harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{c}{2ImT_{1}=\frac12\intd\tauT_{\mu\nu}^{abcd}T_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}b^{\prime}cd*}[P^{\mu\mu^{\prime}}(k_{1})P^{\nu\nu^{\prime}}(k_{2})+Q^{\mu\mu^{\prime}}(k_{1})g^{\nu\nu^{\prime}}}\\{+g^{\mu\mu^{\prime}}Q^{\nu\nu^{\prime}}(k_{2})-Q^{\mu\mu^{\prime}}(k_{1})Q^{\nu\nu^{\prime}}(k_{2})]}\\\end{array}
s=-\frac{1+E^{2}}{1-E^{2}}\frac{\alpha^{\prime}}{2}(k_{0})^{2}-\frac{1+E^{2}}{1-E^{2}}\frac{\alpha^{\prime}}{2}\left(\frac{wR}{\alpha^{\prime}}\right)^{2}-\frac{\alpha^{\prime}}{2h}k^{2}-\frac{2EwR}{1-E^{2}}k_{0}.
\frac{1}{2}(\theta^{2})^{\alpha}{}_{\beta}=-(\theta_{\mathrm{s}}^{2})^{\alpha}{}_{\beta}\,\quad(\theta)^{2}=-3(\theta_{\mathrm{s}})^{2}\,\quad\Big[\Big(1+\frac{1}{2}(\theta)^{2}\Big)^{-1}\Big]^{y}{}_{y}=[\mathrm{det}(1-(\theta_{\mathrm{s}})^{2})]^{-1/2}\.
\hat{q}_{3}\equiv\frac{q_{3}}{\sqrt{2\verteH\vert}}_{,}~~~\hat{m}_{\mu}\equiv\frac{m(0,\mu)}{\sqrt{2\verteH\vert}}_{,}~~~\hat{\mu}\equiv\frac{\mu}{\sqrt{2\verteH\vert}}_{,}~~~\hat{Q}_{1,2}\equiv\frac{Q_{1,2}}{\sqrt{2\verteH\vert}}_{.}
W^{(n,n^{\prime})}(z)=\sum_{l,l^{\prime},m,m^{\prime}}{\calD}_{(n,n^{\prime})(m,m^{\prime})}^{(l,l^{\prime})}V_{h_{l,l^{\prime}}h_{n,n^{\prime}}}^{h_{m,m^{\prime}}}(\cdot,z)\,,
\left({\frac{A(q,p)}{4}}\right)_{N=8}=\pi\left({|(\sum_{i}|Z_{i}|^{4}-2\sum_{i>j}|Z_{i}|^{2}|Z_{j}|^{2}+8|Z_{1}Z_{2}Z_{3}Z_{4}|})|\right)^{1/2}\,
\mathrm{Tr}~(\lambda^{i}~\lambda^{j})\,\quad\mathrm{Tr}~(\bar{\lambda}^{\bar{i}}~\bar{\lambda}^{\bar{j}})\,\quad\mathrm{Tr}~(X^{A}~X^{B})~-~{\frac{1}{6}}~\delta^{AB}~\mathrm{Tr}~(X^{C}~X^{C})\.
\sum_{\rho\inS_{n}}^{}\chi_{R}(\rho)D_{R}(\rho^{-1})=\frac{n!}{d_{R}}I_{R},\qquad\mathrm{and}\qquad\sum_{\sigma\inS_{n}}^{}D_{R}(\sigma\rho\sigma^{-1})=\frac{n!}{d_{R}}\chi_{R}(\rho)I_{R},
\frac{d}{dt}\Bigl(\dot{q}_{i}+\lambda^{2}q_{i}\Bigr)=-\biggl[\lambda^{2}+\frac{2\tilde{T}_{2}(2\tilde{T}_{1}+1)}{(q^{2}+2\eta)^{2}}\biggr](\dot{q}_{i}+\lambda^{2}q_{i})-\frac{1}{q^{2}}(\dot{q}_{k}+\lambda^{2}q_{k})^{2}\,q_{i}\,.
Z_{\mathrm{new}}=\sum_{n_{h}=0}^{\infty}{\frac{1}{n_{h}!}}\otimes\Big(\sum_{N=0}^{\infty}{\frac{1}{N!}}{\calS}_{\mathrm{st}}(N\leftarrown_{h})\otimes\sum_{m=0}^{\infty}{\frac{1}{m!}}{\calS}_{\mathrm{ws}}(n_{h}\rightarrowm)\Big).
{\frac{l_{s}^{3}\,N^{3}}{g_{s}}}E^{3}={l_{1}}^{2}\,{l_{2}}^{2}+{l_{2}}^{2}\,{l_{3}}^{2}+{l_{1}}^{2}\,{l_{3}}^{2}+{\frac{5}{12}}\,({l_{1}}^{2}+{l_{2}}^{2}+{l_{3}}^{2})^{2}
[P,\partial_{\mu}P]=\frac{1}{(1+|u|^{2})^{2}}\left(\begin{array}{cc}{\partial_{\mu}u\bar{u}-u\partial_{\mu}\bar{u}}&{\partial_{\mu}\bar{u}+\bar{u}^{2}\partial_{\mu}u}\\{-(\partial_{\mu}u+u^{2}\partial_{\mu}\bar{u})}&{-(\partial_{\mu}u\bar{u}-u\partial_{\mu}\bar{u})}\\\end{array}\right).
