harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Big[\begin{array}{c}{v}\\{\emptyset}\\\end{array}\Big]\times\Big[\begin{array}{c}{v}\\{\emptyset}\\\end{array}\Big],\qquad\Big[\begin{array}{c}{\emptyset}\\{\emptyset}\\\end{array}\Big]\times\Big[\begin{array}{c}{v}\\{\emptyset}\\\end{array}\Big],\qquad\Big[\begin{array}{c}{v}\\{\emptyset}\\\end{array}\Big]\times\Big[\begin{array}{c}{\emptyset}\\{\emptyset}\\\end{array}\Big]
R_{MN}-\frac{1}{2}G_{MN}R=\frac{1}{4M^{3}}\left\{\partial_{M}\pi\partial_{N}\pi-G_{MN}\left(\frac{1}{2}(\partial\pi)^{2}+V\right)-G_{\mu\nu}\delta_{M}^{\mu}\delta_{N}^{\nu}\vartheta\delta(y)\right\}
\begin{array}{c}{-i\omega\epsilon\Psi^{1}+(\partial_{1}-i\partial_{2})\Psi^{4}+\partial_{3}\Psi^{3}=0,}\\{-i\omega\epsilon\Psi^{2}+(\partial_{1}+i\partial_{2})\Psi^{3}-\partial_{3}\Psi^{4}=0,}\\{-i\omega\mu\Psi^{3}+(\partial_{1}-i\partial_{2})\Psi^{2}+\partial_{3}\Psi^{1}=0,}\\{-i\omega\mu\Psi^{4}+(\partial_{1}+i\partial_{2})\Psi^{1}-\partial_{3}\Psi^{2}=0,}\\\end{array}
\tilde{T}=\left(\begin{matrix}{1}\\{z^{m}}\\{z^{n}}\\\end{matrix}\right)+\left(\begin{matrix}{1}\\{z^{m}}\\{z^{n}}\\\end{matrix}\right)\,\frac{1}{1+\bar{z}^{m}z^{m}+\bar{z}^{n}z^{n}}\,(\bar{z}^{m}f+\bar{z}^{n}g)-\left(\begin{matrix}{0}\\{f}\\{g}\\\end{matrix}\right),
D_{\mu\nu}(p,p_{3})=\frac{1}{p^{2}-p_{3}^{2}+i\epsilon}\left[-g_{\mu\nu}+p_{\mu}p_{\nu}(1-\delta_{p_{3},0})\frac{1}{p_{3}^{2}}+p_{\mu}p_{\nu}\delta_{p_{3},0}(1-\xi)\frac{1}{p^{2}+i\epsilon}\right]
g_{\mu\nu}^{grav}dx^{\mu}dx^{\nu}=\left.\hat{g}_{\mu\nu}\right|_{y=R(t)}dx^{\mu}dx^{\nu}\approx-\Omega_{R}^{2}c^{2}dt^{2}+a^{2}\Omega_{R}^{2}\gamma_{ij}dx^{i}dx^{j}.
\left(\begin{array}{cc}{(1-G^{-1}U^{T}Z^{-1}U)^{-1}G^{-1}}&{-(1-G^{-1}U^{T}Z^{-1}U)^{-1}G^{-1}U^{T}Z^{-1}}\\{-(1-Z^{-1}UG^{-1}U^{T})^{-1}Z^{-1}UG^{-1}}&{(1-Z^{-1}UG^{-1}U^{T})^{-1}Z^{-1}}\\\end{array}\right)
L^{int}=-e(\overline{\psi}_{1}\gamma^{\mu}\gamma^{5}\psi_{1}-\overline{\psi}_{2}\gamma^{\mu}\gamma^{5}\psi_{2})C_{\mu}+ie(\overline{\psi}_{1}\gamma^{\mu}\psi_{1}+\overline{\psi}_{2}\gamma^{\mu}\psi_{2})B_{\mu}
F_{(5)}=4\betadx^{+}\left(d\tilde{x}^{1}d\tilde{x}^{2}d\tilde{x}^{3}d\tilde{x}^{4}+d\tilde{x}^{5}d\tilde{x}^{6}d\tilde{x}^{7}d\tilde{x}^{8}\right).
\nabla^{\dot{A}}G_{A\dot{A}}=-\frac{1}{4}\alpha\left(\nabla_{A}W^{2}\right)\overline{{\nabla}}^{2}\overline{{W}}^{2}-i\alphaW^{2}\nabla^{\dot{A}}\nabla_{A\dot{A}}\overline{{W}}^{2}+{\calO}\left(\alpha^{2}\right)
\delta\alpha_{1}+\gamma\left(-\frac12C_{\;\;bc}^{a}B_{a}^{\mu\nu}A_{\mu}^{b}A_{\nu}^{c}\right)=\partial_{\mu}\left(-\frac12C_{\;\;bc}^{a}\varepsilon^{\mu\nu\lambda\rho}A_{\nu}^{b}A_{\lambda}^{c}\eta_{a\rho}\right).
