harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\psi_{\mu\vertA}\,\leftrightarrow\,\mathrm{~\begin{array}{|c|}\hline{\stackrel{~}{A}}\\\hline{}\\\end{array}~}\,\equiv\,{\bf8}\quad;\quad\chi_{ABC}\,\leftrightarrow\,\mathrm{~\begin{array}{|c|}\hline{\stackrel{~}{A}}\\\hline{\stackrel{}{B}}\\\hline{\stackrel{}{C}}\\\hline{}\\\end{array}~}\,\equiv\,{\bf56}
\varepsilon_{ajk}\left\{L_{k}R_{b},L_{j},R_{3}\right\}=i\hbar\varepsilon_{ajk}\varepsilon_{kjm}L_{m}\left(R_{b}R_{3}-\frac{1}{2}\left\{R_{b},R_{3}\right\}\right)=\frac{1}{2}\left(i\hbar\right)^{2}\varepsilon_{ajk}\varepsilon_{kjm}L_{m}\varepsilon_{b3c}R_{c}=\hbar^{2}L_{a}R_{c}\varepsilon_{b3c}\;.
\left(\begin{array}{cccc}{\hat{p}_{0}-m}&{-\hat{p}_{1}}&{-\hat{p}_{2}}&{-\hat{p}_{3}}\\{\hat{p}_{1}}&{-\hat{p}_{0}-m}&{0}&{0}\\{\hat{p}_{2}}&{0}&{-\hat{p}_{0}-m}&{0}\\{\hat{p}_{3}}&{0}&{0}&{-\hat{p}_{0}-m}\\\end{array}\right)
V(\varphi^{a},\sigma)=J^{a}\varphi^{a}-W(J^{a},\sigma)={\frac{1}{2}}\sigma\varphi^{a}\varphi^{a}+{\frac{Nm_{0}^{2}}{g_{0}}}\sigma-{\frac{N}{2g_{0}}}\sigma^{2}+V_{1-\mathrm{loop}}(\sigma),
f_{\dot{\alpha}_{1}\ldots\dot{\alpha}_{p}\alpha_{1}\ldots\alpha_{q}}^{p+1\ldotsk+q}=\bar{D}_{1\dot{\alpha}_{1}}\ldots\bar{D}_{p\dot{\alpha}_{p}}D_{\alpha_{1}}^{k+1}\ldotsD_{\alpha_{q}}^{k+q}W^{12\ldotsk}\vert_{0}\;,\qquadp\leqk\;,\q\leqN-k
^{(\bar{H})}T^{AB}={\frac{e^{-a\Phi}}{(D-n)!}}\left((D-n)\bar{H}_{~~C_{1}C_{2}\ldots}^{A}\bar{H}^{BC_{1}C_{2}\ldots}-{\frac{1}{2}}g^{AB}\bar{H}^{2}\right)\.
\beta_{e}^{b}=\sum_{o}\beta_{o}^{b}\,U_{-o,e}^{-1}\,,\;\beta_{e}^{c}=\sum_{o}U_{e,-o}\beta_{o}^{c}\,,\;\{\beta_{e}^{b},\beta_{e^{\prime}}^{c}\}_{\star}=\delta_{e+e^{\prime}}\,,
\begin{array}{ll}{\mathrm{(i)}}&{\\L_{\omega_{\alpha}}V_{4}=\Lambda_{\omega_{\alpha}}V_{0}=0}\\{\mathrm{(ii)}}&{\\L_{\omega_{\alpha}}\omega=(\omega_{\alpha},\omega)\Upsilon}\\{\mathrm{(iii)}}&{\\\Lambda_{\omega_{\alpha}}\omega=(\omega_{\alpha},\omega)\mathbbI}\\{\mathrm{(iv)}}&{\\L_{\omega_{\alpha}}\mathbbI=\omega_{\alpha},\\Lambda_{\omega_{\alpha}}\Upsilon=\omega_{\alpha}.}\\\end{array}
\Lambda_{magn}^{10-2F}\tilde{Y}^{11}\lambda^{5}\tilde{q}^{2F-3}\tilde{\bar{q}}^{2F-3}q^{*\;2}\bar{q}^{*\;2}M_{1}\tilde{M}_{2}^{2}\left(\frac{h}{\mu^{2}}\right)^{3}.
ds^{2}=\Sigma\left(\frac{d\tau^{2}}{1-\tau^{2}}-\frac{d\zeta^{2}}{1-\zeta^{2}}\right)-\frac{1-\tau^{2}}{\Sigma}\left(dy-Q\zetadx\right)^{2}-(1-\zeta^{2})\Sigmadx^{2},
\boxed{\begin{aligned}{\mathcal{D}_{L}=\frac{1}{2}\left(1-\gamma^{5}\right)\mathcal{D}=\frac{\partial}{\partial\overline{{\theta}}_{L}}-(\mathrm{i}\gamma^{\mu}\theta_{R})\partial_{\mu}}\\{\mathcal{D}_{R}=\frac{1}{2}\left(1+\gamma^{5}\right)\mathcal{D}=\frac{\partial}{\partial\overline{{\theta}}_{R}}-(\mathrm{i}\gamma^{\mu}\theta_{L})\partial_{\mu}}\\\end{aligned}}
\langle\varphi^{2}\rangle_{M}^{\mathrm{th}}=\frac{1}{2\pi^{2}}\int_{0}^{\infty}\frac{\left(\omega^{2}+2m\omega\right)^{1/2}d\omega}{e^{(m+\omega)/T}-1},\quadT=\frac{1}{2\pi\xi}
{\calM}^{2}\sim\tilde{g}^{2}g_{ij}(n^{i}+A^{ikl}n_{kl})(n^{j}+A^{jrs}n_{rs})+\left(\frac{2\pi}{\tilde{g}}\right)^{2}g^{ij}g^{kl}n_{ik}n_{jl},
\left\langle\Psi_{1}|\Psi_{2}\right\rangle=N\int_{p^{2}=-m^{2}}\frac{d{\bfp}}{p_{0}}\overline{{\psi}}_{1A(l_{1})B(l_{2})\cdotsC(l_{r})}\psi_{2}^{A(l_{1})B(l_{2})\cdotsC(l_{r})}\,
\begin{array}{l}{\left.X_{1}^{+}\right.^{*}=X_{1}^{+}~,~~~~~\left.X_{1}^{-}\right.^{*}=X_{1}^{-}~,~~~~~H_{1}^{*}=-H_{1}~,}\\{\left.X_{2}^{+}\right.^{*}=-X_{2}^{+}~,~~~~~\left.X_{2}^{-}\right.^{*}=-X_{2}^{-}~,~~~~~H_{2}^{*}=-H_{2}~.}\\\end{array}
L_{0}^{m}=q^{1/2}m[(\dot{x}_{\mu}+q^{1/2}i\chi\psi_{\mu})^{2}]^{1/2}+\frac{i}{2}(\psi_{\mu}\;\dot{\psi}^{\mu}-\psi_{5}\dot{\psi}_{5})-i\chi\psi_{5}m.
