harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{rl}{V_{-1/2}^{(j)}}&{=u_{\alpha}^{(j)}S^{\alpha}e^{-\phi/2}e^{i\vec{w}_{j}\cdot\vec{H}}e^{ik\cdotX}}\\{V_{-1}^{(j)}}&{=e^{-\phi}e^{i\vec{v}_{j}\cdot\vec{H}}e^{ik\cdotX}}\\\end{array}
{\hat{H}}=\frac{1}{r^{2}}\frac{d}{dr}r^{2}\left(\begin{array}{c}{\begin{array}{cc}{-1-E_{c}^{2}}&{E_{c}^{2}}\\{E_{c}^{2}}&{1-E_{c}^{2}}\\\end{array}}\\\end{array}\right)\frac{d}{dr}
=\{\chi^{\mu\nu}\}+\omega\left(\begin{array}{cccc}{2\zeta^{0}}&{\zeta^{1}}&{\zeta^{2}}&{(\zeta^{0}+\zeta^{3})}\\{\zeta^{1}}&{0}&{0}&{\zeta^{1}}\\{\zeta^{2}}&{0}&{0}&{\zeta^{2}}\\{(\zeta^{0}+\zeta^{3})}&{\zeta^{1}}&{\zeta^{2}}&{2\zeta^{3}}\\\end{array}\right).
\hat{\varrho}_{c}={\frac{3}{8\pi\hat{G}_{4}}}\left[\left({\frac{\dot{\hat{a}}_{0}}{\hat{a}_{0}}}-{\frac{\dot{\varepsilon}}{\varepsilon}}\right)^{2}-{\frac{\hat{C}c_{0}^{2}}{\hat{a}_{0}^{4}}}\right].
G_{D}[x_{2},x_{1}]=\sum_{n_{1}}\sum_{n_{2}}...\sum_{n_{D-1}}\,e^{i(n_{1}\theta_{1}+n_{2}\theta_{2}...+n_{D-1}\theta_{D-1})}\,G_{n_{1}n_{2}...n_{D-1}}[x_{2},x_{1}],
Q_{\infty}(\xi)=\frac{2\pi}{\ell^{2}\kappa}\left(r_{b}^{3}+(-6N^{2}+\ell^{2})r_{b}-2N(\ell^{2}-4N^{2})+\frac{N^{2}(\ell^{2}-3N^{2})}{r_{b}}\right)
Y_{1}=\left(\begin{array}{cccc}{x_{1}}&{0}&{0}&{\cdots}\\{0}&{x_{1}}&{0}&{\cdots}\\{\vdots}&{0}&{x_{1}}&{\cdots}\\{\vdots}&{\vdots}&{0}&{\ddots}\\{0}&{0}&{0}&{0}\\\end{array}\right).
{\frac{\frac{d}{d\tau}\,\zeta^{\mu}(\tau,{\vec{\sigma}}_{o})}{\sqrt{\eta_{\alpha\beta}\,\frac{d}{d\tau}\,\zeta^{\alpha}(\tau,{\vec{\sigma}}_{o})\,\frac{d}{d\tau}\,\zeta^{\beta}(\tau,{\vec{\sigma}}_{o})}}}={\tilde{U}}^{\mu}(\zeta(\tau,{\vec{\sigma}}_{o})),
{\calA}_{D1}={\frac{z_{12}z_{13}z_{14}}{2\,\prod_{i=1}^{4}y(z_{i})}}\sum_{j=2}^{4}k_{1}\cdotk_{j}f(z_{1},z_{j})\langle\prod_{i=1}^{4}\mathrm{e}^{ik_{i}\cdotX(z_{i},\bar{z}_{i})}\rangle.
G^{\mu\nu\alpha\beta}(x-x^{\prime})=\int{\frac{dp^{4}}{(2\pi)^{4}}}{\frac{{\frac{1}{2}}(\eta^{\mu\alpha}\eta^{\nu\beta}+\eta^{\mu\beta}\eta^{\mu\alpha})-{\frac{1}{2}}\eta^{\mu\nu}\eta^{\alpha\beta}}{p^{2}-i\epsilon}}e^{-ip(x-x^{\prime})}
\gamma_{m}^{N,k,d}-\gamma_{m+1}^{N,k,d}=L_{m}^{N,k,d}-\sum_{d^{\prime}=1}^{d-1}L_{m}^{N,k,d-d^{\prime}}\gamma_{m+(N-k)(d-d^{\prime})}^{N,k,d^{\prime}}.
\frac{m^{2}}{\mu^{2}}=\frac{2\sqrt{2}\displaystyle\sum_{n=2}^{\infty}n^{2}\alpha^{n-2}}{\displaystyle\sum_{n=2}^{\infty}n\alpha^{n-2}}=2\sqrt{2}\frac{4-3\alpha+\alpha^{2}}{(1-\alpha)(2-\alpha)}
\left|\beta_{k}\right|=\frac{\left|3m+i\right|}{\sqrt{\pi}}\frac{2^{n_{s}-3}\left|\Gamma\left(n_{s}\right)\right|}{|n_{1}(1-n_{1})|}z_{S}^{-1}\left(\frac{g_{S}}{g_{1}}\right)^{-l}\left(\frack{k_{s}}\right)^{2-n_{s}}\left(\frac{k_{re}}{k_{s}}\right)^{-n_{1}}.