\Psi_{\delta_{i}}^{R}\equiv\Psi_{l_{i}}^{R},\Psi_{q_{i}}^{R}\qquad\Psi_{l_{i}}^{R}=\left(\begin{array}{c}{\psi_{\nu_{i}}^{R}}\\{\psi_{e_{i}}^{R}}\\\end{array}\right)\qquad\Psi_{q_{i}}^{R}=\left(\begin{array}{c}{\psi_{u_{i}}^{R}}\\{\psi_{d_{i}}^{R}}\\\end{array}\right)
\intd{\bfk}\frac{|{\bfk}|}{(k_{0}^{\mathrm{\scriptsize~eff}})}\Psi({\bfk},{\bfq};\omega)=\frac{\alpha}{\left\{1-\left(g^{2}/4\pi\right)(\xi+1)\int\frac{d{\bfk}^{\prime}}{(k_{0}^{\mathrm{\scriptsize~eff}})^{\prime}}\frac{({\bfk}^{\prime})^{2}}{[\{(k_{0}^{\mathrm{\scriptsize~eff}})^{\prime}\}^{2}-\omega^{2}/4+i\epsilon]}\right\}}
\mathrm{Re}\,\Pi^{\mathrm{(R)}}(p^{0},\vec{p})=\frac{2}{\pi}\,\mathrm{P}\,\int_{0}^{\infty}d\omega\,\frac{\omega\,\mathrm{Im}\,\Pi^{\mathrm{(R)}}(\omega,\vec{p})}{\omega^{2}-(p^{0})^{2}}
{\calO}_{1}{}^{i}{}_{j}\left[\begin{matrix}{\bar{\psi}_{n}^{j\dot{1}}}\\{\psi_{n2}^{j}}\\\end{matrix}\right]=-im_{n}^{(1)}\left[\begin{matrix}{\psi_{n1}^{i}}\\{\bar{\psi}_{n}^{i\dot{2}}}\\\end{matrix}\right]
\zeta_{\calD}(s)=\sum_{j=-\infty}^{\infty}\sum_{N=0}^{\infty}{\frac{(N+1)^{2}}{[({\frac{2\pi}{\beta}}j+i\mu)^{2}+{\frac{N(N+2)+6\xi}{a^{2}}}+m^{2}]^{s}}}
\displaystyle{\frac{\phi^{(2)}(a,b+1,c|u,v,w+1/4)}{\phi^{(2)}(a,b-1,c|u,v,w+1/4)}}=\displaystyle{\frac{\phi^{(0)}(a,b+1,c|u+\eta,v+\eta,w+1/4)}{\phi^{(0)}(a,b-1,c|u+\eta,v+\eta,w+1/4)}}
d_{Q}T_{2}(x,y)^{\mathrm{loops}}\big|_{:u(x)h(y):}=:\!u^{\rho}(x)_{,\sigma}h^{\mu\nu}(y)\!:\,\big(b^{\alpha\beta\rho\sigma}\,\Pi(x-y)_{\alpha\beta\mu\nu}\big)\,.
\bar{q}_{xA}=\left(\begin{array}{ccc|c}{b}&{}&{}&{}\\{}&{\ddots}&{}&{~00000\!}\\{}&{}&{b}&{}\\\end{array}\right),\quad\bar{S}_{AB}=\left(\begin{array}{ccc|ccc}{}&{}&{}&{}&{}&{}\\{}&{{\bf{0}}_{F\timesF}}&{}&{}&{}&{}\\{}&{}&{}&{}&{}&{}\\\hline{}&{}&{}&{}&{}&{}\\{}&{}&{}&{}&{b{\bf1}_{5\times5}}&{}\\{}&{}&{}&{}&{}&{}\\\end{array}\right),
\hat{\tilde{E}}_{i}^{a}(x)\Psi_{\gamma}(A)={\frac{\delta}{\deltaA_{a}^{i}(x)}}\Psi_{\gamma}(A)=\ointds\\delta^{3}(x-\gamma(s))\dot{\gamma}^{a}(s)\mathrm{Tr}(U(0,s)\tau^{i}U(s,1))
F^{\mu\nu}=\left(\begin{matrix}{0}&{E_{x}}&{E_{y}}&{E_{z}}\\{-E_{x}}&{0}&{B_{z}}&{-B_{y}}\\{-E_{y}}&{-B_{z}}&{0}&{B_{x}}\\{-E_{z}}&{B_{y}}&{-B_{x}}&{0}\\\end{matrix}\right)
V_{B}^{(0)}=\frac{T_{0}C_{2}Nf^{2}\beta^{2}}{\lambda^{2}}\left(\frac{M^{2}\lambda^{2}}{f^{2}}-\frac{4}{9}\right)\left(\frac{1}{2}+\frac{1}{1+3\sqrt{1-\frac{2M^{2}\lambda^{2}}{f^{2}}}}\right)
Q_{I}(\theta)\rightarrow\mathrm{e}^{i\beta}Q_{I}(\mathrm{e}^{-i\beta}\theta),\qquad{\overline{Q}}^{I}(\theta)\rightarrow\mathrm{e}^{i\beta}{\overline{Q}}^{I}(\mathrm{e}^{-i\beta}\theta).
