v}}_{t-16}+1.5\\left(+{\\Delta p}_{D}+{\\overline {\\Delta q}}\\right)-0.5{\\Delta x}_{t-1}\\,,", "1642ee17ffab28d42adcacbed7c0fb37": "\\psi _{m}", "1642fc54cad195a9f9506852715243d1": "n\\in N\\cup \\{+\\infty \\}", "16430576533683a9271051b3ad4624ae": "dU=\\delta Q+\\delta W\\,", "16446094d5c28548a6ec20124c0be07f": "{\\boldsymbol {n}}", "1644b01d18c035656d6e57b206074fd7": "x*g", "16450bd602506f0b37d566de5b28afee": "\\forall P\\in {\\text{vbl}}(A)", "1645103e03da3acaba25816554710a35": "{\\frac {1}{\\sqrt {1-\\beta ^{2}}}}", "164515215ac18c3392e1c77f6bb7a837": "E_{n}^{(3)}=\\sum _{k\\neq n}\\sum _{m\\neq n}{\\frac {\\langle n^{(0)}|V|m^{(0)}\\rangle \\langle m^{(0)}|V|k^{(0)}\\rangle \\langle k^{(0)}|V|n^{(0)}\\rangle }{\\left(E_{m}^{(0)}-E_{n}^{(0)}\\right)\\left(E_{k}^{(0)}-E_{n}^{(0)}\\right)}}-\\langle n^{(0)}|V|n^{(0)}\\rangle \\sum _{m\\neq n}{\\frac {|\\langle n^{(0)}|V|m^{(0)}\\rangle |^{2}}{\\left(E_{m}^{(0)}-E_{n}^{(0)}\\right)^{2}}}.", "1645436a004690d079de9d4b1098a409": "=e^{-i2\\pi fT}\\mathrm {sinc} ^{2}(fT)\\ ", "1645ca15abc2c95197ff2f2ca8d6eed7": "=\\int _{P(t_{1},t_{2})}{\\dot {Q}}(t)dt", "16460f0d757edd0a3f7f700687a971ee": "{\\mathcal {L}}_{t}\\left\\{f(t)\\right\\}(s)=F(s)", "16462602ca908d0a89a03d3a156a3d7d": "\\!A", "16463c38031cc857d20f8fb7f4d39bbe": "\\textstyle b_{2}=p(t_{x},a_{(-1,2)},a_{(0,2)},a_{(1,2)},a_{(2,2)})", "16467399d678c1f3c1ebf9748719e110": "{\\frac {\\partial {{T}_{{}^{1}\\!\\!\\diagup \\!\\!{}_{2}\\;}}}{\\partial P}}", "164678f5a92460bc67ae48cd8cc734df": "\\mathbf {p} =(p_{1},p_{2},...,p_{s}),", "16467b631d4b7e72d7968123019c1f52": "z^{\\mu }=x^{\\mu }+iy^{\\mu }", "164685fed16995ed1d1cb60398eddfbd": "{\\ddot {Z}}/Z", "1646e7c616f381772efcb605cba6c83d": "{\\vec {Q}}", "16474519c7480ba37ef273b9fbcc6ab5": "{\\frac {1}{E(\\phi )}}", "16477fd3011ff1bbcdacfcd551410fca": "\\eta =1-\\left({\\frac {{\\mathit {T}}_{1}}{{\\mathit {T}}_{2}}}\\right)", "16479c46a8bec4b2a6377f42a58b4507": "\\omega _{B}", "1647dd85b1d4d58cae1e001d1dec34b4": "{\\frac {\\mathrm {d} \\phi }{\\mathrm {d} t}}\\,", "16484169ec44a6340333ce901d9be6f8": "{\\frac {3}{4}}\\,{\\sqrt {\\frac {\\pi }{2}}}", "16485e8f988cee3cb1d86eeefc0d2c4b": "x(t)\\approx \\int _{-\\infty }^{t}dt'\\,\\chi (t-t')h(t')\\,.", "164935bf3f110ca673a798339531a5e0": "c_{k+1}", "1649394019327a77e24d2a4f059541ee": "\\gamma =\\lim _{n\\to \\infty }\\left\\{{\\frac {\\Gamma ({\\frac {1}{n}})\\Gamma (n+1)\\,n^{1+1/n}}{\\Gamma (2+n+{\\frac {1}{n}})}}-{\\frac {n^{2}}{n+1}}\\right\\}", "164939437939651e6be0b758f8098a69": "z_{i,j}={\\sqrt {(y_{i,j}-y_{i+1,j+1})^{2}+(y_{i+1,j}-y_{i,j+1})^{2}}}", "16495c5742548582fff3ab84d15b3aec": "\\sigma ^{2}=g(r)\\,r\\,d\\theta ", "16496926f12dd2010b5744bbd2a69d1f": "\\sum ({\\overline {X}}-X_{i})=0,", "16497c5a9f8ee922bec9f237aed3ce26": "f({\\textbf {x}}_{1})\\leq f({\\textbf {x}}_{2})\\leq \\cdots \\leq f({\\textbf {x}}_{n+1})", "164986b728adb47ad55fa7ca3ac2a753": "y\\;=\\;y_{0}\\,\\sin(kx-\\omega t)\\;+\\;y_{0}\\,\\sin(kx+\\omega t).\\,", "1649e712c8b0ee50a07ffdac17318102": "\\displaystyle {gJg^{t}=J,}", "164a4d74285de53017b4fede37b869c9": "\\ A-{\\text{vertex}}=-1:1:1", "164a6b7b7e304335bcb246b02ca99655": "\\{k\\}_{k=1}^{M}", "164aab095469b363240d39c73f7459da": "{\\frac {\\partial u}{\\partial y}}", "164ae789a1915c8b331ea8f8293b3498": "\\Lambda (\\alpha ^{i})", "164b3923779554606f2f414e4f1db29a": "h(k_{3})", "164bb2c5e87cb4246ba5b43dfbbac5cb": "f(n)=O(1)\\,", "164bdb06b8b59a8e9e74a494488ebfd8": "(1+2/e)\\approx 1.736", "164c33008d5460f9bbe21283058f42cf": "y_{i}=g(x_{i})+\\sigma \\varepsilon _{i}", "164c5a364542be3f2784eed0ef2c2f26": "{\\begin{aligned}{\\frac {\\partial {\\mathcal {L}}}{\\partial f_{1}}}-\\sum _{i=1}^{n}{\\frac {\\mathrm {d} }{\\mathrm {d} x_{i}}}{\\frac {\\partial {\\mathcal {L}}}{\\partial f_{1,i}}}&=0\\\\{\\frac {\\partial {\\mathcal {L}}}{\\partial f_{2}}}-\\sum _{i=1}^{n}{\\frac {\\mathrm {d} }{\\mathrm {d} x_{i}}}{\\frac {\\partial {\\mathcal {L}}}{\\partial f_{2,i}}}&=0\\\\\\vdots \\qquad \\vdots \\qquad &\\quad \\vdots \\\\{\\frac {\\partial {\\mathcal {L}}}{\\partial f_{j}}}-\\sum _{i=1}^{n}{\\frac {\\mathrm {d} }{\\mathrm {d} x_{i}}}{\\frac {\\partial {\\mathcal {L}}}{\\partial f_{j,i}}}&=0.\\end{aligned}}", "164ced55a7307af6ea4ce518ea4cebe5": "I[0]=0", "164d023a50096f27e480982d74126e86": "J_{0}^{k}(x^{i}\\circ f)=J_{0}^{k}(x^{i}\\circ g)", "164d2d8646ce09e5fc2f8a1ac90806c6": "x(t)=e^{at}\\left(\\cos bt-{\\frac {a}{b}}\\sin bt\\right)", "164d3ae4d5a48f402683064c87c34c81": "\\chi _{+}^{a}", "164d5a0a2f035a9a6d50b91f2bdda4c7": "f_{\\epsilon _{0}}\\,\\!", "164d6fdff9a80685d868f5b638f19cf1": "xN_{2}z", "164d9e2d995ed1383b38a3744490eda4": "\\displaystyle {W=(Z-iI)(Z+iI)^{-1}.}", "164daf5448d54bbcefbde0c8defe3691": "(P^{2})", "164dc5156fc56c268db408b4a370921e": "\\nu =(4,3,2)", "164df731e691a1830185268ae22c0f4b": "S(\\mathbf {x} _{\\text{i}},\\,\\omega _{\\text{i}},\\,\\mathbf {x} _{\\text{r}},\\,\\omega _{\\text{r}})", "164e032dee785a1a7f0d4c368c062564": "=\\lambda \\mathbf {r} _{dx}(n-1)+d(n)\\mathbf {x} (n)", "164e31839cda9817457d5bc0e2d68c7d": "\\scriptstyle {\\mathcal {N}}=\\{(0,0),(0,1),(1,0),(0,-1),(-1,0)\\}", "164e4d24adf147d550c64683e6c7e780": "F(v_{1}\\otimes \\cdots \\otimes v_{n})=f(v_{1},\\ldots ,v_{n}).", "164ebdc38094787328f6096fc110974d": "R_{\\mathrm {load} }", "164fa3034e56d79832412700b1e82ef4": "f_{t}(z)", "164fca470391d1420d94c70c3f30b1ae": "K\\subseteq E", "164fe527970fd1e2c6fdb77c0af38250": "\\langle S\\mid R\\rangle .\\,\\!", "16504c6dfbbdfbc46f9b4e5253cf112f": "\\operatorname {Tr} (Q\\rho )\\geq \\epsilon ", "1650a49e25841a98079bfd84ae2d5825": "D^{k}", "1650c60a66df0651e8a0e7d9106a6188": "w[T]y", "1650ef9cdf8ca25428de17cce9de60ee": "\\partial _{k}", "165131f15d6a3d7c81b6de6c863919cf": "\\left({\\frac {\\pi }{a}}\\right)^{2}+\\left({\\frac {\\pi }{b}}\\right)^{2}+\\left({\\frac {\\pi }{c}}\\right)^{2}", "16516fa91b6fefc568800d4afe70994d": "\\langle \\mu ,\\xi \\rangle ", "1651bc5814e151fa96904c35b425e4f0": "\\Theta _{r}={\\frac {c_{p}(T-T_{e})}{U_{e}^{2}/2}}", "1651c8d0d18ccb5537744513e1d21694": "B=f(P,E)", "1651f955ab36f5cc34d4c8ea42f785a4": "F_{m}^{i}(x)=\\left\\langle 0|{\\mathcal {T}}\\phi _{i}(x_{1})\\phi _{i}(x_{2})|0\\right\\rangle =\\sum _{\\mathrm {pairs} }{\\overline {\\phi (x_{1})\\phi (x_{2})}}\\cdots {\\overline {\\phi (x_{m-1})\\phi (x_{m}}})", "1652e05f528a1bfaab2dff61d8e5fdf9": "M,a\\models \\phi \\to \\psi \\iff \\forall b,c((Rabc\\land M,b\\models \\phi )\\Rightarrow M,c\\models \\psi )", "1652e4db0fa082db1030997d869ac5b8": "D={\\begin{bmatrix}d_{1}&0&0&\\ldots &0\\\\0&d_{2}&0&\\ldots &0\\\\\\vdots &&&d_{s}&0&\\\\0&0&0&\\ldots &0\\\\\\vdots &\\vdots &\\vdots &\\vdots &\\vdots \\end{bmatrix}}_{m\\times n}", "165342509514b58fe5c69e1b3dcf6ffb": "\\eta ={\\frac {P_{\\mathrm {out} }}{P_{\\mathrm {in} }}}", "165396c52a4595b2abe2257f713eb3a4": "\\mathbf {e} _{1},\\dots ,\\mathbf {e} _{n}", "165402a54d9bd5635bc19abf245ea205": "\\Delta H=\\Delta U+\\Delta (PV)\\,.", "165418bce527a40ce68df045e2fc7d77": "{\\frac {d^{2}x(t)}{dt^{2}}}+{\\bar {c}}{\\frac {dx(t)}{dt}}+{\\bar {k}}x(t)=0", "16543ff8b3e39de69197ad3589ef8f84": "0", "166ef4089e5732c873c2526d33e7710c": "\\ell =1,\\quad m=-1,0,+1", "166f06881d2443484d668f428e0edbc7": "r_{C}/r_{A}", "166f4c38702ff86aca210c5e5f5dcc86": "H=c{\\sqrt {D}}", "166f9e86385fd3afd95705014199864c": "\\varphi (\\theta )={\\frac {1}{2}}\\sin(\\theta ).", "166fef34cd9044eae64854011bcb636f": "\\left(s_{1},s_{2}\\right)", "167042c9cc7ab2432700baea81ab4551": "NRx_{i}=2\\rho {\\sqrt {\\frac {D_{i}}{\\pi t}}}", "16705a640ed247a8ac9ccb8b768339fb": "M={\\frac {N\\mu _{B}gs}{V}}", "16708713ee7cafa4138ccb10b0648703": "(\\mu _{(1)}(t),\\dots ,\\mu _{(n)}(t)),", "1670dc45bd7bbc920f5f9f31b4f234df": "x_{i}\\succ x_{j}\\,\\!", "167113cb54f30e1b2d8cd0ca9f4428e1": "h(t)=\\left({\\frac {\\alpha }{\\pi }}\\right)^{\\frac {1}{4}}e^{\\left(-{\\frac {\\alpha }{2}}t^{2}\\right)}", "16713f4c4e8601fb977cc0d9b6b5237e": "{\\mathbf {E}}[H]=\\langle H\\rangle =-{\\frac {\\partial \\log(Z(\\beta ))}{\\partial \\beta }}", "1671d9a186e4d59d6bb343658aed59a0": "\\mathbf {F} (x,y,z)=-x^{2}{\\boldsymbol {\\hat {y}}}.", "1672043fe46b2fec6f20afaade81374f": "Y_{n}={X_{1}+X_{2}+\\cdots +X_{n} \\over n}", "167251a335db6215942049a68dbd32a6": "I_{1}={\\begin{bmatrix}1\\end{bmatrix}},\\ I_{2}={\\begin{bmatrix}1&0\\\\0&1\\end{bmatrix}},\\ \\cdots ,\\ I_{n}={\\begin{bmatrix}1&0&\\cdots &0\\\\0&1&\\cdots &0\\\\\\vdots &\\vdots &\\ddots &\\vdots \\\\0&0&\\cdots &1\\end{bmatrix}}", "1672a332ce03149c177157209d513c22": "\\displaystyle {B(z_{1},z_{2})=(x_{1},y_{2})(x_{2},y_{1})^{-1},}", "1672a615ce788a56391fdc6829c01643": "\\sum _{i=1}^{m}{\\frac {1}{p_{i}}}=1", "1672e4681257f3088399ae0e60791814": "\\scriptstyle u(x,0)=e^{-x^{2}+ik_{0}x}", "16730d28aa7355aa4d176c4f742a70de": "n\\equiv 2k{\\pmod {4k}}.", "16732763560d16f8bb7a9516b4c52008": "\\int {d^{3}k \\over \\left(2\\pi \\right)^{3}}\\left(\\mathbf {\\hat {k}} \\cdot \\mathbf {\\hat {r}} \\right)^{2}{\\exp \\left(i\\mathbf {k} \\cdot \\mathbf {r} \\right) \\over k^{2}+m^{2}}={e^{-mr} \\over 4\\pi r}\\left\\{1+{2 \\over mr}-{2 \\over \\left(mr\\right)^{2}}\\left(e^{mr}-1\\right)\\right\\}", "16740d7be57aa330dfaeaf568c1ea000": "\\prod _{i=1}^{n}x_{i}^{w_{i}\\cdot q}\\leq \\sum _{i=1}^{n}w_{i}x_{i}^{q}", "1674704f332635dbe7cd31559d9f6882": "V_{out}=V_{dd}", "16747abb0f9c30cb7b7b535375ab7c42": "[H^{+}]_{i^{}}", "1674bd0aeb5c1975dceb4da4afb998b0": "A^{ik}{}_{;\\ell }=A^{ik}{}_{,\\ell }+A^{mk}\\Gamma ^{i}{}_{m\\ell }+A^{im}\\Gamma ^{k}{}_{m\\ell }.\\ ", "1674bf5405d7ec0c7ff38416f63514c5": "\\omega _{\\alpha KI}={1 \\over 2}e_{\\alpha }^{J}(\\Omega _{JKI}+\\Omega _{IJK}-\\Omega _{KIJ})", "1674e892010094e397eb445ebd8a276c": "T(A,D)=\\,{\\underset {x\\sim D}{\\operatorname {E} }}[T(A,X)].\\,", "1676287dc26461863a4a5b591267b6c9": "X\\rightarrow Y,Y\\rightarrow Z\\vdash X\\rightarrow Z", "16762b9416c61b0c19f78a50060a16c1": "\\scriptstyle V_{1}", "1676b26ab76d2b643e1ecaa470fccd69": "\\nabla \\times (\\nabla \\phi )=\\mathbf {0} ", "1676ca11a54cd44804f13eba862bf321": "\\Pi _{k}^{EXP}=\\mathrm {coNEXP} ^{\\Sigma _{k-1}^{P}}", "1676cc5d3594a76d211152c373fc7f30": "\\int _{-a}^{a}(a+x)^{m-1}(a-x)^{n-1}\\ dx=(2a)^{m+n-1}{\\frac {\\Gamma (m)\\Gamma (n)}{\\Gamma (m+n)}}", "1676f4da60fdf0e2e8d8b657ca9b2f44": "[H_{k}]={\\frac {1}{C_{k}^{2n}}}\\sum \\limits _{i+j=k}{C_{i}^{n}C_{j}^{n}w_{i}w_{j}[H_{ij}^{\\ast }]},", "1677313b2ca799c5023f98104417c7e9": "\\psi (x)=\\left(1-{\\frac {\\lambda \\mu }{c}}\\right)\\sum _{n=0}^{\\infty }\\left({\\frac {\\lambda \\mu }{c}}\\right)^{n}(1-F_{l}^{\\ast n}(x))", "167738efe7dcee510cd9530f7becb675": "2\\cos({\\frac {2k\\pi }{11}})", "16773fb299ef418cb45caa2a2e1769a8": "E_{CMI}(\\varrho _{A,B})=S(\\varrho _{A})=S(\\varrho _{B})", "1677631ccc17836e8c579eb52f58716e": "Z[J]=\\int {\\mathcal {D}}\\phi e^{i\\int d^{4}x\\left({1 \\over 2}\\partial ^{\\mu }\\phi \\partial _{\\mu }\\phi -{m^{2} \\over 2}\\phi ^{2}-{\\lambda \\over 4!}\\phi ^{4}+J\\phi \\right)}=Z[0]\\sum _{n=0}^{\\infty }{\\frac {1}{n!}}\\langle \\Omega |{\\mathcal {T}}\\{{\\phi }(x_{1})\\cdots {\\phi }(x_{n})\\}|\\Omega \\rangle .", "1677a73721b8a0e1c3e271a16d789274": "i=1,\\dots ,k", "1677cc1936c7182653edf7b7bd6e0ece": "Error=\\left({\\frac {(RealDelta)+i(ImaginaryDelta)}{(RealSum)+i(ImaginarySum)}}\\right)\\div \\left({\\frac {(RealDeltaCal)+i(ImaginaryDeltaCal)}{(RealSumCal)+i(ImaginarySumCal)}}\\right)", "1677e732300edef0ec6a6198b3f80d1e": "-1.3817", "1677fb4aa8e4bc73c50c9c8947d84b54": "z_{\\epsilon }={\\frac {1}{\\epsilon \\epsilon _{0}}}{\\frac {l}{S}}", "1677fc4fc2e375b1a180a50af7982cf3": "N_{m}", "16781f38736f8651413590942dbb7283": "D_{t}T=\\nabla _{{\\dot {\\gamma }}(t)}T.", "1678825430c97f4459e90b6b596867a1": "\\,(1-p+pe^{it})^{n}", "1678fb690b605f4aa115d64302c45008": "T_{l}-T_{t}={\\frac {2}{c}}\\left({\\frac {L}{1-{\\frac {v^{2}}{c^{2}}}}}-{\\frac {L}{\\sqrt {1-{\\frac {v^{2}}{c^{2}}}}}}\\right)", "1679091c5a880faf6fb5e6087eb1b2dc": "6", "16797a1d4ca01f5e58b63a1b014e788b": "H_{\\leq \\beta }:=\\{X\\in 2^{\\omega }:X\\ \\mathrm {has\\ effective\\ Hausdorff\\ dimension\\ } \\leq \\beta \\}", "16798347341b6d506739a559b107dc3a": "V_{t}-{\\frac {V_{1}}{2/N^{2}}}=(1-2/N^{2})\\left(V_{t-1}-{\\frac {V_{1}}{2/N^{2}}}\\right)=(1-2/N^{2})^{t-1}\\left(V_{1}-{\\frac {V_{1}}{2/N^{2}}}\\right)", "16799f3c67e13519813f1287274fd6a3": "{\\begin{aligned}F=&{\\cfrac {1}{2}}\\left[\\Sigma _{2}^{2}+\\Sigma _{3}^{2}-\\Sigma _{1}^{2}\\right]~;~~G={\\cfrac {1}{2}}\\left[\\Sigma _{3}^{2}+\\Sigma _{1}^{2}-\\Sigma _{2}^{2}\\right]~;~~H={\\cfrac {1}{2}}\\left[\\Sigma _{1}^{2}+\\Sigma _{2}^{2}-\\Sigma _{3}^{2}\\right]\\\\L=&{\\cfrac {1}{2(\\sigma _{23}^{y})^{2}}}~;~~M={\\cfrac {1}{2(\\sigma _{31}^{y})^{2}}}~;~~N={\\cfrac {1}{2(\\sigma _{12}^{y})^{2}}}\\\\I=&{\\cfrac {\\sigma _{1c}-\\sigma _{1t}}{2\\sigma _{1c}\\sigma _{1t}}}~;~~J={\\cfrac {\\sigma _{2c}-\\sigma _{2t}}{2\\sigma _{2c}\\sigma _{2t}}}~;~~K={\\cfrac {\\sigma _{3c}-\\sigma _{3t}}{2\\sigma _{3c}\\sigma _{3t}}}\\end{aligned}}", "1679f575a1431c0bdf34817ac3134043": "\\gamma ^{n}\\to Gr_{n}", "167a43013459122478d65d0486e072f3": "{\\frac {d\\theta }{ds}}=2a^{2}s", "167a6ca9bf7204de1d67fe059b0bc8c6": "\\|x+y\\|^{2}=\\|x\\|^{2}+\\|y\\|^{2}+2\\Re \\langle x,y\\rangle .", "167aafc414fe18093b849c7973cbd52d": "\\cdot {\\frac {1+j\\omega C_{C}R_{o}/A_{v}}{1+j\\omega (C_{L}+C_{C})(R_{o}//R_{L})}}\\ .", "167adf021ddd27f51b332943f9a3eaba": "\\operatorname {pf} (A^{2m+1})=(-1)^{nm}\\operatorname {pf} (A)^{2m+1}.", "167b2c0f5dfdb9c26488887f385114ca": "\\mathbf {\\bar {C}} _{k}\\sim \\mathbf {C} _{k}\\,\\mathbf {T} ^{-1}", "167b489781f5d6d30157c7da73b38c45": "T=(diag(A-\\lambda _{n}I))^{-1}.", "167baa44d6110b64adc0d945fdab8764": "|U(T)|\\geq |U(S)\\setminus N(T\\setminus S)|\\geq |U(S)|-|N(T\\setminus S)|>0\\,", "167bec0c3d393d628af84e4d84b5d304": "p=1-e^{-m}", "167bec1b4b56516ff646c8d64be75ce6": "a{\\frac {\\partial \\mathbf {u} }{\\partial \\mathbf {x} }}", "167c277f3f6087e57bf32672bfe6aca7": "\\alpha ={\\frac {(^{18}O/^{16}O)_{Liquid}}{(^{18}O/^{16}O)_{Vapor}}}=1.0098", "167c29112335c2b9314a648f7a212251": "t=t_{1}\\,", "167c4d78881b5b5cbfee437ca788aa4d": "W_{4}", "167cc9f062f4d38b1261890524305caa": "mx+a", "167cd4c82c0fa1b3a0035bcf95f8f0fb": "\\scriptstyle \\mathbf {u} ", "167d333d2d3be390b69d13e4b2b73bd1": "S_{u}(1\\pm S_{u})=\\pm 1(1\\pm S_{u})", "167d4868b1b2c0ae2d4de98781a02181": "w(x)=\\prod _{i=1}^{20}(x-i)=(x-1)(x-2)\\ldots (x-20)", "167dd4e7be820e8d78b1cd65f032d973": "\\sigma _{P}^{2}=n{\\frac {1}{n^{2}}}\\sigma _{i}^{2}+n(n-1){\\frac {1}{n}}{\\frac {1}{n}}{\\bar {\\sigma }}_{ij}", "167ddff2137a87505cb96475e7c2dfb6": "r_{t}=\\exp {X_{t}}\\,", "167e29312938b3dbe217b0dc31b21470": "{\\overline {\\Omega }}", "167e904200ca0672c63498b5c1d7a05b": "\\mathbf {a} _{\\mathrm {i} }\\ {\\stackrel {\\mathrm {def} }{=}}\\ \\left({\\frac {d^{2}\\mathbf {r} }{dt^{2}}}\\right)_{\\mathrm {i} }=\\left({\\frac {d\\mathbf {v} }{dt}}\\right)_{\\mathrm {i} }=\\left[\\left({\\frac {d}{dt}}\\right)_{\\mathrm {r} }+{\\boldsymbol {\\Omega }}\\times \\right]\\left[\\left({\\frac {d\\mathbf {r} }{dt}}\\right)_{\\mathrm {r} }+{\\boldsymbol {\\Omega }}\\times \\mathbf {r} \\right]\\ ,", "167ecc84d75e93d135f13984c1cf1e8c": "|\\pi _{0,\\mathrm {c} }|=2\\left(\\gamma _{0}+Bn\\right).", "167ed8514419d2205f0ff2df3702f235": "\\textstyle [255,223,33]", "167ede0a6f34ccc88cbbedf0f4be1160": "\\phi (f^{n}(\\phi ^{-1}(z)))=e^{2\\pi i\\alpha }z", "167f45cd3a6eee87f6681a44aeacb3ea": "R=df/dn(n=0)", "167f50f8b86df0cc6b109a047bd4569d": "C_{TD}={\\frac {\\epsilon _{0}S_{TD}}{a_{TD}}}\\approx 6\\times 10^{-14}\\;\\mathrm {F} =60\\;\\mathrm {fF} ", "167f56648fab33f229d1e190ebe3a3d0": "{\\alpha \\over \\beta }>0", "167f8a50ff76bb9d711ccd2975c771f4": "Z_{i+1}=\\left\\{{\\begin{array}{ll}S_{i+1}&{\\text{ if }}\\displaystyle \\max _{1\\leq j\\leq i}S_{j}<\\lambda \\\\Z_{i}&{\\text{ otherwise}}\\end{array}}\\right.", "167fe137153670790383cbb917cda28b": "\\sum _{k=0}^{N}a_{k}=1", "167fea3190f7f16b048b48c870d3b509": "h\\in S_{A}", "167ff787438d34206f480eeb06fadb80": "\\ln {\\begin{vmatrix}P\\end{vmatrix}}-\\ln {\\begin{vmatrix}K-P\\end{vmatrix}}=kt+C", "16800f198d2dc25278ad33e4f69593bd": "P=(0,0)", "1680148e7e7eba23f2f80fbe1ce05b38": "S_{n}\\to \\operatorname {Aut} (S_{n})\\to \\operatorname {Aut} (A_{n})", "1680192943391da77061c05a12a56c18": "L_{0}a=na", "16802c9cf36be2ec89e817e794ea7789": "V={\\frac {32}{27}}d^{2}\\ h={\\frac {128}{27}}r^{2}\\ h", "1680cff44ed649c99fb0bb03cb462d9e": "{\\vec {E}},{\\vec {B}}", "1680d665d12fc59d7313f1cfd5f4f9e4": "\\,nR=VP/T", "1680e216449d35e28285c3957d58b177": "s_{1}^{1}", "16812b18f6c4a43dcba3c5a35f30b398": "G\\circ F", "16813d8ec487fd81bb2935acf8a55203": "\\int _{0}^{T}|g(t,0,0)|\\,dt\\in L^{2}(\\Omega ,{\\mathcal {F}}_{T},\\mathbb {P} )", "1681a269f207d171e1d6601426869356": "{}_{0}F_{0}(;;z)=e^{z}", "1681b2d3a0b2c8d2426cd46b28e1a7f8": "{\\hat {x}}(k+1)=A{\\hat {x}}(k)+L\\left(y(k)-{\\hat {y}}(k)\\right)-BK{\\hat {x}}(k)", "1681e73ee89e573a24560a097f72b1fc": "{\\frac {d}{dx}}\\,\\operatorname {arcosh} \\,x={\\frac {1}{\\sqrt {x^{2}-1}}}", "16827544fa24ecd3208034a4ade83805": "\\tan \\theta _{2}={\\frac {\\cos \\chi \\cos \\eta \\sin \\lambda +\\sin \\chi \\cos \\lambda -\\cos \\chi \\sin \\eta \\cot(15^{\\circ }\\times t)}{\\sin \\eta \\sin \\lambda +\\cos \\eta \\cot(15^{\\circ }\\times t)}}", "1682a5545014ce25d84150a004231cb2": "x^{\\ast }", "16834cd14b9c34e277ef1a3d4756547e": "V_{2}=a_{12}V_{1}\\,.", "168372b616ea823492ad1dd8eafbf548": "Re_{M}", "1683a9be417c472ab92116ccd1b18b55": "=0.58{\\overline {3}}{\\text{ t/kg}}", "1683cf4bd6069d993a80978b6a729856": "\\int \\limits _{0}^{1}\\!{\\frac {\\ln \\ln {\\frac {1}{x}}}{1+x+x^{2}}}\\,dx=\\int \\limits _{1}^{\\infty }\\!{\\frac {\\ln \\ln {x}}{1+x+x^{2}}}\\,dx={\\frac {\\pi }{\\sqrt {3}}}\\ln {\\biggl \\{}{\\frac {\\Gamma {(2/3)}}{\\Gamma {(1/3)}}}{\\sqrt[{3}]{2\\pi }}{\\biggr \\}}", "1683f7fce37ca235a8156e67324774d4": "A^{n}\\rightarrow A\\otimes _{B}A", "16848237c9f6badf633047324f061743": "f(x)=1/x-b", "1684a6ada91ba7ba5e8ee8a5d3ac6353": "\\mathrm {C_{j}^{\\alpha }} ", "1684c9a7919d201d3cda31c30d335c93": "(E_{d/s})=(E_{u/p}-Z)=(6.04-2.0)=4.04ft\\,\\!", "16852819d6fc5efe295ef21bcb61a228": "{\\mathfrak {A}}_{P}:=({\\mathcal {P}}\\setminus {\\overline {P}},\\{z\\setminus \\{{\\overline {P}}\\}\\ |\\ P\\in z\\in {\\mathcal {Z}}\\}\\cup \\{E\\setminus {\\overline {P}}\\ |\\ E\\in {\\mathcal {E}}\\setminus \\{{\\overline {P}}_{+},{\\overline {P}}_{-}\\}\\},\\in )", "16854e1386ed8200a5a8fe7f2ba2a275": "\\left\\{(1,1),(-1,-1),(0,-1),(-1,0),(i,-i),(-i,i),\\left(j,j^{2}\\right),\\left(j^{2},j\\right)\\right\\},\\qquad j=e^{\\frac {2\\pi i}{3}}.", "1685689005f40a5e5acb81579434af26": "b=mn+n^{2},", "168596dbae3631ca956cc83c054eddae": "u_{m,n}^{(0)}", "1685a7a0109d194b75b3f235a35f3a5d": "F_{N}=({\\dot {m}}_{air}+{\\dot {m}}_{fuel})v_{e}-{\\dot {m}}_{air}v", "1685ba0f9cc9b8ad02cb300309a6baf7": "V(p,T)\\ ", "1685f9dd49c1dbca222d54b1ffeeefa4": "D_{n}(z)", "16868711d2f80fc5c4b56ed9721a2224": "a_{ik}", "1686cf84ce1b43f43c1ae367d8fac28b": "-jx_{C}=-j{\\frac {1}{\\omega C_{M}}}={\\frac {1}{j\\omega C_{M}}}", "1686d357641f40023539db7d6003a397": "\\displaystyle {J_{f}(0)=|a_{1}|^{2}-|a_{-1}|^{2},}", "16879d73e1e2753ff48c3b9447f47c15": "C_{1},D_{1}", "1687c45aeab91edcb6b520726a653881": "m+1R(x_{i}),\\forall x_{i}\\in W(x_{c}){\\big \\}},\\\\R(x_{c})>t_{threshold}\\end{aligned}}", "1699cfa4c2e15c121d1bc27ebdadca58": "D_{k}\\!", "1699ffa7de42dc727e04687526ebf9de": "y_{j}=x_{N}", "169a4f40a5cf867f1b3fc2789aab42b1": "{\\frac {\\partial c}{\\partial t}}=\\nabla \\cdot \\left[D\\nabla c-uc+{\\frac {Dze}{k_{B}T}}c(\\nabla \\phi +{\\frac {\\partial \\mathbf {A} }{\\partial t}})\\right]", "169a62eb6001e54160963ac839db49f5": "\\cos A={\\frac {\\textrm {adjacentside}}{\\textrm {hypotenuse}}}={\\frac {b}{h}}\\,.", "169a70df525985ea9e8e998d6317fe86": "G*H=\\langle R_{G}\\cup R_{H}\\mid S_{G}\\cup S_{H}\\rangle .", "169a79ea33108c3e7f3c69b39e57d9e5": "{\\boldsymbol {\\tau }}={{d\\mathbf {L} } \\over {dt}}={{d(I{\\boldsymbol {\\omega }})} \\over {dt}}=I{\\boldsymbol {\\alpha }}", "169a85994b624f4ab86ed9a8cda295f3": "I=I_{\\mathrm {f} }+I_{\\mathrm {b} }=\\iint _{\\Sigma }\\left(\\mathbf {J} _{\\mathrm {f} }+\\mathbf {J} _{\\mathrm {b} }\\right)\\cdot \\mathrm {d} \\mathbf {S} =\\iint _{\\Sigma }\\mathbf {J} \\cdot \\mathrm {d} \\mathbf {S} ", "169b2a7f5d167e8ba4c5445dc69bfa73": "W=C_{1}\\left[{\\tfrac {1}{2}}(I_{1}-3)+{\\tfrac {1}{20\\lambda _{m}^{2}}}(I_{1}^{2}-9)+{\\tfrac {11}{1050\\lambda _{m}^{4}}}(I_{1}^{3}-27)+{\\tfrac {19}{7000\\lambda _{m}^{6}}}(I_{1}^{4}-81)+{\\tfrac {519}{673750\\lambda _{m}^{8}}}(I_{1}^{5}-243)\\right]", "169b3f1f4947d42cc7dfda0da5ce654d": "~\\displaystyle r\\approx \\exp \\!\\left(-{\\sqrt {8\\!~K\\!~L}}~\\theta \\right)~", "169b6386685e0a08bbbda63d50f9d4b7": "A[y]=\\int _{x_{1}}^{x_{2}}{\\sqrt {1+[y'(x)]^{2}}}\\,dx\\,,", "169b79051910d72cde0b46dac09c127a": "\\theta (x,y)", "169b8c3071d1b8e0db23132d28cf265f": "xdx+ydy=0.\\,", "169b9f22f8b5f03a6bef4823504c67f1": "{\\mbox{Internal virtual work}}=\\delta \\ \\mathbf {q} ^{T}{\\big (}\\mathbf {k} ^{e}\\mathbf {q} +\\mathbf {Q} ^{oe}{\\big )}\\qquad \\mathrm {(13)} ", "169c28ffb439d5b2b751a1446380e816": "\\sum _{n=1}^{\\infty }{\\frac {1}{n^{p}}},\\!", "169c3d03113bef59c25d02c8cb99afef": "00", "169eb94d047bc07b2754f5f8d44d3c4e": "T(n)=O(n^{2}\\log n)", "169eca1e1b28f5d8137bc45994a46e3b": "|x|_{\\ast }=|x|_{\\infty }^{\\lambda }", "169efdd2169061cf846e8d497305f653": "Q(a)b=\\{a,b,a\\},\\,\\,\\,Q(a,c)b=\\{a,b,c\\},\\,\\,\\,R(a,b)c=\\{a,b,c\\}.\\,", "169eff002b261e5a69da9c8a9680bc23": "\\int _{a}^{b}\\omega (x){\\frac {p_{n}(x)}{x-x_{i}}}dx={\\frac {1}{q(x_{i})}}\\int _{a}^{b}\\omega (x){\\frac {q(x)p_{n}(x)}{x-x_{i}}}dx", "169f167cab2dff26fb4aab2d311cbceb": "C_{n}={\\frac {1}{n+1}}{2n \\choose n}={2n \\choose n}-{2n \\choose n+1}{\\text{ for all }}n\\geq 0.", "169f180acdd94d759304288328d61533": "x=21(4m)+10=84m+10", "169f4bce463a537e62524b0a1493aecd": "a={\\sqrt {2}}", "169f8521200f5d0c433da5fc19ad2746": "\\scriptstyle g_{1},g_{2},g_{3}", "169fccccfac4c03483aee4997242944c": "A_{\\nu ;\\rho \\sigma }-A_{\\nu ;\\sigma \\rho }=A_{\\beta }R^{\\beta }{}_{\\nu \\rho \\sigma }\\,,", "169fe261b465a1e705f84ae1855e4c33": "CT_{min}={\\begin{matrix}max\\\\j=1,M\\end{matrix}}\\lbrace \\tau _{j}\\rbrace ", "169fe46f9e3b26ea660113cc6e77f860": "g_{ac}", "169ff30122149f9639c095347cef88d4": "6^{2}=36=10", "16a01ac89474d5630319eb0fb7571e56": "\\Psi (\\mathbf {r} _{1},\\mathbf {r} _{2}\\cdots \\mathbf {r} _{N},t)", "16a06a3cd4e00155267a56d4aa1c340d": "\\ {\\frac {\\rho _{2}}{\\rho _{1}}}={\\frac {1}{1-{\\frac {2}{\\gamma +1}}\\left[1-{\\frac {1}{M_{x}^{2}}}\\right]}},", "16a089e396133b12b10421e2456f50fc": "(S\\rightarrow T)\\times S", "16a0e235f6503acdf412ccabe44c3d54": "d_{K}^{-p/2}", "16a13757a4124f665161d66e050f425e": "K_{\\text{J}}={\\frac {e}{\\pi \\hbar }}\\,", "16a14abf647efc7e1c4ee6fa1116dc0d": "F={\\frac {A_{\\rm {r}}M_{\\rm {u}}It}{m}}.", "16a154fe4cca457120af19ac743891d8": "\\chi _{1}\\,", "16a1ffec8671aa049b3d8102ea6835e6": "A=(a_{ij})\\in \\mathbb {K} ^{n\\times n}", "16a29a176abc995c7eb433e1d9295ac1": "p=\\alpha ", "16a2a040bc3a10f838621c288e9743bd": "(M\\otimes N)^{\\prime }=M^{\\prime }\\otimes N^{\\prime },", "16a2ba6e6c88546836698c6ca0667288": "S=\\{A_{1}\\lor A_{2},B_{1}\\lor B_{2}\\lor B_{3},C_{1}\\}", "16a2c750fd37f68f3676f0c5d1bfc264": "\\omega _{J,g}(X,Y):=g(JX,Y)\\,", "16a30ca585e448f16cf118208f1e92b1": "\\mathbb {F} =GF(q)", "16a328a0ce8373a0d284c18a264360a6": "g(a_{1},a_{2})", "16a33dc956d9aec298961523a1dcb25c": "f(z_{j})=z_{j+1}\\,", "16a36d421769a95e1bb9308c1fe1e361": "Y_{\\alpha }(z)\\sim {\\begin{cases}{\\frac {2}{\\pi }}\\left[\\ln(z/2)+\\gamma \\right]&{\\text{if }}\\alpha =0\\\\\\\\-{\\frac {\\Gamma (\\alpha )}{\\pi }}\\left({\\frac {2}{z}}\\right)^{\\alpha }&{\\text{if }}\\alpha >0\\end{cases}}", "16a3878e41695554a1447e89432d47cb": "f(t,x,y)", "16a38b734abba3099415026dec1f8cc3": "d_{n}=\\left({\\frac {6\\times 453.59237~\\mathrm {g} }{n\\times \\pi \\times 11.352~\\mathrm {g/cm} ^{3}}}\\right)^{\\frac {1}{3}}=4.2416~\\mathrm {cm} \\times {\\frac {1}{\\sqrt[{3}]{n}}}", "16a3953129d392af21bd5b9da09337b5": "\\textstyle {n={\\frac {1}{2}}n(n-1)},", "16a3bcecb5da65488bbab82da108cd0e": "\\left({\\frac {ds}{dt}}\\right)^{2}=2T", "16a41b243f960f02d199ac609ae53819": "p_{k}<0", "16a4212dc6b7f9b139dd8ca1d8157eca": "M'\\rightarrowtail M\\twoheadrightarrow M''", "16a4640ee412133f070022418d0811f7": "{\\boldsymbol {B}}={\\boldsymbol {F}}\\cdot {\\boldsymbol {F}}^{T}", "16a47d7ee62bad534f75e73892c6b43c": "\\sigma =\\sigma _{y}", "16a48976ff99d162c02117a8970bbc1d": "X=R,G,orB", "16a50da8413e9b786c570ae4d1e9b737": "{\\frac {\\partial \\ln |\\mathbf {U} |}{\\partial x}}=", "16a58513eecc5e27172b60143e8d6ddb": "I_{0}=I_{in}{\\frac {1-R}{1+R}}", "16a59c6c5a0a6afe21215cf9de2f1746": "\\Gamma _{ijk}={\\tfrac {1}{2}}[C_{ik,j}+C_{jk,i}-C_{ij,k}]={\\tfrac {1}{2}}[(\\mathbf {G} _{i}\\cdot {\\boldsymbol {C}}\\cdot \\mathbf {G} _{k})_{,j}+(\\mathbf {G} _{j}\\cdot {\\boldsymbol {C}}\\cdot \\mathbf {G} _{k})_{,i}-(\\mathbf {G} _{i}\\cdot {\\boldsymbol {C}}\\cdot \\mathbf {G} _{j})_{,k}]", "16a5cb2c6cba1f761c791f5a62272fd8": "V(x)>0\\quad \\forall x\\in U\\setminus \\{0\\}", "16a5df436ba13bdc1d9e2b63d6f57614": "\\delta \\circ \\psi =\\psi \\otimes \\psi ", "16a5fc11b09e10089ab035bae5b5aa18": "\\forall x,y\\,(xRy\\to yRx)", "16a70deb1e982ed3e49f2ee8e304a9bb": "\\alpha _{j}=-E[S_{j}-T_{j}]", "16a73ac697d40736ec5f9ae22a421a88": "[0,\\lambda ]=\\{\\alpha \\mid \\alpha \\leq \\lambda \\}\\,", "16a743dc2a6ab476e2710703f7169a3e": "U(S,X_{1},X_{2},\\dots )", "16a79c47a5e344f74fdd77e83e5e116f": "k\\geq 7.", "16a7b8c1735a9904ec407d4bb604ae26": "1+{\\sqrt {2(1+{\\frac {1}{\\sqrt {5}}})}}", "16a7c1cc5b3bf22339897a1884bc48fd": "{\\frac {dF}{dt}}=\\sum _{i}{\\frac {\\partial F(T,V,N)}{\\partial N_{i}}}{\\frac {dN_{i}}{dt}}=\\sum _{i}\\mu _{i}{\\frac {dN_{i}}{dt}}=-VRT\\sum _{r}(\\ln w_{r}^{+}-\\ln w_{r}^{-})(w_{r}^{+}-w_{r}^{-})\\leq 0", "16a7fe571c562813c7f001b7ee55539e": "\\mathbf {J} ={\\begin{bmatrix}A&0\\\\B&C\\\\\\end{bmatrix}}", "16a8292f9c251f63eba17ee3a4ffb0e9": "L_{D}{\\big [}\\rho _{S}(t){\\big ]}={\\frac {1}{2}}\\sum _{\\alpha ,\\beta =1}^{M}b_{\\alpha \\beta }{\\big (}{\\big [}\\mathbf {F} _{\\alpha },\\rho _{S}(t)\\mathbf {F} _{\\beta }^{\\dagger }{\\big ]}+{\\big [}\\mathbf {F} _{\\alpha }\\rho _{S}(t),\\mathbf {F} _{\\beta }^{\\dagger }{\\big ]}{\\big )}.", "16a8903fe77d11bb06ee8b0e74c2bf74": "{V^{2}}/{R}", "16a8985dcd52a6e8e7c8606fbfd999d6": "\\sum _{k=0}^{N-1}\\sin _{k}(i)\\equiv 0.", "16a8b4a76f977ee32c0e6b1056793fbb": "...", "16a8ccafe26e5d98879c8705900e6106": "H(X|Y)=H(X,Y)-H(Y).\\,", "16a8df07ccdcdf3110da7b9d03d20bea": "{2 \\over 3}Cr_{2}O_{3(s)}+{4 \\over 3}Al_{(s)}\\rightarrow {2 \\over 3}Al_{2}O_{3}+{4 \\over 3}Cr", "16a8eca0ea4ba9714d36814c882c5d29": "\\sin(x){\\frac {d^{2}y}{dx^{2}}}+4{\\frac {dy}{dx}}+y=0\\,,", "16a8f4705030acb2cb92aecd5394f911": "\\eta _{c}=\\ell mN/L^{2}", "16a972d00aa8a005ce23e7cff814a943": "{\\hat {\\alpha }}(q,{r_{c}})=\\max \\left\\{\\alpha :\\ {r_{\\rm {c}}}\\leq \\min _{u\\in {\\mathcal {U}}(\\alpha ,{\\tilde {u}})}R(q,u)\\right\\}=\\max _{\\alpha \\geq 0}\\min _{u\\in {\\mathcal {U}}(\\alpha ,{\\tilde {u}})}\\varphi (q,\\alpha ,u)\\quad \\quad \\Box ", "16a9cd46194fa8b7788705f676cf1d74": "f(x)=a+bx+cx^{2}+dx^{3}+\\cdots ", "16a9d8aef04c20112bf117547e6d9b75": "TM\\otimes \\cdots \\otimes TM\\to E", "16a9daeee032b8aa28ed65b393dd1482": "D_{\\gamma }(\\gamma (b)||\\gamma (a))=\\int _{a}^{b}(b-s)g_{\\gamma }(s)ds", "16aa0bc34e6700948541cceae21b5132": "T(*)=A", "16aa0bea3153c70928f6376f5bf2fd64": "X=f(A,B,C,\\dots )", "16aa29af1a2222aa6c4e25b8b424f4e9": "P_{n}=[n]P=(X_{n}:Z_{n})", "16aac161e34573e60f6477dfbaa20035": "{\\hat {H}}={\\tfrac {1}{2}}\\left[{\\frac {{\\mathcal {P}}^{2}}{I_{1}}}+{\\mathcal {P}}_{z}^{2}{\\Big (}{\\frac {1}{I_{3}}}-{\\frac {1}{I_{1}}}{\\Big )}\\right],", "16aaee0be4310bec4d0a90b4bed35e5e": "b_{0}=a_{0}", "16ab025e32e8e64ef4df917da38fe052": "{\\vec {S}}\\cdot {\\vec {J}}={\\frac {1}{2}}(J^{2}+S^{2}-L^{2})={\\frac {\\hbar ^{2}}{2}}[j(j+1)-l(l+1)+s(s+1)],", "16ab1743af9f39327b5022d6f180289d": "f(y)\\geq f(x)", "16ab73b826991d3c50cd61d0770fc0ce": "P=I\\cdot V=I^{2}\\cdot R={\\frac {V^{2}}{R}}\\,", "16abbef98c44386e29d9fdbf34ed42dd": "h_{F}^{(1)}(z)", "16ac1c6d827ea95108985f89842a4bda": "1/x^{2}", "16ac2591ca4b5e895b0da5c506267b60": "(\\mathbf {\\lambda } x.xxx)(\\lambda x.xxx)", "16ac3d8d6743d5e4c80689495c2b7104": "r={\\frac {C}{2\\pi }}.", "16ac52216bc60f5527ae62d3dc4bf60e": "\\ln p(\\mathbf {x} ;A)=-N\\ln \\left(\\sigma {\\sqrt {2\\pi }}\\right)-{\\frac {1}{2\\sigma ^{2}}}\\sum _{n=0}^{N-1}(x[n]-A)^{2}", "16ac58e5a01f207741c6bba9fdcbde0e": "{\\frac {a^{x}}{a-1}}\\,", "16ac8405f3b110a8a9aba8ae0c9b1853": "A({\\boldsymbol {\\eta }})", "16acf7c0df028e5213192de2a13e142a": "{\\begin{matrix}a^{b}&=&\\underbrace {a_{}\\times a\\times \\dots \\times a} \\\\&&b{\\mbox{ copies of }}a\\end{matrix}}", "16ad3c1d98da48e45bd03f2ff4a95085": "{\\hat {\\mathbf {T} }}={\\frac {d{\\hat {\\mathbf {S} }}}{dt}}=-{\\frac {i}{\\hbar }}\\left[{\\frac {\\hbar }{2}}{\\boldsymbol {\\sigma }},{\\hat {H}}\\right]", "16ad52fbafe38fa0a8cb2104979e119e": "I=\\int _{0}^{a}{\\frac {1}{z}}\\,dz-\\int _{0}^{b}{\\frac {1}{z}}\\,dz=\\ln a-\\ln b-\\ln 0+\\ln 0", "16ad9a091cbce331ba8b01551f76ae3b": "\\left(a,q,u\\right)\\succsim \\left(c,s,v\\right)", "16adefc4cacd3b68984078ed4156c60e": "\\scriptstyle P(x_{t}|s_{t})", "16ae57e1f40b9d912fe5a835a276d48e": "K_{i}={\\frac {[A]_{i}}{p_{A}\\,[A]_{i-1}}}", "16aee0aadc2eea05f6a7b5f5ff540bc9": "({\\tfrac {1}{2}}\\pi ,\\pi )", "16af36c55c890e104d43c814f6155c21": "O_{b}:{\\mathfrak {H}}_{b}\\rightarrow {\\mathfrak {H}}_{b}", "16afcd0d566abc717455f061574e3c41": "\\tau =\\sum _{i=0}^{\\infty }{\\frac {t_{i}}{2^{i+1}}}=0.412454033640\\ldots ", "16b001475fa4f62cd9c791a6124b1124": "\\Pi (i\\omega _{n})={\\frac {1}{\\beta }}\\sum _{i\\omega _{m}}{\\frac {1}{i\\omega _{m}+i\\omega _{n}-\\epsilon }}{\\frac {1}{i\\omega _{m}-\\epsilon '}}=-{\\frac {n_{F}(\\epsilon )-n_{F}\\left(\\epsilon '\\right)}{i\\omega _{n}-\\epsilon +\\epsilon '}}", "16b00f56a247758348fd7a99e067926d": "g_{t}(z)={\\sqrt {z^{2}+4t}}", "16b0575a7d218647422e64a1c4ba9964": "{\\hat {K}}\\in L^{\\infty }(\\mathbf {R} ^{n})", "16b08e97689b511cadc975ebdf083d7e": "h(A)=h(B)", "16b0a75bfaf7b9b278dd8756ff2ff0e7": "B\\supset A", "16b0e315e31e18b689a57982fecfb00c": "S(t)\\ :=\\ (S_{1}(t),\\ ...,\\ S_{N}(t))", "16b107be450e66526a158d467b243511": "Dep_{t}", "16b109dc2dec4055ae83fb11e66f7051": "\\scriptstyle {Rc<0.1}", "16b116313055e5c9191d03265ec3ef7d": "={\\frac {a(cf+ed)}{bdf}}={\\frac {acf}{bdf}}+{\\frac {aed}{bdf}}={\\frac {ac}{bd}}+{\\frac {ae}{bf}}", "16b137c62f4d3787762c6cc432de5dee": "E\\supseteq U\\supseteq F", "16b167dd37b3ff3684efb40975ae6701": "{\\hat {\\rho }}^{[?]}=\\sum _{m}\\,{\\mathcal {P}}_{m}{\\hat {\\rho }}_{m}={\\hat {1}}/D,", "16b1a0a1182d3bad56f6778e2d143cf8": "r\\ r=(r\\ r\\to y)", "16b1f9bb7d3e8127c682b8758496704f": "A={\\frac {1}{N}}\\sum _{n=1}^{N}(G_{n})^{2}\\times (D_{1}-D_{2})", "16b21a78d2151d2181f1f90ba6df20f8": "A(\\omega )\\approx {\\frac {R_{0}}{i\\omega L}}", "16b21ea5f7fa343162c6f418b5082e5b": "M_{r}={\\frac {(c-\\beta )(c)_{r}(c+1-\\gamma )_{r}}{(c+1-\\alpha )_{r}(c+1-\\beta )_{r}}}.", "16b2a3f12c31d6683042b74b8d58e1d2": "{\\frac {|-1000|}{|-1|}}=1000,\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad (14)", "16b2af654956187ce493e9dcc3b2ff0c": "\\pi \\colon E\\to B", "16b335bcc0440032e49f1cfe971dc66e": "J=cl{\\{x\\in M\\mid x{\\mbox{ is critical point of }}f_{t}\\}}", "16b367867c939c67d4bedb0ab2f1ca94": "\\left(\\mathbf {ab} \\right)\\times \\mathbf {c} =\\mathbf {a} \\left(\\mathbf {b} \\times \\mathbf {c} \\right)", "16b3afc7bbec72e53bcb3c7a63a2e809": "H_{\\nu }(\\omega )=-e^{-j{(\\nu -2)}{\\frac {\\omega }{2}}}P_{\\nu }(\\sin({\\frac {\\omega }{2}}))", "16b3b302a249f8ae3d2e20cdc10b7879": "xy''+(c-x)y'-ay=0\\,", "16b3c9a0b5fbcd00decec1cb0da0892a": "\\,d^{(p)}", "16b40ea147efb071bdfb78e052c94881": "i\\hbar {\\dfrac {\\partial \\Psi (x,t)}{\\partial t}}=i\\hbar \\left(\\psi _{0}\\exp \\left(-i{\\dfrac {E_{0}t}{\\hbar }}\\right)\\left({c_{0}}'(t)-i{\\dfrac {E_{0}}{\\hbar }}c_{0}(t)\\right)+\\psi _{1}\\exp \\left(-i{\\dfrac {E_{1}t}{\\hbar }}\\right)\\left({c_{1}}'(t)-i{\\dfrac {E_{1}}{\\hbar }}c_{1}(t)\\right)\\right)", "16b410487fc05b66f5df3523c3f08a03": "K_{1}={\\frac {d_{1}+k_{1}}{a_{1}[W_{T}]}}={\\frac {K_{M1}}{[W_{T}]}},", "16b416f663e73abc97691dd5600fe620": "Tr(g^{a})\\in GF(p^{2})", "16b42e4985cdca5c4f7ece787bbae454": "x=(-1)^{\\mathrm {Sign} }(1+M)2^{E-B}", "16b453556ed1b50bfe1a5cd950ae5d17": "{\\begin{alignedat}{2}f(x)&=(a-b)^{2}\\\\&=a^{2}-2ab+b^{2}\\\\\\end{alignedat}}", "16b47caeb49427975a9579db51e50f2e": "S_{n}\\equiv S_{n-j}\\star S_{n-k}{\\pmod {m}},0\\beta \\geq 1.28", "16c0633d8e14b1ef64266f73e020087e": "{\\frac {1}{\\sqrt {f}}}=-2\\log _{10}\\left({\\frac {\\varepsilon }{3.7D_{\\mathrm {h} }}}+{\\frac {2.51}{\\mathrm {Re} {\\sqrt {f}}}}\\right)", "16c06809b1abd50dccc9b0b4452fabf9": "z=0.5", "16c095dc7b230e3acc7413cc95dcb161": "x_{i}\\rightarrow c", "16c0ce42e9ce29ac2677a5b3ce87f60c": "I_{n}^{f}", "16c0d8acf8b9179918431e74424ca42c": "P(x)\\uparrow (\\forall {y}{\\in }\\mathbf {Y} \\,Q(y))\\equiv \\ \\exists {y}{\\in }\\mathbf {Y} \\,(P(x)\\uparrow Q(y)),~\\mathrm {provided~that} ~\\mathbf {Y} \\neq \\emptyset ", "16c10841d7a5a537e6cf660c4e6fb148": "q_{k}<0", "16c1f38a909314fbe3ae79c748e64131": "c={\\sqrt {\\frac {f}{\\rho }}}", "16c22f65f46040a73b02c0c586f8c06f": "g^{-1}(0)=\\{x\\in U\\mid g(x)=0\\in Y\\}\\subseteq U.", "16c25137caeae53bb0457c856bdbfc29": "{\\begin{array}{lcl}E(k)&\\propto &k_{\\perp }^{-5/3};\\\\k_{||}&\\propto &k_{\\perp }^{2/3}\\end{array}}", "16c2aef1bc0d558423d3cac2813d18e8": "A={\\frac {1}{2}}r^{2}\\theta .", "16c2d41dc546deefd4884444e211e0fc": "\\left({\\frac {-3}{\\sqrt {10}}},\\ {\\frac {5}{\\sqrt {6}}},\\ {\\frac {2}{\\sqrt {3}}},\\ \\pm 2\\right)", "16c2d9b8074678da923688e5bd5ac573": "2^{2^{n}}+1", "16c320e56daf3de5e759b678e4d933a2": "\\int _{P_{1}}^{P_{2}}{\\frac {\\mathrm {d} P}{P}}={\\frac {L}{R}}\\int {\\frac {\\mathrm {d} T}{T^{2}}}", "16c33b2aea95d7a05d44e3953b7b7339": "E_{0}=m_{0}c^{2}\\,\\!", "16c3684e2506c2b41e33e0fbfa64b664": "\\forall x\\exists rFxr\\,.", "16c38032f3060fd48736eb5585960948": "\\sin \\theta =\\pm {\\frac {\\sqrt {a^{2}+b^{2}+c^{2}}}{k}}", "16c39a87a1212841ba3b7282866e88d7": "{\\vec {C}}_{4}", "16c3b2d0ab41db51c39adbdb28a649b5": "an^{2}+bn", "16c3da2bfdfd7c4ccd7dbc71bc8586b8": "P_{1}[b/a]", "16c42a5ee061a1af911bca5732810c65": "f:\\bigoplus _{i\\in I}M_{i}\\rightarrow M", "16c440089ff8c668d9424255742f6189": "d=f+p", "16c4ea3612cb5ea98bacc490e87c400c": "\\varphi \\left(\\mathbf {x} \\right)=\\sum _{i=1}^{N}\\left(a_{i}+\\mathbf {b} _{i}\\cdot \\left(\\mathbf {x} -\\mathbf {c} _{i}\\right)\\right)\\rho {\\big (}\\left\\Vert \\mathbf {x} -\\mathbf {c} _{i}\\right\\Vert {\\big )}", "16c514d35bebb256433423143ddad593": "\\textstyle {1 \\over r^{2}}", "16c51c36290d4b8798f7a3fb20911d0b": "{\\mathfrak {sl}}_{2}(\\mathbb {C} ).", "16c530bab59fbeb92f66984ec9e48833": "\\left(r^{2}-a^{2}+c^{2}+x^{2}+y^{2}\\right)^{2}=4r^{2}\\left(x^{2}+c^{2}\\right)", "16c5397e29ad3811cfe1e43e07f74da6": "n^{-1}U\\left(z,{\\hat {\\zeta }}\\right)=n^{-1}\\sum U\\left(z_{j},{\\hat {\\zeta }}\\right)=0", "16c562902f30ec739882c634cd4dc131": "\\scriptstyle \\eta ", "16c5a49a504194bc0e18af85aededdbf": "\\sum _{k=m}^{n}ar^{k}={\\frac {a(r^{m}-r^{n+1})}{1-r}}.", "16c5c5ce108d0c7535c7dcf10da2cd84": "g_{\\mathrm {rev} }\\,", "16c614385de625bdf9326591e0669e95": "G_{0}^{-1}", "16c632ecacdee7be03cc2217008c3049": "\\omega _{1}(t):=\\sup _{s\\leq t}\\omega (s)", "16c65d5e3b8286ef18f318bebbc942ff": "(2\\pi /V)^{d}", "16c6abcd001c113b16487e33f6d7163c": "q\\ ", "16c6bb1e7ff25c5260b286008575c27b": "p=uq+r,\\ ", "16c6bb7496a4e3b1d19d3f0704df42df": "{\\begin{cases}x=e^{\\rho }\\cos \\theta ,\\\\y=e^{\\rho }\\sin \\theta .\\end{cases}}", "16c6bfd6b23c6f45049add0389959756": "T_{R}=0.9", "16c6c5b94de8822436356c7b733fc01f": "90=2\\times 3^{2}\\times 5^{1}", "16c6eda5a517e1f144a37ccc79ff988e": "-1+Y/{\\overline {Y}}=c({\\overline {u}}-u).", "16c7232f28b44c2de2125326950fa11c": "\\Im (z)-\\Re L_{t}(w,w)\\in V\\,", "16c797230fdbc4ea01ae6620b45d0080": "{\\mathbf {y} }{\\sqrt {\\nu /u}}={\\mathbf {x} }-{\\boldsymbol {\\mu }}", "16c7a057cab1116577b57586002d37e3": "1r", "16c8f1c6563a76f46f8afe0fbbf5c843": "1\\cdot \\mathbf {0} =\\mathbf {0} ", "16c94e555cfeb5d52ae72c4432eddb03": "P=V+E+F+C\\,\\!", "16c966bb9d8ce4d1c75a475dd7530a93": "t_{3}", "16ca9e4dd057aab73be1cf835757f8f2": "M_{2}=({\\frac {{\\frac {2}{\\gamma -1}}+{M_{1}}^{2}}{{\\frac {2\\gamma }{\\gamma -1}}{M_{1}}^{2}-1}})^{0.5}", "16cad9ed5a1fd86e4fc3a258234e05ad": "{\\frac {L+1.25\\times {\\sqrt {S}}-9.8\\times {\\sqrt[{3}]{DSP}}}{0.686}}\\leq 24.000\\,metres", "16caf13f5b9a7080370ebef99784919d": "{\\mathit {K}}_{a}\\varphi ", "16cb48aae3ae2294910eae05f66494ac": "f_{\\mathbb {H} }", "16cb9bebe1747819ee672898705388b3": "mP=0", "16cbab002642812cd0809c0d17337c32": "{1 \\over 10^{6}}", "16cbad6b5e2cfb980807cef70cdcfd80": "z=-2", "16cbc5bf48660cfe5af2ad29f8b80020": "W^{\\ast }", "16cbca6493b4098798e6251eb7eb46a2": "O(A|B,C)=\\Lambda (A|B\\cap C)\\cdot \\Lambda (B|A)\\cdot O(A).", "16cc133003f237995ca05ccd1c029bce": "S\\times N", "16cc4117aa11d6fadfe295ec43fda2ec": "kr\\ll 1", "16cc58311e2f72f8c096bc28c6286603": "\\mathbf {e} _{m+1}=\\sum _{i}^{m}\\ c_{i}\\mathbf {e} _{i}.", "16cc6a9cf846e8519da532cde0bdae6a": "\\{X_{i}(\\omega )\\}_{i\\in \\mathbb {N} }", "16ccf71b13078607a3bc5cd8dc0e7330": "1-e^{-\\hbar \\omega _{\\alpha }/k_{B}T}\\approx \\hbar \\omega _{\\alpha }/k_{B}T\\,", "16cd175b544caf24acd4ffb5ee5c0b7a": "S_{M_{j}}\\subsetneq \\cup _{k\\in S}S_{M_{k}}", "16cd3f0aef1db407d98b9f8b3be95743": "w[n]=\\ w_{0}\\left(n-{\\frac {N-1}{2}}\\right),\\ 0\\leq n\\leq N-1.", "16cd68cdabaee2bc69fc456e12d83fd6": "BG/BH", "16cd7a00e1ad8c96bffaef367609ee61": "\\mathbf {X} {\\boldsymbol {\\beta }}=\\mathbf {y} ,", "16cd8cd85e30de3f2180a0ab1c1048b3": "P(\\mathbf {x} ,\\mathbf {t} \\mid \\mathbf {x_{0}} )", "16cd97a7898ef945dda8e8e678d652c3": "\\|\\cdots \\|", "16cdbf9053a8e8ccf2cb8edaf7b6083a": "dl^{2}=t^{2p_{1}}dx^{2}+t^{2p_{2}}dy^{2}+t^{2p_{3}}dz^{2}", "16ce356a35b66ac10b64ac243ab6b4c9": "v_{R_{1}}=v{\\frac {R_{1}}{R_{1}+R_{2}}}\\iff i_{G_{1}}=i{\\frac {G_{1}}{G_{1}+G_{2}}}", "16ce429f1f4fb2b3c8916043cd87c18e": "\\mathbf {y} (x)=U(x)\\mathbf {z} (x)", "16ce5d3553abbb977981758212d402a8": "v^{*}:=\\max _{d\\in D,\\,z\\in \\mathbb {R} }\\{z:z\\leq f(d,s),\\forall s\\in S(d)\\}", "16ce7039ef7ef21de0aa7aab84fc39bc": "4\\Delta x", "16ce72b18f657df6f5805128affa3798": "C(\\{x\\})=\\{x\\}\\cap X", "16ce959d2748e1a3109f216b62ec7b21": "x{\\frac {d^{2}g}{dx^{2}}}+{\\Big (}(l+{\\frac {1}{2}})+1-x{\\Big )}{\\frac {dg}{dx}}+{\\frac {1}{2}}(n-l)g(x)=0", "16ced14ad381d0c55f46a750367db2ed": "f(i)", "16ced8c81e9b88644007c41e5b59e871": "\\left.{\\begin{aligned}|1,1\\rangle &=\\;\\uparrow \\uparrow \\\\|1,0\\rangle &=\\;(\\uparrow \\downarrow +\\downarrow \\uparrow )/{\\sqrt {2}}\\\\|1,-1\\rangle &=\\;\\downarrow \\downarrow \\end{aligned}}\\;\\right\\}\\quad s=1\\quad \\mathrm {(triplet)} ", "16cf0a9a371e7164ea3b6340f91368f5": "\\beta _{\\mathrm {Darlington} }=\\beta _{1}\\cdot \\beta _{2}+\\beta _{1}+\\beta _{2}+1", "16cf15177829b95493295d847b2b4ff6": "k=1,2,\\dots ,b", "16cf1fd6fa42533790803c6a67756d27": "_{k}\\mathbf {E} _{l,m,n}=_{k}\\left[E_{1},E_{2},\\ldots ,E_{11},E_{12}\\right]_{l,m,n}^{T}", "16cf3f93acb1340b11e29743e93d30e5": "\\textstyle P_{X_{r}}=P", "16cf641f02aa0faa1420b417a3285027": "p({\\tilde {x}}|\\mathbf {X} ,\\alpha )=\\int _{\\theta }p({\\tilde {x}}|\\theta )p(\\theta |\\mathbf {X} ,\\alpha )\\operatorname {d} \\!\\theta ", "16cfbfe13e6ad2c3af1781267ebe899d": "e^{z}={}_{1}F_{1}(1;1;z)", "16cfc8da079c902590e0f61f38c7258e": "M(f(z),1)", "16cff04968eb771b3f52a6813c1627ea": "20*log_{10}(2)", "16d00a9e0d3adae598b69752f3ff8582": "\\{f,g\\}_{DB}=\\{f,g\\}_{PB}+{\\frac {c\\epsilon _{ab}}{qB}}\\{f,\\phi _{a}\\}_{PB}\\{\\phi _{b},g\\}_{PB}.", "16d026ccb78f6c6416731bdd70e8dfe6": "f(\\mathbf {0} _{V})=f(0\\cdot \\mathbf {0} _{V})=0\\cdot f(\\mathbf {0} _{V})=\\mathbf {0} _{W}.", "16d047303bd2cfb65e57195a14cbae77": "(1,2,2)_{H}", "16d06cc014affbbee42fcb1a9fd6f1a9": "\\sin(\\alpha )\\simeq \\tan(\\alpha )=y/f", "16d08b744ced0e4ca300155818c83561": "(x,a)\\sim (x,b){\\text{ if }}x\\neq 0.\\;", "16d09d70ab577692fdd225f531f12bbb": "P_{c}^{p}(z_{0})", "16d0bbb234cf39043edbf27486b18997": "\\{v_{k}\\}", "16d0c35deb3ef8acc4e3e5a1f6beab23": "{\\hat {A}}_{n,\\alpha }^{\\dagger }", "16d1cef20c82a52818efbfb619dc03ba": "B((x_{0},x_{1},\\ldots ,x_{n}),(y_{0},y_{1},\\ldots ,y_{n}))=x_{0}y_{0}-x_{1}y_{1}-\\ldots -x_{n}y_{n}", "16d25daca7eddc4c0d1820e886268c6a": "S^{2}\\left|s,m_{s}\\right\\rangle =\\hbar ^{2}s(s+1)\\left|s,m_{s}\\right\\rangle ,", "16d2b578b3ae98e4a1a5cf6bf8fab1b7": "\\jmath ^{2}=+1", "16d2d225f3edaf4ebcdcb1beed2792d1": "D=\\left({\\frac {r_{ij}}{2a_{ij}}}\\right)^{2},a_{ij}={\\sqrt {a_{i}a_{j}}}", "16d2f645b9b6da3e8f2f8c1c40a5201d": "\\ln(-\\ln(1-{\\hat {F}}(x)))", "16d3189c358fed34f9c25dd51b228f95": "{\\mathit {M}}_{\\mathit {s}}\\approx {\\frac {{\\mathit {m}}_{\\mathit {p}}{{\\mathit {t}}_{\\mathit {rec}}}^{3/2}}{\\sqrt {{\\mathit {n}}_{\\mathit {rec}}\\sigma ^{3}}}}.", "16d3241486079b87b3ddeb0cb9108e03": "h(t,s)=0{\\text{ for }}t1\\quad {\\text{(supercritical flow)}}", "16e26bfa366568987cf4db796f8638da": "\\forall A\\in {\\mathcal {A}}:\\Pr[A]\\leq (1-\\varepsilon )x(A)\\prod \\nolimits _{B\\in \\Gamma (A)}(1-x(B))", "16e2ff425646e5ddc2f1a535ee02344f": "\\sigma (x):=x^{p}+p\\delta (x)", "16e310a1ed7e0079ae7f0d1a37fcd43b": "{H}_{ij}^{(n)}:=\\langle \\psi _{in}|{\\hat {H}}_{\\text{JC}}|\\psi _{jn}\\rangle ,", "16e35d16063cd69632515f00ed9fe3f1": "\\mathbf {L} ^{0}=\\mathbf {L} ,\\quad \\mathbf {L} ^{i+1}=[\\mathbf {L} ,\\mathbf {L} ^{i}]\\ ", "16e3790df660a79605a6bdf740d269d0": "x\\leq y\\iff g(x)\\leq g(y)", "16e382ee9c0a96925397b433935c3a01": "\\tau _{1}=C(R_{1}+R_{2})", "16e3959b07a211413a1792253f70d03b": "{\\mathcal {H}}={\\mathcal {H}}_{1}\\otimes {\\mathcal {H}}_{2}\\otimes \\ldots \\otimes {\\mathcal {H}}_{n}", "16e3b1c0636921d75c950eb3310952f3": "\\bigcup _{n}f^{-n}(z)", "16e3c0ac16cd77c7dc8f997e25ea64d0": "H_{n}=\\sum _{k=1}^{n}{\\frac {1}{k}},\\!", "16e3d07447da62c38c3d77771099b56d": "[\\wp '(z)]^{2}=4[\\wp (z)]^{3}-g_{2}\\wp (z)-g_{3},", "16e49b4da73008464b35f6203f8942ce": "B={\\begin{pmatrix}\\lambda _{1}&0&\\cdots &0\\\\0&\\lambda _{2}&\\cdots &0\\\\\\vdots &\\vdots &\\ddots &0\\\\0&0&\\cdots &\\lambda _{n}\\end{pmatrix}}", "16e49c15023b5a27ed9208a42f6c70a6": "{\\boldsymbol {F_{\\dot {\\theta }}}}=-{\\boldsymbol {\\omega \\times }}\\left({\\boldsymbol {\\omega \\times r}}\\right)\\ ,", "16e4c0e1ecb8055b5e485411345bcf11": "D\\zeta =k_{B}T", "16e4eb6dcceb237abc7b6ca6a4e88ad4": "\\sin(\\alpha \\pm \\beta )={\\sqrt {\\sin ^{2}\\alpha -(\\sin \\alpha \\sin \\beta )^{2}}}\\pm {\\sqrt {\\sin ^{2}\\beta -(\\sin \\alpha \\sin \\beta )^{2}}}", "16e4f58d4fb62a8a513367812d04dfae": "R_{\\mathrm {Th} }=R_{1}+\\left[\\left(R_{2}+R_{3}\\right)\\|R_{4}\\right]", "16e4f97b0940534df8715707fc9812f6": "\\left({\\frac {du}{d\\varphi }}\\right)^{2}={\\frac {1}{b^{2}}}-\\left(1-ur_{s}\\right)\\left({\\frac {1}{a^{2}}}+u^{2}\\right)", "16e52377ecc0439f0b23f5cc76b486b8": "z,x,y,u", "16e53a8a0bbef14e25e7ca0b905e5aef": "{\\begin{aligned}k_{i}^{j}=\\delta _{ij}+P_{i,i-1}k_{i-1}^{j}+P_{i,i}k_{i}^{j}+P_{i,i+1}k_{i+1}^{j}\\end{aligned}}", "16e5df79200ad41c38172c97487f4658": "{\\dot {u}}_{n+1}={\\dot {u}}_{n}+(1-\\gamma ){\\Delta }t~{\\ddot {u}}_{n}+\\gamma {\\Delta }t~{\\ddot {u}}_{n+1}.", "16e5f3f88bbe946adb9b8c8a27ee69f5": "(1\\;9\\;13\\;8\\;11)", "16e5f63cd62f3d302473718e0268a1ab": "\\scriptstyle {|n\\rangle }", "16e65f972992070177a4a55062a0a576": "{\\frac {\\Gamma (n+k)}{k!\\Gamma (n)}}={\\binom {n+k-1}{k}}", "16e68d4ccb1adf12c2bef442253896b9": "\\sigma _{P}^{2}=\\mathbb {E} \\left[\\sum _{i=1}^{n}x_{i}^{2}(R_{i}-\\mathbb {E} [R_{i}])^{2}+\\sum _{i=1}^{n}\\sum _{j=1,i\\neq j}^{n}x_{i}x_{j}(R_{i}-\\mathbb {E} [R_{i}])(R_{j}-\\mathbb {E} [R_{j}])\\right]", "16e6ed33d9ac773f84170a78ddd62195": "2^{4n}-1,3^{4n}-2^{4n},\\dots ,(4n)^{4n}-(4n-1)^{4n}", "16e6f8d793c8cd5ce99a43c14b824d77": "{\\hat {\\zeta }}", "16e726c7058178bad641a36d6b3b19b3": "+S_{x}\\otimes S_{x}", "16e74e6424378befa9518a89eab147cf": "u(r)\\ {\\stackrel {\\mathrm {def} }{=}}\\ rR(r)", "16e75f3245ddbed593b5d02616091410": "G/F", "16e785725a7220b7a85526e174790102": "G=(V,E)", "16e79aa2f3c5754429354a0d3ca19f79": "P_{i}^{2}=1,i=1,2,...,2k", "16e7ace3e532a8caf680ee9be480a708": "\\scriptstyle \\lambda /4", "16e7da158482ea8feb9445179d63c0d8": "\\mathbf {u} (t)=-K\\mathbf {y} (t)+\\mathbf {r} (t)", "16e7e5453849788209d61a0081c7bab1": "\\displaystyle \\partial ^{2}W/\\partial i_{m}\\partial i_{n}=\\partial ^{2}W/\\partial i_{n}\\partial i_{m}", "16e901f7a9cd9c3b2ffb79e2eeac3ce4": "0\\leq \\theta \\leq 2\\pi ", "16e90f744846824ef4a53e1c666aa71b": "0<\\alpha _{01}<\\alpha _{02}<\\cdots ", "16e9236ce5c9dc1b390c452daaf7817a": "C_{1}=-{\\frac {mk^{2}}{2\\hbar ^{2}}}H^{-1}-I~,", "16e92a4edb6df477c18ba6024decaf45": "p_{2}=\\Delta x,q_{2}=x_{\\text{max}}-x_{0}\\,\\!", "16e94be175a986910fc60aa7a2a2ac28": "\\operatorname {var} (\\mathbf {b} ^{\\rm {T}}\\mathbf {X} )=\\mathbf {b} ^{\\rm {T}}\\operatorname {var} (\\mathbf {X} )\\mathbf {b} ,\\,", "16e9715b466d97e5bb135373cdb9125b": "\\left(3^{3}\\right)^{3}=27^{3}=19683.", "16e9a29195756bdd601e03d359a8f85f": "{\\begin{aligned}&P_{0}(x)=1\\\\&P_{n}(x)={\\frac {1}{2^{n}n!}}\\ {\\frac {d^{n}(x^{2}-1)^{n}}{dx^{n}}}\\quad n\\geq 1\\\\\\end{aligned}}", "16e9c3f8d9aceb5f85b77d80341ce643": "a_{ij}=\\operatorname {sgn}(r_{j}-r_{i})", "16ea0c3e6859c3d0b32be0ae9e56d506": "T(n)=T(n-1)+(n-1)T(n-2),", "16ea1b135d8b4465523d38135e0f98f2": "A_{i+1}=V_{i}^{R}||W_{i}^{L}", "16ea29d4ffa526805e62b698e00a3456": "N=E_{s}-A_{s}+E_{m}-A_{m}+E_{x}-A_{x}+E_{h}-A_{h}\\,", "16ea3b59c0fd578220c90beba8a2d37b": "\\mathbf {F} =I{\\boldsymbol {\\ell }}\\times \\mathbf {B} \\,\\!", "16ea4948eb2b1b482baab59de49c6297": "\\sum _{i}\\rho _{i}v_{i}=0", "16ea531aa0702f01b1f02b8be48b8437": "(m_{1}+m_{2})T^{2}={\\frac {4\\pi ^{2}(a''/p'')^{3}}{G}}", "16eb00fe0ecb2a0c45e9e3f73910cd8e": "\\sum _{i}dp_{i}\\wedge dq_{i}+\\sum _{j}{\\frac {\\varepsilon _{j}}{2}}(d\\xi _{j})^{2},", "16eb486d49a62370c99319da4e44b5c2": "k_{+}(x,y)=J_{x-{\\frac {1}{2}}}(2{\\sqrt {\\theta }})J_{y+{\\frac {1}{2}}}(2{\\sqrt {\\theta }})-J_{x+{\\frac {1}{2}}}(2{\\sqrt {\\theta }})J_{y-{\\frac {1}{2}}}(2{\\sqrt {\\theta }}),", "16eb95071ac6698f089ac0dd5880a640": "10", "16f3662e7b30d40e58b0dca7fbccbba0": "\\theta =\\theta _{1}+\\theta _{2}", "16f3950f493bb08aab77a9b282fdd8df": "G_{0},...,G_{r}", "16f3bd88ce08f120182fed45a48d9599": "{\\hat {\\rho }}_{\\mathrm {neq} }", "16f3ff6e7d89d829c66893a0084dd097": "a'(\\varphi _{0})\\neq 0", "16f42f4f42aafb415758f210b02fc309": "{T_{lm}=\\left(T_{hi}-T_{co}\\right)-\\left(T_{ho}-T_{ci}\\right) \\over \\ln {\\frac {\\left(T_{hi}-T_{co}\\right)}{\\left(T_{ho}-T_{ci}\\right)}}}", "16f43908b65b3f0d9c9e2b84a16fbb7a": "P(t)=P_{0}2^{t}=P_{0}exp(ln(2)t)", "16f466ac6bd14bbbc95af323ae71414b": "r=s+a=((b-a)/3)+a=(b+2a)/3.\\,", "16f46f2b997aa981614ad208ca053944": "\\left\\langle {dG}/{dt}\\right\\rangle _{\\tau }", "16f4fe576ac571fdd3713b2d99d53750": "e_{2}:=\\sum _{1\\leq i0", "16f9aa34da1d97384ed54233c900279d": "PER\\,={\\frac {Gain\\ in\\ body\\ mass(g)}{Protein\\ intake(g)}}", "16f9b442d02e26ee3ed40f46b46f6887": "z{\\bar {z}}=\\|z\\|^{2}", "16f9ead2c2e45daf6d89916ab375a151": "\\forall {\\mbox{ positive }}\\delta \\cong 0,\\ (|h|\\leq \\delta \\implies |f(x+h)-f(x)|<\\varepsilon ).\\,", "16f9fc2d60a8ccc3772bd576e5fce6be": "\\mathbf {X} _{i}", "16fa2ad938f416ac1e50ed325df8cef9": "E=\\epsilon \\sum _{i=1}^{N}\\sigma _{i}=\\epsilon \\cdot j", "16fa3ba41bb3c2dab8035a38081dacc5": "nI_{n}\\ =\\cos ^{n-1}(x)\\sin x\\ +(n-1)I_{n-2},\\,", "16fa878f90dee208c7a65e7638ebbc68": "P(X|Y_{k})={\\frac {P(Y_{k}|X)P(X)}{P(Y_{k}|X)P(X)+P(Y_{k}|{\\overline {X}})P({\\overline {X}})}}", "16fa9edaca4ee1a8b926461817e6b71c": "U^{*}={\\frac {\\sigma \\epsilon }{2\\rho }}{\\sqrt {\\frac {E}{\\rho }}}", "16fb33e5ce923b1434adf91f2b4ac2cb": "m\\cdot v=m_{nakedeye}-2+2.5\\cdot \\log _{10}(D\\cdot P\\cdot t)", "16fb5918eb70d0d6acc3daed889cd363": "{\\frac {m+n+2}{4}}", "16fb836da2fd9a38138a938d2156502d": "I^{-}(p)=I^{-}(q)\\implies p=q", "16fc00d9c551df704e15cdaa567c0190": "d(m,m')<\\epsilon ", "16fc436cacaccd4f1b8f07e26ead3e13": "({\\mathcal {H}},\\langle \\;,\\;\\rangle )", "16fc72f98c9998f9973af1e73be5c81f": "P_{\\alpha ,i}", "16fc9fad607ac1b49b5068fd050644ec": "K_{a1}={\\frac {\\rm {[HCO_{3}^{-}][H^{+}]}}{\\rm {[H_{2}CO_{3}]}}}", "16fcb2602ba3fd09374ae2137bd519b6": "MPL_{nt,1}=MPL_{nt,2}=1", "16fd5836b763262bfe10b4db1cd3719a": "A\\in \\mathbb {R} ^{|V|}", "16fd677cc5b05f7fcb56da20a7788a0a": "\\mathrm {\\tfrac {u{\\bar {u}}-d{\\bar {d}}}{\\sqrt {2}}} \\,", "16fd9066d9bef6399cfcaab1384912b7": "\\operatorname {isnil} \\equiv \\lambda l.l(\\lambda h.\\lambda t.\\lambda d.\\operatorname {false} )\\operatorname {true} ", "16fd93e97d6c513d305c9205f1e7e2b6": "a(k,k)\\leftarrow \\sum _{i'\\neq k}\\max(0,r(i',k))", "16fd9a6130542c1032b8e74d6614bc1a": "au+bv-(1+c)w+c=0", "16fdb1ddc8b014d044d9c92c00873a52": "\\operatorname {BG} _{p}(a_{n};x)=\\sum _{n=0}^{\\infty }a_{p^{n}}x^{n}.", "16fdb1e52c7be43e270abe7b1450d431": "n_{\\rm {air}}0\\}.", "170a021c18372d5cd5f02b5000ca04b0": "\\mathbf {H_{{2}/{1}}} =\\mathbf {r} \\times m_{2}\\mathbf {v} \\,\\!", "170a3afdaf86e278ca3f41f4516996fd": "B(\\cdot ,\\cdot )", "170a8b2c6c715faa5960e7d9d6c9284e": "G_{1}/G_{0}", "170a9cf30edcf19a6b1316a96bf9aaf4": "{{E^{*}}_{1/2}}^{red}={E_{1/2}}^{red}+E_{0,0}+w_{r}", "170ab05a5e126c1de8cf36946ebf3936": "L^{q,w}", "170ae1795f2fd8686870d54149200ec4": "{\\frac {A_{final}}{A_{initial}}}=R.G.e^{\\frac {-{\\pi }ft}{Q}}", "170b1ffef2e848ab03962b0b5c8da7f7": "H(z)={\\frac {A(z)}{D(z)}}", "170b417a215e4315aaadd0957a34bd06": "\\int \\limits _{-a}^{a}dx\\phi _{m}^{\\mathrm {even} }(x)\\phi _{n}^{\\mathrm {even} }(x)=\\int \\limits _{-a}^{a}dx\\phi _{m}^{\\mathrm {odd} }(x)\\phi _{n}^{\\mathrm {odd} }(x)=\\delta _{mn},", "170b6207d8f5fdd990786f522e1a66d0": "dh/dx", "170b68b2f94b5597788d16c07d30e6e3": "r_{t}^{j}=\\sum _{i=1}^{n_{T}}\\alpha _{ij}s_{t}^{i}+n_{t}^{j},", "170b8a9c81ffc255884bd6b8e913b204": "|i-j|\\geq d", "170bcf66a8e7ee0c0efa559edc068d7b": "x=P_{2k}+P_{2k-1},\\quad y=P_{2k};", "170c08c43fe6ad0ff79fdeccf3c64d07": "(\\varepsilon (h_{(1)})h_{(2)})cv=h_{(1)}c\\otimes h_{(2)}v=h(c\\otimes v)=h(cv)=(h_{(1)}\\varepsilon (h_{(2)}))cv.", "170c15fd5e38683aa5a621230785b640": "H(x)=-\\sum _{m=0}^{N-1}Q_{m}\\ln Q_{m}", "170c4bb02d619f9e397688d607475f9a": "\\scriptstyle {R_{a}}", "170c77d8f686791cfdf50da14b2fa91f": "\\mu +\\Psi ^{\\textrm {T}}X", "170c853ac78dbd16b267ff433c931225": "H_{2}(S(2k+2,n))\\approx H_{2}(S(2,n)),", "170cf3f4ef3d2a4c4e4aeedf6e4a95ca": "\\sum _{k=1}^{N}f_{k}\\times C_{k}+G\\times CM=0", "170d2eecc0f93a4d20868d1b8a87b983": "350={\\sqrt {50^{2}+100c^{2}}}", "170d8791a4ae6face0b12360cfa0ae41": "\\mathrm {STC} ={\\tfrac {1}{n_{s}-1}}\\sum _{i=1}^{T}y_{i}(\\mathbf {x_{i}} -STA)(\\mathbf {x_{i}} -STA)^{T},", "170d90690d0f47817f97d245ab25bef4": "\\ s=\\sigma +j\\omega ", "170dc478c00fce61d2362bfcb05f5ed6": "\\left[W_{\\mu },W_{\\nu }\\right]=-i\\epsilon _{\\mu \\nu \\rho \\sigma }W^{\\rho }P^{\\sigma }.", "170de09171e462c79a0a467657f7657f": "P=(1-R_{C}\\cdot ({\\frac {1}{R_{B}\\cdot 0.96}}+{\\frac {T_{Sync}}{B_{B}}}))\\cdot 100\\%", "170e467f4fd3ffa4f8865f512182be11": "{\\begin{aligned}\\mathbb {P} (y{\\mbox{ received}}\\mid x{\\mbox{ sent}})&{}=(1-p)^{n-d}\\cdot p^{d}\\\\&{}=(1-p)^{n}\\cdot \\left({\\frac {p}{1-p}}\\right)^{d}\\\\\\end{aligned}}", "170eadc653de1095b407b17797e4c8ab": "\\cos \\phi ={\\frac {c}{a}},\\qquad k^{2}={\\frac {a^{2}(b^{2}-c^{2})}{b^{2}(a^{2}-c^{2})}},\\qquad a\\geq b\\geq c,", "170ed7ef177e15746d13cc285477fc3e": "{\\mathbf {g}}=g_{ab}e^{(a)}e^{(b)}~~~~~~~~~~~{\\text{where}}~g_{ab}={\\mathbf {g}}(e_{(a)},e_{(b)})", "170edee006367db5753199b060f56efb": "\\forall s_{1},s_{2}\\in S:(s_{1}{}^{\\bullet }\\cap s_{2}{}^{\\bullet }\\neq \\emptyset )\\to [(s_{1}{}^{\\bullet }\\subseteq s_{2}{}^{\\bullet })\\vee (s_{2}{}^{\\bullet }\\subseteq s_{1}{}^{\\bullet })]", "170f2ff9222438eff1b1fe52504fd318": "\\sum _{i,j=1}^{D+1}x_{i}Q_{ij}x_{j}+\\sum _{i=1}^{D+1}P_{i}x_{i}+R=0", "170fbd5308fd36ad30c7f6dc718a087f": "\\left[p_{j},p_{k}\\right]={\\frac {i\\hbar e}{c}}\\epsilon _{jk\\ell }B_{\\ell }", "171112f2bb94019fcc7f14cf4b455dc2": "n/4", "171223f9e05098de2c6f350469e82847": "v_{i}\\,\\,(i=1,\\dots ,N),", "17125c99aa83f9857286f13ec27952b1": "{\\vec {x}}={\\vec {p}}+s{\\vec {v}}+t{\\vec {w}}", "171264d2bf0638a198285613bc47a0cb": "{\\begin{smallmatrix}{a}\\end{smallmatrix}}", "1712c40a855b15c69ef5b4d83252cd40": "{\\begin{aligned}\\int _{a}^{x}0\\,dt&=F(x)-F(a)\\\\&=F(x)-C,\\end{aligned}}", "17138a41a24d490cc0c203367a68f09e": "L(\\{{\\vec {X}}(n)\\},\\{{\\vec {M}}_{m}({\\vec {S}}_{m},n)\\})=\\prod _{n=1}^{N}{l({\\vec {X}}(n))}.", "1713b0b431803bcde713e1f515966200": "3+\\omega =0+\\omega =\\omega ", "1714710c482c335b73bcb11c5c212afb": "n/p_{i}=n_{i}", "1714d32d8ad484d79bd943949c8844d8": "T_{p}^{\\mathrm {F} }", "1714f289f19fb86b2b94f85b8a0257b8": "{\\begin{matrix}{\\frac {1}{2}}\\end{matrix}}mv^{2}", "17153842d71a270755b28a5437b587ad": "a_{x}=b_{y}c_{z}-b_{z}c_{y}\\,", "171542424903a643d769114933ed18af": "J(C_{k})", "17156bcef6eef317e4800186b470858e": "W=\\mathbf {F} \\cdot \\mathbf {s} .", "171597bdad29d41a5b5271798a4382ef": "0.{\\overline {010}}=11.{\\overline {001}}=1110.{\\overline {100}}={\\tfrac {1}{5}}+{\\tfrac {3}{5}}\\mathrm {i} ", "1715dce24081993c16ef3b5a7a706e8a": "AU_{i}", "1715f5c89a324a37473d6cb0fbbb44d3": "~\\epsilon _{t-1}>0", "1715f71e00ebfb176fb46ac7702fda50": "X\\backslash E", "171604f61e838ee6fed394e7947e75e2": "{\\frac {:{\\neg }F}{{\\neg }F}}", "1716db8d8c4a9d5d541af07bb0b88ed0": "T'_{0}=L'/v", "1716ed2c573f24825e26a9357a01e344": "\\partial _{[\\alpha }F_{\\beta \\gamma ]}=\\nabla _{[\\alpha }F_{\\beta \\gamma ]}=0", "1717328f425a00d6595eecabb83d5060": "p,r\\in M", "171792057920e87b953a09f334d8b305": "\\beta _{\\rho }(\\pi (x_{1},x_{n}))=\\min\\{\\alpha _{\\rho }(x_{i},x_{i+1})|i=1,\\ldots ,n-1\\}", "1717b4fa0d2eb8701e25f6eedc76eda8": "\\mathrm {var} (T)\\geq {\\frac {[\\psi '(\\theta )]^{2}}{I(\\theta )}}", "1717f46ab3798da532df687c515efa8d": "P(n)=\\int _{N=n}^{\\Omega }P(n\\mid N)P(N)\\,dN=\\int _{n}^{\\Omega }\\left[{\\frac {1}{c}}\\right]{\\frac {1}{N\\ln(\\Omega )}}\\,dN", "17181a814248569e5aaf5b4dd08762d2": "F_{5}(a,b)=(x\\to x^{x^{(a-1)}})^{b-1}(a)", "171823b25b82527cbb9a4fe103f56382": "Q_{xz}Q_{xy}Q={\\begin{bmatrix}1&0&0\\\\0&\\ast &\\ast \\\\0&\\ast &\\ast \\end{bmatrix}}.", "1718285dcfbb4375534b24a6ce4b4de9": "104348/33215", "171842d0fb5a27563dc1e74d31c6a86c": "\\ln(4)/\\lambda .\\,", "1718a8d40af715e6d22c93cbf9aedbd9": "I_{z}=I_{x}=I_{y}={\\frac {ms^{2}}{6}}\\,\\!", "17192c7c7385d28cfb715d8114f2a871": "100\\times \\left[{\\sqrt {\\ln({\\rm {bills\\ entered}})}}+\\ln({\\rm {hits}}+1)\\right]\\times [1-({\\rm {days\\ of\\ inactivity}}/100)]", "17195d679bbebdf832eea52c4aeeb67a": "a^{n}+b^{n}=(a^{2^{k}}+b^{2^{k}})(a^{n-2^{k}}-(b^{2^{k}}a^{n-2^{k+1}})+(b^{2^{k+1}}a^{n-2^{k+2}})-\\ldots -(b^{n-2^{k+1}}a^{2^{k}})+b^{n-2^{k}}).\\!", "17196543ea11f4d138e6d964ea599ca3": "k_{B}T={\\Bigl \\langle }q{\\frac {\\partial H_{\\mathrm {pot} }}{\\partial q}}{\\Bigr \\rangle }=\\sum _{n=2}^{\\infty }\\langle q\\cdot nC_{n}q^{n-1}\\rangle =\\sum _{n=2}^{\\infty }nC_{n}\\langle q^{n}\\rangle .", "1719b84b44c5607f08d152e36c1822d6": "F_{66}=2(F_{11}-F_{12})", "1719ce882e9408d8bddf91b0838ed8f7": "\\displaystyle {f_{0}(z)=f(z).}", "1719dbe616e8e0d7595a881a067f694a": "y[n]=\\sum _{k=1}^{K}r_{k}[n]\\cos \\left(\\theta _{k}[n]\\right)", "1719e500cb70ce6fe5324977af326eae": "{\\sqrt {a}}=10^{(\\log _{10}a)/2}.", "1719f1b1de3be5b0a562c541707ad6b3": "y=b\\,y_{frac}", "171a4a620f686059b754a3e628101f97": "\\langle \\psi |\\phi \\rangle ={\\bar {\\psi }}\\phi =\\psi ^{\\dagger }\\gamma _{0}\\phi ", "171a85425401f3a3f74d635d926ed909": "\\Box (\\Box (A\\to \\Box A)\\to A)\\to A", "171adce7d002f6554d326370d75e1f69": "C=D^{2}-{\\frac {dD}{dr}}.", "171b2995d469895708807c5a205ba7f4": "A\\cap B=\\emptyset ,", "171b35751dd1aee0b02241fc538146e0": "\\omega =Uk-\\beta {\\frac {k}{k^{2}+l^{2}}}", "171b840f35653fd7cd771c592ecb12a7": "Num(S)", "171ba604e41847774b6b7660afb06ff2": "S(x)=x\\cup \\{x\\}", "171bbd3dff8eaa3bd8c623e8c31530a1": "E_{c,d}", "171bc67358f9223edc208cc0e7257c8f": "{\\begin{aligned}V_{1}&\\equiv V_{a,1}=\\alpha V_{b,1}=\\alpha ^{2}V_{c,1}\\\\V_{2}&\\equiv V_{a,2}=\\alpha ^{2}V_{b,2}=\\alpha V_{c,2}\\end{aligned}}", "171c0567ea77a56621950fd292f4911f": "\\gamma _{n,\\mathbf {C} }:=(E(\\gamma _{n,\\mathbf {C} })\\to \\mathbf {P} ^{n}(\\mathbf {C} )),", "171c6a5ea8091dec84c5174287871f50": "\\mathbf {v} =\\mathbf {v} _{\\parallel }+\\mathbf {v} _{\\perp }", "171c7af866e00f6506eaaf1e2eb8f63a": "C\\left(F\\right)=F^{\\beta }", "171d0c2ca1f83e496c822c37e7e0cd81": "A\\times M\\to M", "171d229a4e4ab98e709c3654d5c0daed": "n!=1\\times 2\\times \\ldots \\times n", "171d3aa311f3f29d8e99ac8325c918fc": "M_{ij}=\\sum _{x}\\sum _{y}x^{i}y^{j}I(x,y)\\,\\!", "171e15ca81b2d98245ff4885ac3a27e2": "(-1)^{k+1}B_{k}", "171e64fb2f0d2f8e1497a11eba7f7e71": "\\mathbf {A} ={\\begin{bmatrix}\\mathbf {B} _{1}&\\mathbf {C} _{1}&&&\\cdots &&0\\\\\\mathbf {A} _{2}&\\mathbf {B} _{2}&\\mathbf {C} _{2}&&&&\\\\&\\ddots &\\ddots &\\ddots &&&\\vdots \\\\&&\\mathbf {A} _{k}&\\mathbf {B} _{k}&\\mathbf {C} _{k}&&\\\\\\vdots &&&\\ddots &\\ddots &\\ddots &\\\\&&&&\\mathbf {A} _{n-1}&\\mathbf {B} _{n-1}&\\mathbf {C} _{n-1}\\\\0&&\\cdots &&&\\mathbf {A} _{n}&\\mathbf {B} _{n}\\end{bmatrix}}", "171fd7a6983ab7275e7a8c1e6eed4e63": "c=m^{N}\\mod N.", "1720049518c54612d768892ca32048c0": "{\\frac {X_{b}(t)-X_{b}(0)}{X_{b}(\\infty \\,)-X_{b}(0)}}=1-\\exp \\left({\\frac {FDt}{\\beta \\,^{2}f^{2}}}\\right)", "17204bfde38cd457a51d05c7c95fd7c8": "(\\mu _{i,j}:i\\in I,j\\in J)", "1720789a56ce8752d07f0f918f8a64e1": "{\\frac {a-b}{a+b}}={\\frac {2\\sin {\\tfrac {1}{2}}\\left(\\alpha -\\beta \\right)\\cos {\\tfrac {1}{2}}\\left(\\alpha +\\beta \\right)}{2\\sin {\\tfrac {1}{2}}\\left(\\alpha +\\beta \\right)\\cos {\\tfrac {1}{2}}\\left(\\alpha -\\beta \\right)}}={\\frac {\\tan[{\\frac {1}{2}}(\\alpha -\\beta )]}{\\tan[{\\frac {1}{2}}(\\alpha +\\beta )]}}.\\qquad \\blacksquare ", "17207dbd5a04c3d90149864006383014": "{\\frac {f'(x)}{f(x)}}=\\sum _{i}\\left[\\alpha _{i}'(x)\\cdot \\ln(f_{i}(x))+\\alpha _{i}(x)\\cdot {\\frac {f_{i}'(x)}{f_{i}(x)}}\\right].", "17208bc21c168b8dbe30c84f7f5ad525": "\\!\\ S_{m}^{11}=S_{(m^{11}+11m^{9}+44m^{7}+77m^{5}+55m^{3}+11m)}.", "17209e122bf8ceb1729f4dac494bcfc0": "p_{y}={\\frac {\\partial L}{\\partial {\\dot {y}}}}={\\frac {qB}{2c}}x~,", "1720a41a5a5f58f1bfb25ec584cf44a8": "G_{y}", "1720bc203c4dddbe2f4e111c59cc6ffd": "J_{\\rm {IIIc}}=G_{\\rm {IIIc}}=K_{\\rm {IIIc}}^{2}\\left({\\frac {1+\\nu }{E}}\\right)", "17214be14077a7bb9269ef04f22b99d2": "F_{n}=F_{n-1}^{2}-2(F_{n-2}-1)^{2}\\!", "1721587d08a34ccd759115f0420033c7": "d\\Phi =-Ud{\\frac {1}{T}}+{\\frac {P}{T}}dV+\\sum _{i=1}^{s}(-{\\frac {\\mu _{i}}{T}})dN_{i}", "1721752663bd779c01ca03dce6bd54da": "{\\textit {state}}(s,1)\\leftrightarrow {\\textit {state}}(s\\circ {\\textit {open}},0)", "17219536c279def0b89ac9fddadc20d2": "\\rho ={\\frac {MP}{RT}},\\,", "1721dbc9f60370bfc91ebe3b40bb262a": "\\|\\cdot \\|:G\\to \\mathbb {R} ", "1721dc124aaa2e2f5bd410c3aff72717": "F_{44}={\\cfrac {1}{S_{\\mathrm {f} 4}^{2}}}\\ ,\\ F_{55}=F_{66}={\\cfrac {1}{S_{\\mathrm {f} 6}^{2}}}", "17227d892ae518eab12eb3f0e596f1a0": "\\mathbf {w} ", "17229288b2769adcd181b070b07d59f6": "C''(s)=\\Gamma \\,", "17229b02720250cdd7ab37279edaece1": "{\\frac {qu}{p}}=\\left\\lfloor {\\frac {qu}{p}}\\right\\rfloor +{\\frac {r(u)}{p}},", "17235296cc0d7139787e99102bf29481": "G_{i}", "1723f0367131823396fb44455f3891dd": "{2^{2^{2^{2^{2^{2^{2}}}}}}}-3", "172445a742c9032f03e56b101552d995": "R\\to Wa~~~~~~~~~~~~~{\\text{probabilities of generating 4 possible single bases on the right}}", "1724964d2765fa9bf14a4e5b07f5dfdb": "x_{0}=\\left(a+\\omega \\right)^{\\frac {p+1}{2}}\\in \\mathbf {F} _{p^{2}}", "172498163519cd368a518393e93ba7f1": "|0\\rangle \\otimes |1\\rangle ={\\frac {1}{\\sqrt {2}}}(|\\Psi ^{+}\\rangle +|\\Psi ^{-}\\rangle ),", "1724c619c44897dc6700782b3cf55f59": "\\Delta {V}=-\\int _{m_{0}}^{m_{1}}{V_{exh}\\ {\\frac {dm}{m}}}\\,", "1725021a1fd7adfbd3cba077eb93235d": "x^{14}+x^{13}+x^{12}+x^{2}+1", "172509e7156c407cab071232c7056005": "b(X^{*},X)", "17250f10921a8413163fcb1e6e551e58": "\\left({\\frac {b-a}{2}},{\\sqrt {ab}},{\\frac {a+b}{2}}\\right)=\\left({\\frac {b-a}{2}},G(a,b),A(a,b)\\right),", "172552e34863f606b13a9b5d12159573": "\\mathbf {M} =z_{0}\\mathbf {I} +z_{1}\\sigma _{1}+z_{2}\\sigma _{2}+z_{3}\\sigma _{3}", "17257fb51e468cf116b2851ff4fd878a": "\\ell (\\alpha ,x_{\\mathrm {m} })", "1725a9a1f22f0f8f34419e0a68301cf8": "\\displaystyle S=\\int d^{d}x\\left\\{{\\frac {1}{2}}\\left(\\nabla \\phi \\right)^{2}+u\\phi ^{4}\\right\\},", "1725df284746a3494c732dc688b5be96": "\\langle {\\overline {\\psi }}\\gamma _{0}\\psi \\rangle ", "17269924657084fe6f89b4e5cb37ef46": "(xy)^{2}=x^{2}y^{2}[y,x][[y,x],y].\\,", "1726cfafae44140b6606fd400f6a4ea4": "{\\hat {A}}=(A,B)=a_{0}+a_{1}i+a_{2}j+a_{3}k+b_{0}\\epsilon +b_{1}\\epsilon i+b_{2}\\epsilon j+b_{3}\\epsilon k,", "1726fd7e38959a179460d2de6236de2d": "O\\left(M\\cdot N\\cdot \\max(M,N)\\right)", "17274951c9305fb389a1735655fc5528": "Q_{\\lambda }", "172794e66f60a2bd89c30f9058533193": "P^{h}(S)=\\inf \\left\\{\\left.\\sum _{j\\in J}P_{0}^{h}(S_{j})\\right|S\\subseteq \\bigcup _{j\\in J}S_{j},J{\\text{ countable}}\\right\\}.", "1727b9c08154370a391b2e28b8237b03": "\\Gamma (W)", "1728302e5fa24c84432f55cdd4949cbe": "f_{\\pm }", "172885e5b9d0cde8933ea03c41747df4": "\\mathbf {Z} /4\\mathbf {Z} ", "17288667b04c1724e099c79bcc459224": "R[It]=\\oplus _{n\\geq 0}I^{n}t^{n}", "1728f3d25e0cccf13ceb8553926b64e1": "V_{GB}", "17294f613b0fd19f088e163d30eacfaf": "{\\mathcal {B}}", "17296396be5e5c482b8393fbb5ca3058": "\\ \\Delta C_{p}", "1729b8c9500bfbb58a35601690afb2db": "{\\binom {x}{n}}={\\frac {x^{\\underline {n}}}{n!}}={\\frac {x(x-1)\\cdots (x-n+2)(x-n+1)}{n(n-1)\\cdots 2\\cdot 1}}.", "1729d92b1e53c65e265b69bd586bd7ca": "V_{R}(s)={\\frac {R}{R+1/Cs}}V_{in}(s)={\\frac {RCs}{1+RCs}}V_{in}(s)", "172a13fefd9f6ef964da471648ef8d0d": "A_{m}(0,2)=1,2,1", "172a5c063150e87a4986e5332316bb1a": "R\\in \\Gamma ", "172adeae62e42a3f931142a3b6c947cb": "\\mathbf {B} ={\\frac {1}{2}}\\sum _{j=1}^{N}\\mathbf {Q} ^{*}\\left(z_{j}^{2}-z_{j-1}^{2}\\right)", "172af6e98ac8d53d4d370e9880b69aa2": "T_{e}\\equiv \\left[{\\frac {S_{0}(1-\\alpha _{p})}{4\\sigma }}\\right]^{1/4}", "172b188a189ff9160ce3584be3faed94": "-D", "172b3c5b6bfbcaf4e31e2d76c7bbf7fc": "(b)\\subseteq (a)", "172bb0ef0948f11d57b405877894c7ca": "{\\hat {H}}=\\sum _{n=1}^{N}{\\frac {{\\hat {\\mathbf {p} }}_{n}\\cdot {\\hat {\\mathbf {p} }}_{n}}{2m_{n}}}+V(\\mathbf {r} _{1},\\mathbf {r} _{2},\\cdots \\mathbf {r} _{N},t)\\,,\\quad {\\hat {\\mathbf {p} }}_{n}=-i\\hbar \\nabla _{n}", "172bb4f2e3b12c7ad45d389214920339": "\\mathbf {x} _{k+1}=\\mathbf {x} _{k}+{\\Delta t}\\mathbf {v} _{k+1/2},", "172c34e52f2ad58b8aa02d0ea051fd3f": "\\omega _{i,j}=|S_{i}\\cap S_{j}|", "172c3b691d282078cc7565a2dd97eaad": "C_{l}^{m}=(-1)^{m}C_{l}^{m\\ast }\\ .", "172c400b7a3be1beda6c5d248bebd2f6": "\\scriptstyle \\bigcap U_{n}", "172c7724d1ac7fdd5ef14cbe1f5b1cab": "{\\overline {z}}=x-iy", "172c99a29affd5ca98c168e05cb23598": "\\scriptstyle x\\in V\\subset U", "172cae39db5319717355fcf70ebfbc99": "K^{\\Omega (abc)}\\mathrm {rad} (abc)\\ ", "172cd15a60d26551b02a67cb1f85bbc9": "\\langle I\\rangle ", "172d321a9936a53cf15b27645872406b": "\\int _{a}^{b}\\!h(x)e^{Mg(x)}\\,dx\\approx {\\sqrt {\\frac {2\\pi }{M|g''(x_{0})|}}}h(x_{0})e^{Mg(x_{0})}{\\text{ as }}M\\to \\infty \\,", "172d97d4169a28ca8940502f204073a4": "\\prod \\limits _{\\scriptscriptstyle 0\\leq q0", "17315dcde1582930cd13b22ffca1b111": "0\\leq p^{ij}\\leq 1,\\ ", "1731675b9449e566b9a4a3e9461ed119": "\\lambda _{i}=a_{i}+ib_{i}", "17316f6240b76d26354ab4603876984e": "\\psi _{n}''(x)+(2n+1-x^{2})\\psi _{n}(x)=0\\,.", "173191c2af0ab653e50d4d36c8669ef9": "B_{1},\\ldots ,B_{k}", "1731df130fe4bcbf41673ccdf7032cb5": "f(n)=f(n-1)+f(n-2)", "1731e3c1b29b2944b20f9fe530039eae": "{\\frac {dy}{dx}},\\quad {\\frac {df}{dx}}(x),\\;\\;\\mathrm {or} \\;\\;{\\frac {d}{dx}}f(x),", "17327d83679cae8edb03c309813574cb": "{\\vec {s}}_{h}", "1732d134ca3ba11a60ff7a05b91ef57b": "{\\overline {4}}3m\\,", "173319ec51e0cb83d0aa8fdb9114c57b": "{\\hat {f}}_{x}(x)={\\frac {1}{(2\\pi )^{k}}}\\int _{-C}^{C}\\cdots \\int _{-C}^{C}e^{-iu'x}{\\hat {\\varphi }}_{x}(u)du.", "1733515c422ad853ae7bff1a46020627": "F[r]=\\int _{a}^{t}{\\sqrt {r^{2}+r'^{2}}}\\,dt", "17339e49e26410f83248221222de09cb": "S=\\operatorname {logit} (TPR)+\\operatorname {logit} (FPR)", "17341c72e3914addbaf2a060f24f1851": "p_{n}(z_{i})=w_{i},{\\text{ where }}i=0,1,\\ldots ,n.", "1734251c88262ea01fc9f4a0a499327b": "\\zeta _{H}(s,a)", "17343dcfd5c053e4fc0541e05b62e0d1": "K=V_{star}\\sin(i)", "1734748cf2157571a75097c150cf1318": "E_{i}=-\\beta _{ij}{\\frac {\\partial T}{\\partial x_{j}}}\\,", "17349bd2be4e84440fd0fce9024282d6": "\\langle 0|\\varphi (0)|p\\rangle ={\\sqrt {\\frac {Z}{(2\\pi )^{3}}}}", "1734dd9a5bd7daaf776bbf2aec444a9f": "\\hbar =m=e=4\\pi \\varepsilon _{0}=1", "17356bc0b465dc4f005e9667bd4b74bf": "\\{\\{1,1,0,0\\},\\{1,0,1,0\\},\\{1,0,0,1\\},\\{0,1,1,0\\},\\{0,1,0,1\\}\\}.", "173574726983ebaf30966ab11713568d": "\\mathrm {Sh} =f(\\mathrm {Re} ,\\mathrm {Sc} )", "17364994986119dca6dad2664c6820c3": "R_{2}={\\sqrt {\\frac {a^{2}+{\\frac {ab^{2}}{\\sqrt {a^{2}-b^{2}}}}\\ln {\\left({\\frac {a+{\\sqrt {a^{2}-b^{2}}}}{b}}\\right)}}{2}}}={\\sqrt {{\\frac {a^{2}}{2}}+{\\frac {b^{2}}{2}}{\\frac {\\tanh ^{-1}e}{e}}}}={\\sqrt {\\frac {A}{4\\pi }}}\\,\\!", "17365806639070f35b9d795ed6d467c1": "\\delta (a+b)=\\delta (a)+\\delta (b)", "17367a5ea3fa4e299e8f743c94d654d2": "\\lim _{r\\to 1^{-}}\\left|G\\left(re^{i\\theta }\\right)\\right|=\\varphi \\left(e^{i\\theta }\\right)", "1736816c553751b49ff1427556851138": "{\\hat {K}}_{j}(x_{1})f(x_{1})=\\phi _{j}(x_{1})\\int {{\\frac {\\phi _{j}^{*}(x_{2})f(x_{2})}{r_{12}}}dx_{2}}", "1736980c0d884fc7c686aaef2e50cddc": "P_{y}(t_{0}+t)=\\sum _{x}P_{x}(t_{0})K_{x\\rightarrow y}(t)\\,", "1736c96d8462cc67dc802fe429f285e4": "\\sin ^{2}A=1-\\cos ^{2}A", "173729637a572ff803abb41fd008de8c": "E_{x}^{\\rm {HF,SR}}(\\omega )", "17373286ba73f225a11f2f63f53153e4": "=\\sum _{j}Q_{j}e^{-E_{j}it/\\hbar }.", "173732e67770e37f938f5b0b25273def": "1=l_{B}+(1+r)l_{A}a_{B}", "173765c55d3651bf08b53ecfa4360a02": "L={\\frac {b^{2}}{a}}=a-{\\frac {p^{2}}{4a}}", "173766566526cecae92c644f88999c11": "{\\frac {\\partial }{\\partial p}}M_{p}(x_{1},\\dots ,x_{n})\\geq 0", "17378dfaf07e35d67b31fbf2bf3cf9e6": "\\displaystyle {|z_{n}-w|^{2}\\geq (|z_{n}-\\zeta _{n}|^{2}+|\\zeta _{n}-w|^{2})/2.}", "17379b12d232f584364ddec6589f5c8d": "(\\mathbb {N} ^{k},\\leq )", "173809e93ca6c16931824a8ef54422d2": "{\\boldsymbol {\\Omega }}=(0,\\ 0,\\ \\Omega )", "17381f2e1b6f22c064711a5031ff4711": "z(1-z){\\frac {d^{2}w}{dz^{2}}}+(c-(a+b+1)z){\\frac {dw}{dz}}-abw=0.", "173820b37ed0a5a230c612b461393344": "\\langle Ax,y\\rangle =\\langle x,Ay\\rangle \\quad \\forall x,y\\in {\\mathbb {R}}^{n}.", "1738c715ce1d2aafe9af90bfe0f2e1d7": "P_{2}(L)", "173961bd17d03217951462a63890254d": "{\\begin{array}{c}X\\\\Z\\\\I\\\\I\\end{array}}\\left\\vert {\\begin{array}{cccc}X&I&I&I\\\\Z&I&I&I\\\\I&Z&I&I\\\\I&I&Z&I\\end{array}}\\right.", "17396456dd02397800f8119772aa6ced": "\\mathrm {T_{High}} (f)={\\frac {jf/f_{1}}{1+jf/f_{1}}}\\ ,", "17399f7fe04b8f845b32806e6f321047": "\\bigcup _{n}{{\\mathcal {E}}^{n}}\\subseteq RP", "173a12501c478f5b3ff53c81f2bc834d": "\\det(E+(n-i)\\delta _{ij})=0,\\qquad n>1", "173a25c28578e35900262628c795b2fc": "\\forall x\\forall y\\forall z\\left((x,y)\\in R^{+}\\wedge (y,z)\\in R^{+}\\Rightarrow (x,z)\\in R^{+}\\right)", "173a9f5bc66c81d744e870b53ae1debf": "{\\mbox{Assets}}={\\mbox{Liabilities}}+{\\mbox{(Shareholders or Owners equity)}}", "173b4cbdd2b8cca7b037e5d332c0fef8": "V=-{\\mathbf {\\hat {d}}}\\cdot {\\mathbf {E}}", "173b5c7a232c4dcbf8d1dede80d7b3eb": "(I+X)^{-1}", "173b9840ef9471cc14cef7d0ff4bbcd6": "\\displaystyle {[L_{-1},A]=0,\\,\\,\\,[L_{-1},A^{*}]=A,\\,\\,\\,[L_{1},A]=-A^{*},\\,\\,\\,[L_{1},A^{*}]=0.}", "173ba88afa3df16251f954483c103981": "\\omega _{1}\\}", "173bbf5cc516192a8bbccb9f9e0df4bd": "y/a", "173bf0309223a0ac683fbac777c06dc9": "{\\bar {m}}=n-1", "173bf34eee5800f154eb445ed88673f6": "({\\mathcal {M}},s\\models \\phi )", "173c8073bfd1673a2ab083de3d2cf719": "-12>v_{g}>-20{\\mbox{km/h}}", "173ca4505aa6bf1ee817a4f11ebe7a40": "J_{Ay}=-D_{AB}{\\frac {\\partial Ca}{\\partial y}}", "173cb3e04e1fa574fa79207939ae4070": "N=\\left({\\frac {16\\pi k^{3}\\zeta (3)}{c^{3}h^{3}}}\\right)\\,VT^{3}", "173cbfb2d8c11ec7cf732ab3b7bed49d": "k=\\pi /a", "173cc10937fd02da1474c93db88af033": "{\\begin{aligned}\\omega ^{2}&=3-2{\\sqrt {2}},&\\omega ^{3}&=5{\\sqrt {2}}-7,&\\omega ^{4}&=17-12{\\sqrt {2}},\\\\\\omega ^{5}&=29{\\sqrt {2}}-41,&\\omega ^{6}&=99-70{\\sqrt {2}},&\\omega ^{7}&=169{\\sqrt {2}}-239,\\,\\end{aligned}}", "173ce9e659cb9b7dc5327e19db205553": "\\textstyle r\\in \\mathbb {Z} _{q}^{*}", "173d3129b6e5ce018b8182501754c9d2": "{\\mathfrak {gl}}_{n}(F).", "173d7fcd2d8200e05642ce657be2939f": "P={2 \\over 27}\\rho _{a}AC_{L}G^{2}V^{3}", "173d9d565e424853a1738020af455efa": "\\langle x,y\\rangle ", "173db3743590aa57d106899b4c34bcb7": "\\,{\\tfrac {-1}{2}}-i{\\tfrac {\\sqrt {3}}{2}}\\,", "173df0ffd4eb9c70507ef7554236c433": "[\\cdot ,\\cdot ]:{\\mathfrak {g}}\\times {\\mathfrak {g}}\\to {\\mathfrak {g}}", "173e28bfa8f53dd008f8555783f9ab4c": "\\mathbf {X} {\\boldsymbol {\\beta }}=\\ln {\\left({\\frac {\\mu }{1-\\mu }}\\right)}\\,\\!", "173e2dfac89205aefbe5fc527f03f518": "(x-y)^{2}=x^{2}-2xy+y^{2}", "173eb932753a4dc968906d82b84ed4b7": "t_{1}={\\frac {{\\frac {1}{2}}9h}{2m_{e}c^{2}\\alpha ^{6}(\\pi ^{2}-9)}}=1.386\\times 10^{-7}\\;{\\text{s}}", "173eca2b659d6c2c9684731bff40c02e": "(q_{1},...,q_{d(t)})", "173eda91467d59e92b43f2a1d21ee177": "N_{i}", "173f181c4e69501de095ab911751e631": "\\Pr(X=x)=P(X_{1}=x_{1},X_{2}=x_{2},\\ldots ,X_{n}=x_{n}).\\,", "173f385fe2356f7930466dcaa3b8e60b": "{\\begin{aligned}H_{2}\\left(e\\left(d_{ID},u\\right)\\right)&=H_{2}\\left(e\\left(sQ_{ID},rP\\right)\\right)\\\\&=H_{2}\\left(e\\left(Q_{ID},P\\right)^{rs}\\right)\\\\&=H_{2}\\left(e\\left(Q_{ID},sP\\right)^{r}\\right)\\\\&=H_{2}\\left(e\\left(Q_{ID},K_{pub}\\right)^{r}\\right)\\\\&=H_{2}\\left(g_{ID}^{r}\\right)\\\\\\end{aligned}}", "173f642a9f40499f67618728f0a7417e": "\\eta _{\\alpha }", "173fc88448988c94a179d9f91deee880": "z/y=x", "173fea468dfcfa2873ea7d9193925d29": "{\\frac {dF}{dL}}=2k_{\\rm {A}}{\\frac {I\\,I^{\\prime }}{d}}", "174020a4fda27b50f8bdd13ca15a4984": "V=\\oplus _{n>0}V^{n},\\,", "1740360406a39e1bde26622fe5aa9be6": "\\left\\{ie_{1},ie_{2},\\dots ,ie_{n}\\right\\},", "1740434d059957cb8d49720e9b8c4a2b": "{\\begin{aligned}\\mathrm {Ai} (x)&{}={\\frac {1}{\\pi }}{\\sqrt {\\frac {x}{3}}}\\,K_{\\frac {1}{3}}\\left({\\tfrac {2}{3}}x^{\\frac {3}{2}}\\right),\\\\\\mathrm {Bi} (x)&{}={\\sqrt {\\frac {x}{3}}}\\left(I_{\\frac {1}{3}}\\left({\\tfrac {2}{3}}x^{\\frac {3}{2}}\\right)+I_{-{\\frac {1}{3}}}\\left({\\tfrac {2}{3}}x^{\\frac {3}{2}}\\right)\\right).\\end{aligned}}", "1740488d59aadb04e1399f773a1c81ba": "F=UW_{11}\\cdot UW_{21}\\cdot ...\\cdot UW_{33}=\\sum _{i=1,2,3}\\sum _{j=1,3}UW_{i,j}=\\sum _{j=1,3}\\sum _{i=1,2,3}UW_{i,j}", "17406acdebe7b545b729e831eb48edd6": "p^{*}(E)=L_{1}\\oplus L_{2}\\oplus \\cdots \\oplus L_{n}.", "1740a8d7884ce281cf8d3886e330006c": "m=r^{n}-1", "17411bacf6215a744f9ce16a91c3770d": "|p_{k,S}^{C}-p_{k,U}^{C}|<{\\frac {1}{Q(k)}}", "17411c07a625ac4ce3f197f2d3af3688": "\\displaystyle {[v_{m},u_{n}]=m(v,u)\\delta _{m+n,0}I.}", "17419c20cb3bce78262515b00838d114": "\\mathbf {v} \\left(\\mathbf {x} ,t\\right)", "1741ade17578f33c67660b8d2deb7778": "O(\\log \\left(n\\right))", "1742282a2bd8b3ed6dbf6c1c8c811f0b": "p(x,y)\\vdash p(x,y)", "174292d27dc52d87f231ae33de57a704": "\\phi (x^{n})\\!", "1742965f5157fa75f883cd81f720681f": "\\left({\\begin{matrix}r\\\\r^{'}\\\\\\end{matrix}}\\right)_{4}=\\left({\\begin{matrix}r\\\\r^{'}\\\\\\end{matrix}}\\right)_{1}=\\left(M_{1}\\cdot M_{2}\\right)_{4}\\cdot \\left(M_{1}\\cdot M_{2}\\right)_{3}\\cdot \\left(M_{1}\\cdot M_{2}\\right)_{2}\\cdot \\left(M_{1}\\cdot M_{2}\\right)_{1}\\cdot \\left({\\begin{matrix}r\\\\r^{'}\\\\\\end{matrix}}\\right)_{1}", "1742f17f7b567d9760a4e56ac957b47a": "L_{p}(x)={\\frac {\\sum _{k=1}^{n}x_{k}^{p}}{\\sum _{k=1}^{n}x_{k}^{p-1}}}.", "17433526ae602e81c37ba6aaf3b64607": "\\ \\varphi \\sim \\varphi '", "17436d00389ef3801531d628b9511c65": "X\\sim {\\textrm {Exponential}}(1)\\,", "174388fa85a9bfa6cf9cb250994ced64": "f=\\left(1+{\\frac {\\Delta v}{c}}\\right)f_{0}", "174432b9f4bd922f90d241af67db1dc8": "{\\begin{bmatrix}{\\boldsymbol {I}}_{m}&{\\boldsymbol {0}}&{\\boldsymbol {V}}_{1}^{(t)}\\\\{\\boldsymbol {0}}&{\\boldsymbol {I}}_{m}&{\\boldsymbol {V}}_{1}^{(b)}&{\\boldsymbol {0}}\\\\{\\boldsymbol {0}}&{\\boldsymbol {W}}_{2}^{(t)}&{\\boldsymbol {I}}_{m}&{\\boldsymbol {0}}\\\\&{\\boldsymbol {W}}_{2}^{(b)}&{\\boldsymbol {0}}&{\\boldsymbol {I}}_{m}\\end{bmatrix}}{\\begin{bmatrix}{\\boldsymbol {X}}_{1}^{(t)}\\\\{\\boldsymbol {X}}_{1}^{(b)}\\\\{\\boldsymbol {X}}_{2}^{(t)}\\\\{\\boldsymbol {X}}_{2}^{(b)}\\end{bmatrix}}={\\begin{bmatrix}{\\boldsymbol {G}}_{1}^{(t)}\\\\{\\boldsymbol {G}}_{1}^{(b)}\\\\{\\boldsymbol {G}}_{2}^{(t)}\\\\{\\boldsymbol {G}}_{2}^{(b)}\\end{bmatrix}}{\\text{.}}", "17445cc02e4386c0cc3c9f3c395ef829": "c=22", "174468e4c22084ddbd425ccc0ebe19ba": "f:X\\times Y\\rightarrow \\mathbb {R} \\,\\!", "174473646a7721beb9f8452033ac5460": "U(r_{0})={\\frac {U_{0}e^{ikr_{0}}}{r_{0}}}", "174477bd1162f95bbb23a311cde01779": "i_{R}", "1744ede5a55a23cdb01a0953549ad95d": "\\beta =3+{\\sqrt {2}}", "17451fb33613dfc7cbc7c3c12a7a366f": "Q\\;=\\;C\\;A\\;P\\;{\\sqrt {{\\bigg (}{\\frac {\\;\\,k\\;M}{Z\\;R\\;T}}{\\bigg )}{\\bigg (}{\\frac {2}{k+1}}{\\bigg )}^{(k+1)/(k-1)}}}", "1745a02cc4394ce7b2685c55574b08ab": "F={\\frac {Gm_{1}m_{2}}{r^{2}}}\\ ", "1745b4953c00539ba21e44f9abc8cc97": "\\sin A={\\frac {S}{bc}}={\\frac {S}{\\sqrt {S_{A}^{2}+S^{2}}}}\\quad \\quad \\cos A={\\frac {S_{A}}{bc}}={\\frac {S_{A}}{\\sqrt {S_{A}^{2}+S^{2}}}}\\quad \\quad \\tan A={\\frac {S}{S_{A}}}\\,", "1745e9d592c2ab0e11cbd8fe427537b5": "v_{\\infty }-\\varepsilon ", "1745ef2bee802b3b9f5f0c18f44cb788": "{\\hat {r}}_{x}(\\tau )={\\frac {1}{2T}}\\int _{-T}^{T}[x(t+\\tau )-m_{x}(t+\\tau )][x(t)-m_{x}(t)]\\,dt.", "17464456145e52e94d3cd374bcd7d408": "\\alpha =\\pi /2", "174681f99032180611ec2b4174570582": "\\forall x_{1}\\dots \\forall x_{n}\\exists !y\\phi (y,x_{1},\\dots ,x_{n})", "1746cbcc187adb28414504c7f97f42af": "zero:1\\longrightarrow N", "1746f67f00e854d46fdb8878bde5fa18": "V_{\\rm {free}}(\\phi )=m^{2}|\\phi |^{2}", "17475e30b9d3a31adbf5349b24f842b4": "z\\ .", "17476c882b5f21746e6d9caa0e5a4637": "F_{6}=x,S_{6}=\\operatorname {false} ,A_{6}=\\_", "174771634f3bff27e9f60fd2b6299004": "\\psi ={\\begin{cases}\\psi _{1},&{\\mbox{if }}x<-L/2{\\mbox{ (the region outside the box)}}\\\\\\psi _{2},&{\\mbox{if }}-L/2L/2{\\mbox{ (the region outside the box)}}\\end{cases}}", "174776a4d763af020bdc7ba2284d97ee": "{\\begin{array}{rcl}{\\dot {x}}&=&-{\\frac {1}{RC}}x+{\\frac {1}{RC}}A_{c}A_{r}\\sin(\\theta _{r}(t))\\cos(\\theta _{c}(t)),\\\\{\\dot {\\theta }}_{c}&=&\\omega _{c}+g_{v}(c^{*}x)\\\\\\end{array}}", "1747843e12d3afa55996f7758e7ee45e": "y=12x-16.\\,", "1747a47991cb34d6cb09f39dd306afc1": "q_{i}=\\gamma _{i}+{\\frac {\\beta _{i}}{p_{i}}}(y-\\sum _{j}\\gamma _{j}p_{j})", "1747a5ff1f9339b3296f2d77928addd8": "i^{2}=j^{2}=k^{2}=ijk=-1", "1747f350909032aaed074d59b9ea28e0": "\\displaystyle R^{i}", "17482df55c7541f5816a0cf1f0528c60": "p_{k-1}(x)", "17483b8b7cbf9673c9b341a5777c575d": "\\,d^{2}=b^{2}-n^{2}=c^{2}-m^{2}.", "17484ba992e6f05ebc523ee77a0606ae": "\\{f_{a}(x)|0\\leq x\\leq 2^{n-1}\\}", "174859f261d1f7c9c0fd4e4cc4c4b312": "M_{r}={\\frac {c(c+\\alpha )_{r}(c+\\beta )_{r}}{(c+1)_{r}(c+\\gamma )_{r}}}.", "174894e77664dab8849d51fb3b47ece0": "q={\\tfrac {1}{2}}\\,\\gamma \\,p_{s}\\,M^{2},", "1748a220ecd820fb203b06ffb7033737": "{\\begin{matrix}\\varepsilon u^{\\prime \\prime }(x)+u^{\\prime }(x)=-e^{-x},\\ \\ 0\\in \\mathbb {Z} ^{+}", "174ae2d0892b8c69256466a362894c77": "g=d_{1}d_{2}-d_{1}-d_{2}+1", "174b1cbfc72f642f67b903c405071571": "2^{20}", "174b70c87ce3052f3b3933cce3aee74f": "H_{\\Delta }(N)=\\sum _{\\Delta }N(v(\\Delta ))\\epsilon ^{ijk}Tr{\\big (}h_{\\alpha _{ij}}h_{s_{k}}\\{h_{s_{k}}^{-1},V\\}{\\big )}", "174c05740fad5257a6361d6857eb10f5": "I_{\\Delta }=(x_{i_{1}}\\ldots x_{i_{r}}:\\{i_{1},\\ldots ,i_{r}\\}\\notin \\Delta ),\\quad k[\\Delta ]=k[x_{1},\\ldots ,x_{n}]/I_{\\Delta }.", "174c4d96165f80fa7d79b782465b107f": "{\\mbox{vec}}(\\mathbf {H} _{\\textrm {estimate}})\\sim {\\mathcal {CN}}({\\mbox{vec}}(\\mathbf {H} ),\\,\\mathbf {R} _{\\textrm {error}})", "174c52b4a94e91e2b50026a7592ef8a0": "\\kappa =ik_{x}=i{\\sqrt {\\frac {2mE_{x}}{\\hbar ^{2}}}}", "174c58df1106208177f43cad26ae0f89": "\\varphi _{s}(z)=\\int _{K}f_{s}(k\\cdot z)\\,dk", "174c9e03e4cd448e4f736b78ad0f2dcb": "\\{X_{k}\\}_{k}", "174ca2263fe946ae25c9f97fabcb5821": "\\mathbf {n} ={\\begin{pmatrix}-\\mathbf {R} ^{-1}\\,\\mathbf {t} \\\\1\\end{pmatrix}}={\\begin{pmatrix}{\\tilde {\\mathbf {n} }}\\\\1\\end{pmatrix}}", "174cd94d5d88b59ab8c3452a431b4b7e": "(a_{n})_{n\\geq 1}", "174cde27d620166ac1ec839b056b732f": "\\sum _{k=1}^{\\infty }a_{k}^{2}\\,", "174cde985f3d39e5948ab4b16539001c": "b_{jk}", "174ce8d105c077051ca06b49874f3b4a": "\\chi _{\\text{Yates}}^{2}={\\frac {N(\\max(0,|ad-bc|-N/2))^{2}}{N_{S}N_{F}N_{A}N_{B}}}.", "174d360832f8ebb914afc6e652a0bdc3": "a=(1+\\beta ^{2})~\\sigma _{y}^{2}~,~~b=0~,~~c=-{\\cfrac {\\beta ^{2}}{3}}", "174dec5f47173c47d5df2a4716159be4": "{\\boldsymbol {R}}:={\\boldsymbol {\\nabla }}\\times ({\\boldsymbol {\\nabla }}\\times {\\boldsymbol {\\varepsilon }})~.", "174e1812b3055b16729af7f8c2e7215e": "K_{\\rm {w}}=[{\\rm {H_{3}O^{+}}}][{\\rm {OH^{-}}}]=K_{\\rm {eq}}\\cdot [{\\rm {H_{2}O}}]^{2}", "174ecaac70fee70cb474d4460843453c": "\\scriptstyle x_{0}\\;=\\;1", "174ef061d89d32d78550dd2c9750b04a": "D_{KL}(P\\|Q)=D_{KL}(P\\|Q_{\\theta })+\\int _{\\mathrm {supp} P}\\left(\\log {\\frac {\\mathrm {d} Q_{\\theta }}{\\mathrm {d} Q}}\\right)\\mathrm {d} P.", "174ef4d74a2606605b883a3f167a451d": "\\scriptstyle (\\Omega ,{\\mathcal {F}},\\operatorname {P} )", "174eff7dcd101af0dd39189666f975f9": "A'=-\\ln \\left({\\frac {I_{l}}{I_{0}}}\\right)=\\alpha '\\ell =\\sigma \\ell N\\,", "174f10edc73d5e91787b56ada00ac773": "(A\\to B)\\to (\\neg B\\to \\neg A)", "174f69398458629f706bdabe0273dc9c": "10^{10^{122}}", "174f7c96898c9a63cff1391c99d00243": "\\partial _{t}u+\\partial _{x}\\left[{\\frac {u^{2}}{2}}+{\\frac {G}{2}}*\\left({\\frac {bu^{2}}{2}}+{\\frac {(3-b)u_{x}^{2}}{2}}\\right)\\right]=0.", "174fd9ce3a7423e156500a5419fb02b0": "z(r)", "174fe838e8a4cab9a3dd8605464b1193": "\\left\\langle {1 \\over r^{3}}\\right\\rangle ={\\frac {2}{a^{3}n^{3}l(l+1)(2l+1)}}", "17504fd21ddcc77ac88048c5efebff64": "{\\bar {\\theta }}_{\\mathrm {BiasCorrected} }=N{\\bar {\\theta }}-(N-1){\\bar {\\theta }}_{Jack}", "1750547d58e6cbed636c208e7ed77c46": "Y=X_{1}-X_{2}", "1750af558ad9b2dcd88469fb40956e7a": "\\delta =2\\arctan \\left({\\tfrac {d}{2D}}\\,\\right),", "1750b042c42f9c594cb18716b276eb0a": "a^{P-1}\\equiv 1{\\bmod {P}}", "1750c3b4547dddd44b5a3e54de0a5349": "K=K(1/\\xi )", "1750ef56ae16b301c0937525324c8719": "\\operatorname {Cl} _{2}(2\\theta )=2\\,\\operatorname {Cl} _{2}(\\theta )-2\\,\\operatorname {Cl} _{2}(\\pi -\\theta )\\,.\\,\\Box ", "17510d9922345041fc751157475c1cfe": "N={\\text{notional}}", "1751262e393fabde29e37eea9c80d72a": "2^{\\frac {8}{12}}x=2^{\\frac {2}{3}}\\approx 1.5874x", "175136b6b4cfd99a3f456fa7583ef49c": "Z(C_{4})={\\frac {1}{4}}\\left(a_{1}^{4}+a_{2}^{2}+2a_{4}\\right).", "1751451b6dc35f297a20feaa3ecd4f58": "\\mathbb {H} ^{n+1}", "1751a31cc2df3d4594e2a0a4e851125f": "F\\subseteq K\\subseteq E", "1751ce5de61d19323342560c39917f4a": "{\\tilde {\\eta }}={\\frac {\\eta }{h}},", "1751ef48dcc4d8d2ff3b72e77c0ca481": "\\scriptstyle 27", "17520d4d060a3cc2f959e92c458cc7a1": "\\scriptstyle {\\hat {X}}_{t}=X_{T-t}", "17522925d6d23c29bfc7d6055726f041": "{jx,jy,jz}", "175249f0f863294af34f377b2dd61654": "=(1+sT)\\left({\\frac {1-e^{-sT}}{sT}}\\right)^{2}\\ ", "17526fb08909fff52dca340b830143df": "{\\frac {c}{\\Gamma }}={\\frac {c}{\\Gamma _{max}}}+{\\frac {1}{K\\Gamma _{max}}}", "17527299cb76e2a99e2b35913a405244": "=\\int _{\\Lambda ^{m\\mid n}}f\\left(x\\left(y,\\xi \\right),\\theta \\left(y,\\xi \\right)\\right)\\varepsilon {\\frac {\\det \\left(A-BD^{-1}C\\right)}{\\det D}}\\mathrm {d} \\xi \\mathrm {d} y,", "1752755ca5dff8012ab0608cc117721b": "\\int _{\\Omega }|\\nabla u|=\\int _{-\\infty }^{\\infty }H_{n-1}(u^{-1}(t))\\,dt,", "17528c55f1ccc8b31fa4fa0b2797faaf": "{\\boldsymbol {F}}={\\begin{bmatrix}1&\\gamma &0\\\\0&1&0\\\\0&0&1\\end{bmatrix}}.", "1752a3f2513518579a5c5e1e0a2d2a71": "X={\\frac {1}{2}}(X+X^{\\top })+{\\frac {1}{2}}(X-X^{\\top }).", "1752b96aca8a0c7d9c8e8a3573696f76": "\\operatorname {lambda-anon} [\\lambda F.X]=\\operatorname {lambda-free} [X]\\lor \\operatorname {lambda-anon} [X]", "175365a9244a0c319fd719cc7249f5b5": "\\phi _{1}=N_{1}/N\\,", "1753707741af858c26810fff895890e5": "i\\leq k", "175382ee410cdc08711fd0275d240df1": "\\displaystyle {Q(a,c)a^{-1}=(L(a)L(c)+L(c)L(a)-L(ac))a^{-1}=c.}", "17539888aee6adb744e9da8ffa591d18": "\\ h_{n,m}", "1754502ab58151debbe9131124689c8c": "e^{\\pi {\\sqrt {3}}}", "175451350fed58edcb2fabab66691717": "6nK(N-K)(N-n)(5N-6){\\Big ]}", "1754694780d6338ae0e49f9574566b34": "n>N", "17548253c5b40740ab4af63ba95b1f2b": "(Y-a)", "1755166fadbc8ff2ded65af9bde12a0b": "N_{\\lambda }", "17552d8fe194c5daaae6cd4ab21deea1": "\\operatorname {def} [F_{5}]\\land \\operatorname {ask} [S_{5}]\\land FV[A_{5}]\\subset V", "17553f733ecf3df7a552b73c0e31e04c": "c=(k+m)^{2},\\,", "175547908ad406e937971a72d0ae00a0": "\\left|x-{\\frac {p}{q}}\\right|>{\\frac {A}{q^{n}}}", "17555e08fcaec4fca9e2d2668ba56d92": "{\\frac {a}{1}}={\\frac {x}{d}}.", "1755a890cde845f3d1b32367212adc0b": "\\Phi =\\int {J_{N}}\\,{\\rm {d}}t", "1755c1e41132c60a8012914f96c0c12b": "C:P\\rightarrow \\!\\!\\!\\shortmid I\\!N", "1755cc516504aa84bb2999de853350be": "\\mathrm {cos(Z)={\\frac {sin(dec)-sin(lat)\\cdot sin(Hc)}{cos(lat)\\cdot cos(Hc)}}} ", "1755ce438b523466ba5e4f4d0783cd95": "\\sum _{n=1}^{\\infty }{\\frac {{\\bar {H}}_{n}^{(b)}}{(n+1)^{a}}}=\\zeta (a,{\\bar {b}})", "175690cd0939245d7491c5d3d4aaa8ec": "\\omega _{z}", "1756b3eefaa66a19436acbcee1fb6fde": "(E\\rightarrow (I\\land R))", "1757e499c2621580e2a14d865c590d61": "\\ ,", "17581d9b08b5dbbe1d9a19fa13fb8502": "|xy|_{X}", "17582f947c63c9d42cbfb41720f1bbb1": "\\alpha ,\\beta \\in \\{1,2,3\\}.", "175849e9f7c912307cefeef1b7eb0786": "\\lim \\limits _{x\\to 0}{{\\frac {1}{x}}={\\frac {\\lim \\limits _{x\\to 0}{1}}{\\lim \\limits _{x\\to 0}{x}}}}=\\infty .", "1758745a1d746224c2edc0d9e7c012b9": "C_{q}(n,d,w)", "175881293316e72678aab82de418cd35": "B\\cdot U(\\$100)+Y\\cdot U(\\$100)+R\\cdot U(\\$0)>R\\cdot U(\\$100)+Y\\cdot U(\\$100)+B\\cdot U(\\$0)", "175892a3f4a8bf0ab19a58b05e3a0268": "\\scriptstyle z^{n}+a_{n-1}z^{n-1}+\\cdots +a_{1}z+a_{0}", "175897d733004b51f2c1544f684a3f4f": "D_{L}\\!\\,", "1758da9afa5c8280e7490f43cf60bc2a": "\\mathop {\\textbf {y}} :C\\to {\\hat {C}}", "17592756f17022bca13a655f532b760e": "E_{2}=E_{c}=6.00{\\text{ ft}}", "175941cf15dface5b5ccc6598e745f1b": "e/(e-1)-\\varepsilon ", "1759b64f14aa6a420ddfb0d46aff804c": "\\displaystyle {\\iint K(x,y)U(x)V(y)\\,dxdy,}", "175a14eff85f19c7717a85eade25564f": "C(t)-P(t)=S(t)-K\\cdot B(t,T)\\,", "175a1871bf1e4515cdf79890de61bbb7": "x^{a}\\,", "175a72f96874ebbecef1aff430a212ff": "\\rho _{L}\\gg \\rho _{V}", "175a98399a31bad8588050dd66c51af2": "p={\\frac {\\sum _{j}w_{j}}{B}}", "175ada960806d9770bca772f36da1fb2": "{\\mathcal {U}}(\\alpha ,{\\tilde {u}})\\ \\subseteq \\ {\\mathcal {U}}(\\alpha ^{\\prime },{\\tilde {u}})", "175adee607d9e5458be2e70d5c779e13": "(q^{i},{\\dot {q}}^{i})", "175b01aab4742b658515aa97b5b1f46e": "\\scriptstyle {\\mathsf {Boolean}}", "175b09eaffbf4ab560223e56eb2b99f1": "S\\setminus \\{0\\}", "175b357719fd5e327f1b35f9341d8993": "{\\frac {\\partial }{\\partial {z}}}P_{c}^{p}(z_{0})", "175b43734dd6f6e1bfbf1b2e0807af43": "x_{a}({\\hat {t}})={\\frac {1}{2\\pi ^{M}}}\\int _{-\\infty }^{+\\infty }\\!X({\\hat {\\Omega }})e^{(j{\\hat {\\Omega }}^{T}t)}\\,\\mathrm {d} {\\hat {\\Omega }}", "175b77f3130b96d739593010c7e367e1": "\\scriptstyle h\\left(\\varepsilon \\right)\\,=\\,0", "175bc361ec8dd6641b83df29547bd50b": "\\beta \\;", "175be3872bda5b46254c095f0e5a4041": "{\\hat {s}}\\longleftarrow {\\hat {k}}z+{\\hat {l}}", "175c2a7728d40da4abed6466b5750475": "{\\hat {a}}^{\\dagger }{\\hat {b}}^{\\dagger }|0,0\\rangle _{ab}=|1,1\\rangle _{ab},\\,", "175c917933eb5001461d8fb66e8a8d6f": "P(y)\\rightarrow Q(y)", "175cb6e3faeaa0ebffb7242bcb221c66": "\\pi (\\theta )\\,\\!", "175d1fdd05ac104c7e529f3fb33ca0f7": "V=-\\alpha /r", "175d2d37002a91a298e5ef1f04d59760": "\\displaystyle {K(\\mathbf {v} (t)+\\lambda \\mathbf {n} (t),\\mathbf {v} (t))=-{1 \\over 2\\pi \\lambda },}", "175d40d97c68675d5042ddbd1e001446": "S_{z_{l}}", "175d7cdd44d3f83be8ed5ef305ac03c5": "0<\\int _{0}^{1}{\\frac {x^{8}(1-x)^{8}(25+816x^{2})}{3164(1+x^{2})}}\\,dx={\\frac {355}{113}}-\\pi .", "175ec79741187ccb0d653bf0d4496ef5": "f:E\\to R", "175ecc60bca959c190842b52a64fde1b": "\\exists x_{1}\\,\\forall x_{2}\\,\\exists x_{3}\\,\\forall x_{4}:(x_{1}\\lor \\neg x_{3}\\lor x_{4})\\land (\\neg x_{2}\\lor x_{3}\\lor \\neg x_{4})", "175f6b9d8ddc0312e81448a7d7e7a5e4": "K\\subset G", "175faeda2910b65061d640f59c5bc596": "{W_{I}(\\mathbf {r} ,t)}=<\\psi \\mid {E^{(-)}(\\mathbf {r} ,t)}\\cdot {E^{(+)}(\\mathbf {r} ,t)}\\mid \\psi >", "175fb02dc71571bb97cf42967a26105c": "v_{1}", "175fcebc8deacf3546044cc17448eaba": "\\det {\\frac {\\partial }{\\partial (q_{0},p_{0})}}\\Phi _{{\\mathrm {eE} },h}(z_{0})={\\begin{vmatrix}1&h\\\\-h\\cos q_{0}&1\\end{vmatrix}}=1+h^{2}\\cos q_{0}.", "175fd06d917042e11bfaafd50673d327": "\\psi _{0}\\left(\\mathbf {r} ,t\\right)=A_{u}e^{i\\left(\\phi _{u}+2\\pi \\nu t+\\mathbf {k} \\cdot \\mathbf {r} \\right)}", "175ff65b914e27276738ba9ffbe19e3d": "\\Delta k=k_{1}-k_{2}\\,\\!", "176020b4240418af973b5269eb2a6065": "v^{i}{}_{,j}", "17602a036765eef44433837afdb6ad01": "(\\lambda p.(\\lambda q.q\\ p)\\ \\lambda p.\\lambda f.(p\\ f)\\ (p\\ f))\\ \\lambda f.\\lambda x.f\\ (x\\ x)", "17605398a1ab4b49b4c0d1921530c956": "(Z,X-{\\widehat {X}})", "1760b5914bafbb3cf75543953b246a56": "V(y)=\\sum _{i,j}{\\sqrt {|y_{i+1,j}-y_{i,j}|^{2}+|y_{i,j+1}-y_{i,j}|^{2}}}", "1760de4b0e1885f437ef5fe60131ffb0": "d_{k+1}={\\frac {\\pi }{4}}\\sum _{j=0}^{k}{\\frac {d_{j}d_{k-j}}{(j+1)(2j+1)}}", "1760e159d6228519538e3dffe9325aed": "\\eta (x,t)\\,=\\,a\\,\\cos \\,\\left(kx\\,-\\,\\omega t\\right)", "17615a2e243f16be6ecef9feb4ef87db": "\\scriptstyle \\ {\\hat {p}}=0.05", "1761717b355ea511f94484ad463a8f52": "L=\\sum _{i=0}^{\\infty }T^{i}L^{e}", "17618083689bf13313acbd196bb685e0": "={i \\over \\hbar }\\left(HA(t)-A(t)H\\right)+e^{iHt/\\hbar }\\left({\\frac {\\partial A}{\\partial t}}\\right)e^{-iHt/\\hbar }.", "1761e24704946375173598549ecd1012": "-\\log _{2}\\left[1-{\\frac {1}{(k+1)^{2}}}\\right]", "1761f60bdeaf2d88bdec534f514c9336": "\\gamma ={\\frac {1}{\\sqrt {1-{\\frac {v^{2}}{c^{2}}}}}}", "176201a26fe6f1bacb2a114fae9ae83b": "Z_{3}=T_{1}Z_{1}T_{2}Z_{2}-kX_{1}Y_{1}X_{2}Y_{2}", "17620ea387c9667fc488f53cbc0913f6": "\\sup _{f}\\inf _{g}\\int \\int K\\,df\\,dg=\\inf _{g}\\sup _{f}\\int \\int K\\,df\\,dg", "17622817a410c06cb3fda67ad352c896": "{\\frac {\\partial \\phi }{\\partial x}}={\\frac {2x}{r_{0}^{2}}}{\\big (}U+V\\cos \\Omega t{\\big )}.\\qquad \\qquad (8)\\!", "1762491f4275aa9bea2e2db2b2cdb56a": "f+g=(f\\vee g)\\circ \\Psi ", "17627090466ae824e1436544cf7736a3": "(V\\oplus W)_{0}=V_{0}\\oplus W_{0}", "176279e9ba3c959a7c2ad5ac5e3b31c0": "T\\cong R/P_{1}^{a_{1}}\\oplus \\cdots \\oplus R/P_{r}^{a_{r}}\\cong R/Q_{1}^{b_{1}}\\oplus \\cdots \\oplus R/Q_{s}^{b_{s}}", "17628416b3b5fb3e3a9650b13dc37632": "s_{m}(z)=a_{m}\\cdot (2\\cdot m+1)\\cdot (1+z^{(-1)^{m}})", "17628862c345dee13e3753bc9404d951": "M=\\sum _{i=1}^{K}(f_{m}-f_{i})", "176299cad60ea69f96dc88a9a6eb354e": "R(f,{\\hat {f}})=E\\|f-{\\hat {f}}\\|^{2}.\\,", "1762d0e1d148f6f38ba399d6cde28f22": "\\mathbf {\\nabla } \\cdot \\mathbf {D} =\\rho _{f}", "1762f2b3c74cc1e91ebc4f9e3f3651c1": "E=(1/2)mv^{2}", "1762f998dd0adbca0ce21062820cebd2": "{\\begin{aligned}(x_{T}*h)(t)\\quad &{\\stackrel {\\mathrm {def} }{=}}\\ \\int _{-\\infty }^{\\infty }h(\\tau )\\cdot x_{T}(t-\\tau )\\,d\\tau \\\\&\\equiv \\int _{t_{o}}^{t_{o}+T}h_{T}(\\tau )\\cdot x_{T}(t-\\tau )\\,d\\tau ,\\end{aligned}}", "1762faaa922d868294a5ab1609394fa7": "x\\,\\partial _{v}+u\\,\\partial _{x}", "1763106a890a247a6c993683bb6951de": "S_{i+1}=S_{i}+X_{i+1}-1=\\sum _{j=1}^{i+1}X_{j}-i", "17638b2cfef7dad323f09188465ed3bc": "{\\infty \\choose r}_{q}=\\lim _{m\\rightarrow \\infty }{m \\choose r}_{q}={\\frac {1}{[r]_{q}!\\,(1-q)^{r}}}", "17640101c8f06c39d3f1693066b19980": "V_{t}\\,", "176404b457074913779b56dc87b5e733": "f(x)=\\sum _{i=0}^{\\infty }f_{i}^{(\\alpha )}L_{i}^{(\\alpha )}(x).", "17644ab31f043b5d89fa42bf3faf3428": "\\varepsilon _{xx}^{\\mathrm {face} }\\equiv \\varepsilon _{xx}^{\\mathrm {face} }(z)~;~~\\varepsilon _{zx}^{\\mathrm {core} }=\\mathrm {constant} ", "176485ce5fde0a9ef40e9381a49d1b01": "SS_{AB}\\equiv {\\frac {\\sum _{ij}Y_{ij}({\\bar {Y}}_{i\\cdot }-{\\bar {Y}}_{\\cdot \\cdot })({\\bar {Y}}_{\\cdot j}-{\\bar {Y}}_{\\cdot \\cdot })}{\\sum _{i}({\\bar {Y}}_{i\\cdot }-{\\bar {Y}}_{\\cdot \\cdot })^{2}\\sum _{j}({\\bar {Y}}_{\\cdot j}-{\\bar {Y}}_{\\cdot \\cdot })^{2}}}", "1764d984660f71b3ca6fc710e423b397": "Q(t_{1})", "1764f63cabdd74d5073884fc55a2439d": "(X:Y:Z)", "17653a5dd5025b5896481bedea79b2da": "{}+2432902008176640000.\\,\\!", "176569038972377570bd31259dd891a7": "{\\overline {y}}_{x}={\\frac {\\sum _{i}y_{xi}}{n_{x}}}", "17656ba78da00c251e7f970e3fde9dd8": "\\Delta _{\\sigma }(i\\omega _{n})=\\sum _{p}{\\frac {|V_{p}^{\\sigma }|^{2}}{i\\omega _{n}-\\epsilon _{p}}}", "17658e936aa274650caa657853f14be7": "L_{p}=\\sum _{e\\in p}L_{e}", "1765cd3fb116ffe3eba6be27a5b07311": "\\scriptstyle {L\\ll \\lambda /2}", "1765d6e7f7e6cd1d4e1c6cca31874546": "C_{2n}", "1765e8f89e1c7c1b5461b43512fad4fd": "h:P\\to N", "17661d4747c70f6cac9b4dcf367593dc": "c_{2}<4.858.", "176644ed9355fa81569a1d04a2feb34b": "(x_{0},\\lambda _{0})", "17664c8b2270579567ebf180edfa497f": "{\\frac {\\partial }{\\partial \\theta }}\\int _{-\\infty }^{+\\infty }f(x;\\theta )\\,dx={\\frac {\\partial }{\\partial \\theta }}1=0.", "176696e9a382536728fc2404f4975923": "X^{(k+1)}\\gets X^{(k)}-t_{k}t^{(k)}{p^{(k)}}^{T}", "1766a950d5bf3e30e9a8d4e3569dc658": "{\\mathcal {I}}_{X},{\\mathcal {I}}_{Y}", "1766cf8974a990000e9268a5232f7efa": "Tavg=1*p(1)+2*p(2)+3*p(3)+...+n*p(n).", "176745d0595b87280eb0bd893e3d72c4": "F[y]={\\frac {y''}{(1+y'^{2})^{3/2}}}", "176758722b72783cd4d234c82901d1c2": "\\operatorname {cov} (X_{i},X_{j})=-np_{i}p_{j}\\,", "1767d92a794bc11bac54f591d2827040": "\\mathbf {F} =I\\int \\mathrm {d} {\\boldsymbol {\\ell }}\\times \\mathbf {B} ", "1767fd7908e742adc3decc1f50a4e30a": "\\theta _{E}={\\sqrt {{\\frac {4GM}{c^{2}}}\\;{\\frac {d_{LS}}{d_{L}d_{S}}}}},", "176805ccb06ef2f1aad08fb1fa0eda35": "|00\\rangle ", "17681d9f74bcaf4496441d6764fdf9ee": "{\\bar {I}}", "17688b5953b212ce73fac400bd26a9f3": "{\\hbox{GrossEnergyYield}}\\div {\\hbox{EnergyExpended}}=EROEI", "17689202824b609470b454d76bddd6c1": "\\mathbf {T} ^{(\\mathbf {n} )}=\\lambda \\mathbf {n} =\\mathbf {\\sigma } _{\\mathrm {n} }\\mathbf {n} \\,\\!", "1768e97ca4bf9a473b3887ca2cd169fa": "{\\color {white}-}\\nabla ^{2}\\mathbf {H} =\\sigma {\\frac {\\partial \\mathbf {B} }{\\partial t}}.", "1769440372afd113afa3dfe3dbb6bd13": "r_{i}=p_{i}+q_{i}", "1769b1634e330cfd2cfb6c79d2a7b1bf": "x_{11}\\ ", "1769bc7411ec457889728713eb45ba1a": "{\\frac {p}{\\Lambda _{\\chi }}},{\\frac {m_{\\pi }}{\\Lambda _{\\chi }}}.", "1769ce0d7c6ba7e4e3ba4b3484914301": "L_{i}=S_{i}/S_{n}\\,", "1769d22938331cc5da5beda0790a949e": "1,173,696\\,", "1769efe85750b874aa755898ad7e7832": "\\displaystyle {\\frac {1}{\\sqrt {n}}}X_{n}", "1769f59c2429883213f99bde95fc0b31": "P_{m+1}(x)=-\\left((m+1)xP_{m}(x)+(1-x^{2})P_{m}^{\\prime }(x)\\right)", "176a7bf83331ccf31c470c2708c9f27f": "\\cdots \\rightarrow P_{2}\\otimes _{R}B\\rightarrow P_{1}\\otimes _{R}B\\rightarrow P_{0}\\otimes _{R}B\\rightarrow 0", "176aa7194e39eba5f0e5d95556d19286": "\\textstyle \\sigma _{k}=A_{k}(\\Delta )", "176ac6c823791503a45e40e4889baf49": "\\equiv \\!\\,", "176af80d345d94230eb34a21552188e3": "\\mathbf {y} =y_{1},\\ldots ,y_{n-d}", "176b0cdbad28c96c3eb0bb4a353b1b26": "|\\vartheta (x)-x|=O(x^{1/2+\\varepsilon })", "176b2782706ca28a8c15347ec6109d8d": "{\\sqrt {1-\\omega ^{2}\\,r_{0}^{2}}}", "176b3ce05d8c52640ba016c35513d2aa": "\\mathbf {B} =\\nabla \\phi ", "176b65da777007d9f1de53fb6b53d82c": "[accumulation]=[in]-[out]+[generation]", "176ba9414d317d91a6bd4f84229441af": "{\\frac {d\\nu (r)}{dr}}=-{\\frac {dP(r)}{dr}}\\left({\\frac {2}{\\rho (r)c^{2}+P(r)}}\\right)={\\frac {1}{r}}\\left({\\frac {8\\pi Gr^{2}e^{\\lambda (r)}P(r)}{c^{4}}}+e^{\\lambda (r)}-1\\right)\\;", "176bb9b3ad688f704cbf850f4d565b4d": "Z(t)=t-J_{N(t)}", "176bd80f7f7faa81b018f95a7704109f": "M_{\\mathbf {X} }(\\mathbf {t} ):=\\mathbb {E} \\!\\left(e^{\\mathbf {t} ^{\\mathrm {T} }\\mathbf {X} }\\right).", "176c105d84355949ba9bf580facdc8a6": "A_{i}^{\\mu }", "176c73c858e82a8946004b8a3dbe15b0": "\\ \\mathbf {E} _{J}\\cdot \\mathbf {e} _{i}=\\delta _{Ji}=\\delta _{iJ}.", "176c85ab780d0c03c7c09283455378d5": "x_{n+1}=rx_{n}(1-x_{n}).\\,", "176ca1a09fd104ec061ea0866d48a298": "{\\bigg \\}}", "176d3416e8d21df33f0a7564f9da5317": "\\Gamma _{n}", "176d36a7176bf0c35fafb07b294c0f17": "{\\hat {m}}(\\mathbf {s} _{0})", "176e1a28d6bba261ab25fc2f03057bda": "E^{2}\\propto |{\\vec {p}}|^{2}", "176e2d1e783c286e4332660dbb7b9ab4": "(M\\phi )(v)={\\frac {1}{\\deg v}}\\sum _{w:\\,d(w,v)=1}\\phi (w).", "176e3ff49c0c6a81d63c5670fbbc2c36": "(0,1)", "176e73ec56c6c7bc569799ebb258d956": "c_{\\mathrm {fluid} }={\\sqrt {\\frac {K}{\\rho }}}", "176e90fb1e16a8c389165e9a080d6e74": "mc^{2}=hf", "176ed33e2972bb05f049c90b7d3684ae": "var_{ab}(p)=var_{ba}(p)\\geq \\rho _{ab}(p).", "176f2138d490a253db52f22991735fa4": "x={1 \\over 2}", "176f2b311ff7838e01b3897686a9cf42": "r={\\frac {a(e^{2}-1)}{1+e\\cos \\theta }}", "176f88c4442da55bd38348867589f832": "\\,{\\mathbf {U} }_{||}={\\mathbf {V} \\cdot \\mathbf {U} \\over |\\mathbf {V} |^{2}}\\mathbf {V} \\ ,\\quad {\\mathbf {U} }_{\\perp }=\\mathbf {U} -{\\mathbf {U} }_{||}", "176fcbf217e88e2cd699969dba3f9560": "\\lim _{n\\to \\infty }\\ _{nominal}\\alpha =\\pi ", "176fdd253124b72dfe88010ab78a4875": "\\left[\\left.{\\begin{array}{cccc}1&0&0&0\\\\0&0&0&0\\\\0&0&0&0\\\\0&0&0&1\\end{array}}\\right\\vert {\\begin{array}{cccc}0&0&0&0\\\\1&0&0&0\\\\0&1&0&0\\\\0&1&1&0\\end{array}}\\right].", "17703819644d613ad6612c213f82ca45": "P_{A}={\\frac {e^{v_{A}}}{1+e^{v_{A}}}}", "177055d2a03702ee3f24b1c612d51a02": "\\rho _{0}=z_{0}=0", "17705fa5092e3c0e5ecb4eff57649048": "\\sigma _{1}>(\\sigma _{2}=\\sigma _{3}=0)", "17706a5eeda0f3d308d147e4a236e853": "F_{max}=mg+{\\sqrt {(mg)^{2}+2mghk}}=mg+{\\sqrt {(mg)^{2}+2mgEqf}}", "1770742ef37d454091b5a8a77715dc1d": "s_{1}=h_{1}(g_{\\boldsymbol {\\theta }}(z_{1}),\\ldots ,g_{\\boldsymbol {\\theta }}(z_{m}))=\\rho _{1}({\\boldsymbol {\\theta }};z_{1},\\ldots ,z_{m})", "1770c6a3f164b7b8545cdc534c92ed7c": "(\\cos x+i\\sin x)^{n}=\\cos nx+i\\sin nx.", "1770efac20a3b8c84aa0d011327395ec": "d=L+C_{p}", "17712dd223c37c2448a5977f4ee8d2da": "\\displaystyle {\\phi _{\\lambda }=b(\\lambda )^{-1}M_{1}\\Phi _{\\lambda }}", "1771922ba3f2f5259277a0da11c7ce71": "I_{\\rm {sp}}={\\frac {v_{\\rm {e}}}{g_{\\rm {0}}}}", "1771b95c53a26c3ef3b49a8db54a6831": "K(u)=-1,", "1771f03abd4e4befe6bafc39ddf672af": "L\\subset K", "17728c4f74c62fed1eb7dde9001a354c": "T/T_{F}", "1772d4458ea641aea33fb110181d6252": "Q\\times (\\Sigma \\cup \\{\\varepsilon \\})\\times \\Gamma ", "1772d85d55fdafb8073bf06e58643fa3": "12^{m}", "1772f5e3fa387f41c4d370a50f55fb0a": "\\lim _{r\\rightarrow \\infty }\\epsilon _{0}r^{2}E_{r}(r,\\theta ,\\phi )", "1772fe8c31c29d51809d32973cf44367": "\\mathbf {R} =(R_{x},R_{y},R_{z}),\\quad \\mathbf {P} =(P_{x},P_{y},P_{z})\\quad {\\hbox{with}}\\quad P_{\\alpha }\\equiv \\sum _{i=1}^{N}q_{i}r_{i\\alpha },\\quad \\alpha =x,y,z.", "1773482855a272c7f6cbb0016fa025c3": "\\,\\mathrm {st} (u_{H}-u_{K})=0", "177384b1214cbeb26b7ceeae2bf5e1dc": "{\\bar {\\rho \\,\\!c}}", "17738a7e29a06ff14b41547b42cab0ad": "I_{\\alpha -{\\frac {1}{2}}}", "177424a3c885eb3fa306394e3214aba0": "\\nabla \\cdot \\mathbf {u} =\\nabla ^{2}\\phi \\qquad ({\\text{since,}}\\;\\nabla \\cdot \\mathbf {u} _{\\text{sol}}=0)", "177426167a2bcf74f813ef638dac959a": "C_{QL}=e^{2}\\cdot D_{2D}=\\xi \\cdot C_{Q0}", "17744d29ff700a64df8f9c36cb388e05": "ds^{2}=-Fdv^{2}+2dvdr+r^{2}(d\\theta ^{2}+\\sin ^{2}\\!\\theta \\,d\\phi ^{2})\\,,\\;\\;{\\text{with }}F\\,:=\\,{\\Big (}1-{\\frac {M}{r}}{\\Big )}^{2}\\,,", "17744e18bcb7a165d7853fd826590946": "Q\\propto I^{2}\\cdot R", "177489ca0526a65cfabd1ee858229ede": "{\\hat {T}}", "1774b4985d74a9bc5610a4a058b8257d": "{\\ddot {x}}_{\\alpha }(t)={\\dot {v}}_{\\alpha }(t)=F(v_{\\alpha }(t),s_{\\alpha }(t),v_{\\alpha -1}(t))", "17753508725fc4f161632f3e013643fa": "A_{1}+\\cdots +A_{n}", "177577158c966ab6acba2ea262931b40": "B_{k}+{\\frac {(y_{k}-B_{k}\\,\\Delta x_{k})(y_{k}-B_{k}\\,\\Delta x_{k})^{T}}{(y_{k}-B_{k}\\,\\Delta x_{k})^{T}\\,\\Delta x_{k}}}", "1775d28b9d25df4fe462d94a5ebe9b85": "F_{4}^{(1)}(a):={\\begin{bmatrix}1&1&1&1\\\\1&ie^{ia}&-1&-ie^{ia}\\\\1&-1&1&-1\\\\1&-ie^{ia}&-1&ie^{ia}\\end{bmatrix}}{\\quad {\\rm {with\\quad }}}a\\in [0,\\pi ).", "1775f7f0ab98e038920e8dea58d951d8": "s_{1}\\cap s_{2}", "1776b3cd9fe220c1aa21653dd7c74534": "\\lim _{x\\to -\\infty }e^{x}=0.\\,", "1776ed614400a592d1878ad01f6c9453": "F_{R}=dE_{R}/dz\\,", "177710f052a9e58fefe56b52bb2f69c5": "(A\\cap C)B/(A\\cap D)B", "17775e4524ddb0f6401bfc9880e6fcad": "{\\begin{bmatrix}-1&0&0\\\\0&-1&0\\\\0&0&1\\end{bmatrix}}", "1777977b30493e6f7287ea6589b701d3": "\\Psi _{A}(1,2,\\dots ,N_{A})", "177797ba021505cd7e5210e726cbc190": "M_{1}=\\int _{0}^{r_{B}}{\\frac {4\\pi \\rho (r)r^{2}}{\\sqrt {1-2GM(r)/rc^{2}}}}\\;dr\\;", "1778714680e63da209e4e6bfeb82315d": "s_{\\max }\\sim |p-p_{c}|^{-1/\\sigma }\\,\\!", "1778ee59ad7a3daf9b8b87d899f5f671": "\\mathrm {Dir} (\\alpha )", "1778f282ca3754aa74a2cc1eecbd52b7": "{\\frac {1+\\mathrm {Interest} }{1+\\mathrm {Inflation} }}=1+\\mathrm {Real} ", "17790cb45f1c34ae9e690a3ec2eafd04": "\\rho _{He}", "1779a593abb219f5dc33d7bfdc2ca4fd": "\\phi _{n}(t)=a_{n1}e^{-\\alpha _{1}t}+a_{n2}e^{-\\alpha _{2}t}+\\cdots +a_{nn}e^{-\\alpha _{n}t}", "1779f3405a94f64dfe8d80c7721d87be": "\\delta {\\hat {\\mathcal {O}}}", "177a075552116b2c5a897ce885ba1955": "{\\frac {d}{dx}}=i{\\frac {1}{i}}{\\frac {d}{dx}}", "177a11d2ed5d8dd744348a0ef108bcb5": "\\sum _{x:x\\neq y}2^{-D(x,y)}\\leq 1,\\;\\sum _{y:y\\neq x}2^{-D(x,y)}\\leq 1,", "177a732126b9e041aeaa34f4f67b5f28": "{\\Delta }E=A-I\\,", "177adae3f347234e5b924f9bc07c2744": "h^{-1}:B\\to A", "177b2d5b45d574d3ee2c3f97b4ec7c90": "\\det(b_{i+j-2})_{1\\leq i,j\\leq n+1}=\\det(c_{i+j-2})_{1\\leq i,j\\leq n+1}.", "177b606113ef85fa923e9f8862e33686": "\\cdots \\xrightarrow {\\delta _{n}+1} C_{n}(E)\\xrightarrow {\\delta _{n}} C_{n-1}(E)\\rightarrow \\cdots \\rightarrow C_{1}(E)\\xrightarrow {\\delta _{1}} C_{0}(E)\\xrightarrow {\\varepsilon } Z\\rightarrow 0,", "177ba492474125cd459f2c1b25aa1d22": "\\ln \\gamma _{i}^{C}=(1-V_{i}+\\ln V_{i})-{\\frac {z}{2}}q_{i}\\left(1-{\\frac {V_{i}}{F_{i}}}+\\ln {\\frac {V_{i}}{F_{i}}}\\right)", "177bacd5d6b71dfe71960cee0973f27c": "e_{1}+e_{2}+\\ldots +e_{k}=d_{k}", "177bbd1739d89b8478b8cf5f59c8465c": "\\mathbf {S} ={\\frac {1}{2}}\\left[{\\begin{array}{cccc}0&0&1&-1\\\\0&0&-1&1\\\\1&1&0&0\\\\1&1&0&0\\end{array}}\\right]", "177bdabc21ecb8bf905cf74432062a39": "X_{-\\infty }^{-1}", "177c1ae1ba2c169df6b1ea712a78a42e": "\\partial \\circ M\\circ (\\eta \\times \\operatorname {Id} )=M\\circ (\\eta \\times \\partial )", "177c221d2d445974750486c15bbcdad5": "({\\sqrt {a}}x+{\\sqrt {c}})^{2}", "177c30c62f58fc8a3ad3ffbad159a6a7": "h_{k-1},h_{k},v_{i}", "177cb9f826acd274e783e78b93813147": "A=(10+{\\sqrt {{\\frac {5}{2}}(10+{\\sqrt {5}}+{\\sqrt {75+30{\\sqrt {5}})}}}})a^{2}\\approx 17.7711...a^{2}", "177d378a365010537be94217cae9f35d": "|N|={\\frac {E_{Y}/B_{Z}}{dT/dx}}", "177d3caa3dd1021bb0b0dad1b0c145ad": "\\sin \\beta =\\sin \\delta \\cos \\epsilon -\\cos \\delta \\sin \\epsilon \\sin \\alpha ", "177d4cf5ae3516971f738c112d484efc": "\\lambda ({\\boldsymbol {r'}})", "177d55e40e9e8ed721129cd5248484a9": "\\{\\mu \\}", "177e30cf2f489eb4d125da0d514e1290": "R_{\\mu \\nu }-{\\frac {1}{2}}g_{\\mu \\nu }R+\\Lambda g_{\\mu \\nu }={\\frac {8\\pi G}{c^{4}}}T_{\\mu \\nu }\\,.", "177e6387b133e204f24828511a0f6029": "\\scriptstyle {v_{i}}", "177f3996aeb76e98c1222e5965f549d8": "B\\left(\\omega \\right)", "177fb9f61549d60681a22c28297a536f": "\\Lambda _{\\epsilon }(A)=\\{\\lambda \\in \\mathbb {C} \\mid \\exists x\\in \\mathbb {C} ^{n}\\setminus \\{0\\},\\exists E\\in \\mathbb {C} ^{n\\times n}\\colon (A+E)x=\\lambda x,\\|E\\|\\leq \\epsilon \\}.", "177fc0526bf0aa11a448148794b5146d": "V\\oplus iV,", "177ff7a0a8afd8ed925d19883df14b09": "\\mathbf {P} _{A}^{n}", "17802d1c3257f9eb535d022de318b865": "{n \\choose k-1}+{n \\choose k}={n+1 \\choose k}.", "178124b1758d16949e81aa1f44d98eee": "\\lim _{n\\to \\infty }ca_{n}=c\\lim _{n\\to \\infty }a_{n}", "17816666f24174bfe6ad635fcca8c329": "L^{2,h}(D)=L^{2}(D)\\cap H(D)", "178180505bbefefa1b9e2df946db4b7b": "\\tau _{y\\eta }\\,\\!", "17819b8266769abbc6cfdab5b9bd13ec": "C\\to C[W^{-1}]", "1781bffe49946243e024aa08d1875e83": "\\mathbf {x} ^{(k+1)}==\\mathbf {x} ^{(k)}", "178205c9bd089ef5c4753e28023874f3": "f\\,'(x_{*})=0", "17826de19259fe610a2e507dbde16d9a": "1264460", "178290ff5e89e36da41ad880c703ffdc": "{48 \\choose 5}=1,712,304", "1782a68bf03aa518c0133b3a0f719fbe": "\\nabla \\cdot \\mathbf {A} ", "1782b4feb18f4ff4a707d9b654b9a42c": "x>-1", "1782c20027d6396a72ff230f10a56a13": "E_{j}\\ ", "1782c7cf7d537d897cfb4f49d114237c": "\\left[B\\right]=\\left[A\\right]_{0}{\\frac {k_{f}}{k_{f}+k_{b}}}\\left(1-e^{-\\left(k_{f}+k_{b}\\right)t}\\right)+\\left[B\\right]_{0}{\\frac {1}{k_{f}+k_{b}}}\\left(k_{f}+k_{b}e^{-\\left(k_{f}+k_{b}\\right)t}\\right)", "1782d655042b1dde95098b237cbc2198": "\\left\\lfloor {\\frac {a}{n}}\\right\\rfloor \\,", "17830770eb29367b1aa42df9232cc24c": "c\\left({\\text{largest monomial of }}s_{1}\\right)^{i_{n}-i_{n-1}}", "1783285049bdf2b33440c1ef59dbb7d4": "y-p(x)=0.\\,", "17838356b676a558ddab46ff3f3541ec": "\\forall _{f}", "1783a6cef146a7924ae6050425441f7b": "{\\begin{bmatrix}1&0\\\\0&0\\end{bmatrix}}:\\mathbf {b} ", "1783af5d653d2c8f657d293124bc9c16": "{\\tilde {f}}(x):=\\inf _{u\\in U}f(u)+{\\text{Lip}}(f)\\cdot |x-u|,", "1783b5c03568f5881cbeab4c314e2f71": "p_{i}\\leq 1+\\sigma (p_{1}^{\\alpha _{1}}\\dots p_{i-1}^{\\alpha _{i-1}})=1+\\prod _{j=1}^{i-1}{\\frac {p_{j}^{\\alpha _{j}+1}-1}{p_{j}-1}},", "1783c04045ff795c6a08356457e01e5f": "T_{20}=A+{\\frac {Bk_{2}}{k}}", "1783f0a6148a058095693dd0911a6846": "G/\\omega C", "1783f494cee67967e04e7339f7051cfd": "\\varphi _{j}^{n}\\quad ", "1784389e5b4552af6e1f5efc07bcc924": "A={\\frac {G}{3-G}}", "1784993a5398a5edf91ac7b5cdeb7838": "M=0", "17849dc9dc7804ba75421a404bd6584c": "1/4", "1784a093ef579a1471da686ea4f67641": "(11)\\quad {\\hat {\\sigma }}^{2}={\\hat {\\sigma }}_{ab}{\\hat {\\bar {\\sigma }}}^{ab}={\\frac {1}{2}}\\,g^{ca}\\,g^{db}\\,k_{(a\\,;\\,b)}\\,k_{c\\,;\\,d}-{\\Big (}{\\frac {1}{2}}\\,k^{a}{}_{;\\,a}{\\Big )}^{2}=\\,g^{ca}\\,g^{db}{\\frac {1}{2}}\\,k_{(a\\,;\\,b)}\\,k_{c\\,;\\,d}-{\\hat {\\theta }}^{2}\\;.", "1785004d270166b673ae4b9655e64c5c": "{\\boldsymbol {\\mu }}_{S}={\\frac {g_{s}e}{2m}}{\\mathbf {S}}", "178510e30d9d11ffe732c8887f8c3425": "A(t)=4\\int _{0}^{t}W_{s}^{2}\\,\\mathrm {d} s", "17852f3942dcbd7b170f99d534d35a7d": "\\mathrm {Gr} _{c}={\\frac {g\\beta ^{*}(C_{a,s}-C_{a,a})L^{3}}{\\nu ^{2}}}", "178555f4aa2e0953c0346f4daa7b7025": "M_{2}=(0.03000-31.4424x_{D}+30.0717y_{D})/M", "17857fb7e4f853d4ddfedcfa9d2ea7ac": "\\,i>i^{(2)}>i^{(3)}>\\cdots >\\delta >\\cdots >d^{(3)}>d^{(2)}>d", "1785e1d662f804683f08b169ff39baa3": "\\log a=0.6192290,\\log b=0.9618637,\\log c=1.0576927", "17869e223a6460984564ea87da53f358": "\\Psi (w,v)=\\alpha \\left(\\Psi (v,w)\\right)", "1786a94e78f8b9d3ca65b0d26e3afda4": "\\log {\\frac {\\epsilon }{\\epsilon -(1-\\epsilon )\\delta }}~=~-\\log \\left(1-{\\frac {(1-\\epsilon )\\delta }{\\epsilon }}\\right)~\\geq ~\\delta {\\frac {1-\\epsilon }{\\epsilon }}~.", "178717df2d86b8abe6915c8e34d7329c": "\\mathbf {H} _{\\text{Electric dipole}}", "17873da24045b25d5f2b6312a23b573a": "d\\sin \\theta =m\\lambda \\ ,", "17875ea53adbcfcd53aecc2489ac3296": "L_{2}>L_{1}", "1787b1efaef540c5fb7ea8f7c68c9614": "Is=ss^{T}s+nn^{T}s=s\\langle s,s\\rangle +n\\langle n,s\\rangle =s", "1787b77ce65b27ee0f59df5a9cf57624": "V=i\\gamma ^{0}\\gamma ^{1}\\gamma ^{2}\\gamma ^{3}.\\ ", "1787e2f85c4dfde563002fc2390d555f": "\\psi (x)-x>K{\\sqrt {x}}", "1788028c10706464be2da741adf591a4": "\\{P_{i},y_{i}\\}_{i=1}^{n},\\ y_{i}\\in \\{+1,-1\\}", "17886787641252b8a48b1ec11bde58cf": "k_{A^{*}}", "17888738df4aca4408299efc0441466d": "{\\widetilde {\\phi }}_{\\alpha }(x,v^{i}\\partial _{i})=(\\phi _{\\alpha }(x),v^{1},\\cdots ,v^{n})", "1788efbde856bef67011f8e7c220da5f": "t=s[r\\sigma ]_{p}", "1788f99cea96fa8638f44a07a0d24426": "{\\frac {\\partial \\mathbf {S} }{\\partial t}}=\\mathbf {S} \\wedge \\sum _{i}{\\frac {\\partial ^{2}\\mathbf {S} }{\\partial x_{i}^{2}}}+\\mathbf {S} \\wedge J\\mathbf {S} .\\qquad (2)", "1788ff61f604c6d2fd4817d6360b92a2": "dx\\wedge dy=-dy\\wedge dx", "1789285921aaf4d6521e44991781f8bd": "D={\\frac {k_{B}T}{f_{tot}}}", "17896256c1b9e6ef89e3469fc4ad4cae": "^{d_{h}}M_{P_{h}}", "1789830e938bff850f904817801f60dd": "{\\textbf {r}}", "178ae2dac028fcabed5ddc9e65b31a71": "{\\begin{matrix}0&=&{}+p_{12}z_{0}&{}-p_{02}z_{1}&{}+p_{01}z_{2}&\\\\0&=&{}-p_{31}z_{0}&{}-p_{03}z_{1}&&{}+p_{01}z_{3}\\\\0&=&{}+p_{23}z_{0}&&{}-p_{03}z_{2}&{}+p_{02}z_{3}\\\\0&=&&{}+p_{23}z_{1}&{}+p_{31}z_{2}&{}+p_{12}z_{3}\\end{matrix}}", "178b3db30810e8e054a1625de581c2a4": "{\\hat {\\beta }}_{MSM}=\\operatorname {argmax} \\,{\\hat {m}}(x,\\beta )'W{\\hat {m}}(x,\\beta )", "178b4aef597fe388e4ee52a421262826": "Q_{r}n", "1790b22375d4ec762ff92ccc420a7849": "F_{b}=-k_{eq}(x_{1}+x_{2}).\\,", "1790bc121dfa00a2596c8ab82b0c700a": "{\\frac {d\\theta }{dx}}=k.\\,", "17919569f8e772299d2de89183c44ca5": "\\lambda =\\ell (\\ell +1)\\,", "17920c7c691ff7e27dea9c75b8e69535": "\\mathbf {P} =-{\\frac {\\partial G_{1}}{\\partial \\mathbf {Q} }}=-\\mathbf {q} ", "17927b4bf4452d98ac4b4028ae740c58": "1\\rightarrow K\\rightarrow G\\rightarrow H\\rightarrow 1", "17927d9db511c26d4a938508b33ed9dd": "\\mathbf {r} =(r_{1},r_{2},\\dots ,r_{M})", "17928c2d1d3db1ec69e5da3ac7f3ffdf": "D\\approx {\\frac {2}{3}}E^{f}f^{3}+2E^{f}fh(f+h)=2fE^{f}\\left({\\frac {1}{3}}f^{2}+h(f+h)\\right)", "1792a218a0645935327cb8550802aef6": "V={\\frac {-AN\\mu _{0}}{l}}{\\frac {dI}{dt}}", "17931ef55a26de0f4731092727e66ab9": "b'_{i}=1\\,", "179329c50708d1c7519c988b83a75c08": "R_{n}=|m_{n}|\\,", "17932fac05948bbb398143b07c3bbbb0": "H\\left|\\psi (t)\\right\\rangle =i\\hbar {\\partial \\over \\partial t}\\left|\\psi (t)\\right\\rangle .", "1793556bbdd9a64451ce294befbfc1b7": "n=\\infty ", "179392386646efb4d26cf987f20acf02": "\\operatorname {Cov} (X,Y)=\\operatorname {E} [XY]-\\operatorname {E} [X]\\operatorname {E} [Y].", "1793a877238d6da0fe259af3daf8db85": "\\mathrm {COP} _{\\mathrm {heating} }-\\mathrm {COP} _{\\mathrm {cooling} }=1\\,", "1793c4200a4c56163c6cd7f77714a874": "\\mathbf {F} (\\mathbf {r} ,\\mathbf {m} _{1},\\mathbf {m} _{2})={\\dfrac {3\\mu _{0}}{4\\pi r^{5}}}\\left[(\\mathbf {m} _{1}\\cdot \\mathbf {r} )\\mathbf {m} _{2}+(\\mathbf {m} _{2}\\cdot \\mathbf {r} )\\mathbf {m} _{1}+(\\mathbf {m} _{1}\\cdot \\mathbf {m} _{2})\\mathbf {r} -{\\dfrac {5(\\mathbf {m} _{1}\\cdot \\mathbf {r} )(\\mathbf {m} _{2}\\cdot \\mathbf {r} )}{r^{2}}}\\mathbf {r} \\right],", "1793c47ddcfebb7f8155f46e35e4ceb6": "C{\\frac {dV}{dt}}=0", "179402e04ecb6a630ac7344a8954555e": "x*y=y*x\\qquad {\\mbox{for all }}x,y\\in S", "179427f2fbc92ddc7ebac4730f462562": "\\Psi _{id}:{\\mathcal {B}}_{2}\\rightarrow {\\mathcal {A}}_{2}", "17942debc698e3def4d4757acf18ca87": "A_{u}", "1794402f3638b827218820aa44994365": "Q_{p}=\\sum H_{p}-\\sum H_{r}=\\Delta H", "179474ed9fdcb4ad56770fe1228e1a75": "c_{0}", "179537b26ee04a6a1eefe6e8a2ddde69": "{\\vec {A}}_{||}=0", "17953a9b349b2821519c4d445618cd6e": "R={\\frac {\\eta _{j}}{c_{p}}}{\\frac {C_{L}}{C_{D}}}ln{\\frac {W_{1}}{W_{2}}}", "17957d878bf4960e34d492340ca6a20d": "a,u,w", "1795d46584feaf59136988d6dba38568": "<\\cdot ,\\cdot >", "17960f004fee00eab0fd8dc31fbc48a7": "V(\\phi )-0\\end{cases}}", "17a04cd02c73f2dd6ad1b42cbf9f9813": "\\mathbb {R} ^{V}", "17a05bd3f52ed80b340510ee96ee2ccd": "{\\mathit {q}}", "17a05f391c28aea7f21706b9dbdfa424": "\\tau \\sigma ", "17a09206d578b43575f7a0c51640dfd2": "|\\alpha '\\rangle ", "17a09af768b03dcfa184287545e42097": "{D'}", "17a0e5b21c35c5084fd512a64b739918": "C_{y}=-a\\lambda \\ +A_{y}.\\,", "17a11337560bb7c4b9f3c3589b2d0888": "\\mathrm {Si-OH+OH-Si\\ \\xrightarrow {polymerization} \\ Si-O-Si+H{_{2}}O} ", "17a152fc9de829d1970cff4cce1efb74": "\\operatorname {Spec} ", "17a184a43993bddb45ac3ca884a4ac6b": "\\scriptstyle f_{s}/2,", "17a1d6e8a523d4194783fa994c8fc18c": "{\\begin{aligned}&Y+\\sigma (100-l_{1}+\\beta )=X+\\sigma (100-l_{2}+\\beta )\\\\\\Rightarrow {}&X-Y=\\sigma (l_{2}-l_{1})\\end{aligned}}", "17a20a60a422a3defa40e901baafdf6b": "\\theta (X)=-X^{T}", "17a2182ae342971e27f272fea6fc808d": "s(F)=\\infty ", "17a2b4d73340f97c1610cee83578c08b": "\\left.{\\frac {\\partial }{\\partial r_{\\alpha }}}{\\bar {\\rho }}(r_{\\alpha })\\right|_{r_{\\alpha }=0}=-2Z_{\\alpha }{\\bar {\\rho }}(0).", "17a2e0b3ddf682d73a80dc7838b2557b": "p_{1}=1", "17a381949cb658f3d0481ce690136c70": "\\det {\\big (}\\mathrm {adj} (\\mathbf {A} ){\\big )}=\\det(\\mathbf {A} )^{n-1},", "17a3ce8d2e0ad6c889d99c68461f6a08": "a_{j}-a_{i}", "17a40acb3d14a0f613090927710d3cff": "\\mathbb {Z} _{2}", "17a4993817ab6428bb5403cc90569a55": "p(x)=x^{3}-7x+7", "17a4a7ba2a8f61ef70c0982be6a125a9": "(I,\\leq )", "17a4d45d616eba0eb85a700663753697": "A={\\begin{pmatrix}0&M\\\\M&B\\end{pmatrix}}{\\text{,}}", "17a4f92d86b83d69536e600092b5d32c": "i\\hbar {\\frac {\\partial }{\\partial t}}\\psi ={\\widehat {H}}\\psi ", "17a5180712427043192a565defdef006": "\\sigma _{11}=\\left(\\lambda ^{2}-{\\cfrac {1}{\\lambda }}\\right)\\left({\\cfrac {\\mu J_{m}}{J_{m}-I_{1}+3}}\\right)~.", "17a59ae0a855620d0292bcb1063b200a": "F={\\frac {dW_{f}}{dt}}", "17a5d4967911e7e20345d77861224ec9": "|x\\cdot y|\\leq |x|\\cdot |y|", "17a62842bc7e50de218dd3fd1867f110": "\\Pi '", "17a648e35d5b58384e77d0744f15a2c5": "\\displaystyle {\\sum (1-|\\lambda _{i}|)<1}", "17a671a7916725910cd594ad1a71ce23": "\\ \\Phi ", "17a689a53f7d8737b57380c3c7245e11": "disc({\\mathcal {H}}):=\\min _{\\chi :V\\rightarrow \\{-1,+1\\}}disc({\\mathcal {H}},\\chi ).", "17a698156de43e09b45d2228624d26f1": "|a_{i,j}|_{0,\\alpha ;\\Omega },|b_{i}|_{0,\\alpha ;\\Omega }^{(1)},|c|_{0,\\alpha ;\\Omega }^{(2)}\\leq \\Lambda .", "17a6b2f3ab3bae8cacd196c239094d52": "\\mathbf {F} (\\mathbf {x} )=-{\\frac {dU}{d\\mathbf {x} }}~.", "17a6e3786fcee00d26de14049f89ebe1": "\\Phi :\\mathrm {hom} _{C}(F-,-)\\to \\mathrm {hom} _{D}(-,G-)", "17a70d8ced682a0a156863b6d2168ccd": "\\ q\\in M", "17a71577171bd97c71e95a5bbcc87083": "x\\not \\ll y", "17a74e3e623011c3c98da41e4c4888aa": "M=4", "17a81c48246c845c6e16d9fef4e07b5e": "\\epsilon ^{*}-\\epsilon _{\\infty }={\\dfrac {\\epsilon _{0}-\\epsilon _{\\infty }}{1+(i\\omega \\tau _{0})^{1-\\alpha }}}", "17a82ce61a79c3c6ac15944a42512eb7": "m{\\frac {du^{a}}{ds}}=qF^{ab}u_{b},", "17a836b4bbded5a128f4cffb33077b99": "A={\\sqrt {(s-a)(s-b)(s-c)(s-d)}}", "17a8dc0d6d7b993f341d41b392189f51": "|x^{\\rho }|={\\sqrt {x}}", "17a8e639d22844bb61cc5c7b8458ee60": "D=\\{z\\in \\mathbf {C} :|z|<1\\}.", "17a8f8fa883a0c4dcc09cd8ee62f500f": "\\min _{x\\in X}f(x)", "17a92b25b2e3289c37dc5f44a1ccd28e": "\\sum _{P\\in C}{c_{P}[P]}", "17a941d7d1417fc0813745695592cda6": "Q=\\omega {\\frac {\\rm {maximum\\;stored\\;energy}}{\\rm {average\\;power\\;loss}}}=\\omega {\\frac {W_{s}}{P_{l}}},", "17a9bfda729499597088e43c0ea096fa": "\\omega =\\omega _{0}-\\nu ", "17a9c2a57b6050eb05a980849218627a": "\\partial _{i}g_{jk}=(\\Gamma _{ij,k}=0)+\\Gamma _{ik,j}^{*}=\\partial _{i}\\partial _{j}\\partial _{k}\\psi ", "17aa1504fb8fd391c95e17fb2c4081ed": "\\scriptstyle \\delta \\,", "17aa20a885a267996bffe03ac0f92a2e": "g(z)", "17aa75fca3cc0b4ad4ffce5e9538affc": "g^{\\cdot }:(\\mathbb {R} ,+)\\to ({\\widehat {G}},\\cdot ):\\lim _{i}q_{i}\\in \\mathbb {Q} \\mapsto \\lim _{i}g^{q_{i}}", "17aa8209a276e9c96addf6c2450b9adc": "K=-(4/9)/J,\\ ", "17aa82a77167a98516267f83b4d068ec": "k_{1H}", "17aa86dc1a0121a9375d5a602db9caae": "N_{1}0=39", "17aaa2643df92cff6f25e31f91d5f001": "g_{nm}(k)=s(k-mN)\\cdot e^{j\\Omega nk}", "17ab56afce1cbcde6cf56efab9582d5a": "C^{\\omega }(X,Y)\\subset {\\mbox{Hom}}(X,Y)", "17ab83d68ab936d528e1d6c94516b424": "([A_{1},A_{2}],[A_{3},A_{4}],[A_{5},A_{6}])=0", "17ab9b118b7b926df7ff0cd2c06d8ff0": "E=\\sum _{i=1}^{k}\\sum _{p\\in C_{i}}(p-m_{i})^{2},", "17aba57fa3ba1e52436bde651dd534d1": "\\mu S+\\lambda S", "17abd174add9feacc3e0cc97d1006db1": "\\operatorname {E} [\\operatorname {dCov} _{n}^{2}(X,Y)]={\\frac {n-1}{n^{2}}}\\left\\{(n-2)\\operatorname {dCov} ^{2}(X,Y)+\\operatorname {E} [\\|X-X'\\|]\\,\\operatorname {E} [\\|Y-Y'\\|]\\right\\}={\\frac {n-1}{n^{2}}}\\operatorname {E} [\\|X-X'\\|]\\,\\operatorname {E} [\\|Y-Y'\\|].", "17abd966bd70d77a73b47d95d9938945": "V_{a,b}\\subset Y", "17abe57b40fd9d840c7150a52489a424": "\\chi ^{*}", "17ac2f84363af2816dc33f64149a7afe": "{\\mathit {KS}}({\\mathit {PRIMES}})\\leq 167", "17ac68370640535beb63f1bf83f34b59": "x_{i}={\\frac {\\rho _{i}}{\\rho }}\\cdot {\\frac {M}{M_{i}}}", "17ace7e621c20488079195db5269a5e3": "\\mu =\\sin \\varphi ", "17ad6396d0bc8018ae6f3b0cb6605452": "\\int _{S^{2}}f^{*}\\omega =\\langle c_{1}(TM),f_{*}[S^{2}]\\rangle =0", "17adacdc69beb819166653c9eba94116": "\\lambda _{i}\\,\\!", "17aded73356e21e73d7a4bee820dc865": "\\ NEP={\\sqrt {4k_{B}T^{2}G}}", "17ae139928090f55ffd98a1aeef2def6": "((A\\to B)\\to C)\\to (D\\to ((B\\to (C\\to E))\\to (B\\to E)))", "17ae8a5a7840c436d16bccba8948e7a8": "[h,e]v=hev-ehv=2ev", "17aead184440e2929e84471fa1beb937": "SO(2n)\\supset SU(n)", "17b005d3350f6a036823c05e0a250413": "\\beta ={\\frac {2a\\pi ^{2}(2mE)^{1/2}}{h}}", "17b00cad4f9dad1cfae944abdff28ba4": "O({\\sqrt {|S|}})", "17b03549ad2fdaec017297e23f7e16c6": "0\\rightarrow B\\rightarrow X''_{n}\\rightarrow X_{n-1}\\oplus X'_{n-1}\\rightarrow \\cdots \\rightarrow X_{2}\\oplus X'_{2}\\rightarrow X''_{1}\\rightarrow A\\rightarrow 0.", "17b08a7e7f246ac9b0dedae92fb24176": "\\mathbb {C} ^{m}", "17b0ab432dd3922b037c1d0fdf63d83d": "\\operatorname {gr} _{I}R=\\oplus _{n\\in \\mathbb {N} }I^{n}/I^{n+1}", "17b0e656c34493ed5b35d97611e857ca": "\\chi _{\\mathrm {top} }(x,y)=x.\\,", "17b0ef287ca1f56a02f649f56164e0d3": "\\nabla f(x^{*})=\\sum _{i=1}^{m}\\mu _{i}\\nabla g_{i}(x^{*})+\\sum _{j=1}^{l}\\lambda _{j}\\nabla h_{j}(x^{*}),", "17b1036d35e80e298e933c6204c02533": "E_{n,\\mathbf {k} }=E_{n,0}+{\\frac {\\hbar ^{2}k^{2}}{2m}}+{\\frac {\\hbar ^{2}}{m^{2}}}\\sum _{n'\\neq n}{\\frac {|\\langle u_{n,0}|\\mathbf {k} \\cdot \\mathbf {p} |u_{n',0}\\rangle |^{2}}{E_{n,0}-E_{n',0}}}", "17b11950df5099f23d85e7778d2bc537": "a_{(2)}0=0", "17b1364578f4405182b834ad2d41b20e": "g(t)=\\ln M(t)=\\mu t+{\\frac {1}{2}}\\sigma ^{2}t^{2}", "17b1c1f2a7934842f633b148e5db8c94": "q=\\operatorname {sgn}(y)\\left\\lceil \\left|y\\right|\\right\\rceil =-\\operatorname {sgn}(y)\\left\\lfloor -\\left|y\\right|\\right\\rfloor \\,", "17b1fe3ec31b0960e20f3f520c275e25": "R=\\mathbb {Z} \\left[{\\sqrt {-3}}\\,\\,\\right],\\quad a=4=2\\cdot 2=\\left(1+{\\sqrt {-3}}\\,\\,\\right)\\left(1-{\\sqrt {-3}}\\,\\,\\right),\\quad b=\\left(1+{\\sqrt {-3}}\\,\\,\\right)\\cdot 2.", "17b28cacf3d4e06ca759108cd4440a57": "A\\subsetneq \\mathbb {R} ^{n}", "17b292ded2032405051317f9eb1cd065": "F=F^{ab}e_{a}\\wedge e_{b}\\quad (1\\leq a0\\ {\\Big \\}}.", "17c404b5557ce2ba595af555a3900529": "m{\\ddot {x}}_{j}=k(x_{j+1}+x_{j-1}-2x_{j})[1+\\alpha (x_{j+1}-x_{j-1})]", "17c4455eee7df04570466b8813f44fd4": "dt={\\frac {\\mathrm {d} v}{g-{\\frac {kv^{2}}{m}}}}", "17c454d47bc2c11ccffbc4e79285729e": "\\left\\vert \\Phi ^{+}\\right\\rangle ^{BA}\\left\\vert 00\\right\\rangle ^{A}\\left\\vert \\psi \\right\\rangle ^{A}.", "17c48cb446def2f4705f94a428a39eb8": "{\\frac {R_{x}}{R_{g}+R_{y}}}={\\frac {R_{B1}}{R_{B2}}}", "17c4f6323042441ecc73cc0ad3ea1937": "L_{\\text{shd}}\\,", "17c50171dd99868a958e0264ce4810b4": "C_{3}\\,", "17c525d4a9c29902bec0ba6221908fa5": "\\displaystyle {\\frac {(n+2)(n-1)}{2n}}", "17c57b31a421edeefb6e192b2beb0a83": "*2222", "17c5b8360f5cdbf285fa2422a6c8875f": "8r\\leq a+b+c+d\\leq 8r\\cdot {\\frac {R^{2}+x^{2}}{R^{2}-x^{2}}}", "17c5e5068f279fab07a07d35ff3989b1": "\\mathbb {P} (\\vartheta _{s}^{-1}(E))=\\mathbb {P} (E)", "17c60af8fac70fec9c3e9b8ca1aa7530": "\\tan 2A={\\frac {2*{\\frac {3}{7}}}{1-({\\frac {3}{7}})^{2}}}={\\frac {21}{20}};", "17c61e39f9cf40cab4a05059a53d8376": "Y={\\text{Output}}", "17c67b2bab1aebc4e1f7b5eb6fee0117": "q^{(p-1)/2}\\equiv (-1)^{\\sum _{u}\\left\\lfloor qu/p\\right\\rfloor }{\\text{ (mod }}p).", "17c68304c4529a455a48bca26140fca8": "{\\frac {N}{C}}\\geq 1.0", "17c6af449b4cd6a11cea0ab961ad4748": "\\left.\\left[k_{c}(2H+c_{0})+{\\bar {k}}k_{n}\\right]\\right\\vert _{C}=0", "17c6d3da4310dd3a50677c17124d7d1b": "\\textstyle p:2q:r=p_{1}:2q_{1}:r_{1}", "17c6e53aae3864376a8f622096aadc59": "=\\sum _{i}|\\langle \\psi |i\\rangle |^{2}|\\langle i|\\varphi \\rangle |^{2}=\\sum _{i}|\\psi _{i}|^{2}|\\varphi _{i}|^{2}", "17c705e5800fcfa64be2b39ebb2260ab": "1/\\mu ", "17c73c55ae4d30d27532f2f538a13d9a": "\\operatorname {H} [\\mathbf {X} ]=-\\ln \\left(B(\\mathbf {V} ,n)\\right)-{\\tfrac {1}{2}}(n-p-1)\\operatorname {E} [\\ln |\\mathbf {X} |]+{\\frac {np}{2}}", "17c77fdbc4369d09c0eaa4476f071f50": "\\operatorname {arccot} x={\\frac {\\pi }{2}}-\\arctan x", "17c8012f5ad9fda667a889e3100c98c4": "r_{-}", "17c80d41500d6d7a9a9c6b6f98404ae8": "\\int _{0}^{1}\\rho ^{D-1}R_{n}^{(l)}(\\rho )R_{n'}^{(l)}(\\rho )d\\rho =\\delta _{n,n'}", "17c80d92eb4858d09f08dec199b0ef2e": "L=2\\pi /k", "17c87d31e54b48968dfbbdc8a9402b47": "PWV={\\dfrac {\\Delta x}{\\Delta t}}", "17c8942c9ed21522767ba16348df9fc8": "S=\\int d\\tau {\\Big [}{dx \\over d\\tau }p+{dt \\over d\\tau }p_{t}-{\\mathcal {H}}(x,t;p,p_{t}){\\Big ]}", "17c8a1e9f50d9481d42c2e20562a0ffe": "Z=-20+{\\tfrac {2}{3}}x+{\\tfrac {11}{3}}y+{\\tfrac {4}{3}}t", "17c8d71b3cb9e7cc90744c303e545195": "\\textstyle {\\binom {m}{k}}", "17c99259a31002d4a2c3f6808be06073": "x\\neq 0\\neq y", "17c9a81794db833e07f6f584a6a1f5e5": "Cl_{3}^{\\geq }", "17c9f5dfb9a82e9274d56673d21f43d6": "f^{*}=\\prod _{k=1}^{n}(X-\\alpha _{k}^{*})", "17ca14f8aac63565748d7ddfd947bdd9": "\\nu _{m}>\\nu _{c}", "17ca6282072bbed909f3ef091779fc2a": "L_{GD}^{*}", "17cb493a06665211bf75c645fe365aea": "x+y+z\\leq 1,\\,", "17cb5c634ffb0882870156bfc5c6d026": "\\mathbf {S} ", "17cb5ce8cfac29649bd24e8d99371861": "|r|<\\sigma R", "17cb78d49103e560c1607f407a29fd84": "E_{y}=E_{z}=0,E_{x}", "17cba223b15f99c9d3e05f8d81a31f15": "I=\\sigma _{1}\\sigma _{2}\\sigma _{3}", "17cbad5a6ccad13458f01b9984c3ddf0": "\\,\\{A+iB\\}=e^{i\\theta }\\{A+iB\\}", "17cbdb44f384e6ae465e3a6fad5f0e3b": "VCA(64x^{3}+576x^{2}-64x-64,({\\frac {3}{2}},2))", "17cc2f79a63a0b9472935ee3da247399": "{\\text{Var}}(X_{i})={\\frac {K_{i}}{N}}\\left(1-{\\frac {K_{i}}{N}}\\right)n{\\frac {N-n}{N-1}}", "17cc4c2a6905b10f8868fabdde4a8d45": "k\\rightarrow \\infty ", "17cc679caf1c157c79d1b8f5ee6ad837": "{_{4}^{2}}{\\text{S}}^{\\beta }", "17cc6e28087828386c5c61be16e76e64": "\\sum _{i=1}^{n-1}i^{\\varphi (n)}\\equiv -1{\\pmod {n}}", "17cc7292d05c904e019e499b3c54e4b2": "{\\begin{aligned}dy\\,dz&{\\hat {\\mathbf {x} }}\\\\+dx\\,dz&{\\hat {\\mathbf {y} }}\\\\+dx\\,dy&{\\hat {\\mathbf {z} }}\\end{aligned}}", "17cd31ca2ad4c993f41a911921171370": "\\alpha =(K_{eq}\\cdot \\Pi \\sigma _{Products}/\\Pi \\sigma _{Reactants})^{1/n}", "17cd4c40578493bfa013b53eb2a18d1d": "{\\hat {\\rho }}_{k}", "17cdaa30a3eb4051f92a1ad6ec3eea8f": "\\sum _{x,y}z(x,y)<\\infty ", "17cdc70546189b942bd85708596e5d7d": "S_{2}={\\frac {A}{S_{1}}}={\\frac {{\\frac {\\pi ^{2}}{6}}-1}{\\frac {5}{6}}}={\\frac {\\pi ^{2}-6}{5}}=0.77392088021\\cdots ", "17cdf36a23aaa5a7bb966970166c0bc0": "{\\frac {d}{dx}}\\delta (-x)={\\frac {d}{dx}}\\delta (x)", "17ce0156c6dcc475ca036f4501ef3ebf": "\\rightarrow KR_{n-1}^{G}(\\coprod _{j\\in I_{i}}G/H_{j}\\times S^{i-1})\\rightarrow KR_{n-1}^{G}(\\coprod _{j\\in I_{i}}G/H_{j}\\times D^{i})\\oplus KR_{n-1}^{G}(X^{i-1})", "17ce27cb1c765a63f8281d55549733ca": "\\gamma _{x}^{-}:=\\{\\Phi (t,x):t\\in (t_{x}^{-},0)\\}", "17ce668a51bb6d0b0444a59cc73462f8": "{\\tfrac {K_{5}}{K_{4}+{\\text{Reliability}}}}", "17ce9c5b987418f79e631f97db2d47b5": "Q_{0}=\\Delta U+W+W_{0}=W+W_{0}=W_{\\text{total}}", "17ceb532aa3b3c830a707488b50fd0f5": "{\\frac {d\\beta }{dt}}=-r", "17ceb7ba0753f4512e4359bd74969db8": "\\mathrm {d} H=T\\mathrm {d} S+V\\mathrm {d} P.", "17ceca71c3b34dcff1b82d7924668885": "\\psi (\\Omega +\\omega )", "17cf0ad1e9531bb5a463c43fd23f6fde": "\\scriptstyle (\\phi ,\\,\\theta ,\\,\\psi )", "17cf3b08f51ab9b8b1d3f0e6501721f7": "\\alpha =(-i\\tau )^{\\frac {1}{2}}\\exp \\!\\left({\\frac {\\pi }{\\tau }}iz^{2}\\right).\\,", "17cf4602a3e3b1bcbc0e72b28a774fde": "H(s)={\\frac {1}{1+\\alpha s}}", "17cf4afaf1b8e94b4fdaabef8ebd31c1": "\\mathbf {r} =\\mathbf {r} _{1}-\\mathbf {r} _{2}", "17cfc942478b711d8bf4422277d4778b": "Y(t)=R_{n}(t)-{\\frac {R_{n}(x)}{W(x)}}\\ W(t)", "17cfd35a502a12aa5a05c07941a1d7df": "\\Lambda ^{k}(V^{*})\\otimes \\Lambda ^{n}(V)\\to \\Lambda ^{n-k}(V).\\,", "17cfdb75c84613db2b01f5fc846927fb": "{\\omega {L}}={{1} \\over {\\omega }{C}}\\,", "17d010993125eb26c35e811ae20210d5": "k\\in \\mathbb {Z} ^{n}", "17d05096790cbdd8958e957958486f35": "\\beta ,", "17d0762a42b1dba98752b272708d079b": "f(x|a,b)=abe^{-(be^{-ax}+ax)}\\,", "17d08c915c3a43915454ebfb586ea269": "{\\acute {P}}_{t}={\\acute {P}}_{0}+{\\acute {V}}_{0}T_{t}+{\\frac {1}{2}}{\\acute {A}}_{0}T_{t}^{2}", "17d0bcc02bc07c0b74d5f33cbc060307": "m=0,\\dots ,n", "17d0d1ea51e81fb2acbbb612f626e0c9": "k+k_{0}", "17d0e11fa183629e20c5d1d6dc67a09e": "\\left[{\\begin{array}{ccc|c}2&1&-1&8\\\\-3&-1&2&-11\\\\-2&1&2&-3\\end{array}}\\right]", "17d1429efeeffd4856a2b694a256b90f": "B_{n+1}=\\sum _{k=0}^{n}{\\binom {n}{k}}B_{k}.", "17d146961d6fd0989e8a614104483829": "{\\boldsymbol {R}}^{-1}={\\boldsymbol {R}}^{T}", "17d16d4daa010d46d058e890010dc3be": "f(z)=\\sum _{k=0}^{n}f(z_{k}){\\frac {G_{n}(z)}{(z-z_{k})G'_{n}(z_{k})}}", "17d192bcb49d19a39379bd72c95c7302": "\\mathbb {Z} ", "17d1b5e8b5708d0f5ff55acaa623845b": "\\lim _{\\|\\Delta x_{i}\\|\\to 0}F(b)-F(a)=\\lim _{\\|\\Delta x_{i}\\|\\to 0}\\sum _{i=1}^{n}\\,[f(c_{i})(\\Delta x_{i})].", "17d1ef843c6f0b7c5aad448a3add820d": "\\left(e'_{w}\\right)", "17d21ebc44fa5379d9cb9ad36628829a": "X_{k}=p(e^{2\\pi i{\\tfrac {k}{2^{n}}}})", "17d2207c50da3999aa28e73690766eaa": "g(t,V)={\\bar {g}}\\cdot m(t,V)^{p}\\cdot h(t,V)^{q}", "17d27ca75f435cede1f55a245f5b17e0": "y^{e}(\\mathbf {x} )", "17d28213f49f4ce6885e4fb73e47fb92": "R_{tot}", "17d28b9167937faaa8b7b4c2c535d0c8": "p^{2}/(m\\gamma )", "17d2c51392b33917e88fe1af02247eb6": "g={{\\frac {4\\pi }{3}}G\\rho r}", "17d3351614d0cbdb29faca85af012655": "M_{ij}N_{ij}", "17d356e3538088f76021a7460a7aa73c": "k={\\frac {2\\pi }{a}}", "17d397110a74f0a092f88e83498f5b97": "D_{n}=n^{2}+4(n^{2}-n)", "17d39d28ebbdd22d46434f959a8ffd20": "|x|=1,", "17d3c881aafd93da5b5f7fc26e4b4634": "\\mathbf {P} (t)=\\int _{0}^{t}\\mathbf {V} (t)dt=\\int (\\mathbf {A} t+\\mathbf {V} _{0})dt={\\tfrac {1}{2}}\\mathbf {A} t^{2}+\\mathbf {V} _{0}t+\\mathbf {P} _{0}.", "17d3f7c9e9890baaecd46cd73efc65f4": "{-1,1}", "17d40c630ff3797e876aec68840cb050": "f(y,x)=y^{2}+x^{2}-r^{2}", "17d40dcb94593a9fe4c7310f00df1058": "\\delta \\varphi \\approx {\\frac {3\\pi m^{2}c^{2}}{2L^{2}}}\\left({\\frac {4G^{2}M^{2}}{c^{4}}}\\right)={\\frac {6\\pi G^{2}M^{2}m^{2}}{c^{2}L^{2}}}", "17d457e6178abe32066f660033985f3e": "\\mathbb {Z} _{1}", "17d4e88907d03a20732396e906541e4b": "P_{\\text{x,t}}={\\frac {E_{\\text{x,t}}}{R_{\\text{x,t}}[1-T]}}", "17d520aa4f3eafa7af25c085f70ce613": "N>1", "17d5525e412071fd1062277888a78a18": "\\mathrm {P} (C|AB)={\\frac {\\frac {1}{16}}{{\\frac {1}{16}}+{\\frac {3}{16}}}}={\\tfrac {1}{4}}=\\mathrm {P} (C)", "17d5974b1ec85d536ce08ce952a7a1f2": "f(T)\\leq f(S)", "17d597592b8d90e504ea2213fef9ad7e": "\\scriptstyle {\\tau \\rightarrow \\infty }", "17d59f02b8d46a7461189ef24ebf4176": "(-1)^{m}n!\\;[z^{n}][u^{m}]g(z,u)=\\left[{\\begin{matrix}n\\\\n-m\\end{matrix}}\\right]", "17d5e5c0307c4829aa24ba82a0d121fb": "[A,BCDE]=[A,B]CDE+B[A,C]DE+BC[A,D]E+BCD[A,E]", "17d648ea50b87e97505a799d87654435": "={\\frac {1}{A_{n}}}\\int _{{\\vec {r}}'\\in \\mathbb {R} ^{n}}^{}\\left({\\vec {\\nabla }}_{\\vec {r}}\\bullet {}{\\frac {{\\vec {r}}-{\\vec {r}}'}{|{\\vec {r}}-{\\vec {r}}'|^{n}}}\\right){\\vec {F}}({\\vec {r}}')d\\tau '", "17d6987bc53318b1760eb61d1bafe508": "\\left({\\frac {d}{dq}}q-q{\\frac {d}{dq}}\\right)f(q)={\\frac {d}{dq}}(qf(q))-q{\\frac {df(q)}{dq}}=f(q)", "17d6f7f139aaae95eebdbfdfba085d39": "x^{2}+y^{2}=0.", "17d79e89fec3c320ef63926fd3862a72": "Y_{1}={\\frac {1}{T_{1}}}", "17d7f6111f47230fd65ada0707fd1e61": "(x\\cdot f)(v)=xf(v)-f(xv)", "17d7f64b69ae5e67c2ab0b2acd2e2abe": "\\rho :2^{X}\\rightarrow 2^{Y}", "17d808103292821dacc048df5fc5b40a": "{\\boldsymbol {\\sigma }}=-p~{\\boldsymbol {\\mathit {1}}}+2C_{1}\\left[\\sum _{i=1}^{5}i~\\alpha _{i}~\\beta ^{i-1}~(3+\\gamma ^{2})^{i-1}\\right]~{\\boldsymbol {B}}", "17d8413d5243ef4f856d002265462720": "S_{a/\\$}", "17d85532452294ee6899ee5305b835b0": "U_{1}(\\mathbf {r} ,t)=A_{1}(\\mathbf {r} )e^{i[\\varphi _{1}(\\mathbf {r} )-\\omega t]}", "17d86fb528165239ebc895b4b0dc8e23": "\\mu _{Y}^{\\pi }", "17d884a358c3b723f0a9e53102c8bfad": "T_{O}", "17d8bc79fdd575e57161dfa63642a306": "b\\left(\\alpha \\,\\right)={\\frac {\\alpha \\,}{2^{N}}}:0\\leq \\,\\alpha \\,\\leq \\,{2^{N}}", "17d94809652ba924fae9a8de54a13295": "e^{\\Lambda }=1+\\Lambda +{\\tfrac {1}{2}}\\Lambda *\\Lambda +{\\tfrac {1}{3!}}\\Lambda *\\Lambda *\\Lambda +\\ldots ", "17d9c075c38185fc7315e5888f9537e0": "A>0", "17d9f208d6b981f4e7ab37a914cac6a1": "k_{\\lambda }.(v\\otimes w)=k_{\\lambda }.v\\otimes k_{\\lambda }.w", "17da46b46e9e71bdc29ed3087b7f17dd": "\\sup _{x_{1},x_{2},\\dots ,x_{n},{\\hat {x}}_{i}}|f(x_{1},x_{2},\\dots ,x_{n})-f(x_{1},x_{2},\\dots ,x_{i-1},{\\hat {x}}_{i},x_{i+1},\\dots ,x_{n})|\\leq c_{i}\\qquad {\\text{for}}\\quad 1\\leq i\\leq n\\;.", "17da6fbed2142f22acfaaca480d2760c": "H[A]=\\int _{-\\infty }^{\\infty }S(Cr\\lbrace A\\geq t\\rbrace )\\,dt.", "17da84f07ba0cb6220c486de4f46bb6d": "\\Delta Q=Q\\left({\\frac {1}{\\eta }}-1\\right)", "17dab9ed9db2de2a595daf012fbe23f6": "36\\cdot V^{2}={\\begin{vmatrix}\\mathbf {a^{2}} &\\mathbf {a} \\cdot \\mathbf {b} &\\mathbf {a} \\cdot \\mathbf {c} \\\\\\mathbf {a} \\cdot \\mathbf {b} &\\mathbf {b^{2}} &\\mathbf {b} \\cdot \\mathbf {c} \\\\\\mathbf {a} \\cdot \\mathbf {c} &\\mathbf {b} \\cdot \\mathbf {c} &\\mathbf {c^{2}} \\end{vmatrix}}", "17dac61b10d732b54d8450853a3b1804": "\\sum _{i}\\langle (Id\\otimes \\Psi _{i})(M_{i}\\otimes I)(\\rho \\otimes \\omega )(M_{i}\\otimes I),\\;I\\otimes O\\rangle ", "17daff7d1f0fba077955fd83d9738f1d": "\\alpha _{t}=\\sup _{\\phi \\in Z:\\,\\|\\phi \\|_{\\infty }=1}\\|{\\mathcal {E}}_{t}\\phi \\|_{1}.", "17db218b1466f2e08000d399647601ae": "A(x,\\ t)", "17dbe4047de689f1c13508decd4fc23e": "L(5;1)", "17dc50dad6abc4a399913c75bd267f02": "s(n,1/\\epsilon )\\,", "17dc58d253690d82d82ae8d75b50e0c1": "CAT=PL", "17dca03da967dd1094df5beef44a4425": "R_{\\mathrm {ESR} }", "17dcd3c4d1f699c2a07fa3d192e47538": "{\\begin{aligned}\\operatorname {E} \\left[\\operatorname {E} [X|Y]\\right]&=\\sum \\limits _{y}\\operatorname {E} [X|Y=y]\\cdot \\operatorname {P} (Y=y)\\\\&=\\sum \\limits _{y}\\left(\\sum \\limits _{x}x\\cdot \\operatorname {P} (X=x|Y=y)\\right)\\cdot \\operatorname {P} (Y=y)\\\\&=\\sum \\limits _{y}\\sum \\limits _{x}x\\cdot \\operatorname {P} (X=x|Y=y)\\cdot \\operatorname {P} (Y=y)\\\\&=\\sum \\limits _{y}\\sum \\limits _{x}x\\cdot \\operatorname {P} (Y=y|X=x)\\cdot \\operatorname {P} (X=x)\\\\&=\\sum \\limits _{x}x\\cdot \\operatorname {P} (X=x)\\cdot \\left(\\sum \\limits _{y}\\operatorname {P} (Y=y|X=x)\\right)\\\\&=\\sum \\limits _{x}x\\cdot \\operatorname {P} (X=x)\\\\&=\\operatorname {E} [X]\\end{aligned}}", "17dcfdbe98f10e7f081ad8053f904a2a": "{\\begin{aligned}V_{n}(\\mathbb {R} ^{n})&\\cong \\mathrm {O} (n)\\\\V_{n}(\\mathbb {C} ^{n})&\\cong \\mathrm {U} (n)\\\\V_{n}(\\mathbb {H} ^{n})&\\cong \\mathrm {Sp} (n)\\end{aligned}}", "17dd17f2b91bae92b15063867360844f": "E<0", "17dd7e22f2e19c2490d98e12a8f6a395": "f(t,n)=f(t-1,n-1)+f(t-1,n)", "17ddb805ae48564e4909e568dce535f8": "\\left|\\int _{\\Gamma }f(z)\\,dz\\right|\\leq M\\,l(\\Gamma ),", "17ddf3e3567916ddac68e16b0b2dc3b9": "E[Y(T)|X_{t}=x]=E\\left[e^{-\\int _{t}^{T}V(X_{\\tau })\\,d\\tau }u(X_{T},T)+\\int _{t}^{T}e^{-\\int _{t}^{r}V(X_{\\tau },\\tau )\\,d\\tau }f(X_{r},r)dr{\\Bigg |}X_{t}=x\\right]", "17ddf8e61a6ab211fa78338c327c82a9": "{\\begin{cases}{\\begin{bmatrix}1&0\\\\1&3\\end{bmatrix}}{\\begin{bmatrix}a\\\\c\\end{bmatrix}}=x{\\begin{bmatrix}a\\\\c\\end{bmatrix}}\\\\{\\begin{bmatrix}1&0\\\\1&3\\end{bmatrix}}{\\begin{bmatrix}b\\\\d\\end{bmatrix}}=y{\\begin{bmatrix}b\\\\d\\end{bmatrix}}\\end{cases}}", "17de3fcadb0e421107f556d9261471e5": "\\langle k\\rangle =\\lim _{\\alpha \\rightarrow \\infty }\\zeta _{G}(\\alpha ).", "17deef526b086afa3169a9a27d4557f1": "{\\text{N}}\\;=\\;k_{0}\\left[m+\\nu \\theta _{2}+{\\frac {z\\nu \\omega s}{4}}(9+4\\varepsilon c^{2}-11\\omega ^{2}+20\\omega ^{2}c^{2})\\right]\\,", "17df1378f55240630e871c62e792c2a7": "{\\frac {d}{dt}}{\\frac {F(t_{0}+t)-F(t_{0})}{S(t_{0})}}={\\frac {f(t_{0}+t)}{S(t_{0})}}", "17dff42fa1938ffdc5e74c06779b8f81": "2^{m/n}=3\\,", "17e0b3891a3a2f5236fb0167e144365e": "{\\textbf {Vect}}_{K}", "17e108b76635be3f4b27ff31aa8bfb0c": "{\\mathbf {A}}\\;", "17e12308ccf9654d467c52094f918cb7": "z_{k}=\\langle X-\\mu ,\\mathbf {v_{k}} \\rangle ,k=1,2,\\dots ,p", "17e13c57ff1871a28f517738649ad583": "{\\vec {F}}=-\\ C{\\vec {r}}", "17e13ebe4ed9acd37f7e01278383735a": "b(z)={\\frac {\\mu _{x}}{\\sigma _{x}^{2}}}z+{\\frac {\\mu _{y}}{\\sigma _{y}^{2}}}", "17e1824bb144c81c6d3ac0e6e5a648a0": "Y_{10}^{9}(\\theta ,\\varphi )={-1 \\over 512}{\\sqrt {4849845 \\over \\pi }}\\cdot e^{9i\\varphi }\\cdot \\sin ^{9}\\theta \\cdot \\cos \\theta ", "17e193f9695bd4c01f2b52a1a19b8b95": "A(\\rho )=w\\,h", "17e1a23dbbc47a45c6f4dfab217d4b77": "{\\mathfrak {k}}\\oplus i{\\mathfrak {p}}", "17e24d6ff3aab946eb9f4f51370ed0b9": "I_{z}'=\\int _{0}^{aL}dz\\int _{x,y}dxdy\\,\\rho '(x,y,z)\\,r^{2}", "17e284288b0fa14e8f43c9ea9d6b268b": "(n-4)2^{n-3}+1", "17e28d9d301cdae584e6e378659377b7": "\\ \\sigma _{rc}={\\sqrt {PDOP^{2}\\times \\sigma _{R}^{2}+\\sigma _{num}^{2}}}={\\sqrt {PDOP^{2}\\times 2.2^{2}+1^{2}}}\\,\\mathrm {m} ", "17e2b8e42b405da0ba0f3bcb72d7fb1d": "b_{M}=\\infty ", "17e2ef344c869b7ea7ef0536ad36ac7e": "\\varepsilon _{\\alpha +1}", "17e322c32a2e1f442a2373d3a0b702b0": "F_{a}\\propto {\\vec {v}}", "17e35b9d873da2bfe8ca3d676364b8d3": "\\gamma :I\\to M", "17e3962169351fef25a32f844841b242": "h\\ll N", "17e3d5f0f7addd53afb1f2019e738996": "X_{(a,0,c,d)}(u)={\\sqrt {d}}\\cdot e^{i\\pi cdu^{2}}x(du)\\,,", "17e3d855d8e4734cdff491abc2c67ceb": "\\Delta l", "17e42881b74c73353df5ae18d8fb8b11": "f_{W}(t)=\\prod _{d,e\\geq 1}(1-X_{d}^{[d,e]/d}Y_{e}^{[d,e]/e}t^{[d,e]})^{de/[d,e]}=\\sum _{n\\geq 0}D_{n}t^{n}", "17e447c4fff19f1dfcee4cf10417360f": "r=0", "17e452bcb87c8a1b8217d6d23e11324b": "O(n-i)", "17e4d31435ffc486ccd366828de4060e": "D_{\\mathrm {KL} }(P\\|Q)=\\sum _{j}p_{j}\\log {\\frac {p_{j}}{q_{j}}}\\geq 0,", "17e4f3012bab4a66c45e19c94018062e": "a_{ij}={\\frac {1}{x_{i}-y_{j}}};\\quad x_{i}-y_{j}\\neq 0,\\quad 1\\leq i\\leq m,\\quad 1\\leq j\\leq n", "17e503f0700ea0935618b473e566fc9f": "\\{T_{r},p_{r}\\}=const.", "17e52223a61815d0ea2bd345aee6d51d": "q(x,y,z)=d((x,y,z),(0,0,0))^{2}=\\|(x,y,z)\\|^{2}=x^{2}+y^{2}+z^{2}.", "17e522aeebd1905f06c37e69f8664047": "\\mathbf {C} _{ik}=(\\mathbf {A} \\,\\mathbf {B} )_{ik}=\\sum _{j=1}^{N}A_{ij}B_{jk}", "17e54b0e83161c98ed2854b021ad4ecd": "m>\\deg(M(x))", "17e571d37c7867877ed899821c75ecbb": "{\\sqrt {\\frac {2}{\\pi \\left(2k-1\\right)}}}\\,{\\frac {2^{2k-2}\\left(k-1\\right)!^{2}}{\\left(2k-2\\right)!}}", "17e59ccf437c33faae05999c47d52256": "{\\begin{smallmatrix}[{\\frac {Fe}{H}}]=0.5\\end{smallmatrix}}", "17e5b0256f40aef34af8d1cc8d5ff62b": "\\forall s,t:W^{+}[s,t]=W(t,s).", "17e5ea4c140ad5b30d1faf3a121e954f": "\\sum _{i=0}^{n}{\\mbox{metaball}}_{i}(x,y,z)\\leq {\\mbox{threshold}}", "17e61d9f31314bbba0b1bbd8fe0bbcd8": "{\\boldsymbol {\\nabla }}\\cdot ({\\boldsymbol {A}}\\cdot \\mathbf {b} )=({\\boldsymbol {\\nabla }}\\cdot {\\boldsymbol {A}})\\cdot \\mathbf {b} +{\\tfrac {1}{2}}[{\\boldsymbol {A}}^{T}:{\\boldsymbol {\\nabla }}\\mathbf {b} +{\\boldsymbol {A}}:({\\boldsymbol {\\nabla }}\\mathbf {b} )^{T}]", "17e637885b1c4007253313a5aeec81cf": "Y_{i}=\\alpha +\\beta X_{i}+u_{i},", "17e6b2525c9c388d6dd326ee60381750": "D_{1}(P\\|Q)=\\sum _{i=1}^{n}p_{i}\\log {\\frac {p_{i}}{q_{i}}}", "17e6c3efed1d499a57c8409b187c939e": "{\\frac {{\\text{d}}f(x_{1}^{*}(c_{1},c_{2},\\dots ),x_{2}^{*}(c_{1},c_{2},\\dots ),...)}{{\\text{d}}c_{k}}}=\\lambda _{k}^{*}.", "17e707a577e6d93dab45da9b7d77c7e0": "\\sec \\theta \\!", "17e71961946c1870e63767e5d71b4464": "\\ {\\vec {\\mathrm {M} }}_{sail/G}={\\vec {F_{forward}}}\\times distance_{G-E}", "17e71eacd30f8cd4e4375cdf8b30ff0a": "{\\frac {d^{2}y}{dx^{2}}}.", "17e7271ffe23a3f4f90c8f8fec96d3c1": "\\left\\{a_{1}+a_{2}+\\cdots +a_{n}\\right\\}_{n=1}^{\\infty }=O(1)", "17e84e30aa55a84e3d2e51aca9d2dbae": "C(d)=\\sigma ^{2}\\exp(-d^{2}/2\\rho ^{2}).", "17e89e388328fe6f5b8be54f67510a17": "\\tan \\theta ={\\frac {S_{1}/{\\sqrt {n_{1}}}}{S_{2}/{\\sqrt {n_{2}}}}}", "17e8c3dbda94ba325c41c9dc3626a197": "\\chi \\,", "17e8e4449ae48d3c4c33725bf70c768e": "2^{571}", "17e8ec35cedb8027aa7ebe1ca77d0af8": "\\Pi =\\left\\{I,X,Y,Z\\right\\}", "17e913e5835be6f896c1a2c573944c21": "{\\text{s.t.}}={\\begin{cases}g_{1}\\left(x,y\\right)&={\\frac {f_{2}\\left(x,y\\right)}{0.858\\exp \\left(-0.541f_{1}\\left(x,y\\right)\\right)}}\\geq 1\\\\g_{1}\\left(x,y\\right)&={\\frac {f_{2}\\left(x,y\\right)}{0.728\\exp \\left(-0.295f_{1}\\left(x,y\\right)\\right)}}\\geq 1\\end{cases}}", "17e9a133159f21c9c6f879857ef85501": "\\varphi =\\pm \\arccos {(-\\beta /(2{\\sqrt {R}}))}", "17e9c13d22a77ff2682c7bbc86bb2b25": "{\\mathbf {\\hat {n}}}", "17e9ce08b3ecfda0a0c0f6da00595919": "|njm\\rangle ", "17e9cf799d6214d684ddcedaef132457": "XZ\\to YZ", "17e9e68ae2640ae3b793de415fecc265": "\\chi _{\\text{e}}", "17ea10df84753d1d5e066f5beaec91ac": "\\left\\|\\cdot \\right\\|", "17ea482e746464da7ade6f19d04a6100": "\\theta _{0}=\\omega _{r}\\quad {\\text{ if }}\\quad P(\\omega _{r}\\mid \\xi )=\\max _{s=1,2,\\ldots ,R}P(\\omega _{s}\\mid \\xi )", "17ea7cf4504f64dbfca9dcb2e873e4b4": "{\\begin{aligned}d(u\\cdot v)&{}=(u+du)\\cdot (v+dv)-u\\cdot v\\\\&{}=u\\cdot dv+v\\cdot du+du\\cdot dv.\\end{aligned}}", "17ea8c81f955f7ecf305a5a65a492ff3": "~{\\mathsf {4BaCrO_{4}+2Ba(OH)_{2}\\xrightarrow {NaN_{3}} \\ 2Ba_{3}(CrO_{4})_{2}+O_{2}\\uparrow \\ +2H_{2}O\\uparrow }}", "17eafc5969f5dd6acc87a4b5c0ec3618": "x_{n}=(x_{n},x_{n}+1,\\dots ,x_{n}+s-1)", "17eb8e2e179b9bc4c11ac3b9c8d9a498": "(i-1)^{th}", "17eb927e31859ce116f5c64cbcc30d75": "\\alpha =", "17ebd8bd5b8095d735ea6687ef82fdc2": "\\mathbf {\\Pi } _{1}^{1}", "17ec219d55c8e745978ccb3a839409fe": "H\\;1000\\lor \\lnot H\\;1000", "17ec36c91e738aa8ccc78d949b6ea8cb": "e^{j\\pi /2}.\\,", "17ec52c66977b4ee27cc3cb70b2fe91a": "{81 \\over 80}", "17ec6134c938436ab5d31f054fe7c889": "\\cosh(t)\\,", "17ec75419272fa56fe701527393f9d6f": "h_{\\mathrm {FOH} }(t)\\,", "17ec7cd41ca3a8a01a81a7cbf39adc69": "{\\rm {pcf}}(A)=\\{cf(\\prod A/D):D\\,\\,{\\mbox{is an ultrafilter on}}\\,\\,A\\}.", "17ec8a053605926c3943b75c1335d79f": "\\kappa _{2}^{F(n)}-\\gamma _{2}=0\\,,", "17ed423bd00722bfd44f70ac800b9e37": "{\\mathcal {F}}_{t}^{W}.\\,", "17ed59237c2847f885568b7457a7393c": "V(t)=V(0)+\\int _{0}^{t}L_{t}(x^{\\ast }(s),y^{\\ast }\\left(s\\right),s)ds.", "17ed5bc0d01231e7079511b916b3c957": "{\\overline {16}}_{-1H}", "17ed833de1218186e4c981bd4fb9cace": "|\\langle f,g\\rangle |\\leq \\|f\\|_{2}\\|g\\|_{2}\\,,", "17edf411e4e987edb5be66da1089d267": "cov(w_{i},z_{i})+wz=E(w_{i},z_{i})=az(1-z)", "17edfef36bddd2ddf333e14ad7b4f3c5": "0.999^{1000}", "17ee1b1c6e8ebde78abc253ed5d9db76": "\\textstyle -", "17ee82407f59375a680009514e71d0be": "E_{n}(x^{\\mu })", "17ee8b370256539c3ff0c5d37345248d": "F_{f}=\\mu N_{f}\\,", "17eee94126b662ee7d4156e34c11f25a": "\\scriptstyle \\otimes ", "17eefdd872ea45fa01f79c426ed6902b": "{\\widehat {R}}(\\theta ,{\\hat {\\mathbf {n} }})=\\exp \\left(-{\\frac {i}{\\hbar }}\\theta {\\hat {\\mathbf {n} }}\\cdot \\mathbf {J} \\right)", "17ef988738145fa073738ee7a5d90a21": "M\\leq \\left\\lfloor {\\frac {2d}{2d-n}}-1\\right\\rfloor =\\left\\lfloor {\\frac {2d}{2d-n}}\\right\\rfloor -1\\leq 2\\left\\lfloor {\\frac {d}{2d-n}}\\right\\rfloor .", "17efccd691a4cec276dd62d69692cdbb": "W_{1}(\\mu ,\\nu )=\\sup \\left\\{\\left.\\int _{M}f(x)\\,\\mathrm {d} (\\mu -\\nu )(x)\\right|{\\mbox{continuous }}f:M\\to \\mathbb {R} ,\\mathrm {Lip} (f)\\leq 1\\right\\},", "17f08c78c5e3bf040c7b550baf559d5a": "{\\frac {1}{T}}=\\left({\\frac {\\partial S}{\\partial U}}\\right)_{V,N}\\,.\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(3)", "17f0b3bb7c80669befd5d00ff2f3b4f1": "\\{x\\}", "17f0bc7fc1b8f99ef9d45fc0d93687ec": "\\ \\ t=\\tan \\alpha \\ ;\\ \\ t\\in \\mathbb {R} \\ )", "17f124bfd50ee0bb5f6cedeb37230021": "\\nu _{ij}", "17f16f9bc981b8ae04f421aa6fc72918": "k(t)=\\sum _{i=1}^{\\infty }\\lambda _{i}S_{i}\\Phi _{i}(t),00", "17f3977b33d086bcbbb9da8729521343": "A^{\\prime }=A,", "17f3e0655786ef4df373ecbee1773ed2": "\\epsilon _{F}", "17f405fb83dd8bc4825175104470d95a": "\\textstyle {\\bar {M}}_{\\mathrm {e} }=\\left({\\begin{array}{cc}{\\frac {1}{2}}&{\\frac {1}{2}}\\\\-{\\frac {1}{4}}&{\\frac {1}{4}}\\end{array}}\\right)", "17f4149c1f2c00ad85e587fc916fd6dc": "(a){\\text{ }}\\bigcup \\nolimits _{i=1}^{n}{R_{i}=R.}", "17f42511d5c6b54d2bc99422fca646d2": "{\\begin{aligned}f&={\\frac {a-b}{a}},\\qquad e^{2}=f(2-f),\\qquad n={\\frac {a-b}{a+b}}={\\frac {f}{2-f}}\\\\b&=a(1-f)=a(1-e^{2})^{1/2},\\qquad e^{2}={\\frac {4n}{(1+n)^{2}}}.\\end{aligned}}", "17f426970d265ff3f12b55d280187164": "f_{i}({\\vec {x}}+{\\vec {e}}_{i}\\delta _{t},t+\\delta _{t})-f_{i}({\\vec {x}},t)+F_{i}=\\Omega (f)", "17f492cdd3a7eb944da6b39b7507ef79": "g_{ab}={\\frac {\\partial Z^{A}}{\\partial E^{a}}}{\\frac {\\partial Z^{B}}{\\partial E^{b}}}G_{AB}\\ .", "17f4c9f7d2f67ac6806846b512efb63a": "\\alpha (\\omega )={\\frac {4\\pi \\omega }{n_{b}c}}\\chi ''(\\omega )", "17f4d827a7cec674b0d3b8002e546081": "X\\in T_{p}M", "17f53f0181d25d95449981f1a841cb56": "\\delta n^{a}-\\Delta m^{a}=-{\\bar {\\nu }}l^{a}+(\\tau -{\\bar {\\alpha }}-\\beta )n^{a}+(\\mu -\\gamma +{\\bar {\\gamma }})m^{a}+{\\bar {\\lambda }}{\\bar {m}}^{a}\\,,", "17f547bd474a83a51aff9e1b5fbee91a": "\\mathbf {n} _{1}", "17f58a87bc544f5200da2fc46428254c": "\\min(3-1,4-1,2-0,0-(-1),6-3,9-3)=", "17f5939bda952cd3ff4a4d34c8f3b408": "O(d)\\,", "17f59a354839efc176a2773aff3f44e0": "\\pi _{*}W_{v}=v\\,", "17f59d9ac48df9d60f6fe8999d8b683a": "x^{\\left[1\\right]}=x\\log(x)-x", "17f5f6cfe7ccbe4d9ca46a9b9da0027f": "\\sin \\left(x-y\\right)=\\sin x\\cos y-\\cos x\\sin y,\\,", "17f60ef594eb982a99682a230d70d57c": "u=(u_{ij})_{i,j=1,\\dots ,n}", "17f665f57c52c9926c2e9855cdc5bff2": "E\\left\\{\\left|h_{w,ij}\\right|^{2}\\right\\}=1\\quad \\forall i,j", "17f66fbdd5bf0d7626acef952d4baea9": "|0\\rangle ,\\ldots ,|2^{n}-1\\rangle .", "17f67df2f5d9ee3661255e2bec2cb7ba": "h(x)=g(x)-(\\lambda ^{\\prime }-\\lambda )\\int _{c}^{x}(\\varphi _{\\lambda }(x)\\theta _{\\lambda }(y)-\\theta _{\\lambda }(x)\\varphi _{\\lambda }(y))g(y)\\,dy", "17f6942775078d20dc17baeabe217cea": "H_{n}(0)={\\begin{cases}0,&{\\mbox{if }}n{\\mbox{ is odd}}\\\\(-1)^{n/2}2^{n/2}(n-1)!!,&{\\mbox{if }}n{\\mbox{ is even}}\\end{cases}}", "17f6f18308e191fbac91ee26bb7d7e8a": "u(w)=-w^{\\alpha }", "17f71054ef4cfe4d134500b3d0199d3c": "m=\\mathop {\\mathrm {arg\\ min} } _{p\\in M}\\sum _{i=1}^{N}w_{i}d^{2}(p,x_{i})", "17f712769b8bf67b62230eccf552fb7d": "\\mathbf {n} ={\\frac {\\mathbf {i} -\\mathbf {r} }{\\|\\mathbf {i} -\\mathbf {r} \\|}}", "17f73ca0ee9e3525bbf58451e1d8bf5e": "\\alpha _{\\text{pump off}}(\\omega )", "17f76e61dd4b3788991af8773f778192": "<0.24", "17f7b419875e59455c6f5c6f424f14ee": "c(w)={\\frac {1}{A(w)}}{\\frac {d}{dw}}A(w)=\\sum _{n=0}^{\\infty }c_{n}w^{n}.", "17f7cbd51c8256a9a58b5dddad126b5f": "F_{X_{\\gamma }}=-{\\frac {\\partial E}{\\partial X_{\\gamma }}}=-{\\bigg \\langle }\\psi {\\bigg |}{\\frac {\\partial {\\hat {H}}}{\\partial X_{\\gamma }}}{\\bigg |}\\psi {\\bigg \\rangle }.", "17f7f80674eb042d147a005aeaaac5eb": "00)", "18069f4398b00eae525578799415d74f": "{\\partial ^{2}\\mathbf {E} \\over \\partial t^{2}}\\ -\\ c^{2}\\cdot \\nabla ^{2}\\mathbf {E} \\ \\ =\\ \\ 0", "1806c82a6011d980a7ac56004460bc6c": "A={\\frac {V_{o}}{V_{i}}}={\\frac {\\left({\\frac {M}{C}}+a\\left({\\frac {M}{C}}\\right)\\left(\\beta -1\\right)+{\\frac {\\beta +2}{3\\left(\\beta +1\\right)}}\\right)}{\\left({\\frac {N}{C}}+{\\frac {2\\beta +1}{3\\left(\\beta +1\\right)}}\\right)}}", "1806d217fc8c24b7c1180da5fe84471a": "\\langle \\chi |\\partial ^{\\mu }A_{\\mu }|\\psi \\rangle =0", "180708f18711731d870976286cc28396": "{\\begin{aligned}x&={\\frac {a\\sinh v}{\\cosh v-\\cos u}}\\\\y&={\\frac {a\\sin u}{\\cosh v-\\cos u}}\\\\z&=z\\end{aligned}}", "18072fd37c78b5b3fd11544442824fe3": "{\\mathbf {r}}_{0}=h_{1}{\\mathbf {n}}_{1}+h_{2}{\\mathbf {n}}_{2}", "18075ab9eb580438e37c317057fd7a4a": "p=j^{2}\\cdot g", "1807752fa6d342715d0003971f3c0f41": "\\mathbf {NL} \\subsetneq \\mathbf {PSPACE} \\subsetneq \\mathbf {EXPSPACE} ", "1807cd6c50789312c1cc88c6c1a8998d": "\\mathbb {E} (D_{u})=\\int _{-\\infty }^{\\infty }|x'|p(u,x')\\,\\mathrm {d} x'", "1807d8c86d25ce116582a56803c5df0e": "\\Delta m_{\\odot }^{2}\\simeq 8\\times 10^{-5}\\,{\\mbox{eV}}^{2}", "1807e33be3dcfdc55408ee2ef33c235a": "1/K_{p}", "180823239b488c4f827b05ced75230c4": "1<{\\frac {\\theta }{\\sin \\theta }}<{\\frac {1}{\\cos \\theta }}\\implies 1>{\\frac {\\sin \\theta }{\\theta }}>\\cos \\theta \\,.", "180863eaffa00e8b6dff86a9923f8059": "m^{2}=-I,\\quad z=xI+m{\\sqrt {-p}}", "18088d589a8d81b9444df964cae50e2b": "{\\begin{aligned}&{}\\qquad F_{X_{n},X_{n+1},\\dots ,X_{n+N-1}}(x_{n},x_{n+1},\\dots ,x_{n+N-1})\\\\&=F_{X_{n+k},X_{n+k+1},\\dots ,X_{n+k+N-1}}(x_{n},x_{n+1},\\dots ,x_{n+N-1}),\\end{aligned}}", "1809064148dcc02bb7beb9a848be89a9": "1=H'\\int _{0}^{T}\\{x,p\\}dt=H'T\\,", "18095107a92891a2e4f4fd2486db5e67": "2N^{2}", "1809847ce91741ecc9a82882159e044b": "a_{q}=\\min\\{n\\in \\mathbb {N} \\colon n\\geq (5/2)^{q+1},", "180986a5c166b5d8f40456577d106cc0": "{\\frac {\\mathrm {d} F}{\\mathrm {d} t}}-P\\leq 0.", "1809998d6fbca11e75ba1344184616c2": "{\\mathsf {T}}=(0,\\mathbf {v} ).", "1809a9122cb55a13bc0af46771724fbb": "P=\\sum _{ij}|i\\rangle \\langle j|\\otimes |j\\rangle \\langle i|", "1809c3e5e783c9346f41afa25b4d6055": "S(x)=\\sum _{i=\\lfloor x\\rfloor -a+1}^{\\lfloor x\\rfloor +a}s_{i}L(x-i),", "1809c7e7ff9da0b203c92112c6581db2": "\\sigma _{x}=-zE{\\cfrac {\\mathrm {d} ^{2}w}{\\mathrm {d} x^{2}}}", "1809d05376474d84d30f35040b8f201f": "S(\\rho ^{AB})=-\\operatorname {Tr} \\rho ^{AB}\\log \\rho ^{AB}.", "1809e0d2643c824182747d32bf2ce40c": "v={\\frac {3}{2}}\\mathbf {e} _{1}+2\\mathbf {e} _{2}", "1809eeb44977727661f96e4b6976e864": "dP\\propto \\exp {\\left[-{\\frac {{\\mathcal {H}}(\\mathbf {p} _{1},\\ldots ,\\mathbf {p} _{N};\\mathbf {r} _{1},\\ldots ,\\mathbf {r} _{N})}{k_{\\text{B}}T}}\\right]}d\\mathbf {p} _{1},\\ldots ,d\\mathbf {p} _{N}d\\mathbf {r} _{1},\\ldots ,d\\mathbf {r} _{N},", "180a1dde2077bc80fa44e1339484d091": "-i\\hbar {\\boldsymbol {\\nabla }}", "180a2870d4cbbac91b35a6c1813f13c0": "HS_{K}(t)=1\\,.", "180a2a9d56f09498295807239ede8d0a": "|I(t)-I(0)|<\\varepsilon ^{1/(2n)}", "180a33eb34f823e27b4d6f454dbb7e6a": "2\\hbar k", "180a58c005ad2c2e5dea7450f3f3d1bb": "R(t)=Pr\\{T>t\\}=\\int _{t}^{\\infty }f(x)\\,dx\\ \\!", "180aa8b273e58d0c881edca7d4df39b8": "[S]=[S]_{0}(1-k)^{t}\\,", "180c59c3a2a8fb21185fe069086eeea3": "\\left\\{{\\begin{matrix}\\tau _{12}={\\frac {\\Delta g_{12}}{RT}}={\\frac {U_{12}-U_{22}}{RT}}\\\\\\tau _{21}={\\frac {\\Delta g_{21}}{RT}}={\\frac {U_{21}-U_{11}}{RT}}\\end{matrix}}\\right.", "180c8a867d9f782257eafe4e72f56262": "(S_{1},\\ldots ,S_{N})", "180d3f0cdc3ae1cf35627c95601d898b": "r_{1}<\\cdots k", "180eba59ff13816c241ab7b5d6bf1104": "\\mathbf {E} _{l,m}^{(M)}={\\sqrt {l(l+1)}}\\left[E_{l}^{(1)}h_{l}^{(1)}(kr)+E_{l}^{(2)}h_{l}^{(2)}(kr)\\right]\\mathbf {\\Phi } _{l,m}", "180efb12feb4d089b09615f5f21a415c": "\\psi (x)=\\int {{\\frac {d^{3}p}{(2\\pi )^{3}{\\sqrt {2E}}}}\\sum _{\\lambda \\pm 1}{\\left({\\hat {a}}_{p}^{\\lambda }u_{\\lambda }(p)e^{-ip\\cdot x}+{\\hat {b}}_{p}^{\\lambda }v_{\\lambda }(p)e^{ip\\cdot x}\\right)}}\\,", "180f42501da3daa476de45218a3534c3": "pb_{t}", "180fbe7f9e38c992a5df3eb4797cb8db": "{\\boldsymbol {\\varphi }}_{x}'(x)\\equiv \\left({\\frac {\\partial \\varphi _{i}(x)}{\\partial x_{j}}}\\right),\\qquad 1\\leqslant i\\leqslant k,\\quad 1\\leqslant j\\leqslant n.", "180fe72520038e7e96fe41557628018b": "x_{t}=x_{0}(1+r)^{t}", "18101784a9d512383c0dac78f71f0c3b": "y^{[n-1]}=\\left[{\\begin{matrix}y_{1}^{[n-1]}\\\\y_{2}^{[n-1]}\\\\\\vdots \\\\y_{r}^{[n-1]}\\\\\\end{matrix}}\\right],\\quad y^{[n]}=\\left[{\\begin{matrix}y_{1}^{[n]}\\\\y_{2}^{[n]}\\\\\\vdots \\\\y_{r}^{[n]}\\\\\\end{matrix}}\\right],\\quad Y=\\left[{\\begin{matrix}Y_{1}\\\\Y_{2}\\\\\\vdots \\\\Y_{s}\\end{matrix}}\\right],\\quad F=\\left[{\\begin{matrix}F_{1}\\\\F_{2}\\\\\\vdots \\\\F_{s}\\end{matrix}}\\right].", "181043a9bff552b4a3bbb2745ab710ae": "\\mu _{X}=\\mathbb {E} _{X}[\\mathbf {e} _{x}]=\\left({\\begin{array}{c}P(X=1)\\\\\\vdots \\\\P(X=K)\\\\\\end{array}}\\right)", "1810ccc0e3399443049e39f537c570db": "\\Delta W_{\\mathrm {ON} }=-\\Delta W_{\\mathrm {BY} }\\,\\!", "1810e57e32c3de41d6778dec879b9ebe": "\\quad (8)\\qquad \\qquad {{\\partial {\\mathbf {u} }} \\over {\\partial t}}+\\nabla \\cdot {\\mathbf {f} }\\left({\\mathbf {u} }\\right)={\\mathbf {0} }.", "1811ac550cb73c34cb1df992312e8080": "\\mathbb {T} ", "1811be2c89fa4bd425250a9b495af177": "D(d_{\\lambda }f)=\\lambda \\cdot d_{\\lambda }f.", "1811de483b0e564d9e631e7f99564c69": "I_{X}={\\frac {Y_{X}}{Y_{Total}}}I_{T}={\\frac {\\frac {1}{R_{X}}}{{\\frac {1}{R_{X}}}+{\\frac {1}{R_{1}}}+{\\frac {1}{R_{2}}}+{\\frac {1}{R_{3}}}}}I_{T}", "1811f7eae428ee41f5e8bb015d95695a": "\\eta ^{-1}", "181205e7cc6366aaf861ab61194e524c": "MRP_{L}=MC_{L}", "181241214a35d54b2a97ae35a3d34167": "f_{j}(c_{1},c_{2},...,c_{N_{B}})\\,=\\,[j]_{TOT}-[j]-\\sum _{i=1}^{N_{S}}{\\frac {\\nu _{i,j}}{\\gamma _{i}}}\\,K_{i}\\,\\prod _{k=1}^{N_{B}}\\{k\\}^{\\nu _{i,k}}\\,=\\,0", "18124e94c021584f29f6e4621673ba79": "s_{kk}=0\\,\\!", "18125e406a3acc3e377df532a4ed09a6": "(M,H,g)", "1812b83ad7e9470a13d9ab8e08fc8843": "j^{n}(\\kappa )", "1812cfa10c32a5773462a3d2de54216a": "\\sum _{n=1}^{\\infty }x_{n}", "1813496391386430b1fecf04c2db1533": "\\varepsilon \\thicksim N(0,\\sigma ^{2}).\\,", "1813bd3fbc3c2b75ebde60944ee14902": "(X,id_{X})", "1813d71ac32a0e0aabc2af49fbf0bf0a": "\\subseteq ", "1813e6b0253367e59eaf37adbcb09190": "\\left(\\lambda +\\mu \\right){\\frac {\\partial }{\\partial x}}\\left({\\frac {\\partial u_{x}}{\\partial x}}+{\\frac {\\partial u_{y}}{\\partial y}}+{\\frac {\\partial u_{z}}{\\partial z}}\\right)+\\mu \\left({\\frac {\\partial ^{2}u_{x}}{\\partial x^{2}}}+{\\frac {\\partial ^{2}u_{x}}{\\partial y^{2}}}+{\\frac {\\partial ^{2}u_{x}}{\\partial z^{2}}}\\right)+F_{x}=0\\,\\!", "18143f739d4c788fdefe9d0f358e83d5": "(A\\to B)\\to ((B\\to A)\\to (A\\leftrightarrow B))", "1814c9fb56fbc8b6ae3b1a241ea29aad": "Q={\\mbox{Re}}(L)", "1814fcab976450ffcf27d04a82cf5a87": "w(x_{1},x_{2},\\ldots ,x_{n})dx_{1}dx_{2}\\ldots dx_{n}", "18154f1a351aed69411dfc1d74afd945": "f(z)=z^{m}e^{g(z)}\\displaystyle \\prod _{n=1}^{\\infty }E_{p}(z/a_{n})", "181552e6c8d3178a0e5bcfb93e76999b": "\\mathrm {d} \\mathbf {r} =\\mathrm {d} \\rho \\,{\\boldsymbol {\\hat {\\rho }}}+\\rho \\,\\mathrm {d} \\varphi \\,{\\boldsymbol {\\hat {\\varphi }}}+\\mathrm {d} z\\,\\mathbf {\\hat {z}} .", "18156063a1acd4a642204b6bfa284b2d": "{\\frac {4}{3}}\\div {\\frac {704}{576}}={\\color {blue}{\\frac {12}{11}}}", "1815ac41d05d5259e9e54e83d8c22f92": "P(M>a)", "1815b38a57a4b348b7ff676955a34aa4": "B_{\\infty }", "1815e5f6c120502d38f2eed8b0b5e2a1": "(X_{1}\\wedge Y_{1})\\vee (X_{2}\\wedge Y_{2})\\vee \\dots \\vee (X_{n}\\wedge Y_{n}).", "1815efb92ba9df6aa94736e4304db783": "\\log _{b}(a-c)=\\log _{b}a+\\log _{b}(1-b^{\\log _{b}c-\\log _{b}a})", "181646f5cc3c6780e968c8b0760781a5": "G^{(\\alpha )}=\\{e\\}", "1816909600ff52b7c8a6fe5da1894e1e": "\\mu (\\emptyset )=0", "1817b2083540c4801888dd6aebf9b9e6": "{\\tilde {f}}^{*}(\\cdot )=\\sum _{i=1}^{n}\\alpha _{i}k(\\cdot ,x_{i})+\\sum _{p=1}^{M}\\beta _{p}\\psi _{p}(\\cdot )", "1817fa16f68b0ff291d3fd9ce0bae2d1": "{\\frac {2}{3}}\\!\\,", "18184e80f69735e66651ff330ed7ec36": "H(X,Y)=H(X)+H(Y)-I\\left(X;Y\\right)=2", "181862a5ec1bff268df5184dd2489f6b": "\\tau (p)=0{\\bmod {p}}", "18186a80c2c74e2ff966d067f207009c": "{\\vec {w}}", "1818709942d238261498fa72703eaf05": "h_{\\mathcal {D}}(t)", "1818dddfa5de4e4b3f846b29023bd48b": "G:{\\mathcal {D}}\\to {\\mathcal {C}}", "181902e0f73e7f74a3406034e566e102": "(v,N)", "18193226b0d4f87b68420151cd720406": "V(\\mu )=\\tau ^{\\prime }[\\tau ^{-1}(\\mu )]", "1819c280dd2113ed22ff2311f27df264": "\\mathbf {Top} .", "181a27fff2021c81b6f3fcf14f7d796b": "W(I_{1}){\\biggr |}_{I_{1}=3}=0\\quad {\\text{and}}\\quad {\\cfrac {\\partial W}{\\partial I_{1}}}{\\biggr |}_{I_{1}=3}={\\frac {\\mu }{2}}\\,.", "181a3633f90605c2fc1f907b02924640": "\\Omega =q\\theta -(q-2)\\pi .\\,", "181a3829e04b0262ffd2065a7c1c77ec": "|\\psi \\rangle =\\sum _{i}|i\\rangle \\psi _{i}", "181a4a90433d2b6ccbc1896dae157bad": "\\sigma ={\\frac {\\epsilon \\epsilon _{0}\\psi _{0}}{\\lambda _{D}}}", "181a6afe6190dfe00f9471a7e26fd622": "\\csc ^{2}x_{1}+\\csc ^{2}x_{2}+\\cdots +\\csc ^{2}x_{m}={\\frac {2m(2m-1)}{6}}+m={\\frac {2m(2m+2)}{6}}.", "181ad54b8fcdb1066fb2288cfdd61d9e": "\\Delta _{5}", "181b034810968526cce3d1ed24830e5d": "uv^{T}", "181b14f3c64571384be6c9bf3eaba6bc": "r_{p+i\\cdot k}=r_{p+p\\cdot k}=r_{p(k+1)}=0", "181b8d98d4c1465346d92acde3025ceb": "T^{a}{}_{acd}=0", "181b944684b8d793134d351dbcaf0638": "H_{s}(r)=P({N}(b(o,r))=0).", "181c427b49efda95cd62934c8dfc9935": "B=0.07780{\\frac {P_{r}}{T_{r}}}", "181cb3dcccd04c0d46c7a6571810ac27": "\\scriptstyle {\\hat {s}}(t)", "181cb4fa8519e6c0a19fe85dd8d57e2b": "\\mathbf {u} =\\{u[j]\\}", "181ce4d603bde8eb87d56af0620d4bc9": "{\\begin{bmatrix}k_{1}&0\\\\0&k_{2}\\end{bmatrix}}", "181d10b7e3a35994a766a94799fc3111": "\\|\\mathbf {x} -\\mathbf {x} ^{*}\\|\\leq \\delta ;\\,", "181d1319de482194de9ca39c9ff1576d": "\\chi =\\psi _{1}+i\\psi _{2}\\,", "181db2fd298f0302c590f1b3babdc612": "{\\hat {S}}_{N}\\subset C", "181db821b8b2aac77c3e03c5e67efbb0": "p({\\vec {r}})=1/V", "181dcd45678d4499bdbad5e4a29d6d1d": "\\lambda _{2}=(\\mathrm {E} X_{2:2}-\\mathrm {E} X_{1:2})/2", "181ddd0df008d6860b34522ff8b92d58": "\\sigma f(A_{1},\\dots ,A_{n})=f(\\sigma A_{1},\\dots ,\\sigma A_{n})", "181de74002d56eb422c240e61ac0a015": "v_{0^{}}", "181e05b7f24072b3e0fdf492b5d6a85d": "\\scriptstyle d\\geq 3", "181e54f99352a0a0f9f1d1d138f7bd5a": "P(t)={\\begin{cases}0&tt_{o}+t_{p}\\\\\\end{cases}}", "181e72e1f409b60f5322f9008a8ea469": "\\int _{\\mathbb {R} ^{n}}\\prod _{i=1}^{m}f_{i}(x\\cdot u_{i})^{c_{i}}\\,\\mathrm {d} x\\leq \\prod _{i=1}^{m}\\left(\\int _{\\mathbb {R} }f_{i}(y)\\,\\mathrm {d} y\\right)^{c_{i}}.", "181edc009ceec906e6db431b342313c0": "G\\,=\\,\\int _{-h}^{0}\\left({\\frac {{\\text{d}}f}{{\\text{d}}z}}\\right)^{2}\\;{\\text{d}}z.", "181f3a7a07c9c0f763b1028d304d8c4d": "{\\begin{smallmatrix}\\left({\\frac {T_{eq}}{T_{pole}}}\\right)^{4}=\\left({\\frac {7,600}{10,000}}\\right)^{4}=0.33\\end{smallmatrix}}", "181f5015bdd98f043258bdff3ccc8bb7": "\\mathbb {R} ", "181f6a46c34ce06a6bd8e1ea5d7ef213": "H_{i}=V^{i}(H)", "181fab25d30b8f19c9fc01d32cd59774": "T(x_{i},y_{i})=M^{2}\\iint T(Mk_{x}^{i},Mk_{y}^{i})~e^{j(k_{x}^{i}x_{i}+k_{y}^{i}y_{i})}dk_{x}^{i}\\,dk_{y}^{i}", "181fdd3909a3a3a9e3fe179ce6bfe9c5": "|b|^{2}", "18203051a20839c54417d2891036cbbf": "X_{1},\\dots ,X_{n}", "18203483de0584daf8422945759fc3fd": "{\\text{pitch}}_{Z}={\\sqrt {6}}\\cdot {d \\over 3}\\approx 0.81649658d,", "18203e6e345a3a9b7e570767a7aac3d4": "\\mathbb {E} f", "1820552c3b85931361457e1c3ff6ec3e": "{\\tilde {E}}_{n}", "182060f212b58d78b2a6d58b79f3496d": "b_{1}=B/(1+(o_{1}/o_{2})+(o_{1}/o_{3}))", "1820726e9191d57202abd54a4d7a39bc": "Q={\\begin{pmatrix}{-(x_{1}+x_{2}+x_{3})}&{\\pi _{1}x_{1} \\over \\pi _{2}}&{\\pi _{1}x_{2} \\over \\pi _{3}}&{\\pi _{1}x_{3} \\over \\pi _{4}}\\\\{x_{1}}&{-({\\pi _{1}x_{1} \\over \\pi _{2}}+x_{4}+x_{5})}&{\\pi _{2}x_{4} \\over \\pi _{3}}&{\\pi _{2}x_{5} \\over \\pi _{4}}\\\\{x_{2}}&{x_{4}}&{-({\\pi _{1}x_{2} \\over \\pi _{3}}+{\\pi _{2}x_{4} \\over \\pi _{3}}+x_{6})}&{\\pi _{3}x_{6} \\over \\pi _{4}}\\\\{x_{3}}&{x_{5}}&{x_{6}}&{-({\\pi _{1}x_{3} \\over \\pi _{4}}+{\\pi _{2}x_{5} \\over \\pi _{4}}+{\\pi _{3}x_{6} \\over \\pi _{4}})}\\end{pmatrix}}", "18209cace6d67a042a882ca4f3179f21": "2\\beta _{E}(Q)\\ell (Q)", "1820a23a4a3410b7193215760c1d395d": "{\\text{NC}}(S)", "1820ae845d6ddabc083fd26f233ac6e9": "a_{14}-a_{13}", "1820bab280edf6a1691cafd2a067ac5e": "(p-p')^{2}\\approx \\,", "1820e42debc4030e7555fca17b8fe89f": "D=18abcd-4b^{3}d+b^{2}c^{2}-4ac^{3}-27a^{2}d^{2}.\\,", "1821302a913a29379a79c7ddd3758434": "E_{4}\\cong A_{4}", "1821456c71226d0bccd9a057396c8139": "{\\frac {\\mu -\\theta }{\\sigma }},", "18215de4205af8f9584cde4689e1d404": "{\\hat {\\beta }}=(1-{\\bar {x}})\\left({\\frac {{\\bar {x}}(1-{\\bar {x}})}{\\bar {v}}}-1\\right),", "18219425e20e5727d544a843f11d26c2": "k\\ ", "1822125c2e9e2ef8c64b4354c50928fa": "H_{\\mathrm {DR} }^{k}(M)=\\ker d_{k}/\\mathrm {im} \\,d_{k-1}.", "1822d316572396360302ebd0f07c7c0e": "<24>5_{H}{\\bar {5}}_{H}", "1822e3fbda81bca73026581c2ef5de67": "{\\mathsf {PH}}\\subseteq {\\mathsf {BP}}\\cdot \\oplus {\\mathsf {P}}\\subseteq {\\mathsf {P}}\\cdot \\oplus {\\mathsf {P}}\\subseteq {\\mathsf {P}}^{\\sharp P}", "18230599c03e0cbba169ed1c88e13259": "{\\textit {add}}\\in F_{2}", "18236dfdc3f0677bf5145aa77f138cb9": "\\Gamma \\left({\\frac {1}{2}}\\right)=\\left(-{\\frac {1}{2}}\\right)!=\\Pi \\left(-{\\frac {1}{2}}\\right)={\\sqrt {\\pi }},", "182371b59680b2bb3995d8e8961c17fa": "d_{0,1}=\\log \\left(q_{0}^{-1}q_{1}\\right)", "18237fd134c0886a5e2d0289670e0a10": "{\\frac {dP(r)}{dr}}=-{\\frac {G}{r^{2}}}\\left(\\rho (r)+{\\frac {P(r)}{c^{2}}}\\right)\\left(M(r)+4\\pi r^{3}{\\frac {P(r)}{c^{2}}}\\right)\\left(1-{\\frac {2GM(r)}{c^{2}r}}\\right)^{-1}\\;", "1823cf44a3acf6db04f9133807c7897c": "\\mu =GM", "1823f7bf48e8b1b9e556cdf72b9631bf": "L_{\\phi }=e^{\\phi }(\\textstyle {\\frac {1}{2}}e^{-\\phi }\\partial _{\\alpha }\\phi \\partial _{\\alpha }\\phi +\\textstyle {\\frac {3}{2}}e^{\\phi }\\partial _{0}\\phi \\partial _{0}\\phi )\\,", "18243f98d62ee0fd1a1f457d44f2da7a": "\\xi _{0}(\\mathbf {q} )", "1824892d5451c3075554d64665949840": "\\operatorname {cn} (u)={\\frac {2\\pi }{K{\\sqrt {m}}}}\\sum _{n=0}^{\\infty }{\\frac {q^{n+1/2}}{1+q^{2n+1}}}\\cos((2n+1)v),", "1824ac77ba3c5497e869a995d49d0200": "x_{n}=e^{-{\\frac {2n\\pi {\\rm {i}}}{N-1}}}-{\\frac {t}{(N-1)^{2}}}{\\sqrt {\\frac {N}{2\\pi (N-1)}}}\\sum _{q=0}^{N-2}\\psi _{n}(q)_{(N+1)}F_{N}{\\begin{bmatrix}{\\frac {qN+N-1}{N(N-1)}},\\ldots ,{\\frac {q+N-1}{N-1}},1;\\\\[8pt]{\\frac {q+2}{N-1}},\\ldots ,{\\frac {q+N}{N-1}},{\\frac {q+N-1}{N-1}};\\\\[8pt]\\left({\\frac {te^{\\frac {2n\\pi {\\rm {i}}}{N-1}}}{N-1}}\\right)^{N-1}N^{N}\\end{bmatrix}},\\quad n=1,2,3,\\dots ,N-1", "1824d212ba3a9fa8345f51e91d95b859": "\\mu =\\nu _{0}\\leq \\nu _{1}\\leq \\ldots \\leq \\nu _{k}=\\lambda ", "182502d59cbc0ac4a2c39722dfa57368": "(-1+1)>0", "18254041af8bbfe82d842bf5668298e0": "1-\\lambda n(n-1)dt", "18256429c0e252c25805a86b1d8e6de0": "k\\to 0", "1825842e493abf125ade3b211e1b6e69": "\\scriptstyle {\\frac {k}{\\lambda ^{2}}}\\,", "182599009ae897b9003442eb4f24a279": "2^{k}/2", "1825c9d481bf09bd774ef70ead9b5def": "c_{n}(t)\\,", "1825deb23e17e064408129c0448a3298": "x\\in {\\mathbb {R}}^{n}", "1825f7196cf32cb18df1663623af7d88": "y_{j}\\epsilon )\\leq \\rho ", "18514abffdf8c0476fc31bbcc67288d9": "\\nabla \\sigma =(\\mathrm {d} \\sigma ^{\\alpha }+\\omega ^{\\alpha }\\!{}_{\\beta }\\sigma ^{\\beta })e_{\\alpha }", "1851c0714960d2de5b1fc4fe90456ab6": "w(E\\oplus F)=w(E)\\smallsmile w(F)", "1851c157eadb0c9b355f1ba8bd7a0ac2": "{\\mathfrak {g}}\\to {\\mathfrak {gl}}_{V}", "1852050689b1c87aafadf313e46821e8": "=\\left\\vert {\\frac {1-k(N-\\varphi (N))}{Nd}}\\right\\vert ", "185254c23f9d65f50f7cc185b854a1a8": "x^{2}+y^{2}+z^{2}=R^{2}.", "1852867e59486e21c1b1ef2bb36b7318": "\\mathrm {Ad} _{g}\\colon {\\mathfrak {g}}\\to {\\mathfrak {g}}", "18529b6408dd8f7ec05df5cbb594f2eb": "f(x_{0}+\\Delta x)=f(x_{0})+\\nabla f(x_{0})^{T}\\Delta x+{\\frac {1}{2}}\\Delta x^{T}{B}\\Delta x", "1852bd5f3275076ca7061de28284c940": "{\\begin{aligned}-\\gamma ^{\\mu }{\\hat {P}}_{\\mu }+mc&=-\\gamma ^{0}{\\frac {\\hat {E}}{c}}-{\\boldsymbol {\\gamma }}\\cdot (-{\\hat {\\mathbf {p} }})+mc\\\\&=-{\\begin{pmatrix}I_{2}&0\\\\0&-I_{2}\\\\\\end{pmatrix}}{\\frac {\\hat {E}}{c}}+{\\begin{pmatrix}0&{\\boldsymbol {\\sigma }}\\cdot {\\hat {\\mathbf {p} }}\\\\-{\\boldsymbol {\\sigma }}\\cdot {\\hat {\\mathbf {p} }}&0\\\\\\end{pmatrix}}+{\\begin{pmatrix}I_{2}&0\\\\0&I_{2}\\\\\\end{pmatrix}}mc\\\\&={\\begin{pmatrix}-{\\hat {E}}/c+mc&0&{\\hat {p}}_{z}&{\\hat {p}}_{x}-i{\\hat {p}}_{y}\\\\0&-{\\hat {E}}/c+mc&{\\hat {p}}_{x}+{\\hat {p}}_{y}&-{\\hat {p}}_{z}\\\\-{\\hat {p}}_{z}&-({\\hat {p}}_{x}-i{\\hat {p}}_{y})&{\\hat {E}}/c+mc&0\\\\-({\\hat {p}}_{x}+i{\\hat {p}}_{y})&{\\hat {p}}_{z}&0&{\\hat {E}}/c+mc\\\\\\end{pmatrix}}\\end{aligned}}", "1852de6ec5dfa06cad32a00df9ee261d": "\\mathbf {M} =(\\mathbf {M} _{2}-\\mathbf {M} _{1})(\\mathbf {M} _{2}-\\mathbf {M} _{1})^{\\text{T}}", "185320f6d98bb95ba3588dbe2393fe5f": "P_{1},\\ldots ,P_{4}", "18537c930bc137026b2ea3d0813a7b48": "(\\lambda _{\\mathbf {k}}-\\epsilon )C_{\\mathbf {k}}=0", "18538269a292aa77c1d029bc6d0105ea": "|\\varphi \\rangle ", "18538c9da0cd3169a30025c503983ea7": "W_{+}=\\sum _{i:h_{t}(x_{i})=y_{i}}D_{t}(i)", "1853aec857e01032063e88e43ebaec3f": "F_{n}=\\oplus _{i=0}^{n}A_{i}", "1853fb3bbf5f335001f003dbedccfdcd": "J^{\\mu }\\,", "1854264739edc59e21798ba131285dd6": "F_{\\mathbf {X}}", "185435a919237441cf36f1f63c896e2a": "{\\sqrt {{\\sqrt {0.1_{2}}}\\cdot {\\sqrt {10_{2}}}}}={\\sqrt[{4}]{1}}=1", "185449a1785117afe26a58f039212d04": "\\mathbf {\\Pi } =\\varepsilon _{0}{\\frac {\\partial }{\\partial t}}\\mathbf {A} \\ ,", "18545200d5d476d2bc8427bb8da1e0bf": "\\epsilon \\left(t,z\\right)", "18546e49a068f46d81972bd13a9a67de": "(1-q){\\mathcal {L}}[U(x;q)-u_{0}(x;q)]=c_{0}\\,q\\,{\\mathcal {N}}[U(x;q)],", "185475a4564babb661facae2a50a333a": "\\omega =\\pm \\pi /T.\\ ", "1855079f4b10ec37c47495cc078be98f": "b,s,\\beta >0\\,\\!", "1855385b46bd6c0446dfae06dbcc3684": "\\chi _{1}(\\omega )", "185555f8187ea28e0eee70a61b00f772": "r\\approx \\pi {\\sqrt {N}}/4", "185591a9ae38cd83b5044462a10ddf15": "\\lambda _{k}(D_{1})\\leq \\lambda _{k}(D_{2}).", "1855972f21eb11bd3b16e231bc9638f9": "W=Fr\\phi =\\tau \\phi ,", "1856372dfcb39dd3413daaa4c918f846": "z^{*}={\\frac {r^{2}}{\\overline {z-z_{0}}}}+z_{0}", "185679f33acda4cb07f311b5e5c6c129": "\\mathbf {u} =0", "185680cadfcb6e4f750942dae9a7693c": "\\rho =0.065*\\mathrm {D} ^{2}*\\pi /4", "1856add222a611b85b3d0b0080e48e71": "(3)\\qquad A_{a}=\\Phi (\\rho ,z,\\phi )[dt]_{a}\\;,", "1856e10b6af995b7af7b8f7513b7a8e2": "{\\frac {1}{r^{2}}}P_{1}^{1}(\\sin \\theta )\\sin \\varphi ={\\frac {1}{r^{2}}}\\cos \\theta \\sin \\varphi ", "18570c3481ad123007be7bcd04adab83": "{\\dot {y}}=\\mu (1-x^{2})y-x.", "1857464d3aea614c6f299ce9bf719c11": "P_{3}(y)=0", "18574b71aa1fffdc3695c71726b90157": "u\\in L", "18574eb24482def01156684ff5cafec4": "b\\wedge \\left(\\bigvee a_{i}\\right)=\\bigvee \\left(a_{i}\\wedge b\\right)", "1857aa4a5dddf86a1adbf7a7873986b7": "f(s)=\\sum _{n}a_{n}e^{-s|\\omega _{n}|}", "18583cd15b2f403d87db08950a7da0ed": "w(L)", "1858ffa9e5e0e0a36776d25229a263c2": "\\mathbf {\\Lambda } =\\operatorname {diag} (\\lambda _{1},\\lambda _{2},\\dots )", "1858fff801b6db73d147b01c34fd20e5": "W(r)~", "185945601a09320d7c3e192bec764901": "f+g", "18595730605f0e917b8b166fda878841": "*\\to G", "18595d5dac54286e2332c93b5ac49817": "\\prod _{i=1}^{p-1}i^{p-1}\\equiv (-1)^{p-1}\\equiv +1{\\pmod {p}}", "1859a071aa5560a33a5fff6ef385231c": "R_{Hp}={\\frac {1}{pq}}={\\frac {V_{Hp}t}{IB}}", "1859a1f931628a74f6fd532a2e0596ab": "{\\bar {f}}:X\\times _{Z}Y\\rightarrow Y", "1859a405da6a0198f3bd160c94b74887": "III_{1}", "1859b03afd0fd4ac368db995f8ff41a0": "\\scriptstyle ab", "1859cfba978b10c953c0969e4fff638b": "\\sum _{t}\\left\\{[(at+b)-y_{t}]^{2}\\right\\}", "1859ffa58ca953bc04a9e7b04bfd470e": "J_{G}(\\mathbf {x} ^{(0)})", "185a344b8f2b9207948ecfd93a14ccc3": "L(u)=E(u)-\\omega Q(u)+\\Gamma (Q(u)-Q(\\phi _{\\omega }))^{2}\\,", "185a6de473e8ea6a0c322c509739a449": "v_{i}=(0,\\ldots ,0,1,\\ldots ,0,w_{i_{1}},\\ldots ,w{i_{d}})", "185a71b1e8b5ea0f127e8f499994851e": "F(x)={\\begin{cases}p^{2}&{\\mbox{if }}x\\equiv p{\\pmod {p^{3}}}\\\\p^{4}&{\\mbox{if }}x\\equiv p^{2}{\\pmod {p^{5}}}\\\\p^{6}&{\\mbox{if }}x\\equiv p^{3}{\\pmod {p^{7}}}\\\\\\vdots &\\vdots \\\\0&{\\mbox{otherwise}}.\\end{cases}}", "185a7a7dbfb9f18344c8a84cf7e87737": "\\sum _{i=1}^{n-1}{\\frac {\\partial }{\\partial x_{i}}}{\\frac {\\frac {\\partial f}{\\partial x_{i}}}{\\sqrt {1+\\sum _{j=1}^{n-1}({\\frac {\\partial f}{\\partial x_{j}}})^{2}}}}=0", "185aea59b36f8863e71df2730ca694aa": "a=b=1/2", "185b340d0b0f300147c5444b745d5411": "\\operatorname {Tr} ({\\bar {Q}}\\rho )~=~(\\epsilon -\\mu )+(1-\\epsilon +\\mu )\\operatorname {Tr} (Q\\rho )~=~\\epsilon ~.", "185b41b4aa6d23dd8d92382fdb269cdc": "v=\\left|{\\boldsymbol {v}}\\right|=\\left|{\\dot {\\boldsymbol {r}}}\\right|=\\left|{\\frac {d{\\boldsymbol {r}}}{dt}}\\right|\\,.", "185ba22e9e196d8ad29861510c83f703": "I_{A}r(y)", "18625bfe5e21136c4de45bbffe7d96f9": "{\\eta \\over {\\mathit {\\Delta }}}=\\cos \\delta \\sin \\alpha ", "186278def33580aff74015039c35a86a": "{\\Bigg (}{\\frac {17}{p}}{\\Bigg )}_{4}{\\Bigg (}{\\frac {p}{17}}{\\Bigg )}_{4}={\\begin{cases}+1{\\mbox{ if and only if }}\\;\\;p=x^{2}+17y^{2}\\\\-1{\\mbox{ if and only if }}2p=x^{2}+17y^{2}\\end{cases}}", "186301a10c315169618815cfa0f9676d": "F(b)-F(a)=\\int _{a}^{b}f(t)\\,dt.", "1863363aafb2bdcffae32ab04b678990": "\\{\\varnothing ,\\{1\\},\\{0,1\\}\\}.", "18636d2afe15922099975c8eeb99e766": "T_{0}\\,\\!", "186386077e6b916e2b10cf6846e5d46a": "e^{ix}=\\cos x+i\\,\\sin x.", "186389b2a6f4a298a3fcf683b501e6a6": "\\chi (t,\\zeta )={\\frac {|t-\\zeta |^{2}}{4(1-|t|^{2})(1-|\\zeta |^{2})}}", "18639bfc3b6c41225bd852a12786f3a8": "r\\ =\\ (1+i/n)^{n}-1", "1863b0de8f3dfbeeef0096c03e5d7e13": "L_{\\gamma }(X,Y)", "1863eb80e0e8f209e80feab2b46a4a92": "\\rho _{k}=k{\\frac {U_{MN}}{I_{AB}}}", "18641f3cca77893e1c73938c9d01b1e5": "\\tau \\,\\!", "1864472b3e5d7c64a6f1de15a3824c4c": "\\operatorname {Li} _{2}(z)+\\operatorname {Li} _{2}(1-z)={\\frac {{\\pi }^{2}}{6}}-\\ln z\\cdot \\ln(1-z)", "1864ef34a0f34b19a7bb1567e2ffc841": "\\sum \\limits _{i,j=1}^{2n}a_{ij}\\xi _{i}\\xi _{j}", "1864f7e6674885847484937d6b4e7a52": "V=\\bigcup {}_{q\\in Q}V_{q}", "18657a80050f9b39dc140f0c7010defb": "p/m", "1865945d65b5d44c44bc46f94cef8dce": "{\\begin{aligned}C&=S_{0}N(d_{1})-Xe^{-r(T)}N(d_{2})\\\\\\\\d_{1}&={\\frac {\\left[\\ln \\left({\\frac {S_{0}}{X}}\\right)+\\left(r+{\\frac {\\sigma ^{2}}{2}}\\right)(T)\\right]}{\\sigma {\\sqrt {T}}}}\\\\\\\\d_{2}&=d_{1}-\\sigma {\\sqrt {T}}\\end{aligned}}", "1865ed583b301d5bb65c70756b1fca7b": "XAX=X", "1865fe9e1b89182e53afbfe3edf37e62": "{\\begin{array}{cl}&P\\left(Searched|Known\\wedge \\delta \\wedge \\pi \\right)\\\\=&{\\frac {1}{Z}}\\sum _{Free}\\left[\\prod _{k=1}^{K}\\left[P\\left(L_{i}|K_{i}\\wedge \\pi \\right)\\right]\\right]\\end{array}}", "18662a1d455ea4bd72218c341f352cb9": "y={\\sqrt {x(x-1)(x-\\lambda )}}", "186639c8a2a8b44425af82dc6b84c9fc": "v={\\begin{bmatrix}0&1\\\\-1&0\\end{bmatrix}},\\qquad p={\\begin{bmatrix}0&1\\\\-1&1\\end{bmatrix}}.", "18666261315c816a2946cff7d6fe3b26": "\\theta ={[L]^{n} \\over K_{d}+[L]^{n}}={[L]^{n} \\over (K_{A})^{n}+[L]^{n}}={1 \\over ({K_{A} \\over [L]})^{n}+1}", "18666c11ee0dfbf891d77c2337a0c5b9": "|b|={\\ln {\\varphi } \\over 90}=0.0053468\\,", "18668ec6adff0ef97462783657852525": "\\omega =ck.\\,", "1866d848800d1b62d365810bde02c45d": "P_{\\mathrm {L} }", "1866d9c98df8758a6c0a1964aa6fe14b": "\\omega =e^{\\frac {2\\pi i}{N}}", "186712eacb73412fd4a48d9729e619c4": "S_{M}\\;", "1867344aac4679ac6402b8b6d56ec051": "x_{0}=(q(q^{-1}~{mod}~p)r_{p}+p(p^{-1}~{mod}~q)r_{q})~{mod}~N", "1867ef321d4d3366fd96271ff5ae9d10": "H_{G}^{*}(X)=H^{*}(EG\\times _{G}X)", "18681430cfcf9b8e7499013e820e18c8": "A\\vdash B\\to A", "1868212d4409c2055f2ee0de17f15273": "y=x\\tan(\\theta )-g(x/v_{h})^{2}/2", "1868aa091794cb79f9771468cd8699d7": "\\nu _{p}\\left({\\frac {a}{b}}\\right)=\\nu _{p}(a)-\\nu _{p}(b).", "1868c96ee5f6376b4c68420d1f66138f": "WTS(O_{j})=TS(T_{i})", "1868d7b7c25e12ceaa9a407985c5c4b4": "{\\begin{aligned}g_{\\mathrm {Alice} }&={\\begin{pmatrix}1&6&2\\\\6&3&8\\\\2&8&2\\end{pmatrix}}{\\begin{pmatrix}3\\\\10\\\\11\\end{pmatrix}}={\\begin{pmatrix}85\\\\136\\\\108\\end{pmatrix}}\\ \\mathrm {mod} \\ 17={\\begin{pmatrix}0\\\\0\\\\6\\end{pmatrix}}\\ \\\\g_{\\mathrm {Bob} }&={\\begin{pmatrix}1&6&2\\\\6&3&8\\\\2&8&2\\end{pmatrix}}{\\begin{pmatrix}1\\\\3\\\\15\\end{pmatrix}}={\\begin{pmatrix}49\\\\135\\\\56\\end{pmatrix}}\\ \\mathrm {mod} \\ 17={\\begin{pmatrix}15\\\\16\\\\5\\end{pmatrix}}\\ \\end{aligned}}", "18691c10200e9ea26a70494882efe103": "{\\frac {{\\bar {C}}Gz^{-k}}{1+{\\bar {C}}Gz^{-k}}}=z^{-k}{\\frac {CG}{1+CG}}", "18691caddabf3374addff8727ba4e557": "{\\tilde {\\mu }}\\in \\mathbb {R} ", "1869b9ea99aacf415166baed7116e097": "{\\mbox{sn}}(u;k)=-{\\vartheta \\vartheta _{11}(z;\\tau ) \\over \\vartheta _{10}\\vartheta _{01}(z;\\tau )}", "1869d28c749042cfa1c8296c8a4ad012": "f=(f_{0},f_{1},...,f_{d-1})", "186a619c3a85cece584b7762e3d763a6": "{\\frac {e^{i\\theta }-e^{-i\\theta }}{2i}}=x", "186a69818830137b3bfb5b730e1d5096": "{\\frac {dV}{dt}}=\\mu NP+\\rho S-\\mu V", "186a9ed8101469fef65efd76d866a250": "\\mathbf {i} =[i_{1},\\ldots ,i_{M}]^{T}", "186ad2176f601254ef09c9f380d51485": "D={\\text{close}}_{\\text{previous}}-{\\text{close}}_{\\text{now}}", "186bca604231a6fce5b31dca568a19d9": "|\\alpha _{1}\\rangle ", "186c8e87a13bdf57a2481d8f9b0c2aac": "{\\tfrac {4000}{100}}+{\\tfrac {4000}{100}}=80", "186ca497852157dc0dbd3807900348e0": "\\phi _{e}\\in F", "186cb8ae024dfc1ecaec185badf296fc": "f''\\;", "186cc8c31c3bc1746689ef8db4909fba": "\\nabla \\times \\mathbf {g} =-{\\frac {\\partial \\mathbf {H} }{\\partial t}}\\,\\!", "186cfb1f9726d17262aadd50fbbf725a": "{\\begin{aligned}x&=\\rho \\cos \\theta \\sin \\phi \\\\y&=\\rho \\sin \\theta \\sin \\phi \\\\z&=\\rho \\cos \\phi \\end{aligned}}", "186d22e3d612accd890bde850f52fb5d": "U_{f,P}=\\sum _{i=1}^{n}(x_{i}-x_{i-1})M_{i}.\\,\\!", "186d80b5088cae799f56877e01bc556f": "\\arccos x=2\\arctan {\\frac {\\sqrt {1-x^{2}}}{1+x}},{\\text{ if }}-1|V|", "1873d9b7b11aa509eff2fc484ab67eaa": "\\parallel \\cdot \\parallel ", "1873dab299ef284bdbbe09125544493a": "(1-f)p_{01}/(p_{01}+p_{00})", "1873f65817f530e6e476612315feeab4": "F_{r}", "18740f4295b5f655aeae9efb25bfc461": "\\kappa ={\\sqrt {{\\frac {4\\pi e^{2}}{\\epsilon }}{\\frac {\\partial n}{\\partial \\mu }}}}={\\sqrt {\\frac {6\\pi e^{2}n}{\\epsilon E_{f}}}}", "18743e056ddb7395827903e98e8f6c99": "{\\begin{matrix}a_{11}x_{1}+a_{12}x_{2}&\\leq b_{1}\\\\a_{21}x_{1}+a_{22}x_{2}&\\leq b_{2}\\\\a_{31}x_{1}+a_{32}x_{2}&\\leq b_{3}\\\\\\end{matrix}}", "18744044d44ab57577e37ea2b9960a89": "\\log _{10}(3542)=\\log _{10}(10\\cdot 354.2)=1+\\log _{10}(354.2)\\approx 1+\\log _{10}(354).\\,", "1874569290841c7e1de30ce6dca99899": "f'(t')={\\frac {\\mathbf {r} -\\mathbf {r} _{s}(t_{r})}{|\\mathbf {r} -\\mathbf {r} _{s}(t_{r})|}}\\cdot (-\\mathbf {v} _{s}(t'))+c\\geq c-\\left|{\\frac {\\mathbf {r} -\\mathbf {r} _{s}(t_{r})}{|\\mathbf {r} -\\mathbf {r} _{s}(t_{r})|}}\\right|\\,|\\mathbf {v} _{s}(t')|=c-|\\mathbf {v} _{s}(t')|\\geq c-v_{M}>0", "187493889ad89f38f705454689353c02": "T/3", "1874d0265729cd43d493f155abd96cb6": "q={\\frac {pn}{n-\\alpha p}}", "1874d623d539c98b814300c167b581e9": "\\Delta ^{n-1}\\twoheadrightarrow P,", "18755c468fb372c4cddba132750e2004": "U,V\\subset \\mathbb {R} ^{n}", "18757e58a5884d3c1b9a74a7e3992a6f": "C^{1}\\ ", "1875901a60e57216ed47ab4356063ff8": "x\\geq 0", "1875aa150ba914a9608b24632930f58e": "\\delta =\\det \\left({\\begin{bmatrix}A_{1}&B_{1}\\\\B_{1}&A_{2}\\end{bmatrix}}\\right)", "1875ca061052a33c650e9571b6ab4154": "{\\frac {dx}{dy}}={\\frac {1}{dy/dx}}.", "1875fa35976f0599b3118837e7d8d735": "\\sigma =(i_{1}i_{2}\\dots i_{r+1})(j_{1}j_{2}\\dots j_{s+1})\\dots (l_{1}l_{2}\\dots l_{u+1})", "1876368e2bac99e6451d747d17c992a3": "{\\begin{bmatrix}0&2\\\\4&8\\\\\\end{bmatrix}}", "1876a4f175bf5d4f9d04539dc5dc638a": "\\mathbf {a} _{\\text{per}}", "1876b62c5de864e9ce8c9eaaa3a74c77": "\\int \\exp \\left(-{\\frac {1}{2}}\\varphi {\\hat {A}}\\varphi +J\\varphi \\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)", "1876c1097bbb344f2a72eebc3f3b76d0": "Z(t)=X(t)+RY(t)", "1876dc672a0088bf5c1af26e279636c8": "{\\mathbf {x}}^{(m+1)}=({\\mathbf {D}}-{\\mathbf {L}})^{-1}{\\mathbf {U}}{\\mathbf {x}}^{(m)}+({\\mathbf {D}}-{\\mathbf {L}})^{-1}{\\mathbf {k}}.\\quad (8)", "18770c7b15d98ad5e1ff0dbd7d01f259": "\\mathbf {F} =0\\,\\mathbf {e} _{x}-F\\,\\mathbf {e} _{y}+0\\,\\mathbf {e} _{z}\\quad {\\text{and}}\\quad \\mathbf {r} =x\\,\\mathbf {e} _{x}+0\\,\\mathbf {e} _{y}+0\\,\\mathbf {e} _{z}\\,.", "18771d41ccabfc4946af8e1af6c804d9": "H^{n}(G,M)=H^{n}({\\rm {Hom}}_{G}(F,M))", "187726061a7b5d0256f0fb0a1b0782fd": "E_{n}^{(1)}=-{\\frac {1}{2mc^{2}}}\\left(E_{n}^{2}+2E_{n}{\\frac {e^{2}}{a_{0}n^{2}}}+{\\frac {e^{4}}{(l+1/2)n^{3}a_{0}^{2}}}\\right)=-{\\frac {E_{n}^{2}}{2mc^{2}}}\\left({\\frac {4n}{l+1/2}}-3\\right)", "187748e61a55b804310bd7735c5dfcb7": "X_{1}|X_{2}=x_{2}\\ \\sim \\ {\\mathcal {N}}\\left(\\mu _{1}+{\\frac {\\sigma _{1}}{\\sigma _{2}}}\\rho (x_{2}-\\mu _{2}),\\,(1-\\rho ^{2})\\sigma _{1}^{2}\\right).", "18779b51bce5fbab4541556836f02c78": "G=\\left(V,E\\right)", "1877e79bf17c9bb94711116d27d7e21a": "f(A)=\\sum _{i,j}[A]_{ij}f(e_{ij})=\\sum _{i}[A]_{ii}f(e_{11})=f(e_{11})\\operatorname {tr} (A)", "1877f35323191ae2c325de0f825b5104": "I_{i}(x|k,t)=\\int _{L}^{x}M_{i}(u|k,t)du,", "1877f900717497b9eb9dc884068091e3": "Z_{in}^{\\infty }", "1878228ff796bfddfd0fd54947e0d034": "\\mathrm {E} _{\\mathbf {R} }[{\\hat {\\mathbf {R} }}]\\ {\\stackrel {\\mathrm {def} }{=}}\\ \\exp _{\\mathbf {R} }\\mathrm {E} \\left[\\exp _{\\mathbf {R} }^{-1}{\\hat {\\mathbf {R} }}\\right]", "18783a8efbfafd336e5b5e56fc17778a": "R^{-1}", "18784a361e5f96ca27b114b6f796861d": "V={\\frac {1}{6}}\\left(10+8{\\sqrt {5}}+15{\\sqrt {5+2{\\sqrt {5}}}}\\right)a^{3}\\approx 12.3423...a^{3}", "187898bb1a2a862a0999cf86f8f0e5ae": "d\\,", "1878e085b76a1a5654fdf769b7de3ab4": "D^{k+1}F(u)\\{v_{1},v_{2},\\dots ,v_{k+1}\\}=\\lim _{\\tau \\rightarrow 0}{\\frac {D^{k}F(u+\\tau v_{k+1})\\{v_{1},\\dots ,v_{k}\\}-D^{k}F(u)\\{v_{1},\\dots ,v_{k}\\}}{\\tau }}.", "1878f8d8eb6a9f9725df66c327ec4dab": "Ma=Mg-Kv\\,", "18791407f9ee7cc46e94fb7b08edb1fc": "\\partial _{1}(e)=d_{0}(e)-d_{1}(e)=v-v=0.", "1879468c7491a981ddb054d01fa46bb0": "Q_{\\alpha }:=\\kappa ~\\int _{-h}^{h}\\sigma _{\\alpha 3}~dx_{3}", "1879730781338b7cedca8eadb50cc4a4": "p^{\\prime }\\equiv p-{\\frac {\\Lambda }{8\\pi G}}", "1879a2c3bada456b9205d5ffafb423a5": "y_{1}=x_{0}-x_{1}.\\,", "1879e8731a1184ddc30914ba547720a1": "Bonus=(R_{n}-R_{o}-(18+2\\times G));\\ \\ \\ min(Bonus)=0", "187a0d50b45ca04ce2afe3133405d4c0": "g=3", "187a64a9c9654c056a7c99d75714852f": "\\alpha =(\\alpha _{1},...,\\alpha _{n})\\in \\mathbb {R} ^{n}", "187ae22789155aea6a3467b9669ed4d4": "d_{\\lambda }={\\frac {9!}{7\\cdot 5\\cdot 4\\cdot 3\\cdot 2\\cdot 2\\cdot 1\\cdot 1\\cdot 1}}=216.", "187afe26af8552f6f386be9c87f1ccdf": "F(A)=\\{b\\in M\\mid b=f_{\\varphi }(a_{1},\\dots ,a_{n});\\,\\varphi \\in \\sigma ;\\,a_{1},\\dots ,a_{n}\\in A\\}", "187b5a35ea419ef4f511f3bcf4fc6a5d": "U_{i}\\cap U_{j}\\neq \\emptyset ", "187b8c1e8bb4444f20512f326e1f5c9c": "\\delta _{H}(d_{F})=\\delta _{H1}{\\frac {1-\\exp \\left(-d_{F}/\\lambda _{F}\\right)}{1+d_{F}/d_{0}}}.", "187bb5014952afaa4369c9979477e9d1": "b_{4}", "187bb5f36111bc779b615517fdbeda55": "(n-1)(n-2)~r^{-n}~\\cos(n\\theta )", "187bfe98bdbd2ee8e0b484481dba04a2": "P=(N_{\\uparrow }-N_{\\downarrow })/N", "187cd6236f5eedff9399c4c606ea81af": "I[u]=\\int _{D}(\\nabla u)^{T}A(\\nabla u)\\,\\mathrm {d} x", "187cd9936495a2e5cf67dbb7dba8538f": "V=x^{2}y+y^{4}+ax^{2}+by^{2}+cx+dy\\,", "187d62995335efabffe91dac7937321c": "X_{i_{1}},\\dots ,X_{i_{n}}{\\text{ and }}X_{j_{1}},\\dots ,X_{j_{n}}\\!", "187e1a4c3d63ccac36ce283d420330a4": "h_{combined}", "187e526b73ab20fc09c2965b87e443ba": "\\operatorname {Cov} (X_{i}^{2},X_{j}^{2})=2\\langle X_{i}X_{j}\\rangle ^{2}=2M_{ij}^{2}", "187ecb99c8f8d632222c8040ba7c84ad": "J{\\frac {\\partial }{\\partial x^{\\mu }}}={\\frac {\\partial }{\\partial y^{\\mu }}}\\qquad J{\\frac {\\partial }{\\partial y^{\\mu }}}=-{\\frac {\\partial }{\\partial x^{\\mu }}}", "187ef4436122d1cc2f40dc2b92f0eba0": "ab", "187f23dece76c2c8cbb550ab38bfcfc3": "\\Lambda (E)", "187f3958cab7e623394c2973e801bfe6": "P(t)={\\mathcal {P}}_{ab}(c_{a}^{*}c_{b})exp(i\\nu _{1}t)+{\\mathcal {P}}_{ac}(c_{a}^{*}c_{c})exp(i\\nu _{2}t)+c.c.", "187fd4fa8be2256a660a7b18fd4b73a5": "p_{0{\\frac {1}{2}}}\\leftarrow 64x^{3}-448x+448", "187fde02d22edfe777d97a947b269419": "\\pi _{1}=1-\\pi _{0}", "188079d973ad0be8a1013e48e672258c": "\\sum _{i=1}^{k}\\mathrm {n_{i}} ^{\\mathrm {S} }\\,\\mathrm {d} \\mu _{i}\\,+A\\mathrm {d} \\gamma \\,=0\\,.", "18808159c9f664bc9a4d04faa92a8638": "{\\hat {S}}^{2}", "1881135d49064106948ed208a99690ff": "(\\mathbf {a} \\times \\mathbf {b} )\\cdot (\\mathbf {c} \\times \\mathbf {d} ),\\quad (\\mathbf {a} \\times \\mathbf {b} )\\times (\\mathbf {c} \\times \\mathbf {d} ),\\ldots ", "188156c823ead08aa06494871fec8a63": "\\omega _{S}={\\sqrt {\\frac {k_{z}^{2}B^{2}}{\\mu \\rho _{e}}}}", "1881da51ebf6ff6391ae373f1193144c": "\\#_{n,m}={n+m-1 \\choose m-1}={n+m-1 \\choose n}\\,.", "1881e88f33be2eb98dd778054f5c1983": "\\langle x,y\\rangle ", "188213b2b0bd3065d7a200c46d874c09": "\\lim _{t\\rightarrow 1}f(\\gamma (t))=f_{1}(e^{i\\theta })", "188214f7ce4cff74d5fdcf33c75fca8c": "\\Delta f={\\frac {\\partial ^{2}f}{\\partial r^{2}}}+{\\frac {N-1}{r}}{\\frac {\\partial f}{\\partial r}}+{\\frac {1}{r^{2}}}\\Delta _{S^{N-1}}f", "188231a91ed69207a52b0e59b91d3cf9": "x=1\\ ", "188250b55a4085437d24ba37a2bddfd3": "\\displaystyle a=1\\times 5=5", "18825fe9b7c871249babd1c757873dd2": "(a_{ij})", "1882745480f80c63870fe2d96e330a5a": "{\\dot {G}}(t-\\tau )", "188281dd07a8792df31de293dc5a0bda": "\\ \\displaystyle R\\ ", "188286ebc8b3b9c46590c712327d89b9": "c_{\\alpha }=\\pi ^{n/2}2^{\\alpha }{\\frac {\\Gamma (\\alpha /2)}{\\Gamma ((n-\\alpha )/2)}}.", "1882c64a6cf94a30158ffa7334e95c7c": "10^{\\frac {13-1}{2}}=10^{6}\\equiv 1{\\pmod {13}}", "1882c704faa3f93353e9006938a2894e": "\\langle F,R,{\\Vdash }\\rangle ", "1882e4ea827fcae43798331cd797ece6": "4\\pi \\epsilon _{0}", "188341f39fc157faaad493bf5a425cce": "\\textstyle \\int _{0.5}^{1}\\mathrm {\\mathrm {B} } (49582,48871)\\approx .983.", "1883d50bb4fbe1033018f925d816a192": "\\hbar ={\\frac {h}{2\\pi }}", "1883fa6484e08c82ad8c920ad0008bd7": "S\\subseteq A^{\\mathbb {N} }", "1884004e4b00addecc0049d153d2e87d": "E(f(x)\\cdot g(x))=Ef(x)+Eg(x)", "18843a2371bae8637a5b867636c13986": "x,y\\in A_{1}\\cup A_{2}", "188456e85e5258396aa9ebdb7b8178a0": "\\langle f,g\\rangle :=\\int _{\\Omega }(Df(x),Dg(x))\\,dx", "18849c52417407de4d69623d07843c20": "n_{i,t}", "18849d4b2ec3504c6aae3f12ce88893b": "Df=-(pf^{\\prime })^{\\prime }+qf", "1884a44435bf6e23a9face1a0bc52c2c": "\\Theta =\\tan ^{-1}(m/k)", "1884fd5772c3ca09da0a0e70f4f7d693": "{\\sqrt {\\frac {1}{6}}}\\!\\,", "18856861ae0142eef654eb99e18259f3": "\\int _{C_{1}}L(x,y)\\,dx=\\int _{a}^{b}L(x,g_{1}(x))\\,dx.", "188594fb1141ab6ffe1e33ddbbe20c8b": "\\mathrm {VF_{5}+H_{2}O\\ \\xrightarrow {} \\ VOF_{3}+2HF} ", "1885dfefcfe67d59db0ce26605e0304e": "\\left({\\begin{matrix}{\\mathbf {F}}\\\\{\\boldsymbol {\\tau }}\\end{matrix}}\\right)=\\left({\\begin{matrix}m{\\boldsymbol {1}}&-m[{\\mathbf {c}}]\\\\m[{\\mathbf {c}}]&{\\mathbf {I}}_{\\rm {cm}}-m[{\\mathbf {c}}][{\\mathbf {c}}]\\end{matrix}}\\right)\\left({\\begin{matrix}{\\mathbf {a}}_{\\rm {cm}}\\\\{\\boldsymbol {\\alpha }}\\end{matrix}}\\right)+\\left({\\begin{matrix}{m{\\boldsymbol {\\omega }}}\\times \\left({\\boldsymbol {\\omega }}\\times {\\mathbf {c}}\\right)\\\\{\\boldsymbol {\\omega }}\\times ({\\mathbf {I}}_{\\rm {cm}}-m[{\\mathbf {c}}][{\\mathbf {c}}])\\,{\\boldsymbol {\\omega }}\\end{matrix}}\\right),", "1886817aba72d836d299b56d835b6c9f": "\\mathbf {j} _{r}=\\int _{t_{0}}^{t_{1}}\\mathbf {f} _{r}dt", "18868d1ecfe0da570f1be63b1d908784": "v_{s}", "1886acccb9fc762085008aedda7ca779": "Vt=Vg+Vi+Vo", "1886e625b4b5241d722ba825c148a4c8": "s={\\frac {a+b+c+d}{2}}.", "1886eda224719f904cfcc2077619b2d7": "C:y^{2}+h(x)y=f(x)", "1887c9b3a5386c6b83d6711c107dda8e": "P_{D}", "1887d3f69a6d9aaba8eda1127f4cbbfe": "{\\frac {vX}{c+vt}}\\,\\!", "1887ec7b1748201ee8deb2468e0af960": "{}+1206647803780373360x^{6}-3599979517947607200x^{5}\\,\\!", "188849597db9198d230f06e96b725773": "2.2\\times 10^{-6}\\ \\mathrm {seconds} \\,", "188858d46d2c549f8a2da42ac3d316f8": "I(q)\\sim S'q^{-(6-d)}", "1888a16042020813c1d29b0954ca215c": "{\\boldsymbol {v}}\\,", "1888ba03d9fb43a16946a10cb446f645": "\\sigma ^{-1}", "1888d1745398648fa07cbccb8926b1cd": "\\langle \\cdot ,\\cdot \\rangle ", "188904e7e059c0e5a05da77223ff8281": "S\\circ T:=\\{s\\circ t:s\\in S,t\\in T\\}.", "18893eb772c377a31b1746974df39682": "n\\sin \\theta ", "1889a9a8e04e4cf883e27a2f2c6255eb": "\\mathbf {v} _{1}+{\\boldsymbol {\\omega }}_{1}\\times \\mathbf {r} _{1}=\\mathbf {v} _{2}+{\\boldsymbol {\\omega }}_{2}\\times \\mathbf {r} _{2}", "1889f83b287b0314cc26bae74eea41ec": "s(x_{1},x_{2},x_{3},x_{4},\\ldots )=(x_{2},x_{3},x_{4},\\ldots ){\\text{.}}", "188a03f8de5252635c30495fa4a88c33": "L=\\int _{a}^{b}{\\sqrt {\\left|\\sum _{i,j=1}^{n}g_{ij}(\\gamma (t))\\left({d \\over dt}x^{i}\\circ \\gamma (t)\\right)\\left({d \\over dt}x^{j}\\circ \\gamma (t)\\right)\\right|}}\\,dt\\ .", "188a1d39c07fa79aa6c51d2152c031cc": "{\\hat {f}}(r\\alpha )=\\int _{-\\infty }^{\\infty }Rf(\\alpha ,s)e^{-2\\pi isr}\\,ds.", "188a3d326e5e2280b7d046a8554fc9dc": "\\varinjlim A_{i}=\\bigsqcup _{i}A_{i}{\\bigg /}\\sim .", "188a58b793667fd45e61a6bb13a42184": "(v_{x},v_{y},v_{z})", "188aa4ab7ba11a91a978f53a101542ba": "F_{t}-S_{t}", "188ac3fae1e32f569fc6f1b089cd00fa": "\\mathrm {S_{2}F_{2}+3H_{2}SO_{4}\\ \\xrightarrow {80^{o}C} \\ 5SO_{2}+2HF+2H_{2}O} ", "188b05260f017cedba309d25f64fd5cb": "\\mathbf {F} =\\nabla \\phi .\\,\\!", "188b3ffb15876c4642210083458c5ece": "(\\pm 1,\\pm 1,\\pm 1)", "188b625c6f167af25d9a215f6d6ccebb": "{\\mathit {He}}_{n}(x)=\\int _{-\\infty }^{\\infty }(x+iy)^{n}\\,\\mathrm {e} ^{-y^{2}/2}\\,\\mathrm {d} y/{\\sqrt {2\\pi }}\\ .", "188b6a454f3adfa535548223ec0929c5": "x<10", "188b6bf67bc2a40ec15a30200802c54d": "a\\delta \\phi ", "188b710daba82055cf96cb1d4a59630f": "Vol(M,\\lambda g)=\\lambda ^{n/2}Vol(M,g)", "188bcd21b27436ee4a913ac583bd5f41": "A+{\\overline {B_{\\delta }}}:=\\left\\{a+b\\in \\mathbb {R} ^{n}\\left|a\\in A,b\\in {\\overline {B_{\\delta }}}\\right.\\right\\}", "188bf411bbe9a71c900da19bb53db73b": "z_{12}\\,", "188c151e25bc3cea4641cc722994fbc4": "U\\left(x,y\\right)=x^{\\alpha }y^{1-\\alpha }", "188c226ff875898861c69c960928378d": "\\int \\operatorname {arcosh} (a\\,x)\\,dx=x\\,\\operatorname {arcosh} (a\\,x)-{\\frac {{\\sqrt {a\\,x+1}}\\,{\\sqrt {a\\,x-1}}}{a}}+C", "188c886236774d0659c204b94056a07a": "0,\\infty ", "188d4ada5f4a8b9309ac3c0a9ce29200": "\\mathbf {J} ", "188dcd29088d10048de9d208b99de1db": "{\\vec {0}}={\\vec {\\mathrm {M} }}_{sail/G}+{\\vec {\\mathrm {M} }}_{hull/G}", "188de81d3fe7186b3b5d3c48c79dd941": "{\\frac {d{\\mathbf {r} }(s)}{ds}}={\\mathbf {u} }_{1}({\\mathbf {r} }(s))", "188dfaa1f8283faaa0e5dcd3cee46fd6": "B^{\\ast }\\,", "188e0a27226ea35d2c78cbf5c952ad1d": "{\\dot {K}}", "188e65fae6efd0d7733ef6bcb00465c0": "f={\\mathbf {1}}*g", "188e8738bcb05c9402ff373fa7c05f15": "\\beta _{min}=\\kappa _{3}+\\kappa _{4}d_{2}", "188ec85ebad245459752275259b526a7": "dr_{t}=\\theta (\\mu -r_{t})\\,dt+\\sigma \\,{\\sqrt {r}}_{t}dW_{t}\\,", "188ee2a34be84205542f460fa30736a9": "\\alpha :L(G)\\longrightarrow \\mathbb {F} ^{n}", "188f44cd54ff166a66ac4266fc4892e0": "\\sigma (X,X^{*})", "188f5ac1b1ac9c5aa7eec28793155847": "n=2", "188f625f814556e9f8ebbca7cfbaa0be": "y^{*}\\,\\!", "188fafc09550fa0eff705766a7c57d60": "B_{3}(T)", "188fde19b9ab3ea4bb7ed655bc54a8c0": "(J^{\\alpha })(J^{\\beta }f)(x)=(J^{\\beta })(J^{\\alpha }f)(x)=(J^{\\alpha +\\beta }f)(x)={1 \\over \\Gamma (\\alpha +\\beta )}\\int _{0}^{x}(x-t)^{\\alpha +\\beta -1}f(t)\\;dt", "188ffa3164c852925f68f3e80893580e": "P_{1}=(1,0)", "189054031045ff1382c8129b458d36b0": "\\Theta _{d}^{*}={h|h\\in S_{A},j_{h}=d},", "189062606b55eebd5846d99d9782ffb2": "k=0,1,2,\\ldots ,", "18908fff8062dd0029b99bd92ab0a09e": "K(k)={\\frac {2}{1+k1}}K({\\frac {1-k1}{1+k1}})", "18911f15de48873b70bb53c4b7805a11": "{\\frac {2\\lambda }{R}}", "1891ca4d0ef625a3c4c190c917282422": "H^{i}(G/B,\\,L_{\\lambda })", "1892330c3bf4359ba7d85f5477cd1dc5": "L[u]=f,\\,", "1892449a02312a313a23c49baa2a40cf": "\\langle ,\\rangle ':(y,x)\\to \\langle x,y\\rangle ", "1892b1f129a1c9594967558c3433a408": "\\operatorname {not1} \\ \\operatorname {true} =(\\lambda p.\\lambda a.\\lambda b.p\\ b\\ a)(\\lambda a.\\lambda b.a)=\\lambda a.\\lambda b.(\\lambda a.\\lambda b.a)\\ b\\ a=\\lambda a.\\lambda b.(\\lambda x.b)\\ a=\\lambda a.\\lambda b.b=\\operatorname {false} ", "189316d02d660f0f0478948cb7edb95d": "h=f(x_{n})\\ ", "189326fb7b8cd2c93dc280f79a91644a": "{\\tilde {h}}=H_{1}({\\tilde {w}},{\\tilde {s}})", "189331fab3bb02edd66a3b5d9c45217b": "-p\\;{\\frac {p+\\ln(1-p)}{(1-p)^{2}\\,\\ln ^{2}(1-p)}}\\!", "18937b1c3d76aad31e69c077d76dc204": "\\tau _{\\mathrm {n} }=-{\\frac {1}{2}}(\\sigma _{1}-\\sigma _{2})\\sin 2\\theta \\,\\!", "1893ff69e5339a235236ba1e6f7dac13": "S:={\\mathcal {M}}+{\\mathcal {M}}^{*}+\\mathbb {C} 1", "18946e4aeec4d299237daa07fa2387b0": "{\\begin{array}{lcl}a&=&{\\frac {d^{2}x}{dt^{2}}}\\\\&=&{\\frac {d}{dt}}{\\frac {dx}{dt}}\\\\&=&{\\frac {d}{dt}}({\\frac {dx}{dA}}\\cdot {\\frac {dA}{dt}})\\\\&=&{\\frac {d}{dt}}({\\frac {dx}{dA}})\\cdot {\\frac {dA}{dt}}+{\\frac {dx}{dA}}\\cdot {\\frac {d}{dt}}({\\frac {dA}{dt}})\\\\&=&{\\frac {d}{dA}}({\\frac {dx}{dA}})\\cdot ({\\frac {dA}{dt}})^{2}+{\\frac {dx}{dA}}\\cdot {\\frac {d^{2}A}{dt^{2}}}\\\\&=&{\\frac {d^{2}x}{dA^{2}}}\\cdot ({\\frac {dA}{dt}})^{2}+{\\frac {dx}{dA}}\\cdot {\\frac {d^{2}A}{dt^{2}}}\\\\&=&{\\frac {d^{2}x}{dA^{2}}}\\cdot \\omega ^{2}+{\\frac {dx}{dA}}\\cdot 0\\\\&=&x''\\cdot \\omega ^{2}\\\\\\end{array}}", "18956d8d721be71de2aed54a7c3bcc89": "\\exp(-\\beta \\varepsilon ({\\mbox{state}}))=\\prod _{\\mbox{vertices}}\\exp(-\\beta \\varepsilon _{ij}^{k\\ell })", "1895917fcf2f74fd389a9bc370d5bdf6": "I_{spike}=\\sum _{\\mathbf {s} }P(\\mathbf {s} |spike)log_{2}[P(\\mathbf {s} |spike)/P(\\mathbf {s} )]", "1895fe06dd85b03071367dd35f2fb74c": "\\coprod _{i=1}^{N}x_{i}", "189615efad256e6f603839ab9d05a6dc": "2U_{k}", "189640332c952b62d541cc2e00a34d3d": "M_{B}=-11.569\\ kN\\cdot m", "189695234146739ede1f2e9692ae62cb": "(1-\\lambda _{p})", "1896d795094586c954ea9b8a46891783": "NID(x,y)", "18973a90cbf22c78e57b5405c6c144e8": "\\langle T,\\Delta \\phi \\rangle =0", "18975b0a3a106031fc0b45a8659f56a4": "-20\\leq x,y\\leq 20", "18975f661ac54937ed3ebafd74ae1826": "\\exp(-E/k_{B}T)\\,", "18978cc2fc5ae6db3b5102353d6a6950": "\\sin \\theta =\\theta -{\\frac {\\theta ^{3}}{3!}}+{\\frac {\\theta ^{5}}{5!}}-{\\frac {\\theta ^{7}}{7!}}+\\quad \\cdots ", "1897be964cf0e1693a2e83aa9d1f7e3d": "D(0,0)=0;~~D(0,-1)=-1;~~D(1,-1)=-1;~~D\\left({\\frac {3}{8}},-{\\frac {3}{4}}\\right)={\\frac {27}{128}}.", "1897dc95f9896bab096535c9def46084": "{\\frac {x\\sin \\theta }{\\lambda }}=\\pm {\\frac {1}{2}},\\pm {\\frac {3}{2}},...", "1897df0ccae432eb35a4dea99a291f7d": "{\\boldsymbol {\\beta }}(y)\\geq 0,\\mathbb {E} _{Y^{tr}}[{\\boldsymbol {\\beta }}(y)]=1", "1897e60fa0f0a08bd59f3a9aff1f0982": "\\phi _{2}", "189823f8b90513488cb6048ddb2fd60f": "\\cos \\varphi ", "1898df23a13badc3d56f4de3b76b1b19": "(A,\\phi )", "1899006f6f0679e0964cf6f1fc124588": "(b_{1},\\dots ,b_{n})", "189934d1d98999afd63209dba9fc4e82": "{\\breve {\\boldsymbol {\\theta }}}_{i}=\\mathrm {Inv} ({\\boldsymbol {s}},{\\boldsymbol {z}}_{i})", "18997d64d3fb1b8b34b50d1d5d0b2036": "k=1\\ldots \\lambda ", "1899cddb77666d0e852a4209f2462148": "\\scriptstyle {\\frac {-3\\pi }{2}}", "1899e9301923b81e4442da9c926ad330": "\\{a^{n}:n\\geq 1\\}", "1899f1559d1d26066caa707897b30d48": "ModD(y)\\equiv -{\\frac {1}{V}}\\cdot {\\frac {\\partial V}{\\partial y}}=-{\\frac {\\partial ln(V)}{\\partial y}}", "189a3676b44955400e597ca7b79a2aa2": "\\operatorname {P} (\\left|{\\overline {X}}_{n}-\\mu \\right|<\\varepsilon )=1-\\operatorname {P} (\\left|{\\overline {X}}_{n}-\\mu \\right|\\geq \\varepsilon )\\geq 1-{\\frac {\\sigma ^{2}}{n\\varepsilon ^{2}}}.", "189a562f37b5839c495bb968c1059fb9": "Y|X", "189a62adcb589050d3be98be1f765c0e": "I(\\nu ,T)={\\frac {2h\\nu ^{3}}{c^{2}}}{\\frac {1}{e^{\\frac {h\\nu }{kT}}-1}},", "189a6c69916161df3d709f299a64b337": "(-1)^{\\frac {n-1}{2}}\\cdot C_{\\frac {n-1}{2}}\\equiv 2{\\pmod {n}},", "189ac807f51cdb474ee0740d9fd0e893": "\\langle Q\\rangle _{\\psi }=\\int _{-\\infty }^{\\infty }\\,x\\,|\\psi (x)|^{2}\\,dx", "189ae40408bef1b9cad90605bb51f47b": "p_{1}(({\\frac {T_{2}}{T_{1}}})^{\\frac {\\gamma }{\\gamma -1}}-1)\\,", "189b93d19c17dddaccaf7e4fcb4fcaea": "[H^{+}]_{0^{}}10^{b_{0}}", "189bdd68a9c070d450bc218ea2211555": "[[z]]", "189bfb2c2abb79025eb4c6cea0d8cf7b": "{\\begin{aligned}G_{L}&\\to 1\\\\G_{R}&\\to 0\\end{aligned}}", "189c028399d91e0fca282c1955fa491a": "p=0.", "189c4f5febfa6b17825c341346e5a1f7": "{\\begin{aligned}\\phi ({\\mathbf {X} },z_{n+1})=\\int p({\\mathbf {X} }-{\\mathbf {X'} },z_{n+1}-z_{n})\\phi ({\\mathbf {X} },z_{n})\\exp \\left(-i\\sigma \\int \\limits _{z_{n}}^{z_{n+1}}V({\\mathbf {X'} },z')dz'\\right)dX'\\end{aligned}}", "189c5675254ab7ef45079aa3c9af7d09": "x[T\\cup S]y", "189c6cbe80dae0f849c56df4e57b8bb1": "R_{x}(\\tau )=\\int _{-\\infty }^{\\infty }\\left.x(t+\\tau /2)x^{*}(t-\\tau /2)\\right.\\,dt.", "189cdf573cc9e575ff2f7fe0ded02e44": "\\Phi _{i}\\rightarrow \\Phi ", "189ce067cc5b92d3228402efcffd1ab2": "\\cdots \\to H^{1}(V,{\\mathcal {O}}_{V}^{*})\\to H^{2}(V,\\mathbb {Z} )\\to H^{2}(V,{\\mathcal {O}}_{V})\\to \\cdots .", "189cf407f19bf0f6fe6ead4392233c19": "x'>y\\sqsubseteq x:=y+1", "189d33de47b11995edd6014c9250beaa": "{Q_{1} \\over \\ Q_{2}}={\\left({N_{1} \\over N_{2}}\\right)}", "189d65ea689ab8abd354c7c4883f9241": "\\|x_{i}-x_{j}\\|", "189d673d3b7970a55671dc60f60a1774": "i\\leftarrow I", "189e47d3077e96af6c3a115fe1bf9f87": "\\varphi =1/(2\\sin(\\pi /10))=1/(2\\sin 18^{\\circ })\\,", "189e8c39ea43baed4198f243db129874": "A\\oplus \\!B\\oplus C", "189edb1c7606c629725529fd5210fc9d": "\\ {\\hat {x}}(t)=g(t)*y(t)", "189f4ae226ce8e2c1c6f6ed5142138fb": "B_{\\infty }^{p,q}", "189f4c64921adeb80f74c2d4f76e9437": "2\\ \\operatorname {tr} (\\gamma ^{\\mu }\\gamma ^{\\nu }\\gamma ^{\\rho }\\gamma ^{\\sigma })=8\\eta ^{\\rho \\sigma }\\eta ^{\\mu \\nu }-8\\eta ^{\\nu \\sigma }\\eta ^{\\mu \\rho }+8\\eta ^{\\mu \\sigma }\\eta ^{\\nu \\rho }\\,", "189f532fa4d24c07110adbc183c0154f": "m^{2n}\\equiv m^{2n-\\wp (p-1)}{\\pmod {p}}\\!", "189f68a2a3a410a2389708559c3f4d4b": "\\delta _{y}", "189f8851bd75530579c998307e0d67f9": "a\\neq b\\neq c\\neq d,\\alpha =\\beta =\\gamma =\\delta =\\epsilon =\\zeta =90^{\\circ }", "189f9d45cdc1369a54d346d6f86dd900": "S(0)=S(0/1)=[1;2,3,4,5,6,7,\\dots ].", "189fa3d6ec47083f68846211f25c0d71": "{\\mathbb {Q} }(x)", "189fad1b0f1de9a3570cda392fdadab5": "\\operatorname {de-lambda} [x\\ f=\\lambda y.f\\ (y\\ y)]\\equiv x\\ f\\ y=f\\ (y\\ y)", "18a03bd19d746163bac6e05c45218e70": "H=-\\ln \\left({\\frac {\\phi (q)}{2\\pi }}\\right)+2\\sum _{k=1}^{\\infty }{\\frac {(-1)^{k}}{k}}\\,{\\frac {q^{(k^{2}+k)/2}}{1-q^{k}}}", "18a0eaae05c4e17c8d499d0bdff02859": "|\\Psi _{\\epsilon }^{(\\pm )}\\rangle ={\\frac {U_{\\epsilon I}(0,\\pm \\infty )|\\Psi _{0}\\rangle }{\\langle \\Psi _{0}|U_{\\epsilon I}(0,\\pm \\infty )|\\Psi _{0}\\rangle }}", "18a10280ff9fec2cb2035def93074577": "_{C}", "18a157d55c3a810756af692f2a7674c3": "H={\\frac {2}{3}}\\cdot {\\frac {A+B}{2}}+{\\frac {1}{3}}\\cdot {\\sqrt {AB}}.", "18a1b53e3e5951aac453e86ddbc284a7": "(R\\to S)", "18a21ad7167c7ea66d80707a3bc51733": "l={\\frac {\\lambda }{4}}\\ ,", "18a2422c9c7fa6717043797dd738d3e8": "0.5\\ln(1.5+5f^{*})+0.5\\ln(0.75-f^{*})\\!", "18a2450249093366c5ae3bccf41225bd": "{\\mathit {Z}}_{p}=\\{0,1,2,...,{\\mathit {p}}-1\\}", "18a33fea91b4526949a255113206ada0": "=\\operatorname {sgn}(\\tan({\\frac {2\\theta +\\pi }{4}})){\\frac {\\tan \\theta }{\\sqrt {1+\\tan ^{2}\\theta }}}", "18a3d15e003ab3fde06f39ff59e3bce4": "(N/{\\sqrt {T}})", "18a3d979d4ad4037702a8021c2524565": "Y\\to Z,S\\to Z,Y\\to S,S\\to A", "18a4ca9a19f345bf6866427902b62e29": "2V/(V+2)", "18a4fdd9c57ce3f5a140e89d9df15b6f": "\\lambda (s,t_{s})=y", "18a5579c6872193e5f1a96343278c52b": "T_{p}=1-R_{p}\\,\\!", "18a57207cfbee0edf6bd2fff690470cc": "\\sum _{i=0}^{\\infty }\\operatorname {P} [{\\textrm {first\\ }}i{\\textrm {\\ rolls\\ are\\ ties,\\ }}(i+1)^{\\textrm {th}}{\\textrm {\\ roll\\ is\\ 'the\\ point'}}]", "18a59284ce6e3123929431d22a5c3655": "a*=1\\cup a;a*\\,\\!", "18a5bd7a2e6026f4e0495bbc5afd600b": "\\theta _{R}=1.22\\lambda /\\,\\!d", "18a5e913f49e8dc1fa94296720bd647a": "{\\frac {\\langle B,s\\rangle \\Rightarrow \\mathbf {true} }{\\langle \\mathbf {while} \\ B\\ \\mathbf {do} \\ C,s\\rangle \\longrightarrow \\langle C;\\mathbf {while} \\ B\\ \\mathbf {do} \\ C,s\\rangle }}\\quad {\\frac {\\langle B,s\\rangle \\Rightarrow \\mathbf {false} }{\\langle \\mathbf {while} \\ B\\ \\mathbf {do} \\ C,s\\rangle \\longrightarrow s}}", "18a63a8c4991585eb2858222600ca602": "f(\\mathbf {x} )=0", "18a6ee04730b22a17211fe8cad693ac1": "{\\hat {Y}}_{j(i)}\\,", "18a71b3d5f7d5b1af3d595a00049595e": "v_{i,m}-\\varepsilon 1", "18b4729f3b0657e360cb639e7ada72ab": "R^{{\\mathfrak {T}}_{\\Phi }}{\\overline {t_{0}}}\\ldots {\\overline {t_{n-1}}}", "18b4c68bfc760bbf3e3624b9466b59a2": "1\\leq p,q\\leq \\infty .", "18b4da017912c5e9183df31ac6f8370e": "\\sigma _{i,\\,j}", "18b5349300d054f65d2b26943937c3d1": "2x=a(3x^{2}-y^{2})", "18b540028c08b3564e25f146db9ef2ab": "F[G]\\,", "18b5416b92bc741f5f3dd407738866de": "L=2S-{\\frac {400+3.5S}{A}}", "18b5c3583e02eb21216625472e25f88d": "{\\begin{aligned}\\mathbb {E} (I-g(X))^{2}&=\\int _{0}^{1/3}(1-g(3y))^{2}\\,\\mathrm {d} y+\\int _{1/3}^{2/3}g^{2}(1.5(1-y))\\,\\mathrm {d} y+\\int _{2/3}^{1}g^{2}(0.5)\\,\\mathrm {d} y\\\\&=\\int _{0}^{1}(1-g(x))^{2}{\\frac {\\mathrm {d} x}{3}}+\\int _{0.5}^{1}g^{2}(x){\\frac {\\mathrm {d} x}{1.5}}+{\\frac {1}{3}}g^{2}(0.5)\\\\&={\\frac {1}{3}}\\int _{0}^{0.5}(1-g(x))^{2}\\,\\mathrm {d} x+{\\frac {1}{3}}g^{2}(0.5)+{\\frac {1}{3}}\\int _{0.5}^{1}((1-g(x))^{2}+2g^{2}(x))\\,\\mathrm {d} x\\,;\\end{aligned}}", "18b5c459d9acf84628fa39c81c689b23": "X^{*}", "18b7828af58814a03317d2ad8d4901ee": "W=D_{1}\\left({\\tfrac {J^{2}-1}{2}}-\\ln J\\right)+C_{1}~\\sum _{i=1}^{5}\\alpha _{i}~\\beta ^{i-1}~({\\overline {I}}_{1}^{i}-3^{i})", "18b7911727a89109b30e9db96ae16dda": "[B_{1}(t),\\phi (t)]\\Longleftarrow SLR\\Longrightarrow [A_{N}(z),B_{N}(z)]", "18b7bb34a2981efb7b59903eeff7bedd": "\\ where", "18b8533be0dbf96a2eaff6d16df1b38d": "{\\dot {V}}(x)={\\frac {d}{dt}}V(x)\\leq 0", "18b8a4aa0f31e06aa7f8a35dd26f88b0": "|\\rho )", "18b91870f12e40867d62dda1870a0bf4": "Pmo+Pmf=1", "18b94090f21af324926114c559ca4b66": "\\sigma _{\\mathrm {n} }=\\sigma _{x}\\cos ^{2}\\theta +\\sigma _{y}\\sin ^{2}\\theta +2\\tau _{xy}\\sin \\theta \\cos \\theta ", "18b94c714aafaa62984c0b1618036562": "q\\gamma (a,q)={\\frac {q}{d}}\\gamma (a/d,q/d)-\\log d.", "18b9f2fdc605563121d97539fc8d5e0c": "\\forall {\\vec {y}}\\,\\varphi (f({\\vec {y}}),{\\vec {y}})", "18ba02363b7fedc6e301e7d2d930d96f": "\\scriptstyle {p}", "18ba06497be736a00e5d586e7c0eff8a": "f_{n}(z)={\\frac {a_{n}}{b_{n}+z}}.", "18ba0877c3b039bcf8af77f16a2cd37a": "E(\\phi ,k)=\\int _{0}^{\\phi }{\\sqrt {1-k^{2}\\sin ^{2}\\theta }}\\,\\mathrm {d} \\theta =\\sin \\phi \\,F_{1}({\\tfrac {1}{2}},{\\tfrac {1}{2}},-{\\tfrac {1}{2}},{\\tfrac {3}{2}};\\sin ^{2}\\phi ,k^{2}\\sin ^{2}\\phi ),\\quad |\\Re \\,\\phi |<{\\frac {\\pi }{2}}~,", "18ba55b0cd0f940044ec599262597c2c": "\\{(\\emptyset ,{\\mathsf {i}})\\}", "18ba72e0fe9235a80f0a6ab3e98a49dc": "\\langle |\\gamma (n)|^{2}\\rangle ={\\frac {1}{c_{n}}}\\sum _{n\\;\\mathrm {step\\;SAW} }|\\gamma (n)|^{2}=n^{2\\nu +o(1)}", "18ba8aaaf28689458abcfc03fb6497ae": "{\\begin{array}{l}(\\forall L\\subseteq \\Sigma ^{*})\\\\\\quad ({\\mbox{regular}}(L)\\Rightarrow \\\\\\quad ((\\exists p\\geq 1)((\\forall w\\in L)((|w|\\geq p)\\Rightarrow \\\\\\quad \\quad ((\\exists x,y,z\\in \\Sigma ^{*})(w=xyz\\land (|y|\\geq 1\\land |xy|\\leq p\\land (\\forall i\\geq 0)(xy^{i}z\\in L))))))))\\end{array}}", "18ba9473b1f7b96c41e90f5242c156d8": "\\psi _{n}", "18bae66d415041394488d0f810328d9d": "ncp={\\sqrt {n}}{\\frac {\\mu -\\mu _{\\text{baseline}}}{\\sigma }}", "18bb05897a006e4b06c97d63ed9a74f2": "{\\dot {\\theta }}=f(\\theta )+g(\\theta )S(t)", "18bb1b76f70276f343d7526478c27f27": "\\mathbb {E} _{\\theta }\\left[V(x(\\theta ),\\theta )-{\\underline {u}}(\\theta _{0})-\\int _{\\theta _{0}}^{\\theta }{\\frac {\\partial V}{\\partial {\\tilde {\\theta }}}}d{\\tilde {\\theta }}-c\\left(x(\\theta )\\right)\\right]", "18bb1e8fa7618fb780342b2c803e1637": "{\\bar {w}}_{1L}(s,2n+\\gamma _{1L};L)", "18bb41f9fbbfa60a43d5e7350d6c9ae4": "y\\;=\\;2\\,y_{0}\\,\\cos(\\omega t)\\;\\sin(kx).\\,", "18bb8cacf20918a41196f9bb562f8869": "xy+ax^{3}+bx^{2}+cx=d\\,", "18bb8cd057d57f22e2742ea35ba662d7": "V_{3}", "18bbae7591c05f96e5a7d44431e2444c": "\\lim _{x\\to 0^{+}}\\log x=-\\infty ", "18bbae909b0daf82a2969f5f5ef3a589": "\\,\\kappa _{3}", "18bc3ca421bd5a60bdfdb2d8c7cf765f": "V_{n+1}=\\int _{0}^{1}S_{n}r^{n}\\,dr", "18bc6acd4040524ea2270b74ca8b389f": "{\\begin{pmatrix}y_{1}\\\\y_{2}\\end{pmatrix}}={\\begin{pmatrix}m_{1}(x_{1},x_{2},x_{3})\\\\m_{2}(x_{1},x_{2},x_{3})\\end{pmatrix}}", "18bc6bc95462fe455140fd689af4f37f": "a=2r_{2},c=r_{2},d=r_{1}\\,\\!", "18bc82f68262b2a5c4408648bd492194": "2\\pi \\varepsilon a\\sum _{n=1}^{\\infty }{\\frac {\\sinh \\left(\\ln \\left(D+{\\sqrt {D^{2}-1}}\\right)\\right)}{\\sinh \\left(n\\ln \\left(D+{\\sqrt {D^{2}-1}}\\right)\\right)}}", "18bccbfdf90f4137431af6bc072bc5bc": "C_{stat}={\\frac {V_{T}}{P_{plat}-PEEP}}", "18bd7f90e2be815249783b0021fa338f": "(J^{\\alpha }f)(x)={1 \\over \\Gamma (\\alpha )}\\int _{0}^{x}(x-t)^{\\alpha -1}f(t)\\;dt", "18bd8fc9c1add66574b2c6ad352c1e09": "\\mathbf {r} ", "18bdb74993b655d2d9fc8f2e500a1278": "\\Delta _{\\alpha }", "18be17b6f838e77cb27987a8b4b95fe4": "\\left({\\frac {\\partial T}{\\partial V}}\\right)_{S}=-\\left({\\frac {\\partial P}{\\partial S}}\\right)_{V}={\\frac {\\partial ^{2}U}{\\partial S\\partial V}}", "18be2b8a85054984b0979facedfeff38": "W_{i}(x)", "18be2c14dd16f907369b3fb992b42eef": "X_{t}:=W(t+\\tau _{a})-a", "18be7e17673a28353590323423b1cfd3": "\\Rightarrow P=5.25{\\mbox{ big bets }}", "18be99b429a0b6209066c6e48df5d0f4": "{\\frac {\\partial p}{\\partial t}}+\\rho _{0}~c_{0}^{2}~\\nabla \\cdot \\mathbf {v} =0~.", "18bf58ee81d2c796057d130e40e94e3c": "e_{s}(T)", "18bf9dd809926e4de2b43a9eda126d74": "\\|f\\|_{p}=\\sum _{n}|a_{n}|^{p}\\qquad (0", "18c26fbb9faeb672c839801ae8d7ad34": "B^{ij}", "18c2c9e447046792e2c30f4b110ac12d": "2^{K}", "18c2f4b4d5330d21f199aab54a25ca65": "\\zeta ", "18c36012bc30feba7ea4d3f8c40e9070": "2^{\\frac {1}{12}}\\approx 1.0595", "18c456eb149d0b8085c092f13c390655": "\\langle X,Y\\rangle _{A}=\\langle Y,X\\rangle _{A}", "18c5171e237b269287e952f3d3052c6d": "{\\begin{aligned}&W^{2}={\\sqrt {2\\gamma _{2}+4}}-1,\\\\&\\delta =1/{\\sqrt {\\ln W}},\\\\&\\alpha ^{2}=2/(W^{2}-1),\\\\\\end{aligned}}", "18c5bee80a7702628fdb81f660d2d122": "\\gamma =\\int _{0}^{1}{\\frac {1}{1+x}}\\sum _{n=1}^{\\infty }x^{2^{n}-1}\\,dx.", "18c5f9700e93ac23c1ca12fe82793464": "S_{0}", "18c622a400c959474d52682d250a2545": "\\#X<\\nu (W)", "18c6b428497c5dfc34dc2381fefd2a01": "\\scriptstyle \\mathbb {R} ^{2}", "18c6f2a3052cd34e931618a614ee9608": "{\\begin{bmatrix}\\Psi \\end{bmatrix}}^{T}{\\begin{bmatrix}M\\end{bmatrix}}{\\begin{bmatrix}\\Psi \\end{bmatrix}}={\\begin{bmatrix}^{\\diagdown }m_{r\\diagdown }\\end{bmatrix}},", "18c70df624239185d211541116b4658a": "A_{w}(\\mathbb {T} )=\\{f:f(t)=\\sum _{n}a_{n}e^{int},\\,\\|f\\|_{w}=\\sum _{n}|a_{n}|w(n)<\\infty \\}\\,(\\sim \\ell _{w}^{1}(\\mathbb {Z} ))", "18c783c7483df4341ae61e26e46688e7": "A\\geq {\\frac {P^{m}(V)}{2}}\\,\\!", "18c79bc7572a8969200765a7e578384e": "\\div \\!