harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
I(z)=\mathrm{Tr}\left({\frac{z}{z+D_{F}^{\dagger}D_{F}}}-{\frac{z}{z+D_{F}D_{F}^{\dagger}}}\right)=\int_{0}^{\infty}d\lambda{\frac{z}{z+\lambda}}\left({\frac{dn_{+}(\lambda)}{d\lambda}}-{\frac{dn_{-}(\lambda)}{d\lambda}}\right).
1-\frac{1}{8}x^{2}y^{2}+\frac{1}{192}x^{4}y^{4}-\left(\frac{1}{9216}x^{6}+\frac{1}{24x^{2}}\right)y^{6}+o(y^{8})-\frac{4a}{\piy^{2}x^{2}}+...
f^{(n,0)}(r_{1},\dots,r_{n-2},r_{n-1},r_{n}+1)=\frac{2r_{n-1}+1}{2r_{n}+1}\frac{r_{n}+1}{r_{n-1}+1}f^{(n,0)}(r_{1},\dots,r_{n-2},r_{n-1}+1,r_{n}).
\left\{-\delta+\frac{\nabla^{2}}{m}+2\mathrm{Re}\Sigma_{-}\right\}\left[\tilde{\phi}(\left|{\bfr}\right|)e^{\pmi\ell\theta}\right]=V(r)\left[\tilde{\phi}(\left|{\bfr}\right|)e^{\pmi\ell\theta}\right].
{\calR}\left(\lambda,\mu\right)=\left(\begin{array}{cccc}{\frac{\lambdaq-\muq^{-1}}{\lambda-\mu}}&{0}&{0}&{0}\\{0}&{1}&{\frac\mu{\lambda-\mu}\left(q-q^{-1}\right)}&{0}\\{0}&{\frac\lambda{\lambda-\mu}\left(q-q^{-1}\right)}&{1}&{0}\\{0}&{0}&{0}&{\frac{\lambdaq-\muq^{-1}}{\lambda-\mu}}\\\end{array}\right)
2\left(\frac{\Thetag_{m}^{2}\eta}{4\pi}\right)^{2}\left(s_{a}^{(c)}\right)^{2}\intd^{4}xd^{4}y\tilde{\calF}_{\mu\nu}(x){\calD}_{m}^{(4)}(x-y)\tilde{\calF}_{\mu\nu}(y),
J_{z}=J_{(1)}=-[{\frac{1}{\sqrt{2}}}({\bar{\alpha}}_{A}{\bar{\beta}}^{B}+{\bar{\beta}}_{A}{\bar{\alpha}}^{B}){\bar{\pi}}_{B}\omega^{A}+{\frac{1}{\sqrt{2}}}(\alpha_{A^{\prime}}\beta^{B^{\prime}}+\beta_{A^{\prime}}\alpha^{B^{\prime}})\pi_{B^{\prime}}{\bar{\omega}}^{A^{\prime}}],
\Gamma_{N}^{[r,p]}=\Gamma_{N}^{[r,0]}(\{k\};\Lambda)+\sum_{p^{\prime}=1}^{p}\,\left[\frac{g(\nu)}{4\pi}\right]^{2p^{\prime}}\Gamma_{N}^{\left[\left.r,p^{\prime}\right)\right.}(\{k\};\Lambda;\nu)
|\Phi(z,\overline{{z}})\rangle=\phi^{1}(z,\overline{{z}})|0\rangle+\chi^{a}(z,\overline{{z}})\overline{\psi}_{a}|0\rangle+\phi^{2}(z,\overline{{z}})\overline{\psi}^{a}\overline{\psi}_{a}|0\rangle,
m{\frac{d\Omega_{BH}}{dm}}\sim{\frac{mt_{eq}^{1/2}}{m_{p}^{2}t_{f}^{3/2}}}\sim10^{27}\left({\frac{\lambda}{e}}\right)^{1/2}\left({\frac{M}{m_{p}}}\right)^{5/2}\left({\frac{M}{em}}\right)^{\frac{n-1}{2}}\;,
v=\left(\begin{array}{ccccccc}{v_{1}}&{0}&{\ldots}&{0}&{0}&{\ldots}\\{0}&{v_{2}}&{\ldots}&{0}&{0}&{\ldots}\\{\ldots}&{\ldots}&{\ldots}&{\ldots}&{\ldots}&{\ldots}\\{0}&{0}&{\ldots}&{v_{N_{c}}}&{0}&{\ldots}\\\end{array}\right)
S=\int_{0}^{1}\left[-\frac{1}{2e}\left(\dot{x}^{\mu}-i\psi^{\mu}\chi-\epsilon^{\mu\nu\rho\varsigma}b_{\nu}\psi_{\rho}\psi_{\varsigma}+\frac{i\alpha}{2}b^{\mu}\right)^{2}-i\psi_{\mu}\dot{\psi}^{\mu}\right]d\tau\;,
{\frac{\partialW_{0}}{\partialm^{2}}}=-{\frac{AT}{2}}\sum_{n=1}^{\infty}\int{\frac{dk_{1}dk_{2}dk_{0}}{(2\pi)^{3}}}\int_{0}^{\infty}dss^{\nu}e^{-is(-k_{0}^{2}+k_{1}^{2}+k_{2}^{2}+{\frac{n^{2}\pi^{2}}{a^{2}}}+m^{2})}\,.