\begin{array}{c}{\dot{\bfx}={\bfe}_{1},}\\{{\dot{\bfe}}_{1}=k_{1}{\bfe}_{2},}\\{{\dot{\bfe}}_{2}=-k_{1}{\bfe}_{1}+k_{2}{\bfe}_{3},}\\{{\dot{\bfe}}_{3}=-k_{2}{\bfe}_{2}+k_{3}{\bfp}_{3}/c}\\{{\dot{\bfp}}_{3}=-k_{3}c{\bfe}_{3}-k_{2}{\bfp}_{2},}\\{{\dot{\bfp}}_{2}=-k_{1}{\bfp}_{1}+k_{2}{\bfp}_{3},}\\{{\dot{\bfp}}_{1}=-s{\bfp}+k_{1}{\bfp}_{2}+sc_{0}{\bfe}_{1},}\\{{\dot{\bfp}}=0.}\\\end{array}
\tilde{g_{1}}=\frac{\alpha_{1}g_{1}+\beta_{1}}{\overline{{\beta_{1}}}g_{1}+\overline{{\alpha_{1}}}},~\tilde{g_{2}}=\frac{\alpha_{2}g_{1}+\beta_{2}}{\pm\overline{{\beta_{2}}}g_{2}+\overline{{\alpha_{2}}}},~~\mathrm{with}~~|\alpha_{i}|^{2}\mp|\beta_{i}|^{2}=1~~,
L_{0}=\pi\sum_{n}\left[(|n+mH\alpha^{\prime}|+n)^{2}A_{1n}A_{1n}^{\dagger}+(|n-mH\alpha^{\prime}|+n)^{2}A_{2n}A_{2n}^{\dagger}\right]-\frac{\pi}{2}m^{2}\alpha^{2},
f_{\mu\nu}^{\xi}(x)=\partial_{\mu}[n^{A}(x){\calA}_{\nu}^{A}(x)]-\partial_{\nu}[n^{A}(x){\calA}_{\mu}^{A}(x)]-{\frac{1}{g}}f^{ABC}n^{A}(x)\partial_{\mu}n^{B}(x)\partial_{\nu}n^{C}(x),
P=\frac{2M^{2}}{3e^{2}}\left(\frac{1}{B}-(2a+c)\right),\;\;\;Q=-\frac{2M^{2}}{3e^{2}}\left(\frac{2}{B}+(2a+c)\right),\;\;\;R=\frac{M^{2}}{2e^{2}}\left(2m+n-4d\right).
S_{SD}[B,\tilde{B}]=\intd^{5}x\left\{-{\textstyle\frac{1}{4}}m^{2}B^{2}-{\textstyle\frac{1}{4}}m^{2}\tilde{B}^{2}-{\textstyle\frac{m}{12}}\epsilon\tilde{H}B\right\}\,.
\delta\psi_{t}^{1}=\left[\mathcal{\partial}_{t}+\frac{1}{2}V^{\prime}e^{V-W}\gamma_{0}\gamma_{1}+\frac{i}{\sqrt{6}}e^{-W}A_{t}^{\prime}\gamma_{1}+{\frac{1}{4\sqrt{6}}}ge^{2\phi}e^{V}\gamma_{0}\right]\epsilon^{1}.
\left|\left\{\Theta^{4}\left[\begin{array}{ll}{0}&{0}\\{0}&{0}\\\end{array}\right]+\Theta^{4}\left[\begin{array}{ll}{0}&{0}\\{0}&{1/2}\\\end{array}\right]+\Theta^{4}\left[\begin{array}{ll}{0}&{0}\\{1/2}&{0}\\\end{array}\right]+\Theta^{4}\left[\begin{array}{ll}{0}&{0}\\{1/2}&{1/2}\\\end{array}\right]\right\}(0\vert\Omega)\right|^{2}
ds_{10}^{2}=g_{\mu\nu}^{5}dx^{\mu}dx^{\nu}+l^{2}\sum_{i=1}^{3}[d\mu_{i}^{2}+\mu_{i}^{2}(d\varphi_{i}+\frac{2}{\sqrt{3}}A_{\mu}dx^{\mu})^{2}],
<x|A^{\alpha}e^{-tA}|x>=e^{-tm^{2}}\int\frac{d^{d}k}{(2\pi)^{d}\sqrt{g(x)}}\int_{C}\frac{id\lambda}{2\pi}(\lambda+m^{2})^{\alpha}e^{-t\lambda}\sigma^{0}(x,k;\lambda),
\sum_{k}(k-1)\beta_{i}^{k-2}W_{kj}=\frac{\sum_{k}(k-1)\beta_{i}^{k-2}(-1)^{n-k}S_{n-k}(\beta_{1},\dots,\hat{\beta}_{j},\dots,\beta_{n})}{\prod_{s\neqj}(\beta_{j}-\beta_{s})}.
{\binom{1}{1}}{\binom{1}{-1}}{\binom{1}{1}},\,\,\,\,\,\,\,\,\,\,\,{\binom{1}{-1}}{\binom{1}{1}}{\binom{1}{1}},\,\,\,\,\,\,\,\,\,\,\,{\binom{1}{1}}{\binom{1}{1}}{\binom{1}{-1}},
\begin{array}{ll}{A\alpha_{0}=\alpha_{2n+1},\qquad}&{A\alpha_{1}=\alpha_{2n+2},}\\{A\alpha_{2n+1}=\alpha_{1},\qquad}&{A\alpha_{2n+2}=\alpha_{0},}\\{A\alpha_{j}=\alpha_{2n+2-j},\qquad}&{j=2,\ldots,2n.}\\\end{array}
\begin{array}{lcl}{\lambda^{(0)}=\gamma_{1,6}\lambda^{(0)}\gamma_{1,6}^{-1}}&{,}&{\lambda^{(0)}=-\gamma_{\OmegaR,6}\lambda^{(0)^{T}}\gamma_{\OmegaR,6}^{-1}}\\{\lambda^{(i)}=e^{2\piiv_{i}}\gamma_{1,6}\lambda^{(i)}\gamma_{1,6}^{-1}}&{,}&{\lambda^{(i)}=-\gamma_{\OmegaR,6}\lambda^{(i)^{T}}\gamma_{\OmegaR,6}^{-1}}\\\end{array}
\prodU\equivU_{2}U_{4}U_{3}^{-1}\cdotU_{3}U_{5}U_{1}^{-1}\cdotU_{1}U_{6}U_{2}^{-1}\cdotU_{2}U_{6}^{-1}U_{5}^{-1}U_{4}^{-1}U_{2}^{-1}=1.
