harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{rcl}{\chi_{1,i}^{(2,2k+1)}(q)}&{=}&{\displaystyle\frac{1}{(q)_{\infty}}\sum_{n=-\infty}^{\infty}\left(q^{(4k+2)n^{2}+(2k-2i+1)n}-q^{(2n+1)((2k+1)n+i)}\right)}\\{}&{=}&{\displaystyle\frac{1}{(q)_{\infty}}\sum_{n=-\infty}^{\infty}(-1)^{n}q^{n((2k+1)n+2k-2i+1)/2}}\\{}&{=}&{\displaystyle\prod_{n=1\atopn\not\equiv0,\pmi(\mathrm{mod}2k+1)}^{\infty}(1-q^{n})^{-1},}\\\end{array}
4\pi\frac{H^{\prime}(r_{++})}{(r_{++}^{2}+a^{2})}\\delta\left[\frac{(r_{++}^{2}+a^{2})}{Z}\right]=\delta\left[\frac{m}{Z^{2}}\right]-\left(\frac{aZ}{r_{++}^{2}+a^{2}}-\frac{a}{l^{2}}\right)\\delta\left[\frac{ma}{Z^{2}}\right]\,
\sum_{n_{\mu}^{+},n_{\mu}^{-},m_{\mu}^{+},m_{\mu}^{-}=0}^{\sum_{\mu=1}^{M}(n_{\mu}^{+}+n_{\mu}^{-}+m_{\mu}^{+}+m_{\mu}^{-})}\,e^{i\sum_{\mu=1}^{M}\omega_{\mu}(n_{\mu}^{+}-n_{\mu}^{-}-m_{\mu}^{+}+m_{\mu}^{-})}\,\delta\left(\sum_{\mu=1}^{M}(n_{\mu}^{+}-n_{\mu}^{-}+m_{\mu}^{+}-m_{\mu}^{-})\right)\\\.
N=\left(\begin{matrix}{\frac{G+2}{r}-\frac{mp}{2G+1}}&{2\frac{\sqrt{G(G+1)}}{2G+1}mp}\\{2\frac{\sqrt{G(G+1)}}{2G+1}mp}&{-\frac{G-1}{r}+\frac{mp}{2G+1}}\\\end{matrix}\right)\quad\mathrm{and}\quad\bar{N}=\left(\begin{matrix}{-\frac{G}{r}+\frac{mp}{2G+1}}&{2\frac{\sqrt{G(G+1)}}{2G+1}mp}\\{2\frac{\sqrt{G(G+1)}}{2G+1}mp}&{\frac{G+1}{r}-\frac{mp}{2G+1}}\\\end{matrix}\right)
x=(0,\,-(\frac{m_{0}m_{2}}{\Lambda_{1}^{4}\Lambda_{2}^{4}})^{\frac{1}{2}},\,-(\frac{m_{0}m_{2}}{\Lambda_{1}^{4}\Lambda_{2}^{4}})^{\frac{1}{2}},\,(\frac{m_{0}m_{2}}{\Lambda_{1}^{4}\Lambda_{2}^{4}})^{\frac{1}{2}},\,(\frac{m_{0}m_{2}}{\Lambda_{1}^{4}\Lambda_{2}^{4}})^{\frac{1}{2}}).
G_{0}^{(2)}(\lambda,\vec{r}_{1},z_{1},\vec{r}_{3},z_{3})=\frac{\lambda}{2}\intd^{d-1}r_{2}\int_{0}^{L}dz_{2}G_{0}^{(2)}(\vec{r}_{1}-\vec{r}_{2};z_{1},z_{2})G_{0}^{(2)}(\vec{0},z_{2})G_{0}^{(2)}(\vec{r}_{2}-\vec{r}_{3};z_{2},z_{3}).
\phi_{i}(\tau)=\big(\pi_{i}^{\mu}(\tau)+\fraci2\xi_{i}^{\mu}(\tau)\big)\int{d^{3}\sigma\rho_{\mu}(\tau,\vec{\sigma})\equiv\big(\pi_{i}^{\mu}(\tau)+\fraci2\xi_{i}^{\mu}(\tau)\big){p_{s}}_{\mu}}\approx0,
\hat{a}_{i}\hat{a}_{j}\pm\hat{a}_{j}\hat{a}_{i}=0=\hat{a}_{i}^{\dagger}\hat{a}_{j}^{\dagger}\pm\hat{a}_{j}^{\dagger}\hat{a}_{i}^{\dagger},\quad\hat{a}_{i}\hat{a}_{j}^{\dagger}\pm\hat{a}_{j}^{\dagger}\hat{a}_{i}=\delta_{ij}\hat{I}=\pm(\hat{a}_{j}^{\dagger}\hat{a}_{i}\pm\hat{a}_{i}\hat{a}_{j}^{\dagger}).