z_{j}=\prod_{\nu=1}^{M}\frac{\lambda_{\nu}-k_{j}-ic/2}{\lambda_{\nu}-k_{j}+ic/2},\,\,\bar{z}_{j}^{-1}=\prod_{\nu=1}^{M}\frac{\lambda_{\nu}+k_{j}-ic/2}{\lambda_{\nu}+k_{j}+ic/2}
\int{\frac{d^{D}k}{i(2\pi)^{D}}}{\frac{1}{(m^{2}+2p\cdotk-k^{2})^{a}}}={\frac{\Gamma(\epsilon+a-2)}{(4\pi)^{2-\epsilon}\Gamma(a)}}(m^{2}+p^{2})^{2-a-\epsilon},
\left[-\frac{d^{2}}{d\rho^{2}}-\frac{1}{\rho}\frac{d}{\rho}+\frac{s^{2}}{\rho^{2}}+m^{2}+\frac{\lambda}{2}\phi_{\mathrm{wall}}^{2}(\rho-\bar{\rho})+\frac{\sigma}{4!}\phi_{\mathrm{wall}}^{4}(\rho-\bar{\rho})\right]\psi_{n,s}(\rho)=E_{\mathrm{wall}}^{2}(n,s)\psi_{n,s}(\rho)\;,
g^{-1}(x,y)=i\left[\Box+m_{0}^{2}+\frac{\lambda_{0}}{6}(\frac{\phi^{2}}{N}+g(x,x))+\frac{\eta_{0}}{5!}(\frac{\phi^{2}}{N}+g(x,x))^{2}\right]\delta^{D}(x-y)+O(\frac{1}{N})
\begin{array}{rcl}{}&{}&{\delta(i_{h}i_{k}N^{(9)})=7\left\{\partial(i_{h}i_{k}\Omega^{(8)})+6(i_{h}i_{k}{\calN}^{(7)})\partial(i_{h}i_{k}\Lambda^{(3)})\right.}\\{}&{}&{}\\{}&{}&{\left.-15\partial(i_{h}\Lambda^{(3)})(i_{h}i_{k}C^{(4)})^{2}+10\partial(i_{h}i_{k}\Lambda^{(3)})(i_{h}C^{(4)})(i_{h}i_{k}C^{(4)})\right\}\,.}\\\end{array}
ds^{2}=\frac{r^{2}}{l^{2}}\left[-dt^{2}+dx_{1}^{2}+dx_{2}^{2}+\left(1-\frac{r_{0}^{4}}{r^{4}}\right)d\theta^{2}\right]+\frac{l^{2}}{r^{2}}\left(1-\frac{r_{0}^{4}}{r^{4}}\right)^{-1}dr^{2}
\begin{array}{lcr}{J_{0}^{0}=\gamma{{\partial}/{\partial{\gamma}}}-1/2{{\partial}/{\partial{\gamma}}},}\\{J_{0}^{-}={\partial}/{\partial{\gamma}},}\\{J_{0}^{+}={\gamma}^{2}{{\partial}/{\partial{\gamma}}}-{\gamma}{{\partial}/{\partial{\Phi}}}-e^{-2{\Phi}}{\partial}/{\partial{\gamma}}.}\\\end{array}
\left\langle0\right|\alpha_{m}^{l}\left[l^{i-},E^{j-}\right]\alpha_{-m}^{k}\left|0\right\rangle=\frac{m^{2}}{\left(p^{+}\right)^{2}}\left(\delta^{li}\delta^{kj}-\delta^{lj}\delta^{ki}\right)+\frac{m}{\left(p^{+}\right)^{2}}\left(\delta^{lj}p^{i}p^{k}-\delta^{jk}p^{l}p^{i}\right)\tag{60}
\Psi=-\intd^{4}x\;\Bigg\{\alpha(\nabla^{\mu}\bar{\phi}^{\nu})B_{\mu\nu}+\beta(\phi_{\mu}\nabla^{\mu}\bar{\phi}-\bar{\phi}^{\mu}(\pi_{\mu}-{\frac{1}{2}}\nabla_{\mu}\tilde{\phi})+\gamma\bar{\pi}\bar{\phi}+\delta\tilde{\pi}\tilde{\phi}\Bigg\}
\!|h_{3}\rangle=h_{3}({\bfP},\beta,M)|0\rangle_{1}|0\rangle_{2}|0\rangle_{3}\,,\quad\!\!\!|j_{3}^{-I}\rangle=\frac{2}{3{\bfP}^{2}}\sum_{a}\check{\beta}_{a}(\mathrm{i}{\bfP}^{I}\beta_{a}\frac{\partial}{\partial\beta_{a}}+M_{a}^{IJ}{\bfP}^{J})|h_{3}\rangle\,,
\partial_{\mu}\tau\partial^{\mu}a+2i(\mathrm{Im}\,\tau)\partial_{\mu}\partial^{\mu}a-\frac{1}{4}\bar{\tau}^{\prime}F_{\mu\nu}(F^{\mu\nu}+i\widetildeF^{\mu\nu})=-\bar{\eta}\,\sqrt{2}(4\pi)(\nu_{e}+\bar{\tau}\nu_{m})
\Omega_{1}=\intd^{3}x\left(f_{\;bc}^{a}A_{i}^{c}{\calP}^{\left(2\right)ib}\eta_{a}-\frac{1}{2}f_{\;bc}^{a}\epsilon^{0ijk}A_{j}^{b}A_{k}^{c}\eta_{ia}^{\left(2\right)}\right).
\prod_{j\ge1}\bigg(1+\frac{(-1)^{j}x^{3}}{j^{3}}\bigg)=\prod_{j\ge1}\bigg(1+\frac{(-1)^{j}x}{j}\bigg)\bigg(1-\frac{(-1)^{j}x}{j}+\frac{x^{2}}{j^{2}}\bigg),
\left\{x_{\mu},p_{\nu}\right\}^{\ast}\,=\,\frac{(p\cdotx)}{S^{2}}\epsilon_{\mu\nu\gamma}S^{\gamma}\,+\,\frac{2\,b_{\mu}S_{\nu}}{S^{2}}\,-\,\frac{(S\cdotb)S_{\mu}S_{\nu}}{(S^{2})^{2}}
{R_{\mu\nu\lambda}}^{\kappa}={{\bar{R}}_{\mu\nu\lambda}}^{\;\;\;\;\;\;\kappa}+\bar{D}_{\mu}{K_{\nu\lambda}}^{\kappa}-\bar{D}_{\nu}{K_{\mu\lambda}}^{\kappa}-{[K_{\mu},K_{\nu}]_{\lambda}}^{\kappa}{}.