S_{B}=\frac{2g^{2}}{\beta(8\pi)^{2}}\intd^{3}x\left[\left(\frac{4\pi}{g^{2}}\partial_{\mu}\omega\right)^{2}+\left(\partial_{\mu}\sigma+\frac{\theta}{2\pi}\partial_{\mu}\omega\right)^{2}\right]
\hat{L}_{ij}=i\hbar\left[{\alpha_{i}^{a}}^{\dagger}\alpha_{j}^{a}-{\alpha_{j}^{a}}^{\dagger}\alpha_{i}^{a}\right]=i\hbar\left[{\alpha_{i}^{+}}^{\dagger}\alpha_{j}^{+}+{\alpha_{i}^{-}}^{\dagger}\alpha_{j}^{-}-{\alpha_{j}^{+}}^{\dagger}\alpha_{i}^{+}-{\alpha_{j}^{-}}^{\dagger}\alpha_{i}^{-}\right]\\\,
{\cal{V}}_{{\bf{k}},{\bf{k}}^{\prime}}^{\Lambda,\Lambda_{0}}\{\varphi\}=(2\pi)^{D}\delta({\bf{k}}+{\bf{k}}^{\prime})\Sigma^{\Lambda,\Lambda_{0}}({\bf{k}})+{\cal{U}}_{{\bf{k}},{\bf{k}}^{\prime}}^{\Lambda,\Lambda_{0}}\{\varphi\}\;,
\vec{\phi}^{B_{\pm}}[x;x_{0},\alpha,\beta]=(-1)^{\alpha}\left(\frac{1}{4}(1-(-1)^{\beta}\mathrm{tanh}(\sqrt{2}(x+x_{0}))\vec{e}_{1}\pm\sqrt{\frac{1}{2}(1+(-1)^{\beta}\mathrm{tanh}(\sqrt{2}(x+x_{0}))}\vec{e}_{2}\right)\,
{\bf{F}}=\frac{w^{\prime}}{e}\,[-\hat{\tau}_{\varphi}\,dr\wedged\theta+\hat{\tau}_{\theta}\,sin\theta\,dr\wedged\varphi]-\frac{(1-w^{2})}{e}\,\hat{\tau}_{r}\,sin\theta\,d\theta\wedged\varphi
I=\left[{\frac{1}{(n\cdotp_{1})(n\cdot(p_{1}+p_{2}))}}-{\frac{e^{-ip_{1}\cdotp}}{(n\cdotp_{1})(n\cdotp_{2})}}+{\frac{e^{-(p_{1}+p_{2})\cdotp}}{(n\cdotp_{2})(n\cdot(p_{1}+p_{2}))}}\right]\,\bar{I}
\partial_{\alpha}\partial^{\alpha}X-X^{\prime\prime}-3{\calH}X^{\prime}+\frac{\partial^{2}V}{\partial\varphi^{2}}a^{2}X-\varphi^{\prime}[4\Psi^{\prime}+\Xi^{\prime}]-2\Xi\,(\varphi^{\prime\prime}+3{\calH}\varphi^{\prime})=0.
f_{h}=0,\quad\varphi_{h}<\infty,\quad\delta_{h}<\infty,\quadf_{h}^{\prime}\varphi_{h}^{\prime}=\frac{1}{2}\gamma^{2}\frac{dV}{d\varphi}(\varphi_{h})+2\alphae^{2\alpha\varphi_{h}}\frac{\sqrt{R_{h}^{4}+q_{e}^{2}}-R_{h}^{2}}{R_{h}^{2}}\,\,\,.
{\frac{\partial\eta}{\partialz}}={\frac{\sum_{k=1}^{N}\mathrm{e}^{-i\alpha_{k}}f_{k}}{\sum_{j=1}^{N}f_{j}}},\qquad{\frac{\partial^{2}\eta}{\partialz\partialz^{*}}}={\frac{1}{2}}\left[1-\left|{\frac{\partial\eta}{\partialz}}\right|^{2}\right],
[J_{m}^{-},\left(J_{-1}^{+}\right)^{n}]=n\left(J_{-1}^{+}\right)^{n-1}\left(2J_{m-1}^{3}+km\delta_{m-1}\right)+n(n-1)\left(J_{-1}^{+}\right)^{n-2}J_{m-2}^{+}
z^{\sigma(\Lambda)}=z_{1}^{\Lambda_{\sigma(1)}}\cdotsz_{m}^{\Lambda_{\sigma(m)}}z_{m+1}^{\Lambda_{\sigma(m+1)}}\cdotsz_{N}^{\Lambda_{\sigma(N)}}\quad\mathrm{{and}}\quad\theta^{\sigma(1,\ldots,m)}=\theta_{\sigma(1)}\cdots\theta_{\sigma(m)}\,.
\varphi_{q_{2},\{\alpha\}}(z_{i})=z_{1}^{\alpha_{1}-h}z_{2}^{\alpha_{2}-h}z_{3}^{\alpha_{3}-h}z_{4}^{\alpha_{4}-h}z_{12}^{\alpha_{34}-h}z_{23}^{\alpha_{14}-h}z_{34}^{\alpha_{12}-h}z_{41}^{\alpha_{23}-h}z_{13}^{\alpha_{24}-h}z_{24}^{\alpha_{13}-h}\,,
+s^{2}(\Phif_{4}(-s\Box)\Phi+G_{\mu\nu}f_{5}(-s\Box)G_{\mu\nu}+4G_{\mu0}f_{5}(-s\Box)G_{\mu0}\beta^{2}{\frac{\partial}{\partial\beta^{2}}})]exp(-{\frac{\beta^{2}}{4s}}n^{2})
l=\int_{0}^{y_{*}}\sqrt{B(y)}dy={\frac{3}{4}}(B_{0}y_{0})^{1/3}y_{0}(e^{4n}-1)={\frac{3}{2\kappa^{2}}}\left({\frac{1}{\sigma_{I}}}+{\frac{1}{\sigma_{II}}}\right)\,.
\int{\calD}\{v(\tau,x)\}e^{-\frac12\lambda_{1}\int_{0}^{\beta_{T}}d\tau\int_{-\infty}^{\infty}dx\v(\tau,x)\hat{{\calG}}^{(0)}v(\tau,x)}=\prod_{k\inI\!\!R}\prod_{n=-\infty}^{\infty}\left\{\left(\frac{2\pin}{\beta_{T}}\right)^{2}+\mu^{2}(1+k^{2})\right\}^{-\frac12}
{\calD}({\bfx})=\frac{e}{2}\int{d{\bfy}\,\left[\xi({\bf|x-y|})\frac{\partial}{\partialy_{l}}A_{l}({\bfy})+\eta({\bf|x-y|})\epsilon_{ln}\frac{\partial}{\partialy_{l}}\Pi_{n}({\bfy})\right]},
L=-\frac{1}{g^{2}}\left[\frac{1}{4}G_{\mu\nu}^{a}G_{\mu\nu}^{a}+\kappa\left((D_{\mu}\phi)^{\dagger}D_{\mu}\phi+\frac{1}{4}(\phi^{\dagger}\phi-\mu^{2})^{2}\right)\right]
I=\frac{1}{2\kappa^{2}}\intd^{D}X\sqrt{|g|}\left(R-\frac{1}{2}(\nabla\vec{\phi})^{2}-\sum_{I=1}^{k}\frac{s_{I}e^{-\vec{\alpha}^{(I)}\vec{\phi}}}{2(d_{I}+1)!}F^{(I)2}\right),
T_{MN}=\frac{8}{3}\partial_{M}\Phi\partial_{N}\Phi-\frac{4}{3}\left(\partial\Phi\right)^{2}G_{MN}-V\!\left(\Phi\right)G_{MN}-\sum_{i}f_{i}\,\delta\!\left(y-y_{i}\right)g_{\mu\nu}\delta_{M}^{\mu}\delta_{N}^{\nu},
\langle\psi_{1}|\psi_{2}\rangle=c(\hbar)\,\int_{\calM}\mathrm{d}\mu(z,\bar{z})\,\mathrm{exp}(-\hbar^{-1}K_{\calM}(z,\bar{z}))\,{\overline{\psi}_{1}(z)}\psi_{2}(z),
\bar{S_{v}}^{ab}=\frac{1}{(n-1)!^{2}}\varepsilon^{ab}{}_{a_{1}a_{2}\ldotsa_{d-3}a_{d-2}}\theta^{a_{1}}p_{\theta}^{a_{2}}\theta^{a_{3}}p_{\theta}^{a_{4}}\ldots.