\begin{array}{rcl}{g^{2}}&{=}&{\textstyle{\frac{1}{6}}\left\{K^{2}\;-\;\alephL^{2}\right\}\;,}\\{}&{}&{}\\{\alpha^{2}\,+\,\beta^{2}}&{=}&{\textstyle{\frac{1}{2}}\left(K^{2}\;+\;\alephL^{2}\right)\;.}\\\end{array}
\delta\ddot{x}^{\mu}\,=\,\frac{d^{2}}{ds^{2}}\,\deltax^{\mu}\,-\,2\,\ddot{x}^{\mu}\left(e_{0\nu}\,\frac{d}{ds}\,\deltax^{\nu}\right)\,-\,e_{0}^{\mu}\,\frac{d}{ds}\,\left(e_{0\nu}\,\frac{d}{ds}\,\deltax^{\nu}\right)\,{.}
\begin{array}{ccccc}{\chi_{++}:\quad}&{{\bfQ}^{R}=}&{(e_{3}/3,e_{3}/3,e_{1}/3)\quad}&{{\bfQ}^{L}=}&{(e_{3}/3,e_{3}/3,0)}\\{\chi_{-+}:\quad}&{}&{(e_{2}/3,e_{1}/3,e_{3}/3)\quad}&{}&{(e_{3}/3,e_{3}/3,0)}\\{\chi_{+-}:\quad}&{}&{(e_{3}/3,e_{1}/3,e_{2}/3)\quad}&{}&{(e_{2}/3,e_{2}/3,0)}\\{\chi_{--}:\quad}&{}&{(e_{2}/3,e_{2}/3,e_{1}/3)\quad}&{}&{(e_{2}/3,e_{2}/3,0)}\\\end{array}
W_{G}(N,n)=\sum_{N\geqm_{1}\geq\ldots\geqm_{G-1}\geq0}C_{N}^{m_{1}}\,C_{m_{1}}^{m_{2}}\ldotsC_{m_{G-2}}^{m_{G-1}}\,C_{\,m_{G-1}}^{n-m_{1}-\ldots-m_{G-1}}\,,
\lambda_{\pm}(x_{0},r;z;q)\to\frac{1}{2}\left\{\sum_{n=0}^{\infty}\frac{(1-\frac{\rho_{+}}{\rho_{-}})^{n}}{n!}(i\rho_{-}z)^{n}\pm\sum_{n=0}^{\infty}\frac{(1-\frac{\rho_{-}}{\rho_{+}})^{n}}{n!}(i\rho_{+}z)^{n}\right\}
{f^{(12,-12)}(\Omega,\bar{\Omega})={\frac{c}{\sqrt{\Omega_{2}}}}\left({\frac{\partial}{\partial\alpha}}\right)^{24}\sum_{m,n}\int{\frac{dt}{t}}t^{23/2}\left.e^{-t|m+n\Omega|^{2}+\alpha(m+n\bar{\Omega})}\right|_{\alpha=0},}
\int_{G}t_{mn}^{(\rho)}(g)\,dg=\sum_{j=1}^{\ell}P^{(j)m}P_{n}^{(j)},\qquadP^{(j)m}=\eta^{m}(w^{(j)}),\quadP_{n}^{(j)}=\theta^{(j)}(v_{n}).
\Sigma_{div}^{0}={\frac{Ce^{2}}{\beta}}\int{\frac{|{\bfk}|d|{\bfk}|}{(2\pi)}}{\frac{{\calM}({\bfk},p_{0f})}{{\bfk}^{2}+p_{0f}^{2}+{\calM}^{2}({\bfk},p_{0f})}}{\frac{1}{|{\bfk}^{2}-{\bfp}^{2}|}}
\begin{array}{rcl}{\Gamma_{1}}&{=}&{\dot{x}^{2}}\\{\Gamma_{2}}&{=}&{(\dot{x}n)^{2}}\\{\Gamma_{3}}&{=}&{(\dot{x}\dot{n})}\\{\Gamma_{4}}&{=}&{\varepsilon_{abc}\dot{x}^{a}n^{b}\dot{n}^{c}}\\{\Gamma_{5}}&{=}&{\dot{n}^{2}}\\\end{array}
\mathcal{M}^{2}=2\,(\alpha+\beta)\left[\begin{array}{cccc}{b_{2}^{2}}&{-b_{1}\,b_{2}}&{-b_{2}\,a_{2}}&{a_{1}\,b_{2}}\\{-b_{1}\,b_{2}}&{b_{1}^{2}}&{b_{1}\,a_{2}}&{-b_{1}\,a_{1}}\\{-b_{2}\,a_{2}}&{b_{1}\,a_{2}}&{a_{2}^{2}}&{-a_{1}\,a_{2}}\\{a_{1}\,b_{2}}&{-b_{1}\,a_{1}}&{-a_{1}\,a_{2}}&{a_{1}^{2}}\\\end{array}\right]
{\calL}_{kin}=-\frac{k^{\prime}}{4l}\,\partial^{\mu}l\,\partial_{\mu}l+\frac{k^{\prime}}{4l}\,{\calH}^{\mu}{\calH}_{\mu}+(lV-{G})_{i\,\bar{\!\jmath}}\,\partial^{\mu}z^{i}\partial_{\mu}\bar{z}^{\,\bar{\!\jmath}}+\frac{1}{2}{\calH}^{\mu}{\calV}_{\mu}\,,
(-\partial_{i}^{2}+m^{2})\chi_{R}-bm^{2}g^{2}(\{\sigma^{2},\chi_{R}\}+\sigma\chi_{R}\sigma)+\frac34cm^{2}g^{4}(\{\sigma^{4},\chi_{R}\}+\sigma^{2}\chi_{R}\sigma^{2}+\sigma\{\sigma,\chi_{R}\}\sigma)=\Lambda_{a}^{2}\chi_{R}\.