\Pi_{00}^{\Sigma}={\frac{4\alpha}{\beta}}\int_{0}^{1}dx\int{\frac{d^{2}l}{(2\pi)^{2}}}[S_{1}^{\Sigma}+2({\bfl}^{2}+\Sigma^{2}+x(1-x)p_{0b}^{2})S_{2}^{\Sigma}+(1-2x)p_{0b}S_{2}^{*\Sigma}]
M_{N,\,0m}^{\prime\,IJ}=-\frac{1}{\sqrt{m}}\int_{-\infty}^{\infty}d\kappa\,\mu_{1,N}^{IJ}(\kappa)\,\mathscr{P}\,\frac{A_{1}(\kappa)}{\kappa}\,V_{m-1}^{(1)}(\kappa)+\mu_{1,N}^{IJ}(0)\,\xi_{m},
A\rightarrow\left(\begin{array}{ccccc}{}&{A_{1}}&{}&{\ldots}&{}\\{}&{}&{A_{2}}&{\ldots}&{}\\{\vdots}&{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{}&{}&{}&{\ldots}&{A_{\Gamma-1}}\\{A_{\Gamma}}&{}&{}&{\ldots}&{}\\\end{array}\right),
S_{\mathrm{str}}=\sigma\intd^{2}\xi\sqrt{\sfg}+\alpha^{-1}\intd^{2}\xi\sqrt{\sfg}{\sfg}^{ab}(\partial_{a}t_{\mu\nu})(\partial_{b}t_{\mu\nu})+\kappa\intd^{2}\xi\sqrt{\sfg}{\calR}+{\calO}\left(\frac{\sigma}{m_{D}^{4}R^{2}}\right).
E_{0}=-{\frac{1}{2}}\,\sum_{n=-\infty}^{\infty}\sqrt{\left({\frac{(2\,n+1)\,\pi}{L}}\right)^{2}+m^{2}}=-\sum_{n=0}^{\infty}\sqrt{\left({\frac{(2\,n+1)\,\pi}{L}}\right)^{2}+m^{2}}.
ds^{2}=\tilde{f}^{1/2}f^{-1}(-dx_{0}^{2}+dx_{1}^{2})+\tilde{f}^{-1/2}dx_{2}^{2}+\tilde{f}^{1/2}(dx_{3}^{2}+\cdots+dx_{9}^{2})~,
\Box_{^{+}_{-}}^{\lambda}*\phi(x)=_{-}^{+}ie^{^{+}_{-}i\pi(\lambda+n/2)}4^{\lambda}(4\pi)^{n/2}\frac{\Gamma(\lambda+n/2)}{\Gamma(-\lambda)}(t^{2}-r^{2}{^{+}_{-}}i0)^{\lambda-n/2}*\phi(x)
\Pi_{\nu\rho\alpha}^{~~~~\gamma}\Pi_{\sigma\tau\mu}^{~~~~\rho}+\Pi_{\rho\mu\alpha}^{~~~~\gamma}\Pi_{\sigma\tau\nu}^{~~~~\rho}-\Pi_{\sigma\tau~\rho}^{~~\gamma}\Pi_{\mu\nu\alpha}^{~~~~\rho}+\Pi_{\mu\nu~\rho}^{~~\gamma}\Pi_{\sigma\tau\alpha}^{~~~~\rho}=0.
\frac{\partialN^{(\pm)}}{\partialX_{0}}+{\bfv}_{\pm}\cdot\nabla_{\bfX}N^{(\pm)}\simeq\left[Z_{\pm}\Gamma_{\pm}^{(p)}\mp\frac{\partial\omega_{\pm}}{\partialX_{\mu}}\frac{\partialN_{B}^{(\pm)}(X;\pmp_{0},\pm\hat{\bfp})}{\partialP^{\mu}}\right]_{p_{0}=\pm\omega_{\pm}}.
Q_{0}=-\sum_{\kappa=-\infty}^{\infty}\vert\kappa\vert\left\{N_{\kappa}^{+}-N_{\kappa}^{--}\left[\eta_{m\kappa}\left(0\right)-\eta_{-m\kappa}\left(0\right)\right]\frac{1}{\pi}+\left[\eta_{\infty\kappa}\left(\infty\right)-\eta_{-\infty\kappa}\left(\infty\right)\right]\frac{1}{\pi}\right\}\,.
\frac{\partialS}{\partialz_{m}}=\frac{\partial}{\partialz_{m}}\int_{X_{\epsilon}}F~\frac{i}{2}dz\wedged\bar{z}+\frac{\partial}{\partialz_{m}}\int_{\mathbb{C}\setminusX_{\epsilon}}F~\frac{i}{2}dz\wedged\bar{z},
\hat{A}(\xi_{0},\xi):=\langle\xi_{0},\xi|\tilde{\Psi}\rangle=-2^{-2N}(1+\bar{w}w)^{-{\frac{1}{4}}}(\xi_{0}e^{-\sqrt{2}\bar{\mu}_{c}^{\ast}w\mu_{0}^{\ast}}+\mu_{0}^{\ast})e^{-{\frac{1}{2}}\bar{\mu}^{\ast}\varepsilonC\mu^{\ast}-\bar{\xi}M_{0}\xi-\bar{\xi}\lambda}
C_{s}(z)\Phi_{lk}\left(w,\overline{{w}}\right)=\sum_{n=2}^{m}\frac{d_{lk}^{(n)}}{\left(z-w\right)^{n}}\Phi_{lk}^{(n)}\left(w,\overline{{w}}\right)+\frac{1}{z-w}B_{lk}\left(w,\overline{{w}}\right)+...