\\,", "18c7da0dbd1bf29357ccae6f8d85cae6": "a*(b+c)=a*b+a*c", "18c83df3e8f90750ec5e350a55ec8090": "{e}=\\rho _{v}R_{v}T\\,", "18c87d80621e6c261a7575039bf8e7e3": "(\\mathbf {\\hat {x}} ,\\mathbf {\\hat {y}} ,\\mathbf {\\hat {z}} )", "18c8d17bf436f990c16703d01556c1ec": "{\\vec {p}}\\in P", "18c91a49cd224a3e390dcaf993db2f56": "\\scriptstyle A_{0},A_{1},B_{0},B_{1}", "18c932f784acdcfb8c7c81591b255ed5": "H_{1-\\alpha }-H_{\\alpha }=\\pi \\cot {(\\pi \\alpha )}-{\\frac {1}{\\alpha }}+{\\frac {1}{1-\\alpha }}\\,.", "18c9879537ecfdb1a719992752a3e4ec": "\\zeta ^{\\prime }(-1)={\\frac {1}{12}}-\\ln A", "18c9ab7fd3438beaac0f33b87fbf6666": "i\\in D", "18c9b29a5a484f5e37991318459fcca8": "\\succ _{i}^{p}", "18c9ce8c77dc0267a83b6ed5a29c0044": "y=x^{m}", "18c9df2276ffd7a0ab756bdfa1f04efd": "1\\leq \\cdots \\leq A_{-1}\\leq A_{0}\\leq A_{1}\\leq \\cdots \\leq G", "18c9ed4dad017ea46713b5309a8c8677": "k_{0}=1.\\!", "18ca3278a836cd4e54595dd54b77daee": "r_{a}", "18ca374c2173f81f8c19eb161900b95c": "\\Delta g_{h}=\\left[G\\,m_{\\mathrm {Earth} }/\\left(r_{\\mathrm {Earth} }+h\\right)^{2}\\right]-\\left[G\\,m_{\\mathrm {Earth} }/r_{\\mathrm {Earth} }^{2}\\right]", "18cab4412675e6cd3658f0b619095dda": "\\lambda =w\\cdot {\\tilde {\\lambda }}", "18cb04d6c0fe962c0924510db4794da0": "{\\begin{aligned}{\\biggl |}\\bigcup _{i=1}^{n}A_{i}{\\biggr |}&{}=\\sum _{i=1}^{n}\\left|A_{i}\\right|-\\sum _{i,j\\,:\\,1\\leq it_{i}\\}", "18d57e2afd67380f1b7252ccec01b7f1": "{\\boldsymbol {\\nabla \\times }}\\left({\\boldsymbol {\\nabla \\times V}}\\right)={\\boldsymbol {\\nabla }}\\left({\\boldsymbol {\\nabla \\cdot V}}\\right)-\\nabla ^{2}{\\boldsymbol {V}}\\ ,", "18d59220b9efd10dd2d46bb3e83efc0f": "M_{\\alpha }=\\sum _{i_{1}c_{2}", "18d916a53926bede3895d71f6621da3a": "\\forall i,\\lambda _{i}\\geq 0;\\lambda _{0}=0", "18d975ee85323d2a1939a43e15666899": "W_{s}=\\int _{1}^{2}\\sigma _{s}\\,dx.", "18d9962fdb0adb9b0587bda0c33524a2": "n^{2}+1", "18d99b829fb348bc558fc223d5c79acf": "(r,\\theta ,z)", "18d9be35d247c9ea4926c801c4c199b6": "\\sum \\limits _{k=1}^{\\infty }|x_{k}\\,y_{k}|\\leq {\\biggl (}\\sum _{k=1}^{\\infty }|x_{k}|^{p}{\\biggr )}^{\\!1/p\\;}{\\biggl (}\\sum _{k=1}^{\\infty }|y_{k}|^{q}{\\biggr )}^{\\!1/q}{\\text{ for all }}(x_{k})_{k\\in \\mathbb {N} },(y_{k})_{k\\in \\mathbb {N} }\\in \\mathbb {R} ^{\\mathbb {N} }{\\text{ or }}\\mathbb {C} ^{\\mathbb {N} }.", "18d9ddc4fe786f1cd073c3dbc282b5da": "x_{1},x_{2}\\geq 0", "18d9f795cd0d91c53c3acbb7daad604c": "\\Delta t\\rightarrow 0", "18da665ed77201f6c7402d395e55c21c": "\\nabla ^{2}\\Phi ={\\frac {1}{a^{2}\\left(\\sigma ^{2}-\\tau ^{2}\\right)}}\\left\\{{\\frac {\\partial }{\\partial \\sigma }}\\left[\\left(\\sigma ^{2}-1\\right){\\frac {\\partial \\Phi }{\\partial \\sigma }}\\right]+{\\frac {\\partial }{\\partial \\tau }}\\left[\\left(1-\\tau ^{2}\\right){\\frac {\\partial \\Phi }{\\partial \\tau }}\\right]\\right\\}+{\\frac {1}{a^{2}\\left(\\sigma ^{2}-1\\right)\\left(1-\\tau ^{2}\\right)}}{\\frac {\\partial ^{2}\\Phi }{\\partial \\phi ^{2}}}", "18da6fd1d28eafe7341e995408a06d00": "D_{y}\\ f(x)", "18da78172f27d03ad7e9502ee5fb4b79": "g(f(x))=\\sum _{n=1}^{\\infty }{\\sum _{k=1}^{n}b_{k}B_{n,k}(a_{1},\\dots ,a_{n-k+1}) \\over n!}x^{n}.", "18da8dfa42c5e288b3d8a601179e0a59": "\\mathbf {y} (\\cdot )", "18daef71b5d25ce76b8628a81e4fc76b": "y_{i}", "18db1019fc935b332ccd75838c1a05ae": "4K^{2}=(ad)^{2}\\sin ^{2}\\alpha +(bc)^{2}\\sin ^{2}\\gamma +2abcd\\sin \\alpha \\sin \\gamma .\\,", "18db1f7b31478a3839616580ad00ad0d": "C=D\\left[N(d_{+})F-N(d_{-})K\\right]", "18db58092a2eba6265a61b7cce5c3ae3": "{\\begin{matrix}D_{\\mathrm {KL} }(P\\|Q)&=&-\\sum _{x}p(x)\\log q(x)&+&\\sum _{x}p(x)\\log p(x)\\\\&=&H(P,Q)&-&H(P)\\,\\!\\end{matrix}}", "18db5e44caf2ed2fdde04a9dd6a6c41f": "(\\underbrace {x_{n}+x_{n+1}-\\alpha } _{=\\,y})\\alpha -x_{n}x_{n+1}=(x_{n}-\\alpha )(\\alpha -x_{n+1})>0,", "18db600e9b6993dd9ec8642eb24695dd": "\\pi _{k}", "18db90767e8ab871f14664034faf96e7": "g(r)dr", "18dbc544e753ec339db6cb5606b70029": "{\\frac {{\\sqrt {17}}-1}{2}}", "18dbec4959311ce1ed660197e86282b2": "E={\\frac {\\hbar ^{2}k^{2}}{2m}}", "18dbff11522d3c95e274440bdd47deb1": "\\Omega _{-}\\,", "18dc2d7de0b78341c8d30be4ecb7217b": "|N(h)|", "18dc516a4024559888e5f7271fd3f7cf": "T=4{\\sqrt {\\ell \\over g}}F\\left({\\theta _{0} \\over 2},\\csc {\\theta _{0} \\over 2}\\right)\\csc {\\theta _{0} \\over 2}", "18dc5eaca5c946d491477c8aba406a34": "v(\\tau )=\\int {\\frac {r(\\tau )}{r(\\tau )-2GM}}d\\tau ", "18dc8499fdc12a50265acbc280088b5b": "\\mathbb {T} ^{3}", "18dcc10831be4494a9c0fcda3f7bec40": "d_{j}", "18dce07d0b7288f146469052faa6a251": "{u}_{1}^{z}={g}_{1}^{kz}=h^{k}\\,", "18dd082ee2e6b606e52cf983f153f3f4": "B=QTZ^{H}", "18dd11165ce16b83780b7d075019caf7": "d*\\mathbf {F} =*\\mathbf {J} ", "18dd30e1971035d87332bb7a9da52125": "ab\\leq \\int _{0}^{a}f(x)\\,dx+\\int _{0}^{b}f^{-1}(x)\\,dx", "18dd6968964f626cb21125eb5a414ad6": "y={\\frac {Y}{Z}}", "18dd8a5b501b6e52fab6fdca29986aae": "\\int _{a}^{b}f(x)\\,dx\\approx {\\frac {1}{2}}\\sum _{k=1}^{N}\\left(x_{k+1}-x_{k}\\right)\\left(f(x_{k+1})+f(x_{k})\\right).", "18dda4dd3a74b0ad8cba16542145e5bc": "{\\vec {B}}\\!", "18ddbcdfc1287d2e4d37590299c4e28d": "R_{mn}(-ic,i\\xi )", "18ddf017f6cd97b31468ebfc3e167b7b": "001100", "18de282afdf186a5a68e15c900084ce7": "E[{\\textbf {v}}_{k}{\\textbf {v}}_{k}^{T}]={\\textbf {R}}_{k}^{a}", "18de505a0c1d409e45c8075261763358": "cf\\in {\\mathcal {H}}", "18deab078d5dde8e2825954f355e7932": "\\beta =\\arctan \\left({\\frac {b\\sin \\alpha }{a+b\\cos \\alpha }}\\right)+{\\begin{cases}0&{\\text{if }}a+b\\cos \\alpha \\geq 0,\\\\\\pi &{\\text{if }}a+b\\cos \\alpha <0.\\end{cases}}", "18deda0400bf981d62336050161c54f4": "A^{\\mu }={\\hat {P}}^{\\mu }A(r)", "18df69c7e2afa34bf74da608bb2bd466": "{\\mbox{Mortgage Yield: ri such that P}}=\\sum _{n=1}^{N}{\\frac {C(t)}{(1+ri/1200)^{t-1}}}", "18dfdf66424bc93f268b0a52ea1088ef": "\\epsilon _{ij}=d_{ijk}E_{k}\\,", "18e06cc13564d3ae926f420ebf45ebcf": "y^{2}=x(x-a^{n})(x+b^{n}).\\,", "18e08796f10460ab1bc7e5622cd15249": "r^{*}t'+t^{*}r'=0.", "18e0e3c623cb11ea32986af2a0ec35da": "\\scriptstyle \\forall n:\\;\\sum _{j=1}^{J}P_{nj}=1,", "18e0e8c459b7e584339cbbd03089c555": "m\\in M.", "18e0ff95ddcaa085a20abacb0a6c1ec7": "1-e^{-4.7/4.5}", "18e135ba82fd466fb3d91d10eff214d9": "P(v,u\\in K)=P(v\\in K)\\,P(u\\in K)", "18e151b7cdd4f24b150d6dea6516d4dc": "~y", "18e18adb2e9f5c02ad1b703d0a0528cc": "M_{-1}^{2}", "18e1be1f137eecd5e0d08c063ddf1d21": "{\\partial v \\over \\partial y}=e^{x}\\sin y", "18e1c440689fbb0b3331a75502d696bf": "(\\forall k\\in \\mathbb {Q} \\setminus \\{0,-1,-2,\\ldots \\}):f_{k}(x)\\neq 0{\\text{ and }}{\\frac {f_{k+1}(x)}{f_{k}(x)}}\\notin \\mathbb {Q} .", "18e1d074eb6ca6bab87c99b95af11ee9": "d_{1}\\ldots d_{n}", "18e1e571ad0dea2afe023c83368239ae": "R_{CF}\\ll R_{P}", "18e2143a3c1df78caf3a5884732e5fea": "{\\frac {{\\rm {d}}n}{{\\rm {d}}\\lambda }}<0.", "18e232b29e32721fdfdf28d703e4c490": "a=\\alpha r/2", "18e253d4ce939cf1524819f4bd610674": "\\sigma =[1+{\\frac {6.2}{z.n^{2/3}}}]^{-1}", "18e27dd067ddf4ee760b43ba878543d8": "b0", "18e7f09e1ba3ead19d2fc82b4efcd47e": "a^{\\ln x}\\,", "18e833a4416a8cf33f6eb69697cf10f3": "\\{{\\boldsymbol {m}}\\}={\\begin{cases}\\mathbf {m} _{1}=\\mathbf {\\hat {x}} _{1}=[1,0,0]&\\|\\,{\\textrm {to}}\\,{\\textrm {applied}}\\,{\\textrm {tension}}\\\\\\mathbf {m} _{2}=\\mathbf {\\hat {x}} _{2}=[0,1,0]&\\perp {\\textrm {to}}\\,{\\textrm {applied}}\\,{\\textrm {tension}}\\\\\\mathbf {m} _{3}=\\mathbf {\\hat {x}} _{3}=[0,0,1]&\\|\\,{\\textrm {to}}\\,\\mathbf {N} \\end{cases}}", "18e871e3a2f44664dac2bbe3bdde2356": "n=1.5", "18e8aec7b6bb93acabfcc38b5f352dac": "\\ln(xy)=\\ln(x)+\\ln(y).\\!\\,", "18e8af50161ff9bb61d83bf509d4ba78": "\\mu =M_{\\mbox{i}}-M_{\\mbox{f}}\\,.", "18e8cd2673a772049abd316473986c6c": "\\left\\vert -h\\right\\vert ", "18e91a026e88127189fb8fc07e1b1e4f": "A=\\pi D^{2}/4", "18e965e2958c2b0d27d4026a6b39391c": "\\sigma ,\\tau ", "18e9981ff27b4c67567ed89fd97460bd": "{\\overline {q}}_{tt}^{i}=0", "18e9ce193cb916c093dc6103d005c35e": "F_{r}=1;\\;{\\textit {Critical}}", "18e9e4628380071621088ec8c296e8db": "a_{3}b_{1}", "18e9e4d860da1afcb3b2eb5a2ae6b0f3": "x+x^{2}", "18ea38550fd57e7c2a64fa2851d7eed5": "\\pi (x)=\\operatorname {li} (x)+O{\\bigl (}xe^{-{\\sqrt {\\ln x}}/15}{\\bigr )}\\!", "18eacf44f99c3a847cf8833245952bd8": "D_{a}={\\frac {313\\Pi }{10800}}", "18eb3dc33619724182bacdda52edccf2": "H=H_{1}\\otimes H_{2}", "18eb57492e10c8ba84f1907647b1b647": "{\\underline {\\hat {\\mathbf {h} }}}(\\ell )={\\underline {\\hat {\\mathbf {h} }}}(\\ell -1)+\\mu \\mathbf {G} _{2}\\mathbf {\\Phi } _{\\mathbf {xx} }^{-1}(\\ell ){\\underline {\\mathbf {X} }}^{H}(\\ell ){\\underline {\\mathbf {e} }}(\\ell )", "18eb6b29f002e684df90786d417aea35": "U/D>\\pi ", "18ec0f8feaca1b6b0dca2651a8c87ed5": "\\scriptstyle A_{t}", "18ec6089a8de6865d2a39891d930b03a": "\\displaystyle {X}", "18ec6addef1c5059221a87ace99feaa9": "U_{DC}=n{\\frac {\\hbar }{2e}}\\omega ,\\ \\ \\ I(t)=I_{c}J_{-n}(a)\\sin \\phi _{0}.", "18ec7450b1cade397101e5dd9f496e22": "a({\\overline {y}})=1-a(y)", "18ec821f6c25dd91aaeed00426da2ae3": "\\mathbf {x} _{j}", "18ec93e899d470f2998ea9875533a148": "{\\sqrt {p(R_{i})}}", "18ecbaf66226ba4ea4e7604b197113dd": "{\\frac {\\mu _{0}l}{\\pi }}\\left(\\ln \\left({\\frac {d}{a}}\\right)+{\\frac {Y}{2}}\\right)", "18ed0afc43da2399560134b6e8fd05c1": "\\sum F_{\\perp }=F_{n}-F_{w}\\cos \\theta =0\\,", "18ed25604450022943278b745379b09d": "D^{3}", "18ed92d5aa74d11b26b02495c0225105": "\\{a^{i}b^{i}\\;|\\;i\\geq 0\\}", "18edb17c0fe3647a4edb159d9dbe2f62": "\\mathbf {e} _{1}\\wedge \\mathbf {e} _{2}", "18ee1a620604335fcb97eed24e611d9c": "f(T_{1},T_{3})={\\frac {q_{3}}{q_{1}}}={\\frac {q_{2}q_{3}}{q_{1}q_{2}}}=f(T_{1},T_{2})f(T_{2},T_{3}).", "18ee261e8fdaf09a093f4fca95284db9": "\\forall a,b\\in X,\\ aRb\\lor bRa.", "18ee590341dab70c6f0399729deebe52": "0=-\\Delta P2\\pi rdr+\\eta 2\\pi dr\\Delta x{\\frac {dv}{dr}}+\\eta 2\\pi rdr\\Delta x{\\frac {d^{2}v}{dr^{2}}}+\\eta 2\\pi (dr)^{2}\\Delta x{\\frac {d^{2}v}{dr^{2}}}.", "18ee59b780433789750e641ecc86e92f": "\\int \\cos ^{2}{ax}\\;\\mathrm {d} x={\\frac {x}{2}}+{\\frac {1}{4a}}\\sin 2ax+C={\\frac {x}{2}}+{\\frac {1}{2a}}\\sin ax\\cos ax+C\\!", "18ee6ecfee0d63526a15095efea248ff": "({\\mathcal {X}},\\Sigma )", "18ee7743f4ccf7470f652fd2a1a7238b": "t_{2}=[v_{2},v_{3}]\\,", "18ee983cb0b3b556bd69afb048eb4e89": "K_{t}(x)=e^{-tH}\\,,", "18eebd452f2617cf78ffc2dec6ce8c06": "v^{*}\\leftarrow v,x^{*}\\leftarrow x", "18eec6b2df6e8b8bbf3183366172fa07": "\\mathrm {Ref} (\\theta )\\,\\mathrm {Ref} (\\phi )=\\mathrm {Rot} (2(\\theta -\\phi )),\\ ", "18ef107c7fab1ca903a3a8754b7eef0a": "|n(x^{\\mu })\\rangle ", "18ef2138a9fe6ef69fe7d2f5d7b88fa2": "F=f(W(h_{1}),\\ldots ,W(h_{n})),", "18ef2de26637057f0d82d5641dfa5e49": "\\scriptstyle u=u_{0}+u_{1}...+u_{n}", "18ef4309191f5c9b8813f814787c46fa": "\\displaystyle \\alpha ", "18ef54debd2505c76d3ee54354a8d237": "w'_{1}=1", "18efa680e578ea5c08bd4057de44317b": "ST_{x}(\\Box _{m}\\varphi )\\equiv \\forall y(R_{m}(x,y)\\rightarrow ST_{y}(\\varphi ))", "18efc5472201a32ac91ddc05d4daf6a6": "\\operatorname {qri} (C)=\\operatorname {ri} (C)", "18f00026f69ba077a7cc4320bb07b329": "\\|x\\|_{f}{\\overset {\\text{def}}{=}}|f(x)|", "18f13abd48fe832e3323f2e24c87a82b": "\\mathbb {Z} /2\\mathbb {Z} :=\\lbrace 0,1\\rbrace ", "18f147d88109cddb695b05d78d52ff25": "NPSH_{A}=\\left({\\frac {p_{e}}{\\rho g}}+{\\frac {V_{e}^{2}}{2g}}\\right)-{\\frac {p_{v}}{\\rho g}}", "18f1887869ebade600220dbc16f4562a": "G(A,B)", "18f197b8fa5e4c2ba49cea075cc713ca": "\\mu (x\\wedge y)\\mu (x\\vee y)\\geq \\mu (x)\\mu (y)", "18f1bb52ba5aa4d30fad7bdbfad303fd": "\\cup _{n=1}^{\\infty }A_{n}\\in \\Sigma _{0}", "18f1c04cc33cac73730aea1f1d01e305": "m/n", "18f1e3d02ce97d1ecf4ce1d497979c4d": "\\left({\\frac {p_{1}}{p_{0}}}\\right)^{R/c_{p}}={\\frac {T_{1}}{T_{0}}},", "18f1fec2d0d81e43578c0b9957369590": "\\sin ^{2}\\theta _{1}+\\cos ^{2}\\theta _{1}=1\\,", "18f22aeaffef62020b1b657f172811ec": "\\gamma _{0}={\\frac {1}{\\sqrt {1-v_{0}^{2}/c^{2}}}},", "18f23635fdd22c2cf89d8e3b668a7a96": "\\Pr(\\tau _{n+1}\\leq t,X_{n+1}=j|(X_{0},T_{0}),(X_{1},T_{1}),\\ldots ,(X_{n}=i,T_{n}))", "18f265310240b2e3b1d77eb09b736540": "a\\neq 0,", "18f284a405363e95719439b6fa1b9f8a": "(t_{i},s_{i})", "18f2c079d88ea8e87f4ed76ebff0fe76": "{\\mathfrak {p}}={\\mathfrak {g}}_{-}", "18f2c9be3e9dd6426637bea129ddfbf9": "b_{1}=9", "18f2d60a0f21cd6529721272e79c4f00": "{\\frac {de}{dt}}=0", "18f3017ea924138f657aff2cb1bffbe9": "{\\sqrt {\\varphi }}", "18f39b8f8bda0dfedbc8fade1db56137": "\\Gamma (n+1)=n!", "18f3b10a7a29947a625031e81b8df176": "\\eta \\colon X\\to G(X),", "18f3c8f2cb09a2eaf55c0b0ac9ae2ada": "m:G\\times G\\to G,(g,h)\\mapsto gh,\\quad i:G\\to G,g\\mapsto g^{-1},", "18f3cdd8b0f4cc5e1f8d5439ce61eb2f": "\\operatorname {csch} (z)", "18f411624e9ce3015b527bae9d546a29": "\\scriptstyle \\lfloor x\\rfloor ", "18f41580a86278dc12dfd80b1c9b73fa": "\\vert S_{j}\\vert >t", "18f454cfb034d7a0d0586a1045fc556c": "{\\tfrac {1}{p^{2}}}", "18f4550b6bb63d16fad13c06c1db98ef": "{\\tfrac {{f}_{T}}{10}}", "18f46c53d1a1c94e144b56a59b6bbb33": "\\forall ^{p}L:=\\left\\{x\\in \\{0,1\\}^{*}\\ \\left|\\ \\left(\\forall w\\in \\{0,1\\}^{\\leq p(|x|)}\\right)\\langle x,w\\rangle \\in L\\right.\\right\\}", "18f5053e0cb6f1a3411f89ae082f21f4": "\\Delta u=O(k)+O(h^{2}).\\,", "18f50e3141228f645297a63cc81b571f": "s-{\\frac {r^{2}-R^{2}}{s}}", "18f584b183ed0444e6daaf778bdfc47f": "N_{K}(x)=\\left\\{p\\in V\\;:\\;\\forall x^{*}\\in K,\\langle p,x-x^{*}\\rangle \\geq 0\\right\\}.", "18f59f9764a34a4d7435285ceaf8113f": "(z-\\mu )'(Az-A\\mu )=\\operatorname {trace} \\left[{(z-\\mu )'(Az-A\\mu )}\\right]=\\operatorname {trace} \\left[(z-\\mu )'A(z-\\mu )\\right]", "18f5ade42ad7739d82ef2849344d9c9f": "M_{31}", "18f5e3c31625d4ca5e7af7c0e6c5b43d": "\\lambda \\in \\mathbb {R} ", "18f603cdab6dcb7cdc475029a155b7c6": "\\alpha _{i}>0", "18f63800376271ee4b0efe1545744cd6": "f\\,", "18f660f590c9ca42e886d51788fb0467": "\\|f\\|_{\\infty }=\\|f\\|_{\\infty ,S}=\\sup \\left\\{\\,\\left|f(x)\\right|:x\\in S\\,\\right\\}.", "18f6b37b70921418ec756ccf08e2c2db": "e_{f}[n]", "18f6bcb2a23df35bfd6d7b21564b7e65": "10\\log _{10}r.", "18f6c67ffdaa6186ffcf50b9cddc072f": "G=", "18f6d5c060dd41efcc31e6d834d6650b": "\\sum _{n=1}^{\\infty }{\\frac {1}{n^{p}}},", "18f6f55e3f5d7e22553835ad93c7c23a": "d={\\begin{pmatrix}1\\\\2/(1-4)\\\\2/(1-16)\\\\\\vdots \\\\2/(1-[N-2]^{2})\\\\1/(1-N^{2})\\end{pmatrix}}.", "18f71314ac1dc4e2f314e82bcceac49a": "y_{1}=f_{1}(x_{1},x_{2},\\ldots ,x_{n})\\,,\\quad y_{2}=f_{2}(x_{1},x_{2},\\ldots ,x_{n})\\,,\\ldots ,y_{m}=f_{m}(x_{1},x_{2},\\cdots x_{n})", "18f71d2215a6b4a7ec4352c8ebfbdb13": "F=M\\otimes _{A}A_{f}=M_{f}\\qquad G=M\\otimes _{A}{\\hat {A}}.", "18f765e1efc6dff1c9fa8864607252bd": "\\langle {\\mathcal {M}}_{\\rm {Tot}}^{2}\\rangle =\\langle {\\mathcal {M}}_{\\rm {Tot}}(t=0){\\mathcal {M}}_{\\rm {Tot}}(t=0)\\rangle ", "18f76f7414cba8374e6f6ebcbb5a5009": "\\mathbb {Z} /12\\mathbb {Z} ", "18f771d524a5919810aaab8a58eb2e61": "\\mathbf {h} \\cdot \\mathbf {\\hat {y}} ", "18f78f003aa49ff8df3fefef866a7d90": "{\\boldsymbol {y}}\\in \\mathbb {R} ^{k},\\nu >0,\\lambda _{j}>0,\\mu _{j}>0", "18f799f827537f60dd46a8d256ba4282": "i=1,\\ldots ,m", "18f7d20ecf76be3a747691b7b3e1c8e2": "F_{e}-F_{w}\\,=0", "18f83fb5ad488b078566b61892a9ff40": "n-2", "18f89f56a678d37acced735a2d961dae": "k'_{x}=0.332{D_{AB} \\over x}Re_{x}^{1/2}Sc^{1/3}", "18f8a08cd35e5fed6565afbfaa1582b0": "L_{v}", "18f92f56f72a7d8ce15437db8d79ecc7": "\\theta >\\phi \\,", "18f9af01396d2d5def8b17a47882862e": "{\\begin{pmatrix}z&z\\\\z&z\\end{pmatrix}}{\\begin{pmatrix}z&-z\\\\-z&z\\end{pmatrix}}\\equiv z^{2}(1+j)(1-j)\\equiv z^{2}(1+\\varepsilon )(1-\\varepsilon )=0.", "18f9d0bcfb04da11e6954ec61ad78567": "V_{\\sigma ,\\sigma '}", "18f9d28289326992546aa45cdf816044": "k=1/(4\\pi \\epsilon _{0})", "18f9e1145b32ce66c8dfe53ba399373f": "\\scriptstyle \\phi ", "18fa9ab0c7793958866516b87ce04e30": "C_{S}(x)=C_{O}\\left(1-(1-k_{O})e^{-{\\frac {k_{O}x}{L}}}\\right)", "18fafbf8831652b47769858cd64a3235": "\\sum _{k}\\lambda _{k}=1", "18fbae7a8d395ca90ea7a0da7fd7333a": "\\pi (.)", "18fbf71efae89c1195cab30f475216d3": "={\\frac {1}{2}}{\\frac {m}{3}}{\\dot {x}}^{2}+{\\frac {1}{2}}Mv^{2}-{\\frac {1}{2}}kx^{2}-{\\frac {mgx}{2}}-Mgx", "18fc1da9ea816f51ddb2c56aee3a2430": "r\\in \\mathbb {F} _{d}", "18fc2dc5ac82ba13872c570add73a685": "\\displaystyle {P(z)=\\int _{0}^{2\\pi }{1+e^{-i\\theta }z \\over 1-e^{-i\\theta }z}\\,d\\mu (\\theta )}", "18fc7b2c943c858084c4b9086ba49eae": "a_{0}=1\\qquad b_{0}={\\frac {1}{\\sqrt {2}}}\\qquad t_{0}={\\frac {1}{4}}\\qquad p_{0}=1.\\!", "18fc9dc6ec01c64f80ea7a7614b5abce": "f\\left(a_{0},\\dots ,a_{n-1},s\\right)={\\begin{cases}0&\\mathrm {if} \\;\\forall iM_{N}\\,\\!", "1904d27aca1c555d389d975078847239": "A_{ij}=A_{ji}", "1904e29e549219c7d14cf955974e8ea8": "\\pm {\\sqrt {\\frac {1}{5}}}", "1904ea2370b07a7d979157c028604f2d": "{\\frac {\\partial n(x,t)}{\\partial t}}={\\frac {1}{2}}\\int _{0}^{x}K(x-y,y)n(x-y,t)n(y,t)\\,dy-\\int _{0}^{\\infty }K(x,y)n(x,t)n(y,t)\\,dy.", "1904ea7d67b03a8d978e8506424b0fc0": "J_{1}:=\\int _{\\Gamma }\\left(Wn_{1}-n_{j}\\sigma _{jk}~{\\cfrac {\\partial u_{k}}{\\partial x_{1}}}\\right)d\\Gamma ", "19051409c58bc32d1473bb47c5552740": "V(\\phi )=-10|\\phi |^{2}+|\\phi |^{4}\\,", "1905649b6ea7885554c4419ba187f092": "\\operatorname {Tot} (K)^{n}=\\bigoplus \\nolimits _{i-j=n}K^{i,j}", "190566692b1effdb3f849a990c73b854": "Au_{xx}+2Bu_{xy}+Cu_{yy}+\\cdots {\\mbox{(lower order terms)}}=0,", "1905ce7705139b9b165d23a6a3dd4e05": "\\mathbf {b} \\succ \\mathbf {a} ", "1905fc641acdb9bea8d26c943679261d": "b_{t+1}=\\eta ", "1906336c3f4e493cebc6eb9e18e6a89a": "{\\frac {2(2n+5)}{9n(n-1)}}", "190651a250b815e68dd858227c1fbc0e": "{\\underline {P}}(Cl_{3}^{\\geq })=\\{x_{7},x_{9},x_{10}\\}=Cl_{3}^{\\geq }", "190659faa50c094c619f88465cbd31c4": "Q_{\\text{cmb}}", "19065fa8b53dac2165c51708f2507fce": "G(\\mathbb {C} )", "19068b9ba15ee939ab3a1c69306fe055": "[A,A]=0", "1906aaaaa3e678ee299235fe2d6c9804": "\\scriptstyle \\leq 10^{-9}", "1906edafcbaf52ab3d49e826d207d413": "{\\frac {120}{90}}={\\frac {12}{9}}={\\frac {4}{3}}\\,.", "19070998a9b4872fdcaf8c07ec04e5ab": "g=h.", "19070d9f38190632818d17aee69b2ec3": "\\ (1/r)", "19073ae57c6ae431a149f51f528f2004": "\\exists f\\exists g\\forall x_{1}\\forall x_{2}\\phi (x_{1},x_{2},f(x_{1}),g(x_{2}))", "19078e0d21cf083774b72f8bed41493f": "P\\rightarrow \\infty ", "1907a6c841851b3db4cff1d67a62cc96": "(E(t)+E(t-\\tau ))^{2}", "1907b3d79cadae2f7bada01c2d62b4f6": "-{\\frac {\\eta (n_{\\eta }(\\xi _{1})-n_{\\eta }(\\xi _{2}))}{\\xi _{1}-\\xi _{2}}}", "190822760036f293ff506223b47df974": "s=x+iy", "19088f512b8bcc8ba258062ab4c63300": "\\$1.4339\\times {\\frac {(1+5\\%)}{(1+7\\%)}}=\\$1.4071", "190919bf666ac59775ab7bf011c5cb98": "={\\frac {15!}{2}}\\cdot \\left({\\frac {4!}{2}}\\right)^{14}\\cdot 4\\cdot {\\frac {64!}{2}}\\cdot 3^{63}\\cdot {\\frac {96!\\cdot 2}{2\\cdot (4!)^{24}}}\\cdot {\\frac {2^{95}\\cdot 64!\\cdot 2}{2\\cdot (8!)^{8}}}", "190919d2a19f2a0efff59a4f54a97ee7": "\\mathrm {d} U_{cv}=\\mathrm {\\delta } Q+\\mathrm {d} U_{in}+\\mathrm {d} (p_{in}V_{in})-\\mathrm {d} U_{out}-\\mathrm {d} (p_{out}V_{out})-\\mathrm {\\delta } W_{shaft}", "190934e3960941c01c649e915a7fc345": "C(s)={\\frac {rT}{1-e^{-rT}}}={\\frac {s}{1-e^{-s}}}", "1909510a27c3009912a5340daa6cc051": "\\zeta {\\Bigl (}{\\tfrac {1}{2}}+it{\\Bigr )};", "19095d5a3ebdf83a709a60458674391b": "Y_{m_{n}}={\\frac {Z_{m_{n}}}{k^{2}+Z^{2}-Z_{m_{n}}^{2}}}", "1909a078d1c60ee7523a86ecb8342601": "L_{b}", "1909a3f64512211ef691d7b75956252b": "\\sin ^{2}\\theta +\\cos ^{2}\\theta =1\\!", "1909d419aa966ab9ce8853170d3cf71c": "s\\cdot y=1", "190a24aeaa26edd193cde20ece6f649d": "\\operatorname {E} [L]=\\operatorname {E} [v^{K(x)+1}]-P\\operatorname {E} [{\\ddot {a}}_{\\overline {K(x)+1|}}]", "190a429490bdd22373bd5ad2cf1b3cf8": "\\cot \\theta ={\\frac {i(e^{i\\theta }+e^{-i\\theta })}{e^{i\\theta }-e^{-i\\theta }}}\\,", "190a6ae9bde93da07082c39175a31258": "\\scriptstyle P_{T}", "190a828a81b325eb6cf942ba638cd631": "T:x\\mapsto 1-x", "190a889bd2b8d23c6c72e0e02980111b": "\\simeq {\\begin{cases}\\mathbf {R} &{\\mbox{if }}k=0,n-1\\\\0&{\\mbox{if }}k\\neq 0,n-1\\end{cases}}", "190b448fa5759f21444a5c5cc9146475": "\\displaystyle \\|J_{n}^{-1}-J_{n-1}^{-1}\\|_{F}", "190b622c95192e0c61283b638e832c14": "SL^{\\pm }{\\frac {c}{2}}\\end{cases}}", "1916c232538093b85e75c7bbef2b9f1a": "E(Nl,t)=1-e2-(aNl,t)", "1916e884f16ba7cd98984c74753f9d93": "y(x,t)\\,=y_{0}\\cos(kx-\\omega t+\\varphi )", "1916edb7738d3b4298a20c3b8f9cd8f4": "{\\frac {1}{42}}", "19170f32cbe777c9a9cf2aec0a7bca7e": "f^{+}:X^{+}=Proj(\\oplus _{m}f_{*}({\\mathcal {O}}_{X}(mK)))\\to Y", "1917627925acb0bfc171301afe5ccaff": "\\liminf _{n\\to \\infty }{\\frac {p_{n+1}-p_{n}}{\\log p_{n}}}=0.", "191801332c4c8db223789478221abe61": "{\\frac {\\partial |\\mathbf {X^{\\rm {T}}} \\mathbf {A} \\mathbf {X} |}{\\partial \\mathbf {X} }}=", "19183bff564aea419754540d1e8985e7": "{\\frac {2^{n}}{n!}}\\mathrm {vol} (\\mathbb {R} ^{n}/\\Gamma )\\leq \\lambda _{1}\\lambda _{2}\\cdots \\lambda _{n}\\mathrm {vol} (K)\\leq 2^{n}\\mathrm {vol} (\\mathbb {R} ^{n}/\\Gamma ).", "19188b5d7016d022def74c01e8eab114": "p_{i}=x_{i}/N", "1918a9736f6b21d6b9a70d98a215d27a": "m_{L}={\\frac {m_{0}}{\\left({\\sqrt {1-{\\frac {v^{2}}{c^{2}}}}}\\right)^{3}}},\\quad m_{T}={\\frac {m_{0}}{\\sqrt {1-{\\frac {v^{2}}{c^{2}}}}}}", "1919805b955ddc40ad3a8d6a4a702dc5": "{\\mathit {He}}_{n}'(x)=n{\\mathit {He}}_{n-1}(x),\\,\\!", "191a0ee133be76689375af2af2dadd8f": "\\Gamma _{ij}^{k}=(e_{i}\\cdot De_{j})\\cdot e^{k}", "191a357f585c58425ce783a3329006e5": "C_{n}(x_{1},x_{2},\\ldots ,x_{n}):=\\left\\langle \\phi (x_{1})\\phi (x_{2})\\ldots \\phi (x_{n})\\right\\rangle ={\\frac {\\int D\\phi \\;e^{-S[\\phi ]}\\phi (x_{1})\\ldots \\phi (x_{n})}{\\int D\\phi \\;e^{-S[\\phi ]}}}", "191a92911a45010610972b5f343df714": "df=1", "191ad2ce69d3206c7675b8406b3c1009": "\\iiint _{T}f(x,y,z)\\ dx\\,dy\\,dz=\\iint _{D}\\int _{\\alpha (x,y)}^{\\beta (x,y)}f(x,y,z)\\,dzdxdy", "191ad5bc4dec318a1d9486075e1e0bd8": "\\tan(\\gamma +\\pi /2)=1/\\tan \\gamma ", "191adff20517fd0a0a85a18812b5e379": "\\lambda d", "191ae49db56239410cd4378e1b545d62": "|\\eta |.", "191bfc8b6e4ae7102318121c0d4a170e": "L=\\int _{S}F_{\\rm {rad}}\\cdot dS=\\int _{S}{\\frac {c}{\\kappa }}\\nabla \\Phi \\cdot dS\\,.", "191c64bdae5c35fed2281891274cf4c8": "\\pi _{2}=\\pi _{5}=\\mathbb {Z} ", "191c7ad983ddfff9ba73849ded89a9d9": "g_{1mm'}", "191cb8b106b45d57bea9f2bd41dda222": "{du}/{dt}=F(u(t))", "191d0544d85e6b3395250a9665f25ebe": "u\\equiv _{L}v", "191d670b6482c653d233c8d77011a2b3": "\\mathrm {FWHM} \\,=d_{\\mathrm {F} }\\mathrm {ln} (2)\\approx 0.693\\,d_{\\mathrm {F} }...........(22)", "191d9ce5e8e1141122738cb31a576cc2": "A={\\frac {1+{\\frac {1}{4}}(k_{1})^{2}}{1-k_{1}}}", "191e06be3e8e4dbd08734eaa8834d317": "\\mathbf {r} ={\\frac {a}{1+e\\cos \\theta }}\\mathbf {\\hat {r}} \\,\\!", "191e49aac76695b80d44c2ad107aad93": "Decrypt_{\\varepsilon }", "191e94a2627524f3e4f21ffac612b31d": "{\\begin{array}{lcl}minimize:V({\\vec {w}},{\\vec {\\xi }})={1 \\over 2}{\\vec {w}}\\cdot {\\vec {w}}+C_{onstant}\\sum {\\xi _{i,j,k}}\\\\s.t.\\\\{\\begin{array}{lcl}\\forall \\xi _{i,j,k}\\geqq 0\\\\\\forall (c_{i},c_{j})\\in r_{k}^{*}\\\\{\\vec {w}}(\\Phi (q_{1},c_{i})-\\Phi (q_{1},c_{j}))\\geqq 1-\\xi _{i,j,1};\\\\...\\\\{\\vec {w}}(\\Phi (q_{n},c_{i})-\\Phi (q_{n},c_{j}))\\geqq 1-\\xi _{i,j,n};\\\\where\\ k\\in \\left\\{1,2,...n\\right\\},\\ i,j\\in \\left\\{1,2,...\\right\\}.\\\\\\end{array}}\\end{array}}", "191ead2c11f43c73e7f3970b0b1d201a": "R>r+a", "191f030411bd38b4955c373b2820842b": "\\int _{V}{\\boldsymbol {\\epsilon }}^{T}{\\boldsymbol {\\sigma }}^{*}dV", "191f1a8df597bd192c07369a641e91a3": "(A,\\sigma )", "191f7020f5bf3a87b2024dcdd6b90c22": "M_{0}\\equiv \\int d\\zeta \\ {\\frac {\\lambda (\\zeta )}{\\zeta }}", "191f7940711bd6730aefc9a3e3ef9fb6": "\\varepsilon _{Y}=\\mathrm {id} _{TY}.", "192040a481a02c91f4175743c4c1553b": "p=a-b(x+y)", "1920cd19c04008439b17cc72788dcfb0": "{\\mathbf {U}}(x_{0},y_{0})={\\frac {1}{j\\lambda }}\\int \\!\\int {\\mathbf {U}}(x_{1},y_{1}){\\frac {e^{jkr_{01}}}{r_{01}}}\\cos \\theta dx_{1}dy_{1}", "19211978ba7e1567773c82d4c6bedcd8": "S(-1)^{\\oplus n+1}\\to S,e_{i}\\mapsto x_{i}", "1921660d6bbf2dc4ecd5c78858e3be04": "\\operatorname {E} ({\\tilde {Y_{k}}}-{\\widehat {Y_{k}}})=0.", "192175bb9483bf2588cd1ea6eea6bc13": "\\cos B=-\\cos C\\,\\cos A+\\sin C\\,\\sin A\\,\\cos b,", "19218cf103615515c67b3c4e989541f3": "\\max w\\left({T_{ij}}\\right)={\\frac {T!}{\\prod _{ij}{Tij!}}}", "1921bcba2403434dd77df374f622ea16": "\\alpha _{k}", "1921d0566ae938d2ffd6a9d062956775": "\\{h_{n}\\}_{n\\geq 0}", "1921fa2bfe99382ee4140f2d7caf0491": "\\alpha _{i}\\leftrightarrow \\beta _{i}", "192201d9ff83fdf71c35bf91e9de4f54": "{\\dot {\\mathbf {x} }}=\\varphi (\\mathbf {x} ,t)", "19222ddb0696c213078a8cb0948c63b7": "(i_{\\alpha }t)^{i_{1}\\dots i_{r-1}}=r\\sum _{j=0}^{n}\\alpha _{j}t^{ji_{1}\\dots i_{r-1}}.", "19222fec8f1b6be85309fe2382449484": "V\\otimes W", "19223e2e77817e86e125e9328b46017a": "ml{\\ddot {\\theta }}=-mg\\sin \\theta -kl{\\dot {\\theta }}", "192250ef22b00eabb5ff7b832d5236de": "B^{(n)}", "192264f8d32bb758e69f9279d91e4afc": "F(t')\\!", "19228d0eb1ab46c3dcfd608496f2ea2d": "{\\frac {D^{2}}{dt^{2}}}J(t)+R(J(t),{\\dot {\\gamma }}(t)){\\dot {\\gamma }}(t)=0,", "192292e35fbe73f6d2b8d96bd1b6697d": "lm", "1922b8e22bae729ffa04ea46d29ac5b8": "L^{1}", "1922c26b309cf406ae462f001364c213": "L(t_{1}',\\ldots ,t_{n}'):-B", "1922cd45d2eba9f6c9b03187b517e51c": "\\sigma _{K}^{2}=\\sigma _{\\beta _{12}}^{2}+\\sigma _{\\beta _{13}}^{2}-2\\sigma _{\\beta _{12}}\\sigma _{\\beta _{13}}\\rho _{12,13}\\,", "1923029fbd6f645959ca421259dd04d6": "E_{2n}\\left({\\frac {p}{q}}\\right)=(-1)^{n}{\\frac {4(2n)!}{(2\\pi q)^{2n+1}}}\\sum _{k=1}^{q}\\zeta \\left(2n+1,{\\frac {2k-1}{2q}}\\right)\\sin {\\frac {(2k-1)\\pi p}{q}}", "19233bae16418436417653d8c79a8df2": "F(x_{1}+\\Delta x)-F(x_{1})=\\int _{a}^{x_{1}+\\Delta x}f(t)\\,dt-\\int _{a}^{x_{1}}f(t)\\,dt.\\qquad (1)", "192340153c5bcd6aa2a6d178e3bf2492": "G=SO(3)/U(1)", "1923533c88582e19945e82caa101e43d": "{\\hat {a}}_{i}^{(\\eta )}={\\frac {x_{i+}}{\\sum _{j}{\\hat {b}}_{j}^{(\\eta -1)}}},", "19235f840cf32d786c6c2994e209068f": "{\\overset {\\cdot }{x}}=f(x)+B(x)u+h(x)", "19236c254f82397d4ae5c7cbce41b945": "a_{1}=-\\sum _{n=2}^{\\infty }a_{n}.", "19238f4bc02e570e0307d962e220ed03": "H={\\frac {{\\dot {\\theta }}^{2}}{2}}-{\\frac {g}{l}}\\cos \\theta .", "19243a0646263bd27c594dca33db048b": "yxyx\\rightarrow x^{2}y^{2}", "19246823504bc4b4c31674c0fa4b02d6": "R_{0}=0~{\\mbox{since}}~0.0078\\ >=0.02", "192481a8c43bb58d917346eff39e3ad3": "\\scriptstyle {\\frac {\\pi }{2}}", "19249588a3dcaa56d833265b84881d9e": "{\\frac {1}{\\ln(p)}}={\\frac {1}{p-1}}+\\int _{0}^{\\infty }{\\frac {1}{(x+p)(\\ln ^{2}(x)+\\pi ^{2})}}dx\\qquad \\qquad \\forall p>1", "1924b6c9f38d8780fa413e43d85312db": "D=R{\\sqrt {\\theta _{1}^{2}\\;{\\boldsymbol {+}}\\;\\theta _{2}^{2}\\;\\mathbf {-} \\;2\\theta _{1}\\theta _{2}\\cos(\\Delta \\lambda )}};{\\color {white}{\\frac {\\big |}{.}}}\\,\\!", "1924d321b3befac1eb6225d40c28a06c": "V_{\\text{out}}=-(V_{1}+V_{2}+\\cdots +V_{n})\\!\\ ", "19250d87521b8d914daa572fce49e8ba": "{\\tilde {\\phi }}=\\phi -\\phi _{0}", "19253dfa5cc89774dd1d17fc77625115": "\\langle \\mu \\nu |\\lambda \\sigma \\rangle =\\iint \\mathbf {\\chi } _{\\mu }^{A}(1)\\mathbf {\\chi } _{\\nu }^{B}(1){\\frac {1}{r_{12}}}\\mathbf {\\chi } _{\\lambda }^{C}(2)\\mathbf {\\chi } _{\\sigma }^{D}(2)d\\tau _{1}\\,d\\tau _{2}\\ ", "19255638ed5bbbfc2dd2f00ac6437b4a": "\\left({\\frac {\\partial U}{\\partial \\theta }}\\right)-{\\frac {d}{dz}}\\left({\\frac {\\partial U}{\\partial \\left({\\frac {d\\theta }{dz}}\\right)}}\\right)=0", "19259731a0d9038d4878b99f240120b6": "e_{1},\\ldots ,e_{k}", "1925a6ecea62e67108e05837ad42357c": "T_{c}=\\beta ^{-1}", "1925b1bdcaf9d8e96499b51263835338": "\\left\\|A\\right\\|_{p}=\\max \\limits _{x\\neq 0}{\\frac {\\left\\|Ax\\right\\|_{p}}{\\left\\|x\\right\\|_{p}}}.", "1926312deeeff81b8a4fb89ee30f48e5": "1=1^{n}=(p+q)^{n}=\\sum _{k=0}^{n}{n \\choose k}p^{k}q^{n-k}.", "19263a9a067a890e402d5cd545860a1a": "V^{\\infty }", "1926d31e091d7dc30b2c9a3702537101": "U_{\\texttt {name}}\\,", "19272339c815dc47eef0ffe51ab54812": "~|n,\\pm \\rangle ~", "19275a041eee84dfb0180bb6eddef687": "\\lambda ^{n}f(a,b,c)+\\mu \\lambda ^{n-1}\\Delta _{P}f(a,b,c)+{\\frac {1}{2}}\\mu ^{2}\\lambda ^{n-2}\\Delta _{P}^{2}f(a,b,c)+\\dots .", "192780821efae6755e9477fdd4a39223": "f:a\\to b", "1927858e73c05099c08b94db09efef41": "P_{i}=A_{i}\\oplus B_{i}", "19278974a2304842ecd802e7cf16bb3b": "|\\psi ^{T}\\rangle ", "1928392ac0b4a77dd9a968dc6c69cf32": "\\alpha _{R}={\\frac {138}{D}}", "1928c3c876fb36e5e7c18d9a0e1fb117": "k_{\\mathrm {B} }=1\\ ", "1928ccf03ae5338f733965d62b8770c5": "{d\\theta \\over dt}={\\sqrt {{2g \\over \\ell }\\left(\\cos \\theta -\\cos \\theta _{0}\\right)}}", "1928dc58254957913034062e2de8eb3d": "{n! \\over (k-1)!(n-k)!}u^{k-1}(1-u)^{n-k}\\,du+O(du^{2}),", "1928efa279cbaedaac2e470d954032c6": "{\\frac {\\partial ^{\\ell }f_{i}}{\\partial x_{i_{1}}^{\\ell _{1}}\\partial x_{i_{2}}^{\\ell _{2}}\\cdots \\partial x_{i_{n}}^{\\ell _{n}}}}", "1928f2d87af684d81f758680332b2a40": "d[x,x^{*}]=\\max _{a}|u(a)-u^{*}(a)|\\,", "1929052caef667d98743df8761d507b9": "f_{i}(0,0,\\dots ,0)=0", "19298983a27eeab314d1beea9b7d3dde": "\\left[(\\mathbf {AB} )^{\\dagger }\\right]_{ij}=\\left[\\left(\\mathbf {AB} \\right)^{\\star }\\right]_{ji}=\\sum _{k}\\left(\\mathbf {A} ^{\\star }\\right)_{jk}\\left(\\mathbf {B} ^{\\star }\\right)_{ki}=\\sum _{k}\\left(\\mathbf {A} ^{\\dagger }\\right)_{kj}\\left(\\mathbf {B} ^{\\dagger }\\right)_{ik}=\\sum _{k}\\left(\\mathbf {B} ^{\\dagger }\\right)_{ik}\\left(\\mathbf {A} ^{\\dagger }\\right)_{kj}=\\left[\\left(\\mathbf {A} ^{\\dagger }\\right)\\left(\\mathbf {B} ^{\\dagger }\\right)\\right]_{ij}", "19298c2af784855a55e7c066f70bf605": "a^{2}+b^{2}=a^{2}-(ib)^{2}=(a-ib)(a+ib)", "19299660df53a56963d814791baf42ed": "F=G{\\frac {m_{1}m_{2}}{r^{2}}}", "192a40b891d9c20925aeb2261dbdea4f": "S_{0}(P,Q)=\\cdots =S_{d-1}(P,Q)=0.", "192a427f362d1037d7df4509745a8470": "\\int D\\rho \\;\\delta \\left[\\rho -{\\hat {\\rho }}\\right]F\\left[\\rho \\right]=F\\left[{\\hat {\\rho }}\\right],\\qquad (4)", "192a762853fa5f186f124bf445aca1ad": "H(A):=-Tr\\rho _{A}\\log \\rho _{A}", "192a7808925538a267466b5c6916f851": "y(t)=A\\sin(kx-\\omega t)", "192a842434fcee9247b04972d6c6858c": "{\\mathcal {L}}_{\\mathrm {EW} }=\\sum _{\\psi }{\\bar {\\psi }}\\gamma ^{\\mu }\\left(i\\partial _{\\mu }-g^{\\prime }{1 \\over 2}Y_{\\mathrm {W} }B_{\\mu }-g{1 \\over 2}{\\boldsymbol {\\tau }}\\mathbf {W} _{\\mu }\\right)\\psi ", "192a8ca6c41cfdd77c52d52ab92e3641": "\\sum _{j=0}^{p}c_{j,k}z^{j}=[\\operatorname {Res} _{z=\\lambda _{k}}f(z)]\\sum _{j=0}^{p}{\\frac {1}{\\lambda _{k}^{j+1}}}z^{j}", "192ad965c4815ef67c86ed1460766ddb": "\\min _{i=1...n}({\\frac {S_{i}^{T}}{S_{i}^{0}}}).", "192af400fa1b0725c03fbac9e698abdb": "|-\\rangle ={\\tfrac {1}{\\sqrt {2}}}\\left(1,-1\\right)", "192bbaede1fdef1a7b0145d11040ad4a": "{\\begin{matrix}\\underbrace {\\begin{matrix}\\left[{\\begin{matrix}{\\frac {\\partial f}{\\partial x_{1}}}&{\\frac {\\partial f}{\\partial x_{2}}}&...&{\\frac {\\partial f}{\\partial x_{N}}}\\end{matrix}}\\right]\\\\{}\\\\\\end{matrix}} _{\\nabla f^{T}}&\\underbrace {\\begin{matrix}\\left[{\\begin{matrix}v_{x_{1}}\\\\v_{x_{2}}\\\\\\vdots \\\\v_{x_{N}}\\\\\\end{matrix}}\\right]\\\\{}\\\\\\end{matrix}} _{v}&=\\,\\,0\\\\\\end{matrix}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{\\text{is the same as writing}}\\,\\,\\,\\,\\,\\,\\,\\,\\nabla f^{T}\\,\\,\\,\\centerdot \\,\\,\\,\\,v\\,\\,=\\,\\,\\,0.", "192bcd2ad930494b6485f1216526656a": "f(x_{i})\\neq y_{i}", "192bf38ad826c8b799596aa95acb4195": "A{\\frac {dp}{dt}}=L_{p}p.", "192c1b4c97e7cf16034ba365cf8c55a7": "U(t_{k}+\\delta t)", "192c442ae99c13b6c7440ac3e550cad3": "{\\frac {d}{dx}}\\log _{b}(x)={\\frac {1}{x\\ln(b)}}.", "192c572fde9511263584e5b69e4af96e": "\\mathbf {X'} =\\mathbf {wX} ,\\mathbf {y'} =\\mathbf {wy} .\\,", "192c5b4983f3b8987a93c4b25b4842ce": "\\mathbf {N} -1", "192c6d15ee502fdbb42d32464b038da0": "y\\wedge x/y=0=x/y\\wedge y", "192c88685e3c0bc9cc206a7752ceffb6": "\\Omega =2\\pi -2n\\arctan \\left({\\frac {\\tan {\\pi \\over n}}{\\sqrt {1+{r^{2} \\over h^{2}}}}}\\right)", "192cb49bf467fb27b0f009d38960a20b": "H_{0}:\\ m(\\theta _{0})=0", "192cdf9f9b1215e11fd8fc45c007e6c2": "|k\\rangle ", "192d3b3e09258b9b77a60113b4ec1b73": "{\\frac {dx}{dt}}(t_{n})=\\sum _{k=-q}^{q}{\\frac {-i2\\pi k}{T}}X_{k}e^{\\frac {-i2\\pi kt_{n}}{T}},\\quad n=1,\\dots ,N.\\,", "192d4f10a3ad648108d7b6dfc9660d26": "\\lambda \\sigma _{i}+(1-\\lambda )\\sigma '_{i}\\in r(\\sigma _{-i})", "192d5d177f953dbc5942e7947ff67dde": "\\pi F", "192d7d5c5ca46a19f63389a3dd8ee20d": "t=t^{\\prime }=0", "192d84954d9750750c14b37c8f52a887": "S:f\\rightarrow u", "192db13978a4a8572e8c06886ebfa144": "d_{W}", "192db2794b64f15ea81817ea9ff30487": "I_{E}=(\\beta +1)I_{B}+(\\beta +1)I_{CBO}\\,", "192e1f16c897d48f16b1c807064d482a": "\\mu _{z}\\,\\,=\\,\\,\\,{\\rm {E}}\\,[z]\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\sigma _{z}^{2}\\,\\,\\,=\\,\\,\\,{\\rm {E}}\\,\\left[{\\left({z\\,\\,-\\,\\,\\mu _{z}}\\right)^{2}}\\right]", "192e3ceb138349886f6d2f79a846333f": "\\sum F_{\\|}=F_{i}-F_{f}-F_{w}\\sin \\theta =0\\,", "192e6e81db824c6662019be0ce399ea5": "\\left({\\frac {x}{\\lambda }}-f\\ t\\right)=\\left({\\frac {x+\\Delta x}{\\lambda }}-f(t+\\Delta t)\\right)\\ ,", "192ed0507b1fccca48b2aa716d2dc671": "\\textstyle |\\Omega \\rangle ", "192eea1bb2e8cf134bcc35715e0c45e7": "\\Delta E={\\frac {\\beta ^{2}}{1-\\beta ^{2}}}eE_{0}l_{s}e^{z/l_{s}},", "192ef223ed254490b3223bd3f76f791d": "[x^{-1},y]=[y,x]^{x^{-1}}.", "192efa85fde664088378d0d93bcbf5fb": "DPW={\\frac {\\displaystyle \\pi d^{2}}{4S}}-0.58^{*}{\\frac {\\displaystyle \\pi d}{\\sqrt {S}}}", "192f0c0fcb2a45e70b1f15fb29ac9a51": "F\\left({\\dot {x}},x,y,t\\right)=0", "192f379152753d0f6e087e0eb792c051": "x^{2}-{\\frac {1}{2}}", "192f50ec41d691af8f3c88111d47ae56": "\\phi =+1", "19309f2cc286af9ef849b241ca24622f": "y_{k}={\\frac {1}{\\sqrt {N}}}\\sum _{j=0}^{N-1}x_{j}\\omega ^{jk}.", "1930a0a422d84cad426361a5a881cd04": "\\Rightarrow \\left\\|Av_{i}\\right\\|^{2}=\\lambda _{i}\\left\\|v_{i}\\right\\|^{2}", "1930dfe3b9d903d5f359c763a91870a5": "\\Delta E=E_{1}-E_{2}=\\left(y_{1}+{\\frac {q^{2}}{2gy_{1}^{2}}}\\right)-\\left(y_{2}+{\\frac {q^{2}}{2gy_{2}^{2}}}\\right)={\\frac {(y_{2}-y_{1})^{3}}{4y_{1}y_{2}}}", "19310380597f85e281e904af33e77de0": "\\iota :F\\longrightarrow E", "193117293445f89d422795f3e4fdc6b7": "{\\textit {mother}}:{\\textit {animal}}\\longrightarrow {\\textit {animal}}", "19315b8b6eb6ac0c2eb5b79ae702ed70": "\\{(x,y):|x|=|y|\\}", "193161db9089f52be260b4ece5c1cfe7": "J_{1}:=\\int _{-h}^{h}\\rho ~dx_{3}=2~\\rho ~h~;~~J_{3}:=\\int _{-h}^{h}x_{3}^{2}~\\rho ~dx_{3}={\\frac {2}{3}}~\\rho ~h^{3}", "1931882a1554796a61e241aae439cce7": "\\mathbf {\\Lambda } (\\varphi _{1}+\\varphi _{2})=\\mathbf {\\Lambda } (\\varphi _{1})\\mathbf {\\Lambda } (\\varphi _{2})", "1931c7a7475c79456708bac13d9513e6": "~\\omega _{2}", "1931d9f57487f77009123860cc6dc57e": "\\sum _{i=0}^{n}\\lambda ^{n-i}\\left[d(i)-\\sum _{l=0}^{p}w_{n}(l)x(i-l)\\right]x(i-k)=0\\qquad k=0,1,\\cdots ,p", "193203bdd11eec56406cb25632951832": "Z_{1,t}>Z_{2,t}", "1932680bf99159dc4c21366f34de8091": "x^{2}-d(y+1)^{2}=1", "19329487fb29679f9be6282392a0d113": "SL_{3}(\\mathbb {Z} )", "1932e200fb070fed86e65d1af49c1d3c": "e^{-{\\frac {\\Delta G_{F}}{k_{B}T}}}=[V_{\\mathrm {Mg} }'']^{2}", "1932fe953f3822e6b0ca4ac8a5fff2e6": "\\epsilon (\\lambda _{ex})", "19338d854f9df8e21ee89b3484591c89": "\\varphi :V\\otimes V^{*}\\rightarrow \\varepsilon _{H}", "1933bc7f5f194c785408953539480ba1": "a_{p}^{N-1}\\equiv 7^{11350}\\equiv 1{\\pmod {11351}}", "1933f6b65b56a3fbe1ffbae527e7ab61": "\\eta ={\\frac {9}{2}}{\\frac {1}{(1-{\\frac {a^{3}}{b^{3}}})\\mu }}", "1933fef2f568bd948d257717ac87df5b": "p_{1}^{2}=m_{1}^{2}", "19340c3322f86f511a91926308c71a92": "(m+n)r=mr+nr", "19346b8e34b57b09a0bb4f4742e291fb": "s_{i}=S{\\bmod {\\ }}m_{i}", "1934bd15286c328fb04bd0ad2668a7d7": "(B\\oplus C)", "1934eebebfb7e103a4578459790df54c": "dl^{2}=r_{0}^{2}(d\\theta ^{2}+\\sin ^{2}\\theta \\,d\\phi ^{2})", "193585327f6db8ff0db3628dc7987aee": "\\beta ={\\frac {1}{k_{B}T}}={\\frac {1}{\\tau }}", "193609b18eb606ed7fde71157f573e0d": "\\int _{X}\\|f\\|_{B}\\,d\\mu <\\infty .", "193638893fb777e821eaa1d468ec3caa": "[T_{A}^{1}]\\longrightarrow ^{*}T_{B}^{2}", "193651e2eb979ea5e5a24d8f84d0f5fe": "Z_{\\mathrm {p} }=sL_{\\mathrm {p} }+R_{\\mathrm {p} }+{1 \\over sC_{\\mathrm {p} }}", "1936c4ad5c8f3b4a913024eca19e5c2c": "{\\mathfrak {e}}_{7}(\\mathbf {K} )", "1936f0f8638c6b51a4687ce25546a6b8": "week(date)=\\left\\lfloor {\\frac {ordinal(date)-weekday(date)+10}{7}}\\right\\rfloor ", "19376b55f56f17c4cf6bf16b7a7e77b2": "1/28+1/14+1/7+1/4+1/2+1/1=2", "1937807a199f91da25898e2f21180d27": "{\\begin{aligned}\\lambda _{12}&=\\omega _{12}-f\\sin \\alpha _{0}\\int _{\\sigma _{1}}^{\\sigma _{2}}{\\frac {2-f}{1+(1-f){\\sqrt {1+k^{2}\\sin ^{2}\\sigma '}}}}\\,d\\sigma '\\\\&=\\omega _{12}-f\\sin \\alpha _{0}I(\\sigma _{1},\\sigma _{2};\\alpha _{0}).\\end{aligned}}", "1937b3e880c7debe7b1335b5d669f54e": "{\\frac {\\partial (x,y)}{\\partial (\\rho ,\\phi )}}={\\begin{vmatrix}\\cos \\phi &-\\rho \\sin \\phi \\\\\\sin \\phi &\\rho \\cos \\phi \\end{vmatrix}}=\\rho ", "1937caa6d015eda45795912a693a7967": "\\Delta x^{0}=-{\\frac {g_{0\\alpha }\\,dx^{\\alpha }}{g_{00}}}\\equiv g_{\\alpha }\\,dx^{\\alpha }.", "1938902477bca0776aa4829c01ae6a81": "\\operatorname {mr} (G)=\\operatorname {mr} (H)", "1938b1b88119b0925c537b7ecd09ed5d": "\\epsilon _{i}=P(X_{i}=1)-P(X_{i}=0),", "19392441d05c74603ef22c51c9fdb551": "B_{v}=\\log _{2}", "193a066f6522c2f2057198d112fbf80f": "N={\\frac {\\Delta }{2\\pi }}", "193a115c37779fb02577488534b775b2": "00\\}", "1944c8b606cd7e825933e899045e349b": "{\\tilde {a}}_{j,k}=a_{j,k}^{*}", "19452467807fd0cc6c5e1acb5671540b": "P_{r}=P_{t}({\\frac {\\lambda G}{4\\pi d}})^{2}\\times (1-e^{-j\\Delta \\phi })^{2}", "1945549fffae6f3522223c949427b214": "{\\mathbb {C}}", "1945592150dc57ce70dc093104426983": "e^{0}=1", "19455ac5c29b2010ec728029dd68f587": "E_{m}\\rho _{j}E_{n}^{\\dagger }=\\sum _{k}B_{m,n,j,k}\\rho _{k}", "194597a5efd6c389627376ec2fde5cc4": "h(a)=-2b", "19466c1163ddead1f4f9b2441b1cd57d": "j_{elec}^{sat}=j_{ion}^{sat}{\\sqrt {m_{i}/\\pi m_{e}}}=j_{ion}^{sat}\\left(24.2*{\\sqrt {\\mu _{i}}}\\right)", "1946a849341f615050b658e939636ace": "|\\phi _{2}\\rangle ", "19474d0e6e5389c1119755dec508681a": "P{\\text{(hp)}}={\\tau {\\text{(in}}{\\cdot }{\\text{lbf)}}\\times f{\\text{(rpm)}} \\over 63{,}025}", "1947725787968c88314154082ac2060f": "{\\frac {76852}{(1+0.10)^{10}}}", "194796193aeeb8e0d4221dd12a3a4c43": "\\mathrm {P} _{A}", "1948050f2614803a4f4404c0b09fd2ed": "\\lim _{N\\to \\infty }{\\frac {1}{N}}\\ln P(M_{N}>x)=-I(x).", "1948598529f6ea51a587de55a057214a": "{\\omega ^{0}}_{3}=0", "19486cb2649ab19b4ef58d56fd1989df": "{Si(\\pi )}", "19486e37c9a977b1170872f33d11984e": "c_{BE}", "19486f2687c77e4a3415727ae5eb1192": "k0", "1948f70653bd15f1433732002bcd31b8": "B_{\\lambda }(T)={\\frac {2hc^{2}}{\\lambda ^{5}}}{\\frac {1}{e^{\\frac {hc}{\\lambda k_{\\mathrm {B} }T}}-1}}", "19491c8f005eda8dfb0274fd15c3f2ca": "{\\text{Step 2}}", "194953e30316e24fb4420d76b37cafcd": "\\delta /\\pi ", "1949656b6054b1d3db99ae022030f65e": "\\pm {\\frac {\\sqrt {\\csc ^{2}\\theta -1}}{\\csc \\theta }}\\!", "1949dfce6c0997eb45e54c5177fc303b": "Y=3K(1-2\\nu )", "1949fa95340d1b2ad877c66b6ec79483": "{t=t_{0}}", "194a106512985b68e4eac3327944e664": "{\\overline {BC}}", "194a627a74b816b929fc0317031eb63a": "f:M\\to M.", "194a89c2824f82bb0652f01ba6846eb7": "V={\\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\\\a_{21}&a_{22}&a_{23}&a_{24}\\\\a_{31}&a_{32}&a_{33}&a_{34}\\\\a_{41}&a_{42}&a_{43}&a_{44}\\end{bmatrix}}={\\begin{bmatrix}A_{1}&B_{1}\\\\C_{1}&D_{1}\\end{bmatrix}}", "194a90472811968f7abeb8820178108b": "\\langle A,\\oplus ,0\\rangle ", "194ab452466ff815b5ff05275f83b9aa": "\\zeta (-1),\\zeta (-3),\\zeta (-5),\\ldots ,\\zeta (-m)", "194aba5d80daa7cd28cc1d364e9404b5": "s_{a}(t)\\,", "194abc078116f951908caa1ccd7c0ac7": "V[e_{k}]=[e_{k+1}]\\quad {\\mbox{for}}\\quad k=0\\ldots n-1.", "194b273f891dd740717bbf358c22f4af": "H_{n}(x)=(-1)^{n}e^{x^{2}}{\\frac {d^{n}}{dx^{n}}}e^{-x^{2}}\\,\\!", "194b3067b7171e89f6633b9860173a1a": "x\\in \\mathbb {R} ^{2}", "194ba0016440f9cec26a48a3fe2dbc6b": "{52 \\choose 2}{50 \\choose 2}\\div 2=812,175", "194bcfe56baf8ce108926fee5db3e6f4": "A\\parallel _{+}P\\parallel _{-}B", "194c770a6fe09b0e9562c6eba2723cda": "{\\hat {\\theta }}={\\frac {F^{-1}({\\hat {q}}_{2})\\tau _{1}-F^{-1}({\\hat {q}}_{1})\\tau _{2}}{F^{-1}({\\hat {q}}_{2})-F^{-1}({\\hat {q}}_{1})}},\\quad F(x)={\\frac {1}{\\sqrt {2\\pi }}}\\int \\limits _{x}^{\\infty }e^{-v^{2}/2}dw", "194c894c9acf73e811a69934bfa217dc": "\\|f_{n}(x)\\|_{B}\\leq g(x)", "194d0881f0cc91741a4da51dc881bf3d": "x_{i+1}-x_{i}=h\\ \\forall i", "194d38ed3c7df173d586ecf3b2a71044": "\\sin \\theta ={\\frac {H}{L}},", "194d9d1ae7e135a37ece8c0f3b3f2368": "\\mathbb {C} ^{n}", "194dcc7caa45b56e1ab889f01de707a4": "\\operatorname {R} (T)=\\operatorname {R} (T_{0})(1+\\alpha \\Delta T)", "194e3a185efd76ee4c356ad1f1abca0d": "X(f)\\ ", "194e41a4599c3d829537b3d0c9c4dc46": "={\\frac {q_{0}}{q_{1}}}exp(-\\beta \\Delta U){\\frac {\\int ds^{N}exp(-\\beta U_{0})\\delta (U_{1}-U_{0}-\\Delta U)}{q_{0}}}={\\frac {q_{0}}{q_{1}}}exp(-\\beta \\Delta U)p_{0}(\\Delta U)", "194e447c7b2e57806a89eb161999446a": "X\\succeq 0\\Leftrightarrow A-BC^{-1}B^{T}\\succeq 0", "194e544049e383bc2b78c97f145f763f": "\\Delta f(x,y)\\approx {\\frac {f(x-h,y)+f(x+h,y)+f(x,y-h)+f(x,y+h)-4f(x,y)}{h^{2}}}.", "194ea13262c0693cded4e399f0ac7681": "=\\ b_{ij}(-1)^{i+j}|M_{ij}|,", "194ecc34fa7f41fbd809e664212047df": "\\|d\\mathbf {X} \\|^{2}=dX^{\\mu }dX_{\\mu }=c^{2}d\\tau ^{2}=ds^{2}\\,,", "194efeb51dc3cb04870fe56983cc97b4": "\\left[{\\hat {A}},{\\hat {B}}\\right]\\psi =0,", "194f0320e13b47803e90b2dd736e5737": "\\log ^{+}(x)=\\max\\{0,\\log(x)\\}", "194f57407da821bac52ed80cd55c385c": "\\,^{z_{2}=x_{2}y_{1}+x_{1}y_{2}+x_{4}y_{3}-x_{3}y_{4}+x_{6}y_{5}-x_{5}y_{6}-x_{8}y_{7}+x_{7}y_{8}+u_{2}y_{9}+u_{1}y_{10}+u_{4}y_{11}-u_{3}y_{12}+u_{6}y_{13}-u_{5}y_{14}-u_{8}y_{15}+u_{7}y_{16}}", "194f7c71cfbe161d5653093f67932e6b": "{\\mathcal {H}}={\\mathcal {H}}(p,q,t)=\\langle p,{\\dot {q}}\\rangle -L(q,{\\dot {q}},t)", "194f991f3d55ccd3cb374059e0434a39": "(Out)\\quad m[\\;n[\\;out\\ m.A\\mid A'\\;]\\mid {\\overline {out}}\\ m.B\\mid B'\\;]\\Rightarrow _{amb}n[\\;A\\mid A'\\;]\\mid m[\\;B\\mid B'\\;]", "194fd5eb705b174f78cea7e27d58dbe8": "A^{*}A\\,=\\,(QR)^{*}(QR)\\,=\\,R^{*}Q^{*}QR\\,=\\,R^{*}R", "195003a6255123ad9e0a1941ed6dc0d5": "y=\\sum _{r=0}^{\\infty }a_{r}(x-1)^{r+c},", "195009052b7cbca0104a915b7bb7563f": "\\gamma =\\gamma ", "19500bf5729221abf1c8fad5c45f16bb": "{\\dot {V}}(\\mathbf {x} )", "19503ee8ccd1f90942f178c4b12131b5": "M_{1}({\\vec {X}},{\\vec {\\rm {E}}},Y)=\\left[{\\begin{array}{*{20}c}0&0&b\\\\0&0&A\\\\0&0&I\\\\{A^{T}}&I&0\\\\\\end{array}}\\right]", "1950bb0c40ceaec16b377f829b96cff1": "r={\\frac {1}{s}}", "1950ee158c131d36cf7b83596c240212": "\\beta H_{g}=-\\beta \\sum _{(i,j)}J_{ij}\\delta (s_{i},s_{j})-\\sum _{i}h_{i}s_{i}\\,", "1950f10625b96aa4611b31935901fdee": "{\\begin{aligned}c(i,k,X)&:=\\left\\{p\\left(X_{i}|X_{i-k}^{i-1}\\right)\\right\\}\\\\c(i,X)&:=\\left\\{p\\left(X_{i}|X_{-\\infty }^{i-1}\\right)\\right\\}\\end{aligned}}", "19519dcebc223000df9e7ec676788402": "\\sigma _{yy}\\sigma _{zz}-\\sigma _{yz}^{2}", "19522d05daea1699923e26e2794c4aa5": "\\theta _{(l)}\\,{\\hat {=}}\\,0", "195234152f7fb4fdd94ac4b2a3890208": "\\displaystyle R(f)(\\phi )(x)=\\int _{G}f(y)\\phi (xy)\\,dy=\\int _{\\Gamma \\backslash G}\\sum _{\\gamma \\in \\Gamma }f(x^{-1}\\gamma y)\\phi (y)\\,dy.", "195246810f9bfc228bca491859062b14": "\\xi ", "19524783f77b962726c6a3a32e1d969b": "c(x)\\geq 0~~x\\in \\mathbb {R} ^{n},c(x)\\in \\mathbb {R} ^{m}~~~~~~(1)", "19524a01b0bb23fb28170076a61824c4": "\\left\\{{j \\atop k}\\right\\}", "195294d43f24d89a4d84eaf58a604dcf": "\\langle A,\\in \\rangle ", "1952a387272e13895410e766c5ba4dc5": "z{\\frac {d^{2}w}{dz^{2}}}+\\left(C+{\\frac {D}{\\sqrt {D^{2}-4F}}}z\\right){\\frac {dw}{dz}}+\\left({\\frac {E}{\\sqrt {D^{2}-4F}}}+{\\frac {F}{D^{2}-4F}}z\\right)w=0", "1952dd2c9b14fde81357c321fccfe516": "7.76\\times 10^{206544}", "19532796fff88b522980627cbaffa592": "\\mathrm {H=Wb\\ A^{-1}=V\\ A^{-1}s=kg\\ A^{-2}m^{2}s^{-2}} ", "19532812640bd20f43d31023e1071ae3": "\\varphi \\circ \\pi ={\\mathrm {pr} _{1}}\\circ \\psi \\,,", "195371a0f34f791e88e6eb0743000476": "t(d,n)\\leq {\\mathcal {O}}(d^{2}\\log n)", "1953796efcf6b915573c34df836234d5": "\\Gamma =3\\,", "1953a4318c4b5edf02c2289be3a0366a": "\\scriptstyle |\\psi (x,y,z,t_{0})|^{2}.", "1953c85dddb1d67afcd41a23688ef8a4": "\\{|r_{i}\\rangle \\}", "1953f08796bace949f6fdbbdc3f0f6e9": "2t_{d}", "1954b5a14afc4cd694d658a009372c0d": "x_{0}=b", "1954e0047441a6c9527ea1cd1955cd6f": "E\\left[e^{t\\log X}\\right]=\\lambda ^{t}\\Gamma \\left({\\frac {t}{k}}+1\\right)", "19552212ad592fb4fbb8ac7cf48300b3": "\\inf _{N\\geq 1}\\sum _{n=N}^{\\infty }\\Pr(E_{n})=0.\\,", "19552944d265423ce186e9781e0b3411": "\\scriptstyle -{\\frac {11+4{\\sqrt {5}}}{41}}", "195532bbf10ce37f6de16665f92d93a5": "p^{n}\\;", "1955336337eb36003a780cd4a1034492": "\\Delta {s}\\rightarrow 0", "19555dbb9b4922bfca024a893f0c1288": "\\left(b^{2d}+b^{d}+1\\right)^{n},", "1955aee141ceb183b70fd1f2df16acf4": "\\mathbf {s} =[s_{1},s_{2},\\ldots ,s_{N_{s}}]^{T}", "1955b89f7ee3e8a8a8d18615efe03518": "f(a)=0.", "1955ec7f301777ef0a6c6281d15534d4": "a_{j}-a_{i}=K", "19561bb22d07427e063434ddbeaac62e": "\\sum _{n\\geq 1}{\\frac {d(n)^{2}}{n^{s}}}={\\frac {\\zeta (s)^{4}}{\\zeta (2s)}}", "19568cba6a2f4d3bb0cd5b4ae05ac110": "A={\\frac {1}{b}}\\inf _{|\\gamma |\\in [1,a]}(G_{0}(\\gamma )-G_{1}(\\gamma ))>0", "1956940fd568edcef6c60b3723bb951b": "m.", "1956d148f8c77df7bf21265fd957e797": "a\\in \\mathbb {R} ", "1956e7f4c76d3febd1870fbfbaa9988c": "t^{-1}K(x,t;X_{0},X_{1})=K(x,t^{-1};X_{1},X_{0}).", "195716ce17af6b96683e9cfd8c57e49c": "f:X\\rightarrow Z", "19574c28c05ecf3ca919073e93040e30": "\\nabla \\times \\mathbf {H} _{\\text{d}}=0", "19578c52c9799a48fb3138dec36c30f6": "u=u_{1}+\\left(u_{2}-u_{1}\\right)\\,\\sin ^{2}\\left({\\frac {1}{2}}\\varphi +\\delta \\right)", "1957d47e4158cf35010ac79565813d73": "\\{(A1,A2)|(A1,B1)\\}=\\{\\{|\\}|\\{|\\}\\}.", "1957d7a6b83de643498aa23bd6949a47": "{\\vec {q}}=-k{\\vec {\\nabla }}T", "19583fa1a671f0a0810cbe7cd5ee5f75": "{\\begin{bmatrix}L'\\\\M'\\\\S'\\end{bmatrix}}=\\mathbf {M} _{H}{\\begin{bmatrix}X_{c}\\\\Y_{c}\\\\Z_{c}\\end{bmatrix}}=\\mathbf {M} _{H}\\mathbf {M} _{CAT02}^{-1}{\\begin{bmatrix}L_{c}\\\\M_{c}\\\\S_{c}\\end{bmatrix}}", "19585c35847bbd6a71e35ae3299dfdd1": "x_{m-1}\\succ x_{m}", "1958a12f9fac2e513669175780a3641d": "f^{-1}(F(f))=f(F(f))=F(f)", "1958d167a60990afb443fca72ea814b7": "f_{t}:M\\to \\mathbb {R} ", "19591371392c0b82414c07db29ac41f5": "F(A,B)=\\inf _{\\alpha ,\\beta }\\,\\,\\max _{t\\in [0,1]}\\,\\,{\\Bigg \\{}d{\\Big (}A(\\alpha (t)),\\,B(\\beta (t)){\\Big )}{\\Bigg \\}}", "195928339db213e4ca31e731b2fdbd40": "{\\boldsymbol {\\Gamma }}", "1959334d1a3d88cc15fd96ab15d8e030": "\\int _{0}^{t}f(w(s))\\,\\mathrm {d} s=\\int _{-\\infty }^{+\\infty }f(x)L_{t}(x)\\,\\mathrm {d} x", "195973c728080ff7059576a88d4d7ccd": "\\epsilon \\in \\mathbb {C} ", "1959cd91477c06cb6bb761ed8169dd66": "x^{\\prime }[(m-l)\\mod N]", "1959e499d5110d8b4abf857fb94fdb32": "D_{\\mathrm {KL} }(P\\|Q)", "195a0b1b92912ef0c047c789b27dc0dc": "J_{(1)}=J_{(2)}", "195a21c626ce9f44e3e2cc6c10f29ca0": "{\\hat {\\rho }}=\\sum _{i}p_{i}|\\psi _{i}\\rangle \\langle \\psi _{i}|.", "195a5b104f0f0d94cc53f4827ff57648": "\\sigma (x,y)", "195a65e0f5f4e2663c174d1fee2a2fe6": "{\\textit {true}}\\rightarrow {\\textit {open}}(1)\\wedge {\\textit {occludeopen}}(1)", "195a98b1b5ba6b44fb1820f3087d3cee": "\\left(1-{\\frac {a}{2}}\\right){\\sqrt {-(a+1)}}-a\\left(2+{\\frac {a}{2}}\\right)\\arcsin {\\frac {1}{\\sqrt {-a}}}.", "195af63d1c54ae6dad4123723e2a22c3": "{\\frac {\\sum _{n=0}^{\\infty }b_{n}X^{n}}{\\sum _{n=0}^{\\infty }a_{n}X^{n}}}=\\sum _{n=0}^{\\infty }c_{n}X^{n},", "195b2834fd19ddf153b8ee828acde11b": "\\sum _{p|N}{\\frac {N}{p}}+1=N.", "195b3a7fbc45810c7069800c54e755b8": "a={\\frac {a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}}{\\left(a^{2}+b^{2}+c^{2}\\right)^{2}}}\\Delta ", "195b51f3ae98c50f1541bb54a442efcd": "\\sigma _{f}(\\theta )", "195b7f1122557c408cf93fac9770f5c2": "y_{c}={\\begin{cases}-1.1063814x_{c}^{3}-1.34811020x_{c}^{2}+2.18555832x_{c}-0.20219683&1667{\\text{K}}\\leq T\\leq 2222{\\text{K}}\\\\-0.9549476x_{c}^{3}-1.37418593x_{c}^{2}+2.09137015x_{c}-0.16748867&2222{\\text{K}}\\leq T\\leq 4000{\\text{K}}\\\\+3.0817580x_{c}^{3}-5.87338670x_{c}^{2}+3.75112997x_{c}-0.37001483&4000{\\text{K}}\\leq T\\leq 25000{\\text{K}}\\end{cases}}", "195bc176a3eecc0e1cb95ab4f7eea1a8": "p_{\\operatorname {interp} }(r)", "195bcb37cd58a7069bf2b8d723364a21": "\\scriptstyle \\delta t_{{\\text{clock}},i}(t_{i})\\;=\\;\\delta t_{{\\text{clock,sv}},i}(t_{i})\\,+\\,\\delta t_{{\\text{orbit-relativ}},\\,i}({\\boldsymbol {r}}_{i},\\,{\\dot {\\boldsymbol {r}}}_{i})", "195c48cfb7a585c41f637ab76f4fd590": "\\mathbf {e} _{1}(t)={\\frac {\\mathbf {\\gamma } '(t)}{\\|\\mathbf {\\gamma } '(t)\\|}}.", "195c599630049736688b0afbe45f87a2": "{\\mathcal {U}}^{\\times _{X}^{r}}:={\\mathcal {U}}\\times _{X}\\dots \\times _{X}{\\mathcal {U}}.", "195cef8076f3333a5fd5025f74daca4f": "m_{\\mathrm {i} }", "195d2fdbffb939fe4bc792cadd80bf62": "\\exists f_{1},f_{2}\\in C\\;pf_{1}=1+f_{2}", "195e2a44f7c0122d96ac7be568d73207": "1\\ \\mathrm {V} \\cdot 10^{\\frac {-60}{20}}=.001\\ \\mathrm {V} =1\\ \\mathrm {mV} ", "195eaa6f24ce8f721729d5115bae0fd2": "\\theta >0", "195f426da0265c47431bef593a930574": "\\mathbb {P} (A_{1})\\leq \\mathbb {P} (A_{1}).", "195f4d1cc140d1ca55725b434285ee08": "\\left(10-1.37218{\\frac {\\sqrt {2}}{\\sqrt {11}}},10+1.37218{\\frac {\\sqrt {2}}{\\sqrt {11}}}\\right)=\\left(9.41490,10.58510\\right).", "195f4dd692396f9164530339f3c96309": "\\,N(R)=(2\\pi )^{-d}\\omega _{d}VR^{d/2}+{\\frac {1}{4}}(2\\pi )^{-d+1}\\omega _{d-1}AR^{(d-1)/2}+o(R^{(d-1)/2}).\\,", "195f6e3e9d12ba6e65b20f2719df81c1": "a=(a_{1},\\dots ,a_{n})", "195fb9efa4f61e9968b73af9b4aeeef6": "x_{0}\\!", "195fcf1c043c8af2ad00aaeabfd1912d": "\\textstyle \\epsilon ", "1961487550cb1f61914c9800f6957560": "{A}_{18}^{(2)}", "19617304f80da783687c0800e3ac3d7e": "\\theta =\\pi ", "1961a97e2b0aa2c4cba24e56315ca139": "E_{\\text{exch}}=A\\int _{V}\\left((\\nabla m_{x})^{2}+(\\nabla m_{y})^{2}+(\\nabla m_{z})^{2}\\right)\\mathrm {d} V", "1961c355ac953bae13c3727883912638": "P_{\\mathbf {k} }^{\\star }", "1961d722eb0a0f899070cd88293dc19d": "{\\begin{aligned}\\sum _{i\\in S}\\sum _{a\\in A(i)}R(i,a)y^{*}(i,a)\\geq \\sum _{i\\in S}\\sum _{a\\in A(i)}R(i,a)y(i,a)\\end{aligned}}", "1961eeac1fd1559e2cd40a0324eb7411": "{\\overline {\\Gamma ^{k}{}_{ij}}}={\\frac {\\partial x^{p}}{\\partial y^{i}}}\\,{\\frac {\\partial x^{q}}{\\partial y^{j}}}\\,\\Gamma ^{r}{}_{pq}\\,{\\frac {\\partial y^{k}}{\\partial x^{r}}}+{\\frac {\\partial y^{k}}{\\partial x^{m}}}\\,{\\frac {\\partial ^{2}x^{m}}{\\partial y^{i}\\partial y^{j}}}\\ ", "1962304f997d73c67d909135f0a99e1e": "EZ\\mathbf {y} =0", "19627b9cfe9eb538f4d940082c339f43": "{\\hat {b}}", "1962a11c3dec36526b3a35f65d8c3777": "\\int _{E}f(x)\\,dx.", "1962aeb01df827ac7650c224e5687417": "H_{1},\\dots ,H_{p}", "196331f7ca0856bc1404c81620d5ddee": "A=\\{x\\mid \\phi \\}", "19634dba6597efe959b91b31202c96bd": "\\left\\langle Ax,y^{\\star }\\right\\rangle =\\left\\langle x,A^{\\star }y^{\\star }\\right\\rangle ,", "1963bf9d24e3b9be35691d6026aeed3e": "\\Delta C_{t}=0.5\\Delta Y_{t}-0.2(C_{t-1}-0.9Y_{t-1})+\\epsilon _{t}", "1963f8f15b60547359961bf40f5d83a3": "W^{II}(z,x)=E\\left[\\beta ^{II}(Y_{1})\\mid X_{1}=x,Y_{1}0\\},&&{\\text{the heat equation}},\\\\-{\\frac {\\partial u}{\\partial x}}(0,t)&=f(t),&&t>0,&&{\\text{the Neumann condition at the left end of the domain describing the inlet heat flux}},&&\\\\u{\\big (}s(t),t{\\big )}&=0,&&t>0,&&{\\text{the Dirichlet condition at the water-ice interface: setting melting/freezing temperature}},\\\\{\\frac {\\mathrm {d} s}{\\mathrm {d} t}}&=-{\\frac {\\partial u}{\\partial x}}{\\big (}s(t),t{\\big )},&&t>0,&&{\\text{Stefan condition}},\\\\u(x,0)&=0,&&x\\geq 0,&&{\\text{initial temperature distribution}},\\\\s(0)&=0,&&&&{\\text{initial depth of the melted ice block}}.\\end{aligned}}", "1987a88fc39f6b7f2fdf571395212e68": "d-1", "1987b6358e69904a4b2620751195dc61": "[0,\\pi ]\\,", "1987b8342168ae3d754d4a353832e2b3": "H\\subseteq A", "1987d0c8cb6768a4753bf5df2056dfa0": "V_{i}:=\\bigcup _{j1.", "19954eb5893dbb80cfd91d6ebcad2680": "O({\\underline {u}}:G)", "19954f1ee6438a24f1758b43c4c6e606": "\\Gamma ^{k}{\\mathbf {e} }_{k}\\,", "19955180e4f1f7c7c96da365d15b388d": "\\mathbb {P} (\\theta _{up}(X)\\geq \\theta |\\theta )", "1995a69ab16b97dedd15c9ee0abd4ae2": "\\Gamma \\left({\\frac {s}{2}}\\right)\\pi ^{-s/2}\\zeta (s)={\\frac {1}{2}}\\int _{0}^{\\infty }\\left[\\vartheta (0;it)-1\\right]t^{s/2}{\\frac {dt}{t}}", "19964a2c10afc6d91dc66293130ce42a": "z^{N}-1=F_{1}(z)F_{2}(z)F_{3}(z)", "1996a01256151a7de63182612cd5b7e4": "\\alpha (U_{+})", "1996bc29a9052cd839e74c9aba11c599": "IM_{i}=e_{i}^{t}\\left(G_{i}-G\\right)", "19974d9ff467ca51c10ed34d6f3bea26": "(q_{2},p_{2})", "19978c5eae58f5be5c746a2f29a2339f": "{\\underline {\\varphi \\vdash \\lnot \\psi }}\\,\\!", "1997fbf8e16efe1aeeea49a032454b26": "e^{-x}=\\cosh x-\\sinh x", "19983625ce00458e19cb8d2325fad636": "versin(\\theta )=1-cos(\\theta )", "199841974c59ef53d9ebb97ac335a2fc": "HD=DP.", "199848333e004f3702eb338744d818f2": "H=\\sum _{n=0}^{\\infty }E_{n}|n\\rangle \\langle n|", "19989f1685981884934ee90c26d26dcb": "\\mathbf {P} ^{-1}\\mathbf {R} _{1}\\mathbf {P} =\\mathbf {D} ", "1998bc8bcd6e62169bdb4c6f1d826835": "\\displaystyle {G_{0}=KA_{0}K.}", "1998bebc8aab93ad11d7cc346f7b9a17": "f\\colon S\\to S", "1998f5b4ce24c28961cec25bcdcebc20": "VCA(64x^{3}-112x+56,(0,2))", "199911eaf439374bc1ca41fe5bfb2dfe": "{\\tfrac {1+{\\sqrt {5}}}{2}}.\\,", "199967db6c19aa7fe20fbd760e2f261f": "{\\tau }=I\\ {\\alpha }", "19996851bbe16e45b824474b587f7d52": "|({\\mathcal {F}}_{a}f)(at,y)|\\leq Ce^{-\\varepsilon a},", "19997ce9e1d2be894d4ae5d767dabd76": "\\mathrm {Ad} _{g}", "1999c50f4e93857f7436bed20696d9e1": "n^{\\underline {k}}=n\\times (n-1)\\times (n-2)\\times \\cdots \\times (n-k+1),", "1999d0078e9a6e421e00f3b58513d38d": "\\pi /2", "1999f106954f6ff4669d1f460fab14a0": "\\displaystyle {[\\delta ^{h},\\sum a_{\\alpha }\\partial ^{\\alpha }]=(\\delta ^{h}(a_{\\alpha })\\circ R_{h})\\partial ^{\\alpha }.}", "199a3cef9b2a7c0762fa0a6568fef088": "a_{i,j,k}", "199aa29ac20ea76bed3965887ba7c383": "(\\forall {\\alpha <\\kappa ^{+}})A_{\\alpha }\\leq _{f}B\\,", "199ab0fddcf6057c1d12262475af4a8a": "\\Sigma _{n}", "199b0643221b2d0887c987a1462255ee": "{\\mathcal {M}}(U)\\to {\\mathcal {M}}(V)", "199b2da68391c77caa328a519959fdb4": "12.9232102", "199b4394970666ce7528e192cf3b2676": "{\\frac {d\\varphi }{d\\alpha }}=\\int _{a}^{b}f_{\\alpha }(x,\\alpha )\\,dx+f(b,\\alpha ){\\frac {db}{d\\alpha }}-f(a,\\alpha ){\\frac {da}{d\\alpha }}", "199b77219a0793e89f49c7614b42838a": "u_{1},u_{2}\\in (V\\cup \\Sigma )^{*}", "199ba4358939098d08e196df1a2df739": "A\\circ B", "199bc539a17705dd92c959577d1404a5": "K_{i}\\varphi \\implies \\varphi ", "199be30a7ca6070d6bb1c9e9d325ea94": "C_{1}^{-1},C_{1}^{0},C_{1}^{1}", "199c2f04393f16410802bee481d0614c": "\\varphi \\left(\\int x\\,d\\mu _{n}(x)\\right)\\leq \\int \\varphi (x)\\,d\\mu _{n}(x),", "199c2f6f10dd3cfc94ecc4c05929fd30": "{\\frac {\\partial z}{\\partial x}}", "199c34ec680fb1368f84dc9a5d564db0": "~\\sum _{mnp}~{\\frac {1}{k^{2}-\\alpha _{m}^{2}-\\beta _{n}^{2}-\\gamma _{p}^{2}}}~{\\mathbf {G}}_{mnp}~{\\mathbf {J}}(\\alpha _{m},\\beta _{n},\\gamma _{p})~e^{j(\\alpha _{m}x+\\beta _{n}y+\\gamma _{p}z)}~=~{\\mathbf {0}}~~~~~~~~~~~~~~~~(3.3)", "199c4df01bd6170f7d83145e1189cf56": "\\Delta \\mathbf {v} \\,", "199cb56b173080aeff39332c795afc8a": "\\,{{(z_{1}-z_{3})(z_{2}-z_{4})} \\over {(z_{1}-z_{4})(z_{2}-z_{3})}}", "199cbda39eae2b91d55c0c1201cb4fe0": "L_{n}\\left[1/2,1+o(1)\\right]", "199cc2ae9e3f8dac92151ce502826fda": "\\pi ={\\frac {F}{l}}.", "199d664785a7182bcb78907cc6efd426": "F_{5}\\;=\\;({\\frac {f}{900}})^{-n}{\\mbox{ for }}2\\zeta _{b}", "199e770aefdf9e1969110b945e19e6ef": "R/I", "199e9d4ae76d467d01bf27d9a38d7104": "N\\rightarrow U", "199ea3d974838ca329bcc09f0a3e2449": "x_{k+1}=x_{k}-\\tau \\nabla f(x_{k})", "199eae636d5286e6fa5a6d864976d126": "p(m)=-{\\tfrac {1}{8}}", "199eb8d4ba1982878efc883c2a8841e5": "{\\begin{matrix}&&K&&\\\\&\\eta _{A}\\swarrow &\\,&\\eta _{B}\\searrow &\\\\A&&{\\begin{matrix}f\\\\\\longrightarrow \\end{matrix}}&&B\\end{matrix}}", "199eefb242c3eb4ca44ee1dcd612012e": "{\\mathcal {O}}=k[[t]]", "199f077e864bd9f00c7533d0c523c2ad": "\\{x_{t},y_{t}\\}", "199f45229705d0c176d17b4b9d58e129": "(x_{1},x_{2},....,x_{n})", "199f70ea7fa4be0b6421b970bda236d2": "NM(X|k_{0},\\{p_{1},p_{2},p_{3}\\})=0.00465585119998784", "199f78aa20d48a01e124292d955b6357": "w=g_{1}g_{2}\\cdots g_{m}", "199fa642986296f2fbe02a1258402aa1": "\\int _{\\mathbb {R} ^{3}}\\rho =N", "199fb1ee6c70970e322bc7f5c9e0d01c": "e_{\\alpha }^{I}", "199fb9b43bc6e2b346b45c63c01366ec": "\\delta (X,Y)=\\log {\\Bigl (}\\inf\\{\\|T\\|\\|T^{-1}\\|:T\\in \\operatorname {GL} (X,Y)\\}{\\Bigr )}.", "199fdba64b4357d13b0a54611db02c72": "\\mathbf {S} =\\mathbf {E} \\times \\mathbf {H} ,", "199fe0ee55a358eae25d6583a56369a1": "{\\frac {\\partial f(g(u))}{\\partial \\mathbf {x} }}=", "199fe28b14d7de1f22666e319f0c8ecf": "{\\begin{aligned}&\\mu _{1}(g_{2})=-{\\frac {6}{n+1}},\\\\&\\mu _{2}(g_{2})={\\frac {24n(n-2)(n-3)}{(n+1)^{2}(n+3)(n+5)}},\\\\&\\gamma _{1}(g_{2})\\equiv {\\frac {\\mu _{3}(g_{2})}{\\mu _{2}(g_{2})^{3/2}}}={\\frac {6(n^{2}-5n+2)}{(n+7)(n+9)}}{\\sqrt {\\frac {6(n+3)(n+5)}{n(n-2)(n-3)}}},\\\\&\\gamma _{2}(g_{2})\\equiv {\\frac {\\mu _{4}(g_{2})}{\\mu _{2}(g_{2})^{2}}}-3={\\frac {36(15n^{6}-36n^{5}-628n^{4}+982n^{3}+5777n^{2}-6402n+900)}{n(n-3)(n-2)(n+7)(n+9)(n+11)(n+13)}}.\\end{aligned}}", "19a029a162a67f2d310b823f89820339": "{\\mathcal {P}}=\\left\\{\\left({\\frac {74}{511}},{\\frac {81}{511}},{\\frac {137}{511}}\\right),\\left({\\frac {148}{511}},{\\frac {162}{511}},{\\frac {274}{511}}\\right),\\left({\\frac {296}{511}},{\\frac {324}{511}},{\\frac {37}{511}}\\right)\\right\\rbrace ", "19a069f087c731d82fbf373afa97bbf8": "(5)\\quad ds^{2}=-{\\Big (}1-{\\frac {2M}{r}}{\\Big )}dv^{2}+2dvdr+r^{2}(d\\theta ^{2}+\\sin ^{2}\\theta \\,d\\phi ^{2})\\;.", "19a07f42375250eeec2c1f1256cb964d": "\\epsilon =GM\\left({\\frac {2a-r_{a}}{2ar_{a}}}-{\\frac {1}{r_{a}}}\\right)=-{\\frac {GM}{2a}}", "19a09e5535341fae17b8a43797a82c6d": "|0\\rangle _{B}", "19a15e3925d19a6d53e8bf065bce0755": "\\left(1+\\sum _{i=1}^{p}\\phi _{i}L^{i}\\right)X_{t}=\\left(1+\\sum _{i=1}^{q}\\theta _{i}L^{i}\\right)\\varepsilon _{t}\\,.", "19a1db91013fe6f388f9df252069e3b5": "S_{\\mathrm {f} 6}", "19a1f82859cf1cc42f7a174e49a73cf7": "da\\,\\!", "19a236bd33c9d1b25b54078d75438610": "S_{1}={25 \\over 24}\\approx 70.7\\ {\\hbox{cents}}", "19a238b9254d274589701eb7f8d6aa94": "\\langle ,\\rangle :X\\times Y\\to \\mathbb {F} ", "19a25d8b621357c0b7d6bf525e840291": "\\mathbf {r} =\\mathbf {y} -\\mathbf {\\hat {y}} =\\mathbf {y} -H\\mathbf {y} =(I-H)\\mathbf {y} .", "19a279a922e5f4740784161e5a305682": "{\\dot {S}}=-2{\\dot {R}}.", "19a2cd3ecad002bbd85764a966f36e03": "\\mathbf {x} =\\{x_{-M},x_{-M+1},\\ldots ,x_{M}\\}", "19a31e17d0cf3e104a26a165ba89603c": "\\left(\\omega =\\omega _{o}\\right)", "19a350289fe7909ac6ae6aaf7fb21cee": "\\left\\langle N,e\\right\\rangle =\\left\\langle 90581,17993\\right\\rangle ", "19a3d21a29c9e5afbecd9dd04ddad36d": "{\\begin{aligned}&D[p||q]={\\frac {1}{2}}g_{ij}(q)\\Delta \\xi ^{i}\\Delta \\xi ^{j}+{\\frac {1}{6}}h_{ijk}\\Delta \\xi ^{i}\\Delta \\xi ^{j}\\Delta \\xi ^{k}+o(||\\Delta \\xi ||^{3})\\\\&h_{ijk}=D[\\partial _{i}\\partial _{j}\\partial _{k}||]\\\\&\\partial _{i}g_{jk}=\\partial _{i}D[\\partial _{j}\\partial _{k}||]=D[\\partial _{i}\\partial _{j}\\partial _{k}||]+D[\\partial _{j}\\partial _{k}||\\partial _{i}]=h_{ijk}-\\Gamma _{jk,i}\\\\&h_{ijk}=\\partial _{i}g_{jk}+\\Gamma _{jk,i}.\\end{aligned}}", "19a40010111789ab2b002fa427c3f9e1": "{\\begin{pmatrix}a&-m\\\\c&n\\end{pmatrix}}", "19a4350bbcbe6580b14d35a450257d7a": "L=M", "19a439ed08428eba46fcb31fce6737b9": "m_{\\text{S}}={\\sqrt {\\frac {e^{2}}{G(4\\pi \\epsilon _{0})}}}", "19a4a2507d68bf2f9d8cef90ed0c85b8": "\\mathbb {N} ^{\\mathbb {N} }", "19a569e51bc4fb0f04565b126517215b": "\\psi _{klm}(r,\\theta ,\\phi )=N_{kl}r^{l}e^{-\\nu r^{2}}{L_{k}}^{(l+{1 \\over 2})}(2\\nu r^{2})Y_{lm}(\\theta ,\\phi )", "19a5d1215959f1bf660ef0836ccb6b69": "\\beta _{E}(Q)", "19a5dc08814aa09fd5bbeff85538e995": "\\oint _{S}\\mathbf {E} \\cdot \\mathrm {d} \\mathbf {A} ={\\frac {Q}{{\\mathcal {E}}_{0}}}", "19a5e25d843fc40b787024a57e3c49ba": "q_{y}=q_{y}(x,y)", "19a5ec6e8d51a3c90c48aa9548f39c92": "\\ln(L)=-{\\frac {1}{2}}\\ln(|{\\boldsymbol {\\Sigma }}|\\,)-{\\frac {1}{2}}(\\mathbf {x} -{\\boldsymbol {\\mu }})^{\\rm {T}}{\\boldsymbol {\\Sigma }}^{-1}(\\mathbf {x} -{\\boldsymbol {\\mu }})-{\\frac {k}{2}}\\ln(2\\pi )", "19a62cc119b1e2575c9316137fdd2b98": "\\scriptstyle L_{c}", "19a6461776a93d2451ce850c2d8ad598": "{\\color {Blue}~2.15}", "19a73fc3b84f377f6508b7ff40f3e170": "{\\tilde {D}}_{8}", "19a74ef2d7c2f6370559c559e606fff0": "e^{(a-1)\\Theta }f(x)=f(ax)", "19a7a37a20ea2231cb90ba01577ffe5d": "{\\sqrt {E}}=(k^{2}-m^{2})^{1 \\over 4}", "19a7b0f68176cdfcbcaff8e0001493e4": "U(\\alpha S,\\alpha V,\\alpha N_{1},\\alpha N_{2},\\ldots )=\\alpha U(S,V,N_{1},N_{2},\\ldots )\\,", "19a7bdc2300335c921dd2d36f9b8c82c": "{\\bar {\\mathbf {e} }}{}_{j}={\\bar {\\mathbf {e} }}{}_{j}\\left(\\mathbf {e} _{1},\\mathbf {e} _{2}\\cdots \\right)\\quad \\rightleftharpoons \\quad \\mathbf {e} {}_{j}=\\mathbf {e} {}_{j}\\left({\\bar {\\mathbf {e} }}_{1},{\\bar {\\mathbf {e} }}_{2}\\cdots \\right)", "19a7ed3a8acee2707be22f0831d04827": "\\left({\\frac {\\partial (G/T)}{\\partial T}}\\right)_{p}={\\frac {1}{T}}\\left({\\frac {\\partial G}{\\partial T}}\\right)_{p}+G{\\frac {\\partial (T^{-1})}{\\partial T}}={\\dfrac {T\\left({\\dfrac {\\partial G}{\\partial T}}\\right)_{p}-G}{T^{2}}}={\\frac {-ST-G}{T^{2}}}=-{\\frac {H}{T^{2}}}\\,\\!", "19a852d6b5d898d377fbf3324fd276ca": "\\langle R'(X,Y)Z,W\\rangle =\\langle R(X,Y)Z,W\\rangle +\\sum _{j=1}^{k}\\alpha _{j}(X,Z)\\alpha _{j}(Y,W)-\\alpha _{j}(Y,Z)\\alpha _{j}(X,W)", "19a891cd03ec35200ce75ce8dea8a445": "[\\ ]_{+}", "19a8b2ea9738dbc9cf366bb300209209": "z_{1},z_{2},z_{3},\\infty ", "19a8f3dcb49fdee5dc4a48fb7eb82e4a": "x_{\\omega }", "19a93ebc67feceda8f02609cfcc96847": "\\ln \\left[{\\frac {1+e}{1+e_{0}}}\\right]=\\ln \\left[{\\frac {v}{v_{0}}}\\right]=-{\\tilde {\\lambda }}\\ln \\left[{\\frac {p_{c}}{p_{c0}}}\\right]", "19a93edbfbcb14ce3b9f423896dba1c0": "{\\frac {dN}{dt}}", "19a94c143c55b7663bcfe27d51a12cf2": "\\{\\,r^{X_{n}}:n=1,2,3,\\dots \\,\\}", "19a95445d0c61258be64392a9ff1c155": "d/dzId-E/z", "19a96b15cf0c9a45e9314e0d76e5f5f4": "\\mathrm {Interest} =\\mathrm {Principal} \\times \\mathrm {CouponRate} \\times \\mathrm {Factor} ", "19a9725c7555f62b3c35c410036f9b7e": "\\scriptstyle |\\zeta |^{n}\\leq \\|a\\|_{1}\\max\\{|\\zeta |^{n-1},\\cdots ,|\\zeta |,1\\}=\\|a\\|_{1}|\\zeta |^{n-1}", "19a9d41a1b22e947799c020049da0dfc": "{\\begin{aligned}&{\\text{maximize}}&&\\mathbf {c} ^{\\mathrm {T} }\\mathbf {x} \\\\&{\\text{subject to}}&&S\\mathbf {v} =\\mathbf {0} \\\\&{\\text{and}}&&\\mathbf {lowerbound} \\leq \\mathbf {x} \\leq \\mathbf {upperbound} \\end{aligned}}", "19a9da7015c340be722505f039eee1c4": "{\\tfrac {1}{\\sqrt {2}}}(\\pm 1\\pm 1i+0j+0k)", "19a9dff0655da20749d7605f858ed898": "\\varphi (w)=\\exp \\!{\\big \\{}i\\operatorname {Re} ({\\overline {w}}'\\mu )-{\\tfrac {1}{4}}{\\big (}{\\overline {w}}'\\Gamma w+\\operatorname {Re} ({\\overline {w}}'C{\\overline {w}}){\\big )}{\\big \\}},", "19aa9b43a5b37b0e2fb3c76a039e190b": "N\\times H", "19aa9d41967d0ef9a4a149fa8ae6c81c": "P=\\langle \\psi \\vert \\Pi \\vert \\psi \\rangle =\\vert \\langle \\phi \\vert \\psi \\rangle \\vert ^{2}.", "19aad067e6c9338626cc507960677fda": "(e,f)", "19aae4a342457e40d2a76175cf81e309": "(x\\cdot y)^{l}=y^{l}\\cdot x^{l}\\qquad (x\\cdot y)^{r}=y^{r}\\cdot x^{r}", "19ab0a90ac537abbb29aebc8b2cf9cd9": "{\\frac {{\\rm {d}}z}{(z+a)^{3}z^{1/2}}}", "19ab32fc4270d0869b1c02223f69cfd3": "U=L^{3}{\\frac {8\\pi }{h^{3}c^{3}}}\\int _{0}^{\\infty }{\\frac {\\varepsilon ^{3}}{e^{\\beta \\varepsilon }-1}}\\,d\\varepsilon .\\qquad {\\text{(3)}}", "19ab5b91020dd6bff9e9cdd3d0ee0a54": "{d^{2}x^{\\mu } \\over dt^{2}}=-\\Gamma ^{\\mu }{}_{\\alpha \\beta }{dx^{\\alpha } \\over dt}{dx^{\\beta } \\over dt}+\\Gamma ^{0}{}_{\\alpha \\beta }{dx^{\\alpha } \\over dt}{dx^{\\beta } \\over dt}{dx^{\\mu } \\over dt}\\ .", "19ab6cb8f6b9c3e72bd8424a447446a8": "s\\cdot {\\sqrt[{3}]{2}}", "19abfe4c3cc868998f84e98086dc1540": "m'\\,", "19ac0fa5ab97186b57ea5c6bc38b10cd": "P^{\\phi }=\\exp \\left(\\int _{0}^{T}{\\dot {\\phi }}(t)\\,dX_{t}^{\\phi }+\\int _{0}^{T}{\\tfrac {1}{2}}|{\\dot {\\phi }}(t)|^{2}\\,dt\\right)\\,dP.", "19ac1c5586c1bf7a7e142abc86f89793": "u,v,x", "19ac43e8c3ca5e3bf53bed85137d249d": "\\scriptstyle A^{\\ast }", "19acb64bd338ad59c888c5ada9b04cb0": "a_{2}-a_{1}0)", "19c5fd810c785d7c02be5caf7155e63e": "{\\overline {\\Sigma }}_{t}", "19c6144ed8ef656efd227f1039569f17": "\\Leftrightarrow m\\sum _{i=1}^{N}w_{i}(x_{0})-m=0\\Leftrightarrow \\sum _{i=1}^{N}w_{i}(x_{0})=1\\Leftrightarrow \\mathbf {1} ^{T}\\cdot W=1", "19c630f91994382664b4d0a43490e00e": "\\left(E_{x},\\,E_{y},\\,E_{z}\\right)\\propto \\left(\\cos {\\frac {2\\pi }{\\lambda }}\\left(ct-z\\right),\\,\\sin {\\frac {2\\pi }{\\lambda }}\\left(ct-z\\right),\\,0\\right).", "19c63bf027a2b276b972c908673e2fcd": "\\ \\mathbf {e} _{i}=\\alpha _{iJ}\\mathbf {E} _{J}", "19c65dee9dbe1341f19aedb556e4e012": "h(h(h(h(x))))", "19c66269c491582465c0984a7f1badb2": "\\phi _{\\it {nk}}", "19c68888c8a31f0cdf5bc9c4f2a13502": "\\scriptstyle \\mathbb {R} ", "19c6a57b27e96f30d2184428b5cbd1b5": "\\mathbf {F} _{\\mathrm {rad} }={\\frac {\\mu _{0}q^{2}}{6\\pi c}}\\mathbf {\\dot {a}} ={\\frac {q^{2}}{6\\pi \\epsilon _{0}c^{3}}}\\mathbf {\\dot {a}} ", "19c6ea88d2dd2b91545bf32eaa4c7354": "{\\begin{aligned}R_{j}&=\\int _{G}\\varphi (g)gH^{(1)}g^{-1}\\,dg,\\\\R_{j,\\varepsilon }&=\\int _{G}\\varphi (g)gH_{\\varepsilon }^{(1)}g^{-1}\\,dg,\\\\R_{j,\\varepsilon ,R}&=\\int _{G}\\varphi (g)gH_{\\varepsilon ,R}^{(1)}g^{-1}\\,dg.\\end{aligned}}", "19c75c0cadb518b03f76b2ec2c736893": "100^{100^{12}}=10^{2*10^{24}}", "19c7d8a7070d3e0a2848f6ccc987fbf7": "n_{\\mathrm {A} }=n_{\\mathrm {B} }\\times {\\frac {R_{\\mathrm {B} }-R_{\\mathrm {AB} }}{R_{\\mathrm {AB} }-R_{\\mathrm {A} }}}\\times {\\frac {1+R_{\\mathrm {A} }}{1+R_{\\mathrm {B} }}}", "19c80cc2f677c57189bb16ae5cab2f29": "||p({\\hat {x_{0}}}+\\sigma )-p({\\hat {x_{0}}})||", "19c813e392f357145417b8f60b426696": "g_{1}(x)=f(x)^{e}-C_{1}\\in \\mathbb {Z} _{N}[x]", "19c820f1ae9866f44a5b7cef354745af": "P(a,b)=\\int d\\lambda \\cdot \\rho (\\lambda )\\cdot p_{A}(a,\\lambda )\\cdot p_{B}(b,\\lambda )", "19c839db16b077c602f45a2d0dd27959": "\\varepsilon ={\\frac {B^{2}}{4A}}", "19c84108184d89f3a1482920d2fae171": "d\\phi =d\\phi '-\\omega \\,dt.", "19c888ba52a35eab3a5af395c6d1faf4": "v_{5}\\geq 1", "19c92099d3668b7da1a0cf6ddd1dcee0": "|\\psi (X,t)|^{2}", "19c992110e5455b198ab03e8d1ad79bf": "f(z)=\\int _{0}^{1}{\\frac {d}{dt}}f(tz_{1},\\ldots ,tz_{n})dt=\\sum _{i=1}^{n}z_{i}\\int _{0}^{1}\\left.{\\frac {\\partial f(z)}{\\partial z_{i}}}\\right|_{z=(tz_{1},\\ldots ,tz_{n})}dt,", "19c9bf426005882255abb29721a7150a": "H^{ex}={\\frac {b_{1}X_{1}b_{2}X_{2}}{b_{1}X_{1}+b_{2}X_{2}}}\\left({\\frac {\\sqrt {a_{1}}}{b_{1}}}-{\\frac {\\sqrt {a_{2}}}{b_{2}}}\\right)^{2}", "19c9f4929ee9063820dc84f68ff51e2c": "\\Delta ^{\\mathcal {I}}", "19ca14e7ea6328a42e0eb13d585e4c22": "36", "19cba33fcdc47a9f58cec7f2965061e9": "P_{1}.P_{2}\\equiv P_{2}.P_{3}\\equiv P_{3}.P_{1},", "19cbb2503e262ab7024cc88bcd963e1e": "Y_{i}=\\sum _{j=1}^{s}a_{ij}hF_{j}+\\sum _{j=1}^{r}u_{ij}y_{j}^{[n-1]},\\qquad i=1,2,\\dots ,s,", "19cbba1a7e1e68a7b0c695568bcc0189": "{\\sqrt {2}}^{\\sqrt {2}}", "19cc0601bb5dc8fdd4098eb03f50e0b9": "f(t)\\,", "19cc7cb792134fba45b1afd05d094fd2": "ki=j\\,", "19cc903b5cd8b382913a6a74a4202327": "{\\boldsymbol {H}}={{\\boldsymbol {B}} \\over \\mu _{0}}-{\\boldsymbol {M}},", "19cca7437823282823f09e331017607d": "\\scriptstyle <10^{-18}", "19ccc747dd43a70a2eb0fe20d641e764": "{\\overline {\\Delta M}}=-1.382\\sigma ^{2}", "19ccefb55180e7fdd751bcd8c63cda03": "ab>1+{\\frac {3}{2}}\\pi .", "19cd10d86dd5d8353a4df2185f6de2f8": "z'_{0}=1\\,", "19cd1682b899b2c3d73c7fbe7917b7d0": "\\displaystyle P(x)(v)=v+F(v)x=v+v^{q}x", "19ce0b99a4d47b4da8c60d07b1b9e15d": "{\\dot {S}}_{i}\\geq 0", "19ce3f1ab29759bfa4d3be5f8907ff4f": "E_{i}=B_{ij}^{0}D_{j}+{\\tilde {\\delta }}_{ijk}{\\frac {\\partial D_{j}}{\\partial x_{k}}}=B_{ij}^{0}D_{j}+(ie_{ijl}{\\tilde {g}}_{lk}k_{k})D_{j}\\,", "19ce4fbf1ad559a3000196a288479889": "r_{\\text{o}}", "19cebccfc5684b769c284fdf48909b14": "CD=\\tan \\theta \\,", "19cebe3bf9410862bf336407424b43fc": "\\langle x,z\\rangle ", "19cf1f5cf01713eb38f2476ea6794f5d": "-0.060607633\\ldots ", "19cf6eade436c435282f89a1395a26b2": "\\mathbf {d} _{i+1}={\\frac {(1-t)(1+b)(1-c)}{2}}(\\mathbf {p} _{i+1}-\\mathbf {p} _{i})+{\\frac {(1-t)(1-b)(1+c)}{2}}(\\mathbf {p} _{i+2}-\\mathbf {p} _{i+1})", "19cf8dce89a1372496bfd1be8532d42e": "{\\bar {T}}", "19cf9e131a73c44f4a4bcc6c6d340b9a": "\\chi _{\\nu }^{2}", "19cfcda70d2f5ead370e3b01e6ec3c66": "\\min _{x\\in X}\\sup _{y\\in Y}f(x,y)=\\sup _{y\\in Y}\\min _{x\\in X}f(x,y).", "19d016fa223651f47973cb72766e9527": "\\Delta S=S_{final}-S_{initial}", "19d100d2614e0a93db2379ed36e54876": "1800\\lessapprox Re_{crit}\\lessapprox 2400", "19d12e5ba4a943ea376b0525ef4a000d": "(a+b-c)ab=24", "19d130cfcb822cf5a8bc39cba9f19aa5": "{\\begin{aligned}S_{\\psi }&=I-2|\\psi \\rangle \\langle \\psi |\\quad {\\text{and}}\\\\S_{P}&=I-2P.\\end{aligned}}", "19d16bcb2c1f294750496d3c0f6f7555": "S_{z_{U}}", "19d16f72700692802d294b69fa873395": "\\textstyle {f(x)=\\sum _{n=0}^{\\infty }{w^{n}s(2^{n}x)}}", "19d17e3162ab21978a86db1a21dbae90": "x^{2}+bx=a.\\,", "19d18610a6b0c235295aed555c112ba7": "\\psi _{n}(z)", "19d2515790047bf939dd2a786d1d3cc8": "P_{i}\\lambda \\}=m\\{x:\\,\\omega (f-g)(x)>\\lambda \\}\\leq m\\{x:\\,(f-g)^{*}(x)>\\lambda \\}+m\\{x:\\,|f(x)-g(x)|>\\lambda \\}\\leq C\\lambda ^{-1}\\|f-g\\|_{1}.", "19db16814aff96b326320bed21b785cc": "\\left(\\partial ^{2}+m_{0}^{2}\\right)\\varphi (x)=j_{0}(x)", "19db670847403247777e2501e38b2186": "p,q\\vdash (p\\land q)", "19dbbc35c052b7a07c9435928d957f96": "{\\sqrt {\\frac {75.625}{BAF}}}", "19dccc13f5d90e676c4582fbcb18122a": "W'=-p_{n-1}\\,W,", "19dd05a36a09c3e84a348ecc349229b5": "P_{3}=0", "19dd0627ce59cfd646f739d6eb1bdedb": "I_{2}=\\left({\\frac {Z_{1}}{Z_{1}+Z_{2}}}\\right)I", "19dd2025a9538bd04f531c342b0970c5": "h(x)=\\int {{\\frac {1}{v(x)}}dx}.", "19dd5c655a402ccb3d9130e134eb86df": "Z_{\\mathrm {S} }", "19dd745b55ffbd6ab638d5a9e5c2b72a": "{\\mu _{roll}}", "19dd7be9c3492ddb11062df460ae1d13": "e_{\\bar {\\alpha }}=L^{\\gamma }{}_{\\bar {\\alpha }}e_{\\gamma }", "19dd8f409402ed2ebf0a7f7e4bf00cd2": "{\\frac {{\\sqrt {3}}+x}{{\\sqrt {3}}-x}}=2", "19dd93004dcba98538034e183e1129b0": "u\\in \\mathrm {Im} (\\theta )", "19ddfa387dad84759572266b01b283e5": "[v'\\;\\|\\;N\\;\\|\\;[u\\;\\|\\;v\\;\\|\\;M]_{h}]_{m}\\!\\rightarrow \\![w\\;\\|\\;M]_{h}\\;\\|\\;[w'\\;\\|\\;N]_{m}", "19de00ba89c28bd16f9e6e4d89e87a2c": "\\mathbb {E} ^{g}[1_{A}\\mathbb {E} ^{g}[X\\mid {\\mathcal {F}}_{t}]]=\\mathbb {E} ^{g}[1_{A}X]", "19de22b6b6f1eaa2c81473906c2cb0ce": "\\beta _{n}^{\\prime }=\\sum _{j\\geq 0}{\\frac {(\\rho _{1};q)_{j}(\\rho _{2};q)_{j}(aq/\\rho _{1}\\rho _{2};q)_{n-j}(aq/\\rho _{1}\\rho _{2})^{j}\\beta _{j}}{(q;q)_{n-j}(aq/\\rho _{1};q)_{n}(aq/\\rho _{2};q)_{n}}}.", "19de55e31f7adde48f9d4ff1a7b7cd58": "s_{k+1}=\\pm {\\frac {b}{GCD(a,b)}}", "19de5995f93d1366db2003d2f2dc9352": "\\beta (Y,X)", "19de5ea00c082bfabab8ed300916e90d": "P(t)=e^{Qt}=\\sum _{n=0}^{\\infty }Q^{n}{\\frac {t^{n}}{n!}}\\,,", "19de801e778d3555e11f8e65f6cad3d9": "{\\mathfrak {d}}", "19dec9ddba44f4d334bf71b538bf9935": "j_{\\ell }(kr')={\\frac {(kr')^{\\ell }}{(2\\ell +1)!!}}+O((kr')^{\\ell +2})", "19df0bd299379fa1b03b3befb0919996": "\\Theta |n\\rangle ", "19df16fdd3b97d5e761ed662490747b6": "R_{z}={\\frac {1}{Y_{21}}}\\qquad R_{x}={\\frac {1}{Y_{11}-Y_{21}}}\\qquad R_{y}={\\frac {1}{Y_{22}-Y_{21}}}\\,", "19df1c2726ed43128440c1157f72a937": "\\psi ", "19df4ca33417c98bcc9aa9bd7dc3e761": "{\\begin{aligned}Nu&=-b\\partial _{x}u^{2}=-b\\partial _{x}(u_{0}+u_{1}+u_{2}+u_{3}+\\cdots )(u_{0}+u_{1}+u_{2}+u_{3}+\\cdots )\\\\&=-b\\partial _{x}(u_{0}u_{0}+2u_{0}u_{1}+u_{1}u_{1}+2u_{0}u_{2}+\\cdots )\\\\&=-b\\partial _{x}\\sum _{n=1}^{\\infty }A(n-1),\\end{aligned}}", "19df61df78d2023969e529969db8666b": "S={\\frac {2}{\\beta ^{2}(4+5\\alpha ^{2})}}", "19dfc03da5a2401a24bcd161e1c7005b": "\\mathbf {E} _{\\text{Electric dipole}}(\\mathbf {x} ,t)=Z_{0}(\\mathbf {H} _{\\text{Electric dipole}}\\times \\mathbf {n} )", "19dfe1ce8dcf67b391022a230d12ca02": "{\\begin{pmatrix}\\alpha ^{7}+\\alpha ^{-3}x&1\\\\1&0\\end{pmatrix}}{\\begin{pmatrix}x^{6}\\\\\\alpha ^{-7}+\\alpha ^{4}x+\\alpha ^{-1}x^{2}+\\alpha ^{6}x^{3}+\\alpha ^{-1}x^{4}+\\alpha ^{5}x^{5}+(\\alpha ^{7}+\\alpha ^{7})x^{6}+(\\alpha ^{-3}+\\alpha ^{-3})x^{7}\\end{pmatrix}}=", "19dfe2d759ec6d6869da6d2185fc0294": "\\phi (x,z)", "19dffbdddca00c828918d66ba4dc5c72": "{{\\Delta {\\hat {g}}} \\over {\\hat {g}}}\\,\\,\\,=\\,\\,\\,\\,{{{\\hat {g}}\\left({L+\\Delta L,\\,\\,\\,T+\\Delta T,\\,\\,\\,\\theta +\\Delta \\theta }\\right)\\,\\,\\,-\\,\\,\\,{\\hat {g}}\\left({L,\\,\\,T,\\,\\,\\theta }\\right)} \\over {{\\hat {g}}\\left({L,\\,\\,T,\\,\\,\\theta }\\right)}}{\\mathbf {\\,\\,\\,\\,\\,\\,\\,\\,Eq(4)} }", "19e05ebd8991fb1887041c225c155b85": "{\\mathit {EQ_{2}}}", "19e072a673ccaf1a4033f4ad50f09e4f": "cos(\\beta )", "19e0b5b8efdc14659ec37dcc46c83dc0": "\\sigma _{2}=\\left[{\\begin{matrix}0&i\\\\-i&0\\end{matrix}}\\right].", "19e0bdfb900e2453070209e52b1652fd": "{\\mathfrak {T}}_{\\mu }^{\\nu }=T_{\\mu \\gamma }\\,g^{\\gamma \\nu }\\,{\\sqrt {-g}}.", "19e0da4efbbbc66c49565ee1ad98bdec": "\\Phi (r)=k_{e}{\\frac {Q_{1}}{r}}", "19e10bb4977d06dc80bb1a04c601f19f": "\\left(p,q,Tr(g),Tr(g^{k})\\right)", "19e11abbbd15592e0b71681078a54b56": "\\langle \\xi _{i}(t)x_{k}\\rangle =\\langle x_{i}\\xi _{k}(t)\\rangle ", "19e16962430ae761a0a084cf561c0fb7": "{\\hat {B}}_{\\mathbf {q} }^{\\dagger }", "19e1d7a54cc56a652bf6cdf9232f1b08": "x_{i}=w_{i}\\cdot {\\frac {M}{M_{i}}}", "19e1dc236e52a6b58a1b97a3c875f3e3": "e=(e_{1}~e_{2}~\\ldots ~e_{N})^{T}", "19e1f9ce7932cf347eb1e6c8eb2e9844": "f\\star g=m\\circ e^{{\\frac {i\\hbar }{2}}\\Pi }(f\\otimes g),", "19e2174c07384de3e01d224eb444dd85": "1/I+1/O=1/F", "19e2212113b16a50c0b6dca8e2d281f1": "\\sigma (\\mathbf {x} )=\\mathbf {0} ", "19e233c942f346c6d3c194b220ec2dcb": "(R\\bowtie S)\\cup ((R-\\pi _{r_{1},r_{2},\\dots ,r_{n}}(R\\bowtie S))\\times \\{(\\omega ,\\dots \\omega )\\})", "19e2582f3ecf941ec253e61681428bd5": "S_{25}=\\sigma _{put,25}-2\\sigma _{atmf}+\\sigma _{call,25}", "19e25da981a1c93f5b6b1014a8da5d0b": "t_{0.975,11}=2.20", "19e2adc1d3d62258a2e756cc95311b79": "ku", "19e2c87dfcd499d96746ef27b99584a5": "\\Delta H", "19e2d001c09a4f577d39d167abf246ba": "\\langle u,\\ v\\rangle =\\|u\\|\\|v\\|\\cos \\theta \\ ;\\ (-\\pi <\\theta \\leq \\pi )\\ ,", "19e30df859ba67aeef61c8f5d50ff631": "([A,B]:[B,D])\\diamond ([A,C]:[C,D])\\,\\!", "19e35b7c71424608ffca4314bad0f968": "R/i-P", "19e363414ccd6e4acbcb5d2871c81440": "i\\in \\{1,\\dotsc ,n\\}", "19e371803b9249b55ff650318c5f403f": "G(z)={\\begin{cases}\\exp \\left\\{-\\left(-\\left({\\frac {z-b}{a}}\\right)\\right)^{\\alpha }\\right\\}&zt+s) \\over \\Pr(X>t)}=\\Pr(X>s).", "19e615e86e3d41cab8e2947599fb6087": "[Z_{0},\\dots ,Z_{n}]{\\sim }[1,z_{1},\\dots ,z_{n}],", "19e65262ebd183549d4ac03426fedc99": "407=4^{3}+0^{3}+7^{3}", "19e682ea3bbfe7bb2e5f7698d09654cb": "e={\\sqrt {1+{\\frac {2EL^{2}}{k^{2}m^{3}}}}}", "19e6a1351f4593fa6a5da49794269627": "\\scriptstyle (T-T_{\\mathrm {g} })\\,\\leq \\,t\\,<\\,T", "19e6a1d69386ac0d16c7dde15eef91da": "z=z_{2},", "19e6e1c159523f625c67d4492ed1c447": "f_{c}'(z_{0})\\,", "19e6f6f3a2e7fc4953ef59e0428767b8": "\\eta _{\\mathrm {max} }={T_{H}-T_{C} \\over T_{H}}{{\\sqrt {1+Z{\\bar {T}}}}-1 \\over {\\sqrt {1+Z{\\bar {T}}}}+{T_{C} \\over T_{H}}},", "19e7217bfb2e45ed28490e0382fb1b5b": "I_{a}", "19e72e04be6bd147dff922246c7f44c3": "g_{\\mathrm {N} }", "19e750dab3777f80f35c1f2a308d91c8": "S_{ff}(\\ell )=O(\\ell ^{-s})\\quad {\\rm {{as\\ }\\ell \\to \\infty }}", "19e7510288fd3003c4417537f390de4f": "r=\\,\\alpha \\beta ;", "19e761868636d00524df5d7c593b12b7": "\\textstyle i=1,\\ldots ,n", "19e7d50d4f002dee55277c8e4ef278f2": "\\,p\\in [0,1]\\,", "19e863e91f9e87dfcc34b1a02d54acf8": "AB^{+}+h\\nu \\to A+B^{+}", "19e8654ae6dd5270a6a66ccabff752b2": "C{\\frac {dV}{dt}}+V{\\frac {dC}{dt}}=-K\\cdot C+{\\dot {m}}\\qquad (8)", "19e8fe8dc8c51f66a419e33370a8184d": "\\delta \\psi _{out}=\\left(u+{\\frac {\\partial u}{\\partial x}}\\delta x\\right)\\delta y+\\left(v+{\\frac {\\partial v}{\\partial y}}\\delta y\\right)\\delta x.\\,", "19e91a54a86f07291e8fed48b3b39929": "{\\frac {k}{2}}(1+\\ln(2\\pi ))+{\\frac {1}{2}}\\ln |{\\boldsymbol {\\Sigma }}|", "19e92905a38d916db89789f0f2ad9ef7": "e_{i}^{(2)}", "19e97d692138430a6798e7982a6afb4a": "V(t)=Vo(e^{-t/\\tau })", "19e98ae1ed9addc1200ac4848dc319c8": "{\\frac {\\partial ^{2}f}{\\partial x_{i}\\,\\partial x_{j}}}(a_{1},\\dots ,a_{n})={\\frac {\\partial ^{2}f}{\\partial x_{j}\\,\\partial x_{i}}}(a_{1},\\dots ,a_{n}).\\,\\!", "19e9b798261f7e946e5f4224ca8bed43": "{\\frac {dy}{dx}}={\\frac {x}{y}}.", "19ea073c275437e1989504018be9d469": "\\varphi _{0}(\\beta )=\\omega ^{\\beta }", "19ea192b116c0c4596b4e4fa425109e2": "\\varphi (m)=2\\times 4=8\\,", "19ea3a21eded0317d0d66c37967a34bd": "\\langle x,y\\mid x^{m},y^{n},xy=yx\\rangle \\,\\!", "19ea6c34cad73649f5458684f75d4938": "x_{t}=x_{t}^{*}+\\eta _{t}\\,,", "19ea97c73dd38f9fd6de34bbc3643d8e": "\\gamma _{1}=1", "19eacb3d42b4cf08e0cdfd1156aa2a11": "\\{y\\mid yWx\\}", "19eb17812e5818b7937d6aaad1924a31": "{\\boldsymbol {F}}={\\boldsymbol {\\mathit {1}}}+\\gamma \\mathbf {e} _{1}\\otimes \\mathbf {e} _{2}", "19ebf996e999e8465f8f46aacb4871ba": "\\sigma _{y}=\\sigma _{2}\\,\\!", "19ec015a1157fe19cc585137ffc7498c": "\\psi (\\mathbf {r_{1}} ,\\mathbf {r_{2}} ,\\,t)=\\psi _{A}(\\mathbf {r_{1}} ,\\,t)\\psi _{B}(\\mathbf {r_{2}} ,\\,t)", "19ec3635ded4bd76dd08dbcf77c70174": "q_{\\mathit {left}}", "19ec3a462af7814b2ec18c24ccc61713": "\\ P_{r}=S{\\frac {\\lambda ^{2}}{4\\pi }}", "19ec53dfabe89169ed7a8bc65cd88b59": "X\\subseteq E", "19ec5e9b380068ccdeb1484dcb25ed2a": "\\kappa =\\pi =\\varepsilon =0\\,,\\quad \\rho ={\\bar {\\rho }}\\,,\\quad \\tau ={\\bar {\\alpha }}+\\beta \\,.", "19ecaeaf6300fcee068abbd62a647346": "\\sum _{i=1}^{k}c_{i}=\\sum _{i=1}^{k}d_{i}=1", "19ece1047cce719d82cc4d85015d7680": "{\\mathcal {C}}^{k}", "19ecf8797efa6d5fe048d4cf9eeeb878": "{\\tilde {R}}_{n}={\\begin{bmatrix}R_{n}\\\\0\\end{bmatrix}},", "19ed6e5bc03178928ac93ba1c3f37fa4": "[[x,y],z]", "19ed86b1d0789172abc1b1d706b65638": "{\\mathcal {F}}_{x}=\\varinjlim _{U\\ni x}{\\mathcal {F}}(U),", "19edf4beb4324355ecacca062fb1591a": "\\operatorname {Var} (X)=\\sum _{k=0}^{n}{n \\choose k}p^{k}(1-p)^{n-k}(k-np)^{2}=np(1-p),", "19ee71762e0afea39366e1117aa8c7d4": "A=CF", "19ee9cac88d24cff99fbd8fb801c3999": "{\\mathcal {L}}=p_{1}\\partial _{x}+p_{2}\\partial _{y},", "19eecdc3c1f102237a0373c741b27c67": "{\\mathcal {Z}}'\\{x[z]\\}={\\frac {{\\mathcal {Z}}\\{x[z+1]\\}}{z+1}}", "19ef01f138f295a96bea25ef79c3ab01": "\\int _{C}\\mathbf {F} (\\mathbf {r} )\\cdot \\,d\\mathbf {r} =\\int _{a}^{b}\\mathbf {F} (\\mathbf {r} (t))\\cdot \\mathbf {r} '(t)\\,dt=\\int _{a}^{b}{\\frac {dG(\\mathbf {r} (t))}{dt}}\\,dt=G(\\mathbf {r} (b))-G(\\mathbf {r} (a)).", "19ef17a8f2028f4e879aff3b3bc287e1": "{D+\\Delta {P}_{t} \\over D_{i}}={\\Delta {P}_{t} \\over P-\\Delta {P}_{t}}\\left({P-\\Delta {P}_{t} \\over D_{i}}\\right)+{D \\over D_{i}}", "19ef2509d2a6fa6c934d4610a440c3a4": "|R_{2}|\\leq {\\frac {9}{10}}{\\frac {1}{2r+1}}{\\frac {1}{3^{2r+1}}};", "19f03da607592325113d5e0ed71242fb": "\\mathbf {g} ={\\frac {1}{\\mu _{0}c^{2}}}\\mathbf {E} \\times \\mathbf {B} \\,,", "19f044a9f1a4d75e246b101e61f63a95": "\\mid {\\bar {q}}\\mid \\rho _{c}", "1a0e7c104e11587e09cdb6359fbf9b32": "(4*x^{2}+644*x)*(4*x^{2}+644*x+25921)=", "1a0eda170261067891c463698a39840e": "|s_{n+p}-s_{n}|=|a_{n+1}+a_{n+2}+\\cdots +a_{n+p}|<\\varepsilon .", "1a0f19f5d9bb73a80767125a095830c2": "\\mathbf {E} \\left[|{\\hat {x}}_{T}(\\omega )|^{2}\\right]=\\mathbf {E} \\left[{\\frac {1}{T}}\\int \\limits _{0}^{T}x^{*}(t)e^{i\\omega t}\\,dt\\int \\limits _{0}^{T}x(t')e^{-i\\omega t'}\\,dt'\\right]={\\frac {1}{T}}\\int \\limits _{0}^{T}\\int \\limits _{0}^{T}\\mathbf {E} \\left[x^{*}(t)x(t')\\right]e^{i\\omega (t-t')}\\,dt\\,dt'.", "1a0f60f3a01da6dc8131bf7902834cd3": "Z_{\\mathrm {i+1} }=Z_{\\mathrm {0,i} }{\\frac {Z_{i}+jZ_{\\mathrm {0,i} }\\tan(\\beta _{i}l_{i})}{Z_{\\mathrm {0,i} }+jZ_{i}\\tan(\\beta _{i}l_{i})}}", "1a0f96bf5b18d6c2d897fc713bb757c5": "x_{i}=\\sum _{j=0}^{n}a_{ij}x_{j},\\;\\;\\;1\\leq i\\leq n,", "1a0fa318ebc1b3fa6ffac01d86c286ce": "\\mathbf {M} ", "1a1076dd909636723e179faa34a243c9": "a^{n}u[n]", "1a111a571f475971c073db0569639061": "\\vdash A\\Rightarrow B", "1a114ad255499321be443a99fe3223b2": "L_{d}\\ll L_{nl}", "1a1177a1d8ec7c65c09dc01365e8103d": "\\log z", "1a1211130e9ebcd2bb22893c98729af4": "e", "1a12ad8dd212e7bca659a1069c60eaf7": "{\\vec {\\sigma }}=\\sigma _{1}{\\hat {x}}+\\sigma _{2}{\\hat {y}}+\\sigma _{3}{\\hat {z}}\\,", "1a12c3f9274f0a122401680b3f987135": "i=\\{1,2\\}", "1a12d22d7ba0eeb246c640888ec4be84": "\\int _{a}^{b}f(x)\\,h(x)\\,dx=0", "1a132bc2cbba830a7b68f8e698ed0c92": "U(\\rho ,z)=2\\pi a\\int _{0}^{\\infty }\\exp {\\left[-{\\frac {\\rho '}{\\sigma }}\\right]^{2}}J_{0}(2\\pi \\rho '\\rho /\\lambda z)\\rho '\\,d\\rho '", "1a1353b888ed0c40c98d7485d51a56c7": "{\\begin{aligned}{\\frac {1}{v_{0}}}&={\\frac {K_{M}^{A}}{v_{\\max }{[}A{]}}}+{\\frac {K_{M}^{B}}{v_{\\max }{[}B{]}}}+{\\frac {1}{v_{\\max }}}\\end{aligned}}", "1a136410efbfa73a600f642108604711": "a\\wedge b:={\\frac {1}{2}}(ab-ba)=-(b\\wedge a)", "1a13ab2a75388a97891f5da138805178": "a_{10}={\\mathcal {L}}(p_{7})+p_{3}p_{7}+p_{1}p_{9},", "1a13d972a2c8473f075ded6eca465169": "\\wp ", "1a14960d92068132a0bf2b4af919afa4": "\\mathrm {T_{Low}} (f)={\\frac {1}{1+jf/f_{1}}}.\\ ", "1a1531d53a55979d47e156aacec78d8f": "\\nu _{2}=p[\\sigma _{1}^{2}+\\delta _{1}^{2}]+(1-p)[\\sigma _{2}^{2}+\\delta _{2}^{2}]", "1a15595311fac89ba9e35e0705565675": "-y\\,\\partial _{x}+x\\,\\partial _{y}", "1a157ce446c4d4303add1ed2743b9c53": "\\theta =1.220{\\frac {\\lambda }{D}}", "1a158c616a63418bcb38ebd16dfb5a25": "\\sum _{S\\in 2^{N}\\setminus \\{\\emptyset \\}}\\alpha (S)v(S)\\leq v(N).", "1a158e580034a26770fc56570c0bd8b6": "{\\bar {E_{C}}}", "1a159f9c4e423e4bdfc97e70fa40ef43": "5=(1+2i)(1-2i)", "1a15a0b20d03015288edec6e2323ff81": "(\\mu ,\\sigma ^{2})", "1a15eba66eae25d869ef4b46d3e95fc1": "(A.2.b)\\quad \\psi _{,\\,\\rho \\rho }+{\\frac {1}{\\rho }}\\psi _{,\\,\\rho }+\\psi _{,\\,zz}=0", "1a160589f3e43c74f4937249d3696eac": "{\\frac {\\sigma ^{2}}{\\xi ^{2}}}[2\\alpha \\csc(2\\alpha )-(\\alpha \\csc(\\alpha ))^{2}]", "1a162ef8231f6511b998943140a294d5": "n=q-1", "1a168d63b24ed54acd26b68abca7bc3d": "\\alpha _{t}(s,a)", "1a16a59beb0861c9a3e55fc4a5302933": "\\phi -\\beta \\,\\!", "1a16e1d3b1f098c5eac89fad5b0643b1": "z^{2}+w^{2}", "1a16fb018d6deada23b0d6f36f4d4ccf": "S[i,n]", "1a172f0c2eec73530f9c8a46483152e9": "H_{(1)}\\ldots H_{(m)}", "1a1753442ad7ebc5ecf4e02c8bb0c2b0": "\\left({\\frac {\\partial S}{\\partial V}}\\right)_{T}=\\left({\\frac {\\partial p}{\\partial T}}\\right)_{V}", "1a175da2a613413afd8909856e55f324": "f:S\\to \\mathbb {R} ", "1a17f12edaea2b8c11634bfc7e635939": "n_{F}(z)=(e^{\\beta z}+1)^{-1}", "1a183e02d2dd8ea8a94f071e55b3d237": "P(T)\\approx 10^{-21.85}~~~~~~~~~~~(10^{4.3}\\log _{10}(1/\\epsilon )", "1a18da63cbbfb49cb9616e6bfd35f662": "2.3", "1a197f7095cfb3f9dc8dca1c7f53d11d": "{\\frac {v_{\\|}}{\\omega _{c}}}\\,{\\vec {b}}\\times \\left[{\\frac {\\partial {\\vec {b}}}{\\partial t}}+({\\vec {v}}_{E}\\cdot \\nabla {\\vec {b}})\\right].", "1a19a3958183c94db6c796cf1c7cfce5": "Mod(\\sigma ):Mod(\\Sigma ')\\to Mod(\\Sigma )", "1a19d562d550955d6f369224aaaefdda": "{\\frac {\\partial \\phi }{\\partial t}}+\\nabla \\cdot \\mathbf {j} =0", "1a19da4efeaa082b043e74b90523cfa1": "U_{0}(x,y)=e^{jkn\\Delta }e^{-j{\\frac {k}{2f}}[x^{2}+y^{2}]}U_{i}(x,y)", "1a19f25c6c41c70f170606d534a70c36": "\\mathbf {P} =\\left({\\frac {E}{c}},\\mathbf {p} \\right)\\,.", "1a1a263d52179f947b00a2d5e60d5c71": "\\operatorname {d} x", "1a1a7a2af9c4e7a342016c165269a8ab": "\\operatorname {DG} (a_{n};s)=\\sum _{n=1}^{\\infty }{\\frac {a_{n}}{n^{s}}}.", "1a1acd502bd38b524fe9a3514edbd68d": "\\mathbf {M} =\\mathbf {U} \\mathbf {D} \\mathbf {U} ^{-1}", "1a1af3040befd2ea8a0052c09d5cb367": "p\\in F", "1a1ba3480c81a5596fb27544a2ee115f": "w(t)\\sim {\\mathcal {N}}(0,\\sigma ^{2})", "1a1c01c0cafb38c4beaa366617390158": "M({\\vec {X}})=\\left({\\begin{array}{*{20}c}{{\\bar {\\mu }}_{1}+{\\bar {\\mu }}_{2}}\\\\{{\\bar {\\Sigma }}_{1}+{\\bar {\\Sigma }}_{2}}\\\\\\end{array}}\\right)", "1a1c70969df916e13c687dc39945b90a": "\\sum _{k}Y_{ik}a_{k}[n]+Y_{i}^{\\text{sh}}a_{i}[n]=S_{i}^{*}b_{i}^{*}[n-1]\\qquad (n=0,\\ldots ,\\infty )", "1a1cf0c14d19a7c9da326d8c9da1d74e": "{\\mathbb {R}}^{d}", "1a1d180acc07655b2faceaa901393763": "\\mathbf {GOP_{\\nu }} ", "1a1d3891d7ea2f2db154c402cd39a885": "({\\overline {Y}}-Y)/{\\overline {Y}}=c(u-{\\overline {u}})", "1a1d739bbc059614509d400db7ebbaab": "\\beta _{i}s\\equiv w_{i}rs\\equiv w_{i}{\\pmod {q}}.", "1a1da387d9d61a52ef19911a8f5a968a": "=\\left[{\\begin{array}{c c | c c c c | c | c c}1&2&4&7&8&14&3&12&21\\\\4&5&16&28&20&35&6&24&42\\\\\\hline 2&4&5&8&10&16&6&15&24\\\\3&6&6&9&12&18&9&18&27\\\\8&10&20&32&25&40&12&30&48\\\\12&15&24&36&30&45&18&36&54\\\\\\hline 7&8&28&49&32&56&9&36&63\\\\\\hline 14&16&35&56&40&64&18&45&72\\\\21&24&42&63&48&72&27&54&81\\end{array}}\\right].", "1a1e4db88a701533707cfb28adf26251": "V^{AB}=\\sum _{\\ell _{A}=0}^{\\infty }\\sum _{\\ell _{B}=0}^{\\infty }(-1)^{\\ell _{B}}{\\binom {2\\ell _{A}+2\\ell _{B}}{2\\ell _{A}}}^{1/2}\\sum _{M=-\\ell _{A}-\\ell _{B}}^{\\ell _{A}+\\ell _{B}}(-1)^{M}I_{\\ell _{A}+\\ell _{B},-M}(\\mathbf {R} _{AB})\\;\\left[\\mathbf {Q} ^{\\ell _{A}}\\otimes \\mathbf {Q} ^{\\ell _{B}}\\right]_{M}^{\\ell _{A}+\\ell _{B}}", "1a1e8192343516e070a02f53c2e12959": "\\mathbb {Z} [G]", "1a1e8c397346e7ddf74dcb73fc212c9a": "{\\mathcal {E}}=\\omega BA\\sin {\\omega t}", "1a1ec506a61023dc8691d6704389e898": "\\mathbf {J} _{\\alpha }=\\sum _{\\beta }L_{\\alpha \\beta }\\,\\nabla f_{\\beta }", "1a1f8c23f557524f6f963fd4dccfe6f9": "{\\tfrac {1}{2}}ab\\sin(C)", "1a1f9ccafae1fff64519f639e9c5df06": "2k_{\\text{F}}", "1a1fa4952566a429eca578933704d3ef": "\\lim _{n\\to \\infty }diam(C_{n})\\rightarrow 0", "1a1fdcd5797f1fb6a1319277cbbafbef": "p={\\frac {h}{2L}}{\\sqrt {n_{x}^{2}+n_{y}^{2}+n_{z}^{2}}}\\qquad \\qquad n_{x},n_{y},n_{z}=1,2,3,\\ldots ", "1a202ed3612f0fef7ffc7841b7b82c7c": "p_{0}=p+q+\\rho gz\\,", "1a203b4ddb47f654f0f97a816acbae22": "p(\\theta _{1},\\cdots ,\\theta _{m})=C\\prod _{1\\leq i\\leq m}(1-\\sigma \\cos \\theta _{i})\\prod _{1\\leq k0}\\left[{\\frac {1}{w}}\\ln \\left(R\\int _{-\\infty }^{\\infty }k(s)e^{ws}ds\\right)\\right]", "1a23fced934e003b1401139a270c5b7b": "\\scriptstyle x\\;\\in \\;(0,\\,\\infty )", "1a2476517cc63fc8afec181b597a1a38": "F\\star G", "1a2488dcc33fda995f00c0a54d200c51": "I={\\frac {1}{T}}\\int _{0}^{T}p_{\\mathrm {inst} }(t)v(t)\\,dt", "1a24cab6eed461fd75b52a97d07d762f": "Leader_{j}", "1a2594c0b021589ae05bad3d95f83dc8": "S_{i}^{2},n_{i},i=1,2", "1a25aa972d53d71e503d08368a715ad9": "{\\begin{matrix}{r \\choose 3}{4 \\choose 1}^{3}{52-4r \\choose 2}\\end{matrix}}", "1a25de39bf8617ac6e1a53c26679f691": "\\Delta ^{o}{\\overset {S^{1}}{\\longrightarrow }}{\\text{Fin}}_{*}{\\overset {F}{\\longrightarrow }}k{\\text{-}}\\operatorname {mod} .", "1a25fcfce4d11c72822d702ae83c331f": "V\\subset Y", "1a2643b3e60cdec43207da0d0cbfefc8": "LL(\\alpha ,1)=GPD(1,\\alpha ,1).\\,", "1a2647b75933b0328eae3f99f425dd3b": "U=Nk_{B}T^{2}\\left({\\frac {\\partial \\ln Z}{\\partial T}}\\right)_{V}~", "1a26aa505912b7e0fb043ef8b3f1733a": "I_{M-1}", "1a26b7963eb20eb0068bd30e0eb8a198": "\\Gamma (\\gamma )_{0}^{t}e_{\\alpha }(\\gamma (0))=\\sum _{\\beta }e_{\\beta }(\\gamma (t))g_{\\alpha }^{\\beta }(t)", "1a2757da1c9f59abaa141d0e59d0cea5": "\\omega ^{2}=\\omega _{pe}^{2}+{\\frac {3k_{B}T_{\\mathrm {e} }}{m_{e}}}k^{2}=\\omega _{pe}^{2}+3k^{2}v_{\\mathrm {e,th} }^{2}", "1a278dbd840884f28111395f12670b18": "{\\frac {p(x)}{(x-a)}}", "1a279d88b6b11a085b685b8a2c9b5aa5": "\\operatorname {E} [g(X)]=\\int _{-\\infty }^{\\infty }g(x)f_{X}(x)\\,dx", "1a27b65d4a17d6f364230c928295ea64": "G^{33}", "1a286a2615acbff05f1a75b5729813ee": "x=b_{0}+{\\underset {i=1}{\\overset {\\infty }{\\mathrm {K} }}}{\\frac {a_{i}}{b_{i}}}.\\,", "1a28a5cc2aa6aace2a45bec4532237a6": "\\varepsilon _{ij}={\\frac {1}{2}}(u_{j,i}+u_{i,j})\\,\\!", "1a28f3eca45061acd51ea9e7b4fa92c6": "\\Phi (t)=\\phi \\,(t){\\phi \\,}^{-1}(t_{0})", "1a2962da12d6f9ebf6bd4211b30e3137": "{\\mathfrak {h}}\\subseteq {\\mathfrak {g}}", "1a297ccc433d980263f70a9318599a89": "p(G(U_{\\ell }))", "1a2981eb7c3dfb8b3ce7b26baa6f1352": "f:{\\textbf {R}}\\to Y", "1a29b6c4530e41e3701fa1777ea8de7c": "\\displaystyle {D(x+h)\\geq D(x)+f_{\\varepsilon }(h)+|h|^{2}.}", "1a29c002755e05f297c99c693440ea60": "\\log \\left({\\frac {p(r)}{P}}\\right)\\approx {\\frac {R_{critical}}{r}}", "1a29d84310c1c1f5efb6d267e5c724fe": "\\displaystyle s_{i}=s_{i-1}(s_{i-1}-1)+1,", "1a29e4ae2a9c784e24bf7db9259012c7": "{\\text{Area of polygon (on the unit sphere)}}\\equiv E_{n}=\\Sigma -(n-2)\\pi .", "1a29eab4660c57a9984afd9d6316dfaf": "{\\sqrt {\\epsilon _{0}/2}}", "1a2a377d76bb9d8f65ce131a509c7442": "C_{\\bullet }\\mapsto H_{n}(C_{\\bullet })=Z_{n}(C_{\\bullet })/B_{n}(C_{\\bullet })", "1a2a50975b5f0f92f11ed99578ace24e": "{\\bar {V}}\\,", "1a2abe6648e807b90565e8f674fd916b": "\\int _{1}^{\\infty }x^{-\\alpha }\\;(x-1)^{\\alpha -\\beta -1}\\;G_{p,q}^{\\,m,n}\\!\\left(\\left.{\\begin{matrix}\\mathbf {a_{p}} \\\\\\mathbf {b_{q}} \\end{matrix}}\\;\\right|\\,zx\\right)dx=\\Gamma (\\alpha -\\beta )\\;G_{p+1,\\,q+1}^{\\,m+1,\\,n}\\!\\left(\\left.{\\begin{matrix}\\mathbf {a_{p}} ,\\alpha \\\\\\beta ,\\mathbf {b_{q}} \\end{matrix}}\\;\\right|\\,z\\right).", "1a2ae1978c4f7a32d6d8841c1144771c": "p_{f}", "1a2aeea258edc2e2631943480206b928": "D_{\\mu }D^{\\mu }\\phi +AF^{\\mu \\nu }D_{\\mu }\\phi D_{\\nu }(D_{\\alpha }D^{\\alpha }\\phi )=0", "1a2afd5fb3b5d4d3b5be9eb3799ae79a": "\\chi (\\mathbf {S} ^{n})=1+(-1)^{n}={\\begin{cases}2&n{\\text{ even}}\\\\0&n{\\text{ odd}}.\\end{cases}}", "1a2b216da572f46514b14211db41d931": "R=\\left({\\begin{array}{cc}a&0\\\\0&d\\end{array}}\\right)", "1a2b24ba1ab504e66e5da705604e7833": "f(r)=\\int _{0}^{\\infty }F_{\\nu }(k)J_{\\nu }(kr)k\\operatorname {d} \\!k", "1a2b5ab4a01b13e4d6f09a48da699851": "KC({\\mathcal {X}})=\\sum _{i\\neq j}KC(x_{i},x_{j})=2\\sum _{i\\left\\lceil {\\sqrt {kn-{\\frac {1}{2}}}}\\right\\rceil >\\left\\lceil {\\sqrt {kn}}\\right\\rceil ", "1a55cfd5c620ed82c9e8acd1de5cc345": "f(x)=\\exp(x)", "1a55f2f94e125f450f6dc10fd4efc87e": "\\sin \\alpha _{\\mathrm {s} }=\\cos h\\cos \\delta \\cos \\varphi +\\sin \\delta \\sin \\varphi ", "1a56230b94ffdd34834d107134ff55c3": "{\\boldsymbol {F(r)}}={q \\over 4\\pi \\varepsilon _{0}}\\sum _{i=1}^{N}q_{i}{{\\boldsymbol {r-r_{i}}} \\over |{\\boldsymbol {r-r_{i}}}|^{3}}={q \\over 4\\pi \\varepsilon _{0}}\\sum _{i=1}^{N}q_{i}{{\\boldsymbol {\\widehat {R_{i}}}} \\over |{\\boldsymbol {R_{i}}}|^{2}},", "1a568e8c0cad16ab1cb2c9fb56b3d4fc": "\\mathbf {K} _{t},\\mathbf {K} _{s}", "1a568f568605e054f69eb1345f32b2d7": "S(S-1)>8m+20", "1a56e49b4136ae2b142973db3bda0a2e": "(n+1)I_{n}=x^{n+1}e^{ax}-aI_{n+1},\\!", "1a57c2d9c9aa76af684beb70ab97851e": "d(x+a)=dx", "1a582a733c885e7a80817518f785f423": "\\phi ^{\\epsilon }(x)={\\begin{cases}2,&0&|{\\boldsymbol {b_{2}}}|^{2}+|{\\boldsymbol {b_{3}}}|^{2}{\\text{ (favorable, will decompose)}}\\\\|{\\boldsymbol {b_{1}}}|^{2}<&|{\\boldsymbol {b_{2}}}|^{2}+|{\\boldsymbol {b_{3}}}|^{2}{\\text{ (not favorable, will not decompose)}}\\\\|{\\boldsymbol {b_{1}}}|^{2}=&|{\\boldsymbol {b_{2}}}|^{2}+|{\\boldsymbol {b_{3}}}|^{2}{\\text{ (will remain in original state)}}\\end{aligned}}", "1a5e7589d1b67b2582217a61abe05b9e": "E_{k}={\\frac {1}{2}}I\\omega ^{2}={\\frac {1}{4}}mR^{2}\\omega ^{2}", "1a5ea4a4764a3cd2a7488159260c2cec": "Y_{t}(u)-T=Y_{c}(u).", "1a5fa80967ef93cd206235c69222e454": "-\\int \\!\\!\\!\\!\\int \\!\\!\\!\\!\\int _{V}\\nabla \\cdot \\mathbf {j} (\\mathbf {r} ,t)dV+\\int \\!\\!\\!\\!\\int \\!\\!\\!\\!\\int _{V}\\sigma (\\mathbf {r} ,t)dV=\\int \\!\\!\\!\\!\\int \\!\\!\\!\\!\\int _{V}{\\frac {\\partial \\rho (\\mathbf {r} ,t)}{\\partial t}}dV.", "1a60202321d13ec160707b458b4843e5": "OC={\\sqrt {OA^{2}-AC^{2}}}", "1a60a8a28e78e7f8687c68f3810ca48b": "{\\begin{bmatrix}F_{k_{1}}\\\\\\vdots \\\\F_{k_{M}}\\end{bmatrix}}={\\begin{bmatrix}(e^{\\lambda _{1}})^{k_{1}}&\\dots &(e^{\\lambda _{M}})^{k_{1}}\\\\\\vdots &\\ddots &\\vdots \\\\(e^{\\lambda _{1}})^{k_{M}}&\\dots &(e^{\\lambda _{M}})^{k_{M}}\\end{bmatrix}}{\\begin{bmatrix}\\mathrm {B} _{1}\\\\\\vdots \\\\\\mathrm {B} _{M}\\end{bmatrix}},", "1a60b54e92409d9942ddb745668bde54": "{\\hat {q}}_{\\rm {w}}", "1a60f2375c09a4f535e258d31ce6f15e": "\\mathbf {w} _{old}=\\mathbf {w} _{new}", "1a60f59a6950089a686df1c9c861db97": "\\ M_{heel_{max}}>D_{heel}\\times F_{forward}\\times {\\frac {sin(\\beta )}{cos(\\beta )}}", "1a61219ecd70908c2b27d5370742d76e": "f={\\frac {qB}{2\\pi m}}", "1a61e3bbdab6197f36a54dca32cc06c5": "\\log _{10}1000+1=3+1", "1a61fc3a62547c7e7e856e5fedc60324": "f(x_{1})", "1a6248c24cd550e0af12637d660f0e27": "{\\hat {e}}=r_{b}\\cdot {\\hat {r}}+\\cos \\alpha \\cdot ({\\hat {b}}-r_{b}\\cdot {\\hat {r}})+\\sin \\alpha \\cdot {\\hat {r}}\\times {\\hat {b}}", "1a625564ff18c476c6d14a4ca9aad569": "{\\dot {\\gamma }}={\\frac {8v}{d}}={\\frac {8\\left({\\frac {Q}{\\pi r^{2}}}\\right)}{2r}},", "1a62680a10418c04766f451e8b8fd3d6": "{ds}", "1a62a7f47cbbcdf21d7c4bd92c1ef382": "\\oint _{C}f(z)\\,dz=\\oint _{C}{\\frac {1}{(z^{2}+1)^{2}}}\\,dz=\\oint _{C}{\\frac {\\frac {1}{(z+i)^{2}}}{(z-i)^{2}}}\\,dz=2\\pi i{\\frac {d}{dz}}\\left({1 \\over (z+i)^{2}}\\right){\\Bigg |}_{z=i}=2\\pi i\\left({\\frac {-2}{(z+i)^{3}}}\\right){\\Bigg |}_{z=i}=2\\pi i({\\tfrac {1}{4i}})={\\frac {\\pi }{2}}", "1a6306e2eff59a0d4ac22658f931d547": "{\\ddot {u}}_{n}={\\left({\\frac {c}{\\Delta x}}\\right)}^{2}\\left(u_{n-1}\\ -\\ 2u_{n}\\right)", "1a634659de31ceecf01934f5f856d863": "O\\lnot A", "1a6377c12b6296d58f067ea886528e95": "X=\\{n_{i}\\}", "1a63a4784dc7678e17a6b8c2746d7332": "(p',q')\\in R", "1a63ae2b14ef48ddee17fbf5f27ccf57": "P_{i}={\\frac {Y_{i}}{\\sum _{i=1}^{k}Y_{i}}}", "1a63e745c9b2a53946eda467620d3c47": "\\zeta =\\exp \\left({\\frac {2\\pi i}{n}}\\right).", "1a6469bf3cc8e2b0bd9f6609934db745": "\\langle \\Gamma (\\gamma )_{s}^{t}X,\\Gamma (\\gamma )_{s}^{t}Y\\rangle _{\\gamma (t)}=\\langle X,Y\\rangle _{\\gamma (s)}.", "1a647bee1de03e394b83433055b43e77": "P=\\mathbf {Tr} _{\\mathbb {Q} (\\zeta )/L}(\\zeta ^{j})", "1a647cf8bf64c1b5ba54475e698bc332": "W_{im}(X)", "1a648b693f4a80f009e6cd7d2c04d73a": "x{\\frac {B^{y}-1}{B-1}}", "1a64c2793eebda69654632af2b594d7d": "E(Nl,t)=1-(1-p)Nl,t", "1a650918887bebae77d29e2898e366bb": "D_{1}\\equiv (\\exists z_{1}...z_{m+1})\\phi (z_{a_{1}^{1}}...z_{a_{k}^{1}},z_{2},z_{3}...z_{m+1})\\equiv (\\exists z_{1}...z_{m+1})\\phi (z_{1}...z_{1},z_{2},z_{3}...z_{m+1})", "1a655149dd84d1ce4c8423604d4b91c0": "=\\{\\gamma ^{\\mu },\\gamma ^{\\nu }\\}\\gamma ^{\\rho }\\gamma _{\\mu }-\\gamma ^{\\nu }\\gamma ^{\\mu }\\gamma ^{\\rho }\\gamma _{\\mu }\\,", "1a655bf339675674ce1d65dac8c4d7a3": "{\\mathcal {F}}(\\{x_{n}\\})_{k}=X_{k}", "1a655c6a5a5f4c2b16fdac55057f3b15": "{\\begin{aligned}|6x^{4}-2x^{3}+5|&\\leq 6x^{4}+|2x^{3}|+5\\\\&\\leq 6x^{4}+2x^{4}+5x^{4}\\\\&\\leq 13x^{4}\\\\&\\leq 13|x^{4}|\\end{aligned}}", "1a65805ba6780532940834717af63f05": "(\\partial V)_{T}=-(\\partial T)_{V}=-\\left({\\frac {\\partial V}{\\partial P}}\\right)_{T}", "1a65c9370bab7419505cbf3054197b60": "-\\hbar ^{2}c^{2}\\mathbf {\\nabla } ^{2}\\psi +m^{2}c^{4}\\psi =-\\hbar ^{2}{\\frac {\\partial ^{2}}{(\\partial t)^{2}}}\\psi .", "1a660d50a5bde5f2a00182b539e5ee45": "{\\frac {\\Gamma (2+\\nu -b)}{2\\Gamma (2+\\nu -b+a)}}\\left({\\frac {k}{2}}\\right)^{\\nu }e^{-{\\frac {k^{2}}{4}}}\\,_{1}F_{1}\\left(a,2+a-b+\\nu ,{\\frac {k^{2}}{4}}\\right)", "1a6625b838bfc2f2a625943886df5fc4": "\\left({\\begin{smallmatrix}9&15\\\\-10&-10\\\\\\end{smallmatrix}}\\right)", "1a6685e8682360033668f96a1e356ee8": "\\mathbf {a} =(a_{1},\\dots ,a_{n})", "1a66874f4d02e4e1f0e1fb1fd5eb0deb": "Z=n_{i}\\times [Z]_{i}", "1a66fe7666f8c838a98f68842592af79": "\\scriptstyle [-\\infty ,\\,-{\\frac {1}{2}}]\\,\\cup \\,[{\\frac {1}{2}},\\,+\\infty ]", "1a672d1ca022697d18a7c264effb2e55": "m_{1}=m_{2}=m_{3}=\\cdots =m_{n}\\,", "1a675014f12d411e1f6c3e4bbd8cd29a": "A(z)=0.5[P(z)+Q(z)]", "1a67ba8ad67cfa9f00c9acfb8486de3c": "F_{s}=\\mu _{s}N\\,", "1a6808b414be52a7e29392d502dd3f3b": "(1-a)(1-b)(1-c)(1-d)+a+b+c+d\\geq 1.\\,", "1a6870c1806d0999d6f50ff05d9997fc": "x_{0}^{\\mu }=0", "1a68e1921524f3940b32459a97a922a5": "(\\wedge ){\\frac {X\\cup \\{A\\wedge B\\}}{X\\cup \\{A,B\\}}}", "1a68ee730acf9a84d5cdeca0f49b3d4b": "\\eta (\\gamma ,T)", "1a6966dd780373beb846f896352fd840": "{\\frac {r_{01}}{z}}", "1a698ec3c23dd6854144511551a1037b": "\\det {\\boldsymbol {F}}", "1a69c882a15ce6b8e0ff48212cc1d7ca": "T=k[u,z_{2},\\dots ,z_{d}]", "1a69e6851da9ee7d69d5853b8c28c298": "\\tau .", "1a69eeca39dc3f620fc2696ee7cd4506": "\\mathbf {x_{a}} ", "1a69f0fe0579f3c6e4481328710bc443": "|\\alpha _{i}|={\\sqrt {p}}", "1a6a271b54c4eca5c4f826114f8b1153": "c2^{k}.", "1a6a8f5aa0a5827bf2aa808d5457f3b6": "{Gr}(r,{\\mathcal {E}}\\otimes _{O_{S}}k(s))", "1a6a97c01b352176c29df771091402c9": "\\ e_{2}=(0,1)", "1a6ae5c0baad0c27ffbe1ba00f30713c": "{\\begin{array}{cc}{\\begin{array}{rrr}\\\\&1&\\\\2&&\\\\\\\\&&/3\\\\\\end{array}}{\\begin{array}{|rrrr}6&5&0&{\\text{-}}7\\\\&&2&\\\\&4&&\\\\\\hline 6&&&\\\\2&&&\\\\\\end{array}}\\end{array}}", "1a6b1b4f9c5a3d10cb92eba5457a6fca": "(AB+BC)-(AC')=n\\lambda ,\\,", "1a6b321258d385e619bc65e5f3f61b81": "x=\\lambda \\,", "1a6b5bc401b71fea09fdb9d86e4e72eb": "\\operatorname {E} [X]={\\frac {1}{k\\theta }}+{\\frac {1}{k\\theta }}+\\cdots +{\\frac {1}{k\\theta }}={\\frac {1}{\\theta }}.", "1a6b8149214fbc8d38ad67c14f0f8ecc": "\\chi ={\\frac {x}{x_{c}}}\\Rightarrow x=\\chi x_{c}.", "1a6bb2427289563c8608a0e17d35c863": "G^{\\wedge }", "1a6be56064dd18db1a86c91a572d9722": "ds^{2}=[a(u)(x^{2}-y^{2})+2b(u)xy]du^{2}+2dudv+dx^{2}+dy^{2}", "1a6beb77bddb2f66cf392e0dbd85aa69": "(p-1)/2", "1a6c3467aa7fd759747d4a7fb823f2ec": "\\Phi _{D}\\equiv \\int _{S}\\mathbf {D} \\cdot \\mathrm {d} \\mathbf {A} ", "1a6c429fe21d3716bdb6e6b304c02e99": "\\alpha \\approx {\\frac {3\\pi m}{r}}{\\sqrt {\\beta }}={\\frac {3\\pi m}{r}}\\sin(\\theta ).", "1a6c4a7d6e569ed632794aff1e5b1f63": "\\Pi _{2}^{P}", "1a6c5ef3aa87a5a6d13d16673a97359b": "{\\frac {{\\vec {H}}\\{0.133;\\;0.65;\\;0\\}}{\\|{\\vec {H}}\\|}}={\\frac {{\\vec {H}}\\{0.133;\\;0.65;\\;0\\}}{\\sqrt {0.133^{2}+0.65^{2}}}}={\\frac {{\\vec {H}}\\{0.133;\\;0.65;\\;0\\}}{0.668}}=\\{0.20048;0.979701;0\\},", "1a6c7be6c6aaf90360222418445b8eae": "{\\mathfrak {sl}}_{3}(\\mathbf {C} )", "1a6c8740525a46fc102b34d03a1cf2e2": "\\mathbf {v} =(v_{x},v_{y})", "1a6c8e8b7623d6584b5d7f5281a56539": "{100 \\over {\\sqrt {4}}}", "1a6cd631bb5c47f70bd072a4ffc75e9c": "\\displaystyle {\\|\\sum _{i=m}^{n}T_{i}v\\|^{2}\\leq \\sum _{i,j\\geq m}|(T_{i}v,T_{j}v)|.}", "1a6d08f3912bb21c2091322a8e239924": "l(t,s)=0", "1a6d6a8904fb0ea0f4e465ab4f426ff4": "\\Pr[R(x,y)=0]>{\\frac {1}{2}},{\\textrm {if}}\\,f(x,y)=0", "1a6dac261da09965540c1a15d7e0eb52": "f:={\\cfrac {1+2R}{2(1+R)}}|\\sigma _{1}-\\sigma _{2}|^{m}+{\\cfrac {1}{2(1+R)}}|\\sigma _{1}+\\sigma _{2}|^{m}-\\sigma _{y}^{m}\\leq 0", "1a6e4e0f467fc9d39da1758ad2129208": "{\\begin{aligned}A_{mn}&={\\frac {4}{ab}}\\int _{0}^{a}\\int _{0}^{b}\\varphi (x_{1},x_{2})\\sin {\\frac {m\\pi x_{1}}{a}}\\sin {\\frac {n\\pi x_{2}}{b}}dx_{1}dx_{2}\\\\B_{mn}&={\\frac {4}{ab\\omega _{mn}}}\\int _{0}^{a}\\int _{0}^{b}\\psi (x_{1},x_{2})\\sin {\\frac {m\\pi x_{1}}{a}}\\sin {\\frac {n\\pi x_{2}}{b}}dx_{1}dx_{2}\\,.\\end{aligned}}", "1a6ea6c95ed75bdd123541f9886b3f6c": "A_{1}^{x}+B_{1}^{y}=C_{1}^{z}", "1a6eefaf05c3f65caab7b869ff68e205": "i=1,2,\\dots ,m", "1a6fefa5ee8dd24299f086f37a3e090c": "W=2({\\mathcal {L}}(X;{\\hat {\\mu }},{\\hat {d}})-{\\mathcal {L}}(X;{\\hat {\\mu }},1))", "1a6ff258f201b594f40a5b69b3192fc6": "{\\tfrac {360^{\\circ }}{n}}", "1a703131a00e1bcf0abc991e3982641e": "\\rho \\left({\\frac {\\partial \\mathbf {v} }{\\partial t}}+\\mathbf {v} \\cdot \\nabla \\mathbf {v} \\right)=-\\nabla p+\\mu \\nabla ^{2}\\mathbf {v} +\\left({\\tfrac {1}{3}}\\mu +\\mu ^{v})\\nabla (\\nabla \\cdot \\mathbf {v} \\right)+\\mathbf {f} ", "1a70ea4c892391f380b78d9ae28a0a4b": "{\\begin{aligned}\\int \\cos ^{2}x\\,dx\\,&=\\,\\int \\left({\\frac {e^{ix}+e^{-ix}}{2}}\\right)^{2}dx\\\\[6pt]&=\\,{\\frac {1}{4}}\\int \\left(e^{2ix}+2+e^{-2ix}\\right)dx\\end{aligned}}", "1a70f94acac1a1cff56a30b320c15e50": "\\int _{0}^{\\infty }{\\frac {\\sin px}{x}}\\,dx={\\begin{cases}\\pi /2&{\\text{if }}p>0\\\\0&{\\text{if }}p=0\\\\-\\pi /2&{\\text{if }}p<0\\end{cases}}", "1a71226f50ca2b457f76a271d8b46749": "y_{0}\\in \\mathbb {R} ^{m}", "1a7127e19d14f8076463305f808a4fe6": "\\scriptstyle \\{e_{(a)}\\}_{a=1\\dots 4}", "1a71788fbb1b36a1b36a07b94700785a": "\\kappa ={\\frac {8\\pi G}{c^{4}}}", "1a71a6001ed87018a99f82028678afb8": "C={\\frac {n_{c}P(c|{\\vec {y}})-1}{n_{c}-1}}", "1a71a708ba65371d2a498f5d1c48162a": "r\\Pi \\,\\Delta t=\\Delta \\Pi ", "1a71c2d916e5a78c3e8684550bc4abc9": "\\bigtriangleup _{SO,dB}=ISOI_{dBm}-P_{in,f,dBm}", "1a71c8ad897631136005ffa374e15e10": "h_{o}^{2}={\\frac {2L^{2}k_{o}}{rD_{c}C_{o}}}", "1a71c95dce1de5c5c2ae18ca8819d0e7": "1.44\\log _{2}(1/\\epsilon )", "1a72048d807c29547d5382b269caba61": "{\\hat {2}}", "1a723406116be4bce41d9db031b8d0e1": "p_{ij}\\geq 0", "1a72a02ddd1d46ddf960993b0b964940": "n_{e}\\propto e^{e\\Phi /k_{B}T_{e}}.", "1a72abb3e21b33d0c2371a5a902c9061": "Q=K\\left({\\text{Final}}\\right)-K\\left({\\text{Initial}}\\right)=\\left(m({\\text{Initial}})-m({\\text{Final}})\\right)c^{2}", "1a72e1c52b1e0e6314d233b3ba60b794": "{xe^{x} \\over e^{x}-1}-5=0.", "1a732869a01a7e79c5eb16a68a1d3de1": "\\sigma (\\pi )", "1a7335d95d296135763ee0f305faa9b7": "0.99\\cdot 100-0.98(100-x)=x", "1a7392f171ee64b2e179badcd8a2337d": "\\displaystyle {{dg(z) \\over dz}={1 \\over (cz+d)^{2}}.}", "1a739545d2078c6844af26099315532f": "{\\frac {1}{\\sqrt {\\pi }}}\\int w(z)\\,dz={\\frac {\\mathrm {erf} (z)}{2}}+{\\frac {iz^{2}}{\\pi }}\\,_{2}F_{2}\\left(1,1;{\\frac {3}{2}},2;-z^{2}\\right)", "1a73cbf05c151e03d7c64f466053673d": "\\forall x\\,[\\alpha (x)\\land \\gamma (x)]\\leftrightarrow (\\forall x\\,\\alpha (x)\\land \\forall x\\,\\gamma (x)).", "1a73da00df44a7180f2edde6598a5fdd": "\\Omega (o\\mid s',a)", "1a73ec89563d5c404c0a50d028565bee": "\\mathrm {St} ={\\frac {h}{Gc_{p}}}={\\frac {h}{\\rho uc_{p}}}", "1a73f328bad560e4c0fe9f0896a655fb": "x_{1}-x_{2}", "1a7431ece4a4e909c86d5e69dec1d56e": "\\sin t=\\sin(2\\pi k+t)\\,\\!", "1a744e85878a0dde48aa99e18b3e0149": "U={\\frac {1}{2}}\\varepsilon \\int _{V}|\\mathbf {E} |^{2}\\,\\mathrm {d} V\\,,", "1a7452e4104f96d3669bf689f18195b9": "t=(r_{1}\\cdots r_{h})(q_{2}\\cdots q_{n}),", "1a748458bd91e0994006f75c21367da7": "E_{2}=c/a", "1a74909b6cbaa4532a76d83b72c12de0": "\\bigcirc ", "1a74921448294d4a093b21e2b282d879": "\\Delta )=m", "1a74ac7dd63ca38b48896cd693b87f23": "{\\hat {G}}=\\langle N,{\\hat {A}}=S_{1}\\times S_{2}\\times \\dotsb \\times S_{N},{\\hat {u}}=u\\rangle ", "1a75049c643e0fbd930397c02706bc47": "y_{n}=S\\cdot x_{n}^{2}\\,,\\!", "1a7523e278dcbce48c7cf3ee3bb31b1e": "\\chi _{U}^{-1}(1)=U.", "1a75b8deabcb9580d994fb8f28602355": "\\pi ({\\mathcal {C}})", "1a75be367908f94bfb46e0b31f9ccac3": "GB(y;a,b,c,p,q)={\\frac {|a|y^{ap-1}(1-(1-c)(y/b)^{a})^{q-1}}{b^{ap}B(p,q)(1+c(y/b)^{a})^{p+q}}}\\quad \\quad {\\text{ for }}0=k", "1a8797adbc75a653772d2b460945cccd": "\\mathbf {F} (\\mathbf {x} )={\\begin{matrix}{\\frac {1}{2}}\\end{matrix}}{\\frac {dC}{d\\mathbf {x} }}{\\frac {Q^{2}}{C^{2}}}~.", "1a87c43deb6b36b27dedc7fa7c8170ef": "S_{m}=k\\ln \\Omega ,", "1a87d3616b3113145971b343c21d4cec": "-{\\frac {d\\Phi }{dp}}\\left[Ap+Pf(p+\\Phi (p))\\right]+A\\Phi (p)+Qf(p+\\Phi (p))=0.", "1a87df7c2c14481aa21ce0dd48430051": "{\\tfrac {3M-E+S}{6}}", "1a87f4922e4f646da575229a1a6ae1b4": "{\\text{mul}}_{f(A)}f(\\lambda )=\\sum _{\\mu \\in {\\text{spec}}A\\cap f^{-1}(f(\\lambda ))}~{\\text{mul}}_{A}\\mu .\\,", "1a8825691c8f861298868dfc294b7e50": "S_{mk}^{}=\\sigma _{mk}+Rp_{mk}", "1a88a26c7cc184520c0bf97122ce5e68": "f_{n}^{(0)}(x)", "1a89202c3ef48551b193fdf212df77af": "q=\\gcd {(a,b,c)}", "1a8948f5786cc2a2b27c2738a9b5b27b": "\\Omega =\\Omega _{1}\\cup \\Omega _{2}", "1a8949ae238dd2c3c9285cdbf0eeaec3": "V(r)={\\frac {Z_{1}Z_{2}e^{2}}{r}}\\phi (r)\\qquad (1)", "1a894b853f9480eef2091046c3b8105c": "{\\begin{matrix}1\\leq p\\leq n\\end{matrix}}", "1a897370e99b3bdbd2842a3d6e181096": "b_{0}=g_{0}", "1a898f5e52c455a5cc3698003cb95bdb": "\\tau =K\\left({\\frac {\\partial u}{\\partial y}}\\right)^{n}", "1a89bfeaf5f474c01cfdc78223a007f0": "12x\\equiv 20{\\pmod {28}}\\ ", "1a8a066052a033eed9de1056eba7afde": "\\alpha =(\\alpha _{1},\\ldots ,\\alpha _{n})", "1a8a34a6cf1e4282b31ea247ac3936b1": "\\in \\mathbb {F} _{5}", "1a8a36d21d63fa0a8c0f1af7d9aac318": "\\alpha =-{\\frac {\\mu }{kT}}", "1a8a5ae6c2ae1b82c7aa71b1421db69a": "S^{2}=\\{a\\in E^{3}\\colon \\|a\\|=1\\}.", "1a8a7397f830da3fa4e6eb0108a95942": "{\\big .}{\\frac {\\partial E_{\\mathrm {in} }}{\\partial t}}-{\\frac {\\partial E_{\\mathrm {out} }}{\\partial t}}-{\\frac {\\partial E_{\\mathrm {accumulated} }}{\\partial t}}=0", "1a8ad4c88dd861d0bd473ecfe1cfa18f": "\\eta (x)", "1a8b1f992463adee24d67af3a196deea": "X\\Rightarrow Y.", "1a8ba3d288ebb1c6ba3d0eae81c0fdb7": "{\\frac {AH}{AD}}+{\\frac {BH}{BE}}+{\\frac {CH}{CF}}=2.", "1a8c26bf5084e8118743b648ecb4b764": "\\lambda _{\\pm }={3 \\over 2}\\pm {{\\sqrt {5}} \\over 2}.", "1a8ca35b9618b99de19964b07dd81d2a": "F^{\\text{T}}", "1a8cd99f0a4c0b00e1478a10285b74be": "{\\mathfrak {der}}(A)\\oplus {\\mathfrak {der}}(J_{3}(B))", "1a8ce6da442aad48c4f9a2c0954e194c": "A_{0}(x)^{2}-B_{0}(x)^{2}=2m\\left(V(x)-E\\right)", "1a8d1dc4f09158f889f21ddae1829c8d": "{\\mathbf {p} }={\\mathbf {p} }_{1}\\times {\\mathbf {p} }_{2}\\times \\cdots \\times {\\mathbf {p} }_{m}.", "1a8d35fda7748bd4dc345e93bc0c079a": "\\int {\\frac {\\mathrm {d} x}{\\sin ^{n}ax\\cos ax}}=-{\\frac {1}{a(n-1)\\sin ^{n-1}ax}}+\\int {\\frac {\\mathrm {d} x}{\\sin ^{n-2}ax\\cos ax}}\\qquad {\\mbox{(for }}n\\neq 1{\\mbox{)}}\\,\\!", "1a8d5719b61cfd85ebd754c952f4f209": "-{\\frac {m}{2}}\\ln |-{\\boldsymbol {\\eta }}_{1}|+\\ln \\Gamma _{p}\\left({\\frac {m}{2}}\\right)=", "1a8d5f02caa455c7f95570c75992136a": "\\phi (\\omega )=-{\\mathcal {H}}\\lbrace \\alpha (\\omega )\\rbrace \\ ", "1a8d7519b8f5068cacc05537e2ae2da4": "r_{AB}", "1a8d9cc12436e061ce67f8e35f43284a": "C={c_{ij}}", "1a8e11552a6809e21105413af53ddcc9": "\\varphi _{B}=(k_{B}T/q)\\ln(N_{A}/n_{i})\\ ,", "1a8e6971cffd73e684a6ff22c14f422c": "k^{*}=S", "1a8ed6dae1864fd2f4eb7f64df03e09f": "{\\vec {\\mu }}_{S}=g\\mu _{B}{\\vec {S}}=2\\mu _{B}", "1a8f6283ec5209a155dbdf387097ad8e": "{\\mathsf {STUVWXYZ}}\\!", "1a8f718d39b894547d56008f0501e6ce": "\\,\\Theta (N\\,\\mathrm {slog} \\,N)", "1a8f8945a45db50f58a4d474f56060ca": "k_{\\mathrm {spec} }=\\left[\\sin(L,T)\\sin(V,T)-\\cos(L,T)\\cos(V,T)\\right]^{n}=(-\\cos(\\angle (L,T)+\\angle (V,T)))^{n},", "1a8fdbf343d292335103d618023ec6c1": "\\forall x:\\neg P(x)", "1a9043f9be4dab0c9bb6fe9c43d1589d": "\\operatorname {de-lambda} [p\\ f=\\lambda x.f\\ (x\\ x)]\\equiv p\\ f\\ x=f\\ (x\\ x)", "1a90529f34a92813a96700c4294b0afe": "R={\\tfrac {1}{2}}[v,v]", "1a90561678516393a75425bf10db083a": "\\mathrm {d} f=f(d_{p})\\,\\mathrm {d} d_{p}", "1a9071e8efdeaceba95466942bcb5366": "\\left(\\beta -\\delta -{\\tfrac {1}{2}}\\right)=(4\\pm 8)\\times 10^{-12}\\,", "1a907711a467a9b7a1dcfa7d70af30eb": "{\\mathcal {B_{A}}}=({\\mathcal {A}},\\Delta ',\\varepsilon ,\\Phi ')", "1a90808e6fb9d34c17d8e35101921ea8": "\\gamma _{m}={\\underset {\\gamma }{\\operatorname {arg\\,min} }}\\sum _{i=1}^{n}L\\left(y_{i},F_{m-1}(x_{i})+\\gamma h_{m}(x_{i})\\right).", "1a90a6ec2f44d453602421cbfc9e3488": "b_{m}=(a*1)(m)=\\sum _{n\\mid m}a_{n}.\\,", "1a90e02e0bdc29efffba77f2b52939c4": "A_{1}X_{1}", "1a90e28476d96f6ea05b599082b2bbc9": "H_{R}(s)={V_{R}(s) \\over V_{in}(s)}={RCs \\over 1+RCs}", "1a90f1c5fdae4d81c92c70f2fa5ce580": "f_{\\omega }(f_{1}(f_{0}(3)))-2", "1a912d74b53523a6cc7810b3fc595e97": "\\Phi _{S}^{R}=-m(S\\otimes \\Phi _{S}^{R}\\circ P)\\Delta .", "1a918609cb6a66b5316cc7712b123c1e": "f(x)=1/|x-1|", "1a9193d185242c89451cefe024afe9e7": "a\\,{\\mathcal {L}}\\,b\\Longleftrightarrow a^{-1}a=b^{-1}b,\\quad a\\,{\\mathcal {R}}\\,b\\Longleftrightarrow aa^{-1}=bb^{-1}", "1a921f9516272fbf443757f475f7242d": "(\\cos x+i\\sin x)^{n}=\\cos(nx)+i\\sin(nx),n\\in \\mathbb {Z} .", "1a9250d60dce98ddbeecca72f913cd92": "E=\\gamma (\\mathbf {u} )m_{0}c^{2},\\quad \\mathbf {p} =\\gamma (\\mathbf {u} )m_{0}\\mathbf {u} ", "1a9256ddec9b9994ff1d9cec33f734f8": "\\left\\|Cy_{n}-Cy_{m}\\right\\|=\\left\\|(C-I)y_{n}+y_{n}-(C-I)y_{m}-y_{m}\\right\\|", "1a926eecc71b437898a276c1b780ffb5": "J_{F,\\alpha }(\\theta :\\theta ')=\\alpha F(\\theta )+(1-\\alpha )F(\\theta ')-F(\\alpha \\theta +(1-\\alpha )\\theta ')", "1a92a0337571c31ccd3dfab189f0a66c": "\\max _{|z|\\leq 1}(|P^{(k)}(z)|)\\leq {\\frac {n!}{(n-k)!}}\\cdot \\max _{|z|\\leq 1}(|P(z)|).", "1a92f8e9bf58b924ec5b5c4aa397211e": "U=-\\int {\\vec {F}}\\cdot d{\\vec {x}}", "1a931c2def48d4eea352d9a9fbd59e3b": "q^{n-m}", "1a933888906895bacb397ad221330dc9": "i_{n}=1,\\dots ,r_{n}", "1a933ede8b514212b3a6bdef389a1dd7": "E_{\\mathrm {ground} }=D_{\\alpha }\\left({\\frac {\\pi \\hbar }{2a}}\\right)^{\\alpha }.", "1a937a42af3ef6ff6e8b13f3d1c49d1f": "\\chi (A)-\\chi (B)+\\chi (C)=0", "1a9381d08198452c6aeef9dc9d039e2d": "={A \\over \\pi }<\\infty .", "1a93e86e9842175c41e5468dfdf11b6d": "pE=pN/2", "1a940f5d62a46f0aef587132fb0b28b8": "{\\vec {P}}={\\frac {d{\\vec {\\tau }}}{dt}}", "1a942d6e3e7cf96bc0cf07c5f4c30bcd": "{\\begin{array}{ccc}{1 \\over \\sin A}=\\csc A&{\\text{or}}&{1 \\over \\csc A}=\\sin A\\\\\\\\{1 \\over \\tan A}=\\cot A&{\\text{or}}&{1 \\over \\cot A}=\\tan A\\\\\\\\{1 \\over \\sec A}=\\cos A&{\\text{or}}&{1 \\over \\cos A}=\\sec A\\end{array}}", "1a94e3268c0eaf8f14089e3e84e97827": "\\chi _{+}^{\\alpha }={\\frac {x_{+}^{\\alpha }}{\\Gamma (1+\\alpha )}}.", "1a94ed9fc45e5af6704e271bb651baa6": "A=\\{x\\in \\mathbb {R} ^{2}:x_{2}\\geq x_{1}^{2}{\\text{ or }}x_{2}\\leq 0\\}", "1a953cc0fadf728c89ef26fe6c6b091b": "{\\text{Quot}}(A)", "1a955e5540626e58816bb17f281209f1": "d=r\\operatorname {haversin} ^{-1}(h)=2r\\arcsin \\left({\\sqrt {h}}\\right)", "1a95766f6767c6a343957e18e5a4742b": "\\mathrm {SNR} ={\\frac {P_{\\mathrm {signal} }}{P_{\\mathrm {noise} }}}=\\left({\\frac {A_{\\mathrm {signal} }}{A_{\\mathrm {noise} }}}\\right)^{2},", "1a9593007f548b7f31565c4abd411335": "x\\in \\mathbb {R} _{+}", "1a95b81e6a2cb8cd7300d96e399fb31c": "\\Phi (v)=\\left(\\int _{0}^{\\infty }{\\bigl (}t^{-\\theta }J(v(t),t;X_{0},X_{1}){\\bigr )}^{q}\\,{dt \\over t}\\right)^{1/q}<\\infty .", "1a95e7845d318f5307e1823f9e522f4e": "A(x)=\\prod _{i=1}^{n}(x-x_{i})\\quad {\\text{and}}\\quad B(x)=\\prod _{i=1}^{n}(x-y_{i}).", "1a9611b7f0c4e4f44aad57a98e867c97": "f={\\frac {c}{\\lambda }},\\quad {\\text{or}}\\quad f={\\frac {E}{h}},\\quad {\\text{or}}\\quad E={\\frac {hc}{\\lambda }},", "1a9633c244f9c88f511ab4cc2d69d023": "2\\leq n\\in \\mathbb {N} ", "1a96d24b78a5995681e97e696f2065a8": "e", "1a96eae5c008381d0e652f607a58557d": "J=2,1,0\\,", "1a97461b7353373189c0be4a1150d99c": "E(\\mathbf {k} )=E_{0}+{\\frac {\\hbar ^{2}\\mathbf {k} ^{2}}{2m^{*}}}", "1a97a3e30ffa3d15521abef105847057": "\\sum _{i=1}^{N}\\Delta t_{i}=T.", "1a97f660485876e6e2ea92d858b91d7c": "(\\hbar m_{\\ell })", "1a984ae26cdd98f2387b6b61e920ced2": "f^{(n)}(x)=\\lim _{h\\to 0}{\\frac {\\sum _{0\\leq m\\leq n}(-1)^{m}{n \\choose m}f(x+(n-m)h)}{h^{n}}}.", "1a987d1cc9164f7818bea6654c4c44f5": "P(x,y,z)", "1a988b93be6d7a771ba072cb98dcb127": "n_{t+1}=\\int _{-\\infty }^{\\infty }k(x-y)Rn_{t}(y)dy", "1a9896044879d8764e859d413903d919": "\\mathbf {n} _{u}", "1a98b6d80ebdcb60a7721c86f3551f52": "{\\frac {1}{\\pi }}=12\\sum _{k=0}^{\\infty }{\\frac {(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}640320^{3k+3/2}}}\\!", "1a99d7d6b1e3c2fcd1882d7737707aca": "\\operatorname {logit} (P)=\\beta _{0}+\\beta _{1}\\times \\operatorname {logit} (Q)+\\epsilon ", "1a9a075f5f14056ae46f7fe18148e131": "\\left(q,\\rho ,I,\\mathbf {J} ,\\mathbf {P} ,\\mathbf {p} \\right)", "1a9a35e3258bdf25c4e4785fe5a1d974": "{\\frac {s+\\beta }{(s+\\alpha )^{2}+\\omega ^{2}}}", "1a9a752aae65f753a327c946d03efb6d": "(By+\\beta +1)^{n}\\leq B^{n}x+\\alpha \\,", "1a9a76745437118fd8c423c3f5b97a36": "\\phi _{u,v}", "1a9a95bd0521864b1f6d46eb3859717d": "G^{v}=G_{\\psi (v)}", "1a9ab3873881de67769dad4378e52156": "10^{k}", "1a9abb630f4abe69219824994b7890ad": "\\partial x/\\partial \\theta <0", "1a9b19229ca3fe5680330a79335ec60a": "(0,1,2,3....,n-1)", "1a9b46b14a1e4ce15b79d99a8f30d4e1": "k_{a}f_{a}+k_{b}f_{b}+\\cdots +k_{N}f_{N},\\,", "1a9bd525923ce8981e3524656300ba0e": "A={\\frac {1}{\\sqrt {2k(k-\\gamma )}}}{\\sqrt {{\\frac {C}{n-|k|+\\gamma }}{\\frac {(n-|k|-1)!}{\\Gamma (n-|k|+2\\gamma +1)}}{\\frac {1}{2}}\\left(\\left({\\frac {Ek}{\\gamma \\mu c^{2}}}\\right)^{2}+{\\frac {Ek}{\\gamma \\mu c^{2}}}\\right)}}", "1a9bd52b2b494ef364bf20824cad2c06": "v_{2}/c={\\mbox{tanh}}(s_{2})={\\frac {e^{s_{2}}-e^{-s_{2}}}{e^{s_{2}}+e^{-s_{2}}}}", "1a9bdda778ba11f53e2711c7a4a375c6": "I=n{\\begin{bmatrix}0.2&0&0\\\\0&10.1&0\\\\0&0&10.1\\end{bmatrix}}.", "1a9cf77d65498a343549ea09ba214c00": "Q=-W\\;", "1a9d4831368d128c719d803d1911f984": "c_{2}\\in C_{2}", "1a9d650d978ba43940a313836d36ef77": "\\{1,2,\\cdots n\\}", "1a9da1ef8e4b4bd566b330ff883549aa": "I_{r,r-1}={\\begin{bmatrix}I\\\\O\\end{bmatrix}}", "1a9dd50f4606d0c0057fac6200cd691f": "{\\boldsymbol {\\hat {\\beta }}}=(X^{\\mathrm {T} }X)^{-1}X^{\\mathrm {T} }\\mathbf {y} .", "1a9e06fcf8d6146d0e73f567d279a582": "G(\\mu )=G_{0}\\sum _{n}T_{n}(\\mu )\\ ,", "1a9e6c60b1b60e7e47a816dcba44c6b2": "{\\frac {d{\\vec {J_{R}}}}{dt}}={\\frac {\\gamma }{\\hbar }}{\\vec {J}}\\times ({\\vec {B_{0}}}+{\\vec {B_{R}}})-{\\vec {J}}\\times \\omega ", "1a9f3f557cfc6f226a63069d9f568925": "{\\tfrac {52163}{16604}}", "1a9fbde296633dbac2d794dfa832d7e2": "\\forall i\\,[P_{i}\\in R_{i}]", "1aa03c78298365d8bd5e95ec8f57da89": "{\\mathcal {M}}_{\\rm {Tot}}={\\mathcal {M}}_{\\rm {Trans}}+{\\mathcal {M}}_{\\rm {Rot}}.", "1aa081cc2ba4f8142d252c4cec0d1e66": "\\lambda ^{4}", "1aa0b3f8750dea7a4fac798c3d7ed865": "X=\\{2,4,4\\}", "1aa0b69ace93700ecaee285cd2424ea1": "a=2^{r}", "1aa0dcb8bf73775ab5ba718bbed70c70": "\\mathbf {r} ", "1aa12dc7c2e0ab1f2dc9a71409e7cd06": "Fd=C{\\frac {\\operatorname {d} ^{2}\\theta }{\\operatorname {d} t^{2}}}", "1aa1303a9ab04ddd702ab3a9fe021e77": "\\sum _{k=0}^{n}{n \\choose k}k^{n-k}", "1aa16aa11e9490e7f534c4326f4890f8": "y_{0}=0", "1aa194ce5024713d8a0db656df36b8d7": "{\\frac {1}{r^{3}}}P_{2}^{1}(\\sin \\theta )\\cos \\varphi ={\\frac {1}{r^{3}}}3\\sin \\theta \\cos \\theta \\ \\cos \\varphi ", "1aa1bda74f474fad29d9654956ae10f5": "\\beta :=2\\rho h/D", "1aa1f6198ae825527ff76de0187af369": "C_{13}={\\frac {1}{N}}\\sum _{r=1}^{N}Q_{r}(t_{1})Q_{r}(t_{3})=1.", "1aa20d211130ae324333d03bdb26b5dc": "J(n,x)=\\int _{0}^{x}t^{n}{\\frac {e^{t}}{(e^{t}-1)^{2}}}\\,dt.", "1aa21c217a57dfafcea07b7fc80e8b23": "\\sigma _{1}\\,\\!", "1aa2861f962fb86bbbb2d43ff4af92d2": "{\\frac {\\partial {\\overline {\\rho }}}{\\partial t}}+{\\frac {\\partial {\\overline {\\rho }}{\\tilde {u_{i}}}}{\\partial x_{i}}}=0.", "1aa2b8822a696953d20dec92d93271e1": "{\\frac {1}{2}}{\\mathbf {u}}\\cdot \\nabla {\\mathbf {u}}+{\\frac {1}{2}}\\nabla ({\\mathbf {u}}{\\mathbf {u}})", "1aa2c1eaa5a11af7eb7311b86b998c4d": "{\\mathfrak {so}}(n,\\mathbf {R} ).", "1aa2d2eddbda71018bd60c149ebd02bb": "\\rho _{\\text{realized }}={\\frac {2}{n(n-1)}}\\sum _{i\\beta \\geq 0.75", "1aa678fa785dfbffc14987659e60412a": "\\alpha _{1},\\dots ,\\alpha _{n}", "1aa68dd2536c4596fdc9cf97ca5806df": "{\\text{Base amperes }}={\\frac {\\text{base kva * 1000}}{{\\sqrt {3}}*{\\text{base volts}}}}", "1aa6d6592ba6720659b290dd764e1cd7": "{\\pi }/{2N},", "1aa6d783d3063dba6684bf278e7ee136": "\\pi _{i}\\ {\\stackrel {\\mathrm {def} }{=}}\\ {\\frac {\\partial {\\mathcal {L}}}{\\partial (\\partial x_{i}/\\partial t)}}.", "1aa6da72fa4d6bb6f20d91e21e8c1043": "\\log _{2}(n!)\\approx n(\\log _{2}n-\\log _{2}e)", "1aa6dda22ff509a4f9f5c73df9683726": "\\left\\{C_{i}\\right\\}", "1aa6e1ba9fa095f15ad2ce49efa5d744": "\\{\\mathbf {e} _{23},\\mathbf {e} _{31},\\mathbf {e} _{12}\\}", "1aa70620cb0cd0c1b75bba9b2d801ff1": "{\\frac {\\partial E}{\\partial w_{ji}}}={\\frac {\\partial \\left({\\frac {1}{2}}\\left(t_{j}-y_{j}\\right)^{2}\\right)}{\\partial w_{ji}}}\\,", "1aa7082d2797393c491580a00c94e6d9": "f_{X}(x)={\\frac {1}{\\sqrt {2\\pi }}}e^{-x^{2}/2}.", "1aa70e4f96d6b21922eab96291658630": "\\lim _{k\\to \\infty }{\\mathbf {T}}^{k}\\quad (6)", "1aa763d7d571d1e8ebc0f2a07b4ed873": "Z_{eff}=Z-S", "1aa7cf9d18e5fec597b9fa5ed84853dc": "g(a+h)-g(a)=g'(a)h+\\varepsilon (h)h.\\,", "1aa802db0b35a76a2242e7da8504d344": "E(Q)\\leq {\\frac {m_{0}}{m}}\\alpha \\leq \\alpha ", "1aa83bd9d4d6c1a9578db73c73fa45b6": "P_{s}=P_{r}=P=V\\cos \\left({\\frac {\\delta }{2}}\\right)\\cdot {\\frac {2V\\sin {\\left({\\frac {\\delta }{2}}\\right)}}{X}}={\\frac {V^{2}}{X}}\\sin(\\delta )", "1aa8545042c8cc55c9bfd042ea68d783": "F(a_{1},\\ldots ,a_{n})=F(a_{\\boldsymbol {\\pi (1)}},\\ldots ,a_{\\boldsymbol {\\pi (n)}})", "1aa857e72edc5e805a793c7e3d616602": "|u'(x)-(\\pi u)'(x)|\\leq Kh\\|u''\\|_{L^{2}(a,b)}", "1aa85f3d7c5a523cbb53688f030b3774": "\\omega _{2}^{L}", "1aa8f07cc43fd4a3b0e2cd7b4b941fbc": "F(y)={\\frac {\\int _{x_{0}}^{x_{1}}y(x)\\;\\mathrm {d} x}{\\int _{x_{0}}^{x_{1}}(1+[y(x)]^{2})\\;\\mathrm {d} x}}", "1aa8fae67966d47f1298d3fcd8547a57": "\\ \\|x[n]\\|_{\\infty }<\\infty ", "1aa923b192fe5854eb638f1229244a50": "(e^{b\\epsilon }e^{-ar})q(e^{ar}e^{bs\\epsilon })=e^{b\\epsilon }(e^{-ar}qe^{ar})e^{bs\\epsilon })=e^{2b\\epsilon }(e^{-ar}qe^{ar}).", "1aa93e33f7ed6b57746ed181cb547aab": "(\\mathbf {v} _{0},\\mathbf {v} _{1},\\mathbf {v} _{2})", "1aa9518cd94e43191f3ba41046f68b41": "\\Im \\tau >0", "1aa9a41799aa3458645cdf076e47b306": "f(a)+f(b)=f\\left((a+b){\\frac {a}{a+b}}\\right)+f\\left((a+b){\\frac {b}{a+b}}\\right)\\leq {\\frac {a}{a+b}}f(a+b)+{\\frac {b}{a+b}}f(a+b)=f(a+b)", "1aa9dd34bc41c3db620c1af63d236085": "d=67", "1aaa43be82036c17abef5f99eed9ffcd": "\\lim _{n\\rightarrow \\infty }a_{n}=0\\,", "1aaa4ffef6743b01f0eee69e3dd93a78": "\\operatorname {somb} (\\rho )=2J_{1}(\\pi \\rho )/\\pi \\rho .", "1aaa76cbe62e84dcaf2dd492c2f6bf9b": "\\psi _{k}(x)=e^{ikx}u_{k}(x)", "1aaa7883ce81785407d702ee0470e1d3": "E=\\sum _{i}\\left[{\\dot {q}}_{i}{\\frac {\\partial L}{\\partial {\\dot {q}}_{i}}}\\right]-L", "1aaa85d122537e89ab6d3c6bd30b8a74": "x^{*}x=x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}.", "1aaa898a2fcca7ced16a675e9b6ef627": "f(x;\\alpha ,\\beta )=f(1-x;\\beta ,\\alpha )", "1aaabf98cfc7b0b338ac1e17085c2f55": "IC_{p}(X)=\\tau _{\\leq p(n)-n}Ri_{n*}\\tau _{\\leq p(n-1)-n}Ri_{n-1*}\\cdots \\tau _{\\leq p(2)-n}Ri_{2*}{\\mathbb {C} }_{X-X_{n-2}}", "1aab2390bf8013903313e6614f2d6009": "a_{n}z^{n}", "1aab3c3aefb40f201b84c9d3f55516d4": "dN", "1aabcbe06016c902922ecb772d9c9717": "\\pi _{i}(f)", "1aac09608797b7147f80b5305dfc14d3": "K_{\\mathrm {b} }=\\mathrm {\\frac {[HB^{+}][OH^{-}]}{[B]}} ", "1aac0d8ab58b9c1cbfbd805ea6e4e5c7": "\\tau \\;", "1aac384b1eef0ded9c28536e13fb5b07": "d(v)=1\\,,", "1aac416b5b8df7462f817f76847c58a8": "{\\begin{aligned}\\left|{\\frac {f(w)-f(z)}{w-z}}-g(z)\\right|&=\\left|\\int _{z}^{w}{\\frac {g(\\zeta )d\\zeta }{w-z}}-\\int _{z}^{w}{\\frac {g(z)d\\zeta }{w-z}}\\right|\\\\&\\leq \\int _{z}^{w}{\\frac {|g(\\zeta )-g(z)|}{|w-z|}}d\\zeta \\\\&\\leq \\max _{\\zeta \\in [w,z]}|g(\\zeta )-g(z)|,\\end{aligned}}", "1aac601bf88d00188a0f6f3cb8f94770": "{\\frac {1}{2}}\\left(1\\pm |{\\vec {a}}|\\right)", "1aac73b58786c10d55743184579c5e92": "\\Psi =Re^{iS/\\hbar }", "1aacb8a1f06d3f525c2d52dcde17814f": "\\Omega =2\\pi \\left(1-\\cos \\theta \\right)", "1aad288fdab1a2cc333ae1b11fde7171": "\\Gamma ={\\frac {1-y_{T}}{1+y_{T}}}\\,", "1aad2fbcac161490247876059b7b6822": "r_{k}<{\\sqrt {m}}", "1aad642de22f845592b93adcabb1781c": "[1,B^{3}]", "1aad9eb45b1c24b7c14b395757608a58": "\\scriptstyle \\sigma _{1}^{2}", "1aae166bfdcdfb3c41e9474b78098c74": "\\sigma _{1}", "1aae16b5e6fb949662a8cfc55c51da66": "V({\\mathcal {C}})", "1aae6b2cfb9b43aa540fb9096a375bc3": "B^{I}", "1aae8228d2c02a0a841bbbe9f0243b7d": "F_{(\\xi ,\\mu ,\\sigma )}(x)={\\begin{cases}1-\\left(1+{\\frac {\\xi (x-\\mu )}{\\sigma }}\\right)^{-1/\\xi }&{\\text{for }}\\xi \\neq 0,\\\\1-\\exp \\left(-{\\frac {x-\\mu }{\\sigma }}\\right)&{\\text{for }}\\xi =0.\\end{cases}}", "1aaea31c49d95301fafca03d2e175417": "R_{\\mu }", "1aaf45c4d15258a4669acd07e7c709fd": "\\Lambda ^{m\\mid n}", "1aafa2e5b4efd167cb0eb77e0f9eb594": "U_{i}:A\\longrightarrow {\\mathbb {R}}", "1aafc2ced604a9974ed7c15f3f8511f4": "x_{i}=\\tan \\theta _{i}\\,", "1aafc7bb8e0f10aa0d6074ade5b6a9d4": "2^{-2}\\times 0.100_{2}", "1aafd4023807be7de8aee982b3f9a4c8": "{\\frac {BC}{BD}}={\\frac {AC}{AD}}", "1ab023d93061289b8e4dc122b5cb1ace": "{\\begin{bmatrix}c_{2}c_{1}&-c_{2}s_{1}&s_{2}\\\\s_{3}s_{2}c_{1}+c_{3}s_{1}&-s_{3}s_{2}s_{1}+c_{3}c_{1}&-s_{3}c_{2}\\\\-c_{3}s_{2}c_{1}+s_{3}s_{1}&c_{3}s_{2}s_{1}+s_{3}c_{1}&c_{3}c_{2}\\end{bmatrix}}", "1ab02506e653a363608474e8a9dac7de": "\\scriptstyle q\\;=\\;y^{2}", "1ab039c742bc51077ff5aa4ae8f8b8a5": "P=\\sum _{i=0}^{\\infty }{l_{i}}", "1ab0540f1dfcd2c19af04a8d7c838e30": "u_{f}=dl/dt", "1ab06556316daa80b830e50aca6310c0": "p_{n}>{{p_{n-1}+p_{n+1}} \\over 2}", "1ab08df0178c231a421a1d866362b7bb": "b(w)", "1ab0e533205cfd46c57f78c4da18b051": "K={\\frac {a+b}{2}}\\cdot h", "1ab11c6c85ef77b316d793c7b4f98916": "p=8m+5", "1ab13dc8c122d60b750463dd80c4a7f7": "\\deg(f,\\Omega ,p)=\\deg(g,\\Omega ,p)", "1ab16a900ad65dd304e17b7940393417": "\\left.{\\frac {\\mathrm {d} }{\\mathrm {d} t}}I({\\boldsymbol {u}}+t{\\boldsymbol {v}})\\right\\vert _{t=0}=-\\int _{A}\\sigma _{ik}({\\boldsymbol {u}})\\varepsilon _{ik}({\\boldsymbol {v}})\\mathrm {d} x-\\int _{A}v_{i}f_{i}\\mathrm {d} x-\\int _{\\partial A\\setminus \\Sigma }\\!\\!\\!\\!\\!v_{i}g_{i}\\mathrm {d} \\sigma \\geq 0\\qquad \\forall {\\boldsymbol {v}}\\in {\\mathcal {U}}_{\\Sigma }", "1ab2b8ed0b4480f6fe8a50cbe067b350": "N={\\frac {f}{c}}{\\frac {v_{\\mathrm {N} }-v_{\\mathrm {F} }}{v_{\\mathrm {N} }+v_{\\mathrm {F} }}}\\,.", "1ab2ebb77170c5836399cab7b0e8b8fb": "{\\tbinom {n+1}{k}}", "1ab2fb4c84388318d58cce76c8ab9675": "\\displaystyle {AA^{*}=JA^{*}AJ=I}", "1ab32c9f25c8affa819513aa0b30a70b": "i,j,l,k", "1ab370c0dfbac1d5c1f9206d83868632": "\\lambda _{x}^{2}\\leq \\lambda _{y}^{2}\\leq \\lambda _{z}^{2}", "1ab3a5bf4068065989c6064b169147e1": "{\\begin{bmatrix}1\\\\1\\\\0\\end{bmatrix}}\\wedge {\\begin{bmatrix}1\\\\1\\\\1\\end{bmatrix}}y={\\begin{bmatrix}1\\\\1\\\\0\\end{bmatrix}}\\wedge {\\begin{bmatrix}1\\\\1\\\\2\\end{bmatrix}}.", "1ab42883c679fad5383a9cac3f2a4c3a": "\\int _{0}^{\\pi }f(x)\\sin(x)\\,dx={\\bigl (}F'(x)\\sin x-F(x)\\cos x{\\bigr )}{\\Big |}_{0}^{\\pi }.\\!", "1ab45bbf086d8a17c53c13522059d5c2": "\\Delta _{I}", "1ab49de7222bf270920191a26a9cd052": "f+\\lambda g", "1ab4e2d1136854fbb042404772121520": "1>0", "1ab5158115ac415d756a00c9cf5dc169": "\\Delta E", "1ab52d81f116196376dc9f9c549f069b": "i=1,\\dots ,\\min\\{m,n\\}", "1ab5909b949af30693e04c9782baf20f": "\\sum _{i=1}^{4}\\Omega _{i}=2\\sum _{i=1}^{6}\\phi _{i}-4\\pi ", "1ab61f15198d7ed5c0af334198a0d764": "=\\epsilon /2", "1ab65788298b915edaa589fd7507c5bb": "a_{n}\\rightarrow 1", "1ab67afd638a0b530a9e8c8eb7f147db": "C={\\begin{bmatrix}0&1\\\\0&0\\end{bmatrix}}.", "1ab698ea28a21c95680707077e229fa0": "\\alpha =3\\,", "1ab6c02438070224a4e6e42117db9e98": "{\\frac {\\mathrm {d} y}{\\mathrm {d} t}}=f(t,y)", "1ab6dd1ebba23da4c4cf415515c12387": "{\\begin{aligned}\\mathbf {u} _{k}^{(1)}&=\\mathbf {v} _{k}-\\mathrm {proj} _{\\mathbf {u} _{1}}\\,(\\mathbf {v} _{k}),\\\\\\mathbf {u} _{k}^{(2)}&=\\mathbf {u} _{k}^{(1)}-\\mathrm {proj} _{\\mathbf {u} _{2}}\\,(\\mathbf {u} _{k}^{(1)}),\\\\&\\,\\,\\,\\vdots \\\\\\mathbf {u} _{k}^{(k-2)}&=\\mathbf {u} _{k}^{(k-3)}-\\mathrm {proj} _{\\mathbf {u} _{k-2}}\\,(\\mathbf {u} _{k}^{(k-3)}),\\\\\\mathbf {u} _{k}^{(k-1)}&=\\mathbf {u} _{k}^{(k-2)}-\\mathrm {proj} _{\\mathbf {u} _{k-1}}\\,(\\mathbf {u} _{k}^{(k-2)}).\\end{aligned}}", "1ab74872675fa1297f76e8950dc938dd": "\\mathrm {D} _{2}\\cong \\mathrm {A} _{1}\\times \\mathrm {A} _{1},", "1ab76407e86e8a7c491577eb955316b9": "\\nabla \\cdot \\left({\\frac {\\mathbf {s} }{|\\mathbf {s} |^{3}}}\\right)=4\\pi \\delta (\\mathbf {s} )", "1ab77df72ddb65a53b2d82e411c8e206": "\\ C_{L}=2\\pi (A_{0}+A_{1}/2)", "1ab7846b0bfae81bf125c7f8d3cdc9f5": "\\phi ^{n+m}=\\left({\\frac {1+{\\sqrt {5}}}{2}}\\right)^{n+m}\\in O\\left(1.62^{n+m}\\right),", "1ab7bfc3b2401aed340f8f631e5556ce": "\\scriptstyle {\\tau =t_{1}-t_{0}}", "1ab7d6c15619557c80e8fcd6b03964aa": "f(X)+E[u(X)-u(B)]", "1ab8077d9861c4b3e56e863685dc869d": "0=MU_{x}+MU_{y}{\\frac {dy}{dx}}", "1ab85836a785e5828c5764871a1b82b3": "\\{p_{1},p_{2},p_{5}\\}", "1ab8c4df2b32b6f1921373f0eb03deb6": "\\phi _{n}:x\\mapsto n^{k-1}B_{k}(\\langle x\\rangle )", "1ab8f74662094375be833aa7331a5dec": "DR_{T/D}^{S/V}", "1ab918526661afc3a55e5afe89aa0ed4": "{\\text{ }}{\\frac {1}{t}}\\sum _{\\tau =0}^{t-1}\\sum _{i=1}^{N}E[Q_{i}(\\tau )]\\leq {\\frac {B+V(p^{*}-p_{min})}{\\epsilon }}+{\\frac {E[L(0)]}{\\epsilon t}}", "1ab920863900f52cc962a172df58ea63": "d={\\frac {v^{2}}{g}}", "1ab934a547e2bc5d8afa2b41f85fe0df": "\\mu _{2}=\\dots =\\mu _{k}=0", "1ab949e8425c93a96c3625f82709dbc4": "\\left\\langle 0,c\\right\\rangle ", "1ab9808a03a46592b2e540f8159c6df1": "P(H_{n})=\\mathbb {C} P^{n-1}", "1ab98f8bbc41e04448d5356018efee8e": "1_{En}", "1ab990ac5e62b96d59ad474058cdad2e": "f_{*}(u_{n})\\in H_{n}(X)", "1ab9a39ee2d4b6e08dcda32167b23b23": "{\\text{Throughput}}=({\\text{Sales revenue – Direct Material Costs}})", "1ab9cb5b6b65eeb000441c91765659df": "\\lambda _{\\max }", "1ab9ced7035753d114f561b980d5709b": "\\geq 1", "1ab9e9918dcf677575c8aa3ff6123437": "{\\hat {8}}", "1ab9ea8ff2894b022b3f7cd5e6d58e5b": "\\psi \\propto \\sum _{i=1}^{m}A_{i}e^{i{\\mathbf {k}}_{i}\\cdot {\\mathbf {r}}}", "1aba2b3443cfb68c08d0123d11920345": "x''+c_{1}x'+c_{2}=0", "1aba6bc2e8410c9c20c626125896651c": "\\mathrm {Win} ={\\frac {{\\text{runs scored}}^{2}}{{\\text{runs scored}}^{2}+{\\text{runs allowed}}^{2}}}={\\frac {1}{1+({\\text{runs allowed}}/{\\text{runs scored}})^{2}}}", "1aba9a316555b2a1f8f6abef61479bfb": "x\\in {\\mathcal {N}}(X)", "1abac7e5686a1a07099b0bd47416bcd7": "{\\binom {p+q}{q}}-{\\binom {p+q}{q-1}}={\\binom {p+q}{q}}{\\frac {p+1-q}{p+1}}.", "1abaca9a286657bcc5335c98b479783b": "{\\widehat {\\mathbf {J} }}={\\widehat {\\mathbf {L} }}+{\\widehat {\\mathbf {S} }}\\,.", "1abae6323d1a5a6d9ba2d4026013e26b": "{\\begin{aligned}\\int _{0}^{1}f(x)v(x)\\,dx&=\\int _{0}^{1}u''(x)v(x)\\,dx\\\\&=u'(x)v(x)|_{0}^{1}-\\int _{0}^{1}u'(x)v'(x)\\,dx\\\\&=-\\int _{0}^{1}u'(x)v'(x)\\,dx=-\\phi (u,v),\\end{aligned}}", "1abb17dc3174c03be1f2fd795a777250": "\\pi <{\\widehat {1}}", "1abb4c8a30ca619e2395a8d002bbfded": "\\varphi _{\\beta +1}(\\gamma +1)[0]=\\varphi _{\\beta +1}(\\gamma )+1\\,", "1abb9bd1f9b31964775398d8fae5502f": "g_{0}(n)=0", "1abbb88228d6436ed45e5741586a747d": "\\Gamma _{ik}^{1}={\\begin{bmatrix}0&1/r&0&0\\\\1/r&0&0&0\\\\0&0&-\\sin \\theta \\cos \\theta &0\\\\0&0&0&0\\end{bmatrix}}", "1abbe13a6b9520d438676740cdb15158": "Q_{eq}", "1abc03af71306d9db4a1524d034c2e07": "Vz-=Iz.Zz\\,", "1abc86f1f3b1a7ba8fdaba3bf6ec9729": "\\Delta p=p\\otimes K+1\\otimes p", "1abcc653113c3d1f138ce2feef07bc27": "b=2i+(h-i-j)=h+i-j\\,", "1abcec815c3e2c9a6fcac5be41fa230b": "x^{\\prime }=x-vt", "1abcf05eb36512ac8672493b4a410cd9": "\\operatorname {E} [\\,x_{t}(y_{t}-x_{t}'\\beta )/\\sigma ^{2}(x_{t})\\,]=0", "1abcfff16c133734ea77fa457f77aed6": "\\beta _{b}^{n}", "1abd1944ed6f70fd94104c9db8473d11": "S_{n}(x)=xj_{n}(x)={\\sqrt {\\frac {\\pi x}{2}}}\\,J_{n+{\\frac {1}{2}}}(x)", "1abd3d98386280fdf4e48c43896e37f4": "\\ u'w'", "1abd5bc7d61eadc9b29dc764596d11bc": "{\\textbf {K}}_{k}{\\textbf {S}}_{k}=({\\textbf {H}}_{k}{\\textbf {P}}_{k\\mid k-1})^{\\text{T}}={\\textbf {P}}_{k\\mid k-1}{\\textbf {H}}_{k}^{\\text{T}}", "1abdd729b882e1bbb21eabf7f8cdb9b9": "G_{\\epsilon }(V,E)=D\\cap \\mathbb {Z} _{\\epsilon }^{d}", "1abe1e488858561aeec0dce26c295f66": "X\\mathbf {\\operatorname {m} } Y", "1abe21c5f54ab3be3b3a92ce4df7128e": "x=x", "1abe384bf2adb759827fb149cfe66d64": "H+L(\\alpha )\\geq 0.", "1abe44305ac7e4e6521be709b6d05fcf": "\\Omega ^{3}", "1abeba592078e159bfb4b1c2a9bdcb7b": "H=2h", "1abed2f9ab62c9c6df7ada4534217ea9": "{\\text{(6)}}\\qquad \\sigma _{y}(\\varepsilon _{\\rm {p}},{\\dot {\\varepsilon _{\\rm {p}}}},T)={\\begin{cases}2\\left[\\tau _{s}+\\alpha \\ln \\left[1-\\varphi \\exp \\left(-\\beta -{\\cfrac {\\theta \\varepsilon _{\\rm {p}}}{\\alpha \\varphi }}\\right)\\right]\\right]\\mu (p,T)&{\\text{thermal regime}}\\\\2\\tau _{s}\\mu (p,T)&{\\text{shock regime}}\\end{cases}}", "1abf24343749f2bb74c6f78e7b6bca6b": "m_{2}=1\\ ,", "1abf2b520ef1a91db8f2a22012de71e4": "\\phi (e_{i})=e_{j}", "1abf3d7c735ea59569e8588ba93482a4": "S_{0}=\\{0,1,2\\},S_{i+1}=(S_{i}\\times S_{i})+S_{i}", "1abf43d94ea93793febf2ceeaebd2c43": "S_{n}=\\sum _{i=1}^{n}x_{i}", "1abf84996f3f707e1b215bbc2b012f2c": "I_{R}=GV_{R}", "1abfbd1bb9566bf753a1ba13e4676027": "S\\circ F\\in \\mathrm {D} '(V).", "1abff943e15eaf744beb3885cbf89134": "{\\mathcal {G}}=\\{A:\\mathbf {1} _{A}\\in {\\mathcal {H}}\\}", "1ac042b49fb7d8cada7d342e7c4065f2": "\\partial f/\\partial y", "1ac0839b13f4885faded5c2a49afa025": "{\\overline {\\mathbb {D} }}", "1ac08e0fd0cf461755eef45da79d9c66": "S(p)={\\frac {1}{2}}+{\\frac {1}{3}}+{\\frac {1}{5}}+{\\frac {1}{7}}+\\cdots +{\\frac {1}{p}}.", "1ac0b55f1f613743f7271e10b100ad71": "S^{-1}R=Q(R)", "1ac0d601419879e7f42c90f6698f325e": "h_{\\phi }=a\\sigma \\tau ", "1ac10edee329c373b1465dd6491f3595": "P=\\int _{0}^{\\infty }d\\nu \\int _{0}^{\\pi /2}d\\theta \\int _{0}^{2\\pi }d\\phi \\,B_{\\nu }(T)\\cos(\\theta )\\sin(\\theta )=\\sigma \\,T^{4}", "1ac1159bb78e0a96149ecc60295cbb5d": "{\\mathcal {L}}=\\partial _{\\mu }\\phi \\partial ^{\\mu }\\phi -(\\phi ^{2}-1)^{2}", "1ac11e8ba2687f97dc0dce38a5da1ce0": "Z_{1},Z_{2}\\sim {\\mathcal {N}}(0,1)", "1ac167a94601d3bd9407cd841b300125": "{\\frac {\\partial \\ln |a\\mathbf {X} |}{\\partial \\mathbf {X} }}=", "1ac16a82c7efc4d76924cd6e464a3091": "\\{1,2,\\dots ,k^{k}\\}", "1ac1a4f4fcd5a72da08cc1955d65703c": "Q_{i}", "1ac1ed89c9194c9e7779939efaa611d0": "V={\\sqrt {\\frac {(l^{2}+m^{2}-n^{2})(l^{2}-m^{2}+n^{2})(-l^{2}+m^{2}+n^{2})}{72}}}.", "1ac21dcc002aa4bf060ab2c3cba3d029": "-671\\pm 6", "1ac23cbbd1934a654cf3ee328fea1882": "\\phi \\to \\phi \\lor \\chi ", "1ac25ad53a1a66592c89613ae74fab24": "{\\begin{aligned}P(Hypercalcemia~WHOIFPI~by~cancer)=\\\\P(cancer~WHOIFPI)*r_{cancer\\rightarrow hypercalcemia}=\\\\0.002*0.1=0.0002\\end{aligned}}", "1ac29daf7266b56354a14880fcf1ad2a": "\\operatorname {sgn} (0+0i)=0", "1ac2a5282e20b69e09578e7ed63c832d": "\\mu _{\\mathrm {B} }=e\\hbar /2m_{e}", "1ac30352595c745bd666db5058d4b43b": "f'(\\alpha )\\neq 0", "1ac3b73863e00a4ee60189f18d0a802b": "\\lim _{n\\to \\infty }\\operatorname {Pr} (X_{n}\\leq a)=\\operatorname {Pr} (X\\leq a),", "1ac3b7ae1f1053a05eb3af6dfa9efd6d": "-K(T-t)e^{-r(T-t)}N(-d_{2})\\,", "1ac42d09412d03d9e4267d2c9534658f": "P(N(t+h)-N(t)=1)=\\lambda (t)h+o(h)", "1ac451f00486b941e4c498570d8073b3": "\\therefore T={\\frac {c}{v}}\\left(e^{v/\\alpha }-1\\right)\\,\\!", "1ac48a03d67cd2d89e3dad75529af52a": "s(x)=1", "1ac4b7c8c6e5bc02cda050d6b9eac989": "d_{1}(z)=a+(1-a){\\overline {z}}.", "1ac4fe89f4f267d51da6bcf9a386b49e": "L_{\\mathrm {MV} }(x_{0},\\dots ,x_{n})={\\sqrt[{-n}]{(-1)^{(n+1)}\\cdot n\\cdot \\ln[x_{0},\\dots ,x_{n}]}}", "1ac52899339bbbd6dd0c2f01d1e9928c": "(T\\subseteq Y)\\;\\mapsto \\;f^{*}(T)=f^{-1}\\lbrack T\\rbrack ", "1ac54f12ab5718857144a90269e92464": "\\mathrm {SCL} :r_{a,t}=\\alpha _{a}+\\beta _{a}r_{m,t}+\\varepsilon _{a,t}", "1ac59a71348c01c1360c048b7f28c536": "{x_{2} \\choose \\theta _{2}}={\\begin{pmatrix}A&B\\\\C&D\\end{pmatrix}}{x_{1} \\choose \\theta _{1}},", "1ac5da260430003e69811d7015e822d8": "w=\\exp(\\pi iz)", "1ac648ddca94359d2437f0f13aa930d2": "\\beta \\equiv {\\frac {1}{k_{B}T}}", "1ac64d00754e43b4abdda0d7b336fd5b": "2^{2n}+1", "1ac66aa19f2577a001c296ee4c3f84fa": "{\\begin{aligned}\\sum _{n=1}^{\\infty }\\left|\\gamma _{n}-\\beta _{n}\\right|&<\\infty \\\\\\sum _{n=1}^{\\infty }n\\left|\\beta _{n+1}-\\beta _{n}\\right|&<\\infty \\end{aligned}}", "1ac67d3a5300ab0c5022cfb9fedf3f61": "(1,0),(0,1)\\quad ", "1ac722c49b915507c9cff059702cf706": "Z_{L}\\,", "1ac73d4e1e66919fc33c11419b2c6c2c": "\\sin \\theta \\approx 1.22{\\frac {\\lambda }{d}}", "1ac74de873b6051623f6a6666f04e25f": "t\\to \\infty .", "1ac752fd452eea38d6c44306c9522b1e": "x\\in {a,b}", "1ac765d6411a75ad190348f96b70614e": "(M,\\mu ,\\eta )", "1ac783c6c43ba3762d481352146e4538": "\\longrightarrow _{R}^{*}", "1ac7b0f3dd179c5b5b8c9fa073466d46": "h=1-{\\frac {2}{3}}{\\frac {(k+\\lambda )(k+3\\lambda )}{(k+2\\lambda )^{2}}}\\,;", "1ac7c1a17a7c88b812debfa6457dba1f": "{\\begin{aligned}y_{1}&={f_{1}}^{}(x_{1},\\ldots ,x_{n})\\\\y_{2}&=f_{2}^{}(x_{1},\\ldots ,x_{n})\\\\&{}\\ \\vdots \\\\y_{m}&=f_{m}^{}(x_{1},\\ldots ,x_{n})\\end{aligned}}", "1ac7d34981306d1966c327b4a357ab2b": "\\textstyle ID\\in \\left\\{0,1\\right\\}^{*}", "1ac8088c44308012141655c125bafac8": "U_{--}=\\bigcup _{n\\geq 0}\\alpha ^{-n}(U_{-})", "1ac81b8ad1e2335ce45a23887a7ded18": "\\textstyle (X,\\Sigma ).", "1ac84e66b4aa6d5c44163a47022bd90b": "|a'\\rangle \\to \\alpha ={\\frac {|\\alpha '\\rangle }{\\sqrt {\\langle \\alpha '|\\alpha '\\rangle }}}", "1ac899ccce2d417ad6fe6e9fa4bbf9ad": "\\omega =exp(\\theta ^{a}t^{a})", "1ac8cf945037b51f528f5097a2097a51": "|f(x)-f(c)|<\\varepsilon .\\,", "1ac8e3e74660dafa1f6a667bb16ceebc": "D=\\sum _{P\\in C}{n_{P}P}", "1ac9053477ef2c67dd2ead878f32361e": "R^{2}\\equiv 1-{SS_{\\rm {res}} \\over SS_{\\rm {tot}}}.\\,", "1ac93d0427703b0a5ea6255031052d68": "s,e,e_{v}\\in \\mathbb {Z} _{q}", "1ac951819db336b3c16b92d393cde79e": "I(X;Y)=\\int _{Y}\\int _{X}p(x,y)\\log {\\left({\\frac {p(x,y)}{p(x)\\,p(y)}}\\right)}\\;dx\\,dy,", "1ac970b27d2f76d07d9061a3f8bc0557": "\\Phi (B)", "1aca039fca0fe90ce18c019d54503bfb": "{\\hat {H}}", "1aca1c27d02ba4b9425b4132e0f47f1a": "L^{2}Y_{\\ell m}=\\ell (\\ell +1)Y_{\\ell m}", "1aca56b06c43f6675bf5ccaf087b5e62": "a_{i\\pm {\\frac {1}{2}}}\\left(t\\right)=\\max \\left[\\rho \\left({\\frac {\\partial F\\left(u_{i}\\left(t\\right)\\right)}{\\partial u}}\\right),\\rho \\left({\\frac {\\partial F\\left(u_{i\\pm 1}\\left(t\\right)\\right)}{\\partial u}}\\right),\\right]", "1acaac09b7d5de46368f0e9efb4eef75": "B_{ijkl}{\\frac {\\partial ^{2}u_{k}^{(1)}}{\\partial x_{j}\\partial x_{l}}}=\\rho _{0}{\\frac {\\partial ^{2}u_{i}^{(1)}}{\\partial t^{2}}},", "1acaac888369dfac85fb74c88944f023": "O(2^{O({\\frac {k\\log {k}}{\\epsilon ^{2}}}))}dn)", "1acac23443d49bc12be4e4c098717dae": "B_{n}^{1}", "1acac7a29d4fd172140faaf938e385c9": "\\gamma ^{5}=i\\gamma ^{1}\\gamma ^{2}\\gamma ^{3}\\gamma ^{4}=\\gamma ^{5+}.", "1acb598033c89b51148ce7e98c0245b9": "\\,\\gamma =\\arccos \\left({\\frac {a^{2}+b^{2}-c^{2}}{2ab}}\\right)\\,;", "1acbaf9dda30a8d0c95687637b60a8f2": "\\sigma (T)=\\bigcup _{i=1}^{m}F_{i}.", "1acbb9979bdeda693648b7d31f1c01dd": "A_{1},\\ldots ,A_{k}", "1acc02fb7d3da69280926cb4654fbba1": "R_{3},\\,M_{2},\\,M_{3}", "1acc77d5bb5a1645b117522235bb23db": "P_{1}^{0}(x)=x", "1acd18f6a4cd5b9931e49823fbb4cf74": "d\\in {\\mathcal {D}}^{n},r\\in R\\,\\!", "1acd2dc7cbc07aee146fe91daa38e943": "{\\boldsymbol {\\bar {v}}}", "1acd80071923fa3df9af07fae6718a4d": "{\\mathcal {H}}_{h,v}=[\\eta _{h,v}^{2}+(\\beta _{h,v}\\eta '_{h,v}-{\\frac {1}{2}}\\beta '_{h,v}\\eta _{h})^{2}]/\\beta _{h,v}", "1acd988cd6abcf169fd0b14b1e164b70": "\\int {\\frac {\\mathrm {d} x}{\\csc {x}+1}}=x-{\\frac {2\\sin {\\frac {x}{2}}}{\\cos {\\frac {x}{2}}+\\sin {\\frac {x}{2}}}}+C", "1acdbbcba509a96934e167c71ecd73a0": "\\{b_{n}\\}", "1ace0414b12fa88b25c98bfce55ae3ca": "\\mathbf {A} ^{\\mathrm {T} *}\\,\\!", "1ace0a5a3204a493e831125ce4e509e0": "\\langle E(s)\\rangle ", "1ace25909136c74fe8d3d9b1effdcd09": "G(v)=\\omega _{e}(v+{1 \\over 2})-\\omega _{e}\\chi _{e}(v+{1 \\over 2})^{2}\\,", "1aceea56504628d531c9ab8aa9120c44": "\\Delta _{S^{n-1}}f(t,\\xi )=\\sin ^{2-n}t{\\frac {\\partial }{\\partial t}}\\left(\\sin ^{n-2}t{\\frac {\\partial f}{\\partial t}}\\right)+\\sin ^{-2}t\\Delta _{\\xi }f", "1acf1f8b7a4510e7e56d520b6334f701": "\\langle B,+,\\lnot ,1\\rangle ", "1acfc4ccaeb8d5d13372dac7f0798adf": "p={\\sqrt {\\frac {ab^{2}-a^{2}b-ac^{2}+bd^{2}}{b-a}}},", "1ad0438311fecd2dda9d26fc710c3f58": "\\mathbf {v} \\equiv {\\begin{pmatrix}q_{1}\\\\\\vdots \\\\\\vdots \\\\q_{3N-6}\\\\0\\\\\\vdots \\\\0\\\\\\end{pmatrix}}={\\begin{pmatrix}\\mathbf {B} ^{\\mathrm {int} }\\\\\\cdots \\\\\\mathbf {B} ^{\\mathrm {ext} }\\\\\\end{pmatrix}}\\mathbf {d} \\equiv \\mathbf {B} \\mathbf {d} .", "1ad084538eeb6c542fb5188d4237a35e": "z_{1},z_{2}={\\frac {1}{2}}\\left(\\varphi _{1}\\pm {\\sqrt {\\varphi _{1}^{2}+4\\varphi _{2}}}\\right)", "1ad10a950d51012a4abdf8af6a94f7f5": "0\\leq y(t)-r\\quad \\perp \\quad N(t)\\geq 0.", "1ad141546f024caa695127aa78b2d96b": "\\zeta (s)={\\frac {s}{s-1}}-s\\sum _{n=0}^{\\infty }(-1)^{n}{s-1 \\choose n}t_{n}", "1ad1d1c7d3bfe417d5a1e3e7e49a1263": "{\\sqrt {-h}}={\\frac {2{\\sqrt {-G}}}{h^{cd}G_{cd}}}", "1ad208c737ceea78c03c278168ebf1cd": "\\varepsilon ^{\\{i,j\\},\\{p,q\\}}={\\mbox{sgn}}{\\begin{bmatrix}1&2&3&4\\\\i&j&p&q\\end{bmatrix}}", "1ad21099eee186a10725e09281509ecb": "\\leq \\subseteq S\\times S", "1ad246ac836d81dc93091efbfa00d82c": "m=\\int _{0}^{2}{\\int _{x}^{4-x}}_{}{}\\,2x+3y+2\\,dy\\,dx", "1ad250362e34f47ca7d51569614d2f9a": "{\\overline {X}}_{n}\\ \\xrightarrow {a.s.} \\ \\mu \\qquad {\\textrm {when}}\\ n\\to \\infty .", "1ad2e99505fd53dc4849b34b220ad5db": "Y_{p}", "1ad30ef5daf05fcd00b41d70a71773b7": "I(2\\omega )=\\left|\\int {|E(\\omega )|^{2}e^{i\\phi }\\mathrm {d} \\phi }\\right|^{2}", "1ad39da56b90bccfcc436120392f903c": "S(a,b)=(b,a^{b})", "1ad3bd4a3b8e07df0909d4d845c59dd3": "S=F\\cup B", "1ad3de4c753b77af0666a3918886d04c": "(x)_{n}", "1ad3ef5ca8da2adc2a9e7cd396ff0824": "\\supseteq \\!\\,", "1ad40bb74aee47b9e9f178b9a2a01b2d": "F_{G}=m\\cdot g", "1ad42097bf0281b3046a04a2af3c44d7": "k^{3}\\varepsilon ", "1ad458b4f4762e6e73e401659a75d97f": "{\\mathbf {x} }^{n}", "1ad484422d4e1ff54a1b6dd58fccee96": "\\operatorname {ch} (V)={\\frac {\\sum _{w\\in W}\\varepsilon (w)e^{w(\\lambda +\\rho )}}{e^{\\rho }\\prod _{\\alpha \\in \\Delta ^{+}}(1-e^{-\\alpha })}}", "1ad587203f5f3126d22a016b85a195fb": "\\ {\\frac {D_{pitch}\\times lift\\times (sin(\\beta )-{(L/D)_{\\alpha }}^{-1}\\times cos(\\beta ))}{D_{heel}\\times lift\\times (cos(\\beta )+{(L/D)_{\\alpha }}^{-1}\\times sin(\\beta ))}}=10", "1ad59df534c1f6f7133fa06ae5bd4aab": "-\\alpha ", "1ad5ab4c730c076fe255ba9b87850d18": "{\\begin{aligned}P&=1-{\\frac {N-1}{N}}\\cdot {\\frac {N-2}{N-1}}\\cdot \\cdots \\cdot {\\frac {N-n}{N-(n-1)}}\\\\&{\\stackrel {Canceling}{=}}1-{\\frac {N-n}{N}}\\\\&={\\frac {n}{N}}\\\\&={\\frac {100}{1000}}\\\\&=10\\%\\end{aligned}}", "1ad610ebd0add956af81456e922ad8ef": "\\,D", "1ad64ff9d9f71a5fe1b6fad1e0a307a7": "\\mathbf {B} (\\mathbf {r} ,t)={\\frac {1}{r}}\\mathbf {B} _{0}\\sin(\\omega t-\\mathbf {k} \\cdot \\mathbf {r} +\\phi _{0}).", "1ad6613e952c2e85fba379464b7eef03": "{\\mathcal {S}}=\\langle S,\\rightarrow ,\\cdots \\rangle ", "1ad6a19f85d3abaa03514e1f391ac1eb": "\\left\\langle v_{1}\\wedge \\cdots \\wedge v_{k},w_{1}\\wedge \\cdots \\wedge w_{k}\\right\\rangle =\\det(\\langle v_{i},w_{j}\\rangle ),", "1ad7040b9de2599867a04826b5419dc9": "T_{p}(K)=\\mathrm {Gal} (A^{(p)}/{\\hat {K}})\\ .", "1ad710cb18694afa8cc43db1f37b8d0a": "={\\begin{pmatrix}a&-b&-c&-d\\\\b&\\;\\,\\,a&-d&\\;\\,\\,c\\\\c&\\;\\,\\,d&\\;\\,\\,a&-b\\\\d&-c&\\;\\,\\,b&\\;\\,\\,a\\end{pmatrix}}\\cdot {\\begin{pmatrix}p&-q&-r&-s\\\\q&\\;\\,\\,p&\\;\\,\\,s&-r\\\\r&-s&\\;\\,\\,p&\\;\\,\\,q\\\\s&\\;\\,\\,r&-q&\\;\\,\\,p\\end{pmatrix}}.", "1ad71f862b1d76bd199fecbee7a7b198": "T_{t}=", "1ad77b38db007391397f32ad71290205": "\\Delta t'=0\\ .", "1ad7b392bd95dfab8e9468840279d17e": "{\\overline {B_{\\delta }}}:=\\left\\{x=(x_{1},\\dots ,x_{n})\\in \\mathbb {R} ^{n}\\left||x|:={\\sqrt {x_{1}^{2}+\\dots +x_{n}^{2}}}\\leq \\delta \\right.\\right\\}", "1ad7d8b9f2f69a879280e302f824d8da": "\\left({\\frac {\\partial T}{\\partial P}}\\right)_{S}=\\left({\\frac {\\partial V}{\\partial S}}\\right)_{P}", "1ad7f005c0cd028353fa6dd2e8c6bf4e": "\\int (Tf)g", "1ad80504847b080a710a00021959ebf4": "\\left[\\Phi (x),\\Phi (y)\\right]:=\\Phi (x)\\Phi (y)-\\Phi (y)\\Phi (x)", "1ad810342a232a36a57e5314053472dc": "n\\geq 1", "1ad827e8c9d18d30dd832cb313109dc9": "m{\\dot {r}}^{2}=2E-{\\frac {L^{2}}{mr^{2}}}+{\\frac {2GmM}{r}}=2E-{\\frac {1}{r^{2}}}\\left({\\frac {L^{2}}{m}}-2GmMr\\right),", "1ad83aa505c260716d973a5d3e5d476e": "a_{4}+b_{4}+c_{4}=2b_{1}", "1ad87ce2f948498f698911a05b157327": "J_{2}^{2}|j_{2}m_{2}\\rangle =j_{2}(j_{2}+1)\\hbar ^{2}|j_{2}m_{2}\\rangle ", "1ad8fe7cf27c143c862ecc737d61a192": "\\Phi _{\\pm }", "1ad943023f9e4231a37fead0969a5ac3": "(\\mathbb {N} ,\\leq )", "1ad9459b74efd19572aad78729e56b16": "\\phi ,\\ \\phi \\rightarrow \\chi \\vdash \\chi ", "1ad9467ebbcf00fe7fb6fec2167b2ebe": "O(d_{in}\\log p\\log ^{1+\\epsilon }q)", "1ad96c376da3011a6f66f75ae0bcb5ab": "P(x)\\in U", "1ad9b76c75f2cb9cfa49956459d83077": "(P\\cap B)+x\\subseteq K", "1ada27fb55c3ab32cb1d63b3ad9b3399": "\\Box \\lnot K", "1ada7aab630bcf6f89e480b4baf646ac": "b(\\lambda )=f_{\\lambda }(i)=\\int {\\sin \\lambda t \\over \\lambda \\sinh t}\\,dt={\\pi \\over \\lambda }\\tanh {\\pi \\lambda \\over 2},", "1ada834c5b64222b916a7060cdf8f9e6": "\\delta W=(\\sum _{i=1}^{n}\\mathbf {F} _{i})\\cdot {\\dot {\\mathbf {d} }}\\delta t+(\\sum _{i=1}^{n}(\\mathbf {X} _{i}-\\mathbf {d} )\\times \\mathbf {F} _{i})\\cdot {\\vec {\\omega }}\\delta t=(\\mathbf {F} \\cdot {\\dot {\\mathbf {d} }}+\\mathbf {T} \\cdot {\\vec {\\omega }})\\delta t,", "1ada9851b7773942f2ad049381808b54": "\\pi (\\mu ,\\nu )\\leq 1", "1adaff66a44a3afe0e927847259e591d": "\\quad \\eta =\\rho \\sin \\psi ,\\;", "1adb033695d2e51e0690730fa5e82a63": "\\delta _{ext}:Q\\times X\\rightarrow S", "1adb1080e3b2e54604085aa615621ca9": "MC_{L}=30", "1adb3925456c348576c82a866708f398": "x^{4}-y^{4}=z^{2}", "1adb47037809792950943fb40f3a1517": "G(\\chi ):=G(\\chi ,\\psi )=\\sum \\chi (r)\\cdot \\psi (r)", "1adb4bbcfdce4fa99e3411506ef587df": "\\sum _{J_{z'}}\\langle J,J_{z}|{\\vec {\\mu }}_{J}|J,J_{z'}\\rangle \\cdot \\langle J,J_{z'}|{\\vec {J}}|J,J_{z}\\rangle =\\sum _{J_{z'}}g_{J}\\mu _{B}\\langle J,J_{z}|{\\vec {J}}|J,J_{z'}\\rangle \\cdot \\langle J,J_{z'}|{\\vec {J}}|J,J_{z}\\rangle ", "1adb53068c520e0b7b65e6a924b5607e": "IMPL(\\alpha ,\\alpha ')=s^{T}L(u\\otimes v)=1-\\alpha (1-\\alpha ')", "1adb5d76e6c14cef19c702b1cbadb50a": "\\lambda _{D}={\\sqrt {\\frac {\\epsilon _{0}kT_{e}}{n_{e}q_{e}^{2}}}}", "1adb6a5cd3e5ae167cec17f89a817c77": "dz'={\\frac {dz}{(cz+d)^{2}}}", "1adb8d805ba5e60698e25a35a37691ac": "s_{k}={\\frac {\\pi k}{2q+1}},k=-q,\\dots ,-1,0,1,\\dots ,q.", "1adc163714b0581dce311b5686767089": "\\displaystyle {f=v+\\partial _{n}u.}", "1adc4d19bb30cabcc86226a8691870be": "e(P,Q)=e(Q,P)", "1adc5344f42e3e5f0782da68fa3ef848": "={\\frac {-y(x)\\left[p(x)y'(x)\\right]|_{a}^{b}+\\int _{a}^{b}{y'(x)\\left[p(x)y'(x)\\right]}\\,dx+\\int _{a}^{b}{q(x)y(x)^{2}}\\,dx}{\\int _{a}^{b}{w(x)y(x)^{2}}\\,dx}}", "1adc6e3ec105d58d84dd8a7e414600bd": "{\\begin{aligned}g(Z,w)&=91125Z^{6}\\\\&{}\\quad {}+(-133650w^{2}+61560w-193536)Z^{5}\\\\&{}\\quad {}+(-66825w^{4}+142560w^{3}+133056w^{2}-61140w+102400)Z^{4}\\\\&{}\\quad {}+(5940w^{6}+4752w^{5}+63360w^{4}-140800w^{3})Z^{3}\\\\&{}\\quad {}+(-1485w^{8}+3168w^{7}-10560w^{6})Z^{2}\\\\&{}\\quad {}+(-66w^{10}+440w^{9})Z\\\\&{}\\quad {}+w^{12}\\\\[8pt]h(Z,w)=&(1215w-648)Z^{4}\\\\&{}\\quad {}+(-540w^{3}-216w^{2}-1152w+640)Z^{3}\\\\&{}\\quad {}+(378w^{5}-504w^{4}+960w^{3})Z^{2}\\\\&{}\\quad {}+(36w^{7}-168w^{6})Z\\\\&{}\\quad {}+w^{9}\\end{aligned}}", "1adcf2582883c2cadbdca811a6c5391d": "u_{i}^{\\mathrm {T} }u_{j}={\\begin{cases}1,&{\\mbox{if }}i=j,\\\\0,&{\\mbox{if }}ij\\notin E.\\end{cases}}", "1adcf68d8fc50ae80421aeb541eb6970": "\\lambda (T,W)={\\frac {\\int _{-W}^{W}{\\|U(f)\\|}^{2}\\,df}{\\int _{-1/2}^{1/2}{\\|U(f)\\|}^{2}\\,df}}.", "1adcfd50c9e00fbf417ee939945504a1": "|f|", "1add0562d4853ae3e7d487717e311840": "16K^{2}=16(S-p)(S-q)(S-r)(S-s).\\,", "1add12e7dd6cb71bf2db56c642b5537d": "\\operatorname {Pr} (X=x_{k})=p_{k}\\quad {\\mbox{ for }}k=1,2,\\ldots ", "1add3dba0a8521842d4ff1e36249442f": "\\int _{\\gamma }g(\\zeta )d\\zeta =0,", "1add97377ec9dafbb474c7f8508992fd": "r\\equiv \\mathrm {RAD} ~{\\mathbf {cm}}", "1add9df4162f037631f347b31eb1383e": "x\\geqq y", "1add9f2286646ffc9e9923902b69e28b": "d{\\boldsymbol {\\varepsilon }}_{e}=0", "1ade06deda570e2c07559914edbcf850": "K_{s}", "1ade0f8a3afed6749e6ee9c2385386a4": "\\sigma =T", "1ade18fc80eda153811ffe166cf883c0": "{\\vec {e}}_{1}=\\left(1+m/r\\right)\\,\\partial _{r}", "1ade2fe2b9ab15923e8f6392e07b09bb": "{M}", "1ade52b43a6888e9f458dc70296bcdc2": "n_{1}=0,\\,\\,n_{2}=\\pm {\\frac {1}{\\sqrt {2}}},\\,\\,n_{3}=\\pm {\\frac {1}{\\sqrt {2}}},\\,\\,\\tau _{\\mathrm {n} }=\\pm {\\frac {\\sigma _{2}-\\sigma _{3}}{2}}\\,\\!", "1ade53d02d5d0a17297a27f4aecd3282": "B_{4}{\\bar {S}}", "1ade87eaaa384e7322455afd323580e0": "{\\frac {F_{A}}{F_{B}}}={\\frac {a}{b}}", "1ade98012fd87daa808ea7ba82716ee2": "1-e^{-5.0/4.5}", "1adea2b8be39c8108bee671b4cc1a745": "V_{D}\\gg nV_{T}", "1aded4342fe9d330c576dc01a4b4bab4": "\\definecolor {gray}{rgb}{0.9764705882352941,0.9764705882352941,0.9764705882352941}\\pagecolor {gray}g\\mapsto g\\circ h", "1adee0c3551c6f3ec04634a43c34b5a7": "P^{2}=P_{\\mu }\\,P^{\\mu }", "1ae04f9254bd1f61562ead77db876ef7": "\\cdot \\left({\\text{largest monomial of }}s_{n}\\right)^{i_{1}}", "1ae06e1aa73f04423aa9975348ef050c": "f'(x)=(\\ln a)a^{x}=\\lambda f(x)", "1ae09669bbba392bff06eb2207e2bebc": "\\sum _{n=1}^{\\infty }{\\frac {n+1}{n}}\\left({\\frac {1}{2}}\\right)^{n}={\\frac {2}{1}}\\cdot {\\frac {1}{2}}+{\\frac {3}{2}}\\cdot {\\frac {1}{4}}+{\\frac {4}{3}}\\cdot {\\frac {1}{8}}+\\cdots ", "1ae0b142f4e01f6d1f5f86d7c9245cf9": "h_{x}\\leftarrow (A^{T}D_{v}^{-2}A)^{-1}c", "1ae0b23a2363a831199866191d4433ab": "\\,{}^{(-1)}a=0", "1ae0cde92eed5b79c8556648ec6b680b": "\\mathbb {E} \\left(\\left(\\int _{0}^{t}H\\,dM\\right)^{2}\\right)=\\mathbb {E} \\left(\\int _{0}^{t}H^{2}\\,d[M]\\right).", "1ae1189ceb808bad843e4f5d93100fcc": "\\mathrm {not} ~r\\equiv \\mathrm {true} ", "1ae13ce616cb08a5c83105ddf5a44817": "\\Omega _{d}={\\begin{cases}{\\frac {1}{\\left({\\frac {d}{2}}-1\\right)!}}2\\pi ^{\\frac {d}{2}}&d{\\text{ even}}\\\\{\\frac {\\left({\\frac {1}{2}}\\left(d-1\\right)\\right)!}{(d-1)!}}2^{d}\\pi ^{{\\frac {1}{2}}(d-1)}&d{\\text{ odd}}\\end{cases}}", "1ae1a4d556f4fb861e4235bcdbede541": "S=\\langle U,Q,V,f\\rangle ", "1ae261aef27d11634ebcab81d3f8d4e3": "n^{2}*s^{2}", "1ae277cfc394d164e4e632b087787cb8": "f(x_{1})", "1ae2cb1e93a77ab84ee40435e84c4313": "v_{O}", "1ae2d257a35264eeb51c7f6c196fd023": "I[f]=\\mathbb {E} [V(f(x),y)]=\\int V(f(x),y)\\,dp(x,y)\\ .", "1ae2ddebac92c61b4f392152c972c9fe": "A_{1}F_{1\\rightarrow 2}=A_{2}F_{2\\rightarrow 1}", "1ae2f80bd43c7cba007bc6f7781de58a": "(x^{2}+y^{2})^{2}+18a^{2}(x^{2}+y^{2})-27a^{4}=8a(x^{3}-3xy^{2})\\,", "1ae334a3278b1eae94b2bf01cff69fdb": "3\\left\\lceil {\\frac {\\Delta }{2}}\\right\\rceil ", "1ae34f534195b92331cccf570d1f286e": "J_{2}^{2}", "1ae351b7d3143355015b4c5d0c938716": "{R^{2}}_{2}=(C)-(E)", "1ae38954f6cba2eafda4e9c34df8d944": "C_{j}", "1ae3bacaa3c96d25914aa2aabd6b6400": "L_{x}", "1ae3c0d1129eb02888859330b05558bf": "\\sigma _{u}^{2}=\\max(\\sigma _{-}^{2},\\sigma _{+}^{2}),", "1ae435df7b794701ad49b6c179408458": "\\scriptstyle T_{i+1}", "1ae437dda2d55cf38f3167cf58fe257c": "R_{3}^{3}(\\rho )=\\rho ^{3}\\,", "1ae43bd67873b70aa82b58769422583b": "M=\\mathbb {N} ", "1ae47e75a6b333ab3fa002e906cd54dc": "D'=2(B'_{x}C'_{y}-B'_{y}C'_{x}).\\,", "1ae4abe506d112a3cd2d8bc32e33ca75": "\\mathbf {C} =\\left(\\mathbf {J} ^{T}\\mathbf {J} \\right)^{-1}\\mathbf {J} ^{T}", "1ae5443ca64730ce84e57c692982120f": "2k_{1}^{2}>1-\\rho ", "1ae5926370b8019094da892cbbd7f01f": "T\\sin \\theta ={\\frac {mv^{2}}{r}}\\,", "1ae5ef11d69f34d471ab6f9e8bd13751": "\\Sigma _{k}^{EXP}=\\mathrm {NEXP} ^{\\Sigma _{k-1}^{P}}", "1ae5f88f935f3f6a3260fc08318ea62a": "{\\begin{aligned}\\operatorname {E} (X)&=\\mu \\quad \\quad \\quad {\\text{for }}\\,\\nu >1,\\\\{\\text{var}}(X)&={\\frac {1}{\\lambda }}{\\frac {\\nu }{\\nu -2}}\\,\\quad {\\text{for }}\\,\\nu >2,\\\\{\\text{mode}}(X)&=\\mu .\\end{aligned}}", "1ae615f85c2c8a72da6c11e3faa2042d": "\\scriptstyle {a=6.1121\\ \\mathrm {millibar} ;\\quad \\;b=18.678;\\quad \\;c=257.14^{\\circ }\\mathrm {C} ;\\quad \\;d=234.5^{\\circ }\\mathrm {C} .}", "1ae6253da2e5447636536685459f4b67": "r_{1}={\\sqrt {R}}", "1ae6c2974417879c11bc464c52e34304": "S\\subseteq Y", "1ae6c67f72f01dbe1006f899e10e09da": "X_{n+1}=\\sum _{j=1}^{X_{n}}\\xi _{j}^{(n)}", "1ae6d477182e26a40c2fb736336b7f0d": "\\delta _{0}=\\left\\{{\\begin{matrix}1&{\\text{if}}\\ X_{1}=0,\\\\0&{\\text{otherwise,}}\\end{matrix}}\\right.", "1ae6f2bb35b18be6fc7fcd08b1574326": "\\log _{e}(Y)=a_{0}+\\sum _{i}{a_{i}\\log _{e}(I_{i})}", "1ae7082c8d30dcb49a59f41439bfc911": "{\\tilde {H}}_{d}", "1ae7346d959265fa4ed8ec8b5a856305": "{\\begin{pmatrix}1\\\\0\\\\0\\end{pmatrix}},{\\begin{pmatrix}-{\\frac {1}{3}}\\\\{\\frac {\\sqrt {8}}{3}}\\\\0\\end{pmatrix}},{\\begin{pmatrix}-{\\frac {1}{3}}\\\\-{\\frac {\\sqrt {2}}{3}}\\\\{\\sqrt {\\frac {2}{3}}}\\end{pmatrix}},{\\begin{pmatrix}-{\\frac {1}{3}}\\\\-{\\frac {\\sqrt {2}}{3}}\\\\-{\\sqrt {\\frac {2}{3}}}\\end{pmatrix}}", "1ae745ad48763d30ac1f85fada71e162": "a\\cos(x)+b\\sin(x)={\\sqrt {a^{2}+b^{2}}}\\cos(x-\\operatorname {atan2} \\,(b,a))\\;", "1ae78e54b1449ba31b989d59863351fd": "A_{m},A_{e}", "1ae7a7f0027ec06f1da999992b2403b8": "A=A_{0}\\exp(-z/2H)\\,", "1ae7ec45b7aec5668bc80a216f29c88d": "\\int _{0}^{1}\\phi (p)dp=1", "1ae806034c2ba77b4044c196756871ad": "[n:=n+1][n:=n+1]\\ldots \\Phi (n)\\,\\!", "1ae824002506a3f5ce2ca38753bf3fca": "M,a\\models \\lnot \\phi \\iff M,a^{*}\\not \\models \\phi ", "1ae883436e2a70938611aa5d1af0ae9b": "\\textstyle \\mathbf {c} ", "1ae8c0f9b51291d9f997641c6d948c66": "F_{ii}=1-\\sum _{j}C_{ji}", "1ae8c13f1d0b25fce1cbff89f45f9769": "[{\\ddot {U}}]", "1ae8cb5ef015bcabe733ef4c8fc38de8": "(2^{5}/9!!)\\pi ^{4}=(32/945)\\pi ^{4}", "1ae8efcb74e5b61c4d2e52ea85f292df": "{\\bar {X}}_{m}", "1ae8ff0eafb50e0559d4a0ef51f7c463": "{\\frac {df(v)}{dv}}=0", "1ae919ef6ec1d1f5f9b2a5a219aa1601": "V(x,t)", "1ae98698b8f9f9ff679d95a8c43ac71a": "\\tau :[0,1]\\to D", "1aea2dc12c695e4d9b36b9b5a1d0133f": "{\\text{Standard error}}={\\sqrt {\\frac {p(1-p)}{n}}}={\\sqrt {\\frac {p-p^{2}}{n}}}.", "1aea572dc87627e1d16d240540acd41c": "1\\leq \\mathrm {Ra} _{D}\\leq 10^{5}", "1aeae92df63ec0a79495589220062dad": "{\\mathcal {A}}_{\\mu }^{a}", "1aeaf3c261f9acaa8ac8e99bf011ea10": "g_{ij}(z):={\\frac {\\partial ^{2}}{\\partial z_{i}\\,\\partial {\\bar {z}}_{j}}}\\log K(z,z),", "1aeb00464964c160b0ea3840de0ccd00": "\\,d_{x}=l_{x}-l_{x+1}", "1aeb120f6ea3f5113553d4337931b1ec": "V_{base}=1pu", "1aeb1ed45d9f55b1428dad6ca5ed5b44": "P_{D,max}", "1aec2483b0477f359ee72eb858d11ea9": "\\kappa \\geq cf(\\mu )", "1aec5e46df30faddf18262d8160c6572": "BSC=BAC+\\pi /3", "1aec6c8c4e8c7782fa4aae85b0de36e3": "A=(E^{2}-\\omega ^{2}L^{2})^{1/2}/\\omega ", "1aec8bd65246b75bc289689c744323aa": "\\,x=x_{1}x_{2}\\in \\Sigma ^{*}", "1aecb5334e90200e1a0a8411633463c2": "2s2p", "1aecf962e2e222b81a3f7cb8716ff5a2": "\\mathbf {y} ^{\\prime \\prime }=(y_{1}^{\\prime \\prime },\\ldots ,y_{N}^{\\prime \\prime })", "1aed01a08b6e79d1dc616180c500ce40": "-{\\frac {174611}{330}}", "1aed088131190cd9f70e514814c97e6b": "\\delta (\\mathbf {x} -\\mathbf {x'} )", "1aed1b41855e3eee16204d835251eb08": "ds=(1/T)\\,du+(-\\mu /T)\\,d\\rho ", "1aed2ee78c0a5c1f09824c55f049b212": "FAP", "1aed45a53e3cd44a6895a5f4b2da1d4b": "{hS}={{\\frac {(\\rho _{s}-\\rho )}{\\rho }}(D)}\\left(f\\left(\\mathrm {Re} _{p}*\\right)\\right)=RD\\left(f\\left(\\mathrm {Re} _{p}*\\right)\\right)", "1aed7c905c058af93801adf85f46d9f7": "=\\displaystyle {{G_{a}\\lambda ^{2} \\over 4\\pi Z_{\\circ }}E_{b}^{2}}\\,", "1aed7e6dcb00e784d459219bd9149e76": "F^{*}{\\overset {x^{n}}{\\to }}F^{*};", "1aedc54f1f41e3c44632d5c0bdca0542": "T_{E}={\\epsilon \\over k}={h\\nu \\over k}={hc_{s} \\over 2k}{\\sqrt[{3}]{N \\over V}}\\,,", "1aee05778344022ab93538f353a5d7f7": "S=(X,\\delta )", "1aee131c9e35a67ca3309e83f8ff2c27": "\\sigma _{\\log K}={\\frac {\\sigma _{K}}{K}}", "1aee47dc2cebad185137293a664a4605": "E[\\varepsilon _{t}^{2}]=\\sigma ^{2}\\,,", "1aee9b8c8a0bf160241e3d3fa152abd4": "{\\vec {f}}_{2}={\\frac {1}{r}}\\,\\partial _{\\theta }", "1aeebc7bcf55efa7233f61531760078b": "\\mathrm {n_{0}} \\,=\\,e_{rms}{\\frac {\\pi }{\\sqrt {3}}}\\,(2f_{0}\\tau )^{\\frac {3}{2}}", "1aeecb0f67090c42293a6fd8f982ce5a": "\\langle z\\rangle =e^{i\\mu -\\sigma ^{2}/2}", "1aeef03af57d0a3b26888e29399960f7": "A1^{+}", "1aef646d38450e1146d8a855a3311ccf": "{\\begin{aligned}&(x_{1},x_{2},\\dotsc ,x_{n},x_{n+1},\\dotsc )+(y_{1},y_{2},\\dotsc ,y_{n},y_{n+1},\\dotsc )=\\\\&(x_{1}+y_{1},x_{2}+y_{2},\\dotsc ,x_{n}+y_{n},x_{n+1}+y_{n+1},\\dotsc )\\end{aligned}}", "1aef938f991402967c780af0792d9b92": "\\sigma ^{\\mu }\\partial _{\\mu }\\equiv \\sigma ^{0}\\partial _{0}+\\sigma ^{1}\\partial _{1}+\\sigma ^{2}\\partial _{2}+\\sigma ^{3}\\partial _{3}", "1aefa448f9b0ed9db73e879cbb71d5d0": "\\mathbf {V} \\,\\!", "1af0871d9e866330e9ea59cae72d88d2": "v={\\sqrt {G(M\\!+\\!m) \\over {r}}}={\\sqrt {\\mu \\over {r}}}", "1af08bc2d0cd512c8939c6d0570db4f9": "\\Gamma _{0}[0]=0\\,", "1af08db3081b7836689bf2297ceb6819": "\\nabla \\left(u\\cdot \\nabla f(x)\\right)=u^{T}H(x)=H(x)u", "1af15663e8ebe3a5d5d01b8a078db690": "{\\dfrac {\\partial u}{\\partial x}}={\\dfrac {\\partial v}{\\partial y}}", "1af173878deaef50f556e964bb4e2771": "\\kappa _{2}=2Dt,", "1af199c4f7213582e2812f300d60ae6b": "{\\frac {1}{c_{1}}}V\\left({\\frac {1}{\\lambda _{1}}}J\\right)^{k}\\left(c_{2}e_{2}+\\cdots +c_{n}e_{n}\\right)\\rightarrow 0", "1af20ac9e418aefa8e36fd283210e1f1": "|\\nu _{j}\\rangle ", "1af22538e8a7973f9748d6c5bbca0ebd": "cP=0\\,", "1af2564dc766de96dd5e6677bd695b65": "k_{n}|{\\mathcal {Z}}|^{2K_{n}}=o\\left({\\frac {n}{\\log n}}\\right)", "1af25d11f87465ffdaf6a78f602aed98": "x=R_{pullup}={\\sqrt {R_{sensmax}R_{sensmin}}}", "1af2a5e7f1328ae428d623a8529f0f3c": "\\langle A\\rangle _{\\sigma }=\\int a\\;\\mathrm {d} \\langle \\psi |P(a)\\psi \\rangle ", "1af2bbfa7890a063130883af933d16a6": "T^{4},SU(2l+1),T^{1}\\times SU(2l),T^{l}\\times SO(2l+1),", "1af2d1bcd12380ef8ef0f12b7c837d88": "H(k-1)={\\begin{bmatrix}Y(k)&Y(k+1)&\\cdots &Y(k+p)\\\\Y(k+1)&\\ddots &&\\vdots \\\\\\vdots &&&\\\\Y(k+r)&\\cdots &&Y(k+p+r)\\end{bmatrix}}", "1af2de8141afd27e1667da9adeee79b2": "\\forall x(\\phi \\lor \\psi )", "1af32d4335c674f8b2b658bffc259ff1": "x\\not \\in D", "1af32dad0ac1d436a4489c7fe92e05e0": "<0.0062", "1af349c66ef07dec75bc7de287758b66": "P_{n}(x)\\,", "1af3682c0da2ff90a6494748af21f0d2": "\\int _{a}^{b}f(x)\\,dx.", "1af386504f5e4171edd57c6ae4bd4f44": "{dn \\over dt}=an^{2}-bn,", "1af38af214bc4019e1fae1aaad4140c5": "L/k", "1af3a816afaa0d99b385293f1fcc8273": "P={\\begin{pmatrix}{\\frac {1}{2}}&{\\frac {1}{2}}&0&0\\\\{\\frac {1}{2}}&{\\frac {1}{2}}&0&0\\\\0&0&{\\frac {1}{2}}&{\\frac {1}{2}}\\\\0&0&{\\frac {1}{2}}&{\\frac {1}{2}}\\\\\\end{pmatrix}}+\\epsilon {\\begin{pmatrix}-{\\frac {1}{2}}&0&{\\frac {1}{2}}&0\\\\0&-{\\frac {1}{2}}&0&{\\frac {1}{2}}\\\\{\\frac {1}{2}}&0&-{\\frac {1}{2}}&0\\\\0&{\\frac {1}{2}}&0&-{\\frac {1}{2}}\\\\\\end{pmatrix}}", "1af3cecd0523deda46f04a2404178925": "f_{l}(X)", "1af40b4606c6faa8813987603da8d6b6": "b'\\in I^{j}", "1af41556c2fd55bd3afea0318302ae33": "x=-{\\frac {b}{2a}}", "1af432fe417df6053045ad5ab173a4ab": "{\\alpha }={\\frac {d\\omega }{dt}}={\\frac {d^{2}{\\theta }}{dt^{2}}}", "1af455baf8cf53a8d9f71472178234c4": "{\\tfrac {1}{3}}\\left(\\eta '\\right)^{2}=f(\\eta )\\,", "1af471feba26b381c155ad93b3614986": "n^{2}\\,", "1af4ae0db5e582d6f295d4c53389d8ad": "\\displaystyle {[JA,JB]=[A,B].}", "1af4e40ddcbad4cca752eedef2af0fd3": "\\int d^{D}x\\left[(A_{\\mu }^{1})^{2}+(A_{\\mu }^{2})^{2}\\right]\\,,", "1af4f7d743268fd743f3a5bc4c11da86": "\\mathbf {F} _{p^{2}}", "1af58d000230b6ec31191d9d26f529ec": "[\\alpha ]_{D}^{20}=+6.2", "1af58fe9c80b6605f2b3aed805bbc079": "g_{2}(x)=x^{e}-C_{2}\\in \\mathbb {Z} _{N}[x]", "1af59be9b51bd923c5ada20b43588dce": "\\operatorname {E} (\\theta )", "1af59d01d5e2f7161edbe57591415dd4": "(e_{i})", "1af5ea123a721e525ebb311c9ebf901d": "f_{i}\\in {\\mathfrak {g}}_{-\\alpha _{i}}", "1af609ca52ca77c7bbddcef1c9ba62bb": "\\operatorname {length} (E)", "1af651daccfc49790236c9f39855cffa": "{\\begin{aligned}E_{\\mathrm {electrostatic} }^{(1)}=&\\sum _{\\ell _{A}=0}^{\\infty }\\sum _{\\ell _{B}=0}^{\\infty }(-1)^{\\ell _{B}}{\\binom {2\\ell _{A}+2\\ell _{B}}{2\\ell _{A}}}^{1/2}\\\\&\\sum _{M=-\\ell _{A}-\\ell _{B}}^{\\ell _{A}+\\ell _{B}}(-1)^{M}I_{\\ell _{A}+\\ell _{B},-M}(\\mathbf {R} _{AB})\\;\\left[\\mathbf {M} ^{\\ell _{A}}\\otimes \\mathbf {M} ^{\\ell _{B}}\\right]_{M}^{\\ell _{A}+\\ell _{B}},\\end{aligned}}", "1af6bd199ce2aac9a45c123f97659a0e": "P\\propto {\\frac {1}{V}}", "1af6c7540c44ac8e8c44a5e3297c6136": "1-\\left(1-{\\frac {1}{m}}\\right)^{kn}.", "1af7139e781c6797dec3e18ae85b5c7b": "\\scriptstyle n=1,2,3,4\\ldots ", "1af7393e428e1b9ae5792044d259ecb8": "V_{T}", "1af766c105f030c46e281daddcf56e41": "L_{\\rho }(\\gamma )=\\int _{\\gamma }\\rho ^{*}{\\bigl (}f(z){\\bigr )}\\,|f\\,'(z)|\\,|dz|=\\int _{\\gamma ^{*}}\\rho (w)\\,|dw|=L_{\\rho ^{*}}(\\gamma ^{*}).", "1af768e593ae489c4917bfe516aa5c78": "2i+2", "1af7884d818f0bad92cf66873f0b12a2": "{\\tilde {f}}_{1}(\\lambda )={\\tilde {f}}(\\lambda )\\cdot \\varphi _{\\lambda }(w),", "1af7acc84b33c00ecff4867eb8f32cd2": "p=m{\\dot {x}}.\\,", "1af85e561d314cf46f473d4ef8346f53": "x_{(n+1)/2}", "1af8b10681588f6c5cf157b7f6944d73": "L_{f}+L_{c}=12\\,", "1af95adc078fdfccde1eb4b99c6207dc": "\\langle {\\mathcal {P}}(F),\\cap ,\\cup ,-\\rangle ", "1af98d15d06890f93e936957e66704d1": "\\Sigma (W,S)=(X\\setminus \\{0\\})/\\mathbb {R} _{>0}^{\\times }", "1af992a6aba6cfd7cd4af977c36b829f": "r(t)=g(e(t),c(t))\\,", "1af99f41ed5cdd516b3360121fd136eb": "d{\\mathcal {O}}_{s}(x_{0})=\\mathrm {span} (dh_{1}(x_{0}),\\ldots ,dh_{p}(x_{0}),dL_{v_{i}}L_{v_{i-1}},\\ldots ,L_{v_{1}}h_{j}(x_{0})),\\ j\\in p,k=1,2,\\ldots .", "1af9a3f284badda70538449cb633af2a": "c-ab-a_{x}\\neq 0\\quad {\\text{and/or}}\\quad c-ab-b_{y}\\neq 0,", "1af9d4cc61f114b34680210b593cffe2": "z=n{\\frac {\\left(1+{\\frac {1}{\\zeta }}\\right)^{n}+\\left(1-{\\frac {1}{\\zeta }}\\right)^{n}}{\\left(1+{\\frac {1}{\\zeta }}\\right)^{n}-\\left(1-{\\frac {1}{\\zeta }}\\right)^{n}}},", "1af9dcecc465950e25f7153943970180": "r", "1afa62b4df51ba8156571ba39b48b019": "b=2(1-\\nu )=a+1\\,\\!", "1afa8f5325d8a9235e9a108944ba038b": "1.9761", "1afae345de48dcb83decaa2fb09e98eb": "\\displaystyle {\\mathcal {Z}}(z)=\\sum _{n}z^{n}g(n,N)=1+Nz+{\\tfrac {1}{2}}N(N-7)z^{2}+\\cdots ", "1afaff502a27cd99fbb7e1dfe2e464de": "{\\bar {\\delta }}\\delta -\\delta {\\bar {\\delta }}=({\\bar {\\mu }}-\\mu )D+({\\bar {\\rho }}-\\rho )\\Delta +(\\alpha -{\\bar {\\beta }})\\delta -({\\bar {\\alpha }}-\\beta ){\\bar {\\delta }}\\,,", "1afb2ce5e9b16a2d326c9da195e92dc2": "{\\begin{bmatrix}\\mathbf {N} \\\\\\mathbf {M} \\end{bmatrix}}={\\begin{bmatrix}\\mathbf {A} &\\mathbf {B} \\\\\\mathbf {B} &\\mathbf {D} \\end{bmatrix}}{\\begin{bmatrix}\\varepsilon ^{0}\\\\\\kappa \\end{bmatrix}}", "1afb4b51684ca3abc8d0f4d361258c17": "q<_{\\mathcal {O}}p", "1afb6aa26b49a381393f524b4882ce27": "{E}_{3-8}", "1afb87747818374ee07fe60f295dbc7a": "{\\begin{aligned}{\\frac {dm_{12}}{ds_{2}}}&=M_{21},\\\\{\\frac {dM_{12}}{ds_{2}}}&=-{\\frac {1-M_{12}M_{21}}{m_{12}}}.\\end{aligned}}", "1afbbb386ab798c6b552991f69e18130": "f(\\Theta )", "1afc00238bee17ca5a88497e5c931c63": "\\mathbb {R} ^{d}", "1afc21e08db9a8abb8c34d682609ce27": "\\mathbf {c} \\cdot \\mathbf {\\Delta k} =2\\pi l", "1afc24b08861fbf702476f54795b2a63": "M\\subset E^{\\downarrow }.", "1afc32f9ecdea6a79b7c763e1065835f": "\\displaystyle {D(x+h)\\leq |x+h-z|^{2}\\leq |z-x|^{2}+2h\\cdot (x-z)+|h|^{2}=D(x)+f_{0}(h)+|h|^{2}}", "1afc42f49b669ad0d6ef5477115a490e": "c\\!+\\!i\\infty ", "1afc5128582f9baefeebfde2d9aefb45": "k_{x}\\in {\\mathbf {k} }_{x},\\ k_{y}\\in {\\mathbf {k} }_{y}", "1afcfaa63a5afea49f514264538b3a1a": "\\mu _{2}^{'}=k+\\lambda ^{2}", "1afcfda102bde92d02d058d3ae82d1dc": "\\Lambda ^{n}", "1afd7b9bc7095e2960ce839642923fa0": "\\phi (x)=x^{*}(x)", "1afd9c8547614dce607f5dba2c1fd0cf": "G={\\frac {\\cos \\theta }{\\sin \\theta +\\cos \\theta -1}}", "1afe0d11723f75be1089d388b38f68b2": "\\qquad i(\\lambda v)=(i\\lambda )v=(\\lambda i)v=\\lambda (iv)\\qquad ", "1afe2cad8fc2cedde53f95b7c1eb90bb": "\\{\\Phi _{00}", "1afe31c4d8ceddf84df3464c592cfd50": "\\beta _{i}={\\frac {\\sin(\\nu \\phi _{i})}{\\cosh(\\nu \\eta )-\\cos(\\nu \\phi _{i})}}", "1afe49a9fa63c16926a3feb7e0640f32": "F(t)=\\int _{0}^{t}E(t)\\,dt\\qquad E(t)={\\frac {dF(t)}{dt}}", "1afe83854fb470a16e06fc2c306e175e": "dS/S=\\rho /\\sigma \\cdot (G\\cdot M/r^{2}-\\omega ^{2}\\cdot r)\\cdot dr", "1afe83a709a1a412122aade17092fe25": "f^{-1}(n)=m", "1afeeaca605ea59476fe74c104c489d1": "f_{i}=\\pi _{i}\\circ f", "1afef125f6afcab921f2dc3748a15ffb": "\\int _{0}^{\\infty }e^{-ax^{b}}dx={\\frac {1}{b}}\\ a^{-{\\frac {1}{b}}}\\,\\Gamma \\left({\\frac {1}{b}}\\right).", "1aff20ed758ea7a989f9f154a4f24541": "h_{t}(x,y)=\\sum _{i=0}^{\\infty }\\exp(-\\lambda _{i}t)\\phi _{i}(x)\\phi _{i}(y).", "1aff2c5530b9ce3b9ccc580904206b22": "IV_{1}", "1aff676d2e4456ca58de4a9549636141": "K(\\sigma )=1+3\\langle JX,Y\\rangle ^{2}", "1aff74ad01db1471c065dfd06c510874": "f(x)=\\delta (x-x_{0}),", "1affff94ac44e6254d43802aade7dd7a": "h_{jk}\\in C^{\\infty }(U,\\mathbb {C} )", "1b0038ae6e0a54ae4f634238411efc32": "{\\frac {1}{n}}x+{\\frac {1}{n}}l", "1b00858e0e33eefa62983374a555f13f": "-a_{4}=5\\sum _{n}{\\frac {n^{3}q^{n}}{1-q^{n}}}=5q+45q^{2}+140q^{3}+\\cdots ", "1b00aa69ce82836826f738b97a135e4b": "\\lfloor n/r\\rfloor ", "1b00ae279f7b96a86886ca6f15d165b1": "{\\frac {B_{in}^{2}}{2\\mu _{0}}}\\sim {\\frac {\\rho V_{out}^{2}}{2}}", "1b00bc8e537c9b2193aba3c95e56c935": "{\\frac {137,200\\ \\mathrm {N} }{(285\\ \\mathrm {kg} )(9.807\\ \\mathrm {m/s^{2}} )}}=49.1", "1b00d919da1dc670605fb472e0c6b429": "n_{-}\\,", "1b0116561d756b52af5539658a6a42fc": "\\operatorname {Var} [S-T]>0", "1b018e24dc0b76b2f2b2d1ea33c857bc": "\\mathbf {O_{j}} =\\mathrm {diag} (b_{*,o_{j}})", "1b019cb970b7e8eb584a06d91f73c401": "{\\mathfrak {g}}\\to G", "1b01a8552a79f513b0119c0dd04813ae": "{\\boldsymbol {\\varepsilon \\digamma \\varkappa \\varpi }}\\!", "1b0216b9681891f44a24a44e28f3d477": "\\|a\\mathbf {v} \\|=|a|\\|\\mathbf {v} \\|", "1b023c8a1c0f4e6bb39d9c33b59a6d6b": "e^{-S_{F}}X", "1b024fce65a4a1b71a19e3f722ce6dbc": "\\ r={\\mathcal {F}}^{-1}\\{R\\}", "1b025d92fd92b3e06c33a1875cdc9a70": "{G^{\\prime \\prime }(\\omega )=\\ }[G^{\\prime }(\\omega )G_{0}-{G^{\\prime }(\\omega )}^{2}]^{-{1 \\over 2}}", "1b0268237ddc2e8d876edc88cd3f8d09": "\\chi _{0.99}^{2}", "1b030a0c8c2c9ce33437b7e8ec3d6c34": "\\cot \\beta _{\\rm {2}}", "1b0333e98862ae2000bb04c4cfcbac58": "T[i]=-1", "1b0430bedb8015c29413b5091ca48eb9": "D_{n,0}=\\left[{n! \\over e}\\right]", "1b04546c2176a0f4bbd2ea67c4b705ea": "1/4-\\varepsilon ", "1b045fe6d5677587f838df87afd7b302": "f(\\theta )=f(0)", "1b04817e489e244106a6be4aaacbed01": "{\\frac {1-6p(1-p)}{np(1-p)}}", "1b04bb9d91cbcfd506c84bcbf3e6cbba": "\\scriptstyle -0.9(2.0)\\times 10^{-10}", "1b04cd43f4af6a7904f2a9e2add46a71": "(3/4)^{N}", "1b04d4b8d95be6e4d0231344c1b80613": "\\left(A^{-1}\\right)^{T}=\\left(A^{T}\\right)^{-1},", "1b050f0576b87200d049e7e65e341bdd": "u=0", "1b05190451f5afe29e2a16ff0a55eb2d": "\\ C_{2}^{3}(3)={\\frac {16}{49}}", "1b052cd3b241da58d06cf7ed003e803e": "I_{lm}\\ {\\stackrel {\\mathrm {def} }{=}}\\ \\int d\\mathbf {r} ^{\\prime }{\\frac {\\rho (\\mathbf {r} ^{\\prime })}{\\left(r^{\\prime }\\right)^{l+1}}}{\\sqrt {\\frac {4\\pi }{2l+1}}}Y_{lm}^{*}(\\theta ^{\\prime },\\phi ^{\\prime })", "1b05392e4237526b07a8c891e9225476": "iF=\\nabla \\wedge \\nabla ", "1b054d93d3165ef3eae57a0c1a8bc21b": "{\\frac {1}{k^{2}}}=\\left({\\frac {R}{C}}\\right)\\left.{\\frac {dC}{dr}}\\right|_{r=R}", "1b055e10cf989e212d72b842cec4ccd7": "P(x)=1/|S|", "1b0579088363d0cbcbb03faec2fd656b": "p_{Z}(z)={\\frac {1}{\\pi }}{\\frac {\\beta }{(z-\\alpha )^{2}+\\beta ^{2}}},", "1b05822c92b77d143c8d71ab41e8eee2": "\\liminf _{n\\to \\infty }{\\frac {g_{n}}{\\log p_{n}}}=0", "1b058d255444fb05d6146751280865a7": "\\zeta ={\\frac {1}{2{\\sqrt {K_{p}K_{v}RC}}}}", "1b058f5ee96d93ddcbec00c3907ece91": "\\;\\varrho _{A_{1}\\ldots A_{m}}^{T_{A_{k_{1}}\\ldots A_{k_{l}}}}", "1b059ef5acffd2bc1150d3ff659a8140": "p_{i}e_{k-i}=r(i)+r(i+1)\\quad {\\text{for }}1{\\tfrac {1}{4}}", "1b0e89ed6ef04d7559592cf9e6c07b3f": "\\exp \\left(t+{\\frac {z}{t}}\\right)=\\sum _{n=-\\infty }^{\\infty }t^{n}{\\mathcal {C}}_{n}(z).", "1b0e9a9a0efa483da2349413a7db5ffc": "a_{2}=V_{2}^{+}", "1b0eb889c60178c7a00bfa6e5a378d0e": "{\\displaystyle {\\tilde {\\varphi }}_{t}(k)}", "1b0f1f8c09bb40a7ad3e8d7777c205aa": "FDR=Q_{e}=\\mathrm {E} \\!\\left[Q\\right]=\\mathrm {E} \\!\\left[{\\frac {V}{V+S}}\\right]=\\mathrm {E} \\!\\left[{\\frac {V}{R}}\\right]", "1b0f51fd5dddd69b87ba8dd3e2ce584d": "\\mathbb {A} _{k}^{n+1}", "1b0f5dc4eccc85b21ee2c35db8c58961": "\\scriptstyle {\\bar {x}}", "1b0f9a6fc397e23f3f4249156b108dcc": "\\partial {\\bar {\\partial }}\\omega =i{\\text{Tr}}F(h)\\wedge F(h)-i{\\text{Tr}}R^{-}(\\omega )\\wedge R^{-}(\\omega ),", "1b0fb89907c90b0cfd071a0cdd7ea0dc": "p<0.1", "1b0fd7c8fbe6c3931bb3d345bb459303": "\\langle e_{n},e_{m}\\rangle =\\delta _{mn}", "1b0fd9efa5279c4203b7c70233f86dbf": "-10", "1b0fe23e62dc07b05a69b73d1bd071fa": "G\\subset {\\mathbb {C} }^{n}", "1b0fe25146802dc106c54db9bc5fdd13": "F_{1}(x)=1\\,", "1b0ff8a1440da24f36b42de8c0012e0a": "\\epsilon (v)>\\epsilon (u)", "1b105b1e9533074584d7bc91d314181d": "\\sigma _{B}", "1b10710d91eb975193c12955abd2076a": "2.1)\\ {\\text{Stock A}}\\ -={\\text{Flow}}\\ ", "1b108299b9df85ab77ac0ace61b2f6c6": "C_{D}^{(\\beta )}(\\{x\\})", "1b10a896736a697cd7094aac70a40b46": "\\operatorname {let} M\\operatorname {in} N", "1b10bad2218174608c7e730723bcf53d": "\\left(0,\\ 0,\\ \\pm 1,\\ 0,\\ 0\\right)", "1b10e53be4f6a6819579125899cd553a": "[L_{i},L_{j}]=i\\hbar \\varepsilon _{ijk}L_{k}.", "1b10e63097337b87541c40654ec099d7": "Dyad:V_{2}\\otimes V_{2}\\to V_{2}", "1b114a065e8ab5d12728de23768cc065": "1s^{2}", "1b116c483fcaf1a450b647b7013f9082": "\\sum _{i=0}^{n}i^{3}=\\left({\\frac {n(n+1)}{2}}\\right)^{2}={\\frac {n^{4}}{4}}+{\\frac {n^{3}}{2}}+{\\frac {n^{2}}{4}}=\\left[\\sum _{i=1}^{n}i\\right]^{2}", "1b119ba70111eab26fa5b4fed8be139e": "m={\\frac {\\Delta y}{\\Delta x}}", "1b11aad7da804360e2f924b70f733bd6": "{\\frac {d\\rho _{gg}}{dt}}=\\gamma \\rho _{ee}+{\\frac {i}{2}}(\\Omega ^{*}{\\bar {\\rho }}_{eg}-\\Omega {\\bar {\\rho }}_{ge})", "1b11eb9de4c96790aa8dd236ac790c24": "L-k+1", "1b1209e26e79b8a78f0a945c9abf3533": "v(t)=-v_{\\infty }\\tanh \\left({\\frac {gt}{v_{\\infty }}}\\right),", "1b122c7cf5af5b1e264fa2b2f73f32f8": "\\nabla ^{2}\\mathbf {E} ={\\frac {1}{c^{2}}}{\\frac {\\partial ^{2}\\mathbf {E} }{\\partial t^{2}}}", "1b127b7ac6ed1d2cc682f414fadf8b0d": "A_{zz}", "1b128a5e47b8cadcd9d0ff3fdb6bea60": "D\\leq {\\frac {\\log {(n-1)}}{\\log(k/\\lambda )}}+1", "1b132921f3be55f89eead5bcf306a1c9": "{\\text{bias}}_{I}(X)=\\left|\\Pr _{x\\sim X}\\left(\\sum _{i\\in I}x_{i}=0\\right)-\\Pr _{x\\sim X}\\left(\\sum _{i\\in I}x_{i}=1\\right)\\right|=\\left|2\\cdot \\Pr _{x\\sim X}\\left(\\sum _{i\\in I}x_{i}=0\\right)-1\\right|\\,,", "1b133abdb241471bdf2d07238fb6a4c9": "n![z^{n}]g_{m}(z)=\\left[{\\begin{matrix}n\\\\m\\end{matrix}}\\right].", "1b1428001688c80e573382c49c3909c6": "f(x,y)=0=Ax+By+C", "1b14afd9d15b235d067ee09a9e52dbd2": "V=L^{3}", "1b14c6675b6864645688b221ff428d67": "{\\begin{array}{lcr}A={\\begin{bmatrix}1&-2&2\\\\2&-1&2\\\\2&-2&3\\end{bmatrix}}&B={\\begin{bmatrix}1&2&2\\\\2&1&2\\\\2&2&3\\end{bmatrix}}&C={\\begin{bmatrix}-1&2&2\\\\-2&1&2\\\\-2&2&3\\end{bmatrix}}\\end{array}}", "1b14ffc1a2dfe5f6bfdaee7845fceb91": "\\mathrm {D} _{\\mathsf {C}}=2", "1b15262bb34d66f685defeedea57672a": "I\\propto \\sum _{E_{f}-eV}^{E_{f}}|\\psi _{n}(0)|^{2}e^{-2\\kappa W}", "1b152b3c29c380b16d50209c76e62c0b": "\\left(1\\right)", "1b1539e8c5d91bc9f00cb6f3dbe1380e": "\\exists x\\in t", "1b155d1e83ad160155b2fb2210c1f77b": "\\alpha ={\\frac {k}{c_{p}\\rho }}\\,\\!", "1b1566ddd2bae1fd84f8547c9ff572b1": "\\forall {\\underline {y}}\\in Y\\forall \\epsilon >0\\exists {\\underline {x}}\\in X", "1b157ca049809d68e0e5b45d9492f3a5": "\\left\\{S_{1},\\ S_{2},\\ S_{3},\\dots \\right\\},\\quad S_{n}=\\sum _{k=1}^{n}a_{k},", "1b15a6408f9e08a508bca7eaa092c69f": "\\alpha _{\\mathrm {sun} }=\\displaystyle {\\frac {\\int _{0}^{\\infty }\\alpha _{\\lambda }I_{\\lambda \\mathrm {sun} }(\\lambda )\\,d\\lambda }{\\int _{0}^{\\infty }I_{\\lambda \\mathrm {sun} }(\\lambda )\\,d\\lambda }}", "1b15ed4e55d56f85b19efef1526ea0b1": "R^{5}", "1b165ec0353c6d96260293c1ddef44b1": "{\\mathcal {M}}_{X}", "1b17215df910f275ee469afd6b7e757f": "{\\tilde {B}}_{t}=\\int _{0}^{t}\\operatorname {sgn} {\\big (}{\\hat {B}}_{s}{\\big )}\\,\\mathrm {d} {\\hat {B}}_{s}=\\int _{0}^{t}\\operatorname {sgn} {\\big (}X_{s}{\\big )}\\,\\mathrm {d} X_{s},", "1b174a16164187f32b5dc73fe2e8d8bd": "\\sigma _{11}=2C_{1}\\left(\\lambda ^{2}-{\\cfrac {1}{\\lambda }}\\right)\\left[\\sum _{i=1}^{5}i~\\alpha _{i}~\\beta ^{i-1}~I_{1}^{i-1}\\right]~.", "1b174f1c12b1644b3ecc6993071b66a3": "I_{R2}(=I_{E})={\\frac {V_{R2}}{R2}}={\\frac {V_{Z}-V_{BE}}{R2}}.", "1b175bb3eaf1726400f845a802b7b6d4": "1\\leq aF(x,t)\\implies F(x',t')>F(x,t')", "1b21919945f5543d35a7e0eb3a7a91b7": "(\\alpha ',\\beta ',f')", "1b21ad3c59bcd4efec61f76a90129960": "U(r+\\bigtriangleup r,w)=U(r,w)\\exp([-bw+iH(bw)]\\bigtriangleup r)\\quad (1.5)", "1b21fcb18c994f5454d00136aa5e09cb": "M=\\{M_{1},M_{2},\\dots ,M_{m}\\}", "1b21fcdc827eaafe07a3bcae544005d3": "|\\phi (t)|", "1b22953c54e2b44e5f9cce5cbd322258": "\\scriptstyle -{\\frac {df(E)}{dE}}={\\frac {1}{4kT}}\\operatorname {sech} ^{2}{\\tfrac {E-\\mu }{2kT}}", "1b22975cf1cd9da405329c03dbe730e8": "l\\cdot w\\cdot h", "1b22aca4fdd9e66bc63bb0af7bc6667b": "2^{2}q", "1b231df3e6bc982fad35499758cd755a": "(n,k)=(n,\\delta n)\\Rightarrow k=\\delta n", "1b2348bbf1dd243e00397ab347b3692e": "{\\begin{aligned}u_{i}&=x_{i}-\\delta _{iJ}X_{J}=x_{i}-X_{i}\\\\{\\frac {\\partial u_{i}}{\\partial X_{K}}}&={\\frac {\\partial x_{i}}{\\partial X_{K}}}-\\delta _{iK}\\\\\\end{aligned}}", "1b2378c4fe01c9bf699c919383f18b7f": "\\langle x+y,z\\rangle =\\langle x,z\\rangle +\\langle y,z\\rangle ", "1b23e007aad71555a2966913ddddb764": "D(c,\\rho )", "1b244b225115a4f82e3839b82107f51b": "disc({\\mathcal {H}})\\leq C{\\sqrt {t\\log m}}\\log n", "1b24866f630c1d4ec0678944aa57a558": "{e}^{-\\Theta /bT}", "1b2491207a778f51461693338fefeaf2": "n=\\left\\{0,1,2,\\dots \\right\\}", "1b24b243af9d0f8ab7d830c0ae82dc33": "H(k[\\Delta ];x_{1},\\ldots ,x_{n})=\\sum _{\\sigma \\in \\Delta }\\prod _{i\\in \\sigma }{\\frac {x_{i}}{1-x_{i}}}.", "1b24f7977473be435ca421be58d3cfbe": "F_{\\alpha \\beta }=\\left({\\begin{matrix}0&E_{x}/c&E_{y}/c&E_{z}/c\\\\-E_{x}/c&0&-B_{z}&B_{y}\\\\-E_{y}/c&B_{z}&0&-B_{x}\\\\-E_{z}/c&-B_{y}&B_{x}&0\\end{matrix}}\\right)\\,", "1b25dcf600c55da38f936b7f89980efe": "2^{\\operatorname {LS} (K)}", "1b26002baa591e65e5367c52fab2e293": "{\\frac {P}{A}}={\\frac {2\\pi h}{c^{2}}}\\left({\\frac {kT}{h}}\\right)^{4}\\int _{0}^{\\infty }{\\frac {u^{3}}{e^{u}-1}}\\,du.", "1b262206c61572286b81d69802c6513b": "\\phi ^{\\prime }", "1b26465fdda19c334798e17d2b12c012": "A\\in 2^{\\Omega }", "1b26b257e441b257a962f30a4a9a6dcc": "E_{A}(x)\\geq E_{B}(x)", "1b26c0e4b5e30055cd3fc6bd3bd77540": "f^{-1}(\\operatorname {int} '(A))\\subset \\operatorname {int} (f^{-1}(A))", "1b26d21e5198a479a3fee07e069c7cbc": "i(V)", "1b26e3e896b5448109552db32f2be8d3": "{\\begin{aligned}&\\cos {\\frac {\\theta }{2}}=\\operatorname {sgn} \\!\\!\\left(\\!\\!\\pi \\!+\\!\\theta \\!+\\!4\\pi \\!\\left\\lfloor \\!{\\frac {\\pi \\!-\\!\\theta }{4\\pi }}\\!\\right\\rfloor \\!\\right)\\!\\!{\\sqrt {\\frac {1+\\cos \\theta }{2}}}\\\\\\\\&\\left(\\mathrm {or} \\,\\,\\cos ^{2}{\\frac {\\theta }{2}}={\\frac {1+\\cos \\theta }{2}}\\right)\\end{aligned}}", "1b274cbe937c84ed14ed978d8834ffaa": "\\sum _{m=1}^{\\infty }{\\frac {y}{m^{2}+y^{2}}}=-{\\frac {1}{2y}}+{\\frac {\\pi }{2}}\\coth(\\pi y)", "1b2765839de3715dbb3d37931c270d8e": "{\\begin{pmatrix}({\\bar {3}},1)_{\\frac {1}{3}}\\\\(1,2)_{-{\\frac {1}{2}}}\\end{pmatrix}}", "1b276ac1fe6d4425a75cdb365a367406": "1.25{\\sqrt {H}}", "1b28486709f46a3ac30f5a4cdc6b3141": "dU=\\delta Q+\\delta W", "1b28aced7e96c3f8b82023a6a2bef154": "\\lambda _{i}=\\lambda {\\text{ for }}0\\leq i}}\\exp \\left(-{\\frac {KA^{2}}{4}}\\cdot \\operatorname {sock} \\right),", "1b2a3668a465885eed04e4509594ba55": "{\\boldsymbol {\\eta }}=\\mathbf {X} {\\boldsymbol {\\theta }}", "1b2a8d80369b97c3086dd8d2361ed876": "XY={\\frac {|a^{2}+c^{2}-b^{2}-d^{2}|}{2p}}.", "1b2aa540bc5a9d35ac3dbd2e36cf1652": "u(c)={\\frac {c^{1-\\theta }-1}{1-\\theta }}\\,", "1b2af3b11deb3956ffed253d7bb2cca5": "\\left\\langle M_{B}\\left(l_{B}\\right)\\right\\rangle \\sim l_{B}^{d_{B}}", "1b2af99c6eb76e81f85b75383bf8e6ee": "\\omega _{pl}^{2}(q)={\\frac {2\\pi e^{2}nq}{\\epsilon m}}", "1b2b32c7df5f76141684bb30d39399e0": "y=\\rho _{0}-\\rho \\cos \\theta ", "1b2b45d45abd23bf45bf8b83d83858a1": "Q_{ab}{}^{c}=Q_{(ab)}{}^{c}", "1b2b4cf5107b7a7ad9566dce8dfbad22": "\\omega _{f}(t):=\\sup\\{d_{Y}(f(x),f(x')):x\\in X,x'\\in X,d_{X}(x,x')=t\\},\\quad \\forall t\\geq 0.", "1b2b7abdacade6bfee8532c14df40f73": "u_{2}={\\frac {\\lambda }{2}}+{\\frac {1}{r^{2}}}\\left({\\frac {p^{2}}{2m}}-{\\frac {1}{2}}mgz\\right).", "1b2b7fcfa3d6ceddab3a6be1df824430": "(R^{-1}P)", "1b2b88b65adbdc568e55a7df404cc854": "x=\\cos y\\,\\!", "1b2bf2869b0b2c84ba7c50ec6198ee12": "\\cos \\left({\\frac {\\pi }{5}}\\right)=\\cos 36^{\\circ }={{\\sqrt {5}}+1 \\over 4}={\\frac {\\varphi }{2}}", "1b2c0a0a81fd5346d9ce68b30226bc5d": "f\\in BV_{\\varphi }([0,T];X)\\iff \\mathop {\\varphi {\\mbox{-Var}}} _{[0,T]}(f)<+\\infty ", "1b2c2a7cee33b6fb0a68211f8500a107": "x^{4}-10x^{2}+1,\\ ", "1b2c8fcf35b342643acb33167d8db4dd": "{\\frac {\\mu ^{2}}{x}}.", "1b2c9883e732ab514175f8543c910230": "\\|Tx\\|^{2}=\\langle T^{*}Tx,x\\rangle =\\langle TT^{*}x,x\\rangle =\\|T^{*}x\\|^{2}", "1b2d2a835c5ac47d32c91fee49c9dcb9": "F\\left(\\mathbf {x} ^{(0)}\\right)=0.5((-2.5)^{2}+(-1)^{2}+(10.472)^{2})=58.456", "1b2d6445b054f48a72dd810423cd4e01": "\\tau =I\\alpha \\,", "1b2d8c73ecf1b97144e0f4b489873868": "{\\tilde {H}}(t)=\\sum _{t_{i}\\leq t}{\\frac {d_{i}}{n_{i}}},", "1b2de5396f196794b727393419fed9dc": "f(n)>g(n)\\,\\!", "1b2e2cf401b1d188124328c5ea88af01": "y\\geq 0.138+0.580x", "1b2e49c26b804dc9ac8b6343f9fdec8f": "{\\hat {\\mathbf {e} }}^{2}", "1b2e5739a139fe87db706788cc8e1e16": "\\mathbf {R} =\\mathbf {N} -\\mathbf {L} ", "1b2eb5418feaee2d3f95ec87ee7b7464": "\\sum _{i=0}^{n}p_{i}(z)f^{(i)}(z)=0", "1b2ecdaf6a6c88fa2bc82edfad84ec6e": "0<\\phi \\left[|m|\\right]<1", "1b2eddf3577dc84a9fa476308eecd83c": "h_{n}={V_{n} \\over cos(i_{n})}\\left({T0_{n+1} \\over 2}-\\sum _{j=0}^{n-1}{h_{j}{\\sqrt {{1 \\over V_{j}^{2}}-{1 \\over V_{j+1}^{2}}}}}\\right)", "1b2f08e4d4e535a01a4040874fc86a5b": "X=Y\\cdot Z", "1b2f0a509545e54f01c59a2bf2a6dfe1": "P=c_{s}^{2}\\rho ,", "1b2f36e274431ebd48a2baa36c32c1b4": "\\scriptstyle {\\frac {V}{R}}", "1b2f4f4447872b11bd2b0b2382056302": "\\mathbf {B} =\\mathbf {U} \\mathbf {S} \\mathbf {V} ^{T}", "1b2f688f7c8b1a82bc4585cda2c1bd81": "m_{\\text{density}}^{*}=g^{\\frac {2}{3}}{\\sqrt[{3}]{m_{x}m_{y}m_{z}}}", "1b2f9141e3069258db3931fba8ad271f": "u(a)={\\frac {1}{2\\pi }}\\int _{\\mathbf {T} }{\\frac {1-|a|^{2}}{|a-e^{i\\theta }|^{2}}}f(e^{i\\theta })\\,\\mathrm {d} \\theta ", "1b2fd60cd9ac6b079fd721837a9b8a77": "{\\frac {dF}{dx}}=f(x).", "1b3019d392e3ee0dc65eee83cec274af": "{\\bar {y}}={\\frac {M_{01}}{M_{00}}}", "1b30706240f0355ce91e70838a143c19": "(s)", "1b30d4cd598edf69006b735efd94fa02": "\\lambda =(\\lambda _{1},\\ldots ,\\lambda _{m_{2}})", "1b30ef2a31be91ea72f6f8d7b2d4fb9c": "x+h", "1b3117d018f75ed45c50fdc849645054": "\\Lambda (A_{1}),\\Lambda (A_{2}),\\ldots ,\\Lambda (A_{n})", "1b3134184af0d97cf98e98550f657c7d": "\\displaystyle {\\partial _{n}u|_{\\partial \\Omega }=0,}", "1b313f382aeab7621a804a829996c451": "E=\\int d^{3}x\\,\\left[{\\frac {1}{2}}{\\overrightarrow {D\\varphi }}^{T}\\cdot {\\overrightarrow {D\\varphi }}+{\\frac {1}{2}}\\pi ^{T}\\pi +V(\\varphi )+{\\frac {1}{2g^{2}}}\\operatorname {Tr} \\left[{\\vec {E}}\\cdot {\\vec {E}}+{\\vec {B}}\\cdot {\\vec {B}}\\right]\\right]", "1b31d878869a7a2a2243f0a281b4c244": "{K \\choose k}\\cdot {N-K \\choose n-k}.", "1b323c6fb303ace1a6fb6e38d644c47b": "{\\frac {\\partial {\\mathcal {H}}(q,p)}{\\partial p}}={\\frac {p(t)}{L}}=-{\\dot {q}}(t)\\ ", "1b32515c2316ef92ea433626a6489a7a": "M_{v}-N_{u}=L\\Gamma ^{1}{}_{22}+M(\\Gamma ^{2}{}_{22}-\\Gamma ^{1}{}_{12})-N\\Gamma ^{2}{}_{12}", "1b329308e0009389e60936f954966404": "\\beta =-\\eta _{2},", "1b329aa84296d7d276fa8f082f5bc636": "{\\frac {1}{m!}}\\sum _{k=0}^{m}(-1)^{k}\\left[{m+1 \\atop k+1}\\right]B_{n+k}=A_{n,m}.", "1b32b33339b988be580e35465349943e": "F(n+1)", "1b32d078a58b149739c80c76f5c3d1f5": "P_{G}", "1b3375ba38a555c30a12f04439c7443c": "I(0)<\\delta {\\text{ and }}I(T)>K\\,", "1b3385b9d76a7193c4b9bd94d6be5833": "{\\mathsf {H}}(\\cdot )", "1b33b27a9ff57ef2e67c799b57359f42": "2g", "1b33c4b25aeb798428694efad45ac72a": "e=\\sum _{k=0}^{\\infty }{\\frac {1}{k!}}", "1b3402a67b5c39eb34bffd4740445683": "K(x)=h(x)e^{{\\boldsymbol {\\eta }}\\cdot \\mathbf {T} (x)}", "1b34543b731e888cb39ff70888596175": "\\scriptstyle \\varphi ={\\frac {1}{2}}(1+{\\sqrt {5}})\\,\\!", "1b34680bfad4fd69ba62366786bb78c3": "I_{N}", "1b34cea2960b5b23a77f94ec29e5985b": "k\\{\\tau _{p}\\}.\\,", "1b34ece2df19d366b8d32e1d039a844c": "\\rho _{Actual}\\,\\!", "1b350f9b23871bccf35b479ae3fb0c5f": "r_{i}(x_{j})", "1b3551633296b23d0445a4219f4cee14": "\\delta +1", "1b355b909deb67616eaa466f2b4e65dc": "(2)\\quad t=u+r+2M\\ln {\\Big (}{\\frac {r}{2M}}-1{\\Big )}\\qquad \\Rightarrow \\quad dt=du+{\\Big (}1-{\\frac {2M}{r}}{\\Big )}^{-1}dr\\;,", "1b356209d3fef39971420b5bd851f84e": "Z={\\frac {1}{{i}\\omega C_{b}+1/R_{b}}}", "1b35c6928b246f99a9b16e5679126109": "h(x)\\rightarrow 0+h_{0}\\delta (x)", "1b35da8305a59462f6fe75a96bfc485f": "uw=vw", "1b35e567cebf847f3c717a84ab46d37d": "\\textstyle \\mathbf {b} _{1}", "1b36bb13c11fbd30a693ff6645dd9d11": "{\\begin{aligned}{\\frac {dy}{dx}}&={\\frac {dy}{dz}}\\times {\\frac {dz}{dx}}=-{\\frac {dy}{dz}}=-y'\\\\{\\frac {d^{2}y}{dx^{2}}}&={\\frac {d}{dx}}\\left({\\frac {dy}{dx}}\\right)={\\frac {d}{dx}}\\left(-{\\frac {dy}{dz}}\\right)={\\frac {d}{dz}}\\left(-{\\frac {dy}{dz}}\\right)\\times {\\frac {dz}{dx}}={\\frac {d^{2}y}{dz^{2}}}=y''\\end{aligned}}", "1b370d5d59a83a035f2e61c1dabab139": "\\operatorname {club} (\\kappa )", "1b3714cce3f5e6ebffe57748205e733c": "{\\textit {open}}\\circ {\\textit {on}}", "1b3723e0b4f1c8a81ad5ab64ae0fa23e": "t=|H|_{x}|\\,\\!", "1b372f792f4d1decf9048f8c4ee6a5dd": "{\\tilde {J_{n}}}=-{\\frac {J_{n}}{\\mu \\ R^{n}}}", "1b3731fca4937dbd72f9c627836df1e2": "X\\in C,Y\\in F", "1b373a2a965f31e05343e2b85a556dbe": "m_{5}(x)=x^{2}+x+1,\\,", "1b3763b4d746c4ef40c240595fd50baf": "O(2^{2k^{2}}+n+m)", "1b376fbd71b27090316ce8a7ae5497a1": "T(n)=2T\\left({\\frac {n}{2}}\\right)+O(1)", "1b377f93f1a3cf825444d7e86eb1cee1": "\\mathrm {Li} (x)=\\int _{2}^{x}{\\frac {1}{\\ln t}}\\,\\mathrm {d} t=\\mathrm {li} (x)-\\mathrm {li} (2).", "1b37a63f2416c0e53c0a897c01d232c4": "q=\\left(\\left[qx1,qy1\\right];\\left[qx2;qy2\\right]\\right)", "1b3823f10bac3c0a7d8ce5fa8c4129a4": "E_{\\pm }=a\\pm |\\mathbf {r} |", "1b383b961d55892e6b952c3adea3f635": "a_{0j}", "1b388f6e849e75c8a72152dfbb8715e8": "f_{\\epsilon }(x)=\\mathbb {E} {\\big [}f_{\\epsilon }(X_{\\tau })\\,|\\,X_{0}=x{\\big ]}", "1b38aed9648ae9d27b9e27801d10f1ab": "P_{\\mathrm {L} }={1 \\over 2}{{|V_{\\mathrm {S} }|^{2}R_{\\mathrm {L} }} \\over {(R_{\\mathrm {S} }+R_{\\mathrm {L} })^{2}}}\\,\\!", "1b38f75b0e72c028b4030f714aac5d83": "{\\dot {\\varepsilon _{\\rm {p}}}}=\\left[{\\frac {1}{C_{1}}}\\exp \\left[{\\frac {2U_{k}}{k_{b}~T}}\\left(1-{\\frac {\\sigma _{t}}{\\sigma _{p}}}\\right)^{2}\\right]+{\\frac {C_{2}}{\\sigma _{t}}}\\right]^{-1};\\quad \\sigma _{t}\\leq \\sigma _{p}", "1b3934fab43f480b55b91fcd4fc81893": "\\displaystyle {g_{a}(z)={z-a \\over 1-{\\overline {a}}z}.}", "1b398ca551ec519ba826c135203136f8": "k<\\beta ,", "1b39ca16b2a0f5f6c6a9f49b017d7d8c": "s=-{\\dfrac {q^{2}-a_{2}+a_{1}}{a_{1}q-a_{2}+1}}{\\bmod {\\ell }}", "1b3a394794c85d511c66dd960cdfb070": "f(U)\\neq f(V)\\vee P.", "1b3b029a8a42c7c0b8d89a7ff05ae745": "~\\alpha (0)=|\\alpha (0)|\\exp(i\\sigma )", "1b3b0c41edb32817ed869d11402df2f7": "\\Gamma _{i}", "1b3b0d39eace66eb6a6e5e6816d21d79": "f(X_{t})=f(X_{0})+\\int _{0}^{t}f'(X_{s})\\,dX_{s}+{1 \\over 2}\\int _{0}^{t}f''(X_{s})\\,d[X]_{s}.", "1b3b211925112c8a732895669fe4f002": "\\epsilon ={\\frac {\\sigma }{E}}+\\alpha {\\frac {\\sigma _{0}}{E}}\\left({\\frac {\\sigma }{\\sigma _{0}}}\\right)^{n}", "1b3b3584117fefe62a76ad3395cbdb08": "\\textstyle \\left\\{O_{i}\\right\\}", "1b3b5316829b718d24ae9b76afc9b65f": "M=S^{3}/\\Gamma ", "1b3c1a40f9cb094d47e8c6f9b0df773f": "\\circ ", "1b3cc0bbd8f0a2f68c3c22a2a0ef865b": "a=-\\omega ^{2}", "1b3cf6b61727d01f863a13b7db408732": "\\{X\\mid \\exists m\\phi (X,n,m)\\}", "1b3d000397579bb97b3a5e1776e1af53": "X\\,\\sim Bin(n,p)", "1b3d5826f934b5011047068d49f9cf15": "{\\frac {dx}{dt}}", "1b3d7a732d58fe6e43a547b840c12641": "y_{1},y_{2},...,y_{t}", "1b3d98a676ae1d627fa50d03b8e66857": "(z_{1},t_{1})(z_{2},t_{2})\\cdots (z_{n},t_{n})", "1b3dc69c97f688744bc79c7ed28fd36a": "{s_{1}}<{s_{2}}", "1b3e05e131b73cf457ff1731e746c43c": "F_{L}=L\\otimes _{k}F", "1b3e3ac75f1b8361df481220890f7585": "s_{Tx}\\,", "1b3ea1017e85ec576d2a82dc6ec24c7f": "x(t)=-1+e^{-t\\left(x_{0}+1\\right)}", "1b3eb1b5d7c7f832d9b42c55e5fc86af": "OLD(T_{i})", "1b3eec6d8313a155d8a6ff0fe0d7a255": "[{\\textrm {CO}}_{2}]_{eq}={\\frac {[{\\textrm {H}}^{+}]_{eq}^{2}}{[{\\textrm {H}}^{+}]_{eq}^{2}+K_{1}[{\\textrm {H}}^{+}]_{eq}+K_{1}K_{2}}}\\times {\\textrm {DIC}},", "1b3f3b4ccd035da3eb6fd2afa1f1d423": "O(k/\\log N)", "1b3f4b49ed4628101af2fbf5f092dbdc": "Z_{0}={\\sqrt {L/C}}", "1b3f905457325b5fe3c22e9666469fe0": "{\\hat {s}}(t),", "1b3fc1bac9f0239d5f484cbc612ad43e": "s_{j}", "1b3fd21fb65b59e433321883e7a81bf5": "|j_{1}m_{1}\\rangle ", "1b402923d0746dc6d27a1afa5ff5f03b": "{E'}^{2}-(p'c)^{2}=(m_{0}c^{2})^{2}\\,.", "1b40637a8330003a9cd61bc5d0760966": "t:={\\frac {M_{1}-M_{2}}{SD_{\\text{within}}/{\\sqrt {\\frac {n_{1}n_{2}}{n_{1}+n_{2}}}}}},", "1b40c5455185ed19cc57b87fffb65193": "|a|:={\\begin{cases}a,&{\\text{if }}a\\geqslant 0,\\\\-a,&{\\text{otherwise}}.\\end{cases}}", "1b41195424fdfedc75e4caccb6d6932d": "{S=\\ln \\Omega }\\ ", "1b413646eb2008295c35ec8f603240ea": "P(\\lambda x,\\lambda ^{2}t)=\\lambda ^{m}P(x,t){\\text{ for }}\\lambda >0.\\,", "1b4148d654a27793ad826f017779f467": "\\Gamma _{a}~,~\\Gamma _{a_{1}a_{2}}", "1b41cc724e4d1bb15eb70eebf87ca378": "(x_{n},0)>(0,1)", "1b41d1c0c4512f8042ca7e985bd521fd": "{\\rm {{dK=K\\left({\\frac {BFP-dIBB-dHB}{BFP-IBB-HB}}\\right)}\\,}}", "1b421dd6318ea30ee8682c0f2c59cd3e": "=(z+i{\\sqrt {5}})(z-i{\\sqrt {5}})", "1b4237b50df5b053d3f63da329acc13c": "\\psi (\\mathbf {r} )=\\phi _{n}(z){\\frac {1}{\\sqrt {A}}}e^{i(k_{x}\\cdot {x}+k_{y}\\cdot {y})}u(\\mathbf {r} ).", "1b4279687adfeaa02fdd2bfec5fe4638": "(-{\\vec {\\infty }},{\\vec {\\infty }})\\,", "1b429003910d26d98f5568f1d9696d7b": "P_{3}=(X_{3},Y_{3},Z_{3},ZZ_{3})", "1b42b9e56c650c87bf0c7f6e201b6d19": "I\\subset \\mathbb {R} ,\\;c\\in I", "1b43008ad84e9f9265fd27f818562daa": "\\operatorname {drop-params} [(\\lambda N.S)\\ L,D,V,R]\\equiv (\\lambda N.\\operatorname {drop-params} [S,D,F,R])\\ \\operatorname {drop-formal} [D[N],L,F]", "1b432187eac2a64b8eafbf3d484181bc": "{}_{p}F_{q}(a_{1},\\dots ,a_{p};b_{1},\\dots ,b_{q};z),\\vartheta \\;{}_{p}F_{q}(a_{1},\\dots ,a_{p};b_{1},\\dots ,b_{q};z)", "1b4329fb23abc4a14893d4fd446c6eb1": "R_{h}\\rightarrow {\\frac {bh}{b}}=h", "1b432b35835c583d857dea6ba33d82c6": "\\delta _{\\beta }^{\\alpha }\\,A^{\\beta }=A^{\\alpha }\\,", "1b4399ac0fc9fd174a9bded78a61608c": "v_{0},v_{1},...", "1b43c5e851bef33b50a224ae72a362fa": "x\\Rightarrow _{m}^{ac}y", "1b43cbbc95fd5231faf001c79f604285": "T_{\\rm {F}}", "1b43fafb317533d3532d38613cd54b72": "(n_{x},n_{y},n_{z})=(3,4,7)", "1b4408ddec06de485fef9b8deb30571e": "\\mu (({\\tilde {X}}\\cup {\\tilde {Y}})\\cap {\\tilde {Z}}).", "1b441e71205bc8c052e778ea96e6faf0": "v={\\begin{bmatrix}1\\\\0\\\\\\vdots \\\\0\\end{bmatrix}}.", "1b4447b5544fdbb0d0eeb55ec0458be8": "\\omega _{x{\\overline {\\lor }}y}^{A}=\\omega _{x}^{A}\\;{\\overline {\\sqcup }}\\;\\omega _{y}^{A}\\,\\!", "1b44f492708bf1379b984f6e5a4d1ab6": "a_{i}:{\\mathbf {S} }\\rightarrow {\\mathbf {S} }", "1b45283d035db8ea8e09a6477dd9ead0": "{\\Omega }/{\\sqrt {seconds}}={\\Omega }(s^{-1/2})", "1b45644ccd90ebdfafe1e4c6788b3960": "{\\textbf {m}}=-1+X^{3}-X^{4}-X^{8}+X^{9}+X^{10}", "1b456ead54371bf51c51ce05b19be298": "{\\frac {d\\nu }{d\\lambda }}={\\frac {d\\nu }{d\\mu }}{\\frac {d\\mu }{d\\lambda }}\\quad \\lambda {\\text{-almost everywhere}}.", "1b45d958b7370013a0ebb1a7eccad35f": "O(g(x))", "1b45e859fa657432f287c46240dcd33a": "\\forall t,s\\in [a,b],K_{X}(s,t)=\\mathrm {E} [X_{s}X_{t}].", "1b46097e090cd78100d9e5316dc22cd1": "F(x;s,\\theta )=F(x/s;1,\\theta ),\\!", "1b464d960c55c91a4a76daf6c958de82": "z_{n}=\\left({\\frac {\\pi }{2}}+n\\pi \\right)^{-1}", "1b47b53cd6d4ec9cc239fd5b8cb77217": "{\\mathcal {E}}_{T}:={\\mathcal {E}}\\otimes _{O_{S}}O_{T}", "1b47baeb7aad9b53d39f85cfa8fd68fe": "G_{0}=G", "1b47bc0cd1fc5ae0dbb5cb24777207a1": "{\\begin{matrix}\\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\\\10{\\mbox{ multiplied copies of }}10\\end{matrix}}", "1b47f2139ffb5eaff8ca580c27d33b1a": "\\pi _{ij}=\\pi _{i+}\\pi _{+j}", "1b47f56bf2bcc0b15e87f80c4ce571e5": "|z|<1", "1b4807f906c7c3d1d50b86d4a838ee27": "\\mu _{app}", "1b485d1acae11a3b1a9b6738596255e4": "[a_{0};a_{1},a_{2},a_{3},\\,\\ldots ]=\\lim _{n\\to \\infty }[a_{0};a_{1},a_{2},\\,\\ldots ,a_{n}].", "1b488e5e33d2760243ddc093082c75f6": "\\Delta f={\\frac {\\partial ^{2}f}{\\partial x^{2}}}+{\\frac {\\partial ^{2}f}{\\partial y^{2}}}+{\\frac {\\partial ^{2}f}{\\partial z^{2}}}=0.", "1b48ea9ca6e26f197682482ed59aaac6": "{\\frac {f(t)}{1-F(t)}}=p+qF(t)", "1b490e9a944f7e6e06bf0c5fc13812bc": "\\phi :\\mathbb {D} \\rightarrow D", "1b491b4a89ee669759b0a60d706fc152": "p_{s}=(1-tanh(s)^{2})", "1b4929e68b16fee477efd52fe0ede1de": "\\beta =\\omega {\\sqrt {LC}}", "1b4977a0055292ac2003ed4d6fef9609": "f(x)=1/x^{2}", "1b49803cfd0027456248b46bc70ec0c8": "\\alpha \\rightarrow 1", "1b499f2a7a708ad6a2c28505c128c5af": "h\\in H", "1b49adbd471f5cab0dd2d987adbf306e": "\\varphi (ST)={\\rm {f}}(\\{\\langle Se_{n},e_{n}\\rangle \\lambda (n,T)\\}_{n=0}^{\\infty })=v_{\\varphi ,T}(\\{\\langle Se_{n},e_{n}\\rangle \\}_{n=0}^{\\infty })", "1b49e6abcbda3ae40e96c9ce0b7690b3": "S\\subseteq T", "1b4a14adec41007ea6e5375a9ddeaa4c": "I_{t}=K_{t}-K_{t-1}\\,", "1b4a2a402bba276b957d024ec3420636": "u\\in U_{\\alpha }\\cap U_{\\beta }\\cap U_{\\gamma }.", "1b4a5e9f9368e622cee01761693ae308": "\\Box \\phi ={\\frac {\\rho }{\\epsilon _{0}}}", "1b4a6c716ee0fe03a390a9efa14589dd": "\\limsup P(n)/\\log n=2", "1b4a951adf43a75ec87f6238921b9124": "H_{0}+W_{DE}(t)", "1b4aac870f6357bc8ddff4c729d0b04c": "\\scriptstyle d_{ij}^{*}", "1b4abca54260bd0cc661ea6ea92a042c": "\\rho _{t}^{\\mathrm {ent} }(X)={\\frac {1}{\\theta }}\\log \\left(\\mathbb {E} [e^{-\\theta X}|{\\mathcal {F}}_{t}]\\right).", "1b4acfb4fc5569995071577574d5206e": "-{\\frac {d^{2}}{dx^{2}}}", "1b4b77779c688edee9a5ea643ed58935": "{\\begin{aligned}red=n-dof=n-(n+m-p)=p-m,\\end{aligned}}", "1b4ba3cb2f01720d82e6ff42f5a6ec84": "P(R_{t}|\\lambda )=\\exp(-\\lambda t)\\,", "1b4bc213275944e02ecbb63dcaca3fd5": "\\mu _{1},\\mu _{2},\\ldots ,\\mu _{t}", "1b4be3506b06bc47595b9c28c1c6870b": "r-R=0", "1b4be3778caa7728d3ab9e49bc8e72c3": "{\\frac {d\\ln k}{dT}}={\\frac {\\Delta E}{RT^{2}}}", "1b4c2e7e05508488a05be6513e9acbe3": "{\\dot {q}}={\\partial H \\over \\partial p}\\,", "1b4c552f30e1067f9215425f8b955d43": "{\\overline {A+B}}={\\overline {A}}\\cdot {\\overline {B}}\\iff {\\overline {A\\cdot B}}={\\overline {A}}+{\\overline {B}}\\iff {\\overline {AB}}={\\overline {A}}+{\\overline {B}}\\iff {\\overline {A+B}}", "1b4c5d4144aee6c054ed5cd1055a9c19": "\\theta _{\\text{c}}=\\arcsin \\left({\\frac {n_{2}}{n_{1}}}\\right)", "1b4c85c07f032f6196cc69fe47162079": "X\\colon (\\Omega ,{\\mathcal {F}},\\mathbb {P} )", "1b4cc5606b16d3c2f88f20faf90ceca4": "(5)\\qquad D\\sigma =\\sigma (\\rho +{\\bar {\\rho }})+\\Psi _{0}=-2\\sigma \\theta _{(l)}+\\Psi _{0}\\,,", "1b4cf0cd01ca9c8d65c2cfb1504b642f": "\\mathbf {v} ={\\frac {d\\mathbf {r} }{dt}}={\\dot {r}}(\\cos \\varphi ,\\ \\sin \\varphi )+r{\\dot {\\varphi }}(-\\sin \\varphi ,\\cos \\varphi )", "1b4d003146297fb4f3be4fd494526149": "d_{M}(x_{n_{k}},y_{n_{k}})<{\\frac {1}{n_{k}}}\\wedge d_{N}(f(x_{n_{k}}),f(y_{n_{k}}))\\geq \\varepsilon _{0}.", "1b4d0651ef806ab5744190328623be43": "\\left(\\gamma ^{0}\\right)^{\\dagger }=\\gamma ^{0}\\,", "1b4d6251fd64b973ee623eb795ad2a47": "\\lVert q\\rVert ={\\sqrt {qq^{*}}}={\\sqrt {q^{*}q}}={\\sqrt {a^{2}+b^{2}+c^{2}+d^{2}}}", "1b4dad9912359203c05e07e1c5e50728": "U(t)=\\exp \\left({-{\\frac {i}{\\hbar }}\\int _{0}^{t}H(t')\\,dt'}\\right),", "1b4dc54a182a7cb2bfa19f47631ac17b": "\\hbar ^{2}\\,(1/2)\\,(1/2+1)=(3/4)\\,\\hbar ^{2}", "1b4dd6e049bbb7a0f16f52b72a6572c8": "c>d", "1b4dfb5f9d10b23544b634743cfceaa1": "\\Lambda (x)={\\frac {L(\\theta _{0}|x)}{L(\\theta _{1}|x)}}={\\frac {f(x|\\theta _{0})}{f(x|\\theta _{1})}}", "1b4e3e0d4b4d4d08055963947bf7647b": "p_{i}=P(X=i)", "1b4e63855c02338acdccb214086a8da5": "\\displaystyle U\\sim \\epsilon \\epsilon 'a\\omega \\sin \\phi ", "1b4e7585bcd85536c0533fd4bf6627b8": "X\\sim {\\textrm {GB1}}(a,1,1,b)\\,", "1b4f61b72b387fd5c9532ba654e379d7": "x_{1}=1.42>{\\sqrt {2}}", "1b4f957b738a66b3788d74d02273fec7": "\\{K_{AB},A,\\mathbf {N_{B}'} \\}_{K_{BS}}", "1b4fa43f0c88a661a2a15987ee12bff1": "v=V_{max}{\\frac {[S]}{K_{M}+[S]}}", "1b4fb5ad8e17e6094bed78c8d9285d66": "2^{3}(1+x)^{3}p({\\frac {{\\frac {3}{2}}+2x}{1+x}})=8x^{3}+4x^{2}-4x-1", "1b4fe2190ffbf0dfce4ce9629f72a29a": "Y_{x}=Kx^{\\log _{2}(b)}", "1b500c1bb411497a99e4630bd770ab8f": "V(t)=\\ M(q(t))I(t)", "1b502c3a73a7fdee82caa2ef41704254": "w=1/3", "1b50878e6d4f005a7e6ff25c4d90064a": "t\\mapsto (a\\cos t,a\\sin t,bt),t\\in [0,T]", "1b5097d55f7ad1576b2ad3c383712785": "\\langle r^{2}\\rangle \\approx {\\frac {3k_{\\rm {B}}T}{m}}t^{2}=\\langle v^{2}\\rangle t^{2}.", "1b50bf999080f951478f832d0890ab62": "{\\boldsymbol {T}}", "1b50e809e15adc21f66b4190985678b5": "\\Re \\left[\\mathrm {Bi} (x+iy)\\right]", "1b5159dad7eb824f669e3bcfbc42b358": "d_{k,n}=a_{k,n}-k/m_{n}", "1b5161108cfa5abe8069d6bca469b6d5": "{\\frac {1}{2}}\\left|{\\frac {f^{\\prime \\prime }(x_{n})}{f^{\\prime }(x_{n})}}\\right|0\\}\\,", "1b735602edd399667deac52c541bd290": "{\\mathfrak {M}}(d)", "1b735b48cf704e8029cab1e20af3d078": "\\int _{0}^{\\pi }\\sin \\theta \\,d\\theta \\int _{0}^{2\\pi }d\\varphi \\;R_{\\ell }^{m}(\\mathbf {r} )^{*}\\;R_{\\ell }^{m}(\\mathbf {r} )={\\frac {4\\pi }{2\\ell +1}}r^{2\\ell }", "1b7368f46bb45dd22ef25c46d6f54619": "{\\begin{cases}{\\dot {\\mathbf {x} }}=f_{x}(\\mathbf {x} )+g_{x}(\\mathbf {x} )z_{1}\\\\{\\dot {z}}_{1}=f_{1}(\\mathbf {x} ,z_{1})+g_{1}(\\mathbf {x} ,z_{1})\\overbrace {{\\frac {1}{g_{1}(\\mathbf {x} ,z_{1})}}\\left(u_{a1}-f_{1}(\\mathbf {x} ,z_{1})\\right)} ^{u_{1}(\\mathbf {x} ,z_{1})}\\end{cases}}", "1b7370c73eb8409047acd351389d740c": "\\lim _{p\\to \\infty }{\\left(\\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}\\right)^{\\frac {1}{p}}}=\\max _{i=1}^{n}|x_{i}-y_{i}|.\\,", "1b73b46902f9af4b1078f9decf3c9431": "\\psi _{1}(z)={\\frac {d^{2}}{dz^{2}}}\\ln \\Gamma (z)", "1b74464353515988215bd5bf2abb0a72": "A={\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix}}", "1b74657b0b8ce073588540d39aed374d": "u/y", "1b7490913900363d51207a4fcb174457": "R={\\frac {(\\rho _{s}-\\rho )}{\\rho }}=1.65", "1b74b6d01d07b242dcbb75969c8ece95": "\\langle n\\rangle ", "1b74be694c6066749910c82f1cfdf08b": "0^{(\\omega )}", "1b74dc569e10c78567281a506e9461aa": "C_{3}=36\\ \\mathrm {pF} \\,", "1b751afbd2349bcb84a75e0942e51086": "K[[x,y]]/(x^{2},xy)", "1b7537beff8c524c217c5f1321e00e19": "e^{-{\\frac {2\\pi i}{N}}}", "1b756b041148bd177b3712638a3f0af1": "\\limsup _{T\\rightarrow \\infty }{\\frac {1}{T}}\\log \\left({\\frac {Z_{\\pi }(T)}{Z_{\\nu }(T)}}\\right)\\leq 0", "1b7570d1931c89df7bdd6120615e01d9": "\\forall X,Y\\in L^{0}({\\mathcal {F}}_{T})", "1b75767a344d24b547a3a740f05863c1": "\\sin E'={\\frac {y}{b}}\\ .", "1b757c17f789d57d1b9d842b06f4eaf8": "{n \\choose k}_{2}", "1b75b3fabd2029c79b5dec5720e6323a": "(p-1)/2p", "1b75f8b00d53e9fff5826221d62557d4": "(1-x)(1-x^{2})(1-x^{3})\\dots =1-x-x^{2}+x^{5}+x^{7}-x^{12}-x^{15}+x^{22}+x^{26}+\\dots .", "1b75fa0bf829aa8736aa82aa3c12e657": "J\\propto \\Psi ^{\\delta }", "1b762e083205175dcb99111ae0ebdd00": "{\\mathit {n_{q}}}=t,q=0,1,...,p-1", "1b763d485c59cddde353d9f6cea6834c": "=1-{\\frac {1}{(1+e^{\\mathrm {log} (e^{y}-1)})^{\\theta }}}", "1b764acd730c804b795b63ea90e0b13c": "(w\\ll \\Delta ^{1/2})", "1b764ad88cde290846778f96df1b2ed9": "p\\colon E\\to B", "1b76663e3241b2a3ac0f22f0008ae1f2": "Q_{lm}\\ {\\stackrel {\\mathrm {def} }{=}}\\ \\int d\\mathbf {r} ^{\\prime }\\rho (\\mathbf {r} ^{\\prime })\\left(r^{\\prime }\\right)^{l}{\\sqrt {\\frac {4\\pi }{2l+1}}}Y_{lm}^{*}(\\theta ^{\\prime },\\phi ^{\\prime })", "1b767ef177de610ed1d759cf6fe484d8": "B_{z/2}=0", "1b76da0b0550e492d5065c9bf3304fba": "\\,\\varphi =\\ t-\\arctan(\\ t)", "1b76e99344a80d5b7b89d132066e2c9b": "f,g_{1}\\ldots g_{m}:\\mathbb {R} ^{n}\\rightarrow \\mathbb {R} ", "1b76f3e5a33306fad957baab52fa5b80": "t{\\begin{Bmatrix}2,p\\end{Bmatrix}}", "1b770f7c8775617e4cb88819c8ff370f": "{\\begin{aligned}&R\\cos \\left(\\alpha \\right)=R_{0}\\sin \\left(l\\right)\\\\&R\\sin \\left(\\alpha \\right)=R_{0}\\cos \\left(l\\right)-d\\\\\\end{aligned}}", "1b7733fea96aba283e1b6cd6affba3d0": "\\lim _{x\\to \\infty }u'(x)=1.004.", "1b7735a9e93908622c9d66fcdb533079": "\\Delta m=111,132-559\\cos 2\\varphi .", "1b774e01dcddf51f1f0a9c15f8dc5162": "t_{1}'<0", "1b776dc3cdd65c6b8d92d255476372de": "V(I)=\\{P\\in \\operatorname {Spec} \\,(A)\\mid I\\subseteq P\\}", "1b77c7997a606e649b4eb4fcb9e89a20": "\\sigma _{12}=\\sigma _{21}", "1b7816047af5232d8bc125c49d4c9883": "e^{it{\\hat {M}}_{E}}=\\lim _{n\\rightarrow \\infty }[e^{it{\\hat {M}}_{E}/n}]^{n}=\\lim _{n\\rightarrow \\infty }[1+it{\\hat {M}}_{E}/n]^{n}.", "1b782558c9fe4f9fbdf04185745495e1": "r_{\\rm {max}}", "1b7857e0d4b5d61a948fb961d284fff2": "{}_{2}H_{2}(a,b;c,d;1)=\\sum _{-\\infty }^{\\infty }{\\frac {(a)_{n}(b)_{n}}{(c)_{n}(d)_{n}}}={\\frac {\\Gamma (d)\\Gamma (c)\\Gamma (1-a)\\Gamma (1-b)\\Gamma (c+d-a-b-1)}{\\Gamma (c-a)\\Gamma (c-b)\\Gamma (d-a)\\Gamma (d-b)}}", "1b78730de93934d43eca8453af483816": "M=m_{1}+m_{2}.\\,", "1b78a2ba2cf8935925fc806e008eb57b": "E_{1,{\\text{thr}}}={\\frac {(m_{a}c^{2}+m_{b}c^{2}+m_{c}c^{2})^{2}-m_{1}^{2}c^{4}-m_{2}^{2}c^{4}}{2m_{2}c^{2}}}", "1b78c0858e819a50f597dc98350f8034": "{\\dot {\\,}}\\!\\,", "1b7906ac4c24db91d350e6fb03bdec98": "\\left(0,\\ \\pm 1,\\ 0,\\ 0,\\ 0\\right)", "1b79461b3b244d0166e065bd8dfc71a2": "T:K\\to L", "1b79523a0b0bef2ec987bd9bfb9435f6": "\\int Z_{n}^{m}(\\rho ,\\varphi )Z_{n'}^{m'}(\\rho ,\\varphi )d^{2}r={\\frac {\\epsilon _{m}\\pi }{2n+2}}\\delta _{n,n'}\\delta _{m,m'}", "1b79539cc938bf4486effbbd51d2b564": "n=PQ", "1b795a7e3d4498259bd0eaaa4d690fc9": "g\\left(\\sum _{b\\in B}l_{b}b\\right):=\\sum _{b\\in B}f\\left(l_{b}^{\\sigma ^{-1}}b\\right)=\\sum _{b\\in B}l_{b}f(b)", "1b7a15f2c94a6171f41f9e3318261f5a": "\\scriptstyle K_{a}>K_{b}>0,", "1b7a1fe49217687bd95fd5b4153543fe": "a\\uparrow b=a^{b}", "1b7a3f3b4150a0c946e3ebc7311300c6": "\\Omega _{n}R^{n-1}\\sinh ^{n-1}{\\frac {r}{R}}\\,", "1b7a983bd2b06d95dac83700eb5bab7b": "D_{R\\delta }^{2}(\\mathbf {X} ,\\mathbf {0} )\\left(\\left(\\mathbf {X,0} \\right):\\Omega \\to \\mathbb {R} ^{2}\\right)", "1b7ab232531f34313df95a803e1e48e3": "\\exp _{2}^{i-2}(n^{O(1)}))^{\\Sigma _{j}^{\\rm {P}}}", "1b7acce0697412647bb92c42729c5c62": "g=G{\\frac {m}{r^{2}}}", "1b7ad3c0ff1ba15b03f7ff013ed02510": "S=\\sum _{k}\\sum _{j}r_{k}W_{kj}r_{j}\\,", "1b7afc034cd1fa6c19102ed668dd9365": "\\pi ({\\tilde {\\gamma }}(t))=\\gamma (t).", "1b7b1eff5898e3b799c58f0c34df4309": "\\eta _{Optics}", "1b7b9f64be4dc475dfb3f269abb52d38": "(m_{i,j})_{i,j=1}^{k}", "1b7bd384811acae63af2e05877b3655f": "\\sigma _{\\rm {e}}", "1b7c05daef90262f2ca226e6a3cd8927": "max_{i,j}p_{i,j}\\leq T", "1b7c22e4097190e5b451fe489abb0806": "G_{p}^{(n)}=\\left\\langle 0|{\\mathcal {T}}{\\mathopen {:}}v_{i}(y_{1}){\\mathclose {:}}\\dots {\\mathopen {:}}v_{i}(y_{n}){\\mathclose {:}}\\phi _{i}(x_{1})\\cdots \\phi _{i}(x_{p})|0\\right\\rangle ", "1b7c339be6ab6ae47ca404b83f8eb1dc": "\\eta (x,t)=\\eta _{2}+H\\,\\operatorname {cn} ^{2}\\left({\\begin{array}{c|c}\\displaystyle {\\frac {x-c\\,t}{\\Delta }}&m\\end{array}}\\right),", "1b7c82c51c132dc31a920a4b44acdc8e": "{\\frac {\\text{supports}}{{\\text{supports}}+{\\text{opposes}}}}\\times 100", "1b7cede1bebd106339192982aa1c13b6": "S(\\mathbf {q} )", "1b7d097cfaccb7213fd2574f25f593d8": "RF=(2W+X-Y){\\sqrt {8}}", "1b7d27c3d2842da2b7d340f681586262": "(J^{n}f)(x)={1 \\over (n-1)!