\left[\frac{\partial^{2}\}{\partialr_{*}^{2}}-\frac{\partial^{2}\}{\partialt^{2}}-\left(1-\frac{2M}{r}\right)\,\frac{2M}{r^{3}}\right]\Big(r\phi(r,t)\Big)-\left(1-\frac{2M}{r}\right)\,V_{,\phi}=0,
H_{~~~~\mu\nu~\alpha\beta}^{-1\sigma~~\rho}H_{\rho~~\omega}^{~\alpha\beta~\eta\kappa}=\frac{1}{2}\delta_{\omega}^{\sigma}\left(\delta_{\mu}^{\eta}\delta_{\nu}^{\kappa}-\delta_{\mu}^{\kappa}\delta_{\nu}^{\eta}\right)
D_{\nu}\equiv\int_{\gamma_{\rho}^{+}}{f(z)\frac{\bar{H}_{\nu}^{(1)}(z)}{\bar{J}_{\nu}(z)}dz}-\int_{\gamma_{\rho}^{-}}{f(z)\frac{\bar{H}_{\nu}^{(2)}(z)}{\bar{J}_{\nu}(z)}dz}=-\pi{\mathrm{Res}}_{z=0}f(z)\frac{\bar{Y}_{\nu}(z)}{\bar{J}_{\nu}(z)}
\begin{array}{ccc}{\deltax^{A}=\lambdax^{A}~~~}&{~~~\delta\theta^{i}=\textstyle{\frac{1}{2}}\lambda\theta^{i}~~~}&{~~~\delta\tilde{\theta}^{\tilde{\imath}}=\textstyle{\frac{1}{2}}\lambda\tilde{\theta}^{\tilde{\imath}}}\\\end{array}
I={\frac{1}{16\piG}}\intd^{4}x\sqrt{-g}\biggl\{R-{\frac{1}{2}}\int_{0}^{1}d\xi_{1}\int_{\xi_{1}}^{1}d\xi_{2}\left[\partial\sigma(\xi_{1})-\partial\sigma(\xi_{2})\right]^{2}-\int_{0}^{1}d\xi\;e^{-\sigma(\xi)}F(\xi)^{2}\biggr\},
L_{f}=\frac{1}{2e}(\dot{x}_{\mu}-iv\epsilon_{\mu\nu\lambda}\xi^{\nu}\xi^{\lambda})^{2}-\frac{e}{2}m^{2}+2i\numv\theta_{1}\theta_{2}-\frac{i}{2}\xi_{\mu}\dot{\xi}{}^{\mu}+\frac{i}{2}\theta_{a}\dot{\theta}_{a}
V_{eff}=-{\frac{m^{2}}{4\pi}}+\theta(\mu^{2}-m^{2}){\frac{m^{2}}{2\pi}}[ln{\frac{\mu+\sqrt{\mu^{2}-m^{2}}}{m}}-{\frac{\mu}{m}}\sqrt{{\frac{\mu^{2}}{m^{2}}}-1}].
S_{\Psi}=\int\!dt\{p_{i}\dot{q}^{i}+\pi_{a}\dot{\lambda}^{a}+{\overline{{\calP}}}_{a}\dot{\calC}^{a}+{\overline{{\calC}}}_{a}\dot{\calP}^{a}-H_{m}+\{Q,\Psi\}\}.
\zeta(s;\widehat{\omega},\widehat{m})={\frac{1}{(2\widehat{\omega})^{s}}}{\frac{1}{\Gamma(s)}}\int_{0}^{\infty}du\,u^{s-1}\,{\frac{e^{\displaystyle-{\frac{u}{2}}\left(1-{\frac{\widehat{m}}{\widehat{\omega}}}\right)}}{1-e^{-u}}}\,Y_{{\bf\mathrm{S}}^{2}}\left({\frac{u}{2\widehat{\omega}}}\right).
\tilde{R}(\alpha,\beta;p)=\frac{1}{2}\int_{-\infty}^{\infty}dse^{4\pisp}R(\alpha,\beta;s),~~~~R(\alpha,\beta,s)=-i\int_{-i\infty}^{i\infty}dpe^{-4\pisp}\tilde{R}(\alpha,\beta,p),
{\calN}_{M}[{\calH}_{\vec{x}}]={\frac{1}{2}}\partial_{t}\phi_{x}\partial_{t}\phi_{x}+{\frac{1}{2}}\partial_{\vec{x}}\phi_{\vec{x}}\partial_{\vec{x}}\phi_{\vec{x}}+{\calN}_{M}[V(\phi_{\vec{x}})]-{\frac{1}{2}}I_{0}[M^{2}]+{\frac{1}{4}}M^{2}I_{1}[M^{2}]\;,
M=\left(\begin{array}{cccc}{M_{1}\epsilon}&{}&{}&{}\\{}&{M_{2}\epsilon}&{}&{}\\{}&{}&{\ddots}&{}\\{}&{}&{}&{M_{g}\epsilon}\\\end{array}\right),\qquad\epsilon=\left(\begin{array}{cc}{0}&{1}\\{-1}&{0}\\\end{array}\right)\tag{A.6}
\begin{array}{cc}{e^{A}(z)=\mathrm{d}x^{A}-i\bar{\theta}_{i}\gamma^{A}\mathrm{d}\theta^{i}~~~~}&{~~~f^{A}(\tilde{z})=\mathrm{d}y^{A}-i\bar{\phi}_{i}\tilde{\gamma}^{A}\mathrm{d}\phi^{i}}\\\end{array}
\langleN\rangle_{T}^{(4)}=\left[\frac{m^{3}T}{12\pi}\sum_{n=-\infty}^{\infty}\frac{m^{2}-5((2n+1)\piT)^{2}}{[m^{2}+((2n+1)\piT)^{2})]^{4}}\right]\intd^{2}x\,(\epsilon^{ij}\,\epsilon^{abc}\,\hat{n}^{a}\,\partial_{i}\hat{n}^{b}\,\partial_{j}\hat{n}^{c})(\partial_{k}\hat{n}^{d}\,\partial_{k}\hat{n}^{d})
\prod_{n=1}^{\infty}\frac{\left(1-q^{6n}\right)^{2}\left(1-q^{6n-3}\right)^{2}\left(1+q^{6n-\frac{3}{2}}\right)^{2}\left(1+q^{6n-\frac{9}{2}}\right)^{2}}{\left(1+q^{6n-\frac{1}{2}}\right)^{2}\left(1+q^{6n-\frac{5}{2}}\right)^{2}\left(1+q^{6n-\frac{7}{2}}\right)^{2}\left(1+q^{6n-\frac{11}{2}}\right)^{2}}.