\phi(x)\,\star\,\chi(x)=\phi\left({\frac{1}{i}}{\frac{\partial}{\partialk^{\mu}}}\right)\chi\left({\frac{1}{i}}{\frac{\partial}{\partiall^{\mu}}}\right)e^{ir^{\mu}(k,l)x_{\mu}}\Big|_{k^{\mu}=l^{\mu}=0}\,.
\begin{array}{c}{B^{-}B^{+}-QB^{+}B^{-}~=~Q^{-N_{B}}}\\{B^{-}B^{+}-Q^{-1}B^{+}B^{-}~=~Q^{N_{B}}}\\{Q^{N_{B}}B^{+}Q^{-N_{B}}~=~QB^{+}}\\{Q^{N_{B}}B^{-}Q^{-N_{B}}~=~Q^{-1}B^{-}}\\{Q^{N_{B}}Q^{-N_{B}}~=~Q^{-N_{B}}Q^{N_{B}}~=~1}\\\end{array}
\left({\frac{\dot{a}}{a}}\right)^{2}={\frac{(p-3)^{2}}{16}}\left[\bar{\calE}^{2}\bar{\calP}_{\phi}^{-2{\frac{p-1}{p-3}}}a^{2{\frac{p+1}{p-3}}}-{\frac{1}{a^{2}}}\right],
I^{(d)}\rightarrow\Gamma\left(2-l+\varepsilon\right)\left(\frac{-\pim^{2}}{q^{2}l}\right)^{\frac12}\left(1-\frac{q^{2}}{4m^{2}}\right)^{l-\varepsilon-\frac{3}{2}}(1+O(1/l)).
T_{\mu\mu}={\frac{1}{2}}\left(B+e\,\phi^{*}\,\phi\right)\left(B-e\,\phi^{*}\,\phi\right)\,\quadT^{\pm}=2\left(D_{1}\phi^{*}\pmiD_{2}\phi^{*}\right)\left(D_{1}\phi\pmiD_{2}\phi\right)\,
r(\rho)=\frac{(ga^{3})^{1/2}}{2}\sqrt{\frac{3}{5}}\Big[\frac{2}{3}\rho_{\:\quad2}^{2/3}F_{1}\big[-\frac{9}{20},\frac{1}{4},\frac{11}{20};\frac{l^{10/3}}{\rho^{10/3}}\big]+3l^{3/2}\frac{\Gamma\left(-\frac{9}{20}\right)\Gamma\left(\frac{3}{4}\right)}{10\Gamma\left(\frac{3}{10}\right)}\Big],
\begin{array}{ccc}{\Gamma_{ij}^{A}}&{=}&{-a_{k}\partial^{A}a_{l}\eta_{ij}^{kl},}\\{}&{}&{}\\{\Gamma_{jA}^{i}}&{=}&{a^{k}\partial_{A}a_{l}\eta_{kj}^{li},}\\{}&{}&{}\\{\Gamma_{BC}^{A}}&{=}&{\tilde{\Gamma}_{BC}^{A},}\\\end{array}
{\tilde{c}}_{\alpha\beta}^{\enspace\enspace\gamma}=A_{\alpha}^{\delta}A_{\beta}^{\sigma}(A^{-1})_{\mu}^{\gamma}c_{\delta\sigma}^{\enspace\enspace\mu}+(A^{-1})_{\delta}^{\gamma}(A_{\alpha}^{\sigma}E_{\sigma}A_{\beta}^{\delta}+A_{\beta}^{\sigma}E_{\sigma}A_{\alpha}^{\delta})
\delta[{\frac{1}{12}}\sum_{\mu}({\frac{1}{\alpha_{\mu}}}-\alpha_{\mu})\lambda_{\mu}+{\frac{1}{6}}\sum_{i}({\frac{1}{\alpha_{i}}}-\alpha_{i})\lambda_{i}]+{\frac{1}{6}}\sum_{\mu}\delta\alpha_{\mu}\lambda_{\mu}+{\frac{1}{3}}\sum_{i}\delta\alpha_{i}\lambda_{i}.