\frac{i}{2}TrD^{-1}G=\frac{1}{12N}\intd^{D}x\left(\lambda_{0}+\frac{\eta_{0}\phi^{2}}{10N}\right)[G_{aa}(x,x)]^{2}+\frac{2\eta_{0}}{6!N^{2}}\intd^{D}x[G_{aa}(x,x)]^{3}+cte.
S=\intdtdx\,\left(\varphi_{x}\psi_{t}-\alpha_{x}\beta_{tx}-\varphi_{x}\psi_{x}^{3}-{\frac{2}{\gamma(\gamma-1)}}\varphi_{x}^{\gamma}+2\alpha_{x}\beta_{xx}\psi_{x}+{\frac{2}{\gamma}}\alpha_{x}\alpha_{xx}\varphi_{x}^{\gamma-2}\right)
tT\;\;=\;\;\frac{1}{p!q!}T_{\nu_{1}...\nu_{q}\mu_{1}...\mu_{p}}dx^{\nu_{1}}\wedge...\wedgedx^{\nu_{q}}\otimesdx^{\mu_{1}}\wedge...\wedgedx^{\mu_{p}}\,\in\,X^{q,p}
Z_{4}^{0}=\left(\begin{matrix}{0}&{\lambda_{1}}&{0}&{0}&{0}&{0}&{0}&{0}\\{-\lambda_{1}}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{\lambda_{2}}&{0}&{0}&{0}&{0}\\{0}&{0}&{-\lambda_{2}}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{\lambda_{3}}&{0}&{0}\\{0}&{0}&{0}&{0}&{-\lambda_{3}}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}&{0}&{\lambda_{4}}\\{0}&{0}&{0}&{0}&{0}&{0}&{-\lambda_{4}}&{0}\\\end{matrix}\right)\,
Z_{N}=\frac1{h^{3N}N!}\inte^{-\betaH^{(0)}}\left(1-\betac^{-2}H^{(1)}\right)\prod_{a}d^{3}x_{a}d^{3}p_{a}\equivZ_{N}^{(0)}+c^{-2}Z_{N}^{(1)},
\sum_{\alpha}\frac{k_{3}}{\left|{\bfy}-{\bfy}_{\alpha}\right|^{6}}\longrightarrow\int_{-\infty}^{\infty}\frac{k_{3}dz}{(\hat{r}^{2}+z^{2})^{3}}=\frac{\tilde{k}_{3}}{\hat{r}^{5}},
\int_{0}^{T}d\tau\psi_{\mu}\psi_{\nu}\omega_{\mu\nu}=-T\int_{0}^{T}d\tau(\psi\cdot\dot{x})^{\cdot}(\psi\cdot\dot{x})+T\int_{0}^{T}d\tau(\psi\cdot\dot{x})(\dot{\psi}\cdot\dot{x})+{\frac{T}{4}}\int_{0}^{T}d\tau\dot{x}\cdot\ddot{x}
Z_{\;\;C}^{B}=\left(\begin{array}{ccc}{\mathbf{0}}&{\delta_{\;\;c}^{b}}&{-\delta_{\;\;c}^{b}}\\{-\delta_{\;\;c}^{b}}&{\mathbf{0}}&{\delta_{\;\;c}^{b}}\\{\delta_{\;\;c}^{b}}&{-\delta_{\;\;c}^{b}}&{\mathbf{0}}\\\end{array}\right)\,,Z_{\;\;d}^{C}=\left(\begin{array}{c}{-\delta_{\;\;d}^{c}}\\{-\delta_{\;\;d}^{c}}\\{-\delta_{\;\;d}^{c}}\\\end{array}\right)\,.
\left|\Psi\right\rangle=\sumC_{\mu_{1}...\mu_{m}}\alpha_{-n_{1}}^{\mu_{1}}...\alpha_{-n_{m}}^{\mu_{m}}\left|0;k\right\rangle,[\alpha_{m}^{\mu},\alpha_{n}^{\nu}]=m\eta^{\mu\nu}\delta_{m+n}
[D_{\mu},G^{-1}(\phi)F^{\mu\nu}]=J^{\nu}\quad(a)\quad,\qquad\partial_{\mu}\partial^{\mu}\phi={\frac{\partialW}{\partial\phi}}-{\frac{1}{4}}F_{\mu\nu}^{a}F_{a}^{\mu\nu}{\frac{\partialG^{-1}(\phi)}{\partial\phi}}\quad(b)
{\calL}=-{\frac{1}{4}}\mathrm{F}_{\mu\nu}\mathrm{F}^{\mu\nu}+\bar{\psi}i{\hbar}{\gamma}^{\mu}{\partial}_{\mu}\psi+e{\hbar}\bar{\psi}_{\mathrm{R}}{\gamma}^{\mu}{\psi_{\mathrm{R}}}A_{\mu},
\mathrm{Re}\,\chi_{T}^{\mathrm{(TM)}}(\omega)=\frac{\hbarA}{4\pi^{2}}\left(\frac{k_{B}T}{\hbar}\right)^{5}\int_{0}^{\Omega}dv\frac{\left(\frac{\Omega}{2}-v\right)^{2}}{\sqrt{v(\Omega-v)}}\int_{0}^{\infty}du\,\frac{u^{2}}{e^{\frac{u}{2}}-1}\left[1+{\calO}(\Omega^{2})\right].