{\bf70}\,\stackrel{Usp(4)\,\times\,SU(4)\,\times\,U(1)}{\longrightarrow}\,\left({\bf1}+{\bar{\bf1}},{\bf1},{\bf1}\right)\,\oplus\,\left({\bf1},{\bf5},{\bf6}\right)\,\oplus\,\left({\bf1},{\bf1},{\bf6}\right)
\sum_{l=0}^{\infty}\gamma_{l}^{I}\sum_{n=0}^{\infty}\frac{u^{n}}{n!}G_{n}^{I}(\lambda,l;V)+u\sum_{l=0}^{\infty}\gamma_{l}^{({I\negthickspaceI})}\sum_{n=0}^{\infty}\frac{u^{n}}{n!}G_{n}^{I\negthickspaceI}(\lambda,l;V)+\,\mathrm{contributions~of~higher~towers}
\inte^{i\vec{q}\cdot\vec{\varrho}}\frac{\vec{q}^{2}}{\vec{q}^{2}+\mu^{2}}\frac{d^{2}\vec{q}}{(2\pi)^{2}}=\delta^{2}(\vec{\varrho})-\frac{\mu^{2}}{2\pi}K_{0}(|\mu|\varrho)~~,
\begin{array}{c}{\frac{1}{2}\left[\begin{array}{cc}{1}&{0}\\{0}&{1}\\\end{array}\right]}\\{\pm\frac{1}{2}\left[\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right]\mathrm{~or~}\pm\frac{1}{2}\left[\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right]\mathrm{~or~}\pm\frac{q_{i}}{2}\left[\begin{array}{cc}{0}&{1}\\{-1}&{0}\\\end{array}\right]}\\\end{array}
\zeta^{(1)}\cdot\tilde{\zeta}^{(2)}\\zeta^{(2)}\cdot\tilde{\zeta}^{(3)}\\zeta^{(3)}\cdot\tilde{\zeta}^{(4)}\\zeta^{(4)}\cdot\tilde{\zeta}^{(1)}\\frac{1}{s^{2}t^{2}},
R_{ab}(\xi)=\left(\begin{array}{cccc}{1}&{0}&{0}&{0}\\{0}&{\frac{\xi^{-1}-\xi}{q^{-1}\xi^{-1}-q\xi}}&{\frac{q^{-1}-q}{q^{-1}\xi^{-1}-q\xi}}&{0}\\{0}&{\frac{q^{-1}-q}{q^{-1}\xi^{-1}-q\xi}}&{\frac{\xi^{-1}-\xi}{q^{-1}\xi^{-1}-q\xi}}&{0}\\{0}&{0}&{0}&{1}\\\end{array}\right)\,,
\begin{array}{ll}{\alpha_{1}={\frac{1}{2}}(e_{1}-e_{2}-e_{3}-e_{4}-e_{5}-e_{6}+\sqrt2e_{7}),}&{\alpha_{2}=e_{1}+e_{2},}\\{\alpha_{3}=-e_{1}+e_{2},}&{\alpha_{4}=-e_{2}+e_{3},}\\{\alpha_{5}=-e_{3}+e_{4},}&{\alpha_{6}=-e_{4}+e_{5},}\\{\alpha_{7}=-e_{5}+e_{6},}&{\alpha_{0}=-\sqrt{2}e_{7}.}\\\end{array}
S=\frac{1}{g_{0}^{2}}\left(\frac{1}{2\alpha^{\prime}}\left<I\circ\Phi(0)Q_{B}\Phi(0)\right>+\frac{1}{3}\left<f_{1}\circ\Phi(0)f_{2}\circ\Phi(0)f_{3}\circ\Phi(0)\right>\right)\,
{\vec{Q}}\rightarrow{\vec{Q}}^{N}\equiv\left(\begin{matrix}{t_{(1,1,1)}^{0}}\\{t_{(1,1,15)}^{1}}\\{t_{(1,1,15)}^{2}}\\{t_{(1,1,15)}^{3}}\\{0}\\{\dots}\\{0}\\{{\bar{t}}_{{\bar{(}1,1,1)}}^{0}}\\{{\bar{t}}_{{\bar{(}1,1,15)}}^{1}}\\{{\bar{t}}_{{\bar{(}1,1,15)}}^{2}}\\{{\bar{t}}_{{\bar{(}1,1,15)}}^{3}}\\{0}\\{\dots}\\{0}\\\end{matrix}\right)
T_{\mu\nu}=\nabla_{\mu}\varphi\nabla_{\nu}\varphi+\left[\left(\zeta-\frac{1}{4}\right)g_{\mu\nu}\nabla_{\rho}\nabla^{\rho}-\zeta\nabla_{\mu}\nabla_{\nu}-\zetaR_{\mu\nu}\right]\varphi^{2},
t_{\mu\nu}^{(\pm)}(p,k)\;=\;\sum_{n=-\infty}^{+\infty}\,\mathrm{tr}\left[\frac{1}{i\not\!p+\omega_{n}^{(\pm)}}\gamma_{\mu}\frac{1}{i(\not\!p+\not\!k)+\omega_{n}^{(\pm)}}\gamma_{\nu}\right]
\begin{array}{rcl}{(f_{1}\starf_{2})\starf_{3}}&{=}&{f_{1}\star(f_{2}\starf_{3}),}\\{(f_{1}\starf_{2})^{\dagger}}&{=}&{f_{2}^{\dagger}\starf_{1}^{\dagger},}\\{f_{1}\starf_{2}}&{=}&{f_{2}\starf_{1}|_{\theta\rightarrow-\theta},}\\{\displaystyle\intd^{n}x~f_{1}\starf_{2}}&{=}&{\displaystyle\intd^{n}x~f_{2}\starf_{1},}\\\end{array}
\mathrm{sch}V(\Lambda)=\left(\frac{\prod_{\beta\in\Phi_{odd}^{+}(g/r)}(e^{\frac{\beta}{2}}-e^{-\frac{\beta}{2}})}{\prod_{\alpha\in\Phi_{even}^{+}(g/r)}(e^{\frac{\alpha}{2}}-e^{-\frac{\alpha}{2}})}\right)\sum_{c\inC}\mathrm{sgn}(c)\mathrm{sch}U(c\bullet\Lambda).