J_{1}=-\left(\begin{array}{ccc}{D^{-1}}&{0}&{0}\\{0}&{0}&{\frac{\textstyle{1}}{\textstyle{q_{xx}}}}\\{0}&{-\frac{\textstyle{1}}{\textstyle{q_{xx}}}}&{\frac{\textstyle{1}}{\textstyle{q_{xx}}}D\frac{\textstyle{r_{x}}}{\textstyle{q_{xx}}}+\frac{\textstyle{r_{x}}}{\textstyle{q_{xx}}}D\frac{\textstyle{1}}{\textstyle{q_{xx}}}}\\\end{array}\right)
\cdot{_3F}_{2}\left\{\begin{array}{l}{-n_{1},-l+|m|,l+|m|+\delta_{1}+\delta_{2}+1}\\{|m|+\delta_{1}+1,-n+|m|+1}\\\end{array}\biggr|1\right\}E_{n_{1}n_{2}}^{lm}
\delta\psi_{\alpha}^{(a)}=F_{\mu\nu}\gamma^{\mu\nu}\epsilon^{(a)}+2i(\partial_{\mu}\phi_{m})\gamma^{\mu}(\Sigma_{m})^{aa^{\prime}}\epsilon^{a^{\prime}}+2i(\partial_{\mu}X_{I})\gamma^{\mu}(\Sigma_{I})^{aa^{\prime}}\epsilon^{a^{\prime}}a,
G_{y}^{y}=-\frac{p+1}{p-1}\Lambda_{p}e^{2U}+\frac{p+1}{2}\Big[2(n-1)U^{\prime}V^{\prime}+p(U^{\prime})^{2}\Big]+\frac{(n-1)(n-2)}{2}\Big[(V^{\prime})^{2}-e^{2V}\Big]
E(n_{b},m_{b};A^{(I)}):=\Big\langle\prod_{b=1}^{N}\prod_{i=1}^{n_{b}}\overline{{\psi}}^{(b)}({x}_{i}^{(b)})P_{+}\psi^{(b)}({x}_{i}^{(b)})\prod_{j=1}^{m_{b}}\overline{{\psi}}^{(b)}({y}_{j}^{(b)})P_{-}\psi^{(b)}({y}_{j}^{(b)})e^{ie\sum_{I=1}^{N}(A_{\mu}^{(I)},J_{\mu}^{(I)})}\Big\rangle_{0}^{\theta}\;.
{\cal{A}}_{lowenergy}^{relativ}\,=\,\int^{\infty}d^{3}kI\left(k^{0},\vec{k},w(p),\vec{p}\right)=\int_{0}^{\Lambda}d^{2}k\int_{-\infty}^{\infty}dk^{0}I+\int_{\Lambda}^{\infty}d^{2}k\int_{-\infty}^{\infty}dk^{0}I
L_{\mathrm{St}}^{(n)}=({\bar{\chi}}_{n}^{1*}+(-)^{n+1}{\bar{\pi}}_{n+1}^{1*}\sigma_{-}+{\bar{\chi}}_{n+2}^{3*})\omega_{n}+(-)^{n}{\bar{\chi}}_{n+1}^{1*}\sigma_{-}\chi_{n+2}^{3}\,;
\hat{S}_{\lambda\lambda^{\prime}}=\left[\nabla_{\lambda}\;,\;\nabla_{\lambda^{\prime}}\right]\hat{1}+\frac{1}{2}\left(\nabla_{\lambda}\hat{E}_{\lambda^{\prime}}-\nabla_{\lambda^{\prime}}\hat{E}_{\lambda}\right)+\frac{1}{4}\left(\hat{E}_{\lambda}\hat{E}_{\lambda^{\prime}}-\hat{E}_{\lambda^{\prime}}\hat{E}_{\lambda}\right)
\partial_{\phi}W={\frac{({\frac{1}{3}}h_{1}C-h_{0}D)\phi^{2}+{\frac{2}{3}}(h_{1}B-h_{0}C)\phi+h_{1}A-{\frac{1}{3}}h_{0}B}{(A+B\phi+C\phi^{2}+D\phi^{3})^{4/3}}}\,
\phi(r)=\int_{r}^{\infty}dr\,{\frac{1}{\sqrt{1-a+({\frac{r^{p-1}}{(p-2)c_{p}}})^{2}}}},\qquady(r)=\int_{r}^{\infty}dr\,\sqrt{\frac{a}{1-a+({\frac{r^{p-1}}{(p-2)c_{p}}})^{2}}}.
{\frac{d\omega_{n}^{F}}{dm}}={\frac{1}{\sqrt{k^{2}+m^{2}}}}\left(m+k_{n}{\frac{dk_{n}}{dm}}\right)={\frac{1}{\sqrt{k^{2}+m^{2}}}}\left(m+{\frac{1}{L}}{\frac{k_{n}^{2}}{m}}\left(\delta^{\prime}(k)+{\frac{\theta^{\prime}(k)}{2}}\right)\right)
e^{\phi}=\frac{8f^{\prime}\bar{f^{\prime}}}{(1-f\bar{f})^{2}}=\frac{8|w_{12}|^{2}}{(y_{2}\bar{y}_{2}-y_{1}\bar{y}_{1})^{2}},~~~~f(z)=\frac{y_{1}}{y_{2}}
\begin{array}{c}{\left\{Q_{\alpha}^{a},Q_{\beta}^{b}\right\}=\left(S\right)_{\alpha\beta}^{ab}}\\{\left(S\right)_{\alpha\beta}^{ab}=\delta^{ab}\gamma_{\alpha\beta}^{\mu}\,\,P_{\mu}+\sum_{p=0,1,,2,\cdots}\gamma_{\alpha\beta}^{\mu_{1}\cdots\mu_{p}}\,\,\,Z_{\mu_{1}\cdots\mu_{p}}^{ab}.}\\\end{array}
\frac{1}{g_{h}^{2}}=\frac{1}{g_{h}^{2}}+\frac{b_{0}}{8\pi^{2}}\mathrm{ln}\frac{M}{M^{\prime}}-\sum_{i}\frac{t_{2}(i)}{8\pi^{2}}\mathrm{ln}Z_{i}(M,M^{\prime}).