}\\int _{0}^{x}(x-t)^{n-1}f(t)\\;dt,", "1b7d724abed9413f1199be7002f07a70": "\\displaystyle \\gamma (z)=\\gamma _{-}(z)^{-1}\\gamma _{+}(z),", "1b7d955c8da7f8acd3bcf74a0b8c7850": "g_{v}=0", "1b7e1e089f8753271b38f59863c395e2": "{\\tilde {r}}^{2}=\\pm 1", "1b7edb89e3ef2335e8520393ad42259b": "m\\!\\left(a\\right)", "1b7f768767c24b69e9847bd5a21605c3": "{\\text{h}}=E_{1}-y_{2}", "1b7f7db9071ffe6cd26422c9cc12efa3": "=2\\eta ^{\\mu \\nu }\\gamma ^{\\rho }\\gamma ^{\\sigma }\\gamma _{\\mu }-4\\gamma ^{\\nu }\\eta ^{\\rho \\sigma }\\,\\quad ", "1b7f8cd1802f3169842aa431387f8375": "d_{xy}=N_{2}^{c}{\\frac {xy}{r^{2}}}=-{\\frac {i}{\\sqrt {2}}}\\left(Y_{2}^{2}-Y_{2}^{-2}\\right)", "1b7fcaa2e80f3cc459ba13babb1338cb": "{\\frac {1}{10}}", "1b80171f969d96935530a2b127c14332": "p:G_{0}\\to M", "1b80823027034aa6a1ca8f974c00863f": "=10^{-23}{\\frac {\\mathrm {erg} }{\\mathrm {s} \\cdot \\mathrm {cm^{2}} \\cdot \\mathrm {Hz} }}", "1b809a07021b54600a36b5a3e4fa1850": "\\displaystyle L=\\mu _{0}N^{2}A/l.", "1b80bc8d1448b3cbae3fcf9ef043613d": "(Th)(w)=\\lim _{\\varepsilon \\rightarrow 0}-{1 \\over \\pi }\\iint _{|z-w|\\geq \\varepsilon }{h(z) \\over (z-w)^{2}}\\,dxdy.", "1b80ccb2d4c422d730f4aa47fd783fd5": "B^{2}-n", "1b80f9b12175158ad3ded9ac0026534a": "N(T)={\\frac {1}{\\pi }}\\mathop {\\mathrm {Arg} } (\\xi (s))={\\frac {1}{\\pi }}\\mathop {\\mathrm {Arg} } (\\Gamma ({\\tfrac {s}{2}})\\pi ^{-{\\frac {s}{2}}}\\zeta (s)s(s-1)/2)", "1b8118e5a29ee40996e1365a66bdb848": "2^{w}", "1b81437c098fa9628dba16678b880e9e": "g_{p}d.\\,", "1b85f74371eca8f93aca0edac979c194": "Y_{(j-{\\frac {1}{2}},{\\frac {1}{2}})jm}=\\left({\\begin{array}{c}{\\sqrt {\\frac {j+m}{2j}}}Y_{j-{\\frac {1}{2}},m-{\\frac {1}{2}}}\\\\{\\sqrt {\\frac {j-m}{2j}}}Y_{j-{\\frac {1}{2}},m+{\\frac {1}{2}}}\\end{array}}\\right)", "1b864433866ef59c2742e830e37ecab9": "\\det(M)", "1b87236b320458c9f43f9264176074b9": "\\textstyle {\\frac {2}{2-1}}=4", "1b8729c7d13c458fea49216876f07d87": "|\\mathbf {C} |^{p+1}", "1b873842c06a9e4e8f9966b5de2ebd7e": "g(x)=1", "1b8748df044433752e67451aaf47cc8e": "\\xi ={\\sqrt {\\frac {\\hbar ^{2}}{2m|\\alpha |}}}", "1b874b93eafc45ae8405387284583a2b": "x\\times y={\\frac {1}{2}}(xy-yx).", "1b87548baf8b937bf17fe8b0028fbedd": "(8)\\qquad Q_{1}=C\\;A_{2}\\;{\\sqrt {2\\;{\\frac {Z\\;R\\;T_{1}}{M}}{\\bigg (}{\\frac {k}{k-1}}{\\bigg )}{\\bigg [}(P_{2}/P_{1})^{2/k}-(P_{2}/P_{1})^{(k+1)/k}{\\bigg ]}}}", "1b875b2c0ff5e0e4d60b42ad73afc5d7": "J(\\mathbf {w} )={\\frac {\\mathbf {w} ^{\\text{T}}\\mathbf {S} _{B}\\mathbf {w} }{\\mathbf {w} ^{\\text{T}}\\mathbf {S} _{W}\\mathbf {w} }},", "1b87d4b994b3b7efd92b9978a7b91fe6": "\\operatorname {Var} (y_{1})=\\sigma ^{2}", "1b8862873c3ba5af1cae7bf97ed24da4": "N(s)={\\frac {1}{\\pi }}\\operatorname {Arg} \\xi (1/2+i{\\sqrt {s}})", "1b886b42c6e32bf155d8bd311e9deb9d": "\\mathbb {F} {q^{n}}\\to \\mathbb {F} _{q^{n}}", "1b88cb287e992918a7fe73097849d157": "T_{env}", "1b88f86f59d82a9fb80a5d934d507647": "\\pi (x)={\\rm {Li}}(x)+O\\left(x\\mathrm {e} ^{-a{\\sqrt {\\ln x}}}\\right)\\quad {\\text{as }}x\\to \\infty ", "1b899005d745ac8a3be24520dc235ee9": "{\\begin{array}{rcl}y_{1}'&=&y_{2}\\\\y_{2}'&=&y_{3}\\\\&\\vdots &\\\\y_{n-1}'&=&y_{n}\\\\y_{n}'&=&F(x,y_{1},\\cdots ,y_{n}).\\end{array}}", "1b8991d3a0692e7c568ab956587c289c": "X({\\mathbf {a}})", "1b89aa48b2fe8b2316d378e3a6856897": "f(z)=F(z){\\overline {F({\\overline {z}})}}", "1b89e3633cfbb7f178ae571f5112b033": "\\kappa _{n}=2^{n-1}(n-1)!\\,k", "1b89ef0da3fd9a5e3e638c79d2cd8d42": "\\displaystyle r\\approx r_{0}\\!\\left({\\frac {\\ell }{L}}C,\\!~K\\sin(\\theta )\\right)", "1b8a33c2ae28ec4042bf8fc01661b131": "G(\\zeta )={\\frac {\\tfrac {e^{\\zeta }}{4}}{3+\\zeta +{\\frac {\\tfrac {e^{\\zeta }}{8}}{3+\\zeta +{\\frac {\\tfrac {e^{\\zeta }}{12}}{3+\\zeta +\\ldots }}}}}}", "1b8a9ed8b39f57e8029cf33e1e263147": "\\forall p:\\forall q:{\\mathcal {B}}(p\\to q)\\to ({\\mathcal {B}}p\\to {\\mathcal {B}}q)", "1b8ab6e8102bc2b91763316423e516f1": "{\\frac {\\delta L}{\\delta \\psi }}=0", "1b8b1e87d8a0a0190df5f3456d9c3d8d": "-\\sum _{i\\in I}p_{i}\\ln q_{i}\\geq -\\sum _{i\\in I}p_{i}\\ln p_{i}", "1b8b252ef3b59b90eda3acccba6157a2": "p={\\cfrac {\\mu J_{m}}{\\lambda (J_{m}-I_{1}+3)}}~.", "1b8b63fdd7dfe788966556a03e6ae788": "U=\\textstyle {\\frac {1}{2}}m\\omega ^{2}\\langle x^{2}\\rangle =e^{2}E^{2}/4m\\omega ^{2}", "1b8b807e27f114c5da16d417fa95c38a": "\\omega _{n}={\\frac {2n\\pi }{\\beta }}", "1b8bbc1b1ea773af28b9d12b017b8156": "{\\sqrt {x^{2}+c}}=-{\\frac {t^{2}-c}{2t}}+t={\\frac {t^{2}+c}{2t}}", "1b8bcebdef594b8fade4901e48532218": "{\\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}})\\exp \\left[{\\frac {-z_{i}q\\Psi ({\\vec {r}})}{k_{B}T}}\\right]", "1b8bf69b0ecb33dd72ce92a6d1350d51": "2^{p}", "1b8c0e669f5722f2adef6a9c76a0eaa7": "{\\tilde {f}}(d)=\\left\\{{\\frac {1}{2}}\\cdot \\left[1-N\\left(z<{\\frac {w\\cdot shared(d)}{\\sqrt {\\theta }}}\\right)+N\\left(z<{\\frac {-w\\cdot shared(d)}{\\sqrt {\\theta }}}\\right)\\right]-{\\frac {d}{n}}\\right\\}^{2}", "1b8c0fd3c14439e8dc4dddc9c194cb87": "A(x)=\\sum _{n=0}^{\\infty }A_{n}x^{n}", "1b8cd9c110785bf8c22c2e6d4e97d44b": "\\not \\sim ", "1b8cfb60d7230535d61d09197f9cf2ab": "x\\langle y\\rangle \\cdot P\\;\\vert \\;x(v)\\cdot Q\\longrightarrow P\\;\\vert \\;Q[^{y}\\!/\\!_{v}]", "1b8d01e89556aa40e3d4989cf8433d90": "\\scriptstyle {\\vec {x}}(t)", "1b8d1ca08cb3d0016a33712a5046de57": "{\\text{Posterior Probability}}(p=x|s,f)={\\frac {x^{s-1}(1-x)^{n-s-1}}{\\mathrm {B} (s,n-s)}},{\\text{ with mean = }}{\\frac {s}{n}},{\\text{ (and mode= }}{\\frac {s-1}{n-2}}{\\text{ if }}10,", "1bbeaa8ed797729d76f0ea23e7ffa3b2": "(\\mathbf {H} _{i})_{i\\in I\\cup \\{0\\}}", "1bbec9c72a30ba825f8fdf8712e1dd5a": "{\\begin{aligned}&{}\\quad \\mathrm {I} (aX_{u}+bX_{v},cX_{u}+dX_{v})\\\\&=ac\\langle X_{u},X_{u}\\rangle +(ad+bc)\\langle X_{u},X_{v}\\rangle +bd\\langle X_{v},X_{v}\\rangle \\\\&=Eac+F(ad+bc)+Gbd,\\end{aligned}}", "1bbf1dac71916ec739ce62419036abc3": "a={\\frac {g-2}{2}}=0.00116592080(54)(33)", "1bbfa93467d0b1215d9e91c16bfcca5d": "e^{i\\theta }=\\cos(\\theta )+i\\sin(\\theta )", "1bbff955e91db43588c36b697e816f0d": "{\\textrm {Vanna}}={\\frac {\\partial {\\mathcal {V}}}{\\partial S}}", "1bc0111e74b5890679ae2b76aecc1a89": "G_{i}E_{i}=E_{i}G_{i}=l^{-1}E_{i}{\\text{ and }}E_{i}G_{i\\pm 1}E_{i}=lE_{i}.", "1bc04a77a265ef88dcae82e76ac1175e": "\\rho _{n}(x_{1},\\ldots ,x_{n})=\\det(K(x_{i},x_{j})_{1\\leq i,j\\leq n})", "1bc07fbecdda3200bf8bf1facbeb6342": "q=(s,t_{s},t_{e})\\in Q_{A}", "1bc082d30fad094884f7944bf4f545fd": "F={\\frac {Q_{1}Q_{2}}{r^{2}}}", "1bc09488f4474b40c8d7034be69237ee": "(a_{1},b_{1},c_{1},d_{1})", "1bc095facd7aa3d5f146d1d4e451668c": "e^{j\\omega t}", "1bc0d5d71244b2fbad5828cd30d084f6": "E_{7}\\supset SU(8)", "1bc0de05c2e6e5a09bc5d8eb891a17c9": "15+10+10+10", "1bc119c840bd7ca7b72fcf641a8eda11": "{\\frac {\\sqrt {2}}{2}}", "1bc1d245ed07f1edd2d2cce819cf8ce8": "|\\psi (t)\\rangle =\\sum _{n}c_{n}^{A}(t)e^{-iE_{n}t/\\hbar }|\\phi _{n}\\rangle ", "1bc20f4d7e96716314c87512afc8e4e0": "\\Pi _{(x:A)}B", "1bc23524400091e2ab502d09e8ba0a9c": "s(-2)=s(-1)=0", "1bc247733faee428bfb487bdac42dbc0": "\\ y", "1bc259dc28d0fac41eb104afcb51e3a7": "u_{g}=-{g \\over f}{\\partial Z \\over \\partial y}", "1bc28fe541df9356b27da251e53977b9": "\\mathbf {A} (\\mathbf {r} ,t)=\\int {{\\delta \\left(t'-{{\\left|\\mathbf {r} -\\mathbf {r} '\\right|} \\over c}-t\\right)} \\over {\\left|\\mathbf {r} -\\mathbf {r} '\\right|}}{\\mathbf {J} (\\mathbf {r} ',t') \\over c}d^{3}r'dt'", "1bc2ae67c035b63e923c0d3aafee04e8": "{1 \\over 4\\pi }\\int _{\\theta =0}^{\\pi }\\int _{\\varphi =0}^{2\\pi }Y_{\\ell }^{m}\\,Y_{\\ell '}^{m'*}d\\Omega =\\delta _{\\ell \\ell '}\\,\\delta _{mm'}.", "1bc2d021afd8533fb390a0755879a3d2": "1/{\\sqrt {T}}", "1bc3a93d5290eae711b84e134a837a77": "{}_{2}Q_{3}=0", "1bc3d6195f9df90cf9487debc8e7ec97": "{\\ddot {\\mathbf {r} }}=a_{r}{\\hat {\\boldsymbol {r}}}+a_{\\theta }{\\hat {\\boldsymbol {\\theta }}}", "1bc3da55df6b9d3cfd405a27e0277b69": "\\delta [(h(S,X),S),(U,S)]\\leq \\varepsilon ", "1bc41df5b6ae524fb5b675215a381ea8": "\\ C_{rr}={\\sqrt {z/d}}", "1bc42f3abe50d807e52bb7172bd0285a": "({\\bar {I}}_{1}:=J^{-2/3}I_{1}~;~~{\\bar {I}}_{2}:=J^{-4/3}I_{2}~;~~J=1)~.\\,\\!", "1bc439f9155578498a3b415bbfc0caf4": "w_{m}(x)=min\\{w(y)|y\\in \\mathbb {Z} ,y\\equiv x{\\pmod {m}}\\}", "1bc43d4e951cefc8ebe56cc32154ba4a": "\\Rightarrow A(ab\\bullet {}b\\bullet {}abbabb,abbab\\bullet {}b\\bullet {}abb,abbabbab\\bullet {}b\\bullet {})\\Rightarrow A(\\epsilon ,\\epsilon ,\\epsilon )\\Rightarrow \\epsilon ", "1bc497a86e587fbe9483f246053ec777": "{\\text{Re}}[{}_{1}F_{1}(\\alpha ;\\alpha +\\beta ;it)]={\\text{Re}}[{}_{1}F_{1}(\\alpha ;\\alpha +\\beta ;-it)]", "1bc4ca19c5596cae065d65fdce61ae5b": "{\\frac {dM}{dt}}=-{\\frac {M-M_{e}}{\\tau }}", "1bc50df3342e17d7d9600b97dccfa40e": "\\delta {\\vec {A}}_{||}=0", "1bc5866b2cb82e351a0f9af0d513b2cc": "f(z_{0})=0", "1bc59abd07973e988bba58467b3ed671": "f(q)\\propto q^{a}(1-q)^{b}", "1bc5ecc2b5ccbe4899e44f4baa692179": "SCR=\\left({\\frac {l_{\\mathrm {sum~of~solid~core~pieces} }}{l_{\\mathrm {tot~core~run} }}}\\right)\\times 100", "1bc602e024fb5d8cce515046c5301091": "y\\in colgroups", "1bc681fc533ecae560d765f7cb5ee721": "k={\\begin{matrix}{\\frac {\\omega }{c}}\\end{matrix}}n_{1}", "1bc6948dac8209fc147b46fe7801080b": "|p'\\rangle ", "1bc6cfd8076d0b863b69179d88f7e1eb": "A\\otimes _{B}A", "1bc6dafd86926517273eb37143b2aa2b": "H(q,u,p,t)=\\langle p,u\\rangle -L(q,u,t)", "1bc6eac9bb4f76beb126b2b0f33504f9": "{\\frac {x+z-y}{2y}}+{\\frac {y+z-x}{2x}}+{\\frac {x+y-z}{2z}}\\geq {\\frac {3}{2}};", "1bc6f95da3fe9457cdfca20647403589": "{\\mathfrak {so}}_{16}", "1bc731481c2b613c2a616e48c66ce518": "f_{1}={\\frac {1+\\sin(k_{1}x)}{2}}", "1bc772367f585f98792bba00c2413ec0": "{\\frac {x}{\\ln x}}<\\pi (x)<1.25506{\\frac {x}{\\ln x}}\\!", "1bc7a8eecd75843b5adf929abef0c88a": "t\\mapsto f(t)", "1bc80bba59e7a4e7432d77ada13f7fa2": "\\sum _{n=1}^{\\infty }a_{n}(e^{-s})^{n},", "1bc83122008ede41abea59f585259090": "M_{yy}=f_{1}(x)={\\frac {4M_{0}}{\\pi }}\\sum _{m=1}^{\\infty }{\\frac {1}{2m-1}}\\,\\sin {\\frac {(2m-1)\\pi x}{a}}\\,.", "1bc8880c8534e60bab13d6fa7c544573": "(\\Omega ,{\\mathcal {F}},\\mathbb {P} ):=\\left(C_{0}(\\mathbb {R} ;\\mathbb {R} ^{d}),{\\mathcal {B}}(C_{0}(\\mathbb {R} ;\\mathbb {R} ^{d})),\\gamma \\right).", "1bc8a62ae7924077fffe9c4191c83a7c": "[T,b_{n}]=-nb_{n-1}.", "1bc8ab8c837a94663c834603776a07da": "{\\frac {\\lambda +\\alpha \\beta (1+3\\beta +2\\beta ^{2})}{\\left(\\lambda +\\alpha \\beta (1+\\beta )\\right)^{\\frac {3}{2}}}}", "1bc8f608669061203b8d5768133e5df2": "[AFO]=[FCO],[AFO]={\\frac {1}{2}}ACO={\\frac {1}{2}}[ABO]=[ADO]", "1bc94c6d7c3b09999d3eb4081f357914": "(n_{x},n_{y},n_{z})=(3,2,7)", "1bc9ad24e86219fb6be0016d070af7e5": "p(O_{fg}|I,I_{t})", "1bca5cdf5481f06a29a91a4ac16e7aee": "R_{ab}\\,=R^{c}{}_{abc}", "1bcad372191a195aeadb257a348f6d29": "C\\approx \\ l(b(C_{000},C_{010},C_{100},C_{110}),b(C_{001},C_{011},C_{101},C_{111}))", "1bcaf55db1313514b84b90dc2aad93ce": "{\\begin{aligned}E&={\\frac {-Aa_{1}+a_{2}}{B}}\\\\F&=-i{\\frac {A^{2}a_{1}-Aa_{2}+2a_{1}B}{B{\\sqrt {A^{2}+4B}}}}\\\\\\theta &=a\\cos \\left({\\frac {A}{2{\\sqrt {-B}}}}\\right)\\end{aligned}}", "1bcb0986cca78a9f4d13bb5c6405898d": "{\\mathsf {0123456789}}\\!", "1bcb13e174b66706c64aff4605ee7f29": "\\Gamma (x,\\alpha +1,1)", "1bcb5fc443a4ef1238c836ec8fb16bec": "{\\hat {1}}", "1bcb6e4e498db7dee7bd4003415f1a1f": "\\ p=\\rho \\cdot g\\cdot z", "1bcb7bf7c91933bcaf4cb1e2aa72988a": "{\\begin{aligned}\\operatorname {sc} (u)&={\\frac {\\operatorname {sn} (u)}{\\operatorname {cn} (u)}}\\\\[8pt]\\operatorname {sd} (u)&={\\frac {\\operatorname {sn} (u)}{\\operatorname {dn} (u)}}\\\\[8pt]\\operatorname {dc} (u)&={\\frac {\\operatorname {dn} (u)}{\\operatorname {cn} (u)}}\\\\[8pt]\\operatorname {ds} (u)&={\\frac {\\operatorname {dn} (u)}{\\operatorname {sn} (u)}}\\\\[8pt]\\operatorname {cs} (u)&={\\frac {\\operatorname {cn} (u)}{\\operatorname {sn} (u)}}\\\\[8pt]\\operatorname {cd} (u)&={\\frac {\\operatorname {cn} (u)}{\\operatorname {dn} (u)}}\\end{aligned}}", "1bcc51e0181cb040d09cf67d0dae83b1": "\\alpha _{0}=\\sum _{i=1}^{K}\\alpha _{i}.", "1bcc614b6a9911ffd3b21dd15c38bd2b": "\\phi (t_{1},\\dots ,t_{n})", "1bcc6bb41717e20699eefacf42a13b94": "p=mv", "1bcc883a5cfe902e818db11ddcf383aa": "{\\mathsf {(CH_{2}CH_{2})O\\ \\xrightarrow {200\\ ^{o}C,\\ Al_{2}O_{3}} \\ CH_{3}CHO}}", "1bcc97a4960578435bfb82efeb18074a": "b,\\beta ,s", "1bccc32572670e48d124adf891f7f656": "\\varepsilon \\;=\\;{\\frac {2r-1}{(r-1)^{2}}}=\\;{\\frac {a^{2}-b^{2}}{b^{2}}}\\;", "1bccd5177620e7bea397b13173c521ce": "{\\frac {1}{2}}=\\nu _{2}(a)-\\nu _{2}(b),", "1bcceb1739ade831ff36ab935b1f4a89": "{\\frac {d{M}}{dt}}=\\gamma {M}\\times {B}-{\\frac {M_{x}{\\vec {i}}+M_{y}{\\vec {j}}}{T_{2}}}-{\\frac {(M_{z}-M_{0}){\\vec {k}}}{T_{1}}}", "1bcd1d24c31c92fb81001c13784e8450": "k=n-m", "1bcd98c227e8fad247c90cc53757f4fc": "(h,f(x),b\\oplus h(x))", "1bcda3ba30d77e5af56e20a09c22c46e": "\\mathbf {B} (t)=(1-t)\\mathbf {B} _{\\mathbf {P} _{0},\\mathbf {P} _{1},\\mathbf {P} _{2}}(t)+t\\mathbf {B} _{\\mathbf {P} _{1},\\mathbf {P} _{2},\\mathbf {P} _{3}}(t){\\mbox{ , }}t\\in [0,1].", "1bcda555b639307502313373bfa46365": "f_{xx}(a,b)", "1bcdc50c69e3ae5a3df98f77af261ef9": "k\\colon X\\to Z", "1bcdcdfe7fe7c15733da2f9063433d5a": "\\pi _{*}(\\alpha )=\\pi _{*}(g\\,dt\\wedge dx_{j_{1}}\\wedge \\dots \\wedge dx_{j_{k-1}})=\\left(\\int _{0}^{1}g(\\cdot ,t)\\,dt\\right)\\,{dx_{j_{1}}\\wedge \\dots \\wedge dx_{j_{k-1}}}.", "1bcdfa302fbbfccf2f080a31c84d402f": "\\partial u^{j}/\\partial x^{i}(p)", "1bce001a90b4d5fc4782ba842b63508c": "R={\\frac {0.61\\lambda }{\\mathrm {NA} }}\\approx {\\frac {\\lambda }{2\\mathrm {NA} }}", "1bce2786340f97240df47054c15abc60": "P(\\partial _{t}):=a_{m}\\partial _{t}^{m}+\\cdots +a_{1}\\partial _{t}+a_{0},\\;a_{m}\\neq 0.", "1bcea7efe500a2118662f1ce388b3179": "\\sin \\theta _{m}>1", "1bcec32e763b100d3fc6081331122dac": "Q=I_{\\mathrm {RMS} }^{2}X={\\frac {V_{\\mathrm {RMS} }^{2}}{X}}", "1bcee5ef91e3358fb98a5e59d1495308": "I_{\\mathcal {Q}}(0)\\in Q", "1bcf079fcee1fc6eea3d5b191e256ac6": "\\mathbf {X} =[\\mathbf {X_{1}} \\mathbf {X_{2}} ]", "1bcf26dde8ccec121c574ffa808bf666": "i=i_{L}+i_{C}\\,", "1bcf5a5015afc1e7230f1352b941bbfe": "y(t)=A_{c}\\cos \\left(2\\pi \\int _{0}^{t}f(\\tau )d\\tau \\right)", "1bcf9619feedd85ccda146090dae2b41": "\\tau ={\\sqrt {-3502}}", "1bcfa8d639c0827f66bc7348405d7069": "\\liminf _{n\\to \\infty }{\\frac {p_{n+1}-p_{n}}{\\log p_{n}}}=0", "1bcfda5bf0cc77ec49adf2ec96177a1f": "v={\\frac {ds}{dt}}", "1bcfead2eb189a758dfcb71e8c26e9cb": "{\\frac {y\\cdot r}{x}}\\cdot {\\frac {y^{2}}{x^{2}}},\\quad {\\frac {y\\cdot r}{x}}\\cdot {\\frac {y^{2}}{x^{2}}}\\cdot {\\frac {y^{2}}{x^{2}}},\\quad \\cdots ", "1bcfece10d705dc1a01a12d1eaacb181": "{\\text{Area of triangle (on the unit sphere)}}\\equiv E=E_{3}=A+B+C-\\pi ,", "1bd010da361d94202bde8e5fc899d075": "b^{-1}", "1bd01cd6ede07179a46ee938af84c0a4": "c_{\\nu _{j}}", "1bd01e5b7dcb94c8635f13ec4afe6e63": "\\mathbf {a} ={\\frac {\\mathrm {d} \\mathbf {u} }{\\mathrm {d} t}}\\,\\!", "1bd04dc50382dd83e46aaa28e08c415e": "(0\\mid c_{2})\\in C_{1}\\mid C_{2}", "1bd086ed753e265ebf843dbf7a3109a8": "{\\hat {q}}\\psi (q)=q\\psi (q)", "1bd0d928c27718f7d4d1d2783642bd51": "H=H_{1}\\otimes \\cdots \\otimes H_{n}", "1bd0e2fb802e234db043dc71ab762f7d": "\\beta _{c}={\\frac {1+{\\sqrt {1-4c}}}{2}}", "1bd0e726e5223868d9e94828d998abbd": "{\\frac {\\partial f}{\\partial {\\boldsymbol {S}}}}:{\\boldsymbol {T}}=\\left({\\frac {\\partial f_{1}}{\\partial {\\boldsymbol {S}}}}+{\\frac {\\partial f_{2}}{\\partial {\\boldsymbol {S}}}}\\right):{\\boldsymbol {T}}", "1bd0eaa80ccce6a7a5a74cbf4b357a99": "\\log {\\left({\\frac {k}{k_{0}}}\\right)}=mY\\,", "1bd0f17521e98fdd86d15f23ed355e9a": "D(\\theta ,\\phi )", "1bd0f2ee48485d88ef3073a09c2f441c": "\\ln {(1+x^{\\alpha })}\\leq \\alpha x\\quad {\\rm {for}}\\quad x\\geq 0,\\alpha \\geq 1\\,", "1bd1038286a54f23d9b9b31b939d8b4f": "(2h)^{-1}\\operatorname {sech} (\\pi x/2h)", "1bd1171c2d31c99af30803780fc1ad9a": "{\\text{Minimize}}={\\begin{cases}f_{1}\\left(x,y\\right)&=x^{2}-y\\\\f_{2}\\left(x,y\\right)&=-0.5x-y-1\\\\\\end{cases}}", "1bd13e85d6fef0d794a6b76627e093d5": "C_{OX}", "1bd14d5d9f4779f2b343d3dfbe5021ad": "\\lambda _{u}/\\gamma ", "1bd1533b3f14e293084ada0f9342d52d": "{\\mathit {Q}}", "1bd17ae62259f77c4c2993e0fef35e40": "d=36", "1bd1b20f5d71d9ad04975eba148ac0c9": "dr=\\left(\\pm 1-{\\sqrt {r_{g} \\over r}}\\right)d\\tau ,", "1bd1d7859f18b53a71912db48664941e": "G_{1}=1-\\sum _{k=1}^{n}(X_{k}-X_{k-1})(Y_{k}+Y_{k-1})", "1bd21153a5cd43e02348425800884962": "x^{2}=(2{\\sqrt {a^{2}-b}})x^{0}+2a", "1bd27edb6aa60ae131675f69bb4603b2": "r_{1}(q_{2})", "1bd2eacbfeff61915b919add5b9cc378": "{\\begin{aligned}P&={\\frac {V^{2}}{X-Xc}}\\sin(\\delta )\\\\Q&={\\frac {V^{2}}{X-Xc}}(1-\\cos \\delta )\\end{aligned}}", "1bd31d6528f61f805d65e2a1b09ed43f": "K(X_{i},X_{j})=k_{i}k_{j}.", "1bd3381018944fbae7c135405b4b1e4c": "yp_{y}+p-c_{y}=0", "1bd33b4d38275d250db687544f0dcd1a": "\\mathbf {H} =-\\nabla U.", "1bd37145ba1d6e0dacf97875354d9a72": "y=2A\\cos \\left({\\frac {\\phi }{2}}\\right)\\sin \\left(kx-\\omega t+{\\frac {\\phi }{2}}\\right)\\,\\!", "1bd3a0484883ca6deaada8395a8f6e85": "ER", "1bd3a0a036394af6f652a839a394e50d": "\\quad \\min \\limits _{D,X}\\sum _{i}\\|x_{i}\\|_{0}\\qquad {\\text{subject to }}\\quad \\forall i\\;,\\|Y-DX\\|_{F}^{2}\\leq \\epsilon .", "1bd3a67216de1df16947669396c731b8": "\\mathbf {u} \\cdot \\mathbf {v} =u^{i}v_{i}=u_{i}v^{i}=g_{ij}u^{i}v^{j}=g^{ij}u_{i}v_{j}", "1bd3bc12f14e732ede664b4533bd16f7": "\\alpha (t):=v(t+1)", "1bd3dd08b9e17023952d66db5ed54828": "k_{AXU}AXUHash\\,", "1bd44540e3583032f8a8da76b9596f5d": "\\mu _{4}=(k)(k+2)\\,", "1bd4917950a5f106cd4d60ae2e453bf5": "E_{ij}^{c}=x_{i}p_{j}", "1bd4a7dd52e10f575b981566b4a9dc42": "T\\left(n\\right)=\\Theta \\left(f(n)\\right)", "1bd4e0a45f9e6595c5c9892a70749e74": "i(v)=enS_{z}{\\frac {1}{4\\pi }}\\int \\limits _{\\sqrt {2eV/m}}^{\\infty }f(v)dv\\int \\limits _{0}^{\\zeta }v\\cos \\vartheta 2\\pi \\sin \\vartheta d\\vartheta ", "1bd50aca6a19eafdf6b066a5482b5ffc": "{\\vec {v}}\\times {\\vec {w}}", "1bd53667c6094ad36b2fefaff9a3a198": "\\scriptstyle f/f_{s}", "1bd5b24ca53af2660ac41e52d8d89c30": "\\log _{b}\\colon H\\rightarrow \\mathbf {Z} _{n},", "1bd5c321bd2c9f7d11ca953696c871ad": "{\\widehat {\\mathbb {C} }}", "1bd5fc8bf62f545461553d4db1260ae8": "{\\Big (}c\\,\\log n/{\\sqrt {n}}{\\Big )}\\,2^{n}", "1bd6025bc4c7c1e1d455879c82787003": "{\\mathbf {P} }^{'}=[{\\mathbf {A} }\\;|\\;{\\mathbf {a} }_{4}]", "1bd666e2402482fc78b41259f2379f3f": "x_{i}=x_{j}={\\frac {1}{n}},\\forall i,j", "1bd6b82b3156fb27a41e45aaceb5fb62": "\\sum _{p}1/(p^{s}\\log p)", "1bd72a5be0b1aa8bdbe43e863ad2d174": "{\\boldsymbol {a}}_{C}=-2{\\boldsymbol {\\Omega \\times v}}=2\\,\\omega \\,{\\begin{pmatrix}-v_{u}\\\\0\\\\v_{e}\\end{pmatrix}}\\ .", "1bd732526c90c0bb02936ffafe3e1245": "\\oint {\\frac {\\delta Q}{T}}\\leq 0,", "1bd7360eabfb28d378d3887e08f8f140": "\\omega _{H}=2\\pi f_{H}={\\frac {B_{0}|e|}{m}}", "1bd73b6a902126b9426a518b15fe164e": "M_{2}=P_{1}'|P_{2}|\\dots ", "1bd75746e65a23e91c62fca4a042fead": "L*(x,\\,y)=x", "1bd77db2d6340fccfcfcdf471b48d92c": "PCAB", "1bd81eefdfdb041fac841ae5aa460083": "\\sigma ^{2}(q_{k})", "1bd826fdb9af2ee7b85c6728d14ed907": "\\xi _{1}<{\\sqrt {2}}\\lambda <\\xi _{2}", "1bd82d9fef5d54a32abe5c6d00d9222c": "{\\tilde {D}}{\\tilde {D}}^{T}/\\left(N-1\\right)", "1bd8a5072c62d68eb86590ac1b4f302e": "h\\cdot (ab)=(h_{(1)}\\cdot a)(h_{(2)}\\cdot b)", "1bd8ef232b476a728243e7e57a22160c": "cov\\left[\\sum _{jN+1}^{(j+1)N}u_{i},\\sum _{k=jN+1}^{(j+1)N}X_{k}\\right]\\neq 0.", "1bd938626cd53edc67aba10ed8d58f87": "k^{p}", "1bd9e32fbaad8de1230a622253b21d71": "\\nabla \\cdot \\mathbf {E} =4\\pi \\rho ", "1bda13b8625684afc4a1e834834579ef": "p_{2}={\\frac {q_{1}-q_{2}}{2q_{4}}}\\ ,", "1bdab83c0032f4b872140bec5652375e": "{\\mathcal {S}}={\\mathcal {A}}\\boxtimes _{n=1}^{N}\\mathbf {U} _{n}", "1bdab9b05e05baef1023dde257789634": "I+A\\,d\\theta ~,", "1bdb2988d0a26f6a36c94786a3180521": "x(t)=e^{A(t_{1}-t_{0})}x_{0}+\\int _{t_{0}}^{t_{1}}e^{A(t_{1}-\\tau )}Bu(\\tau )d\\tau ", "1bdb8d319354b7cfbf552d2f9fc8aa9f": "dU=T\\,dS-PA\\,dx+mg\\,dx", "1bdbc508611dee197a73f9d6c8f6366b": "f({\\textbf {x}}_{1})\\leq f({\\textbf {x}}_{r})0", "1bdf7884034eda04ad2367a8f0cf9dae": "={\\frac {\\operatorname {cov} (\\beta ,\\theta )}{\\sqrt {(\\operatorname {var} [\\beta ]\\operatorname {var} [\\theta ])}}}.{\\frac {\\sqrt {\\operatorname {var} [\\beta ]\\operatorname {var} [\\theta ]}}{\\sqrt {(\\operatorname {var} [\\beta ]+\\operatorname {var} [\\epsilon _{\\beta }])(\\operatorname {var} [\\theta ]+\\operatorname {var} [\\epsilon _{\\theta }])}}}", "1bdf88aa90697f1e367a6834cffadf41": "Q_{j}\\in \\mathbb {R} ^{1}", "1bdfb537c38c6ca2008cbfe5df779c5c": "{n \\choose (n+k)/2}", "1bdfe670c1db0224349c9d3cee020cfc": "-{\\text{E}}^{2}(\\tau |b,s,\\beta ),\\quad \\beta \\neq 1", "1be0108445274b64e746144b367919e8": "L=-\\partial _{x}^{2}+u\\,", "1be03680765d1758574e1a99d121e1bf": "X_{j}", "1be0b4087d78d6c2bffe2f70b8460fb5": "{r_{m}}", "1be0c936ac35a1237f3d91a5bab13e94": "\\qquad \\qquad b_{\\kappa ,\\alpha }^{\\dagger }={\\frac {1}{N^{1/2}}}\\sum _{\\kappa _{p},\\alpha }e^{i({\\boldsymbol {\\kappa }}_{p}\\cdot \\mathbf {x} )}\\mathbf {s} _{\\alpha }({\\boldsymbol {\\kappa }}_{p})\\cdot [({\\frac {m\\omega _{p,\\alpha }}{2\\hbar }})^{1/2}\\mathbf {d} (\\mathbf {x} )-i({\\frac {1}{2\\hbar m\\omega _{p,\\alpha }}})^{1/2}\\mathbf {p} (\\mathbf {x} )].", "1be10656885c257a4c98af160b7e8170": "2^{-{\\frac {1}{\\alpha }}}", "1be112a328f3a4276fcd11289e021d22": "\\left({\\frac {\\partial A}{\\partial y}}\\right)_{x,z}\\!\\!\\!=\\left({\\frac {\\partial B}{\\partial x}}\\right)_{y,z}", "1be11ba7e5433c85d4f8e1a13572fe8f": "\\lim _{n\\rightarrow \\infty }p_{jj}^{(n)}={\\frac {C}{M_{j}}}", "1be144c34fd0853d08077d686b5e2088": "M=n+1", "1be155fc754d0ae7841b97a5bbbebd86": "\\int _{0}^{\\delta }t^{\\lambda +n}e^{-xt}\\,dt={\\frac {\\Gamma (\\lambda +n+1)}{x^{\\lambda +n+1}}}+O\\left(e^{-\\delta x}\\right)", "1be16d00e39d89d873b58a7b77e1d808": "(i,j)\\leq (i',j')\\iff i\\leq i'\\qquad {\\textrm {and}}\\qquad j\\leq j'", "1be182f7fa88172265721fcf91b8581b": "{\\tfrac {3}{2}}\\scriptstyle {\\sqrt {2}}", "1be184b6963cbaf5745122e24b7d65c7": "1.\\mu _{8,4}(p_{4})=\\alpha _{8}(p_{4})", "1be1ae1b8153d616f18e7cc7843e1b9f": "c_{d_{0}}\\;", "1be1b640abdda07d8f3090291f0ca40f": "s,t\\in V(G)", "1be1dfd33d881912b75eb5a462ec2c89": "\\textstyle (x\\pm 1,y\\mp 1)", "1be21a6bc22b3ad4ec5eb48be6c8fc23": "l_{A}", "1be30633e9258f7e4604829638482361": "z_{1}(x,y)={x^{2}-y^{2} \\over 2}", "1be336e794138c4c30f91c0dd4fe8100": "={T \\over 2}\\int \\mathrm {d} ^{2}\\sigma {\\sqrt {-h}}\\left(\\Lambda \\Lambda ^{-1}\\right)h^{ab}g_{\\mu \\nu }(X)\\partial _{a}X^{\\mu }(\\sigma )\\partial _{b}X^{\\nu }(\\sigma )={\\mathcal {S}}", "1be36aeb33833bf58795e6b3f642dbe5": "\\,q", "1be38d91a56c5b60de469bfa43491807": "\\{C_{i}\\}_{i\\in I}", "1be39fccae0fc210477ad8978cd96012": "(x^{3}+Ax+B)((x^{3}+Ax+B)^{\\frac {q^{2}-1}{2}}-\\theta (x))", "1be3dfd47cb6e9cc23030e7854f25ef6": "1/p^{2}", "1be3e9656794404dce8d35f83552d1ac": "t^{a}(d,n)", "1be3fe099395a4ab475678f1602c795c": "\\mathbf {a} =a_{1}{\\vec {r}}_{u}+a_{2}{\\vec {r}}_{v}", "1be3fed02676c5e01c601bf209a0af3c": "\\mathbf {X} ^{1/2}", "1be40b9e096dc090f91585f6114d8dca": "y_{i}=\\sum _{k}M_{ik}\\otimes x_{k}\\in EndPol\\otimes Pol", "1be44330461d19cbe0f5003716b2f5c8": "u\\in V", "1be46696105293817ab91de9ab176038": "{\\mathcal {O}}(n+1)", "1be4ca7b3d5cd777eee7a50db7f74011": "f_{i}(\\theta )", "1be50391e28ecd508f471224325bb583": "f_{t}\\left(x,t\\right)", "1be558f7c415d9e857e4b27511cbe55a": "{\\text{ORL}}(\\mathrm {dB} )=10\\log _{10}{P_{\\mathrm {i} } \\over P_{\\mathrm {r} }}", "1be56a018a8e1ffcea6e3b2f143b5f79": "\\ (1-\\eta ^{2}){\\frac {d^{2}Y_{mn}(c,\\eta )}{d\\eta ^{2}}}-2(m+1)\\eta {\\frac {dY_{mn}(c,\\eta )}{d\\eta }}+\\left(c^{2}\\eta ^{2}+m(m+1)-\\lambda _{mn}(c)\\right){Y_{mn}(c,\\eta )}=0,", "1be59e92cc2818029c7fed4a024b46a1": "\\sin ^{2}(18^{\\circ })+\\sin ^{2}(30^{\\circ })=\\sin ^{2}(36^{\\circ }).\\,", "1be5aa8d4cea80201a31a16b8a9666c7": "\\operatorname {pd} _{R}M=1", "1be5b382180319353ccbbda39c4bd7c3": "f(z+\\omega )=f(z)\\ ", "1be614c4e1b209ddfef534ec9247b48c": "\\mathbb {C} \\circledast X\\cong X\\cong X\\circledast \\mathbb {C} ,", "1be64bd809d160c2874421c1083792a7": "\\scriptstyle A\\oplus B", "1be669931bdb641d83b5b4fc38c7bb3c": "\\sum _{i=1}^{k+1}\\delta _{i}\\!\\int _{z_{i-1}}^{z_{i}}g_{j}(z)\\,dz=0{\\text{ for }}1\\leq j\\leq k.", "1be676dbcb38c53f32cbc0b9f5553026": "F_{\\ell }", "1be69f76d50b0f2c4edcfcba8ac43fc4": "\\mathbf {A} =\\|\\mathbf {A} \\|\\left(\\cos \\alpha \\ {\\hat {\\mathbf {i} }}+\\cos \\beta \\ {\\hat {\\mathbf {j} }}+\\cos \\gamma \\ {\\hat {\\mathbf {k} }}\\right)\\ ,", "1be6f70858bb4e92745f4c48eae7bbcc": "\\Psi =\\sum _{i=1}^{N}c_{i}\\Psi _{i}.", "1be6fb612b389126b539545989ef67b7": "x=-a", "1be6fc2ce2a6d196c4e363adb61ca5be": "y\\rightarrow y/r", "1be72ec0830c1e6af36794185ef386f6": "2+1\\twoheadrightarrow 3", "1be7511f2ef4e5b483710b80fd3e2b71": "\\Gamma _{12}(l,m,0)=\\iint U(l,m,P_{1})U^{*}(l,m,P_{2})\\,dS", "1be7594e4a1f09b7de7cbfb6605c20bb": "{\\begin{aligned}\\Theta _{i}&=\\sphericalangle (\\mathbf {p} _{i},\\mathbf {k} ),\\\\\\Theta _{f}&=\\sphericalangle (\\mathbf {p} _{f},\\mathbf {k} ),\\\\\\Phi &={\\text{Angle between the planes }}(\\mathbf {p} _{i},\\mathbf {k} ){\\text{ and }}(\\mathbf {p} _{f},\\mathbf {k} ),\\end{aligned}}", "1be7ad08853563968d35e7b9c5466afd": "n!\\;\\;-\\;\\;{n \\choose 1}(n-1)!\\;\\;+\\;\\;{n \\choose 2}(n-2)!\\;\\;-\\;\\;{n \\choose 3}(n-3)!\\;\\;+\\;\\;\\cdots \\;\\;\\pm \\;\\;{n \\choose n}(n-n)!", "1be7fa233dcda7831ce592bede114223": "P_{r}=\\left[P_{t}{{G^{2}\\lambda ^{2}} \\over {{(4\\pi )}^{3}R^{4}}}\\right]\\left[{\\frac {c\\tau }{2}}\\right]\\left[{\\frac {\\pi R^{2}\\theta ^{2}}{4}}\\right]\\eta =\\left[P_{t}\\tau G^{2}\\lambda ^{2}\\theta ^{2}\\right]\\left[{\\frac {c}{512(\\pi ^{2})}}\\right]{\\frac {\\eta }{R^{2}}}", "1be8629b9f429ff89175d33603d85241": "\\int _{0}^{1}\\ln ^{2n}\\left({\\frac {x}{1-x}}\\right)\\,dx=(-1)^{n+1}(2^{2n}-2)\\beta _{2n}\\pi ^{2n}", "1be8d255bf6b3efa7625c373f92aa19e": "N_{0}", "1be8f15b1c555b7c1ae119f5b616658b": "\\delta /2\\pi =n\\,\\!", "1be921bc3bdeeda4c4e7e2703a027e8e": "PSL(2,\\mathbb {R} ).", "1be9d935c62cd1e7b99778baf951744c": "\\operatorname {Gal} (\\mathbf {F} _{q^{f}}/\\mathbf {F} _{q})", "1be9e3fd25a7da54adcd22f67fa89429": "x_{n}=-\\cos \\left({\\frac {\\pi }{N}}(n+{\\frac {1}{2}})\\right)", "1be9f9f8e460d33db1f1089254e78a39": "e^{ix}=\\cos x+i\\sin x", "1bea86b9e0729f0e7efe21dde34ffba3": "V\\left(x\\right)", "1bea8ba4ceaf923082ab0d43fbe96908": "S[t]", "1bea95bc1fb13d3d185e67411b368d41": "\\operatorname {sgn} \\left(nx-k\\right)={\\begin{cases}-1&nxk.\\end{cases}}", "1beac5384ab16f9f67515dd08b57b177": "(x)O({\\underline {x}}:G)", "1beada934b50b5a8190c03add75c909b": "N^{*}(x)", "1beadd88d70982f74ea530acda32c043": "p_{n-1}/q_{n-1}", "1beb163b90b89f0ca8e4b0ed3d6ac705": "\\sum _{q\\leq Q}\\max _{y0.", "1bf06980cb4eb314ea1b4bcf3354fc2c": "{\\frac {\\langle E,s\\rangle \\Rightarrow V}{\\langle L:=E\\,,\\,s\\rangle \\longrightarrow (s\\uplus (L\\mapsto V))}}", "1bf0b779c9001024285ab7454a249f59": "2^{n}+1", "1bf12be07abaf625d933facb962b2eb6": "G_{N}=1=c", "1bf15896187ff0f9d7d66b07432c8f1a": "\\left[B\\right]=\\left\\{{\\begin{matrix}\\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}\\;\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{\\text{otherwise}}\\\\\\end{matrix}}\\right.", "1bf198096556483c6f2b1079f9206dac": "{\\frac {1}{q}}+{\\frac {1}{p}}=1,", "1bf1c13cf84920da719f1860fd9d6d3f": "x\\oplus x\\oplus x=x\\oplus x", "1bf21b2e6f60c278eae7d5e03e39de43": "\\Gamma (N)=\\left\\{{\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}}\\in SL_{2}(\\mathbf {Z} ):c\\equiv b\\equiv 0,a\\equiv d\\equiv 1{\\pmod {N}}\\right\\}.", "1bf28f20e2f530cc5813719c6daa75cb": "(m/q)P_{p}=uq(m/q)P_{p}=umP_{p}=0\\,", "1bf292a3e14620ef9316194dab31d354": "{\\frac {x^{3}-12x^{2}-42}{x-3}}=x^{2}-9x-27-{\\frac {123}{x-3}}", "1bf2d677e8320c91e78baa12d1a52721": "\\int \\sec {ax}\\,\\mathrm {d} x={\\frac {1}{a}}\\ln {\\left|\\sec {ax}+\\tan {ax}\\right|}+C", "1bf3305525def18f3230ab86d8a9e485": "\\Phi _{n}(x)=\\prod (x-\\zeta )", "1bf383803d7a344c74729c9494a374d8": "\\ln \\ I(R)=\\ln \\ I_{0}-kR^{1/n},", "1bf38bc5d5af0aa152208f67f2ffef00": "\\gamma ={\\sqrt {ZY}}", "1bf3b44da47ce3985f33a759a52e8a02": "y_{n}", "1bf3ca070beb57c0d5d9c85a3ff0b88f": "A\\in M(m,n;\\mathbb {K} )", "1bf417e737d12f0763d9cfea4aec48a8": "B=Y", "1bf41f16dfd33ceec7d09e91e2f90de8": "\\lambda _{i}\\equiv l_{i}/L_{i}", "1bf45066aff20080dc142253ad6a5f89": "{\\frac {\\partial n(x_{i},t)}{\\partial t}}={\\frac {1}{2}}\\sum _{j=1}^{i-1}K(x_{i}-x_{j},x_{j})n(x_{i}-x_{j},t)n(x_{j},t)-\\sum _{j=1}^{\\infty }K(x_{i},x_{j})n(x_{i},t)n(x_{j},t).", "1bf49a39c7156b5b62ded5cc5e7674dd": "\\displaystyle x(s)=x_{0}+\\sum _{t}{s^{|t|} \\over |t|!}\\alpha (t)\\delta _{t}(0).", "1bf49f5d69bf7910161aefc94f7d2f0b": "i\\hbar {\\frac {d}{dt}}\\rho _{I}(t)=\\left[H_{1,I}(t),\\rho _{I}(t)\\right].", "1bf4a25fbcef1e257fa670665497dbcf": "{f_{2} \\over f_{1}}={p_{2} \\over p_{1}}", "1bf4acdcd001c0df98fe6f63f1f2d5fd": "P_{0}=(x_{0},y_{0},z_{0})", "1bf4d10c9c7255077d5b7209bb8c3c1a": "Y\\sim HN(\\sigma )", "1bf4df1aff97f5dc6a294225bb309604": "\\mathbf {B} _{\\|}", "1bf4f9175b2e7b4284a8ccf17ab6406c": "\\mathbb {E} [X\\,|\\,{\\mathcal {G}}]+c\\leq \\mathbb {E} {\\Bigl [}\\liminf _{n\\to \\infty }(X_{n}+c)^{+}\\,{\\Big |}\\,{\\mathcal {G}}{\\Bigr ]}\\leq \\liminf _{n\\to \\infty }\\mathbb {E} [(X_{n}+c)^{+}\\,|\\,{\\mathcal {G}}]", "1bf51fd0e2af1f4e78ee3124cabccb37": "\\mathbb {C} c.", "1bf53e0cb19c70923b9d853769fa07ac": "\\scriptstyle \\mathbf {R} _{R}", "1bf55568f54820efd0d8d3d3639c5cda": "P_{\\mbox{rej}}", "1bf574a0bf5334ddadcacec3c594b642": "clearance\\ ratio\\ of\\ X={\\frac {C_{x}}{C_{in}}}", "1bf5762089537f2be13b54cecf8643a7": "{\\vec {v}}_{n}={\\tfrac {{\\vec {x}}_{n+1}-{\\vec {x}}_{n-1}}{2\\Delta t}}", "1bf58115c16d44e451e2b5f745c4cdeb": "{\\dot {\\rho }}=-{i \\over \\hbar }[H,\\rho ]+\\sum _{i=1}^{N^{2}-1}\\gamma _{i}{\\big (}A_{i}\\rho A_{i}^{\\dagger }-{\\frac {1}{2}}\\rho A_{i}^{\\dagger }A_{i}-{\\frac {1}{2}}A_{i}^{\\dagger }A_{i}\\rho {\\big )}.", "1bf59e2ed0faf854d0b53249221ba72c": "Q=A.f(K,L)", "1bf5eaea06b6f27fc3f1f5fac4bedeae": "(sx_{0},sy_{0};s^{2}t_{0})", "1bf6bd3b2c78a00259dec0f21a719486": "kM_{p}\\omega ^{2^{p-2}}=0.", "1bf6befa83331a40e84e4ab90663ec7b": "\\neg c\\vee d,", "1bf6fb07acdcac1b6857170e69490930": "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}", "1bf6fea172e8f53ec71c328f1773e763": "X=\\left\\{x_{1},\\dots x_{n}\\right\\}\\subset A", "1bf6ffb816890e357bb896d986081897": "G^{(n)}\\neq \\{e\\}", "1bf77084f13b7b646c64e8cb6391a11e": "J^{k}={\\begin{bmatrix}J_{m_{1}}^{k}(\\lambda _{1})&0&0&\\cdots &0\\\\0&J_{m_{2}}^{k}(\\lambda _{2})&0&\\cdots &0\\\\\\vdots &\\cdots &\\ddots &\\cdots &\\vdots \\\\0&\\cdots &0&J_{m_{s-1}}^{k}(\\lambda _{s-1})&0\\\\0&\\cdots &\\cdots &0&J_{m_{s}}^{k}(\\lambda _{s})\\end{bmatrix}}", "1bf7a75a7e48320092acd6ca45bfd430": "T\\gg T_{D}", "1bf7eb0f9d5aace4bc3b6189193edaff": "x^{2}+3x-10=0.\\,", "1bf7f73bece5d5fce7bbb9b7508ec692": "P_{2}^{-1}A_{2}P_{2}={\\begin{bmatrix}0&B_{3}\\\\0&A_{3}\\end{bmatrix}}", "1bf82d713f0474bcd0e347fad5614540": "N_{c}=3", "1bf837fed349552284b3c3c3df8906df": "\\displaystyle {\\partial _{\\overline {z}}f=0,}", "1bf838a198d6e93c0e32da9b1b817459": "\\sum _{i=0}^{n}a_{i}x^{i}", "1bf866429b2a54fd84f0aee2ae662c51": "g_{1},\\cdots ,g_{n}", "1bf8d8d6a9a099c337f117ba80365582": "\\varphi (x)\\leq \\varphi (x+y)-(D\\varphi )(x)\\cdot y.\\,", "1bf8dadfbb854d882936ec2879ec1a21": "S(\\theta _{0},\\theta _{i})={\\frac {(\\theta _{0}-\\theta _{i})L_{f}}{t^{1/2}}}", "1bf8e3390c743e096bf374ffa4ddca25": "\\{f_{i}\\}_{0}^{N-1}", "1bf94a7242a891b1b716b2be9752f3d8": "{\\boldsymbol {m}}", "1bf94ab2452a1c00a90ffdf53f4021c1": "\\left|{\\begin{matrix}u_{x}&u_{y}\\\\v_{x}&v_{y}\\end{matrix}}\\right|=u_{x}v_{y}-u_{y}v_{x}.", "1bf95797fdcec75939f4d30d01976f96": "{\\mathbf {x}}=(1,1)", "1bf9b0f97c36784c9777cf501bafe8a4": "\\ R_{i}", "1bf9cf49aeb5e41fd877ef6a6054046c": "x\\#y\\;\\to \\;y\\#x", "1bfb454cb8ec46ca8b62a9592e71968b": "\\sigma _{\\mathrm {n} }", "1bfb4d9609e159a3a98469dd54f7e25f": "{\\begin{pmatrix}1&0&0&0\\\\0&1&0&0\\\\0&0&1&0\\end{pmatrix}}", "1bfb576fcb9f1cfc973c6e67d94d083f": "\\int _{-\\infty }^{\\infty }1\\cdot e^{2\\pi ix\\xi }\\,d\\xi =\\delta (x)", "1bfb8f8268a57cd59f5baee24e661a77": "S\\ =\\ \\mathbf {true} \\ |", "1bfba8f3e6a79196eea583801d10fe00": "\\omega _{i_{1}\\cdots i_{k}}", "1bfbc252a8042e91426cc7d65a3c22e2": "1/kT", "1bfc27d98d7ddc7cee933b74e192bcec": "\\limsup _{n\\to \\infty }1_{X_{n}>0}>0", "1bfc2f5638af87d5f561ca965d6d8132": "{\\frac {dq_{2}}{d\\sigma }}={\\frac {\\partial P}{\\partial u_{2}}}\\quad \\quad {\\frac {du_{2}}{d\\sigma }}=-{\\frac {\\partial P}{\\partial q_{2}}}", "1bfc95b7db00c43fc91d7b382873d65a": "f_{y}", "1bfc9a47684e89400f8d71be994d632d": "S=(\\alpha -2^{1/\\alpha }[\\alpha -1])({\\frac {\\alpha -2}{\\alpha }})^{1/2},", "1bfcf24be75b3a202003b3a12d80edc4": "\\pm {\\sqrt {3/5}}", "1bfd0b8e3736ffff7cae934d5053738b": "P_{V}\\approx 0.64700", "1bfd163ce0059b9a475b6088f420c593": "{\\dot {x}}=Ax+Bu,", "1bfd1b65a8a1ca265d2e256bf9595e69": "(1~2)", "1bfd6527cc6feb106cc15616f8eb33af": "Gx+Cx=Bu", "1bfd995632a44da79e17f7be8c449bb3": "\\Gamma (s,x)=\\int _{x}^{\\infty }t^{s-1}\\,e^{-t}\\,{\\rm {d}}t,\\,\\!\\qquad \\gamma (s,x)=\\int _{0}^{x}t^{s-1}\\,e^{-t}\\,{\\rm {d}}t.\\,\\!", "1bfdee66bea8333f33d815506c94e5cd": "I=\\int _{0}^{1}{\\frac {1}{1+x}}\\,\\mathrm {d} x", "1bfe02982bce0776c1fc7570aea7b149": "{{\\Gamma ,\\;x{:}1{\\to }\\tau _{1}\\;\\vdash \\;e:\\tau _{2}{\\to }\\tau _{3}} \\over {\\Gamma \\vdash \\kappa x{:}1{\\to }\\tau _{1}\\,.\\,e\\;:\\;\\tau _{1}\\times \\tau _{2}\\to \\tau _{3}}}", "1bfe0e73558c831835b2fd1adddfcbd9": "\\Phi (\\mathbf {r} )={\\frac {q}{4\\pi \\varepsilon r}}\\sum _{l=0}^{\\infty }\\left({\\frac {r^{\\prime }}{r}}\\right)^{l}P_{l}(\\cos \\gamma )", "1bfe2d324b29d527a1f275a18fe75284": "R_{\\alpha }=\\sum _{i=1}^{N}r_{i,\\alpha }.", "1bfe3322286a935319deebe894821cf1": "\\mathbf {log_{2}} ", "1bfe3dfdab29a5d4fd77abe31df08416": "\\eta \\in \\Omega ^{q}(M,{\\mathfrak {g}})", "1bfe66ecc330b132953d08d2a66981ec": "f_{n}\\in {\\mathcal {L}}^{1}(\\mu )", "1bfe7dafaf05184e938a2ba95a7eeb1a": "F=\\mathbf {Q} [\\alpha ],\\alpha ={\\sqrt[{3}]{m}}", "1bfe80701f41249750cf8bd84b6231c6": "\\det {\\mathsf {f}}={\\underline {\\mathsf {f}}}(I)I^{-1}", "1bfecfb756a695b24a50c41856a030db": "{\\mathfrak {P}}^{6}", "1bfeedaa9ac34f9efee3ae79fbd70271": "s_{d}", "1bff036e16c55ad582bc9eea9151aa92": "M=(X,d)", "1bff67e994e50e4ab0e9f7186aafefd4": "{\\begin{aligned}i&\\leftarrow 1\\\\r_{1}&=r\\\\G_{1}&=(H^{H}H+\\sigma ^{2}I_{N_{r}})^{-1}H^{H}\\\\k_{1}&=\\arg \\min \\left\\|(G_{1})_{j}\\right\\|^{2}\\\\\\end{aligned}}", "1bff719ee29804967eecf1a1a4b530fe": "\\mu _{X}", "1bffc09e811a7e8b411db8af32cd3b80": "g(n)y\\\\\\end{cases}}", "1c035ebd32780e9598c129db0080f0e1": "|H_{rrc}(f)|={\\sqrt {|H_{rc}(f)|}}", "1c03a88506a096ad5e0093f4709bdbf3": "y\\leftarrow \\lambda (s);", "1c0414627a0735be19bd579fbff98b69": "D=R{\\sqrt {(\\Delta \\phi )^{2}+(\\cos(\\phi _{m})\\Delta \\lambda )^{2}}}{\\color {white}{\\frac {\\big |}{.}}}\\,\\!", "1c043ea664f5eeae49a122668816fdc2": "[2](....([2]([2]([2]([2]([2]P+[k_{(l-1)}]P)+[k_{(l-2)}]P)+[k_{(l-3)}]P)+\\dots )\\dots +[k_{1}]P)+[k_{0}]P=[2^{l}]P+[k_{(l-1)}2^{l-1}]P+\\dots +[k_{1}2]P+[k_{0}]P", "1c04b57af911f4322b635dde5cc7218d": "C_{1}\\approx ", "1c04dfcb2339a32e464a2dfdd28ee0a8": "\\mu =\\sigma /\\xi ,", "1c05096cd8253c71618496a0d6cada8e": "T^{**}=T", "1c057bb27e7d822d124d1eb212b95728": "D_{A}\\subseteq D_{\\mathcal {A}}", "1c059324aad3fbf9c3d0899a20368f79": "w(s(x)-x)\\geq i+1", "1c05b3032149efc238bf8e123466e1a0": "\\scriptstyle y[n]=x[nM].", "1c0619d09229237f572ba6bd8f584a31": "{\\hat {T}}_{f}", "1c0659a20edbc8831b3e86ecdf22bab6": "s\\left\\{{\\begin{array}{l}5\\\\3\\\\2\\end{array}}\\right\\}", "1c0690daf307465960195d4cb564c003": "\\omega =\\sum _{i_{1}<\\cdots 0\\right)\\equiv Prediction\\\\\\left(k<0\\right)\\equiv Smoothing\\end{array}}\\end{cases}}\\end{cases}}", "1c292d1b5f4509e592cf514474627b37": "\\{0,1\\}^{\\omega }", "1c297e4b5a0eb679bc0e359b5e8ca21b": "K_{0}-K_{1}={\\frac {E}{c^{2}}}{\\frac {v^{2}}{2}}", "1c298f636304df472f3afe17835056cc": "{{\\mathit {l}} \\over {\\mathit {l}}^{\\prime }}=1.", "1c2996b132a5e1a99ab974a7847bb4ac": "[S:T]\\mapsto \\left[{\\frac {1}{(a_{0}S-b_{0}T)}}:\\ldots :{\\frac {1}{(a_{n}S-b_{n}T)}}\\right]", "1c299de5ca9a68113e73838956143b6e": "\\varepsilon _{d,e}(n)\\ll \\mu _{\\infty }", "1c29c75443c7353fa8c0e751fe7147e9": "D(E(m_{1},r_{1})^{k}\\mod n^{2})=km_{1}\\mod n.\\,", "1c2a30042227ef28e6f80a0357486a17": "W=p_{me}V_{d}", "1c2a7a71e9919fa300337686e14c401a": "g_{obs}", "1c2b18f667fd7890f3cf77fbe3a6b2d3": "\\Lambda _{\\mathit {g}}={\\frac {\\sum _{{\\mathit {g}}=1}^{\\mathit {G}}{\\overline {\\lambda _{\\epsilon ,g}}}}{\\mathit {G}}}", "1c2b590c64bf541a3776f1f172cc55db": "1/(1-c(1-t)+m)", "1c2b764adae2f300ae3ec5af7955b7ec": "t\\in \\mathbb {N} ", "1c2c2362531608056be5fd4f52d34593": "z^{2}-z-2", "1c2c24ee2eb5178071df9cffe3fc076b": "X_{0}\\rightarrow Hc+X(t)", "1c2c731b59e264d7418cf2e67ff13a80": "[\\mathbf {\\omega } ]_{\\times }=\\left[{\\begin{array}{ccc}0&-\\omega _{z}&\\omega _{y}\\\\\\omega _{z}&0&-\\omega _{x}\\\\-\\omega _{y}&\\omega _{x}&0\\end{array}}\\right]={\\frac {d\\mathbf {A} }{dt}}\\mathbf {A} ^{\\mathrm {T} }", "1c2c79bba234d498032a8fa0c99b0249": "\\scriptstyle k\\;\\in \\;\\mathbb {N} ", "1c2d1ea3189036f4e7c69fb2aaead855": "TR(Q)=P(Q)\\times Q", "1c2d71e7f7350aa94366105ee126c3ec": "(1-k)\\psi (k)+\\ln {\\frac {\\Gamma (k)}{\\lambda }}+k", "1c2d73cfb375053f0e8ac0d545939f5c": "A_{11}U_{1}=F_{1}-A_{1\\Gamma }U_{\\Gamma },\\qquad A_{22}U_{2}=F_{2}-A_{2\\Gamma }U_{\\Gamma },", "1c2d7e686e3fbb8df4e3d04d670bbe89": "r_{o1}\\leq r\\leq r_{o2}:B(r)={\\frac {\\mu _{o}I}{2\\pi r}}\\left({\\frac {r_{o2}^{2}-r^{2}}{r_{o2}^{2}-r_{o1}^{2}}}\\right)", "1c2d8136bdf41e80805ad06d618aebb1": "\\lim _{k,\\ n\\to \\infty }\\int _{\\Omega }|{f}_{k}(x)-{f}_{n}(x)|^{p}\\,dx=0", "1c2da119093f203b8199820fed981ca5": "m=y/k_{0}", "1c2e506e6b9cf80dec6b872f2a8df579": "\\nabla \\varphi ", "1c2e523fa41945de6b573c3f2bdaf0ac": "\\{\\mathbf {r} _{i},i=1\\ldots N\\}", "1c2e6d1e5251188006726e21fb28c9bf": "i_{\\Sigma }(U_{g})", "1c2e99e91c7591f266382696af1cd0ea": "{\\dot {\\vee }}", "1c2ecc1e484af46e354c703b9b883320": "Pr[a|x,y]={\\frac {e^{\\frac {S(a)}{T}}}{Z}}", "1c2ef73a3ca30e01e84925abc81d20b1": "p=3{\\sqrt[{3}]{uv}}", "1c2fa9d25e93b49cb5ef04dfd369e034": "{\\frac {1}{\\pi }}={\\frac {2{\\sqrt {2}}}{9801}}\\sum _{k=0}^{\\infty }{\\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}", "1c30425f594774f625f07b91e0c74f44": "x_{1}^{(m)}", "1c30562b13bbd6274b768336b8ad404c": "\\alpha =2\\beta \\,\\!", "1c3078a6d060a4be8480bc5786c83050": "\\operatorname {Morse} (M)\\subset \\operatorname {Func} (M)\\,", "1c30ef10d44fc702b8ec7b47505d274d": "{\\frac {dN(t)}{dt}}=((b-aN(t))-(d-cN(t)))N(t)", "1c3124c4a6828ae5c34e380d1fe2aa6e": "S\\subseteq U", "1c3142900f48dba597d93d2353b416f3": "{\\partial F \\over \\partial \\tau }={1 \\over 2}{\\partial \\over \\partial \\mu }\\left[(1-\\mu ^{2}){\\partial F \\over \\partial \\mu }\\right]", "1c3197309f2484c3acc03cb9cb5e23c8": "N\\ll 1", "1c31c4f07766a89d9ea719eb295714c6": "{\\tilde {A}}_{2+}", "1c322c8b009a068add19f34f14ccbcae": "{\\mathcal {I}}(\\theta )={\\frac {n}{\\theta (1-\\theta )}},", "1c32337809de0fb306e63a187716e426": "(\\Lambda (A):A\\in \\mathbb {B} _{b}(S))", "1c323c8ca15a4c35cb36b4da70fe0ff7": "y^{2}(a+x)=x^{2}(a-x)", "1c3249898004ed7c0dbf777e27170afd": "{\\begin{array}{lcl}\\#\\mathbb {W} ^{k}&=&{\\text{number of words generated by topic }}k\\\\B&=&\\sum _{v=1}^{V}\\beta _{v}\\\\\\end{array}}", "1c324cad92f5b0e361fb8aabfd970fbb": "n_{x}=\\cos {\\theta (z)}", "1c3256d49054045ca0fc464c35a5dfba": "\\dim P({\\mathbb {M}})=24", "1c3267b5eeb9bee4f051357c232fa0a9": "f_{i}\\,", "1c3272b41fbbb0766c15f8a53642fb84": "\\mathrm {argmax} ", "1c32745198714b41fac2325dffbda74e": "{\\frac {dy_{c}}{dx}}=\\left\\{{\\begin{array}{ll}\\displaystyle {{\\frac {2m}{p^{2}}}\\left(p-{\\frac {x}{c}}\\right)},&0\\leq x\\leq pc\\\\\\\\\\displaystyle {{\\frac {2m}{(1-p)^{2}}}\\left(p-{\\frac {x}{c}}\\right)},&pc\\leq x\\leq c\\end{array}}\\right.", "1c32cc86a864e97cdcbd295d388a031e": "{\\begin{aligned}\\epsilon _{j}^{n}&=e^{at}e^{ik_{m}x}\\\\\\epsilon _{j}^{n+1}&=e^{a(t+\\Delta t)}e^{ik_{m}x}\\\\\\epsilon _{j+1}^{n}&=e^{at}e^{ik_{m}(x+\\Delta x)}\\\\\\epsilon _{j-1}^{n}&=e^{at}e^{ik_{m}(x-\\Delta x)},\\end{aligned}}", "1c32da2676256c493882db195b9bb467": "{\\begin{aligned}Prob(choosing\\,1)&=Prob(U_{n}\\;1", "1c3e4b496b916e02089dbce96fc9f352": "P_{1},\\ldots ,P_{g}", "1c3e5b01cf3fb0746fbdfaef6196bdc5": "\\,p\\,", "1c3ebc7ce19cf99522c07cbfd56cd4b7": "\\langle x,x\\rangle =0,", "1c3ed3cad133be512c832aa31c66e756": "-\\Delta G_{v}", "1c3ee470a49884f06eebbcc89cf037db": "\\alpha _{F}={\\frac {\\mu _{\\theta }-\\mu _{ref}}{\\mu _{ref}^{2}*(\\theta -\\theta _{ref})}}", "1c3ef2e566aeeff55d60174956f6df09": "\\Gamma (n)=(n-1)!\\,", "1c3f6b03df3e5052c5fa146f834629b9": "\\mathbf {E} (X_{n+1}-X_{n}\\mid X_{1},\\ldots ,X_{n})=0", "1c40b838dc2d0ea913fbc87f8736ffc9": "(1-\\epsilon )\\int _{\\psi }(|\\psi \\rangle \\langle \\psi |)^{\\otimes t}d\\psi \\leq \\sum _{i}p_{i}(|\\phi _{i}\\rangle \\langle \\phi _{i}|)^{\\otimes t}\\leq (1+\\epsilon )\\int _{\\psi }(|\\psi \\rangle \\langle \\psi |)^{\\otimes t}d\\psi ", "1c40ba79ab3adfea1caefb39bf16beb3": "H^{s}\\left(E\\backslash \\bigcup _{j}U_{j}\\right)=0\\ {\\mbox{ or }}\\sum _{j}\\mathrm {diam} (U_{j})^{s}=\\infty .", "1c40d31668eae673d5e93a765605a29b": "e=\\pi ^{2}/3", "1c41229e2b74fe6a119548ffc4e2be66": "Ne^{-0.69(\\Delta \\ln \\Delta )^{1/3}}", "1c4143e1a048dcc6bf4042a06c8d4cc7": "{\\frac {d^{2}\\eta }{d\\theta ^{2}}}+\\beta ^{2}\\eta ={\\frac {1}{2}}\\eta ^{2}J^{\\prime \\prime }(u_{0})+{\\frac {1}{6}}\\eta ^{3}J^{\\prime \\prime \\prime }(u_{0})+\\cdots ", "1c41a54117126c2d3e9fddd89835744d": "K\\approx S^{2}=(\\Delta \\omega _{r}/\\Delta _{d})^{2}>1.", "1c41c5d4f3efd10f91738280b1ce6404": "(P\\lor \\lnot Q)\\land (\\lnot P\\lor Q)", "1c424ae50281fefe6d4ee5c6fc203d1a": "{\\frac {d}{d\\tau }}\\left[\\left(1-{\\frac {r_{s}}{r}}\\right){\\frac {dt}{d\\tau }}\\right]=0,", "1c428fe2154ada309d28ab3fdbf27691": "\\displaystyle {T(t)=e_{t}(T)}", "1c42fe52256483ab41bb7b17ae8384fe": "S(x,T)=\\sum _{\\rho :|\\Im \\rho |\\leq T}{\\frac {x^{\\rho }}{\\rho }}\\ .", "1c43187e6537884cf6df72751ffd3d88": "t_{0}={(E(Y)-u_{0})}/{(S/{\\sqrt {n}})}", "1c43543eaf7dd5e788958ac606719245": "{\\text{RR}}={\\frac {1}{N^{2}}}\\sum _{i,j=1}^{N}\\mathbf {R} (i,j).", "1c43a44932f4d872a0822895d8d60002": "\\displaystyle {P=\\bigcup _{\\sigma \\in W_{\\lambda }}B\\sigma B,}", "1c43e15a6a5759ff701a87602a7e2a68": "e^{-\\phi }=\\gamma (1-\\beta )=\\gamma \\left(1-{\\frac {v}{c}}\\right)={\\sqrt {\\frac {1-{\\tfrac {v}{c}}}{1+{\\tfrac {v}{c}}}}}.", "1c43f5ea701b38b0911d9f759851eab5": "(u+du,v)", "1c4401173d3ad39aac1763e8af2899d9": "\\rho (x,y)=\\sum _{n=1}^{\\infty }\\,2^{-n}\\,{\\frac {\\left|\\langle x-y,x_{n}\\rangle \\right|}{1+\\left|\\langle x-y,x_{n}\\rangle \\right|}}", "1c448d9a95cb8443ad3aebe570f84a1c": "\\epsilon \\neq 0", "1c4538f141eea8fca6edcae46b84952c": "f:\\mathbb {R} ^{+}\\to \\mathbb {R} ", "1c45eb5ee0916e325a82f4e65b562e42": "{\\hat {X}}_{i}(z^{n})=f\\left(z_{i-k},\\ldots ,z_{i+k}\\right)", "1c45ebb05173f6e426ded4a1daf8a971": "\\operatorname {gr} _{I}(R)=\\oplus _{0}^{\\infty }I^{k}/I^{k+1}", "1c462dfa9fb82a30c577e6071ece6058": "w_{k}=\\prod _{j\\neq k}\\left[{\\frac {1}{(x_{k}-x_{j})^{2}+4\\sigma ^{2}}}\\right]\\left[1-{\\frac {2\\sigma }{(x_{k}-x_{j})}}i\\right].", "1c465b7374b42f36c9793deba30b1424": "\\quad (6)\\qquad D=3\\pi \\mu dV\\qquad \\qquad {\\text{or}}\\qquad \\qquad C_{d}={\\frac {24}{Re}}", "1c46cfa94fcc2df31d3b02d6f005e381": "K\\geq \\delta \\,.", "1c471e07976adc044ea76befbdf83d49": "dN_{A}", "1c479d86edcf59cac399b3c7ce809692": "P_{\\delta }=\\{\\,\\epsilon \\tau \\mid 0\\leq \\epsilon \\leq \\delta ,\\tau \\in \\Sigma \\,\\}", "1c480bfc50af9298b9c1375718c80d5f": "X^{2}-X=T^{-1}", "1c4815f29e0ab19d0b2189ecba95b328": "\\left\\langle r_{1},r_{2}\\mid (r_{1})^{2}=(r_{2})^{2}=(r_{i}r_{j})^{3}=1\\right\\rangle .", "1c486b8ceb456485b79590f7f74ea4a3": "R(k_{sp})=-m((A+3B\\phi _{in}^{2})k_{sp}^{2}+\\kappa k_{sp}^{4})={\\frac {m(A+3B\\phi _{in}^{2})^{2}}{4\\kappa }}={\\frac {1}{t_{sp}}}", "1c487b6309929f01684384c1f6996f4c": "\\mathrm {Spec} (R)\\,", "1c48a42f3cb0847676c6a9037010b507": "\\lambda =n(n+1)", "1c48b8143edc09b9af7372a2f64ed0a8": "n!\\,", "1c48db78ec96cfa89ec7d1be5f76f092": "\\mathbf {{\\tilde {W}}=WB} ", "1c49520f0d8adc0fe1a6a84e6602f009": "\\cdots \\to A_{n+1}\\xrightarrow {d_{n+1}} A_{n}\\xrightarrow {d_{n}} A_{n-1}\\xrightarrow {d_{n-1}} A_{n-2}\\to \\cdots \\xrightarrow {d_{2}} A_{1}\\xrightarrow {d_{1}} A_{0}\\xrightarrow {d_{0}} A_{-1}\\xrightarrow {d_{-1}} A_{-2}\\xrightarrow {d_{-2}} \\cdots ", "1c4988f6d3eb6e846b3e993a74985cc4": "\\langle p?\\rangle q\\equiv p\\land q\\,\\!", "1c499338dc9e5c0944eadda56ce89a0c": "(M,\\{\\cdot ,\\cdot \\}_{M})", "1c499c81c22c9e35d499955324741447": "{\\textrm {Spec}}(K)", "1c49bd60b17ecb7b07f5421cc0c30f41": "N(q,n)={\\frac {1}{n}}\\sum _{d|n}\\mu (d)q^{\\frac {n}{d}},", "1c49ea22300f7dfdc46f257bc0f7cca8": "r={\\frac {\\bar {y}}{\\bar {x}}}={\\frac {\\sum _{i=1}^{n}y}{\\sum _{i=1}^{n}x}}", "1c4a005185dc1442ee5ed80d2cdef37b": "a^{d}\\not \\equiv 1{\\pmod {n}}{\\text{ and }}a^{2^{r}\\cdot d}\\not \\equiv -1{\\pmod {n}}{\\text{ for all }}r\\in [0,s-1]", "1c4a2e4e903c1e6d7407b5b6b04857bc": "{\\mathcal {O}}_{Y}(V)", "1c4a680612d4325954d3598f4f5ccda3": "\\sigma _{zx}=-{\\frac {\\partial ^{2}\\Phi _{zx}}{\\partial y\\partial y}}-{\\frac {\\partial ^{2}\\Phi _{yy}}{\\partial z\\partial x}}+{\\frac {\\partial ^{2}\\Phi _{xy}}{\\partial z\\partial y}}+{\\frac {\\partial ^{2}\\Phi _{yz}}{\\partial x\\partial y}}", "1c4ae5cd5e03ce9525c477cdcc43e697": "a\\geq 1,b\\geq 1,(a,b)\\neq (1,1)", "1c4bcce1e6eaa460149ddee222bcae1b": "\\pi _{1}(B)", "1c4c2cafb71629c1d53036c979c00745": "A={\\frac {\\theta }{2}}(R^{2}-r^{2})", "1c4c7e5b289b597cdb93d142c5c314b6": "\\;\\sum _{k}p_{k}=1.", "1c4c82613dea79522895a92aaf101488": "\\{x\\}\\;", "1c4c8752888327e81aa3eb6b08f1a267": "{\\overset {\\ \\ \\uparrow }{\\leftarrow }}", "1c4d15dbd8927e28500ebd372157a666": "2\\int _{M}^{\\infty }G(u,t)\\,du=2\\int _{\\frac {M}{\\sqrt {t}}}^{\\infty }G(v,1)\\,dv<\\varepsilon .", "1c4d4639dab2e33c8cfb92a5114f34d9": "{\\frac {}{}}LC", "1c4d6a41ac38a2b6917ce63f2532368a": "(X_{1},\\ldots ,X_{n})", "1c4d7e42642bd2735940546c805c0d46": "\\mathbf {S} \\mathbf {v} =\\sum _{k}\\langle \\mathbf {v} \\mid \\mathbf {e} _{k}\\rangle \\mathbf {e} _{k}", "1c4d9c0cedd24d24519f6b736e6fac0d": "{\\hat {h}}(\\xi )={\\hat {f}}(-\\xi ).", "1c4dcd3ff8ccfb15e4baeb9a7a4a7445": "{\\frac {df}{dx}}=-w^{2}{\\frac {df}{dw}}", "1c4df245f03e161511e62aaaf07d095e": "=e(p_{1},u_{0})-e(p_{1},u_{1})", "1c4e1f5790311980dcc9e27da292725c": "S=S(E({\\boldsymbol {r}}))", "1c4e69e5c6d5cac17caf1f5892daa30f": "\\lim _{q\\rightarrow 1}{\\frac {1-q^{n}}{1-q}}=n,", "1c4e7194da4aa7c2a1b7cf24a4c9e6c8": "\\lim _{|n|\\rightarrow \\infty }{\\hat {f}}(n)=0", "1c4e9bfc751d2af604b2c4dd3d25f05a": "\\theta _{p1}", "1c4eda9c7c048b179b530b3d13e6f9cf": "b(1)e^{\\beta (1)}+b(2)e^{\\beta (2)}+\\cdots +b(N)e^{\\beta (N)}=0.", "1c4edb839c44be4902670b37ab7dd9bf": "=10^{\\left(\\log _{10}\\left({\\sqrt {2\\pi n}}\\left({\\frac {n}{e}}\\right)^{n}\\right)\\right)}", "1c4ef1790cdaf2a5b7979c108bb5591f": "\\mu _{Y\\mid x}=\\mathbb {E} _{Y\\mid x}[\\phi (Y)]=\\int _{\\Omega }\\phi (y)\\ \\mathrm {d} P(y\\mid x)", "1c4efd4bab61a242856978b7696d8b31": "\\ln(K)", "1c4f811b3373e9e19a72b4b8f4129d53": "\\tan {\\frac {\\pi }{60}}=\\tan 3^{\\circ }={\\tfrac {1}{4}}\\left[(2-{\\sqrt {3}})(3+{\\sqrt {5}})-2\\right]\\left[2-{\\sqrt {2(5-{\\sqrt {5}})}}\\right]\\,", "1c4f818eeadcda33a3dfc8087f7b50b5": "j\\geq p", "1c4f9544f45764cb99c98aa9d1971d26": "\\forall g\\in G\\;\\;\\exists k\\in \\mathbb {N} ", "1c4fa11dd36a7a148d168289d83e8ad5": "\\nu _{-}", "1c4fc8813eab2b90cf9d115c6af601c9": "\\mathrm {sf} (n)=\\prod _{k=1}^{n}k!=\\prod _{k=1}^{n}k^{n-k+1}=1^{n}\\cdot 2^{n-1}\\cdot 3^{n-2}\\cdots (n-1)^{2}\\cdot n^{1}.", "1c4fea42b65c8a992897a3ab09b57be5": "(-\\omega _{1},-\\omega _{2})", "1c4ff757428239961f07611397dbf2be": "Div^{0}(K)", "1c4ff8d63ca74605d1903b21a1c2037e": "2T={\\begin{pmatrix}p_{\\alpha }&p_{\\beta }&p_{\\gamma }\\end{pmatrix}}\\;\\mathbf {g} ^{-1}\\;{\\begin{pmatrix}p_{\\alpha }\\\\p_{\\beta }\\\\p_{\\gamma }\\\\\\end{pmatrix}},", "1c4ffc545b59d3484e489fb0a64051e6": "\\Phi '", "1c500c16e7634d3875f5122c6a7c1e6a": "code_{i-1}", "1c501ad62472068a5a3efdadecdca821": "p_{n}(x)y^{(n)}(x)+p_{n-1}(x)y^{(n-1)}(x)+\\cdots +p_{0}(x)y(x)=r(x).", "1c50a4a02391a82a7e5052559d914e65": "\\textstyle a:={\\frac {k_{f}}{k_{i}}}", "1c50b879bdeb42283abf63f666c15c69": "\\langle \\psi _{2}|\\mu _{z}|\\psi _{1}\\rangle =\\left(\\mu _{z}\\right)_{21}=\\int \\psi _{2}^{*}\\mu _{z}\\psi _{1}\\,\\mathrm {d} \\tau .", "1c50ef63bad5614f30f8222f9855c841": "w=3", "1c510886acbe8d3685ed0412a5b29643": "\\theta (t)={\\frac {\\log \\Gamma \\left({\\frac {2it+1}{4}}\\right)-\\log \\Gamma \\left({\\frac {-2it+1}{4}}\\right)}{2i}}-{\\frac {\\log \\pi }{2}}t,", "1c510fe172f7f0d649512631b74d078d": "\\Delta (x_{1}\\wedge \\dots \\wedge x_{k})=\\sum _{p=0}^{k}\\sum _{\\sigma \\in Sh_{p,k-p}}\\operatorname {sgn} (\\sigma )(x_{\\sigma (1)}\\wedge \\dots \\wedge x_{\\sigma (p)})\\otimes (x_{\\sigma (p+1)}\\wedge \\dots \\wedge x_{\\sigma (k)}).", "1c51329b5519ac641e96d11751ca1703": "{\\begin{aligned}\\mathrm {d} \\sigma &=\\left(\\sum _{i=1}^{2}{\\frac {\\partial u}{\\partial x^{i}}}\\mathrm {d} x^{i}\\wedge \\mathrm {d} x\\right)+\\left(\\sum _{i=1}^{2}{\\frac {\\partial v}{\\partial x^{i}}}\\mathrm {d} x^{i}\\wedge \\mathrm {d} y\\right)\\\\&=\\left({\\frac {\\partial {u}}{\\partial {x}}}\\mathrm {d} x\\wedge \\mathrm {d} x+{\\frac {\\partial {u}}{\\partial {y}}}\\mathrm {d} y\\wedge \\mathrm {d} x\\right)+\\left({\\frac {\\partial {v}}{\\partial {x}}}\\mathrm {d} x\\wedge \\mathrm {d} y+{\\frac {\\partial {v}}{\\partial {y}}}\\mathrm {d} y\\wedge \\mathrm {d} y\\right)\\\\&=0-{\\frac {\\partial {u}}{\\partial {y}}}\\mathrm {d} x\\wedge \\mathrm {d} y+{\\frac {\\partial {v}}{\\partial {x}}}\\mathrm {d} x\\wedge \\mathrm {d} y+0\\\\&=\\left({\\frac {\\partial {v}}{\\partial {x}}}-{\\frac {\\partial {u}}{\\partial {y}}}\\right)\\mathrm {d} x\\wedge \\mathrm {d} y\\end{aligned}}", "1c5172aeb28d4aeb3cd5087153da0386": "E=\\sum _{i}^{N}\\varepsilon _{i}-V_{H}[\\rho ]+E_{\\rm {xc}}[\\rho ]-\\int {\\delta E_{\\rm {xc}}[\\rho ] \\over \\delta \\rho (\\mathbf {r} )}\\rho (\\mathbf {r} )d\\mathbf {r} ", "1c51f6cbf795392ea96452762265415a": "{\\begin{aligned}&p=\\operatorname {prox} _{f}(x)\\Leftrightarrow x-p\\in \\triangledown f(p)&(\\forall (x,p)\\in \\mathbb {R} ^{N}\\times \\mathbb {R} ^{N})\\end{aligned}}", "1c5217190a44289fc4a9eb9e66e80002": "{\\frac {\\partial }{\\partial z}}={\\frac {1}{2}}\\left({\\frac {\\partial }{\\partial x}}-i{\\frac {\\partial }{\\partial y}}\\right)\\quad ,\\quad {\\frac {\\partial }{\\partial {\\bar {z}}}}={\\frac {1}{2}}\\left({\\frac {\\partial }{\\partial x}}+i{\\frac {\\partial }{\\partial y}}\\right)\\ .", "1c529cec043f9393c5b9eeff1ee05313": "{\\begin{bmatrix}ct'\\\\x'\\\\y'\\\\z'\\end{bmatrix}}={\\begin{bmatrix}\\gamma &0&-\\beta \\gamma &0\\\\0&1&0&0\\\\-\\beta \\gamma &0&\\gamma &0\\\\0&0&0&1\\\\\\end{bmatrix}}{\\begin{bmatrix}c\\,t\\\\x\\\\y\\\\z\\end{bmatrix}},", "1c52eda493d450bd1d581d69967a7440": "{\\frac {\\log 4}{3}}n\\leq ({\\sqrt {2n}}+1)\\log 2n\\;.", "1c53068008783089636dc7dd4c4db0d1": "K{\\overset {\\underset {\\mathrm {def} }{}}{=}}{\\frac {P_{d}'}{P_{a}+P_{d}}}", "1c531023f400b197b6326eee95d0fb57": "d(1)+d(2)+\\dots +d(n)", "1c5334b07353ad26d30f5ea19c5d13d3": "\\delta n^{a}=\\mu m^{a}+{\\bar {\\lambda }}{\\bar {m}}^{a}-({\\bar {\\alpha }}+\\beta )n^{a}\\,,", "1c53a2dc3676eae0a0a1d7bfeaadafec": "A_{i}\\;=\\;min(w_{T},w_{R},w)\\;x\\;min(h_{T},h_{R},h)", "1c53b7f9f523bf2260227916bf59a065": "Y=\\sum _{i=1}^{2}X_{i}.", "1c544dbb59f8edc73f78a75a00058d64": "c_{11}={\\frac {E(1-\\nu )}{(1-2_{\\nu })(1+\\nu )}}", "1c54740a278e3e6754c6613d37a45dd9": "=\\ln(1.23456)+\\ln(10^{2})\\,\\!", "1c5489955c368efe305c6502fa9ef9d1": "F={\\frac {\\text{explained variance}}{\\text{unexplained variance}}},", "1c54eba980b1dad6b86fd8dcc67a4370": "{\\begin{bmatrix}1&2&3\\\\0&-6&7\\end{bmatrix}}^{\\mathrm {T} }={\\begin{bmatrix}1&0\\\\2&-6\\\\3&7\\end{bmatrix}}", "1c54fa1ca18a7f95a9f9a47c415850dd": "\\left(T_{\\rm {matter}}\\right)_{ab}=\\phi \\,\\rho \\,u_{a}\\,u_{b}", "1c5588099e6cfe62a3c92aee15b30623": "W_{1-2}=\\int PdV\\,,\\quad PV=P_{1}V_{1}=C", "1c55c133bea8aba8e8387e50d98ccace": "W{\\stackrel {f}{\\ \\to \\ }}X{\\stackrel {g}{\\ \\to \\ }}Y{\\stackrel {h}{\\ \\to \\ }}Z", "1c55eb549799ee2b94ae0aff2cc131d1": "[x_{a},y_{a}]", "1c561612f36ddb819edba5c11fb547b4": "\\csc ^{2}x=\\cot ^{2}x+1\\,", "1c56351cf3e3e978c052916892fec228": "\\mathrm {CG_{p}} =\\sum _{i=1}^{p}rel_{i}=3+2+3+0+1+2=11", "1c563d20d5649f7f8e0bba17756dcc0c": "p\\mapsto \\deg(f,\\Omega ,p)", "1c56bea9894c33e806353adcca923dc4": "B_{\\mathrm {cris} }\\otimes _{K_{0}}H_{\\mathrm {dR} }^{\\ast }(X/K)\\cong B_{\\mathrm {cris} }\\otimes _{\\mathbf {Q} _{p}}H_{\\mathrm {{\\acute {e}}t} }^{\\ast }(X\\times _{K}{\\overline {K}},\\mathbf {Q} _{p})", "1c56c39632626797ea585e6ab90cb55f": "\\sum D=d1+d2+d3+...=W", "1c5719558eaa6a8d98d3d19967bbb2d9": "(U(t),u(t))", "1c5763b9e8eb7f55dc116b43fea94aa3": "F_{[\\alpha \\beta ;\\gamma ]}=0", "1c57ae6cbc9b53c0d31db055651dad64": "S=S_{1}\\times S_{2}\\times \\dotsb \\times S_{n}", "1c57b752b116101480666cfbdf066ef6": "{\\frac {1}{P_{n}}}={\\frac {2k_{t,d}+k_{t,c}}{{k_{p}}^{2}[M]^{2}}}R_{p}+C_{M}+C_{S}{\\frac {[S]}{[M]}}+C_{I}{\\frac {[I]}{[M]}}+C_{P}{\\frac {[P]}{[M]}}+C_{T}{\\frac {[T]}{[M]}}", "1c57bfff308992577309393944f18144": "{\\tfrac {8}{11}}={\\tfrac {1}{2}}+{\\tfrac {1}{22}}+{\\tfrac {1}{6}}+{\\tfrac {1}{66}}.", "1c581ea484d263fd6e80c4956016cb62": "{\\frac {d}{dt}}\\left({\\frac {\\partial T}{\\partial {\\dot {\\varphi }}_{r}}}\\right)={\\frac {d}{dt}}\\left(Y{\\dot {\\varphi }}_{r}\\right)={\\frac {1}{2}}F{\\frac {\\partial Y}{\\partial \\varphi _{r}}}-{\\frac {\\partial V}{\\partial \\varphi _{r}}}.", "1c586082f47470b4924982809196cf78": "\\Omega (M,{\\mathfrak {g}})", "1c58710a9ae54b5391bc956b4ad0b739": "w_{12}", "1c58c24c0cace2a7636b63b4a21b2f38": "\\partial \\Omega .", "1c58c49a8c16c94c9dd36d35dc2fefd1": "u(r,t)", "1c59bab178620da6ded38dab4e9da1e7": "A,H", "1c59f6acef572ff8ecb140f2da10b619": "1/\\phi ^{2}\\,", "1c5a0a5eef9692c3c64c564d6f3e7bda": "{\\frac {1024}{729}}", "1c5a1220efecab7294c9da0decf2e8dd": "H=-{\\frac {\\hbar ^{2}}{2m}}\\nabla ^{2}", "1c5a59d3392d3ae548f2bc5f90d3affe": "d={\\boldsymbol {\\bar {v}}}t\\,.", "1c5b7210fa6af8b417c2aaf4029442e2": "Y\\sim N\\left(\\nu \\sin \\theta ,\\sigma ^{2}\\right)", "1c5b755b740cc6ec6c6c79ca5fab5faf": "\\lambda _{a'}{\\frac {}{}}", "1c5b8ad0b38072954978be014250a473": "Q(a)=2L(a)^{2}-L(a^{2}).\\,", "1c5b9ac51b36f47a2a0133af347d784b": "x_{a}=x_{0}(u_{a}\\cos \\omega t+v_{a}\\sin \\omega t)", "1c5bd0d2cefbac52e9efdc1d23eb4d0f": "V^{\\prime }(t)=L_{t}(x^{\\ast }(t),y^{\\ast }\\left(t\\right),t)=y^{\\ast }\\left(t\\right)", "1c5be04ffe5ce630ec04bc1e9d4d301e": "\\mathbf {B} _{i}={\\frac {\\rho _{i}}{a_{i}}}=\\left\\langle \\exp \\left(-{\\frac {\\psi _{i}}{k_{B}T}}\\right)\\right\\rangle ", "1c5c38c58a819eec741eaa412529eb61": "(F_{T},G_{T},\\eta ,\\mu _{T}):C\\to C_{T}", "1c5c3c3deb192f2386f158c1db09b1c7": "\\nabla ^{\\perp }u(x,y)=(-{\\frac {\\partial u}{\\partial y}},{\\frac {\\partial u}{\\partial x}})", "1c5c4d8ad4a109cdb8b02589c504c5e0": "\\scriptstyle f_{s}/2\\ >\\ |f|,\\,", "1c5c95e86bcc29aa01b84a2fdd0a982f": "\\sum _{i=1}^{n}\\min(|x_{i}-y_{i}|,q-|x_{i}-y_{i}|).", "1c5cc9e3cc8428c278542f2db5ae0367": "\\omega _{s(g=2)}={\\frac {eB}{mc\\gamma }}", "1c5d311069ad014b707416c9e5b634a4": "\\color {BurntOrange}{\\text{BurntOrange}}", "1c5d4fb0da7e6cab54ee95e3876644d3": "x_{2}=-10^{0.2192318+0.2706462}=-3.08943", "1c5df1a76b5213b6806eed9d517febbe": "{\\sqrt {\\pi }}xe^{x^{2}}{\\rm {erfc}}(x)\\sim 1+\\sum _{n=1}^{\\infty }(-1)^{n}{\\frac {(2n)!}{n!(2x)^{2n}}}\\ (x\\rightarrow \\infty ).", "1c5e43ea86c386691c991c8393e0a4ab": "\\gamma =\\;F_{\\mathrm {r} }/F_{\\mathrm {M} }=\\beta _{\\mathrm {r} }/\\beta _{\\mathrm {M} }...........(39)", "1c5f0856f920300aaf6c5fa98fa25f8f": "\\beta :=1/\\alpha ", "1c5f435fb535264e196d35a96c4f37bc": "{\\frac {v_{\\mathrm {N} }-v}{v_{\\mathrm {N} }}}={\\frac {c}{d}}", "1c5f96fbaddf8646b1b61f37bee394cc": "{\\begin{aligned}\\mu \\nabla ^{2}\\mathbf {u} -{\\boldsymbol {\\nabla }}p+\\mathbf {f} &=0\\\\{\\boldsymbol {\\nabla }}\\cdot \\mathbf {u} &=0\\end{aligned}}", "1c5fab45d7e7ed8040730c315c282c6b": "X\\in L^{p}", "1c601b274dff34ba5cddd90c78098d2e": "(\\mathrm {Tr} \\;G)^{2}=\\left(\\sum _{i=1}^{r}\\lambda _{i}\\right)^{2}\\leq r\\sum _{i=1}^{r}\\lambda _{i}^{2}\\leq n\\sum _{i=1}^{m}\\lambda _{i}^{2}", "1c60654fc884a3c9f6f68df340a9a3c7": "-\\left(r^{2}+\\alpha ^{2}+{\\frac {r_{s}r\\alpha ^{2}}{\\rho ^{2}}}\\sin ^{2}\\theta \\right)\\sin ^{2}\\theta \\ d\\phi ^{2}+{\\frac {2r_{s}r\\alpha c\\sin ^{2}\\theta }{\\rho ^{2}}}d\\phi dt", "1c6084fcb37821198a37c9aac724c79d": "\\,\\sim ", "1c611a3c6dcdaa03254da9d0b48a317a": "\\mathbb {R} ^{n}\\to \\mathbb {R} ^{m}", "1c611cb07379069e04f9cb683b9769d0": "H_{max}={v^{2} \\over 2g}", "1c612194910bba6b902a07432daf99e3": "\\ F_{propulsive}=drag\\times cos(\\beta )", "1c61331dc90773c0639c886bcd3f0d15": "\\mathrm {Eu} ={\\frac {p_{\\mathrm {upstream} }-p_{\\mathrm {downstream} }}{\\rho V^{2}}}", "1c615f65a308eeed3c6141be0884f993": "10\\log _{10}2=3.010...\\approx 3;", "1c616152acb6c113829bc3d22571a857": "\\int _{X}\\left[\\sum _{i=1}^{n}\\chi _{E_{i}}(x)b_{i}\\right]\\,d\\mu =\\sum _{i=1}^{n}\\mu (E_{i})b_{i}", "1c61ba3178870926ac25a4323e2f688d": "n_{l}", "1c61bdda2415b95c99a794d20d458cd4": "|P_{1}(A)|<|P(A)|", "1c6291276f74cc51f5e77a71d39b2703": "[K_{r}:K_{r-1}]\\cdots [K_{2}:K_{1}][K_{1}:F]", "1c62b5a83ad7e7c61dcec7d2170d889d": "{\\begin{aligned}N_{\\alpha \\beta ,\\beta }&=J_{1}~{\\ddot {u}}_{\\alpha }^{0}\\\\M_{\\alpha \\beta ,\\alpha \\beta }+q(x,t)&=J_{1}~{\\ddot {w}}^{0}-J_{3}~{\\ddot {w}}_{,\\alpha \\alpha }^{0}\\end{aligned}}", "1c64054c43378c4d9ae6023cd7bcd11e": "A+B\\;{\\stackrel {k}{\\rightharpoonup }}\\;2B", "1c642e2367be346bf1fc473d5bd88995": "\\pm d_{0}.d_{1}d_{2}d_{3}\\dots \\times b^{n},", "1c64cc685525b2e35162727c9479858e": "B(x)=B(x+1)", "1c64d561964fe18cd5da2af686eb6205": "\\mu ^{\\Delta }(A)=\\lambda (\\rho ^{-1}(A)),", "1c64e2e5ab36fa13078dd411c0b87c55": "{\\frac {1}{\\mu }}", "1c64ff65f0a138b630bc81e5d89a53a7": "\\rho =3.1\\times 10^{-8}\\alpha ^{-7/10}{\\dot {M}}_{16}^{11/20}m_{1}^{5/8}R_{10}^{-15/8}f^{11/5}{\\rm {g\\ cm}}^{-3}", "1c652a36cc6d37cfcc80d493acd40005": "r\\left({{\\vec {x}};q}\\right)", "1c653d402a0c4964779b7add171c5de1": "t_{initial}=0", "1c6602184b14b54f7c8c77575b9dcd0d": "-K\\leq k\\leq K", "1c66051886d92a7d30efc78da0516de5": "\\nu (\\phi )", "1c6628f3d3a789a3e0337c89ce034933": "\\pi _{2}(x)<(2C_{2}+\\varepsilon ){\\frac {x}{(\\log x)^{2}}}", "1c667b5f5eb8b7b1573c85df7942eced": "Pe_{l}{\\text{≤}}-2", "1c668a16384ae736fa0f458acd2fbeab": "{c \\over \\Gamma }={{1 \\over {B_{\\text{ADS}}\\Gamma _{\\text{max}}}}+{c \\over \\Gamma _{\\text{max}}}}", "1c66bdd57870d22b3b88245ca36dc616": "\\partial L'=(\\partial L-\\operatorname {int~im} \\phi )\\;\\cup _{\\phi |_{S^{p}\\times S^{q-1}}}(D^{p+1}\\!\\times \\!S^{q-1}).", "1c66f77c2b23c82672259e6acf547044": "\\prod _{p^{k}|24}f(p^{k})=f(2)f(3)f(4)f(8).\\ ", "1c670dd6401a5f99e70f67fcf18604f6": "H(x_{k})", "1c6738f054f7c3c42610e55869877f3b": "{\\mathcal {F}}\\{f\\cdot g\\}={\\mathcal {F}}\\{f\\}*{\\mathcal {F}}\\{g\\}", "1c6762424783508ee56aca6114cf840e": "\\beta =1", "1c677b588b3ee01de2e6238a50dbdfa4": "(17)\\quad \\theta _{(\\ell )}=-(\\rho +{\\bar {\\rho }})={\\frac {r-2M(v)}{r^{2}}}\\,,\\quad \\theta _{(n)}=\\mu +{\\bar {\\mu }}=-{\\frac {2}{r}}\\;.", "1c67f492e89efef7536cc587cdd9d729": "\\mathbf {N} \\left(\\mathbf {u} \\right)\\equiv N\\left(u_{1}\\right)\\otimes \\cdots \\otimes N\\left(u_{n}\\right).", "1c6879bc7895b3c10f3fd741c9794930": "\\sum _{j=1}^{i-1}a_{ij}=c_{i}\\ \\mathrm {for} \\ i=2,\\ldots ,s.", "1c68a22f881267ed0b83ac703bd4de16": "\\angle MHA=\\theta =\\angle ZHA-\\zeta '\\approx \\angle ZHA-\\zeta =90^{\\circ }-{\\frac {1}{2}}(\\varphi _{H}+\\varphi _{A})+\\delta ", "1c68b5c5ce310279c0365d8b30a78399": "c=M_{4,1}\\,", "1c68e8cf9e85a95d36ee5e277c357228": "\\scriptstyle S^{+}", "1c690c7ff32ffce96cf98c0e665ca69f": "l^{2}=r^{2}+x^{2}-2\\cdot r\\cdot x\\cdot \\cos A", "1c691959efb790722254fb9c4248fdd2": "\\Delta (v)=1\\otimes v+v\\otimes 1", "1c6972d39651ec7e1dba31e3afe7b583": "{\\begin{aligned}\\pi &=\\sum _{k=0}^{\\infty }\\left[{\\frac {1}{16^{k}}}\\left({\\frac {4}{8k+1}}-{\\frac {2}{8k+4}}-{\\frac {1}{8k+5}}-{\\frac {1}{8k+6}}\\right)\\right]\\\\&=P\\left(1,16,8,(4,0,0,-2,-1,-1,0,0)\\right)\\end{aligned}}", "1c69a4620c0d68697f23fdcd60af8893": "W=F_{2}(\\alpha _{1},\\alpha _{2},\\alpha _{3}.....\\alpha _{k})=0", "1c69d7afa8c525697e88b79b2dcd72d9": "{\\begin{cases}\\Theta ''(\\theta )+\\nu ^{2}\\Theta (\\theta )=0\\\\r^{2}R''(r)+rR'(r)-\\nu ^{2}R(r)=0\\end{cases}}", "1c69e1002e35c7f7f902d105198c609b": "{\\bar {y}}={\\frac {1}{n}}\\sum _{i=1}^{n}y_{i}", "1c69e1ec27d9b129f6ef93946e6aca76": "{\\mathcal {V}}(G)", "1c69ed435f3e2b312a2c89598557892f": "n(\\omega _{j})", "1c6a00ccd6683fb3660cf782aa572b51": "g_{n}=p(N_{-},n)", "1c6a7adf7c2ec75b937b4cccfed35a91": "{\\hat {H}}_{\\text{JC}}=\\hbar \\omega {\\hat {a}}^{\\dagger }{\\hat {a}}+\\hbar \\omega {\\frac {{\\hat {\\sigma }}_{z}}{2}}+{\\frac {\\hbar \\Omega }{2}}\\left({\\hat {a}}{\\hat {\\sigma }}_{+}+{\\hat {a}}^{\\dagger }{\\hat {\\sigma }}_{-}\\right),", "1c6ae411d247e441a289a913fc02e2f1": "{\\gamma =\\sum _{k=2}^{\\infty }(-1)^{k}{\\frac {\\left\\lfloor \\log _{2}k\\right\\rfloor }{k}}={\\frac {1}{2}}-{\\frac {1}{3}}+2\\left({\\frac {1}{4}}-{\\frac {1}{5}}+{\\frac {1}{6}}-{\\frac {1}{7}}\\right)+3\\left({\\frac {1}{8}}-{\\frac {1}{9}}+{\\frac {1}{10}}-{\\frac {1}{11}}+\\dots -{\\frac {1}{15}}\\right)+\\dots }", "1c6aef6220c2cf4351c45767effb9026": "\\theta :T^{*}M\\otimes \\Omega ^{p}M\\rightarrow \\Omega ^{p+1}M", "1c6b0ca5e03a886e7a0d1741331e929f": "A_{v}=\\prod _{i\\in v}\\sigma _{i}^{x},\\,\\,B_{p}=\\prod _{i\\in p}\\sigma _{i}^{z}.", "1c6b2ba181612dba150a315b6347bdc7": "1/4\\leq p_{1}\\leq 2/3", "1c6ba617904a799bfa9062d749845c7b": "t_{i}({\\hat {\\theta }})=\\sum _{j\\in I-i}v_{j}(x_{I-i}^{*}(\\theta _{I-i}),\\theta _{j})-\\sum _{j\\in I-i}v_{j}(x_{I}^{*}({\\hat {\\theta }}_{i},\\theta _{I}),\\theta _{j})", "1c6cfb5075461ab5e2a3bc8372013a86": "{\\bar {\\psi }}(\\mathbf {x} ,\\tau )=\\mathrm {e} ^{K\\tau }\\psi ^{\\dagger }(\\mathbf {x} )\\mathrm {e} ^{-K\\tau }.", "1c6d0cbab0de2ada403c82149c6a0a2f": "\\operatorname {MSE} ({\\overline {X}})=\\operatorname {E} (({\\overline {X}}-\\mu )^{2})=\\left({\\frac {\\sigma }{\\sqrt {n}}}\\right)^{2}={\\frac {\\sigma ^{2}}{n}}", "1c6d24c6b7f55dd92938207a410900dd": "1\\,+\\,{\\frac {1}{4}}\\,+\\,{\\frac {1}{16}}\\,+\\,{\\frac {1}{64}}\\,+\\,\\cdots .", "1c6d53d479dc823b9a5f16a668020cf3": "q=2\\int _{0}^{\\pi }{\\frac {I(\\alpha )}{I(0)}}\\sin \\alpha \\,d\\alpha .", "1c6d843e390cc7ec217f046f3fcf1eb6": "\\inf _{Q_{Y|X}(y|x)}I_{Q}(Y;X)\\ {\\mbox{subject to}}\\ D_{Q}\\leq D^{*}.", "1c6dc09098c96cf690cb9beadbb06a0f": "V(r)={\\frac {-Ze^{2}}{r}}", "1c6dc490ee2368603e2732ba7f386055": "f_{n}(z)=z+{\\frac {1}{\\rho n^{2}}}{\\sqrt {z}},\\qquad \\rho >{\\sqrt {\\frac {\\pi }{6}}}", "1c6dcc1aca556dcc6d3d48e509e391fc": "{\\frac {H_{n}}{P_{n}}}", "1c6dffbd1ec7425930842b204a3fea31": "s^{2}={\\frac {1}{2N}}\\left\\{\\sum _{n=1}^{N}(x_{n,1}-{\\bar {x}})^{2}+\\sum _{n=1}^{N}(x_{n,2}-{\\bar {x}})^{2}\\right\\}", "1c6e4c7ea9187799fcf4027de228b70e": "m=f_{sc}/f_{ref}", "1c6e55d5abe7c4ef4168fcd59c3db538": "\\ T={\\frac {I_{l}}{I_{0}}}=e^{-\\sigma \\ell N}=e^{-\\alpha '\\ell }.", "1c6e747eb9def5c640a74525104ac27a": "S_{m}(n)={1 \\over {m+1}}\\sum _{k=0}^{m}(-1)^{k}{m+1 \\choose {k}}B_{k}\\;n^{m+1-k}.", "1c6f09f07c06ae3128af4f5bb6557f22": "K_{\\alpha }(z)\\sim {\\begin{cases}-\\ln(z/2)-\\gamma &{\\text{if }}\\alpha =0\\\\\\\\{\\frac {\\Gamma (\\alpha )}{2}}\\left({\\frac {2}{z}}\\right)^{\\alpha }&{\\text{if }}\\alpha >0.\\end{cases}}", "1c6f5120a7b01c7bd288c93b64cb3cc5": "\\displaystyle {a^{b+c}=(a^{b})^{c}.}", "1c6f85857795410fcd52bf6f0fe79144": "L_{2}=sh/(h+s)=h/(h/s+1)=h", "1c6f91ff6d4fb5c67f510420dd9045d9": "{x}\\geq {0}\\,", "1c6fac357e8f6e53c4673f3db3091f0a": "x^{2}y,\\,\\!", "1c700ae5dcf9c1305a41e496acf47e3f": "m(x)\\sim 1/x,\\,", "1c702ab65cf4affa4e895efd8f6c7ece": "F({\\underline {x}},-)", "1c70377b92bdfef9317e4b334d30d2a8": "{\\frac {dA}{dt}}={\\tfrac {1}{2}}r^{2}{\\frac {d\\theta }{dt}}.", "1c704bd22fa98873b2a8070a9bf4a041": "{\\mbox{Slip}}={\\frac {NP-V_{a}}{NP}}=1-{\\frac {J}{p}}", "1c705cfecb670563cc60061396f14de4": "\\Omega =-kT\\ln {\\Big (}\\sum _{\\rm {microstates}}e^{\\frac {\\mu N-E}{kT}}{\\Big )}", "1c706071f28fbd72fc368322c4b30488": "p_{k}=n{\\frac {{\\dot {x}}_{k}}{\\sqrt {{\\dot {x}}_{1}^{2}+{\\dot {x}}_{2}^{2}+{\\dot {x}}_{3}^{2}}}}=n{\\frac {dx_{k}}{\\sqrt {dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}}}=n{\\frac {dx_{k}}{ds}}", "1c708e1e013f604acc1893dd78feec3b": "x_{n_{3}}\\geq x_{n_{2}}.", "1c7109d5368f7c71b0494b40d337ae06": "{{\\sin(\\theta )} \\over {4\\left[{1\\,\\,\\,+\\,\\,\\,{1 \\over 4}\\sin ^{2}\\left({\\theta \\over 2}\\right)}\\right]}}\\,\\,\\,\\approx \\,\\,\\,{\\theta \\over 4}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\Rightarrow \\,\\,\\,\\,\\,\\,\\,\\,{\\theta \\over 4}\\,\\,\\Delta \\theta \\,\\,\\,=\\,\\,\\,{{\\theta ^{2}} \\over 4}{{\\Delta \\theta } \\over \\theta }", "1c712ddaa8457a7926d7129d93d8c9c0": "dy=a\\,dx", "1c7180361eacf57d714d4399e28bf909": "(\\kappa \\leq \\mu )\\rightarrow ((\\kappa +\\nu \\leq \\mu +\\nu ){\\mbox{ and }}(\\nu +\\kappa \\leq \\nu +\\mu )).", "1c7186dbfb3f611312b8825f6c1f9a72": "\\varphi (g_{1}g_{2})=\\varphi (g_{1})\\circ \\varphi (g_{2})\\quad {\\text{for all }}g_{1},g_{2}\\in G\\,\\!", "1c71a3b383e32d4fdaeabb1e43e883b3": "\\chi (M\\cup N)=\\chi (M)+\\chi (N)-\\chi (M\\cap N).", "1c71a523b2398cbd176b796ff63f6feb": "\\textstyle f_{C}(c)=0.2", "1c71f2c078e7194c56a9499e600270b3": "y(x)=x\\cdot y'+(y')^{2}\\,\\!", "1c724b96ba5973b9023107c7fb44acde": "{\\mathcal {F}}^{-1}({\\mathcal {F}}f)(x)=g(x)", "1c7266bda6399771f14154b48959630b": "S_{5}={\\begin{pmatrix}1&1&1&1&1\\\\1&2&3&4&5\\\\1&3&6&10&15\\\\1&4&10&20&35\\\\1&5&15&35&70\\end{pmatrix}}.", "1c72b6206da332c036fd47a4a90fbfea": "\\mathbf {I} =\\Delta \\mathbf {p} =\\int _{t_{1}}^{t_{2}}\\mathbf {F} \\mathrm {d} t\\,\\!", "1c72b9453cd275f8176a4259a2799580": "3^{F_{n}-1}\\equiv 1{\\pmod {F_{n}}}", "1c72c13da4eee1c61daddad8b3828f7b": "M_{\\mu }\\to M_{\\lambda }", "1c72c52a434cdb758d0fb79567cf0448": "\\int {\\frac {dx}{\\sqrt {x^{2}+c}}}=\\int {\\frac {{\\frac {t^{2}+c}{2t^{2}}}dt}{\\frac {t^{2}+c}{2t}}}", "1c72f4b271e661bc0736d3a5dab40b7e": "(\\cdot ,\\cdot ,\\cdot ):Q\\times Q\\times Q\\to Q", "1c738fcd9733a432602ff6defd139a37": "\\displaystyle {[J_{m},J_{n}]={c \\over 3}m\\delta _{m+n,0}}", "1c739c6ff0405d1d347ea49f7748a29b": "\\scriptstyle \\mathbf {\\hat {n}} ", "1c73b5822c23dd7396dfcb36a450fd59": "{\\frac {\\mathrm {d} \\mathbf {L} _{i}}{\\mathrm {d} t}}={\\boldsymbol {\\tau }}_{E}+\\sum _{i\\neq j}{\\boldsymbol {\\tau }}_{ij}\\,\\!", "1c73b6ffebeaf3cbfc5df82ff58cf1fa": "{\\bar {G}}", "1c73c110dded50675fbcf1565c392b80": "\\mathbf {F} \\cdot \\mathbf {r} =0", "1c73fcf3c87d83e82c59038f41e8c671": "m_{\\mathrm {eff} }\\leq m", "1c74234e407004c3b89313435efadc47": "\\left({\\widehat {E}}-c{\\boldsymbol {\\alpha }}\\cdot {\\widehat {\\mathbf {p} }}-\\beta mc^{2}\\right)\\left({\\widehat {E}}+c{\\boldsymbol {\\alpha }}\\cdot {\\widehat {\\mathbf {p} }}+\\beta mc^{2}\\right)\\psi =0\\,,", "1c745f2ad0a65bf4634fa2187c63b019": "\\aleph _{1}\\times 2^{\\aleph _{0}}\\,=2^{\\aleph _{0}}.\\,", "1c747a0c90dfbc859316f497b8808fba": "C\\subseteq \\kappa ", "1c7498d3fd21d52291b85e86451296ff": "k_{F}=(3\\pi ^{2}N_{e}/V)^{1/3}", "1c74df82dd4854589387348363a7a3e0": "S_{\\text{CGHS}}={\\frac {1}{2\\pi }}\\int d^{2}x\\,{\\sqrt {-g}}\\left\\{e^{-2\\phi }\\left[R+4\\left(\\nabla \\phi \\right)^{2}+4\\lambda ^{2}\\right]-\\sum _{i=1}^{N}{\\frac {1}{2}}\\left(\\nabla f_{i}\\right)^{2}\\right\\}", "1c74f010aea750d8b5099df7e065fdf8": "a-{\\sqrt {a^{2}-b^{2}}}", "1c74f5def7f7f6d1fd206d987a896857": "\\lambda =B\\left({\\frac {n^{2}}{n^{2}-4}}\\right)\\qquad \\qquad n=3,4,5,6", "1c752c7271436df7057aa75f56dc5dd0": "MarginalCost(MC)={\\frac {\\ dC}{\\ dQ}}", "1c758c905658f1dbb33cc6de313998f1": "\\ \\{x:A(x)\\},", "1c761199cc445e9b249285cf6bf5a98f": "\\Psi (x)\\approx C_{0}{\\frac {e^{i\\int \\mathrm {d} x{\\sqrt {{\\frac {2m}{\\hbar ^{2}}}\\left(E-V(x)\\right)}}+\\theta }}{\\sqrt[{4}]{{\\frac {2m}{\\hbar ^{2}}}\\left(E-V(x)\\right)}}}.", "1c767451119de5d3c9aae08e244eae9b": "\\gamma _{total}\\left(d\\right)", "1c76cde5362619ed6d443a8778d7d68f": "E=E_{KIN}+E_{POT}", "1c779a57da1668aa058a883a56e8b6ca": "\\int _{-\\infty }^{\\infty }|f(\\xi +i\\eta )|^{2}\\,d\\xi <\\infty .", "1c77e6aa877e041ff7d34240fd9ecf4c": "f(a)=\\min _{x\\in R\\setminus \\{0\\}}g(xa)", "1c7840262592bcfeca27aac4b10d4969": "d(k)", "1c784b849e900375636d7cdba245a599": "{\\mathcal {F}}\\{f\\}", "1c786d45b40ca94e9a858276bd971f4c": "z\\rightarrow z^{3}-3zx^{2}+zy^{2}+z_{0}", "1c78b486fa89d4f71edbbd0d53d214dc": "ub", "1c78e51631375cfe500f96f321d063a3": "\\omega _{pl}^{2}={\\frac {4\\pi e^{2}N}{\\epsilon L^{3}m}}", "1c78ea100bb0b6822a900c6f5053c40d": "C_{\\mathrm {p} }", "1c7919ee51bf9c43612d9a6aa4d9d68b": "\\displaystyle S(4,2)=\\left\\{(1111),(1112),(1122),(1222),(2222)\\right\\}", "1c795b5da70841d856b4410f4af87290": "T\\vdash \\varphi ", "1c795d8bf9c86273efb69c4d580a8457": "s(n)", "1c79ab541f8c525efd74cdb3f18a361e": "{\\begin{cases}{\\dot {u}}(t)=Au(t);\\\\u(0)=u_{0}.\\end{cases}}", "1c79c561f7aa073bb69217a07ffdcffc": "\\,\\alpha ", "1c7a25de0c4ef3098e5a0d92002e9e0b": "\\{N_{L/K}(x)|x\\in {\\mathfrak {b}}\\}.", "1c7a7eda90d34f821a7834455648f510": "\\int _{-\\infty }^{\\infty }L(\\Delta f)d\\Delta f={\\frac {f_{\\Delta }}{\\pi }}\\int _{-\\infty }^{\\infty }{\\frac {d\\Delta f}{f_{\\Delta }^{2}+\\Delta f^{2}}}=\\left.{\\frac {1}{\\pi }}\\tan ^{-1}({\\frac {\\Delta f}{f_{\\Delta }}})\\right|_{-\\infty }^{\\infty }=1", "1c7ae0212b67b6dc7104070fe8a2aaba": "M(f)<0\\,\\Rightarrow \\,L(f^{-1})\\leq -{\\frac {1}{M(f)}},", "1c7b1295a3ca164f6f10cb3ab5ae9458": "\\partial /\\partial t", "1c7b5f4e1a5f1092da4dfebdf41af0ee": "{\\tfrac {10}{3}}\\div 5", "1c7b783ea8c31c2816a193adc996de6c": "b'=\\tau (b,a,o)", "1c7bfccc43787f838cf05e40103a5d91": "\\displaystyle {e_{W}=\\det(I-W^{*}W)^{1/4}f_{W}}", "1c7c0cad838a82350b38d91c1e1d7ad5": "y=b+r{\\frac {2t}{1+t^{2}}}.\\,", "1c7c13712c27246ed8b722bf28b7324b": "{1 \\over 1-\\alpha z}=\\prod _{j=1}^{\\infty }\\left({1 \\over 1-z^{j}}\\right)^{M(\\alpha ,j)}", "1c7c139dfb84d36a62ad85bd61550790": "(7,4,3)", "1c7c2038f8c25bd16ace98a12f447c54": "V_{YY}", "1c7c23455301f7b488f798bcc3af5f0c": "B_{\\text{op}}=B", "1c7c6fcce368ca6f327aaa7ade047006": "{\\frac {1}{24}}\\left(n^{6}+3n^{4}+12n^{3}+8n^{2}\\right).", "1c7c84a53deafa61a815efed32a0c28c": "(\\beta _{2},\\;\\lambda _{2})", "1c7c877f919bc8e523de0c79f8482b7d": "10^{1/30}=1.079775", "1c7ca8a78509e148b0da0cafce863b28": "h_{\\alpha \\beta ,\\gamma }\\eta ^{\\beta \\gamma }={\\frac {1}{2}}h_{\\beta \\gamma ,\\alpha }\\eta ^{\\beta \\gamma }\\,,", "1c7cab2cde93564526ac136aec9c505f": "N'=\\pi ^{-1}(N),", "1c7cb00334a8fe64ac7cf9a120cae5b5": "\\mathrm {^{239}_{\\ 94}Pu\\ \\xrightarrow {2(n,\\gamma )} \\ _{\\ 94}^{241}Pu\\ {\\xrightarrow[{14.35\\ yr}]{\\beta ^{-}}}\\ _{\\ 95}^{241}Am\\ \\xrightarrow {(n,\\gamma )} \\ _{\\ 95}^{242}Am\\ {\\xrightarrow[{16.02\\ h}]{\\beta ^{-}}}\\ _{\\ 96}^{242}Cm} ", "1c7cd00d98876082e5f6d8d8d04ea7f5": "f_{\\mathbf {v} }\\left(v_{x},v_{y},v_{z}\\right)\\,dv_{x}\\,dv_{y}\\,dv_{z}.", "1c7d013f78c102398b93496d91ad3671": "={\\frac {2}{3}}", "1c7da5980eaba46dd7471584b1fb5de9": "\\land ((T_{2}=[F_{2},S_{2},A_{2}]::[F_{1},S_{1},A_{1}]::R", "1c7dae5409aeeedcb6aa2e4c2d1ada71": "F_{j}", "1c7e17a14a4bd186001307ac352818d1": "\\sigma _{t}^{2}", "1c7e814a9045da29bc091492b08d32c1": "C={\\begin{bmatrix}1/2&0\\\\0&1/7\\\\\\end{bmatrix}}{\\begin{bmatrix}11\\\\13\\\\\\end{bmatrix}}={\\begin{bmatrix}11/2\\\\13/7\\\\\\end{bmatrix}}.", "1c7e889d05243af857153c401b236617": "\\lambda =2\\pi /k", "1c7efd1a42ffd08bae9b2235d2754060": "y:\\Delta ^{n}\\rightarrow Y\\,", "1c7f3f0a8f462f5f400fd91704c75d68": "\\left[{\\begin{array}{cc|c}5.291&-6.130&46.78\\\\0.00300&59.14&59.17\\\\\\end{array}}\\right].", "1c7fa447397b3403dc2f4b810cca18ea": "\\phi _{n}(0)=1", "1c7fc52761eb224fa431c5eab519f17b": "\\xrightarrow {C4H} ", "1c7ffb424e9c698c8289752ec032e0cf": "\\psi _{1}(\\varepsilon _{\\Omega _{2}+1})", "1c801983acd85eb12c45df8969ffcbc8": "E\\{({\\hat {x}}-x)(y-{\\bar {y}})^{T}\\}=0", "1c804e0352c2fd04dd6923b2dd622089": "R_{n}^{2}(\\xi ,x)>1", "1c807797a1f5e7f6062275fb795f5d50": "n_{t}(x)", "1c80e7ac4762c7585993604b5e3bb2e9": "P_{d}(x,y,z)=0", "1c815001366cf21910e7c76ade62bb11": "|0{\\rangle }", "1c81771509cdf9517c7d05eacf348dde": "\\scriptstyle \\lfloor \\lambda \\rfloor ", "1c81a3b249a7cd61bcddab88e6de8f6f": "{\\overline {AD}}^{\\,2}={\\overline {AB}}^{\\,2}+{\\overline {BC}}^{\\,2}+{\\overline {CD}}^{\\,2}\\ .", "1c820b60ce4540c5339ea64d2d90bd46": "^{2}9", "1c8212910e2d491904e9a5bcf0d80d4e": "{\\Big (}({\\mathcal {M}},s)\\models AF\\phi {\\Big )}\\Leftrightarrow {\\Big (}\\forall \\langle s_{1}\\rightarrow s_{2}\\rightarrow \\ldots \\rangle (s=s_{1})\\exists i{\\big (}({\\mathcal {M}},s_{i})\\models \\phi {\\big )}{\\Big )}", "1c8213b9d9bbfd3480b71550dd87e5ed": "\\ L_{0}=a", "1c827500761de941cba99239aaf76484": "\\phi =(f\\circ \\xi ^{-1})^{-1}=\\xi \\circ f^{-1}", "1c828fc0c76726d11ee41c69e2c09dc4": "\\sigma (f(x))=f(x+1)", "1c82b70aacf66befdb20dd9e36393fbf": "M={\\begin{pmatrix}a_{11}&a_{12}&a_{13}\\\\a_{21}&a_{22}&a_{23}\\\\a_{31}&a_{32}&a_{33}\\end{pmatrix}},", "1c82fd822d8f2c860f6fe9a1909b3b71": "(r_{i}r_{j})^{k}", "1c834430b9871100b53bffd979309472": "{\\begin{bmatrix}N\\\\M\\end{bmatrix}}", "1c837f53ad1173411db02ee49173a61e": "k_{i}(k_{i}-1)", "1c83d2c7ec448e1586fced85203ad83c": "\\Delta \\langle {\\hat {B}}{\\hat {B}}\\rangle ", "1c8414cf43dc06fb11810afe58fc570e": "{\\tfrac {2}{7}}\\scriptstyle {\\sqrt {30-3{\\sqrt {2}}}}", "1c847d4cf6ac267949f1919c24e212c2": "R^{d-1}", "1c848169dba57c91ed187e3f7faa480a": "1={\\frac {(k_{2}-k_{1})k_{1}}{k_{2}\\cdot k_{1}}}+{\\frac {(k_{3}-k_{2})k_{1}}{k_{3}\\cdot k_{2}}}+\\dots +{\\frac {(k_{n}-k_{n-1})k_{1}}{k_{n}\\cdot k_{n-1}}}+{\\frac {1\\cdot k_{1}}{k_{n}}}", "1c84ca98188b58c3f258c5658e82c6d2": "1-[-0.625\\log _{2}(0.625)+-0.375\\log _{2}(0.375)]\\approx 4.6\\%.", "1c8518c1bda4a2652bd187c5a1107743": "{B}_{7}^{(1)}", "1c852c5de4573e4e63fc4f853f888362": "0=(x^{\\alpha }T^{\\mu \\nu }-x^{\\mu }T^{\\alpha \\nu })_{,\\nu }.\\!", "1c85557f3478c8868cf71f227382ea81": "\\sin x=\\sum _{n=0}^{\\infty }{\\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}=x-{\\frac {x^{3}}{3!}}+{\\frac {x^{5}}{5!}}-\\cdots \\quad {\\text{ for all }}x\\!", "1c85bd366dda9578bca1bfa13468a4e2": "V(z)={\\frac {2z}{1+{\\sqrt {1+4z}}}}", "1c85f1028949810c6dffc85bb32fd6e6": "B_{k}=B'_{0}=(1+r_{k})B_{0}-p_{k}", "1c86023efc384cc5c1dd5b88fcf0baec": "{\\begin{aligned}I_{3}\\cdot R_{3}&=I_{1}\\cdot R_{1}\\\\I_{x}\\cdot R_{x}&=I_{2}\\cdot R_{2}\\end{aligned}}", "1c8604ae4056da7feb0a94276fc4d8f0": "\\Delta f\\approx {\\frac {c}{2nl\\cos \\theta }}", "1c8643fa8c04255cfcafb144796c658a": "b=2", "1c864b26839eacd2aa5df39ded60a9d9": "\\displaystyle \\Omega (k)", "1c86543803e2095a2064ee70cd5776c3": "[*F,*G]^{IJ}=-[F,G]^{IJ}\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;Eq.7", "1c8703e7c502bf1efa7273435fe78032": "\\theta _{(l)}", "1c8714e724a7f0cc344c30d4c738d4f8": "f_{xx}(x,y)\\approx {\\frac {f(x+h,y)-2f(x,y)+f(x-h,y)}{h^{2}}}\\ ", "1c8717bf743dc20b47fd28420f8baaa3": "\\delta \\mathbf {u} (\\lambda r)", "1c87957ed09e68858cc37b66ec7c07a8": "\\left|{\\widehat {f}}(n)\\right|\\leq {{\\rm {var}}(f) \\over 2\\pi |n|}", "1c880ddca12fd76766ff725309096805": "{\\mathit {a}}", "1c880f6fff7af3578e6bd5531378df0d": "{\\boldsymbol {v}}\\in \\mathbb {R} ^{n}", "1c8864318c57f63ae29a56c532a2453a": "a^{n}+b^{n}", "1c8881c2b72d487bc18d53cd7346a05e": "\\Lambda _{min}\\subseteq \\Lambda \\subseteq \\Lambda _{max}", "1c8896865e21230f8e056c2dae680164": "name_{i}", "1c88c4abbc6007b0cf5e13958f616db9": "{\\mathcal {F}}_{t}", "1c8916911546684ecb5b0b598164bfe0": "C^{2+\\delta }", "1c8965dd4db91534cbb67d6973226576": "\\mathbf {E} (z,t)=\\mathrm {Re} (\\mathbf {E} _{0}e^{i\\omega (({\\tilde {n}}z/c)-t)})", "1c89991159992ffc7304ed59ded82efa": "\\langle \\mathbf {v} ,\\mathbf {w} \\rangle ", "1c89ce97cc18cd81508173797a6e6264": "q_{\\alpha }", "1c8a2370746031bd1040ab77818d7a5d": "J(x,t)={\\frac {i\\hbar }{2m}}(\\psi {\\frac {\\partial \\psi ^{*}}{\\partial x}}-{\\frac {\\partial \\psi }{\\partial x}}\\psi )", "1c8a5bebf295c421d007960e4cf5c258": "1-\\zeta _{m}", "1c8a7d35ef26d89684fce7ad05bb9e4d": "k={\\frac {2\\pi }{\\lambda }}", "1c8b02d6bdd89c337bf1b4781f48e573": "P*V^{\\gamma }=\\operatorname {constant} =1.58\\times 10^{9}=P*100^{7/5}", "1c8b53f09d0c114797831f239ade0e47": "u_{70}(\\mathbf {r} )={\\bar {u}}_{lh}(\\mathbf {r} )=\\left|{\\frac {3}{2}},-{\\frac {1}{2}}\\right\\rangle ={\\frac {1}{\\sqrt {6}}}|(X-iY)\\uparrow \\rangle +{\\sqrt {\\frac {3}{2}}}|Z\\downarrow \\rangle ", "1c8b5678830b0f2a6eda1271b3a626a0": "\\Delta v=v_{\\mathrm {g} }\\left(1-{\\tfrac {v_{\\mathrm {c} }}{v_{\\mathrm {g} }}}\\right)\\approx v_{\\mathrm {g} }", "1c8b5802df8a5f185235bd4446d655c0": "U(y,\\xi )\\,", "1c8bd988c72b27e2355be2bbdbe7e584": "\\mathbf {T} (5)=5\\,", "1c8beb3b0b0c1514f36716363c3c2f1b": "(\\lambda x.x\\ x)\\ (\\lambda x.\\lambda f.f\\ (x\\ x\\ f))", "1c8c85cde8a69ff8604b10d5fe1a4ca5": "\\delta _{2s}(n)={\\frac {\\pi ^{s}n^{s-1}}{(s-1)!}}\\left({\\frac {c_{1}(n)}{1^{s}}}+{\\frac {c_{4}(n)}{2^{s}}}-{\\frac {c_{3}(n)}{3^{s}}}+{\\frac {c_{8}(n)}{4^{s}}}+{\\frac {c_{5}(n)}{5^{s}}}+{\\frac {c_{12}(n)}{6^{s}}}-{\\frac {c_{7}(n)}{7^{s}}}+{\\frac {c_{16}(n)}{8^{s}}}+\\dots \\right)", "1c8c99796fcd4a0ab2d3c328dc851f9a": "{\\biggl (}{\\frac {1-p}{1-pz}}{\\biggr )}^{\\!r}{\\text{ for }}|z|<{\\frac {1}{p}}", "1c8ca73486fe326c54f1559358ddfecf": "X\\cap W_{i}", "1c8cee48c07a1cdbfdbaa79748cb2e90": "y_{1i}", "1c8d2ef4fc5764198bb6df8299263ee3": "\\alpha =1\\mathrm {m} ^{-1}", "1c8d401a5790a50d57caa5233aefa6d4": "\\cos(wz)+i\\sin(wz)\\,", "1c8d8ce34f2148797866bbb422292d95": "C(N)\\leftrightarrow \\forall x\\lnot C(x)", "1c8dd4a654b61c6941618bf1534ba829": "\\int _{\\mathbf {R} }{\\frac {\\sin x}{x}}dx", "1c8efcab01cf7143097d467c4625e859": "{\\tilde {Fr_{1}}}", "1c8f3f00e1925efd56f1d0b6aa4347c3": "(c\\delta \\tau )^{2}=(c\\delta t)^{2}-(\\delta x)^{2}.\\,", "1c8fbeae1557bc0ac8a45d7afc837a93": "P_{dBW/m^{2}/Hz}=10\\log _{10}(P_{Jy})-260", "1c8fcd356afec21c05421124b8862ad0": "M(n,m+1,p)=M(M(n,1,p),m,p)", "1c902748eb82600ffe44ad79cc82ce7a": "{\\frac {\\partial \\mathbf {\\hat {z}} }{\\partial \\varphi }}=\\mathbf {0} .", "1c902eb0605242d70cadbe2bc78f2467": "{\\begin{aligned}p(n,k,m)&=1-{(m-nk-1)! \\over m^{n-1}(m-n(k+1))!}\\end{aligned}}", "1c90436cf9c162977e8089732c3440a4": "\\varphi (p)=2^{k}", "1c9058bd156b8bceef2935bcb68818bf": "n\\in \\mathbb {N} ^{*}", "1c90a40df84e0f1c1e897026fc30e897": "R/n=\\tan \\theta \\,", "1c90e57d7bc9bd980d44fd8965ff5ca3": "k=8", "1c90e87927def654d6dfda90ea92fda4": "\\varphi (x_{1},x_{2},x_{3},x_{4},x_{5})=\\varphi (x_{5},x_{2},x_{3},x_{4},x_{1})+\\varphi (x_{1},x_{5},x_{3},x_{4},x_{2})+\\varphi (x_{1},x_{2},x_{5},x_{4},x_{3}).", "1c90fba63b8486db2db0ac0f13399e80": "{\\mbox{Copper Loss}}=I^{2}\\cdot R\\cdot t", "1c910bea2d3a2b0796c8508e570a902c": "p_{1}=1,", "1c91b2f11123ab24bedf87567864c9b5": "\\displaystyle z={\\frac {G(x)^{5}}{xH(x)^{5}}}", "1c91b6f7877b71a3d51eebeb61b1d6a1": "e_{t}^{2}", "1c91c91f48e191b0c9c7803995d05726": "{\\text{NPV}}={\\frac {\\text{number of true negatives}}{{\\text{number of true negatives}}+{\\text{number of false negatives}}}}={\\frac {\\text{number of true negatives}}{\\text{number of negative calls}}}", "1c91e1280eb5b3626c029e2d88ab3b76": "L_{\\rho }(\\gamma )", "1c9214abf423bd36c4239e71ba55fee8": "P=I/G", "1c924f5e1db39797bea5db87dd7e80b8": "f(\\mathbf {X} )\\sim {\\mathcal {N}}(\\mathbf {m} ,\\mathbf {K} ),", "1c9252d5c532fcfa9d50175a938c66c7": "\\ell _{2}", "1c9261848713b3e7c2c609c5fe39548a": "t^{-n-1}", "1c92a23649ed4c9789b0a04246cd5e5d": "{\\frac {d^{2}\\mathbf {x} _{\\mathrm {A} }}{dt^{2}}}=\\mathbf {a} _{\\mathrm {AB} }+{\\frac {d\\mathbf {v} _{\\mathrm {B} }}{dt}}+\\sum _{j=1}^{3}{\\frac {dx_{j}}{dt}}{\\frac {d\\mathbf {u} _{j}}{dt}}+\\sum _{j=1}^{3}x_{j}{\\frac {d^{2}\\mathbf {u} _{j}}{dt^{2}}}.", "1c92b3817f410c97e751dae1e51eb544": "t\\in \\lbrack 0,1]", "1c92c395255065be3acc86f2db8b0f18": "g_{m}={\\frac {I_{C}}{V_{T}}}\\left({\\textrm {in}}\\ {\\frac {\\mathrm {mA} }{\\mathrm {mV} }}\\right)", "1c92c7ef37dab7c528d9200dc57951a1": "r_{k+1}=0", "1c92d8ba252e2ff0a6ffcea8ed6f2190": "{\\textbf {W}}^{-1}", "1c930d3f99626629cfb7bc70c72a7871": "i\\hbar \\epsilon g\\partial _{g}U_{\\epsilon }(t_{1},t_{2})=H_{\\epsilon }(t_{1})U_{\\epsilon }(t_{1},t_{2})-U_{\\epsilon }(t_{1},t_{2})H_{\\epsilon }(t_{2}).", "1c9316ac43eba22c820cd8d05dee28d6": "{\\begin{aligned}x&={\\frac {a\\sinh v\\cos \\phi }{\\cosh v-\\cos u}}\\\\y&={\\frac {a\\sinh v\\sin \\phi }{\\cosh v-\\cos u}}\\\\z&={\\frac {a\\sin u}{\\cosh v-\\cos u}}\\end{aligned}}", "1c935e25bb7bd4ba1ad8cbf5a67bf173": "k\\mid l\\implies f(n,k)\\mid f(n,l)", "1c93667590f379fe888f55c064ae3c6a": "{\\text{var}}\\,[Y(\\mu ;t)]=a\\mu ^{p}t^{2-d}\\,\\!", "1c93a490ffc51046d58fbd22f9d48256": "I_{cr}=I_{p}/N_{cr}", "1c93b5ef40f1a929e9f061424bbafc69": "c^{2}{d\\tau }^{2}=\\left(1-{\\frac {r_{s}}{r}}\\right)c^{2}dt^{2}-\\left(1-{\\frac {r_{s}}{r}}\\right)^{-1}dr^{2}-r^{2}\\left(d\\theta ^{2}+\\sin ^{2}\\theta \\,d\\varphi ^{2}\\right),", "1c93fc7a6403093fdb794859c65cd6af": "\\sum _{n}|a_{n}|^{p}<\\infty ", "1c93fdbbf342d2d131689f2a5492a9b7": "K_{t}(x,y)=K_{t}(x-y)={1 \\over {\\sqrt {2\\pi it}}}e^{-i(x-y)^{2} \\over 2t}\\,.", "1c94057e24b6e3c36d1c799fb6ec5e9b": "CD_{\\text{wave}}={\\frac {24V}{L^{3}}}", "1c940d18986659f3a362e3b582d26422": "i\\leq j", "1c944c31217a8d5559c152cafb5f1a4f": "{\\frac {1}{\\sigma {\\sqrt {2\\pi }}}}\\,e^{-{\\frac {(x-\\mu )^{2}}{2\\sigma ^{2}}}}", "1c944d3a52acd084050af2ce9ff0e655": "V_{A},V_{B}:\\Sigma \\mapsto \\mathbb {R} ", "1c94e9ee1711656d0dc527c6ef08f12d": "f(x_{0}+2,y_{0}+3/2)-f(x_{0}+1,y_{0}+1/2)=A+B=\\Delta y-\\Delta x", "1c955d3f150225f23e6d0dcec41478e7": "x^{4}=a", "1c95700138cb221d5added03a8c70015": "\\sum _{n=1}^{\\infty }1^{n}", "1c95a1fad0c57258bf1819bee7212683": "\\beta ^{*}\\theta =\\beta ^{*}(\\sum _{i}p_{i}\\,dq^{i})=\\sum _{i}\\beta ^{*}p_{i}\\,dq^{i}=\\sum _{i}\\beta _{i}\\,dq^{i}=\\beta .", "1c95e241a89bb33fd581bd8610cdde77": "P_{load}\\,", "1c95e7392296825bddad0404bfe05a07": "{\\frac {dx}{ds}}=a", "1c95edcecaa3a950fcaf841590eca415": "W_{i_{1}..i_{k}j_{1}...j_{k}}=V_{(i_{1}..i_{k})(j_{1}...j_{k})}", "1c9605e969bb3f5c8c80d74071bba478": "{\\sqrt {I_{L1}^{2}+I_{L2}^{2}+I_{L3}^{2}-I_{L1}*I_{L2}-I_{L1}*I_{L3}-I_{L2}*I_{L3}}}", "1c96bf233e7a9087addf7fddd590e073": "\\left({\\frac {am+Nb}{|k|}},{\\frac {a+bm}{|k|}},{\\frac {m^{2}-N}{k}}\\right)", "1c96f72e1b6a4afd406b04ac8501209e": "\\{e\\}\\!", "1c9766fc4685abae490b7eadb6bb29d7": "\\left\\{{\\begin{matrix}N&{\\mbox{if }}a=e^{i2\\pi k/N}\\\\{\\frac {1-a^{N}}{1-a\\,e^{-i2\\pi k/N}}}&{\\mbox{otherwise}}\\end{matrix}}\\right.", "1c97b0cf1433df7d02b752a961b38f4d": "t={\\frac {\\mu _{y}z-\\mu _{x}}{\\sqrt {\\sigma _{y}^{2}z^{2}-2\\rho \\sigma _{x}\\sigma _{y}z+\\sigma _{x}^{2}}}}", "1c985a311fe6854e1bf2225b3e1e479d": "{\\frac {kT}{m}}", "1c988e8755ae3b64b1bfbfc7ae4347c5": "a.b", "1c98abcebf708f057c7735ddfe3bf252": "\\lambda _{3}", "1c98f86c5fb58734f117ca79657fd754": "I_{sp}/c", "1c98fb9836626c6feb3a9b692c25bc6b": "\\int x^{m+n}\\left(2a\\,B(m+1)-A\\,b(m+n(2p+1)+1)+(b\\,B(m+1)-2\\,A\\,c(m+n(2p+1)+1))x^{n}\\right)\\left(a+b\\,x^{n}+c\\,x^{2n}\\right)^{p-1}dx", "1c99396fe1e7076b433ce55dc1a84fad": "A=\\int _{\\infty }^{-\\infty }y(T)x'(T)dT=\\int _{\\infty }^{-\\infty }TPR(T)FPR'(T)dT=\\int _{-\\infty }^{\\infty }TPR(T)P_{0}(T)dT=\\langle TPR\\rangle ", "1c9957f8c25dad33fd3e05db74e039d1": "{\\frac {dD}{dr}}\\leq 0.", "1c99968db24ffddae0aa89e1ddba57dc": "\\scriptstyle \\mathbf {J} \\;=\\;0", "1c99974e6b4cc2b15e75506341ed3b37": "{\\overline {D}}_{\\hat {\\dot {\\alpha }}}X=0", "1c9a842ee9ad3c637b71018a7a51bf23": "D(y)", "1c9a8f941d8f259af16672f3a21f79e1": "p^{\\mu }=-i{\\partial \\over \\partial x_{\\mu }}", "1c9aa43424adf8d05120f44a1b5b0866": "u(x_{i})=u_{i}", "1c9af354e628d5361597afc6238a7af1": "=2p_{T1}p_{T2}(\\cosh(\\eta _{1}-\\eta _{2})-\\cos(\\phi _{1}-\\phi _{2})).\\,", "1c9b1d75901ea83338820ba3f26b9414": "({\\mathcal {F}}f)(\\xi )", "1c9b28fa6a62e4f679cbdcc68255afbd": "\\mathrm {Hom} (\\mathrm {colim} F,N)\\cong \\mathrm {lim} \\,\\mathrm {Hom} (F-,N)", "1c9b442b89f521da79549d4d571f30a0": "\\left\\vert \\psi \\right\\rangle ^{A}", "1c9b841e06b401ef5d0967e2153df858": "0.{\\overline {45}}", "1c9baa79f8f2d8e973c9269e0b5ddde3": "{\\begin{aligned}&A_{j_{0}}(x)~,\\\\&[A_{j_{0}}(x),A_{j_{1}}(x)]~,\\\\&[[A_{j_{0}}(x),A_{j_{1}}(x)],A_{j_{2}}(x)]~,\\\\&\\quad \\vdots \\quad \\end{aligned}}\\qquad 0\\leq j_{0},j_{1},\\ldots ,j_{n}\\leq n", "1c9bac504188b2152665436fb2ac6dcf": "\\Phi _{R}(\\mathbf {r} ,t)={\\tilde {\\Phi }}_{L}(\\mathbf {r} ,t)", "1c9bbd54e308aec6a21619347b83639a": "S_{h}^{p}(\\Omega _{h})=\\{v_{|\\Omega _{e_{i}}}\\in P^{p}(\\Omega _{e_{i}}),\\ \\ \\forall \\Omega _{e_{i}}\\in \\Omega _{h}\\}", "1c9bcf491c515eece22a8d7783e1ead2": "g(x)^{q}", "1c9be3ee8886693a30f34e670e5a43e6": "\\langle \\cdot \\rangle ", "1c9c15c0d2a33410277fbf10ab53db9c": "F(b)-F(a)=\\sum _{i=1}^{n}\\,[F'(c_{i})(x_{i}-x_{i-1})].", "1c9c3fdd1d73cf00b078d94336f6bd6b": "\\sigma \\ =\\ 0", "1c9c88b480a6054d92374045a1450acc": "{\\frac {a}{\\sin A}}={\\frac {c}{\\sin C}}.", "1c9ca2b01e5e7189cd537ececc519e88": "\\tan E={\\frac {\\sin E}{\\cos E}}={\\frac {{\\sqrt {1-e^{2}}}\\sin \\theta }{e+\\cos \\theta }}\\ .", "1c9cb84425b520512f4e6e88a9d56f50": "A^{\\text{T}}=F^{\\text{T}}C^{\\text{T}}", "1c9ccae34809bb9e4997c31189b9f515": "z(t)=t.\\,", "1c9cdcf12b2d951923f539f93de579f8": "\\mu ={\\frac {\\exp {(\\mathbf {X} {\\boldsymbol {\\beta }})}}{1+\\exp {(\\mathbf {X} {\\boldsymbol {\\beta }})}}}={\\frac {1}{1+\\exp {(-\\mathbf {X} {\\boldsymbol {\\beta }})}}}\\,\\!", "1c9d246ca12efceb95d9a8dff4ab90f0": "D\\colon K(\\!(X)\\!)\\to K(\\!(X)\\!)", "1c9db336f635ec280e505d27bed0f426": "M_{\\lambda }=\\{v\\in M;\\,\\,\\forall \\,h\\in {\\mathfrak {h}}\\,\\,h\\cdot v=\\lambda (h)v\\}.", "1c9db949e178c9969de0565ead550e01": "A+B:=\\{\\,a+b\\in \\mathbb {R} ^{n}\\mid a\\in A,\\ b\\in B\\,\\}.", "1c9dd5e318b571f3a58075685b58721b": "\\phi (n)=n-\\sum _{i=1}^{r}{\\frac {n}{p_{i}}}+\\sum _{1\\leq i0", "1cb266746bc89ca135ae5cd90d588cac": "e:{\\textit {Eve}}", "1cb2b472910ab2f9145e3ce23f9276a8": "f_{max}", "1cb2d3fcd417cc48cba7e3fe824d8c4d": "{\\vec {v}}_{\\rm {inertial}}={\\frac {v_{\\|}}{\\omega _{c}}}\\,{\\vec {b}}\\times {\\frac {d{\\vec {b}}}{dt}},", "1cb2e0fe79c829bb6d883a3a81e684e1": "x_{\\lambda }", "1cb30670494b9a39dd3155194767e368": "|{\\mathcal {A}}|\\subseteq |{\\mathcal {B}}|", "1cb392867eefe35628f6900f5a3f7aba": "\\langle n|H|n\\rangle =E_{0}=E_{i}-U\\ .", "1cb43ab803bc4fcf0458cdc0220fb1bc": "\\|f\\|^{2}=\\sum _{n=0}^{m-1}{|a_{n}|^{2}}+\\|R_{m}\\|^{2}", "1cb485cfb0886bf602848cf9464ed20d": "\\scriptstyle A\\;\\subseteq \\;\\{\\lim _{n}a_{n}:\\,\\forall n\\;\\geq \\;0,\\,\\ a_{n}\\in A\\}", "1cb4a2503f05c07ea4ebe9eb8b283c95": "s(x)=\\min _{n\\in {\\mathbf {Z}}}|x-n|", "1cb536c18ef0f7d7957882aea091a61c": "\\leq _{c}", "1cb539cf31a9bc49ac9a1ccbc12b552f": "E_{2}=E_{1}+\\hbar \\omega ", "1cb561e2a7b9a4dcf8e787631e2d543a": "M={\\frac {qB}{c}}\\left({\\begin{matrix}0&1\\\\-1&0\\end{matrix}}\\right),", "1cb6185d789b84280012a1e2c1a4caae": "PA=\\bigcup _{n}PA_{n}", "1cb61f6cac13eef6084f2336b4232846": "\\pi _{i}P_{ij}=\\pi _{j}P_{ji}\\,,", "1cb69307b0779287bd3a69f93b367e70": "\\displaystyle \\delta (\\xi )", "1cb6c310d2c2934b9b91d17ca7b37abf": "\\alpha \\equiv ", "1cb7049065a0f3067920a95ecdde8dac": "\\Delta _{k}(x)={\\mathcal {O}}\\left(x^{\\alpha _{k}+\\varepsilon }\\right)", "1cb706122d4a127ce184f937c69c4be9": "\\theta =\\pm 2\\ \\mathrm {atan} \\left({\\sqrt {\\frac {1-C}{1+C}}}\\right).", "1cb731ffffcba6391745197b9efe509d": "\\det A_{33}=0", "1cb7ccd2a282e73d862063571c8226e6": "\\cot {\\frac {\\gamma }{2}}\\sin {\\frac {a-b}{2}}=\\tan {\\frac {\\alpha -\\beta }{2}}\\sin {\\frac {a+b}{2}}.", "1cb8a3872985bd0131ff7d7806f37db4": "-\\alpha -1", "1cb8ae633a26b6181b0ca6e88b3d81d4": "R[e_{1},\\ldots ,e_{k}]/\\left\\langle \\{e_{i}e_{i}|1\\leq e_{i}\\leq e_{j}\\leq l\\}\\right\\rangle ,", "1cb8afd87fc11473fe98ebfdfbd306da": "\\scriptstyle {\\dot {R}}", "1cb8c8bbf8c66366dde4b17c0f835666": "C_{V,m}", "1cb90320c4bbd6ed430ced9542e09e76": "h\\mathbb {Z} ", "1cb9372b8daebabd798e972344b47b36": "n_{1},\\ldots ,n_{k}", "1cb983f53e57343e2f6370899b4d7282": "E_{act}", "1cba6786cc8f541030caa7244f3e6430": "r=r_{0}\\cos(\\varphi -\\gamma )+{\\sqrt {a^{2}-r_{0}^{2}\\sin ^{2}(\\varphi -\\gamma )}}", "1cba6aeca5622f7ce19ea07e918f9aa8": "{\\begin{aligned}e^{\\pi {\\sqrt {19}}}&\\approx x^{24}-24;x^{3}-2x-2=0\\\\e^{\\pi {\\sqrt {43}}}&\\approx x^{24}-24;x^{3}-2x^{2}-2=0\\\\e^{\\pi {\\sqrt {67}}}&\\approx x^{24}-24;x^{3}-2x^{2}-2x-2=0\\\\e^{\\pi {\\sqrt {163}}}&\\approx x^{24}-24;x^{3}-6x^{2}+4x-2=0\\end{aligned}}", "1cba7f343e73b1666ece4c0cb999540d": "\\,CD", "1cba900507099f1394ee8482d9f49f2d": "u_{i}^{2}=48", "1cbab575f605089f277cae2dc59a6d6c": "B_{5}", "1cbac16f4efe4579cc742613501efb47": "f(\\alpha ^{-1}(y))=\\alpha ^{-1}(y+1)\\,.", "1cbac314be0c2537a9e5d16882368af0": "\\times \\sum _{r=0}^{\\ell -s}{\\ell -s \\choose r}{\\ell +s \\choose r+s-m}(-1)^{\\ell -r-s}e^{im\\phi }\\cot ^{2r+s-m}\\left({\\frac {\\theta }{2}}\\right)\\ .", "1cbaf38e845f301fdf5302a032c8dc98": "f:{\\mathcal {Z}}^{2k+1}\\to {\\mathcal {X}}", "1cbb092423afe4c132ebe4567a1221ac": "\\{A,B\\}=AB+BA", "1cbb6a0c917aec4a216cd9d1a7f008d8": "\\psi (\\Omega ^{2})=\\phi _{2}(0)", "1cbb6b81beb9b71a9b17ce72a078b7be": "\\Delta m^{2}<0", "1cbb7e672946849e053ea4a59ba4adf9": "\\mathbf {\\Psi } _{10}=-{\\sqrt {\\frac {3}{4\\pi }}}\\sin \\theta \\,{\\hat {\\mathbf {\\theta } }}", "1cbc42372b8f385a270129f40d3ceb41": "x_{i}=y_{i}\\cdot z_{i}", "1cbc4db7d1a3530507323b0f04eb00c5": "v={\\begin{matrix}{\\frac {1}{2}}\\end{matrix}}\\cdot (v_{1}+v_{2})", "1cbc50fff51d6206e043395083ed818e": "u_{I}^{\\alpha }:U^{k}\\to \\mathbf {R} \\,", "1cbc7c2ae0f92a1a4ea670af843ff776": "\\left[1+\\left({\\frac {x-\\mu }{\\sigma }}\\right)^{1/\\gamma }\\right]^{-\\alpha }", "1cbc9459a63fb82dbea22b1fedb7b5d0": "{\\frac {{\\dot {A}}(t)}{A(t)}}", "1cbc99dcc76739a152d2346c0454a903": "z=4.467\\times \\log _{10}[\\alpha 2macroglobulin(g/L)]-1.357\\times \\log _{10}[Haptoglobin(g/L)]+1.017\\times \\log _{10}[GGT(IU/L)]+0.0281\\times [Age(years)]", "1cbcc67ccff7ece2d4bf172a373ca61f": "\\arg {z}=\\{\\operatorname {Arg} z+2\\pi n:n\\in \\mathbb {Z} \\}", "1cbcddb8c6d3a87ee51c468ed772f8e0": "\\displaystyle \\pi ^{-\\delta }\\Gamma (\\delta +1)\\left|{\\frac {\\boldsymbol {\\nu }}{2\\pi }}\\right|^{-n/2-\\delta }", "1cbdf7c5ed9bd6ba0ab007419195da48": "\\dim(R/P)+\\dim(R/Q)=\\dim(R)\\ ", "1cbdf9c177c2d0933865ccb355ced749": "\\langle i,j\\rangle ", "1cbe0fb258b2a216e0accf76ee525a84": "|z|=1/(t-\\epsilon )>R", "1cbe65f54d92fb7a7358e16c40572f51": "T_{2}={\\frac {1}{4}}\\sum _{i,j}\\sum _{a,b}t_{ab}^{ij}{\\hat {a}}^{a}{\\hat {a}}^{b}{\\hat {a}}_{j}{\\hat {a}}_{i},", "1cbe6ec4bf4eb89865a7434d6ff87fab": "A^{-1}[i]", "1cbea6a7320b6cc31e964dbe512128ee": "\\mathrm {St} ={f \\over U}{C^{3}}", "1cbef2a2a9ae468d92c38e9de6e33b35": "-S=\\{-s:s\\in S\\}", "1cbf23be176975393ecabffa2dab73b8": "c+d\\,\\mathrm {i} =s\\cdot (\\cos(\\psi )+\\mathrm {i} \\sin(\\psi ))=s\\cdot \\mathrm {e} ^{\\mathrm {i} \\psi }", "1cbf4b8617ae5542fdea66c1fb111aac": "\\Pi _{1}=P(q_{1}+q_{2}(q_{1})).q_{1}-C_{1}(q_{1})", "1cbf4d434e3bcf063cb73efb4a652edb": "n\\equiv {c \\over c'}\\approx \\left(1-{2\\Phi \\over c^{2}}\\right).", "1cbfe66b8fa0b227c1a43fd7414c37f1": "J\\subseteq A", "1cc060f045ded3bc5bab0dd42e9cb72b": "V_{1}=P_{11}Q_{1}+P_{12}Q_{2}+P_{13}Q_{3},", "1cc07aeda7870026cd806aa21bae633e": "i_{1},i_{2},\\ldots ,i_{n}", "1cc0878180d3266d09d8927f7e23a6b1": "{a\\pi \\over 5}\\ {b\\pi \\over 5}\\ {c\\pi \\over 3}", "1cc098e1b55764f2f0b192a23f5ea6ef": "(-\\Delta )^{-\\alpha /2}f(x)", "1cc0c0a24f2576cbe53d268966f8d942": "\\scriptstyle i,j,k", "1cc0efbfef307be7e7b959ddcee5715b": "A(\\alpha ,\\beta \\,|\\,z)=G_{p+2,\\,q}^{\\,q-m,\\,p-n+1}\\!\\left(\\left.{\\begin{matrix}-a_{n+1},-a_{n+2},\\dots ,-a_{p},\\alpha ,-a_{1},-a_{2},\\dots ,-a_{n},\\beta \\\\-b_{m+1},-b_{m+2},\\dots ,-b_{q},-b_{1},-b_{2},\\dots ,-b_{m}\\end{matrix}}\\;\\right|\\,z\\right).", "1cc1162706b268c0d1033138c4762469": "y_{Q21}=1.00+j1.52\\,", "1cc1585ee701f5338f50e49c6383c469": "\\vartheta _{11}", "1cc1896313f1e17e38e59979c1055ddc": "{{\\frac {1}{\\varrho }}{\\frac {\\partial p}{\\partial z}}}=-g.\\qquad (3)", "1cc18a8a1efe46c7faa9f15b4f04e124": "\\Phi _{4}", "1cc2076be935cc3893f05d7dd7c8a9a2": "{\\frac {X_{s}}{X_{r}^{'}}}\\approx {\\frac {0.4}{0.6}}", "1cc24d7843f23d50d16562d326fa25ec": "r(\\emptyset )=0", "1cc250ab164bfba7b16daaacdc8d7fbb": "-m_{1}/(m_{1}+m_{2})", "1cc2a1b2e89ad52fa91a8e69ea064176": "k\\varphi (N)<\\varphi (N)d", "1cc2d642ae02fa633ff953fdd4eae851": "\\varphi _{a_{1}X_{1}+\\ldots +a_{n}X_{n}}(t)=\\varphi _{X_{1}}(a_{1}t)\\cdot \\ldots \\cdot \\varphi _{X_{n}}(a_{n}t).", "1cc355103aee2cd8a91686dfb9e13305": "(42/9!)\\prod _{p}\\left((1-p^{-1})^{4}(1+4p^{-1}+p^{-2})\\right)", "1cc3ddf38986dbd80fdbb180af426e1a": "I(t)\\to 0", "1cc3ed73d449afce1c25c8344ab61a97": "\\mathbf {a} \\cdot (\\mathbf {b} \\times \\mathbf {c} )=(\\mathbf {a} \\times \\mathbf {b} )\\cdot \\mathbf {c} ", "1cc3ff30e30d539a35dc81eb33a97281": "f\\,'", "1cc422427571db0aa3bb835d0311c7ea": "D(s)=1+kG(s)", "1cc46c0b4d052113fcc067fdeab855bf": "b_{n}\\to \\infty ", "1cc4cd649adc27e8b546ad8b5fc47770": "TM=\\bigsqcup _{x\\in M}T_{x}M=\\bigcup _{x\\in M}\\left\\{x\\right\\}\\times T_{x}M=\\bigcup _{x\\in M}\\left\\{(x,y)\\vert \\;y\\in T_{x}M\\right\\}.", "1cc4dd87545adf85076e1b3fb381f96d": "{\\begin{aligned}R_{F}(x,y,z)&=R_{F}(A(1-\\Delta x),A(1-\\Delta y),A(1-\\Delta z))\\\\&={\\frac {1}{\\sqrt {A}}}R_{F}(1-\\Delta x,1-\\Delta y,1-\\Delta z)\\end{aligned}}", "1cc4e93534d5e2271a2c8e3b11d6bdc8": "\\sum _{i=1}^{2}\\left({\\frac {1}{2}}\\omega ^{2}Y_{i}^{2}M_{i}\\right)=\\sum _{i=1}^{2}\\left({\\frac {1}{2}}K_{i}Y_{i}^{2}\\right)", "1cc5eb97f8145ab4b4bfdb4dcc3ee936": "BWT[i]=S[A[i]-1]", "1cc5fb6d3b10cf0b4029e23d46fa7fc0": "[0,2\\pi ]", "1cc77ce43189651af6fecded877ab079": "PGL(n,K)=\\mathbf {P} (GL(n+1,K)),", "1cc7d54323b032d8b8e28e4cb52bc7aa": "+1.0000000000+1.1913785723\\cos(x)-0.0793018558\\cos(2x)-0.2171442026\\cos(3x)-0.0014526076\\cos(4x)", "1cc856eaa4864355bc7aa0fb29ae5615": "{\\hat {\\beta }}=(X^{T}X)^{-1}X^{T}y\\ .", "1cc8a9e0e8873dd05558fd7c41b3f0bd": "F_{13}=\\sin {(\\alpha )}\\sin {(\\theta )}d\\alpha \\wedge d\\phi ", "1cc8c15587cd0351848b438d37b4d7c9": "a_{m}", "1cc8cb1735d017609f416303c2001bda": "\\tau _{1}={\\frac {K_{p}K_{v}}{\\omega _{n}^{2}}}", "1cc8dbd0c67f648d4cbaa62a06caa3b0": "Q~", "1cc8e279567ce5de5e131f8766ecc402": "\\mathbf {A} \\ast \\mathbf {B} =(\\mathbf {A} _{ij}\\otimes \\mathbf {B} _{ij})_{ij}", "1cc8f926734abc0689153376616c4415": "u=cu_{+}+(1-c)u_{-}\\,", "1cc904f35a8e49ba6d91450f4234369c": "(\\mathbf {a} \\cdot \\mathbf {b} )^{2}-(\\mathbf {a} \\wedge \\mathbf {b} )^{2}=\\mathbf {a} ^{2}\\mathbf {b} ^{2}", "1cc93e320814dbaa255fc5169c1a2b90": "\\sigma _{CS}=\\sigma _{SC}=E(\\cos \\theta \\sin \\theta )-E(\\cos \\theta )E(\\sin \\theta )\\,", "1cc9819f58b6f7bc0edc8b0b14f2a17e": "\\mathrm {d} (\\mathbf {p} ,\\mathbf {q} )=\\mathrm {d} (\\mathbf {q} ,\\mathbf {p} )={\\sqrt {(q_{1}-p_{1})^{2}+(q_{2}-p_{2})^{2}+\\cdots +(q_{n}-p_{n})^{2}}}={\\sqrt {\\sum _{i=1}^{n}(q_{i}-p_{i})^{2}}}.", "1cc99f1904970eee572f4af94eb33df9": "{10}^{\\,\\!4\\cdot 2^{80}}", "1cc9a401b56e2abfee9ece6e4ded7c90": "0\\leq \\varepsilon _{n+2}\\leq \\min \\left\\{{\\frac {\\varepsilon _{n+1}^{2}}{2}},{\\frac {\\varepsilon _{n+1}}{2}}\\right\\}", "1cc9fe7ce0659e1a37d7da9b0995ef57": "\\mathbf {z} _{k+1}^{\\mathrm {T} }\\mathbf {r} _{k}=0,", "1ccaa0caff8c07504f2e17efe436eee3": "Z_{22}={Y_{11} \\over \\Delta _{Y}}\\,", "1ccae03d4ec9c20db7dd2217110b298d": "\\mu (a_{m}/a_{0})=\\mu (a_{M}/a_{0})", "1ccaf86acc085dac3283262263f5d8e3": "R_{a}={\\frac {Z_{1}Z_{2}}{R_{b}}}\\,", "1ccb2fd4de8445c82de205e329c265d5": "N\\,\\!", "1ccb420d7a9e7bfe23bab3ed06f52f86": "f(x)=x^{2}+2", "1ccb56303db4b96362d87746292c87bd": "\\Delta =11-d", "1ccb88e6ee7760657b5deef01fe44993": "{\\frac {\\partial \\mathbf {g(u)} }{\\partial x}}=", "1ccbe99d6d11036b50d874766f2dc52d": "\\phi (x)=\\int _{V}G(x,x')\\rho (x')\\ d^{3}x'+\\int _{S}\\left[\\phi (x')\\nabla 'G(x,x')-G(x,x')\\nabla '\\phi (x')\\right]\\cdot d{\\hat {\\sigma }}'.", "1ccc01d3c0998832c2378069d983090b": "2\\pi \\!", "1ccc0a479afff13788f6cb71495bfd6e": "\\gamma =\\alpha +i\\beta \\,", "1ccc4bfd098e47a4765e886f6c93c518": "\\Delta p_{\\text{B}}(W)", "1cccd264fcb55821fc47b052ee6473bf": "\\alpha _{p}={\\begin{cases}{\\frac {1}{\\sqrt {M}}},&p=0\\\\{\\sqrt {\\frac {2}{M}}},&1\\leq p\\leq M-1\\end{cases}}\\qquad \\alpha _{q}={\\begin{cases}{\\frac {1}{\\sqrt {N}}},&q=0\\\\{\\sqrt {\\frac {2}{N}}},&1\\leq q\\leq N-1\\end{cases}}", "1ccd30acf2008c6f39d121aa6d39a4da": "G(i\\omega _{n})=G_{ii}(i\\omega _{n})=\\sum _{k}{\\frac {1}{i\\omega _{n}+\\mu -\\epsilon (k)-\\Sigma (k,i\\omega _{n})}}", "1ccd391c7a32c629fbd0f1374d5f869f": "{\\mathit {n}}\\times {\\mathit {n}}", "1ccd53a382f5e10570851976edcd9dfe": "\\mathrm {bind} :(E\\rightarrow \\mathrm {M} \\,A)\\rightarrow (A\\rightarrow E\\rightarrow \\mathrm {M} \\,B)\\rightarrow E\\rightarrow \\mathrm {M} \\,B=m\\mapsto k\\mapsto e\\mapsto \\mathrm {bind} \\,(m\\,e)\\,(a\\mapsto k\\,a\\,e)", "1ccd8cbc9f5a7e9d2ffc3c999affc700": "[t_{0}..t_{1}]", "1ccd900875bba9c7d559acdee0b4fb9d": "t_{1}\\dots t_{k}", "1ccda985f3150fb4bd15f727a19c992e": "1/{\\sqrt {2\\pi }}", "1ccdd092bfc31bc1293527c0b16cd7be": "\\ln R=B/T+\\ln r_{\\infty }", "1ccde18c954e788ef8622714daeee0cd": "V_{n}^{p}(R)={\\frac {(2\\Gamma ({\\frac {1}{p}}+1)R)^{n}}{\\Gamma ({\\frac {n}{p}}+1)}}.", "1cce07fd798a0442264d36d53e24ad74": "j=l+1/2", "1cce16180bd8804d26c6edbf9a6e23e2": "{\\widehat {J}}_{j}(1)", "1cce9bf6cac32b87a3bbbdc7d0d05e72": "A(x,t)=A_{0}\\left(1-[1-{\\Lambda }(t)]{x \\over {L}}\\right)^{2}", "1ccea00c097b218c202eaa8e7271fb1c": "M_{p}", "1cceb86f8c64f5d0a71ea5c16f51c8c8": "\\alpha =3-{\\sqrt {2}}", "1cceec254196e85c5f3f8122ee8d5ef9": "10_{3}", "1ccf05ad152eb0c5c90f509a2c2e762d": "\\nabla _{v}w", "1ccf3aafa41fa9e9817790bdb6bbfe21": "\\textstyle i\\neq j", "1ccf4de9a09566ed93bfc633441c2906": "\\Delta E=p\\Delta V\\,\\!", "1ccf6ee6b96969b20880ea6f7d4f6057": "a\\wedge S\\wedge a", "1ccf9d49b2635c61d18bb9d4195ed51e": "\\Phi \\left(x\\right)=\\left(x+\\mathrm {Ker} F_{i}\\right)_{i\\in I}", "1cd04755bef80e06c4f019e1df39705f": "{\\frac {\\sin ^{2}\\theta }{\\cos ^{2}\\theta }}+{\\frac {\\cos ^{2}\\theta }{\\cos ^{2}\\theta }}={\\frac {1}{\\cos ^{2}\\theta }}\\!", "1cd0542b8a4213c3651d6382bc564e72": "\\alpha \\,=\\,\\sum _{j=1}^{n}\\,{\\frac {\\sigma -\\sigma _{e}}{\\sigma _{e}+L_{j}(\\sigma -\\sigma _{e})}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(3)", "1cd09da875db2eb61ac45ea99f671405": "\\gamma _{xy}={\\frac {\\partial u_{x}}{\\partial y}}+{\\frac {\\partial u_{y}}{\\partial x}}\\,\\!", "1cd0ff1e99d9c944d46c03ef6f61bafd": "\\displaystyle {\\dot {q}}^{1},\\,\\ldots ,\\,{\\dot {q}}^{n}", "1cd11c858b4834a5be54b0f8437a042b": "{\\boldsymbol {\\alpha }}'={\\boldsymbol {\\alpha }}", "1cd1a221fbeeda54d8a08614860cd1db": "z\\in E-\\{y\\}", "1cd1e2e06eb41e3936b9217aa690b7f2": "\\Box \\pi _{1}", "1cd1ff934752811a90d7e66379f28338": "\\sigma _{1}(A)\\geq ...\\geq \\sigma _{n}(A)", "1cd20690433dfc37b3cbb0f8684ac1c2": "f^{1}(\\theta )", "1cd23684becc903ec5c255244fe6a199": "p_{1},\\cdots ,p_{n}", "1cd25a5fc2fb9ebee169900e0e98606a": "p\\leftarrow {\\hbox{not }}q", "1cd2d9d41e52e99601b1f76c468d1750": "u=\\left[{\\begin{array}{c}u_{0}\\\\u_{1}\\\\...\\\\u_{Ne}\\\\\\end{array}}\\right]", "1cd322f30a6aeac3cb47ee07ace5baa2": "{\\frac {a_{0}}{2}}+\\sum _{n=1}^{\\infty }\\left\\{a_{n}\\cos(nx)+b_{n}\\sin(nx)\\right\\}.\\,", "1cd326ed099fd1fa2a8dfd1040bc0bef": "\\lambda ^{2}-2\\lambda \\cosh({\\sqrt {r}})\\cos({\\sqrt {r}})+1=0.", "1cd36e87b5c3913d05abfabc7e1b903d": "{\\mathbf {r} }_{0}", "1cd3b99a075c7bbe95dff2598110f6e7": "\\varphi (t;\\mu ,c)=\\exp \\left[~it\\mu \\!-\\!|ct|^{\\alpha }~\\right],", "1cd3c693132f4c31b5b5e5f4c5eed6bd": "pk", "1cd3ca12f87a9155303b836c586f501c": "\\forall n\\in {}^{*}\\mathbb {N} ,{}^{*}\\!\\!\\sin n\\pi =0", "1cd3efe5e1b77b80ad181cb5e4f1ebf3": "{\\vec {r}}(t+dt)={\\vec {r}}(t-dt)+{\\vec {v}}(t+{\\frac {dt}{2}})\\,dt", "1cd3f31c8b808dd3a48192c5d374370e": "{\\dot {\\mathbf {q} }}={\\frac {d\\mathbf {q} }{dt}}~.", "1cd424f15e3be6952346fa87e390dbf2": "k=({\\vec {N}}\\cdot {\\vec {H}})^{n}=({\\vec {N}}\\cdot (({\\vec {L}}+{\\vec {E}})/2))^{n}=({\\vec {N}}\\cdot ((\\{-0.6+{\\frac {\\sqrt {3}}{2}};\\;0.8+0.5;\\;0+0\\})/2))^{3}=({\\vec {N}}\\cdot ((\\{0.266;\\;1.3;\\;0\\})/2))^{3}=", "1cd4373a3d973f6625a8be848b1f5202": "\\operatorname {var} (X)=\\mu ^{2}{\\frac {\\nu +L+1}{L\\nu }}.", "1cd46ed590b2194ee3e1f91fc947b976": "\\scriptstyle {D(ab)=D(a)b+(-1)^{|a|}aD(b)}", "1cd47fc412c528599d95b10f1d859e56": "P\\sim |p-p_{c}|^{\\beta }\\,\\!", "1cd487965b1ed24da3ec7d8b94f777f5": "X\\subseteq \\mathbb {R} ^{n}", "1cd4b5fcad87d31f6b59dd482f5ac8cb": "[u^{m}]g(z,u)|_{u=1/u}|_{z=uz}=[u^{m}]\\left({\\frac {1}{1-z}}\\right)^{u}={\\frac {1}{m!}}\\left(\\log {\\frac {1}{1-z}}\\right)^{m},", "1cd4d0951222a1c6d57e05ef3c64e449": "m{\\ddot {x}}(t)=-Kx(t)-\\mathrm {B} {\\dot {x}}(t)", "1cd60907f570de5a87b8908eeac809a6": "p\\in \\mathbb {C} ", "1cd64c21a3b2e95af070ab72218fa049": "4x^{2}+16-16x", "1cd65ff01699dd11eb3dfe5b54ead3c7": "f\\,", "1cd7507f4d63278c051fdc6d44cc320c": "\\beta _{s}", "1cd75e09ec880cf6ee2951b369365b9e": "P(R_{i})", "1cd77b2bb17c91f1dd5cbcde6618fad7": "\\rho (\\tau _{s})\\propto \\exp \\left\\{-{\\frac {\\left(\\tau _{s}-2.1s\\right)^{2}}{4s}}\\right\\}.", "1cd7ec5fde8e5c0801bcac12f73c28c8": "{\\frac {3}{5}}N", "1cd82b14c99e016b291c66efbdd309fa": "(-\\infty ,\\infty )", "1cd885cedacc70751adbbef0c940749d": "\\gamma ={\\frac {\\lambda _{PN}-\\lambda _{NN}}{(\\lambda _{NP}-\\lambda _{NN})-(\\lambda _{PP}-\\lambda _{PN})}},", "1cd8c453b83f2419b13b01d9265fd7a4": "|[0,1]|=|{\\mathcal {P}}(\\aleph _{0})|\\,", "1cd8ee2a55797ac390dc1156e40d05bb": "{\\hat {f}}(\\nu )={\\mathcal {F}}\\{f(x)\\}.", "1cd9138c1db89e13e94da8a5016fafcc": "\\omega _{B}={\\sqrt {\\frac {1}{L_{B}C_{B}}}}={\\frac {\\alpha ^{2}}{2}}\\cdot {\\frac {m_{0}c^{2}}{\\hbar }}={\\frac {2\\pi c}{\\lambda _{B}}},\\ ", "1cd91ae4b965b62ef44654d90c159fde": "\\textstyle (E_{1}-E_{2})=0", "1cd92fb7962b7d320479e660c6906598": "{\\sqrt {2}}X(0)={\\sqrt {\\frac {h}{2\\pi }}}\\;{\\begin{bmatrix}0&{\\sqrt {1}}&0&0&0&\\cdots \\\\{\\sqrt {1}}&0&{\\sqrt {2}}&0&0&\\cdots \\\\0&{\\sqrt {2}}&0&{\\sqrt {3}}&0&\\cdots \\\\0&0&{\\sqrt {3}}&0&{\\sqrt {4}}&\\cdots \\\\\\vdots &\\vdots &\\vdots &\\vdots &\\vdots &\\ddots \\\\\\end{bmatrix}},", "1cd94bf033c877c13c30af03158c5785": "B^{2}X_{t}=X_{t-2}\\,,", "1cd99a20e031887eddfbbf0f13dc167f": "A+n\\to B^{*}+\\gamma ", "1cd99bd071e595c0f0d56fad91ad578f": "|G|^{2}", "1cd9dcadee7afefb384b421277f67489": "(\\neg A\\to C)\\to ((B\\to C)\\to ((A\\to B)\\to C))", "1cd9ddec625f256fa0eb2351577c9e25": "D{\\overline {D}}", "1cd9e0326aabfbdf163502d4ba0fa2ce": "\\Delta y_{t-k}", "1cdaad81bd3a9b5bcc8062a2740a125a": "\\int _{0}^{a}\\sin {\\frac {m\\pi x}{a}}\\,{\\text{d}}x={\\frac {a}{m\\pi }}(1-\\cos m\\pi )\\quad {\\text{and}}\\quad \\int _{0}^{b}\\sin {\\frac {n\\pi y}{b}}\\,{\\text{d}}y={\\frac {b}{n\\pi }}(1-\\cos n\\pi )\\,.", "1cdabbbc5aeb71136feb65014a5f23b1": "{\\mathcal {H}}\\left(\\mathbf {q} ,\\mathbf {p} ,t\\right)=\\mathbf {p} \\cdot {\\dot {\\mathbf {q} }}-{\\mathcal {L}}\\left(\\mathbf {q} ,{\\dot {\\mathbf {q} }},t\\right)\\,,", "1cdabff88cce339aff58dc31d64699fc": "2^{\\mathfrak {c}}=\\beth _{2}>{\\mathfrak {c}}", "1cdac79582f2f6beac46655f65887e55": "\\scriptstyle \\tan(\\alpha /2)=\\sin \\alpha /(1+\\cos \\alpha )", "1cdaef02128c2d5b400d891252f604c9": "\\textstyle {\\binom {n}{k}}={n! \\over (n-k)!k!}", "1cdbc922ee64437602c70cdc96536629": "\\times \\!\\,", "1cdbe8a4b1159a7d40b486c20d070fcf": "n+m", "1cdbf789246d6f65618b03de148e4be0": "H(s)={\\frac {P(s)}{Q(s)}}={G\\cdot \\displaystyle \\sum _{m=0}^{M}{b_{m}s^{m}} \\over s^{N}+\\displaystyle \\sum _{n=0}^{N-1}{a_{n}s^{n}}}", "1cdc33bffa0035bd9e28d912e751902d": "{\\mathcal {L}}_{V^{r}}(\\theta )", "1cdc759be8124ad9e6a8740f89d90387": "[\\mathrm {j} _{k},\\mathrm {j} _{l}]\\equiv \\mathrm {j} _{k}\\mathrm {j} _{l}-\\mathrm {j} _{l}\\mathrm {j} _{k}=i\\hbar \\sum _{m}\\varepsilon _{klm}\\mathrm {j} _{m},\\quad \\mathrm {where} \\quad k,l,m\\in (x,y,z)", "1cdcf69d432207336eb5b5adb0592c43": "K\\in F_{n}", "1cdd0149b9bb34251d059c4e5377c674": "\\lambda /(1+\\lambda )", "1cdd3c61a09d223e435dcfd791327ecd": "B\\cap C=\\emptyset ", "1cdd6d84f016b0d8e0adc923266b2347": "\\varphi (x)\\sim {\\sqrt {Z}}\\varphi _{\\mathrm {in} }(x)\\quad \\mathrm {as} \\quad x^{0}\\rightarrow -\\infty ", "1cddc5b19a6d4bd574a44601c2a9f48d": "\\lambda \\phi (x)-\\int _{a}^{b}K(x,y)\\phi (y)\\,dy=f(x).", "1cddcff75aed7e23f1722644f199fc05": "(x,y,z)\\mapsto (x-2(xz+y^{2})y-(xz+y^{2})^{2}z,y+(xz+y^{2})z,z)", "1cdde264f9d885c0538c0ca09d62c843": "u(t,x,y,z)={\\frac {t}{4\\pi }}\\iint _{S}\\varphi (x+ct\\alpha ,y+ct\\beta ,z+ct\\gamma )d\\omega ,\\,", "1cde93af7086982a4107c892bf2e6830": "\\scriptstyle {{\\widetilde {f}}(x)=f(-x)}", "1cdf190afc8d25133df9bd083884d909": "2^{r-1}", "1cdf461d8cadd88aa7b166761ea3a149": "(U;z^{A},y^{a})", "1cdfe650245cf3a6f5fb647a5913f16c": "d(x,z)\\leq d(x,y)+d(y,z).", "1ce015ccf21b95ea19abc8fedd03b2e0": "S_{N}(x)", "1ce030c95233754dc5c8d41c864d6a7f": "\\displaystyle {{\\widehat {\\delta _{h}f}}(m,n)=h^{-1}(e^{-ihn}-1){\\widehat {f}}(m,n)=-\\int _{0}^{1}ine^{-inht}\\,dt\\,\\,{\\widehat {f}}(m,n).}", "1ce03cac9d873314495641b5c46d455a": "M=EI\\kappa =EI{\\frac {\\mathrm {d} ^{2}w}{\\mathrm {d} x^{2}}}", "1ce083a8c349416531d476ac432d5ed2": "\\mathbf {k} \\cdot \\mathbf {x} -k_{0}t", "1ce09f0913db43ca57cccaebc264976f": "\\tan \\delta ={\\frac {\\sum _{i}a_{i}\\sin \\delta _{i}}{\\sum _{i}a_{i}\\cos \\delta _{i}}}.", "1ce0c055b6c21f755cd037310fd10439": "n_{1}=1/2,n_{2}=-1/2,n_{3}=1/2.\\ ", "1ce0cdaf66b9bf4b74d1f118c30080ef": "{\\overline {X}}", "1ce0fdd7f80fafc38fda2607a030f179": "\\scriptstyle b(k):\\mathbb {N} \\rightarrow \\mathbb {R} ", "1ce1124a62c94540c3fa52466204221f": "E_{\\infty }", "1ce11b33802040e197b34bf394f8ac6e": "\\langle \\psi \\psi ^{*}\\rangle =\\langle {\\mathcal {T}}_{\\tau }\\psi (\\tau =0^{+})\\psi ^{*}(0)\\rangle =-G_{\\eta }(\\tau =0^{+})=-{\\frac {1}{\\beta }}\\sum _{i\\omega }G(i\\omega )e^{-i\\omega 0^{+}}", "1ce17f52a3da9f0cac5c05d849bea323": "({\\mathcal {X}},{\\mathcal {A}})", "1ce18e9eb07124ee80c9fcc310058e3f": "W_{c}=\\int \\limits _{0}^{\\infty }e^{A\\tau }BB^{T}e^{A^{T}\\tau }d\\tau ", "1ce1c81e90d06b8d11e838885ed1e381": "270^{o}", "1ce2098b2c9422a038ff3dda1691d035": "\\sum _{\\gamma \\in \\Gamma }H(\\gamma (z)).", "1ce214a6276be7d69ebeb3fb8bc82c94": "p=S_{kl}", "1ce215ed1f4788e6e001d8966093ba78": "t_{l}={\\bar {t}}", "1ce21b9bfb0496c711676169d9203a41": "e\\times m=\\psi .", "1ce2400a58ddd50f4f1d738e2e9664e8": "H_{i}=\\{y\\in (\\mathbb {F} _{2})^{d}\\mid y_{i}=0\\}", "1ce240eb4d6eb4e7a06d67f0451f463b": "\\sigma ^{*}=\\left({\\frac {1}{2.48\\rho ^{*}}}\\right){\\Bigg [}\\log \\left({\\frac {k_{s}}{k_{\\text{CH3}}}}\\right)_{B}-\\log \\left({\\frac {k_{s}}{k_{\\text{CH3}}}}\\right)_{A}{\\Bigg ]}", "1ce253abfa7195d2847feed751306a4a": "c=15.2", "1ce2681b830182c7dbf9cb8f2c253ba7": "{\\mathcal {A}}=\\sum _{j+k\\leq n}a_{jk}\\partial _{x}^{j}\\partial _{y}^{k}={\\mathcal {L}}\\circ \\sum _{j+k\\leq (n-1)}p_{jk}\\partial _{x}^{j}\\partial _{y}^{k}", "1ce27fbc45637312633e0d319aa4c219": "c_{p}={\\frac {\\Omega (k)}{k}}\\,", "1ce2957d5c7ae9fec475b8c46433f3f0": "\\ O(kg)=O(g)", "1ce2ab1565da9257186d6d781fa5d2ae": "\\scriptstyle I_{L_{\\text{max}}}", "1ce2af8f7af2409385d6d6a0194375f3": "P=(-1:-1:1)", "1ce2c059f82c35a93a738da9c6384e0d": "M_{n,R}", "1ce2e225eacd40e6f260535b3a26bf7d": "\\tau _{nuc}={\\frac {\\mbox{total mass of fuel available}}{\\mbox{rate of fuel consumption}}}\\times {\\mbox{fraction of star over which fuel is burned}}={\\frac {MX}{\\frac {L}{Q}}}\\times F", "1ce2e339cb54454547e3e7fb1ca12b16": "{\\mathcal {B}}_{t}=(\\lfloor n(t+1)\\rfloor )_{n\\geq 1}", "1ce2fb3c0bc43df43b8fdf58ae46ec05": "f(x)=b^{x}.\\,", "1ce3eba5df9f486625aea47d3b77c63e": "{\\bar {w}}={\\sum _{i=1}^{n}{p_{i}w_{i}}}~~~~~~~~~~(2)", "1ce409f8edd134fc37e504bc10fdf81d": "\\mathrm {Ro} ={fL^{2} \\over \\nu }=\\mathrm {St} \\,\\mathrm {Re} ", "1ce4300a2940e7ecadebdfd13ebd3690": "\\operatorname {Re} (\\lambda _{k})=\\operatorname {Re} \\left(1+\\alpha _{-2}e^{i2\\pi k(N-2)/N}+\\alpha _{-1}e^{i2\\pi k(N-1)/N}+\\alpha _{1}e^{i2\\pi k/N}+\\alpha _{2}e^{i4\\pi k/N}\\right)", "1ce447f7f87af00c7f626731ce643be9": "\\mathbf {x} =(x,y,z)", "1ce45f59098ba4a7ed6bf15c314b711a": "{\\bar {Q}}_{7}", "1ce480cea2943b84c93dc05886f859d7": "\\vdash _{\\vec {s}}", "1ce4d16bb9365ec9bee39910f6435e15": "\\sum _{i=0}^{j-1}{\\frac {1}{s_{i}}}=1-{\\frac {1}{s_{j}-1}}={\\frac {s_{j}-2}{s_{j}-1}}.", "1ce4f6cc0332b04d06101ea472baf7c1": "u={\\frac {\\gamma }{2}}x^{2}-{\\frac {\\gamma ^{2}}{2}}\\left({\\frac {x^{5}}{5!}}\\right)+{\\frac {11\\gamma ^{3}}{4}}\\left({\\frac {x^{8}}{8!}}\\right)-{\\frac {375\\gamma ^{4}}{8}}\\left({\\frac {x^{11}}{11!}}\\right)+\\cdots ", "1ce5140c3855f1bcf480d6c70594b877": "k={\\frac {t_{r}-t_{M}}{t_{M}}}", "1ce5576158df1955460f5f4a96dacee0": "\\eta _{1},\\ldots ,\\eta _{b}", "1ce5630bcb590535a9327bcbd3865a51": "S(P,f,g)=\\sum _{i=0}^{n-1}f(c_{i})(g(x_{i+1})-g(x_{i}))", "1ce57d11f9b40e15fff91bec1c6ca43d": "D0", "1ceeef23130c31175b2c9cd34cce205e": "{\\mathcal {N}}(\\mu _{1},\\sigma _{1}^{2})", "1cef9417759ba8bf07cb1050433cebb6": "T_{V,0}", "1cefaeb1fd125eec86f7ad85dde4239b": "{\\pi \\over 3}\\ {\\pi \\over 5}\\ {4\\pi \\over 5}", "1cefb365e65368c8ac7f843f4de4aa4a": "x=\\int _{0}^{\\infty }v(t)\\,{dt \\over t},", "1ceff20a10441bedd341aaa90dd541ec": "L/T^{2}", "1ceff5f76263a1bb77621183fea19c9a": "V_{i}=({\\bar {n}}-1)/(n_{F}-n_{C})", "1cf01d999318a69d5b7aa91bf285193e": "P_{2}+P_{3}=2+5=7", "1cf04dd62248019272dedd8f3a473b34": "C_{\\text{sr}}", "1cf0d7a19e485e062479c0764f4b172c": "\\pi _{\\mathbf {Q} }\\colon {\\mathbf {Q} }\\to M\\,", "1cf115f1bbe885df6427c6c712a48ab0": "(6,10,3)", "1cf135419cd9d294d0eb7d895ff91656": "{\\star {\\mathbf {J}}}=-\\rho dt+j_{x}dx+j_{y}dy+j_{z}dz", "1cf1819449da0dda035ac5a0b98e214a": "[S^{2},S_{i}]=0,\\quad [J^{2},J_{i}]=0", "1cf18829a73835a00db8c2035cf3cb94": "V_{D}\\approx 0.30V", "1cf1c03e9683d9d613db148f1f8d6c85": "\\operatorname {Pr} (Y_{i}=y_{i}\\mid \\mathbf {X} _{i})={p_{i}}^{y_{i}}(1-p_{i})^{1-y_{i}}=\\left({\\frac {1}{1+e^{-{\\boldsymbol {\\beta }}\\cdot \\mathbf {X} _{i}}}}\\right)^{y_{i}}\\left(1-{\\frac {1}{1+e^{-{\\boldsymbol {\\beta }}\\cdot \\mathbf {X} _{i}}}}\\right)^{1-y_{i}}", "1cf1c96acf610a8d1b74c56ccc34f670": "\\mathbf {o} ", "1cf1d91afafb2698aafb9325fb3093f2": "F_{3}(p,Q)={\\frac {p}{Q}}", "1cf23d2e5df5bb09142dda7c9d61b5c8": "Z_{\\Lambda }^{\\Phi }({\\bar {\\omega }})", "1cf27ed6cc67f92bb14ae5d4259c671d": "{\\mathcal {S}}_{n}({\\mathcal {B}}_{1\\cdots n},{\\mathcal {D}}_{1\\cdots n})", "1cf29347652ea8d52429ea825172febc": "\\beta _{k}:M_{n_{k}}(C(\\mathbb {T} ))\\;\\rightarrow \\;M_{n_{k+1}}(C(\\mathbb {T} )).", "1cf2ef7480a1663c36331c9739b7d053": "\\left\\langle {\\tilde {H}}\\right\\rangle =\\left\\langle H\\right\\rangle \\,", "1cf326e9182840bfece8444098a54864": "=([2.6m+7.8-2.6m+0.2])\\ {\\bmod {\\ }}7", "1cf34e5dd31d2479b54e1cb71f245068": "d_{1}={\\frac {4\\alpha (2-\\alpha )}{(1+\\alpha )(3-\\alpha )}},\\qquad \\alpha ={\\frac {MG}{2Rc^{2}}}", "1cf36eb340053a174f7e7790b097e17b": "\\Phi (\\tau +16)\\,", "1cf3e126f8b3c6d59a6779a879a2b87a": "g_{i}(\\mathbf {x} ,z_{1},\\ldots ,z_{k})\\neq 0", "1cf3fb150f32e804edce72891ef1e672": "f(z)=\\sum _{n=-\\infty }^{\\infty }a_{n}(z-c)^{n}", "1cf4387829ed522a24d38654e5dd5c47": "\\nabla _{\\mu }T_{M\\;\\nu }^{\\;\\mu }={\\frac {1}{8\\pi }}{\\frac {\\nabla _{\\nu }\\phi }{\\phi }}\\Box \\phi ,", "1cf455285d2e7a4b60c283029308a6f0": "\\deg(h^{2})\\leq 2g", "1cf516c2f98b957b6da0c5797c929279": "{\\mathfrak {q}}_{1}\\subseteq {\\mathfrak {q}}_{2}\\subseteq \\cdots \\subseteq {\\mathfrak {q}}_{m}", "1cf539d6135f019489aa2eea29490b0f": "H^{\\bullet }(X;R)=\\bigoplus _{k\\in \\mathbb {N} }H^{k}(X;R).", "1cf56c97d4c0fa4d8bdddb3e2a46f76b": "\\Omega _{m}=0.22,\\,", "1cf601be5b401e2e48bc6402d60a60a8": "\\zeta '(-1,x)=\\psi (-2,x)+{\\frac {x^{2}}{2}}-{\\frac {x}{2}}+{\\frac {1}{12}}", "1cf60238995c56da9882fd097c30c0bd": "{\\frac {d}{dx}}\\int _{0}^{x}t^{3}\\,dt={\\frac {d}{dx}}F(x)-{\\frac {d}{dx}}F(0)={\\frac {d}{dx}}{\\frac {x^{4}}{4}}=x^{3}.", "1cf6397b8a31655406688d7725836657": "C_{\\mathfrak {st}}^{\\lambda }C_{\\mathfrak {uv}}^{\\lambda }\\equiv \\phi _{\\lambda }(C_{\\mathfrak {t}},C_{\\mathfrak {u}})C_{\\mathfrak {sv}}^{\\lambda }\\mod A(<\\lambda )", "1cf65087c4906cbcf20b638409689c9a": "\\lim _{N\\rightarrow \\infty }p=c", "1cf65e8a140011ebf78b8454671cbdd0": "n_{0}\\equiv a\\mod m", "1cf6b25a231270f5f28700065c4e274d": "s={\\tfrac {1}{2}}(a+b+c)", "1cf725da6a7a4a5c579f48c3a99815cf": "R_{S}=R_{H}\\left(1+{\\frac {\\cos(\\theta )\\sin(\\alpha )^{2}-\\cos(\\alpha )\\sin(\\theta )\\sin(\\alpha )}{\\cos(\\alpha )\\cos(\\theta -\\alpha )}}\\right)\\sec(\\alpha )\\,", "1cf731faae065d45ebbc13e2c7c715a0": "\\sum _{n\\geq 1}\\tau (n)q^{n}=q\\prod _{n\\geq 1}(1-q^{n})^{24}=\\eta (z)^{24}=\\Delta (z),", "1cf73e6c36aae5fd4e13eec1213377ef": "O(2k)\\neq SO(2k)\\times \\{\\pm I\\}", "1cf75149b9422cba89f2425a2bcab13a": "f(x_{1},x_{2},\\dotsc )=x_{1}", "1cf7e37ff57ca733fc3b734ff6aedd27": "f(a,b)=ab", "1cf908a3f39d126231dc2a0188de1ffb": "f(n)\\leq 3n-3", "1cf93f995f6f916485e0070c15f43dec": "x\\not \\in x", "1cf967455cabaf16644f53878baaf4de": "O(f(k))+n^{3}", "1cf97c9c847a8b54b24badde907650fc": "2\\log _{2}(n+1)", "1cf991a9bdf72b1e77edd3e431aa9c1d": "E=0", "1cf9a39837609bbcd177490e800d6658": "v=r\\omega _{rad}", "1cf9af29f60beca43656e9cad0fd964f": "\\nabla _{\\beta }", "1cfa165c65aa313723b46a93744e107e": "\\langle \\tau ^{n}\\rangle =\\Gamma (n)\\int _{0}^{\\infty }d\\tau \\,t^{n}\\,\\rho (\\tau )", "1cfa5cd3d571616b47f464db167449a2": "{\\vec {h}}_{3}={\\frac {\\sqrt {1-2m/r}}{\\sqrt {1-3m/r}}}\\,\\partial _{\\phi }-{\\frac {\\sqrt {m/r^{3}}}{{\\sqrt {1-2m/r}}\\,{\\sqrt {1-3m/r}}}}\\,\\partial _{t}", "1cfa601ba3730bb8056ce65fc98279bb": "d^{2}(\\ln d)^{O(1)}n^{2}\\mathbb {F} _{q}", "1cfa671e18be131c7c0f0e60a17972c3": "F_{\\phi }(\\mathbf {r} ,\\alpha ,\\beta )=-{\\frac {3\\mu _{0}}{4\\pi }}{\\frac {m_{2}m_{1}}{r^{4}}}\\sin(2\\phi -\\alpha -\\beta )", "1cfab2140efbdd5af94682fc6fc2aa66": "\\int _{I}\\rho (\\gamma (t))\\,d{\\mathrm {length} }_{\\gamma }(t)", "1cfabe79d22d83569089441eabba1408": "(\\leq ):R\\times R\\rightarrow {\\mathbb {B}}", "1cfaea59874dead9782c6df0b761ce7f": "u\\left(x\\right)=x_{1}+\\theta \\left(x_{2},...,x_{L}\\right)", "1cfb4b62c3fabfea351cba30357798ec": "\\lim _{N\\rightarrow \\infty }P_{N}({\\overline {R}})=2N{\\overline {R}}\\,e^{-N{\\overline {R}}^{2}}.", "1cfba28a1a5c4897824baaa5a083c448": "\\sigma {\\dot {\\sigma }}\\leq -\\mu |\\sigma |^{\\alpha }", "1cfc07378c97b94a5a1786150e83076a": "\\mathrm {Ann} _{R}(N)\\,", "1cfc0aa03d7aa8ce83a4ea0c373b971d": "\\lim _{n\\to \\infty }{\\frac {f(n)}{g(n)}}=1", "1cfc931107decdbed8a865001aba9843": "{\\frac {V_{dd}}{2}}", "1cfd00088c529e6d258215ef50d7e59f": "\\scriptstyle \\sim ", "1cfd4b2437b9e4a2ff0a7f36ba5d578e": "\\scriptstyle A\\,\\!", "1cfd9accc927798abb4d608bbaa80325": "\\ln(-\\eta )-\\ln k", "1cfe28c7956121e2b307e8b1835de656": "J_{ij}>0", "1cfea071e94b76c58b5358e49570bea7": "\\tau ={VQ \\over It},", "1cfea84fb23cd83eeea02873f4f9b9af": "Q_{t}(W_{t})", "1cff150e60dc33dbf91b52a000ce1d34": "\\epsilon _{p}\\,\\!", "1cff204ad548756e580ee0236dc7b24c": "p(k)", "1cff66606d5aa45c44413e71941c95ca": "D_{i}={\\frac {({\\hat {\\beta }}-{\\hat {\\beta }}^{(-i)})^{T}(X^{T}X)({\\hat {\\beta }}-{\\hat {\\beta }}^{(-i)})}{(1+p)s^{2}}}.", "1cff6d11b4f6770f15830df4a34a59fb": "R=VSV^{-1},\\,", "1cff7f1273ec23bab95cf613aa13f66b": "\\ln {\\frac {q_{i}}{p_{i}}}={\\frac {q_{i}}{p_{i}}}-1", "1cff9b06913ce7a14a720da1e7bdf753": "R={\\frac {r_{\\mathrm {o} }}{\\sin(\\beta _{\\mathrm {o} })}},", "1d000b5118c83dec82d15e759fe7fe49": "\\sum _{p}\\nu _{p}(n)=\\Omega (n).\\;", "1d005c6c722588708966f5fd576a1b7b": "C_{01}=C{11}=0", "1d005cb5fe95e6315c899350ce8635e4": "v={\\sqrt {rg\\left(\\sin \\theta -\\mu _{s}\\cos \\theta \\right) \\over \\cos \\theta +\\mu _{s}\\sin \\theta }}={\\sqrt {rg\\left(\\tan \\theta -\\mu _{s}\\right) \\over 1+\\mu _{s}\\tan \\theta }}", "1d0070647c4f8e37197c120d1211e0e3": "E'\\rightarrow {\\mbox{Div}}^{0}(E')\\to {\\mbox{Div}}^{0}(E)\\rightarrow E\\,", "1d00b3fea8988ffc59fb7df622e2893a": "P(X_{1}=X_{2})", "1d00d7385620b9bab21da659cd57717b": "\\cos \\theta ={\\frac {g_{ij}U^{i}V^{j}}{\\sqrt {\\left|g_{ij}U^{i}U^{j}\\right|\\left|g_{ij}V^{i}V^{j}\\right|}}}.", "1d00e7dce692e8dc3f6877f035e3a616": "OR", "1d01486ae6512712dd2e838821363980": "c(c-1+\\gamma )=0.", "1d0189a358bb5596d2094f68f8802358": "\\phi ::=p|\\neg p|\\phi \\lor \\phi |\\phi \\land \\phi |{\\mathcal {P}}_{\\sim \\lambda }(\\phi {\\mathcal {U}}\\phi )|{\\mathcal {P}}_{\\sim \\lambda }(\\square \\phi )", "1d01a98ba02aaa48fcd47b7149ac6222": "\\,C\\bullet D:=\\sum _{r,s}\\langle \\langle C\\rangle _{r}\\langle D\\rangle _{s}\\rangle _{|s-r|}", "1d02078fc5d3856634ab73a252bfca84": "c_{i}^{fit}", "1d021fb48c03fc0011d4a58820b0a984": "T_{0}T_{p}M\\cong T_{p}M", "1d022b8a2e3fee501dd4e3f6f9d2b779": "m{\\ddot {\\mathbf {r}}}\\cdot \\delta {\\mathbf {r}}=m\\sum _{j}\\left[\\sum _{i}\\left[{\\mathrm {d} \\over \\mathrm {d} t}\\left({\\dot {r_{i}}}{\\partial {\\dot {r_{i}}} \\over \\partial {\\dot {q_{j}}}}\\right)-{\\dot {r_{i}}}{\\partial {\\dot {r_{i}}} \\over \\partial q_{j}}\\right]\\right]\\delta q_{j}", "1d024df1d507b2f709e2eaa6552e442e": "\\rho (\\mathbf {k} ,x)", "1d029df1f7a0d87a1f3ea15adb0d001b": "\\{(x_{1},y_{1}),\\dots ,(x_{n},y_{n})\\}", "1d02c143ba4ec966a7581562630c547f": "B={\\frac {\\ell b^{2}+mb+n}{(b-c)(b-a)}};", "1d031f5fd5b71ebb7a40e6420eedd760": "{\\dot {x}}={\\frac {\\mathrm {d} x}{\\mathrm {d} t}}=v", "1d039297310d25a38770a0bb01d5273b": "{x}(t+1)\\approx \\varphi (t)=\\varphi [\\varphi (t-1)]", "1d039423b51ea148024ae30fe89fc0eb": "y=a\\ {\\frac {\\sin \\sigma }{\\cosh \\tau -\\cos \\sigma }}", "1d03a3efae4a4440309e2476cd68fea0": "\\Delta =e^{D}-1=\\sum _{k=1}^{\\infty }{\\frac {D^{k}}{k!}}.", "1d03eb69a5b982ca0d68147841c973ed": "\\scriptstyle f_{\\text{partial}}(2,3)", "1d041058b22e4094e4be1b2a3e6e4653": "\\zeta ^{\\prime }(a,z)\\ {\\stackrel {\\mathrm {def} }{=}}\\ \\left[{\\frac {d\\zeta (s,z)}{ds}}\\right]_{s=a}.", "1d046cdeb1d2a0b59685c72cc5e7cf1e": "\\gamma ={\\frac {\\omega F}{\\sqrt {2E_{i}}}}", "1d0486297fc1f0519981865d661ae52c": "{\\begin{bmatrix}0&-0.80&-0.60\\\\0.80&-0.36&\\;\\;\\,0.48\\\\0.60&\\;\\;\\,0.48&-0.64\\end{bmatrix}}\\qquad \\left({\\begin{aligned}&{\\text{rotoinversion:}}\\\\&{\\text{axis }}(0,-3/5,4/5),{\\text{ angle }}90^{\\circ }\\end{aligned}}\\right)", "1d04ab28a442296ed5b91981b3f24e34": "SV=EDV-ESV", "1d04b87647563f698b40f902bf3be219": "{\\mathbf {A}}=A{\\mathbf {\\hat {n}}}", "1d04bee1e6333486c62cd23320def217": "\\delta V_{xc}[\\rho ](\\mathbf {r} )={\\frac {\\delta V_{xc}[\\rho ]}{\\delta \\rho }}\\delta \\rho =f_{xc}(\\mathbf {r} t,\\mathbf {r'} t')\\delta \\rho (\\mathbf {r'} )", "1d04edc71ddab1778583bcdd223eee54": "\\{P(x),\\neg P(c)\\}", "1d053bb69a72439e60cad0ca3d48fa56": "{\\mathit {STOP}}", "1d0554c602996b48acdc2601cedceb15": "{\\hat {\\theta }}_{n}\\ \\xrightarrow {p} \\ \\theta _{0}", "1d0562814a2d1571b3917d1b3bd4cbeb": "V_{y}=V\\cos \\theta ", "1d058e0b16481515f9584d013188fe7d": "m_{\\mathrm {p} }\\,", "1d05ad00cd73c54cd23c63c6fb28cf7b": "A(H)", "1d05c5dab7063bff697b15f0134444dd": "\\mathbb {R} ^{mn}", "1d06aa5b77fd46ae43227dca7b73ccc3": "\\mu _{5}=\\kappa _{5}+10\\kappa _{3}\\kappa _{2}\\,", "1d06ab96b1bc72408795689983f7ad71": "\\gamma _{BY}={\\frac {1}{2\\pi \\alpha }}\\ ", "1d06ae8429d08eb73ceb64a7bebd8466": "2+log(w)/log(2)", "1d07081b4d542b7be7aefc5b32035e72": "\\alpha (t_{0})=p;\\,", "1d072ef3ecb054838e1abc7d9db1e8f7": "I_{0}(\\lambda )", "1d07af0d67adc3a5d5615df1cb4e188c": "B_{3}=\\langle b_{3}^{*},b_{3}^{*}\\rangle ={\\begin{bmatrix}{\\frac {-6}{14}}\\\\{\\frac {9}{14}}\\\\{\\frac {-3}{14}}\\end{bmatrix}}{\\begin{bmatrix}{\\frac {-6}{14}}\\\\{\\frac {9}{14}}\\\\{\\frac {-3}{14}}\\end{bmatrix}}={\\frac {126}{196}}={\\frac {9}{14}}", "1d087e01ccfd95fd1bb1b1c8988b6200": "H^{(\\lambda )}(X)=\\sum _{m=1}^{n}\\left[\\prod _{\\kappa =0}^{\\lambda -1}(\\alpha _{m}-s_{\\kappa })\\right]^{-1}\\,P_{m}(X)\\ .", "1d08abebe95ee454cff00e30ec2ed04a": "F[x,y]=\\int _{E}xy\\,dt", "1d08ba687006178661bd7a7400347034": "{\\vec {r}}(\\theta ,\\phi )=(\\cos \\theta \\sin \\phi ,\\sin \\theta \\sin \\phi ,\\cos \\phi ),\\quad 0\\leq \\theta <2\\pi ,0\\leq \\phi \\leq \\pi .", "1d08e9b8066e77b4d3df6437d07ab1dd": "0={\\frac {\\mathrm {d} ^{2}\\psi }{\\mathrm {d} \\eta ^{2}}}+({\\frac {2mEl^{2}}{\\hbar ^{2}}}-{\\frac {2m^{2}gl^{3}}{\\hbar ^{2}}}-{\\frac {2m^{2}gl^{3}}{\\hbar ^{2}}}\\cos(\\eta ))\\psi ", "1d08edf206479de251ec1b3ac29d3970": "\\cos \\theta _{xy}={\\frac {\\langle x,y\\rangle }{\\|x\\|\\|y\\|}}.", "1d08eef584c96ec937cbef62be2b5b18": "X_{m,\\delta }(i)={x_{i},x_{i+\\delta },x_{i+2\\times \\delta },...,x_{i+(m-1)\\times \\delta }}", "1d0903a376aeda2858f95a2b34e21d77": "{\\begin{pmatrix}1&2&3&\\cdots &n\\\\1&2&3&\\cdots &n\\end{pmatrix}}.", "1d09105b167b9bbf3528f5e7f1fcb6ac": "\\textstyle a\\mapsto \\sum f(a_{i})g(b_{i}).", "1d099d6fa0cc669495c7140ec758a3f0": "\\Delta \\tau _{xy}^{k}={\\begin{cases}Q/L_{k}&{\\mbox{if ant }}k{\\mbox{ uses curve }}xy{\\mbox{ in its tour}}\\\\0&{\\mbox{otherwise}}\\end{cases}}", "1d09bb12779c08bace7b3993b11d9a79": "Z_{r}=\\{(A,W)\\mid A(k^{n})\\subseteq W\\}", "1d09d05816c90dc60b668987aeff61df": "\\langle \\theta ,\\phi |l,m\\rangle =Y_{l,m}(\\theta ,\\phi )", "1d0a1b14ad47f6ae076ca39ae71204b5": "O(P_{r}+P_{s})", "1d0a1e229efe244255ef3a39e7f515bd": "a_{1}=b_{1}", "1d0a2997bcaed0081207b7016227b9f4": "MPGe", "1d0a4578303784dbdcbaccc5ec00389d": "V_{A}(a)+V_{B}(a)=0", "1d0a83572fe667d5ed596a2ec3173019": "\\textstyle \\sum _{i=1}^{n}x_{i}{\\hat {\\varepsilon }}_{i}=0.", "1d0aa1ecfacc5ba465979b175c8b0955": "d_{\\text{star}}", "1d0adc654ce9be12c177436514a0e366": "\\ F={\\frac {Nb}{r}}", "1d0ae0aaa70d341c07e688a300930dfc": "{\\hat {f}}(\\xi )=\\int _{\\mathbf {R} ^{n}}f(x)e^{-2\\pi i\\xi \\cdot x}\\,dx", "1d0b03be98d6579a21d0251bc91f1d03": "A_{2N}0", "1d1582046e8d36fd5ce799a30216337d": "x_{0}\\,", "1d15a403c240f27b5dcb1a6a1ccf8c24": "{\\mathcal {L}}(\\phi )={\\frac {1}{2}}(\\partial _{t}\\phi )^{2}-{\\frac {1}{2}}(\\partial _{x}\\phi )^{2}-{\\frac {1}{2}}m^{2}\\phi ^{2}-V(\\phi ),", "1d15dbeb4c94fd96cdd91e59d043eae6": "K(x,y)=\\langle \\varphi (x),\\varphi (y)\\rangle ", "1d165b280947110da81c36101e38cc3f": "{\\begin{aligned}K_{l}&={\\text{lower strike price}}\\\\K_{u}&={\\text{upper strike price}}\\\\C_{n}&={\\text{net credit per share}}\\\\N&={\\text{number of shares per options contract}}\\end{aligned}}", "1d16bb9e5363ad5ff192208223914626": "{\\begin{aligned}\\operatorname {var} \\left[\\ln \\left({\\frac {1}{X}}\\right)\\right]&=\\operatorname {var} [\\ln(X)]=\\psi _{1}(\\alpha )-\\psi _{1}(\\alpha +\\beta ),\\\\\\operatorname {var} \\left[\\ln \\left({\\frac {1}{1-X}}\\right)\\right]&=\\operatorname {var} [\\ln(1-X)]=\\psi _{1}(\\beta )-\\psi _{1}(\\alpha +\\beta ),\\\\\\operatorname {cov} \\left[\\ln \\left({\\frac {1}{X}}\\right),\\ln \\left({\\frac {1}{1-X}}\\right)\\right]&=\\operatorname {cov} [\\ln(X),\\ln(1-X)]=-\\psi _{1}(\\alpha +\\beta ).\\end{aligned}}", "1d16d8fa7ed9455943cfd7290fa2a99d": "\\mathbf {v_{d}} ", "1d16e6fb0f673014cb357bb9eccbdcc3": "\\delta _{X}(\\varepsilon /2)\\leq {\\tilde {\\delta }}_{X}(\\varepsilon )\\leq \\delta _{X}(\\varepsilon ),\\quad \\varepsilon \\in [0,2].", "1d16efe78a4740b4259635d2a339490b": "I=\\{1,\\ldots ,n\\}", "1d16f95987cd040456c2c25067376ecf": "\\pi /4{\\sqrt {2}}", "1d1721a2abc5d2837988edc518ce65a5": "\\scriptstyle A\\otimes _{K}L", "1d175fdb3ce7c9f111f1c66ef135d7eb": "{\\frac {{\\sqrt[{3}]{108+12{\\sqrt {69}}}}+{\\sqrt[{3}]{108-12{\\sqrt {69}}}}}{6}}", "1d17d27be0541975ee753678f711844b": "{\\mathcal {L}}_{X}T_{ab}=0", "1d17d685bdc9b4bd3ef4055466d82e3d": "\\omega _{k}={\\sqrt {2\\omega ^{2}(1-\\cos(ka))}}.\\ ", "1d183d57a828a3bb3ca7567a9f163b3f": "P=200000", "1d18a6e569fe5adf17271f3ea5bd8385": "\\left(\\nabla ^{2}-{1 \\over {c}^{2}}{\\partial ^{2} \\over \\partial t^{2}}\\right)\\mathbf {E} \\ \\ =\\ \\ 0", "1d18be8c729ed0c03e30093596f091e6": "\\tau _{\\mathrm {bulk} }", "1d18d50801ca9cc67ca08cdd963123d3": "U_{kl}^{AB}", "1d18e10b70af19ccf8c4112c49b3336b": "\\sigma _{\\mathrm {T} }", "1d19250a02b3155af1c561e6ff38dc17": "(S,d)", "1d192b41bbfb0fb5c4bff3a91da05d3f": "T-\\lambda ", "1d192bb209fa2f0f0cfdbe23eb00310b": "10^{18}", "1d196fe3d16a4493c2e69e69499e4e49": "\\mathbf {B} (t)=\\mathbf {T} (t)\\times \\mathbf {N} (t)={\\frac {\\mathbf {r} '(t)\\times \\mathbf {r} ''(t)}{\\|\\mathbf {r} '(t)\\times \\mathbf {r} ''(t)\\|}}.", "1d1a81c66e1d2c4f2e054e585951fef6": "{\\rm {VRR}}\\,\\,\\,=\\,\\,{\\alpha \\over {2\\,\\,-\\,\\,\\alpha }}", "1d1adab8af1bec934d272c17ba78f036": "d^{3}k", "1d1b1e144dcef8174bd5666f38ade04f": "T(\\omega )=0", "1d1b5506d787a1f0f9c4eb9a53a41ddb": "1+0{\\sqrt {2}}=1.0", "1d1b5c9a5ab264843dd3be53564b121f": "e={\\hat {x}}-x", "1d1b67a4315090635fdc1b0ab810c96c": "\\cos \\gamma =1-{\\frac {c^{2}}{2a^{2}}}", "1d1b67d7612bb0ca44695a80d17d4753": "v_{1}\\leq v_{2}\\leq v_{3}\\leq \\dots \\leq v_{N}", "1d1c906e20011d74be0c2ee83abaf921": "[0..1]", "1d1d4dc3dd666b887ba8cba6cc215a37": "R/(q_{i})", "1d1d9610e0f6cb66cbb43923a733020b": "\\ R_{l}(r)=r^{l}\\sum _{p=1,P}c_{p}A(l,\\alpha _{p})\\exp(-\\alpha _{p}r^{2})", "1d1db6bcc58465ede4a89d0d6b82815e": "{\\frac {a^{x}}{\\ln a}}\\,", "1d1dc2fafef279934f2a51b4ac4d59dd": "(v,w)", "1d1e2f4b9431bafc7b3471e0dbf0f8a2": "{n+1 \\choose k}_{2}={n \\choose k-1}_{2}+{n \\choose k}_{2}+{n \\choose k+1}_{2}", "1d1e6aaaa43d48bcb97c42991f9dc978": "{\\frac {d}{ds}}{\\Big |}_{s=0}{\\mathcal {S}}(\\gamma _{s})={\\Big |}_{a}^{b}\\alpha _{i}X^{i}-\\int _{a}^{b}g_{ik}({\\ddot {\\gamma }}^{k}+2G^{k})X^{i}dt,", "1d1ed0671a92ee771eb1c1d9e5277e39": "{\\big \\{}\\mathbf {F} _{\\alpha }{\\big \\}}_{\\alpha =1}^{M}", "1d1f36b0adc3a84df596f5e91c9b815e": "\\mathbf {R} =n_{1}\\mathbf {a} _{1}+n_{2}\\mathbf {a} _{2}+n_{3}\\mathbf {a} _{3}", "1d1f4de02833556752efbb3a17e6df36": "\\eta (2i)={\\frac {\\Gamma \\left({\\frac {1}{4}}\\right)}{2^{{11}/8}\\pi ^{3/4}}},", "1d1fc588f515ba1ed0950a81b58511dc": "\\eta ^{R}:G\\to {\\mathfrak {g}}\\otimes {\\mathfrak {g}}", "1d200ec1a1ca4eac6094ccd7ed425069": "C_{MX}^{\\phi }=\\left[{\\frac {3}{\\sqrt {pq}}}\\right]\\left(p\\mu _{MMX}+q\\mu _{MXX}\\right).", "1d206774fa859376863653284d771a6a": "\\phi (x,y,z)=\\left({\\frac {x}{a}}\\right)^{2}+\\left({\\frac {y}{b}}\\right)^{2}+\\left({\\frac {z}{c}}\\right)^{2}-1=0", "1d2071b227874a89041a480925185758": "X(t)", "1d208ff8689c77f7bfcb9495d1a788fd": "c={\\sqrt {\\frac {N_{rr}^{*}}{\\rho h}}}", "1d20d35a3896ac35d7ed51f16dfe0155": "{\\mathcal {ABCDEFGHI}}\\!", "1d20e87c46153a58bd076c6e7e02ed55": "\\beta '", "1d20e9a24e4e64512e51afef852e5bc6": "J_{0}^{k}({\\mathbb {R} },M)_{p}", "1d2118af8c9c571fec6305fff4b35c9b": "T\\cup M", "1d2177b052eaed03c61ace7c1fa54613": "\\theta =2", "1d21d03dadb8329d94da8bb76f7ad49a": "R_{f}(-\\tau )=R_{f}^{*}(\\tau )\\,", "1d223191de06a2371757e9757343129d": "\\pi =T^{a_{1}}M^{a_{2}}L^{a_{3}}g^{a_{4}}\\,", "1d223f52e789290f1117815490b53287": "(X\\perp \\!\\!\\!\\perp Y)\\,|\\,W", "1d22b15b9985b38212a8e8a713ced177": "\\mathbf {H} ={\\begin{pmatrix}a_{1}&c-id\\\\c+id&a_{2}\\end{pmatrix}}", "1d22bc54639ff8a8581254bf211b0230": "{\\vec {q}}_{r}^{A}", "1d239f79c9f7e82de16b560d96c9e162": "z\\rightarrow z+s,\\ \\ \\infty \\rightarrow \\infty \\quad ,", "1d23cc09d50f87e679b7935ed6459ba0": "Re(p)<0", "1d23f07abd11ceb4ee03406801920fd0": "\\color {Goldenrod}{\\text{Goldenrod}}", "1d23fe052b5300f3e02bd97241675abc": "\\scriptstyle z\\,=\\,2\\pi ni{\\text{ for }}n\\,=\\,\\dots ,\\,-1,\\,0,\\,1,\\,\\dots .", "1d2402dc131d1a3153c9e9ed520d67f2": "{\\mbox{Impairment Cost}}={{\\mbox{Recoverable Amount}}-{\\mbox{Carrying Value}}}", "1d2410c062f6dbb0e30df6a396937426": "\\textstyle |S_{11}|^{2}+|S_{12}|^{2}=1", "1d243aa59a13db8030d5bd9ce261eb70": "P_{2}^{2}-4Q_{2}=(P+2)^{2}-4(P+Q+1)=P^{2}-4Q=D", "1d24703d5697cb2cc2a53e21dca00503": "a_{n}\\ ", "1d247098f8d5b1e465ddb3aea319e8a4": "{\\begin{bmatrix}1&2&3&4&5&6&input\\\\1&0&0&1&0&1&flag\\ bits\\\\1&3&6&4&9&6&segmented\\ scan\\ +\\end{bmatrix}}", "1d2480ccb042a88e9a02225b784344ed": "T_{n}(y)", "1d24acc1fd4dfe3aed2e137a0c8fc5b9": "P={\\frac {A_{x}}{{\\ddot {a}}_{x}}}", "1d252e2846757f89d5debf77be43e868": "D\\lambda -{\\bar {\\delta }}\\pi =(\\rho \\lambda +{\\bar {\\sigma }}\\mu )+\\pi ^{2}+(\\alpha -{\\bar {\\beta }})\\pi -\\nu {\\bar {\\kappa }}-(3\\varepsilon -{\\bar {\\varepsilon }})\\lambda +\\Phi _{20}\\,,", "1d2559a184122bca9aa0e8a0f42951df": "\\varphi =\\theta -\\theta _{k}", "1d257c8c6d304f5f82f314447adbdb76": "\\chi (z_{1},\\ldots ,z_{n})=({\\bar {z}}_{1},\\ldots ,{\\bar {z}}_{n})", "1d25b957cb614d7ba68f68a392f09ee1": "0/x=0", "1d2630a70db48afa6a8e27d4e0b7eef1": "M_{2413\\oplus 35142}=M_{241379586}={\\begin{bmatrix}&1&&&&&&&\\\\&&&1&&&&&\\\\1&&&&&&&&\\\\&&1&&&&&&\\\\&&&&&&1&&\\\\&&&&&&&&1\\\\&&&&1&&&&\\\\&&&&&&&1&\\\\&&&&&1&&&\\end{bmatrix}}", "1d2636c91e25cadc0349b96c98ace41e": "\\lim _{n\\to \\infty }{\\frac {\\ln(g(n))}{\\sqrt {n\\ln(n)}}}=1", "1d2640cdbd600cbdd72fe6915e8ca26f": "\\displaystyle \\left\\langle {\\begin{matrix}3\\\\4\\end{matrix}}\\right\\rangle ={3+4-1 \\choose 3-1}={3+4-1 \\choose 4}={\\frac {6!}{4!2!}}=15", "1d268cc2ef7f20c744884a9eb3690f00": "{\\boldsymbol {\\Omega }}={\\begin{pmatrix}0&-\\omega _{\\text{z}}&\\omega _{\\text{y}}\\\\\\omega _{\\text{z}}&0&-\\omega _{\\text{x}}\\\\-\\omega _{\\text{y}}&\\omega _{\\text{x}}&0\\\\\\end{pmatrix}}", "1d26cbe9a42f8d62c5797957ef636062": "d(x,z)\\leq d(x,y)+d(y,z)", "1d26f6727e69643cd634718b396fe9f4": "{\\begin{array}{rcl}{\\mathcal {G}}[r]\\rho &\\equiv &{\\frac {r\\rho r^{\\dagger }}{\\operatorname {Tr} [r\\rho r^{\\dagger }]}}-\\rho \\\\{\\mathcal {H}}[r]\\rho &\\equiv &r\\rho +\\rho r^{\\dagger }-\\operatorname {Tr} [r\\rho +\\rho r^{\\dagger }]\\rho \\end{array}}", "1d27a39a83f24f03b07f92fc1cd1b9d8": "i_{1}=i_{r}\\ ", "1d281a0422f9404c3e31b82139063c9c": "b_{T(V)}(T(v),T(w))=b_{V}(v,w).", "1d28c23070e40220d6e6469a8907bb41": "\\varphi =\\psi \\circ \\iota ", "1d28e9e4fde2fdcbc5ddf5e775da4a8e": "{\\vec {a}}={{\\vec {F}} \\over m}", "1d29601709b8f5db6ebfb0a8f499c594": "f(t|\\theta )=\\theta f_{0}(\\theta t)", "1d29e8837c3c095758938f04277c7b24": "R={3L_{\\text{s}}\\phi _{\\text{d}} \\over \\phi _{\\text{s}}}", "1d29fecebb067f6c48648988d35a9170": "0=2(Q-M)+2QY,\\,\\!", "1d2a08776d6865edf674c7483274b0ab": "\\partial _{xx}\\left(-\\partial _{xx}\\psi \\right)=-\\partial _{xxx}\\phi \\,", "1d2a75a22aa751c6bd56e7d8a1dee4e2": "e^{X}=e^{A+N}=e^{A}e^{N}.\\,", "1d2ab395697ecff7df4765ecfbbc1a75": "\\forall x,sp(S,P)\\Rightarrow Q", "1d2ad734ad8942abc34f835b99d9f8fd": "\\int _{[A]_{i}}^{[A]_{f}}{\\frac {d[A]}{[A]}}=\\int _{t=0}^{t}-kdt", "1d2b0795a73e0fe83a86d3dbe925fbc4": "(\\log ^{12}(n))", "1d2b0af30d37632489c41b7d100a657d": "{}^{\\mathrm {N} }{\\boldsymbol {\\omega }}^{\\mathrm {B} }", "1d2be4138e36e3f0174a09e105e59ce5": "P_{n}=P_{a}({\\sqrt[{12}]{2}})^{(n-a)}", "1d2bfdbb955b25644e406437ace85ec8": "\\mathbf {u} \\otimes \\mathbf {v} =\\mathbf {u} \\mathbf {v} ^{\\mathrm {T} }={\\begin{bmatrix}u_{1}\\\\u_{2}\\\\u_{3}\\\\u_{4}\\end{bmatrix}}{\\begin{bmatrix}v_{1}&v_{2}&v_{3}\\end{bmatrix}}={\\begin{bmatrix}u_{1}v_{1}&u_{1}v_{2}&u_{1}v_{3}\\\\u_{2}v_{1}&u_{2}v_{2}&u_{2}v_{3}\\\\u_{3}v_{1}&u_{3}v_{2}&u_{3}v_{3}\\\\u_{4}v_{1}&u_{4}v_{2}&u_{4}v_{3}\\end{bmatrix}}.", "1d2c10da76029714f21d877c8bc5a76c": "R_{\\text{earth}}", "1d2c3603b7999012d70ac5524af2a7c3": "T_{tot}={\\frac {3}{8}}+0.01+{\\frac {1.1\\times 8}{2}}\\left({\\frac {1}{2.5}}-{\\frac {1}{1}}\\right)", "1d2c54c954311d74b323c8e0c7b89271": "ES_{1}", "1d2c859cd62301bae4a3faa504591742": "H^{1}(M,{\\mathbb {Z} _{2}})", "1d2cc93fbd9cac7967b8fd1d8515b00d": "{Q_{f}}={Q_{water}}{\\sqrt {\\frac {1}{Sg}}}", "1d2ce798c79c32ec5a818af7916b952b": "F(x,y)=c\\,", "1d2cfe729015cab697f270ff1888209a": "p_{i}(e^{t}-1)+1", "1d2d4195485cd005cfcfd848e68fac35": "\\cos ^{n}\\theta ={\\frac {1}{2^{n}}}{\\binom {n}{\\frac {n}{2}}}+{\\frac {2}{2^{n}}}\\sum _{k=0}^{{\\frac {n}{2}}-1}{\\binom {n}{k}}\\cos {((n-2k)\\theta )}", "1d2d8ca6b51a04f6e1af12388213a3c4": "I(X;Y|Z)\\geq 0", "1d2dbf930a6b8e7cedc1ab80ef0a0413": "\\varphi (x)=2\\cdot \\sum _{k=0}^{N-1}h_{k}\\cdot \\varphi (2\\cdot x-k)", "1d2e4a75af5e2ca37f6cb36da7b84503": "\\displaystyle {g(x,y)={\\begin{pmatrix}E&F\\\\F&G\\end{pmatrix}}}", "1d2ea3322e2c8dad27ddcf5b7f9dee18": "\\scriptstyle {\\Delta p}", "1d2ea57ef07652f3d99a8ff58b4cf62c": "k[t]/(t^{n})", "1d2eddd7ba651206d34777e23dca96de": "-e({\\boldsymbol {r}}_{\\text{SO}}\\cdot {\\boldsymbol {E}})", "1d2f075daabf3af437d7c9e767beac46": "{\\begin{bmatrix}M\\end{bmatrix}}={\\begin{bmatrix}1&0\\\\0&1\\end{bmatrix}}", "1d2f2a8410f934caccc8a8c490bcde1d": "\\rho _{C}^{2}=1-D(Y|X)/D(Y)", "1d2f4325ad8a9eb376e910194c05f7ce": "\\operatorname {de-let} [M]\\ \\operatorname {de-let} [N]", "1d2fd37f34f4719d4d7913085b671de4": "{\\mathcal {L}}(x)", "1d30b0ebb0b7bb4c870cc60572f6cee3": "{\\frac {F_{\\rm {el}}}{kT}}={\\frac {\\pi ^{2}}{24N^{2}a^{5}}}\\int _{0}^{\\infty }\\left\\{-z^{3}{\\frac {{\\rm {d}}\\phi (z)}{{\\rm {d}}z}}\\right\\}{\\rm {d}}z", "1d30efc186ca4fea7fa73770343b0f3b": "{\\frac {k_{i-1}}{x_{i}-x_{i-1}}}+\\left({\\frac {1}{x_{i}-x_{i-1}}}+{\\frac {1}{x_{i+1}-x_{i}}}\\right)\\ 2k_{i}+{\\frac {k_{i+1}}{x_{i+1}-x_{i}}}=3\\ \\left({\\frac {y_{i}-y_{i-1}}{{(x_{i}-x_{i-1})}^{2}}}+{\\frac {y_{i+1}-y_{i}}{{(x_{i+1}-x_{i})}^{2}}}\\right)", "1d30f17eeb1407150f3c23cfd825bc81": "{\\mathbf {f} \\cdot \\mathbf {u} }={d \\over dt}\\left(\\gamma mc^{2}\\right)={dE \\over dt}", "1d319b7464fcdeee5495d34cbc509caa": "(f,g)\\!", "1d326adfe9f8565083aab6c5b9228c7a": "K=\\Gamma +\\Theta ", "1d327044e41b0e626ea100800e2d3e40": "s_{1}s_{2}=1-2\\sin(\\pi /18)", "1d3290610438676fa6ccb055ceb357ba": "1<{\\frac {1}{p}}+{\\frac {1}{r}}", "1d32a269c403d8433b519ae62633924c": "X={\\frac {s}{y_{\\mathrm {atm} }}}={\\frac {R_{\\mathrm {E} }}{y_{\\mathrm {atm} }}}{\\sqrt {\\cos ^{2}z+2{\\frac {y_{\\mathrm {atm} }}{R_{\\mathrm {E} }}}+\\left({\\frac {y_{\\mathrm {atm} }}{R_{\\mathrm {E} }}}\\right)^{2}}}-{\\frac {R_{\\mathrm {E} }}{y_{\\mathrm {atm} }}}\\cos \\,z\\,.", "1d32b4f78d324eb6a203be804ff8685e": "X^{\\ast }(t)", "1d32c8887a589d26705465a877b77786": "{\\begin{cases}\\pi _{r}:J^{r}(\\pi )\\to M\\\\j_{p}^{r}\\sigma \\mapsto p\\end{cases}}", "1d3393a593a8d010c17164c091fc835d": "\\epsilon _{N}(t)=\\sum _{k=N+1}^{\\infty }A_{k}(\\omega )f_{k}(t)", "1d339c99c60488442eec9d99082b8544": "(A,B)=\\{\\varphi (a):a\\in A\\}\\cup \\{\\varphi (b)\\cup \\{0\\}:b\\in B\\}.", "1d33b07b5bee96afc9984dcc816496e4": "j=A+B,A+B-1,...,|A-B|,", "1d33bce3f353bfc95cbe375bd1cacf54": "\\int _{0}^{\\infty }{\\frac {\\sin b\\,\\omega }{\\omega }}\\,d\\omega =\\int _{0}^{b\\,\\infty }{\\frac {\\sin b\\,\\omega }{b\\,\\omega }}\\,d(b\\,\\omega )=\\int _{0}^{\\operatorname {sgn} b\\times \\infty }{\\frac {\\sin x}{x}}\\,dx=\\operatorname {sgn} b\\int _{0}^{\\infty }{\\frac {\\sin x}{x}}\\,dx={\\frac {\\pi }{2}}\\,\\operatorname {sgn} b", "1d340e6277c8fc93c281886c2168e9ff": "\\delta _{d}(\\lambda )=\\sigma _{i}\\beta _{i}^{*}(\\lambda )\\,\\Delta \\sigma _{i}", "1d3436db5be542b178d90641c39cca00": "\\exp(J)=I+{\\frac {e^{n}-1}{n}}J,", "1d34418628f60dfc415247a9c153ea2e": "\\langle r^{k}\\rangle =\\int _{0}^{\\infty }P(r)r^{k}dr=3a^{k}\\int _{0}^{\\infty }x^{k+2}e^{-x^{3}}dx\\,,", "1d345316ad424b3749d50afe8ddf5986": "l^{2}=(r_{2}-r_{1})^{2}+d^{2}", "1d34b2af147c3fc1da4824e49cee7df6": "S=\\langle A,R\\rangle ", "1d34b7bfbd1bb58bca83ceb881a648de": "V=2.357Y^{0.343}-1.52", "1d34e9b5adf6dcba4e191356dfeb497d": "h_{i}^{\\ast }", "1d3507f9f362d005e060af7db94b5895": "\\sigma =\\left\\langle v^{0},v^{1},\\dots ,v^{k}\\right\\rangle .", "1d354c8a9e98c3186010e9385b6b7bf9": "\\nabla ^{2}\\Phi (\\mathbf {r} )=\\left(\\sum _{j=1}^{N}{\\frac {n_{j}^{0}\\,q_{j}^{2}}{\\varepsilon _{r}\\varepsilon _{0}\\,k_{B}T}}\\right)\\,\\Phi (\\mathbf {r} )-{\\frac {1}{\\varepsilon _{r}\\varepsilon _{0}}}\\,\\sum _{j=1}^{N}n_{j}^{0}q_{j}", "1d3642c30ab2c5484f8350169b4563ce": "X_{t+s}-X_{t}=\\int _{t}^{t+s}\\mu (X_{u},u)\\mathrm {d} u+\\int _{t}^{t+s}\\sigma (X_{u},u)\\,\\mathrm {d} B_{u}.", "1d365b3f9f13c0609211a70a86b00f83": "{\\mathbf {w} }_{i}", "1d36b3a4b0d4a5a9b50d8cd88a9797f9": "\\lambda /r_{0}", "1d36e954629388450816271e727b296d": "\\int _{-\\infty }^{\\infty }e^{-{1 \\over 2}x^{2}}\\,dx={\\sqrt {2\\pi }}.", "1d3770d156411dd4c88e005d627cd180": "F_{2}-1", "1d37a8eebccbb9658f0445293da44a53": "R\\bowtie R=R", "1d37b6c07075a4c9e96e7aad36b0f9ec": "_{q=p\\ \\Rightarrow \\ q'p=qp'\\,}\\!", "1d37d33fe0039d0a8ec30888ee1a6093": "(1000{\\bar {1}}000{\\bar {1}}0)_{\\text{NAF}}", "1d384225a77cf17f948f472379227899": "2^{14}{\\text{NO}}_{3}^{-}\\rightarrow {^{14}}{\\text{N}}_{2}{\\text{O}},", "1d387249e9f5f25c7ebe255c27423383": "arg(\\rho _{n}(c))\\,", "1d388a6cfe0ec30ae232e93ddf6f498a": "{\\frac {\\partial \\mathbf {j} _{s}}{\\partial t}}={\\frac {n_{s}e^{2}}{m}}\\mathbf {E} ,\\qquad \\mathbf {\\nabla } \\times \\mathbf {j} _{s}=-{\\frac {n_{s}e^{2}}{m}}\\mathbf {B} .", "1d38c4832f2571b5302710d65162fa8b": "i_{1}\\circ p_{1}+\\cdots +i_{n}\\circ p_{n}", "1d39446ddef5b06cb81be7a7ebe2ccac": "x=16/5", "1d39e8a1e4641edb811735775e09a251": "n_{x},n_{y},n_{z}", "1d3a28d9aeb7bbb01e922d3f6c1b5d60": "B\\mathbb {Z} ", "1d3a87c5e30b58ca492fec586a514952": "E2\\pi rh={\\frac {\\lambda h}{\\varepsilon _{0}}}\\quad \\Rightarrow \\quad E={\\frac {\\lambda }{2\\pi \\varepsilon _{0}r}}", "1d3a8c1c45dcd259e461eb235cc01eb4": "H^{\\alpha }", "1d3a9bcc849732ca8aa8ebd418fad8c4": "\\mathrm {d.f.} ={\\frac {(s_{1}^{2}/n_{1}+s_{2}^{2}/n_{2})^{2}}{(s_{1}^{2}/n_{1})^{2}/(n_{1}-1)+(s_{2}^{2}/n_{2})^{2}/(n_{2}-1)}}.", "1d3ad27025247c575180b1e2b240759d": "(x_{3},y_{3})", "1d3b01955fdea74780e1a4edf6c17c1f": "f={\\frac {14.227}{Re}}", "1d3b7ac8e4cbd8240a762f35358e57d8": "\\prod _{x}{\\frac {x+1}{x}}=Cx", "1d3c60d2089d78ba278af2440fd2e193": "Y_{i}=C+(B_{0}+B_{1}T_{i})\\sin(2\\pi \\omega T_{i}+\\phi )+E_{i}", "1d3c782b4cf65afc51aaaacb5e19991d": "S\\otimes T:V\\otimes W\\rightarrow X\\otimes Y", "1d3c85f182100b4c7c6fc89c6be06234": "\\left({\\frac {^{13}C}{^{12}C}}\\right)_{sample}=\\left(1+{\\frac {\\delta ^{13}C}{1000}}\\right)\\left({\\frac {^{13}C}{^{12}C}}\\right)_{PDB}", "1d3d3f9328743930a4c139944f9c077c": "\\Delta \\mathbf {w} _{n}", "1d3d5a0a1ebc552467c21311862b1f4a": "{\\mathbf {y} }\\sim N({\\boldsymbol {\\theta }},\\sigma ^{2}I).\\,", "1d3d85263048ff73d9e59682b0c8bf44": "\\Gamma ", "1d3d87beda749a3cfd36e08c70c49689": "f=\\lambda x.x\\to y", "1d3d942080e4730c027537aac281ee7a": "a_{i}=F(x_{i})", "1d3daf0a9c786b400ce07505c8d87344": "\\displaystyle {N_{+}=N_{\\sigma }\\cdot M_{\\sigma }.}", "1d3db74a7915d264c94efcdc1deb7b3b": "\\left(1+{\\frac {x}{n}}\\right)^{n}=e^{x}\\left(1-{\\frac {x^{2}}{2n}}+{\\frac {x^{3}(8+3x)}{24n^{2}}}+\\cdots \\right),", "1d3deb12b13167802707b7f44b5f7fb4": "C_{v}\\ll C_{0},", "1d3e2e1c3773321d16b9d5209399784a": "[0,1]^{2}\\subset \\Re ^{2}", "1d3e96bbf9c719eaa5997ada8c11b4e9": "\\,\\tau ", "1d3f02d09cebcfb3ac14e5d79b2550c3": "f(i)=f_{i}", "1d3f08959ad89a735f7a8bd9cdd66cd8": "{\\boldsymbol {v}}_{T}", "1d3fbf50383d9ed5abcd8f1f90482c3f": "\\mathbf {a} \\times \\mathbf {b} =\\det {\\begin{bmatrix}\\mathbf {i} &\\mathbf {j} &\\mathbf {k} \\\\a_{1}&a_{2}&a_{3}\\\\b_{1}&b_{2}&b_{3}\\\\\\end{bmatrix}}", "1d3fd31d4040a5ae75362c68e32d0127": "~G(k_{x},k_{y})", "1d4028c806916cd2b17d10a69d4060f0": "\\varphi ^{*}", "1d40d5538760bdfbe2064fcccef3e685": "{\\mathcal {O}}\\left({A^{m-2}}\\right)", "1d40e87acdff39128077caa9341a519f": "\\min\\{2\\times 2^{d-1-r},2^{d-r}\\}=2^{d-r}.", "1d40f2a8de06149d8c9bf6bdd21d28c5": "D_{m}", "1d40f91c0e72a2d0c91d31b3a5e0d3ff": "0<\\varepsilon ,\\varepsilon _{1}<1", "1d4163f47a7280ec1bf4458eee4ea5a8": "{\\frac {-y_{2}}{f}}={\\frac {x_{2}}{x_{3}}}", "1d41c663447da5010acb3c15f1b56a2d": "\\pm {\\sqrt {1-\\sin ^{2}\\theta }}\\!", "1d41e6f55521cdba4fc73febd09d2eb4": "1.", "1d4248f05f5db5ffe09587b0c4f70f8a": "={\\frac {0.00198_{TTT}+0.1584_{TFT}}{0.00198_{TTT}+0.288_{TTF}+0.1584_{TFT}+0.0_{TFF}}}={\\frac {891}{2491}}\\approx 35.77\\%.", "1d427c8367456193613932dda2e3ba5b": "[\\alpha ]_{\\lambda }^{T}={\\frac {\\alpha }{l\\times \\rho }}", "1d43b20acb41616e49414788d25e5cbb": "\\varepsilon \\left[M\\right]\\leq {\\frac {\\left\\|f\\right\\|_{\\mathrm {B} ,p}^{2}}{{\\frac {2}{p}}-1}}{{M}^{1-{\\frac {2}{p}}}}", "1d43bb992f4fe5cdf3855c74e50ba30f": "J(g)=R_{emp}(g)+\\lambda C(g).", "1d43cf3e9cc8956a19279855442a7082": "{\\bar {V}}", "1d4423fdfa639d8e41ae4b5bc7f77930": "{\\begin{cases}y_{t}=g(x_{t}^{*})+\\varepsilon _{t},\\\\x_{t}=x_{t}^{*}+\\eta _{t}.\\end{cases}}", "1d443da7a6187301dc4a51d1c2f143a1": "(1-x^{2})^{\\alpha +1/2}\\,", "1d4448d69d246bdb3569ccaa133e6d8c": "\\mathbb {C} ^{*}=\\operatorname {GL} _{1}(\\mathbb {C} )", "1d44e82aaa2153d7a6863720f1b2c633": "f(x_{0}+0)", "1d4511c16a6accc47fadf4d30028e150": "{\\phi }~{\\alpha }~{\\frac {Q}{{N}{D^{3}}}}\\ ", "1d452d075fcc6d09f630b1db8ddcaac7": "{\\tilde {Z}}[{\\tilde {J}}]\\sim \\int {\\mathcal {D}}{\\tilde {\\phi }}\\prod _{p}\\left[e^{-(p^{2}+m^{2}){\\tilde {\\phi }}^{2}/2}e^{-g{\\tilde {\\phi }}^{4}/4!}e^{{\\tilde {J}}{\\tilde {\\phi }}}\\right].", "1d453013429955e91f67836f27a9b9f8": "X", "1d4563ad8f05a81145fc0dc117173bb3": "\\operatorname {Cov} (X,Y)=\\operatorname {E} (XY)-\\mu _{X}\\mu _{Y}.", "1d45a21b927abbd3006925b74888ab7f": "n=1b_{1}b_{2}b_{3}\\dots b_{m}", "1d45c7dfc919755222e2dda68568ffee": "L_{k+1}(z)={\\frac {(1-z_{k+1}z^{-1})}{(1-z_{k}z^{-1})}}L_{k}(z),\\quad k=0,1,...,N-1", "1d45d439a89e67a6b198fdb8b6078aee": "m_{L}=1", "1d45e093a1233f41208d78517154e06c": "x\\times 1=x\\times S(0)=(x\\times 0)+x=0+x=x", "1d45f1e23075edec79cd564d71e12434": "f(u)=\\left\\{{\\begin{array}{ll}{\\frac {841}{108}}u+{\\frac {4}{29}},&u\\leq (6/29)^{3}\\\\\\\\u^{1/3},&u>(6/29)^{3}\\end{array}}\\right.", "1d45feee96031bc04aee40ca77147aea": "p-p_{\\infty }\\approx -\\rho U^{2}.", "1d464372c2f6fa56541e4fd51e96622a": "dG=\\pi dS\\mathrm {NA} ^{2}", "1d464403210356102cc98246113e82e5": "{\\frac {e^{-\\mu n}(\\mu n)^{n-1}}{n!}}", "1d464fed4a2d9d927ae50a3f73135e57": "\\lim _{t\\to \\infty ,F_{e}\\to 0}{\\frac {1}{t}}\\ln \\left({\\frac {p(\\beta {\\overline {J}}_{t}=A)}{p(\\beta {\\overline {J}}_{t}=-A)}}\\right)=-\\lim _{t\\to \\infty ,F_{e}\\to 0}AVF_{e},\\quad F_{e}^{2}t=c.\\,", "1d465a1b7417fbdae9686f5baaafbc40": "\\ Pxy\\rightarrow Exy.", "1d465eac11c2923d28416d7cf61c23c6": "\\ C2:A(AB)=A(B).", "1d46a97c5cdb3a0f34d54f1abe33a5b7": "{(f(x^{*})-f(x))/f(x^{*})<\\epsilon ,\\qquad \\forall {x}\\,\\in X}", "1d46c3a6f3af40613bd1dda44b6dca25": "W_{R}(K)", "1d470f5ffaea4f02f9110fb88761e3da": "|x_{n}-0|\\leq x_{N+1}={\\frac {1}{\\lfloor 1/\\epsilon \\rfloor +1}}<\\epsilon ", "1d4734bc6db8002e08e28eec76ab0b7a": "\\{1,2,3\\}", "1d477f59ca62afb7461d567fee79597f": "v={\\partial g \\over \\partial y}~.", "1d4797b7552fe8177b57f559dec9cd75": "\\scriptstyle {b^{2}=a^{2}(1-k^{2})}", "1d479fa9eda9073d7ca5349b14c702aa": "u^{k+1}={\\overline {u}}^{k}-{\\frac {I_{x}(I_{x}{\\overline {u}}^{k}+I_{y}{\\overline {v}}^{k}+I_{t})}{\\alpha ^{2}+I_{x}^{2}+I_{y}^{2}}}", "1d47cea66c20b89dfd6d8405fb88b55c": "I^{+}(p)=I^{+}(q)\\implies p=q", "1d47ee00add2712ecbb662a0761059d9": "\\zeta (s)=2^{s}\\pi ^{s-1}\\ \\sin \\left({\\frac {\\pi s}{2}}\\right)\\ \\Gamma (1-s)\\ \\zeta (1-s)\\!,", "1d47f6bfb593b3bacb00a81bb09a556c": "p_{i}(q_{i},{\\dot {q}}_{i},t)={\\frac {\\partial {\\mathcal {L}}}{\\partial {{\\dot {q}}_{i}}}}\\,.", "1d48201eed3a959fa34361f9f3e31adb": "1,0,1,0,\\ldots ,\\,", "1d4823040c20fc3503ef2ed71f1c5ef4": "Y[\\mathrm {iso} ]=c_{11}+c_{12}-2({\\frac {c_{12}^{2}}{c_{11}}})", "1d486bef7b21da51b2fd579356bfed3c": "g(x)=x^{2}+x+1", "1d48918b52a759841acbeb52ba8ffafa": "\\dots ,W_{1},W_{0},W_{-1},\\dots \\dots ", "1d48acc6256ae2f04176755fe33bc267": "{\\frac {d^{2}f}{dx^{2}}}\\ \\cdot \\ {\\frac {1}{\\left[{\\sqrt {1+[f'(x)]^{2}}}\\ \\right]^{3}}}=0\\,,", "1d48ad52203ce2db0dcff7720a4ab428": "\\pm {\\frac {\\sqrt {\\sec ^{2}\\theta -1}}{\\sec \\theta }}\\!", "1d48e49bf3bcd6eb886b1ea6e15d1dfd": "S(T\\rho ||T\\sigma )\\leq S(\\rho ||\\sigma ),", "1d48ec5d0e91220d61ae0a60c712115b": "\\nabla ^{2}={\\frac {\\partial ^{2}}{{\\partial x}^{2}}}+{\\frac {\\partial ^{2}}{{\\partial y}^{2}}}+{\\frac {\\partial ^{2}}{{\\partial z}^{2}}}", "1d490f95c5ce690b24b275c6534c5c6b": "y^{2}-x^{2}=1", "1d492e2555c61e41c66ebe4bd3445d96": "\\mu _{x,\\lambda }\\rightharpoonup \\theta \\;{\\mathcal {H}}^{m}\\llcorner P", "1d493fe8c05826374f9f4a7e0796c0e1": "S(T)=CT^{A}{\\exp \\left({\\frac {-B}{T}}\\right)}", "1d494c8741553c6005078f497c4de429": "{\\tfrac {9}{64}}", "1d49717c355548d0e46ffaca66c79ac1": "E^{*}(\\mathbb {C} \\mathbf {P} ^{\\infty })=\\varprojlim E^{*}(\\mathbb {C} \\mathbf {P} ^{n})=\\varprojlim R[t]/(t^{n+1})=R[\\![t]\\!],\\quad R=\\pi _{*}E=\\oplus \\pi _{2n}E", "1d49888f7d173effb405fd6d82bcad40": "S^{3}/\\Gamma ", "1d49f6b0447a7c24f4d25bbecbd29f8c": "S^{-1}S", "1d4a0d4c0554b9adede9ab4f4770cda9": "\\operatorname {E} (\\theta |y)=\\alpha '\\beta '={\\frac {{\\bar {y}}+\\alpha }{1+1/\\beta }}={\\frac {\\beta }{1+\\beta }}{\\bar {y}}+{\\frac {1}{1+\\beta }}(\\alpha \\beta ).", "1d4a71f32d5b082fe2c8d398373ba946": "d(\\mathbf {u} ,\\mathbf {v} )=\\cosh ^{-1}\\left({\\frac {\\mathbf {u} }{\\|\\mathbf {u} \\|}}\\cdot {\\frac {\\mathbf {v} }{\\|\\mathbf {v} \\|}}\\right)", "1d4a807a66bf7a94bed524002019c136": "{\\hat {\\pi }}^{ij}(t,x^{k})\\to -i{\\frac {\\delta }{\\delta g_{ij}(t,x^{k})}}", "1d4b75ec6ca01707473e5fdea231a12d": "(1+\\alpha ^{2})", "1d4c8c8c8dc9be9f621bbe1573bfbaf5": "{\\alpha }=0", "1d4ca999c879306d9bc62e3d0286bc93": "{\\tbinom {5}{5}}", "1d4cced60293ce8255ecf5a7c6af8023": "T>\\mathrm {X} _{1-\\alpha ,k-1}^{2}", "1d4d1b1ff131021be62024be4cf7428c": "x_{k}=P_{k}x_{k-1},", "1d4d8f777bfc9d82f0406139bd10a18c": "h(x)=\\sum _{n=0}^{\\infty }(\\sum _{k=0}^{n}f_{k})x^{n}", "1d4df73dba8e5102b370c589be328316": "K=\\mathbb {C} ", "1d4df8918f52e9e7b47f6352913e1a3e": "a_{65}^{(12)}", "1d4e14a615189406689300061fde8f26": "L(P,t)=\\#(\\{x\\in \\mathbb {Z} ^{n}:Ax\\geq tb\\}).", "1d4e6a165aec4d835bac5a99e3aaa280": "k>0\\!", "1d4e742c33c518eb95ba0c7b5eb2f602": "f(a\\mathbf {i} +b\\mathbf {j} +c\\mathbf {k} )={\\frac {1+\\mathbf {i} +\\mathbf {j} +\\mathbf {k} }{2}}(a\\mathbf {i} +b\\mathbf {j} +c\\mathbf {k} ){\\frac {1-\\mathbf {i} -\\mathbf {j} -\\mathbf {k} }{2}}", "1d4e77f8edc6910f1ceb23746d41e466": "T_{\\mu \\nu }\\,=\\,-{\\frac {1}{\\mu _{0}}}(F_{\\mu \\alpha }g^{\\alpha \\beta }F_{\\beta \\nu }\\,-\\,{\\frac {1}{4}}g_{\\mu \\nu }\\,F_{\\sigma \\alpha }g^{\\alpha \\beta }F_{\\beta \\rho }g^{\\rho \\sigma })\\,", "1d4edddb3776c42416cc6e868f64d55d": "\\mu _{x}=\\sum _{y\\neq x}\\mu _{xy}", "1d4eee0ed1882a60f5479952eb627a6e": "log_{e}(1+x)", "1d4f4565bf4e38019ef4bd68abbdf54b": "x\\cdot 1", "1d4ffec1abf84d616a5fccc0b82ce5ef": "\\Delta V=-\\int _{r_{1}}^{r_{2}}\\mathbf {E} \\cdot d\\mathbf {r} \\,\\!", "1d500c527c4bafef0c2509f414a36ea7": "\\delta _{\\text{int}}:Q\\times \\Sigma _{\\text{int}}\\to Q", "1d506411839ec4d34d04f651d368fc1b": "G\\triangleright G_{2}(q)',\\ p^{d}=q^{6}{\\text{ and }}p=2.", "1d506772e9b0365fae9d33937050c36c": "D=i_{D}\\circ j^{k}:J^{k}(E)\\rightarrow F", "1d507a41e72e98114f2899a68e757758": "z=(x-\\mu )/{\\sqrt {2}}\\sigma ", "1d50f2b88311cf645dc2b9b531dfb0c2": "\\phi ^{1}={\\begin{bmatrix}1\\\\0\\end{bmatrix}}\\quad \\quad \\phi ^{2}={\\begin{bmatrix}0\\\\1\\end{bmatrix}}\\,", "1d5166eef62bebb51c438fa18f4564f4": "U(\\mathbf {r} ,t)=A_{o}e^{i(\\mathbf {k} \\cdot \\mathbf {r} -\\omega t)}e^{i\\varphi }", "1d51c8441aba0bd976918896d08cdf33": "{\\mathit {dr}}(n)>0\\Leftrightarrow n>0.", "1d51d95895854e7ce977bb4b4549f40c": "H(A,X)=H(A)+\\mathbb {E} _{a\\sim A}{\\big [}H(X|A=a){\\big ]}", "1d51e47aee3c00362bcd110f0b15da60": "\\scriptstyle Z", "1d51ee6cbfc93d4c1ef4df9388ab7656": "\\Pr(|{\\overline {X}}-\\mathrm {E} [{\\overline {X}}]|\\geq t)\\leq 2\\exp \\left(-{\\frac {2t^{2}n^{2}}{\\sum _{i=1}^{n}(b_{i}-a_{i})^{2}}}\\right),\\!", "1d522af0612abeb7211c7277a784cc28": "{\\kappa }a>>1", "1d524d950b3cce35265078a236975954": "{\\widehat {E}}=i\\hbar {\\frac {\\partial }{\\partial t}}\\,,\\quad {\\widehat {\\mathbf {p} }}=-i\\hbar \\nabla \\,,", "1d527a31bc42ee12f6cb40f9cc8119fd": "f(p_{1},p_{2},\\ldots ,p_{n})=-\\sum _{j=1}^{n}p_{j}\\log _{2}p_{j}.", "1d52b324fdabd94caa8ce7f2932c99b1": "\\mathbf {K} '", "1d52c92dd60ef9f0d2057df6151b1e8d": "e^{z}={\\cfrac {1}{1+{\\cfrac {-z}{1+{\\cfrac {z}{2+{\\cfrac {-z}{3+{\\cfrac {2z}{4+{\\cfrac {-2z}{5+{}\\ddots }}}}}}}}}}}}", "1d5329f203d7a63255e2e5db5a4112b1": "f(B)=2\\times {\\frac {e^{-{\\frac {B}{2}}}}{2\\pi }}\\int _{0}^{\\frac {\\pi }{2}}\\,dt", "1d53d949b390f882462fcbe467410386": "{\\tfrac {1}{6}}=2^{-1}3^{-1},\\;\\;{\\tfrac {3}{4}}=2^{-2}3^{1},\\;\\;\\gcd({\\tfrac {1}{6}},{\\tfrac {3}{4}})=2^{-2}3^{-1},\\;\\;\\operatorname {lcm} ({\\tfrac {1}{6}},{\\tfrac {3}{4}})=2^{-1}3^{1}.\\;\\;", "1d53f3a57a091df3f5ade1f1a82b7a0c": "{\\begin{aligned}x&=\\sigma \\tau \\\\y&={\\tfrac {1}{2}}\\left(\\tau ^{2}-\\sigma ^{2}\\right)\\\\z&=z\\end{aligned}}", "1d54619180cd6a0c14d8e9f4ac419a1e": "\\{x\\},\\langle x\\rangle ", "1d54b43dd99935b88fc2b4d3b088c422": "y^{2}={\\frac {1-{\\sqrt {1-4U^{2}}}}{2}},", "1d54b5fd261d00aa57786d0e9df9ff09": "{\\mathfrak {P}}^{18}", "1d54ed8072dd721e45418d5262d4ecef": "{\\frac {x}{{\\sqrt {1-x^{2}}}\\arccos(x)}}", "1d55069e1b3a1643950d574c45ad0255": "\\beta (\\phi )", "1d5515054c9a9e0e7ac990a1eaaf1e87": "{\\dot {Q}}(x+dx)={\\dot {Q}}(x)+d{\\dot {Q}}_{conv}.", "1d55179592f8269c60366d4d88ab7ae2": "1-{\\frac {\\alpha }{m}}", "1d55415060896863055888932cf4fe64": "a_{jk}=\\Vert T_{j}T_{k}^{\\ast }\\Vert ,\\qquad b_{jk}=\\Vert T_{j}^{\\ast }T_{k}\\Vert .", "1d55585e0a170f790fa9cf37a5c7bf88": "J_{k}(n)=n^{k}\\prod _{p|n}\\left(1-{\\frac {1}{p^{k}}}\\right)=n^{k}\\left({\\frac {p_{1}^{k}-1}{p_{1}^{k}}}\\right)\\left({\\frac {p_{2}^{k}-1}{p_{2}^{k}}}\\right)\\ldots \\left({\\frac {p_{\\omega (n)}^{k}-1}{p_{\\omega (n)}^{k}}}\\right).", "1d557dcab13f9dd21c96f0d41637d596": "{\\frac {dK(t)}{dt}}=I^{g}(t)-D(t)=I^{n}(t)", "1d558400d437662fca851d765c760faf": "{\\begin{aligned}f(t)&{}={\\frac {d}{dt}}\\Pr(T_{x}\\leq t)={\\frac {d}{dt}}\\Pr(X_{t}\\geq x)={\\frac {d}{dt}}(1-\\Pr(X_{t}\\leq x-1))\\\\\\\\&{}={\\frac {d}{dt}}\\left(1-\\sum _{u=0}^{x-1}\\Pr(X_{t}=u)\\right)={\\frac {d}{dt}}\\left(1-\\sum _{u=0}^{x-1}{\\frac {(\\lambda t)^{u}e^{-\\lambda t}}{u!}}\\right)\\\\\\\\&{}=\\lambda e^{-\\lambda t}-e^{-\\lambda t}\\sum _{u=1}^{x-1}\\left({\\frac {\\lambda ^{u}t^{u-1}}{(u-1)!}}-{\\frac {\\lambda ^{u+1}t^{u}}{u!}}\\right)\\end{aligned}}", "1d55ac14cb6a40e57ae7e2b51265adbe": "K_{1}=\\oplus H_{\\alpha }", "1d55f0a7b672133457c8cdea68050323": "{\\hat {H}}={\\hat {H}}^{\\dagger }.", "1d55f10c9df658c98268925d074ca2e0": "\\Delta h\\,=\\,h_{1}\\,-\\,h_{2}\\,=\\,-\\,{\\frac {1}{g}}\\,{\\frac {A_{1}}{A_{2}}}\\left(1\\,-\\,{\\frac {A_{1}}{A_{2}}}\\right)\\,v_{1}^{2}.", "1d5613061b14dc1cb2960aa3db5afdae": "\\scriptstyle M_{\\text{A}}", "1d56258109956b026d767af1cd300dee": "Vd_{F}=Vd_{T1}+Vd_{T2}+Vd_{T3}+...+Vd_{Tn}\\,", "1d5631b985746dafa7e04d0310b95b34": "\\mathbf {H^{r}} ", "1d5675825f91c3465a267c659bd3ea49": "~D(x)=ue^{x}+v~", "1d56f87a4031f10036022423ee209ad7": "\\!E_{\\mathrm {h} }/k_{\\mathrm {B} }", "1d5713b7fac39021d0c83c67243ddcb0": "v_{p}={\\sqrt {2\\mu /r_{p}+v_{\\infty }^{2}}}\\,", "1d5721cf09e47f5f5ed4d9b91b45ac57": "S=\\{s_{1},s_{2},\\dots \\},", "1d57c1cd49d1bdb80fac5ce2cc88102a": "\\displaystyle {(Rv,v)=|(Rv,v)|=\\sum |(R_{i}v,R_{j}v)|.}", "1d581721d86ead4ec5b3356b640ad503": "e^{\\frac {-E_{a}}{k_{B}T}}", "1d5875904f4d060f31bef3a06f7b3af7": "{\\dot {m}}", "1d58aacbca61067d3a49e219ab45bcc3": "G\\approx 0.8650{\\rm {\\ cm}}^{3}{\\rm {g}}^{-1}{\\rm {hr}}^{-2}.", "1d58f3169c11bb5c1e58cb861ecf17c5": "\\scriptstyle \\psi ^{\\dagger }(x)", "1d58f82b73910c13c8fbc7ff4a224220": "\\alpha _{F}", "1d590407211c73321619d8ff9f69f772": "\\lambda _{z,n}={\\frac {2\\pi \\,H}{\\alpha _{n}}}={\\frac {2\\pi \\,H}{\\sqrt {{\\frac {\\kappa H}{h_{n}}}-{\\frac {1}{4}}}}}.", "1d596daea8bb14160d6be0ca21907552": "{\\boldsymbol {\\sigma _{1}}}", "1d5996df077e0eb3bc551915ce350b15": "{\\sqrt {2}}m", "1d5a11b49d0db1090a8b06b07b9e5d0c": "\\scriptstyle \\eta =a\\,\\cos \\,2\\pi \\left({\\frac {x}{\\lambda }}-{\\frac {t}{T}}\\right)", "1d5a8eed291c76081bc5f3837bf6b03b": "\\Phi (\\mathbf {r} )={\\frac {q}{4\\pi \\varepsilon r^{\\prime }}}\\sum _{l=0}^{\\infty }\\left({\\frac {r}{r^{\\prime }}}\\right)^{l}\\left({\\frac {4\\pi }{2l+1}}\\right)\\sum _{m=-l}^{l}Y_{lm}(\\theta ,\\phi )Y_{lm}^{*}(\\theta ^{\\prime },\\phi ^{\\prime })", "1d5ad01bf19db90e246090e2edd640e2": "B_{3}(t)=\\sum _{i=0}^{n}z_{i}b_{i,n}(t){\\mbox{ , }}t\\in [0,1]", "1d5ad60e3f3b96e4c163f7c2d3e17079": "{\\begin{aligned}a'&=e^{-{\\frac {1}{2}}{\\pi b}}\\cos {\\frac {\\pi a}{2}}\\\\b'&=e^{-{\\frac {1}{2}}{\\pi b}}\\sin {\\frac {\\pi a}{2}}\\end{aligned}}", "1d5add4da2f1cf4a6d634e0364c18eb6": "Fp(1-p)\\!", "1d5aea1a16e1ca0318afebf8afd5dab2": "f(x)-R(x)=c_{m+n+1}x^{m+n+1}+c_{m+n+2}x^{m+n+2}+\\cdots ", "1d5b319daed3968632474c3925152ebf": "{\\frac {J_{X_{t}}}{X_{t}}}={\\frac {J_{n}}{n}}={\\frac {1}{n}}\\sum _{i=1}^{n}S_{i}\\to \\mathbb {E} S_{1}", "1d5bcddd8f72c9fdd5792ecf040b4190": "\\mathbf {H} ^{n}", "1d5bdb4e4f0a2930dbde370fe4329761": "\\rho ({\\vec {R}},{\\vec {R}}^{'})=1-{6\\sum _{i=1}^{N}(r_{i}-r_{i}^{'})^{2} \\over N(N^{2}-1)}", "1d5c293884de00260dc3401c9a937af6": "\\mathbf {n} /\\mathbf {N} =(n_{1}/N_{1},\\ldots ,n_{d}/N_{d})", "1d5c7a7ceaa34162179ad5899a42355b": "{\\begin{aligned}\\cos a&=\\cos b\\cos c+\\sin b\\sin c\\cos A\\\\&=\\cos b\\ (\\cos a\\cos b+\\sin a\\sin b\\cos C)+\\sin b\\sin C\\sin a\\cot A\\\\\\cos a\\sin ^{2}b&=\\cos b\\sin a\\sin b\\cos C+\\sin b\\sin C\\sin a\\cot A.\\end{aligned}}", "1d5c7da00713ff53d2b5572b4a4d3886": "e^{-cx}=a_{o}{\\frac {\\displaystyle \\prod _{i=1}^{\\infty }(x-r_{i})}{\\displaystyle \\prod _{i=1}^{\\infty }(x-s_{i})}}\\qquad \\qquad \\qquad (3)", "1d5cfc8aee4cdfdfae5e607679fc7d46": "\\Pr \\left[L\\leq x\\right]={\\frac {2}{\\pi }}\\arcsin \\left({\\sqrt {x}}\\right),\\qquad \\forall x\\in [0,1].", "1d5d4b7f9a4e70be1153d2ca59cbb619": "B_{MX}^{\\phi }=\\lambda _{MX}+I\\lambda '_{MX}+\\left({\\frac {p}{2q}}\\right)\\left(\\lambda _{MM}+I\\lambda '_{MM}\\right)+\\left({\\frac {q}{2p}}\\right)\\left(\\lambda _{XX}+I\\lambda '_{XX}\\right)", "1d5db6f070c64386ff6beb61f5b003d3": "f(z)={u_{1}(z) \\over u_{2}(z)},", "1d5df386b22a85673fb715c105056b35": "\\pm {\\sqrt {b^{2}-4ac}}", "1d5e405aaec56a6bd177dc0bc7b4dc03": "{\\frac {dz}{d\\zeta }}={\\frac {4n^{2}}{\\zeta ^{2}-1}}{\\frac {\\left(1+{\\frac {1}{\\zeta }}\\right)^{n}\\left(1-{\\frac {1}{\\zeta }}\\right)^{n}}{\\left[\\left(1+{\\frac {1}{\\zeta }}\\right)^{n}-\\left(1-{\\frac {1}{\\zeta }}\\right)^{n}\\right]^{2}}}.", "1d5e9b163626c6491d87617cc1b5cd9c": "{\\frac {\\partial {L}}{\\partial y}}={\\frac {\\mathrm {d} }{\\mathrm {d} t}}{\\frac {\\partial {L}}{\\partial {w}}}", "1d5efd5057833752abb9e5b4b4b9c476": "P^{-1}", "1d5f0987dc5ddb9f0e562b71ad1328f3": "\\xi \\to 0", "1d5f13a32cfc9f9dbf439cb713150287": "\\langle (x\\wedge y)\\rangle _{1}=0", "1d5f4ebbc17e65fd05d1f31e27819617": "P_{1}(x)=f(a)+f'(a)(x-a)\\ ", "1d5fcd4dc23c54451926d7dc44dc3de7": "\\Delta f=h", "1d5fec8f558e051bc01b16d69f8f114b": "2^{519}=1014=-5{\\pmod {1019}}", "1d5ffb0049a570d315d44b167a1691c6": "\\Phi (z=0,t)", "1d6003c5e88533a10ad268d0847541b1": "\\|f\\|_{B}=\\sup _{x\\in X}|f(x)|", "1d605122a27e4d8dc338b46db6dd0738": "\\Sigma _{1}={\\begin{bmatrix}0&0&0&\\cdots &0&1\\\\1&0&0&\\cdots &0&0\\\\0&1&0&\\cdots &0&0\\\\0&0&1&\\cdots &0&0\\\\\\vdots &\\vdots &\\vdots &\\ddots &\\vdots &\\vdots \\\\0&0&0&\\cdots &1&0\\\\\\end{bmatrix}}", "1d6058c3ce9f887f333fb0ded070b74e": "\\tau ={\\frac {T_{r}}{T}},\\delta ={\\frac {\\rho }{\\rho _{r}}}", "1d60842db666ff59bdc22df52c73a05a": "{\\bar {\\psi }}=\\psi ^{\\dagger }\\gamma ^{0}", "1d60b1442a9d5eacf8dd1eabf79f9c7c": "n_{y}=\\sin {\\theta (z)}", "1d6116c41063af5b064a7022b2389df8": "H[m]=\\sum _{i=1}^{N}\\sum _{j=1}^{N}\\Lambda (x_{i}-m_{j},m_{i}-m_{j},x_{i}-x_{j},i-j)", "1d61b934f3739f8624d0b16e35fca486": "Mq_{1}q_{2}q_{3}\\cdots q_{n}", "1d61ee0df5b4f14b76dad026d0ce7237": "(m,n)", "1d624ad71e522bbb7f350285861d662c": "U_{n,n}", "1d6294ec0ec341770a7e42e92f09ebae": "1\\in \\mathbb {H} ", "1d62aebfe58e35809090178c7bd470e3": "d=(1+v^{2})(u-t)(1+tu)", "1d62bd4ccdce8ac859106a51fdacd1b7": "\\int {\\frac {\\operatorname {li} (x)}{x}}\\,dx=\\ln x\\,\\operatorname {li} (x)-x", "1d62bf433344920508bcf728e23a0bba": "\\displaystyle {[a,b^{2},c]=2[a,b,c]b.}", "1d62dac31efa32b7bc1107ac057ad35e": "V_{\\mathrm {i} }=Ve^{-\\gamma x'}\\,\\!", "1d631eccc872cf5df8adc79a656c9543": "\\mathbf {a} \\mathbf {b} =\\sum _{i=1}^{7}f_{i}(\\mathbf {a} )g_{i}(\\mathbf {b} )w_{i}", "1d632e41fe2aa21c5e497bc4ea7a5fa3": "N_{z}", "1d6334d2b826f324c19a806a337916ac": "|\\psi |^{2}\\approx 1+2~\\mathrm {Re} \\left({\\frac {f(\\theta )}{z}}e^{ik(x^{2}+y^{2})/2z}\\right).", "1d6339c4e9579e94a94eab0ac041c057": "{\\Pr }_{\\theta ,\\varphi }(u(X)<\\theta z,x>y", "1d854f5053c4bf83e5d61e2f492903bd": "\\nabla ^{2}\\mathbf {H} =\\lambda ^{-2}\\mathbf {H} \\,", "1d855628a292b6f30e79cc9a2b97c3e7": "\\Pi =\\mathrm {surface\\ pressure} ", "1d8566f83d829f889db1c6a75fff2f44": "{\\frac {|\\nu -\\mu |}{\\sigma }}\\leq {\\sqrt {0.6}},", "1d8576b6520a3bdd8529045e2881751b": "-1.0185", "1d85e15e8c42d4c860536e0ed0a1414d": "({\\hat {\\theta }}_{i},\\theta _{i})", "1d85ed6726aabd57598d193483c4d8d8": "^{14}{\\text{NO}}_{3}^{-}+{^{15}}{\\text{NO}}_{3}^{-}\\rightarrow {^{14}}{\\text{N}}^{15}{\\text{N}}{\\text{O}},", "1d86318750e84e56e8b35ee8ddb90f44": "{\\frac {\\partial {\\overline {\\rho }}{\\tilde {u_{i}}}}{\\partial t}}+{\\frac {\\partial {\\overline {\\rho }}{\\tilde {u_{i}}}{\\tilde {u_{j}}}}{\\partial x_{j}}}+{\\frac {\\partial {\\overline {p}}}{\\partial x_{i}}}-{\\frac {\\partial {\\overline {\\sigma _{ij}}}}{\\partial x_{j}}}=-{\\frac {\\partial {\\overline {\\rho }}\\tau _{ij}^{r}}{\\partial x_{j}}}+{\\frac {\\partial }{\\partial x_{j}}}\\left({\\overline {\\sigma }}_{ij}-{\\tilde {\\sigma }}_{ij}\\right)", "1d865e3b55875f1ea75c2e130391a393": "Q=(m_{\\text{n}}-m_{\\text{p}}-m_{\\mathrm {\\overline {\\nu }} }-m_{\\text{e}})c^{2}", "1d873a0bdc91561d3a2249a8d7bc680b": "{\\mathcal {H}}^{(0)}=-{\\frac {\\hbar ^{2}}{2m}}\\left(\\nabla _{1}^{2}+\\nabla _{2}^{2}\\right)-{\\frac {e^{2}}{r_{1}}}-{\\frac {e^{2}}{r_{2}}}", "1d87851171785ace9925fe50a74c6035": "A\\otimes _{R}B", "1d8787b3e1d78a2432637c07a70bb4f2": "\\{(C\\otimes 1)\\Delta (C)\\}", "1d87e563779e35c308b3e14105676e62": "\\tau _{D}", "1d8801f11813b5ab1cff84d42b3f8248": "\\forall x,y\\in H:\\qquad \\lnot (x\\vee y)=\\lnot x\\wedge \\lnot y.", "1d882c8740ea7acc672e223eb0a5d08b": "K,m", "1d88764db0920f9142118ab7152c8edd": "d^{2}F_{x}=kII'dsds'\\left[\\left({\\frac {1}{r^{2}}}\\left({\\frac {\\partial r}{\\partial s}}{\\frac {\\partial r}{\\partial s'}}-2r{\\frac {\\partial ^{2}r}{\\partial s\\partial s'}}\\right)+r{\\frac {\\partial ^{2}Q}{\\partial s\\partial s'}}\\right)\\cos(rx)+{\\frac {\\partial Q}{\\partial s'}}\\cos(xds)-{\\frac {\\partial Q}{\\partial s}}\\cos(xds')\\right]", "1d88b7cfeb6250482de2665829a48d24": "\\displaystyle m_{1}=m_{2}=1", "1d891016a1af1c3d2629a64edd89186e": "t=x^{0}", "1d891b1f7023bfee3bfdc9a37fac6adc": "{\\cfrac {\\Gamma \\vdash f=g\\qquad \\Delta \\vdash x=y}{\\Gamma \\cup \\Delta \\vdash f(x)=g(y)}}", "1d892645821c09b9ed596974b6743d3b": "{\\widehat {1}}", "1d899cab1f6d6425a8953a423247442f": "\\forall a,b\\in F,x,y,z\\in {\\mathfrak {g}}", "1d89c6558230056446b622fe8270d797": "C_{G}\\varphi =\\psi \\wedge E_{G}(C_{G}\\varphi )", "1d8a8e11d5c23997d3d1f96ac8c8eb16": "B=R_{3}C_{5}+R_{1}C_{5}+R_{1}R_{3}C_{5}/R_{4}\\,", "1d8b1160213ecd78f58542e6940e688d": "H_{0}=\\mathbf {P} ^{2}/(2m)+V(R)", "1d8b1d86e6fac43aa734ce3ff485bb22": "\\Omega <\\Omega _{gp}", "1d8ba0e908e6cf5ad2b9867f2a67e2c9": "P(x_{1},\\ldots ,x_{n})=\\prod _{1\\leq ic", "1de5ad497ee53dbb833a1d08983e7454": "\\|X_{t}\\|_{p}=\\|X_{t}\\|_{L^{p}(\\Omega ,{\\mathcal {F}},\\mathbf {P} )}=\\left(\\mathbf {E} \\left[|X_{t}|^{p}\\right]\\right)^{\\frac {1}{p}}.", "1de5d075d6ef848ba4e376da21c3bdd1": "\\int _{E}\\varphi \\leq M\\mu (A),", "1de5ecffd06da46aa1d53dd092ebae15": "I_{SSP}=|f'_{i}f_{i}f_{z}\\chi _{iiz}^{(2)}|^{2}", "1de5f5c3f8ea1d74e467b2556cdd7c7f": "r(t)=0", "1de5fa297e36c4631ad16ab29e156896": "P(M2)/P(M1)=1", "1de64e2f0836654d868a134e60ed8069": "f(x_{1},\\ldots ,x_{n})=\\exp {\\Big (}{\\mathord {-}}\\textstyle {\\frac {1}{2}}\\displaystyle \\sum _{i=1}^{n}x_{i}^{2}{\\Big )}.", "1de664ec707d90bc57f381b34df6493f": "n-1,2n-1,\\dots ,2^{k}n-1", "1de7048a4a0cf6a9727ac843cbdc7d88": "D_{2}=\\sum _{P\\in E}{d_{P}[P]}", "1de7249ae2c2037f27c16de0d8a8e562": "\\alpha d\\geq 1\\qquad \\qquad (16)", "1de73570f74e6abebac3149f4bc81a5c": "(Eq.5){\\text{ }}E[Y_{i}(\\alpha ^{*}(t),\\omega (t))]\\leq -\\epsilon {\\text{ }}\\forall i\\in \\{1,...,K\\}", "1de74c386fba4dd379391103c5504b73": "wrap(H(M*share_{i}))", "1de74cec8715f07eaa1bb70ace22a931": "\\xi \\notin \\Sigma _{x}(f)\\iff \\exists \\phi \\in {\\mathcal {D}}_{x},\\ \\exists V\\in {\\mathcal {V}}_{\\xi }:{\\widehat {\\phi f}}|_{V}\\in O(V)", "1de762a65de4dac349ca69a2704cb53a": "a>0", "1de7ba39d7d0f39048e0c25d57fb4c43": "h^{2}", "1de7bba5087b35cd2bd086c543be2a0a": "{\\text{DOR}}^{\\prime }={\\frac {26/13}{12/48}}={\\frac {2}{1/4}}=8", "1de7d5a40868131086194e5b5cf22fc7": "2^{w}-2+{n \\over w}", "1de8d94eb7eb0f6b16eace0a9e5d29a6": "B(x,r)", "1de8ee568ae3d9835bbc9bb6e0922b84": "N_{E},", "1de8f604fb15e570931224cef7eb781d": "\\log _{b}{\\sqrt[{p}]{x}}={\\frac {\\log _{b}(x)}{p}}\\,", "1de902ca15957f704df8261c38059f7e": "\\mu _{1}+\\mu _{2}", "1de94aead55527d330f705167ab22c08": "\\pi ^{p,q}:E^{k}\\rightarrow \\Omega ^{p,q}.", "1dea274073a62beb01020a77de46f3e9": "\\int _{0}^{T}{\\partial f \\over \\partial W}(W_{t},t)\\circ \\mathrm {d} W_{t}+\\int _{0}^{T}{\\partial f \\over \\partial t}(W_{t},t)\\,\\mathrm {d} t=f(W_{T},T)-f(W_{0},0).", "1dea2fcd967e24d4f117f5a329bcfbd7": "\\underbrace {a+\\cdots +a} _{n{\\text{ summands}}}=0", "1deaac1da29e3d14f16afe72ff798c9d": "|\\psi _{A,B}^{\\lambda }\\rangle ", "1deb083d5023ba7d31e5395102fe07dc": "R_{o}={\\frac {1}{h_{o}A}}", "1debabc8374a2ef134e215b6a6bd0548": "(x_{k}\\ ,\\ y_{k})", "1debbe54dba95801e3faa292b2ca6be6": "{\\mathcal {O}}(-(n+1))", "1debd78014583b732b1f3cf1cc5c09e6": "\\sin(\\phi )\\,\\partial _{\\theta }+\\cot(\\theta )\\,\\cos(\\phi )\\,\\partial _{\\phi }", "1dec1a166251074116da41fcdab6a53d": "\\displaystyle {V(z)=\\int _{0}^{z}-U_{y}dx+V_{x}dy,}", "1dec495631d797b12539f985590dad8f": "\\alpha _{i}=w_{i}^{0}/w_{i},\\alpha _{j}=w_{j}^{0}/w_{j}", "1dec6a2bf7b1dcca28934c607dc6231c": "\\alpha :2^{N}\\setminus \\{\\emptyset \\}\\to [0,1]", "1dec9b8bb460b92746c53c327fd0714a": "G=\\operatorname {Gal} (L/\\mathbf {Q} )\\cong (\\mathbb {Z} /p\\mathbb {Z} )^{\\times },", "1decb1be6422b2f79cde017fb0d16066": "p(C,F_{1},\\dots ,F_{n})\\,", "1decbdaaa6c63fddefa9b4dec5c0bc67": "\\ LiPE=pIC_{50}-LogP", "1decc1582611559c6fc0de82d3f8cc28": "\\sin {\\frac {\\pi }{5}}=\\sin 36^{\\circ }={\\tfrac {1}{4}}{\\sqrt {2(5-{\\sqrt {5}})}},\\,", "1dece67cba02763af5161628c8319557": "\\mathbf {z} \\in [\\mathbf {x} ]", "1decf65a72cb7ac2505dcb1b81c6ba3b": "0<\\alpha \\leq 2", "1ded8e027258a7c5af6b41de2bbbd312": "0=(C_{\\beta I}^{\\;\\;\\;K}e_{K}^{\\alpha }e_{J}^{\\beta }-C_{\\beta J}^{\\;\\;\\;K}e_{I}^{\\beta }e_{K}^{\\alpha })e_{\\gamma }^{I}e_{\\delta }^{J}", "1ded92344130202f59bf76d2654bc6f5": "-\\infty ~{\\mbox{dB}}", "1dedb09e29bee9dbce2484e86436078f": "U_{coul}={{k\\,Z_{1}\\,Z_{2}\\,e^{2}} \\over r}", "1dee6dfdf7227bf0dba3870ccf39db7c": "(p_{1}(x)y^{\\prime })^{\\prime }+q_{1}(x)y=0\\,", "1deeca0bf079b2938a04f3d2320e4710": "\\phi (x,t)", "1def01a15617e8735b7ceac252d4257d": "\\{-S_{\\nu },-S_{\\nu }+1,\\ldots +S_{\\nu }-1,+S_{\\nu }\\}", "1df06e81064e3628e7ea7991f0efdcdf": "\\sum _{i}b_{i}\\beta S(c_{i})=\\varepsilon (a)\\beta ", "1df0bffb6b51ce9b4706f71bf988047f": "Y_{lm}(\\theta ,\\phi )\\,", "1df107efd8d34de0c4d18ba3eafe211c": "\\alpha _{i}\\,", "1df136faa1dc83d592e49a5c6c2af9f7": "+\\beta _{3}{\\left({\\frac {\\left[{1-\\exp \\left({-m/\\tau _{2}}\\right)}\\right]}{m/\\tau _{2}}}-\\exp \\left({-m/\\tau _{2}}\\right)\\right)}", "1df17e122bd0e48415de424bca436227": "K=K_{0}\\subset K_{1}\\subset \\cdots \\subset K_{\\infty },", "1df181eaa1bb40a0067c06ead197170d": "v_{i}", "1df183675d8ffd7c88b3cec87a4298fb": "{v_{R}^{3}}-{\\frac {1}{3}}\\left({1+{\\frac {8T_{R}}{p_{R}}}}\\right){v_{R}^{2}}+{\\frac {3}{p_{R}}}v_{R}-{\\frac {1}{p_{R}}}=0", "1df190750410b63a67a7363bc4111116": "\\sigma ^{2}\\mathbf {V} ", "1df19b202443af9a31a2e550d82b500c": "Q_{\\alpha }:=\\kappa ~\\int _{-h}^{h}\\sigma _{\\alpha 3}~dx_{3}\\,.", "1df19bac7fe0bd2d0b5bcf1135b0b708": "\\infty {\\text{ for }}1<\\alpha \\leq 2", "1df1ced8e3c3ba4edff179fc7437709a": "|F|^{2}", "1df20f26154730a6d82bca09dce5224e": "\\aleph _{\\alpha +1}", "1df22c28794d53578c6ca0cd22042322": "e(aP,bP)", "1df24731e63176f329736029af75b458": "y_{p}+c_{1}y_{1}+\\cdots +c_{n}y_{n}", "1df2ab3c50118f24be33fba5d440e80c": "n=\\sum _{i,j}n_{i,j}\\,", "1df2c056cf8e52de12585851b645fe2c": "(a+b)^{2}=a^{2}+2ab+b^{2}\\,", "1df30cf48b5f32023ec8eafbb42fbc5d": "\\mathbb {R} ^{n},", "1df3165a9a59617b013ae8d9d11df21e": "T^{ik}\\quad i\\neq k", "1df31fbb8d73ff712d9ee67af78c8233": "\\phi \\ =\\!\\left({Q \\over {ND^{3}}}\\right).\\,", "1df32c525d59a2a66ce9adc6c2180489": "\\operatorname {E} (X^{m})=k(k+2)(k+4)\\cdots (k+2m-2)=2^{m}{\\frac {\\Gamma (m+{\\frac {k}{2}})}{\\Gamma ({\\frac {k}{2}})}}.", "1df33a96013c92f9fe241b0a590b84a5": "{\\begin{aligned}G(0)&=0\\\\G(n)&=n-G(G(n-1)),\\quad n>0.\\end{aligned}}", "1df35a922fade3b3959e05b9c95ee7af": "C_{R}={\\sqrt {\\frac {mgh_{\\text{after}}}{mgh_{\\text{before}}}}}", "1df35ad86dbda1c7e2d633cd454fd549": "\\textstyle X=\\{X^{+},X^{-}\\}", "1df37b4fb42a199fcbfa74d5b53a23bf": "\\ \\sum _{w\\in V}f(u,w)=\\mathrm {excess} (u)\\geq 0", "1df393e4d06908549a854db9a7321736": "f_{A,B,C,D}=A'BC'D'+AB'C'D'+AB'C'D+AB'CD'+AB'CD+ABC'D'+ABCD'+ABCD.\\ ", "1df39db061257815e95ad24357573b57": "D={\\frac {\\pi }{8}}\\lambda ^{2}\\nu _{m}", "1df4154727320375999c5308abba1fae": "Q\\to \\mathbb {R} ", "1df41652d340541fb57804fa1eda22b0": "y=mx+b\\,\\!", "1df4347163c1ab391950554ba0080cbb": "\\mathbf {S} =\\mathbf {\\hat {n}} S", "1df4369acaf6c7a982a4816188b7fe2b": "ax^{2}+bx+c=0", "1df459705e2c4ee6d34c123e8e7f9e4a": "(\\wp ^{\\prime })^{2}=4(\\wp -e_{1})(\\wp -e_{2})(\\wp -e_{3}).", "1df461fcf54bd774d1636134ff74dcc4": "h(x,y)=k+xh_{1}(x,y)+yh_{2}(x,y)", "1df46e7600d3b91caafb3e729fd45157": "{\\frac {{x'}^{2}}{a^{2}}}+{\\frac {{y'}^{2}}{b^{2}}}=1.", "1df4757a0d45090031e682efe5d82112": "\\sum _{i=1}^{n}(Y_{i}-{\\hat {\\mu }}(x_{i}))^{2}+\\lambda \\int _{x_{1}}^{x_{n}}{\\hat {\\mu }}''(x)^{2}\\,dx.", "1df47b82d28942c9874ab4498956826c": "{\\mathit {q_{i}}}", "1df4b45e4433fa83c2fe1a62bd59e9aa": "r_{\\mathrm {outer} }={\\frac {a^{2}}{r_{s}}}\\left(1+{\\sqrt {1-{\\frac {3r_{s}^{2}}{a^{2}}}}}\\right)", "1df4c2f1262424302507253c0fb9688a": "b=N/K", "1df505889f0be5b811854368ce12f6ac": "{\\mathit {WER}}={\\frac {S+D+I}{N}}", "1df5713c50393d5a6d45de18d062efaf": "F^{+}", "1df599af0ed840284f23532e7dc2a9b2": "{\\begin{matrix}Proc&::=&{\\textit {STOP}}&\\;\\\\&|&{\\textit {SKIP}}&\\;\\\\&|&e\\rightarrow {\\textit {Proc}}&({\\text{prefixing}})\\\\&|&{\\textit {Proc}}\\;\\Box \\;{\\textit {Proc}}&({\\text{external}}\\;{\\text{choice}})\\\\&|&{\\textit {Proc}}\\;\\sqcap \\;{\\textit {Proc}}&({\\text{nondeterministic}}\\;{\\text{choice}})\\\\&|&{\\textit {Proc}}\\;\\vert \\vert \\vert \\;{\\textit {Proc}}&({\\text{interleaving}})\\\\&|&{\\textit {Proc}}\\;|[\\{X\\}]|\\;{\\textit {Proc}}&({\\text{interface}}\\;{\\text{parallel}})\\\\&|&{\\textit {Proc}}\\setminus X&({\\text{hiding}})\\\\&|&{\\textit {Proc}};{\\textit {Proc}}&({\\text{sequential}}\\;{\\text{composition}})\\\\&|&\\mathrm {if} \\;b\\;\\mathrm {then} \\;{\\textit {Proc}}\\;\\mathrm {else} \\;Proc&({\\text{boolean}}\\;{\\text{conditional}})\\\\&|&{\\textit {Proc}}\\;\\triangleright \\;{\\textit {Proc}}&({\\text{timeout}})\\\\&|&{\\textit {Proc}}\\;\\triangle \\;{\\textit {Proc}}&({\\text{interrupt}})\\end{matrix}}", "1df61e7f2446684a7a300e9420d62011": "[n/2]", "1df620affaeb14fc3171571997bdf3dc": "P\\left(S^{t}|S^{t-1}\\right)", "1df64bfbca56349ab1f424cdf971290a": "f\\left(\\perp \\right)=\\perp ", "1df664e1d99be05226de273dea7ef62e": "L_{O}\\;=\\;L_{U}\\;-\\;4.78\\;(\\log _{10}f)^{2}\\;+\\;18.33\\;\\log _{10}f\\;-\\;40.94", "1df675be3ba9e7f7d83b3d5d80c00f1f": "(s,t)", "1df6dd794032e3223d41e87b5a03b6c7": "2^{\\kappa }=\\kappa ^{+}", "1df6fe382decdbe5d1e57a57f9781bd9": "n=1,2,\\dots ", "1df76a4943737376230bff97483321f5": "p-q={\\frac {r'}{r}}", "1df7965db6cdd504e4fcc0bd5d525600": "X_{\\mu >\\epsilon }", "1df7a88342668cf09c9c7f585750bbe6": "I(F_{a};C)=\\sum _{v_{i}\\in F_{a}}\\sum _{c_{j}\\in C}p(v_{i},c_{j})\\log {\\frac {p(v_{i},c_{j})}{p(v_{i})\\,p(c_{j})}}", "1df7e018ccee4b10f078eabf749776bb": "\\scriptstyle \\Delta f\\,=\\,{\\frac {1}{T_{U}}}\\,\\approx \\,{\\frac {B}{N}}", "1df7e831297ee6ad1755f34604a7d6ae": "m=e/h^{k}.\\,", "1df82a65af6404e5ecc02b6996d52e0d": "{\\begin{matrix}S\\\\S(4,1,2)_{H}({\\bar {4}},1,2)_{H}\\\\S(1,2,2)_{H}(1,2,2)_{H}\\\\(6,1,1)_{H}(4,1,2)_{H}(4,1,2)_{H}\\\\(6,1,1)_{H}({\\bar {4}},1,2)_{H}({\\bar {4}},1,2)_{H}\\\\(1,2,2)_{H}(4,2,1)_{i}({\\bar {4}},1,2)_{j}\\\\(4,1,2)_{H}({\\bar {4}},1,2)_{i}\\phi _{j}\\\\\\end{matrix}}", "1df83c22566e7ce784d12b63f75e2d7b": "q({\\boldsymbol {x}})=f({\\boldsymbol {x}})/g({\\boldsymbol {x}})", "1df891ee336daa1d480e0f4e4bcfe5c1": "\\scriptstyle {A_{0}'\\sim A_{0}^{k^{2}}}", "1df8a4a4e0f8a6c31bf31d4c1c55c4b7": "{\\begin{aligned}(f*g_{N})[n]&=\\sum _{m=0}^{N-1}f[m]\\ g_{N}[n-m]\\\\&=\\sum _{m=0}^{n}f[m]\\ g[n-m]+\\sum _{m\\,=\\,n+1}^{N-1}f[m]\\ g[N+n-m]\\\\&=\\sum _{m=0}^{N-1}f[m]\\ g[(n-m)_{\\bmod {N}}]\\equiv (f*_{N}g)[n]\\end{aligned}}", "1df904db52397b02f7affb126532d43e": "e^{-At}\\mathbf {y} '-e^{-At}A\\mathbf {y} =e^{-At}\\mathbf {b} ", "1df9b2496ff2902ab27959eda4522ac5": "C(a,b,\\xi )", "1dfa44dac19fbe129a5c3445c64f935e": "{\\frac {ad}{bd}}={\\frac {cb}{db}}", "1dfa982f9fd6291a0dae769ba5d92300": "f(x)={\\begin{cases}\\sin {\\frac {5}{x-1}}&{\\text{ for }}x<1\\\\0&{\\text{ for }}x=1\\\\{\\frac {0.1}{x-1}}&{\\text{ for }}x>1\\end{cases}}", "1dfab721eb29fe296bb2117503cdf833": "\\eta _{IJ}=e_{\\beta I}e_{J}^{\\beta }", "1dfac909c5b4f67502bee95250077e2d": "{\\frac {\\partial {G}}{\\partial {w_{ij}}}}=-{\\frac {1}{R}}[p_{ij}^{+}-p_{ij}^{-}]", "1dfb693903c04376fbba72707d0c5b51": "\\exists x(\\forall y((\\phi (y)\\land y=a)\\leftrightarrow y=x)\\land a=x)", "1dfbd026fd0f0c0533329becc24a8f41": "\\int \\operatorname {csch} \\,x\\,dx=\\ln \\left|\\tanh {x \\over 2}\\right|+C,{\\text{ for }}x\\neq 0", "1dfbd5f936359002bd59ec4e9495c4b3": "\\!ds_{3}^{2}", "1dfc0dc62ab0b72188128f273546f6b4": "O(n!)", "1dfc1ac114077ccfb54c2beaffaa43a9": "\\mathbf {Q} _{1}={\\begin{pmatrix}1\\;0\\;0\\;0\\;0\\;0\\;1\\;0\\;0\\;0\\;0\\;1\\;1\\;1\\;0\\;1\\;1\\;0\\;0\\;0\\;0\\\\0\\;1\\;0\\;0\\;0\\;0\\;1\\;1\\;0\\;0\\;0\\;0\\;1\\;0\\;0\\;0\\;0\\;0\\;1\\;1\\;1\\\\0\\;0\\;1\\;0\\;0\\;0\\;0\\;1\\;1\\;0\\;0\\;0\\;0\\;1\\;1\\;1\\;0\\;1\\;0\\;1\\;1\\\\0\\;0\\;0\\;1\\;0\\;0\\;0\\;0\\;1\\;1\\;0\\;0\\;0\\;1\\;0\\;0\\;1\\;1\\;1\\;1\\;0\\\\0\\;0\\;0\\;0\\;1\\;0\\;0\\;0\\;0\\;1\\;1\\;0\\;1\\;0\\;1\\;1\\;0\\;1\\;1\\;1\\;1\\\\0\\;0\\;0\\;0\\;0\\;1\\;0\\;0\\;0\\;0\\;1\\;1\\;0\\;0\\;1\\;0\\;0\\;1\\;1\\;0\\;1\\end{pmatrix}},", "1dfc3aab6c6a1ad0be7d45c4f80c0d15": "(m-1)", "1dfc489688a6d018cf9f3f8d4b36daf7": "\\Pr[X_{t}\\in A|X_{0}=x]=P_{t}(x,A).", "1dfc984cd186c14bc2f982051249426c": "C=-2\\log _{10}\\left({\\varepsilon \\over 3.7D}+{2.51B \\over {\\mbox{Re}}}\\right)", "1dfca7cf06f97da3caffdd92493ae95b": "C_{k}=2\\sin \\left[{\\frac {(2k-1)}{2n}}\\pi \\right]\\qquad \\mathrm {k=odd} ", "1dfcd0d40e2df0f9246a6a50eec94d07": "\\prod _{k=0}^{n}(k+1)^{(-1)^{k+1}{n \\choose k}},", "1dfd2d833baf110820ff695854938b91": "f(x)={\\frac {p(x)}{q(x)}}", "1dfd442197164527a1c17663f87cc910": "\\Delta \\tau =Gb\\epsilon ^{\\tfrac {3}{2}}{\\sqrt {c}}", "1dfd84a3a4e325e59d305dae73c9abfa": "=\\sum _{k=1}^{d}{\\dot {q}}_{k}\\ {\\boldsymbol {e_{k}}}+\\sum _{k=1}^{d}\\sum _{j=1}^{d}\\sum _{i=1}^{d}q_{j}\\ {\\Gamma ^{k}}_{ij}{\\boldsymbol {e_{k}}}{\\dot {q}}_{i}\\ ", "1dfe00188d41c81852565054e5ffc450": "A\\to I\\otimes A\\xrightarrow {\\eta ^{l}} (A\\otimes A^{l})\\otimes A\\to A\\otimes (A^{l}\\otimes A)\\xrightarrow {\\epsilon ^{l}} A\\otimes I\\to A", "1dfe0367b2af745caabd0c4ecb6227f9": "T_{i_{1}i_{2}\\dots i_{r}}=T_{i_{\\sigma 1}i_{\\sigma 2}\\dots i_{\\sigma r}}.", "1dfe3ddd6a20e3a5597cdd6ac92a8500": "B_{k}", "1dfe43ffdcc981625d06c7bcd72d071f": "T_{1}", "1dfe54c9c1f45d7314aaf7a613e24288": "f(\\chi )=\\chi ^{\\frac {3}{2}}", "1dfecb246f5e3c96b396c40915f3cbcd": "-{\\frac {d^{2}}{dx_{e}^{2}}}f_{e}(x_{e})=k^{2}f_{e}(x_{e}).\\,", "1dfed80b15ba628ac3285509c581caf3": "p={1 \\over 50}=0.02", "1dff07f05201bd3b498721c16963c2d8": "T_{n}", "1dff0bbf926bdd57b1cd92e37054479a": "{\\begin{aligned}\\sum _{i=1}^{n}w_{i}\\log(x_{i})&\\leq \\log \\left(\\sum _{i=1}^{n}w_{i}x_{i}\\right)\\\\\\log \\left(\\prod _{i=1}^{n}x_{i}^{w_{i}}\\right)&\\leq \\log \\left(\\sum _{i=1}^{n}w_{i}x_{i}\\right)\\end{aligned}}", "1dff316f66d4c78c2d42d7cf911c1d76": "\\mathbf {\\Psi } (\\mathbf {x} )=\\sum _{\\ell =0}^{\\infty }\\sum _{m=-\\ell }^{\\ell }f_{\\ell m}(kr)\\mathbf {X} _{\\ell m}(\\theta ,\\phi )", "1dff618f0b9e2a63a20fd24a0b366d29": "90^{2}", "1dff826c9863d64159569d231dcbf583": "{\\frac {2a+b}{a}}={\\frac {a}{b}}\\equiv \\delta _{S}\\,.", "1dffcc3ca5837e9b45989b58f60d1f3c": "J_{1}^{1}Q\\to Q", "1e005a5e1667ad463b52df02be54c3d1": "z\\notin L\\implies \\Pr \\nolimits _{x}[\\phi (x,D_{n}(x,z),z)]\\leq {\\tfrac {1}{3}}", "1e0080f795a3f79218c6108fd24e5849": "\\int _{0}^{\\infty }x^{n}e^{-\\alpha x}dx={\\frac {n!}{\\alpha ^{n+1}}}.", "1e00b1d5a74ff4fb1dc7f4c860e906cb": "G_{\\infty }", "1e00d7935e25062f1c069773721987b7": "\\sigma _{\\text{avg}}={\\tfrac {1}{2}}(\\sigma _{x}+\\sigma _{y})", "1e00f5ee4127339bd403b4427c22ec0f": "\\textstyle \\lambda _{\\diamond N}=\\lambda (a_{\\diamond }\\mid A^{c})", "1e0150a7acdbc56d21b6d3543b44185e": "[0,T]", "1e017ccc1bb70de3e10e0d384224f4d7": "k(x,y)\\geq 0\\,\\,\\forall x,y", "1e01c0c5257f32a264d1ce400440c13d": "||\\cdot ||_{\\mathcal {H}}", "1e020145714349ed65e0141f34fb0ff6": "\\lambda \\ {\\stackrel {\\text{def}}{=}}\\ {\\frac {\\text{stress}}{\\text{strain}}}", "1e020b4cef70e46efcfa7eca6e284ce5": "vv_{i}\\delta _{i\\ell }=\\operatorname {trace} D_{i}D_{\\ell }=\\sum _{k=0}^{n}\\mu _{i}p_{i}(k)p_{\\ell }(k),\\quad (13)", "1e0229daa8b36edd93b7441685c7a21f": "{\\frac {d}{dt}}\\left.{\\!\\!{\\frac {}{}}}\\right|_{t=t_{1}}(F_{t,t_{0}}^{*}\\eta _{t})_{p}=\\left(F_{t_{1},t_{0}}^{*}\\left({\\mathcal {L}}_{X_{t_{1}}}\\eta _{t_{1}}+{\\frac {d}{dt}}\\left.{\\!\\!{\\frac {}{}}}\\right|_{t=t_{1}}\\eta _{t}\\right)\\right)_{p}", "1e0243c22cf1cea7557e901cc1a648a6": "L={\\frac {d^{2}}{dx^{2}}}+2{\\frac {d}{dx}}-3", "1e024eeb367ce692449ac013dcfb4376": "{\\rm {}}_{}^{2}\\Pi _{\\rm {g}}", "1e02be62dd84e9d143fb56aaf17cff93": "{\\sqrt {\\frac {1}{2}}}\\!\\,", "1e02c940a2ddb996551d42f499f4a217": "\\mathrm {n} +{}_{\\ 90}^{232}\\mathrm {Th} \\rightarrow {}_{\\ 90}^{233}\\mathrm {Th} \\xrightarrow {\\beta ^{-}} {}_{\\ 91}^{233}\\mathrm {Pa} \\xrightarrow {\\beta ^{-}} {}_{\\ 92}^{233}\\mathrm {U} +\\mathrm {n} \\rightarrow {}_{\\ 92}^{232}\\mathrm {U} +2\\mathrm {n} ", "1e02d3c90ea320fab39d11231b0ae156": "{\\frac {\\partial \\rho u_{i}}{dt}}+{\\frac {\\partial \\rho u_{i}u_{j}}{\\partial x_{j}}}=-{\\frac {\\partial P}{\\partial x_{i}}}+{\\frac {\\partial \\tau _{ij}}{\\partial x_{j}}}+S_{i},", "1e03062508b649616d73a0ecc6e8455c": "g=\\lambda x.f(x,x_{0})", "1e0393e4c92649f72485c1d972890876": "\\sum a_{n}^{2}", "1e03a25b2cd0c2bb73c7bde586f6c7c9": "\\bigoplus M_{i}", "1e04093b501bd763008d3a2af51d768e": "\\Psi _{\\sigma }(t)=c_{\\sigma }\\pi ^{-{\\frac {1}{4}}}e^{-{\\frac {1}{2}}t^{2}}(e^{i\\sigma t}-\\kappa _{\\sigma })", "1e04185298ca24bf4ee6fb138346d74d": "5x^{2}+8xy+5y^{2}=\\mathbf {x} ^{T}A\\mathbf {x} =(S^{T}\\mathbf {x} )^{T}D(S^{T}\\mathbf {x} )=1\\left({\\frac {x-y}{\\sqrt {2}}}\\right)^{2}+9\\left({\\frac {x+y}{\\sqrt {2}}}\\right)^{2}.", "1e042185015b14c1aeebe79b35ede84b": "\\tau =(T*Q)\\div (w*I)", "1e04d386db9d622844a019c53f0f74d1": "p=", "1e04f2bdeaf9585249f17f55593413e1": "\\sum _{n=-\\infty }^{\\infty }x_{1}[n]x_{2}^{*}[n]\\quad =\\quad {\\frac {1}{j2\\pi }}\\oint _{C}X_{1}(v)X_{2}^{*}({\\tfrac {1}{v^{*}}})v^{-1}\\mathrm {d} v", "1e059d412a4c83c6a304e8279bd045bd": "\\varphi =2\\arctan t,\\,", "1e05ae2024bca910ca19d06ed9d6dd70": "=e_{\\gamma }^{I}(\\nabla _{\\alpha }\\nabla _{\\beta }-\\nabla _{\\beta }\\nabla _{\\alpha })V_{I}", "1e06d30fd85493b647f5c0993d7c215e": "\\beta _{\\mu \\nu }^{G}=\\beta ^{\\Phi }=0,", "1e06f1b5b8b527c142781dcd56a0246f": "\\int x^{2}\\arccos(a\\,x)\\,dx={\\frac {x^{3}\\arccos(a\\,x)}{3}}-{\\frac {\\left(a^{2}\\,x^{2}+2\\right){\\sqrt {1-a^{2}\\,x^{2}}}}{9\\,a^{3}}}+C", "1e07361523c6c42f2d4e89854088a5b1": "\\left|{\\frac {BD}{DC}}\\right|=\\left|{\\frac {BF}{CK}}\\right|,\\,\\left|{\\frac {AE}{EC}}\\right|=\\left|{\\frac {AF}{CK}}\\right|", "1e0739add05675a5f8b1eb0ed4648284": "\\sum _{m}\\Lambda _{m}=I.", "1e07454f3a79a2e1babcac294b38d943": "n_{i1}", "1e0759765959983006db68634ebe6f1b": "\\cot ^{2}\\theta =\\csc ^{2}\\theta -1\\!", "1e0785f6e45e681b6994e5d2dedf646c": "h([f])([x])=f(x).", "1e07b95d54ae61ef99004a92f8fe1ee0": "F^{\\,ab}{}_{;b}={4\\pi \\over c}\\,J^{\\,a}", "1e07f918931f07d8bccbb8fa8d1aed5c": "S^{-1}E\\to QC", "1e07ff3e7fbfaa7a9cce21bc9bd4911a": "\\lim _{k\\to \\infty }{\\tilde {A}}^{k}=0.", "1e0810557b7fb9379a2d2da216988d9c": "{\\mathit {A}}", "1e0829840a8ebcea8ab634477c64baed": "{\\begin{aligned}\\operatorname {erfc} (x)&=1-\\operatorname {erf} (x)\\\\&={\\frac {2}{\\sqrt {\\pi }}}\\int _{x}^{\\infty }e^{-t^{2}}\\,dt.\\end{aligned}}", "1e083fdd1771fc156b9671e5e51827ba": "\\displaystyle f(x-a)\\,", "1e08448969430624ea5d6774d8634cc8": "F={\\sqrt {1-\\alpha }}", "1e085d0da474ac3a6750d1dfe8793853": "{\\mathcal {C}}=\\sum _{n\\geq 1}\\sum _{G\\in \\operatorname {Cl} (S_{n})}c_{G}(X^{n}/G)", "1e0878d8784d3ac9b439c9a49e759a90": "x^{\\prime }=\\gamma x^{*},\\quad y^{\\prime }=y,\\quad z^{\\prime }=z,\\quad t^{\\prime }={\\frac {t}{\\gamma }}-\\gamma x^{*}{\\frac {v}{c^{2}}}", "1e08cd27033ec496b4afbaeb61a5ba21": "t={\\frac {4k_{0}k_{1}e^{-ia(k_{0}-k_{1})}}{(k_{0}+k_{1})^{2}-e^{2iak_{1}}(k_{0}-k_{1})^{2}}}", "1e08d45ded57c82199afb08ee697d96b": "\\scriptstyle X\\;=\\;F\\,\\otimes \\,I", "1e090c855460d203f6f724c183b93a14": "{d \\over dt}\\left\\{X\\right\\}=\\left\\{A\\right\\}+\\left\\{X\\right\\}^{2}\\left\\{Y\\right\\}-\\left\\{B\\right\\}\\left\\{X\\right\\}-\\left\\{X\\right\\}\\,", "1e091611d92c98ad709051b6026a5382": "f(tx)=f(tx+(1-t)\\cdot 0)\\leq tf(x)+(1-t)f(0)\\leq tf(x)", "1e0936f3c664fdefd77bca2f8160c845": "\\lambda (\\rho ,\\theta )", "1e093896a811989d6495e6dceca6b695": "\\eta :k\\to K[G]~{\\text{by}}~\\eta (1)=\\sum _{X\\in G_{0}}\\mathrm {id} _{X}", "1e094a7e906873267a27b015aa06412e": "K_{a}={\\frac {[Ab-Ag]}{[Ab][Ag]}}", "1e09caee46b6e9e3d6bcda917808afe7": "\\zeta =[a_{0};a_{1},a_{2},\\dots ,a_{k},{\\overline {a_{k+1},a_{k+2},\\dots ,a_{k+m}}}],\\,", "1e09d5e72498868ecd01e65f2b13dbf1": "P_{\\gamma }(u)=\\gamma \\|\\nabla u\\|_{0}+\\|u-f\\|_{p}^{p}=\\gamma \\#\\{i:u_{i}\\neq u_{i+1}\\}+\\sum _{i=1}^{n}|u_{i}-f_{i}|^{p}", "1e0a4fdd7c0d0022dfe3fa292da0e43e": "\\ k>n", "1e0a66fa5070650229fb4133311a2a27": "1\\!-\\!\\sigma te^{-\\sigma ^{2}t^{2}/2}{\\sqrt {\\frac {\\pi }{2}}}\\!\\left({\\textrm {erfi}}\\!\\left({\\frac {\\sigma t}{\\sqrt {2}}}\\right)\\!-\\!i\\right)", "1e0ae887a8bc1a75eac1e908db71148e": "\\sigma ^{\\mu \\nu }={\\frac {i}{2}}[\\gamma ^{\\mu },\\gamma ^{\\nu }].", "1e0bce26efa9df31ee455c350132b69f": "t\\,J_{\\nu }(ut)", "1e0c472dce5fac8838a1ec2a8339fd8f": "C_{\\ell }^{m}", "1e0c515fbbf48571172dd9db1ae76adb": "{\\frac {R^{2}}{4}}\\!", "1e0c82468a72255a243355e65210b715": "|f|=1,e^{\\pm 1},e^{\\pm 2},\\ldots ", "1e0cdfa223e8363cf8d29d60bba85686": "Y_{9}^{0}(\\theta ,\\varphi )={1 \\over 256}{\\sqrt {19 \\over \\pi }}\\cdot (12155\\cos ^{9}\\theta -25740\\cos ^{7}\\theta +18018\\cos ^{5}\\theta -4620\\cos ^{3}\\theta +315\\cos \\theta )", "1e0d350de3eb8486a13405f1c6a9e2fe": "V_{n}\\,", "1e0d4bae6b43d66468848fb1b84199fb": "|\\psi _{+}\\rangle ={\\frac {1}{\\sqrt {2}}}{\\begin{pmatrix}e^{i\\phi }\\\\1\\end{pmatrix}}={\\frac {1}{\\sqrt {2}}}(e^{i\\phi }|\\psi _{1}\\rangle +|\\psi _{2}\\rangle )", "1e0d7f9f483b334c9338bbfa826c6902": "\\lambda =\\int \\limits _{S}\\,{\\vec {B}}\\cdot dS", "1e0d98dda5b53d7513c3c04358354a11": "K_{c}={\\frac {k_{+}}{k_{-}}}={\\frac {\\{S\\}^{\\sigma }\\{T\\}^{\\tau }}{\\{A\\}^{\\alpha }\\{B\\}^{\\beta }}}", "1e0db62a6c407ed0a9441ed95214d962": "\\zeta =\\sinh \\mu ", "1e0df5844efe59994ca77852a7fbe497": "s=-2", "1e0e8cb1678ea775e999fdbd18401dbf": "\\displaystyle (u_{xt}/u)_{tt}-(u_{xt}/u)_{xx}+2(u^{2})_{xt}=0", "1e0ea187e1aea82b76bcca184cb26a68": "a(a+c)=b^{2}", "1e0eb18249c0be2e7a41d28444a93fdc": "{\\frac {1}{\\rho }}{\\frac {dP}{dr}}=-{\\frac {Gm}{r^{2}}}", "1e0f064f1e21869a987dcab0d49af0b5": "\\lim _{\\gamma \\rightarrow 0}\\ln(Y)=\\ln(A)+\\alpha \\ln(K)+(1-\\alpha )\\ln(L).", "1e0f0cb2f38147b79cb6a2f0fbcec918": "f\\colon M\\to \\mathbf {C} ,", "1e0f6976c889a7e664bbd6a1b016495e": "{\\tfrac {12}{5}}", "1e0f89617fa4cf0b9e3f39bec7a5b120": "(x+{\\sqrt {p^{*}}})=(x+{\\sqrt {p^{*}}})(x-{\\sqrt {p^{*}}},q)=(cq,q(x+{\\sqrt {p^{*}}}))", "1e105ca1edf10a02f877a447a1eb4657": "N(h)", "1e10c82d417500bb233ebcbff5bf8b48": "b_{V}\\colon V\\times V\\to K.", "1e10d39f8f01bc372e71f42b8679a619": "{\\frac {1}{L_{\\mathrm {net} }}}=\\sum _{i}{\\frac {1}{L_{i}}}\\,\\!", "1e10f7272d7b7efeefc06d7af864565f": "\\{Y_{ij}:i=1,\\dots ,n;j=1,\\dots ,n\\}", "1e10f8d6287a173e44f5c08666d326c7": "b^{k}\\,", "1e11170bdeedd8d7c10a8c33d09805f0": "{\\hat {B}}\\rightarrow \\langle {\\hat {B}}\\rangle ", "1e113a9765e3544edb5801e4ef809d35": "\\left\\langle \\chi _{i},\\chi _{j}\\right\\rangle ={\\begin{cases}0&{\\mbox{ if }}i\\neq j,\\\\1&{\\mbox{ if }}i=j.\\end{cases}}", "1e11406e6bec7febca1885f80ff4150f": "6{\\frac {2}{3}}", "1e11537148f70bfcd4ae23170eee4bdb": "\\scriptstyle t\\,\\in \\,\\mathbb {R} ^{1}", "1e11b90628e8702455229e8f1ec5a4ea": "J_{n}=-{\\frac {\\cos {ax}}{(n-1)x^{n-1}}}-{\\frac {a}{n-1}}I_{n-1}\\,\\!", "1e1261b0a987624ef3c639af9223629f": "=\\int _{P(t_{1},t_{2})}C_{T}^{(V)}(V,T)\\,{\\dot {V}}(t)\\,dt\\,+\\,\\int _{P(t_{1},t_{2})}C_{V}^{(T)}(V,T)\\,{\\dot {T}}(t)\\,dt", "1e12695db456351073ba964c90ec65b8": "{\\mathfrak {m}}:={\\mathfrak {n}}\\cap R", "1e12697a9253051abbab0bd59341b281": "{\\frac {d\\eta }{dT_{H}}}(T_{\\mathrm {opt} })=0", "1e126d744d10e8cda59bbad027aa1d76": "\\!b", "1e127d1d7caf74f19411c3ad6c748447": "x={\\frac {m}{\\ell }}\\cosh {t};\\quad y=m\\sinh t;\\quad -\\pi {mv^{2} \\over r}.", "1e1e0321b4da4f26bccc41b71c61cb26": "{\\begin{array}{ll}&(\\mathbf {a} \\times \\mathbf {b} )_{i}={\\mathbf {e} _{i}\\cdot \\mathbf {a} \\times \\mathbf {b} }=\\varepsilon _{\\ell jk}{(\\mathbf {e} _{i})}_{\\ell }a_{j}b_{k}=\\varepsilon _{\\ell jk}\\delta _{i\\ell }a_{j}b_{k}=\\varepsilon _{ijk}a_{j}b_{k}\\\\\\Rightarrow &{\\mathbf {a} \\times \\mathbf {b} }=(\\mathbf {a} \\times \\mathbf {b} )_{i}\\mathbf {e} _{i}=\\varepsilon _{ijk}a_{j}b_{k}\\mathbf {e} _{i}\\end{array}}", "1e1e5eccf4a5ea7c3fd1741bdc2a7623": "R_{\\text{E}}", "1e1e62245868f79465d0394d369e4f48": "u(c)=ln(c)", "1e1e69b2b3974ba174f3736736e15421": "S_{0}=V_{0}U_{0}+V_{1}U_{1}", "1e1e801aeb11bb244603a70750d4334b": "{\\frac {\\partial y}{\\partial \\mathbf {x} }}=\\left[{\\frac {\\partial y}{\\partial x_{1}}}{\\frac {\\partial y}{\\partial x_{2}}}\\cdots {\\frac {\\partial y}{\\partial x_{n}}}\\right].", "1e1e8d5205f8b4bd6bb74bc5b114b14b": "{\\ddot {x}}\\to 0", "1e1f05888ba72db97143e254de930780": "A_{\\rho }=G\\left(R_{C2}//R_{L}\\right)\\ .", "1e1faa105a807c9426dc22241e9c8569": "({\\bar {3}},1)_{\\frac {1}{3}}", "1e1fcaab26b776d30b942ba55db4022b": "h\\nu =2\\mu _{n}B\\pm 2d_{n}E", "1e1fdaf362331e5dde393bbcf0e588d9": "u_{1},u_{2},\\dots \\in H^{1}(\\Omega )", "1e207f198eaa72d1d2a243f8a6dc13dc": "-{\\boldsymbol {e}}_{k}\\,a\\,{\\frac {\\cosh \\,{\\bigl (}k\\,(z+h){\\bigr )}}{\\sinh \\,(k\\,h)}}\\,\\sin \\,\\theta \\,", "1e20a1265e3ff10d80ec18008aadb1ad": "{\\frac {R_{r}^{'}}{s}}={\\frac {R_{r}^{'}(1-s)}{s}}+R_{r}^{'}", "1e20b86c6079c96584c82cb26dacce30": "\\log(\\operatorname {E} (Y|\\mathbf {x} ))={\\boldsymbol {\\theta }}'\\mathbf {x} ,\\,", "1e20c98e9a40aec4e34c003313b50d56": "(B_{t})_{t\\geq 0}", "1e212f2bd617078f8ece73468e1996b7": "I=Ft\\;", "1e215061a23606dce6d47f3eddaf53ac": "f(n,1)=1", "1e21c820c3d405d35f77578e3a764ae8": "M\\geq 2", "1e22436e93648bf46b139758d94363a9": "{\\hat {\\nu }}={\\hat {\\alpha }}+{\\hat {\\beta }}=3{\\frac {({\\text{sample excess kurtosis}})-({\\text{sample skewness}})^{2}+2}{{\\frac {3}{2}}({\\text{sample skewness}})^{2}-{\\text{(sample excess kurtosis)}}}}{\\text{ if (sample skewness)}}^{2}-2<{\\text{sample excess kurtosis}}<{\\tfrac {3}{2}}({\\text{sample skewness}})^{2}", "1e22c0a29a0f1efe1152ef209a85f29c": "\\mathbf {r} =(r_{1},r_{2},\\ldots ,r_{d})", "1e22cf3958a1e11bd4b597e8b9276584": "y=R2(x)", "1e22e102462d26e31173e264726a8b4b": "(S_{1}'\\cup S_{2}')''=(S_{1}''\\cap S_{2}'')'=(S_{1}\\cap S_{2})'.", "1e22f5cb8e9ca7f4a0cfb2b12c591e59": "\\mathbf {r} _{\\bot }=\\mathbf {r} -\\mathbf {r} _{\\parallel }", "1e235263c4f9532c6ff84e5d62b1fd78": "\\emptyset \\neq F\\neq \\mathbf {P} ^{(1)}", "1e2378037e79a15246de537d17f45e09": "\\ m_{i}=a_{i}M_{i}", "1e239f0f11b7ecdc27e614c12cbe461b": "\\log(k)-\\log(k_{0})=N^{+}", "1e23b655e511a3419dba719163cf9155": "{\\text{GC}}", "1e240e97012e638df064e1ec777733d4": "{\\frac {\\zeta (2s)}{\\zeta (s)}}=\\sum _{n=1}^{\\infty }{\\frac {\\lambda (n)}{n^{s}}}.", "1e2438a01cd72fa352a6a2220699a7be": "2+\\epsilon ", "1e2440eaeae55cbc52c42c6b9bdb8ee5": "a\\geq -1", "1e24756c0182c0072b66095ff4016b79": "e^{-}+p\\to \\nu _{e}+n", "1e249431948c84852ec2bf57c9cfc7db": "VP(\\alpha (t),\\omega (t))+\\sum _{i=1}^{K}Q_{i}(t)Y_{i}(\\alpha (t),\\omega (t))\\leq C+\\inf _{\\alpha \\in A}[VP(\\alpha ,\\omega (t))+\\sum _{i=1}^{K}Q_{i}(t)Y_{i}(\\alpha ,\\omega (t))]", "1e24ae3e82e399abf54b0db66ea780f3": "{\\frac {\\mu _{k}}{\\sigma ^{k}}}", "1e24b57a60fa0c2ee1ec42a041e0b459": "L(\\mathrm {dB} )=-20\\log |\\tau |\\ ", "1e24d7cbc803f83656cb43dfc048cbf0": "C^{*}={dS^{*} \\over d\\ln T}", "1e24ecdf242587ba77f74f0013b80aeb": "V_{(xu)}\\ {\\stackrel {\\mathrm {def} }{=}}\\ \\rho ^{i}(x,u){\\frac {\\partial }{\\partial x^{i}}}+\\phi ^{\\alpha }(x,u){\\frac {\\partial }{\\partial u^{\\alpha }}}\\,", "1e24f050f6b5031f08488b37a6747ca4": "A=L_{x}L_{y}", "1e25927467b6edc075e12ab885b7ca14": "M=T\\oplus P", "1e2598c2436ff79f9b7eda55cb72d6da": "\\scriptstyle (A_{1},A_{2},\\dots ,A_{n})", "1e25a28bcc613e69aa14c846f5e20967": "b_{\\mu }", "1e25b19c863180be3e08a84947d8be4b": "y=X\\beta +\\varepsilon ,\\,", "1e25c93c2e525e8f77d124c13b5ee4a2": "n=\\underbrace {1+1+\\cdots +1} _{n\\ {\\rm {times}}}", "1e25df665765ebc102d573435b6954a2": "\\alpha \\in S_{0}", "1e262a89a8c9a8c0972fc856219ad291": "2^{n^{2}}", "1e2760f04e9da338f7c8016af27e2132": "0\\leq j\\leq n", "1e27b8e81380cae5d9d3f663129d7247": "\\int _{a}^{b}f(x)\\,dx\\approx \\sum _{i=1}^{n}w_{i}f(x_{i}).", "1e27d9f62821ae2dc7c972ed836d8a54": "\\gamma _{P}(Q)=6/10", "1e27e8280ef2841621c6d02b6bc2d30e": "u_{n\\mathbf {k} }(\\mathbf {r} )=e^{-i\\mathbf {k} \\cdot \\mathbf {r} }\\psi _{n\\mathbf {k} }(\\mathbf {r} )", "1e27f8dff6b94cf7501a78d279c810e0": "f(\\mathbf {r} )", "1e27fa611b45d34533cc3f3640e5ff86": "(1+x)^{n}=\\sum _{r=0}^{\\infty }{n \\choose r}x^{r}.\\qquad ", "1e289ef3e74d4da0bf3522a08e3dd560": "Qg", "1e28bf2618ad0b447ac840c5b2a760ec": "X=(0,2)", "1e28eb7fdb064b732eedfb52ed545db1": "\\delta _{1}", "1e28f5ccd0a57b77be5be2345d3a4404": "H_{2}(S(2k+1,n))\\approx H_{2}(S(1,n))", "1e294daf7eb32c0569c072418b7ec80c": "{\\mathcal {L}}_{xy}^{4}", "1e297312635c9b63a628670bad30ff7d": "{\\frac {1+{\\scriptstyle {\\frac {2}{3}}}z+{\\scriptstyle {\\frac {1}{6}}}z^{2}}{1-{\\scriptstyle {\\frac {1}{3}}}z}}", "1e29a4c9c1cafa1e76cf817422645ae7": "\\ v_{1}=u_{1}", "1e29c0f2be0df7aa866c998b11f15078": "a=-2c,a\\in \\mathbb {R} ", "1e29e391c0913065b89ad63d8374d4f1": "\\scriptstyle E_{K}^{T}", "1e29f2abcb5c7dd05f5f58e3563205eb": "\\sum I(X_{i};X_{j(i)})", "1e2a5aff951b17199d1b84fcb15911de": "\\psi _{1}(\\Omega )=\\varepsilon _{\\Omega 2}", "1e2a8d3722ec9ec0bfba6308421eaadf": "A_{j,k}:=a_{j,k}-{\\overline {a}}_{j.}-{\\overline {a}}_{.k}+{\\overline {a}}_{..},\\qquad B_{j,k}:=b_{j,k}-{\\overline {b}}_{j.}-{\\overline {b}}_{.k}+{\\overline {b}}_{..},", "1e2aae8ca2d5a3610033c709151ad943": "0<\\lambda _{1}<\\lambda _{2}\\leq \\lambda _{3}\\cdots \\leq \\lambda _{k}\\leq \\cdots ", "1e2b2cce6a896b3f6f54a7f2051b3de4": "{\\begin{aligned}{\\tilde {G}}(A,B)\\cap {\\tilde {G}}(B,A)=\\emptyset \\end{aligned}}", "1e2b810ac17403f923606c61b7b33345": "\\displaystyle \\alpha _{0}:=\\arg \\min _{\\alpha }f(x_{0}+\\alpha \\Delta x_{0})", "1e2bf581085e56cb474e467c368bc5b0": "u_{5}={\\tfrac {(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+ax_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2})x_{13}-2x_{5}(x_{1}x_{9}+x_{2}x_{10}+x_{3}x_{11}+x_{4}x_{12}+bx_{5}x_{13}+x_{6}x_{14}+x_{7}x_{15}+x_{8}x_{16})}{c}}", "1e2c08643b32f74eed20a4ab0f43eb5d": "\\Lambda ^{0}{\\stackrel {d_{0}}{\\longrightarrow }}\\Lambda ^{1}{\\stackrel {d_{1}}{\\longrightarrow }}\\cdots {\\stackrel {d_{n-1}}{\\longrightarrow }}\\Lambda ^{n}", "1e2c45d1e61b80301668f46997ed6108": "{\\frac {\\xi (s)}{\\xi (0)}}={\\frac {\\det(H+s(s-1)+1/4)}{\\det(H+1/4)}}.", "1e2c4f3382d0e45a90e50c4470b05569": "\\Psi (x,t+\\tau )=\\int G(x,x',\\tau )\\Psi (x',t)dx'", "1e2c648a07414987f099fad84449e0c5": "P=D^{-1}A", "1e2c7bbfbf004c872ff4dac95436e45e": "F(t)=t^{3}+t^{2}+t+1.", "1e2c8e3cb41f7c53eb8acdf16a79a5b4": "\\sigma ({\\mathfrak {G}}^{2}\\oplus {\\mathfrak {G}}^{2}\\oplus {\\mathfrak {G}}^{2})=5/6", "1e2cbb00091ac276f03054a02f3c5096": "P_{1},\\dots ,P_{q}", "1e2cbfe20d37e7a02b0b7b3111920d9e": "\\mathbf {HM} ={\\begin{bmatrix}x_{1}^{1}&\\cdots &x_{n}^{1}&|&f(\\mathbf {x} ^{1})\\\\\\vdots &\\ddots &\\vdots &|&\\vdots \\\\x_{1}^{hms}&\\cdots &x_{n}^{hms}&|&f(\\mathbf {x} ^{hms})\\\\\\end{bmatrix}}.", "1e2cedcbd7996f95bce9efb56a675f5b": "V={\\begin{pmatrix}v_{1,1}&\\cdots &v_{1,n}\\\\\\vdots &\\ddots &\\vdots \\\\v_{j,1}&\\cdots &v_{j,n}\\end{pmatrix}}", "1e2d2a07db97dfd80a4a6ea7d9d7f3dd": "S^{1}:=\\{e^{i\\theta }:\\theta \\in [0,2\\pi ]\\}=\\{z\\in \\mathbb {C} :|z|=1\\}", "1e2d487248a85db8776b1bd299804741": "{\\dot {n}}_{k}", "1e2d4d3dc451d79c29471f15ae12231e": "{\\frac {4}{3}}\\pi r^{3}.", "1e2d529e5f026dbb066eaf336b996cc1": "{x_{1},x_{2},\\ldots ,x_{m}}", "1e2d6bbe28c18249f5e8f73f6d871709": "{\\mathcal {M}}=i{\\sqrt {\\frac {2\\omega _{p}}{Z}}}\\int \\mathrm {d} (x^{0})\\partial _{0}\\int \\mathrm {d} ^{3}xf_{p}(x){\\overleftrightarrow {\\partial }}_{0}\\eta (x)", "1e2d97211b830e72db51d66a5fff5028": "|CK|=R\\sin u=r\\sin \\theta .\\,", "1e2d99fe6e049f27fc2f36efe005c72b": "\\Gamma _{ik}^{2}={\\begin{bmatrix}0&0&1/r&0\\\\0&0&\\cot \\theta &0\\\\1/r&\\cot \\theta &0&0\\\\0&0&0&0\\end{bmatrix}}", "1e2db417da24999c108a7b1298feac0c": "E^{(+)}={\\mathcal {E}}_{1}exp(-i\\nu _{1}t)+{\\mathcal {E}}_{2}exp(-i\\nu _{2}t)", "1e2dd21ff5726cdb39fe594204144f6d": "d\\ell ^{*}(t)=|f\\,'(\\gamma (t))|\\,d\\ell (t)", "1e2e8ac2df3afd1c70b4671ee45973ef": "y_{0}=x_{0}+x_{1}\\omega ^{k}\\,", "1e2e9beb95534b1b064b66b6f12a53f5": "d(h)", "1e2ead32b82cf738fd51e053419d7b5f": "H_{n}(x)=2^{n}\\,U\\left(-{\\frac {n}{2}},{\\frac {1}{2}},x^{2}\\right)", "1e2ebe6ec2e5271bed3cd936a70753cc": "{\\dot {d}}(t)=Hd(t)", "1e2ed9c70078369aa32adf7258d91d59": "{\\bigl (}{\\begin{smallmatrix}\\\\~\\;0&4\\\\-1&5\\end{smallmatrix}}{\\bigr )}", "1e2f22562bc5e1e31c398ba8dec69ea4": "I_{12}={\\frac {V_{12}}{|Z_{\\Delta }|}}\\angle (30^{\\circ }-\\theta )", "1e2fae9a8c75fcdfcc9778a3de1e5b51": "\\ggg ", "1e2fbbc6d76a75e54f2f7124366b6e1d": "\\textstyle \\phi \\,\\!", "1e2fd236b0f1f6afe242c308a4f20107": "2-2g-n", "1e2ff085202e0f73bc3a9c1b217ceefe": "\\left[{\\begin{matrix}T_{1}^{(\\mathbf {n} )}&T_{2}^{(\\mathbf {n} )}&T_{3}^{(\\mathbf {n} )}\\end{matrix}}\\right]=\\left[{\\begin{matrix}n_{1}&n_{2}&n_{3}\\end{matrix}}\\right]\\cdot \\left[{\\begin{matrix}\\sigma _{11}&\\sigma _{12}&\\sigma _{13}\\\\\\sigma _{21}&\\sigma _{22}&\\sigma _{23}\\\\\\sigma _{31}&\\sigma _{32}&\\sigma _{33}\\\\\\end{matrix}}\\right].", "1e304c87731dfd154242c607c033a040": "c\\,\\!=2.9979\\times 10^{8}", "1e30583a668faf2830dde67886eae589": "p\\leq {\\sqrt {N}}", "1e30e4cb4f5a8339aced72cd68e7dab0": "\\left(ay_{1}\\right)v''+\\left(2ay_{1}'+by_{1}\\right)v'+\\left(ay_{1}''+by_{1}'+cy_{1}\\right)v=0.", "1e310372187f4d8a364537ce9f3dea8e": "\\sum _{k=1}^{\\infty }(-1)^{k}k^{1/k}.", "1e311d2af722d32aabe125cc7d4d09a0": "{\\begin{aligned}{\\overline {B}}_{0}\\pi _{0}&=\\pi _{0}\\\\\\quad \\left(\\mathbf {e} ^{\\text{T}}+\\mathbf {e} ^{\\text{T}}\\left(I-\\sum _{i=1}^{\\infty }{\\overline {A}}_{i}\\right)^{-1}\\sum _{i=1}^{\\infty }{\\overline {B}}_{i}\\right)\\pi _{0}&=1\\end{aligned}}", "1e3123ca721afdd8faf375e578def3d5": "{\\frac {dY}{dt}}=\\gamma .\\beta .X.Y-\\delta .Y", "1e313c8fb893ceb573a970f301c28a5d": "\\Delta _{0}>0", "1e31517a3fadc96248187953035b8d21": "\\mathbf {E} (\\mathbf {x} ,t)=-\\mathbf {\\nabla } \\phi (\\mathbf {x} ,t)-{\\frac {\\partial \\mathbf {A} (\\mathbf {x} ,t)}{\\partial t}}", "1e315926814501ae561ac55649ca2c2b": "B_{\\alpha }^{\\alpha }", "1e3175ad60733025780ecf44a00c3c12": "g(n)+h(n)", "1e320272dee87c54a0d0c11f37f010c4": "(x_{i},y_{i})", "1e32120bc32641948a6e1f23bce22a57": "\\,r_{\\max }-a=a-r_{\\min }", "1e322e66675741f9385158d461c51179": "\\{1,2,\\cdots ,n\\}", "1e32803809b74ee00277d188d53f31b5": "V_{L}=V_{\\{U,A\\}}+DT=E_{U}+DT=E_{L|T=0}+D+DT=E_{L|T>0}+D", "1e3283825ac3b269511bcebc810ed162": "E=\\sum _{n=1}^{\\infty }{\\frac {1}{2^{n}-1}}", "1e329fd14bfc1dbca6de9d44ef95c8fe": "n(n-1)", "1e32c08b5ab608d74bda7449ae33c40b": "1\\to \\Gamma \\to G\\to H\\to 1.", "1e32c7c40639e530a915f8a019184464": "{\\bar {W}}=\\int _{A}^{B}\\mathbf {F} \\cdot d(\\mathbf {r} +\\epsilon \\mathbf {h} )=\\int _{t_{0}}^{t_{1}}\\mathbf {F} \\cdot (\\mathbf {v} +\\epsilon {\\dot {\\mathbf {h} }})dt.", "1e32c88788e71ab442caebe09b5209dd": "c\\|u-u_{n}\\|^{2}\\leq a(u-u_{n},u-u_{n})=a(u-u_{n},u-v_{n})\\leq C\\|u-u_{n}\\|\\,\\|u-v_{n}\\|.", "1e33fc59e44f93c771bd7e3571abf63a": "{\\frac {T_{2}}{T_{1}}}={\\left({\\frac {V_{1}}{V_{2}}}\\right)^{\\gamma -1}}=r^{\\gamma -1}", "1e34158aea636fcba832e750ea990bd8": "x=z^{\\lambda }", "1e34da68df8b8d139f8715b6adb9210a": "{\\frac {\\operatorname {Li} _{s}(e^{it})}{\\zeta (s)}}", "1e34f0b9e24019efa08dcc2aa64d62dd": "Q_{h}\\,\\!", "1e34f6dd45bcfc2c9ca0c6cc148f6e3a": "1,2,3,\\ldots ,100\\,.", "1e352d77c0676af89b75de1fe497afd2": "\\zeta _{p}", "1e35427d0615a090fd1922a361bedaeb": "{\\frac {1}{\\lambda }}={\\frac {E_{\\text{i}}-E_{\\text{f}}}{12398.4\\,{\\rm {\\mbox{\\AA} }}\\,{\\text{eV}}}}=\\left({\\frac {12398.4}{13.6}}\\,{\\rm {\\mbox{\\AA} }}\\right)^{-1}\\left({\\frac {1}{m^{2}}}-{\\frac {1}{n^{2}}}\\right)=R_{\\text{H}}\\left({\\frac {1}{m^{2}}}-{\\frac {1}{n^{2}}}\\right)", "1e355379e5cabc37440ddeaee6cf96f3": "f\\cdot \\delta _{\\xi }=f(\\xi )\\cdot \\delta _{\\xi }", "1e358be5d3c4865d739eb5e215fe732a": "f(x;\\sigma )={\\frac {1}{2\\pi \\sigma ^{2}}}\\int _{0}^{2\\pi }\\,d\\phi \\int _{0}^{\\infty }dr\\,\\delta (r-x)re^{-r^{2}/2\\sigma ^{2}}={\\frac {x}{\\sigma ^{2}}}e^{-x^{2}/2\\sigma ^{2}},", "1e35af1dd7be0ef842b1465e0d2e18ca": "\\langle X,Y\\rangle =\\int _{\\mathcal {T}}X(t)Y(t)dt", "1e35d4b37940830eafff801767ea7878": "\\beta \\left(f\\right)", "1e366ca512b63b7ae4b43a53a09e7ca9": "2R_{1}={\\frac {2R_{b}R_{c}}{R_{T}}}", "1e3685905926003c67063d929b728f8f": "\\Delta G=RT\\ln {K_{\\rm {d}} \\over c^{\\ominus }}", "1e368b367b562e2b24baf9e7e472fef9": "K_{\\mathrm {max} }=h\\left(f-f_{0}\\right).", "1e3744b2efc3fa7fd0c96e919c64f59b": "\\displaystyle J_{n-1}", "1e37a3de24daabb18be66a06504edb8d": "k=0,\\ldots ,N/4-1", "1e37d2243ffa0c3e07b9ec3a15de1b3c": "H^{-1}=-H", "1e380e2e57747292097b7c4b1a500a4f": "\\psi (x)=\\sum _{p^{n}\\leq x}\\ln p=\\sum _{n=1}^{\\infty }\\theta (x^{1/n})=\\sum _{n\\leq x}\\Lambda (n).", "1e381e17eab6ada0fb074d04edba421c": "{\\bar {t}}_{i}", "1e382f269c34a8b0815ff928991765fa": "H_{\\text{kin}}={\\tfrac {1}{2}}m|\\mathbf {v} |^{2}={\\tfrac {1}{2}}m\\left(v_{x}^{2}+v_{y}^{2}+v_{z}^{2}\\right),", "1e38b469b80cfc6dbc4e93eeef8b141e": "\\rho _{SB}=\\rho _{S}\\otimes \\rho _{B}", "1e38bf1e82fb4056e643090ae7c2616d": "{\\begin{aligned}\\sigma =\\pi /E\\lambda \\end{aligned}}", "1e38c7344e689cd8a9f036f723587855": "\\textstyle {\\mathcal {Z}}=e^{-\\Omega /(kT)}", "1e39067b4f7fe5e485ff761c957f6398": "H_{j\\pm 1,j}^{}\\neq 0", "1e390ae7b04ced5cf31a0388edabf6e1": "{\\frac {dI}{dt}}=\\beta IS-\\nu I", "1e392c8c9b7fc3bef75bfa231d7f057d": "\\{\\langle A\\rangle \\}", "1e3963276740f08d71278f0cdfca2e96": "{\\bar {x}}=E\\{x\\}", "1e3a8381a429dfcc8ec660242f159644": "\\left({\\frac {m_{2}(t_{2})^{0.5}}{p_{2}}}\\right)", "1e3ab99ffdbff3af3048bed72735b709": "\\mathbf {\\hat {\\Omega }} ", "1e3b56bc6834a8007fa28f870a3bfe4b": "\\det(I+tA)=1\\!