\delta_{\kappa}A^{ij}=\frac12Str\left(\left(\begin{array}{cc}{0}&{\left(\not{X}_{k}\kappa^{(i}\right)\varepsilon^{j)k}}\\{-\left(\tilde{\kappa}^{(i}\not{X}_{k}\varepsilon^{j)k}\right)}&{0}\\\end{array}\right)\partial_{\tau}t\left(\Theta\right)t^{-1}\left(\Theta\right)\right).
ds_{\mathrm{BTZ}}^{2}=-{\frac{(\rho^{2}-\rho_{+}^{2})(\rho^{2}-\rho_{-}^{2})}{\rho^{2}\ell^{2}}}dt^{2}+\rho^{2}\left(d\phi-{\frac{J}{2\rho^{2}}}dt\right)^{2}+{\frac{\rho^{2}\ell^{2}}{(\rho^{2}-\rho_{+}^{2})(\rho^{2}-\rho_{-}^{2})}}d\rho^{2}.
\Phi_{L}\left(p^{\mu}\right)=\Lambda_{L}\left(\dot{p}^{\mu}\top^{\mu}\right)\Phi_{L}\left(\dot{p}^{\mu}\right)=e^{i\vec{J}\cdot(\vec{\vartheta}+i\vec{\varphi})}~\Phi_{L}\left(\dot{p}^{\mu}\right).
\alpha_{0}^{25}=\frac{1}{2}\sqrt{2\alpha^{\prime}}\left[\frac{n}{R}-\frac{R}{\alpha^{\prime}}m\right]\\,\\\bar{\alpha}_{0}^{25}=\frac{1}{2}\sqrt{2\alpha^{\prime}}\left[\frac{n}{R}+\frac{R}{\alpha^{\prime}}m\right]\.
Z(\beta\,;R(S_{\nabla}^{{\bfC}}),\langle\cdot\,,\cdot\rangle_{H},\langle\cdot\,,\cdot\rangle)=\pi^{\zeta}\,e^{-\frac{i\pi}{2}((\frac{2\theta}{\pi}\mp1)\zeta\,\pm\,\eta)}\,|\beta|^{-\zeta}\tau(M,\chi,\langle\cdot\,,\cdot\rangle_{H})^{-1}
\left[12uv{\calC}+{\frac{1}{3}}P{\calU}+D^{2}(1-a^{i}\partial_{i})\right]{\calF}_{l}=0=\left[12uv{\calC}+{\frac{1}{u}}P{\calV}+D^{2}(1-a^{i}\partial_{i})\right]{\calF}_{l},
L_{gh}=-\left(\bar{c}^{\mu}~\bar{\chi}^{\mu}~\bar{\eta}^{\mu}\right)\left(\begin{array}{ccc}{\omega_{\mu\sigma}\triangle_{~\nu}^{\sigma}}&{0}&{0}\\{0}&{(f_{1}+f_{2}+4f_{3})\zeta_{\mu\nu}}&{0}\\{0}&{0}&{\varsigma_{\mu}^{~\alpha}\epsilon_{\alpha\sigma\lambda\nu}}\\\end{array}\right)\left(\begin{array}{c}{c^{\nu}}\\{\tilde{\chi}^{\nu}}\\{\tilde{\eta}^{\sigma\lambda\nu}}\\\end{array}\right)
G_{\mu\nu}=\left(\begin{array}{ccc}{-g(1-e^{2})}&{0}&{-geb}\\{0}&{g(1-e^{2})+\frac{g^{2}b^{2}}{g_{2}}}&{0}\\{-geb}&{0}&{g_{2}+gb^{2}}\\\end{array}\right),
S_{\mathrm{H}}=\intd^{4}x\sqrtg\left(-{\frac{M_{\mathrm{P}}^{2}}{16\pi}}{\calR}+f_{0}^{-2}H_{\mu\nu\lambda}^{2}\right)-{\frac{M_{\mathrm{P}}^{2}}{16\pi}}\int_{\partialV}d^{3}S(K-K_{0})\.
{\mathcalL}^{2\!\times\!3}=\frac{1}{2\,g^{2}}\sum_{i=1,2}\sum_{a=1}^{3}\big(\dot{a}_{i}^{a}+\epsilon^{abc}\:a_{0}^{b}\:a_{i}^{c}\big)\big(\dot{a}_{i}^{a}+\epsilon^{ade}\:a_{0}^{d}\:a_{i}^{e}\big)-{\mathcalV}^{2\!\times\!3}(a_{i}^{a})\;,
Q_{Z}=Q_{1}+\frac{121}{12\sqrt{105}}S^{\prime\prime}-\frac{2}{3}\sqrt{\frac{2}{3}}(J\;S^{\prime})-\sqrt{\frac{2}{3}}(J^{\prime}\;S)+\frac{1}{3}\sqrt{\frac{35}{3}}(J\;J\;S)-\frac{1}{3}\sqrt{\frac{35}{3}}(T\;S)
K_{\phi}^{ij}(\nu)=\frac{(-1)^{i+j}}{4}\left(\frac{(\nu-2\omega)^{2}}{\omega^{4}}\Gamma_{\phi}^{ij}(\nu-\omega)+\frac{(\nu+2\omega)^{2}}{\omega^{4}}\Gamma_{\phi}^{ij}(\nu+\omega)\right)\,.
