harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
m_{el}^{2}\;=\;-ie^{2}\frac{1}{\beta}\,\mathrm{tr}(I)\,\sum_{n=-\infty}^{+\infty}\int\frac{d^{2}p}{(2\pi)^{2}}\,\frac{\partial}{\partialp_{j}}\frac{p_{j}}{{\mathbfp}^{2}+{\tilde{\omega}_{n}}^{2}}
I_{\pm}=\frac{\pi}{\vert\lambda\vert^{2}\verta\vert^{2}}-\frac{\pi(1+\vert\beta\vert^{2}\verta\vert^{2})}{\vert\lambda\vert^{2}\verta\vert^{2}(1+\vert\beta\vert^{2}\verta\vert^{2}+4\vert\lambda\vert^{2}\verta\vert^{7/2})},
(\kappa_{4})^{4}U_{OM}=\frac{3|h|^{2}}{4dJ^{2}}\Bigg\{\beta\Big[e^{-J\betax}-e^{-J\beta(1-x)}\Big]^{2}+\frac{2}{3}\frac{J}{V}\beta^{2}\Big[e^{-J\betax}x+e^{-J\beta(1-x)}(1-x)\Big]^{2}\Bigg\}\;.
\frac{1}{2}\bigg(\bigg|\frac{\alpha_{{k}}}{\beta_{{k}}}\bigg|-\bigg|\frac{\beta_{{k}}}{\alpha_{{k}}}\bigg|\bigg)=\frac{\Omega_{{k}}}{|\Lambda_{{k}}|},\quad|\alpha_{{k}}|^{2}+|\beta_{{k}}|^{2}=1,
\sqrt{\nabla_{\mu}l\nabla^{\mu}l}\epsilonf_{1}+\epsilonf_{1}\sqrt{\nabla_{\mu}l\nabla^{\mu}l}+\epsilon^{2}f_{1}^{2}+\sqrt{\nabla_{\mu}l\nabla^{\mu}l}\epsilon^{2}f_{2}+\epsilon^{2}f_{2}\sqrt{\nabla_{\mu}l\nabla^{\mu}l}+\dots
\frac{\partialL_{1}}{\partial\mu}=\frac{1}{2}\left(\sum_{n}\frac{\partial\beta_{n}}{\partial\mu}z_{n}^{2}+\sum_{B}\frac{\partial\beta_{B}}{\partial\mu}z_{B}^{2}\right)=k_{1}[(1-\mu)t-ib]^{\frac{1}{1-\mu}}.
\left(\begin{array}{ccccccc}{\hat{n}_{4}}&{\hat{n}_{5}}&{\hat{n}_{6}}&{\hat{n}_{7}}&{\hat{n}_{8}}&{\hat{n}_{9}}&{\hat{n}_{10}}\\{1}&{0}&{0}&{0}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}&{1}\\\end{array}\right),
F_{3}(\alpha,\alpha^{\prime},\beta,\beta^{\prime};\gamma|x,y)=\sum_{j,k=0}^{\infty}\frac{x^{j}y^{k}}{j!k!}\frac{(\alpha|j)(\alpha^{\prime}|k)(\beta|j)(\beta^{\prime}|k)}{(\gamma|j+k)}
\{{\calM}{}_{,}^{\otimes}v(y)\}=\Bigl(-{\calM}T_{\rho}\;{\calH}_{\rho\sigma}(0,y)\;+\;T_{\rho}{\calM}\;{\calH}_{\rho\sigma}(\ell,y)\Bigr)\otimesT_{\sigma}v(y)\quad.
A(r)={\frac{3}{2}}(r^{2}+l^{2})\,,\qquadU(r)={\frac{3r^{4}+8l^{2}r^{2}+6l^{4}}{6(r^{2}+l^{2})^{2}}}\,,\qquadC(r)=\frac14U\,r^{2}\,.
\left(\begin{array}{ccc}{2x_{1}^{2}-1}&{2x_{1}x_{2}}&{2x_{1}x_{3}}\\{2x_{1}x_{2}}&{2x_{2}^{2}-1}&{2x_{2}x_{3}}\\{2x_{1}x_{3}}&{2x_{2}x_{3}}&{2x_{3}^{2}-1}\\\end{array}\right)
S_{0}=\intd^{4}x\left\{\intd^{2}\theta\,\frac{1}{4}W^{\alpha}W_{\alpha}+\intd^{2}{\overline{\theta}}\,\frac{1}{4}{\overline{W}}_{\dot{\alpha}}{\overline{W}}^{\dot{\alpha}}+\intd^{4}\theta\,m^{2}V^{2}\right\}\,.
Q_{\perp}^{-}=\left(\frac{ig2^{-1/4}\sqrt{L}}{\pi}\right)\sum_{n,n_{\perp}}\frac{n_{\perp}}{\sqrt{n}}\left(A^{\dagger}(n,n_{\perp})B(n,n_{\perp})-B^{\dagger}(n,n_{\perp})A(n,n_{\perp})\right)\,.
L=\frac{H}{2}\epsilon_{i_{1}j_{1}}Q_{1}^{i_{1}}D_{t}Q_{1}^{j_{1}}+\frac{H}{2}\epsilon_{{\tilde{i}}_{1}{\tilde{j}}_{1}}Q_{1}^{{\tilde{i}}_{1}}D_{t}Q_{1}^{{\tilde{j}}_{1}}+\frac{H}{2}\epsilon_{i_{2}j_{2}}Q_{2}^{i_{2}}D_{t}Q_{2}^{j_{2}}+\frac{H}{2}\epsilon_{{\tilde{i}}_{2}{\tilde{j}}_{2}}Q_{2}^{{\tilde{i}}_{2}}D_{t}Q_{2}^{{\tilde{j}}_{2}}
(m-\ell)(1-12\epsilonm^{2})=\frac{\nu}{\alpha}\frac{PP^{\prime}}{{\calL}^{2}}-\biggl(1+\left.\frac{\nu}{\alpha}\frac{P^{\prime}}{{\calL}}\right|_{0}\biggr)\frac{1}{M^{4}{\calL}}.