{{\calG}^{-1}}_{\mua}^{\nub}\eta_{\nub}^{\mathrm{L,L}}=\frac{1}{2}\left(k^{2}-\Lambda_{l}+\frac{1}{2r^{2}}(N-1)(N-2)-\frac{N+d}{N+d-2}\Lambda\right)\eta_{\mua}^{\mathrm{L,L}},
\mathbb{X}^{M}(\mathbf{z},\overline{{\mathbf{z}}})=\sqrt{\frac{2}{\alpha^{\prime}}}X^{M}(z,\overline{{z}})+i\theta\psi^{M}(z)+i\overline{{\theta}}\widetilde{\psi}^{M}(\overline{{z}})~.
l_{p}^{-2}ds^{2}=\hat{H}^{-1/3}\left[-\hat{f}dt^{2}+\sum_{i=1}^{5}(dx^{i})^{2}+l_{s}^{-4}\hat{H}\left(AdR^{2}+\frac{A}{K^{3}}dv^{2}+KR^{2}d\Omega_{3}^{2}\right)\right]
S_{\mathrm{int}}=-{\frac{1}{2}}\sum_{{\bfx},{\bfx^{\prime}}}(\theta{\calB}({\bfx},t)-{\frac{1}{2}})V({\bfx}-{\bfx}^{\prime})(\theta{\calB}({\bfx}^{\prime},t)-{\frac{1}{2}})
{R^{\mu}}_{\nu\rho\lambda}=\partial_{\rho}\Gamma_{\nu\lambda}^{\mu}-\partial_{\lambda}\Gamma_{\nu\rho}^{\mu}+\Gamma_{\nu\lambda}^{\omega}\Gamma_{\omega\rho}^{\mu}-\Gamma_{\nu\rho}^{\omega}\Gamma_{\omega\lambda}^{\mu}.
V\Big[\begin{array}{cc}{\alpha_{1}+b_{3}}&{\alpha_{2}}\\{\beta_{1}+b_{4}}&{\beta_{2}}\\\end{array}\Big]\propto\prod_{f\inF+b_{1}+b_{2}}\Theta^{1/2}\Big[\begin{array}{cc}{(\alpha_{1}+b_{3})(f)}&{\alpha_{2}(f)}\\{(\beta_{1}+b_{4})(f)}&{\beta_{2}(f)}\\\end{array}\Big].
{\calT}_{\alpha}^{\dagger}=[\Omega,\bar{C}_{\alpha}^{\dagger}]=T_{\alpha}^{\dagger}+\bar{C}_{\gamma}^{\dagger}U_{\beta\alpha}^{\dagger\gamma}C^{\beta}(-1)^{\varepsilon_{\alpha}}+C^{\dagger\beta}\bar{C}_{\gamma}^{\dagger}\bar{U}_{\alpha\beta}^{\dagger\gamma}(-1)^{\varepsilon_{\beta}}+C^{\dagger\beta}\bar{U}_{\beta\alpha}^{\gamma}\bar{C}_{\gamma}(-1)^{\varepsilon_{\alpha}}.
Z_{\mathrm{orb.}}=\mathrm{Tr}_{\mathrm{untw^{\prime}d}}\left({\frac{(1+R_{25})}{2}}q^{L_{0}-{\frac{1}{24}}}{\bar{q}}^{{\bar{L}}_{0}-{\frac{1}{24}}}\right)+\mathrm{Tr}_{\mathrm{tw^{\prime}d}}\left({\frac{(1+R_{25})}{2}}q^{L_{0}-{\frac{1}{24}}}{\bar{q}}^{{\bar{L}}_{0}-{\frac{1}{24}}}\right)
ds^{2}=-{\frac{(\rho^{2}-\rho_{+}^{2})(\rho^{2}-\rho_{-}^{2})}{l^{2}\rho^{2}}}dt^{2}+\rho^{2}\left(d\varphi-{\frac{\rho_{+}\rho_{-}}{l\rho^{2}}}dt\right)^{2}+{\frac{l^{2}\rho^{2}}{(\rho^{2}-\rho_{+}^{2})(\rho^{2}-\rho_{-}^{2})}}d\rho^{2}\,,
\frac{1}{|f|^{2}}=\frac{1}{1+|\beta|^{2}r^{2}+|\lambda|^{2}r^{4}}-\frac{\lambda\gamma^{*}z^{2}+\lambda^{*}\gammaz^{*2}}{(1+|\beta|^{2}r^{2}+|\lambda|^{2}r^{4})^{2}}+O(\vert\gamma\vert^{2}).
\prod_{i=1}^{m_{a}^{\prime}}\psi_{2}^{(a)}({y^{\prime}}_{i}^{(a)})\prod_{i=1}^{n_{a}}\overline{{\psi}}_{1}^{(a)}(x_{i}^{(a)}\!+\hat{\tau})\prod_{i=1}^{m_{a}}\psi_{2}^{(a)}(y_{i}^{(a)}\!+\hat{\tau})\prod_{i=1}^{n_{a}^{\prime}}\overline{{\psi}}_{1}^{(a)}({x^{\prime}}_{i}^{(a)})\Big\rangle\;.
\tilde{g}=-{\calF}(R)\,dt^{2}+{\calF}(R)^{-1}dR^{2}+R^{2}d\Omega^{2},\qquad{\calF}(R)=1-\frac{2M}{R}+\frac{Q^{2}}{R^{2}}-H^{2}R^{2}
(B_{i}^{\Lambda_{1}\dots\Lambda_{i}\Lambda_{i+1}\dots\Lambda_{n}})^{\dagger}Q^{\Lambda_{1}\dots\Lambda_{i}\Lambda_{i+1}\dots\Lambda_{n}}B_{i}^{\Lambda_{1}\dots\Lambda_{i+1}\Lambda_{i}\dots\Lambda_{n}}=Q^{\Lambda_{1}\dots\Lambda_{i+1}\Lambda_{i}\dots\Lambda_{n}}\.
\phi_{P}=\left({\frac{X^{2}}{X^{1}}}\right)_{P}={\frac{Y_{im}^{2}}{Y_{im}^{1}}}\,\qquad\phi_{M}=\left({\frac{X^{2}}{X^{1}}}\right)_{M}={\frac{Y_{re}^{2}}{Y_{re}^{1}}}\,
\hat{H}:\\\-\frac{\hbar^{2}}{2}\left[\partial_{r}^{2}+\frac{1}{r}\partial_{r}+\frac{1}{r^{2}}\partial_{\theta}^{2}+\partial_{z}^{2}\right]\,+\,\frac{1}{2}\omega^{2}r^{2}\\\.