({\calD}_{\mu}{\mit\Phi})^{A}:=(\partial_{\mu}\delta^{AC}+gf^{ABC}{\calA}_{\mu}^{B}){\mit\Phi}^{C}=(D_{\mu}{\mit\Phi})^{A}+gf^{AbC}A_{\mu}^{b}{\mit\Phi}^{C}.
\begin{array}{c}{\widehat{T}_{1}=\Box-m^{2}\quad,\quad\widehat{T}_{2}=\bigtriangleup_{1}-\delta_{1}^{^{\prime}2}\quad,\quad\widehat{T}_{3}=\bigtriangleup_{2}-\delta_{2}^{^{\prime}2}}\\{\widehat{T}_{4}=d-n\qquad,\qquad\widehat{T}_{5}=\overline{{d}}-m}\\\end{array}
\overline{{x}}=\left[\frac\alpha{1-\alpha^{2}+6\gamma(\gamma-1)}\frac{12\gamma}{\alpha+2\gamma}\left(\frac{g_{i}^{N}-\gamma/\alphaf_{i}^{N}}{g_{i}^{N}+\gamma/\alphaf_{i}^{N}}\right)\right]\.
A_{N}=c\int\frac{\displaystyle\prod_{a=1}^{N}d\xi_{a}}{d^{3}F(\xi_{1},\xi_{2},\xi_{3})}\frac{\displaystyle\prod_{c=3}^{N}d\theta_{c}}{\xi_{1}-\xi_{2}}\langle0|~V_{T}^{+(-1)}(\xi_{1};k_{1\mu})V_{T}^{-(-1)}(\xi_{2};k_{2\mu})\prod_{c^{\prime}=3}^{N}{\bfV}_{\mathrm{vec}}(\xi_{c^{\prime}},\theta_{c^{\prime}};k_{c^{\prime}\mu},\zeta_{c^{\prime}\mu})~|0\rangle~.
\Phi=U_{2}^{-1}({\bfr})\left(\begin{matrix}{t_{4}-{\frac{1}{2y_{R}}}}&{0}&{0}&{0}\\{0}&{t_{2}+{\frac{1}{2y_{R}}}}&{0}&{0}\\{0}&{0}&{t_{2}-{\frac{1}{2y_{L}}}}&{0}\\{0}&{0}&{0}&{t_{1}+{\frac{1}{2y_{L}}}}\\\end{matrix}\right)U_{2}({\bfr})+\cdots
\hat{W}_{N_{r}}(t)=\sum_{N_{r}^{\prime},N_{r}^{\prime\prime}}Y_{p,N_{r}^{\prime}}(t_{1})A_{N_{r}^{\prime\prime}N_{r}^{\prime}}^{-1}\sum_{p=a,b}\int_{C_{p}^{(r)}}\chi_{N_{r}^{\prime}}(t_{1})dt_{1}E_{p,N_{r}}^{(-2)}(S_{\sigma}(t_{1},t)).
G_{ij}=\left(\begin{array}{cc}{2\omega\left(e^{-2\phi}+\frac{\kappa}{4}\right)}&{e^{-2\phi}+\frac{\kappa}{4}(1-\omega)}\\{e^{-2\phi}+\frac{\kappa}{4}(1-\omega)}&{-\frac{\kappa}{2}}\\\end{array}\right)
\sqrt{-G}G^{M_{1}N_{1}}\cdotsG^{M_{D-p-1}N_{D-p-1}}\tilde{F}_{N_{1}\dotsN_{D-p-1}}={\frac{1}{(p+1)!}}\epsilon^{M_{1}\dotsM_{D-p-1}N_{1}\dotsN_{p+1}}F_{N_{1}\dotsN_{p+1}}.
\textstyle\mathcal{J}^{2}=C^{\infty}(M)\otimes\sum(\{\hat{\pi}(\mathfrak{a}),\hat{\pi}(\mathfrak{a})\}+[\hat{\pi}(\mathfrak{a}),[\dots[\hat{\pi}(\mathfrak{a}),\hat{Y}_{\perp}^{2}]\dots]])~.
\partial_{2}\left(\frac{B}{R}\sqrt{\frac{R^{2}+\Pi^{2}}{B^{2}+R^{2}}}\right)=0\,,\qquad\partial_{2}\left[2y^{\underline{{i}}}\frac{R^{4}-\Pi^{2}B^{2}}{R^{2}\sqrt{(R^{2}+\Pi^{2})(R^{2}+B^{2})}}\right]=0\,.
{\calL}_{\mathrm{eff}}=\frac{1}{2}\int\!\mathrm{d}^{2}\theta\mathrm{d}^{2}\bar{\theta}\,\bar{\Phi}\Phi+\left\{\frac{m}{2}\int\!\mathrm{d}^{2}\theta\,\Phi^{2}+\mathrm{H.c.}\right\}\,.
A_{1}(x)=-\Theta_{1}^{-1\mathrm{t}}\stackrel{1}{\Pi}\lambda_{\alpha_{0}}^{|0|}[r_{p_{0}}^{|0|}]^{\alpha_{0}}dA_{0}^{p_{0}}(x)+\lambda_{\alpha_{1}}^{|1|}A_{1}^{\alpha_{1}}(x)\,.