\begin{matrix}{f_{2}}&{f_{1}f_{2}}&{f_{1}^{3}}&{f_{0}^{2}f_{1}^{2}}&{f_{0}^{5}}\\{f_{1}}&{f_{0}f_{2}}&{f_{0}f_{1}^{2}}&{f_{0}^{3}f_{1}}\\{f_{0}}&{f_{1}^{2}}&{f_{0}^{2}f_{1}}&{f_{0}^{4}}\\{{}}&{f_{0}f_{1}}&{f_{0}^{3}}\\{{}}&{f_{0}^{2}}&{f_{0}^{2}f_{2}}\\\end{matrix}
T_{\mu\nu}^{(PL)}=-\frac{\kappa}{2\pi}\{\partial_{\mu}\psi\partial_{\nu}\psi-2\nabla_{\mu}\nabla_{\nu}\psi+g_{\mu\nu}[2\nabla^{2}\psi-\frac{1}{2}(\nabla\psi)^{2}]\}\mathrm{,}
T=\frac{1}{4\pi}r_{+}^{\frac{a^{2}-1}{a^{2}+1}}\left[\frac{1+a^{2}}{(1-a^{2})\beta^{2}}-\frac{Q^{2}(1+a^{2})e^{2a\phi_{0}}}{\beta^{4}r_{+}^{2}}\right],
\left.\frac{\partial}{\partial\zeta}R_{i,i+1}(\zeta)\right|_{\zeta=1}\proptoh_{i,i+1}P_{i,i+1}+\mathrm{const.},\qquad\left.\frac{\partial}{\partial\zeta}K_{i}(\zeta;r)\right|_{\zeta=1}\proptof_{i}\sigma_{i}^{z}+\mathrm{const.},
D^{IJ}=u_{a}^{A}(\gamma^{IJ})^{AB}{\frac{\partial}{\partialu_{a}^{B}}}+w_{\dot{a}}^{\dot{A}}(\gamma^{IJ})^{\dot{A}\dot{B}}{\frac{\partial}{\partialw_{\dot{a}}^{\dot{B}}}}+v_{i}^{[I}{\frac{\partial}{\partialv_{i}^{J]}}}\;.
{\frac{2e^{\left({\frac{\calH}{\hbar}}+{\frac{1}{2}}\right)\tau}}{e^{\tau}+1}}=\sum_{n=0}^{\infty}{\frac{\tau^{n}}{n!}}E_{n}\left({\frac{\calH}{\hbar}}+{\frac{1}{2}}\right)\quad(|\tau|<\pi).
\vec{b}_{DES}^{\,\prime}=R_{z}(\phi^{\prime}-{\frac{\pi}{2}})R_{x}({\frac{\pi}{2}}-\theta^{\prime})R_{z}(2v)R_{z}(-\theta_{2}^{\prime})R_{x}(\phi_{2}^{\prime})R_{z}(\theta_{2}^{\prime})R_{z}(v)\hat{x},
\begin{array}{ccc}{\left\{c_{r}^{a},\overline{{c}}_{s}^{b}\right\}=2\delta_{rs}\eta^{ab}}&{r,s\neq0}&{}\\{\left\{d_{r}^{a},\overline{{d}}_{s}^{b}\right\}=2\delta_{rs}\eta^{ab}}&{}&{}\\\end{array}
\left[\frac{d^{2}}{d\varrho^{2}}+\widetilde{E}-\mathrm{sgn}\left(\widetilde{\beta}\right)\,\widetilde{\lambda}\varrho^{\widetilde{\beta}}-\frac{\left(\widetilde{l}+\widetilde{\nu}\right)^{2}-1/4}{\varrho^{2}}\right]\widetilde{u}_{l}(\varrho)=0\;,
({\calQ}_{k})_{\alpha}^{m}=\left(\begin{matrix}{({\calQ}_{k})_{\alpha}^{p}}\\{({\calQ}_{k})^{*\alphap}}\\\end{matrix}\right)\,\qquad({\calQ}_{\bar{k}})_{\alpha}^{\hat{m}}=\left(\begin{matrix}{({\calQ}_{\bar{k}})_{\alpha}^{\hat{p}}}\\{({\calQ}_{\bar{k}})^{*\alpha\hat{p}}}\\\end{matrix}\right)\,\qquadp,\hat{p}=1,\dots,{\frac{N}{2}}\.
{\calE}_{as}^{spin}=C_{s}\sum_{\nu=\frac12}^{\infty}\int_{m_{e}}^{\infty}dk[k^{2}-m_{e}^{2}]^{1-s}\frac{\partial}{\partialk}\sum_{j,n}^{3}Y_{j,n}\frac{t^{j}}{\nu^{n}}-{\calE}_{div}^{spin},
J_{nn^{\prime\prime}}(\hat{q}_{\perp})\equiv\sum_{m=0}^{\mathrm{min}(n,n^{\prime\prime})}\frac{n!n^{\prime\prime}!}{m!(n-m)!(n^{\prime\prime}-m)!}[i\mathrm{sgn}(gH)\hat{q}_{\perp}]^{n+n^{\prime\prime}-2m},
I_{1}(B)=-\frac{eB}{{\pi}^{\frac{3}{2}}\beta}\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}\int_{0}^{\infty}dx\;e^{-\frac{x^{2}}{4}-\frac{1}{x^{2}}\beta^{2}k^{2}m^{2}}\frac{1}{e^{\frac{2\beta^{2}eBk^{2}}{x^{2}}}-1}\;.
\langle~\delta_{S}{\bar{Q}}_{\dot{\alpha}}~\rangle~=~\epsilon^{\beta}~\sum_{k}k_{\beta{\dot{\alpha}}}~\sum_{n_{k}^{B}=0}^{\infty}~n_{k}^{B}~\left(~\rho_{n_{k}^{B},n_{k}^{B}}^{B}~\rho_{00}^{F}~-~\rho_{n_{k}^{B}-1,n_{k}^{B}-1}^{B}~\rho_{11}^{F}~\right)~~.
\times\sum_{{\bf{q}}_{1}\neq{\bf{q}},{\bf{0}}}n_{{\bf{k}}-{\bf{q}}/2+{\bf{q}}_{1}}e^{i\mathrm{~}\theta({\bf{k}}+{\bf{q}}_{1}/2,-{\bf{q}}+{\bf{q}}_{1})}e^{-i\mathrm{~}\theta({\bf{k}}-{\bf{q}}/2+{\bf{q}}_{1}/2,{\bf{q}}_{1})}
{\calO}=(-)^{q-1}\,p\,[\xi^{i_{1}^{*}}...\xi^{i_{p-1}^{*}}V_{i_{1}^{*}...i_{p-1}^{*}i_{p}^{*}}^{j_{1}...j_{q}}g^{i_{p}^{*}j_{q+1}}{\bar{\psi}}_{j_{1}}...{\bar{\psi}}_{j_{q+1}}].