ds^{2}=-\frac{(r^{2}+r_{-}^{2})(r_{+}^{2}-r^{2})}{r^{2}}dt^{2}+\frac{r^{2}}{(r^{2}+r_{-}^{2})(r_{+}^{2}-r^{2})}dr^{2}+r^{2}(d\phi+\frac{r_{+}r_{-}}{r^{2}}dt)^{2}
\mathrm{d}\mu=e^{-8\pi^{2}/g^{2}}\,(M_{\mathrm{PV}})^{n_{b}-\frac{1}{2}n_{f}}\,(2\pi)^{-n_{b}/2}\,\prod_{i}\mathrm{d}\eta_{i}\,\left\{\mathrm{Ber}\,\left\langle\frac{\partial\Phi(\eta)}{\partial\eta_{j}}\bigg|\frac{\partial\Phi(\eta)}{\partial\eta_{k}}\right\rangle\right\}^{\frac{1}{2}}\,,
\epsilon=e^{-{\frac{g\etat}{2\sqrt{2}}}\epsilon_{ij}}H^{-1/8}\left[\sqrt{f^{1/2}+(f-1)^{1/2}}-\eta\sqrt{f^{1/2}-(f-1)^{1/2}}\gamma_{\overline{{0}}}\epsilon^{ij}\right]e^{{\frac{\eta}{2}}\gamma_{\overline{{0\thetar}}}\epsilon_{ij}}e^{-{\frac{1}{2}}\gamma_{\overline{{\phi\theta}}}}(1-\gamma_{\overline{{r}}})\tilde{\epsilon}_{0},
\int_{0}^{1}d\lambda\,s(\lambda)=\int_{x_{0}^{+}}^{x_{1}^{+}}d\tilde{x}^{+}s_{+}(\tilde{x}^{+})=\int_{x_{1}^{+}}^{x_{0}^{+}}d\tilde{x}^{+}\left(-s_{+}(\tilde{x}^{+})\right)\leq\frac{1}{4}\left(A_{\mathrm{qu}}(0)-A_{\mathrm{qu}}(1)\right)\,.
\left(\begin{array}{c}{\lambda_{1}^{ij}}\\{\lambda_{2}^{ij}}\\{\lambda_{3}^{ij}}\\{\lambda_{4}^{ij}}\\\end{array}\right)=\left(\begin{array}{c}{\lambda_{1}^{ij}}\\{\lambda_{2}^{ij}}\\{\lambda_{3}^{ij}}\\{0}\\\end{array}\right)+\left(\begin{array}{c}{0}\\{0}\\{0}\\{\lambda_{4}^{ij}}\\\end{array}\right)\;.
X_{\mu\nu\lambda\sigma}=-C_{[\mu\nu}C_{\lambda\sigma]}-A_{[\mu}^{i}A_{\nu}^{i}C_{\lambda\sigma]}-f_{ijk}A_{[\mu}^{i}A_{\nu}^{j}A_{\lambda}^{k}C_{\sigma]}-A_{[\mu}^{i}A_{\nu}^{i}A_{\lambda}^{j}A_{\sigma]}^{j}\quad.
\begin{array}{l}{\displaystyle\sigma(x^{\mu},x^{\nu})=\frac{1+{\bfx}{\bfx}^{\prime}}{1+\xi^{0}\xi^{0}}~,}\\{\displaystylex^{0}x^{0}=\frac{\xi^{0}\xi^{0}}{1+\xi^{0}\xi^{0}}(1+{\bfx}{\bfx}^{\prime})~,}\\\end{array}
\propto\intdx^{+}dp_{i}^{+}dp_{j}^{+}dp_{k}^{+}g(p_{i}^{+},p_{j}^{+},p_{k}^{+})\delta(p_{i}^{+}+p_{j}^{+}-p_{k}^{+})\phi_{i}(x^{+},p_{i}^{+})\phi_{j}(x^{+},p_{j}^{+})\phi_{k}(x^{+},p_{k}^{+})
\tilde{W}=\frac{1}{2}\intd^{4}x\hat{A}_{a}(g^{\mu\nu}(-\partial_{\mu}\delta_{ac}+\omega_{\mua}^{c})(\partial_{\nu}\delta_{cb}+\omega_{\nuc}^{b})+R_{ab})\hat{A}_{b}\equiv\frac{1}{2}\int\hat{A}\triangle_{EM}\hat{A}
ds^{2}=(1-\lambda)^{\frac{2}{1+\alpha^{2}}}\lbracedr^{2}+r^{2}d\theta^{2}+r^{2}sin^{2}\thetad\varphi^{2}\rbrace-{\frac{dt^{2}}{(1-\lambda)^{\frac{2}{1+\alpha^{2}}}}}
[{\calD}\mu]={\calD}B^{\mu}{\calD}\lambda{\calD}\rho{\calD}\xi\prod_{\beta=0}^{3}\{\delta(\Gamma_{\beta}[B^{0}+\xi,B^{i},\lambda,\rho])\}det\mid\{\tilde{\Omega}_{\alpha},\Gamma_{\beta}\}\mid.
L_{G}=\frac{1}{2}\dot{x}^{\alpha}g_{\alpha\beta}([g])\dot{x}^{\beta}+\frac{1}{2}\left(\dot{x}^{\mu}+\dot{x}^{\alpha}A_{\alpha}^{\mu}(g)\right)g_{\mu\nu}(g)\left(\dot{x}^{\nu}+A_{\beta}^{\nu}(g)\dot{x}^{\beta}\right)
{\calL}_{\mathrm{TA}}=-{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}+{\frac{1}{12}}H_{\mu\nu\rho}H^{\mu\nu\rho}+{\frac{m}{4}}\epsilon^{\mu\nu\rho\sigma}B_{\mu\nu}F_{\rho\sigma}\:.