\frac{\partial^{2}{\calL}}{\partialE_{i}\partialE_{j}}=(1+E_{c}^{2})\delta_{ij},\frac{\partial^{2}{\calL}}{\partial\partial_{i}y\partial\partial_{j}y}=-(1-E_{c}^{2})\delta_{ij},\frac{\partial^{2}{\calL}}{\partialE_{i}\partial\partial_{j}y}=-E_{c}^{2}\delta_{ij},\frac{\partial^{2}{\calL}}{\partial{\dot{y}}^{2}}=1
\begin{array}{l}{\tilde{{\cal{H}}}^{(1)}=-\eta^{1}(\pi-\phi^{\prime}-\lambda)-\eta^{2}(\phi^{\prime\prime}+2\lambda^{\prime}-\pi^{\prime})}\\{\tilde{{\cal{H}}}^{(2)}=\frac{1}{2}\eta^{1}\eta^{1}+\eta^{1}\eta^{2^{\prime}}-\eta^{1^{\prime}}\eta^{2}-\eta^{2}\eta^{2^{\prime\prime}}.}\\\end{array}
\begin{array}{|l|c|l|c|}\hline{\mathrm{Genus}}&{\mathrm{Curvature}}&{\mathrm{Group}}&{\Lambda}\\\hline{0}&{1}&{SO(1,2)}&{-1}\\{1}&{0}&{ISO(1,1)}&{\,\,\,\,0}\\{\geq2}&{-1}&{SO(2,1)}&{\,\,\,\,1}\\\hline{}\\\end{array}
\Psi^{\prime\prime}+[3{\calH}-2\frac{\varphi^{\prime\prime}}{\varphi^{\prime}}]\Psi^{\prime}+[4{\calH}^{\prime}-4{\calH}\frac{\varphi^{\prime\prime}}{\varphi^{\prime}}]\Psi-\partial_{\alpha}\partial^{\alpha}\Psi=0.
{\frac{e^{2}}{2}}f^{adc}f^{bde}\int_{x}\int_{y}\sum_{k}{\frac{\delta^{ce}P_{ij}(k)}{2\omega_{k}}}e^{ik(x-y)}G(x,y)E_{1i}^{a}(x)E_{1j}^{b}(y)
\begin{array}{l}{z_{0}=i\sqrt{\alpha_{1}-\alpha_{2}}\;e^{-i(\alpha_{1}+\alpha_{2})\sigma}}\\{z_{1}=i\sqrt{-\alpha_{1}-2\alpha_{2}}\;e^{i\alpha_{1}\sigma}}\\{z_{2}=\pmi\sqrt{\alpha_{2}+2\alpha_{1}}\;e^{i\alpha_{2}\sigma}}\\\end{array}
H_{Q-Q}=-\,\frac{1}{2}{\int}d{\bfr}{\calK}_{0}^{b}({\bfr})\left(\frac{1}{\partial^{2}}\right){\calK}_{0}^{b}({\bfr})=\frac{1}{2}{\int}d{\bfr}d{\bfx}\,{j}_{0}^{b}({\bfr}){\calF}^{ba}({\bfr},{\bfx}){j}_{0}^{a}({\bfx})\,.
\phi(u,v)=-4v^{2}\frac{\partial}{\partial\epsilon}\sum_{m,n=0}^{\infty}\frac{Y^{m}v^{n+\epsilon}}{m!\Gamma(n+1+\epsilon)}\frac{\Gamma(n+3+\epsilon)\Gamma(n+m+3+\epsilon)\Gamma(n+m+4+\epsilon)}{\Gamma(2n+m+6+2\epsilon)}|_{\epsilon=0}
D[\varphi]=\frac{c_{d+1}}{2}\int_{\partialA}d^{d}x\int_{\partialA}d^{d}x^{\prime}[\varphi_{0}(\mathbf{x})-\varphi_{0}(\mathbf{x}^{\prime})]^{2}\frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|^{d+1}}\mathrm{~.}\tag{2.18}
{\calB}^{+}{}_{\pm\vec{p},h=-1}=\alpha_{\vec{p}}\;(b_{\pm\vec{p},h=-1}^{(0)+}-d_{\mp\vec{p},h=1}^{(0)})-\beta_{\vec{p}}\;(d_{\pm\vec{p},h=1}^{(0)+}-b^{(0)}{}_{\mp\vec{p},h=-1}),
\frac{2\mathrm{i}\pi^{2}}{\sqrt{-\lambda\left(k_{12}^{2},k_{23}^{2},k_{31}^{2}\right)}}\;\left\{\mathrm{Cl}_{2}\left(2\theta_{12}\right)+\mathrm{Cl}_{2}\left(2\theta_{23}\right)+\mathrm{Cl}_{2}\left(2\theta_{31}\right)\right\}
\frac{L}{2}=x(U=\infty)=\frac{R^{2}\sqrt{1-f^{2}(l_{\psi},l_{\phi_{1}})}}{\sqrt{U_{0}^{2}+\left(\frac{r_{0}}{2}\right)^{2}}}I_{1}(l_{\psi},l_{\phi_{1}})
{\frac{g(\tau,\vec{\sigma})}{\gamma(\tau,\vec{\sigma})}}=g_{\tau\tau}(\tau,\vec{\sigma})-\gamma^{{\check{r}}{\check{s}}}(\tau,\vec{\sigma})g_{\tau{\check{r}}}(\tau,\vec{\sigma})g_{\tau{\check{s}}}(\tau,\vec{\sigma}).