\stackrel{(p)}{B_{\alpha}^{a}}(x)=-\frac{1}{4\pi}\intd^{3}x^{\prime}\epsilon_{\alpha\beta\gamma}\stackrel{(p)}{J^{\beta\gamma}}(x^{\prime})\frac{(x-x^{\prime})^{\gamma}}{|x-x^{\prime}|^{3}}
\begin{array}{cc}{\left.\phi_{\omega}(s)\right|_{s=0}=0,}&{\left.\phi_{\omega}(s)\right|_{s=s_{0}}=0,}\\{}&{}\\{\left.\phi_{\omega}(s)\right|_{s=a_{1}^{+}}=\left.\phi_{\omega}(s)\right|_{s=a_{1}^{-}},}&{\left.\phi_{\omega}^{\prime}(s)\right|_{s=a_{1}^{+}}=\left.\phi_{\omega}^{\prime}(s)\right|_{s=a_{1}^{-}}.}\\\end{array}
{\calW}\,(\varphi_{1},\varphi_{2})=\frac{m^{2}}{4\lambda}\,\varphi_{1}-\frac{\lambda}{3}\,\varphi_{1}^{3}+\lambda\,\varphi_{1}\varphi_{2}^{2}+\frac{\mu}{2}\,\varphi_{2}^{2}\,,
\left(\,X_{\bfm}\,\right)_{\bfkl}\equiv\intd^{2}\sigma\,e^{-2\pii{\bfk}\cdot\sigma}\,X_{e^{2\pii{\bfm}\sigma}}\,e^{2\pii{\bfl}\cdot\sigma}=-4\pi^{2}{\bfm}\times{\bfl}\,\delta_{{\bfm}\,({\bfk-l})}\,,
S=\intd^{4}x\sqrt{|g_{\scriptstyleE}|}\left[\hat{R}(\hat{g}_{E})-{\textstyle\frac{1}{2}}\left(\Im\mathrm{m}\hat{\lambda}\right)^{-2}\partial_{\hat{\mu}}\hat{\lambda}\partial^{\hat{\mu}}\bar{\hat{\lambda}}+{\textstyle\frac{1}{4}}\hat{F}^{I}\,{}^{\star}\tilde{\hat{F}^{I}}\right]\,.
D_{\mathrm{H}}^{2}[\Phi,\Phi^{\prime}]=\sum_{x}\sum_{\alpha=0}^{3}\left[((\phi_{\alpha})_{x}-(\phi_{\alpha}^{\prime})_{x})^{2}+((\dot{\phi}_{\alpha})_{x}-(\dot{\phi}_{\alpha}^{\prime})_{x})^{2}\right]\,,
\Phi_{j_{1}\,j_{2}\,\dots\,j_{k}}^{i_{1}\,i_{2}\,\dots\,i_{k}}\equiv\mathrm{Tr}\left(u^{(i_{1}}\,v_{(j_{1}}\,u^{i_{2}}\,v_{j_{2}}\,\dots\,u^{i)_{k}}\,v_{j_{k})}\right)
{}_{N}\!\langle\varrho_{n}^{\prime},\bar{\varrho}_{n}^{\prime}|\varrho_{n},\bar{\varrho}_{n}\rangle_{N}=\prod_{\mu=0}^{p}\delta(\mathrm{Re}\varrho_{n\mu}^{\prime}-\mathrm{Re}\varrho_{n\mu})\delta(\mathrm{Im}\varrho_{n\mu}^{\prime}-\mathrm{Im}\varrho_{n\mu})~.
\tilde{O}={\frac{1}{c_{V}(G)}}\left[J^{a}\bar{J}^{b}\phi^{ab}+(1+{\frac{4}{k}})J^{a}\bar{J}\phi^{a3}+(1+{\frac{4}{k}})J\bar{J}^{b}\phi^{3b}+(1+{\frac{4}{k}})^{2}J\bar{J}\phi^{33}\right],
\lambda_{CW}\left(X(\frac{p_{1}}{q_{1}},\ldots,\frac{p_{n}}{q_{n}})\right)=\frac{1}{12}\frac{P}{H}\left(2-n+\sum_{i=1}^{n}p_{i}^{-2}\right)-\frac{1}{12}\left[3\mathrm{sign}\left(\frac{H}{P}\right)+\sum_{i=1}^{n}\left(12s(q_{i},p_{i})-\frac{q_{i}}{p_{i}}\right)\right].
\Lambda\partial_{\Lambda}L(\phi,\Lambda)=-{\frac{1}{2}}\intdp(2\pi)^{4}\Lambda{\frac{\partial_{\Lambda}K(p^{2}/\Lambda^{2})}{p^{2}+m^{2}}}\left\{{\frac{\delta^{2}L}{\delta\phi(p)\delta\phi(-p)}}+{\frac{\deltaL}{\delta\phi(p)}}{\frac{\deltaL}{\delta\phi(-p)}}\right\}
\bar{R}_{\mu\nu\beta}^{\lambda}=\bar{\Gamma}_{\mu\nu,\beta}^{\lambda}-\bar{\Gamma}_{\mu\beta,\nu}^{\lambda}+\bar{\Gamma}_{\mu\nu}^{\alpha}\bar{\Gamma}_{\alpha\beta}^{\lambda}-\bar{\Gamma}_{\mu\beta}^{\alpha}\bar{\Gamma}_{\alpha\nu}^{\lambda}.