(k_{1}-k_{2})_{\mu}T_{\mu}^{AP}=-4mi(k_{1}-k_{2})^{2}\left\{I_{log}(m^{2})-\left(\frac{i}{(4\pi)^{2}}\right)Z_{0}(m^{2},m^{2},(k_{1}-k_{2})^{2};m^{2})\right\},
\intdx\left(\,{\frac{1}{2}}\,\tilde{g}^{ij}{\frac{\delta^{2}\Psi}{\delta\phi^{i}\delta\phi^{j}}}\,+\,{\frac{1}{2}}\,g_{ij}\partial_{x}\phi^{i}\partial_{x}\phi^{j}\Psi\,\right)\,=\,E\Psi\,.
|\mathrm{Im}\,\Psi(\beta^{\prime})|={\frac{|\mathrm{Im}\,Z(g^{\prime})|}{|1-\beta^{\prime}|^{1/2}}}\sim{\frac{1}{\sqrt{2}}}{\frac{e^{1/(4g^{\prime})}}{|1-\beta^{\prime}|^{1/2}}}(1+O(g^{\prime})),
D_{\theta_{\mu}}=(2\pi\alpha^{\prime})^{2}(T_{11}T_{22}-T_{12}^{2})+\frac{1}{4}\theta_{\mu}^{2}~~~,~~~\tilde{T}_{\theta_{\mu}}^{-1}=\frac{2\pi\alpha^{\prime}}{D_{\theta_{\mu}}}\left(\begin{matrix}{T_{22}}&{-T_{12}}\\{-T_{12}}&{T_{11}}\\\end{matrix}\right)~.
i\Pi^{\mu\nu}(p)=-2g^{2}\int_{0}^{1}~dx\int\frac{d^{3}l}{(2\pi)^{3}}[\frac{1}{(l^{2}+M^{2})^{2}}-\frac{1}{2}\frac{(e^{\tilde{p}l}+e^{-\tilde{p}l})}{(l^{2}+M^{2})^{2}}]{\mathit{M}}^{\mu\nu}~~~,
\begin{array}{lcr}{\sum_{j}q_{a}^{j}\varphi_{j}^{+}\overline{{\varphi}}_{j}^{+}}&{=-i\xi_{a}^{++}\qquad}&{(a)}\\{\sum_{j}q_{a}^{j}(\varphi_{j}^{+}\overline{{\varphi}}_{j}^{-}+\varphi_{j}^{-}\overline{{\varphi}}_{j}^{+})}&{=-2i{\xi_{a}^{0}}\qquad}&{(b)}\\{\sum_{j}q_{a}^{j}\varphi_{j}^{-}\overline{{\varphi}}_{j}^{-}}&{=-i{\xi_{a}^{--}}\qquad}&{(c).}\\\end{array}
L(x,\dot{x},\{q_{n},\dot{q}_{n}\},t)=\frac{M}{2}\dot{x}^{2}-V(x,t)+\sum_{n}\left\{\frac{m}{2}(\dot{q}_{n}^{2}-\omega_{n}^{2}q_{n}^{2})+c_{n}q_{n}F_{n}(x,t)\right\},
F_{B}(\beta)=F_{B}(2^{N}\beta)+\sum_{k=1}^{N}\int_{\calS}\frac{d^{2}\tau}{\tau_{2}^{2}}\tau_{2}^{-\frac{d-2}{2}}\left[\frac{\bar{\theta}_{4}^{4}}{\bar{\eta}^{12}}(z_{v}-z_{o})-\frac{\bar{\theta}_{3}^{4}}{\bar{\eta}^{12}}(z_{v}+z_{o})\right]\theta_{2}\left(0\left|\frac{i(2^{k}\beta)^{2}}{2\pi^{2}\tau_{2}}\right.\right)
H_{sph}=2\pi\left\{\frac14\frac{N^{2}}{r}f^{\prime2}+\frac{\eta^{2}}{2}\left[rh^{\prime2}+\frac{N^{2}}{r}h^{2}(1-f)^{2}\right]+\frac{\lambda\eta^{4}}{4}r(1-h^{2})^{2}\right\}.
\epsilon^{\mu\alpha\beta...}~\{X_{\mu},1/\sqrtg\}_{PB}\{X_{\alpha},X_{\beta}\}_{PB}=2\epsilon^{\mu\alpha\beta...}\left(\dot{X}_{\mu}\dot{X}_{\alpha}X_{\beta}^{\prime}(1/\sqrt{g})^{\prime}-X_{\mu}^{\prime}X_{\beta}^{\prime}\dot{X}_{\alpha}(1/\sqrt{g})\dot{}\right),
\int_{\calM}d^{4}x(\frac{\delta\Sigma}{\delta\varphi^{a}}-f^{abc}\bar{\varphi}^{b}\frac{\delta\Sigma}{\deltab^{c}})=\int_{\calM}d^{4}xf^{abc}(\tau^{b\mu}A_{\mu}^{c}+D^{b}c^{c}),
\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\piG_{N}}{3}\left[\frac{1}{2}\dot{\phi}^{2}+V_{eff}(\phi)+\rho_{matter}\right]-\frac{K}{a^{2}}+\frac{\Lambda_{4}}{3}+\frac{\mu}{a^{4}}+O(\epsilon^{2}),
\begin{array}{lcr}{S}&{={\frac{m}{4\pi}}\int_{M}{d^{3}x}a_{\mu}\partial_{\nu}a_{\rho}\epsilon^{\mu\nu\rho}}\\{}&{={\frac{m}{4\pi}}\int_{M}{d^{3}y}b_{\mu}\partial_{\nu}b_{\rho}\epsilon^{\mu\nu\rho}}\\\end{array}
D^{(3)}\equiv\left|\begin{array}{c}{{\;1\;\;\;\;c_{12}\;\;\;c_{13}\;}}\\{{c_{12}\;\;\;\;1\;\;\;\;c_{23}\;}}\\{{c_{13}\;\;\;c_{23}\;\;\;\;1\;\;}}\\\end{array}\right|=1-c_{12}^{2}-c_{13}^{2}-c_{23}^{2}+2c_{12}c_{13}c_{23}.