D_{-\frac{1}{2}~\frac{1}{2}}^{\frac{1}{2}}=\frac{\partial}{\partial\xi_{2}^{*}}~(\xi_{1}^{*}\psi_{1}+\xi_{2}^{*}\psi_{2})=\psi_{2}=-e^{i(\gamma-\alpha)/2}~sin(\frac{\beta}{2})
\sigma_{F}\,\longleftrightarrow\,(-1)^{\sumJ+\sumj}\:\:\left\{\begin{array}{ccc}{j_{1}}&{j_{2}}&{j_{3}}\\{J_{1}}&{J_{2}}&{J_{3}}\\\end{array}\right\}\,\doteq\,(-1)^{\sumJ+\sumj}\:\{6j\}(\sigma_{F}),
B_{\mu\nu}^{a}=\frac{1}{2}\eta_{BC}^{a}e_{\mu}^{B}e_{\nu}^{C},\\,bar{B}_{\mu\nu}^{a}=\frac{1}{2}\bar{\eta}_{BC}^{a}\bar{e}_{\mu}^{B}\bar{e}_{\nu}^{C},
D\rho^{++}=-{\frac{1}{2}}E^{\underline{{B}}}F_{\underline{{Bm}}}u^{++\underline{{m}}}=e^{+q}W^{\underline{{\mu}}}v_{\underline{{\mu}}q}^{+}-{\frac{i}{2(D-2)}}e^{++}v_{\dot{q}}^{+\underline{{\nu}}}\bigtriangledown_{\underline{{\nu}}}W^{\underline{{\mu}}}v_{\underline{{\mu}}\dot{q}}^{-}
\int_{{\bfr}(t^{\prime})={\bfr}^{\prime}}^{{\bfr}(t^{\prime\prime})={\bfr}^{\prime\prime}}\;{\calD}^{(N)}{\bfr}(t)\equiv\left[{\mathcalC}(M,\epsilon,\hbar,D)\right]^{N}\,\left[\prod_{k=1}^{N-1}\int_{\mathbb{R}^{D}}d^{D}{\bfr}_{k}\right]\;,
Z_{F}[C]=\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n!}\int\prod_{i=1}^{n}{\calD}{\bfX}_{i}exp\{-\sum_{i=1}^{n}[\intdt(M\sqrt{\dot{\bfX}_{i}^{2}}+i{\bfC}\cdot\dot{\bfX}_{i})-\frac{i}{2}\Phi[\frac{\dot{\bfX}_{i}}{|\dot{\bfX}_{i}|}]-i\frac{\alpha}{2}\Theta_{ii}]+i\alpha\sum_{i<j}\Theta_{ij}\}\;,
\sum_{r=0}g^{r}(-1)^{r}\epsilon_{(r)}^{\vec{\delta}dh}\,{\calT}_{(r)}^{\vec{\delta}}({\bfr})=-\delta_{d,h}-{\textstyle\frac{g}{1+{\calW}}}\epsilon^{\deltadh}\,{\calT}^{\delta}+{\textstyle\frac{g^{2}}{1+{\calW}}}\epsilon^{{\delta}(1)db}\epsilon^{b{\delta}(2)h}\,{\calT}^{{\delta}(1)}\,{\calT}^{{\delta}(2)}\;,
\begin{array}{c}{F(u_{0},v)=\alpha+\bar{\beta}\quad\leftrightarrow\quadv=W(\alpha+\bar{\beta},u_{0}),}\\{\bar{F}(u,v_{0})=\bar{\alpha}+\beta\quad\leftrightarrow\quadu=\bar{W}(\bar{\alpha}+\beta,v_{0}),}\\\end{array}
\widetilde{\Psi}[\widetilde{\varphi},\widetilde{x}]\equiv\mathrm{e}^{\frac{7+d}{32}(\widetilde{\varphi}(0)+\widetilde{\varphi}(1))}\Psi[{\textstyle\int\mathrm{e}^{\frac{1}{2}\widetilde{\varphi}}},\widetilde{x}\circ\gamma[\widetilde{\varphi}]^{-1}]\;\;\;.
R=-\frac{2}{C^{2}}\left(\frac{A^{\prime\prime}}{A}+3\frac{B^{\prime\prime}}{B}+3\frac{{B^{\prime}}^{2}}{B^{2}}+3\frac{A^{\prime}B^{\prime}}{AB}-\frac{A^{\prime}C^{\prime}}{AC}-3\frac{B^{\prime}C^{\prime}}{BC}\right)=-h^{\prime\prime}(r)-6\frac{h^{\prime}(r)}{r}-6\frac{h(r)}{r^{2}}.
{\frac{1}{k!}}\sum_{a_{1},\dots,a_{k}}Tr_{F}\{t_{F}^{a_{1}}t_{F}^{a_{2}}\cdotst_{F}^{a_{k}}\}t_{(i_{1})}^{(a_{1}}t_{(i_{2})}^{a_{2}}\cdotst_{(i_{k})}^{a_{k})}
D_{\mu\nu}(p)=\frac{i}{p^{2}-M^{2}}\left[-m^{2}g_{\mu\nu}+p_{\mu}p_{\nu}\frac{m^{2}-\xi\kappa^{2}}{p^{2}+\xim^{2}}+i\kappa\epsilon_{\mu\nu\lambda}p^{\lambda}\right]\frac{1}{\kappa^{2}},
B^{a}=\left(\begin{array}{cccc}{c_{1}^{a}}&{\ldots}&{0}&{0}\\{\vdots}&{\ddots}&{\vdots}&{0}\\{0}&{\ldots}&{c_{l}^{a}}&{0}\\{0}&{0}&{0}&{Y_{n}^{a}}\\\end{array}\right).
\frac{x_{3}}{x_{1}}\stackrel{\rho}{\rightarrow}\frac{x_{13}x_{4}}{x_{3}x_{14}},~~\frac{x_{3}}{x_{2}}\stackrel{\rho}{\rightarrow}\frac{\omegax_{23}x_{4}}{x_{3}x_{24}},~~\frac{x_{4}}{x_{2}}\stackrel{\rho}{\rightarrow}\frac{x_{23}}{x_{24}},~~\frac{x_{14}x_{23}}{x_{13}x_{24}}\stackrel{\rho}{\rightarrow}\frac{x_{2}}{x_{1}}
\Lambda_{\pm}(t,r;q):=\frac{1-q}{2}\left\{\sum_{n=0}^{\infty}\frac{(q\frac{\rho_{+}}{\rho_{-}};q)_{2n}}{(q;q)_{2n+1}}\biggl(i\frac{(1-q)\rho_{-}}{2}\biggr)^{2n}\pm(\rho_{+}\leftrightarrow\rho_{-})\right\}.