\langleN\rangle=\left\{\begin{array}{lc}{\hphantom{-}\frac{1}{2}(1-F),}&{-1<A<\infty}\\{}&{}\\{-\frac{1}{2}(1+F),}&{-\infty<A<-1}\\{}&{}\\{-\frac{1}{2}F,}&{A^{-1}=0}\\\end{array}\right\},\qquad0<F<\frac{1}{2}
\Gamma_{k}[\phi]=\intd^{d}x\left\{U_{k}(\rho)+\frac{1}{2}Z_{k}(\rho)\,\partial_{\mu}\phi^{b}\partial_{\mu}\phi^{b}+\frac{1}{4}Y_{k}(\rho)\,\partial_{\mu}\rho\partial_{\mu}\rho+\dots\right\},
0=\frac{4k}{DM^{D+2}\tilde{b}_{0}^{D+4}}+\frac{4}{D+2}\frac{V\left(\tilde{b}_{0}\right)}{M^{D-2}\tilde{b}_{0}^{D}}-\frac{2}{D(D+2)}\frac{V^{\prime}\left(\tilde{b}_{0}\right)}{M^{D-2}\tilde{b}_{0}^{D-1}}
\XiX=\left(\begin{array}{cc}{m}&{b}\\{\bar{b}}&{n}\\\end{array}\right)\left(\begin{array}{cc}{M}&{B}\\{\bar{B}}&{N}\\\end{array}\right)=\left(\begin{array}{cc}{mM+b\bar{B}}&{mB+bN}\\{\bar{b}M+n\bar{B}}&{\bar{b}B+nN}\\\end{array}\right),
\delta(t)=\delta(t_{0})+\frac{k}{6H^{2}a^{2}(t_{0})}+\frac{\dot{\delta}(t_{0})}{3H}-\frac{k}{2H^{2}a^{2}(t_{0})}e^{-2H(t-t_{0})}+\left[\frac{k}{3H^{2}a^{2}(t_{0})}-\frac{\dot{\delta}(t_{0})}{3H}\right]e^{-3H(t-t_{0})}
=\;2^{N}\prod_{a=1}^{N}\sum_{n_{a}=0}^{\infty}\frac{(\beta^{(a)})^{2n_{a}}}{(2n_{a})!}\prod_{b=1}^{N}\sum_{q_{b}=0}^{2n_{b}}{\binom{2n_{b}}{q_{b}}}\left\langle\prod_{c=1}^{N}\left[\chi_{+}^{(c)}[Q^{W}]\right]^{q_{c}}\left[\chi_{-}^{(c)}[Q^{W}]\right]^{2n_{c}-q_{c}}\right\rangle_{Q^{W}}.
\begin{array}{lll}{SU\left(2\right)\subsetG:}&{\varepsilon^{mn}}&{\varepsilon^{12}=-1}\\{SU\left(3\right)\subsetG:}&{\varepsilon^{\dot{m}\dot{n}\dot{r}}}&{\varepsilon^{\dot{1}\dot{2}\dot{3}}=1}\\{SU\left(2\right)\subsetH:}&{\varepsilon^{\dot{m}\dot{n}}}&{\varepsilon^{\dot{1}\dot{2}}=1}\\{SO\left(7\right)^{c}:}&{\varepsilon^{abcdefg}}&{\varepsilon^{1234567}=-1\,.}\\\end{array}
\int_{1}^{\lambda}{t\left[J_{\nu}(tz)\bar{Y}_{\nu}(z)-Y_{\nu}(tz)\bar{J}_{\nu}(z)\right]^{2}dt}=\frac{2}{\pi^{2}zT_{\nu}^{AB}(\lambda,z)}\,,\quadz=\gamma_{\nu,k},
{}^{({\calC}_{H}^{H})}\Psi_{Y,X}\circS^{2}\circ{}^{({\calC}_{H}^{H})}\Psi_{X,Y}={}^{\calC}\Psi_{Y,X}\circ(S^{2}\otimesS^{2})\circ{}^{\calC}\Psi_{X,Y}
u_{(1)}(y)=-\frac{1}{54\hat{T}^{1/3}}\frac{5(\hat{z}_{+}(y)^{2/3}+\hat{z}_{-}(y)^{2/3})+11\hat{T}^{1/3}}{(\hat{z}_{+}(y)^{2/3}+\hat{z}_{-}(y)^{2/3}+\hat{T}^{1/3})^{5}}.
j_{\theta}^{\mu}=-\frac{\theta}{\alpha}\Phi\partial^{\mu}\phi+\theta\varepsilon^{\mu\nu\alpha\beta}\varepsilon_{abcd}\varphi_{a}\partial_{\nu}\varphi_{b}\partial_{\alpha}\varphi_{c}\partial_{\beta}\varphi_{d}L_{1}\equiv\thetaj_{D}^{\mu}
\sum_{n_{1},n_{2}}\Bigl((\frac{4p_{1}.p_{3}}{s+m^{2}}+1)|C(n_{1},n_{2})|^{2}+(\frac{4p_{1}.p_{3}}{t+\tilde{m}^{2}}+1)|\tilde{C}(n_{1},n_{2})|^{2}\Bigr)
\zeta_{\mu\nu}(s,x|L_{b})=\frac{2^{N-3}\Gamma\left(\fracN2\right)}{\pi^{\fracN2+1}N}a^{2s-N}\left[\left(1-\fracN2\right)a^{-2}I_{N}(s-1)-m^{2}I_{N}(s)\right]g_{\mu\nu}^{{\mathbbH}^{N}}(x).