\left<T^{\mu}{}_{\nu}\right>=\frac{1}{r^{4}}\left(\begin{array}{cccc}{-A}&{0}&{0}&{0}\\{0}&{-A}&{0}&{0}\\{0}&{0}&{3A}&{\kappaB/r^{2}}\\{0}&{0}&{\kappaB}&{-A}\\\end{array}\right),
\int\!dx\,_{R}\langle\,\psi_{0}(x)\midH-E\mid\psi_{0}(x)\,\rangle_{R}\mid\psi(y)\,\rangle+\int\!dx\,_{R}\langle\,\psi_{0}(x)\midH\mid\Psi_{Q}(x,y)\,\rangle=0
S_{SD}[\vec{B}_{m},V_{mn}]=\intd^{5}x\left\{-{\textstyle\frac{1}{4}}m^{2}\vec{B}_{m}^{T}\vec{B}_{m}-{\textstyle\frac{1}{4}}{\calF}_{mn}{\calF}_{mn}-{\textstyle\frac{m}{8}}\epsilon\vec{B}_{m}^{T}\eta\mathcal{D}\vec{B}_{m}\,\right\}\,,
\begin{array}{rl}{\langled\psi^{\prime},\psi\rangle=}&{\displaystyle\int_{G}d\psi^{\prime}\wedge*\psi=(-1)^{q}\int_{G}\psi^{\prime}\wedged*\psi=(-1)^{q+(q-1)(r-q+1)}\int_{G}\psi^{\prime}\wedge*(*d*\psi)}\\{\equiv}&{\displaystyle\int_{G}\psi^{\prime}\wedge*\delta\psi=\langle\psi^{\prime},\delta\psi\rangle\quad,}\\\end{array}
S_{\mathrm{bulk}}=-{\frac{1}{2\kappa_{(10)}^{2}}}\intd^{10}x\sqrt{-G}\Biggl[e^{-\Phi}\Bigl(R-d\Phi^{2}+{\frac{1}{12}}dB^{2}\Bigr)+\sum{\frac{1}{2k!}}F_{(k)}^{2}\Biggr]
Ae(n_{1}n_{2}nk;i)=\sum_{n_{1}^{\prime}n_{2}^{\prime}n^{\prime}k^{\prime}i^{\prime}}A(n_{1}^{\prime}n_{2}^{\prime}n^{\prime}k^{\prime};i^{\prime}|n_{1}n_{2}nk;i)e(n_{1}^{\prime}n_{2}^{\prime}n^{\prime}k^{\prime};i^{\prime})
\left(\lambda_{\alpha\beta}^{Fg_{1},g_{2}}\right)^{-1}\,\left(\lambda_{\alpha\beta}^{Rg_{1},g_{2}}\right)\,\left(\kappa_{\alpha\beta}^{g_{1}g_{2}}\right)\:=\:\left(\kappa_{\alpha\beta}^{g_{2}}\right)\,\left(g_{2}^{*}\kappa_{\alpha\beta}^{g_{1}}\right)\,\left(h_{\alpha}^{g_{1},g_{2}}\right)\,\left(h_{\beta}^{g_{1},g_{2}}\right)^{-1}
u(x)=-2\partial_{x}^{2}\,\mathrm{ln}\,\theta[\delta]\left({\frac{x}{2\omega}},0|\left(\begin{array}{ll}{{\frac{1}{N}}{\frac{\omega^{\prime}}{\omega}}}&{{\frac{1}{N}}}\\{{\frac{1}{N}}}&{{\frac{1}{N}}{\frac{\widetilde\omega^{\prime}}{\widetilde\omega}}}\\\end{array}\right)\right),\;U_{2}=0,
S_{F}^{-1}=\not\!\!p_{L}\left[1-\Sigma_{L}({\vec{p}}_{L}^{2},{\vec{p}}_{\perp}^{2})\right]+\not\!\!p_{\perp}-\left[m+\Sigma_{0}({\vec{p}}_{L}^{2},{\vec{p}}_{\perp}^{2})\right].
f_{1}^{\mu\nu}+f_{2}^{\mu\nu}={\frac{1}{\sqrt{2N}}}{\overline{f}}_{+}^{\mu\nu}\mathrm{diag}\left(I_{N_{1}},I_{N_{2}}\right)+{\frac{1}{\sqrt{2N}}}{\overline{f}}_{-}^{\mu\nu}\mathrm{diag}\left(-\sqrt{\frac{N_{2}}{N_{1}}}I_{N_{1}},\sqrt{\frac{N_{1}}{N_{2}}}I_{N_{2}}\right)~.
u_{j,\alpha}^{(\infty)}(n)=\frac{P_{j}(-\hbar\tilde{\partial}_{t})\tau(n)}{\tau(n+1)^{\frac{1}{2}-\alpha}\tau(n)^{\frac{1}{2}+\alpha}}~,~~u_{j,\alpha}^{(0)}(n)=\frac{P_{j}(-\hbar\tilde{\partial}_{\bar{t}})\tau(n+1)}{\tau(n+1)^{\frac{1}{2}-\alpha}\tau(n)^{\frac{1}{2}+\alpha}}~.