\begin{array}{c}{\phi^{\prime\prime}(u)=0}\\{1+c_{wzw}-26+6\alpha^{\prime}(\phi^{\prime})^{2}=0}\\{-\frac{1}{2}\alpha^{\prime}T^{\prime\prime}-2T+\alpha^{\prime}\phi^{\prime}T^{\prime}=0}\\\end{array}
\phi_{m}=N_{m}e^{{\frac{5}{4}}k\rho}\left[J_{3/2}\left({\frac{2m}{k}}\right)Y_{5/2}\left({\frac{2m}{k}}e^{{\frac{k}{2}}\rho}\right)-Y_{3/2}\left({\frac{2m}{k}}\right)J_{5/2}\left({\frac{2m}{k}}e^{{\frac{k}{2}}\rho}\right)\right]
0=\frac{\delta_{L}}{\deltaC^{\alphaa}}\mathbf{S}_{m}^{a}(\Gamma_{m})=-\mathbf{\tilde{Q}}_{m}^{a}\frac{\delta_{L}}{\deltaC^{\alphaa}}\Gamma_{m},\qquad0=\frac{\delta_{L}}{\deltaC^{\alphaa}}\mathbf{D}_{A}(\Gamma_{m})=\mathbf{\tilde{Q}}_{A}\frac{\delta_{L}}{\deltaC^{\alphaa}}\Gamma_{m}.
r\,\frac{dg_{\parallel}}{dr}=-\frac{g_{\perp}^{2}}{f_{\parallel}(g_{\parallel},g_{\perp})}\,\qquad\qquadr\,\frac{dg_{\perp}}{dr}=-\frac{g_{\parallel}\,g_{\perp}}{f_{\perp}(g_{\parallel},g_{\perp})}\.
+{\frac{\pi^{{\frac{5}{2}}}}{4}}(\Phi_{0}a)^{4}(aT)^{2}\sum_{n=1}^{\infty}n^{2}\int_{0}^{1}d\alpha\alpha^{-{\frac{3}{2}}}(1-\alpha)^{-{\frac{3}{2}}}\Psi({\frac{3}{2}},2,{\frac{4\pi^{2}a^{2}T^{2}n^{2}}{\alpha(1-\alpha)}})
X_{8}={\frac{1}{(2\pi)^{4}}}\left({\frac{1}{48}}\mathrm{Tr}F^{4}-{\frac{1}{192}}\mathrm{Tr}F^{2}\mathrm{Tr}R^{2}\right)+{\frac{1}{128}}p_{1}^{2}(R)-{\frac{1}{96}}p_{2}(R)\,
\Gamma_{\mathrm{inv}}=\intd^{4}x\,\Big(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}\theta^{\rho\sigma}\Big(F_{\mu\rho}F_{\nu\sigma}F^{\mu\nu}-\frac{1}{4}F_{\rho\sigma}F_{\mu\nu}F^{\mu\nu}\Big)\Big)+O(\theta^{2}),
ds^{2}=\alpha^{\prime}\left[\frac{u^{2}}{R^{2}}\left(dx_{+}dx_{-}+dx_{1}^{2}+dx_{2}^{2}-\frac{u^{4}}{R^{4}}dx_{-}^{2}\right)+\frac{R^{2}}{u^{2}}\left(du^{2}+u^{2}d\Omega_{5}^{2}\right)\right],
{\calL}=\left\{\frac{1}{8T(R)g^{2}}\,\int\!\mathrm{d}^{2}\theta\,\mathrm{Tr}\,W^{2}+\mathrm{H.c.}\right\}+\frac{1}{4}\int\!\mathrm{d}^{2}\theta\mathrm{d}^{2}\bar{\theta}\,\bar{Q}e^{V}Q\,,
L_{\mathrm{int}}^{\prime}=-{\frac{g}{2}}\Bigl(\psi_{1}^{\dagger}(t)\psi_{2}(t)e^{2\sqrt{2}\piip_{X}}\-\\psi_{1}^{\dagger}(l+t)\psi_{2}(l+t)\Bigr)\+\\mathrm{h.c.}.
\begin{array}{lll}{b_{-}=-\frac{4}{3}\,}&{}&{T_{-}=-2\,e^{+\frac{4}{3}d_{1}}\,}\\{b_{+}=+\frac{4}{3}\,}&{}&{T_{+}=-2\,e^{-\frac{4}{3}d_{2}}\,}\\\end{array}
\vec{\pi}(\vec{\pi}_{\mathrm{V}})=-\sqrt{1-\vec{\pi}_{\mathrm{V}}^{2}}\;\vec{\pi}_{\mathrm{V}_{0}}+\sqrt{1-\vec{\pi}_{\mathrm{V}_{0}}^{2}}\;\vec{\pi}_{\mathrm{V}}+\vec{\pi}_{\mathrm{V}_{0}}\times\vec{\pi}_{\mathrm{V}}.
R_{1/2,1/2}^{\mathrm{(s_{0},s_{1})}}(u)=f_{q}(u)\cdot\left(\begin{array}{cccc}{1}&{{}}&{{}}&{{}}\\{{}}&{\frac{q^{-1}(1-u^{s_{1}+s_{0}})}{1-q^{-2}u^{s_{1}+s_{0}}}}&{\frac{u^{s_{1}}(1-q^{-2})}{1-q^{-2}u^{s_{1}+s_{0}}}}&{{}}\\{{}}&{\frac{u^{s_{0}}(1-q^{-2})}{1-q^{-2}u^{s_{1}+s_{0}}}}&{\frac{q^{-1}(1-u^{s_{1}+s_{0}})}{1-q^{-2}u^{s_{1}+s_{0}}}}&{{}}\\{{}}&{{}}&{{}}&{1}\\\end{array}\right)\,,
\{x_{n}\}=\mathrm{\small~\{\frac{11}{120},\frac{11}{120},\frac{1}{8},\frac{17}{40},\frac{71}{120},\frac{71}{120},\frac{19}{24},\frac{19}{24},\frac{37}{40},\frac{131}{120},\frac{131}{120},\ldots\}~}
\Big(\frac{1}{\tau^{2}}\Big)^{\frac{1}{N}\sum_{a,b=1}^{N}(n_{a}-m_{a})(n_{b}^{\prime}-m_{b}^{\prime})-\sum_{a=1}^{N}(n_{a}-m_{a})(n_{a}^{\prime}-m_{a}^{\prime})}=:\Big(\frac{1}{\tau^{2}}\Big)^{E}\;.