S_{f}~=~\intd^{4}x~h\psi_{f}^{\dagger}\left[\left(\frac{1}{2}mv^{\mu}v_{\mu}+V-\frac{1}{2m}\partial^{\mu}\partial_{\mu}-\mu\right)-i\left(\frac{1}{2}\partial^{\mu}v_{\mu}+v^{\mu}\partial_{\mu}\right)\right]\psi_{f}.
\left\{-\sum_{i=1}^{3}\frac{\partial^{2}}{\partialx_{i}^{2}}+\sum_{\stackrel{i,j=1}{i\neqj}}^{3}\left({\frac{\omega^{2}}{8}}\left(x_{i}-x_{j}\right)^{2}+{\frac{2\alpha}{\left(x_{i}-x_{j}\right)^{2}}}+\Omega\left(\frac{x_{i}-x_{j}}{\sqrt{2}}\right)\right)\right\}\psi=E\psi~,
T=T_{\alpha}^{~\beta}\delta^{\alpha}\otimes\delta_{\beta}=T_{i}^{~j}d^{i}\otimes\delta_{i}+T_{i}^{~a}d^{i}\otimes\partial_{a}+T_{b}^{~j}\delta^{b}\otimes\delta_{j}+T_{a}^{~b}\delta^{a}\otimes\partial_{b}.
\frac{d^{2}\overline{{\phi}}(x)}{dx^{2}}=V^{\prime}(\overline{{\phi}})\quad\longrightarrow\quad\frac{d}{dx}\left[\frac{1}{2}\left(\frac{d\overline{{\phi}}}{dx}\right)^{2}\right]=\frac{d}{dx}V(\overline{{\phi}}).
{\calM}_{T}\equiv{\frac{1}{T_{2}}}\left(\begin{matrix}{1}&{T_{1}}\\{T_{1}}&{|T|^{2}}\\\end{matrix}\right),\\\\\{\calM}_{U}\equiv{\frac{1}{U_{2}}}\left(\begin{matrix}{1}&{U_{1}}\\{U_{1}}&{|U|^{2}}\\\end{matrix}\right).
{\calS}_{0}=\int\frac{d\omegadk}{(2\pi)^{2}}d^{2}z\left[\Psi_{L}^{\dagger}\left(\omega-k\sigma^{3}+{\calD}\right)\Psi_{L}+i\Psi_{R}^{\dagger}\left(\omega+k\sigma^{3}-{\calD}\right)\Psi_{R}\right],
\begin{array}{l}{[{\bf1}_{2}\oplus{\bf\overline{{6}}}_{3/2}\oplus{\bf\overline{{15}}}_{1}\oplus{\bf20}_{1/2}\oplus{\bf15}_{0}\oplus{\bf6}_{-1/2}\oplus{\bf1}_{-1}]}\\{{}~~~~\oplus~[{\bf1}_{1}\oplus{\bf\overline{{6}}}_{1/2}\oplus{\bf\overline{{15}}}_{0}\oplus{\bf20}_{-1/2}\oplus{\bf15}_{-1}\oplus{\bf6}_{-3/2}\oplus{\bf1}_{-2}],}\\\end{array}
\begin{array}{l}{D_{a\alpha}\hat{w}_{\mu\nu}(z)=2(\hat{\bar{\rho}}_{a}(z)\gamma_{[\mu}\gamma_{\nu]})_{\alpha}\,,}\\{{}}\\{D_{a\alpha}\hat{\lambda}(z)=2\hat{\bar{\rho}}_{a\alpha}(z)\,,}\\{{}}\\{D_{a\alpha}\hat{t}{}_{b}{}^{c}(z)=2(\delta_{ab}\delta^{cd}-\delta_{a}{}^{c}\delta_{b}{}^{d})\hat{\bar{\rho}}_{d\alpha}\,.}\\\end{array}
\sigma^{l}=\pi\Omega_{n+1}\left[{\frac{\Gamma({\frac{n}{2}}+1)}{\Gamma({\frac{n}{2}}+l+1)\Gamma({\frac{n+2l}{2b}})}}\right]^{2}{\frac{2l+n}{n}}{\binom{l+n-1}{l}}S\left({\frac{T}{b}}\right)^{{\frac{n+2l}{b}}-1}\left({\frac{\omega}{2}}\right)^{{\frac{n+2l}{b}}+2l-1}.
F_{n}(x,q)=\frac{1}{2}\frac{\partial^{2}P_{n-1}(x,q)}{\partialx^{2}}-\frac{1}{2}\sum_{n^{\prime}=1}^{n-2}\frac{\partialP_{n^{\prime}}(x,q)}{\partialx}\frac{\partialP_{n-n^{\prime}-1}(x,q)}{\partialx}
\zeta(s,x|L_{b})={\frac{a^{2s-N}}{2^{N-1}\pi^{\fracN2}\Gamma\left(\fracN2\right)}}\sum_{n=0}^{\fracN2-1}c_{2n+1}^{N}\left({\frac{1}{2}}n!(a^{2}b)^{n+1-s}{\frac{\Gamma(s-n-1)}{\Gamma(s)}}-2H_{n}(s;a\sqrtb)\right).