\begin{array}{l}{{\displaystyle{\partial_{k}R(u,v)\,-\,\sum_{j=1}^{r}\,\partial_{j}A_{k}(u)\otimesH_{j}}}}\\{{\displaystyle{~-\,\Bigl[\,R(u,v)\,,\,A_{k}(u)\otimes1\,+\,1\otimesA_{k}(v)\,\Bigr]\,+\,\sum_{j=1}^{r}\,A_{j}(u)\otimes[H_{j}\,,A_{k}(v)]~=~0~.}}}\\\end{array}
\hat{P}^{\mu\nu}(k)=-\frac{e^{2}}{(2\pi)^{\frac{5}{2}}}\intd^{3}p\;\theta(p_{0})\delta(p^{2}-m^{2})\theta(k_{0}-p_{0})\delta[(k-p)^{2}-m^{2}]j^{\mu\nu}(k,p),
T_{ab}{}^{r}=2\big(({\calC}\Gamma^{r_{1}})_{ab}\,X^{r}{}_{r_{1}}+\frac{1}{2!}({\calC}\Gamma^{r_{1}r_{2}})_{ab}\,X^{r}{}_{r_{1}r_{2}}+\frac{1}{5!}({\calC}\Gamma^{r_{1}\cdotsr_{5}})_{ab}\,X^{r}{}_{r_{1}\cdotsr_{5}}\big)\,.
|u\rangle_{\tau}=N(u)\cdot\frac{\sqrt{u}}{\sqrt{1-u\tau}}\cdotexp\Bigl(\sum_{m=1}^{\infty}\left(\frac{m-u}{m+u}\right)a_{m}^{\dagger}\tilde{a}_{m}^{\dagger}e^{2m\tau}\Bigr)exp\Bigl(-\frac{u\hat{q}^{2}}{4(1-u\tau)}\Bigr)|0\rangle\,.
c(1)~=~4[(p_{1}\!\cdot\!p_{2})(\epsilon_{1}\!\cdot\!\epsilon_{2})(p_{3}^{+}p_{4}^{-}p_{5}^{+}p_{6}^{-})(\epsilon_{3}^{-}\epsilon_{4}^{+}\epsilon_{5}^{-}\epsilon_{6}^{+})].
=\langleo\midT\left(\overline{{C}}^{a_{1}}(x_{1})\,\deltaA^{\mu_{2},a_{2}}(x_{2})\,\partial_{\mu_{3}}A^{\mu_{3},a_{3}}(x_{3})\,\partial_{\mu_{4}}A^{\mu_{4},a_{4}}(x_{4})\right)\mido\rangle
|\varphi_{k}\dot{\varphi}_{k^{\prime}}-\dot{\varphi}_{k}\varphi_{k^{\prime}}|=|\dot{\varphi}_{k}|^{2}|\varphi_{k^{\prime}}|^{2}+|\dot{\varphi}_{k}|^{2}|\varphi_{k^{\prime}}|^{2}+\frac12(\dot{\varphi}_{k}\varphi_{k}^{*}+\varphi_{k}\dot{\varphi}_{k}^{*})(\dot{\varphi}_{k^{\prime}}\varphi_{k^{\prime}}^{*}+\varphi_{k^{\prime}}\dot{\varphi}_{k^{\prime}}^{*})+2
\mathrm{F}^{T}\mathrm{M}\mathrm{F}=\left(\begin{array}{cc}{{\tilde{\calF}}}&{{\calF}}\\\end{array}\right)e^{\phi}\left(\begin{array}{cc}{e^{-2\phi}+C^{(0)\,2}}&{C^{(0)}}\\{C^{(0)}}&{1}\\\end{array}\right)\left(\begin{array}{c}{{\tilde{\calF}}}\\{{\calF}}\\\end{array}\right)\,,
\partial_{t}^{2}\left(\begin{array}{c}{\varphi}\\{{(\pi_{\chi}+{\frac{\kappa}{2}}\varphi)/\mu}}\\\end{array}\right)=\left(\begin{array}{cc}{\nabla^{2}-\mu^{2}-{\kappa^{2}}}&{\kappa\mu}\\{\kappa\mu}&{\nabla^{2}-\mu^{2}}\\\end{array}\right)\left(\begin{array}{c}{\varphi}\\{{(\pi_{\chi}+{\frac{\kappa}{2}}\varphi)/\mu}}\\\end{array}\right)
{\frac{d^{2}y}{d\lambda^{2}}}+{\frac{hh^{\prime}}{2}}\left({\frac{dt}{d\lambda}}\right)^{2}-{\frac{1}{2}}{\frac{h^{\prime}}{h}}\left({\frac{dy}{d\lambda}}\right)^{2}-hy\sum_{j}\left({\frac{dx^{j}}{d\lambda}}\right)^{2}=0,
T_{+p-2}^{-q}=-2i\gamma_{q\dot{q}}^{i}\psi_{-2\dot{q}}^{2+}\gamma_{p\dot{p}}^{i}\psi_{+2\dot{q}}^{1-},\quadT_{+2-2}^{+q}=-\frac{i}{16}\nabla_{+p}T_{+p-2}^{-q}.
\left(\begin{array}{cc}{u}&{i\alpha}\\{i\beta}&{\bar{u}}\\\end{array}\right)\sim\left(\begin{array}{cc}{e^{2\piir_{+}}u}&{ie^{2\pir_{-}}\alpha}\\{i\betae^{-2\pir_{-}}}&{\bar{u}e^{-2\piir_{+}}}\\\end{array}\right)\,.
T\simeq\frac{2}{\sqrt{\pi}}\,c\,(x^{0})^{2+\alpha}x^{1}\,,\qquad\dot{T}\simeq\frac{2(2+\alpha)}{\sqrt{\pi}}\,c\,(x^{0})^{1+\alpha}x^{1}\,,\qquadT^{\prime}\simeq\frac{2}{\sqrt{\pi}}\,c\,(x^{0})^{2+\alpha}\,.
\begin{array}{ll}{M_{n}^{d}(w,z_{k},\eta)=\displaystyle\int_{0}^{\infty}dy\;y^{d/2}\displaystyle\frac{1+r(y)+yr^{\prime}(y)}{(P(y)+w)^{n}}}\\{\Bigg\{y(1+r(y)+yr^{\prime}(y))(\etar(y)+2yr^{\prime}(y))\bigg(\displaystyle\frac{n}{P(y)+w}\bigg)}\\{-2\eta(r(y)+yr^{\prime}(y))-4y(2r^{\prime}(y)+yr^{\prime\prime}(y))\Bigg\}}\\\end{array}
\mathcal{S}=\int\frac{1}{2l_{11}^{9}}\left(d^{11}x\sqrt{-g}R-\frac{1}{2\cdot4!}d\mathcal{C}\wedge*d\mathcal{C}\right)+\int\frac{\sqrt{2}}{2^{7}\cdot3^{3}}\mathcal{C}\wedged\mathcal{C}\wedged\mathcal{C}\.