w^{\prime\prime}(x)=\left(-\frac{\lambda^{2}}{(1-x^{3})^{2}}+\frac{l(l+6)}{4x^{2}(1-x^{3})}+\frac{5-46x^{3}+5x^{6}}{16x^{2}(1-x^{3})^{2}}\right)w(x)\,
=2{\left({\frac{e^{-\varphi/2}}{\psi_{2}}}\right)^{\prime}}^{\frac{1}{2}}\partial_{z}{\left({\frac{e^{-\varphi/2}}{\psi_{2}}}\right)^{\prime}}^{-1}\partial_{z}{\left({\frac{e^{-\varphi/2}}{\psi_{2}}}\right)^{\prime}}^{\frac{1}{2}}\cdot1=-2e^{\varphi/2}\left(e^{-\varphi/2}\right)^{\prime\prime}=-2\psi^{-1}\psi^{\prime\prime},
\begin{array}{ll}{a^{\dagger}=\frac{x/m-ip_{x}}{\sqrt{2}},}&{a=\frac{x/m+ip_{x}}{\sqrt{2}},}\\{b^{\dagger}=\frac{y/n-ip_{y}}{\sqrt{2}},}&{b=\frac{y/n+ip_{y}}{\sqrt{2}}.}\\\end{array}
ds^{2}({\bfY},{\bfY}^{\prime})=\sum_{\mu=1}^{4}\bigl[(g_{1}+g_{1}^{2}X)m_{\mu}^{\prime}+(g_{2}+g_{2}^{2}X)n_{\mu}n_{\mu}^{\prime}+g_{1}g_{2}X(m_{\mu}^{\prime}+n_{\mu}m_{\mu}^{\prime})\bigr],
\omega(\zeta(u))E_{\alpha}[\zeta|u]\omega^{-1}(\zeta(u))=\frac{2}{\bar{N}}\epsilon_{\alpha\beta\mu\lambda}\dot{\zeta}^{\beta}(u)\int\delta\etadt\tilde{E}^{\mu}[\eta|t]\dot{\eta}^{\lambda}(t)\dot{\eta}^{-2}(t)\delta(\eta(t)-\zeta(u)),
\tilde{m}\simN_{f}(q+\tilde{q}){\frac{\Lambda^{3}}{\xi^{2}}}\left[{\frac{m}{\Lambda}}\left({\frac{\xi}{M_{P}}}\right)^{q+\tilde{q}}\right]^{N_{f}/N_{c}}=N_{f}(q+\tilde{q}){\frac{<\lambda\lambda>}{\xi^{2}}},
\tilde{R}(\tilde{g})=e^{U}\tilde{R}(\tilde{\gamma})+(d-2)e^{U}D^{2}U-{\frac{1}{4}}(d^{2}-5d+8)e^{U}(DU)^{2}-{\frac{1}{4}}e^{3U}V_{ij}V^{ij}\,
\langle0|O_{i}^{V}O_{j}^{\widehatV}-O_{i}^{\widehatV}O_{j}^{V}|0\rangle=\int_{V}\!\!d^{d}y_{1}\int_{\widehatV}\!\!d^{d}y_{2}\left(f_{ij}\left(|\vec{y}_{1}-\vec{y}_{2}|\right)-f_{ij}\left(|\vec{y}_{2}-\vec{y}_{1}|\right)\right)=0,
H=\frac{\displaystyle\sum_{k=1}^{N}\left(\frac{p_{k}^{2}}{2\mu_{k}}+2\beta_{k}^{2}x_{k}^{2}\right)}{\displaystyle1+\sum_{k=1}^{N}\frac{x_{k}}{a_{k}}\sqrt{1+\frac{x_{k}^{2}}{a_{k}^{2}}}},\\\\\begin{array}{l}{\displaystyle\beta_{k}^{2}=\frac{\alpha^{2}\mu_{k}m_{k}^{2}}{8},}\\{\displaystyle\beta_{1}^{2}a_{1}^{2}=\ldots=\beta_{N}^{2}a_{N}^{2}}\\\end{array}
P_{L}^{\prime}=({\rho}\frac{\tilde{m}}{2}+(G_{\bot}-B_{\bot}-\frac{1}{4}A_{\bot}^{t}C_{\bot}A_{\bot}){\tilde{n}}-\frac{1}{2}A_{\bot}^{t}C_{\bot}{\tilde{l}},\;{\tilde{l}}+A_{\bot}{\tilde{n}})
\langlem^{2}\rangle={\frac{\tilde{\Sigma}}{\tilde{\rho}}}={\frac{\left(e^{2h}-1\right)^{2}\left(3+(2+\sqrt{7})e^{2h}\right)^{2}}{\left(1+(-1+\sqrt{7})e^{2h}+(2+\sqrt{7})e^{4h}\right)^{2}}}\,
f\starg=\sum_{j=0}^{N}\sum_{s=0}^{\infty}\frac{\kappa^{s}}{s!}\sum_{t=0}^{s}(-1)^{t}\left(\begin{matrix}{s}\\{t}\\\end{matrix}\right)[\partial_{x_{j}}^{s-t}\partial_{p_{j}}^{t}f][\partial_{x_{j}}^{t}\partial_{p_{j}}^{s-t}g]\,.
c_{-}\intd^{3}x\intd^{3}x^{\prime}\phi_{-}(x)|x-x^{\prime}|^{-2h_{+}}\phi_{-}(x^{\prime})+c_{+}\intd^{3}x\intd^{3}x^{\prime}\phi_{+}(x)|x-x^{\prime}|^{-2h_{-}}\phi_{+}(x^{\prime})
I=\frac{1}{16\piG}\intd^{10}x\sqrt{g}\left[e^{-2\phi}\Big(R+4\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{12}H_{3}^{2}\Big)-\sum_{p}\frac{1}{2(p+2)!}F_{p+2}^{2}\right]+I_{WZ}
\Big\langle\prod_{a=1}^{N}\prod_{i=1}^{n_{a}^{\prime}}\overline{{\psi}}^{(a)}({x^{\prime}}_{i}^{(a)})P_{+}\psi^{(a)}({x^{\prime}}_{i}^{(a)})\prod_{i=1}^{m_{a}^{\prime}}\overline{{\psi}}^{(a)}({y^{\prime}}_{i}^{(a)})P_{-}\psi^{(a)}({y^{\prime}}_{i}^{(a)})\Big\rangle\;,
|\hat{\bboxp}j_{3}({\mathsf{s}}_{R}j_{R})^{-}\rangle=\frac{i}{2\pi}\int_{-\infty_{II}}^{+\infty_{II}}d{\mathsf{s}}|\hat{\bboxp}j_{3}({\mathsf{s}}j_{R})^{-}\rangle\frac{1}{{\mathsf{s}}-{\mathsf{s}}_{R}}.