Q_{+}^{I}=\frac{\partial}{\partial\theta_{I}^{+}}-i\eta^{IJ}\theta_{J}^{+}\frac{\partial}{\partial\sigma^{+}},\\\\Q_{-}^{I^{\prime}}=\frac{\partial}{\partial\theta_{I^{\prime}}^{-}}-i\eta^{I^{\prime}J^{\prime}}\theta_{J^{\prime}}^{-}\frac{\partial}{\partial\sigma^{+}},
M_{e}l=\frac{1}{3\sqrt{6}}\left(\sqrt{(1+a^{2}l^{2})^{2}+12a^{2}l^{2}}+2(1+a^{2}l^{2})\right)\left(\sqrt{(1+a^{2}l^{2})^{2}+12a^{2}l^{2}}-(1+a^{2}l^{2})\right)^{\frac{1}{2}}.\nonumber
\frac{1}{U^{3}}\partial_{U}\left(\frac{U^{5}f(U)}{R^{2}}\partial_{U}G(U,U^{\prime},x_{i})\right)+\frac{R^{2}}{U^{2}}\partial_{ii}G(U,U^{\prime},x_{i})=\frac{\sqrt{f(U)}}{R^{3}U^{2}}\delta(U-U^{\prime})\delta^{(3)}(x_{i})
\begin{array}{c}{q^{*}(t)\varphi_{i+}^{\alpha}(t)=\sqrt{E_{i}(t)}\varphi_{i-}^{\alpha}(t)}\\{q(t)\varphi_{i-}^{\alpha}(t)=\sqrt{E_{i}(t)}\varphi_{i+}^{\alpha}(t)\.}\\\end{array}
\left\{\omega-m\left(\gamma_{1}^{0}+\gamma_{2}^{0}+\gamma_{3}^{0}\right)\right\}\eta_{r}C_{22}\left|\Phi_{r}^{a}\right>=\frac{1}{2}\left\{C_{22}\hat{O_{1}}+C_{21}\hat{O_{2}}C_{12}\right\}\eta_{r}C_{22}\left|\Phi_{r}^{a}\right>
\begin{array}{l}{(\beta_{2}/\beta)b_{n-1}-\alphab_{n}-(s+n)a_{n}+a_{n-1}-(i\nu-2)a_{n}=0\mathrm{~,}}\\{(\beta_{1}/\beta)a_{n-1}-\alphaa_{n}-(s+n)b_{n}+b_{n-1}+(i\nu+2)b_{n}=0\mathrm{~.}}\\\end{array}\tag{A6}
a^{\dagger}(1)a^{\dagger}(2)|0\rangle=\langle\phi_{s}|a_{1}^{\dagger}a_{2}^{\dagger}|0\rangle|\phi_{s}\rangle+\langle\phi_{a}|a_{1}^{\dagger}a_{2}^{\dagger}|0\rangle|\phi_{a}\rangle.
S_{dilaton}=\intd^{5}X\sqrt{-G}\biggl(-\frac{1}{4}e^{-\lambda\pi/2\sqrt{3M_{Pl}^{3}}}G^{MN}G^{PQ}F_{MP}F_{NQ}+\frac{1}{2}(\partial\phi)^{2}+\frac{1}{2}(\partial\pi)^{2}-V(\phi,\pi)\biggr),
I_{\mathrm{bulk}}+I_{\mathrm{surf}}=\frac{1}{16\piG}\int_{{\calM}}d^{n+1}x\sqrt{-g}\left(R-\frac{n(n-1)}{l^{2}}\right)+\frac{1}{8\piG}\int_{\partial{\calM}}d^{n}x\sqrt{\gamma}K\,.
\begin{array}{lll}{\left(E_{4s}-2Q-2Q^{-1}\right)A(k_{1},k_{2})}&{=}&{A(k_{1}-1,k_{2})+A(k_{1}+1,k_{2})}\\{}&{}&{}\\{}&{+}&{A(k_{1},k_{2}-1)+A(k_{1},k_{2}+1)}\\\end{array}
Z_{\;\;\alpha_{1}}^{\alpha_{0}}\frac{\deltaZ_{\;\;\beta_{2k+1}}^{\alpha_{2k}}}{\delta\Phi^{\alpha_{0}}}\approxC_{\;\;\alpha_{1}\beta_{2k+1}}^{\gamma_{2k+1}}Z_{\;\;\gamma_{2k+1}}^{\alpha_{2k}}+C_{\;\;\alpha_{1}\beta_{2k+1}}^{\gamma_{2k-1}}A_{\gamma_{2k-1}}^{\;\;\alpha_{2k}},\;k=1,\cdots,b,
\frac{\partialA_{ij}}{\partialq^{k}}-\frac{\partialA_{kj}}{\partialq^{i}}=\frac12\frac{\partial}{\partial\dot{q}^{j}}\left(\frac{\partialB_{i}}{\partial\dot{q}^{k}}-\frac{\partialB_{k}}{\partial\dot{q}^{i}}\right).
\left[\Omega_{1},\Omega_{1}\right]^{*}=\frac{1}{3}f_{\;a\left[d\right.}^{e}f_{\;\left.bc\right]}^{a}\intd^{3}x\left(\epsilon^{0ijk}A_{i}^{d}A_{j}^{b}A_{k}^{c}\eta_{e}\right).
{\calH}_{can}=\Pi^{-}\partial_{+}A_{-}+\Pi^{i}\partial_{+}A_{i}-{\calL}_{Weyl}=\frac12\Pi^{2}+\frac12(\partial_{i}A_{j})^{2}+\Pi\partial_{i}A_{i}-A_{i}J^{i}-A_{-}J^{-}
e^{ik_{1}\cdotX}e^{ik_{2}\cdotX}\sim\left(\tau-\tau^{\prime}\right)^{2\alpha^{\prime}G^{\mu\nu}k_{1\mu}\cdotk_{2\nu}}\times\left[e^{-\frac{i}{2}\Theta^{\mu\nu}k_{1\mu}\cdotk_{2\nu}}\right]\timese^{i\left(k_{1}+k_{2}\right)\cdotX}+\cdots,
\left(1-\bar{P}{\frac{1}{\bar{P}^{\dagger}\bar{P}}}\bar{P}^{\dagger}\right)_{\mu\nu,\lambda\rho}(x,x^{\prime})=\bar{h}_{\mu\nu}^{m}(x)\bar{M}_{mn}^{-1}\bar{h}_{\lambda\rho}^{n}(x^{\prime})
G_{5}^{5}=\frac{1}{4b^{2}}\frac{(V^{\prime})^{2}}{V^{2}}+\frac{1}{4a^{2}}\frac{\dot{V}^{2}}{V^{2}}-\frac{\alpha^{2}}{6V^{2}}=\frac{\alpha^{2}}{3C^{6}}t^{-2\delta}+\frac{\delta^{2}}{4a_{0}^{2}C}t^{\delta-3}.