\left(\begin{array}{cc}{a^{\prime\prime}}&{b^{\prime\prime}}\\{c^{\prime\prime}}&{d^{\prime\prime}}\\\end{array}\right)=\left(\begin{array}{cc}{a}&{b}\\{c}&{d}\\\end{array}\right)\left(\begin{array}{cc}{a^{\prime}}&{b^{\prime}}\\{c^{\prime}}&{d^{\prime}}\\\end{array}\right)
{\calS}(\Gamma)\equiv\int\left\{\partial_{\mu}c\frac{\delta\Gamma}{\deltaA_{\mu}}+B\frac{\delta\Gamma}{\delta\bar{c}}+\frac{\delta\Gamma}{\delta\underline{{Y}}}\frac{\delta\Gamma}{\delta\underline{{\varphi}}}+\underline{{q}}\frac{\delta\Gamma}{\delta\underline{{\hat{\varphi}}}}\right\}+\chi\partial_{\xi}\Gamma=0\;.
{\bfA}\star{\bfB}({\bfx})=e^{\frac{i}{2}\theta^{ij}\partial_{i}^{(1)}\partial_{j}^{(2)}}{\bfA}({\bfx}_{1}){\bfB}({\bfx}_{2})\vert_{{\bfx}_{1}={\bfx}_{2}={\bfx}},
(AB)_{ik}^{ac}(x,z)=\intd^{3}yA_{ij}^{ab}(x,y)B_{jk}^{bc}(y,z),~~~~~~~~~~\lambdaO\lambda=\intd^{3}xd^{3}y\lambda_{i}^{a}(x)O_{ab}^{ij}(x-y)\lambda_{j}^{b}(y)
V(\Phi,\chi)=-{\frac{m^{2}\Phi^{2}}{2}}+{\frac{\lambda_{1}\Phi^{6}}{6M_{\mathrm{P}}^{2}}}+{\frac{m^{2}\Phi_{0}^{2}}{3}}+{\frac{1}{2}}g^{2}\Phi^{2}\chi^{2}\,
\left.+\frac{2\lambda}{Mc_{0}}\left(l_{1}^{2}-\frac{1}{4}\right)q^{+}(\frac{1}{2},l_{1})\right]\psi(l_{1})+\left(l_{1}+\frac{1}{2}\right)\sqrt{(l_{1}+J)(l_{1}-J-1)}\psi(l_{1}-1)=0,
\frac{1}{u^{n}}\sum_{m_{1},\ldots,m_{n}=1}^{\infty}\left(C^{\prime}(0)F(0)+\frac{m_{1}F_{1}^{\prime}(0)+\ldots+m_{n}F_{n}^{\prime}(0)}{m_{1}+\ldots+m_{n}}\right)(\frac{v-1}{u})^{m_{1}+\ldots+m_{n}-n}.
\frac{d}{dr}\left(e^{(\Gamma-\Lambda)/2}(\Gamma^{\prime}-\phi^{\prime})\,r^{2}-\frac{\alpha^{\prime}e^{\phi}}{g^{2}}e^{(\Gamma-\Lambda)/2}[(\phi^{\prime}-\Gamma^{\prime})\,(1-e^{-\Lambda})+e^{-\Lambda}\phi^{\prime}\Gamma^{\prime}r]\right)=0
K_{\mu}\left(\begin{matrix}{r_{k}}\\{s_{k}}\\\end{matrix}\right)=\left(\begin{matrix}{r_{k}}\\{s_{k}}\\\end{matrix}\right)+(\,r_{k}\,q_{\mu}\,\,-s_{k}\,p_{\mu}\,)\left(\begin{matrix}{p_{\mu}}\\{q_{\mu}}\\\end{matrix}\right)\,.
\{\Psi^{i},\Psi^{k}\}=\frac{R_{vac}^{2}\sum_{s=1}^{N}|\Pi^{s}|^{2}}{4(\sum_{s=1}^{N}|\Pi^{s}|^{2}+R_{vac}^{2})^{3}}(\Pi^{i}\Pi^{*k}-\Pi^{*i}\Pi^{k}).
\chi_{1}^{}\chi_{3}^{-1}\chi_{5}^{}\chi_{7}^{-1}\ldots\chi_{2N-3}^{}\chi_{2N-1}^{-1}=\chi_{2}^{}\chi_{4}^{-1}\chi_{6}^{}\chi_{8}^{-1}\ldots\chi_{2N-2}^{}\chi_{2N}^{-1}\;.
\frac{1}{2}\left(\frac{W^{\prime}}{\sigma}\right)^{\prime}r+\left(\frac{W}{\sigma}\right)^{\prime}+\sigma\left(wr\Phi^{2}-\frac{(1-f^{2})^{2}}{r^{3}}e^{-2\Phi}\right)+4\beta\left(\frac{W^{\prime}w\Phi^{\prime}e^{-2\Phi}}{\sigma}\right)^{\prime}=0\;,
\begin{array}{l}{[{H}_{\pi_{\alpha}}]:={\dot{u}}^{\alpha}(x)-\frac{\partial{H}}{\partial\pi_{\alpha}},}\\{[{H}_{u^{\alpha}}]:={\dot{\pi}}_{\alpha}(x)+\frac{\partial{H}}{\partialu^{\alpha}}-\nabla_{a}(\frac{\partial{H}}{\partial(\nabla_{a}u^{\alpha})}).}\\\end{array}
\sum_{\vec{k}}\left\{{\frac{d}{dt}}\left[\delta\Pi\\varphi(\vec{k},\,t)\right]-\left[\delta\dot{\Pi}+\delta\varphi\\omega^{2}(\vec{k})\right]\\varphi(\vec{k},\,t)+\left[-\delta\Pi+\delta\dot{\varphi}\right]\\Pi(\vec{k},\,t)\right\}=\dot{\Omega}\,
P_{(2)}(q)=\frac{1}{N!}\sum_{\mathrm{perm}(q_{1}\dotsq_{N})}\left(4\sum_{i,j}(Q_{i}\cdotQ_{j})\left(\frac{1}{N+1}\right)-\sum_{i}Q_{i}^{2}\left(\frac12+\frac{3}{N+1}\right)\right).