S_{WZNW}(g;k)={\frac{-k}{4\pi}}\int[\mathrm{Tr}|g^{-1}\mathrm{d}g|^{2}~+~{\frac{i}{3}}\mathrm{d}^{-1}\mathrm{Tr}(g^{-1}\mathrm{d}g)^{3}],
i^{n}{\epsilon}_{(n)}^{{\vec{\delta}}ba}L_{(n)}^{\vec{\delta}}={\calL}_{(n)}^{ba}={\sfA}_{n}\left(L^{a}L^{b}+L^{b}L^{a}\right)+{\sfB}_{n}{\delta}_{ab}+i{\sfC}_{n}{\epsilon}_{abj}L^{j}\,,
\left(\Gamma_{eff}^{PV(1)}\right)_{3}=-{\frac{iM}{12}}\intd^{2}x\,B\left[\beta(\nabla^{2}\hat{A}_{0}){\frac{\partial}{\partiala}}+\nabla^{2}\right]{\frac{\partial}{\partialM^{2}}}\left({\frac{\Gamma(a+\beta\hat{A}_{0},M)}{M}}\right)
{\alpha_{h}}^{(h-p)}(x)=(-1)^{p}\sum_{r_{1}=0}^{h-1}\sum_{r_{2}=0}^{h-1-r_{1}}\cdots\sum_{r_{p}=0}^{h-1-r_{1}-\cdots-r_{p-1}}{\frac{(h-1)!}{(h-1-r_{1}-\cdots-r_{p})!}}x^{h-1-r_{1}-\cdots-r_{p}}e^{-x}\,,
T_{i_{1}\cdotsi_{N-r}}^{(N)}={\frac{1}{\sqrt{(N-r)!}}}\;k\;\varepsilon_{i_{1}\cdotsi_{N-r}j_{1}\cdotsj_{r}}\;\hat{k}^{j_{1}}\tilde{D}\hat{k}^{j_{2}}\cdots(\tilde{D}^{r-1}\hat{k}^{j_{r}})\,.
Q=\left(\begin{array}{cccc}{0}&{-x_{3}}&{-x_{2}}&{-x_{1}}\\{x_{3}}&{0}&{-x_{1}}&{x_{2}}\\{x_{2}}&{x_{1}}&{0}&{-x_{3}}\\{x_{1}}&{-x_{2}}&{x_{3}}&{0}\\\end{array}\right).
\tilde{\Psi}_{\vec{k}a}\;=\;\Psi_{\vec{k}a}\:+\:\sum_{n=1}^{\infty}(-1)^{n}\;s^{(n)}\:{\cal{B}}\:\cdots\:s^{(2)}\:{\cal{B}}\:s^{(1)}\:{\cal{B}}\:\Psi_{\vec{k}a}
{\calL}=\intd^{4}\theta\,\left(\mathrm{tr}\left(\Phi^{\dagger}\Phie^{V}\right)-c\,\mathrm{tr}V\right)+\left(\intd^{2}\theta\;\mathrm{tr}\left(\Phi_{0}\Phi^{T}J\Phi)\right)+\mathrm{h.c.}\right).
{\calG}_{a}^{(\pm)}\equiv\frac{L}{2}\eta_{ab}\epsilon^{ij}(\partial_{i}A_{j}^{(\pm)b}-\partial_{j}A_{i}^{(\pm)b}+\epsilon_{\mathrm{~}cd}^{b}A_{i}^{(\pm)c}A_{j}^{(\pm)d})\approx0,
S_{M}=-M\intd\tau+\frac{\kappa}{2}\intd^{4}y\sqrt{-g}s_{\mu\nu}(y)T^{\mu\nu}(y)+\intd^{4}y\sqrt{-g}s_{\mu\nu}(y)s_{\rho\sigma}(y){\tilde{M}}^{\mu\nu\rho\sigma}(y)+\ldots,
\mathcal{L}_{\,\mathrm{kin}}=-\frac{i}{2}\bar{\psi}_{M}\gamma^{MNP}\mathcal{D}_{N}\psi_{P}+\frac{i}{2}\mathrm{Tr}\,\left(\bar{\lambda}\gamma^{M}\mathcal{D}_{M}\lambda\right)-\mathrm{Tr}\left|\mathcal{D}_{M}A\right|^{2}\,
\sigma_{abs}^{\phi_{a}}\rightarrow\frac{r_{1}^{6}r_{5}^{6}}{(r_{1}^{2}+r_{5}^{2}+\sqrt{r_{1}^{4}+r_{5}^{4}-r_{1}^{2}r_{5}^{2}})^{2}}\omega^{2}T_{L}^{3},
\partial_{i}\beta_{j}^{a}-\partial_{j}\beta_{i}^{a}+\beta_{i}^{b}\partial_{b}\beta_{j}^{a}-\beta_{j}^{b}\partial_{b}\beta_{i}^{a}={\frac{1}{2}}\gamma_{b}^{a}\left(\xi_{ij}^{b}-\xi_{ji}^{b}-\partial_{i}\rho_{j}^{b}+\partial_{j}\rho_{i}^{b}-\nu_{ic}^{b}\rho_{j}^{c}+\nu_{jc}^{b}\rho_{i}^{c}\right)\,\,\,.
\sum_{e,f}G_{ef}^{ab}(p_{1},p_{2};+)G_{cd}^{ef}(p_{1},p_{2};-)=\sum_{e,f}G_{ef}^{ab}(p_{1},p_{2};-)G_{cd}^{ef}(p_{1},p_{2};+)=\delta_{c}^{a}\delta_{d}^{b}.