{\tilde{\mathcal{F}}_{Q}}={\frac{1}{4\pi\alpha^{\prime}}}\int{d^{2}}\tilde{s}(\xi){d^{2}}\tilde{s}(\xi^{\prime}){\tilde{J}_{Q}}(\xi){\tilde{G}_{N}}(\xi,\xi^{\prime}){\tilde{J}_{Q}}(\xi^{\prime}),
U_{\mathrm{eff}}=-\frac{\alpha}{4\pi^{2}}\frac{1}{\sqrt{x(1-x)x^{\prime}(1-x^{\prime})}}\left(\frac{<j^{\mu}J_{\mu}>_{ex}}{Q^{2}}+\frac{<c^{\mu}C_{\mu}>_{an}}{M^{2}}\right)\,.
L_{F}=\bar{\psi}i\gamma^{\mu}(\partial_{\mu}-ieA_{\mu}-iB_{\mu})\psi-{\frac{1}{2g}}\midB_{\mu}\mid^{2}-m\bar{\psi}\psi-{\frac{pe^{2}}{4}}\midA_{\mu\nu}\mid^{2}+{\frac{qe^{2}}{2}}\epsilon_{\mu\nu\lambda}A^{\mu}A^{\nu\lambda}+J_{\mu}A^{\mu}.
u_{{m}}^{{{f}}}={\frac{1}{32}}\left[v_{{A}}^{{\hat{a}}}\left(C\Gamma_{{m}}\right)v_{{\hat{b}}{A}}+v^{{\hat{a}}{\dot{A}}}\left(C\Gamma_{{m}}\right)v_{{\hat{b}}{\dot{A}}}\right](\gamma^{f})_{{\hat{a}}}^{{{\hat{b}}}},
\breve{\Phi}(\omega;\lambda)\simeq{\frac{2\omega^{D-2}}{(4\pi)^{(D-1)/2}}}\sum_{n=0}^{\infty}\left[{\frac{\breve{a}_{n}(\lambda)}{\Gamma\left({\frac{D-1}{2}}-n\right)}}\omega^{-2n}+{\frac{\breve{b}_{n}(\lambda)}{\Gamma\left({\frac{D-2}{2}}-n\right)}}\omega^{-(2n+1)}\right]~~~.
\int_{-m/2}^{m/2}{g_{+}^{2}\,ds}=\int_{-m/2}^{m/2}{\left(s^{2}-\frac{m^{2}}{4}\right)^{k-1}e^{2rs}\,ds}\approxe^{mr}\int_{-\infty}^{0}e^{2ru}u^{k-1}(m+u)^{k-1}\,du
n_{0}=\int\frac{d^{2}p}{(2\pi)^{2}}\left(\frac{1}{e^{\beta(\omega-\mu)}-1}-\frac{1}{e^{\beta(\omega+\mu)}-1}\right)=\frac{T^{2}}{2\pi}[g_{2}(x,-r)-g_{2}(x,r)]\;,
\left(\begin{array}{cc}{\{\Phi_{0},\Phi_{0}\}}&{\{\Phi_{0},\Phi_{1}\}}\\{\{\Phi_{1},\Phi_{0}\}}&{\{\Phi_{1},\Phi_{1}\}}\\\end{array}\right)={\calD}\delta(z-z^{\prime})={\calD}\delta(x-x^{\prime})\delta(\theta-\theta^{\prime})
{\calG}^{a}=\int_{\calM}d^{4}x\left(\frac{\delta}{\deltac^{a}}+\frac{1}{2}f^{abc}\chi_{\mu\nu}^{b}\frac{\delta}{\deltaB_{\mu\nu}^{c}}+f^{abc}\bar{\varphi}^{b}\frac{\delta}{\delta\eta^{c}}+f^{abc}\bar{c}^{b}\frac{\delta}{\deltab^{c}}\right)
\begin{array}{rcl}{\pm\,\alpha_{kl}\!\!}&{=}&{\!\!\pm\,(e_{k}-e_{l})\quad(1\leqk<l\leq6)}\\{\pm\,\alpha_{78}\!\!}&{=}&{\!\!\pm\,(e_{7}-e_{8})}\\{\pm\,\beta_{kl}\!\!}&{=}&{\!\!\pm\,(e_{k}+e_{l})\quad(1\leqk<l\leq6)}\\{\pm\,\gamma_{\pm,\dots,\pm}\!\!}&{=}&{\!\!\pm\,{\frac{1}{2}}\left(e_{8}-e_{7}+{\displaystyle{\sum_{k=1}^{6}}}\,(-1)^{s(k)}\,e_{k}\right)}\\\end{array}~,
\left(\Psi,\Phi\right)=\int_{-\infty}^{\infty}dx\;\int_{-\infty}^{\infty}dy\;\Psi^{\ast}\left(x,y\right)\Phi\left(x,y\right)=\left(\widetilde{\Psi},\widetilde{\Phi}\right)=\int_{-\infty}^{\infty}d\xi\int_{-\infty}^{\infty}d\eta\;\widetilde{\Psi}^{\ast}\left(\xi,\eta\right)\widetilde{\Phi}\left(\xi,\eta\right)\;.