\left|B_{(j,n)}^{k}\right>=\sum_{\left(j^{\prime},n^{\prime}\right)}\frac{S_{(j,n),(j^{\prime},n^{\prime})}^{k}}{\sqrt{S_{(0,0)(j^{\prime},n^{\prime})}^{k}}}\left|\left.\left(j^{\prime},n^{\prime}\right)\right>\right>,
\nabla_{a}T^{ab}[g,\phi]=0\;\qquadT^{ab}[g,\phi]={\frac{2}{\sqrt{-g}}}{\frac{\deltaS[g,\phi]}{\deltag_{ab}}}=\partial_{a}\phi\partial_{b}\phi-{\scriptstyle{\frac{1}{2}}}(\partial\phi)^{2}g_{ab}
\begin{array}{lll}{l_{1}=l_{23}/N,}&{l_{2}=l_{13}/N,}&{l_{3}=l_{12}/N}\\{l_{1}^{\prime}=l_{45}/N,}&{l_{2}^{\prime}=l_{15}/N,}&{l_{3}^{\prime}=l_{14}/N}\\{l_{1}^{\prime\prime}=l_{46}/N,}&{l_{2}^{\prime\prime}=l_{26}/N,}&{l_{3}^{\prime\prime}=l_{24}/N}\\{l_{1}^{\prime\prime\prime}=l_{56}/N,}&{l_{2}^{\prime\prime\prime}=l_{36}/N,}&{l_{3}^{\prime\prime\prime}=l_{35}/N}\\\end{array}
g(z)=\frac{(x^{6}z;x^{4},x^{4})_{\infty}(x^{2}z^{-1};x^{4},x^{4})_{\infty}}{(x^{8}z;x^{4},x^{4})_{\infty}(x^{4}z^{-1};x^{4},x^{4})_{\infty}}.
<\Psi_{1}|\Psi_{2}>=N\int_{-\infty}^{\infty}\frac{d^{2}\vec{p}}{p^{0}}\int_{0}^{2\pi}\frac{d\varphi}{(p,n)}\overline{{\Psi_{1}(p,\varphi)}}\Psi_{2}(p,\varphi),
\deltaZ^{\underline{M}}E_{\underline{M}}^{\underline{\alpha}}=\deltaz^{M}e_{M}^{A}(e_{A}^{N}\partial_{N}Z^{\underline{M}}E_{\underline{M}}^{\underline{\alpha}})\equiv\deltaz^{M}e_{M}^{A}E_{A}^{\underline{\alpha}},
1,\{{\gamma}_{i_{1}}\},\{{\gamma}_{i_{1}i_{2}}=\frac{1}{2}({\gamma}_{i_{1}}{\gamma}_{i_{2}}-{\gamma}_{i_{2}}{\gamma}_{i_{1}})\},\{{\gamma}_{i_{1}i_{2}i_{3}}\},...,\{{\gamma}_{i_{1}...i_{7}}\},{\gamma}_{9}={\gamma}_{1}....{\gamma}_{8}.
\hat{Q}_{1}(\bar{z})=\bar{z}_{1}\frac{\partial}{\partial\bar{z}_{1}}+\bar{z}_{2}\frac{\partial}{\partial\bar{z}_{2}},\,hatQ_{2}(\bar{z})=\bar{z}_{1}\frac{\partial}{\partial\bar{z}_{1}}-\bar{z}_{3}\frac{\partial}{\partial\bar{z}_{3}}.
S^{(k)}=\frac12\intdt\left[g_{pq}d_{\tau}Y^{p}d_{\tau}Y^{q}+g_{pq}i\bar{\alpha}^{p}\gamma^{0}D_{\tau}\alpha^{q}+\frac{1}{12}R_{pqrs}(\bar{\alpha}^{p}\alpha^{r})(\bar{\alpha}^{q}\alpha^{s})\right]\,
Z_{\mathrm{new}}=\sum_{N=0}^{\infty}{\frac{1}{N!}}\otimes\Big(\prod_{n=1}^{N}\int[dY_{n}^{a}]\Big)\otimes\sum_{h=0}^{\infty}{\frac{1}{h!}}\otimes\sum_{a_{1},\cdots,a_{h}=1}^{N}{\frac{h!}{m_{1}!m_{2}!\cdotsm_{N}!}}
|\Omega_{\theta}>=\sum_{\ell=-\infty}^{+\infty}\,e^{i\ell\theta}\,e^{-\sum_{n=1}^{\infty}A_{+,n}^{\dagger}A_{-,n}^{\dagger}}\,|n_{+}=\ell,n_{-}=\ell;0>_{\Phi}\,
C\,\left(\chi_{s_{1}}^{i},\chi_{s_{1}}^{\bar{i}}\bigm|\chi_{s_{2}}^{j},\chi_{s_{2}}^{\bar{j}}\bigm|\chi_{s_{3}}^{l},\chi_{s_{3}}^{\bar{l}}\right)=C^{k}\left(s_{1},s_{2},s_{3}\right)\,C_{\,jli}\left(s_{2},s_{3}\bigm|s_{1}\right)_{\bar{j}\,\bar{l}\,\bar{i}}\,
S={\frac{1}{4\pi\alpha^{\prime}}}\int_{D}d^{2}z\partial_{a}X^{i}\partial_{a}X^{j}G_{ij}+{\frac{1}{4\pi\alpha^{\prime}}}\int_{D}dzd\bar{z}J^{i}(z)\bar{J}^{j}(\bar{z})B_{ij},
M_{2}(t,u)=\Bigr(-{\frac{3}{16}}+{\frac{1}{2}}u-{\frac{1}{2}}u^{2}\Bigr)t^{2}+\Bigr({\frac{5}{8}}-{\frac{1}{2}}u\Bigr)t^{4}-{\frac{7}{16}}t^{6},
I^{(6)}=\intd^{6}x\sqrt{-g^{(6)}}(e^{2\psi-2\phi}(R^{(6)}+3(D\psi)^{2}-8D_{\alpha}\psiD^{\alpha}\phi+4(D\phi)^{2})-\frac{1}{12}H^{(6)2}),
ds^{2}=\frac{1}{H_{1}\tilde{H}_{1}}dt^{2}-\frac{H_{1}}{\tilde{H}_{1}}ds_{1}^{2}-\frac{\tilde{H}_{1}}{H_{1}}ds_{1}^{2}-H_{1}\tilde{H}_{1}ds_{3}^{2}\quad,\quade^{-2\phi}\sim1\.