\frac{1}{A(q^{2})}=1-\frac{e^{2}(\xi-\xi_{0})}{16q}+\left(\frac{e^{2}(\xi-\xi_{0})}{8\piq}\right)^{2}-\frac{1}{3}\left(\frac{e^{2}(\xi-\xi_{o})}{8\piq}\right)^{4}+\ldots
\left[{\frac{1}{\alpha}}{\frac{d^{2}}{dr^{2}}}+{\frac{2}{\alpha}}{\frac{1}{r}}{\frac{d}{dr}}-{\frac{2}{\alpha}}{\frac{1}{r^{2}}}\right]b_{0}+M\left({\frac{1}{\alpha}}-1\right){\frac{da_{0}}{dr}}=M^{2}b_{0},
\left\{i\left[V_{a}^{\prime\mu}\gamma^{a}\left(\partial_{\mu}-ieA_{\mu}\right)+\gamma^{a}C_{ab}^{\primeb}+\frac{i}{4}\omega_{\mu}^{\primecd}\varepsilon_{abcd}\gamma^{5}\gamma^{b}V^{\primea\mu}\right]-\frac{h}{\sqrt{2}}\varphi\frac{e^{\gamma\phi/M_{p}}}{\chi^{3/2}}\right\}\Psi^{\prime}=0
S[\psi^{*},\psi]=\int[Dx^{\mu}(s)]\,\psi^{*}[x(s)]\left[-\hbar^{2}\left(\int_{0}^{1}ds\sqrt{x^{\,\prime\,2}}\right)^{-1}\int_{0}^{1}{\frac{ds}{\sqrt{x^{\,\prime\,2}}}}{\frac{\delta^{2}}{\deltax^{\mu}(s)\\deltax_{\mu}(s)}}+m^{4}\right]\psi[x(s)]\nonumber\,
\xi^{\prime}\equiv{\frac{2}{\hbar}}(x-x^{\prime}),\quad\xi\equiv\tau-{\frac{2}{\hbar}}(x-x^{\prime}),\quad\eta^{\prime}\equiv{\frac{2}{\hbar}}(p^{\prime}-p),\quad\eta\equiv\sigma-{\frac{2}{\hbar}}(p^{\prime}-p),
\Psi_{\mathrm{Pfaff+qholes}}=\left\langle\sigma(v_{1})\mathrm{e}^{{\frac{i}{2\sqrt{q}}}\varphi(v_{1})}\sigma(v_{2})\mathrm{e}^{{\frac{i}{2\sqrt{q}}}\varphi(v_{2})}\prod_{i=1}^{N}\chi(z_{i})\mathrm{e}^{i\sqrt{q}\varphi(z_{i})}\,\mathrm{e}^{-i\sqrt{q}\intd^{2}z^{\prime}\,\bar{\rho}\varphi(z^{\prime})}\right\rangle.
\hat{S}[\hat{B}]=\intd^{3}x[-\frac{1}{8\pi}\epsilon^{\mu\nu\lambda}(\hat{B}_{\mu}*\partial_{\nu}\hat{B}_{\lambda}+\frac{2i}{3}\hat{B}_{\mu}*\hat{B}_{\nu}*\hat{B}_{\lambda})+\frac{1}{2g^{2}}\hat{B}^{\mu}\hat{B}_{\mu}]+O(\frac{1}{m}).
S_{0}={\frac{1}{\pi}}\intd^{2}x[e^{-2\phi}(2\partial_{+}\partial_{-}\rho-4\partial_{+}\phi\partial_{-}\phi+\lambda^{2}e^{2\rho})+{\frac{1}{2}}\sum_{i=1}^{N}\partial_{+}f_{i}\partial_{-}f_{i}]\\.
Q\bar{Q}Q=\left(\begin{array}{cc}{-u_{1}d_{1}d_{2}+u_{2}d_{1}^{2}}&{-u_{1}u_{2}d_{1}+u_{1}^{2}d_{2}}\\{u_{2}d_{1}d_{2}-u_{1}d_{2}^{2}}&{u_{1}u_{2}d_{2}-u_{2}^{2}d_{1}}\\\end{array}\right)
\varepsilon\propto(E_{\mu}^{a})^{2}+(H_{\mu}^{a})^{2}\propto\left(\frac{rg^{\prime}-g}{r^{2}}\right)^{2}+\frac{2\langlef^{2}\rangleg^{2}}{r^{4}}+\frac{2\langle{f^{\prime}}^{2}\rangle}{r^{2}}+\frac{\langle(f^{2}-1)^{2}\rangle}{r^{4}}
k_{2}^{\nu}T_{\rho\sigma\mu\nu}^{(1)abcd}(p_{1},p_{2};k_{1},k_{2})=C_{1}k_{1\mu}S_{\rho\sigma}^{(1)}+C_{1}\Gamma_{\rho\mu\sigma}^{(1)}(p_{1},k_{1},q_{1})
z_{\tau}^{\mu}(\tau,\vec{\sigma})=\sqrt{\frac{g(\tau,\vec{\sigma})}{\gamma(\tau,\vec{\sigma})}}l^{\mu}(\tau,\vec{\sigma})+g_{\tau\check{r}}(\tau,\vec{\sigma})\gamma^{\check{r}\check{s}}(\tau,\vec{\sigma})z_{\check{s}}^{\mu}(\tau,\vec{\sigma}).
\Omega_{AB}(q)=\left(\begin{array}{cc}{0,}&{-q^{-1/2}}\\{q^{1/2},}&{0}\\\end{array}\right)\qquad\mathrm{and}\qquad\Omega^{AB}(q)=\left(\begin{array}{cc}{0,}&{q^{-1/2}}\\{-q^{1/2},}&{0}\\\end{array}\right).
\alpha_{(\kappa)\hat{\imath}}=(\underbrace{0,\dots,2(D-\kappa-3)\beta_{D-\kappa-1},-2(D-\kappa-1)\beta_{D-\kappa},0,\dots,0}_{k=D-3\mathrm{~terms}}\,;\underbrace{0,\dots,0}_{q\mathrm{~terms}})\,,
\sum_{l}\int_{0}^{\infty}\,drr\,\psi_{\scriptscriptstyle{l,L}}^{\ast}(k,r)\,\psi_{\scriptscriptstyle{l,L^{\prime}}}(k^{\prime},r)=\delta_{\scriptscriptstyle{L,L^{\prime}}}\,\delta(E-E^{\prime})\,.