\\,", "1e3b7caedf11e130b16ffebe3415189b": "P(\\imath \\mu )", "1e3b960faf537820a0d297654df8bfae": "\\%K=100*(C-L5)/(H5-L5)", "1e3bd8cbcbcc9dc95145f2d5934819e6": "(R\\bowtie S)\\cup (\\{(\\omega ,\\dots ,\\omega )\\}\\times (S-\\pi _{s_{1},s_{2},\\dots ,s_{n}}(R\\bowtie S)))", "1e3be70ea009f7621f47bf2c2ffa0242": "P(x)=(x^{2}+ux+v)\\left(\\sum _{i=0}^{n-2}b_{i}x^{i}\\right)+(cx+d).", "1e3bfe87e8f1e4d9bc363f5b21336572": "|z|>|a|", "1e3c62d96bafbcbefcdcf2e76a6d021b": "{\\frac {\\partial x^{i}}{\\partial X^{\\alpha }}}=F_{~\\alpha }^{i}", "1e3c830c91f80cde3f1c3ba50bfa1f77": "\\mathbf {F} _{jk}=-\\nabla _{\\mathbf {r} _{k}}V=-{\\frac {dV}{dr}}\\left({\\frac {\\mathbf {r} _{k}-\\mathbf {r} _{j}}{r_{jk}}}\\right),", "1e3c91b3ec49933f2bfc4844769c0e72": "\\mathbf {\\nabla =\\partial /\\partial r} ", "1e3cdb7374d2bebeef8e35d478809387": "a_{ij}=g_{i}\\ \\log(\\mathrm {tf} _{ij}+1)", "1e3ce89d36b72bce86120bef0f2a34ee": "{Y_{i}=\\beta _{0}+\\sum _{j=1}^{p}{\\beta _{j}X_{i,j}}+\\varepsilon _{i}},", "1e3d22673f29dd81ee82c4d391c306c1": "\\lim _{x\\to A_{x}}y(x)=C={\\frac {1}{2}}\\ A_{x}\\left({\\frac {y'(0)+{\\sqrt {{y'(0)}^{2}+1}}}{1-{\\frac {V_{t}}{V_{d}}}}}-{\\frac {1}{(y'(0)+{\\sqrt {{y'(0)}^{2}+1}})\\ (1+{\\frac {V_{t}}{V_{d}}})}}\\right)", "1e3d68248046f7494b085aaad411de2d": "\\sum _{k=0}^{\\infty }(-1)^{k}k!=\\sum _{k=0}^{\\infty }(-1)^{k}\\int _{0}^{\\infty }x^{k}\\exp(-x)\\,dx", "1e3dbd2c487ac922d4bf423893170fb4": "\\cos {5x}=16\\cos ^{5}x-20\\cos ^{3}x+5\\cos x\\,", "1e3dd379db944b6a68599c109a8a1ebf": "2\\cdot A_{5}\\cong 2I;", "1e3ddc86694466e49b567756823f16c7": "2\\cdot \\$1\\cdot {\\sqrt {100\\cdot 18/38\\cdot 20/38}}\\approx \\$9.99", "1e3e248876093a873e743eb428d1046a": "\\tau _{E}=rC", "1e3e4ff16f6baa4cd3550bab5f8a5346": "I=I_{0}e^{V/V_{T}}", "1e3e7ebd591a19b526595b0d33e25e83": "{\\boldsymbol {\\xi }}\\geq 0", "1e3ebbbb84d762b678c967b4a48e66fc": "=\\sum _{i=0}^{\\infty }\\sum _{j=0}^{\\infty }a_{i}b_{j}(x-c)^{i+j}", "1e3edc773e96664b21e08f9f57e3909a": "\\varepsilon _{t}\\,", "1e3efa9f6dbc97637e6c81f6e3e4609c": "\\scriptstyle \\mathbb {P} \\left(\\{\\max(X,Y,Z)\\leq W\\}\\right)=\\mathbb {P} \\left(\\{X\\leq Y\\}\\cap \\{Z\\leq W\\}\\right).", "1e3f45e9978768b2964485a638b34b09": "K_{\\mathrm {w} }=[\\mathrm {H_{3}O} ^{+}][\\mathrm {OH} ^{-}]\\,", "1e3f6b6fb268d6d6846610de88a98517": "H_{n}({\\widetilde {X}},{\\widetilde {K}})", "1e3fab1f7f9cfbcd9075e8ed644b104c": "10\\cdot \\log _{10}0.000001=10\\cdot (-6)=-60", "1e3faeeeb7bbc004fbb6cddde64c14ad": "a_{N}.", "1e40088aeb26790910f939c4109b95d0": "\\nabla \\times \\mathbf {E} =-{\\frac {\\partial \\mathbf {B} }{\\partial t}}", "1e402d31333e6e9e107ec5c391aa4983": "L_{n}={\\begin{pmatrix}1&&&&&0\\\\&\\ddots &&&&\\\\&&1&&&\\\\&&l_{n+1,n}&\\ddots &&\\\\&&\\vdots &&\\ddots &\\\\0&&l_{N,n}&&&1\\\\\\end{pmatrix}}.", "1e4048c06b5051839aacbeab2292a8b1": "z/2", "1e406e15eb2d26271a2492ecb59fe4cb": "C={\\begin{pmatrix}0&1&0&0&0&0&0&0\\\\1&1&0&0&0&0&0&0\\\\0&0&0&0&0&0&0&-1\\\\0&0&1&0&0&0&0&-4\\\\0&0&0&1&0&0&0&-4\\\\0&0&0&0&1&0&0&2\\\\0&0&0&0&0&1&0&4\\\\0&0&0&0&0&0&1&0\\end{pmatrix}}.", "1e4093c2ec402e5000b88665cc90e8dc": "S_{\\mathrm {TeVeS} }=\\int \\left({\\mathcal {L}}_{g}+{\\mathcal {L}}_{s}+{\\mathcal {L}}_{v}\\right)d^{4}x.", "1e40ce4bdd56981553fb0cf000e5f009": "{\\boldsymbol {\\epsilon }}", "1e4181c897a8e179bdef6c606dfd511c": "\\,=xy", "1e41a3d7f7a85636818103a782f69b69": "\\langle {\\overline {\\Psi }}\\Psi \\rangle =\\int {\\frac {m}{(k^{2}+m^{2})^{1/2}}}{\\frac {2d^{3}k}{(2\\pi )^{3}}},\\quad 0\\theta >0", "1e4b2a81708c10a3c0a5ec189e88ce85": "d_{\\varrho }\\times d_{\\varrho }\\,", "1e4b34a3445892cc736bdfb854cf71fa": "11^{3}+12^{3}+13^{3}+14^{3}=20^{3}", "1e4b64ce049b4974dcaaf4a303b65cbb": "{\\begin{aligned}{\\mbox{E}}\\left[T\\right]&={\\begin{cases}\\mu {\\sqrt {\\frac {\\nu }{2}}}{\\frac {\\Gamma ((\\nu -1)/2)}{\\Gamma (\\nu /2)}},&{\\mbox{if }}\\nu >1;\\\\{\\mbox{Does not exist}},&{\\mbox{if }}\\nu \\leq 1,\\\\\\end{cases}}\\\\{\\mbox{Var}}\\left[T\\right]&={\\begin{cases}{\\frac {\\nu (1+\\mu ^{2})}{\\nu -2}}-{\\frac {\\mu ^{2}\\nu }{2}}\\left({\\frac {\\Gamma ((\\nu -1)/2)}{\\Gamma (\\nu /2)}}\\right)^{2},&{\\mbox{if }}\\nu >2;\\\\{\\mbox{Does not exist}},&{\\mbox{if }}\\nu \\leq 2.\\\\\\end{cases}}\\end{aligned}}", "1e4b80ad8cb0639e6472d95fcb4334e3": "k_{z}={\\frac {n\\pi }{D}}", "1e4c077dac47dd0ca2aeaf76e91ef292": "{\\frac {D_{F}X}{ds}}={\\frac {DX}{ds}}-(X,{\\frac {DV}{ds}})V+(X,V){\\frac {DV}{ds}},", "1e4c3ad17c402f7881a63b4de5119c1e": "\\exists \\mathbf {I} \\,(\\emptyset \\in \\mathbf {I} \\,\\land \\,\\forall x\\in \\mathbf {I} \\,(\\,(x\\cup \\{x\\})\\in \\mathbf {I} )).", "1e4c6ef14d4285258cb4cf256ba54e97": "{\\dot {\\gamma }}(0)", "1e4c8c2dbd7f101468a3d4d37c713ded": "R_{n}(x)=f(x)-p_{n}(x)={\\frac {f^{(n+1)}(\\xi )}{(n+1)!}}\\prod _{i=0}^{n}(x-x_{i})", "1e4ca7bfe6703f29bc91e6fd3d105638": "\\left(T_{p},\\Omega _{p}\\right)", "1e4d249f488e2aed941eeb594249467a": "GBWP_{\\omega >>{\\omega _{c}}}={A_{1}}(\\omega )\\cdot \\omega \\approx const.", "1e4d424194f50c262230c1da7d3fae4a": "{\\begin{aligned}A(1,2)&=A(0,A(1,1))\\\\&=A(0,A(0,A(1,0)))\\\\&=A(0,A(0,A(0,1)))\\\\&=A(0,A(0,2))\\\\&=A(0,3)\\\\&=4.\\end{aligned}}", "1e4d4d92c1af37c4fdd8f4aa22faaac7": "|{\\hat {v}}(\\zeta )|\\leq C_{m}(1+|\\zeta |)^{N}e^{H({\\text{Im}}(\\zeta ))}", "1e4d6beda591b5ec1a4faedcf669aeb8": "\\{e\\}(n)<_{\\mathcal {O}}3\\cdot 5^{e}", "1e4defbb027a039f2cc6f2d895f3c8cc": "dU=TdS-pdV", "1e4dff0e53f26b1ebc00a0b6ac2e2dcc": "\\alpha ^{-\\infty }=0", "1e4e0c1917dd0daa3b23b875c152f9a1": "x^{2}-2x+1=0", "1e4e13fb4446ade6e2b28182eda82a16": "r_{a}({\\mathfrak {u}},{\\mathfrak {s}})\\in R", "1e4e1c787ac9716ed6d775c9b46f1cfd": "\\sum _{i=1}^{\\infty }\\omega _{f}(\\theta ^{i}a)<\\infty ", "1e4e247239569ce6ffccc95412afa71d": "\\{\\{1,2,3\\},\\{4,5\\}\\}", "1e4e53d1de46b0c0385c9f92a7d57486": "L^{p_{1}}(\\mu _{1})", "1e4e63ad1646c182e276da11670ed597": "{\\hat {\\theta }}(\\theta )=\\theta ", "1e4ea6c5aed1f01961fa5c49ceab2e0e": "\\displaystyle i^{n}{\\frac {d^{n}{\\hat {f}}(\\nu )}{d\\nu ^{n}}}", "1e4eb1780a2cae2798ee7401b1a49974": "(I-\\alpha A^{T})^{-1}", "1e4eecbe642a29d96f004cfdd9b7de77": "\\delta =2\\arcsin({\\tfrac {d_{\\mbox{act}}}{2D}})", "1e4eef7558b91d7d68e50db23b2a8d5c": "\\Gamma (t;\\gamma ,\\lambda )", "1e4efd5777cf6575b29ceb94268c7f75": "\\Lambda _{i}(x_{0})", "1e4fa5a2f7e75dee62260b2b41e06b92": "|r+k|^{2}", "1e4fd27ec4108fb1b194a3e68bd3b9fd": "{\\frac {1}{(36+n)}}", "1e4fe334e1d0dadd55071989f7b4122c": "\\|A\\|_{\\text{max}}=\\max\\{|a_{ij}|\\}.", "1e4ff68109a469de5724197f2907022a": "\\ell \\,\\!", "1e507617c88afcc15b3ad58f77d93889": "\\Psi \\to e^{-\\Lambda }(\\Psi +Q_{B})e^{\\Lambda }", "1e5091e28a538015bcfa1654ba21ca42": "e^{i\\pi /4}{\\begin{pmatrix}1&0\\\\0&i\\end{pmatrix}}", "1e509e2224c03f1491ddbae75742305e": "P\\left(S^{0}\\wedge O^{0}\\right)", "1e50afbd1d095a4b30d39231713df069": "\\delta _{pen}", "1e50f561f62b6d9911310897f42b1b7d": "K_{\\text{L}}(x,x')=x^{T}x'", "1e5107910c05c6026d81c9cadcf55080": "\\displaystyle {\\|a^{2}\\|=\\|a\\|^{2},\\,\\,\\,\\|a^{2}\\|\\leq \\|a^{2}+b^{2}\\|.}", "1e5125525e34d539dad2e5826a0855c5": "z(\\mathbf {x} )=z_{B}+{\\text{nonnegative terms corresponding to nonbasic variables}}", "1e514639f323f456e033fdd8b7a3b961": "\\gamma _{k}", "1e51567b447d675a61e4d3f346f48e85": "\\Delta \\sigma =2\\arcsin \\left({\\sqrt {\\sin ^{2}\\left({\\frac {\\Delta \\phi }{2}}\\right)+\\cos {\\phi _{1}}\\cos {\\phi _{2}}\\sin ^{2}\\left({\\frac {\\Delta \\lambda }{2}}\\right)}}\\right).\\;\\!", "1e5176e8dc3ffc13a5b1c575a3ef66c1": "{\\begin{matrix}B\\left(\\cos ^{2}\\theta \\ -\\ \\sin ^{2}\\theta \\right)\\ -\\ 2\\left(A\\ -\\ C\\right)\\sin \\theta \\cos \\theta \\ &=&0\\\\\\\\B\\cos 2\\theta \\ -\\ \\left(A\\ -\\ C\\right)\\sin 2\\theta &=&0\\\\\\\\B\\cos 2\\theta &=&\\left(A\\ -\\ C\\right)\\sin 2\\theta \\\\\\\\\\cos 2\\theta &=&{\\frac {\\left(A\\ -\\ C\\right)\\sin 2\\theta }{B}}\\\\\\\\\\cot 2\\theta &=&{\\frac {A\\ -\\ C}{B}}\\end{matrix}}", "1e517786a0e67442b0e893d616545145": "\\sum _{i,j=1}^{n}c_{i}c_{j}K(t_{i},t_{j})\\geq 0", "1e519606e956f1a22a01608b9d209ff6": "f^{\\prime }", "1e51c5bbc73626240b4653e7ce21a196": "F_{A_{CO_{O}}}", "1e51d804270b348c8dc7f76e1ce40781": "{\\mathbf {v}}={\\frac {\\mathrm {d} {\\mathbf {r}}}{\\mathrm {d} t}}", "1e51f69d124674835d343bce4e9317ef": "\\varepsilon _{zz}^{\\mathrm {face} }=\\varepsilon _{xz}^{\\mathrm {face} }=0~;~~\\varepsilon _{zz}^{\\mathrm {core} }=\\varepsilon _{xx}^{\\mathrm {core} }=0", "1e51f9af3dbb6da3de414a41fe36c0a7": "\\omega _{p}^{2}={4\\pi ne^{2} \\over m}", "1e524043d819073e5c99baa905aa5ae0": "{\\dot {Q}}_{A}", "1e52e54235a832a1cb5487cc6d6eced3": "\\epsilon \\circ \\eta \\colon K\\to K", "1e53088e4e8de854d6388882d6bfafcc": "{\\tilde {Q}}", "1e53191b63d73e90b087796538b00d08": "S\\subset \\mathbb {C} ", "1e534c5017df637c225a19d2dab347af": "\\Rightarrow (\\lambda _{j}-\\lambda _{i})v_{j}'v_{i}=0", "1e53541040158ce823175a992e94b3d1": "\\{p_{i}\\}", "1e53bf640a7fccf0641120be0fe6b4d5": "\\|f(v)\\|=\\|v\\|", "1e53d393f73a027f66e24444419c8cec": "R(0)=Q(0)-P(0)=0", "1e549d985ee4c7496664c23e8bd864c6": "Z=(-1)^{n}\\left(|a|\\cdot |2a|\\cdot |3a|\\cdot \\cdots \\cdots \\left|{\\frac {p-1}{2}}a\\right|\\right).", "1e550efc3e079ab4a53032a92120f630": "\\phi _{3}(0)", "1e559cf7727cb99e66dc2a8895ec93d8": "F[\\rho ]=\\int f({\\boldsymbol {r}},\\rho ({\\boldsymbol {r}}),\\nabla \\rho ({\\boldsymbol {r}}))\\,d{\\boldsymbol {r}},", "1e55c86eb8461e6037b49096e60aaac4": "H={\\frac {|0\\rangle +|1\\rangle }{\\sqrt {2}}}\\langle 0|+{\\frac {|0\\rangle -|1\\rangle }{\\sqrt {2}}}\\langle 1|", "1e561d65475b286cda62e86aa46b1455": "\\,\\gamma \\,", "1e56337a9807469ffcbf8cfbb789132e": "\\lambda _{2}={\\text{L-scale}}.", "1e56ab39ac37d9e52a28ab759d6a5c9d": "\\|f\\|^{2}=\\langle f,f\\rangle =\\int \\langle f,x\\rangle \\langle x,f\\rangle \\,dx=\\int f^{*}(x)f(x)\\,dx", "1e56e7eb4fdad20212e427be90489579": "c_{p}={\\frac {c_{p0}}{\\sqrt {{M}^{2}-1}}}.", "1e56f07219f471ac26749e4a5a07972d": "{\\frac {\\delta {\\mathcal {S}}}{\\delta x(t)}}=0", "1e570eca215ae91746ddf576aabd7506": "\\,g_{1}+g_{2}+g_{3}", "1e571e9b12107359edef3b6421538e4f": "[ab,c]=[ba,c]", "1e5746ef63bd8b4d9d5ce55f97a3fcf0": "\\{a_{1},\\cdots ,a_{l}\\}", "1e58550590c5e7360a977a8843de23c2": "A={\\begin{bmatrix}3&1/7\\\\4&-1/7\\end{bmatrix}}{\\begin{bmatrix}5&0\\\\0&-2\\end{bmatrix}}{\\begin{bmatrix}3&1/7\\\\4&-1/7\\end{bmatrix}}^{-1}={\\begin{bmatrix}3&1/7\\\\4&-1/7\\end{bmatrix}}{\\begin{bmatrix}5&0\\\\0&-2\\end{bmatrix}}{\\begin{bmatrix}1/7&1/7\\\\4&-3\\end{bmatrix}}.", "1e58846148e5819f8129066addc73cbf": "{\\frac {S_{n}}{\\sqrt {n\\log \\log n}}}\\ \\xrightarrow {p} \\ 0,\\qquad {\\frac {S_{n}}{\\sqrt {n\\log \\log n}}}\\ {\\stackrel {a.s.}{\\nrightarrow }}\\ 0,\\qquad {\\text{as}}\\ \\ n\\to \\infty .", "1e58e44b4471a36ec24119e084077aad": "(Y_{t})", "1e5905dac89d15fc7561f6e74088f630": "\\left({\\frac {\\partial L}{\\partial {\\dot {\\mathbf {q} }}}}{\\dot {\\mathbf {q} }}-L\\right)T-{\\frac {\\partial L}{\\partial {\\dot {\\mathbf {q} }}}}{\\frac {\\partial \\phi }{\\partial \\epsilon }}.", "1e59a5887a7ae4f6a0388d082005bef7": "\\{\\gamma ^{\\mu }={\\frac {1}{\\gamma _{\\mu }}}\\}", "1e59be739bec1935e1de3a198e463e2b": "{\\begin{aligned}\\int _{a}^{x}{\\frac {f^{(k+1)}(t)}{k!}}(x-t)^{k}\\,dt=&-\\left[{\\frac {f^{(k+1)}(t)}{(k+1)k!}}(x-t)^{k+1}\\right]_{a}^{x}+\\int _{a}^{x}{\\frac {f^{(k+2)}(t)}{(k+1)k!}}(x-t)^{k+1}\\,dt\\\\=&\\ {\\frac {f^{(k+1)}(a)}{(k+1)!}}(x-a)^{k+1}+\\int _{a}^{x}{\\frac {f^{(k+2)}(t)}{(k+1)!}}(x-t)^{k+1}\\,dt.\\\\\\end{aligned}}", "1e59ca70316288f7068f793f6f65f09b": "x_{2}=b(x,t)", "1e5a15341a53a855acf9e2ed7ef87c48": "g_{k}=2\\sin \\left[{\\frac {(2k-1)}{2n}}\\pi \\right]\\qquad \\mathrm {k=1,2,3,\\ldots ,n} .", "1e5a745950640ec83e2db28fefba5713": "\\mathbf {J} (\\mathbf {x} )e^{i\\omega t}", "1e5a7aa7fa68e5da8e2037688fea874a": "{\\bar {\\psi }}\\mapsto {\\bar {\\psi }}e^{-i\\Lambda }", "1e5a99226dcb6668c2ad852cdeaf0b58": "\\scriptstyle A_{k}\\,\\subseteq \\,A", "1e5ae0790d23df2ab8ef0deeeda01572": "C=\\{\\theta _{0}u_{0}+\\dots +\\theta _{k}u_{k}|\\theta _{i}\\geq 0,0\\leq i\\leq k,\\sum _{i=0}^{k}\\theta _{i}=1\\}", "1e5b90c28bf733fcbc42234d9d2c2e57": "\\displaystyle u^{\\circ }\\ ", "1e5ba4374603d680da6650dccca5efc1": "\\exists x\\colon F(x)", "1e5bbac4af7b8387f6972b60f9451f2e": "f.\\,", "1e5bca6acba54a125db59c97dae06728": "F_{-X}", "1e5c0187d71ff6b16bcb2cf3b3c07266": "v_{max}", "1e5c05a80c9511f22e15fd520d6d62c9": "x=x_{0}-\\delta ", "1e5c08d41d18ca7ec201a49f599bf1d7": "D_{m'm}^{s}(\\alpha ,\\beta ,\\gamma )\\equiv \\langle sm'|{\\mathcal {R}}(\\alpha ,\\beta ,\\gamma )|sm\\rangle =e^{-im'\\alpha }d_{m'm}^{s}(\\beta )e^{-im\\gamma },", "1e5c312fe8d876897fd931933d471c01": "{\\frac {F_{1}}{F_{2}}}={\\frac {k_{1}}{k_{2}}}={\\frac {c_{2}}{c_{1}}}", "1e5c39d3fa6723a61e41ef9730184404": "\\,{\\begin{aligned}(n-1)^{x}(n-1)!&\\leq \\Gamma (n+x)\\leq n^{x}(n-1)!\\\\(n-1)^{x}(n-1)!&\\leq (x+n-1)(x+n-2)\\cdots (x+1)x\\Gamma (x)\\leq n^{x}(n-1)!\\\\{\\frac {(n-1)^{x}(n-1)!}{(x+n-1)(x+n-2)\\cdots (x+1)x}}\\leq \\Gamma (x)&\\leq {\\frac {n^{x}(n-1)!}{(x+n-1)(x+n-2)\\cdots (x+1)x}}\\\\{\\frac {(n-1)^{x}(n-1)!}{(x+n-1)(x+n-2)\\cdots (x+1)x}}&\\leq \\Gamma (x)\\leq {\\frac {n^{x}n!}{(x+n)(x+n-1)\\cdots (x+1)x}}\\left({\\frac {n+x}{n}}\\right)\\\\\\end{aligned}}\\,", "1e5c506eafe162ad5ae6ec998e20de21": "A={\\begin{pmatrix}A_{11}&A_{12}\\\\A_{21}&A_{22}\\end{pmatrix}}.", "1e5cca92761444cf1f0af1d409169ae1": "2\\sum |a_{n}|", "1e5d15d224c79e93e43cf38cfda3f702": "\\operatorname {sgn} \\cdot R_{i}", "1e5d33039d6915bc2c092dd3c2c569a5": "N=\\Lambda ", "1e5d834efde5c5a6035bb70686966c1c": "{\\frac {{4 \\choose d}2^{d}}{80}}", "1e5e187c4dc39fafb616cae64fbd60f4": "M_{i,j}={\\begin{cases}1&(x_{i},y_{j})\\in R\\\\0&(x_{i},y_{j})\\not \\in R\\end{cases}}", "1e5e452bcbe2e7ab4b4e8a31124f1699": "(b^{2}+{{a^{2}} \\over 2})\\pi ", "1e5e98a1d443ce997009f4a934899bb5": "\\lambda =\\lambda ^{e}\\cdot F\\lambda ^{g}\\,", "1e5f4524228e7fbe821213ee3ea49ff4": "\\ \\theta ", "1e5f59030bba1744e19b56d70954ec83": "\\displaystyle ax+by=1,", "1e5f68e0199b14f2fcf1dd97b4e2e023": "X,\\,P", "1e5f8e371e5f2702dd614118c73a18c4": "\\varphi (\\mathbf {r} ,t)=\\int {{\\delta \\left(t'-{{\\left|\\mathbf {r} -\\mathbf {r} '\\right|} \\over c}-t\\right)} \\over {\\left|\\mathbf {r} -\\mathbf {r} '\\right|}}\\rho (\\mathbf {r} ',t')d^{3}r'dt'", "1e5fa6bc8b78c30f20e2881cda58c960": "a=E(X)-{\\sqrt {3V(X)}}", "1e5fcd4e3e504765906bf72c15c72526": "S=125348=1\\;1110\\;1001\\;1010\\;0100_{2}=1.1110\\;1001\\;1010\\;0100_{2}\\times 2^{16}\\,", "1e5fe5f8e0559984d1b9c83487a88896": "(a+b+c)^{3}=a^{3}+3a^{2}b+3a^{2}c+b^{3}+3ab^{2}+3b^{2}c+c^{3}+3ac^{2}+3bc^{2}+6abc", "1e6090e11edd7143eb8873c5a471f504": "\\zeta (s)=1+{\\frac {1}{2^{s}}}+{\\frac {1}{3^{s}}}+{\\frac {1}{4^{s}}}+\\cdots ", "1e60ac5b6a6c68387adbf24423080499": "{A_{\\mathrm {v} }}=g_{21}={\\begin{matrix}{v_{\\mathrm {out} } \\over v_{\\mathrm {in} }}\\end{matrix}}{\\Big |}_{i_{out}=0}", "1e60c5dbe3c4ceadbcdd83c3da14f717": "L\\in \\mathbf {H} _{n}", "1e60c992dbf0b1058a651aefa5756197": "{D}_{8}^{(1)}", "1e60cbfdbc8e58ee4666f8347c45772d": "{\\frac {\\partial }{\\partial t}}\\int _{V}|\\Psi |^{2}\\mathrm {d} V+", "1e611d330e969808712c01bfa73373aa": "\\left(x-\\left[a+{\\sqrt {b}}\\right]\\right)\\left(x-\\left[a-{\\sqrt {b}}\\right]\\right)=(x-a)^{2}-b.", "1e6135086e34aadbecfb6c2db3657684": "\\phi \\in [-\\pi /2,\\pi /2]", "1e6139569ed78daa97e465c4c47f98de": "e^{x}=\\sum _{n=0}^{\\infty }{\\frac {x^{n}}{n!}}.", "1e614e8c622dcee85c766fcb66f836a2": "b_{0}\\,", "1e619c7acca2c56de9ed36d8cb38da57": "b>a^{k}.", "1e6232d9749cf454e8b1e01c3aac263c": "\\gamma ^{\\mu }\\gamma ^{\\mu }=\\eta ^{\\mu \\mu }\\,", "1e6239456a8cc478884590c65ca4a49e": "\\sum _{k}\\left|y_{k}-x_{k}\\right|<\\delta ", "1e625395b869eab77e08995e859ec6c2": "U_{\\text{B}}-U_{\\text{A}}\\ll kT", "1e626c8462d7a9509166e454df2195d9": "\\Delta G_{vap}=\\Delta H_{vap}-T_{vap}\\times \\Delta S_{vap}=0", "1e628035d8c3452ec5f5956ce7932ee3": "\\Pr(E^{{\\tilde {P}}(x)}(x)\\in W(x))\\geq \\Pr({\\tilde {P}}(x)\\leftrightarrow V(x)\\rightarrow 1)-\\kappa (x).", "1e628e069c87a62f83812cc4d2efdbb2": "{\\stackrel {\\nabla }{\\mathbf {T} }}={\\frac {\\partial }{\\partial t}}\\mathbf {T} +\\mathbf {v} \\cdot \\nabla \\mathbf {T} -((\\nabla \\mathbf {v} )^{T}\\cdot \\mathbf {T} +\\mathbf {T} \\cdot (\\nabla \\mathbf {v} ))", "1e629c3ae6802b986ae4e40f5a5b78d4": "\\langle F,R,V\\rangle ", "1e62b4784321049a13156262c19eb09e": "\\|f\\|_{L^{2}(\\mathbf {R} ^{d})}\\leq Ce^{C|S||\\Sigma |}{\\bigl (}\\|f\\|_{L^{2}(S^{c})}+\\|{\\hat {f}}\\|_{L^{2}(\\Sigma ^{c})}{\\bigr )}~.", "1e63062d3a6a7a15572d3dcdd3f1917d": "\\sum _{s}Z_{s}eB\\int dv_{||}d\\mu d\\varphi h_{s}\\left({\\vec {R}}\\right)=\\sum _{s}{\\frac {Z_{s}^{2}e^{2}n_{s}\\phi }{T_{s}}}", "1e632750f338be384b9c593e0b9ba0d5": "{\\begin{matrix}{\\frac {1}{1}}\\end{matrix}}", "1e6340e8a3c403489533caa0a82995fc": "y_{1}(x)=x+1\\,", "1e634bf1ea3b0f28ef7fbbd749f5b062": "(G/H)_{H}", "1e635cbb83f09f791a47ed0b52ff3f2b": "T_{K}\\phi (x)=\\sigma _{i}\\phi _{i}(x)", "1e63bdaecd47c11e2ee78040582906c7": "{\\vec {S}}^{\\dagger }", "1e6438c55f008afca0ec21157d9fa91c": "r_{O}={\\frac {V_{A}+V_{CE}}{I_{C}}}\\approx {\\frac {V_{A}}{I_{C}}}", "1e6470a189d249ba285f55eff7a7b3f1": "{}_{3}F_{2}(a,b,c;1+a-b,1+a-c;1)={\\frac {\\Gamma (1+{\\frac {a}{2}})\\Gamma (1+{\\frac {a}{2}}-b-c)\\Gamma (1+a-b)\\Gamma (1+a-c)}{\\Gamma (1+a)\\Gamma (1+a-b-c)\\Gamma (1+{\\frac {a}{2}}-b)\\Gamma (1+{\\frac {a}{2}}-c)}}.", "1e64a37925a67754d47b4e9dc3c3cc27": "u''+p(x)u'+q(x)u=0\\,", "1e64eda87b479596e4c27cb7031c73c2": "p_{1}^{n_{1}}+1\\leq {\\mathfrak {M}}(d)\\leq d+1.", "1e653e2c49f14f81b60205a54383a208": "\\mathbf {\\zeta } =79", "1e655c0d4e30ebee2f566a68094ff520": "V=C_{2}\\times C_{2}", "1e656f2a69ca438488679478c18a08e2": "\\displaystyle {f_{\\overline {z}}=\\mu f_{z},}", "1e659c51cfd47a2e9f68e157448c1af7": "f({\\frac {1}{n}}\\sum _{i=1}^{n}f^{-1}(x_{i})))=\\exp({\\frac {1}{n}}\\sum _{i=1}^{n}\\log x_{i})={\\sqrt[{n}]{x_{1}\\cdots x_{n}}}", "1e65aac7dec0a23e038a7178178647d3": "r_{b}=\\langle {\\hat {d}}|{\\hat {c}}\\rangle -\\langle {\\hat {f}}|{\\hat {a}}\\rangle ", "1e65b4d6d965fcd8d83565cd5f540322": "R_{p}\\leq 0.2", "1e65fb8201c846adcdb178db36b85680": "{\\frac {3}{8}}\\pi a^{2}", "1e66110d881ae2168db84d6165f63282": "y=\\pm \\left({\\sqrt {2}}+1\\right)\\left(x-2\\right)", "1e664a6988ea2aeacaf377d679115846": "i_{\\mathrm {r} }=-i_{\\mathrm {i} }\\,\\!", "1e666f542e40880d8117b2e32c5f58da": "V=\\{I_{1},I_{2},\\ldots ,I_{n}\\}", "1e669c437f4f44ed5c3df28aa00728cb": "\\qquad \\sum _{j\\in J}v_{j}\\,x_{j}\\ \\geq \\alpha \\,v_{i}\\,", "1e66a07e9888bda94396dae58a9afab3": "\\varphi _{\\lambda }(e^{X})={\\chi _{\\lambda }(e^{X}) \\over \\chi _{\\lambda }(1)}.", "1e66f932c5d60285740041dbb7887110": "\\delta =\\alpha _{y}-\\alpha _{x}.", "1e66fb47d18b9cd9db82bd08af3e3fa2": "w_{min}={\\frac {Q}{q_{max}}}={\\frac {150}{45.5}}=3.30{\\text{ ft}}", "1e671a558c1fd5d1ecd30bc4e907c6c6": "IMM(s)=\\{i\\in D|ta_{i}(s_{i})=ta(s)\\}", "1e6743a673a02d3aa6df71cffd9e1467": "{\\frac {d\\eta }{dt}}=-\\eta \\nabla _{h}\\cdot {\\vec {v}}_{h}-\\left({\\frac {\\partial \\omega }{\\partial x}}{\\frac {\\partial v}{\\partial z}}-{\\frac {\\partial \\omega }{\\partial y}}{\\frac {\\partial u}{\\partial z}}\\right)-{\\frac {1}{\\rho ^{2}}}{\\vec {k}}\\cdot (\\nabla _{h}p\\times \\nabla _{h}\\rho )", "1e67498c9f66d1b7dbf98263daf51d86": "\\omega +2", "1e67824fa95e97cf11ed7e152bcd8f34": "m_{c}(z)", "1e67931e0bb08cd4aa009dd0d33a3da5": "{\\tfrac {1}{2}}\\div {\\tfrac {3}{4}}={\\tfrac {1}{2}}\\times {\\tfrac {4}{3}}={\\tfrac {1\\cdot 4}{2\\cdot 3}}={\\tfrac {2}{3}}", "1e6796dd3b22411de7aff0d03ed9504d": "y_{i}=\\sum _{j=1}^{2}a_{ij}x_{j}+\\sum _{j=1}^{2}b_{ij}y_{j}", "1e67eb0e840c08faf67f937b328a70ca": "{\\mathcal {L}}^{-1}\\left\\{{\\frac {1}{s+a}}\\right\\}\\,*\\,{\\mathcal {L}}^{-1}\\left\\{{\\frac {1}{s+b}}\\right\\}=e^{-at}\\,*\\,e^{-bt}=\\int _{0}^{t}e^{-ax}e^{-b(t-x)}\\,dx={\\frac {e^{-at}-e^{-bt}}{b-a}}.", "1e6813370e2e3c30585362cc6ea39a66": "(Sg)(z)=2Q(z)", "1e68393bec7f6c54ac3b093c91eca86d": "{\\frac {n}{2}}\\langle V_{\\mathrm {TOT} }\\rangle _{\\tau }=\\left\\langle \\sum _{k=1}^{N}\\left({\\frac {1+{\\sqrt {1-\\beta _{k}^{2}}}}{2}}\\right)T_{k}\\right\\rangle _{\\tau }=\\left\\langle \\sum _{k=1}^{N}\\left({\\frac {\\gamma _{k}+1}{2\\gamma _{k}}}\\right)T_{k}\\right\\rangle _{\\tau }\\,.", "1e68ce3487dc77148d543eaee8ea98ba": "H_{j\\gamma }^{'}=\\left\\langle u_{j0}\\right|{\\frac {\\hbar }{m_{0}}}\\mathbf {k} \\cdot \\left(\\mathbf {p} +{\\frac {\\hbar }{4m_{0}c^{2}}}{\\bar {\\sigma }}\\times \\nabla V\\right)\\left|u_{\\gamma 0}\\right\\rangle \\approx \\sum _{\\alpha }{\\frac {\\hbar k_{\\alpha }}{m_{0}}}p_{j\\gamma }^{\\alpha }.", "1e68f3c3bc3a2f57ae71b355e9b97509": "{\\sqrt {}}", "1e6926bdfaa69e759514d2ffe9a0d913": "P_{0}(y)=((-a/b)y)P_{1}(y)+c,", "1e69589c6af42e8428c244ec0fa39fb7": "{\\begin{aligned}Prob(choosing\\,5)&=Prob(U_{n}>d)\\\\&=Prob(\\varepsilon >d-\\beta z_{n})\\\\&=1-{1 \\over 1+exp(-(d-\\beta z_{n}))}\\end{aligned}}", "1e69745e09d506d2ef26efedec1555a4": "n^{1/k}", "1e6a270bbc2e1939ea5d07bd54824309": "\\ p{\\textbf {r}}\\cdot {\\textbf {g}}+{\\textbf {f}}\\cdot {\\textbf {m}}", "1e6a73d75db921c729655dfd65c4bfb0": "{\\frac {1}{2}}mv^{2}", "1e6aa8ce268f794b5fb70b85dd5eac6c": "(x_{1}x_{2}-Ny_{1}y_{2}\\,,\\,x_{1}y_{2}-x_{2}y_{1}\\,,\\,k_{1}k_{2}).", "1e6add065eaeb84b24416409275cff3d": "g(p_{i},q_{i},t)", "1e6ae43ee27bc8d9b445f6a4c2cd2785": "\\left|\\langle v|U(t)|\\psi _{0}\\rangle \\right|^{2}", "1e6ae83f2fee94e07ae3092f83754ea2": "C\\otimes C", "1e6aeaa9e1123305e73497f7dd3cf29e": "m_{1}>m_{2}>m_{3}>\\cdots >m_{n}.\\,", "1e6aebdb27572591e955109a44f56085": "A\\approx {360^{\\circ } \\over 2\\pi }\\cdot {M \\over 1000}\\approx 0.0573\\times M", "1e6af492fc535cd50f9473573526d5d0": "H^{q}(B^{\\bullet })/H^{q}(A^{\\bullet })\\cong \\ker d_{0,q}^{1}:H^{q}(C^{\\bullet })\\rightarrow H^{q+1}(A^{\\bullet })", "1e6b0b6ab3635f47cadf5d5a904ae452": "f=m\\circ e", "1e6b498af6527dbdc6dd238b0769ac51": "D\\neq -4", "1e6b5b1b8090ad0dae695e69f873d5d7": "{\\begin{pmatrix}w_{1}\\\\w_{2}\\\\w_{3}\\\\\\end{pmatrix}}={\\frac {1}{1+{\\frac {u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}}{c^{2}}}}}\\left\\{\\left[1+{\\frac {1}{c^{2}}}{\\frac {\\gamma _{\\mathbf {u} }}{1+\\gamma _{\\mathbf {u} }}}(u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3})\\right]{\\begin{pmatrix}u_{1}\\\\u_{2}\\\\u_{3}\\\\\\end{pmatrix}}+{\\frac {1}{\\gamma _{\\mathbf {u} }}}{\\begin{pmatrix}v_{1}\\\\v_{2}\\\\v_{3}\\\\\\end{pmatrix}}\\right\\}", "1e6b962ad166b958db642d1c8f9bb2ce": "{\\sqrt {B}}", "1e6b9d3c5e221072d0cf678046ccbb3c": "\\operatorname {E} [\\,x_{t}(y_{t}-x_{t}'\\beta )\\,]=0", "1e6bad2d26e82f4ec22c0ec1459693a1": "e^{X}=\\sum _{k=0}^{\\infty }{1 \\over k!}X^{k}.", "1e6c8d4af9f706609017fdf575df3936": "\\gamma =(3\\pi ^{2})^{2/3}{\\frac {\\hbar ^{2}}{2m}}", "1e6c90ab968cdee11dfa4eda42d573b1": "A_{\\mu }\\,dx^{\\mu }", "1e6ccffc63598fd20a34d06cbf6cfe7d": "\\textstyle E_{2}", "1e6d676a68b43a677542bfb15c2e3d22": "\\phi (\\rho )", "1e6d69ede2647a75797c95245ce41590": "M_{i,j}^{p,q}", "1e6d8eaee0c0359ab430aa0d60a4755b": "\\gamma _{k}(a)", "1e6d9ce93f5b5229c3073e1754b38e51": "f^{2}(\\theta ^{2}(t))", "1e6df665d4baaa1b5e239311c9cc9550": "{\\mbox{recall}}={\\frac {|\\{{\\mbox{relevant documents}}\\}\\cap \\{{\\mbox{retrieved documents}}\\}|}{|\\{{\\mbox{relevant documents}}\\}|}}", "1e6e0a04d20f50967c64dac2d639a577": "1100", "1e6e3432d141e09032cb6127fb1892ec": "\\{[-\\infty ,a[:a\\in \\mathbb {R} \\cup \\{\\pm \\infty \\}\\}", "1e6e63df7e2ac7cf7da2de654081f717": "f(x;h)={\\frac {e^{hx}f(x)}{\\int _{-\\infty }^{\\infty }e^{hx}f(x)dx}}.\\,", "1e6e975c355631d08f98ab1b56bc4c0b": "\\langle \\cdot ,\\cdot \\rangle _{H}=\\langle \\cdot ,\\cdot \\rangle _{G}", "1e6ec2d5200ad13c03cb2f08a59ec4eb": "\\lnot \\;\\exists \\;x", "1e6f02e8fed70befb3f11f21b57feaef": "\\phi -\\xi \\,\\!", "1e6fb6df8558239f4c1e7f5daaa8898e": "F[\\rho ]=E[\\rho ]+{\\frac {1}{\\beta }}S[\\rho ],", "1e6fdcc12bc90d0e69cad8f2dc5aff90": "\\sigma \\,", "1e7070c60d0e2863e29d9c2df0690ef7": "f''(x_{n})(a-x_{n})\\!", "1e707431667377ca0c2d700e0c68e90e": "\\Sigma \\left|a_{n}\\right|^{2}\\neq \\Sigma \\left|b_{n}\\right|^{2}\\,", "1e707a6d712ce45fb70d0708804cec98": "0\\leq p\\leq \\infty ", "1e70c470db2cb9b251c37e6551abd0ae": "\\alpha >k", "1e70d4038de3ca07fb37ecc6cb9dda8c": "{\\frac {-b}{2a}}+i{\\frac {\\sqrt {-\\Delta }}{2a}}\\quad {\\text{and}}\\quad {\\frac {-b}{2a}}-i{\\frac {\\sqrt {-\\Delta }}{2a}},", "1e714b5d06631bc8d6d3fc3b94e9dfad": "0.3333...\\triangleq \\lim _{n\\to \\infty }\\sum _{i=1}^{n}{\\frac {3}{10^{i}}}", "1e714e7dc6b10400f2ca0bb9fa812586": "\\nu ={\\frac {\\mathrm {tr} ({\\tilde {S}}^{2})+[\\mathrm {tr} ({\\tilde {S}})]^{2}}{{\\frac {1}{n_{1}}}\\left\\{\\mathrm {tr} ({\\tilde {S_{1}}}^{2})+[\\mathrm {tr} ({\\tilde {S_{1}}})]^{2}\\right\\}+{\\frac {1}{n_{2}}}\\left\\{\\mathrm {tr} ({\\tilde {S_{2}}}^{2})+[\\mathrm {tr} ({\\tilde {S_{2}}})]^{2}\\right\\}}}.", "1e715e612b0cdcd77062a419885a9903": "dI=-k_{O}C_{L}dV\\;", "1e71ae53ab23cd5e20eec152e26bdfed": "u_{0}\\equiv 1/r_{0}", "1e720784691b8fa4b97de5b85bd30740": "p_{t}=+ip^{2}q-{\\frac {i}{2}}p_{xx}", "1e7283225ce5499c242a102b3114df13": "A_{R}=\\{X\\in L_{d}^{p}:0\\in R(X)\\}", "1e72a2bf7dd3ff3f7d07f61f78fff740": "\\varepsilon _{r}=1", "1e72bbfe15ab0ee1d2867751fa481ad1": "(M,0,+)", "1e72e9ab043b82daed58a3fecdc65530": "y_{i}^{(\\lambda )}={\\begin{cases}{\\dfrac {y_{i}^{\\lambda }-1}{\\lambda (\\operatorname {GM} (y))^{\\lambda -1}}},&{\\text{if }}\\lambda \\neq 0\\\\[12pt]\\operatorname {GM} (y)\\log {y_{i}},&{\\text{if }}\\lambda =0\\end{cases}}", "1e7312c49039bcb28334287d601dc600": "(2f(x))^{-2}", "1e734701cabcb9ef0341bfbcc078099f": "\\mathrm {R{-}COOH\\ +\\ H_{2}O_{2}\\longrightarrow \\ R{-}COOOH\\ +\\ H_{2}O} ", "1e736c36a77187bef111184d72efc65c": "S_{xx}\\,=\\,\\left(2\\,{\\frac {c_{g}}{c_{p}}}\\,-\\,{\\frac {1}{2}}\\right)\\,E\\,", "1e73748946bdea5a3a9598725d447cf7": "f(x,y)=\\left(1+\\left(x+y+1\\right)^{2}\\left(19-14x+3x^{2}-14y+6xy+3y^{2}\\right)\\right)", "1e73851613c1e4ea3349cce62371d06f": "\\mathbf {v} ^{\\infty }", "1e7395cf4c6edc4feb9c28cfb7b5605b": "\\alpha \\,\\!", "1e73f1c54aef1f67330ddd4d06f3de9b": "\\Delta {\\tilde {x}}_{j+1,j}\\geqslant {\\Bigl [}2(\\Delta x_{\\rm {meas}})^{2}+{\\Bigl (}{\\frac {\\hbar \\vartheta }{2M\\Delta x_{\\rm {meas}}}}{\\Bigr )}^{2}{\\Bigr ]}^{1/2}\\geqslant {\\sqrt {\\frac {3\\hbar \\vartheta }{2M}}}\\,,", "1e73fe9f68e2fffb020194ffc4973230": "t=\\left\\lfloor {\\frac {1}{2}}(d-1)\\right\\rfloor ", "1e7424fb7d3295afda4baff6a394af07": "\\;_{j}\\psi _{k}\\left[{\\begin{matrix}a_{1}&a_{2}&\\ldots &a_{j}\\\\b_{1}&b_{2}&\\ldots &b_{k}\\end{matrix}};q,z\\right]=\\sum _{n=-\\infty }^{\\infty }{\\frac {(a_{1},a_{2},\\ldots ,a_{j};q)_{n}}{(b_{1},b_{2},\\ldots ,b_{k};q)_{n}}}\\left((-1)^{n}q^{n \\choose 2}\\right)^{k-j}z^{n}.", "1e743608ddcde5fca35c2ddfbe320b2f": "x=r_{k},y=s", "1e745ce7a94e1520a9971a9e632da226": "\\iota :S\\rightarrow M", "1e745e0ad4f266a9731796cfcdaad451": "\\operatorname {pf} (\\lambda A)=\\lambda ^{n}\\operatorname {pf} (A).", "1e74ba795769d7ac9e53d80449a89211": "\\ell =\\log s=\\log h+O(\\log(\\epsilon ^{-1}))", "1e7500d608224a00aeddde704d355629": "x(j_{p}^{1}\\sigma )\\,", "1e750e88eb7b47f58efe94f2d6e8ccc9": "\\sum _{i}n_{i}\\,{\\rm {d}}\\mu _{i}=0", "1e75163d385267ee084f40ced40434d6": "T={\\frac {n_{2}\\cos \\theta _{\\text{t}}}{n_{1}\\cos \\theta _{\\text{i}}}}\\left|t\\right|^{2}", "1e7545758d4279e834d03fb03793ff56": "y'(t)=f(t,y(t))\\,", "1e757a9802187d75c3c899bacbec3449": "E^{(2)}=\\sum _{l=1}^{\\infty }E_{l}^{(2)}=\\sum _{l=1}^{\\infty }-{\\frac {Q^{2}\\alpha _{l}}{2R^{2l+2}}}", "1e75b72b8c5de5758b91c64b2b22d8bc": "V=I.(R'x+L'v)+L'x{\\frac {{\\text{d}}I}{{\\text{d}}t}}", "1e75fd0d939080ce35dbfdbe436a50a8": "H=(h\\nu -\\mu )a^{\\dagger }a.\\,", "1e76c16988f7c31f3453c8f97a8973b7": "\\int _{a}^{b}e^{nf(x)}\\,dx\\geq \\int _{x_{0}-\\delta }^{x_{0}+\\delta }e^{nf(x)}\\,dx\\geq e^{nf(x_{0})}\\int _{x_{0}-\\delta }^{x_{0}+\\delta }e^{{\\frac {n}{2}}(f''(x_{0})-\\varepsilon )(x-x_{0})^{2}}\\,dx=e^{nf(x_{0})}{\\sqrt {\\frac {1}{n(-f''(x_{0})+\\varepsilon )}}}\\int _{-\\delta {\\sqrt {n(-f''(x_{0})+\\varepsilon )}}}^{\\delta {\\sqrt {n(-f''(x_{0})+\\varepsilon )}}}e^{-{\\frac {1}{2}}y^{2}}\\,dy", "1e76d3ca183f72dced9912dfecfc0307": "\\displaystyle {\\frac {1}{|a|}}{\\hat {f}}\\left({\\frac {\\omega }{a}}\\right)\\,", "1e77254ca5fcadd2ce551a5d9a582c0b": "K_{M}", "1e775635789aa04cfde0076b5309757e": "\\mathbf {F} _{ext}=m_{rocket}(t){\\frac {\\mathrm {d} \\mathbf {V} }{\\mathrm {d} t}}+\\mathbf {V} (t){\\frac {\\mathrm {d} m_{rocket}}{\\mathrm {d} t}}", "1e777dda72b6d3aa51998a46db2b7705": "\\mathrm {I} ", "1e777f6242e02301be04ee07183ef18d": "F_{2}=A'B+A'C+A'E.\\,", "1e781fc216d96f315ab12b8fcdda607e": "du/dt=f(u)", "1e788ff9085a1e3b210ea92c78cf3632": "{\\mbox{DL}}=({\\mbox{C}}-{\\mbox{DW}}){\\bmod {7}}", "1e78a9aba621c1c0d22c97d014a12fe5": "|\\zeta (\\sigma )^{3}\\zeta (\\sigma +it)^{4}\\zeta (\\sigma +2it)|\\geq 1", "1e78ae39fce06be746293234e6a76604": "\\xi _{b_{min}}^{d}(k,0)=\\xi _{f_{min}}^{d}(k,0)=x^{2}(k)+\\lambda \\xi _{f_{min}}^{d}(k-1,0)\\,\\!", "1e78aed60e0426874a9441ac0e49cce9": "x|U|", "1e796a0b3f8b5304b315a469e03d207e": "{\\mathcal {Q}}_{\\alpha }^{t}=\\{Q=P\\,\\vert _{{\\mathcal {F}}_{t}}:{\\frac {dQ}{dP}}\\leq \\alpha _{t}^{-1}\\mathrm {a.s.} \\}", "1e7a04bf0828dde479ede2a9786cb8cf": "b=gb_{1}", "1e7a10f3033a4a5115ced31dbc7be2ea": "b^{2}+2b+a", "1e7a3dca7e341ed0a9e192b8e1a41bbd": "F(f,x,min,max)", "1e7a583a2c2f40dc1e0ce0ad44f43fd8": "\\delta (x-\\alpha )={\\frac {1}{2\\pi }}\\int _{-\\infty }^{\\infty }dp\\ \\cos(px-p\\alpha )\\ .", "1e7a76e54b24d1fd8be98ae26d2aaf02": "f_{2}(x)>g_{2}(x)", "1e7a95084764435f336d82520870da56": "c=m^{2}-n^{2}\\,", "1e7ade3c8b3e50be8b010c2d3ea788eb": "{d^{2}\\mathbf {h} \\over d\\tau ^{2}}+R\\mathbf {h} =0", "1e7b00feebe46b1c1626493867c4eccf": "\\oint ", "1e7b7d2802cb330b1eeaddab728cfb46": "z'=z\\ ,", "1e7b7e3fd5b47c9570dedbb50136a6cb": "+\\;1\\;2", "1e7bc1632394649623a661545ce1a7cf": "\\mathbf {P} \\neq \\mathbf {NP} ", "1e7bc9a2ab0d5d89a02786140dd211e6": "\\langle E\\rangle ={\\frac {1}{V}}\\int _{\\Omega }H({\\boldsymbol {r}}){\\frac {e^{-\\beta H({\\boldsymbol {r}})}}{Z}}\\,d{\\boldsymbol {r}}", "1e7bebf86d156f7ad96432f008a6e35c": "\\color {Dandelion}{\\text{Dandelion}}", "1e7c07ccc754819426431c9497d34914": "\\,C\\,", "1e7c0c3ed3b4738b500cb336f2fb412d": "f(x)=\\sum _{n=0}^{N}c_{n}\\phi _{n}(x)", "1e7c6bbe7dddffd8b881e4d0c8bc33e0": "\\tau _{1}\\subseteq \\tau _{2}", "1e7cd2a0dee459160c9f2bcc1235bc1b": "M_{A}=\\{A(x):\\mu _{A}(x)=\\mu _{A}(0)\\}.", "1e7ce94b292b3d89bc8519e65cc3439c": "G(n,t)={\\frac {1}{\\sqrt {2\\pi t}}}e^{-{\\frac {n^{2}}{2t}}}", "1e7d174b547879e98eb9401a87dedd23": "p\\ \\sim \\ a{\\mathcal {N}}(0,\\,\\sigma _{1}^{2})+(1-a){\\mathcal {N}}(0,\\,\\sigma _{2}^{2})", "1e7d1b003e3176219b2ae5ebbeaf1031": "Z={\\frac {m_{1}z_{1}+m_{2}z_{2}}{m_{1}+m_{2}}}", "1e7d5b1120971bec627357369217cb27": "\\mathbb {E} _{k-1}\\,\\mathbf {X} _{k}=\\mathbf {0} \\quad {\\text{and}}\\quad \\mathbf {X} _{k}^{2}\\preceq \\mathbf {A} _{k}^{2}", "1e7d6aeb607c3dede585d9c31e2c0b84": "\\partial _{\\alpha }", "1e7d7566c106841d6cfe9b1fd747e964": "\\varphi (L)X_{t}=\\theta (L)\\varepsilon _{t}\\,.", "1e7da01d39e5161e28caad04c027d09f": "\\alpha _{A,B,C}", "1e7e437b6bda6ebbf2bb803bf0d8b527": "{\\mathcal {Q}}", "1e7e5f5a7518cb92b3ab9738cbb66472": "\\Phi _{F}=", "1e7e737f03162a66cf88ce8d6f914a7f": "q-1", "1e7ea31ae782c651874ff5025183cbb4": "\\displaystyle -13.26~{\\mbox{dB}}", "1e7ec45e0a6b809a1c7c568779937091": "y_{1},...,y_{m}", "1e7f545b52a76f19ae73c16220ecdddb": "{\\mathcal {L}}_{i}^{2}", "1e7f8a9cc31a5c86bea281fcf5469160": "\\surd \\!\\,", "1e800e0087577d6240a8fdc1d55e77b4": "\\mu =-k_{B}T\\ln(Q_{N+1}/Q_{N})=-k_{B}T\\ln \\left({\\frac {V/\\Lambda ^{d}}{N+1}}\\right)-k_{B}T\\ln {\\frac {\\int ds^{N+1}\\exp[-\\beta U(s^{N+1})]}{\\int ds^{N}\\exp[-\\beta U(s^{N})]}}=\\mu _{id}(\\rho )+\\mu _{ex}", "1e808aad9132c1e4c6a9ae9599a71dd6": "(c-v)", "1e80b4619b3906efb16c182dfed52079": "g=(1-hi)/2,\\quad g'=(1+hi)/2", "1e8113563f6efbe6fc7582915bd9d196": "B\\subseteq A\\,\\!", "1e81560541aba927c9901960f577c97d": "c(V):=\\sum _{i=0}^{n}c_{i}(V),", "1e8174308ccec52a8ed5a958ed5a5286": "E(R_{i})=\\alpha +\\beta _{1}F_{1}+...+\\beta _{N}F_{N}.\\,", "1e8174f459bbbf902d2b5fe5e9d2f9fb": "\\log {\\lambda }\\,{\\sqrt {\\frac {n\\,d_{2}}{4}}}", "1e81c86f8bfe80f31b02c4a26b4b47ef": "PX^{r}XY=PY^{r}YX", "1e81f9ac6ddbbaaad4c6546eaa992115": "{\\tilde {\\boldsymbol {a}}}", "1e822c1b8bc59590eba1cd3cb398345f": "T_{M}", "1e82876a796b9b9c6bf6343c3d250800": "{\\sqrt {(x-x_{0}\\cos \\omega t)^{2}+y^{2}}}-L=0\\,\\!.", "1e82a85e0675eb78e9b164eed57fd98c": "\\displaystyle x^{2}-7y^{2}=1.", "1e82dbe1cd5035ffb372877e90e383db": "S_{-}=S_{x}-i\\cdot S_{y}", "1e82f1965e874d393899f37a32735133": "(1-|\\xi |^{2})_{+}^{\\delta }", "1e833757db81cc56d508d7e92493ba05": "s\\to \\infty ", "1e835d236501c8b76c71ab325a52ce43": "Z=i\\omega L+{\\frac {1}{i\\omega C}}=i\\left(\\omega L-{\\frac {1}{\\omega C}}\\right)", "1e8362535fbd55ce411d3435ae049fa1": "{\\frac {\\alpha \\in \\Gamma }{\\Gamma \\vdash \\alpha }}\\qquad \\qquad {\\text{Assum}}", "1e83a0cf3dc85f38e57fcb9ffe0a2e27": "f(t)f(t^{-1})", "1e8412230afd0e7398ffa44be708db25": "C(s)=\\left(K_{P}+K_{I}{\\frac {1}{s}}+K_{D}s\\right).", "1e8430cd4400644e3be887d094e1f718": "k_{1}\\lambda /NA.", "1e84bea511a3aad79dba2270cf88357d": "2.9\\times 10^{-13}\\ \\mathrm {seconds} \\,", "1e84e6be586b9aa4feee7cfa46717109": "\\{z_{i}\\}=\\{-1,-1,-1,0,0,0,1,1,1\\}", "1e84e7aebef9db2a5b1cf5dfd0412660": "|f(z)-g(z)|<|f(z)|+|g(z)|\\qquad \\left(z\\in \\partial K\\right)", "1e852e0f33b4828c0b6f4c97ad83de37": "P=SD", "1e85405ccdb068dfc9f572ad1e2e99df": "f:{\\widehat {\\mathbb {R} }}\\to {\\widehat {\\mathbb {R} }},\\quad p\\in {\\widehat {\\mathbb {R} }}.", "1e855871a2cd6171c400620ec84b4907": "\\mu _{k}=\\mu \\star \\cdots \\star \\mu ", "1e85aeb8635edcbda752900f27545be5": "{\\overrightarrow {Vr}}={\\overrightarrow {\\omega }}\\times {\\overrightarrow {r}}", "1e85e17009873544fc0de38506cbc46e": "R(l)=P", "1e85f001be5638f058db9553abc27b4b": "{\\rm {Var}}\\left[{\\bar {x}}\\right]\\,\\,\\,=\\,\\,\\,{\\rm {E}}\\left[{{{s^{2}} \\over {\\gamma _{1}}}\\left({{\\gamma _{2}} \\over n}\\right)}\\right]\\,\\,\\,\\,=\\,\\,\\,{\\rm {E}}\\left[{{{s^{2}} \\over n}\\left\\{{{n\\,\\,-\\,\\,1} \\over {{n \\over {\\gamma _{2}}}-\\,\\,1}}\\right\\}}\\right]", "1e8645d11348a002175e879941581a47": "{\\hat {x}}=i\\hbar {\\frac {d}{dp}}", "1e866d5f281eac8d5c308adc87f8ff79": "\\omega ={2\\pi f}={\\frac {qB}{\\gamma m_{0}}}={\\frac {\\omega _{0}}{\\gamma }}={\\omega _{0}}{\\sqrt {1-\\beta ^{2}}}={\\omega _{0}}{\\sqrt {1-\\left({\\frac {v}{c}}\\right)^{2}}}", "1e86942d94320f52ecec0ef9a3c240e6": "a\\in X", "1e869f65393143c1fa4e48406908c913": "{\\begin{aligned}\\det P&=h_{\\mbox{e}}\\cdot g_{\\mbox{o}}-h_{\\mbox{o}}\\cdot g_{\\mbox{e}}\\\\\\exists A\\ A\\cdot P&=I\\iff \\exists c\\ \\exists k\\ \\det P=c\\cdot \\delta \\rightarrow k\\end{aligned}}", "1e86a59aaeacba51ef67c5d782567489": "\\mathrm {Ai} (x)\\sim {\\frac {e^{-{\\frac {2}{3}}x^{3/2}}}{2{\\sqrt {\\pi }}x^{1/4}}}.", "1e86b2387fd7153b65b59f62a0a84da6": "\\operatorname {d} f\\triangleq \\partial _{a}f\\operatorname {d} x^{a}", "1e86f20d2b9663939272a08b5fe03347": "\\Diamond \\equiv \\lnot \\Box \\lnot ", "1e875930916055e38a1aa149e49e8e0e": "\\sum \\mathbf {F} ", "1e87739c52c29e47366dcf1c2e2ba400": "\\mathrm {Var} [X]={\\frac {1}{\\lambda ^{2}}},", "1e878283d8545e84fa5ae0c64ad5c558": "a(x-y)", "1e87afc305aef174158abf0e9ebdef40": "\\beta =(-27-8y-9w+6w^{2}-18yw-11yw^{2})/23.\\ ", "1e87b01155ef0c8a49aa0db71e773285": "\\pi _{jk}", "1e87c89899dbdbb094664e5069042f64": "A(t)=[{\\rm {TransmitterRelease}}(t)/{\\rm {TransmitterRelease}}(0)]-1,", "1e8805fea863aeee2e770a679221d945": "[2]P_{1}", "1e881f1147e5a7d5a0c70f0dd278cc79": "\\mathrm {cov} (\\epsilon _{1},\\epsilon _{2})=0", "1e885daabb04b3785157337cd8035b53": "X'", "1e88d273f7b53f586c2c007fa3eaf27a": "x_{n}\\to c", "1e88ee1d296e6d1b55a0590a0b6df693": "\\phi [J;k]={\\frac {\\delta W_{k}}{\\delta J}}[J]", "1e88f0c74292063f438ebd9ee1dd3a3c": "{\\hat {\\xi }}^{i}=({\\hat {x}}^{1},...,{\\hat {x}}^{n},{\\hat {p}}_{1},...,{\\hat {p}}_{n})\\in Op(L^{2}(\\mathbb {R} ^{n})).", "1e890adfba7893718038ec806bf6f474": "{\\frac {v(t)^{2}}{2}}-{\\frac {v_{0}^{2}}{2}}={\\frac {GM}{r(t)}}-{\\frac {GM}{r_{0}}}.\\,", "1e8926511aacf32a13d69853fccb8b21": "\\textstyle a={\\frac {1}{2}}", "1e8935a0ff187de7a0d29e64f8368fe0": "H(V)=-\\sum _{j=1}^{C}P'(j)\\log P'(j)", "1e893d796502e82a6397cc05a0b14270": "\\nabla y_{t}=a_{0}+a_{1}t+\\delta y_{t-1}+u_{t}\\,", "1e898ee02cc457f607e6775a6b41c35a": "{\\frac {1}{\\Phi (\\varphi )}}{\\frac {d^{2}\\Phi (\\varphi )}{d\\varphi ^{2}}}=-m^{2}", "1e8a3b0c83f3b1356bb3d0e338f1be1b": "|s\\rangle ={\\frac {1}{\\sqrt {N}}}\\sum _{x=1}^{N}|x\\rangle ", "1e8a6e39c7d8b2e7cf9244b3fe00206c": "a^{2}-b^{2}=(B+A)(B-A)\\,", "1e8a6f1eba76f41565744b2b5a60faf1": "[X,O]", "1e8a77e39c48f2a34cbad4b5bde534be": "\\sum _{n=-\\infty }^{\\infty }e^{inx}=", "1e8a9b76ba915e36d15fb02748c98b43": "\\operatorname {tr} (A^{k})=\\sum _{i}\\lambda _{i}^{k}", "1e8aa01b05bb3acf8fc94e10f42e2849": "x(x+1)(x-3)(x+2)(x-2)", "1e8adead525ae74ded132a535815e836": "N=53", "1e8b05e8eb3c10c67aea261922c6a515": "\\int _{t_{n}}^{t_{n+1}}f(t,y(t))\\,\\mathrm {d} t\\approx {\\tfrac {1}{2}}h{\\Big (}f(t_{n},y(t_{n}))+f(t_{n+1},y(t_{n+1})){\\Big )}.", "1e8b25ae6007d18dc5fb9c7d2a327acc": "y_{c}=(C_{1}x+C_{2})e^{-bx/2}\\,\\!", "1e8b776274ed98a563cb2f33261db947": "\\mathbf {A} \\in \\mathbb {R} ^{n\\times d_{f}},\\mathbf {B} \\in \\mathbb {R} ^{n\\times d_{g}}", "1e8bb92fd39a8197600aa6a6ab063543": "a=\\epsilon ", "1e8bdfb26cee0d20342689d4d42c18a7": "\\Phi \\left(R_{\\mu \\nu }-{\\frac {1}{2}}g_{\\mu \\nu }R\\right)+\\left(g_{\\mu \\nu }\\Box -\\nabla _{\\mu }\\nabla _{\\nu }\\right)\\Phi +{\\frac {1}{2}}g_{\\mu \\nu }V(\\Phi )=\\kappa T_{\\mu \\nu }", "1e8c46590be47a55c12810c111ad5388": "{\\sqrt {1000}}", "1e8c9455378cb019a1adc4640bf6b82f": "H^{(4)}=x\\partial _{x}-y\\partial _{y}-2y'\\partial _{y'}-3y''\\partial _{y''}-4y'''\\partial _{y'''}-5y''''\\partial _{y''''}.", "1e8cb71fdce2f50477e39b5a6193c381": "S(\\forall )", "1e8d5d88133019a244b3b3f7e2ca62ca": "\\rho =|\\psi |^{2}", "1e8d7c15a990ff1bb7e77997832e2959": "{\\frac {T\\left(x\\right)-T_{o}}{T_{L}-T_{o}}}={\\frac {e^{\\left(Ax\\right)}-1}{e^{\\left(AL\\right)}-1}}", "1e8d903de5c6c2c4f790bb10e0ab6c3d": "{1 \\over 1-u/c}\\,.", "1e8dbf14472f2ff22e6193c24c67e193": "{\\frac {10}{\\sqrt {a}}}", "1e8de0dca01123264ecf4e9fd548bd0a": "b=0.26\\ V_{c}", "1e8e20e14406b61c060c3b0a4549731d": "E_{n}^{(3)}={\\frac {V_{nk_{3}}V_{k_{3}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{3}}}}-V_{nn}{\\frac {|V_{nk_{3}}|^{2}}{E_{nk_{3}}^{2}}}", "1e8e33292fa3d52911c13ea72a3ed4e8": "{\\dot {\\textbf {S}}}(t)=-{\\textbf {S}}(t){\\textbf {A}}-{\\textbf {A}}^{\\text{T}}{\\textbf {S}}(t)+{\\textbf {S}}(t){\\textbf {B}}{\\textbf {R}}^{-1}{\\textbf {B}}^{\\text{T}}{\\textbf {S}}(t)-{\\textbf {Q}}", "1e8e410f33f8835cebf75cbfe47dc6ed": "C_{\\alpha -1}(x)+C_{\\alpha +1}(x)=2{\\frac {dC_{\\alpha }}{dx}}\\!", "1e8e7508cbde21c325411e7ac541a594": "{\\dot {r}}_{1}=\\lambda _{1}r_{1}", "1e8e804a8abb60c2ea93b6e4c2ee346a": "D_{q}(f(x))={\\frac {d_{q}(f(x))}{d_{q}(x)}}={\\frac {f(qx)-f(x)}{(q-1)x}}", "1e8ee50f289d1eb2915c2df6ce794358": "{\\frac {1}{\\eta }}{\\frac {\\Delta P}{\\Delta x}}={\\frac {d^{2}v}{dr^{2}}}+{\\frac {1}{r}}{\\frac {dv}{dr}}", "1e8fb5fe03bb77c216f73e2da35f99c9": "{\\frac {dG(\\mathbf {r} (t))}{dt}}=\\nabla G(\\mathbf {r} (t))\\cdot \\mathbf {r} '(t)=\\mathbf {F} (\\mathbf {r} (t))\\cdot \\mathbf {r} '(t)", "1e8fecec6574defe9a47a25d5e6ece2f": "f(a)=s", "1e90718d1c8f722d2b1382f2988d200f": "P=\\left({\\frac {8}{47}}\\times {\\frac {4}{46}}\\right)+\\left({\\frac {8}{47}}\\times {\\frac {8}{46}}\\right)={\\frac {96}{2162}}\\approx 0.0444", "1e909059821760764bc9a81e277e2b54": "\\beta _{1}\\leq \\alpha ", "1e909fdc9a50cd6c3cc08459f767b893": "02\\,\\!", "1eac97b0bd15177655ce2ac021e5bc78": "\\sum _{k=0}^{\\infty }{\\frac {(2k)!z^{2k+1}}{2^{2k}(k!)^{2}(2k+1)}}=\\arcsin z,|z|\\leq 1\\,\\!", "1eacbcc0397be0e4a907e3ce1a198110": "{\\text{ROL}}(W)_{j}^{i}=W_{j-i{\\pmod {4}}}^{i}", "1ead2716db7df7dcc1709a6a33ec4f4e": "p_{k}^{\\prime }(t)=\\lambda _{k-1}p_{k-1}(t)+\\mu _{k+1}p_{k+1}(t)-(\\lambda _{k}+\\mu _{k})p_{k}(t){\\text{ for }}k\\leq K\\,", "1ead44e5ad8e6c3bea2a31f2dc914472": "\\scriptstyle PG(4,q)", "1eadba1eac75e1625f3b97758c9d387f": "|a_{ab}|=\\sin ^{2}y\\,", "1eadcb31337236828c385ff9a720cbd7": "\\tan x+\\cot x=2", "1eade162f25e730eba454664e8edb94d": "\\phi (e^{-4\\pi })={\\frac {e^{\\pi /6}\\Gamma \\left({\\frac {1}{4}}\\right)}{2^{{11}/8}\\pi ^{3/4}}}", "1eae2486887be456286befc09daedd27": "entry(O_{r})", "1eae3b5da849864d11e3a01f1a639cd3": "\\lambda =0\\;", "1eae8d8bf99a09bf2866cadc7ef1032b": "{\\hat {\\beta }}={\\frac {\\Gamma ({\\frac {n-1}{2}})}{\\Gamma ({\\frac {n-2}{2}})}}{\\sqrt {\\frac {2}{n-1}}}{\\frac {\\bar {D}}{s_{D}}}.", "1eaea362a7da9eab5dfbf953006d0b3d": "\\mathbf {X} _{1}{\\boldsymbol {\\Theta }}\\mathbf {X} _{2}'", "1eaeb4e469a7bcd8303e4c5e7bd0d7b1": "{\\begin{bmatrix}a1&a2&a3&a4\\\\b1&b2&b3&b4\\\\c1&c2&c3&c4\\\\d1&d2&d3&d4\\end{bmatrix}}", "1eaede3fb376df81307243db8085fb51": "\\Omega =(\\mathbb {R} ^{n})^{T}", "1eaf00ae3a22e4ffa8fbf0c72614014d": "\\sigma ^{2}>0", "1eaf1750ac8f09f965f5fa1f4dc78cb3": "x_{1},x_{2},...,x_{n+2}", "1eaf2b478b94a8e39838dec596a9678f": "g_{1}(x)\\geq 0,\\dots ,g_{k}(x)\\geq 0", "1eaf6a8a048a168eebb15c592fc60938": "\\alpha ={\\frac {1}{V}}\\left({\\frac {\\partial V}{\\partial T}}\\right)_{p}.", "1eafa4076a4f6cd5f516fb25fd29219d": "\\kappa =0.76/{\\sqrt {2}}", "1eafb8c762aad9f9e0904f7182cd6602": "f(t)=ae^{it}", "1eafbb3bc6ac35501dac29c6dffa7972": "B:V\\times V\\rightarrow K:(u,v)\\mapsto B(u,v)", "1eaffe0a843e0a227878c0cddff2590b": "d(\\mathbf {X} \\otimes \\mathbf {Y} )=", "1eb0319c222eeeb48f74082365c147bb": "10^{\\frac {1}{10}}\\,", "1eb0847d06faf9748dfb3de3414c1b9e": "f_{v}(k,r)\\approx f_{0}(E,E_{Fp},T_{p})", "1eb0ab4694f9f819624bd2c8b267f9ff": "{\\overline {u_{i}u_{i}}}", "1eb0b99262ae6f45bf59fd05dcfc5bf5": "B\\to bB|a", "1eb0fd27e19297521d5675eab1493b6b": "G=\\xi \\sin y.\\,", "1eb188bbaf4e8189b0e279e691d157c0": "\\omega ^{2}=\\Omega ^{2}(k).\\,", "1eb18bb9ec47e7ef72520976183ac651": "\\mathbf {M} ={\\begin{pmatrix}1&1\\\\0&-1\\end{pmatrix}};\\quad \\Longrightarrow \\quad \\mathbf {M} ^{2}={\\begin{pmatrix}1\\times 1+1\\times 0&1\\times 1+1\\times -1\\\\0\\times 1-1\\times 0&0\\times 1-1\\times -1\\end{pmatrix}}={\\begin{pmatrix}1&0\\\\0&1\\end{pmatrix}}=\\mathbf {I} ", "1eb1a638125a45b12b1920cc39313b7b": "v", "1eb1e2e6fbebe0f85954f10e242e7b94": "y_{n}=\\Phi (\\ h_{n-1},t_{n-1},y(t_{n-1})\\ )\\ y_{n-1}\\quad ", "1eb252f7f2c44e4209ab920309ccd6d0": "A(\\tau )=\\int _{-\\infty }^{+\\infty }I(t)I(t-\\tau )dt", "1eb270cf3d338409d9e60187f92390fa": "{\\frac {N_{i+1}}{D_{i+1}}}={\\frac {N_{i}}{D_{i}}}{\\frac {F_{i+1}}{F_{i+1}}}.", "1eb27c23d4ff90e3f25a86fb74533d6b": "{\\vec {x}}(t)", "1eb27d73010feab8ae80aba1c2868504": "n_{\\gamma }=0.243\\left({\\frac {k_{\\text{B}}T}{\\hbar c}}\\right)^{3},", "1eb29e3cdf958fac9cc6aadcd673407e": "U'\\left(c-G(l)\\right)\\left({\\frac {dc}{dl}}-G'(l)\\right)=0", "1eb2ba37b7483956db17dbc186067fa3": "\\max\\{\\mathrm {Eigenvalues~of~} {\\mathbf {c}}\\}>1", "1eb2e6e46266c3257075ba1a6f397dac": "P_{N}=P_{0}(1+r)^{N}-c(S)", "1eb30d26c30426eefd6e5a992d5385ac": "p_{xy}(x,y)=\\textstyle \\sum _{i=1}^{3}\\sum _{j=1}^{3}a_{ij}ix^{i-1}jy^{j-1}", "1eb39a201897580c41fe452696be65e8": "r^{2}=k_{1}^{2}+k_{2}^{2}+\\cdots +k_{n}^{2}", "1eb3ad11f243fc51325908e63c21f4b3": "K(z)=Ae^{\\psi (-2,z)+{\\frac {z^{2}-z}{2}}}", "1eb3af657da6b3b9ce9a7a20c6bd5548": "m\\,", "1eb3c24d6fe5ad4c15ff82422cbc3fbd": "{\\begin{cases}x'=\\gamma \\left(x-vt\\right)\\\\t'=\\gamma \\left(t-{\\frac {v}{c^{2}}}x\\right)\\,\\end{cases}}", "1eb3e51e34ea37fd500d8b1dc6a094d0": "a_{1},\\ldots ,a_{m}\\in {\\mathbb {C} }^{d}", "1eb403d789e2ed4931912221521a01d0": "V_{BE1}=V_{T}\\ln \\left({\\frac {I_{C1}}{I_{S1}}}\\right)\\ .", "1eb462ffcdc77f555c772f3667aaa5f0": "{\\sqrt {S}}\\approx A-{\\frac {P^{2}}{2A}}", "1eb4747b7534c0d36614bc45dc3a2de4": "=P_{1}.P_{2}", "1eb47b89c91d71aa2f79919d19291104": "M(n)>(4-6^{-1/2})n/5", "1eb4ad517ed799a98fda4d0804e77f64": "{\\mathbf {y}}'={\\mathbf {A}}{\\mathbf {y}}+{\\mathbf {f}}(x),\\qquad \\qquad \\qquad \\qquad \\quad (5)", "1eb4b0483cbd69e03b061c51698c5826": "\\gamma <\\alpha ", "1eb4d9edfbd8070c1939a09236c9ac84": "p_{a}={\\frac {2aT}{a^{2}+b^{2}-c^{2}}},", "1eb4e4be2223b428703eef6c7124088c": "B=Q\\,\\Delta t", "1eb541989515fb3004afde202e9f96a1": "{\\mathfrak {d}}", "1eb5523dd5141a3528c44fe436670141": "\\int {\\frac {1}{x^{2}+a^{2}}}\\,dx={\\frac {1}{a}}\\arctan {\\frac {x}{a}}\\,\\!+C", "1eb55750136e4b81e9ca596c930dadbd": "\\textstyle 0.5\\left({\\frac {PRF\\times C}{Transmit\\ Frequency}}\\right)", "1eb604f4b3d67b9579f88b6101c82d0f": "\\gamma (x^{\\prime }+\\beta ct^{\\prime })", "1eb63d7bb002ce211e87dcb8fdacfa6c": "Q=\\left({\\frac {V\\times I\\times 60}{S\\times 1000}}\\right)\\times \\mathrm {Efficiency} ", "1eb67b6313a6bd419e37b9f470230f12": "\\displaystyle {\\frac {\\mathrm {d} ^{2}X}{\\mathrm {d} t^{2}}}={\\frac {c^{2}}{2}}\\varepsilon \\gamma _{00}", "1eb69130f20bcad84623b374650bd4c0": "\\textstyle {\\tilde {\\beta }}", "1eb6b4dc0e030aa9d9a0fead410245e6": "(x_{3},y_{3},0)", "1eb6ee0f1d09829a14aa4c231804ebcb": "{\\overset {\\square }{\\boldsymbol {\\tau }}}={\\dot {\\boldsymbol {\\tau }}}+{\\boldsymbol {\\tau }}\\cdot {\\boldsymbol {\\Omega }}-{\\boldsymbol {\\Omega }}\\cdot {\\boldsymbol {\\tau }}", "1eb713fb2232eb9765474f4445e1f516": "{\\begin{aligned}\\int \\sec \\theta \\,d\\theta &=\\tanh ^{-1}\\!\\left(\\sin \\theta \\right)=\\sinh ^{-1}\\!\\left(\\tan \\theta \\right)=\\cosh ^{-1}\\!\\left(\\sec \\theta \\right).\\end{aligned}}", "1eb717a0ad95e460119838cd78cdb560": "E=\\alpha U.", "1eb72583ff9c2e5311bb49e6a74743b2": "g(\\gamma )={\\frac {3}{2\\gamma }}(1+{\\frac {1}{2\\gamma ^{2}}}\\sinh ^{-1}(\\gamma )-{\\frac {\\sqrt {1+\\gamma ^{2}}}{2\\gamma }})", "1eb73a36533500517ef53038b85bb075": "a={\\sqrt {\\frac {m}{b^{3}}}}={\\sqrt {2^{2}\\times 5^{2}}}=10\\,,", "1eb7e11a305ed75dfbdb2e9ec6bd4abd": "c^{2}={\\frac {1}{1-{\\frac {r_{s}}{r}}}}\\,{\\frac {E^{2}}{m^{2}c^{2}}}-{\\frac {1}{1-{\\frac {r_{s}}{r}}}}\\left({\\frac {dr}{d\\tau }}\\right)^{2}-{\\frac {1}{r^{2}}}\\,{\\frac {L^{2}}{m^{2}}}\\,,", "1eb7fd7fc72fa6ebc4a1f2c5372dc736": "e(g^{x},g^{y})=e(g,g^{z})", "1eb822c1536ddd256f247f2c18394b57": "\\mathbf {A} {\\hat {e}}_{1}\\ ,\\ \\mathbf {A} {\\hat {e}}_{2}\\ ,\\ \\mathbf {A} {\\hat {e}}_{3}.", "1eb8422215acbc85d7e07cb68b8ebb79": "\\psi _{j}^{\\alpha }(\\mathbf {r} _{j})=\\psi _{N_{\\alpha }+j}^{\\beta }(\\mathbf {r} _{N_{\\alpha }+j}),\\ \\ \\ 1\\leq j\\leq N_{\\beta }.", "1eb88ef5f847daf8fe938dc577168c22": "z\\leftarrow w", "1eb908b2f3d928c4be9c6a7021d71df4": "x^{\\frac {3}{2}}", "1eb9418dc7a627f01fdb6559de364dae": "c_{i+1,\\sigma }", "1eb97e787a29b7eba3f3e7952baffa2a": "H(X)=-\\int _{X}f(x)\\log f(x)\\,d\\nu (x)=-\\sum _{x\\in X}f(x)\\log f(x).", "1eba2348fa963a9d5633506a55f96a69": "I_{J}=0,2\\mu ", "1eba63d2398920deec0b5cb407f6ccdc": "X^{\\mu }\\rightarrow X^{\\mu }+\\delta X^{\\mu }", "1eba6caacde14b058d866140c4becda5": "V_{\\omega }", "1eba74430ba4c3a6e3c64b2f2a53008c": "{\\text{fmap}}:(A\\rightarrow B)\\rightarrow (M\\rightarrow A)\\rightarrow M\\rightarrow B=f\\mapsto g\\mapsto (f\\circ g)", "1eba79fad5125d7a595e26aed932f2fc": "{\\tilde {\\psi }}\\in L^{2}(\\mathbb {R} )", "1ebab854b80c84d7bf85081c61e15bb1": "{{258\\cdot {{42} \\over {43}}} \\over {49 \\choose 6}}", "1ebad413d19b19c7d0993da142e34b53": "\\scriptstyle V(u,\\Omega )<+\\infty ", "1ebaffa1e997c3467a93fb42f6cd5f3f": "{\\mbox{SAIDI}}={\\frac {\\mbox{sum of all customer interruption durations}}{\\mbox{total number of customers served}}}", "1ebb03d778b6d5cc34693b39fcd8521b": "g_{k,n}(z)=z+\\varphi _{k,n}(z)", "1ebb1bee05b6f39e75c62132bda2d9e3": "{\\mathfrak {H}}(k;\\gamma ,\\infty )={\\begin{pmatrix}k&(1-k)\\gamma \\\\0&1\\end{pmatrix}}.", "1ebb1bf30a0eb0253fccf64abf0340a0": "\\sum _{-m\\leq i\\quad \\, \\atop j\\leq m}\\langle A(i,j)h_{i},h_{j}\\rangle \\geq 0.", "1ebb3364f0e9db61ef1e91d37fd2fe6c": "{\\scriptstyle A}", "1ebb4f51b91b59ad3b759bc796368346": "{\\overline {h}}={{\\overline {p}}C_{p} \\over {\\overline {\\rho }}R}.", "1ebba5c39adb8e3bd296222db8bfb921": "F_{n}{\\mathbb {R}}^{2}", "1ebbac688acaee729f998a9e6366f104": "{\\begin{aligned}\\mathbf {U} (\\mathbf {x} ,t)&=\\mathbf {x} -\\mathbf {X} (\\mathbf {x} ,t)\\qquad &{\\text{or}}&\\qquad U_{J}=\\delta _{Ji}x_{i}-X_{J}=x_{J}-X_{J}\\\\\\nabla _{\\mathbf {x} }\\mathbf {U} &=\\mathbf {I} -\\nabla _{\\mathbf {x} }\\mathbf {X} =\\mathbf {I} -\\mathbf {F} ^{-1}\\qquad &{\\text{or}}&\\qquad {\\frac {\\partial U_{J}}{\\partial x_{k}}}=\\delta _{Jk}-{\\frac {\\partial X_{J}}{\\partial x_{k}}}=\\delta _{Jk}-F_{Jk}^{-1}\\,.\\end{aligned}}", "1ebbf6f08e5c59420c4b625fe9c3438b": "d{\\tilde {\\rho }}(\\rho )=0", "1ebc54845e26c3f8c0e3f3d16d1ba0b2": "\\scriptstyle e'=(v,u)", "1ebc96c7171c6c6e3dd2693c232f979b": "\\int _{\\gamma }f(z)\\,dz=0", "1ebc9b9ab89e548d244f790a3d49a1ff": "(x-1)-{\\frac {1}{2}}(x-1)^{2}+{\\frac {1}{3}}(x-1)^{3}-{\\frac {1}{4}}(x-1)^{4}+\\cdots ,\\!", "1ebdbced25bfd2da459ee8e5e392cb8b": "\\scriptstyle I", "1ebe36cdc84b91a06059e4216dc77cd4": "(r+\\mathrm {d} r,\\,\\theta +\\mathrm {d} \\theta ,\\,\\varphi +\\mathrm {d} \\varphi )", "1ebe9460cbcb56b1c2745a1811380bf4": "D_{H}=D", "1ebeeeddf6cfbc1739f288b5a1abf54a": "\\langle 0,...,k\\rangle ", "1ebefbc1820dde5c50e0ee06e695c605": "F=-kx\\,", "1ebfb6ba9c25312cf01cb8f62ae61e5c": "I_{n}\\otimes \\Phi ", "1ebfeda33a028acc66c6867405ad96f1": "c={\\sqrt {\\frac {C}{\\rho }}}\\,", "1ec00546f44bc0f23e184ac04e8faa45": "u^{c}", "1ec0091eff0f72f0b43f941266c5d50a": "i^{0}=i^{1-1}=i^{1}i^{-1}=i^{1}{\\frac {1}{i}}=i{\\frac {1}{i}}={\\frac {i}{i}}=1\\,", "1ec021f4c3dd9dcb9f443e1681fdf9a8": "\\sum _{x}\\log d_{x}", "1ec029cc60e4f40018998a544cb2abda": "\\gamma ^{(j)}\\,", "1ec07abf40f29e3bad8c3ed0c7ce13a9": "B^{\\prime }", "1ec0a3265956fdcb7000186655bc5179": "d+2w(u)\\geq 2d", "1ec0f4b4a81fd6310c27e26073cae36a": "{\\dot {\\sigma }}={\\dot {x}}_{1}+{\\dot {x}}_{2}={\\dot {x}}+{\\ddot {x}}={\\dot {x}}\\,+\\,\\overbrace {a(t,x,{\\dot {x}})+u} ^{\\ddot {x}}", "1ec119c66c955d3dd5b0eeb7427c0014": "(10{\\bar {1}}1100{\\bar {1}}10)_{s}", "1ec1290f8a4a4c061afa10121e93d1ac": "T^{00}=\\rho ,", "1ec18a4052c11adfb09f4a75a3975532": "\\mathbf {i} _{k}", "1ec19cc751ac5590b83da3c41d6b509a": "\\gamma (t)=\\gamma '\\ast \\lambda (t)", "1ec1a0912d95a892b4bab948cd8ce556": "\\Phi _{G}=\\left\\{V_{1}^{G},\\dots ,V_{k}^{G}\\right\\}", "1ec1b0da7355babafa9b7e836e8e7cc4": "\\int dq\\rho _{0}\\left(q\\right)f\\left(q\\right)\\delta \\left(A\\left(q\\right)-a\\right)=\\int dq\\rho _{0}\\left(q\\right)F\\left(A\\left(q\\right)\\right)\\delta \\left(A\\left(q\\right)-a\\right)=F\\left(a\\right)\\int dq\\rho _{0}\\left(q\\right)\\delta \\left(A\\left(q\\right)-a\\right).", "1ec25ab34fbd021319e99b0e2e78f4ae": "-{\\dot {q}}_{\\rm {ext}}={\\boldsymbol {\\nabla }}\\cdot (\\kappa {\\boldsymbol {\\nabla }}T)+\\mathbf {J} \\cdot \\left(\\sigma ^{-1}\\mathbf {J} \\right)-T\\mathbf {J} \\cdot {\\boldsymbol {\\nabla }}S", "1ec291bf65047f349d3a68b24ddd4713": "0\\mod q_{\\ell }", "1ec2b08201603b79ba8eaf478495c4ae": "\\scriptstyle {\\mathcal {F}}", "1ec2b4e835eb0b5c2f0911b3d98a6fac": "{\\widetilde {u}}(x)={\\frac {u(x)}{u_{0}}}.", "1ec3411f57192614871b8895b9063a4c": "R(Q)/R_{F}(Q)=\\left|{\\frac {1}{\\rho _{\\infty }}}{\\int \\limits _{-\\infty }^{\\infty }{e^{iQz}\\left({\\frac {d\\rho _{e}}{dz}}\\right)dz}}\\right|^{2}", "1ec37fcc002d088d99d7035fcbf66f5b": "R^{i}\\,\\!", "1ec39ae7c2acd408004153069552825c": "x\\otimes y\\rightarrow y\\otimes x", "1ec3da35329c1ddd2e2daeb9a3f39cbd": "\\operatorname {E} [(X)_{n}]=\\lambda ^{n}.", "1ec3f969df401edf119afc601685afe7": "\\lambda _{0}(t)={\\frac {c_{0}}{y_{0}-a_{0}t}}", "1ec50ef80f5b75f1be0bfb4a313d1999": "\\sigma ^{a}", "1ec525e1383c756df033f0aae02897ea": "w=w_{1}w_{2}\\cdots w_{k}\\in Y^{+}", "1ec53c7bc6a4f251d05b39e8658ccd8c": "P_{L1}=V_{L1}I_{L1}=V_{P}I_{P}\\sin \\left(\\theta \\right)\\sin \\left(\\theta -\\varphi \\right)", "1ec583308d8b08c6057294571484abd9": "y(t+h)\\approx y(t)+hy'(t)\\qquad \\qquad ", "1ec5b9681a2294e1e67deca9429992ad": "s_{2}(t)={\\overset {\\cdot }{s}}_{1}(t)+\\alpha _{2}(t)s_{1}^{\\gamma _{2}}(t)", "1ec5c265d552445b7fb34c244df285d1": "{\\tilde {C}}(u)=W(u)", "1ec683e05e7921e95e358f72f778dd17": "\\operatorname {E} [X]={\\frac {a}{3}}", "1ec7099c3d74c9e4b4e046dbd5aba758": "\\sum _{\\lambda \\vdash n}(t_{\\lambda })^{2}=n!", "1ec77202edb98270c964ba0be7bd7d32": "\\left(A\\partial _{x}+B\\partial _{y}+C\\partial _{z}+{\\frac {i}{c}}D\\partial _{t}\\right)\\psi =\\kappa \\psi ", "1ec78147bdba402eb81e5d03a81195f7": "A_{q}(n,d)", "1ec78c301719f36037769309903cb376": "M_{xy}'(t)=M_{xy}'(0)e^{-t/T_{2}}", "1ec79c69bb84d5219a85d59bf2585e4b": "V_{T}\\,", "1ec79e91d92df04471972727d4bb9404": "10^{-6}", "1ec801116fc1d9b1f2147a635cadcb7c": "\\mathrm {[H^{+}]^{3}} +(K_{a}+C_{b})\\mathrm {[H^{+}]^{2}} -(K_{w}+K_{a}C_{a})\\mathrm {[H^{+}]} -K_{a}K_{w}=0", "1ec83761b4b7c997bfd5dbed0c07e279": "h^{eff}=h+Jzm", "1ec850722475d8e580ae1a17de338859": "A_{i}\\rightarrow A_{j}\\gamma ", "1ec88b186241729643610e9889038694": "(\\mu _{ab}^{*}(t))", "1ec8a5108db98ecbe0d0278c0b78e878": "D_{26}=\\langle a,b|a^{2}=b^{13}=1,aba=b^{-1}\\rangle .", "1ec8aa82f6aad4cc49960e44b75c0a16": "\\forall H\\in \\mathbf {H} ,\\forall v\\in H", "1ec8cbe560e8240917a675c75d97b089": "\\pi _{1}{\\big (}PSO(2){\\big )}=\\mathbf {Z} ,", "1ec8d7af63e281a0d5407febbd4416b1": "\\prod _{i\\in I}U_{i}", "1ec8d7eff29ecd71a82e2a4fb6b63275": "\\|\\tau _{v+h}f-\\tau _{v}f\\|_{p}=o(1),", "1ec8f0085668d675f0c970e4bbec9280": "\\delta _{\\vec {\\xi }}h=\\delta _{\\vec {\\xi }}g-\\delta _{\\vec {\\xi }}\\eta ={\\mathcal {L}}_{\\vec {\\xi }}g={\\mathcal {L}}_{\\vec {\\xi }}\\eta +{\\mathcal {L}}_{\\vec {\\xi }}h=\\left[\\xi _{\\nu ;\\mu }+\\xi _{\\mu ;\\nu }+\\xi ^{\\alpha }h_{\\mu \\nu ;\\alpha }+\\xi _{;\\mu }^{\\alpha }h_{\\alpha \\nu }+\\xi _{;\\nu }^{\\alpha }h_{\\mu \\alpha }\\right]dx^{\\mu }\\otimes dx^{\\nu }", "1ec8f4d6e0bbcc7da8cc8b45bc8753c1": "\\lim _{n\\rightarrow \\infty }f_{n}(x)={\\begin{cases}0,&x\\in [0,1)\\\\1,&x=1.\\end{cases}}", "1ec911c2b681b9eef78e1a55795a6d9d": "F_{max}=mg+{\\sqrt {(mg)^{2}+F_{0}(F_{0}-2m_{0}g){\\frac {m}{m_{0}}}{\\frac {f}{f_{0}}}}}", "1ec91515d3d64d3f7811f9d115134b0c": "\\aleph ", "1ec956b9b4dcc57a48a67a7a81be2255": "(s_{i}f)(t_{0},\\dots ,t_{n+1})=f(t_{0},\\dots ,t_{i-1},t_{i}+t_{i+1},t_{i+2},\\dots ,t_{n+1})\\,", "1ec9a25a6812b49dbb3712f0a283146f": "\\sum a_{ij}b_{ij}", "1eca0bffac8452289ff430574fc62790": "\\mathbf {u} (\\mathbf {X} ,t)\\,\\!", "1eca421f0ee2a202ab9a72b8c31fcb44": "\\kappa _{2}", "1ecacddd93add60feb6cc079a64f2859": "S_{xx}(\\omega )=\\lim _{T\\rightarrow \\infty }\\mathbf {E} \\left[|{\\hat {x}}_{T}(\\omega )|^{2}\\right].", "1ecb1b8b55f12b5843e4f10be4ea0242": "I=\\mathbf {J} \\cdot \\mathrm {d} \\mathbf {S} ", "1ecb2d11982ea453cf1163ec464a5531": "x_{13}", "1ecba53ee71ad3416181255101a4b593": "{\\frac {d}{dk}}\\Gamma _{k}={\\frac {1}{2}}\\operatorname {Tr} \\left[\\left({\\frac {\\delta ^{2}\\Gamma _{k}}{\\delta \\phi \\delta \\phi }}+R_{k}\\right)^{-1}\\cdot {\\frac {d}{dk}}R_{k}\\right]", "1ecbb1dedea742667ad6b1433e573773": ",\\left[s(nT)\\cdot e^{-j2\\pi {\\frac {B}{2}}Tn}\\right],", "1ecbb1ee49a73ef8de2c078073396750": "e^{z}=1+{\\cfrac {2z}{2-z+{\\cfrac {z^{2}}{6+{\\cfrac {z^{2}}{10+{\\cfrac {z^{2}}{14+\\ddots }}}}}}}}", "1ecbd0049570d70901dae7506e9b5ed7": "\\lim _{a\\to \\infty }\\int _{-2a}^{a}xf(x)\\,dx,\\!", "1ecc0dc5e22a9589cc0c664537248e3b": "E(y)=\\sigma {\\sqrt {2/\\pi }}\\exp(-\\mu ^{2}/2\\sigma ^{2})+\\mu \\left[1-2\\Phi (-\\mu /\\sigma )\\right],", "1ecc0fb85f9c89e1cb406b758fead9b1": "\\mathbf {v} ={\\dot {\\mathbf {x} }}=\\left[{\\dot {\\mathbf {p} }},{\\dot {\\mathbf {r} }}\\right]", "1ecc6552a9bdc3dc9d7e5907abcff95d": "{\\begin{aligned}{\\overset {\\circ }{\\boldsymbol {\\sigma }}}&=J^{-1}~{\\boldsymbol {F}}\\cdot ({\\dot {J}}~{\\boldsymbol {F}}^{-1}\\cdot {\\boldsymbol {\\sigma }}\\cdot {\\boldsymbol {F}}^{-T})\\cdot {\\boldsymbol {F}}^{T}+J^{-1}~{\\boldsymbol {F}}\\cdot (J~{\\dot {{\\boldsymbol {F}}^{-1}}}\\cdot {\\boldsymbol {\\sigma }}\\cdot {\\boldsymbol {F}}^{-T})\\cdot {\\boldsymbol {F}}^{T}\\\\&+J^{-1}~{\\boldsymbol {F}}\\cdot (J~{\\boldsymbol {F}}^{-1}\\cdot {\\dot {\\boldsymbol {\\sigma }}}\\cdot {\\boldsymbol {F}}^{-T})\\cdot {\\boldsymbol {F}}^{T}+J^{-1}~{\\boldsymbol {F}}\\cdot (J~{\\boldsymbol {F}}^{-1}\\cdot {\\boldsymbol {\\sigma }}\\cdot {\\dot {{\\boldsymbol {F}}^{-T}}})\\cdot {\\boldsymbol {F}}^{T}\\end{aligned}}", "1ecc6f8f35c8b73b295f2401531692b0": "{4 \\choose 1}^{4}-{3 \\choose 1}=253", "1ecc96509281ceb9a14ca7752aea73a5": "\\cosh \\delta =\\left|{\\frac {a^{2}+b^{2}-c^{2}}{2ab}}\\right|.", "1eccab7596a75d9efad6b16b8a4ccbdd": "~1/\\beta =k_{B}T", "1ecd27c57b10d1a1d5595f8bc0b753de": "{\\mathcal {L}}\\left\\{f^{(n)}(t)\\right\\}=s^{n}\\cdot {\\mathcal {L}}\\left\\{f(t)\\right\\}-s^{n-1}f(0^{-})-\\cdots -f^{(n-1)}(0^{-}),", "1ecd4b9ba395716a2e4d946246995f66": "\\lim _{k\\to \\infty }{\\frac {G^{k}}{k!}}=0", "1ecd513e7722d5d91e6ca38bf7406ce8": "C={\\begin{pmatrix}0&1&0&0\\\\0&0&1&0\\\\0&0&0&1\\\\1&0&0&0\\end{pmatrix}}", "1ecd9c40df5730b41e9171c3879f1a72": "P=\\rho k_{B}T-{\\frac {2}{3}}\\pi \\rho ^{2}\\int _{0}^{\\infty }dr{\\frac {du(r)}{dr}}r^{3}g(r)", "1ecda1ec9e56eed9b868b17661d5faaa": "C_{n}={\\frac {i^{n}}{2^{n}n!}}m\\circ \\Pi ^{n}.", "1ecddf43eb7e714c44e6c40eda1368ea": ",z", "1ecdedab2122c26ebb4abcc1cd1f6355": "\\sum _{n=1}^{\\infty }\\lambda (n)\\,{\\frac {q^{n}}{1-q^{n}}}=\\sum _{n=1}^{\\infty }q^{n^{2}}", "1ece566f374b1e5a56e7b527d662b1a7": "P_{d}=h^{0}(X,K_{X}^{d})=\\operatorname {dim} \\ H^{0}(X,K_{X}^{d}).", "1ece9307e8a144b0407bbf299c982297": "\\;\\Omega _{R}(s_{1})=2\\;\\Omega _{R}(s_{2})", "1ece9e67f6debc628f87c4940f7a213f": "P_{0}(z)=1;~~Q_{0}(z)=1;", "1eced98c2a6b6999908a2ef56fdd1a3c": "D=s\\Delta p_{0}\\cos \\alpha _{m}", "1ecf17e2f655476074d1a3e8601fbeeb": "\\sum _{n=1}^{\\infty }{\\frac {(-1)^{n}}{n(4n^{2}-1)}}=\\ln 2-1.", "1ecf20ba5dc5e9823517fc42523001c5": "\\lim _{t\\to \\infty }T^{t}", "1ecf270f0290445050f81ac3b4a5d0e4": "\\int _{-\\infty }^{\\infty }{\\frac {\\sin(\\pi x)}{\\pi x}}\\,dx=\\mathrm {rect} (0)=1\\,\\!", "1ecf547d838b4d85fcded4b9d529ac84": "g(z)=\\exp \\left(\\sum _{d|m}{\\frac {z^{d}}{d}}\\right).", "1ecf927a1a35e6df99036eb95154335e": "A\\times A:=\\{xy:x,y\\in A\\}", "1ecfb6e44370f0b6b4eb4d2878e5f5e1": "U=-\\int _{r_{0}}^{r}\\mathbf {F} \\cdot \\,d\\mathbf {r} =-\\int _{r_{0}}^{r}{\\frac {1}{4\\pi \\varepsilon _{0}}}{\\frac {q_{1}q_{2}}{\\mathbf {r^{2}} }}\\cdot \\,d\\mathbf {r} ={\\frac {q_{1}q_{2}}{4\\pi \\varepsilon _{0}}}({\\frac {1}{r_{0}}}-{\\frac {1}{r}})+c", "1ecfd85482f88c8586259a19bbd37772": "z_{0}\\neq 1", "1ed0239331fe8d32570400c9b178a854": "\\{\\{M,O\\},O\\}_{M=0}=0", "1ed0ea414ba01502814842052d5de139": "f(w)=w/\\phi (w)", "1ed0f83a527955cdd8d14fb0e648cbec": "\\sigma _{xx}={\\frac {-\\mu b}{2\\pi (1-\\nu )}}{\\frac {y(3x^{2}+y^{2})}{(x^{2}+y^{2})^{2}}}", "1ed15447914f5bf8ee54dee88ce7bab7": "K={\\tfrac {1}{2}}|x_{1}y_{2}-x_{2}y_{1}|.", "1ed166e6ef6ff34b5afe5f77083cf66c": "g_{3}={\\tfrac {1}{2}}(1+(\\eta ^{2}-2)j+(3-\\eta ^{2})ij).", "1ed18a8ab3bc29b703218e07354ee43a": "\\zeta (10)=1+{\\frac {1}{2^{10}}}+{\\frac {1}{3^{10}}}+\\cdots ={\\frac {\\pi ^{10}}{93555}}=1.000994...\\dots \\!", "1ed196f2325656a83ec23cc61c7e24fc": "A=\\sum _{n}a_{n}|e_{n}\\rangle \\langle e_{n}|", "1ed1f129f7f85ec8f52eecc00ba8a013": "\\exists xRx", "1ed208f30969a8a5fd0197a2e4521373": "R=I\\cos \\theta +[\\mathbf {k} ]_{\\times }\\sin \\theta +(1-\\cos \\theta )\\mathbf {k} \\mathbf {k} ^{\\mathsf {T}}", "1ed262cb444f95b8a45106a84b13660d": "c_{i}^{A},c_{j}^{B}", "1ed297019b270c06d14408dadd755f0c": "\\Delta U(t)\\Delta V(t)=d[U,V]", "1ed2e6ac8d74840a5cc110e1d760b2dc": "\\mathrm {K_{a}\\,=\\,{\\frac {[H^{+}\\,][A^{-}\\,]}{[HA]}}} ", "1ed34692f7bdfacfd9f09f6b5c3cdb52": "P(M1)=P(M2)=1/2", "1ed346930917426bc46d41e22cc525ec": "\\phi ", "1ed376d481fa7f58d46d052807858828": "C_{Q0}=8\\pi \\alpha \\cdot C_{QY}\\ ", "1ed389a4c15ffbc7673138ed2131d981": "\\partial _{t}^{k}u|_{t=0}=\\phi _{k}(x),\\qquad 0\\leq k\\leq m-1,", "1ed39bcd371df8a7ee38ac8351925aae": "\\beta =x_{\\eta }x_{\\xi }+y_{\\xi }y_{\\eta }", "1ed3dd6f9e7bfd66ed48b6dbf817c918": "M=E\\cdot K^{-1}", "1ed3f3127aa57f10fc1e3c53a03bd840": "\\mathbf {H} _{\\alpha -1}(x)+\\mathbf {H} _{\\alpha +1}(x)={\\frac {2\\alpha }{x}}\\mathbf {H} _{\\alpha }(x)+{\\frac {{(x/2)}^{\\alpha }}{{\\sqrt {\\pi }}\\Gamma (\\alpha +{\\frac {3}{2}})}}", "1ed442173a51dc0b9c3cf2e7c5352f61": "w=h_{1}h_{2}\\cdots h_{n}", "1ed4d1b05034e33c314cf915ce670b75": "m_{n}=E(X^{n})={\\frac {1}{\\zeta (s)}}\\sum _{k=1}^{\\infty }{\\frac {1}{k^{s-n}}}", "1ed52deb0d185243d6eaec665ec3652d": "F_{ij}=G(M_{i}^{\\beta _{1}}M_{j}^{\\beta _{2}}/D_{ij}^{\\beta _{3}})\\eta _{ij}^{\\ }", "1ed55474ef550ce4b7b3cf47aa5a87cf": "S2=\\sum _{j=1}^{k}|High[j]-High[j-2]|+|Low[j]-Low[j-2]|", "1ed5867197fe41edbc4177ed4c5fada7": "\\textstyle {\\overline {U}}", "1ed5ca65c4195b2b1c8bded40794f6bd": "\\theta \\log {\\tan \\theta }-\\int _{0}^{\\tan \\theta }{\\frac {\\log x}{1+x^{2}}}\\,dx", "1ed62ba1ea3ede7d4731810251d61eb6": "|b|_{\\ast }\\leq 1", "1ed77893fbc1f3afe712f5db696748e1": "{\\frac {\\partial H}{\\partial u_{t}}}=p-\\lambda _{t+1}-2{\\frac {u_{t}}{x_{t}}}=0", "1ed86f05e38d3a425946046e3ac59b5e": "E(\\delta (X))=\\sum _{x=0}^{\\infty }\\delta (x){\\frac {\\lambda ^{x}e^{-\\lambda }}{x!}}=e^{-2\\lambda },", "1ed872aa38c852cc1e1f37b2b44cc8be": "{\\partial u \\over \\partial \\lambda }=8\\pi hc\\left({hc \\over kT\\lambda ^{7}}{e^{hc/\\lambda kT} \\over \\left(e^{hc/\\lambda kT}-1\\right)^{2}}-{1 \\over \\lambda ^{6}}{5 \\over e^{hc/\\lambda kT}-1}\\right)=0,", "1ed894a22a52f869cbb866bc50d6f4a9": "(X^{n}(j),Y^{n})", "1ed8c30a30ec29b18d747f1d6c52c49f": "\\scriptstyle {\\widehat {\\varepsilon }}_{i}", "1ed99c2dd202f44ce8a65fc27f48e8c3": "{\\frac {\\$10,000}{\\$40,000}}=0.25=25\\%,", "1ed9be740d876001d18445bf0f9b7069": "cdt=\\gamma d\\tau ", "1eda154dc6063bfa4c1a9f1d0aa7bffd": "a_{32}", "1eda90079cca7409a9e415de34dbc8db": "v\\otimes w\\in V\\otimes W", "1edaa21a993e6df47a4872f64a335cee": "E=\\mathbb {Z} ^{2}", "1edaaf4a82237b5d111baaec4357d25c": "R[[X]]", "1edada44ebf674c8caaeb29c756caaea": "X_{m}", "1edc06c025db330b27d6b616c9ee2f1c": "f\\not \\in {\\sqrt {I}}", "1edc146a69b97f24fd3d42b4f099c15d": "W_{2}=m_{2}g", "1edc4dc654eb77a7ae71e2d2713b2392": "\\nu =\\,", "1edc799361f3e8c3c40286fa9c4bc033": "AF\\phi \\equiv \\phi \\lor AXAF\\phi ", "1edccfe03fc23a9364699d8f19d62a07": "f(X)=g(X^{p^{n}})", "1edd09ab5f2737166b43ccea97bf3845": "=2\\pi \\cdot I_{max}\\int \\limits _{0}^{\\pi /2}\\cos(\\theta )\\sin(\\theta )\\,\\operatorname {d} \\theta ", "1edd211800a09dac211e661dfae8eb98": "W.L", "1edd5630ed74cfd4167cd5e6f3511ee6": "-v(A-0)", "1edd6c1727192ed6cefb1d1e7e397967": "f:{\\mathbb {R} }\\rightarrow {\\mathbb {R} }", "1edd7293daa48a3827508d749f619f79": "\\varphi _{\\alpha }(x)=\\inf _{y\\in H}{\\frac {1}{2\\alpha }}\\|y-x\\|^{2}+\\varphi (y).", "1edd8076f9f5d6bb20d01f2aa1293fda": "c={\\sqrt {\\frac {B}{\\rho _{0}}}}", "1edd9321f7d0513cdde995c33dfb15d1": "g=g_{0}(r_{0}/r)^{2}\\,", "1edd95e5747d6d9484585ea941426b81": "e(P_{\\infty ,i})", "1edd9b126c2fb9b4869a84cf4f0f02e1": "\\kappa _{f}(k,i)={\\frac {\\delta (k,i)}{\\xi _{b_{min}}^{d}(k-1,i)}}", "1eddfdf60538c4f0d3dd59006ad2a9c4": "\\mathrm {Aff} (n)\\simeq \\mathrm {GL} (n)\\rtimes \\mathbb {R} ^{n}", "1ede65dd57d62e9140674f7bd1ea5662": "\\varepsilon =\\varepsilon _{r}\\varepsilon _{0}=(1+\\chi _{e})\\varepsilon _{0}.", "1ede801a3909cb17e110112a6ec20a02": "\\chi ^{1}={\\begin{bmatrix}0\\\\1\\end{bmatrix}}\\quad \\quad \\chi ^{2}={\\begin{bmatrix}1\\\\0\\end{bmatrix}}\\,", "1edf00e4b6254b847d8e47b2c1cddb2f": "\\mathbf {A} =A_{+}\\mathbf {e} _{+}+A_{-}\\mathbf {e} _{-}+A_{0}\\mathbf {e} _{0}", "1edf39bdff607c52cec40bb94869f57e": "{\\boldsymbol {F}}({\\boldsymbol {S}})={\\boldsymbol {F}}_{1}({\\boldsymbol {F}}_{2}({\\boldsymbol {S}}))", "1edf6235ff0aa6691866148877ad3395": "{\\bar {m}}", "1edf66f6f28accded7d527ba25b85227": "dA=r^{2}\\sin \\theta \\,d\\theta \\,d\\phi .", "1edf7a10cb75a5ed6d5b1357e53c88ae": "Y:M\\to \\mathbb {R} ^{n}", "1edf84c6ee3304391923017efd607558": "Y=\\bigoplus _{i=1}^{\\infty }\\mathbb {R} ", "1edfa143a11d4b9bc5e821f7a399814a": "\\left[{\\begin{array}{c}R\\\\G\\\\B\\end{array}}\\right]=\\left[{\\begin{array}{ccc}255/R'_{w}&0&0\\\\0&255/G'_{w}&0\\\\0&0&255/B'_{w}\\end{array}}\\right]\\left[{\\begin{array}{c}R'\\\\G'\\\\B'\\end{array}}\\right]", "1edfc958d5a196148a1cb43373a2ccf0": "{\\frac {dy}{dx}}=-1,", "1edff01f1e3949fc2a9f60b46008155b": "(q_{t}^{i}-\\Gamma ^{i})\\partial _{i}", "1edffe9debd22ecd5ea5124ce339e908": "\\int _{0}^{2\\pi }|f(h(re^{i\\theta }))|^{p}\\,d\\theta \\leq \\int _{0}^{2\\pi }|f(re^{i\\theta })|^{p}\\,d\\theta .", "1ee0430c5ff75ede5c4fafc29a89bd08": "\\lambda _{j}=(-1)^{j}{\\binom {n-k-j}{k-j}}", "1ee0447fde38a4aa7f88d413c6f0efd3": "SM_{3}(levol,endo,exo)=RE", "1ee06e198eb28d5cac674daadb9de5f2": "\\operatorname {Ber} (X)=\\det(A)\\det(D)^{-1}", "1ee07d263b25e3dcda7057e9deb3360e": "{\\underline {\\underline {{\\boldsymbol {A}}_{2}}}}", "1ee0ac9be1d0328a58dca551ee263bcc": "|t_{\\lambda }-t_{\\lambda -1}|<{\\tfrac {1}{2}}\\,|t_{\\lambda -1}|", "1ee109bff381e763045eb7bdc8a4792b": "M_{t}=p(W_{t},t)", "1ee12efc153c609465e4909f6fdc8a6f": "W_{1}=1,\\quad W_{2}=b,\\quad W_{3}=c,\\quad W_{4}=d.", "1ee147abfa6420fb97621c10756a699c": "2g(\\nabla _{X}Y,Z)-g([X,Y],Z)+g([X,Z],Y)+g([Y,Z],X)", "1ee150b7ad7e504033b554dc3310db12": "(a+b)'=a'+b'=b'+a'=(b+a)',", "1ee16b1070b825a921e89eb3242a1739": "C_{\\mathrm {Artin} }=\\prod _{q\\ \\mathrm {prime} }\\left(1-{\\frac {1}{q(q-1)}}\\right)=0.3739558136\\ldots ", "1ee17a051f9512465a3beaba1e26bcee": "{\\sqrt {\\pi }}=\\left(-{\\frac {1}{2}}\\right)!=\\int _{-\\infty }^{\\infty }{\\frac {1}{e^{x^{2}}}}\\,dx=\\int _{0}^{1}{\\frac {1}{\\sqrt {-\\ln x}}}\\,dx", "1ee1df7fcc344f3c320827ad3bf8507c": "\\ldots 999=9+9(10)+9(10)^{2}+9(10)^{3}+\\cdots ={\\frac {9}{1-10}}=-1.", "1ee1ee02d11ae3ac38869733bda55607": "J_{n+1}=2J_{n}+(-1)^{n}\\,,", "1ee252f4f0c5f94fdc1d55ac1da82bd1": "-\\infty ", "1ee2e9faa290b6678b2f43a98646ab0e": "\\phi _{x}(t)", "1ee2ecc57219e2c075c1fa46b77322cf": "10\\log(|H_{NEXT}(f)|^{2})={\\begin{cases}-59.2+4\\log(f)dB&f<20KHz\\\\-42.2+14\\log(f)dB&f>=20KHz\\end{cases}}", "1ee2fe985288c2aeb44987b8cc68e4e6": "p=4^{a}(8b+7)", "1ee30fe57fb3d6e3a88f99c939dcc7a8": "\\lnot (A\\vee \\lnot C)", "1ee362b96fe261ee6c1b843359900103": "k_{B}", "1ee3826e912aeb916b230416bda12077": "u\\neq s", "1ee42ca6404c21d6a2e35ba1a4aa8ed4": "\\Phi _{,abe}g^{ef}\\Phi _{,cdf}=\\Phi _{,ade}g^{ef}\\Phi _{,bcf}\\,", "1ee43acafda81d3e23e7c1747a8e0949": "F=m\\,a", "1ee43e4f7458d3f23849169ce032385f": "P={\\frac {\\varepsilon _{0}}{2}}E^{2}", "1ee463a70ffcc5b1473fb0adf84a6c8a": "(a/r)<1", "1ee46d33ce7b7811b8174e3084ddbd66": "\\psi _{p}(a)={\\frac {\\partial \\log \\Gamma _{p}(a)}{\\partial a}}=\\sum _{i=1}^{p}\\psi (a+(1-i)/2),", "1ee4d32ea5d6ce61083cedc8fb49c1fa": "S\\otimes _{R}S\\to S", "1ee4e23d128fdc058979a7c272bb6a18": "f={\\frac {i}{g}}", "1ee4e6d404e15f7e69516faf0dd74473": "A(\\omega )={\\frac {i\\omega CR_{0}}{1+i\\omega CR_{0}}}", "1ee50c13ab1fd6caa44fd1dfe008b00c": "\\nu (i)\\!", "1ee520a293a30663d966f36ddcc76c75": "{\\Bigg \\lbrack }{\\frac {\\rho _{2}}{\\rho _{1}}}{\\Bigg \\rbrack }", "1ee525229971f852cd26319e93c691d4": "{\\frac {\\partial N_{1}}{\\partial t}}={\\frac {\\partial }{\\partial x}}[(N_{2}D_{1}+N_{1}D_{2}){\\frac {\\partial N_{1}}{\\partial x}}]", "1ee57128bcf4db92a06c0662c7fb6233": "V_{\\text{A}}", "1ee5bed5e674968e100df7851c46a966": "\\Omega _{\\alpha \\beta }^{\\;\\;\\;\\;IJ}", "1ee5c1448db5e8b23058cf86d4aabd2f": "w={\\frac {1}{2}}\\left[q+c\\left(e^{Q}-1\\right)\\right]", "1ee5df290249b90a32de833a16db6d03": "|a(u,v)|\\leq C\\|u\\|\\|v\\|", "1ee5e987f66b3cfc2885688fafc96054": "43.2\\ \\mathrm {MHz} 1", "1ee6d9a5fa503d195d2e3b5bd5ee65ca": "m_{i}\\equiv 0{\\pmod {p_{j}}},\\;{\\text{for}}\\;i\\neq j,", "1ee6ee63ba880b9c4f012300e626a495": "x_{i}^{*}(p_{1},p_{2},m):=\\arg \\max\\{\\,\\!u(x_{1},x_{2}){\\mbox{ }}:{\\mbox{ }}p_{1}x_{1}+p_{2}x_{2}=m\\}{\\mbox{ for }}i=1,2", "1ee7168df412ac4a6756f690c84b2177": "n=m=0", "1ee758e2534b094fc9d70b930d8dbfef": "F={\\frac {\\pi ^{2}EI}{(Kl)^{2}}}", "1ee7f61b5986d7f4101df415be88215a": "L_{I}=10\\log _{10}{\\frac {I_{\\mathrm {rms} }}{I_{\\mathrm {ref} }}}", "1ee82323ad342a2205a10be3af98414d": "{\\begin{bmatrix}1&0&0&X\\\\0&1&0&Y\\\\0&0&1&Z\\\\0&0&0&1\\end{bmatrix}}", "1ee82e35f66fd2b76b7727d5eb7b88bb": "{\\rm {Tr}}\\,e^{F}e^{R}\\geq {\\rm {Tr}}\\,e^{F+R}\\geq e^{f}.", "1ee87af34e9524c8647fe97472ae25e1": "-F\\cos \\left({\\frac {\\pi y}{b}}\\right)", "1ee896764b7b9a94a4065790aafe0ec5": "6:m\\ ", "1ee89e16442cb637bf606cd711160855": "{\\hat {x}}_{U}(k)", "1ee96b94aac8db4b64643e165fd6f4cd": "\\lambda _{0}", "1ee99ad169b9c53b420a06eadbf9dcd7": "G(k)\\leq k(3\\log k+11).\\,", "1ee9afe1e0c02256726c3cedea348b74": "(\\star )\\qquad (x+2)^{2}-2(x+1)^{2}+x^{2}=2.", "1ee9d94057e7d7605bfc63415d02976b": "J=\\det({\\boldsymbol {F}})=\\det({\\boldsymbol {V}})\\det({\\boldsymbol {R}})=\\det({\\boldsymbol {V}})=\\lambda _{1}\\lambda _{2}\\lambda _{3}~.", "1ee9de4d3e0a76a925ef6a6e996a4d70": "k_{\\mu }=\\left({\\frac {\\omega }{c}},-k_{x},-k_{y},-k_{z}\\right).\\,", "1eea1b3e499a0d98b30ce32867363093": "e^{k}k^{O(\\log k)}\\log |V|", "1eea91b5b846092859e6ff642bd6f07a": "g_{ij}[\\mathbf {f'} ]=\\sum _{k,\\ell =1}^{n}{\\frac {\\partial x^{k}}{\\partial y^{i}}}g_{k\\ell }[\\mathbf {f} ]{\\frac {\\partial x^{\\ell }}{\\partial y^{j}}}.", "1eeaed5a852812f46ce2c3ace34b0218": "x_{n+1}=x_{n+2}=\\cdots =x_{m}=\\alpha .", "1eeb7b99f63813eaeb5f4a8d46a18e1c": "4\\arctan {\\frac {1}{5}}-{\\frac {\\pi }{4}}", "1eec10ca260348b69eba8ad49caff2ee": "{\\mathfrak {g}}^{*}", "1eec2c4878a6131fca529f588c64e913": "d=1\\,\\!", "1eec2f60d7698cae0e16c9d44d844992": "V:\\Sigma \\rightarrow \\omega _{1}.", "1eed6793fbbe439f3999805296cf03ff": "N=\\int _{\\sigma (N)}\\lambda dP(\\lambda ).\\,", "1eed99fbfa4589fc221c3ccc318b4adb": "r~\\ln r~\\sin \\theta \\,", "1eedc893a4328896c81612f6ff33547c": "\\sin ^{2}\\theta +{\\frac {b}{\\sqrt {ac}}}\\sin \\theta \\cos \\theta \\pm \\cos ^{2}\\theta =0.", "1eee927fa7d08c21e1326fab2cd1690a": "\\mathrm {im} (f_{k})=\\mathrm {ker} (f_{k+1})", "1eeebec4cd4013af45cdf78207df27af": "\\sigma _{a,A}^{2}", "1eeef773e753f748faa0e30b1b7470a0": "f^{n}(x)=f^{m}(x)", "1eeefbbedc91b784caee8137d78c8ba8": "\\Omega \\in \\Gamma (\\Omega ^{2}M\\otimes {\\text{Hom}}(E,E)).", "1eef0c80d2a41e21642c910860dcdd69": "{\\begin{aligned}i\\sin(2\\delta )&=&-i{\\frac {\\hbar ^{2}\\pi q}{maV}}\\end{aligned}}", "1eef0cefa3d1e95a4b85ae78f149c65d": "\\{l^{a},n^{a},m^{a},{\\bar {m}}^{a}\\}", "1eef2ae82273c69e5edfafd0f27cd167": "2^{o({\\sqrt {k}})}n^{O(1)}", "1eef41f7a8ad9841878e0db789a753af": "\\mu =\\mu ^{o}+RT\\ln C", "1eef82f7c74c576d995e1aad6e42b4a1": "E_{n},F_{n},H_{n}", "1eefa25bdcadaa35cf74332d38fc6b6d": "|X|^{2}", "1eefbe153094631c685a2f2a3181dbba": "T(\\mathbf {x} )={\\mathcal {F}}(\\mathbf {x} ^{*})/{\\sqrt {N}}", "1eeff2a905146fd96a38404f20be4b0e": "A\\wedge (B\\vee D)\\wedge (B\\vee E).", "1ef02a90fa9c68748d3491ca5daacf51": "n_{P}\\in \\mathbb {Z} ", "1ef04040bcc33c0e19a494092beac50d": "L_{N}=40+10\\log _{2}(N)", "1ef05e70666c31cf854c01348d923bef": "g(z)=\\exp \\left(z+{\\frac {1}{2}}z^{2}\\right).", "1ef07bd84f2b87f2a6259421fb1fdc2c": "g(v_{1}\\otimes v_{2}\\otimes \\cdots \\otimes v_{k})=gv_{1}\\otimes gv_{2}\\otimes \\cdots \\otimes gv_{k},\\quad g\\in GL_{n}.", "1ef08264fd9d75cbc7eadccf77eca478": "l,d", "1ef09bc7962e10cf53f533c5c150a4b6": "\\tau =F/A", "1ef0a5385296f6aed49d2dad9ff81ed1": "x_{0}=-{\\frac {r_{m}}{E}}\\quad (11)", "1ef0b5f5cc89a40a75bbef1d6276a37f": "[[Category:Canadahighwayinfoboxtemplates|{PAGENAME}]]", "1ef0d39a58e0abfa1047fef542c2bcd5": "\\ pH", "1ef0d93e70b9bb2bce782faf50c85441": "\\,\\Re (s)=1/2\\,", "1ef11aaf01f76caf91ebb91be3b81453": "A=\\bigcap _{k=0}^{\\infty }H_{k}", "1ef131c6360538eb2157797e477b8535": "DX\\ {\\stackrel {\\mathrm {def} }{=}}\\ \\mathrm {d} X+{\\mathbf {A}}X", "1ef161eb7eb9cc7abef753e7a8085d8f": "(\\|f+g\\|_{p})", "1ef1662438bc782c8447683b1505ef93": "p(w_{i}\\vert C)\\,", "1ef17e2c27f55803da91fd23d04fcf9d": "V_{\\mathrm {CE} }", "1ef227ba79cfbacab970539602c573a2": "\\log {\\frac {1+z}{1-z}}={\\cfrac {2z}{1-{\\cfrac {{\\frac {1}{3}}z^{2}}{1+{\\frac {1}{3}}z^{2}-{\\cfrac {{\\frac {3}{5}}z^{2}}{1+{\\frac {3}{5}}z^{2}-{\\cfrac {{\\frac {5}{7}}z^{2}}{1+{\\frac {5}{7}}z^{2}-{\\cfrac {{\\frac {7}{9}}z^{2}}{1+{\\frac {7}{9}}z^{2}-\\ddots }}}}}}}}}}\\,", "1ef22f9271cf28bb14c41e743a1d46bb": "x_{32}=0\\,", "1ef241eede58d1e6c65db40b92d979e6": "I=\\int _{0}^{\\frac {\\pi }{2}}{\\frac {1}{\\sqrt {a^{2}\\cos ^{2}(\\theta )+b^{2}\\sin ^{2}(\\theta )}}}\\,d\\theta ", "1ef28a62d145a3adf85031bd5b0e0e30": "f(0)^{2}=f(0)^{2}+f(0)^{2}.\\,", "1ef2cac2d529634ee6d3e2d37a783113": "|j_{1}-j_{2}|\\leq j_{3}\\leq j_{1}+j_{2}.\\,", "1ef314b17a6445ca6d4fa39931c36b18": "\\mathbb {C} ^{p+2}", "1ef3624527a866311b2ec0f5eea3085f": "\\psi ({\\vec {\\theta }})={\\frac {2D_{ds}}{D_{d}D_{s}c^{2}}}\\int \\Phi (D_{d}{\\vec {\\theta }},z)dz", "1ef38b6e6615f3349dd0b332928345d3": "{{i}_{B}}={{i}_{B1}}={{i}_{B2}}\\approx {{i}_{B3}}", "1ef3cf04952cb236fc8816e505ea60e6": "v_{i}^{t}", "1ef41038f1f51fa9c3aa6f6a3aef0906": "\\nu =\\gamma /(2-\\eta )", "1ef4277ed1d8e21c07dc689651d3f3e6": "\\displaystyle V_{\\mathrm {rms} }=V_{\\mathrm {peak} }.", "1ef463d9d90207455224b6e8602d8447": "S\\subset \\Delta ", "1ef484e79db2e006dcb9790efe20d7ec": "\\{G_{j},O\\}_{G_{j}=C_{a}=H=0}=\\{C_{a},O\\}_{G_{j}=C_{a}=H=0}=\\{H,O\\}_{G_{j}=C_{a}=H=0}=0", "1ef49a79254e216f1bcd6e6054f52568": "L_{g}(\\psi )(x)=\\psi ''(x)+g(x)\\psi (x)", "1ef49ba6e88d1ba9677bf9fa307070eb": "[HG_{eq}]", "1ef4c65d7f3dbbadc1ce6ece9276f5c9": "\\mathbf {D_{p}h} (\\mathbf {p} ,u)", "1ef60ce19ec5dc53a45b70e2bde872a3": "_{2}F_{1}(-n,\\alpha ;\\alpha +\\beta ;1-e^{t})\\!", "1ef6239e9c13b74e03815653a0a30ea2": "\\sigma (\\pi )=-1", "1ef6497c374eeec5201083094fb7eedb": "Tr(h)\\in GF(p^{2})", "1ef66a8c7baefeae1c9b1514657dc7d6": "{\\text{Weight density}}={\\frac {\\text{weight}}{\\text{volume}}}", "1ef6d366eacf543eda392737f58ec070": "\\lor \\!\\,", "1ef6eb217fecb5fb96bf076f74c144d6": "T_{1/2}={\\frac {-0.693(6.9\\times 10^{21})}{-3200{\\text{ s}}^{-1}}}=1.5\\times 10^{18}{\\text{ s or 47 billion years}}", "1ef72658d38a6aabc91b10b883e43707": "\\{f_{n}(x)\\}", "1ef7dce99947d616b7c793cbf99adbc1": "PE", "1ef8053640bf0ef38885ecfcbdae2a67": "\\pm (\\cosh a+j\\sinh a)", "1ef8072925c45d1b3981ea0cdd79131c": "x=n\\cdot \\pi ", "1ef87c8080a59242121546bea9bc9427": "x_{k-1}=a_{k-1}", "1ef89c48072cb255897aa3f247027eb7": "89^{2}", "1ef9431a2275300913280d155e0e241c": "\\rho =0.23V_{p}^{0.25}.", "1ef949555dc8084976eb10ff4a6c039c": "m_{e}\\,", "1ef95eed3cd3ce2a9555803b643ba077": "4\\pi /10", "1ef963168911972b7276caca19b787b6": "\\sigma _{i}:\\Delta ^{n}\\to X", "1ef9a04b89fb016dc4a4e359d2b3b627": "p_{e}=0.5\\,\\operatorname {erfc} \\left({\\sqrt {\\frac {E}{N_{o}}}}\\right).", "1ef9b98b337e22327a68c68ec75423aa": "{\\rm {BW}}_{Q}", "1efa05f20b2a63e1424093803ff4babf": "T_{2^{p}-1}=1+{\\frac {(2^{p}-2)\\times (2^{p}+1)}{2}}=1+9\\times T_{(2^{p}-2)/3}", "1efa51216eec3754a4c0dced6f32e4cf": "\\oint _{C}f(z)\\,dz=2\\pi i\\sum _{k=1}^{n}\\operatorname {Res} (f,z_{k})\\,,", "1efa6e1c6e83cd5ec8c7c4994f9cb3e7": "\\pi (G)^{\\prime }", "1efa8c001160d38b4e98079bdd8612fc": "\\sum _{n=-\\infty }^{\\infty }x[n]\\cdot e^{-i\\omega n},", "1efaa7095258dd303271242a3f6eb977": "\\epsilon {k^{2}}", "1efad6da2812baf6aec9090867729714": "\\gamma (z)={\\frac {1}{\\pi }}\\,{\\frac {z^{q}-z}{(z^{q}-z)^{2}-1}}", "1efb22fe96de8ff0215a81f08fc77e2c": "\\Delta \\subset \\mathbf {C} \\subset {\\widehat {\\mathbf {C} }},", "1efb71712325cc3e0a13d0979c38f21c": "V[q]=A_{4}", "1efba7a029eb1fd9ef3880e62005812e": "[\\Gamma _{}e_{1}(\\phi _{2}-\\phi _{1})/\\delta x)-q_{A}]+[\\Gamma _{e_{2}}(\\phi _{3}-\\phi _{2})/\\delta x)-\\Gamma _{w_{2}}(\\phi _{2}-\\phi _{1})/\\delta x)]+[\\Gamma _{e_{3}}(\\phi _{4}-\\phi _{3})/\\delta x)-\\Gamma _{w_{3}}(\\phi _{3}-\\phi _{2})/\\delta x)]+[q_{B}+\\Gamma _{w_{4}}(\\phi _{4}-\\phi _{3})/\\delta x)]=q_{B}-q_{A}", "1efbb68627093dd8d725a6990a3a1f92": "E\\to \\mathbb {Z} _{2}", "1efbb7486d34ae867af95c3e001c7923": "z\\,\\,=\\,\\,x^{2}\\,y\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{{\\partial z} \\over {\\partial x}}\\,\\,=\\,\\,2x\\,y\\,\\,\\,\\,\\,\\,\\,\\,\\,{{\\partial z} \\over {\\partial y}}\\,\\,=\\,\\,x^{2}", "1efbcd77389f6824e6f369a4a2582bb8": "\\mathbf {y} (t)=\\left(I+DK\\right)^{-1}C\\mathbf {x} (t)+\\left(I+DK\\right)^{-1}D\\mathbf {r} (t)", "1efbf07a71c9649aa3fafff0c0c1597b": "\\omega _{P}={\\sqrt {\\frac {ne^{2}}{{\\varepsilon _{0}}m^{*}}}}", "1efbf11bcac83d29bc87a2252a8b0b34": "pq-qp={h \\over 2\\pi i}I", "1efce36899292a978860d1767ba537c4": "(y-\\beta )^{v}", "1efd17c5334eb10033d6c8d85d7102a1": "Z_{\\mathbf {x} }^{(\\ell )}(\\mathbf {y} )={\\frac {1}{c_{n,\\ell }}}C_{\\ell }^{(\\alpha )}(\\mathbf {x} \\cdot \\mathbf {y} )", "1efd1906a27397ce248b5084aa4b6114": "\\scriptstyle {(TS)^{\\prime }=T^{\\prime }\\!S+TS^{\\prime }}", "1efd2f442f734bd1941a1564a717fd88": "-{\\frac {1}{N}}{\\frac {\\partial ^{2}\\ln {\\mathcal {L}}(\\alpha ,\\beta ,a,c|Y)}{\\partial \\beta ^{2}}}=\\operatorname {var} [\\ln(1-X)]=\\psi _{1}(\\beta )-\\psi _{1}(\\alpha +\\beta )={\\mathcal {I}}_{\\beta ,\\beta }=\\operatorname {E} \\left[-{\\frac {1}{N}}{\\frac {\\partial ^{2}\\ln {\\mathcal {L}}(\\alpha ,\\beta ,a,c|Y)}{\\partial \\beta ^{2}}}\\right]=\\ln(\\operatorname {var_{G(1-X)}} )", "1efd5556e50645e2635406aa5d2efea8": "{\\begin{pmatrix}1&2&\\ldots &j\\\\a_{1}&a_{2}&\\ldots &a_{j}\\end{pmatrix}}.", "1efd67cb3a20180ec9a3acbcc9027e52": "{\\bar {d}}=k{D_{v}\\centerdot R\\centerdot \\nu _{m}\\centerdot \\epsilon \\over D_{s}\\centerdot N\\centerdot \\nu _{l}\\centerdot C_{s}}", "1efd9a635b42115e7c7d97615365932c": "z={\\boldsymbol {\\phi }}(y)", "1efddb3de3857e020d408e024131ed06": "\\psi (t)=e^{-2\\pi it}", "1efe007ba269e986c5c89e0e0f97c2e4": "f,g\\colon I\\rightarrow \\mathbf {R} ", "1efe05c9929d61163fd26e129d738e8a": "(n,2n,n/2)_{2}", "1efe5b315515cb3fc28697f8e5387d9a": "{\\text{dBSWL}}={\\text{dBSPL}}+10\\,\\log _{10}\\left(4\\pi r^{2}\\right)", "1efeafb3273a3444e33378721bee739a": "\\Delta G_{fus}=\\Delta H_{fus}-T\\times \\Delta S_{fus}<0", "1efedfa53eecc0c70040f7548268358c": "{\\begin{aligned}{\\vec {a}}\\cdot {\\vec {\\sigma }}&=(a_{i}{\\hat {x}}_{i})\\cdot (\\sigma _{j}{\\hat {x}}_{j})\\\\&=a_{i}\\sigma _{j}{\\hat {x}}_{i}\\cdot {\\hat {x}}_{j}\\\\&=a_{i}\\sigma _{j}\\delta _{ij}\\\\&=a_{i}\\sigma _{i}\\end{aligned}}", "1eff35924a0a7427b46c41783b29482e": "\\beta =(k_{B}T)^{-1}", "1eff5d5532fb11b974ce63997c6ab09f": "\\scriptstyle {\\hat {Q}}_{n}(\\theta )", "1eff6f9ba863f1cc47e7778835eabcc2": "V_{R_{2}}=R_{2}\\cdot I", "1eff9f154dc2deb2631f7684b3aa6aac": "{\\mathfrak {m}}_{K}^{n}", "1effdb4188635e43ed906aa0a042c9cf": "6x^{2}+x", "1f000654eded2cbd11ca1b30f4587df0": "\\kappa _{\\varepsilon },\\lambda _{\\varepsilon }\\in \\mathbb {N} ", "1f003ac607460f29a62af072bc9c09a3": "\\mathbf {\\hat {a}} \\,\\!", "1f0084145af7183c4329d81c4b36887a": "{\\overline {\\mu }}", "1f00a3726d12ca656512fc80ee56a598": "i\\pm s_{1},\\ldots ,i\\pm s_{k}\\mod n", "1f00a874d59f8c770904806aef90c41f": "\\int {\\frac {\\delta Q}{T}}", "1f00c737b07bd25d2a17f037c778d9ea": "g_{B}", "1f00f2d25f982aa43bbf74de09d32bdc": "\\mu _{4}", "1f010219c12d38ad5804fce31ce83588": "{\\rm {{det}\\left(\\alpha _{i-j}^{q^{j}}\\right)\\neq 0\\quad (i,j=0,1,\\ldots ,r-1).}}", "1f0117b50a395695cc3b64c5cbda290f": "{\\overline {GM_{L}}}", "1f016e2922c4a0c78ee36579a303d5e3": "\\tau =RC={\\frac {1}{2\\pi f_{c}}}", "1f01c6c162b4191653b4b3d7b322f6ad": "i=\\max\\{{\\lfloor }{\\lceil }\\log _{b}m{\\rceil }/k_{m}{\\rfloor },{\\lfloor }{\\lceil }\\log _{b}n{\\rceil }/k_{n}{\\rfloor }\\}.", "1f021fc28ce12eceaed7b63e536ffdf0": "\\mu (x)\\in \\mathbb {Z} _{2}", "1f023407b09aaee18b9e667bcbe6c32e": "a_{1}=ta_{2},b_{1}=tb_{2},c_{1}=tc_{2}", "1f024fdb79e82573d2228a17a2f035f1": "(\\mathbf {a} \\cdot (\\mathbf {b} \\times \\mathbf {c} ))=\\varepsilon _{ijk}a^{i}b^{j}c^{k}", "1f025434136e300ca3fde75a26f500a8": "\\|y-x\\|=\\|-1(x-y)\\|=|-1|\\|x-y\\|=\\|x-y\\|", "1f02cb7a20a8040b4334cb20b500371d": "B_{\\lambda }(T)=B_{\\nu }(T)\\left|{\\frac {d\\nu }{d\\lambda }}\\right|.", "1f02d30d6d5256e978b7e9e0b627abad": "(\\hbar ^{2}n(n+1))", "1f02f33b2d6bb0795cc3fe463002ba11": "T_{x}M=P", "1f02fefb8b21f8e363974a76ffb7b7a4": "{\\sqrt {(x_{n}+j\\omega )\\cdot (x_{n}-j\\omega )}}={\\sqrt {x_{n}^{2}+\\omega ^{2}}}", "1f03158808dda5b2a28536079b8b5a8e": "Q_{50}\\,\\!", "1f032b3ce3ca33ff496ce91d5f767fb7": "{\\Bigl (}\\prod _{j\\neq i}^{n}n_{j}{\\Bigr )}+1\\quad ?", "1f03375653efd91fc9bb2ee25d25d991": "{\\textbf {j}}", "1f035115f01c45a324885efc424e7663": "2(2^{n+1}-1)+1=2^{n+2}-1", "1f038e50ee8a54cfd39e18fe2a70016a": "T_{o-}^{TE}=Hcos({\\frac {m\\pi }{a}}y)e^{jk_{xo}(x-w)}\\ \\ \\ \\ \\ \\ \\ \\ (25)", "1f039f0d70d5de5474a6a025e4b95232": "E(Z_{n}^{2})=1", "1f03c5fb4f7cc4d3211d4621b0babc8a": "\\sec(x)\\tan(x)", "1f03df4826f6535122cf138ccf358189": "\\delta S=0.", "1f03e39c60ff1a05f4cefad1c188a25a": "{\\begin{aligned}C\\sum _{n=0}^{\\infty }{\\frac {2^{n+1}}{\\ln F_{n}}}&={\\frac {C}{\\ln 2}}\\sum _{n=0}^{\\infty }{\\frac {2^{n+1}}{\\log _{2}(2^{2^{n}}+1)}}\\\\&>{\\frac {C}{\\ln 2}}\\sum _{n=0}^{\\infty }1\\\\&=\\infty ,\\end{aligned}}", "1f041e9b64e564523f75291d663a4d70": "L,M,R", "1f0496978b7c523822db641b6eabdf61": "f({\\boldsymbol {x}})=y_{\\mathcal {P}},\\ \\forall {\\boldsymbol {x}}\\in {\\mathcal {P}},", "1f04afe4d38ceb532641f994e26a421d": "\\sinh k,\\cosh l,\\tanh m,\\coth n\\!", "1f04e69159e3fcc91bb192edd5464858": "v^{2}=1,", "1f04ebf4f97b823896c132dd40a60215": "{}_{s}\\lceil X\\rceil _{N}", "1f0500df195dab3ad70d51fb53eef2d2": "g\\Delta \\rho L^{2}/\\mu ", "1f057274abfd9304c46facb1657190fd": "e=\\lim _{n\\to \\infty }{\\frac {n}{\\sqrt[{n}]{n!}}}", "1f05d57705c44d935b0aa3fe94ddde68": "C_{i}=MX_{i}-P_{i}t", "1f05fbde3719e63aef76b645cb302ba5": "C_{\\ell }^{m}(x,y,z)=\\left[{\\frac {(2-\\delta _{m0})(\\ell -m)!}{(\\ell +m)!}}\\right]^{1/2}\\Pi _{\\ell }^{m}(z)\\;A_{m}(x,y),\\qquad m=0,1,\\ldots ,\\ell ", "1f060825ab4836bc9fcf50cf260d201a": "L>0", "1f065facd0383670a3191f42592ee725": "p^{-n}H(t)={\\frac {t^{n}}{n!}}H(t).", "1f0696288950daa954db60fd2f824ae5": "y=Ae^{-1/x}", "1f06b2e690571c83ee7560b0a6bdef88": "C=1/(2\\lambda n)", "1f06e7326edc99094e54d8f40c8b9b9c": "\\nabla \\times \\mathbf {H} =\\mathbf {J} _{\\mathrm {tot} }", "1f06f4a45dd68ea7380a2365e1ddc441": "{\\sqrt[{2}]{E}}/\\rho ", "1f07489d3b5117f6c843b6e274a3be25": "\\delta \\times \\left(x\\times \\operatorname {p.v.} {\\frac {1}{x}}\\right)=\\delta ", "1f0757ca5effe6dbdd0e3ebe6315c0df": "\\omega _{r}^{2}=\\left({\\frac {c^{2}r_{s}}{2r_{\\mathrm {outer} }^{4}}}\\right)\\left(r_{\\mathrm {outer} }-r_{\\mathrm {inner} }\\right)=\\omega _{\\varphi }^{2}{\\sqrt {1-{\\frac {3r_{s}^{2}}{a^{2}}}}}", "1f07688e9eba01a0b7de63c1aaa97b23": "\\sum _{n=N}^{\\infty }f(n)", "1f077f11860d444f2540b4a9e38a425c": "{K_{1},\\dots ,K_{8}}={3,1.52183,-0.7607+0.8579i,-0.7607-0.8579i,1,1,0,0},", "1f07ba9362086bc3911f13dc46801f6f": "Q\\to M", "1f07c918c1596b1661e1e2ed2b5df2d9": "K\\in {\\mathcal {K}}^{k}", "1f085faff7a7245811a3abe4465cae4f": "V=iR", "1f088e03d24d8126303e28fae0dd535f": "\\alpha =\\sin ^{-1}(e)=\\cos ^{-1}\\left({\\frac {b}{a}}\\right)=2\\tan ^{-1}\\left({\\sqrt {\\frac {a-b}{a+b}}}\\right);\\,\\!", "1f08a053f7a87ba537cbaa078b1c3c21": "\\phi -\\phi _{0}\\,", "1f08cc3575b84272b143d7949ae6ca26": "\\alpha _{D}={\\frac {116}{D}}", "1f090d50661700e511ba18b9ed50b1b9": "{\\mathfrak {P}}^{94}", "1f0915acdc81986a1afbbc69bf7ae883": "I_{r}\\,", "1f093dffb289d91bb99dabc5ac554786": "\\alpha _{i_{1}}\\ldots \\alpha _{i_{K}}=\\beta _{i_{1}}\\ldots \\beta _{i_{K}}.", "1f094363ce760dff0653e9df2164ca3c": "(x-\\alpha )", "1f09c25c5247c1eaf121df644ca42f8c": "\\theta \\,", "1f0a1363fc89fc5d89439c81643b43ee": "I_{2}={{}^{\\star }C}_{abcd}\\,C^{abcd}", "1f0a43647c13586708fbe223ef7e4a7b": "\\sum C_{i}", "1f0a83c5067ef7ce2bad08832aa44515": "[A,B]=-[B,A]", "1f0aa218526653a68a6b8e4bf6cf1ec9": "{\\textit {VendingMachine}}\\left\\vert \\left[\\left\\{{\\textit {coin}}\\right\\}\\right]\\right\\vert {\\textit {Person}}\\equiv \\left({\\textit {coin}}\\rightarrow {\\textit {choc}}\\rightarrow {\\textit {STOP}}\\right)\\Box \\left({\\textit {card}}\\rightarrow {\\textit {STOP}}\\right)", "1f0ab62f04bc58686fa2792d9f54ce37": "K(x,y;T)=\\langle y;T|x;0\\rangle =\\int _{x(0)=x}^{x(T)=y}e^{iS[x]}Dx\\,", "1f0ad6b12251675fd1cd347ed4a75d3e": "re^{i\\theta }={\\tfrac {b}{2}}(e^{i\\rho }+e^{i\\lambda })=b\\cos({\\tfrac {\\rho -\\lambda }{2}})e^{i{\\tfrac {\\rho +\\lambda }{2}}}.", "1f0af275f0e3eadd85a8079af7554a7e": "y=f\\circ x", "1f0b3a1b46852789ebe48485aa4c1a7d": "\\mathbf {V} _{r}", "1f0b4c0f1e368e8221e44946ef71253c": "a_{i,k}", "1f0b62b42686598457d4e526ba2e422a": "(X\\to UA_{i})_{I}", "1f0c02c6cddbfdec1726d2600e3ba510": "\\pi \\approx {\\frac {2l\\cdot n}{th}}.", "1f0c0c5d028cbbc4a091b5c55d91852a": "E_{y}={\\frac {k_{o}^{2}\\varepsilon _{r}-k_{z}^{2}}{k_{o}^{2}-k_{z}^{2}}}[-{\\frac {k_{z}}{\\omega \\varepsilon _{o}\\varepsilon _{r}}}{\\frac {m\\pi }{a}}(A\\ e^{-jk_{x\\varepsilon }w}+B\\ e^{jk_{x\\varepsilon }w})-jk_{xo}(C\\ e^{-jk_{x\\varepsilon }w}+D\\ e^{jk_{x\\varepsilon }w})]e^{-jk_{xo}(x-w)}cos({\\frac {m\\pi }{a}}y)e^{-jk_{z}z}\\ \\ \\ \\ \\ \\ (40)", "1f0c2db70e0a325864589dedd7d4cde0": "x_{n+1}=x_{n}-{\\frac {g(x_{n})}{g'(x_{n})}}", "1f0c3e0e13b0288d850817a4e42f74a8": "C(r,z)=G_{1}(0,z){\\frac {\\delta (r)}{2\\pi r}}+G_{2}(r,z),\\qquad (12)", "1f0c46369932f7c599890d7879a9c143": "g_{j}(x)\\leq 0,j\\in J={1,\\dots ,m}", "1f0c47394c86781199452029a96d0941": "E={\\begin{pmatrix}0&1&0&\\cdots &0\\\\0&0&1&\\ddots &\\vdots \\\\\\vdots &\\ddots &\\ddots &\\ddots &0\\\\0&\\cdots &0&0&1\\\\0&\\cdots &0&0&0\\end{pmatrix}},\\qquad \\Delta ={\\begin{pmatrix}-1&1&0&\\cdots &0\\\\0&-1&1&\\ddots &\\vdots \\\\\\vdots &\\ddots &\\ddots &\\ddots &0\\\\0&\\cdots &0&-1&1\\\\0&\\cdots &0&0&-1\\end{pmatrix}},", "1f0c52e43e173d595dafb708be9450b1": "d_{f}={\\frac {\\lambda }{\\sin \\theta }}", "1f0c67f18e607266d583cb4e37c5b67c": "C_{ib}", "1f0c7758c8636b86afe67b24dfeada88": "A\\in R", "1f0cb456a5300604254f0cf4545de981": "(x,y,z)=\\left(r\\cos \\phi ,r\\sin \\phi ,2{\\sqrt {ar}}\\cos {\\frac {\\phi }{2}}\\right),", "1f0cc8ee9c54cbf62b346f2b77546d68": "\\lambda \\geq 5", "1f0cdee6fbc0e83b9d98379de50c1f0c": "y_{j}\\succsim _{j}x_{j}", "1f0cf6d8670b1392997cde57ae1370d0": "\\scriptstyle e^{x}=\\sum _{n=0}^{\\infty }{\\frac {x^{n}}{n!}}", "1f0d073946036c64527daf08d7005aa2": "{\\begin{aligned}U^{2}V^{2}+V^{2}W^{2}+W^{2}U^{2}&=z^{2}x^{2}y^{4}+x^{2}y^{2}z^{4}+y^{2}z^{2}x^{4}=(x^{2}+y^{2}+z^{2})(x^{2}y^{2}z^{2})\\\\[8pt]&=(1)(x^{2}y^{2}z^{2})=(xy)(yz)(zx)=UVW,\\end{aligned}}", "1f0d4385d1f315ed430a48e6bb97fc39": "i,\\;j", "1f0d81af885552f90dfc35eb2010b240": "S_{G}", "1f0dc57b8567c1ff562d4401cd8ad78a": "\\mathrm {N\\,m} \\,", "1f0ddb961f3b8a85c16f52ccb8cac7df": "\\left|m_{1}-m_{0}\\right|\\theta _{0}", "1f3b0d6372efc310d19d3da4d9598e2c": "R_{2}^{2}(\\rho )=\\rho ^{2}\\,", "1f3b61e91aa24a058a0c44d34329d4e3": "\\rho ={\\sqrt {x^{2}+y^{2}}}={\\frac {\\cosh \\tau -\\cos \\sigma }{a\\sinh \\tau }}.", "1f3b88055acc2cbfee9a93ae871ccef3": "\\scriptstyle [m,\\,1.35m]", "1f3b9963c18a50a20be0e4f76889e67f": "(t_{1},t_{0},p)\\in D(X)", "1f3bc08a1906d6e143352b3d35dfb43a": "A\\mathbf {x} ", "1f3bd215db6702294038f002385005fc": "\\textstyle \\left\\lfloor {\\frac {n}{p^{2}}}\\right\\rfloor ", "1f3c2767b1f2e88c7d18a42f45979a6c": "P_{c}:\\mathbb {C} \\to \\mathbb {C} ", "1f3c290b8064255881ad4a0aad7979cd": "(\\,s_{i}\\to s_{i}\\cdot \\epsilon _{i}\\quad \\,J_{i,k}\\to \\epsilon _{i}J_{i,k}\\epsilon _{k}\\,\\quad s_{k}\\to s_{k}\\cdot \\epsilon _{k}\\,)\\,.", "1f3ca10444992ce792db696a63c1ca05": "|ab|<1.", "1f3cbb67c8d96699e4018225bef4d60a": "={\\frac {1}{\\eta }}\\;G_{q+\\sigma ,\\,p+\\tau }^{\\,n+\\mu ,\\,m+\\nu }\\!\\left(\\left.{\\begin{matrix}-b_{1},\\dots ,-b_{m},\\mathbf {c_{\\sigma }} ,-b_{m+1},\\dots ,-b_{q}\\\\-a_{1},\\dots ,-a_{n},\\mathbf {d_{\\tau }} ,-a_{n+1},\\dots ,-a_{p}\\end{matrix}}\\;\\right|\\,{\\frac {\\omega }{\\eta }}\\right)=", "1f3cc75b7fa36af13ca0cc3af6b6a35a": "2e^{2}/c\\epsilon _{0}m=8\\pi /137", "1f3cdc34448e6565e762d4ced875d96b": "\\gamma _{\\mathrm {rad} ,0}", "1f3d4624d792d00e1ab2798c64395cac": "\\operatorname {E} [X]=\\int _{\\Omega }X\\,\\mathrm {d} P=\\int _{\\Omega }X(\\omega )P(\\mathrm {d} \\omega )", "1f3d5637c7b8d794919b3a06cea60e44": "\\cos \\theta _{W}={\\frac {m_{W}}{m_{Z}}}", "1f3d71fa4cdc4c4099b532bc1f4b5588": "\\sin ^{2}\\theta \\cos ^{2}\\theta ={\\frac {1-\\cos 4\\theta }{8}}", "1f3d72fb5326777ad5cfeb1e5cc90440": "\\displaystyle {V(\\psi )=U(D(\\varphi )+S(\\psi ))|_{\\Omega },}", "1f3deeaa18f337548cd5c3c97253ce60": "r_{m}=a\\cdot \\left(1+{\\frac {1}{{\\sqrt {5}}{}}}\\right)\\approx 1.44721\\cdot a", "1f3e10978c3c8ca6fa28f2ee456d53b0": "D_{KL}(f_{\\theta }\\|f_{\\theta '})\\leq \\beta .\\,", "1f3e2a1050d96cf50b27ec001a13c40d": "e_{1}", "1f3e4cf1bd075aef01ebdd178383b837": "{\\hat {\\rho }}=\\sum _{i}P_{i}|\\psi _{i}\\rangle \\langle \\psi _{i}|", "1f3e52a86740c86d5e1c517c2538a9fd": "Select(IMM(s))", "1f3eae811de99bc27d17d4d84a1b2b6a": "\\mathrm {1\\,Fr=1\\,statcoulomb=1\\,esu\\;charge=1\\,cm{\\sqrt {dyne}}=1\\,g^{1/2}\\cdot cm^{3/2}\\cdot s^{-1}} ", "1f3f1817f5c049fd90ea4a4108dd3fe7": "P_{\\mathbf {k} }{\\bar {P}}_{\\mathbf {k} }=0.", "1f3f379e16dab43a0c25b19c455470bc": "Z_{CO}^{1}", "1f3f6a56f9245d79f9c4e8b49d50547a": "t^{-2}", "1f3f9dafa983e3fdd412115b9628dc9c": "TSR={(Price_{end}-Price_{begin}+Dividends)}/{Price_{begin}}", "1f3fb04ef4c6a63473fd68fb3c2ff371": "a_{C}", "1f3fba608646fd938773e3e5d0690858": "{\\rm {Ci}}(x)={\\frac {\\sin x}{x}}\\left(1-{\\frac {2!}{x^{2}}}+{\\frac {4!}{x^{4}}}-{\\frac {6!}{x^{6}}}\\cdots \\right)-{\\frac {\\cos x}{x}}\\left({\\frac {1}{x}}-{\\frac {3!}{x^{3}}}+{\\frac {5!}{x^{5}}}-{\\frac {7!}{x^{7}}}\\cdots \\right)", "1f3fe37d57b5d75151520b4b34bc7955": "{\\mbox{Gross Rental Yield}}={\\frac {\\mbox{Monthly Rent x 12}}{\\mbox{House Price}}}{\\mbox{ x }}100\\%", "1f4094778137452f8468dae050771d8f": "\\langle \\chi |\\psi \\rangle =\\int \\limits _{R}dx\\,\\langle \\chi |x\\rangle \\langle x|\\psi \\rangle =\\int \\limits _{R}dx\\,\\chi (x)^{*}\\psi (x)\\,.", "1f409c9d5128288757505262a36c22f1": "(a\\ b\\ c)", "1f40c763ceeeeeb1983e8b5bdfa161d6": "\\pi _{a_{1},\\ldots ,a_{n}}(R\\cup P)=\\pi _{a_{1},\\ldots ,a_{n}}(R)\\cup \\pi _{a_{1},\\ldots ,a_{n}}(P).\\,", "1f410e06ccf74c48eede05b6c2e91e1c": "\\scriptstyle {\\varepsilon _{\\circ }}", "1f4121af8984d0ebd7fa79d32be87b0e": "|\\mathbf {V} |={\\frac {\\hbar }{2}}{\\frac {\\nabla \\rho }{m\\rho }}", "1f4133da43e7b595d5a9fc119ada2db4": "t\\in \\{1,\\ldots ,r\\}", "1f418d379721166253dc07fafea9de67": "{\\mathbf {A}}", "1f4193490d801c7996a882ca0bb3fc86": "r_{1}={\\frac {k_{11}}{k_{12}}}\\,", "1f41db851d3307e8af992e525012e296": "{\\mathcal {C}}_{\\varepsilon }", "1f424e16f292e4f25fec4f3755f6b2c8": "\\lim _{t\\to 0}K(t,x,y)=\\delta (x-y)=\\delta _{x}(y)", "1f4296b4900a3ea65bb380f9581ab03b": "\\textstyle M^{1}", "1f42a693986145cd1ff0f482061cac60": "\\Delta \\mathbf {b} ", "1f42c93c4f5f3eec4ba70733b656dd31": "E_{internal}=E_{cont}+E_{curv}", "1f42d996e23374db76583a24734ccac8": "L={\\frac {\\mathrm {d} ^{3}}{\\mathrm {d} x^{3}}}", "1f4326a426629934888b77136e0091ae": "0={\\frac {d}{dq}}\\left[\\ln {\\frac {dt}{dq}}+\\ln w\\right]", "1f43a63a03f9e8fe1beca54e344ae88a": "{\\hat {R}}_{x}^{\\alpha }", "1f43e0ded98cec51b0f5c5e3c70c7de7": "L(\\cdot )", "1f44364091ee843b40ae19679b18fa04": "X=x_{\\bar {t}}^{q}", "1f444c951dd6595039d509a4c29c3537": "{\\begin{bmatrix}1&-1&0&0&0&0&0\\\\0&1&-1&0&0&0&0\\\\0&0&1&-1&0&0&0\\\\0&0&0&1&-1&0&0\\\\0&0&0&0&1&1&0\\\\-{\\frac {1}{2}}&-{\\frac {1}{2}}&-{\\frac {1}{2}}&-{\\frac {1}{2}}&-{\\frac {1}{2}}&-{\\frac {1}{2}}&{\\frac {\\sqrt {2}}{2}}\\\\0&0&0&0&1&-1&0\\\\\\end{bmatrix}}.", "1f446f46fdec7813d84134cfdb6e473e": "I_{2}\\,", "1f45204f4e062aaf3426ef96ad73fa94": "\\displaystyle \\operatorname {e} ^{-a|x|}\\,", "1f45b131da0475894ef4251a14655476": "\\operatorname {gl.dim} R\\leq n\\Rightarrow \\operatorname {pd} _{R}k\\leq n\\Rightarrow \\operatorname {Tor} _{n+1}^{R}(-,k)=0\\Rightarrow \\operatorname {pd} _{R}-\\leq n\\Rightarrow \\operatorname {gl.dim} R\\leq n,", "1f45dade52b12e5f2940932a6bc48842": "P(S)\\,\\!", "1f4650482e9b45cf1daf841afccf3606": "c_{0}=\\lambda _{J}\\omega _{p}", "1f465f44a79303480395bf7a8c0eecee": "K_{c}={\\frac {[CO_{2}]}{[CO]^{2}}}", "1f469cbe9eac6bdcab228b014d4575a2": "x_{n}=f(x_{n-1}).", "1f46a06bc6bd9d1c680fec7f3d741581": "E_{cont}", "1f46d7acb595e6ae81bb41b111ff6207": "A\\leftarrow B\\rightarrow C", "1f47e9d92cd2291ea9ee5f59f9cc2519": "\\nabla ^{2}\\varphi '-\\mu _{0}\\varepsilon _{0}{\\frac {\\partial ^{2}\\varphi '}{\\partial t^{2}}}=\\Box ^{2}\\varphi '=-{\\frac {\\rho }{\\varepsilon _{0}}}", "1f48aa385a9d223f4976c3ba2a97e691": "e^{+}e^{-}\\to e^{+}e^{-}\\mu ^{+}\\mu ^{-}", "1f48e973d6a9075dbaaf41a9e85f034e": "t=0", "1f4912bb9695e3a49fe4dd26f27bf7d3": "\\tau _{GL}={\\frac {\\pi }{\\hbar (T-T_{c0})}}", "1f492f938b0ebf705bab51e7d3fd4493": "x_{k}^{(L)}", "1f495127b0b25bf21fa02fe4ca6724ef": "m_{f}", "1f495ef0b3c1242c76ef57d9089861f2": "X_{i+1}=X_{i}-{f(X_{i}) \\over f'(X_{i})}=X_{i}-{1/X_{i}-D \\over -1/X_{i}^{2}}=X_{i}+X_{i}(1-DX_{i})=X_{i}(2-DX_{i})", "1f49882545df5064b8291982a190dbb4": "\\lambda (V):=\\oplus _{i=1}^{d_{\\lambda }}V_{i}\\simeq \\mathbb {C} ^{d_{\\lambda }}\\otimes M_{\\lambda }", "1f498ec83e89f2e24c108ad374add6d6": "U=-\\mu _{o}\\int \\limits _{V}{{\\vec {M}}\\cdot {\\vec {H}}\\,dV}\\,\\!", "1f4996abc4d629767fda76768e8f6dcb": "s=a\\sinh {\\tfrac {x_{2}}{a}}-a\\sinh {\\tfrac {x_{1}}{a}}.\\,", "1f499d5cf581c80f9cd49e02f19f1c12": "c_{1}\\neq 0", "1f49a248f82d08eacacc767bdd0530d9": "\\scriptstyle {\\boldsymbol {r}}_{\\text{rec}}", "1f49da4fe243c9c917c6a908b103cb01": "f:X\\to {\\mathbb {F} }", "1f49f3c06a9c5c0c22ddce381601071e": "r\\neq {\\textbf {Q}}", "1f49fb8e184a29d979f2f7e6053139dd": "G={\\frac {E}{2(1+\\nu )}}", "1f4a232959727d93b19ca9ba4c05c3de": "\\mathbf {Q_{A}} ", "1f4a4bd67aee269a610747c82c36f2c0": "{C}/{r}=2\\pi ", "1f4aa87ff18be3eb68d92e481da60979": "\\forall a\\in A,L(a)={\\mathit {undec}}", "1f4af700e15cc80d29f62724c7f11015": "{\\overline {u}}_{i}", "1f540901e420b1ac5f7d7ea7e3f8bce6": "p(0,0,t,y)\\,dy:=P(W_{t}^{+}\\in dy)=2{\\sqrt {2\\pi }}{\\frac {y}{t}}\\phi _{t}(y)\\Phi _{1-t}(0,y)\\,dy.", "1f54379eeba326e08190e38ebcd6afa6": "1RM={\\frac {100\\cdot w}{101.3-2.67123\\cdot r}}", "1f54a6a12d87458b298ca5403b2071cc": "Y=\\beta _{60}+\\beta _{61}X+\\beta _{62}Mo+\\beta _{63}XMo+\\beta _{64}Me+\\beta _{65}MeMo+\\varepsilon _{6}", "1f54f78a933ec5aaf6d1156971b7f40f": "100\\uparrow \\uparrow n", "1f5572dd81fa4bff2fd57f6e7d51e9fe": "f(z)=\\int _{0}^{z}{\\frac {\\sin(\\zeta ^{\\rho })}{\\zeta ^{\\rho }}}d\\zeta ,", "1f559df95ed5b5d0d01cdbd0dc629275": "AMF={\\frac {1.55L_{c}+{\\frac {80.2}{R}}-.012S}{1.55L_{c}}}", "1f55e5a38b4ce09135b4d303e612648f": "k=\\omega {\\sqrt {\\mu \\epsilon }}", "1f561e138e1028e3e74607b612bf3ea7": "\\alpha t0", "1f739322941c6c0a3988da8753d9e1fe": "{\\frac {\\partial c}{\\partial \\zeta }}+c=0", "1f7433251704ab78402297095d1b75a2": "\\ell ={\\sqrt {\\frac {\\hbar }{2m\\omega }}}", "1f7435e186e1674a8ec46647bc7e071b": "0\\leq B_{k}\\leq 1", "1f7454b690cfd499a92b31e637154250": "(1337;12,3)", "1f74b35153d4b8589d59a970040474fc": "{\\frac {\\mathbf {r} -\\mathbf {r} '}{|\\mathbf {r} -\\mathbf {r} '|^{3}}}=-\\nabla \\left({\\frac {1}{|\\mathbf {r} -\\mathbf {r} '|}}\\right)", "1f75068ed26ed32ec9c58d7286beadb4": "b'_{\\nu ,n}(x)=n\\left(b_{\\nu -1,n-1}(x)-b_{\\nu ,n-1}(x)\\right).", "1f750d5ef3c153fc96d9bc2535f039b8": "\\Psi _{0}={\\frac {1}{4}}\\,\\left(\\left(H_{xx}-H_{yy}\\right)+2i\\,H_{xy}\\right).", "1f75711854a34370646d90bfe893eae2": "l_{t,j}", "1f75b46c230d976c7b4dbdf5b0e97eae": "f:\\Omega \\to \\mathbb {R} ^{n}\\,", "1f7604f572d60d019fc02094aa89752b": "{\\mathfrak {P}}^{77}", "1f76101821e0fdcc44c8005b074a06c8": "E_{u/p}=y_{1}+{\\frac {v^{2}}{2g}}\\,\\!", "1f7687e0136b2939ff35ca53d68d9059": "\\{a,b,c,d,e\\}", "1f7690b98656f2c8c186057d9144cea6": "\\pi _{a}", "1f76a005b92071a645846a3e0d47f29c": "-T\\left({\\frac {\\partial p}{\\partial V}}\\right)_{T}\\left({\\frac {\\partial V}{\\partial T}}\\right)_{p}^{2}=-T\\left({\\frac {-p}{V}}\\right)\\left({\\frac {R^{2}}{p^{2}}}\\right)=R", "1f76e2404b9fa24edbfbcbea997d0658": "\\phi (t,x,y,z)=Ctze^{tx-yz}+A\\sin(3\\omega t)\\left(x^{2}z-By^{6}\\right)=0", "1f76f920c90927cc350a43475ff62403": "\\left({\\frac {\\hbox{Total points}}{\\hbox{Total rides}}}\\right)\\times 4", "1f76fb19d415dc27bc9b2c96e86d4275": "Z_{2}", "1f7759a6d875cc94b8a4a01bd3cdbb2c": "\\to _{G}", "1f77f89be58905adacbd429ac2e90cd8": "{\\vec {J}}({\\vec {r}},t)=-D\\nabla \\Phi ({\\vec {r}},t)", "1f780e7515b4875df8af45756b64d680": "\\left\\langle q(x)[F]\\right\\rangle +i\\left\\langle Fq(x)[S]\\right\\rangle =\\left\\langle q(x)[F]\\right\\rangle +i\\left\\langle F\\partial _{\\mu }j^{\\mu }(x)\\right\\rangle =0.", "1f784ec5434f8f9533d0f3113207e37d": "\\exists a({\\text{Phil}}(a))", "1f78b4a727501b38facda7f4ba43de6e": "I_{\\Delta }={\\frac {I_{0}}{2}}\\left[\\left(1-\\sin \\left(\\delta _{0}\\sin \\omega t\\right)\\right)10^{-A_{-}}+\\left(1+\\sin \\left(\\delta _{0}\\sin \\omega t\\right)\\right)10^{-A_{+}}\\right]", "1f7980eefdde26f3d798185efbf24822": "\\pi _{5}=z_{0}", "1f79816dc0aed43748bab3dbe522ab92": "a={\\frac {5l(3l^{5}-4m)}{m^{2}+l^{10}}}\\qquad b={\\frac {4(11l^{5}+2m)}{m^{2}+l^{10}}}", "1f79919fa4cd374d5fbbd38cb1f91f7f": "F=\\mathbb {Q} ({\\sqrt {m}})", "1f79bd220adebedde86d8f480387ab23": "x(t)=a\\cos(t),\\,", "1f79cff37d49f5ecfcd7a2ed0515b658": "Z=SJ_{z}S^{-1}\\ ,", "1f7a01f320e61049f9caab900de6c8c5": "\\ a\\,", "1f7a03a30225fa2a6f4c7a30a01aba3c": "\\lim \\sup {\\frac {\\varphi (n)}{n}}=1,", "1f7a0475def51abcfc24d4bae3c59e65": "[W_{t},W_{t}]=t", "1f7a0af9cc469b34a2ecb0900b054c4e": "AUC_{k,k}", "1f7a2076398a463bf7b804eae94d14c2": "H_{n}={\\frac {(1+{\\sqrt {2}})^{n}+(1-{\\sqrt {2}})^{n}}{2}}.", "1f7a2282c1b87361b78d0e482dfb3a6b": "h_{UV}(x)=s_{V}\\circ s_{U}^{-1}(x),\\quad x\\in U\\cap V.", "1f7a8242f444344f1d38b9bf8ec4f2bf": "m(x,y)=g\\{m(x,y')\\mid y'\\in f(x)\\}.", "1f7ab1be359da8cec58ac3fa12ea7380": "\\ C_{min}", "1f7aebd96c654d4801987febb871dc94": "F_{d}=-6\\pi r_{p}\\mu V_{r}.", "1f7b25d25ec49d743c2a2874fcf6829c": "\\mathbb {T} =[0,\\infty )", "1f7b3522f3b3bed9e913fdd88d46589b": "[Hf](x)=-{\\frac {1}{2}}{\\frac {{\\mbox{d}}^{2}}{{\\mbox{d}}x^{2}}}f(x)+{\\frac {1}{2}}x^{2}f(x).", "1f7b4a736a7b7602983079aee9570a62": "\\Sigma _{n}^{0}", "1f7b599a15d834071d22b23c416307b9": "a_{2}={\\frac {a_{1}\\cos \\beta }{1-\\sin ^{2}\\beta \\,\\cos ^{2}\\gamma _{1}}}-{\\frac {\\omega _{1}^{2}\\cos \\beta \\sin ^{2}\\beta \\sin 2\\gamma _{1}}{(1-\\sin ^{2}\\beta \\cos ^{2}\\gamma _{1})^{2}}}", "1f7b9318648e593fadf6354cad799415": "p_{dyn}", "1f7cce85dac307ff99fa7256bbfcacfe": "\\mu _{\\alpha }", "1f7cd82ae9a9d3216218fb12dab4cfa9": "{\\frac {1-G(tx)}{1-G(t)}}{\\xrightarrow[{t\\to +\\infty }]{}}x^{-\\theta },\\quad x>0", "1f7cf91278a3e855c94c6522e53d061a": "\\lambda _{\\delta }(n)", "1f7d08374c1a84af1d31c26bd6dd107e": "L_{U}\\;=\\;69.55\\;+\\;26.16\\;\\log _{10}f\\;-\\;13.82\\;\\log _{10}h_{B}\\;-\\;C_{H}\\;+\\;[44.9\\;-\\;6.55\\;\\log _{10}h_{B}]\\;\\log _{10}d", "1f7d961980f72a9b177c1e1d18684079": "[T]_{\\beta }^{\\gamma }", "1f7dd5ddf86f117be08408867eeb7b22": "S_{\\mathrm {total} }={\\frac {\\left(v_{\\mathrm {m} }Ns\\right)}{V}},\\qquad (5)", "1f7dde3d771e96b3f67fc88478c03c75": "\\theta ^{*}=g\\left(\\theta ^{*}\\right)", "1f7e1d0f45f1ca1997d2a0570b01fd5d": "\\textstyle <1", "1f7e34bed65711fcdf7f01154c81ff52": "\\sigma _{x}\\sigma _{p}=\\hbar \\left(n+{\\frac {1}{2}}\\right)\\geq {\\frac {\\hbar }{2}}~.", "1f7e7016cbf09526b85723fc5dfa2bc3": "x\\wedge (y\\vee z)=(x\\wedge y)\\vee (x\\wedge z)", "1f7e9f621d27f705126be23079eab4bf": "{\\begin{aligned}\\alpha _{i}&={\\frac {{\\boldsymbol {p}}_{i}^{\\mathrm {T} }{\\boldsymbol {r}}_{i}}{{\\boldsymbol {p}}_{i}^{\\mathrm {T} }{\\boldsymbol {Ap}}_{i}}}{\\text{,}}\\\\{\\boldsymbol {x}}_{i+1}&={\\boldsymbol {x}}_{i}+\\alpha _{i}{\\boldsymbol {p}}_{i}{\\text{,}}\\\\{\\boldsymbol {r}}_{i+1}&={\\boldsymbol {r}}_{i}-\\alpha _{i}{\\boldsymbol {Ap}}_{i}\\end{aligned}}", "1f7ee3ae4faccfd528c14e6afaa11370": "=\\gamma ^{0}\\gamma ^{\\mu n}\\gamma ^{0}\\dots \\gamma ^{0}\\gamma ^{\\mu 2}\\gamma ^{0}\\gamma ^{0}\\gamma ^{\\mu 1}\\gamma ^{0}", "1f7efb11a5062e35c4c103407d504274": "sp(S_{1};S_{2}\\ ,\\ R)\\ =\\ sp(S_{2},sp(S_{1},R))", "1f7f1d6e89b60f2be92eeb4072c300cb": "a(n+4)\\equiv a(n){\\pmod {10}},", "1f7f73a7d98c9557ebb18edcfe4c51ce": "f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\\cdots +a_{2}x^{2}+a_{1}x+a_{0}\\,", "1f7f73c7e8ed6da874ec067f99c09cd5": "A_{1}\\lor \\cdots \\lor A_{k}\\lor C", "1f7fd21193f25f0c5c04f5a4f5eeb128": "\\mathbf {E} ^{a}\\left[f{\\big (}B_{\\sigma _{k}}{\\big )}\\right]", "1f800f5a94616fd62bd0bd168c6af7d2": "1\\ \\mathrm {Gy} =1\\ {\\frac {\\mathrm {J} }{\\mathrm {kg} }}=1\\ {\\frac {\\mathrm {m} ^{2}}{\\mathrm {s} ^{2}}}", "1f801258c264b85e8e0bb3eea2ebda9b": "\\ell '=\\alpha (\\ell )", "1f802ac05865de81ee4ebea7e9a923c4": "|x\\rangle \\ {\\stackrel {\\mathrm {def} }{=}}\\ {\\begin{pmatrix}1\\\\0\\end{pmatrix}}", "1f80925efd94cb3fc32d5e819f21b731": "\\omega _{F0n}={\\frac {\\omega _{F0}}{n}}.\\ ", "1f80a18c00065c19dbcf57589a0358aa": "|P(V_{\\omega +\\alpha })|", "1f80bbe06d1bc10023d855d9751c0f25": "X^{*}=X\\cup \\{\\infty \\}", "1f80cd05ff7868722abdc53bd90fef4c": "T_{f^{-1}P}(X)", "1f8101037e3afb9c21a8883823bf88f4": "{\\textbf {a}}=p{\\textbf {r}}\\cdot {\\textbf {g}}+{\\textbf {f}}\\cdot {\\textbf {m}}{\\pmod {q}}", "1f815e0e42cb81964f6e99531cd64e46": "{\\overline {\\rho _{f}}}", "1f819b6c431402316e13dfda49cdf745": "\\{[M(OH)]^{(z-1)+}\\}=K_{1,-1}\\{M^{z+}\\}\\{H^{+}\\}^{-1}", "1f81d5eb0fe2f45b2f8d8c2ad4c63d12": "0\\varepsilon }K(x-y)f(y)\\,dy.", "1f91795b5c570049cdcaf73add8215d1": "\\forall x(x\\in a\\leftrightarrow x\\in A)", "1f917e19e02c2345838ab694e3ee6dd9": "\\displaystyle {\\sqrt {\\frac {\\pi }{a}}}\\cos \\left({\\frac {\\nu ^{2}}{4a}}-{\\frac {\\pi }{4}}\\right)", "1f91c52f1a198771670ab18fed186555": "N=(V,E)", "1f91f6ce62d1ce9ba7447981981447e6": "x\\wedge \\bigvee Y=\\bigvee \\{x\\wedge y\\mid y\\in Y\\}", "1f9211e8cf33d0a4fa93afe3a059243e": "{\\begin{bmatrix}{1 \\over \\eta }\\\\-\\left({a-\\lambda _{-} \\over c\\eta }\\right)\\end{bmatrix}}", "1f92448d4e6d4ad647b427349e4480a9": "U=", "1f926e96e3354f4a635403dc1fca133c": "(p+1)\\times (p+1)", "1f9271778b3f16601ff4b29d26b2b610": "I=ms^{2}", "1f9296350287268701819ffbeef8105a": "\\displaystyle x=x_{0}", "1f92f4e75e6b9ccd79395801a27b4cee": "PG_{a}={\\tfrac {4}{3}}PG_{v}.", "1f9351fbf04b3aef582230fec73e233e": "N\\sim \\operatorname {Poisson} (\\lambda ),", "1f93bad7415d79432555fcd6a857ac2b": "\\displaystyle {\\sigma (a,T,b)=(b,-T^{t},a).}", "1f93d5afff4b28e0f0c52056368781b2": "F_{eachAnchor}=F_{load}{\\frac {Sin(\\alpha )}{Sin(2\\alpha )}}\\,", "1f93db021b5b139c283ca168bbf204d0": "\\oint _{\\gamma }(v\\,dx+u\\,dy)=\\iint _{D}\\left({\\frac {\\partial u}{\\partial x}}-{\\frac {\\partial v}{\\partial y}}\\right)\\,dx\\,dy", "1f941ff8f996e4eb9e49d2b87ceef78f": "\\scriptstyle 0\\;\\leq \\;\\alpha \\;\\leq \\;1", "1f94fb38335d4138271b1794010e4750": "\\displaystyle A=B=C=60^{\\circ }", "1f95691b519aef3c48c1663e47d963a4": "[{\\widehat {a}},{\\widehat {a}}^{\\dagger }]=1", "1f95d9c308e1cbee4861bf63edd51bd2": "td(E)=\\prod Q(\\alpha _{i})", "1f95de6207444b162046eac435c087d1": "{\\mathfrak {sl}}(3,\\mathbb {C} )", "1f95fc1ca06160b80e6d2a3734882aa6": "\\ln(f(x))=\\ln(g(x)h(x))=\\ln(g(x))+\\ln(h(x))\\,\\!", "1f9640b2dc1da48b14e9028afaac496f": "y=g_{2}(x)", "1f964de0e7de97915329fad8482efbb9": "\\cos {\\frac {\\pi }{5}}=\\cos 36^{\\circ }={\\frac {{\\sqrt {5}}+1}{4}}={\\frac {\\varphi }{2}}", "1f966d3a5742d7653b2b5d2cb80d7ec3": "r({\\text{in}},{\\text{out}})", "1f967e815b21989b041a773b59242310": "{\\frac {[A]}{[B]}}={\\frac {[A]_{0}}{[B]_{0}}}e^{([A]_{0}-[B]_{0})kt}", "1f96b648c52c26eefd587a1ffb8325a5": "{\\mathcal {H}}|0\\rangle =(-Js^{2}-g\\mu _{B}Hs)N|0\\rangle ", "1f9719b14c24ef318f1bf7f1e2cf3a24": "\\left({\\begin{smallmatrix}X&X&X&\\cdot &\\cdot &\\cdot &\\cdot &\\\\X&X&\\cdot &X&X&\\cdot &\\cdot &\\\\X&\\cdot &X&\\cdot &X&\\cdot &\\cdot &\\\\\\cdot &X&\\cdot &X&\\cdot &X&\\cdot &\\\\\\cdot &X&X&\\cdot &X&X&X&\\\\\\cdot &\\cdot &\\cdot &X&X&X&\\cdot &\\\\\\cdot &\\cdot &\\cdot &\\cdot &X&\\cdot &X&\\\\\\end{smallmatrix}}\\right)", "1f974c9f11aaf2340f59ee57960191bc": "\\Delta (n)=O\\left({\\frac {1}{n^{2}}}\\right).", "1f978226deaa49b61ad98f59ec684582": "{\\bar {I}}_{1}=I_{1}/J", "1f97ded0269589e6de9986986596b429": "CO_{2}", "1f97e6939ad24f53f41347befef01b3b": "\\left\\{{n \\atop k}\\right\\}\\sim {\\frac {\\sqrt {n-k}}{{\\sqrt {n(1-G)}}\\ G^{k}\\ (v-G)^{n-k}}}\\left({\\frac {n-k}{e}}\\right)^{n-k}\\left({n \\atop k}\\right)\\quad \\forall k,11\\\\0&{\\text{if}}\\ r\\leq 1\\end{cases}}", "1fa926410c869d167ee022e2f5dd7520": "M_{m}", "1fa92a141bd7196d5a899706bca57b7b": "V_{OUT}=V_{S}\\cdot {\\frac {R_{L}}{R_{L}+R_{S}}}", "1fa964bd3d7810755e38cb5ac8bdc92c": "~\\langle n\\rangle =\\langle {\\hat {a}}^{\\dagger }{\\hat {a}}\\rangle =|\\alpha |^{2}~", "1fa98d52609723bc930372993cdaa8fc": "T\\colon V\\to W", "1fa9b26f7365a8ded43a38c4ee11b156": "{\\frac {2+2\\!\\times \\!0+3\\!\\times \\!0+4\\!\\times \\!0+5\\!\\times \\!0+6\\!\\times \\!3}{11}}={\\frac {20}{11}}=1+{\\frac {9}{11}}", "1fa9e78917b3d3493ca593691fc15c27": "SDecode\\,", "1faa2c93a3ec023bb2c9223048f3f494": "E_{v}/hc=\\omega _{e}(v+1/2)-\\omega _{e}\\chi _{e}(v+1/2)^{2}\\,", "1faa6aa0bd93ed5bbcefc7841caa78b2": "{\\mathcal {I}}=\\int \\rho \\,(\\partial _{x}\\ln \\rho )^{2}dx=-\\int \\rho \\,\\partial _{x}^{2}\\ln \\rho \\,dx", "1faa858ca2abe2b3aa9f96d65ee58641": "\\nabla _{a}G^{ab}=G^{ab}{}_{;a}=0.\\ ", "1fab2c9874da1b07df366b9da5d7b4d0": "\\theta =np", "1fab35819992d248f017e9a25a932ada": "hv=fg^{-1}v=\\sum _{b\\in B}l_{b}^{\\sigma }b", "1fab5d7c97b9c2c85134e1f0a7de8e26": "\\lambda =\\lambda _{0}{\\sqrt {\\frac {c-v}{c+v}}}\\,\\!", "1fac0a296ee042dd60c7d010fbe2c43d": "X_{\\alpha }={\\frac {c-b}{a-b}}", "1fac195ae37518af8c49062eabf906d9": "\\mathbf {v} =(v[1],v[2],...,v[n])", "1fac2717165a3b978ff96b0dca2daa57": "\\Omega ={\\begin{pmatrix}0&I_{n}\\\\-I_{n}&0\\\\\\end{pmatrix}}.", "1fac55fda14af9b0f523d440f6b3e67f": "\\,(x_{2},\\;f(x_{2}))\\,", "1fac7dadd89836f0f6067e29eb8b9dad": "\\scriptstyle \\left(0\\right)_{i\\in I}", "1fac9c3e8d14989d754cccf7959a65c6": "\\scriptstyle p_{2m}\\,", "1facc2c2ff757ec505f683cb05e83a83": "(3)\\qquad \\kappa \\,{\\hat {=}}\\,0\\,,\\quad {\\text{Im}}(\\rho )\\,{\\hat {=}}\\,0\\,,\\quad {\\text{Re}}(\\rho )\\,{\\hat {=}}\\,0\\,,\\quad \\sigma \\,{\\hat {=}}\\,0\\,,\\quad R_{ab}l^{a}l^{b}\\,{\\hat {=}}\\,0.", "1fad09f744892580d9e2a7011552f358": "\\phi '", "1fad52e9ab3aaca935a52e04f39c3998": "I(z)=I_{in}e^{\\gamma _{0}(\\nu )z}", "1fad6fb55ae33616839fd6efaff3d65f": "\\ {\\textbf {f}}\\cdot {\\textbf {f}}_{p}=1{\\pmod {p}}", "1fad78a534b0f4bce77cc199ea12e8d6": "\\min\\{\\,d(x,y):x\\in {\\mathcal {A}},\\,y\\in {\\mathcal {B}}\\,\\}.", "1fada30449a302274c8154bca665b89a": "|AE|=|BD|,\\,\\alpha =\\beta ,\\,\\gamma =\\delta ", "1fadb14ce7b0c4d3ff72761ac56602e6": "\\mathrm {d} \\varphi _{x}\\left({\\frac {\\partial }{\\partial u^{a}}}\\right)={\\frac {\\partial {\\widehat {\\varphi }}^{b}}{\\partial u^{a}}}{\\frac {\\partial }{\\partial v^{b}}},", "1fadba73d16884e8eb9d0d5759c2a3ce": "{\\hat {\\mathbf {J} }}=J(r)\\mathbf {\\Phi } _{lm}", "1fae8ae4cd24bdf1d07032bfa1aec3a1": "R={\\frac {\\sum _{\\mathrm {all\\ reflections} }\\left|F_{o}-F_{c}\\right|}{\\sum _{\\mathrm {all\\ reflections} }\\left|F_{o}\\right|}}", "1faeb5a85963286c305ea018ec8f80cf": "u:\\mathbb {R} _{+}^{L}\\rightarrow \\mathbb {R} _{+}\\ .", "1faecc4dab1b59b0ffb508741106b414": "{\\frac {d}{dt}}\\left({\\vec {r}}(t)\\right)=\\nabla g", "1faee40204cf10774ef0e6e5a36f674f": "S_{1}S_{2}S_{1}", "1faef6f25aaf90eb0b18b5660c863d70": "10\\uparrow ^{10\\uparrow ^{3\\times 10^{5}}10}10\\!", "1faf146dfd7cb3cfdaa429a2e1180c31": "{\\boldsymbol {\\nu }}", "1faf2087d8c8e3b01e9b74327d36cbcc": "\\iint _{D}^{}{\\left({\\frac {\\partial (\\varphi f)}{\\partial x}}+{\\frac {\\partial (\\varphi g)}{\\partial y}}\\right)dxdy}=\\oint _{C}^{}{-\\varphi gdx+\\varphi fdy}", "1faf9a5495803e0f69c960caf661a4ac": "Z\\to Z'", "1fafdbccfb571d7aa2644d01c407a521": "E_{2}-E_{1}={\\tfrac {1}{2}}\\,(p_{2}+p_{1})\\,\\left({\\tfrac {1}{\\rho _{1}}}-{\\tfrac {1}{\\rho _{2}}}\\right)={\\tfrac {1}{2}}\\,(p_{2}+p_{1})\\,(v_{1}-v_{2})", "1fb00bdb877b46c5733e41cb124cae04": "f(z)=z+1+\\exp(-z)", "1fb0266c45408f54306c7de03baa455d": "f_{!}:{\\text{Mod}}_{R}\\leftrightarrows {\\text{Mod}}_{S}:f^{*}", "1fb035acb226f98a6db5e42ac9b1f597": "V_{m}", "1fb07bcb9c06a1aebd89952d9feaf4c2": "X_{1}={\\sqrt {\\frac {R_{2}R_{1}^{2}}{R_{1}-R_{2}}}}", "1fb08ef1a34b9faa675756e053c6dde1": "\\delta (x)\\,\\!", "1fb0b2f1bc03d2d85ddf720fbab3510c": "\\int ds{\\sqrt {v}}", "1fb0c65e1513d515f6c134c4311db6d8": "|\\mathbf {r} -\\mathbf {r} _{s}(t_{2})|=c(t-t_{2})", "1fb0d028e0c7b8573681a7f42c2d4efa": "-{\\frac {1}{2}}(\\mathbf {a} -\\mathbf {x} )^{2}+{\\frac {1}{2}}\\rho ^{2}=0", "1fb0d9e574d4a8dfa17b15c0a4ed38b4": "Y_{s}=Y_{N_{t}}+\\alpha [P_{t}-E\\left(P_{t}|\\Omega _{t-1}\\right)]", "1fb1796c7637dee6e153e4443b16aeb0": "{\\mathbf {X} }_{t_{1},\\ldots ,t_{k}}=(\\mathbf {X} _{t_{1}},\\ldots ,\\mathbf {X} _{t_{k}})", "1fb1919af4234e6802dec9dc716a6ac9": "S:X\\to W", "1fb1c3dd21b6d638e029aabcab98f5ee": "S^{\\mathrm {W} Z}(\\gamma )=\\int _{B^{3}}\\gamma ^{*}c.", "1fb1ecd403869dc2743712b600e83e80": "E(t)dt", "1fb2751e36aa7ee7a58a656b96882674": "b^{n}+a_{n-1}b^{n-1}+\\cdots +a_{1}b+a_{0}=0.", "1fb27adeb836505973906ae13194b614": "Y^{c}=\\max(0,Y)", "1fb28683a5575f143e9315835c154793": "D=\\sum _{k=1}^{M}d_{k}", "1fb29d291d9073c4adbc5091ac0ea9fd": "{\\frac {d}{dz}}\\left[z^{-b_{1}}\\;G_{p,q}^{\\,m,n}\\!\\left(\\left.{\\begin{matrix}\\mathbf {a_{p}} \\\\\\mathbf {b_{q}} \\end{matrix}}\\;\\right|\\,z\\right)\\right]=-z^{-1-b_{1}}\\;G_{p,q}^{\\,m,n}\\!\\left(\\left.{\\begin{matrix}\\mathbf {a_{p}} \\\\b_{1}+1,b_{2},\\dots ,b_{q}\\end{matrix}}\\;\\right|\\,z\\right),\\quad m\\geq 1,", "1fb2aafe1caf3f07796d6bdb8d2f21f8": "\\sigma :\\mathbb {R} ^{n}\\mapsto \\mathbb {R} ^{m}", "1fb326aec8788f6df25829f701154fa0": "\\textstyle {x<0}", "1fb33067cb5fa847c62d7b356bd8d099": "2[x,x]=0.", "1fb3315e27fbbad8db2c3146f4910096": "i=1\\ldots n", "1fb35632d73d869b3f21afb0dc1256a8": "\\,\\sigma _{j,k}", "1fb382a46b80e13ce933747251179910": "\\operatorname {Riesz} (x)=O(x^{e})\\qquad ({\\text{as }}x\\to \\infty )", "1fb38ac2aa147f6aca29ee7ffaa9675e": "S_{uvw}", "1fb39d7cebf6a4dbd7490988e3f93c57": "{\\mathbf {r}}'=x'{\\mathbf {\\hat {x}}}+y'{\\mathbf {\\hat {y}}}", "1fb3b304980d61ce94691724464e7737": "N^{\\pm }(X)\\sim {\\frac {A_{\\pm }}{12\\zeta (3)}}X+{\\frac {4\\zeta ({\\frac {1}{3}})B_{\\pm }}{5\\Gamma ({\\frac {2}{3}})^{3}\\zeta ({\\frac {5}{3}})}}X^{\\frac {5}{6}}", "1fb3c3f4c5b1d00cda1e19f92e455d84": "\\varphi _{n}\\ ", "1fb4649654c08362e432d4122f0d9e10": "\\lambda _{c}={\\sqrt {\\frac {\\gamma }{\\rho g}}}", "1fb4a6fbea22c7ff968bd40ae9e3b158": "\\int _{0}^{+\\infty }x^{\\alpha }e^{-x}f(x)\\,dx.", "1fb4aa001bc075df9e1572b9849aa430": "{\\textrm {VD}}={\\frac {\\textrm {DS}}{{\\sqrt {\\left({\\frac {\\textrm {NHR}}{\\textrm {NVR}}}\\right)^{2}+1}}\\cdot {\\textrm {CVR}}\\cdot \\tan {\\frac {1}{60}}}}", "1fb528d8993fc05cfb59d882c12b24e4": "{\\frac {\\det \\left(-{\\frac {d^{2}}{dx^{2}}}+V_{1}(x)-m\\right)}{\\det \\left(-{\\frac {d^{2}}{dx^{2}}}+V_{2}(x)-m\\right)}}={\\frac {\\psi _{1}^{m}(L)}{\\psi _{2}^{m}(L)}}", "1fb540e9c9a2f2e282f38c6ab3b51a2c": "\\ A^{*}\\vee B^{*}", "1fb5520dcd1b01e2bfab33e2ff16093b": "\\lim _{\\underset {h\\in \\mathbf {R} }{h\\to 0}}{\\frac {f(z_{0}+ih)-f(z_{0})}{ih}}={\\frac {1}{i}}{\\frac {\\partial f}{\\partial y}}(z_{0}).", "1fb5a8da55877e07a8d0d5a5efc8bc4e": "{\\mathcal {O}}_{s}", "1fb6035d496acc3a2cc1a66f2a1743b6": "w\\colon E\\rightarrow \\mathbf {R} ^{+}", "1fb623471d5eab2b9ad3de8b3549dfd0": "\\chi _{\\text{2}}(2\\omega )={\\frac {\\varepsilon _{0}^{2}m\\zeta _{2}}{N^{2}q^{3}}}\\chi _{\\text{1}}(\\omega )\\chi _{\\text{1}}(\\omega )\\chi _{\\text{1}}(2\\omega )", "1fb62380d61606d120fcda6f8c418a78": "=V_{\\text{d}}\\cdot k_{\\text{e}}={\\frac {D}{AUC}}", "1fb63781a11b5a567349fba3a81d7e6e": "{\\frac {k(t)}{k}}=s.A-n", "1fb64da16298cade5031507606c4d8f7": "\\psi (x,y,z)=C\\cdot Ai({{-eEx-\\epsilon +\\hbar ^{2}k_{y}^{2}/2\\mu _{y}+\\hbar ^{2}k_{z}^{2}/2\\mu _{z}} \\over {\\hbar \\theta _{x}}})", "1fb66244dedcb6d5448434977ea56f76": "X_{i}Y_{i}", "1fb68aa8ed8484819dde02aab6e1f900": "s(N-1)", "1fb69f73c9269e2b8b2cd0bfe4be00a5": "f_{0}^{(k)},\\ldots ,f_{N^{(k)}-1}^{(k)},\\,", "1fb6ff3edc72f426caac14ac4e3a4554": "\\supseteqq ,\\nsupseteqq ,\\supsetneqq ,\\varsupsetneqq \\!", "1fb7194838c79a33be3bffd36b785baa": "{\\cfrac {A+T}{G+C}}", "1fb727587954bb88da496f93270c3b9f": "K_{p}=\\mathrm {{\\frac {(p_{T})^{2}(x(NO_{2}))^{2}}{(p_{T})(x(N_{2}O_{4}))}}={\\frac {(p_{T})(x(NO_{2}))^{2}}{(x(N_{2}O_{4}))}}} ", "1fb729604087c74338e709f171ee54bd": "{\\frac {D}{\\sqrt {2E}}}", "1fb76e2ebc0eae759b6a4a36f544c646": "p(b)\\leftarrow ", "1fb79f6304c9b4d8127c513fed28f459": "\\langle O(n),O(n)\\rangle ", "1fb7e280baed28452bb0a22572bb067e": "\\left\\{x_{1},x_{2},x_{3}\\right\\},", "1fb7e803f0886d2bc4e45844e7ea2fef": "\\int {\\frac {x}{R}}\\;dx={\\frac {R}{a}}-{\\frac {b}{2a}}\\int {\\frac {dx}{R}}", "1fb83c04d48fa6fbebce4ac9e113390f": "F_{2}=F_{load}{\\frac {Sin(\\alpha )}{Sin(\\alpha +\\beta )}}\\,", "1fb85b7836468f8b53e9515000f7468d": "G", "1fb8edf718f6e25b6e0c2a8fcbc8fb6c": "a_{11}={\\mathcal {L}}(p_{5})+p_{3}p_{5}+p_{2}p_{7}+p_{1}p_{8},", "1fb914ba5bd1e318e3c585e0449ad6aa": "{\\begin{aligned}j&=\\sum _{i=1}^{3}{\\frac {\\partial L}{\\partial {\\dot {x}}_{i}}}Q[x_{i}]-f\\\\&=m\\sum _{i}{\\dot {x}}_{i}^{2}-\\left[{\\frac {m}{2}}\\sum _{i}{\\dot {x}}_{i}^{2}-V(x)\\right]\\\\&={\\frac {m}{2}}\\sum _{i}{\\dot {x}}_{i}^{2}+V(x).\\end{aligned}}", "1fb94faa966d0d6666e65f6ac14d0a13": "\\mu _{n}=(-nq)\\mu _{n}(-{\\frac {1}{nq}})=-\\sigma _{n}R_{Hn}", "1fb9d1774ad1d205d4638da8187490a3": "c\\equiv z^{Q}", "1fba55d0040df49a890f8b8a3a05cf6f": "{\\vec {D_{\\beta }}}=|{\\vec {C_{2}}}.{\\vec {X_{\\beta }}}-{\\vec {X}}|", "1fba648f51eefc37c7f8ad89472f9562": "A={\\frac {M}{\\rho }}{\\frac {n^{2}-1}{n^{2}+2}},", "1fba8a94ec81415fbeaefc7ab437b827": "g={\\frac {Q^{*}}{L\\rho }}", "1fbaae231d01ec1cafacff645b3fce46": "\\mathbf {C} _{f}", "1fbab1bd9145501b6784a0f9f6480a83": "\\Sigma _{n}=\\left\\{(x,y,u_{n}(x,y))\\in \\mathbb {R} ^{3}\\left|-{\\frac {\\pi }{2}}2", "1fc0821f7fe8bd240fa2bf466df4e057": "{\\frac {\\pi }{3{\\sqrt {2}}}}\\simeq 0.74048.", "1fc0940dc26a177aab0d82003f11ab03": "(S\\,y)''=S\\,y''+2\\,S'\\,y'+S''\\,y", "1fc0bc211ee47dee1ec609f450ad9b4a": "a_{i}\\in I^{i}", "1fc0dd3998a2de27dd5dc13fe32ac9ab": "\\cos \\pi z=\\prod _{q\\in \\mathbb {Z} ,\\,q\\;{\\text{odd}}}\\left(1-{\\frac {2z}{q}}\\right)e^{2z/q}=\\prod _{n=0}^{\\infty }\\left(1-{\\frac {4z^{2}}{(2n+1)^{2}}}\\right)", "1fc1523c7e6de14066b43a40f1c05c8d": "\\scriptstyle \\{|0\\rangle _{A},|1\\rangle _{A}\\}", "1fc164dc85d156f67eb71706d0d982cb": "\\mathbf {n} ={\\frac {\\mathbf {r} _{1}\\times \\mathbf {r} _{2}}{|\\mathbf {r} _{1}\\times \\mathbf {r} _{2}|}}.", "1fc19afb794c4b2368ae082be5dd92c5": "\\omega =\\omega ^{r}\\,\\cap \\,\\omega ^{l}", "1fc1ae3cd4e91f49a4cf9ff4c2ba014f": "\\mathbf {v} \\in V", "1fc1d1474faa1718ebfa13afd7e0a213": "\\nabla ^{2}V=-{\\frac {\\rho }{\\varepsilon _{0}}}", "1fc1f7b000b1faa032f4aba0365049f7": "-g^{(k)}", "1fc26eae19dc1a77f822d6cfd408177f": "H=-{\\frac {\\hbar ^{2}}{2m}}\\int d^{3}\\!r\\ \\phi ^{\\dagger }(\\mathbf {r} )\\nabla ^{2}\\phi (\\mathbf {r} )+{\\frac {1}{2}}\\int \\!d^{3}\\!r\\int \\!d^{3}\\!r'\\;\\phi ^{\\dagger }(\\mathbf {r} )\\phi ^{\\dagger }(\\mathbf {r} ')U(|\\mathbf {r} -\\mathbf {r} '|)\\phi (\\mathbf {r'} )\\phi (\\mathbf {r} ).", "1fc2e24908f22c141e31d784ba6c981e": "E\\supseteq K\\supseteq F", "1fc2eae4b74acbd862dd18c59301ade4": "{\\frac {z^{3}-{\\tfrac {1}{3}}a}{z^{2}-z^{3}}}.", "1fc33df1a678c1b88ebf1296f44e6706": "\\displaystyle {Q_{r}(\\theta )={2r\\sin \\theta \\over 1-2r\\cos \\theta +r^{2}}.}", "1fc3846a47807c5850a84319a75e9b16": "L(T)", "1fc3a72bbaae4e22c94971ef3ef2b87f": "C_{i}\\left(\\mathbf {r} ,t\\right)", "1fc3cc59c43a5922b75132c2c4bdd247": "\\,\\,f={\\frac {\\omega }{2\\pi }}{\\text{.}}\\,\\!", "1fc3f9dcad3b87a2966b40f5fd357b94": "r_{i}=r-{\\frac {nh}{2}}", "1fc4629200f55e2110fb4c571263cebf": "{\\mathbf {j}}=\\rho {\\frac {{\\mathbf {\\nabla }}S}{m}}", "1fc4e4f82497357b11205a031af26f0e": "X_{f}[g]:=(f,g).", "1fc4e7e27edb61a19b11a83a971d82f7": "F_{1}=(z^{N/2}-1)", "1fc500923d2353f29a0bac9484bcbbfb": "a_{m,n}={\\frac {2n+2}{\\epsilon _{m}\\pi }}\\langle G(\\rho ,\\varphi ),Z_{n}^{m}(\\rho ,\\varphi )\\rangle ,\\quad b_{m,n}={\\frac {2n+2}{\\epsilon _{m}\\pi }}\\langle G(\\rho ,\\varphi ),Z_{n}^{-m}(\\rho ,\\varphi )\\rangle .", "1fc504ff79d40d107a9462846763a793": "2E={\\dot {\\mathbf {s} }}^{\\mathrm {T} }\\mathbf {G} ^{-1}{\\dot {\\mathbf {s} }}+\\mathbf {s} ^{\\mathrm {T} }\\mathbf {F} \\mathbf {s} ", "1fc51e2b44083e5b687bf632543991c6": "\\lambda _{k}=0", "1fc5ef9f424597d11cc4df7b51cb4b40": "\\scriptstyle i({\\hat {x}}{\\hat {v}}\\,-\\,{\\hat {v}}{\\hat {x}})\\;=\\;2i({\\hat {x}}\\,\\wedge \\,{\\hat {v}})", "1fc5faa3e5665e1e49335257346e88f1": "G={\\frac {E}{2(1+\\nu )}}\\,\\!", "1fc60e48b417ee760896887baecb5cfe": "L\\propto \\sigma ^{3.1}", "1fc6739b4b3eda38d449d4fdb9eb29f3": "{\\mbox{Quick (Acid Test) Ratio}}={{\\mbox{Cash and Cash Equivalent}}+{\\mbox{Marketable Securities}}+{\\mbox{Accounts Receivable}} \\over {\\mbox{Current Liabilities}}}", "1fc6a5a8a0be46bdd5cdde820e524e43": "u^{3}=a^{3}({\\tfrac {u}{a}})^{3}=a^{3}({\\tfrac {u}{a}})({\\tfrac {v}{u}})({\\tfrac {b}{v}})=a^{3}({\\tfrac {b}{a}})=2a^{3}", "1fc6b6f3884a3c5f19269335c1f5fafa": "a(u,u)\\geq c\\cdot \\|u\\|^{2}", "1fc6d8b994af7bde5fadfb8e78e06d0e": "\\left(\\mathbf {A} \\mathbf {B} \\right)^{-1}=\\mathbf {B} ^{-1}\\mathbf {A} ^{-1}.", "1fc6e6f313532186a8d96cc6910bf25e": "M({\\hat {x}})", "1fc75db554dc991d1f8cf5668c582b79": "(c,b)\\in R", "1fc7d6d5b5d7964c051d68d889978731": "\\cos E=\\varepsilon +{\\frac {1-\\varepsilon ^{2}}{1+\\varepsilon \\cdot \\cos \\theta }}\\cdot \\cos \\theta ", "1fc80270fd84bfd02eece8812c33b2c9": "\\mathbb {E} _{X^{n}}\\left\\{{\\text{Tr}}\\left\\{\\Pi _{\\rho _{X^{n}},\\delta }\\ \\rho _{X^{n}}\\right\\}\\right\\}\\geq 1-\\epsilon .", "1fc839208f85d5dce5778ed901c3139f": "y=f(k)", "1fc86ac1032be25eca633b7544f777eb": "Z=\\int Dx\\,e^{{\\rm {i}}{\\mathcal {S}}[x]/\\hbar }", "1fc8819fc98b84e1deacffebf4bf4b59": "{\\textbf {M}}_{k}=[{\\textbf {F}}_{k}^{-1}]^{\\text{T}}{\\textbf {Y}}_{k-1\\mid k-1}{\\textbf {F}}_{k}^{-1}", "1fc89784c87362b7983a165bb98a7fb8": "{\\frac {4}{5}}-{\\frac {2}{5}}=0.4=40\\%", "1fc8a40e4104f198e87e696887ea80e4": "\\mathbf {a} ", "1fc8d41944f3f64afdae3b54c2d2ecaf": "\\mathrm {M} ^{+z}+z\\mathrm {e} ^{-}\\rightleftharpoons \\mathrm {M} .", "1fc99ac427a2fc26dd414a641b9fcc51": "\\operatorname {MIAE} ({\\mathbf {H}})=\\operatorname {E} \\,\\int |{\\hat {f}}_{\\mathbf {H}}({\\mathbf {x}})-f({\\mathbf {x}})|\\,d{\\mathbf {x}}.", "1fc9daa3ef27c43d9726b587a03c04d6": "{\\mathit {H}}({\\mathit {n}},\\mathbb {C} _{p})", "1fca04efa8462cf833826246202ebe71": "e^{3}(e+2)(a+1)^{2}+1-o^{2}", "1fca1ed477c05ac9ba4fa12a7a5e07ae": "a+b{\\sqrt {-1}},a,b\\in \\mathbf {Z} ", "1fca2047eb5ddba83d5a487b46762bf6": "-{\\frac {\\mathrm {d} ^{2}\\mathbf {r} (t)}{\\mathrm {d} t^{2}}}m={\\frac {\\partial V[\\mathbf {r} (t)]}{\\partial x}}\\mathbf {\\hat {x}} +{\\frac {\\partial V[\\mathbf {r} (t)]}{\\partial y}}\\mathbf {\\hat {y}} +{\\frac {\\partial V[\\mathbf {r} (t)]}{\\partial z}}\\mathbf {\\hat {z}} ,", "1fcadb08bff81d8ce0420b1afbf65ac0": "S_{i+\\nu }+\\Lambda _{1}S_{i+\\nu -1}+\\cdots +\\Lambda _{\\nu -1}S_{i+1}+\\Lambda _{\\nu }S_{i}=0.", "1fcaeaab429d24583c738c40b4ad5a94": "p_{k,i}^{\\mathcal {M}}<{\\frac {1}{2}}+{\\frac {1}{Q(k)}}", "1fcbd3c93e923b362d00abe9173f37ca": "k[x_{0},...,x_{n}]", "1fcc7d66dd0358306585155cca8dd467": "\\sum _{n\\leq x}d(n)=x\\log x+(2\\gamma -1)x+o(x)", "1fccb59f752c8d14c6c45b5bfa6e6987": "r*e", "1fccc31fd9fb43d716f47f1a5e980228": "\\operatorname {coth} (z)", "1fcd5af5f402845603fcd016135f1ba7": "-{\\frac {d[A]}{dt}}=k[A]^{n}", "1fcde9c7cb11774abe82bfaacd66d365": "\\langle O(1),poly(n)\\rangle ", "1fce27f4bdb1769982216f30a0e34430": "\\mathrm {E} \\left[{\\hat {A}}_{1}\\right]=\\mathrm {E} \\left[x[0]\\right]=A", "1fce774d8fba992688a16f1e59db178c": "\\mathbf {a} =(a_{1},a_{2},a_{3}).", "1fcebc486bde79742426965170807f63": "y=0\\,", "1fcec14a18d4361fc19b6029a400f985": "F=[S-PV(Div)]\\cdot (1+r)^{(T-t)}\\ ", "1fcee716e2996ef968f74e4f38db6427": "\\scriptstyle E_{\\vec {p}}", "1fcef9f6803f17930b733c743e4c50c6": "{\\frac {d\\theta _{i}}{dt}}=\\omega _{i}+{\\frac {K}{N}}\\sum _{j=1}^{N}\\sin(\\theta _{j}-\\theta _{i}),\\qquad i=1\\ldots N", "1fcf001307506fb97c7d1b64fb64191b": "x_{k}\\neq 0", "1fcf115fe35ee54ac21ed724e675b078": "O(2^{k}m)", "1fcf22b3d5d987dca010507dc8b5d278": "\\scriptstyle {E/c^{2}}", "1fcf323a90975ddd11f9a2e5f497ffe2": "x(t)=x_{1}(t){\\begin{bmatrix}1&e^{-j\\omega \\Delta t}&\\cdots &e^{-j\\omega (M-1)\\Delta t}\\end{bmatrix}}^{T}", "1fcf70ef1a0772e4085892530938a676": "L_{x}(\\mathbf {x} ,\\sigma _{D})", "1fcf841ba6cb2c8ae44b2ca642079582": "q_{k}", "1fcf8b73b74b29e57619233527678428": "\\,i=0.12", "1fcfaa7dc3845b6f3ea2e9da6ea7872a": "V_{n}^{2}=kT/2C", "1fcfd80ca2ff94600509ff85ea85f28a": "k=Ae^{-E_{a}/(RT)}", "1fcfe57fca607ad295918e02809dc58c": "n_{1}=0", "1fd0040e4536f962319411ad4a97ad39": "0=A_{21}n_{2}+B_{21}n_{2}\\rho (\\nu )-B_{12}n_{1}\\rho (\\nu )\\,", "1fd02ff915a31059b6326fdee5c760e7": "n\\in {\\mathbb {N} },\\;n>1", "1fd04858035a621bd3c4d4f9ac6e3f84": "X^{2}+Y^{2}=1\\,", "1fd089f0f4cece02f59759415980140d": "R(t)=P(\\{T>t\\})=\\int _{t}^{\\infty }f(u)\\,du=1-F(t).", "1fd08f6f5c641d939dc5d558f161bca9": "m_{2}\\ ", "1fd0f4ce52d05dcefaed9324d2f95569": "33^{8}+1549034^{2}=15613^{3}\\;", "1fd164fcc71fcbb2514cbfb400eb21dc": "u_{xx}+u_{yy}=0", "1fd17283506e24d39281068cc05afbf2": "\\kappa =\\Omega ", "1fd1ba4abf5af186a5fd797f9450974b": "z_{1},z_{2}\\in \\mathbb {H} ", "1fd1c35df6b07d9227eba1f95e975e49": "\\epsilon _{0}", "1fd1d5ccd8873172d395d866d529bc6b": "e_{i}e_{j}={\\Bigg \\{}{\\begin{matrix}-1&i=j,\\\\-e_{j}e_{i}&i\\not =j\\end{matrix}}", "1fd1dc9845ddbbed5201319bc00d29f7": "\\delta J(y)(h)=\\lim _{\\varepsilon \\to 0}{\\frac {J(y+\\varepsilon h)-J(y)}{\\varepsilon }}=\\left.{\\frac {d}{d\\varepsilon }}J(y+\\varepsilon h)\\right|_{\\varepsilon =0},", "1fd20e5a539570a179ce3d65f7a2daac": "d(1-2\\varepsilon )>d/2\\,", "1fd21fe376a3332e8fd67ed25d32f4bb": "\\deg(F)", "1fd244f6feab5985d8b3f8bc348441da": "{\\begin{bmatrix}A&B\\\\B&C\\\\\\end{bmatrix}},", "1fd2a1293dd3850017ed2b95a3957f2c": "\\pm {\\sqrt {1/3}}", "1fd2c5228de66642b72d9a2e747dfdb2": "V=k^{n}", "1fd2e0b1caa9af835977c93b0c9f9a42": "{\\boldsymbol {\\beta }}^{(s+1)}={\\boldsymbol {\\beta }}^{(s)}+\\Delta ;\\quad \\Delta =-\\left(\\mathbf {J_{r}} ^{\\top }\\mathbf {J_{r}} \\right)^{-1}\\mathbf {J_{r}} ^{\\top }\\mathbf {r} .", "1fd32fc57b08616399f84f6546a4dcf1": "\\eta _{II}={\\frac {\\eta _{th}}{\\eta _{th,rev}}}", "1fd38986e807462ae37dad3b58dd6795": "{\\;\\,dS^{\\alpha } \\over d\\tau }={e \\over m}{\\bigg [}{g \\over 2}F^{\\alpha \\beta }S_{\\beta }+\\left({g \\over 2}-1\\right)u^{\\alpha }\\left(S_{\\lambda }F^{\\lambda \\mu }U_{\\mu }\\right){\\bigg ]}\\;,", "1fd399a2880edd5fcf798968ffb57e92": "Q_{\\mathrm {max} }=P", "1fd3b9dc3137a01f24a070e90e55d483": "0.0235\\times W^{0.51456}\\times H^{0.42246}", "1fd41058e9fad1ba04b393195d340fce": "{\\tilde {C}}_{1}", "1fd45e3467029291005ada5ed15dbc46": "A\\left(t,z+h\\right)", "1fd5410dc84ea247a8bfc1c2e3c2f632": "d_{f}(\\alpha )=\\mu (\\{x\\in X:|f(x)|>\\alpha \\}).", "1fd5e94b586ef6429747b5914d475480": "{\\hat {H}}'_{0}=(\\alpha \\cdot p+\\beta m)(\\cos \\theta -\\beta \\mathbf {\\alpha } \\cdot {\\hat {p}}\\sin \\theta )^{2}=(\\alpha \\cdot p+\\beta m)e^{-2\\beta \\mathbf {\\alpha } \\cdot {\\hat {p}}\\theta }=(\\alpha \\cdot p+\\beta m)(\\cos 2\\theta -\\beta \\mathbf {\\alpha } \\cdot {\\hat {p}}\\sin 2\\theta )", "1fd60c9a78e8faa71fc216a281d63384": "\\lambda _{1}=e^{\\beta J}\\cosh \\beta h+{\\sqrt {e^{2\\beta J}(\\sinh \\beta h)^{2}+e^{-2\\beta J}}}", "1fd630296b1f93f433df34dc7c273ade": "\\sigma _{i0}", "1fd69d9a090ba0001b0e5394959b494f": "a_{m}={\\frac {p_{m}}{N}}\\sum _{n=0}^{N-1}u(x_{n})(-1)^{m}\\cos \\left({\\frac {m\\pi }{N}}(n+{\\frac {1}{2}})\\right)", "1fd6c6fa8a7e525affc2c3317ab3491c": "\\textstyle v(x)=x^{i}a(x)+x^{i+g(2l-1)+r}", "1fd7302c96142ceef8c7ee57cdfaae6b": "\\scriptstyle {\\mathcal {Z}}", "1fd73412b32c4c19c2c03c8a74d7499e": "s_{2}=(1-u_{\\min })^{-{\\frac {1}{a}}}k", "1fd73ffdce10723eb2c8b591234164f5": "{\\frac {\\partial L}{\\partial q_{k}}}=0\\quad \\Rightarrow \\quad {\\frac {d}{dt}}{\\frac {\\partial L}{\\partial {\\dot {q}}_{k}}}=0\\quad \\Rightarrow \\quad {\\frac {dp_{k}}{dt}}=0\\,,", "1fd744d93512f9948b4c5f6d47013971": "{\\mathit {i}}^{2}=(\\mathbf {e_{1}e_{2}e_{3}} )^{2}=\\mathbf {e_{1}e_{2}e_{3}e_{1}e_{2}e_{3}} =\\mathbf {e_{1}e_{2}e_{3}e_{3}e_{1}e_{2}} =\\mathbf {e_{1}e_{2}e_{1}e_{2}} =-1\\ .", "1fd8110c514c213b06482dd214af81e2": "{\\begin{matrix}\\sin 0&=&\\sin 0^{\\circ }&=&{\\sqrt {0}}/2&=&\\cos 90^{\\circ }&=&\\cos \\left({\\frac {\\pi }{2}}\\right)\\\\\\\\\\sin \\left({\\frac {\\pi }{6}}\\right)&=&\\sin 30^{\\circ }&=&{\\sqrt {1}}/2&=&\\cos 60^{\\circ }&=&\\cos \\left({\\frac {\\pi }{3}}\\right)\\\\\\\\\\sin \\left({\\frac {\\pi }{4}}\\right)&=&\\sin 45^{\\circ }&=&{\\sqrt {2}}/2&=&\\cos 45^{\\circ }&=&\\cos \\left({\\frac {\\pi }{4}}\\right)\\\\\\\\\\sin \\left({\\frac {\\pi }{3}}\\right)&=&\\sin 60^{\\circ }&=&{\\sqrt {3}}/2&=&\\cos 30^{\\circ }&=&\\cos \\left({\\frac {\\pi }{6}}\\right)\\\\\\\\\\sin \\left({\\frac {\\pi }{2}}\\right)&=&\\sin 90^{\\circ }&=&{\\sqrt {4}}/2&=&\\cos 0^{\\circ }&=&\\cos 0\\end{matrix}}", "1fd82f5b59dc829eaeda90a9084ebd81": "\\;(m-1)", "1fd8397e614bf2466baea635d1c07767": "(\\mathbf {F} _{2})^{n}", "1fd9103246379056b637fea02a7446bc": "P(X)={Q(X) \\over E(X)}", "1fd91969442deaa96407f8f6fe20185a": "l={\\frac {1}{2}}\\lambda _{d}={\\frac {1}{2}}k\\lambda _{0}={\\frac {1}{2}}k{\\frac {c}{f}}", "1fd9bb5b485ab782ebaa65b3c1aeb01e": "H(x,p;V)=K(p)+\\varphi (x;V)", "1fd9e04643bdb6e5eb06c88fa8a13c0d": "M_{A}{\\text{ and }}M_{B}", "1fda275a1dd0f7d854b5fdb6cbb47944": "0\\leq r0", "1fee71f3df550cd0f911ddda8dc3e5f0": "G=\\epsilon D", "1fee89eb9f6defa9a339bba48af66e89": "y\\sinh R\\,.", "1feed622f38bffb5bafc73abc37ce300": "\\lim _{x\\to 0}{\\frac {1-\\cos x}{x}}=0", "1fef1620c1f394615594661919b65820": "N_{\\cdot }(\\omega )=\\sum \\limits _{i=1}^{Z(\\omega )}\\delta _{X_{i}(\\omega )}(\\cdot )", "1fef4ccc8a592606893fc1d5ac15a50a": "R\\approx 0.17{\\text{ mm}}", "1fef59cc5cb36eb2912ac0b60f9b74ce": "{\\frac {d\\Omega }{dt}}={\\frac {2GS}{c^{2}a^{3}(1-e^{2})^{3/2}}}={\\frac {2G^{2}M^{2}\\chi }{c^{3}a^{3}(1-e^{2})^{3/2}}}", "1fef637cd581bc96ca8b90d9540b65a6": "C_{H_{2}O}=V-C_{osm}", "1fef648eb6b8c5df6940415461b7bd5b": "[\\![[a]\\phi ]\\!]_{i}=\\{s\\in S\\mid \\forall t\\in S,(s,t)\\in R_{a}\\rightarrow t\\in [\\![\\phi ]\\!]_{i}\\}", "1fef666ee59b638703edd5761c1e3234": "d\\leq wt({\\boldsymbol {c'}})", "1fef8ad8da4ddb479acc614014398798": "{\\kappa }a<1", "1fefab0afb5f96cf49a2a297c10ad0af": "\\psi ^{(\\operatorname {Sha} )}(t)=2\\cdot \\operatorname {sinc} (2t-1)-\\operatorname {sinc} (t),", "1fefb51f912faf4d7b5492909002bebd": "\\Delta p_{i}\\,\\Delta q^{i}", "1fefef4e73e239627356fc5754d0e513": "x+y+z", "1ff0149dcb540c9f2aa8d5c2c896830a": "{\\mathbf {\\lambda }}", "1ff0495fb838d78a99b66c4eebd8d063": "(L+R)+(L-R)=2L", "1ff0c7beaca6fe0b96e9aae622d7bb39": "\\langle A\\rangle _{\\sigma }=\\sigma (A)", "1ff0fcba6806d7847a8c41a0de3230b1": "z={\\frac {S}{\\sqrt {\\operatorname {VAR} (S)}}}={\\frac {40}{\\sqrt {185.212}}}=2.939", "1ff1259634a6ac4d0b5a3c1eef269d21": "\\Psi ^{\\dagger }", "1ff12a5acda775c5bab92daa74008547": "h,x\\,", "1ff14003d9a42832730c63e2c0a57762": "c_{\\text{man}}", "1ff18d25b2eb29d2363ede64da7925a0": "{2\\pi }", "1ff1cdb5504d68ae31a9b49260bab342": "A\\models R(a_{1},\\ldots ,a_{n})", "1ff1d97415ecb181916778609427f577": "\\delta {\\boldsymbol {B}}\\ ,", "1ff1de774005f8da13f42943881c655f": "24", "1ff1f5d3d9f35a3ea9919c13026866b0": "\\mathbf {} A", "1ff202b3c9d393e51fabe205889065f6": "E_{\\mathrm {st} }=\\sigma \\int dx\\,dy\\,\\left[{\\sqrt {1+\\left({\\frac {dh}{dx}}\\right)^{2}+\\left({\\frac {dh}{dy}}\\right)^{2}}}-1\\right]\\approx {\\frac {\\sigma }{2}}\\int dx\\,dy\\,\\left[\\left({\\frac {dh}{dx}}\\right)^{2}+\\left({\\frac {dh}{dy}}\\right)^{2}\\right],", "1ff22ff94ade4fbf7b49c3ee44b36ea2": "M_{cycles}", "1ff232ea2e2dcdb4f6d57e7d3bf956ff": "x\\approx y", "1ff255db0229ad6fbc10596d4fa61116": "IC=\\sum {\\frac {f_{i}(f_{i}-1)}{n(n-1)}}", "1ff2ec97b1eef19f42854107729dcffc": "x=x_{1}2^{n-1}+x_{2}2^{n-2}+\\cdots +x_{n}2^{0}.\\quad ", "1ff31fa359f487e13b71f5ac7d32f216": "{\\mathcal {P}}={\\frac {\\Phi }{NI}}", "1ff3538d6e1da90eeb7dc7eb5773060b": "T_{ij}={\\cfrac {1}{2}}(S_{i}^{+}S_{j}^{-}+S_{i}^{-}S_{j}^{+})", "1ff3572ed7b400d7b8d032be1bf0527f": "\\Sigma _{j}p_{j}f(r_{j}),", "1ff37ed118824272399e65808644ec23": "{\\mathfrak {gl}}(V).", "1ff3840c1a8a45266a65ac3c4e7bee6f": "\\sin \\theta =\\left({\\frac {m\\lambda _{0}}{n\\Lambda }}\\right)", "1ff3bf454807c8dc19037254634ba246": "r=r_{0}A^{1/3}\\,\\!", "1ff3f4bb5aa85f4a91c784c1db0346de": "\\ell {\\ddot {\\theta }}-g\\sin \\theta ={\\ddot {x}}\\cos \\theta ", "1ff413f5d8bed6830140433b661c2d43": "\\Delta {\\vec {p}}_{\\mathrm {avg} }=\\langle \\Delta {\\vec {p}}\\rangle _{\\Omega }", "1ff446a62cd2afe18c54b1769177a85f": "Q=({\\sqrt {F+T}}-{\\sqrt {F}})^{2}", "1ff499587e86790b65b0446f871850a5": "{\\text{s.g.}}={\\frac {140}{130+{\\text{degrees Baumé}}}}", "1ff4aa19f042337a731b322c618ae897": "\\Delta t_{i=1}={\\frac {\\Delta S_{i=1}}{\\left({\\frac {\\Delta S}{\\Delta t}}\\right)_{i=1}}}={\\frac {130.5{\\text{ ft}}^{3}}{2.9{\\text{ ft}}^{3}/{\\text{ s}}}}=44.9{\\text{ s}}", "1ff4df1af6cc3ec92ea0790fd0597fb4": "\\Phi (e)\\,\\!", "1ff4e1b0e340bfdb4666f52eefdfce87": "\\left(\\pm {\\frac {(\\varphi +1)}{2}},\\pm {\\frac {1}{2}},0\\right)", "1ff4e7c4ea49e4f89fcea2a90968d87f": "A^{-1}", "1ff501e87fcaa0c9540d2579cc518901": "\\alpha =-k\\cot(kL/2)", "1ff5162360bdfc6fa7e0e8d160a87919": "\\mathbf {R} =\\left(d{\\mathcal {M}}+{\\frac {1-d}{N}}\\mathbf {E} \\right)\\mathbf {R} =:{\\widehat {\\mathcal {M}}}\\mathbf {R} ", "1ff530e35d29a87a18afb6b3e08eb4a8": "{\\mathcal {S}}\\subset {\\mathcal {S}}_{drs}", "1ff60dc22ce0d4fde64f6d5b9f5f0f1b": "S_{xyz}", "1ff6384a373ab9fcaba03f902b643b4a": "f^{(k)}", "1ff63c25d28601b1dc2fefde50759425": "\\psi \\rightarrow \\gamma _{5}\\psi ", "1ff66da1a15e819564d488926c87eb35": "ds^{2}=g_{ij}(q)dq^{i}dq^{j}\\,", "1ff69a9028511a2cbf16f4a10ec64c5b": "b=40", "1ff6a3df32b4eb55ddf93e67f25bd1ed": "r_{443}", "1ff70241875e2a9e46eac25ae203a35c": "\\int \\csc {ax}\\,\\mathrm {d} x=-{\\frac {1}{a}}\\ln {\\left|\\csc {ax}+\\cot {ax}\\right|}+C", "1ff763ea8c3c373f046f13cd9bb77620": "\\sum _{j=1}^{n}{g_{j}y_{j}}+\\sum _{i=1}^{m}{h_{i}s_{i}}", "1ff7703188f686aa337b6b9225a066c5": "N(|fg|)\\leq {\\bigl (}N(|f|^{p}){\\bigr )}^{1/p}{\\bigl (}N(|g|^{q}){\\bigr )}^{1/q}.", "1ff7af9a810bd160a7334c3de425ed9d": "\\nu _{2}", "1ff7b0f5809662e4d8c79d69b6501f74": "{\\mathfrak {P}}^{34}", "1ff7d920c350dc17d36721e9b0afef9b": "{\\mathcal {F}}\\left\\{\\mathbf {x\\cdot y} \\right\\}_{k}\\ {\\stackrel {\\mathrm {def} }{=}}\\sum _{n=0}^{N-1}x_{n}\\cdot y_{n}\\cdot e^{-{\\frac {2\\pi i}{N}}kn}", "1ff8d33a7184bc39ba3e430744d7fccf": "S_{v}={\\begin{bmatrix}1&0&0&0\\\\0&1&0&0\\\\0&0&1&0\\\\0&0&0&{\\frac {1}{s}}\\end{bmatrix}}.", "1ff8db085ecbde5d955cdce9ac55bf54": "S={\\begin{bmatrix}1&1&1\\end{bmatrix}}", "1ff91f1c77c49a8eece699a459554f49": "\\{f_{1}\\circ \\iota ,f_{2}\\circ \\iota \\}=-\\{f_{1},f_{2}\\}\\circ \\iota ", "1ff9d567c2cb356621879748e6d73740": "\\lambda _{r}=r^{-1}{\\tbinom {n}{r}}^{-1}\\sum _{x_{1}<\\cdots 2.", "200f32e7a4295b416a6483b18cfcaa4a": "Re(\\alpha )=Re(\\beta )=p^{11/2}.", "200f3e833bde508a7fc92bc788de32d5": "k/2-1", "200f97de129bc399309c550f6befc5fc": "F_{11}={\\cfrac {1}{X_{\\mathrm {f} }X_{\\mathrm {f} }^{\\prime }}}\\ ,\\ F_{22}=F_{33}={\\cfrac {1}{Y_{\\mathrm {f} }Y_{\\mathrm {f} }^{\\prime }}}", "200fb59701b9bd0a8b763bf6abffd673": "I_{h}={\\frac {1}{12}}m\\left(w^{2}+d^{2}\\right)", "200fe503a74540f60f6c8ef9b63c41be": "S=-{\\frac {V_{left}-V_{right}}{T_{left}-T_{right}}}", "201022e258ff15130972d727002647a3": "\\,\\phi (0,{\\bar {c}})=0~~{\\textrm {for}}~{\\textrm {all}}~{\\bar {c}},", "20102365216dff092a1e49ad64ab1560": "\\theta _{1}={\\begin{bmatrix}0&0&0&0\\\\1&0&0&0\\\\0&0&0&0\\\\0&0&1&0\\\\\\end{bmatrix}}\\qquad \\theta _{2}={\\begin{bmatrix}0&0&0&0\\\\0&0&0&0\\\\1&0&0&0\\\\0&-1&0&0\\\\\\end{bmatrix}}\\qquad \\theta _{1}\\theta _{2}=-\\theta _{2}\\theta _{1}={\\begin{bmatrix}0&0&0&0\\\\0&0&0&0\\\\0&0&0&0\\\\1&0&0&0\\\\\\end{bmatrix}}.", "201033ebe336e21b40e85354b56eb722": "\\mathbf {abcdefghijklm} \\!", "20103d49e434ed6da124ad94f13d10fd": "\\sin(\\theta )=|P|\\psi \\rangle |={\\sqrt {G/N}}", "20106470c24912c405cb864f34729a09": "{\\acute {R}}={\\acute {R}}_{\\alpha }^{\\alpha }=8\\pi {G \\over {c^{4}}}\\left({A \\over 2}{\\acute {T}}_{\\alpha }^{\\alpha }+{B \\over 2}{\\acute {T}}\\delta _{\\alpha }^{\\alpha }\\right)=8\\pi {G \\over {c^{4}}}\\left({A \\over 2}+2B\\right){\\acute {T}}", "20111eed844ef5aaca21baaf6764cec7": "y_{s}(x)=0", "2011d4d37e456bacc31fa7ae7e4d8939": "f={v \\over {\\sqrt {\\|v\\|_{2}^{2}+e^{2}}}}", "2011e57b4da471e895fd0b082258456e": "T(G)={\\frac {3\\delta (G)}{\\tau (G)}}", "20122d059ed1a20a6f75460236c6004b": "r_{q}=y^{((q+1)/4)^{L}}~mod~q", "20123137a0c82b01eda55afa45e2d7aa": "\\omega ={\\frac {d\\theta (t)}{dt}}.\\,", "2012b89ffaa36af607737b3a2fdaac3d": "p(\\eta |y)=\\int p(\\eta |\\theta )p(\\theta |y)\\;d\\theta .", "2012d3be38fe508842908bf182923326": "{\\begin{aligned}{\\mathcal {F}}\\{\\operatorname {tri} (t)\\}&={\\mathcal {F}}\\{\\operatorname {rect} (t)*\\operatorname {rect} (t)\\}\\\\&={\\mathcal {F}}\\{\\operatorname {rect} (t)\\}\\cdot {\\mathcal {F}}\\{\\operatorname {rect} (t)\\}\\\\&={\\mathcal {F}}\\{\\operatorname {rect} (t)\\}^{2}\\\\&=\\mathrm {sinc} ^{2}(f).\\end{aligned}}", "2012db2e145e2ba9a0281677d78333fd": "{\\bar {\\psi }}_{\\rm {D}}", "2013318ff8c29061e4b3f55fb741685a": "Y(s)=e^{-\\int _{t}^{s}V(X_{\\tau })\\,d\\tau }u(X_{s},s)+\\int _{t}^{s}e^{-\\int _{t}^{r}V(X_{\\tau },\\tau )\\,d\\tau }f(X_{r},r)dr", "201333d839f4c5c86404ce011dd177ef": "{\\vec {H}}=-\\nabla U,", "2013375db1e9bcfafec7e8e15a0ab741": "\\left({\\sqrt {1/55}},\\ {\\sqrt {1/45}},\\ 1/6,\\ {\\sqrt {1/28}},\\ {\\sqrt {1/21}},\\ -{\\sqrt {5/3}},\\ 0,\\ 0,\\ 0,\\ 0\\right)", "2013e7f910b3ad7a8ca424c6c595f49f": "C=1/B\\,\\!", "201400d073866463c42632818d600ec1": "K_{i}", "20140cad4d60856d0523dc485a53491a": "\\Omega +x", "20146a71790cb22bf57a9dc7a6635d42": "{\\mathbf {u}}\\in U", "20147c33cd3f60bf25a075a94a30e831": "\\forall b\\in B", "2014ae35f262614f8c25f371f442733b": "{\\big (}\\sigma _{H}(\\omega ){\\big )}^{2}=e^{\\pm i\\pi }=-1\\qquad {\\text{for }}\\omega \\neq 0", "2014fdc1cbe9447487bdf23d005fd891": "x^{4}+ax^{3}+bx^{2}+cx+d=0,", "20155d866107391be25917b0e0f51c03": "V_{0}\\subset L^{2}(\\mathbb {R} )", "20156b59cc794e6fffaecbdafdca48ce": "G(z)=1+\\sum _{n\\geq 1}\\left({\\frac {1}{|S_{n}|}}\\right)g(z)^{n}=\\sum _{n\\geq 0}{\\frac {g(z)^{n}}{n!}}=\\exp g(z).", "20158603ea6b9c587746cb1690128eac": "\\sum _{y}K_{x\\rightarrow y}=1\\,", "20158a4ffa5d497f0bc3ae9c02e50af6": "{\\begin{matrix}{5 \\choose 3}=10\\end{matrix}}", "2015cf0fc2bf47fdce8ecba5e5d0dc4e": "0\\to W\\to V\\to V/W\\to 0", "2015d9bf0c6335d7be587d7aaf11c963": "a_{t}", "2015e468b5d574c5d4461b86714c6e15": "\\left(D^{2}+{\\frac {b}{m}}D+\\omega _{0}^{2}\\right)y=0,", "2016356256defe42c3cc28fb8395f1f9": "{z}\\geq {0}\\,", "20168cc271fd1cc842852737e2eb3c4c": "d=\\partial +{\\bar {\\partial }}", "2016a704d85bb3fd22de1c1e8c84c932": "D_{\\Gamma }:J^{1}Y\\to _{Y}T^{*}X\\otimes _{Y}VY,\\qquad D_{\\Gamma }=(y_{\\lambda }^{i}-\\Gamma _{\\lambda }^{i})dx^{\\lambda }\\otimes \\partial _{i},", "2016c67940ecaf7dd07f0a9ff50cdba7": "m(\\varphi )=a{\\big (}1-e^{2}{\\big )}\\,\\Pi (\\varphi ,e^{2},e).", "2016dcded84431e5dcffe74b504077f7": "\\displaystyle {|T_{r}f|\\leq f^{*}.}", "201758c0875f4fd664aaff970658e00f": "{\\mathcal {L}}_{2}^{2}", "20178e73e38b59447c1df6e7e591191b": "{\\mathcal {O}}_{X,x}", "2017e5f52020adfe2439467b8edd154e": "|\\phi |_{p}=(\\int |\\phi |^{p})^{1/p}<\\infty ", "2017ea3a8c7580ee97b5c3b4d613d343": "A\\rtimes M", "20184613b9b2e6b97ec1f92477b708ea": "s(\\theta _{i})", "20188c84a6d5f200f682b89222904584": "V_{An}", "201897cfea5e1475b846b4b43c71d40b": "{\\ \\atop {\\ }}{{\\underbrace {a^{a^{\\cdot ^{\\cdot ^{a}}}}} } \\atop n}", "2018aa53f275c90824a0645ec65c0d67": "S_{\\mathrm {FN} }=\\;-b{\\phi }^{3/2}/\\beta ...........(47)", "2018d4c3d1d6bf445544371983da3319": "H_{2}(\\mathrm {A} _{n},\\mathbf {Z} )=\\mathbf {Z} /2", "2018ee47b429e6afa4a5aa175dde10c5": "U\\ ", "20196b4cb80ce92f9bfb43f9512c85c6": "\\delta ={\\frac {\\sigma _{\\varepsilon }^{2}}{\\sigma _{\\eta }^{2}}}.", "20197ba1d2398678eac31c9b6140e8ea": "\\mathbf {u} _{n}(s)=\\left[y'(s),\\ -x'(s)\\right]\\ ,", "2019ff2e43966b2b4e29ce21465ce09f": "A_{4}={\\frac {\\Gamma (-1+{\\frac {1}{2}}(k_{1}+k_{2})^{2})\\Gamma (-1+{\\frac {1}{2}}(k_{2}+k_{3})^{2})}{\\Gamma (-2+{\\frac {1}{2}}((k_{1}+k_{2})^{2}+(k_{2}+k_{3})^{2}))}}", "201a3b8a9fbfecf4786a2fbe17a0b453": "Q_{i}={\\mathbf {1}}'\\otimes \\dots \\otimes {\\mathbf {1}}'\\otimes Q\\otimes {\\mathbf {1}}\\otimes \\dots \\otimes {\\mathbf {1}}", "201a43085a9f350e2ab654ba5b1c6296": "\\vartriangle \\!\\,", "201a95ae50fc9c25d2a745afedc7c44d": "q^{-1}+20q-62q^{3}+\\dots ", "201a977cc7cc9c097ccacb28d3a5423f": "[\\mathrm {AB} ]^{\\ddagger }=K^{\\ddagger \\ominus }[\\mathrm {A} ][\\mathrm {B} ]", "201ab2acc2532c51366486edb54cb680": "=\\iiint _{V}\\left(\\psi \\nabla ^{2}\\varphi -\\varphi \\nabla ^{2}\\psi \\right)dV\\,\\!", "201ae8beaa5802350b0b9b40d154d4c3": "\\scriptstyle (2\\pi )^{-1}", "201b8de1971547855998ee521b5eab17": "u(w)=\\log(w)", "201bc2e3922b02dc235b91b60c756a0f": "\\theta _{i}\\theta _{j}=-\\theta _{j}\\theta _{i}\\qquad \\theta _{i}x=x\\theta _{i}.", "201be10430645ea2b9dd61d8778a75b6": "f(k)={\\boldsymbol {\\tau }}{T}^{k-1}\\mathbf {T^{0}} ,", "201c1fde84fd061bb99a5884a945a854": "8\\,{\\bmod {\\,}}5=3", "201c3c93f5d5ea6871ff56b38c1b214a": "F(x)=\\sum _{i=1}^{k}G_{i}(x)'G_{i}(x).", "201c5344a19c395a763e135ef02f2408": "{\\text{Mean time between failures}}={\\text{MTBF}}={\\frac {\\sum {({\\text{start of downtime}}-{\\text{start of uptime}})}}{\\text{number of failures}}}.\\!", "201c7e241ef26d91eca7d251c1a792e9": "\\|\\mathbf {v} \\|\\geq 0", "201c85f01b16ec4217870763f4256d6d": "\\scriptstyle {\\hat {v}}", "201c9062d348ec6953a2500e2ece47c8": "z\\in I^{*}", "201ca9a59e64c984b4e8eee84d9e69aa": "{\\vec {e}}_{0}=\\partial _{t},\\;{\\vec {e}}_{1}=f(r)\\,\\partial _{z},\\;{\\vec {e}}_{2}=f(r)\\,\\partial _{r},\\;{\\vec {e}}_{3}={\\frac {1}{r}}\\,\\partial _{\\phi }-h(r)\\,\\partial _{t}", "201d2179e81e6eabf33675ccb4b71768": "\\textstyle a_{\\diamond }", "201d31d0d9f83384accff0b4e0d69ca1": "Oxy\\rightarrow Cxy.", "201d423d29f6ec052d70ed3cf7782bbe": "r_{xy}={\\frac {\\sum \\limits _{i=1}^{n}(x_{i}-{\\bar {x}})(y_{i}-{\\bar {y}})}{(n-1)s_{x}s_{y}}}={\\frac {\\sum \\limits _{i=1}^{n}(x_{i}-{\\bar {x}})(y_{i}-{\\bar {y}})}{\\sqrt {\\sum \\limits _{i=1}^{n}(x_{i}-{\\bar {x}})^{2}\\sum \\limits _{i=1}^{n}(y_{i}-{\\bar {y}})^{2}}}},", "201d544169dfc667c3e1215c648949b4": "\\mu _{\\infty }", "201d9f9cca290b4fafd5c4b1e30cc642": "\\Psi :L(H_{B})\\rightarrow L(H_{A})", "201db00595dbe04b7342f371d3b59af6": "{\\lambda }_{L}={\\frac {1}{3V}}\\sum _{q,j}v\\left(q,j\\right)\\Lambda \\left(q,j\\right){\\frac {\\partial }{\\partial T}}\\epsilon \\left(\\omega \\left(q,j\\right),T\\right),", "201dbb2fde10937103359420fe8d8b84": "{\\begin{array}{|rlr|rlr|rlr|}\\hline \\alpha &\\mathrm {alpha} &1&\\iota &\\mathrm {iota} &10&\\varrho &\\mathrm {rho} &100\\\\\\beta &\\mathrm {beta} &2&\\kappa &\\mathrm {kappa} &20&&&\\\\\\gamma &\\mathrm {gamma} &3&\\lambda &\\mathrm {lambda} &30&&&\\\\\\delta &\\mathrm {delta} &4&\\mu &\\mathrm {mu} &40&&&\\\\\\varepsilon &\\mathrm {epsilon} &5&\\nu &\\mathrm {nu} &50&&&\\\\\\mbox{\\stigma} &\\mathrm {stigma\\ (archaic)} &6&\\xi &\\mathrm {xi} &60&&&\\\\\\zeta &\\mathrm {zeta} &7&o&\\mathrm {omicron} &70&&&\\\\\\eta &\\mathrm {eta} &8&\\pi &\\mathrm {pi} &80&&&\\\\\\vartheta &\\mathrm {theta} &9&\\mbox{\\koppa} &\\mathrm {koppa\\ (archaic)} &90&&&\\\\\\hline \\end{array}}", "201dbc14304a12449ea74847bb74f4cc": "\\mu _{A}(x)=1", "201df202f2c20ac904dd728e83149379": "f_{P}=2u_{P}+t_{P}+\\delta _{P},\\,", "201df2c079672eecd6fb5f7bbc470ce8": "Q_{j}={\\frac {\\mathrm {d} }{\\mathrm {d} t}}\\left({\\frac {\\partial T}{\\partial {\\dot {q}}_{j}}}\\right)-{\\frac {\\partial T}{\\partial q_{j}}}=-{\\frac {\\delta T}{\\delta q_{j}}}=\\sum _{i=1}^{n}\\mathbf {F} _{i}\\cdot {\\frac {\\partial \\mathbf {r} _{i}}{\\partial q_{j}}},", "201e449b5c0ad4f72e4c75be4e871ba9": "\\sigma _{Age=Weight}(Person)", "201e4c97bb7492d7a1486234cf3b7951": "(xy\\equiv zu\\land xy\\equiv vw)\\rightarrow zu\\equiv vw.", "201e5d97363081be8d7d9ba3950979b7": "\\scriptstyle P(G,t)=P(G_{1},t)P(G_{2},t)\\cdots P(G_{c},t)", "201ec3d1014d1b23827e94d1f12f1628": "\\cos(a)\\sin(b)+\\cos(b)\\sin(a)=\\sin(a+b)", "201edf47f9bc92b0d80b898f3a0f3325": "y_{I}=B\\left({1-e^{-\\tau }}\\right)=B\\left({1-e^{-t/\\epsilon }}\\right).\\,", "201fafdb2155f6d76c5c1b07165ea248": "H(0)=PDQ^{T}", "201fef0a1b6e41c52bbb3f413281698c": "{\\boldsymbol {\\gamma }}_{A}=n", "20206a1cf0576e5a53dcf98e1b224f4e": "\\|f(x+h)-f(x)\\|=\\left\\|\\int _{0}^{1}(Df(x+th)\\cdot h)\\,dt\\right\\|\\leq \\int _{0}^{1}\\|Df(x+th)\\|\\cdot \\|h\\|\\,dt\\leq M\\|h\\|.", "2020fc9835e7d2c3452e01c81324cc88": "\\mu -{\\frac {1}{2q}}\\operatorname {E} |X-\\mu |\\leq x_{q}\\leq \\mu +{\\frac {1}{(2-2q)}}\\operatorname {E} |X-\\mu |", "202132bc21a6efe9877e3c1b66c1c4c2": "L(u)=(a_{1},a_{2})", "20213a5d9d8c020f03389a3f258f6dce": "n(n+1)/2", "20213ccbbfc03eec6a9af8f5f8a7eacc": "S_{B}(1)>S_{L}(1)", "2021807fc420a850ee0c8c970872304c": "{\\overline {\\rho _{f}}}\\cong \\rho ", "2021be6537e8cc3008de7f024495dbcd": "{\\mathfrak {H}}\\,", "2021c518b2aeba3432e062299defff7f": "x(w).w(y_{1}).\\cdots .w(y_{n}).[P]", "2021ecb257f4e84d3a7f7ae10302330f": "{\\frac {a+b^{n}}{n}}=x", "2022087cbb38390ede0e909a2d83d004": "\\qquad \\qquad S_{\\mathrm {vib} }=-{\\frac {\\partial F_{\\mathrm {mix} }}{\\partial T}}=3Nk_{\\mathrm {B} }T\\int _{0}^{\\omega }\\{{\\frac {\\hbar \\omega }{2k_{\\mathrm {B} }T}}\\mathrm {coth} ({\\frac {\\hbar \\omega }{2k_{\\mathrm {B} }T}})-\\mathrm {ln} [2\\mathrm {sinh} ({\\frac {\\hbar \\omega }{2k_{\\mathrm {B} }T}})]\\}D_{p}(\\omega )d\\omega ,", "20221eb0dfd040cecfb1adf15012b96a": "F_{X}", "2022b5245c9db75248b3e6d92b875b8e": "P_{ij}={\\frac {\\partial V_{i}}{\\partial Q_{j}}}", "2023780fd7b69a828d64e91f54dba30b": "R_{WM}", "20237af8291aa5af64eaca03645a2b81": "\\kappa (P)=\\lim _{Q\\to P}{\\sqrt {\\frac {24\\left(s(P,Q)-d(P,Q)\\right)}{s(P,Q)^{3}}}}", "2023b8707ce7b23a12d7c93feb59c3a8": "\\Delta i\\ =\\ -2\\pi \\ {\\frac {J_{3}}{\\mu \\ p^{3}}}\\ {\\frac {3}{2}}\\ \\cos i\\ e_{g}\\ (1-{\\frac {5}{4}}\\ \\sin ^{2}i)", "2023d76b1604e1566a570f76915d16a6": "E_{2}={\\frac {I}{I_{max}}}", "202410d9b87c4632cc661c7040dfcbf0": "_{ordinal}\\alpha \\geq \\rho ", "20247f49f235be7930c80fa6ff8dccb7": "F'(x)=f(x).\\ ", "202481cace524e514625905a6bd4c310": "~n", "2024877e575c02cce04cd613b0493aa9": "Naturalincrease_{t}=Births_{t}-Deaths_{t}", "2024a6c2fed6ca688781b0e72a19b8c3": "\\lVert L_{n}\\rVert _{\\infty }={\\overline {\\Lambda }}_{n}(T)=\\max _{-1\\leq x\\leq 1}\\lambda _{n}(T;x),", "2024cb82c08f7f013d91d0045d60d531": "\\mu \\in {\\mathbb {R} }^{d}", "202512f55d8eb4bd14e2216cc032fc34": "{\\mathcal {B}}=\\{U_{1}\\times \\cdots \\times U_{n}\\ |\\ U_{i}\\ \\mathrm {open\\ in} \\ X_{i}\\}.", "20254ab2405c961827d66348819dfdea": "\\Gamma =KK^{T}", "202568a675719177e4ea5ca5dc10ab24": "{\\begin{aligned}N&=q^{\\left\\lfloor {\\frac {n}{2}}\\right\\rfloor }A_{n}\\left({\\frac {\\pi }{2}}\\right)\\\\&=q^{\\left\\lfloor {\\frac {n}{2}}\\right\\rfloor }{\\frac {\\left({\\frac {p}{q}}\\right)^{n+{\\frac {1}{2}}}}{2^{n}n!}}\\int _{0}^{1}(1-z^{2})\\cos \\left({\\frac {\\pi }{2}}z\\right)\\,dz.\\end{aligned}}", "20257d7bd7864e94ecd986c9d56fc5b8": "{\\tbinom {n}{c}}", "202590a2bb0742584ca0c87d0d1b034a": "a_{2}=\\sum x_{i}x_{j}={\\frac {t_{1}^{2}-t_{2}}{2}}\\qquad {\\text{ where }}ib\\end{cases}}", "20364188d7f21997427137fea2884c4a": "M(t)={\\frac {2}{\\pi }}\\int _{0}^{\\pi }e^{Rt\\cos(\\theta )}\\sin ^{2}(\\theta )\\,d\\theta ", "20364c0769e8d234355c9c55028ca7eb": "b_{i}={\\frac {a_{n-1}\\times {a_{n-2i}}-a_{n}\\times {a_{n-2i-1}}}{a_{n-1}}}.", "203672b557ed05d928d335b344dfcc53": "(N,M,d,\\gamma ,\\alpha )\\,", "203691d3dbce41e8b3844a9f6c11ada6": "f(x)={\\frac {1}{2}}\\int _{\\mathbb {R} }\\left({\\frac {x-t}{|x-t|}}+{\\frac {t}{|t|}}\\right)d\\mu (t).", "2036a808ee21bbcf2c59ee19ef445c5f": "T_{a}^{a}=0", "2036e0c98c776a5749c1fc8710897765": "x\\in \\mathbb {R} ^{+}", "203747583974154343acc18e4c2344af": "{\\begin{aligned}P_{0}(s)&=-\\alpha \\beta ,\\\\P_{1}(s)&=(2-\\gamma )s^{2}+(\\alpha +\\beta -1)s,\\\\P_{2}(s)&=s^{3}-s^{2}.\\end{aligned}}", "20376defb6278d75a6b8164aa0678f36": "S_{C}-\\ ", "2037d7c874e96e2590141ddfa40006dd": "P(R_{NP},\\theta _{1})\\geq P(R_{A},\\theta _{1})\\iff P(R_{NP}\\cap R_{A}^{c},\\theta _{1})\\geq P(R_{NP}^{c}\\cap R_{A},\\theta _{1}).", "2037e6d50299db3ae33cc7163565a0f2": "\\Phi (\\mathbf {r} )={\\frac {1}{4\\pi \\varepsilon }}\\sum _{l=0}^{\\infty }\\sum _{m=-l}^{l}Q_{1lm}\\left({\\frac {1}{r^{l+1}}}\\right){\\sqrt {\\frac {4\\pi }{2l+1}}}Y_{lm}(\\theta ,\\phi )", "2037e9d9cd494a1c0f1d41b0ad55d3d6": "n_{T}=n_{1}+n_{2}+\\cdots ", "20380359266cad80d4ffc3471f9aac31": "K_{n}(R)\\oplus K_{n-1}(R)\\hookrightarrow K_{n}(R)\\oplus K_{n-1}(R)\\oplus NK_{n}(R)\\oplus NK_{n}(R)", "2038417a58ce7c85ce4d285f97276f2b": "E(x,y,z,t)=\\psi (x,y,z)exp(-j\\omega t)", "203882e7318bff85cd4be8de76e9c597": "m_{r}\\rightarrow m", "20388f2486a5e230651bd183756ca3cf": "\\scriptstyle \\alpha =\\alpha _{j}.\\,", "2038b6513d55bf09ecdc4fee846133fb": "9x+1={\\frac {9n^{2}+9n+2}{2}}={(3n+2) \\choose 2}", "2038b79d7cd36c70ce0391dac4eb5d75": "\\mathbf {z} _{i}=\\mathbf {x} _{i}^{k}=V_{k}^{T}\\mathbf {x} _{i}\\rightarrow ", "2038dc3228f50bbaa92add3ad546ab6a": "\\mathrm {colim} F=\\mathrm {Lan} _{E}F", "2038dfb91e03138996b554d71d12f3ca": "{\\Big [}1+\\int d^{3}x\\lambda ^{j}(x){\\hat {G}}_{j}{\\Big ]}\\Psi (A)=\\Psi [A+D\\lambda ]=\\Psi [A],", "20393c3c1b996c28b22be809f0c6174f": "I(y)=I_{0}\\exp(-\\mu y)\\,", "20393ca9ddc076dcff0304ea82d86d71": "AFC={\\frac {FC}{Q}}.", "20399d4c636d9f981f8aa5514893fe58": "\\mu ={\\sqrt {2+{\\sqrt {2}}}}", "2039ed44e4cb974ccd148ac7a0217ed3": "-\\Delta f=\\lambda f.", "203a010a29f65f259129ceace16bf7c4": "P(k,k')={\\frac {2\\pi }{\\hbar }}|\\langle \\ k'|H_{1}|k\\rangle |^{2}\\delta (E_{k'}-E_{k})", "203a2ddbdfd61eb9b495bd814bbc1e9e": "\\cos(2i+1){\\frac {\\pi y}{2}}", "203a5514c05960964289d7ed500356a9": "(g,gs),", "203ad18d956014e6b38e110b924d9d4e": "2\\pi rh\\,\\!", "203af86679d37b77967d2f6b69eca672": "S\\,\\cap \\,U=T\\,\\cap \\,U.", "203b72e01aead089a31554ab7f19520e": "K,", "203bc9f691ad42bb8a61de285930a8ec": "\\mathrm {Re} _{D}\\,\\!", "203be82fb8a0190c92a25c7d25e66956": "\\mu _{1}\\neq 0", "203c0af8a744562d4fbd246a0c51abfc": "y_{it}=X_{it}\\mathbf {\\beta } +\\alpha _{i}+u_{it}", "203c6c0ed52fc22493566593542f1b10": "\\nabla ^{2}\\phi (x)=0", "203c9d98329c8d8813a748ed7a6ccf51": "m=[P_{1}\\to Q_{1},\\ldots ,P_{r}\\to Q_{r}].", "203cd67e50bbe55a394401476ba32ad0": "{\\bar {T}}_{\\text{fixed}}={\\frac {-4N_{e}(1-p)\\ln(1-p)}{p}}", "203d1c981d2f1a9e483ecc4670f963cc": "\\int {\\frac {dx}{x^{2}-a^{2}}}", "203d3156a6ab6b2e0cb9434710695dbf": "nK_{X'}", "203d42e4ae0cdda20ed9898e0f46ecb3": "\\tau =1/ar", "203d4a8f1e080479c8212983c98b4eaa": "s^{2}=\\sigma ^{2}+\\nu ^{2}", "203d50c3bd0305720f25937edb8333ed": "y_{k}=\\Delta \\cdot \\left(k+{\\tfrac {1}{2}}\\right)", "203d592ad1127f6fafd9086e8f8048e5": "V_{nm}(x,y)=R_{nm}(x,y)e^{jm\\arctan({\\frac {y}{x}})}", "203da0c9ff18113bb2d218578f6d208e": "pK_{w}=-\\log([\\mathrm {H} ^{+}][\\mathrm {OH} ^{-}])", "203dbf08881b4384d10c4d75caa9c949": "\\cos(wz)+i\\sin(wz){\\text{ is one value of }}\\left(\\cos z+i\\sin z\\right)^{w}.\\,", "203dceaa71c9b022d3a263e5323a9202": "E_{1}=E_{2}=5.06ft", "203df8fbc04f32d79455b4dfede33d96": "{\\begin{aligned}\\epsilon (f)&={\\Big [}1-G(f)H(f){\\Big ]}{\\Big [}1-G(f)H(f){\\Big ]}^{*}\\,\\mathbb {E} |X(f)|^{2}\\\\&{}-{\\Big [}1-G(f)H(f){\\Big ]}G^{*}(f)\\,\\mathbb {E} {\\Big \\{}X(f)V^{*}(f){\\Big \\}}\\\\&{}-G(f){\\Big [}1-G(f)H(f){\\Big ]}^{*}\\,\\mathbb {E} {\\Big \\{}V(f)X^{*}(f){\\Big \\}}\\\\&{}+G(f)G^{*}(f)\\,\\mathbb {E} |V(f)|^{2}\\end{aligned}}", "203e17ba75294f65b427c448cd80dac9": "\\left|x-{\\frac {p}{q}}\\right|={\\frac {|cq-dp|}{dq}}\\geq {\\frac {1}{dq}}", "203e512d62eb42752be4d5705f8b183f": "S^{n}\\Rightarrow _{f}...\\Rightarrow _{f}A^{2n}\\Rightarrow {g}A^{2n}\\Rightarrow {h}...\\Rightarrow {h}S^{2n}\\Rightarrow {k}S^{2n}", "203e5226dbc89b1e10a193132149d1d2": "\\mathbf {j} _{1}\\cdot \\mathbf {S} _{1}=\\mathbf {j} _{2}\\cdot \\mathbf {S} _{2}", "203ebca743b5d98f495a7486f5d8f578": "x+0=x\\land x\\cdot 0=0", "203ec63042dd375a3b29ea01a9c78c78": "{\\tfrac {D\\rho }{Dt}}={\\tfrac {\\partial \\rho }{\\partial t}}+\\mathbf {u} \\cdot \\nabla \\rho =0", "203ecab60dda5c2f66a4f4d1929e3784": "[M+H]^{+}\\,", "203ef25b3c7ebcf377cf9efa0daecd75": "\\scriptstyle V\\times V\\times V\\to \\mathbf {R} ,", "203f25bf91b5f66a9445f085ca5470e0": "{\\frac {|AB|}{|BD|}}={\\frac {|AC|}{|DC|}}", "203f50efaf747583be4a2fe1573fb274": "i\\in W", "203f5a19123d1fc58ec1322c177738d6": "f\\colon X\\to Y", "203fe3f061619ecd7f4145314a1294e9": "{\\mbox{Current ratio}}={\\frac {\\mbox{Current Assets}}{\\mbox{Current Liabilities}}}", "203ff16ffef753b59319494d8d37d701": "{\\hat {x}}_{2}=x_{2}(1+\\delta _{2})", "2040284354875e6ee18b159421143ff7": "{\\begin{array}{rrrrl}&&4B_{0}&&=1\\\\2A_{0}&&&+2B_{1}&=0\\\\-4A_{0}&&&&=0\\\\&-2A_{1}&+2B_{0}&&=0\\\\\\end{array}}", "204059374481b7692bcc7df238f2a9ff": "\\varphi _{\\ast }(v)=w", "2040a4632a008550f80d79fa9d6e745e": "z^{2}{\\frac {d^{2}y}{dz^{2}}}+z{\\frac {dy}{dz}}+(z^{2}-\\nu ^{2})y=z^{\\mu +1}.", "2040ae2a83e7d05572da70dfd15d1641": "I_{{(Q)}_{[\\epsilon ]}}\\varpropto \\epsilon ^{\\tau _{(Q)}}", "2040bb4b583f25fcec49f290f50fc306": "\\mathbf {p} _{k+1}:=\\mathbf {z} _{k+1}+\\beta _{k}\\mathbf {p} _{k}", "2040cef9f1ee0e832073b7d2d8f74af2": "\\Psi =24\\pi ^{3}{\\frac {a^{2}}{\\tau ^{2}c^{2}(1-\\epsilon ^{2})}}", "2040f215e04ca3a38f2415924a2ef7cb": "\\Box \\phi -{\\frac {\\partial }{\\partial t}}\\left({\\frac {1}{c^{2}}}{\\frac {\\partial \\phi }{\\partial t}}+\\nabla \\cdot \\mathbf {A} \\right)=-\\left({\\frac {mc}{\\hbar }}\\right)^{2}\\phi \\!", "204207c8d7d1f183d2f17e098b467a8f": "H=H_{0}+\\sum _{\\mathbf {k} \\sigma }E_{\\mathbf {k} \\sigma }\\gamma _{\\mathbf {k} \\sigma }^{\\dagger }\\gamma _{\\mathbf {k} \\sigma }", "2042c57a8594fd481a86ed09150e7940": "a(i,k)\\leftarrow \\min \\left(0,r(k,k)+\\sum _{i'\\not \\in \\{i,k\\}}\\max(0,r(i',k))\\right)", "2042cb759e460d4f7e1a7b64636695e1": "|G:H|={\\frac {|G|}{|H|}}", "2042dc926cdcb316e1ca51acac530ff2": "k^{3}+k^{2}+k-1=0", "20430349f06c24e96da01a2d87d4e1eb": "\\quad \\min \\limits _{D,X}\\{\\|Y-DX\\|_{F}^{2}\\}\\qquad {\\text{subject to }}\\quad \\forall i,\\|x_{i}\\|_{0}=1.", "20430f2a60485445bbe717043289b53d": "\\Delta (y,E(m))", "2043ae3195102845d4282184b4ce309d": "(x_{1},x_{2},\\ldots ,x_{n}):{\\mathsf {T}}_{1}\\times {\\mathsf {T}}_{2}\\times \\ldots \\times {\\mathsf {T}}_{n}", "2043c650fe9b7badf7a84526c59003ad": "f_{0}={\\frac {v_{p}}{p}}", "204461f3a954991bd6a23e6cbca5bc3f": "d^{O(n^{2})}", "2044be5b4190df0f3a5329d67cd81b2d": "C_{q}", "2044ef98cee1ccb5a305af5666b15315": "\\lambda (L(B))\\geq \\gamma (n).d", "20451f27f52a9eeecfa03f72097c964e": "\\int \\cdots \\int _{T}\\;f(x_{1},x_{2},\\ldots ,x_{n})\\;dx_{1}\\!\\cdots dx_{n}", "204525c222ecda99e85600541180aa96": "n_{r}=4", "204561e6ba8a7e1b6f6ba708e381f6a6": "\\Vert Tf\\Vert _{L^{2}}\\leq {\\sqrt {\\alpha \\beta }}\\Vert f\\Vert _{L^{2}}", "20457b2d9f82e369552c95dd95a02f2b": "K=K_{c}\\cdot [H_{2}O]\\,", "20458d655602f1000157cdccaa8cf6c2": "T\\wedge ", "2045ae15f2dd4be5335c1a311bf8133e": "\\zeta (1+iy)=0", "2045f467a8140abc9cbc9aee0ef38b3c": "\\lim _{x\\to c}{\\frac {f(x)}{g(x)}}=L.", "2045fac6dfd26e6847d378cfcc725653": "I(v)", "2046329e7f7efc6218449be7d6b228cd": "\\mathbf {w} (\\mathbf {X} _{A})", "2046418f021f57a61b365b767031423d": "\\ \\ ", "20468fd7ea8d02f9e6325ee347fc593b": "\\mathbb {C} \\setminus \\{0\\}", "20474e637aea89a205b98de37327ce55": "{\\bar {3}}", "20477a5c69344d9880d5646859d2e942": "z=a\\ \\sigma \\ \\tau ", "20478c9ffc87050bb7c3ef6777c5231b": "\\textstyle {\\frac {1}{2}}", "2047cd0dae5279273852c51490304e2b": "P_{avg}=I_{rms}V_{rms}\\cos \\phi =I_{rms}^{2}\\operatorname {Re} (Z)=V_{rms}^{2}\\operatorname {Re} (Y^{*})", "2047ead45ca6f21887ab2103559ee6cd": "\\ddots \\,\\!", "2048368ed761bd149a142bada7b69b4c": "\\iiint _{T}\\rho ^{4}\\sin \\theta \\,d\\rho \\,d\\theta \\,d\\phi =\\int _{0}^{\\pi }\\sin \\phi \\,d\\phi \\int _{0}^{4}\\rho ^{4}d\\rho \\int _{0}^{2\\pi }d\\theta =2\\pi \\int _{0}^{\\pi }\\sin \\phi \\left[{\\frac {\\rho ^{5}}{5}}\\right]_{0}^{4}\\,d\\phi =2\\pi \\left[{\\frac {\\rho ^{5}}{5}}\\right]_{0}^{4}\\left[-\\cos \\phi \\right]_{0}^{\\pi }={\\frac {4096\\pi }{5}}.", "20483ccf1d525f4e7244f45e44beff88": "C\\ell ^{\\,i}(V,Q)C\\ell ^{\\,j}(V,Q)=C\\ell ^{\\,i+j}(V,Q)", "20487f3fc4870604823b57d84585d521": "(a+b\\,x)^{m}(c+d\\,x)^{n}(e+f\\,x)^{p}", "2048f874b7f49cc705aa00f16fd02a5b": "y=y_{0}+u(y_{1}-y_{0})=y_{0}+u\\Delta y\\,\\!", "2049d730c41156679db0639b8996a25d": "{\\frac {\\sum _{i=1}^{n}p_{i}f(x_{i})}{\\sum _{i=1}^{n}p_{i}}}-f\\left({\\frac {\\sum _{i=1}^{n}p_{i}x_{i}}{\\sum _{i=1}^{n}p_{i}}}\\right)\\leq {\\frac {\\sum _{i=1}^{n}p_{i}f(2a-x_{i})}{\\sum _{i=1}^{n}p_{i}}}-f\\left({\\frac {\\sum _{i=1}^{n}p_{i}(2a-x_{i})}{\\sum _{i=1}^{n}p_{i}}}\\right).", "2049f39b58752335ac7596cd4103ac5e": "\\langle [u],[v],[w]\\rangle =\\{[{\\bar {s}}w+{\\bar {u}}t]\\mid ds={\\bar {u}}v,dt={\\bar {v}}w\\}.", "2049fa62070c05f0282a98e6bae5511c": "0\\leq \\lambda _{0}\\leq \\lambda _{1}\\leq \\cdots \\rightarrow \\infty .", "204a117c43be2990f56f146e6e36a10b": "{\\tilde {\\mathbf {x} }}_{0}=(0,0,0)", "204a2ecdcd5df7e5397090158d38688b": "{\\boldsymbol {r_{j}}}={\\frac {1}{m_{0j}}}\\sum _{k=1}^{j}m_{k}{\\boldsymbol {x_{k}}}\\ -\\ {\\boldsymbol {x_{j+1}}}\\ ,", "204aa5563c205da00017aed8ee13bff3": "q_{0}", "204ab3355e47207871f7e205000a5b8b": "y_{k+1}", "204abe0fe4062fc3f086e6e2bfe6e8c0": "\\mathrm {\\frac {1\\,statvolt}{1\\,abvolt}} =\\mathrm {\\frac {1\\,stattesla}{1\\,gauss}} =c", "204ad303624cc8d3a95b287fdb591759": "s_{i+128}=s_{i}+s_{i+7}+s_{i+38}+s_{i+70}+s_{i+81}+s_{i+96}", "204ae41c9f2d97a460fe17fb5ea3bd30": "\\alpha _{s}=g^{2}/4\\pi ", "204af65a59a3302947657da139d1d531": "U>0", "204b4a555f494b07495342b6dd29439e": "{\\mathcal {D}}.", "204baab08e71d8a8906d6189187d6974": "RD'={\\sqrt {\\left({\\frac {1}{RD^{2}}}+{\\frac {1}{d^{2}}}\\right)^{-1}}}", "204bce9220cd8af6db2d5a1b2d0eb9be": "x>2.5", "204c67cda0406debb25bfbf373a76395": "1+2+3+...+34+35+36=666", "204c9aaba2398fcf04c876537c08cb07": "[-2U,2U]", "204d3d55e29ce977037d3582d193a451": "D_{F_{1}+\\lambda F_{2}}^{q}(p,q)=D_{F_{1}}^{q}(p,q)+\\lambda D_{F_{2}}^{q}(p,q)", "204d66232acb3b331a33ed086427ae11": "p=d+1", "204d752083b4f8cf5e72f022da431f2a": "M_{y}^{\\alpha \\beta \\mu }(x)=M_{0}^{\\alpha \\beta \\mu }(x)+y^{\\alpha }T^{\\beta \\mu }(x)-y^{\\beta }T^{\\alpha \\mu }(x)\\,,", "204dba7f4676afc3860775970826a607": "\\mathrm {af} (n)=\\sum _{i=1}^{n}(-1)^{n-i}i!", "204dd132ab2f7058d6f5d4c2db308d76": "{\\frac {1}{2m}}\\left({\\frac {\\mathrm {d} S_{z}}{\\mathrm {d} z}}\\right)^{2}+U_{z}(z)+{\\frac {1}{2m\\left(\\sigma ^{2}+\\tau ^{2}\\right)}}\\left[\\left({\\frac {\\mathrm {d} S_{\\sigma }}{\\mathrm {d} \\sigma }}\\right)^{2}+\\left({\\frac {\\mathrm {d} S_{\\tau }}{\\mathrm {d} \\tau }}\\right)^{2}+2mU_{\\sigma }(\\sigma )+2mU_{\\tau }(\\tau )\\right]=E.", "204e0c6532cd435b9ed7472232525770": "\\nabla ^{2}\\mathbf {A} -{\\frac {1}{c^{2}}}{\\frac {\\partial ^{2}\\mathbf {A} }{\\partial t^{2}}}=\\mu _{0}e\\psi ^{\\dagger }{\\boldsymbol {\\alpha }}\\psi ", "204ebb982944bbbe234738f0f182b210": "{\\frac {f_{t}-f}{f_{t}}}\\cong {\\frac {1}{2}}({\\frac {f_{L}}{f_{t}}})^{2}\\propto {\\frac {1}{L^{4}}}", "204ed65a6066a366af330efbb1b6aa54": "\\varepsilon _{c}<\\varepsilon _{3}<\\varepsilon _{1}<\\varepsilon _{t}\\,", "204ee47f6b4f19e278483ba4368661ed": "g(n,1)=(X_{1})^{n}{\\text{ for }}n=0,1,\\cdots ,N.", "204eff4ca49030cd6f1699fa62ea15d3": "k=1\\,", "204f229f1526ee197915660c00172d38": "\\csc A=\\cot A\\cdot \\sec A\\ ", "204f3129d219241b43702105b4a4132d": "(-1,1,0,\\ldots ),", "204f9e3a6df76f200dc59fe06bde7e03": "E(R_{i})", "204fc3dff9532da29132e59a6ec38452": "y=e^{-3x}\\left(\\int 2e^{3x}\\,dx+\\kappa \\right).\\,", "204ffe9e6ef2e939447ffa8c47b31117": "T_{\\textrm {inv}}={\\frac {2a}{bk_{B}}}={\\frac {27}{4}}T_{c}", "205004edee9808a164afb3df3700f192": "a_{n}{\\frac {d^{n}x(t)}{dt^{n}}}+a_{n-1}{\\frac {d^{n-1}x(t)}{dt^{n-1}}}+\\ldots +a_{1}{\\frac {dx(t)}{dt}}+a_{0}x(t)=\\sum _{k=0}^{n}a_{k}{\\frac {d^{k}x(t)}{dt^{k}}}=Af(t).", "20505ab4007e928f63cf5ca43ad33782": "\\deg ^{+}(v).", "20507138f83081613cb2501cb625cea8": "K={\\frac {W_{i}-W_{f}}{24^{2}}}", "20507620d3551f1aaf80633f91297c47": "Z_{\\rho }", "2050a14457b7612ba7933b13eaea90e2": "\\left(1-{\\frac {2}{n}}\\right)180", "2050d0571d5055fca960ce1f83039126": "F(x',t)\\geq F(x,t)\\implies F(x',t')\\geq F(x,t')", "2050fd4181d055173dfd65ec559c1777": "f_{0}(z)=z^{2}", "20513588752806e9bfb2ed1bc964b694": "G=Z/2", "20514493f67e911591b7b3bc7fac78a6": "j\\left(e^{{\\frac {2}{3}}\\pi i}\\right)=0", "2051bd927e73bb3db18273fdc7a64a60": "\\;\\langle N,v\\rangle \\;", "2052810d60550741aab58c3fc2edf189": "\\textstyle {\\int _{0}^{1}dx=1},", "2052bb19c4653bc524568888f7899081": "u_{s}=0", "2052c78f76daae16a03d205108038d25": "\\psi \\ ", "2052dbb2e849f42de8efe914a950b565": "L(n)=\\sum _{k=1}^{n}\\lambda (k),", "20533123b3fcf98d6fac64bc80c56124": "\\scriptstyle X\\,+\\,Y\\;\\sim \\;\\mathrm {Erlang} (k_{1}\\,+\\,k_{2},\\,\\lambda )\\,", "20536c3ed8571e0c5126720acd8a4706": "f(\\mathbf {x} _{i})", "20537467225a9a19279f80961c13128b": "{\\begin{aligned}dg(t)&=\\left({\\frac {\\partial g}{\\partial t}}+\\mu {\\frac {\\partial g}{\\partial S}}+{\\frac {1}{2}}\\sigma ^{2}{\\frac {\\partial ^{2}g}{\\partial S^{2}}}+h(t)\\int _{\\Delta g}(\\Delta g\\eta _{g}(\\cdot )\\,d{\\Delta }g)\\,\\right)dt+{\\frac {\\partial g}{\\partial S}}\\sigma \\,dW(t)+dJ_{g}(t).\\end{aligned}}", "2053a0ea36275c2f63fa6a919e6a173b": "C_{1},C_{2}\\in {\\mathcal {C}}", "2053b64241fe97c69cff5ad5b63c4a39": "p_{n}={\\frac {100}{S_{N}}}\\left(S_{n}-{\\frac {w_{n}}{2}}\\right)", "2053d93b55d0eb582645dffb756c1e74": "{\\bar {V}}_{t}=r_{t}+\\sum _{i=0}^{\\infty }\\gamma ^{i+1}r_{t+i+1}", "20542f4e8343028344b17de6e34792fa": "(A.1)\\quad \\theta _{(\\ell )}:=h^{ab}\\nabla _{a}l_{b}\\;,", "2054cd77dfaaf3a71e4ec3772945e6d5": "H_{n}=f^{n}(S_{0})", "2054ddef5ff1df5a5e8c1d50a812fefe": "L(x)", "20551ac845e9324b14111cda5898b060": "S_{fg}(\\ell )={\\frac {1}{2\\ell +1}}\\sum _{m=-\\ell }^{\\ell }f_{\\ell m}g_{\\ell m}^{\\ast }", "20554bde52fe1dec5de93aba0fb4f06a": "\\mathrm {d} U={\\frac {\\partial U}{\\partial S}}\\mathrm {d} S+{\\frac {\\partial U}{\\partial V}}\\mathrm {d} V+\\sum _{i}\\ {\\frac {\\partial U}{\\partial N_{i}}}\\mathrm {d} N_{i}\\ =T\\,\\mathrm {d} S-p\\,\\mathrm {d} V+\\sum _{i}\\mu _{i}\\mathrm {d} N_{i}\\,", "2055a02c19f5f22260c8171133f9040b": "\\tan y=x\\,\\!", "2055ce02dfb28c4f27f3380f091ddc6b": "z\\in \\mathbb {R} ^{n}", "2055facda651ef041290496532fe596c": "\\scriptstyle {\\bar {c}}\\not =0", "20564c16f30faa276c1b566a7b2526b9": "\\left|{\\frac {f(0)}{f(1)}}\\right|=\\left|{\\frac {e\\alpha -1}{\\alpha -e}}\\right|=1.", "205658bd0aac797daddf05c5faf88368": "n^{2}+{\\frac {7}{4}}T\\left(\\left\\lfloor {\\frac {1}{2}}n\\right\\rfloor \\right)+T\\left(\\left\\lceil {\\frac {3}{4}}n\\right\\rceil \\right)", "205666098a1de1940139b86d6e249010": "E=\\gamma (\\mathbf {u} )mc^{2}", "2056e378a89734e28902601172873c26": "E\\approx {\\frac {\\pi Q}{R}}+{\\frac {\\pi }{3}}\\lambda \\sigma _{0}^{4}R^{3}", "2056fe3cfd812cca3ef8e6d81dfe7821": "h(x,0)=x", "2057444b6785795febd5c824279015a0": "S_{0}=\\epsilon ,S_{k}=V_{k}S_{k-1},\\,k\\geq 1,", "20574c2c244f53789533eebcc7d722ed": "\\rho ':{\\bar {V}}\\otimes V\\rightarrow L", "2057fb04ead3ad75b331a38b5dc60710": "\\deg(h)\\leq g", "205815fb0908beec0e4fdd62939363f3": "\\forall \\varepsilon >0\\,\\,\\exists \\delta >0", "20582dc932a1afff55ccb94246448d18": "L_{k}(m-1)\\approx {\\frac {m-1}{m}}L_{k}(m)", "2058a496d4d2dc7fcb88a886e222012b": "C:y^{2}+h(x)y=f(x)\\in K[x,y]", "2059266a7c166658db3899e6a76c7231": "{\\begin{matrix}\\\\f_{s}(0)=\\lim _{h\\to 0}{\\frac {f(0+h)-f(0-h)}{2h}}\\\\\\\\f_{s}(0)=\\lim _{h\\to 0}{\\frac {f(h)-f(-h)}{2h}}\\\\\\\\f_{s}(0)=\\lim _{h\\to 0}{\\frac {\\left\\vert h\\right\\vert -\\left\\vert -h\\right\\vert }{2h}}\\\\\\\\f_{s}(0)=\\lim _{h\\to 0}{\\frac {h-(-(-h))}{2h}}\\\\\\\\f_{s}(0)=0\\\\\\end{matrix}}", "2059b60122713e7c81b0928af2e20cd1": "(x_{n})_{n\\geq 0}", "205a145f1e14c8745b9be077b7563952": "{{i}_{c1}}", "205a23b7bcfb4fb4267f9380ac41ab79": "y=x^{32}+x^{26}+x^{23}+x^{22}+x^{16}+x^{12}+x^{11}+x^{10}+x^{8}+x^{7}+x^{5}+x^{4}+x^{2}+x^{1}+x^{0}", "205a2fe5e15d3e5f38376ea01cb06d34": "(f)=nP+n{\\overline {P}}-2nO", "205a45bcbb1927c5deb31e5f7dbd8406": "\\mathbb {Q} ({\\sqrt[{3}]{2}})=\\{a+b{\\sqrt[{3}]{2}}+c{\\sqrt[{3}]{4}}\\in \\mathbb {A} \\,|\\,a,b,c\\in \\mathbb {Q} \\}", "205aad06bd09fcb5edec890d9ea3be56": "\\mathbf {z} ={\\boldsymbol {\\eta }}+\\mathbf {W} _{\\delta }^{-1}(\\mathbf {y} -{\\boldsymbol {\\mu }})", "205aaf7c16957725177f1f6b6c46806e": "[\\omega ]\\in H_{dR}^{p}(M)", "205ad2f6a0ff4d1930feb1ac26e1fcbc": "\\textstyle n=\\mathrm {LCM} (9,31)=279", "205b7801d2d1c683bee12560d59679e6": "x+y", "205b9653225df73515b0f66c3344dc3e": "V=\\left({\\frac {kba}{3}}\\right)\\left({\\frac {Tn}{P}}\\right)", "205ba2211474d3b0c7180d3e29f082f7": "q^{\\mathrm {II} }", "205ba9dea3392f4ff5f3395f5d0849e9": "(P\\land (Q\\lor R))\\leftrightarrow ((P\\land Q)\\lor (P\\land R))", "205bd15e7103b85e0697fc4a787783f0": "\\mu l_{i}du_{i}=dr_{i}", "205bdef5c5df475e646a6087adffe3b9": "a+(+\\infty )=+\\infty ", "205c152a628418c40f42cc6d4a2de52b": "e^{[-a_{1}+a_{2}]}\\sum _{i=0}^{\\lfloor x\\rfloor }\\sum _{j=0}^{[i/2]}{\\frac {a_{1}^{i-2j}a_{2}^{j}}{(i-2j)!j!}}", "205c805f50846e03136e47c04e5b7638": "{R_{B}}", "205cbe965c3f779f59983c5b2f1b5e08": "{\\text{Accuracy}}={\\frac {tp+tn}{tp+tn+fp+fn}}\\,", "205d1dedd9aa680e4933f92b47b24421": "x_{p}(t)={\\frac {Q_{t}.e^{-At}-R_{t}.e^{-Bt}}{P}}", "205d2a2609ddb5dd05e8daa2dbcbfaa1": "{\\boldsymbol {\\Omega }}={\\dot {\\boldsymbol {R}}}\\cdot {\\boldsymbol {R}}^{T}", "205d4b8ec12621104046523548638d3b": "P\\in E(K).", "205d562b0cd80d24b9b8c008c634358f": "{\\bar {c}}_{k_{0}}(s;L)", "205d61cdc5551457dace656369b87957": "\\ln \\left({\\frac {4}{\\pi }}\\right)", "205d832ce327e2acad75558a3c308914": "{\\mathit {V}}", "205d8911e249c5139ec76303508bcbc3": "GG^{T}=G^{T}G=I_{2}.", "205dd32b23d692d77d5aeb8386bacb78": "{\\frac {49}{64}}", "205dd432cb13231391c269fc1d666dc6": "Z_{0}={\\frac {E}{H}}=\\mu _{0}c_{0}={\\sqrt {\\frac {\\mu _{0}}{\\varepsilon _{0}}}}={\\frac {1}{\\varepsilon _{0}c_{0}}}", "205dff55cfcd6394bcb95b10d2fafea3": "{\\frac {a+{\\sqrt {b}}}{c}}", "205e707d9306d90601465607cb15130b": "\\pi (x\\,|\\,\\theta )", "205e7babefed275b957af261449bd284": "g_{\\mu \\nu }=\\eta _{\\mu \\nu }+h_{\\mu \\nu }\\;", "205ecdabb6e07c16fdb01089230a0ae5": "a=a_{\\mathrm {then} }", "205ed822adf8d4270adf45b6b06a936f": "k_{0}^{2}=\\omega _{p}^{2}+{\\vec {k}}^{2},", "205ee73e08578d5214f3277847bdf752": "f(\\tau )\\propto \\tau ^{k},\\quad \\tau \\approx 0{\\text{.}}", "205eea09c4c50d88ea1a0d64d968350d": "\\mathbf {F} =\\mathbf {F} _{1}+\\mathbf {F} _{2}=2({\\frac {\\mathbf {B} +\\mathbf {D} }{2}}-\\mathbf {A} )=2(\\mathbf {E} -\\mathbf {A} ),", "205f482d85d3197002a4090d6397252f": "g(z)=z+b_{1}z^{-1}+b_{2}z^{-2}+\\cdots ", "205f62a21471a1ee45a17a8dd774a1c4": "f'=f\\left({\\frac {u'}{u}}+{\\frac {v'}{v}}\\right).", "205f6b95f6eb654f95a8a1b765f4d9e9": "{\\mathcal {L}}\\{f*g\\}=({\\mathcal {L}}\\{f\\})({\\mathcal {L}}\\{g\\})", "205f7c4cb76a8d7a3ffed74fceab3f56": "JMJ=M',", "205fab0af12124d9aee0b08d8fb2bb6d": "v_{j}^{T}\\sum _{i}v_{i}^{T}xv_{i}", "205fac105ae50467cd7a88f130566933": "F=-CU^{2}\\left({\\frac {2k}{rd^{2}}}\\right)\\theta ", "205fd7dbb011fd50e2f18e8ed22e205c": "w\\left(\\sigma _{i},\\sigma _{j},\\sigma _{k},\\sigma _{l}\\right)=\\exp \\left(-\\epsilon \\left(\\sigma _{i},\\sigma _{j},\\sigma _{k},\\sigma _{l}\\right)/k_{B}T\\right).", "20602149bc620308c95b7f1bf547f553": "RC=\\left({\\frac {(2.4C+A)\\;(3C+B)}{9C}}\\right)-.9C", "20606089822f65139f3b4fb5d159d596": "S_{0}=v^{2}\\,h+{\\tfrac {1}{2}}\\,g\\,h^{2}.", "20607b924915f2ad0f21bed0a36543b3": "cos(\\theta _{E})=e^{-(d1+at)/T_{1}}", "206090b3503a2efab7d30e7a8edc7e8f": "(\\eta ,\\sigma )\\in P_{1}\\times \\Sigma ", "2060b80d3753bdb5ebb8f103d73bea70": "{\\cfrac {\\partial ^{2}\\varepsilon _{11}}{\\partial x_{2}^{2}}}-2{\\cfrac {\\partial ^{2}\\varepsilon _{12}}{\\partial x_{1}\\partial x_{2}}}+{\\cfrac {\\partial ^{2}\\varepsilon _{22}}{\\partial x_{1}^{2}}}=0", "2060fea633b9f238d10e6e0256e59cd6": "\\qquad \\qquad m_{j}{\\frac {d^{2}\\mathbf {d} (jl,t)}{dt^{2}}}=-\\sum _{j'l'}{\\boldsymbol {\\Gamma }}{\\binom {j\\ j^{\\prime }}{l\\ l^{\\prime }}}\\cdot \\mathbf {d} (j^{\\prime }l^{\\prime },T),", "2061500006bc9c20d7feeeb35b006b77": "\\mathrm {CaCO_{3}+H_{2}CO_{3}\\longrightarrow Ca^{2+}+2\\ HCO_{3}^{-}} ", "20617328aa6593d5b7a83981483de541": "\\delta K=\\int _{0}^{T}\\int _{\\Omega ^{0}}\\int _{-h}^{h}{\\cfrac {\\rho }{2}}\\left[2\\left({\\frac {\\partial u_{1}}{\\partial t}}\\right)\\left({\\frac {\\partial \\delta u_{1}}{\\partial t}}\\right)+2\\left({\\frac {\\partial u_{2}}{\\partial t}}\\right)\\left({\\frac {\\partial \\delta u_{2}}{\\partial t}}\\right)+2\\left({\\frac {\\partial u_{3}}{\\partial t}}\\right)\\left({\\frac {\\partial \\delta u_{3}}{\\partial t}}\\right)\\right]~\\mathrm {d} x_{3}~\\mathrm {d} A~\\mathrm {d} t", "2061b5417b4830010ba7230945d07c2b": "{\\mathfrak {sl}}_{2}\\cong {\\mathfrak {so}}_{3}\\cong {\\mathfrak {sp}}_{1}", "20632063e7966f0d8be28872d9e7ec41": "[0.16,0.88]", "20636c8f6cb428ec06e12eb768b6b5a6": "\\mu _{0}{\\vec {J}}={\\frac {1}{r}}{\\frac {d}{d\\theta }}B_{z}{\\hat {r}}-{\\frac {d}{dr}}B_{z}{\\hat {\\theta }}", "2063afb4ceb2f7be9e69d670e24c35a3": "\\min _{\\hat {x}}\\max _{(\\Delta ,x)\\in G}\\left\\{\\left\\|{\\hat {x}}\\right\\|^{2}-2{\\hat {x}}^{T}x+\\operatorname {Tr} (\\Delta )\\right\\}", "2063b0e291fb57c35bfcaffc3512f791": "w(i,j)={1 \\over Z(i)}e^{-{{\\lVert v({\\mathcal {N}}_{i})-v({\\mathcal {N}}_{j})\\rVert }_{2,a}^{2} \\over h^{2}}},", "2063c1608d6e0baf80249c42e2be5804": "value", "2063d0affeef5d6910cbbcf34e7aaa03": "(a,b)=\\omega ((-1)^{\\operatorname {ord} (a)\\operatorname {ord} (b)}b^{\\operatorname {ord} (a)}/a^{\\operatorname {ord} (b)})^{(q-1)/n}", "2064304e1ccd34ba21a833d7c0e53cac": "c_{s}={\\sqrt {k_{B}(ZT_{e}+\\gamma _{i}T_{i})/m_{i}}}", "2064ab0ee8c30521856289fb49b77303": "\\tau \\in (0,1)", "2064bee67c6253602fa1641939864f98": "R=I\\cos \\theta +[\\mathbf {k} ]_{\\times }\\sin \\theta +(1-\\cos \\theta )\\mathbf {k} \\mathbf {k} ^{\\mathsf {T}}", "2064decc737bdae581967090428344e1": "(-1)^{\\text{signbit}}\\times 2^{-16382}\\times 0.{\\text{significandbits}}_{2}", "20651f27bdf0de3917790d052683c999": "\\ Q_{T}P_{W}Q_{T}\\psi _{n}=\\lambda _{n}\\psi _{n},", "2065257cf6e0dcc272dcdd78ed42d498": "{\\mathcal {F}}\\left[{\\frac {dW(t)}{dt}}\\right](\\omega )=i\\omega {\\mathcal {F}}[W(t)](\\omega )", "2065300734dd5a1601e0aa3f77b6a5e1": "{\\begin{aligned}\\rho {\\frac {\\partial ^{2}u_{i}}{\\partial t^{2}}}&=\\partial _{i}\\lambda \\partial _{k}u_{k}+\\partial _{j}\\mu \\left(\\partial _{i}u_{j}+\\partial _{j}u_{i}\\right)\\\\&=\\lambda \\partial _{i}\\partial _{k}u_{k}+\\mu \\partial _{i}\\partial _{j}u_{j}+\\mu \\partial _{j}\\partial _{j}u_{i}\\end{aligned}}", "206531fd76f97729d7a2fafb7c89b8fd": "A_{1},A_{2},\\dots .\\,", "20659aa667a2bb32f3f657ebb9ac94ae": "{\\frac {\\operatorname {Li} _{s}(e^{t})}{\\zeta (s)}}", "2065a40804b013b6991324c6e537ee43": "\\scriptstyle M_{\\text{B}}(H)", "2065ccddc9d2e5a5775dba303175c899": "S_{-1}(x,y)", "2065dc32759240079154c80c68197d3b": "\\lfloor \\,\\rfloor ", "20662f05b5c034e0b13f26ebfb8ec126": "\\lim _{k\\to \\infty }c_{2}(k)=-2.157782996659\\ldots .", "20663615ea3c66d0f3574986dd4d451a": "E_{\\mathrm {h} }={\\hbar ^{2} \\over {m_{\\mathrm {e} }a_{0}^{2}}}=m_{\\mathrm {e} }\\left({\\frac {e^{2}}{4\\pi \\epsilon _{0}\\hbar }}\\right)^{2}=m_{\\mathrm {e} }c^{2}\\alpha ^{2}={\\hbar c\\alpha \\over {a_{0}}}", "206685a8b2ab441b326a9ae36f4b6a55": "{documentation}", "2066b304fa942ebd4a361706d666449f": "Poss(a,s)", "2066c1f1a8900c8221e8af51f739270e": "{\\text{Pressure}}={\\frac {\\text{force}}{\\text{area}}}={\\frac {\\text{weight}}{\\text{area}}}={\\frac {{\\text{weight density}}\\times \\!\\,{\\text{volume}}}{\\text{area}}}", "20673b94d2fc45249d711e9537a5e199": "n\\geq k.", "20678e52a5a38e7cac895a95032560c8": "\\tau _{b}=\\rho gh\\sin {\\left(\\alpha \\right)}\\,", "2067ada9a78d139b52366b7d1fcde7f2": "f(\\theta ):\\Theta \\rightarrow X", "2068415ef79ba0ebccd64a69eb4e39c8": "\\left({\\frac {2}{3}}\\right)^{6}\\times 2^{4}", "2068f32facfa7b1aa7c05c8ebaca5598": "\\mathbb {R} ^{s}", "206933123ea016e7a32f62a8228bed70": "2\\cos(\\theta )\\in [-2,2]", "2069341942491d969789bebf869bb515": "Q={\\frac {1}{2m}}\\sum _{ij}\\sum _{r}\\left[A_{ij}-{\\frac {k_{i}k_{j}}{2m}}\\right]S_{ir}S_{jr}={\\frac {1}{2m}}\\mathrm {Tr} (\\mathbf {S} ^{\\mathrm {T} }\\mathbf {BS} ),", "206961687cca6a6d73589655b9c519cc": "{\\acute {x}}^{\\mu }", "2069624b1edb6f4f49638ecba24b62bb": "{\\begin{matrix}LU\\end{matrix}}", "206967bde31f59e3738700c424e6aa60": "{\\vec {A}}\\cdot {\\vec {p}}", "206985becba81fae5c16d0a337205f03": "\\vert \\psi \\rangle ", "206994b52dd2efafb56f5c7331505dc5": "p^{2}-p+1=r^{2}+2rkq+k^{2}q^{2}-r-kq+1=r^{2}-r+1+q(2rk+k^{2}q-k)=q(1+2rk+k^{2}q-k)", "2069b2da1dd33fee288586b45874989a": "z^{\\rho }\\;G_{p,q}^{\\,m,n}\\!\\left(\\left.{\\begin{matrix}\\mathbf {a_{p}} \\\\\\mathbf {b_{q}} \\end{matrix}}\\;\\right|\\,z\\right)=G_{p,q}^{\\,m,n}\\!\\left(\\left.{\\begin{matrix}\\mathbf {a_{p}} +\\rho \\\\\\mathbf {b_{q}} +\\rho \\end{matrix}}\\;\\right|\\,z\\right),", "206a5ce347a6319110f01671ec55eda4": "1\\leq n\\leq m^{\\varepsilon }", "206a879d3092a3a69c6333762fa43017": "{\\overline {M}}\\not =0", "206ac5421ae2d42d7eec63d13e565774": "(pq)^{*}=q^{*}p^{*},\\quad (pq)^{\\star }=p^{\\star }q^{\\star },\\quad (q^{*})^{\\star }=(q^{\\star })^{*}.", "206ad198647ed7b157cf69f8ddae5815": "U_{s}U_{\\omega }", "206b07dd0662a648b57cf38257dad50b": "\\left({\\sqrt {\\frac {\\nu }{\\nu +1}}}\\mu ,\\,{\\sqrt {\\frac {2\\nu }{2\\nu +5}}}\\mu \\right)", "206b0d88fdb7f78ddb24fa76ed96d514": "\\textstyle A>0", "206b5818102fa109c532a00771827df5": "\\left({\\frac {1}{\\sqrt {10}}},\\ {\\frac {1}{\\sqrt {6}}},\\ {\\frac {1}{\\sqrt {3}}},\\ \\pm 1\\right)", "206b8e2aaa06147783d6b2b1a74fe5e7": "Y(\\mathbf {x} )=\\int _{S^{n-1}}Z_{\\mathbf {x} }^{(\\ell )}(\\mathbf {y} )Y(\\mathbf {y} )\\,d\\Omega (y)", "206bbd1a2a64e8b7fe2b83e7be3ddf0a": "\\nabla {\\boldsymbol {v}}({\\boldsymbol {x}},t)", "206bc708461fb2d643b849a51e05891d": "\\forall a,a\\nrightarrow a", "206c14f3c74acb981b8a97bd2f9c6778": "a_{\\ell }\\geq 0", "206c3e312d7eca5ffef26210af670ba2": "(By+\\beta )^{n}-B^{n}y^{n}\\leq B^{n}r+\\alpha .", "206c4038a3708ae85c43a6908d95a0b4": "(A\\land \\neg B\\land \\neg C)\\lor (\\neg D\\land E\\land F)", "206c439059aa2f3b67e467160bd4fd32": "\\|u\\|_{BV}:=\\|u\\|_{L^{1}}+V(u,\\Omega )", "206c6ac133a0bbdea3c46da831e54a1e": "\\sum {c(S_{i})\\cdot x_{i}}\\leq B", "206c78301070322bb1a720756bb9233a": "\\{a^{\\dagger }(\\mathbf {k} ),a^{\\dagger }(\\mathbf {l} )\\}=0,", "206d5d2d5d9e396b2bb2d4cb7adcb550": "1_{5}\\rightarrow (1,1)_{1}", "206d6bab922b99c880f70866794b2981": "{\\bar {6}}_{H}", "206d6fd394df9d600c8d4f2543bc7b2b": "(\\operatorname {artanh} \\,x)'={1 \\over 1-x^{2}}", "206dec5cf451ab5a8246de91d8423db2": "L_{q}=0,L_{qq}\\geq 0,|L_{qq}|\\geq |L_{pp}|.", "206df40ba048f9db44de55a465eb0558": "\\mathrm {SL} (n,\\mathbb {C} )\\times \\mathrm {SL} (n,\\mathbb {C} )", "206e195a0f979781f522f00ccfd20dc9": "2^{f(k)}\\cdot {\\text{poly}}(n)", "206e5823b03c7d1e7e410c9bbdf37a3c": "V_{1},V_{2},\\ldots ,V_{k}", "206e7271cd45995cabb08ce22fcfa34b": "\\mathrm {OSR} \\,=\\,{\\frac {f_{s}}{2f_{0}}}\\,=\\,{\\frac {1}{2f_{0}\\tau }}", "206ead868b0cca3b1a4361b7fdb1ab7c": "n={\\frac {{\\sqrt {8x+1}}-1}{2}}.", "206f456fce9dbe75a05e706c2209fc19": "{\\vec {r}}=(\\sigma _{1},\\sigma _{2},...,\\sigma _{N})", "206f631c216cbe73637e2c316dd93e9e": "x^{4}+360*x^{3}-270000*x^{2}+20736000*x+1866240000=0", "206f852748d691594e2c56c687a1a019": "-\\left|y\\right\\vert \\leq {\\frac {x^{2}y}{x^{2}+y^{2}}}\\leq \\left|y\\right\\vert ", "206f9a4c66837577bcd76cb05206aa92": "|p\\rangle ", "206fb2258d45d00879d85a9adcc03109": "\\Delta p_{\\text{B}}(x)=D_{1}x+D_{2}", "206ff7f7b3a92ebbe4e47954fe123ea9": "n=(n_{1},\\cdots ,n_{k})", "207037226fa58e4d57d7a77bc9e0302b": "ABCA^{-1}B^{-1}C^{-1}", "20707bf68f4fb4abb0e76ad879d53443": "\\theta =45^{\\circ }\\,\\!", "2070bc588b47e00a8993466a73acbe87": "T={\\frac {e^{-2\\int _{x_{1}}^{x_{2}}dx{\\sqrt {{\\frac {2m}{\\hbar ^{2}}}\\left(V(x)-E\\right)}}}}{\\left(1+{\\frac {1}{4}}e^{-2\\int _{x_{1}}^{x_{2}}dx{\\sqrt {{\\frac {2m}{\\hbar ^{2}}}\\left(V(x)-E\\right)}}}\\right)^{2}}}", "2070bd63aa1c00592882b4552d3c6259": "{\\overline {P}}(Cl_{3}^{\\geq })=Cl_{3}^{\\geq }", "2070c9d3875256af2db4f9dd7714abd8": "{\\frac {dS(t)}{S(t)}}=\\mu \\,dt+\\sigma (t)\\,dz_{1}(t)", "2070cb4e8fa2d2ff1f81c2dc25c99357": "\\pi :N\\to N", "2070df98a5a32adb6a58c24fe0711daa": "\\tau _{ind}\\left(\\omega \\right)=-\\Gamma _{cap}(\\omega )", "20714ab30e31ded7a10b95dda6e5e641": "\\pi _{n}", "207152a8f129786e2397aaf321a1300e": "{\\mathcal {Q}}_{\\mathrm {Hur} }=\\mathbb {Z} [\\eta ][i,j,j'].", "2071617678698e26842fc35370cbf2b0": "G=\\oplus G_{n}", "207169934204c3b2fe19369e648ea9f8": "\\lambda _{3}=1,", "20716d61b83df5b4491e8af347e446ae": "d[x,y]=xdy-ydx~", "20719dba0b11ea8c7110f78a229023ad": "J_{X_{t}}", "2071e4ac842502f8495b6a9295aa9890": "{\\frac {(1-z^{3}/6)}{(z-z^{2}/2)^{2}}}+c", "20721b29492ad3e86ee8c4c252994e60": "\\beth _{0}", "207289a61ac1a0b28ac396b19be58e3d": "\\mathrm {tr} (\\Lambda )=\\lambda _{1}+\\lambda _{2}+\\lambda _{3}", "2072a3bd5ae2ccadfe1c5823abf8cea1": "D(t)={\\begin{cases}1/m&{\\text{if }}|t|\\leq 1\\\\0&{\\text{otherwise}}\\end{cases}}", "2073054630cb7e10828f84bd2e27b7a7": "Q_{n}={\\frac {\\pi }{n\\tan {\\tfrac {\\pi }{n}}}}.", "2073266755beab8dcc86a74a2cf0815f": "[p_{1}]-[p_{0}]", "207382741d990214940a479473be1455": "\\mathrm {Force} =\\mu {\\frac {d^{2}Q}{dt^{2}}}", "207387eaeefe20c9b2db223df88ecfe5": "g_{ij}(q^{i},q^{j})={\\cfrac {\\partial x_{k}}{\\partial q^{i}}}{\\cfrac {\\partial x_{k}}{\\partial q^{j}}}=\\mathbf {b} _{i}\\cdot \\mathbf {b} _{j}", "2073e1a8b5f6bbfd69ef66bc1378e427": "M(q)", "2073eb7da1bbb5c8a78332c8cb5adb8b": "\\mathbf {P} ={\\hat {\\mathbf {a} }}+\\alpha \\mathbf {e} _{\\infty }", "2073f901fe08a2b05a0e7a4bd5ad6850": "\\langle \\rho F,\\leq \\rangle =\\langle F,R\\rangle /{\\sim }", "207414d039b69d01f9cac3feea2fa810": "f(x)\\rightarrow u^{T}\\nabla f(x)", "20744fde0fdbe04fdf0431ac73fb66e1": "V_{nk}", "2074550af4223c91d90914db05822ee6": "\\phi _{0}(q)=\\sum _{n\\geq 0}{q^{n^{2}}(-q;q^{2})_{n}}", "20745c1d00357ecc3fc2bd9c036ad3e6": "g(r)=h(r)=\\sin(r)", "20746c5bb5a448a879fef06531fd81d8": "{\\begin{pmatrix}\\cdots &\\cdots &\\cdots &\\cdots &\\cdots \\\\\\cdots &M_{ij}&\\cdots &M_{il}&\\cdots \\\\\\cdots &\\cdots &\\cdots &\\cdots &\\cdots \\\\\\cdots &M_{kj}&\\cdots &M_{kl}&\\cdots \\\\\\cdots &\\cdots &\\cdots &\\cdots &\\cdots \\end{pmatrix}}={\\begin{pmatrix}\\cdots &\\cdots &\\cdots &\\cdots &\\cdots \\\\\\cdots &a&\\cdots &b&\\cdots \\\\\\cdots &\\cdots &\\cdots &\\cdots &\\cdots \\\\\\cdots &c&\\cdots &d&\\cdots \\\\\\cdots &\\cdots &\\cdots &\\cdots &\\cdots \\end{pmatrix}}", "20748d644d61c6b2e159d34b7bb8602c": "qSC_{p}", "2074d39ef94b3dd4c74e082594324705": "[a_{1}\\cdots a_{m},b_{1}\\cdots b_{n}]=\\sum _{i,j}(-1)^{i+j}[a_{i},b_{j}]a_{1}\\cdots a_{i-1}a_{i+1}\\cdots a_{m}b_{1}\\cdots b_{j-1}b_{j+1}\\cdots b_{n}", "2074eba49c67c835d4a3f9d123543181": "h_{\\lambda }(X_{0})", "2074f85dbbc2bc49e580c90bb4faa6f1": "{\\dot {A}}=(\\sigma -\\sigma _{0})A-|A|^{2}A", "2075641805a5a19e5148a44be83ceb16": "({\\mathfrak {k}},{\\mathfrak {p}})", "207566963d710aa1e1649d5b692647cc": "{\\hat {\\gamma }}_{3}", "207598acc692f8e1ffca65d23bbbeb9e": "{\\frac {1}{2}}[(\\kappa +1)\\theta ~\\sin \\theta +\\{1+(\\kappa -1)\\ln r\\}~\\cos \\theta ]\\,", "20759e21ff650169d9316fa62c5a6415": "{\\begin{aligned}&{\\frac {\\partial ^{2}M_{11}}{\\partial x_{1}^{2}}}+{\\frac {\\partial ^{2}M_{22}}{\\partial x_{2}^{2}}}+2{\\frac {\\partial ^{2}M_{12}}{\\partial x_{1}\\partial x_{2}}}+{\\frac {\\partial }{\\partial x_{1}}}\\left(N_{11}\\,{\\frac {\\partial w}{\\partial x_{1}}}+N_{12}\\,{\\frac {\\partial w}{\\partial x_{2}}}\\right)+{\\frac {\\partial }{\\partial x_{2}}}\\left(N_{12}\\,{\\frac {\\partial w}{\\partial x_{1}}}+N_{22}\\,{\\frac {\\partial w}{\\partial x_{2}}}\\right)=P\\\\&{\\frac {\\partial N_{\\alpha \\beta }}{\\partial x_{\\beta }}}=0\\,.\\end{aligned}}", "2075b9dae17795ff2f98b7b33aeb8c8a": "P^{2}", "2075d27e08213919ca36d7e20259c2c4": "{\\it {n}}", "2075d5ae7df6b71f3af7407be8430e25": "\\varphi :U\\to \\mathbf {R} ^{n}", "2075ddd04285d19e8a3c75c4512126c5": "(s^{2}-s_{1}^{2})(s^{2}-s_{2}^{2})(s^{2}-s_{3}^{2}).\\qquad (2)", "2076531fb3d1ad20a62189f5a7cc599c": "\\,e^{t^{\\mathrm {T} }\\mu +{\\frac {1}{2}}t^{\\mathrm {T} }\\Sigma t}", "2076a8c0fd580b12f5d4e122293a0672": "Z\\to Y", "2076d0174cd9eeab2f1214c44ebfe3b9": "\\{x|x<2\\}", "20771de5872107097a8b16ef61dc7c84": "\\mathrm {L\\Pi f} ", "2077245a1790b5d220de388d5fdc3a18": "{\\begin{array}{lll}\\omega (0)&=W_{0}(1)&\\approx 0.56714\\\\\\omega (1)&=1&\\\\\\omega (-1\\pm i\\pi )&=-1&\\\\\\omega (-{\\frac {1}{3}}+\\ln \\left({\\frac {1}{3}}\\right)+i\\pi )&=-{\\frac {1}{3}}&\\\\\\omega (-{\\frac {1}{3}}+\\ln \\left({\\frac {1}{3}}\\right)-i\\pi )&=W_{-1}\\left(-{\\frac {1}{3}}e^{-{\\frac {1}{3}}}\\right)&\\approx -2.237147028\\\\\\end{array}}", "207775be4940d5b794e994c9886ceabd": "T_{f}x", "2077a4df35530fa21c75805d81eb3822": "(u,v)\\in E", "20781fbb3c21bd05b096c1f105d7b2c1": "T_{B}={\\frac {KV}{k_{B}\\ln \\left({\\frac {\\tau _{m}}{\\tau _{0}}}\\right)}}", "20782b093c43ae61ddfa354b15d14a61": "P_{\\pm }", "20782b8c73d7645b74a76f7d240ad4c5": "\\sinh x=-{\\rm {i}}\\sin {\\rm {i}}x\\!", "2078432ee5ce779cf504b669e32e6c83": "\\scriptstyle \\mathbf {J} =0", "20784f49791eb12824dffcbbe64d1879": "f[x_{0},\\dots ,x_{n}]x^{n}", "20785b1fd4010c141d303552d0d56b4c": "V=\\{a,b\\}", "20787d0db6fe6ce4e643f6163582b8b1": "E_{lm}^{(1)}", "20788323e35116f5b75c24b873f87182": "H(s)={\\frac {K\\omega _{0}^{2}}{s^{2}+{\\frac {\\omega _{0}}{Q}}s+\\omega _{0}^{2}}}.", "20788529a74289c9aa5830c206e5e389": "\\phi (t)\\rightarrow x_{1}\\quad \\mathrm {as} \\quad t\\rightarrow +\\infty ", "20788fc6e749c259bd34532804fa3931": "p_{X}(x)=\\mathbb {P} (x\\in X)", "207901533929b2bb847c5bec50369024": "\\left\\{{\\begin{pmatrix}a&b\\\\0&c\\end{pmatrix}}\\ :\\ a,b,c\\in \\mathbb {C} \\right\\}", "207911b636e27a8100352b0a5865afea": "{\\tfrac {3(K-\\lambda )}{2}}", "207958891181c4c70f6dc50f54589c42": "\\mathrm {-C(=O)-\\left(CH_{2}\\right)_{12}-CH_{3}} ", "20797bbaeafe6f03dc99b2407c05ff82": "x^{q^{i-1}}\\mod f^{*}", "2079ce51aaff670dee5c28e7b6f2b617": "\\textstyle p", "2079d20e2f6c414ecca721f4e32a19bf": "C_{\\ ab}^{c}", "207a144a3a9c97669246f0c5b0775f38": "x_{l}", "207a2029feaca2d5b411a99129e09fe8": "{\\begin{cases}\\overbrace {\\begin{bmatrix}{\\dot {\\mathbf {x} }}\\\\{\\dot {z}}_{1}\\\\{\\dot {z}}_{2}\\end{bmatrix}} ^{\\triangleq \\,{\\dot {\\mathbf {x} }}_{2}}=\\overbrace {\\begin{bmatrix}f_{x}(\\mathbf {x} )+g_{x}(\\mathbf {x} )z_{2}\\\\z_{2}\\\\0\\end{bmatrix}} ^{\\triangleq \\,f_{2}(\\mathbf {x} _{2})}+\\overbrace {\\begin{bmatrix}\\mathbf {0} \\\\0\\\\1\\end{bmatrix}} ^{\\triangleq \\,g_{2}(\\mathbf {x} _{2})}z_{3}&\\qquad {\\text{ ( by Lyapunov function }}V_{2},{\\text{ subsystem stabilized by }}u_{2}({\\textbf {x}}_{2}){\\text{ )}}\\\\{\\dot {z}}_{3}=u_{3}\\end{cases}}", "207a62a838ea9503c8526235c3f260c4": "I_{c}=K_{L}a^{2}\\pi {\\frac {U_{str}}{L}}", "207b37f465033f6f26a6586c48d143e7": "I_{5}=\\int \\cos ^{5}xdx.\\,\\!", "207b9ac161ce3de9b4eb309be8b9ebd5": "p_{H}(x|\\alpha )={\\displaystyle \\int \\limits _{\\theta }p_{F}(x|\\theta )\\,p_{G}(\\theta |\\alpha )\\operatorname {d} \\!\\theta }", "207bdc22da9732fe0057ade0c4ff2912": "R_{\\sigma \\mu \\nu }^{\\rho }=\\partial _{\\mu }\\Gamma ^{\\rho }{}_{\\sigma \\nu }-\\partial _{\\sigma }\\Gamma ^{\\rho }{}_{\\mu \\nu }+\\Gamma ^{\\alpha }{}_{\\sigma \\nu }\\Gamma ^{\\rho }{}_{\\alpha \\mu }-\\Gamma ^{\\alpha }{}_{\\mu \\nu }\\Gamma ^{\\rho }{}_{\\alpha \\sigma }", "207bea04dc6e1fb45324059a327c5413": "\\scriptstyle {\\boldsymbol {\\omega }}_{s}", "207c36c980f7436de6f3cbda26ff14b4": "T_{G}=T_{H}\\cdot T_{H'}", "207c95d1571012ca5f2b969c28c854b3": "quot={\\frac {V}{2s+1}}", "207cb816461e0107ac882ae191f7b577": "m_{i+1}={\\frac {m_{i}}{2}}={\\frac {m}{2^{i+1}}},", "207cbde45d6daf5462111dc4909fbea9": "{\\text{Per-unit volts}}={\\frac {\\text{volts}}{\\text{base volts}}}", "207cce2fa09af07a07d4b07a18fd18bb": "[a\\cdot D,\\,b\\cdot D]F=-({\\mathsf {S}}(a)\\times {\\mathsf {S}}(b))\\times F.", "207cee98846317c6585e709825debb95": "F_{\\omega }={\\underset {1\\leq i}{\\bigcup }}F_{i}", "207d00ce1e082e33e804f9581229e848": "q(x)", "207d04ab5fc1d23d0005098e2c08be74": "\\gamma _{p}~=~k_{0}~\\cos \\theta _{0}~+~{\\frac {2p\\pi }{l_{z}}}~~~~~~~~~~~~~~~~~~~~~~(2.2c)", "207d9f091422a706b30b61fbb183dd43": "f(n,\\mathrm {lcm} (a,b))=\\mathrm {lcm} (f(n,a),f(n,b))", "207da16cc8f9d9fa927af4cceb570577": "T_{i}^{i}", "207db55304371765030f0620bac2e291": "E_{6},E_{7},E_{8},F_{4},", "207df3eb785318293fed2c850b270633": "P(b)", "207e5272447a09e113d098df65d1220b": "X^{-}", "207e746526e5222ecbf49a6452561d50": "\\langle \\partial _{t}u,v\\rangle =\\langle \\partial _{x}\\left(-{\\frac {1}{2}}u^{2}+\\rho \\partial _{x}u\\right),v\\rangle +\\langle f,v\\rangle \\quad \\forall v\\in {\\mathcal {V}},\\forall t>0", "207e7b7bce2323d6b9ac928479641dc5": "c_{2}=1.4380\\times 10^{-2}{\\text{m·K}}", "207eb0107485f9dcae3e744c56b4b740": "{\\begin{bmatrix}0&0&0&1\\\\0&0&1&0\\\\1&0&0&0\\\\0&1&0&0\\end{bmatrix}}\\qquad ({\\text{permutation of coordinate axes}})", "207ec0782f5d52153e5c67480a137c58": "a_{0,i}=H(X_{i})", "207eca66b1e44d27964af57d63bd77c4": "x(2x+3)", "207ed1b47adb503f5641942bbd58345a": "g:X'\\to P", "207ed940dad9d4051b5db5f65856fd2d": "{\\mathfrak {gl}}_{n}(\\mathbb {C} )", "207eea647354ee56fa464951d421977c": "\\langle x,y\\rangle :=y^{*}Mx", "207eea7ca4cab14e6ba251be11386ae4": "r={\\frac {t}{\\sqrt {n-2+t^{2}}}}.", "207f0eb0a76c4d6e97c08be9a4bb364c": "p\\in M\\,,", "207f357a51ff79289b5baec16d8d1e9d": "\\varepsilon -A", "207f56123e92280bae7a2c0074908549": "u,v,\\dots ", "207f576a883d318726577787d6486ea0": "f(X\\cup \\{x\\})-f(X)\\geq f(Y\\cup \\{x\\})-f(Y)", "207f5a50db1cfe9e29ded884d412b99f": "\\langle (1\\;2)\\rangle =\\{id,\\;(1\\;2)\\}", "207f8f62269e90c1360d8f78e99262b2": "(i,a,j)*(k,b,n)=(i,ap_{jk}b,n)", "207f90ce614be95ba66ee1e78ff7044c": "{\\frac {dx^{\\mu }}{d\\tau }}", "207fcfc23446f43162e637c77dde949a": "n^{(\\lambda )}=4\\,.", "207fd2354f3cb9caaba67edf2a151876": "\\{\\mathbf {e} _{1},\\mathbf {e} _{2},\\mathbf {e} _{3}\\}", "207fff85ab28770d611c5b760dc1cce1": "\\displaystyle {f^{2}-(Hf)^{2}=(f_{+}+f_{-})^{2}+(f_{+}-f_{-})^{2}=2(f_{+}^{2}+f_{-}^{2})=-2iH(f_{+}^{2}-f_{-}^{2})=-2H(f(Hf)).}", "2080029f4d2e6a034da928677cb4bf69": "\\scriptstyle n\\log _{2}n", "208080d6d26cc44588610c941efa5a2e": "(\\mathbb {Z} S)_{n}=\\mathbb {Z} \\langle S_{n}\\rangle ,", "20809753072d8216d2d267a6975d3f59": "{\\begin{bmatrix}1&0&0&0&1&1&1&1\\\\1&1&0&0&0&1&1&1\\\\1&1&1&0&0&0&1&1\\\\1&1&1&1&0&0&0&1\\\\1&1&1&1&1&0&0&0\\\\0&1&1&1&1&1&0&0\\\\0&0&1&1&1&1&1&0\\\\0&0&0&1&1&1&1&1\\end{bmatrix}}{\\begin{bmatrix}x_{0}\\\\x_{1}\\\\x_{2}\\\\x_{3}\\\\x_{4}\\\\x_{5}\\\\x_{6}\\\\x_{7}\\end{bmatrix}}+{\\begin{bmatrix}1\\\\1\\\\0\\\\0\\\\0\\\\1\\\\1\\\\0\\end{bmatrix}}", "2080e5fb6a39b4b63f56e0ae19e42b59": "A\\ ", "20810b9d5d8241560758acb05ec26a10": "\\ {a}\\sin \\theta =n\\lambda ", "2081129cb58749c4c1fd729f2e9c0316": "\\textstyle P+B=k\\cdot (W+M)\\,\\ ", "20815a319b71e93ce06725170fb27b5a": "\\mathrm {S} =\\left\\{\\mathbf {x} \\in \\mathbb {Z} _{0+}^{c}\\,:\\,\\sum _{i=1}^{c}x_{i}=n\\right\\}", "20816b42df35c3c31c997e2cc29298f9": "\\lim _{x\\rightarrow 0}{}^{n}x={\\begin{cases}1,&n{\\text{ even}}\\\\0,&n{\\text{ odd}}\\end{cases}}", "2081ad12a23f4b0338e234e9ccc335d2": "{\\frac {1}{2}}m{\\dot {r}}^{2}=E-U_{\\text{eff}}(r),", "2081b8a37cccaf4c1fccb8cc98fb8181": "\\int _{1}^{e}{\\frac {1}{t}}\\,dt=1.", "2081c4b1f0106ed4730bc6c98d3534fa": "\\sum _{j\\in J}r_{ij}x_{j}=m_{i}", "2083002807c96abfd625edbbf1d8aa8b": "[F]:={\\big \\{}([X],[F(X)]):[X]\\in [\\mathbf {T} ]{\\big \\}},", "20832663121267cf1e870a93012b852d": "R=1+{\\frac {1}{|a_{n}|}}\\max\\{|a_{0}|,|a_{1}|,\\dots ,|a_{n-1}|\\}", "20832717ce694af3178f64db70967e96": "{(x_{i},y_{i})_{i=1}^{n}}", "2083ff843480dc3cf763b67d9c1e84e6": "\\mu _{n}=\\{x\\in K\\mid x^{n}=1\\}", "20845714172d35a29ea1354b1cd28128": "ta(s)", "20847b3e450b7099c538aa18704dcd3d": "\\;\\Psi (f)=\\sum _{i}f_{i}F_{i}.", "208483e1b22f0faac3568c16f4d0a830": "f_{pegtop}(a,b)=(1-2b)a^{2}+2ba", "2084ee7ae2ffd8c3b03beec4c26148fb": "A(c)=\\alpha ", "208506a77740ceb65deabe81b8830464": "{\\sqrt {\\pi /2}}", "20856da3bd9d0da62cc5ed62b53c5915": "|6x^{4}-2x^{3}+5|\\leq 13\\,|x^{4}|.", "2085774286f7871235c13a7175e0f500": "W(\\mathbf {x} _{w}^{(k-1)})", "2085d15bd12658fa6b669fd344bb4d3e": "\\beta (S)=USU^{*}\\,", "20860728c901636fb62b8c24a9fae162": "\\Delta W>0", "20862345fd71eea49f2f0c77b4cd9022": "\\varphi (\\cdot ,t)", "20865956f9e89cba1d9a2d3e93a071c9": "O(\\log ^{5}n)", "20867301d8255889fb1e2d828c4040a1": "|E_{+}\\rangle ", "20868fa29dfc38ac154b8ef762766b41": "P_{1}", "208694c8fef4503dc4b4bb316d72aa36": "P\\in {\\mathcal {P}}", "20872ae883c008735d4ba2f01caa2645": "N(E)=8{\\sqrt {2}}\\pi m^{3/2}E^{1/2}/h^{3}\\,\\!", "20874875b73599d964f317c9131b7c9d": "T_{\\text{S}}={\\sqrt {\\frac {c^{4}e^{2}}{G(4\\pi \\epsilon _{0}){k_{\\text{B}}}^{2}}}}", "2087c4d19d3950cdf80c1bd41e55d685": "E=\\int _{\\mathbb {R} ^{n}}[\\vert \\nabla \\theta \\vert ^{2}+f(\\theta )]\\,d^{n}x", "2087c90f4a5ea9520f71a5adf6a10671": "(w_{j_{1}},v_{i2})", "2087dee0ddeb74165914a0235a1dedc0": "q=||\\mathbf {u} ||", "2088415d40f5bb6d98f073b2adc5d0f8": "D_{r}=0\\%", "2088a4950220c4f835240ee5f8f2ae39": "A={\\frac {\\ell a^{2}+ma+n}{(a-b)(a-c)}};", "2088ce368027c19e7bbcc881bc17096b": "4(n/4)^{2^{d}}", "2088ee0289517c52ce0909b2eb04aeb1": "\\,_{2}F_{1}(a,b;c-1;z)-\\,_{2}F_{1}(a,b+1;c;z)={\\frac {(b-c+1)az}{c(c-1)}}\\,_{2}F_{1}(a+1,b+1;c+1;z)", "2088f5efc9e499ec688808ab22345f66": "|x-y|\\cdot |u-v|=0\\!", "20893a79ab3703cc4306a163d28457f3": "C_{t}=C_{0}+(C_{ss}-C_{0})*(1-e^{-k_{e}t})\\,", "208962ff8e78316fb4bfac7fb1477315": "|e_{d}|=\\alpha ", "208977d2d0f39115cfb566943cc7bae1": "\\Pi (h)=-{dW \\over dh},", "2089d8bf9f740c545594ad0084769d20": "R_{\\text{b}}", "208a2e1e25a1d2471a9ec43d12904f0a": "\\nu (H)", "208aa0b1bf7a5c9bdc99942a5a975c8e": "{\\overrightarrow {C_{i}^{\\alpha }C_{i+2}^{\\alpha }}}", "208aac8531e8b12f465db7687ba06536": "v_{t}=(1/m)\\sum _{i=1}^{m}v_{i}", "208ab7ff1846bd299e38eafff2074ab5": "v_{3}'=-q_{2}^{*}\\,v_{1}+i\\,\\xi \\,v_{3}.", "208abb92643ee22b27ebe8e0c41cfe96": "\\operatorname {de-lambda} [M\\ N]\\equiv \\operatorname {de-lambda} [M]\\ \\operatorname {de-lambda} [N]", "208ae34312e0b8353e975021e5c0c117": "\\scriptstyle a/0=\\infty ", "208afe57c5c87b32248a9ba3ff7a902f": "\\ell ^{1}({\\mathbf {Z} })", "208b1a2faf3b3ce6d1771f818d553f79": "2^{5/12}={\\sqrt[{12}]{32}}", "208c0ca2c177cc62481f57d1ea37e8b3": "\\mathrm {GF} (p^{m_{i}})", "208c1a15c01e03fbaf20b24adb9ec893": "(\\xi _{i},\\zeta _{i})", "208c205b3a6a8dcdded0c9965d794513": "\\zeta _{T}(s)=\\operatorname {Tr} (T^{s})=\\sum {\\mu _{i}^{s}}", "208c463516de35fd37c9c05690481754": "\\mathbf {x} \\,\\!", "208c8871e5341088d747ebfa33200ccb": "I=1.1\\times I_{\\mathrm {o} }\\times 0.56^{(AM^{0.715})}\\,", "208cf1f183f6deb761a946d3f9f54d83": "{\\hat {\\Gamma }}(G,H)", "208d043b83da5b0f7a5e20c0d8365066": "{}_{\\ 81}^{208}\\mathrm {Tl} \\xrightarrow {\\beta ^{-}\\ } {}_{\\ 82}^{208}\\mathrm {Pb} \\ \\mathrm {(3\\ m,\\ 2.6\\ MeV)} ", "208d20f4755e1076b7887452984c035d": "c_{s}", "208d41fe62ce2fd1c59a919be8cdc5da": "V_{t}=V_{d}", "208d62271248374409db31e13a8236ab": "(X_{i},Y_{j})", "208d7198d8fe154f9818bf70544cd742": "\\|f\\|_{p,w}=\\sup _{t>0}~t\\lambda _{f}^{\\frac {1}{p}}(t)", "208dabbf44304ced4c124225fc8dcec0": "x={\\cfrac {1}{1+{\\cfrac {a_{2}}{b_{2}+{\\cfrac {a_{3}}{b_{3}+{\\cfrac {a_{4}}{b_{4}+\\ddots }}}}}}}}={\\cfrac {1}{1-{\\cfrac {r_{1}}{1+r_{1}-{\\cfrac {r_{2}}{1+r_{2}-{\\cfrac {r_{3}}{1+r_{3}-\\ddots }}}}}}}}\\,", "208db2e0f67134f3139d31e6d42a2d32": "(b_{14}-a_{14})+(b_{15}-a_{15})", "208de4a9da7c299c8297125594a9abcc": "p^{*}={\\frac {pL}{\\mu U}}", "208dfb484df3a17a2283ddf04dd68fef": "\\lambda _{n}", "208e54fbbe29178500b9cff00644664b": "\\mathbf {x} ={\\begin{pmatrix}x_{1}\\\\\\vdots \\\\x_{n}\\end{pmatrix}}\\mathbf {r} ={\\begin{pmatrix}r_{1}\\\\\\vdots \\\\r_{t}\\end{pmatrix}}", "208e74d334ffb4457c1d81e6d19665b7": "({X}+i{P})", "208eceaa97e484a14d2ead1ef0d5c170": "u_{z}({\\boldsymbol {x}},z,t)\\,", "208f14f63c4e5e316cd158fc16021a3a": "z=C_{1}-C_{2}x,w=2C_{2}y", "208f606cd2ac0c09099a90673fba102e": "K_{H}(x')=[x'-x_{0}(T)][1-\\exp(-\\beta u_{0})]", "208f9b6d94b8c1ba746970adebfe72be": "\\alpha _{-{\\sqrt {s}}}\\beta _{\\sqrt {s}}\\alpha _{\\sqrt {s}}(x)", "208fa5fb0c777e3607d8aaae87f04e46": "r_{a,t}=\\alpha +\\beta r_{b,t}+\\varepsilon _{t}", "208faa0748518dd4a15ec03052c3ef40": "ds^{2}=\\left({\\frac {2}{1+x^{2}+y^{2}}}\\right)^{2}\\,\\left(dx^{2}+dy^{2}\\right)", "208faf50c7d0d6ff541a5acb72dfe9a0": "x\\,,", "208fe5ba27bf9b429f3e9038eb37f418": "P_{g}", "208fef0aa47c42303fc43171e9208bd6": "G^{-1}=G,G^{0}=G_{0}", "209014186ceb22097909ea08902ad592": "\\log P=A-{\\frac {B}{T}}", "2090ccc580c1018be58d0e475731ed4b": "I_{N}={\\begin{bmatrix}1&0&\\dots &0\\\\0&1&\\dots &0\\\\\\vdots &\\vdots &\\ddots &\\vdots \\\\0&0&\\dots &1\\end{bmatrix}}", "2090dfd4ebff94e15276443d70bce100": "{\\frac {\\Delta Y}{\\Delta L}}.", "2091197a46f36821862906517d736b23": "f(x_{i},{\\boldsymbol {\\beta }}+{\\boldsymbol {\\delta }})\\approx f(x_{i},{\\boldsymbol {\\beta }})+J_{i}{\\boldsymbol {\\delta }}\\!", "20913802caafee6a96a2bd74a09d4404": "\\Delta H_{uv}^{*}", "20914a8b61d19586f9299be423c38444": "\\beta ={\\frac {2\\pi }{\\lambda }}", "20916dabf96955e868230d6e1bcf2256": "\\arcsin x", "20917dd7710034fbb4b8225d13f919d0": "\\lambda _{n}={2L \\over n}\\,,", "2091903a94ea18141c35943959df7409": "R_{2}", "209190b60ca73a4441c11ae1414a845c": "{\\hat {\\beta }}={\\frac {{\\tfrac {1}{T}}\\sum _{t=1}^{T}(z_{t}-{\\bar {z}})(y_{t}-{\\bar {y}})}{{\\tfrac {1}{T}}\\sum _{t=1}^{T}(z_{t}-{\\bar {z}})(x_{t}-{\\bar {x}})}}\\ .", "20919f1863f9c6cb0e51c24baba6adc6": "s-c", "2091a08f9ec1dc2d7c9a5439c4ce0ba0": "a_{p}^{N-1}\\equiv 2^{11350}\\equiv 1{\\pmod {11351}}", "2091a58e31371b3fb52ca32c254a2d1e": "x_{R}^{k}", "2091dd23b300678ba522157d608e8b56": "{\\overline {b}}", "20922aeebb8b2897957287c81f9eecbf": "A(\\lambda _{i})", "209260bc81cdac9309a866a1bd753b29": "orb_{G}(f).", "2092ba9d89943bba987adcf439d63ae2": "1)\\ {\\text{Flow}}=\\sin(5t)", "2092bc5439fcc9101275015c68cfe80b": "x\\leftarrow w\\rightarrow y", "2092db726c76d96d6498bbafd938b1b3": "T\\colon X\\to Y", "2092fb7abc34b184470968282565e956": "w=e^{\\pi {\\mathrm {i} }z}\\,", "209300826abe7aaadad6f689b8e3a3cc": "(\\epsilon \\times c\\times h)", "20935a593906878a804568b25caa17c5": "R_{\\mathrm {eq} }=R_{1}+R_{2}+\\cdots +R_{n}.", "2093fa8bfe7f315d11a92ca192012fb6": "d{\\vec {S}}", "209413243fe2fab39f5a44771dcc5eb1": "\\scriptstyle \\Delta G\\ <\\ 0", "2094deafa65af634815619d9a44c1d61": "g_{ij^{*}}=K_{ij^{*}}={\\frac {\\partial ^{2}}{\\partial z^{i}\\partial {\\bar {z}}^{j^{*}}}}K", "20950550c4f35405aa7e23ce1326af79": "x_{1}...x_{n}", "2095511883bc4525333f9c17a5859189": "{A_{final}}", "20955fbbb092dbb1b32bcc23dcb8eb5a": "h(t-\\tau )={\\frac {1}{m\\omega _{d}}}e^{-\\varsigma \\omega _{n}(t-\\tau )}\\sin[\\omega _{d}(t-\\tau )]", "2095c87259ad731c60cc913b472d1632": "\\operatorname {fact} (n)={\\begin{cases}1&{\\mbox{if }}n=0\\\\n\\cdot \\operatorname {fact} (n-1)&{\\mbox{if }}n>0\\\\\\end{cases}}", "2095dde5a62bdcf100ad3708be0e0846": "x_{1},x_{2},\\dots ,x_{m}\\,", "2095e0c85dee4edec84f5de072e87cb8": "c\\leq a+b", "20961389e3333aee40f8c9be05205c36": "=-x_{0}y_{0}+x_{1}y_{1}+\\cdots +x_{n}y_{n}+x_{n+1}y_{n+2}+x_{n+2}y_{n+1}.", "20964233f5adf84ae8fbfc6caa650166": "P_{\\,e\\,CO_{2}}", "20965d489ba2a92f1914aadab977f2ce": "-\\ln R_{\\text{OC}}", "20967b3bbdf971d29f86509ca172cf0b": "i>k>\\ell >j", "2096db7ddc796bf5e596b3b5cc5b0bec": "||x||<\\delta ", "2097e0fe2c8c767a6a4f5e125b261346": "{\\frac {\\partial h}{\\partial t}}+h\\left({\\frac {\\partial u}{\\partial x}}+{\\frac {\\partial v}{\\partial y}}\\right)-K_{E}T=0,", "209868cdaac76ca50991af2141e79dbd": "\\,\\operatorname {Cl} _{2}(\\theta )-2\\,\\operatorname {Cl} _{2}(\\pi -\\theta )+2\\,\\operatorname {Cl} _{2}(\\pi )", "2098be7861f7fa2e2776049e3034990f": "\\rho =2\\pi a{\\frac {V}{I}}", "2098e6223965e8601f9d526c38ca6d55": "E=nhf,\\quad {\\text{where}}\\quad n=1,2,3,\\ldots ", "2099c0bb59279cbba0ab1911d587e7c8": "(4)\\qquad F_{EW}^{2}M_{\\pi T}^{2}\\cong {\\frac {g_{ETC}^{2}\\langle {\\bar {T}}T{\\bar {T}}T\\rangle _{ETC}}{M_{ETC}^{2}}}\\cong {\\frac {16\\pi ^{2}F_{EW}^{6}}{\\Lambda _{ETC}^{2}}}\\,.", "209ab57524e199179d5a350e411bbd0c": "0\\leq \\pi _{j}\\leq 1.", "209ad8d77bacb41213ac2bd354b6fde6": "\\left|M_{1}\\right|=\\left|M_{2}\\right|=1", "209ad9e23cfcdfbb4fe713bbde018816": "G(1+z)=\\exp \\left[{\\frac {z}{2}}\\log 2\\pi -\\left({\\frac {z+(1+\\gamma )z^{2}}{2}}\\right)+\\sum _{k=2}^{\\infty }(-1)^{k}{\\frac {\\zeta (k)}{k+1}}z^{k+1}\\right]=", "209ae5901db49699801a059981e07f58": "C_{1}|s-t|\\leq |f(s)-f(t)|\\leq C_{2}|s-t|", "209af6ce94b98a80e92fb1c100604c90": "A(T)\\propto (T-T_{c})^{\\alpha }", "209b69c1589a664d3dbfecfed453d313": "\\{e\\}(n)<_{\\mathcal {O}}\\{e\\}(n+1)", "209b6e1499b9f66a28c7999b4a632eb2": "(l/2)\\sin \\theta ", "209b778dbd6169b713f78105b1fd63e2": "\\Delta B_{k}=B'-B=(\\lambda _{k}rB-\\lambda _{k}p)=\\lambda _{k}\\Delta B\\;", "209c3a11e14f5cd7681e9cee9a804c4d": "{\\tfrac {1}{2}}+it", "209c8c791cf564ee2481069eaf517270": "m_{\\text{H}}", "209cd04713b086a562ae4089ac7c3e02": "{161+77 \\over 2}", "209cd1db751ab1d3b7e3fd49f86ed040": "|JKM\\rangle =(D_{MK}^{J})^{*}\\quad \\mathrm {with} \\quad M,K=-J,-J+1,\\dots ,J", "209cdfd9890c7cb6448afe3d5175ea7f": "N^{b}a\\mathrm {inf} _{b}=m/r^{2}\\,", "209d3d43d72ce1cf85533c545f3b502a": "t=B\\left({\\frac {V}{A}}\\right)^{n}", "209d42169b8361e22d2a77f90e7cbd41": "\\|f\\|:=[f,f]^{1/2},\\quad f\\in V", "209d77700b3e63f2670d8e070361c295": "G\\colon D\\to C", "209db172a2921ab26bd2470781425de4": "\\left(2{\\sqrt {\\frac {2}{5}}},\\ 2{\\sqrt {\\frac {2}{3}}},\\ {\\frac {-2}{\\sqrt {3}}},\\ 0\\right)", "209dc3fa4d7c07f0d500ae8fd4129de4": "|Gx|=|G:G_{x}|.