L_{Weyl}^{pzm}=\intd^{3}{x}\left[-\partial_{i}a_{-}\partial_{+}a_{i}+\frac12(\partial_{+}a_{-})^{2}-\frac14(\partial_{i}a_{j}-\partial_{j}a_{i})^{2}+j^{\perp}a_{\perp}+j^{-}a_{-}\right]\;,
\Gamma\lbrackA\rbrack=\!-2\!\int_{0}^{\infty}{\frac{dT}{T}}\mathrm{e}^{-m^{2}T}\int{\calD}x\,{\calD}\psi\,\mathrm{exp}\left[-\int_{0}^{T}\!\!\!d\tau\left({\frac{1}{4}}{\dot{x}}^{2}\!+\!{\frac{1}{2}}\psi\dot{\psi}\!+\!ieA_{\mu}\dot{x}^{\mu}\!-\!ie\psi^{\mu}F_{\mu\nu}\psi^{\nu}\right)\right]\nonumber\,
N_{(i,\bar{i})(j,\bar{j})(k,\bar{k})}^{t}=N_{ijk}\bar{N}_{\bar{i}\bar{j}\bar{k}}\;\;\;\;,\;\;\;Y_{(i,\bar{i})(j,\bar{j})(k,\bar{k})}^{t}=Y_{ijk}\bar{Y}_{\bar{i}\bar{j}\bar{k}}\;\;\;\;\;.
B=\mid\psi_{1}\mid\left(\begin{matrix}{1}&{-(\frac{\psi_{2}}{\psi_{1}})^{*}}\\{(\frac{\psi_{2}}{\psi_{1}})}&{1}\\\end{matrix}\right)=\frac{1}{\sqrt{1+{\midz\mid}^{2}}}\left(\begin{matrix}{1}&{-z^{*}}\\{z}&{1}\\\end{matrix}\right)
\begin{array}{c}{H^{p}\left(S\capT,{\calE}^{\vee}|_{S\capT}\otimes{\calF}|_{S\capT}\otimes\Lambda^{q-m}\tilde{N}\otimes\Lambda^{top}N_{S\capT/T}\right)}\\{H^{p}\left(S\capT,{\calE}|_{S\capT}\otimes{\calF}^{\vee}|_{S\capT}\otimes\Lambda^{q-n}\tilde{N}\otimes\Lambda^{top}N_{S\capT/S}\right)}\\\end{array}
S_{min}=\bar{\sigma}^{\dot{\alpha}}\bar{D}_{\dot{\alpha}}D_{\alpha}\sigma^{\alpha}\+\sum_{i=1}^{\infty}\sigma_{A_{i}}^{*}D_{\beta}\sigma^{(\betaA_{i})}+\sum_{i=1}^{\infty}\bar{\sigma}^{(\dot{\beta}\dot{A}_{i})}\bar{D}_{\dot{\beta}}\bar{\sigma}_{\dot{A}_{i}}^{*}\,,
H_{\pm}=\left(\begin{array}{cc}{0}&{ie^{\mpi\theta}\left(\partial_{r}\pm\mathrm{B}\right)}\\{-ie^{\pmi\theta}\left(-\partial_{r}\pm\mathrm{B}\right)}&{0}\\\end{array}\right)\qquad\mathrm{B}=-\frac{i}{r}\partial_{\theta}-\frac{\kappa}{r}
\hat{\nabla}_{i}=i\sqrt{V}P_{i}+\frac{i}{2}V\sqrt{V}\varepsilon_{ijk}\Sigma_{j}^{*}B_{k}\,,\quad\hat{\nabla}_{4}=\frac{i}{\sqrt{V}}P_{4}-\frac{i}{2}V\sqrt{V}\vec{\Sigma}^{*}\cdot\vec{B}\,.
min_{C(\cdot)}\left(\intdt\left(\frac1C\frac{\partialC}{\partialt}+\frac1C\frac{\sigma^{2}}{2}S^{2}\frac{\partial^{2}C}{\partial^{2}S}-r_{b}(1-\fracSC\frac{\partialC}{\partialS})\right)^{2}-\frac{1}{\beta_{2}}\deltas_{g}\right)\.
Z=\bigl(\prod_{y,\mu}\,\sum_{\tilde{n}_{y},\tilde{m}_{y,\mu}}\bigr)\prod_{y}\delta^{(P)}(\triangle_{\mu}\tilde{m}_{y,\nu}-\triangle_{\nu}\tilde{m}_{y,\mu})\prod_{y,\mu}\tilde{B}(\triangle_{\mu}\tilde{n}_{y}-Q\tilde{m}_{y,\mu}).
ds^{2}={\frac{1}{\sqrt{H_{A}}}}\left(-dt^{2}+dz_{A}^{2}\right)+\sqrt{H_{A}}dx_{\perp}^{2}\,,\qquad\mathrm{where\\}H_{A}=1+\left({\frac{r_{A}}{|x_{\perp}|}}\right)^{7-A}\,.
P^{i}=\intd^{2}x\left[\varphi\nabla_{i}\dot{\varphi}+{\frac{1}{\mu^{2}}}({\pi_{\chi}}+{\frac{\kappa}{2}}\varphi)\nabla_{i}({\dot{\pi}_{\chi}+{\frac{\kappa}{2}}\dot{\varphi}})\right]
H_{N}=\frac{1}{2m}\sum_{j=1}^{N}\left[\left(-i\frac{\partial}{\partial{\bfr}_{j}}-e\sum_{k\neqj}{\bfA}({\bfr}_{jk})\right)^{2}-2\sigmae\sum_{k\neqj}B(r_{jk})\right]+V,
A^{a\varrho}\rightarrow\frac{1}{\partial_{\eta}\partial^{\eta}}\partial^{\varrho}\partial_{\sigma}A^{a\sigma}=-\frac{1}{2}\frac{\partial^{\varrho}}{\partial_{-}}A_{-}^{a}\;\;,\;-\frac{1}{\partial_{\eta}\partial^{\eta}}\partial_{-}\partial^{\varrho}A_{+}^{a}
\begin{array}{ll}{A=A^{1}+iA^{2},}&{A^{*}=A^{1}-iA^{2}}\\{\gamma=\gamma^{1}+i\gamma^{2},}&{\gamma^{*}=\gamma^{1}-i\gamma^{2}}\\{\gamma_{\pm}=\gamma_{0}\pm\gamma_{3}}&{}\\\end{array}
\Gamma_{A}\equiv\left(\begin{matrix}{0}&{(1-C^{2})A^{-1}}\\{A}&{0}\\\end{matrix}\right)\,=\sum_{n=1,3,5}\Gamma^{i_{1}}\cdots\Gamma^{i_{n}}F_{i_{i}\cdotsi_{m}}(X^{a^{\prime}},\theta^{\alpha^{\prime}},h_{ijk},a)\,,
f_{k}^{(2)}(t)=-\frac{1}{4(\omega_{k}^{0})^{2}}\int_{0}^{t}\!\mathrm{d}t^{\prime}\int_{0}^{t^{\prime}}\!\mathrm{d}t^{\prime\prime}V(t^{\prime})V(t^{\prime\prime})+O((\omega_{k}^{0})^{-3})\;.