\mathrm{\Pi_{\mu\nu}^{af}(p)=\frac1{3!}\int\!\!\int\frac{d^{4}q~d^{4}k}{(2\pi)^{8}}\,V_{\mu\lambda\sigma\rho}^{abcd}\,G_{\rho\alpha}^{dg}(q)\,G_{\sigma\beta}^{ch}(p-k-q)\,G_{\lambda\xi}^{be}(k)\,V_{\alpha\beta\xi\nu}^{ghef}~,}
\mathcal{Q}_{\mathrm{BRST}}\star\Psi=0,\qquad\overline{{\Psi}}\star\mathcal{Q}_{\mathrm{BRST}}=0,\qquad\mathcal{Q}_{\mathrm{BRST}}\star\mathcal{Q}_{\mathrm{BRST}}=\left(\Psi\star\overline{{\Psi}}\right)_{-2},
{}_{2}F_{1}(-\ell+2ikr_{0}\;,\;-\ell\;;\;-2\ell\;;\;z)={\textstyle{\frac{\Gamma(\ell+1)\Gamma(\ell+1+2ikr_{0})}{\Gamma(2\ell+1)\Gamma(1+2ikr_{0})}}}\;{}_{2}F_{1}(-\ell+2ikr_{0}\;,\;-\ell\;;\;1+2ikr_{0}\;;\;1-z)
\langlen_{B}\rangle=\left(\frac{\mu_{1}}{\mu}\right)^{2}\langlen_{PV}\rangle=\sum_{n,n_{1}}\frac{Zn}{(2n+2n_{1}+1)!n!n_{1}!}\frac{(g/\mu)^{2n}(g/\mu_{1})^{2n_{1}}}{(16\pi^{2})^{n+n_{1}}}\,.
A=\frac{1}{16\pi}\int_{0}^{\beta}dt\int_{S_{\perp}}d^{2}x_{\perp}\int_{x_{1}}^{x_{2}}dx\sqrt{f}R=\frac{\betaS_{\perp}}{16\pi}\left[\frac{f^{\prime}}{\sqrt{f}}\right]_{x_{1}}^{x_{2}}
\bar{H}_{\mu\nu\lambda}^{\prime}=\frac{1}{2}e^{4\phi_{5}/3}\sqrt{\hat{g}_{5}}\epsilon_{\mu\nu\lambda\alpha\beta}\hat{X}^{\alpha\beta}-\Bigl(V_{\mu}H_{\nu\lambda}^{\prime}+cyclic~permutations\Bigr)
\begin{array}{l}{S=\displaystyle\intd\tau\left\{\frac{1}{2e}\Pi^{\mu}\Pi_{\mu}+n^{\mu}\dot{z}^{\mu}-\phi(n^{2}+1)\right\},}\\{\qquad\Pi^{\mu}\equiv\dot{x}^{\mu}-i(\bar{\theta}\Gamma^{\mu\nu}\dot{\theta})n_{\nu}-\xin^{\mu},}\\\end{array}
U^{(tr)}=e^{24\pi^{2}f_{G}(S,T)/b_{0}-(b+2c)/b_{0}}k^{-b/b_{0}}\left(\frac{c}{32\pi^{2}N_{f}}\right)^{3c/b_{0}}\prod_{i}M_{i}(T)^{-3c/N_{f}b_{0}}
ds^{2}=(H_{1}H_{2}...H_{n})^{-4t/r}\eta_{\mu\nu}dy^{\mu}dy^{\nu}+(H_{1}H_{2}...H_{n})^{4u/r)}[\sum_{i=1}^{n}H_{i}^{-2/r}dz_{i}^{m_{i}}dz_{i}^{m_{i}}+dx^{\gamma}dx^{\gamma}],
\partial_{m}\partial_{n}\gamma^{ij}=-\partial_{m}\Gamma_{nj}^{i}-\partial_{m}\Gamma_{ni}^{j}+\Gamma_{im}^{h}\Gamma_{hn}^{j}+\Gamma_{jm}^{h}\Gamma_{hn}^{i}+\Gamma_{mh}^{i}\Gamma_{nh}^{j}+\Gamma_{mh}^{j}\Gamma_{nh}^{i}
\int_{1}^{\infty}\frac{dx}{\left[(x+s)^{2}+a^{2}\right]^{z}}=\frac{(1+s)^{1-2z}}{2z-1}\,F\left(z,z-\frac{1}{2};z+\frac{1}{2};-\left(\frac{a}{1+s}\right)^{2}\right).
L_{0}(x_{I}(\tau))=\overline{{\psi}}_{I}(\tau)i\overleftrightarrow{\partial_{\tau}}\gamma_{\mu}\psi_{I}(\tau)\frac{x_{\tauI}^{\mu}(\tau)}{2\sqrt{-x_{\tauI}^{2}}}-\sqrt{-x_{\tauI}^{2}(\tau)}\,\overline{{\psi}}_{I}(\tau)\,m\,\psi_{I}(\tau)
V_{GS}(T^{m},\bar{T}^{m})={\frac{2(h+\bar{h})-(T^{m}+\bar{T}^{m})(\partial_{T^{m}}h+\partial_{\bar{T}^{m}}\bar{h})}{(T^{m}+\bar{T}^{m})\eta_{mn}(T^{n}+\bar{T}^{n})}}
Q(x,y)=-{\frac{(x+y)^{3}}{2}}\partial_{y}H=A+B\left[\mathrm{ln}\left({\frac{(x+y)^{2}}{\rho_{0}^{2}y}}\right)+{\frac{x^{2}+4xy-3y^{2}}{2y^{2}}}\right]
q_{2}\,=\sum_{n=1}^{N}\frac{-\sqrt{2m}\left(\alpha_{n_{0}}^{\prime}\right)^{*}\,e^{\eta_{n}^{*}}}{1\,+\,e^{R_{n}\,+\,\eta_{n}\,+\,\eta_{n}^{*}}},\tag{3.7}
(1+\frac{K}{2f_{5}})dx^{+}dx^{-}+\frac{R_{s}^{2}}{R^{2}}(h+\frac{K}{4f_{5}})dx^{-}dx^{-}+\frac{R^{2}}{R_{s}^{2}}\frac{K}{4f_{5}}dx^{+}dx^{+}
{\frac{i}{\hbar}}\overline{C}^{a}exp{\frac{i}{\hbar}}\Big(S_{_\Sigma}+\hbarM_{1}\,+\,J^{A}\,\phi_{A}\Big)\,\,=\,\,{\frac{\partial}{\partialC^{a}}}\,exp{\frac{i}{\hbar}}\Big(S_{_\Sigma}+\hbarM_{1}\,+\,J^{A}\,\phi_{A}\Big)
<\chi(Z_{1}){\bar{\chi}}(Z_{2})>={\frac{1}{Z_{12}}}+{\frac{\theta_{12}\bar{\theta}_{12}}{2Z_{12}^{2}}}\;,\;\;<\xi(Z_{1}){\bar{\xi}}(Z_{2})>={\frac{1}{Z_{12}}}+{\frac{\theta_{12}\bar{\theta}_{12}}{2Z_{12}^{2}}}