\\!", "209ddfc6b9725cf89cc12a22fd2505ad": "\\min \\left\\{D+\\lambda \\cdot R\\right\\}", "209e0435487b47557f289debb4f40904": "3\\rightarrow \\infty ", "209e08a8880a47d34233388fdf123285": "{\\frac {\\partial }{\\partial t}}{\\Bigl (}\\rho \\,h\\,{\\tilde {U}}{\\Bigr )}\\,+\\,{\\frac {\\partial }{\\partial x}}\\left(\\rho \\,h\\,{\\tilde {U}}^{2}\\,+\\,{\\frac {1}{2}}\\,\\rho \\,g\\,h^{2}\\,+\\,S_{xx}\\right)\\,=\\,\\rho \\,g\\,h\\,{\\frac {\\partial d}{\\partial x}}\\,", "209e7cba885bd45ad022868b7e0068ab": "x\\,{\\bmod {\\,}}y=x-y\\left\\lfloor {\\frac {x}{y}}\\right\\rfloor .", "209e7f62d93a6b720b53c8238cfe33a7": "\\mathbf {\\hat {n}} \\,\\!", "209e989a13dc3c8a4905a664732d1b59": "\\Gamma \\vdash B", "209eea4efe796cd199c34e40e2b3ad0a": "{\\frac {\\partial }{\\partial t}}\\left(\\nabla ^{2}\\psi \\right)+{\\frac {\\partial \\psi }{\\partial y}}{\\frac {\\partial }{\\partial x}}\\left(\\nabla ^{2}\\psi \\right)-{\\frac {\\partial \\psi }{\\partial x}}{\\frac {\\partial }{\\partial y}}\\left(\\nabla ^{2}\\psi \\right)=\\nu \\nabla ^{4}\\psi ", "209efd9f67a854ace01461d5fbb434fc": "g\\left({\\tfrac {\\pi }{2k}},s\\right)=0", "209f8221aa83586d6cbefa8904fa375e": "u\\rightarrow \\langle Wu,u\\rangle ", "209f8c323e8c9814e29704d7069a64fc": "g_{\\lambda ,\\mu }", "209f913967396f03eee9fe65b329e5a6": "L(\\mathbf {x} ,s)=G(s)\\otimes I(\\mathbf {x} )", "209fe01340e6ff7f8c5101179362a7f0": "{\\frac {d\\theta _{i}}{dt}}=\\omega _{i}+Kr\\sin(\\psi -\\theta _{i})", "20a0032877c145184fefabfc60f11bd0": "\\Gamma \\left(a,x\\right)", "20a043d0cdb93b03d7bad2aeb8a8ee36": "(2/T)\\tan(\\omega T/2)\\ ", "20a053769e404267a2ae12156ac8dafa": "a\\in P(A)", "20a0638015bc9fdf42a79dd9eb3e4ba3": "c=12k,n_{0}=1", "20a07672786cc81927503b82d25bcad0": "d={\\frac {L}{1-e\\cos \\theta }}", "20a0904afcde68e76cca984559fbe70e": "g:\\mathbb {R} \\to \\mathbb {R} ", "20a0b741e487b744ca07836a9fe38d81": "f(z)=az^{2}+bz+c", "20a0b7e630f0a1cf0af6344ac11f62df": "\\sigma =\\sigma _{s}{\\begin{Vmatrix}{\\dfrac {1}{1+\\beta ^{2}}}&{\\dfrac {-\\beta }{1+\\beta ^{2}}}\\\\{\\dfrac {\\beta }{1+\\beta ^{2}}}&{\\dfrac {1}{1+\\beta ^{2}}}\\end{Vmatrix}}", "20a119fabb991c8890a168765b20da50": "\\omega _{\\mu }\\xi ^{\\mu }=0", "20a12d82dbaaf5e08ad45b32441a8dd9": "\\ d[x,x^{*}]", "20a157c4bbe3d768becd7f7a7a0293da": "a_{n+1}={\\frac {a_{n}+{\\frac {2}{a_{n}}}}{2}}.", "20a1669b2a01139280a8d3587e683ae4": "{\\begin{aligned}&F_{r}=J_{3}\\ {\\frac {1}{r^{5}}}\\ 2\\ \\sin \\lambda \\ \\left(5\\sin ^{2}\\lambda \\ -\\ 3\\right)\\\\&F_{\\lambda }=-J_{3}\\ {\\frac {1}{r^{5}}}\\ {\\frac {3}{2}}\\ \\cos \\lambda \\ \\left(5\\ \\sin ^{2}\\lambda \\ -1\\right)\\end{aligned}}", "20a1909eeafdc138c7fa99b12684226f": "f(\\mathbf {x} )=f(x_{1},x_{2},\\dots ,x_{N})=\\sum _{i=1}^{N/2}\\left[100(x_{2i-1}^{2}-x_{2i})^{2}+(x_{2i-1}-1)^{2}\\right].", "20a1a63a1b2852836a29bc68db220567": "2^{\\mathfrak {c}}", "20a1b334b1266b6e4506a69af1020ea1": "D_{\\mathrm {KL} }(P\\|Q)\\geq 0,\\,", "20a1d58ffcc7687155924937659432c4": "Ld=(Nlex/N)*100", "20a1e1d8cfad3f526ad7c27d2ed4e240": "J_{2}\\,\\!", "20a227b6d92677d03653bb6114b03213": "5\\cdot 2\\quad {\\text{or}}\\quad 5\\,.\\,2", "20a22dc9d88cd27ea9374b693ff0379d": "q^{n-1}+1", "20a230756ee08c0172213ee817f5ad80": "0\\to (I^{n}M\\cap M')/I^{n}M'\\to M'/I^{n}M'\\to M/I^{n}M\\to M''/I^{n}M''\\to 0,", "20a272e26348790364d474801503037c": "\\alpha =(\\alpha _{1},\\dots ,\\alpha _{N})\\in \\mathbb {N} ^{N},", "20a28def80a071ab78c7f5adceb3dec3": "\\textstyle k_{f}", "20a2c2b5c075bcd541ccbc5cd98e3b9f": "M\\leq {\\frac {5N_{2}}{6N}}\\log _{2}N_{2}", "20a30b85a6d73094e7182ac2da584d71": "\\left({\\frac {\\partial U}{\\partial x}}\\right)_{y}=T\\left({\\frac {\\partial S}{\\partial x}}\\right)_{y}-P\\left({\\frac {\\partial V}{\\partial x}}\\right)_{y}", "20a310c2d1b01188e45a7cfcbb4015a5": "m_{a}\\mathbf {u} _{a}+m_{b}\\mathbf {u} _{b}=\\left(m_{a}+m_{b}\\right)\\mathbf {v} \\,", "20a32afb81ddb586733e34d97b4102d9": "H^{\\dagger }W^{\\mu \\nu }W_{\\mu \\nu }H/\\Lambda ^{2}", "20a33039235de3fbcafec0890c0e1f23": "A(P)={\\text{true}}\\,", "20a33076c14333cca00dae49ab668c7e": "X_{k}={\\frac {(-1)^{k}}{2}}x_{N-1}+\\sum _{n=0}^{N-2}x_{n}\\sin \\left[{\\frac {\\pi }{N}}(n+1)\\left(k+{\\frac {1}{2}}\\right)\\right]\\quad \\quad k=0,\\dots ,N-1", "20a34d575190cd2f81fbb0fceac31ca2": "({\\sqrt {2}}/2,{\\sqrt {2}}/2)", "20a35deccc43bc3649a30db2a05459c5": "{\\bar {\\mu }}_{i}=\\mu _{i}+z_{i}F\\Phi ", "20a3ea17c77f9cc20f66e0fd5947ab39": "l_{2}\\,", "20a41d766611e10616be2d515661c6ca": "\\forall {x}{\\in }\\mathbf {X} \\,P(x)\\to \\ P(c)", "20a42a5dde88287861c70c71083e04c7": "{\\begin{aligned}\\det(I+A)=\\sum _{k=0}^{\\infty }{\\frac {1}{k!}}\\left(-\\sum _{j=1}^{\\infty }{\\frac {(-1)^{j}}{j}}\\mathrm {tr} (A^{j})\\right)^{k}\\,,\\end{aligned}}", "20a42ee1a5e60ad864ff5b9ff00d70ee": "10^{-15}", "20a44d27f648ac287bb99a4fac421306": "\\left({\\frac {du}{d\\varphi }}\\right)^{2}=r_{s}\\left(u-u_{1}\\right)\\left(u-u_{2}\\right)\\left(u-u_{3}\\right)", "20a457360e28737ac4b5e8ad4da5a04e": "x_{2}=l_{2}\\|r_{2}", "20a4691fdac93a7c2752215d7583fbd8": "P_{i}={\\frac {\\partial L}{\\partial {\\dot {Q}}_{i}}}.", "20a46a3e73e1b7f12946865fac62c6b4": "\\zeta \\,", "20a46d50bf5985db93fed1b6d8132d8a": "x^{2}=x+1\\,.", "20a4e88e0dcf3ec01fde74300eee2c17": "\\varphi (y,x_{1},\\ldots ,x_{n})\\,,", "20a5584ee629eebc4e7cc21cf7e10e76": "argmax_{x_{i}}U_{i}(x_{i})-w_{i}", "20a559c816dd998ed51efc6eeafdcf6f": "{\\frac {\\partial ^{\\alpha }u}{\\partial t^{\\alpha }}}=K(-\\triangle )^{\\beta }u.", "20a579522374bfc5abecb9104f95fa4f": "\\beta (3)\\;=\\;{\\frac {\\pi ^{3}}{32}},", "20a5814ca881712da464550aa9f1e4b0": "\\gamma _{\\mathrm {n} }", "20a5843ff628cb806e9cbaa6e4564f72": "\\displaystyle d=|W|^{-1}\\prod _{\\alpha >0}\\alpha .", "20a593338e6c44cb42fc416c9595fb9c": "f(x)=a^{x}{\\bmod {N}},", "20a6a6d5c2b2a5d46d69823465819be1": "\\theta (t)t\\mathrm {e} ^{-\\gamma t}", "20a750aa7e547a07ca92d82beaab7762": "H^{s,p}(\\mathbb {R} ^{n}):=\\left\\{f\\in L^{p}(\\mathbb {R} ^{n}):{\\mathcal {F}}^{-1}\\left(1+|\\xi |^{2}\\right)^{\\frac {s}{2}}{\\mathcal {F}}f\\in L^{p}(\\mathbb {R} ^{n})\\right\\}", "20a7bcc8c96e0dd950a5b609684a01cf": "w\\mapsto R(w+1)/(w-1)", "20a7d3666e8b3881d5f647d2cbb68960": "S(t)=\\Pr(T>t)=\\int _{t}^{\\infty }f(u)\\,du=1-F(t).", "20a7f792c1d57dbffad5b060d69dd636": "p(t)=p_{0}+p_{1}t+p_{2}t^{2}+\\cdots +p_{n}t^{n}.", "20a7fa65c56a8e19071add5fab21d4d0": "({\\hat {b}})", "20a828cfd0116783d64e4ba6b53a1093": "h(n)=\\sum _{alltiles}distance(tile,correctposition)", "20a88ac3888a899d399311da6d255677": "P={\\frac {\\sigma }{\\rho _{0}}}", "20a897e52793d8a753747acc6a49ccae": "\\pi (s')P(s',s)=\\pi (s)P(s,s')\\,.", "20a8a0d25273d0aa5c6cd9f9a593150d": "\\left\\vert t_{0}\\right\\vert >t_{a/2,n-1}", "20a8b67b292fffe9c5c0497dd4c88d4f": "{\\frac {1}{\\sqrt {2}}}\\left(\\left|\\uparrow \\downarrow \\right\\rangle -\\left|\\downarrow \\uparrow \\right\\rangle \\right)", "20a8c5af26c8467c25578b3c290f7d3c": "c(e)", "20a8f1d6bfce22843ae66bc937214aff": "0\\leq \\left\\|x-\\sum _{k=1}^{n}\\langle x,e_{k}\\rangle e_{k}\\right\\|^{2}=\\|x\\|^{2}-2\\sum _{k=1}^{n}|\\langle x,e_{k}\\rangle |^{2}+\\sum _{k=1}^{n}|\\langle x,e_{k}\\rangle |^{2}=\\|x\\|^{2}-\\sum _{k=1}^{n}|\\langle x,e_{k}\\rangle |^{2},", "20a900fd9e7b00c046e5c3383cc6043b": "\\sum _{i,j,k}n_{i,j,k}=n", "20aabe43c0d85ccfa2e0569c14a4f2fa": "r(\\chi )", "20ab11ea5dac33395e8bcd91b5139a35": "t\\!:\\!1=()", "20ab5788949fb57a8079d320118b75fc": "Y_{1,0}=\\omega _{e}", "20aba0e07a6973414bbb80a9bc38ec44": "Z_{\\mathrm {L} }=Z_{\\mathrm {S} }^{*}.", "20abe4fe92c6ff63ce8662baa381b4e8": "RS_{i}=h_{i}^{t}\\left(g_{i}-G_{i}\\right)", "20abf4a461caa1aaf33cdb4ba7f2f817": "T\\!", "20ac1091b574264f19428c5e8a16cf18": "\\scriptstyle \\{E_{n}\\}_{n\\in \\mathbb {N} }", "20ac2604c5f01012e0ae891f821889e2": "K_{i}={\\frac {K_{p}}{T_{i}}}", "20ac642354edd3b8c533da348a2a7d4f": "\\tau _{0}", "20aca1d0e1d213f44a364112c73b3733": "\\mathrm {Var} (A)/\\mathrm {Var} (P)", "20acb8dc532569a72c5300bb12c2a64b": "\\Omega _{e}", "20ad0beb992b77b965467a7ce969fe7b": "{\\dot {Q}}/T,", "20ad1a1315fb26f3855b0fa1ecf448f8": "\\mathbb {S} \\subset {\\mathcal {P}}(X)", "20ad3def259e01a1988d612f560b9261": "(2\\pi )^{-p/2}\\det(\\Sigma )^{-1/2}", "20ad3ee341af55b0f7e85294b37858cf": "\\pi _{1}>{\\frac {1}{2}}(\\pi _{0}+\\pi _{1})", "20ad5e53e4d83d825c1b82028c029bf9": "S_{2}\\wr S_{4}", "20ad69def1d8850663d3f7ff3c6bad87": "{\\tilde {u}}(x,z,t)", "20ae02bcb7f2aca2f71c7f3222859cf5": "f''(x)>0\\,\\!", "20ae0ca384c2b6db18f2015758f20e44": "\\Delta _{1}^{0}", "20ae67cc67cece0d01cc618deda067c4": "\\mathrm {NaCl} _{(s)}\\leftrightarrow \\mathrm {Na} _{(aq)}^{+}+\\mathrm {Cl} _{(aq)}^{-}", "20ae81b3ec202d2f45a2b086df3ad10e": "K\\otimes _{\\mathbf {Q} }\\mathbf {R} ", "20aedc820017e944714601d421c802bc": "F_{n+1}(x,y+1)=F_{n}(F_{n+1}(x,y),F_{n+1}(x,y)+y+1),\\ n\\geq 0.\\,", "20aef2db5978094f6a4f342cf37bb8ff": "\\nabla \\times \\mathbf {F} (\\mathbf {r} )=\\mathbf {0} {\\text{.}}", "20af07b3b4880fac6c9ad98a5f167fc5": "\\ell _{2}={\\frac {x-x_{0}}{x_{2}-x_{0}}}\\cdot {\\frac {x-x_{1}}{x_{2}-x_{1}}}={\\frac {x-2}{5-2}}\\cdot {\\frac {x-4}{5-4}}={\\frac {1}{3}}x^{2}-2x+{\\frac {8}{3}}\\,\\!", "20af3400ab1736816d0f9182f38bdb4f": "\\prod _{r=1}^{m}(k_{r}^{+})^{\\lambda _{r}}=\\prod _{r=1}^{m}(k_{r}^{-})^{\\lambda _{r}}\\,.", "20af5b95dee8298982ba48d50301b8d7": "\\sum _{i}p_{i}=1", "20af69574ebf361b959ad65f73e21c5e": "K,D,\\epsilon ", "20af8d60b7ef72d5e54813776f6ec9e6": "n_{i,\\downarrow ,\\uparrow }", "20af94dc731f882792321b75bcaa5f40": "\\ln \\left({\\frac {1}{3}}\\right)={\\frac {-1}{RC}}{\\frac {T}{2}}", "20b0a85c57598e8a84c691b40efab1c2": "\\textstyle P(K)\\geq 1-\\varepsilon .", "20b0af8cc41c4a218b9df8a52c44ae63": "\\sum _{m=1,3,5,\\ldots \\leq n}{\\frac {2}{m^{2}}}S_{n,m}=\\sum _{m=2,4,6,\\ldots \\leq n}{\\frac {2}{m^{2}}}S_{n,m}\\quad \\left(n>2\\ {\\text{is even}}\\right).\\ ", "20b0f2beecf44c57d7f48b8f7425cea7": "{\\frac {\\partial }{\\partial {\\boldsymbol {A}}}}\\left({\\boldsymbol {A}}^{-T}\\right):{\\boldsymbol {T}}=-{\\boldsymbol {A}}^{-T}\\cdot {\\boldsymbol {T}}^{T}\\cdot {\\boldsymbol {A}}^{-T}", "20b1029d22d53d8c55b969ae3fac4ab1": "^{99m}Tc", "20b139859ee6ebb1959a029f89b72c69": "\\omega _{3}^{0}\\approx 2.018236", "20b199e0fa0e7ccf734ae90645c6274b": "E(x,v)={\\tfrac {1}{2}}|v|^{2}+V(x)", "20b1d149bb56e5a0a98a6daf5eb85d0a": "x^{2}=a^{2}", "20b2922f9886239ebd293c2c3aaf72ad": "\\Delta _{3}", "20b2ea8f6d11644de7777aaec3ddfab4": "S\\circ \\theta ", "20b31398cd6cb929f79cd77b4fc4dc8e": "{\\frac {d^{2}y}{dx^{2}}}+[a-2q\\cos(2x)]y=0.", "20b31b2a6b9cc5155ed8be0a34b2267a": "v_{e}\\approx {\\sqrt {k_{B}T_{e}/m_{e}}}", "20b32f3e1f9c08ff1061937a3bc6f9a4": "\\sigma _{e}", "20b339b1d01b344d8425414a7361b189": "I_{O}", "20b3434c2b08d6836f709fac78cb5535": "\\delta W_{b}", "20b38cad687f3a42223a6b76ee618584": "\\mathrm {P} =V_{\\mathrm {rms} }^{2}={\\frac {A^{2}}{2}}", "20b3b43fa0a5232467c736032ae5872f": "{\\frac {x-x_{0}}{x_{1}-x_{0}}}", "20b3c312e73d36724d0da0236da891e0": "|O_{Jac(C)}(2\\Theta _{C})|\\cong \\mathbb {P} ^{2^{2}-1}", "20b5a430be9d21dbf6426a473694c8e3": "Q(z,u,v)=\\exp \\left({\\frac {v}{u}}\\left({\\frac {zu}{1}}+{\\frac {z^{2}u^{2}}{2}}+{\\frac {z^{3}u^{3}}{3}}+{\\frac {z^{4}u^{4}}{4}}+{\\frac {z^{5}u^{5}}{5}}+\\cdots \\right)\\right)", "20b5b39134571ced128a48a17c99239b": "\\log n=H_{0}\\geq H_{1}\\geq H_{2}\\geq H_{\\infty }", "20b5bc423a9465ea9e92b3b548de636e": "\\log _{e}(4)=1.386\\ldots ", "20b5d88e4ae28e345fdc41a7981795be": "(x-y)^{2}<0", "20b620923ab918a6f2b7a0eb419f8fc4": "c_{3}", "20b651ac31475acd10f2960db5a7b906": "D_{\\mathrm {EO} }", "20b681f83c861cda77a086148916d0c8": "c=c(r)\\,=\\,{\\frac {1-(r-1)^{2}}{4}}", "20b695481bc167da71c5cd2e1f37798a": "\\ t^{*}", "20b6cb72922664013f0cdccae982f913": "{\\mathcal {F}}", "20b6d9304641233858e5a753de073b9d": "{\\mathcal {D}}=\\left\\{(\\mathbf {x} _{i},y_{i})\\mid \\mathbf {x} _{i}\\in \\mathbb {R} ^{p},\\,y_{i}\\in \\{-1,1\\}\\right\\}_{i=1}^{n}", "20b6db30395adbefd322cfcdfbe75568": "L=\\{[a,b]|a,b\\in R\\}\\cup \\{[a]|a\\in R\\}\\cup \\{[\\infty ]\\}", "20b6e64a7fa44c24f451be4d922f083e": "((a),[c,d])\\in I\\Longleftrightarrow a=c", "20b749114df2b1c07fbc352b74f080e9": "f^{-1}[0]", "20b7a52f8f715aaef45597e9c69389aa": "(p(x)-\\beta )", "20b805bd59e258906141ccdfb2b14da5": "f(\\mathbf {x} ),\\quad \\mathbf {x} =(x_{1},x_{2},...,x_{d})", "20b822946db0592437bac132e7c8c2e1": "i_{0},i_{1},...,i_{n-1}", "20b892eb9ba2d63443af1b8b0cea3a42": "B={\\frac {u^{2}}{1024}}\\left\\{256+u^{2}\\left[-128+u^{2}(74-47u^{2})\\right]\\right\\}", "20b8a05d21a9e5375159943d884cdc3e": "a_{W}=D_{w}+\\max(F_{w},0)", "20b8c531784bd9b94acd3847a085f5d9": "t.i\\in T_{r}", "20b8cd5baaf102c114df2b6d230219a7": "\\tau (y)", "20b8d3b7b3a7299ab26c0faea4c1c6dc": "({\\mathbf {x}},t)\\mapsto (G{\\mathbf {x}},t)", "20b94052ecc1f9710fd7cb8860090a52": "\\lambda _{j}=-{\\frac {4}{h^{2}}}\\sin({\\frac {\\pi (j-{\\frac {1}{2}})}{2n+1}})^{2}", "20b9a5134bfb3eff46e780f9be0aee43": "\\liminf P(n)/\\log n=1", "20b9c4f885244e3fb95721fe0cede08d": "Q>0", "20ba34dd04da9010b0f8a9e48a0ead5d": "H^{i}(j_{x}^{!}C)\\neq 0", "20ba8e458cd798886cdf940d0c03f0fe": "\\kappa <\\operatorname {cf} (2^{\\kappa })\\,", "20babae90780fc6b00ddff0308d30750": "v_{\\text{out}}=\\alpha _{1}(A_{1}\\sin \\omega _{1}t+A_{2}\\sin \\omega _{2}t)+\\alpha _{2}(A_{1}^{2}\\sin ^{2}\\omega _{1}t+2A_{1}A_{2}\\sin \\omega _{1}t\\sin \\omega _{2}t+A_{2}^{2}\\sin ^{2}\\omega _{2}t)+\\ldots \\,", "20bb0f2f1c51c05f8d7fe46d1cfe15fa": "V_{w}=f^{-1}(w)", "20bb162142fabbf69624ad8edf5f4909": "\\mathbf {E(r)} ={\\frac {1}{4\\pi \\varepsilon _{0}}}\\int {\\frac {\\rho (\\mathbf {r'} )\\left(\\mathbf {r} -\\mathbf {r'} \\right)}{\\left|\\mathbf {r} -\\mathbf {r'} \\right|^{3}}}\\mathrm {d^{3}} \\mathbf {r'} ", "20bb45adbec97d06c584996d95e61f5b": "v=\\left(\\left(g^{u_{1}}y^{u_{2}}\\right){\\bmod {\\,}}p\\right){\\bmod {\\,}}q", "20bbe5d77ead81de0f42a83de89e1551": "{\\mathcal {B}}=A_{0}B_{0}+A_{0}B_{1}+A_{1}B_{0}-A_{1}B_{1}", "20bc064261b7a7ef94a243c2d3055af1": "{\\frac {Z}{r}}", "20bc4de03db1afd161ee457adbbe77d5": "g={\\tfrac {1}{2}}mw_{i}w_{i}", "20bc7929074736fce1ec19d3c4014dd8": "k_{2}=\\gamma /J_{2}", "20bc9b46b67fd66d7dbaa60eeb4211a1": "G_{1}=\\langle U,D,L^{2},R^{2},F^{2},B^{2}\\rangle ", "20bca72a4078dda835a09d04a42b6ac7": "\\psi \\rightarrow e^{-i\\sigma _{z}\\omega _{r}t/2}\\psi ", "20bcb091288350de78be686f978e0af7": "P_{A}O_{2}=P_{I}O_{2}-{\\frac {V_{T}}{V_{T}-V_{D}}}(P_{I}O_{2}-P_{E}O_{2})", "20bd0cde5d9b9ca34b7eac666c36f489": "j=(3/2)N-2.\\!", "20bd73670af509549323edec0ecc2908": "\\left\\lfloor 2^{2^{2^{\\cdot ^{\\cdot ^{2^{\\omega }}}}}}\\!\\right\\rfloor \\scriptstyle {\\text{= primes:}}\\displaystyle \\left\\lfloor 2^{\\omega }\\right\\rfloor \\scriptstyle {\\text{=3,}}\\displaystyle \\left\\lfloor 2^{2^{\\omega }}\\right\\rfloor \\scriptstyle {\\text{=13,}}\\displaystyle \\left\\lfloor 2^{2^{2^{\\omega }}}\\right\\rfloor \\scriptstyle =16381,\\ldots ", "20bd79c7e8fc0332f739c4b23024405d": "\\pi _{1}(\\mathbb {R} ^{3}\\backslash K).", "20bda037fbe16c7c412958bd780c75b2": "vs^{2}=(\\mathbf {y} -\\mathbf {X} {\\hat {\\boldsymbol {\\beta }}})^{\\rm {T}}(\\mathbf {y} -\\mathbf {X} {\\hat {\\boldsymbol {\\beta }}}),{\\text{and}}\\;v=n-k,", "20bdca70df2cb0f14692db9a49459d13": "dz=dx+idy=ds(\\cos \\phi +i\\sin \\phi )=ds\\,e^{i\\phi }\\qquad \\Rightarrow \\qquad d{\\bar {z}}=e^{-i\\phi }ds.", "20be63df70211320461d7c28274fa158": "{\\mathcal {B}}(X_{\\sigma }^{*},Y_{\\sigma }^{*})", "20bece015bc0b4532a910251c85d944d": "K", "20bed1a202baffaf13072a180c841a75": "Y=c_{1}X_{1}+\\cdots +c_{N}X_{N},", "20bed5031a50dde7a2a01a4527c8a532": "\\textstyle 2", "20bf01a92dbb20e410671da659b63f42": "{\\mathcal {S}}=\\int _{t_{1}}^{t_{2}}\\int {\\mathcal {L}}(\\mathbf {r} ,t)\\mathrm {d} ^{3}\\mathbf {r} \\mathrm {d} t.", "20bf4ae8960323bcc6729f73a0f12e4f": "k(\\cdot ,\\cdot )", "20bf643f86666317010a2c125c415722": "\\int {\\frac {mx+n}{ax^{2}+bx+c}}\\,dx={\\begin{cases}\\displaystyle {\\frac {m}{2a}}\\ln \\left|ax^{2}+bx+c\\right|+{\\frac {2an-bm}{a{\\sqrt {4ac-b^{2}}}}}\\arctan {\\frac {2ax+b}{\\sqrt {4ac-b^{2}}}}+C&{\\text{(for }}4ac-b^{2}>0{\\mbox{)}}\\\\[12pt]\\displaystyle {\\frac {m}{2a}}\\ln \\left|ax^{2}+bx+c\\right|-{\\frac {2an-bm}{a{\\sqrt {b^{2}-4ac}}}}\\,\\mathrm {arctanh} {\\frac {2ax+b}{\\sqrt {b^{2}-4ac}}}+C&{\\text{(for }}4ac-b^{2}<0{\\mbox{)}}\\\\[12pt]\\displaystyle {\\frac {m}{2a}}\\ln \\left|ax^{2}+bx+c\\right|-{\\frac {2an-bm}{a(2ax+b)}}+C&{\\text{(for }}4ac-b^{2}=0{\\mbox{)}}\\end{cases}}", "20bf7d15202b1aee628da0448e04249e": "ID(x,y)=\\min\\{|p|:p(x)=y\\;\\&\\;p(y)=x\\},", "20bfa168ce610d00f699745357feac58": "U(a,L)^{\\dagger }A(x)U(a,L)=S(L)A(L^{-1}(x-a)).", "20bfa47aa60668c81d7b30f2bed6f111": "x_{i+1}=x_{j+1}", "20bfeaa797f4eea5ac0f3b4e63251947": "\\sigma _{\\hat {X}}^{2}", "20c0096d84b3cc07999e442af53c51bc": "\\sum _{n=1}^{\\infty }f_{n}(x)", "20c0315b91a6c38d14a7027727ea99df": "{d \\over dt}\\left\\{X_{1}\\right\\}=\\left\\{A\\right\\}+\\left\\{X_{1}\\right\\}^{2}\\left\\{Y_{1}\\right\\}-\\left\\{B\\right\\}\\left\\{X_{1}\\right\\}-\\left\\{X_{1}\\right\\}+D_{x}\\left(X_{2}-X_{1}\\right)\\,", "20c035aa42b5810ac4b8697a376b1ffa": "{\\hat {S}}", "20c0a109edb71ec82aa27d1ab30d2b10": "f_{i}:X_{i+1}\\to X_{i}", "20c0c2a43c0526784fd7990f0cace32e": "\\min J_{2}(w,b,e)={\\frac {\\mu }{2}}w^{T}w+{\\frac {\\zeta }{2}}\\sum \\limits _{i=1}^{N}{e_{c,i}^{2}},", "20c14e48ea891a030419ad510a091f3d": "J_{G}={\\begin{bmatrix}3&\\sin(x_{2}x_{3})x_{3}&\\sin(x_{2}x_{3})x_{2}\\\\8x_{1}&-1250x_{2}+2&0\\\\-x_{2}\\exp {(-x_{1}x_{2})}&-x_{1}\\exp(-x_{1}x_{2})&20\\\\\\end{bmatrix}}", "20c1b341b3e5f08cd3aabce814a68fb2": "C(r,z)=\\int _{0}^{\\infty }G(r'',z)r''\\left[\\int _{0}^{2\\pi }S\\left({\\sqrt {r^{2}+r''\\,^{2}-2rr''cos\\phi ''}}\\right)\\,d\\phi ''\\right]dr''\\qquad (4)", "20c2a133fc1344dcbe39ac49205bc6f0": "{3\\ln \\left(3\\right)}-{\\pi {\\sqrt {3}} \\over 3}", "20c313bc356ccfa9910fe693f4ad7e90": "E={\\frac {(\\hbar k)^{2}}{2m}}", "20c337d75e048c578e64dd7edf0b9c52": "Z(S_{n})=\\sum _{j_{1}+2j_{2}+3j_{3}+\\cdots +nj_{n}=n}{\\frac {1}{\\prod _{k=1}^{n}k^{j_{k}}j_{k}!}}\\prod _{k=1}^{n}a_{k}^{j_{k}}", "20c3436d03583003315ea675d5937609": "S_{N}f", "20c408bbe4c0287d4208f9021733e784": "w(m+n)\\leq w(m)w(n),\\quad w(0)=1", "20c4506ca1a74fd02ee0db3bce103947": "K(u,v)={-4 \\over a^{2}b^{2}\\left(1+{4u^{2} \\over a^{4}}+{4v^{2} \\over b^{4}}\\right)^{2}}", "20c49e9494e4d9ac6129823e949bb499": "|c_{11}|\\geq |c_{12}|+|c_{13}|", "20c4d77dedb38bf0037b4541b7678bfb": "[abababbca]_{D}", "20c536c76ec00fc5629ff6706225106f": "r_{i}\\in R", "20c57af8d1b0993791bf95f30dd376e3": "{x_{1}+x_{2}+\\cdots +x_{n} \\over n}\\geq {\\sqrt[{n}]{x_{1}x_{2}\\cdots x_{n}}},", "20c57b8eaa12808def61c7485cc6f513": "\\mathrm {d} E=-\\mu _{0}M_{s}\\int _{V}(\\mathrm {d} \\mathbf {m} )\\cdot \\mathbf {H} _{\\text{eff}}\\,\\mathrm {d} V", "20c5919ba51759e1facf7e26c315eeeb": "{\\frac {1}{(1-P)+{\\frac {P}{S}}}}={\\frac {1}{(1-0.3)+{\\frac {0.3}{2}}}}=1.1765", "20c5a5694e53f4089d67ceabe80d4185": "\\varepsilon \\ {\\stackrel {\\mathrm {def} }{=}}\\ {\\frac {hc}{2L}}n,", "20c5b2c7c9c1212a92cbe33244a0ba3c": "\\alpha (u)e^{\\pi H(z,u)+H(u,u)\\pi /2}\\ ", "20c5cc83f43c6278e02526dc2b68d7d9": "{\\frac {\\mathrm {d} \\mathbf {M} }{\\mathrm {d} t}}=-\\gamma \\mu _{0}\\mathbf {M} \\times \\mathbf {H_{eff}} +{\\frac {\\alpha }{M}}\\left(\\mathbf {M} \\times {\\frac {\\mathrm {d} \\mathbf {M} }{\\mathrm {d} t}}\\right)", "20c67f1c30d70d528f7f47c6379cfd46": "T\\Delta S_{int}-\\Delta H\\geq 0\\,", "20c67f6f4616c675e0c79631c6758535": "A_{v}={\\begin{matrix}{\\frac {v_{out}}{v_{S}}}={\\frac {R_{L}}{R_{S}}}\\end{matrix}}", "20c6ddd38f53084219eb44ebb5db226b": "p,\\,q\\geqslant 0,\\,p+q=1,\\,p=q", "20c72299ca7a678df883523be23130ef": "\\ln 2=3\\ln \\left(1+{\\frac {1}{4}}\\right)+\\ln \\left(1+{\\frac {3}{125}}\\right)={\\cfrac {6}{9-{\\cfrac {1^{2}}{27-{\\cfrac {2^{2}}{45-{\\cfrac {3^{2}}{63-\\ddots }}}}}}}}+{\\cfrac {6}{253-{\\cfrac {3^{2}}{759-{\\cfrac {6^{2}}{1265-{\\cfrac {9^{2}}{1771-\\ddots }}}}}}}}.", "20c7804ad15bd45ec87e4f57cfd0455d": "r=n_{1}/n_{2}", "20c79511c10764bf5b3c841563c1ad54": "{\\vec {p}}_{i}", "20c7e667ae456bb7f99e6432344c57d8": "\\varphi (t)=\\int _{\\mathbf {R} }g(t+\\theta ){\\overline {g(\\theta )}}d\\theta .", "20c8017f0ec6b1f09ec5d3ada1d387fb": "S=F", "20c855fce927bc79b3005831c5e396b6": "e^{-\\alpha t}", "20c8ce4143446376fb48d1766507c0b4": "I\\mapsto \\Gamma _{I}", "20c8f80b4052c67d08f2fde427397642": "T'(t)=-\\lambda \\alpha T(t),", "20c90d0049e2805935fa6624a7851ff7": "K={\\frac {\\det II}{\\det I}}={\\frac {LN-M^{2}}{EG-F^{2}}}.", "20c92a5711dc3cfc4cbb239400989e45": "440{\\rm {{Hz}\\cdot ({\\sqrt[{12}]{2}})^{-19}\\approx }}", "20c935729081116936e1028c9a0ac22f": "\\textstyle \\left({\\frac {m_{k+2}-m_{k+1}}{\\sigma _{k+1}}}\\right)^{T}\\!C_{k}^{-1}{\\frac {m_{k+1}-m_{k}}{\\sigma _{k}}}\\approx 0", "20c980b08233e5e56e3618389d84c401": "{\\mathcal {F}}", "20c9ae9cc379ee8869915a7a59cbac3a": "a/b\\geq 0.3", "20ca18f64d855fafba1388f9754fa9ff": "f(i)=v_{i}", "20ca2061bd29f4acd103fbb51febee03": "(-1)^{n}\\,Z\\left(g_{n}\\right)>0", "20ca601153d0257d8301cf041dcc75b8": "\\int \\sec ^{2}{x}\\,\\mathrm {d} x=\\tan {x}+C", "20ca9a3d4b18cda9f2435c5ef0db734e": "R_{N}^{k}(n)", "20caa0c4e7dc6781e52cea13ab24c431": "\\scriptstyle BV_{loc}(\\Omega )", "20caa2174db477f7e70f1e0521411b1d": "H=T", "20cb0bd46f91db8d4035ea9410549614": "\\alpha {\\hat {x}}-i\\beta {\\frac {\\partial }{\\partial x}}", "20cb2b8185895d37704f276ca7ffa4c9": "{\\widehat {\\mathbf {S} }}={\\frac {\\hbar }{2}}{\\boldsymbol {\\sigma }}", "20cba01b64876ff7813889d00f872ef3": "\\operatorname {logit} ^{-1}(\\alpha )={\\frac {1}{1+\\operatorname {exp} (-\\alpha )}}={\\frac {\\operatorname {exp} (\\alpha )}{\\operatorname {exp} (\\alpha )+1}}", "20cbabe7b7b66592f9bb126bc3bba84c": "2s=2^{1}+2^{2}+2^{3}+\\cdots +2^{63}+2^{64}.", "20cc7e4af4ff3c37a73652b3daffe414": "A=\\sum _{k=0}^{r}a_{k}D_{x}^{k}", "20cc86b07fda93e7f5beaf4d818cb916": "K\\geq 4r^{2}", "20cc95ecc649d54bfac200a9de4c9d3a": "{\\begin{aligned}I(x,y)&={\\frac {1}{e}}\\cdot \\lim _{(\\xi ,\\eta )\\to (x,y)}{\\sqrt[{\\xi -\\eta }]{\\frac {\\xi ^{\\xi }}{\\eta ^{\\eta }}}}\\\\[8pt]&=\\lim _{(\\xi ,\\eta )\\to (x,y)}\\exp \\left({\\frac {\\xi \\cdot \\ln \\xi -\\eta \\cdot \\ln \\eta }{\\xi -\\eta }}-1\\right)\\\\[8pt]&={\\begin{cases}x&{\\text{if }}x=y\\\\[8pt]{\\frac {1}{e}}{\\sqrt[{x-y}]{\\frac {x^{x}}{y^{y}}}}&{\\text{else}}\\end{cases}}\\end{aligned}}", "20ccb3f2890a1717c1a82836c908f6ed": "[-1,1]\\subseteq \\mathrm {Supp} (\\rho )", "20cceca1ea9ebf5e96033ac7c935ef44": "\\Gamma _{L}=\\left(Z_{L}-Z_{0}\\right)/\\left(Z_{L}+Z_{0}\\right)", "20cd0007c70777b8a741bfcf078ddd0c": "|\\mathbf {a} |=|\\mathbf {r} (t)|\\left({\\frac {\\mathrm {d} \\theta }{\\mathrm {d} t}}\\right)^{2}=r{\\omega }^{2}\\ ", "20cdaf79975da27f6d7805988de90cc4": "D^{\\mathrm {BW} }=\\bigotimes _{r=1}^{2j}\\left[D_{r}^{(1/2,0)}\\oplus D_{r}^{(0,1/2)}\\right]\\,.", "20cdbb1f9187cacde999eaf2e22872ed": "n,f=0,1", "20cde1c4bdf2968a7179cef9df71c509": "f(x)=\\Omega _{-}(g(x))", "20ce7fa8448ea84ee6b413a6cee13a39": "\\ell _{f}", "20ce92eb8aa8c082a577c8bdceff718e": "\\lim _{t\\to \\infty }\\sup _{x\\geq t}f(x)^{2}~-~f(1)^{2}=\\limsup _{t\\to \\infty }\\int _{1}^{t}(f(x)^{2})'\\,\\mathrm {d} x", "20ce97c91c124e7cd5c89325fe05d341": "\\scriptstyle \\{x_{1},\\,x_{2},\\,\\ldots ,\\,x_{N}\\}", "20ceb395355c2b74b998a7ce3c3ebdee": "\\sigma _{xx}=0~,~~\\sigma _{yy}=0~,~~\\sigma _{zz}=0~,~~\\tau _{zx}=0~,~~\\tau _{yz}=\\mu (z)\\,{\\frac {dV}{dz}}\\,\\exp[i(kx-\\omega t)]~,~~\\tau _{xy}=ik\\mu (z)V(k,z,\\omega )\\,\\exp[i(kx-\\omega t)]\\,.", "20ced19c091d56592b7672ffd717c229": "J_{\\mu }=-v^{2}\\partial _{\\mu }\\theta ~.", "20cf05f6d27c0e78cd8fcebea0c361ee": "\\angle (U'MV')", "20cf129cd0789619a8de032f0a11be24": "C=Q/V", "20cf2bfd7a8e8ae19dd02f2b94f71fe0": "{\\mathcal {G}}_{\\alpha _{1}\\ldots \\alpha _{n}|\\beta _{1}\\ldots \\beta _{n}}^{(n)}(\\tau _{1}\\ldots \\tau _{n}|\\tau _{1}'\\ldots \\tau _{n}')=\\langle T\\psi _{\\alpha _{1}}(\\tau _{1})\\ldots \\psi _{\\alpha _{n}}(\\tau _{n}){\\bar {\\psi }}_{\\beta _{n}}(\\tau _{n}')\\ldots {\\bar {\\psi }}_{\\beta _{1}}(\\tau _{1}')\\rangle ", "20cf53a41ee43600b05be1321ab55c83": "H_{{\\frac {1}{4}},2}=16-8G-{\\tfrac {5}{6}}\\pi ^{2}", "20cf7f599291741f768ab0a868bbd6b0": "\\rho ={\\frac {s}{1+c}},", "20cf8f453b412b5182e12b4d296219b6": "{\\textbf {E}}=k{\\frac {q}{r^{2}}}{\\hat {r}}", "20cf8fbe230129cb7649e46075914b6e": "0\\leq \\alpha ,\\gamma \\leq 2\\pi ", "20cf940f82b8c291365cc57df94a8855": "a_{m}={\\frac {1}{(n^{2}-4\\cdot m^{2})\\cdot a_{m-1}}}", "20cf9e4d0af1a4d443c4b7557e3a4801": "\\beta =-1/2", "20cfadf5d2ce6a1eb1905b9a5680ef87": "\\cos x/x", "20d030ef843b639bb3abfc4fd2c0aaac": "\\{0,1\\}^{d}", "20d03ff568918a64596dcebf9b67e243": "1+r={\\frac {M_{2}-C_{2}}{M_{1}}}", "20d0481557295b669184875b9819bedb": "a\\in \\mathbb {A} ", "20d06d7aa143953677590f94d3c1f746": "{\\rm {tr}}\\left(\\left({\\frac {\\partial g(\\mathbf {U} )}{\\partial \\mathbf {U} }}\\right)^{\\rm {T}}{\\frac {\\partial \\mathbf {U} }{\\partial X_{ij}}}\\right)", "20d077c69f6963ac0ba5da366ce2a923": "a+ar+ar^{2}+ar^{3}\\cdots ", "20d0851abebbc3844348605389aebf6a": "\\sum _{k=0}^{40}-{\\frac {1}{2k+1}}{\\frac {\\sin(2k+1)x}{\\left|\\sin(2k+1)x\\right|}}", "20d095e37702a33ea5751dee71ab3c5d": "U[]\\to \\epsilon ", "20d0cb9db0e0158dc37a127e6dc5ecc6": "{\\frac {d\\mathbf {E} _{1s}}{d\\zeta }}=\\zeta -\\mathbf {Z} =0", "20d0e274357ecad4683762bd23f2af91": "\\kappa \\leq \\beth _{\\alpha }(\\mu ).", "20d159120deba8433025b88ad0412f2f": "{\\tilde {K}}", "20d15e139938616554a8b836dc0bf312": "l_{j}(x):=\\prod _{\\begin{smallmatrix}i=0\\\\j\\neq i\\end{smallmatrix}}^{n}{\\frac {x-x_{i}}{x_{j}-x_{i}}}", "20d172080f660ac3d58312127d2d6a3f": "i,j,k", "20d177acdb4cbbe32c7efbb92200bafc": "log_{e}(1+x)=x-{\\frac {1}{3}}x^{3}+{\\frac {1}{5}}x^{5}-{\\frac {1}{7}}x^{7}+........", "20d17e960003bc8d6f7e253c9cc9cb5d": "F(x)+R(x)", "20d19303a5605bb10b28d7ff01de48df": "N_{c}={\\frac {30}{\\pi }}*{\\sqrt {\\frac {g}{\\delta _{st}}}}", "20d193762effc4b034c3a40cac7fe965": "\\{p_{1},\\ldots ,p_{n}\\}", "20d19aced1cf6b50e7396ea98b01ec4b": "2+{\\frac {3}{4}}=2{\\tfrac {3}{4}}", "20d268de25571167bcaf59c0e1620db8": "e\\cdot p_{cv}={1 \\over V}\\int _{v}e^{i(k_{p}+k-k')\\cdot r}{u^{*}}_{ck'}(r)e\\cdot (p+\\hbar k)u_{vk}(r)d^{3}r", "20d293a9d465c6550229b114a3167474": "A^{k}", "20d2c204a22cf4fcea332d24ced5da12": "-\\pi \\leq \\theta \\leq \\pi ", "20d2d4f4ae963686eeab302a75c4f6d0": "H:V\\longrightarrow G", "20d2da23cedade95311c2bbad8a20081": "(xy)z=e", "20d2ead06a023e5482b1d98175f488d9": "{\\sqrt {\\sum _{x\\in R}\\left[F(x+h)-G(x)\\right]^{2}}}", "20d2f0395dde46c301322dc09a22c049": "A(Z)", "20d2f715850fc500d5539e505954c2d0": "{\\hat {1}},{\\hat {2}},{\\hat {3}}...", "20d2ffc9f78da23d6e917cf339b84f95": "(\\lambda b.(\\lambda v.b(v)(b)))", "20d393454bfe0f00c745866e9d92a118": "{q_{v}}", "20d43b66a2e94d95b5f023e9766e48b8": "(\\mathbf {A} \\lor \\mathbf {B} )\\land \\mathbf {C} =0,", "20d454fa9095f9e40799379ee6767546": "|h(x)-h^{*}(x)|=O(\\log h^{*}(x))", "20d4a7120fbc6352ee55ca710b0ffef4": "(x^{15}+3x^{14}-16x^{13}-50x^{12}+94x^{11}+310x^{10}-257x^{9}-893x^{8}+366x^{7}+1218x^{6}-347x^{5}-717x^{4}+236x^{3}+128x^{2}-56x+4).", "20d4c9addfa527f384b56406b8050346": "\\mathrm {d} s^{2}=-(1+2\\Phi )\\mathrm {d} t^{2}+\\alpha ^{2}(1-2\\Psi )\\delta _{ij}\\mathrm {d} x^{i}\\mathrm {d} x^{j}\\,", "20d509482e96c8775462390fd8b82213": "\\mathbf {A} \\mathbf {B} -\\mathbf {B} \\mathbf {A} \\neq \\mathbf {0} .", "20d52abc09173f86219fb55221cbe607": "\\varepsilon =\\omega ^{\\varepsilon },\\,", "20d53966e8be0803570e7a0ea0cfbf7e": "\\left(T^{(\\mathbf {n} )}\\right)^{2}=T_{i}^{(\\mathbf {n} )}T_{i}^{(\\mathbf {n} )}=\\left(\\sigma _{ij}n_{j}\\right)\\left(\\sigma _{ik}n_{k}\\right)=\\sigma _{ij}\\sigma _{ik}n_{j}n_{k}.", "20d541b011ff78bc440d2cfbf8ad887b": "R=V/I.\\,\\!", "20d54e562a8b26ae2d07db119a34415d": "r^{n}~\\sin(n\\theta )\\,", "20d5560f2544cbe795e33e94283944e7": "\\Omega ^{p,q}\\xrightarrow {\\overline {\\partial }} {\\mathcal {F}}^{p,q+1}\\xrightarrow {\\overline {\\partial }} {\\mathcal {F}}^{p,q+2}\\xrightarrow {\\overline {\\partial }} \\cdots \\,", "20d560b50a6e9a9a6c9a295e42e00a29": "\\displaystyle iu_{t}+L_{1}u=\\phi u", "20d5613f94bff9d298999ad004515694": "{\\begin{matrix}\\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\\\\\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\\\10{\\mbox{ multiplied copies of }}10\\end{matrix}}", "20d5c45fa36239071fc39f9da0b64b56": "\\langle \\sigma v\\rangle ", "20d5e793fe2bae035c61ebc175ca100f": "P=(2w+t+x)/(2g)", "20d6404b6699189eaace5e299743a287": "X=\\{x_{1},...,x_{n}\\}", "20d64e44f690a99cab045a75f4945c4e": "H_{n}=\\sum _{1\\leq k\\leq n}k^{-1}", "20d66aff35587e970f1594bf04d9ab2a": "r=k_{2}C_{S}", "20d6ce2bfe89316407fe33f37f5f4057": "\\kappa _{0}=\\left(1-e^{2}\\right)^{-1}.", "20d6d9712e7752c095c696b1ae8546d3": "\\log _{2}(n)", "20d6ec8acf2f9cb3251c8627b07cf434": "z(2,{\\sqrt {3}})={\\frac {1}{2\\cdot 2\\cdot {\\sqrt {3}}}}{\\sqrt {(2)^{2}+({\\sqrt {3}})^{2}}}={\\frac {1}{4{\\sqrt {3}}}}{\\sqrt {7}}\\,,", "20d6fa8706af2f0b4d35728d4f246a59": "[A,B]_{D}=[A,B]+d\\langle A,B\\rangle ", "20d75a4511a598975f5b08db2d51a778": "E={\\frac {\\hbar ^{2}\\pi ^{2}n^{2}}{2mL^{2}}}={\\frac {n^{2}h^{2}}{8mL^{2}}}.", "20d767482f2b6d8a9ba0ae2795986dfd": "K=2000", "20d79abe87b4fe200a9f4fb8c99809e5": "f_{X}(x|\\theta )=\\exp \\left(\\eta (\\theta )\\cdot T(x)-A(\\theta )+B(x)\\right)", "20d7a7d2b7b46e18837cec0b98de23a1": "p(x)\\in {\\mathbb {R}}_{+}", "20d7d7fb0c9eb438a0c0231c6f03c370": "\\operatorname {Br} (k)=\\mathbb {Q} /\\mathbb {Z} ", "20d7e92c66c971b964b9d516481faa72": "N=\\sum _{i=1}^{c}m_{i}", "20d81c5cd67e4e6607548b46f849d5c4": "\\operatorname {corr} (U,V)", "20d832093010c3145e67518f2d373212": "\\scriptstyle -{\\frac {2}{5}}", "20d867527ce0138b640d18c18998a01f": "d\\alpha |S|\\,", "20d8b96be2eb751c30ee5f07ef79b9da": "{\\frac {\\partial V}{\\partial t}}", "20d8fc0c8edfc3f982a3a87a913b9193": "(a,b,c,d,\\ldots ,y,z)=(a,b)\\cdot (b,c,d,\\ldots y,z)", "20d902e1c18ed79cb5014a9cffc3b7af": "y-p(x)", "20d9062d72216393b3f4e3f4712d8794": "{\\mathbb {R}}\\times M", "20d911b659d6fe0e2ff505a55afa4a41": "\\mathbf {a} \\cdot \\mathbf {b} =a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}+a_{4}b_{4}+a_{5}b_{5}+a_{6}b_{6}.", "20d9395ee2c26340c3472ea724441d14": "l_{i}", "20d99ad067aa972f3caa30d66ad0571e": "r\\leftarrow min(L(e)-i,m+16)-16", "20d9db3f2cde18e3bad04d5f0e0041b9": "\\omega >5", "20d9f8e44e17b9fffc6c1d5c2699d050": "\\ell (s)=m", "20da2ba13fd330de7fd25dbc076eba9d": "{\\underline {F}}=F_{z}{\\underline {z_{o}}}=L^{TM}(z)T^{TM}(x,y){\\underline {z_{o}}}\\ \\ \\ \\ \\ \\ (11)", "20da3f2895d89a7b6d88a22ff2c862d5": "{\\mathfrak {c}}", "20da8f5e5ea8d8a3919479d21d7e600e": "\\cos(y)=x\\ \\Leftrightarrow \\ y=\\arccos(x)+2k\\pi {\\text{ or }}y=2\\pi -\\arccos(x)+2k\\pi ", "20dab6b69cfe9451cb4270bbf30e917a": "a=q^{h+k-1}{\\frac {\\sin h\\alpha }{\\sin \\alpha }}=q^{k}\\cdot \\sum _{0\\leq i\\leq {\\frac {h-1}{2}}}(-1)^{i}{\\binom {h}{2i+1}}p^{h-2i-1}(q^{2}-p^{2})^{i},", "20db790a20fa81c5222dbd2535251d58": "", "20dbc545ed110cf28853e2cffff404fa": "\\operatorname {Var} (X)=\\operatorname {E} [(X-\\operatorname {E} (X))^{2}].", "20dbd027f37eb9ae2ae420388f05976c": "S_{2}(x,y)", "20dc355b5839b6667f4f68bfda9e7f3b": "|f|:=f^{+}+f^{-}", "20dc50a57ecb7832263b29328a648e47": "\\chi _{k}^{2}\\!", "20dca66d2ad86a50c3dd4e109943a45a": "\\psi _{i}\\circ \\psi _{j}^{-1}\\in {\\rm {Aff}}({\\mathbb {R}}^{n})", "20dcbf7b35b5b91f95d171ebd7c15668": "{\\begin{aligned}d\\mathbf {x} ^{2}-d\\mathbf {X} ^{2}&=d\\mathbf {X} \\cdot \\mathbf {C} \\cdot d\\mathbf {X} -d\\mathbf {X} \\cdot d\\mathbf {X} \\\\&=d\\mathbf {X} \\cdot (\\mathbf {C} -\\mathbf {I} )\\cdot d\\mathbf {X} \\\\&=d\\mathbf {X} \\cdot 2\\mathbf {E} \\cdot d\\mathbf {X} \\\\\\end{aligned}}\\,\\!", "20dcf6e9c4706a97d4dcc5740057631f": "(\\lambda _{k})^{n/2}\\sim {\\frac {(2\\pi )^{n}k}{\\omega _{n}Vol(M)}}.", "20dd02ee0c556fe3161f2c8cd88b6b6b": "{\\begin{smallmatrix}\\alpha =\\arccos \\left({\\frac {c}{a}}\\right)\\,\\,\\!\\end{smallmatrix}}", "20dd07091086404ca1ad129bbdc56ff0": "2mK_{\\mathrm {NR} }(p)={i \\over (p_{0}-m)-{{\\vec {p}}^{2} \\over 2m}}", "20dd67fbd6baf6ab5f921defa3d1fab5": "\\pi _{1}*u_{i}(x_{1i})+\\pi _{2}*u_{i}(x_{2i})", "20ddb8f45d952b2b89f6e83b42f6e106": "b_{2}=0.51465", "20dde788ca784c85072583be4f1c1bc2": "G/H", "20de10b3bb137e5e4a140200a998eab0": "SH={{0.622p_{(H_{2}O)}} \\over {p-0.378*p_{(H_{2}O)}}}.", "20de642efff12648d361ef057e0e6f22": "{\\begin{aligned}0&\\leq (x-y)^{2}\\\\&=x^{2}-2xy+y^{2}\\\\&=x^{2}+2xy+y^{2}-4xy\\\\&=(x+y)^{2}-4xy.\\end{aligned}}", "20dec4a500cd77c0439ab9a7e9d0d593": "P_{hs}={\\frac {R\\,T}{V_{m}}}\\,{\\frac {1+\\eta +\\eta ^{2}-\\eta ^{3}}{(1-\\eta )^{3}}}", "20ded0edb3b9ea15104fafcb11aa63e1": "\\lambda _{1}=\\lambda ~;~~\\lambda _{2}={\\cfrac {1}{\\lambda }}~;~~\\lambda _{3}=1", "20df44ef93547d846dff479b1e50221c": "O(N^{K+1}\\,K\\,T)", "20df632a6718c25c2ba1d19116c506f4": "r_{O}={\\frac {v_{ce}}{i_{c}}}{\\Bigg |}_{v_{be}=0}", "20e0297ca80d4aec3ae36dc9ca05f984": "{\\frac {\\partial ^{2}u}{\\partial t^{2}}}-c^{2}{\\frac {\\partial ^{2}u}{\\partial x^{2}}}=0", "20e031c66d65a43ca472dd567f875046": "H^{i}(K,V)\\times H^{2-i}(K,V^{\\prime })\\rightarrow H^{2}(K,\\mathbf {Q} _{p}(1))=\\mathbf {Q} _{p}", "20e068b26e271fa140694d1fd26e47fe": "P_{k}(x)=f(a)+f'(a)(x-a)+{\\frac {f''(a)}{2!}}(x-a)^{2}+\\cdots +{\\frac {f^{(k)}(a)}{k!}}(x-a)^{k}", "20e081d6fb099f1f8a5ba23fbb4d02a1": "D_{\\alpha \\beta }:=\\int _{-h}^{h}x_{3}^{2}~C_{\\alpha \\beta }~dx_{3}\\,.", "20e0b52e6d3e923b93e49d6e76ae95ef": "s\\models _{K}f_{1}\\land f_{2}", "20e0bf7c2a3e16c86607cfbc5936c365": "\\mathrm {*} ", "20e0dcdef543077c4f381bdc541f60c8": "\\tau _{y}", "20e0df549f6996e9f1b5a006e70e74b0": "(\\exists x(x^{2}=1))\\land (0=x)", "20e0ec70a6a972a34857e889c7107fbf": "a^{\\dagger }(\\phi _{i})", "20e18000db1a74a9ad1701f6525c66d8": "\\left\\{A_{i}:i\\in I\\right\\}", "20e203719b50e9c60a5beb721a2abcd3": "p_{X}(x)", "20e3b3320ec256387c12e3a417c8e15b": "a^{2}=(c-d)^{2}+h^{2},", "20e3d8da75328aac4f408a5453232f12": "\\varphi \\left(n\\right).", "20e454e6e64e161875b69f7f7d4167bf": "2\\cdot 3\\cdot 5\\cdot p+1", "20e4771cda51db739d2329c563d59a2e": "\\Pi ^{1}(f_{1},f_{2})=\\{f_{1},f_{2}\\}={\\frac {\\partial f_{1}}{\\partial q}}{\\frac {\\partial f_{2}}{\\partial p}}-{\\frac {\\partial f_{1}}{\\partial p}}{\\frac {\\partial f_{2}}{\\partial q}}~,", "20e4b1ce4cef9b637a7966fda43fdef5": "\\mathbb {F} _{5}[x]/(f)", "20e4bd895f17797637f5fd8a9283346a": "a<^{d}(A_{i})a'\\iff (a<_{i}a')", "20e4ddef904130f6591f68da27414d00": "F_{n}(x)={\\frac {1}{n}}\\sum _{i=1}^{n}\\mathbf {1} _{\\{X_{i}\\leq x\\}},\\qquad x\\in \\mathbb {R} .", "20e4e0380e23aa4f093495700facc65d": "y_{i}={\\frac {\\sin(2^{i}x)}{2i}}{\\text{ where }}0\\leq i\\leq n-2{\\text{ and }}i\\in \\mathbb {N} .", "20e4fbc46786f1717f5bdb2ddbfd2bde": "bc0", "20f0c094631ad021ea89a7c1d38a4c7f": "An=m", "20f0c768a98d8b235bf0632aab83776e": "\\|g\\|=\\left(\\int _{a}^{b}g^{*}(x)g(x)\\,dx\\right)^{1/2}", "20f1393be4efac6e7e12e5bead09c36d": "f(U)\\,", "20f1857ecb28af03362514f373883caf": "\\scriptstyle {\\frac {p}{q}}", "20f1a09a3c069fdf0fe3e5bfab14068a": "K_{\\text{SE}}(x,x')=\\exp {\\Big (}-{\\frac {|d|^{2}}{2l^{2}}}{\\Big )}", "20f1a2931818d3d66250da25dad395b6": "\\tan 2\\theta _{0}={\\frac {K\\sum _{k=0}^{K-1}\\sin 2\\omega t_{k}-2\\left(\\sum _{k=0}^{K-1}\\cos \\omega t_{k}\\right)\\left(\\sum _{k=0}^{K-1}\\sin \\omega t_{k}\\right)}{K\\sum _{k=0}^{K-1}\\cos 2\\omega t_{k}-{\\big (}\\sum _{k=0}^{K-1}\\cos \\omega t_{k}{\\big )}^{2}+{\\big (}\\sum _{k=0}^{K-1}\\sin \\omega t_{k}{\\big )}^{2}}},", "20f1a322fdf387ae8d55344af2420f8e": "M^{+\\bullet }", "20f1e3ed4c495dc14fab17ea99c757db": "\\gamma _{\\mathrm {la} }\\ >\\ 0\\ >\\ \\gamma _{\\mathrm {ls} }-\\gamma _{\\mathrm {sa} }", "20f1fd21cc52a413261f5785db47d395": "K_{eq}^{A}=K_{eq}^{A,0}\\mathrm {e} ^{-(\\Delta H_{ad}^{0}\\,\\alpha _{T}\\,\\theta /k\\,T)}", "20f20e28d803a40b26b36c30a091f9e0": "(a*(a+1))*(a+2)", "20f215e7f6d0ce2da52b45114602170b": "\\langle p\\rangle ", "20f21ddfebdc5fe8545d810077835346": "s_{pm}=i\\,\\mathrm {cd} (w,1/\\xi )\\,", "20f22d577843609e922439f5cc25199e": "\\varphi (x)=k_{0}n(x)L=k_{0}L[n+n_{2}I(x)]", "20f257992ca15946e44f09e6a1c75a70": "{\\begin{aligned}\\iint _{D}&{\\sqrt {({\\vec {r}}_{u}\\cdot {\\vec {r}}_{u})({\\vec {r}}_{v}\\cdot {\\vec {r}}_{v})-({\\vec {r}}_{u}\\cdot {\\vec {r}}_{v})^{2}}}\\,du\\,dv\\\\&\\quad =\\iint _{D}{\\sqrt {EG-F^{2}}}\\,du\\,dv\\\\&\\quad =\\iint _{D}{\\sqrt {\\operatorname {det} {\\begin{bmatrix}E&F\\\\F&G\\end{bmatrix}}}}\\,du\\,dv\\end{aligned}}", "20f27b2c7fc9ea3ef1929069fec0c678": "G\\not =H.", "20f28e5eab4848360edb8d12ed7e10af": "\\cos {(\\omega t+\\phi )}\\ ", "20f2b5df65abf7eabf1a33e774da9b1e": "V^{2}-U^{2}=\\left(1-{\\frac {r}{2GM}}\\right)e^{r/2GM}", "20f312ee72f7c11b18aa61f8bec45901": "{c}_{1},{c}_{2}\\ =0\\ or\\ 1", "20f318bcdff4f6671c214a5879c6f2e5": "\\omega :TM\\to T^{*}M,", "20f3289d8ea77e7fa9e383bca697fe8f": "{\\color {Red}{\\tfrac {3}{m}}}", "20f32ebdcb84b5c527f71f06f65e47b3": "{\\frac {6(\\alpha ^{3}+\\alpha ^{2}-6\\alpha -2)}{\\alpha (\\alpha -3)(\\alpha -4)}}{\\text{ for }}\\alpha >4\\,", "20f337d4f696c19aef8092a04a3d2bcd": "-10\\leq x,y\\leq 10", "20f345c02d0d0cfe94e015beea799a87": "\\scriptstyle kE_{b}", "20f389c6c2e5772002e158267812736f": "L_{\\infty }=0", "20f3a2bf12102a1b2bea413f120bc892": "T_{0}=T_{1}\\left({\\frac {p_{0}}{p_{1}}}\\right)^{R/c_{p}}\\equiv \\theta .", "20f3b0ad27d8efefa70322e562bed881": "{~\\wedge \\!\\!\\!\\!\\!\\!\\bigcirc ~}", "20f3b928b89c294fd4d77e5207cdb2e5": "C_{m}=\\lambda _{p}\\alpha \\beta t", "20f3c6f714eca082ebe481740c044aaa": "\\Delta \\mathbf {B} =\\mu _{\\Delta }\\Delta \\mathbf {H} .", "20f40b40f35d779133c4ce09a9b04a33": "0.0003207\\times weight{\\mbox{(g)}}^{(0.7285-0.0188\\log _{10}{weight\\mathrm {(g)} })}\\times H^{0.3}", "20f462cf01b3cd1486e2f778344f188b": "n_{ij}", "20f48d7a2997abd00ab3ce9150044f8e": "g\\approx Gm_{s}/R^{2}", "20f4f84a176d1a289dfe885b7329230d": "{\\frac {AF}{FB}}={\\frac {AF'}{F'B}}.", "20f4fc9845a6d15afc7325d0c8de7100": "\\delta _{D}(k,i)=\\lambda \\delta _{D}(k-1,i)+{\\frac {e(k,i)e_{b}(k,i)}{\\gamma (k,i)}}", "20f565b0e375e4530e9e06572d540815": "W_{i}(x_{j})=e^{-{\\frac {(r_{i}(x_{j})+s)^{2}}{c}}}", "20f582adbd88049d88a7e3fc28b77abd": "S_{i+1}(c)={\\begin{cases}T(N_{i}(c)),&c\\in I\\\\S_{i}(c),&c\\in \\mathbb {Z} ^{k}\\setminus I\\end{cases}}", "20f5bb7768a2835572ec8afd83908156": "\\scriptstyle {\\mathcal {P}}=\\left\\{P=\\{x_{0},\\dots ,x_{n_{P}}\\}|P{\\text{ is a partition of }}[a,b]\\right\\}", "20f63e25158b1a65da45158667578138": "x(\\theta )=r(k+1)\\cos \\theta -r\\cos \\left((k+1)\\theta \\right)\\,", "20f65f5ce12aba395d6bde91fad3fbcc": "2n-3j-3=0", "20f68045237ddb0c784f763d2dfcbcc7": "\\Pr(X=x)=\\sum _{y}\\Pr(X=x,Y=y)=\\sum _{y}\\Pr(X=x|Y=y)\\Pr(Y=y),", "20f69ee8b55dfa42fbcd834a08e51e6d": "\\displaystyle {E=E_{0}(e)\\oplus E_{1/2}(e)\\oplus E_{1}(e),}", "20f6c3cf5ba601cb6b721de9000c9fcf": "-\\omega _{p}", "20f708c6b908a93c04614f84ec9755f8": "A={\\begin{bmatrix}0&1\\\\0&0\\\\\\end{bmatrix}},", "20f72e3a694f33b819326f512e033e8a": "\\mu ={\\frac {1}{n}}\\sum _{v\\in V}f(v)", "20f733ab3007d15c80c69b09ba23cb7a": "\\sum _{j=1}^{p}a_{j}+\\sum _{j=1}^{q}b_{j}=\\sum _{j=1}^{m}c_{j}+\\sum _{j=1}^{n}d_{j},", "20f75d80400b9fad74bf07698f1f233c": "\\scriptstyle a^{2}\\,+\\,2", "20f794b4581f8eb8790f9de4506a79be": "\\!v_{1}", "20f7a05bbd8892a2883a8187a92a8a59": "\\Gamma ({\\tfrac {1}{8}})\\approx 7.5339415987976119047", "20f7d74d908ea49f9ae41a4fe8f99407": "\\mu =\\cos(\\theta )", "20f7e12a8a2fde1243dba86c544f4a16": "\\nu _{e},\\nu _{\\mu },\\nu _{\\tau }", "20f7e76830a65a841b88e32259f08c02": "A^{-1}[j]", "20f7ee512c044c85911c9c71c1abd4c1": "\\leq \\sum _{a^{n},b^{n}\\in T_{\\delta }^{\\mathbf {p} ^{n}},\\ b^{n}\\neq a^{n}}\\Pr \\left\\{E_{a^{n}}\\right\\}2^{-\\left(n-k\\right)}", "20f865f3d198abbeedbde0c70f886dcb": "f^{-1}(I)=\\{x\\in X\\,|\\,f(x)\\in I\\}.", "20f86b6a2b331ecfbcc72e9da3cfbb78": "|\\lambda -a_{ii}|=\\left|{\\frac {\\sum _{j\\neq i}a_{ij}x_{j}}{x_{i}}}\\right|\\leq \\sum _{j\\neq i}\\left|{\\frac {a_{ij}x_{j}}{x_{i}}}\\right|\\leq \\sum _{j\\neq i}|a_{ij}|=R_{i}", "20f88010d58f526bc934d1be41e9f6a2": "{\\begin{array}{rcl}{\\dot {x}}&=&Ax+b\\varphi (\\theta _{e}),\\\\{\\dot {\\theta }}_{e}&=&\\omega _{e}+g_{v}(c^{*}x).\\\\\\end{array}}\\quad x(0)=x_{0},\\quad \\theta _{e}(0)=\\varphi _{0}.", "20f8927b47392d78410d77c1f9fea521": "\\mathbf {B} (\\mathbf {r} ,t)={\\frac {\\mu _{0}}{4\\pi }}\\left({\\frac {qc({\\boldsymbol {\\beta }}\\times \\mathbf {n} )}{\\gamma ^{2}(1-\\mathbf {n} \\cdot {\\boldsymbol {\\beta }})^{3}|\\mathbf {r} -\\mathbf {r} _{s}|^{2}}}+{\\frac {q\\mathbf {n} \\times {\\Big (}\\mathbf {n} \\times {\\big (}(\\mathbf {n} -{\\boldsymbol {\\beta }})\\times {\\dot {\\boldsymbol {\\beta }}}{\\big )}{\\Big )}}{(1-\\mathbf {n} \\cdot {\\boldsymbol {\\beta }})^{3}|\\mathbf {r} -\\mathbf {r} _{s}|}}\\right)_{t_{r}}={\\frac {\\mathbf {n} (t_{r})}{c}}\\times \\mathbf {E} (\\mathbf {r} ,t)", "20f8c201a44fe1396fdaf93ca731c6a6": "e^{2\\pi i(\\pm \\alpha +k)/N}", "20f955c07e2b78ce401cb5957f562829": "{\\begin{aligned}\\lambda &={\\frac {3x_{p}^{2}+a}{2y_{p}}}\\\\x_{r}&=\\lambda ^{2}-2x_{p}\\\\y_{r}&=\\lambda (x_{p}-x_{r})-y_{p}\\end{aligned}}", "20f9785ef496bbc909e3bdb4cbb672a5": "\\beta >0.", "20f9cc840114df83f38f208b082a24f9": "\\delta \\rightarrow 0", "20fa156064ac8e04951081b1b13a24de": "\\rho =\\mathrm {Diag} (1-T,T/(d-1),\\dots ,T/(d-1))\\,", "20fa6aa9e7878378c2d71004716b15f7": "a+Ls_{1}+Ls_{2}...+Ls_{D-k}", "20fafdce54af27770f7b3ef6278b5812": "{\\begin{aligned}\\sum f_{1}&=F_{1}+\\int _{\\alpha }^{\\beta }\\left[\\sigma _{rr}(a,\\theta )~\\cos \\theta -\\sigma _{r\\theta }(a,\\theta )~\\sin \\theta \\right]~a~d\\theta =0\\\\\\sum f_{2}&=F_{2}+\\int _{\\alpha }^{\\beta }\\left[\\sigma _{rr}(a,\\theta )~\\sin \\theta +\\sigma _{r\\theta }(a,\\theta )~\\cos \\theta \\right]~a~d\\theta =0\\\\\\sum m_{3}&=\\int _{\\alpha }^{\\beta }\\left[a~\\sigma _{r\\theta }(a,\\theta )\\right]~a~d\\theta =0\\end{aligned}}", "20fb4d8ede3411cc607eda7243facd2d": "{\\bar {t}}\\phi (P)=2{\\bar {q}}P", "20fb4da8a6c902f16c462cdc9262eab6": "A\\Vert c_{jk}\\Vert _{l^{2}}^{2}\\leq {\\bigg \\Vert }\\sum _{jk=-\\infty }^{\\infty }c_{jk}\\psi _{jk}{\\bigg \\Vert }_{L^{2}}^{2}\\leq B\\Vert c_{jk}\\Vert _{l^{2}}^{2}\\,", "20fb90202322cf9249d954b2c9dfe49c": "T_{j}={\\begin{cases}0,&{\\text{if }}f'(j){\\text{ is an integer}};\\\\\\min \\left({\\frac {1}{||f'(j)||}},{\\sqrt {U}}\\right),&{\\text{if }}||f'(j)||\\neq 0;\\\\\\end{cases}}", "20fba7355d3343ae5512ac81dbc86711": "\\textstyle B\\subset \\mathbb {R} ", "20fbf4588c21b11d252391ab537df1f8": "\\operatorname {dCov} _{n}^{2}(X,Y):={\\frac {1}{n^{2}}}\\sum _{j,k=1}^{n}A_{j,k}\\,B_{j,k}.", "20fc23b3c8c153837ab352af17737d64": "2^{2^{k}}", "20fc9bd90179cc1e0efed843ad24baaa": "\\mu _{0}<{\\frac {x+m}{2}}\\pm k|x-m|", "20fcafe0b441c06f880bebb0bf710f03": "as_{k}+bt_{k}=r_{k},", "20fcc2b0b0db33ad9173adf8c846b2b6": "i,j\\in [n]", "20fcfc90e076ffd4c3383e1cf53e54ed": "f_{j}(x)=x^{m_{j}}", "20fd09ec99f9adf3093962f9014bd228": "\\forall a\\forall b\\;a\\vee (a\\wedge b)=a", "20fdd72c3baf5f669923c415149b7bca": "{1 \\over 2}\\ln(a/b)+\\ln 2=\\ln {\\bigl (}2{\\sqrt {a/b}}{\\bigr )}.", "20feaada7405d1e5d23377966f803dd9": "v\\in C", "20fed57f6034f18c49ae6be9d1b5f2d5": "I_{1}=\\lambda _{1}^{2}+\\lambda _{2}^{2}+\\lambda _{3}^{2}=\\lambda ^{2}+{\\cfrac {1}{\\lambda ^{2}}}+1~;~~I_{2}={\\cfrac {1}{\\lambda _{1}^{2}}}+{\\cfrac {1}{\\lambda _{2}^{2}}}+{\\cfrac {1}{\\lambda _{3}^{2}}}={\\cfrac {1}{\\lambda ^{2}}}+\\lambda ^{2}+1", "20fedcb1ee65fa5b298b9f81ff297d9d": "e(n)=\\|x(n)-{\\widehat {x}}(n)\\|\\,", "20ff348f7148c598a3116aab8a6efbc7": "{\\frac {36}{12}}=3", "20ff553787d7e49312e41b129a86394f": "1.00U(\\$1{\\text{ M}})<0.89U(\\$1{\\text{ M}})+0.01U(\\$0{\\text{ M}})+0.1U(\\$5{\\text{ M}}),\\,", "20ff593c848a70c0dabb59b9ad133748": "p(n)\\approx {n^{2} \\over 2m}", "20ff69eab81f54bda6a85e7a571105ae": "\\delta (w,C)>\\delta ", "20ffc407a7df1e1a33a10467ebab7083": "{\\frac {\\mathrm {d} P}{\\mathrm {d} T}}={\\frac {PL}{T^{2}R}}.", "20ffc5333da43150cce288425448048e": "{\\frac {1}{\\lambda }}=RZ^{2}\\left({\\frac {1}{n_{1}^{2}}}-{\\frac {1}{n_{2}^{2}}}\\right)", "21002a065f857b236bdeb31eeb5903f9": "N,T", "21005d7e8daa3cbad69503aa1690b27b": "\\scriptstyle \\mathbf {F} _{\\mathrm {B} }", "21008212015dff44f37d38b5323be66c": "A(G(U_{\\ell }))", "210085fa8d344c8d675a5a95c14160b0": "\\mu (E)=1\\,", "2100a14a01f7575769649a657dc173e8": "\\lambda ={\\frac {4(\\beta ^{2}-1)}{h^{2}}}.", "2100cba1a2d20db5e948a8ee6b375efc": "m=30,s_{\\Lambda }=72.82", "2101799bfb33d9ecb0ddfb6d9a945c04": "U\\in SU(2)", "2101a0dae11357eadf8218353debbb13": "\\sum _{i=1}^{N}", "2101c52ab5c2165a9ab05e436e5c1225": "\\ F_{propulsive}=-drag\\times cos(\\beta )", "2101e755479497032bdb2be4540a733d": "\\forall n:\\sum _{i_{n}=1}^{I_{n}}w_{n,i_{n}}(x_{n})=1", "2101faf5535187deafb25b1f29820a5b": "N-1", "210213e0226d4a0a771d39cad2f8be36": "\\int _{-\\infty }^{\\infty }{\\frac {e^{ix}}{1+x^{2}}}\\,dx=2\\pi i\\,\\operatorname {Res} (f,i)\\,.", "210232a31406fa33daaa96c8891b893d": "f'(r)^{-1}", "21024fda36099e6e51162b85632c4050": "f(\\partial X)=f(X)\\cap \\partial Y", "21025654afc9c9d35b1c6c001c0ab0c2": "\\ v_{i}={\\sqrt {\\frac {2GMd}{r^{2}}}}\\ ", "21025746c635a505f874340700583b32": "f\\in \\Lambda ^{0}(M)", "21029821e28fff21e7e513a03cd7935b": "\\mathrm {Var} [X_{i}]={\\frac {\\alpha _{i}(\\alpha _{0}-\\alpha _{i})}{\\alpha _{0}^{2}(\\alpha _{0}+1)}},", "2102dd1b2044d44f90fa82e962aedbb4": "A=\\Gamma +\\sigma ", "21031b3f34f0ba738d4b950e92c371fd": "B({\\mbox{G}})=\\mu _{r}H({\\mbox{Oe}})", "21033766da7e8eb0f8fec8d49a59366e": "h(p,u)=x(p,e(p,u)),\\ ", "21033b6683ac17eff06663bdc84ad8b4": "E=\\langle H\\rangle =\\int _{\\Omega _{r}}\\psi _{\\mathbf {k}}^{*}({\\mathbf {r}})[T+V]\\psi _{\\mathbf {k}}({\\mathbf {r}})d{\\mathbf {r}}", "2103c5d935de1b88534eba03240c2f86": "[x]E[y]", "2103fa777b761eb5f96147eb13b9ea1c": "\\psi (\\Omega ^{\\Omega }\\psi (0))", "210406315148712b30c14738a90c7912": "{\\frac {1}{\\sqrt {\\lambda }}}=-2\\log(({\\frac {\\varepsilon }{3.71D}})+({\\frac {7}{Re}})^{0.9})", "21043b06a7c2486a7f60484d7b6ed109": "\\Delta _{3}={\\begin{vmatrix}I_{1}\\left(ka\\right)&kaI_{0}\\left(ka\\right)-I_{1}\\left(ka\\right)&-kaK_{0}\\left(ka\\right)-K_{1}\\left(ka\\right)\\\\I_{0}\\left(ka\\right)&I_{0}\\left(ka\\right)+kaI_{1}\\left(ka\\right)&-K_{0}\\left(ka\\right)+kaK_{1}\\left(ka\\right)\\\\{\\frac {\\mu _{A}}{\\mu _{B}}}I_{1}\\left(ka\\right)&{\\frac {\\mu _{A}}{\\mu _{B}}}kaI_{0}\\left(ka\\right)&-kaK_{0}\\left(ka\\right)\\end{vmatrix}}", "2104539ef0681ef8b631148125964af5": "A\\ {\\mathsf {Type}}", "21046019a83639b34b1a4554f090dae6": "m:=(k(T{\\bmod {R}})){\\bmod {R}}", "2104a6c38162f12e8ee445ee880a95fd": "\\int \\mathrm {versin} (x)\\,\\mathrm {d} x=x-\\sin {x}+C", "2104c757c287b58721a5eadd66dd6e74": "AT=T_{\\rm {a}}+0.33e-0.70ws-4.00", "2104cef2f88129d565f97bce6dc33772": "{\\dot {\\mathbf {x} }}(t)=A\\mathbf {x} (t)+BK\\mathbf {y} (t)", "21055bff5955e4cc9c27d48e936cab73": "XX'\\,", "21055e9940bad315fe988f32326b4532": "A={\\frac {I^{right}-I^{left}}{I^{right}+I^{left}}}", "21056027de74fcbd53667fce78c80309": "G=P^{(\\pm )}G+P^{(\\mp )}G", "210575ab02fd679c06facac66677c8c9": "\\Box A^{\\alpha }=\\mu _{0}J^{\\alpha }", "210581cd06642c70a8e15d8b62a96ed1": "\\mathbf {b} ^{i}\\cdot \\mathbf {b} ^{j}={\\begin{cases}g^{ii}&{\\text{if }}i=j\\\\0&{\\text{if }}i\\neq j,\\end{cases}}", "21060f2b7d0257a90489be69c28de392": "\\kappa _{0}", "210614a1117e89a255b9d0b39516b604": "L_{s}", "21063e45d6cd23e1a9bd54efe94fefaf": "\\left\\langle \\mu _{z}\\right\\rangle =\\mu L(\\mu B\\beta ),", "2106559595e7e9ff37bce9ae5a4e9dfd": "G(K_{1},K_{2},\\ldots ,K_{n})={\\sqrt[{n}]{K_{1}K_{2}\\cdots K_{n}}}=\\left(\\prod _{i=1}^{n}K_{i}\\right)^{1/n}", "210698eaa5ec36e1aa91641fed266fcc": "\\rho ={\\frac {1}{h^{n}C}}{\\frac {1}{W}}f({\\tfrac {H-E}{\\omega }}),", "2106e21acdeca804d32fce51647b5337": "a_{2}b_{2}a_{2}^{-1}b_{2}^{-1}", "2106e48ad038fad267b57b1dd6ea9bce": "|\\mathbf {W} |", "2106f45529bc655f7837b9cfc58b5e52": "n_{i,j}\\geq n_{i,j+1}\\quad {\\mbox{and}}\\quad n_{i,j}\\geq n_{i+1,j}\\,", "21071ec7d32c3de41502a24fdad955f6": "\\nu (\\emptyset )=0,\\nu (\\Omega )=1).", "210738af164bceaefde27ec1c8849854": "{\\begin{aligned}U(\\rho ,\\omega ,z)&\\propto \\int _{0}^{\\infty }\\int _{0}^{2\\pi }A(\\rho ')e^{-i{\\frac {2\\pi }{\\lambda z}}(\\rho \\rho '\\cos \\omega \\cos \\omega '+\\rho \\rho '\\sin \\omega \\sin \\omega ')}\\rho 'd\\rho 'd\\omega '\\\\&\\propto \\int _{0}^{2\\pi }\\int _{0}^{\\infty }A(\\rho ')e^{-i{\\frac {2\\pi }{\\lambda z}}\\rho \\rho '\\cos(\\omega -\\omega ')}d\\omega '\\rho 'd\\rho '\\end{aligned}}", "2107779565348eb246bd0cd86956f98a": "q=e^{2i\\pi \\tau }", "210791c540797d0752f4374e61a3dfc2": "{\\begin{aligned}2\\cdot R_{*}&={\\frac {(28.6\\cdot 0.70\\cdot 10^{-3})\\ {\\text{AU}}}{0.0046491\\ {\\text{AU}}/R_{\\bigodot }}}\\\\&\\approx 4.31\\cdot R_{\\bigodot }\\end{aligned}}", "2107ac18348daad232de0dcbff57ec35": "NSE_{i}=h_{i}^{t}\\times G", "2107bb7fb91501a501bc1a39117fdb0a": "X^{2}\\backslash \\left\\{y~\\backepsilon ~y\\succ x\\right\\}=\\left\\{y~\\backepsilon ~x\\succcurlyeq y\\right\\}", "21082734df9c3874201e90b00aaf7d32": "{\\dot {x}}(t)=Ax(t)+Bu(t)", "21084b46aa1ea7f5ed5684b74d102a84": "a_{0}+a_{1}x+a_{2}x^{2}+\\cdots +a_{n-1}x^{n-1}", "210852a558482b5b517bed4854f026ee": "TE=\\omega ={\\sqrt {\\operatorname {Var} (r_{p}-r_{b})}}={\\sqrt {{E}[(r_{p}-r_{b})^{2}]-({E}[r_{p}-r_{b}])^{2}}}", "21086ffb2a96c5ce628d0bab4a880c04": "{\\mathit {prob}}_{\\mathit {before}}(\\psi \\rightarrow \\phi )=\\langle \\phi |\\rho _{\\mathit {sys}}|\\phi \\rangle =\\langle \\phi |\\psi \\rangle \\langle \\psi |\\phi \\rangle ={|\\langle \\psi |\\phi \\rangle |}^{2}=\\sum _{i}|\\psi _{i}^{*}\\phi _{i}|^{2}+\\sum _{ij;i\\neq j}\\psi _{i}^{*}\\psi _{j}\\phi _{j}^{*}\\phi _{i}", "2108ca3cc3e0f456e3d02d406e218e9d": "\\|A\\|", "2108e0e0a702e6a6e2c10721c6525a78": "E(X,U_{D})", "2108f821a8320e949b80836c467fa065": "Y_{i}=I({\\boldsymbol {X}}=i),", "210908c31a22a5686e1bf1d6927ee932": "f(x)={\\mathcal {F}}^{2}f(-x)", "210912d7ada6f25132d0b99ece462deb": "{\\mathbb {Z}}+\\alpha \\cdot {\\mathbb {Z}}", "210999d61bacf4ccb392353212aaa396": "P_{k+1}=P_{k}+\\left(1-P_{k}\\right)u_{k}\\otimes {v_{k}^{*}A\\left(1-P_{k}\\right) \\over v_{k}^{*}A\\left(1-P_{k}\\right)u_{k}}.", "210a2008c014aa4224a8879073870164": "|S_{N}|=\\sum _{i=1}^{\\infty }|S_{i}|\\,1_{\\{N=i\\}}.", "210a252876d793288f2fa87d4584d801": "\\epsilon =\\hbar \\omega =h\\nu ", "210a64e0047f7efdd5b2e2fed364a783": "M(p)\\leq {\\sqrt {|a_{0}|^{2}+|a_{1}|^{2}+\\cdots |a_{n}|^{2}}}\\,.", "210a92bee716e5e7ce6793cd30d98945": "\\mathrm {P} (A_{m}\\cap A_{n})=\\mathrm {P} (A_{m})\\mathrm {P} (A_{n})", "210adbc7aca929cfe25208c13de793bc": "\\sigma _{T}=\\sigma {\\sqrt {T}}.\\,", "210afcc5dabc5d571a560937efc7bbdf": "\\operatorname {str} (A^{-1}TA)=(-1)^{|i'|}(A^{-1})_{j}^{i'}T_{k}^{j}A_{i'}^{k}=(-1)^{|i'|}(-1)^{(1+|j|+|k|)(|i'|+|j|)}T_{k}^{j}(A^{-1})_{j}^{i'}A_{i'}^{k}=(-1)^{|j|}T_{j}^{j}=\\operatorname {str} (T).", "210b858ee9912f329ee30b89ee11377e": "\\delta ^{13}C={\\Biggl (}{\\frac {{\\bigl (}{\\frac {^{13}C}{^{12}C}}{\\bigr )}_{sample}}{{\\bigl (}{\\frac {^{13}C}{^{12}C}}{\\bigr )}_{standard}}}-1{\\Biggr )}*1000\\ ^{o}\\!/\\!_{oo}", "210be343b12fd63b62fa20d1bae83423": "X_{g}\\in T_{g}G", "210c2bad6a6071e6d3d354d214810bc2": "x_{n}=\\sum _{k=1}^{L-1}a_{k}x_{n-k}", "210c50e20fd80389a61bb47811bc02c4": "\\mathbf {\\mathit {n}} <\\mathbf {\\mathit {d}} ", "210cb4d4b54d55fc8e199f2a9325180a": "v=v_{0},v_{1},....,v_{n-1}", "210d19383a60789aee5cd4279ac40f7c": "I(x)={\\frac {k}{t_{1}-t_{0}}}\\int _{t_{0}}^{t_{1}}\\Psi \\Psi ^{\\mathrm {*} }\\,dt", "210d3b1eff8734130cbbc88f5f8a7781": "ReticIndex=ReticCount*{Hematocrit \\over NormalHematocrit}", "210d440b51e4a0d583dda554afca940c": "C=\\left[\\sum _{i=1}^{n}a_{i}^{\\frac {1}{s}}c_{i}^{\\frac {(s-1)}{s}}\\ \\right]^{\\frac {s}{(s-1)}}", "210d8ee6f28e6db6da836357fb8b3b23": "T(z)=\\sum L_{n}z^{-n-2}.", "210dabdf79eaa352193ae8c434ca4999": "\\ g_{\\phi }=\\left(9.7803267714~{\\frac {1+0.00193185138639\\sin ^{2}\\phi }{\\sqrt {1-0.00669437999013\\sin ^{2}\\phi }}}\\right)\\,{\\frac {\\mathrm {m} }{\\mathrm {s} ^{2}}}", "210dce79f6446af4a23630f02d0c89ea": "|T_{j}\\cap T_{k}|>{\\frac {2t}{d^{2}}}]\\leq n^{2}\\cdot n^{-4}=n^{-2}", "210dcf7c0826f72842e034eefcef60e4": "f_{n}(x)=\\sin(n\\pi x)\\quad n=1,2,\\ldots ", "210e25391dc4d1fa712937abc3502fc3": "{\\begin{aligned}t&={\\frac {c}{g}}\\operatorname {arctanh} \\left({\\frac {cT}{X}}\\right)\\;{\\overset {X\\,\\gg \\,cT}{\\approx }}\\;{\\frac {c^{2}T}{gX}}\\\\X&\\approx {\\frac {c^{2}T}{gt}}\\;{\\overset {T\\,\\approx \\,t}{\\approx }}\\;{\\frac {c^{2}}{g}}\\end{aligned}}", "210e7fbb7f07dc95138df019481df201": "y=b+r\\,\\sin t\\,", "210efea0d341988ec7e5c6d3427dba91": "\\lim _{t\\to \\infty }{\\frac {1}{t}}m(t)=1/\\mathbb {E} [S_{1}].", "210f450adcd4b85d828fceec2646ca18": "\\left(\\bigcup A_{\\alpha }\\right)^{C}", "210f4634b0ea12fe19ad659528cd57a0": "p_{1}:H\\rightarrow G", "210f5e8a28ec8ccadfc74756c048e84c": "f'\\circ g=T(h)\\circ f", "210f9a4f8aa2e5d730f8d7b30f74af11": "\\lim _{r\\to 0}{\\frac {1}{\\mu {\\big (}B_{r}(x){\\big )}}}\\int _{B_{r}(x)}f(y)\\,\\mathrm {d} \\mu (y)=f(x)", "210fa56cfae1e6d423a7029786c411c4": "\\mathrm {Gamma} \\left(\\alpha =3,\\theta =2/3\\right)", "210fb18336e9db0a0c4f43ecec3af608": "\\zeta (\\theta ,\\tau )", "210fb4ef99cd876a142d5be659f19551": "c_{1}(t')={\\dfrac {\\mu _{01}\\epsilon _{0}}{i\\hbar }}\\int _{0}^{t'}\\mathrm {d} t\\exp \\left(-i{\\frac {E_{0}-E_{1}}{\\hbar }}t\\right)\\exp({i\\omega t})=\\int _{-\\infty }^{+\\infty }\\mathrm {d} t\\exp \\left(-i{\\frac {E_{0}-E_{1}}{\\hbar }}t\\right)H\\left({\\frac {t}{t'}}-{\\frac {1}{2}}\\right)\\exp \\left(i\\omega t\\right)", "2110031e1734e005d86fcd041e2227b4": "(X_{\\infty },d_{\\infty })=\\lim _{\\omega }(X_{n},d_{n})", "21108df1fdaa3d9d4d6537ddf33a8935": "x^{2}+{\\frac {b}{a}}x=-{\\frac {c}{a}}.", "2110c431de90c55f41f318401085fd33": "ck{\\frac {x^{c-1}}{(1+x^{c})^{k+1}}}\\!", "2110eb159e2837a4c5ba8607afd39842": "\\eta =(d-a+r)/2c", "21111d16b98301c1a7689d454b01d926": "R=q_{1}+\\left({\\frac {q_{2}'}{q_{2}}}\\right)", "211173d48327977749523fd0cf1cd1db": "\\delta =\\omega ", "2111a4ca6b798d17e1e24a71e2133417": "\\nu _{2,1}(\\mathbb {R} _{-},\\mathbb {R} _{+})", "2111debd4daefa01aa7fd538468aff13": "{\\sqrt {\\frac {4\\alpha }{\\pi }}}\\,", "2111f5cb30941292d805e3f23c37b59d": "\\beta _{0}^{(P)}={\\frac {-1}{V_{0}^{(P)}}}\\left({\\frac {\\partial V}{\\partial P}}\\right)_{T}={\\frac {0.4343C}{V_{0}^{(P)}(B+P)}}", "21127b7c198b4b23cffaf6e165d5811d": "\\left\\{K_{t}(\\omega )\\right\\}_{t=0}^{T}", "211280d6ac65cbea1979a6605899eb13": "(H_{\\mathrm {sat} }-H_{0})\\cdot \\lambda \\cdot k'=(T_{0}-T_{\\mathrm {wb} })\\cdot h_{\\mathrm {c} }", "211281577c395023b46990d7232b99ff": "s=-\\alpha \\pm {\\sqrt {\\alpha ^{2}-{\\omega _{0}}^{2}}}", "21128eb55b23661e0bd31443607b7008": "\\rho _{A}", "2112d62ca10eb93d2f30104298fbefab": "\\scriptstyle 0\\leq i\\leq n", "2112d8b4f21da3344c70855f72d46566": "\\beta \\left(t\\right)", "21132b004174eb9f6d2f8270c7840198": "B_{H}", "21137d59846cb3308e7ca2b1df56613c": "B_{p}=\\left\\langle s_{p}(t),h_{p}(t)\\right\\rangle ", "21138ddf8bfc90594a31b76e7613f041": "x(t)=\\sum _{n=-\\infty }^{\\infty }x(nT)\\cdot \\mathrm {sinc} \\left({\\frac {t-nT}{T}}\\right),", "2113bf8814c95c6b906bd14e90814d2b": "\\textstyle \\prod ", "2113de175e87a01dbc1a3e9a2de21f24": "\\epsilon _{\\rm {XC}}(n_{\\uparrow },n_{\\downarrow })", "211417d5fc075d7dd97ce20375a1c7d2": "\\mathbf {b} \\in [\\mathbf {b} ]", "21148ab8c96c67f17ae1a377d3d25804": "\\ \\beta \\approx 90\\deg ", "21149001bffe394775cc14d867c6a0d5": "S_{1}=\\{v_{k+1},v_{k+2},\\ldots ,v_{n}\\}", "21149c37d56deec56d426d926e6bf21a": "{\\mathbf {x}}_{i}=(\\underbrace {0,\\ldots ,0} _{i},1,\\underbrace {0,\\ldots ,0} _{k-i})\\ \\ {\\text{for all}}\\ 0\\leq i\\leq k.", "21149f5c3e297243c120e0b5ba67a9cb": "\\displaystyle {|f(z)-f(w)|\\leq C|z-w|^{d},}", "2114b0474c49a7b51407aefb6bd0fa0c": "c_{4}", "2114ced7573391d9d333a7445ed4e84b": "\\scriptstyle \\mathrm {N} ", "21151a45612d9ca246743b9ed723d972": "{\\frac {\\Gamma ({\\frac {\\nu +1}{2}})}{{\\sqrt {\\nu \\pi }}\\,\\Gamma ({\\frac {\\nu }{2}})}}={\\frac {(\\nu -1)(\\nu -3)\\cdots 5\\cdot 3}{2{\\sqrt {\\nu }}(\\nu -2)(\\nu -4)\\cdots 4\\cdot 2\\,}}.", "21154b597796752cfca7b95438a0a101": "g>1", "21155319d91de83485dcb50ed8921daf": "x^{2}=H(mU)\\mod n", "211583217c1b986c221fbba8ff28a722": "k_{1}\\rightarrow k_{2}", "2115c611d7d0b89269dbac416a156588": "n=n'", "2115e4cdb2479fe88369518753500fcb": "r_{1}\\approx -{\\frac {b}{a}},", "2116b38bafca8f9ed4a322933759117e": "E_{snake}^{*}=\\int \\limits _{0}^{1}E_{snake}(\\mathbf {v} (s))\\,ds=\\int \\limits _{0}^{1}(E_{internal}(\\mathbf {v} (s))+E_{image}(\\mathbf {v} (s))+E_{con}(\\mathbf {v} (s)))\\,ds", "2116d3d06ec5a6087b916394b5b3fdb2": "C\\circ C([x])=C([x]).", "21170fe882e70504bd1ab64a6c66ead1": "{\\mathcal {Q}}", "2117792ab55e0c69b9083d88bbd58990": "D_{\\mathrm {KL} }(Q||P)=\\sum _{\\mathbf {Z} }Q(\\mathbf {Z} )\\log {\\frac {Q(\\mathbf {Z} )}{P(\\mathbf {Z} \\mid \\mathbf {X} )}}.", "2117b6f5085f8ddd319b323b304d263c": "P_{i}=\\Pr(\\varepsilon _{i}-\\varepsilon _{a}>V_{a}-V_{i})", "2117c2b079a80ae8c736299613fbef07": "U\\rightarrow X", "2117c36a7d4ffb499a7711f2f728394a": "b+1", "21181731231c92f29b583c210be2c22b": "U_{p}=-{\\frac {1}{4\\pi }}\\iiint \\limits _{V}\\left({\\frac {\\nabla ^{2}\\cdot \\mathbf {U} }{R}}\\right)dV_{Q}", "21189b731b9643e7c074925dec2b0982": "2\\mathbf {A} =2{\\begin{pmatrix}a&b\\\\c&d\\\\\\end{pmatrix}}={\\begin{pmatrix}2\\!\\cdot \\!a&2\\!\\cdot \\!b\\\\2\\!\\cdot \\!c&2\\!\\cdot \\!d\\\\\\end{pmatrix}}={\\begin{pmatrix}a\\!\\cdot \\!2&b\\!\\cdot \\!2\\\\c\\!\\cdot \\!2&d\\!\\cdot \\!2\\\\\\end{pmatrix}}={\\begin{pmatrix}a&b\\\\c&d\\\\\\end{pmatrix}}2=\\mathbf {A} 2.", "2119519f8c668c4f3d888a7652803e04": "\\gamma _{1}^{1}", "2119ded08945b62cc65af8ac978d99ac": "\\operatorname {build-param-lists} [q\\ q,D,V,T_{6}]", "2119e2a2916bb0681bb1c78e628d87a4": "(p_{\\alpha })", "211a0fcc6ae7e91141dea389445255dc": "(1+{\\sqrt {-163}})/2", "211a5f1e2be060cf6061b48659e97e7f": "p_{1}^{2}=S\\cdot q_{1}^{2}\\pm 1", "211a8d29e41f913abbaea0f6fa595518": "\\sigma \\colon \\{1,\\cdots ,n\\}\\longrightarrow \\{1,\\cdots ,n\\}", "211a91193c9bb9f557c997884842f72c": "\\scriptstyle k\\left[M\\right]/\\mathrm {Ker} F_{i}\\,\\to \\,F_{i}\\left(k\\left[M\\right]\\right)\\;=\\;k", "211a9403e14d8709c63a3914bc3598d8": "p(\\theta |M)", "211aa45515c41464c690f98b0693d281": "u_{\\nu }(T)={8\\pi h\\nu ^{3} \\over c^{3}}{1 \\over e^{h\\nu /k_{\\mathrm {B} }T}-1}.", "211adc5c7e0543cc8ee753574846e3c7": "\\mu _{s}(\\sigma )=-kT\\ln \\int p_{s}(\\sigma ')e^{-{\\frac {E_{int}(\\sigma ,\\sigma ')-\\mu _{s}(\\sigma ')}{kT}}}d\\sigma '", "211ade7224e0efc1d390159956191b56": "2^{b}-M=16-10=6", "211b1d1d7e5573449c1f9ad4d1af78f6": "q\\leq n", "211b390281a508a10972c1154a2b54b3": "d(x,y)=d(z,x)", "211b77134cd9624c3fb9f750f5a8ceac": "O(f(x))", "211b7ac1f9be504c00a742ec71212848": "{\\hat {S_{i}}}\\otimes {\\hat {B_{i}}}", "211ba1f30e0c0f90dc20e2dcd3fee427": "{\\begin{aligned}\\left({\\frac {\\partial A_{z}}{\\partial y}}-{\\frac {\\partial A_{y}}{\\partial z}}\\right)&{\\hat {\\mathbf {x} }}+\\\\+\\left({\\frac {\\partial A_{x}}{\\partial z}}-{\\frac {\\partial A_{z}}{\\partial x}}\\right)&{\\hat {\\mathbf {y} }}+\\\\+\\left({\\frac {\\partial A_{y}}{\\partial x}}-{\\frac {\\partial A_{x}}{\\partial y}}\\right)&{\\hat {\\mathbf {z} }}\\end{aligned}}", "211c0c63f4ec05cb32817be102b178a3": "{\\textbf {R}}={\\textrm {E}}[{\\textbf {v}}_{k}{\\textbf {v}}_{k}^{\\text{T}}]={\\begin{bmatrix}\\sigma _{z}^{2}\\end{bmatrix}}", "211c11414e7f4f449cabdc9f8698f5c2": "\\,{}^{x}a\\approx a^{x}", "211c18dc3fcdc6b621185ddef385bf17": "n>A", "211c23f12fd466e20ea54fa89fd4b660": "f(\\zeta ,{\\bar {\\zeta }})={\\frac {1}{2\\pi i}}\\iint _{D}\\varphi (z,{\\bar {z}}){\\frac {dz\\wedge d{\\bar {z}}}{z-\\zeta }}", "211c39650123980b1f1139685d464aa5": "2{\\frac {\\ddot {a}}{a}}+\\left({\\frac {\\dot {a}}{a}}\\right)^{2}+{\\frac {kc^{2}}{a^{2}}}-\\Lambda c^{2}=-{\\frac {8\\pi G}{c^{2}}}p.", "211c4899f0dd793bba64a02586f50b49": "\\{\\alpha (f_{n}),\\alpha (f_{m})\\}=0\\quad {\\mbox{and}}\\quad \\alpha (f_{n})^{*}\\alpha (f_{m})+\\alpha (f_{m})\\alpha (f_{n})^{*}=\\langle f_{m},f_{n}\\rangle I.", "211cef4edee330569392fe4dc2bd34a7": "b\\cap a", "211d0301a34fe1b26ab49b576989dd94": "\\chi \\,\\!", "211d88963272883183a29c021a4473c4": "T(0,a)={\\frac {1}{2\\pi }}\\arctan(a)", "211e1ef71689b2d0a2f96e0341a13dd7": "\\mathrm {error} {\\bigl (}x(t_{0}+3\\Delta t){\\bigl )}=6\\,O(\\Delta t^{4})", "211e2de732456fdc7d133c2592822b49": "n_{0}", "211e6a6dfe4c5d6a30825a9f727eedcb": "{\\text{excess kurtosis}}={\\frac {6}{3+\\nu }}\\left({\\frac {(2+\\nu )}{4}}({\\text{skewness}})^{2}-1\\right){\\text{ if (skewness)}}^{2}-2<{\\text{excess kurtosis}}<{\\tfrac {3}{2}}({\\text{skewness}})^{2}", "211e7ecf33ea40db9f17c31c40dbb71b": "{\\begin{aligned}DR(n)&{}={\\begin{cases}9,&{\\mbox{if }}SOD(n)\\mod 9=0\\\\SOD(n)\\mod 9,&{\\mbox{ otherwise}}\\end{cases}}\\\\&{}=(n-1)\\mod 9+1\\end{aligned}}", "211e8e03586e3ce8f3111a2d989c8c80": "{D^{\\mathrm {eff} }}=f", "211e9d86525447163ec551304527acb9": "\\mathbf {G_{1}} ", "211e9e1b9a1deaebe40b5327408325a6": "{\\frac {\\pi }{4}}\\approx 1-{\\frac {1}{3}}+{\\frac {1}{5}}-\\cdots +(-1)^{(n-1)/2}{\\frac {1}{n}}+(-1)^{(n+1)/2}f_{i}(n+1)", "211ee22d6bb06cc02eeb32323149c639": "t_{\\text{score}}={\\frac {({\\widehat {\\beta }}-\\beta _{0}){\\sqrt {n-2}}}{\\sqrt {{\\text{SSR}}/\\sum _{i=1}^{n}\\left(x_{i}-{\\overline {x}}\\right)^{2}}}}.", "211ef1c36db15621421a0bb8966542f9": "T[g\\sigma ]\\to bT[\\sigma ]", "211f75c471bdb4d01bf0e20fdcafa052": "\\pi ^{-}+C\\to {\\bar {\\Sigma }}^{-}+K^{0}+{\\bar {K}}^{0}+K^{-}+p^{+}+\\pi ^{+}+\\pi ^{-}+{\\hbox{nucleus recoil}}", "211f75e4e049ce88f909ed2d6121741f": "M^{i}", "211f76b374349d65654ac466c2151c65": "R=L-\\mathbf {p} \\cdot \\mathbf {\\dot {q}} \\,,", "211f795a4e11243aa46499fbb4fa3148": "a_{02}-{\\mathcal {L}}(a_{30}\\omega ^{2}+a_{21}\\omega +a_{12})=p_{3}(a_{30}\\omega ^{2}+a_{21}\\omega +a_{12})-\\omega p_{8}.", "211f8ad93197385ab3022f269af0182b": "|(Ax,y)|\\leq \\|x\\|\\cdot \\|y\\|.", "2120b9eb2815b7508011a866620a5090": "={\\widehat {a}}+(\\delta \\alpha ^{*}{\\widehat {a}}-\\delta \\alpha {\\widehat {a}}^{\\dagger }){\\widehat {a}}+{\\widehat {a}}(\\delta \\alpha {\\widehat {a}}^{\\dagger }-\\delta \\alpha ^{*}{\\widehat {a}})+O(\\delta \\alpha ^{2},(\\delta \\alpha ^{*})^{2})", "2120d770928c7c5926578c3ef8bea274": "{\\textbf {G}}(s)={\\frac {{\\textbf {P}}(s)}{{\\textbf {Q}}(s)}}=K{\\frac {(s-a_{1})(s-a_{2})\\cdots (s-a_{n})}{(s-b_{1})(s-b_{2})\\cdots (s-b_{m})}}", "2121151af0ca2705fa39bc3024216e5b": "v,v'\\in \\mathbb {R} ^{d}", "2121e7a6dc8ed64fc93b6c002a6e20b8": "f_{\\pm }(z)", "2121ecf663aa5ebb7e4e7346588c451d": "E=\\alpha -\\pm 1\\times \\beta ", "21220b38dfb7eda7c4e2e2c5bf1ba2ee": "\\lambda _{B}={\\frac {4\\pi a_{B}}{\\alpha }}-\\ ", "212249d4697805c6577cce97b2441033": "\\cos \\left({\\frac {x}{2}}\\right)=\\pm {\\sqrt {{\\tfrac {1}{2}}(1+\\cos x)}}", "21225d4856fbd75e5bf47e907766ee94": "g_{DS}={\\frac {\\partial I_{DS}}{\\partial V_{DS}}}", "2122cec3f6e5240bb424e4f99d9ec79a": "\\bigcup \\nolimits _{n,m}\\left\\{x\\in X\\ :\\ \\sup \\nolimits _{T\\in F_{n}}\\|Tx\\|_{Y}\\leq m\\right\\}", "2122f738725ca53a7844f2be811c500e": "\\sum _{n=k}^{\\infty }\\left[{\\begin{matrix}n\\\\k\\end{matrix}}\\right]{\\frac {z^{n}}{n!}}={\\frac {\\left(\\log(1+z)\\right)^{k}}{k!}}", "21235a0884c91abd5366b2f39d43c170": "W={\\frac {1}{\\sqrt {8}}}{\\begin{bmatrix}\\omega ^{0}&\\omega ^{0}&\\omega ^{0}&\\ldots &\\omega ^{0}\\\\\\omega ^{0}&\\omega ^{1}&\\omega ^{2}&\\ldots &\\omega ^{7}\\\\\\omega ^{0}&\\omega ^{2}&\\omega ^{4}&\\ldots &\\omega ^{14}\\\\\\omega ^{0}&\\omega ^{3}&\\omega ^{6}&\\ldots &\\omega ^{21}\\\\\\omega ^{0}&\\omega ^{4}&\\omega ^{8}&\\ldots &\\omega ^{28}\\\\\\omega ^{0}&\\omega ^{5}&\\omega ^{10}&\\ldots &\\omega ^{35}\\\\\\vdots &\\vdots &\\vdots &\\ddots &\\vdots \\\\\\omega ^{0}&\\omega ^{7}&\\omega ^{14}&\\ldots &\\omega ^{49}\\\\\\end{bmatrix}}", "212369dee6600287df0b32fbbc41526f": "x_{n}\\in C^{*}(\\theta _{n})", "21237eea68173ccbd84d3e49a3a4b533": "\\varphi :(\\mathbb {R} ^{n}\\times \\mathbb {R} )\\times \\mathbb {R} \\to \\mathbb {R} ^{n}\\times \\mathbb {R} ;\\qquad \\varphi ({\\boldsymbol {x}}_{0},t_{0},t)=(\\varphi ^{t,t_{0}}({\\boldsymbol {x}}_{0}),t+t_{0})", "21239a5b1e10cd0db525d4bbaf54495f": "\\scriptstyle \\cos \\theta =\\cos \\left(\\theta +2\\pi k\\right).", "2123cdd736e0847396860bd9ab07fb23": "\\rho _{L^{*}}={\\frac {\\sum _{j=1}^{l}L_{j}(\\phi _{j}\\rho _{w}+(1-\\phi _{j})\\rho _{g})}{L^{*}}}", "2123d2f52593debfd7975686ee23ebce": "\\textstyle f(X)=\\sum _{k}f_{k}X^{k}\\in R[[X]]", "212446327682b0b29d2bf3d2aa9f40fb": "s,h\\models P-\\!\\!\\ast \\,Q", "21244b0dc08c3ec734a7f3a4fc809137": "\\displaystyle {Tf(w)={1 \\over 2\\pi }\\int _{\\partial \\Omega }\\partial _{n}(\\log |z-w|)f(z)={1 \\over 2}\\Re (Hf)(w).}", "2124732b29a8cf7e1c7b8a0f10ff7513": "=-\\left(A_{1}\\mathbf {e_{1}} +A_{2}\\mathbf {e_{2}} +A_{3}\\mathbf {e_{3}} \\right)=-(\\star \\mathbf {A} )\\ .", "21247db6e87679d507efabe47d0fe495": "\\psi '_{W}(t)+t\\psi _{W}(t)\\approx 0", "2124d8e264ed693811271631c7fa8114": "{\\frac {1}{2}}+(\\pi _{1}-\\pi _{0})/(2^{1+B(n)+2B(n)G(n)}),", "21253cd82d899d67c25399e1441426bb": "S_{q}=-\\lim _{x\\rightarrow 1}D_{q}\\sum _{i}p_{i}^{x}", "212540b9663b79dae6a1756f12ca8f53": "{\\begin{aligned}({\\mathcal {L}}_{X}T)^{\\alpha _{1}\\cdots \\alpha _{r}}{}_{\\beta _{1}\\cdots \\beta _{s}}=X^{\\gamma }T^{\\alpha _{1}\\cdots \\alpha _{r}}{}_{\\beta _{1}\\cdots \\beta _{s},\\gamma }&-\\,X^{\\alpha _{1}}{}_{,\\gamma }T^{\\gamma \\alpha _{2}\\cdots \\alpha _{r}}{}_{\\beta _{1}\\cdots \\beta _{s}}-\\cdots -X^{\\alpha _{r}}{}_{,\\gamma }T^{\\alpha _{1}\\cdots \\alpha _{r-1}\\gamma }{}_{\\beta _{1}\\cdots \\beta _{s}}\\\\&+\\,X^{\\gamma }{}_{,\\beta _{1}}T^{\\alpha _{1}\\cdots \\alpha _{r}}{}_{\\gamma \\beta _{2}\\cdots \\beta _{s}}+\\cdots +X^{\\gamma }{}_{,\\beta _{s}}T^{\\alpha _{1}\\cdots \\alpha _{r}}{}_{\\beta _{1}\\cdots \\beta _{s-1}\\gamma }\\,.\\end{aligned}}", "2125bf68acdb12d921a6f98810f19499": "B_{\\mathrm {v} }=\\log _{2}{\\frac {B}{NK}}\\,.", "2125ffc6864ab39b3c34a712115a945f": "{\\mathbf {j}}_{{\\rm {n}},\\,i}=n_{i}\\mathbf {u} _{i}", "21266c7a829c230ef5cdc467bbfcdd49": "{1 \\over q(z)}={1 \\over z+iz_{\\mathrm {R} }}={z \\over z^{2}+z_{\\mathrm {R} }^{2}}-i{z_{\\mathrm {R} } \\over z^{2}+z_{\\mathrm {R} }^{2}}={1 \\over R(z)}-i{\\lambda \\over \\pi w^{2}(z)}.", "2126bbe09c2f409b8fbd517a757ae841": "s=r-{\\sqrt {r^{2}-\\ell ^{2}}}", "2126c397d14354598fea1232133f900d": "{\\frac {d\\varphi }{ds}}={\\frac {\\cos ^{2}\\varphi }{a}}\\,", "2126c71158ed6f42ed70125278a13808": "f\\sim \\operatorname {GP} (m(\\cdot ),C(\\cdot ,\\cdot )),", "2126d1446d273b5be35e0826cabc727d": "\\displaystyle [a_{0};a_{1},a_{2},\\ldots ,a_{k}-1,2].", "212763f9db7d33a473fcc1e6bf513ea1": "a<\\Im (s)\\varepsilon )\\\\\\operatorname {Pr} (X\\leq a-\\varepsilon )&\\leq \\operatorname {Pr} (X_{n}\\leq a)+\\operatorname {Pr} (|X_{n}-X|>\\varepsilon )\\end{aligned}}", "21283c2d43001596bd3772536a86c10b": "12^{2}+33^{2}", "2128831dd136ed12ac08450dc1eb8ed6": "\\pi _{2}(x)\\sim 2C_{2}{\\frac {x}{(\\log x)^{2}}}.", "2128932afe1ea5d1a376cd1e04cdaeeb": "\\langle F,\\phi \\rangle =\\int \\int f(x+iy)\\phi (x,y)\\,dx\\,dy,", "2128e3cbaf09a616dfa6d6213cb75e7d": "{\\tilde {R}}=e^{-2\\varphi }\\left(R+2(n-1)\\triangle \\varphi -(n-2)(n-1)\\|\\nabla \\varphi \\|^{2}\\right)", "212920f6b2f02ec197cf6c6dbdce97ee": "\\psi ({\\mathbf {r}})=N_{x}N_{y}N_{z}e^{i(k_{x}x+k_{y}y+k_{z}z)}", "212998b2ff69c216de369f89adf41533": "A\\in \\Sigma _{0}", "2129d0a9ce8d45ced11f68400c0b65be": "{\\sqrt {6}}", "212a06a728f5a87715d96892dec70902": "K_{ijk}={\\frac {1}{2}}(T_{ijk}-T_{jki}+T_{kij}),", "212a3567df6fe6fe11caf388a08cd95a": "\\varepsilon >0", "212a5f55a2cd108a969af1ca8d6e602c": "I_{D}=\\Pi -\\Pi _{0}-T\\,", "212a83d39f05a6c6e8398d561bffa3e1": "\\mathbf {J} :P\\to {\\mathfrak {g}}^{*}", "212abcc00be63b38e4cd48603ee59f7d": "m_{i}", "212ad034d8f5a3267775b134271474ce": "A\\oplus B=(A^{c}\\ominus B^{s})^{c}", "212b4ac94f46affa0d3f4e4c79366ccf": "\\tan[\\arccos(x)]={\\frac {\\sqrt {1-x^{2}}}{x}}", "212b5b6eb6cfc4b0566d4170b619e313": "{3 \\over 2}\\cdot {1 \\over 2}\\cdot {3 \\over 2}\\cdot {3 \\over 2}\\cdot {1 \\over 2}\\cdot {3 \\over 2}\\cdot {1 \\over 2}\\cdot {8 \\over 5}={81 \\over 80}", "212b7c3587327c94815ef6e6308e30ed": "\\beta \\mapsto \\lfloor \\beta \\rfloor ", "212bb69afb2b28c05db2f77492c35827": "E[|\\eta |^{q}]<\\infty ", "212bf522bb938b1b30a08776ca7c5702": "F=-kx", "212c21acd2153d472c5d4d7cea240804": "W\\triangleq 2{\\sqrt {V}}", "212c5bc6aec0d7d0ef21a9eb168dc147": "h\\left((1-\\lambda )x+\\lambda y\\right)\\geq M_{p}\\left(f(x),g(y),\\lambda \\right),", "212c65e23fb48bbd1117d6a196787694": "x^{2}+y^{2}=1.\\,", "212c97b4c8663c693e903ad8397db815": "x_{0}+tx\\in A", "212ce081962da99ceaf0f174e6dea9e4": "\\lbrace \\left\\langle x,y,z\\right\\rangle \\mid \\phi _{x}(y)=z\\rbrace ", "212d15fb7bf1e166c56ef4d659cf0e3a": "{\\mathfrak {H}}(\\beta ;\\gamma )={\\begin{pmatrix}1+\\gamma \\beta &-\\beta \\gamma ^{2}\\\\\\beta &1-\\gamma \\beta \\end{pmatrix}}", "212d2883c38dfe911872d14954ae3922": "\\scriptstyle {}^{n}z", "212d4cd8917d593fbb5f373e0f120cd0": "\\eta (2)={\\pi ^{2} \\over 12}", "212d511ae81118fe6ed289359ad8fca9": "Q=NX_{S}", "212d920902289b751d63f2f09f473c01": "\\eta _{11}=\\langle \\mathbf {e} _{1}{\\bar {\\mathbf {e} }}_{1}\\rangle =\\langle \\mathbf {e} _{1}(-\\mathbf {e} _{1})\\rangle _{S}=-1,", "212dcafb83b41e3df531dfccb65bfb3e": "p\\in K", "212dce1da2f0bd80366b5e1de813ac04": "g^{abc}", "212ddf6d0ae36a842fc3e390da024d9d": "{\\frac {\\phi \\left(r\\right)}{r}}=\\phi \\left({\\frac {1}{r}}\\right)", "212df6022ee1798525c5b59d475949dc": "\\lim _{t\\to \\infty }f(t)=\\lim _{s\\to 0}{sF(s)}", "212e0f9b1a3b926410e12b0857d9d1d3": "j_{0}(x)={\\frac {\\sin x}{x}}", "212e28a0426d14377c80f95b032e2d82": "1,2,\\dots ,d", "212e2ff8a0bea5f2efa58d0e6c1c5b3d": "Rate\\approx 4mm/s", "212ef7c5e874ed5afbabb177adb1b599": "e_{f}(k,i)\\,\\!", "212f038037d2d4b09431a7cd618abe5a": "d^{3}", "212f453914b387160ac98d737b9c7005": "I_{1}x^{2}+I_{2}y^{2}+I_{3}z^{2}=1,", "212f4dbe3ff9046736bf2a9996af85d5": "I_{ii}\\cdot I_{ij}=I_{ji}\\cdot I_{jj}\\ \\ (i0,\\alpha >0,\\beta >0\\,\\!.", "213e590462f19fe6ea0b3f2a89ebd9e6": "w(\\cdot )", "213e5fc1c033082d04358907bd04252c": "e>1\\,\\!", "213f1f4415a4710e51d06683d3b6d534": "G={\\frac {1}{a}}+{\\frac {n-1}{b}}", "213f4fb5a8e8c7dbd4eea72703ad72b6": "F_{n}={\\frac {\\varphi ^{n}-(-\\varphi )^{-n}}{\\sqrt {5}}}", "21401031c7b4952c7eaf66699024c74c": "[T]=[Z_{1}][X_{1}][Z_{2}][X_{2}]\\ldots [X_{n-1}][Z_{n}],\\!", "21401732cf37f63c4e284e8ee1803c08": "V_{bi}", "21403f41e6740242b98efdf83a6c9728": "\\psi ^{\\ast }", "21404f4fc13796efcc7c147e9e18daf6": "P_{1}\\in \\{0,1\\}=\\{{\\mbox{AB}},{\\mbox{BA}}\\}\\,", "21409bc14c5e30c7c3c772a2c6ef9c70": "\\operatorname {Aut} (A_{n})=\\operatorname {Aut} (S_{n})=S_{n}", "2140f74715e50750afb1b1fc403c43f1": "{\\frac {1}{n_{1}^{i_{1}}n_{2}^{i_{2}}\\cdots n_{k}^{i_{k}}}}", "2140f981496f532d911daf78ec6cb3ac": "\\mathbf {x} (\\cdot )", "214127a972fef1ba141d369d3e3e8e59": "\\Pi _{1}\\cdots \\Pi _{N}", "21416146b1d79fe6c1e3d8f75db8ff6e": "\\rho (x_{1},x_{2})\\geq 0", "21416eb08876f081761ab76cc5c3bc86": "{b}_{eq}\\,", "21417e9440ad5a08913a33d94e698d0c": "\\scriptstyle r(S-\\{k\\})=\\sum _{i,j\\in \\{S-\\{k\\}\\},i0,", "21660fbc2c6c800dffad4c713ac6bb2c": "P({\\bar {W}})=1-p=q", "21667337d35b3088f0886e094a0735ae": "q(\\mathbf {s} )", "216688c25dc0048065a97a4bf41a3468": "\\geq \\gamma ", "21669de2ac24f04eaf1e71bf4fc0a816": "r_{k}={\\frac {\\pi ^{2}kRn_{k}^{2}}{16p^{2}\\left(1-{\\frac {p^{2}}{6n^{2}}}\\right)}}", "2166e53e332c832ba939cf34ea2066ba": "\\sigma _{\\infty }=\\sigma _{f}=\\sigma _{m}", "2166e5c72213000c38110dc49d281755": "(n_{c}-n_{b})(n_{a}-n_{b})", "2166eb389329df913db3c4a0da8ad491": "a=\\gamma ^{x}\\in \\langle \\gamma \\rangle {\\text{ with }}0\\leq x<\\omega ", "2166f84f98663924d0bbc97bb9a74a28": "\\scriptstyle t,", "21670535a673053399a24a3c90dac59b": "A=A_{1}\\oplus A_{2}", "2167964f4466bfa4db38a8751fb64270": "{\\begin{bmatrix}1&1\\\\0&1\\end{bmatrix}}.", "2167d4ef0d12b26b1fc3849840467956": "d_{f}={\\frac {2\\lambda z}{W}}", "2167e8074c05c593950e63e729adaf1e": "\\ln \\left(F/K\\right){\\Big /}{\\sqrt {\\tau }}.", "2167e809dd41963976e7f077c6def3c5": "e^{\\theta \\mathbf {e} _{12}}=e^{i\\theta }=\\cos {\\theta }+i\\sin {\\theta },", "2168f809c48d845c008727a64b82317d": "{\\mathcal {L}}={\\frac {m}{2}}\\mathbf {\\dot {r}} \\cdot \\mathbf {\\dot {r}} +e\\mathbf {A} \\cdot \\mathbf {\\dot {r}} -e\\phi \\,\\!", "21692a6d1653c6178f6ed80a91c20417": "I_{n}=\\int {\\frac {\\sin {ax}}{x^{n}}}dx\\,\\!", "216962e12dcab274b3b0442a17e13ac2": "\\partial \\Omega _{D}\\cup \\partial \\Omega _{N}=\\partial \\Omega ", "216981343002548ccd2ca27ad4c78afb": "\\mathbf {z} (0)=\\mathbf {z} _{0},", "21698d76abdd91c9c3642cceeca0159d": "\\mathbf {PSPACE} \\subseteq \\mathbf {EXPTIME} \\subseteq \\mathbf {EXPSPACE} ", "2169974babd649b56bc24b639c16650b": "{\\partial }\\!\\!\\!/", "216a0253bf5eb89f199bb229c4443461": "x_{i}=x_{i}\\left(y,\\xi \\right),\\ i=1,...,m;\\ \\theta _{j}=\\theta _{j}\\left(y,\\xi \\right),j=1,...,n", "216a1d6f59fbbf8fd59a8ce9d5856626": "\\Delta \\epsilon =\\int v\\,d(\\Delta v)=\\int v\\,adt", "216a24e1d9bafa2ab9a7bf67f3ce58d9": "|B'|\\leq w(X)\\,", "216a73eaac72e61b567454572f8e37b1": "\\mathbf {A} \\times \\mathbf {B} =\\mathbf {-B} \\times \\mathbf {A} ", "216af02b7bd7129ef08358c9fb766254": "\\Theta ({\\sqrt {n}})", "216c2c3e54e824efc60ad36ac56e81fa": "E_{\\mu \\nu }", "216c34d4761fbb92ab296f45178aae1b": "x_{i}+\\sum {\\bar {a}}_{i,j}x_{j}={\\bar {b}}_{i}", "216d158cc28f3a40bcf1351169cb14aa": "|A_{ij}X'-a_{ij}T'|=0", "216d35e5170732b9f0554582122689af": "q4", "217bc58072358947de65f93d2a3d6512": "{\\mathcal {J}}_{i,i}=J_{i,i}=a_{i-1},\\,i=1,\\ldots ,n", "217bd2cdd730c9e010ee477318dfa2cf": "{\\begin{matrix}{12 \\choose 1}{4 \\choose 2}{11 \\choose 2}{4 \\choose 1}^{2}\\end{matrix}}", "217be54fadc95f6fe4a637f6689a1c39": "\\psi (X)=\\left\\{{\\begin{matrix}X&\\mathrm {if} \\quad 0\\leq X<1\\\\1+\\psi (\\ln X)&\\mathrm {if} \\quad X\\geq 1\\end{matrix}}\\right.", "217c294495dd5c85768b52127fd0ef52": "\\lambda ^{*}={\\frac {\\frac {\\partial u(x_{1}^{*},x_{2}^{*})}{\\partial x_{1}}}{p_{1}}}={\\frac {\\frac {\\partial u(x_{1}^{*},x_{2}^{*})}{\\partial x_{2}}}{p_{2}}}", "217c7f0433ae5ea45963384a9802b834": "{\\begin{pmatrix}1\\\\f_{1}(t,x_{1},x_{2},\\ldots ,x_{n})\\\\f_{2}(t,x_{1},x_{2},\\ldots ,x_{n})\\\\\\vdots \\\\f_{n}(t,x_{1},x_{2},\\ldots ,x_{n})\\end{pmatrix}}", "217ca48c96862ca304bb6e11932fd305": "{\\begin{aligned}\\Pr(C_{i}=c|C_{1},\\ldots ,C_{i-1})&{}={\\begin{cases}{\\dfrac {\\theta +|B|\\alpha }{\\theta +i-1}}&{\\text{if }}c\\in {\\text{new block}},\\\\\\\\{\\dfrac {|b|-\\alpha }{\\theta +i-1}}&{\\text{if }}c\\in b;\\end{cases}}\\end{aligned}}", "217ceedd1a815bcbf02f345358158df3": "n^{*}", "217cf6883d5315ae356bdc232176ba89": "m(m-1)\\cdots (m-k+1){\\frac {x^{m}}{x^{k}}}", "217e064d9a83d54005f3059c663c8150": "b_{n}=\\sum _{r>0,r|(m,n)}r^{k-1}a_{mn/r^{2}}.", "217e367c9c26e887ca561010e123ced5": "C_{n}={\\frac {1}{n+1}}{\\binom {2n}{n}}", "217e6ecadea770b3576db741285e9f30": "=(P^{+}\\Omega _{\\alpha \\beta })^{IJ}.", "217e725ccc2519a857f93afb871d805c": "a={\\frac {u^{2}}{v}}+v,\\ \\ b={\\frac {u^{2}}{w}}+w,\\ \\ c={\\frac {u^{2}}{v}}+{\\frac {u^{2}}{w}}-(v+w)", "217eb5768e3734fc04b42e6f9e6056a9": "\\pi \\approx \\left(9^{2}+{\\frac {19^{2}}{22}}\\right)^{1/4},", "217eda8f9466e8bd352d2721bc6f56a3": "{\\tilde {v}}_{i}=\\left\\{{\\begin{array}{lcl}v_{1}+v_{n}-v_{0}&&i=0\\\\v_{i+1}+v_{i-1}-v_{i}&\\qquad \\qquad &0\\\\v_{n-1}+v_{0}-v_{n}&&i=n\\\\\\end{array}}\\right.", "217f0daf17ff768519e1719db0e9991a": "(\\varphi ,\\partial _{\\mu }\\varphi )", "217f791a974496a62399f2af77740933": "\\ N(r)\\ =a_{0}+a_{1}*r+a_{2}*r^{2}+...+a_{t}*r^{t}.", "217f871736e88ae84650c9a14575aa1e": "|0\\rangle \\ ", "217fcbfee83c5f2161c4648b0e3a2892": "u_{\\epsilon }(x)", "217fceb9e055913453cdc5df16f098bc": "\\scriptstyle f\\,\\circ \\,\\phi ^{-1}:\\;\\mathbb {C} \\,\\rightarrow \\,\\mathbb {C} ", "218002f80c2146d160f0a971b56299c5": "\\mathbf {a} \\times \\mathbf {b} =[\\mathbf {a} ]_{\\times }\\mathbf {b} ={\\begin{bmatrix}\\,0&\\!-a_{3}&\\,\\,a_{2}\\\\\\,\\,a_{3}&0&\\!-a_{1}\\\\-a_{2}&\\,\\,a_{1}&\\,0\\end{bmatrix}}{\\begin{bmatrix}b_{1}\\\\b_{2}\\\\b_{3}\\end{bmatrix}}", "21800b32a1b608fb5d507ce2d302a460": "\\lambda /4", "2180271cc6dced8cef8da9e9c2083ef9": "D_{Y}(f_{*}M)\\cong f_{!}(D_{X}(M)).", "21808daa1d65eea33b42649cd40f749d": "F(U)\\rightarrow \\prod _{i}F(U_{i}){{{} \\atop \\longrightarrow } \\atop {\\longrightarrow \\atop {}}}\\prod _{i,j}F(U_{i}\\cap U_{j}).", "2180a228cca6f7bf19cd3e895bc5a29e": "{\\bar {e}}\\,", "2180b0d941e4eb8fddc3a3ff67f1070b": "X_{\\mathbf {k} }=\\sum _{\\mathbf {n} =0}^{\\mathbf {N} -1}e^{-2\\pi i\\mathbf {k} \\cdot (\\mathbf {n} /\\mathbf {N} )}x_{\\mathbf {n} }", "2180b76803a964de63e337b4627ca6bc": "(a\\times b)\\times c=a\\times (b\\times c)", "2180bb9c32e453402e673ddd5eef141c": "\\mathbf {W} _{\\mu }", "2180c5b619fac6dd8af3882d1f447dd3": "C^{1}([a,b])", "2180c88e88976ba8f197f7463f2e71e5": "x\\in L\\Leftrightarrow \\#\\{r\\in \\{0,1\\}^{T}\\mid f(x,r)=1\\}\\geq 2^{T-1}", "2180f11cad5ad387710f6b6570c32909": "p\\in S'", "218154caad3c5613d38c37bd42d6abc4": "If=\\lim _{n\\to \\infty }Ih_{n}", "218239dc837f5b391c72492f17a6c453": "p(x)=\\sum _{i=0}^{\\infty }f^{2}(\\lambda _{i})\\phi _{i}^{2}(x)", "2182d241246465cc9703604be20703fd": "\\scriptstyle T_{1}^{1}(V)\\rightarrow \\mathrm {End} (V)", "2183053d0fa6fb9108a5a4683dacf8b7": "{\\text{Observe: }}\\omega (t),Q_{1}(t),...,Q_{K}(t)", "21833b669d3b5876b6b5f6f9363165ce": "A^{+}=A^{-1}", "21838ae802ab1ae5b2df6aadd31a76d9": "R_{\\mathrm {ads} ,i-1}=k_{i}P\\Theta _{i-1}", "2183d8f3770be285177b65cb8269f615": "\\langle F\\rangle ", "21846d0110acae240f582d2d5d0e758e": "V_{b}\\,", "21846dc6f689a02ac3eda9491286f24f": "a+R0,\\end{aligned}}", "21905692161563b3412cba070633fa5d": "|z|^{2}<{1 \\over 2}(1+(z\\cdot z)^{2})<1", "2190729126e01a6e0bca46ba317aaf95": "E\\in {\\mathcal {F}}", "2190772e7cee0d1c38929df30a209acc": "c=M_{2,2}=i\\,", "2190f11ad8d6c85164aa477d9af4a153": "Z=\\sum _{n=0}^{\\infty }\\left({\\frac {4}{125}}\\right)^{n}{\\frac {(33n+4)\\left({\\frac {1}{2}}\\right)_{n}\\left({\\frac {1}{3}}\\right)_{n}\\left({\\frac {2}{3}}\\right)_{n}}{(n!)^{3}}}\\!", "21914e895a78d22fdfc8697ea6767aa2": "2\\pi \\int _{a}^{b}yh(y)\\,dy", "21914f5c5cb29d69e4b5a09057f8d88f": "A=0.7\\,162\\,162\\,162\\,\\cdots .", "21916701de34e75d8627a7f2d04451b5": "b(v)\\sim 1+\\beta v^{2}/c^{2}\\,", "2191b48b16a1ac22f0b98df02d86e23c": "\\!\\ S_{m}^{n}=K_{n}S_{m}+K_{(n-1)}", "2191ec67979ff4d85c86941d9c099727": "\\,\\lambda _{i}\\geq 0", "21926fbdb516b0d13b6babd728522e50": "\\nu =\\mu /\\rho ", "2192a006517c647900bd7a805a778976": "\\pi _{k}\\circ \\delta _{a}=id_{a}", "2193148b619ac10a6f31cb8ec9dc9cfc": "r={\\frac {\\hbar }{\\sqrt {2m(V_{0}-E)}}}", "21931fdd29a1b3cf9fb338534b73d30a": "N={\\begin{pmatrix}0&1\\\\0&1\\end{pmatrix}}", "219323bb72cf06f8fbfc5060a07b4ab1": "{\\overrightarrow {\\mu }}=g^{(l)}{\\overrightarrow {l}}+g^{(s)}{\\overrightarrow {s}}", "21932895058c780ca922d1c4cae29a90": "S(\\phi )=\\int {\\mathcal {L}}(\\phi )dxdt=\\int L(\\phi ,\\partial _{t}\\phi )dt", "219332220478cdc3913ce2b6ab9160cc": "{29^{2}\\equiv 2^{1}\\cdot 11{\\pmod {91}}}", "21936b63da64f2186685a392e5bb4da2": "f_{1}=S_{1}(Y-f_{2}),f_{2}=S_{2}(Y-f_{1})", "21938e8ebc7552abdaa866f2d9bd1b28": "d(u\\cdot v)=v\\cdot du+u\\cdot dv\\,\\!", "2193c0ffd6e3180ea5062649c074b8aa": "0\\leq a_{n}\\leq 1/n", "2193dc8df6dab6ef728ddaf403ae568f": "{\\sqrt {2GM/r}}", "2193e0c0223c798179faec9d4f1dc2c8": "S(\\mathbf {q} )={\\frac {1}{N}}\\left|\\sum _{j=1}^{N}\\mathrm {e} ^{-i\\mathbf {q} \\mathbf {R} _{j}}\\right|^{2}", "21941ad54f38291b6dcd99693d60311d": "f_{2}=f_{2}(z)=\\sum _{j=0}^{\\infty }{\\frac {a(j)}{b(j)}}{\\frac {z^{j}}{j!}},", "2194555f48693165cc98db004f19c8f6": "\\simeq H_{\\mathrm {dR} }^{k}(S^{n-1}).", "219468df07c99ba2b66370ee4901b586": "C_{\\Psi }", "21947a886f4cc608dcffca2281029477": "D_{MLD}(y)=\\arg \\min _{c\\in C}\\Delta (c,y)", "21948c8acb91cdb55fa17cb8f63e172e": "T_{q}^{*}Q", "21949897e63154d0901b52494ec5cf44": "Z(T):=\\sum _{n=1}^{\\infty }\\exp \\left({\\frac {-E(n)}{k_{B}T}}\\right)=\\sum _{n=1}^{\\infty }\\exp \\left({\\frac {-E_{0}\\log n}{k_{B}T}}\\right)=\\sum _{n=1}^{\\infty }{\\frac {1}{n^{s}}}=\\zeta (s)", "2194c107cf0eca36caa33fdb998cf399": "(A_{3},E',F',G',A_{0})=(A_{3},H'',I'',J'',A_{0})", "2194fd3f7ed676cabc87a0ef7592c060": "P(|X-m|\\geq ks)\\leq {\\frac {1}{[N(N+1)]^{1/2}}}\\left[\\left({\\frac {N-1}{k^{2}}}+1\\right)\\right]", "2195605d639e96a5729795e7a72a9da9": "\\,r=a\\cdot \\sec \\alpha ={\\frac {a}{\\cos \\alpha }}", "2195654fc4d972af24f6bf4d8fadec86": "c\\leftarrow g_{1}^{x}remP", "21956742389d9d22c22317c84fc9cf95": "\\mathbf {L_{z}} =m\\hbar .", "2195ad440ca8575c79d0ba37346586d5": "4K^{2}=(pq+rs)^{2}-{\\frac {1}{4}}(p^{2}+q^{2}-r^{2}-s^{2})^{2}", "2195b3f6edffdf08bcb55afe484c7855": "m{\\ddot {x}}+q\\left({\\frac {\\partial A_{x}}{\\partial t}}+{\\frac {\\partial A_{x}}{\\partial x}}{\\dot {x}}+{\\frac {\\partial A_{x}}{\\partial y}}{\\dot {y}}+{\\frac {\\partial A_{x}}{\\partial z}}{\\dot {z}}\\right)=-q{\\frac {\\partial \\phi }{\\partial x}}+q\\left({\\frac {\\partial A_{x}}{\\partial x}}{\\dot {x}}+{\\frac {\\partial A_{y}}{\\partial x}}{\\dot {y}}+{\\frac {\\partial A_{z}}{\\partial x}}{\\dot {z}}\\right)", "2195cc57caeddee2548ec4f9445860a1": "\\det(\\exp(A))=\\exp(\\mathrm {tr} (A))\\,", "21960fdf21e7764f37e80137585d270f": "\\mu =m", "219612d73ea46ad66f68d40ac42608d2": "2^{n}\\,n!\\,{\\sqrt {\\pi }}", "21969ea72c67bd9f8fcc0517eab6fb92": "m_{\\mathrm {Earth} }=5.9736\\times 10^{24}\\,\\mathrm {kg} ", "21970cb4c9f7e16003fb4f6bb5c5e2c6": "y^{2}=x^{3}+ax^{2}+16ax", "2197457d64b9da6dbfeac1ee45355738": "{\\begin{aligned}\\operatorname {Cov} (z',z'A')&=E\\left[\\left(z'-E(z')\\right)\\left(z'A'-E\\left(z'A'\\right)\\right)'\\right]\\\\&=E\\left[(z'-\\mu ')(z'A'-\\mu 'A')'\\right]\\\\&=E\\left[(z-\\mu )'(Az-A\\mu )\\right].\\end{aligned}}", "219769a87487526f8865e0e39a5273fb": "0\\leq p_{i}\\leq 1", "219777140f6fe25f388e0b9ded74a35c": "s_{p}:M\\to M,h'K\\mapsto h\\sigma (h^{-1}h')K", "2197831c783e534c8a0f59a63ae6d248": "\\varepsilon _{n}={\\frac {x_{n}}{\\sqrt {S}}}-1", "2197c8ec7856b0314cbf2a728655065c": "[a,bc]=abc-bca=abc-bac+bac-bca=[a,b]c+b[a,c]\\,", "2197d5bceba5fe732952fc70e386f45b": "{\\frac {AE}{EC}}={\\frac {DE}{EB}}={\\frac {AD}{BC}}.", "2197f06949db5d28e82bee14a5915942": "{\\boldsymbol {v}}={\\frac {d}{dt}}{\\boldsymbol {r}}=\\sum _{k=1}^{d}{\\dot {q}}_{k}\\ {\\boldsymbol {e_{k}}}+\\sum _{k=1}^{d}q_{k}\\ {\\dot {\\boldsymbol {e_{k}}}}", "2197f09034ad29709154d05d4693ca6b": "n_{k}=\\lfloor k\\phi \\rfloor =\\lfloor m_{k}\\phi \\rfloor -m_{k}\\,", "219802318bd299f9ad24b80b40198274": "\\psi (f)=\\int _{X}f(x)d\\mu (x)\\quad ", "21982bd70bd6e0cd6f10a64ad3df1ac7": "J_{p}^{k}(M,N)", "219848d8701d5ed6d5812669f35b7359": "\\displaystyle z(\\alpha _{1},\\beta _{1},\\gamma _{1},\\delta _{1})+(1-z)(\\alpha _{2},\\beta _{2},\\gamma _{2},\\delta _{2})", "2198520b18bf1d54d1776639d54efe00": "{\\frac {\\sin A}{a}}\\,=\\,{\\frac {\\sin B}{b}}\\,=\\,{\\frac {\\sin C}{c}}\\!", "219882225a16b72a8cd923b5a26cd66b": "{\\frac {(a_{1}+a_{2}+\\cdots +a_{n})!}{a_{1}!a_{2}!\\cdots a_{n}!}}.", "2198c38fed3d7958b10f26dd2b997702": "\\lim _{\\theta \\to 0}\\left({\\frac {\\cos \\theta -1}{\\theta }}\\right)=\\lim _{\\theta \\to 0}\\left({\\frac {-\\sin ^{2}\\theta }{\\theta (\\cos \\theta +1)}}\\right)=\\lim _{\\theta \\to 0}\\left({\\frac {-\\sin \\theta }{\\theta }}\\right)\\times \\lim _{\\theta \\to 0}\\left({\\frac {\\sin \\theta }{\\cos \\theta +1}}\\right)=(-1)\\times {\\frac {0}{2}}=0\\,.", "2198db912e7df085b4fbdff0c7f8f01c": "\\cap ", "21994c70f5a75ee3e495452ebb848f43": "B'=SA'\\cap OB.", "2199e77fe8985fdc76b2e0f353c06e6a": "\\gamma '=-{\\frac {1}{r}}\\mathbf {r} \\cdot \\mathbf {Q} \\alpha _{j}sc_{1}+\\alpha _{j}'{\\big [}-ac_{0}+2b\\alpha _{j}s^{3}{\\bar {c}}_{3}+{\\frac {1}{2}}\\gamma \\alpha _{j}s^{4}c_{2}^{2}{\\big ]}", "219a2a169da4f5d9c0a48dc5cd7e155e": "\\displaystyle {H^{*}=JUHU^{*}J}", "219a5d496f8fbe1fccb81110f6439f08": "Y=\\mathrm {constant} +\\mathrm {error} ", "219ae189251bb9b8fae5e14581243ba6": "1.09", "219b1a6aa471120bbcaec3177d90d70b": "\\left(\\left(\\left(1\\times 2\\right)\\times 3\\right)\\times 4\\right)\\times 5=120", "219b9e4cdcc5810b06cbfe5c1a466094": "(X,Y)=\\left({\\sqrt {\\frac {2}{1-z}}}x,{\\sqrt {\\frac {2}{1-z}}}y\\right),", "219bf11299b2d7ba9c98962f38670996": "A(x):=\\sum _{n\\leq x}a(n).", "219bf5e4ef25edbc50ce770c1ae9c5db": "\\mathrm {\\Lambda } ^{\\!\\otimes }\\!\\left({H(A)}\\right)=\\mathrm {Re} \\ \\mathrm {\\Lambda } ^{\\!\\otimes }\\!\\left({A}\\right)", "219c3a03696e437dab545b6bc8824a60": "u={\\frac {x}{B}}", "219c3a44c547a6d2478694ab5ce49a4b": "\\left\\{{\\frac {\\varphi (n)}{n}},\\;\\;n=1,2,\\cdots \\right\\}", "219c3b87d8bbdf44aa782b6a0dcc0e28": "k_{a;b}=0", "219c446833ddcc2b4adbd7892ee711ee": "b(t)=b1\\times e^{\\frac {-t}{\\tau }}", "219c44b0752226bd7fa1d9d976370f07": "\\displaystyle {C={\\begin{pmatrix}{1 \\over 2}I+T_{K}&S\\\\H&{1 \\over 2}I-T_{K}^{*}\\end{pmatrix}}.}", "219cb6630f157dd3cc997f173aa4c405": "\\mathrm {sys\\pi } _{1}\\leq 6\\;\\mathrm {FillRad} (X),", "219cdd6a0fbfb8200eb7d345cf471291": "\\{B_{\\theta }:\\theta \\in {\\rm {pcf}}(A)\\}", "219d1394a5750927b6b96d842caff51e": "{\\bar {G_{i}}}", "219d311774dda5f7d41dee43795919de": "f_{\\max }", "219d32caa42c38ae364187299de27674": "\\geq H_{q}^{-1}({\\frac {1}{2}}-\\varepsilon )", "219d4767458bdcb419b19aa038473c1d": "\\nu =\\sum _{j=1}^{q}b_{j}-\\sum _{j=1}^{p}a_{j}.", "219da01a766fa0c263ae5be8a84311bf": "\\int _{V}\\Delta \\varphi \\,dV=\\int _{V}q\\,dV,", "219db81e380381a0af3354fd274187b4": "\\mathbf {c} =\\mathbf {F} ^{-T}\\mathbf {F} ^{-1}\\,\\!", "219de227675053bc12f6dca9936d0113": "\\delta (t/a)=|a|\\delta (t)\\,", "219e2f3e8418557860664004661edf98": "\\mathbf {E} '=\\mathbf {U} \\,\\mathbf {S} '\\,\\mathbf {V} ^{T}", "219e43d163fc31f6e3862a10a7b22c11": "U(1)^{k}", "219e624547fd0f23ce41fecdd2913e14": "\\Phi \\vdash \\lnot \\phi ", "219eb343cc35d8e05e64f66df1e8058c": "C(\\alpha )_{n}", "219eb7ec52f9d5f67c38cca1c985cd64": "E_{\\text{fusion}}", "219eddc2f3dbb165221efe47feed094b": "\\scriptstyle {\\mathcal {P}}", "219ee16176deb0d6415dc361d70959a2": "(a^{p})^{q}", "219f03cc5d2577d07cd7548fcd7f588b": "{\\mbox{mex}}(\\left\\{1,2,3\\right\\})=0", "219f06b96c9ac36bc7b15218cdc766c0": "\\mathbf {a\\times b} =\\mathbf {c} \\Leftrightarrow \\ c^{m}=\\sum _{i=1}^{3}\\sum _{j=1}^{3}\\sum _{k=1}^{3}\\eta ^{mi}\\varepsilon _{ijk}a^{j}b^{k}", "219f5122ac51e237892a779e9e604abe": "\\ G(\\tau )=G(0)\\exp(-\\tau /\\tau _{B})+G(\\infty )", "219f775cb652d64218be939388e2b4e5": "2\\cos ^{2}(A/2)=1+\\cos A", "219f7e9eeab9f437b53a8dc3488dc52b": "\\mathbf {x} =\\left({\\begin{array}{c|c}\\mathbf {R} &\\mathbf {t} \\\\\\hline \\mathbf {0} &1\\end{array}}\\right)\\mathbf {x} '", "219fa2b389b3fbaccde6adc972cb6dc5": "2^{7}\\ln(2)=2^{7}\\sum _{k=1}^{\\infty }{\\frac {1}{k2^{k}}}\\,.", "219fac52e3d4591ff08ec9f9022e11b8": "\\exists {}HO^{i}", "219fad12ff3320f7e420d483f8a46940": "|1-\\omega \\lambda _{j}|<1", "21a0217e626766f8b415278366a7b6ce": "\\scriptstyle BV_{\\varphi }([0,T];X)", "21a05a87645983bef4aacd9878b63e8a": "F_{X}(x)=\\int _{-\\infty }^{x}f_{X}(u)\\,du,", "21a05bde42ad77a1873d79e0a4e08f3e": "\\eta \\left\\{\\eta _{b}Ev/x+P\\right\\}=\\left\\{WC_{rr1}v+{\\frac {1}{2}}\\rho C_{d}Av^{3}\\right\\}", "21a06a1a7de8ecfebd96b4581f51acdf": "V=\\{w\\in W:\\chi (w)=w\\}.", "21a0733e2b73149cea1c3986ea696462": "g_{uc}(\\langle 0\\rangle )=\\varepsilon ", "21a0b82edd1eedf49dd4f3eb4e6f51cc": "{\\tilde {H}}", "21a0bcaa65ef95104444f8cdae75d880": "\\langle u,Au\\rangle \\leq -\\pi ^{2}\\|u\\|^{2}", "21a0c8bfcbd982d292d737bead64ef3b": "M\\times \\{i\\}", "21a0d87d4f3d97af5fd2f61706d82be5": "{\\vec {x}}={\\vec {x}}(s,t)", "21a10423e8b4d72dfd61f42f29584eb6": "{}_{2}F_{1}(0,b;c;z)=1", "21a11e1dfff2741aa9860f3f8e3b4ba8": "dV_{t}=r\\left(V_{t}-{\\frac {\\partial f}{\\partial S}}S_{t}\\right)\\,dt+{\\frac {\\partial f}{\\partial S}}\\,dS_{t}.", "21a18c9aff0f8639bcdc41560aaf6676": "B(\\mathbf {x} _{1},\\|\\mathbf {h} _{0}\\|)", "21a20f6e1cf26e09ca079e3637c781c8": "{\\dot {Q}}^{\\mathrm {T} }=-{\\dot {Q}}.", "21a22d84d38af9a58deb1813f6f4f223": "A_{\\alpha },B_{\\alpha }", "21a254f696171178f3710aaa5520bfb3": "\\phi _{\\bar {z}}\\neq 0", "21a2716caab13db9a186b8d89913b682": "\\mathbf {J} =\\mathbf {J} _{\\mathrm {f} }+\\mathbf {J} _{\\mathrm {b} }", "21a273d78ad462e392de05819a8346f3": "P_{4}=\\left[{\\begin{matrix}1&\\alpha \\\\0&1\\end{matrix}}\\right]", "21a27a014af615af8ca081ddca28d907": "r_{n}=a_{0}.a_{1}a_{2}\\cdots a_{n}", "21a2b2e5e1cfc105ff09d51c3762050b": "v({\\vec {p}},s)={\\sqrt {E+m}}{\\begin{bmatrix}{\\frac {{\\vec {\\sigma }}\\cdot {\\vec {p}}}{E+m}}\\chi ^{(s)}\\\\\\chi ^{(s)}\\end{bmatrix}}\\,", "21a2cb0a67a5b3c2d27e28e6085589b8": "2+3X=a+bX\\,", "21a391b8e38d3d324bb6f9fd1d9d38b2": "K(x,t;x',t')=\\langle x|{\\hat {U}}(t,t')|x'\\rangle ", "21a3cad64734cbdf9a67fb16cdcc72b1": "e_{t}", "21a3e817665d005a1846fd4e26541527": "[f]_{x}=\\{g:X\\to Y\\mid g\\sim _{x}f\\}.", "21a4bf60d4e339e630e37d9559b3cfbc": "m_{x,0}", "21a5234ddafd9efc17eb04b8941c9f4f": "q=0,1,...,7", "21a53a20a0a6172c51ad8a45ad7ad09d": "P_{D}(d_{i},x,R)", "21a584010929a6f7bc32e9add73348ee": "10^{n}=2^{n}\\cdot 5^{n}\\equiv 0{\\pmod {2^{n}\\mathrm {\\ or\\ } 5^{n}}}", "21a5928563662602aeed5efc90682ef6": "\\operatorname {Ti} _{2}(\\tan \\theta )=\\int _{0}^{\\tan \\theta }{\\frac {\\tan ^{-1}x}{x}}\\,dx", "21a598deb8b0ac865d65dfec564006e2": "\\kappa (V)\\geq \\kappa (V_{w})+\\kappa (W).", "21a5bedfeb46006a7a45ffe902459694": "\\nabla \\times (\\nabla \\times {\\vec {\\psi }})=\\nabla (\\nabla \\cdot {\\vec {\\psi }})-\\nabla ^{2}{\\vec {\\psi }}=-\\nabla ^{2}{\\vec {\\psi }}", "21a5c8a26ec2aae23ed1a3cd86c8a853": "\\Pi _{xy}^{(0)}=\\sum _{i}{\\vec {e}}_{ix}{\\vec {e}}_{iy}f_{i}^{eq}=p\\delta _{xy}+\\rho u_{x}u_{y}\\,\\!", "21a5e27ef44f94e5bf0ae1ff9ed8e7ef": "T={\\frac {CD}{V}}", "21a5ef591ceb9f3ee4345203d42b121b": "\\int _{-1}^{1}{\\sqrt {1-x^{2}}}\\,dx={\\frac {\\pi }{2}}", "21a616bf4c681017567121c3b38b79ea": "{\\frac {N+\\Xi }{2}}=1128.5~\\mathrm {MeV} /c^{2}", "21a642a5a7a88dca190ea846eed208b1": "{\\begin{aligned}|f(z)|-|f(0)|&\\leq |f(z)-f(0)|\\leq {\\frac {2r}{R-r}}\\sup _{|w|\\leq R}\\operatorname {Re} (f(w)-f(0))\\\\&\\leq {\\frac {2r}{R-r}}\\left(\\sup _{|w|\\leq R}\\operatorname {Re} f(w)+|f(0)|\\right),\\end{aligned}}", "21a666f271e6798ed49f5c9769cf9059": "\\eta ={\\frac {\\epsilon ^{2}}{54}}+O(\\epsilon ^{3})", "21a699fba656bff051113323238e0471": "s={\\frac {((y_{1}-y_{3})(x_{2}-x_{3})-(y_{2}-y_{3})(x_{1}-x_{3}))^{2}}{((y_{1}-y_{3})^{2}+(x_{1}-x_{3})^{2})((y_{2}-y_{3})^{2}+(x_{2}-x_{3})^{2})}}.\\,", "21a70d47bb5899bc506c5b9868c0d3d5": "{\\begin{aligned}{\\partial ^{2}\\mathbf {E} \\over \\partial t^{2}}-{c_{0}}^{2}\\cdot \\nabla ^{2}\\mathbf {E} \\;&=\\;0\\\\{\\partial ^{2}\\mathbf {B} \\over \\partial t^{2}}-{c_{0}}^{2}\\cdot \\nabla ^{2}\\mathbf {B} \\;&=\\;0\\end{aligned}}", "21a72f9348451ecde99da6d28f886ea6": "{\\frac {CN}{\\log ^{A}N}}\\int _{0}^{1}|S(\\alpha )|^{2}\\;d\\alpha \\ll {\\frac {N^{2}}{\\log ^{A-1}N}}", "21a77a08a8b5fc8c27eee72bd195b1b3": "dL_{j}(t)={\\begin{cases}L_{j}(t){\\sigma }_{j}(t)dW^{Q_{T_{p}}}(t)-L_{j}(t)\\sum \\limits _{k=j+1}^{p}{\\frac {\\delta L_{k}(t)}{1+\\delta L_{k}(t)}}{\\sigma }_{j}(t){\\sigma }_{k}(t){\\rho }_{jk}dt\\qquad jp\\\\\\end{cases}}", "21a7a1c172b43bd471c7d51fcf171250": "\\int _{\\Gamma }f_{j}~n_{j}~d\\Gamma =\\int _{A}{\\cfrac {\\partial f_{j}}{\\partial x_{j}}}~dA", "21a7bcd8dc635202ea92dfbc60189d5a": "\\Rightarrow _{l}^{ac}aSSS\\Rightarrow _{l}^{ac}aaSS\\Rightarrow _{l}^{ac}aaaS\\Rightarrow _{l}^{ac}aaaa", "21a7c1e709f78e098c19f4103aedaed6": "U(1)=e^{iq\\theta }", "21a8159f7a1ed14d87666f82b1249bc7": "P_{\\mathbf {v} ^{K}}(\\mathbf {x} )=\\langle \\prod _{i=1}^{K}\\delta (x_{i}-\\mathbf {s} \\cdot \\mathbf {v} _{i})\\rangle _{\\mathbf {s} }", "21a82c1170f096a900f5968cb974e946": "f_{\\tau }(x)=e^{{\\frac {1}{2}}i\\tau x^{2}}", "21a870c7f3e21a18cbd4e17d50307cba": "\\lambda x.x", "21a89f2090efe32537fab73de5fe6576": "q'^{0}=q'^{0}(q^{0},q^{k}),\\quad q'^{i}=q'^{i}(q^{0},q^{k}),\\quad {q'}_{0}^{i}=\\left({\\frac {\\partial q'^{i}}{\\partial q^{j}}}q_{0}^{j}+{\\frac {\\partial q'^{i}}{\\partial q^{0}}}\\right)\\left({\\frac {\\partial q'^{0}}{\\partial q^{j}}}q_{0}^{j}+{\\frac {\\partial q'^{0}}{\\partial q^{0}}}\\right)^{-1}.", "21a8a09f4db0fe4cd2937c3cde905084": "E(M,K)", "21a8ac2e9b9c2bbb2d33c99fd8004136": "{\\hat {\\mu }}={\\bar {X}}={\\frac {1}{n}}\\sum _{i=1}^{n}X_{i}.", "21a8b82e294ace0af8d6dfc9b3c106bc": "K_{O}(x)=\\sup _{\\xi \\not =0}{\\frac {\\langle G(x)\\xi ,\\xi \\rangle ^{n/2}}{|\\xi |^{n}}},\\quad K_{O}(x)=\\sup _{\\xi \\not =0}{\\frac {\\langle G^{-1}(x)\\xi ,\\xi \\rangle ^{n/2}}{|\\xi |^{n}}}.", "21a92feea1ac714a8879fbb7571af88d": "A={\\begin{pmatrix}1&2\\\\3&4\\end{pmatrix}}", "21a961d2422c57fda332346d822d982e": "(A(\\mathbb {R} ),\\circ ,+)", "21a9a8c563a40cfa8232139660aaa54f": "E=E^{0}+{RT \\over F}\\ln {a_{\\mathrm {H} ^{+}} \\over (p_{\\mathrm {H} 2})^{1/2}}", "21a9ba04fdb60d769a6fbadfb0ef6d43": "K_{n}={\\frac {2\\lambda }{d}}", "21a9eb087eddd40f26e4d9ea1fd52abb": "{\\frac {\\partial \\psi }{\\partial t}}+u_{x}{\\frac {\\partial \\psi }{\\partial x}}=0", "21aa65a2cc27e6986d2463788197ef3d": "\\operatorname {\\widehat {MSPE}} (L)=\\sum _{i=1}^{n}\\left(y_{i}-{\\widehat {g}}(x_{i})\\right)^{2}-{\\widehat {\\sigma }}^{2}\\left(n-2\\operatorname {tr} \\left[L\\right]\\right).", "21aa98ce5e5122192256ca834e3b16df": "{\\begin{aligned}\\nabla _{\\mathbf {a} }\\det(A)&=\\mathbf {b} \\times \\mathbf {c} \\\\\\nabla _{\\mathbf {b} }\\det(A)&=\\mathbf {c} \\times \\mathbf {a} \\\\\\nabla _{\\mathbf {c} }\\det(A)&=\\mathbf {a} \\times \\mathbf {b} .\\end{aligned}}", "21aaa585089a8af662a881765d53eaaf": "Z=\\infty \\,\\!", "21aacba5b2d5a4edb5f71e36df15e37b": "\\Lambda _{\\chi '}{}^{\\psi }\\,", "21ab045de04ee078ecaed9e2cf771bff": "\\mathbf {\\nabla } ^{2}\\psi -{\\frac {1}{c^{2}}}{\\frac {\\partial ^{2}}{\\partial t^{2}}}\\psi =\\left({\\frac {mc}{\\hbar }}\\right)^{2}\\psi ", "21ab116f87935a8ab62a5822eff38fa1": "Q=UA\\Delta T.", "21ab432bee0d12ab6048fbcf82534fd0": "\\operatorname {ad} (xy)=\\operatorname {ad} (x)\\operatorname {ad} (y)", "21ab989245a35ab4aac4d0b8347b5729": "E=E^{0}+s\\log _{10}[A]", "21abe8380a871d73c31889902896ff5a": "(\\theta _{xi},\\theta _{yi})", "21abf108f5ac054b37011bc2d56119f7": "{\\hat {\\textbf {Q}}}={\\textbf {Q}}+\\varepsilon {\\textbf {Q}}^{0}", "21abf34a02364751b5cdf2a937985fd9": "D_{0}(L\\ 1s\\rightarrow \\psi ^{*})=const\\ \\vert \\langle L\\ 1s\\vert \\mathbf {r} \\vert \\psi ^{*}\\rangle \\vert ^{2}=\\alpha ^{2}\\ const\\ \\vert \\langle L\\ 1s\\vert \\mathbf {r} \\vert L\\ np\\rangle \\vert ^{2}", "21ac784dadad918e06ec61ecac9100dc": "{\\begin{array}{ll}\\nabla ^{2}f&={\\frac {\\partial ^{2}f}{\\partial x^{2}}}+{\\frac {\\partial ^{2}f}{\\partial y^{2}}}\\\\\\nabla ^{2}f&={\\frac {f\\left(x+h,y\\right)+f\\left(x-h,y\\right)+f\\left(x,y+h\\right)+f\\left(x,y-h\\right)-4f(x,y)}{h^{2}}}-4{\\frac {f^{(4)}(x,y)}{4!}}h^{2}+\\cdots \\\\\\nabla ^{2}f&={\\frac {f\\left(x+h,y\\right)+f\\left(x-h,y\\right)+f\\left(x,y+h\\right)+f\\left(x,y-h\\right)-4f(x,y)}{h^{2}}}+O\\left(h^{2}\\right)\\\\\\end{array}}", "21ac7c6ccca0262e51fe5f90ee13f8b0": "{\\begin{aligned}P(S,t)&=Ke^{-r(T-t)}-S+C(S,t)\\\\&=N(-d_{2})Ke^{-r(T-t)}-N(-d_{1})S\\end{aligned}}\\,", "21acc3c91f0859755da1a796bbd2b59f": "\\xi {\\stackrel {q}{\\longrightarrow }}{\\acute {\\xi }}", "21acdb4b4a82954016549e00c3b258fe": "\\forall x\\exists y\\;Rxy", "21acec853474dc91e8a731c4802a3ff6": "\\displaystyle {\\frac {P(I|c_{f})}{P(I|c_{b})}}>{\\frac {\\lambda _{Ac_{b}}-\\lambda _{Rc_{b}}}{\\lambda _{Rc_{f}}-\\lambda _{Ac_{f}}}}{\\frac {P(c_{b})}{P(c_{f})}}", "21acf1cb957157441bc6225b7a3b618d": "R(t)=st,", "21ad0bd836b90d08f4cf640b4c298e7c": "bb", "21ad1e7bcddc9ac0a73707b2c67fa180": "\\ h(t)=\\Pi \\left({\\frac {t}{T}}\\right).", "21ad46960bd03ad6a13bd66fb151dd20": "\\mathbf {G} ^{\\circ {\\frac {1}{2}}}", "21ad58214bf92fc8248308f411dfe06b": "{\\mathcal {R}}_{m}", "21ad6ffe12ed760ae44e0a2f022fcf5d": "D={\\frac {n(m-a)}{s}}", "21adb1e2fa95ca7917e3501971e9d33d": "\\ Ax^{2}+Bx+C=0", "21aded5bc02cb46da611b6d06728cfd1": "\\partial {X}/\\partial {n}", "21adf2555d341667416ddafb147528e9": "J={\\frac {Lf\\cdot {\\sqrt {S}}}{\\sqrt[{3}]{D}}}\\leq 3,2", "21ae44554c8a9abd918b378db07bfc26": "R_{3}={\\sqrt[{3}]{a^{2}b}}\\,\\!", "21ae4b9ec57cd4f9466ee9d227cd7dec": "f\\circ h=h", "21ae5be16635dceac0663f61c8166ed2": "C(x)={\\cfrac {Q_{x}E^{f}f(f+2h)}{2D}}", "21aec3f46816de8f2cf79cd830fa9355": "{\\boldsymbol {r_{2}}}", "21aec79060493aefb649c1f18a089364": "\\gamma ={\\frac {\\mu \\,\\mu _{N}}{hI}}\\,", "21aeea5542d054ef5319ba5408b22919": "Pxy\\leftrightarrow \\forall z[Ozx\\rightarrow Ozy].", "21aef3b32ae97d3e6e8ff4d095fb3d56": "O(E)", "21aef423aa6dcc39d9de66e65dd84fd3": "I={\\cfrac {ah^{2}}{4}}~;~~S=0.5ah", "21af4c38641ef6027157fda1aa11ebeb": "{\\frac {1}{2}}(1+Z)^{2}\\,\\left(-{\\frac {1}{2}}+2\\,Z-{\\frac {1}{2}}\\,Z^{2}\\right)", "21af72b899e10fe2b905482164c5521f": "c_{c}=1", "21afbc1472f1730a357f8fb094fec734": "|Q\\mathbf {x} |=|\\mathbf {x} |.", "21afc3b9dd9406c8d0805dd2134f7ac5": "V_{N\\setminus \\{b_{i}\\}}^{M}-V_{N\\setminus \\{b_{i}\\}}^{M\\setminus \\{t_{j}\\}}", "21afe9fbbff33112073d4503058c61de": "\\ \\lambda _{\\max }=s^{\\mathrm {H} }R_{v}^{-1}s,", "21b04446dad94958c45d30dd528a0992": "U_{A}=\\{x\\in V:\\quad ||\\varphi ||_{A}<1\\},\\qquad A\\in {\\mathcal {A}},", "21b07a37faaf6965b72f89cbd8251d94": "\\Delta T_{s}=~\\lambda ~\\Delta F", "21b0de2803e0673a3130518b97dd4b6b": "{\\tilde {\\boldsymbol {\\theta }}}", "21b104594c0d5568d9366efe574c413f": "X\\rightarrow \\mathbf {S} ^{3}\\rightarrow K(\\mathbf {Z} ,3).\\ ", "21b17b93b982731158ed26218d82f304": "z={\\sqrt {\\frac {n-3}{1.06}}}F(r)", "21b17cb041aaa50a306879929f6fff25": "N/\\Gamma ", "21b18ded62ac8435bfe537a692b177ac": "{dy \\over dx}={\\frac {-1}{1+x^{2}}}", "21b1c45277875087ae9ce842c35abbe4": "T_{1}\\times \\ldots \\times T_{n}", "21b242289444ad94ca1dc6922d859997": "\\Sigma \\subset \\mathbb {R} ^{3}", "21b26028bbb53216ae5978a061c255e6": "\\mathrm {0.91{\\overline {6}}} ", "21b322eb587ea005d92ee7cbf31b6953": "{\\mbox{(3) }}\\bigcap _{n=0}^{\\infty }T^{-n}{\\mathcal {K}}=\\{X,\\varnothing \\}", "21b3570a016496d218587902e82b01ae": "\\alpha =\\beta {(\\beta +1)^{-1}}", "21b373d65f9f091033b97d828e58c582": "A,A\\mid (B\\mid C)\\vdash C.", "21b3864e99cdc2b18e22c2034108ae77": "x_{min,i}n^{2}/8", "21b83bda2f7d7ceeb49a469f14dd308e": "U(\\phi ^{*}\\phi )", "21b8576ef395661df94261210e7bd739": "u''_{i}={\\frac {u_{i+1}-2u_{i}+u_{i-1}}{h^{2}}}", "21b859709111590884742608e81ec288": "{\\frac {D\\Gamma }{Dt}}=0", "21b87a1b4f67cdd4c7b08ac8f9ca88eb": "\\mathrm {pH} =14+\\log {\\frac {C_{b}V_{b}-C_{a}V_{a}}{(V_{a}+V_{b})}}", "21b87b2162f412e1d531342e20e9a70f": "|1/2,-1/2\\rangle \\;|1/2,-1/2\\rangle \\ (\\downarrow \\downarrow )", "21b8fce671acf5fa4690193ad7ef3461": "[0,t]", "21b90015ce3b6040c6cd306292f6f5a5": "\\ v_{z}=v_{g}\\cdot \\left({\\frac {z}{z_{g}}}\\right)^{\\frac {1}{\\alpha }},02", "21bbc98e29201c146415a0bfa1e9e375": "b\\neq 0", "21bc13fe5056aab09be15cf1f0d74d26": "(499^{2}/113)x^{7}-212x+3^{4}", "21bc59c4af87c0e60e8a9934d3248eb4": "\\ e={\\frac {\\Delta L}{L}}={\\frac {\\ell -L}{L}}", "21bc83cf4a53b207df3036528243c16c": "\\Delta x=\\Delta y=\\Delta z", "21bca4849a6c32af37f07134830fa8f2": "\\Sigma A", "21bd12bf50083f0383866bad81d7cd00": "\\displaystyle \\cos {A}+\\cos {B}+\\cos {C}={\\frac {3}{2}}", "21bd3325ed1c6660b473dd500b8d64b6": "N_{v}(\\mu _{2}-\\mu _{1})={\\frac {df}{dc}}", "21bd4ac9ae8f37ea9dfd40fde2480f4e": "{\\frac {\\Delta _{h}[f](x)}{h}}-f'(x)=O(h)\\quad (h\\to 0).", "21bd6f944f2d2c8f0be7887d57d8924c": "{\\rm {ATIME}}(f,j)", "21bd7c80ad49a3a667d0b949a3cdb2f1": "EIw(L/2)={\\dfrac {PbL^{2}}{48}}-{\\cfrac {Pb}{12}}(L^{2}-b^{2})=-{\\frac {Pb}{12}}\\left[{\\frac {3L^{2}}{4}}-b^{2}\\right]", "21be2b16da3ee028399208a558ea6cfd": "\\lambda (\\mathrm {lcm} (a,b))=\\mathrm {lcm} (\\lambda (a),\\lambda (b))", "21be923a2b2132284994c5bd8b97120c": "\\mathrm {NAG} _{v}", "21be99bf30cd4c16e58b13b348ec83f3": "\\mu \\leqslant x\\leqslant \\mu -\\sigma /\\xi \\,\\;(\\xi <0)", "21bec8a1b7392f36d5ba2eb7cab517be": "\\scriptstyle C_{-1}\\;=\\;E_{A}(I)", "21bf2d7fcc1241739f999554ef066363": "\\operatorname {tr} {\\boldsymbol {E}}^{q}", "21bf37951cc8e00646b1d53b0508924b": "\\lim _{z\\rightarrow a}(z-a)^{m+1}f(z)=0", "21bf8cc923343da81207d352ea5d376b": "\\langle \\exp[-i\\mathbf {q} (\\mathbf {R} _{j}-\\mathbf {R} _{k})]\\rangle =\\langle \\exp(-i\\mathbf {q} \\mathbf {R} _{j})\\rangle \\langle \\exp(i\\mathbf {q} \\mathbf {R} _{k})\\rangle =0", "21bfa6203e00d318d5f83337e2565f99": "-\\,\\Gamma ^{d}{}_{b_{1}c}T^{a_{1}\\ldots a_{r}}{}_{d\\ldots b_{s}}-\\cdots -\\Gamma ^{d}{}_{b_{s}c}T^{a_{1}\\ldots a_{r}}{}_{b_{1}\\ldots b_{s-1}d}.", "21bfaca9f12f961ce907de3b07b92119": "{\\vec {\\sigma }}{\\vec {p}}=\\sigma _{1}p_{1}+\\sigma _{2}p_{2}+\\sigma _{3}p_{3}={\\begin{bmatrix}p_{3}&p_{1}-ip_{2}\\\\p_{1}+ip_{2}&-p_{3}\\end{bmatrix}}", "21bfc0f40564c8e7a0a0d326169fe067": "{\\ddot {\\delta \\mathbf {r} }}=\\mathbf {\\ddot {r}} -{\\boldsymbol {\\ddot {\\rho }}}.", "21bfdc09a752a2146a235c75c1e6d1be": "p_{\\text{H}}", "21bfe20e958d7e594a1f7bdf86d904bd": "C_{XX}(t,s)=C_{XX}(s-t)=C_{XX}(\\tau )\\,", "21bff0735aa0f83cf7b2e82cbc71d956": "\\psi (\\omega )={\\frac {K}{2\\pi c}}(\\hbar \\omega _{max}-\\hbar \\omega )", "21c06467164694b0c2371a1dcbe47877": "\\mathbf {F} _{jk}=0", "21c0841b6ad2cf73baf36e63225c9f02": "\\lceil \\log _{2}(r)\\rceil ", "21c0b34ea58c1f739cb7a3d3e0ea31d1": "f(k;\\mu ,\\mu )\\sim {1 \\over {\\sqrt {4\\pi \\mu }}}\\left[1+\\sum _{n=1}^{\\infty }(-1)^{n}{\\{4k^{2}-1^{2}\\}\\{4k^{2}-3^{2}\\}\\cdots \\{4k^{2}-(2n-1)^{2}\\} \\over n!\\,2^{3n}\\,(2\\mu )^{n}}\\right]", "21c0d1834bc2e077338ee7cc5e756b90": "a={\\frac {p}{1-\\varepsilon ^{2}}}.", "21c1192485826411f9832876565d9ac4": "\\mathbf {\\hat {b}} ", "21c1369312cbb505f1c5c9ccd1a31245": "\\cup _{i=1}^{R}U_{i}=\\cup _{j=1}^{C}V_{j}=S", "21c13fc1d626e728e3b9e43133ca9fd0": "{\\frac {1}{2\\alpha (25812.807)(299792458)}}\\ ", "21c15de1c01ac39cccbd399f0f67060c": "{\\mathcal {P}}_{\\geq 1}(S)\\,.", "21c1c06705b8dbf9a41f88b96fd53efb": "{\\mathbf {F}}_{12}=-{\\mathbf {F}}_{21}.\\!\\,", "21c1d7ad02e551a3577c6bc909aed682": "\\mu \\!", "21c1de103aa1aa8d7206d41bc13effd6": "\\textstyle {\\frac {1}{2}}(-\\textstyle \\sum _{i=1}^{j}e_{i}+\\textstyle \\sum _{i=j+1}^{8}e_{i})", "21c1e0489fb47e3c0fc1afe02d937719": "\\alpha \\to 0", "21c1e7526266604679965449f6671ee9": "K_{y}=K_{2}=i\\left.{\\frac {\\partial {\\widehat {B}}(\\varphi ,{\\hat {\\mathbf {e} }}_{y})}{\\partial \\varphi }}\\right|_{\\varphi =0}=i{\\begin{pmatrix}0&0&1&0\\\\0&0&0&0\\\\1&0&0&0\\\\0&0&0&0\\\\\\end{pmatrix}}\\,,", "21c1e76b34325a824daf403598ab3ca2": "K({\\frac {\\sqrt {(a^{2}-b^{2})}}{a}})={\\frac {2a}{a+b}}K({\\frac {a-b}{a+b}})", "21c22b677573f3d63c5869a242da7c3c": "\\left|\\Gamma (w)\\right|^{2}+\\left|\\tau (\\omega )\\right|^{2}=1", "21c24539e3545e49a9442443cd8ffff7": "{\\hat {V}}(t)", "21c254aa9d46621286aba6e057dc00ae": "h(\\lambda )=e_{\\lambda }", "21c271f7447e2223eba7afc8ae421604": "\\eta (a,b,c):F(a,b,b)\\rightarrow G(a,c,c)", "21c2e59531c8710156d34a3c30ac81d5": "Z", "21c31e49b1198d4503c0f45dd58bcb94": "a_{m}^{T}", "21c320134d8391a110980e7d36b5ece7": "\\Phi (0\\leqslant \\Phi \\leqslant 2\\pi )", "21c3cb737a47ce4168b5c0969839eab2": "G(f:X\\to TY)=\\mu _{Y}\\circ Tf\\;", "21c3f076a0f7021d825a5e331bfc42f7": "1.\\ {\\frac {\\partial }{\\partial \\theta }}{\\frac {\\partial u/\\partial x_{k}}{\\left|\\partial u/\\partial t\\right|}}>0\\ \\forall k", "21c3fbf8e4d82135c471f8b84222bc3d": "H_{5,s}=H_{f}+x_{5,s}H_{fg}\\,", "21c46aa00609bbf6c4f04e3f48016d4f": "d_{LS}\\neq d_{S}-d_{L}", "21c496a5c0465f6ed18fea3adbd8edce": "K=-{{\\ddot {\\varphi }} \\over \\varphi },\\,\\,K_{m}={-{\\dot {\\psi }}+\\varphi ({\\dot {\\psi }}{\\ddot {\\phi }}-{\\ddot {\\psi }}{\\dot {\\varphi }}) \\over 2\\varphi }.", "21c51974d9f075258474465849fa0b78": "y^{2}=x^{3}-mx", "21c5405de8f9fdcf289788ff27c3a3b9": "Z=N+P", "21c54cc667bb7e76babb6d01c610d6ad": "P\\phi _{y}=y\\phi _{y}.\\;", "21c5a20a62d0add53666d1c6b1ce0570": "S_{n}=a_{i}b_{i}+\\cdots +a_{n}b_{n}", "21c628c474b8a15e12145bf86cbb7ea8": "-{\\tfrac {1}{24}}(b-a)^{3}f''(a)+O((b-a)^{4})\\quad {\\text{and}}\\quad {\\tfrac {1}{12}}(b-a)^{3}f''(a)+O((b-a)^{4}),", "21c640ff51ce2908e16105e6a6a6d49c": "\\mathbf {M} _{\\text{in}}=\\mathbf {Q} \\mathbf {M} ,", "21c64c32fa2bfe330b51871455345f58": "(t_{1},t_{2},\\ldots ,t_{n})", "21c67960cb7b2b08abc7b835db8948f9": "{\\frac {1}{\\rho }}{\\frac {\\,d\\,P}{\\,d\\,X}}-{\\tfrac {\\,f}{2\\,D}}\\,W^{2}+\\left({\\frac {2-\\beta }{2}}\\right){\\frac {\\,d\\,W^{2}}{\\,d\\,X}}\\,=\\,0", "21c685c73142fd23b47dac7f5fea115e": "-1<\\beta \\leq -0.75", "21c68f31bf6db00547183f0985d533b4": "({\\boldsymbol {\\mu }},{\\boldsymbol {\\Sigma }}^{-1})\\sim \\mathrm {NW} ({\\boldsymbol {\\mu }}_{0},\\lambda ,{\\boldsymbol {\\Psi }}^{-1},\\nu )", "21c701d4913b4baa6a9d259a4c28897c": "D={\\begin{vmatrix}h_{3}&h_{4}&h_{5}&h_{6}\\\\h_{1}&h_{2}&h_{3}&h_{4}\\\\1&h_{1}&h_{2}&h_{3}\\\\0&0&1&h_{1}\\end{vmatrix}}.", "21c755a92838b5266375443f943e66c8": "{\\hat {H}}_{\\textrm {ph}}", "21c7571812e736d745ba0a095483c464": "{\\textrm {efficiency}}={\\frac {w_{cy}}{q_{H}}}={\\frac {q_{H}-q_{C}}{q_{H}}}=1-{\\frac {q_{C}}{q_{H}}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(4)", "21c77b9a7835730e5cd63cbbdce5bb09": "K=\\sup _{z\\in D}|K(z)|={\\frac {1+\\|\\mu \\|_{\\infty }}{1-\\|\\mu \\|_{\\infty }}}", "21c785ced9aa6216ba4f7da5898f647a": "\\mathbf {y} _{1}=\\mathbf {x} _{1}-{\\boldsymbol {\\Sigma }}_{12}{\\boldsymbol {\\Sigma }}_{22}^{-1}\\mathbf {x} _{2}", "21c7be7dfd22e403ead3ddc3fcc71b9a": "\\mathbf {Q} (i,{\\sqrt[{4}]{1+2i}})/\\mathbf {Q} (i)", "21c7d89f1ea9018a256a242de3f9cbe8": "\\sum _{\\delta \\mid n}\\delta ^{s}J_{r}(\\delta )J_{s}\\left({\\frac {n}{\\delta }}\\right)=J_{r+s}(n)", "21c7e04ab228d557038b1c86f1a7d3f4": "\\Gamma _{bc}^{a}=0", "21c83c4b80485ef168fb5580c9c1048b": "\\langle f(z)\\rangle =\\int _{0}^{2\\pi }p_{w}(\\theta ')f(e^{i\\theta '})d\\theta '", "21c84aeeb81773fdedf0a0921c333eda": "(k-k')^{2}=\\,", "21c85dd4693a3bb121e7f5a07ab95440": "O(\\theta _{ex})", "21c86e6a06ad1fb5314eef5b829b2d3a": "_{s.3.right\\,}\\!", "21c8e45ae2a20305a3879255bb250aff": "h^{b}", "21c95d4c8936a39bb904563a8fcdbfdd": "i=1,\\ldots ,k", "21ca490699c289310b46d98921017cad": "R_{A}^{\\infty }=\\inf\\{r\\geq 1\\mid \\exists n\\in \\mathbb {Z} ^{+},R_{A}(x)\\leq r,\\forall x,|x|\\geq n\\}.", "21ca5d5ab1034997251573bd3cb96022": "z\\mapsto z+\\lambda _{q},", "21ca69461b1164668edff077d8e6c879": "{N\\hbar \\omega \\over V}={\\mathcal {E}}_{c}={\\frac {\\mid \\mathbf {E} \\mid ^{2}}{8\\pi }}", "21ca7f31773c8f7cbae3c19fa5a26659": "\\,\\mu _{h}", "21cac1c56e3ca5d185185164169844b0": "{\\mbox{P-to-W}}={\\frac {|\\mathbf {a} (t)||\\mathbf {v} (t)|}{|\\mathbf {g} |}}\\;", "21cacdf2a61d99a9e876d6ea5eef4c98": "\\nabla \\cdot {\\mathbf {A} }(\\mathbf {r} ,t)=0\\,.", "21cb16b322901cf5a7ea71c15e02570f": "B(T)=\\exp \\left(\\int _{0}^{T}r(u)\\,du\\right)", "21cb21c7c1de5b4c61da90794c62bc1a": "X\\,", "21cb349c4eb81423924e4725169705ad": "{\\begin{aligned}&\\int \\limits _{I_{x}\\subset \\Omega _{x}}f(x)e^{\\lambda S(x)}dx\\equiv \\sum _{k=1}^{K}\\int \\limits _{I_{x}\\subset \\Omega _{x}}\\rho _{k}(x)f(x)e^{\\lambda S(x)}dx\\\\&\\xrightarrow {\\lambda \\to +\\infty } \\sum _{k=1}^{K}\\int \\limits _{{\\text{a neighborhood of }}x^{(k)}}f(x)e^{\\lambda S(x)}dx=\\left({\\frac {2\\pi }{\\lambda }}\\right)^{n/2}\\sum _{k=1}^{K}e^{\\lambda S(x^{(k)})}\\left[\\det \\left(-S_{xx}''(x^{(k)})\\right)\\right]^{-1/2}f(x^{(k)}),\\end{aligned}}", "21cb523daa8742d2e3c2d1616a82f40e": "f'(x)={\\begin{cases}1&{\\text{if }}x=0,\\\\1+2\\,x\\,\\sin \\left({\\frac {2}{x}}\\right)-2\\,\\cos \\left({\\frac {2}{x}}\\right)&{\\text{if }}x\\neq 0.\\end{cases}}", "21cbbf3f2fec9b685f5f4f9d0080018c": "k=2^{r}-r-1", "21cbdbb49c9593eda7f8a5b8b7f91beb": "(F_{1})", "21cbe545595a3a3bb4abd093cae5c23d": "\\scriptstyle \\prec ", "21cbf60b5cf640edeab2e133d099672f": "w(a):=1", "21cc02743caf32a7473a553a60deaeb8": "g'(x)\\neq 0", "21cc0296c4ffdcda0def415a578628f0": "{\\frac {1+{\\tfrac {1}{x}}}{1-{\\tfrac {1}{x}}}}", "21cc35251dc254d546249bec179e1c18": "[a]_{E}", "21cc5121dfaa1c274dad0b91604c058b": "\\sigma _{\\mathrm {theoretical} }={\\sqrt {\\frac {E\\gamma }{r_{o}}}}", "21cc77c693e86268948a6bdf84a4ac15": "\\pi (x)=\\sum _{p\\leq x}1", "21cc879ea266df2c7eb756c8e89e19a8": "(X,\\mu )", "21ccb026eca7fcd7ef9199d311cc5971": "{\\frac {\\alpha }{\\beta }}=\\,{\\alpha }\\times {\\frac {1}{\\beta }}", "21ccca36f04c9d7feb2a6144829511c8": "\\scriptstyle 230V\\times {\\sqrt {2}}", "21cd291945e68dde6823d75a7bb11263": "x=v\\cos u,y=v\\sin u,z=cu.\\,", "21cd68a66073cb64978d45ac19a7436d": "{\\textbf {G}}(s)={\\frac {{\\textbf {N}}(s)}{{\\textbf {D}}(s)}}={\\frac {s^{4}+n_{1}s^{3}+n_{2}s^{2}+n_{3}s+n_{4}}{s^{4}+d_{1}s^{3}+d_{2}s^{2}+d_{3}s+d_{4}}}", "21cd74b937cc73b4f874e866b30a6a53": "x=t-{\\tfrac {b}{3}}", "21cd74ef05532869557ee641b0fdb453": "{\\boldsymbol {D}}=\\varepsilon {\\boldsymbol {E}}\\ ,", "21cdb0fe2644150136453d08f15b691f": "(x_{i}-{\\bar {x}})^{2}", "21cdd99a0d0275336bfd1f0b40bfcd37": "\\eta ^{T}{\\textbf {x}}", "21ce4100c74705354653e363cdbbef62": "(k-p')^{2}=\\,", "21ce51f55520bb24e2e8921a42df2263": "\\mathbf {C} _{N}", "21ce96636b51870a329ae7ee42a54301": "\\textstyle N", "21cea3548576bea6c96918ed857a23df": "\\int _{a}^{b}f(x)\\,dx=\\lim _{c\\to b^{-}}\\int _{a}^{c}f(x)\\,dx", "21ceb0d41549ee8b5c7a87c2a28913d7": "df_{p}=\\sum _{i=1}^{n}{\\frac {\\partial f}{\\partial x^{i}}}(p)(dx^{i})_{p}.", "21cecf7312747f554c9f27205170bd9a": "|\\mathbf {a} |={\\frac {GM}{x^{2}}}.", "21cf170e4f4944aab8f399d58ce7ba2c": "[N_{i},p_{0}]=Dp_{i}", "21cf46d9d1ff93c724548378ad3d67b9": "f(\\mathbf {x} )=\\psi ^{i}(\\mathbf {x} )", "21cf7234f16ef41bfe752433f61842e7": "{\\ddot {\\mathbf {R} }}=0", "21cfd9d4dcb76da6be683980afe33c22": "(\\ell +s)<(P+q)", "21d02a4967bc8e159ffbf819214540b4": "W_{k}(m)={\\frac {L_{k}\\left(m-1\\right)+1}{\\mu _{k}}}.", "21d088cac9c105200619b0e25e0ddb26": "c={\\frac {2}{\\alpha }}\\ ", "21d0a88940bcd2d935ef0e7bd357931a": "2+{\\cfrac {1}{2+{\\cfrac {1}{2+{\\cfrac {1}{2+{\\cfrac {1}{\\ddots }}}}}}}}", "21d0aafe1f8d3a118e9982abc0ccf3ce": "X_{n}(x)", "21d0bad3ad4c2b0ed56a639a6a2efe7c": "U({\\mathfrak {h}})", "21d0e50b69b614b110a6c83f6a6b4ae8": "h_{0}", "21d0f74cc012c3c9c601e6a1688b3735": "\\sim p(X,Y)\\leftarrow {\\hbox{not }}p(X,Y)", "21d12e9bed5387438c53b1d46cedc77a": "\\rho (\\mathbf {r} ,t)=\\psi ^{*}(\\mathbf {r} ,t)\\psi (\\mathbf {r} ,t)=|\\psi (\\mathbf {r} ,t)|^{2}", "21d194eed3d57fc32553dc8caae15eb0": "\\tau _{q}(r)=-\\pi ^{s_{p}(r)}\\prod _{0\\leq i\\nu _{\\rm {yx}}", "21e20f5b08399f2bfbb34f42da1e3e23": "{\\bar {y}}_{i}={\\bar {y}}(Ti,\\tau )\\,", "21e2a83e2af3b9027ce11e1f3f0cff5d": "{\\mathbf {j}}\\times {\\mathbf {B}}=j_{z}({\\hat {\\mathbf {z}}}\\times {\\mathbf {B_{\\perp }}})+{\\mathbf {j_{\\perp }}}\\times {\\hat {\\mathbf {z}}}B_{z}", "21e2c0c0472b331622877accbe29b91b": "2n", "21e2ff6dafe11cd84a15ab5ca6e87d35": "b=\\nu \\sigma ^{2}/2", "21e32273ce601b8acabf7dc65ec8b185": "n_{r}", "21e3547c407da5c382577451a75758f0": "a_{k}n^{k}+a_{k-1}n^{k-1}+\\cdots +a_{2}n^{2}+a_{1}n+a_{0}", "21e36780a20412f14d86f6464c026c2c": "i\\hbar {\\frac {\\partial }{\\partial t}}\\Psi (\\mathbf {r} ,t)=-{\\frac {\\hbar ^{2}}{2m}}\\nabla ^{2}\\Psi (\\mathbf {r} ,t)+V(\\mathbf {r} ,t)\\Psi (\\mathbf {r} ,t)", "21e37710fe723c7024f42ad99689ac46": "F^{-1}{\\dot {F}}={{\\rm {id}}-\\exp -{\\rm {ad}}g(t) \\over {\\rm {ad}}g(t)}\\cdot {\\dot {g}}(t),", "21e3a1b406f440431e62942027c10444": "\\gamma _{xy}=\\theta ", "21e3a548c696935211633e68c9ee3ee9": "\\int x^{m}\\operatorname {arcsec}(a\\,x)\\,dx={\\frac {x^{m+1}\\operatorname {arcsec}(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)", "21e3f1aabb00f4a43571fa5183f39987": "Ext_{\\mathbb {Q} [\\mathbb {Z} ]}(H_{1}(X;\\mathbb {Q} ),\\mathbb {Q} [\\mathbb {Z} ])\\simeq Hom_{\\mathbb {Q} [\\mathbb {Z} ]}(H_{1}(X;\\mathbb {Q} ),[\\mathbb {Q} [\\mathbb {Z} ]]/\\mathbb {Q} [\\mathbb {Z} ])", "21e3fa339c256536b5b8f4e3c5dbe5dd": "x_{t}|x_{t-1}", "21e43c09d2e82c67da128f8fae7f4665": "t=1,...,T", "21e4b1c6304d92c1868e92aa6249b2f7": "{\\tilde {\\omega }}={\\sqrt {\\frac {I_{1}\\sin \\delta }{I_{2}}}}\\,{\\sqrt {|{\\dot {\\psi }}|\\Omega }}\\,,", "21e4e65f1864235e4de1d73732a866d9": "{\\mathcal {V}}_{i}", "21e50d603bf0437d8d015a9f0daa3a42": "g[m_{1}:b_{1}:1]_{L}.g[m_{2}:b_{2}:1]_{L}=g^{-1}(g[m_{1}:b_{1}:1]_{L}\\times g[m_{2}:b_{2}:1]_{L})", "21e58db0cc0dd23a1c12ad1d9b78bbb5": "Hom(A,B)\\otimes Hom(B,C)\\rightarrow Hom(A,C)", "21e625c55747befa134186bf825f041a": "\\pi (a,b)=\\displaystyle \\sum _{k=1}^{q}P_{k}(a,b).w_{k}", "21e63896d757575b7e8e3e375c98041f": "{\\mathcal {L}}_{X}g=0\\,.", "21e6474fe1fe6adfd37a5f88b706b39b": "DGS_{{\\sqrt {n}}\\gamma (n)/\\lambda (L^{*})}", "21e65807162540f4366a63c2ea32e567": "f(2,1)=(2)-2(1)+2=2-2+2=2", "21e65da76bbea608758e4b3975bc9d3e": "x^{p/q}={\\sqrt[{q}]{x^{p}}}", "21e690c7e12f19b7a4f6a9af031dcaa5": "Q=\\left({\\frac {V\\times I\\times 60}{S\\times 1000}}\\right)\\times {\\mathit {Efficiency}}", "21e70074dc992a23b1ba28c5521ed3a1": "k_{\\rm {C}}\\epsilon _{0}=1", "21e70e478a8de65a8e6f5a84e0c60036": "\\log |\\{x\\}|=0", "21e71656ba372de0586dc86d483bf2c9": "\\{X_{1},\\ldots ,X_{n}\\}.", "21e7dc01138446b187a289582101b3fd": "\\left(\\Pi ,\\Lambda \\right)", "21e81452b36b188644972ae10823d164": "G^{(k,k+1)}(t):=\\log(\\mu _{(k)}(t)/\\mu _{(k+1)}(t)),", "21e851bdae85c7399c902cd949acaffd": "rE_{g}", "21e897c34f94928b35c74795069a310b": "(w,x,y,z)", "21e8e5bc8dc4cd709a6e0883c4492a1f": "X(\\cdot )", "21e8f0cc7489f6353d3c8f470355704e": "\\Gamma ,R", "21e912297d672900424fb5399eafd195": "\\{p_{\\perp }^{2}+\\Phi (x_{\\perp })\\}\\Psi =b^{2}(w^{2},m_{1}^{2},m_{2}^{2})\\Psi \\,,", "21e9481059b59d0f3dcd27d2566e79aa": "\\mathbf {y} _{j}^{*}", "21e95751c1bda66db33f3eda534b443c": "K(z,w)=A(w)\\Psi (zg(w))=\\sum _{n=0}^{\\infty }p_{n}(z)w^{n}", "21e99750d94034f933e392918f6bae4a": "\\{M,M\\}=0", "21e9b4cfdaa86e2a2f8312f7e83c8026": "g^{2}/(4\\pi )", "21e9c8bcd59a3047981a64534fe99c7d": "\\textstyle {{\\frac {\\log(32)}{\\log(4)}}={\\frac {5}{2}}}", "21e9cd269752010b61d2d071fde00d38": "{\\frac {p(x)}{q(x)}}\\,", "21e9cd9d46715d1282c7a6d9eae99f0b": "\\lambda _{\\epsilon ,g}=(CV_{\\epsilon ,g})^{2}=\\left({\\frac {\\sigma _{\\epsilon ,g}}{\\mu _{\\epsilon ,g}}}\\right)^{2}", "21ea0ff38e34b3ebaa31cf86e3bc5862": "{\\mathfrak {so}}(3,3)\\cong {\\mathfrak {sl}}(4,\\mathbb {R} )", "21ea68c33e83f2a59e03195928ae4a8a": "17\\zeta (7)-10\\zeta (2)\\zeta (5)", "21ea8b26e1aaadd3be289c4ba772ca67": "\\left.{\\frac {d}{dt}}f(\\gamma (t))\\right|_{t=0}.", "21eaa0bd4da3fa13459974c651c9dfb6": "\\Sigma _{I,b}", "21eaf78fb8776f3439a573a383143150": "\\delta _{t}", "21eb418a27252c36a64aa5a98e271c33": "\\Omega ^{2}(k)={\\frac {g\\,k(\\rho -\\rho ')}{\\rho \\,\\coth(kh)+\\rho '\\,\\coth(kh')}},", "21eb58e47aabfdc803791bd20ca80db3": "U=b", "21eb6d4d75eddcd1f00cb5d41baf1cf8": "{\\sigma _{T}^{2}}", "21eb7c7821fbeae4ed45c4e11d1302af": "y=f(x_{1},\\dots ,x_{n}),\\,", "21ebcef62c355619930aff670065aa44": "\\mathbf {R} _{x}(n)\\,\\mathbf {w} _{n}=\\mathbf {r} _{dx}(n)", "21ebd8909c12a33d9af1122b7c9b2a83": "\\hbar /E_{\\mathrm {h} }", "21ebf7cef931ca55aa2c54896d5c8fee": "{\\hat {T}}_{1}", "21ebffcd9bf01b4bd9333cb5d1ac506e": "|{\\Psi }\\rangle =\\sum _{\\alpha _{k}}\\lambda _{{\\alpha }_{k}}^{[k]}|{\\Phi _{\\alpha _{k}}^{[1..k]}}\\rangle |{\\Phi _{\\alpha _{k}}^{[k+1..N]}}\\rangle ", "21ec2250bb4ac62aa9cf39947574abb1": "[N_{i},N_{j}]=-i\\epsilon _{ijk}M_{k},", "21ecb996a8c58670fc837e8947a4738e": "|p'_{R_{0}}(x,y)-p(x,y)|<0.1", "21ecd620a26c35f2131d96e1352664c0": "\\ R={\\frac {\\mathbf {G} _{a}\\circ \\mathbf {G} _{b}^{*}}{|\\mathbf {G} _{a}\\mathbf {G} _{b}^{*}|}}", "21ecf75eff332c6b984b25654374e82b": "\\det \\left({\\begin{array}{*{20}c}a&b\\\\c&d\\end{array}}\\right)=\\left\\lbrace {\\begin{array}{*{20}c}-cb&{\\text{if }}a=0\\\\ad-aca^{-1}b&{\\text{if }}a\\neq 0\\end{array}}\\right..", "21ed281c3c1a46b17604d7a2a334f0e3": "R=2{\\sqrt {\\frac {\\nu }{\\alpha }}}.", "21ed6b7e4066a1a82734fa6c9d82de37": "{10 \\choose 1}{4 \\choose 1}=40", "21ed9ed557da313b63ec7f6540423c25": "\\nabla \\times \\mathbf {B} ={\\frac {1}{c^{2}}}\\left({\\frac {1}{\\epsilon _{0}}}\\mathbf {J} +{\\frac {\\partial \\mathbf {E} }{\\partial t}}\\right)", "21edd6a56a792599782c03b790062358": "\\beta _{2},\\beta _{3},\\ldots ", "21ee199ddc4d248a43f348ef621f31ab": "e^{s_{3}}={\\sqrt {\\frac {c+u_{1}}{c-u_{1}}}}", "21ee2bbe52ab65fa45a793b13a19fa1c": "\\exp _{a}^{[x]}", "21ee3b0904fd08b42a4f0d1f526663c3": "S_{x}(t,f)=\\int _{-\\infty }^{\\infty }X(f+\\alpha )\\,e^{-\\pi \\alpha ^{2}/f^{2}}\\,e^{j2\\pi \\alpha t}\\,d\\alpha ", "21ee592e825a4e1dc861d3305dd7c1b6": "\\{O_{3},O_{7},O_{10}\\}", "21ee6c0ce8b50c639460f4363d64fc6a": "S_{i}+", "21eeb7f27fafff850e2879be42c2812e": "m=\\{p_{i_{1}},p_{i_{2}},...,p_{i_{n}}\\}", "21eeeaaae03e5f62b56f4b234c535337": "\\Psi ({\\mathbf {r}},t)=\\psi ({\\mathbf {r}})e^{\\alpha t}", "21eeeafe03259c10670306d25432863d": "F(y)=2\\int _{y}^{\\infty }{\\frac {f(r)r\\,dr}{\\sqrt {r^{2}-y^{2}}}}.", "21ef120a20ad19cdbe6df748e4095a90": "\\int x^{2}\\arcsin(a\\,x)\\,dx={\\frac {x^{3}\\arcsin(a\\,x)}{3}}+{\\frac {\\left(a^{2}\\,x^{2}+2\\right){\\sqrt {1-a^{2}\\,x^{2}}}}{9\\,a^{3}}}+C", "21ef28ecab8df9c1ef7c69512f5c63a9": "\\Phi (x)=1-\\Phi (-x)", "21ef6015c37d59a5b99052fe71db2704": "{\\stackrel {\\mathbf {v\\times B} }{}}", "21f00a5a3bb19d3c08967e8c8d656c5b": "F\\left(x,y,y',y'',\\ \\cdots ,\\ y^{(n-1)}\\right)=y^{(n)}", "21f00c1ae631f17e60fec725359bf0ae": "f(t)={\\frac {x_{1}+\\cdots +x_{n}+t}{n+1}}-({x_{1}\\cdots x_{n}t})^{\\frac {1}{n+1}},\\qquad t>0.", "21f051bda58ab781c99d9e78c9d6f369": "S_{z}={\\frac {\\hbar }{2}}\\sigma _{z}={\\frac {\\hbar }{2}}{\\begin{bmatrix}1&0\\\\0&-1\\end{bmatrix}}", "21f058f0e0d49f66a0d61db52288b53e": "{\\tfrac {1}{20}},\\,{\\tfrac {1}{10}},\\,{\\tfrac {3}{20}},\\,{\\tfrac {1}{5}},\\,{\\tfrac {1}{4}},\\,{\\tfrac {3}{10}},\\,{\\tfrac {7}{20}},\\,{\\tfrac {2}{5}},\\,{\\tfrac {9}{20}},\\,{\\tfrac {1}{2}},\\,{\\tfrac {11}{20}},\\,{\\tfrac {3}{5}},\\,{\\tfrac {13}{20}},\\,{\\tfrac {7}{10}},\\,{\\tfrac {3}{4}},\\,{\\tfrac {4}{5}},\\,{\\tfrac {17}{20}},\\,{\\tfrac {9}{10}},\\,{\\tfrac {19}{20}},\\,{\\tfrac {1}{1}}", "21f0ba6062febcb2d1f7bb42495fb01b": "e_{\\lambda }=\\sum _{\\mu \\uparrow \\lambda }e_{\\mu },", "21f0bf2686f22020a22b90198be2891d": "{\\mathcal {C}}_{x}^{k}\\,", "21f0e8f4b03d775ed256a52422ec5143": "\\Sigma =\\{0,1\\}", "21f1210cdf6c001738c78fc2496f8a3e": "\\inf \\theta \\leq 27/82.", "21f14abb6be09d790a6cf7f4c9445f08": "|n_{{\\mathbf {k} }_{l}}\\rangle ", "21f1a519c9345d5b1d6829b0cbaf6435": "m_{k}={\\frac {1}{k+1}}\\sum _{i=0}^{k}a^{i}b^{k-i}.\\,\\!", "21f1af307837d7582360f4dc4c83d343": "s_{\\lambda }s_{\\mu }=s_{\\lambda \\mu }", "21f1b0b3c18ef8a4fb2875d937c89f5f": "a_{n}=\\sum _{k=0}^{n}k!\\left\\{{\\begin{matrix}n\\\\k\\end{matrix}}\\right\\}", "21f1f173facad48671acbe78a72df3e1": "VC(x)_{z'}0:~~~~~~~~R={\\frac {M}{2E}}(\\cosh \\eta -1)~,~~~~~~~~(\\sinh \\eta -\\eta )={\\frac {(2E)^{3/2}(t-t_{B})}{M}}~;", "22022562a563391c724ddde602fa2086": "ITGAP={\\frac {(r-g)(b_{t}-\\sum _{i=1}^{\\infty }({\\frac {1+g}{1+r}})^{i}pb_{t+i})}{1+g}}", "22025301a52cbe9f4e26b2098c4a7a4c": "{\\mathit {d}}_{H}^{RC}({\\mathit {p}},{\\mathit {q}})={\\mathit {d}}_{H}({\\mathit {p}},{\\mathit {q}}^{RC})", "2202ca535b0da497c61f58deb49dc13b": "\\tau _{(12)}(v\\otimes w)=w\\otimes v", "2203061a5961bb1c8930c42039ce7967": "E_{\\mathrm {int} }=-\\sum _{i=1}^{3}F_{i}\\int \\rho (\\mathbf {r} )r_{i}d\\mathbf {r} \\equiv -\\sum _{i=1}^{3}F_{i}\\mu _{i}=-\\mathbf {F} \\cdot {\\boldsymbol {\\mu }}", "220311a8754754d5b9cea9d4d0a6e4d6": "{\\frac {2\\times S\\uparrow \\times \\ S\\downarrow }{S\\uparrow +\\ S\\downarrow }}", "220315c41c00e02c99b4a1db2b752916": "E_{av}=E_{0}+{\\frac {3}{5}}E_{F}", "2203413082fcdf541b0f2a2e2ae2e676": "p_{L}(y)=(y-1)(y-m-1)\\cdots (y-[n-1]m-1),", "22034964c1ebe6228ad0ef8ecbe55798": "|T|<2(1-\\varepsilon )\\gamma n\\,", "220357cdfce1c39f1f6ae23a3ffb324c": "U_{9}(x)=512x^{9}-1024x^{7}+672x^{5}-160x^{3}+10x.\\,", "22036bf382317be0cf3f0fe97438311b": "\\Delta ={\\frac {D-2}{2}},", "2203afc382f0fd3dbf33b1e3c4f829be": "g(r)=-{\\frac {GM}{r^{2}}}.", "2203b25fe01105d229917494c693c44e": "y^{i}", "2203ccab854940fbb4a69a75fafb7c0c": "|{\\bar {\\psi }}\\rangle =U(R)|\\psi \\rangle ", "22044d076d49f14e3a53df7106374ddd": "I_{m,n}=\\int {\\frac {\\sin ^{m}{ax}}{\\cos ^{n}{ax}}}dx\\,\\!", "220493f3b19884c7e297e2239c96e51b": "(\\mathbf {A} ^{\\mathrm {T} })^{\\mathrm {T} }=\\mathbf {A} \\quad \\,", "2204a224e95c42057b6d7daa1982b25b": "d={\\frac {-(\\mathbf {l} \\cdot (\\mathbf {o} -\\mathbf {c} ))\\pm {\\sqrt {(\\mathbf {l} \\cdot (\\mathbf {o} -\\mathbf {c} ))^{2}-\\mathbf {l} ^{2}((\\mathbf {o} -\\mathbf {c} )^{2}-r^{2})}}}{\\mathbf {l} ^{2}}}", "2204a3e2080fc9c09eff788f9fb870ac": "\\mathrm {Gr} _{D}={\\frac {g\\beta (T_{s}-T_{\\infty })D^{3}}{\\nu ^{2}}}\\,", "2204c2b4d60ae2fd5279ec54cfaed2a4": "2x", "22050a8034e3b7aa0b6c1c0d7e8282bc": "\\cdot :S\\times S\\rightarrow S", "22052c5ec2eb75c3ed60007dc407074a": "\\lim _{h\\to 0}{(6+h)}=6+0=6.", "22059e96225d83824a489c15e16df795": "x'_{v_{i1}j}=x_{ij}", "2205a844de8489299dfcdc94854ffcd3": "\\int \\limits _{0}^{2\\pi }\\left(-{\\hat {t}}\\ \\left({\\frac {p}{r}}\\right)^{2}\\ {\\frac {3}{2}}\\ \\left(3\\ \\sin ^{2}i\\ \\sin ^{2}u\\ -\\ 1\\right)\\ -\\ \\left(2\\ {\\hat {r}}-{\\frac {V_{r}}{V_{t}}}\\ {\\hat {t}}\\right)\\ \\left({\\frac {p}{r}}\\right)^{2}\\ 3\\ \\sin ^{2}i\\cos u\\ \\sin u\\right)du", "2205eb6d84c9e4fdb31bffeb3985caff": "V=\\{{{v}_{1}},...,{{v}_{n}}\\}", "2206201a5e45eef63325349495157b49": "\\scriptstyle 7{\\frac {3}{4}}", "22062280a22f5bb83e61c8c505c99730": "m_{\\mathrm {solvent} }", "220683701f0349cee2f1486988bc81e5": "t.i", "22069d1b5b6cfd8c40c5a36a09ff9e93": "\\Delta \\phi =\\phi _{2}-\\phi _{1}", "2206a5f55d5ca3673011e8950515c717": "1\\;\\xrightarrow {} \\;A\\;\\xrightarrow {f} \\;B\\;\\xrightarrow {g} \\;C\\;\\xrightarrow {} \\;1", "2206ada777980db3afedddfc359a6804": "\\varepsilon (t)=\\int _{u(0)}^{u(t)}{\\frac {F^{h}(u)}{m}}\\mathrm {d} u=(1-a){\\frac {k_{i}}{m}}\\int _{0}^{t}z(\\tau ){\\dot {u}}(\\tau )\\mathrm {d} \\tau ", "2206d80622823ef284615384c2571f9a": "\\sum _{i=0}^{n}a_{i}p^{i}", "220706312b4b2735e34e8976ade8e853": "\\Delta _{\\pi }", "220735b2831beee988dbf3e81cc873b3": "p=3", "22073a4470ab419111518edb4ff18e62": "O(n\\log \\log n)", "2208041b7c34dcf146ce994216b2e83f": "\\mathrm {tr} (\\mathbf {AB} )=\\mathrm {tr} (\\mathbf {BA} )", "22084ea77de533cc1ca20d79d438022d": "X=A\\sum _{i=0}^{m-1}a_{i}.", "220876ed22fec26384299279d7c0c462": "\\theta ^{*}\\approx \\theta _{0}+{\\mathcal {J}}^{-1}(\\theta _{0})V(\\theta _{0}).\\,", "22088400ede476760e73ab87d3e80d48": "{\\mathbf {A} }_{11}^{-1}{\\mathbf {A} }_{12}|{\\mathbf {A} }_{22\\cdot 1}\\sim MN_{p_{1}\\times p_{2}}({\\mathbf {\\Psi } }_{11}^{-1}{\\mathbf {\\Psi } }_{12},{\\mathbf {A} }_{22\\cdot 1}\\otimes {\\mathbf {\\Psi } }_{11}^{-1})", "2208dd816285a4d3689ca38bb789159a": "t'=t{\\omega }\\,\\!", "2208f21c6385673f7f5757329a5ac544": "\\mathbf {r} \\cdot \\mathbf {\\hat {n}} =\\left|\\mathbf {r} \\right|\\cos \\alpha \\,\\!", "22091444e3da60d0a52fcc4b2cee6d70": "\\chi ^{\\prime \\prime }(\\mathbf {Q} ,\\omega )", "220984254487f8707d685479a3761c25": "h(x_{0}+\\Delta x)=f(x_{0}+\\Delta x)g(x_{0}+\\Delta x)=f(x_{0})g(x_{0})+\\Delta fg(x_{0})+f(x_{0})\\Delta g+\\Delta f\\Delta g", "2209cdd73bf13391bbd4ceb8bf75b1a0": "\\int _{0}^{\\infty }{\\frac {x^{3}}{e^{x}-1}}\\,dx={\\frac {\\pi ^{4}}{15}}", "2209e21d3d3c62419b29c1bcf5fca29f": "C_{(-)}^{T}=C_{(-)};~~~C_{(-)}^{2}=1", "220a0c5ce505a28b7d9dfb9fa53d69ed": "F_{n}(z)=f\\circ f\\circ \\cdots \\circ f(z)\\to \\alpha ,", "220a97a21b885785e62eadc7a4c78236": "{\\frac {\\partial \\chi }{\\partial t}}=\\nabla \\cdot (D\\,\\nabla \\chi +D_{T}\\,\\chi (1-\\chi )\\,\\nabla T)", "220abf6bc46dcc1c63937fc38f315032": "2\\Pi _{2}\\left(\\prod _{p|n;p\\geq 3}{\\frac {p-1}{p-2}}\\right)\\int _{2}^{n}{\\frac {dx}{\\ln ^{2}x}}\\approx 2\\Pi _{2}\\left(\\prod _{p|n;p\\geq 3}{\\frac {p-1}{p-2}}\\right){\\frac {n}{\\ln ^{2}n}}", "220ac14f14c1a1efed582072f8d77770": "10/81=0.{\\overline {123456790}}", "220b19101c8bbdf0a36f62a9c641c913": "\\mathbf {e} ^{i}(c_{1}\\mathbf {e} _{1}+\\cdots +c_{n}\\mathbf {e} _{n})=c_{i},\\quad i=1,\\ldots ,n", "220b5ed3b6746762a49bf602e16d9d26": "\\cot {\\frac {\\pi }{10}}=\\cot 18^{\\circ }={\\sqrt {5+2{\\sqrt {5}}}}", "220b812198ad2aec784cbe90738b880c": "b_{0}\\centerdot 1+b_{1}\\centerdot 1+b_{2}\\centerdot 1+b_{3}\\centerdot 1=0", "220ba13e9bec9ee194f6ab5c635eb034": "{\\mathcal {L}}={\\mathcal {L}}_{1}+{\\frac {1}{2}}\\,I_{2}\\Omega ^{2}\\cos ^{2}\\delta +{\\frac {d}{dt}}(I_{2}\\alpha \\Omega \\cos \\delta )\\,,", "220bb215b8cc6dbe2c8c8244941d3fb1": "x_{n}={\\tfrac {2}{\\pi }}sin^{-1}(y_{n}^{1/2})", "220c0ff010dbd6b4eaccd8c260f9ffe5": "x_{i-{\\frac {1}{2}}}\\ ", "220c2d191a1f2377194b2c2bdd0619d2": "K(z)=\\exp \\left(\\psi ^{(-2)}(z)+{\\frac {z^{2}-z}{2}}-{\\frac {z}{2}}\\ln(2\\pi )\\right)", "220cc04c478ff42bc4a41bcd9f586e27": "{\\hat {U}}(t,t_{0})=1-i\\int _{t_{0}}^{t}dt'{\\hat {V}}(t')", "220cf0b6aa780d764d26d6292f1de38c": "a=-b", "220d0eaba897c0f6628aae07cafb3076": "L=\\int _{a}^{b}{\\sqrt {\\left[r(\\varphi )\\right]^{2}+\\left[{{dr(\\varphi )} \\over {d\\varphi }}\\right]^{2}}}d\\varphi ", "220d1237794470250148ee37a0af3ef1": "0<\\Im {z}<\\beta ", "220d21a517172a2a589add2f409355d4": "g_{N}>0", "220d6b2c0d44810aaab1e548204997f6": "\\scriptstyle {i=1,\\ldots ,m}", "220d7a1030e205f9570c702431c103a2": "\\pi ={\\cfrac {4}{1+{\\cfrac {1^{2}}{2+{\\cfrac {3^{2}}{2+{\\cfrac {5^{2}}{2+{\\cfrac {7^{2}}{2+{\\cfrac {9^{2}}{2+\\ddots }}}}}}}}}}}}={\\cfrac {4}{1+{\\cfrac {1^{2}}{3+{\\cfrac {2^{2}}{5+{\\cfrac {3^{2}}{7+{\\cfrac {4^{2}}{9+\\ddots }}}}}}}}}}=3+{\\cfrac {1^{2}}{6+{\\cfrac {3^{2}}{6+{\\cfrac {5^{2}}{6+{\\cfrac {7^{2}}{6+{\\cfrac {9^{2}}{6+\\ddots }}}}}}}}}}", "220d8023f37720b8da0754d367daf08b": "P(A_{1},\\ldots ,A_{n})\\iff P(*A_{1},\\ldots ,*A_{n})", "220d918ef21cbfe62ad0d92710d44c97": "\\sigma _{x}^{2}\\sigma _{y}^{2}/[\\sigma _{x}^{2}+\\sigma _{y}^{2}]", "220d927e869c2390c1fda7eea6361023": "a_{k}:=|L\\ \\cap \\Sigma ^{k}|", "220dbe8a0972557222940bbbf80d7103": "{\\hat {R}}={\\frac {1}{N}}\\sum \\limits _{i=1}^{N}R_{i}", "220deedeea68edced24ebf77f4e3efd3": "X:=\\bigcup _{k=1}^{m}3\\,B_{j_{k}}", "220ec1fd6364da4809a5bbd4d695f0bc": "\\!{\\mathcal {A}}\\models _{X[F/x]}^{+}\\phi ", "220eebf8f8c8c5b7044a42816ab5f56c": "condition_{j}", "220f251b894c9f00202cad338d1be09a": "\\theta _{S}", "220f4e8081048e24f3aa6b2114d1174b": "s(x)=\\sum _{i=1}^{n}\\chi _{E_{i}}(x)b_{i}", "220f5f2dab41dbddf09457b5491942dc": "\\textstyle c_{n}=n+1", "220f6cae28ee5cb53474ce99bac13c05": "\\beta _{F0}", "220f9cd35d0f2f9b80d13d73bcda009b": "{\\frac {\\partial }{\\partial t}}\\left(\\nabla ^{2}\\phi -\\phi \\right)+\\left[\\phi ,\\nabla ^{2}\\phi \\right]-\\left[\\phi ,\\ln \\left({\\frac {n_{0}}{\\omega _{ci}}}\\right)\\right]=0.", "220fbc0f092419ba112d734a5369e9d8": "\\ {\\begin{array}{rrcl}&P^{*}(F^{*})^{T}&=&J^{*}\\sigma ^{*}\\\\\\Rightarrow &P^{*}(QF)^{T}&=&JQ\\sigma Q^{T}\\\\\\Rightarrow &P^{*}F^{T}Q^{T}&=&QJ\\sigma Q^{T}\\\\\\Rightarrow &P^{*}F^{T}Q^{T}&=&QPF^{T}Q^{T}\\\\\\Rightarrow &P^{*}&=&QP.\\end{array}}", "220fc943a0224aa48cda73ea86e998f6": "p\\colon C\\to X\\,", "220fe190634db161b0a7d244646394a8": "[p]\\mapsto [m]", "2210248fff97dbc40ca58394b52d9036": "\\int _{-N}^{N}f(x)\\,dx", "221056de6a39bf400060292db53a4fba": "\\bigcup _{n}B_{n}=B.", "2210862e586d661e35708384d9c3c957": "\\displaystyle {[L_{m},\\,J_{n}]=-nJ_{m+n}}", "22109003a1884d96d9f69b7f7e5735c2": "w_{2}w_{4k-1}.", "2210cc7648fa8173829e59639e012641": "L(s_{r})=aL(s),\\qquad L(s_{\\ell })=a^{-1}L(s).", "2210e5a25f7eecce23c67b527913b105": "{\\mathbf {z} }", "221120ba3536e487a32e92676b14f911": "{\\begin{aligned}dA_{\\mathbf {x}}\\,dA_{\\mathbf {y}}&{}={\\begin{bmatrix}1&0&d\\phi \\\\d\\theta \\,d\\phi &1&-d\\theta \\\\-d\\phi &d\\theta &1\\end{bmatrix}}\\\\dA_{\\mathbf {y}}\\,dA_{\\mathbf {x}}&{}={\\begin{bmatrix}1&d\\theta \\,d\\phi &d\\phi \\\\0&1&-d\\theta \\\\-d\\phi &d\\theta &1\\end{bmatrix}}.\\\\\\end{aligned}}", "221127e1633dc0a4863bb235db9bd976": "{\\frac {\\partial \\rho }{\\partial t}}+\\nabla \\cdot \\left(\\rho \\mathbf {u} \\right)=0", "221139977c206ec99f734c5b6d17818e": "\\displaystyle {\\omega (g,h)=\\Omega (g,h)\\beta (gh)^{-1}\\beta (g)\\beta (h),}", "22113d887e01888316f5755413ee79d1": "{\\tilde {U}}={\\text{constant}}", "22116896dfcf2ea54f842e9a9e489a83": "|w|\\leq \\tau ", "221193c150f7ebad479701e16265e561": "\\|S_{1}\\|\\cdot \\|S_{2}\\|<1", "2211d6f3b3b84f1ffc9d5bb45a3f6827": "(x_{i})_{i\\in I}", "221208aa5328f227392e786def848f87": "(\\neg A\\to A)\\to A", "221208e49f32879e4044303eb88d09fe": "f_{c}={\\frac {1}{2\\pi RC}}={\\frac {1}{2\\pi \\tau }}", "2212099388254949ff008d0ec8a0dd91": "[T^{i},T^{j}]=2i\\epsilon ^{ijk}T^{k}", "2212512dabfa2a117c1943024ce6e01b": "{[G(x)]}^{n}=G{(a_{n}x+b_{n})}\\,", "221299193f96c540ae9f38cf64ac7dfa": "3\\mid a_{1}^{2}+b_{1}^{2}\\,", "2212bfd884f6df6f8a643fbd2a8219d9": "\\min \\ ||\\mathbf {v_{w}} -\\mathbf {v_{d}} ||^{2}\\qquad s.t.\\quad \\mathbf {S} \\cdot \\mathbf {v_{d}} =0", "2212d0c6199e14737dcacf7a66394234": "{\\dot {\\mathbf {x} }}(t)=\\mathbf {A} (t)\\mathbf {x} (t)+\\mathbf {B} (t)\\mathbf {u} (t)", "221334506ba2dcb828d221e71ed0f507": "x+t", "2213a573866cc3a01015b285a0497dec": "Sf=\\sum _{i\\in J}\\langle f,\\phi _{i}\\rangle \\phi _{i}", "2213acc02bfb76c74b0e9a37d13509dc": "y_{j}\\not =c_{j}", "2213b5ce4b9b042f46a3d6cdd88a7584": "2\\times {\\sqrt {6}}", "2213be3a40642d54539be537adf83cd0": "\\mathrm {Ein} (z)=\\int _{0}^{z}(1-e^{-t}){\\frac {dt}{t}}=\\sum _{k=1}^{\\infty }{\\frac {(-1)^{k+1}z^{k}}{k\\;k!}}", "2213e0878e8ac143520a0dcdd18c04f8": "{\\mathcal {M}}\\subseteq {\\mathcal {A}}", "2213e1b4cbd940603a6702ebe6ba6ae4": "3r", "2214485fc65a1dcd92b453e7c67d3795": "n={\\frac {pAl}{RT}}", "2214d4a7928dcb6ea83559bd8705ef2b": "\\,{\\frac {e^{h-\\lambda e^{h}}\\lambda ^{k}}{k!}}", "2215026cf1f70449f7e2952be00fe7a4": "\\scriptstyle (x_{1}-{\\overline {x}},\\;\\dots ,\\;x_{n}-{\\overline {x}}).", "2215093743db6194cf6ada00469b56ee": "{\\begin{aligned}{\\frac {dM_{xy}'(t)}{dt}}=i\\gamma B_{xy}'M_{z}(t)\\end{aligned}}", "221515b5aa06aceed08af2d973b2821d": "\\mu (a_{m}/a_{0})ma_{m}=GmM/r^{2}=GMm/r^{2}=\\mu (a_{M}/a_{0})Ma_{M}", "2215b59c88015fb83a27d020c7876665": "\\mathrm {\\Lambda } (A\\otimes B)=\\mathrm {\\Lambda } ^{\\!\\otimes }\\!\\left({A\\otimes B}\\right).", "2215ceecf3e5d33ad40a470d13d10143": "u_{z}={\\frac {u_{*}}{\\kappa }}\\left[\\ln \\left({\\frac {z-d}{z_{0}}}\\right)+\\psi (z,z_{0},L)\\right]", "2215d77738a7148e3887d4852e22fcdf": "D\\in \\Delta ", "221609d2bf909d45d661979c396c4f16": "Q={\\frac {\\text{gap}}{\\text{range}}}={\\frac {0.177-0.167}{0.189-0.167}}=0.455.", "221624388bab536d75aa96e7edd5931a": "I_{o}=I_{e}\\left({\\frac {q^{4}}{m^{2}c^{4}}}\\right){\\frac {1+\\cos ^{2}2\\theta }{2}}=I_{e}7.94.10^{-26}{\\frac {1+\\cos ^{2}2\\theta }{2}}=I_{e}f", "2216667551c62a6524a5d2127500a86f": "\\mu (x)", "221671d8a139ff2b0c8d6c3cd2e80dac": "{\\frac {dv}{dz}}", "22169b1420143c14ead6e48bfcb9c955": "\\mathbf {u} _{i}=-L_{i}{\\hat {\\mathbf {x} }}_{i}.\\,", "2216c9f8ff7ef6e6e0757d44d224f7a2": "E_{1}(\\mathbf {R} )", "22171e2d949bacb18801616495bf984c": "s_{ij}>\\tau ", "22172fff96e170b3c10015784f31f1c2": "\\mathrm {MA} _{compound}=\\mathrm {MA} _{1}\\mathrm {MA} _{2}\\ldots \\mathrm {MA} _{n}\\,", "22174ae7212cdb08f779c3f416f611c0": "{{\\mathbb {R}}^{2n+1}}", "221814f0e8a26cc03151e44c36139876": "f(\\mu )=|{\\partial \\mu ^{(1)} \\over \\partial \\mu }|f(\\mu ^{(1)})=f(\\mu +b)", "221825347b108bed93476c58a740404c": "(k-1)", "221839f750ef50666ab1b7fabf2bcd71": "C_{diffmap}=(m|F_{obs}|-D|F_{calc}|)exp(2\\pi i\\phi _{calc})", "22185b2c60d34219fadd22e4825f2055": "|\\ \\rangle \\!\\,", "221871dd6f60317e2b09fa62f6cfc24c": "{\\begin{aligned}x_{n+1}&=x_{n}+f(t_{n},v_{n})\\,\\Delta t\\\\[0.3em]v_{n+1}&=v_{n}+g(t_{n},x_{n+1})\\,\\Delta t\\end{aligned}}", "2218befd39fdaf83cbf2992a51196237": "p\\leq \\gcd \\left(x-y,n\\right)\\leq n", "2218c392302ec9a2b0bd03bb94556e90": "{\\mathcal {L}}_{V^{2}}(\\theta _{1})\\,", "2218d976a5d4315bc4d5ee15cab7b526": "=y+b(\\alpha -a)-\\delta (\\beta -b)\\ ", "2219018b5eb6a0a9eee1ba29c4257304": "{\\hat {\\eta }}=\\left(\\theta _{1},\\theta _{2},\\ldots ,\\theta _{k}\\right)", "221945ee4f93c359c2e5ed2fffea0053": "f\\left({\\frac {x}{n}}\\right)={\\frac {1}{n}}f(x)\\ ", "22194ac0a68811d159a39093a6c7b5a9": "\\pi _{1}(S)", "221975c3903b34e463f15f43a56ba925": "\\phi _{U}=-V_{\\infty }\\cdot \\mathbf {n} ", "22198b86edb5e7de28851ce9f8b8283e": "\\sigma _{i}=2C_{1}J^{-5/3}\\left[\\lambda _{i}^{2}-{\\cfrac {I_{1}}{3}}\\right]+2D_{1}(J-1)~;~~i=1,2,3", "221993df96d492ff7f5903c9283b3914": "\\scriptstyle T_{\\max }", "2219a9f7865e881a74b4ee585db08b16": "{\\frac {\\Delta z}{z}}\\,\\,\\,\\approx \\,\\,\\,{\\frac {a\\,\\Delta x_{1}+\\,\\,\\,b\\,\\Delta x_{2}}{a\\mu _{1}+\\,\\,b\\mu _{2}}}", "2219beee0b45b55be029df4d479009f1": "\\omega _{J}={\\frac {q}{\\hbar }}\\cdot V.\\ ", "2219ef8a39b20e9eddde42a61835e452": "w_{i}=\\mathrm {weight} \\left(a_{i}\\right),1\\leq i\\leq n", "221a116083f87d4f06615bb781ff8d0f": "\\psi {\\mathcal {U}}\\phi ", "221ade70cc2786150d39b2f8db658d17": "{\\text{Cov}}(X,Y)\\leq {\\text{Cov}}(X^{*},Y^{*})", "221af9da4d5dffa6d63bc209d7fbcc9e": "(ab)(ac_{1}c_{2}\\dots c_{r})(bd_{1}d_{2}\\dots d_{s})=(ac_{1}c_{2}\\dots c_{r}bd_{1}d_{2}\\dots d_{s})", "221b95feacf04f368578e81b7dced556": "m1_{R}=m", "221beccb58823ee171a9b857d5d2d82e": "i\\hbar |d\\Psi (t)/dt\\rangle ={\\hat {H}}|\\Psi (t)\\rangle ,", "221befdf6eae02b1e4d49c3132467ad6": "\\mu _{1}=\\mu _{1}^{+}-\\mu _{1}^{-}", "221c0543f6730384203a254d367997d3": "P(x)=\\langle \\mathbf {x} ,W\\mathbf {x} \\rangle ", "221c58ef9b428b01ffbf6f2452941bbb": "x^{\\alpha +1}\\,e^{-x}\\,", "221c5c2b55a969010749ab3a78145b76": "\\nabla _{\\mathbf {X} }y(\\mathbf {X} )={\\frac {\\partial y(\\mathbf {X} )}{\\partial \\mathbf {X} }}", "221c6947ebece069ce42d39104fde809": "I_{r}^{'}", "221c786b331969eef91620ccf284b91a": "\\left(\\int _{\\mathbb {R} }|g(y)|^{2\\beta }\\,dy\\right)^{1/2\\beta }\\leq {\\frac {(2\\alpha )^{1/4\\alpha }}{(2\\beta )^{1/4\\beta }}}\\left(\\int _{\\mathbb {R} }|f(x)|^{2\\alpha }\\,dx\\right)^{1/2\\alpha }.", "221c7fb8e8f49035f169fb9e3b18715f": "\\Pi ", "221c830ead58aba253bcc8c0c6ca8315": "d_{H}(S,T)=\\max\\{\\sup\\{d(s,T):s\\in S\\},\\sup\\{d(t,S):t\\in T\\}\\}", "221c8e862c3cc667a9665abfc938c0b1": "{\\binom {m}{k}}", "221ca05ac39e48b3f5bb48470ca50cce": "\\sigma _{m}^{2}=E_{\\pi }[\\sigma _{f}^{2}(\\theta )]+E_{\\pi }[\\mu _{f}(\\theta )-\\mu _{m}],", "221ccf451d1e86a2ada2b1d27fc2da3c": "\\mathrm {return} :A\\rightarrow E\\rightarrow \\mathrm {M} \\,A=a\\mapsto e\\mapsto \\mathrm {return} \\,a", "221cd0cf08ec47534f9ec4a2f85fa263": "\\tau _{Wall}={\\frac {D\\Delta P}{4L}}", "221d16fe871541dfe85913346464e16b": "I_{\\mathrm {NMDA} }(t,V)={\\bar {g}}_{\\mathrm {NMDA} }\\cdot B(V)\\cdot [O]\\cdot (V(t)-E_{\\mathrm {NMDA} })", "221d34767145014b134a6473c37783cf": "z=n-c\\,", "221db8ccb6aab742201b2882f3f99bb0": "\\displaystyle {f_{-}=D(\\varphi )|_{\\Omega }+S(\\psi )|_{\\Omega },\\,\\,\\,\\,\\,f_{+}=D(\\varphi )|_{\\Omega ^{c}}+S(\\psi )|_{\\Omega ^{c}}.}", "221df38694e658cd5dbaed60e0ce0f39": "\\operatorname {E} [\\,y_{t}|x_{t}\\,]=\\int g(x_{t}^{*},\\beta )f_{x^{*}|x}(x_{t}^{*}|x_{t})dx_{t}^{*},", "221e173faf90c030933b36de017f8c59": "\\int _{N}^{M+1}f(x)\\,dx=\\sum _{n=N}^{M}\\underbrace {\\int _{n}^{n+1}f(x)\\,dx} _{\\leq \\,f(n)}\\leq \\sum _{n=N}^{M}f(n)", "221e1cdcadaf602988e259936603aba6": "{\\mathfrak {B}}(V_{q})=k[x]/(x^{n})", "221e883a0f7f6f1fb3a57690de684c90": "x^{1/n}", "221e8aff5b8660c28babe46e4013e440": "\\{t,r,\\theta ,\\phi \\}", "221e8b5819c99d5fa3748d8244f1d9da": "sW(s,\\xi )=-{\\frac {d}{d\\xi }}W(s,\\xi )+U(s),", "221ea1d11882ec6de661d2cfa95dec6c": "{}=x_{i}y_{j}-x_{j}y_{i}.\\,\\!", "221ecb1d2c96d59c24f22beca73b517d": "(X^{2},\\mu \\otimes \\mu ,T\\times T)", "221eecf8c8a6f69ec2eefc1cb14c9444": "|\\delta {\\vec {x}}|\\approx |\\delta {\\vec {x}}_{0}|e^{\\lambda _{1}t}.", "221efb49aa5d6137bd51717b3d1f7900": "\\Gamma (s+1)=\\int _{0}^{\\infty }x^{s}e^{-x}\\,dx", "221f2fe2d639a951158966101b301a96": "x\\rightarrow \\infty ,", "221f772f52e35588c3cc6de43af981af": "\\mathbf {b} _{i}={\\cfrac {\\partial \\mathbf {x} }{\\partial q^{i}}}={\\cfrac {\\partial \\mathbf {x} }{\\partial x_{j}}}~{\\cfrac {\\partial x_{j}}{\\partial q^{i}}}=\\mathbf {e} _{j}~{\\cfrac {\\partial x_{j}}{\\partial q^{i}}}", "221fa9fd4a0f43e4289e27fa49206858": "w_{\\mathrm {eff} }", "221fca41660e44a19daae729f90480ea": "J=\\det({\\boldsymbol {F}})=1", "221fcae732f0b7d83af3b140a6b84d0e": "\\lceil {\\text{slog}}_{e}(-x)\\rceil =-1", "221fe2b1a2a50054c7daa69e1efead29": "r_{\\mathrm {min} }={\\frac {1}{u_{2}}}=A(1-e)", "221ff9000d1eef9d1f6152d888048616": "\\forall \\alpha .(\\alpha \\rightarrow \\alpha )\\rightarrow \\alpha \\rightarrow \\alpha ", "2220759a127017880918765ba56d6a9a": "\\liminf _{n\\to \\infty }x_{n}:=\\sup _{n\\geq 0}\\,\\inf _{m\\geq n}x_{m}=\\sup\\{\\,\\inf\\{\\,x_{m}:m\\geq n\\,\\}:n\\geq 0\\,\\}.", "2220883d6d94e5a745ed87c456f28fb5": "p(t)={\\frac {v^{2}(t)}{R}}", "2220a969492b1e2236a06c268e8691c9": "Cl(p+8,q)", "2220bca035dc1ca4b3c43a0465cf7de4": "\\,\\!T[n]", "2220db6065a8fbbafcdddbe44f43c16e": "{\\frac {\\left({\\frac {K^{-}+{\\bar {K}}^{0}}{2}}\\right)^{2}+\\left({\\frac {K^{+}+K^{0}}{2}}\\right)^{2}}{2}}={\\frac {3\\eta ^{2}+\\pi ^{2}}{4}}", "2220e37f566a5e0ec392c53d1be00d22": "[{\\mathbf {k}},{\\mathbf {k}}+d{\\mathbf {k}}]", "22212958906cc4ee62de0b991b64c031": "M_{\\lambda }\\to M_{\\mathfrak {p}}(\\lambda )", "2221a16c5d2f0e18546ddc0d6908f91d": "Y=f(\\beta _{1}^{\\top }X,\\ldots ,\\beta _{k}^{\\top }X,\\varepsilon )\\quad \\quad \\quad \\quad \\quad (1)", "22221b657ee7ad9ecd2bfe6b9571d60e": "R_{a}=1.22\\,wf,", "22221d7f2e51c00d35e8eefee65090ef": "ds^{2}=-\\left(\\alpha ^{2}-\\beta _{i}\\beta ^{i}\\right)\\,dt^{2}+2\\beta _{i}\\,dx^{i}\\,dt+\\gamma _{ij}\\,dx^{i}\\,dx^{j}", "22230e27d6cea1a1def99a921859833b": "\\displaystyle {\\mathrm {Tr} _{\\mathbf {R} }\\,XY=\\mathrm {Tr} _{\\mathbf {R} }\\,YX,\\,\\,\\,\\mathrm {Tr} _{\\mathbf {R} }\\,(XY)Z=\\mathrm {Tr} _{\\mathbf {R} }\\,X(YZ).}", "222396d6b6003dfca2c070ef98dfeb07": "\\rho (x)={\\frac {e^{-{\\frac {x^{2}}{2}}}}{\\sqrt {2\\pi }}}", "2223e0180eae55db3545442c9e49689a": "x^{\\ast }\\left(p\\right)", "22249858a0839ff24ae59955c4af7686": "(x+y)_{n}=\\sum _{k=0}^{n}{n \\choose k}(x)_{n-k}~(y)_{k}~,", "2224ddcd6c0dc550e8b1a3fdc924d474": "f(x)=c\\,,\\,\\,c\\neq 0", "2224e658ccf23b1b58dd91759f1a29db": "\\omega ^{2^{p-2}}+{\\bar {\\omega }}^{2^{p-2}}=kM_{p}", "2225254da29c2022cc60e8ad66d647d4": "f(\\phi /c^{2})=\\exp(-\\phi /c^{2}-(\\phi /c^{2})^{2}/2)\\,", "2225583ba281dbe31d86257406b1155b": "\\sigma _{xy}=-{\\frac {\\partial ^{2}C}{\\partial x\\partial y}}", "222562ff9046bc9b7c04c5686fe36932": "c=1\\ ", "2225a02cdd4daf072d2a9ef5b91e1383": "I_{x,A}:F_{x,A}\\to F", "2225a33c170f19586f1ee1de06e36962": "p\\!", "2226aa2c8cb2fae9564e61b68d62da1d": "{\\begin{aligned}x(t)&=e^{-\\alpha t}{\\mathcal {L}}^{-1}\\left\\{{s \\over s^{2}+\\omega ^{2}}+{\\beta -\\alpha \\over s^{2}+\\omega ^{2}}\\right\\}\\\\[8pt]&=e^{-\\alpha t}{\\mathcal {L}}^{-1}\\left\\{{s \\over s^{2}+\\omega ^{2}}+\\left({\\beta -\\alpha \\over \\omega }\\right)\\left({\\omega \\over s^{2}+\\omega ^{2}}\\right)\\right\\}\\\\[8pt]&=e^{-\\alpha t}\\left[{\\mathcal {L}}^{-1}\\left\\{{s \\over s^{2}+\\omega ^{2}}\\right\\}+\\left({\\beta -\\alpha \\over \\omega }\\right){\\mathcal {L}}^{-1}\\left\\{{\\omega \\over s^{2}+\\omega ^{2}}\\right\\}\\right].\\end{aligned}}", "2226d8be45d510354555f60a9dabbcf3": "T_{d}\\subseteq T", "2226e9cfddef0a4c2521be342f24b47d": "{\\begin{matrix}{r \\choose 2}\\end{matrix}}", "2226efbbc3e8bd017487e77e040141f2": "R(u,v)w=\\nabla _{u}\\nabla _{v}w-\\nabla _{v}\\nabla _{u}w-\\nabla _{[u,v]}w.", "2226f830d7912cb8ecb96a2eeaaa3aba": "\\partial _{t}(\\rho e)+\\partial _{i}(\\rho eu_{i})+p\\partial _{i}u_{i}=0\\,", "2226f8bb8106ac95fb2db9b191fe6deb": "\\prod _{i=1}^{s}x_{i}^{\\nu _{i}}=K", "2227111c676c40339debd4a010708a7a": "\\theta (f,b)=0", "22275308eaa32ab19c032fa7594bbc09": "E(X^{n})=\\sum _{k=1}^{m}\\left\\{{\\begin{matrix}n\\\\k\\end{matrix}}\\right\\}.", "22279224d3434f44f995197df53c43c9": "{\\tau }", "2227d091e1a568ba04be669214b32cf7": "a,b\\in H_{1}(S)", "2227d8afc43d38f338fadea18383d84b": "R_{X}(t,s)=E\\{X[t]X[s]\\}", "2227fc6b1daac62daf62a1253b870030": "DF:U\\times X\\to Y", "222809caa99a6ac65909bdb8821ccc74": "x_{0},y_{0},z_{0}", "22281defad6e2d58653198aea6fce8fd": "y,t", "2228562089c383a77ce74771d157b5eb": "T(g)\\cdot T(h)=f_{g}\\cdot f_{h}=f_{g*h}=T(g*h).", "2228ac3354ce4b278faae0fcf1ae05e9": "y=1.80-j3.90\\,", "22290fe6a2e26be2ed37aa263a664fa4": "ds^{2}=-dt^{2}+\\alpha ^{2}\\sinh ^{2}(t/\\alpha )dH_{n-1}^{2},", "222924df98b78878ce8dd47d80d7711d": "\\int _{\\gamma }f(z)\\,dz=F(\\gamma (b))-F(\\gamma (a)).", "22292f1f0079317e05c9463ddec0d10b": "a_{2,1}x_{1}+a_{2,2}x_{2}=b_{2}", "222940f2e774445e0755a45550549e2e": "n=(-1+{\\sqrt {-163}})/2", "2229498613991f866abcc16b06bbdadf": "m(\\mathbf {x} )", "2229819e8cb2bd103c6c16257adde5d4": "{\\frac {r}{2}}{\\frac {\\cos(\\pi /n)}{n}}", "222a2db67ea0ee77a516fc42bcd41be2": "symb(L)\\equiv \\sum _{i+j=n}r_{i,j}(x,y)X^{i}Y^{j}", "222a3fa8402831cba5b5b30101950ad6": "A(\\lambda )={SRM \\over 12.7}(0.018747e^{-{(\\lambda -430) \\over 13.374}}+0.98226e^{-{(\\lambda -430) \\over 80.514}})", "222a5c4abd0a41e794c06da5fbed1f1f": "{\\frac {2}{\\pi }}\\arctan \\!\\left[\\exp \\!\\left({\\frac {\\pi }{2}}\\,x\\right)\\right]\\!", "222a689be9f378cc408cefbe9098e2e7": "{\\hat {\\sigma }}^{2}={\\frac {\\|{\\hat {r}}\\|^{2}}{{\\hbox{tr}}\\left((I-H)'(I-H)\\right)}},", "222aa6b0ef9632880a3e99be90e3a960": "\\operatorname {d} U\\propto \\operatorname {d} T", "222ac3de6b1502e116388d9614079d7b": "\\mathbf {\\pi } ^{(k)}=\\mathbf {x} (\\mathbf {U\\Sigma U} ^{-1})(\\mathbf {U\\Sigma U} ^{-1})...(\\mathbf {U\\Sigma U} ^{-1})", "222ae801471de7c2fb00b40944059e26": "v=|\\Delta \\lambda |c/\\lambda _{0}\\,\\!", "222aec8b2aa2268f9a7f9b27a1ccfc0a": "\\mathrm {Yield} =2\\left[\\left({\\frac {a-b}{cd}}+1\\right)^{6}-1\\right].", "222b1c1713b0aaea1eac6ce4e6012064": "{\\hat {x}}_{free}", "222b2d694042ec4ca03110a9fe562534": "\\left(h\\right)", "222b869619a9316a8a4cd6d277e71fd0": "{\\frac {d}{dx}}\\log \\left(\\kappa \\right)={\\frac {\\kappa '}{\\kappa }}", "222bec494dbefef35a81f2f7f509bb14": "\\tan \\psi ={\\frac {u'}{f}}\\sin \\theta .", "222c4b94c1ea472a9bb798958f79783b": "\\pi 12.5^{2}", "222c712cd0c12d2369bf9d4f1f0f17e9": "(\\log f)'=f'/f", "222ca92183d49f526b200cb797382583": "\\delta (a)", "222cc9d39d988b23f191ea9b230f1364": "{\\frac {1}{y}}{\\frac {dy}{dx}}={\\frac {f'(x)}{f(x)}}", "222cd21492d718a11bb6b44d70e22ef0": "537^{2}\\mod 84923=33600=2^{6}\\cdot 3\\cdot 5^{2}\\cdot 7", "222cfdeee227a3bdb45b5f8c24f9f80c": "D\\subset \\Omega ", "222d574a209171c87ef83aafdc6015f0": "\\chi _{\\lambda }", "222d8343c991fefb0ad3d04e509dbcdf": "Target\\ Height>{\\frac {\\left(Target\\ Range-{\\sqrt {2\\times H\\times Re}}\\right)^{2}}{2\\times Re}}", "222e2caf9c7b49d3432466e360eceba6": "{\\bar {b}}", "222e3759bff6266c3657d2605dd0d3c3": "M_{i}=\\sum _{j=1}^{n}a_{ij}\\left(v{\\frac {\\partial u}{\\partial x_{j}}}-u{\\frac {\\partial v}{\\partial x_{j}}}\\right)+uv\\left(b_{i}-\\sum _{j=1}^{n}{\\frac {\\partial a_{ij}}{\\partial x_{j}}}\\right),", "222e8d8a0206971bcfef33d5df0cd5b3": "K={\\frac {[{\\text{M}}({\\text{OH}})]}{[{\\text{M}}]K_{\\text{w}}[{\\text{H}}]^{-1}}}", "222e928942f6b98b9735c3507a5f1218": "\\Psi (1,2)-\\Psi (2,1)\\neq 0", "222ead7e5a28ba3bfa54297e8cec7848": "R_{\\text{C}}\\,", "222f9c69d670649db60ab306d8067676": "{\\frac {\\partial }{\\partial t}}p_{\\varepsilon }(x,t)=D\\Delta p_{\\varepsilon }(x,t)-{\\frac {1}{\\gamma }}\\nabla (p_{\\varepsilon }(x,t)F(x))", "222fb9cd78c40c4159dfe7037be035ea": "A=A\\left(t\\right)", "2230281fd061be3809247518cc5475c3": "A^{*}=A^{T}", "22304c1694a706284cd8fa43a39d21d6": "e^{i\\pi /2}", "2230aa3e24e99e12d7e98baa3fcaf1cb": "k={\\sqrt {n}},\\,", "223129f0f6dee1a483eea56e6c1772ab": "G=U+pV-TS,", "2231d00269d93bf209149ab3e4d7d954": "\\mathbf {P} (X\\geq a)=\\mathbf {P} \\left(e^{tX}\\geq e^{ta}\\right)\\leq {\\frac {E[e^{tX}]}{e^{ta}}}={\\frac {E[\\prod _{i}e^{tX_{i}}]}{e^{ta}}}.", "2232150810ebf9eafff772a344ee2afe": "f(x_{1},a,g(z_{1}),y_{1})\\{x_{1}\\mapsto x_{2},y_{2}\\mapsto y_{2},z_{1}\\mapsto z_{2}\\}=f(x_{2},a,g(z_{2}),y_{2})", "2232207c8c8080c643581dddad302d3e": "{I}^{2}=I\\times I", "223266d636e7de07caa11d1932c67960": "B\\Sigma _{\\infty }^{+}\\simeq \\Omega _{0}^{\\infty }S^{\\infty }", "22329c7bcc696978d61a533c5e731931": "\\theta _{e}=T_{e}\\left({\\frac {p_{0}}{p}}\\right)^{\\frac {R_{d}}{c_{pd}}}\\approx \\left(T+{\\frac {L_{v}}{c_{pd}}}r\\right)\\left({\\frac {p_{0}}{p}}\\right)^{\\frac {R_{d}}{c_{pd}}}", "2233034547204d23736f9d678ccc29cc": "U_{n+m}=U_{n}U_{m+1}-QU_{m}U_{n-1}={\\frac {U_{n}V_{m}+U_{m}V_{n}}{2}}\\,", "22330c6e44f0b4606fe2c256ba4dc504": "_{P}(f)", "223331028c1cd2de584aed1c07802329": "\\mathbf {E} _{1s}=", "2233e573678ec41f964e2c7a84713e27": "c=\\hbar =1", "2233e9aa2a260a799da251119f19a591": "a_{1}={\\frac {1}{2}}{\\textrm {ln}}{\\frac {W_{+}(p\\wedge c)+1}{W_{-}(p\\wedge c)+1}}", "22340c794846330532f7134892854875": "{\\begin{aligned}\\mathbf {L} ^{2}Y&=\\lambda Y\\\\L_{z}Y&=mY\\end{aligned}}", "2234183167c70d3c9d29467fff0a5f82": "\\,q_{x+t}", "22344b37955faff24a5f0d8738f6bd79": "d\\theta .", "22345e775d02d325fb3868cd8605d792": "\\log _{10}({\\frac {N}{S}})=-k\\cdot \\log _{10}(r)+m", "22347b673da12b1762a8be43132a7457": "{\\hat {f}}:{\\hat {M}}\\to {\\hat {N}},\\quad ", "2234d561f3d703ad89ea8fbf38149f5b": "{\\mathfrak {H}}(G)", "2234d9213618f3a9d0ad9562eb2689b1": "r_{0}\\in R_{n}", "22350f234515f924b1d780028a105000": "\\psi (x_{1},x_{2},...,x_{N})", "2235e2a6524268c02c325a2b5f421847": "\\succcurlyeq ~\\subseteq ~X^{2}", "2235ea211c48c8890b062a889642da43": "{\\mathfrak {P}}({\\mathfrak {C}}_{\\operatorname {odd} }({\\mathcal {Z}}))\\left({\\mathfrak {P}}_{\\geq 1}({\\mathfrak {C}}_{2}({\\mathcal {Z}}))+{\\mathfrak {P}}_{\\geq 1}({\\mathfrak {C}}_{4}({\\mathcal {Z}}))+{\\mathfrak {P}}_{\\geq 1}({\\mathfrak {C}}_{6}({\\mathcal {Z}}))+\\cdots \\right)", "223635019e1f1f9f08d57ceb7ba1f0e9": "{\\frac {M\\mu _{0}}{B+\\lambda M}}={\\frac {C}{T}}", "2236418f3f6e3ce8823154e3a2cb642d": "g(x)=f(x)\\cdot h(x)", "223663e6160588369676df1c64a0da13": "J_{\\mu \\nu }={\\mbox{tr}}\\left[\\rho {\\frac {L_{\\mu }L_{\\nu }+L_{\\nu }L_{\\mu }}{2}}\\right].", "2236fa96dbe0616c465504d8304e6695": "pa_{i}", "22374d9016e74189ca3c8d07e708791a": "{\\mathrm {G} \\mathrm {L} }(n,\\mathbb {R} )", "22376fa0b6c15326ef87486a34a0c3f5": "\\mu _{\\operatorname {eff} }=K\\left({\\frac {\\partial u}{\\partial y}}\\right)^{n-1}", "22378879221d36a8a7cb55080aa89325": "(A.1.a)\\quad \\psi _{,\\,\\rho \\rho }+{\\frac {1}{\\rho }}\\psi _{,\\,\\rho }+\\psi _{,\\,zz}=\\,(\\psi _{,\\,\\rho })^{2}+(\\psi _{,\\,z})^{2}+\\gamma _{,\\,\\rho \\rho }+\\gamma _{,\\,zz}", "2237b7669af7f66884a4633a7d148a3a": "\\scriptstyle i,j", "2238fa09b11be0d1ec1ef9566cd71e63": "\\displaystyle \\mathbf {F} ", "2239b1aa4eafd9521e10ab1c772c88e7": "\\alpha t=\\pi ", "223a2ab31fa3378e0704f97a28a23490": "{3,2,1}\\,\\!", "223a32b4cae9183f2aa207f44678539f": "{\\boldsymbol {N}}=J~{\\boldsymbol {F}}^{-1}\\cdot {\\boldsymbol {\\sigma }}", "223a44e9ac64f3b34b57a3fd52c33ac6": "z=y/|c|", "223a4c7fe5a38eeb7c156f661da89fe5": "I(0)", "223a4da3924e1261bb886c1cebb6a2b4": "p\\cdot (x_{i}+\\Sigma _{k}x_{k}^{*})\\geq r=p\\cdot (x_{i}^{*}+\\Sigma _{k}x_{k}^{*})", "223a6c161200973fb2499c7eb4cb404a": "U_{0}[\\mathbf {R} ]={\\frac {k_{B}T}{4R_{g0}^{2}}}\\int _{0}^{1}ds\\left|{\\frac {d\\mathbf {R} (s)}{ds}}\\right|^{2}.", "223a9a209f36205a1a3d5ed5109dfb09": "{\\frac {d[ES]}{dt}}=k_{f}[E][S]-k_{r}[ES]-k_{\\mathrm {cat} }[ES]", "223ad665207f596361afa1cbbc7c2b30": "D_{u}={\\frac {1}{m_{u}(m_{u}-1)}}\\sum _{i=1,j=1}^{m}{_{metric}}\\delta _{c_{iu}k_{ju}}^{2}", "223aef8d300309527c240820d1f79c06": "Q(A)", "223b26bb5784686cb86485c52065d727": "\\{\\{\\emptyset \\},\\emptyset \\}", "223b87a72aff030cdb1b3f7c469b1a48": "\\mathbf {C} ", "223b9082743bc4ca199916e7da490ee4": "Con:=\\{\\emptyset \\}\\cup \\{\\{n\\}\\mid n\\in \\mathbb {N} \\}", "223bac8cfcb457c56e07e83ffaadb980": "n_{1}=2n_{11}+n_{12}", "223c2862459819ff4c33649c11ef8c98": "{\\tilde {E}}_{6}", "223c39c524bdf2bf4c8bbf904ab164b1": "(\\alpha +1)_{n}", "223c5f932d52e55a9006250b9ef75e79": "p_{ij}", "223c63cbec2d9e24b2fe53154b3ea80e": "h_{f}={\\frac {8fLQ^{2}}{g\\pi ^{2}d^{5}}}", "223cfbcb711b638187c911a76985e998": "{\\frac {t}{V_{L}}}={\\frac {\\eta \\cdot \\alpha _{\\text{c}}\\cdot c\\cdot {\\frac {\\left(E_{\\text{crit}}-E\\right)}{E_{\\text{crit}}}}}{2\\cdot \\left(\\Delta P_{H}+P_{e}\\right)\\cdot A^{2}}}\\cdot V_{L}", "223d0261369101390bbd7bd804a9249e": "v(0)=0", "223d2a7a0dd96168b21b6359e9e19e39": "\\int _{a}^{b}f(t)\\,\\mathrm {d} t:=\\lim _{n\\to \\infty }\\int _{a}^{b}\\varphi _{n}(t)\\,\\mathrm {d} t,", "223d38f172bcfb9fa351581d51d00178": "F^{-1}(\\alpha /2;k,1)\\leq \\mu \\leq F^{-1}(1-\\alpha /2;k+1,1),", "223d4771230e69468777a3c416955562": "r_{t}=.95r_{c}", "223d62b222458cfb4428eb7071ab68bb": "P_{\\infty }=p'[1-\\exp(-\\langle k\\rangle P_{\\infty })].\\,", "223d7ffd94124c09d7297fea2a547837": "J^{\\mu }=c{\\bar {\\psi }}\\gamma ^{\\mu }\\psi ", "223dcfe974611928ed0043b1b06f6e50": "m_{\\mathrm {H_{2}O} }=\\left({\\frac {120.{\\mbox{ g }}\\mathrm {C_{3}H_{8}} }{1}}\\right)\\left({\\frac {1{\\mbox{ mol }}\\mathrm {C_{3}H_{8}} }{44.09{\\mbox{ g }}\\mathrm {C_{3}H_{8}} }}\\right)\\left({\\frac {4{\\mbox{ mol }}\\mathrm {H_{2}O} }{1{\\mbox{ mol }}\\mathrm {C_{3}H_{8}} }}\\right)\\left({\\frac {18.02{\\mbox{ g }}\\mathrm {H_{2}O} }{1{\\mbox{ mol }}\\mathrm {H_{2}O} }}\\right)=196{\\mbox{ g}}", "223de349ec8fb021a9a62be621abcb01": "T_{\\mathbf {v} }\\mathbf {p} ={\\begin{bmatrix}1&0&0&v_{x}\\\\0&1&0&v_{y}\\\\0&0&1&v_{z}\\\\0&0&0&1\\end{bmatrix}}{\\begin{bmatrix}p_{x}\\\\p_{y}\\\\p_{z}\\\\1\\end{bmatrix}}={\\begin{bmatrix}p_{x}+v_{x}\\\\p_{y}+v_{y}\\\\p_{z}+v_{z}\\\\1\\end{bmatrix}}=\\mathbf {p} +\\mathbf {v} ", "223e2733e86a820e720bbf1e576fa8f7": "{\\frac {\\partial u}{\\partial x}}+{\\frac {\\partial v}{\\partial y}}=0", "223e60e37db7f0ff6159878c4fbda615": "{\\mathcal {G}}", "223eac722791eb38fab27c68e9e818c7": "A_{\\mu \\nu }^{}", "223ec0ca02ec3f40feda5b84931cfbac": "(W\\mathbf {r} )\\cdot \\mathbf {s} =*(*L^{*}\\wedge \\mathbf {r} \\wedge \\mathbf {s} )=*(\\omega \\wedge \\mathbf {r} \\wedge \\mathbf {s} )=*(\\omega \\wedge \\mathbf {r} )\\cdot \\mathbf {s} =(\\omega \\times \\mathbf {r} )\\cdot \\mathbf {s} ", "223ec47cc11dec206b411abe0d3438d4": "y\\in F_{q}^{2k}", "223f66531b1616f96e05d245abd53226": "\\sum _{k}a_{k}m_{k}.\\,\\!", "223f9097f15d3718a97e71f249d1156c": "0\\leq P(x_{1},x_{2},\\dots )\\leq 1", "223fbfbe052c81e79519eae973e5031f": "(b-a)^{2}+4{\\frac {ab}{2}}=(b-a)^{2}+2ab=a^{2}+b^{2}.\\,", "22402084bcd1328735d8b7f88915b3d2": "{\\begin{aligned}I[f]&=\\int _{\\Omega }{\\mathcal {L}}(x_{1},x_{2},f,f_{,1},f_{,2},f_{,11},f_{,12},f_{,22},\\dots ,f_{,22\\dots 2})\\,\\mathrm {d} \\mathbf {x} \\\\&\\qquad \\quad f_{,i}:={\\cfrac {\\partial f}{\\partial x_{i}}}\\;,\\quad f_{,ij}:={\\cfrac {\\partial ^{2}f}{\\partial x_{i}\\partial x_{j}}}\\;,\\;\\;\\dots \\end{aligned}}", "22405e2588aeae52551d82e3357cd205": "V_{W}={\\frac {1}{3}}(RA+LA+LL)", "22409d008af3f0f56c6bf526efb1a7ac": "\\,X(s)", "2240a3a2f59af5a5ca1daaa0b5609aa7": "{\\begin{array}{lcl}b'_{i}x_{i}+c'_{i}x_{i+1}=d'_{i}\\qquad &;&\\ i=1,\\ldots ,n-1\\\\b'_{n}x_{n}=d'_{n}\\qquad &;&\\ i=n.\\\\\\end{array}}\\,", "2240ce32f133de4aea4a324fae607b6f": "{\\text{SSMD}}={\\frac {X_{i}-{\\bar {X}}_{N}}{s_{N}{\\sqrt {2(n_{N}-1)/K}}}},", "224113b8b6965461c6778a9ac583ac49": "C_{N}^{\\infty }(\\mathbb {R} ^{n})", "2241516c7bc19c2347a15f953aa0ccbb": "H\\geq T^{{\\frac {27}{82}}+\\varepsilon }", "224159ef585a875eb3fe1e83992d7b54": "g\\otimes t^{n}", "22417f146ced89939510e270d4201b28": "{\\frac {1}{5}}", "22419ac61847201a249ca1f20b856584": "\\lambda _{o}\\,", "2241a76302b85300767fddee76c569ff": "c=55000", "2241b0b2f93a790ac4f2be9c3817d86b": "x=1.61633...", "2241d424233d02badfaf8419e5b7cc99": "m\\ll M\\ll m_{0}+t", "2241e369c36cd20ae3c19afcd2046949": "h(t)={\\frac {f(t)}{R(t)}}={\\frac {\\lambda e^{-\\lambda t}}{e^{-\\lambda t}}}=\\lambda .", "2242813bbb729d8c0d26394043414885": "\\pi _{\\ast }^{S}=\\bigoplus _{k\\geq 0}\\pi _{k}^{S}", "2242b62a9f3cc176a289724bf656bb71": "2c^{2}{\\frac {d\\tau }{dq}}\\delta {\\frac {d\\tau }{dq}}=-2r^{2}{\\frac {d\\varphi }{dq}}\\delta {\\frac {d\\varphi }{dq}}\\,.", "2242bbc8a52f22e15b530681bf33cc5e": "\\langle x_{i}\\rangle =\\operatorname {E} (x_{i})", "2242e31e940d691621bc4180904b3fa0": "^{\\;}H(\\xi )=H(\\star q(\\xi ,\\tau ))", "2242e3a216462f3e09b1b7ab646fb358": "\\varphi (y)=a_{0}\\cos {\\frac {\\pi y}{2}}+a_{1}\\cos 3{\\frac {\\pi y}{2}}+a_{2}\\cos 5{\\frac {\\pi y}{2}}+\\cdots .", "2242f747e29686a564b1d02dad55a501": "(s,0)", "224329cbd2e73a03613eae039e07ca19": "\\Delta U=Q-W\\,.", "22438d0bd5a3515a3417a4f1bbc49a33": "\\omega (P)", "2243b8e5a4963aea64fe054e030a37bc": "(\\pi ,V)", "2244052c5e8b25a58c0fd1624c90e3c5": "{\\begin{aligned}\\mathrm {Area} (r)&{}=\\int _{0}^{r}2\\pi t\\,dt\\\\&{}=\\left[(2\\pi ){\\frac {t^{2}}{2}}\\right]_{t=0}^{r}\\\\&{}=\\pi r^{2}.\\end{aligned}}", "22440e3b3c7bb02fa840ffc896c293f9": "u(x)\\geq u(y)", "22441de56f68f79e82d5fd259aa0be1a": "U^{(k-1)}\\mathbf {x} _{w}^{(k-1)}=\\mathbf {x} ^{(k-1)}", "224461e6e20eeee4e93682a56ca9ed1f": "\\alpha (x_{1},\\ldots ,x_{n}):=(\\alpha x_{1},\\ldots ,\\alpha x_{n})", "224497d30d971b29ae596ffb2eb66c32": "\\left(-{\\sqrt {2}},{\\sqrt {2}}\\right)\\cap \\mathbf {Q} =\\left\\{x\\in \\mathbf {Q} :x^{2}\\leq 2\\right\\}\\,", "22449d6ad757b0c2d9357ef8dfb82aa1": "u=\\,", "2244af6c06adde0b8f2bbabf8fbbd7dd": "(1){\\cfrac {(2){\\cfrac {(1){\\cfrac {C_{1}(1,3)\\qquad {\\color {red}{C_{8}}^{*}}}{C_{3}(2,3,5)}}\\qquad C_{4}(1,-2)}{C_{7}(1,3,5)}}\\qquad (4){\\cfrac {C_{5}(-1,4)\\qquad C_{6}(-1,-4)}{C_{8}(-1)}}}{C_{9}(3,5)}}", "2244f86344bb8e0e36e2d71ddda5dc1d": "{\\frac {1}{4\\pi }}\\,", "22450705f389f291b9067609a71e524c": "\\chi (M\\#N)=\\chi (M)+\\chi (N)-2.\\,", "22451bea04a5f3aa2064a2332894c757": "\\mathbf {p} \\times \\mathbf {\\epsilon ^{1}} (\\mathbf {p} )=-ip_{0}\\mathbf {\\epsilon ^{1}} (\\mathbf {p} ),", "22457e91c28c2b794eddafacfef67e6b": "\\operatorname {Var} (\\theta )=\\sigma ^{2}={\\frac {n-1}{n}}\\sum _{i=1}^{n}({\\bar {\\theta }}_{i}-{\\bar {\\theta }}_{\\mathrm {Jack} })^{2}", "2245afcd303302e719ebacfba93f012a": "\\left\\{y~\\backepsilon ~x\\succ y\\right\\}", "22463d9c1bc15c257b2f293199a5d33f": "{\\hat {y}}={\\arg \\min }_{l\\in \\mathbf {Y} }\\|{\\vec {\\mu }}_{l}-{\\vec {x}}\\|", "224699c6511b7daa5bef4e8592df21e8": "b_{2}=V_{2}^{+}", "2246f6c1e31a1a1d205ca85f2865980c": "\\displaystyle {\\|F_{n}\\|_{2}^{2}=2^{n}n!{\\sqrt {\\pi }},}", "22474115ab11dc153654afa1ebf9470a": "\\cos(\\theta -\\alpha )=\\cos(\\theta )\\cos(\\alpha )+\\sin(\\theta )\\sin(\\alpha )", "2247880f9af2adc7d58c0347b06f0e1e": "\\lambda _{B}\\,", "22478ae47519f744dc6486756c4f0fc6": "\\,Z", "2247dcb235fa744046fbb98801b82f69": "{\\text{average queue (}}Q_{\\text{avg}}{\\text{)}}={\\frac {{\\text{total delay experienced by }}m{\\text{ vehicles}}}{\\text{duration of congestion}}}={\\frac {TD}{(t_{2}-t_{1})}}", "2247f35d207dd0f781fe9ab1e72aebeb": "x={\\frac {A_{k}\\zeta _{k+1}+A_{k-1}}{B_{k}\\zeta _{k+1}+B_{k-1}}}\\,", "2247fa6106d68a9de4fc57932e875f0c": "{\\frac {d^{2}x^{\\alpha }}{{d\\tau }^{2}}}=-\\Gamma _{\\beta \\gamma }^{\\alpha }{\\frac {dx^{\\beta }}{d\\tau }}{\\frac {dx^{\\gamma }}{d\\tau }}\\,.", "2248836a69373c04ae26d5ffed2f3c04": "{\\mathfrak {a}}_{0}=i{\\mathfrak {a}}", "224897321fbf64a276e08322476804a7": "\\Delta f=f_{x}\\Delta x+f_{y}\\Delta y+\\cdots ", "224902b3cdbcd914cbe6eedcbfcdc53a": "F(\\rho _{1},\\rho _{2})=\\left[{\\mbox{tr}}({\\sqrt {{\\sqrt {\\rho _{1}}}\\rho _{2}{\\sqrt {\\rho _{1}}}}})\\right]^{2}", "224974d4a0bc74dffaa3b99e64dd2968": "\\left\\langle \\mu _{z}\\right\\rangle ={1 \\over ZB}\\partial _{\\beta }Z.", "224988b42bde97f9049e354e4730d05e": "\\mathrm {Ass} _{R}(R/J)\\,", "224a052c1b83e5445af51c14a293aba6": "3\\times 4", "224a371dff3f0e6e0893041fb23ded0c": "D_{1}-D_{2}", "224a6a92aa2c5776682a8555a7ca3901": "\\alpha S\\subseteq S", "224a9787a32d8d94a16b1cce4f9f1c62": "f(x)={\\begin{cases}e^{-1/x^{2}}&\\mathrm {if} \\ x\\not =0\\\\0&\\mathrm {if} \\ x=0\\end{cases}}", "224ab7ca239f7f9f114abb12aa3f898f": "{\\frac {\\partial \\left({\\overline {u_{i}}}+u_{i}'\\right)}{\\partial x_{i}}}=0,", "224add98dbb4102205b7cc52696c238c": "\\mathbf {J} _{\\mathrm {P} }={\\frac {\\partial \\mathbf {P} }{\\partial t}}\\ ,", "224ae3da68337fd43644c9708b5a7acb": "M0", "2252270c2bcb6ac92270bdad39dcd3bd": "P(G|T)={\\frac {P(T|G)P(G)}{P(T)}}={\\frac {0.5\\times 0.4}{0.8}}=0.25.", "22526c8adc68ebc06217902d4953d382": "q=e^{i\\lambda }", "2252921b4e948fa1887c763a38683576": "(5)\\;h_{j}={\\frac {y_{1}{\\sqrt {1+8Fr_{1}^{2}}}-3y_{1}}{2}}", "2252b7e7efc31bb535d1a0d6a2410b52": "\\left({\\frac {2}{3}}\\cdot 3\\right)+\\left({\\frac {1}{3}}\\cdot 5\\right)={\\frac {11}{3}}", "225307b866e46c96a58bd7cf8180a612": "N_{R}\\equiv {\\frac {r}{\\sqrt {L\\cdot {l_{p}/3}}}}<1", "2253167e685877214053c59fdd4d7382": "P_{r}={\\frac {\\eta C_{p}}{M\\kappa }}.", "22533fd98da46c0a90133768edfd4492": "A=B=2000", "22538fec3670ec24af4efef1cc5bffae": "\\lim _{\\varepsilon \\to 0}\\int _{\\varepsilon }^{1}x^{-1}\\cos \\left(x^{-1}\\log x\\right)\\,dx", "2253a8014cfc390789d836412bcb5475": "\\mathrm {\\Lambda } (A\\otimes B)=\\mathrm {Co} (\\mathrm {\\Lambda } ^{\\!\\otimes }\\!\\left({A\\otimes B}\\right)).", "2253caead7a72c8416ce0f8161c14145": "\\scriptstyle {\\mathcal {N}}", "2253ce8932d0ba3c7946661746d7d699": "P_{f}", "22540e32376525bceef75d2743196324": "*,", "22542cbebe63b6383f14ae041a9a7866": "C{out}", "22544ac13fbd4046fc510862b025da5b": "q^{n}/(1-q^{n})=\\mathrm {Li} _{0}(q^{n})", "22545c621dbc9000b8d1f449365ace27": "L(\\theta )=\\prod _{T_{i}\\in unc.}\\Pr(T=T_{i}|\\theta )\\prod _{i\\in l.c.}\\Pr(TT_{i}|\\theta )\\prod _{i\\in i.c.}\\Pr(T_{i,l}0", "22608ba7a68432367466a1be451b81bb": "g_{1},\\,g_{2}\\ldots \\,g_{n}", "226154735e81aae3518edb0f0325adaf": "\\bigoplus _{i=1}^{k}A=\\mathbb {N} .", "2261952b6cfe72cac2524f034717f079": "\\pi \\,{\\hat {=}}\\,\\alpha +{\\bar {\\beta }}\\,,\\quad \\varepsilon \\,{\\hat {=}}\\,{\\bar {\\varepsilon }}\\,,\\quad {\\bar {\\mu }}\\,{\\hat {=}}\\,\\mu \\,.", "22619593c0fbc1e140dcb640797c3864": "I_{SN}", "2261c7733c26e6528897e54351fba28f": "\\displaystyle {L_{n}=-\\pi \\left(ie^{in\\theta }{d \\over d\\theta }\\right)}", "2261f7f97154a49e8f3cb88b3fe36a1f": "\\int _{A}f(x)\\,d\\mu \\,\\!", "22622153c9b67a263ee317eff42fdc66": "{\\mathcal {C}}^{\\mathbf {2} }", "2262516b6e53309bd21d5ef437e42add": "v=v_{N}", "22626a8fcb1e214bb21713aaddfce9d3": "B_{n-1}", "22629ab7e8c584608cf26d2d7d4f1bd1": "{\\widehat {\\Omega }}={\\widehat {\\Omega }}^{\\dagger }", "2262c3350a6b3901ba193a5df2002184": "\\mu (A)>0\\,", "2262d6762e030e102c2ae0d8069ea924": "n^{2}/4", "2262dad4d16e5fe0c9acb45b3ad0a340": "\\displaystyle {f_{i}(x)=e^{-x^{2}/2},}", "2263734592a53171cd4c3698ec6861d9": "F(t)=E_{t}\\left\\{S(T)\\right\\}", "226392170cb6694015de6e52b4439687": "{\\frac {dy}{dx}}=y(1-y).", "2263993744d371b7781cb26a879af2f5": "P(P(X_{1}^{n}(i'))>P(X_{1}^{n}(i)))\\,", "226410410a2cf571745c0ceee7e2db66": "f(x,n)\\geq k", "2264124205ac059db59fdef14bd6b4ad": "W_{21}", "22643d03eebf368665338b9135e0b354": "{\\frac {\\mu _{i}}{k_{0}}}p_{0}=p_{i}", "226485b2a4d52c54299bb4d81807acff": "E[X;P]=\\int _{a=-\\infty }^{+\\infty }a\\,P(X\\in [a;a+da])", "2264869f1f8f3e104112b5733112fce1": "\\tanh {\\frac {\\hbar \\Omega (n)\\beta }{2}}=4{\\frac {\\omega }{\\Omega }}{\\sqrt {n+1}}", "2264abf871c877cb8ff7fbc20badfbf1": "y\\in [0,1/2),x\\in [1/2,1]", "2264cea93c5894cfe2028f03999ff4a5": "\\alpha (L_{n})=L_{n}+{1 \\over 2}J_{n}+{c \\over 24}\\delta _{n,0}", "2264e8fc1e460956e73d941e58a03bf5": "f_{n}(z)", "2265081cfabbbdfb0fe32b2470788b3a": "{\\frac {c(a_{1}+n)\\dots (a_{p}+n)}{d(b_{1}+n)\\dots (b_{q}+n)(1+n)}}", "22653600bd600eeaea111eec9e3dbea9": "B+B'", "2265ea919196a64603cce82ab29cb0dc": "f={\\frac {P(z)}{Q(z)}}", "2265f94210d0bb3ed52a92bf469753a9": "e^{T}e=(My)^{T}(My)=y^{T}M^{T}My=y^{T}MMy=y^{T}My.\\,", "2266859dfa2a82c894311a81fa37c0a5": "\\sum _{n=0}^{\\infty }{\\tbinom {n+k}{k}}x^{n}={1 \\over (1-x)^{k+1}}.", "22669228e8b758bacf747d8edccb2cb9": "={\\frac {y''(s)}{x'(s)}}=-{\\frac {x''(s)}{y'(s)}}\\ ,", "2266b19beb55aedf4eeabc06ecff01d6": "\\textstyle \\nu (\\varepsilon )", "2266d496aee9eeafa9fac38168432991": "g_{0}={\\frac {-K_{P}K_{V}K_{C}K_{M}}{s^{3}L_{M}M}}\\,", "2266f958d5b6d1a5e70c49bc6cbfd749": "\\Leftrightarrow \\operatorname {gl.dim} R<\\infty \\Leftrightarrow \\operatorname {gl.dim} R=\\dim R.", "2267285faeaaab31fdef7c2076a7a2fa": "C_{2}=138\\ \\mathrm {pF} \\,", "22673c3a677cc01c9657579ad724d73c": "\\sum _{j\\in \\mathbf {S} }p_{ij}V(j)\\leq V(i)-1", "22674641d6c678893d803618d7c44f7f": "Q[{\\mathcal {L}}(x)]=\\partial _{\\mu }f^{\\mu }(x)", "2267d584dc5a275b5e2759d0e2ee6ec7": "z\\leq {\\frac {\\mu }{\\mu +x}}", "22681addb9f61e033b57a4e9a241f269": "((\\land _{\\epsilon <\\delta }{(A_{\\delta }\\implies A_{\\epsilon })})\\implies (A_{\\delta }\\implies \\land _{\\epsilon <\\delta }{A_{\\epsilon }}))", "226823a43503af76deed62f3dae9aef3": "{\\overline {x}}={\\frac {x_{1}+\\cdots +x_{n}}{n}}", "22683886b351a6d6e86df531e6f92511": "p\\gamma =\\ell \\pi \\,", "226849c3c1225387d930ec1d7f68843c": "x=a{\\sqrt {\\left(\\sigma ^{2}-1\\right)\\left(1-\\tau ^{2}\\right)}}\\cos \\phi ", "226882a5358b2995c80cc5c52caf5cb9": "\\displaystyle {A=(T+I)(T-I)^{-1}.}", "2268a4ebb90af21d37c61d23669b55bb": "\\sum _{ij;i\\neq j}\\psi _{i}^{*}\\psi _{j}\\phi _{j}^{*}\\phi _{i}", "2268b534827bf910c7adf0adc132dfcd": "=\\Pr[B]\\Pr[A|B]+\\Pr[B^{c}]\\Pr[A|B^{c}]", "226901b0a97afaa088dc0d1b03c8544b": "T_{(0,0)}M_{1}=\\mathbb {R} \\times \\{0\\}", "22694bd165f4573e4de384414870ba22": "D=\\sum _{i=1}^{r}P_{i}-r\\infty _{2}", "22697bea86be9f5fa4bd1c7a01057a3b": "\\lim _{y_{0}\\to 0}\\arccos {\\sqrt {\\frac {2gy_{0}+v^{2}}{2gy_{0}+2v^{2}}}}={\\frac {\\pi }{4}}", "22698d55e75769bbf77c6a4740784ace": "F_{l}(x)={\\frac {1}{\\mu }}\\int _{0}^{x}\\left(1-F(u)\\right){\\text{d}}u.", "2269b7011477c8198ccce0be6831621b": "{\\begin{bmatrix}a_{11}&a_{12}\\\\a_{21}&a_{22}\\end{bmatrix}}", "2269ce8576a07bf1200c38d76b9583b3": "i_{1}\\leq i_{2}\\leq ...\\leq i_{n}", "2269df51c696d2c936a8f39d437a8b6e": "{\\begin{matrix}abc\\\\a/bc\\\\b/ac\\\\c/ab\\\\a/b/c\\end{matrix}}", "226a3611c04c0195c3eeefeea9874979": "\\lim _{x\\to -\\infty }a^{x}=0", "226a3a8ab5a08cf7e07a32f7934e295a": "y\\gg x", "226a6fa4e14c89a07f20590279625b22": "{\\hat {p}}\\pm z^{*}{\\sqrt {\\frac {{\\hat {p}}\\left(1-{\\hat {p}}\\right)}{n}}}", "226a80326ba7b969338188a6eec43084": "x=\\pm 1.\\,", "226aae9407d7e4e227264578fee61f04": "1_{X}", "226b1835b5bc6785ca974568a3ebcaf1": "-\\left({\\frac {dt}{dx}}\\right)^{-1}{\\frac {d^{2}t}{dx^{2}}}=f(x)", "226b330635b5795c5a4c35d7adafb340": "H^{*}(M,\\mathbb {K} )", "226b4102c3a771a9eea412a5b0b2a4b7": "S:X\\rightarrow D", "226b4a6dd11c3e3a357ba20363ff2eca": "g_{n,k}(r){\\text{ and }}f_{n,k}(r)", "226b7338fedfbeffa9c3f1704ad8ca1e": "{\\text{Level 1:}}\\ \\ 266=2+2+\\dots +2\\ \\ {\\text{(with 133 2s)}}", "226ba5c7020271363249ed79de8b28aa": "{1 \\over 2(1-p^{-2})(1-p^{-4})\\cdots (1-p^{1-n})}", "226bf0c0a9ff13ad1ba07d51f7146ad6": "\\int _{-\\infty }^{\\infty }pdp,", "226bf15cf2f1805a0e75266274ca7bb4": "\\mathbf {a} \\otimes \\mathbf {b} =a_{i}\\mathbf {e} _{i}\\otimes b_{j}\\mathbf {e} _{j}=a_{i}b_{j}\\mathbf {e} _{i}\\otimes \\mathbf {e} _{j}", "226bf73809883928549b784b38b05ab1": "det\\;q^{(2)}={\\epsilon ^{3ab}\\epsilon ^{3cd}q_{ac}q_{bc} \\over 2}", "226c04efecfa7d5b71fd9f7e577074ed": "\\sum v(t)\\times \\Delta t", "226c282e3406665df3099cc7846c4e4d": "{\\tilde {A}}_{2n}", "226c5c2d4f3900230fe868998f914776": "\\langle ab\\rangle _{2}=a\\wedge b\\,", "226c82c7c67baf94f37b8665e6e44855": "i=\\{1\\dots k\\}", "226ca979c6ec4b377cf65fbbc9cfcb98": "s=\\sin \\theta ", "226ced0d1f1d6a992ab8d73a50553199": "{}^{q}\\!D={1 \\over {\\sqrt[{q-1}]{\\sum _{i=1}^{R}p_{i}p_{i}^{q-1}}}}", "226d14875dbf4bd2bc5ad4e186a9ea8f": "B=\\mu H+\\chi {\\sqrt {\\varepsilon \\mu }}E", "226d6149db2d22ff1f3fe7f8de4335d5": "\\rho :\\pi _{1}B\\rightarrow Homeo\\left(F\\right)", "226d81cd7e156aa12d7cf723ec95b732": "{\\begin{aligned}\\mathbf {F} &=\\left(\\mathbf {p} \\cdot \\nabla \\right)\\mathbf {E} +{\\frac {d\\mathbf {p} }{dt}}\\times \\mathbf {B} \\\\&=\\alpha \\left[\\left(\\mathbf {E} \\cdot \\nabla \\right)\\mathbf {E} +{\\frac {d\\mathbf {E} }{dt}}\\times \\mathbf {B} \\right],\\\\\\end{aligned}}", "226dad79459a4965dc74922b2c8806f9": "\\psi _{1}(\\alpha )={\\frac {d^{2}\\ln \\Gamma (\\alpha )}{\\partial \\alpha ^{2}}}=\\,{\\frac {\\partial \\psi (\\alpha )}{\\partial \\alpha }}", "226dc2c78e429964a36315c25003b36b": "\\Delta p=0", "226ddae13c247a83e462106a4b91a3f2": "{\\text{IE}}_{\\text{M}}", "226e4e424a35e70a3b73010c8332f8b2": "(-\\infty ,+\\infty )", "226e4e7ddbaeaaa707ecec449b3643df": "K={\\sqrt {(s-a)(s-b)(s-c)(s-d)-abcd\\cdot \\cos ^{2}\\left({\\frac {\\alpha +\\gamma }{2}}\\right)}},", "226e72f3c8524592b874e5608bfbd47e": "f(1)=e", "226e7afd0e1a288e6dc4629fd3ca157a": "\\left\\vert C\\cup D\\right\\vert +\\left\\vert C\\cap D\\right\\vert =\\left\\vert C\\right\\vert +\\left\\vert D\\right\\vert \\,.", "226e9249ecc997c2861fa3fcd71276fe": "\\,{\\overline {A}}_{x}\\!=E[v^{T}]=\\int _{0}^{\\infty }v^{t}f_{T}(t)\\,dt=\\int _{0}^{\\infty }v^{t}\\,_{t}p_{x}\\mu _{x+t}\\,dt,", "226eade4534836ddc41cfa424762e203": "{\\overline {C}}_{1},\\dots ,{\\overline {C}}_{m}", "226f41bebdc17c042b30b9f30a446172": "s\\mid _{p}", "226f88dfa19b49c359e6a8b05a4b95b5": "{\\mathcal {I}}(\\alpha )=\\operatorname {E} \\left[\\left({\\frac {\\partial }{\\partial \\alpha }}\\ln {\\mathcal {L}}(\\alpha |X)\\right)^{2}\\right],", "226f8c7d3fd00a687c4898cea554f1f0": "|p_{z},i+1_{x,y};\\downarrow \\rangle ", "226facaa77bdc83dd7a591b81eab77c7": "P\\Lambda ={\\frac {\\sum _{g=1}^{G}{\\left({\\frac {A_{g}}{\\overline {A}}}-1\\right)^{2}}}{G}}", "226fde4c9564579c4397f1b5b9d101f1": "l=gm,\\quad k=nf,\\quad h=j[1]i,\\quad ig=u[1]n,\\quad fj=mv.", "226ff57c061c7c2d84c5c79cc3b5800c": "Var(\\epsilon )=\\sigma ^{2}", "227063a0613b40bee6f0fdc8000e546f": "\\mathbf {a} \\odot \\mathbf {b} ", "2270a2c8191552ee95b50f15ce1c8521": "{\\begin{aligned}u_{r}=u_{1}&={\\cfrac {F_{1}}{4\\pi \\mu }}\\left[1-(\\kappa +1)\\ln |x_{1}|\\right]\\\\u_{\\theta }=u_{2}&={\\cfrac {F_{2}}{4\\pi \\mu }}\\left[1+(\\kappa +1)\\ln |x_{1}|\\right]\\end{aligned}}", "2270a978930a622d7f7f71ab6d8328e2": "{\\text{Im}}(b_{\\lambda })\\cong \\bigwedge ^{\\mu _{1}}V\\otimes \\bigwedge ^{\\mu _{2}}V\\otimes \\cdots \\otimes \\bigwedge ^{\\mu _{k}}V", "2270abb8d03f3f3dfd8d156106ffca77": "\\lambda ^{*}(A)=\\inf {\\Bigl \\{}\\sum _{B\\in {\\mathcal {C}}}\\operatorname {vol} (B):{\\mathcal {C}}{\\text{ is a countable collection of boxes whose union covers }}A{\\Bigr \\}}.", "2271023011b790d9e76063ed69f3d310": "\\sum _{i=1}^{K}N_{i}=N", "2271087912c89cf4a1598cf92424a0c4": "x_{marker}={x_{marker\\_orig} \\over x_{orig}}\\cdot x_{scaled}-{marker\\_size \\over 2}+x_{adjust}", "227121311da87d0e3c70dbe4de52c297": "s={\\sqrt[{3}]{1880}}\\approx 12+1/3", "2271277f219f0c9d043d20c52b347204": "\\mathbf {P} \\,=\\,(\\cos(s),\\,\\sin(s))\\,\\Rightarrow \\,\\mathbf {F} \\,=\\,2\\sin(s)\\mathbf {i} +5\\cos(s)\\mathbf {j} \\,.", "22717e1e758fd2673ed8df3c42fb1898": "(\\mathbb {C} P^{n},\\Sigma ,\\{U_{\\sigma _{1}},U_{\\sigma _{2}},\\dotsc ,U_{\\sigma _{p}},\\})", "2271b96386d7bae7ff90ed544fbded4d": "nym", "2271e6168d7774017270b743b36505a6": "\\left({\\frac {2D^{\\mathrm {face} }}{S^{\\mathrm {core} }}}\\right){\\cfrac {\\mathrm {d} ^{4}w}{\\mathrm {d} x^{4}}}-\\left(1+{\\frac {2D^{\\mathrm {face} }}{D^{\\mathrm {beam} }}}\\right){\\cfrac {\\mathrm {d} ^{2}w}{\\mathrm {d} x^{2}}}={\\frac {M}{D^{\\mathrm {beam} }}}-{\\cfrac {q}{S^{\\mathrm {core} }}}", "227257860c77fa00f2f99cbd3394882c": "A={1 \\over 2}A_{0}e[e^{i(k_{p}\\cdot r-\\omega t)}+e^{-i(k_{p}\\cdot r-\\omega t)}]", "2272f2e6b87d10518445ebdd9145108c": "D=\\varepsilon E-i\\kappa {\\sqrt {\\varepsilon \\mu }}H", "227318fc4085a330ead6c7306364680e": "\\rho =2\\pi a{\\frac {V}{I}}", "2273248e0465f9d3b0ae75cc6309c3a2": "(Sx)_{n}=x_{n+1}", "227369ab78112212e471d8d14bcd256e": "\\Pr\\{(M_{n}-b_{n})/a_{n}\\leq z\\}\\rightarrow G(z)\\propto \\exp \\left[-(1+\\zeta z)^{-1/\\zeta }\\right]", "22737ab7c5a33e22fae42a02622594b7": "\\sum _{p=0}^{q-1}\\zeta (s,a+p/q)=q^{s}\\,\\zeta (s,qa)\\ .", "22739d947baf1fe8db6cd0af5b975188": "\\Gamma =+1", "2273a299c2452e3a37bb1bf05f9b544c": "\\mathrm {error} {\\bigl (}x(t_{0}+n\\Delta t){\\bigr )}={\\frac {n(n+1)}{2}}\\,O(\\Delta t^{4})", "2273cb4d87e3bb44fbd63a81fe86bd13": "O(b^{3d/4})", "227430bed634bfed57d72512c96579df": "X\\times X_{\\mathcal {G}}^{*}\\to {\\mathcal {F}}", "22743bf12516741749f26f3c5f4de552": "B=\\mu _{0}\\mu _{r}((1+\\chi _{0})(H_{ext}+H_{exc})+N_{e}(H_{ext}+H_{exc})^{3})", "22745f534bd8eefc03d911ac4d0eaef2": "M_{ij}={\\frac {\\mu _{0}}{4\\pi }}\\oint _{C_{i}}\\oint _{C_{j}}{\\frac {\\mathbf {ds} _{i}\\cdot \\mathbf {ds} _{j}}{|\\mathbf {R} _{ij}|}}", "227467a6e2e9997bd58cac601460f305": "-mg\\ell \\theta =I\\alpha ", "227518080d8aa94cba952ad8d8993a99": "D_{n}=\\langle x,y\\mid x^{n}=y^{2}=(xy)^{2}=1\\rangle .", "2275313af06aac1aa13e170975b089f9": "\\beta =-2e^{i\\theta }-e^{-2i\\theta }", "22754160dbb35f8af7a58dfa77019184": "r\\times n/w", "22759181021b36d7f0d67d9e51c255d3": "C\\subseteq N", "2276514e5ab82e283c11603081bbb011": "H_{q}", "2276c1144ab01cd3f25f4431378801da": "k_{o}^{2}=\\left({\\frac {2\\pi }{\\lambda _{o}}}\\right)^{2}=k_{xo}^{2}+k_{y}^{2}+k_{z}^{2}=-\\left|k_{xo}\\right|^{2}+k_{y}^{2}+\\beta ^{2}\\ \\ \\ \\ (1)", "227728bdb14346248f49f78a229f9a92": "FA={\\frac {\\sqrt {3((\\lambda _{1}-\\mathbb {E} [\\lambda ])^{2}+(\\lambda _{2}-\\mathbb {E} [\\lambda ])^{2}+(\\lambda _{3}-\\mathbb {E} [\\lambda ])^{2})}}{\\sqrt {2(\\lambda _{1}^{2}+\\lambda _{2}^{2}+\\lambda _{3}^{2})}}}", "2277301ed8f09dec845731b78c8de40e": "p_{\\mathrm {av} }={\\frac {F}{A}}\\approx {\\frac {1}{2}}E^{*}h'", "227734aacbe93673bc7690d5db079ffb": "{\\begin{aligned}{\\frac {\\Delta ^{2}F(P_{0})}{\\Delta _{1}P^{2}}}&={\\frac {\\Delta F'(P_{0})}{\\Delta _{1}P}}={\\frac {{\\frac {\\Delta F(P_{1})}{\\Delta _{1}P}}-{\\frac {\\Delta F(P_{0})}{\\Delta _{1}P}}}{\\Delta _{1}P}},\\\\[10pt]&={\\frac {{\\frac {F(P_{2})-F(P_{1})}{\\Delta _{1}P}}-{\\frac {F(P_{1})-F(P_{0})}{\\Delta _{1}P}}}{\\Delta _{1}P}},\\\\[10pt]&={\\frac {F(P_{2})-2F(P_{1})+F(P_{0})}{\\Delta _{1}P^{2}}};\\end{aligned}}", "22776e4755070319688ad67d2ba31792": "\\displaystyle f(x)={\\begin{cases}0.5x&{\\text{for }}x\\in Z,\\\\0.5+0.5x&{\\text{for }}x\\in (0,1)\\setminus Z\\end{cases}}", "2277d2411368b9b4f49069698fb3d2ca": "|ix|_{B}=B(ix,ix)=i^{2}B(x,x)=-|x|_{B}", "2277f1a42853d774c23fdbfe2b9c77c0": "\\partial ^{2}w/\\partial x^{2}=0", "227859f1a2a285796c8205901726c7b6": "y_{i}'(t)=f_{i}(t,y_{1}(t),y_{2}(t),...)", "227943546052d74a24059ca3d69a49f2": "\\left[\\nabla ^{2}-k_{0}^{2}\\right]\\phi (r)=-{\\frac {Q}{\\epsilon _{0}}}\\delta (r)", "22794cb2b733cc8ebb9d611c61d681c1": "\\nabla \\times \\left(f(r)\\mathbf {Y} _{lm}\\right)=-{\\frac {1}{r}}f\\mathbf {\\Phi } _{lm}", "2279689f6a085fbee4812ab8d1d59f02": "x_{21}=p_{2}q_{1}-D", "2279a6aa7461805a50f4a48e95e98c50": "B_{\\text{op}}:=\\{|\\omega _{k}\\rangle \\}_{k=0}^{N-1}", "2279b1c42b4353ba60defb0f1a3099e7": "[\\cdot ]:A\\longrightarrow B", "2279ccee0f36d7513f6d5377fa63fc41": "a=0~mod~p_{1}", "2279f3c6247904961abd5a25d1b935af": "\\lfloor x\\rfloor =\\max \\,\\{m\\in \\mathbb {Z} \\mid m\\leq x\\},", "227a357c6842c0dcff5a79898f1919d7": "\\mu _{2,1}={\\frac {\\langle b_{2},b_{1}^{*}\\rangle }{B_{1}}}={\\frac {{\\begin{bmatrix}4\\\\5\\\\4\\end{bmatrix}}{\\begin{bmatrix}1\\\\1\\\\1\\end{bmatrix}}}{3}}={\\frac {13}{3}}(>{\\frac {1}{2}})", "227a37229b074c000759054e905d3373": "u_{n{\\boldsymbol {k}}}({\\boldsymbol {r}})", "227a3b89d3ccabbeea3231109e2120df": "P(X<(1-\\delta )\\mu )\\leq e^{\\frac {-\\delta ^{2}\\mu }{2+\\delta }}.", "227a596608db91cda8b7badb9d00e318": "\\lim \\inf _{n}{\\frac {f(x_{n})-f(x)}{x_{n}-x}}>\\lim \\sup _{n}{\\frac {f(x'_{n})-f(x)}{x'_{n}-x}}.", "227a8acaef192b432d8a9c025dba9806": "{\\mathcal {}}BP_{*}/I_{n}", "227af28dda4008a745cdf602ff14759f": "i=p_{1}", "227b22b0b6222ff30e6a91a78770436a": "{\\begin{array}{lcl}s_{n}&=&s_{n-1}^{2}-2\\\\&=&\\left(\\omega ^{2^{n-1}}+{\\bar {\\omega }}^{2^{n-1}}\\right)^{2}-2\\\\&=&\\omega ^{2^{n}}+{\\bar {\\omega }}^{2^{n}}+2(\\omega {\\bar {\\omega }})^{2^{n-1}}-2\\\\&=&\\omega ^{2^{n}}+{\\bar {\\omega }}^{2^{n}},\\\\\\end{array}}", "227ba3fd670437fe111640f3f14ad637": "\\Omega \\subseteq ^{d}", "227c25f6f71eb0791cd1709f7d0ea089": "\\operatorname {dist} (t)=\\infty ", "227cbbb1283b7e00e04c126a76868c41": "\\psi ={\\begin{pmatrix}\\psi _{+}\\\\\\psi _{-}\\end{pmatrix}}={\\begin{pmatrix}\\psi _{+\\uparrow }\\\\\\psi _{+\\downarrow }\\\\\\psi _{-\\uparrow }\\\\\\psi _{-\\downarrow }\\end{pmatrix}}", "227cc397230b4109597580cb85b55db8": "f(x)\\leq _{T}f(y)", "227cca83ba68bff68e953b5e70fb8106": "S(Y)", "227ce333769088e212af5de5bbf00220": "{\\frac {\\partial \\mathbf {u} }{\\partial \\mathbf {x} }}{\\frac {\\partial \\mathbf {g(u)} }{\\partial \\mathbf {u} }}{\\frac {\\partial \\mathbf {f(g)} }{\\partial \\mathbf {g} }}", "227d83c49d2731dfd04a990c284358a5": "G(\\mathbf {M} )", "227dc4637de62614a5fe6ced410526af": "X'={\\frac {\\partial X}{\\partial \\sigma }},", "227df9cde82b3f98fac8ed3d583aa95a": "(\\mathbb {Z} _{N})^{\\times }", "227e005edb52415812f6f527a829ae19": "u_{2}(x)", "227e2321f71de511a85dd64c13a90dec": "{\\begin{aligned}p(\\mu ,\\sigma ^{2}|\\mathbf {X} )&\\propto p(\\mu ,\\sigma ^{2})\\,p(\\mathbf {X} |\\mu ,\\sigma ^{2})\\\\&\\propto (\\sigma ^{2})^{-(\\nu _{0}+3)/2}\\exp \\left[-{\\frac {1}{2\\sigma ^{2}}}\\left(\\nu _{0}\\sigma _{0}^{2}+n_{0}(\\mu -\\mu _{0})^{2}\\right)\\right]{\\sigma ^{2}}^{-n/2}\\exp \\left[-{\\frac {1}{2\\sigma ^{2}}}\\left(S+n({\\bar {x}}-\\mu )^{2}\\right)\\right]\\\\&=(\\sigma ^{2})^{-(\\nu _{0}+n+3)/2}\\exp \\left[-{\\frac {1}{2\\sigma ^{2}}}\\left(\\nu _{0}\\sigma _{0}^{2}+S+n_{0}(\\mu -\\mu _{0})^{2}+n({\\bar {x}}-\\mu )^{2}\\right)\\right]\\\\&=(\\sigma ^{2})^{-(\\nu _{0}+n+3)/2}\\exp \\left[-{\\frac {1}{2\\sigma ^{2}}}\\left(\\nu _{0}\\sigma _{0}^{2}+S+{\\frac {n_{0}n}{n_{0}+n}}(\\mu _{0}-{\\bar {x}})^{2}+(n_{0}+n)\\left(\\mu -{\\frac {n_{0}\\mu _{0}+n{\\bar {x}}}{n_{0}+n}}\\right)^{2}\\right)\\right]\\\\&\\propto (\\sigma ^{2})^{-1/2}\\exp \\left[-{\\frac {n_{0}+n}{2\\sigma ^{2}}}\\left(\\mu -{\\frac {n_{0}\\mu _{0}+n{\\bar {x}}}{n_{0}+n}}\\right)^{2}\\right]\\\\&\\quad \\times (\\sigma ^{2})^{-(\\nu _{0}/2+n/2+1)}\\exp \\left[-{\\frac {1}{2\\sigma ^{2}}}\\left(\\nu _{0}\\sigma _{0}^{2}+S+{\\frac {n_{0}n}{n_{0}+n}}(\\mu _{0}-{\\bar {x}})^{2}\\right)\\right]\\\\&={\\mathcal {N}}_{\\mu |\\sigma ^{2}}\\left({\\frac {n_{0}\\mu _{0}+n{\\bar {x}}}{n_{0}+n}},{\\frac {\\sigma ^{2}}{n_{0}+n}}\\right)\\cdot {\\rm {IG}}_{\\sigma ^{2}}\\left({\\frac {1}{2}}(\\nu _{0}+n),{\\frac {1}{2}}\\left(\\nu _{0}\\sigma _{0}^{2}+S+{\\frac {n_{0}n}{n_{0}+n}}(\\mu _{0}-{\\bar {x}})^{2}\\right)\\right).\\end{aligned}}", "227e31dc9b5fba5b17e457149c943543": "\\sum _{k=1}^{n}k={\\frac {n(n+1)}{2}}", "227e569fcfd998e36f71dc6add69e4f4": "\\Pr[A]\\cdot \\Pr \\left[{\\overline {A_{t}}}\\right]\\leq e^{-t^{2}/4}\\,,", "227e5999bd72ae167f9df0a54e2dae93": "{\\boldsymbol {\\chi }}+\\mathbf {T} (x){\\chi }", "227e71e02805ad1a3af1a39dcfb114de": "S_{\\mathrm {sat} }={\\frac {78}{H_{\\mathrm {sat} }}},", "227e9e6ea96659f752771b4ec095b788": "{\\frac {\\sqrt {3}}{3}}", "227ec2f6235ca10a60b3dbdcf2d6a2d6": "d\\nu (x)={\\frac {1}{2\\pi \\sigma ^{2}}}{\\frac {\\sqrt {(\\lambda _{+}-x)(x-\\lambda _{-})}}{\\lambda x}}\\,\\mathbf {1} _{[\\lambda _{-},\\lambda _{+}]}\\,dx", "227eefcde3831848c9f061c203a54a8e": "(m-n)\\sigma ^{2}", "227f0a2b17854530d758163f006c4c17": "\\displaystyle {\\nabla f=(\\partial _{x}f,\\partial _{y}f).}", "227f51a03add5697607dd2e43d252d10": "\\alpha =\\beta =1/2", "227f9708f96cc5a8b10b059eae7fae23": "\\int _{\\underline {\\theta }}^{\\overline {\\theta }}\\left({\\frac {\\partial V}{\\partial x}}(x,\\theta )-{\\frac {1-P(\\theta )}{p(\\theta )}}{\\frac {\\partial ^{2}V}{\\partial \\theta \\partial x}}(x,\\theta )-{\\frac {\\partial c}{\\partial x}}(x)\\right)p(\\theta )d\\theta =0", "227fc08ff898b88a9d6372b0b4119f14": "PAI={\\frac {Y_{2}-Y_{1}}{T_{2}-T_{1}}}", "227fe3b6d887ce0652cf6b31c79be01f": "\\mathbf {h} =\\mathbf {r} \\times \\mathbf {\\dot {r}} ", "22802944eafccd9c716ba44dc2b72404": "l,j,m_{\\text{l}},m_{s}", "2280564bd195de2b6fa87c324114f7b3": "H=-kA{\\frac {\\mathrm {d} T}{\\mathrm {d} x}}.", "2280a32694681e20f981c0ac7da9e8e8": "\\mathbf {1} ", "22810e81f0ece5fca221b4646654ca62": "B=\\{b_{ij}\\}", "228110fca3070ee8a51a4cfe19af18d2": "{\\mathcal {A}}\\times {\\mathcal {B}}", "22814532056645082db9285e61829350": "\\gamma (1,p,v)=\\exp _{p}(v)\\ ", "22814b0c0ef80d4859ad64adf4df1cee": "\\chi =(1/3)\\chi _{||}+(2/3)\\chi _{\\perp }", "22817cdefe557180f3bfc1d7f26e2a5d": "x\\mapsto Ax+b", "2281ab85abb157e602df0522a63947c5": "{\\Delta T}", "2281f9451273e0946363b14f9d515097": "0\\longrightarrow L_{\\bullet }{\\stackrel {f}{\\longrightarrow }}M_{\\bullet }{\\stackrel {g}{\\longrightarrow }}N_{\\bullet }\\longrightarrow 0,", "22822211393ae7ba4f3ba47a4aaf16d7": "P\\cdot p\\approx 0,", "2282547b896d6479da1e7395d0228350": "{\\begin{smallmatrix}\\rho _{\\odot }\\end{smallmatrix}}", "2282d786e48974120f13d5e110c9af75": "{\\begin{aligned}P(\\{1,2\\}\\mid 2)&={\\frac {P(\\{1,2\\})/2}{P(\\{1,2\\})/2+P(\\{2,4\\})/2}}\\\\&={\\frac {P(\\{1,2\\})}{P(\\{1,2\\})+P(\\{2,4\\})}}\\\\&={\\frac {1/3}{1/3+2/9}}=3/5,\\end{aligned}}", "2282fe81a8a5b19e5ed8e55d872e9e37": "\\mathrm {RED} \\subseteq P", "228309397e41304acbe86da5cf74c117": "R=\\emptyset ", "22831c74c8457fc7914fdd0ed1ae3a7d": "\\displaystyle F_{2}(q)=\\sum _{n\\geq 0}{q^{n(n+1)} \\over (q^{n+1};q)_{n+1}}", "228384324495e0007a20814a0ac67d58": "R\\not \\equiv 0{\\pmod {2\\pi }}", "2284588ce1230ed395de263a7a11e3bd": "x=t", "22847c5fc0529b8aa1499e73649153ed": "+\\omega _{p}", "228496ce41925056aed635886c4801d4": "U|a\\rangle ", "2284b1ec3ab27148142b9eb254e7c65d": "\\textstyle \\ {\\mbox{M}}^{-}+HX{\\overset {k_{term}}{\\longrightarrow }}{\\mbox{M-H}}+{\\mbox{X}}^{-}", "2284dfb06618756726306db417c3b1ba": "H^{i}(X,{\\mathcal {L}}^{-1})=0", "2284fb75e2414865ccbe78ad02e7b38a": "L(k,\\theta )=\\prod _{i=1}^{N}f(x_{i};k,\\theta )", "228506be5ba2d72337b0cb82e61c4878": "{\\begin{bmatrix}\\langle {\\hat {d}}|{\\hat {a}}\\rangle &\\langle {\\hat {e}}|{\\hat {a}}\\rangle &\\langle {\\hat {f}}|{\\hat {a}}\\rangle \\\\\\langle {\\hat {d}}|{\\hat {b}}\\rangle &\\langle {\\hat {e}}|{\\hat {b}}\\rangle &\\langle {\\hat {f}}|{\\hat {b}}\\rangle \\\\\\langle {\\hat {d}}|{\\hat {c}}\\rangle &\\langle {\\hat {e}}|{\\hat {c}}\\rangle &\\langle {\\hat {f}}|{\\hat {c}}\\rangle \\end{bmatrix}}", "228522fb5ddacb55fbfee30aa764859e": "\\delta _{i}\\equiv \\mu -x_{i}", "22852db1843bb875b101dba86b3014be": "{\\tfrac {n}{\\sigma ^{2}}}{\\hat {\\sigma }}^{2}|X=(\\varepsilon /\\sigma )'M(\\varepsilon /\\sigma )\\ \\sim \\ \\chi _{n-p}^{2}", "2285337eabb02aaf78fdfb0f0604125f": "Q\\propto I^{2}\\cdot R", "2285a65da4c02fbc98f8fff91cc9171d": "{\\frac {\\partial (x,y)}{\\partial (r,\\theta )}}={\\begin{pmatrix}\\cos \\theta &-r\\,\\sin \\theta \\\\\\sin \\theta &r\\,\\cos \\theta \\end{pmatrix}}", "2285ac68ad27595ee79cb73e480038e2": "Y_{3}=T_{1}Y_{2}Y_{1}T_{2}-Z_{1}X_{2}X_{1}Z_{2}=4{\\sqrt {15}}", "2285baaa031d4539b75da963410965e0": "B={V \\over 3000}", "2285d625ec7ca05677d53bd0b7a5f0d0": "\\displaystyle \\Delta _{G}=H^{-1}\\circ \\Delta _{A}\\circ H-\\|\\rho \\|^{2},", "2286627ca88b2665bc202d5f909cd00f": "u_{1}^{2}+u_{2}^{2}=(b_{1}^{2}+b_{2}^{2})^{2}(b_{3}^{2}+b_{4}^{2})", "2286693e822d82d351a9d673f3507d0e": "A=\\sum {\\frac {1}{a_{i}}}", "228690d6e971e8a5254bbff3e0eed6d1": "\\wp (z;\\Lambda )=-\\zeta '(z;\\Lambda ),{\\mbox{ for any }}z\\in \\mathbb {C} ", "22875afd90ad1c2b92df9ff60e8de80b": "\\textstyle g'(\\alpha )-{\\frac {k}{\\alpha }}g(\\alpha )=0", "22879e773b76f063d272c50e129955e5": "0.75>\\lambda >0.5", "2287fc14c08365ade485c15ecd6fe355": "syn(x_{v})=s'-s", "22882ac2d2f6949a9bdcdae163004c07": "{\\frac {\\operatorname {d} }{\\operatorname {d} \\!\\theta }}\\,\\tan \\theta =1\\times {\\frac {1+\\tan ^{2}\\theta }{1-0}}=1+\\tan ^{2}\\theta .", "228847ace1dcd7bc400a6cbd6fd8efbd": "\\psi (\\beta )<\\delta ", "22884d574b92407e456ac7f99fdefa0a": "x_{t+2}=Ax_{t+1}+Bx_{t}", "228881f61fb647f466f0136bc32ac2d0": "x_{n+1}=x_{n}^{2}{\\bmod {M}}", "22890114671a8e131fb24cbdf9dd8c4e": "i[{\\hat {H}}'_{0},{\\hat {v}}_{i}']=i[\\beta p_{0},\\beta v_{i}]=0", "2289ba90f290ac31612326dc5a85d6d7": "a=g{m_{1}-m_{2} \\over m_{1}+m_{2}}", "2289c700810a20d629484f7348490277": "|y\\rangle \\ {\\stackrel {\\mathrm {def} }{=}}\\ {\\begin{pmatrix}0\\\\1\\end{pmatrix}}", "2289cdb87c8d9405b62f9e5b05f78893": "\\det {\\begin{bmatrix}|\\mathbf {v} |^{2}&v_{x}&v_{y}&1\\\\|\\mathbf {A} |^{2}&A_{x}&A_{y}&1\\\\|\\mathbf {B} |^{2}&B_{x}&B_{y}&1\\\\|\\mathbf {C} |^{2}&C_{x}&C_{y}&1\\end{bmatrix}}=0.", "2289f25989e23ff385128410a3ea839d": "{\\begin{aligned}d\\log(Y)&={\\frac {1}{Y}}\\,dY-{\\frac {1}{2Y^{2}}}\\,d[Y]\\\\&=dX-{\\frac {1}{2}}\\,d[X].\\end{aligned}}", "228a0e2d73ee6203c784a64e07329d11": "\\scriptstyle {\\mathbf {n} =(n_{1},n_{2},\\ldots ,n_{m})}", "228a4e11b53e500007ff622b8f453f29": "L=rp\\sin \\theta =rmv\\sin \\theta \\,,", "228ad9eaef032864e1c7edfea2a83ad1": "2^{-D^{\\epsilon }(\\rho ||\\sigma )}\\geq {\\frac {\\epsilon -\\delta }{\\epsilon }}~.", "228b208e2ac025be451a6553086f76ce": "(2,n)", "228b66e923d2392c41bc9cfdccbd710f": "10\\uparrow \\uparrow (n+1)", "228bb5b0c2039c8bd09b435b92a94674": "{\\boldsymbol {a}}={\\frac {d}{dt}}{\\boldsymbol {v}}=\\sum _{k=1}^{d}{\\dot {v}}_{k}\\ {\\boldsymbol {e_{k}}}+\\sum _{k=1}^{d}v_{k}\\ {\\dot {\\boldsymbol {e_{k}}}}\\ .", "228bcf5a2258bb4fed05b6fbc5f44ea7": "{\\underline {P}}(Cl_{1}^{\\leq })=\\{x_{1},x_{5}\\}", "228c16e5f1d9dd19534ab8b18fa869e9": "{\\mathfrak {sl_{2}}}", "228c1d23540424a876a8aa7ecd264c71": "\\mathbf {A\\cdot B} =-A^{0}B^{0}+A^{1}B^{1}+A^{2}B^{2}+A^{3}B^{3}", "228c4d4b19a627aec27b93846e1f8627": "\\mathbf {a} ={\\begin{pmatrix}10&8&9&6&3&5\\\\\\end{pmatrix}}", "228c4e84b7cb4d5c726aae1067ff8eb9": "\\forall j\\in [n]{\\text{, }}|S_{M_{j}}|\\geq w_{\\min }", "228cb04c8c17ffd5817982ff9f06021b": "{\\mathfrak {D}}=\\{d\\}", "228cc1099bb99471c9e663dc3179923a": "q\\in {\\mathbb {C}}", "228cd14961ec9120661c673a1ceeeda3": "C_{in}^{\\alpha _{1}}", "228ce10760234445865375e58ea6518a": "X_{A}={\\frac {n_{A}(t=0)-n_{A}(t)}{n_{A}(t=0)}}=1-{\\frac {n_{A}(t)}{n_{A}(t=0)}}={\\frac {100-10}{100}}=0.9=90\\%", "228d1770f0d3e778245ad74459489fc9": "{f_{st}={\\frac {m_{ox,0}}{{sm}_{fu,1}+m_{ox,0}}}}", "228d2c8af9e8c6eb744b7eac6902d632": "R[\\Delta ^{n}]:=R[x_{1},...,x_{n+1}]/(\\sum x_{i}-1)", "228d7376d3b7de0ae7e87873f2fe5b49": "{\\begin{aligned}&(\\lambda ,\\mu ,\\nu )\\\\&\\lambda \\operatorname {rk} E", "22a2218d6501ef08e25255ecb6ef7cc6": "H1=0.541266*P", "22a2368d74c955d340f9d5c19933a038": "D=D_{max}\\cdot (P_{max}-P)", "22a24d0a8fee23437196d4726128ec2d": "Q_{C}={\\frac {X_{C}}{R_{C}}}={\\frac {1}{\\omega _{0}CR_{C}}}", "22a270b99d811f74be44e37fce3d4075": "[x_{0},\\ldots ,x_{n};f],", "22a2c0be1acdb616c4ab144870516fad": "Z_{\\mathrm {in} }=Z_{0}{\\frac {Z_{L}+iZ_{0}\\tan({\\pi \\over 2})}{Z_{0}+iZ_{L}\\tan({\\pi \\over 2})}}=Z_{0}{\\frac {iZ_{0}\\tan({\\pi \\over 2})}{iZ_{L}\\tan({\\pi \\over 2})}}={\\frac {{Z_{0}}^{2}}{Z_{L}}}", "22a30be03491a5692ec41f1e349a146c": "\\lambda _{tot}(t_{sc})=\\sum _{i}\\lambda _{i}.", "22a37cb64bafddb3d7949e57a1823655": "mkT", "22a40e0d7a0856ffb1f6a8ebdbe5aa08": "V={\\frac {4p}{4-(p-2)(q-2)}},\\quad E={\\frac {2pq}{4-(p-2)(q-2)}},\\quad F={\\frac {4q}{4-(p-2)(q-2)}}.", "22a4709ac96af95e41cb78d6a82bc438": "{\\frac {1}{\\sqrt {2}}}", "22a47cbe71bd434887b9a5829eda6400": "n_{1}\\times n_{2}", "22a48ba1c4f204f3d96be834d354f8e1": "P\\,", "22a50bf3e8f2ebe23a22c5120c9dbd11": "{\\hat {R}}", "22a53da2ba6bc1f0d5993c9204d1dfb4": "|f(x)-f(y)|\\leq M\\,|x-y|^{\\alpha },\\quad x,y\\in [a,b]", "22a59ab9300a67e7239c0808e90e0c32": "\\sigma =\\sigma _{R}+\\sigma _{I}", "22a5e4487011b3fb5a47053cb7595504": "\\lambda ^{(n)}", "22a62e7c3050e9c2e75e65b185f67d37": "X,Y\\sim {\\textrm {N}}(0,1)\\,X,Y", "22a64f1493b9a4f61c3db7745a7a2c51": "f(x)-p_{n}(x)=f[x_{0},\\ldots ,x_{n},x]\\prod _{i=0}^{n}(x-x_{i})", "22a68463505ebd43886cb03513291958": "\\eta _{\\mu \\nu }", "22a69378ea2160345fdf1d500bb4e6b1": "\\Gamma _{ij,k}^{(\\alpha )}+\\Gamma _{ik,j}^{(-\\alpha )}=\\Gamma _{ij,k}^{(0)}+\\alpha T_{ijk}+\\Gamma _{ik,j}^{(0)}-\\alpha T_{ijk}=\\partial _{i}g_{jk}", "22a6b88432bbca1acd71e971b0d4371a": "B_{\\alpha }^{\\alpha }=-1/R", "22a6f6afd068c21597507178eeea9fad": "A=(2\\pi r)(2\\pi R)=4\\pi ^{2}Rr.\\,", "22a7042b12b52418b61e4dc5b41bcc0d": "\\operatorname {perm} A\\leq \\prod _{i=1}^{n}(r_{i})!^{1/r_{i}}.", "22a755a82d173d75f4e52d622665c138": "\\psi _{2,m}", "22a761d68fa745d8a4cc71ca8dfbcb2d": "a^{\\dagger }a\\psi _{0}=0=\\left({\\frac {\\hat {H}}{\\hbar \\omega }}-{\\frac {1}{2}}\\right)\\,\\psi _{0}=\\left({\\frac {E_{0}}{\\hbar \\omega }}-{\\frac {1}{2}}\\right)\\,\\psi _{0}.", "22a769b60d470fc6c37f1f9cd6dc22e8": "\\exp _{10}^{2}(8.56784)", "22a7c7266fe187fc829c235da96ead3d": "term(x,j)={\\begin{cases}\\left\\{\\right\\},&j\\geq \\#input\\\\\\left\\{j+1\\right\\},&j^{th}{\\mbox{ element of }}input=x\\\\\\left\\{\\right\\},&{\\mbox{otherwise}}\\end{cases}}", "22a7d94e48f27443084943725603ecb7": "G={\\frac {G_{1}\\Delta _{1}}{\\Delta }}={\\frac {-y_{21}R_{L}}{1+R_{in}y_{11}+R_{L}y_{22}-y_{21}R_{L}y_{12}R_{in}+R_{in}y_{11}R_{L}y_{22}}}\\,", "22a8234e4fdd3bc701326a4bef77f4c5": "1-t+t^{2}-\\cdots +(-t)^{n-1}={\\frac {1-(-t)^{n}}{1+t}}", "22a8244308bcb651c842bd0a205748ba": "-\\mathbf {X} ^{-1}\\mathbf {A} ^{\\rm {T}}\\mathbf {X} ^{-1}", "22a8a2128364a8ea862aa20f13cd15a2": "{2x^{2}-x+4}=0.\\,\\!", "22a9b1494679181dc52202d0ba1f31ec": "w-M", "22a9c6dcbd0208a13a6c3d91d2a157c5": "\\bot =I^{*}", "22aa2bff8aa681ee2891df8ce6a31c63": "C_{14}=C_{15}=C_{24}=C_{25}=C_{34}=C_{35}=C_{46}=C_{56}=0~.", "22aa50b172e16f7e74061d1dd4f356c3": "{\\begin{aligned}\\gamma =\\lim _{n\\to \\infty }\\left(\\ln n-\\sum _{p\\leq n}{\\frac {\\ln p}{p-1}}\\right)\\end{aligned}}.", "22aac8d36955638219f72323821d785e": "{\\begin{aligned}&(\\varphi +\\psi )(x)=\\varphi (x)+\\psi (x)\\\\&(a\\varphi )(x)=a\\left(\\varphi (x)\\right)\\end{aligned}}", "22ab80d98be6ba3a9c4a490313739aaf": "aq_{n}-bp_{n}", "22abccf2370a96166c5f2d79376223fa": "D<{\\frac {10^{k}}{n}}-1,", "22ac1b7a739599100f9f72a046c8f90d": "p({\\boldsymbol {\\theta }},{\\boldsymbol {\\phi }}|{\\rm {{data})}}", "22ac23194a1cabb43ad2478cbaa41224": "z_{}^{}", "22ac714ffb9af4d2a92b5b3dd9691da4": "r\\in V\\,\\!", "22accd3047daa3dab24e04f158c97c46": "\\int _{-\\infty }^{\\infty }\\!dA\\,e^{-{\\frac {1}{2}}A^{2}+Af}={\\sqrt {2\\pi }}e^{-{\\frac {f^{2}}{2}}}", "22acdfc0b7b024b89a91859e542dcde3": "\\displaystyle {W(z,w)=e^{-izw/2}U(z)V(w)}", "22acedc7ba733d3aeb517e79de401799": "XXZ", "22acf4791c3a74b310d3605ecd18bcce": "\\scriptstyle h(\\tau )", "22acfffa5abb07d762bc57ab5ac340b1": "g(x)=e^{-\\lambda _{0}-1-\\lambda (x-\\mu )^{2}}", "22ad456fb0d74b9f48f29688f43aec1c": "k_{e}q_{1}q_{2}/m", "22ad47f404ea32dd988a4fbdaad50d50": "{\\begin{cases}2ab=1\\!\\\\a^{2}-b^{2}=0\\!\\end{cases}}", "22ad78c2ee68841d4520da3fbf7bf1da": "f(x)=\\int _{-\\infty }^{\\infty }{\\hat {f}}(\\xi )\\ e^{2\\pi i\\xi x}\\,\\mathrm {d} \\xi ,", "22ae54f18ab71732cb2a3530a4901017": "\\operatorname {lift} _{\\tau }(e)", "22ae666a3cd3a651f61269ca0506defd": "t=F_{{\\hat {\\theta }}_{n}}(x)", "22aed3510591eeac0f958dc9bebea90f": "p_{i}(m)=m(i)", "22aeed46d7274e42d16f9e3e251512b7": "w{\\sqrt {\\theta }}/{\\delta }", "22af4663c8b81b46f33b115b45ce76a9": "T=1.41", "22af6cbed482a4c8f6de941833c0ee91": "T_{[\\alpha \\beta \\mu ]}=0", "22af7d1f2e55c8916a0850fa940b0947": "k_{\\rm {on}}\\,[{\\rm {R}}]\\,[{\\rm {L}}]=k_{\\rm {off}}\\,[{\\rm {RL}}]", "22af8ec879e356c2ac01d4c4d5dcd9d6": "\\exp(\\gamma l)\\,", "22b0085b5fd016929fb8cb89ab9d549a": "\\{(ab^{n})^{n}|n\\geq 0\\}", "22b022754506b390a3b5414fcebfad9b": "(x,a)\\sim (x,b){\\text{ if }}x<0.\\;", "22b04348dc60a3508c31118ba163533c": "{\\overline {OB}}", "22b04a52da7d54bcb03d1c2182adbcff": "{\\frac {d}{dx}}\\chi _{2}(x)={\\frac {{\\rm {arctanh\\,}}x}{x}}.", "22b0787b820d7b94ac706a6749339ac9": "t_{1}}", "22b1b7527bd6b270d0da9f1796dce2aa": "~\\Phi _{n}(x)=1+x+x^{2}+\\cdots +x^{n-1}=\\sum _{i=0}^{n-1}x^{i}.", "22b1bb1a8179fdeae8d7b9c2c1467671": "open(door,result(opens,s))", "22b1c2e66eb01489c2c231414c8a6884": "(f*g)+\\varepsilon =h\\,", "22b1f7d85314571dc1e1dcfd40154e8e": "p(\\pm x_{1},\\pm x_{2},\\ldots ,\\pm x_{n})=0.\\,", "22b1fc7b4f996437e16a0b7b318a2dc6": "m_{\\text{red}}", "22b218e8955210ea5e10226c2b5d9e06": "\\,NA", "22b21e43056af1cc82aa19c1b4ae2b52": "{\\hat {\\beta }}-\\beta ", "22b24e1d84ff18af22bb23323caf1731": "\\lim _{\\Delta x\\to 0}{\\frac {\\Delta h}{\\Delta x}}", "22b297e10a9746252315abecfd3ff4d0": "(\\lambda \\leq \\lambda ')\\leftrightarrow (s_{\\lambda '}\\leq s_{\\lambda })", "22b2f497188e759b7e75d8ad879a1c30": "D=R-r,\\,", "22b3028f41872d929a5faadc56f66a1b": "c\\subseteq \\left(c'\\right)^{+}", "22b31c9ebdf0c1ee047224a011a1cd8c": "Q\\,\\!", "22b369532f78351aaf1e4e642e6e584d": "V(x)=V_{0}*\\Theta (x)=\\left\\{{\\begin{aligned}0&\\quad x<0\\\\V_{0}&\\quad x>0\\end{aligned}}\\right.", "22b36edb7824baece8731d66b6ec6eec": "{\\frac {{\\text{d}}[{^{b_{j}}_{a_{j}}}S_{j}^{\\beta _{j}}]}{{\\text{d}}t}}=\\sum _{i}x_{b_{ji}}[{\\text{k}}_{2(i)}C_{i}-{\\text{k}}_{1(i)}E{\\overline {S}}_{i}]\\qquad \\qquad (8a)", "22b46f9b460bd11b80a712f9c181a096": "{\\hat {S}}_{N}", "22b474e48c46f805f571fc9bdff89efe": "\\;\\Lambda ^{k}[\\psi ]=sup_{\\phi \\in S_{k}}|\\langle \\psi |\\phi \\rangle |^{2}", "22b4c68332930d02a2f874460da513ea": "E_{r}", "22b4f59cb8348c7d6373fa0d0e3ed055": "(\\mathrm {C_{6}H_{4}(CH_{3})_{2}} )", "22b505bc46bc4a9efa6c680256705f04": "V_{\\beta }^{I}", "22b59ed7a72604f34413115e7f046960": "\\scriptstyle {\\mathbf {r} }_{1}(t)", "22b5abefcd0a9eb941640b1947621dd9": "\\alpha ^{2}u_{j,iij}=0\\,\\!", "22b5b5cecc8d2260f24543201e571c8b": "{\\Big (}({\\mathcal {M}},s)\\models \\phi _{1}\\land \\phi _{2}{\\Big )}\\Leftrightarrow {\\Big (}{\\big (}({\\mathcal {M}},s)\\models \\phi _{1}{\\big )}\\land {\\big (}({\\mathcal {M}},s)\\models \\phi _{2}{\\big )}{\\Big )}", "22b60909b0f6cfc7b72d6f1b63f670c3": "{\\hat {\\textbf {x}}}_{t-i\\mid t}", "22b63c0948778288c4c9f3860b411df5": "{\\rm {tr}}((\\mathbf {B} -{\\hat {\\mathbf {B} }})^{\\rm {T}}\\mathbf {X} ^{\\rm {T}}{\\boldsymbol {\\Sigma }}_{\\epsilon }^{-1}\\mathbf {X} (\\mathbf {B} -{\\hat {\\mathbf {B} }}))={\\rm {vec}}(\\mathbf {B} -{\\hat {\\mathbf {B} }})^{\\rm {T}}{\\rm {vec}}(\\mathbf {X} ^{\\rm {T}}{\\boldsymbol {\\Sigma }}_{\\epsilon }^{-1}\\mathbf {X} (\\mathbf {B} -{\\hat {\\mathbf {B} }}))", "22b641fa5ac4714d0d111eec5786aa0d": "\\ell _{0}(x)={x-x_{1} \\over x_{0}-x_{1}}\\cdot {x-x_{2} \\over x_{0}-x_{2}}\\cdot {x-x_{3} \\over x_{0}-x_{3}}\\cdot {x-x_{4} \\over x_{0}-x_{4}}={1 \\over 243}x(2x-3)(4x-3)(4x+3)", "22b647628f488d35fc7a07e821cbcff6": "V:=L({\\underline {a}}_{1},\\ldots ,{\\underline {a}}_{r})", "22b68bd688762b5f6a2136e8c974e779": "g=-G\\cdot M/r^{2}+\\omega ^{2}\\cdot r", "22b6e3dd03cb9a8688980c1a8fbcca0d": "\\pi (g)f(x)=f(g^{-1}x),", "22b6f54eaa543e0735135e60d5b13cab": "-c{\\frac {df}{dX}}+{\\frac {d^{3}f}{dX^{3}}}+6f{\\frac {df}{dX}}=0,", "22b71634db2e7a17be047da9b0301089": "{\\sqrt {ax^{2}+bx+c}}\\,\\!", "22b78845d32790f6b041bee362863a7c": "V_{S}", "22b7bb0c699f5c82b02fd8b51e3812ec": "X_{r}=\\{(x,t):0\\leq t\\leq r(x),x\\in X\\}/(x,r(x))\\sim (fx,0).", "22b7d404620033d69e1c4929f3fd672e": "{\\hat {A}}{\\hat {A}}^{*}=(A,B)(A^{*},B^{*})=(AA^{*},AB^{*}+BA^{*})=(1,0).\\!", "22b7dc4a2e7161229f7fa237d1689d4b": "{\\frac {\\mathrm {d} \\mathbf {A} }{\\mathrm {d} t}}={\\frac {\\partial \\mathbf {A} }{\\partial t}}+(\\mathbf {v} \\cdot \\nabla )\\mathbf {A} ", "22b7e63eac22c6b06c4070e776f628ba": "f(c_{1})=(1.5)^{3}-(1.5)-2=-0.125", "22b80d566556e632e7ee5467dcab4c44": "(+\\alpha ,-\\alpha )", "22b812eac0dc95c05573f232e51a8355": "Y_{i+1,2}=40,692\\times Y_{i,2}{\\pmod {2,147,483,399}}", "22b867147f4fe11a2d41a9ef13f6ca09": "W\\in \\alpha ", "22b888cb5dbe74050a5416436a29bb57": "-{\\sqrt {\\frac {1}{6}}}\\!\\,", "22b8959dca36615c493537b1bc760a05": "\\|Tx\\|=\\|T^{*}x\\|", "22b8ae2f554cb5cd4dd83d8b79db3107": "V_{\\,t}", "22b8c33fed8bd7cf6a22988a9d172de2": "{}_{j}{\\bar {P}}_{L}^{l}(\\theta )={\\sqrt {{\\frac {2L+j-1}{2}}{\\frac {(L+l+j-2)!}{(L-l)!}}}}\\sin ^{\\frac {2-j}{2}}(\\theta )P_{L+{\\frac {j-2}{2}}}^{-(l+{\\frac {j-2}{2}})}(\\cos \\theta )", "22b8eefb59e47bfc8edc7277baa9fe1c": "{\\begin{array}{rccrcrcrcr}{\\color {BrickRed}P}{\\color {RoyalBlue}Q}&{=}&&({\\color {BrickRed}2x}\\cdot {\\color {RoyalBlue}2x})&+&({\\color {BrickRed}2x}\\cdot {\\color {RoyalBlue}5y})&+&({\\color {BrickRed}2x}\\cdot {\\color {RoyalBlue}xy})&+&({\\color {BrickRed}2x}\\cdot {\\color {RoyalBlue}1})\\\\&&+&({\\color {BrickRed}3y}\\cdot {\\color {RoyalBlue}2x})&+&({\\color {BrickRed}3y}\\cdot {\\color {RoyalBlue}5y})&+&({\\color {BrickRed}3y}\\cdot {\\color {RoyalBlue}xy})&+&({\\color {BrickRed}3y}\\cdot {\\color {RoyalBlue}1})\\\\&&+&({\\color {BrickRed}5}\\cdot {\\color {RoyalBlue}2x})&+&({\\color {BrickRed}5}\\cdot {\\color {RoyalBlue}5y})&+&({\\color {BrickRed}5}\\cdot {\\color {RoyalBlue}xy})&+&({\\color {BrickRed}5}\\cdot {\\color {RoyalBlue}1})\\end{array}}", "22b93da3890cf46eb0b150f7878b9071": "{\\sqrt {n/2-3/8}}", "22b95bcb1de3e047408bc7a83934005d": "\\log {\\frac {k_{x,sol}}{k_{x,80\\%EtOH}}}=mY", "22ba0642369aad29f9fb30cbb3ee20f5": "T_{se}", "22ba10e403d2fbb63084c676966d5f63": "x_{\\mathrm {a} }(t)={\\begin{cases}\\ \\ e^{j|\\omega |t}\\cdot e^{j\\theta },&{\\mbox{if}}\\ \\omega >0,\\\\\\ \\ e^{j|\\omega |t}\\cdot e^{-j\\theta },&{\\mbox{if}}\\ \\omega <0.\\end{cases}}", "22ba3df1ffa88853a725f9aa70271d54": "S_{m}=C_{V}\\ln {\\frac {T}{T_{0}}}+R\\ln {\\frac {V_{m}}{V_{0}}}.", "22ba7f3c9b64390cc173abcb7b2bba40": "0\\leq deg(D_{x})+v_{1}(D)\\leq g", "22ba8403045255f335e95f0a6991e146": "\\phi =\\pm \\varphi ", "22badfb1e212c287a4db5c70237313f3": "\\scriptstyle \\theta <{\\frac {1}{4}}", "22baeba5a637ed9bc00f744931dee7c7": "{\\frac {a+b}{a}}={\\frac {a}{b}}\\equiv \\varphi ", "22bafd6a4d2255a6c8b96ac73b92ea79": "t\\in [0,1]", "22bb7366806b4ad3012332ef01c5577e": "h(x)\\leq 0", "22bba63018a0bb86116fe1da65478529": "f_{\\ell }={\\frac {S_{\\ell }-1}{2ik}}={\\frac {e^{2i\\delta _{\\ell }}-1}{2ik}}={\\frac {e^{i\\delta _{\\ell }}\\sin \\delta _{\\ell }}{k}}={\\frac {1}{k\\cot \\delta _{\\ell }-ik}}\\;.", "22bbc5c7044ea97ef7f07eff75d01f27": "z\\ ", "22bbd54d13c77f28dc92a2401e67eab0": "\\,e^{it\\mu -{\\frac {1}{2}}\\sigma ^{2}t^{2}}", "22bbede97f45426877830300a702e52d": "I=30/300=0.1", "22bc40bf712ae7f4908381369fdea7db": "x_{1},x_{2}\\in R^{d}", "22bc4cbf3b3f4b737baac878df8336bd": "d\\Omega _{0}={\\frac {dS_{0}}{|{\\vec {r}}-{\\vec {r_{0}}}|^{2}}}{\\frac {{\\hat {n}}_{0}^{s}.({\\vec {r}}-{\\vec {r_{0}}})}{|{\\vec {r}}-{\\vec {r_{0}}}|}}.", "22bc4e390a8a9014b293f0826d32cfcd": "Q_{D}(l_{A}a_{B}+l_{B})l_{D}", "22bc9d624fbca98ab53fef969b68e2d4": "{\\frac {T}{P}}={\\frac {x_{p}-x_{f}}{x_{f}-x_{t}}}", "22bcaeefcaa1b7e712df20e3f8c556f7": "{\\mathfrak {sp}}(2n,\\mathbb {C} )", "22bd077637a7a9fd6c42e5cd7339a59c": "f'(x)\\,=\\,Df(x)=na_{n}x^{n-1}+\\cdots +2a_{2}x+a_{1}", "22bd0e466382096a4d54bc510ac07c7a": "{\\tilde {M}}=|M_{1}|e^{i\\theta _{1}}e^{-i\\phi _{1}}+|M_{2}|e^{i\\theta _{2}}e^{-i\\phi _{2}}", "22bd11870acfeecb092e5489a752655f": "\\mathbb {C} \\left({\\textbf {g}}\\otimes {\\textbf {n}}\\right){\\textbf {n}}=0.\\qquad {(4)}", "22bd69f86c3300dd9fae5fd90260c5d6": "a_{i},_{J}u_{i},_{J}=\\sum a_{n}bu_{n}b-\\Delta V_{u}{\\frac {P_{I},_{J}-P_{I-1},_{J}}{\\partial x_{u}}}+S\\Delta V_{u}", "22bd6b9028fd636ae7c7645c70f381f7": "Select:2^{D}\\rightarrow D", "22bd7f9b5e766d9bf85195216db60d95": "{\\begin{aligned}\\sup _{y\\in [-\\varepsilon ,\\varepsilon ]}|Q_{1-\\varepsilon }(y)|&\\leq \\varepsilon ^{-1}.\\\\\\sup _{y\\notin (-\\varepsilon ,\\varepsilon )}|Q_{1}(y)-Q_{1-\\varepsilon }(y)|&\\to 0.\\end{aligned}}", "22bdb2e60be4a709b885eed926c4f3fd": "\\|a-b\\|_{1}=\\sum _{i}|a_{i}-b_{i}|", "22be5ef5aa61a0c1449c19deadf13b71": "f(x_{1})=y_{1}", "22be842f9527b4b0ef3b7085b5486b7b": "\\left[a,b\\right]", "22bf33e6cde821d4237c35cc7713611f": "\\mathbb {R} _{--}", "22bf9624469967c4fafbc0759ca05313": "A_{1}A_{2}\\ldots A_{n}", "22bfe044e763ff1cbb0ee1d54b4d4cd2": "Q_{b}^{(i)}", "22c02fcb3c26b705ce285fab39fe0698": "G\\hookrightarrow U", "22c062b79ff14c547953fc25d7aff1af": "\\scriptstyle |\\zeta |\\;\\leq \\;1", "22c073e1631e10ad78dddf1c6681f398": "Y^{(n+1)}(\\xi )=f^{(n+1)}(\\xi )-{\\frac {R_{n}(x)}{W(x)}}\\ (n+1)!