e_{2}=\frac{4}{A}\bigg(\frac{\dot{A}}{A^{2}}\bigg)^{\dot{\}}\bigg[1-\frac{12\alpha}{M^{2}}\frac{1}{A^{2}}\bigg(\bigg(\frac{A^{\prime}}{A}\bigg)^{2}+\bigg(\frac{\dot{A}}{A}\bigg)^{2}\bigg)\bigg],
\frac{\partial^{2}{Z_{0}}_{l}(r)}{\partialr_{*}^{2}}-V(r){Z_{0}}_{l}(r)=-\frac{4\piq_{s}m_{0}f(r)Y_{l}(0)}{r^{(d-2)/2}}(\frac{dt}{d\tau})_{r_{0}}^{-1}\delta(r-r_{0})\,,
C(\lambda_{t})\Phi^{\prime}\left(\underline{M}_{0}C(\lambda_{t}){\frac{(1+|t|)^{\hat{N}}}{|t|^{r-{\frac{1}{2}}}}}\right)(1+|t|)^{\hat{N}}[(1+\rho)(1+|w|)(1+\rho^{\prime})(1+|w^{\prime}|)]^{N}\tag{\mathrm{A}22}
L=\frac{1}{2e}(\dot{x}_{\mu}-iv\epsilon_{\mu\nu\lambda}\xi^{\nu}\xi^{\lambda})^{2}-\frac{e}{2}m^{2}-2\numv\theta^{+}\theta^{-}-\frac{i}{2}\xi^{\mu}\dot{\xi}_{\mu}+\frac{i}{2}\theta_{a}\dot{\theta}_{a}.
\frac{1}{2}\frac{\partial}{\partial\phi}\left(\frac{H^{3/2}(\phi)}{2\pi}\frac{\partial}{\partial\phi}\left(\frac{H^{3/2}(\phi)}{2\pi}\pi_{j}(\phi)\right)\right)+\frac{\partial}{\partial\phi}\left(\frac{V^{\prime}(\phi)}{3H(\phi)}\,\pi_{j}(\phi)\right)+3H(\phi)\cdot\pi_{j}(\phi)=\lambda_{j}\,\pi_{j}(\phi)\.
\begin{array}{c}{[\,E_{\gamma}\otimesE_{-\gamma}\,,\,H_{j}\otimes1\,]~=~-\,\gamma_{j}\;E_{\gamma}\otimesE_{-\gamma}~,}\\{[\,E_{\gamma}\otimesE_{-\gamma}\,,\,1\otimesH_{j}\,]~=~\gamma_{j}\;E_{\gamma}\otimesE_{-\gamma}~,}\\\end{array}
\overline{{\psi}}(\hat{x},x_{2})=\sum_{n}[\overline{{\psi}}_{L}^{(n)}(\hat{x})\phi_{n}^{\dagger}(x_{2})+\overline{{\psi}}_{R}^{(n)}(\hat{x})\widetilde{\phi}_{n}^{\dagger}(x_{2})].
\frac{\partial\rho(u)}{\partial\tau}=\frac{\partialu}{\partial\tau}\frac{d\rho}{du}-\sum_{r=1,2}\alpha_{r}\frac{\partialZ_{r}}{\partial\tau}g_{1}(u-Z_{r})-\sum_{r=1,2}\frac{i\alpha_{r}}{4\pi}\left(g_{1}^{\prime}(u-Z_{r})+{g_{1}}^{2}(u-Z_{r})\right),
S=\intdt\frac{dx}{2\piR^{\prime}}\frac{1}{2g}\left[\mathrm{Tr}\;\dot{Y}^{i}\dot{Y}^{i}+\mathrm{Tr}\;\dot{A}^{1}\dot{A}^{1}-\mathrm{Tr}\;(\partial_{1}Y^{i}-i[A^{1},Y^{i}])^{2}+\frac{1}{2}\mathrm{Tr}\;[Y^{i},Y^{j}]^{2}\right]
\begin{array}{c}{r=\delta^{-1}(R+\nablar+\frac1{i\hbar}r\circr),}\\{\sigma^{-1}(a)=a+\delta^{-1}(\nabla\sigma^{-1}(a)+\frac1{i\hbar}[r,\sigma^{-1}(a)])}\\\end{array}
H=\left(\begin{array}{cc}{2}&{2}\\{2}&{2}\\\end{array}\right)=2\left(\begin{array}{c}{1}\\{1}\\\end{array}\right)\left(\begin{array}{cc}{1,}&{1}\\\end{array}\right)=4uu^{T},\quadu=\frac{1}{\sqrt{2}}\left(\begin{array}{c}{1}\\{1}\\\end{array}\right).
\left(D^{({\mathbfm}}{}_{{\mathbfj}}{\bar{D}}_{{\mathbfk}{\mathbfl}}+{\bar{D}}^{({\mathbfm}}{}_{{\mathbfj}}D_{{\mathbfk}{\mathbfl}}\right)T^{{\mathbfi}){\mathbfj}{\mathbfk}{\mathbfl}}=-8i\partial_{\alpha\dot{\alpha}}[D_{\mathbfk}^{\alpha},{\bar{D}}_{\mathbfl}^{\dot{\alpha}}]T^{{\mathbfm}{\mathbfi}{\mathbfk}{\mathbfl}}.
F_{\bar{M}\bar{N}}^{\left[mn\right]_{+}}=V_{[\bar{M}}^{m}V_{\bar{N}]}^{n}+\frac12\varepsilon_{\,\,\,\,\,\,\,\,\,pq}^{mn}V_{[\bar{M}}^{p}V_{\bar{N}]}^{q}.
\rho_{t}\left(qy_{3}y_{21}-q^{-1}y_{21}y_{3}\right)=\rho_{t}\left(qy_{13}y_{2}-q^{-1}y_{2}y_{13}\right)=-\frac{q^{\frac{1}{2}}\,t}{q-q^{-1}},\qquad\rho_{t}\left(y_{23}\right)=0,
\begin{array}{rcl}{m_{0}^{2}}&{=}&{\displaystyle\frac{1}{2a^{2}}L^{\alpha\beta}L_{\alpha\beta}}\\{}&{=}&{E^{2}-{\bfP}^{2}-\displaystyle\frac{{\bfL}^{2}}{a^{2}}~,}\\\end{array}
\Psi_{\epsilon}(\alpha)\equiv\frac{\mu^{\epsilon}}{2R}\sum_{n=-\infty}^{\infty}\int\frac{d^{d-\epsilon}p}{(2\pi)^{d-\epsilon}}\,\left[p^{2}+\left(\frac{2\pin}{R}\right)^{2}+m^{2}\right]^{\alpha}.