gIg^{-1}=\left(\begin{matrix}{I_{xx}}&{I_{x\check{y}}}&{I_{x\check{x}}}&{I_{xy}}\\{I_{\check{y}x}}&{I_{\check{y}\check{y}}}&{I_{\check{y}\check{x}}}&{I_{\check{y}y}}\\{I_{\check{x}x}}&{I_{\check{x}\check{y}}}&{I_{\check{x}\check{x}}}&{I_{\check{x}y}}\\{I_{yx}}&{I_{y\check{y}}}&{I_{y\check{x}}}&{I_{yy}}\\\end{matrix}\right)
\sum_{l=0}^{\infty}(l+1/2)a_{l}^{n}=\int_{-1}^{1}\prod_{j=1}^{n-1}dx_{j}\big(f(x_{1})\underbrace{\widehat{f}(x_{1},x_{2})\cdot...\widehat{f}(x_{n-2},x_{n-1})}_{(n-2)terms}f(x_{n-1})\big)
G(z,z^{\prime},t)=\alpha\sum_{n}\frac{e^{in(\phi-\phi^{\prime})}}{2\pi}\int_{o}^{\infty}\frac{r^{(\alpha-1)}}{\alpha}J_{\frac{|n|}{\alpha}}(\frac{k}{\alpha}r^{\alpha})(k^{2})^{-ict/l_{P}}kdk\frac{{r^{\prime}}^{(\alpha-1)}}{\alpha}J_{\frac{|n|}{\alpha}}(\frac{k}{\alpha}{r^{\prime}}^{\alpha}).
T_{4}=\\sum_{i=1}^{N}\left[a_{i}{\left(\partial_{z}^{2}\phi_{i}\right)}^{2}+\b_{i}{\left(\partial_{z}\phi_{i}\right)}^{4}\right]+\sum_{i<j}c_{ij}{\left(\partial_{z}\phi_{i}\right)}^{2}{\left(\partial_{z}\phi_{j}\right)}^{2}\.
\frac{\delta\Sigma}{\delta\bar{c}^{a}}-\partial_{\mu}(\sqrt{g}g^{\mu\nu}\frac{\delta\Sigma}{\delta\Omega^{a\nu}})=-\partial_{\mu}[s(\sqrt{g}g^{\mu\nu})A_{\nu}^{a}]-\partial_{\mu}[\sqrt{g}g^{\mu\nu}\psi_{\nu}^{a}],
\langle\prod_{i}P_{e_{i}}(\vec{x}_{i})\rangle=\frac{\intdA_{\mu}d\psid\bar{\psi}e^{-\int_{0}^{1/T}F_{\mu\nu}^{2}/4+\bar{\psi}(\gamma\cdotD+m)\psi}\prod_{i}e^{ie_{i}\int_{0}^{1/T}d\tauA_{0}(\tau,\vec{x}_{i})}}{\intdA_{\mu}d\psid\bar{\psi}e^{-\int_{0}^{1/T}F_{\mu\nu}^{2}/4+\bar{\psi}(\gamma\cdotD+m)\psi}}
\epsilon^{ij}\nabla_{i}^{(p,k)}{\calA}_{(p,k)j}=-\sum_{(p^{\prime},k^{\prime})\neq(p,k)}\beta_{pp^{\prime}}\delta^{2}({\bfr}_{k}^{(p)}-{\bfr}_{k^{\prime}}^{(p^{\prime})})\,.
\begin{array}{rcl}{\mathrm{Tr}\,[\Gamma_{s}(P^{a})\Gamma_{s}(P_{b})]=\delta^{a}{}_{b}\,,\,\,\,\,}&{\Rightarrow}&{\Gamma_{s}(P^{a})={\textstyle\frac{-i}{2g}}\gamma^{a}\,,}\\{}&{}&{}\\{\mathrm{Tr}\,[\Gamma_{s}(M^{ab})\Gamma_{s}(P_{cd})]=\delta^{ab}{}_{cd}\,,\,\,\,\,}&{\Rightarrow}&{\Gamma_{s}(M^{ab})=-{\textstyle\frac{1}{2}}\gamma^{ab}\,.}\\\end{array}
\tilde{\rho}_{\mathrm{w}}(x)\;=\;-{\frac{\mathrm{Im}\,f_{\mathrm{w}}(x+i\epsilon)}{\pi\sqrt{x}}}\;=\;{\frac{1}{8\pi^{2}\beta}}\int_{0}^{\infty}{\frac{dy}{x+y}}\sqrt{{\frac{y}{x}}}\sqrt{{\frac{(b^{2}-x)(x-a^{2})}{(b^{2}+y)(y+a^{2})}}}\;.
\frac{1/k}{(z-w)^{2}}<s,p^{+}|p^{+}f(w)\left(:e^{-i\frac{2u(w)}kt_{2}}:\right)\left(:e^{\frac{2u(w)}k}:\right)\left(:e^{-it^{-}X^{+}}:\right)p^{-}|s,p^{-}>
\int_{\Sigma}F\wedge\bar{G}=\int_{M_{1}^{d-2}}F\wedge\Psi_{1}+\int_{M_{2}^{d-2}}F\wedge\Psi_{2}+\int_{\partial\sigma_{1}\subset\cap_{\sigma_{1,2}}}F\wedge(\Psi_{1}-\Psi_{2}),
K=\frac{1}{2}\frac{\partial^{2}A}{\partial\phi^{i}\partial\phi^{j}}\dot{\phi}^{i}\dot{\phi}^{j}+i\frac{\partial^{2}A}{\partial\phi^{i}\partial\phi^{j}}(\bar{\psi}_{a}^{i}\dot{\psi}^{aj}+\psi^{ai}\dot{\bar{\psi}_{a}^{j}}),
W_{\mathrm{T}}^{(0)}=\intd^{4}x\,\mathrm{tr}\Bigr\{\lambda^{\mu\nu}F_{\mu\nu}-2\chi^{\mu\nu}D_{\mu}\psi_{\nu}+2\etaD^{\mu}\psi_{\mu}+2\bar{\phi}D^{2}\phi+2\bar{\phi}\{\psi^{\mu},\psi_{\mu}\}\Bigr\},
{\frac{\partial^{R}{\calH}}{\partial\Pi_{\bar{c}}^{a}}}-\partial_{t\mu}{\frac{\partial^{R}{\calH}}{\partial\left[{\partial_{t\mu}\Pi_{\bar{c}}^{a}}\right]}}+\partial_{n}\bar{c}^{a}=0,\qquad{\frac{\partial^{R}{\calH}}{\partial\bar{c}^{a}}}-\partial_{t\mu}{\frac{\partial^{R}{\calH}}{\partial\left[{\partial_{t\mu}\bar{c}^{a}}\right]}}+\partial_{n}\Pi_{\bar{c}}^{a}=0.