\begin{array}{rcl}{}&{}&{\ddot{\phi}+4\,a^{-1}\dot{a}\dot{\phi}=0,}\\{}&{}&{a^{-1}\ddot{a}+\frac{1}{3}\,\dot{\phi}^{2}+\frac{1}{12}\Lambda=0,}\\{}&{}&{a^{-1}\ddot{a}+3\,a^{-2}\dot{a}^{2}+\frac{1}{3}\Lambda=\tilde{\Lambda}a^{-2}.\,}\\\end{array}
T_{1}=-{\frac{i}{2}}\left(\begin{matrix}{0}&{1}\\{1}&{0}\\\end{matrix}\right)~~~;~~~T_{2}={\frac{1}{2}}\left(\begin{matrix}{0}&{-1}\\{1}&{0}\\\end{matrix}\right)~~~;~~~T_{3}=-{\frac{i}{2}}\left(\begin{matrix}{1}&{0}\\{0}&{-1}\\\end{matrix}\right)\.
U=\left(\begin{array}{cc}{e^{v/\sqrt{3}}M}&{-e^{v/\sqrt{3}}M\sigma}\\{-e^{v/\sqrt{3}}\sigma^{T}M}&{e^{-2v/\sqrt{3}}+e^{v/\sqrt{3}}\sigma^{T}M\sigma}\\\end{array}\right)\,
\frac{\theta_{3}^{8}(\tau)\theta_{4}^{8}(\tau)}{\eta^{32}(\tau)}=\frac{1}{\eta^{16}(2\tau)},\;\;\;\frac{\theta_{2}^{8}(\tau)\theta_{3}^{8}(\tau)}{2^{8}\eta^{32}(\tau)}=\frac{1}{\eta^{16}(\frac{\tau}{2})},\;\;\;\frac{\theta_{2}^{8}(\tau)\theta_{4}^{8}(\tau)}{2^{8}\eta^{32}(\tau)}=e^{\frac{8\pii}{3}}\frac{1}{\eta^{16}(\frac{\tau+1}{2})}.
\int\frac{d^{3}\xi}{(2\pi)^{3}}C_{0}(x,\xi/|\xi|;-1)|\xi|^{-1}e^{i\xi.(x-y)}=\frac{1}{4\pi}\left\{-im\frac{1}{|z|}+eA_{\alpha}(x)\frac{z_{\alpha}\not\!z}{|z|^{3}}\right\},
M=\left(\begin{matrix}{1}&{\frac{g_{12}g_{33}-g_{13}g_{23}}{g_{22}g_{33}-g_{23}^{2}}}&{\frac{g_{13}}{g_{33}}}\\{0}&{1}&{\frac{g_{23}}{g_{33}}}\\{0}&{0}&{1}\\\end{matrix}\right)\left(\begin{matrix}{\frac{1}{\sqrt{g_{22}g_{33}-g_{23}^{2}}}}&{0}&{0}\\{0}&{\sqrt{g_{22}-\frac{g_{23}^{2}}{g_{33}}}}&{0}\\{0}&{0}&{\sqrt{g_{33}}}\\\end{matrix}\right).
W=\sum_{e_{0,1,2,5,6}}a_{e_{0,1,,2,5,6}}{\calX}_{0}^{e_{0}}{\calX}_{1}^{e_{1}}{\calX}_{2}^{e_{2}}{\calX}_{3}^{{\frac{e_{1}+e_{2}}{2}}}{\calX}_{4}^{-{\frac{e_{1}+e_{2}}{2}}+2[6e_{0}-(2e_{5}+3e_{6})]}{\calX}_{5}^{e_{5}}{\calX}_{6}^{e_{6}}{\calX}_{7}^{6e_{0}-(2e_{5}+3e_{6})},
f^{(n,0)}(r_{1},\dots,r_{n-2},r_{n-1},r_{n}+2)=\frac{2r_{n-1}+2}{2r_{n}+2}\frac{r_{n}+2}{r_{n-1}+2}f^{(n,0)}(r_{1},\dots,r_{n-2},r_{n-1}+2,r_{n}).
\begin{aligned}{U(q)=\sum_{i=1}^{N}\frac{q_{i\,xx}^{2}}{4(h-q_{i\,x})}}&{{}-\frac{1}{2}\sum_{j\neqi}\bigl[(h-q_{j\,x})q_{i\,xx}-(h-q_{i\,x})q_{j\,xx}\bigr]\zeta(q_{i}-q_{j})+{}}\\{{}}&{{}+\frac{1}{2}\sum_{j\neqi}\bigl[(h-q_{j\,x})^{2}(h-q_{i\,x})+(h-q_{j\,x})(h-q_{i\,x})^{2}\bigr]\wp(q_{i}-q_{j}).}\\\end{aligned}
\Theta_{-b/2}^{++}(z_{1})V_{\alpha}(z_{2})\sim|z_{12}|^{b\alpha}C_{+}(\alpha)\Theta_{\alpha-b/2}^{++}(z_{2})+|z_{12}|^{b(Q-\alpha)}C_{-}(\alpha)\Theta_{\alpha+b/2}^{--}(z_{2}),
{\calL}_{\mathrm{red}}^{\Psi}={\frac{1}{2}}J_{0}{\frac{1}{\vec{\partial}^{2}-M^{2}}}J_{0}+J_{i}\left(\delta_{ij}-{\frac{\partial_{i}\partial_{j}}{\vec{\partial}^{2}-M^{2}}}\right)V_{j}+\bar{\Psi}^{R}(i\not{\partial}-m)\Psi^{R}~.