=0", "22c092b03d5cf415fecd20a7832f513b": "\\pi :Y\\longrightarrow X\\ ", "22c0dea095c6efd1d8603fa8dda8456a": "R_{d}/100", "22c0fa0bbf2a22a2a32c4a16b7acb329": "\\mathbf {r} (0)+\\left({\\frac {s^{2}\\kappa (0)}{2}}+{\\frac {s^{3}\\kappa '(0)}{6}}\\right)\\mathbf {N} (0)+\\left({\\frac {s^{3}\\kappa (0)\\tau (0)}{6}}\\right)\\mathbf {B} (0)+o(s^{3})", "22c1331bd91a23f7e9ed38e5aa35e4be": "{x_{1}}^{e_{1}}\\cdots {x_{n}}^{e_{n}}", "22c13fd01ba06a4d700000f2495eef6a": "-{\\sqrt {\\frac {4}{15}}}\\!\\,", "22c17ed039c41f337db4b60659671c13": "f_{\\mathrm {\\omega } }=N{\\omega }^{2}r.", "22c198f267e4b618677adcd0bdb675f0": "\\int _{A}dA\\nabla f=\\oint _{\\partial A}dxf", "22c1ad678d3c6547e0ec61b6d3b8efc3": "r={\\frac {l}{1+e\\cos \\theta }},", "22c1b95d0b51bf3b5d08fc1f455aa2e2": "vm_{pq\\mu \\gamma }=\\sum _{i=2}^{images}\\sum _{x=1}^{M}\\sum _{y=1}^{N}U(i,\\mu ,\\gamma )C(i,p,g)P_{i_{xy}}", "22c1c5fe8131c74196a508aaab6eab9e": "\\mu _{k}={\\frac {b_{k}^{*}Ab_{k}}{b_{k}^{*}b_{k}}}", "22c1d034b2416df93506e1b66b4d05f2": "\\xi \\mapsto \\theta =\\theta \\left(y,\\xi \\right)", "22c1e12cf94b7c16ca4d04c3403bd807": "{\\hat {E}}\\left\\{\\mathbf {x} (n)\\,e^{*}(n)\\right\\}={\\frac {1}{N}}\\sum _{i=0}^{N-1}\\mathbf {x} (n-i)\\,e^{*}(n-i)", "22c1e4c2633d9b563f2c5c1f3ac2f52d": "\\hbar \\ ", "22c204a666cbbe332d51b68b15d2b2d8": "T(t,r)=t-\\int {\\frac {\\sqrt {2m/r}}{1-2m/r}}\\,dr=t+2{\\sqrt {2mr}}+2m\\log \\left({\\frac {{\\sqrt {r}}-{\\sqrt {2m}}}{{\\sqrt {r}}+{\\sqrt {2m}}}}\\right)", "22c274b8e98bdb49fdb95f8bd9e352bf": "q_{\\mathrm {1} },q_{\\mathrm {2} }", "22c2afaa9d18cdbc249dd99607477df0": "f(x)g(y)=f(x')g(y')", "22c32dce6b76f3d06f2aa5eecdbaaf63": "S^{\\sigma }(1853)=dr(1853)=8.\\,", "22c3717ba1dfac337b183f3b9949b9a3": "S(t)=(r_{1},r_{2},\\cdots ,r_{N})", "22c3bbaf6078188688c09d8fd47f39d2": "C_{n}\\left(Y\\right)", "22c3fda0bab16352dbf2135dd00710c9": "m_{(3,1,1)}(X_{1},X_{2},X_{3})=X_{1}^{3}X_{2}X_{3}+X_{1}X_{2}^{3}X_{3}+X_{1}X_{2}X_{3}^{3}", "22c41e4cb82486019b10e43932785f0a": "dY=de^{-\\int _{t}^{s}V(X_{\\tau })\\,d\\tau }u(X_{s},s)+e^{-\\int _{t}^{s}V(X_{\\tau })\\,d\\tau }\\,du(X_{s},s)+de^{-\\int _{t}^{s}V(X_{\\tau })\\,d\\tau }du(X_{s},s)+d\\int _{t}^{s}e^{-\\int _{t}^{r}V(X_{\\tau })\\,d\\tau }f(X_{r},r)dr", "22c45d34d43506fde9b57bc2fecfa313": "\\operatorname {var} (r)={\\frac {1}{s_{x}^{2}+m_{x}^{2}}}\\left[(s_{y}^{2}-s_{x^{2}[y^{2}/x^{2}]})-(s_{x[y/x]})^{2}+2m_{y}s_{x[y/x]}-{\\frac {s_{x}^{2}}{m_{x}^{2}}}(m_{y}-s_{x[y/x]}^{2})\\right]", "22c471f4d0c633bc09ca7b5da07eba7c": "F^{*}(H)=O_{p}(H)", "22c4b22afee2c15de89a176f0f3be2d7": "Q^{-}(2n+1,q)(n\\geq 2)", "22c50e2263d50cc11c9b087099987602": "\\theta =n(\\lambda -\\lambda _{0})", "22c523fa8e1c05573e7d47deacb79ef8": "{\\frac {1}{\\zeta (s)}}=\\sum _{n=1}^{\\infty }{\\frac {\\mu (n)}{n^{s}}}\\!", "22c59af425ae2fc25b9ce9fa8522347a": "\\mathbf {A} ^{*}", "22c5c97c10c6ce627cf54510bc15d9ce": "{\\frac {P\\to Q,P}{\\therefore Q}}", "22c60cff0fe91b977f01efe3015c845d": "{\\boldsymbol {U}}={\\boldsymbol {S}}\\cdot {\\boldsymbol {T}}", "22c774ffaf3b9d43995af95f849ee0cf": "{\\mathfrak {a}}^{e}", "22c7ad124dcee12145c7d3eb3bccffab": "A_{v}=\\log _{2}", "22c7e18f2ea59e4c2564d09f31d2f96a": "P({\\text{ill}})=1\\%=0.01{\\text{ and }}P({\\text{well}})=99\\%=0.99.", "22c7e32877d4d61963168e0846fa0f66": "{\\frac {d^{2}\\theta }{dt^{2}}}+1=0.", "22c826d0f93b82d4487b4d1c7287d5bd": "{\\hat {\\boldsymbol {x}}}", "22c85dce207d7deb6e0c71bd5cd20983": "\\mathbf {y} ={\\begin{pmatrix}y_{1}\\\\y_{2}\\\\\\vdots \\\\y_{n}\\end{pmatrix}},\\quad \\mathbf {X} ={\\begin{pmatrix}\\mathbf {x} _{1}^{\\rm {T}}\\\\\\mathbf {x} _{2}^{\\rm {T}}\\\\\\vdots \\\\\\mathbf {x} _{n}^{\\rm {T}}\\end{pmatrix}}={\\begin{pmatrix}x_{11}&\\cdots &x_{1p}\\\\x_{21}&\\cdots &x_{2p}\\\\\\vdots &\\ddots &\\vdots \\\\x_{n1}&\\cdots &x_{np}\\end{pmatrix}},\\quad {\\boldsymbol {\\beta }}={\\begin{pmatrix}\\beta _{1}\\\\\\beta _{2}\\\\\\vdots \\\\\\beta _{p}\\end{pmatrix}},\\quad {\\boldsymbol {\\varepsilon }}={\\begin{pmatrix}\\varepsilon _{1}\\\\\\varepsilon _{2}\\\\\\vdots \\\\\\varepsilon _{n}\\end{pmatrix}}.", "22c8881ed41502f4c10ff52e7a77be8b": "j(p)-j(q)=\\left({1 \\over p}-{1 \\over q}\\right)\\prod _{n,m=1}^{\\infty }(1-p^{n}q^{m})^{c_{nm}}", "22c8b6d45aa503fe11828c07d8f72070": "\\left({\\frac {\\partial T}{\\partial N_{j}}}\\right)_{V,S,\\{N_{i\\neq j}\\}}=\\left({\\frac {\\partial \\mu _{j}}{\\partial S}}\\right)_{V,\\{N_{i}\\}}", "22c8e0ca89de0a991eb80ba8e4909788": "\\scriptstyle B'(t)", "22c91e66de1ec0110c6e56792d6bd354": "\\scriptstyle f(b)", "22c93f1db08e62f9be885995800a23f7": "{\\begin{aligned}x&=a_{1}+n_{1}\\,x_{1}\\\\&\\cdots \\\\x&=a_{k}+n_{k}\\,x_{k}\\end{aligned}}", "22c9cadd7be326b089a02a690bcfae93": "(AC_{X}A^{T}+C_{Z})^{-1}", "22ca0bf4f8a252846e1230ab2317ee27": "\\ell ^{2}(\\mathbb {Z} ,\\mathbb {R} )", "22ca1e4bf74cbeaf4a3e975fd61d2a00": "\\chi _{j}", "22ca4f25f895d9eff1b217a46c51e235": "72\\,{\\text{Hz}}\\,", "22ca8242cf7a3701611fcdab4e8dfa64": "{v_{A}^{2} \\over 2}-gd+{P_{A} \\over \\rho }=\\mathrm {constant} ", "22ca90cf38b94a0cacdcfcbb81888d65": "w(n)=\\alpha -\\beta \\;\\cos \\left({\\frac {2\\pi n}{N-1}}\\right),", "22caaf8a6d3b83a60147cf1aeebd2a2c": "[x_{k-1},x_{k+1}]", "22cacd68e86fdaf2c4e89036b65cf7fa": "RE", "22cb1d066965711886f7f32d7e339f5d": "R={-1 \\over \\beta -\\alpha }=-P", "22cb5b460e26ec3f84212b29890c8444": "\\mathbf {Z} [{\\sqrt {-1}}]", "22cb7353becd1d379c0665375bf3ed08": "\\int _{-\\infty }^{\\infty }t^{m}\\,\\psi (t)\\,dt=0.", "22cb79a41a510659669eda0a6beb14ae": "\\sigma _{t}:V\\to P(\\lbrace 1,2,\\dots ,n\\rbrace )", "22cc17e596c8a7f56c0a0cf91e52da14": "||f||=\\int _{I}{\\frac {|f(x)|}{1+|f(x)|}}dx.", "22cca3c2763975bc466d7c093f75ccdf": "(M,x)\\to (N,y).", "22cce6485be59b549cfa47d345cb4658": "i=1,2,3", "22ccfd23882d7b133ea545e86ec9b379": "[B(t,t_{0}),B^{\\dagger }(t,t_{0})]=t-t_{0}", "22cd026d16ab1560ebc7ea257a07119e": "E(Y)=f(x)", "22cd966c78cf0bdc86e33a6f65e2cd3d": "P_{n+1}P_{n-1}-P_{n}^{2}=(-1)^{n},", "22cda15fa012b0aa16b56e2b782b5a8e": "\\sigma _{it},i=1,2,3", "22cdab75fa4f8f4fa943c380088c7e83": "(18)\\quad k^{c}\\nabla _{c}{\\hat {\\sigma }}_{ab}=-{\\hat {\\theta }}{\\hat {\\sigma }}_{ab}+{\\widehat {C_{cbad}k^{c}k^{d}}}\\;,", "22cdfba0a1ebeb4a28e9923cae52f3da": "{\\tilde {\\lambda }}", "22ce083a837c03f6ac00fa32f090faa4": "1,\\ldots ,m", "22ce395e620847f4bb68a699a4e3b054": "A-BD^{-1}C.\\,", "22ce56f9641b574163434fc04ae313e9": "{\\sqrt {2}}=1.41421356237\\dots \\,", "22ce89a67ea7c0606f279b68246bdfdb": "\\vartheta \\,", "22ceb3021cef9c0c6440cd66db26518f": "0\\leq i\\leq 2^{k}", "22cec9fb06f2632dae2e18f9dc341597": "{52 \\choose 5}\\times 5=12,994,800", "22cee76ced886e62267b5ccdf7c6944c": "\\langle W,\\subseteq \\rangle ", "22ceeec583e2a37b64d3b902df5b159f": "p\\mid m_{i}-\\left(i+1\\right)\\cdot m", "22cf2a620dbbf9604a6987d3f92cd549": "\\iota _{\\Omega _{\\alpha }}\\omega =\\alpha ", "22cf60311c48f84042ec15829ca57e33": "[m_{1}:b_{1}:1]_{L}\\cap [m_{2}:b_{2}:1]_{L}=g([m_{1}:b_{1}:1]_{L}\\times [m_{2}:b_{2}:1]_{L})", "22cf6d90476d6b87655fb83d74ee08f5": "x_{i}\\ ", "22cf8f2060388a102b3b19f20eff55fe": "f_{d}", "22cfba13b9e3941f5d42ba93aeafc2f0": "a={\\sqrt {\\gamma \\,R\\,T \\over m_{m}}}", "22cfcaee6fdd930609e7ef0912a8b2bd": "SV=\\bigcup _{x\\in V}S_{x}V\\subset T^{*}V", "22d0625861255a03788c33b708a968bb": "\\theta (y)", "22d06c72798fbff71fd04ff95a0b4847": "a_{n},b_{n}\\,", "22d0a945eea27eb58d7dc0db5269e8a7": "\\varepsilon =\\varepsilon _{r}\\times 8.854\\times 10^{-12}", "22d0b62b6c49ba34a65562e2c95468a4": "{\\overset {\\circ }{\\boldsymbol {\\sigma }}}=J^{-1}~\\phi _{*}[{\\dot {\\boldsymbol {S}}}]", "22d0f55ea4fe03fcd11235288cfd7192": "\\eta _{\\mu \\nu }^{a}={\\begin{cases}\\epsilon ^{a\\mu \\nu }&\\mu ,\\nu =1,2,3\\\\-\\delta ^{a\\nu }&\\mu =4\\\\\\delta ^{a\\mu }&\\nu =4\\\\0&\\mu =\\nu =4\\end{cases}}.", "22d10b0fa35414f1135574f773e217f3": "G(x,y)=\\sum _{i=1}^{n}{w_{i}f(x_{i},y_{i})}", "22d130a1e88fa413d5125b364911a41e": "\\scriptstyle (m|k)\\;=\\;\\Pr(M\\;=\\;m\\mid K\\;=\\;k)", "22d220cf1140901d1acfea1561ce5bb8": "({\\frac {1+2\\gamma }{3}})G_{M},", "22d263f15b992f2d921d0f468f32d67c": "\\mathbb {D} _{8}\\times \\mathbb {Z} _{2}", "22d28c8a344b73518482174da583d4b0": "\\times _{Ke}", "22d2bd888aca987a88ca290ec2ce84d2": "\\textstyle g^{(k)}", "22d2c1a2560de0cb328f9e1b5ab3fa93": "L=v+\\sum _{x,y\\in P\\cap \\mathbb {Z} }\\mathbb {Z} (x-y)\\subseteq \\mathbb {Z} ^{d}", "22d347d27a47be575a278c8b08ae610e": "|jk>=e^{ik\\cdot r}u_{jk}(r)", "22d3661276d7c08d8dfcaad397db53f0": "Y\\Gamma =X\\mathrm {B} +U.\\,", "22d389a801fe4e36114bf4d7be3253e9": "\\displaystyle {f(D_{r})\\subseteq g(D_{r}).}", "22d3b80f23edf0208833afb47a9664e1": "P(x)=x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1,", "22d3e1c778f6f0b7d7999b957f076456": "\\psi (x)=\\int _{a}^{b}T(x,y)\\phi (y)\\,dy", "22d3f3e6a790b98c678c3737cb89dcf2": "VO_{2}={\\frac {FR\\cdot (F_{in}O_{2}-F_{ex}O_{2})}{1-F_{ex}O_{2}}}", "22d418f8a8b2c6f1f211dc9dc86facab": "(\\mathbf {a\\times b} )\\mathbf {\\cdot } (\\mathbf {c} \\times \\mathbf {d} )\\ ,", "22d45f795da3d935ba8f360cc95caa2b": "V_{T-j+1}(k)", "22d522d3cb8903ab87de21df83c457f9": "1\\rightarrow {\\rm {Homeo}}_{0}(X)\\rightarrow {\\rm {Homeo}}(X)\\rightarrow {\\rm {MCG}}(X)\\rightarrow 1.", "22d5692a36f28033482dfbb336031aef": "\\Phi =2m_{w}S+S^{2}+2\\varepsilon _{w}A-A^{2}.", "22d57784214e1849ca39755700509021": "f:Y\\to X", "22d5787ce98c757f51265c4189835526": "H_{n,m}=\\sum _{k=1}^{n}{\\frac {1}{k^{m}}}.", "22d58f986a91a64a65c36b719ec53b51": "R^{\\mathrm {irr} }(M)", "22d5bab3c27751760ca6706d617bc025": "\\mathrm {Hm^{-1}=kg\\ A^{-2}m\\ s^{-2}} ", "22d5c218958f1c20cbb1ad0935b7da31": "\\scriptstyle L^{+}", "22d5e330c821503795a8c584386b2435": "e[n]=\\mathbf {g} ({\\hat {s}}[n])-{\\hat {s}}[n]", "22d62ff86c45846a9f3aad19d1d93496": "x=\\pm {\\textstyle {\\frac {R}{2}}}", "22d6b321d799f9745fb385204412a88a": "\\eta ={\\frac {P_{m}}{P_{e}}}", "22d7398bdcfabffac1dfb2667f2a0327": "2/2^{M}=2/m", "22d769999ddd33eaac5fd3b22662715c": "f(t)=\\pi ^{-1/4}e^{(-t^{2}/2)}", "22d7910dba3761b7279fee9afb9e1882": "L(G)=\\{ww^{R}:w\\in \\{a,b\\}^{*}\\}", "22d7a1e557c0ca842e1b75ba6d2c7531": "t_{g}=\\sum _{i=1}^{n}a_{i}\\otimes _{B}g(b_{i})", "22d7e9cd9df3c67c815d73aaba7142ed": "2,\\ 4,\\ 4,\\ 4,\\ 5,\\ 5,\\ 7,\\ 9.", "22d80b96c76d1e7a52bb57b85c1d6a0e": "\\Psi _{m}", "22d816cdaffe473e9e39f69e723f64a8": "F_{g}=sin\\theta gM", "22d8730cbb2430d3d6fb7d0cfac07eaf": "\\bot \\to P", "22d89488220b08661427036e3fff70b2": "{\\text{ Enrichment ratio}}=\\left({\\frac {\\text{Protein concentration in the foam}}{\\text{Protein concentration in the initial feed}}}\\right)", "22d8f9e35eab2b8be32667c4bf0da1f4": "n_{j}=1", "22d9bb2875d7a70aeb68696096f3b9b2": "k=0", "22da033f3890bc136cba5580f84b96c9": "\\scriptstyle \\|z\\|\\;=\\;1.", "22da2023f8f367eeb6206a82ca1ca4ad": "f_{!}", "22daa808fe379d6d394a9dbb0c249e2b": "{\\vec {w}}={\\begin{pmatrix}w_{ATM}\\\\w_{RR}\\\\w_{BF}\\end{pmatrix}}", "22dae327fda5db8fec6b131d011aabc2": "(\\ln(R))^{2}", "22daf8bc7f5ca04eb9e9ce418cf5912f": "x{\\stackrel {*}{\\leftrightarrow }}y", "22db1031dcb36bf530e4f9b3b2c35c36": "d\\varepsilon _{p}>0", "22db457b2fae003aa8dea3e5abc5f98b": "{\\sqrt {\\frac {3}{7}}}", "22db8860461b91290d24c916adad2638": "(p,q'){\\overset {\\alpha }{\\rightarrow }}(p',q')", "22dc00aa64c364f678244faa2f73df38": "E_{0}=mc^{2}\\,", "22dcb10dfd8956fcfdbc2778bfc020df": "a(r)={\\frac {F(r)}{m_{2}}}=-g(r)", "22dcd2539d1d56be8a89fa0435b91851": "{\\hat {X}}_{t}(\\omega )=\\sum _{k=1}^{N}A_{k}(\\omega )f_{k}(t)", "22dd3045a0e13f527d39bbca5e01e3bf": "V_{1},\\ldots ,V_{e}\\subset {\\mathbf {K} }^{n}", "22dd4413280a578900d2d0dfb281e1f2": "E_{k}(t=0)=0{\\frac {}{}}", "22dd54aa750673b490ed6f91c78015ea": "[*:*:*:\\dots :*].", "22ddbf62f98628ea786257aff5b7a79b": "isopen", "22de1dad9c0e2ca378424e69bd58ca84": "\\mathbf {R} ^{n-1}", "22de1f94595109c91ceec4427839ebab": "D_{mn}(x,\\alpha )=D_{m}(D_{n}(x,\\alpha ),\\alpha ^{n})\\,.", "22de2f6b95a4696194cab0645cfb0f4f": "N(x+iy)=x^{2}+y^{2}", "22de5f19c5b9a3350a930bd1769b8a78": "[\\hbar ]=M^{1}L^{2}T^{-1}\\ ", "22dee644ebbdcf03082ee5a318d5ef1a": "C_{Z}^{-1}", "22def76b0c5b26c0fe458372886e3b5a": "\\sum _{a{\\bmod {n}}}\\chi (a)=0\\ ", "22df3bfa159427841071fc5bde856d8d": "\\mu :{\\mathcal {R}}^{1}\\times \\Omega \\to \\mathbb {R} ", "22df64cfced466dc281f190c8b118999": "P(q|{\\vec {y}})={\\frac {1}{{\\sqrt {2\\pi }}\\sigma _{q}}}\\exp \\left\\lbrace -{\\frac {\\left[q-{\\bar {q}}({\\vec {y}})\\right]^{2}}{2\\sigma _{q}}}\\right\\rbrace ", "22dfafcd62c3550fd5a4a1a72b89cc19": "r={\\sqrt {\\frac {(-a+b+c)(a-b+c)(a+b-c)}{4(a+b+c)}}};", "22dffc78863472e96fc21f12a59ad781": "e/n", "22e01032d5fcaa51fd1709f479866b20": "S^{u}", "22e031cfce65a2cb553499b57e7ec119": "\\mathrm {1\\ Square\\ of\\ Land} =({\\frac {\\mathrm {77\\ acres} }{\\mathrm {3\\ Squares\\ of\\ Land} }})\\cdot 1=25.41\\ acres", "22e03bff9f647ce776eab446cf8e6905": "\\int _{-\\pi }^{\\omega }X(e^{i\\vartheta })d\\vartheta \\!", "22e0638796dbbb922223fef51aaaa483": "S={\\begin{bmatrix}T1&T2\\\\R(A)&\\\\&R(A)\\\\&W(A)\\\\&Com.\\\\R(A)&\\\\W(A)&\\\\Com.&\\end{bmatrix}}", "22e0920e1aff68d5f69d8eb76cf5a7b4": "\\rho ^{AB_{j}^{L}}", "22e0ac2f652a61a657ac17bf856c7c27": "n=1+2\\pi {\\frac {Nf(0)}{k^{2}}},", "22e0cff7e3d0aae59f6778f95c3fa552": "\\langle C,X^{*}\\rangle _{\\mathbb {S} ^{n}}=\\langle b,y^{*}\\rangle _{\\mathbb {R} ^{m}}.", "22e0e32d8bbb4f7340a07d4501a3b134": "\\log _{10}", "22e12092f9902162c1c6582e543c92f3": "q_{ult}={\\frac {2}{3}}c'N'_{c}+\\sigma '_{zD}N'_{q}+0.5\\gamma 'BN'_{\\gamma }\\ ", "22e192ed29440b29adf4c7ab579b8beb": "f^{*}(\\varphi )=\\varphi \\circ f\\,", "22e1d7251d0b26b9f07d387c23012d60": "\\eta .\\,", "22e1f708b0c0894783ee62418f70f5ea": "C={\\frac {1}{n}}R^{1/6}", "22e1f9a10a44975974b865c1a64f2eb7": "y(t)=h(t)=\\sum _{n=0}^{N-1}{\\rho _{n}e^{j\\phi _{n}}\\delta (t-\\tau _{n})}", "22e200fea1113dbe93603000cc7532b2": "{\\frac {\\partial (\\rho ,\\theta ,\\phi )}{\\partial (x,y,z)}}={\\begin{pmatrix}{\\frac {x}{\\rho }}&{\\frac {y}{\\rho }}&{\\frac {z}{\\rho }}\\\\{\\frac {xz}{\\rho ^{2}{\\sqrt {x^{2}+y^{2}}}}}&{\\frac {yz}{\\rho ^{2}{\\sqrt {x^{2}+y^{2}}}}}&-{\\frac {\\sqrt {x^{2}+y^{2}}}{\\rho ^{2}}}\\\\{\\frac {-y}{x^{2}+y^{2}}}&{\\frac {x}{x^{2}+y^{2}}}&0\\\\\\end{pmatrix}}", "22e2539e7481867fecb346fa2d0d069b": "P(n)=\\int _{N=n}^{N=\\infty }P(n\\mid N)P(N)\\,dN=\\int _{n}^{\\infty }{\\frac {k}{N^{2}}}\\,dN", "22e2744501d61a9e6d970545cf6564f7": "f(x_{1})=-0.931596", "22e286e3641bc6ec4931d833727b2e10": "P_{\\mathbf {k} }=P_{\\mathbf {k} }P_{\\mathbf {k} }=P_{\\mathbf {k} }P_{\\mathbf {k} }P_{\\mathbf {k} }=...", "22e28ce56139a3f95d4451b40101eba1": "ST_{i+1}", "22e32525978de2afcb523c07dc5392cf": "y(a)=A", "22e34fbd2ad7b4e2a51eb56bb09fbe73": "{\\mathcal {P}}={\\mathfrak {P}}({\\mathfrak {C}}({\\mathcal {Z}})).", "22e3e0b04c69568c816aee6360c6556a": "{\\hat {\\beta }}_{FE}", "22e4725b1acb9670542fdb7c895dcd7b": "i=1,2,\\ldots ,r", "22e47b98a0c45aa63d843362b64b9542": "\\!{\\mathcal {A}}\\models _{X}\\phi ", "22e489002c01094d900991d247371f2e": "r_{1},r_{2}\\in \\{pino,exo,phago\\}", "22e491b93131c5b703d9f91a45f37aed": "f_{2}(z)={\\frac {i}{\\sqrt {3}}}z+\\lambda ", "22e4e3bb019bbf06a2a17e7c7e70c1ee": "F_{i}={\\frac {1}{2}}\\times \\rho \\times S\\times C_{i}\\times V^{2}", "22e570efeb540a531b7e0db25305935f": "H(x)=H_{0}(x)+xf_{0}", "22e5a6f72a75b47314dd93e2d36e433c": "R_{j}=\\sum _{i=1}^{b}R(X_{ij})", "22e5cf28a04f45dee59f0181a1d9d8a2": "(S,O,P)", "22e61eb97def5e6b43a38244a4cb761e": "{\\begin{aligned}y&={\\frac {E}{(\\gamma )_{-\\gamma }}}\\sum _{r=1-\\gamma }^{\\infty }{\\frac {(\\alpha )_{r}(\\beta )_{r}}{(1)_{r}(1)_{r+\\gamma -1}}}x^{r}+Fx^{1-\\gamma }\\sum _{r=0}^{\\infty }{\\frac {(1-\\gamma )(\\alpha +1-\\gamma )_{r}(\\beta +1-\\gamma )_{r}}{(2-\\gamma )_{r}(1)_{r}}}{\\Biggl (}\\ln(x)+\\\\&\\qquad \\qquad +{\\frac {1}{1-\\gamma }}+\\sum _{k=0}^{r-1}\\left({\\frac {1}{\\alpha +k+1-\\gamma }}+{\\frac {1}{\\beta +k+1-\\gamma }}-{\\frac {1}{2+k-\\gamma }}-{\\frac {1}{1+k}}\\right){\\Biggr )}x^{r}.\\end{aligned}}", "22e6c3f0ee1f8905e041d6c21c92cc26": "g_{\\textrm {eff}}", "22e6d5746e6ab192c386725cf9ed26fc": "\\nabla ^{2}\\omega _{\\varphi }={\\frac {1}{r}}{\\frac {\\partial }{\\partial r}}\\left(r\\,{\\frac {\\partial \\omega _{\\varphi }}{\\partial r}}\\right)+{\\frac {\\partial ^{2}\\omega _{\\varphi }}{\\partial z^{2}}}-{\\frac {\\omega _{\\varphi }}{r^{2}}}=0.", "22e6fb33c18e1bf8c36d363cb929a188": "[{\\tfrac {\\mathrm {g} }{\\mathrm {m} ^{3}}}]", "22e790427895c1318b592dfb13a0415a": "E_{\\gamma }'", "22e798ea25efd6ba1b134cc5d658d0e9": "0\\times \\infty ", "22e7b39854553cc52020035ba6860ca5": "{\\frac {d^{2}Y}{dx^{2}}}={\\frac {1}{h^{2}}}\\left({2a_{2}+6a_{3}z}\\right)", "22e80712e1ef6ee60d30f2781c99625f": "\\mathbf {B} =\\nabla \\times \\mathbf {A} ", "22e997671c8b8d0fc6e123375bd85df0": "\\sin \\delta '={\\frac {y_{1}'}{d\\,'}}={\\frac {y_{1}}{-c\\,t_{1}'}}={\\frac {y_{1}}{-\\gamma \\cdot (c\\,t_{1}-\\beta \\cdot x_{1})}}={\\frac {y_{1}}{-c\\,t_{1}\\cdot \\gamma \\cdot \\left(1-\\beta \\cdot {\\frac {x_{1}}{c\\,t_{1}}}\\right)}}={\\frac {\\frac {y_{1}}{-c\\,t_{1}}}{\\gamma \\cdot \\left(1+\\beta \\cdot {\\frac {x_{1}}{-c\\,t_{1}}}\\right)}}={\\frac {\\sin \\delta }{\\gamma \\cdot (1+\\beta \\cdot \\cos \\delta )}}", "22e999a52dc7430687a131e54ae312d2": "\\min |\\nabla ^{2}\\lambda _{s}|", "22e99b2bf4aa1f5139945247d9d691cf": "\\rho :{\\mathcal {L}}\\to \\mathbb {R} \\cup \\{+\\infty \\}", "22e9a1fd370ba9921a4acf82ac96f684": "\\liminf _{n\\to \\infty }J(u_{n})\\geq J(u_{0})", "22e9aa730e42e918d65058d10d04c5c9": "C^{\\infty }(M)\\to \\Omega ^{1}(M)", "22ea539b47a9cd3bd00b4f15fc100b60": "D=\\{(x,y)\\in \\mathbf {R} ^{2}\\ :\\ x\\geq 0,y\\leq 1,y\\geq x^{2}\\}", "22eaaf4d1af87378b86b3b3f68ad6230": "K_{s}={\\frac {G\\cdot T\\cdot d}{a}}", "22eb067ded0491350b556a4c3d47339c": "v_{t}={-\\operatorname {d} [M\\cdot ]/\\operatorname {d} t}=2k_{t}[M\\cdot ]^{2}", "22eb11bb1f6326c98bb47a97490c0b7a": "a^{2}-b^{2}=(a+b)(a-b)\\,\\!", "22eb193dd71821ac481fd5a7b3c35b28": "{\\mathit {b}}_{1}{\\mathit {b}}_{3}{\\mathit {b}}_{5}...{\\mathit {b}}_{2n-1}", "22eb33dabe332a8d56b2e8c59d922217": "i\\colon X\\hookrightarrow \\mathbf {R} ^{m}\\cong \\mathbf {R} ^{m}\\times \\left\\{(0,\\dots ,0)\\right\\}\\subset \\mathbf {R} ^{m}\\times \\mathbf {R} ^{N-m}\\cong \\mathbf {R} ^{N}.", "22eb3b7a8e4ca76647ba5ad73118f58b": "C_{r}=t", "22eb527a1eaa4ce9a4005f4ad53aefcb": "({256-Priority)}/{256}", "22eb5f099b94df3465627a6c6d43ac17": "U(\\mathbf {r} )\\propto {\\int _{\\text{Aperture}}A(\\mathbf {r'} )e^{-i\\mathbf {k} \\cdot (\\mathbf {r'} -\\mathbf {r} )}dr'}={\\int _{\\text{Aperture}}a_{0}(\\mathbf {r'} )e^{i\\mathbf {(k_{0}-k)} \\cdot (\\mathbf {r'} -\\mathbf {r} )}dr'}", "22eb666e418ec71e5684e65f968e1ca7": "{\\text{Corr}}_{r}(X,Y)={\\frac {E[XY]}{\\sqrt {EX^{2}\\cdot EY^{2}}}}.", "22ebcc0f00cf6301feaced0cfd484ea0": "G/(1+GH)", "22ebffaa068f8d9e635a7f1e485a6703": "J^{\\alpha }=\\rho _{0}U^{\\alpha }=\\rho {\\sqrt {1-{\\frac {u^{2}}{c^{2}}}}}U^{\\alpha }", "22ecaedf4651bf6d3fb6fa434968946b": "M'/IM'", "22ecb84d5c753f26be914c51a31ff86c": "f(x)={\\begin{cases}n&{\\text{if }}x\\in \\left[n,n+{\\frac {1}{n^{4}}}\\right],\\\\0&{\\text{else.}}\\end{cases}}", "22ece8703114e57f45d62da7bf13ff36": "H_{d}^{L}", "22ed815ec63bca38eee6db9b2ee0f6ec": "\\langle X,({\\mathbf {e} }\\cdot g)^{*}\\omega \\rangle =\\langle [d(\\mathbf {e} \\cdot g)](X),\\omega \\rangle ", "22ed847edd18a81b80c3740b474d419c": "\\displaystyle {\\Phi _{-}=\\partial _{z}D(\\varphi )|_{\\Omega }\\in A^{2}(\\Omega ),\\,\\,\\,\\Phi _{+}=\\partial _{z}D(\\varphi )|_{\\Omega ^{c}}\\in A^{2}(\\Omega ^{c}).}", "22ed949936c7631f7ba8bb3284e52b2d": "b_{8}=b_{9}", "22ee096edc2ef98efbed688e1ebf59f7": "MV_{i}", "22ee438dd5b1b796564a32eaa1945728": "(x^{2}-4\\alpha )y''+3xy'-n(n+2)y=0.\\,", "22ee57f531d782c4879a12d9d2f9e637": "{\\text{area}}(\\Delta )=v_{\\mathrm {interior} }+{v_{\\mathrm {boundary} } \\over 2}-1", "22ee87e2422ab33407db87ed858c7865": "{\\begin{matrix}{1 \\choose 1}{3 \\choose 1}{44 \\choose 1}\\end{matrix}}", "22eeba46963601c5e3d47f2f76a0e5c3": "52!/(13!)^{4}", "22eee2a7796e208e35c837e0f267fb9b": "{\\begin{bmatrix}{\\dfrac {a_{22}}{\\Delta \\mathbf {[a]} }}&{\\dfrac {-a_{12}}{\\Delta \\mathbf {[a]} }}\\\\{\\dfrac {-a_{21}}{\\Delta \\mathbf {[a]} }}&{\\dfrac {a_{11}}{\\Delta \\mathbf {[a]} }}\\end{bmatrix}}", "22eee3168be4b62b963096d4fcbbd920": "c_{i}u(x_{i})", "22ef00d9b773564414497cd5ed0b8f34": "G_{Newton}", "22ef086fb5099e38cce2c95badc3ca05": "\\mathrm {On} (\\mathrm {box} ,\\mathrm {table} )", "22ef0ab2ecfc80f99751536f12372062": "K_{\\text{joint}}={\\frac {\\Delta M}{\\Delta \\theta }}", "22ef3e19c49e6f387bd23164fbd94bff": "{\\mathbf {} }+P(t)C'(t)W^{-1}(t)C(t)P(t),", "22ef3fa14d2c529866b606a2ce8d0f2b": "P_{K}(p^{0},p^{1},q)={\\frac {C(f(q),p^{1})}{C(f(q),p^{0})}}", "22ef68e8e6c6ac5245c71f4a59132a76": "x_{n}=x_{n-1}-f(x_{n-1}){\\frac {x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}}={\\frac {x_{n-2}f(x_{n-1})-x_{n-1}f(x_{n-2})}{f(x_{n-1})-f(x_{n-2})}}", "22f1243f5a04fe18c6bdf57e74e423fa": "{\\begin{matrix}{48 \\choose 5}=1,712,304\\end{matrix}}", "22f125c2773a73ac8d13eda027963b48": "a_{1},\\ldots ,a_{n}", "22f18ffb924fac1950c89b1acb04bbba": "xu\\sim yv\\,", "22f19570a3c98d06252ea2d62ef7dd70": "{\\begin{matrix}V&=&C+i\\theta \\chi -i{\\overline {\\theta }}{\\overline {\\chi }}+{\\frac {i}{2}}\\theta ^{2}(M+iN)-{\\frac {i}{2}}{\\overline {\\theta ^{2}}}(M-iN)-\\theta \\sigma ^{\\mu }{\\overline {\\theta }}v_{\\mu }\\\\&&+i\\theta ^{2}{\\overline {\\theta }}\\left({\\overline {\\lambda }}+{\\frac {1}{2}}{\\overline {\\sigma }}^{\\mu }\\partial _{\\mu }\\chi \\right)-i{\\overline {\\theta }}^{2}\\theta \\left(\\lambda +{\\frac {i}{2}}\\sigma ^{\\mu }\\partial _{\\mu }{\\overline {\\chi }}\\right)+{\\frac {1}{2}}\\theta ^{2}{\\overline {\\theta }}^{2}\\left(D+{\\frac {1}{2}}\\Box C\\right)\\end{matrix}}", "22f199833017f92f64fb5db5064f58ef": "\\int x^{2}e^{cx}\\;\\mathrm {d} x=e^{cx}\\left({\\frac {x^{2}}{c}}-{\\frac {2x}{c^{2}}}+{\\frac {2}{c^{3}}}\\right)", "22f1acd1a4fbf71dc20b7ec840ba10a1": "B_{5}", "22f26060bb20519f5b0e5903562ad2e0": "\\scriptstyle D(E)", "22f264899eb897c3c2b86c1841d2748b": "h_{V}(F)", "22f288dff7f7fd02841a6f9faf866615": "{\\sqrt {re^{i\\theta }}}\\,=\\,{\\sqrt {r}}\\,e^{i\\theta /2}", "22f315c35d1b1baae66ac8ea06b4cb62": "H=-{\\boldsymbol {\\sigma _{1}}}\\cdot {\\mathbf {B}}-{\\boldsymbol {\\sigma _{2}}}\\cdot {\\mathbf {B}}-{\\boldsymbol {\\sigma _{1}}}\\cdot {\\boldsymbol {\\sigma _{2}}}", "22f36059ec66f6fccbbf213adcf799ce": "M\\setminus X=(M^{\\ast }/x)^{\\ast }", "22f36e0df2b5695fd40510ba95e2ff11": "i\\in [0,n]", "22f401e22c11f23ecc63c154e4242f8f": "{\\frac {1}{c}}{\\sqrt {(hf-hf'+m_{e}c^{2})^{2}-m_{e}^{2}c^{4}}}>{\\frac {hf-hf'}{c}}.", "22f44c8d97bb37e28b0fd9862e733667": "b=|S^{*}|", "22f45b29e686e354046350faedc38e23": "\\sum _{a=1}^{f}\\chi (a){\\frac {te^{at}}{e^{ft}-1}}=\\sum _{k=0}^{\\infty }B_{k,\\chi }{\\frac {t^{k}}{k!}}.", "22f46facc109a4f629ea7810679b6d37": "{\\begin{aligned}&f({\\boldsymbol {x}})=\\sum _{|\\alpha |\\leq k}{\\frac {D^{\\alpha }f({\\boldsymbol {a}})}{\\alpha !}}({\\boldsymbol {x}}-{\\boldsymbol {a}})^{\\alpha }+\\sum _{|\\beta |=k+1}R_{\\beta }({\\boldsymbol {x}})({\\boldsymbol {x}}-{\\boldsymbol {a}})^{\\beta },\\\\&R_{\\beta }({\\boldsymbol {x}})={\\frac {|\\beta |}{\\beta !}}\\int _{0}^{1}(1-t)^{|\\beta |-1}D^{\\beta }f{\\big (}{\\boldsymbol {a}}+t({\\boldsymbol {x}}-{\\boldsymbol {a}}){\\big )}\\,dt.\\end{aligned}}", "22f4c43698aaffeb960a7431f56e7f2b": "\\gamma (t)=(t,f(t))", "22f5ad893189686c8162b5fe4a52f4a1": "\\left({\\begin{smallmatrix}1&-2\\\\-1&-2\\\\\\end{smallmatrix}}\\right)", "22f65ff3cfbd5e2581bde6550f97ed04": "\\theta _{2}=\\theta _{4}\\;", "22f6672d27ebd439ab873803bd5eb74f": "x\\div 2", "22f6e8f374dba554beb0f2253b3522d0": "K_{p}^{*}(s;\\theta ,\\lambda )={\\begin{cases}\\lambda \\kappa _{p}(\\theta )[(1+s/\\theta )^{\\alpha }-1]&\\quad p\\neq 1,2\\\\-\\lambda \\log(1+s/\\theta )&\\quad p=2\\\\\\lambda e^{\\theta }(e^{s}-1)&\\quad p=1\\end{cases}}", "22f6fa494752b44996195ef3e7460c7f": "\\cdots \\rightarrow \\mathrm {Tor} _{2}^{R}(M,B)\\rightarrow \\mathrm {Tor} _{1}^{R}(K,B)\\rightarrow \\mathrm {Tor} _{1}^{R}(L,B)\\rightarrow \\mathrm {Tor} _{1}^{R}(M,B)\\rightarrow K\\otimes B\\rightarrow L\\otimes B\\rightarrow M\\otimes B\\rightarrow 0", "22f705717a70e84ebedcab10d55f3a48": "\\omega _{1}=R_{1}", "22f80facb2dca684c48d3ecf5b5e830a": "\\Delta \\ R(n)", "22f852dd1a0d559fad4e7285d87dbbe9": "RQ=\\sin \\alpha \\sin \\beta \\,", "22f865b7424640766aa415fd04678c44": "{V^{2} \\over 2}\\left({g \\over g+2}\\right)", "22f890ed84298aa3ed42a31142816ef9": "x\\neq 0", "22f8c6d9223b2aec1ca4596fcea846fd": "{\\widehat {H}}={\\frac {{\\widehat {\\mathbf {p} }}\\cdot {\\widehat {\\mathbf {p} }}}{2m}}+V(\\mathbf {r} ,t)", "22f8da1d27bb6ded691a104bfe83278d": "r_{n}={\\sqrt {n\\lambda f+{\\frac {n^{2}\\lambda ^{2}}{4}}}}", "22f938662ef8f0191ea1e88788b80bea": "\\eta =\\zeta +f", "22f9aec3b1a0b58958335d25543ca153": "M'|_{\\sigma }\\models _{\\Sigma }\\varphi ", "22fa0023b1cce6f0cc59dcf34f49c978": "G^{(n)}:=[G^{(n-1)},G^{(n-1)}]", "22fa92a4f599d9d4e6e8f647af631f65": "B=T_{b}P_{b}^{'}+E_{b}\\,", "22fad7b1c3f32ec9d4b77f3c3e39a055": "V_{\\mathit {Au}}", "22fbb20a50589c272943586195928d7c": "{\\frac {-b+{\\sqrt {\\Delta }}}{2a}}\\quad {\\text{and}}\\quad {\\frac {-b-{\\sqrt {\\Delta }}}{2a}},", "22fbfe2dd949cade4ee4367d509a541c": "\\operatorname {Cov} (X_{i}Y_{i},X_{j}Y_{j})=\\langle X_{i}X_{j}\\rangle \\langle Y_{i}Y_{j}\\rangle =M_{ij}N_{ij}", "22fc0f000d103985b67453ee5ea6de9e": "P(A)\\geq P(t\\in \\cup _{k=1}^{\\infty }[U_{k},V_{k}])\\to 1\\quad {\\text{as}}\\quad t\\to \\infty ", "22fc2a9064fdf57fe13242c7953dbbf3": "\\zeta (a,a)={\\tfrac {1}{2}}{\\Big [}(\\zeta (a))^{2}-\\zeta (2a){\\Big ]}", "22fc3804bc17c2e60fc92057d0c8f681": "\\mathrm {E} (e_{t}e_{t}')=\\Omega \\,", "22fc5185530c55ddeade7b62048fdd85": "\\langle f,g\\rangle =\\int _{0}^{1}f(t){\\overline {g(t)}}w(t)\\,dt.", "22fc9da9357cf962536f136d35d1d0fe": "K:={\\mathsf {Quot}}(D)", "22fc9ee0ce97e758226989f934a0b2ce": "(u_{1},u_{2},e,v)", "22fcb20dddcf288e461ba1a0f6ee14ae": "I(\\cdot ,t):\\Omega \\rightarrow \\mathbb {R} ", "22fdbb398361c988e1426aa3b8003667": "(\\forall x\\,D(x)\\Rightarrow M(x))\\land \\neg (\\exists y\\,M(y)\\land B(y))\\Rightarrow \\neg (\\exists z\\,D(z)\\land B(z))", "22fdbbe8801aef3fc41b45b6e50ecc61": "\\mathbf {x} =\\mathbf {0} ", "22fdd2644056adc3082414529a73cb51": "F=\\sum _{i}X_{i}X_{i}^{T}", "22fe010f477e0c8fc8e0387ea420851c": "x^{n}(x^{3}-x-1)=-(x^{3}+x^{2}-1)", "22fe10ce60d8fdc529a8871cc21852ff": "\\exists {n}{\\in }\\mathbb {N} \\,{\\big (}Q(n)\\;\\!\\;\\!{\\wedge }\\;\\!\\;\\!P(n,n,25){\\big )}", "22fe172a4e348397ad14e33e28f3c42b": "\\sin {\\frac {\\pi }{10}}=\\sin 18^{\\circ }={\\frac {{\\sqrt {5}}-1}{4}}={\\frac {\\varphi -1}{2}}={\\frac {1}{2\\varphi }}", "22fe3cdddae451c56ce053bf01b5f677": "{Z_{0}}^{2}-Z_{0}\\delta Z={\\frac {\\delta Z}{\\delta Y}}", "22fe80e01877c307adcb1259a6be52c5": "{\\mathfrak {P}}^{79}", "22feba68f74656cc2728f2680f70653d": "\\Phi (\\mathbf {u} )=R_{l}(u)Y_{lm}", "22ff393a1512745b7e8047514e231915": "\\phi (p)", "22ff5fab92f1b68f3929d883c7ccc134": "r_{1}e_{1}+r_{2}e_{2}+\\cdots +r_{n}e_{n}=0_{M}", "22ff82baf2e3c2d7457ae7c44a243bcf": "\\mathrm {p} I={\\frac {\\mathrm {p} K_{1}+\\mathrm {p} K_{2}}{2}}", "22ff96923684c09029dd90763f969478": "\\scriptstyle f_{n}(x)={\\frac {1}{n}}\\sin(nx)", "22ff9ce6ad7f1d3fc56099e93bf0883b": "(1+n)^{x}=\\sum _{k=0}^{n}{n \\choose k}x^{k}=1+nx+{x \\choose 2}n^{2}+higher\\ powers\\ of\\ n", "22ffb27c9be319dbdbecd3f4e05fa7cb": "sk_{n}(K):=i^{*}i_{*}K.", "230012c6f99aca97efed4e1bcef70310": "{\\begin{aligned}\\operatorname {var} (X)&=\\operatorname {E} (\\operatorname {var} (X\\mid Y))+\\operatorname {var} (\\operatorname {E} (X\\mid Y))\\\\&={\\frac {1}{9}}\\operatorname {var} (X)+\\operatorname {var} \\left\\{{\\begin{matrix}1/6&{\\mbox{with probability}}\\ 1/2\\\\5/6&{\\mbox{with probability}}\\ 1/2\\end{matrix}}\\right\\}\\\\&={\\frac {1}{9}}\\operatorname {var} (X)+{\\frac {1}{9}}\\end{aligned}}", "23004d0620341a3d9071989768839a2c": "\\langle R(u,v)w,z\\rangle =\\langle \\mathrm {I} \\!\\mathrm {I} (u,z),\\mathrm {I} \\!\\mathrm {I} (v,w)\\rangle -\\langle \\mathrm {I} \\!\\mathrm {I} (u,w),\\mathrm {I} \\!\\mathrm {I} (v,z)\\rangle .", "230061e941713013a0db9fc929beba0e": "y^{k}(0)", "2301009e66e55e01b984dc8c6f5fbba0": "v_{i}=\\left\\{{\\begin{matrix}-({\\sqrt {5}}-1)/2&{\\mbox{with prob. }}({\\sqrt {5}}+1)/(2{\\sqrt {5}})\\\\({\\sqrt {5}}+1)/2&{\\mbox{with prob. }}({\\sqrt {5}}-1)/(2{\\sqrt {5}})\\end{matrix}}\\right.", "230177a498dcf374b3b53957fa8e05b1": "x\\mapsto \\lceil x\\rceil ", "23019b2c16f2bbf38ea3614b93e0cc7d": "\\operatorname {E} [X]=\\int _{0}^{\\infty }P(X\\geq x)\\;\\mathrm {d} x", "2301abbe24cd34957bd2da20069e26dd": "z_{pj}^{new}=z_{pj}^{old}+\\eta (x_{ij}^{p}-z_{pj}^{old})", "2301bb6eca121fd14c03eb848b9131e7": "{\\begin{pmatrix}z&\\pm z\\\\\\pm z&z\\end{pmatrix}}\\equiv z(1\\pm j)\\equiv z(1\\pm \\varepsilon )", "2301d21430babc6749805421a309085d": "{\\begin{bmatrix}-1&0\\\\1&0\\end{bmatrix}}:\\mathbf {b} ", "23021d7deda8dd985feea3361ddff3c5": "{\\begin{cases}{\\frac {b^{2}\\left(-(n-2){\\zeta (n-1)}^{2}+(n-1)\\zeta (n-2)\\zeta (n)\\right)}{(n-2){(n-1)}^{2}{\\zeta (n)}^{2}}}&{\\text{if}}\\ n>3\\\\{\\text{Indeterminate}}&{\\text{otherwise}}\\ \\end{cases}}", "23026a02c604089bf625625c47df7c4d": "{\\mathcal {L}}(\\theta |x)/{\\mathcal {L}}({\\hat {\\theta }}|x).", "2302867fe83a7e4e3e837097afcaa2d7": "\\lim _{x\\to a}f(x)\\,", "23029a3f3167126c932fae635bf27942": "{\\begin{aligned}\\mathbb {A} :\\ \\sum _{i\\in \\mathbb {A} }2^{i}&=n-k,\\\\\\mathbb {B} :\\ \\sum _{j\\in \\mathbb {B} }2^{j}&=\\left\\lfloor {\\dfrac {k-1}{2}}\\right\\rfloor ,\\\\\\end{aligned}}", "2302a1ebf61ffd94d4f2b54fbfc8fc0d": "E(x)=x^{i}+x^{k}=x^{k}\\cdot (x^{i-k}+1),\\;i>k", "2302aad96d1bb3751d6830750da33d80": "Hf:=p.v.{\\frac {1}{\\pi }}\\int _{-\\infty }^{\\infty }{\\frac {f(y)}{x-y}}dy", "2302ac9939a0aed0905c827aad7f6fc3": "C\\vee D", "2302f615a22721d010362b96785d9224": "{\\text{dBW}}=10\\log \\left({\\frac {\\text{power out}}{1\\,\\mathrm {W} }}\\right)", "23047fe13afc543e209a67d1cb8c234e": "\\delta {\\mathit {u}}=({{\\mathit {c}}_{v}})({\\delta {T}})", "230511aef09db5df73f8cee151aedd5d": "Y_{10}^{4}(\\theta ,\\varphi )={3 \\over 256}{\\sqrt {5005 \\over 2\\pi }}\\cdot e^{4i\\varphi }\\cdot \\sin ^{4}\\theta \\cdot (323\\cos ^{6}\\theta -255\\cos ^{4}\\theta +45\\cos ^{2}\\theta -1)", "230524ed32654bb0474613f749cd1160": "R_{\\text{C}}", "23054e4b7f917c799ae6e50126cdfe3a": "(\\mathbb {Z} /n\\mathbb {Z} )^{\\times }", "2305a1f45a161a61d14a96715fb63c2e": "{\\frac {dx}{ds}}={\\frac {1}{\\frac {ds}{dx}}}={\\frac {a}{\\sqrt {a^{2}+s^{2}}}}\\,", "2305c5f77d759d096f6baa1eb0e63f36": "x=10^{7}", "2305e2ea419df87e959238e900e1c0de": "M_{i}={\\frac {v_{i}}{a}}={\\frac {1}{a}}{\\frac {\\partial \\Phi }{\\partial x_{i}}}.", "2305e50866edb8b78d64ad4d722deda3": "\\ P_{t}", "2305e7121098d4e0453f0e966a5e2efb": "{\\frac {1}{1828}}", "23060848695a4e86616b04628c690052": "f(x)=a(x-x_{1})(x-x_{2})\\,\\!", "230683d7d3c44607b309d1398a9bfb3d": "k=\\alpha +\\beta ", "230690b6009633aef7f3952d13f5bbe5": "\\phi (1,0,\\alpha )", "2306afe17e5a963a6db8f0c5bfef940a": "x\\mapsto P(x)", "2307257f77f2f6f7b529dbd0b716bc8c": "2\\theta ", "230725c8a902fece069278555e44b20e": "f(E;\\beta )={\\frac {e^{-\\beta E}}{{\\mathcal {Z}}(\\beta )}}\\Omega (E),", "2307b3524d2b98ad095260b2a07d26dd": "-\\operatorname {dn} ^{2}(u)+m_{1}=-m\\;\\operatorname {cn} ^{2}(u)=m\\;\\operatorname {sn} ^{2}(u)-m", "2307c3b04fae9951bd06cb895aa5596d": "dist(a,F_{l})=min_{x:W^{\\prime }x=\\gamma }\\|x-a\\|_{F}^{2}=\\|W(W'W)^{-1}(W^{\\prime }x-\\gamma )\\|_{F}^{2}=\\|W^{\\prime }x-\\gamma \\|_{F}^{2}.", "2307d7a384963a0d4ba7d43c108a25ee": "SiO_{2}(s)+H_{2}(g)\\rightleftharpoons SiO(g)+H_{2}O(g)", "2307fd207601cbdd65c9657ac82d3af7": "E_{\\textrm {out}}=A\\,(E^{+}-E^{-})=A\\,(E_{\\textrm {i}}-E_{\\textrm {r}})", "23080dd8dabe6f5f3f1257f3afa1f431": "(\\nabla \\cdot {\\vec {E}}(x)-\\rho (x))|\\psi \\rangle =0.", "230870d60568d3489294d8a442c03a50": "\\alpha ={\\frac {648000}{\\pi }}{\\frac {D}{d}}", "23087ec4b16e9e22b7a74a9ee85cbf3f": "{\\overline {u}}(x,y)", "2308ce4093a9f53fb7c79dcc57b69421": "\\mathbf {J} _{u}", "2308f08f54c2aa0e02fa012e8ba23e69": "u(x)=\\int _{0}^{\\ell }f(s)G(x,s)\\,ds,", "230907278d4536f7774c88169d4cebfa": "X^{2}\\sim \\chi _{k}^{2}", "23098ff0969bf5f743a51dc144bb14b6": "{\\begin{bmatrix}a&b\\\\b&a\\end{bmatrix}}.", "2309dcac9d120bf6dace40b5da2f8657": "|P_{c}^{n}(0)|", "2309f5d7da7451c2ebb83f0492a53060": "\\mu _{1}={\\frac {2}{\\bar {x}}}\\left[1+{\\sqrt {1+2(c^{2}-1)}}\\right]^{-1}", "230a19c4f2a98b247a96bf2bad958849": "B_{0}", "230a2f63233afc6a69916e7b48655457": "\\beta _{i}={\\frac {cov(r_{i},r_{M})}{\\sigma ^{2}(r_{M})}}\\qquad (2)", "230a92030ee28292aecceec8b8e648c7": "{\\dot {x}}=n/R", "230a96b33c823b586f9a118fda5589a0": "{\\frac {1}{k}}+{\\frac {1}{k}}={\\frac {1}{k}}+{\\frac {1}{k+1}}+{\\frac {1}{k(k+1)}}.", "230aab57abce22285fb273b1b2f1b0a4": "t>0", "230ac4f533ceebb6a21f5cf31d45d8d0": "\\{(1\\otimes C)\\Delta (C)\\}", "230af5f8816bfde0a5fbfb0da04aca77": "l(E^{2})", "230afda05fea500e61a98b5bf6ed7d31": "\\,t_{i}", "230b05027bba27a1a7e0b39522c9fae7": "I_{\\mathrm {xy} }=I_{\\mathrm {yz} }=I_{\\mathrm {zx} }=0.\\,\\!", "230b0a094fed614a9837462f91313e9b": "\\!f(k;\\lambda )=\\Pr(X=k)={\\frac {\\lambda ^{k}e^{-\\lambda }}{k!}},", "230b0c186f7270d21df0a906be2e708a": "x_{\\mathrm {per} }(t)", "230bba38bf8502715eab8532e6161075": "\\psi :G\\to \\mathbb {C} ^{\\times }", "230c5be17836808745534341c2b013e4": "a=\\omega _{a}+\\ln(\\omega _{a})", "230c78e77f33a4c022acbfcf22deafe3": "t=0", "230c934da5d4b5b490ed13d934ce7dcf": "X_{t}=\\int _{0}^{t}Y_{s}\\,dW_{s}.", "230ca2b209963f5f766cdac6016305fb": "GDGTratio-1=\\left({\\tfrac {[GDGT-2]}{[GDGT-1]+[GDGT-2]+[GDGT-3]}}\\right)", "230cbd34cd6d77898ee190e915cae0af": "V_{m}^{2}=2{\\frac {C^{2}}{C^{2}-1}}{\\frac {P_{settl}-p_{m}}{\\rho }}\\approx 2{\\frac {\\Delta p}{\\rho }}", "230cf541b93aad9f46dcd2f8768b5697": "c=\\cos \\theta ", "230d00fbfda6756e68062a7308f3c74c": "K(p\\mid x)\\approx 0", "230d0baf751e4d052a3ca8c2fa716bc0": "{\\begin{matrix}{13 \\choose 4}{4 \\choose 2}\\end{matrix}}", "230d459bbe09ecc9b72c1649b749dfe9": "3f\\leq 2e", "230d862bc744aa2593ff6284d77fbb66": "\\eta _{a\\mu \\nu }\\eta _{a\\mu \\nu }=12\\ ,\\quad \\eta _{a\\mu \\nu }\\eta _{b\\mu \\nu }=4\\delta _{ab}\\ ,\\quad \\eta _{a\\mu \\rho }\\eta _{a\\mu \\sigma }=3\\delta _{\\rho \\sigma }\\ .", "230e0fa157fca83b162e2aa9ea804a1c": "x+x^{5}\\,", "230e7414692f4ac8812c78fe0021a2b4": "\\{\\mathbb {I} _{\\{x\\}}\\mid x\\in X\\}", "230f0c736c8c262e3b210008b4e2b84b": "{\\underline {\\underline {{\\mathsf {S}}_{k}}}}", "230f1a588f3ad7335bd21a06826c0a89": "\\mathbf {S} =n\\mathbf {U} ", "230f3ee5eeb9715975662d97dbe3834d": "3.04=y_{2}+{\\frac {q_{2}^{2}}{2gy_{2}^{2}}}", "230f86cde757defd574865449bd4cccb": "n=\\lim _{x\\rightarrow +\\infty }(f(x)-mx)=\\lim _{x\\rightarrow +\\infty }\\left({\\frac {2x^{2}+3x+1}{x}}-2x\\right)=3", "230fe39110734f96f8a1339f996e96c1": "{\\dot {n}}_{3}", "231014c4b811712d23ae0065656fa636": "d\\mathbf {S} =d\\mathbf {r} \\mathbf {\\times } d\\mathbf {l} ", "23107cbfd471282bfcd256c03b4c0a5f": "|\\mathbb {F} |^{m}", "2310b7efec383853fd5ebf4450fb32e4": "t_{1/2}={\\frac {\\ln(2)}{\\lambda _{e}}}", "2310bf4db2d4df990d3968cf24341b61": "A\\star B\\simeq \\Sigma (A\\wedge B)", "2310f4d1db9b79c9823ef2921fdfbe89": "\\vdots \\!\\,", "2310fcb903a834976a8cb874163cea36": "Term\\rightarrow Term\\,*\\,Factor\\,|\\,Factor", "231139902b600f0a2667f05e171178a5": "{\\begin{matrix}\\underbrace {3_{}^{3^{{}^{.\\,^{.\\,^{.\\,^{3}}}}}}} \\\\7{,}625{,}597{,}484{,}987{\\mbox{ multiplied copies of }}3\\end{matrix}}", "23114540a0d686317aec37435cc4fabe": "G=2\\pi /d.", "23115a683378406a49f2ecc06c3cb215": "\\lambda (I-P)=\\gamma \\,,", "23118956d0ef63833d33f87979517f00": "S=\\lim _{\\delta \\to 0}\\sum _{k=1}^{m}f(P_{k})\\,\\operatorname {m} \\,(C_{k})", "231193c9a440a576e96d31fcb7d3a840": "22x34x16.5cm^{3}", "23119775abd0f5e44d5d6d464dc9c5b5": "{\\sqrt {8}}", "2311bc09829951d345e0e2134f73d92a": "{\\underline {P}}(Cl_{t}^{\\leq })", "2312318a3d6f0523ddeb199d81b979c2": "g:U\\to V", "231235977c18c870b95ad93f2d891c19": "b_{r}=a\\cdot p^{n}", "2312387d03bbc97d0603638ba7aea397": "I_{z}=I_{x}+Ar^{2},", "23128b8cb7328e6e876d2b58cc568f80": "g^{xy}", "23128e728469b331b523c975833e6b26": "({\\boldsymbol {\\mu }},{\\boldsymbol {\\Lambda }})\\sim \\mathrm {NW} ({\\boldsymbol {\\mu }}_{0},\\lambda ,\\mathbf {W} ,\\nu )", "2312c4994315c260141cb9f1566ad9aa": "d(df)=d^{2}f=0", "2312e2593eb0306fbfc855ff56a3a3fc": "\\lnot (P\\lor Q)", "23135c83bec9892a858f5df36b570e15": "\\scriptstyle F_{n}", "23135fe7433de2cb77eb245c58b94009": "{\\hat {A}}", "2313b8be1c82dc40fc40b928a8ae5af4": "\\,\\,\\sigma _{ij}=-P\\,\\delta _{ij}+2\\mu {\\dot {\\varepsilon }}_{ij}+\\lambda {\\dot {\\varepsilon }}_{kk}\\delta _{ij}", "2313c6bd3548a7c09f99089c9d819e8c": "\\lambda _{1},\\dots ,\\lambda _{n}", "2313ec33c5a56d1926036bf6ccaf8496": "{\\begin{aligned}\\lambda ={\\frac {h}{\\sqrt {2mE}}},\\qquad \\lambda [{\\textrm {nm}}]={\\sqrt {\\frac {1.5}{E[{\\textrm {eV}}]}}}\\end{aligned}}", "23145d60e7030c6dcbe596d2b2bc3b46": "0_{K}\\,", "2314f3fd7a034e8881916792e583d3ad": "y_{1}=\\ell \\cos \\theta \\,", "2314f8332cc30a42b15c5c68c040b3c2": "\\ni _{X}", "231519a78c94ecac2754031288439eea": "{\\begin{bmatrix}A&B/2&D/2\\\\B/2&C&E/2\\\\D/2&E/2&F\\end{bmatrix}}", "2315b140e19d399613f103c678f54c43": "r_{1},\\ p_{2}", "2315cbb6356aacb11611c3589a61a27e": "S=-(n_{\\mathrm {s} }-n_{\\mathrm {\\bar {s}} }),", "2315d8baf06210e5c406fb036f8a36b7": "A_{1}~r~\\cos \\theta \\,", "2316258b9a321d49e079a1f92b7c220e": "R(t)={\\frac {R_{+}-gR_{-}}{1-g}}", "23165550808dc05ba62fd9793ba25b08": "\\lim _{N\\rightarrow \\infty }\\left(I+{\\frac {i}{\\hbar }}{\\frac {\\Delta \\mathbf {r} }{N}}\\cdot {\\widehat {\\mathbf {p} }}\\right)^{N}=\\exp \\left({\\frac {i}{\\hbar }}\\Delta \\mathbf {r} \\cdot {\\widehat {\\mathbf {p} }}\\right)={\\widehat {T}}(\\Delta \\mathbf {r} )", "2316b6ae497378cbbde1e9c6a3d62223": "(a_{1}+\\cdots +a_{n})p=a_{1}x_{1}+\\cdots +a_{n}x_{n}", "2316d6de164515ad7cc41181d980b13a": "[x,zy]=[x,y]\\cdot [x,z]^{y}", "23170ce210a776a1b748591e7bea019d": "\\ f_{c}", "231713988facdd83b119e7d2199f52bf": "c^{2}\\ =a^{2}+b^{2}-2ab\\cos(\\gamma )", "231757a21070724a0c33ceaecc78f31a": "a(\\sigma +\\tau )", "2317793a8de61ab32c0f17adff9ea8d4": "x,y", "2317978c7fba5d4b31c782cd418ac89d": "\\scriptstyle \\angle PCP'\\ =\\ 60^{\\circ }", "2317a1ccc9c873b1b1fe3ab54f532ccc": "x^{3}-tx^{2}+(t-3)x+1", "23180b39cb9ea45e15f5c4c5b297a328": "\\{\\mu _{\\ell }\\}_{1\\leq \\ell \\leq \\omega }", "23180d3bfa1b6e008cc678c31c92829c": "{\\frac {d^{2}\\eta }{d\\theta ^{2}}}+\\eta =\\eta J^{\\prime }(u_{0})+{\\frac {1}{2}}\\eta ^{2}J^{\\prime \\prime }(u_{0})+{\\frac {1}{6}}\\eta ^{3}J^{\\prime \\prime \\prime }(u_{0})+\\cdots ", "2318a1b670d8d9d3932392055f1b5815": "\\ v_{z}", "2318e923074452066aefd71555d84015": "\\left\\langle \\sum _{j=1}^{n}b_{j}K_{y_{j}},\\sum _{i=1}^{m}a_{i}K_{x_{i}}\\right\\rangle =\\sum _{i=1}^{m}\\sum _{j=1}^{n}{\\overline {a_{i}}}b_{j}K(y_{j},x_{i}).", "2318eee43508b4fc2c38bbfae4f83830": "\\left(c,t,y\\right)\\succsim \\left(d,q,y\\right)", "2318fd28280e98162d1997aef2c47266": "\\Phi (x,z,t)=\\beta x-\\gamma t+\\varphi (x-ct,z),", "231913f7b50c942bea5dcba554f7136e": "\\mathbb {G} (k,n)", "23195da5a5101c716bdf110938e63e3f": "\\forall \\epsilon >0\\,\\underbrace {\\forall x\\in \\mathbb {R} \\,\\,\\exists \\delta >0} \\,\\forall h\\in \\mathbb {R} \\,\\left(\\,|h|<\\delta \\,\\to \\,|f(x)-f(x+h)|<\\epsilon \\,\\right)", "23195e81cb3f118faed4391f01e55ef6": "\\mathbb {Z} ^{k}", "2319741eb6cbaadad952a3a5f8eaf937": "(\\hbar ^{2}j(j+1))", "23198942fb12e0346d3c3a0c7a568a29": "x_{n}=\\sum _{k}[a_{k}\\cos(2\\pi \\nu _{k}n)+b_{k}\\sin(2\\pi \\nu _{k}n)].", "231a5e59501c03138fc7b97fa1d64f99": "{\\dot {\\boldsymbol {U}}}\\cdot {\\boldsymbol {U}}^{-1}=\\left[{\\begin{array}{ccc}{\\dot {\\lambda }}_{X}/\\lambda _{X}\\\\&{\\dot {\\lambda }}_{Y}/\\lambda _{Y}\\\\&&{\\dot {\\lambda }}_{Z}/\\lambda _{Z}\\end{array}}\\right]=U^{-1}{\\dot {U}}", "231a7d9c3ebb0b43d27e1ef688dbf41a": "\\beta _{j}^{+}(r_{m}^{+}-r_{f})_{t}", "231aa2bff58513c68b5ae63332d4b5f6": "\\beta _{FB}={\\frac {1}{G_{\\infty }}}\\ ,", "231aad2be32ac2295b4e74ebf73435e8": "E_{g}(T)=E_{g}(0)-{\\frac {\\alpha T^{2}}{T+\\beta }}", "231abf4361067da96e3fa12983b3590f": "\\scriptstyle {\\mathrm {R} =_{\\mathrm {def} }\\{x\\,|\\,x\\notin x\\}}", "231ad25673de638082c402391d702ed5": "{\\begin{aligned}\\ln(n!)-{\\tfrac {1}{2}}\\ln(n)&={\\tfrac {1}{2}}\\ln(1)+\\ln(2)+\\ln(3)+\\cdots +\\ln(n-1)+{\\tfrac {1}{2}}\\ln(n)\\\\&=n\\ln(n)-n+1+\\sum _{k=2}^{m}{\\frac {(-1)^{k}B_{k}}{k(k-1)}}\\left({\\frac {1}{n^{k-1}}}-1\\right)+R_{m,n},\\end{aligned}}", "231b61a8f020db4bc50dfff33143ada9": "{\\frac {dy}{dx}}={\\frac {r'(\\varphi )\\sin \\varphi +r(\\varphi )\\cos \\varphi }{r'(\\varphi )\\cos \\varphi -r(\\varphi )\\sin \\varphi }}.", "231b8bbeb4fa45885c97d90d3d2637be": "t>0", "231ba258c315092ab9e8413697d3cf6b": "h(w;\\rho )=\\rho \\,\\exp(-\\rho \\,w)\\,.", "231be161dcd193963413b740ed6b93f3": "\\pi \\,Q=0\\,.", "231c01def816b7f8d44ef229646388a3": "{\\mathcal {N}}={\\frac {\\left\\langle \\mathrm {MFT} \\right|\\left.\\mathrm {RPA} \\right\\rangle }{\\left\\langle \\mathrm {MFT} \\right|\\left.\\mathrm {MFT} \\right\\rangle }}", "231c1815bcf79ba04aef591f3b115baf": "\\mathbf {Z} /n\\mathbf {Z} ", "231c7156745b50c7d196cbd4452c836b": "\\int \\!x^{-1}\\,dx=\\ln |x|+C,", "231caefb182582246d50c67eaebe9f85": "y^{2}=x^{3}+ax+b{\\pmod {N}}", "231cc60965f06073a43ef55ab620830f": "\\scriptstyle O({\\frac {y\\log ^{2}y}{\\log \\log y}})", "231cee4f7f919f94f548fd81679d9bbe": "{\\begin{bmatrix}\\sigma _{11}\\\\\\sigma _{22}\\\\\\sigma _{12}\\end{bmatrix}}={\\cfrac {E}{1-\\nu ^{2}}}{\\begin{bmatrix}1&\\nu &0\\\\\\nu &1&0\\\\0&0&1-\\nu \\end{bmatrix}}{\\begin{bmatrix}\\varepsilon _{11}\\\\\\varepsilon _{22}\\\\\\varepsilon _{12}\\end{bmatrix}}", "231cf45da6abaad8b308bc5789e33732": "y={\\frac {x-\\mu '}{|c|}}", "231d18b7b72dcf05cf206d6e3faa4746": "f(x)>-\\infty ", "231d1a2d41cf6dfdd4908db8e68db7a9": "{\\dot {\\alpha }},{\\dot {\\beta }},\\dots ", "231d368ae058d2a789709f907f5bd2b1": "d:=1-{\\frac {H_{M}-H(X)}{H_{M}}}={\\frac {H(X)}{H_{M}}}", "231d53659de6e3ce9aff3cfc3fe599e7": "E\\to {\\mbox{Pic}}^{0}(E).", "231d94a8c0bf34e69b4bb12314f0ef2e": "{\\frac {\\partial \\mathbf {y} }{\\partial \\mathbf {x} }}={\\begin{bmatrix}{\\frac {\\partial y_{1}}{\\partial x_{1}}}&{\\frac {\\partial y_{2}}{\\partial x_{1}}}&\\cdots &{\\frac {\\partial y_{m}}{\\partial x_{1}}}\\\\{\\frac {\\partial y_{1}}{\\partial x_{2}}}&{\\frac {\\partial y_{2}}{\\partial x_{2}}}&\\cdots &{\\frac {\\partial y_{m}}{\\partial x_{2}}}\\\\\\vdots &\\vdots &\\ddots &\\vdots \\\\{\\frac {\\partial y_{1}}{\\partial x_{n}}}&{\\frac {\\partial y_{2}}{\\partial x_{n}}}&\\cdots &{\\frac {\\partial y_{m}}{\\partial x_{n}}}\\\\\\end{bmatrix}}.", "231e1cafb9ca5b0db1ff4680a8ebf6c4": "{\\mbox{E}}(x)=(-x\\mod {m})+1", "231e4262985d8d8725ee11abe226833d": "d_{2}\\sigma =2\\sigma /{\\sqrt {\\pi }}.", "231e70c2ef91a82daf1e552748f86400": "{\\bar {e}}^{ch}=\\left[{\\bar {g}}_{F}+\\left(a+{\\frac {b}{4}}-{\\frac {c}{2}}\\right){\\bar {g}}_{O_{2}}-a{\\bar {g}}_{CO_{2}}-\\,{\\frac {b}{2}}{\\bar {g}}_{H_{2}O(g)}\\right]\\,\\left(T_{0,}p_{0}\\right)+a{\\bar {e}}_{CO_{2}}^{ch}+\\,\\left({\\frac {b}{2}}\\right){\\bar {e}}_{H_{2}O(l)}^{ch}-\\,\\left(a+\\,{\\frac {b}{4}}\\right){\\bar {e}}_{O_{2}}^{ch}", "231e8e41e749b6b7c9d92b5162bdb26b": "U=TS-PV+\\sum _{i}\\mu _{i}N_{i}\\,", "231f33bb0acc36647826b75a3af1289a": "R_{0}^{0}=T_{0}^{0}-{\\frac {1}{2}}T,\\ R_{\\alpha }^{\\beta }=T_{\\alpha }^{\\beta }-{\\frac {1}{2}}\\delta _{\\alpha }^{\\beta }T,", "231f3e688aee8ae9a16497a902b4ed93": "q=\\int _{t_{1}}^{t_{2}}\\iint _{S}{\\mathbf {J}}\\cdot {\\mathbf {\\hat {n}}}{\\rm {d}}A{\\rm {d}}t", "231fec80038de71293915cb76e01f49e": "j", "2320809b6aad6116b4d965b8ebe426dc": "r^{\\ell }\\,{\\begin{pmatrix}Y_{\\ell m}\\\\Y_{\\ell -m}\\end{pmatrix}}=\\left[{\\frac {2\\ell +1}{4\\pi }}\\right]^{1/2}{\\bar {\\Pi }}_{\\ell }^{m}(z){\\begin{pmatrix}A_{m}\\\\B_{m}\\\\\\end{pmatrix}},\\qquad m>0.", "232095999b139073422402e7a7d99180": "d_{pore}", "2320b34c7e52ecd5fecf070c57c721d4": "A_{0}\\leq B\\leq A_{\\infty }", "2320f8341b90e8fc3d126f0d9589ce16": "\\sin \\alpha =\\cos U_{1}\\sin \\alpha _{1};\\,\\,\\,\\,\\cos ^{2}\\alpha =(1-\\sin \\alpha )(1+\\sin \\alpha )", "232106f2ffc5cb3ab802c5db61abfda6": "{\\Gamma (q+1) \\over s^{q+1}}", "23210a6f7a6269e7143a1479fb4d5bf9": "2\\;\\mathrm {H_{3}O^{+}} \\;+\\;2\\;\\mathrm {e^{-}} \\quad {\\overrightarrow {\\leftarrow }}\\quad \\mathrm {H_{2}} \\;+\\;2\\;\\mathrm {H_{2}O} ", "232154ce43270546a2f51a381610508f": "V_{C}=I_{C}X_{C}\\,\\!", "23219e49231e79f7a2038c40eacd101b": "C^{(k)}(f)\\leq {\\bigg \\lfloor }{n \\over k}{\\bigg \\rfloor }+1.", "2321a22c502ea202bfbb0899f9c6db8a": "\\mathbf {Q} _{M\\times 1}=\\mathbf {B} _{R}\\mathbf {R} _{N\\times 1}+\\mathbf {B} _{X}\\mathbf {X} _{I\\times 1}+\\mathbf {Q} _{v\\cdot M\\times 1}\\qquad \\qquad \\qquad \\mathrm {(5)} ", "2321d873510c23f952b659d98d72b3be": "{\\frac {1}{\\exp(\\psi (x))}}={\\frac {1}{x}}+{\\frac {1}{2\\cdot x^{2}}}+{\\frac {5}{4\\cdot 3!\\cdot x^{3}}}+{\\frac {3}{2\\cdot 4!\\cdot x^{4}}}+{\\frac {47}{48\\cdot 5!\\cdot x^{5}}}-{\\frac {5}{16\\cdot 6!\\cdot x^{6}}}+\\dots ", "2321e4a6beb8f7f8e2fc3db2cda337b3": "W\\sin \\theta =n\\lambda ,n=0,\\pm 1,\\pm 2,.....", "23221d115f2d2dfdb561ebb8d0381213": "\\rho ({\\vec {x}})", "2322b5d74b4c6531231b44907c63e19d": "\\textstyle f:\\mathbb {R} \\to \\mathbb {R} ", "2322bed50680b12610e1de3ed1bb6e49": "e^{1+4\\pi in-4\\pi ^{2}n^{2}}=e", "2322c35a6584d4de80833c06cbf5e660": "t^{p_{j}}\\Delta x^{j}", "2322e378cc10be5269f3aa394173f611": "L={\\frac {\\left[T_{0}\\right]}{\\left[R_{0}\\right]}}", "23231d7551563a696689ebcf26c3ad89": "R_{S}={\\frac {v_{Bullet}^{2}\\sin(2\\theta )}{g}}\\,(1-\\cot(\\theta )\\tan(\\alpha ))\\sec(\\alpha )\\,", "2323224adc9eba3042ff8bc091e13d5a": "A\\leq X", "23233b069fbf87cc459239ac7a9a3d12": "X\\times [0,1]", "23238f7945b0132ba8d0a626f430f231": "x^{\\rho }=x\\backslash e\\qquad xx^{\\rho }=e", "2323e6383dae6e770d2a04cfd26efb62": "u_{i}(s_{i}^{\\prime },s_{-i})\\geq u_{i}(s_{i},s_{-i})", "232401d163456315f4e43684c23a90ad": "p^{2}+2pq+q^{2}=1\\,", "232410dff222adec1a01b5622968330d": "{\\sqrt {1+u}}=(1+u)^{1/2}=1+{\\frac {u}{2}}-{\\frac {u^{2}}{8}}+\\cdots ", "2324874785f036ca1c0544ed044b93b9": "\\langle j_{1}\\,m_{1}|j_{2}\\,m_{2}\\rangle =\\delta _{j_{1},j_{2}}\\delta _{m_{1},m_{2}}.", "2324e37190effb1ac8ec120eb30e66c8": "{\\frac {\\partial }{\\partial {\\overline {z}}}}w(z)=0.", "2324e442204d6804a4f366eaf7c0bf61": "{\\frac {\\pi ^{2}}{16}}\\cong 0.61685.", "232541ce34c356ff4c4a89d3d9c02a54": "{\\mathcal {D}}_{L^{p}}", "23254e1da8500aa9f42f028a48c31059": "{\\hat {\\gamma }}_{ij}(t,x^{k})\\to \\gamma _{ij}(t,x^{k})", "2325c4299726b2882245ed19b4a729f9": "\\mathrm {^{249}_{\\ 98}Cf\\ +\\ _{1}^{2}D\\ \\longrightarrow \\ _{\\ 99}^{248}Es\\ +\\ 3\\ _{0}^{1}n\\quad (_{\\ 99}^{248}Es\\ {\\xrightarrow[{27\\ min}]{\\epsilon }}\\ _{\\ 98}^{248}Cf)} ", "2325e7f41bebcbc36375d5ad894c49df": "S^{n},n\\geq 1", "232610533a5221611a29fcab57a94949": "\\beta =\\nu Bd\\,", "2326261f97ce716dcaa54130a44800fd": "\\pi _{1}\\geq 2\\pi _{0}", "232626674bf22554c40ddae795431c9c": "M=B\\times F", "2326280d96e91557660e34dfae7d3741": "E_{7}^{\\mathbb {C} }", "2326293cdaefca46aa48a0b9e2019063": "{\\widehat {\\theta }}\\;=\\;{\\frac {\\log(1-P_{a})-\\log(1-P_{b})}{\\log(b)-\\log(a)}}.", "232644f3a5cdd6c6c291ef64e680bdc0": "J_{mB-sV}", "232679d7b0b4e10c1d7036944b9bb054": "A\\mathbf {x} =\\mathbf {b} ,", "2326c4ef49073348a7bfc3e4664dc43f": "C_{c}\\,", "2326f5a7d8d423517cc9e11687f759a4": "x,y\\mapsto \\Omega (A(x),y)", "232713107d94d28733937b2e8a4640d4": "dz=h(x,t)\\,dt+\\eta dv", "232752419ea9a788f263f491c73d5ebf": "f(r)=r^{2}-a\\equiv 0\\,{\\bmod {p}}", "23276174fbe6520075ed81f47fd75424": "f(x)=-\\int _{D}L_{X}f(y)\\,G(x,\\mathrm {d} y).", "232842acbb0ea982cb93ef74b5ab9bbd": "\\sum _{w\\in N^{(t)}(u)\\cup \\{u\\}}{\\frac {1}{d(w)+1}}\\leq (d'(u)+1){\\frac {1}{d'(u)+1}}=1", "23289f855511140daf15c53e505e12a7": "\\lim _{x\\rightarrow \\infty }\\pi (x)/\\operatorname {li} (x)=1\\!", "2328b9a25b41ea4f5a2cb4c7165b86ed": "\\Psi _{g}", "23293447e60df54b74844d7a2a8fea02": "\\bot ^{*}", "23293d8e1539bc098ed6697f3f787376": "m_{\\pi }^{2}=\\lambda m_{q}F", "2329b355331544ba5d62ec7c483fa962": "\\omega _{-}", "2329b6d5250a30af4fa11ec2cfd30405": "L(I,E)", "2329d83d3aabf668ed1dbfbdd68e1583": "[[en:Category:LittleEgypt", "232a239af70024857e58402597bddb62": "u_{1}=\\Re \\left\\{F(z)\\;{\\text{e}}^{-i\\,\\Omega \\,t}\\right\\},", "232a257a6a66bb71c16f041814f97f5e": "{\\frac {\\gamma N}{2}}", "232a5f881029483ce28d22701afc69d3": "R(z;\\tau )=\\sum _{\\nu \\in Z+{\\frac {1}{2}}}(-1)^{\\nu -{\\frac {1}{2}}}\\left({\\rm {sign}}(\\nu )-E\\left[\\left(\\nu +{\\frac {\\Im (z)}{y}}\\right){\\sqrt {2y}}\\right]\\right)e^{-2\\pi i\\nu z}q^{-{\\frac {1}{2}}\\nu ^{2}}", "232aacc2dea01263736e030917b4f7fd": "p\\in \\partial G", "232aee5d5ef373c4a399cf849c7af49e": "\\ {\\frac {dM^{2}}{M^{2}}}={\\frac {\\gamma M^{2}}{1-M^{2}}}\\left(1+{\\frac {\\gamma -1}{2}}M^{2}\\right){\\frac {4f}{D_{h}}}dx", "232ba1b1b52154ea1f7ada0cd59236c3": "F(x,y,z)={\\text{constant}}", "232bb26c60bdf9c1588703dfe41bea96": "\\rho \\rightarrow r^{-5}", "232c11ad705f64366c244a5c14d00d0f": "{\\cfrac {f_{i+1/2}^{n+1/2}-{\\cfrac {f_{i}^{n}+f_{i+1}^{n}}{2}}}{(1/2)*\\Delta t}}={\\cfrac {g_{i+1}^{n}-g_{i}^{n}}{\\Delta x}}.\\,", "232c15632fc5873535b7cec2ab1cc6a6": "X_{1}\\,\\!", "232cd3d624c8c047a1ed85c10f781d4e": "L_{pq}", "232d74eef31de47e4f6fc05ed827c066": "\\Gamma ={\\frac {Z_{2}/Z_{1}-1}{Z_{2}/Z_{1}+1}}", "232d799240386ad2cef225c9658cb41f": "|2b^{2}-a^{2}|\\geq 1", "232e25eac629a6caf56137617e048dc5": "\\left\\{{\\mathcal {B}}f\\right\\}(s)=\\left\\{{\\mathcal {M}}f(-\\ln x)\\right\\}(s)", "232e692208ba68e806e728df049b3426": "\\sum _{j}(h_{i}-\\lambda \\gamma _{j})\\delta O_{j}=0\\,\\!", "232eb7cc7a84426b9c8f9018f864488f": "-u''=f\\,", "232ec1e29d38b3e23c7069719e5a683d": "\\ I(r,d)={\\frac {1}{2}}\\eta I_{b}pS\\int \\limits _{0}^{d}\\ln \\left[1+\\left({\\frac {r}{h}}\\right)^{2}\\right]\\exp\\{\\alpha (d-h)\\}\\,dh", "232ee56e6a8db37f94fb4a1092584774": "\\lambda \\neq \\nu ", "232f3642e7d22258a083c3201f5d89ed": "{\\bar {x}}={Cm+\\sum _{i=1}^{n}{x_{i}} \\over C+n}", "232f86038a9523597db834a2e07d0d49": "a\\uparrow b", "232fe9c5351bbdd433e6372d99059bb5": "S={\\sqrt[{3}]{K_{sp} \\over 4}}", "232ff693e1d0c7801103fbcd774244f0": "R={\\frac {8\\eta \\Delta x}{\\pi r^{4}}}.", "2330121f5b60d4059618793cd46a36de": "J=J_{\\mathrm {S} }\\,e^{-{d/\\delta }}", "23304e6f4636f8928748ef37e228ac3a": "A_{1}\\cup A_{2}", "23305b935bc7ae09773e2fdda5e9e903": "\\nabla _{\\ell }A^{ik}={\\frac {\\partial A^{ik}}{\\partial x^{\\ell }}}+\\Gamma ^{i}{}_{m\\ell }A^{mk}+\\Gamma ^{k}{}_{m\\ell }A^{im},\\ ", "2330f832f570abb9f1e74b16c1db8dc2": "\\sum _{s\\ni e}x_{s}\\geq 1.", "2331169b1529cf60b773c4e32b559307": "\\lambda _{i}{\\frac {\\partial J}{\\partial \\lambda _{i}}}=\\lambda _{1}\\lambda _{2}\\lambda _{3}=J", "23317a24fe445dee548eb78d67331c37": "A\\land (B\\lor \\lnot B)", "2331ac7cad0ef331aad5a86b6fcd97ec": "[{\\mathcal {L}}_{X},d]=0", "2331b73e435e1e964b2a624977d6f02a": "R(n_{1},\\ldots ,n_{k})", "2332013279f13f9456d4c5f762874fee": "{\\hbox{ }}^{t}P_{i}\\rightarrow \\,^{t}S(R)^{-1}\\,^{t}P_{i}\\,^{t}S(R)=(CS(R)C^{-1})\\,^{t}P_{i}(CS(R)C^{-1})^{-1}", "23323329fc56bd5f967123243ce410c1": "{\\mathbf {e} }'_{i}={\\frac {\\partial }{\\partial {x'}^{i}}}={\\frac {\\partial x^{j}}{\\partial {x'}^{i}}}{\\frac {\\partial }{\\partial x^{j}}}={\\frac {\\partial x^{j}}{\\partial {x'}^{i}}}{\\mathbf {e} }_{j}", "233265f2755d63ebe4c53956ce590b57": "\\neg q", "2333045d4b886c7f7a0e88e015b50404": "\\oplus ,\\lnot ,", "2333143cc15e1f0f4a5c90dc5c1128ad": "\\pi _{A}=\\pi _{T}={(1-\\pi _{GC}) \\over 2}", "23332375d2d6411d73209f584ea0d1cb": "b={\\sqrt[{2}]{r_{a}\\cdot r_{p}}}", "23337dd98563f83f8f813676de53f783": "\\Lambda ^{1,0}M", "2333ad596f03a73f3ae4e7c3bcd7deb1": "\\beta (t)", "23341d52b446cf32ee4690f0ca940433": "i\\ll _{\\equiv }j", "233480c8487464adeb1ece918e1efa26": "g(a,b)={\\begin{cases}{\\frac {a^{3}}{a^{2}+b^{2}}}&{\\mbox{ if }}(a,b)\\neq (0,0)\\\\0&{\\mbox{ if }}(a,b)=(0,0).\\end{cases}}", "2334c8dbde2064f1233818452783f9e7": "{\\begin{matrix}l_{1,1}x_{1}&&&&&=&b_{1}\\\\l_{2,1}x_{1}&+&l_{2,2}x_{2}&&&=&b_{2}\\\\\\vdots &&\\vdots &\\ddots &&&\\vdots \\\\l_{m,1}x_{1}&+&l_{m,2}x_{2}&+\\dotsb +&l_{m,m}x_{m}&=&b_{m}\\\\\\end{matrix}}", "2334cd758d520a39a1145f8e772b88eb": "\\scriptstyle I'_{2}=-I_{2}", "233506ca9bbed051e0c7b94610c61272": "\\mu (t)=\\sigma (t)-t.", "2335460affc1028c6adcf25fd95bfa3f": "0\\leq \\{x\\}<1.\\;", "23356e38dd2b2a9abee9b9bba5194521": "\\gamma _{\\mathrm {SG} }=\\gamma _{\\mathrm {SL} }-\\gamma _{\\mathrm {LG} }\\cos \\theta _{\\,}", "23357f61f717162a44f9298b93024e63": "{\\hat {g}}={\\hat {a}}\\,", "2335b9effd79e7fd6f4d72d92fc70cdf": "F=m{\\frac {dv}{dt}}=ma,", "2335c44ebf82aafa079bfa8d44ad5a9d": "\\inf _{x\\in X}[F(x,0)]-\\sup _{y^{*}\\in Y^{*}}[-F^{*}(0,y^{*})]", "23360ad55340de0bf869970c25c30920": "={1 \\over p+1}T\\left(\\sum _{j=0}^{p}{p+1 \\choose j}b^{j}n^{p+1-j}\\right)=T\\left({(b+n)^{p+1}-b^{p+1} \\over p+1}\\right).", "23362b6a9cf8baeda38a0c42f5546aa9": "\\mathbf {J} ={\\frac {\\nabla \\times \\mathbf {B} }{\\mu _{0}}}", "2336381d9797c76cd95c41cd1058dbda": "{\\tfrac {1}{4}}x+{\\tfrac {1}{12}}y=1", "2336428ac0c3fd74dfb8a24194470967": "H_{\\alpha }^{(2)}(x)={\\frac {J_{-\\alpha }(x)-e^{\\alpha \\pi i}J_{\\alpha }(x)}{-i\\sin(\\alpha \\pi )}}.", "2336615428eab2afccc2ac25e8f6fa82": "\\mathrm {If} \\;X,Y\\in \\mathbf {L} ,\\;\\mathrm {then} \\;\\rho (X+Y)\\leq \\rho (X)+\\rho (Y).", "23366b866af9998f0e0f9d71e384afd8": "|n,\\pm \\rangle ~", "2336b323371eeb24c77850ebdd813597": "\\ x^{2}-92y^{2}=1", "233724c5adf28da47784390134db3c66": "LP", "233739a45217375ed25dff7f40981967": "{\\text{maximum speedup }}\\leq {\\frac {2}{1+0.25\\cdot (2-1)}}=1.60", "233748e19aa603307b32a1ecb5f5c5e3": "\\ \\Delta S", "233751764b03943325d8c8105c80103e": "E=1-\\left({\\frac {\\alpha }{P}}+{\\frac {1-\\alpha }{R}}\\right)^{-1}", "2337afb449890b668b909a691aff05ce": "\\epsilon \\left|{\\frac {dQ}{dx}}\\right|\\ll Q^{2},", "2337c4458facd9d85d41861fe6cfa24f": "G_{ij}=\\int _{t_{0}}^{t_{f}}\\ell _{i}(\\tau ){\\bar {\\ell _{j}}}(\\tau )\\,d\\tau .", "2337f8f5bc8c054982e4095745934447": "{\\vec {J}}={\\frac {\\vec {E}}{\\rho }}", "23388d3e952d73c823bd73d2e36c193f": "c_{i}(0)", "2338b165cb0c6d8e81795427c696178a": "\\mathbf {A} ^{-1}={\\begin{bmatrix}a&b\\\\c&d\\\\\\end{bmatrix}}^{-1}={\\frac {1}{\\det(\\mathbf {A} )}}{\\begin{bmatrix}\\,\\,\\,d&\\!\\!-b\\\\-c&\\,a\\\\\\end{bmatrix}}={\\frac {1}{ad-bc}}{\\begin{bmatrix}\\,\\,\\,d&\\!\\!-b\\\\-c&\\,a\\\\\\end{bmatrix}}.", "2338b4622b5f09bb0224354019972fe8": "P_{A}=A(A^{\\mathrm {T} }A)^{+}A^{\\mathrm {T} }", "2338bde451058e64de2ffb3e0f6ffaed": "x=(x_{1},\\dots ,x_{k})\\in F^{k}", "2338cf290c8adf956d23f809731161a3": "\\mathbf {a} =\\mathbf {E} _{\\text{g}}+\\mathbf {v} \\times \\mathbf {B} _{\\text{g}}-{\\frac {(\\mathbf {E} _{\\text{g}}\\cdot \\mathbf {v} )\\mathbf {v} }{c^{2}}}\\,,", "23390b320c5e0587c6c69e017578349c": "q<1", "23390e8880564338ee3d0000189447bc": "\\approx 2.3632718012073547031\\,,", "23394a6d0cacaaf92fd60c71c6ec6684": "P_{4}=1", "23399feab2e14145b433823229fc2f47": "b(z)", "2339a20c078567400faaed5e64ca3862": "H={\\dot {x}}_{1}p_{1}+{\\dot {x}}_{2}p_{2}-L", "2339af4300318a95889ce984b5e0da2c": "v={\\text{constant}}", "233a45460285415f4db07c8cd4db0a64": "\\sum _{j}n_{j}(\\mu _{j}^{\\ominus }+RT\\ln a_{j})=\\sum _{k}m_{k}(\\mu _{k}^{\\ominus }+RT\\ln a_{k})", "233a4b9703803a852db5d6e79a74712f": "M^{k}=f^{-1}(1,0,\\dots ,0)\\subset S^{n+k}", "233a9ef42f2d3a8ad026c7e6e0b8abf4": "\\square _{x}={\\frac {\\partial ^{2}}{\\partial t^{2}}}-\\nabla ^{2}", "233b30a7af1fd107aa48e7c51002ef25": "{\\boldsymbol {\\sigma }}={\\cfrac {1}{J}}~{\\boldsymbol {F}}\\cdot {\\boldsymbol {S}}\\cdot {\\boldsymbol {F}}^{T}={\\cfrac {2}{J}}~{\\boldsymbol {F}}\\cdot {\\cfrac {\\partial W}{\\partial {\\boldsymbol {C}}}}\\cdot {\\boldsymbol {F}}^{T}", "233b34e613ad045674818ff0463b7bfe": "[n=2^{m},n-1-\\lceil {d-2}/2\\rceil m,d]_{2}", "233b7351292b6bdf37f49680619f2abd": "w^{0}\\in \\mathbb {R} ^{d}", "233b93e9880422468f3de127f5d35650": "\\Pr(W=n)={\\frac {k}{n}}{\\frac {e^{-\\mu n}(\\mu n)^{n-k}}{(n-k)!}}", "233ba484d99cff1c9c5bf1f5c2d3f1ac": "Q={\\begin{pmatrix}-\\lambda &\\lambda \\\\\\mu &-(\\mu +\\lambda )&\\lambda \\\\&\\mu &-(\\mu +\\lambda )&\\lambda \\\\&&\\mu &-(\\mu +\\lambda )&\\lambda &\\\\&&&&\\ddots \\end{pmatrix}}", "233c0ffc8923672a3fbb1b236ff4b2bf": "R(t)", "233cb03365463a02b3dc2e27888e1c34": "m'_{0},m'_{1}", "233cd4d0d439b88f954ec5c1386eb31d": "\\textstyle {{n \\choose m}={n \\choose n-m}}", "233ce6c358743a7a3cf222891e69d780": "\\sum _{k=1}^{n}k^{p}={1 \\over p+1}\\sum _{j=0}^{p}{p+1 \\choose j}B_{j}n^{p+1-j}={1 \\over p+1}\\sum _{j=0}^{p}{p+1 \\choose j}T(b^{j})n^{p+1-j}", "233cf01fbd0ad3dfe5f6cad2df5ba448": "\\neg P\\rightarrow Q", "233cf5fa593c8789337a5f5acf10fe03": "=\\,\\!", "233cf870a12a7529768a7652736dd420": "Z=\\int _{p^{2}\\leq \\Lambda ^{2}}{\\mathcal {D}}\\phi \\exp \\left[-S_{\\Lambda }[\\phi ]\\right].", "233cf9693d17043e811ffd6b68a7e14c": "f_{\\mathrm {BE} }(E)={\\frac {1}{\\exp \\left({\\frac {E-\\mu }{k_{B}T}}\\right)-1}}", "233d12c1b1057ed9f9190079f9f81ff7": "\\Psi (\\omega )", "233d46fef141ec6815e57014337d4872": "FAD", "233de11b2feb965d4b6a724f67ff8412": "\\sin(\\arcsin x)=x\\quad {\\text{for}}\\quad |x|\\leq 1", "233de1dc6a21d2b5d5c78e19305b8de5": "\\theta _{a}=\\theta (x)|_{x=ja}\\,", "233e358a243eb6ea560c49ff206c227c": "kx^{\\prime 2}/z", "233e480656c9a5c6a42ab6ce57062b00": "\\tau _{ax}", "233e7af80710717c2f009f9fb5573a93": "z_{T}={\\frac {Z_{T}}{Z_{0}}}=0.35+j0.65\\,", "233ec80ab66c64260d74c1ab87efb94d": "G\\circ G':{\\mathcal {C}}\\to {\\mathcal {E}}.", "233ed2fb96370017307e35365abb1c5a": "R_{1}^{-1}Q_{1}^{-1}XQ_{1}R_{1}", "233ed40fba1ece94870ad675746ed120": "s(10011)=00111", "233f33c80532b1878f4ba200a82e6c88": "{\\begin{aligned}\\ln \\,\\operatorname {var_{G(1-X)}} &=\\operatorname {E} [(\\ln(1-X)-\\ln G_{(1-X)})^{2}]\\\\&=\\operatorname {E} [(\\ln(1-X)-\\operatorname {E} [\\ln(1-X)])^{2}]\\\\&=\\operatorname {E} [(\\ln(1-X))^{2}]-(\\operatorname {E} [\\ln(1-X)])^{2}\\\\&=\\operatorname {var} [\\ln(1-X)]\\\\&\\\\\\operatorname {var_{G(1-X)}} &=e^{\\operatorname {var} [\\ln(1-X)]}\\\\&\\\\\\ln \\,\\operatorname {cov_{G{X,(1-X)}}} &=\\operatorname {E} [(\\ln X-\\ln G_{X})(\\ln(1-X)-\\ln G_{(1-X)})]\\\\&=\\operatorname {E} [(\\ln X-\\operatorname {E} [\\ln X])(\\ln(1-X)-\\operatorname {E} [\\ln(1-X)])]\\\\&=\\operatorname {E} \\left[\\ln X\\ln(1-X)\\right]-\\operatorname {E} [\\ln X]\\operatorname {E} [\\ln(1-X)]\\\\&=\\operatorname {cov} [\\ln X,\\ln(1-X)]\\\\&\\\\\\operatorname {cov} _{G{X,(1-X)}}&=e^{\\operatorname {cov} [\\ln X,\\ln(1-X)]}\\end{aligned}}", "233f44fd7bc7220ba7e705f6430c2947": "D_{192}", "233f4ea7ce4ce7eeb467ecb1c7c7a65a": "Z_{4}^{2}", "233f509eb47fd05032c61682f1b722f2": "{\\mathfrak {sp}}_{n},", "233f7b2492df8fce12e24c77d47c0ff7": "h:S^{d+1}\\rightarrow S^{d+1}", "233fab408950ad3f1f4c8aa993c8a09a": "c\\equiv b^{e}\\equiv d^{\\left|e\\right|}{\\pmod {m}}", "233fb88eb9865a2ea95f18e5475c8190": "(p,q)\\mapsto qf(p/q)", "233fdc21188d5d2d0c7a751119ee20dd": "-ds^{2}=-(c\\,dt)^{2}+dx^{2}+dy^{2}+dz^{2},\\ ", "23406d7586dd6f3fab7f26cdb8bc56e5": "i{d\\psi _{n} \\over dt}=c^{*}\\psi _{n+1}+c\\psi _{n-1}", "23409caafc0a7acd1c1a5e1bfdc61fd3": "{\\begin{matrix}\\lim \\limits _{x\\to p}&(f(x)+g(x))&=&\\lim \\limits _{x\\to p}f(x)+\\lim \\limits _{x\\to p}g(x)\\\\\\lim \\limits _{x\\to p}&(f(x)-g(x))&=&\\lim \\limits _{x\\to p}f(x)-\\lim \\limits _{x\\to p}g(x)\\\\\\lim \\limits _{x\\to p}&(f(x)\\cdot g(x))&=&\\lim \\limits _{x\\to p}f(x)\\cdot \\lim \\limits _{x\\to p}g(x)\\\\\\lim \\limits _{x\\to p}&(f(x)/g(x))&=&{\\lim \\limits _{x\\to p}f(x)/\\lim \\limits _{x\\to p}g(x)}\\end{matrix}}", "2340ab547795f525e876ea2aa43afd35": "\\tau *_{c}", "2340c61cc12792eabc36261b46a564ee": "w^{1}=~w", "23410b34b4eed832e5dc2e74999989ce": "N({1 \\over 2}\\ln {{1+\\rho } \\over {1-\\rho }},{1 \\over n-3})", "23412e26eaf7dbe3d75bf54113c30660": "D_{\\nu }", "2341bef88c41a7ddd194710c806d18c3": "\\displaystyle {\\int _{\\partial \\Omega }gk=0,}", "2341ec116e8c14cf06d6242f8dba1fde": "P\\subset TM", "2342172b3413abe4d07f038e9b364d96": "c=2,\\phi =-20^{\\circ }", "2342592b6bf34e67ab5c919d09811d24": "k=7l+3", "2342b48312bd302535c591a695c604ce": "\\psi _{5,7}=1", "2342b4aef789ab838dc8e31f0e7bc6a3": "l_{l}\\,", "2342c04efb9091b1882f8d86de1ec83e": "q_{jk}={\\frac {\\sum _{i=1}^{N}w_{i}}{\\left(\\sum _{i=1}^{N}w_{i}\\right)^{2}-\\sum _{i=1}^{N}w_{i}^{2}}}\\sum _{i=1}^{N}w_{i}\\left(x_{ij}-{\\bar {x}}_{j}\\right)\\left(x_{ik}-{\\bar {x}}_{k}\\right).", "2342e2937aa9c7a2bede7aaf54f1bdf0": "L(P)=\\sum _{i=0}^{n}|a_{i}|.\\,", "2342f6a1e9a6ec09cb6666eb10c65cfa": "\\nabla _{\\mu }\\phi \\;", "2342f863ee74d3b15b6a9b6d441d0d15": "{\\bar {\\mathcal {M}}}", "23432b43afaed9c5e08413111781379d": "{\\begin{aligned}2\\cdot R_{*}&={\\frac {(91\\cdot 2.06\\cdot 10^{-3})\\ {\\text{AU}}}{0.0046491\\ {\\text{AU}}/R_{\\bigodot }}}\\\\&\\approx 40\\cdot R_{\\bigodot }\\end{aligned}}", "2343ae36a4aef6c9c78fd7893c6a1ff3": "\\scriptstyle n=9", "2344b5cdce19cc822937215901c0474d": "\\textstyle R_{k}=A_{k+1}A_{k}^{-1}", "23451b9fd03698800326e3e3d9528c48": "{\\boldsymbol {\\nabla }}P", "2345397ce9a108564892fc0ad8ad2e50": "F=1-{138 \\over 141.2}=0.023", "2345ad2ff3f824cf10dce7fea2dee3d6": "\\scriptstyle P\\left({{a}{|}{A,\\lambda }}\\right)", "2345b6bcd2bb6331a0e16a0d6e76c195": "H_{\\alpha }=D_{\\alpha }(-\\hbar ^{2}\\Delta )^{\\alpha /2}+V(\\mathbf {r} ),", "23467fa733d75a9d3a6368e6aef0d216": "\\mathbf {H} :={\\begin{pmatrix}1&1&0&1&1&0&0\\\\1&0&1&1&0&1&0\\\\0&1&1&1&0&0&1\\\\\\end{pmatrix}}_{3,7}.", "2346f1c4cf0c69cf0b3bb07ef2a32e2f": "H|\\Psi _{E}\\rangle =E|\\Psi _{E}\\rangle ", "234701287ef75a9fc06b34fed1c10647": "N=2^{40},~N=2^{42},~N=2^{44}", "23471ed40851377bdb5e5dc7e1d79465": "\\lim _{n\\to \\infty }F_{1,n}(z)", "2347290b3b5b09e01a1a948ea5856d2a": "|c_{n}|\\geq \\sum _{k=0}^{n}{\\frac {1}{n+1}}=1", "23473bb1c2488766e85dec0082631dcc": "|-3|\\geq |1|+|2|", "234747638eede29658077df51b44a1bd": "\\lambda (t)={\\frac {R(t_{1})-R(t_{2})}{(t_{2}-t_{1})\\cdot R(t_{1})}}={\\frac {R(t)-R(t+\\triangle t)}{\\triangle t\\cdot R(t)}}\\!", "23475d68badb023184833bdb56865946": "u=\\sum _{i=1}^{n}{\\frac {1}{k_{i}^{2}}}+2\\sum _{i=1}^{n}\\sum _{jED_{B}", "2348591ff3dca719242cee8fdb317635": "+S_{z}\\otimes I", "2348b63eaeb088d0664d5c8bf5e0e90f": "c_{3}=5.37941,\\,\\!", "23491e0f43a649c1d36334cec1b4e16a": "{\\frac {\\ln \\zeta (s)}{s}}=(1-e^{\\Theta (s)})^{-1}g(s)", "23493f0c83e580864617f5f0539440e5": "\\partial H/\\partial x_{3}=-\\partial L/\\partial x_{3}", "2349669b34f3919642db317737d5b83a": "{\\varepsilon _{0}}^{\\omega ^{\\omega }}", "2349a21aa47e05e5357b5477a9be0c28": "\\delta _{n+k}^{h}=O(h^{p+1})\\quad {\\mbox{as }}h\\to 0.", "2349baad5a455edaf15b953f1109edd0": "\\,\\varepsilon \\in (0,1)", "234a01199629059ae39994975f3fc672": "ID(x,y)", "234a30d800ce9fcb08c900c0af6d8c24": "u\\smile (v_{1}+v_{2})=u\\smile v_{1}+u\\smile v_{2}.", "234a7c2fc724e523fd7e5316bf668eae": "a-x_{1}=a_{0}", "234aa4d9049f32d3d45fb4f881b03de0": "f(t)+(p-r(t)){f'(t) \\over r'(t)}=t-i+{p-\\sinh(t)+i(1+p\\sinh(t)) \\over \\cosh(t)}=t-i+(p+i){1+i\\sinh(t) \\over \\cosh(t)}.", "234ab41f40d521af361c69e06036c78d": "{1 \\over {\\sqrt {7}}}", "234ac57e05424fa1be970a1e6fd3d12a": "[Gf](x)=\\sum _{n=1}^{\\infty }{\\frac {1}{(x+n)^{2}}}f\\left({\\frac {1}{x+n}}\\right).", "234afa2f2f61de3e3046b5fb45f1e80a": "EMA(m_{0},n)={\\frac {2}{n+1}}\\left[m_{0}-EMA(m_{1},n)\\right]+EMA(m_{1},n)", "234b184ecc674deea1dec0186a96087f": "\\mathbf {G} _{b}(u,v)=\\mathbf {G} _{a}(u,v)e^{-2\\pi i({\\frac {u\\Delta x}{M}}+{\\frac {v\\Delta y}{N}})}", "234b5a65a80b5eba5a6c669cc69793b5": "s\\in S,r_{1},r_{2}\\in R", "234ba6bcc30b4055ba58372d588bb60d": "={\\frac {2\\pi }{\\hbar }}Z_{DP}^{2}{\\frac {\\hbar \\omega _{q}}{2V\\rho c^{2}}}({\\frac {kT}{\\hbar \\omega _{q}}})\\sum _{k}\\delta _{k',k\\pm q}\\delta [E(k')-E(k)\\pm \\hbar \\omega _{q}]", "234baa95c6797c787cccadc28b777ed2": "Y_{5}^{-5}(\\theta ,\\varphi )={3 \\over 32}{\\sqrt {77 \\over \\pi }}\\cdot e^{-5i\\varphi }\\cdot \\sin ^{5}\\theta ", "234bb2baf02732d5dc6a5c99b2c63e64": "\\lambda _{3}^{2}=\\lambda _{2}", "234be622bf8a2780f3af13636d96cace": "{\\tilde {u}}(x)=\\prod _{i=1}^{L}x_{i}^{\\lambda _{i}}", "234c401016cac18ecf029eac4f77bccf": "\\sum _{i=1}^{m}r^{-\\ell _{i}}", "234c9253117921d5c6e9596f3d052e99": "|F(a)|\\leq \\|F_{a}\\|\\|F\\|=\\exp(|a|^{2}/2)\\|F\\|.", "234ca5cbf0da3ed2871f75588e6c5a61": "BQC", "234cd21913a2b229d4b6762cc206ca16": "p+2a^{2}", "234ce3a3a5f25164b277fd3c0358c6e4": "\\varphi (y)", "234d00f67b1e6f07c05b396f93052b63": "\\left(\\mathbb {Z} _{2}\\right)^{2n}=\\left\\{\\left(\\mathbf {z,x} \\right):\\mathbf {z} ,\\mathbf {x} \\in \\left(\\mathbb {Z} _{2}\\right)^{n}\\right\\}.", "234d045b51cb63e629f76bf64dfcf160": "{\\mathbb {Q}}(q)", "234d1483579df1cf3580be575c75b6a4": "\\mathbf {J_{r}} ={\\frac {\\partial r_{i}({\\boldsymbol {\\beta }}^{(s)})}{\\partial \\beta _{j}}}", "234d36aa7a269d91e4145eb15886a48a": "Q\\mathbf {x} =(\\alpha ,0,\\cdots ,0)^{T}.\\,", "234dd936588c26b061e493bd9893eb5c": "y=vt\\sin \\theta -{\\frac {1}{2}}gt^{2}", "234e24b907b6b4f01f5f92d85bdbf3e2": "fg(s)={\\begin{cases}f(s)&0\\leq s\\leq |f|\\\\g(s-|f|)&|f|\\leq s\\leq |f|+|g|\\end{cases}}", "234e3086c911173bf299e0acb1425e43": "p_{0},\\dots ,p_{s-1}", "234e499e88b87ec1734605deefcb49b9": "g\\colon S^{l}\\to X,\\,", "234e8a186be07ff7d9345a83c3db0dcc": "K=\\mathbf {C} ,V=\\mathbf {C} ^{n},", "234ed327f2b88768ab89e1d82ba6e4f8": "x\\asymp y", "234eed3258ea9f33d4e5e309ee30ecd5": "{k_{n}}", "234f1d51dadb82902864301f413e2a69": "M=M_{r}\\left[\\sin(k\\phi ){\\hat {\\rho }}-\\cos(k\\phi ){\\hat {\\phi }}\\right]", "234f3ca5f473d2bfc6fbec3010a3884a": "\\mu '_{2}=\\kappa _{2}+\\kappa _{1}^{2}\\,", "234f6d5069f37733fc2f0c34d3d279da": "a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2}-d_{1}d_{2}", "234f8093911ead8601518be7507bcbda": "\\eta _{Y}(y)(t)=y\\otimes t\\quad {\\text{for }}t\\in X.", "2350366b7c4ec7cc0dfe7223007a2385": "\\ell =\\ell ^{(1)}=\\log p(X^{(e)}={\\frac {1}{p}}X^{(m)}", "235041d5e2608942c2c6b197647f4b12": "IV=\\int _{0}^{t}\\sigma _{s}^{2}ds,", "235045dbfe14a1a9536141c7de5fbf21": "E=KV\\left(1-\\gamma ^{2}\\right)=KV\\sin ^{2}\\theta ,", "2350602ebf365f745426f31e86dbc8cd": "I\\approx {\\frac {nV_{T}}{R}}W\\left({\\frac {I_{S}R}{nV_{T}}}e^{V_{s}/(nV_{T})}\\right)", "2350748743ebdcc9885503739fa90e4c": "\\varphi _{1},\\ldots ,\\varphi _{p}", "23508d585f85400287f68b56a2588d40": "{\\frac {\\varphi ^{n}}{\\sqrt {5}}}\\,.", "2350a843c8f43af69be3f86a933d7b7c": "y(t+\\delta )", "2350c68d803e3cbaf529df86bc075ffe": "\\{92,19,\\mathbf {101} ,58,\\mathbf {101} ,91,26,78,10,13,\\mathbf {2} ,\\mathbf {101} ,86,85,15,89,89,25,\\mathbf {2} ,41\\}\\qquad (N=20)", "2350e285ff44ad0829ad9ffba1477913": "={\\frac {1}{\\rho }}x'(s)\\ .", "2350fcc56d715c9f0009672d0af44a87": "\\int _{c}^{d}f(x)\\,dx\\leq \\int _{a}^{b}f(x)\\,dx.", "23510dcc19d85fe5480876cd69bd24d9": "\\langle v,w\\rangle _{\\Phi }=(\\Phi (v))(w)=[\\Phi (v),w].\\,", "2351172b4e1c20a5423cfb36f18fa0c1": "C=\\alpha _{1}\\alpha _{2}\\beta _{1}+\\alpha _{1}\\alpha _{2}\\beta _{2}+\\alpha _{1}\\beta _{1}\\beta _{2}+\\alpha _{2}\\beta _{1}\\beta _{2}\\,", "235164d2cf3431287af070ce1cf2a341": "i_{\\mathrm {F} _{SO}(M)}^{\\ast }(T\\mathrm {F} M)=T\\mathrm {F} _{SO}(M)\\oplus {\\mathcal {M}}(\\mathrm {F} _{SO}(M))\\,,", "23518477a30967adff51b953abfc869d": "U=e^{iH\\Delta t}", "2351c247b6ca644c057234899c9ab831": "{\\begin{aligned}J_{Y}=\\sum \\limits _{l}\\sum \\limits _{u}{\\frac {{Y_{u}}^{2}}{r}}\\end{aligned}}", "2351e0244ac705b29b075f0da3cf1273": "k=2\\pi /\\lambda ", "2352042fcd550f1c30df3ad3a162ff9c": "\\theta _{1}(x)=x+1\\,", "235220b37b7ef719f379a94c2fd29b76": "j:A\\to A", "235223d6c51d418e4a2f025021fc5a27": "{(-g_{\\mu \\nu }{\\dot {x}}^{\\mu }{\\dot {x}}^{\\nu }){d \\over d\\tau }(g_{\\lambda \\nu }{\\dot {x}}^{\\nu }+g_{\\mu \\lambda }{\\dot {x}}^{\\mu })+{1 \\over 2}(g_{\\lambda \\nu }{\\dot {x}}^{\\nu }+g_{\\mu \\lambda }{\\dot {x}}^{\\mu }){d \\over d\\tau }(g_{\\mu \\nu }{\\dot {x}}^{\\mu }{\\dot {x}}^{\\nu }) \\over -g_{\\mu \\nu }{\\dot {x}}^{\\mu }{\\dot {x}}^{\\nu }}=g_{\\mu \\nu ,\\lambda }{\\dot {x}}^{\\mu }{\\dot {x}}^{\\nu }\\qquad \\qquad (5)", "235229cf71f83fb6d1cdd43e507d4756": "\\mathbb {R} ^{G}", "23522e6a4fd4e6a2396a9d78edcd943b": "\\textstyle \\{(1,\\alpha ),(0,1)\\}", "2352370de300e377d936eebeb6a19fca": "s_{\\lambda }=-{\\frac {1}{\\lambda }}\\sum _{i=1}^{m}\\log u_{i}.", "23523f57588abc942a605b3d8189349b": "=\\pi \\,r^{2}\\,h\\,", "2352440e134e788a22913b9413152c51": "k\\approx {\\frac {\\Phi }{\\tau }}\\left({\\frac {V}{S}}\\right)^{2}", "2352a1c6ea85d3da98e16dc1454014a7": "\\Delta f_{\\text{Lichte}}=-0.75C_{s}(u_{\\max }\\lambda )^{2},", "2352bc178fd5518eafe59b217cb70038": "\\phi ={\\frac {n_{\\rm {C_{2}H_{6}}}/n_{\\rm {O_{6}}}}{(n_{\\rm {C_{3}H_{1}2}}/n_{\\rm {O_{5}6}})_{st}}}={\\tfrac {1}{0.286}}=3.5", "2352df0abb89e8e75697e39058eb45c6": "F(x_{i})-F(x_{i-1})=F'(c_{i})(x_{i}-x_{i-1}).\\ ", "23530a5ed42f5b3e5c368d8552aa4047": "{\\textit {open}}(t)", "2353580fd98524d7ac4503f04bf0fb6e": "\\alpha _{T}", "2353a1995f42a9d31d92b156c912116b": "H(X|Y)\\leq H(X)\\,", "2353b33d05b46ebd075cb80fa0b85df2": "\\nabla \\times \\mathbf {B} ={\\frac {1}{c}}{\\frac {\\partial \\mathbf {E} }{\\partial t}}+{\\frac {1}{c}}\\mathbf {J} \\,", "2353e0b2c403513744a1b4a9f0b4e846": "{\\tfrac {223}{71}}", "2353fee5e449a2af9c07990fd016f096": "1\\leq i\\leq M,\\ 1\\leq j\\leq N,\\ 1\\leq k\\leq P", "23540abc37a11f15ce6f9ea51c0317cc": "a=(m^{2}-n^{2})\\,,\\ b=2mn\\,,\\ c=(m^{2}+n^{2})", "23541a8b0eb5625d82c596e27e8d9092": "{\\hat {\\beta }}_{i}", "23549fa4187a90ea50a65765eb129e0d": "e_{n}=(0,0,\\ldots ,1)\\,", "235516b7c7eeb9656ab251af3da5d9f0": "\\forall y(R(x,y)\\rightarrow ST_{y}(\\Box p))", "23553a3efed3e14076b7d2711c12c59f": "TE=PE+KE", "23553a42f3592ec43606472c3a0b69a4": "k={\\frac {2\\pi }{\\lambda }}={\\frac {2\\pi \\nu }{v_{\\mathrm {p} }}}={\\frac {\\omega }{v_{\\mathrm {p} }}}", "23555dbf85c1ea8daecbb06576281135": "{\\hat {\\sigma }}={\\sqrt {{\\frac {1}{n-1.5}}\\sum _{i=1}^{n}(x_{i}-{\\bar {x}})^{2}}}", "2355710c7695014f70afed018c833287": "{\\hat {z}}\\,", "2355c7d2486944f160451a088f0c0093": "a\\uparrow \\uparrow b", "2355cb8b5650d0dae2b9ed76d4fb816b": "c^{2}\\!\\left(1-{\\frac {3}{\\chi ^{2}}}+{\\frac {\\chi \\phi (\\chi )}{\\Psi (\\chi )}}\\right)-\\mu ^{2}", "2355cf1586fe371e5e9ba793b8f4cc1d": "\\forall X\\left[\\emptyset \\notin X\\implies \\exists f\\colon X\\rightarrow \\bigcup X\\quad \\forall A\\in X\\,(f(A)\\in A)\\right]\\,.", "2355df5ba6e2d5374a59e708dd274537": "\\,B_{n}\\equiv B_{n}(0)\\,", "23563cc5158ad32029640e4654069d46": "{\\begin{aligned}R(1)&=1~;\\ S(1)=2\\\\R(n)&=R(n-1)+S(n-1),\\quad n>1.\\end{aligned}}", "2356526b18d2bf2b63f7aae655b3e6eb": "p(Y_{i}|X_{i};\\theta )", "23566566f5c6e1bbe000952182c6a765": "S_{n+1}=4S_{n}\\left(8S_{n}+1\\right)", "235713ba0e6e557501874ebb6bd5ed98": "\\sum _{k=0}^{\\infty }{\\frac {z^{2k+1}}{2k+1}}=\\operatorname {arctanh} z,|z|<1\\,\\!", "23576f84cf7a8eb541fb1d375f83e843": "~c~", "235787cb80f6c57b68fcad12be72c73a": "\\subseteq [D\\rightarrow D^{'}]", "2357906cde9be9fbeeeddad94ead2195": "\\Psi {\\bar {\\Psi }}=1", "2357937b260fcb196b683fc955f0d85d": "\\operatorname {str} [M^{2}]=\\sum _{s}(-1)^{2s}(2s+1)\\operatorname {tr} [m_{s}^{2}].", "2357e0797ceb7b6442606d72127f77a6": "R=Y\\left({P-c}\\right)-Ytd", "2357eb40d94bb16a2924d28d49878ff4": "A,A\\to B\\vdash B.", "235857b6eea7dee4025536f1c3b70606": "M_{2}=2^{m}M+r_{2}", "235862adb1aee17962054e1f56979856": "U=cNT,", "23587ed20409fa2d2991cdb38c2e369a": "p_{G}(t),\\ p_{C}(t),\\ \\mathrm {and} \\ p_{T}(t)", "2358af47b78c047fc06818896464288e": "{\\frac {\\partial V}{\\partial p_{1}}}=-\\lambda x_{1}^{m}", "23592977b41d94ac274b7994df5f73f1": "V(x)\\approx {\\frac {1}{2}}x^{2}V^{(2)}(0)={\\frac {1}{2}}kx^{2}", "23594482424dd1ca0fa546997a50edcf": "3r_{s}", "235951f6537cab516ef388c374ecaf85": "H_{t}=c(t,Y_{t})+\\gamma (t,Y_{t})\\cdot {\\mbox{noise}}.", "2359530b0cfdb29b89ccad184e6a7f84": "\\kappa (z)=\\lim _{N\\rightarrow \\infty }{\\mathcal {Z}}(z)^{1/N}=1+z-3z^{2}+\\cdots ", "23595c5b094be7af9ee0cb827749cd8c": "{AE}_{8}", "2359b305d824ee6e76797bc5893cd2dc": "g^{(k+1)}\\in \\mathbb {R} ^{n}", "2359e542ad7475044ada953081f084c2": "f\\circ m=g\\circ m", "2359fb39d00cab953824a9b705ca394d": "\\tau \\in \\mathbb {H} ", "235a07e42d1a1df80cd847a0995ecf01": "f_{n}={\\frac {d\\Delta x}{4\\rho _{i}Lc}}{\\frac {V_{n-1}^{e}-2V_{n}^{e}+V_{n+1}^{e}}{\\Delta x^{2}}}", "235a317a7ea55294b48a6628e1a866d3": "r(\\alpha ,\\beta )={\\frac {\\sum _{i}(j_{i}^{\\alpha }-{\\bar {j^{\\alpha }}})(k_{i}^{\\beta }-{\\bar {k^{\\beta }}})}{{\\sqrt {\\sum _{i}(j_{i}^{\\alpha }-{\\bar {j^{\\alpha }}})^{2}}}{\\sqrt {\\sum _{i}(k_{i}^{\\beta }-{\\bar {k^{\\beta }}})^{2}}}}}.", "235aa52652b0ea705fcb82b459b713b4": "\\nabla \\cdot {\\mathbf {A} }+{\\frac {1}{c^{2}}}{\\frac {\\partial \\varphi }{\\partial t}}=0", "235ae6b195bb2da952f164bfab68f5af": "{\\hat {i}}", "235afc8e5e8f2df2733e9f963c692f9d": "\\sin {\\frac {\\pi }{3}}=\\sin 60^{\\circ }={\\tfrac {1}{2}}{\\sqrt {3}}\\,", "235b22b61b91378858950314fc1aeb24": "\\Gamma ^{\\alpha }{}_{\\beta \\gamma }\\,", "235b5d5ac9cc93b581a03a111099e3a6": "H_{max}=\\sum ({\\frac {x_{ij}}{X+Y}}log{\\frac {X+Y}{x_{ij}}}+{\\frac {x_{kj}}{X+Y}}log{\\frac {X+Y}{x_{kj}}})", "235b6050567deb4869e17c67a67d8fa0": "E_{bw}", "235c00252b00294922d25475ea14b751": "\\mathbb {D} ^{q}f", "235c1346c13200a614d3de059559eb3e": "V_{o}=NkT_{o}/P_{o}", "235c5146ab110558897640c34dad7d97": "\\textstyle j", "235c6c3907e3665a528ff5e2eef98828": "{\\begin{aligned}{\\mathcal {H}}&=\\mathbf {P} \\cdot {\\dot {\\mathbf {r} }}-L\\\\&={mc^{2} \\over {\\sqrt {1-\\left({\\frac {\\dot {\\mathbf {r} }}{c}}\\right)^{2}}}}+e\\phi \\\\&={\\sqrt {c^{2}(\\mathbf {P} -e\\mathbf {A} )^{2}+(mc^{2})^{2}}}+e\\phi \\end{aligned}}", "235c96b10c31fbc9850f675b3b40c1ac": "m=(n'-n)", "235cb7d36b84f7025dd7368a261c66c1": "\\displaystyle w(3,3)", "235ce462121ca365eeebb50a07dae474": "{\\frac {1}{\\beta }}\\log \\left(\\int _{\\mathbb {R} }|g(y)|^{2\\beta }\\,dy\\right)\\leq {\\frac {1}{2}}\\log {\\frac {(2\\alpha )^{1/\\alpha }}{(2\\beta )^{1/\\beta }}}+{\\frac {1}{\\alpha }}\\log \\left(\\int _{\\mathbb {R} }|f(x)|^{2\\alpha }\\,dx\\right).", "235cfa13c27f1cf0d8c4f73f9e26d6e1": "\\mathbf {\\bar {A}} _{l}|j\\rangle =g_{l}\\mathbf {\\tilde {U}} |j\\rangle ,\\quad \\forall {l}.", "235d06e215c331069f42afdf1e358a81": "Q(t_{2})=U\\left(t_{2}-t_{1}\\right)Q(t_{1})U^{-1}\\left(t_{2}-t_{1}\\right)", "235d4fc18a47f1c0af61502e8a77ab12": "{\\overline {\\mathcal {H}}}", "235d6de4f1342359794904a98d7b075a": "f(x)=-\\omega \\phi (x)+\\int K(x,y)\\phi (y)\\,dy", "235e1d04c4d21aaa94da65ab69723ee1": "{{MU^{L}} \\over {MU^{Y}}}={{dY} \\over {dL}}", "235e56112ae68064b3e62bb0568189de": "\\scriptstyle M_{V}=m_{V}-5\\log _{10}\\left({\\frac {100}{\\mathrm {parallax\\ in\\ milliarcseconds} }}\\right)", "235e775d3ea79387b5672b378cb376b8": "n_{e}=N\\nu _{A}-M-z", "235eabe6f46f60ca11d8531057325b48": "R_{2}={\\frac {R_{1}X_{1}^{2}}{R_{1}^{2}+X_{1}^{2}}}", "235ee020747b9898945419fb2013d380": "\\Omega =W^{T}MW", "235fc4f0cc15c1d797335d94366b5af3": "x_{1}=(1.786737601482363+1.786737578486707)/2=1.786737589984535", "235ffe7e77710e69a63177379eb46e70": "0.10266547\\ldots ", "23604a79b07ed670fbbe5ea69e85a21e": "\\mathrm {area} (g)-{\\tfrac {\\sqrt {3}}{2}}\\mathrm {sys} (g)^{2}\\geq 0.", "2360d41faa0467daa4123eb94e627830": "\\omega \\,=\\,{\\frac {2\\pi }{T}}\\,=\\,2\\pi \\,f.\\,", "23615a2a3cae58cfc4abf1efe72c9d70": "\\langle Pu,u\\rangle \\geq 0,\\forall u\\in V", "23617856b7344c2616ae5b177e91be6b": "\\square =\\langle {\\bar {\\partial }}\\partial \\rangle _{S}=\\langle \\partial {\\bar {\\partial }}\\rangle _{S}", "23619bdeaf491eae74c8c0b8ee2d96b0": "{\\pmod {\\mathfrak {p}}},", "2361b568349137ab9f6f9e9b755b60b8": "c=\\sum _{i=1}^{n}\\alpha _{i}\\beta _{i}.", "236202cab115a5c3f9f16a3fbc54d8fa": "cov\\left[u_{i},\\sum _{k=jN+1}^{(j+1)N}X_{k}\\right]=0\\quad {\\text{ for all }}i.", "23620a0d966ef0a180663e677f989219": "\\tau '_{ij}\\equiv \\rho \\,{\\overline {u'_{i}\\,u'_{j}}},\\,", "2362325508d65682119cd54a8300f765": "{\\frac {2\\pi ^{n/2}}{\\Gamma (n/2)}}", "23624eb2b836a2ebb457a2066df37a28": "\\textstyle {\\bigcap _{k=1}^{\\infty }I^{k}M=0.}", "2362f200efa80e8e069234c06e176812": "(p_{1}+\\cdots +p_{n})^{c}", "2362f28c5a8a2b22e2e10ac974286df0": "\\left({\\sqrt {1/45}},\\ -4/3,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0\\right)", "23631343b1457b33689f556b7fa29516": "\\varphi _{\\alpha }(\\beta )<\\varphi _{\\gamma }(\\delta )\\,", "23633e7e4f60dcc9b42883e83148f2de": "{\\tilde {\\mathsf {T}}}", "236385978fcc1f9676435034e18da0d8": "\\phi \\in (0,\\pi )", "2363cf525f777cda62557af32c64b075": "\\varnothing =\\mathrm {cl} (\\varnothing )", "2363d8d614e9d4e921a7554ce1f0146e": "\\Phi (-x)=1-\\Phi (x)", "23647e2b2afb00b78d0df9577ef46854": "\\int _{0}^{\\infty }t^{a}e^{-xt}\\,dt={\\frac {\\Gamma (a+1)}{x^{a+1}}}", "23648d3f1d8baedfc570e43faab317bb": "T_{\\text{cool}}", "236536186ff53fbf31de6fa822e385fa": "2^{2^{h-2}}", "23664b4fba104db53e7d3eaefdf74c2c": "{\\text{Gal}}({\\overline {\\mathbf {Q} }}_{p}/\\mathbf {Q} _{p})", "2366728f0a3f337be4645682c8984bc9": "I_{L1}+I_{L2}*\\cos {\\frac {2}{3}}\\pi +j*I_{L2}*\\sin {\\frac {2}{3}}\\pi +I_{L3}*\\cos {\\frac {4}{3}}\\pi +j*I_{L3}*\\sin {\\frac {4}{3}}\\pi ", "2366947f34186f36e13ae124aad86b09": "\\sigma _{k}=\\gamma _{k}\\gamma _{0}", "2366bd5446dc3b6a239076a083472e37": "{\\mathcal {E}}(u)=\\int (A\\nabla u,\\nabla u)\\;\\mathrm {d} x,", "2366d570e9dfccfa473963f9c6ed4c74": "{\\widehat {\\delta }}_{ab}=\\delta _{a}\\circ \\delta _{b}", "2366ea6eef15986d41d31b72266ef715": "\\tan {\\theta \\over 2}={\\frac {\\cos(\\pi /q)}{\\sin(\\pi /h)}}.", "2367104f17127148a69711afc06422dc": "e^{-b{\\mathrm {Log} [x]}^{2}}x^{-2-a}\\left(-{\\tfrac {2b}{a}}+\\left(1+a+2b\\mathrm {Log} [x]\\right)\\left(1+{\\tfrac {2b\\mathrm {Log} [x]}{a}}\\right)\\right)", "2367138ffe7e282e2a6ae2cd8685e4d8": "n_{z}^{e}", "236779514b8ad71053b6ca13af512fb3": "f(x)={\\frac {1}{2{\\sqrt {3}}}}\\quad {\\text{if}}\\quad |x|<{\\sqrt {3}}", "2367926ea7250ce863c566930ac3bd3d": "(\\mathbf {A} \\mathbf {B} )\\lambda =\\mathbf {A} (\\mathbf {B} \\lambda )", "2367dc82da759521a88eaf98587d8e45": "\\mathbf {p} =\\gamma (\\mathbf {u} )m_{0}\\mathbf {u} ", "236819c437d8b1cd0383b13798b2535b": "\\operatorname {build-param-lists} [g\\ q\\ p\\ n,D,V,K_{1}]", "23689dc4a677a22115dac83b43008b11": "{\\frac {\\partial A_{ij}^{-1}}{\\partial A_{kl}}}~T_{kl}=-A_{ik}^{-1}~T_{kl}~A_{lj}^{-1}\\implies {\\frac {\\partial A_{ij}^{-1}}{\\partial A_{kl}}}=-A_{ik}^{-1}~A_{lj}^{-1}", "23691b0a9421c76fd0952811483c3cf2": "\\mathrm {x} \\approx {\\sqrt {.20\\times (1.8\\times 10^{-9})}}=1.9\\times 10^{-5}", "2369665ca29a60404bd292319f39bd8c": "\\sum _{i=1}^{n}\\mathrm {NegativeBinomial} (n_{i},p)\\sim \\mathrm {NegativeBinomial} \\left(\\sum _{i=1}^{n}n_{i},p\\right)\\qquad 00", "23756c5030d08070ac91df03b7fdf5c7": "x\\in \\mathbb {R} ^{N}", "237575586a56b316e56ee84559734faf": "\\left({\\frac {p_{1}}{p_{2}}}\\right)^{\\frac {1}{\\gamma }}", "237579ab9744ae9f71b788e67b3124ac": "p_{1},p_{2}", "23758582a8255b75590b7cb4b6e2d6a0": "[01{\\overline {1}}2]", "23758dd51f91d093c8627e71bea89b1d": "(\\mathbf {F} ^{+})_{+}", "237591bc7e6ea76fe0d65161a4060cc0": "{\\frac {n-5\\pm \\Delta _{n}}{2-2n}}", "2375a5deb49069f214314679f307be3a": "\\alpha _{i}\\nmid \\alpha _{i+1}", "2375a71d49a71c8a4f4f53000318c51c": "A_{0}=|0\\rangle \\langle 0|+{\\sqrt {\\eta }}|1\\rangle \\langle 1|", "2375b0c83441ed229b84914eb93c9420": "B={\\begin{bmatrix}0&a&b\\\\-a&0&c\\\\-b&-c&0\\end{bmatrix}}.\\qquad \\operatorname {pf(B)} =0.", "2375ed3e6c9ca2eb8b716aecf32c52a3": "m{\\frac {\\partial ^{2}x(t)}{\\partial t^{2}}}", "2376435019508a9eefbf970a93e540e6": "P_{\\alpha }\\,", "2376946ea2c96e1271fe1c701383d7e6": "{\\hat {Z}}(x)", "23769a8d77dd49f082b6b3dfe5f0cc8d": "P\\lor Q", "2377355ca640f8a52026b876b620faa0": "{\\frac {t}{x}}={\\frac {S}{S-L}}\\Rightarrow S={\\frac {L}{1-x}}", "2377543493a57fd7b5022e6b2d2b447a": "s_{1}(T)\\geq s_{2}(T)\\geq \\cdots s_{n}(T)\\geq \\cdots \\geq 0", "23776b94da51114c679f9f0cb970350c": "\\{x_{1},\\ldots ,x_{k}\\},\\{x_{k+1},\\ldots ,x_{n}\\}.", "23778b1022fc8bb9bcd84fafc7159666": "T_{B}", "2377c945680222d8ac6084fc0501055d": "b\\in e_{2}", "2377d1b826c82fe2a508d2544d0960d8": "{\\tilde {\\psi }}(\\eta ,\\tau )={\\frac {1}{\\sqrt {2\\pi }}}\\int {\\psi (\\xi ,\\tau )e^{i\\eta \\xi }d\\xi }={\\sqrt {2\\pi }}e^{i\\tau }\\left[{\\frac {1+2i\\tau }{\\sqrt {1+4\\tau ^{2}}}}\\exp \\left(-{\\frac {|\\eta |}{2}}{\\sqrt {1+4\\tau ^{2}}}\\right)-\\delta (\\eta )\\right]", "2377e8b25b3edd80cd8d32c69588aae1": "{}^{\\perp }", "2377f2b7f6e882d65432133c1674bd3f": "J=\\sum _{i}|X_{i}-LM_{i}R^{T}|^{2}", "2378182c67dfb9efc306ea55080dbebc": "Z=\\Phi (\\beta )-\\Phi (\\alpha )", "23782db3097d425aad3a7de0bcfb39c6": "1=|\\psi _{R}|^{2}+|\\psi _{L}|^{2}.", "23785c83f96ce16d3ef39c928607b10a": "y'=x\\sin \\left(-\\Omega t\\right)+y\\cos \\left(-\\Omega t\\right)", "23786eb10c4d9583aa62e2be47214bff": "\\rho :G\\rightarrow \\ker \\,f", "237897ffeb8ab2371f1fb350cc3de54c": "\\ w_{r}>{\\frac {l_{e}}{l_{us}}}w_{u}", "2378df30463c6263f6c597191efd3b24": "\\sigma _{i}^{2}=h(z_{i}'\\gamma )", "2379f2d9fcb8804f8f4abb661817aeb2": "(a\\pm {\\sqrt {a^{2}-4}})/2", "2379ffdee680960ff908060d756e6db5": "\\Delta F=F(B)-F(A)", "237a2dfc9d74b20f86a58afa5f636cad": "\\operatorname {let-combine} [\\operatorname {let} p:\\operatorname {de-lambda} [p\\ f]=\\operatorname {let-combine} [\\operatorname {let} x:\\operatorname {de-lambda} [x=\\lambda x.f\\ (x\\ x)]\\operatorname {in} f\\ (x\\ x)]\\operatorname {in} p]", "237a3fc7110eaf54a6abda52d0700abe": "H_{x}=-\\int |\\psi (x)|^{2}\\ln(|\\psi (x)|^{2}\\cdot \\ell )\\,dx=-\\left\\langle \\ln(|\\psi (x)|^{2}\\cdot \\ell )\\right\\rangle ", "237a57b8d8eecd8d201e9608c89e4f02": "I\\otimes _{R}M\\to R\\otimes _{R}M\\cong M", "237af5f18d650c377d23ebd2837d6567": "R_{\\mu \\nu }^{\\lambda \\sigma }=e_{a}^{\\lambda }e_{b}^{\\sigma }R_{\\mu \\nu }^{ab}", "237b1e5150b595330a18876a348aec8a": "\\delta (A,0)=\\arg \\max _{\\delta \\in {NS}}U_{A}(\\delta )", "237b49ed1be171cae222e45e319c353a": "{\\frac {4}{\\tbinom {52}{13}}}=0.000000000629908\\%=1:158753389900", "237bbb45046cc6f091ce32c38f1394e5": "_{0}=-2.90", "237bc5f379ba03d325cbd7e68dc587d1": "\\mu ^{\\otimes n}(A_{n}(s,t))\\leq {\\frac {{\\bigl (}\\mu (I_{s,t}){\\bigr )}^{n}}{n!}}", "237be621b09d243f5efd72fdd9d8dc4c": "\\pi \\cong \\pi _{1}(M)\\cong \\pi _{1}(W)\\cong \\pi _{1}(N).", "237bf0471d8776da81fee9ce62a99972": "{\\begin{aligned}{\\frac {dH}{dt}}&=rH-cHP\\\\{\\frac {dP}{dt}}&=acHP-mP\\\\\\end{aligned}}", "237c0cd81f3b0024201d79f3d9ecd726": "c=1-\\zeta _{n}^{2}+x_{i}^{2}\\zeta _{n}^{2}/\\xi ^{2}", "237c6c591bfd76cb00fd29d266dc33e5": "{\\tfrac {DGxDF}{FH-DG}}", "237c8d751559b361bc60246dcf6a700e": "L(u\\otimes v)=D(N\\otimes I)(u\\otimes v)=D(Nu\\otimes v)=D(Nu\\otimes NNv)=", "237ccaf4172f5eee73b9914a166293ed": "a_{2}b_{2}", "237cd679ed94572b222ae88fe5d3027a": "V=\\{V_{\\gamma }|\\gamma <\\delta <\\beta \\}", "237d08720d6bd10135017a72d0fab0fb": "\\Omega ^{0}(M,E)=\\Gamma (E).\\,", "237d817de718a94e4e290a836db315e5": "u=1/r", "237dccd3133f35c013d7673ca2a706e8": "f(U)", "237dd051b49fa19fb73e8afe6f3beb07": "arg(z(z-a))=const.", "237e74255e59d6d0e534c2d26942fb38": "C(v)=\\int \\limits _{0}^{v}{}xf(x)dx", "237e8cbfd0e5837c9b0b05422147a950": "\\mu ({\\bar {S}}_{2t})\\,\\!", "237ea2504d2bb6bea643d21912ae9164": "{\\dfrac {\\mbox{b.hp.}}{{\\mbox{b.hp.}}+{\\mbox{avg. f.hp.}}}}\\times 100", "237f0134f94505e054a0e4afcfe5c2d7": "{\\rm {E}}[z]\\,\\,\\,=\\,\\,\\,\\mu _{z}\\,\\,\\approx \\,\\,\\,\\,a\\mu _{1}+\\,\\,b\\mu _{2}", "237f1ba24374434f8078fb3f372497d4": "R_{U}=\\mathrm {C} _{E}(\\mathrm {C} _{E}(R_{U}))\\,", "237f5daac464adafffb85d6975de4fc3": "h/b\\geq 1", "237f6174f1e520ad0dc3f11738a2b7a5": "o(dtdu)", "237f94b37825bf506d3a1dff916aeddb": "0=\\nabla _{\\vec {\\theta }}\\tau ={\\vec {\\theta }}-{\\vec {\\beta }}-\\nabla _{\\vec {\\theta }}\\psi ({\\vec {\\theta }})", "23801f0b589df9a8994604fa47a70629": "F\\times _{G}B\\to B", "2380987640578a06dbcaace09c71d566": "\\psi (s+1)=-\\gamma -\\sum _{n=1}^{\\infty }{\\frac {(-1)^{n}}{n}}{s \\choose n}", "2380c1c885fee165e5b39b18a2b0fd02": "i^{n}=i^{n{\\bmod {4}}}\\,", "238116692f4ee6f75942fdbe61ba1ec5": "\\displaystyle {Cf(w)=T_{\\Omega }F(w),}", "238126cb40b224b4aef6af5e51f29227": "\\mathrm {Hom} (U\\otimes V,W)\\cong \\mathrm {Hom} (U,\\mathbf {Hom} (V,W)).", "2381484424253e9d6c57aa5192e74a3d": "\\lim _{n\\to \\infty }a_{n}=\\operatorname {st} (a_{H})", "238180514e48be9bf0f85ba9a7ec5df0": "\\int _{X}^{\\oplus }H_{x}\\ d\\nu (x)\\rightarrow \\int _{X}^{\\oplus }H_{x}\\ d\\mu (x).", "23819be0599b381a8c0e87d3b0626ed2": "K(x_{i},x_{j})=\\left\\langle {\\phi (x_{i}),\\phi (x_{j})}\\right\\rangle ", "2382172c93f1ac204fa7c10f2f9b683d": "{\\sqrt {\\gamma _{n}}}", "2382846a7a5785b46866da2e1ca4dfec": "{\\tfrac {d({\\hbox{Victims}})}{d({\\hbox{Time}})}}", "2382b6abb273218ca5d4671ded8fe172": "p(x_{i})=y_{i}\\qquad {\\mbox{for all }}i\\in \\left\\{0,1,\\dots ,n\\right\\}.", "2382c4917af5ed37cb90605c3766a491": "h=h_{1}\\cup h_{2}", "2382ed508ccfd013f65c11775a121625": "h_{rf}={\\frac {14.2(1-\\nu ^{2})}{\\rho _{w}g}}{\\frac {\\sigma _{t}^{2}}{Y}}", "23834e353307c767c28711b41823b8d9": "\\star \\mathrm {d} t\\wedge \\mathrm {d} x=-\\mathrm {d} y\\wedge \\mathrm {d} z", "238351f95190b407b5d9fb859e20452a": "fg=f\\circ g=(1\\ 2\\ 4)(3\\ 5)={\\begin{pmatrix}1&2&3&4&5\\\\2&4&5&1&3\\end{pmatrix}}.", "2383bd43d68a54dcbd5502106997681e": "2\\pi \\,", "2383ce0ca57d0ff95d4a67f8914e19a5": "\\lambda _{\\mathrm {impurity} }", "2383f2fa68a5acd665dceb30749c31c0": "X\\cup E", "2384172053ad501905269dc99e8f1d7f": "x_{Ai}", "23841a994404a10914aef93d4dc2bd74": "{\\mathcal {A}}\\setminus (\\{A\\}\\cup \\Gamma (A))", "23844370a14cda68defd9dceb8e35c08": "{\\vec {x}}_{A}", "2384515c6f2b85927f5384e3051df992": "x^{(n)}={\\frac {\\Gamma (x+n)}{\\Gamma (x)}},", "23846db5d05a3b92d77a743039357fcf": "x'=\\sum _{i=1}^{k}e_{i}p_{1}^{e_{1}}\\cdots p_{i-1}^{e_{i-1}}p_{i}^{e_{i}-1}p_{i+1}^{e_{i+1}}\\cdots p_{k}^{e_{k}}=\\sum _{i=1}^{k}e_{i}{\\frac {x}{p_{i}}}.", "23848cef89ce4cbeb31a6eb8b3910be0": "T^{2}\\sim \\nu pF_{p,\\nu -p+1}/(\\nu -p+1),", "2384b8df829cd86139562bcae6ac6912": "\\pi \\rho _{S}(x)^{2}=2\\pi x-\\pi x^{2}.\\,", "2384d2709593fcabb11aa6609035d169": "{\\begin{pmatrix}a_{1}\\\\b_{1}\\end{pmatrix}}={\\begin{pmatrix}T_{11}&T_{12}\\\\T_{21}&T_{22}\\end{pmatrix}}{\\begin{pmatrix}b_{2}\\\\a_{2}\\end{pmatrix}}\\,", "2384d8e539390199adca8dec11680d5c": "\\int _{G}f(g)\\,dg=\\int _{S}\\int _{K}f(x\\cdot k)\\,dx\\,dk=\\int _{S}\\int _{K}f(k\\cdot x)\\Delta _{S}(x)\\,dx\\,dk.", "238526af1095d1d92c03cb5704755132": "\\alpha ,\\beta ", "2385cbeaa2abe7575d167ac4075664de": "\\textstyle {-{\\frac {\\partial }{\\partial x}}\\left(EI{\\frac {\\partial ^{2}u}{\\partial x^{2}}}\\right)}\\,", "2385fde6b300ad088e0c2711c9da8a74": "f:X\\to Y", "238615d0ac8432bb816b02baa36ea904": "f:X\\to {\\mathbb {F} }", "238645f3fe21752b7ae115372af76d48": "(x_{1},y_{1},z_{1})", "23864fdbf6ba66eaa4a2f5101797b807": "w_{p}\\cdots w_{q}", "23868a17ccbec489b6bee19f0efdc750": "\\mathbf {AB} ={\\begin{pmatrix}a&b\\\\c&d\\\\\\end{pmatrix}}{\\begin{pmatrix}\\alpha &\\beta \\\\\\gamma &\\delta \\\\\\end{pmatrix}}={\\begin{pmatrix}a\\alpha +b\\gamma &a\\beta +b\\delta \\\\c\\alpha +d\\gamma &c\\beta +d\\delta \\\\\\end{pmatrix}}\\,,", "2386a28918264fbc7e3276cf6dbc8ce8": "\\textstyle \\ I_{max}={\\underset {f}{max}}{\\underset {C}{max}}I(C,F_{f})", "2386a3fde7097a0f9550ae4cacd201ca": "k\\,", "2386ae014238cad8ddde254421155123": "...\\Rightarrow SS\\Rightarrow _{f}^{ac}AAS\\Rightarrow _{f}^{ac}AAAA\\Rightarrow _{g}^{ac}AAAA", "2386f69294fb579d19a8f23a7326a80b": "T_{T}", "23874fcd9a755f0cb8c441c0b638723a": "{{\\sqrt {z}} \\over \\tanh({\\sqrt {z}})}=\\sum _{k\\geq 0}{2^{2k}B_{2k}z^{k} \\over (2k)!}=1+{z \\over 3}-{z^{2} \\over 45}+\\cdots ", "23876585065cf839a9d9c54b0fac8fa9": "\\min \\left\\{d_{i}\\right\\}=\\left|m_{q}-M_{n}\\right|-d_{q}", "23876a350e1516e8360aee7a14a88b79": "{\\rm {HCO_{3}^{-}+H^{+}\\rightarrow H_{2}CO_{3}\\rightarrow CO_{2}+H_{2}O}}", "2387946cfc90e40a61f381d6509241ef": "\\geq {\\tfrac {1TeV}{c^{2}}}", "238853b38d39941942795591e260d2c6": "S=\\{(x,y,z)\\in \\mathbf {R} ^{3}|x^{2}+y^{2}+z^{2}=1\\}.", "238855f99bc46041d67124d358a109dc": "{\\frac {dT_{n}}{dx}}=nU_{n-1}\\,", "238864cdb6c51a514c83eba92384d9ee": "C_{ijkl}=K\\,\\delta _{ij}\\,\\delta _{kl}+\\mu \\,(\\delta _{ik}\\delta _{jl}+\\delta _{il}\\delta _{jk}-\\textstyle {\\frac {2}{3}}\\,\\delta _{ij}\\,\\delta _{kl})\\,\\!", "2388672dc3566e4f4633f7c4c3de12a6": "P(S_{i})=P(X_{1}^{n}(i))\\,.", "2388d4d89b206844a3a394ed42d7e7bf": "\\coprod f_{\\alpha }:\\coprod X_{\\alpha }\\to Y", "2388de8f5034cdb4e6cab155850aa63d": "\\mathbf {e} _{\\text{y}}\\times \\mathbf {e} _{\\text{x}}=-\\mathbf {e} _{\\text{z}}\\,\\quad \\mathbf {e} _{\\text{z}}\\times \\mathbf {e} _{\\text{y}}=-\\mathbf {e} _{\\text{x}}\\,\\quad \\mathbf {e} _{\\text{x}}\\times \\mathbf {e} _{\\text{z}}=-\\mathbf {e} _{\\text{y}}", "238957a918afc3365e5e323c018e81e5": "g_{ab}=-g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\\theta \\theta }d\\theta ^{2}+g_{\\phi \\phi }d\\phi ^{2}\\,,", "2389b49c96b5d85c40421b6f9822025f": "{\\textbf {z}}_{n}", "2389b5a8925c6beffa1b119c2bb770e0": "\\omega _{k}=\\pm {\\sqrt {\\beta _{k}-{\\frac {1}{4}}}}", "2389c100b5dda2fd31576fac04f31556": "\\psi (\\Omega ^{\\omega })=\\phi _{\\omega }(0)", "2389ddfd028c6cd7ff3adb88a080a44b": "{\\sqrt[{39}]{92}}", "238a3262a795dd7faf2d5d8ea056982b": "{}=-20\\log \\left|1+j{\\omega \\over {\\omega _{\\mathrm {c} }}}\\right|=-10\\log {\\left(1+{\\frac {\\omega ^{2}}{\\omega _{\\mathrm {c} }^{2}}}\\right)}", "238a3a42dbfa6d81eebb6f7a95c52a3c": "\\alpha ^{\\prime }=1+{\\frac {[I]}{K_{i}^{\\prime }}}.", "238a85eb4bda03d1a04693c67aea5fe3": "R(\\theta ,d')=R(\\theta ,d)-(n-2)^{2}\\mathbb {E} _{\\theta }\\left[{\\frac {1}{|\\mathbf {X} |^{2}}}\\right]", "238aaffb94a970fdcbb4af65c56bea91": "\\scriptstyle \\psi x\\,=\\,x+{\\frac {x^{2}}{2^{2}}}+{\\frac {x^{3}}{3^{2}}}+\\cdots +{\\frac {x^{n}}{n^{2}}}+\\cdots ", "238ac359e88c840a09058d93876324bf": "H_{R}=\\alpha ({\\boldsymbol {\\sigma }}\\times {\\mathbf {p}})\\cdot {\\hat {z}}", "238aca5bcf24c08bae38ce4c34911a54": "{\\begin{smallmatrix}R_{R}={\\left({\\frac {d_{R}}{2}}\\right)}={\\left({\\frac {3.740}{2}}\\right)}=1.870AU\\end{smallmatrix}}", "238b200b5118a4cc0861756b00125757": "d^{4}p\\to \\mu ^{4-d}d^{d}p", "238b920eb9099971ebcd2818f464368c": "x\\equiv _{qc}y", "238bdc3e65bb66614ddacc39edbd8b9d": "Y=\\{Y_{1},\\ldots ,Y_{s}\\}", "238be49f72833929caf8ae9920d83178": "n(\\lfloor {\\frac {rB}{r+1}}\\rfloor -\\lfloor {\\frac {B}{r+1}}\\rfloor )", "238beb6701b94915256eccc0a727af78": "S(x)=-\\int p_{x}(u)\\log p_{x}(u)du", "238c250f1aa3d0266edb98bf95910304": "|R_{i}|\\geq {\\sqrt {N-1}}", "238c251f2e4a9d0d6d67bd0a05939e84": "O(k^{2})", "238c25fc2c22e400d1a424d62c116d49": "a=1-\\gamma ", "238c2ae511378acdf3d80257b24aa2c5": "P{\\underline {\\lor }}Q", "238c4ca8ae138e02783454e6728a0d4c": "\\ e_{t}", "238c9ad82f55cc6ac411519d3cf05738": "\\cos(\\theta _{2}+\\theta _{4})+\\sin \\theta _{2}\\sin \\theta _{4}=\\cos \\theta _{2}\\cos \\theta _{4}\\,", "238d30203e50860341d9c8edf5b4266c": "T>0", "238d4d00dd73cbde448fcce08074ac68": "{\\mathcal {P}}\\,\\exp ", "238d4f6abb27ad17e50c8c742730d8a7": "\\sin \\alpha =\\left({\\frac {w}{l}}\\right)", "238d6fea37f08fcff762402d8d887de2": "k(z,y)=G_{p+2,\\,q}^{\\,m,\\,n+2}\\!\\left(\\left.{\\begin{matrix}1-\\nu +iz,1-\\nu -iz,\\mathbf {a_{p}} \\\\\\mathbf {b_{q}} \\end{matrix}}\\;\\right|\\;y\\right),", "238d7aed8f22662e832d56ca109c0669": "A_{\\mathrm {T} }=A^{n}\\ ", "238d7c3eb38f1f63a6956416977dd17f": "(n,k)=(255,223)", "238daed21dea66f05db354ce045740c9": "\\scriptstyle w_{1},\\dotsc ,w_{n}", "238dd3e454c07523ddacafe5918cbe5c": "S_{1}(A_{n})=S(A_{n}).", "238e1e63379bc49db327f9386c2da30e": "\\mu _{1}\\equiv E[W^{1}]=g_{1}(\\theta _{1},\\theta _{2},\\dots ,\\theta _{k}),", "238e215539bc986852cd4cfe25aca0c8": "\\mathbf {a} \\cdot \\mathbf {b} .", "238e280f09ea50bfb37c6c03c4967374": "\\delta ={\\frac {V_{i}-V_{o}}{V_{o}}}D", "238e58962004992c54f3db69d239e45b": "{\\begin{bmatrix}1&1&1&1\\\\1&-i&-1&i\\\\1&-1&1&-1\\\\1&i&-1&-i\\end{bmatrix}}", "238e60693cccdab5e4844673d0b6565a": "(2)\\ \\ H_{n}(x)=(-1)^{n}e^{x^{2}}{\\frac {d^{n}}{dx^{n}}}e^{-x^{2}}={\\bigg (}2x-{\\frac {d}{dx}}{\\bigg )}^{n}\\cdot 1,", "238e6ae69670fa2008d54c4e37bded3b": "M_{G}(x):=\\sum _{k\\geq 0}(-1)^{k}m_{k}x^{n-2k}.", "238e8cc76ba7adfb1211da47d36f1db5": "{k'}^{2}=m_{1}\\,\\!", "238e984bfd53f92e48fe40526c14ba2f": "\\ v={\\frac {V}{m}}\\ ={\\rho }^{-1}", "238ea6b940d3a3d5abdc962e4ca0bdaa": "x^{2}+y^{2}+z^{2}=0", "238ecf0920fda7c5cc8326b6d2f6a48f": "\\Gamma \\subset V\\times W", "238f05d594779801bc7f1a7cfdb0cd7f": "\\phi =\\phi _{0}e^{\\frac {-x}{\\lambda _{D}}}", "238f6e7476de18fd893637ef736be09e": "a=1-2\\nu \\,\\!", "238f70cbb76eb9971a22df56c7253e52": "A\\mapsto {\\rm {Tr}}f(A)=\\sum _{j}f(\\lambda _{j}),", "23908618399eeea28692428caa459379": "x_{k}^{*}:=x_{j}^{*}+\\left(b^{*}-x_{j}^{*}A\\right)P_{k}A^{-1},", "239098b65fded7e8d3ac5307c68e5336": "\\scriptstyle \\sum _{n}|c_{n}|^{2}|\\psi _{n}\\rangle \\langle \\psi _{n}|", "23909d5f27eefc3f6b24a7f82507b07f": "y_{k+1}=n-\\sum _{i=1}^{k}y_{i}", "2390aa392ed52d52147ba520bdec82e5": "{\\mathbf {J}}=-\\rho dt+j_{x}dx+j_{y}dy+j_{z}dz", "2390b4a08d441ad9adb1b76b742a9b57": "\\Rightarrow _{S\\to XX}\\ XXS\\ \\Rightarrow _{S\\to XX}\\ XXXX", "239103f41782fb2d7d5e43f9d27a22f0": "\\left(\\left|{{H}_{WCM}}\\left(s\\right)\\right|=2.4\\right)", "23915299bcbc02470f6d56e5050cf5ae": "x^{2}z^{2}>xy^{2}z>x^{3}>z^{2}", "239158be952fd5b200781d73d04c1e14": "F_{\\rm {fluid}}=-k_{B}T_{e}\\nabla n_{e},", "2391a32feb98c643e952e6404103389a": "p'_{G}-p'_{L}=g\\eta \\left(\\rho _{G}-\\rho _{L}\\right)+\\sigma \\eta _{xx},\\qquad {\\text{on }}z=0.\\,", "2391bbeeb5b0869378b98bd1b0e4c141": "{\\frac {\\partial \\mathbf {y} }{\\partial x}}", "2391c76ff1b8acbf14b99a46a439afa4": "[\\wp '(z)]^{2}|_{z=0}\\sim {\\frac {4}{z^{6}}}-{\\frac {24}{z^{2}}}\\sum {\\frac {1}{(m\\omega _{1}+n\\omega _{2})^{4}}}-80\\sum {\\frac {1}{(m\\omega _{1}+n\\omega _{2})^{6}}}", "2391cf7a07ccaf0c37f4a7e691755e5d": "t\\to \\infty ", "2391ef400234f70fd83e0eac6a57a2c0": "\\left|{\\frac {\\sqrt {x^{2}+y^{2}}}{r}}\\right|^{p}+\\left|{\\frac {z}{h}}\\right|^{p}\\leq 1", "2391f65e83f43fc79933e459137a88c4": "\\,h_{1}=0", "2392948d1f505aaa5ad677305cc81019": "e=\\sum _{k=1}^{\\infty }{\\frac {k^{n}}{B_{n}(k!)}}", "2392c3be9fd69cf5e77eccfc8f16fda3": "={\\frac {T(x,t)-T_{\\infty }}{T(0,t)-T_{\\infty }}}", "2392ec97fe6e7f855d788933c202c36d": "\\displaystyle \\alpha =-\\operatorname {sgn} (a_{k+1,k}^{k}){\\sqrt {\\sum _{j=k+1}^{n}(a_{jk}^{k})^{2}}}", "2392f0dce629068d9184a545432c7c52": "\\varphi ^{N}", "239335c749997f502334b305a7771f63": "IG(T,a)=H(T)-H(T|a)", "23936a48ad55b7d6583f95ee5ec3b66c": "\\partial f_{i_{m}}/\\partial x_{j_{n}}", "2393b884af08b2044143851934cedaff": "X\\times _{S}P\\to P", "2393e9ae74251d903ac59710283e6df0": "r={\\sqrt {\\frac {A}{\\pi }}}", "239417505db3134d8402553ded71e953": "\\gamma _{c}(A)=1", "23941c0e5146df2c92402295ed7ad5df": "-1+j0", "23942fdca9dba114b4644be5dab1edea": "v_{s}(\\mathbf {r} ,t)=v_{\\rm {ext}}(\\mathbf {r} ,t)+v_{J}(\\mathbf {r} ,t)+v_{\\rm {xc}}(\\mathbf {r} ,t).\\,", "23943ab5eddecf19de0515a8c9d3736a": "\\left(\\mathbb {R} ,{\\mathfrak {B}}\\right)", "239463091cd28199463a3ed2f4a3590a": "Z\\ {\\stackrel {\\mathrm {def} }{=}}\\ \\sum _{k=1}^{N}m_{k}\\left|{\\frac {d^{2}\\mathbf {r} _{k}}{dt^{2}}}-{\\frac {\\mathbf {F} _{k}}{m_{k}}}\\right|^{2}", "2394716fa39131e443b4267df83bec7e": "{\\mathcal {U}}\\leftarrow {\\mathcal {U}}-s'", "23948fdc171ad44f7311e78f0cc253d8": "k(\\mathbf {x_{i}} ,\\mathbf {x_{j}} )=\\varphi (\\mathbf {x_{i}} )\\cdot \\varphi (\\mathbf {x_{j}} )", "2394a5883d598fd4f2f11222b5617cc0": "\\max _{\\mu \\in {\\mathcal {M}}_{X}}\\,\\inf _{\\lambda \\in A_{Y}}k(\\mu ,\\lambda )=\\inf _{\\lambda \\in A_{Y}}\\,\\max _{\\mu \\in {\\mathcal {M}}_{X}}k(\\mu ,\\lambda ).", "2394af39a0188d6347ac0e8ebe3bab16": "{\\textrm {Beta}}\\left(\\alpha _{1},\\sum _{i=2}^{K}\\alpha _{i}\\right)", "2394cf96e6b78ac1050249265c29c7ab": "E[A]=\\int _{0}^{\\infty }Cr\\lbrace A\\geq t\\rbrace \\,dt-\\int _{-\\infty }^{0}Cr\\lbrace A\\leq t\\rbrace \\,dt.", "2394dba51aa9e08f4b4632f2d4bf59f1": "Q^{n}(c)=0,n=1,2,3,...", "23951ca44c73dfbd0ddc9928c64ce935": "Q(x'\\mid x_{t})\\approx P(x')\\,\\!", "239520e7273cec8c3f84cf32958223bd": "\\chi _{E}", "239524b42c6cd49b5a89d78d3034a69c": "logBCF=1.119logKoc-1.579", "2395a0e2858e528bc86da62d4975bb4a": "{\\begin{cases}x\\equiv &s_{i_{1}}\\ {\\bmod {\\ }}m_{i_{1}}\\\\&\\vdots \\\\x\\equiv &s_{i_{k}}\\ {\\bmod {\\ }}m_{i_{k}}\\\\\\end{cases}}", "2395d95eb0d33660a2e4633fe768c60d": "2g/R=0.3086", "2395fec912343299bf6ca0cbcf2406f6": "-{\\frac {1}{\\lambda ^{k}}}", "2396067fdff30ce1346a1ac7e0229601": "m^{a}\\partial _{a}=\\Omega \\partial _{r}+\\xi ^{3}\\partial _{y}+\\xi ^{4}\\partial _{z}\\,:=\\,\\delta \\,,", "23960e7bd640dd34bec78ed78df96b74": "2^{32}-1", "239693968f7b490432a60823494e2381": "\\gamma _{\\alpha }=\\lim _{n\\to \\infty }\\left[\\sum _{k=1}^{n}{\\frac {1}{k^{\\alpha }}}-\\int _{1}^{n}{\\frac {1}{x^{\\alpha }}}\\,dx\\right],", "2396eaa516b5621ecd34523c0ccaef23": "{\\frac {({\\frac {f_{m}}{N}})-{\\frac {1}{K}}}{{\\frac {N}{K}}{\\frac {(K-1)}{N}}}}={\\frac {M}{N(K-1)}}", "23974652112a6f1319fa3c24e0f1cbea": "e^{i\\pi }=-1.\\!", "2397510aaaee5caa79c250cc987ac54e": "\\mu (x,G)", "239826a0a6a1c947d34924ba94b7aef8": "\\displaystyle {\\gamma =\\gamma _{1}\\gamma _{2}\\left(1+{\\beta _{1}{\\overline {\\beta _{2}}} \\over \\alpha _{1}\\alpha _{2}}\\right)^{1/2},}", "239859719908f5b818416fb06d6b3018": "p_{2},\\ldots ,p_{n}", "23986d9b2a622a418cb84ad397a8df80": "h_{t}", "239891422d5b3bb67af33d93c9b76bb9": "(\\partial S)_{V}=-(\\partial V)_{S}={\\frac {C_{P}}{T}}\\left({\\frac {\\partial V}{\\partial P}}\\right)_{T}+\\left({\\frac {\\partial V}{\\partial T}}\\right)_{P}^{2}", "23996d75b8ea74deaa0f05c057b2ed08": "{\\begin{aligned}|\\mathbf {A} |&={\\begin{vmatrix}A^{0}+A^{3}&A^{1}-iA^{2}\\\\A^{1}+iA^{2}&A^{0}-A^{3}\\end{vmatrix}}\\\\&=(A^{0}+A^{3})(A^{0}-A^{3})-(A^{1}-iA^{2})(A^{1}+iA^{2})\\\\&=(A^{0})^{2}-(A^{1})^{2}-(A^{2})^{2}-(A^{3})^{2}\\end{aligned}}", "239a0d67ec7c87ba303b64d0f2f6102b": "\\delta U+\\delta V_{\\mathrm {ext} }=\\delta K", "239a330b376765b9a6539f2f958e753e": "{\\overline {\\Omega }}_{\\theta \\phi }=\\Omega _{\\theta \\phi }/\\sin \\theta =1/2", "239a649ca1f3611c6756dae314fffc6b": "{\\dot {V}}(x)=V'(x)f(x)=\\mathrm {sgn} (x)\\cdot (-x)=-|x|<0.", "239a945679982606cb34e373d12f39cf": "(-1,0),", "239bda9e973ed934b8b9190b21e5ab40": "[W_{T}]=[W]+[W']+[WE_{1}]+[W'E_{2}]", "239c179c407a5968cd05a75cc50ff292": "T_{Chorok}=T_{Dhuhr}-T(0.833)", "239c244fb3678b77ad4946105b79dd2f": "PK\\{(x):y_{1}=g_{1}^{x}\\land y_{2}={(g_{2}^{a})}^{x}g_{2}^{b}\\},", "239c42d35e7c4548370c82ddf2cbce99": "\\scriptstyle \\theta ", "239c89327a02417de78b0f12f9dbbe66": "\\mathbf {1} _{A\\times B}(x,y)=\\mathbf {1} _{A}(x)\\cdot \\mathbf {1} _{B}(y).", "239c97996db1251ae54ba9641466897d": "{\\frac {q}{\\omega }}\\ll 1", "239ce1c5e25a42de1d9befab1a20d68f": "2x(m+\\lambda )c_{mk}^{\\lambda }(x;k)=(m+2\\lambda )c_{mk+1}^{\\lambda }(x;k)+mc_{mk-1}^{\\lambda }(x;k)", "239cf037248aa80419b280e7247a9951": "U(-n,-2n,z)", "239d03df6eba1d0826665bafded86cd5": "Y_{i}=\\alpha +\\sum _{j=1}^{p}f_{j}(X_{ij})+\\epsilon _{i}", "239d0a7df79a33d003515d8154e55a93": "K_{\\text{Matern}}(x,x')={\\frac {2^{1-\\nu }}{\\Gamma (\\nu )}}{\\Big (}{\\frac {{\\sqrt {2\\nu }}|d|}{l}}{\\Big )}^{\\nu }K_{\\nu }{\\Big (}{\\frac {{\\sqrt {2\\nu }}|d|}{l}}{\\Big )}", "239d80b28cf56b7451ad4a8215e694fd": "|r|", "239dd9a0fb8f15d164c9276b6b9abcf2": "\\mu _{ij}", "239e598b0761292677c59856acffd419": "S=S_{\\sigma }(\\sigma )+S_{\\tau }(\\tau )+S_{z}(z)-Et", "239e6aabd39cc7d05cbfefd79f33acaa": "H_{b}=5L/9", "239e8301daacc33676597ad1076edd54": "{\\text{max}}(X)", "239eef8d93c4dbbdcf7288f3b518928e": "y=r\\sin \\phi .", "239f29cf2cb510a6a635a1f26dde7a2a": "{\\frac {a}{b}}<{\\frac {c}{d}}", "239f582295e0212effea57f1d78fb944": "\\mathrm {SR} (K)\\leq {\\frac {\\pi }{\\sqrt {8}}},", "239fd97453551fa7922d8e122a6ae591": "-\\nu ", "239fe35f187fc4e957f54b892987f260": "f(x+b)", "23a04f8a07f13d42a8482b7079861ac8": "\\{n+1,n{\\stackrel {.}{-}}m,\\lfloor n/m\\rfloor ,n^{m}\\}", "23a086f0edf8d1ad3deffad53608e8f1": "f={\\frac {1}{2\\pi {\\sqrt {LC}}}}.", "23a0a79f2d63407554c2be91a50a2172": "z_{i}=\\lambda {\\bar {x}}_{i}+\\left(1-\\lambda \\right)z_{i-1}", "23a1310d1120d016f48912b033305226": "{1 \\over \\rho }{\\partial \\left(\\rho A_{\\rho }\\right) \\over \\partial \\rho }+{1 \\over \\rho }{\\partial A_{\\phi } \\over \\partial \\phi }+{\\partial A_{z} \\over \\partial z}", "23a151eebcbede0cb7c833bbbe1449a3": "P\\left(Searched|Known\\wedge \\delta \\wedge \\pi \\right)", "23a153d91b7dd0bc7f3b7a7c602d0c1f": "t=-0.5,", "23a1967d654f959dd43f22bb255d330b": "\\scriptstyle {{\\bar {k}}\\sim \\ln N}", "23a1a539958ae0123a278e164ab77f30": "{\\frac {\\left|{\\mathbf {\\Psi } }\\right|^{\\frac {\\nu }{2}}}{2^{\\frac {\\nu p}{2}}\\Gamma _{p}({\\frac {\\nu }{2}})}}\\left|\\mathbf {X} \\right|^{-{\\frac {\\nu +p+1}{2}}}e^{-{\\frac {1}{2}}\\operatorname {tr} ({\\mathbf {\\Psi } }\\mathbf {X} ^{-1})}", "23a1daa7cd08229dfcf67f642ab33607": "E_{t}", "23a25c3fced257b61da4af1bd5eaf276": "C_{k}=Hg_{n_{1}}g_{n_{2}}\\cdots g_{n_{j}}", "23a298bf814fdb25f587cbb703ddc7c2": "-{\\sqrt {2}}~\\xi ~\\sin \\phi +\\rho [\\cos \\theta -\\cos(\\theta +2\\pi /3)]-\\rho \\sin \\phi [\\cos \\theta +\\cos(\\theta +2\\pi /3)]={\\sqrt {6}}~c~\\cos \\phi ", "23a2b356551c0b040af48b08dc08ae11": "I_{\\mathrm {Alice} },I_{\\mathrm {Bob} }\\in GF(p)", "23a2f3546bcfc25675fcefce51056465": "PC_{x}=C_{x}=QC_{x}.", "23a31c52414c99221ef910fce5b36399": "{\\vec {n}}_{i}\\cdot {\\vec {p}}_{0}=d_{i},\\ i=1,2,3,", "23a343321e5dd03c5fcfeae6cdeb5299": "S(K)=(P,B,I)", "23a3441ddcb52730f26d485b7599d638": "Ty=y.\\,", "23a3605478ef02cbbb98871a555610d3": "-K_{22}^{-1}K_{21}x_{1}=x_{2}", "23a3650f37741dfa7394f4711ea081c3": "{\\mathbb {Z}}", "23a36a1dc6ce08dbf01396df6cdc1217": "\\left\\{{B_{n}:n=1,2,3,\\ldots }\\right\\}", "23a375a1ae51d74da9ee54c67369a2ce": "H_{2}A\\rightleftharpoons A^{2-}+2H^{+}:\\beta _{D}={\\frac {[A^{2-}][H^{+}]^{2}}{[H_{2}A]}}=K_{1}K_{2}", "23a3897e299ad31ee0a49fef114f7b89": "{\\vec {B}}={\\rm {curl\\,\\,}}{\\vec {A}}=\\left\\{{\\frac {\\partial A_{3}}{\\partial x_{2}}}-{\\frac {\\partial A_{2}}{\\partial x_{3}}},{\\frac {\\partial A_{1}}{\\partial x_{3}}}-{\\frac {\\partial A_{3}}{\\partial x_{1}}},{\\frac {\\partial A_{2}}{\\partial x_{1}}}-{\\frac {\\partial A_{1}}{\\partial x_{2}}}\\right\\},{\\text{ or }}\\Phi _{B}={\\rm {d}}\\mathbf {A} .", "23a3afef797162db2d239fca85e0ad42": "^{A}_{Z}E", "23a3c81abab62099dcfb6c08284b8331": "={1 \\over (j+1)}\\left(g^{(l)}{1 \\over 2}\\left(j(j+1)+l(l+1)-s(s+1)\\right)+g^{(s)}{1 \\over 2}\\left(j(j+1)-l(l+1)+s(s+1)\\right)\\right)", "23a4493c718a73272da4e3ba1248ed60": "(a+b\\sigma _{1}\\sigma _{2})^{*}=a+b\\sigma _{2}\\sigma _{1}=a-b\\sigma _{1}\\sigma _{2}\\,", "23a45ad10669944a029db1d16895b8dc": "\\omega _{i}", "23a48eb292114d7fabcc83c653e46cfb": "g(r,r^{\\prime })=g(r^{\\prime },r),\\,", "23a4c65d87d1725a311b299b7f458550": "x=\\forall +", "23a4ce747750e06ece8d17d70a2f1c78": "\\varphi \\rightarrow \\lambda ^{-1}\\varphi .", "23a51b36d5d18a407d6b0dd347f07e12": "\\lambda =-\\sigma =1\\,", "23a51ca43e95162f2d61f3b929d4ec7a": "a\\otimes (1-a)\\,", "23a53888f9da74144627bc2466a21718": "(-1)^{k}{\\frac {\\partial }{\\partial u}}\\left[{\\left({\\frac {d}{dt}}\\right)}^{2k}H_{u}\\right]\\geq 0,\\,k=0,1,\\cdots ", "23a559d727323acfb796d87142eef2aa": "{\\begin{Bmatrix}x\\end{Bmatrix}}={\\begin{bmatrix}\\Psi \\end{bmatrix}}{\\begin{Bmatrix}q\\end{Bmatrix}}", "23a57408645e14fdc1ad9521daabfae0": "\\mathbf {B} =e^{-i\\omega t}\\sum _{l,m}{\\sqrt {l(l+1)}}\\left[a_{E}(l,m)\\mathbf {B} _{l,m}^{(E)}+a_{M}(l,m)\\mathbf {B} _{l,m}^{(M)}\\right]", "23a63f4431ad1392176c6b0dfc6936d0": "L_{-1}^{2}\\Psi =0,", "23a673f00eeac011bf6fa07a729444c1": "B_{2}(r,\\mu )={\\frac {1+{\\frac {1}{2}}\\left[2r^{\\mu +2}-\\left(r+1\\right)^{\\mu +2}-\\left|r-1\\right|^{\\mu +2}\\right]}{2\\left(1-2^{\\mu }\\right)}}.", "23a6a55c2e1f7574ea763601a3a53b77": "({\\text{extend}}\\,f)\\circ ({\\text{extend}}\\,g)={\\text{extend}}\\,(f\\circ ({\\text{extend}}\\,g))", "23a6bbb53a41138cdbec7eb5a61b1e19": "{\\mathfrak {J}}(a)_{n}=b_{n}.", "23a6e0f6cd66756fdfb28d181da13ce2": "\\Pi _{\\rho ,\\delta }^{n}", "23a747a1bb2060b23c65a7d930dd6a5e": "\\displaystyle {P=i{d \\over dx},\\,\\,\\,Q=x}", "23a76a73c8fcc330078c8915ad481662": "{\\begin{array}{cc}P_{j}(d_{j})=\\left\\{{\\begin{array}{lll}{\\frac {|d_{j}|}{p_{j}}}&{\\text{if}}&|d_{j}|\\leq p_{j}\\\\\\\\1&{\\text{if}}&|d_{j}|>p_{j}\\\\\\end{array}}\\right.\\end{array}}", "23a76f078afcdf1abac5e68b57be2cd0": "A(f)=2.0+20\\log _{10}\\left(R_{A}(f)\\right)", "23a79b700e61c75945a8ca545c360d39": "{\\bar {R}}_{P}", "23a7dd4a99d3dab7ff18ddab1590f332": "p:P\\to X", "23a867e8ff309993734c69271e021ae3": "\\mathrm {CR} (x)=\\sum _{i=-1}^{2}f_{i}b_{i}(x)", "23a8b60c55364e140f2bf980986ff433": "T_{i}={\\frac {\\mu _{i}-{\\bar {X}}_{i}}{S_{i}/{\\sqrt {n_{i}}}}}{\\text{ for }}i=1,2\\,", "23a8f63f4604e8422942a3c194f245dd": "S:R^{m}\\to R^{m}", "23a94169ff080fcf979759b5152be9fe": "\\{1,\\ldots ,n\\}", "23a98e177f47b98e0e69c4dd9c5b073a": "\\sigma _{xx}+\\sigma _{xz}-\\sigma _{xy}", "23a9d6498ebd2e2516325125b95f5487": "{\\mathfrak {P}}^{117}", "23a9e65fcebd8b93cb2f34f72e470751": "\\displaystyle {{\\mathfrak {h}}_{i}={\\mathfrak {h}}\\cap {\\mathfrak {g}}_{i},}", "23a9f81480e531b259106bb768b50d8a": "|c|=b", "23aa193272e880d05e5edb1bf28c9a43": "\\Phi _{pre}={\\frac {{\\frac {1}{2}}m_{i}c_{s}^{2}}{Ze}}=k_{B}(T_{e}+Z\\gamma _{i}T_{i})/(2Ze)", "23aa2b30caa48a902b5f1b098c90f2c3": "S^{2}=c^{2}\\left.{\\frac {\\partial p}{\\partial e}}\\right|_{\\rm {adiabatic}}.", "23aa406cd9eac1a3eabf5e352cc1ab1e": "\\theta _{a}\\,", "23aaea169d1d875537281c7b2c857d8e": "\\log(C_{N})/N", "23aafbb9fed33af6b165eb862ffefcf3": "(\\neg P\\lor Q)", "23aaffb330da7c647a08f30396a229cf": "16y^{5}-20y^{3}+5y-1=0\\,", "23abd9baea9c20c3bbfcc3399c734a1b": "f_{i}g_{j}=0", "23abf0416e59bee199a8119efffae661": "GJ{\\frac {d^{2}\\theta }{dy^{2}}}=-M'", "23ac45d769a17e4b37a2bf2ca94b4b4c": "\\Delta L=Ed_{31}=\\left({\\frac {d_{31}L}{t}}\\right)U_{x}", "23ac91a8667937c895c3d6cdc2b8f33b": "\\beta _{j}+\\delta \\beta _{j}\\,", "23ac9895c8117cde10077d87166f9d67": "\\mathbb {P} ^{n}\\to \\mathbb {P} ^{N}", "23ac989c5916fa0f1327d642ed8edf63": "z=x+iy\\ ", "23ac996bad415c332fbb59ed77c514c2": "({\\text{ }})", "23aca7bc64ac6120c5f43348a53d4d50": "\\langle X-\\alpha _{i}\\rangle ", "23acab630dbfc6af384824c9ee7fe984": "O\\left(\\exp \\left(\\left({\\tfrac {64n}{9}}\\log(2)\\right)^{\\frac {1}{3}}\\left(\\log(n\\log(2))\\right)^{\\frac {2}{3}}\\right)\\right)", "23acaedd883a80eb5bed45d6d23267ee": "A(K\\backslash G/K).", "23acd870f145b90a0347c61dc7a0ddf5": "|S(P,f,g)-A|<\\varepsilon \\,", "23acf87fa1a3e56dce02aef0ac7b9399": "U=(Y,Z)", "23ad0ee06b2c26cf10aad272dc8b5229": "end\\,while", "23ad95ed53baf9237f7365e441e73623": "T_{ant}", "23adb4af8f148e6bf490477f73c76f36": "d={\\frac {D}{m}}={\\frac {130}{18.6}}\\approx 35", "23adf8420e4622106eb134378be472de": "{\\frac {R_{1}R_{2}+R_{1}R_{3}+R_{2}R_{3}}{R_{1}}}=R_{a},", "23aea4b7d12cd13a89aeb3e58dd5de73": "2^{i}\\operatorname {value} (x_{i})", "23aef064683dda7b6efc577b136c2646": "\\alpha =2\\,\\!", "23af098b65939cb9b1a8d5f9f7d2893f": "\\Delta S_{vap}={\\frac {\\Delta H_{vap}}{T_{b}}}", "23af122fc1d9de834d75d743f71af921": "H=(V,E)", "23af3eb99906c8ae50c67bd1522bc519": "\\land ,\\lor ,\\lnot ", "23af853bd144b951de6349d131bd5057": "m_{1}=\\left\\lceil m/2\\right\\rceil ", "23afc5da73974ff83aa0207d14fb141d": "=\\mathbf {R} _{x}^{-1}(n)", "23afec566cf78d7a392b9c8b88d85570": "Q(2n,q)", "23b0cad8bd941a5f2b9119a05d70ca75": "k_{h}", "23b0e2a288ff19edf3eb766ae9837f7e": "SU(N_{f})_{L}^{3}", "23b10a65a6c1b6dee97a31cd4f03436d": "\\sigma _{gt}", "23b111ab578d31773fad406694483bcd": "\\omega _{1}\\times \\omega ", "23b1b2dd06a03b694c8a9d2daea99758": "\\;{}_{1}F_{1}(a+1;b;z)-\\,{}_{1}F_{1}(a;b;z)={\\frac {z}{b}}\\;{}_{1}F_{1}(a+1;b+1;z)", "23b20ef3b27490f589843505e708242a": "\\ker(L)=\\left\\{v\\in V:L(v)=0\\right\\}{\\text{,}}", "23b212925dda1686cdc401f390653ce2": "\\{n,l,m\\}", "23b22ecd3125f01a95e03114d1fe3515": "{\\mathcal {A}}_{k},{\\mathcal {B}}_{k},{\\mathcal {C}}_{k}", "23b271603807fb72ed74507183d7faea": "\\mathbb {Z} _{p_{i}}", "23b2d65e7203d77e0c82610240710588": "3\\rightarrow 3", "23b31d3f31d2735be679aa3b5f7a9c27": "{\\ddot {x}}+2\\zeta \\omega _{0}{\\dot {x}}+\\omega _{0}^{2}x=0.\\,", "23b3421a91ee73f84e5dd44ce2743d6d": "{\\begin{pmatrix}3&-4\\\\4&-7\\end{pmatrix}}{\\begin{pmatrix}x\\\\y\\end{pmatrix}}.", "23b353ef9d1a4c4c6485b898e7f82590": "{\\begin{aligned}\\oint _{\\partial V}{\\mathsf {L}}(dS;x)&=\\oint _{\\partial V}\\langle F(x)dSI^{-1}\\rangle \\\\&=\\oint _{\\partial V}\\langle F(x){\\hat {n}}|dS|\\rangle \\\\&=\\oint _{\\partial V}F(x)\\cdot {\\hat {n}}|dS|\\end{aligned}}", "23b374d014fd7adc2e0c6f3d74d05e9d": "R_{c}\\simeq \\ell {\\sqrt {s}}\\simeq {\\frac {\\sqrt {s}}{n\\sigma }}", "23b380b4c4720500069c5178f7341f19": "={\\frac {2v^{2}\\cos ^{2}\\theta }{g}}{\\sqrt {\\left({\\frac {\\sin \\theta }{\\cos \\theta }}-m\\right)^{2}+m^{2}\\left({\\frac {\\sin \\theta }{\\cos \\theta }}-m\\right)^{2}}}", "23b3d324b0c3346906af9f52a9777b33": "L_{x}\\times L_{y}", "23b402ff95ceb6034741c161086af6c2": "R/P_{i}^{a_{i}}", "23b4091f193cd663f97501b87d0bd379": "{\\frac {1}{\\mu _{0}\\varepsilon _{0}}}=c_{0}^{2}.", "23b42c80589a9a4843f6ff06f5ea079f": "Q={\\frac {M\\omega }{\\Gamma }}\\,", "23b4ab61064190df13af7b7a3dc32681": "\\left\\vert \\det {\\left[{\\frac {\\partial {\\bar {x}}^{\\iota }}{\\partial {x}^{\\gamma }}}\\right]}\\right\\vert ={\\sqrt {-{g}}}\\,,", "23b4cf6ce31c03eb8ad55ac3b9a12702": "{\\mathbb {N} }=\\{0,1,2,\\ldots \\}", "23b4e9baf4059782e0fe4bcca4624132": "{\\begin{aligned}q_{0}(x)&{}=\\int _{\\mathbb {R} }\\!{\\frac {t^{3}-x^{3}}{t-x}}\\rho (t)\\,dt\\\\&{}=\\int _{\\mathbb {R} }\\!{\\frac {(t-x)(t^{2}+tx+x^{2})}{t-x}}\\rho (t)\\,dt\\\\&{}=\\int _{\\mathbb {R} }\\!(t^{2}+tx+x^{2})\\rho (t)\\,dt\\\\&{}=\\int _{\\mathbb {R} }\\!t^{2}\\rho (t)\\,dt+x\\int _{\\mathbb {R} }\\!t\\rho (t)\\,dt+x^{2}\\int _{\\mathbb {R} }\\!\\rho (t)\\,dt\\end{aligned}}", "23b55fe0f610bd2c1f56cf3d0dd9e5b5": "\\int _{K}f(x+cy)\\,\\mathrm {d} x\\geq \\int _{K}f(x+y)\\,\\mathrm {d} x.", "23b583065f5b7336c728011ccd7375b2": "s=2", "23b58def11b45727d3351702515f86af": "", "23b6559edfe9ac80f3036952ef4475ef": "\\mathrm {Factor} ={\\frac {\\mathrm {Days} (\\mathrm {Date1} ,\\mathrm {Date2} )}{\\mathrm {DiY} }}", "23b6882b77ee4423111844725fea3211": "\\varepsilon >0", "23b6cf6cb393df40b34b28d11a0ae402": "s=1-s\\rho ", "23b6ded51212dde5ce0120d1b04f8ee4": "D^{2n}", "23b6fd17d8b54b9d376cd838bdc9a667": "n\\in \\mathbb {Z} _{>0}", "23b70914f040c18f76b9ce8e79b26b7a": "{\\hat {H}}=-{\\frac {\\hbar ^{2}}{2\\mu }}\\nabla ^{2}", "23b75d836de111c58a9222754d56232a": "B_{i}\\rightarrow A_{ij}", "23b7c8d325e72db6c0f77bba87c6f7c3": "f^{-1}(x)={\\sqrt {\\pi }}{\\frac {d^{1/2}}{dx^{1/2}}}N(x)", "23b7e6f3b6ef6687fad60e3ebcf0e538": "\\scriptstyle [m,\\,4.5m]", "23b82211d1500bafb08c9592e48fb022": "x=DL(a)", "23b8bb554bc3f9301d0ab92a9d70ce98": "\\Re (s+a)<0.", "23b8d4d489e0d6f9cbe2019ca2970b45": "\\mathrm {TAS} =39M{\\sqrt {T}}", "23b8f234501b668d0a2b631279f8cd7c": "{\\hat {H}}=-{\\frac {\\hbar ^{2}}{2m}}{\\frac {\\partial ^{2}}{\\partial x^{2}}}+{\\frac {m\\omega ^{2}}{2}}x^{2}", "23b8fba3208ac35982d80854b03836b8": "\\ F_{p}(z,f)", "23b90b58292a238cba5b94ce50a920a9": "{\\frac {d\\Sigma (t)}{dt}}=a\\Sigma (t)+\\Sigma (t)a^{\\top }+\\sigma ^{\\top }\\sigma -\\Sigma (t)c^{\\top }\\eta ^{-\\top }\\eta ^{-1}c\\Sigma (t).", "23b9145e0b96b523fb4fb0ebad89862a": "\\tan \\theta \\simeq \\theta ,\\quad \\theta \\ll 1", "23b935a9e8ba7b787a2fed9e032731c7": "\\rho v_{i}v_{j}", "23b99a350c9539dbf7f07a6d88b303a4": "{\\sqrt {b^{2}-4ac}}=b\\ {\\sqrt {1-{\\frac {4ac}{b^{2}}}}}\\approx b\\left(1-{\\frac {2ac}{b^{2}}}+{\\frac {2a^{2}c^{2}}{b^{4}}}+\\cdots \\right).", "23ba1317325b796879752bebaddf491f": "(\\cosh x+\\sinh x)^{n}", "23ba1d2ca255fc8b583d89e3438d96d3": "\\Gamma (n)=(n-1)!", "23ba4a06c16b5af59c7ea899eb018d74": "\\ {F_{p}}", "23ba9e54564986150c054cc03ce19851": "o_{0},o_{1},\\dots ", "23baf1c8533a27d7e0b71354215df8be": "~\\sigma _{\\rm {e}}~", "23bb3977222abdc31ab56dc750dcd81a": "{\\frac {j_{\\odot }}{j_{c}}}=\\left({\\frac {R_{\\odot }}{r_{c}}}\\right)^{2}\\approx 10\\%", "23bb5ace61d39fe73bb071c7b681b0c8": "\\left(I_{1}=\\int _{0}^{\\infty }{\\frac {1}{s^{2}+1}}\\,ds={\\frac {\\pi }{2}}\\right)=\\left(I_{2}=\\int _{0}^{\\infty }\\sin t\\,{\\frac {1}{t}}\\,dt\\right){\\text{, provided }}s>0.", "23bb7f57f852b9ad4a15d4172ca7f51c": "\\forall x\\in \\mathbb {N} .x\\div 0=0", "23bb838830357a14004d0a3d4c5ee9cb": "y_{2}(t).\\,", "23bbb5f94412bba42ad2d9146c557128": "i\\in {1,\\dots ,p}", "23bbc6767ef583c5c110f0d8600fc225": "\\rho =F\\cot ^{n}({\\frac {1}{4}}\\pi +{\\frac {1}{2}}\\phi )", "23bbda368ef2588119b20bfa2f0e7862": "(R\\rightarrow S)", "23bc026f54dc6c8dcdb51cdec3484dd4": "{\\overline {\\psi }}=\\psi ^{*}\\gamma ^{0}", "23bc4ea7524a65c4bd221c7a5fe3060a": "\\{1,1/\\ln 2,1.38\\times 10^{-23}\\}", "23bc67683a24998f3a433faa44e375aa": "(p,q)(r,s)=(pr-\\gamma s^{*}q,sp+qr^{*})\\,", "23bc91d4d8a0e8019afc06d91b1d5a32": "\\operatorname {core} (A)\\neq \\operatorname {core} (\\operatorname {core} (A))", "23bcc90039d834be67867ed3c50c3014": "=\\mathbf {J} _{\\mathrm {M} }+\\mathbf {J} _{\\mathrm {P} }\\ ,", "23bcf0fcbfbbbcb548cef20f0d777ce1": "Z=P(D|M)", "23bd5b653b621f2e3ec719b3f714137b": "W_{t}^{2}-t=V_{A(t)}", "23bdbca5262c205034b42c05238e547a": "\\scriptstyle \\mathbf {A} _{2}", "23bdc504737ab44ddc76420a05a0dc51": "g\\in \\operatorname {GL} (V).", "23be1cf7d1cce64e15bf457a6799241d": "U={\\begin{bmatrix}0&3\\\\0&0\\\\\\end{bmatrix}}.", "23be41a6881417ca0ae511ad34a155d2": "[{\\mathcal {L}}_{X},i_{Y}]\\alpha =[i_{X},{\\mathcal {L}}_{Y}]\\alpha =i_{[X,Y]}\\alpha ,", "23be5ec342503d75ec343bc8ab875dcb": "T;", "23be6d112e3978dd56e59b7b77d32cf7": "1-{\\frac {1}{2}}\\left({\\frac {c}{R}}\\right)^{2}+O\\left({\\frac {1}{R^{4}}}\\right)=\\left[1-{\\frac {1}{2}}\\left({\\frac {a}{R}}\\right)^{2}+O\\left({\\frac {1}{R^{4}}}\\right)\\right]\\left[1-{\\frac {1}{2}}\\left({\\frac {b}{R}}\\right)^{2}+O\\left({\\frac {1}{R^{4}}}\\right)\\right]{\\text{ as }}R\\to \\infty \\ .", "23be8d9c45c361293949eaaa4f0fc589": "ds={\\sqrt {1+\\left({\\frac {dy}{dx}}\\right)^{2}}}dx", "23be92cc1bd7e5281672bfd49ba0fce5": "\\phi ,\\psi \\in {\\rm {Fm}}", "23becc99d590ed7e80a82f683279d9e5": "C_{\\rho }(x,y)\\geq \\lambda ", "23bed8a531cf6e2aa30e4edb506de660": "|\\psi _{s'}^{b'}\\rangle ", "23bf0b74c7792113e1315e4b0c327f55": "|S-S(D)|\\leq kL,\\quad |S-S(\\gamma )|\\leq kL,", "23bf0b74e80cefeaaf6e21373d906360": "c=A{|S_{2}-S_{1}| \\over S_{2}}{f \\over S_{1}-f}\\,.", "23bf3694cdeeb4e0e6fd9a33fbeb863e": "V=V_{1}", "23bf484ade61fac15f9ba5ef09e7f6d1": "{\\mathbf {E}}(\\alpha _{m},\\beta _{n},\\gamma _{p})~=~{\\frac {jk\\eta }{k^{2}-\\alpha _{m}^{2}-\\beta _{n}^{2}-\\gamma _{p}^{2}}}~{\\mathbf {G}}_{mnp}~{\\mathbf {J}}(\\alpha _{m},\\beta _{n},\\gamma _{p})~~~~~~~~~~~~~~~~~~~~~~~~~(3.1)", "23bf904c0087862e9e96c68a2e0574a1": "A={\\frac {1}{4}}(60+{\\sqrt {10(190+49{\\sqrt {5}}+21{\\sqrt {75+30{\\sqrt {5}}}}}}))a^{2}\\approx 33.5385...a^{2}", "23bf95b1e11954528cbf52e4b91e5f85": "b(f)b^{*}(g)+b^{*}(g)b(f)=\\langle f,g\\rangle ,\\,", "23bf99ccc3bbd46e0635c7260bc979c3": "{\\frac {\\partial {\\boldsymbol {\\mathit {1}}}}{\\partial {\\boldsymbol {A}}}}:{\\boldsymbol {T}}={\\boldsymbol {\\mathsf {0}}}:{\\boldsymbol {T}}={\\boldsymbol {\\mathit {0}}}", "23bfb08773af94a0fc5d4cd6a3fe49d6": "\\quad {\\text{if}}\\quad A\\subseteq B\\quad {\\text{then}}\\quad P(A)\\leq P(B).", "23bfe92930440fd633290fd303ad4dca": "{\\mathcal {GW}}^{-1}(\\mathbf {\\Psi } ,\\nu ,\\mathbf {S} )={\\mathcal {W}}^{-1}(\\mathbf {\\Psi } ,\\nu )", "23bfef6efe19f38f7b263d0b05451f90": "PV=nRT", "23c059d3aff1ac5ac14f9e2fce20dedc": "g=h^{-1}\\circ f\\circ h", "23c165292d248a6a2cc4b6e2e0a15a51": "(\\Lambda ^{n}A)(v_{1}\\wedge \\dots \\wedge v_{n})=\\det(A)\\cdot v_{1}\\wedge \\dots \\wedge v_{n}.", "23c1966daecfca8fa2e3356303d4c332": "\\psi (\\Omega ^{\\Omega ^{2}+\\Omega 3})", "23c1a48ad9fbe3ad64d7aa5385aa1e2e": "\\Sigma _{b}={\\frac {1}{C}}\\sum _{i=1}^{C}(\\mu _{i}-\\mu )(\\mu _{i}-\\mu )^{T}", "23c1d6e3d23eff609b8d40053726f979": "c(E)c(F)=1.", "23c1eebec72f1d26b3301755984c99af": "\\lambda =\\pi _{1}^{\\alpha _{1}}\\pi _{2}^{\\alpha _{2}}\\pi _{3}^{\\alpha _{3}}\\dots ", "23c2219ceec497e49ae22eb2293ade74": "2\\pi \\ a{\\sqrt {\\frac {a}{\\mu }}}\\,", "23c23f37519259cc0488f451412cf7b8": "T_{1L}=A-{\\frac {Bk_{1}}{k}}e^{-kL}", "23c24870014daf855f7f8b128123fda2": "{\\operatorname {d} \\over \\operatorname {d} x}e^{f(x)}=f'(x)e^{f(x)}", "23c24d0093409e12701e17c328ef0021": "-{\\frac {p{\\dot {r}}}{r^{2}}}=-\\varepsilon \\sin \\theta \\,{\\dot {\\theta }}", "23c264b6f6e6a07ae77dd2361052edbf": "\\epsilon =\\lambda _{\\mathbf {k}}\\pm |U_{\\mathbf {k}}|", "23c279316b512b1b58b126b8f6ee4478": "{\\begin{aligned}\\int _{\\gamma }|\\mathbf {x} |^{\\alpha -1}\\mathbf {x} \\cdot d\\mathbf {x} &={\\frac {1}{\\alpha +1}}\\int _{\\gamma }(\\alpha +1)|\\mathbf {x} |^{(\\alpha +1)-2}\\mathbf {x} \\cdot d\\mathbf {x} \\\\&={\\frac {1}{\\alpha +1}}\\int _{\\gamma }\\nabla (|\\mathbf {x} |^{\\alpha +1})\\cdot d\\mathbf {x} ={\\frac {|\\mathbf {q} |^{\\alpha +1}-|\\mathbf {p} |^{\\alpha +1}}{\\alpha +1}}\\end{aligned}}", "23c283d7c7b2825417f5c1a14e1a326a": "D_{F}^{y}(x,y)={\\tfrac {1}{2}}(x-y)^{T}Q(x-y)", "23c28afed27aef38002f04431081d759": "f^{-1}\\left(Z\\right)", "23c2a750e8fa642f81831a37a6e29cc4": "\\nabla (\\rho _{i}v_{i})", "23c2bf51098c5a3eaf30b5cf67a08690": "{\\text{End}}(X)", "23c32f7b4e87e1501a1919b688a597b1": "\\tan \\theta ={\\sqrt {{\\mathit {l}} \\over {\\mathit {l}}^{\\prime }}}", "23c384b02f4434e913e60342303544e4": "\\theta _{i}\\sim N(\\varphi ,\\tau ^{2})", "23c39258ce7abbc878ee06c9032b700e": "S^{+}=U^{+}", "23c39f47c90d0ff8fdc98599843b3f1b": "F=F_{3}(p,Q).", "23c468dd1827fd95bc5896543f9ea09c": "\\lim _{k\\rightarrow \\infty }\\phi (x,k)=f(x)", "23c4a4cdc7ec77a3051417cba3f0adad": "\\epsilon \\,\\!", "23c4f7f4f6c497d66282ecead1c3fe89": "[2]P=(0,a_{3})=-P", "23c4fdd03f5ce7d27dccf925a15650bf": "J_{k}(n)\\star 1=n^{k}\\,", "23c548c956f11a965860a2d604a44a01": "L={\\frac {\\alpha \\pi r}{180}}.\\,\\!", "23c562f36d97050019d1ad9ebf8b598f": "\\left[{\\begin{matrix}a&b\\\\b&c\\end{matrix}}\\right]", "23c592dc2d084b509b4b478ba107308b": "\\log P", "23c597174ff34699ce82b9efd1667324": "P_{\\ell }(\\mathbf {x} \\cdot \\mathbf {y} )={\\frac {4\\pi }{2\\ell +1}}\\sum _{m=-\\ell }^{\\ell }Y_{\\ell m}^{*}(\\theta ',\\varphi ')\\,Y_{\\ell m}(\\theta ,\\varphi ).", "23c5e9916fa1537ddd64310b9711af84": "H(f)", "23c615a2a27a4fa59b2dafbf99d2f587": "600\\,", "23c63758b6ac4633019e26628b9a045a": "\\scriptstyle f\\;=\\;t_{B}/(t_{A}\\,+\\,t_{B})\\;=\\;0.25", "23c6690a32ac89060a7d6db76c7b9e00": "\\log _{R}\\colon R\\to \\mathbb {C} ", "23c66a1c57b4b7dcabe64dcf376931ef": "r=f_{2}(\\theta )-f_{1}(\\theta +2\\pi ),\\ r=f_{2}(\\theta )-f_{1}(\\theta -2\\pi ),\\ \\dots ", "23c6a6b2aa19db9a65ed2a6b127b85f7": "\\scriptstyle \\beta >0", "23c6afa5185a0ce8befd144048f9cff7": "S\\subseteq \\Omega ", "23c6e52ef0e6dd1b810d94294b8814d0": "T'=0", "23c7619f9591397e0693cb905481b0f5": "\\langle 1,2\\rangle ", "23c7b7066a6682d5b74dba6c61d9353a": "\\lim _{(x,y)\\to (0,0)}{\\frac {x^{2}y}{x^{2}+y^{2}}}=0", "23c7e6f1043b05ab8e990b65072c70e1": "-{\\frac {\\hbar ^{2}}{2m}}{\\frac {\\partial ^{2}\\psi _{n}(z)}{\\partial z^{2}}}+U(z)\\psi _{n}(z)=E\\psi _{n}(z)", "23c83f25c5adef8eba735a863fb3b260": "{\\overline {\\mathsf {f}}}", "23c8bbdcc111df29633ca3286fdac77e": "v_{1}=30Hz", "23c8cf99d0de441831078d97d1767974": "\\;ord_{P}(G)=-\\deg(G)", "23c8e0c37863d9787a6eef8ec8b36595": "\\phi ={1 \\over {\\sqrt {2}}}\\left({\\begin{array}{c}\\phi ^{+}\\\\\\phi ^{0}\\end{array}}\\right)\\;,", "23c921d15b1eed7f54fd608bd050f2af": "\\varphi _{\\delta }(E)=\\inf {\\biggl \\{}\\sum _{i=0}^{\\infty }p(A_{i})\\,{\\bigg |}\\,E\\subseteq \\bigcup _{i=0}^{\\infty }A_{i},\\forall i\\in \\mathbb {N} ,A_{i}\\in C_{\\delta }{\\biggr \\}}.", "23c93efb376fe6e04cc6679ea7334630": "{\\begin{aligned}{\\frac {\\delta F[\\varphi (x)]}{\\delta \\varphi (y)}}&{}=\\lim _{\\varepsilon \\to 0}{\\frac {F[\\varphi (x)+\\varepsilon \\delta (x-y)]-F[\\varphi (x)]}{\\varepsilon }}\\\\&{}=\\lim _{\\varepsilon \\to 0}{\\frac {e^{\\int (\\varphi (x)+\\varepsilon \\delta (x-y))g(x)dx}-e^{\\int \\varphi (x)g(x)dx}}{\\varepsilon }}\\\\&{}=e^{\\int \\varphi (x)g(x)dx}\\lim _{\\varepsilon \\to 0}{\\frac {e^{\\varepsilon \\int \\delta (x-y)g(x)dx}-1}{\\varepsilon }}\\\\&{}=e^{\\int \\varphi (x)g(x)dx}\\lim _{\\varepsilon \\to 0}{\\frac {e^{\\varepsilon g(y)}-1}{\\varepsilon }}\\\\&{}=e^{\\int \\varphi (x)g(x)dx}g(y).\\end{aligned}}", "23c93f662a57841a6fa19e6541235a3b": "\\scriptstyle {\\hat {\\beta }}_{2}", "23c99dd46138f40f5a11fa6a8393bace": "\\sigma _{c}=\\limsup _{n\\to \\infty }{\\frac {\\log |a_{n+1}+a_{n+2}+\\cdots |}{\\lambda _{n}}}.", "23c9b37fc12b486b33472f6b4b3ee67a": "(\\psi *\\mu )*\\alpha \\equiv \\psi *\\alpha ", "23c9dd77a36cd09b6b9cc1652764e5b1": "\\forall x\\exists y\\exists z((\\lnot x=y)\\land xRy)\\land ((\\lnot x=z)\\land zRx)", "23ca036df52a0f463097dbd900fe2db9": "\\rho '={M_{i}\\rho M_{i}^{\\dagger } \\over {\\rm {tr}}(M_{i}\\rho M_{i}^{\\dagger })}", "23ca2799f321c43c6040f19abe563a70": "I(\\theta )=I_{0}\\left({\\frac {2J_{1}(ka\\sin \\theta )}{ka\\sin \\theta }}\\right)^{2}=I_{0}\\left({\\frac {2J_{1}(x)}{x}}\\right)^{2}", "23ca45b083741524ef60d5bba26241cd": "G=({\\frac {X}{E}})*R", "23ca8327f2391b57a0efc9af4d6d2dd5": "{\\begin{matrix}x=a\\sec t+h\\\\y=b\\tan t+k\\\\\\end{matrix}}\\qquad \\mathrm {or} \\qquad {\\begin{matrix}x=\\pm a\\cosh t+h\\\\y=b\\sinh t+k\\\\\\end{matrix}}", "23ca8707a3713f7ecdce63873914898c": "c_{1}=\\cos \\phi '\\qquad s_{1}=\\sin \\phi '", "23cb6ca4c2ccdf8d7ea726f2145aadde": "\\psi _{jk}(x)=2^{\\frac {j}{2}}\\psi (2^{j}x-k)\\,", "23cb7c2267ceb3b88c6590e0afb6363c": "x={\\begin{bmatrix}k_{1}\\\\k_{2}\\end{bmatrix}}c_{1}e^{\\lambda _{1}t}+{\\begin{bmatrix}k_{3}\\\\k_{4}\\end{bmatrix}}c_{2}e^{\\lambda _{2}t}.", "23cbb311f5bdd17fa67cfe566f03971b": "K_{\\infty }", "23cc1d479117eb4f2b7595c429a6838a": "\\{1\\}=G_{0}\\leq G_{1}\\leq \\cdots \\leq G_{k}=G", "23cc3455946bd067502400723ba748d6": "\\ MRS_{xy}=MU_{x}/MU_{y}.\\,", "23cc3cf75618fe3af7094ff8fbde2fd7": "\\textstyle _{2}", "23cc7fb43c8e28748be8b74e934a7405": "i", "23d348b1bb1082c3b8f29399e4ea57d1": "\\xi (B)", "23d373c00ae4d31d07131cd67bb1cb1e": "\\gamma _{x}(0)=x\\,", "23d37ee4c7752085ce47f0e08867d86e": "A^{*}A=(UP)^{*}UP=PU(PU)^{*}=AA^{*}.\\,", "23d3951ceb825305e9cdb59f6db21820": "D=3", "23d429e839a9f1e759eb789e303b9777": "\\tau _{\\mathrm {n} }\\,\\!", "23d436f386aa1c3879c3e6b928ed5a2f": "c_{ab}^{\\mathrm {opt} }(t)", "23d4677b4f913df994eb085f2af76f9a": "{\\frac {kg\\cdot m^{2}}{s}}=J\\cdot s.", "23d489baff18c52e301316fe66e0399f": "2\\cdot n", "23d4c274f1466e7a2aa4b978566e9fc1": "(F_{n}-1)/2", "23d4edadc63deda482784bcbe3f219b2": "{d \\over dz}V(a,z)=[L_{-1},V(a,z)]=V(L_{-1}a,z),\\,\\,[L_{0},V(a,z)]=(z^{-1}{d \\over dz}+\\alpha )V(a,z)", "23d4efad9cb40c1e5b4cf17bcffcd1c6": "R+\\delta \\leq 1+{\\frac {1}{n}}", "23d549d108ac186706cfd25e7bd3fd99": "x^{4}-2s(t^{2}+1)x^{2}+s^{2}t^{2}(t^{2}+1)", "23d5798ed0ba2cab790a634bfe5d8ff7": "f_{w}(z)={\\frac {1}{1-wz}}", "23d5b4ec9c8ed6d5fabbedc26af854e7": "\\left(V_{i}-V_{o}\\right)DT-V_{o}\\delta T=0", "23d5d35448c76214b072a4574b261c6a": "R_{x\\rightarrow y}={K_{x\\rightarrow y}(dt)-\\delta _{xy} \\over dt}\\,", "23d5d5902487a8ef1d4d4ce0cc26b6d6": "G^{(i)}\\neq G^{(j)}", "23d628ba6e7d5d6e226f573c92b65df5": "x=s+n\\ ", "23d65004d7f63b6cb23e01d2c0ffa366": "x\\in L\\iff \\exists t_{1},t_{2},\\dots ,t_{|r|}\\,\\forall r\\in R\\bigvee _{1\\leq i\\leq |r|}(M(r\\oplus t_{i}){\\text{ accepts}}).", "23d6a3dd8d83e20ff7f35d3ab53b8256": "c_{7}=0.000687678,\\,\\!", "23d6b9df3f2e80bc5ffc16bb97e02a71": "\\scriptstyle {\\frac {U}{V}}\\;\\sim \\;\\mathrm {Pareto} (1,\\,n)", "23d6eefab6856ed4358d2b9441680a8e": "|\\det(A)|=|\\det(R)|={\\Big |}\\prod _{i}r_{ii}{\\Big |},", "23d71a6b56b5391a141bb8d367ffbf48": "\\delta _{\\nu _{1}\\dots \\nu _{p}}^{\\mu _{1}\\dots \\mu _{p}}=\\sum _{\\sigma \\in {\\mathfrak {S}}_{p}}\\operatorname {sgn}(\\sigma )\\,\\delta _{\\nu _{\\sigma (1)}}^{\\mu _{1}}\\cdots \\delta _{\\nu _{\\sigma (p)}}^{\\mu _{p}}=\\sum _{\\sigma \\in {\\mathfrak {S}}_{p}}\\operatorname {sgn}(\\sigma )\\,\\delta _{\\nu _{1}}^{\\mu _{\\sigma (1)}}\\cdots \\delta _{\\nu _{p}}^{\\mu _{\\sigma (p)}}.", "23d739962942fc054c55a89cd67e9d6b": "\\{(-,+,+,+);l^{a}n_{a}=-1\\,,m^{a}{\\bar {m}}_{a}=1\\}", "23d82421eca87bfbc476709b38271e5d": "h=S+1\\mathrm {{\\tfrac {BTU}{lb}}\\;} Wt'", "23d866926e56d1a284f29e1a6190294e": "A\\cdot \\neg B\\cdot \\neg C+A\\cdot B\\cdot C\\,", "23d8a7ee54a72bb853d21a58b7d1f591": "S(r_{ij})={\\begin{cases}0,&{\\mbox{if }}r_{ij}\\leq R_{L}\\\\{\\frac {(r_{ij}-R_{L})^{2}(3R_{U}-R_{L}-2r_{ij})}{(R_{U}-R_{L})^{2}}},&{\\mbox{if }}R_{L}\\leq r_{ij}\\leq R_{U}\\\\1,&{\\mbox{if }}R_{U}\\leq r_{ij}\\end{cases}}", "23d8b8d110bde0cd6d2bd8ae7e32607c": "x_{1}=0.0+106/1121=0.09455842997324", "23d8c75253607767632b097c2e5ffb97": "(1,2)_{\\frac {1}{2}}", "23d8ccc41660f6590d657c10229bf389": "\\Gamma (s)=\\int _{0}^{\\infty }t^{s-1}\\,e^{-t}\\,{\\rm {d}}t=\\lim _{x\\rightarrow \\infty }\\gamma (s,x)", "23d8eaae7fdb5e514a3937b4974d508e": "E=-\\sum _{ij}J_{ij}S_{i}S_{j}\\,", "23d90a6f0d8da5d137980ba2a3110305": "3^{6}(1\\times 2\\times 3\\times 4\\times 5\\times 6)\\equiv (1\\times 2\\times 3\\times 4\\times 5\\times 6){\\pmod {7}}.\\,\\!", "23d919c5c2b9f223dcdad8e4e6bd8d34": "R=\\rho _{1}-\\rho _{2}={\\frac {Q(x)Q(x^{5})}{Q(x^{3})^{2}}}.", "23d9609b985033ce76fec3fffc3633d8": "r_{s}<1", "23d973454eb4411090d7caa2e562715e": "-j1.52={\\frac {-j}{\\omega L_{m}Y_{0}}}={\\frac {-jZ_{0}}{2\\pi fL_{m}}}\\,", "23d997d4996e0b119cdbb7ef6531051c": "V=({\\frac {\\sqrt {3}}{2}})a^{3}\\approx 0.866025...a^{3}", "23da1c930bd68cabcda4a09b5794ad24": "{}^{n}i", "23da2548db5f0c6be9a7f74924bb6bc0": "{\\overline {x}}=\\left(x_{1}+\\cdots +x_{n}\\right)/n", "23da5aa14c1980e5e97d34b2f13df592": "\\Phi =0", "23dab08094c604888829a5ec46d6f668": "C_{0}=I", "23dae17f89257c3a092a043dc7dd1be4": "n\\in N", "23db1b7eda16b0d2612259db4a694f74": "R={\\sqrt {(E^{2}-P^{2})\\,s^{2}+2\\,s\\,{\\sqrt {(E^{2}-P^{2})\\;R_{0}^{2}-L^{2}+R_{0}^{2}}}}}", "23db274b4bbbabc08f31b9010282d9b5": "\\displaystyle H_{k+1}=B_{k+1}^{-1}", "23db6b295827d9e0550a2e70a566bc23": "{\\frac {1}{1-x}}=\\sum _{n=0}^{\\infty }x^{n}=1+x+x^{2}+x^{3}+\\cdots ,", "23dc0259bef08d9fe92fec01cd597a30": "r_{min}", "23dc2c615dc734f7d54bff5766375c1f": "q\\in [0,\\infty )", "23dc4c955c65ff558a575af49e4fe16d": "\\varprojlim _{i\\in I}A_{i}={\\Big \\{}{\\vec {a}}\\in \\prod _{i\\in I}A_{i}\\;{\\Big |}\\;a_{i}=f_{ij}(a_{j}){\\mbox{ for all }}i\\leq j{\\mbox{ in }}I{\\Big \\}}.", "23dc89fb40cf563663c82871b6047fa0": "\\int _{0}^{1}n(x)dx", "23dcd751d57c61c044b858a78891b740": "E_{hh}", "23dd20f26e81a98748a108e77865a369": "S_{R}f=\\int _{-R}^{R}{\\hat {f}}({\\xi })e^{2\\pi ix\\xi }\\,d\\xi ", "23dd262e42fcb11ceb6d5d047ac7c9e3": "{\\frac {\\sin {\\theta }}{v}}={\\frac {1}{v}}{\\frac {dx}{ds}}={\\frac {1}{v_{m}}}", "23dd2f81505a1251726e530eda0f6b7c": "{\\mbox{rank}}_{+}(A)=\\min\\{q\\mid \\sum _{j=1}^{q}R_{j}=A,\\;{\\mbox{rank}}\\,R_{1}=\\dots ={\\mbox{rank}}\\,R_{q}=1,\\;R_{1},\\dots ,R_{q}\\geq 0\\},", "23dd2f927191e44f73f4d781ed3dcf4c": "\\omega (t)", "23dd469c4426142d58c4042044a3d620": "f(\\theta _{m})", "23dd9612439c9135b55719bba1c80165": "P\\lor (Q\\land R)", "23ddbc15d0e477a2e23e2208974b5d24": "N_{0}=\\oplus _{i\\geq 1}(M_{i}\\ominus N_{i})\\quad \\oplus \\quad \\oplus _{j\\geq 0}(N_{j}\\ominus M_{j+1})\\quad \\oplus R.", "23ded4c3f6dae981aea9b1ad7949f3e3": "D=0", "23df15e889065fea6de753e93a33600d": "S_{m}=\\int d^{4}x{\\sqrt {g}}G_{N}L_{m}\\;", "23df1c2774dc2fcdc6d8929f0ab1998c": "T(p)=Ap+o\\!", "23df1f25c482a1f9f005d6d28900efcc": "1\\,+\\,2\\left({\\frac {1}{8}}\\right)\\,+\\,4\\left({\\frac {1}{8}}\\right)^{2}\\,+\\,8\\left({\\frac {1}{8}}\\right)^{3}\\,+\\,\\cdots .", "23df3a6fff1a01d6ec00692b0deab731": "\\mathbb {Z} _{N}^{*}", "23df6d2c6361af6323c3ff0636fe535d": "(X,A)", "23dfaa15305ed565e20cd0db432e9357": "N=\\prod _{l\\in S}l>4{\\sqrt {q}}.", "23e0399205ca41b972912e899bf07e02": "{\\begin{aligned}c^{2}&{}=(a-b\\cos \\gamma )^{2}+(-b\\sin \\gamma )^{2}\\\\c^{2}&{}=a^{2}-2ab\\cos \\gamma +b^{2}\\cos ^{2}\\gamma +b^{2}\\sin ^{2}\\gamma \\\\c^{2}&{}=a^{2}+b^{2}(\\sin ^{2}\\gamma +\\cos ^{2}\\gamma )-2ab\\cos \\gamma \\\\c^{2}&{}=a^{2}+b^{2}-2ab\\cos \\gamma \\,.\\end{aligned}}", "23e089b13385e1e642c5ee3a4bdfd271": "4\\pi ^{2}a^{3}/T^{2}=\\mu ", "23e0a1aad2df9118e737b91371ef9b49": "(f\\ast g)(a)=\\sum _{b\\in G}f(ab^{-1})g(b).", "23e0d1abd0f4c120a04dd9fbec8c343d": "Int1\\,", "23e10a53e00dddee62ab390541c84a23": "y^{2}+\\left(x-a\\coth \\tau \\right)^{2}={\\frac {a^{2}}{\\sinh ^{2}\\tau }}", "23e19a5e67f634103bab55a89e40af40": "\\cdot \\left({\\text{largest monomial of }}s_{n-1}\\right)^{i_{2}-i_{1}}", "23e1a7473140f07ec726d8c2a56be341": "L+R", "23e1b428b0fc5151e814bb5aecd1cbf4": "d={{d}_{p}}+{{d}_{n}}={\\sqrt {{\\frac {2\\varepsilon }{q}}{\\frac {{{N}_{A}}+{{N}_{D}}}{{{N}_{A}}{{N}_{D}}}}\\left(\\underbrace {{V}_{bi}} _{\\text{ built-in voltage}}-\\underbrace {V} _{\\begin{smallmatrix}{\\text{external applied}}\\\\{\\text{voltage}}\\end{smallmatrix}}\\right)}}", "23e1d14f7ed944578835806021118038": "I=2\\pi i{\\frac {\\exp({\\tfrac {\\pi i}{4}})}{-1+i}}\\left({\\frac {17}{4}}-5^{\\frac {3}{4}}2^{\\frac {1}{4}}\\right)=2\\pi 2^{-{\\frac {1}{2}}}\\left({\\frac {17}{4}}-5^{\\frac {3}{4}}2^{\\frac {1}{4}}\\right)", "23e235bf17f43bad9cbba7d643410429": "(\\alpha \\mathbf {I} -A)R_{\\alpha }g=g.", "23e2421d9e55c3e325e80695c3febcb8": "\\pi ^{ab}/tor", "23e305f5c59cfde43b557424c099f31f": "A\\rightarrow B:\\{N_{A},A\\}_{K_{PB}}", "23e34142844417943efef1fc7b90b4e6": "(X,Y)", "23e39697bf142d80db47f28147d96238": "\\scriptstyle \\,s_{8,4}=0.7853982...(+4.7\\times 10^{-8})", "23e39b6ca29aa8919650c25d9c13c784": "{\\mathbf {v}}_{1}=\\left(-{\\frac {5}{9}}\\right){\\mathbf {v}}_{2}+\\left(-{\\frac {4}{9}}\\right){\\mathbf {v}}_{3}+{\\frac {1}{9}}{\\mathbf {v}}_{4}.", "23e3dca93afb4313a95dddb2168027d9": "w=g_{1}g_{2}\\cdots g_{n}", "23e409125d0dec1b5e8de58c0471a991": "L_{1},\\ldots ,L_{m}", "23e4466cef9c96eeb0957ac8bdc22166": "A(\\theta ,|\\varepsilon _{1}|)={\\frac {|\\varepsilon _{1}|+1}{|\\varepsilon _{1}|-1}}{\\frac {4}{1+\\tan {\\theta }/|\\varepsilon _{1}|}}", "23e48292968e5312bc26785270f6dfd8": "\\Delta f=\\mathrm {tr} (H(f))\\,\\!", "23e488a0e115510162ebf0de68f86fe5": "T(X_{1}^{n})=\\left(\\prod _{i=1}^{n}{x_{i}},\\sum _{i=1}^{n}{x_{i}}\\right)", "23e48f687e5e6d8396b088bc24a62aa3": "F_{3}(x)=-{\\frac {7}{12}}x-{\\frac {5}{12}}x^{3}", "23e49ae1e02da2126813c8a2b7883a62": "TAT^{-1}", "23e4cb5ae1fa2cc6a7392eba7f8fa43a": "B=X^{T}NX=\\int _{-1}^{1}\\int _{-1}^{1}[x_{1}^{j}x_{2}^{i}\\delta (x_{1},x_{2})]_{i,j=0}^{i,j=m-1}dx_{1}dx_{2}", "23e4cef02c3a564e22687b8dd764d166": "G(n)={\\begin{cases}0&{\\text{if }}n=0,-1,-2,\\dots \\\\\\prod _{i=0}^{n-2}i!&{\\text{if }}n=1,2,\\dots \\end{cases}}", "23e56d5fa289f90d21c09761263ff3fe": "E(x)=x+e_{0}", "23e583672624c06585e951e656d20cfe": "D=({\\frac {16T_{\\max }}{\\pi {\\tau }_{\\max }}})^{1/3}", "23e63a25f7824e2986f9f43ed10cf207": "\\mathrm {Ref} (\\theta )={\\begin{bmatrix}\\cos 2\\theta &\\sin 2\\theta \\\\\\sin 2\\theta &-\\cos 2\\theta \\end{bmatrix}}.", "23e6510a92f69fd27b75330c38cdc75c": "\\varepsilon _{\\mu \\nu \\rho \\sigma }", "23e6586a97e32cca35d8ab3252372f3a": "V(x)\\rightarrow -V(x)", "23e669d6ad693609e0e8f00a802d066d": "10\\uparrow \\uparrow (7.21\\times 10^{8})", "23e680d0444b05718f4ef44806aa398d": "\\left[{\\frac {d}{d\\nu }}K_{\\nu }\\left({\\sqrt {ab}}\\right)\\right]_{\\nu =p}", "23e693f316b16953ff7b32c0edd2374f": "F(i)=\\sum _{j=1}^{i}f(j).", "23e6fe08f9235acfffcc2f35b7323d4f": "{\\vec {b}}\\cdot {\\vec {c}}=\\Vert {\\vec {b}}\\Vert \\Vert {\\vec {c}}\\Vert \\cos \\theta ", "23e7447a67cc1c81672b56dd0124e598": "\\left\\langle {\\tfrac {-1+\\mathrm {i} {\\sqrt {7}}}{2}},Z_{2}\\right\\rangle ", "23e77967ec81ef64052d2a3a5e7c3941": "\\min _{g}\\sum _{i=1}^{n}w_{i}(g(x_{i})-f(x_{i}))^{2}", "23e7b16af5dc92102362735e2ca02351": "f(0)=f(1)", "23e7cd22bbd57dad770596c468508ecd": "\\mathbf {r} =\\mathbf {i} -2\\left(\\mathbf {i} \\cdot \\mathbf {n} \\right)\\mathbf {n} ", "23e7f62effdf1421d013678a23dd6659": "\\sum _{k=0}^{\\infty }|a_{k}|^{2}r^{2k}\\leq {M_{r}}^{2}", "23e84d859854635723fb981e89a57b9d": "\\left.+{\\frac {|V_{nk_{1}}|^{2}}{E_{k_{1}n}^{2}}}\\left({\\frac {3|V_{nk_{2}}|^{2}}{4E_{k_{2}n}^{2}}}-{\\frac {2|V_{nn}|^{2}}{E_{k_{1}n}^{2}}}\\right)-{\\frac {V_{k_{2}k_{3}}V_{k_{3}k_{1}}|V_{nk_{1}}|^{2}}{E_{nk_{3}}^{2}E_{nk_{1}}E_{nk_{2}}}}\\right]|n^{(0)}\\rangle ", "23e8b9d5ba66e70b9908c75c554fcdc8": "NI=B\\left({\\frac {L_{\\mathrm {core} }}{\\mu }}+{\\frac {L_{\\mathrm {gap} }}{\\mu _{0}}}\\right)\\qquad \\qquad \\qquad \\qquad (1)\\,", "23e8e5e378c9b93c40d2764e452093e4": "{\\tilde {D}}_{5}", "23e92ab7b76eb6aac23963e0878f58a1": "\\forall i\\;\\alpha _{i}+c_{i}>1", "23e94a5b63b9b7094898ecb49f0863b9": "(x,{\\sqrt {1+\\langle x,x\\rangle }})\\in R^{n+1}", "23e9629c607ede03f40f01c4cea9d240": "M+X^{+}+A\\to MX^{+}+A", "23e97042c4556020f81aac26ef95e508": "\\mathbf {a} _{A}=\\mathbf {a} _{B}+2{\\boldsymbol {\\Omega }}\\times \\mathbf {v} _{\\mathrm {B} }+{\\frac {d{\\boldsymbol {\\Omega }}}{dt}}\\times \\mathbf {x} _{\\mathrm {B} }+{\\boldsymbol {\\Omega }}\\times \\left({\\boldsymbol {\\Omega }}\\times \\mathbf {x} _{B}\\right)\\ .", "23e972dc5b9e5bf3522364d5e3f768d8": "\\mathbf {M} {\\vec {v}}_{1}=\\sigma _{1}{\\vec {u}}_{1}", "23e9977507f20de084570a97152b3af8": "y(n_{1},n_{2})=\\sum _{r_{1}=0}^{N_{1}-1}\\sum _{r_{2}=0}^{N_{2}-1}{a(r_{1},r_{2})x(n_{1}-r_{1},n_{2}-r_{2})}-\\sum _{l_{1}=0}^{M_{1}-1}\\sum _{l_{2}=0}^{M_{2}-1}{b(l_{1},l_{2})y(l_{1},l_{2})}", "23ea0a754ade8b53333d854ad775b358": "\\mathbb {C} P^{\\infty }", "23ea4cbdd18453356a82bfb9216bf0c9": "{\\frac {\\partial (\\mathbf {a} \\cdot \\mathbf {u} )}{\\partial \\mathbf {x} }}={\\frac {\\partial \\mathbf {a} ^{\\rm {T}}\\mathbf {u} }{\\partial \\mathbf {x} }}=", "23ea65cd896afe45430888242b5e61d7": "N={\\frac {n+\\Delta }{2}}\\,", "23ea6dea311ba937d7450552d8beaaf2": "p_{N}={\\frac {1}{1+e^{-\\Delta G/RT}}}", "23ea7d9b55a3bdb471880901cc5b7758": "\\ d\\theta /dt", "23ea985f6cd0a19f97cc6e3f1f4b64f2": "\\displaystyle {\\iint S(\\varphi _{1})_{x}S(\\varphi _{2})_{x}+S(\\varphi _{1})_{y}S(\\varphi _{2})_{y}=\\int _{\\partial \\Omega }S(\\varphi _{1})\\varphi _{2}.}", "23eb297c3a6ee517ff8f26e031907e69": "{\\tilde {R}}", "23eb8bffcb179a47fe2ea4c5bf3dc0b3": "w^{f}(f^{*})I(Y;Z)", "23fae11cdff623c92b7990de4ea689c4": "y\\cdot a=b", "23fb5e1f04d1bbf245f9436fa5b401af": "OLD(T_{i}).\\mathrm {add} (O_{j},WTS(O_{j}))", "23fbdee7ae3251bc2a9a4387e84c1730": "X_{1}\\oplus X_{2}\\oplus \\cdots \\oplus X_{r}\\cong Y_{1}\\oplus Y_{2}\\oplus \\cdots \\oplus Y_{s}", "23fc24180ed54848815b5f089cb85e77": "m,E", "23fc4dab9dcc62c777316418c76498f0": "\\mathbf {A} ={\\begin{bmatrix}\\mathbf {A} _{1}&0&\\cdots &0\\\\0&\\mathbf {A} _{2}&\\cdots &0\\\\\\vdots &\\vdots &\\ddots &\\vdots \\\\0&0&\\cdots &\\mathbf {A} _{n}\\end{bmatrix}}", "23fca4b3ca08964ff17e0ccda7ec165e": "{\\frac {\\partial u}{\\partial \\varphi }}=-{\\frac {\\partial u}{\\partial x}}r\\sin \\varphi +{\\frac {\\partial u}{\\partial y}}r\\cos \\varphi =-y{\\frac {\\partial u}{\\partial x}}+x{\\frac {\\partial u}{\\partial y}}.", "23fd65060978db561c7263597635bb4f": "\\mathbb {Z} [x]/\\langle f\\rangle ", "23fdd249f8a14a37ce304f95c445d21b": "W_{d}={\\frac {\\pi \\;d^{2}\\;\\sigma _{f}^{2}\\;l_{d}}{24\\;E_{f}}}", "23fe60ab88b9d0143b905bfe4312ab76": "P_{F_{4}}(x)=(1+x^{3})(1+x^{11})(1+x^{15})(1+x^{23})", "23fec5a7dd51d840c9c9001300fc9815": "\\int \\!\\!\\!\\!\\int \\!\\!\\!\\!\\int _{V}\\left(\\mathbf {\\nabla } \\cdot \\mathbf {F} \\right)dV=", "23fed1bf4ae66d6f433e2976253a6c36": "f_{j}(\\mathbf {x} )=f_{j}(\\mathbf {\\phi } +\\Omega ^{-1/2}\\mathbf {\\xi } )=f_{j}(\\mathbf {\\phi } )+\\Omega ^{-1/2}\\sum _{i=1}^{N}{\\frac {\\partial f'_{j}(\\mathbf {\\phi } )}{\\partial \\phi _{i}}}\\xi _{i}+O(\\Omega ^{-1}).", "23ff1f724367fb5e7e84d4a4119fb0e2": "\\theta _{1}=q\\theta +\\theta _{0}=q\\varphi +\\theta _{0}", "23ff39fb1f877cc41825a04a56ff796b": "{\\begin{aligned}\\xi &=\\int f_{3}(x)E^{2}~dx,\\\\[6pt]u&=\\left(y+{\\dfrac {f_{2}(x)}{3f_{3}(x)}}\\right)E^{-1},\\\\[6pt]E&=\\exp \\left(\\int \\left(f_{1}(x)-{\\frac {f_{2}^{2}(x)}{3f_{3}(x)}}\\right)~dx\\right)\\end{aligned}}", "23ff4f88d24b2ab879022e08470371d1": "(2n+1)\\times {(2n^{2}+2n+3) \\over 3}", "23ff52fec673aefb0fe63a8a5a515835": "K_{s}+(K_{1}\\cup K_{t})(s,t\\geq 1)", "23ffddee7308c3074e598b33ea734cb9": "\\int _{B_{r}(x)}|u(y)-u_{x,r}|^{2}dy\\leq Cr^{n+2\\alpha },", "2400327ee0a9ab5f6ac42bc5d9f68fa3": "\\psi _{1}^{*}\\mu \\psi _{2}", "240134b8dfc597471cebc9632476d8b5": "g(x)=2x^{2}+x", "24014aad976921599876393ccb4505b2": "N(T+H)-N(T)\\geq cH\\log T", "24017038961c2e19b400d9d48161f2fc": "\\psi (x)|N;x_{1}...,x_{N}\\rangle =\\delta (x-x_{1})|N-1;x_{2}...,x_{N}\\rangle +\\delta (x-x_{2})|N-1;x_{1},x_{3}...,x_{N}\\rangle +\\ldots \\,", "240173ccb9bbf446b512851fa10562cf": "{\\frac {\\mbox{Net Profit}}{\\mbox{Net Sales}}}", "24018524cadd986ddc0c4a5ac8ae1bc3": "\\|x-y\\|^{2}=\\langle x-y,x-y\\rangle =\\langle x,x\\rangle -\\langle x,y\\rangle -\\langle y,x\\rangle +\\langle y,y\\rangle .\\,", "2401a92796de212edebb573a3f9bb570": "2{_{1}^{1}}{\\text{S}}+{\\text{E}}{\\overset {\\xrightarrow {{\\text{k}}_{1(3)}} }{\\xleftarrow[{{\\text{k}}_{2(3)}}]{}}}{\\text{C}}_{3}\\xrightarrow {{\\text{k}}_{3(3)}} {_{2}^{2}}{\\text{P}}+{\\text{E}},", "2401f2f849d12bf941087bce37b05e47": "h^{*}Y\\subset Y", "24022c769eecbaedaafc669396c108f8": "\\scriptstyle r({\\boldsymbol {r}}_{i},\\,{\\boldsymbol {r}}_{\\text{rec}})/c\\,+\\,(t_{i}\\,-\\,t_{\\text{rec}})\\,+\\,\\delta t_{{\\text{atmos}},i}\\,-\\,\\delta t_{{\\text{meas-err}},i}\\;=\\;0", "24025cc13aa5ee724bda1e2ee758faac": "R_{1}={\\frac {1}{2!}}f^{\\prime \\prime }(\\xi _{n})(\\alpha -x_{n})^{2}\\,,", "2402a79d9775fe2647dc4c3f7ff5d08e": "W_{ij}(\\tau ;L)=\\sum _{n=0}^{\\infty }w_{ij}(\\tau ,2n+\\gamma _{ij};L).", "2402a9d2129977cf71ca984aace5945c": "\\in [0,1]", "24032c923c6a222208b06b4fcbc32f4e": "{\\epsilon _{F}}_{p}", "24033353660ebc9cdcd08b42a4b48053": "\\Delta x=S\\cdot {\\frac {d}{D}}={\\frac {S}{D}}\\cdot d", "24034ed2a38db3e976df517686f2ca99": "F_{\\varphi }=mr{\\ddot {\\varphi }}+2m{\\dot {r}}{\\dot {\\varphi }}\\ .", "24036210a715b56461455780d4328a8e": "Q_{p}-p\\Delta V\\;", "2403a5949bf4e2a162f178426657a24e": "K_{0}(kz)\\,", "2403c7d6aadc2e805495cee7a9b4a01b": "h(\\gamma )=\\int _{0}^{1}\\log \\,(Q'(p;\\gamma ))\\,\\mathrm {d} p=\\log(\\gamma )\\,+\\,\\log(4\\,\\pi ).\\!", "2403c99af623e6d14539dcd1322f91d6": "C_{V}=3Nk\\left({\\epsilon \\over kT}\\right)^{2}{e^{\\epsilon /kT} \\over \\left(e^{\\epsilon /kT}-1\\right)^{2}}", "2403e2473d422a58a37da1e30257a255": "\\left|\\psi (t+s)\\right\\rangle =U(t)\\left|\\psi (s)\\right\\rangle ", "240410838ce7ff09ca6f36b495ce9419": "\\mathbb {K} ({\\mathfrak {g}}^{*})", "240462904ee0fb4460a569d0bfb64833": "y=A'y_{1}+B'y_{2}.", "2404d6607ee5fd25ec09349c0b8d79a4": "(x^{2}+y^{2})x=2ay^{2}", "2404e5f41f5b9d922c6b25ad9adc2d79": "y_{1}\\cdot x_{1}=b", "24057184845d7591124f3a972dd8fc9a": "\\Im Z>0", "24059f8d3f14fbb9a073f7c7695a972c": "H(z)={\\frac {1}{3}}+{\\frac {1}{3}}z^{-1}+{\\frac {1}{3}}z^{-2}={\\frac {1}{3}}{\\frac {z^{2}+z+1}{z^{2}}}", "2405cafa6b9395667a87d54422165fdc": "\\operatorname {Li} _{2}(1)={\\frac {{\\pi }^{2}}{6}}", "2405d1c907533cea98024ef7160926f3": "u(D)=u(E)\\ ", "2405d1e7aeea61f12b8e810e9c2eca12": "=SP-PV", "2405e4da3da2906003cd5a1d3c2e0adf": "\\left(a^{2}(-a^{2}+b^{2}+c^{2}),\\;b^{2}(a^{2}-b^{2}+c^{2}),\\;c^{2}(a^{2}+b^{2}-c^{2})\\right),\\,", "2406296eb2187f4e21db33e97bd039b4": "q\\leq \\sigma (n/p_{k}^{\\alpha _{k}})", "24067d95feb5392a997c89d626a35b1a": "{\\begin{aligned}x&=r(\\sin \\phi +\\phi )+C_{x}\\\\y&=r(\\cos \\phi )+C_{y}\\end{aligned}}", "24068064c988504afa7fe67630781aa5": "x_{2}^{1}+x_{2}^{2}\\leq \\omega _{2}^{tot}", "24071a7dd978247f26f5e21ebfb26a3f": "(\\psi _{j})_{j\\in J}", "24074b0aded9c4d45dd5f0f65d027202": "\\mathrm {ZnO+CO\\rightarrow Zn} (vapour)+\\mathrm {CO} _{2}", "2407a35c78f62f9a191c3e908b5a57af": "{\\tfrac {a}{c}}-{\\tfrac {b}{d}}\\cdot {\\tfrac {a}{c}}={\\tfrac {a(d-b)}{cd}}", "2407c25545c0b1d8bb34048625ae4d8e": "{\\bigl (}L^{p_{0}}(R,\\Sigma ,\\mu ),L^{p_{1}}(R,\\Sigma ,\\mu ){\\bigr )}_{\\theta }=L^{p}(R,\\Sigma ,\\mu )\\ \\ {\\text{if}}\\ \\ {\\frac {1}{p}}={\\frac {1-\\theta }{p_{0}}}+{\\frac {\\theta }{p_{1}}},", "24084c207c0e3e676fa8096de415fbe7": "\\scriptstyle \\psi ", "2408886c8483c573f783bc106b63664f": "He^{-\\alpha L/2}=B\\cos(kL/2)", "2408cb4d8ee3e0022e9cc798ff9cdc0c": "A_{m}^{n}", "2408ea4b733afbff3c9c0d237012453d": "\\theta _{i,1\\dots V};", "2408ee21a1fa3d4c111be443bc5995ab": "e^{\\operatorname {Log} (z)}=z", "2409490d9a3ebf3bdd6c260322ef4beb": "\\xi \\rightarrow 0", "240995c3564979561dc6c553378a32ed": "\\textstyle v_{\\infty }", "2409e0cc6c54f2521b9e2662fe74abbb": "\\rho {\\frac {d{u}}{d{t}}}=\\mu \\,\\Delta u-\\nabla p+\\Lambda [\\Upsilon (V-\\Gamma {u})]+\\lambda +f_{thm}(x,t)", "2409f4ea11c6c5105d5142778466be1a": "H\\psi =E\\psi ", "240a0160550f4bbeb9314b34bdaac144": "=\\delta \\int d^{4}x\\;e\\;e_{M}^{[\\gamma }e_{N}^{\\beta ]}C_{\\gamma }^{\\;\\;\\;MK}C_{\\beta K}^{\\;\\;\\;\\;N}", "240ac9bf44a953a5c69f54afe5c26e4b": "c_{\\mathrm {d} }\\,,", "240b014a3e2b839d3fde8e6f2771b15c": "\\lambda _{A}^{\\dagger }=\\lambda _{A}^{-1}:A\\rightarrow I\\otimes A", "240b9031d3ea0805ea08027ce0da089b": "P(G,{\\mathcal {X}},{\\mathcal {Y}})=\\{(A,g)\\in {\\mathcal {Y}}\\times G:g^{-1}A\\in {\\mathcal {Y}}\\}", "240bbf26af031154a783e1e50f18a824": "{\\frac {{\\text{d}}E}{{\\text{d}}t}}=-\\sum _{i}{\\frac {{\\text{d}}C_{i}}{{\\text{d}}t}}\\qquad \\qquad (8d)", "240bc0e838354eaa0340663feba3372e": "{\\text{DSPACE}}\\left(\\left(\\log n\\right)^{2}\\right)", "240bcde4ee17a54e39df97f90d3a91c7": "{\\alpha \\over 3}\\left((x-\\beta )^{3}+(\\beta -a)^{3}\\right)", "240be3b61f4a2fd8112c881bced19f8c": "B[t]", "240bee42bcd5d5bf642f9d695d73273a": "m>n", "240bf90ae5c4e2821bb0cc785672f2a0": "dA=h_{\\mu }h_{\\nu }d\\mu d\\nu =a^{2}\\left(\\sinh ^{2}\\mu +\\sin ^{2}\\nu \\right)d\\mu d\\nu =a^{2}\\left(\\cosh ^{2}\\mu -\\cos ^{2}\\nu \\right)d\\mu d\\nu ={\\frac {a^{2}}{2}}\\left(\\cosh 2\\mu -\\cos 2\\nu \\right)d\\mu d\\nu ", "240bfee86a84fd31cb1c595d292afce9": "|x_{2,i}-x_{1,i}|", "240c542a2de8ecff5d3dc6e6930a2e1b": "{dR \\over dx}=g=-{dV \\over dx}\\,", "240ca5d9ac67864959486d6164bb291a": "{\\vec {x}}=(x_{1},\\ \\ldots ,\\ x_{j},\\ \\ldots ,\\ x_{k})", "240cb752542bdd47e05557913bc1a2b1": "{\\frac {\\partial {\\bigg (}a-(q_{1}+q_{2}){\\bigg )}}{\\partial q_{i}}}\\cdot q_{i}+a-(q_{1}+q_{2})-{\\frac {\\partial C_{i}(q_{i})}{\\partial q_{i}}}=0", "240cca0f4ba92b82a32460272d802110": "{\\hat {\\lambda }}=median(Y_{1}^{2},Y_{2}^{2},\\ldots Y_{L}^{2})/0.456", "240cd6157888502d48efd27cad607796": "1-\\lambda =1-\\sum _{i=1}^{R}p_{i}^{2}=1-1/{}^{2}D", "240d21a3e55502570dea21946efdccbc": "\\lim _{\\Delta v\\rightarrow (\\Delta )}h_{\\alpha _{ij}}=I+{1 \\over 2}F_{ab}s_{i}^{a}s_{j}^{b}", "240d6ba63629fe52b91247e6495e9879": "\\alpha \\in \\mathbb {C} ", "240dfc8139c07b05ccdd3a5214b30c23": "p=\\sum _{i=0}^{n}10^{i}a_{i}", "240e9eb92207d1cc19b88ac26ae02e89": "\\delta _{n}(\\varepsilon )=\\varepsilon ^{n}", "240ed1caf4bfaead0e0a7ed2ac2c163c": "\\textstyle {\\mathcal {F}}", "240f20dd1ffdf90209159cbfd6a00eb1": "\\scriptstyle {\\frac {T}{4}}\\,", "240f35b706a7942e1d4bbaa9a69dcc45": "~\\beta +\\theta ~", "241011517ef934025e8046adb0d12150": "\\cosh \\tau ", "24102b8810e709c22e5fd469aeb382e4": "{\\boldsymbol {\\omega }}:={\\frac {1}{2}}[{\\boldsymbol {\\nabla }}\\mathbf {u} -({\\boldsymbol {\\nabla }}\\mathbf {u} )^{T}]", "24104125c56ad21393cf61aef16679cb": "(\\sigma (\\mathbf {x} ))(n)=x_{n+1}", "2410a721b635895e0ac56e581383749e": "\\alpha ^{k}\\neq 1", "2411211a4a69c1448a58cb0dd61b4cda": "b+q{\\sqrt {-1}}", "24113a23216377402ea721cf190d51dc": "\\exists k\\in \\mathbb {N} ", "241160933a4a8cf96721871427e08599": "\\operatorname {sink} [(\\lambda p.p)\\ (\\lambda f.\\lambda x.f\\ (x\\ x)),X]", "2411864754bbb6e5022835d9845e615e": "\\int _{Y}\\left(\\int _{X}|f(x,y)|\\,{\\text{d}}x\\right)\\,{\\text{d}}y", "24118873fb2c93ac1bd1c2f28a8a403c": "Happens(a,t_{1})", "2411b1411112cde09ae1a1955e0bd016": "f(t,z)=\\left(2t,{\\tfrac {1}{4}}z+{\\tfrac {1}{2}}e^{it}\\right).", "2411ca58d417a38ae2cf8b2aa8a68ef0": "d(f(x),f(y))<1/n", "24121a01794db7941870290c800103ae": "A(x)=\\sum _{n=-\\infty }^{\\infty }a_{n}e^{inx}", "24129487a0710fea4d16212dad58d52a": "\\langle \\rangle ", "2412be3c6a0e9b799c10c8518f456cd8": "\\left[{\\begin{smallmatrix}\\;\\,\\,2&-3\\\\-1&\\;\\,\\,2\\end{smallmatrix}}\\right]", "241379628a0d7176065b5a90525513ad": "\\left\\lfloor ~\\right\\rfloor ", "2413cc8e1087319b5805fc163f9dcb8d": "O(d)", "24146412282a6f4d924c894dfa0c78c8": "\\operatorname {E} [H(X)]=\\sum _{i=1}^{n}w_{i}\\operatorname {E} [H(X_{i})],", "24146b4863f968ca22d68077bd962523": "\\!w=p/\\rho ", "24146fae61cef46dcc98b85d1511eee2": "\\tau (X^{*},X)", "24147dce52ab7a70fab349e1189a1fd3": "{\\begin{aligned}\\sin x&=x-{\\frac {x^{3}}{3!}}+{\\frac {x^{5}}{5!}}-{\\frac {x^{7}}{7!}}+\\cdots \\\\&=\\sum _{n=0}^{\\infty }{\\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}},\\\\\\cos x&=1-{\\frac {x^{2}}{2!}}+{\\frac {x^{4}}{4!}}-{\\frac {x^{6}}{6!}}+\\cdots \\\\&=\\sum _{n=0}^{\\infty }{\\frac {(-1)^{n}x^{2n}}{(2n)!}}.\\end{aligned}}", "24149d372a375f03fdc210b4c5b73f2f": "288\\cdot V^{2}={\\begin{vmatrix}0&1&1&1&1\\\\1&0&d_{12}^{2}&d_{13}^{2}&d_{14}^{2}\\\\1&d_{12}^{2}&0&d_{23}^{2}&d_{24}^{2}\\\\1&d_{13}^{2}&d_{23}^{2}&0&d_{34}^{2}\\\\1&d_{14}^{2}&d_{24}^{2}&d_{34}^{2}&0\\end{vmatrix}}", "2414a7564965f430f73abda3413f8691": "\\alpha \\in \\mathbb {R} ", "2414a91c52961be6980ea4413cad3a1d": "\\sin(\\alpha \\pm \\beta )=\\sin \\alpha \\cos \\beta \\pm \\cos \\alpha \\sin \\beta ", "24150779bbb609f0a576e9ac392c13a6": "H[{\\vec {\\sigma }}]", "24155d52f61ea4f32cc8773ec814c18f": "n^{2}=1-{\\frac {X}{1-{\\frac {{\\frac {1}{2}}Y^{2}\\sin ^{2}\\theta }{1-X}}\\pm {\\frac {1}{1-X}}\\left({\\frac {1}{4}}Y^{4}\\sin ^{4}\\theta +Y^{2}\\cos ^{2}\\theta \\left(1-X\\right)^{2}\\right)^{1/2}}}", "2415617337a0c3166705cd79ae82b48c": "e_{1}e_{2}=e_{12}.", "24158b912b74355de1137f8c382faa43": "K[2(1-p)-\\Delta ]=(N-K)(2p+\\Delta )\\!", "2415ebc2189743f85b6261915877c515": "P=-P", "24164ff9fec86e44f7fc76b93f9225e2": "2A=1.\\,", "2416671ff27d8feb73fb4a9c9a84e204": "1{\\mbox{ year}}=10^{0}{\\mbox{ year}}=10^{0+7.50}{\\mbox{ seconds }}=10^{0.50+7}s=3.16*10^{7}s", "2416ea9ae300408b9a44ab8290980e4f": "P(i)={\\rm {tr}}(\\rho F_{i}),\\;", "2417104e69f3f67f7b49cd5bc9a47edf": "(A,B,C,D,E,F);", "2417ac592cb91d9bbf701d7c6fecb4ed": "F_{\\mu \\nu }=K_{\\nu ;\\mu }-K_{\\mu ;\\nu }\\;", "2417bbdb4cf4a028325e4672ada77d5c": "H^{H^{}}", "24181f9b52fb80ea5ae1328be6c21ee1": "\\{\\Gamma _{a}~,~\\Gamma _{b}\\}=2\\eta _{ab}I_{N}", "24184a0d99632378ab37c1380d5f4cec": "ax+by=\\gcd(a,b).", "2418b341ccfaf67090455c4d096a3c5d": "(ax+b)", "2418c6d324f6bf7a6fba298607222305": "{\\frac {d\\omega }{dt}}\\propto {\\frac {2\\pi }{\\hbar }}|\\langle \\psi _{f}|{\\hat {H}}'|\\psi _{i}\\rangle |^{2}\\rho _{f}=|M_{fi}|^{2}\\rho _{f}", "2418df3231cefa6b45e218a949f8be46": "\\rho ,\\phi ,z\\,\\!", "2419b93ef3604c5a3264af97a499c7d0": "L\\in \\mathbb {R} \\;", "2419f6ec37e18bef68fe519fed46bb58": "\\displaystyle \\left\\{2,3\\right\\}", "241aa15ca419f2b7a55a3d9768e491a7": "H(\\sigma )=-J\\sum _{}\\sigma _{i}\\sigma _{j}-h\\sum _{j}\\sigma _{j}.", "241aa9aa5d4399e2b72b1d2905628e03": "{\\text{Hom}}_{{\\mathcal {H}}_{s}(T)}({\\mathcal {Y}},{\\mathcal {Z}})\\to {\\text{Hom}}_{{\\mathcal {H}}_{s}(T)}({\\mathcal {X}},{\\mathcal {Z}})", "241acd54d1978b594b634c2a178c525d": "|\\beta A_{\\text{OL}}(f_{\\text{0 db}})|=1.\\ ", "241adb70191c5fdcf8696c8440c80a59": "1,2,6,24,120,600,4200,28560,257040,2207520,24282720,258128640,\\ldots ", "241b0850aaf82d4a063421daaf8930e7": "u(x(\\theta ),t(\\theta ),\\theta )\\geq {\\underline {u}}(\\theta )\\ \\forall \\theta ", "241b274a394e50ea293729c4f6c7573c": "({\\hat {A}}\\cdot {\\hat {B}})", "241b2b804deca23af2cd4e5bee12d971": "\\xi \\colon E\\rightarrow X", "241b326f29c91a4985c3c29e11defa23": "\\vartheta ^{*}", "241b6843cbc504c15aa56bcd7ada3c4c": "q'_{x}(a,b)=q'_{y}(a,b)=p_{d-1}(a,b)=0,", "241b6ae687af83d9105af1d39fe65871": "L={\\frac {\\kappa }{\\sigma T}}={\\frac {\\pi ^{2}}{3}}\\left({\\frac {k_{B}}{e}}\\right)^{2}=2.44\\times 10^{-8}\\,\\mathrm {W\\,\\Omega \\,K^{-2}} .", "241b93e940b673843d90cd22ff5769ae": "T_{n}\\left(e^{x}\\right)=e^{-e^{x}}{\\frac {d^{n}}{dx^{n}}}\\left(e^{e^{x}}\\right)", "241bc0e67997a62f6f01e5ffaee2ca94": "(-0)\\cdot (-0)=+0\\,\\!", "241c08a511cdb97665c315fea0ca70b7": "\\Delta x_{0}=-\\nabla _{x}f(x_{0})", "241cc27e21b71612d68e06bd075e7c3b": "{\\begin{matrix}4\\times 3&=&\\underbrace {4+4+4} &=&12\\\\&&3{\\mbox{ copies of }}4\\end{matrix}}", "241cd16f723e93837a908a7ab5cc744f": "x-x_{0}=-\\lambda (x_{P}-x_{0})", "241ceb844d4f74769f81f803fd5019c2": "{\\frac {F(n,k+1)}{F(n,k)}}", "241cfb1fae34d452e8a2c3f516b0ba09": "\\Lambda (f)\\geq 0\\,", "241d1e24afc5d646b2b85d67180c0877": "\\mathrm {B} (x,y)=2\\int _{0}^{\\pi /2}(\\sin \\theta )^{2x-1}(\\cos \\theta )^{2y-1}\\,\\mathrm {d} \\theta ,\\qquad \\mathrm {Re} (x)>0,\\ \\mathrm {Re} (y)>0\\!", "241d5eac62cecf2ddf71c3c1309b7a7d": "{\\frac {\\Gamma (s,x)}{x^{s}}}\\rightarrow -{\\frac {1}{s}}", "241d5fe1c7b6ae59665e690935915df3": "\\lim _{n\\to \\infty }x_{n}=T\\left(\\lim _{n\\to \\infty }x_{n-1}\\right).", "241d8be2f6acb2b26a4c527ddda29912": "2^{m},2^{m}-m-1,4", "241d970020e5a85935611e1a21ca2492": "R=\\epsilon {\\frac {\\gamma lv_{eff}}{2g\\omega d}}", "241db4d9abc9b28fa107f33829dbaa91": "(K_{i}\\varphi \\land K_{i}(\\varphi \\implies \\psi ))\\implies K_{i}\\psi ", "241dfe447bdcc6a628b26b03b373f89a": "{\\frac {1}{\\sqrt {2^{n+1}}}}\\sum _{x=0}^{2^{n}-1}|x\\rangle (|f(x)\\rangle -|1\\oplus f(x)\\rangle )", "241e3d7f47575ba4f78c407e52624196": "\\alpha =\\pi /2", "241eaa1fc195495d2413cd6340253152": "I(X;Y)=\\mathbb {E} _{X,Y}[SI(x,y)]=\\sum _{x,y}p(x,y)\\log {\\frac {p(x,y)}{p(x)\\,p(y)}}", "241ee7772718f1dbba6132708283bb46": "\\rho _{g}", "241f07d6ab890aaddbe1be15afa3ee31": "\\sigma (\\theta )=\\epsilon _{0}{\\frac {\\partial V}{\\partial r}}{\\Bigg |}_{r=R}={\\frac {-q(R^{2}-p^{2})}{4\\pi R(R^{2}+p^{2}-2pR\\cos \\theta )^{3/2}}}", "241f1f977405d69c414b305dd1231605": "{\\textit {ADJ}}", "241f2c2ece90e3aaea33cb68974338ae": "SU(3)_{L}\\times SU(3)_{R}", "241f4d180074e6128af716201733bfe6": "C={\\frac {D(N+R)}{D+T}}-R", "241f58916abd8d14ee53a4e93ac1523f": "N=\\langle X,<,(h_{i})_{i<\\omega }\\rangle ", "241f63ea68af23a32f4dfd57235903da": "|e^{+}e^{-}\\rangle \\to {\\frac {1}{4}}\\left(-3|c^{+}\\rangle |c^{-}\\rangle +i|c^{+}\\rangle |d^{-}\\rangle +i|d^{+}\\rangle |c^{-}\\rangle -|d^{+}\\rangle |d^{-}\\rangle -2|\\gamma \\rangle |\\gamma \\rangle \\right).", "241f8c44e56ef5aad194ce849daa7b8b": "\\,(e^{z})^{n}=e^{nz},n\\in \\mathbb {Z} ", "241fc0dbf8a2026a91d27f1ff562c508": "\\langle j_{1}m_{1}j_{2}m_{2}|JJ\\rangle ", "241fd430d256b028d54827ce01d696b5": "\\phi _{sl,v}={\\frac {1}{1+SG_{s}({\\frac {1}{\\phi _{sl,m}}}-1)}}", "241fe717afac381d2f1c2bb2e8cc47bd": "v_{x}={\\frac {-K}{\\mu }}{\\frac {du_{e}}{dx}}", "24201fa264f16618654527c9397e61c7": "\\displaystyle {K_{r}(e^{i\\theta })=\\sum _{n\\in \\mathbf {Z} }r^{|n|}e^{in\\theta }={1-r^{2} \\over 1-2r\\cos \\theta +r^{2}}.}", "2420985db3b2cf97ef88f5afeeda328f": "\\{|\\phi _{i}\\rangle \\}", "2420a564c10620d51603591e2e229c68": "Y_{AC}=Y_{ref}+c{dC_{l} \\over dC_{z}}+c{dC_{n} \\over dC_{x}}", "24212f98da731bb5089af174b48eae8d": "{\\boldsymbol {F}}=m(({\\ddot {r}}-r{\\dot {\\theta }}^{2}){\\boldsymbol {\\hat {r}}}+(r{\\ddot {\\theta }}+2{\\dot {r}}{\\dot {\\theta }}){\\boldsymbol {\\hat {\\theta }}})", "242207bb1e4afe948d672b22fa2f266a": "\\forall \\theta \\in \\Theta ,T(F_{\\theta })=\\theta ", "24221c77f2c8e840a8feeede5c9c46c6": "Ln=10\\,Laborers\\cdot 0.1\\,{\\frac {Ph}{Laborer}}=1\\,Ph", "24222b71f652dede388e6f6a2f4d8598": "\\omega .\\ ", "242264d45a6f335037e99fd55e9270c0": "F=hA_{s}\\left(T(t)-T_{a}\\right),", "2422a40f372d5458139b57d1c17d7996": "\\textstyle \\mathbb {R} ^{2}", "2422e38ab4ab81b9ffc5b1719c200094": "e_{1}=\\{e_{2}\\}", "242333ca9ea668feee4365b2a5801f2e": "H'", "242382e94d383195c580445532bf54bf": "f(z)={\\frac {z+i}{iz+1}}", "242424100ca47448be646d0a449fb59f": "v=\\sum _{i=1}^{m}\\alpha _{i}u_{i}\\otimes v_{i}", "242453842814c9f382714eaed0796558": "\\left({\\frac {p^{2}}{2m}}+V\\right)\\vert \\psi ^{0}\\rangle =E_{n}\\vert \\psi ^{0}\\rangle ", "2424748e5ec008cbe69b8435d94acc86": "\\lambda \\Delta t", "242481206124a6391cd827193a9a6666": "r=a{\\frac {\\sin[(q-1)\\theta +\\theta _{0}]}{\\sin[q\\theta +\\theta _{0}]}}=a{\\frac {\\sin[(1-q)\\theta -\\theta _{0}]}{\\sin[((1-q)-1)\\theta -\\theta _{0}]}}", "24249186e96870782319d236ac5ab372": "=\\int {\\underline {A}}(a,\\lambda ){\\underline {B}}(b,\\lambda )[1\\pm {\\underline {A}}(a^{\\prime },\\lambda ){\\underline {B}}(b^{\\prime },\\lambda )]\\rho (\\lambda )d\\lambda ", "2424a5d08456b0aeddc3dd59e07345e0": "L_{\\theta }", "2424dfe11a036b8bfcfe32e55bcb5e13": "\\operatorname {Ext} _{\\mathbb {Z} }^{1}(Q,N)", "2424fd19d256658f663a44eb19f4ca37": "\\ {\\mathcal {L}}=h^{\\mathrm {H} }ss^{\\mathrm {H} }h+\\lambda (1-h^{\\mathrm {H} }R_{v}h)", "242510caefe1265ff8011beef0009775": "\\scriptstyle Cr\\sigma (2r)", "24254816db548f97c3f720d80e4e21ac": "\\ {\\hat {X}}(f)", "24256265347d650c27922ed7e60354d2": "{\\begin{aligned}0&\\leq \\operatorname {var} (X_{1}+\\cdots +X_{n})\\\\&=\\operatorname {var} (X_{1})+\\cdots +\\operatorname {var} (X_{n})+\\underbrace {\\operatorname {cov} (X_{1},X_{2})+\\cdots \\quad {}} _{\\text{all ordered pairs}}\\\\&=n\\sigma ^{2}+n(n-1)\\operatorname {cov} (X_{1},X_{2}).\\end{aligned}}", "2425e19dbb6f681de91e8d855649d8a3": "A_{i+1}:=A_{i}+A_{i}\\left(I-AA_{i}\\right);", "2425ffea73f5aa3fe13b298ec0f2d548": "(1+x)(1+qx)\\cdots (1+q^{m+n-1}x)", "24265d43c429b6f3e99087e635bd8322": "{\\frac {S(t)}{M(t)}}=E_{Q}\\left[\\left.{\\frac {S(T)}{M(T)}}\\right|{\\mathcal {F}}(t)\\right]\\qquad \\forall \\,t\\leq T.", "24266eddfd108c33abf8e9683c99d8f5": "|S|\\geq \\gamma 'p", "24269bbebfe003c4f208928fbb15bd9a": "{\\mathfrak {e}}_{8}\\cong {\\mathfrak {so}}_{16}\\oplus \\Delta _{+}^{128}.", "2426a99211d9cfab74f51ebbb25d9bcf": "n^{k}", "2426e1d85a8a79af4cace140015a067b": "\\alpha _{t}(E)=U_{t}^{*}EU_{t}.", "2426e73b8bff0fbb8226d97deeadc28a": "0\\leq i\\operatorname {E} (X))", "242852df083c52259fe2b6c1aaffff1e": "\\mathbf {R} _{i}=\\mathbf {R} +\\mathbf {r} _{i}", "2428892f474a249918e07e46ac641219": "s_{g}=g\\sin \\zeta ", "2428ad6532e7ae9342ea8a3cb9cd1561": "\\Gamma =(V,E,s,t)", "2428cf94fb92da6c2fe5f558d428e71f": "P(O_{j}|A)", "24291822e17b06250ca5c5d6a1b9839e": "i_{V_{s}}", "242923e989687b37952059bf2d304ad5": "d{\\boldsymbol {\\sigma }}:d{\\boldsymbol {\\varepsilon }}_{p}\\geq 0\\,.", "242949e33aa6cbdad3f260b738f3fb1e": "|(j_{1},(j_{2}j_{3})J_{23})JM\\rangle =\\sum _{m_{1}=-j_{1}}^{j_{1}}\\sum _{M_{23}=-J_{23}}^{J_{23}}|j_{1}m_{1}\\rangle |(j_{2}j_{3})J_{23}M_{23}\\rangle \\langle j_{1}m_{1}J_{23}M_{23}|JM\\rangle ", "2429662bc6bd92759a97555705a475bf": "\\int _{0}^{\\infty }{\\frac {dx}{1+e^{nx}}}={\\frac {1}{n}}\\ln 2;\\int _{0}^{\\infty }{\\frac {dx}{3+e^{nx}}}={\\frac {2}{3n}}\\ln 2.", "24297a1d3877eb702c4fb9738668c724": "\\sum _{n=N}^{M-1}|a_{n+1}-a_{n}|", "24297bfad56825c23b85fe0b3d840f35": "T_{\\delta }^{X^{n}}", "24299185be4aeb562884f784e8a2f1f2": "u\\equiv v", "2429979ed7dcea52dc8ec15d47ef8419": "{\\mathbf {j}}", "2429b8369df78c15000b25fff4e2b2fa": "\\ln \\lambda =-a_{1}{\\frac {Z}{\\sqrt {E}}}+a_{2}", "2429c530ad389f58ca8dfa8c14029ce4": "s_{1}=3u", "242a07516b50fdfd0a849361e768ea96": "Y=a_{0}+a_{1}z+a_{2}z^{2}\\cdots +a_{k}z^{k}.", "242a323c8cf39cd383a6d6e43d223116": "||n(t)||=1", "242a5a0dbf1d010d7cba18256aee9e13": "{\\Big [}{\\big [}{\\mbox{nuclear}}{\\big ]}{\\big [}{\\mbox{physic(s)}}{\\big ]}{\\Big ]}{\\Big [}{\\mbox{-ist}}{\\Big ]}", "242a806131579109a40ea247f4738568": "Z={\\frac {Z_{0}}{\\sqrt {1-\\left({\\frac {f_{c}}{f}}\\right)^{2}}}}\\qquad {\\mbox{(TE modes)}},", "242a899af774fd774f98dc2244ccbbbe": "\\{h^{\\lambda }\\}", "242a90af18bf216720c75144e27e0fa0": "\\omega 3", "242aa8dd622ddba1ed9f92e3124fd7c5": "E_{k}=\\alpha +2\\beta \\cos {\\frac {k\\pi }{(n+1)}}", "242b0ca05590373248074786044d9ca9": "V_{\\theta }(r,t)={\\frac {\\Gamma }{2\\pi r}}\\left(1-\\exp \\left({\\frac {-r^{2}}{r_{c}^{2}(t)}}\\right)\\right),", "242b144c97e372558f9fe4a63224107f": "{^{h}\\!P}(x_{0},x_{1},\\cdots x_{n})=x_{0}^{d}P\\left({\\frac {x_{1}}{x_{0}}},\\cdots {\\frac {x_{n}}{x_{0}}}\\right),", "242b18b1a08f6f3ce757c6d01374f206": "{\\mathfrak {P}}({\\mathfrak {C}}_{\\operatorname {odd} }({\\mathcal {Z}})){\\mathfrak {P}}_{\\operatorname {even} }({\\mathfrak {C}}_{2}({\\mathcal {Z}})){\\mathfrak {P}}_{\\operatorname {even} }({\\mathfrak {C}}_{4}({\\mathcal {Z}})){\\mathfrak {P}}_{\\operatorname {even} }({\\mathfrak {C}}_{6}({\\mathcal {Z}}))\\cdots ", "242b21c0dbb30219f25255908b1f82d0": "e^{\\int ^{x}P(\\lambda )\\,d\\lambda }", "242b72f236e229cd9fbec5120ab5f383": "w\\not \\in L\\Rightarrow \\Pr[V\\leftrightarrow Q{\\text{ accepts }}w]\\leq {\\tfrac {1}{3}}", "242ba00b2d535fe0336bea8b63b44bf6": "\\lim _{t\\to \\infty }{\\frac {G(t)}{t}}=\\infty ", "242bc13cbbb878c6eaa143e6fd8766fc": "{\\frac {dx}{d\\varphi }}={\\frac {dx}{ds}}{\\frac {ds}{d\\varphi }}=\\cos \\varphi \\cdot a\\sec ^{2}\\varphi =a\\sec \\varphi \\,", "242bf38b6700d710f1652ff2e0e3f956": "l=ct", "242c223bfeb2a43c7d19f46aefcc8f51": "{\\text{shortcut distance}}=m,\\,", "242c5c46ef10c431acd3863dbf9f88b9": "\\pi _{1}(S^{1}\\times D^{2})\\cong \\pi _{1}(S^{1})\\cong \\mathbb {Z} ,", "242ca47e01a4730b1bd7d9359053424d": "tr([\\sigma (X),\\sigma (Y)])=tr(1)", "242cbccc536df14351262be70c5e0b2b": "(p\\vdash q)\\vdash (p\\to q)", "242cfab32f6406e5f29f186fd60fa8cf": "rR={\\frac {abc}{2(a+b+c)}}.", "242d0b2fed4787636bd8580b15ffd33b": "x^{n}", "242d1e34520df9ea564ed1e96f605c2e": "b(z)=\\sum _{n=-N}^{N}b_{n}z^{-n}", "242d3aa7363acea77309a43bcd66ce71": "\\delta \\rightarrow 0", "242d9831e6119ccd5638c7d736e0314c": "F(t)=\\Pr(T\\leq t)=1-S(t).", "242e2965f0cdcdc179a3e89cc7f15ed2": "\\left[-\\pi <\\theta <\\pi \\right]", "242e2d9ab287449110b3042bd205d04f": "W+E_{1}{\\overset {a_{1}}{\\underset {d_{1}}{\\rightleftharpoons }}}WE_{1}{\\overset {k_{1}}{\\rightarrow }}W'+E_{1}", "242e78a1c09de76f4a8640840c940419": "e^{s*2k\\pi i}", "242e9a5304d0556cfa19bdbf1c9351c6": "h^{2}={\\frac {b}{r}}={\\frac {t^{2}}{r}}", "242ecc1e1cbb9a9abd76d500128ad62d": "\\sum _{n=0}^{\\infty }U_{n}(x){\\frac {t^{n}}{n!}}=e^{tx}\\left(\\cosh(t{\\sqrt {x^{2}-1}})+{\\frac {x}{\\sqrt {x^{2}-1}}}\\sinh(t{\\sqrt {x^{2}-1}})\\right).\\,\\!", "242f17df8ba2a9a9ffab6072b45d1914": "N_{2}+3H_{2}\\leftrightharpoons 2NH_{3};K={\\frac {{f_{NH_{3}}}^{2}}{f_{N_{2}}{f_{H_{2}}}^{3}}}", "242f38c7cf1bae1a1aa52a4d4ec434a2": "s\\geq t", "242f4ec3930d49d5b4fb6f24840cc7df": "={\\frac {1}{2}}{\\sqrt {\\pi }}\\,", "242f9e8954acf29d57f61cfdf72d6ba0": "\\kappa ^{+}\\leq \\kappa \\,", "242fb4eabf10782385559744a34e6ed9": "f(x)={2 \\over \\pi R^{2}}{\\sqrt {R^{2}-x^{2}\\,}}\\,", "242fcbd26fc724c44d99b7c0aa3d6255": "(i\\omega -\\xi )^{-3}", "242fd799bd3cf89b2e6559cd759161ff": "\\delta ={\\rho \\over \\rho _{SATP}}", "242fec0f6f4d66625e876bae4fd5986b": "\\ PV\\ =\\ {1-e^{(-rt)} \\over r}", "242ff8a89b48fd3dea9c0cbe7f325217": "c=2.\\xi .\\omega .m,\\;k=\\omega ^{2}.m", "243025d4bd16bb2d631c4c237c9e2188": "C_{K}'=C_{T}'\\setminus \\{U\\},", "2430bcde069033ed91e8c1f08b45af47": "w=f(z)", "2430e8cac753cd80ea633c4620547221": "U(S,V,N)={\\hat {c}}_{V}Nk\\left({\\frac {N\\Phi }{V}}\\,e^{S/Nk}\\right)^{1/{\\hat {c}}_{V}}", "2430fe39e225818bd957ed4cb53eb770": "loaded", "2431182f37baf8fe883c2b2d878adcc4": "u(\\mathbf {x} ,t)=\\int _{\\mathbf {R} ^{n}}\\Phi (\\mathbf {x} -\\mathbf {y} ,t)g(\\mathbf {y} )d\\mathbf {y} .", "24314abb2689c5f3ec005d7a28f38df5": "\\ AFG(p)", "243162513db9068187d17df7eb3f5c30": "\\lambda \\in {\\mathfrak {h}}^{*}", "24319e2ecb7a53b3423e5757b71299cd": "t\\mapsto Q(t)", "2431b9f60c225fda83dc6a505db3d52f": "x^{2}+y^{2}+z^{2}=3xyz", "2431c81bde187d7a90cad3006728bfd3": "\\mathbf {F} =\\mathbf {B} -\\mathbf {A} =(B_{x}-A_{x},B_{y}-A_{y},B_{z}-A_{z}).", "2431d2c7cab36c00c3365201ceecc7f8": "\\mathrm {[Fe(H_{2}O)_{6}]^{2+}} +\\mathrm {[[Co(H_{2}O)_{6}]]^{3+}} \\rightleftharpoons \\mathrm {[Fe(H_{2}O)_{6}]^{3+}} +\\mathrm {[Co(H_{2}O)_{6}]^{2+}} ", "24322ff156ea659506ba7cd79f95367e": "V^{2}={\\frac {2kb(a+b)}{Ma}}", "24324be2b5fc384366011d9e66b09385": "2g\\,\\!", "2432510585320e8ef5261eef8e3e2230": "(t',R')", "2432845effe541006ad4a21224531332": "\\alpha (s)", "24328c00062b34932a316adbb4cf8fe4": "{\\frac {}{}}|I|C,", "24347c396fd889e7296136560780553e": "E=3h\\nu -{\\frac {3}{4}}{\\frac {h\\nu \\alpha ^{2}}{R^{6}}}", "2434909c1229f430c7f314f6408a3806": "V(\\alpha )={\\frac {V_{0}V_{90}}{V_{0}\\sin ^{2}\\alpha +V_{90}\\cos ^{2}\\alpha }}", "24349f7f2ed7850d4a617488de293dd3": "F_{\\text{avg}}", "2434f9a3ed45bcb781a653613282e848": "\\epsilon _{i+1}=-{\\epsilon _{i}}^{2}\\,.", "243598ddf46dd59abef780e6e59a335b": "\\int \\mathbf {\\Phi } _{lm}\\cdot \\mathbf {\\Phi } _{l'm'}^{*}\\,\\mathrm {d} \\Omega =l(l+1)\\delta _{ll'}\\delta _{mm'}", "24359c25dbbae53bc7aa028591b1301f": "c(\\psi )=\\int _{G}\\vert \\langle \\psi |U(g)\\psi \\rangle \\vert ^{2}\\;d\\mu (g)", "2435b4ecaa7db0d307d6e9d7ed6d860d": "\\mathbf {r} =[x^{1},\\ x^{2},\\ \\dots \\ ,x^{n}]\\ .", "2436035702ebfa512c151e3db6ce3818": "\\phi _{c_{1},c_{2},t}=\\exists m_{1}(\\phi _{c_{1},m_{1},\\lceil t/2\\rceil }\\wedge \\phi _{m_{1},c_{2},\\lceil t/2\\rceil }).", "24360894c15f3d07fa45ca0df04bd422": "{\\mathcal {L}}\\left\\{f^{(n)}\\right\\}=s^{n}{\\mathcal {L}}\\{f\\}-s^{n-1}f(0)-\\cdots -f^{(n-1)}(0).", "24361916e44e4a5b85bd83722de77ee5": "\\gcd(n,q-1)", "24362966659f2ef47654f464801f8753": "x^{\\mu }(\\tau )", "2436cdeee6481d15d6d69b56478984c2": "{}-2{\\boldsymbol {\\Omega }}\\times \\mathbf {v} _{B}", "243709dfcaf238130d8a5c741921d12a": "\\left({\\frac {\\gamma }{\\delta }},{\\frac {\\alpha }{\\beta }}\\right)", "24371d52e3e877b36285ec4a768a173e": "\\sinh ^{-1}\\left({\\frac {L}{2a}}\\right)={\\frac {k}{2a}}", "24373250a2769355d4267adb69c0b86b": "x_{0}\\in M", "2437591c91057ac8301b055d43ea5268": "g(n,k)", "2437637a4dd5be8d46a156da6f10977f": "\\sum _{j=1}^{n}\\ln(j)\\approx \\int _{1}^{n}\\ln(x)\\,{\\rm {d}}x=n\\ln(n)-n+1.", "24378fecc92de5e4ab5e0159d6fc6617": "u+v=\\int \\left({\\frac {du}{dx}}+{\\frac {dv}{dx}}\\right)\\,dx\\quad {\\mbox{(2)}}", "24379644b3a1cd8f206e34bff3a13513": "{\\frac {d[X_{i}]}{dt}}=0", "2437c9c3f8c71cbf3b8abb2040095788": "BRET={\\frac {{SV}_{dose}}{{SV}_{background}}}\\cdot 365\\,", "24380e87f075e06d8b4e2e8b7405daf6": "A={\\begin{bmatrix}19&3\\\\-2&26\\end{bmatrix}}", "2438952cdb0ccf66aa8ddac532a4448e": "PR=0.829PIMP+25SOIL+0.078UCWI-20.7", "2439719d95fa43252410c291ce1c1382": "{\\begin{aligned}\\cos(nx)&=\\mathrm {Re} \\{\\ e^{inx}\\ \\}=\\mathrm {Re} \\{\\ e^{i(n-1)x}\\cdot e^{ix}\\ \\}\\\\&=\\mathrm {Re} \\{\\ e^{i(n-1)x}\\cdot (\\underbrace {e^{ix}+e^{-ix}} _{2\\cos(x)}-e^{-ix})\\ \\}\\\\&=\\mathrm {Re} \\{\\ e^{i(n-1)x}\\cdot 2\\cos(x)-e^{i(n-2)x}\\ \\}\\\\&=\\cos[(n-1)x]\\cdot 2\\cos(x)-\\cos[(n-2)x]\\ \\end{aligned}}", "24398ad1035a912f50ef81f8ec23b571": "TCPI_{BAC}={BAC-EV \\over BAC-AC}", "2439aadd2bb46509e581c407202a1036": "z(\\tau )=\\exp[\\tau (D_{T}+D_{V})]z(0).\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (4)", "243a755f44e23db09ff326871c09868f": "C_{3}H_{8}+5O_{2}+18.55N_{2}\\to 3CO_{2}+4H_{2}O+18.55N_{2}", "243a993106f5075714c0660b46722a63": "A\\otimes B(L^{2}(G))", "243aabb75efbee694c96c1e8e8959901": "A:={\\begin{bmatrix}a&b\\\\c&d\\end{bmatrix}}.", "243abdafc8064e8d50323d6df0b853d5": "P,Q", "243af2c054114524134b019fecfa6e56": "B^{2}-4AC", "243b601b937fe5a0435dbc7c4ce1464b": "{\\sqrt {(x^{2}+a^{2})(x^{2}+b^{2})}}=2x{\\sqrt {t^{2}+\\left({\\frac {a+b}{2}}\\right)^{2}}}", "243bc6e21cbf3cf3325ce698d9963049": "f\\colon PG(V)\\to PG(W).", "243c1697d6a4e8a0a7cf8e401254b394": "V={\\begin{bmatrix}T&D_{T^{*}}\\\\\\ D_{T}&-T^{*}\\end{bmatrix}}.", "243c23e2bae2da6ee1821bf6425a64ff": "\\operatorname {head} \\equiv \\operatorname {first} ", "243c6d50e0d4cede5228158592982342": "\\Phi (m,n)=\\sum _{k=0}^{+\\infty }P_{k}(m,n)", "243c912695da5a4d625b19b4efc92e57": "{\\mathbf {x} }_{r}(0)=H(0)E({\\mathbf {x} }(0)).", "243cd43194b9adc20721f3042fd7866e": "R_{\\text{score}}={\\frac {C(A,B)}{\\min(n_{A},n_{B})}}", "243d004a44b4d0fb0416f544abc9fcca": "x_{i}=(d'_{i}-c'_{i}x_{i+1})/b'_{i}\\qquad ;\\ i=n-1,n-2,\\ldots ,1.", "243d966e78d91d39f375d05c64724211": "\\left[{\\begin{smallmatrix}2&-1\\\\-2&2\\end{smallmatrix}}\\right]", "243dbfc2d884c043fcb510c397236781": "\\sum _{i,j=1}^{n}Q_{i,j}x_{i}x_{j}+\\sum _{i=1}^{n}P_{i}x_{i}+R", "243dfd8ea41852d0851558c81abc7498": "\\arctan z=z-{\\frac {z^{3}}{3}}+{\\frac {z^{5}}{5}}-{\\frac {z^{7}}{7}}+\\cdots ", "243e49aa369f9b49c8dd0e42018d8f8f": "P_{\\mathrm {max} }=0.36\\,\\mathrm {kg\\,m^{-3}} \\cdot h\\cdot r\\cdot v^{3}", "243f0a3d4598abbc7ca89beb29c137df": "{\\frac {\\mathrm {d} ^{2}y}{\\mathrm {d} x^{2}}}-(2-k^{2})y+2y^{3}=0", "243f3ecbe9fc68ee30ddb3b195e47b0a": "{\\begin{aligned}\\Pr(Y_{i}=1)&={\\frac {e^{{\\boldsymbol {\\beta }}_{1}\\cdot \\mathbf {X} _{i}}}{\\sum _{k=1}^{K}e^{{\\boldsymbol {\\beta }}_{k}\\cdot \\mathbf {X} _{i}}}}\\,\\\\\\Pr(Y_{i}=2)&={\\frac {e^{{\\boldsymbol {\\beta }}_{2}\\cdot \\mathbf {X} _{i}}}{\\sum _{k=1}^{K}e^{{\\boldsymbol {\\beta }}_{k}\\cdot \\mathbf {X} _{i}}}}\\,\\\\\\cdots &\\cdots \\\\\\Pr(Y_{i}=K)&={\\frac {e^{{\\boldsymbol {\\beta }}_{K}\\cdot \\mathbf {X} _{i}}}{\\sum _{k=1}^{K}e^{{\\boldsymbol {\\beta }}_{k}\\cdot \\mathbf {X} _{i}}}}\\,\\\\\\end{aligned}}", "243f680e6aec259838dd2bfdf4ffb03f": "0\\neq I\\subset B", "2440130bca4cf4e5d5fb084a38542e03": "z\\in {\\text{int}}(\\Pi _{A})", "244047292af4d3e23737741e96b33aa4": "f=F(x)", "2440b202faa88ee2121a65cc6f1078e0": "\\delta =\\arg \\max _{\\delta '\\in {NS}}\\{\\pi (\\delta ')\\},", "2440bd96fa11f3157389f599e61af0ab": "{\\mathcal {B}}_{X,D}", "2440c0e92da317521c66c34c334f72d7": "\\displaystyle [a_{0};a_{1},a_{2},\\ldots ,a_{k}+1]", "2440c76f978106801754f5704eac7677": "\\scriptstyle f_{\\mathrm {image} }(0)=f\\,", "2440e52818e53754435521bd0a671098": "t^{2}g_{ij}(y)\\,dy^{i}\\,dy^{j}+2\\rho \\,dt^{2}+2t\\,dt\\,d\\rho ,\\,", "2441a1576fb6f0154cc9bfb54969425f": "D_{\\text{wave}}=-{\\frac {1}{2\\pi }}\\rho U^{2}\\int _{0}^{\\ell }S''(x)\\mathrm {d} x\\int _{0}^{x}S''(x_{1})\\ln(x-x_{1})\\mathrm {d} x_{1}", "244267c4ed7833da3e0931c4b73bd1ee": "f(t)=\\int _{0}^{\\infty }{\\hat {f}}^{c}\\cos(2\\pi \\nu t)d\\nu +\\int _{0}^{\\infty }{\\hat {f}}^{s}\\sin(2\\pi \\nu t)d\\nu ,", "2442843ca6d60f18bb03e41622b6bd6d": "X\\sim \\mathrm {GH} (1,\\alpha ,\\beta ,\\delta ,\\mu )\\,", "24428d786877a897d17450ee310d0e97": "t\\mapsto f(s,t)", "2444992556a3cbce4e2c8a165c19de8e": "B\\in U", "2444f161d3357ef9adb644f5fb640e19": "\\sin \\alpha _{1}\\cos \\beta _{1}=\\sin \\alpha _{2}\\cos \\beta _{2}.", "244587ed3606ffd53995be138c333f51": "x\\in \\mathbb {R} ^{*}", "2445b10c68a38304f4659bcf751ca109": "SD\\approx {\\frac {\\mathrm {ES} }{1^{\\prime \\prime }}}={\\frac {1\\,{\\mbox{AU}}}{({\\tfrac {1}{60\\times 60}}\\times {\\tfrac {\\pi }{180}})}}={\\frac {648\\,000}{\\pi }}\\,{\\mbox{AU}}\\approx 206\\,264.81{\\mbox{ AU}}.", "2445c3058a8435ec01d1d9afcba8b807": "\\;\\pi _{i.}={\\sum _{j}O_{ij} \\over N}\\;", "2445d101cfdc5a52dbd9bcbfa52f0e92": "{\\begin{bmatrix}V_{2}\\\\I'_{2}\\end{bmatrix}}={\\begin{bmatrix}1&-R\\\\-sC&1+sCR\\end{bmatrix}}{\\begin{bmatrix}V_{1}\\\\I_{1}\\end{bmatrix}}", "2445e21e060c798f60b2267a28a84779": "{\\text{(1)}}\\qquad \\Delta U=\\alpha RnT_{2}-\\alpha RnT_{1}=\\alpha Rn\\Delta T", "2445eb98d73933994ded7abd95eb2744": "{\\mathcal {F}}^{-1}({\\mathcal {F}}f)(x)=f(x)", "2445f64e7b38fe3e2a55f729b70da344": "{\\begin{pmatrix}1&0&0&0\\\\0&1&0&0\\\\0&0&-1&0\\\\0&0&0&-1\\end{pmatrix}}\\quad ", "244652dd7c78a2a5ba1e017f1551b888": "\\left(b^{d\\left(n+1\\right)}+b^{d}+1\\right)^{n},", "24467b3b75a804536b3763ada09f5cdc": "\\left[{\\hat {A}},{\\hat {B}}\\right]={\\hat {A}}{\\hat {B}}-{\\hat {B}}{\\hat {A}}", "244693c6e2c5dab007b514b422d741cd": "\\lambda x.~e", "2446f40c44d0c2c2d5160c9da98a62a7": "\\langle a_{n}\\rangle _{n=0}^{\\infty }\\subset \\mathbb {N} ", "2447296319d6618f48dc970147c79613": "[{\\overline {C}},{\\overline {S}}]", "2447543af6d49189acf0dd0e4aa290c0": "a(f(x))", "244778145f264d691c2cea0c39bc3578": "{\\dot {x}}=v(x).\\,", "2447f5eca0cfbcc8d5bf5f083fa0a427": "T^{{\\hat {j}}{\\hat {k}}}={\\frac {q^{2}\\sin(\\omega u)^{2}}{4\\pi }}\\,\\left[{\\begin{matrix}1&1&0&0\\\\1&1&0&0\\\\0&0&0&0\\\\0&0&0&0\\end{matrix}}\\right]", "244801ab19702a917aec76fc6185555f": "T=\\{x\\in L_{\\beta }:x\\in S\\wedge \\Phi (x,z_{i})\\}=\\{x\\in L_{\\gamma }:x\\in S\\wedge \\Phi (x,w_{i})\\}", "24483c9ef7c2d7e4dffa8631de4dc5b7": "a_{rel}=a_{0}\\,{\\sqrt {1-v^{2}/c^{2}}}", "2448700b9f3718a00ce54a2ba83c097c": "\\langle X+\\xi ,Y+\\eta \\rangle ={\\frac {1}{2}}(\\xi (Y)+\\eta (X))={\\frac {1}{2}}(\\epsilon (Y,X)+\\epsilon (X,Y))=0", "2448dcb128af25515be8da1c63e06b79": "\\sin {\\frac {\\pi }{6}}=\\sin 30^{\\circ }=\\cos {\\frac {\\pi }{3}}=\\cos 60^{\\circ }={1 \\over 2}\\,,", "2449100368528312c338b371fe47282e": "{\\begin{bmatrix}0&-1\\\\1&1\\end{bmatrix}}", "24491083c736320c8cc2eaf94a35423c": "Tail^{+}(X)\\gtrdot Head^{*}(Y)", "244933cfba8c3170f0b78c358a308214": "Q_{H}\\ ", "24497f94a02864da45e42b4d0aa18105": "y'=0", "2449b2b12012224dbe9f384565e9d9a6": "\\Theta (n)", "2449d7a4ddc3683a4ea3e26113008cac": "\\!r", "2449e763ba410d15e19d157dca2b84fb": "E_{2}-E_{1}=h\\nu ", "244a4edba2e64de6d63c462e112457ea": "f(\\mathbf {x} )={\\frac {1}{2}}\\mathbf {x} ^{\\mathrm {T} }\\mathbf {A} \\mathbf {x} -\\mathbf {x} ^{\\mathrm {T} }\\mathbf {b} ,\\quad \\mathbf {x} \\in \\mathbf {R} ^{n}.", "244a5e6267e626cf954c9c5f07ff11a7": "|\\eta |<=1", "244a828ce9d2a1be8337b80ed6d7ac53": "y_{i,j}\\in Y", "244a9c5de5fbf9edb1fd963d048235b9": "\\sum _{i=0}^{n-1}i2^{i}=2+(n-2)2^{n}", "244aa9fde966071f8506747a10e3bbc4": "\\scriptstyle \\left(1\\,-\\,{\\frac {t}{\\lambda }}\\right)^{-k}\\,", "244af61415aef8b1b0e09b7be8edca0a": "x_{r}\\geq b_{i}-\\sum _{k=1}^{r-1}a_{ik}x_{k}", "244b514554bd62c441f1c7c3f5c22489": "{\\dot {\\mathbf {f} }}", "244b9167481853192f86172ee1e494bb": "P_{\\mu }(\\tau )", "244bbdc27565643bc21136fe5dc90d3e": "D[x_{0},\\ldots ,x_{n}]f", "244bfdd19c862ff8a1def77e44ff0938": "A=LU_{3}U_{1}P", "244c3860eecf01974210e62004dc6c1c": "MRRT=A(B-C)-D-E", "244c4c56d63caa6e162fc0f66b2cc2b4": "E_{0,1}=510,260*{\\frac {510,000}{510,260}}*{\\frac {10.060}{510,260}}", "244c9b45fd4b6d1ec882ba14017ed53f": "\\mu (t)=1", "244cc30fc4411b8d6f3726697f7e496f": "child_{i}", "244cfde86976377e8abddef73f8ea3fd": "S_{k}=\\{1,2,\\ldots ,n_{k}\\}.", "244d0e81de5d4d54781156359a0f7196": "n=\\left\\lceil {\\frac {n}{m}}\\right\\rceil +\\left\\lceil {\\frac {n-1}{m}}\\right\\rceil +\\dots +\\left\\lceil {\\frac {n-m+1}{m}}\\right\\rceil ,", "244d469c6900dc9c105dccd7052a886b": "\\textstyle \\mathbf {b} _{2}", "244d7aa2b96b5914f99b5e42b7a4bf15": "expr=A_{i}", "244e37af0beebc59ffe40ccc01c155ce": "\\operatorname {E} ((\\delta _{1}(X)-\\theta )^{2})\\leq \\operatorname {E} ((\\delta (X)-\\theta )^{2}).\\,\\!", "244e49456af1cc5706c9d6fcb4ab2472": "HV=0.0018544\\times {\\tfrac {L}{d^{2}}}", "244eef0c57f18ff9556a81e16c13b47f": "Z=\\int e^{{\\bar {\\psi }}M\\psi +{\\bar {\\eta }}\\psi +{\\bar {\\psi }}\\eta }D{\\bar {\\psi }}D\\psi =\\int e^{({\\bar {\\psi }}+{\\bar {\\eta }}M^{-1})M(\\psi +M^{-1}\\eta )-{\\bar {\\eta }}M^{-1}\\eta }D{\\bar {\\psi }}D\\psi =\\mathrm {Det} (M)e^{-{\\bar {\\eta }}M^{-1}\\eta }", "244f54ae45e7cee8bf4f6232400dbc65": "\\tan \\alpha _{0}={\\frac {\\sin \\alpha _{1}\\cos \\phi _{1}}{\\sqrt {\\cos ^{2}\\alpha _{1}+\\sin ^{2}\\alpha _{1}\\sin ^{2}\\phi _{1}}}}.", "244f80b6384ad586f65b49e353452512": "\\sum _{i,j}de_{i}de_{j}'u_{x}u_{r}=-dqdq'(w_{x}w'_{r}+w'_{x}w_{r})", "244f8a8950b08319ca83df2b979ef548": "{\\frac {p_{2}}{p_{1}}}={\\frac {2\\gamma M_{1}^{2}}{\\gamma +1}}-{\\frac {\\gamma -1}{\\gamma +1}}", "244f9eb65594e5b633f7824843bc823d": "X_{i}={\\frac {n_{i}(t=0)-n_{i}(t)}{n_{i}(t=0)}}=1-{\\frac {n_{i}(t)}{n_{i}(t=0)}}", "244fed5b7e7ce4f9789c0e721d13e353": "\\textstyle K", "24503f7a99422feb310621692bc83d45": "\\,_{92}^{238}\\mathrm {U} \\ +\\,_{12}^{24}\\mathrm {Mg} \\to \\,_{104}^{259}\\mathrm {Rf} \\ +3\\,_{0}^{1}\\mathrm {n} ", "245046eb150ed2c547e7d7fd2e76d785": "\\int _{-1}^{1}{\\frac {\\mathrm {d} x}{x}}{\\ }\\left({\\mbox{which}}\\ {\\mbox{gives}}\\ -\\infty +\\infty \\right).", "24505b9fa4f63b41dee0e32a2f76bae4": "{\\rm {E}}_{1}(z)=\\int _{1}^{\\infty }{\\frac {\\exp(-zt)}{t}}\\,{\\rm {d}}t\\qquad ({\\rm {Re}}(z)\\geq 0)", "24508b7adec3c6b74d857690f2d0f155": "\\sum a_{i,j}(x)\\xi _{i}\\xi _{j}\\geq \\lambda |\\xi |^{2}", "245095537e35690bc3ab52834617f330": "x(t)\\rightarrow X^{*}(s)\\rightarrow x^{*}(t)", "24509e79859838cd4f18abf9c4b8b24f": "50i+8", "2450dbe33f813053f861d027e07433d5": "{\\frac {p}{q}}=\\Omega ", "2450eb54d756a3c3aa5f1047f05ae24f": "\\mu =\\phi _{U}-\\phi _{L}", "2450ff67454e08a2f200b0fa45e46ad1": "e>1", "24510848c8c770ba84dc1b49662d8abf": "L(A)=\\bigcup _{\\alpha }L_{\\alpha }(A)\\!", "24512306ec1d0e218647f19389ba8ca7": "\\Delta W={\\sqrt {e^{3}F \\over 4\\pi \\epsilon _{0}}},", "245137fd8f8adedf3cfcb35f6007a9a0": "\\Delta _{g}u=0", "2451a2737600e06be25d25ea911745f0": "{\\frac {a}{b}}\\cdot {\\frac {c}{d}}={\\frac {ac}{bd}}.", "2451cdf7638b8a5ee073f85cec264a0d": "s_{0}\\in S", "2451d7c7b1b3db49966f2e7cb7705fb1": "{\\begin{matrix}{4 \\choose 2}{3 \\choose 1}^{2}{9 \\choose 3}{4 \\choose 1}^{3}\\end{matrix}}", "2452185351ff69c60b0d1d1c91fe369d": "V={\\frac {H-E(H)}{SD(H)}}", "24522dcf5c8109082a791413ee40dfb6": "\\rho =\\sum _{i}\\rho _{i}{\\frac {V_{i}}{V}}.\\,", "2452dd6fd20532beca558815574e25bf": "2-m^{2}={\\sqrt {2+(2-M^{2})}}\\,,", "2452fee413f58bb9509e88d80d4b9f8d": "T_{1}", "245322caae7a2d3c852e000db88953f8": "\\lambda _{k}=\\left({\\frac {1}{(k-{\\frac {1}{2}})\\pi }}\\right)^{2},\\qquad k\\geq 1", "2453932be4f298a60996cb10e00df32a": "A=F_{1}^{-1}F_{2}", "2453b7e9c144fd28cedf8891935348e8": "~x_{1}\\leftrightarrow ...\\leftrightarrow x_{n}", "2453bacd77eac08e9d8b8b385f62e9f6": "{\\begin{aligned}\\lim _{\\beta \\to 0}{\\text{median}}=\\lim _{\\alpha \\to \\infty }{\\text{median}}=1,\\\\\\lim _{\\alpha \\to 0}{\\text{median}}=\\lim _{\\beta \\to \\infty }{\\text{median}}=0.\\end{aligned}}", "2454070f8b93297ec10fe81246271641": "\\pi ={\\frac {P_{o}-P_{e}}{1-P_{e}}}", "2454075020a3066631a1428a19ad9b53": "\\{U_{\\lambda }:\\lambda \\in \\Lambda \\}", "24547d2a00606bb3e907ac0eb0479ba3": "[n]\\times [n-1]\\times \\cdots \\times [2]\\times [1]", "2454db539b568a96c228d8da5d8118f3": "e^{2}(t)=s^{2}(t+\\alpha )-2s(t+\\alpha ){\\hat {s}}(t)+{\\hat {s}}^{2}(t),", "2454e6f9d47ed8a8828b5c41faff111e": "\\mathbb {E} [\\langle \\varphi (x),\\varphi (x')\\rangle ]=\\langle x,x'\\rangle ", "2454eb77baff6695c637a9e13543de62": "{\\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})\\;.\\end{aligned}}", "24551f19411355b0536faf9cbefd0b30": "{\\mathfrak {T}}_{\\beta }^{\\alpha }=\\left(\\det {\\left[{\\frac {\\partial {\\bar {x}}^{\\iota }}{\\partial {x}^{\\gamma }}}\\right]}\\right)^{W}\\,{\\frac {\\partial {x}^{\\alpha }}{\\partial {\\bar {x}}^{\\delta }}}\\,{\\frac {\\partial {\\bar {x}}^{\\epsilon }}{\\partial {x}^{\\beta }}}\\,{\\bar {\\mathfrak {T}}}_{\\epsilon }^{\\delta }\\,,", "2455a8a7d97ec8b95a1a30edd357d41c": "p_{k}(x_{1},x_{2},\\dots ,x_{n})=\\sum _{i=1}^{n}x_{i}^{k}\\,.", "2456b7e422105920fa68d4bc3f26dc7c": "=\\mu _{0}rN^{2}\\left[\\left(1+{\\frac {1}{32w^{2}}}+O\\left({\\frac {1}{w^{4}}}\\right)\\right)\\ln(8w)-1/2+{\\frac {1}{128w^{2}}}+O\\left({\\frac {1}{w^{4}}}\\right)\\right]", "2456d4989e097c16eaf2840d4ed575be": "x^{10}+x^{9}+x^{5}+x^{4}+x+1", "2456e3c6455cf5f0ef7699d8402c593c": "{\\breve {L}}", "2457137e54d3deb8f40af46c2b91d0f6": "\\Delta S=2", "245718652e75023ed42583f4aa9eae43": "-1.6326", "24576f92e39e9f1414a0a3274f54dfd1": "F_{i}\\,", "245788b7873f368fccc0e60527b506aa": "\\mathbf {b} :G\\rightarrow \\mathbf {Bohr} (G).", "24578a31f08ed1c309dc19d26ac2ff56": "v\\;:\\;2^{N}\\to \\mathbb {R} ", "2457912d4aedc72628414f6e3af49956": "\\displaystyle A(s,\\lambda )\\xi _{0}=c_{s}(\\lambda )\\xi _{0},", "2457beb505c56430bb2766f1e6560319": "u(0,t)=0", "2457de0e8d9c6ff5bdee4d8b16d22a52": "\\mathbf {F} =d\\mathbf {A} ", "2457eb40a864b7ec5938b8529a840a30": "D^{*}={\\frac {D_{n}D_{p}(n+p)}{pD_{p}+nD_{n}}}", "24581b6da3ebcd3b67aedf87eb3064ed": "A={1 \\over 2}\\int r^{2}\\,d\\theta ", "24586132f0c6282a02427606e8edc229": "f(x_{1}^{|q|},x_{2}^{|q|},x_{3}^{|q|},\\ldots ).", "24587072e08dad9a498c6368faf629d3": "P=2L/\\pi D", "2458ada6c84b426854fc3cba2b496658": "\\mathrm {cd} \\left((2m-1)K\\left(1/z\\right),{\\frac {1}{z}}\\right)=0\\,", "2459347edf2a3aa48d415d1ab6dbefeb": "v_{\\perp }", "2459559fef7bb6a2ba28485295b759d0": "f(f^{-1}(L))\\neq L", "24595899053c8b3129863642a3af6fd1": "F_{1},F_{2},\\dots ,F_{b}", "245a0d7f5a309d7fba90767f2954da9a": "{\\frac {\\partial ^{3}f}{\\partial x^{3}}}(0,r_{o})\\left\\{{\\begin{matrix}<0,&\\mathrm {supercritical} \\\\>0,&\\mathrm {subcritical} \\end{matrix}}\\right.\\,\\,", "245a11fe34daba1bd187173380903e51": "{\\mathbf {S}}={\\frac {c}{4\\pi }}{\\mathbf {E}}\\times {\\mathbf {B}}.", "245abd5ab96370bdebeb42ea6ae350ff": "\\operatorname {st} (x)\\leq \\operatorname {st} (y)", "245ac8a4c70fda0d388b7a28f87e812b": "0^{o}\\leq \\theta \\leq 180^{o}", "245ace04106ef045600bd5e5aff6088f": "X_{i}", "245ad35e07ae8c1bf8735e60e89b903e": "\\Omega ^{\\infty -n}", "245ae94ccc26532e160fa6595447b49a": "l=2\\pi R", "245affc8fbd8b1fdf069c46737a42317": "S_{21}=S_{31}=-3\\,{\\text{dB}}=10\\log _{10}({\\frac {1}{2}})", "245b4065edc742bbee4211427948e0a6": "M,g", "245b4df5eddfa06a01b6271cc6b24e49": "A={\\tfrac {1}{2}}\\cdot p\\cdot r.", "245b73f1cf7bd93ee112f210c45f709b": "F=eE\\,\\exp(-i\\omega t)", "245d1a3a3b91ccec6df63ecda744d4dc": "R={\\sqrt {E^{2}+9\\lambda ^{2}+2E\\lambda }}", "245d621658e06a1539bfa71238254da7": "p_{1}(w)=1/w", "245d6e9f87ec4aedcf399c1c593b0643": "\\nabla _{[v,w]}(m)=[\\nabla _{v},\\nabla _{w}](m)", "245deed2b19c7bc7eee605f65d4778e3": "R_{m}", "245e1186ebbe95f8c367e7c53cc219ee": "W_{1-2}=\\int PdV=0", "245e30ffc38fd76a0e66469c65fedab8": "\\Delta E_{C}", "245e49e03c1aff7dfa8f672583ef0787": "\\operatorname {Corr} [dz_{1}(t),dz_{2}(t)]=\\rho \\,dt", "245e87fef88092abbcee43101a49ec1a": "\\Delta u=u_{xx}+u_{yy}", "245ec15c7f91ba3eac42fa7c359b4593": "{\\begin{aligned}X_{i}&={\\bar {M}}+({\\bar {X}}-{\\bar {M}})+(X_{i}-{\\bar {X}})\\\\Y_{i}&={\\bar {M}}+({\\bar {Y}}-{\\bar {M}})+(Y_{i}-{\\bar {Y}})\\\\Z_{i}&={\\bar {M}}+({\\bar {Z}}-{\\bar {M}})+(Z_{i}-{\\bar {Z}})\\end{aligned}}", "245fdf0a63da0b7785a2053075468879": "v={\\sqrt {rg\\left(\\sin \\theta +\\mu _{s}\\cos \\theta \\right) \\over \\cos \\theta -\\mu _{s}\\sin \\theta }}={\\sqrt {rg\\left(\\tan \\theta +\\mu _{s}\\right) \\over 1-\\mu _{s}\\tan \\theta }}", "245fe2514d99ad59089374bedfe13d63": "(-ae,0)\\,\\!", "24605685c7a717e67bd6cba22d58c86b": "\\gamma (1-1/k)+\\ln(\\lambda /k)+1\\,", "2460d413afee164e62afdabf88dea1a1": "p_{1\\infty }\\leftarrow p(x+1),M_{1\\infty }\\leftarrow M(x+1)", "24614cce729e2ce03e2cc2616ebc7833": "{\\color {Blue}~2.10}", "246151b993bb0d33f3ce10c84bb3ce42": "k=k_{2}K_{1}", "246163c5d68d70388f2ad8c9de1d6574": "\\cup _{n=1}^{\\infty }A_{n}\\in D", "2461a7355d77e296c24b1ea113d92f50": "\\scriptstyle H_{\\mathrm {norm} }=0", "2461b178936988bc7debd4270f341ac0": "H(f)={\\begin{bmatrix}{\\frac {\\partial ^{2}f}{\\partial x^{2}}}&{\\frac {\\partial ^{2}f}{\\partial x\\,\\partial y}}\\\\[10pt]{\\frac {\\partial ^{2}f}{\\partial y\\,\\partial x}}&{\\frac {\\partial ^{2}f}{\\partial y^{2}}}\\end{bmatrix}}.", "2461b7f5d1624dcb655db12b9c5844cc": "(a,0)", "24627148c3db6e7db25302e606d45201": "G^{\\alpha \\gamma }+\\Lambda \\mathrm {g} ^{\\alpha \\gamma }=(\\mathrm {const} )T^{\\alpha \\gamma }~", "246275f41e3f6d3a6f1ca6f426a56eb0": "THD={\\frac {\\sqrt {{a_{2}}^{2}+{a_{3}}^{2}+..}}{a_{1}}}", "2462874e9db0669fa8628425e3ba8aa4": "\\Box {\\bar {h}}^{\\alpha \\beta }=-16\\pi T^{\\alpha \\beta }\\,", "24629c2297fda97bb844dcb324dbae4b": "\\mathrm {Br} ={\\frac {\\mu U^{2}}{\\kappa (T_{w}-T_{0})}}=\\mathrm {Pr} \\,\\mathrm {Ec} ", "2462af80e4e5654179d9901d27e86570": "[L_{z},X]=iY\\,", "2462bd0db7e8e91137d13dc7222e5bc6": "J:X\\times Y\\to \\mathbb {R} \\cup \\{+\\infty \\}", "2462e861bff7cdbccf0bac8a204b035a": "G\\times H", "2462ea1b25d1a7d2fc4cd5554dbf07d3": "{\\begin{bmatrix}1&0\\\\0&0\\end{bmatrix}}", "2462ef886428ee7d949936d41281f003": "\\bot ", "246314289bb4d3cae42bd012459358dd": "{\\begin{aligned}\\operatorname {E} \\left[\\ln ^{2}(X)\\right]&=(\\psi (\\alpha )-\\psi (\\alpha +\\beta ))^{2}+\\psi _{1}(\\alpha )-\\psi _{1}(\\alpha +\\beta ),\\\\\\operatorname {E} \\left[\\ln ^{2}(1-X)\\right]&=(\\psi (\\beta )-\\psi (\\alpha +\\beta ))^{2}+\\psi _{1}(\\beta )-\\psi _{1}(\\alpha +\\beta ),\\\\\\operatorname {E} \\left[\\ln(X)\\ln(1-X)\\right]&=(\\psi (\\alpha )-\\psi (\\alpha +\\beta ))(\\psi (\\beta )-\\psi (\\alpha +\\beta ))-\\psi _{1}(\\alpha +\\beta ).\\end{aligned}}", "24634e1eb7a9a49d8175fb7d4720f78b": "\\Delta Y", "246352fe547c6354c96b1fb0cbd30bb1": "D(1-\\epsilon )\\leq deg(x)\\leq D(1+\\epsilon )", "24637ead895f29c197c74d17a7250db7": "{\\begin{cases}0&{\\text{if }}q>p\\\\0,1&{\\text{if }}q=p\\\\1&{\\text{if }}q0{\\mbox{ }}\\forall {\\mbox{ }}x\\in \\left(x_{0},x_{1}\\right]", "2468e2042f49e1dbaa66c9d3c000614c": "\\lim _{h\\to 0}{\\frac {f(a+h)-f(a)-f'(a)\\cdot h}{h}}=0,", "2468f34f7bcfc04b8dc4e8b02e82d3d8": "x\\vee y=y\\wedge x", "246a2ad6ed979571de053aaa546b0bc9": "f(x,\\lambda )=0\\,", "246a45d39169e57be898a4f7c0266da8": "P(A_{n}^{\\epsilon })>1-\\epsilon ", "246a5a9681223e6f86c751d2ccc75e8d": "\\sum _{n=-\\infty }^{+\\infty }{E_{n}}e^{j({\\omega }t-{k_{n}}z)}", "246a7af4b0002703c1e2f2893c4f0749": "(R_{i})_{i\\in I}", "246ac82fd099529153b2f77dcbb34aa0": "\\textstyle {\\text{Var}}[{N}(B)]", "246ad60b77c59cc8d50ba80f069880b6": "f\\left(K_{\\alpha }\\right)=\\left(3.29\\times 10^{15}\\right)\\times 3/4\\times \\left(Z-1\\right)^{2}", "246b422897f30671287899e9b10bae05": "K_{x\\rightarrow y}(T)=\\sum _{x(t)}\\prod _{t}K_{x(t)x(t+1)}\\,", "246b56c160a1cf80eac88068e2c80052": "T_{J}>3", "246b9d3adbbf81c7a7e6de83e021f1f8": "\\{x_{1},\\dots ,x_{n}\\}", "246baba6b98e66c88c532355f0b442ca": "(-0.50,1.25);", "246bd5f89d1e5369386ad10faa3a7514": "f=v/\\lambda ", "246bf7fa6f68f91c8fbacbaafd71292a": "{\\frac {\\partial p}{\\partial r}}={\\frac {\\rho V^{2}}{L}},", "246bfa1954a3377f0870eca10268dc94": "f_{\\bullet }", "246c34c550a7977ad4c582df9b57f0b6": "Q_{k}", "246ccc14b1b11a00fb8d4970dd1cad22": "\\mathbf {j} _{s}", "246cd0290a6d2a5035216c453a5736ce": "R(X,L)=\\bigoplus _{d=0}^{\\infty }H^{0}(X,L^{d}).", "246cda3e2d79544b723caf66a11c140c": "P_{2}=(2:{\\sqrt {17}}:1)", "246d69873133ce38e4f388cfe0366489": "\\scriptstyle {\\sqrt {n}}", "246db1bb1d0e97de1a270dd9ad493c22": "(-\\alpha ,+\\alpha )", "246e0e4d80939d910c75df8b16e682ec": "\\mathbb {P} ^{x}\\{X_{\\tau -}-X_{\\tau }0", "248fac604a7b2866118d8ad60aa44cb5": "V_{\\mathrm {rms} }={\\sqrt {{\\frac {1}{T}}\\int _{0}^{T}{v^{2}(t)dt}}}.", "248fe2b4c32922f410d2c4254b78ab9e": "A_{1}(\\omega )", "248ff36395cbf1144dcd1f862261999d": "\\|\\varphi \\|_{A}=\\sup _{x\\in A}|\\varphi (x)|,", "24907c8f307e5dee7921fac642e00f3b": "P(V)={\\frac {K_{0}}{K_{0}'}}\\left[\\left({\\frac {V}{V_{0}}}\\right)^{-K_{0}'}-1\\right]\\,.", "24908481f44c18214b59a7a808b67842": "m\\in (-\\infty ,\\infty )", "2491378d33154a96a2c9af449897f8b5": "(m)", "24913b81a5bfa4e88180f16b257b19f1": "Q=k_{f}-k_{i}", "24916ff2fc7f6d6ab6b27242c06383ff": "(2x+7i)(2x-7i)", "249192a9e27a59ea5fa7eaaa62d2430b": "{\\begin{matrix}\\sum _{i=0}^{t}{\\binom {n}{i}}<|C|\\\\\\end{matrix}}", "2491ae3fc417527afa536d03fe9d89eb": "{\\begin{aligned}\\mathbf {A} &=A_{x}\\mathbf {e} _{x}+A_{y}\\mathbf {e} _{y}+A_{z}\\mathbf {e} _{z}\\\\&=A_{x}{\\begin{pmatrix}1\\\\0\\\\0\\end{pmatrix}}+A_{y}{\\begin{pmatrix}0\\\\1\\\\0\\end{pmatrix}}+A_{z}{\\begin{pmatrix}0\\\\0\\\\1\\end{pmatrix}}\\\\&={\\begin{pmatrix}A_{x}\\\\0\\\\0\\end{pmatrix}}+{\\begin{pmatrix}0\\\\A_{y}\\\\0\\end{pmatrix}}+{\\begin{pmatrix}0\\\\0\\\\A_{z}\\end{pmatrix}}\\\\&={\\begin{pmatrix}A_{x}\\\\A_{y}\\\\A_{z}\\\\\\end{pmatrix}}\\end{aligned}}", "2491bc398e209b15f0fdb9e2c5ba9b5d": "A=(1-\\kappa )\\left[{\\begin{array}{c c }1&0\\\\0&1\\end{array}}\\right]-\\gamma \\left[{\\begin{array}{c c }\\cos 2\\phi &\\sin 2\\phi \\\\\\sin 2\\phi &-\\cos 2\\phi \\end{array}}\\right]", "2491dcb2cb7af403d6e2919067ee5b67": "x\\ {\\stackrel {\\mathrm {def} }{=}}\\ qe^{-D(t)}", "2491edac6a2e000320344a81ed495041": "G'/N'", "2491f248be5dc4ce53c7e4c71ef5f2c5": "G_{Q}", "24920da9de402da836513cd42905fd2c": "1/0.02=0.02x/0.02", "249242eb3951489052d4817a186bc3d7": "i_{c}", "2492a2642e37b60a57d624cd2a7b0481": "{\\frac {\\mathbf {v} _{k+1/2}-\\mathbf {v} _{k-1/2}}{\\Delta t}}={\\frac {q}{m}}\\left(\\mathbf {E} _{k}+{\\frac {\\mathbf {v} _{k+1/2}+\\mathbf {v} _{k-1/2}}{2}}\\times \\mathbf {B} _{k}\\right),", "2492d92018bbf90beebdad43cfd94db0": "C^{SL}", "2493b6785c2b61271276d9d95c922e45": "\\lambda _{m}\\neq \\lambda _{n}", "2493de0b8b5a0cafed15dec56532e0e5": "g(t)=\\int _{a}^{b}K(t,s)f(s)\\,\\mathrm {d} s", "24945af137fce862851d363c5b3f815e": "\\langle Y(m),m\\rangle =0,\\qquad \\forall m\\in \\mathbf {S} ^{2}.", "249475e5352e7697fed5f70afe08b16c": "\\phi ^{A,{\\bar {x}},{\\bar {a}}}=\\{(x_{1},\\dots ,x_{n})\\in A^{n}\\colon A\\models \\phi [{\\bar {x}},{\\bar {a}}]\\}", "24953a2ff725454733f859ab05e22e89": "x>1", "24953cb957250615795fde69e9eaa485": "w(n)=w(n-i_{1})w(n-i_{2})...w(n-i_{k}){\\text{ for }}n\\geq \\max(i_{1},...i_{k})\\,.", "24955ccde15784b0553f7b017688f522": "\\%{\\mbox{ change in }}y={\\frac {y_{2}-y_{1}}{(y_{2}+y_{1})/2}}.", "2495b620f9fe1e74ee63a12dbd0505af": "\\rho ={\\frac {M(G)}{M(G\\setminus e)}}", "249618d56fde8a84831bcba20d0335d1": "\\mathbf {y} _{i}^{\\rm {T}}", "24965242343cfdc15070483df7dd5ba0": "(I_{2}-I_{3})/3", "24967e4c685321c2de26d045eafe6eac": "I_{n}[w]={\\frac {1}{n}}\\sum _{i=1}^{n}V(\\langle w,x_{i}\\rangle ,y_{i})\\ .", "249686a76141115e53bddd9d3b343cac": "h_{t}(x,x)={\\frac {1}{4\\pi t}}+{\\frac {s(x)}{12\\pi }}+O(t).", "2496ad327ba72d3eba715f38ad0a1c9f": "T_{F}", "2496e19090045bb9934209147bc15ef7": "q\\to 1", "2496ecc23ba1b56e9e24d721c68edab4": "{\\frac {c_{metal}}{c_{air}}}={\\frac {f\\lambda _{metal}}{f\\lambda _{air}}}={\\frac {\\lambda _{metal}}{\\lambda _{air}}}={\\frac {L}{d}}\\,", "2496f5b9e347f307d1b0968e0814e7f8": "P_{D-}+{\\frac {1}{2}}\\rho v_{D,~z}^{2}=P_{\\infty }+{\\frac {1}{2}}\\rho v_{w,~z}^{2}=P_{D-}+{\\frac {1}{2}}\\rho (v_{\\infty }(1-a))^{2}", "24971138d67f6c0ea3e617ee86c3615f": "\\cos(a)\\cos(b)=[\\cos(a+b)+\\cos(a-b)]/2", "249716552568c05f7088d49aeccc2076": "\\left|{\\frac {a}{b}}\\right|={\\frac {|a|}{|b|}}\\ ", "2497400a9ba7423587c3556b3f98913a": "e^{-{\\frac {\\Omega }{kT}}}=\\sum _{N_{1}=0}^{\\infty }\\ldots \\sum _{N_{s}=0}^{\\infty }\\int \\ldots \\int {\\frac {1}{h^{n}C}}e^{\\frac {\\mu _{1}N_{1}+\\ldots +\\mu _{s}N_{s}-E}{kT}}\\,dp_{1}\\ldots dq_{n}", "2497a006e5d4ffa2c1f8cd28d6d95968": "L\\in \\Sigma _{2}-{\\mathsf {SIZE}}(n^{k})", "249808a73c8d68bfebb903d6671d9f23": "A={\\begin{bmatrix}6&5&0\\\\5&1&4\\\\0&4&3\\\\\\end{bmatrix}}", "2498173b86ccb56a03f008bcaeccb27d": "\\delta _{ext}(q,x)=((\\ldots ,(s_{i}',t_{si}',t_{ei}'),\\ldots ),b)", "249885905d0e164b83a0e8e50faf3d72": "\\mathrm {A+B\\longrightarrow AB} ", "24989ee522c37a12fb2cf2fbf5b21d6f": "m_{1}...n", "24991f14f1c5bb692e681b48cf98c3b5": "z^{\\star }=a-bh", "24998f188b7fe55cfc8e1a74b15f4491": "P=I^{2}R\\,", "2499930d8c19771316386635407f0685": "{\\sqrt {\\Delta }}=1.786737578486707", "2499ea491442016d7e606662980fd27f": "y=Ce^{0.85t}", "2499f36a8dfdc34b857a32c5a89bff7d": "\\phi ^{+}=0", "249a19de06ae2179da1a1edbd07c3585": "17.93^{-n}", "249a6f7a124c5e629a59e212ecf9fc3e": "\\varphi _{k-p}^{n+1}-\\varphi _{k-p-1}^{n+1}={\\gamma _{p}\\left({\\varphi _{k}^{n}-\\varphi _{k-1}^{n}}\\right)}<0,\\quad \\quad (7)", "249accf4fffcd97ed6fad6da05ec2a6a": "\\dim \\operatorname {ker} (AB)\\leq \\dim \\operatorname {ker} (A)+\\dim \\operatorname {ker} (B)", "249acd69b59226e13718e66b07941082": "\\tau _{23},\\tau _{12},\\tau _{31}", "249b0568ea087a98d6ea67843bea5b1f": "\\omega _{r}={\\sqrt {\\alpha ^{2}-{\\omega _{0}}^{2}}}", "249b09615be83fe41f1928ffdc946984": "A+B+A...\\rightarrow ABA...", "249b4c584ee566c0dbe934080d723381": "\\left\\{E_{i}\\right\\}_{i\\in \\mathbf {N} }", "249c1ed19cc4cfd4387c799673b5dd60": "i=1,\\cdots ,n\\,\\!", "249c2d6e4dec0a904576fde16bf34822": "e_{i}\\in B", "249c4f25cdc5525e3ad7de912960294b": "{\\frac {\\mbox{d}}{{\\mbox{d}}t}}\\left[\\delta (x-vt)m{\\frac {{\\mbox{d}}w(vt,t)}{{\\mbox{d}}t}}\\right]=-\\delta ^{\\prime }(x-vt)mv{\\frac {{\\mbox{d}}w(vt,t)}{{\\mbox{d}}t}}+\\delta (x-vt)m{\\frac {{\\mbox{d}}^{2}w(vt,t)}{{\\mbox{d}}t^{2}}}\\ .", "249c7e060ed36f85f4c29c575da881df": "f_{U}(x)=1\\,", "249cdbfbd582d76f1cae60545bb42535": "6/\\pi ^{2}", "249d61a1c9812ef9aa72fccafe2c3c5c": "{\\begin{matrix}{4 \\choose 1}{3 \\choose 1}{9 \\choose 1}{4 \\choose 3}\\end{matrix}}", "249da23ffe87a313ab1ed0fb22cbabae": "{\\vec {\\Omega }}", "249da942355f2a9352ba5ea6e9c5f1a7": "\\operatorname {cl} (\\varnothing )=\\varnothing \\!", "249e18c2d3d0f24ed8c3d9369218bf2f": "{\\frac {dG(s)H(s)}{ds}}=0{\\text{ or }}{\\frac {d{\\overline {GH}}(z)}{dz}}=0", "249e6aa25aac245d991375fa480857d0": "\\int _{a}^{b}f(x)\\,dx", "249e78ff656ca5fac54b5641b976abd2": "\\tan ^{2}+1", "249e7bc44bca05c93afdd803e9e30736": "E\\in V^{n}", "249eee7e901b96a04c07fe271c64419e": "{\\overline {\\varepsilon }}=\\varepsilon ({\\underline {E}},{\\overline {P}})={\\frac {\\overline {P}}{{\\underline {E}}A}}", "249f05209600c740604e68878055057f": "(x,u,w)\\ {\\stackrel {\\mathrm {def} }{=}}\\ (x^{i},u^{\\alpha },w_{i}^{\\alpha })\\,", "249f4c688e26cebec24dda77c94052ca": "r\\left\\|A\\right\\|_{\\alpha }\\leq \\left\\|A\\right\\|_{\\beta }\\leq s\\left\\|A\\right\\|_{\\alpha }", "249f7f11c19dfd71a4ac13915b1b1a07": "\\mathbb {E} {\\bigl [}|X|^{p}1_{\\{U=0\\}}{\\bigr ]}=\\mathbb {E} {\\bigl [}1_{\\{U=0\\}}\\underbrace {\\mathbb {E} {\\bigl [}|X|^{p}{\\big |}\\,{\\mathcal {G}}{\\bigr ]}} _{=\\,U^{p}}{\\bigr ]}=0,", "24a11bee8257982e773660bdb81269f2": "k^{h}", "24a1366c88a506ebe02c15977d9087aa": "\\displaystyle {|f(x)|\\leq C\\left(\\int |{\\widehat {f}}(t)|^{2}(1+t^{2})^{s}\\,dt\\right)^{1 \\over 2}=C((I+Q^{2})^{s}{\\widehat {f}},{\\widehat {f}})^{1 \\over 2}\\leq C^{\\prime }\\|{\\widehat {f}}\\|_{(s)}=C^{\\prime }\\|f\\|_{(s)}.}", "24a15b1db2548d1b0fcb5b4635f9c4f2": "K={e^{-\\Delta G/RT}}", "24a15eedee98a63ecd810b75a2485668": "A({\\boldsymbol {\\theta }})", "24a17d2e2b000c187e42eafb8faf1059": "\\rho =\\rho (x)", "24a1dd1f7257a670e1ade13ed3b5911f": "\\ell =I-{\\frac {1}{k}}A", "24a1e5c13bb0e805cdd787d0d9b273fb": "V^{*}:H\\rightarrow K", "24a30534e1fe74568cb9e8e02833e17d": "R_{ijk}^{l}", "24a30ae32a63fdf7ef771bba6df75b18": "\\lim _{P\\to \\infty }{\\frac {1}{{\\frac {1-\\alpha }{P}}+\\alpha }}={\\frac {1}{\\alpha }}", "24a357ee515b42f211bac808553e0691": "\\rho _{0}={\\frac {v_{3}^{2}}{{\\sqrt {v_{1}^{2}+v_{3}^{2}}}{\\sqrt {v_{2}^{2}+v_{3}^{2}}}}}.", "24a3e44064471d8b817d8cd5e82bc327": "X_{1}^{n}=(X_{1},\\ldots ,X_{n})", "24a443043ad5cfe44f3e8bf46d900075": "{\\mathbb {P}}H^{0}(L^{N})", "24a472a6b170218c5ec1d2c4104784ee": "\\log _{2}(x^{2})=2\\log _{2}(x).\\,", "24a4a511e6dd27921fa4b30560e20ad5": "D={\\frac {|(\\mathbf {r} _{1}-\\mathbf {r} _{0})\\cdot \\mathbf {n} |}{|\\mathbf {n} |}}={\\frac {|\\mathbf {r} _{1}\\cdot \\mathbf {n} -\\mathbf {r} _{0}\\cdot \\mathbf {n} |}{|\\mathbf {n} |}}={\\frac {|\\mathbf {r} _{1}\\cdot \\mathbf {n} +a_{0}|}{|\\mathbf {n} |}}={\\frac {|a_{1}x_{11}+a_{2}x_{21}+\\dots +a_{N}x_{N1}+a_{0}|}{\\sqrt {a_{1}^{2}+a_{2}^{2}+\\dots +a_{N}^{2}}}}", "24a4b0b887e1b2dbd2331de88598a6d3": "{\\sqrt {(350^{2}-50^{2})/100}}=c", "24a4d5e671be5bb7026ad86a68d14220": "l\\,\\!", "24a50daac817f05c417d95eb1594d01c": "q_{k}\\leftarrow Aq_{k-1}\\,", "24a515992482e3b61870c98e7a3d11c7": "\\{b^{n}a^{m}b^{2n}:n\\geq 0,m\\geq 0\\}", "24a5b2f2acd079f75981e6d679984669": "\\scriptstyle {AB=Rc-Rt=h(Y,X)}", "24a5b831b2a036aae4037979b2684412": "a\\times b\\times c", "24a5d8fdaf168d10d65281b502d81b77": "(E,\\rho )", "24a616dd6d3f439d19fc73f8c5730c8b": "E_{1ss}=y_{1ss}+{\\frac {q_{1}^{2}}{2gy_{1ss}^{2}}}", "24a6423c7547bf00b2965d8006744abb": "f^{(n)}(c)", "24a715c68f0e7aa01f37ea50d5f5097e": "\\,{\\mbox{R}}(z,dt)={\\mbox{T}}_{x}(-ydt){\\mbox{T}}_{y}(xdt)", "24a78ab07f629474b1adfa6d8a28c2b6": "y_{tt}=c^{2}y_{xx}", "24a79614c9dea1f695109be16950b771": "p_{ij}^{k}", "24a79bd440c8de8936305ba1d446559b": "{\\frac {{\\text{d}}C_{1}}{{\\text{d}}t}}={\\text{k}}_{1(1)}{^{0}_{2}}SE-({\\text{k}}_{2(1)}+{\\text{k}}_{3(1)})C_{1}", "24a7a95f596acb7970f244c548f9243d": "\\mathbf {e} \\,", "24a8042b1facc2ce188626f45ea0c332": "E_{\\mathrm {nonbonded} }=\\sum _{i>j}f_{ij}\\left({\\frac {A_{ij}}{r_{ij}^{12}}}-{\\frac {C_{ij}}{r_{ij}^{6}}}+{\\frac {q_{i}q_{j}e^{2}}{4\\pi \\epsilon _{0}r_{ij}}}\\right)", "24a8315094adbb2a7669d46774e04007": "\\varepsilon _{x}={\\frac {\\partial u_{x}}{\\partial x}}\\,\\!", "24a855e468f4680ca98638a05ae8405c": "\\mathrm {E} [\\,G_{F}(t_{1})G_{F}(t_{2})\\,]=F(t_{1}\\wedge t_{2})-F(t_{1})F(t_{2}).", "24a88d099bf4d379d88f05eefe00577a": "T_{cold}", "24a8a9e4491a57822186f76fea9d33f0": "F_{total}=F_{1}+{\\frac {F_{2}-1}{G_{1}}}+{\\frac {F_{3}-1}{G_{1}G_{2}}}+{\\frac {F_{4}-1}{G_{1}G_{2}G_{3}}}+...+{\\frac {F_{n}-1}{G_{1}G_{2}...G_{n-1}}}", "24a8fec9119c2ada2e64a219eaa84200": "e^{ia({\\hat {n}}\\cdot {\\vec {\\sigma }})}=I\\cos {a}+i({\\hat {n}}\\cdot {\\vec {\\sigma }})\\sin {a}\\,", "24a9cba1145c9db7a6ff6ddb6d04f4a7": "{\\mbox{AC}}=\\bigcup _{i\\geq 0}{\\mbox{AC}}^{i}", "24a9eb07c65a72ffea1be24501ee8ee0": "(C\\otimes y)_{j}\\ =Y_{j}=\\sum _{i=-(m-1)/2}^{i=(m-1)/2}C_{i}\\,y_{j+i}\\qquad {\\frac {m+1}{2}}\\leq j\\leq n-{\\frac {m-1}{2}}", "24aa249550e0e3ff769e970ee6935284": "\\kappa _{\\mathit {ri}}=\\kappa _{i}/\\kappa ", "24aa669497bfafeeefd673619c4092b3": "u\\colon C\\to J(C),u(p)=\\left(\\int _{p_{0}}^{p}\\omega _{1},\\dots ,\\int _{p_{0}}^{p}\\omega _{g}\\right){\\bmod {\\Lambda }}.", "24aa964ae15fb4eb6571e1409ccd553c": "(x-az)^{2}+(y-bz)^{2}-r^{2}z^{2}=0,", "24aa966d5cc7017e292a62f7346b33b8": "H(\\varepsilon )", "24aaaef7c95258440ec788ae63e3a9f5": "L_{c}", "24aacc78f1b57600cf0ce46ef027d918": "\\mathbf {j=} {\\frac {\\mathbf {v} }{\\mathrm {v} }},\\qquad \\mathbf {v} =\\alpha D_{\\alpha }|\\mathbf {p} ^{2}|^{{\\frac {\\alpha }{2}}-1}\\mathbf {p,} ", "24aafc6683975951198e6f23019e9487": "n(n-1)f[n+1]+3nf[n+2]-4f[n+3]-3nf[n+1]-f[n+2]+2f[n]=0", "24aafe7bd251499482422dfe3e244486": "\\mathrm {nDCG_{p}} ={\\frac {DCG_{p}}{IDCG_{p}}}", "24ab0b3c20d6ea64b5c6c9233b77142a": "S(q)=1", "24ab5ae3904669f68999ec38eacafe27": "P=\\left[{\\frac {u_{i}^{*}u_{i}-v_{j}^{*}v_{j}}{1-f_{i}^{*}f_{j}}}\\right]_{i,j=0}^{n-1}", "24abb4502a39fc5003509e54f618ff18": "I^{*}", "24abdbbf72ebeef64a1626b7c9a1e27e": "1.34\\times {\\sqrt {\\mbox{LWL}}}", "24ac30a2296609d4fcf5c5d6d4e9f2d1": "1-s\\rho =s\\,", "24ac87f341319a196ff395579e2ffb8d": "{{V}_{bi}}={\\frac {kT}{q}}\\ln \\left({\\frac {{{N}_{A}}{{N}_{D}}}{{{p}_{0}}{{n}_{0}}}}\\right)", "24ad323f18c8a9f7f4949cb23d9a2c58": "\\sigma =\\nabla \\times \\nabla \\times \\Phi ", "24ad65ff63b2e8e09ec3f313600e1b55": "{\\begin{bmatrix}1&1&1&1\\\\1&1&-1&-1\\\\1&-1&0&0\\\\0&0&1&-1\\end{bmatrix}}", "24ad896dccf6fd94eb18478f521558ff": "S^{ij}=g^{ik}~S_{k}^{~j}=g^{jk}~S_{~k}^{i}=g^{ik}~g^{jl}~S_{kl}", "24adaa6bd51b1b184e6ae5f5a7b57496": "\\displaystyle {F_{m}(t,x)={t^{m} \\over m!