\sigma(1;2)=16\pi^{3}\hbar\sqrt{\omega_{1}^{2}-k_{\|}{}_{1}^{2}}\;\Theta(\omega_{1}^{2}-k_{\|}{}_{1}^{2})\,\delta(\omega_{1}+\omega_{2})\delta^{(2)}({\bfk}_{1}{}_{\|}+{\bfk}_{2}{}_{\|})\left[1+2{\overline{n}}(|\omega_{1}|)\right]
e^{-{\frac{1}{4}}\int_{\Sigma}(fBf-gDg+2fAg)}=e^{-{\frac{i}{2}}\int_{\Sigma}fg}e^{-{\frac{1}{2}}\int_{\Sigma}g\hat{m}g}e^{-{\frac{1}{4}}\int_{\Sigma}(f-i\hat{m}g)B(f-i\hat{m}g)}
\frac{1}{\rho}k(\rho\lambda)^{2}=\left.\frac{1}{\rho}\frac{a(\rho)c_{1}(\rho\lambda)^{2}+b(\rho)c_{2}(\rho\lambda)^{-2\gamma}}{a(\rho)c_{3}+c_{4}b(\rho)(\rho\lambda)^{-2\gamma}}\right.
\Lambda_{W}^{(0)}(\zeta;r)=1,\qquad\Lambda_{W}^{(1)}(\zeta;r)=\Lambda(\zeta;r)=\frac{\Theta_{q^{4}}(r\zeta^{2})\Theta_{q^{4}}(q^{2}r\zeta^{-2})}{\Theta_{q^{4}}(r\zeta^{-2})\Theta_{q^{4}}(q^{2}r\zeta^{2})}\zeta^{-2},
T^{N+1}\equiv\left\{t=t\left(\theta_{1},\cdots,\theta_{N+1}\right)=\left.\left(\begin{matrix}{e^{i\theta_{1}}}&{}&{}\\{}&{\ddots}&{}\\{}&{}&{e^{i\theta_{N+1}}}\\\end{matrix}\right)\right\vert\theta_{1},\cdots,\theta_{N+1}\in{\bfR}\right\}\.
\sigma_{\mathrm{abs}}={\pi^{2}R^{7}}(\omega^{2}+16\pi^{2}T_{L}^{2})(\omega^{2}+16\pi^{2}T_{R}^{2})\omega{\frac{e^{\frac{\omega}{T_{H}}}-1}{(e^{\frac{\omega}{2T_{L}}}-1)(e^{\frac{\omega}{2T_{R}}}-1)}}\.
\left[\frac{1+S({\lambda_{s}^{\prime}}^{2}e^{-i\gamma}+{\lambda_{s}^{\prime}}^{-2}e^{i\gamma})}{1+S({\lambda_{s}^{\prime}}^{-2}e^{-i\gamma}+{\lambda_{s}^{\prime}}^{2}e^{i\gamma})}\right]^{N/4}=\prod_{\stackrel{r=1}{r\not=s}}^{l}\frac{{\lambda_{r}^{\prime}}^{2}e^{-i\gamma}-{\lambda_{s}^{\prime}}^{2}e^{i\gamma}}{{\lambda_{r}^{\prime}}^{2}e^{i\gamma}-{\lambda_{s}^{\prime}}^{2}e^{-i\gamma}}\,,
\Bigl[\varepsilon({\bfr},\omega)\,\omega^{2}\delta_{jm}-\mathrm{rot}_{jl}\mathrm{rot}_{lm}\Bigr]D_{mk}(\varepsilon,{\bfr},{\bfr}^{\prime},\omega)=-\omega^{2}\delta({\bfr}-{\bfr}^{\prime})\delta_{jk}
\mathcal{L}_{F}=\overline{{\Psi}}_{L}i\gamma^{\mu}D_{\mu}\Psi_{L}+\overline{{\Psi}}_{R}i\gamma^{\mu}\partial_{\mu}\Psi_{R}-f\left(\overline{{\Psi}}_{L}\Phi\Psi_{R}+\mathrm{h.c.}\right)\,,
(-\frac{1}{2}\frac{d^{2}}{{(dx^{1})}^{2}}+\frac{1}{2}{(\Lambda_{1}^{3})}^{2}(x^{1})^{2}+\frac{1}{(x^{1})^{2}}(8(m^{1})^{2}+4m^{1}+\frac{3}{8}))X_{1}(x^{1})=4E_{I}^{1}X_{1}(x^{1}),
\langle\lambda\vert\xi\rangle=\psi_{\xi}(\lambda)=N\left(1-\vert\beta\vert\lambda^{2}\right)^{-1/2}\left(1-\sqrt{\vert\beta\vert}\lambda\right)^{i\xi/2}\left(1+\sqrt{\vert\beta\vert}\lambda\right)^{-i\xi/2}
\mathcal{T}_{1,\kappa}^{\lambda}F_{n}^{\mathcal{O}}(\theta_{1},\ldots,\theta_{n})\simF_{\kappa}^{\mathcal{O}^{\prime}}(\theta_{1},\ldots,\theta_{\kappa})F_{n-\kappa}^{\mathcal{O}^{\prime\prime}}(\theta_{\kappa+1},\ldots,\theta_{n})\,\,,
I(w^{\prime},\eta^{\prime},n^{\prime},a^{\prime};w,\eta,n,a)=\frac{1}{2}\int_{-\infty}^{\infty}\,dx^{+}\int_{-\infty}^{\infty}\,dx^{-}\int_{-Y_{0}}^{Y_{0}}\,dy\psi_{w^{\prime},\eta^{\prime},n^{\prime}}^{a^{\prime}}(x)\psi_{w,\eta,n}^{a}(x)\,.