P_{12}\cdotsP_{N-1N}G_{\varepsilon}^{(nl)}(\zeta_{2},\cdots,\zeta_{N},\zeta_{1}q^{-2})=\varepsilonr(\zeta_{1})D_{1}G_{\varepsilon}^{(nl)}(\zeta_{1},\cdots,\zeta_{N}),\,
S_{2}^{(0)}=\varphi^{I}D_{1II^{\prime}}\varphi^{I^{\prime}}+\varphi_{Ia}^{*}D_{2\alpha}^{I}c^{\alphaa}+\bar{\varphi}_{I}D_{3\alpha}^{I}B^{\alpha}+\varepsilon^{ab}c_{\alphab|a}^{*}D_{4\beta}^{\alpha}B^{\beta}.
\gamma_{0}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)~,\qquad\gamma_{1}=\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right)~,\qquad\gamma_{5}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right),
\begin{array}{ccc}{\partial^{\mu}C_{n\mu}^{F}=0,}&{\quad\partial_{\mu}b_{n}^{F\mu}=0,}&{\quad\partial_{\mu}\bar{C}_{n}^{F\mu}=0,}\\{\partial^{\mu}C_{n\mu}^{B}=0,}&{\quad\partial_{\mu}b_{n}^{B\mu}=0,}&{\quad\partial_{\mu}\bar{C}_{n}^{B\mu}=0.}\\\end{array}
+{\sum_{m=1}^{\infty}}{\sum_{\alpha>0}}\frac{\alpha^{2}}{2}\left[E_{\alpha}^{(m)}\otimesE_{-\alpha}^{(-m)}+E_{-\alpha}^{(m)}\otimesE_{\alpha}^{(-m)}\right],\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\psi_{-}^{\underline{m}}=\sqrt{8}\lambda_{-}^{\underline{\alpha}}\sigma_{\underline{\alpha}\underline{\beta}}^{\underline{m}}\theta^{2\underline{\beta}},\\,psi_{+}^{\underline{m}}=\sqrt{8}\lambda_{+}^{\underline{\alpha}}\sigma_{\underline{\alpha}\underline{\beta}}^{\underline{m}}\theta^{1\underline{\beta}};
0=-\intd^{N^{2}}Xd^{N^{2}}Y\,\sum_{\alpha=1}^{N^{2}}\left[\frac{\partial}{\partialX_{\alpha}}\mathrm{e}^{-S}\frac{\partial}{\partialX_{\alpha}}+\frac{\partial}{\partialY_{\alpha}}\mathrm{e}^{-S}\frac{\partial}{\partialY_{\alpha}}\right]\mathrm{e}^{K\cdot\Psi}.
\tilde{M}_{tN}=\left(\begin{matrix}{A}&{D}&{0}&{\dots}&{\dots}&{0}&{E}\\{E}&{A}&{D}&{0}&{\dots}&{\dots}&{0}\\{0}&{E}&{A}&{D}&{\dots}&{\dots}&{\dots}\\{\dots}&{0}&{E}&{A}&{\dots}&{\dots}&{\dots}\\{\dots}&{\dots}&{\dots}&{\dots}&{\dots}&{\dots}&{\dots}\\{\dots}&{\dots}&{\dots}&{\dots}&{\dots}&{\dots}&{D}\\{D}&{0}&{\dots}&{0}&{\dots}&{E}&{A}\\\end{matrix}\right).
g_{\mu\nu}^{(1)}=\left(\begin{array}{cc}{l_{1}^{2}}&{\frac{1}{2}(l_{1}^{2}+l_{2}^{2}-l_{3}^{2})}\\{\frac{1}{2}(l_{1}^{2}+l_{2}^{2}-l_{3}^{2})}&{l_{2}^{2}}\\\end{array}\right).
\begin{array}{cc}{G_{-}(i_{5})^{-1}=C^{-1}G_{+}(i_{5})^{t}C~~~~}&{~~~~K_{-}(z;i_{5})^{t}=\left(\begin{array}{cc}{\tilde{\gamma}{}^{0}}&{0}\\{0}&{{\calE}}\\\end{array}\right)K_{+}(z;i_{5})^{\dagger}\left(\begin{array}{cc}{\gamma^{0}}&{0}\\{0}&{\bar{{\calE}}}\\\end{array}\right)}\\\end{array}
q^{2}\xi_{11}(p,q)-p.q\xi_{20}(p,q)={\frac{1}{2}}\left\{{\frac{-Z_{0}(p+q)^{2};m^{2})}{2}}+{\frac{Z_{0}(p^{2};m^{2})}{2}}+q^{2}\xi_{10}(p,q)\right\}
M_{U\\scriptstyle\mathrm{min}}^{E_{8}-E_{6}}\approx5.49\times10^{17}\timesg_{U}^{E_{8}-E_{6}}\times\frac{1}{\sqrt{1+0.08\times\left(g_{U}^{E_{8}-E_{6}}\right)^{2}}}\,\mathrm{GeV}\,.
\Pi_{ij}^{\Sigma}={\frac{4\alpha}{\beta}}\int_{0}^{1}dx\int{\frac{d|{\bfl}|}{(2\pi)}}\left[{\frac{-(p_{b}^{2}\delta_{ij}-p_{i}p_{j})}{p}}{\frac{\partialS_{1}^{\Sigma}}{\partialp}}|{\bfl}|+(1-2x){\frac{\betap_{0b}}{2}}{\frac{\partialS_{1}}{\partial\omega}}\delta{ij}\right]