(\lambda-3)v+\phi^{2}v^{\prime}=\frac{(N-1)v^{\prime}}{(2+2V^{\prime})^{3/2}}+\frac{v^{\prime}+\phi^{2}v^{\prime\prime}}{(2+2V^{\prime}+4\phi^{2}V^{\prime\prime})^{3/2}}
T_{1}=\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{0}&{0}\\{0}&{0}&{0}\\\end{array}\right)\;,\qquadT_{2}=\left(\begin{array}{ccc}{0}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{0}\\\end{array}\right)\;,\qquadT_{3}=\left(\begin{array}{ccc}{0}&{0}&{0}\\{0}&{0}&{0}\\{0}&{0}&{1}\\\end{array}\right)\;
\begin{array}{c}{\rho_{i}=g_{i}\delta^{4}\left(x-z_{i}\right),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\rho_{ij}=g_{ij}\delta^{4}\left(x-z_{ij}\right),}\\{1\leqj\leql.}\\\end{array}
\left[\bar{H}_{0}\left(x_{0}\right),b_{a}\left(x\right)\right]\eta_{2}^{a}\left(x\right)+\int\mathrm{d}^{D-1}x^{\prime}\;\left[j\left(x\right),\pi_{a}^{i}\left(x^{\prime}\right)\right]\partial_{i}\eta_{2}^{a}\left(x^{\prime}\right)=\partial_{i}k^{i}\left(x\right).
\begin{array}{l}{W[J,{\eta},\bar{\eta},M,K,L]=-\mathrm{ln}Z[J,{\eta},\bar{\eta},M,K,L]}\\{{\Gamma}[A_{\mu}^{a},B^{a},c^{a},\bar{c}^{a},K_{\mu}^{a},L^{a}]=W[A_{\mu}^{a},B^{a},c^{a},\bar{c}^{a},K_{\mu}^{a},L^{a}]-\left(A^{\mua}J_{\mu}^{a}+B^{a}M^{a}+\bar{\eta}^{a}c^{a}+\bar{c}^{a}{\eta}^{a}\right)\,,}\\\end{array}
\begin{array}{c}{_{N}\left\langlea\right|t^{(N)}(\lambda)\left|b\right\rangle_{N+1}=_{N}\left\langlea\right|t^{(N)}(\lambda)e^{i\omegaA}\left|b\right\rangle_{1}=}\\{=e^{i\omegab}\,_{N}\left\langlea\right|t^{(N)}(\lambda)\left|b\right\rangle_{1}}\\\end{array}
{x_{t}^{(1/3)}}^{\prime}(u)x_{t}^{(1/3)}(v)-{x_{t}^{(1/3)}}^{\prime}(v)x_{t}^{(1/3)}(u)=x_{t}^{(1/3)}(u+v)[\wp^{(1/3)}(v)-\wp^{(1/3)}(u)],
(1i):~~(-\lambda_{0})^{2}\frac{1}{4}\int_{C}d^{4}\eta\int_{C}d^{4}\eta^{\prime}D_{c}(\eta_{1}-\eta)[D_{c}(\eta-\eta^{\prime})]^{2}D_{c}(\eta^{\prime}-\eta^{\prime})D_{c}(\eta-\eta_{2})
w_{2}={\frac{\partialW}{\partialJ_{2}}}={\frac{Q^{2}m}{\pi\epsilon_{0}J_{2}}}\intdr(2mE-{\frac{J_{1}^{2}}{4\pi^{2}r^{2}}}-{\frac{Q^{2}m}{\pi\epsilon_{0}}}ln{\frac{r}{r_{0}}})^{-{\frac{1}{2}}}~
ds^{2}={\frac{x^{2}}{R_{1}^{2}}}(-dt^{2}+dz^{2})+R_{1}^{2}{\frac{dx^{2}}{x^{2}}}+R_{1}^{2}d\Omega_{3}^{2}+H_{5}^{\prime}(y)(dy^{2}+y^{2}d{\Omega_{3}^{\prime}}^{2}),
(f\starg)(x)=\int\frac{d^{4}k}{(2\pi)^{4}}\int\frac{d^{4}p}{(2\pi)^{4}}\,\mathrm{e}^{-\mathrm{i}(k_{\mu}+p_{\mu})x^{\mu}}\,\mathrm{e}^{-\frac{\mathrm{i}}{2}\theta^{\mu\nu}k_{\mu}p_{\nu}}\tilde{f}(k)\,\tilde{g}(p)~,
\begin{array}{c}{\Upsilon_{t}=\epsilon_{(1)c}\;\epsilon_{(2)c}...\epsilon_{(t)c}\Rightarrow}\\{\{\Upsilon_{t},p^{2}\}=-(a_{(1)c}\;\epsilon_{(2)c}...\epsilon_{(t)c}+\;...\;+\epsilon_{(1)c}...\epsilon_{(t-1)c}\;a_{(t)c})\equiv-A_{t}p^{2}.}\\\end{array}
\begin{array}{lcl}{\mathrm{SU}(8/1)}&{\longrightarrow}&{~\mathrm{SU}_{V}(4)\otimes\mathrm{SU}_{S}(4/1)\otimes\mathrm{U}^{a}(1)}\\{}&{\longrightarrow}&{~\mathrm{SU}_{V}(4)\otimes\mathrm{SU}_{S}(4)\otimes\mathrm{U}^{a}(1)\otimes\mathrm{U}^{b}(1)}\\{}&{\longrightarrow}&{~\mathrm{SU}_{V+S}(4)\otimes\mathrm{U}(1),}\\\end{array}
\left(C_{\mathbf{N}\,\,K}^{\mathbf{P}}\,\,\,\right)^{2}\,\propto\gamma_{i}^{\left(A\right)}\left(a,b,c\right)\mathcal{N}_{i}\left(a,b,c\right)\mathcal{N}_{n+1-i}\left(\overline{{a}},b,c\right).