\begin{aligned}{e\,\delta\Big[e_{r}{}^{\mu}e_{s}{}^{\nu}R_{\mu\nutu}\Big]\,S^{rstu}}&{{}=4\,e\Big(\bar{\epsilon}\Gamma^{\mu}\psi_{[r}\Big)\,R_{s]\mutu}S^{rstu}}\\{}&{{}\quad+2\,\Big(\frac{1}{2}T_{rs}{}^{\nu}S^{rstu}-T_{r\lambda}{}^{\lambda}S^{r\nutu}\Big)\delta\omega_{\nutu}}\\{}&{{}\quad-2\,e\Big(D_{\mu}S^{rstu}\Big)e_{r}{}^{\mu}e_{s}{}^{\nu}\,\delta\omega_{\nutu}\,,}\\\end{aligned}
\lambda_{2}^{(k+2)}=-{\frac{a_{s}}{a_{k+1}}}\left({\frac{2}{\lambda_{2}^{(k+1)}}}+{\frac{k}{\lambda_{3}^{(k+1)}}}\right)\quad,\quad\lambda_{3}^{(k+2)}=-{\frac{a_{s}}{a_{k+1}}}{\frac{k+1}{\lambda_{2}^{(k+1)}}}\quad.
p_{i}\left|\begin{array}{cc}{[m]_{n}}\\{(m)_{n-1}}\\\end{array}\right\rangle=\hbar(m_{in}+n-i)\left|\begin{array}{cc}{[m]_{n}}\\{(m)_{n-1}}\\\end{array}\right\rangle\equiv\hbarp_{in}\left|\begin{array}{cc}{[m]_{n}}\\{(m)_{n-1}}\\\end{array}\right\rangle
<\phi_{\eta}(x_{1},\tau_{1})\ldots\phi_{\eta}(x_{n},\tau_{n})>_{\eta}\equiv\intd\eta(x,t)e^{-\int\eta^{2}/\Omega}\phi_{\eta}(x_{1},\tau_{1})\ldots\phi_{\eta}(x_{n},\tau_{n}).
\left(\begin{matrix}{\tilde{x}^{++}}\\{\tilde{x}^{--}}\\\end{matrix}\right)=\left(\begin{matrix}{e^{\alpha}}&{0}\\{0}&{e^{-\alpha}}\\\end{matrix}\right)\left(\begin{matrix}{x^{++}}\\{x^{--}}\\\end{matrix}\right)\,\,,
\delta_{04}=\int_{r_{\mathrm{min}}}^{\infty}dr\left[\sqrt{D_{4}(E^{2}-p_{11}^{2})-{\frac{J^{2}}{r^{2}}}}-\sqrt{E^{2}-p_{11}^{2}-{\frac{J^{2}}{r^{2}}}}\right]
\begin{array}{l}{\displaystyle\omega^{ij}\frac{\deltaS}{\delta\omega^{ij}}=\{X^{A},X^{B}\}\{X_{A},X_{B}\}-\frac{\alpha^{2}}{2}=0\,}\\{\displaystyle-\frac14\frac{\deltaS}{\deltaX^{A}}=\{X^{B},\{X_{B},X_{A}\}\}=0\,.}\\\end{array}
A_{\mu\nu}^{\lambda}=\frac{1}{2}\sum_{\kappa=0}^{3}\left(\frac{\partialg_{\nu\kappa}}{\partialx^{\mu}}+\frac{\partialg_{\mu\kappa}}{\partialx^{\nu}}-\frac{\partialg_{\mu\nu}}{\partialx^{\kappa}}\right)g^{\kappa\lambda}.
\frac{dP}{d\Omega_{8}}=2\kappa^{2}T_{0i}\hat{k}_{i}={\frac{2\kappa^{2}T_{0}}{105^{2}\,\Omega_{8}^{2}}}\omega^{8}\,{\frac{1}{3!}}\,\left[J^{ijk}J_{ijk}^{*}-3\,\hat{k}_{l}\hat{k^{m}}J^{lij}J_{mij}^{*}\right]
\begin{array}{ccccccc}{\bigcirc_{0}}&{}&{}&{}&{}&{}&{}\\{}&{\diagdown}&{}&{}&{}&{}&{}\\{}&{}&{\oplus_{2}}&{-}&{\oplus_{3}}&{-\dots-}&{\oplus_{d-1}}\\{}&{\diagup}&{}&{}&{}&{}&{}\\{+_{1}}&{}&{}&{}&{}&{}&{}\\\end{array}
\bar{A}_{k}^{a}=\frac{2}{L}\,\vartheta\:\delta^{a3}\delta_{k0}+\frac{2}{L}\,\vartheta\:\epsilon^{a3b}\:\bar{\varphi}_{k}^{b}+\mathrm{i}\,k\:\frac{2\pi}{L}\,\bar{\varphi}_{k}^{a}\;,