}\\cdot \\psi \\left({t \\over \\varepsilon _{m}}\\right)\\cdot f_{m}(x),}", "24adbe30df874788f8cb8a05201eff45": "H=H_{0}+gV", "24adf935c53fd37ad66fd6f27cd7d5ad": "{\\mathcal {O}}_{k}/{\\mathfrak {p}}", "24ae1144aee8e5a8f3c7178e5b037e29": "\\scriptstyle =(1.2\\pm 2.2)\\times 10^{-43}", "24ae25188fa7a1403baf9d625166cf6f": "|f|_{C^{0,\\alpha }}=\\sup _{x\\neq y\\in \\Omega }{\\frac {|f(x)-f(y)|}{|x-y|^{\\alpha }}},", "24ae4b8be5bb0b0fbbd227549427d439": "n\\times n\\,", "24ae7848ddf8493dff50fcd63721a21d": "{\\begin{bmatrix}1&B_{0;1,2}\\\\B_{0;2,1}&1\\end{bmatrix}}{\\begin{bmatrix}y_{1,t}\\\\y_{2,t}\\end{bmatrix}}={\\begin{bmatrix}c_{0;1}\\\\c_{0;2}\\end{bmatrix}}+{\\begin{bmatrix}B_{1;1,1}&B_{1;1,2}\\\\B_{1;2,1}&B_{1;2,2}\\end{bmatrix}}{\\begin{bmatrix}y_{1,t-1}\\\\y_{2,t-1}\\end{bmatrix}}+{\\begin{bmatrix}\\epsilon _{1,t}\\\\\\epsilon _{2,t}\\end{bmatrix}},", "24aeb1bbe0a85f8165835190b05c716f": "\\gamma ={\\frac {1}{\\sqrt[{}]{1-{\\frac {v^{2}}{c^{2}}}}}}", "24aeb2a72a3293beb348aa59ebaac536": "\\mathrm {Re} \\to \\infty ", "24aef4304dfaee6a7a5eaac7491e9b6f": "[y_{j},y_{j+1}]", "24af5322ad0079f3db48eef6e8190130": "P_{A}O_{2}=F_{i}O_{2}(P_{atm}-P_{H_{2}O})-{\\frac {P_{a}CO_{2}}{0.8}}", "24aff3c26c898ef818e63c20cbe93d65": "{\\bar {k}}_{\\alpha }(s)={\\frac {s{\\bar {\\psi }}_{\\alpha }(s)}{1-{\\bar {\\psi }}_{\\alpha }(s)}}", "24b0c35c03f00eeb7945be343bd8827e": "\\alpha \\in K", "24b0c67ec78db0f84243cfef40f3431b": "\\mathbf {F_{L}} ", "24b0c99d5a15453372ae301d7d172706": "A=\\mathrm {diag} (z_{1},\\dots ,z_{n})+{\\begin{pmatrix}1\\\\\\vdots \\\\1\\end{pmatrix}}\\cdot \\left(w_{1},\\dots ,w_{n}\\right).", "24b0cbe10c7ac27d956405e412924d9f": "0=\\lambda _{0}\\leq \\lambda _{1}\\leq \\lambda _{2}\\cdots ,", "24b11757e7c581a4eb0a9b26789bb0d7": "we^{w}={\\frac {I_{S}R}{nV_{T}}}e^{(V_{s}+I_{s}R)/(nV_{T})}", "24b12535ad9f3951b69180dba8ed91a0": "\\Theta (n+z)", "24b137769c59638a8b6dc6e8a3387e93": "{\\textrm {LWL}}", "24b14c8e1f1a9d5509d94061b588ef03": "1={1 \\over \\Gamma (s)}\\int _{0}^{\\infty }e^{-t}\\,t^{s-1}\\,\\mathrm {d} t\\qquad ({\\textrm {Re}}(s)>0)\\,,", "24b1b2e995a24cdff8e24baf095affa5": "\\operatorname {*} {\\arg \\min }_{\\hat {x}}\\max _{x\\in Q}\\left\\|x-{\\hat {x}}\\right\\|^{2}.", "24b1d4672b02ee071d7754ef81b072b0": "R(y)", "24b246725cf6911efcb5021c22a08667": "\\gamma _{\\nu }", "24b28756001eed7b00fb3b1b103a6972": "{\\frac {\\partial u}{\\partial t}}+\\nabla \\cdot \\left({\\boldsymbol {v}}u-D\\nabla u\\right)=f,", "24b2aa616e53ba1f76754338d671eb7e": "E(|x-\\mu |)=b\\,", "24b2fdde8920609c4fd71f9885e4eb5e": "T=T_{0}-\\alpha y\\,,", "24b31fb2a6e90ebc37ce5fc98c425733": "C_{(-)}^{*}=C_{(-)}", "24b32ecb6235989bf18164742203fe2b": "\\pi _{3}E=\\pi _{5}E=\\cdots ", "24b396886f16889928536fa6e4656bd8": "\\mathbb {E} _{Z_{1},\\cdots ,Z_{k-1}}{\\lVert x_{k}-x\\rVert ^{2}}\\leq \\left(1-\\mathbb {E} _{Z_{1},\\cdots ,Z_{k-1}}\\left|\\left\\langle {\\frac {x_{k-1}-x}{\\lVert x_{k-1}-x\\rVert }},Z_{k}\\right\\rangle \\right|^{2}\\right){\\lVert x_{k-1}-x\\rVert ^{2}}.", "24b39e635daa9bd6215d812a91fee642": "p^{k}>(p^{k}-1)=(p-1)(p^{0}+p^{1}+...+p^{k-1})\\geq \\sigma _{1}(w)(p^{0}+p^{1}+...+p^{k-1})", "24b3a8d369da4efdac42fdb0b50059a2": "d(X,Y)=\\mu (X\\,\\triangle \\,Y)", "24b3bfc531bf7328c942fbfa7c7a9e3c": "Y\\sim \\Gamma (\\beta ,1)\\,", "24b3f3219178e0c85f4835adfe4b76ba": "G_{s}(L/K)=\\{\\sigma \\in G\\,:\\,v_{L}(\\sigma a-a)\\geq s+1{\\text{ for all }}a\\in {\\mathcal {O}}\\}.", "24b4140897713f6edb65a504b638e318": "{\\mathbb {R} }^{3}", "24b425f1edb88d255ba074dbb28b9f56": "\\pi _{1},\\pi _{2},\\ldots ,\\pi _{n}", "24b44f491a25097b86f06fd8bb94af27": "\\pi _{3}={\\frac {ak_{b}T\\varepsilon _{0}}{q^{2}}}", "24b493c54f79ba1b99db14a4c0df7da2": "\\Delta \\lambda '=\\lambda _{2}'-\\lambda _{1}'", "24b4b07899c733c26e5d57f69ef9407d": "f(X)=\\sum _{n\\geq 0}a_{n}X^{n}.", "24b4b53fc3b78062cec9c41c82ad37f2": "O=\\{X\\in 2^{\\omega }|\\exists n(X(n)=1)\\}", "24b4fa5e6ee7a1f6bb5083ed874a0ef5": "{\\frac {d[P]}{dt}}=k_{\\mathrm {cat} }[ES]", "24b505136a199a9f9d6089be89f2db23": "(a_{1},b_{1})\\circ (a_{2},b_{2})=(a_{1}\\ast a_{2},b_{1}\\cdot b_{2})", "24b50a5d8711145c1b7e851201a4f3d9": "x(t-T)", "24b5247734788329276b614ba95a1612": "{\\widehat {T}}", "24b57c1f4b2ca24a4902229f7002a7cc": "\\operatorname {CAT} (\\ell )", "24b58c4a77c90cb84f2ff9bdaec29d84": "|a_{\\max }|\\Delta t<\\Delta x/2\\,", "24b5be5ad64e63aa50bca7fa046f399a": "\\omega (r_{t})=\\left[f(r_{t}|M_{t}=m^{1});\\dots ;f(r_{t}|M_{t}=m^{d})\\right].", "24b63acac7a9337befca09cfe12774c6": "\\|x+y\\|\\geq \\|x\\|+\\|y\\|\\;\\forall x,y\\in V", "24b6c570798307c10a1c4555649ab6a5": "{\\frac {d\\mathbf {y} }{dx}}=\\left({\\frac {dy_{1}}{dx}},{\\frac {dy_{2}}{dx}},\\ldots ,{\\frac {dy_{n}}{dx}}\\right)", "24b6e9480fb7028e53033cfea3ff4682": "{\\dbinom {n}{m}}(n-m)!", "24b7953c35aa07650844de907ec18c0c": "J(u):=\\int _{\\Omega }|\\nabla u|^{2}\\,dx", "24b7c69dff795ac056de9fa8a759894b": "\\mathrm {proj} _{0}\\,(\\mathbf {v} ):=0", "24b7e5d884c263515f87bb2a994193d9": "{\\textbf {y}}(t)={\\textbf {G}}(\\infty ){\\textbf {u}}(t)", "24b8179390cbe81d12c66d96cf39f2c5": "_{2}{\\text{H}}_{2}", "24b821dcf1efc2ea1794eb8069959b71": "n(r)=br^{-3/2}\\exp {(-cr)}/V_{\\xi }", "24b8a39ea1e03e97811c6d4a1bb665b8": "\\lim _{s\\rightarrow 1+}P(X\\in A)\\,", "24b8cbe5a7bf12b2db849f10244222f1": "\\textstyle (A_{1},A_{2},\\dots )", "24b93cff3eb3737ebabe2e2633b1553a": "\\deg(u')\\leq g", "24b946056d8467e2d0af4355560e30ee": "u^{*}=Qu", "24b96ccdadbe944372ee4820efef5178": "{\\begin{aligned}x_{i}={\\frac {\\displaystyle 1+\\sum \\limits _{j=1}^{i-1}\\prod \\limits _{k=1}^{j}\\gamma _{k}}{\\displaystyle 1+\\sum \\limits _{j=1}^{N-1}\\prod \\limits _{k=1}^{j}\\gamma _{k}}}\\qquad {\\text{(1)}}\\end{aligned}}", "24b970e0f23fe457bd12aa39c1c9654c": "{\\frac {P_{Rx}}{P_{Tx}}}\\;=\\;{\\frac {c_{0}F_{g}}{d^{\\gamma }}}", "24b9b87b9227d78edc8830883a21fc2a": "{\\Big (}({\\mathcal {M}},s)\\models EG\\phi {\\Big )}\\Leftrightarrow {\\Big (}\\exists \\langle s_{1}\\rightarrow s_{2}\\rightarrow \\ldots \\rangle (s=s_{1})\\forall i{\\big (}({\\mathcal {M}},s_{i})\\models \\phi {\\big )}{\\Big )}", "24b9f46a106a80650c22f6207104dd48": "B=[b_{ij}]", "24b9f5ff78481c5490044dd3fd1cc20d": "y_{t}=\\alpha +X_{t}\\beta +\\varepsilon _{t},\\,", "24ba8ff1842c2cab6cb6a26be956d77f": "E_{x}={\\frac {k_{o}^{2}\\varepsilon _{r}-k_{z}^{2}}{k_{o}^{2}-k_{z}^{2}}}[{\\frac {-jk_{xo}k_{z}}{\\omega \\varepsilon _{o}\\varepsilon _{r}}}(A\\ e^{jk_{x\\varepsilon }w}+B\\ e^{-jk_{x\\varepsilon }w})+{\\frac {m\\pi }{a}}(C\\ e^{jk_{x\\varepsilon }w}+D\\ e^{-jk_{x\\varepsilon }w})]e^{jk_{xo}(x+w)}sin({\\frac {m\\pi }{a}}y)e^{-jk_{z}z}\\ \\ \\ \\ \\ \\ \\ \\ (45)", "24baa138ce2df06c511010f55149feba": "L_{1}=\\ln(-x)", "24bad83487ba662d25dc953467c2907d": "r_{SOI}", "24bb0621fafd46721e3e4a7c5eced3ee": "f(x,y)=\\zeta (\\alpha (x,y),\\beta (x,y),\\gamma (x,y))=\\zeta (\\alpha ,\\beta ,\\gamma )=e^{\\alpha }[\\sin(3\\beta )-\\cos(2\\gamma )]\\,.", "24bb2719a34aa1245dcc03508501a913": "u(x,t)\\,", "24bb4f0638626718b281d971771cf7a3": "Y{\\stackrel {\\mathrm {d} }{=}}\\sum _{k=1}^{n}Y_{n,k},\\,", "24bb7c5899d7b92fba748392f46e3d98": "H^{2}\\times R", "24bb978698041e0766e5222ec0ea2d4f": "\\left(1-{\\frac {t}{\\lambda }}\\right)^{-1}\\,\\exp\\{\\mu t+{\\frac {1}{2}}\\sigma ^{2}t^{2}\\}", "24bbbf66a1261ac1164939a97a6dc6a8": "Formingtime=[L+n(d)]/V", "24bbd0b3e4d56d260993cba2f4754f06": "~A=A_{0}{\\frac {U+s}{1+p+s}}~", "24bc04db033f74eb5e39c7b3e93d0fea": "{\\mathcal {G}}_{n}", "24bc05d66a2dfe8527271f192ed842ca": "U_{nit}={\\bar {\\beta }}X_{nit}+(\\sigma \\eta _{n}X_{nit}+\\varepsilon _{nit})", "24bc3c8968c24096637441060dd9e1db": "\\exp \\left({-i\\omega t}\\right)=\\cos(\\omega t)-i\\sin(\\omega t),", "24bc4eab5e7c9cd9a0f2729889dd04b1": "{\\mathfrak {gl}}_{n}(F)", "24bc8101c171957a7afa41b517f20d63": "U_{\\nu }~d\\nu ={\\frac {8\\pi h\\nu ^{3}}{c^{3}}}~{\\frac {1}{e^{h\\nu /kT}-1}}~d\\nu ", "24bc8b45cc9515c2d88487e9f0b11be0": "y\\in \\{0,1\\}^{k}", "24bcfd8fc713e9ef5503a2f39bcb070b": "N_{n}={\\frac {q^{n}}{n}}+O\\left({\\frac {q^{n/2}}{n}}\\right).", "24bd13ca8544708ecf12e110c2f59713": "\\mathrm {VIF} ={\\frac {1}{1-R_{i}^{2}}}", "24bd55218e20f5d425c2514396c9475c": "m>{\\sqrt[{4}]{q}}", "24be0ace00b62ac1d1a34be3a56c95ca": "O(mn\\cdot \\log(mn))", "24be585e8cf4999121c4b9a8596cc1fa": "L*P*(1-R+R*R-R*R*R+\\cdots )=1", "24be9b8f1355f2bf5bfecc5e34a81517": "(mv^{2}/2)", "24bece2ffba23c79e598e90b8b2b4a5a": "{\\begin{aligned}(Q_{1}Q_{2})^{T}(Q_{1}Q_{2})&{}=Q_{2}^{T}(Q_{1}^{T}Q_{1})Q_{2}=I\\\\\\det(Q_{1}Q_{2})&{}=(\\det Q_{1})(\\det Q_{2})=+1.\\end{aligned}}", "24bed801f103c9657f661c9409cc923b": "-x\\,", "24bf0202fbf5e59462d2be6a194bbd7f": "c_{i}={\\frac {\\left(p^{2}+\\left(1-e^{2}\\right)z^{2}\\kappa _{i}^{2}\\right)^{3/2}}{ae^{2}}}.", "24bf8825218e19ce3174a9b53bb255fc": "{\\frac {\\pi }{4}}=4\\arctan {\\frac {1}{5}}-\\arctan {\\frac {1}{239}}", "24c009e3c295ddbb6dd02c8315c0a30f": "({\\mathfrak {k}},{\\mathfrak {p}})", "24c0619297000e3bae5daa73eb0b9fef": "\\mathrm {ENOB} ={\\frac {\\mathrm {SINAD} -1.76}{6.02}}", "24c0659911c578501609d7fc0c76f71d": "\\nabla \\cdot {\\boldsymbol {\\mathsf {T}}}", "24c06e851945e06335d239267288c10e": "K_{TE}={\\frac {X_{m}}{X_{s}+X_{m}}}", "24c078a13d168b16b0e02a719bae63f9": "\\pi _{0}(X),", "24c1362cd48718caf806bc95ea379b0e": "{\\frac {d\\tau }{dt}}={\\frac {1}{\\gamma (\\mathbf {v} )}}", "24c13af59c9b0b68aae83135bb470957": "6\\pi R\\eta {\\frac {\\partial x(t)}{\\partial t}}", "24c15e4235314183c77878ad115266c3": "s_{n}=\\sum _{k=1}^{r}a_{k}s_{n-k}", "24c1ad5da6e1aed092d9f790085f97e0": "\\mu _{A}^{*}", "24c29d944e732b4b942998e88f8f5da0": "(a,b)<^{d}(A_{i}\\times B)(a',b')\\iff a<^{d}(A_{i})a'\\lor (a=a'\\ \\land \\ b<^{d}(B)b')", "24c2a81e85ef85fa2b705bb745a7df20": "\\mu :A\\otimes A\\to A,\\qquad \\eta :I\\to A,\\qquad \\delta :A\\to A\\otimes A\\qquad \\mathrm {and} \\qquad \\varepsilon :A\\to I", "24c2b25c22df82cbb06ec8151d0afe1b": "R_{;l}-R^{n}{}_{l;n}-R^{m}{}_{l;m}=0.\\,\\!", "24c32aeec391b2003e3bba31372424e7": "\\sum _{i=0}^{n-1}F_{2i+1}=F_{2n}", "24c3eb73d7da000365eee0af15b9c545": "\\beta \\ =G\\,M/R\\,{c}^{2}", "24c42af13a3834012b17b85ae0494863": "y\\in L", "24c43ce76343d2934a0c50725c63b5de": "X_{i}={\\frac {n_{i}(t=0)+\\int _{0}^{t}{\\dot {n}}_{i,{\\text{in}}}(\\tau )d\\tau -n_{i}(t)}{n_{A}(t=0)+\\int _{0}^{t_{\\text{end}}}{\\dot {n}}_{i,{\\text{in}}}(\\tau )d\\tau }}", "24c471c83f0366ad7e69f2e1b9f3a9b3": "{\\textbf {a}}=a_{0}+a_{1}X+a_{2}X^{2}+\\cdots +a_{N-2}X^{N-2}+a_{N-1}X^{N-1}", "24c49a4deebaf691030a91d8d841206e": "{\\mathcal {H}}=2-e_{1}-e_{2}", "24c501341139788bc6c224ad419c3fa5": "G(x)=xK_{\\frac {2}{3}}(x)", "24c50bed2164421ecd6c68ad30e4ac3b": "K_{3}={\\frac {1.25\\times (GT+10000)}{10000}}", "24c538e715b1412d067a2e8b6aa886b1": "f:M\\to \\mathbb {R} ", "24c5982c7bb706ae7fd100f7b121ff68": "x=({\\textbf {A}}^{\\mathrm {T} }{\\textbf {A}})^{-1}{\\textbf {A}}^{\\mathrm {T} }\\phi ", "24c5e3cc2ad9098da423c0128b22d96b": "1\\Rightarrow (3,1,1)", "24c65104eff33337589e8fb013b1a9ba": "J_{n}^{-1}=J_{n-1}^{-1}+{\\frac {\\Delta {\\vec {x}}_{n}-J_{n-1}^{-1}\\Delta {\\vec {F}}_{n}}{\\Delta {\\vec {F}}_{n}^{T}\\Delta {\\vec {F}}_{n}}}\\Delta {\\vec {F}}_{n}^{T}", "24c65bb14b5c3fb29ab36e460036f9b0": "{\\mathcal {N}}=3", "24c6916d4af0fa7318550fd45840efba": "{\\binom {\\alpha }{\\beta }}={\\binom {\\alpha _{1}}{\\beta _{1}}}{\\binom {\\alpha _{2}}{\\beta _{2}}}\\cdots {\\binom {\\alpha _{n}}{\\beta _{n}}}={\\frac {\\alpha !}{\\beta !(\\alpha -\\beta )!}}", "24c696b8e54e0f2fd066ff1a07671784": "f_{k-q}-f_{k}=f_{k}-{\\vec {q}}\\cdot \\nabla _{k}f_{k}+\\cdots -f_{k}\\simeq -{\\vec {q}}\\cdot \\nabla _{k}f_{k}", "24c6d63c42123501905bacb333fb4cfe": "LL(\\alpha ,\\beta )\\to L(\\alpha ,\\alpha /\\beta ).", "24c71577c69bc60531e52b93bca4d205": "s_{x}=0", "24c77c7193a63959ff5559007f44c3f7": "S={\\cfrac {\\pi r^{3}}{4}}={\\cfrac {\\pi d^{3}}{32}}", "24c795eb1ea4acd2196677895b464e80": "{\\begin{aligned}S(\\beta _{1},\\beta _{2})=&\\left[6-(\\beta _{1}+1\\beta _{2})\\right]^{2}+\\left[5-(\\beta _{1}+2\\beta _{2})\\right]^{2}\\\\&+\\left[7-(\\beta _{1}+3\\beta _{2})\\right]^{2}+\\left[10-(\\beta _{1}+4\\beta _{2})\\right]^{2}\\\\&=4\\beta _{1}^{2}+30\\beta _{2}^{2}+20\\beta _{1}\\beta _{2}-56\\beta _{1}-154\\beta _{2}+210.\\end{aligned}}", "24c7be430ce43ffd7f8a68f81fa60c1c": "R^{2n}", "24c846bc486b0866b3fe3baa11a8336c": "\\varepsilon _{i}:\\ {\\vec {n}}_{i}\\cdot {\\vec {x}}=d_{i},\\ i=1,2", "24c8538e32f233a9a540389ab367b02e": "\\|x\\|", "24c856463af06ddf255f0a562a204930": "\\displaystyle {C_{k}=\\sup _{n}\\sum _{m}{(1+n^{2})^{k-1/2} \\over (1+m^{2}+n^{2})^{k}}<\\infty ,\\,\\,\\,c_{k}=\\inf _{n}\\sum _{m}{(1+n^{2})^{k-1/2} \\over (1+m^{2}+n^{2})^{k}}>0.}", "24c8594555b16056e42e5ff17e999a78": "C=\\Sigma X_{i}", "24c865567dce8c0be1ed3c757d851551": "t_{prt}=1.5", "24c881539adac6252bc3a25fe9874c42": "b=a{\\sqrt {e^{2}-1}}.", "24c8a5615d9c6cad23d79d448c7b848c": "E_{n}(x)", "24c8b4ccba82809cd0b19f9ed4d9e04a": "x\\mid y", "24c8b71eb3c8121becf7796ea3150bfd": "\\Re ", "24c91c9077e792ce00390be740e2ac86": "{\\mathbf {u}}\\cdot {\\mathbf {v}}=\\left(Q{\\mathbf {u}}\\right)\\cdot \\left(Q{\\mathbf {v}}\\right)\\,", "24c96602e63ed1085ee85286841c7812": "s\\in {\\mathcal {S}}", "24c98e0a64ae0022152f49e86de95e8d": "Y=C+I+G", "24c9d04f199a1b2dfc9231d2b3b28971": "Ns,\\end{cases}}", "24daf38045457445fdfdf14ea6f6d8f2": "[k]b\\,\\!", "24db79e02d60a103be48a8741621a148": "\\mathbb {X} ^{I}", "24dba8162dd15a07f4fc686acddd0364": "y_{1}(x)=c_{1}\\ln x+c_{2},\\qquad x\\in I,", "24dbca283bf50aac7c83a5ff91dcff5c": "S(\\rho )N", "24dbcdc7e16a64c1ca05b49a723b3fbb": "FDR\\leq {\\frac {m_{0}}{m}}q", "24dc004b2a162849cc94a209a4cc8179": "S_{1},S_{2},S_{3},S_{4},S_{5},\\ldots ", "24dc658c7d1a81ddde3724364a495837": "H=\\alpha H_{\\text{norm}}", "24dc6fc32961a2176df0aa3a52ab18d9": "\\omega \\in S_{0}", "24dc7610f253c0163a9c87083a8e01fe": "(\\mathbf {M} _{*})_{j}={\\frac {1}{l}}\\sum _{k=1}^{l}k(\\mathbf {x} _{j},\\mathbf {x} _{k}).", "24dcaae077391f9d6a58d9d523f66f18": "D_{1}=6P_{1}+4P_{2}-5D_{\\infty _{1}}-5D_{\\infty _{2}}", "24dcc5d4dcd3ca613a2f9a3ca80e6c45": "\\langle \\psi |F(A)|\\psi \\rangle \\,.", "24dd5a270bfda8301ed421642ad8c0b0": "\\Sigma _{s}(\\mathbf {r} ,E'\\rightarrow E,\\mathbf {\\hat {\\Omega }} '\\rightarrow \\mathbf {\\hat {\\Omega }} ,t)dE^{\\prime }d\\Omega ^{\\prime }", "24de032c4429b1e400821dd8d2d09fdc": "0\\leq \\phi <2\\pi ", "24de0f931509f7b5e1dab5e2772f7ba1": "P\\longrightarrow M", "24de86cb6f4854a91c3241f87455ed83": "T\\approx {\\frac {M}{M_{contact}}}", "24de89f6fe6f897e7e9937d0b0c0968d": "\\int _{0}^{\\frac {\\pi }{2}}\\sin ^{n}(x)\\,dx=\\int _{0}^{\\frac {\\pi }{2}}\\sin ^{n-2}(x)\\left[1-\\cos ^{2}(x)\\right]\\,dx", "24deb65c9483db51a191d620be5b85d4": "{\\text{ or if }}a_{i}\\neq b_{j}", "24ded52ad667983821193712d754cf03": "{\\mathcal {H}}[t]={\\dot {\\vec {x}}}[t]\\cdot {\\vec {P}}\\,[t]-{\\mathcal {L}}[t]=c{\\sqrt {m^{2}c^{2}+{\\left({\\vec {P}}\\,[t]-e{\\vec {A}}[{\\vec {x}}[t],t]\\right)}^{2}}}+e\\phi [{\\vec {x}}[t],t]\\,.", "24defe516f8f7de3587d5c420a828a15": "K_{i}(X)\\otimes \\mathbf {Q} =0\\ \\,i>0.", "24df0e3e4d4467501d969f8e52cab983": "z=1000", "24df22f27c601f112dd3e91419d580e8": "{\\mathbf {e} }_{i}=\\partial /\\partial x^{i}", "24df36c3111d445a3efa5b459f96911a": "s_{1}=m\\log k+1/a\\sum _{i=1}^{m}\\log u_{i}", "24df482628dbea25d86d464caf520193": "f(k)\\cdot |x|^{c}>|x|^{c+1}", "24df948cfd7a531ac26e5bb882ea659d": "t_{i}\\cdot t_{j}=SR((t_{i},t_{j}),(s_{i},s_{j}),O)", "24e01848941ab242ea0770788cf1b8f5": "u\\in C^{\\infty }(\\mathbb {H} )", "24e05b3963753071a519405aef2b57a5": "00", "24e6a539fecbe62603a707f51a16e864": "e^{i\\pi /4}", "24e70836845f5642d842c4f13772a410": "c\\in (a,b)", "24e779109890ff62e67e281d145c6ef0": "(a_{2},b_{2},c_{2},d_{2})", "24e7cba57e66fed366d96c2b17eaf2f8": "a2)\\Rightarrow a\\leq 2", "24ecc899bd14b3aa417c00485013e94a": "\\mu \\,", "24ecda7ae483893d9b5f7067c0762f5e": "\\nu _{i}+\\rho _{n}\\leq \\mu _{i}\\leq \\nu _{i}+\\rho _{1}\\,", "24ed0825e3e1581782a9e2f9eec52704": "\\scriptstyle n_{i}\\,", "24ed39c417b81b294aeb7f4254d8064d": "\\zeta (s_{1},s_{2},\\ldots ,s_{n})=\\sum _{k_{1}>k_{2}>\\cdots >k_{n}>0}k_{1}^{-s_{1}}k_{2}^{-s_{2}}\\cdots k_{n}^{-s_{n}}.\\!", "24ed45b7698c9f29f3784e1be345b643": "{\\begin{matrix}a\\uparrow b=a^{b}=&\\underbrace {a\\times a\\times \\dots \\times a} \\\\&b{\\mbox{ multiplied copies of }}a\\end{matrix}}", "24ed48f676744bfa8e9ea9c6392b24df": "\\exists b\\in B", "24ed4bfc9a0a0b6dbd672bc9e95c153a": "\\mathrm {Var} (Z(x_{0}))=\\mathrm {Var} ({\\hat {Z}}(x_{0}))+\\mathrm {Var} \\left({\\hat {Z}}(x_{0})-Z(x_{0})\\right).", "24ed528ee2191bdc8bb3484035b1e8c5": "83^{\\mathrm {o} }", "24edb2ea71428b08548c56890a88df9b": "A[i,j]", "24edce33157cd57ec63d505a4cfa4e7b": "\\delta _{S}=2+{\\cfrac {1}{2+{\\cfrac {1}{2+{\\cfrac {1}{2+\\ddots }}}}}}\\,.", "24ee2d6f1184b980ffbc9c4a9e2683b5": "(\\forall x\\phi )\\lor \\psi ", "24eeb3d50c0aa4c9ac79ff1e040f748e": "dM={\\frac {R^{2}/2d\\theta sin^{2}{\\theta }}{\\pi R^{2}}}M={\\frac {sin^{2}{\\theta }}{2\\pi }}Md\\theta ", "24ef13e9fd9fbae1b5a1a824f2148ab6": "E(B|A=a,A>B)=E(A/2|A=a,A>B)=a/2.", "24ef65f019cdeff3dd168dbbc1cffda4": "\\textstyle {\\{e_{1},\\ldots ,e_{n}\\}}", "24ef74c7fc988f1595bd0dcf26fad3ae": "x=x_{0}+x_{1}\\epsilon _{1}+x_{2}\\epsilon _{2}+{}", "24ef86dc89c61f8eefb663cc113a45f5": "{\\tfrac {\\partial I}{\\partial t}}", "24efb8b3d47995de686703e4a61ca4b6": "8911=7\\cdot 19\\cdot 67\\qquad (6\\mid 8910;\\quad 18\\mid 8910;\\quad 66\\mid 8910).", "24efbc381c64afe6caab140825745c2c": "B_{\\lambda }", "24efc5444fdc314cad0efbd7a2b32136": "{\\text{Geom}}(p)={\\text{NB}}(1,\\,1-p).\\,", "24f011e35eb6f32491e5dade56a07292": "\\mathbf {E} [W(h)]=0,", "24f064825d181debea866c76cc0a8dff": "\\scriptstyle 1,000,000{\\sqrt {N}}", "24f068c5eda18182c4e987e30361dde5": "\\tau _{c}", "24f09494d50a573dc0b88e5606762fee": "A(t)=\\mathbf {U} _{n}^{T}\\mathbf {A} ", "24f0ded95131e41d741b2305a48b89b1": "=M*V*{\\ddot {s}}E+B*V*{\\dot {s}}E+pE", "24f1319f9fcdeb0929fcfb97c311efad": "\\pi -\\pi ", "24f152d852ee24e9eda729027991a6d4": "\\chi (n)=\\left({\\frac {n}{p}}\\right),\\ ", "24f15e0ed871f58aa67cdf8b404d88cd": "r={\\frac {{\\frac {1}{n-1}}\\sum _{i=1}^{n}(X_{i}-{\\overline {X}})(Y_{i}-{\\overline {Y}})}{s_{X}s_{Y}}}", "24f15fe91fe77af4a20095de542bd9b8": "\\ f_{tuning}(t)=K_{o}\\cdot \\ v_{in}(t)", "24f16ae6ff5964def1d42c687bea8fdd": "R=\\lceil \\log _{2}M\\rceil ", "24f20d0a0d4685f592bc3f345b494f43": "A(t):=e^{iHt/\\hbar }Ae^{-iHt/\\hbar }.", "24f2409d3cea6f8b2b12eaff194b2f58": "{\\text{return}}\\colon A\\to A^{?}=a\\mapsto {\\text{Just}}\\,a", "24f25cdcd2eb36e86f10afc40fb322c9": "a\\wedge I=0", "24f28446139ad8828f346d0732af0f6b": "\\Delta \\mathbf {A} \\equiv \\nabla ^{2}\\mathbf {A} ", "24f2851d3a81304b4217edeeeac4604d": "R_{\\infty }={\\frac {m_{\\rm {e}}c\\alpha ^{2}}{2h}}\\Rightarrow m_{\\rm {e}}={\\frac {2R_{\\infty }h}{c\\alpha ^{2}}}", "24f28becfa0d33c285b226a2013cd1a0": "b>a>c", "24f30e7175d822fc09d4649525bb8af5": "\\mathbf {Q} _{v}", "24f32180dfcdc43e3a1a30438ae32832": "D\\!\\!\\!\\!/\\psi _{i}=-\\lambda _{i}\\psi _{i}.", "24f35934b2810ea6bde280fbe6c63277": "k_{r}^{2}=k_{x}^{2}+k_{y}^{2}+k_{z}^{2},\\,k_{x}={\\frac {m\\pi }{l}},\\,k_{y}={\\frac {n\\pi }{w}},\\,k_{z}={\\frac {p\\pi }{h}}\\,k_{xy}^{2}=k_{x}^{2}+k_{y}^{2}", "24f38fb2e400574684c81ad625e782ba": "bh+l(a+b+c)", "24f3a172a9966bdeb92d7d2f277d9ca3": "{\\hat {a}},{\\hat {c}}", "24f3a38f4b3207edb37da23fad6e21da": "\\mathbb {P} \\left[A\\cap B\\right]=\\mathbb {P} \\left[A\\right]\\mathbb {P} \\left[B\\right],\\qquad \\forall A\\in {\\mathcal {I}}_{X},\\,B\\in {\\mathcal {I}}_{Y},", "24f401c1e4aab19ec1a825ef2715cadf": "\\operatorname {ad} _{g}", "24f4209d94383ad4b096b27533faa5bd": "C_{d}=\\left({\\frac {s}{c}}\\right)\\left({\\frac {\\Delta p_{0}}{\\rho W_{1}^{2}/2}}\\right)", "24f469f44fc90664c7c15a6979a6e265": "\\mathbf {x} (t_{i})", "24f4b31421e34fe2f8e4866cd53d18d5": "{\\overrightarrow {\\operatorname {div} }}\\,(\\mathbf {\\underline {\\underline {\\epsilon }}} )={\\begin{bmatrix}{\\frac {\\partial \\epsilon _{xx}}{\\partial x}}+{\\frac {\\partial \\epsilon _{xy}}{\\partial y}}+{\\frac {\\partial \\epsilon _{xz}}{\\partial z}}\\\\[6pt]{\\frac {\\partial \\epsilon _{yx}}{\\partial x}}+{\\frac {\\partial \\epsilon _{yy}}{\\partial y}}+{\\frac {\\partial \\epsilon _{yz}}{\\partial z}}\\\\[6pt]{\\frac {\\partial \\epsilon _{zx}}{\\partial x}}+{\\frac {\\partial \\epsilon _{zy}}{\\partial y}}+{\\frac {\\partial \\epsilon _{zz}}{\\partial z}}\\end{bmatrix}}", "24f4f0adfd92d6b4868ef9af99277f5a": "{\\boldsymbol {\\mu }}_{Y|X}={\\boldsymbol {\\mu }}_{Y}+{\\boldsymbol {\\Sigma }}_{YX}{\\boldsymbol {\\Sigma }}_{XX}^{-1}\\left(\\mathbf {x} -{\\boldsymbol {\\mu }}_{X}\\right)", "24f50554a2dfde596785f645117caf0f": "9801{\\sqrt {2}}/4412", "24f564060c02ae068cf53f47d4424bb3": "{V}={I}{R}.\\ ", "24f571c9812028d15a47c7fbc0d75d6f": "J(v)={\\frac {1}{2}}\\int _{\\Omega }{|\\nabla v|^{2}\\,d\\mu }-{\\frac {1}{p+1}}\\int _{\\Omega }{|v|^{p+1}\\,d\\mu }", "24f5a8d494fa9048e5efbd42f26f53ce": "t_{e}", "24f5da9722db93967591a4d0dd3b17b8": "BP*(BP)", "24f64fabf10e083a750b85667f6b2b72": "\\ O[s(t)]=Gs(t)-D_{3}s^{3}(t)+\\ldots ", "24f68a56ed9f3b6ff81063d2e272d703": "x^{(k+1)}=D^{-1}(b-Rx^{(k)})", "24f6af6b5d380b3a471cab99fbde1c35": "d=2^{w-1},2^{w-1}+1,\\dots ,2^{w}-1", "24f6f76b6b7b8d44d6a1d90ab63e5db0": "\\mu \\approx {\\frac {-2c}{b-{\\sqrt {b^{2}-4ac}}}}\\,", "24f73ee335ad46e8862dc2b1320ea152": "\\mathbf {e} _{i}\\mathbf {e} _{j}+\\mathbf {e} _{j}\\mathbf {e} _{i}=2\\delta _{ij}", "24f7973de403049504bf2409dba2715e": "9^{\\frac {1}{2}}=3", "24f7f47d0ec0de18bb9511ddbe1633a1": "{\\mathcal {N}}(np,\\,np(1-p)),", "24f8ccfcc865ccebd9890eaa4fc90b13": "f(\\theta )=\\sum _{n}a_{n}e_{n}(\\theta ),\\quad a_{n}=\\langle f,e_{n}\\rangle ", "24f914094f00026d223831b9e4eb870d": "{\\frac {1}{3}}=e^{{\\frac {-1}{RC}}{\\frac {T}{2}}}", "24f939bf6084ab078b3aa9365a4f17bf": "_{lex}b", "24f98c870c2a31493ef530b1fea423bf": "d_{i}\\colon G_{i-1}\\rightarrow G_{i}", "24f9d0cdc32eb430e6849f098541b043": "m_{rel}", "24fa9b05afcf0704f480a3da2de0301f": "s=\\log _{32}31\\approx 0.991", "24fadd8e528f493243017a74f972c80a": "P(f)(z)={\\frac {1}{2\\pi }}\\int _{0}^{2\\pi }\\mathrm {Re} {\\frac {e^{it}+z}{e^{it}-z}}f(e^{it})\\,\\mathrm {d} t.", "24fb1491d717328be73a8231f64817fb": "2^{101}", "24fb18ba5fd008eb2f8ea515343988f2": "x\\#y\\;\\to \\;(x\\#z\\;\\vee \\;y\\#z)", "24fb5c2ede9d8cae2f20a109b11dd155": "Z_{\\text{lower}}(x)=Z(x)-{\\frac {1}{2}}T(x)", "24fb71cf064857488a308f649e7ab6b2": "\\delta s_{i}\\equiv s_{i}-m_{i}", "24fba1c9c992fe9b15a43c056089b5c9": "a_{i}=\\mathrm {sin} {\\frac {(2i-1)\\pi }{2n}},", "24fbf281eb28e940e9ec7ddaadc8cca3": "f_{c}(z)=z*z+c", "24fc27b22b1e2d7bd039a9af4c9d4dd3": "{\\hat {\\boldsymbol {\\beta }}}", "24fc8725e617037c6e3951fc0de87745": "\\int _{L}\\phi _{A}^{2}~dV=\\int _{R}\\phi _{B}^{2}~dV", "24fc8fd5506c0aaa6d4ccef215528a5d": "d=8", "24fc904db28c965c8d15ab65a2a9cea0": "\\pi _{n}(t)", "24fc98a1b981211033b41407bdbbbcbc": "E=0.0050\\,", "24fd1ff7d4b6cd13a58a1af078614fe2": "T^{-1}(E)=E\\,", "24fd4ddd4f9975086c50b5b9271b2493": "(Sv)(ds)=\\int _{0}^{1}f'_{t\\mu (I)}(\\mu (ds))dt", "24fd8820615f690fb770fbb9e301ad19": "\\Diamond \\Box p\\rightarrow \\Box \\Diamond p", "24fd9276cf3c7dbaeebe5812da978f4f": "{\\begin{array}{lllll}\\langle &a,b,c,d,e,p,q,r,t,k&|&&\\\\&p^{10}a=ap,&pacqr=rpcaq,&ra=ar,&\\\\&p^{10}b=bp,&p^{2}adq^{2}r=rp^{2}daq^{2},&rb=br,&\\\\&p^{10}c=cp,&p^{3}bcq^{3}r=rp^{3}cbq^{3},&rc=cr,&\\\\&p^{10}d=dp,&p^{4}bdq^{4}r=rp^{4}dbq^{4},&rd=dr,&\\\\&p^{10}e=ep,&p^{5}ceq^{5}r=rp^{5}ecaq^{5},&re=er,&\\\\&aq^{10}=qa,&p^{6}deq^{6}r=rp^{6}edbq^{6},&pt=tp,&\\\\&bq^{10}=qb,&p^{7}cdcq^{7}r=rp^{7}cdceq^{7},&qt=tq,&\\\\&cq^{10}=qc,&p^{8}ca^{3}q^{8}r=rp^{8}a^{3}q^{8},&&\\\\&dq^{10}=qd,&p^{9}da^{3}q^{9}r=rp^{9}a^{3}q^{9},&&\\\\&eq^{10}=qe,&a^{-3}ta^{3}k=ka^{-3}ta^{3}&&\\rangle \\end{array}}", "24fdb18c78112712708683fe92101c65": "a,b,c,", "24fdeee659f6f28a73c00424cfd299c5": "H_{\\gamma _{i}}", "24fe0020dbf0309ca4c054343282d962": "G(E/F)", "24fe1566a087e4efbff1404c153c356b": "u\\in H^{r}(E,E\\setminus E_{0};\\mathbf {Z} )", "24fed37420cd2363018dfdd6a5c4ff89": "\\int (d+e\\,x)^{m}(A+B\\,x)\\left(a+b\\,x+c\\,x^{2}\\right)^{p}dx={\\frac {B(d+e\\,x)^{m}\\left(a+b\\,x+c\\,x^{2}\\right)^{p+1}}{c(m+2p+2)}}\\,+\\,{\\frac {1}{c(m+2p+2)}}\\,\\cdot ", "24ff01c6ea93f337e37a627c8682b82c": "U_{P}=U_{R}", "24ff36361545cd7977d15fdf9a2f3126": "p(v;\\theta )=exp[C(v)+\\theta ^{i}F_{i}(v)-\\psi (\\theta )]", "24ff5b6517fdee6e49b054eddb2adc93": "T^{*}={\\frac {T-T_{B}}{T_{T}-T_{B}}}", "24ff69661169374ae12d9075e8d5086c": "v={\\sqrt {\\frac {GM}{r-r_{S}}}}", "24ff77d01658ab0096e85c40a6949343": "|\\lambda _{1}|d(y,z)-\\epsilon .", "250fdfa0018cc5ef4fb3852670864921": "\\langle \\mathrm {axis} ,\\mathrm {angle} \\rangle =\\left({\\begin{bmatrix}a_{x}\\\\a_{y}\\\\a_{z}\\end{bmatrix}},\\theta \\right)=\\left({\\begin{bmatrix}0\\\\0\\\\1\\end{bmatrix}},{\\frac {\\pi }{2}}\\right)", "251049edc52e2dd12c45479e16e7de42": "P=K\\rho ^{1+1/n}", "25104ae3e0ab8b0531e1b65f3a82bb8f": "t\\mapsto e^{tX},\\qquad t\\in \\mathbb {R} ", "251055b97d941e8309198c918c4fb527": "h\\;", "2510c39011c5be704182423e3a695e91": "h", "2510cc52d1d6da1c57a9b461ebaa595f": "\\delta _{i}f={f-f\\circ s_{i} \\over \\alpha _{i}}.", "2510ef50e6f55ad28e7bd866550b2fcf": "\\Delta v=\\pm 1,\\Delta J=0", "2510f269804c9225fd2fb7438c2ea4a0": "\\det(M)={\\begin{vmatrix}a_{11}&a_{12}&a_{13}\\\\a_{21}&a_{22}&a_{23}\\\\a_{31}&a_{32}&a_{33}\\end{vmatrix}}=a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{31}a_{22}a_{13}-a_{32}a_{23}a_{11}-a_{33}a_{21}a_{12}.", "25119edbd6a4e77842e2bdd4b8373f59": "{\\text{Power (MW)}}=0.85\\times 62.5\\times 2800\\times 480\\times 1.356", "2512191aec19c0c4d31150b3a29fc730": "G(k)=({\\frac {B_{1}^{+}+B_{1}^{-}}{B_{1}^{-}-B_{1}^{+}}}){\\frac {1}{\\gamma _{1}}}", "2512cf8fe0b277c46ac2f9cbfd873ab4": "\\mathrm {d} U=\\delta Q-\\delta W.", "2512f106c50874d9df041ec7287a5fc0": "=adx+b\\sigma '(x)dx+c\\sigma ''(x)dx+e\\sigma '''(x)dx\\,", "2512f2c5c25acddc510d885fb78549e2": "V=abc{\\sqrt {1+2\\cos(\\alpha )\\cos(\\beta )\\cos(\\gamma )-\\cos ^{2}(\\alpha )-\\cos ^{2}(\\beta )-\\cos ^{2}(\\gamma )}}.", "251312d089d129e7601aa491ca46f8a8": "\\geq 1-n^{-d}-n^{-2}\\geq 1-{\\frac {1}{n}}", "2513161aff07686e16706b6e53bc78cf": "u*=1-{\\frac {(\\delta +\\alpha +\\beta )\\sigma }{s}}", "2513941340243f23c94bbb5611fa7d4e": "\\textstyle P(\\theta >0.5\\mid k,n)", "2513a9d5a0c040c4a27e1d8b41cfccec": "x_{j}=\\sum _{k}X_{k}\\cdot z^{k\\cdot j}=X_{1}z^{1\\cdot j}+\\cdots +X_{n}\\cdot z^{n\\cdot j}", "2513bfa346dd311530deaebc461e40c4": "{1.17741\\times 2^{N/2}}={1.17741\\times 2^{64}}", "2513c01addac92d01146cc8ac1d15aa0": "\\pi _{k}^{S}", "2513d98b7890af62450c5bfa0c650b66": "L_{4k+2},", "2513e49a55a44df7f603976a3d1a0f50": "AA^{+}=I_{m}\\,\\!", "2513e546a51dc571bda3d6b600f65685": "T_{d}", "2513fd76ca54388d2af360a50e1b482f": "{\\frac {L}{D}}\\gtrsim 10", "2514049e919f1aafa67f512ee5ab402d": "k_{s}", "251410f3e334a087035bb64ee6aeb148": "V(G)\\setminus S", "25142b8236261abe3815a9a221a0dd8d": "m\\{x:\\,|Tf(x)|>a\\}\\leq m\\left\\{x:\\,|Tf_{a}(x)|>{\\tfrac {a}{2}}\\right\\}+m\\left\\{x:\\,|Tf^{a}(x)|>{\\tfrac {a}{2}}\\right\\}\\leq 4a^{-2}\\|T\\|^{2}\\|f_{a}\\|_{2}^{2}+Ca^{-1}\\|f^{a}\\|_{1}.", "2514443c4335be6fe0739dbe636b9a3f": "[SU(3)\\times SU(2)\\times U(1)]/\\mathbb {Z} _{6}", "25144a95720d66e8b8b37a94f7c0ea52": "c_{0}(-\\infty )^{n-2}\\,", "2514e999798ed17212f9b7e877a9bb0f": "u=\\bigvee D", "2514ee206e2d9b11c81e37e721b959fd": "{\\frac {d}{dt}}\\left(\\mathbf {p} \\times \\mathbf {L} \\right)=-mf(r)r^{2}\\left[{\\frac {1}{r}}{\\frac {d\\mathbf {r} }{dt}}-{\\frac {\\mathbf {r} }{r^{2}}}{\\frac {dr}{dt}}\\right]=-mf(r)r^{2}{\\frac {d}{dt}}\\left({\\frac {\\mathbf {r} }{r}}\\right)", "25159730f437291c74260dde3aa49f1e": "H(X)\\gets \\lg(S)", "2515b51a8baa4ba411e7daba3ebf8ebf": "\\ell _{OB}\\cdot \\ell _{OD}=y(y+2x)=y^{2}+2xy", "25160192b44f50bc7d38f3f7508e9a1d": "\\left[P\\left(\\eta \\right)+p'\\left(0\\right)\\right]=-\\sigma \\eta _{xx}.", "25162f2714050ac3a4030f1de2eedeb8": "{\\begin{aligned}Q_{0}&=0\\\\Q_{k}&=Q_{k-1}+{\\frac {k-1}{k}}(x_{k}-A_{k-1})^{2}=Q_{k-1}+(x_{k}-A_{k-1})(x_{k}-A_{k})\\\\\\end{aligned}}", "2516755bb6b32de0e005e9a8a9c70940": "[a;\\sigma ,\\tau ]={\\frac {\\theta _{1}(\\pi \\sigma a,e^{\\pi i\\tau })}{\\theta _{1}(\\pi \\sigma ,e^{\\pi i\\tau })}}", "2516dd3c734e1795c809446c2ec0eb19": "\\chi (\\Sigma )=-26", "25172e21f15f3198e0699f00d35f206a": "{\\frac {1}{4}}\\left({\\ddot {h}}_{{\\hat {\\theta }}{\\hat {\\theta }}}-{\\ddot {h}}_{{\\hat {\\phi }}{\\hat {\\phi }}}\\right)=-R_{{\\hat {t}}{\\hat {\\theta }}{\\hat {t}}{\\hat {\\theta }}}=-R_{{\\hat {t}}{\\hat {\\phi }}{\\hat {r}}{\\hat {\\phi }}}=-R_{{\\hat {r}}{\\hat {\\theta }}{\\hat {r}}{\\hat {\\theta }}}=R_{{\\hat {t}}{\\hat {\\phi }}{\\hat {t}}{\\hat {\\phi }}}=R_{{\\hat {t}}{\\hat {\\theta }}{\\hat {r}}{\\hat {\\theta }}}=R_{{\\hat {r}}{\\hat {\\phi }}{\\hat {r}}{\\hat {\\phi }}}\\ ,", "2517369c29f4d583fb115f4cdc3c7b46": "H_{jk}=2\\sum _{i=1}^{m}\\left({\\frac {\\partial r_{i}}{\\partial \\beta _{j}}}{\\frac {\\partial r_{i}}{\\partial \\beta _{k}}}+r_{i}{\\frac {\\partial ^{2}r_{i}}{\\partial \\beta _{j}\\partial \\beta _{k}}}\\right).", "25179ea592df9c7eb7b55172ec0d0e03": "\\forall g_{1},g_{2}\\in G\\;\\;f(g_{1}g_{2})=f(g_{1})f(g_{2})", "2517a9579d887eb51869981920f99e44": "TE=\\omega ={\\sqrt {\\operatorname {E} [(r_{p}-r_{b})^{2}]}}", "2518177c1e1a44b330e3671a3abeaa7d": "l\\in L", "2518301a3461fe4cf1555427ec80a424": "E_{\\gamma '}=hf'\\!", "25184a2d8fc43075e4fa3e72198f6f6c": "dJ_{S}(t)", "2518d30b2602ca9fde4a416076c761c8": "(2D)^{2}=d^{2}(1+\\cos \\alpha )^{2}+p^{2}", "2518db83877cc10c4ae9d8ba96918bae": "F_{p,n-p}", "2518fe5277d1dd3ebf3508810ed3c5a4": "x(t+1)=Ax(t)", "25197ab000bc8117d287a84dc51fbfb2": "V(e(u,P),P)=u", "25198a9d78486aa67e265053095fa762": "\\displaystyle \\beta _{n}", "2519b299f4a20a25a371fafeb9d0cb44": "\\textstyle (b_{n})_{n\\geq 0}", "2519d32fe90b632c473ca2c1e1e49e8b": "K(BG)\\cong R(G)^{\\wedge }.", "2519ef98bec9314266fc7d111d9fce0a": "2.44\\lambda \\cdot (f/\\#)", "251a0977e76f3fced3f85b7c6e9d0134": "{\\dot {x}}=f(x,r)\\,", "251a367597bb5bfbf269db7214ec1e0d": "E_{\\nu }=J_{\\nu }\\cdot W\\cdot \\rho ", "251a3de35a5b50f3356dd7e4454a7f50": "\\epsilon =\\hbar \\omega ", "251aaa1e171349ddc454ac5134bb63b3": "{\\hat {H}}\\rightarrow -{\\hat {H}}", "251ad222abc2421b9e7cf63f4e605e12": "X\\to {\\mathcal {M}}_{fg}", "251b0166793505d39e34da77651ea565": "p\\to p", "251b2cc19ca3c4aeaafc1250df847c40": "V_{a}=\\,", "251b811a6dd69d7d9faa4bd7ce5d9b8a": "n=\\sum _{i=1}^{c+1}\\varphi ^{i}(n),", "251bd15dbe5045f451baa2c4a04c7b7c": "\\omega ={\\frac {E_{+}-E_{-}}{\\hbar }},", "251bf594f207311dd2152b890c1184e3": "\\{1,\\dots ,n\\}", "251c074f11a2adec72228d7b4bf822d1": "{\\overline {r}}m:=rm\\,", "251c78ad41e62b76bb30af46e86baa31": "\\mathbf {W} ", "251c88f7760037e7b10c274ce0989501": "\\{\\to ,\\bot ,\\Box \\}", "251cb0150094f9a642fc36a608b3b3c7": "\\Omega ^{^{+}}", "251cc7df3616c65649d4791db844f58c": "A\\times A\\leq 2", "251ccaf6ff19a7f1e7444ba012c8be7c": "r''(x)=-3R_{max}\\{[4x(1-x)]^{-5/4}(1-2x)^{2}+2[4x(1-x)]^{-1/4}\\}", "251d2762a233ed83330de44b18b14722": "\\mathbf {\\hat {\\beta }} _{\\mathtt {GLS}}=\\left(\\mathbf {q} ^{\\mathbf {T} }\\cdot \\mathbf {C} ^{-\\mathbf {1} }\\cdot \\mathbf {q} \\right)^{-\\mathbf {1} }\\cdot \\mathbf {q} ^{\\mathbf {T} }\\cdot \\mathbf {C} ^{-\\mathbf {1} }\\cdot \\mathbf {z} ", "251d357c35814dcd39e7769210076a06": "f(k;\\rho )\\approx {\\frac {\\rho \\,\\Gamma (\\rho +1)}{k^{\\rho +1}}}\\propto {\\frac {1}{k^{\\rho +1}}}.\\,", "251d66bf3dae93705a252e5337edf362": "x\\subseteq y\\subseteq d(R)\\,", "251dfe163b89f528e1dd5e1a20e060a8": "a:=b+c", "251e00f8d165c93f69778ead938a36e9": "\\,(1-p+pe^{t})^{n}", "251e354587e0c5d29579d3830444b949": "L={\\sqrt {D\\tau _{\\mathrm {bulk} }}}", "251e5e78943b0ddcdcd4993e102d9104": "{Y \\over R}", "251eaea5b19d866d0d2953a0ee0cf341": "2^{n+m}", "251edaa02ee7eb6557a2c2b952093c2f": "\\lim _{\\lambda \\to \\lambda _{0}}{\\frac {B(\\lambda )-B(\\lambda _{0})}{\\lambda -\\lambda _{0}}}", "251f430c3024b6864ccf68e3a975a8a5": "m_{9}", "251f439513749e51bb87010ca526a4cf": "J_{\\alpha }(z)\\sim {\\frac {\\exp \\left(-i\\left(z-{\\frac {\\alpha \\pi }{2}}-{\\frac {\\pi }{4}}\\right)\\right)}{\\sqrt {2\\pi z}}}{\\text{ for }}0<\\arg z<\\pi ", "251f43dff90d27c060706c621e2f837e": "2^{L}", "251f460eca2ce7f8d63bacceb3dca659": "{\\frac {\\partial ^{2}\\Psi }{\\partial t^{2}}}=c^{2}\\nabla ^{2}\\Psi -\\left({\\frac {mc^{2}}{\\hbar }}\\right)^{2}\\Psi \\,\\!", "251f5260cb3388faaaa7b0054a810b4a": "U^{2}V^{2}=0\\,", "251fc379065ae9668fa68b1c9cd61675": "\\int x\\,\\operatorname {arcoth} (a\\,x)dx={\\frac {x^{2}\\,\\operatorname {arcoth} (a\\,x)}{2}}-{\\frac {\\operatorname {arcoth} (a\\,x)}{2\\,a^{2}}}+{\\frac {x}{2\\,a}}+C", "252038d1a6b80e3c0afc8ece6f19c9b3": "\\operatorname {var} [X]={\\frac {\\alpha \\beta }{(\\alpha +\\beta )^{2}(\\alpha +\\beta +1)}}\\!", "252053821c08ccbdcebf8936c5bdd7a3": "s-1", "2520978a04c536e26f52d4044dc0bef8": "\\sum _{x}f(x)\\Delta g(x)=f(x)g(x)-\\sum _{x}(g(x)+\\Delta g(x))\\Delta f(x)\\,", "252098e684a8b6077cac50a27f937add": "|Y|<|X|", "2520ed8a38f75d2bdeb14869af2fb45c": "K_{6}\\,", "2521125ef5a8f0cf0d1a5263f62b0b89": "Q^{-1}\\sum _{x}\\sum _{y}\\omega ^{xy}\\left|y\\right\\rangle \\left|f(x)\\right\\rangle .", "25211c4704deb25a33a7772ca8695418": "\\omega ,\\omega ^{\\omega },\\omega ^{\\omega ^{\\omega }},\\dots ", "2521514c197f9e53fb3d7e6ec51eef63": "-\\mathbf {j} _{r}", "252169e59f5c3823a3f0bf1ac0f88753": "P\\leq q^{H_{q}(H_{q}^{-1}({\\frac {1}{2}}-\\varepsilon ))\\cdot 2k}=q^{({\\frac {1}{2}}-\\varepsilon )\\cdot 2k}={\\frac {q^{k}}{q^{2\\varepsilon k}}}", "2521a31207feeebd95b31e1d72fa1696": "\\theta =2\\arccos {\\frac {d}{R}}=2\\arcsin {\\frac {c}{2R}}", "2521c07da56242d9c2647134dfeb1a75": "G_{IC}={\\frac {3E^{*}\\delta ^{2}t^{3}}{16L^{4}}}", "2521fdab5ed578f00c634ab23460357b": "\\gamma ''=\\gamma _{1}\\ast \\cdots \\ast \\gamma _{j-1}'\\ast \\gamma _{j+1}\\cdots \\gamma _{m}", "25223af2ca995f8f4201f0ebcaf9a3af": "I_{L}={A \\over \\omega L}={A \\over 2\\pi fL}.", "252244098f0ff5efcc6bdb8c17bc4a28": "e_{i}+\\omega _{i}\\geq d", "2522569284e3b4ef914a360cd45d95bc": "\\chi _{\\mathrm {red} }^{2}={\\frac {\\chi ^{2}}{\\nu }}={\\frac {1}{\\nu }}\\sum {\\frac {(O-E)^{2}}{\\sigma ^{2}}}", "25227b50ca3cad66143c4ead83307970": "{\\hat {n}}_{\\nu _{j}}|n_{\\nu _{j}}\\rangle =n_{\\nu _{j}}|n_{\\nu _{j}}\\rangle ", "2522a2600e1ec72ca00a8ef6ee6a6888": "{\\frac {\\partial {\\rm {tr}}(\\mathbf {AXB} )}{\\partial \\mathbf {X} }}={\\frac {\\partial {\\rm {tr}}(\\mathbf {BAX} )}{\\partial \\mathbf {X} }}=", "252307d8397acf15be49b316720e622f": "g(x)=x", "2523358aa2e2d98a86baf120d342b69c": "{\\mathcal {V}}^{{\\mathcal {A}}^{op}}", "252367bed4ec751f1969f4f038b004d5": "x_{1}", "25236f1df7abf414351793e89b557190": "f(x)\\to g(y)", "252387f90c875205b3da15bb1845d285": "H=-\\zeta {\\epsilon _{ijk}F_{ab}^{k}{\\tilde {E}}_{i}^{a}{\\tilde {E}}_{j}^{b} \\over {\\sqrt {det(q)}}}+2{\\zeta \\beta ^{2}-1 \\over \\beta ^{2}}{({\\tilde {E}}_{i}^{a}{\\tilde {E}}_{j}^{b}-{\\tilde {E}}_{j}^{a}{\\tilde {E}}_{i}^{b}) \\over {\\sqrt {det(q)}}}(A_{a}^{i}-\\Gamma _{a}^{i})(A_{b}^{j}-\\Gamma _{b}^{j})=H_{E}+H'", "2523ae54c47852a8cc196f81711645d1": "\\lambda _{n}={n \\choose 2}{\\frac {1}{N_{e}\\tau }}", "2523cdb202c2dee9ce97c9f68096fd27": "C(t)-P(t)+D(t)=S(t)-K\\cdot B(t,T)\\,", "2523d9d37898495692dfcb3b20f07f1e": "A^{2}S^{2}\\left|F(\\mathbf {q} )\\right|^{2}", "2523fd01e090cb397b54b00e16e7dda8": "{\\frac {dL}{dt}}=[P,L]", "252409d7bc136e9c7263ca5fd1565704": "\\{\\to ,\\neg \\}", "25241c06b7b309ae269afc7db958e1aa": "I=\\iint z^{2}\\;dy\\;dz,", "252457dad3d625c6a447b665098dcdf1": "\\left|{\\alpha \\choose k}\\right|^{2}=\\prod _{j=1}^{k}\\left|1-{\\frac {1+\\alpha }{j}}\\right|^{2}\\leq \\left({\\frac {1}{k}}\\sum _{j=1}^{k}\\left|1-{\\frac {1+\\alpha }{j}}\\right|^{2}\\right)^{k}.", "252489a766af284a4a602a98084c9eae": "a\\ ", "25248e6b08e40a1e228e9241d20f109b": "L_{\\rho }(\\gamma )=L_{\\rho ^{*}}(\\gamma ^{*})", "2524b581e7629fb9cb4a427c1c287eeb": "X_{t}^{\\tau }=X_{0}+\\sum _{s=0}^{\\tau \\land t-1}(X_{s+1}-X_{s}),\\quad t\\in {\\mathbb {N} }_{0},", "2524d282675f03f34a85fc9dbfdcb75c": "p_{0}\\subset p_{1}\\subset \\ldots \\subset p_{d}", "2524e5a4284c8f3c0baf719ae69cbef6": "{\\mathcal {L}}_{2}^{2}:L=Lclm(l_{2}^{(1)},l_{1}^{(1)});", "25250b2c61331aca9522bc76895d7bdc": "b={\\frac {fm_{\\mathrm {s} }}{N}}{\\frac {x_{\\mathrm {d} }}{D}}\\,.", "25252c6d01e06067af9005756c27c31b": "\\kappa \\neq \\lambda \\mu ", "25255508ab4a7f16f62de203ccbe96f9": "{\\boldsymbol {\\sigma }}=\\left[{\\begin{matrix}\\sigma _{11}&\\sigma _{12}&\\sigma _{13}\\\\\\sigma _{21}&\\sigma _{22}&\\sigma _{23}\\\\\\sigma _{31}&\\sigma _{32}&\\sigma _{33}\\end{matrix}}\\right]", "2525893fd2e0f6506aa355bd57863b1c": "(r{\\bar {r}}+b{\\bar {b}}+g{\\bar {g}})/{\\sqrt {3}}.", "2525e1b5206f5300e3fb8aeea46590b7": "{\\bar {N}}_{4}", "2526147fdba451085a02683a7fd5f2b4": "V_{\\mathrm {st} }=\\sigma \\iint dx\\,dy\\;\\left[{\\sqrt {1+\\left({\\frac {\\partial \\eta }{\\partial x}}\\right)^{2}+\\left({\\frac {\\partial \\eta }{\\partial y}}\\right)^{2}}}-1\\right]\\approx {\\frac {1}{2}}\\sigma \\iint dx\\,dy\\;\\left[\\left({\\frac {\\partial \\eta }{\\partial x}}\\right)^{2}+\\left({\\frac {\\partial \\eta }{\\partial y}}\\right)^{2}\\right],", "2526a5c6396d902e770a177c0b18fab9": "\\left[\\Pi _{i}(\\mathbf {r} ,\\ t),\\ A_{j}(\\mathbf {r'} ,\\ t)\\right]=-i\\hbar \\delta _{ij}\\delta (\\mathbf {r-r'} )\\ ,", "2526b82881f1a421d7934fcf01b3b787": "C=e^{-r_{DOM}T}\\Phi (d_{2})\\,", "2526c157039ee718758651ec0fce38b2": "\\Box \\phi \\to \\phi ", "2526d419fa11a3531cecdf869096acb4": "{\\frac {\\mathrm {d} \\sin \\theta }{\\mathrm {d} s}}={\\frac {\\mathrm {d} }{\\mathrm {d} s}}{\\frac {y'(s)}{\\sqrt {x'(s)^{2}+y'(s)^{2}}}}", "2526e28b43e7c658a133d9da7c209352": "\\Psi (x)=\\langle x|\\phi \\rangle ", "2526f4f31e4527c5b0e174883525089e": "P(\\Omega )=1.\\,", "25271175c4285c948207d70b9797233f": "[\\![\\neg \\phi ]\\!]_{i}=S\\smallsetminus [\\![\\phi ]\\!]_{i}", "252724839704aed8ab35a4916428087a": "r_{D}", "25272eeea61c6f971332fa0406d01314": "4n+1", "25273a18977833db3dc0f0fea7963624": "\\{x_{k}\\}", "2527e6b3f4bcfb1164781f59df730464": "\\alpha ={\\frac {n_{i}}{n_{n}}}", "25280223de731b21e5d287cbd3e36648": "a=n-1+2/n", "25282363f65988c29a4a04c9fff0e950": "Q_{P}=\\left({\\frac {mP}{2\\pi \\beta \\hbar ^{2}}}\\right)^{P/2}\\int \\cdots \\int {\\mathrm {d} }x_{1}\\cdots {\\mathrm {d} }x_{P}e^{-\\beta \\Phi _{P}(x_{1}\\cdots x_{P};\\beta )}", "25282978c6dd325321c01dfd5a45a6e0": "6\\cdot m\\ ", "2528442474fe24e441e26de82836d125": "z_{t+k-1}", "25285c6a18af2c7fde66decb5a9db7dd": "{\\sqrt {4k_{B}\\cdot T\\cdot B\\cdot R}}", "25289c2360eb3a15723cf04c639a786c": "\\sigma _{i}=1^{\\otimes i-1}\\otimes {\\check {R}}\\otimes 1^{\\otimes n-i-1}", "2528a2c43c74a683cbe8ad03f948c2e7": "\\Omega \\approx \\int _{0}^{\\infty }\\ln \\left(1-ze^{-\\beta E}\\right)\\,dg.", "2528d54c4e828896ebc6a74a813d38a5": "[(\\mathbb {Z} _{3}^{7}\\rtimes \\mathrm {S} _{8})\\times (\\mathbb {Z} _{2}^{11}\\rtimes \\mathrm {S} _{12})]^{\\frac {1}{2}}", "2529216eb18546a71fb3f90a82aea019": "C{\\dot {=}}D", "252944da9a860bd83d6af5b5d7c4ee52": "{\\mathfrak {gl}}_{n}\\times {\\mathfrak {gl}}_{m}", "252954a8a456f85e7692878f53bbeb82": "{\\begin{bmatrix}0&0&0\\\\0&1&0\\\\0&0&0\\end{bmatrix}}", "252969cde74f8930674f9d397242126f": "G_{\\theta }\\subset \\mathrm {proj} _{\\theta }F", "2529801c16412170af4539dfbb463ac5": "({\\bar {x}}-2\\sigma /{\\sqrt {n}},{\\bar {x}}+2\\sigma /{\\sqrt {n}}).", "252994c9859d4faddb09a4b2f7728a89": "y_{i}=wx_{i}+b", "2529b01e560826ba25853b5702096a96": "y_{1}(x),\\ldots ,y_{n}(x)", "2529c353780da79136fea0d5d0d77391": "O(\\log ^{6}q)", "252a0b496f9e3fe4a85f134165d47db6": "r=k[sucrose]\\,", "252a1000ca3cef3841e2e4cb377e855b": "a_{1}=-\\sum x_{i}=-t_{1}", "252a28040fd777ba9cbf861d21291f18": "{\\frac {\\Gamma (2n+2\\alpha )\\Gamma (1/2+\\alpha )}{n!\\,2^{n}\\,\\Gamma (2\\alpha )\\Gamma (n+1/2+\\alpha )}}\\,", "252a7bb801067ae1570b5756e38f57f4": "\\scriptstyle {\\mathcal {S}}", "252aef9c3bdf27e8b38a235263134762": "v'_{p}=Q_{pq}u_{q}", "252b44b53debbdfe35ab1c20631f92ce": "y'={\\frac {y}{y_{c}}}", "252b57b47e5b225f3fb591572d84809b": "I(f)", "252b9745d226c4589cdaa6f40d5a8532": "\\Box n=-\\nabla ^{2}(|u|_{}^{2})", "252babdcb94e84237a256fc257a2713d": "\\mathbf {x} ^{\\prime }", "252bbc8b495c965ffeecea51ae4c2711": "\\scriptstyle X\\,\\sim \\,{\\text{Frechet}}(1.7,1)", "252bd6db39ce21c2ec843cd88eb7c749": "\\operatorname {dim} B/{\\mathfrak {m}}_{A}B\\geq \\operatorname {dim} B-\\operatorname {dim} A.", "252c1ccab5087fbab854a56b73ac0c6e": "\\int ^{x}{\\frac {P_{1}(\\lambda )}{P_{2}(\\lambda )}}\\,d\\lambda +\\int ^{y}{\\frac {Q_{2}(\\lambda )}{Q_{1}(\\lambda )}}\\,d\\lambda =C\\,\\!", "252c4fcd10a6f3cf1d88d43fe76b3d78": "\\mathbf {F} =m\\mathbf {a} =m\\mathbf {A} +m\\mathbf {a} '", "252c61ff0ddc7290061310f2549a7105": "\\operatorname {let} x:\\operatorname {get-lambda} [x,x\\ f=f\\ (x\\ f)][x:=x\\ x]\\operatorname {in} x[x:=x\\ x]", "252c702ee4cd7552bb08de67d2c190ed": "{\\frac {d}{dx}}\\,x^{\\left[n\\right]}=n\\,x^{\\left[n-1\\right]}", "252c72fc62729e136eaee52b384afcee": "f(xy)", "252c81ff3ea78fe65494aa34b37b9151": "\\epsilon _{i}\\epsilon _{j}=\\delta _{ij}\\epsilon _{i}", "252c8d879fa2f4f34de5d38408180875": "-1<\\sin x<0", "252c8f52673beded4e9bf339c40f4fa4": "{\\tilde {\\nu }}/cm^{-1}={\\frac {1}{\\lambda /cm}}={\\frac {\\nu /s^{-1}}{c/cm\\ s^{-1}}}={\\frac {\\nu /s^{-1}}{2.99792458\\times 10^{10}}}", "252c902f64d80aaa0464b5ed3393d042": "w''\\in W", "252cae61fa653fa4cd005710a212e658": "d={1 \\over 2}at^{2}", "252cb2aa03a67003c67dc927b2433d33": "E={mv^{2} \\over 2}", "252cbdbb1d3ff4ed38eb6476a70902b8": "B(x;r)\\cap B(y;s)", "252cc0500ec93eb00a839062e0def483": "E=\\sum _{x}\\left[F(Ax+h)-(\\alpha G(x)+\\beta )\\right]^{2}", "252cd2d871b205f894cc37e338395ebf": "x^{2}-y^{2}=1", "252d3754c0db62a55b9e25c870a524a5": "n\\times m", "252d3991d2d5d6f74785c3beda3f81cd": "P(E_{\\gamma },\\theta )={\\frac {1}{1+(E_{\\gamma }/m_{e}c^{2})(1-\\cos \\theta )}}", "252d7a61a0b0ade9656f37755ebcfb95": "D=2(A_{x}(B_{y}-C_{y})+B_{x}(C_{y}-A_{y})+C_{x}(A_{y}-B_{y})).\\,", "252dfadab7ba2f60caa5966ec6892679": "\\mathbf {f} =-\\nabla P", "252e10c14628d682991fa536a296b58b": "I_{\\mathcal {R}}", "252e248c96703881aaae9b0b53ce1618": "{\\frac {\\partial ^{2}U}{\\partial s\\,\\partial p}}<0,", "252e2ab7c953c90e6d04370b2574119e": "ds^{2}=g_{\\alpha \\beta }dx^{\\alpha }dx^{\\beta }", "252e2de7f1f06ca485765d556aaf36e5": "\\varepsilon =0", "252e45c62b0e80987c3c01c7f09cf41f": "V(t)={\\dot {X}}(t).", "252e80401de4d6d945f335fae289b773": "{\\textit {dog}}\\subseteq {\\textit {mammal}}", "252ea420a8ac99da86b1cf61ceaab992": "H^{1}(k^{\\text{al}}/k)", "252ed1ace626b1b0da066475c5e53670": "h_{z}'(:,:,-1)={\\begin{bmatrix}+1&+2&+1\\\\+2&+4&+2\\\\+1&+2&+1\\end{bmatrix}}\\quad h_{z}'(:,:,0)={\\begin{bmatrix}0&0&0\\\\0&0&0\\\\0&0&0\\end{bmatrix}}\\quad h_{z}'(:,:,1)={\\begin{bmatrix}-1&-2&-1\\\\-2&-4&-2\\\\-1&-2&-1\\end{bmatrix}}", "252ee2a99a895f36d84ac0c53c192c96": "{\\Bigg [}{\\frac {\\pi }{\\theta }}{\\Bigg ]}=\\left[{\\frac {\\theta }{\\pi }}\\right],", "252f14ed53109d707e1aed7496c91d21": "\\min(c(A,B)-f(A,B),c(B,C)-f(B,C),c(C,D)-f(C,D))=", "252f1c48f3ddfa3167b1e2c47b0f2d2b": "P(recalling~m_{ab})~=~P(similarity(a,m_{ab})>criterion)", "252f5df1a5605c0cd21e9e9e47314ee8": "D=X\\oplus Y\\oplus Z", "252f643d7d66d08b72c830f7d62254ce": "l=\\rho V\\Gamma \\!", "252f9846458ef401bb19ec13240228ae": "x\\mapsto xe^{m}", "252faf0f410727ffa46828abaa601cfe": "\\!\\,p=x^{2}+3y^{2}{\\text{ if and only if }}p=3{\\text{ or }}p\\equiv 1{\\pmod {3}}.", "2530008dfead9e203878cbe7a5f8ae38": "\\kappa \\geq \\aleph _{1}", "25307939f61f821cbaced6f49840afda": "\\Gamma (5-3i)\\approx 0.0160418827+9.4332932897i.", "2530d677b034a008acdda4faf8d81cbd": "{\\begin{pmatrix}1&0\\\\{\\frac {n_{2}-n_{1}}{R_{2}n_{1}}}&{\\frac {n_{2}}{n_{1}}}\\end{pmatrix}}{\\begin{pmatrix}1&t\\\\0&1\\end{pmatrix}}{\\begin{pmatrix}1&0\\\\{\\frac {n_{1}-n_{2}}{R_{1}n_{2}}}&{\\frac {n_{1}}{n_{2}}}\\end{pmatrix}}", "2530ef679c88e7e49285b0ae412a9032": "{\\,\\!256^{256^{256^{256^{257}}}}}", "25311b98169e49fadc9ff483131e3515": "\\nabla ^{2}\\varphi -{\\frac {1}{c^{2}}}{\\frac {\\partial ^{2}\\varphi }{\\partial t^{2}}}={\\frac {1}{\\varepsilon _{0}}}e\\psi ^{\\dagger }\\psi ", "253173b63bd4bbc42e1afa5ca6e82331": "m,n\\in {\\mathcal {H}}_{L}", "2531c96acddc48a80b8dbdff154162b2": "\\sum _{n=0}^{\\infty }\\log(n+a)", "2531cabd3d99df8dbcfb1750b13cfcdd": "2\\theta _{0}", "2531cdfb8e31efa0a6a36a283c0e820f": "\\eta ^{\\alpha \\beta }", "2531e2dcb98cf5efa58a6f3a80da9564": "g=2", "2531f34408f46344aac5c58a14b3f1e9": "P\\vee (\\neg P)", "253220f1715e47c49285ad1a29950104": "\\lVert f-L_{n}(f)\\rVert _{\\infty }\\leq (1+\\lVert L_{n}\\rVert _{\\infty })\\inf _{p\\in P_{n}}\\lVert f-p\\rVert ", "2532964bdf9169751be561fdf6860288": "(x^{2}+3x-4)y^{[3]}-(3x+1)y^{[2]}+2y=0", "2532accc186b203e263542dbf707955b": "\\textstyle \\Gamma _{\\mathrm {f} }(V)", "2532b315a0da83e2c33f97856f49e435": "\\int dx|x+\\epsilon \\rangle \\langle x|\\psi \\rangle =\\int dx|x\\rangle \\langle x-\\epsilon |\\psi \\rangle =\\int dx|x\\rangle \\psi (x-\\epsilon )", "25334350eac8dc810ec1af74079de9a2": "F(t)=Kt+\\psi \\,\\Delta \\theta \\ln \\left[1+{F(t) \\over \\psi \\,\\Delta \\theta }\\right].", "253375e09f2c8034e3460d54c4255e5b": "H_{\\mathrm {top} }(u)=\\lim _{n\\rightarrow \\infty }{\\frac {\\log p_{u}(n)}{n\\log k}}\\ .", "2534671c2fca22d204122bc754559a6d": "\\int _{-1}^{1}(1-x)^{\\alpha }(1+x)^{\\beta }P_{m}^{(\\alpha ,\\beta )}(x)P_{n}^{(\\alpha ,\\beta )}(x)\\;dx={\\frac {2^{\\alpha +\\beta +1}}{2n+\\alpha +\\beta +1}}{\\frac {\\Gamma (n+\\alpha +1)\\Gamma (n+\\beta +1)}{\\Gamma (n+\\alpha +\\beta +1)n!}}\\delta _{nm}", "25347dc20efb4a3c4c12f1138af37b68": "\\psi _{0}=0", "2534bd1325eff468c3a34599f00936fd": "\\,F_{m}", "2534dd5abe10b08435a57f96cd95e5d1": "\\ \\sigma _{1}\\leq \\sigma _{y}\\,\\!", "253501fd7bf03d0cf3ccbff7396a900a": "\\langle n!\\rangle ", "2535035f5e606db6255481e0dde7f8d6": "\\beta =1\\,\\!", "25350f3497f973fbd51840327267016c": "N+1", "2535276ba191907b194e8a3ae0927c85": "x_{k}=g(x_{k-1})+w_{k}\\,", "2535bc8bb3cc7d5f8b5b7443f9bf7d51": "E(t)=E_{0}\\sin \\omega _{0}t", "2535c595455c9896782f36c405813169": "K=\\int _{0}^{\\frac {\\pi }{2}}{\\frac {d\\varphi }{\\sqrt {1-k^{2}\\sin ^{2}\\varphi }}}", "25362dc7326f52f03903c7e90ef5d097": "T_{f}+K\\;", "25362f222a54592f07bb2e5f696ce2eb": "\\Lambda (x)=\\prod _{k=1}^{\\nu }(1-xX_{k})=1+\\Lambda _{1}x^{1}+\\Lambda _{2}x^{2}+\\cdots +\\Lambda _{\\nu }x^{\\nu }", "25366d983b0ad35728b8754d889c425c": "F_{tot}=0=F_{el}+F_{f}+F_{ret}", "25367509c8f05f84cfaccebdf7afd2e7": "{\\begin{bmatrix}N_{11}\\\\N_{22}\\\\N_{12}\\end{bmatrix}}=\\left\\{\\int _{-h}^{h}{\\begin{bmatrix}C_{11}&C_{12}&0\\\\C_{12}&C_{22}&0\\\\0&0&C_{66}\\end{bmatrix}}~dx_{3}\\right\\}{\\begin{bmatrix}u_{1,1}^{0}\\\\u_{2,2}^{0}\\\\{\\frac {1}{2}}~(u_{1,2}^{0}+u_{2,1}^{0})\\end{bmatrix}}", "253681d8085cb847dd94aaa4eec929cb": "Profit=Py", "2536aabb4f39977e0779588f6f1829d4": "{\\textbf {Y}}_{k\\mid k}={\\textbf {Y}}_{k\\mid k-1}+\\sum _{j=1}^{N}{\\textbf {I}}_{k,j}", "25370ad2289ae803d06fcae9391cd014": "{\\begin{aligned}\\mu _{X\\cup Y}&={\\frac {1}{N_{X\\cup Y}}}\\left(N_{X}\\mu _{X}+N_{Y}\\mu _{Y}-N_{X\\cap Y}\\mu _{X\\cap Y}\\right)\\\\\\sigma _{X\\cup Y}&={\\sqrt {\\frac {[N_{X}-1]\\sigma _{X}^{2}+N_{X}\\mu _{X}^{2}+[N_{Y}-1]\\sigma _{Y}^{2}+N_{Y}\\mu _{Y}^{2}-[N_{X\\cap Y}-1]\\sigma _{X\\cap Y}^{2}-N_{X\\cap Y}\\mu _{X\\cap Y}^{2}-[N_{X}+N_{Y}-N_{X\\cap Y}]\\mu _{X\\cup Y}^{2}}{N_{X\\cup Y}-1}}}\\end{aligned}}", "25371db8886a432545653efa3890f916": "club", "253743d2bc60c0a9403d1e12ed243a44": "\\alpha =4", "2537730551a67b48e154fe9b0ccfbd80": "\\gamma =\\sinh ^{-1}{\\sqrt {ZY}}", "2537760f4407b5b66ef4ab8e35a5d554": "\\operatorname {Li} _{2}(\\rho ^{6})=4\\operatorname {Li} _{2}(\\rho ^{3})+3\\operatorname {Li} _{2}(\\rho ^{2})-6\\operatorname {Li} _{2}(\\rho )+{\\tfrac {7}{30}}\\pi ^{2}", "25377e3dda4936694799023001ad3dec": "\\Pi ^{\\pm }", "2537a7dd4b3151e22d524b26f76a0a1e": "\\rho =\\rho _{\\text{f}}-\\nabla \\cdot \\mathbf {P} .", "2537b7d80d015adb0c918ed22a7facd6": "2^{n^{2}}(1-1/2)(1-1/2^{2})\\cdots (1-1/2^{n}).", "25380848e46b356b9f45ce651454e882": "P_{2}=(x_{2},y_{2})", "25388fba62f375b638bb63a32976f4df": "\\int f^{+}\\,d\\mu ", "253891b0d9cbe442b08b712402e79d4e": "{\\widehat {\\lambda M}}=00{\\widehat {M}}", "25389b2aabac718ce6248d3461e2823f": "S^{2}\\times S^{1}", "2538a035e8c3b9705615e139ff07c510": "{\\begin{aligned}V_{1}^{2}&=V_{a}^{2}(1+a)^{2}+4\\pi ^{2}r^{2}(1-a')^{2}\\\\{\\mbox{d}}D&={\\frac {1}{2}}\\rho V_{1}^{2}C_{D}{\\mbox{d}}A={\\frac {1}{2}}\\rho C_{D}[V_{a}^{2}(1+a)^{2}+4\\pi ^{2}r^{2}(1-a')^{2}]b{\\mbox{d}}r\\end{aligned}}", "2538a9c77d7e8d43edecef4f12aa39d0": "S_{21}={2Z_{0}Z_{21} \\over \\Delta }\\,", "2538bf118cd2548f1a6daf2a9949b594": "a_{\\lambda }=\\sum _{g\\in P_{\\lambda }}e_{g}", "2538cca9cd259bc00089dacddcdc8025": "{\\mathfrak {H}}^{2}", "2538d233fcde1fb7cbb81704f0934eba": "\\psi _{i_{1}\\dots i_{\\ell }}", "2538d5218e0536176ced4dbfd005a600": "{\\bar {\\sigma }}", "253918fe42259276aea497527ea0bb7e": "U_{i}\\cap U_{j}\\neq \\phi ", "25392f574efe9e01337a78abbd44e9f5": "D_{ep}", "25395b9e7b08ab3941b209acd96d0c52": "M^{\\prime \\prime }", "2539b0eaef09c39961bdae3e3217b6f9": "{\\bar {\\nabla }}", "253a0b21d77247ef84ff9a7b14a76702": "T\\Delta S_{SA}\\,", "253a169f1411d70ded3b77fdcd427fb2": "G_{\\infty },G_{i},\\tau _{i}", "253a20d5692a7f9b580ebc19794d7d48": "\\textstyle {{\\frac {\\log(4)}{\\log(3)}}+{\\frac {\\log(2)}{\\log(3)}}={\\frac {\\log(8)}{\\log(3)}}}", "253a6f7c3fb04b3a6c6c8b561a330349": "(q/m){\\vec {E}}=\\partial {\\vec {v}}_{s}/\\partial t+(1/2){\\vec {\\nabla }}v_{s}^{2}.", "253b44ec81de66df3fce13fb21ba782e": "C:V^{*}\\otimes V\\rightarrow k", "253c9a2f87701bb0871fc4a79c63a908": "f={O(k)}", "253cb43a575f663ff78d74eb7de7ae0e": "\\mathbb {N} _{0}", "253d57f637ccdc30b513f8e6c0d99a48": "2^{j+1}-1", "253d6f7cacdc7af9cb80d8fd2708ac61": "\\mathbf {\\hat {T}} (\\lambda )", "253d6f830af505a128e8bcd73a930d94": "\\varphi _{\\alpha }\\colon U_{\\alpha }\\to {\\mathbf {R} }^{n}", "253d8fc2cee6ead59eec545ca7baedc7": "\\nabla _{\\mathbf {X} }\\mathbf {u} \\,\\!", "253db92a514fff4d73f63034ea46635a": "\\int _{\\mu ^{\\circ }}^{\\mu }{d\\mu }=\\int _{P^{\\circ }}^{P}{{\\bar {V}}dP}", "253de0d24465abd2294d7aa234b4bfaf": "\\int \\cdots \\int _{\\mathbf {D} }\\;f(x_{1},x_{2},\\ldots ,x_{n})\\;dx_{1}\\!\\cdots dx_{n}", "253df31451013693e7659b47131a5fc5": "D_{1}+D_{2}=7P_{1}+9P_{2}", "253e23bb5e72f125b0954f85fc47908b": "A^{e}", "253e6c1496f2c5124c07897e9fed7677": "\\Delta =1-(L_{1}+L_{2})\\,", "253e93545d645b46e10a771f9c2eefa0": "d_{\\text{f}}\\,\\!", "253f1ac21faf86ab8306d4dc4826b893": "3:4:5\\ ", "253f30871a50c26d798cdb6ce6ff13a2": "v_{i}=\\mathbb {I} _{H_{i}}", "253f4cecf653a35d4de16a4ac62b6509": "100_{2}", "253f6102780979e9b25919ecd38271a4": "s=60\\times m", "253f8cc4950445407a2a5296e9fa1c3b": "g={\\frac {G}{\\gcd(K,G)}}", "25400f376be4435bd631d544653a1b60": "\\psi _{g}^{(\\pm )}(t)=\\int dE\\,e^{-iEt}g(E)\\psi ^{(\\pm )}", "2540416e5de096e5d1ccce004eaae434": "\\mu :=G", "254049e4635d1a954b2007cf1b082ab7": "e^{(a)}(e_{(b)})=e_{\\mu }^{(a)}e_{(b)}^{\\mu }=\\delta _{(b)}^{(a)},", "25408015549da6c15b025ef8dcfc1b79": "g\\,", "25408898feac32ce94ffe011afc1262d": "\\scriptstyle \\approx R_{0}", "25417c76d43dea5b5bc5c8cefe5c8463": "\\,m_{\\mathrm {e} }c^{2}", "2541c5915ac83d5626e7931ecb438d1b": "{\\hat {c}}_{p}", "25421de4e7036ed8c1d1a2aa8091e506": "Ly(t)=f(t)", "25427e5e912393af18f27e6988bbb6ca": "{\\begin{bmatrix}x&y&0&0&1&0\\\\0&0&x&y&0&1\\\\....\\\\....\\end{bmatrix}}{\\begin{bmatrix}m1\\\\m2\\\\m3\\\\m4\\\\tx\\\\ty\\end{bmatrix}}={\\begin{bmatrix}u\\\\v\\\\.\\\\.\\end{bmatrix}}", "2542877c9bb97829cc57b7d798c0db2a": "\\lim _{N\\to \\infty }{\\frac {\\#\\{p\\leq N:\\alpha \\leq \\theta _{p}\\leq \\beta \\}}{\\#\\{p\\leq N\\}}}={\\frac {2}{\\pi }}\\int _{\\alpha }^{\\beta }\\sin ^{2}\\theta \\,d\\theta .", "2542b9eddf5af4037bc0d427d1e0a407": "{W_{I}(x,t)}={\\frac {\\hbar c}{2}}\\mid \\sum _{k}{\\sqrt {k}}c_{k}e^{i(kx-\\omega t)}\\mid ^{2}", "2542d7e6654f326ee8c97bc7230bc506": "\\varepsilon _{ijk}\\varepsilon ^{imn}=\\delta _{j}{}^{m}\\delta _{k}{}^{n}-\\delta _{j}{}^{n}\\delta _{k}{}^{m}", "2542f19f3bf1b32558f05c5436c85b9b": "\\sum _{n=1}^{N}\\sum _{c=1}^{N}Q_{n}^{(c)}(t)\\left[\\sum _{b=1}^{N}\\mu _{nb}^{(c)}(t)-\\sum _{a=1}^{N}\\mu _{an}^{(c)}(t)\\right]", "2543246ad1e6be543dabc46a11e6f599": "F=(F_{1},\\dots ,F_{n}):\\mathbb {R} ^{2n+1}\\to \\mathbb {R} ^{n}", "2543559ca21eef709226ff0896efbb03": "dV=a^{3}\\sinh \\mu \\ \\sin \\nu \\ \\left(\\sinh ^{2}\\mu +\\sin ^{2}\\nu \\right)d\\mu d\\nu d\\phi ", "2543b9d7533c3a9bc39460920aa4cca1": "(g,h)\\circ (g',h')", "2543f79eff7ecf0087b750b553da8aa5": "C_{P}", "254437e58ff9923caab3e866110e2043": "(u,\\xi )", "254440538938893a89ed3c138951ae29": "P_{i};P_{j}", "2544489079fd4bd046ff34b4754be610": "J_{x,y}\\sim |x-y|^{-\\alpha }", "2544b5206106cf08739b1acc6498003f": "0={\\frac {\\partial }{\\partial x}}[D_{1}{\\frac {\\partial C_{1}}{\\partial x}}+D_{2}{\\frac {\\partial C_{2}}{\\partial x}}-C\\nu ]", "2544ba8a3d6678eefbba525723e6f776": "|\\gamma '^{2}(t)|\\,\\!", "2544bcf8ddaed0661c85307a03ec8610": "d=2\\,", "2544c3e5a330516efba8b6641f41d349": "\\textstyle (x,y\\pm 1,z\\mp 1)", "2544ecc769e36b14357e27bf30e0a7c2": "M(u_{1},u_{2},\\dots ,u_{n})", "2544fc061f344ca9b9d7ef1c3e03044a": "G=\\int _{0}^{T}k(t)x(t)dt", "25456431c0c15f8f14d28aa3c9f19a46": "y=r\\cos \\theta +{\\frac {1}{32}}\\varepsilon r^{3}\\cos 3\\theta +{\\frac {1}{1024}}\\varepsilon ^{2}r^{5}(-21\\cos 3\\theta +\\cos 5\\theta )+{\\mathcal {O}}(\\varepsilon ^{3})", "254592628f751cea5ccb8819aab5bea5": "={\\frac {1}{N}}(\\mathbf {X*Y_{N}} )_{k},\\,", "25463b2b8e495ed00f3f4823d3fc03f3": "AF=\\operatorname {min} \\{100+4N_{W}+2N_{D}+N_{R},150\\}", "25468819f701e41188307ca0f02f0fa7": "\\left[{\\frac {\\partial f}{\\partial \\left(\\nabla ^{(i)}\\rho \\right)}}\\right]_{\\alpha _{1}\\alpha _{2}\\cdots \\alpha _{i}}={\\frac {\\partial f}{\\partial \\rho _{\\alpha _{1}\\alpha _{2}\\cdots \\alpha _{i}}}}\\qquad \\qquad {\\text{where}}\\quad \\rho _{\\alpha _{1}\\alpha _{2}\\cdots \\alpha _{i}}\\equiv {\\frac {\\partial ^{\\,i}\\rho }{\\partial r_{\\alpha _{1}}\\,\\partial r_{\\alpha _{2}}\\cdots \\partial r_{\\alpha _{i}}}}\\ ,", "25468eb68b750545294cdc2a53c02f38": "x=(x_{1},...,x_{n})", "2546c0adbf3dd8999fcd67606f809ae4": "(x_{1},x_{2})\\succeq (y_{1},y_{2})", "2546c8e6b888eb37c53df2004700566c": "{\\frac {d}{dt}}(pe^{(1-q)t}+(1-p)e^{-qt})=(1-q)pe^{(1-q)t}-q(1-p)e^{-qt}", "254742ebce7ea7ac552bd164a3583a63": "A-B\\in {\\mathcal {R}}", "25474f22e3282ef1d9d9f4ba593f6952": "M_{x}=\\int _{0}^{2}(xy^{2}+y^{3}+y^{2})|_{x}^{4-x}\\,dx", "2547777c940eaf010b8607935ce496eb": "2\\leq k\\leq n", "25477b2c139abf95020e3c2ffc584e08": "u_{1}(\\mathbf {x} ,z_{1})={\\frac {1}{g_{1}(\\mathbf {x} ,z_{1})}}\\left(\\overbrace {-{\\frac {\\partial V_{x}}{\\partial \\mathbf {x} }}g_{x}(\\mathbf {x} )-k_{1}(z_{1}-u_{x}(\\mathbf {x} ))+{\\frac {\\partial u_{x}}{\\partial \\mathbf {x} }}(f_{x}(\\mathbf {x} )+g_{x}(\\mathbf {x} )z_{1})} ^{u_{a1}(\\mathbf {x} ,z_{1})}\\,-\\,f_{1}(\\mathbf {x} ,z_{1})\\right)", "2547ada68717d8a20610002a185f0a9f": "\\left|\\alpha -{\\frac {p}{q}}\\right|\\leq {\\frac {A}{q^{n}}}\\leq A<\\min \\left(1,{\\frac {1}{M}},\\left|\\alpha -\\alpha _{1}\\right|,\\left|\\alpha -\\alpha _{2}\\right|,\\ldots ,\\left|\\alpha -\\alpha _{m}\\right|\\right)", "2547b213cb02f569a046f088414c0024": "(x_{ij})", "2547e4ad6c49d3508def1b1e6cdc420c": "B^{-1}A", "254819d2ba4db43e243b6af833bfccb1": "w/(\\delta *{\\sqrt {\\theta }})", "254842ca03be8af623146ed7b5685b1f": "1\\leq m\\leq n", "25485a3effeaa7f5c6316aab41dd447b": "\\displaystyle {g(a,b)=(ga,(g^{t})^{-1}b).}", "2548efd01db09ab2a4d253d3819743e4": "?\\left({\\frac {{\\sqrt {3}}-1}{2}}\\right)={\\frac {2}{7}}.", "25490cd0e5045f30483952aeaf911f0d": "x+a_{0}", "254914c90b584b9f3eb12cc91a989301": "t\\in \\mathbb {T} ", "25492e0dc919fcf95c86690ebd5d7a40": "0\\leqslant r\\leqslant 1", "25494f91159b4900496f57cfda71141f": "{\\begin{aligned}LOD=Z&=\\log _{10}{\\frac {\\mbox{probability of birth sequence with a given linkage value}}{\\mbox{probability of birth sequence with no linkage}}}=\\log _{10}{\\frac {(1-\\theta )^{NR}\\times \\theta ^{R}}{0.5^{(NR+R)}}}\\end{aligned}}", "25495af5f12e144da18a998fa075774b": "\\mathrm {e} ^{tA}x(0)", "2549b9b685b6b5e5eb1887f856b760d0": "R_{AB}=0\\,", "2549f9df53fa89a44d956dc199327c4a": "\\phi =\\sin ^{-1}{\\frac {4A}{\\pi \\Delta T_{s}\\Delta d}}", "254a3de2cb6b1b0a54111edab7211819": "{\\begin{aligned}df(t,X_{t})=\\left({\\frac {\\partial f}{\\partial t}}+\\mu _{t}{\\frac {\\partial f}{\\partial x}}+{\\frac {1}{2}}\\sigma _{t}^{2}{\\frac {\\partial ^{2}f}{\\partial x^{2}}}\\right)dt+\\sigma _{t}{\\frac {\\partial f}{\\partial x}}\\,dB_{t}.\\end{aligned}}", "254a999a0efc83bd121c16d43123f81a": "\\nabla \\cdot \\mathbf {B} =\\mu _{0}\\rho _{m}", "254ab56f8b99c2eb239fcc3ff44ab0a8": "{\\vec {1}}=(1,\\ldots ,1)", "254b152ced5c3e1b201c711d5248d339": "V_{\\beta }/V", "254b5d753d392780c7959ef9449100af": "\\cos \\theta +j\\sin \\theta =e^{j\\theta },\\,", "254b78de4a280030e89ddb70e179baf3": "c_{BE}\\simeq 1.18.", "254bbbb0979358af22e17ce8aa103b92": "{\\mathcal {H}}(p,x,t)={\\frac {1}{2}}p^{2}+K\\cos(x)\\sum _{n=-\\infty }^{\\infty }\\delta (t-n)", "254bd4d1093d85cfef105d13b3e65327": "\\int {\\frac {1}{x}}\\,dx,", "254bff34b7fed0f8cc2e6fba9ca77d74": "N(n)={n \\choose 2}p", "254c91ec208b3eb6a8e11c6f2245658c": "d(S)=y\\,", "254cdd7028c03c9ef71cca068827617e": "\\displaystyle {\\mathrm {Ad} (K)\\cdot X\\rightarrow {\\mathfrak {t}},}", "254dad8d7cd9fe82604b9004162cf653": "S={\\sqrt {1-x(1-{\\frac {\\rho _{L}}{\\rho _{G}}}}})", "254dae1eda9bf5d7eecfe42817525457": "d_{\\Gamma }(\\alpha ,\\beta )=\\sup _{t}\\ d_{X}(\\alpha (t),\\beta (t))", "254de19311d9fbea26cf072f8cbbe131": "{\\frac {1}{|B|}}\\int _{B}f(y)\\,\\mathrm {d} y-f(x)={\\Bigl (}{\\frac {1}{|B|}}\\int _{B}{\\bigl (}f(y)-g(y){\\bigr )}\\,\\mathrm {d} y{\\Bigr )}+{\\Bigl (}{\\frac {1}{|B|}}\\int _{B}g(y)\\,\\mathrm {d} y-g(x){\\Bigr )}+{\\bigl (}g(x)-f(x){\\bigr )}.", "254e238e7727536b116f56e0eefa5535": "R(u,v)w=\\nabla _{u}\\nabla _{v}w-\\nabla _{v}\\nabla _{u}w", "254e2b1b88631d117378bb2bfff605a4": "\\omega (a'_{1})p", "254e48412f5342d5be659e4bf68f72f5": "g(y)=\\int _{0}^{\\infty }f(x)K_{iy}(x)\\,dx", "254e56bfbe171aaaa265920bd0c7d0d6": "\\alpha _{2}(t_{1},t_{2})=\\left(\\beta _{1}(t_{1},t_{2}),t_{2}\\right)=\\left({\\frac {1}{(1+t_{1}^{2})(1+t_{2}^{2})}},t_{1}-{\\frac {2t_{1}}{(1+t_{1}^{2})(1+t_{2}^{2})}},{\\frac {t_{1}t_{2}}{(1+t_{1}^{2})(1+t_{2}^{2})}},t_{2}\\right).", "254e5d8374714d8f581c8a7b78e41854": "\\mathbf {S} _{B}\\mathbf {w} ", "254e6411045bcd2fbcdad9098a15a91b": "\\ F(K,L)=AK^{b}L^{1-b}", "254e6b185f8b00bbaa6b21612328cdfe": "\\phi _{0}=1", "254e8ef8081a17541120c662cf9a7e1f": "H\\left|a\\right\\rangle =E_{a}\\left|a\\right\\rangle .", "254e9fb4e38dd529fc08d47f26abc522": "x[T]z", "254eac4804ea2ee88171e6f51368dc4d": "4\\arctan {\\frac {1}{5}}-\\arctan {\\frac {1}{239}}={\\frac {\\pi }{4}}", "254efe90f4986539485f5b6363a6dc9f": "{\\tilde {6}}", "254f24909aebee6dfabd35153cbd9bbc": "\\scriptstyle T=2", "254f2f4f5954954aee0b8569bd94f4ec": "\\sin(\\theta )=0", "254f3f7199dae27be7c539fd9807cd7b": "x\\mapsto (x,1)", "254f7b28c63701c72e615aeff70c2e33": "\\int _{0}^{x}dx=\\int _{I_{O}}^{I}{\\frac {dI}{C_{O}-{\\frac {k_{O}I}{L}}}}", "254f8bd4bb5cc2b69c99253578351797": "ik=-j\\,", "254fcdef530d12899f858adf6da29c6a": "a=mq+r", "255016e591501ca3eac1fab91e82f8c3": "V_{br}", "255112d5251e3baaf3ad04e674ef3fad": "f(x)=\\Omega (g(x))\\ (x\\rightarrow a),", "25517d097f49f0086fb6956ef48268df": "[2n+p+q+z-e]^{2}-", "25518944cc9299566187e85cc2e0d29e": "(Tf)(x)=\\int _{x}^{x+1}f(u)\\,du", "25519360adc624d0e5f4fd4fe4d69e8a": "a_{3}=b_{2}-b_{3}", "2551a96d9646aa3bfe0145656d72887f": "H(t,\\xi )=G(t,\\xi )\\chi _{[0,\\infty )}(t)", "2551ae8977d9c721256076dd6d20ee9f": "i\\in \\{1,\\dots ,n\\}", "2551c422d8cb2efa9555d6da4dd6215a": "\\Lambda \\alpha .\\lambda x^{\\alpha }.\\lambda f^{\\alpha \\to \\alpha }.fx", "25526fec68a6dce76bbe531ddae0eabb": "q\\ {\\stackrel {\\mathrm {def} }{=}}\\ -{\\frac {{\\ddot {a}}a}{{\\dot {a}}^{2}}}", "255270250b7d852a6257712264503781": "\\mathrm {I} _{A}=\\mathrm {tr} (\\mathbf {A} )=A_{11}+A_{22}+A_{33}=A_{1}+A_{2}+A_{3}\\,", "2552bd7d1b25f4f54e7667c59e720d2a": "\\gamma \\in \\{0.1,0.2,0.5,1.0\\}", "2552c49d20963bcd22f1e9a856e67016": "H(X_{1,2})", "255368a486b0e2019a696af262cc2c67": "L(x,v)={\\frac {1}{2}}\\left({\\frac {v-b(x)}{\\sigma }}\\right)^{2}", "2553744408c159f00229055366efeeaa": "\\mathrm {HFC} =\\sum _{i=0}^{N-1}i|X(i)|", "25537f472b26e867b4cf339b1bfd31f4": "\\mathrm {sgn} (\\rho _{n})=(-1)^{\\lfloor n/2\\rfloor }=(-1)^{n(n-1)/2}={\\begin{cases}+1&n\\equiv 0,1{\\pmod {4}}\\\\-1&n\\equiv 2,3{\\pmod {4}}\\end{cases}}", "2553ada18a2b780dc5768c89c651e7d6": "D_{nr}", "2553b244e537f9fa55460b6fb989a422": "\\Gamma \\left({1 \\over 2}\\right)={\\sqrt {\\pi }}\\!", "2553b2f5761b4f1305d909c354f387ed": "v_{v}", "2553eea9c775f908412b5132f7fdbe44": "{\\begin{alignedat}{1}\\mathbf {c} _{3}&=-3\\mathbf {c} _{1}+5\\mathbf {c} _{2}\\\\\\mathbf {c} _{5}&=2\\mathbf {c} _{1}-\\mathbf {c} _{2}+7\\mathbf {c} _{3}\\\\\\mathbf {c} _{6}&=4\\mathbf {c} _{2}-9\\mathbf {c} _{3}.\\end{alignedat}}", "25540be1099e6835d8987bb62ab43404": "L={\\begin{bmatrix}l_{1,1}&&&&0\\\\l_{2,1}&l_{2,2}&&&\\\\l_{3,1}&l_{3,2}&\\ddots &&\\\\\\vdots &\\vdots &\\ddots &\\ddots &\\\\l_{n,1}&l_{n,2}&\\ldots &l_{n,n-1}&l_{n,n}\\end{bmatrix}}", "25540cf68f86fc419ba56aa6b748859c": "p_{k}={\\frac {p}{r}}r_{k}", "2554545d7be77a533307c317b3c99ae5": "{\\mathfrak {g}}^{\\mathrm {ss} }", "25547b0b89ce9564e6b4b2ea2a6eeba6": "{\\hat {b}}\\,{\\hat {b}}^{\\dagger }={\\hat {b}}^{\\dagger }\\,{\\hat {b}}+1.", "2554a2bb846cffd697389e5dc8912759": "\\theta ", "2554bebde0a0860a1c27a6ed145d4ab0": "\\varphi \\left(\\int _{-\\infty }^{\\infty }x\\,f(x)\\,dx\\right)\\leq \\int _{-\\infty }^{\\infty }\\varphi (x)\\,f(x)\\,dx.", "2554c54efee3e60ea7632340791da4f7": "{\\mathcal {N}}({\\boldsymbol {\\mu }},\\,{\\boldsymbol {\\Sigma }})", "2554dba2c9241485a946f34144842493": "R=B_{e}R_{f}\\,", "2554e13fe0a10be188bddee5c632d7ea": "\\phi (E_{i})\\subseteq F_{1-i}.", "255519ed046428549e0f8063e4a97fb7": "{\\widehat {\\theta }}(X)={\\frac {(a+n)\\max {(\\theta _{0},x_{1},...,x_{n})}}{a+n-1}}.", "25554e391c2a86f22a6d715ccc5cb7c5": "{\\mathcal {F}}_{i}\\,\\!", "25554ed2e5e6b0730ea58b85bc6e4b0e": "s_{ij}^{unsigned}", "25557dbb7481e76da40b6abcba01c3de": "\\mathrm {im} (f)=\\ker(\\mathrm {coker} f)", "2555af831cf60e7901e5fd3b604032ae": "\\sigma \\leq \\tau ", "2555c81396a078d56243496affce6023": "\\cos \\Theta =\\cos \\varphi \\cos \\varphi '+\\sin \\varphi \\sin \\varphi '\\cos(\\theta -\\theta ').", "2555d74a0a10e4a19d92045a28da8771": "\\sigma _{ij}", "2555e7525be0e42250212b462c30a58a": "{\\underbrace {\\partial {\\overline {hu}} \\over \\partial t} }_{\\begin{smallmatrix}{\\text{Change in}}\\\\{\\text{x mass flux}}\\\\{\\text{over time}}\\end{smallmatrix}}+\\underbrace {{\\partial \\over \\partial x}\\left({\\overline {hu^{2}}}+{1 \\over 2}{k_{ap}g_{z}h^{2}}\\right)+{\\partial {\\overline {huv}} \\over \\partial y}} _{\\begin{smallmatrix}{\\text{Total spatial variation}}\\\\{\\text{of x,y momentum fluxes}}\\\\{\\text{in x-direction}}\\end{smallmatrix}}=\\underbrace {-hk_{ap}\\operatorname {sgn} \\left({\\partial u \\over \\partial y}\\right){\\partial hg_{z} \\over \\partial y}\\sin \\phi _{int}} _{\\begin{smallmatrix}{\\text{Dissipative internal}}\\\\{\\text{friction force}}\\\\{\\text{in x-direction}}\\end{smallmatrix}}-\\underbrace {{u \\over {\\sqrt {u^{2}+v^{2}}}}\\left[g_{z}h\\left(1+{u \\over r_{x}g_{x}}\\right)\\right]\\tan \\phi _{bed}} _{\\begin{smallmatrix}{\\text{Dissipative basal}}\\\\{\\text{friction force}}\\\\{\\text{in x-direction}}\\end{smallmatrix}}+\\underbrace {g_{x}h} _{\\begin{smallmatrix}{\\text{Driving}}\\\\{\\text{gravitational}}\\\\{\\text{force in}}\\\\{\\text{x-direction}}\\end{smallmatrix}}", "2555fb92144d39b35afea3c41e3ec718": "U=\\sum _{i=1}^{n}w(p_{i})v(x_{i})", "2556271457018ddab6679e152f1fc70a": "\\left[A\\right]", "25562ca8b050ce10d8e57352978f764e": "\\mathbf {X} (s)", "25566ae6804f5deb1adfabccee5a5ec8": "D\\in \\mathrm {Div} ^{0}(C)", "25567ad13be8225351a8d1c5bf12667d": "{\\tfrac {1}{24}}", "25569259ba6b7958edc520f623bb8617": "\\left({\\frac {\\partial \\Omega }{\\partial x}}\\right)_{E}=-\\sum _{Y}Y\\left({\\frac {\\partial \\Omega _{Y}}{\\partial E}}\\right)_{x}=\\left({\\frac {\\partial \\left(\\Omega X\\right)}{\\partial E}}\\right)_{x}\\,", "255696048a5e1a0ae5ae75f2f86729ef": "\\mathrm {I} _{\\mathrm {O} ,\\mathrm {P} }\\,", "2556caf5470671f5f3a77453ede1dcf2": "h\\ =\\ 2{\\sqrt {\\frac {\\gamma }{g\\rho }}}", "2557031a57d67ebb7bb70042942550e0": "v_{e}={\\sqrt {\\frac {2GM}{r}}}={\\sqrt {\\frac {2\\mu }{r}}}={\\sqrt {2gr\\,}}", "25576aa54ce14b81a1c15b50f261d135": "\\mu (x;t,s)=\\int _{\\xi \\in \\mathbb {R} ^{k}}(\\nabla I)(x-\\xi ;t)\\,(\\nabla I)^{T}(x-\\xi ;t)\\,w(\\xi ;s)\\,d\\xi ", "255795cd9acac1e89b36fdec81d7c7a2": "\\gamma _{n}=-\\Omega _{n}(1-\\delta _{n}).\\,", "2557b6328a9fb8ce2995639a657b853d": "p={\\frac {8ac-3b^{2}}{8a^{2}}}\\qquad \\qquad {\\color {white}.}", "255830b5efaaf6264a7a2022281bbd0f": "\\displaystyle \\left({\\frac {i}{2\\pi }}\\right)^{n}{\\frac {d^{n}{\\hat {f}}(\\xi )}{d\\xi ^{n}}}\\,", "255854475f333de326c5de5d49900238": "{\\rho _{air}}", "25588471e26e64b53bd43a86904c30b2": "K:\\mathbf {M} \\to \\mathbf {C} ", "255895d445fba419e08eca0b027c9a07": "\\operatorname {div} (\\rho \\phi \\upsilon )", "255914d86a24ad1606afa3a62df33683": "F(t)=Ae^{i\\omega t}+Be^{-i\\omega t}\\,.", "2559422cb58b669fc8c56a566bc13b09": "\\Delta Q(P(t_{1},t_{2}))\\ ", "255a1aa8e3493f351e72862cda7f3ba8": "\\mathbf {Z} =\\{Z_{1}\\dots Z_{n}\\}", "255a63c38928e6f9575c86526ee81be8": "{\\mathcal {C}}_{XY}^{\\pi }=\\mathbb {E} _{XY}[\\phi (X)\\otimes \\phi (Y)]", "255a8b6e871fa8defd03be7befc32ffd": "s=k\\ln(1/p)", "255a99ac60ad3c5f6e4e178a7b12e6c6": "\\ldots ,", "255aa235e6845bb5e8334d4a77248b96": "DR_{D}^{S}", "255aa5ce12bea936f4447c696a34332b": "|b|", "255b10c452c2af014953b528fe9eb352": "\\mathbf {r} _{1}", "255b4cd0abc731563ce510e10b05c040": "G_{S_{N}}(z)=\\operatorname {E} (z^{S_{N}})=\\operatorname {E} (z^{\\sum _{i=1}^{N}X_{i}})=\\operatorname {E} {\\big (}\\operatorname {E} (z^{\\sum _{i=1}^{N}X_{i}}|N){\\big )}=\\operatorname {E} {\\big (}(G_{X}(z))^{N}{\\big )}=G_{N}(G_{X}(z)).", "255b6ebfa66372ab0b7935076936f442": "\\rho w_{E}={\\hat {\\mathbf {k} }}\\cdot (\\nabla \\times \\tau )/f\\ ", "255bca5d39927a33cd568c4258f7d713": "q^{2n+1}+1", "255bcd90a789eeac8077592a3198277e": "{\\frac {{\\sqrt {i}}+i{\\sqrt {i}}}{i}}{\\text{ and }}{\\frac {{\\sqrt {-i}}-i{\\sqrt {-i}}}{-i}}.", "255c0c87232256a1c7b19a7616e2981b": "q=e^{\\pi i\\tau }", "255c1fae8c9cfdb62d40554320e2be1f": "{\\begin{aligned}{\\mathbf {A}}+{\\mathbf {B}}&={\\begin{bmatrix}a_{11}&a_{12}&\\cdots &a_{1n}\\\\a_{21}&a_{22}&\\cdots &a_{2n}\\\\\\vdots &\\vdots &\\ddots &\\vdots \\\\a_{m1}&a_{m2}&\\cdots &a_{mn}\\\\\\end{bmatrix}}+{\\begin{bmatrix}b_{11}&b_{12}&\\cdots &b_{1n}\\\\b_{21}&b_{22}&\\cdots &b_{2n}\\\\\\vdots &\\vdots &\\ddots &\\vdots \\\\b_{m1}&b_{m2}&\\cdots &b_{mn}\\\\\\end{bmatrix}}\\\\&={\\begin{bmatrix}a_{11}+b_{11}&a_{12}+b_{12}&\\cdots &a_{1n}+b_{1n}\\\\a_{21}+b_{21}&a_{22}+b_{22}&\\cdots &a_{2n}+b_{2n}\\\\\\vdots &\\vdots &\\ddots &\\vdots \\\\a_{m1}+b_{m1}&a_{m2}+b_{m2}&\\cdots &a_{mn}+b_{mn}\\\\\\end{bmatrix}}\\\\\\end{aligned}}\\,\\!", "255c4330d01b0b960275e3fab7899c15": "a=\\left({\\frac {3}{4\\pi n}}\\right)^{1/3}.", "255c478b3e67ab21cc2282bf6b14cbf4": "J_{k}(n)=n^{k}\\prod _{p|n}\\left(1-{\\frac {1}{p^{k}}}\\right).\\,", "255c7dceecbc1fd5827f7bdba0944d49": "\\sigma _{\\mathrm {min} }\\,\\!", "255c82804707a93fc92e0a055994b178": "\\left|k_{1},\\ldots ,k_{n}\\ \\mathrm {in} \\right\\rangle ={\\sqrt {2\\omega _{k_{1}}}}a_{\\mathrm {in} }^{\\dagger }(\\mathbf {k} _{1})\\ldots {\\sqrt {2\\omega _{k_{n}}}}a_{\\mathrm {in} }^{\\dagger }(\\mathbf {k} _{n})|0\\rangle ", "255c9428c27429532fd1ba66bd8945d9": "{{{({{\\partial \\over \\partial t}+{{\\overrightarrow {V_{g}}}\\cdot \\nabla }})({-\\partial \\Phi \\over \\partial p})}-\\sigma \\omega }={kJ \\over p}}", "255ceb11ed822fdd54a4980da3934cc5": "\\scriptstyle V_{\\mathrm {P-P} }", "255d141fd458c93122200ca8e8be9030": "{\\frac {x^{2}}{a^{2}}}-{\\frac {y^{2}}{b^{2}}}=0.", "255d19f3f02489ecd08f33c414a91645": "A_{ij}=\\left\\{k\\in \\{1,2,\\dots ,p-2\\}:\\left({\\frac {k}{p}}\\right)=(-1)^{i}\\land \\left({\\frac {k+1}{p}}\\right)=(-1)^{j}\\right\\},", "255d35062a53d3a1845483ebd237b15c": "dN_{R}", "255e14f48ee5491cdd51a252c9eda89c": "U_{ii}", "255e8abf7d6b2e6efa26d2757f821e2b": "\\mathbf {E} _{8}\\supset \\mathrm {O} (16)", "255e8ca75b82f396f815953a05f9831b": "g_{Y}={F_{A}}A/{Y}*g_{A}+\\alpha *g_{K}+(1-\\alpha )*g_{L}", "255e9f3db3e2810fcf2ed16d8a6a4807": "[L_{u,v},L_{w,x}]:=L_{u,v}\\circ L_{w,x}-L_{w,x}\\circ L_{u,v}=L_{w,\\{u,v,x\\}}-L_{\\{v,u,w\\},x}", "255ec49ebf3d6ad790a9336d4f19a8dd": "{\\begin{aligned}\\left[{\\frac {\\alpha }{\\pi }}\\right]_{2}&=\\pm 1\\equiv \\alpha ^{\\frac {\\mathrm {N} \\pi -1}{2}}{\\pmod {\\pi }}\\\\&={\\begin{cases}+1{\\text{ if }}\\gcd(\\alpha ,\\pi )=1{\\text{ and there is a Gaussian integer }}\\eta {\\text{ such that }}\\alpha \\equiv \\eta ^{2}{\\pmod {\\pi }}\\\\-1{\\text{ if }}\\gcd(\\alpha ,\\pi )=1{\\text{ and there is no such }}\\eta .\\end{cases}}\\end{aligned}}", "255f401be8c1801f07cf6b7e315edbb4": "A^{i}=0", "255f4ce023d93837dc2790ae2eeaad81": "\\mathbf {X} _{2}", "255fd2c1bedf5334a708a63a2027c18c": "\\left({\\frac {2mn}{n^{2}+m^{2}}},{\\frac {n^{2}-m^{2}}{n^{2}+m^{2}}}\\right)", "2560232f9ad271eefce568640ec41b55": "\\sup _{f\\in {\\mathcal {F}}}\\vert f(x)-Pf\\vert <\\infty .\\forall x", "25607e2945a13ef784b465c4349695b5": "{\\mathrm {i} }", "2561086530f0c6483d1b3dc415ff3753": "\\pi \\alpha ", "2561131dc7d4c7c80cf5aad09544bf19": "Y(s)=U(s)(e^{-s}+s-1)/s^{2}", "25613c08c6b68a9704ffc406287fdbd2": "\\kappa =0", "25616bc4ff7a0e11ba9fb588b8f82f3a": "H(P)={\\underset {i}{\\max }}\\,|a_{i}|\\,", "2561826a22ce9e570b348d2cc7ac9cd0": "{r \\over \\sin 3\\theta }={a \\over \\sin 2\\theta }\\!", "2561b2044908d7496b015ccbbc7c7ee2": "g_{n}<(\\log p_{n})^{2}-\\log p_{n}", "2561e2121d99e378b780f48a2e74234f": "BC_{n}.", "2561f35154361734b7646bba2a94a916": "\\int _{-\\infty }^{+\\infty }f(x)dx=\\int _{0}^{1}{dt \\over t^{2}}\\left(f\\left({\\frac {1-t}{t}}\\right)+f\\left(-{\\frac {1-t}{t}}\\right)\\right)\\;.", "25626ddc70ac76d8b42d4dd908f48145": "w_{1}=w_{2}=1", "25628bde18547f3964d5413e189daf9e": "du\\cdot dv=u'v'(dx)^{2}=0\\,\\!", "2563198fdfafae0dbdc71cbec9000b60": "q_{1}", "256352aa9a66cdcd602cc204c46ad920": "{t}", "25639e2275027414b46b9188790c4936": "\\scriptstyle \\langle ", "2563e3348b3e6389fb2522e752e0fa5e": "g_{00}>0.\\,", "256413eb906f3a97cd3e605f5ee7700e": "\\pi _{n-1}(A)\\to \\pi _{n-1}(X)", "25649c3a234023c1126b3c20a5093e7d": "X_{1},X_{2},X_{3}", "25652162bba3af6ecd25aa64a9878ce0": "\\prod _{k=1 \\atop \\gcd(k,m)=1}^{m}\\!\\!k\\ \\equiv {\\begin{cases}-1{\\pmod {m}}&{\\text{if }}m=4,\\;p^{\\alpha },\\;2p^{\\alpha }\\\\\\;\\;\\,1{\\pmod {m}}&{\\text{otherwise}}\\end{cases}}", "25655081f5b649195d0d605aadf84b7c": "e_{q}^{x}=\\sum _{n=0}^{\\infty }{\\frac {x^{n}}{[n]_{q}!}}.", "2565530a65b6f96f25845fe829642a29": "[F^{[l]},F^{[l']}]=0", "25655bfb0b3ddecad59e2941e83a723f": "\\phi (x)\\equiv x^{q}{\\text{ (mod }}\\beta )\\,\\!", "25656b59a2a1984d6e8afb9bf7d588f0": "T_{j;i}^{i}=0.", "25656d52d660036362e969afd45ea4d1": "\\Box \\Diamond A", "2565f0a9b6ad9c5ad3bc7d2224acb9ab": "\\{\\mathbf {\\tilde {e}} _{k}\\}", "256603ffe6f86b3a2b13aacc67873492": "e=1\\,\\!", "25661fd7ddbd2cb22ced5d996d36c536": "f(p,q,p_{c})=q+M\\,p\\,\\ln \\left[{\\frac {p}{p_{c}}}\\right]\\leq 0", "25664e3edaaa20446da9dcd4aab2bd94": "{N \\choose n_{1}}", "2566a5265afa21d30c33510c4b7b1193": "({\\textbf {x}},y)=(x_{1},x_{2},x_{3},...,x_{k},y)", "2566e01a66863fa49cad3c96850c652c": "{\\begin{cases}\\displaystyle {\\frac {d{\\vec {x}}_{P}}{dt}}={\\vec {u}}_{P}({\\vec {x}}_{P},t)\\\\[1.2ex]{\\vec {x}}_{P}(t_{0})={\\vec {x}}_{P0}\\end{cases}}", "256725d40a9342b793c90b69d18cc13b": "\\lim _{x\\to 0^{+}}{\\frac {1}{x}}=+\\infty ", "25678557805d88847a0e87ad678ad2d3": "{\\frac {L_{vap}}{T_{boiling}}}\\approx 87-88{\\frac {J}{Kmol}}", "256791f454ba8ad9aaf418612da842a1": "V_{\\text{out}}=V_{\\text{in}}\\left({\\frac {1}{\\beta +1/A_{OL}}}\\right)", "2567b60415bd43aaa741dc8b9b638e14": "J_{1}=\\int _{\\Gamma }\\left(W\\delta _{1j}-\\sigma _{jk}~{\\cfrac {\\partial u_{k}}{\\partial x_{1}}}\\right)n_{j}d\\Gamma ", "2567e41466de3a8d4495b674b2a20550": "\\sum _{i=1}^{\\infty }\\lambda _{i}|e_{i}(t)e_{i}(s)|\\leq \\sup _{x\\in [a,b]}|K(x,x)|^{2},", "2567f61cd0cf997933a4498197d14e26": "\\sigma _{3}^{}", "25680201bc240752fb5165b1efbff89f": "\\mathbf {\\Pi } ", "256836f64a563df684913be7c29b2ccd": "g^{(2)}(0)=0", "25686f4b7adb7313fc37bfdb01478272": "(u_{r},u_{\\phi })", "25687cf457ad00866c89a51f3bf828ee": "\\mathbf {C} \\sim W_{p}^{-1}(\\mathbf {V} ^{-1},n)", "2568a96eec6be9f969228757abcb44d8": "{\\bar {\\epsilon }}_{sh}(t,t_{0})=-\\epsilon _{sh\\infty }\\ k_{h}\\ S(t),~~~k_{h}=1-{h_{e}}^{3}~~~~~", "2568dab807674550805ec161ca308527": "C({\\mathcal {X}})", "2568f2cf5a729cfa7c05be91d0c423c0": "0=\\iiint \\limits _{V}\\left({\\frac {\\partial \\rho }{\\partial t}}+\\nabla \\cdot \\mathbf {J} \\right)dV.", "2568f6507d084b139cbbc1c2c47d6079": "{-dE \\over dt}={\\sigma _{t}B^{2}V^{2} \\over c\\mu _{o}}", "25694fb3fed429a87fc65b0571eff26d": "k\\in \\{1,...,(p-1)\\}", "25698c7b0873e2903575dce42d8ec0ea": "R\\to R,x\\mapsto uxu^{-1}", "256a3c5fec7439e324970d15eb002fc6": "{\\pi \\over p}\\ {\\pi \\over q}\\ {\\pi \\over r}", "256a52017aefc4b3e12e8b1fb59da9b1": "{\\boldsymbol {\\rm {S}}}=\\{A,B\\}", "256a5a2319ac8c8c0feb1d649b9af51a": "x^{4}+x^{3}+x^{2},\\quad x^{4}+x^{3}+x+1,\\quad x^{4}+x,\\quad x^{4}+x^{2}+1.", "256a5a99ef7e9fb789684a153a523388": "0\\rightarrow \\operatorname {Tor} (K^{*},K^{*})^{\\sim }\\rightarrow \\operatorname {K} _{3}(K)_{ind}\\rightarrow \\operatorname {B} _{2}(K)\\rightarrow 0", "256a5db732a59e61ca5a7221059db921": "v={\\frac {P_{F}+P_{C}}{2}}-P_{P}", "256a9a3e9e20f0593bfe25dc4d095b43": "|SA|", "256aad94d33dcce3e113e24009b69bd5": "I_{D}", "256b2a04d5d2068cd7e07d1944fb7799": "\\gamma (0)=\\gamma (1)", "256b588196ff499cef9d989070b1538d": "na+mb=0\\;", "256b87c9dd69b58688ee5fe48b812611": "e^{-|x|}", "256bda0aeae7109348d945b6d2b2ab18": "{\\sqrt {(2^{AV})}}", "256bed78c559f82e94a4589e0e8a15cc": "N_{n}", "256c0c5dc46ac8604a68c4581884823b": "{\\frac {d\\mu }{dx}}=(e^{\\int p(x)dx}){\\frac {d}{dx}}(\\int p(x)dx)", "256c30e11797273e7f7f047426ad4c41": "{\\frac {a_{i}+b_{i}}{2}}.", "256c3f5f9db8f8161b596d669c0a751a": "q_{m}=(C_{d})(E_{v})(Y)\\left[{\\frac {\\pi }{4}}\\right](d)^{2}{\\sqrt {2\\rho \\Delta P}}", "256c58751184deac8ca43869b6263bee": "M=", "256cd2fd9f0a2dd10189796a403581bc": "\\tau =N_{AO}\\int {\\frac {1}{(-r_{A})V_{R}(1-\\delta _{A}f_{A})}}\\,df_{A}", "256cf2559101d6fa422cd63ed6960933": "v_{A}\\gg c", "256dfd235839c8fe602f1d61b422e2bc": "F(u)={\\frac {1}{2}}\\int _{a}^{b}\\!u'(x)^{2}\\,dx-\\int _{a}^{b}\\!u(x)f(x)\\,dx.", "256e3cfb9430a14b6f8d5cf745b62eb8": "\\delta [n]", "256e72fb9339d35b8d2900be31070c83": "a^{2}+b^{2}=1", "256eb7d8fd6cd53b89c086b170135da2": "n\\mathrm {LiCoO_{2}} \\leftrightarrows \\mathrm {Li} _{1-n}\\mathrm {CoO_{2}} +n\\mathrm {Li^{+}} +n\\mathrm {e^{-}} ", "256ef35a588d032cd55ec94e1b174297": "P=p(x,D)=\\sum _{|\\alpha |\\leq k}a_{\\alpha }(x)D^{\\alpha }.", "256f38fab870148921b032022d16a3a3": "g(t|x)={\\frac {f(x|t)g(t)}{f(x)}}={\\frac {1}{t(\\ln(T)-\\ln(x))}}\\quad {\\text{for all }}t>x.", "256f7794ae0474224f1fabc58859a65b": "p({\\textbf {x}}_{k}\\mid {\\textbf {x}}_{0},\\dots ,{\\textbf {x}}_{k-1})=p({\\textbf {x}}_{k}\\mid {\\textbf {x}}_{k-1})", "256f794a5147cc866416157f9bcbeb0a": "y_{0}=y", "256fa059915945369657d31b52f2b751": "q_{2}=30.0{\\text{ ft}}^{2}/s", "256fb976024a94d6a34e0458d71a7059": "s_{ij}\\,\\!", "256ff1583b33a0b93e4bb748d8bfa199": "V=V^{1}+V^{2}", "2570601a00fb7bed607190dd8d0b41f9": "\\operatorname {Var} ({\\overline {x}})=\\sigma ^{2}/n", "257099ff91578e85efd8ccdc7dc36c11": "\\sum \\left(\\Phi \\left(r_{i}(x_{j})+\\alpha h(x_{j})y_{j}+s-t\\right)-\\Phi \\left(r_{i}(x_{j})+s\\right)\\right)=0", "25709aa3d8d3531cc691b771c7b5773f": "f\\colon Y\\rightarrow X", "2570c2f79c151da6f194fc1c2cc483fa": "{\\sqrt {-1}}\\cdot {\\sqrt {-1}}=(-i)\\cdot (-i)=-1", "2570e6cf9a01f0b85b85f7cc39a8e7e7": "V_{out}=V_{2}\\,", "2570fb2ddc32a7655a586d096fcc2601": "\\sum _{n=-\\infty }^{\\infty }\\left|a_{n}\\right\\vert ^{2}<\\infty ,", "25716c8d0b9dc065c9a1f600c4796897": "L=\\int d^{3}x\\,N\\gamma ^{1/2}(K_{ij}K^{ij}-K^{2}+{}^{(3)}R)", "2571a85a1d3d0b8b26f8f16f6581ddc4": "a^{p-1}\\equiv 1{\\pmod {p}}.", "2571c5d1f72bf84de9f44f6e70ca2895": "{\\vec {e}}_{0}", "2572112c0c0545e1265d7aa59f0e804f": "A(\\alpha )={\\frac {1}{2}}\\int _{0}^{\\alpha ^{2}}dx{\\frac {x}{(x+\\theta _{E}^{2}{(1-\\beta ^{2}))}^{2}}}", "257246acb84acb9b2ad3473fc093fd3b": "g^{ij}A_{j}={\\cancel {g}}^{i{\\cancel {j}}}A_{\\cancel {j}}=A^{i}\\,,", "257267ec594513755e6479afc6a1f1df": "|r_{1}-r_{2}|0\\,", "259473a2166ee1a54585f08d75d8facf": "f(u_{n})\\to f(u_{0}){\\mbox{ as }}n\\to \\infty .", "2594be7a2c65de6fab78cf240239f11a": "{\\frac {\\mathrm {d} U}{\\mathrm {d} t}}={\\dot {Q}}-p{\\frac {\\mathrm {d} V}{\\mathrm {d} t}}+P.", "2594c427d5b4fb0cede2634481fcdd22": "[0,0,1,10000]", "2594df56ff847e95bcf5384264beffa0": "X+\\xi \\mapsto X+\\xi +i(X)\\alpha ", "259501d5dad9632809330ccec6e6c173": "dy/dt=ay^{2/3}", "25952b317143375572bf0d5fc08f8c1f": "D(p||q)+D(q||r)-D(p||r)=\\sum (p-q)(\\log ;p-\\log ;q)", "2595623bce5282d9baa8d6de13409b07": "({\\text{vector length}})^{2}=B_{X}^{2}+B_{Y}^{2}+B_{Z}^{2}\\,", "25957117245b2aa70ff2fa7f609cfc29": "r/s,r\\in R,s\\in R^{\\times }", "2595736a37536ee2c322154c85c34dc7": "(x_{4},y_{4})\\,", "25958882b8538051b2a78425045f6073": "S^{n}=D^{n}/S^{n-1}", "259599a5651e0579b8be314eb1db9c2b": "2^{cn}", "2595bb9d21e06c0196d21c95f805aa71": "\\delta ^{3}\\left(\\mathbf {r} -\\mathbf {r} '\\right)=-{\\frac {1}{4\\pi }}\\nabla ^{2}{\\frac {1}{\\left|\\mathbf {r} -\\mathbf {r} '\\right|}},", "2595ce4b8a99b4006355d425b39f55bb": "\\!\\tau (x)", "2595cfb91a0f53a8d0956953fc7e6b8c": "y:{\\mathcal {P}}(S)\\rightarrow 2^{S}", "2596565c1b36cc258bbd08037575219b": "(v_{0},\\ldots ,v_{n-1})", "2596a2d2fcef3007c2e0783aca180b56": "Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0,", "2596ca7c3fa57c89d937c89ff6237ce9": "g({\\boldsymbol {\\eta }})", "2596f457f57060553cf0737fe2ed27c6": "\\forall x:P(x)\\Rightarrow Q(x)", "25972673cf0f310e877ec635b4cce313": "{\\bar {L}}_{5}", "259755d00ed7b4d2323961431d2ae35e": "{\\vec {F}}_{L}", "2597b32b36935a132d199355d58ca58a": "f_{n}={\\frac {n}{2L}}{\\sqrt {\\frac {T}{\\mu }}}", "2597c0b0a1464ed51ad1ec27ea6bc046": "81.9\\pm 0.5", "25985b0c0d9cb7007f71e16153a20040": "\\lambda r^{2}-(\\lambda +\\mu +s)r+\\mu =0.", "25986f008505072f83781287db5a60a0": "IR={\\frac {E[R_{p}-R_{b}]}{\\sigma }}={\\frac {\\alpha }{\\omega }}={\\frac {E[R_{p}-R_{b}]}{\\sqrt {\\mathrm {var} [R_{p}-R_{b}]}}}", "259881d3201258f509f2425ac5eef713": "deg(p_{i})", "259896284211e022f1dc9ba91ca59cb4": "|\\mathrm {O} (2n+1,q)|=2q^{n}\\prod _{i=0}^{n-1}(q^{2n}-q^{2i}).", "2599298e21387bb12606afe0bd537318": "N_{2}(k)=2\\pi k", "25994f51b2443f84f9ce1bfe0f66e943": "{\\hat {\\Sigma }}={\\frac {1}{T}}\\sum _{t=1}^{T}{\\hat {\\epsilon }}_{t}{\\hat {\\epsilon }}_{t}^{'}", "25995ec689b2617916c0d9f65d4df25b": "a_{1}n_{2}[n_{2}^{-1}]_{n_{1}}+a_{2}n_{1}[n_{1}^{-1}]_{n_{2}}\\equiv a_{1}\\times 1+a_{2}\\times 0\\times [n_{1}^{-1}]_{n_{2}}\\equiv a_{1}{\\pmod {n_{1}}}", "2599af27f8acd90072f1e8b8d13121a5": "y_{i}^{*}=\\beta x_{i}+u_{i},u_{i}\\sim N(0,\\sigma ^{2})\\,", "2599b488fe46b0904e8847aa2b41e787": "\\Delta G_{ad}=\\Delta G_{p}+\\Delta G_{c}", "259a4d8d9adb4bf649336c4e11857c79": "{\\frac {C_{13}^{2}C_{4}^{2}C_{4}^{2}\\cdot C_{11}^{1}C_{4}^{1}}{C_{52}^{5}}}={\\frac {78\\cdot 6\\cdot 6\\cdot 11\\cdot 4}{2{,}598{,}960}}={\\frac {123{,}552}{2{,}598{,}960}}\\approx 4.75\\%", "259a67c52216b72aabbfe1cce1ed03f4": "\\{H,H,H,\\dots \\}", "259a76a6eb182367a3af4cdbaf5ca4d5": "x_{z}^{2}=\\xi ^{2}/x_{p}^{2}", "259a7b00a69fca141f4d6b5684d7d2c6": "~n_{s,j}~", "259b3c532c7ebdf9eeff05b7c2b14a9d": "q_{n}(x)=\\int _{\\mathbb {R} }\\!{\\frac {p_{n}(t)-p_{n}(x)}{t-x}}\\rho (t)\\,dt.", "259b51965d63fcde3aa92b8123d10824": "\\lambda (X,f,\\alpha )=(X,f,\\lambda \\alpha )", "259b5fc1b83ad7a2701579e6cefcd07c": "\\alpha ={\\frac {\\mbox{number of de-excitations via electron emission}}{\\mbox{number of de-excitations via gamma-ray emission}}}", "259bf5c87105bb02ca50410840ce94cc": "f(x)\\,\\!", "259c16c17d1c524fb919fb29b0d19e2a": "j=1,\\ldots ,n", "259cb9e09299038e2c6b59b59b4f342c": "{q}={F(k)}", "259cd84c7e5abdc44922d2a5a8d611cd": "r\\in \\mathbb {R} _{+}", "259d73361aa7312ce9c1d7e113cbc9ad": "W(x)={\\begin{cases}(a+2)|x|^{3}-(a+3)|x|^{2}+1&{\\text{for }}|x|\\leq 1\\\\a|x|^{3}-5a|x|^{2}+8a|x|-4a&{\\text{for }}1<|x|<2\\\\0&{\\text{otherwise}}\\end{cases}}", "259df63ece171d871ac28eec06cd2bff": "\\tau H_{n-i-1}M", "259df87b8ef012696e77cf59e26f8e84": "\\mathbf {O} ={\\begin{bmatrix}1&x_{F}&y_{F}\\\\1&x_{G}&y_{G}\\\\1&x_{H}&y_{H}\\end{bmatrix}}.", "259e1dd16454b6c16cb820bf203cb73e": "{\\hat {M}},{\\hat {N}}", "259e38b65a86aea4501c0a3c02ff098a": "{\\vec {j}}_{s}={\\hbar \\over {4\\pi }}g^{\\uparrow \\downarrow }{1 \\over M_{S}^{2}}\\langle {\\vec {M}}(t)\\times {{d{\\vec {M}}(t))} \\over {dt}}\\rangle ", "259e554e785b3d12cc363922b941c294": "u,v,x,y", "259ef820d1596b7dd83b706e83629595": "t{\\text{ and }}t-\\Delta t", "259f0e46103d040d01d9e2bf94ecaebf": "\\land ", "259f25e0fe4d9d8cdd9bde31b6677d42": "C_{v}\\,\\!", "259f35031eb7561091713eca2a6847e9": "\\scriptstyle O(N\\log N)", "259f9b91dbcdc1567f079bef86ce0526": "\\ \\displaystyle \\ {\\mathcal {U}}(\\alpha ^{*},{\\tilde {u}})\\ ", "259f9d6cd8e591971a46d5ee64224772": "P={\\frac {P_{max}}{3}}", "259fa4d96872e723316c3536350c69b2": "1-\\epsilon ", "259fabdff41a6272665d1b9090335447": "\\mathbf {C} V=\\ker c\\cap \\ker b.", "259fae9a12508a84f35cea2f6ad649ee": "P_{E}~dE={\\frac {1}{N}}\\,\\left({\\frac {f}{(\\hbar \\omega \\beta )^{3}}}\\right)~{\\frac {1}{2}}{\\frac {\\beta ^{3}E^{2}}{\\Phi }}\\,dE", "25a0089b5d82307945c55a0842526127": "55_{11}\\ ", "25a02cd84fc2db9c0a26a146751cfdba": "F[x,y]=\\int _{a}^{t}{\\sqrt[{3}]{x'y''-x''y'}}\\,dt", "25a07727e3646a693e1e3a4f5de16a28": "E(x)=k/\\lambda ,\\,E(\\ln(x))=\\psi (k)-\\ln(\\lambda )", "25a0903e8caa68c62daecfc0e6884635": "g(t)-G(t)\\approx 0", "25a090dad425532d05bb79b2ee8b4818": "L_{4k+2}(\\mathbb {Z} )=\\mathbb {Z} _{2}", "25a0c9a6afcc3945e60901ae32ffecdd": "x(u)=r\\cos u;\\qquad z(u)=r{\\sqrt {2}}\\left[E(u,{\\frac {1}{\\sqrt {2}}})-{\\frac {1}{2}}F(u,{\\frac {1}{\\sqrt {2}}})\\right]{\\text{ for }}u\\in [0,{\\frac {\\pi }{2}}]\\,", "25a0ef5fb7e73104fe0c2220cd15d721": "P(G,4)\\neq 0", "25a21b9f575914a01deedf0e4a927e19": "M_{n}=E{\\bigl (}f|\\Sigma _{n}{\\bigr )}", "25a22f0a3134167043d5c720b8bc2f71": "\\sum _{j=1}^{n}\\ M_{ij}({\\boldsymbol {q}}){\\ddot {q}}_{j}+\\sum _{j,k=1}^{n}\\Gamma _{ijk}{\\dot {q}}_{j}{\\dot {q}}_{k}+{\\frac {\\partial V}{\\partial q_{i}}}=\\Upsilon _{i}\\ ;i=1,...,n\\ ,", "25a25ae65beac79c88c741b3a9212108": "\\mathbb {P} _{\\mathbf {k} }^{n}", "25a2766c207b0ec2811dc40d4434bfaa": "\\mathbf {y} (k)=\\mathbf {C} (k)\\mathbf {x} (k)+\\mathbf {D} (k)\\mathbf {u} (k)", "25a2af99383a033c55be403b39aa184a": "M^{(n)}(B\\times ,\\dots ,\\times B)=E[{N}(B)({N}(B)-1)\\dots ({N}(B)-n+1)],", "25a2b59487fd039c61aac9ca3b74dd69": "\\operatorname {E} [|S_{N}|]\\leq \\sum _{n=1}^{\\infty }\\sum _{i=n}^{\\infty }\\operatorname {E} [|X_{n}|\\,1_{\\{N=i\\}}]\\leq \\sum _{n=1}^{\\infty }\\operatorname {E} [|X_{n}|\\,1_{\\{N\\geq n\\}}],", "25a2f45bf56d3271d707630a550b688e": "\\ \\epsilon _{d}", "25a373da14ac41b26a34db638e790242": "Z=V\\rho \\ ", "25a3b4b1848928f25e0b66aad10aa919": "\\left(a\\right)", "25a3cb6cac366e81f9da57a8f35e6b03": "L(n,2)=(n-1)n!/2", "25a3d207d5eb113a3cf19ce8303db051": "p(\\mathbf {X} |\\mathbf {M} ,\\mathbf {U} ,\\mathbf {V} )={\\frac {\\exp \\left(-{\\frac {1}{2}}\\,\\mathrm {tr} \\left[\\mathbf {V} ^{-1}(\\mathbf {X} -\\mathbf {M} )^{T}\\mathbf {U} ^{-1}(\\mathbf {X} -\\mathbf {M} )\\right]\\right)}{(2\\pi )^{np/2}|\\mathbf {V} |^{n/2}|\\mathbf {U} |^{p/2}}}", "25a44a250490727df635042c6bd98e36": "H^{p}(Y,{\\rm {R}}^{q}f_{*}{\\mathcal {F}})\\implies H^{p+q}(X,{\\mathcal {F}})", "25a4666732ff7c5e4c83a326afbbe8a4": "{\\hat {\\beta }}=1", "25a4c77c966b4817588312d9ed956450": "\\sigma _{1}>0", "25a5d03acecdbe195b198608346808da": "d_{\\Lambda }=d_{\\lambda _{1}}\\cdots d_{\\lambda _{k}},\\qquad d_{\\lambda }=\\partial _{\\lambda }+y_{\\lambda }^{i}\\partial _{i}+\\cdots ,", "25a5d9103a6eae8cc70d729dccaa49fd": "\\lceil mx\\rceil =\\left\\lceil x\\right\\rceil +\\left\\lceil x-{\\frac {1}{m}}\\right\\rceil +\\dots +\\left\\lceil x-{\\frac {m-1}{m}}\\right\\rceil ,", "25a5e1be3b89b1fb9b69c905c958944e": "Initiates(a,f,t)\\equiv [a=open\\wedge f=isopen\\wedge HoldsAt(haskey,t)]\\vee \\cdots ", "25a6377f043661c6808cd71d5bda40c7": "q=3", "25a66bf1d5decd9de746a4c83fba2b0e": "\\mathbf {v} (\\mathbf {x} ,t)={\\dot {\\mathbf {u} }}(\\mathbf {x} ,t)", "25a6b584eb0f72084bf0d6b8192691f0": "\\prod _{i\\in I}A_{i}=\\bigcap _{i\\in I}\\pi _{i}^{-1}(A_{i})", "25a6ecc95e99fc4750aba27923364d60": "i=2,\\dots ,99", "25a6f763ddf2c60ddf79f03b7633fde0": "1-365!/((365-n)!\\cdot 365^{n})", "25a7149bee46c56eec772bc7da8c210d": "O\\left(e^{(\\ln N)^{1/3}(\\ln \\ln N)^{2/3}(C+o(1))}\\right)", "25a77e87b658f3a8afef8ac3e8842a5b": "a_{i}\\in \\mathbb {C} ", "25a7851359f1ba2c2ad013c8a6794b02": "\\exists C_{1}>0,\\ \\exists C_{2}>0,\\ \\forall \\mathbf {x} ,\\mathbf {y} \\in \\mathbb {R} ^{n}:C_{1}d(\\mathbf {x} ,\\mathbf {y} )\\leq {\\sqrt {q(\\mathbf {x} -\\mathbf {y} )}}\\leq C_{2}d(\\mathbf {x} ,\\mathbf {y} ).", "25a7e04a96ee3fac9b7479ffdf429eac": "{\\overline {\\omega ^{2}}}=\\omega ", "25a7eaeea04d826bff41777a9b460936": "E_{n}=-me^{4}/8\\epsilon _{0}^{2}h^{2}n^{2}=13.61eV/n^{2}\\,\\!", "25a7f2d95acf15663de89dcb69d82567": "(P(n))_{n\\in \\mathbb {N} }", "25a7fbd008c5b05bc741a24750bffbb1": "\\omega _{E}\\,", "25a81dde30d0e639428aaaafdc6c3e25": "{\\begin{aligned}P(no~disease~in~population)=1-P(PH~in~population)-\\\\P(cancer~in~population)-P(other~conditions~in~population=0.997\\end{aligned}}", "25a886b1ed0f1ab38d66ceef1c0e0168": "\\left[L\\left(\\alpha ^{A},{\\alpha ^{A}}_{,\\nu },x^{\\mu }\\right)-L\\left(\\phi ^{A},{\\phi ^{A}}_{,\\nu },x^{\\mu }\\right)\\right]={\\frac {\\partial }{\\partial x^{\\sigma }}}\\left({\\frac {\\partial L}{\\partial {\\phi ^{A}}_{,\\sigma }}}\\right){\\bar {\\delta }}\\phi ^{A}+{\\frac {\\partial L}{\\partial {\\phi ^{A}}_{,\\sigma }}}{\\bar {\\delta }}{\\phi ^{A}}_{,\\sigma }={\\frac {\\partial }{\\partial x^{\\sigma }}}\\left({\\frac {\\partial L}{\\partial {\\phi ^{A}}_{,\\sigma }}}{\\bar {\\delta }}\\phi ^{A}\\right)\\,.", "25a8afe7b836ac8b794dd22503cef68c": "\\beta \\ =\\arccos \\left({\\frac {a}{c}}\\right)\\,", "25a8b802657bed9d27681ab643e52118": "A_{8}", "25a8b8693d7a146f37a2273437a74c7c": "\\mu _{c}", "25a8cd9626fd02f8fdf65de5d151c5b0": "1\\leq r4.6\\times 10^{26}\\ \\mathrm {years} \\,", "25ac029a9f81a19f308f15ceb12fa475": "AB=r^{n}-1", "25ac580f519817e6fcaa31d349888b61": "X'=(X_{1},\\cdots ,X_{i}+X_{j},\\cdots ,X_{K})\\sim \\operatorname {Dir} \\left(\\alpha _{1},\\cdots ,\\alpha _{i}+\\alpha _{j},\\cdots ,\\alpha _{K}\\right).", "25ac910d0eda99893ee18a71e7b91e55": "j\\in \\mathbb {Z} ", "25acce8af3450e1843178ad127aa28ba": "E_{1}>0,E_{2}>0,E_{3}>0,G_{12}>0,G_{23}>0,G_{13}>0", "25acff9ab7e7616274ae2b993ef7aca1": "\\int \\left|\\cos {ax}\\right|\\,dx={2 \\over a}\\left\\lfloor {\\frac {ax}{\\pi }}+{\\frac {1}{2}}\\right\\rfloor +{1 \\over a}\\sin {\\left(ax-\\left\\lfloor {\\frac {ax}{\\pi }}+{\\frac {1}{2}}\\right\\rfloor \\pi \\right)}+C\\;", "25ad100b8ab14e7403ad00ff0dc29d70": "\\textstyle f^{-1}", "25ad10834aaa3f14dd15ad5d0671de19": "{\\vec {F}}={\\frac {q_{1}q_{2}{\\vec {r}}}{4\\pi \\epsilon _{0}r^{3}}}", "25ad588dfdb4871d864287e633cfe807": "\\mathbf {E} (\\mathbf {r} ,t)=g(\\phi (\\mathbf {r} ,t))=g(\\omega t-\\mathbf {k} \\cdot \\mathbf {r} )", "25ad8b2cef744ba6d169c5a779761c50": "g^{(1)}(\\tau )=e^{-i\\omega _{0}\\tau -(|\\tau |/\\tau _{c})}", "25ad90f64c93d166bc730e898cb9d4ef": "{M}={\\frac {q^{2}}{gy}}+{\\frac {y^{2}}{2}}", "25adbfaa950f162fd40dc3c6b7c63d43": "T_{\\mp 1}^{(1)}", "25ae7ed1c28cc50b492a0d7f905e9ae4": "{\\dot {x_{i}}}=x_{i}[f_{i}(x)-\\phi (x)],\\quad \\phi (x)=\\sum _{j=1}^{n}{x_{j}f_{j}(x)}", "25ae8cac059f96485f346a2be5805212": "n_{\\text{G}}", "25aec4ef2ab12a03780de97f74b04f87": "\\omega ^{2}\\equiv \\left(\\omega _{o}^{2}+{\\frac {\\mu BH_{k}}{mL_{e}^{2}(B+H_{k})}}\\right)=\\omega _{o}^{2}\\left(1+{\\frac {\\mu BH_{k}}{kL_{e}^{2}(B+H_{k})}}\\right)", "25af01576ff96df1f8f74cc55a1cc913": "p_{a}(z)", "25af54de5611f36f0c598577ee8daafd": "A^{T}X+XA-XBR^{-1}B^{T}X+Q=0\\,", "25af810aa748842731df94a4b0e9aa06": "z", "25afb5b3bbcf1be6e9f7a7bc1ff5e95e": "(a\\cdot b)\\cdot c=a\\cdot (b\\cdot c)", "25afb7203c03c65b50e979a3c2ea46ff": "a_{ji}", "25aff1a98fbb6453f0d2c81f32ad70b8": "N_{2}\\,", "25b00365e84b70670f681371316fa8cb": "\\Delta f={\\frac {1}{\\sqrt {|g|}}}{\\frac {\\partial }{\\partial \\xi ^{i}}}\\!\\left({\\sqrt {|g|}}g^{ij}{\\frac {\\partial f}{\\partial \\xi ^{j}}}\\right)=0,\\qquad (g=\\mathrm {det} \\{g_{ij}\\}).", "25b011dcc8e1f70c761b4244c135fae0": "{\\frac {1}{\\sqrt {2\\pi }}}e^{-x^{2}/2}", "25b075e7e914b9dbfb20c590f9b924a3": "H=H_{0}+{\\frac {\\hbar }{4m_{0}^{2}c^{2}}}{\\bar {\\sigma }}\\cdot \\nabla V\\times \\mathbf {p} ", "25b0be86daad501d2ee3200059e54061": "0=\\sum \\limits _{n=0}^{\\infty }{\\frac {({\\sqrt {\\mu }}\\varsigma ^{\\mu })^{n}}{\\sqrt {n!}}}(E_{\\mu }^{(n)}(-\\mu |\\varsigma |^{2\\mu }))^{1/2}|n>,", "25deee057750ce13907359f2f0996b84": "\\mathbf {q} =-{\\frac {\\partial G_{3}}{\\partial \\mathbf {p} }}", "25df18ba239d20c062d5bebc102b51dc": "HP_{R_{n}}(k)={{k+n} \\choose {n}}={\\frac {(k+1)\\cdots (k+n)}{n!}}\\,.", "25df4e8bbac576f1bff9a90fa85291f4": "(d_{\\mathfrak {g}}\\alpha )(X)=d(\\alpha (X))-i_{X^{\\#}}(\\alpha (X))", "25df5b4729a1f1fc283169879cd78e95": "(d_{1},d_{2},p_{1}^{r},p_{2}^{r})", "25df7a91dac1fae2c27b5a4049a28b03": "H(s)={\\frac {R_{1}/(1+sC_{1}R_{1})}{R_{1}/(1+sC_{1}R_{1})+R_{2}+1/(sC_{2})}}", "25dfd15c1ddf32a42872aecb4a84bdff": "g(X_{i},X_{j})=\\eta _{ij}\\,", "25dff6299ac80287ec301ad3d9162861": "{\\frac {Z'}{Z_{0}}}={\\frac {Z_{0}}{Z}}", "25dff90858f5546dcac1139bc72b02fc": "\\mathbf {p} \\times \\mathbf {\\epsilon ^{1}} (\\mathbf {p} )", "25e06f0578d77de682fcb6ccabc09e57": "{\\begin{matrix}{r \\choose 3}{4 \\choose 1}^{3}\\end{matrix}}", "25e0742bdeb6a95431a067175c08b5a6": "|s_{2}|=8", "25e093c59df3f7799a0fc2ac4bc7d125": "g_{\\varepsilon }(x)=f(x)+\\varepsilon \\eta (x)", "25e09869f741efbb23117fd65a997f55": "GFR={\\frac {U_{[{\\text{creatinine}}]}\\times {\\dot {V}}}{P_{[{\\text{creatinine}}]}}}", "25e09c195d10cf43c35d6a81bf90605a": "E_{B}(v,V)", "25e0b3939da142f651dbe6fa41bbd9e8": "0\\leq x_{i}\\leq 1", "25e10282ddfae9363a4585b081a755e4": "x={a \\over 2}+b\\cos \\theta +{a \\over 2}\\cos 2\\theta ,\\,y=b\\sin \\theta +{a \\over 2}\\sin 2\\theta .", "25e1137601c1c685384cd511fd22cc79": "{\\frac {2\\pi }{45}}", "25e15fb1ba6947f77b7844252441ba50": "t\\neq 1\\Longrightarrow s\\leq t^{2}", "25e160cae17e639db0cab486502321f1": "{\\frac {\\Delta P}{P}}={\\frac {\\Delta {\\mathit {B}}}{\\mathit {B}}}+{\\frac {\\Delta {\\mathit {A}}}{\\mathit {A}}}\\left({\\mathit {1}}+{\\frac {\\Delta {\\mathit {B}}}{\\mathit {B}}}\\right)", "25e1d3ef3ef39a75ae15060c220658ad": "T_{eff}", "25e1f43e4cfc39568d0044de10c3ed7f": "-L{dI \\over dt}=R_{2}\\cdot I", "25e1fd7d8507cd11d1ddd6fefdb26970": "\\phi ^{-1}(S)=\\{x\\mid \\phi (x)\\in S\\}\\in \\mathrm {REC} (M)", "25e21916c510a451a3d032c664d4ce0f": "\\mathbf {T} ^{n}{\\begin{bmatrix}0\\\\{\\vec {b}}^{n-1}\\\\\\end{bmatrix}}={\\begin{bmatrix}t_{0}&\\dots &t_{-n+2}&t_{-n+1}\\\\\\vdots &\\ &\\ &\\ \\\\t_{n-2}&\\ &\\mathbf {T} ^{n-1}&\\ \\\\t_{n-1}&\\ &\\ &\\ \\\\\\end{bmatrix}}{\\begin{bmatrix}\\ \\\\0\\\\\\ \\\\{\\vec {b}}^{n-1}\\\\\\ \\\\\\end{bmatrix}}={\\begin{bmatrix}\\epsilon _{b}^{n}\\\\0\\\\\\vdots \\\\0\\\\1\\\\\\end{bmatrix}}.", "25e224e926b0f85e79372bbbc8fb09c8": "\\xi \\,", "25e2c0c6aa8227f65923457925d531fe": "{\\vec {R}}_{1}\\times {\\vec {R}}_{2}=A\\left({\\vec {r}}_{1}\\times {\\vec {r}}_{2}\\right)", "25e2c7173ddf3bc45afaf3232cf31e6a": "t^{(p-1)(q-1)/2}{\\frac {1-t^{p+1}-t^{q+1}+t^{p+q}}{1-t^{2}}}.", "25e3f33d7f862bcbc49eaf854761adea": "X_{\\mathrm {L} }=2\\pi fL", "25e4162bfc616a3af471c655d32c7a13": "0.3rad/s{\\sqrt {Hz}}", "25e429d17ebcf0858fb256609818c180": "\\left({\\frac {\\partial w}{\\partial x_{1}}}\\right)^{2}~,~~\\left({\\frac {\\partial w}{\\partial x_{2}}}\\right)^{2}~,~~{\\frac {\\partial w}{\\partial x_{1}}}\\,{\\frac {\\partial w}{\\partial x_{2}}}\\,.", "25e4acb15a3c12e809c61253c2399441": "\\langle I\\rangle _{e}", "25e56358cfeee42568285723578129b9": "Q(\\eta )=\\int _{-\\pi }^{\\pi }{\\frac {I(w)}{f(w;(1,\\eta ))}}\\,dw", "25e583ffbc6eefd5c6ea68fdba535f44": "{\\text{Minimize}}={\\begin{cases}f_{1}\\left(x,y\\right)&=x\\\\f_{2}\\left(x,y\\right)&=\\left(1+y\\right)\\exp \\left(-{\\frac {x}{1+y}}\\right)\\end{cases}}", "25e59010356db818689bf1c0b08876b0": "\\lambda f.\\operatorname {let} x=\\lambda x.f\\ (x\\ x)\\operatorname {in} x\\ x", "25e5c301c7c78b7ea0199980a287cd79": "{P}_{\\alpha }", "25e61144ef0afb77d87baec5e56e065f": "{\\widetilde {\\phi }}_{\\alpha }\\colon \\pi ^{-1}(U_{\\alpha })\\to \\mathbf {R} ^{2n}", "25e64b24ddeb89dac4cd34dc76dd08e8": "{\\underline {\\land }}", "25e6574b331aa9685cf6b73ab6b727ef": "J^{r}Y", "25e72971f523d97359091f9b69447429": "\\mu _{p},\\mu _{q},\\nu ", "25e76f75d8c479e282a6a6a3ef18da10": "\\mathbf {V} \\setminus \\{X_{i},X_{j}\\}", "25e7be4ee0124b0dc6c281c7851a4a4b": "dW\\propto dQ", "25e7d150e6eb867b0f59ecb10c4370a0": "E_{C}^{\\rm {PBE}}(\\omega )", "25e7da7705c2c086bee78f9df9e37d33": "\\phi (x)={\\frac {e^{-{\\frac {x^{2}}{2}}}}{\\sqrt {2\\pi }}}", "25e80ff375bc1c922cd5ab6e8bf68cdc": "\\eta ={\\frac {RT_{c}}{P_{c}}}\\Phi {\\frac {1}{V_{m}}}.", "25e84141a9e268f0ee8a38106cc343b0": "R_{\\alpha \\beta }=0", "25e882a5ed62c729c512f1783e022bce": "\\left\\|{\\frac {f+g}{2}}\\right\\|_{L^{p}}^{p}+\\left\\|{\\frac {f-g}{2}}\\right\\|_{L^{p}}^{p}\\leq {\\frac {1}{2}}\\left(\\|f\\|_{L^{p}}^{p}+\\|g\\|_{L^{p}}^{p}\\right).", "25e8e0e034f606b7047a5b6cf1d5eff7": "00", "25f6b9f54baafde2b1693692627eaff8": "H_{d}(z)\\ ", "25f6c0db710c5ccf0780858f0ce230a8": "A-BC^{-1}B^{T}", "25f6c98ca8211955bf8e9ceec6055ae5": "{\\begin{aligned}\\pi (\\theta |\\mu ,M)&=\\operatorname {Beta} (M\\mu ,M(1-\\mu ))\\\\&={\\frac {\\Gamma (M)}{\\Gamma (M\\mu )\\Gamma (M(1-\\mu ))}}\\theta ^{M\\mu -1}(1-\\theta )^{M(1-\\mu )-1}\\end{aligned}}", "25f6fb3d9c4cb2dfb7dc0229cdb5fb56": "f(qm)=f(q)m", "25f77b14bfc643155b563641ac27bb50": "f_{1}(d)", "25f7874321d6a78f6b5ba112fb5953b9": "\\displaystyle (1-e^{2\\pi i\\alpha })(1-e^{2\\pi i\\beta })\\mathrm {B} (\\alpha ,\\beta )=\\int _{C}t^{\\alpha -1}(1-t)^{\\beta -1}\\,dt.", "25f792cf700d49fe24efa94280c2bbf9": "\\mu _{X}^{\\pi }", "25f7c8647f0efc88bd753bbbc6ac2b6c": "(1/2)b\\pm (c^{2}-a^{2})/b", "25f7e414d76106f3f7c99b439bbdcb6e": "\\chi _{e}(\\Delta t)=0", "25f7f3448a7d08e290e55ed407b4793f": "V_{2}", "25f83ceb5ce70937f0f0a51c2296a2b6": "h:H^{n}(C;G)\\rightarrow {\\text{Hom}}(H_{n}(C),G).", "25f8763c5af5fa164c23046c9d98e909": "f(x;\\alpha ,\\beta ,c,\\mu )={\\frac {1}{\\pi }}\\Re \\left[\\sum _{n=1}^{\\infty }{\\frac {(-q)^{n}}{n!}}\\left({\\frac {i}{x-\\mu }}\\right)^{\\alpha n+1}\\Gamma (\\alpha n+1)\\right]", "25f8b8807f99b07cc3d94b8a24f4399e": "v_{0.5}\\neq 0", "25f91780f1181b22a9342596a0c53a52": "(\\mathbf {a} \\cdot \\mathbf {b} )^{2}-(\\mathbf {a} \\wedge \\mathbf {b} )^{2}", "25f93d53f7a72c019d91a984c072d432": "\\Pr\\{h_{a,b}(x)=h_{a,b}(y)\\}\\leq 1/m", "25f966a5374dc451d5c386d0778e2f78": "{\\widehat {p}}=\\left(1+{\\frac {1}{n}}\\sum _{i=1}^{n}k_{i}\\right)^{-1}.\\!", "25f995ab96a6af776e7d445a51f784ec": "u(0)=u^{1,0}{\\mbox{ in }}\\Omega ", "25f99b25d676f4240ab633d2c6ad2085": "{\\frac {AP}{BP}}={\\frac {AC}{BC}}.", "25f9ab91477595c9e9b5d6658e662cf6": "R_{4,2}=36r^{4}-56r^{3}+21r^{2}", "25f9ae96a5021937aa517e982dc0e192": "\\limsup _{n\\to \\infty }d(S\\upharpoonright n)=\\infty ,", "25fa2375c2a54ff028d553e6e083db68": "\\mu =2.394", "25fa44563151151076269d2eab7bbdbd": "\\prod _{d|n}f(d)\\;", "25fa5f13570947da66b0d20e90b33724": "q\\approx kd", "25fab1e2a6439f91ff9f1d1c67fb446e": "v_{\\mathrm {rms} }={\\sqrt {{3RT} \\over {M_{m}}}}", "25fb0beeab6bbc745ef65576317fbf6d": "++", "25fb1fed93190afb86d6e7f2b056e11e": "\\mathbf {j} =\\sigma \\mathbf {E} ", "25fb3646e2c1c65a58906e818c7adb1a": "p_{m}f_{m}", "25fb896cad16acce5dc92116490fcd55": "{\\frac {\\partial V[e(u,P),P]}{\\partial Y}}{\\frac {\\partial e(u,P)}{\\partial p_{i}}}+{\\frac {\\partial V[e(u,P),P]}{\\partial p_{i}}}=0", "25fbb65d93d398451ffbca8cd824dd16": "fg=\\Lambda (f,g)+\\Lambda (g,f).", "25fc133c9da87a7b1fe4c7dfadc67ed9": "A_{z}(\\mathbf {r} ,t)={\\frac {\\mu _{0}}{4\\pi }}\\int _{\\Omega }{\\frac {j_{z}(\\mathbf {r} ',t_{r})}{|\\mathbf {r} -\\mathbf {r} '|}}\\,{\\rm {d}}^{3}\\mathbf {r} '", "25fc2f91140cc449aed7efdc615fb73f": "F(t_{11},\\ldots ,t_{1n})\\vee \\ldots \\vee F(t_{k1},\\ldots ,t_{kn})", "25fc45816f5e341d2ac808d19d264aa7": "{e^{x} \\over \\cos x}=c_{0}+c_{1}x+c_{2}x^{2}+c_{3}x^{3}+\\cdots \\!", "25fc99a8798c1d6b6dfa0fb518cad15b": "H=H^{*},H^{-1}=H^{T},{\\text{i.e. }}HH^{T}=I", "25fd034ca05e19ef700a47a747d3dac1": "{\\frac {1}{m_{\\text{red}}}}={\\frac {1}{m_{1}}}+{\\frac {1}{m_{2}}}\\,\\!", "25fd5591f13dd0a7d509ad4227844d0d": "\\mu (Av_{1},\\ldots ,Av_{n})=|\\det A|\\mu (v_{1},\\ldots ,v_{n}),\\quad A\\in GL(V).", "25fd677a8aae44daac8afba5c182dbc3": "{\\widehat {\\beta }}_{\\mathrm {IV} }=(Z^{\\mathrm {T} }X)^{-1}Z^{\\mathrm {T} }y\\,", "25fd6e4e3f7f67f9753102bfc1ad1d95": "(z_{1},z_{2},z_{3})\\in \\mathbf {C} ^{3},\\qquad (z_{1},z_{2},z_{3})\\neq (0,0,0)", "25fda1a4b7dee5ff36b4ece1f48e4b3d": "\\Re \\{W\\}", "25fda6be1e958dd7c93ac388b9718558": "{\\text{PPI}}={\\frac {{\\text{savings}}\\times {\\text{probability of success}}}{{\\text{cost}}\\times {\\text{time of completion}}}}", "25fdaadb54fd40b61bda97535323b838": "d_{m'm}^{j}(\\beta )=\\langle jm'|e^{-i\\beta J_{y}}|jm\\rangle ", "25fe2f4be9536e006ae2efea18e84a0d": "K>K_{c}\\approx 0.971635\\dots ", "25fe940e08e8c622c0b69d7db24bfd1b": "U_{in}=cos2\\theta +90", "25ff2038fbecfef7255539b176b731c5": "W_{E}(2)={\\frac {\\sigma ^{2}(c_{11}+2c_{22})}{2c_{11}}}", "25ff8153bc9215f993d5cb26530e8225": "D_{Q}=\\int _{-\\infty }^{\\infty }\\int _{-\\infty }^{\\infty }P_{X,Y}(x,y)(x-y)^{2}\\,dx\\,dy=\\int _{-\\infty }^{\\infty }\\int _{-\\infty }^{\\infty }Q_{Y|X}(y|x)P_{X}(x)(x-y)^{2}\\,dx\\,dy.", "25ffdb224a66d8a81311e294590b3237": "b_{ii}=0.0778\\cdot {\\frac {R\\cdot T_{c,i}}{P_{c,i}}}", "2600a1802f049e1ae2a1129a11a11ca6": "F_{t}|_{A}=f_{t}", "2600e9829be252823634ff453498a654": "z^{\\mathrm {T} }Nz={\\begin{bmatrix}1&-1\\end{bmatrix}}{\\begin{bmatrix}1&2\\\\2&1\\end{bmatrix}}{\\begin{bmatrix}1\\\\-1\\end{bmatrix}}={\\begin{bmatrix}-1&1\\end{bmatrix}}{\\begin{bmatrix}1\\\\-1\\end{bmatrix}}=-2\\not >0.", "2601322cc2051cbf1a2b7966b39a83cc": "(4q^{2n}(q^{2n}-1)/(q-1),q^{2n-1}(1+2(q^{2n}-1)/(q+1)),q^{2n-1}(q^{2n-1}+1)(q-1)/(q+1))", "2601335cf248e1f383c00415cbad2b59": "(x+1)^{\\deg(p)}p({\\tfrac {1}{x+1}})", "26014d72fb73347ccc6522901634bb80": "m_{j}\\omega _{j}=m_{k}\\omega _{k}.", "2601d8ffb7521c0457a2f4938716e9c2": "E_{\\pm }(n)=\\hbar \\omega \\left(n+{\\frac {1}{2}}\\right)\\pm {\\frac {1}{2}}\\hbar \\Omega (n),", "2601e22d221336f196364e7992bc8504": "\\displaystyle {\\hat {f}}(x)\\,", "2601e35d8c91429bee12effc91c4eaa8": "F_{c,v}(p)=p-2tv.\\,", "26022e62f04ca7b7cd7f30bbb1a693b0": "0\\to D{\\overset {x}{\\to }}D\\to D/xD\\to 0", "26024536a6ab8cb7a119b1f2e415f2da": "NEXP\\subseteq EXP", "2602b62af20a5c97cc109e074b2c6c9a": "dH\\neq 0", "2602da02f1e0dd5d90a6066fca6a4d1b": "f_{t}:\\Omega _{0}\\to \\Omega _{t},{\\mbox{ for }}0\\leq t\\leq t_{0}.", "2603471ca85409a34b02eeacafd6168f": "\\mathrm {area} (D_{r})=-{\\frac {1}{2}}\\,\\Re \\int _{0}^{2\\pi }\\sum _{n=-1}^{\\infty }\\sum _{m=-1}^{\\infty }m\\,r^{n+m}\\,a_{n}\\,{\\overline {a_{m}}}\\,e^{i\\,(m-n)\\,\\theta }\\,d\\theta \\,.", "2603a0e8267c601830d36280f7707fdc": "{\\mathbb {C}}\\times H/\\Gamma _{n}", "2603a57935aa077f2c7de0149542e2fe": "{\\dot {\\mathbf {p} _{i}}}={\\dot {m}}_{i}\\mathbf {v} _{i}+m_{i}{\\dot {\\mathbf {v} }}_{i}", "2603ac16346b83991a9ffab96d858f22": "D=4", "2603ce233f95dbba95fde3c6f63a261e": "I_{x}=\\int _{A}y^{2}\\,\\mathrm {d} A=\\int _{-b/2}^{b/2}\\int _{-h/2}^{h/2}y^{2}\\,\\mathrm {d} y\\,\\mathrm {d} x=\\int _{-b/2}^{b/2}{\\frac {1}{3}}{\\frac {h^{3}}{4}}\\,\\mathrm {d} x={\\frac {bh^{3}}{12}}", "2603cf4793d36a2d46544facc8d04582": "\\ \\ t\\ (\\ ", "2604223c49bde495937d595925c317f7": "P\\,=\\,f(G_{ij})", "2604293556409495ef9a034e3aa4401c": "\\chi (S')=N\\cdot \\chi (S)-\\sum _{P\\in S'}(e_{P}-1)", "26043f7118c8a0956bf41843f40d0b98": "\\nabla ^{2}\\nabla ^{2}w=0", "2604974757011ea2c086cc62cebba4b7": "\\ \\Delta G(T)=\\Delta H-T\\Delta S", "26051e1e5bb31458e6da232b0e8b3af7": "X(t)=((k+\\cos(t))\\cos(t),(j+\\cos(t))\\sin(t),\\sin(t))\\,", "260545a96c730fc4c721b4007c45594f": "(U,z_{i})", "2605a29607e49db6bfd6c0a476467d7b": "\\beta \\,\\,\\,\\,\\approx \\,\\,\\,\\,{\\frac {a}{2n}}\\left[{\\begin{array}{l}\\left({\\alpha \\left({\\alpha -1}\\right)\\mu _{1}^{\\alpha -2}\\mu _{2}^{\\beta }}\\right)\\sigma _{1}^{2}+\\\\\\left({\\beta \\left({\\beta -1}\\right)\\mu _{1}^{\\alpha }\\mu _{2}^{\\beta -2}}\\right)\\sigma _{2}^{2}+\\\\\\left({2\\,\\alpha \\,\\beta \\,\\mu _{1}^{\\alpha -1}\\,\\mu _{2}^{\\beta -1}}\\right)\\sigma _{1,2}\\\\\\end{array}}\\right]", "2605f657072d8970af099eb548b31503": "\\int _{-\\infty }^{\\infty }|q\\rangle \\langle q|dq=1", "2605fb05b0ab77a103c004518d8cac08": "d:Hom_{n}(A,B)\\rightarrow Hom_{n+1}(A,B)", "26061eb021150a379495d3897bc1754e": "Tr\\,60\\times 18(P9)LH", "260627e8347f059ca3f373fb2becdbd1": "\\scriptstyle \\leq 1\\times 10^{-6}", "2606b581c7a5d5d4b40cff188c6a9a9d": "\\pi _{0}(B^{+}C)", "260733f816b891e2e4c3782d4e2d7317": "\\left({\\frac {{\\partial }u}{{\\partial }t}}+u{\\frac {{\\partial }u}{{\\partial }x}}+v{\\frac {{\\partial }u}{{\\partial }y}}+w{\\frac {{\\partial }u}{{\\partial }z}}\\right)=-{\\frac {1}{\\rho }}{\\frac {{\\partial }P}{{\\partial }x}}+{\\nu }\\left({\\frac {{\\partial ^{2}}u}{{\\partial }x^{2}}}+{\\frac {{\\partial ^{2}}v}{{\\partial }y^{2}}}+{\\frac {{\\partial ^{2}}w}{{\\partial }z^{2}}}\\right)\\,\\!", "26073a43b1a44a43666c01a26c2ef6a9": "\\Phi (z,s+1,a)=-\\,{\\frac {1}{s}}{\\frac {\\partial }{\\partial a}}\\Phi (z,s,a).", "260771353e750e04adbff9d451eddcaf": "{\\begin{aligned}h&={\\frac {1}{4}}\\sum _{i}p_{i}(\\theta )\\;d(\\log p_{i}(\\theta ))\\;d(\\log p_{i}(\\theta ))\\\\&={\\frac {1}{4}}\\sum _{jk}\\sum _{i}p_{i}(\\theta )\\;{\\frac {\\partial \\log p_{i}(\\theta )}{\\partial \\theta _{j}}}{\\frac {\\partial \\log p_{i}(\\theta )}{\\partial \\theta _{k}}}d\\theta _{j}d\\theta _{k}\\end{aligned}}", "2607a0783fe83a1332bfd46825a2e267": "\\displaystyle f*g\\,", "2607a9cdd0478c6e3e74210dfecfdf40": "\\omega ^{2}\\,=\\,g\\,k.", "2607b4e8c424fe06d9d8758ba9011ed3": "\\theta ,\\sigma ,\\nu ,\\mu _{p},\\mu _{q}", "26083d438c762cdbfd9549b1369322c4": "\\alpha _{l}=\\left[{\\frac {q^{2}}{\\mu \\omega ^{2}}}\\right]\\left[{\\frac {(2l-1)!!}{l}}\\right]\\left({\\frac {\\hbar }{2\\mu \\omega }}\\right)^{l-1}", "260855aa0bd47eb2299090579cf71f7f": "n=0", "26087b0b676d811f5163ec265e83fc53": "\\scriptstyle {I\\geq 1}", "2608a6214ce69434fb63e7b047040415": "F_{A}\\;=\\;F_{BH}\\;F_{BG}\\;F_{MH}\\;F_{MG}\\;F_{F}", "2608df8a7b610c08d47d6114f9e7fc7f": "A.a=f(\\alpha _{j1}.a_{1},\\ldots ,\\alpha _{jm}.a_{m})", "2609040706534c6fe5afc2aa4f167b25": "\\mu \\propto \\lambda ^{-2}", "2609163838730964b8cb9690068c2c6e": "{\\frac {(\\Pi _{1})^{0.5}}{(\\Pi _{2})^{0.75}}}=", "2609419f2c6acbb9b25149979e163172": "D_{s,t}", "2609b57414c8d5582997504db19617ec": "\\scriptstyle -V_{CC}", "260aba52b702c01d89bc65ec0e76c3f0": "{\\mathcal {E}}_{p}/e", "260ad813b782b3619de8882732d1ad7a": "A/\\Phi ", "260af3e5b7bd8509be9d318a8401a9a6": "2\\circ 4=F_{3+4}+F_{3+2}=13+5=18.", "260b07d287366cf70efe68becbc02457": "f(\\sup(\\mathbb {M} ))=\\sup(f(\\mathbb {M} ))", "260b0ec1ed0067264f62c96965442a9a": "\\eta (1)=\\lim _{n\\to \\infty }\\eta _{2n}(1)=\\lim _{n\\to \\infty }R_{n}({\\frac {1}{1+x}},0,1)=\\int _{0}^{1}{\\frac {dx}{1+x}}=\\log 2\\neq 0.", "260b1c8924c2baf353aafc35b52abfab": "{\\boldsymbol {r}}(0)={\\boldsymbol {r}}_{0}\\in \\mathbb {R} ^{2N}", "260b2f81e867503921875ad7a8a53eed": "\\lim _{i\\to \\infty }", "260b307f4d300644a2affb7698d385e7": "[Q=R^{2}]\\sim \\chi ^{2}(N)\\ .", "260b57b4fdee8c5a001c09b555ccd28d": "\\omega ", "260b6099fd117588a809e4fb924e1d63": "n=", "260b628da799118dd8118462c0e00d29": "q=ae(-1)+af(+k)+ag(-j)+be(-k)+bf(-1)+bg(+i)+ce(+j)+cf(-i)+cg(-1)\\,", "260b6e3c94e05305f5643bcec244259c": "\\rho (\\vartheta )=\\rho _{0}+\\Delta \\rho \\cdot cos^{2}\\vartheta ", "260b8cee97f646cec505bd3d6380db1e": "f(\\Phi ,I)=0", "260bfa6adbe8062be00d63d7ac624873": "x_{2}^{T}\\omega x_{2}=0", "260c74541d883220ee7ae41939c49721": "\\gamma :Y\\longrightarrow Z\\ ", "260c8863416e86ac180747e2d0d1f332": "c\\cdot (1-\\sum _{j}h_{j})=\\sum _{j}h_{j}\\cdot \\Phi (-j)", "260ca71cafb4af9e777f33f4c99d25b6": "R=ar^{2}+br+c", "260d0632e390b1284ef8fb500a64340a": "0\\sum _{i=1}^{n}w_{i}", "26195a0f50e1cae3446ac73d5c0a1634": "\\mathbf {X} (s)=(s\\mathbf {I} -A)^{-1}B\\mathbf {U} (s).\\,", "2619724bf426aa59f4c41b26983a5120": "F_{E}=-e|E(r)|", "2619783d56716cc9c5f41857a499613d": "g{\\frac {\\partial \\eta }{\\partial x}}=fv", "2619a66d5263460e0277381a556d5327": "L^{2}(G)", "261a15fc7febb966e4c1d4bde12f00da": "Z", "261a22e15384daf71dd5b8abc33391cd": "r(\\cdot )", "261a70c47bce4988014b555fd3e33d49": "{\\hat {P}}^{\\mu })", "261aad7982a137a6759c3e3b58245bfe": "D_{\\mathrm {JS} }={\\tfrac {1}{2}}D_{\\mathrm {KL} }\\left(P\\|M\\right)+{\\tfrac {1}{2}}D_{\\mathrm {KL} }\\left(Q\\|M\\right)\\,\\!", "261ab7b4ab65de1afdc9126decea58bc": "\\Delta =0", "261ad7ab7aa923709ac9a0878237945a": "L(\\mu ,\\lambda )=\\left({\\frac {\\lambda }{2\\pi }}\\right)^{\\frac {n}{2}}\\left(\\prod _{i=1}^{n}{\\frac {w_{i}}{X_{i}^{3}}}\\right)^{\\frac {1}{2}}\\exp \\left({\\frac {\\lambda }{\\mu }}\\sum _{i=1}^{n}w_{i}-{\\frac {\\lambda }{2\\mu ^{2}}}\\sum _{i=1}^{n}w_{i}X_{i}-{\\frac {\\lambda }{2}}\\sum _{i=1}^{n}w_{i}{\\frac {1}{X_{i}}}\\right).", "261adf4ac0a0c093ad6040cb2b82f1d9": "\\mathbf {\\nabla } \\cdot (\\epsilon \\mathbf {\\nabla } \\phi )=0", "261aea3796669e93390b54299964c7ab": "x_{1}^{1}=x_{1}^{0}+u_{1,1}^{0}+u_{2,1}^{0}=8+2+3=2\\in Z_{11}", "261b250b1ce246d05e89e1dd906b8a91": "\\mathrm {DOF} \\approx {\\frac {2Hs^{2}}{H^{2}-s^{2}}}{\\text{ for }}s0)={\\frac {f(k;\\lambda )}{1-F(0)}}={\\frac {\\lambda ^{k}e^{-\\lambda }}{k!\\left(1-e^{-\\lambda }\\right)}}.", "261d0f23d71a972fe61c98929fae3e55": "U(1)_{Y}", "261d0f36f095e319dda2395a80a063a7": "\\sum _{k=0}^{n}a_{k}\\sim n", "261d260048a0aa6acc1a5c7018c911e9": "a\\land 1=a", "261d33fc846bac78f451c85ab0a9f360": "{\\bar {\\theta }}_{i}^{}(t_{n}^{})", "261d91a209af577aca43e0f506c6ba14": "\\geq 7", "261dec347e7132f55de8d744b2c7815e": "{\\bmod {\\,}}n", "261eae0c170eebc56303490a063f656f": "y=x\\tan \\theta -{\\frac {gx^{2}}{2v^{2}}}(1+\\tan ^{2}\\theta )", "261eef8e83eaa0ea15ebee22c8aee0a1": "pt\\sqcup _{S^{n-1}}pt", "261f07894c7a50fc332e0664b7689927": "P(e,e',T)", "261f0cd12c623cdd8bdb63fd4dabbb54": "H_{\\mathrm {e} }(\\mathbf {r,R} )\\;\\chi (\\mathbf {r,R} )=E_{\\mathrm {e} }\\;\\chi (\\mathbf {r,R} )", "261f512c31a17e6242b18461daa988e7": "\\langle j_{1}j_{2};m_{1}m_{2}|j_{1}j_{2};jm\\rangle =(-1)^{j-j_{1}-j_{2}}\\langle j_{1}j_{2};-m_{1},-m_{2}|j_{1}j_{2};j,-m\\rangle ", "261f55e79db500135590dc6661da947c": "{\\mathbf {v}}=(v_{1},v_{2},\\ldots ,v_{n})\\,", "261f5ccd650465e45d7a1d7d06154ae9": "p_{k}=P[x\\in I_{k}]=\\int _{b_{k-1}}^{b_{k}}f(x)dx", "261f71bd601d4ee8032c97caedb44dd0": "{\\begin{matrix}x_{1}\\geq 0\\\\x_{2}\\geq 0\\end{matrix}}", "261f99f678a722399b2b755f55e5a87a": "\\Phi ^{ext}(\\mathbf {q} _{i})", "261ff01d39006cacc0a93c032ab4be63": "{\\hat {p}}=-i\\hbar {\\frac {\\partial }{\\partial x}}\\,\\!", "26201e0d230b32c6986f07b30057524b": "{\\frac {3h}{10}}", "262069c0b77d451cffdd65004c1799e7": "\\det(A-\\lambda B)=0", "262069fe1b77c6d95206e1de6003586d": "q=\\beta :\\alpha =OB:OA\\ ", "2620872d7d8ba4fa9124131fbcdd9fd1": "y(x)=1", "2621084503b1e4c7ae6f480c10d568d1": "{\\mathcal {L}}_{X}f=\\sum _{\\{i\\}}\\lambda _{i}e_{i}^{*}(f)\\otimes e_{i}.\\,", "2621709eccc6edfbc2f88d3447b1ebaa": "{\\frac {1}{2}}{\\frac {\\partial ^{2}a}{\\partial \\xi ^{2}}}+i{\\frac {\\partial a}{\\partial \\zeta }}+N^{2}|a|^{2}a=0", "2621724eb0dde08e3a0419662eb90e70": "{\\begin{aligned}m''&=m'r^{e}{\\pmod {n}}\\\\&=(m^{e}{\\pmod {n}}\\cdot r^{e}){\\pmod {n}}\\\\&=(mr)^{e}{\\pmod {n}}\\\\\\end{aligned}}", "2621893e5ebaf67fef801abaa0ae36e6": "x=r_{1}\\cos(\\omega _{1}t)+r_{2}\\cos(\\omega _{2}t),y=r_{1}\\sin(\\omega _{1}t)+r_{2}\\sin(\\omega _{2}t),\\,", "2621901c2308423e78207b0b84618da9": "\\Psi _{n}(t)=\\psi _{n}(t)e^{i\\theta _{n}(t)}e^{i\\gamma _{n}(t)}.", "2622ad12ceda7ef263d5b0950ffe8514": "\\displaystyle A(s_{1}s_{2},\\lambda )=A(s_{1},s_{2}\\lambda )A(s_{2},\\lambda ),", "2622d7cfb442fea585ec9c8f93cff16b": "p\\Psi =0", "2622e9b348f05f74e3e4bb9b7d213dc8": "RM", "262332df709a278045b9ed154c9c2588": "=\\mu N-NkT\\,", "26237469281bb04986256179d98d39e3": "\\sup _{z\\in K}|f(z)|\\leq C_{K}\\|f\\|_{L^{p}(D)}.", "26238485c5ad19f2bb0c2c9e9323a8d7": "\\left\\langle \\cdot \\right\\rangle _{t}", "2623d2f1c321db000ac891e074436d4b": "(14.d)\\quad 2\\psi _{,\\,\\rho }\\psi _{,\\,z}=2e^{-2\\psi }\\Phi _{,\\,\\rho }\\Phi _{,\\,z}", "2623d49f9338a3601a1f941d40e03431": "{\\rm {Pr}}{\\Big (}{\\hat {f}}(x)-w(x)\\leq f(x)\\leq {\\hat {f}}(x)+w(x){\\Big )}=1-\\alpha ,", "2623efb05a8a39e42ccb4ef058c71923": "\\scriptstyle i\\,=\\,1,\\dots ,n", "26240362df5500d91bf11b5e9e4e67ee": "\\pi _{n}(x)\\sim 2C_{n}{\\frac {x}{(\\ln x)^{2}}}\\sim 2C_{n}\\int _{2}^{x}{dt \\over (\\ln t)^{2}}", "26242bea98bb2f8a596fbe05601de6bf": "n\\approx 2.5{\\sqrt {\\frac {1}{\\sum _{i=1}^{k}{\\frac {1}{c_{i}}}}}}", "2624346167ec52a931cf45725f44a872": "\\scriptstyle {\\sqrt {20}}\\ =\\ {\\sqrt {4}}{\\sqrt {5}}\\ =\\ 2{\\sqrt {5}}", "262449ee5fb5d7363c6767b028300fd9": "\\mathrm {Power} _{ext}", "2624af45ed1dc4c051ed5f7bb72db550": "\\tau _{N}=\\tau _{0}e^{\\frac {KV}{k_{B}T}}", "2624b76ef060439d628c7c1ca57cfb2d": "b\\equiv x^{2}\\mod n", "2624da2470d717aca6614b1f7c0ec6bf": "((P\\leftrightarrow Q)\\leftrightarrow R)\\leftrightarrow (P\\leftrightarrow (Q\\leftrightarrow R))", "2624f7c891830f07ef43b705864ad7af": "(\\mathbf {S} )\\,\\mathrm {d} (ab)=a\\,\\mathrm {d} b+\\mathrm {d} a\\,b\\,.", "262795eaf1590876a9ebce8f96753e2c": "\\cdot :V\\times V\\rightarrow \\mathbb {R} ", "2627afcd51f49d9a2c776f3e3432f3b3": "g_{\\star }\\left(x^{*}\\right):=\\inf \\left\\{\\left.\\left\\langle x^{*},x\\right\\rangle -g\\left(x\\right)\\right|x\\in \\mathbb {R} ^{n}\\right\\}", "2627d0e830c07ca2e61b306fc82c8f1d": "\\chi ({\\mathcal {F}}):=\\sum _{i\\in \\mathbf {Z} _{0}^{+}}(-1)^{i}\\,{\\rm {rank}}\\,(H^{i}(X,{\\mathcal {F}})).", "2627e3d7d3c3ce3b0630af60c312fe86": "\\Box \\,\\mathbf {E} =0,", "2627fbc92df012f8c513367733f89aad": "|f(z)|\\leq {\\frac {2Ar}{R-r}}", "26282668fe59eb6d3739b73f225ac791": "S_{N}f(0)=0={\\frac {-{\\frac {\\pi }{4}}+{\\frac {\\pi }{4}}}{2}}={\\frac {f(0^{-})+f(0^{+})}{2}}", "2628895fa7e68bc1ac8f5dd2fb380ea8": "{\\bigg |}{\\partial A \\over \\partial z}{\\bigg |}\\ll |kA|", "2629298f1a7855098111d2ef159611de": "L=\\phi ^{2}\\,\\eta _{ab}\\,{\\dot {u}}^{a}\\,{\\dot {u}}^{b}", "262957b9975ec187a6116a788e78f7f1": "(A+\\lambda \\operatorname {diag} (A))\\Delta x=b", "26296222ee64bf84e3820fd15687c38b": "\\tau \\mathbb {Z} ", "262a3bd0318d5034272a8f904d6fad24": "\\mathbf {x} _{1}", "262a4584304c4546e261209782f1c4f3": "\\operatorname {Ext} _{R}^{n}(A,B)=(R^{n}G)(A).", "262a772700d605b6e749091a80ccee25": "\\delta (\\rho \\mathbf {x} )=\\delta (\\mathbf {x} ).", "262a7cb37bd4bfdde88b03e865f1b918": "\\beta _{p}=(1-p_{\\infty })/\\tau _{p}", "262ac0d0a5152e06d017d837891c2e15": "{\\begin{aligned}{\\mathcal {F}}^{-1}\\{S(f)\\}&=\\int _{-\\infty }^{\\infty }\\left(\\sum _{n=-\\infty }^{\\infty }S[n]\\cdot \\delta \\left(f-{\\frac {n}{P}}\\right)\\right)e^{i2\\pi fx}\\,df,\\\\&=\\sum _{n=-\\infty }^{\\infty }S[n]\\cdot \\int _{-\\infty }^{\\infty }\\delta \\left(f-{\\frac {n}{P}}\\right)e^{i2\\pi fx}\\,df,\\\\&=\\sum _{n=-\\infty }^{\\infty }S[n]\\cdot e^{i{\\tfrac {2\\pi nx}{P}}}\\ \\ {\\stackrel {\\mathrm {def} }{=}}\\ s_{\\infty }(x).\\end{aligned}}", "262af2e0176f997b94e4e82076618023": "I_{x}(M)/I_{x}^{n}(M)", "262b11e130004bdaf776aa9984e2b2dc": "C={\\frac {1}{2\\pi fR}}={\\frac {1}{6.28*5000*10}}=3.18*10^{-6}", "262c7dd89f0c5792a84310bf193cf11f": "\\langle P\\rangle ={\\mathcal {E}}I_{\\mathrm {rms} }\\cos \\phi \\,\\!", "262c95dfaa6fee4f7f2fe531098eebd2": "\\mathrm {li} (x^{1/2})/2", "262ce00955eccdce97c2cc50caca7291": "{\\frac {1}{1-{\\frac {a}{p}}}}H(t)=\\sum _{n=0}^{\\infty }a^{n}p^{-n}H(t)=\\sum _{n=0}^{\\infty }{\\frac {a^{n}t^{n}}{n!}}H(t)=e^{at}H(t).", "262d5ec61d4727236470a56c2e8433ef": "\\mu _{i}", "262d8fd63e656ffa6e41f9907f2bc635": "\\mathrm {Cdim} X=\\inf _{Y\\in {\\mathcal {G}}}\\dim _{H}Y", "262dfb07bd84ad8f918c1a4035f3232b": "w\\Vdash A", "262e016a156bc7286277d35342224383": "F=N\\varepsilon _{0}+k_{B}T\\sum _{\\alpha }\\log \\left(1-e^{-\\hbar \\omega _{\\alpha }/k_{B}T}\\right)", "262e042acef0bb91653712ba0609f5dd": "E_{\\mathrm {t} cS}(\\varepsilon ,t,T)", "262e0afc75c8a9fc536a7dce57e6ebe1": "B_{1}", "262e75aaa58a61e45bb7097b830f478d": "\\scriptstyle d(x,\\,A)\\;=\\;\\inf\\{d(x,\\,a)\\;|\\;a\\,\\in \\,A\\}", "262ec9ad165d37027c5a31654c4ffdaa": "\\int _{a}^{\\infty }f(t)\\;dt", "262eebaa13a0c4361a4c30b6d5aa3608": "R_{\\theta }", "262f1e8b6eb40a3e90f7dd4288437008": "P(x\\rightarrow x')=g(x\\rightarrow x')A(x\\rightarrow x')", "262f21c02905d92fd7cde701deb1b086": "s^{2}(\\vartheta )", "262f265eb5cfb857e8baa0c83a050d28": "\\mathrm {lift} (X\\Rightarrow Y)={\\frac {\\mathrm {supp} (X\\cup Y)}{\\mathrm {supp} (X)\\times \\mathrm {supp} (Y)}}", "262f4266de2347543f3e5c86ee5c77b3": "{\\begin{aligned}U_{f,P_{n}}&=\\sum _{k=1}^{n}f(x_{k})(x_{k}-x_{k-1})\\\\&=\\sum _{k=1}^{n}{\\frac {k}{n}}\\cdot {\\frac {1}{n}}\\\\&={\\frac {1}{n^{2}}}\\sum _{k=1}^{n}k\\\\&={\\frac {1}{n^{2}}}\\left[{\\frac {(n+1)n}{2}}\\right]\\end{aligned}}", "262f57469ad842e72459d47449608b04": "-1.5913", "262f5d08b5ea34d4a82039f839bd7a67": "p_{i}:=p_{i+1}q_{i+1}-p_{i+2}{\\text{ for }}i\\geq 0.", "262fedc550e771be210750af0ac3e3c4": "1+x^{-1}+x^{-2}+\\cdots ", "26302f0e1c87ba0db4847c24ed9cea8a": "{\\frac {\\partial r_{i}}{\\partial \\beta _{j}}}", "26306baf9f1d96b3b320c15f63275c92": "Z=\\sum _{i}g_{i}e^{-\\varepsilon _{i}/kT}", "2630807dbdafa8aaedeb4d2c13f49c40": "\\nabla \\times \\left(\\mathbf {v\\times F} \\right)=\\left[\\left(\\mathbf {\\nabla \\cdot F} \\right)+\\mathbf {F\\cdot \\nabla } \\right]\\mathbf {v} -\\left[\\left(\\mathbf {\\nabla \\cdot v} \\right)+\\mathbf {v\\cdot \\nabla } \\right]\\mathbf {F} \\ .", "263092b85467e0396b858eb86b921f46": "m_{p}\\,", "26309d9c15677ba25b9737386ab12d11": "{\\mathfrak {d}}", "2630bdd130b8f08da440faac635c86e2": "\\dim X\\leq h^{1,0}", "2630d50cc3b82b44a8e74f59778aed27": "\\Box A^{\\alpha }=\\mu _{0}J^{\\alpha }", "2631162c92ff1b1e31a061124a8c1907": "h_{ij}(z)", "26311c5eaa94a8a9e9e710c21788c541": "\\sum _{k\\mathop {=} a}^{b}f(k)=\\int _{[a,b]}f\\,d\\mu ", "263136042d889b43a694d24249234674": "G_{F}/G_{N}=10^{-32}", "26316c9f3590bf95b86da67699382e0d": "S(x)\\Lambda (x)=\\sum _{j=0}^{\\infty }\\sum _{i=0}^{j}s_{j-i+1}\\lambda _{i}x^{j}.", "2631c53c624694323b2b5a61e078f6ab": "\\delta _{n}(\\varepsilon )", "26323adf064e40aed7415437058397a6": "e^{x}=1+x+{\\frac {e^{\\xi }}{2}}x^{2}<1+x+{\\frac {e^{x}}{2}}x^{2},\\qquad 01\\\\\\infty &{\\text{otherwise}}\\ \\end{cases}}", "263fb2c3864020d3ac31681d43dd580f": "{\\frac {dm_{L}}{dt}}=\\rho _{L}Au_{L}=\\rho _{L}(4\\pi R^{2})u_{L}", "26403ec6d537fa31f63e294b44831734": "lb", "2640a34096ef4392606817e6cc9f8443": "\\Delta _{H^{n-1}}f=\\Box f\\left(x/q(x)^{1/2}\\right)|_{H^{n-1}}", "2640c8a3fe3838d64f95cdd31b786f65": "{\\tfrac {1}{2}}ab", "2640e33d330298a4cb9a2ccc5602c87f": "(x,y)\\in \\mathbb {R} ^{2}", "2641beafbf73b933e19a94c776189cc7": "\\rho _{X,Y}={\\mathrm {cov} (X,Y) \\over \\sigma _{X}\\sigma _{Y}}={E[(X-\\mu _{X})(Y-\\mu _{Y})] \\over \\sigma _{X}\\sigma _{Y}}", "2641ceaf2d4487af403228a918690909": "a_{ij}=a_{ji}.\\,", "2641d746823af32a05f9739287ed0a92": "\\mathrm {ind} (D)=\\prod _{i=1}^{r}p_{i}^{m_{i}}\\ ", "26423f225b030a35616ff13f74676f87": "S(T)={\\frac {C}{\\exp \\left({\\frac {c_{2}}{AT+B}}\\right)-1}}", "26424abc355235f5fca9ee7c22f4ba6e": "3^{3}-1=2\\cdot 3^{2}+2\\cdot 3+2", "26425187ea71acf17552a351519e6bf9": "f_{c}:\\mathbb {\\mathbb {C} } \\to \\mathbb {\\mathbb {C} } \\,", "26427cff6cfbbd418c7c9942403790f1": "\\nu _{1}=\\nu _{2}\\left(1+{\\frac {GM}{rc^{2}}}\\right).", "2642967a858d4825c7531a3ebd993403": "\\det(L_{f}R_{g})=\\sum _{S\\in {\\tbinom {[n]}{m}}}\\det((L_{f})_{[m],S})\\det((R_{g})_{S,[m]}),", "2642a4d37f8a0996f38f2417d545815a": "j^{-i}(\\alpha )", "2642d19b27180c52c08c15902f43a744": "[{\\hat {\\xi }}^{k},{\\hat {\\xi }}^{l}]=-i\\hbar I^{kl}.", "2642e4307a0773c5dfbacac7870a9691": "e^{-i\\omega t}", "26437ff269e7eb50c8c7ae8029657e24": "A_{i}\\rightarrow A_{j}\\rightarrow A_{k}", "26439c75afe493f6dc75e044c39be13a": "\\int {\\frac {\\delta Q}{T}}\\geq 0", "26447a666b41f929a983ec99f48ceb77": "K(x,y)=K(\\|x-y\\|)", "2644bb116b8be92f1f26619adcd2ce30": "O({\\sqrt {n^{2/3}}})=O(n^{1/3})", "2645273ab5c9c2a3dbf0331bb908f733": "{52 \\choose 4}\\times 48=12,994,800", "264536e8fdb3dc30cd764c727b8fecbc": "\\Gamma (b-1)/\\Gamma (a)", "26459813a522c7d3ebaf532ab41117c8": "\\sigma \\ ", "2645c5865ba60b5386a9ba0c0145ee77": "\\Delta t/T", "26461eef607ef249a04022df09903825": "{\\begin{aligned}\\Psi _{n{\\boldsymbol {k}}}&=\\mathrm {e} ^{i{\\boldsymbol {k}}\\cdot {\\boldsymbol {r}}}u_{n{\\boldsymbol {k}}}({\\boldsymbol {r}}).\\end{aligned}}", "2646482daced2a9e6658246ece6c0efe": "\\lim _{n\\rightarrow \\infty }\\prod _{i=1}^{n}{a_{i} \\over 2}={\\frac {2}{\\pi }}", "26467577c5b093d9fda14fa7a3b81502": "\\displaystyle c_{g}={\\frac {\\partial \\Omega }{\\partial k}}", "264688b4fc0958a9fe78e931ed5653ed": "2\\,z", "26469e99f0610d934cab50182188d647": "\\ R=|Z|\\cos {\\theta }\\quad ", "26470d3baf589e9b274e302e20219f9a": "{\\begin{aligned}\\Pr \\left[\\left|-{\\frac {1}{n}}\\log p(X_{1},X_{2},...,X_{n})-{\\overline {H}}(X)\\right|>\\epsilon \\right]&\\leq {\\frac {1}{n^{2}\\epsilon ^{2}}}\\mathrm {E} \\left[\\sum _{i=1}^{n}\\left(\\log(p(X_{i})\\right)^{2}\\right]\\\\&\\leq {\\frac {M}{n\\epsilon ^{2}}}\\to 0\\ {\\mbox{as}}\\ n\\to \\infty \\end{aligned}}", "264731e0e5881b171c226914acb8dfd6": "{\\mathfrak {P}}^{17}", "264732d643f446fa4b258b456ef247b6": "L_{k}=\\pi _{k+1}(X)", "26473cd19fca54957c900301221b0cf0": "|\\downarrow \\rangle ", "26475ca5ca4e4cf82e0cff1083c1183d": "1.5\\ {\\mbox{mol}}\\,C_{6}H_{6}\\times {\\frac {15\\ {\\mbox{mol}}\\,O_{2}}{2\\ {\\mbox{mol}}\\,C_{6}H_{6}}}=11.25\\ {\\mbox{mol}}\\,O_{2}\\ ", "2647e2a1163e35a351dab718086a95c2": "\\mathbf {x} \\,\\!", "264884439b70ab09a86bc848421c6de6": "[0,1]", "2648e871cc14f061a0ce481f7d98c9df": "\\phi _{x}={\\frac {qx}{6EI}}(3L^{2}-3Lx+x^{2})", "264933c95573c19829593f8cb905ad80": "\\phi (x)e^{-i\\omega t},", "2649a41cfa0b81e904a431f5b7a2172e": "{\\widehat {f}}(a/p)", "2649b01c12a3e7387d7247870e07d7f8": "e^{i}\\cdot e_{j}=\\delta ^{i}{}_{j},", "264ad77a8edf4f221cc5662094c9bfa9": "\\langle A(a)B(b)\\rangle +\\langle A(a')B(b')\\rangle +\\langle A(a')B(b)\\rangle -\\langle A(a)B(b')\\rangle ={\\frac {4}{\\sqrt {2}}}=2{\\sqrt {2}}>2", "264b188f5344ce4571c749312061f4f9": "a^{2}+ba-ab-b^{2}\\,\\!", "264b5033da304d28fb4c2cd9c78fbe02": "\\sum _{i+j=r+s}a_{i}b_{j}.\\,", "264b81e24425cc061bb466634ae925f5": "z^{z^{z^{\\cdot ^{\\cdot ^{\\cdot }}}}}\\!", "264b9592a542352eced142a5045369b2": "\\Phi _{3}\\left(\\mathrm {R} _{i+1}\\right)", "264bd75a6e18dc8d7822bc5b5dc45da5": "\\eta =2\\cos({\\tfrac {2\\pi }{7}})", "264bded76d581e781f4e6955792284c2": "Tf", "264bef9ea70f0ec07479b13c30755100": "\\mathbf {l} ^{2}=1", "264c2c065d309877d5e44f939386dafd": "q=1-p=0.046\\,", "264c71cbb3812112f80a6db21b8e54a3": "B=\\lbrace g\\ {\\bmod {\\ }}f:g\\in I\\rbrace ", "264c761f560646414bce67f9cc678398": "0\\left({\\tfrac {Clutter\\ Power}{Subclutter\\ Visibility}}\\right)", "265c78ef9b5fef418ad6f502fc67b673": "l\\varpropto \\langle \\rho \\rangle ^{-1}", "265ce0fb455bf1fefcf9779fd71e728d": "anJ_{n}=\\sin {ax}\\cos ^{n-1}{ax}+a(n-1)J_{n-2}\\,\\!", "265d027ce8920c351a1c593242b6ede3": "\\phi _{\\infty }=-E_{\\infty }z=-E_{\\infty }r\\cos \\theta \\ .", "265d1a1c5f0575be7bc8ecce88a59122": "L^{2}(\\mathbb {R} )", "265d1eb8abe09c91a472927365f15542": "g=\\sum \\limits _{\\mu ,\\nu }g_{\\mu \\nu }\\mathrm {d} x^{\\mu }\\otimes \\mathrm {d} x^{\\nu }\\quad {\\text{with}}\\quad g_{\\mu \\nu }={\\begin{pmatrix}1&0&0\\\\0&1&0\\\\0&0&1\\end{pmatrix}}", "265d2b550f8eccfb61a9aee79d7aa516": "\\pi _{n}", "265ed59ea6f3b2aaacbe46d7bef6adcb": "\\left({\\frac {a}{n}}\\right),", "265f4d60ea930f18f3091783ced65602": "f(Y)\\equiv {\\frac {Y}{1-e^{-Y}}}-{\\frac {Y}{2}}", "265f61cddda429d01585ec31dae9d026": "m_{r}=m-\\rho {\\mathcal {V}}", "265f75dbc4bf5fff33ad62292a6024ac": "1+6\\left({1 \\over 2}n(n-1)\\right)", "265f99a1d3b62521ae509705c7446998": "\\mathbf {M} ^{+}=\\mathbf {V} {\\boldsymbol {\\Sigma }}^{+}\\mathbf {U} ^{*}", "265fc62311e3b50eafa66b2231f0649d": "{\\frac {1}{2\\pi i}}{\\frac {d\\alpha }{\\alpha }}.", "266000dfca90694e4b3b6ad9b79f7779": "\\psi \\left({\\frac {1}{2}}\\right)=-2\\ln {2}-\\gamma ", "26600e1f68a9766ada7cc0d6d20f6915": "\\{0,1\\}^{k}\\rightarrow \\{0,1\\}^{n}", "26604eb2daf4750222b52b54fdb37205": "R_{E}{\\frac {r_{E}+R_{E}}{r_{\\pi }+2R_{E}}}", "2660689d4124d12dcdecff670868c221": "S_{K}=-\\int d^{4}x\\,{\\sqrt {-g}}\\omega \\left({\\frac {1}{4}}B_{\\mu \\nu }B^{\\mu \\nu }+V(K)\\right)\\;", "2660b7bb33c8e129fd15e9bfb98f2efa": "\\kappa (X_{1},\\dots ,X_{n})=\\sum _{\\pi }(|\\pi |-1)!(-1)^{|\\pi |-1}\\prod _{B\\in \\pi }E\\left(\\prod _{i\\in B}X_{i}\\right)", "26613b523d46878898a60f99574a0050": "100\\uparrow \\uparrow 2=10^{200}", "2661c3970881e9fbbaa2cc0df080bf78": "x=379;239", "2661f6d687884cf90fc3b10845bba5dd": "(x_{3},y_{3})", "26620f2598189af535ed9aec5f74a924": "m\\to \\infty ", "266213c3e961aa81bbde92c9c27a3fb2": "r+s=(z_{0}+w_{0})1_{G}+(z_{1}+w_{1})a+(z_{2}+w_{2})a^{2}\\,", "26621d0e2dce9e56c4bb28e4476d906b": "\\displaystyle \\zeta =m_{p}/m_{0}", "26626b19b0afed24c614ae0f3313c426": "\\Delta x'={\\frac {\\Delta x}{\\gamma }}", "26629485120e6c54325171a23a7c0830": "\\int _{X}^{\\oplus }T_{x}d\\mu (x)", "26629951d34eb84641f4d20ad0c96166": "|\\pi (x)-\\operatorname {Li} (x)|<{\\frac {1}{8\\pi }}{\\sqrt {x}}\\log(x),\\qquad {\\text{for all }}x\\geq 2657.", "266381f1d1b849f6fe37e8ba7afd8d64": "W={\\frac {N!}{n_{1}!\\,n_{2}!\\,\\dotsb \\,n_{m}!}}", "26639cf045bb53b25e0029a7c654f721": "f(y)>f(x)", "2663c4b7f1ad1f76970ad1e0932490a0": "S=\\int dt\\,d^{d}x\\left[\\psi ^{*}(i\\hbar {\\frac {\\partial }{\\partial t}}+\\mu )\\psi -{\\frac {\\hbar ^{2}}{2m}}\\nabla \\psi ^{*}\\cdot \\nabla \\psi \\right]-{\\frac {1}{2}}\\int dt\\,d^{d}x\\,d^{d}y\\,V({\\vec {y}}-{\\vec {x}})\\psi ^{*}({\\vec {x}})\\psi ({\\vec {x}})\\psi ^{*}({\\vec {y}})\\psi ({\\vec {y}})", "2663e06beeafd394c3965843c577c57f": "g(x)=\\ln {\\frac {\\pi (x)}{1-\\pi (x)}}=\\beta _{0}+\\beta _{1}x,", "2663f0e5efb0800f0ed77a11d111e585": "B=A", "2664d768d3d5f2961c39e421b7a7b33d": "x=(x_{1},\\dots ,x_{n})", "26650c11c3812f14ae80aa98a730bc63": "d(x,y)=\\sum _{n}|x_{n}-y_{n}|^{p}.\\,", "26652e370814596003842fd41d7242ea": "\\tau ", "26653965d428ac175a444bc5d62795af": "(aX^{2}+Y^{2})Z^{2}=Z^{4}+dX^{2}Y^{2}", "26657c41115dddb2eaf517e8c3d44dfc": "{\\frac {a}{r}}=1+(e-e^{3}/8)\\cos M+e^{2}\\cos 2M+{\\frac {9}{8}}e^{3}\\cos 3M+...", "2665a800ffc06ad0e9d266eeff2ce3b4": "(\\mathbf {u} ,\\lambda )", "266617d5d5cfe4ca48db29497b66f8cd": "S_{r}(m)\\approx S_{r_{1},r_{2}}(m)", "26664eaf6f30370a448568daf55900d4": "W_{1-2}={\\frac {P_{2}V_{2}-P_{1}V_{1}}{n-1}}", "266664f562edd776813555382bcd56c7": "{\\frac {a_{z}}{a_{x}}}={\\frac {\\pi \\nu }{(1-\\nu )}}.{\\frac {d}{\\lambda }}", "26669912de04c3c98d23a7fdd3d623ad": "E_{k}\\varpropto {\\frac {\\sigma _{t}}{\\rho }}m", "2666b1788e867a84c7c7392b63b1f598": "q=e", "266714d3e6cceed45a7bc03b80bbf882": "\\pi _{k}={\\frac {(\\lambda /\\mu )^{k}e^{-\\lambda /\\mu }}{k!}}\\quad k\\geq 0", "26672648aff2a51db4e0eb78b9c7f9fb": "w\\in \\Sigma _{\\epsilon }", "266727659f714ec57bb630d4422a8fdc": "\\left(x_{0},y_{0}\\right)", "26679b2cee4234caebb63a48ddc78bc7": "C(h){\\frac {\\partial h}{\\partial t}}=\\nabla \\cdot K(h)\\nabla h", "2667da0d0b9152a85eb33aebc4a129d3": "\\psi _{1}(0)", "2668486d116e526ad9ba5ad18a93ecc3": "{\\mathfrak {g}},", "2668507a9c768c2e7d0cc92e11596c26": "\\mathbf {r} _{i}", "266853e925b3410402dded85501a25d5": "{\\begin{aligned}T&=5\\;months=.4167\\;years\\\\\\end{aligned}}", "26685b0a9379ff4005a398579768f596": "[D_{1},D_{2}]=D_{1}\\circ D_{2}-D_{2}\\circ D_{1}.", "2668be1101347b8b2866b7cd9dc6fa5b": "S_{n},", "2668c3c4f7f719c3957f1b19be8b287e": "{\\overline {\\mu }}=m-s\\left({\\frac {\\gamma _{1}}{2}}\\right)^{1/3}", "2668f106c4340d1e398fdc6c1c7c4dae": "\\textstyle c>0", "2669345227314a151c2995fd5774d9ef": "{\\mathit {(p-1)}}{\\mathit {p}}^{k-1}", "2669541c1094a7ebe9f3a0661b4be109": "\\vert A\\vert ", "2669d6f77e06eb6142928af9990bc9cf": "\\gg ", "266a00226eac234cc217d7b94054e217": "\\sum _{i=1}^{K}q_{i}=P", "266a143e770ffca8cadd1c3324fd1634": "n=3,", "266a261ef188444921d723b8551b9173": "(ax+by)v=a(xv)+b(yv)", "266a9ba52dac10cfed2b1a9b42770dfb": "S_{12}={-2Z_{0}Y_{12} \\over \\Delta }\\,", "266abe80af4999d44cf72ce102be83e1": "\\mathrm {exp} (iz)=e^{iz}=\\cos(z)+i\\sin(z),\\,", "266ad3fe947dbf1f84901b7162ca6079": "\\Delta p_{\\text{B}}(x)", "266aefa39e04bac79ce325f426bba103": "T_{\\mathrm {w} }\\,", "266b581d5be1b2d1d3fa8aa6f1f9a2cf": "\\max _{N}\\left|\\sum _{n=1}^{N}\\left({\\frac {d}{n}}\\right)\\right|>{\\frac {1}{7}}{\\sqrt {d}}\\log \\log d", "266b72702bc4fc011d9891c69b94ee1f": "x\\pm t", "266b87a0c07c8fed3b8bf891dd5466b4": "\\alpha _{c}=1.02056", "266be3a8836b65a8a27c91346b7e8790": "|\\alpha \\rangle ", "266c0273df772915e068ff8d1e5927da": "\\theta =\\exp {(-\\beta u)}", "266c494ec5fd7f48718fd94a39b0dedd": "\\int _{a}^{b}{\\frac {f(t)}{t-x}}\\,dt", "266c5d7fcf32909c05bbd7d24f77ad5c": "\\omega (a_{n})=\\lim \\sup a_{n}-\\lim \\inf a_{n}.", "266cae595f409e0e2bfb0c329d821806": "-D_{2}{\\frac {dC_{2}}{dx}}", "266cf824cbaa72b4fa91e6387799e939": "{\\hat {f}}^{s}", "266cf8b127ca08bbb10f8e6b9e658485": "aaSbb", "266d730251e5d573139d8f350429a831": "\\sum _{n=1}^{\\infty }c_{n}\\sin {\\frac {n\\pi x}{L}}", "266d8a30639b362857b3aa51fd1f36da": "{\\begin{bmatrix}1&3\\\\1&0\\\\1&2\\end{bmatrix}}+{\\begin{bmatrix}0&0\\\\7&5\\\\2&1\\end{bmatrix}}={\\begin{bmatrix}1+0&3+0\\\\1+7&0+5\\\\1+2&2+1\\end{bmatrix}}={\\begin{bmatrix}1&3\\\\8&5\\\\3&3\\end{bmatrix}}", "266da83adb524171b27816969121dea7": "\\forall x(p(x)\\rightarrow P(x))\\wedge \\neg \\forall x(P(x)\\rightarrow p(x))", "266e1c7527186ea3b4463792f3566073": "\\lim _{(x,y)\\to (0,0)}\\left|y\\right\\vert =0", "266e6027841ee7c4464563f356ebb3c2": "~\\mathrm {2AmF_{3}+3Ba\\ \\xrightarrow {1150-1350^{\\circ }C} \\ 3BaF_{2}+2Am} ", "266e71cb2f4d9f16b67f47774e5fe387": "{\\underline {E\\Gamma }}", "266e84bb62d9bee123e5f5176133a5b6": "({\\mathcal {U}},{\\mathcal {S}})", "266e96d8a9bface34d5fbc576a05f051": "\\max \\sum _{(i,j)\\in E}{\\frac {1-\\langle v_{i},v_{j}\\rangle }{2}},", "266eb6a0c2c2cec33e943ebe07b6c660": "{\\frac {d\\mathbf {L} _{\\text{in}}}{dt}}\\ {\\stackrel {\\mathrm {def} }{=}}\\ {\\frac {d}{dt}}\\left(\\mathbf {I} _{\\text{in}}\\cdot {\\boldsymbol {\\omega }}\\right)=\\mathbf {M} _{\\text{in}}", "266f46c1c2685eb93ec3077c7cf9d174": "D_{k}=C_{k}\\cap C_{k}^{1}\\cap C_{k}^{2}\\cap \\dots ", "266f52938f84a1b3df9287ac61011078": "S_{0}(\\lambda ),S_{1}(\\lambda ),S_{2}(\\lambda )", "266fe0709f0fbabd96ac39549fa95787": "12n-1", "266fff00a1066271ccb1d448ff16c281": "{\\frac {\\beta -1}{r}}", "26704c52602491a16218a00bccc218a9": "{\\frac {8388613}{25165824}}\\,", "2670a221ce80dcaadfc3fd6499513163": "r={\\frac {1}{4(1+{\\sqrt {n}})^{2}}}\\,", "2670bd1be44e0717f23d36652647d704": "O(N\\log {N})", "267110e216b0283e2bdb8a5b0e3fa875": "9\\times 5", "26712969fd3cbb868b40550717f7ab28": "(t_{2},p_{2})", "2671410613b2065a35c0d4dbbb518c1b": "C_{n}=C_{2}\\prod _{q|n}{\\frac {q-1}{q-2}}.", "267148ca6923ee5341b8226de6341734": "{\\mathbf {F}}={\\mathbf {E}}+Ic{\\mathbf {B}}=E^{k}\\sigma _{k}+IcB^{k}\\sigma _{k}", "26714b3711a9cb13f2bd270c64371d2d": "e^{z}={\\cfrac {1}{1-{\\cfrac {z}{1+z-{\\cfrac {{\\frac {1}{2}}z}{1+{\\frac {1}{2}}z-{\\cfrac {{\\frac {1}{3}}z}{1+{\\frac {1}{3}}z-{\\cfrac {{\\frac {1}{4}}z}{1+{\\frac {1}{4}}z-\\ddots }}}}}}}}}}.\\,", "2671ae1eb456013380c59ef9d1cc4344": "{\\frac {F_{m}-F_{0}}{F_{m}}}", "2671d32b1d3ecedbd9787b2aa1c6c0bf": "\\theta _{r_{i}}(x){\\big |}_{x=-j\\infty }=\\angle (-{\\mathfrak {Re}}[r_{i}],-\\infty )=\\lim _{\\phi \\to \\infty }\\tan ^{-1}\\phi ={\\frac {\\pi }{2}}\\quad (10)\\,", "26725ec9a70a1e106f3c61f019935deb": "N^{b}a_{b}=\\left({\\frac {\\partial g_{tt}}{\\partial r}}c^{2}\\right)/\\left(2g_{tt}{\\sqrt {g_{rr}}}\\right)={\\frac {m}{r^{2}{\\sqrt {1-{\\frac {2m}{rc^{2}}}}}}}", "26726077c418e810d31060589b2e9662": "c=\\log _{b}a", "26729eae84c73103f076624bafd107cd": "\\mathbf {C} _{xx}^{0}=\\int _{\\Delta }x^{2}\\,dA=\\int _{x=0}^{1}x^{2}\\int _{y=0}^{1-x}\\,dy\\,dx=\\int _{0}^{1}x^{2}(1-x)\\,dx={\\frac {1}{12}}", "2672c2e2fa771f55e21b493f17200615": "\\theta _{n}={\\frac {2\\pi n}{q}},", "2672e2763e9dd40c08fcddcdd675bcc9": "\\alpha _{12}=-1", "26733e216c33c8a7691c09a8333c04c3": "{\\begin{pmatrix}1\\\\0\\\\0\\end{pmatrix}},{\\begin{pmatrix}-{\\frac {1}{3}}\\\\{\\frac {\\sqrt {8}}{3}}\\\\0\\end{pmatrix}},{\\begin{pmatrix}-{\\frac {1}{3}}\\\\-{\\frac {\\sqrt {2}}{3}}\\\\z_{2}\\end{pmatrix}},{\\begin{pmatrix}-{\\frac {1}{3}}\\\\-{\\frac {\\sqrt {2}}{3}}\\\\z_{3}\\end{pmatrix}}", "26737c7a2bcbdd49ac7c22fcad0b08ff": "\\omega _{x}=(b,d,u,a)\\,\\!", "2673935bf54d9ab04129f8fd316037dc": "Y{\\stackrel {\\pi }{\\to }}X", "2673df4a49c1e899b298259778795b29": "|\\psi \\rangle \\rightarrow \\operatorname {e} ^{-i\\omega _{l}t\\left(a^{\\dagger }a+\\sigma ^{\\dagger }\\sigma \\right)}|\\psi \\rangle ", "2674663231c00e5720fe175729d4e705": "B_{\\nu }(T)={\\frac {2h\\nu ^{3}/c^{2}}{e^{\\frac {h\\nu }{kT}}-1}}\\approx {\\frac {2h\\nu ^{3}}{c^{2}}}\\cdot {\\frac {kT}{h\\nu }}={\\frac {2\\nu ^{2}kT}{c^{2}}}.", "2674f3010db1eb5588dde7e6975f60e6": "P_{c}=\\,", "26752ed6517fdb33649bfedbeecdf05a": "\\scriptstyle {7({1/3})^{s}+6({1/3{\\sqrt {3}}})^{s}=1}", "267572b8963bb529092a22aaf9b536b8": "{\\mathfrak {g}}_{P}:=P\\times _{G}{\\mathfrak {g}}.", "2675b3728b908ca397d9e5fae3c22af5": "2x(x^{2}+y^{2})+x^{2}(2x+2y{\\frac {dy}{dx}})=2a^{2}y{\\frac {dy}{dx}}", "2675ce6fcff505e0923703eafff80b43": "\\left({\\frac {1-w_{i}{\\overline {w_{j}}}}{1-z_{i}{\\overline {z_{j}}}}}\\right)_{i,j=1}^{N}", "2675ff27c7ea1646962c44ba00e126e2": "z<0", "26766491ee80d0b6e3a24da8acbdfb0e": "i^{*}i:\\Phi \\subset H=H^{*}\\to \\Phi ^{*}.", "26766df637276caed72d0a589c0b3fad": "u_{n,i+1/2}={\\frac {u_{n,i}+u_{n,i+1}}{2}},\\quad u_{n+1/2,i}={\\frac {u_{n,i}+u_{n+1,i}}{2}},u_{n+1/2,i+1/2}={\\frac {u_{n,i}+u_{n,i+1}+u_{n+1,i}+u_{n+1,i+1}}{4}}.", "2676768089581d43d30da54d02355ec7": "-Q_{Coble}", "26768ad9076d1fd679d255852be7a2d7": "\\theta =\\arcsin \\left({\\frac {\\text{opposite}}{\\text{hypotenuse}}}\\right).", "2676a74097e6c3471b165eb3abae0598": "\\cos ^{4}\\theta ={\\frac {3+4\\cos 2\\theta +\\cos 4\\theta }{8}}\\!", "2676adec50bb7794c1f00cbb607d6069": "\\tau :\\ z^{\\tau }", "2676b06af354fb3b834b0ba0b4a2e13d": "{\\partial {\\overline {u}} \\over \\partial x}+{\\partial {\\overline {v}} \\over \\partial y}=0", "2677086aefc732eab0acaac7bde0698b": "{\\frac {\\partial h_{s}}{\\partial t}}+\\left(v_{||}{\\hat {b}}+{\\vec {V}}_{ds}+\\left\\langle {\\vec {V}}_{\\phi }\\right\\rangle _{\\varphi }\\right)\\cdot {\\vec {\\nabla }}_{\\vec {R}}h_{s}-\\sum _{s'}\\left\\langle C\\left[h_{s},h_{s'}\\right]\\right\\rangle _{\\varphi }={\\frac {Z_{s}ef_{s0}}{T_{s}}}{\\frac {\\partial \\left\\langle \\phi \\right\\rangle _{\\varphi }}{\\partial t}}-{\\frac {\\partial f_{s0}}{\\partial \\psi }}\\left\\langle {\\vec {V}}_{\\phi }\\right\\rangle _{\\varphi }\\cdot {\\vec {\\nabla }}\\psi ", "2677a78da638bd86011fd4bced2d20a7": "g(C):=dim_{k}\\Gamma (C,\\Omega _{C}^{1})", "2677be747b73393907eb13c451af030a": "\\rho =T_{\\mu \\nu }\\delta ^{\\mu \\nu }", "2677c183da96e57fd50fe8a3156c72e2": "R^{2}-r^{2}", "267835087891678e9f8ffe46bd69783b": "x(u)=a\\mathrm {sn(u,k)} +(a/k)((1-k^{2})u-E(u,k))", "26785473d6d245a0e7def889c530101f": "\\left(\\mathbf {H} (z_{1},\\ldots ,z_{k},\\ldots ,z_{n})-\\mathbf {H} (z_{1},\\ldots ,z'_{k},\\ldots ,z_{n})\\right)^{2}\\preceq \\mathbf {A} _{k}^{2},", "267905407294192856d94d0fb7a3b2b0": "\\varphi _{\\beta }(\\gamma +1)", "26794630b04f565641b4c9576677fa61": "2k", "267957c68d0ff420619cdb14dcf37dc9": "\\langle u{\\bar {v}}\\rangle _{S}.", "267960fd5b43b833707186f15a03ac9d": "{\\begin{array}{cccccccccccccccccc}&&&&&&&&&1&&&&&&&&\\\\&&&&&&&&{\\frac {1}{2}}&&{\\frac {1}{2}}&&&&&&&\\\\&&&&&&&{\\frac {1}{3}}&&{\\frac {1}{6}}&&{\\frac {1}{3}}&&&&&&\\\\&&&&&&{\\frac {1}{4}}&&{\\frac {1}{12}}&&{\\frac {1}{12}}&&{\\frac {1}{4}}&&&&&\\\\&&&&&{\\frac {1}{5}}&&{\\frac {1}{20}}&&{\\frac {1}{30}}&&{\\frac {1}{20}}&&{\\frac {1}{5}}&&&&\\\\&&&&{\\frac {1}{6}}&&{\\frac {1}{30}}&&{\\frac {1}{60}}&&{\\frac {1}{60}}&&{\\frac {1}{30}}&&{\\frac {1}{6}}&&&\\\\&&&{\\frac {1}{7}}&&{\\frac {1}{42}}&&{\\frac {1}{105}}&&{\\frac {1}{140}}&&{\\frac {1}{105}}&&{\\frac {1}{42}}&&{\\frac {1}{7}}&&\\\\&&{\\frac {1}{8}}&&{\\frac {1}{56}}&&{\\frac {1}{168}}&&{\\frac {1}{280}}&&{\\frac {1}{280}}&&{\\frac {1}{168}}&&{\\frac {1}{56}}&&{\\frac {1}{8}}&\\\\&&&&&\\vdots &&&&\\vdots &&&&\\vdots &&&&\\\\\\end{array}}", "267969f90e236db9822e2ba2082b4c37": "A_{21}\\in \\mathbb {R} ^{(n-1)}", "26797144bdcd6c58ce21907642dc9aaf": "0=t_{0}", "26803535adfc0ccb2717ea148d80a1f1": "\\ ax^{2}+bx+c=0", "268040e2f4f91da62ada109f02b8690e": "N_{R_{3}}", "26804af7f94f530583605ba8f6f34298": "a_{0}x_{0}+\\ldots +a_{n}x_{n}=0", "268053fda3c06027a1727ed3bad40129": "H_{n}(M,M\\setminus \\{p\\};\\mathbb {Z} )", "268066e471c5bbf7fc3eee31d5284f4f": "u>0", "2680b78fa14ef53189341c7647c8aca8": "{\\hat {\\mu }}", "2680f716949e39bccf78df6bf345f68e": "x(t)\\in \\mathbb {R} ^{n}", "2681e7877ba15c04280d0d864386e16f": "\\sigma \\in \\Gamma _{m},\\,\\!", "268218361b4c7c68e71482d88f745745": "p_{i}=q_{i}", "26822f6702c369ee454256fe4b9e3515": "\\lambda :B\\rightarrow {\\mbox{End}}\\,A_{B},\\ \\lambda (b)(a)=ba", "268260d6ac67f93e8d9562524b86d16f": "f:X_{1}\\rightarrow X_{2}\\,", "26826df017dfdfb620e9f946b73599ff": "r\\to \\infty ", "26828d204611db6ec7b39b07f84ebd81": "\\lnot (A\\vee B)", "26829e29c2f429b82e3d73743bf4f162": "a,b\\geq 0.\\,", "2682beb0f7eb9b3f8bed0612a1868a3e": "d\\phi :{\\mathfrak {g}}\\to {\\mathfrak {gl}}(V)", "2682ea9b359d6d66b32b9b4f3820c548": "n\\sin \\alpha ", "268354384b548acee2f4aa185c9e0709": "w_{2^{i}}", "26842c97d9db3b5d24e7e649f3da4182": "\\lambda =2\\pi d", "2684306b928eb1de2297a464b6ce64d8": "L_{ij}=S_{ij}+X_{i}P_{j}-X_{j}P_{i}", "268439c1c4bb5b17015bd276de464a11": "p\\rightarrow q", "26846903055b0502be5302302e622f98": "\\varepsilon =\\omega ^{\\varepsilon }", "2684e73f5b0f1c54849b4853cbf8ce79": "{\\boldsymbol {lb}}\\leq {\\boldsymbol {\\beta }}\\leq {\\boldsymbol {ub}}", "2684f315468882817a189207c4b7d653": "n={\\frac {ck}{\\omega }}", "26852d19627b23cfe7c23d5844e50cc3": "T=\\{x_{i},y_{i}\\}", "26857864600298db48bd1c8beb318aa1": "A,C,n", "268582d85d37b7b852bbde44dd4f42c3": "\\alpha \\in \\mathbb {F} _{q^{k}}-\\{0\\}", "26860f75d6e16b71372774d5ddaf0b4f": "\\mu (\\phi )={\\displaystyle {\\frac {\\pi }{2}}{\\frac {m(\\phi )}{m_{p}}}}\\,\\!", "26867ea512c4bcf9d97fbda362dd3233": "1\\leq i0,\\quad ax>0{\\mbox{)}}\\\\-{\\frac {2}{\\sqrt {b}}}\\mathrm {artanh} \\left({\\frac {S}{\\sqrt {b}}}\\right)&{\\mbox{(for }}b>0,\\quad ax<0{\\mbox{)}}\\\\{\\frac {2}{\\sqrt {-b}}}\\arctan \\left({\\frac {S}{\\sqrt {-b}}}\\right)&{\\mbox{(for }}b<0{\\mbox{)}}\\\\\\end{cases}}", "26a192124d2b772107e72bf7deb069a3": "P\\oint _{\\mathrm {surface} }\\mathbf {q} \\cdot d\\mathbf {S} =P\\int _{\\mathrm {volume} }\\left(\\nabla \\cdot \\mathbf {q} \\right)dV=3PV,", "26a1f4dc9c611cfad9a1234fb1dca3d3": "{A}_{8}^{(2)}", "26a23e2b73be519406b32d5c398d4fc6": "{\\mathcal {I}}\\left(\\beta ,\\theta \\right)={\\text{diag}}\\left({\\mathcal {I}}\\left(\\beta \\right),{\\mathcal {I}}\\left(\\theta \\right)\\right)", "26a26e5369baf000281c72fc6ddaa045": "{\\mathcal {O}}(m),\\,", "26a28b8255a8329189dd4c4467c0f5a9": "y^{2}=x^{2}(a^{2}-x^{2})", "26a2a2b0f4e538fd17fbaa7b169c11d2": "R\\rightarrow \\infty ", "26a2a4f695952e6bf6fbc1cc60a45490": "Sc=\\nu /D_{AB}=1", "26a2df4e86602be51adbbe82ac3b1a4a": "C\\subset \\langle F\\rangle ", "26a3121b682435a5e492c652ce4223ab": "x_{\\mathrm {FOH} }(t)\\,", "26a364d355ba4c16768a318817d53feb": "d\\leq r8\\!", "26ab4749b4dafd87fd0902093e560a65": "\\scriptstyle {\\vec {F}}", "26ab61fa8024f78da62a405a1e574d8f": "\\Delta ^{n}", "26ab7b049445e8aa150a013929849c58": "(3,{\\bar {3}},1)", "26abbd3b085205a290dadc2d700b5e18": "\\operatorname {recc} (A)\\cap \\operatorname {recc} (B)", "26abc361ab9c3bf557e489b770982d35": "M(t)", "26abedbeb8af48df427a153933f6498e": "\\exp[ik(x-v_{ph}t)-\\gamma t]", "26ac10be0d4e70ad04c8951000575476": "a_{12}={\\frac {1}{x_{1}-x_{0}}}", "26ac99baf405f378717596eecc8ac783": "P", "26ac9eece1cf4b16d04117ed135de65e": "K\\subseteq \\alpha /A\\leftrightarrow \\exists H:\\exists J:[\\langle H,J,K\\rangle \\in R_{1}\\wedge H\\subseteq B\\wedge J\\subseteq \\alpha /B].", "26acb6a2c3ee11b18e0fb4f2926790d8": "{1 \\over 2(1-p^{-2})(1-p^{-4})\\cdots (1-p^{2-2t})(1-p^{-t})}", "26acd95bf33dfe594ddf1e16892519f9": "{SU(2)_{L}\\times SU(2)_{R}\\times U(1)_{B-L} \\over \\mathbb {Z} _{2}}.", "26ad5a81cae7043365de5f4906a7be8e": "K_{p,q}", "26adbc57d90f6c048126a8c06dd7b3e6": "\\!{\\mathcal {A}}\\models _{X}^{-}\\phi ", "26ae260615c61bbe6e0783851926a0c5": "{\\frac {1-6pq}{pq}}", "26ae4241f36eeeea983277b3e9c91535": "\\phi (\\{\\mathbf {r} _{i}\\},t+T)=\\phi (\\{\\mathbf {r} _{i}\\},t).", "26aeb065deae5a41ef3bf16a6681e88e": "P(A_{K}/A_{L})=\\prod _{j=1}^{n}(a_{Kj}/a_{Lj})^{w_{j}},{\\text{ for }}K,L=1,2,3,\\dots ,m.", "26aec6374089dff58d82ee0b3bd97a64": "\\operatorname {gr} _{I}R=\\oplus _{0}^{\\infty }I^{n}/I^{n+1}", "26af2e336c085d2dbf29088557efc483": "\\mathrm {action\\_delta} \\times \\mathrm {mk\\_max\\_speed} \\times \\left({\\frac {i}{\\mathrm {mk\\_time\\_to\\_max} }}\\right)^{\\frac {1000+\\mathrm {mk\\_curve} }{1000}}", "26af694d60b9f98179dd9b880d8c34a3": "\\ (2f_{a}-f_{b}),(2f_{a}-f_{c}),(2f_{b}-f_{a}),(2f_{b}-f_{c}),(2f_{c}-f_{a}),(2f_{c}-f_{b})", "26afca5530aad74bd80a8f10ab64fda7": "\\psi ={\\frac {1}{\\sqrt {2}}}(1,1),", "26aff2a6e27e70966ab6951a4191f2f3": "u(t,r)={\\frac {1}{r}}\\left[F(r-ct)+G(r+ct)\\right],", "26b018af791aa77b9a7ff0c645ab1d48": "\\Gamma ,{\\mathcal {A}},{\\mathcal {B}},\\Delta \\vdash {\\mathcal {C}}", "26b056081d50fb5e264ced364aa29190": "g(q_{i},p_{j})", "26b09a1294c21a17d6315956e25fa39c": "f_{p}\\leq 2B\\,", "26b0eba40be66dbcf3bd0f6e6928dfde": "\\int \\csc ^{2}x\\,dx=-\\cot x+C", "26b118de71be211ee4d6d1c1041c0d1b": "s={\\frac {(a_{1}b_{2}-a_{2}b_{1})^{2}}{(a_{1}^{2}+b_{1}^{2})(a_{2}^{2}+b_{2}^{2})}}.\\,", "26b171eca8dfdbeb0e9bc9ec52779160": "\\mathbf {[T]} ={\\begin{bmatrix}1&0\\cdots 0\\\\T_{21}&T_{22}\\cdots T_{2n}\\\\\\cdot &\\cdots \\\\T_{n1}&T_{n2}\\cdots T_{nn}\\end{bmatrix}}", "26b17225b626fb9238849fd60eabdf60": "+", "26b18a8e53fae3d6181ef230c56c1896": "ns^{n}", "26b19c07bd6bb27371bdfdd34f78a24b": "\\mathbf {e} _{1}(t),\\ldots ,\\mathbf {e} _{n}(t)", "26b1bd36dc827f561251174c04d194ee": "|x-y|^{2}=(x-y)\\cdot ({\\overline {x}}-{\\overline {y}})=(x-y)\\cdot \\left({1 \\over x}-{1 \\over y}\\right)=-{(x-y)^{2} \\over {xy}}\\ .", "26b21b408b4a25089933eb08c700a95b": "0\\leq y\\leq \\pi ,y\\neq {\\frac {\\pi }{2}}\\,", "26b259fb0497216f29e0eacc0d84900f": "f(x_{1},\\ldots ,x_{n})/\\alpha ", "26b262ed7e40e133b188b43e298f0a22": "p(t)=\\int _{0}^{\\infty }h(\\tau )q(t-\\tau )\\,d\\tau ", "26b267d0c8457ca611c2ad1479263e9d": "f_{X,Y}(x,y)", "26b27098b70ac6082624063de3b23859": "y_{i}=RC\\,\\left({\\frac {x_{i}-x_{i-1}}{\\Delta _{T}}}-{\\frac {y_{i}-y_{i-1}}{\\Delta _{T}}}\\right)", "26b2846a2e8c4c2b2a1d464f08df13db": "{\\frac {dQ^{D}/Q^{D}}{dP^{D}/P^{D}}}", "26b2883c80bf87e565acc1c9830cf00d": "{\\frac {d}{dx}}u(x)+\\int _{x_{0}}^{x}f(t,u(t))\\,dt=g(x,u(x)),\\qquad u(x_{0})=u_{0},\\qquad x_{0}\\geq 0.", "26b33544488bd972dd8a3eb29f0f3b24": "g_{o}", "26b365d36b8467351ea01f96fac79cbe": "T\\rightarrow 0,\\ \\ e^{-K},e^{-L}\\rightarrow 0", "26b3a09059d77ae3e8307d7c536e14f4": "\\Delta {v_{i}}", "26b43a02bd1f0533f8442312cf7010ea": "(X_{i},Y_{i},)", "26b450466710ed7cb561c21e9f334c44": "\\gamma _{i}(t)", "26b478285a574ae2d540e73a99358d20": "P={\\begin{bmatrix}0&0&1/2&0&1/2\\\\0&0&1&0&0\\\\1/4&1/4&0&1/4&1/4\\\\0&0&1/2&0&1/2\\\\0&0&0&0&1\\end{bmatrix}}.", "26b48baa6b7e307ff5452508c507be82": "h=\\left(q+\\left\\lfloor {\\frac {13(m+1)}{5}}\\right\\rfloor +K+\\left\\lfloor {\\frac {K}{4}}\\right\\rfloor +\\left\\lfloor {\\frac {J}{4}}\\right\\rfloor +5J\\right)\\mod 7,", "26b520ac7006c593761e51d463332ae9": "(\\xi _{\\nu })_{\\nu =1}^{k}", "26b543f629b812602f41558e8d2530d6": "Var(X)=V(\\mu )=\\mu ", "26b5566703d20e9c19598f1cdc1a8841": "|\\psi _{E}\\rangle =l|0_{E}\\rangle +m|1_{E}\\rangle ,\\quad l,m\\in \\mathbb {C} .", "26b568e4192a164d5b3eacdbd632bc2e": "kp", "26b57ccb4491dcbaf736d40cc5e859be": "{\\gamma _{\\alpha \\beta }}^{\\chi }=-{\\gamma _{\\beta \\alpha }}^{\\chi }", "26b5e39b6db5357b35357301250234d1": "A\\otimes _{B}\\cdots \\otimes _{B}A", "26b5e64f4e0e9123340a1e229e22ef65": "{\\mathcal {L}}_{j}", "26b62d6c9f347c44b7bf44aac3da8695": "x\\in {\\mathcal {O}}_{L}", "26b66a702fedc65f0c3454b7687c20d3": "3^{3^{3}}+3", "26b69c66794bc0cdaba6a53212fb7d13": "{\\frac {1}{4}}S_{0}(1-\\alpha _{p})+\\epsilon \\sigma T_{a}^{4}-\\sigma T_{s}^{4}=0", "26b6b931b6c258428e315810692e400f": "\\delta -1", "26b6d603363104a2d0ebc1f11bc2f845": "\\quad \\partial _{\\eta }=t^{\\gamma /2}\\partial _{y}.", "26b6f64c665c370db1ddfb73093401dc": "(x_{2}-x_{1})", "26b7b741c50febab49ae56981f53546a": "\\Omega Y", "26b7c6ee3e38c6d0e153032ca062358d": "\\displaystyle E_{A}=K_{1}\\sin ^{2}\\theta +K_{2}\\sin ^{4}\\theta +K_{t}\\sin ^{3}\\theta \\cos \\theta \\sin 3\\phi ", "26b7c897c6ffe20104b64da201ab6344": "TK_{R}'^{}", "26b7d778161ca7bf181776a786f50b71": "a_{i}^{T}x\\leq b_{i}", "26b7ff56473b53ec24426e699af78654": "x_{1}.\\exists x_{2}.R(x_{2})", "26b8131e3c3b5e4ed9afd0e3bf1aedb0": "\\tau _{i}^{2}=1", "26b82bc76d3d4fd7a9213b88d9be7033": "\\scriptstyle [m,\\,1.82m]", "26b87bf5f94e3f4fee01d77f111d86cb": "\\pi _{1}(X)=\\mathbb {Z} ^{n}", "26b88c4ab183ea019a6adde46ead0335": "M_{KL}", "26b8b493123418f571776ca5070ccd02": "p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\\cdots +a_{2}x^{2}+a_{1}x+a_{0}", "26b993b6f84a9de4db810dc30b912578": "\\mathrm {T} ", "26b99d3cb8a610400af9f885724c959d": "(\\mathbf {v} \\cdot \\nabla )\\,\\mathbf {v} ", "26b9eb8c2784c194bb8c61fb925183e9": "f(x)=\\int _{\\mathbb {R} ^{n}}\\int _{\\mathbb {R} ^{n}}e^{2\\pi i(x-y)\\cdot \\xi }\\,f(y)\\,dy\\,d\\xi .", "26b9f13e24da67ea84cbb44c13336024": "\\cap W=\\cap _{S\\in W}S=\\emptyset ", "26ba0cd4dd68d1717bdeb32504c0a3b2": "\\langle \\mathbf {No} ,\\mathrm {<} ,b\\rangle ", "26ba123829294a9d299e0939621ba0f5": "p_{\\alpha ,\\beta }(\\varphi )=\\sup _{x\\in \\mathbf {R} ^{n}}|x^{\\alpha }D^{\\beta }\\varphi (x)|", "26ba39b2f3aaead20e454e3629e8c97a": "H=-c_{0}-2\\sum _{n=1}^{\\infty }c_{n}\\phi _{n}", "26ba4ce0c3c27bb14b5edf62e1c8b2cc": "D_{E}+H_{E}\\approx H_{E};", "26baa2666f336904159f50f42c0a8f52": "p={\\frac {2\\pi r_{A}}{N_{A}}}.", "26bc16dd3f32eb059837c61c14ff24e4": "p_{\\mathrm {tot} }=p_{\\mathrm {rad} }+p_{\\mathrm {gas} }=\\rho c_{\\rm {s}}^{2}", "26bc1d47df866aaadd42a8e018c9730f": "-2\\sum _{i}r_{i}{\\frac {\\partial f(x_{i},{\\boldsymbol {\\beta }})}{\\partial \\beta _{j}}}=0,\\ j=1,\\ldots ,m", "26bc32c6863d0276dcd5fb12fd678378": "\\sum _{p}{\\frac {1}{p-1}}={{\\frac {1}{3}}+{\\frac {1}{7}}+{\\frac {1}{8}}+{\\frac {1}{15}}+{\\frac {1}{24}}+{\\frac {1}{26}}+{\\frac {1}{31}}}+\\cdots =1.", "26bc46534ac19146f9fe821a78434e36": "{\\frac {d\\theta }{dt}}=-\\alpha _{\\mathrm {max} }\\left[\\sin 2\\theta -\\sin 2\\theta _{\\mathrm {eq} }\\right]", "26bc4f3bbbcfbe4272ef19f160f3ce62": "{\\sqrt {\\frac {\\gamma \\pi }{2}}}", "26bc5477295cb67ac2ac6ac7debe3a0b": "\\Omega _{1}=\\{\\lnot \\},", "26bc7511d65f1a966d2f435c51b4aa70": "{\\dot {y}}={\\frac {\\partial y}{\\partial x_{j}}}{\\dot {x}}_{j}+{\\tfrac {1}{2}}{\\frac {\\partial ^{2}y}{\\partial x_{k}\\,\\partial x_{l}}}g_{km}g_{ml}.", "26bc8b6679140e0c85541070555f194c": "Q_{yz}Q_{xz}Q_{xy}Q={\\begin{bmatrix}1&0&0\\\\0&1&0\\\\0&0&1\\end{bmatrix}},", "26bcdb5fa720fc4c43f0e2737252631a": "\\left\\{{n \\atop K_{n}}\\right\\}\\geq \\left\\{{n \\atop K_{n}+1}\\right\\}>\\cdots >\\left\\{{n \\atop n}\\right\\}.", "26bd14b18ea311b141277f8f308a2c54": "v_{n1}=v_{n2}=0", "26bd2a2b77f032d99c0ee4da15cc17ae": "\\,y=\\sum _{j}w_{j}x_{j},", "26bd2a9995658194c0b0ff8697fd7abd": "f_{y}(x)=f(x,y)", "26bd4c6ff77b543f7b6334eb700114e3": "{\\frac {6(90-\\pi ^{4})}{5(\\pi ^{2}-6)^{2}}}\\,", "26bd563a67f2fa50d5d5e1771cf6e730": "\\arccos(-{\\frac {7}{11}})", "26bd8a80ec13a31bffa6f99ad69dec36": "{\\frac {\\theta \\vdash \\phi \\quad \\theta \\vdash \\psi }{\\theta \\vdash \\phi \\wedge \\psi }}", "26bdd98bd26985cabb09e265fc66654a": "T_{\\parallel }", "26be0f3b464716780a48233cff7f468e": "\\Phi (\\operatorname {probit} (p))=p", "26be3afe504e0e33ee08871209b348a9": "2\\alpha =2\\zeta \\omega _{0}={\\frac {\\omega _{0}}{Q}}={\\frac {C_{1}+C_{2}}{R_{2}C_{1}C_{2}}}.", "26be536dfe382a1923e9dd120d99e314": "x_{i,0}", "26beac9e817f135acec7c603ddbeb111": "\\psi ={\\begin{pmatrix}\\psi _{\\uparrow }\\\\\\psi _{\\downarrow }\\end{pmatrix}}", "26bf421cf1549e80620dc23654201bf0": "V_{2n}=V_{n}^{2}-2Q^{n}\\,", "26bf4a980b9ba1af8415680853c3e09f": "(I_{r},G)=FP", "26bf63c8e5ff3fff07ed47a54dc54b55": "H_{5}(a,b)=a\\uparrow \\uparrow \\uparrow {b}\\,\\!,", "26bf6ea4b6a9fc3a70db9a887f75b205": "\\pi /4=\\left(\\prod _{p\\equiv 1{\\pmod {4}}}{\\frac {p}{p-1}}\\right)\\cdot \\left(\\prod _{p\\equiv 3{\\pmod {4}}}{\\frac {p}{p+1}}\\right)={\\frac {3}{4}}\\cdot {\\frac {5}{4}}\\cdot {\\frac {7}{8}}\\cdot {\\frac {11}{12}}\\cdot {\\frac {13}{12}}\\cdot {\\frac {17}{16}}\\cdots ", "26bf879162c8f396c9f96602acd7c584": "f:V\\to S^{3}", "26bfb06404127074c87aed13b12d5dbb": "L^{2}=\\Delta \\xi ^{2}+\\Delta \\eta ^{2}+\\Delta \\zeta ^{2}.", "26bfb88e25141378608867135b6736af": "i\\,=\\,1,2,\\dots ,n.", "26bff523176aaff5a81cfbe4a79a9dd0": "n/(n+1)", "26c02addc6319fd035ca4416674da534": "\\varphi ={\\frac {1+{\\sqrt {5}}}{2}}", "26c04c4f8decaf9cfc60bcdee12a8a3b": "[D(d)\\wedge \\neg D(f(d))]", "26c1c2fa95c6ba5eb3b7085295c084c0": "Q=I_{3L}+I_{3R}+{\\frac {B-L}{2}}", "26c1d2bcd06cb14b59f06851520c3f00": "(X,d,\\mu )", "26c1e30789bc8e8dd563c56f177de79b": "(a\\uparrow ^{n})^{k}b", "26c1f98a352eab1db56ec6532992421a": "dM={\\frac {2\\pi R^{2}\\sin \\theta }{4\\pi R^{2}}}M\\,d\\theta =\\textstyle {\\frac {1}{2}}M\\sin \\theta \\,d\\theta ", "26c2338890453f4d7a04a8353fc88bbb": "\\mathbf {B'} ", "26c2454298e583986270da432b139c8e": "\\mathbf {f} \\ ", "26c25d4ae008288393f7a9264c554dc2": "{\\frac {1}{2}}\\rho W^{2}NcC_{x}=4\\pi \\rho \\left[(a'\\Omega r)^{2}+U_{\\infty }^{2}a(1-a)\\right]r", "26c273d3a925e42cfcb4f11157f63f88": "\\mathrm {tr} (A)=\\log(\\det(\\exp(A))).\\,", "26c2be9e04e04af3e0e4cc804f9c9a9d": "g(x)=\\sum _{k}c_{n,k}f_{n,k}^{(0)}(x)", "26c2d23bef79e0e9febe6a4854522327": "\\,{\\begin{aligned}\\Gamma (x+1)&=\\lim _{n\\rightarrow \\infty }x\\cdot \\left({\\frac {n^{x}n!}{(x+n)(x+n-1)\\cdots (x+1)x}}\\right){\\frac {n}{n+x+1}}\\\\\\Gamma (x)&=\\left({\\frac {1}{x}}\\right)\\Gamma (x+1)\\end{aligned}}\\,", "26c2f26fda59f51040cbdf34c5ac0279": "Y_{CPE}=Q_{0}(\\omega i)^{n}", "26c30fb1f5f05ea0ec7e530e1633c014": "r={{\\ell ^{2}} \\over {m^{2}\\gamma }}{{1} \\over {1-e}}", "26c3160169bcaf62013f30ff9977aefd": "{\\frac {w}{(w-w')}}", "26c329ef8376fca1edddae81491cce2d": "-w\\ ", "26c36aab2464d1ef4f7436c3d7254c0e": "=z^{2}-(i{\\sqrt {5}})^{2}", "26c373554caa28c5cd630ef6e528bc38": "{\\frac {V_{in}}{V_{A}}}\\sim {\\frac {1}{S^{1/2}}}", "26c3c623197f1fd0789a626dbdc02730": "Q=Q_{xy}^{-1}Q_{xz}^{-1}Q_{yz}^{-1}.", "26c3fa9aaea37b9938d6dc02286bccf5": "\\geq 1-\\epsilon ,", "26c4e812364ebac03a62da6aea219421": "A_{2n-1}\\to C_{n}", "26c4f5da45981859f4b0ba9f29b999cb": "\\langle \\phi (k_{1})\\phi (k_{2})...\\phi (k_{n})\\rangle ", "26c4fb5180ecff0561c5bb3d2f1034a1": "\\ MRS_{xy}\\geq 0", "26c51f87b6a18b74e3c2841d550dfa3d": "|V_{cb}|", "26c58301a4757c7072b1df5f508573f6": "\\mathbf {{\\hat {f}}_{0:t-1}} ", "26c5b3ed8e42984f507a6220bd975d31": "{\\mathbf {A} }({\\mathbf {r} })={\\frac {\\mu _{0}}{4\\pi }}{\\frac {{\\mathbf {m} }\\times {\\mathbf {r} }}{r^{3}}},", "26c5c8d27426796c1f4fd7757c8b30ff": "\\,I_{k}=1", "26c5e28d191a17bf1884c7d2a53e9d8a": "E(v)=0", "26c5ebf6dcfc09ec6b90519d2019a7b9": "F(2)+F(5)", "26c660287a296abc741bcf6f516c0f5e": "\\log {C_{t}}=\\log {C_{w}}+m\\log {\\phi }\\,\\!", "26c6873692b59f40cac54c1cb920b74d": "[\\partial _{\\nu }[\\partial _{\\sigma }V_{\\nu }]-\\Gamma ^{\\rho }{}_{\\sigma \\nu }\\partial _{\\sigma }V_{\\rho }-\\Gamma ^{\\rho }{}_{\\mu \\nu }\\partial _{\\mu }V_{\\rho }-\\Gamma ^{\\rho }{}_{\\mu \\sigma }\\partial _{\\rho }V_{\\nu }]-[\\partial _{\\mu }[\\Gamma ^{\\rho }{}_{\\sigma \\nu }V_{\\rho }]-\\Gamma ^{\\alpha }{}_{\\mu \\nu }\\Gamma ^{\\rho }{}_{\\alpha \\sigma }V_{\\rho }-\\Gamma ^{\\alpha }{}_{\\mu \\sigma }\\Gamma ^{\\rho }{}_{\\alpha \\nu }V_{\\rho }]", "26c69c8d54faf7a5e80f28c019bbd8dd": "\\,A+iB=[i\\phi _{1}-\\phi _{2}]+[A_{1}+iB_{1}]\\mathbf {i} +[A_{2}+iB_{2}]\\mathbf {j} +[A_{3}+iB_{3}]\\mathbf {k} \\quad ", "26c7e7dab645083b9bd006831a24c706": "{\\text{failed}}", "26c7ef913dfaa7cd923dbe81b7fe310b": "C_{xy}\\leq 1", "26c82ace22ee147093712760c9b8cc82": "\\phi _{j}^{-}", "26c833bc04fb96df970d8559a5772984": "k_{\\rm {C}}=k_{1}=k_{\\rm {E}}", "26c8461f3396e3b5d7e975b1cdb66063": "\\beta \\,\\,\\,\\approx \\,\\,\\,{{3\\,k} \\over {\\mu _{T}^{2}}}\\,\\left({{\\sigma _{T}} \\over {\\mu _{T}}}\\right)^{2}\\,\\,\\,\\approx \\,\\,\\,30\\,\\,\\left({{\\sigma _{T}} \\over {\\mu _{T}}}\\right)^{2}{\\mathbf {\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,Eq(19)} }", "26c84cdf3743e8221c058e68be6d64eb": "\\int \\!\\int _{D}g(x,y)\\,\\mathrm {d} x\\,\\mathrm {d} y=\\int _{0}^{R}z\\left(\\int _{0}^{2\\pi }g({\\mathit {xc}}+z\\cos t,{\\mathit {yc}}+z\\sin t)\\,\\mathrm {d} t\\right)\\,\\mathrm {d} z", "26c86b4a0b5f3990941fde4859a2d25a": "\\triangle ADC", "26c8abd4645b3f81caf2153c293f3a5a": "f(x_{1},a,g(z_{1}),y_{1})", "26c924f41e6624aeae0d090149e67db5": "I_{R}\\leq RM_{R}\\int _{0}^{\\pi }e^{-aR\\sin \\theta }\\,d\\theta =2RM_{R}\\int _{0}^{\\pi /2}e^{-aR\\sin \\theta }\\,d\\theta \\,.", "26c96c0e80f62091460174baee9f7e6f": "f^{n}-g^{n}=d_{B}^{n-1}h^{n}+h^{n+1}d_{A}^{n},", "26c98462beeb585d0b3f172a6590a218": "c_{ijk}", "26c9b915984f83831ec787c7c58b8c48": "M\\succeq 0", "26c9ccafebf2eddd3146fd187d7269ee": "X^{L,R}", "26c9d04dbe5b72fae1f62238cde54372": "B_{n}=-\\sum _{k=1}^{n+1}{\\frac {(-1)^{k}}{k}}{\\binom {n+1}{k}}\\sum _{j=1}^{k}j^{n}", "26c9d13539db28ab2728ed8411e1b1fb": "\\Delta A\\,\\Delta B\\geq {\\frac {1}{2}}{\\sqrt {\\left|\\left\\langle \\left[{A},{B}\\right]\\right\\rangle \\right|^{2}+\\left|\\left\\langle \\left\\{A-\\langle A\\rangle ,B-\\langle B\\rangle \\right\\}\\right\\rangle \\right|^{2}}},", "26ca11ec1b23655f704c7d7e416aec98": "G=1-2\\left(\\int _{0}^{1}L(F)dF\\right)={\\frac {1}{2\\alpha -1}}", "26ca6d2e27d6a9257723c40e719d666c": "{\\begin{aligned}V_{n}(R)&=(2\\pi )V_{n-2}(R)\\cdot \\left(-{\\frac {R^{2}}{n}}(1-(r/R)^{2})^{n/2}\\right){\\bigg |}_{r=0}^{r=R}\\\\&={\\frac {2\\pi R^{2}}{n}}V_{n-2}(R),\\end{aligned}}", "26ca710c9dfcd9f91f257f0461f3a91b": "\\{1,x^{s_{1}},x^{s_{2}},x^{s_{3}},\\dots \\}\\,", "26cab0e72cc514a66a509d0adbad194c": "c_{i}={\\frac {c_{i,T_{0}}}{(1+\\alpha \\cdot \\Delta T)}}", "26cab7402a83eb4790ba2b807b1879ce": "\\scriptstyle S^{1}\\times I", "26cac5fc2feaa56340efdfa429cc4cb1": "b\\in \\mathbb {R} ^{m}", "26cacd725a4e95b6e717929764a27e50": "A=sd.\\,", "26cb488b84bccd0e8277e1b66456c0f6": "{\\frac {e^{\\mu z}\\gamma ^{\\lambda }}{({\\sqrt {\\alpha ^{2}-(\\beta +z)^{2}}})^{\\lambda }}}{\\frac {K_{\\lambda }(\\delta {\\sqrt {\\alpha ^{2}-(\\beta +z)^{2}}})}{K_{\\lambda }(\\delta \\gamma )}}", "26cbc9fec2e36b71cf6b0cda07107566": "\\kappa ^{-1}R^{2}", "26cbcef7525376da91e891f75ee6357b": "\\eta (z)=\\sum _{n=1}^{\\infty }\\chi (n)\\exp({\\tfrac {1}{12}}\\pi in^{2}z),", "26cbf6ff44380f11896307ab788d49ed": "{\\frac {p}{1+e}}", "26cc56dfd078a74cf504b62440f829d0": "J_{\\nu }(z)=\\sum _{k=0}(-1)^{k}{\\frac {z^{k}}{k!}}I_{\\nu +k}(z);", "26cc67f35fa2131b6387e660c283a103": "\\Omega ^{1}={\\mathbb {C}}.{\\rm {d}}x,\\quad ({\\rm {d}}x)f(x)=f(x+\\lambda )({\\rm {d}}x),\\quad {\\rm {d}}f={f(x+\\lambda )-f(x) \\over \\lambda }{\\rm {d}}x", "26cc860955c9d73a7e0b449459fa79ae": "k={\\frac {2\\pi }{\\lambda }}={\\frac {2\\pi f}{v}}={\\frac {\\omega }{v}},", "26cc864f03d3833569b0ffeeee578ef0": "\\bigvee \\varnothing =0", "26cc89ff187186bcfacb69de349e397b": "w(\\mathbf {R} )", "26ccb135f4ce36c2013e6a6ee083bb37": "-0.7,", "26cd3d4129c87f3cabb842b12d3be904": "W(L,t)", "26cd7d5932f7bf04540ea95da3877be4": "\\Phi _{3}(x)=x^{2}+x+1", "26cd813845709cc1954aef23d6038bbb": "n_{\\mathrm {A} }=n_{\\mathrm {A*} }{\\frac {R_{\\mathrm {A*} }-R_{\\mathrm {A*B} }}{R_{\\mathrm {A*B} }-R_{\\mathrm {B} }}}\\times {\\frac {R_{\\mathrm {B} }-R_{\\mathrm {AB} }}{R_{\\mathrm {AB} }-R_{\\mathrm {A} }}}", "26cdb8feaa3c5ba862ba03289b2571fc": "{\\begin{aligned}x_{1}(t)&=\\log {\\frac {(\\lambda _{1}-\\lambda _{2})^{2}(\\lambda _{1}-\\lambda _{3})^{2}(\\lambda _{2}-\\lambda _{3})^{2}a_{1}a_{2}a_{3}}{\\sum _{j0\\!", "26d27c7fadca9cbd95de96a0880130e4": "{\\bar {\\phi }}_{{\\bar {x}}{\\bar {x}}}+{\\bar {\\phi }}_{{\\bar {y}}{\\bar {y}}}+{\\bar {\\phi }}_{{\\bar {z}}{\\bar {z}}}=0", "26d2be6c9e8375aea57694a9cdf89831": "\\lambda _{f}\\otimes \\cdots \\otimes 1+\\sum _{i=2}^{n}1\\otimes \\cdots \\otimes \\triangle _{i}\\otimes \\cdots \\otimes 1,", "26d2f491715729c3072cc8a7fd245173": "\\mathbf {C} ={\\frac {\\mathbf {b} \\times \\mathbf {d} -\\mathbf {b} \\times (\\mathbf {b} \\times \\mathbf {d} )}{2\\mathbf {b} \\cdot \\mathbf {b} }}.", "26d30a2131b371f4322b9bd6723145e3": "H(f)=(x_{i+1}\\partial f/\\partial x_{i+1}-x_{i-1}\\partial f/\\partial x_{i-1})x_{i}\\partial /\\partial x_{i}+(y_{i-1}\\partial f/\\partial y_{i-1}-y_{i+1}\\partial f/\\partial y_{i+1})y_{i}\\partial /\\partial y_{i}", "26d34f122c7d90a3b9b52a8fe00c7c92": "X\\times I", "26d3619925e090880f2e99d7da74f696": "A\\subseteq B\\implies B\\subseteq B^{*}\\subseteq A^{*}", "26d37a8a294d396b98d9f1cfec827623": "{\\begin{aligned}{\\underline {\\int _{a}^{b}}}cf(x)&=c{\\overline {\\int _{a}^{b}}}f(x)\\\\{\\overline {\\int _{a}^{b}}}cf(x)&=c{\\underline {\\int _{a}^{b}}}f(x)\\end{aligned}}", "26d3c40a1487faf12fe2b7fae709a426": "{\\sqrt {6}}", "26d3db983b439aa3b433b487a41999b8": "\\mathrm {E} (T)=-{\\frac {1}{\\theta }},\\quad \\mathrm {var} (T)={\\frac {1}{\\theta ^{2}}}", "26d448d6cc5261500f4ae877a1fd27c9": "y=-x-1", "26d46389ba7ae04c0fa330e7dd7f88fc": "\\sigma _{y}=\\sigma _{y,0}+{k \\over {d^{x}}}", "26d4795ee1f02913f0dd9c0f4a0a68f5": "(h=2\\,r)", "26d47f8115ac4e9c767b6961ad3fdad0": "g_{j+1}[n]", "26d4848460d3d1e96ed7052c1ae9b421": "=\\zeta (s+1)\\left(n+{\\tfrac {1}{2}}+{\\frac {U_{2}(n)}{2^{s+1}}}+{\\frac {U_{3}(n)}{3^{s+1}}}+{\\frac {U_{4}(n)}{4^{s+1}}}+\\dots \\right)-{\\tfrac {1}{2}}\\zeta (s),", "26d49e1d13f868ac8cb28218942077a7": "\\phi (x)", "26d4ce98ec92e12a3f71a50aa6b3978e": "d(u,v)\\leq 1,", "26d4dd4dcc849e48edffbc71442271de": "\\ G_{f}(V,E_{f})", "26d4e0bfb74c9e35e3da2bd2c88b8235": "P(1)", "26d5074361e2c4b2188df4fa71f51625": "\\lim _{n\\rightarrow \\pm \\infty }f^{n}(x)=p.", "26d5579325275b5591a648f133dc2805": "Q\\;=\\;C\\;A\\;{\\sqrt {\\;2\\;\\rho \\;P\\;{\\bigg (}{\\frac {k}{k-1}}{\\bigg )}{\\Bigg [}\\,{\\bigg (}{\\frac {\\;P_{A}}{P}}{\\bigg )}^{2/k}-\\;\\,{\\bigg (}{\\frac {\\;P_{A}}{P}}{\\bigg )}^{(k+1)/k}\\;{\\Bigg ]}}}", "26d55e289e10e467792f12ed2ece04ad": "p(1-p)\\,", "26d57c4ab1ccef5f55f53712d50ae45f": "rm=0", "26d5cf1de3b457150077701d163bcaca": "x,t", "26d60f2701dae40fe225d043c1197f3d": "k_{2(i)}", "26d6109507c3f0ab30dfe509458ad8ec": "T_{x}f(C)\\subseteq T_{x}X", "26d615700ddecfc9c12815300cf35b5a": "\\Phi .", "26d63478e01213a317db1123c14d2759": "y=f(u)", "26d727f0c595159b3cbe27eaf91ee461": "\\mathbf {P} (t)=[A(t)]\\mathbf {p} +\\mathbf {d} (t).", "26d736f90c5c8777f42c5ef00164ab33": "B_{S}=\\{U\\cap S:U\\in B\\}", "26d7886bece13f655ba16f1b580ccee4": "{\\hat {\\mathbf {x} }}_{1}\\cdot {\\hat {\\mathbf {x} }}_{2}=0", "26d7a2da120e72d0bef2f7ee4ffed5a8": "K(X_{i}\\mid \\pi _{i})", "26d82089ceaf9fdcae25102efdbe2ec8": "(1-B^{-P})(B^{U+1})", "26d8388f3b948b6afd966d17039c60d6": "{\\mathcal {E}}(\\rho _{AB})={\\mathcal {S}}(\\rho _{A})={\\mathcal {S}}(\\rho _{B})", "26d8432ab80efedc6705a4391a640bd4": "r_{n}\\leq x0\\ \\mathrm {and} \\ \\det \\left(\\Delta _{n}^{(1)}\\right)>0", "26ddda519f97d1507efe14ad5c810730": "\\scriptstyle G_{ab}", "26de084b9f19f609a5d7f3886820a922": "\\mathbf {x} ^{k}", "26de7de5687111d53bd53ed4d710e1f1": "r\\to \\infty \\,\\!", "26de8b574a447838373c78574c0ab451": "x_{n/2}", "26decdd7aea1e4e23bbfa64044fc831a": "{\\frac {\\partial u}{\\partial t}}=\\alpha \\left({\\partial ^{2}u \\over \\partial x^{2}}+{\\partial ^{2}u \\over \\partial y^{2}}+{\\partial ^{2}u \\over \\partial z^{2}}\\right)+{\\frac {1}{c_{p}\\rho }}q.", "26df04bf3fef7ea19d441d2dc9b48e0f": "R_{p}=k_{p}\\cdot [M]\\left({\\frac {f\\cdot k_{d}\\cdot [I]}{k_{t}}}\\right)^{1/2}", "26df06ede6ad4afb7342e5814dece2de": "H=\\int d^{3}x{N \\over {\\sqrt {det(q)}}}{\\Big (}{\\tilde {\\pi }}^{2}+{\\tilde {E}}_{i}^{a}{\\tilde {E}}^{bi}\\partial _{a}\\varphi \\partial _{b}\\varphi +det(q)V(\\varphi ){\\Big )}+N^{a}{\\tilde {\\pi }}\\partial _{a}\\varphi ", "26df5203238f27fc19ee78dec8fd77bb": "{\\frac {d}{dt}}{\\begin{bmatrix}u\\\\v\\\\w\\\\\\end{bmatrix}}={\\begin{bmatrix}0&-\\delta &0\\\\\\delta &0&\\kappa E\\\\0&-\\kappa E&0\\end{bmatrix}}{\\begin{bmatrix}u\\\\v\\\\w\\\\\\end{bmatrix}}", "26df63a08a30deb38870e7ea7c000f55": "\\exists {\\bar {x}}.\\forall y.y\\prec {\\bar {x}}\\leftrightarrow F(y)", "26df6ea796628ea196fb25f8d9869493": "V_{O_{2}}", "26dfca3201528440a91f205ec1e13c99": "{\\frac {3^{3}2^{0}+3^{2}2^{3}+3^{1}2^{4}+3^{0}2^{6}}{2^{7}-3^{4}}}={\\frac {211}{47}}", "26dfdc5e5326c37114784ed8d6951f8c": "n=n_{1}N_{2}+n_{2}N_{1}\\mod N,", "26dfe60a54a40b3dd014993eefcd4c0e": "s\\in \\prod _{i=1}^{l}\\mathbb {Z} ^{k}", "26e01c7bb3dc63aad40ef5438cecaef1": "A_{i}^{\\lambda }", "26e0435e3126a2b9e3701bd9c2bfbea4": "y^{2}-az^{2}=P(x),\\,", "26e043923c7b7a8f24d0b0125f3f623a": "B\\rightarrow A:\\{N_{B}\\}_{PK(A)}", "26e045cfb77418927feb2515d43e1887": "A_{T}", "26e05bf63d1897a5effec88e8d98b303": "t<{\\frac {n}{4}}", "26e07223e303dcde222bc539155d0e21": "A^{c}", "26e0e053f40e15adf2d2d1f0d1c00ccf": "\\left\\{{A^{i}}\\right\\}_{i=1}^{n}", "26e10b64500f0121685a2b920221c96c": "q={\\frac {1}{2}}(1+3w)", "26e1370d041e272b870b0e96bb172882": "\\triangleleft \\!\\,", "26e1fccd2f6bdd7343c15dc6ee7b1e6c": "I_{p}=0.5({\\frac {I_{d}-1}{M_{u}-1}})", "26e209bc935a766e62f7ceca48094f01": "S\\propto \\nu ^{-\\alpha }.", "26e26fea3e978ec91940d8c7ca593edf": "x=a{\\frac {\\sin[mp+\\theta _{0}]\\cos np}{\\sin[(m-n)p+\\theta _{0}]}}=a+a{\\frac {\\cos[mp+\\theta _{0}]\\sin np}{\\sin[(m-n)p+\\theta _{0}]}}={a \\over 2}+{a \\over 2}{\\frac {\\sin[(m+n)p+\\theta _{0}]}{\\sin[(m-n)p+\\theta _{0}]}}\\!", "26e28bece6aace617f417e0b6ec5cad3": "(A\\mid (B\\mid C))\\mid [([(B\\mid D)\\mid (A\\mid D)]\\mid (D\\mid B))\\mid ((C\\mid B)\\mid A)]", "26e291b3c44b02ce7303d5e08227534f": "\\mathrm {OffensiveRating} ={\\frac {\\mathrm {PointsScored*100} }{\\mathrm {Possessions} }}", "26e2bcd4bfc1c7b297b775e6389cb94d": "{\\frac {\\pi }{4}}=12\\arctan {\\frac {1}{49}}+32\\arctan {\\frac {1}{57}}-5\\arctan {\\frac {1}{239}}+12\\arctan {\\frac {1}{110443}}", "26e3d8bac39f9313d584a6025bd7544d": "_{2}", "26e3dcf458e2d4832999973519646098": "x_{2}={\\frac {1-{\\sqrt {-31}}}{4}}.\\,\\!", "26e5003e405d5a92b49cb064a2b2b783": "a(u,v)=f(v)\\quad \\forall v\\in V,", "26e523249449fcf19edb3f4eccc6bd38": "z=\\min\\{c(x):x\\in X\\subseteq \\mathbf {R} ^{n}\\}", "26e52f01a8e6c1d3a243c6d26ee449f7": "\\chi (\\mathbf {R} )=e^{-{\\sqrt {k}}R^{2}}.", "26e59f612b72670f328973e0245e0645": "{\\frac {I}{4\\omega ^{2}}}.", "26e5f521c0a935db49eb5c7312aebd6b": "\\ Y=AK^{\\alpha }L^{1-\\alpha }", "26e5f67a2917800ec75553fa3fee7d22": "\\Delta Q=0", "26e616fbee356831cc434555b6a3dd00": "R(x)=b_{n-1}x^{n-1}+b_{n-2}x^{n-2}+\\cdots +b_{1}x+b_{0}", "26e634477c7a1285bb21c5df84371894": "k_{i}", "26e658af79b935eeadbe4520e7b4ca16": "P(X=4)=f(4;50,5,10)={{{5 \\choose 4}{{45} \\choose {6}}} \\over {50 \\choose 10}}={5\\cdot 8145060 \\over 10272278170}=0.003964583\\dots .", "26e6e1759b0f8ac1c98af0938476eb50": "\\lim _{r\\rightarrow 1-}(1-r)\\sum _{n=1}^{\\infty }{\\frac {na_{n}r^{n}}{1-r^{n}}}=A.\\,", "26e6e8e39fb2a664c38774cbea8a5885": "q_{z}", "26e6ebbb242690b03732a7dced6967bb": "{\\dot {v}}_{4}={1 \\over C_{4}}({{v1-v4} \\over R_{6}}-{{v4-v3} \\over R_{2}})", "26e78e5fcf969e68fec94e6f5d5a8a01": "\\prod _{i\\in I}A_{i}", "26e7bf580eac82cf856068189f3fafa0": "\\scriptstyle {\\hat {\\mathbf {u} }},\\,{\\hat {\\mathbf {v} }},\\,{\\hat {\\mathbf {w} }}", "26e7da6f5d5897f027a0d18f0965d342": "opens", "26e858b6e02518cfb1d28704e85bdf56": "Xf(y)={d \\over dt}f(y+tX)|_{t=0}.", "26e8b3a8726777e943bef7f3ebc86dbc": "S\\circ op\\circ {\\overline {op}}=S", "26e8eda7095a007887bdedb5c6a5ff62": "v(x)=-4x+12", "26e8f04ccedbe8469265e977baca5e67": "\\rho _{X_{i}X_{j}\\cdot \\mathbf {V} \\setminus \\{X_{i},X_{j}\\}}=-{\\frac {p_{ij}}{\\sqrt {p_{ii}p_{jj}}}}.", "26e9a323631bf4ba50594e52da23544c": "=1", "26e9b2bbf149fd536bd0cd06ed99dcc6": "\\tau _{W}(m,\\mu ;\\nu )=-\\mathrm {sgn} \\langle y,m\\rangle s(\\langle x,m\\rangle ,\\langle y,m\\rangle )+\\mathrm {sgn} \\langle y,\\mu \\rangle s(\\langle x,\\mu \\rangle ,\\langle y,\\mu \\rangle )+{\\frac {(\\delta ^{2}-1)\\langle m,\\mu \\rangle }{12\\langle m,\\nu \\rangle \\langle \\mu ,\\nu \\rangle }}", "26e9fe8b4f05888550a5fb7d50a7fbea": "{\\overline {V}}=V", "26ea01bef16e9643cc3c62de2bfb40bc": "R={\\frac {\\log M}{n}}\\,\\!", "26ea1866bf5112e2e317ebc1faedf882": "n_{1}=1", "26ea89aab768e8af4da1a7d15c73ca32": "E[\\delta _{i}^{2}]=\\sigma ^{2}", "26ea9f584282d16897a67849d52c7e1b": "Q_{t}\\cdot Cv_{O_{2}}", "26eac4311a8a8fd7faf2867e6dfc1379": "\\chi _{m}={\\frac {\\partial M}{\\partial H}}={\\frac {N_{\\text{A }}}{3k_{B}T}}\\mu _{\\mathrm {eff} }^{2}{\\text{ ; and }}\\mu _{\\mathrm {eff} }=g_{J}{\\sqrt {J(J+1)}}\\mu _{B}", "26eb916c153b5db8077c9661a02db25b": "\\left(A_{z}\\right)_{m'n',mn}=\\delta _{n'n}\\left(J_{z}^{(m)}\\right)_{m'm}\\,\\quad \\left(B_{z}\\right)_{m'n',mn}=\\delta _{m'm}\\left(J_{z}^{(n)}\\right)_{n'n}", "26ebcd8b76999b03bb99289a7b6442ac": "{\\begin{aligned}M_{x}\\beta =-\\beta M_{x}\\,,\\\\M_{y}\\beta =-\\beta M_{y}\\,,\\\\M_{x}^{2}=M_{y}^{2}=M_{z}^{2}=I\\,,\\\\M_{x}M_{y}=-M_{y}M_{x}={\\rm {i}}M_{z}\\,,\\\\M_{y}M_{z}=-M_{z}M_{y}={\\rm {i}}M_{x}\\,,\\\\M_{z}M_{x}=-M_{x}M_{z}={\\rm {i}}M_{y}\\,.\\end{aligned}}", "26ebf6664153aeafc5dce5630c0a7c04": "{\\begin{aligned}\\tan x&=1{\\frac {x}{1!}}+2{\\frac {x^{3}}{3!}}+16{\\frac {x^{5}}{5!}}+272{\\frac {x^{7}}{7!}}+7936{\\frac {x^{9}}{9!}}+\\cdots \\\\\\sec x&=1+1{\\frac {x^{2}}{2!}}+5{\\frac {x^{4}}{4!}}+61{\\frac {x^{6}}{6!}}+1385{\\frac {x^{8}}{8!}}+50521{\\frac {x^{10}}{10!}}+\\cdots \\end{aligned}}", "26ec68acfc259413ea1f513bfce572f9": "n(\\omega _{k,s})={\\frac {1}{\\exp(\\hbar \\omega _{k,s}/k_{B}T)-1}}", "26ec6b68521a53951c7f1c0028226757": "\\rho (q)=\\sum _{n\\geq 0}{q^{2n(n+1)} \\over \\prod _{1\\leq i\\leq n}(1+q^{2i-1}+q^{4i-2})}", "26ed352f3dcaf6fc8e653ab91a56cb1d": "A_{n}(x)=\\sum _{m=0}^{n}A(n,m)\\ x^{m}.", "26ed85588662bf409ab009d1d739368b": "\\mu :A\\otimes A\\to A", "26edac09bc64d341c7f2b0e3652e3bc8": "\\alpha _{1}", "26edbf29b7347287ca128d904d033598": "\\phi ^{2}={\\frac {\\chi ^{2}}{n}}", "26ee0751225c6ba17dedccc89db80797": "{\\begin{aligned}G_{2}&={\\frac {k_{4}}{k_{2}^{2}}}\\\\&={\\frac {n^{2}\\,((n+1)\\,m_{4}-3\\,(n-1)\\,m_{2}^{2})}{(n-1)\\,(n-2)\\,(n-3)}}\\;{\\frac {(n-1)^{2}}{n^{2}\\,m_{2}^{2}}}\\\\&={\\frac {n-1}{(n-2)\\,(n-3)}}\\left((n+1)\\,{\\frac {m_{4}}{m_{2}^{2}}}-3\\,(n-1)\\right)\\\\&={\\frac {n-1}{(n-2)(n-3)}}\\left((n+1)\\,g_{2}+6\\right)\\\\&={\\frac {(n+1)\\,n\\,(n-1)}{(n-2)\\,(n-3)}}\\;{\\frac {\\sum _{i=1}^{n}(x_{i}-{\\bar {x}})^{4}}{\\left(\\sum _{i=1}^{n}(x_{i}-{\\bar {x}})^{2}\\right)^{2}}}-3\\,{\\frac {(n-1)^{2}}{(n-2)\\,(n-3)}}\\\\&={\\frac {(n+1)\\,n}{(n-1)\\,(n-2)\\,(n-3)}}\\;{\\frac {\\sum _{i=1}^{n}(x_{i}-{\\bar {x}})^{4}}{k_{2}^{2}}}-3\\,{\\frac {(n-1)^{2}}{(n-2)(n-3)}}\\end{aligned}}", "26ee4310fbee994ad4d3751e2b8bd3f8": "U_{n}(R)=0=A_{n}J_{0}\\left(\\alpha n^{1/2}i^{3/2}\\right)+{\\frac {iC_{n}}{\\rho n\\omega }}\\,.", "26ee4c3a61a5d968fb1d07411d44c764": "t_{1},t_{2},\\dots ,t_{n},", "26ee688d727ea0c771dbdf3f456895bd": "d_{2}", "26ee7195db770c6621a9090b7adc6c83": "f\\left(\\mathbb {N} _{1}\\right)\\subseteq A", "26ee7f33cb176694db35573ad5220b3b": "z={\\frac {Z}{X+Y+Z}}=1-x-y", "26eeda31559ef28700564b8fb5bca59b": "P_{TNL}", "26ef0f8a1ea18d2fe5d2e63e5df2f01a": "{\\mathbf {M}}=\\chi _{m}{\\mathbf {H}}\\,", "26ef35925b56766a1a91018a9a3e7a2d": "{n^{2}}/8", "26ef9a600660c8eff36adcc3e88179f0": "\\partial M", "26efc5bfcfb2548faba05e445f18d5c3": "{}_{2}F_{1}(a,b;c;z)=(1-z)^{-b}{}_{2}F_{1}\\left(b,c-a;c;{\\tfrac {z}{z-1}}\\right)", "26efd2da36397fbf91a9d93c63c3e00c": "\\{a_{1},\\ldots ,a_{k}\\}", "26effacf274c4c68c2ac4cb31f3a2827": "k_{0}\\neq 1", "26f08bbd852c1de2a68309f4e67a29c5": "\\epsilon \\sigma T", "26f0c21bb5156021bdb7ada4ded246f2": "\\int \\arcsin(x)\\,dx=x\\arcsin(x)+{\\sqrt {1-x^{2}}}+C", "26f0c4532677d807b3fd5e263f9bb071": "={\\frac {{\\textrm {Li}}_{\\alpha }(z)}{\\zeta (\\alpha )}}\\,\\tau ^{\\alpha }", "26f0dba3deea96431da624e45d46ecf1": "{\\dot {u}}_{a}\\ll \\omega u_{a}", "26f10ff3baf0882fc83c7354aa92ff46": "\\mathbf {*3\\cdot 24} .\\ \\ \\vdash .\\thicksim (p.\\thicksim p)", "26f14a67fda683a9425a3b4f481a501d": "\\ M_{y}={\\frac {W-u_{1}+u_{2}}{a_{2}}}.", "26f167f212bd8f42ae4b8ee42e3ddf14": "{\\begin{bmatrix}a_{11}&a_{12}&a_{13}\\\\a_{21}&a_{22}&a_{23}\\\\a_{31}&a_{32}&a_{33}\\\\\\end{bmatrix}}={\\begin{bmatrix}l_{11}&0&0\\\\l_{21}&l_{22}&0\\\\l_{31}&l_{32}&l_{33}\\\\\\end{bmatrix}}{\\begin{bmatrix}u_{11}&u_{12}&u_{13}\\\\0&u_{22}&u_{23}\\\\0&0&u_{33}\\\\\\end{bmatrix}}.", "26f179bae6dfee8b90dd5ef2e3b808b1": "\\left.p_{0}({\\vec {r}})=\\int _{\\Omega _{0}}{\\frac {d\\Omega _{0}}{\\Omega _{0}}}\\left[2p({\\vec {r_{0}}},v_{s}t)-2v_{s}t{\\frac {\\partial p({\\vec {r_{0}}},v_{s}t)}{\\partial (v_{s}t)}}\\right]\\right|_{t=|{\\vec {r}}-{\\vec {r_{0}}}|/v_{s}},\\qquad \\quad (4),", "26f1993b461107644b85c8f4d28b5f4b": "\\log ^{-1}(\\delta ^{-1})", "26f19ca14f39a3efc5c38f244ec3f59e": "\\epsilon _{i}=-{\\frac {1}{\\beta }}\\ln {\\eta _{i}}", "26f1c92918f36841630308e55602e2d8": "\\langle c(\\mathbf {x} ,{\\mathit {t}})\\rangle =\\sum _{i=1}^{m}{\\boldsymbol {\\psi }}_{i}(\\mathbf {x} ,{\\mathit {t}})", "26f1e7f13d94b8ab0fdee1e67877e845": "\\scriptstyle \\tau _{p}^{n}(x)", "26f27b8985b5109ed7b21507986e5502": "a_{b}^{*}=\\varepsilon _{r}\\left({\\frac {m}{\\mu }}\\right)a_{b}", "26f292d0f3f53c5577abe946d98ff641": "{\\textit {closedoor}}(t)", "26f2ab25e693b6175bd49632afaadb9b": "|G_{\\nu }(\\pi )|=1", "26f32361e1d541b933f6f2c2f5c8ba7f": "E=\\angle zcx", "26f33cb7f2916ef99e7f93e87e853d5f": "{\\frac {d\\beta }{dt}}={\\frac {Y_{\\beta }}{mU}}\\beta -r", "26f35c46cb8057f1f68f51cfd52bdd8e": "u_{1},\\cdots ,u_{n}", "26f36f9158a9dccd6efb839b110f1253": "C(u,v)=-C(v,u)_{}^{}", "26f38028f22c5673f7d44b957f23d7ed": "(-\\infty ,\\theta ].", "26f3ca65cbccdbec877f1b8557519832": "\\mathbf {n} \\,", "26f3ce86edceef0706a016012d3ee38c": "{4}a_{A}{\\frac {((A/2)-Z)^{2}}{A}}", "26f3f8c54e57639ba8e1f347da7d53bb": "[X,[Y,Z]]+[Z,[X,Y]]+[Y,[Z,X]]=0.\\,", "26f40fbd8dc75261c300e264c47175a8": "\\left[{\\frac {L}{p}}\\right]_{3}=\\left[{\\frac {M}{p}}\\right]_{3}=1", "26f417d9ce1533efeac796cfb03a5782": "\\Pr(\\chi _{g-1}^{2}\\geq K)", "26f439f8d5cf82407894e3c041b59c3f": "f(x;r,\\lambda )=\\lambda /\\Gamma (r)(\\lambda x)^{r-1}\\mathrm {e} ^{-\\lambda x}I_{[0,\\infty ]}(x)", "26f44b68e6ccc6644cc407025bf3f6a8": "\\Upsilon _{i}", "26f46941f67564404a18a8481513ff29": "\\{P(x_{1},\\ldots ,x_{l})\\}", "26f4c8246f090e860ce12422f4ab12da": "X\\in {\\mathcal {X}}", "26f516d40acc1aa13d7eaabefd8f0362": "\\scriptstyle 2\\pi ft\\,+\\,\\varphi ", "26f54cdc052769bd3b084de1248f6807": "\\aleph =2^{\\aleph _{0}}", "26f55b1ee3c8ccfeaa610c1279a94414": "{\\mathcal {T}}\\left\\{A(x)B(y)\\right\\}:=\\theta (x_{0}-y_{0})A(x)B(y)\\pm \\theta (y_{0}-x_{0})B(y)A(x),", "26f5bfd7253d740dc120503f0593f5c4": "\\left({\\frac {a}{2}}\\right)={\\begin{cases}0&{\\mbox{if }}a{\\mbox{ is even,}}\\\\1&{\\mbox{if }}a\\equiv \\pm 1{\\pmod {8}},\\\\-1&{\\mbox{if }}a\\equiv \\pm 3{\\pmod {8}}.\\end{cases}}", "26f5dc5cc334cfdc99bf2b0dab015b54": "2eV=\\hbar {\\frac {\\partial \\phi }{\\partial t}}", "26f5de97c92b0973d3e1f71e08892164": "h(x;t)", "26f629916ff48f187f646ab41287e184": "x_{n+1}=f(x_{n})", "26f67b2c0c8c620904a393aea234c7f2": "\\kappa (y)", "26f6b925a99ff4e977a40898337267bc": "n=1,\\,2,\\,3\\,...", "26f6c03682a499770d1fe17ce4e5bced": "\\mathbf {R} =\\sum _{i=1}^{N}\\mathbf {r} _{i}", "26f6e4f6e647998ec07a536f0f1c2767": "K^{*}=K", "26f6f8c5261675473f6d17fbcade95c8": "{\\dot {x}}=f(x)+\\sum _{j=1}^{m}g_{j}(x)u_{j}", "26f740429f9b45dc49585a246d5e496a": "Cl_{X}(S)", "26f74836981a83726edf260d34c20733": "\\Phi :M\\to \\operatorname {Hom} _{R}(N,L)", "26f78c73c39a57184aeb02a8c1c3b5bf": "P(in)", "26f7a6df39a1f4fb15523a6acecf34ee": "[\\Gamma L,\\Gamma L]\\subset \\Gamma L", "26f7b570ed83fedde09930e84d1e8824": "71\\cdot 36\\cdot 990=2,530,440\\,", "26f83ce8b1b8d64fcbb6f763641a0392": "{\\mathcal {H}}_{C}", "26f859a24717c23cbac7f287d83584e1": "F_{i}=1+x^{2^{i}}", "26f8ce20c7d8193d0f4473f5a987fe0e": "{\\begin{pmatrix}1&1&1&1&1&1&1&1\\\\1&0&1&0&1&0&1&0\\\\1&1&0&0&1&1&0&0\\\\1&1&1&1&0&0&0&0\\\\1&0&0&0&1&0&0&0\\\\1&0&1&0&0&0&0&0\\\\1&1&0&0&0&0&0&0\\\\\\end{pmatrix}}", "26f90c37a9bdc0c73da3e844a9e052ce": "B^{2}-4AC,\\,", "26f917583380478b1ed798e53434a501": "Q(X)", "26f996c53f99c3b39f3a2906a7837a06": "R_{D}", "26f9bf91d862ce576b3a38a3079f796b": "KS(xx)\\leq \\ell (x)+66", "26fa1a688d94abb4aae215d3fce67334": "{\\frac {\\lambda }{4!}}\\phi ^{4}", "26fa309b0588dc6624bcf44ee18e3fc3": "{\\frac {dN}{dt}}=rN\\left(1-{\\frac {N}{K}}\\right)", "26fa6cd1d8fd5c5972bf80acf55081d2": "\\;S_{R}=k\\ln \\Omega _{R}", "26faa26d869d709e90181d129777daee": "\\lambda m,p,q.(\\lambda g.\\lambda n.(n\\ (g\\ m\\ p\\ n)\\ (g\\ q\\ p\\ n)))\\ \\lambda x.\\lambda o.\\lambda y.o\\ x\\ y", "26fad89e746b1074b40ac4b76d536f25": "E=hcG(v)", "26fafdbd7bbecc8e496fe34d922344dc": "C_{\\mathrm {linear} }", "26fb94687b8c2b9295c76f0cecd07084": "\\textstyle \\ell \\left(v\\right)=0", "26fbaba6b7470aeec676e1205227f869": "{\\frac {\\alpha -1}{\\alpha +\\beta -2}}\\leq {\\text{median}}\\leq {\\frac {\\alpha }{\\alpha +\\beta }},", "26fbb924fe45c49baea75a6d7b6652c5": "SO(10)\\supset SU(5)\\supset SU(3)\\times SU(2)\\times U(1)", "26fbef2fe823a0516645b0ac6846d8be": "m_{\\mathrm {PbO} }=\\left({\\frac {200.0{\\mbox{ g }}\\mathrm {O_{2}} }{1}}\\right)\\left({\\frac {1{\\mbox{ mol }}\\mathrm {O_{2}} }{32.00{\\mbox{ g }}\\mathrm {O_{2}} }}\\right)\\left({\\frac {2{\\mbox{ mol }}\\mathrm {PbO} }{3{\\mbox{ mol }}\\mathrm {O_{2}} }}\\right)\\left({\\frac {223.2{\\mbox{ g }}\\mathrm {PbO} }{1{\\mbox{ mol }}\\mathrm {PbO} }}\\right)=930.0{\\mbox{ g}}", "26fc3f4f62cd5c0be10374b1dd7cf722": "p_{M}", "26fc41e0b44218380d2dad640c61a1b1": "(\\mathbf {0} ,0)", "26fc457eb3db576befaa90506e411a1b": "\\mu _{1}=k_{1}=a_{1}+2a_{2}", "26fc6254e962f329dafd5cd72d2329d5": "\\sigma ={\\frac {NPSH}{H}}", "26fc75e6623d39201877a23b2e314909": "V_{f}={\\frac {4}{3}}\\pi p_{f}^{3}({\\vec {r}}).", "26fcc76c3fe0c945f54524665c04f685": "F^{D}=-2\\sigma _{0}m^{2}a^{2}\\left[\\cos ^{-1}\\left({\\cfrac {1}{m}}\\right)+{\\frac {1}{m^{2}}}{\\sqrt {m^{2}-1}}\\right]", "26fcdc804ab481605a2ee739330b9995": "B'\\subset B", "26fd32bc1691648785b96930792288da": "M_{K}={\\sqrt {|D|}}\\left({\\frac {4}{\\pi }}\\right)^{r_{2}}{\\frac {n!}{n^{n}}}\\ .", "26fd5ec5b0177251689a2e5810aa18c1": "G=\\langle N,E\\rangle ", "26fd608e1a013135e7286babdb72de9c": "\\sigma _{xy}=\\nu e^{2}/h", "26fd738f1230b209710db1faea32e532": "{\\begin{aligned}\\left|\\mu -m\\right|=\\left|\\mathrm {E} (X-m)\\right|&\\leq \\mathrm {E} \\left(\\left|X-m\\right|\\right)\\\\&\\leq \\mathrm {E} \\left(\\left|X-\\mu \\right|\\right)\\\\&\\leq {\\sqrt {\\mathrm {E} ((X-\\mu )^{2})}}=\\sigma .\\end{aligned}}", "26fd892b6929e8f16e658824838a12c1": "H={50 \\choose 2n}\\times (2n-1)!!,", "26fda9fbf4b01770e9b045ce809b28ee": "{\\dot {m}}=\\rho AC", "26fdae23a1790cbefc06c0bdfe8cdd8c": "v={\\sqrt[{4}]{GMa_{0}}}", "26fdcb0d3d378a325ce844bbd442d978": "J_{0}\\supset J_{1}\\supset \\cdots ", "26fdd7d135ba9bcc508fb06ae1100cc7": "K_{x}\\;=\\;0\\quad {\\text{ if and only if }}\\quad f(x)=0\\quad \\forall \\;f\\in H.", "26ff1202aeba011f3471243966982224": "z_{1}=0", "26ff2690c198b417816ba1316f38d1f7": "1.096854\\pm 0.000004", "27000e61f0442c6ca3efc00d32188f5a": "A=-\\log _{10}{\\mathcal {T}}\\ =-\\log _{10}\\left({I \\over I_{0}}\\right)", "27005d51c739fde9326e42ca13589d00": "x_{n}x_{m}^{-1}\\in U", "27007f402ce8752fe81fd8151efbd8ef": "\\scriptstyle EAC={\\frac {NPV}{A_{t,r}}}", "2700ac479255997949615e5b2dfc4fc8": "\\Delta g\\neq x_{B}-x_{A}", "2700d7a0980474ab18cdaf8979f1f93d": "{\\frac {33.33\\%+50.00\\%}{2}}=0.417", "2700e8b3a4ec9965e6e7bc147689dcae": "\\left[{\\begin{array}{*{20}{c}}\\bullet &x\\\\y&\\bullet \\end{array}}\\right],\\left[{\\begin{array}{*{20}{c}}x&y\\\\\\bullet &\\bullet \\end{array}}\\right],\\left[{\\begin{array}{*{20}{c}}y&x\\\\\\bullet &\\bullet \\end{array}}\\right]", "2700fc75fc6ed607525f27b462082ba0": "f:A\\rightarrow B", "27011ac77ba433e0a02943702c0f5ae2": "x_{i}x_{j}=x_{j}x_{i}", "2701603777d27808f7b28936e2233991": "{\\stackrel {\\mathfrak {p}}{}}", "2701bdca83dc42ed0ced6e5975b65f43": "U'", "2701d73149129f48b690985a997d5d5a": "V_{i}\\rightarrow u_{i}", "27020fcb2803efe218bbb26fb4cb1424": "p^{D}(r)={\\begin{cases}-{\\cfrac {\\sigma _{0}}{\\pi }}\\cos ^{-1}\\left[{\\cfrac {2-m^{2}-{\\cfrac {r^{2}}{a^{2}}}}{m^{2}\\left(1-{\\cfrac {r^{2}}{m^{2}a^{2}}}\\right)}}\\right]&\\quad \\mathrm {for} \\quad r\\leq a\\\\-\\sigma _{0}&\\quad \\mathrm {for} \\quad a\\leq r\\leq c\\end{cases}}", "27027264e544fa6cc41b3dd2d1d6bba7": "s_{0}\\in \\Sigma ^{*}", "270284e9d8219ea073f422bae443d856": "(s_{k},t_{k})", "2702b4c643237cf1f8273fd1eddce998": "\\cos(\\pi /7)", "2702c04394874331ffacefa0ed3a6be5": "{s}={M}{\\lambda }+{q}\\,", "2702d78abd1b0c358616113da1d117d3": "X|{\\mathfrak {D}}_{0}", "2702f9f9109bec7289a08bddd4952c8a": "r^{2}~{\\cfrac {d^{2}R}{dr^{2}}}+r~{\\cfrac {dR}{dr}}+{\\cfrac {r^{2}\\omega ^{2}\\rho _{0}}{\\kappa }}~R=\\alpha ^{2}~R~;~~{\\cfrac {d^{2}Q}{d\\theta ^{2}}}=-\\alpha ^{2}~Q", "270314605c8f4802e903259100eaecae": "(1-L)X_{t}=X_{t}-X_{t-1}=\\Delta X.", "2703290fbe8c589b13561a55a1cd8b97": "\\sigma (\\Omega )=-\\Omega ", "27036c71ba699259df2d3c560f647675": "={\\frac {31!}{2}}\\cdot 60^{31}", "27043cf2b1341d557147b4cd7024bb45": "Na|n\\rangle =(n-1)a|n\\rangle .", "270475d41d9194a65a211c637f6dd729": "r\\in I^{+}(q)", "2704a81a01e96b81a9dd3471ea90577d": "u_{t}+uu_{x}+(c^{2}/\\rho )\\rho _{x}=0", "270555012642b902e1336f99a32d1c25": "2x^{2}+3x+1=0", "27056a1579a7bb8ac91056d20f21ab60": "D(E(m)+e)", "27057a6556fecc0df357910852969a6f": "r^{\\ddagger }={\\frac {E_{a}}{m_{1}}}+r_{1}", "2705d975e49eb93a40c99af99f893c94": "f(g(a+h))-f(g(a))=f(g(a)+g'(a)h+\\varepsilon (h)h)-f(g(a)).", "2705fdd6d92f123055bf01749569f715": "k({\\bar {y}})", "2706a002ceff469953fef0e33cf59ef6": "|x|\\ll 1\\ ", "2706bf6fba5d6d10a55e0e1e23ce8ab7": "S_{\\rm {semi}}={\\frac {\\sigma _{\\rm {C}}S_{\\rm {C}}+\\sigma _{\\rm {V}}S_{\\rm {V}}}{\\sigma _{\\rm {C}}+\\sigma _{\\rm {V}}}},\\quad \\sigma _{\\rm {semi}}=\\sigma _{\\rm {C}}+\\sigma _{\\rm {V}}", "2706e644566c335e00fab859772f896e": "=\\mathbf {a} _{AB}\\ +\\mathbf {a} _{B}\\ ,", "27071423b9a403c9ffa1895ef3fefb6e": "c=1", "270723cdccc0f3dc59350031a1f16d2c": "((a\\vee n)\\wedge (\\neg n\\vee b))", "2707787db23e3a0c6c8e85807a6dd229": "\\langle a_{1},a_{2},a_{3}\\ldots \\rangle ", "2707962508b28f176355391b6b8ccf59": "a^{2}+|D|b^{2}=4N\\,", "2707c95949326626c91b956e135e33ce": "\\psi _{R}={\\begin{pmatrix}I_{2}&0\\\\0&0\\end{pmatrix}}\\psi ,\\quad \\psi _{L}={\\begin{pmatrix}0&0\\\\0&I_{2}\\end{pmatrix}}\\psi .", "2707cdce01d26f2442db43ae75a8a086": "{XX' \\over YY''}={AX \\over CY},", "2708a7ce5aa8bd216396727399ff0e4c": "C^{b}", "2708deeea96bf410e230562632495ef7": "\\epsilon =1+\\chi _{\\text{e}}", "2708e78d7629165e264d14628dd8a3fb": "\\displaystyle {[{\\mathfrak {g}}_{p},{\\mathfrak {g}}_{q}]\\subseteq {\\mathfrak {g}}_{p+q}}.", "2708eef135e71e2e7226a20a3c04cdeb": "\\varepsilon _{3}'''={\\frac {1}{E}}\\sigma _{3}", "2708f90a9e3c4dd1fb7c8a71dbfc9cbc": "i=(k^{2}+4ik-5)c_{3}+(-k^{2}+4ik+5)c_{4}", "27095c895734ad8643adc90f0d7ac56f": "S_{i}^{+}\\longrightarrow S_{i}^{+}e^{i\\theta _{i}}", "2709665e0a487a0e385e332b0a7f27a7": "{\\frac {1}{2}}\\log(1+P/N)", "27099b4b2a33741eb51b64121b947485": "J=(1,1,1,...,1)^{T}", "2709b9a1bbb18623580ffc86a4bd452c": "F_{B}{\\big (}u^{*}+{\\frac {\\partial u^{*}}{\\partial x}}dx{\\big )}-F_{A}u^{*}\\approx {\\frac {\\partial u^{*}}{\\partial x}}\\sigma dV+u^{*}{\\frac {\\partial \\sigma }{\\partial x}}dV=\\epsilon ^{*}\\sigma dV-u^{*}fdV", "270a5de65dedb213ba1ea386512f791c": "L[t]=T[t]-V[t]={1 \\over 2}m{\\dot {\\vec {x}}}[t]\\cdot {\\dot {\\vec {x}}}[t]-m\\zeta [{\\vec {x}}[t],t].", "270a67b48931b2e948a36c185afa99fc": "[I_{C}^{B}]=[Q][\\Lambda ][Q^{T}],", "270a8f32a847e623f200eb1092806bf1": "\\varepsilon _{\\color {Orange}{\\color {Orange}{2}}\\color {BrickRed}{1}\\color {Violet}{3}\\color {RedViolet}{4}}=-\\varepsilon _{\\color {BrickRed}{1}\\color {Orange}{\\color {Orange}{2}}\\color {Violet}{3}\\color {RedViolet}{4}}=-1", "270aa4b97e71defc9ce4b8b761704a65": "A'_{\\mu }(x)=\\partial _{\\mu }f(x)", "270aae64e82be75653345dd6955a621a": "S(A,P,z)=X\\cdot W(z)\\cdot \\left({1+O\\left((\\log z)^{-b\\log b}\\right)}\\right)+O\\left(z^{b\\log \\log z}\\right).", "270ac04ef8471368d16d23db66e65ae8": "\\sigma =1/{\\sqrt {2\\mu \\omega }}", "270ada5119a0137c94b83acac6b5f5c5": "\\displaystyle {\\psi =T\\psi +{1 \\over 2}\\psi -(\\lambda -{1 \\over 2})S\\varphi =-(T\\psi -{1 \\over 2}\\psi )+(\\lambda +{1 \\over 2})S\\varphi }", "270b45447668908d2c43583dea36cf38": "\\displaystyle {H=A+iB.}", "270b567948a0aa5966f0202415c0c807": "\\varphi (x)=\\langle x_{n}^{2},\\ldots ,x_{1}^{2},{\\sqrt {2}}x_{n}x_{n-1},\\ldots ,{\\sqrt {2}}x_{n}x_{1},{\\sqrt {2}}x_{n-1}x_{n-2},\\ldots ,{\\sqrt {2}}x_{n-1}x_{1},\\ldots ,{\\sqrt {2}}x_{2}x_{1},{\\sqrt {2c}}x_{n},\\ldots ,{\\sqrt {2c}}x_{1},c\\rangle ", "270b6bc72975b59f6a764e887522bc0a": "C={\\frac {f}{16}}\\cos ^{2}\\alpha {\\big [}4+f(4-3\\cos ^{2}\\alpha ){\\big ]}\\,", "270c10868af5da5d4ccd445df7379e42": "\\delta (X_{1},X_{2},\\ldots ,X_{n})", "270c2bdda0bc4086d603e6985fc854c5": "X_{k+1}=2X_{k}-X_{k}AX_{k}.", "270c3ec8bb0d2394a93a44d22e2d9619": "G=S_{0}\\cup S_{1}\\cup S_{2}", "270c46da70a7caa1e4128dac9c67dd49": "ab^{3}c^{-1}ca^{-1}c\\;\\;\\longrightarrow \\;\\;ab^{3}\\,a^{-1}c.", "270c5326bef04fc8380220384f593f41": "{\\hat {\\textrm {t}}}_{i}", "270c64a980bc0e8443c55d3ccdfd652f": "A_{o}^{FI}=\\left({\\frac {Total\\ Time}{Total\\ Time+Fault\\ Isolation\\ Down\\ Time}}\\right)", "270c6934da2438b1807df18f7f9981c1": "k\\propto \\exp \\left({\\frac {-\\Delta ^{\\ddagger }G^{\\ominus }}{RT}}\\right)", "270d34e5fd341708d731b9dcffdd465d": "\\,={\\mbox{R}}(z,t)[{\\mbox{R}}(z,dt)-1]/dt", "270d9abdf2eaff03751d5c8838706ec4": "S(t^{-})", "270e333d86cd24092f053b605a5eb786": "\\Sigma _{k}^{*}", "270ed143b3fca7cdbaf9b684af626abc": "I=-qnv_{x}tW", "270edb587a53fe341c3fc3ee6ec6b089": "\\psi (x)=0", "270ee9c6bed722c22a54bd5d4025f1ec": "x*y=x+y-k", "270ef91f3c83cc0967b567d93b9e6e61": "\\operatorname {P} [E_{1}\\cap (E_{1}\\cup E_{2})]=\\operatorname {P} \\left[E_{1}\\mid E_{1}\\cup E_{2}\\right]\\operatorname {P} \\left[E_{1}\\cup E_{2}\\right]", "270f2f3792ee9a744b0098929fdd1a50": "\\mathbf {x} =(x_{1},x_{2},...,x_{n})", "270f388654d61c59bcbe2339d97bfd31": "(\\sinh x)'=\\cosh x={\\frac {e^{x}+e^{-x}}{2}}", "270f3adf243ab096a429fd145065be5d": "\\int _{G}F(g)\\,dg={1 \\over |W|}\\int _{\\mathfrak {a}}F(e^{X})\\,|\\delta (e^{X})|^{2}\\,dX.", "270f4a678da5c90d5ab807ce09807149": "E={\\Delta h_{f}+\\Delta h_{m}}", "270f7659761ea30f892569bde98806a8": "W:\\mathbb {R} \\rightarrow \\mathbb {R} ^{+}", "270f8016114715c9801b2896d2378c24": "W=C_{e}A^{T}C_{Z}^{-1}", "270fe5f6ef0acb3e3f811e1423e023cd": "{\\dot {f}}", "271022ddb7e70d6a9628ee9bf8c20d42": "(A{\\mathbf {x} })_{k}=\\sum \\limits _{l=1}^{n}A_{kl}x_{l}", "2710235820156c8b766a3d96c015d5a4": "(p\\lor \\neg p)\\land (q\\lor \\neg q)\\land (r\\lor \\neg r).", "27104399a18330853a1eb19a96eff9da": "\\mu >0", "27104f404ed2fb511d82f2fe8d6e2187": "\\sum _{n=0}^{m}1_{\\{N=n\\}}E^{n}=\\prod _{j=1}^{m}(1_{A_{j}^{\\mathrm {c} }}I+1_{A_{j}}E)", "271088698edc1bb87a961f3e8695be43": "\\ F_{t}", "2710905829de4b69ace1e10ae40a753f": "\\scriptstyle (k-1)", "2710dc64d680e5f96cfb56d025717fe8": "T={\\frac {\\beta A}{\\rho cd^{3}}}", "2710ed90aa9d1a6303e17239ce32c113": "{\\mathcal {V}}(x)", "271103eb78d216d814c50e82e661cc4c": "~T~", "27110e52d1e55c68aba818eb2e4bc5df": "c=\\nu \\lambda ", "2711367841d9e106c018d446887951e3": "[x,y]=z,\\quad [x,z]=0,\\quad [y,z]=0", "2711ba43786e7fc2c890529be11dc6d9": "n_{\\mathrm {dB} }", "27122d9947ad2b554dcca97527e860dc": "={\\frac {4}{3}}\\pi \\rho _{p}r_{p}^{3}{\\frac {V_{t}^{2}}{r}}.", "2712c102eb5f6ec3919e12f4bfd131e0": "{}-p^{6}+28p^{4}r-16p^{3}q^{2}-176p^{2}r^{2}-80p^{2}sq+224prq^{2}-64q^{4}", "2712c552ef453cfd40b586fabbed0da2": "M_{(i,j),k_{1}}=M_{(i,j),k_{2}}=1", "2712d70c435029c9b96cf7fe0b3a5af7": "{\\hbox{ad}}_{g}(R_{g}^{*}\\omega )=\\omega ", "271333d7aa2bed9564a9281a698ab316": "z\\in {\\mathcal {Z}}", "27133668b08c8e94e71f4beddb4ab6e1": "|g_{1}\\rangle ", "27138029e6edfe27251aa78b745d042a": "Y=Z\\,\\!", "27141c6f42f5d19abb786e3a39c7f839": "N=N_{0}\\,e^{-t/\\tau },", "2714380d2a3edbacf1f64854bc8992c5": "O(p*s)", "271476fdd7a1572b71f3b161ee4184c3": "F_{\\mathrm {b} }={\\frac {\\Delta \\omega }{\\omega _{0}}}", "27152207b5aed9d9f94787930812cdb1": "p=mv.", "2715eef5eb01f99922e7aaa18fd7c6f5": "f^{(i)},i\\in \\{0,...,n-1\\}", "2715f1e2f6baa87dcdb8e860fe0ce260": "\\ln y=\\ln y_{0}+rS+\\beta _{1}X+\\beta _{2}X^{2}", "271638047b21bec11fe5f12c96abea1a": "G(k)\\leq k\\log k+k\\log \\log k+Ck.", "2716d3bb8e097bc71156dea958a479b3": "\\gamma _{\\text{m}}", "27172809b72d72bfd29ecb9547047834": "\\mathbf {F} _{l}=-{\\boldsymbol {\\nabla }}\\Phi =-{\\frac {1}{4\\pi }}{\\boldsymbol {\\nabla }}\\int _{V}{\\frac {{\\boldsymbol {\\nabla }}'\\cdot \\mathbf {F} }{\\left|\\mathbf {r} -\\mathbf {r} '\\right|}}\\mathrm {d} V'", "27178175a444521f96e0ba9b070f3275": "m_{\\mathrm {em} }=\\int {1 \\over 2}E^{2}\\,dV=\\int \\limits _{r_{e}}^{\\infty }{\\frac {1}{2}}\\left({q \\over 4\\pi r^{2}}\\right)^{2}4\\pi r^{2}\\,dr={q^{2} \\over 8\\pi r_{e}},", "271831f34e8f14fe4b59c4e82cce24e3": "T_{\\pi ,\\lambda }:[0,1]\\rightarrow [0,1],", "271856303c9419b6ec323bc5529e6ac9": "at+b", "27188a376951ffc4d0134815805884ef": "\\beta _{2}^{(0)}=\\beta _{2}", "27190e3f0498615ef381de55b46a8056": "M_{1}{}^{2}=m_{1}^{2}+2m_{w}S+S^{2},", "2719ae0fd018fefdb833bbb6a2138286": "(d\\sigma /d\\Omega )_{\\rm {Th}}", "2719baa2e157e11e1e115ec6492bdd87": "S=\\int _{k}{1 \\over 2}k^{2}|\\phi (k)|^{2}", "2719ee323ad10e2e12b1c6519db9b4ac": "X(z^{-1})", "271a3fc65144601a177368f8b0121522": "\\Sigma \\to X", "271afe7ed7fb34aabfa67e5e09956295": "\\mathbf {J} =\\mathbf {J} _{\\mathrm {b} }+\\mathbf {J} _{\\mathrm {f} },", "271b43fbc04895ae627d2e48f35b6b62": "f_{\\omega }(3)-2", "271b604dbc38adcb1913438fb7e9c1a2": "\\sigma _{i}^{*}(a)=\\left({\\frac {1}{C-1}}\\right){\\text{Gain}}_{i}(\\sigma ^{*},a)=0", "271bbce94f3e39f7f951d6c77c24a7e2": "H_{n}(x)=(H+2x)^{n}\\,\\!", "271c55dec1713a74b129cd9c00ab5ed6": "{\\frac {\\sqrt {n+1}}{n!{\\sqrt {2^{n}}}}}", "271cabfb4c4b5a83d963deb152a4eee9": "\\kappa ={\\frac {a}{s^{2}+a^{2}}}", "271d1f67d105cacfeb98a901b002e46e": "x[n]={\\frac {1}{2\\pi }}\\int _{-\\pi }^{+\\pi }X(e^{j\\omega })e^{j\\omega n}d\\omega .", "271d27f52ec2722277cacb0941c9b61b": "0={\\frac {1}{n}}\\sum _{i=1}^{n}\\nabla _{\\!\\theta }\\ln f(x_{i}|\\theta _{0})+{\\Bigg [}\\,{\\frac {1}{n}}\\sum _{i=1}^{n}\\nabla _{\\!\\theta \\theta }\\ln f(x_{i}|{\\tilde {\\theta }})\\,{\\Bigg ]}({\\hat {\\theta }}-\\theta _{0}),", "271d7049d6dae75a87206d5c99e22561": "\\delta _{\\odot }=-23.44^{\\circ }\\cdot \\cos \\left[{\\frac {360^{\\circ }}{365}}\\cdot \\left(N+10\\right)\\right]", "271d7a9fcf6a6eaa16f1c6f9684d9bc5": "n\\to \\infty .", "271d90ed2edaa04fba4126811f274c0f": "=V", "271dc98c738720b23af59fcf8afa089c": "A=(10+{\\sqrt {{\\frac {5}{2}}(10+{\\sqrt {5}}+{\\sqrt {75+30{\\sqrt {5}}}})}})a^{2}\\approx 17.7711...a^{2}", "271dec5b34962227e8e94175c57db4e0": "Z_{r}^{p,q}", "271e0e2865ff6a72a19b550043f8d9b2": "p_{n-i}(z)\\,", "271e78c8b2c5ff19e7402d9af42d95e9": "\\lfloor x\\rceil ", "271ec3320b67630e0396ebe32a417967": "(e^{ar}e^{-br\\epsilon })^{-1}=", "271edd37909014ccb24a5277d5fb497b": "\\textstyle \\lbrace {n \\atop k}\\rbrace ", "271ee017284df0fa305a66e0d9d3c351": "\\mathbf {{\\hat {Q}}{\\hat {T}}} (\\lambda )|q\\rangle =\\mathbf {\\hat {T}} (\\lambda )(\\mathbf {\\hat {Q}} +\\lambda )|q\\rangle =\\mathbf {\\hat {T}} (\\lambda )(q+\\lambda )|q\\rangle =(q+\\lambda )\\mathbf {\\hat {T}} (\\lambda )|q\\rangle ", "271efef3c48c27e20228b06587fd8a71": "dy\\wedge dz", "271f152f11da730c7c6a0d139484e370": "f(g)=\\sum _{\\sigma \\in \\Sigma }d_{\\sigma }\\operatorname {tr} ({\\hat {f}}(\\sigma )U_{g}^{(\\sigma )})", "271f34ea27b67c987755acb5056ca8c1": "(L/2\\pi )^{d}", "271f3a5312853160bc812ca05a707243": "{\\text{volume}}={\\frac {6\\cdot L^{2}\\cdot {\\frac {L}{2}}}{3}}=L^{3},", "271f5aee3cb1e9825f52e96c86c81f1e": "f\\colon X\\rightarrow T", "27202c197a32aa14c1cc0fab50393a82": "dE=F\\cdot dx", "272046b75ab4366ee647e77a8f0c0952": "G_{1}\\ast G_{2}", "272090c2bf557823e164225c97327437": "(a,b,c,d)=(xz,yz,z^{2},xy).\\,", "2720b99a1d016ed36f887f52cbbe063d": "{\\frac {GM_{\\text{vir}}}{R_{\\text{vir}}}}\\approx \\sigma _{\\text{max}}^{2}", "272193a81c5c72bbf2a89d8f7d475f7c": "\\!\\,I=I_{o}e^{\\frac {-t}{\\tau }}", "27219bcaad63d305920bcb58027e779d": "S={\\frac {m(m^{3}+1)}{2}}.", "2721a535d23241f23e10717ae00154cd": "\\gamma =2-2\\alpha ", "2721cd572bfad45985a6e055fc2c5ecb": "\\nabla _{\\mathbf {u}}{\\mathbf {v}}=D_{\\mathbf {u}}{\\mathbf {v}}+\\Gamma (\\varphi )\\{{\\mathbf {u}},{\\mathbf {v}}\\}", "2722507f492b9f44ae9c9abc0188b28e": "\\rho (a,b)=d(p^{-1}(a),p^{-1}(b))", "2722a27f161295a14d8d129a7aadad9b": "\\tan(\\arcsin x)={\\frac {x}{\\sqrt {1-x^{2}}}}", "272309cb7aa31f4ac8ffb4f013657f64": "\\tau _{xy}", "27232eb3a713273f57e5859d98768d0b": "f({\\mathcal {Z}}(M))\\subseteq {\\mathcal {Z}}(N)\\,", "27238506c1549f5079b6c7b31a03c708": "\\left(J_{z}^{(m)}\\right)_{m'm}=m\\delta _{m'm}\\,\\quad \\left(J_{x}^{(m)}\\pm iJ_{y}^{(m)}\\right)_{m'm}=m\\delta _{a',a\\pm 1}{\\sqrt {(a\\mp m)(a\\pm m+1)}}", "2723ae0b4c42b64fe8fb495322da10b2": "f(x)g(y)", "2723fe64199bf849b28f7b6e7e5fbd67": "f=f(E)", "27246e52868d656102bf905ef9aab6e3": "{\\frac {\\partial y}{\\partial \\mathbf {x} }}", "27247403fb09ec1edee645ad9e07e454": "G=H-TS_{int}\\,", "2724a23b16ca720546476383e9bc8610": "\\phi _{x}", "27250c680d6309c7511d97267f74686f": "X'_{\\beta }", "272564c8cdaba01a36380357c60dea2c": "D(\\cdot )", "2725725bf31da5361bed83ac9888b95b": "g^{\\alpha \\beta }{}_{,\\beta }=k\\,g^{\\mu \\nu }{}_{,\\gamma }\\eta _{\\mu \\nu }\\eta ^{\\alpha \\gamma }\\,.", "272594d6858105a512ee9b10d837b7d4": "Af(x)=\\sum _{i}b_{i}(x){\\frac {\\partial f}{\\partial x_{i}}}(x)+{\\tfrac {1}{2}}\\sum _{i,j}\\left(\\sigma (x)\\sigma (x)^{\\top }\\right)_{i,j}{\\frac {\\partial ^{2}f}{\\partial x_{i}\\,\\partial x_{j}}}(x),", "27259969b444266306de543d25651403": "C(v)={\\begin{cases}m,{\\mbox{ if }}v\\equiv 0\\;\\;({\\mbox{mod}}N)\\\\\\\\mc,{\\mbox{ otherwise }}\\end{cases}}", "2725a6b17d894abf6c4b6ea71c596f5b": "S=\\{o_{1},\\ldots ,o_{n}\\}", "2725bcf5aa3be2c12ee559ab48f468b3": "\\ d[\\mathbf {x} (1),\\mathbf {x} (1)]=\\max _{a}|u(a)-u^{*}(a)|=00)", "272dfe092bae4beafab716f38be6aa67": "\\det {\\begin{pmatrix}A&0\\\\C&D\\end{pmatrix}}=\\det {\\begin{pmatrix}A&B\\\\0&D\\end{pmatrix}}=\\det(A)\\det(D).", "272e2b19c6c85a1876428fa4e340d7da": "d=n-k+1", "272e458d3dca9d8c009a89f5645c59f2": "T_{ij}=\\rho v_{i}v_{j}-\\sigma _{ij}+(p-c_{0}^{2}\\rho )\\delta _{ij},", "272e51dcc3dd38589dab55a249bc1426": "x=-{\\sqrt {1-y^{2}}}", "272e96042e6302c6415cf814f1be2600": "\\delta (B)\\leq c", "272f259c6011fb0b291e266456dda3a9": "\\langle E_{1}(t)E_{2}^{*}(t-\\tau )\\rangle =\\langle (E_{a1}(t)+E_{b1}(t))(E_{a2}^{*}(t-\\tau )+E_{b2}^{*}(t-\\tau ))\\rangle ", "272f9580a3fcb461e00aa20343002b99": "T_{m}=T_{1}+(T_{2}-T_{1}){e_{2} \\over (e_{2}+e_{1})}", "272fbe3e4089c58e62a042322f7e555e": "H_{G}^{*}(M)=H_{\\text{dr}}^{*}(M_{G})", "272ff88e896857905b1d384ab3f07072": "\\langle x,\\,p|\\Psi (t)\\rangle ", "27301936ef3fd7c774fe94213d1effcf": "l(\\theta )=\\alpha ", "27302d8fd3387cca567f2067f4b4f2f9": "q_{1}''(q^{c})=0", "27305e00f744bed2c72b98c86bde971d": "{\\partial u \\over \\partial t}=\\left({\\partial ^{2}u \\over \\partial x^{2}}+{\\partial ^{2}u \\over \\partial y^{2}}\\right)=(u_{xx}+u_{yy})=\\Delta u", "273068d9988f024a06942438c8e99be0": "N(u)\\setminus \\{v\\}", "273068e6b8868e1b95b9176187588792": "a_{1}x_{1}+\\cdots +a_{k}x_{k}=0.", "27308c709962c69aa6fd62d24482a381": "K{\\big (}{\\tfrac {1}{4}}({\\sqrt {6}}-{\\sqrt {2}}){\\big )}=2^{-{\\frac {7}{3}}}3^{\\frac {1}{4}}\\pi ^{-1}\\Gamma {\\big (}{\\tfrac {1}{3}}{\\big )}^{3}", "2730980eadd097914e11cf8a762bf03f": "\\star du=dv,", "2730a4212af0788b28e3ea2b2eeeb574": "(J^{2}-J_{z}^{2})=(J_{x}^{2}+J_{y}^{2})", "273142f91a8bfde342d386ddfa02c8d2": "(19)\\quad ds^{2}=-{\\Big (}1-{\\frac {2M(v)}{r}}+{\\frac {Q(v)}{r^{2}}}{\\Big )}dv^{2}+2dvdr+r^{2}(d\\theta ^{2}+\\sin ^{2}\\theta \\,d\\phi ^{2})\\;.", "2731456537f2255917a445dceb1e9fdb": "\\gamma _{\\tau }", "27315dfd3971aab09d6a8e52151253d8": "A_{eff}={\\frac {P_{o}}{PFD}}\\,", "2731a8db5befb60c2d7825282eacd372": "S\\in {\\mathcal {A}}", "273224ac9b0fbb35a97d89c586171e8f": "{\\begin{aligned}\\operatorname {Re} \\left\\{{\\frac {d}{dt}}(Ae^{i\\theta }\\cdot e^{i\\omega t})\\right\\}&=\\operatorname {Re} \\{Ae^{i\\theta }\\cdot i\\omega e^{i\\omega t}\\}\\\\[8pt]&=\\operatorname {Re} \\{Ae^{i\\theta }\\cdot e^{i\\pi /2}\\omega e^{i\\omega t}\\}\\\\[8pt]&=\\operatorname {Re} \\{\\omega Ae^{i(\\theta +\\pi /2)}\\cdot e^{i\\omega t}\\}\\\\[8pt]&=\\omega A\\cdot \\cos(\\omega t+\\theta +\\pi /2)\\end{aligned}}", "27326b1b2e401285ac8f33cce12aded5": "V(t)^{2}/Z", "27326c064bb748e01d8469e2eee6b94b": "\\mu _{1},\\mu _{2}", "27326eaea95493aedd18697142882df0": "{DF}_{T}", "2732ccc1df31bcec9df77697931274fa": "\\displaystyle {(f,g)_{S}=(Sf,g).}", "273376e6e51dab1a756368e033fe92b3": "\\sum _{n=0}^{\\infty }(an+b).", "2733a5527c58793650f09341d82595ce": "R_{s}={\\frac {1}{\\int _{0}^{x_{j}}\\mu qN(x)dx}}", "2733f6cd7314219f0318616ed8166651": "E[h(y)]=\\int _{-\\infty }^{+\\infty }{\\frac {1}{\\sqrt {\\pi }}}\\exp(-x^{2})h({\\sqrt {2}}\\sigma x+\\mu )dx", "273409400dc642485c4a2be9d987681f": "f\\rightarrow \\langle h,f(T)h\\rangle ", "27342c9a10f65aab8a44d6ff2301e363": "C_{abcd}=R_{abcd}-{\\frac {2}{n-2}}(g_{a[c}R_{d]b}-g_{b[c}R_{d]a})+{\\frac {2}{(n-1)(n-2)}}R~g_{a[c}g_{d]b}", "27348e00eee50bc77c7521cffe028c51": "b(z)=b_{0}+{\\tbinom {n}{1}}b_{1}z+{\\tbinom {n}{2}}b_{2}z^{2}+\\dots +b_{n}z^{n}", "27349e3432e336d849b069608d2909bf": "g\\left(\\alpha \\,\\right)={\\mbox{isolated guessing; address space is re-randomized after each attempt}}\\,", "2734ce623bc12ed8eb6eafec22ab46d4": "{\\overline {\\theta }}", "2734fd6cfb087caf20b85c37f0ef1b0c": "f(n)\\in \\Omega (g(n))", "273513b017764885c376cff21b5ecd82": "(N-1)(N-2)", "27354ffafa010686959ef57bf0510760": "\\varphi _{n}=\\sup\\{|f(x)|:x\\in F_{n},\\;\\|x\\|\\leq 1\\}", "273595a80ccb2f9dfc683b33a2f8fcd4": "(A+B+C)x^{2}-(3A+2B+C)x+2A=1.\\,", "2735c1f8e2b446903d0489951800dcaf": "\\Pr _{\\mathcal {S}}\\left\\{E_{a^{n}}^{\\dagger }E_{b^{n}}\\in N\\left({\\mathcal {S}}\\right)\\right\\}={\\frac {2^{n+k}-1}{2^{2n}-1}}\\leq 2^{-\\left(n-k\\right)}.", "2735f5bfd8a9ed0062df80a0c0960db1": "\\int f\\,dV=\\int _{-\\infty }^{\\infty }f(t)\\,\\sinh ^{2}t\\,dt.", "27361797574a55a7a543e6dce7f7ad2e": "A\\rightarrow \\alpha ", "2736996057a533311c4b64ab342bdc49": "\\gamma [L+1]", "2736c5b9264fd174b425f09803c70c04": "e(Z-1)", "2736d327efadfbd0e9d3d5fe41cfd660": "g_{1},g_{2}", "2736e11c017eec93b2735d395fff92e5": "I<10^{-3}", "2736fa5d4d16619198139bda55a6df0b": "Velocity=0.0", "27370f4a0d4637879cfbfb3c5573d2f5": "\\left(1+\\beta \\sin(\\Omega t)\\right)Ae^{i\\omega t}=Ae^{i\\omega t}+{\\frac {A\\beta }{2i}}\\left(e^{i(\\omega +\\Omega )t}-e^{i(\\omega -\\Omega )t}\\right).", "2737364a2bd27c93d28d95b771bfb090": "\\mathrm {F} (E)\\times _{\\rho }V", "2737725072e41660df24fc564a017d57": "n!=(n)(n-1)...(2)(1)", "273774476b615c4956b4fc8fdd9a47be": "\\log(xy)=\\log x+\\log y", "2737b027901e1b6811cc275c4db39e97": "\\{\\gamma \\,_{m}\\}", "2737c4d15de5fe830975d5795cafdb56": "\\color {Black}{\\tfrac {\\infty }{m}}\\infty ", "2737ce399344c73e17c42185c211b3f8": "A_{(\\alpha }B_{|\\beta |}{}_{\\gamma )}={\\dfrac {1}{2!}}\\left(A_{\\alpha }B_{\\beta \\gamma }+A_{\\gamma }B_{\\beta \\alpha }\\right)", "273802aae2355f16e5f5f43ca18074ad": "n_{a}(\\mathbf {k} )", "2738439e6db96030205f02509efc0547": "a_{2}'=a_{0}\\oplus a_{1}\\oplus a_{2}\\oplus a_{6}\\oplus a_{7}\\oplus 0=0\\oplus 1\\oplus 0\\oplus 1\\oplus 1\\oplus 0=1", "2738695e271ec60d8ca1fe16da705bc4": "{\\ddot {r}}={\\frac {d^{2}r}{dt^{2}}}", "2739107d20f9b46d9169acd038d6e59e": "A^{\\mathfrak {n}}", "2739214e3cb8b4e6ce30b84a42cdcbe3": "{\\begin{aligned}{\\bar {I_{L}}}&=\\left({\\frac {1}{2}}I_{L_{max}}DT+{\\frac {1}{2}}I_{L_{max}}\\delta T\\right){\\frac {1}{T}}\\\\&={\\frac {I_{L_{max}}\\left(D+\\delta \\right)}{2}}\\\\&=I_{o}\\end{aligned}}", "27395b7ab42c3383bd8ba4d3976a6646": "{\\rm {i}}{\\partial \\over \\partial t}\\psi _{t}=-{\\nabla ^{2} \\over 2}\\psi _{t}\\,", "273a383345e167ee1791232c40eaf917": "v(t)", "273ad1a55db5eb251418161c4734fe02": "Height(t)>k", "273b0dfa8d54fbb6c3ed1d2bc26ee2d7": "\\therefore ,\\because ,\\And \\!", "273b1b32341db7227180543aa12f06d5": "{\\begin{matrix}{\\frac {3}{51}}={\\frac {1}{17}}\\end{matrix}}.", "273b41404f9e076c3264efb39726825a": "W_{w}", "273b55d642a822a5bf1bd829e454fcf4": "\\Rightarrow {\\dfrac {7385}{33}}", "273b947d255f328e21163da6e3af9da3": "d.f.\\cong {e^{\\left(\\ln {\\frac {N-1}{2n}}\\ln {\\frac {(2n+1)(N-1)}{4}}\\right)}}^{-{\\frac {1}{2}}}", "273bceda822e194e1b1eccdb6f67e815": "{\\text{true}}", "273c8b2ab7bb3acf9ec2d790c88ee1ef": "lb_{computed}", "273c9d6047355050192260495b413941": "\\gamma _{2}=1", "273ca24b1aa43b0a863a2b80415a2c21": "x\\prec {\\bar {y}}", "273d4557819e10ea0dd94ba8e91bf251": "\\mathrm {DOF} =4Nc", "273d89150f20417c299b6d41216a2511": "F_{t}=\\ -J_{3}\\ {\\frac {1}{r^{5}}}\\ {\\frac {3}{2}}\\ \\left(5\\ \\sin ^{2}i\\ \\sin ^{2}u\\ -1\\right)\\ \\sin i\\ \\cos u", "273da6412bb76f3f49287e4db7c8f5f9": "\\lim _{n\\rightarrow \\infty }\\mu _{n}(X)=\\mu (X)", "273dff67eba5fe343aac9946c70d1a83": "\\varphi '(a_{i})=-{\\frac {A_{i}e^{-\\kappa a_{i}}(1+\\kappa a_{i})}{a_{i}^{2}}}=-{1 \\over 4\\pi \\varepsilon _{r}\\varepsilon _{0}}{z_{i}q \\over a_{i}^{2}}=\\varphi _{sp}'(a_{i})", "273e187a61dd341f2b2d96cef3d9ecdb": "\\psi _{n}(x)=x^{n}\\,h(x),\\qquad x\\in \\mathbb {R} ,", "273e3bd7999abb12af0c0bdaf3ec1bf7": "mU{\\frac {d\\gamma }{dt}}=-Z_{u}u", "273e3f766f178f55f3108f429c02a840": "A_{(\\alpha \\beta )\\gamma \\cdots }", "273ed8b39a916ed779b7d0f616a9192c": "\\lambda >0", "273ef2b4be84c3ab034dbed66af5e48b": "\\chi ^{2}={(|b-c|-1)^{2} \\over b+c}.", "273efb966e0a9faf69e0cf69d21eaea4": "{\\begin{bmatrix}X\\\\Y\\\\Z\\end{bmatrix}}^{B}={\\begin{bmatrix}c_{x}\\\\c_{y}\\\\c_{z}\\end{bmatrix}}+(1+s\\times 10^{-6})\\cdot {\\begin{bmatrix}1&-r_{z}&r_{y}\\\\r_{z}&1&-r_{x}\\\\-r_{y}&r_{x}&1\\end{bmatrix}}\\cdot {\\begin{bmatrix}X\\\\Y\\\\Z\\end{bmatrix}}^{A}", "273f38d10aeb8f535c3bad342e1e4252": "RMR^{-1}.", "273f75a47297087306deba5501531c59": "A(i)>A(j)", "273f765deba06af4b937fedff2c2fb3e": "\\sup _{V}u\\leq C\\inf _{V}u", "273ff250d6ed9ff94e2ea774603f8a01": "Y^{k}\\ ", "27403c2daa55b104ef80b186167ca301": "x_{i}=y_{i}\\ ", "27406d836990b1d394906916c59c8028": "z_{t}=y_{t}-y_{t-1}", "2740b4ab0224dba921f390db5d3a7f5b": "\\int _{\\mathbb {R} ^{n}}e^{-x^{T}Ax+v^{T}x}\\,dx={\\sqrt {\\frac {\\pi ^{n}}{\\det {A}}}}\\exp({\\frac {1}{4}}v^{T}A^{-1}v)\\equiv {\\mathcal {M}}\\;.", "2741d556b96698c7ec7672166f996a65": "p\\gg \\Lambda ", "2741f9b0051c0b0418fb36daa0559c26": "m_{\\ell }=-2,-1,0,1,2", "27422b7a94e97b3fecaf9f1f9195bfcf": "\\{S\\mid T\\subseteq S\\subseteq A{\\mbox{ and S totally ordered}}\\}", "274235d3c8c0f0991d1f6a54d34065e4": "{\\begin{matrix}({\\mathcal {A}}B)(\\psi )=\\\\(\\forall \\phi :\\phi _{0}=\\psi \\to B(\\phi ))\\end{matrix}}", "2742f5baabd3c896d37d1d18124405cc": "bA_{4}=A_{1}\\cup A_{2}\\cup A_{4}", "2743611decaf752d841c21b28fc798e4": "{\\tilde {Y_{k}}}", "274372371f2edd16f3b4d357c074d957": "i_{V_{s}}+i_{R}=0", "274392175c4682d9c471ce92ef255f65": "\\mathbb {D} (A,B)", "2743922d0685f249a012ab2ffe852c6b": "{\\bar {r}}={\\frac {1}{nk}}\\sum _{i=1}^{n}\\sum _{j=1}^{k}r_{ij}", "2743e38f46e4b83c5e2516babff4eb20": "\\digamma ={\\frac {3FL}{2d}}", "27440a41978b7d37992788a5f4c26f05": "\\displaystyle {\\mathrm {dim} \\,{\\mathfrak {k}}-\\mathrm {dim} \\,{\\mathfrak {k}}_{a}=\\mathrm {dim} \\,{\\mathfrak {m}}-\\mathrm {dim} \\,{\\mathfrak {m}}_{a},}", "2744584198cf6b5801d281dd3e75e745": "\\mathbf {r} _{j}", "274464ffdd99a069f6d52e7f017e90cc": "{\\begin{aligned}\\sigma (x,a)&=0\\\\\\sigma (x,x)&=\\sigma ^{2}(x)\\\\\\sigma (x,y)&=\\sigma (y,x)\\\\\\sigma (ax,by)&=ab\\,\\sigma (x,y)\\\\\\sigma (x+a,y+b)&=\\sigma (x,y)\\\\\\sigma (ax+by,cw+dv)&=ac\\,\\sigma (x,w)+ad\\,\\sigma (x,v)+bc\\,\\sigma (y,w)+bd\\,\\sigma (y,v)\\end{aligned}}", "2744931aa0c18c50218f0195578f4c14": "\\Delta {\\overline {u}}", "2744b8272939c5e4f88f6dc77c12c90d": "{\\bar {w}}_{1L}(s,2n+\\gamma _{1L};L)=\\delta _{n0}{\\bar {\\Gamma }}_{1L}(s)+\\sum _{c=1}^{[L/2]}(-1)^{c+1}{\\bar {w}}_{1L}(s,2(n-c)+\\gamma _{1L};L){\\bar {h}}(s,c;L).", "27451f893b3b7d536f0c01862879971e": "\\mathbf {u} _{1},\\ldots ,\\mathbf {u} _{n}", "27457b029b382d6371057fe7f2a27801": "T_{3}=(T_{1}Y_{2})^{2}+(Z_{1}X_{2})^{2}=16", "2745a719cb8d9bb76b107d35fcd04b0a": "(f_{j}f_{k})(g)=f_{j}(g)f_{k}(g)", "2745b7de130e087b3d5a0e765dfe4c87": "{\\frac {L^{\\alpha }}{1-\\left({\\frac {L}{H}}\\right)^{\\alpha }}}\\cdot \\left({\\frac {\\alpha }{\\alpha -2}}\\right)\\cdot \\left({\\frac {1}{L^{\\alpha -2}}}-{\\frac {1}{H^{\\alpha -2}}}\\right),\\alpha \\neq 2", "2745e451109e98fe2cfc022ec0c5bb06": "J^{\\mu ,\\nu }(z)=\\sum _{k\\geq 0}{\\frac {(-z)^{k}}{\\Gamma (k\\mu +\\nu +1)k!