\theta_{\xi_{j}}(x_{j0},t)=\left(\frac{m_{j}}{2\pi\hbarb_{j2}}\right)^{\frac12}exp\left[-\frac{im_{j}}{2\hbarb_{j2}}\left(b_{j1}x_{j0}^{2}-2x_{j0}\xi_{j}\right)\right].
(\Sigma_{\ReA_{l}}^{s})^{2}=\frac1{4R_{B}^{2}}\sum_{k\ge1}kC_{lk}^{2}\quad\mathrm{and}\quad(\Sigma_{\ImA_{l}}^{s})^{2}=\frac1{4R_{B}^{2}}\sum_{k\ge1}kS_{lk}^{2}.
s=\frac{1}{(1+k)|\mathrm{sn}(\psi)|}\mathrm{tn}^{-1}\left[\left(\frac{1+k}{1-k}\right)^{1/2}\frac{t\;r_{-}-r_{+}}{t\;r_{+}-r_{-}},\frac{2k^{1/2}}{1+k}\right]\nonumber
\tilde{T}\mapsto\tilde{T}_{\star}=\left(\begin{matrix}{1}\\{z^{m}}\\{z^{n}}\\\end{matrix}\right)+\left(\begin{matrix}{1}\\{z^{m}}\\{z^{n}}\\\end{matrix}\right)\star\,\frac{1}{1+\bar{z}^{m}\starz^{m}+\bar{z}^{n}\starz^{n}}\,\star(\bar{z}^{m}\starf_{\star}+\bar{z}^{n}\starg_{\star})-\left(\begin{matrix}{0}\\{f_{\star}}\\{g_{\star}}\\\end{matrix}\right),
\int_{-\infty}^{\infty}\Phi_{z^{\prime}}^{\ast\;\mu}(\eta)\Phi_{z}^{\mu}(\eta)\;d\eta=\delta(z-z^{\prime})\,,\;\int_{-\infty}^{\infty}\Phi_{z}^{\ast\;\mu}(\eta^{\prime})\Phi_{z}^{\mu}(\eta)\;dz=\delta(\eta-\eta^{\prime})\,.
\rho=\bar{\gamma}+\frac{1}{2G^{55}}\tilde{\calH}^{\nu\rho}\gamma_{\nu\rho}\gamma^{5}-\frac{1}{8G^{55}}\epsilon_{\mu\nu\rho\lambda\sigma}\tilde{\calH}^{\mu\nu}\tilde{\calH}^{\rho\lambda}\gamma^{\sigma5}.
\tilde{\Omega}^{\,^{\bullet}}({\mathcalA})\:=\\displaystyle\bigoplus_{k=0}^{\infty}\\tilde{\Omega}^{k}({\mathcalA})\;,\quad\tilde{\Omega}^{k}({\mathcalA})\:=\\Omega^{k}({\mathcalA})\big/K^{k}\.
\left(F,G\right)=\intd\tau\left(\frac{\delta^{R}F}{\delta\phi_{A}(\tau)}\frac{\delta^{L}G}{\delta\phi_{A}^{*}(\tau)}-\frac{\delta^{R}F}{\delta\phi_{A}^{*}(\tau)}\frac{\delta^{L}G}{\delta\phi_{A}(\tau)}\right),
G_{k_{2},k_{3}}^{(2)}[\lambda]=\delta_{h_{2},h_{3}}\left(\sum_{m=0}^{k_{2}+k_{3}}D_{(h_{2},h_{3};k_{2}+k_{3}-m)}\frac{(-2)^{m}}{m!}\lambda^{m}\right)\mathrm{e}^{-\lambda(h_{2}+h_{3})}\,,
f_{\alpha\beta}^{i}f_{\gamma\delta}^{i}\gamma^{\alpha\beta\delta\gamma}=f_{\alpha\beta}^{i}f_{\gamma\delta}^{i}\gamma^{\alpha\beta}\gamma^{\gamma\delta}+2f_{\alpha\beta}^{i}f_{\alpha\beta}^{i}=-16C_{2}(H,T_{i})+2R
\delta_{00}=\int_{r_{\mathrm{min}}}^{\infty}dr\left[\sqrt{E^{2}-p_{11}^{2}-{\frac{J^{2}}{r^{2}}}+{\frac{q_{0}}{r^{7}}}(E\mpp_{11})^{2}}-\sqrt{E^{2}-p_{11}^{2}-{\frac{J^{2}}{r^{2}}}}\right]
\begin{array}{l}{{U}_{k}{\calA}_{k}^{\dagger}={\calA}_{k}^{\dagger}\left({U}_{k}+1\right),}\\{{U}_{k}{\calA}_{k}={\calA}_{k}\left({U}_{k}-1\right),}\\{{\calA}_{k}^{\dagger}{\calA}_{k}=F\Big(m_{k},{U}_{k}\Big)F\Big(m_{N},H-\sum_{\ell=1}^{N-1}{U}_{\ell}\,+1\Big),}\\{{\calA}_{k}{\calA}_{k}^{\dagger}=F\Big(m_{k},{U}_{k}+1\Big)F\Big(m_{N},H-\sum_{\ell=1}^{N-1}{U}_{\ell}\Big).}\\\end{array}
\frac{1}{r^{2}-a^{2}}\partial_{r}(r-a)^{2}\partial_{r}\phi+\big[\omega^{2}H-\frac{r+a}{r-a}\frac{n^{2}}{4a^{2}}-\frac{\ell(\ell+1)}{r^{2}-a^{2}}\big]\phi=0.
D_{\mu}D_{\mu}\Phi^{(A)}-g_{1}\eta^{2}\Phi^{(A)}\left(\frac{1}{2}\mathrm{Tr}[\Phi^{(A)}]^{2}-1\right)-g_{2}\eta^{2}\Phi^{(B)}\left(\frac{1}{2}\mathrm{Tr}[\Phi^{(1)}\Phi^{(2)}]-c\right)=0,