\begin{array}{rcl}{\left({\calV}_{m,1}\otimes{\calV}_{1,2}\right)_{\mathrmf}}&{=}&{{\calV}_{m,2}\,,}\\{\left({\calV}_{m,1}\otimes{\calV}_{1,3}\right)_{\mathrmf}}&{=}&{{\calV}_{m,3}\,,}\\{\left({\calV}_{m,2}\otimes{\calV}_{1,3}\right)_{\mathrmf}}&{=}&{{\calR}_{m,2}\,,}\\{\left({\calV}_{m,3}\otimes{\calV}_{1,3}\right)_{\mathrmf}}&{=}&{{\calV}_{m,3}\oplus{\calR}_{m,1}\,,}\\\end{array}
\begin{array}{c}{\{J_{0}^{a}(x)\;,J_{1}^{b}(y)\}=i\epsilon^{bcd}\;\{J_{0}^{a}(x)\;,J_{0}^{c}(y)n^{d}(y)\}=}\\{=i\epsilon^{bcd}\;(\epsilon^{acf}\;J_{0}^{f}(x)n^{d}(y)+\epsilon^{adf}\;J_{0}^{c}(y)n^{f}(x))\;\delta(x-y)=}\\{=i\;(\;-J_{0}^{b}(x)n^{a}(x)+J_{0}^{a}(x)n^{b}(x))\;\delta(x-y)=\epsilon^{abc}\;J_{1}^{c}(x)\;\delta(x-y)}\\\end{array}
W\,=\,\frac{\rho_{0}}{2\pi}\intdt\int_{0}^{\pi}d\sigma\left[\dot{{\bfx}}^{2}\,-\,a^{2}{\bfx}^{2}\,-\,\epsilona^{2}\left(a^{-2}\ddot{{\bfx}}\,-\,{\bfx}^{\prime\prime}\right)^{2}\right],
p_{+}p_{-}=\sum_{m\ge0}\big(\frac{1}{4}p_{-}^{2}[\overline{x}_{m}^{i}\mu_{ij}^{2}x_{m}^{j}]+\overline{p}_{m}^{i}\delta_{ij}p_{m}^{j}+m^{2}\overline{x}_{m}^{i}\delta_{ij}x_{m}^{j}\big),
\biggl\{a_{\eta^{\prime}}(p^{\prime\,\mu}),\,a_{\eta^{\prime\prime}}^{\dagger}(p^{\prime\prime\,\mu})\biggr\}\,=\,i\,(2\,\pi)^{3}\,2\,p_{0}\,\delta^{3}(\bbox{p^{\prime}}-{\bbox{p^{\prime\prime}}})\,\delta_{\eta^{\prime},-\eta^{\prime\prime}}\quad,
h_{\i}^{\underline{{a}}}\,\dot{\xi}_{\alpha}^{\underline{{a}}}-\varepsilon_{ij}\,B_{\alpha}^{\underline{{a}}}\,\xi_{\alpha}^{\underline{{a}}}\,\dot{x}^{j}-{\calE}_{j;\alpha}^{\underline{{a}}}\,\xi_{\alpha}^{\underline{{a}}}=0.
L=\intd^{2}{\bfr}({\frac{\kappa}{4}}\epsilon^{\alpha\beta\gamma}A_{\alpha}F_{\beta\gamma}+i\psi^{\ast}D_{t}\psi-{\frac{1}{2}}|{\bfD}\psi|^{2}+{\frac{1}{2\kappa}}(\psi^{\ast}\psi)^{2})
\rho(z)dz\;=\;{\frac{dz^{2}}{8\pi\beta}}{\frac{\sqrt{16\beta-(z^{2}-a^{2})}}{\sqrt{z^{2}-a^{2}}}}\;=\;{\frac{1}{8\pi\beta}}{\frac{dx}{\sqrt{x}}}\sqrt{16\beta-x}\;,
\intd^{3}x\;\frac{1}{2\piq_{I}}\varepsilon^{\mu\nu\lambda}\partial_{\mu}\bar{\omega}_{I}\partial_{\nu}H_{\lambda}^{I}=\frac{1}{q_{I}}\intd^{3}x\;K_{I}^{\mu}H_{\mu}^{I}\;\;,
\tau_{RL}(k,\lambda_{RL}(k,\mu),\mu)=\biggr[\lambda_{RL}^{-1}(k,\mu)+\frac{2}{\pi}\sum_{i=0}^{L}\frac{k^{2(L-i)}\mu^{2i+1}}{2i+1}+ik^{2L+1}\biggr]^{-1},
\rho(x)-\PiE=-p_{2}=\frac{ET_{2}^{2}}{\Pi}\frac{1}{1+\left[1+\frac{E^{2}T_{2}^{2}}{\Pi^{2}(E^{2}-1)}\right]\mathrm{sinh}^{2}\left(\frac{\sqrt{E^{2}-1}}{T_{0}}x\right)}.
ds_{str.}^{2}=-\left(1-\frac{2\eta}{r}\right)^{\frac{-m-\sigma}{\eta}}dt^{2}+\left(1-\frac{2\eta}{r}\right)^{\frac{\sigma-m}{\eta}}dr^{2}+\left(1-\frac{2\eta}{r}\right)^{1+\frac{\sigma-m}{\eta}}r^{2}d\Omega^{2}