}}.", "2745ef042f8cd42fa6f627c5f0e62cbd": "{}_{2}F_{1}(a,b;c;z)=1+{\\frac {ab}{c\\,1!}}z+{\\frac {a(a+1)b(b+1)}{c(c+1)\\,2!}}z^{2}+{\\frac {a(a+1)(a+2)b(b+1)(b+2)}{c(c+1)(c+2)\\,3!}}z^{3}+\\dots \\,", "2745f778411586078eb847365808f476": "\\mu :T^{2}\\to T\\,", "2746271b2ccb4ac7ec7409546bdcd631": "|z|=a>1", "2746325ce40ce297977ef99cf076bedc": "\\,R\\,", "2746f6ad41814755ddf43e9bff638fdd": "v_{xy}=0", "274701d8c17341ed11d68088a3da3895": "P_{i}=(x_{i},v(x_{i}))", "27470832e77b40587fcdba53c5009d50": "f(\\sigma ^{*})=\\sigma ^{*}", "274708f04fe3333db664eb753745c82d": "{\\boldsymbol {\\nabla }}\\times ({\\boldsymbol {\\nabla }}\\times {\\boldsymbol {\\varepsilon }})={\\boldsymbol {0}}", "2747deb4bda32d52514dbcb1fcbc98fa": "{\\text{s.t.}}={\\begin{cases}g_{1}\\left({\\boldsymbol {x}}\\right)&=x_{1}+x_{2}-2\\geq 0\\\\g_{2}\\left({\\boldsymbol {x}}\\right)&=6-x_{1}-x_{2}\\geq 0\\\\g_{3}\\left({\\boldsymbol {x}}\\right)&=2-x_{2}+x_{1}\\geq 0\\\\g_{4}\\left({\\boldsymbol {x}}\\right)&=2-x_{1}+3x_{2}\\geq 0\\\\g_{5}\\left({\\boldsymbol {x}}\\right)&=4-\\left(x_{3}-3\\right)^{2}-x_{4}\\geq 0\\\\g_{6}\\left({\\boldsymbol {x}}\\right)&=\\left(x_{5}-3\\right)^{2}+x_{6}-4\\geq 0\\end{cases}}", "2748bb44615a7859b19a958469200f1b": "3.{\\overline {3}}", "2748be1ecae5019bb08f353c3230702b": "\\mathbf {3} \\otimes \\mathbf {\\overline {3}} =\\mathbf {8} \\oplus \\mathbf {1} ", "2749264821871f76e74e56829e430476": "\\alpha =i", "2749c4111d84fb1911141689ca95c933": "f_{x}:E_{x}\\to E'_{f_{0}(x)}", "2749e8ff2fc48c656451643749bc72bc": "IMA={\\frac {F_{out}}{F_{in}}}.", "2749fe8e710a06b7301c082e27dbd803": "\\mathrm {spt} (13n+6)\\equiv 0\\mod (13)", "274a210151565a0b89a12fba5470b55a": "L(j)", "274a4de4dba5deab523aad21aeb4e6aa": "{\\frac {\\partial C}{\\partial t}}+\\mathbf {v} \\cdot \\nabla C=0.", "274a5f0569d1e219981d6147ec062425": "t\\rightarrow \\pm \\infty ", "274a6367f06056ff9ef9597c14588c44": "f(500)=500({\\sqrt {501}}-{\\sqrt {500}})=500(22.3830-22.3607)=500(0.0223)=11.1500", "274a90691bb0359c7d12ef37a056faef": "DN_{c}=1+(DN_{1}-1)+(DN_{2}-1)+...+(DN_{n}-1),\\ ", "274ab8e8dc930b3609d8248a47a5d111": "{\\big .}U={\\frac {kA}{\\Delta x}},\\quad ", "274ac86b27a6521c4b0c711bd91514c5": "~\\sigma _{\\rm {ep}}~", "274b4810b6ea396de824bcdea4c5a773": "\\omega =2\\pi /86164", "274b5d0c8b33506551742b2564f9c661": "\\det T_{h}=(-1)^{\\lfloor {\\frac {b-a+1}{4}}\\rfloor }\\cdot h_{a}\\cdot h_{b}\\cdot \\mathrm {res} (h_{\\mathrm {e} },h_{\\mathrm {o} })", "274b905ffd111cb886d605eb67de88c2": "G_{V_{1},E_{1}}\\square H_{V_{2},E_{2}}\\rightarrow J_{(V_{1}V_{2}),(E_{2}V_{1}+E_{1}V_{2})}", "274bb8be45054e2093b55dee1569bc81": "\\mathbf {\\hat {n}} \\cdot \\mathbf {\\hat {v}} =\\cos(\\theta )\\,", "274c7beee2ccab7eefceb36038ce4f12": "y_{k}={\\mathcal {P}}_{D}(x_{k}+p_{k})", "274c96707e6dd23c702e2cc6d2c3a942": "D=v_{1}f_{1}+v_{2}f_{2}", "274d1181b531f1fa4be4a7eb87c03d40": "(x+1)^{3}p({\\frac {1}{x+1}})=512x^{3}+512x^{2}-128x-64", "274d451e072af48bd0a82bd5dd0a47f2": "\\exists x\\forall y(\\phi (y)\\iff y=x\\land \\psi (y))", "274d4cc0ebee8feefb24798cb04dfb55": "\\mathbf {\\hat {e}} _{i}\\cdot (\\mathbf {\\hat {e}} _{j}\\times \\mathbf {\\hat {e}} _{k})=\\varepsilon _{ijk}", "274d95b0d5faeb0e766fc5486c6acb38": "-1,1\\,", "274dc5087b22c2c1b19fb3e2ee80f32e": "k[x_{1},\\ldots ,x_{d}]", "274dc9c31e7eabf01ae5d1cc4f618a3c": "y=S/N", "274dd0431919a6dd17e19820a505db13": "x_{1}(t)={\\begin{cases}\\cos(\\pi t);&t<10\\\\\\cos(3\\pi t);&10\\leq t<20\\\\\\cos(2\\pi t);&t>20\\end{cases}}", "274dfdba46ac7ef6561fca7ec765267f": "\\ max(m,n)", "274e072c81900310c789143e8ec677b1": "J_{\\alpha }(x)=\\sum _{m=0}^{\\infty }{\\frac {(-1)^{m}}{m!\\,\\Gamma (m+\\alpha +1)}}{\\left({\\frac {x}{2}}\\right)}^{2m+\\alpha }", "274e3682289056f732f253205550354f": "\\Delta E_{00}^{*}={\\sqrt {\\left({\\frac {\\Delta L'}{k_{L}S_{L}}}\\right)^{2}+\\left({\\frac {\\Delta C'}{k_{C}S_{C}}}\\right)^{2}+\\left({\\frac {\\Delta H'}{k_{H}S_{H}}}\\right)^{2}+R_{T}{\\frac {\\Delta C'}{k_{C}S_{C}}}{\\frac {\\Delta H'}{k_{H}S_{H}}}}}", "274e9493a3a437dc2fcae487471409c9": "w_{i}={\\frac {m_{i}}{m_{tot}}}", "274ea7dd52614b381a01dc942fd2e7d8": "Q(\\omega )\\,", "274ed9633cd0696c350a505ae4eb3039": "x(\\theta )=(R-r)\\cos \\theta +r\\cos \\left({\\frac {R-r}{r}}\\theta \\right)", "274f97d6fd93dd1926bf998e332bcb4e": "\\mathbf {E} =\\mathbf {K} '^{\\top }\\;\\mathbf {F} \\;\\mathbf {K} ", "27507d53b6f4eaa2c507cdc12e17b4d2": "E_{d}>1\\!\\ ", "2750fc7ed3ddda1d2dcca7e74183c879": "\\ldots f(x,q_{k})\\geq r_{k}\\,\\!", "27516f7d1bc941a1810087cf9796f8df": "W=-\\int _{1}^{2}\\mathbf {F} \\cdot \\mathrm {d} \\mathbf {s} ", "27518edcf5038ba208f50e61859b125d": "u\\,J_{\\nu }(ut)", "275201561e2fcc42c147997417eb7625": "\\exp \\left(-\\sum d_{i}\\right)", "27520466c0dcb2fdd0acd6bd77193739": "16*x^{4}+5152*x^{3}+518420*x^{2}+16693124", "27526e1672d2f4f9db8ca0a5d04959dc": "\\eta :I\\to M", "2752c313e4c3583391429ea73e5882c6": "u\\cdot d", "2752fc873e24e06963596584eb15e9a7": "{\\mathcal {F}}(U)\\to {\\mathcal {G}}(U)", "2752fe31fd48c9ad40cd6781534c3ad8": "P_{2}(0)=P_{2}(1)=0.", "27536adacd8ee817df66722e0483d870": "\\operatorname {Var} (X_{i})=\\sigma ^{2}", "27537cd086a8e9f62bba71321bd87403": "X_{B}", "27539b46a71ca477e033401acbe8e22e": "{\\frac {\\delta l}{\\delta t}}={\\frac {\\sum P}{8r^{2}\\eta l}}(r^{4}+4\\epsilon r^{3})", "2753a06f3f52c90fd3e8fc243fada346": "A={\\frac {p\\cdot q}{2}},", "2753a7be181ccde924a436eba34ef1f0": "A(n,m)=\\sum _{k=0}^{m}(-1)^{k}{\\binom {n+1}{k}}(m+1-k)^{n}.", "2753c1390f7d25bb15a1b61f29e832c6": "x_{2}=KI\\oplus x_{1}", "27540f6921a0646a06570dde652bb017": "=9000\\times 1.1304\\times 1.1304\\times 1.1304=13000", "275488a53dfd39c8fa0f7d365a6820b2": "b/a", "275496a85f248b8d094a345332fd4266": "P(M,S)=P(M,S_{0})+P(M,S_{1})+\\dots +P(M,S_{n})", "2754ef5a26aa5b053d28cf609ec2f48d": "B_{0}=3.12\\times 10^{-5}\\ {\\textrm {T}}", "2754fa3e07b2d98d2c58c79f20512b58": "\\;p(t)=Pr^{t}-A{\\frac {r^{t}-1}{r-1}}", "2754fadb9f556c033589c1e5ce85e695": "\\displaystyle L_{p}(1-n,\\chi )=(1-\\chi (p)p^{n-1})L(1-n,\\chi )", "275517a7bfa1c9f113d2e1f197fa8700": "\\textstyle (c_{n})_{n\\geq 0}", "2755714f8654906e9829a74c412e3464": "\\Gamma ^{\\mu }{}_{\\alpha \\beta }\\ ", "2755735b2dd7b15a651370c88d3a8d29": "\\chi =B\\rightarrow A", "2755b22abce254d35f89293ff1ba6247": "t_{(1-\\alpha )}^{*}", "2755f5717fbcdcf457d033da65def3f2": "\\epsilon =I_{\\mbox{linear}}+I_{\\mbox{nonlinear}}=0", "27560cccbd150a50a2cf3be11f61c062": "(s^{2}+0.5176s+1)(s^{2}+1.4142s+1)(s^{2}+1.9319s+1)", "2756438f419aab48ab30ac35a1e0090b": "m^{e-1}{\\text{ (mod }}p{\\text{)}}=1", "275667c283fa17dfe2e5b03c1c9875eb": "-\\csc ^{2}(x)", "2756ae341c28ecc5ac73293da66c326e": "\\mu _{B}B/k_{B}T", "27571223560bb0486eddf5e2b505179f": "{\\tfrac {BF}{AF}}", "27571c4b401b45a4691ccbcce0b83cb6": "P(X=x)", "27577bb7e608b77faf24300e302085bd": "D={\\frac {1}{\\ell \\nu }}", "2757f10aaa345ef4178642164895d8d8": "{\\frac {\\rm {d}}{{\\rm {d}}z}}\\,\\mathrm {erf} (z)={\\frac {2}{\\sqrt {\\pi }}}\\,e^{-z^{2}}.", "275828025cde6a5e68b33960e1e9481d": "b={\\frac {I}{AM}}.", "275850a434439f4c91561fc949b9b4e0": "\\mathbf {H} ", "2758a232ea11474a55b32d55425abc4b": "\\varphi {\\Big (}\\varphi (x){\\Big )}={\\rm {e}}^{x}", "275927a68c49aa1e97f8897df962caf3": "F(t_{1},t_{2},\\dots )/(p,q,\\dots ).", "27592bcdffd7f2e1bc027f89075caf46": "\\left(3{\\sqrt {\\frac {2}{5}}},\\ \\pm {\\sqrt {6}},\\ \\pm {\\sqrt {3}},\\ \\pm 1\\right)", "27596623be4332515c975fc35488a568": "R={\\frac {\\sum _{i=1}^{K}\\sum _{j=1}^{L}R_{ij}}{\\sum _{i=1}^{K}n_{i}}}", "2759f89dbe6f3b7665df5cf686c9adb1": "(G1+G2)/2", "275a1b7d253c6554ef51fd2cc2b53492": "ELR=\\sum _{a=a_{E}}^{\\infty }\\left[M(a)+m(E,a_{E},a)\\right]S(E,a_{E},a)-\\sum _{a=a_{E}}^{\\infty }M(a)S(0,a_{E},a)", "275adc3d2dd05b8a995f3cc55f7d3c9a": "\\mathbf {\\nu } (x)=\\int \\theta (x)=-{\\frac {5}{12}}x^{4}+{\\frac {75}{6}}x^{3}-1406.25x(m)", "275b2f471d49a7f76745136620f4e43b": "x^{7}+x^{6}+x^{5}+x^{3}+x", "275b9b1a4ed62e142f27b4565df3cab5": "A(t)=A(0)e^{gt}", "275ba6890cc5db6dc8db920a1caf425d": "\\Delta vV_{\\text{esc}}", "275bb6ebbe2a815ef757d9cfe4d0e006": "r_{t}=\\mu +{\\bar {\\sigma }}(M_{1,t}M_{2,t}...M_{{\\bar {k}},t})^{1/2}\\epsilon _{t},", "275beccbaa7ee56fbe17692df9c282f0": "(A,d)", "275c0fa76f168671c9728ee6b225506c": "Z_{t}=1+\\int _{0}^{t}Z_{s}\\,dX_{s}.\\,", "275c9a28afb8e92b2f3d0085f6c45224": "B=\\sqcup _{\\lambda }B_{\\lambda }", "275c9b0124c604ab7bc5603feccfb615": "p\\leq 2k-2", "275cf1574a89936fbebf4d8d827b9082": "P^{\\prime }(X_{1},X_{2},X_{3},X_{4},X_{5},X_{6})=P(X_{6}|X_{5})P(X_{5}|X_{2})P(X_{4}|X_{2})P(X_{3}|X_{2})P(X_{2}|X_{1})P(X_{1})", "275d1db172f7bad9992a841b1c5d0100": "\\beta (i)", "275d2ad3cc8d2ac9f7c6372fbb672fbe": "-2\\leq k\\leq 2{\\pmod {8}}", "275d33d92153da2dd486a049d5a1d776": "{{\\hat {\\mathcal {H}}}_{\\text{KS}}}=-\\sum \\limits _{\\mathit {i}}^{N}{{{\\gamma }_{\\mathit {i}}}\\cdot {{\\hat {\\vec {I}}}_{\\mathit {i}}}\\cdot {{\\hat {\\mathbf {K} }}_{\\mathit {i}}}\\cdot {\\vec {B}}}", "275d3f445c1cbdb2efb83ac8406c9d27": "\\mathbf {V} _{i}=\\mathbf {V} +{\\frac {d{\\mathcal {R}}}{dt}}{\\mathcal {R}}^{T}{\\mathcal {R}}\\mathbf {r} _{io}", "275d5864e98f807904d403068dfc2a57": "{\\widehat {\\boldsymbol {\\theta }}}_{LS}", "275d93aed63268d7f17314861156435b": "\\theta ={\\frac {\\pi }{2}}", "275daf7d20ada2812241b6fbe246aae9": "d\\omega (X,\\mathbb {D} )/dH^{1}", "275dc431f9270317c68595220d6b8730": "v_{j}", "275de3118360796a2e523dbed9f4c044": "\\mathbf {v'} _{r}\\cdot \\mathbf {\\hat {n}} =-e\\mathbf {v} _{r}\\cdot \\mathbf {\\hat {n}} ", "275dfd3a7281b1ba81bdd8ab1be29465": "V_{D}", "275e5785cd9aa8c6141e54d6378fe0de": "\\displaystyle \\operatorname {Tr} (R(f))=\\int _{\\Gamma \\backslash G}K_{f}(x,x)\\,dx.", "275e68530f43b1233e17955c496f6be3": "\\neg A\\lor B\\iff \\neg A\\lor (\\neg \\neg B)", "275effecf23e17e575a1e5ef0b480385": "S:\\{0,1\\}^{8}\\to \\{0,1\\}^{8}", "275f1d59d7ef4ee5d1a381e9b83ccce7": "\\scriptstyle f/f_{s},", "275f3d8e81cd44f0fc70179f4852e725": "R_{i}\\,=\\,{\\sqrt {(x_{i}-x)^{2}+(y_{i}-y)^{2}+(z_{i}-z)^{2}}}", "275f48850d587181d5e397125b162a14": "\\Pr(X\\geq a)=\\Pr(\\Phi (X)\\geq \\Phi (a))\\leq {\\frac {{\\textrm {E}}(\\Phi (X))}{\\Phi (a)}}.", "275f6b0cbab448a7a6f8e52da71aa7bf": "{\\check {R}}=T\\circ R", "275fa8e2c161ed4a15cf65b6fb619f5a": "2(2n+1)XP_{n}(X)=-P_{n+1}(X)+(2n+1)P_{n}(X)-n^{2}P_{n-1}(X).", "27600dac5d54ba3e39941d5cc6cfa3da": "n=\\pi \\left({\\sqrt[{3}]{m}}\\right)", "2760555d69e9fc19df02ab85f2440a3b": "R_{\\alpha }[f](s)=R[f](\\alpha ,s)", "2760aa4b069c475548ea095563ab2a96": "{\\begin{aligned}\\varphi \\colon {\\mathcal {X}}&\\to \\mathbb {R} ^{\\mathcal {X}}\\\\\\varphi (x)&=k(\\cdot ,x)\\end{aligned}}", "2760c769ad326e9af6a83ddf5a51131b": "{\\frac {dS}{dt}}=\\mu N-\\mu S-\\beta (t){\\frac {I}{N}}S", "2760ceb1f2f894726e40ce3387b99a0f": "{\\begin{cases}\\;[~]_{{\\mathcal {S}}(\\sigma )}&{\\mbox{if }}P=\\sigma (~)\\\\\\;[{\\mathcal {T}}(R)]_{{\\mathcal {S}}(\\sigma )}&{\\mbox{if }}P=\\sigma (R)\\\\{\\mathcal {T}}(Q)\\;\\|\\;{\\mathcal {T}}(R)&{\\mbox{if }}P=Q\\,\\mid \\,R\\\\\\end{cases}}", "27614dca15e222a355daab25528a0eab": "\\{<_{1},<_{2},\\ldots ,<_{n}\\}", "27614e270bda489e4eee7038e76f74bf": "p(c+\\rho z).", "27618de2d512b4918e032c24b793ab71": "\\sigma \\cdot (x_{1},\\ldots ,x_{n})=(x_{\\sigma ^{-1}(1)},\\ldots ,x_{\\sigma ^{-1}(n)}).", "2761f9529f1ee0b20b58ce5ecddff9e1": "Ce^{ax}\\!", "2762084c8ab6713405b97010c9bd8c4d": "{\\overline {X_{n}}}:={\\frac {X_{1}+\\cdots +X_{n}}{n}}", "27625e2398b270762ec395c1fcec2d30": "{\\hat {G}},", "276269c9fbc9a3c07e05ab12ae4600cb": "{\\frac {\\partial }{\\partial t}}(\\rho \\phi )+\\nabla (\\rho \\mathbf {u} \\phi )\\,=\\nabla (\\Gamma \\cdot \\operatorname {grad} \\phi )+S_{\\phi }", "276279c8d1c2bd8f187c15b0834ec189": "\\Delta {{\\vec {p}}_{1,2}}=-\\Delta {{\\vec {p}}_{2,1}}", "27628bb77f94f52fc32b0ab3b77d8bdd": "\\mathbf {d} _{z}", "2762993a69d1b1c0c0a90fc7c63ac7e4": "x^{13}\\,", "2763295b50ce1bdac5f3a61b469d2def": "V\\approx {\\frac {1}{2}}\\sum _{s=1}^{3N-6}f_{s}q_{s}^{2}.", "276367f3edb021ea5e9463ed509ffbc3": "{\\text{PCSA}}_{2}={{\\text{muscle volume}}\\cdot cos\\Phi \\over {\\text{fiber length}}}={{\\text{muscle mass}}\\cdot cos\\Phi \\over {\\rho \\cdot {\\text{fiber length}}}},", "276427364d6f1b07ed0a19d3c673e174": "rR''(r)+R'(r)-KrR(r)=0.\\,", "27643d2c5e4581732750efc2253347ec": "f_{L}(x;\\mu ,\\sigma )=\\prod _{i=1}^{n}\\left({\\frac {1}{x}}_{i}\\right)\\,f_{N}(\\ln x;\\mu ,\\sigma )", "276457088f5bcc11f3ab6724fa1ed6a5": "I_{1}\\left(mr\\right)K_{1}\\left(mr\\right)\\rightarrow {1 \\over 2}\\left[1-{1 \\over 8}\\left(mr\\right)^{2}\\right].", "276466d4d0dfb02721775d26bde78090": "{\\mathfrak {M}}(K)=({\\mathcal {P}},{\\mathcal {Z}};\\parallel _{+},\\parallel _{-},\\in )", "276487f23a58ca0e6e32a6ebd539494a": "f(i,Y_{i-1},Y_{i},X)", "2764b53c84193b755ab4ca2decc87d74": "H=\\cup _{g\\in \\Gamma }\\,g{\\overline {F}}", "2764c01f9eb9624e8d01af0c07988c64": "{\\mathfrak {B}}(V_{1})=k[x]/(x^{2},y^{2},xyxy+yxyx)", "27653afcc6a53702237297f96ebc8153": "Y(L_{-1}a,z)={\\frac {d}{dz}}Y(a,z)=[Y(a,z),T]", "27657fab3a9b5731ceb5db440fb1914a": "S=Y\\log(p)+(Y-1)(\\log(1-p))", "2765802181072b3aa2be59dae8c72b0d": "b'", "2765ccf3a286be18c4c6f6ea271d9e83": "u_{0,k}=k\\pi ", "2765fbff363c1babf12f12163b9485c3": "\\,\\,\\,=-k_{B}\\,\\sum _{i}dp_{i}(-E_{i}/k_{B}T-\\ln Z)", "27660d020719153b246cb01e754e1b10": "{1 \\over 2(1-p^{-2})(1-p^{-4})\\cdots (1-p^{2-2t})(1+p^{-t})}", "27661d60c1191af5e5b8f3cb22618dc3": "\\{(\\mathbf {a_{i}} ,\\mathbf {b_{i}} )\\}", "27665faa8edd1df61e6d1c1e6766e57c": "\\sigma _{ph,\\omega }[I_{ph,\\omega }(\\omega _{ph},T)-I_{ph}(\\omega _{ph},\\mathbf {s} )]+{\\dot {s}}_{ph,i}.", "27666f9e2305c1fb80c6489f31036d2d": "\\mathbf {R} ^{T}\\,\\mathbf {R} =\\mathbf {I} ", "2766aede88f76ff1a4159b47b28620bb": "x^{2}y", "2766eb0d7b795c0563f89611547f28cc": "\\epsilon \\sigma A{(T_{sur}^{4}-T_{0}^{4})}+kA{\\frac {(T_{1}-T_{0})}{\\Delta {x}}}+{\\frac {e_{0}}{2}}A\\Delta {x}=0", "27671e790706777bc1031f3d8903da9a": "f\\left({\\frac {1}{\\frac {1}{r_{n}}}}\\right)=0", "276790d304068427af9e377fe64da34a": "\\lim _{x\\to \\infty }{{\\frac {A}{x-1}}+{\\frac {Bx+C}{x^{2}+x+1}}}={\\frac {A}{x}}+{\\frac {Bx}{x^{2}}}={\\frac {A+B}{x}}.", "27680532e871ad86f92138e3feb018eb": "\\pm \\left({\\sqrt {\\frac {5}{2}}},\\ -{\\sqrt {\\frac {3}{2}}},\\ 0,\\ \\pm 2\\right)", "27681508c78678529ab81bc43ca12e97": "\\phi \\land \\chi \\to \\chi ", "27681b7d5d9808e45b033b9734fff00e": "\\{\\varnothing ,\\{\\varnothing \\}\\}", "276822ed44ee8d235b176d650d8e1ecb": "f\\longmapsto \\mathrm {Hom} (A,f)=[\\![\\mathrm {Hom} (A,X)\\ni g\\mapsto f\\circ g\\in \\mathrm {Hom} (A,Y)]\\!]", "276824969ee0b642db5425f2a835ccde": "\\mathbf {v} _{0}", "27686af236aa56f334c6af2aef5b53eb": "L\\approx 0.318131505204764135312654+1.33723570143068940890116{\\!~{\\rm {i}}}", "27688d8a5c35fe922e19ec29eb4bd21b": "\\tau >1", "2768ad4f672cd88cee064df228ac2a6f": "{\\mathcal {O}}({\\sqrt {\\omega }})", "2768cfbdcda8d3f953a301764a286d9b": "\\gamma _{i}=0", "276925d9f494e4ae53bea5184a3ebba4": "\\theta |_{y=0}={\\frac {d\\theta }{dy}}{\\biggl |}_{y=L}=0", "27695ef13053008baedb4fde723531d0": "Q=CLH^{n}", "2769617597ca4aac74079cf7e76d6685": "\\lambda ={\\frac {g}{\\pi }}T^{2}tan\\beta ,", "2769fa447dac799fd4d54d734fd5cfd3": "{\\begin{matrix}\\\\{\\begin{bmatrix}1\\\\4\\\\2\\\\-3\\end{bmatrix}},{\\begin{bmatrix}7\\\\10\\\\-4\\\\-1\\end{bmatrix}}\\mathrm {and} {\\begin{bmatrix}-2\\\\1\\\\5\\\\-4\\end{bmatrix}}\\\\\\\\\\end{matrix}}", "276a36a8de51acc04265b78e5055945b": "\\mathbf {F} _{c}=2\\omega {\\frac {{\\rm {d}}r}{{\\rm {d}}t}}{\\mathbf {\\hat {e}}}_{\\theta }=2\\omega v{\\mathbf {\\hat {e}}}_{\\theta }\\,\\!", "276a99a28ea953338346c1131112576a": "M_{A}=0\\ kN\\cdot m", "276b2aff8dfe17e21e0a27ae2ee8ea2d": "\\langle \\alpha ',j'm'|T_{q\\pm 1}^{(k)}|\\alpha ,jm\\rangle ", "276b4a6542c0ae3cce06cc323f4150e9": "\\pi _{2}=\\varepsilon _{r}", "276b641f3c57e4dfe79b3378fb70a2ef": "\\approx 0.95\\times 10^{10}\\!\\left({\\sigma \\over 200\\,\\mathrm {km\\,s} ^{-1}}\\right)^{\\!3}\\!\\!\\left({\\rho \\over 10^{6}\\,M_{\\odot }\\,\\mathrm {pc} ^{-3}}\\right)^{\\!-1}\\!\\!\\left({m_{\\star } \\over M_{\\odot }}\\right)^{\\!-1}\\!\\!\\left({\\ln \\Lambda \\over 15}\\right)^{\\!-1}\\!\\mathrm {yr} ", "276b647a7050a689e389a2658420a5e9": "\\sum _{n=1}^{\\infty }|z_{n}|^{2}", "276b6c4692e78d4799c12ada515bc3e4": "ana", "276b882cb9f2ecca73a495e1f0da925b": "{\\overline {xy}}={\\tfrac {1}{n}}\\textstyle \\sum _{i=1}^{n}x_{i}y_{i}\\ .", "276b954f2a31deade9d3b18082bff936": "321=(2)(13)", "276ba7f587cc2e4cdea0a452c2e6e0f4": "(I_{3})", "276bc0f6849abfd539fbb2906a1287a0": "{\\frac {{}_{2}F_{1}(a+1,b;c+1;z)}{{}_{2}F_{1}(a,b;c;z)}}={\\cfrac {1}{1+{\\cfrac {{\\frac {(a-c)b}{c(c+1)}}z}{1+{\\cfrac {{\\frac {(b-c-1)(a+1)}{(c+1)(c+2)}}z}{1+{\\cfrac {{\\frac {(a-c-1)(b+1)}{(c+2)(c+3)}}z}{1+{\\cfrac {{\\frac {(b-c-2)(a+2)}{(c+3)(c+4)}}z}{1+{}\\ddots }}}}}}}}}}", "276bcdef15805a4aedbae90eb2849b26": "(0,1/n,1)", "276be6df76fb6dcd2a37ea9d10f2d98c": "p_{l}+\\rho _{0}gz_{l}-p_{fl}=p_{r}+\\rho _{0}gz_{r}-p_{fr}", "276bf0578d572d7ad8f089c47c6f7216": "\\ \\Pi ", "276bf6d36b19e5e075886d31e5421f34": "(Q_{2},P_{2})", "276c36dc91f0227dd1dd6de28156b820": "z={{\\ln(x)-\\ln(\\mu _{g})} \\over \\ln \\sigma _{g}}={\\log _{\\sigma _{g}}(x/\\mu _{g})}.\\,", "276c524b8fbdc720ad7a3b06fc310ea7": "PostCaP~ICO_{all}=0.00875+0.0002+0.0005+0.0014=0.01085", "276c82a5468967167eeceacd6137cdbd": "\\operatorname {Li} _{s}(1)=\\zeta (s)\\qquad ({\\textrm {Re}}(s)>1)\\,.", "276d1346964c30b18876e6cbed120ca6": "\\displaystyle {T(x)=T_{1}x_{1}+\\cdots +T_{N}x_{N},}", "276d15c5c74a9631e81dd6f31b47c3d1": "M(S)=M\\ /\\ \\sim _{S}.", "276d5c4039b4f274b98e1cb3c0b0484e": "x(t)=vt\\cos \\theta ", "276f3e7717e238a776cff0e656955855": "s_{4}(x)={\\frac {3}{2}}x^{2}+{\\frac {1}{16}}x^{4};", "277071d79ee0b5cc4dfa8846c52c613a": "\\ C_{M}(1/4c)=-\\pi /4(A_{1}-A_{2})", "2770865b917b1e47b31e439571cf2267": "{\\mathit {S}}", "2770f92b9acd33b84fe536a84419d9ae": "{\\mathcal {T}}", "27715c32915f62d045488e5b2a0517d7": "\\lim _{x\\rightarrow c}{\\frac {f(x)}{g(x)}}", "27718aa0995792066ccdaf545221f3d2": "\\,K_{i},K_{j}", "2771ba70fa13f13a15cd82a20b2c4108": "\\nabla ^{2}\\mathbf {A} -{1 \\over c^{2}}{\\partial ^{2}\\mathbf {A} \\over \\partial t^{2}}=-\\mu _{0}\\mathbf {J} ", "2771fe1b76e8424110362d809d10cbc3": "f:X'\\to X", "27723048a786c1ccd2c59a484b7612cf": "w_{j}(k+1)", "27726364d29e7c390d4e2cc48e696941": "[H,P_{i}]=0\\,\\!", "27735626145ee7efdfb52259f1eb3745": "\\phi _{1},\\lambda _{1}", "2773873bc183412daaf58b90a80f4c7b": "\\mathbf {M} =\\sum _{i}\\mathbf {A} _{i}=\\sum _{i}\\sigma _{i}U_{i}\\otimes V_{i}^{\\dagger }", "2773d3f3207e769483bf759171d05c90": "\\langle a,b\\rangle =(-1)^{w(a)w(b)}{\\frac {a_{0}^{w(b)}\\prod _{i,j>0}(1-a_{i}^{j/(i,j)}b_{-j}^{i/(i,j)})^{(i,j)}}{b_{0}^{w(a)}\\prod _{i,j>0}(1-b_{i}^{j/(i,j)}a_{-j}^{i/(i,j)})^{(i,j)}}}", "2773f41894cd947b9f5937684719d2f2": "~G(x,y,z)~", "2774038c84cfa89a32ac99ec96a8e725": "X_{\\mu }=(ct,-x,-y,-z)", "27740890e14f3b355b2a0ede6b0afa4a": "[x_{\\mu },x_{\\nu }]=i\\theta _{\\mu \\nu }", "277409f676982d122fabc2809b14321f": "[f(x)]'\\,\\!", "2774990487f5973714ce57f25f278727": "H_{p}(B(S^{-1}S)^{0})\\subset H_{p}(B(S^{-1}S))=H_{p}(BS)[\\pi _{0}(BS)^{-1}]=H_{p}(BS)[e^{-1}].", "2775086897bc112fbcedbde39eeaeec3": "{\\frac {(z_{i}-z)}{R_{i}}}", "277551180bc42abbcec1a6dee5435059": "{\\textit {Person}}=({\\textit {coin}}\\rightarrow {\\textit {STOP}})\\Box ({\\textit {card}}\\rightarrow {\\textit {STOP}})", "277553df7f8b3fca553fffc9b493c849": "(U-c)^{2}\\left({d^{2}{\\tilde {\\phi }} \\over dz^{2}}-\\alpha ^{2}{\\tilde {\\phi }}\\right)+\\left[N^{2}-(U-c){d^{2}U \\over dz^{2}}\\right]{\\tilde {\\phi }}=0,", "27756ecb2a661a0d820fab830157c678": "Q_{c}=Q_{t}-Q_{s}", "27757d2c07ae8bb420b7f07a22390d88": "P_{\\text{fusion}}=n_{A}n_{B}\\langle \\sigma v_{A,B}\\rangle E_{\\text{fusion}}", "2775a5f8640069e88d98163fc5215fc4": "\\langle E\\rangle =k_{B}T^{2}{\\frac {\\partial \\ln Z}{\\partial T}}.", "27766270a8a2b056aafcbef3d5291e8c": "f(r)\\equiv 0{\\pmod {p^{k}}}", "2776d992d78fb95984a7dbbd9ff024a2": "\\Diamond p\\leftrightarrow \\lnot \\Box \\lnot p", "27771121ff8c496a29efb698016f20ac": "C_{n}^{(\\alpha +m)}(x)={\\frac {\\Gamma (\\alpha )}{2^{m}\\Gamma (\\alpha +m)}}\\!\\ C_{n+m}^{(\\alpha )[m]}(x).", "27778c9dc8a5ab5164523800c0e511ff": "x^{p-1}-1=0", "2777a9c48348dc2e0c8d818023ac6873": "\\mu _{p^{\\infty }}=\\left\\{\\exp \\left({\\frac {2\\pi im}{p^{k}}}\\right):m,k\\in \\mathbb {Z} \\right\\}", "2777cf1cf8193b9df55bd36f4b118448": "w_{ni}^{*}={\\frac {1}{k^{*}}}\\left[1+{\\frac {d}{2}}-{\\frac {d}{2{k^{*}}^{2/d}}}\\{i^{1+2/d}-(i-1)^{1+2/d}\\}\\right]", "2778282001f60831df0f501e37e0542c": "{\\frac {d}{dr}}\\sum _{k=0}^{\\infty }r^{k}=\\sum _{k=0}^{\\infty }kr^{k-1}={\\frac {1}{(1-r)^{2}}}", "277867cc0b3b9c2c5a405749f9303cda": "\\ w_{r}<{\\frac {l_{e}}{l_{us}}}w_{u}", "2778b188ef3f90ab1ed493da5f51979e": "\\alpha _{h}:\\mathrm {M} (a,b,c)\\rightarrow \\mathrm {M} (-h^{-1}b,ha,c-ab)", "2778d9a8e2d5fed1ffb2445b446fd76d": "{\\widehat {H}}_{\\alpha }", "2778f7b14a855985ad597a65abc2a579": "b=f-g", "2779269f045f8bbda797e5afbb852bef": "T_{n}=N_{n-1}=3\\cdot 4^{n-1}={\\frac {3}{4}}\\cdot 4^{n}\\,.", "27793261375ecdf7dd47d7a25fe07a16": "r=r(t)", "27799505fdd23814dedab3b11e8345d6": "R=\\mu _{1}+{\\sqrt {S_{1}}}X.", "2779ce5eec21976ed46a207ca9f9e334": "f_{\\mathbf {v} }d^{3}v=f_{\\mathbf {p} }\\left({\\frac {dp}{dv}}\\right)^{3}d^{3}v", "277a24d5c56626eeef4a229908a01905": "u^{2}=\\cos ^{2}\\alpha {\\frac {a^{2}-b^{2}}{b^{2}}}\\,", "277a860ea9ad635a55eddbc0f6c61eec": "p_{d-1}(a,b)\\neq 0,", "277a8c4a4283993d6c5fd438cce284ae": "\\Pi \\left(V\\right)", "277a9622da034f38ec117ed3e8b3ec63": "\\kappa (q_{f})=\\kappa (q_{r})", "277ab670bf31c63bb078b21122494a64": "y={{a-{1 \\over T}} \\over c}", "277af8b5c5c2e7ca75dee13cef3758ab": "Q=q^{k}", "277b9d9777b3bccf36df1f678d7b4548": "P={\\begin{bmatrix}1&0&0&0\\\\0&1&0&0\\\\0&0&0&0\\\\0&0&0&1\\end{bmatrix}}", "277ba5faec4f25552f01dab091ba87d8": "\\scriptstyle \\rho =\\mu (\\{\\eta :\\eta (x)=1\\})", "277bb4c832f6f6fdc7d60c554bd2100b": "Y={1 \\over 3}(n_{u}+n_{d}-2n_{s}).", "277bec74e1b0363a35a7d0ffa25f101e": "M^{\\bullet +}", "277c01cef39ce9b909029c3fdf5b76d9": "c_{n}\\Delta _{x}^{\\frac {n+1}{2}}\\int \\int _{S^{n-1}}\\varphi (y)|(y-x)\\cdot \\xi |\\,d\\omega _{\\xi }\\,dy=c_{n}\\Delta _{x}^{\\frac {n+1}{2}}\\int _{S^{n-1}}\\,d\\omega _{\\xi }\\int _{-\\infty }^{\\infty }|p|R\\varphi (\\xi ,p+x\\cdot \\xi )\\,dp", "277c363162bd39dad34a80f8c3bbe897": "h:H\\rightarrow \\mathbb {C} ", "277c77db3b85305869ed6ae26ddf123e": "{\\binom {p+q}{q}}-2{\\binom {p+q-1}{q-1}}={\\binom {p+q}{q}}{\\frac {p-q}{p+q}}", "277c8cf393b13ebd0a34ae9d57495f86": "\\{x\\mapsto h(a,y),z\\mapsto b\\}", "277d9451dc0a15827eb130f81600a1bd": "M=6n-\\sum _{i=1}^{j}\\ (6-f_{i})=6(N-1-j)+\\sum _{i=1}^{j}\\ f_{i}", "277dd201ac2206d5ee43956fdb2ad69b": "j_{1}=1", "277dd68fe14717dcff20e76ca03772e4": "Gm", "277e5fee4b7a2c3f392e0caf6d455e1b": "{\\begin{array}{r}{\\text{motor}}\\\\SE\\end{array}}\\;{\\overset {\\textstyle \\tau }{\\underset {\\textstyle \\omega }{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\\rightharpoonup \\!\\!\\!|}}}\\;{\\text{wheel}}", "277ef04e7cde4aaf291aab9c523674e3": "P\\rightarrow P(T)", "277ef62e7aa17e0a4f5e3060a56e4ae6": "K\\leq {\\sqrt {(s-a)(s-b)(s-c)(s-d)}}", "277f78988007fb392744f8b41daee855": "\\pi ^{+}\\rightarrow \\mu ^{+}\\rightarrow e^{+}", "277fa65207488c36e26a21e963dc7f5e": "\\ 0\\leq x<8", "277fccceea698fb9a3678a88a008fe38": "E_{\\sigma }", "278009087b515eb9e9effda62ab3a86b": "\\prod _{n>0}(1+q^{n-{\\frac {1}{2}}}z)(1+q^{n-{\\frac {1}{2}}}z^{-1})=\\left(\\sum _{l\\in \\mathbb {Z} }q^{l^{2}/2}z^{l}\\right)\\left(\\prod _{n>0}(1-q^{n})^{-1}\\right).", "278042aa3dc61abd7e4acf4b0d7421ea": "\\omega =\\eta =\\epsilon =0\\;", "27806f29a817b174524f75eaea5b4e31": "1\\to V\\to V\\rtimes _{\\rho }G\\to G\\to 1.", "278096b54b53e37123816a46e70ed0a8": "F[\\mu _{j}]=U-TS-\\mu _{j}N_{j}\\,", "2780a1eb8c1cfc7839c8f8f32f7fd3d8": "{\\begin{aligned}\\Pr(Y_{i}=1)&={\\frac {1}{Z}}e^{{\\boldsymbol {\\beta }}_{1}\\cdot \\mathbf {X} _{i}}\\,\\\\\\Pr(Y_{i}=2)&={\\frac {1}{Z}}e^{{\\boldsymbol {\\beta }}_{2}\\cdot \\mathbf {X} _{i}}\\,\\\\\\cdots &\\cdots \\\\\\Pr(Y_{i}=K)&={\\frac {1}{Z}}e^{{\\boldsymbol {\\beta }}_{K}\\cdot \\mathbf {X} _{i}}\\,\\\\\\end{aligned}}", "2781100a20d1601ba6de2b39322653cc": "\\|fg\\|_{1}\\leq {\\frac {1}{p}}+{\\frac {1}{q}}=1,", "27814e7a1c31a1afc1f321fce695bd37": "{\\frac {2}{\\sigma ^{3}\\lambda ^{3}}}\\left(1+{\\frac {1}{\\sigma ^{2}\\lambda ^{2}}}\\right)^{-3/2}", "2781afb6c92c8b845208de487f380d0c": "N_{s}=10^{-28/2.5}\\times {\\frac {S_{0}(V)\\times B}{h\\nu }}", "2781eefcce7b29c76f0ecd9107ae518a": "y/x\\leq 1.", "278230924d309fdc65f33cec3b68c0f0": "p(A_{k})", "278277eb98a0e3006b46980c5ac55547": "\\lambda _{i}=\\lambda {\\text{ for all }}i.\\,", "2782a65fb858cea722d94af15bb6055f": "\\exists x(\\forall y(F(y)\\leftrightarrow y=x)\\land G(x))", "2782d8137a4a79615ffdc479e097d40f": "[\\mathrm {O} /\\mathrm {Fe} ]=\\log _{10}{\\left({\\frac {N_{\\mathrm {O} }}{N_{\\mathrm {Fe} }}}\\right)_{\\mathrm {star} }}-\\log _{10}{\\left({\\frac {N_{\\mathrm {O} }}{N_{\\mathrm {Fe} }}}\\right)_{\\mathrm {sun} }}", "2782dc80168e214f664b0cd83bdaba27": "\\scriptstyle {\\theta =0}", "2782f462365c766ced3f3d5c4780f38f": "C_{1}\\ ", "27830add9dbbf2042e6da9411b974160": "x=a\\,", "27830e6e8d3eff22e76251e95a043b72": "\\operatorname {H} ^{2}(\\mathbb {C} \\mathbf {P} ^{\\infty };R)\\simeq \\operatorname {H} ^{2}(\\mathbb {C} \\mathbf {P} ^{1};R)", "278319e7836052d3cae89c209faf3ff1": "\\Phi _{E}(Q_{d},k)\\geq k(d-\\log _{2}k)", "2783403db5224fa06fe0a182ba793308": "L(\\mathbf {q} ,\\mathbf {\\dot {q}} ,t)=T(\\mathbf {q} ,\\mathbf {\\dot {q}} ,t)-V(\\mathbf {q} ,\\mathbf {\\dot {q}} ,t)", "27845a4e11b6879d8122a0a5b4abbe56": "I_{N}:\\{\\mathbb {X} \\subseteq \\mathbb {R} ^{n}\\}\\rightarrow \\{{\\text{newMin}},..,{\\text{newMax}}\\}", "278496314a4cb5c510a96a538033909d": "\\phi _{2},\\lambda _{2}", "278499dda22b50cdb53ffe795d2c472d": "{(21\\cdot 29)^{2}\\equiv 2^{1}\\cdot 7^{1}\\cdot 11^{2}{\\pmod {91}}}", "2785026c4ed4574b65009dcb2485167a": "\\log \\left(|H(j\\omega )|\\right)=\\log \\left(|H(j\\infty )|\\right)+{\\mathcal {H}}\\lbrace \\arg \\left[H(j\\omega )\\right]\\rbrace \\ ", "2785288d99838cbe6a9d5d92a1021f8f": "\\Gamma =-1", "2785467333afc9ec461a225a5b5f94fd": "Q_{1}=I-{2 \\over {\\sqrt {14}}{\\sqrt {14}}}{\\begin{pmatrix}-1\\\\3\\\\-2\\end{pmatrix}}{\\begin{pmatrix}-1&3&-2\\end{pmatrix}}", "27855269317242574206360c52adc400": "\\rho \\Phi +(8\\pi G)^{-1}a_{0}^{2}F(|\\nabla \\Phi |^{2}/a_{0}^{2})", "27861a049634da5c376d4f0ee717a2cf": "(x+y)^{2}=x^{2}+2xy+y^{2}.", "278624858ec5139c481d5eeab6b033a1": "log_{a}(x)=b", "2786498c7e07749eb4a7022d71fee2a4": "n_{0}+n_{1}\\zeta +n_{2}\\zeta ^{2}+...+n_{m-1}\\zeta ^{m-1}\\ ", "27865f16aa96098e5f083f8dd77a0858": "{\\frac {\\partial c}{\\partial t}}=D{\\frac {\\partial ^{2}c}{\\partial z^{2}}}+sg{\\frac {\\partial c}{\\partial z}}.", "2786b220fb78cf6e30e21f26bf0bb48e": "h_{i}=\\left|{\\frac {\\partial \\mathbf {r} }{\\partial q_{i}}}\\right|", "2786b62736b5a7859ed30c0d06008d50": "z_{1},z_{2},z_{3},z_{4}", "2786d392fc756a183f2a4a347b160331": "z=we^{w}", "2786d63a3193f49bbd1d588b0671d7d7": "\\mu ={\\sqrt {2}}\\,{\\frac {\\Gamma ((k+1)/2)}{\\Gamma (k/2)}}", "2786f63b0af6caa8cf83113c4b654b30": "{\\mathtt {rec}}\\ v=e_{1}\\ {\\mathtt {in}}\\ e_{2}\\ ::={\\mathtt {let}}\\ v={\\mathit {fix}}(\\lambda v.e_{1})\\ {\\mathtt {in}}\\ e_{2}", "278714589a64c20353b8acb7f99afc38": "\\pi /2", "278724a1c6095bdbffc5ba8b8e28c8ad": "y=f(a)+M(x-a)", "278766b6672e787ba80de2ded169b523": "P({ax,by}{|}{AX,BY})={\\begin{cases}{\\frac {1}{2}},&{\\mbox{if }}x\\oplus y=XY\\\\0,&{\\mbox{otherwise}}\\end{cases}}", "27877e43e6809e202f73605d1ae6300d": "{\\begin{aligned}f(x,y)\\approx &\\,{\\frac {f(Q_{11})}{(x_{2}-x_{1})(y_{2}-y_{1})}}(x_{2}-x)(y_{2}-y)\\,+\\\\&\\,{\\frac {f(Q_{21})}{(x_{2}-x_{1})(y_{2}-y_{1})}}(x-x_{1})(y_{2}-y)\\,+\\\\&\\,{\\frac {f(Q_{12})}{(x_{2}-x_{1})(y_{2}-y_{1})}}(x_{2}-x)(y-y_{1})\\,+\\\\&\\,{\\frac {f(Q_{22})}{(x_{2}-x_{1})(y_{2}-y_{1})}}(x-x_{1})(y-y_{1})\\\\=&\\,{\\frac {1}{(x_{2}-x_{1})(y_{2}-y_{1})}}{\\Big (}f(Q_{11})(x_{2}-x)(y_{2}-y)\\,+\\\\&\\,\\qquad \\qquad \\qquad \\qquad \\;\\;f(Q_{21})(x-x_{1})(y_{2}-y)\\,+\\\\&\\,\\qquad \\qquad \\qquad \\qquad \\;\\;f(Q_{12})(x_{2}-x)(y-y_{1})\\,+\\\\&\\,\\qquad \\qquad \\qquad \\qquad \\;\\;f(Q_{22})(x-x_{1})(y-y_{1})\\quad {\\Big )}\\end{aligned}}", "278787e660237506b2acd60564befb65": "0.00303993", "278793f3bb0863943f1e3c0c1f124505": "\\mid \\uparrow \\rangle \\to \\mid \\downarrow \\rangle ", "27879488452eee2a1b5ff65c794acfc1": "\\varepsilon \\cdot {\\bigl (}(M-m)+(b-a){\\bigr )}=K\\varepsilon ,", "2787a64c0db89fee93b7696cb340035d": "j_{\\mathrm {F} }[\\,=z_{\\mathrm {S} }d_{\\mathrm {F} }D_{\\mathrm {F} }]", "2787d3aa73e809b30000da5417abc2bd": "x_{n}=\\ell +a^{n}+b^{n}", "2787dc05d00aa076f0f534bd7c26a415": "x=g(z,u)", "2787f66b0a4fe60163c5dec33ccca6a7": "{\\begin{bmatrix}1&0\\\\-sC&1\\end{bmatrix}}", "278842ba20f0911f76d9c3e3bf1addfe": "\\{F,G\\}={\\frac {\\partial _{r}F}{\\partial z^{i}}}\\omega ^{ij}(z){\\frac {\\partial _{l}G}{\\partial z^{j}}}.", "2788914065c0778587e2c22849860121": "\\operatorname {Spec} (B\\otimes _{A}R)\\to \\operatorname {Spec} R", "278891d667d681e9d0fba2c786ef0749": "PA_{n}", "2788b6a89313f71ec28a36b3c3e7898d": "q(t)f(t)", "2788e071b64971b7e4edd78982ce711b": "\\|T\\|", "2788fceede6fb2f41860d100e2fde661": "\\in D", "27896d8fa05b9ade38dd6f1566a38e0b": "\\left\\{\\mathbf {k} \\in \\mathbb {Z} _{0+}^{c}\\,:\\,\\forall i\\ k_{i}\\leq K_{i},\\sum _{i=1}^{c}k_{i}=n\\right\\}", "2789ae5a6db7b750b3329c848e7d1677": "J_{i}=-\\Gamma _{,i}.\\,", "2789be610f2235c8d9bf2a408d619011": "|w|<1", "2789c618b572916756f8a7efde45ef76": "Y=U", "2789d39a74487af568da6d2bc6440058": "{\\frac {d[C]}{dt}}=k_{1}[A]", "2789f9cc1afbe55db73eec0a7dde3588": "\\Delta (x)=\\left\\vert \\psi (x)-x\\right\\vert <{\\sqrt {x}}\\log ^{2}(x)/(8\\pi )", "278a3e63434453b1c123458fd8ca8558": "\\mathbf {e} \\,\\!", "278a5c41893b234bc7ce50f94ddaa587": "p_{\\mathrm {\\varphi } }=mr^{2}{\\dot {\\varphi }}", "278ab54f4842959cabca9f63296aa637": "{dx \\over d\\tau }=\\lambda {p \\over m},\\;\\;\\;\\;{dp \\over d\\tau }=-\\lambda m\\omega ^{2}x;\\;\\;\\;\\;\\;\\;{dt \\over d\\tau }=\\lambda ,\\;\\;\\;\\;{dp_{t} \\over d\\tau }=0,", "278ae7eb6a62d45166f28b78b0db9b61": "{\\begin{aligned}\\Phi (p,t)&={\\frac {1}{\\sqrt {2\\pi \\hbar }}}\\int \\limits _{-\\infty }^{\\infty }dx\\,e^{-ipx/\\hbar }\\Psi (x,t)\\\\&\\upharpoonleft \\downharpoonright \\\\\\Psi (x,t)&={\\frac {1}{\\sqrt {2\\pi \\hbar }}}\\int \\limits _{-\\infty }^{\\infty }dp\\,e^{ipx/\\hbar }\\Phi (p,t).\\end{aligned}}", "278b02c71884316642d17c880d397ab5": "\\theta _{0}=\\varphi ,\\,", "278b04bace20474a4fb53ed31bcdfa11": "mP\\neq 0", "278bc796fc1eb57e965e35bad1636342": "{\\frac {1}{0}}=\\infty ", "278bcacaa64e0c1de4f54215f17885c0": "\\operatorname {Id} *1=\\sigma _{1}=\\sigma ", "278c413fc153007da628afabde022232": "D(a)={\\begin{pmatrix}D(a)_{11}&D(a)_{12}&\\cdots &D(a)_{1n}\\\\D(a)_{21}&D(a)_{22}&\\cdots &D(a)_{2n}\\\\\\vdots &\\vdots &\\ddots &\\vdots \\\\D(a)_{n1}&D(a)_{n2}&\\cdots &D(a)_{nn}\\\\\\end{pmatrix}}", "278c480989bb273d9b1928e3c6e5a5be": "\\qquad \\qquad \\langle \\varphi \\rangle =\\langle \\varphi \\rangle _{\\mathrm {o} }+\\sum _{i}\\sum _{\\alpha }{\\frac {\\partial \\langle \\varphi \\rangle }{\\partial d_{i\\alpha }}}|_{\\mathrm {o} }d_{i\\alpha }+{\\frac {1}{2}}\\sum _{i,j}\\sum _{\\alpha ,\\beta }{\\frac {\\partial ^{2}\\langle \\varphi \\rangle }{\\partial d_{i\\alpha }\\partial d_{j\\beta }}}|_{\\mathrm {o} }d_{i\\alpha }d_{j\\beta }+{\\frac {1}{6}}\\sum _{i,j,k}\\sum _{\\alpha ,\\beta ,\\gamma }{\\frac {\\partial ^{3}\\langle \\varphi \\rangle }{\\partial d_{i\\alpha }\\partial d_{j\\beta }\\partial d_{k\\gamma }}}|_{\\mathrm {o} }d_{i\\alpha }d_{j\\beta }d_{k\\gamma }+...+", "278c4c35ddf978e5ed9edd1a076ad7d8": "g:\\pi ^{ab}\\to \\pi ^{ab}/tor", "278c64c28fc40fc2da7a8c54a01d7d4e": "\\to \\,", "278c93854dce3f4aaa2062d37ed7db84": "\\Xi _{i}(t)", "278d16f919cbecd1f344de14c397f03a": "p=4^{a}(8b+7)=(2^{a})^{2}((8b+6)+1)", "278d1bef0e46890e914e6480716141af": "\\lambda _{D}=(kT/4\\pi ne^{2})^{1/2}=7.43\\times 10^{2}\\,T^{1/2}n^{-1/2}\\,{\\mbox{cm}}", "278d4af61479db825fcdc1232439828b": "R(N_{1},N_{3})", "278d5464a3ad83f50cd6a14afe7b268c": "\\lambda _{k}\\rightarrow 0", "278d8c3103ca7e8384e0843e88214a5b": "\\scriptstyle {\\hat {s}}", "278da0421a6ba0a7cd3acb612cda1c94": "X_{1}\\vee \\cdots \\vee X_{k-1}\\vee Z", "278da48d6363aef31fcd284eb3230389": "TPR(T)=\\int _{T}^{\\infty }P_{1}(T)dT", "278dc0c0edbd70a79deaf77fb4117bc8": "\\alpha (\\gamma )=2(\\sinh ^{-1}(\\gamma )-{\\frac {\\gamma }{\\sqrt {1+\\gamma ^{2}}}})", "278dd91adf3e671184af92c29ca567d5": "Vol_{q}(pn,n)\\leq q^{H_{q}(p)n}", "278dfef6de4292a2fb7f26d718968870": "{\\frac {\\mathbf {\\Psi } }{\\nu -p-1}}", "278e097ec5efd59ec0ef3ad16f9f4bfc": "R\\times S", "278e10d973aff035eaf74fa0fd2421be": "[x]_{Q}=\\{Q_{1},Q_{2},Q_{3},\\dots ,Q_{N}\\}", "278e36c188bf833338cacfae1c15a6f7": "{}_{3}^{5}", "278e82059adbbd5921a129414f5323da": "{\\overline {d}}(A)=\\limsup _{n\\rightarrow \\infty }{\\frac {n}{a_{n}}}", "278e8ebb29da7d1dfd79d4a3e49d7d56": "T_{TF}[n]=C_{F}\\int n({\\vec {r}})n^{2/3}({\\vec {r}})d^{3}r=C_{F}\\int n^{5/3}({\\vec {r}})d^{3}r\\ ", "278e9c7cf97ff5bca9c568f16c8b58dd": "W_{i}^{L}||W_{i}^{R}=W_{i}", "278f07fc5f5be794993672bbc5cfc167": "[x_{1},x_{1}]", "278f49c68eaf3c21246738a88f60963a": "{\\frac {dG}{dt}}=2T+\\sum _{k=1}^{N}\\mathbf {F} _{k}\\cdot \\mathbf {r} _{k}=2T-nV_{\\text{TOT}}.", "279032e265d6f19c137eea07601d0b62": "F_{2}\\sigma _{2}+F_{3}\\sigma _{3}+F_{22}\\sigma _{2}^{2}+F_{33}\\sigma _{3}^{2}+F_{44}\\sigma _{4}^{2}+2F_{23}\\sigma _{2}\\sigma _{3}\\leq 1", "27903ea4c673d6da9760181ab3565712": "\\omega \\in L({\\mathcal {G}},\\infty )", "279044e5d399eac4972990945c1bc199": "|t-1|\\,", "27906170850e5338f2680b069d89ed94": "{\\begin{matrix}\\times &d&e&f&g\\\\a&ad&ae&af&ag\\\\b&bd&be&bf&bg\\\\c&cd&ce&cf&cg\\end{matrix}}", "27906689b24df409980c177214bbb54f": "x^{3}=y^{2}", "27907c3735f3cb3a7554dbc87516cbba": "{\\begin{array}{c|ccc}0&0&0&0\\\\1/2&5/24&1/3&-1/24\\\\1&1/6&2/3&1/6\\\\\\hline &1/6&2/3&1/6\\\\\\end{array}}", "279082712e06a4e84a1ef17f047d7268": "I=\\int _{0}^{eV}\\rho _{S}\\left(r,E\\right)\\rho _{T}\\left(r,E-eV\\right)T\\left(E,eV,r\\right)\\,dE\\ ,\\qquad \\qquad (5)", "2790a140944185518dda5f808f6714dc": "\\nabla G(\\theta ,\\phi )=\\left({{\\partial }G \\over {\\partial }\\theta },{{\\partial }G \\over {\\partial }\\phi }\\right)\\!\\left(\\theta ,\\phi \\right)=(0,0),\\,", "2790e3e7089f798657f49a9d0d4c1e92": "d(f,g)=\\sup _{x\\in X}d(f(x),g(x))", "27910c515720cbf2ebeecb4807bd18b7": "{\\mathit {k_{t}}}", "2791535868f5a81e79af88fe6cf17736": "\\ A\\vee B:=(A\\rightarrow B)\\rightarrow B.", "27916dc0137bebe168302a3f4c8b27cf": "\\phi ^{1}=Re{\\phi }", "2791938e0e42cfd7f491c8dc89472b04": "h(i+1)\\geq {\\text{max }}(i,h(i)){\\text{ for all}}1\\leq i\\leq n-1.", "27925d253a84ba533da9712b3adb140d": "b+{\\frac {aW}{1-\\gamma }}>0.", "27927abcdc1b5f2a5532be3975233415": "\\Omega _{z}", "2792ca8f343c1af025648623daa2f3e3": "\\lambda _{1},\\,\\lambda _{2},\\,\\dots ,\\,\\lambda _{M}", "2792f3248e6fab6e7a8243919f8b9a79": "\\Gamma ^{\\lambda }{}_{\\alpha \\beta }={\\frac {1}{2}}g^{\\lambda \\tau }\\left({\\frac {\\partial g_{\\tau \\alpha }}{\\partial x^{\\beta }}}+{\\frac {\\partial g_{\\tau \\beta }}{\\partial x^{\\alpha }}}-{\\frac {\\partial g_{\\alpha \\beta }}{\\partial x^{\\tau }}}\\right)", "2793202d75b2f767f816f29bd32baea0": "n\\equiv 3{\\pmod {6}}", "279354cf67df39c6278075b103118bfb": "P_{0}=\\int _{R}^{\\infty }g\\rho dr", "2793a6a7a73f4aac4c4e432f9546353e": "(X_{t_{i+1}}+X_{t_{i}})/2.", "2793d8f16e891373a4006cb39d821be5": "\\sigma (p_{S_{i}})=\\alpha _{i}(p_{S_{i}})\\prod _{v_{k}\\operatorname {adj} v_{i}}\\mu _{k,j}(p_{S_{k}\\cap S_{i}})(2).", "2793e6b7b72be69fe3e1bc2e29b526c3": "b\\mapsto (F^{p}H^{k}(X_{b},\\mathbf {C} ))_{p}.", "27940520375a3749ad77c0219c017d79": "M_{PAW}={\\frac {(R)(T_{i})(P_{I})+[60-(R)(T_{i})](PEEP)}{60}}", "27942f80a0bf2a74e414b70181805efd": "a_{ji}\\in R", "27951c1b59c5c18b99439d61b7f90765": "\\sum _{k}n_{k}^{(-n)}+\\alpha _{k}=A+\\sum _{k}n_{k}^{(-n)}=A+N-1", "2795375d3c455eb85839237dec9f5d24": "{\\textrm {B}}(n,\\lambda /n)", "27956635477bf9342ec0a93147ba1b8d": "{\\begin{aligned}&\\lim _{\\alpha =\\beta \\to 0}G_{(1-X)}=0\\\\&\\lim _{\\alpha =\\beta \\to \\infty }G_{(1-X)}={\\tfrac {1}{2}}\\end{aligned}}", "279581f5551f8c88b73f46a2e171bf1d": "\\mathbf {E_{T}} =\\mathbf {E_{0}} e^{i(\\mathbf {k_{T}} \\cdot \\mathbf {r} -\\omega t)}=\\mathbf {E_{0}} e^{i(xk_{T}\\sin(\\theta _{T})+zk_{T}\\cos(\\theta _{T})-\\omega t)}", "2795b6c64e68214badc28fe466884913": "d(a\\mathbf {X} )=", "2795ff0ee8570ad8abccbb8b4adc7370": "2^{n^{k}}", "27961e6dadfdaa121bcc6f1266c8eae0": "\\;U_{R}(s_{i})", "279632db6704008436fb7729f03962ef": "{\\mathbf {j}}={\\frac {1}{m}}\\mathrm {Re} \\left(\\Psi ^{*}{\\mathbf {\\hat {p}}}\\Psi \\right)", "2796384a6eb9df635b956bfcf17ea87d": "\\mathbb {C} ^{n+1}", "27964916d0a9d03db97ccda5fa6f89cd": "\\ AB=BA", "279669cfce7b50380c69c2f7726b61af": "f_{*}:P(X_{1})\\rightarrow P(X_{2})\\,", "279678fbb2ebca8a5941b2929e0a6bc7": "\\scriptstyle {x\\in S_{*}:x<0}", "279688d4413ecbfda9518757ed89969a": "\\sigma _{i}", "2796ae9874431344b7f0393b612ff4aa": "\\left({\\frac {x}{a}}\\right)^{2}-\\left({\\frac {y}{b}}\\right)^{2}=1", "2796af5074a7f27ecccd3cd17e165d53": "c^{2}", "2796c393553c1ae31239afc9217421fa": "\\Pr[p_{i}=0]\\leq {\\frac {1}{2}}", "2796ce5b2e690868654778e01c06caf2": "{\\sqrt {n(n+1)}}", "2796f963954b8fa957de06a9ebe0b57e": "{(-\\Delta )^{\\frac {1}{2}}u=f},", "279701cc8fc997aa92967ed4a7efc10d": "{\\tilde {Z}}(s)=\\left({\\frac {\\lambda }{\\lambda +s}}\\right)^{n}.", "27970285948903787962b5a5dd620273": "{\\begin{aligned}(r+c-1)(r+c+\\alpha +\\beta -1)+\\alpha \\beta &=(r+c-1)(r+c+\\alpha -1)+(r+c-1)\\beta +\\alpha \\beta \\\\&=(r+c-1)(r+c+\\alpha -1)+\\beta (r+c+\\alpha -1)\\end{aligned}}", "27973c6f638ca1138e089c0c808513f1": "a_{\\alpha }", "27973d540e8c9f23fc9e9c5c5655c206": "\\ a^{*}={\\dot {v}}^{*}={\\dot {Q}}v+Qa+{\\ddot {c}}+{\\dot {\\Omega }}(x^{*}-c)+\\Omega (v-{\\dot {c}}),", "279745c4603d4a2b7326742e9e5ad7e5": "a\\cdot W_{0}^{a}\\cdot {\\text{E}}[R_{1}^{a}\\cdot R_{2}^{a}\\cdots R_{T}^{a}],", "279774a05fb144a502945a240531c90e": "\\delta W_{s}", "279799552be553615e434ab17245584a": "f(a,b)=b", "2797eb44363ca064e6727a6dbb4ac9b2": "{\\begin{aligned}\\sum _{x}{\\frac {\\delta H}{\\delta p(x)}}\\,\\phi (x)&{}=\\left[{\\frac {d}{d\\epsilon }}H[p(x)+\\epsilon \\phi (x)]\\right]_{\\epsilon =0}\\\\&{}=\\left[-\\,{\\frac {d}{d\\varepsilon }}\\sum _{x}\\,[p(x)+\\varepsilon \\phi (x)]\\ \\log[p(x)+\\varepsilon \\phi (x)]\\right]_{\\varepsilon =0}\\\\&{}=\\displaystyle -\\sum _{x}\\,[1+\\log p(x)]\\ \\phi (x)\\,.\\end{aligned}}", "27981a0c2bd554cba6e7035c359028da": "\\Delta u=f", "2798492849fd554d789c1cb2f9338bb9": "Z=n_{i}\\times (c_{ij}\\times [Z]_{j})=(n_{i}\\times c_{ij})\\times [Z]_{j}", "279852f50a749fd08b9548a8474ed695": "\\Pr(|x\\leq m-k\\sigma |)\\leq {\\frac {1}{k^{2}}}{\\frac {\\sigma _{u}^{2}}{\\sigma ^{2}}}.", "27985b3610e8c739e119ac305044aff4": "f(S_{x,i},t)", "27991710535b65e6025686c1e8670158": "M_{i}(x|1,t)={\\frac {1}{t_{i+1}-t_{i}}}", "27996dffc97f69742e5f5ae1ac57c6d2": "f_{n-1}\\circ d_{n}=e_{n}\\circ f_{n}", "2799ab38e367ec92f11784aa84f0cf54": "\\Delta S=R*(a-{\\frac {c}{T^{2}}}).", "2799d3215bc1b5dc7f4cdd1a8c7a044a": "mc^{2}", "2799e0879627f8907c4f98efe9abdf88": "T=\\{x_{2}^{2}-x_{1}^{2},x_{2}(x_{3}-x_{1})\\}", "279a252ee34ee8ee2f8473a55587090d": "\\Gamma _{0}(N)=\\left\\{{\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}}\\in SL_{2}(\\mathbf {Z} ):c\\equiv 0{\\pmod {N}}\\right\\}", "279a3c1a9cb094e1deb57d50bc499e9f": "m+S(n)=S(m)+n", "279a541fc719ff2c042c75e6718e47b8": "\\Gamma ={\\frac {V_{R}}{V_{F}}}={\\frac {B\\exp(-\\gamma l)}{A\\exp(\\gamma l)}}=C\\exp(-2\\gamma l)\\,", "279aa2547e65651c954067b66e2ce868": "{13 \\choose 1}{4 \\choose 3}{12 \\choose 1}{4 \\choose 2}{11 \\choose 2}{4 \\choose 1}^{2}=3,294,720", "279aa49cb2e8d29aeb5181bc3c80e9e0": "{\\frac {52!}{(52-5)!}}={\\frac {52!}{47!}}=52\\times 51\\times 50\\times 49\\times 48=311,875,200", "279aa94e1cc07bc98744a6fe5cac570b": "k=\\left({\\sqrt {\\left|\\Re \\{Z_{p}\\}\\right|}}\\right)^{-1}\\,", "279ae870aeb864a102f0ee59af7ef490": "\\mu ={\\frac {n\\alpha }{\\alpha +\\beta }}=n\\pi \\!", "279aff2fa8a7a986117a23e93ffc621c": "f(x)=x^{2}+3x+4+{\\frac {2x^{6}-4x^{5}+5x^{4}-3x^{3}+x^{2}+3x}{(x-1)^{3}(x^{2}+1)^{2}}}", "279b6672e4cfb15e41a1e4deebfc9f22": "(x_{n}^{(1)})_{n\\geq 0}=(0,1,2,3,4,\\dots )", "279c1af2d4950cf38a2364984484afaa": "\\lambda _{k}\\rightarrow \\infty ", "279c52d08b4fb24e43173940c137a3c7": "=\\mathbf {u} _{\\rho }\\left[-{\\frac {v^{2}}{r}}\\right]+\\mathbf {u} _{\\theta }\\left[{\\frac {\\mathrm {d} v}{\\mathrm {d} t}}\\right]\\ ", "279cd731931c0e2318aca0de26c7cc54": "x={\\frac {1}{k_{1}+{\\frac {1}{k_{2}+\\cdots }}}}", "279cf7547a58b656db0e05bc2c2934c4": "a(n,k,X(\\Omega ))", "279d1a6893a98cd638d8a8c4a7ce632d": "p=p_{\\rm {A}}^{\\star }x_{\\rm {A}}+p_{\\rm {B}}^{\\star }x_{\\rm {B}}+\\cdots ", "279d1d55f61d7f5854542f7df1aa20a7": "A\\times B", "279d520ebed641c0cd1854227a3e304d": "F(X,Y)=P(X)+Y*Q(X)", "279d91e3a85738e1a99827cc2fb1e478": "x_{i_{j}}\\in X_{i_{j}}", "279d9621fdedfe6f13cb322ea107e862": "e^{-x^{2}},", "279dac11169af97009e127f834b6b195": "9\\times x", "279db70eb0feab5767eae729a98ce527": "\\Delta (\\tau ^{k})=\\sum _{i+j=k}\\tau ^{i}\\otimes \\tau ^{j}", "279dc6c6dc1a7d5362f6ecb4e0ed828b": "a_{n}:={\\frac {2}{L}}\\int _{0}^{L}f(x)\\cos \\left({\\frac {2\\pi nx}{L}}\\right)\\,dx", "279de18dce0d9a45538934526c4ac45b": "V_{\\text{cc}}", "279ebb36693495fef0683bb02f9c05c5": "s\\rightarrow t", "279f0b9e99f9c8b30282f11ded4b7136": "{\\dot {\\theta }}=\\omega ^{2}-\\sigma ^{2}-{\\frac {\\theta ^{2}}{3}}-{E[{\\vec {X}}]^{m}}_{m}", "279f3aca7e1f966251a3f205d6da9de7": "\\sigma _{x}^{2}(-1)=\\lambda \\sigma _{d}^{2}(-1)=\\epsilon \\,\\!", "279f5cd10832a8d1f30ec7b3b9ea961a": "r_{FOR}", "279f9912912de25162dfeb20b4c1f2c0": "{\\tfrac {3\\pi }{2}}", "279fa09409eafe6144cb3ca00d16f2a3": "\\varepsilon _{1}\\neq 0", "27a021b65d522849d7d2753577fa01b3": "\\sum _{n=-\\infty }^{\\infty }x(n\\cdot MT)\\ e^{-i2\\pi fn(MT)}={\\frac {1}{MT}}\\sum _{k=-\\infty }^{\\infty }X\\left(f-{\\tfrac {k}{(MT)}}\\right).", "27a050edfb4cd6af1c12ff2084077909": "\\mu _{\\mathrm {JT} }=\\left({\\partial T \\over \\partial P}\\right)_{H}={\\frac {V}{C_{\\mathrm {p} }}}\\left(\\alpha T-1\\right)\\,", "27a15290a1f6bc2149e7ebd8d435d3a6": "f\\sim g\\quad ({\\text{as }}n\\to \\infty )", "27a17f736f9b10ea1f9fe2a00c82dfa9": "\\displaystyle x-y", "27a19f5b299251143745fc275925313c": "h={v_{0}^{2}\\sin ^{2}(\\theta ) \\over {2g}}", "27a1b185ddb89e5586285d9373526702": "\\Phi _{o}", "27a1ca1e814fd66c7b8c00777f0698fb": "n\\geq \\ell +1", "27a1f5a6c607a8978b1a7847d7af1b6d": "{\\tilde {e}}_{i}B\\subset B\\cup \\{0\\}", "27a247e7b6344dd7a5441367b899777f": "{\\overline {X}}", "27a28ee9a7c46f034a67670a9d8f7684": "S\\rightarrow ab", "27a2fa3776d5c9dc1265f32ea8494057": "m^{2}+1", "27a3299f3e86235eb2d3447a0111c5dd": "I(X:Y)\\leq S(\\rho )-\\sum _{i}p_{i}S(\\rho _{i})", "27a3388c165427c902bd205be3df6845": "\\iota =(-8-16y-18w+12w^{2}+10yw+yw^{2})/23", "27a356297ad9e4bac3d1c40d9f827b63": "R_{n}=R_{o}+(W-L)\\times 16+D*0.04\\ ", "27a37c4fe93e5908db2dc03e0f1d6f18": "H_{n}\\left(p_{1},p_{2},\\ldots \\right)=H_{n}\\left(p_{2},p_{1},\\ldots \\right)", "27a3872b281ce0ac5eb15f4719960df1": "M_{rot}=\\left({\\begin{matrix}\\cos \\theta &\\sin \\theta \\\\-\\sin \\theta &\\cos \\theta \\\\\\end{matrix}}\\right)", "27a399a8ccbc92b409c763abf8f35f22": "|U|=|V|", "27a3ac0f49e13bf96c7de04cc47a5012": "D\\approx P\\mathrm {e} ^{-G}\\approx \\mathrm {e} ^{-G}.\\qquad \\qquad (7)", "27a3bd0109bf5414d13052cf84b5db8a": "\\epsilon _{i}\\in \\{\\pm 1\\},t\\in [-1,1]", "27a3ffaeb0cc60ab591414f32ba29abb": "2^{\\mathfrak {c}}=\\beth _{2}", "27a4634bc931edbca23e3c16df7d2f56": "{\\boldsymbol {\\sigma }}*", "27a4741dc69513bd2fdaf2bce8de5355": "\\Delta \\subseteq K^{\\times }/(K^{\\times })^{n},\\,\\!", "27a47eb93c74bbe3df1c6e3413d51ed0": "|\\sigma _{ef}|^{2}", "27a4bafa06a43269b6741fa3bcf63ca3": "\\Delta G^{0}=-287kJ", "27a510e920511aee7430a0f676899b6f": "L\\geq 3", "27a57511a2df4d705df5b9c7ef28e2d8": "{\\mbox{If }}\\alpha \\equiv \\eta ^{n}{\\pmod {\\mathfrak {a}}}{\\mbox{ then }}\\left({\\frac {\\alpha }{\\mathfrak {a}}}\\right)_{n}=1.", "27a5892346ed9715d2852b2309022bf5": "\\Phi (z,s,q)=\\sum _{k=0}^{\\infty }{\\frac {z^{k}}{(k+q)^{s}}}\\!", "27a595ea4c3e3e0397ceccd58a465601": "X_{b}", "27a5d4e6447556cd8c23dbd067ea4092": "6\\rightarrow \\infty ", "27a6341c2863995eef4fc01feed13597": "t\\equiv {\\frac {T-T_{c}}{T_{c}}}", "27a65ac1b4863f3eb9a56d95157dbd01": "V(\\rho ,\\varphi ,z)=\\sum _{n=0}^{\\infty }\\sum _{r=0}^{\\infty }\\,A_{nr}J_{n}(k_{nr}\\rho )\\cos(n(\\varphi -\\varphi _{0}))e^{-k_{nr}|z-z_{0}|}", "27a6e74689cbdd9dcec0b4832f918b71": "M_{k}\\equiv \\int d\\zeta \\ \\lambda (\\zeta )\\zeta ^{k}", "27a7579d0309d06241abeb0bccb4ac99": "A\\times B", "27a79e7b529d1c2537b65ad971f64ac6": "G=(\\{S,A,X\\},\\{a\\},S,\\{f,g,h,k,l\\})", "27a7ca65090df9d37e1a58a92d2a120b": "\\sigma _{f}\\approx a{\\frac {\\sigma _{A}}{A\\ln(10)}}", "27a80f01d2e612e23e2906929bab23f6": "P_{\\ell }(\\cos \\theta )", "27a869566bd3162b0e5815cb9c79fec8": "\\sum _{n=1}^{\\infty }{\\frac {(-1)^{n}}{n(9n^{2}-1)}}=2\\ln 2-{\\frac {3}{2}}.", "27a88b9153e137a48135d5a0734f9639": "{mv^{2} \\over r}={mg\\tan \\theta }", "27a89e2e3385601726b145dbb57c330b": "t=(N+1)\\tau ", "27a8a828cafff3320cc5c71bac2d9d82": "\\Rightarrow \\int {\\frac {dx}{x}}=\\int k\\,dt", "27a8ebad4e6b9550dd062968667d229c": "f(x)={\\frac {a_{0}}{2}}+\\sum _{n=1}^{\\infty }\\,[a_{n}\\cos(nx)+b_{n}\\sin(nx)].", "27a91cd57a2c707d8973bdbd5a22f4a6": "f\\colon \\coprod _{i}Y_{i}\\to X", "27a97831c0ed4004ddd04c3b1a962c7f": "i{\\bar {\\partial }}\\Psi \\mathbf {e} _{3}\\rightarrow (i{\\bar {\\partial }}\\Psi )R\\mathbf {e} _{3}+(e{\\bar {\\partial }}\\chi )\\Psi R", "27a9a02a29335319f5104184ef93789d": "(\\mathbf {F} ,\\mathbf {G} )=\\int \\mathbf {F} ^{*}\\cdot \\mathbf {G} \\,dV", "27a9ff394d53cc33da67bc3b592df069": "\\nabla ^{LC}g=0", "27aa222bd2e56b43d428a769bb91f451": "q-\\mu {\\ddot {w}}-(EIw_{xx})_{xx}=0", "27aa289889bd651838d87a1863679bd6": "\\mathbf {B} =(B_{x},B_{y})", "27aa4d0b8c7f3bfbd86f04c285c7fab9": "\\mu ={\\frac {mM}{m+M}}", "27aac050c82759a25e222b09b89b6c9a": "R_{E}\\,", "27aac0e35e6d150d4155bed236c5d77b": "{\\begin{aligned}x_{i}={\\frac {1-r^{-i}}{1-r^{-N}}}\\quad \\Rightarrow \\quad x_{1}=\\rho ={\\frac {1-r^{-1}}{1-r^{-N}}}\\qquad {\\text{(2)}}\\end{aligned}}", "27aacc49b6ded542c74e53dbb60b7a96": "-ln(1-X)\\sim {\\textrm {Exponential}}(b)\\,", "27aaf22833c0b52e1bee9fa3d0852baa": "{\\sqrt {1-v^{2}/c^{2}}}", "27aaf79646b6395a3f0fcce815f008e2": "(m,k+1)", "27aafacee7dca7a5b0f0e046f1b53389": "\\phi \\not \\in Y", "27ab591d01b4a69a03332d2d6dd8a87e": "(\\mathbf {u} \\times \\mathbf {v} )\\times \\mathbf {w} \\neq \\mathbf {u} \\times (\\mathbf {v} \\times \\mathbf {w} )", "27ab84052867f0f4b180754747d25304": "K=mv^{2}/2", "27aba3533827bf2818737f694b1705cd": "{\\mathfrak {p}}_{1}\\subset \\cdots \\subset {\\mathfrak {p}}_{n}={\\mathfrak {p}}'_{n}\\cap A", "27ac0cb30064f5d1bfaac43025994aef": "m=\\left\\lfloor \\left(\\left\\lfloor {\\frac {m}{r}}\\right\\rfloor +1\\right)r\\right\\rfloor ", "27ac38419a218269236e224b213187bf": "n\\sigma ^{2}.", "27ac61084e65eb3b8fa1b990d77038ab": "I=\\{\\mathrm {milk,bread,butter,beer} \\}", "27acac66e6d10fb206e6f72b749111a1": "L=10\\cdot 0.5=5", "27acc1746d27841b372a18c0f53e8726": "\\mathrm {M} ={\\frac {v}{v_{\\text{sound}}}}", "27ad202dffe45e9624f67be7142747d8": "v_{LZ}={{\\frac {\\partial }{\\partial t}}|E_{2}-E_{1}| \\over {\\frac {\\partial }{\\partial q}}|E_{2}-E_{1}|}\\approx {\\frac {dq}{dt}}", "27ad663f56e987dc8af4ffa51f23be46": "\\langle a|b\\rangle ", "27ae082b3a967141f2b263a0b3651fa0": "\\{(i,x)~|~x\\in r(i)\\}\\subseteq I\\times X", "27ae76f1088e9028683775e574a5c224": "0\\lambda .", "27af462209ce360802ebefefa87ed397": "g^{-1}([y-\\varepsilon ,y+\\varepsilon ])", "27afa964a6ce77d50cc84c699e5aa9dd": "\\nu _{t_{1}\\dots t_{k}}\\left(F_{1}\\times \\dots \\times F_{k}\\right)=\\nu _{t_{1}\\dots t_{k}t_{k+1},\\dots ,t_{k+m}}\\left(F_{1}\\times \\dots \\times F_{k}\\times \\mathbb {R} ^{n}\\times \\dots \\times \\mathbb {R} ^{n}\\right).", "27afaf7478927436d0aa59f999497118": "\\mu _{w}\\approx \\lambda /4", "27afc85f9af832ef6e93301afefda923": "n=0,\\ldots \\infty ", "27b00dc5eac3d437f3662585c2411cd8": "\\mathrm {Hom} _{{\\mathcal {C}}/{\\mathcal {R}}}(X,Y)=\\mathrm {Hom} _{\\mathcal {C}}(X,Y)/R_{X,Y}.", "27b019a630a74005270466f918da8f83": "L_{\\text{aligned}}", "27b0a40f069250c87998d1f3f5fc4773": "G_{i,j}={\\begin{cases}1&{\\text{if }}S_{i}\\leftrightarrow S_{j}\\\\0&{\\text{otherwise}}\\end{cases}}", "27b0bf02fa820b274ba3902e5e9066f1": "c=Tr(g)", "27b0d4f0319b09a84eea89c2dac88a33": "\\left({\\begin{matrix}{\\frac {4}{3}}\\end{matrix}}/{{\\begin{matrix}({\\frac {9}{8}})\\end{matrix}}^{2}}={\\begin{matrix}{\\frac {256}{243}}\\end{matrix}}\\right)", "27b10b370ee57be9f3511514dec3603d": "{{\\mathbf {\\ }}g}", "27b10e9039ed092d9f5a246a596e4c34": "{\\hat {f}}\\,{\\hat {f}}^{\\dagger }=1-{\\hat {f}}^{\\dagger }\\,{\\hat {f}}.", "27b14bd802425593dd76d6620fd0631a": "2^{4}\\cdot 3^{2}\\cdot 5\\cdot 7", "27b158888ad28de51445a67fdcc16bbe": "c\\in \\mathbb {R} ", "27b17637d3243fffe576793c03e342ce": "\\partial _{t}^{2}\\psi =\\Delta \\psi -{\\frac {1}{2}}f'(\\psi ),\\qquad \\psi =\\psi (x,t),\\quad x\\in \\mathbb {R} ^{n}.", "27b1a66802f5f1b25164cf0288fff5fe": "S_{\\sqrt {3}}=\\{x=-\\alpha +i\\beta \\ \\ |\\ \\ \\alpha >0\\ {\\text{ and }}\\ |\\beta |\\leq {\\sqrt {3}}\\,|\\alpha |\\}", "27b1e8d69b3976a62776f4f5739f3d5b": "=-2\\left[-\\omega v\\left(\\sin \\alpha -\\omega t\\cos \\alpha \\right),\\right.", "27b2100657a3e547ccfe19854dd5f256": "0={\\frac {\\mathrm {d} ^{2}\\psi }{\\mathrm {d} \\eta ^{2}}}+({\\frac {2mE^{2}}{\\hbar ^{2}}}-{\\frac {2m^{2}gl^{3}}{\\hbar ^{2}}}-{\\frac {2m^{2}gl^{3}}{\\hbar ^{2}}}\\cos(\\eta ))\\psi ", "27b24b1c6e9b103638dd8d17202f7e67": "Fc", "27b27afcd86c4bd2dff0134d4e09597d": "V{\\textbf {y}}={\\textbf {x}}", "27b2ba25830e40e1de7776305d00b107": "e_{i}^{2}=\\gamma _{1}+\\gamma _{2}z_{2i}+\\dots +\\gamma _{p}z_{pi}+\\eta _{i}.", "27b3048beb4ac9f5f207912a6671b06b": "f\\colon X\\rightarrow \\mathbb {P} ^{n},\\ x\\mapsto [a_{0}(x):\\dotsb :a_{n}(x)],", "27b340836a051ddc222e5ba0c78e9e9d": "\\mathbf {\\hat {e}} _{i}", "27b3ac694e515eb0a2bcfabf7798619d": "{\\frac {1}{l}}+{\\frac {1}{m}}+{\\frac {1}{n}}=1.", "27b3c87564726cedbf9fce2a239ef7a4": "{\\begin{aligned}y'&=y\\\\z'&=z\\end{aligned}}", "27b43ece74a399fdb21cfe0d3e893e2d": "{\\frac {\\partial \\sigma }{\\partial \\varepsilon }}={\\frac {\\partial }{\\partial \\varepsilon }}(E\\varepsilon )=E>0", "27b43efaa927397aaf229b2b2f454c03": "f(x)={\\begin{cases}0&{\\text{if }}x=0,\\\\x+x^{2}\\sin \\left({\\frac {2}{x}}\\right)&{\\text{if }}x\\neq 0.\\end{cases}}", "27b44237a08f43c6f622236f07f2fc93": "|E(\\mathbb {F} _{q})|=q", "27b4e6152e680c2e851eefd84b3d8992": "{\\frac {d}{dt}}A=(i\\hbar )^{-1}[A,H]+\\left({\\frac {\\partial A}{\\partial t}}\\right)_{\\mathrm {classical} }.", "27b5a39e249a52de631a969a229dd1f4": "00{\\mbox{)}}\\,\\!", "27d65eb589a4cebc66d7890fcc934454": "{\\mathrm {Sp} }(n,{\\mathbb {R} })", "27d693676731525613f731e4a6d556b8": "\\varepsilon _{tot}=\\varepsilon _{1}=\\varepsilon _{2}", "27d72e9bc3bf7027e606906e49099c57": "z=T(x)={\\begin{bmatrix}z_{1}(x)\\\\z_{2}(x)\\\\\\vdots \\\\z_{n}(x)\\end{bmatrix}}={\\begin{bmatrix}y\\\\{\\dot {y}}\\\\\\vdots \\\\y^{(n-1)}\\end{bmatrix}}={\\begin{bmatrix}h(x)\\\\L_{f}h(x)\\\\\\vdots \\\\L_{f}^{n-1}h(x)\\end{bmatrix}}", "27d79633bd5cb642f1d263d044919d93": "EU(n)=\\left\\{e_{1},\\ldots ,e_{n}\\ :\\ (e_{i},e_{j})=\\delta _{ij},e_{i}\\in {\\mathcal {H}}\\right\\}.", "27d7a7c2c02b7f3740a2989881e7bf0f": "{\\boldsymbol {\\sigma }}={\\begin{bmatrix}{\\tfrac {4C_{1}}{3}}\\gamma ^{2}&2C_{1}\\gamma &0\\\\2C_{1}\\gamma &-{\\tfrac {2C_{1}}{3}}\\gamma ^{2}&0\\\\0&0&-{\\tfrac {2C_{1}}{3}}\\gamma ^{2}\\end{bmatrix}}", "27d80789574cb352c76cf0551a1ccc8f": "\\scriptstyle {A=B+}{\\tfrac {PL}{2}}", "27d8079815fe6ec88887e8fca83e2bae": "nRT\\ln {\\frac {V_{2}}{V_{1}}}\\;", "27d80fc352b727c9aed82973c3131d79": "\\approx 15.04", "27d81d5a29a86b950d70802eccc48801": "H_{\\mu }(x)=\\sum _{\\mathbf {p} }{{\\sqrt {p_{0}}} \\over {\\sqrt {2V}}}\\left\\{\\left[Q_{R}(\\mathbf {p} )\\epsilon _{\\mu }^{1}(\\mathbf {p} )-Q_{L}(\\mathbf {p} )\\epsilon _{\\mu }^{2}(\\mathbf {p} )\\right]e^{ipx}\\right.", "27d82a35d65ae2ca34825d1fb937b701": "e_{i}\\in {\\mathfrak {g}}_{\\alpha _{i}}", "27d83137d6d827b497bbc7a7b62d3bc5": "S_{e}=Lx*(AdjFactor*UFP)^{\\frac {Entropy}{1.2}}", "27d8918145e7dc16d7bf4adc7bfbbb92": "\\lim _{x\\rightarrow a}f'(x)", "27d89261c2b7ea56a1232aaaef86faee": "\\gamma =0.8", "27d8c3268a69c7b04a6c09c634ff8785": "Q={\\sqrt {\\frac {x_{1}^{2}+x_{2}^{2}+\\cdots +x_{n}^{2}}{n}}}", "27d93479cad69a6f99cbfa3fbeb482f8": "\\{it+ju+kv:t,u,v\\in \\mathbb {R} \\}.", "27d966046f5d5711a230b1a7429c4d82": "|p-p_{c}|\\,\\!", "27d9d9ea0ebbe84b5d40dc11977bcf4f": "{\\rm {REACH}}_{\\rm {out}}[S]={\\rm {GEN}}[S]\\cup ({\\rm {REACH}}_{\\rm {in}}[S]-{\\rm {KILL}}[S])", "27d9f6043ebe6e6260e926fef15ce43f": "Q(z-x)", "27d9fb1ffc621b4f9fd6645320932159": "E[|\\xi |^{p}]<\\infty ", "27da571b27422598b7d08b39dad06ced": "p(A)=A^{n}+c_{n-1}A^{n-1}+\\cdots +c_{1}A+(-1)^{n}\\det(A)I_{n}=0~,", "27da58578c907649da175ac03922661f": "A=270^{\\circ }+\\arctan \\left({\\frac {\\left({\\frac {\\partial z}{\\partial x}}\\right)}{\\left({\\frac {\\partial z}{\\partial y}}\\right)}}\\right)-90^{\\circ }\\left({\\frac {\\left({\\frac {\\partial z}{\\partial y}}\\right)}{\\left|{\\frac {\\partial z}{\\partial y}}\\right|}}\\right)", "27da6ada9031d408cb884cfc6a16690e": "{\\hat {x}}={\\frac {x}{|x|}}", "27da8afa442611c608dd049d764da04b": "L\\in {\\text{NSPACE}}\\left(f\\left(n\\right)\\right)", "27daa58a04d1b7f298a11f5dd2184195": "U", "27dab1f9acacac39a41902c1122fb6db": "\\!\\forall x\\phi ", "27dac40d6a46ef32506cd3a86b133e9d": "\\gamma _{\\text{gw}}", "27daf37517119ac992837c5a64776027": "v_{v}^{2}/2g", "27dafa448bd8ce45a1724284101a2321": "{\\boldsymbol {\\mathrm {A} \\mathrm {B} \\Gamma \\Delta \\mathrm {E} \\mathrm {Z} \\mathrm {H} \\Theta }}\\!", "27db16b136dd4e0e121965c38b3ffe33": "+1/3", "27db75a3b79d095b2898588229b1da1d": "v^{2}=Q(v)\\,", "27db9d8bb8ce04ed17ff82b58ec4f8e9": "(\\phi ,\\theta )=\\left(2\\arccos(R/2),\\Theta \\right).", "27dba00e378b6fdc3bf50453c5033406": "T={\\frac {1}{2}}\\left|\\det {\\begin{pmatrix}x_{A}&x_{B}&x_{C}\\\\y_{A}&y_{B}&y_{C}\\\\1&1&1\\end{pmatrix}}\\right|={\\frac {1}{2}}{\\big |}x_{A}y_{B}-x_{A}y_{C}+x_{B}y_{C}-x_{B}y_{A}+x_{C}y_{A}-x_{C}y_{B}{\\big |},", "27dba611aef816d0ef0d6c63a51c1a22": "\\tau =\\int _{R}^{\\infty }k\\rho dr", "27dc57829c9dac76b5e1c07fe5fe27fa": "P^{(L)}=L\\,P", "27dcb3702426feb7fde72b3fc0008972": "({\\mathcal {F}}g)(y)={\\frac {1}{\\varepsilon ^{n}}}e^{-{\\frac {\\pi }{\\varepsilon ^{2}}}|x-y|^{2}}.", "27dcc1404a9f6470a7c3beaadbdc0e0c": "V(r)", "27dd4cff71772ef8f006b36c14615f60": "\\ -{\\frac {\\mu }{2a}}+{\\frac {\\mu }{R}}={\\frac {\\mu (2a-R)}{2aR}}.", "27dd5e8368a931dc6195ff9ec55921e4": "a(x,z)", "27dd630c8cb6aefd909f276b2ee737ec": "U_{t}=e^{itA}\\quad t\\in \\mathbb {R} ", "27dda408c0dbb6e027a2e684c536cbf1": "\\sum _{i=1}^{k}\\mathrm {n_{i}} ^{\\alpha }\\,\\mathrm {d} \\mu _{i}\\,=\\sum _{i=1}^{k}\\mathrm {n_{i}} ^{\\beta }\\,\\mathrm {d} \\mu _{i}\\,=0\\,.", "27ddabb6aa6a2f3efb914342507355f5": "\\int _{0}^{2\\pi }\\!I^{2}\\,d\\omega t=...=\\pi ", "27ddda85ea020143b8119c8e8b398df3": "I\\cong A_{5}", "27dddef5a301d4de6df69ddfa5026fd0": "\\sigma =\\sigma _{f}+\\sigma _{b}\\,.", "27de3b8c9e7b4df1706c75523b53443f": "Y=\\{y_{i}|i=1,2,\\cdots ,k\\}\\,\\!", "27deb2a762eab7f9105efa39c8931066": "Z(G)\\hookrightarrow G{\\overset {\\sigma }{\\to }}\\operatorname {Aut} (G)\\twoheadrightarrow \\operatorname {Out} (G).", "27dec493c1d9c2f38faca1a145fe9554": "a_{S}", "27ded278381e8c673da9be04c9ff99d1": "\\lambda \\ ={\\frac {hm^{2}}{m^{2}-n^{2}}}", "27e03e3bd9f598ce965c88b553764f60": "\\zeta ^{2}+\\zeta +1=0", "27e056cfdcbe531778630615bff26fa0": "m=(-15,-26)U^{-1}=(3,-7).\\,", "27e0af5dd1a2311dd4e2758c96f45b6c": "{\\boldsymbol {\\nabla }}{\\boldsymbol {T}}", "27e0b277aa8b4c6143c169fbb88147d8": "\\langle (1\\;2\\;3)\\rangle =\\{id,\\;(1\\;2\\;3),(1\\;2\\;3))\\}", "27e0da99896208032d6c68cc7bae46e3": "x={\\frac {a\\sin t\\cos t}{t}}", "27e0ea8fc4b5a17d7fa08ed089249b2e": "{\\begin{aligned}{\\rm {Pr}}_{y_{1},\\dots ,y_{m}}(\\exists z&A(x,y_{1}\\oplus z)=\\dots =A(x,y_{m}\\oplus z)=0)\\\\&\\leq \\sum _{z\\in \\{0,1\\}^{m}}{\\rm {Pr}}_{y_{1},\\dots ,y_{m}}(A(x,y_{1}\\oplus z)=\\dots =A(x,y_{m}\\oplus z)=0)\\\\&\\leq 2^{m}{\\frac {1}{(3m)^{m}}}\\\\&<1.\\end{aligned}}", "27e0f96563277444ec54a666352ff058": "\\models _{\\mathrm {P} }\\phi ", "27e14658c241aa9f117b9dbb33e297a5": "S={\\frac {a}{\\sqrt {2}}}+a+{\\frac {a}{\\sqrt {2}}}=(1+{\\sqrt {2}})a", "27e18af31e9b5a97a4f7aa4e19515f22": "\\cot {\\frac {\\pi }{8}}=\\cot 22.5^{\\circ }={\\sqrt {2}}+1\\,", "27e1b955e80670f9116b6bb439df7869": "\\mathbf {a} =\\mathbf {a} _{x}+\\mathbf {a} _{y}+\\mathbf {a} _{z}=a_{x}\\mathbf {i} +a_{y}\\mathbf {j} +a_{z}\\mathbf {k} ,", "27e22067ccdacf738a731e51b933bfd5": "Q=\\left({\\frac {1}{Q_{c}}}+{\\frac {1}{Q_{d}}}\\right)^{-1}\\,", "27e289d40ec0a3b00255dbb0148c4f6e": "f_{\\alpha }(g)=\\lim _{n\\rightarrow \\infty }F_{\\alpha ,z}(g^{n})/n", "27e29960e16256defede6f6e0697ccf7": "P_{n}", "27e3579f027677c27cf69744d5204be6": "a^{2}\\neq 1", "27e38b6d01778665c5311345519fb5ac": "\\mathrm {kT/q} ", "27e3bceec491b2e7e1b2aebc45825852": "v_{s}^{2}={\\gamma _{e}T_{e0} \\over 1+\\gamma _{e}(k\\lambda _{De})^{2}}\\sum _{i}{Z_{i}^{2}f_{i} \\over {\\bar {Z}}m_{i}}", "27e3d18eaa7acc19d4cf6158dedb8fcb": "\\langle x,y\\rangle =\\delta (x-y),", "27e4529acb3935a8f800239c133d7e48": "A_{ij}~", "27e499647c0f21f38a54b1462733ad4f": "\\Delta ^{\\prime }(x)", "27e4aab1a54227b6174b6142d888ac8b": "g(0)=g(1)=1", "27e4e19e734012cbd24593a8c8c5ce29": "A_{11}", "27e4e39e413da1a2bda38dc183f8fa68": "({\\mathcal {O}}_{X})_{p}", "27e5a88671ca0fe1314d45683376fad2": "{\\dfrac {Pbx^{2}}{2L}}-{\\cfrac {Pb}{6L}}(L^{2}-b^{2})=0", "27e5b72facc4de10d5a8a0151d8e3e66": "1\\leq \\sum _{k=1}^{\\infty }\\lambda (V)\\leq 3.", "27e5ea63e4485fb63d807072b21b6214": "~{\\rm {Re}}(x)\\gg 1~", "27e641be47bab4e97f54b08c11780d9b": "a={\\sqrt {\\frac {\\gamma p}{\\rho }}}={\\sqrt {(\\gamma -1)\\left[H-{\\frac {1}{2}}\\left(u^{2}+v^{2}+w^{2}\\right)\\right]}}.", "27e6c4f8d7629a0970b30449bca4cd67": "g_{\\text{I}}", "27e6dfad6fec76276e25e2c61682faa6": "\\mathbf {J} =\\mathbf {L} +\\mathbf {S} ", "27e736508c1666c43d43dde5d3199e7f": "{\\frac {\\Delta \\left(\\omega /2\\right)}{-\\Delta f}}\\approx \\eta \\omega J_{F}^{\\,\\prime }", "27e743a371802c6f5ef6abdfe71a02f8": "\\Omega ={\\frac {\\pi }{12hours}}", "27e7b1346e011cf2896a04df87832534": "n_{2}", "27e7c9600e0c4edda297d058215417a7": "{\\begin{aligned}{\\mbox{Capital account}}&={\\mbox{Foreign direct investment}}\\\\&+{\\mbox{Portfolio investment}}\\\\&+{\\mbox{Other investment}}\\\\&+{\\mbox{Reserve account}}\\\\\\end{aligned}}", "27e7e01f1bd6b7f9b9b0f2d33368d08d": "D(X)=R(X-\\mathbb {E} [X])", "27e8314f8bde178770636924dae3e9a8": "\\delta ={\\frac {2\\pi \\omega }{\\omega _{\\text{res}}(\\theta )}}", "27e864e7c006cdc151f1c46e1e0d4131": "r={\\frac {w\\cos \\left(\\theta \\right)}{\\delta \\cos \\left(\\phi \\right)}}", "27e870c56cdb8c8f4097b7f8c4d76d7d": "\\{\\cdot ,\\cdot \\}", "27e89e226ca1bc86dcbd5ae1b613ce5d": "{\\frac {\\mathrm {d} ^{3}u}{\\mathrm {d} x^{3}}}+{\\frac {1}{2}}u{\\frac {\\mathrm {d} ^{2}u}{\\mathrm {d} x^{2}}}=0", "27e8ba661396ac3961350c5f512b93cb": "\\Phi ,\\Psi ", "27ea1040b1e7109e708e9cd74bdbd001": "f(x)=f(a)+\\int _{a}^{x}g(t)\\,dt", "27ea1dd7d65f5777ba487de4b56a90d6": "\\displaystyle {U_{x}=-V_{y},\\,\\,U_{y}=V_{x}.}", "27ea225d5a2c2504e3a9c6f4fd037e57": "\\mathbf {v} \\cdot \\nabla \\mathbf {v} ", "27ea3b2a24beff39ae9e38b8bca5f5c0": "S=\\sum _{i=1}^{m}r_{i}^{2}", "27ea673590abb25964c0d59178086f25": "t=\\rho \\sinh \\sigma .", "27eaba7588be9ecdc8a11fb052f660ef": "P_{\\sigma \\,\\circ \\,\\pi }", "27eac782422adb62c41a6f3c2c99a5d1": "2^{4}", "27eb99a77cab658667371cf42f73a8a9": "{\\begin{aligned}|C_{n}-AB|&={\\biggl |}\\sum _{i=0}^{n}a_{n-i}(B_{i}-B)+(A_{n}-A)B{\\biggr |}\\\\&\\leq \\sum _{i=0}^{N-1}\\underbrace {|a_{\\underbrace {\\scriptstyle n-i} _{\\scriptscriptstyle \\geq M}}|\\,|B_{i}-B|} _{\\leq \\,\\varepsilon /(3N){\\text{ by (3)}}}+{}\\underbrace {\\sum _{i=N}^{n}|a_{n-i}|\\,|B_{i}-B|} _{\\leq \\,\\varepsilon /3{\\text{ by (2)}}}+{}\\underbrace {|A_{n}-A|\\,|B|} _{\\leq \\,\\varepsilon /3{\\text{ by (4)}}}\\leq \\varepsilon \\,.\\end{aligned}}", "27ebe01457588b35c369065a897d327f": "R={\\sqrt {ax^{2}+bx+c}}", "27ebf14ee602621ce4c15e2b2b1b814f": "\\Delta \\varphi ", "27ec49f37d75272912635455af6a321e": "i_{\\alpha }\\circ i_{\\beta }=-i_{\\beta }\\circ i_{\\alpha }.", "27ec60fd5c6d10afa2b5ac64a18ec247": "p_{\\theta }(x)={\\frac {\\theta e^{-x}}{(1+e^{-x})^{\\theta +1}}}", "27ec9719ca96a0a1cacce9f19e39a900": "\\vartheta (z;\\tau )=\\prod _{m=1}^{\\infty }\\left(1-\\exp(2m\\pi i\\tau )\\right)\\left(1+\\exp((2m-1)\\pi i\\tau +2\\pi iz)\\right)\\left(1+\\exp((2m-1)\\pi i\\tau -2\\pi iz)\\right).", "27ecccaed73a596c0a8b168171d13de7": "\\langle A\\rangle ={\\frac {1}{n}}\\int Af\\,d^{3}p", "27ed54230b571ff8c3d655e7fa719ef8": "{\\boldsymbol {\\epsilon }}={\\tfrac {1}{2}}\\left[{\\boldsymbol {\\nabla }}\\mathbf {u} +({\\boldsymbol {\\nabla }}\\mathbf {u} )^{T}\\right]\\qquad \\implies \\qquad \\epsilon _{jk}={\\tfrac {1}{2}}\\left({\\cfrac {\\partial u_{k}}{\\partial x_{j}}}+{\\cfrac {\\partial u_{j}}{\\partial x_{k}}}\\right)~.", "27eda47b4f20d504e8f8e200bf9071bd": "w''/w'=-2u'/u", "27eda7899fa7635139ec9cbd013cfa01": "u_{tyt}=u_{tty}=u_{xxy}+u_{yyy}", "27edb1441562c5d0490e8e6d7d512869": "g=G=|g\\rangle ", "27edeec0c3a8ede087b8f5ce2d0227fd": "t/X_{t}", "27ee4f0f4a2fd47728d5d2dfa8f9067f": "r_{1},\\dotsc ,r_{n}", "27ee690567bc40e124b828243f559f29": "{\\frac {\\partial ^{2}S}{\\partial \\beta _{j}\\partial \\beta _{k}}}", "27eea3d72f23bcb97741b5d178a264bd": "\\scriptstyle {\\vec {r}}", "27ef2dc21d7d62304916b3fcc6bbff70": "((b^{-1}\\,{\\bmod {\\,}}n)\\,(b\\,{\\bmod {\\,}}n))\\,{\\bmod {\\,}}n=1", "27ef2fb1e6019af1d64369ef4ce13121": "\\Gamma (E\\otimes \\Lambda ^{p}T^{*}M)=\\Gamma (E)\\otimes _{\\Omega ^{0}(M)}\\Gamma (\\Lambda ^{p}T^{*}M)=\\Gamma (E)\\otimes _{\\Omega ^{0}(M)}\\Omega ^{p}(M),", "27ef639d78050675826ab112ed73f033": "\\operatorname {build-param-lists} [g\\ m\\ p\\ n,D,V,K_{2}]\\land \\operatorname {build-param-lists} [g\\ q\\ p\\ n,D,V,K_{1}]", "27efcfd6bdb886eef71ae47a819dcb2e": "\\gamma \\!_{polymer}", "27f00b1280b4f7c0716159a1eaa39998": "1\\leq i\\leq r", "27f02682fb2cf882c5e16c7acb2fee08": "V\\otimes _{k}F", "27f05b13ae778bcb82ca97a51cfa61db": "s_{iw}=1", "27f0aeb7a9260d882114347e0bee9eaf": "g:\\mathbb {R} ^{n+m+1}\\to \\mathbb {R} ^{m}.", "27f0e5db21cd9bc45d5b87992f27d9a8": "\\scriptstyle {\\pi (B)}", "27f0eb52024c5a525b8d3bc3f82e9a09": "\\exp(o(n))m^{O(1)}", "27f11357c767502e831ee0c0babf89aa": "f^{2}(x)=(f(x))^{2}", "27f13a5616ca23f0308f9b3044083826": "\\left(\\pi _{A}=\\pi _{G}=\\pi _{C}=\\pi _{T}={1 \\over 4}\\right)", "27f13e0ce509291bc3d8580a201e44a4": "n_{s}\\sim s^{-\\tau }\\,\\!", "27f19a540bce340378a4383bbcbe734f": "{\\widehat {O1QP1}}={\\widehat {O2QP2}}", "27f29139b38518f98e8ddc2a258d7ae7": "\\lambda >\\xi ", "27f314f4819f7261b9313dd748cd9d27": "2{\\tfrac {2}{5}}", "27f31ebb9d94ab3f0fa68afe9ef3eba7": "r=3", "27f364891b77402222dfab50ed3727d6": "{\\mathcal {A}}(\\omega )={\\overline {\\bigcup _{B}\\Omega _{B}(\\omega )}},", "27f3815f127e8e63b069cd528c0c46f7": "IV_{0}=0", "27f39869a1b44022e8da8f307a844189": "\\Pr(X_{1}\\leq x_{1},X_{2}\\leq x_{2},...,X_{n}\\leq x_{n})=\\int \\prod _{i=1}^{n}F(x_{i}|\\theta )\\,dP(F).", "27f3dd3c963b4ce4817ac47c78998fcd": "a\\lor (b\\lor c)=(a\\lor b)\\lor c", "27f3f92ce9a006f9140ae1d5c367be78": "{\\begin{matrix}(13-r)\\times 4=52-4r\\end{matrix}}", "27f4057b5217c010d2b43c5c15599952": "\\gamma ={\\sqrt {1-\\beta ^{2}}}", "27f425e03aedae73d4dc8bf19b29f35d": "P_{i}=d_{ijk}\\sigma _{jk}+\\mu _{ijkl}{\\frac {\\partial \\epsilon _{jk}}{\\partial x_{l}}}", "27f4292ee91d9061c0f99ea5c45e42d1": "\\xi _{1}={\\frac {1}{\\sqrt {2\\alpha _{1}}}},\\xi _{2}={\\frac {1}{\\sqrt {2\\alpha _{2}}}}", "27f43bb53abf108238c535f0b978f2a7": "n(A)", "27f49cf94e949a7f823f42ef57d6a38d": "q_{k}\\sim (k+1)^{1-\\tau }\\mathrm {e} ^{-(k+1)/\\kappa }", "27f4a8bbc9e658d3b154d82c12f6a799": "r_{5}=(A\\to a,\\emptyset ,\\{S\\})", "27f4ac54592545affda7458f1003269b": "{\\hat {g}}(n)={(-1)^{n+1} \\over \\pi n}", "27f4baf5f01bcd4d7b8a7603bd9b743d": "Lu=f\\left(x,y\\right),\\ \\ \\left(x,y\\right)\\in \\Omega ", "27f4be53df95a1871a12c3f3c7615c26": "\\scriptstyle {\\boldsymbol {\\omega }}dt", "27f50c0a9859482a7eecbc889f8baa73": "E_{\\text{red}}=E_{\\text{red}}^{\\ominus }-{\\frac {RT}{zF}}\\ln {\\frac {a_{\\text{Red}}}{a_{\\text{Ox}}}}", "27f560ca74d15c4ce23ce0e099e7b949": "\\Delta _{\\text{2D}}", "27f609ba5015ce0cef1995a4f3532398": "36^{2}+37^{2}+38^{2}+39^{2}+40^{2}=41^{2}+42^{2}+43^{2}+44^{2}", "27f60fc38013d2fd533cae68931a0e5d": "\\varrho \\mu \\gamma \\angle '", "27f618831cd285cf86733674f0cacd55": "S(E)=\\sum _{i}\\oint p_{i}\\,dq^{i}", "27f63557b67cf9474676551a40758386": "A_{q}(n,d)", "27f68a9083d1c31cade081485e6f6f4a": "z=\\pm {\\frac {j}{2}}", "27f6a12cf284e9947becc82315602b29": "\\iint _{D}f\\,dx\\,dy+\\int _{C}g\\,ds=0.\\,", "27f6afd553845ddc99d9db733d24cf8c": "1\\leq i\\leq n", "27f831c4337484a3eec9f4a9ee24ca10": "x_{c}", "27f83baf58075463929b3bc4d4f7c71b": "{\\frac {4^{n}}{2n+1}}\\leq {2n \\choose n}\\leq 4^{n}{\\text{ for all }}n\\geq 1", "27f8b21afdb7fbabae2ce2b29b412930": "{\\text{Break-even(in Sales)}}={\\frac {\\text{Fixed Costs}}{{\\text{C}}/{\\text{P}}}}.", "27f8d74467a8158a29c133af6d172af4": "\\lim _{x\\rightarrow 0^{-}}{1 \\over 1+2^{-1/x}}=0.", "27f8dcfaa08c45d85f41c9c269c3f0f0": "\\mathbf {\\Omega } ^{-1}=\\mathbf {G} '\\mathbf {G} ", "27f9020ab606dc5e945bbe73ff3a5917": "f_{c}(z).\\,", "27f93195e00b293a5d48892a6723a7dd": "X(0)", "27f999f4adeb24f85e4a2a5c3fb4b41f": "\\left\\{t'_{i,m+j},i=1\\ldots m,\\;j=1\\ldots n\\right\\}", "27f9a52862060d3c27e5813d25601d96": "{\\begin{aligned}g_{jk}(\\theta )=4h_{jk}^{\\mathrm {fisher} }&=4h\\left({\\tfrac {\\partial }{\\partial \\theta _{j}}},{\\tfrac {\\partial }{\\partial \\theta _{k}}}\\right)\\\\&=\\sum _{i}p_{i}(\\theta )\\;{\\frac {\\partial \\log p_{i}(\\theta )}{\\partial \\theta _{j}}}\\;{\\frac {\\partial \\log p_{i}(\\theta )}{\\partial \\theta _{k}}}\\\\&=\\mathrm {E} \\left[{\\frac {\\partial \\log p_{i}(\\theta )}{\\partial \\theta _{j}}}\\;{\\frac {\\partial \\log p_{i}(\\theta )}{\\partial \\theta _{k}}}\\right]\\end{aligned}}", "27f9a87d164b6d49cc7780547b4dcf66": "(a_{i},b_{i},c_{i},d_{i},e_{i})", "27f9e340f9ad3918d61ab9980d52c130": "{\\mathcal {D}}=\\{(x;a;r)\\}", "27fa384a85afbde1b89c109635f880fa": "\\ k_{b}", "27fa6cac97d0ab80f03c90de3961c045": "T_{11}^{*}:={\\cfrac {T_{11}^{\\mathrm {eng} }}{\\alpha -\\alpha ^{-2}}}~;~~\\beta :={\\cfrac {1}{\\alpha }}", "27fa6dee32669f9bcbe7b1a7aa15e711": "\\scriptstyle \\Vert \\;\\Vert _{L^{\\infty }(\\Omega )}", "27fad19a03a94b6acf1e4b65d560fafd": "\\displaystyle \\int _{\\mathbf {R} ^{n}}f(\\mathbf {x} )e^{-i\\mathbf {x} \\cdot {\\boldsymbol {\\nu }}}\\,d^{n}\\mathbf {x} ", "27fada207a1c27e876075a912465983c": "\\displaystyle {{\\mathcal {K}}={\\mathcal {H}}\\oplus {\\mathcal {H}}.}", "27faed787a905e1a574b7141762e307f": "{\\sqrt {n(P+N)}}", "27fb09eba9204db1db850bc094f1063d": "{\\begin{aligned}{\\frac {1}{\\lambda }}\\,\\int _{0}^{\\lambda }\\eta ^{2}\\;{\\text{d}}x&={\\frac {1}{\\lambda }}\\int _{0}^{\\lambda }\\left\\{\\eta _{2}+H\\,\\operatorname {cn} ^{2}\\left({\\begin{array}{c|c}\\displaystyle {\\frac {\\xi }{\\Delta }}&m\\end{array}}\\right)\\right\\}^{2}\\;{\\text{d}}\\xi ={\\frac {H^{2}}{\\lambda }}\\int _{0}^{\\lambda }\\operatorname {cn} ^{4}\\left({\\begin{array}{c|c}\\displaystyle {\\frac {\\xi }{\\Delta }}&m\\end{array}}\\right)\\;{\\text{d}}\\xi -\\eta _{2}^{2}\\\\&={\\frac {\\Delta \\,H^{2}}{\\lambda }}\\int _{0}^{\\pi }\\cos ^{4}\\,\\psi \\,{\\frac {{\\text{d}}\\xi }{{\\text{d}}\\psi }}\\;{\\text{d}}\\psi -\\eta _{2}^{2}={\\frac {H^{2}}{2\\,K(m)}}\\int _{0}^{\\pi }{\\frac {\\cos ^{4}\\,\\psi }{\\sqrt {1-m\\,\\sin ^{2}\\,\\psi }}}\\;{\\text{d}}\\psi -\\eta _{2}^{2}\\\\&={\\frac {1}{3}}\\,{\\frac {H^{2}}{m^{2}}}\\,\\left[\\left(2-5\\,m+3\\,m^{2}\\right)+\\left(4\\,m-2\\right)\\,{\\frac {E(m)}{K(m)}}\\right]-{\\frac {H^{2}}{m^{2}}}\\,\\left(1-m-{\\frac {E(m)}{K(m)}}\\right)^{2}\\end{aligned}}", "27fb6f05171808ff6304a72ab273eaa6": "12\\,", "27fba3fd905517efb91ddf247cb5ec32": "S(\\Psi )={\\tfrac {1}{2}}\\langle \\Psi |Y(i)Y(-i)Q_{B}|\\Psi \\rangle +{\\tfrac {1}{3}}\\langle \\Psi |Y(i)Y(-i)|\\Psi *\\Psi \\rangle \\ ,", "27fbad4d812e76b47303a6b00a3ffbf5": "(a^{2}-b^{2})^{2}=(A+B)^{3}(A+B-4h)\\,", "27fbce8c68ff35a097081bc9cc25c5db": "\\left\\lceil \\log _{2}{\\frac {1}{\\frac {1}{4}}}\\right\\rceil +1", "27fbece4c8c5d1d75d1bf153ef43a950": "i,j,\\dots ", "27fc1509fcb9340ee056a0a82115156f": "|I_{i}(\\alpha _{k})|\\leq |\\alpha _{k}|e^{|\\alpha _{k}|}F_{i}(|\\alpha _{k}|)", "27fc2a6e19a02f734cbe2f8bd4bc2672": "\\gamma \\in \\Gamma ", "27fc3399be3d91097ffd0952c1e71a61": "T\\vdash _{\\mathcal {S}}\\alpha (\\phi ,{\\vec {\\chi }})", "27fc3746bbf605c6aba5e54ebb4a5a42": "f_{s}={R \\over N}.", "27fc75a8139f2b6842b2349f0128da47": "\\vdash \\Gamma \\ {\\mathsf {Context}}", "27fca5bab3d70edcfb77ef603dd0d78c": "[H,H]", "27fca613ea0dfaaf9e48fc5acb29f52e": "{\\frac {f_{o}}{f_{s}}}={\\frac {c}{c\\pm v}},", "27fcb267ccc84d61483bf7e04a68ad63": "C_{j}=P_{j}\\oplus O_{j}", "27fcb9d89098ecff906ff3aa5f3348da": "A.a", "27fd3c9bc0d7e3d18322268862237740": "E(T^{k})=\\nu ^{\\frac {k}{2}}\\,\\prod _{i=1}^{\\frac {k}{2}}{\\frac {2i-1}{\\nu -2i}}\\qquad k{\\text{ even}},\\quad 0m+n\\mid X>m)=\\Pr(X>n).", "2800ad59002cdf687b646b5e335a3032": "M_{\\phi }(X)=-\\delta \\sum _{s=1}^{S}\\phi _{s}X_{s:S}", "2800fc565c33ed23870778ca086b81aa": "A=\\{x\\in E|\\varphi (x)>a\\}", "28013e3a784e4bfc0682f7980f772d63": "{\\frac {a}{b}}\\,{\\bmod {\\,}}n={\\frac {a\\,{\\bmod {\\,}}n}{b\\,{\\bmod {\\,}}n}}", "28015def75a790a459ce4df1732371ea": "{\\frac {\\partial }{\\partial t}}\\rho (x,p;t)=\\langle \\Psi (t)|x,\\,p\\rangle {\\frac {\\partial }{\\partial t}}\\langle x,\\,p|\\Psi (t)\\rangle +\\langle x,\\,p|\\Psi (t)\\rangle \\left({\\frac {\\partial }{\\partial t}}\\langle x,\\,p|\\Psi (t)\\rangle \\right)^{*}", "28018a31d4880aab81125eaa55970b96": "\\langle \\langle \\tau _{e}\\rangle \\rangle ", "2802066d41a4e848f94121b26e06a178": "10\\cdot \\log _{10}", "28023398c861b18571c4114666826084": "\\scriptstyle p^{\\mu }\\;=\\;\\{\\pm {\\sqrt {m^{2}+{\\vec {p}}^{2}}},\\,{\\vec {p}}\\}", "28024c884e786bd0e2974e7e9f044c6e": "x\\in \\{1,\\dots ,k\\}", "28029c5f1a8512ea7efb18cabbb404f6": "S=\\{(x,y)\\in \\mathbb {R} ^{2};-N-{\\frac {1}{2}}\\leq x\\leq N+{\\frac {1}{2}},\\vert \\alpha x-y\\vert \\leq {\\frac {1}{N}}\\}", "2802f947a473c21551e968b94746b06e": "\\Omega =[a_{1},b_{1}]\\times [a_{2},b_{2}]\\times ...\\times [a_{N},b_{N}]\\subset R^{N}", "2803016d9871a3f1a263456a1afd84fa": "\\mathrm {Aut} (\\mathrm {S} _{6})=\\mathrm {S} _{6}\\rtimes \\mathrm {C} _{2}.\\ ", "28033ba00f7438ce839471781027a772": "\\{0\\}\\subset F_{0}\\subset F_{1}\\subset \\cdots \\subset F_{i}\\subset \\cdots \\subset A", "28034a11a3de1ca566826ad371113968": "\\scriptstyle \\mathrm {tr} (\\cdot )", "2803934d96b376a6b582474cd3468897": "P=P^{*}\\,\\!", "2804140db39734ac81b83dcff4e714b8": "P(z)|_{z=\\beta _{m}}=0", "2804aaca3f5dced813f6f0a3723e4b66": "\\rho \\circ \\kappa =\\operatorname {id} _{\\ker \\,f}", "2804ae2d240ced2f3f613120c674da50": "{\\frac {1}{c^{2}}}{\\frac {\\partial ^{2}u}{\\partial t^{2}}}(t,x)=\\Delta u(t,x)\\quad \\mathrm {for} \\quad (t,x)\\in \\mathbb {R} ^{+}\\times \\mathbb {R} ^{n},", "28051e3719a3cc90d5c4cb6104456b47": "{1 \\over 2(1-p^{-2})(1-p^{-4})\\cdots (1-p^{2-n})(1-p^{-n/2})}", "280632de07af20a4f93f65ee3186abfd": "\\theta \\rightarrow -\\theta ", "28063499b41a40a52f1fe38faf5a14e6": "H_{\\mathbf {k} }=e^{i\\mathbf {k} \\cdot \\mathbf {r} }He^{-i\\mathbf {k} \\cdot \\mathbf {r} }", "2806854ac8d69074d5a019bed52140ed": "MRS_{xy}=MU_{x}/MU_{y}\\,", "280695c08e2577b6f3688f92ceeaa3a5": "A\\wedge B\\equiv A\\mathbin {\\And } (A\\rightarrow B).", "2806c14e63b316fdab07605959cd87df": "I=S{\\sqrt {t}}\\ +At", "2806c80bdf23a1b0ffd3b40aec05049e": "X^{\\text{op}}", "2806cf41b068979bad724cd437ca3ac5": "O_{j}=\\ E_{K}(I_{j})", "2806dd52c577a8cb88eee4dc76e86dba": "P(R_{NP},\\theta )=P(R_{NP}\\cap R_{A},\\theta )+P(R_{NP}\\cap R_{A}^{c},\\theta ),", "2806f8e8724b5f7fc1ffaf7db56c7d0a": "{\\tilde {f}}({\\vec {v}})=f_{\\gamma }v^{\\gamma }", "2807046d163521aa66cff3da9b423dbc": "G\\cong \\mathrm {SO} (2,\\mathbb {Q} ).", "280713008b5a86c4303bb3445ad33a51": "\\mathbf {y} ={\\textbf {vec}}(\\mathbf {Y} )", "2807182680f4e5e92c78c39a5778f4e9": "{\\begin{aligned}0.95&=1-\\alpha =P(-z\\leq Z\\leq z)=P\\left(-1.96\\leq {\\frac {{\\bar {X}}-\\mu }{\\sigma /{\\sqrt {n}}}}\\leq 1.96\\right)\\\\[6pt]&=P\\left({\\bar {X}}-1.96{\\frac {\\sigma }{\\sqrt {n}}}\\leq \\mu \\leq {\\bar {X}}+1.96{\\frac {\\sigma }{\\sqrt {n}}}\\right)\\end{aligned}}.", "28076aa4a2eec33aa5cd4bb8d54342d4": "\\sum _{i=0}^{L}q_{i}S_{y}[{i+r}]=0\\forall r", "280788e5a5322a52e66ccb30ebe86eaa": "E_{m}=\\log _{2}\\left(l\\right):\\beta \\,=1,l\\geq \\,1", "2807c995216e38f050312743227225c8": "L+R\\ \\leftrightarrow \\ L\\!\\cdot \\!R", "2807d68c7b2e036120e44086db1c62f0": "C=v^{i}N_{i}\\,", "2807decebf4d8e2037ea37f0461b6f7f": "n(d)=n_{0}e^{-d/\\langle d\\rangle }dD", "280850314e7623dbb55528c4a6eb1159": "[D(f(d))\\wedge \\neg D(f(f(d)))]", "28086ba41ce50eaa4f08c378b1624075": "\\max \\nolimits _{m_{j+1}}*\\Pr[V{\\text{ accepts }}w{\\text{ starting at }}M_{j+1}].", "28087a724bd01324e5a162431b68789f": "\\mathbf {x} ={\\begin{bmatrix}x_{1}&x_{2}&\\dots &x_{m}\\end{bmatrix}}.", "28087c504e209ae70ca4f6370e0c5713": "Chow^{eff}(k):=Split(Corr(k))", "2808fe6abd57d1f3cfdbff5cc347001a": "\\eta =k\\left|{\\frac {du}{dy}}\\right|^{n-1}", "28091d1b5d3294c1f4463a0a94a55cba": "K=-{\\frac {1}{2e^{\\sigma }}}\\Delta \\sigma ,", "28093612c48b29eeebf0c895b2463f18": "(a\\wedge b)\\cdot (c\\wedge d)=(a\\cdot c)(b\\cdot d)-(a\\cdot d)(b\\cdot c).\\,", "280969f675fa9cfde8c0ef9e45b33aa1": "S_{\\beta \\gamma }^{\\;\\;\\;\\;\\beta }=0", "280977f5eb0a373a76facb445b5166ee": "X(s)", "2809a048691bc27c0cac99e5ab15be2f": "H=(V,E)", "2809acf39536b82b5470530e6aed78bc": "\\alpha \\ominus \\beta ", "280a3d3635b953300c3590c60697534f": "\\nu (t)=\\arg \\max _{p\\in \\mathbb {R} ^{n}}(p'\\alpha (t)-{\\tfrac {1}{2}}p'\\alpha (t)p)\\qquad {\\text{ for all }}0\\leq t<\\infty ", "280a525278fde87ef845c2877fa5d1a7": "\\nabla \\times \\mathbf {B} =\\mu _{0}\\mathbf {j} \\ ", "280a84120fd9cd5508ac31ba7149eed3": "\\scriptstyle {L}", "280ab03a3d7fae4506ec3159faae0cb9": "C_{ab}^{*}={\\sqrt {a^{*2}+b^{*2}}}", "280acd0b7dc27e1c15be6d96208c483b": "y={\\frac {Y}{Z}}", "280b9d871a2779ae274c9fdf2f92a7d9": "\\int d\\mathbf {Q} d\\mathbf {P} =\\int Jd\\mathbf {q} d\\mathbf {p} ", "280bdbe1d2750a51b97892b7e0b6f36c": "{\\begin{bmatrix}x\\\\y\\end{bmatrix}}", "280c0a986f03b7daa88d99c832d8d6af": "\\left[1+{\\frac {x-\\mu }{\\sigma }}\\right]^{-\\alpha }", "280c83b9d178ee52cdcb96fe32305d76": "B_{\\theta }=-B_{0}\\left({\\frac {R_{E}}{r}}\\right)^{3}\\sin \\theta ", "280cab3bc6f8ecda6ea25059c0a91bd1": "(\\leftarrow \\backslash )\\quad {Z\\leftarrow \\Delta Y\\Delta '\\qquad X\\leftarrow \\Gamma \\over Z\\leftarrow \\Delta \\Gamma (X\\backslash Y)\\Delta '}", "280cff47b4bf53ba29b780aed56d20f3": "{\\mathsf {f}}(I)\\propto I", "280d0bd4d8017c71a48ebbd88ddde576": "x\\mapsto x^{q}", "280d1021809f3eb41545fead32394ed6": "{\\frac {C}{i}}+FVA={\\frac {C}{i}}(1+i)^{n}", "280d520fa6238d7473aded92a1cab757": "N\\to M", "280d6a71e6cda81773a33482fe7d3438": "\\mathbf {p} _{\\rm {j}}=m_{j}\\mathbf {v} _{\\rm {j}}\\,\\!", "280d890533c1eee90fa363d000bac2b9": "\\phi \\ ", "280d8afbba1dae2ced2afe18b3c0e9e4": "U=mV,", "280dcb9822efc6175734bbb9733a3e3f": "P={E(1-b) \\over {k_{e}-br}}\\,", "280de26ecaae81e6af2475a0aed4d71f": "|\\gamma '|(t):=\\lim _{s\\to 0}{\\frac {d(\\gamma (t+s),\\gamma (t))}{|s|}},", "280df504cc6af5335768fc2291c7afb0": "{\\begin{bmatrix}R(i+1,j)\\\\S(i,j+1)\\\\y(i,j)\\\\\\end{bmatrix}}={\\begin{bmatrix}A_{1}&A_{2}&B_{1}\\\\A_{3}&A_{4}&B_{2}\\\\C_{1}&C_{2}&D\\\\\\end{bmatrix}}{\\begin{bmatrix}R(i,j)\\\\S(i,j)\\\\u(i,j)\\\\\\end{bmatrix}}", "280e6ccedb10278d5a745ab3dc8c469b": "(H-\\mu N)\\psi _{\\alpha }^{\\dagger }|n\\rangle =(E_{n}+\\xi _{\\alpha })\\psi _{\\alpha }^{\\dagger }|n\\rangle .", "280edd3c20bcc641aba370599b4623b9": "f(\\prod x_{i})=\\sum f(x_{i})", "280ee16d36bd690683ee2315b3d5819c": "d_{1},\\ldots ,d_{h}", "280ee3a23290913b9b4830f41e2094e9": "\\left|{\\begin{pmatrix}\\langle \\omega |\\omega \\rangle &\\langle \\omega |s\\rangle \\end{pmatrix}}(U_{s}U_{\\omega })^{r}{\\begin{pmatrix}0\\\\1\\end{pmatrix}}\\right|^{2}=\\sin ^{2}\\left((2r+1)t\\right)", "280f50172ffd5f0a0856dbaa895b0b90": "\\Phi ({\\vec {r}},t)", "280f59d83993f00bdac5721ff792b0ab": "T=t,M=m,L=\\ell ,g=\\ell /t^{2}.\\,", "280f5c08423ff11b7f24e9cdf23398e2": "Z=X\\cup Y^{\\phi }", "280f77ac6f4a90318854c2fa58da0222": "\\mathrm {AmCl_{3}\\ +\\ \\ H_{2}O\\ \\longrightarrow \\ AmOCl\\ +\\ 2\\ HCl} ", "280f797410c8cc0234c2eed13dc4039f": "\\cos \\theta {\\hat {x}}-i\\sin \\theta {\\frac {\\partial }{\\partial x}}", "280f95b2e4395bebecb1daf3eba09a19": "h_{x}(\\alpha )=\\min _{S}\\{\\log |S|:x\\in S,K(S)\\leq \\alpha \\}", "280fb4db74a540a8e6e5b2dc3c448254": "g_{0}=W_{N}(f_{0})\\,", "280fba0f49355074d59c01012952b506": "A,A'\\in {\\mathcal {A}}", "2810eedc2569564b48c1e808d7004834": "\\zeta /r=\\epsilon ^{\\theta }", "28112b4dfed2fd9257085836248439c1": "P_{n}(k\\rho )", "28113eddf896e0ade3f150896800ebef": "\\scriptstyle A\\;\\Rightarrow \\;B", "281171842bfc5a9c47475ebd89f33976": "uv\\in L\\Leftrightarrow vu\\in L", "281177396b89e2b58f7f3f097afc0a19": "E=\\hbar \\omega ={\\frac {\\hbar ^{2}k^{2}}{2m}},", "281187fe095a9609cf54b2b3e17d3b1e": "C_{p}=\\left({\\frac {\\partial H}{\\partial T}}\\right)_{p}", "28119b1609ab96728f02dd7d1d747bb9": "t-0={1 \\over g}\\left[{\\ln(1+\\alpha v^{\\prime }) \\over 2\\alpha }-{\\frac {\\ln(1-\\alpha v^{\\prime })}{2\\alpha }}+C\\right]_{v^{\\prime }=0}^{v^{\\prime }=v}={1 \\over g}\\left[{\\ln {\\frac {1+\\alpha v^{\\prime }}{1-\\alpha v^{\\prime }}} \\over 2\\alpha }+C\\right]_{v^{\\prime }=0}^{v^{\\prime }=v}", "28125304c0b88c2636dd9c38a4400f0a": "f^{-1}\\left(\\,f(C)\\,\\right)=f^{-1}\\left(\\,{\\tfrac {9}{5}}C+32\\,\\right)={\\tfrac {5}{9}}\\left(\\left(\\,{\\tfrac {9}{5}}C+32\\,\\right)-32\\right)=C{\\text{, for every }}C{\\text{.}}", "281262e8067bda6e4fe9319a97c1ed4a": "{\\frac {u\\cdot a}{2}}+{\\frac {s\\cdot a}{2}}+{\\frac {t\\cdot a}{2}}={\\frac {h\\cdot a}{2}}", "2812630fa96c0cd3644f04b61a6dd230": "(E_{11}+E_{22})", "28127894d229b31def8883d8fe770849": "p_{0}={\\cfrac {3F}{2\\pi a^{2}}}={\\cfrac {1}{\\pi }}\\left({\\cfrac {6F{E^{*}}^{2}}{R^{2}}}\\right)^{1/3}", "2812862617caab2c46d3fe75e6037132": "{\\frac {2}{{\\frac {1}{60}}+{\\frac {1}{40}}}}=48", "2812aca5dcca0561b9bde7b0e7a255e1": "\\sim {\\frac {34}{9}}N\\log _{2}N", "2812c143d1f0b96be9fb3f08e3539777": "\\ln(2)/\\lambda ", "2812d5753ee5836c2e89f0285de173b8": "\\int _{0}^{\\pi /2}\\sin ^{2m}x\\,dx=\\int _{0}^{\\pi /2}\\cos ^{2m}x\\,dx={\\frac {1\\times 3\\times 5\\times \\cdots \\times (2m-1)}{2\\times 4\\times 6\\times \\cdots \\times 2m}}{\\frac {\\pi }{2}}\\ \\ m=1,2,3,\\ldots ", "28134c72b74d855489e3cae58c8096d6": "100{\\frac {\\text{rise}}{\\text{run}}}", "2813724c8a547017eff1575e9f11d674": "\\scriptstyle \\tau =\\inf\\{t>0\\,:\\,X(t)<0\\}", "28138b68372ed79835d80a83dab014b1": "1-\\alpha \\approx 0.15", "28139d800f1e0a7d2e3d3002eda9bb49": "\\sum _{k=d}^{n}{n \\choose k}{k \\choose d}=2^{n-d}{n \\choose d}", "2813e55405aa218b929fed2903e5a5c3": "i\\,\\partial _{t}u+\\nabla ^{2}u=V(u)u", "28142ba52dba61cfd7f87558fe61be34": "(1-p_{x})p_{y}", "28145a278d17e03c5a0b34a9e532b9e3": "S\\rightarrow \\mathbb {R} ", "281467171fdeab86696bac11b8ca89ab": "iq>0\\\\\\pi /2&{\\text{ if }}00\\end{cases}}", "2816ead63939515d1c49d5b9ad1e2bbf": "\\Delta _{0}=1,\\Delta _{1},\\ldots ,\\Delta _{n}=\\det A.", "28176e6a04d460a132e3e7a0680401a6": "\\sin {\\frac {\\pi }{n}}={\\frac {16(n-1)}{5n^{2}-4n+4}}.", "2817e279c78754a12fc2aef82ce2c670": "T_{p}(N)", "28183a54575d473f0cb5982ed500f01d": "\\mathrm {STA} _{w}={\\tfrac {T}{n_{sp}}}\\left(X^{T}X\\right)^{-1}X^{T}\\mathbf {y} .", "2818bc0f1e8388585ba001a01f78168a": "n4700m_{e}", "2818de80d899be761fb6deed6bbf8e23": "r^{n}=a^{n}\\sin(n\\theta )\\,", "2818e99ff4a319fbab58c1c5747dabc9": "V/m_{0}", "2819143673ac99247451cda13d1d531e": "|P(s)C(s)|\\gg 1", "28196e5c13728253f927973617493ebb": "M={d_{i} \\over d_{o}}={h_{i} \\over h_{o}}={f \\over d_{o}-f}={d_{i}-f \\over f}", "2819ce0013cc2efdd2c40a35f5d416fb": "\\scriptstyle {c\\,=\\,3\\times 10^{8}\\,m/S}", "2819e818f5c92e1c2e51a1e43dc212b1": "\\cos \\theta ={\\frac {A_{11}+A_{22}+A_{33}-1}{2}}", "281a0e5efb84ad78a412138b9fc11857": "\\Gamma \\vdash _{\\mathrm {F} S}A", "281a35a65047ce911106efe80d16e4f5": "m-n", "281a58ae1cd9d848b0c4f876a3c33b15": "F_{k}=\\rho \\int _{A}\\sum _{i}({\\tfrac {1}{2}}u_{i}^{2}-u_{i}v_{i})n_{k}\\,\\mathrm {d} S.\\qquad (3)", "281a70c20b16a38d7781189936e1ac9f": "E=mc^{2}", "281a89b63c44bf403aaf2e3fde944034": "(n_{1},\\ldots ,n_{d/2})", "281b75a4775396656b19a7f00debb026": "\\mathbb {Z} [i]=\\{a+bi\\mid a,b\\in \\mathbb {Z} \\},\\ {\\mbox{where}}\\ i^{2}=-1.", "281b767d43b900a7f3965b1d572eee16": "E={1 \\over 2}m_{\\mathrm {e} }v^{2}-{Zk_{\\mathrm {e} }e^{2} \\over r}=-{Zk_{\\mathrm {e} }e^{2} \\over 2r}.", "281bee3735e89cd59262dc1a5e4a44c1": "f(x)=\\mathbf {E} ^{x}{\\big [}f{\\big (}X_{\\tau _{D}}{\\big )}{\\big ]}-\\int _{D}L_{X}f(y)\\,G(x,\\mathrm {d} y).", "281c22885742aa5f9bc649b2fc6f13a4": "\\sigma _{xx}-{\\frac {\\sigma _{xz}^{2}}{\\sigma _{zz}}}+|\\sigma _{xy}-{\\frac {\\sigma _{yz}\\sigma _{xz}}{\\sigma _{zz}}}|", "281c7dd75248c5070ef01fd0eb54140b": "2f_{\\Delta }\\,", "281c9aaad475ee64e68124d68b47f010": "t_{4}[\\beta ]=t_{2}[\\beta ]", "281cd41514f1c24fa2dbe44c3e3f3f4b": "3-5\\equiv -2,\\ -2+6\\equiv 4{\\pmod {6}}\\,", "281d033605797b20a134f4e803525aa7": "D_{n}=\\{(X_{1},Y_{1}),\\ldots ,(X_{n},Y_{m})\\}", "281d1818fcfa34029f148efe673ea40e": "K_{f}", "281d692def23e015c2b2190656763759": "\\mathbf {F} _{q}=q\\left(\\mathbf {E} +{\\frac {\\mathbf {v} _{q}}{c}}\\times \\mathbf {B} \\right)\\,", "281d6e900867a49b21e3394f8312541c": "{\\begin{aligned}E&=\\sum _{k=0}^{\\infty }{\\frac {1}{2^{k+1}}}\\cdot \\min(2^{k},W)\\\\&=\\sum _{k=0}^{L-1}{\\frac {1}{2^{k+1}}}\\cdot 2^{k}~+~\\sum _{k=L}^{\\infty }{\\frac {1}{2^{k+1}}}\\cdot W\\\\&={\\frac {L}{2}}~+~{\\frac {W}{2^{L}}}\\,\\,.\\end{aligned}}", "281d7e8ef9853b8080dcbe41aad538cf": "{\\begin{aligned}&\\int \\limits _{0}^{2\\pi }{\\hat {t}}\\ {\\left({\\frac {p}{r}}\\right)}^{3}\\ \\sin ^{3}u\\ du\\ =-{\\hat {g}}\\ \\int \\limits _{0}^{2\\pi }\\ {\\left({\\frac {p}{r}}\\right)}^{3}\\ \\sin ^{4}u\\ du\\ +{\\hat {h}}\\int \\limits _{0}^{2\\pi }\\ {\\left({\\frac {p}{r}}\\right)}^{3}\\ \\sin ^{3}u\\ \\cos u\\ du\\ =\\\\&-{\\hat {g}}\\ \\left(\\int \\limits _{0}^{2\\pi }\\ \\sin ^{4}u\\ du\\ +\\ 3\\ {e_{g}}^{2}\\ \\int \\limits _{0}^{2\\pi }\\ \\cos ^{2}u\\ \\sin ^{4}u\\ du\\ \\ +\\ 3\\ {e_{h}}^{2}\\ \\int \\limits _{0}^{2\\pi }\\ \\sin ^{6}u\\ du\\ \\right)\\\\&+{\\hat {h}}\\ 6\\ e_{g}\\ e_{h}\\ \\int \\limits _{0}^{2\\pi }\\ \\cos ^{2}u\\ \\sin ^{4}u\\ du=\\\\&-{\\hat {g}}\\ \\left(2\\pi \\left({\\frac {3}{8}}\\ +\\ {\\frac {3}{16}}\\ {e_{g}}^{2}\\ +\\ {\\frac {15}{16}}\\ {e_{h}}^{2}\\right)\\right)+{\\hat {h}}\\ \\left(2\\pi \\left({\\frac {3}{8}}\\ e_{g}\\ e_{h}\\right)\\right)\\end{aligned}}", "281db53d244447933f726620ac6fd7c0": "x\\in (-\\infty ,\\infty )", "281dc3696d54353a521f22da74918f90": "X'^{n}=B_{n}(X)\\,", "281de83336672b5c72dc53e363d25dbd": "x\\to 4(x-{\\tfrac {1}{2}})^{2}", "281dfd461ed86b23764c77df036f238c": "s=v_{0}t+{\\frac {1}{2}}at^{2}", "281e189f0280eb27b50110850f6adc17": "f(3)=3+1", "281e67bdb4474965b8e7df2234a16467": "f_{i}(x)\\leq 0,i=1,\\ldots ,m", "281e7d736c5392618178650dc68a3cbf": "v=v'", "281e9dd962f4605c477a3f4578ca80d3": "(\\mathrm {d} G)_{T,p}=\\sum _{i}\\mu _{i}\\,\\mathrm {d} N_{i}\\,", "281f727db8f6c1ae35e00d43f64d8d26": "K_{\\rm {I}}=2\\sigma {\\sqrt {\\frac {a}{\\pi }}}\\,.", "281f89c0b9969c84003dfc12da6d393a": "Q,L:\\mathbb {R} \\to \\mathbb {R} ", "281fa7dba612f89cc36bfcd6504909f1": "\\partial _{\\phi },\\;\\;\\sin(\\phi )\\,\\partial _{\\theta }+\\cot(\\theta )\\,\\cos(\\phi )\\partial _{\\phi },\\;\\;\\cos(\\phi )\\,\\partial _{\\theta }-\\cot(\\theta )\\,\\sin(\\phi )\\partial _{\\phi }", "281fb4623f1eea000f2093f7f9b6c1b9": "x_{k+2}=(2^{32}+6\\cdot 2^{16}+9)x_{k}=[6\\cdot (2^{16}+3)-9]x_{k}\\,", "281fc83a911feffe51f18c681ca881c8": "r_{3}=r_{1}+2^{-k}\\,\\log {|s|}", "281ffc88e975c71953e5139af5d36453": "\\mu _{k}^{'}=\\sigma ^{k}2^{k/2}\\,\\Gamma (1\\!+\\!k/2)\\,L_{k/2}(-\\nu ^{2}/2\\sigma ^{2}).\\,", "28204b91e2d6b472484979781042c627": "(r_{1}+r_{2})\\otimes s=(r_{1}\\otimes s)\\oplus (r_{2}\\otimes s)", "28208ac9405fd04bfe26f04a82ec387e": "\\{w_{1},w_{2},\\ldots ,w_{n}\\}", "28209e455abe9ea5e8dcaa0d421e14da": "z^{-\\alpha }f(z)", "2820c85eb5f3b978ce16452386181120": "x^{2}+y^{4}", "2820e508856bce1a5550abf04b58625b": "2\\cos \\left({\\frac {2\\pi }{17}}\\right)=\\zeta +\\zeta ^{16}\\,", "2820eccbb4bdc8b1a33750b29ba9e912": "p(a/2)<2p(a)", "2821130ecb31647a5f5a06ff6dff0ec1": "s_{0}^{2}", "2821732e30912d6e8b5d0f65a8c2d00f": "x=\\pm {\\sqrt[{m}]{a^{n}}}", "2821992cbb4373430866e32ce07a9405": "M\\simeq 10^{26}", "2821b251f6115ff2e06973617d0ce4ba": "R_{\\mathrm {ab} }=R_{a}R_{b}({\\frac {1}{R}}_{a}+{\\frac {1}{R}}_{b})={\\frac {R_{a}R_{b}(R_{a}+R_{b})}{R_{a}R_{b}}}=R_{a}+R_{b}", "2821cb8255fd755e0cc0360058c76bcf": "d={\\frac {\\sin \\alpha \\,\\sin \\beta }{\\sin(\\alpha +\\beta )}}\\,l={\\frac {\\tan \\alpha \\,\\tan \\beta }{\\tan \\alpha +\\tan \\beta }}\\,l", "28220cdd2ae5553d11f151d1b6215939": "y^{calc}=cy^{obs}", "282225ee9806cb0783a6fc8f100b385a": "{\\tilde {\\psi }}\\upharpoonright _{\\alpha }", "28223ff05a6ea1a63fdec9528aceb64c": "f_{n}(x)={\\frac {\\sin(n^{2}x)}{n}}", "28223ffe334074fc595ed8d6caf08baa": "Beam=LOA^{2/3}+1", "282253d80209604ef46ae039d29cf5fb": "p=a_{0}+a_{1}x+\\cdots +a_{n}x^{n},\\quad a_{n}\\not =0,", "28228fcdbd1463e5e5a16cd0c1025eca": "{\\mathfrak {e}}_{8}(\\mathbf {K} )", "2822bd0cb10b0f399659da47b50dc162": "{\\begin{aligned}{\\tilde {I}}_{N}[f]=A_{1}\\sum _{i=1}^{6}f(a_{i}^{1})&+A_{2}\\sum _{i=1}^{12}f(a_{i}^{2})+A_{3}\\sum _{i=1}^{8}f(a_{i}^{3})\\\\&+\\sum _{k=1}^{N_{1}}B_{k}\\sum _{i=1}^{24}f(b_{i}^{k})+\\sum _{k=1}^{N_{2}}C_{k}\\sum _{i=1}^{24}f(c_{i}^{k})+\\sum _{k=1}^{N_{3}}D_{k}\\sum _{i=1}^{48}f(d_{i}^{k}),\\end{aligned}}", "28237807fc4d6f2305b63a8f0e57f1b8": "\\scriptstyle \\leq 5\\times 10^{-23}", "28239bacd4ca052f4ead09b06a8801e9": "\\int _{0}^{\\theta }\\operatorname {Cl} _{2m}(x)\\,dx=\\zeta (2m+1)-\\operatorname {Cl} _{2m+1}(\\theta )", "28240e1d02b0db1a4040189ebfef3196": "\\ C", "28241226029bf2f89e0c1b139f1bb56f": "ax^{2}+bx+c\\,,", "28246c22d62808930b5f473c188dd456": "\\;-\\log q_{j}.", "2824a348a786343ddc38182c6cc693f1": "H_{t-1}=K+A'H_{t}A-A'H_{t}C(C'H_{t}C+R)^{-1}C'H_{t}A,\\,", "2824ca4de4430067aa4c914f9768f3e1": "{{V}_{TH}}", "2824f21de871d1118a3efac23d4ca430": "{\\frac {1}{2}}\\!\\,", "2825151dd7b28833ca68f978f5757e5d": "f={\\frac {2r_{N}\\Delta r_{N}}{\\lambda }}", "28252280e4a6a2bbe22c9d2f568dd923": "{\\frac {\\partial S}{\\partial x_{k}}}=\\int {\\frac {\\partial L}{\\partial x_{k}}}\\,dx_{3}=\\int {\\frac {dp_{k}}{dx_{3}}}\\,dx_{3}=p_{k}", "28252bc6e93be53331ee998ac1e426a5": "E=D^{2}=9", "282540820ff74ff47d353f3f72714902": "\\left({\\frac {\\Delta \\;h}{T}}\\right)=\\left({\\frac {u}{\\sqrt {T}}}\\right)\\cdot \\left({\\frac {\\Delta \\;v_{w}}{\\sqrt {T}}}\\right)", "2825b5dd089ac4f5817f8bdcb807e7b9": "{\\mathbf {x} }^{\\mathrm {T} }(T)F{\\mathbf {x} }(T)", "2825b5f578c8fd7c338fc3d80329b10c": "f(x)={\\frac {x^{k-1}\\exp(-{\\frac {x}{\\theta }})}{\\theta ^{k}\\Gamma (k)}}", "2825c4004b22329b10f9cc0210977591": "|d_{0}|>0", "2825c4d31e015e354eb3b409fdbe9333": "\\sum _{l=1}^{M}S_{l}(1+\\mathrm {APR} /100)^{-t_{l}}=\\sum _{k=1}^{N}A_{k}(1+\\mathrm {APR} /100)^{-t_{k}}", "2825dd5825664ca99c78d9c5c5aafc59": "\\alpha ={\\frac {d\\log(\\lambda F_{\\lambda })}{d\\log(\\lambda )}}", "2825fda4cda7c4b4283b921167a002fb": "\\Omega _{n,\\mu \\nu }(\\mathbf {R} )=i\\sum _{n'\\neq n}{\\langle n|(\\partial H/\\partial R_{\\mu })|n'\\rangle \\langle n'|(\\partial H/\\partial R_{\\nu })|n\\rangle -(\\nu \\leftrightarrow \\mu ) \\over (\\varepsilon _{n}-\\varepsilon _{n'})^{2}}.", "282642668875973e68fac1bfa91e1227": "C_{1},C_{2}\\mapsto C_{1}\\otimes C_{2}", "282654f74c924ef27bfa39f49305af61": "\\left\\langle {dG}/{dt}\\right\\rangle _{\\tau }=0", "28265d45ded171762eef09cf0766d542": "y(u)=-\\mathrm {cn(u,k)} +(a/k)\\mathrm {dn(u,k)} ", "2826620b910812fe94d8a009e4ef4bd6": "f_{1}\\,", "282667de5cb89611fb86409738a84bb7": "P_{\\mathrm {VWAP} }", "28269d24a5e789d9173414362d012617": "\\mathrm {Diff} _{k}", "2826d392bcafcf2fb02e4eeaa7b90603": "Y_{21}={-Z_{21} \\over \\Delta _{Z}}\\,", "28279454da0fc28ffa53194de06f99ab": "Z(\\cdot )", "2827b73f9cedd981dc3a0cb74f1d5fb4": "\\left[1-{\\left({\\frac {\\partial z}{\\partial x}}\\right)}_{y}{\\left({\\frac {\\partial x}{\\partial z}}\\right)}_{y}\\right]dz=\\left[{\\left({\\frac {\\partial z}{\\partial x}}\\right)}_{y}{\\left({\\frac {\\partial x}{\\partial y}}\\right)}_{z}+{\\left({\\frac {\\partial z}{\\partial y}}\\right)}_{x}\\right]dy.", "2827c0460c2a0f427899d74046d67100": "r=C_{S}^{2}K_{1}K_{2}C_{A}C_{B}", "2827ef02f32f9ad88b2cdb012cf505d1": "{\\frac {\\partial \\Psi }{\\partial t}}(x,t)=\\mu (1-x){\\frac {\\partial {\\Psi }}{\\partial x}}(x,t);\\qquad \\Psi (x,0)=x^{i},\\quad \\Psi (1,t)=1.", "2827f2c33fe59d31c37b7f8822c4d446": "\\left\\{1,...,P-1\\right\\}", "282805a4b296689136c18281442b235e": "\\cos \\sigma =-{\\frac {4a^{2}-d_{1}^{2}-d_{2}^{2}}{2d_{1}d_{2}}}", "2828183aab2fe2abcb6066a505d8cc71": "(a_{21}w\\cdot x_{1}+Ka_{22}\\cdot wx_{2})(1+g)=Kx_{2}", "28283eb26f3b84805a1bcbec1e475d46": "\\ell _{A}\\colon A\\rightarrow \\Sigma _{A}", "2828d562ec89efba33b07eb7412af5d0": "\\left[{\\hat {f}}_{i}^{\\dagger },{\\hat {f}}_{j}^{\\dagger }\\right]_{+}=0", "28292cb917e5527a549d86fe36011484": "{\\text{2. }}\\omega \\notin B:P(\\omega |B)=0", "28297a01714f309f41f03b92c934281e": "{\\vec {\\pi }}_{E}={{\\tilde {r}}_{E}}^{-1}", "282a0fa350c861e987b2d054c1a59169": "AJ=kJ", "282a24cc78a7f580f0f0598dd4622b22": "r={\\frac {mv_{e}rn\\hbar 4\\pi \\epsilon _{0}}{mZe^{2}}}", "282aa95583fb772109d3b9d9c59354c5": "L_{y}'=z'p_{x}'-x'p_{z}'=\\gamma (V)[(zp_{x}-xp_{z})+V(p_{z}t-zE/c^{2})]=\\gamma (V)[L_{y}-V(mz-p_{z}t)]", "282b182ad629456543a036fff3d7daf1": "z_{11}=-z_{22}", "282b581d27f8bb09fa048f5f22f96db9": "\\Delta H_{p}", "282b72fc2e97581553379bb1aee2aba9": "{\\frac {(i\\omega )^{2}}{((i\\omega )^{2}-\\xi ^{2})^{2}}}", "282b9c5a5e060cc36ae36b9f6b3921b3": "P_{n}(-x)=(-1)^{n}P_{n}(x).\\,", "282be2b28622fdff2108ba09a009c856": "x_{n+1}=x_{n}-{\\frac {f(x_{n})}{g(x_{n})}}", "282c7e565887bd16e68d098dd12ca91f": "y_{i}=f(x_{i};\\beta )\\cdot TE_{i}\\cdot \\exp \\left\\{{v_{i}}\\right\\}", "282c8b894d92b2f571ce5ff246465abd": "\\{x\\mid x\\in B\\wedge \\phi \\}", "282ce3e0be0ea2c075b358e722938c17": "{\\frac {X-np}{\\sqrt {np(1-p)}}}", "282cfca5cf7b4b5ad8376ff6c7562cf3": "\\mathbb {E} [f(x)]=\\sum _{j=0}^{\\infty }f(j){\\frac {\\lambda ^{j}}{(j!)^{\\nu }Z(\\lambda ,\\nu )}}.", "282d1b3b6b7c775fb8571bad71d88fb8": "5+5+3+7=20", "282d223b7195128ecbb0623e783b053f": "{\\begin{aligned}m=s'\\cdot r^{-1}{\\pmod {n}}\\end{aligned}}", "282d4fca6d0c63bc6fd7078452c1a037": "\\int x^{n}\\Phi (x)\\,dx={\\frac {1}{n+1}}\\left(\\left(x^{n+1}-nx^{n-1}\\right)\\Phi (x)+x^{n}\\phi (x)+n(n-1)\\int x^{n-2}\\Phi (x)\\,dx\\right)+C", "282d7b02daf8ca7b4818dd13afb16172": "T_{12}={\\frac {S_{11}}{S_{21}}}\\,", "282e2c4eb6b2a897cfc5562516edfadd": "3-n", "282e460e3eaa80d5c744970759cdd4f0": "P(event)={\\frac {\\text{number of outcomes in event}}{\\text{number of outcomes in sample space}}}", "282e48e0982c2a2b03b868e6e19c2882": "E\\psi =-\\hbar ^{2}\\left[{\\frac {1}{2\\mu }}\\left(\\nabla _{1}^{2}+\\nabla _{2}^{2}\\right)+{\\frac {1}{M}}\\nabla _{1}\\cdot \\nabla _{2}\\right]\\psi +{\\frac {e^{2}}{4\\pi \\epsilon _{0}}}\\left[{\\frac {1}{r_{12}}}-Z\\left({\\frac {1}{r_{1}}}+{\\frac {1}{r_{2}}}\\right)\\right]\\psi ", "282eb9aecddbf4051be94a714b45d867": "yRx", "282f45a4eacb40f0f3098ee43999346e": "{2\\sigma \\over {\\bar {\\delta }}}", "282fba55f27ef23de22362dacbf5e00d": "\\ \\tau =Fk", "282feeb6714ee76ff0a1f7b6f5f91551": "0,1,..,m-1", "283004b30f9f239a582bbe4bb6cf4b7e": "{\\dfrac {\\partial {\\vec {u}}}{\\partial t}}=\\nu \\partial _{y}^{2}{\\vec {u}}", "28308a1c6a71a8b874a17a516d0db48e": "\\sum _{n=0}^{\\infty }{a_{n} \\over n!}t^{n}", "2830c53827f4b9ecaaf52120547a7c2f": "\\Rightarrow y[n]+{\\frac {1}{4}}y[n-1]-{\\frac {3}{8}}y[n-2]=x[n]+2x[n-1]+x[n-2]", "2830e9afdffdb79bd779e67fc8ed3a88": "n_{A}=n_{B}=10", "28310a55ffd1c04b3fa6842a2ba62cf7": "2\\lim \\limits _{\\tau \\rightarrow +\\infty }\\langle T\\rangle _{\\tau }=\\lim \\limits _{\\tau \\rightarrow +\\infty }\\langle U\\rangle _{\\tau }", "28311e4d513f284d05fb138d09f18c07": "p^{2}=m^{2}", "28312ac12818caa3729bba0dc7b6ff3f": "g^{x_{1}}", "28313d45fa5075ba2c52991d6219d47a": "i\\equiv g^{j}{\\bmod {p}}", "2831a72437535d7e3b161f09debd7bea": "L=\\oint p\\mathbf {n} \\cdot \\mathbf {k} \\;\\mathrm {d} A,", "2831ad611236f92b164627e9b1679687": "\\displaystyle {\\frac {(\\alpha +2)_{n}}{n!}}{}_{3}F_{2}(-n,n+\\alpha +2,(\\alpha +1)/2;(\\alpha +3)/2,\\alpha +1;t)", "2831b3626b729f14ffc0871fbc3dd5b6": "u=u_{0},v=v_{0}", "2831dae3dfe4647c94b5572d1eed7fe3": "M+i\\Gamma ", "2831e20b5046e457e343e70fbd5d1f3e": "TSS=(y-{\\bar {y}})^{T}(y-{\\bar {y}})=y^{T}y-2y^{T}{\\bar {y}}+{\\bar {y}}^{T}{\\bar {y}}.", "283215c0662bd9d94369fcab9f1f16cf": "q_{1}0\\,", "2840faa55b733382f6d059c6a96dd9ee": "T={\\frac {\\epsilon }{\\delta S}}={\\frac {2\\times 10^{-23}J}{70\\times 10^{-23}J/K}}={\\frac {1}{35}}K", "284126970e43f7ffb46f9127dd30e421": "P={1,2,3,\\ldots ,n}", "284163c26c8a616741409b8060472d13": "\\mathbb {E} [r|\\theta ^{\\ast },a^{\\ast },x]", "2841aef7a5af7a0c9430c0f088fe9c47": "RR^{-1}=kN+1", "2841b11a2b2bd36a84fb102cc80cb037": "{\\frac {DS^{\\mu \\nu }}{ds}}+u^{\\mu }u_{\\sigma }{\\frac {DS^{\\nu \\sigma }}{ds}}-u^{\\nu }u_{\\sigma }{\\frac {DS^{\\mu \\sigma }}{ds}}=0", "2841d1f57bcad856ca8c9dff4edaed7f": "G={\\frac {a(a+1)}{2}}.", "2841d7351508128398154618b9804ce4": "{\\frac {n}{m}}={\\frac {m}{n-m}}.\\qquad (*)", "284228c433b4a2388e4fe048388a2c34": "\\xi =A_{x}f_{x}(z)e^{i(\\omega t-kx)}\\quad \\quad (1)", "28424d5404816c8b7eb01a9c2cb8b8a6": "|m_{I},m_{J}\\rangle ", "28425fa768271bdf6b3e27bf8d94ebbe": "\\sum _{i}p_{i}(\\log p_{i}-\\sum _{j}(\\log q_{j})P_{ij})\\geq \\sum _{i}p_{i}(\\log p_{i}-\\log(\\sum _{j}q_{j}P_{ij})", "284276368e293ce1d801f6ac0ce25589": "U_{\\text{eff}}(\\mathbf {r} )={\\frac {L^{2}}{2mr^{2}}}+U(\\mathbf {r} )", "28429bb4a67f9fdd116462b10fede3f3": "{\\begin{bmatrix}a&b\\\\c&d\\end{bmatrix}}={\\begin{bmatrix}1&\\lambda z\\\\0&1\\end{bmatrix}}.", "2842de00abe300fbaff649e74c25f3e3": "IPxy\\leftrightarrow (Pxy\\land (Czx\\rightarrow Ozy)).", "284305b44f9b2c4f57193fe8bd17706d": "C=\\lbrace U_{\\alpha }:\\alpha \\in A\\rbrace ", "284307c2182414dc93d37f673e36933c": "G_{2}\\setminus G_{1}", "28432a9919394851ab7cfe2d9fbc9765": "RR={\\tfrac {1}{5}}", "284395965384ad307f8acf12c5017a9d": "\\operatorname {Li} _{2}(e^{i\\theta })=\\zeta (2)-\\theta (2\\pi -\\theta )/4+i\\operatorname {Cl} _{2}(\\theta )", "2844051ec28da950fe01fcbe59e38cc4": "\\ln(c_{T-j})+bV_{T-j+1}(Ak^{a}-c_{T-j})", "28441f65f29e9cd84256c4e41ea72e9e": "\\mathbf {n} \\,\\!", "28445999aa4b87bb4fde9c67cd92eab2": "{\\begin{array}{cc}{\\begin{array}{rr}\\\\&3\\\\{\\text{-}}1&\\\\\\\\\\end{array}}&{\\begin{array}{|rrrr}1&{\\text{-}}12&0&{\\text{-}}42\\\\&&&\\\\&&&\\\\\\hline 1&&&\\\\\\end{array}}\\end{array}}", "284499281807875f8f358e48ac647bfd": "r\\in (0,1)", "2844aba8b65a4c2b47378788082a892a": "x:G\\to Y", "2844d2ebc068ccf99556f5454e23c8d8": "h(t)=h_{0}\\sin(\\omega t)", "2844d30f147ee7c01c5e607ee4fae014": "L_{\\text{i}}(\\mathbf {x} ,\\,\\omega _{\\text{i}},\\,\\lambda ,\\,t)", "28450326cc4d39ed0a4b0804f77b4323": "A',S',T'", "28454c2dfb3ca842e861debef3ab4ea0": "f(x)=\\sin(2x)", "284555eabf8f5a0b52d48b5e7eb73293": "P=\\left\\langle \\psi '\\right|{\\boldsymbol {\\mu }}\\left|\\psi \\right\\rangle =\\int {\\psi '^{*}}{\\boldsymbol {\\mu }}\\psi \\,d\\tau ,", "2845650f3467218190805286ce04287c": "{\\frac {\\pi _{1}}{\\pi _{2}}}", "2845710b9b89463d603c6ef9e94a0617": "\\succ _{v}^{p}", "28458041c206e033109a9fb1baec9ca3": "Cl_{t}^{\\geq },t\\in T", "2845860934de4f42775606cc78d3c176": "\\triangle \\delta =-\\beta \\cdot \\sin(\\delta -\\alpha )", "284593eee75e69b2d6421edc5382ee69": "\\mathbf {B} (t)=(1-t)\\mathbf {B} (t)+t\\mathbf {B} (t)", "28459a7631fbd31468e0dab7c52c0751": "(\\log 5-\\log 3).", "2845c55255ebc9076875c397054b9356": "\\scriptstyle \\mathbf {F} ", "2845e91b1773d0d63ac5c8bdf58417de": "H_{p+q}(LM\\times LM)", "28462d62b8f34681c3102a8225af4ebe": "x_{i}={\\frac {g_{i}}{g_{0}}}(u_{1},\\ldots ,u_{d})", "28463ca1f2f2ae194927fe2c395fdf5e": "\\left|S_{k}-L\\right\\vert \\leq \\left|S_{k}-S_{k+1}\\right\\vert =a_{k+1}.\\!", "2846601d9a14d87b9f4eaec14b9b7bae": "(\\lambda ~{\\boldsymbol {\\mathit {1}}}+{\\boldsymbol {A}})^{T}\\cdot \\left[{\\frac {\\partial I_{1}}{\\partial {\\boldsymbol {A}}}}~\\lambda ^{2}+{\\frac {\\partial I_{2}}{\\partial {\\boldsymbol {A}}}}~\\lambda +{\\frac {\\partial I_{3}}{\\partial {\\boldsymbol {A}}}}\\right]=\\det(\\lambda ~{\\boldsymbol {\\mathit {1}}}+{\\boldsymbol {A}})~{\\boldsymbol {\\mathit {1}}}~.", "284661569ec69cb58d73827c0bf7185b": "h(X,Y)=-\\int _{Y}\\int _{X}f(x,y)\\log f(x,y)\\,dx\\,dy", "2846711728a14c311e4e0903f534b01c": "\\lim _{x\\searrow 0}{\\frac {e^{-1/x}}{x^{m}}}=0,", "2846a528b93c0234575ced2eafe3cd8b": "\\Delta {\\hat {z}}\\ =\\ {\\frac {r^{2}}{\\mu }}\\left[\\ 2\\ F\\int \\limits _{0}^{\\pi }\\sin u\\ du\\right]\\quad {\\hat {h}}\\times {\\hat {z}}=\\ {\\frac {r^{2}}{\\mu }}\\ 4\\ F\\ \\quad {\\hat {g}}", "2846d20f5eaaafcd08d5d6adb30a17ff": "{\\hat {a}}_{j}{\\hat {a}}_{j}^{\\dagger }", "28472aed5e1239b54f3d16b3bc3ab3ba": "1/distance", "284780e49866c5b42d7eae735c87400b": "1/4\\left(1/4-1/4e^{-4/3t}\\right)\\ ", "284790fad2154fb9332b066aff36bdb3": "\\partial _{i}:={\\frac {\\partial }{\\partial x^{i}}}", "2847a52983f7ba1fd700d124cff6de4b": "G:=\\left\\{\\left.{\\begin{pmatrix}a&b\\\\0&1\\end{pmatrix}}\\ \\right|\\ a>0,\\ b\\in \\mathbf {R} \\right\\}.", "2847f314b7826dcee416abfb3f02fb0c": "{\\frac {\\mathrm {s} }{\\mathrm {m} }}\\times {\\frac {\\mathrm {m} }{\\mathrm {s} }}", "28481f773b834e7f8a4a0ea3ccace4d7": "{\\frac {s}{1-e^{-s}}}=1+s_{e}", "284890c881bd6f2c0f92f4c1444fe6ab": "R(0,m)", "2848d1c0a2914f6cc2b7bbef19aea479": "{{\\textbf {x}}_{t+1}}=A_{i_{t}}{\\textbf {x}}_{t},\\quad A_{i_{t}}\\in \\{A_{1},\\dots ,A_{m}\\}", "28490a38a6e9cf83fb4ebdd34f96b564": "\\rho =Aa^{-3}+Ba^{-4}+Ca^{0}\\,", "28491b99d2da0d0b821b768753d41a9d": "B<{\\tfrac {1}{M}}\\cdot {\\tfrac {1}{2T}},", "284926c45f7465a039b1d4fd5bedc9d0": "S_{RBB}(n)=S_{RRB}(n)={\\tfrac {1}{2}}3^{n}-{\\tfrac {1}{2}}", "28492fde94aabf7f86a39ee66b6092cf": "\\omega ^{k}{}_{ij}:={\\mathbf {u} }^{k}\\cdot \\left(\\nabla _{j}{\\mathbf {u} }_{i}\\right)\\,,", "2849437afdc4314e32f366a312729b5c": "\\sum _{\\text{cyclic}}\\sin A={\\frac {S}{2Rr}}={\\frac {s}{R}}\\quad \\quad \\sum _{\\text{cyclic}}\\cos A={\\frac {r+R}{R}}\\quad \\quad \\sum _{\\text{cyclic}}\\tan A={\\frac {S}{S_{\\omega }-4R^{2}}}=\\tan A\\tan B\\tan C\\,", "28498eeabbf20326ae32c85db600ad6b": "496=2^{4}(2^{5}-1)=1^{3}+3^{3}+5^{3}+7^{3}", "284a3b4595c993374b7c0da777bc13b0": "\\Pi (x)=\\sum _{p^{k}\\leq x}{\\frac {1}{k}}.", "284a3c2f37ea20a2318ca10ccb62dde5": "\\mathbf {e} _{ij}=\\mathbf {e} _{i}\\otimes \\mathbf {e} _{j}", "284a984fc35ba2a8642aa7e19347be59": "O(\\log |V|)", "284b3567801d1127ea5db52b3fe71319": "F_{i}", "284b3b4a1dfd5b757a138ae6ce43c3e9": "A\\rtimes G", "284b4ca8fc22248876abb4330bf9149b": "({\\ddot {r}},\\ {\\ddot {\\theta }})", "284b6f4c95696bd001a0efadce74e8f0": "{\\begin{aligned}{\\biggl |}\\bigcup _{i=1}^{n}A_{i}{\\biggr |}&=\\sum _{k=1}^{n}(-1)^{k+1}\\left(\\sum _{1\\leq i_{1}<\\cdots 0\\right\\}", "285657e37d61def36f931193630183cb": "\\textstyle g", "2856693c073922f59799e321862e3ad1": "a_{i}^{\\prime \\dagger }", "2856afbc1abd54fd0121d1dd08547784": "\\displaystyle dE=-\\iota _{H}d\\alpha ", "2856bd6d0e04d44a64c851ef98def6c9": "(13)\\quad \\quad \\;u_{s}\\left(\\rho _{2}u_{2}-\\rho _{1}u_{1}\\right)=\\left(\\rho _{2}u_{2}^{2}+p_{2}\\right)-\\left(\\rho _{1}u_{1}^{2}+p_{1}\\right)", "2856d04d586c50a1f56e66edd3e30b9d": "z\\in \\Omega ", "28570ecdcc326c36c408fc7cb4468f78": "\\mu _{u\\to v}(x_{v})=f_{u}(x_{v})", "28572ab1864ef462bc504258cefe27eb": "\\Lambda _{Roy}=max_{p}(\\lambda _{p})", "2857a1840370c99b8e0ecfb0db210e6b": "\\rho (\\mathbf {r} ,t)=N\\sum _{s_{1}}\\cdots \\sum _{s_{N}}\\int \\ \\mathrm {d} \\mathbf {r} _{2}\\ \\cdots \\int \\ \\mathrm {d} \\mathbf {r} _{N}\\ |\\Psi (\\mathbf {r} _{1},s_{1},\\mathbf {r} _{2},s_{2},...,\\mathbf {r} _{N},s_{N},t)|^{2}.", "2857b4fd34e4e5eb74c88b872110da5f": "o_{1}", "28580724e256f4ed9298cf6418346c73": "\\tau ,\\tau '", "28580757bd34c4107f68598d5a55ee02": "k=ap", "28584ee9397b91dd00ece5d7c9eaf1ad": "\\mathbf {a} \\wedge \\mathbf {b} .", "28584f74407126559e4b4fc19fc72bdd": "{\\frac {DV}{Dt}}=-{\\frac {1}{\\rho }}{\\frac {\\partial p}{\\partial s}}-KV", "2858bba5687ac80ed5391eb909efdd32": "T(n,1)=1,\\;T(1,k)=1,\\;n\\geq k:T(n,k)=-\\sum \\limits _{i=1}^{k-1}T(n-i,k),\\;n==Notablecategorizationoddities==*'''Category:TheLeagueofExtraordinaryGentlemen'''under:**'''Category:Alternatehistorycomics'''**'''Category:Steampunkcomics'''*'''Category:100Bullets'''under:**'''Category:Crimecomics'''*'''Category:SinCity'''under:**'''Category:Crimecomics'''*'''Category:Bone(comics)'''under:**'''Category:Fantasycomics'''*'''Category:ConantheBarbariancomics'''under:**'''Category:Fantasycomics'''*'''Category:TheDarkTowercomics'''under:**'''Category:Fantasycomics'''**'''Category:Post-apocalypticcomics'''**'''Category:Sciencefictioncomics'''**'''Category:Westerncomics'''*'''Category:Elfquest'''under:**'''Category:Fantasycomics'''*'''Category:Ozcomics'''under:**'''Category:Fantasycomics'''*'''Category:TheSandman'''under:**'''Category:Fantasycomics'''**'''Category:Horrorcomics'''*'''Category:WarhammerFantasycomics'''under:**'''Category:Fantasycomics'''**'''Category:Warcomics'''*'''Category:ArmyofDarknesscomics'''under:**'''Category:Horrorcomics'''**'''Category:Zombiesincomics'''*'''Category:CthulhuMythoscomics'''under:**'''Category:Horrorcomics'''*'''Category:DanDare'''under:**'''Category:Sciencefictioncomics'''*'''Category:RoboCopcomics'''under:**'''Category:Sciencefictioncomics'''*'''Category:TheStandcomics'''under:**'''Category:Post-apocalypticcomics'''*'''Category:JamesBondcomics'''under:**'''Category:Spycomics'''*'''Category:Warhammer40,000comics'''under:**'''Category:Warcomics'''", "2865c613c99aaa3d69551405dfc218ba": "\\ {\\dot {c}}+\\Omega (x^{*}-c)=0,", "2865caa8a46c6717434c908272c2f810": "x\\div 9", "2865d444b3b35c5bcc305c75e3fd29c8": "p_{mk}^{}", "2865eda9ce5e110adea2b60a14de4b71": "Q_{m_{A}}^{\\ell _{A}}", "2865efcf42bdde4488c4e4e4ef817385": "{\\begin{aligned}1&={\\binom {t}{1}}-{\\binom {t}{2}}+\\cdots +(-1)^{t+1}{\\binom {t}{t}}\\\\&=|\\{A_{i}\\mid 1\\leq i\\leq t\\}|-|\\{A_{i}\\cap A_{j}\\mid 1\\leq iM)<\\varepsilon ,\\ \\forall n.", "2869dd2bcec667c9306f171c159630b1": "C_{2}=\\{U^{3}=-I\\},", "286a08da142b39864018779678bb07de": "x\\geq [x]", "286a0fe07f8fc1f50ec995d93ddda362": "M_{57,885,161}\\approx 5.81\\times 10^{17,425,169}\\approx 10^{10^{7.2}}=(10\\uparrow )^{2}7.2", "286a12120a9296205c7ce4da3f05003f": "\\,pL+(1-p)N\\,", "286a1398fcc0759a3ed7f98cb6e6b8de": "f^{-1}([c,c'])", "286a1f471e752a8643dbe53f15a46946": "C_{in}^{\\alpha }(x)+C_{in}^{\\alpha }(y)=(x,\\alpha x)+(y,\\alpha y)=(x+y,\\alpha (x+y))=C_{in}^{\\alpha }(x+y)", "286a232b5107c68f7cb2c78d233a3f8e": "k_{v}", "286a2ff13eafdc3542dfedef94320557": "\\rangle ", "286a42733f3ff6c6a156d7ec4c9aa10b": "{\\hat {F}}(x)", "286a7c43e4c18da745808373aa193819": "f:k\\to j", "286adb3a7c288faf021ea7ae95aecefe": "(f(x))^{j}=\\sum _{k=0}^{\\infty }M[f]_{jk}x^{k}.", "286b02707b1a36ca99a961b53fae4ae9": "N!=\\Gamma (N+1)", "286b4e6e67ee000352458410bcd9b96a": "\\int _{E}f\\,d\\mu =\\int _{E}f\\left(x\\right)\\,\\mu \\left(dx\\right)", "286b89fffa11ef2b83c2eb8068f9807e": "T_{1},T_{2}", "286c097ff48cad0fb038a7ac9918956a": "A_{\\mathrm {vdB} }", "286c57016f11e68e131e21ead158f853": "\\liminf _{x\\rightarrow c}{\\frac {f(x)}{g(x)}}=\\limsup _{x\\rightarrow c}{\\frac {f(x)}{g(x)}}=L", "286c7c031bc5e164dd0cf8fcc4ca1dee": "\\not =\\varnothing ", "286d1e0b3e88ca0c9b4e019e9ca79cd3": "(\\lambda \\lambda \\lambda 662(\\lambda \\lambda 6(\\lambda 1(26)3)({\\underline {15}}(51(\\lambda 1))(5(\\lambda 1)1))))(12(\\lambda \\lambda \\lambda 312)))1(\\lambda \\lambda 2)))))(3(1(\\lambda \\lambda \\lambda \\lambda 9(1(\\lambda 51(\\lambda 154)))", "286d27874401cea6ac9301ac3d53ce91": "G={\\frac {2}{\\pi }}\\int _{0}^{1}{\\frac {dx}{\\sqrt {1-x^{4}}}}", "286d2cf24aa5a675fa729356990d08d9": "r_{1}=r,r_{2}=-r\\,\\!", "286d3bbc9892c343673303a6f66d5bf7": "A\\bullet B=(A\\oplus B)\\ominus B,\\,", "286d6570ea75b46d39d8c4239542542f": "w=w(r,t)", "286d8cdac8faa121e4c3c846a117c00f": "v={\\frac {\\kappa \\,\\Delta P}{\\mu \\,\\Delta x}}", "286e0e80e1c2db9f4dde113eeb5f5bf4": "\\mathbf {v} ={\\frac {\\mathrm {d} \\rho }{\\mathrm {d} t}}\\mathbf {u} _{\\rho }+\\rho \\mathbf {u} _{\\theta }{\\frac {\\mathrm {d} \\theta }{\\mathrm {d} t}}=v_{\\rho }\\mathbf {u} _{\\rho }+v_{\\theta }\\mathbf {u} _{\\theta }=\\mathbf {v} _{\\rho }+\\mathbf {v} _{\\theta }\\ .", "286f19a46fc199ab18ee0a7421f09438": "{\\mathcal {F}}_{n}", "286f40e05e4f9a74be01f2a63bd5da45": "2^{i}n-1", "286f4a171dbe56dd41892b3fc4badd13": "\\mathbf {T} ^{n}{\\vec {b}}^{n}={\\hat {e}}_{n}.", "287078a3da80f9b3a39007425724e4b9": "\\left(a\\csc \\left(A+{\\frac {\\pi }{6}}\\right),b\\csc \\left(B+{\\frac {\\pi }{6}}\\right),c\\csc \\left(C+{\\frac {\\pi }{6}}\\right)\\right)", "2870da910b1ad6292810b1c3a7181ba8": "[Z,\\Omega X]", "287128bf557db73ad6efaa80e7577070": "{}^{1}i=i", "2871896333dbd34312a604b2625c8f9c": "{\\hat {H}}_{s}", "28718d182c2578ea2c81a53baec06e79": "{\\frac {\\epsilon }{k}}={\\frac {1.000+0.945(c-1)+0.134(c-1)^{2}}{1.023+2.225(c-1)+0.478(c-1)^{2}}}", "28720155915560e4635f8e9edcf4ad97": "P(W|L)={\\frac {0.75\\cdot 0.50}{0.75\\cdot 0.50+0.15\\cdot 0.50}}={\\frac {5}{6}}\\approx 0.83,", "28726f6b66ad687fa43aab0083e2dbe3": "(N,D,\\lambda )", "2872a8a59bedb00b19f0f4ec8a4c1cc1": "X_{n}^{\\beta _{\\gamma +1}}", "2872b50c8f6dab67e5f70c4d75d9a13d": "(a_{1},\\ldots ,a_{n})=((a_{1},\\ldots ,a_{n-1}),a_{n}).\\!", "2872ec6e85009daa0ee74e6240a1fc6c": "{\\vec {W}}_{\\mu }", "2873411454166a7fdc9c5a23220e267f": "{\\begin{smallmatrix}M_{v}\\ =\\ m+5(\\log _{10}{\\pi }+1)\\ =\\ -1.47+5(\\log _{10}{0.37921}+1)\\ =\\ 1.42\\end{smallmatrix}}", "2873a7bc7615707d281da7ce3e828fa3": "f\\wedge g", "2873edfc8eb06138b5cff10e375d037e": "ce^{i\\psi }-a={\\tfrac {b}{2}}(e^{i\\rho }-e^{i\\lambda })=ib\\sin \\alpha e^{i\\theta }.", "287435c44c7b5aaae57c91c0c8aa41bd": "d^{*}=d(d-1)-2\\delta -3\\kappa .\\,", "287457257b63a341593b1b58c8f859bf": "\\;=F_{id}(N,V,T)+F_{ex}(N,V,T)", "287493795d19a0e23be24874d365df98": "L(i)", "2874bb2c3a9cc6217705eead6273d30e": "{\\tilde {C}}_{n}", "2874c284675cd977c0144210092eadde": "\\sigma _{y}^{2}={\\frac {1}{n}}\\sum _{i=1}^{n}\\left(y_{i}-{\\overline {y}}\\right)^{2}", "28751ae47c2cb13551948f3880aa0e2d": "F(\\mathbf {x} +\\mathbf {h} )\\approx F(\\mathbf {x} )+\\mathbf {h} \\left({\\dfrac {\\partial }{\\partial \\mathbf {x} }}F(\\mathbf {x} )\\right)^{T}.", "287534998454db10659f41e200838084": "\\varepsilon _{ijk}A_{j}\\nabla _{k}", "2875548d7438f9df84bdcbefe7012b19": "G(t)-x_{f}(t)", "287563016c5fcf69aea236ee94452d9d": "{\\frac {\\partial T}{\\partial t}}+u{\\frac {\\partial T}{\\partial x}}+v{\\frac {\\partial T}{\\partial y}}+w{\\frac {\\partial T}{\\partial z}}=Q", "28757c54c5480274b2cdf06ddd955555": "\\sin A={\\frac {a}{c}}{\\text{; }}\\sin B={\\frac {b}{c}}", "287581670c3a4e8adf7c9990901ece45": "{\\mathcal {E}}_{\\leftrightarrow }", "2875c41487d51965c29a1ad097af15af": "\\scriptstyle D^{2}\\times S^{1}", "2875e17f4a5214f15921cb23c6b9e69a": "{\\begin{aligned}{\\text{E}}(e^{-tx})={\\begin{cases}\\displaystyle \\beta ^{s}{\\frac {sb}{t+sb}}{\\ }{_{2}{\\text{F}}_{1}}(s+1,(t/b)+s;(t/b)+s+1;1-\\beta ),&\\beta \\neq 1;\\\\\\displaystyle {\\frac {sb}{t+sb}},&\\beta =1.\\end{cases}}\\end{aligned}}", "28760e9d47256f96866f1180fca8de57": "\\operatorname {E} [X_{ik}^{2}]=\\operatorname {E} [\\operatorname {E} [X_{ik}^{2}|\\vartheta ]]=\\operatorname {E} [s^{2}(\\vartheta )+(m(\\vartheta ))^{2}]=\\sigma ^{2}+v^{2}+\\mu ^{2}", "28761eacdc2d801a4e73edeb52d9eb2b": "vRPM=50/[0.5/SSDIOPS(Iwrite)]+[0.5/SSDIOPS(Iread)]", "287642cba9efe66a80d442a1eb9e6c29": "{\\cfrac {{\\cfrac {pq\\qquad p{\\overline {q}}}{p}}\\,q\\qquad {\\cfrac {qr\\qquad {\\overline {p}}{\\overline {q}}}{{\\overline {p}}r}}\\,q}{r}}\\,p", "2876539253e404436c3f51e5c24b4bc2": "\\sigma =1-{\\frac {1.98}{z(1-\\phi _{2}\\cot \\beta _{2})}}", "2876c7f02f54b79183bd1396a64fdf07": "Q(I=7/2)=8\\left({\\frac {4(1+7P^{2}+7P^{4}+P^{6})}{11+35P^{2}+17P^{4}+P^{6}}}\\right)^{-1}", "2876ce6c6f72aa36a8fc5485a58d270c": "{\\mathcal {F}}(t)", "28772a35288c945522e5b411b3fbcdd9": "\\mathbf {F} _{i}=\\langle F_{i},R_{i},V_{i}\\rangle ", "28772f6544a39018e7e4179e09e46f55": "\\rho =x", "28783a7f113b32800178fc5b84d34f6f": "d(\\mathbf {p} ,\\mathbf {q} )=\\sum _{i=1}^{n}{\\frac {|p_{i}-q_{i}|}{|p_{i}|+|q_{i}|}}", "287891642756800c9447d46cad10015d": "\\varphi _{j}^{n}\\ ", "2878dece2fdc3e5a688901cc90a50de6": "\\mathbf {l} _{a}+(\\mathbf {l} _{b}-\\mathbf {l} _{a})t,\\quad t\\in \\mathbb {R} ", "28790af6471dd9c6d4ff451d660b51d5": "A={\\frac {1}{2}}b\\times {\\frac {\\sqrt {3}}{2}}b={\\frac {\\sqrt {3}}{4}}b^{2}", "28790c5ba309dfe563a9a6e3a4593620": "r={\\frac {1}{s}}", "28793b4addb5b2c1014c5eea8b41434f": "P(X_{N}-X_{0}\\leq -t)\\leq \\exp \\left({-t^{2} \\over 2\\sum _{k=1}^{N}c_{k}^{2}}\\right).", "28793c1c9f01c1428df2753ec7fbd5c1": "c_{0}+c_{1}x+\\cdots +c_{n-1}x^{n-1}", "287945a8879d85a7bc550633107adfec": "x^{n}=1", "2879a1dbfba7a1d8e079430911020af8": "c_{1},\\ldots ,c_{N}", "2879afd2d0e6ca3e4c7183af7961ec9c": "\\sin(A+\\pi /3):\\sin(B+\\pi /3):\\sin(C+\\pi /3).", "2879bf34712c8bf592ec1b9098c14655": "\\{\\mathrm {X} _{1},\\dots ,\\mathrm {X} _{4}\\}", "2879c0e4b4982f1d10802d7bb6558b79": "(x_{\\mathrm {u} },\\ y_{\\mathrm {u} })", "2879cb2b74dc2006d295272465b1cae8": "M/P=L(i,Y)", "287a0a75b8ec132318b77f7b907b2dcb": "Z_{2}\\rightarrow 1", "287ac9701a263b0f7ff6a5400c610d90": "F(z)=\\sum _{n=0}^{+\\infty }c_{n}z^{n},\\quad |z|<1", "287ae932718e3fb63658825213c805e4": "X^{\\omega }", "287b2535fff4e23932fb7ff7b74af6e0": "{\\widetilde {X}}^{(n)}(t)={\\begin{cases}\\int _{a}^{x}{\\widetilde {X}}^{(n-1)}(t)y_{0}(t)^{2}w(t)\\,\\mathrm {d} t&n{\\text{ odd}},\\\\-\\int _{a}^{x}{\\widetilde {X}}^{(n-1)}(t)p(t)^{-1}y_{0}(t)^{-2}\\,\\mathrm {d} t&n{\\text{ even}}\\end{cases}}", "287b81f1878fc90975a0f1c67e1e8225": "S(q)=1+\\rho \\int _{V}\\mathrm {d} \\mathbf {r} \\,\\mathrm {e} ^{-i\\mathbf {q} \\mathbf {r} }g(r)", "287bf1714c331d2110554f7c2f83fb8e": "\\Gamma (1)=1", "287c46af37ba6987b96a433dbfe0df3e": "L_{1}\\cap L_{2}=L_{2}\\cap L_{3}=L_{3}\\cap L_{1}.", "287c47b0bfc7d89fff7e9a36d23c73d9": "d_{H}", "287c95eedb857a122992b2c177c311d7": "{\\sqrt[{n}]{x_{1}\\cdot x_{2}\\cdots x_{n}}}.", "287ccb6e6bb4d2453a18e2fe8956879d": "{\\mathcal {U}}(\\alpha ,{\\tilde {u}})=\\left\\{u:\\ |u-{\\tilde {u}}|\\leq \\alpha {\\tilde {u}}\\right\\},\\qquad \\alpha \\geq 0", "287cdbd2f3922c9d49db9202ed01e741": "W_{\\lambda }=\\int (1-F_{\\lambda }/F_{0})d\\lambda ", "287d10d5e6a413c68afea0683409befe": "\\tan {\\frac {7\\pi }{60}}=\\tan 21^{\\circ }={\\tfrac {1}{4}}\\left[2-(2+{\\sqrt {3}})(3-{\\sqrt {5}})\\right]\\left[2-{\\sqrt {2(5+{\\sqrt {5}})}}\\right]\\,", "287d116e3e48f6a7ae2ae0494217ecd0": "B_{ji}", "287d178fcc2d4bc7b592d31a905d2fee": "\\int (u\\pm v)\\,dx=\\int u\\,dx\\pm \\int v\\,dx", "287d5b49c0427ceb28c77529d91eae84": "S\\subseteq R", "287dad0e675fe3db165ee6ad14037314": "F\\to G", "287e27875bd6e45f1ede90cb8edd4976": "{\\frac {\\pi }{C\\,{\\sqrt {2}}}}\\,{\\frac {1}{\\sqrt {p}}}\\sim {\\frac {\\sqrt {\\pi }}{2}}\\,{\\frac {1}{\\sqrt {p}}}", "287e3e2e88971dcd86f8850aeaeaf986": "\\mathbf {q} =(q_{1},q_{2},...,q_{s}),", "287e41fa5a3ed61b5ce2617cd7c3b335": "T_{\\mathrm {2e} }", "287e79f54ba68a25aff64db6ec948a19": "B={\\frac {NI\\mu }{L}}\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad (3)\\,", "287eb3cd1b2cfea33cbbdf1297783be9": "\\gamma (t)={\\begin{pmatrix}t\\\\t^{2}\\end{pmatrix}}", "287f560acfb8d0684a5c3b1f903986f2": "{\\begin{Bmatrix}p,q,r\\end{Bmatrix}}", "287fd91e561e08718940e001af5cbc58": "x^{2}-680x+96000", "2880ef97acc3f8e59e3374553a4b5231": "[L_{ij},P_{k}]=i\\hbar [\\delta _{ik}P_{j}-\\delta _{jk}P_{i}]", "2880f439a7504c5a5527a5683cc4f87a": "r=3.5,3.51,...,3.9", "288149dbad49e8a6122590d773c0d8fc": "1+\\alpha (\\epsilon )", "288188bba1c9a32874e1bbafabb8bc2d": "g:X\\times \\left[0,1\\right]\\rightarrow \\mathbb {R} ^{K}.", "28818b01362a90b8e6d119e6f19e1c5b": "G_{x},", "28818e35f17db8587af52589136b6964": "R=k/n=1-r/(2^{r}-1)", "2881e21e9442022a9608e748e56d58b3": "E(N_{B})/|B|=\\lambda |B|/|B|=\\lambda ", "28827087c40d708a235bd41e90d17d28": "s_{\\lambda }s_{\\mu /\\nu }=\\sum _{\\lambda +\\omega (T_{\\geq j})\\in P}s_{\\lambda +\\omega (T)}", "28829c28370832e24c294853abfd38fe": "dB_{0}/dt", "2882a4d4e8e34bca0fd367e54412e896": "c_{e}\\in K", "2882aa6c39644f6f83701de027213cdd": "{\\overline {\\Gamma (z)}}=\\Gamma ({\\overline {z}})\\;\\Rightarrow \\;\\Gamma (z)\\Gamma ({\\overline {z}})\\in \\mathbf {R} .", "2882e7d04732ae5350ea1b0474c4c7ed": "1-{\\frac {c}{h}}", "2882efc8086632f12f05fe87fb4c0b13": "H\\sim {p^{2} \\over 2m}+V(r)+{e \\over m}A\\cdot p", "2882fbf1372dd8c0221dc92cc8e1f3b0": "\\gamma (v')\\gamma (v)-v\\delta (v')\\gamma (v)=-v'\\gamma (v')\\delta (v)+\\gamma (v')\\gamma (v)\\,", "288309381a9450b6cedad50d0d712178": "P=\\{(a,b)\\in \\mathbb {Z} ^{2}:1\\leq a\\leq N;1\\leq b\\leq 2N^{2}\\}", "288387f5a34cb9438abd9ad648c5fb7d": "r^{2}\\theta =k\\,", "2883aebe263b70b65aa5d5e3cd14c338": "\\delta (x-x')=\\sum _{n=0}^{\\infty }\\Psi _{n}^{\\dagger }(x)\\Psi _{n}(x').", "2883c2ff6d6fd372e7fc5aef08e64210": "V(p_{0})=0", "2884257bc7637641cffdd6e2f3e7c4ca": "\\displaystyle \\int _{G/U}(Mf)\\cdot F=\\int _{G_{0}/K_{0}}f\\cdot (M^{*}F).", "288432b2f7a0392546e5b7621aae7101": "\\scriptstyle {2-\\log(a)/\\log(b)}", "28845640c351ed877261edaff3e8335e": "z={\\rm {{tanh}^{-1}r={\\frac {1}{2}}\\ln {\\frac {1+r}{1-r}}}}", "2884733f7282d5d05a97c5b04cb27cc3": "{{\\mathit {l}} \\over {\\mathit {l}}^{\\prime }}=1", "28848bda21344cbc03dbbb15eba37313": "{\\textbf {f}}_{p}=1+2X+2X^{3}+2X^{4}+X^{5}+2X^{7}+X^{8}+2X^{9}{\\pmod {3}}", "288493872e7c220bc24f478aece573de": "p_{4}(x)=-{\\tfrac {3}{16}}", "2885368c979dd4105be0cb074f1d8a32": "\\!\\{v_{1},v_{2},v_{3}\\}", "28854d5d90290a869c6b9322af6a62ee": "g\\prec h\\quad \\Longleftrightarrow \\quad eg\\,\\,\\omega ^{r}\\,\\,eh\\,,\\,\\,\\,gf\\,\\,\\omega ^{l}\\,\\,hf\\,", "28858a6b49b224e96d8d6ddc645236da": "S_{\\nu }", "2885d29322b1439be5500ebfd072e43a": "Spin(11,\\mathbb {C} )", "2885df70387032b488fb87b0c9e53980": "{\\sqrt[{y}]{x}}=x^{1/y}", "2885fbb8b21dad521828d020c70b5568": "x^{\\ast }", "2885ff17a17e8ff9429703cbb779f425": "\\,t_{Mn,O}", "28862e88d23e036188940923acde8e44": "T_{m}(0)=0", "2886b17d111307ca168c5354c3f55eb1": "\\|(L+I)u\\|_{(-1)}=\\sup _{\\|v\\|_{(1)}=1}|((L+I)u,v)|=\\sup _{\\|v\\|_{(1)}=1}|(u,v)_{(1)}|=\\|u\\|_{(1)}.", "28872c3df4882de21b1dae06aaa2c4a9": "A_{2}=\\{29.89,\\ 29.93,\\ 29.72,\\ 29.98,\\ 30.02,\\ 29.98\\}", "2887bb21b20a9521a53cde222f5459e4": "~a|\\alpha (t)\\rangle =\\alpha (t)|\\alpha (t)\\rangle ", "2887ffea1da7d78c22a9353e4ee64476": "u_{i}0)\\\\&=\\Pr({\\boldsymbol {\\beta }}\\cdot \\mathbf {s_{n}} -e_{n}>0)\\\\&=\\Pr(-e_{n}>-{\\boldsymbol {\\beta }}\\cdot \\mathbf {s_{n}} )\\\\&=\\Pr(e_{n}\\leq {\\boldsymbol {\\beta }}\\cdot \\mathbf {s_{n}} )\\\\&=F_{e}({\\boldsymbol {\\beta }}\\cdot \\mathbf {s_{n}} )\\end{aligned}}", "28a0aea70335430ff7c1d59952eb06f3": "{\\frac {\\partial \\varepsilon }{\\partial P}}={\\frac {1}{L}}{\\frac {\\partial u}{\\partial P}}={\\frac {1}{EA}}>0", "28a108ddeb980b5839a72626307cd6b1": "\\displaystyle {Hf_{\\pm }=\\pm if_{\\pm }.}", "28a12aaebc464bf7f355e7d9f1d432e1": "{\\frac {A(A-1)}{2}}", "28a2240d45c97ef3c8d305fd39d9611f": "V_{\\alpha }:=\\bigcup _{\\beta <\\alpha }{\\mathcal {P}}(V_{\\beta })", "28a22b14eb433428ede245820690485d": "2^{1-s}\\,\\Gamma (s+1)\\,\\eta (s)=2\\int _{0}^{\\infty }{\\frac {x^{2s+1}}{\\cosh ^{2}(x^{2})}}\\,dx=\\int _{0}^{\\infty }{\\frac {t^{s}}{\\cosh ^{2}(t)}}\\,dt.", "28a244e16b11086aec5b5f5b02cce52e": "y_{t}=a+w_{0}x_{t}+w_{1}x_{t-1}+w_{2}x_{t-2}+...+{\\text{error term}}", "28a25713723148648277ae60cd2ae70d": "\\mathrm {ROOH+ROO{^{\\cdot }}\\ \\longrightarrow {}\\ ROOH+Q{^{\\cdot }}OOH\\ \\longrightarrow {}\\ ROOH+QO+^{\\cdot }OH} ", "28a28f32687c353c47c1a1fa98ecb731": "f(t)=\\sum _{n=1}^{\\infty }{p_{n}\\,'(0) \\over n!}t^{n}.", "28a2cc66a7b1dd190d8739118ec2456f": "\\Delta S\\geq 0", "28a36c36a7b6cf70b3c27f4c6498eedd": "{\\frac {T_{2}}{T_{1}}}={\\frac {1+{\\frac {\\gamma -1}{2}}M_{1}^{2}}{1+{\\frac {\\gamma -1}{2}}M_{2}^{2}}}={\\frac {(1+{\\frac {\\gamma -1}{2}}M_{1}^{2})({\\frac {2\\gamma }{\\gamma -1}}M_{1}^{2}-1)}{\\frac {(\\gamma +1)^{2}M_{1}^{2}}{2(\\gamma -1)}}}", "28a400d8a32f368b2370ed8dbe3e5742": "f(R,t)={\\frac {4}{9}}\\left({\\frac {3}{3+\\rho }}\\right)^{\\frac {7}{3}}\\left({\\frac {1.5}{1.5-\\rho }}\\right)^{\\frac {11}{3}}\\exp \\left(-{\\frac {1.5}{1.5-\\rho }}\\right)\\rho <1.5", "28a416b0591ee084dd6ad42d9ae281db": "k>m>0", "28a4358e9cc4d30d1d7d176d03fc0485": "uv^{i}xy^{i}z", "28a44724238fb2d5e107a9ca13327dac": "p^{2}", "28a48dd058f5b93a6fd51ac867a963b9": "\\displaystyle {Q(a)R(b,a)=R(a,b)Q(a)=2Q(Q(a)b,a),}", "28a4acef15cb47fcb71b6c4b5f6a087b": "\\bigcup {\\mathcal {S}}", "28a511bd345ff77ee207e6267d0f60a4": "\\sigma (E_{1},E_{2},\\ldots )", "28a558672bf11580d700153b4dbdcbe3": "\\sum _{k=-\\infty }^{k=+\\infty }{h_{k}=-1}", "28a58ce99c998ec80377b0084ae39bb6": "|x|_{\\ast }=|x|_{p}^{c}", "28a5909a166165424cf1deb5523ce4ab": "{\\begin{matrix}T(\\phi )=T_{hold},&\\phi \\in [\\phi _{hold},\\phi _{intf}]\\\\T(\\phi )=T_{load}\\exp(-\\mu \\phi ),&\\phi \\in [\\phi _{intf},\\phi _{load}]\\\\\\phi _{intf}=\\log(T_{load}/T_{hold})/\\mu &\\end{matrix}}", "28a5cb229fc2b2d80847cf580a5334a5": "Y_{lm}(\\theta ,\\phi )", "28a5d263ab37c04aff26e10cef131159": "\\pi _{4}=a^{3}I", "28a63a49a1531b2004bcb609132b6fd6": "D(n)={\\bigl (}1+o(1){\\bigr )}\\exp \\left({\\frac {\\pi {\\sqrt {8\\log n}}}{{\\sqrt {3}}\\log \\log n}}\\right).", "28a6c9c7fafbb8cd16e2053a1b71e1ac": "{\\begin{smallmatrix}3&1\\end{smallmatrix}}\\quad {\\begin{smallmatrix}3\\\\1\\end{smallmatrix}}\\quad {\\begin{smallmatrix}2&1&1\\end{smallmatrix}}\\quad {\\begin{smallmatrix}2\\\\1\\\\1\\end{smallmatrix}}\\quad {\\begin{smallmatrix}1&1&1\\\\1\\end{smallmatrix}}\\quad {\\begin{smallmatrix}1&1\\\\1\\\\1\\end{smallmatrix}}", "28a6eed312909b540d6dc9948fbf8043": "\\Delta /\\lambda \\to \\mathbf {J} ^{T}\\mathbf {r} ", "28a7192dfd2f4221c2280bbce03b823a": "\\,_{(X)}\\Gamma _{\\alpha \\beta }^{\\mu }={\\cfrac {C^{\\mu \\gamma }}{2}}\\left({\\frac {\\partial C_{\\alpha \\gamma }}{\\partial X^{\\beta }}}+{\\frac {\\partial C_{\\beta \\gamma }}{\\partial X^{\\alpha }}}-{\\frac {\\partial C_{\\alpha \\beta }}{\\partial X^{\\gamma }}}\\right)", "28a77ce91cd222adee7b246d7e6961f7": "\\omega _{0}={\\sqrt {\\frac {k_{c}}{m+m_{A}}}}", "28a7caed5ad631f4829c68ff128d4d67": "ds^{2}=d{\\mathbf {x}}\\cdot d{\\mathbf {x}}=g(d{\\mathbf {x}},d{\\mathbf {x}})", "28a80487f015493e17f128a4d8315303": "U_{3}={\\begin{bmatrix}\\lambda _{1}&z_{1\\,2}&z_{1\\,3}&\\cdots &z_{1\\,n}\\\\0&\\lambda _{2}&z_{2\\,3}&\\cdots &z_{2\\,n}\\\\0&0&\\lambda _{3}&\\cdots &z_{3\\,n}\\\\\\vdots &\\vdots &\\vdots &&\\vdots \\\\0&0&0&\\cdots &\\lambda _{n}\\end{bmatrix}}", "28a828b6c2a018913febb2f8608c0833": "{\\text{period of }}{\\tfrac {1}{p^{m}}}\\neq {\\text{ period of }}{\\tfrac {1}{p^{m+1}}}", "28a85f41b7e440fc40265f6665a032c8": "[w]^{-1}{\\begin{bmatrix}s\\\\t\\end{bmatrix}}[w]={\\begin{bmatrix}s\\\\-ws+t\\end{bmatrix}}={\\begin{bmatrix}1&0\\\\-w&1\\end{bmatrix}}{\\begin{bmatrix}s\\\\t\\end{bmatrix}}", "28a85f612fe1ceb158a78659e547adff": "(W(tf))_{t\\in \\mathbb {R} }", "28a875195e159317e4607813969ffa7d": "\\left\\{6,{3 \\atop 3}\\right\\}", "28a89c9700a0b80c5ca1d1c909c27f90": "\\left(-r,0,\\pm {\\sqrt {R^{2}-(r+a)^{2}}}\\right);\\quad \\left(0,\\pm r,\\pm {\\sqrt {R^{2}-(r-a)(r+a)}}\\right);\\quad \\left(+r,0,\\pm {\\sqrt {R^{2}-(r-a)^{2}}}\\right).", "28a8bd68171f25888dd7398d05b47e89": "X^{\\sharp }=g^{jk}X_{ij}\\,dx^{i}\\otimes \\partial _{k}.", "28aa42bc74779623be657b1a2a98d6af": "\\theta _{r_{i}}(x){\\Big |}_{x=-j\\infty }^{x=j\\infty }=-\\pi \\,", "28aa893b2b548c4e6878c6254adb21ef": "{\\mathsf {(CH_{2}CH_{2})O+HNO_{3}}}\\rightarrow {\\mathsf {HO\\!\\!-\\!\\!CH_{2}CH_{2}\\!\\!-\\!\\!ONO_{2}\\ {\\xrightarrow[{-H_{2}O}]{+\\ HNO_{3}}}\\ O_{2}NO\\!\\!-\\!\\!CH_{2}CH_{2}\\!\\!-\\!\\!ONO_{2}}}", "28ab1f0d75625a70a7a1ce8dbcc3441e": "\\Gamma \\vdash A,\\Delta ", "28ab282fa60e420606e527dd145df0ef": "p_{mk}", "28ab58046e070814096863faf21afc62": "(\\nabla \\cdot \\mathbf {T} )_{j}=\\nabla _{i}T_{ij}", "28ab691ed97e590459baed5de7ecf76a": "-{\\frac {2}{3}}\\!\\,", "28abc630b7646482bb321849e1f9c34b": "\\scriptstyle \\beta l=\\pi ", "28ac4b7cd5d2866dadb8739de7672d8b": "I(\\lambda )\\equiv \\int \\limits _{I_{x}}f(x)e^{\\lambda S(x)}dx=\\left({\\frac {2\\pi }{\\lambda }}\\right)^{n/2}e^{\\lambda S(x^{0})}\\prod _{j=1}^{n}(-\\mu _{j})^{-1/2}\\left[f(x^{0})+O\\left(\\lambda ^{-1}\\right)\\right],\\qquad \\lambda \\to +\\infty ,", "28ac5dc56ae513b58701fa54760224b3": "(p\\times 1)", "28ac7f7c3a6e9dc7b4d23ae4157419b1": "\\Delta Q(t)", "28acfa532fdc51ed7ea9db4f107298db": "(\\mathbf {\\lambda } x.z)((\\lambda w.www)(\\lambda w.www))", "28ad4f627695b5a7c563a43ae9bdf1bc": "\\Phi _{x}(k)", "28ad7ffa2d3ade85771e5200774d94be": "\\theta _{B}", "28ad90166337a5ef4cc3541db97cd01f": "p,q>0", "28ad9f6878f4499d8a155ffafd85376a": "{\\cfrac {\\partial \\mathbf {b} _{i}}{\\partial q^{j}}}=\\Gamma _{ij}^{k}~\\mathbf {b} _{k}", "28adef701946f6edd7a2823653e18098": "\\int |D_{N}(t)|\\,dt>c\\log N+O(1)", "28ae049b4b3acd7eb817e3b59f718fb6": "\\sum _{k}|V_{ik}|^{2}=\\sum _{i}|V_{ik}|^{2}=1", "28ae1a5231e18296b559d2e3efe4e5d7": "(\\theta _{2}=0)", "28ae691713048c742a88452c090aa904": "\\phi (q)=\\sum _{n\\geq 0}{(-1)^{n}q^{n^{2}}(q;q^{2})_{n} \\over (-q;q)_{2n}}", "28aeb5f9d1709c5f74f2e2589681d7b6": "v\\in \\mathbb {R} ^{J}", "28aed79e47133025e96e21ea75724f76": "A=-27q", "28af2c24e43ad3464a98d876d1fd9440": "e(g^{x}PK,p_{SK}(x))=e(g,g)", "28af5813a0714e012dfabc885f6a107c": "{\\frac {\\operatorname {E} [|V^{S}-V^{B}|]}{V}}", "28af5913d08b318946eff48978e34d74": "\\mu \\alpha .[\\alpha ]u\\;\\triangleright _{c}\\;u", "28af99ca5113abb8ccd5b9fefc2e78c3": "J_{\\mathrm {drift} }(x)=\\mu \\,F(x)\\,\\rho (x)=-\\rho (x)\\mu {\\frac {dU}{dx}}", "28b0036ec2336ddd2250f488b1a7dac3": "{\\begin{aligned}{\\boldsymbol {\\nabla }}\\cdot \\mathbf {v} &={\\cfrac {\\partial v_{i}}{\\partial x_{i}}}\\\\{\\boldsymbol {\\nabla }}\\cdot {\\boldsymbol {S}}&={\\cfrac {\\partial S_{ki}}{\\partial x_{i}}}~\\mathbf {e} _{k}\\end{aligned}}", "28b045ff2150137538d36419232ec822": "\\int _{0}^{\\infty }x^{s-1}f(x)\\,dx=\\Gamma (s)\\phi (-s)\\!", "28b0b02364dda204ea262886e24c1310": "\\operatorname {succ} \\ n\\ f\\ x=f\\ (n\\ f\\ x)", "28b11043a4d6a27cf18b864d246e377c": "R_{ab}={\\frac {2\\Lambda }{n-2}}\\,g_{ab}.", "28b12f3cad9f805ee1b9ffb9c64aa0c4": "C_{3}\\,\\!", "28b13d8e7cab68f952838e9da42d0f6f": "H(\\epsilon )=-\\epsilon \\cdot \\log \\epsilon -(1-\\epsilon )\\cdot \\log(1-\\epsilon )", "28b16e43e42d74406e65d1b0865a92f6": "\\lceil \\chi _{c}(G)\\rceil =\\chi (G)", "28b2142b164673a57523c1ec04ac70f2": "1-2x+3x^{2}-4x^{3}+\\cdots ={\\frac {1}{(1+x)^{2}}}.", "28b271012f2a23f67747425e6761498f": "R'=\\langle R\\rangle (1+1/e_{\\langle R\\rangle })", "28b2b45ca95d89009acc6da6a84fdc5a": "S_{3}={v}(\\alpha ^{3})", "28b2f4552d914e2d199eed87b15d41f8": "C_{k}=\\left\\{{\\begin{array}{lr}\\mathbb {Z} ^{2}&0\\leq k\\leq n\\\\0&{\\text{otherwise}}\\end{array}}\\right.", "28b3644f18f7b4b8f4a585a5db6132e8": "A\\times P", "28b3822cba12d64ec1e8b28f05769110": "X_{1,2}^{*}=X_{3}/(1-X_{1}-X_{2}),X_{4}/(1-X_{1}-X_{2}),\\ldots ,X_{k}/(1-X_{1}-X_{2}).", "28b3e72ae6a190dc1972734e9012d660": "\\Delta f=0\\,", "28b42858df9b90cc3c3fe31d01a2f1e5": "a_{i}\\equiv a_{j}{\\pmod {\\gcd(n_{i},n_{j})}}\\qquad {\\text{for all }}i{\\text{ and }}j", "28b42f8d23471cfee2622d8dcaa21457": "\\mathbf {P} =\\bigoplus _{i}{D_{i}}=\\mathbf {D} _{0}\\;\\oplus \\;\\mathbf {D} _{1}\\;\\oplus \\;\\mathbf {D} _{2}\\;\\oplus \\;...\\;\\oplus \\;\\mathbf {D} _{n-1}", "28b4392ea6c6cfb147dcf03c5287aac3": "\\gamma _{3}\\approx 0.9952717", "28b4945770d6eb2452ccb987631e5dfa": "\\int _{0}^{\\pi }f(\\cos \\theta )\\sin(\\theta )\\,d\\theta =a_{0}+\\sum _{k=1}^{\\infty }{\\frac {2a_{2k}}{1-(2k)^{2}}}.", "28b4fb80b2cb7ac657433a4533a8f791": "T_{k,n}=T_{k,n-1}+T_{k-1,k-n}\\quad {\\text{for }}k\\geq n>0.", "28b53c55e4b4a8aa0e3185dff146c629": "B={\\frac {1}{\\omega _{\\mathrm {d} }}}(\\zeta \\omega _{0}x(0)+{\\dot {x}}(0)).\\,", "28b5430d9bc55ec25af7a9a7adbc9422": "\\scriptstyle \\Rightarrow ", "28b59d0bacbcc089cf5c618301ca1a06": "{S}=\\left[{\\begin{matrix}-\\beta _{1}&\\beta _{1}&0&0&0&0\\\\0&-\\beta _{1}&\\beta _{1}&0&0&0\\\\0&0&-\\beta _{1}&0&0&0\\\\0&0&0&-\\beta _{2}&\\beta _{2}&0\\\\0&0&0&0&-\\beta _{2}&\\beta _{2}\\\\0&0&0&0&0&-\\beta _{2}\\\\\\end{matrix}}\\right].", "28b5a997f757c9cdd323087ae2d4f789": "Ar/3", "28b61b964a071d9810f7c155153bb395": "\\left[{\\frac {\\mathbf {E} \\left[e^{tX_{i}}\\right]}{e^{tq}}}\\right]^{m}=\\left[{\\frac {pe^{t}+(1-p)}{e^{tq}}}\\right]^{m}=[pe^{(1-q)t}+(1-p)e^{-qt}]^{m}.", "28b620c932cd787f1c778280868054d3": "(T_{x}f)(y):=f(y-x),", "28b62e9202264ceed8bdf728986de1fb": "k_{cat}\\approx k_{2}", "28b68db57fef06260fe0897e8b97a8b4": "{\\begin{aligned}&\\lim _{x\\to 0}{\\frac {\\sin x}{x}}=1,\\\\[10pt]&\\lim _{x\\to 0}{\\frac {1-\\cos x}{x}}=0.\\end{aligned}}", "28b690ae93159f62319293fad317a164": "y=(t+1)\\inf \\mathrm {supp} X-\\mu (X),", "28b6a1dab720d2d83ee6ddda3a3d27c6": "\\ (u,v)\\in E", "28b6b6d603667291e4e4081bb3bd689e": "M_{CB}^{f}={\\frac {qL^{2}}{12}}={\\frac {1\\times 10^{2}}{12}}=+8.333\\ kN\\cdot m", "28b6d5807e83fc7affde05e0b2cb1c06": "s_{p}={\\sqrt {\\frac {p\\,(1-p)}{n}}}", "28b7539dc5966c0342a7b1a63244f5a1": "R[\\varphi ]=\\iiint _{D}r(X)\\varphi (X)^{2}\\,dx\\,dy\\,dz.\\,", "28b7707502b10fc2a6df36cf73d2f0a4": "\\textstyle \\mathbf {r} ", "28b787f1bbf117ee55e45b572084b840": "U={\\frac {1}{4\\pi \\varepsilon }}\\sum _{l=0}^{\\infty }\\sum _{m=-l}^{l}Q_{1lm}\\int d\\mathbf {r} \\ \\rho _{2}(\\mathbf {r} )\\left({\\frac {1}{r^{l+1}}}\\right){\\sqrt {\\frac {4\\pi }{2l+1}}}Y_{lm}(\\theta ,\\phi )", "28b7b29fd6b4fd86838fb7150f2509e9": "\\omega ^{2}=|k|\\left({\\frac {\\rho -\\rho '}{\\rho +\\rho '}}\\,g+{\\frac {\\sigma }{\\rho +\\rho '}}\\,k^{2}\\right),", "28b7c2f8fe999b0996c0bfe0201ad649": "1+\\sum _{k=1}^{\\infty }\\left(\\prod _{r=0}^{k-1}{\\frac {\\alpha +r}{\\alpha +\\beta +r}}\\right){\\frac {t^{k}}{k!}}", "28b80a684dc23eb94544adb278ceae87": "\\phi (\\alpha ,\\beta )=\\alpha -{\\frac {(2\\alpha +\\beta -3)^{2}}{3(\\alpha +\\beta -2)}}", "28b879780ab0472f166462988d485a6e": "{\\frac {P(t)}{P_{0}}}={\\frac {1}{2}}={\\frac {1-e^{-r(T-t)}}{1-e^{-rT}}}", "28b87b9ac8e02a5ae7946b03463eeb0f": "e^{it}=\\cos t+i\\sin t,", "28b895637058979e494eeb1b21607c5a": "\\mathrm {succ} :\\mathbb {N} \\to \\mathbb {N} ", "28b8a26d7f0397d613226b9b5f91bf26": "X_{(m+1)}", "28b8c557df53bc4583a5fd427e843509": "E=E^{0}+{\\frac {RT}{z_{i}F}}\\ln \\left[a_{i}+\\sum _{j}\\left(k_{ij}a_{j}^{z_{i}/z_{j}}\\right)\\right]", "28b90931b04d7d48fa55a43c85b51ffb": "m{\\sqrt {\\langle v^{2}\\rangle }}\\,\\!", "28b90e5875b7e6ead67900601a63ddb3": "m{\\overline {\\Psi ^{\\dagger }}}\\rightarrow m{\\overline {(\\Psi R_{0})^{\\dagger }}}=m{\\overline {\\Psi ^{\\dagger }}}R_{0},", "28b93b4e37f39ca8eb5420f88cb09029": "\\ y=(d/2){\\sqrt {(\\xi ^{2}+1)(1-\\eta ^{2})}}\\cos \\phi ,", "28b9a066c594489b73e63c644652a468": "=Q\\left({\\frac {\\nu }{2}},{\\frac {\\tau ^{2}\\nu }{2x}}\\right)", "28b9c3643584bbd557ee327db6ca45f6": "M_{\\mathrm {left} }", "28ba4b1c47dc94db458fddb4a406ed19": "\\Gamma _{\\mu \\nu \\alpha }=\\{_{\\mu \\nu \\alpha }\\}+S_{\\mu \\nu \\alpha }+{\\frac {1}{2}}C_{\\mu \\nu \\alpha }", "28baf1b07c1c21b18f00d07764929aca": "x=L\\cot(\\theta )", "28baf2bf0eda6cd1e2df7d17b7eb457b": "(\\omega ^{2}+m^{2})^{-1}", "28bb59a115934bed928b08a7e655a150": "X\\times _{S}T\\to T", "28bb8c90a2ff86f10508973b70b84f50": "{\\frac {dN}{d\\theta }}={\\frac {1}{2\\pi \\sigma ^{2}}}e^{-{\\frac {\\theta ^{2}}{2\\theta _{0}^{2}}}}", "28bbb18ab6a0ab92a1db71071fac6bdd": "M_{PAW}={\\frac {(P_{high}*T_{high})\\,+(P_{low}*T_{low})}{T_{high}+T_{low}}}", "28bbf08592798bf0f19d3d0ae99c9144": "e_{x}=E[K(x)]=\\sum _{k=0}^{\\infty }k\\,Pr(K(x)=k)=\\sum _{k=0}^{\\infty }k\\,\\,_{k}p_{x}\\,\\,q_{x+k}.", "28bc61b908d05c94c2429f4c1a908ef0": "d=2mn(m^{2}-n^{2}),\\,", "28bd34e95174fe595cb535b5d4484c5f": "\\Psi (x,y)=x\\cdot \\rho (u)+O\\left({\\frac {x}{\\log y}}\\right)", "28bdc367b47d1e6c434f115aca7eb86e": "\\lim _{n\\to \\infty }\\int _{S}f_{n}\\,d\\mu =\\int _{S}f\\,d\\mu ", "28bdcf12dce597687c5b9211402b3b27": "\\nabla \\mathbf {v} ", "28be0b2ff0ab1254fe24437ea657174e": "P0", "28c4b79c00453a1edeac1c47cbfb66b5": "\\mathrm {v} =\\alpha D_{\\alpha }p^{\\alpha -1}", "28c563c8d2a6c96d45e652210435c7f4": "u_{n}(x):=(x-1)^{-2n}P_{n}(x)", "28c56e33539bec9a846149fe3d4090ed": "v_{x}>0\\ ", "28c5bf7f78b9219ef528407815ad9b1b": "\\leq r", "28c5eac946471f68eefb01f7a53b1844": "x_{3}", "28c602df45800225c950a013ddc09202": "L_{n+1}(\\pi _{1}(X))", "28c619cc0d686353b6fd04ba4c1d6408": "g=q\\circ h", "28c633da6754fcbb4ae86fe4aedddf3a": "\\xi =\\prod _{j=2}^{\\infty }\\left({1-{\\frac {1}{d_{j}}}}\\right)\\ .", "28c655e9cdc8468134db0d6198f229d3": "\\beta _{F}=\\beta _{F0}\\left(1+{\\frac {V_{\\text{CB}}}{V_{\\text{A}}}}\\right)", "28c68b70ab9de8b616fb4cecb80217d0": "\\sum _{n=1}^{\\infty }{\\frac {\\varphi (n)q^{n}}{1-q^{n}}}={\\frac {q}{(1-q)^{2}}}", "28c6e908f5f6724899dd4ab9850b2ac7": "\\bigsqcup ", "28c71c0152449f92252035c3be84621d": "n=(t/d_{m})^{2}am^{(b-2)}", "28c7595dd3198b1ca9e0aa9f61eab6bc": "{e}^{-{\\frac {1}{e}}}\\color {white}..........\\color {black}", "28c7c1aff5b5edc11e28efeb0787fd39": "\\displaystyle {A_{\\varepsilon }=(R+\\varepsilon I)^{-1}SK(R+\\varepsilon I)^{-1}.}", "28c7c64780580ce4287069efca6fa2e4": "F_{\\alpha \\beta }=\\eta ^{\\gamma \\mu }F_{\\alpha \\beta \\gamma \\mu }", "28c7cd83ad9cee2297835637032cc83a": "\\left({\\frac {\\mathrm {d} S_{\\phi }}{\\mathrm {d} \\phi }}\\right)^{2}+2mU_{\\phi }(\\phi )=\\Gamma _{\\phi }", "28c7ead189650c5e6eff0fb302d4c178": "|{\\vec {n}}|", "28c8505bd610ce17dabf08f7016e970a": "2e+k+1", "28c85e81e97bdc597fa00c11c56b41b4": "G=G'+iG''", "28c89a0cd693a550a97357ba14a00ac9": "\\Phi _{V}={\\frac {\\pi r^{4}}{8\\eta }}{\\frac {\\Delta p^{\\star }}{\\ell }}", "28c89aa2ca416b00536c8118f1125852": "{\\begin{aligned}f_{m}^{(k)}(n)&={\\frac {m(m-1)\\cdots (m-k+1)}{n(n+1)\\cdots (n+k-1)}}f_{m}(n)\\\\&=m(m-1)\\cdots (m-k+1){\\frac {f_{m}(n)}{f_{k}(n)}},\\end{aligned}}", "28c89ca9fb324c3ff8eef65eb8444439": "\\operatorname {Log} z:={\\text{ln }}r+i\\theta =\\ln |z|+i\\operatorname {Arg} z=\\operatorname {ln} {\\sqrt {x^{2}+y^{2}}}+i\\operatorname {atan2} (y,x).", "28c8c22f82b5ace4474f888372f7366b": "\\Pr \\left\\{E[f(X_{1},X_{2},\\dots ,X_{n})]-f(X_{1},X_{2},\\dots ,X_{n})\\geq \\varepsilon \\right\\}\\leq \\exp \\left(-{\\frac {2\\varepsilon ^{2}}{\\sum _{i=1}^{n}c_{i}^{2}}}\\right)", "28c8f13981b150bcab7900a156d0a1bb": "\\lambda _{B}=\\hbar /(m_{e}k_{B}T_{e})^{1/2}", "28c8fdb55cbbeb686581d8f9b4e83c65": "x\\vee y", "28c93829ef1a3dc9e402cf5430fe5bf1": "Q_{m_{B}}^{\\ell _{B}}", "28c9943ff90d9822baf1b68d23009cd1": "(b_{n}<0)", "28c9a9c918b323e3d97bff7726d09d78": "k(\\mathbf {x} _{i},\\mathbf {x} _{t})", "28c9cb88abe6073de58c67c1de224203": "15^{2}+21^{2}=225+441=666.", "28c9fd978710a3aa6bb7f70eded23ab1": "\\langle s+{\\sqrt {\\lambda _{i}}}p_{i}|s+{\\sqrt {\\lambda _{j}}}p_{j}\\rangle =\\langle s|s\\rangle +{\\sqrt {\\lambda _{i}}}\\langle s|p_{i}\\rangle +{\\sqrt {\\lambda _{j}}}\\langle s|p_{j}\\rangle +{\\sqrt {\\lambda _{i}\\lambda _{j}}}\\langle p_{i}|p_{j}\\rangle =1+0+0+{\\sqrt {\\lambda _{i}\\lambda _{j}}}\\cos {\\omega _{ij}}=1+\\cos {\\omega _{ij}}", "28ca45c80daf202a121cea4ffc764ccf": "\\cos(kt)={\\frac {\\exp(ikt)+\\exp(-ikt)}{2}}={\\frac {z^{k}+z^{-k}}{2}}", "28ca4990e68c146a6b71da9a78abebc4": "\\Theta ((|E|+|V|)\\log |V|)", "28ca90f36d10138610a089bec6c49603": "{\\frac {1}{n!}}\\sum _{k=0}^{\\lfloor x\\rfloor }(-1)^{k}{\\binom {n}{k}}(x-k)^{n}", "28cab36c24bc9a734e2d113daf524385": "a\\left(\\sum _{j=1}^{n}u_{j}e_{j},e_{i}\\right)=\\sum _{j=1}^{n}u_{j}a(e_{j},e_{i})=f(e_{i})\\quad i=1,\\ldots ,n.", "28cb2324757fca72a867d4e12ff9fdf3": "\\tan \\psi ={\\frac {m+1}{m}}\\tan \\theta \\,.", "28cb43a74d53b47717fa0e0cd5302fd1": "{\\mathcal {F}}(\\mathbf {x} )\\approx {\\mathcal {S}}\\boxtimes _{n=1}^{N}\\mathbf {w} _{n}(x_{n}),", "28cb4b0766fb8b454a52b03f54477d9b": "e=4", "28cb57b0978a8dbe7b0daf72ae4400bb": "A_{1}\\times A_{2}\\times \\cdots \\times A_{n}", "28cb67cb9650d61b3ab336e684e6db48": "\\sin(\\delta _{1}+\\delta _{2})\\sin(\\delta _{2}+\\delta _{3})=\\sin(\\delta _{1})\\sin(\\delta _{3})+\\sin(\\delta _{1}+\\delta _{2}+\\delta _{3})\\sin(\\delta _{2}).\\,", "28cba20a0c17aa20794963b3c834b062": "G=G_{0}^{}+G_{0}\\Sigma G.", "28cba6de8441f20c37eb88b1a4da9f9b": "{\\frac {n_{x}+n_{y}-p-1}{(n_{x}+n_{y}-2)p}}t^{2}\\sim F(p,n_{x}+n_{y}-1-p;\\delta ),", "28cbcb3bb68f4932682359edaa98e7d0": "{\\vec {P}}_{1,4}", "28cbcc4aab9c2a147e76cbc0c8ddb500": "f(0)=(1,0,\\dots ,0).", "28cbec68a6b311f39d8fa027d71ea7f7": "\\Pi (k_{i})={\\frac {\\eta _{i}k_{i}}{\\displaystyle \\sum _{j}\\eta _{j}k_{j}}},", "28cc3797f616b7542a97179c74438356": "u_{1}={\\frac {(ax_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})x_{5}-2x_{1}(bx_{1}x_{5}+x_{2}x_{6}+x_{3}x_{7}+x_{4}x_{8})}{c}}", "28ccb80471f2fe36a2ba1cb2fc78b45f": "\\scriptstyle R_{ws}[-i]\\,=\\,R_{sw}[i]", "28cd1257ee5de94d6fbc0f0ea89829dc": "\\gamma _{0}=\\gamma ", "28cd90cc994d482cbbab9f256d7bac4e": "f(t)g(t)", "28cdb69bf439f116e22c57bb36c0f32f": "{\\begin{aligned}&\\mathbf {\\hat {A}} \\psi =\\mathbf {\\hat {A}} \\psi (\\mathbf {r} )=\\mathbf {\\hat {A}} \\left\\langle \\mathbf {r} \\mid \\psi \\right\\rangle =\\left\\langle \\mathbf {r} \\mid \\mathbf {\\hat {A}} \\mid \\psi \\right\\rangle \\\\&\\left(\\sum _{j=1}^{n}\\mathbf {e} _{j}{\\hat {A}}_{j}\\right)\\psi =\\left(\\sum _{j=1}^{n}\\mathbf {e} _{j}{\\hat {A}}_{j}\\right)\\psi (\\mathbf {r} )=\\left(\\sum _{j=1}^{n}\\mathbf {e} _{j}{\\hat {A}}_{j}\\right)\\left\\langle \\mathbf {r} \\mid \\psi \\right\\rangle =\\left\\langle \\mathbf {r} \\mid \\sum _{j=1}^{n}\\mathbf {e} _{j}{\\hat {A}}_{j}\\mid \\psi \\right\\rangle \\\\\\end{aligned}}\\,\\!", "28cdecab88496bf35ee30677dc26a20f": "\\lnot \\alpha ", "28ce006791acc547ef12975e97d03440": "H_{A}(s)={k_{A}\\cdot s^{4} \\over (s+129.4)^{2}\\quad (s+676.7)\\quad (s+4636)\\quad (s+76655)^{2}}", "28ce3f4dbe770799c2295b0366ed26a5": "i_{\\mathrm {i} }={\\frac {v_{\\mathrm {i} }}{Z_{0}}}=Iu(\\kappa t-x)", "28ce75dc16591d4b552dc37c94ecea89": "{\\frac {100-95}{95}}=5.26\\%", "28cedf53540543e78462fdc49019fb82": "\\operatorname {Var} (X|Y)", "28cf0975824c61a963ba357235ea6ddc": "Q_{u,v}(\\alpha ,\\beta )x^{u}y^{v}", "28cf3ccd501f08c436bcf478d18b60dd": "\\mathrm {H_{2}O+CO_{2}\\longrightarrow H^{+}+HCO_{3}^{-}\\longrightarrow H_{2}CO_{3}} ", "28d0002cc23364df1a800fbbfd02e58e": "[P,Q]=PQ-QP=-i\\hbar ,\\,", "28d02db8f9cf7c0537fac9666ff53223": "Q\\equiv \\max _{p\\in [0,1]}\\;{\\Big \\{}\\;H_{2}(\\eta \\,p)-H_{2}((1-\\eta )\\,p)\\;{\\Big \\}}\\;", "28d03866159d72ada6e5afcc82194a1a": "{\\boldsymbol {\\sigma }}={\\cfrac {1}{J}}\\left[-p~{\\boldsymbol {\\mathit {1}}}+2\\left(C_{1}+{\\bar {I}}_{1}~C_{2}\\right){\\bar {\\boldsymbol {B}}}-2~C_{2}~{\\bar {\\boldsymbol {B}}}\\cdot {\\bar {\\boldsymbol {B}}}-{\\cfrac {2}{3}}\\left(C_{1}\\,{\\bar {I}}_{1}+2C_{2}\\,{\\bar {I}}_{2}\\right){\\boldsymbol {\\mathit {1}}}\\right]\\quad {\\text{where}}\\quad {\\bar {\\boldsymbol {B}}}=J^{-2/3}\\,{\\boldsymbol {B}}\\,.", "28d0444cbac239d581aa0b475ab95c80": "\\tan(z)=\\sum _{k=0}^{\\infty }\\left[\\left({\\frac {-1}{z-(k+{\\frac {1}{2}})\\pi }}-{\\frac {1}{(k+{\\frac {1}{2}})\\pi }}\\right)+\\left({\\frac {-1}{z+(k+{\\frac {1}{2}})\\pi }}+{\\frac {1}{(k+{\\frac {1}{2}})\\pi }}\\right)\\right]", "28d07eb76cebafadbcddbece0388494d": "\\mathbf {x} \\in \\mathbf {R} ^{m},\\mathbf {y} \\in \\mathbf {R} ^{n}\\ :\\qquad \\left\\langle \\mathbf {x} |A|\\mathbf {y} \\right\\rangle ={}^{t}\\mathbf {x} A\\mathbf {y} ", "28d0b784ce50f818da13317af4642156": "\\pi r^{2}h\\;", "28d18665fabd87e3247a75de13a106a7": "\\mu =m_{i}/m_{p}", "28d1cbd88f29f0b63c26c224e909e8bf": "{\\begin{aligned}U&{}=(A-{\\mathbf {i}}I)(A+{\\mathbf {i}}I)^{-1}\\\\A&{}={\\mathbf {i}}(I+U)(I-U)^{-1}\\end{aligned}}", "28d1f06fd4b27dd7c51107aad497ab86": "\\mathrm {Hom} (\\mathrm {Gr} ,\\mathrm {Gr} )", "28d243b04dd6708200122c614a114d7b": "{\\frac {\\partial V}{\\partial S}}", "28d2647a12e43c3e8cb5d7dec8e77356": "\\scriptstyle {\\sqrt {1}}0", "28d2844e9467acfcc62ccf5e37a7add4": "f\\approx f^{0}+{\\frac {\\partial f}{\\partial a}}a+{\\frac {\\partial f}{\\partial b}}b", "28d28669a693b48e36824e48e032d3b5": "{\\mathfrak {a}}={\\mathfrak {p}}_{1}{\\mathfrak {p}}_{2}\\dots {\\mathfrak {p}}_{g}.", "28d292a93ad6b61b67b324228df0628c": "\\nabla \\cdot \\mathbf {J} +{\\partial \\rho \\over \\partial t}=0.\\,", "28d299cd29fdd90d84afd49e80a40289": "T(f)=Pm(f)P,", "28d2e245d62b9aaeb77521a2d6a42c37": "0={\\frac {dS}{db'}}({\\hat {\\beta }})={\\frac {d}{db'}}{\\bigg (}y'y-b'X'y-y'Xb+b'X'Xb{\\bigg )}{\\bigg |}_{b={\\hat {\\beta }}}=-2X'y+2X'X{\\hat {\\beta }}", "28d386998e75e7820dc75a998c825c0c": "B_{\\mu }B^{\\mu }\\mp b^{2}=0.", "28d3984d5d86f6c759ce3472ee14e223": "\\ d[\\mathbf {x} (1),\\mathbf {x} (3)]=|u(2)-u(4)|=5>r", "28d3f513fbc1ffc400e1ac62ce40e7c8": "A\\in {\\mathcal {F}}", "28d3fa169349d84254463df586e69fa4": "X^{\\nu }=(X^{0},X^{1},X^{2},X^{3})=(ct,x,y,z).", "28d3ff7a00a2a0093577c6231e26050a": "F=true", "28d41a5033c281880a8435443133f857": "P(d\\sigma |\\eta )=\\otimes _{k\\in G}p_{k}(d\\sigma _{k}|\\eta )", "28d43ec89932150e1c5291dae277c928": "v(1),v(2),\\ldots ,v(d)", "28d476a23968b2ab20414ea6c44bfe55": "X'{\\widehat {\\otimes }}_{\\varepsilon }X", "28d5013b0a47deea517d5736b5880d31": "\\det S''_{ww}({\\boldsymbol {\\varphi }}(0))=\\left[\\det {\\boldsymbol {\\varphi }}'_{w}(0)\\right]^{2}\\det S''_{zz}(0)\\Longrightarrow \\det {\\boldsymbol {\\varphi }}'_{w}(0)=\\pm 1.", "28d5256eb61c0054aac4d6f733e0b6f5": "|L|", "28d526a033db0b7760b660efbe0d06b8": "\\displaystyle {\\mathfrak {f}}(L/K)=\\prod _{\\mathfrak {p}}{\\mathfrak {p}}^{{\\mathfrak {f}}(L_{\\mathfrak {p}}/K_{\\mathfrak {p}})}.", "28d537631badbd24f1061bb566250c5e": "\\beta =180^{\\circ }-(\\lambda +\\iota )", "28d549a92404c71e6b0150f24cfd488d": "{\\rm {4\\;Cyt\\,c_{red}+O_{2}+8\\;H_{matrix}^{+}\\rightarrow 4\\;Cyt\\,c_{ox}+2\\;H_{2}O+4\\;H_{intermembrane}^{+}\\!}}", "28d5bb1760b3b528ad3799b57b587ac5": "\\mathbb {Z} _{n}\\!\\,", "28d5c441661cab6e884957e65ce8f69e": "f(z)f(1-z)={\\pi \\over \\sin(\\pi z)}\\,\\!\\,\\,\\,", "28d6486f8767c909c034381ea8c3ab48": "h\\ :=\\ f(h,\\ m_{i})", "28d667fb566e56b2ac3989b622eb52fc": "|{\\bar {h}}_{n}|^{2}", "28d71df4347bc4f693941d665e372687": "{\\frac {\\cos ^{2}\\left({\\frac {\\pi }{q}}\\right)}{\\sin ^{2}\\left({\\frac {\\pi }{p}}\\right)}}+{\\frac {\\cos ^{2}\\left({\\frac {\\pi }{r}}\\right)}{\\sin ^{2}\\left({\\frac {\\pi }{s}}\\right)}}", "28d72a8a977bc6a683245b46a02e9a05": "\\tau =\\ r_{m}c_{m}\\,", "28d72fc1c22e76644ac880b80ef85c83": "G_{k+1}=G_{k}", "28d75f06d7a7ff000953a2e67ed7e280": "i_{R}={\\frac {v_{E}}{R_{2}}}+(\\beta +1)i_{B}\\ .", "28d7772add19980f3717650b6a1717c5": "d\\,\\ ", "28d7c6372fcd5994049e0df3a9f0a490": "{\\begin{aligned}b_{1}x_{1}+c_{1}x_{2}&=d_{1};&i&=1\\\\a_{i}x_{i-1}+b_{i}x_{i}+c_{i}x_{i+1}&=d_{i};&i&=2,\\ldots ,n-1\\\\a_{n}x_{n-1}+b_{n}x_{n}&=d_{n};&i&=n.\\end{aligned}}", "28d7e6c627af51ae5e3dc7d13e42d38b": "\\left\\{{\\overline {D}}_{\\dot {\\alpha }},{\\overline {D}}_{\\dot {\\beta }}\\right\\}=F_{{\\dot {\\alpha }}{\\dot {\\beta }}}=0", "28d7fa312ba6b57eae3c374dd4a47f4e": "\\sim 10^{89}\\,\\!", "28d8c698f4c891c8e9f498c08c99489a": "C(1,1)=1\\,", "28d92b6f3f7c50cdc6095a33c4f29514": "f(t)={\\frac {\\Delta x}{\\sqrt {4\\pi Dt^{3}}}}\\sim t^{-3/2},", "28d92c7e2cd156877779149fed889a84": "\\nabla _{y}\\cdot \\left(A({\\vec {y}})\\nabla w_{j}\\right)=-\\nabla _{y}\\cdot \\left(A({\\vec {y}}){\\vec {e}}_{j}\\right).", "28d9d4b3fde95c6b94ceb397e5398326": "\\psi \\nabla ^{2}\\phi -\\phi \\nabla ^{2}\\psi =\\nabla \\cdot \\left(\\psi \\nabla \\phi -\\phi \\nabla \\psi \\right)", "28d9e034e50e7f293df13ae8273dc668": "\\mathbf {f} (\\mathbf {v} )=\\mathbf {f} _{1}(\\mathbf {v} )\\times \\mathbf {f} _{2}(\\mathbf {v} )", "28da01697074a1dcb4c44691d7d46a3f": "x_{1}x_{2}x_{5}=213", "28da0b2a60d1d778b7254ee9b0c59c36": "(a,b)\\,", "28da5d2b6211555eb46aca2138f79d68": "y'", "28dad983f9e23004ea831ed535ec9a8f": "E^{*}=E", "28db353127e9ec3071296e2a93243a49": "I_{1}(\\mathbf {r} )={\\sum \\limits _{i}\\gamma _{i}\\cdot \\exp(-\\left|\\mathbf {r} -\\mathbf {r} _{i}\\right|/\\varepsilon )}", "28db55b59688d9fa1e76b39c4685b856": "{\\overline {x_{i}-a_{i}}}", "28dc04dbd6e0d5356c224522dfb3dcba": "\\mu _{t}=\\rho {\\tilde {\\nu }}f_{v1}", "28dc0d8885edab31b148abbc16107ed3": "1_{\\{\\tau _{X}\\leq T\\}}", "28dc2326da0088bef4fc8ba7c3b54bec": "J(A,B)={\\frac {|A\\cap B|}{|A\\cup B|}}={\\frac {TP}{TP+FP+FN}}", "28dc628c16e626812035292a35656cb3": "\\alpha =0.54,\\;\\beta =1-\\alpha =0.46,", "28dc8bc4541ce9bff87f7237a83b94fb": "\\Gamma '\\vdash C:D", "28dc930e7c69157c9b0876863eee407c": "q_{1}", "28dcc18b5802f5df1ac6f8fc6a26470e": "\\mathbf {E} (\\mathbf {r} ,t)={\\frac {1}{4\\pi \\epsilon _{0}}}\\left({\\frac {q(\\mathbf {n} -{\\boldsymbol {\\beta }})}{\\gamma ^{2}(1-\\mathbf {n} \\cdot {\\boldsymbol {\\beta }})^{3}|\\mathbf {r} -\\mathbf {r} _{s}|^{2}}}+{\\frac {q\\mathbf {n} \\times {\\big (}(\\mathbf {n} -{\\boldsymbol {\\beta }})\\times {\\dot {\\boldsymbol {\\beta }}}{\\big )}}{c(1-\\mathbf {n} \\cdot {\\boldsymbol {\\beta }})^{3}|\\mathbf {r} -\\mathbf {r} _{s}|}}\\right)_{t_{r}}", "28dd800d51e26dea97c620a5e4859c5c": "S_{k}={\\frac {\\displaystyle \\sum _{1\\leq i_{1}<\\cdots 0\\\\M(n)&=n-F(M(n-1)),\\quad n>0.\\end{aligned}}", "28e00059d4c25537a714d7be81b6bf1f": "\\mathrm {0.41{\\overline {6}}} ", "28e076a6ab479db26b01109c13a29df9": "\\mathbf {\\mathit {C}} ", "28e087909eef108bd6c9cb7060297370": "\\mathrm {T} (v_{i})=w_{i}", "28e0aca9c25422ad78047c497f82627e": "P(s)={\\frac {1}{Z}}e^{-E(s)/kT},", "28e0ff7958c2f125d9998fe9eae0cc83": "V_{TN}=V_{TO}+\\gamma ({\\sqrt {|{V_{SB}+2\\phi _{F}|}}}-{\\sqrt {|2\\phi _{F}|}})", "28e12dd467effc871ee03f0e6fcf564a": "\\!{\\mathcal {A}}\\models _{X}^{-}\\phi \\vee \\psi ", "28e1558cf428043d0daaade25923438b": "\\|Ww\\|_{(1)}\\leq C(\\|VWw\\|_{(0)}+\\|W^{2}w\\|_{(0)})\\leq C\\|(\\Delta -V^{2}-A)w\\|_{(0)}+C\\|(WV+B)w\\|_{(0)}\\leq C_{1}\\|Lw\\|_{(0)}+C_{1}\\|w\\|_{(1)}.", "28e162a6418dd5f7edc5895d55ba436b": "z,w", "28e166a43cf0012124de5a19bc0a99c9": "\\textstyle 3.\\ Check\\ which\\ a(\\theta )\\ \\epsilon span\\{{\\mathbf {E}}_{s}\\}\\ or\\ {\\mathbf {P}}_{A}a(\\theta )\\ or\\ P_{\\mathbf {A}}^{\\perp }a(\\theta ),\\ where\\ {\\mathbf {P}}_{A}\\ is\\ a\\ projection\\ matrix.", "28e185494a796b8af017271e5893494b": "k\\in {\\mathbb {N}}", "28e1bb8f85d27e2fcc0af0ef9ad1a74d": "d(x,y)=d(y,x)", "28e1f27d1f838dbe5700d79acb5f6aad": "i^{2}=-1.", "28e20014de881c3696bec844c941c7a2": "\\textstyle G_{1}", "28e2121ee734ce066f7799b15d6be9ab": "\\chi _{V}", "28e218199ff1d5b2bfc52b11029c6898": "{\\underset {\\alpha >0}{\\lim _{\\alpha \\to 0}}}\\|I^{\\alpha }f-f\\|_{p}=0", "28e281b189e72449fcbc1a707f63a878": "H^{*}(M;o(M))", "28e2b9aee164a85fc8add169fb38fb28": "k=\\left({\\frac {E-E_{0}}{c_{k}}}\\right)^{1/p}\\ ,", "28e31dcb3e39ce0c0b29d60a2156d88f": "-8y^{3}-42y^{2}+72y+378", "28e4058b15e9acc845ad842139379729": "\\mathbb {P} ^{1}\\times \\mathbb {P} ^{1}", "28e41309882ddb824db739780d2ecba1": "\\mu _{\\mathrm {N} }={{e\\hbar } \\over {2m_{\\mathrm {p} }}}", "28e44d5c96883b6bc6e9814567e87def": "\\epsilon /D", "28e512a882025a421ca855d03924af28": "-J,-(J-1)\\cdots 0\\cdots +(J-1),+J", "28e59f3d592d74cc4452eb26f653d9b8": "u_{s}^{\\mathrm {topface} }(x,z)=-\\left(z-h-{\\tfrac {f}{2}}\\right)~{\\cfrac {\\mathrm {d} w_{s}}{\\mathrm {d} x}}", "28e5d910a6734e9209c07a7a37ae4785": "\\left({\\begin{array}{cc}+&-\\\\+&-\\end{array}}\\right),\\quad \\left({\\begin{array}{cc}+&+\\\\-&-\\end{array}}\\right),\\quad \\left({\\begin{array}{cc}-&+\\\\-&+\\end{array}}\\right),\\quad \\left({\\begin{array}{cc}-&-\\\\+&+\\end{array}}\\right).", "28e624a508df728e0273abb1e6394210": "{\\mathbf {y}}(x)", "28e6504280202e340435c4409bd5fa3d": "f(0)=a", "28e71e5fd2f8e19d7a0e518814e83c16": "\\Delta (w,C)=\\min _{x}\\{w,C(x)\\}", "28e77ed4e9ed74c2034ffe46a8b72018": "\\prod _{1\\leq i\\neq j\\leq n}(1-t_{i}/t_{j})^{a_{i}}", "28e78032188ba926bb889e202bd108e8": "ab\\leq a", "28e7d11582dc85c5b92920e705bb93f0": "E_{m}", "28e7e81229f7a91e1d8f8f053510cdee": "S_{mk}^{}=\\sigma _{mk}+p_{mk}/R", "28e82c22b9fa89388865172311e899a1": "E_{3}=2q_{4}q_{3}", "28e90ba64dc301c559ce754e1211a38e": "\\scriptstyle \\sum _{p|N}{\\frac {1}{p}}+{\\frac {1}{N}}=1", "28e92e148f2d5d3214fffc67cd7a4772": "H_{0}={\\rm {p}}K_{\\rm {a}}+\\log {{c_{\\rm {B}}} \\over {c_{\\rm {BH^{+}}}}}", "28e93d307001615f55e1706a2c7f4795": "\\Rightarrow D_{s}^{2}={\\frac {C}{C_{s}}}D^{2}", "28e9634b77bc25a1cca9783a5b4e0acb": "{\\text{Gain}}=\\ln \\left({\\frac {P_{\\mathrm {out} }}{P_{\\mathrm {in} }}}\\right)\\,\\mathrm {Np} ", "28e9d67574a03b995e5f2bf1954096ec": "e_{k}(t):=e^{2\\pi ikt}\\,", "28e9e7985215109604037e2159faf2d8": "\\sin \\left({\\frac {x}{2}}\\right)=\\pm {\\sqrt {{\\tfrac {1}{2}}(1-\\cos x)}}", "28e9e8aad305db2dd028ac959505e30c": "~\\alpha ", "28e9fd68184ad0aee9c687964f68ed92": "c=(64/9)^{1/3}", "28e9fe25681eb18184ad9f1010968a1e": "\\lambda _{i}^{(m)}", "28ea2617536b390d269bd319e07e06f6": "T_{e}G", "28ea300596fd7db0e48bc310ac89717c": "\\scriptstyle (x_{0},y_{0})", "28ea323dad136b15aae0eef007970f6a": "P=|I|\\cdot A_{\\mathrm {surf} }=|I|\\cdot 4\\pi r^{2}\\,", "28eaaf87fe74d9a5f1d491724ec01d65": "-\\cot \\phi ={\\frac {a}{b}}\\tan t={\\frac {\\tan \\theta }{(1-g)^{2}}}={\\frac {\\tan \\theta }{1-e^{2}}},", "28eab5b5254aad71ceee7be78678935d": "{\\mathcal {L}}_{V^{r}}\\,", "28eac0884358a673e9b31ab6cadaa6e6": "\\rho _{i}={\\frac {\\lambda _{i}^{+}}{\\mu _{i}+\\lambda _{i}^{-}}}", "28eae72477860d091f533dd5d9717d18": "{\\mathcal {E}}", "28eb2e70cd20b1c82bc81c0859e1f785": "0\\leq \\beta \\leq \\alpha N", "291792e8f76fbcb9e8d29ace6b730643": "a^{*}=\\left(-1\\right)^{\\frac {a-1}{2}}a.", "29183bddab1919643bed88cca776cc00": "\\Delta \\otimes \\Delta ^{*}\\cong \\bigoplus _{p=0}^{k}\\Gamma _{2p}.", "29183e69afc20b9728933f28cc19fe7e": "S=k_{B}\\ln W\\!", "29184a014c10f35fc201af66ccd3e7c2": "g_{2}=-4(e_{1}e_{2}+e_{2}e_{3}+e_{3}e_{1})", "2918887cc0d46f6afe29dfbf97a7d447": "(x/w\\cdot y)\\backslash z=y\\backslash (w/x\\cdot z)", "2918909c75eb97848d7a7f2258b26448": "b_{ij}=(x_{j}-y_{i})A_{j}(y_{i})B_{i}(x_{j})\\,", "291890b22d98c6955542fa29af6446ed": "\\tan(\\arccos x)={\\frac {\\sqrt {1-x^{2}}}{x}}", "2918a18f29fe7fbc7b60f549fe476c3c": "{{S}_{2}=\\varphi (5+3\\varphi )=5\\varphi +3\\varphi ^{2}=5\\varphi +3(1-\\varphi )=3+2\\varphi }", "29191da0cb92f259d688539792c4f80f": "\\varphi =45", "29195bc86c096d4d8498c9cdb6d57af3": "K_{M}^{N+1}", "2919644b6749bc78c9e450c9eb34340d": "\\sum _{\\stackrel {1\\leq k\\leq n}{\\gcd(k,n)=1}}\\gcd(k-1,n)=\\varphi (n)d(n),", "2919fc5ff9456db0606e9ddae9431498": "\\ E_{z}(k_{z})={\\frac {\\Delta }{2}}(1-\\cos(k_{z}d))", "291a14abc0caca27f61efc7d4d7fb249": "\\max _{x}\\sum _{i=1}^{L}{\\alpha _{i}}\\ln x_{i}{\\text{ s.t. }}\\sum _{i=1}^{L}p_{i}x_{i}=w", "291a4ddb86419421a5f566fb0d7de29b": "n-Tm", "291a6193e0b6283e1b5f839e68d61a09": "_{metric}\\delta _{ck}^{2}=_{metric}\\delta _{kc}^{2}", "291a70545c58742062ae8e0ff4c9f655": "\\forall X_{i}\\exists Y_{j}:X_{i}\\leftrightarrow Y_{j}", "291a83de0d29f5071a5042ad101ed16e": "T_{b}={\\frac {Ic^{2}}{2k\\nu ^{2}\\Delta \\nu }}", "291aa2eb092693df0941c2c9edc0dce5": "\\Gamma _{\\sigma ,K\\sigma }(x,y)=I*({\\frac {1}{2\\pi \\sigma ^{2}}}e^{-(x^{2}+y^{2})/(2\\sigma ^{2})}-{\\frac {1}{2\\pi K^{2}\\sigma ^{2}}}e^{-(x^{2}+y^{2})/(2K^{2}\\sigma ^{2})})", "291b25b6531fca9f96afd4b4d26e3426": "PVA\\,=\\,A\\cdot {\\frac {1-{\\frac {1}{\\left(1+i\\right)^{n}}}}{i}}\\ =\\ 1000\\cdot {\\frac {1-{\\frac {1}{\\left(1+.07\\right)^{20}}}}{.07}}\\ =\\ 1000\\cdot {1-0.258 \\over .07}\\ =\\ 1000\\times 10.594\\ \\approx \\ \\$10,594", "291b84328b4e61c02ee348cd74580d21": "-\\tan t={\\frac {b}{a}}\\cot \\phi ={\\sqrt {(1-e^{2})}}\\cot \\phi =(1-g)\\cot \\phi ={\\frac {-\\tan \\theta }{\\sqrt {(1-e^{2})}}}=-{\\frac {a}{b}}\\tan \\theta .", "291ba2dacaa33653b062ed432db5fa57": "z_{k}'=z_{k}-{\\frac {F(z_{k})}{F'(z_{k})}}=z_{k}-{\\frac {1}{{\\frac {p'(z_{k})}{p(z_{k})}}-\\sum _{j=0;\\,j\\neq k}^{n}{\\frac {1}{z_{k}-z_{j}}}}}", "291c02c2774b8c7fa2befe6db00b023e": "{\\boldsymbol {\\Sigma }}", "291c455ae03646abfc80a97c512bd96f": "R>0", "291c4f7a3117a2744cd981b9df679563": "A=C", "291ccaa5b8fb4d075679baa3139b6bd2": "p+\\rho v^{2}/2+\\rho gy=p_{\\mathrm {constant} }\\,\\!", "291cfb38f92413d5596c642023287da3": "\\ -{\\frac {d[A]}{dt}}=2k[A]^{2}", "291d6229001a92633bd6e7ca3bdbe434": "J\\equiv \\langle g_{1},\\ldots ,g_{q}\\rangle ,", "291d7b2ae4db902bea996c4de1a084ee": "150{\\frac {ml}{kg}}", "291d7d48c952d0f238538ef1e2c93697": "\\langle \\cdot |\\cdot \\rangle _{E}", "291e167621422fcc8c46acd4cfd1edcd": "n\\in \\omega ,f(n)\\leq g(n)", "291eb52763bc2636c46ded92c760d422": "(n-1){\\frac {s^{2}}{\\sigma ^{2}}}\\sim \\chi _{n-1}^{2}.", "291f0ce6b6884782c8ce3e33548ed9de": "\\mathbf {\\gamma _{5}} ", "291f14cc88d426dbf70889d95d379b3d": "L(v,q)={\\tfrac {1}{2}}\\langle v,Mv\\rangle -V(q)", "291f3c6dc729008ef269839f2e2f5ac6": "\\mu _{M_{J}}", "291f4bb5ed11b6767c2c66152be73284": "{\\frac {\\sqrt {\\pi }}{2}}e^{z^{2}}\\operatorname {erf} (z)={\\cfrac {z}{1-{\\cfrac {z^{2}}{{\\frac {3}{2}}+{\\cfrac {z^{2}}{{\\frac {5}{2}}-{\\cfrac {{\\frac {3}{2}}z^{2}}{{\\frac {7}{2}}+{\\cfrac {2z^{2}}{{\\frac {9}{2}}-{\\cfrac {{\\frac {5}{2}}z^{2}}{{\\frac {11}{2}}+{\\cfrac {3z^{2}}{{\\frac {13}{2}}-{\\cfrac {{\\frac {7}{2}}z^{2}}{{\\frac {15}{2}}+-\\ddots }}}}}}}}}}}}}}}}.", "291f847b9ce3439d881e0b0fdb89fd3e": "|{\\hat {f}}(a)|=2^{n/2}", "291f88cb6b44aede0762227b53eb4aa0": "x_{i+1}^{p}\\equiv x_{i}{\\pmod {p}}", "291fc2d6174cc1faa23a3eb29bce16ac": "\\nabla _{x,y,\\lambda }\\Lambda (x,y,\\lambda )=0", "291fe8013e5a5df54ae0a3e599c72f48": "\\log(n/w)", "292040bd0b93ae1610955a6bc08a0c70": "\\mathrm {sinc} \\,x={\\frac {\\sin \\pi x}{\\pi x}}.", "2920631e1a42471196209d2b3298e0d4": "\\lfloor x\\rfloor +\\lfloor -x\\rfloor ={\\begin{cases}0&{\\mbox{ if }}x\\in \\mathbb {Z} \\\\-1&{\\mbox{ if }}x\\not \\in \\mathbb {Z} ,\\end{cases}}", "2920c2d255bf53bc6d66507415c4270f": "{\\mathcal {C}}^{i}", "2920d23b3055384057ce288be7ffea59": "U_{t+1}(N_{i,j})=U_{t}(N_{i,j})+2-\\sum _{N\\in G(N_{i,j})}V_{t}(N)", "2921806082f591b3304e32c2d8cdca65": "{2\\omega _{i} \\over d}", "292195c0c5b9f1bf51e736135819a43f": "{\\mathfrak {g}}\\otimes _{\\mathbb {F} }{\\overline {\\mathbb {F} }}", "2921a820a9cd16690efd0dccbbf27f8d": "{\\mathcal {O}}_{X,x}\\otimes \\kappa (f(x))", "2921d0631d6f88e3143486e8fd577639": "S_{i,t}=0", "29224bf896f66036b2262b17a4a0bfe1": "\\mathbf {n} (\\alpha )=(\\cos \\alpha ,\\sin \\alpha ).", "2922758edbc30cc3fff9fdc1acfa4deb": "x(N_{i})\\rightarrow y.\\,", "29229a6ae7e7cabbe087fa75f585ebeb": "y(1)=1", "292330170a1b29a428cb1a599fcffa05": "\\mathbf {U} \\mathbf {I} \\mathbf {V} ^{*}", "29233df7e8990bbb7330db3388142952": "G^{-1}={\\frac {1}{d}}\\left[\\left(d+1\\right)I-{\\mathcal {I}}\\right]", "29237f68f3e06b1e743edd4e18221fbc": "\\mathrm {AgOCN+2HNO_{3}+H_{2}O\\longrightarrow } ", "2923beed5e42cd6c2ca1a820cdf0edbd": "E_{7}\\,", "292416f0bae8c1ee6758eeb4058b4c84": "\\lambda x+(1-\\lambda )s", "292449054898c976f1e1f40986f47c5b": "\\mu ={G}(m_{1}+m_{2})\\,\\!", "29245b2a513768a2e41a9280a1d1235b": "\\sigma =\\uparrow ,\\downarrow ", "292471e8d2c018f25050961c03b90c9b": "-{\\frac {\\hbar ^{2}}{2m}}({\\frac {d^{2}}{dx^{2}}}\\psi +{\\frac {d^{2}}{dy^{2}}}\\psi )=E\\psi ", "2924a494c04f46fc75b69e8aa4f69484": "\\theta _{k}[0]=\\phi _{k}.\\,", "29251f74c9dbe0812baf49c5cacd2acd": "{\\frac {\\mathrm {d} x_{i}}{\\mathrm {d} t}}=F_{i}(x)-V_{i}(x)", "292541333210954cdea9a3db307b47ee": "a=b=L/2", "292546742ecb79a7a09526f138e40efb": "\\mathbf {\\Delta } _{n}^{1}", "2925779ecee11918465920d51fcd3b64": "\\displaystyle \\times J_{n/2+\\delta }(|{\\boldsymbol {\\omega }}|)", "2925867c0ec7b5f66b5d214f43e6ec13": "x\\in (a,b)", "292592fd60abc577bf03f55ada5a555b": "\\rho {\\sqrt {\\pi }}\\,\\mathrm {e} ^{-\\rho ^{2}x^{2}/4}=\\int \\mathrm {e} ^{isx-s^{2}/\\rho ^{2}}\\,\\mathrm {d} s,\\quad \\rho >0.", "2925bbac3f6291dc9db9cce6904c3bb2": "\\sigma _{yy}", "29262b3fbb8cd0eb7cb6e71126877d03": "50^{2}", "29268545bda778f7b461c44e25c25e7d": "{\\frac {du_{i}}{dt}}=-{\\frac {1}{\\Delta x_{i}}}\\left[F_{i+{\\frac {1}{2}}}^{*}-F_{i-{\\frac {1}{2}}}^{*}\\right]+{\\frac {1}{\\Delta x_{i}}}\\left[P_{i+{\\frac {1}{2}}}-P_{i-{\\frac {1}{2}}}\\right].", "2927545276cfd81445e1ce3b04aea858": "\\ EI{dy \\over dx}\\ =\\int _{0}^{x}M(x)dx+C_{1}", "292764cf6c9366b6633b42eaa8c9ecb2": "\\sin x^{\\circ }-{\\frac {4x(180-x)}{40500-x(180-x)}}", "29277be4418837b26e2ff9f2c651e7ce": "\\int _{0}^{\\infty }e^{-\\omega x}\\;G_{p,q}^{\\,m,n}\\!\\left(\\left.{\\begin{matrix}\\mathbf {a_{p}} \\\\\\mathbf {b_{q}} \\end{matrix}}\\;\\right|\\,\\eta x^{2}\\right)dx={\\frac {1}{{\\sqrt {\\pi }}\\omega }}\\;G_{p+2,\\,q}^{\\,m,\\,n+2}\\!\\left(\\left.{\\begin{matrix}0,{\\frac {1}{2}},\\mathbf {a_{p}} \\\\\\mathbf {b_{q}} \\end{matrix}}\\;\\right|\\,{\\frac {4\\eta }{\\omega ^{2}}}\\right),", "29278cf709c8e5f1acd01b2e6b00a90a": "S=\\int _{\\mu _{b}}^{\\mu _{a}}Ld\\mu =-\\int _{\\mu _{b}}^{\\mu _{a}}\\rho d\\mu ,", "2927a3ff535ff07d901538c6a88f4d3a": "f_{0}\\ ", "2927aeeb198e7f8a99a04a8df1e44d5e": "V_{\\pm }=kx", "2927ca2f2959c0b840c6720c10a25e75": "\\langle B,F\\rangle ", "2928318549a2643477988da5667f5225": "q={\\frac {1}{2}}\\cdot \\left(1+{\\frac {\\sqrt {\\sum _{i=1}^{M}N_{i}}}{\\sum _{i=1}^{M}{\\sqrt {N_{i}}}}}\\right)", "2928b402f6b86569b4d363f685c6e726": "{\\begin{aligned}Y&={\\sqrt {y}}(1_{\\!N}-S)(1_{\\!N}+S)^{-1}{\\sqrt {y}}\\\\&={\\sqrt {y}}(1_{\\!N}+S)^{-1}(1_{\\!N}-S){\\sqrt {y}}\\\\\\end{aligned}}", "2928e82409da933a00f98c7f598a0164": "G(n^{2};x)=\\sum _{n=0}^{\\infty }n^{2}x^{n}={2 \\over (1-x)^{3}}-{3 \\over (1-x)^{2}}+{1 \\over 1-x}={\\frac {x(x+1)}{(1-x)^{3}}}.", "2929392adb0e92275cb920bddde62f66": "x\\mapsto [x]_{\\sim }", "2929747478093b1e66eae476a683144e": "\\mathbf {v} ={\\mathbf {\\hat {e}}}_{r}{\\frac {\\mathrm {d} r}{\\mathrm {d} t}}+r\\omega {\\mathbf {\\hat {e}}}_{\\theta }", "2929ec8b803840e5180435aef7529190": "G=pe+PE-kT[\\ln(n!N!)-\\ln(n-p)!p!(N-P)!P!]", "2929f07151df351cefe0f74fc9bfd0f7": "\\alpha \\in [0,1]", "292a3606dc937039e115a232428e574c": "\\phi \\lor \\psi ", "292a884afb7f0decbe3298d8ea524373": "\\mathbf {F} _{1}=\\int _{A}(\\sigma _{11}\\mathbf {e} _{1}+\\sigma _{12}\\mathbf {e} _{2}+\\sigma _{13}\\mathbf {e} _{3})\\,dA", "292b04e199c59262149feb6231df25b5": "\\omega _{1},\\omega _{2}\\in \\mathbb {C} ", "292b752275ba39830c3b4f79bedabebe": "\\mathbf {I} =\\int _{V}\\rho (\\mathbf {r} )\\left(\\left(\\mathbf {r} \\cdot \\mathbf {r} \\right)\\mathbf {E} -\\mathbf {r} \\otimes \\mathbf {r} \\right)\\,dV,", "292b899320b1048641e3ecbea5fc0c76": "v'=\\sum _{i=1}^{d}p_{i}P_{i}v", "292bdbed75aef7772141d0c068934bdb": "\\mathrm {Cliff} (V,q)", "292beeee007be9bd949047435c0e4c89": "x\\mapsto \\alpha \\cdot f(x)", "292c27ca08795e01cd3ff092eefce474": "s={\\frac {\\sigma +\\tau }{\\sqrt {2}}},\\qquad \\qquad t={\\frac {\\sigma -\\tau }{\\sqrt {2}}}.", "292c6cae649bcee60dba1806ffdf1a5f": "CAS=a_{0}\\left[\\left({\\frac {q_{c}}{P_{0}}}+1\\right)\\times \\left(7\\left({\\frac {CAS}{a_{0}}}\\right)^{2}-1\\right)^{2.5}/\\left(6^{2.5}\\times 1.2^{3.5}\\right)\\right]^{(1/7)}", "292c85ea2f9c2db0624de19cb762988f": "{\\begin{aligned}\\int _{0}^{T}e^{-xt}\\phi (t)\\,dt&=\\sum _{n=0}^{N}{\\frac {g^{(n)}(0)\\ \\Gamma (\\lambda +n+1)}{n!\\ x^{\\lambda +n+1}}}+O\\left(x^{-\\lambda -N-2}\\right)+O\\left(x^{-1}e^{-\\delta x}\\right)\\\\&=\\sum _{n=0}^{N}{\\frac {g^{(n)}(0)\\ \\Gamma (\\lambda +n+1)}{n!\\ x^{\\lambda +n+1}}}+O\\left(x^{-\\lambda -N-2}\\right)\\end{aligned}}", "292c8b98ea452964edada5557c5a03b6": "x\\mapsto x+b", "292c9448644c38f64d5fb8a87201b5d6": "c=1+{\\frac {A}{I}}.\\qquad (4)", "292cd856afcdd36c910d6e7e58b65416": "\\Delta (t)=-2t+5-2t^{-1},\\,", "292d75eed6ccba9c98646d0e1346575f": "n\\neq 1\\,\\!", "292dacc09337aa51119d053c2a3f8f5f": "\\sigma =\\sigma _{0}\\,", "292de4c9ed4553d1b09583c87c3b3315": "V_{r}={\\frac {2}{9}}{\\frac {r_{p}^{2}}{\\mu }}{\\frac {V_{t}^{2}}{r}}(\\rho _{p}-\\rho _{f})", "292e7322fad40ca84cdb01d4189e6543": "G_{\\alpha \\beta }^{a}=\\partial _{\\alpha }{\\mathcal {A}}_{\\beta }^{a}-\\partial _{\\beta }{\\mathcal {A}}_{\\alpha }^{a}\\mp g_{s}f^{abc}{\\mathcal {A}}_{\\alpha }^{b}{\\mathcal {A}}_{\\beta }^{c}\\,,", "292e7e012700b8e5573c4b8b2f99024f": "d\\geq n", "292e81656c9df1ed02ec70d77eb4ace2": "f'(x_{*})=0", "292e983f978119219ed398ffea9b0e78": "f(x)\\sim \\sum _{n=0}^{\\infty }a_{n}\\varphi _{n}(x)\\ (x\\rightarrow L)", "292eb625322c318c6c7a920dd21f6fef": "ee_{\\text{product}}=ee_{\\text{max}}ee_{\\text{auxiliary}}(1+\\beta )/(1+g\\,\\beta )", "292ecd51b122246edd2a81cca44a4623": "s_{j,t}^{*}={\\frac {1}{2}}(s_{j,t}+s_{j,t-1})", "292eebd5b0cb0c7e4c602c751d96665e": "t\\in [0,\\infty )", "292efef686b2da669c0ac372b703804a": "A_{0}", "292f23aa44a37df3400c79c80cb70b40": "\\kappa ={\\frac {x'y''-y'x''}{(x'^{2}+y'^{2})^{3/2}}}", "292f413b7616f72f863d57fb9a82a8c3": "{\\frac {S_{2}-S_{1}}{n}}=C_{p}\\ln \\left({\\frac {T_{2}}{T_{1}}}\\right)-R\\ln \\left({\\frac {T_{2}V_{1}}{T_{1}V_{2}}}\\right)=C_{v}\\ln \\left({\\frac {T_{2}}{T_{1}}}\\right)+R\\ln \\left({\\frac {V_{2}}{V_{1}}}\\right)", "292fbf846a8feab13426b4beeaaef38b": "r_{s}={\\frac {2GM(r)}{c^{2}}}\\;", "292feac8b55587c7dad121f05b52c69c": "{\\text{(1)}}\\qquad bD{\\frac {\\mathrm {d} ^{4}w_{x}}{\\mathrm {d} x^{4}}}=q_{1}(x)-n_{1}(x){\\cfrac {d^{2}w_{x}}{dx^{2}}}-{\\cfrac {dn_{1}}{dx}}\\,{\\cfrac {dw_{x}}{dx}}-{\\frac {1}{2}}{\\cfrac {dn_{2}}{dx}}\\,{\\cfrac {d\\theta _{x}}{dx}}-{\\frac {n_{2}(x)}{2}}{\\cfrac {d^{2}\\theta _{x}}{dx^{2}}}", "2930008b32df801c90a18bb8c32d5845": "\\tan \\alpha ={\\frac {{\\dfrac {\\partial u_{y}}{\\partial x}}dx}{dx+{\\dfrac {\\partial u_{x}}{\\partial x}}dx}}={\\frac {\\dfrac {\\partial u_{y}}{\\partial x}}{1+{\\dfrac {\\partial u_{x}}{\\partial x}}}}\\quad ,\\qquad \\tan \\beta ={\\frac {{\\dfrac {\\partial u_{x}}{\\partial y}}dy}{dy+{\\dfrac {\\partial u_{y}}{\\partial y}}dy}}={\\frac {\\dfrac {\\partial u_{x}}{\\partial y}}{1+{\\dfrac {\\partial u_{y}}{\\partial y}}}}\\,\\!", "29306b904bfcd45a546bc9781d805ae0": "\\forall n0\\,", "2932ed5765ad13137ae767f1adf9789a": "w=w(r)", "293313058418b73a68e81127b1e145ee": "(a+b\\mathbf {i} )(c+d\\mathbf {i} )=ac+ad\\mathbf {i} +b\\mathbf {i} c+b\\mathbf {i} d\\mathbf {i} =ac+ad\\mathbf {i} +bc\\mathbf {i} +bd\\mathbf {i} ^{2}=(ac-bd)+(bc+ad)\\mathbf {i} ", "293396ec32af55d813b7bcbb49c4cb2d": "{\\tilde {C}}^{+}\\rightarrow {\\tilde {\\ell }}^{+}\\nu ", "2933dbea20923654d3f9acc2f7094509": "{\\varphi }_{{\\lambda }_{1}}[Int({I}^{2})]\\cap {\\varphi }_{{\\lambda }_{2}}[Int({I}^{2})]=\\varnothing ", "2933e2e70df46551b8d6dc0a9f41eb14": "{\\tfrac {1}{2}}+{\\tfrac {1}{3}}+{\\tfrac {1}{16}}", "29342c18a7d38bef495a97641e4ed166": "{\\big (}X_{1},\\dots ,X_{i},Y_{i+1},\\dots Y_{r}{\\big )}\\,", "2934473ecbee019919b5e153d22c945e": "n_{th}", "293454042c53cc0c8389d194c51005a7": "x=\\gamma \\left(1+v/c\\right)x'.", "29348a459966fb8609b1d9e6f402750a": "{\\tfrac {1-\\rho }{\\rho }}", "29348f35a86422145870950cb6367435": "\\Delta x_{i}=x_{i,1}-x_{i,0}", "2934adcb166ff11727d04057cd5e1833": "(x+5)^{2}\\,=\\,x^{2}+10x+25.\\,\\!", "2934dca8e0b8b5d09b97bd4e8e5f5492": "X_{1},...,X_{n}\\,", "2935119c2f415dd3cc9562cd11a4f6ec": "k_{\\text{surf}}", "293557b587cbef2634b0f217ac6b64fe": "\\sum _{i,j=1}^{5}a_{ij}x_{i}x_{j}=0", "29356aaa101db61bda55d37de74cb839": "(p,\\omega _{2},q')\\in \\Delta ", "29359d3c3855e09b30cfac86a7049be1": "\\|x\\|_{K(X_{0},X_{1})}=\\sup _{m\\geq 1}{\\Bigl \\|}\\sum _{n=1}^{m}a_{n}K(x,b_{n}/a_{n};X_{0},X_{1})\\,y_{n}{\\Bigr \\|}_{Y}<\\infty ,", "2935fdc45bc7bb1275a6a5912dfece94": "\\operatorname {excosec} (\\theta )", "293659b48715d04088158dcfc276ae79": "{\\frac {\\sin(z)}{z}}", "29367371c91a10923b32d1a8368bd98a": "{\\frac {4+0}{10}},{\\frac {4+3}{10}},{\\frac {4+6}{10}},{\\frac {4+12}{10}},{\\frac {4+24}{10}},{\\frac {4+48}{10}},.....", "2936a850adb90c22c60649f23860a8ee": "d(x)=\\mathrm {GCD} (p(x),(x^{2l-1}+1))", "2936dec4a9eeced481fad24368cda712": "M/N", "29373cca99c2680ad8d8c7ca709a4209": "x=\\cos(at)-\\cos(bt)^{j}", "293781c753971e39e67fa655672737f8": "(0.008\\cdot t)", "29378edd6c1829a96c0f8a0bff7a625e": "C_{V}=\\left({\\frac {\\partial E}{\\partial T}}\\right)_{V}=Mk_{B},", "2937e3057aa2a4701e78e4b9ce43e79c": "e^{+}e^{-}\\to \\eta ^{\\prime },~~a_{0},~~f_{0},~~a_{2},~~f_{2}", "293800d9d0a269bc58ba7372ca7ccac0": "E=\\,h\\,\\nu ", "293808a6148faf157159bd0c927dd2b6": "V_{\\omega C}\\,", "293835a349723dd4872cbc522c002d3f": "\\nabla \\cdot (\\nabla \\times \\mathbf {A} )=0", "29384bd416e67af13fc145dd6e7afad4": "{\\frac {\\pi }{8}}", "2938c35a5fb79f44224bcefe6ef10447": "{\\frac {\\partial X_{i}}{\\partial P_{Y}}}>0", "2938e5d8a1353ddb36bb4471d83e9abe": "\\scriptstyle \\left\\lceil \\log _{2}(n)\\right\\rceil ", "29390f651578055ea5089667bb0373eb": "{\\frac {1}{18}}={\\frac {1}{2}}-{\\frac {1}{3}}-{\\frac {1}{3^{2}}}.", "2939419410b658835778371c60ec1a96": "{\\frac {dQ^{N}}{dQ}}={\\frac {M(0)}{M(T)}}{\\frac {N(T)}{N(0)}}.", "293963d987a6836b57663debb90039ab": "K-L", "2939afe3f2265d04bbe1069815c8e8bc": "EQUI(\\alpha ,\\alpha ')=1-XOR(\\alpha ,\\alpha ')", "2939b545d04021022641daf80f9060a8": "F={\\frac {P}{{\\frac {1}{2}}v}}={\\frac {2P}{v}}", "293b01aa1e7b100d17f18474ff77facb": "u(t,z)", "293b098f4d185a5687c93e0a92386ac3": "k_{1}=hf(t_{n},y_{n}),\\,", "293b20bbe2eb2e5807164601025d81bf": "u(0,x,y)=0,\\quad u_{t}(0,x,y)=\\phi (x,y),\\,", "293b227a3f3489c0f003726872d08340": "\\operatorname {E} (Y)=\\operatorname {E} (N)\\operatorname {E} (X),", "293b481a43ca82f0083b9024f89d2290": "T_{CHUR}=({\\frac {1}{\\lambda }})ln\\left[1+{\\frac {\\left({\\frac {^{143}Nd}{^{144}Nd}}\\right)_{sample}-\\left({\\frac {^{143}Nd}{^{144}Nd}}\\right)_{CHUR}}{\\left({\\frac {^{147}Sm}{^{144}Nd}}\\right)_{sample}-\\left({\\frac {^{147}Sm}{^{144}Nd}}\\right)_{CHUR}}}\\right]", "293ba9537bd511c0af996c508214445b": "Q_{1}={\\frac {1}{2}}\\left[(p-iW)b+(p+iW^{\\dagger })b^{\\dagger }\\right]", "293bd1c8ca8f04858eb6b84292abd331": "P(A>O_{j})", "293bd5976bd48dcdcc71b06592ddca25": "\\vdash (p\\lor \\neg p)", "293bda3b898ecf872fa26c966ad39b77": "m_{ox}={\\frac {f_{st}-f}{f_{st}}}\\left(m_{ox,0}\\right),00}\\exp \\left({\\rm {tr}}(-\\mathbf {SW} -\\mathbf {S^{-1}Z} )\\right)|\\mathbf {S} |^{-\\delta -{\\frac {1}{2}}(p+1)}d\\mathbf {S} ,", "294570c6004ae1a20dca25fcd67eff63": "(u_{0},\\lambda _{0})\\,", "29459ee97e9c09ec5dd9f29f691115b0": "\\mathbf {\\hat {T}} (\\varepsilon )|\\psi \\rangle =|\\psi _{\\varepsilon }\\rangle ", "29465d167bea75dee5c7daec109de55c": "\\delta ^{\\dagger }\\circ (\\psi \\otimes 1_{A})", "29469ac4fbd9fcfeec9390cef4026c35": "RTI_{20}={\\frac {h}{b}}\\times 2924", "29472298d50a0ac838c1c7e187db05d4": "1/{\\sqrt {8\\pi }}", "29476b753ea559cf052cad95e916c541": "\\Sigma =\\{a_{1},a_{2},\\ldots ,a_{n}\\}", "2947c6e4d0f4f9adea4c9afea63ccbff": "\\mathrm {u} (t)=\\mathrm {MV} (t)=K_{p}{e(t)}+K_{i}\\int _{0}^{t}{e(\\tau )}\\,{d\\tau }+K_{d}{\\frac {d}{dt}}e(t)", "2947f2fcefb34d657bffe9ca17a2f4f6": "x=2^{5}-2^{1}-2^{0}", "2948442ee0119a40952e6a90b6b40b01": "\\scriptstyle {\\mathcal {R}}", "2948453c2caad99ed613daf451f989ed": "g_{k}(X_{k})=\\sum _{X_{\\bar {k}}}g(X_{1},X_{2},\\dots ,X_{n})", "29489e883805e0e8831c822a25d120fd": "2+\\left\\lfloor {{n}\\,\\varphi }\\right\\rfloor -\\left\\lfloor {\\left({n+1}\\right)\\,\\varphi }\\right\\rfloor ", "29491fab17df6efb1955549b53655bb2": "-x^{4}+15245x^{2}-6262506.25=0", "294938c000fdaf9ed70ded86fdd73585": "|H_{jk}|=1{\\quad {\\rm {for\\quad }}}j,k=1,2,\\dots ,N", "294946ab4c45219f74c629d5acfa9d28": "{\\hat {H}}_{\\textrm {qp}}", "29496677a4f449513f71015be966c41e": "e^{-2\\pi }", "2949b7f0b10306de3524dfb5634d6a55": "\\{y_{i}\\}_{i=1}^{m}", "2949cd7538ff2fb58368ad1a4fa94f31": "(z_{1},z_{2},z_{3})\\equiv (\\lambda z_{1},\\lambda z_{2},\\lambda z_{3});\\quad \\lambda \\in \\mathbf {C} ,\\qquad \\lambda \\neq 0.", "294a08b6d51d7ec0c28e4371dc0b3232": "z_{i}:=Pz_{i-1}", "294a0f1f0425e0ff841e884db4b552bc": "\\scriptstyle Q_{ext}", "294a37b6e652cff58db912851813b5cd": "{\\mathsf {G}}(b,c)", "294a4794d7549882922c15b9c774d191": "\\tan \\theta =\\sin(\\lambda +\\iota )\\tan(15^{\\circ }\\times t)", "294a7f82eb911ddf59387507a3e0d761": "\\operatorname {ad} _{X}Y=[X,Y]", "294b0425012747bd6c131a798f0cf1ff": "J=-{\\frac {i\\hbar }{2m}}(\\phi ^{*}\\nabla \\phi -\\phi \\nabla \\phi ^{*})", "294b19c767ebb747f42cb9e128902c03": "\\scriptstyle n\\;>\\;m", "294b3d1285766e6ba5fa89bb5a2ee7ce": "\\pi _{3}\\colon {\\mathbb {R} }^{3}\\to {\\mathbb {R} }", "294b7f3aefbd7a1172e1469d8fd37801": "(m+1)(l+1)+d{\\begin{pmatrix}l+1\\\\2\\end{pmatrix}}>n", "294b8a72d22cd0e577b3ba80fa3d69c7": "{\\dot {x}}_{1}=f_{1}(x_{1},x_{2}),\\,", "294b8e7dc7d8731d91e833d536452198": "{\\frac {\\sin A}{\\sin _{K}a}}={\\frac {\\sin B}{\\sin _{K}b}}={\\frac {\\sin C}{\\sin _{K}c}}\\,.", "294b8e86731883d20832410b3c947bcb": "v(t)={\\dot {x}}(t)", "294bc70d90db955ef14b799dcc50a3f9": "\\left\\vert {\\text{supp}}\\left(\\mathbf {A} \\right)\\right\\vert ", "294c0d388c64bacf88e7c8a3630b6823": "{\\begin{aligned}\\chi _{k\\mid k-1}^{0}&={\\textbf {x}}_{k\\mid k-1}^{a}\\\\[6pt]\\chi _{k\\mid k-1}^{i}&={\\textbf {x}}_{k\\mid k-1}^{a}+\\left({\\sqrt {(L+\\lambda ){\\textbf {P}}_{k\\mid k-1}^{a}}}\\right)_{i},\\qquad i=1,\\dots ,L\\\\[6pt]\\chi _{k\\mid k-1}^{i}&={\\textbf {x}}_{k\\mid k-1}^{a}-\\left({\\sqrt {(L+\\lambda ){\\textbf {P}}_{k\\mid k-1}^{a}}}\\right)_{i-L},\\qquad i=L+1,\\dots ,2L\\end{aligned}}", "294c339936e52cae9fcbf151825bdc5d": "y_{i+1}=y_{i}+{\\frac {h}{2}}(f(t_{i},y_{i})+f(t_{i+1},{\\tilde {y}}_{i+1})).", "294c35d6ea23aae50e3de6c3426c51c1": "m(\\theta )=C_{0}\\,\\theta +b_{1}(\\theta )\\sin \\theta .", "294cd75e54a893eb61d83a8246b47cec": "r\\,", "294ce890c4a3b9529d80d402420ecd46": "r^{2}", "294ceda3a27dc78c24a1bb45d9efb479": "|z-b|^{2}-|b|^{2}+c,\\,\\!", "294db25a677c9b8b563e179477d9385b": "q!", "294ddf436ecf167bdc980d9511dd8d43": "S_{11}={\\frac {b_{1}}{a_{1}}}={\\frac {V_{1}^{-}}{V_{1}^{+}}}", "294e0631e0d7a600549b185b99d57e98": "\\scriptstyle \\Vert {\\boldsymbol {\\phi }}\\Vert _{L^{\\infty }(\\Omega )}\\leq 1", "294e0e67e3482c95b9ed2e52e05bde93": "R_{\\mathrm {g} }^{2}={\\frac {I}{A}},", "294e3e2c69994dc359eeddb5a11dc359": "dC=\\lim _{\\delta x\\to 0}(C\\delta x)=Cdx", "294e96e04778ca3569d0520f93ecedb6": "A\\hookrightarrow A\\oplus B\\twoheadrightarrow B", "294ebac37bab6950ad26d870d57046de": "c_{1}=1,\\qquad c_{2}=-{\\tfrac {2}{3}},\\qquad c_{3}={\\tfrac {2}{3}}", "294eec6a8177af125ddede54d40a2937": "{\\vec {S}}_{t}={\\vec {S}}\\wedge {\\vec {S}}_{xx}.", "294f09cf3c59786e3399a916e1a14c18": "S_{i}^{\\ell }", "294f1053bc456b1b7207f9d2efd8eede": "{\\dot {\\psi }}", "294f390713e0756aa86134e59db62a19": "\\{|f_{k_{0}}^{i}\\rangle ,|f_{k_{1}}^{i}\\rangle \\}", "294f4a83a3a842a9e9929f4fd67c8610": "m=-\\infty ", "294f56b2d15043c283f1754b2d825976": "y=t(t^{2}-3)=t^{3}-3t.\\,", "294f64140e6f3cf0e161ee432dc3ce59": "{\\text{for all process }}i:\\sum _{j=1}^{n}a_{ij}{\\dot {\\rho _{j}}}=0\\;,", "294fc7ac225a8db4604137240961bfc6": "s={\\sqrt {\\frac {Ns_{2}-s_{1}^{2}}{N(N-1)}}}.", "294fcfe0b57e5322891a3f209537fdf1": "{\\widetilde {P}}\\to X", "2950463d03c66e1fa373b5ea101d77dc": "S^{n-1}\\to S^{n-1}", "29504708ed38e907c2cad2a24354a291": "\\omega _{3}=1/\\omega _{2}", "2950688a80fc216750d6fff5a31c6247": "K\\colon \\Omega \\to 2^{M}", "2950a03ca2476416f98a235bf40917af": "p(x_{1},\\ldots ,x_{k})=0\\,", "2950b19264901201b8adc0948993265c": "{\\text{API gravity}}={\\frac {141.5}{\\text{SG}}}-131.5", "2950bfdce098187358e6a52b274d2790": "{\\frac {1}{\\pi }}\\sum _{n=1}^{\\infty }{\\frac {\\sin 2\\pi nx\\cdot \\ln {n}}{n}}=\\ln \\Gamma (x)-{\\frac {1}{2}}\\ln \\pi +{\\frac {1}{2}}\\ln \\sin \\pi x-(\\gamma +\\ln 2\\pi )(1-2x)\\,,\\qquad 0a+\\varepsilon )\\\\&\\leq \\operatorname {Pr} (X\\leq a+\\varepsilon )+\\operatorname {Pr} (Y-X\\leq a-X,\\ a-X<-\\varepsilon )\\\\&\\leq \\operatorname {Pr} (X\\leq a+\\varepsilon )+\\operatorname {Pr} (Y-X<-\\varepsilon )\\\\&\\leq \\operatorname {Pr} (X\\leq a+\\varepsilon )+\\operatorname {Pr} (Y-X<-\\varepsilon )+\\operatorname {Pr} (Y-X>\\varepsilon )\\\\&=\\operatorname {Pr} (X\\leq a+\\varepsilon )+\\operatorname {Pr} (|Y-X|>\\varepsilon )\\end{aligned}}", "295df0ec140b4cdc8cf61bd54c48105f": "\\eta _{\\mathrm {Carnot} }=1-{\\frac {T^{0}}{T_{H}}}", "295dfbc881d2a48c5c49cea6beed3dc8": "{\\hat {b}}\\,{\\hat {b}}^{\\dagger }", "295e03c55d097de3e1ccbfdd619854ad": "{\\frac {d^{2}L(F)}{dF^{2}}}={\\frac {1}{\\mu \\,f(x(F))}}\\,", "295e0c5e7f0c44f3793d2260322e32db": "B_{k+1}(\\mathbf {x} _{k+1}-\\mathbf {x} _{k})=\\nabla f(\\mathbf {x} _{k+1})-\\nabla f(\\mathbf {x} _{k}).", "295e204c52e638376dc2db1d4cbcc9f1": "\\epsilon _{n}={\\frac {1}{p-1}}\\sum _{a=1}^{p-1}\\omega (a)^{n}\\sigma _{a}^{-1}.", "295e2ec4a59763f1a58397d118b975b8": "{\\frac {\\partial k_{i}}{\\partial t}}={\\frac {\\delta k_{i}+2(1-\\delta )m}{2t}}", "295e3dcccad50cf2440a4ff0f39f0279": "\\scriptstyle a_{1}b_{1},a_{2}b_{2},\\dots a_{d}b_{d}", "295e71e017bdd5e1fe188812c7f70db8": "g_{\\alpha \\beta }=g_{\\beta \\alpha }", "295ec05211ca7f6177d387e788f268a8": "{\\big (}{\\Big (}{\\bigg (}{\\Bigg (}\\dots {\\Bigg ]}{\\bigg ]}{\\Big ]}{\\big ]}", "295edb9fd0d6c5468a69dadcc17f25da": "c\\in \\mathbb {R} ^{n}", "295f17a6289edd9b7d4eab641f60d944": "M_{n}={\\frac {\\sum M_{i}N_{i}}{\\sum N_{i}}},\\quad M_{w}={\\frac {\\sum M_{i}^{2}N_{i}}{\\sum M_{i}N_{i}}},\\quad M_{z}={\\frac {\\sum M_{i}^{3}N_{i}}{\\sum M_{i}^{2}N_{i}}},\\quad M_{v}=\\left[{\\frac {\\sum M_{i}^{1+a}N_{i}}{\\sum M_{i}N_{i}}}\\right]^{\\frac {1}{a}}", "295f4a7c5bde861af018e4f626a50438": "f'(x)=3x^{2}.\\!", "295f614533305b7f150f6b290a2af846": "\\displaystyle {\\int _{\\partial \\Omega }|K(z,w)|\\,|dw|\\leq C}", "295f6c62cb1b54de8f3fb7ca42b88a06": "a\\triangleright b=ta+(1-t)b", "295fd4c5adffdee4925878733143937d": "aX_{n}+bY_{n}\\ \\xrightarrow {p} \\ aX+bY", "295ff21f17ea0559297bd7754bc4a01f": "\\langle X,\\phi _{i}\\rangle ", "29600f9b306a2ea0b9ece242d13d8d80": "\\oint _{\\partial \\Omega }{\\boldsymbol {A}}~ds=0\\quad \\implies \\quad \\int _{AB}{\\boldsymbol {A}}\\cdot d\\mathbf {X} +\\int _{BA}{\\boldsymbol {A}}\\cdot d\\mathbf {X} =0", "29602944c0c89012ec41166b72856b53": "u_{tt}-u_{xx}=0.\\,", "2960a358606b6aaf12f4e71a38b9002f": "TRR=(Viewthroughs+Clicks)/Impressions", "2960c5e9428ac317772fae258a28128f": "\\left|a\\right|", "2960ce2a06d503771d547d45edf8da10": "g_{b}(n)=\\sum _{k=0}^{L-1}d_{k}(n)b^{-k-1}.", "2960d4135902c2432b7ae490e7f60d65": "\\mathbf {k} ", "2961246ae270353b46fc19632ce63e47": "dS=\\delta Q/T", "296150779bdd21a883cfb284bce270aa": "2\\times (6+6+9+9)=60", "29615af881af39504a58085cd32cb5a3": "\\lambda _{i}(x_{0})=\\log \\Lambda _{i}(x_{0}).\\,", "2961713c49fcd407a54dea778690502d": "X_{t_{2}}-X_{t_{1}},X_{t_{3}}-X_{t_{2}},\\dots ,X_{t_{n}}-X_{t_{n-1}}", "29619a9a3bf838c9caf7560af0b451df": "{\\textbf {k}}-{\\textbf {k}}_{0}={\\textbf {G}}_{\\textrm {hkl}}", "2961d26b07d61318b281bc7d776182c5": "{u}_{1}(\\mathbf {q} )", "2961f649a7553fc301387f44307ede76": "\\alpha =0", "296205b780755bbf462ddbe87aac8a00": "{\\mathcal {H}}={\\mathcal {H}}({\\boldsymbol {q}},{\\boldsymbol {p}},t)", "29622e4f38cbdc7d0f0a9e79176ce0a3": "a+bS(\\mathbf {r} ,i)", "2962302aedbf0aa2240ddc7b8b216780": "\\int _{0}^{1}x^{\\alpha -1}(1-x)^{\\beta -1}dx={\\frac {\\Gamma (\\alpha )\\Gamma (\\beta )}{\\Gamma (\\alpha +\\beta )}}", "29629a5fcb9dcaf4a2573fee5ef9651d": "I={\\frac {V}{R}}", "29629d0d283e07d875be97e5206761cf": "I[w_{t}]-I[w^{\\ast }]", "2962a03148088e0580bdcb1f63566944": "0\\left(c,y\\right)", "2969230a7f3bf08cf0aeef095ce34023": "\\sum _{v}d(v)=2e\\,", "29692464bf2cf6a91fe12b7d66ce42fb": "\\{k~|~(\\exists n\\in \\mathbf {N} )\\land (k=2n)\\}", "2969479e73423a8bedaf3bf99748b288": "t\\mapsto (t^{2}-1,t^{3}-t)", "2969866d4a448913d6117dc576806e3f": "{\\frac {P(X_{1}^{n}(i'))}{P(X_{1}^{n}(i))}}\\geq 1\\,", "2969b560aa1c2010b4df11a53a624e5d": "{\\Big (}\\pi \\models [\\phi _{1}U\\phi _{2}]{\\Big )}\\Leftrightarrow {\\Big (}\\exists n\\geqslant 0:{\\big (}\\pi [n]\\models \\phi _{2}\\land \\forall 0\\leqslant k{\\frac {\\alpha }{2}}\\right\\}\\bigcap \\left\\{\\theta {\\Big |}P\\left[\\mathrm {Bin} \\left(n;\\theta \\right)\\geq X\\right]>{\\frac {\\alpha }{2}}\\right\\}", "2979c79d42096abf62f6e0889e2107e2": "(\\beta _{1},\\;\\lambda _{1})", "297a15d7926199106390913dff3a73ca": "\\Psi _{BETA}(\\omega )", "297abf4d30459b64c62e0be4b042f62c": "\\chi (\\mathbf {x} )", "297b0f0b06db01aa54e2ceb195e11ab0": "\\left({\\frac {a}{n}}\\right)\\left({\\frac {a_{i}}{n}}\\right)=\\left({\\frac {a\\cdot a_{i}}{n}}\\right),", "297b20fe497e7cb284f25e5766c15d31": "F_{0}=1", "297b726923b944213b62a0b4130a60f5": "{\\overrightarrow {\\mathrm {OA} }},{\\overrightarrow {\\mathrm {OB} }},{\\overrightarrow {\\mathrm {OC} }},{\\overrightarrow {\\mathrm {OX} }}", "297b7d57565992652023428b852dd926": "4\\pi \\,", "297b7e83ef1555425016ca004bf9e124": "d={\\sum _{t=2}^{T}(e_{t}-e_{t-1})^{2} \\over {\\sum _{t=1}^{T}e_{t}^{2}}},", "297bbeaa8cfdc35046ffc213b4ebb490": "N\\approx 1", "297be72dc3830cc9f687aef239b8901f": "\\textstyle v(z)", "297befa90860fe809026918e45d1d54b": "h\\leq \\eta ,\\,", "297c3872a104e93b38fc118af395adac": "L^{-1}\\xi (x)=\\int dx\\int \\mathrm {d} x\\int \\mathrm {d} x\\;\\;\\xi (x)", "297c57d3ea9627dcac8a0a12305d084f": "\\Omega =\\sum _{i=1}^{n}\\left(Ad\\lambda _{i}-g_{i}D\\left(\\sum _{j=1}^{r_{i}}T_{j}^{(i)}\\right)g_{i}^{-1}\\right)", "297cafa3373d074f69ec9ceb2c790956": "(1+2^{-f(n)}2^{B})^{G}-1<{\\frac {1}{6}}2^{-n^{c}},", "297ce6e10c399f2c9c9f5b3c8656c231": "{\\hat {\\rho }}", "297cefd50fdb0c3102c89b79f92847c2": "b_{i}\\ ", "297d33f1e34cd9d9733cf0f22763149d": "s={\\tfrac {1}{2}}(a+b+2c)", "297d6c92233468dda0b24805a6467086": "2^{340}\\equiv 1{\\pmod {341}}", "297e170b111298ab1a67ee0991969ce6": "X=T^{-1/2}+{\\frac {1}{2}}+{\\frac {1}{8}}T^{1/2}-{\\frac {1}{128}}T^{3/2}+\\cdots ", "297e50c28b654cdd93bbb047f8d7c636": "\\Phi _{D}=Q", "297e5b8e87867f9f851c1373cd360125": "\\operatorname {ker} f:=\\{(a,a')\\in A\\times A:f(a)=f(a')\\}{\\mbox{.}}\\!", "297e6c5ec950a2c5b5703847ef69f6ff": "\\left(\\nabla _{X}Y\\right)(m)=d_{m}Y(X)+\\langle X(m),Y(m)\\rangle m", "297ec4016762982ea16ad7b60a49c121": "N_{1}~", "297f1fa5c65487468338851aa8b5b0aa": "\\exists ", "297f281068a6eaeb5cd75cda5fa838ba": "x,y\\in A^{\\ast }", "297f5e350d4ebcd69990829993bcba07": "\\,dS", "297fdeac90a89bfb26d9e46ce4d9f59e": "\\mathbb {Q} ^{\\omega }", "298046673e8ba6ef3112885a578d4ae6": "\\Phi (z/{\\sqrt {2}})", "2980695d161e7cbba71edff2ffe5f41d": "d(x,a)<\\delta .", "2980732210cf4ca66f48fc94ff3fb888": "(2~4)(1~2~3)(4~5)(2~4)=(1~4~3)(2~5).\\ ", "29809adc8854c5935ee265f13888294c": "{\\frac {\\partial f}{\\partial A}}=-2A\\sum _{i}\\sum _{j\\in C_{i}}p_{ij}\\left(x_{ij}x_{ij}^{T}-\\sum _{k}p_{ik}x_{ik}x_{ik}^{T}\\right)", "2981128a7d0da9a031bec84fb1279241": "f(z)=z^{m}e^{\\phi (z)}\\prod _{n=1}^{\\infty }\\left(1-{\\frac {z}{u_{n}}}\\right).", "2981c9cdfa92cefc4fc5bb30f1fefb8b": "\\textstyle v\\in P_{k-1}", "29820b9e4fe9e84ecb5ae65facf83700": "{\\begin{aligned}\\lim _{n\\to \\infty }\\operatorname {Pr} \\left(\\left|X_{n}-c\\right|\\geq \\varepsilon \\right)&\\leq \\limsup _{n\\to \\infty }\\operatorname {Pr} \\left(\\left|X_{n}-c\\right|\\geq \\varepsilon \\right)\\\\&=\\limsup _{n\\to \\infty }\\operatorname {Pr} \\left(X_{n}\\in B_{\\varepsilon }^{c}(c)\\right)\\\\&\\leq \\operatorname {Pr} \\left(c\\in B_{\\varepsilon }^{c}(c)\\right)=0\\end{aligned}}", "2982129bf60db5a7a818fcc3cffcfe7e": "M_{\\alpha }\\prec _{K}N", "298225341f913dc3131efd66de412501": "{\\begin{aligned}\\{P_{X},P_{Y}\\}(q,p)&=\\sum _{i}\\sum _{j}\\left\\{X^{i}(q)\\;p_{i},Y^{j}(q)\\;p_{j}\\right\\}\\\\&=\\sum _{ij}p_{i}Y^{j}(q){\\frac {\\partial X^{i}}{\\partial q^{j}}}-p_{j}X^{i}(q){\\frac {\\partial Y^{j}}{\\partial q^{i}}}\\\\&=-\\sum _{i}p_{i}\\;[X,Y]^{i}(q)\\\\&=-P_{[X,Y]}(q,p).\\end{aligned}}", "29827a050ec2ec492f8a284ee4b37ad8": "B_{I}M", "2982c190ff52ec994e56f740e79cfd2c": "\\mathbf {P} ={\\begin{bmatrix}p_{1},\\,p_{2},\\,\\ldots ,\\,p_{20},\\,p_{20+1},\\,\\ldots ,\\,p_{20+\\lambda }\\end{bmatrix}}^{\\mathbf {T} },\\,\\,\\,(\\lambda 0", "29abc36f3a21655bedcf9866fb753dfb": "S[\\gamma ]\\geq r\\int _{a}^{b}|\\theta '(t)|\\,dt\\geq r|\\theta (b)-\\theta (a)|.", "29abd0026e992df3eb99d32327f05047": "t_{i}=s_{j}", "29ac0a962cb46af35d366942a64b9592": "Q{\\mathbf {v}}=\\lambda {\\mathbf {v}},\\,\\!", "29ac0da90074e77d80f59d3abbe0f647": "\\rho (L)=\\{\\lambda \\in \\mathbb {C} |\\lambda {\\mbox{ is a regular value of }}L\\}.", "29ac5e34ee9208af7d5778a4c2f76cc7": "P_{r}={{P_{t}G_{t}} \\over {4\\pi r^{2}}}\\sigma {{1} \\over {4\\pi r^{2}}}A_{eff}", "29acc5c30288c32a4df628d6890e617f": "P,aP,bP,cP", "29ad2e125e9c54c6a0886b0128f5446a": "{\\vec {x}}\\in \\mathbb {R} ^{d}", "29ad3b5cbd560ab226939c72ef8898a1": "f(N+{\\tfrac {1}{N}})-f(N)=N^{2}+2+{\\tfrac {1}{N^{2}}}-N^{2}=2+{\\tfrac {1}{N^{2}}}", "29adc6f138008786753940782d7767e4": "\\operatorname {Spec} A", "29adf134fcdf1ac67eb7e7f1e9bf856c": "\\displaystyle {\\hat {f}}(\\nu _{x},\\nu _{y})=", "29ae56505caed64e4a633ac1f2c103d8": "{\\frac {r_{1}(t)-a_{1}}{dr_{1}(t)/dt}}={\\frac {r_{2}(t)-a_{2}}{dr_{2}(t)/dt}}=\\cdots ={\\frac {r_{n}(t)-a_{n}}{dr_{n}(t)/dt}}", "29ae5c157afa814e817986d828310e7e": "e_{1}(t)={\\dot {\\gamma }}(t)/|{\\dot {\\gamma }}(t)|", "29aeda5edc922fd393d4491f984a6029": "\\sigma (\\mathbf {x} )", "29aee2f5a89c54a9c930558cc3ed0048": "\\lfloor x\\rfloor =\\left\\lfloor x+{\\frac {k'-1}{n}}\\right\\rfloor \\leq x<\\left\\lfloor x+{\\frac {k'}{n}}\\right\\rfloor =\\lfloor x\\rfloor +1.", "29af18978a7f9242822cf79cd30ee44e": "\\Delta \\tau _{v}=-{\\frac {1}{2c^{2}}}\\sum _{i=1}^{k}v_{i}^{2}\\Delta t_{i}", "29af5ac141fd817074b23b97edefb0df": "c(r)=e^{-\\beta w(r)}-1+\\beta [w(r)-u(r)]\\,=g(r)-1-\\ln y(r)\\,=f(r)y(r)+[y(r)-1-\\ln y(r)]\\,\\,({\\text{HNC}}).", "29af847b0ebfc95fb0195320e799f4a3": "A={\\cfrac {6~c~\\cos \\phi }{{\\sqrt {3}}(3+\\sin \\phi )}}~;~~B={\\cfrac {2~\\sin \\phi }{{\\sqrt {3}}(3+\\sin \\phi )}}", "29afbc9975444371e8c48fb436cb0830": "F[y]=\\prod y", "29afc5ae50efa09eaf732a35a9f0bcd7": "player\\leq score", "29affef422d55808197d8a961e9598d8": "N:=\\bigcup \\nolimits _{n}\\left\\{p\\in \\mathrm {Supp} (\\mu ):(T_{0}U_{n})(p)\\sigma _{i+1}\\}", "29b6518844fd31c574e5bfbf2684e2ca": "V(A\\lor B,0)\\Leftrightarrow V(A,0)\\ and\\ V(B,0)", "29b675d2ebb243e9c32090e18b031bac": "{\\frac {1}{\\tau ^{*}}}={\\frac {p\\mu _{p}\\tau _{p}+n\\mu _{n}\\tau _{n}}{\\tau _{p}\\tau _{n}(p\\mu _{p}+n\\mu _{n})}}.", "29b6925e7a5686a742ea70fe47a8688d": "\\mathbf {P} (\\omega )=\\varepsilon _{0}\\chi (\\omega )\\mathbf {E} (\\omega ).", "29b6ab9cc222880da4dec87b34a046be": "O(A_{1}:A_{2})={\\frac {P(A_{1})}{P(A_{2})}}", "29b6dc638161d180edfb11032e5573d4": "{\\frac {\\partial ^{2}F_{~\\alpha }^{m}}{\\partial X^{\\beta }\\partial X^{\\rho }}}={\\frac {\\partial ^{2}F_{~\\alpha }^{m}}{\\partial X^{\\rho }\\partial X^{\\beta }}}\\implies {\\frac {\\partial F_{~\\mu }^{m}}{\\partial X^{\\rho }}}\\,_{(X)}\\Gamma _{\\alpha \\beta }^{\\mu }+F_{~\\mu }^{m}~{\\frac {\\partial }{\\partial X^{\\rho }}}[\\,_{(X)}\\Gamma _{\\alpha \\beta }^{\\mu }]={\\frac {\\partial F_{~\\mu }^{m}}{\\partial X^{\\beta }}}\\,_{(X)}\\Gamma _{\\alpha \\rho }^{\\mu }+F_{~\\mu }^{m}~{\\frac {\\partial }{\\partial X^{\\beta }}}[\\,_{(X)}\\Gamma _{\\alpha \\rho }^{\\mu }]", "29b6e122577d3441925b26874f745868": "A_{\\sigma }{\\hat {\\boldsymbol {\\sigma }}}+A_{\\tau }{\\hat {\\boldsymbol {\\tau }}}+A_{\\phi }{\\hat {\\mathbf {z} }}", "29b7342da0e216a9d99a222895d1620b": "Z_{K}\\,", "29b754ae9147c0635423b21816875c1b": "(\\beta _{1}^{\\top }X,\\ldots ,\\beta _{k}^{\\top }X)", "29b777239effd2bc18a36620d27c7252": "\\pi _{k+1,k}:J^{k+1}(\\pi )\\to J^{k}(\\pi )", "29b81101892c51a370a761e93b5026e2": "\\Sigma ^{x}={\\begin{pmatrix}\\sigma _{1}^{2}&{\\text{cov}}_{12}&{\\text{cov}}_{13}&\\cdots \\\\{\\text{cov}}_{12}&\\sigma _{2}^{2}&{\\text{cov}}_{23}&\\cdots \\\\{\\text{cov}}_{13}&{\\text{cov}}_{23}&\\sigma _{3}^{2}&\\cdots \\\\\\vdots &\\vdots &\\vdots &\\ddots \\\\\\end{pmatrix}}", "29b82d2bfe884b230ecba64782f336a9": "\\int _{0}^{\\infty }e^{-st}f(t)\\,dt,\\quad \\int _{-\\infty }^{0}e^{-st}f(t)\\,dt", "29b88c27df6aea000d29635d7f0192ce": "\\beta _{0}^{(1)}=\\beta _{0}^{(0)}(1-t_{0})+\\beta _{1}^{(0)}t_{0}=\\beta _{0}(1-t_{0})+\\beta _{1}t_{0}", "29b9c3b31d0a78aa123273f664b8710f": "l={\\sqrt {(d-r_{3})^{2}+(r_{2}-r_{1})^{2}}}.", "29ba2e66ff6d8038690f7fc9cbbfd052": "{\\binom {x}{k}}", "29ba32847ba5f357f4dc7e880316dc1e": "\\mathbf {P} (0)", "29ba48a400b99bc1af1a042ef2271a3e": "\\prod _{p{\\text{ prime}}}{\\frac {1}{1-p^{-s}}}={\\frac {1}{1-2^{-s}}}\\cdot {\\frac {1}{1-3^{-s}}}\\cdot {\\frac {1}{1-5^{-s}}}\\cdot {\\frac {1}{1-7^{-s}}}\\cdots {\\frac {1}{1-p^{-s}}}\\cdots ", "29ba6d08667038089a1d5b7759bdfe80": "{\\frac {\\partial }{\\partial t}}\\left(u_{e}+u_{m}\\right)+\\nabla \\cdot \\left(\\mathbf {S} _{e}+\\mathbf {S} _{m}\\right)=0,", "29ba837f6917c895e85d2ac0530a3712": "L={\\frac {\\lambda }{i}}", "29bad86b3d1e30ce154cfdf6edfbc7cd": "Q^{n+1}", "29bae56585ca839094d976ab6e20a8f8": "\\sigma _{\\mathrm {discr} }(T)=\\sigma (T)\\setminus \\sigma _{\\mathrm {ess} }(T).", "29bb6dbc797a1d4733c1a383bdc43ec0": "\\|f\\|_{H^{\\infty }}=\\sup _{z\\in \\mathbf {H} }|f(z)|.", "29bc15db3f59e793965861197514c7c6": "{\\frac {\\partial {\\bar {u_{i}}}}{\\partial x_{i}}}=0", "29bc1e76ad879b2010bf9e616feb8d62": "{\\vec {Y}}", "29bc1f2c10458af64198bfbfddb4db12": "{}^{4}i=i^{\\left({}^{3}i\\right)}", "29bc535dc12c70c70d2634b0469e51bd": "({\\overline {A}}\\vee {\\overline {B}}\\vee {\\overline {C}})\\wedge (A\\vee C)\\wedge (B\\vee C)", "29bccee4c2e0ccdeeebd9cbf0dc3efa6": "{\\boldsymbol {\\sigma }}\\approx 4C_{1}\\left({\\boldsymbol {\\varepsilon }}-{\\tfrac {1}{3}}\\mathrm {tr} ({\\boldsymbol {\\varepsilon }}){\\boldsymbol {\\mathit {1}}}\\right)+2D_{1}\\mathrm {tr} ({\\boldsymbol {\\varepsilon }}){\\boldsymbol {\\mathit {1}}}", "29bce09baa6903ac6347ae366bdbaf08": "|\\Im (z)|>2", "29bd1155882b2200f0e4c900bf0013e2": "\\kappa <\\lambda \\,", "29bd1c430e6310b139175821685db160": "{BC}^{2}+{AC}^{2}={AB}^{2}\\ .\\,\\!", "29bd1dff25998fe3eb500c626b782925": "X(y+d,t)-X(y,t)", "29bd2535fe70780a9b40325ea03cbc20": "\\mathrm {C^{\\beta }} ", "29bd68a263458b967201104f57cf62aa": "|D\\rangle ", "29bdda7a25b4b34bc63f2504fcdcd45f": "[n]_{q}={\\frac {1-q^{n}}{1-q}}.", "29bddacb40c90e334700868b889400aa": "C_{n}^{2}", "29be3503303f17e56d238129eacab0ba": "Z_{i}\\sim {\\mathcal {N}}(0,1)", "29becc720fb9ef8335a19e17843eb9ce": "\\sum _{\\Omega (n)=2}{\\frac {1}{n^{2}}}\\approx 0.1407604", "29bece9915c73278a45289e922cd4f1b": "x_{q}^{(q-1)/2}\\equiv 1{\\pmod {q}}", "29bed172e783845d4feb8f1d116d628b": "\\|\\varphi \\|=\\sup _{\\|x\\|\\leq 1}|\\varphi (x)|.", "29bf0599e94766343ee7d0fff43780d4": "r_{p}=(1-e)a", "29bf28029d7ea6bab8e8063369d72e5c": "\\Phi (x,t):={\\frac {1}{\\sqrt {4kt}}}\\exp \\left(-{\\frac {x^{2}}{4kt}}\\right)", "29bf432841549d61bd87ac91336fa451": "C^{i}{}_{jkl}", "29bf6831dbf3e375977ec61bb3130224": "\\coprod ", "29bfbac3406724739d5d5a41832ee372": "f(T)={\\frac {1}{2\\pi i}}\\int _{\\Gamma }\\left(\\sum _{n\\geq 0}{\\frac {T^{n}}{\\zeta ^{n+1-k}}}\\right)d\\zeta ", "29bfe6c6b36c1ce31ca3fbcfb6f7a021": "\\det {\\begin{pmatrix}u+kv\\\\w\\\\\\end{pmatrix}}=\\det {\\begin{pmatrix}u\\\\w\\\\\\end{pmatrix}}+k\\det {\\begin{pmatrix}v\\\\w\\\\\\end{pmatrix}}", "29c0c3443dde5e4b3d2f1c1de9f67ec3": "\\{Z_{T}\\mid {\\text{T a finite valued stopping time}}\\}", "29c0d7c30f806c4c3b52dc3418158680": "A\\approx 1.2824271291\\dots ", "29c103f223ec9867b9db77b4842cfd20": "[t]_{q}={\\frac {q^{t}-1}{q-1}}.", "29c11585248d2b9ba404b7356b81d7ac": "^{\\top }", "29c1590bcfe1c907116a45a9f499bdb4": "\\displaystyle {gZ=(AW+B)({\\overline {B}}W+{\\overline {A}})^{-1}.}", "29c15a6652271b30a3f23e05303dd6ac": "-V_{YY}^{-1}", "29c1db4ec652f87c2cbe138ff66bd9e3": "w\\in \\{0,1\\}^{*}", "29c27250401a3bd055667e64bc4fe4d3": "G(\\mathbf {k} ,z)", "29c30dda15ca3ad33ff32c1cde78d049": "H^{3}=\\{q\\in M:q(q^{*})=1\\}\\!", "29c3746d395b0d422cc0e02bc412523f": "\\Delta v/c=10^{-5}", "29c4690d7aefe8e74c25c0efd2926f8e": "e^{-i\\omega t}\\phi _{\\omega }(x)\\,", "29c4c8e3126ea02838b76dac205f8d77": "x+x^{3}/3!+...", "29c4e6a51a97c34fa1ec0719e0bfe3fe": "\\left(1/b\\right)\\ln\\{\\beta \\left[\\left(1/2\\right)^{-1/s}-1\\right]+1\\}", "29c4e7235bbbd18b6b9ea346959a18e7": "y=x^{2}.\\ ", "29c4fc430ed5674f0534ba379e026551": "K_{\\max }={\\underset {(i,j)\\in S}{\\max }}K_{ij}\\,", "29c537605144f52e39729ef506b75531": "{\\frac {W_{Sieve}}{W_{Total}}}", "29c5578e78c66cece596aa101d366b32": "\\ Pxx.", "29c5c9eea7260b1be7aac58b135097c2": "E={\\frac {Z}{2\\,{\\sqrt {n}}}}\\,={\\frac {4.4172}{2\\,{\\sqrt {12000}}}}\\,=0.0202", "29c605e88b9e17013e81fd0222cff374": "*\\exp(1)*x^{11}+{\\frac {150349}{6227020800}}*\\exp(1)*x^{13}+", "29c62cec19b0ffa0852baf4677d211d5": "\\sum _{i}u_{i}", "29c6553e3657ba1d462d5f9f83480167": "\\lambda (\\mathbf {x} ,\\mathbf {y} )", "29c662f78660c43b2d1ff32c3919e0ae": "f(aa)", "29c6777fae17c1fe92b9001c10b41a54": "s_{a}^{*}(t),", "29c6b5f8c89dc12061c9159c6fb941de": "\\scriptstyle \\mathbf {X} \\sim {\\mathcal {W}}_{p}({\\mathbf {V} },m)", "29c6f3841f966081742c048709bee263": "p(r,\\theta )=\\ln(I/I_{0})=-\\int \\mu (x,y)\\,ds", "29c6f3f470ffd452cdda2fa6b1efd83c": "\\scriptstyle \\log _{e}({\\frac {760}{101.325}})-12.4379\\log _{e}(T+273.15)-{\\frac {6340.514}{T+273.15}}+95.14704+1.412918\\times 10^{-05}(T+273.15)^{2}", "29c6f6209e92255f1d58bf3a7aa957af": "\\partial R(w)", "29c7075d62fc2d8ceeed999e838205dc": "{\\mathbf {P}}=\\epsilon _{0}\\chi _{e}{\\mathbf {E}}\\,", "29c71d1792b60186fa5f1ef207a52d98": "\\mathbb {F} ^{\\binom {l+d}{d}}\\rightarrow \\mathbb {F} ^{|\\mathbb {F} |^{l}}", "29c759b95a7a4d013fdf52a0062b0ef4": "-{\\frac {\\partial \\phi }{\\partial x}}", "29c75bae1175f1b21628102213b1b229": "\\ln \\left({\\frac {\\varepsilon _{k}}{\\varepsilon _{0}}}\\right)\\equiv \\ln \\left({\\frac {\\varepsilon _{0}'}{\\varepsilon _{0}}}\\right)=2(k-1+x)\\delta _{0}\\Omega _{0}.", "29c7bfcb43d2800dc72a6507a663efe6": "W(K)=W(k)\\oplus \\langle \\pi \\rangle \\cdot W(k)", "29c7ceffd70e0ddeea0d8208c7e7376a": "100+x", "29c8b7410bc311393d83d22792a3ba33": "a_{1},a_{2},a_{3},\\ldots ", "29c8cffcddbd71cce6f4912a2270a9cf": "P_{S/A}(n_{1},\\cdots n_{N}\\rightarrow m_{1},\\cdots m_{N})\\equiv {\\bigg |}\\langle m_{1}\\cdots m_{N};S/A\\,|\\,n_{1}\\cdots n_{N};S/A\\rangle {\\bigg |}^{2}", "29c8fecb3f6d12a471adaac1de2901a5": "\\prod _{i=n-k+2}^{n}m_{i}1.", "29e301a547559e8104024bc29b9704aa": "\\rho _{X,Y}={\\frac {E(XY)-E(X)E(Y)}{{\\sqrt {E(X^{2})-(E(X))^{2}}}~{\\sqrt {E(Y^{2})-(E(Y))^{2}}}}}.", "29e30e5fcafe92ead661b3a8c1718512": "(x-3)(x-1)^{3}(x^{3}+2x^{2}-3x-5)(x^{3}+2x^{2}-x-1)(x^{4}+x^{3}-7x^{2}-6x+7)(x^{4}+x^{3}-5x^{2}-4x+3).\\ ", "29e3814be957e0db0bafd72c9113ecb3": "(R", "29e3d59df5bd9b0b118c858976c8b292": "{\\dot {z}}(t)={\\dot {u}}(t)\\left\\{A-\\left[\\beta \\operatorname {sign} (z(t){\\dot {u}}(t))+\\gamma \\right]|z(t)|^{n}\\right\\}", "29e4096d282c57bc0556fab0b6b30c77": "r_{\\pm }:=M\\pm {\\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}", "29e4b109debcb015655b1f118536c7b9": "\\left({\\frac {\\pi }{a+2\\delta }}\\right)^{2}+\\left({\\frac {\\pi }{b+2\\delta }}\\right)^{2}+\\left({\\frac {\\pi }{c+2\\delta }}\\right)^{2}", "29e4d5355935120a7024e3beacc0d37e": "{\\bigl (}{\\begin{smallmatrix}\\\\~\\;2/3&4/3\\\\-1/3&7/3\\end{smallmatrix}}{\\bigr )}", "29e52023719b292d6b1616d9d6f4a311": "P^{1-\\gamma }T^{\\gamma }=C", "29e52ae7e29e7c0e78e6bc15861ff6cd": "\\mathbf {A} ={\\displaystyle \\bigotimes \\limits _{i=0}^{\\infty }}\\ A_{i},", "29e55ecd8c15b5c71f9df7d98343a7ab": "{\\frac {dy}{dx}}=\\tan \\varphi .", "29e5c1eaba7e88efad0bc3ee3529ff7d": "Q(x_{i},f(x_{i}))", "29e5ce9ea76409d81bdf3c1013f44ca6": "0\\leq \\alpha \\leq 1", "29e5f410bbec6dcedd52fa5398862888": "\\alpha =2-\\nu d\\,\\!", "29e64c4b30ff6900f33237a4d950a8ad": "pB_{p-1}\\equiv -1{\\pmod {p}}.", "29e6a86a081a3345883b7b36e2e9834d": "\\lim _{|z|\\to \\infty }f(z)=0", "29e6ac2362597ecf6518e6630a3d993d": "\\forall p((p\\land \\Diamond Kxp)\\Rightarrow Kxp)", "29e6ed858202b4a55808817f4d35383f": "f(x)=\\left\\{{\\begin{matrix}\\beta x&x\\in S_{0}\\\\x^{2}&x\\in S_{1}\\\\\\alpha x&x\\in S_{2}\\end{matrix}}\\right.", "29e740b43dad747a1610bfc1d290feb1": "\\sigma _{x'}=\\sigma _{y'}=\\sigma _{z'}", "29e79ab629d8b9458ccaaad5cc653317": "e^{[-a_{1}+a_{2}]}\\sum _{j=0}^{[n/2]}{\\frac {a_{1}^{n-2j}a_{2}^{j}}{(n-2j!)j!}}", "29e7a116098ff9d1ba835cdb66b5e45c": "\\lVert e\\rVert =\\lVert e^{*}\\rVert =e^{*}e=0.", "29e83a118576ae33c3193037d219223a": "(V,cl)", "29e843aff61b1b0b5420077eaf68c3e6": "\\|f\\|_{p,w}\\leq \\|f\\|_{p}", "29e9228cc13c4a6251ed718578e025ee": "p({\\vec {r}})", "29e94c844ca81ebcbde16976c1b58ed3": "Z^{T}QZ", "29e9588dd63153a7e7449d95415b0e2b": "\\rho (X)=\\mathbb {E} [-X]", "29e9dcb6982fac0cf29b881c8dc498a9": "a{}_{(n)}b\\,\\!", "29ea29c98ee95a8a0ffac04837f513a1": "\\cos(\\mathbf {X} )", "29ea3d1a2f25cfe972b4944f3498577d": "J(x_{1},y_{1})", "29ea80fa1357ae0fbf430b56279bace6": "\\nabla (FG)=\\nabla FG+{\\dot {\\nabla }}F{\\dot {G}}", "29ea887347d76cb0159ec2a543021615": "S(A_{n})", "29eaa4edcdafcb3b75e273fdc32fb4f4": "{\\hat {H}}={\\frac {{\\hat {p}}^{2}}{2m}}+V(x,t)\\,,\\quad {\\hat {p}}=-i\\hbar {\\frac {\\partial }{\\partial x}}", "29eabb735634a8ea6c082aad32092c9c": "g_{H}(v,w)=g(v,w),\\quad v,w\\in H", "29eabc8b6ffc6b3ee625a20621b8a461": "u_{b}^{\\mathrm {face} }(x,z)=-z~{\\cfrac {\\mathrm {d} w_{b}}{\\mathrm {d} x}}", "29eb38de1ea5fee2d669d4bb4eab405d": "d(i,j)", "29eb74c7f0eb73017d3cec973e83bb49": "i{\\frac {\\partial \\psi }{\\partial t}}+\\Delta \\psi +\\psi \\ln |\\psi |^{2}=0.", "29ec0d1a9c528185adb7f7b4ca06048f": "\\displaystyle {\\pi (R)f(t)=Q(t)^{-1}f(t).}", "29ec7059829975ba85edee0812b82d90": "\\gamma \\approx 1", "29ec7f83e0e2721cd9e0ede57a1d74b8": "\\zeta _{\\pm }=z\\pm {\\frac {L}{2}},", "29ec85a7369f34aa0041f1afb29c85c5": "Y=\\{x_{1}=\\cdots =x_{n}\\}", "29ec92bd04c53faa4008de00fea3d507": "A=L\\times W", "29ed926ca57bede248f975b0c6e19bfe": "b_{1}={\\frac {h_{1}\\rho _{c}}{\\rho _{m}-\\rho _{c}}}", "29edd219e48e5b86eaed08a48a57bf35": "\\frown \\ :H_{p}(X;R)\\times H^{q}(X;R)\\rightarrow H_{p-q}(X;R)", "29ee1c2488d3abd1fb84c6245b2b8347": "c=h/y_{\\mathrm {atm} }", "29ee530a4b63960bcf57dfa0ba882aa3": "Tt", "29ee537b48661644b126d39a44ddc264": "\\|u\\|_{L^{2r}}\\leq Cr\\|u\\|_{L^{r}}^{1/2}\\|\\nabla u\\|_{L^{2}}^{1/2}.", "29ee58e857450851cfdd2daee17f5120": "\\omega =y_{n}", "29ee8d099b18a26660fdaea5e220d87f": "r_{1}(\\theta _{1})+r_{2}(\\theta _{2})=a\\,", "29ef1da92b3f32be82a51653ab10fb3a": "\\left({\\frac {u}{v}}(p_{1}u'v-p_{2}uv')\\right)'=\\left(q_{2}-q_{1}\\right)u^{2}+\\left(p_{1}-p_{2}\\right)u'^{2}+p_{2}\\left(u'-v'{\\frac {u}{v}}\\right)^{2}.", "29ef2276ca4617121e4f8a31f19a9c37": "\\Omega ^{1}={\\mathbb {C}}.{\\rm {d}}t,\\quad ({\\rm {d}}t)f(t)=f(qt)({\\rm {d}}t),\\quad {\\rm {d}}f={f(qt)-f(t) \\over q(t-1)}\\,{\\rm {dt}}", "29ef6869c9df08aaf7e28c34657a31ca": "S^{3}(A_{n})=S(S(S(A_{n}))),", "29efa36295b617dcbe4d98deec8a0048": "d=gh+p=(5/3)(1)+2=11/3", "29efa716c94f7676c6cc56ea3df29588": "\\,0K_{\\alpha },\\,", "29f2347aadd8ded83428ae17017d77c0": "10^{10^{10^{1000}}}", "29f24a19117e7ef8f03fdbd7599953ab": "A={\\frac {n}{2}}(\\cot {\\frac {\\pi }{n}}+{\\sqrt {3}})a^{2}.", "29f26741d36e1ee12e6e7f748820cbc8": "r_{ij}={\\frac {x_{ij}}{\\sqrt {\\sum _{i=1}^{m}x_{ij}^{2}}}},i=1,2,...,m,j=1,2,...,n", "29f26cf9fc8aa164f8880b9a1c202a31": "a_{1},\\dots ,a_{n}", "29f2a960eeb3d9620740cda8349a656a": "{\\dot {\\sigma }}", "29f2b5ecf05fbd7c5099a693390a85f8": "f(r+tp^{k})\\equiv 0\\,{\\bmod {p^{k+1}}}\\,", "29f2c8c1632f8a5ac61637aeef1243f8": "n_{1}/n_{2}", "29f2f1d0c0c5ad8f1b185ce5375cfa81": "\\int _{a}^{b}\\psi (x){\\overline {K(x,y)}}\\,dx={\\overline {\\lambda }}\\psi (y).", "29f356e1c2b9207a9dd6515b5e7a3896": "\\rho \\left({\\frac {\\partial \\mathbf {v} }{\\partial t}}+\\mathbf {v} \\cdot \\nabla \\mathbf {v} \\right)=-\\nabla p+\\mu \\nabla ^{2}\\mathbf {v} +\\mathbf {f} +(\\mu +\\mu ^{v})\\nabla (\\nabla \\cdot \\mathbf {v} )", "29f3953a4b44af7d68d5aa6fa89cbd55": "P(d|q)={\\frac {P(q|d)P(d)}{P(q)}}", "29f3d17687737b2a44357e7738cf17bb": "\\scriptstyle \\sum _{j=1}^{s}(1/x_{j})+(1/(x_{1}\\cdots x_{s}))=1", "29f405a6ec408716f9ccafba985baeec": "K(a,b;m)=\\sum _{d\\mid \\gcd(a,b,m)}d\\cdot K\\left({\\frac {ab}{d^{2}}},1;{\\frac {m}{d}}\\right).", "29f42142e0855daebbcf74bf58faae83": "D\\left({\\frac {\\mathrm {D} u_{x}}{\\mathrm {D} t}}+u_{z}^{2}\\right)=-D{\\frac {\\partial p}{\\partial x}}+\\nabla ^{2}u_{x}", "29f46a5548103f58714aa1bcd07b896c": "\\scriptstyle {\\sqrt {m}}", "29f46f717e18759f92191dd657a050cd": "\\nabla _{X}:\\Gamma (E)\\to \\Gamma (E)", "29f487be7e9393c3c17a8d182febe245": "A[[t]]", "29f4e47bed6b7b7d90221f60a67493c6": "w_{j}^{k}=v^{i}{\\frac {\\partial ^{2}y^{k}}{\\partial x^{i}\\partial x^{j}}}+v_{j}^{i}{\\frac {\\partial y^{k}}{\\partial x^{i}}}.", "29f510d8efa68f06bc0f66f50d2243ec": "A,B_{1},\\dots ,B_{m},C_{1},\\dots ,C_{n}", "29f59b948eac0bfef780d13305f02639": "\\operatorname {X} (u,v)=\\langle \\operatorname {x} (u),\\operatorname {y} (u)\\cos(v),\\operatorname {y} (u)\\sin(v)\\rangle \\,", "29f5f916cffe93380adb55d45a611f48": "b={\\frac {1}{2}}\\,{\\frac {(V-Z_{0}^{*}I)}{\\sqrt {\\left|\\Re \\{Z_{0}\\}\\right|}}}\\,", "29f64c0cb2484b3afe1f4b13c6333a41": "0\\rightarrow \\varprojlim A_{i}\\rightarrow \\varprojlim B_{i}\\rightarrow \\varprojlim C_{i}\\rightarrow \\varprojlim {}^{1}A_{i}", "29f6697d8f34b3077722bdaf8ef13edf": "\\Sigma _{1}^{0,Y}", "29f6936b753d2f1dbef865e00931c202": "{\\text{size of effect}}=\\left|{\\frac {\\text{second value − first value}}{\\text{first value}}}\\right|", "29f6a421b93ebad35c234a806eec931c": "\\Theta (G)=\\sup _{k}{\\sqrt[{k}]{\\alpha (G^{k})}}=\\lim _{k\\rightarrow \\infty }{\\sqrt[{k}]{\\alpha (G^{k})}},", "29f756aaa65d9704f53667fead1686f0": "B_{n}(x)=-{\\frac {n!}{2^{n-1}\\pi ^{n}}}\\sum _{k=1}^{\\infty }{\\frac {1}{k^{n}}}\\cos \\left(2\\pi kx-{\\frac {\\pi n}{2}}\\right),01", "29fdafb9742bde3e49fbb1f6a287120f": "v_{S}", "29fdcea51732a605db59c3a3a642066a": "2x=-4", "29fe3dc612102ac9b5b745d6326fb757": "n(\\omega )", "29fe6ceff060711e7b783b5640996757": "P_{i}=E_{K}(C_{i-1})\\oplus C_{i}", "29fe9336063076aff6b3e12571cdc281": "(x+y)^{n+1}=x(x+y)^{n}+y(x+y)^{n},\\,", "29fea4557dabf715e003f641b49f8800": "\\forall x\\in \\mathbb {R} :\\;g(x)=f(2^{l-k}x)", "29fedeef1251f2ee6e7443e71a2b37c1": "\\int _{0}^{\\frac {\\pi }{2}}\\sin ^{n-2}(x)\\cos ^{2}(x)\\,dx