\psi_{1}(p^{\mu})={\calC}\psi_{2}^{\ast}(p^{\mu})=\left(\begin{matrix}{-i(E^{0}+iB^{0})}\\{E^{1}+iB^{1}}\\{E^{2}+iB^{2}}\\{E^{3}+iB^{3}}\\\end{matrix}\right)\,\,,\,\,\psi_{2}(p^{\mu})={\calC}\psi_{1}^{\ast}(p^{\mu})=\left(\begin{matrix}{-i(E^{0}-iB^{0})}\\{E^{1}-iB^{1}}\\{E^{2}-iB^{2}}\\{E^{3}-iB^{3}}\\\end{matrix}\right)\,,
\frac{1}{2}\Big(\partial_{\mu}\varphi^{(1)}\Big)^{2}\;+\;\frac{1}{2}\Big(\partial_{\mu}\varphi^{(2)}\Big)^{2}\;+\;\frac{1}{2}\Big(\varphi^{(1)}\Big)^{2}\frac{e^{2}N}{\pi+gN}\;.
\begin{aligned}{\}&{{}\pmi\partial_{\bar{z}}\psi_{\pm}(z,\bar{z})=}&{m\overline{{\psi}}_{\pm}\mp}&{{}2g\psi_{\pm}\overline{{\chi}}_{+}\overline{{\chi}}_{-}\,}\\{\}&{{}\pmi\partial_{z}\overline{{\psi}}_{\pm}(z,\bar{z})=}&{m\psi_{\pm}\mp}&{{}2g\overline{{\psi}}_{\pm}\chi_{+}\chi_{-}\.}\\\end{aligned}
\tilde{P}(x,y)\;=\;P(x,y)\:-\:\intd^{4}z\;\left[s(x,z)\:{\cal{B}}_{z}\:P(z,y)\:+\:P(x,z)\:{\cal{B}}_{z}\:s^{*}(z,y)\right]\:+\:{\cal{O}}({\cal{B}}^{2}),
{R=e^{-2A_{5}}+e^{-2A_{6}}+e^{-2A_{7}}-\frac{1}{2}\left(e^{2(A_{5}-A_{6}-A_{7})}+e^{2(A_{6}-A_{7}-A_{5})}+e^{2(A_{7}-A_{5}-A_{6})}\right).}
{\calT}_{\mu\nu}^{\alpha\beta}\epsilon^{ij}\epsilon^{kl}=2{\itg_{1}}^{2}\bigg(F_{\rho\sigma}F_{\lambda\gamma}+{\frac{1}{\sqrt2}}\hat{\theta}_{5}\big(F_{\rho\sigma}\tilde{F}_{\lambda\gamma}+F_{\lambda\gamma}\tilde{F}_{\rho\sigma}\big)+\tilde{F}_{\rho\sigma}\tilde{F}_{\lambda\gamma}\bigg)\epsilon^{ij}\epsilon^{lk}\,.
F_{lm}^{c}=\partial_{l}{\calC}_{m}^{\dagger}-\partial_{m}{\calC}_{l}^{\dagger}-\frac{ig}{\sqrt{L}}({\calB}_{l}^{\dagger}{\calA}_{m}^{\dagger}-{\calB}_{m}^{\dagger}{\calA}_{l}^{\dagger}).
S^{\prime}=-\intd^{5}\sigma\left(e^{-\phi}\sqrt{-G}\sqrt{1+e^{2\phi}z_{1}+e^{4\phi}(\frac{z_{1}^{2}}{2}-z_{2})}+\frac{\epsilon_{\mu\nu\lambda\sigma\tau}e^{\phi}C^{\mu}\tilde{{\calH}}^{\nu\lambda}\tilde{{\calH}}^{\sigma\tau}}{8(1+C_{1}^{2})}\right)+\int\Omega^{\prime},
e^{-i\vec{\xi}_{P}\cdot{^5\!\vec{P}}}\,:\quad\begin{array}{lcl}{Z^{0}}&{\to}&{Z^{0}+\omega\,\vec{\xi}_{P}\cdot\vec{Z}+\frac{1}{2}\,\omega^{2}{{\xi}_{P}}^{2}\,(Z^{0}+Z^{5})}\\{Z^{5}}&{\to}&{Z^{5}-\omega\,\vec{\xi}_{P}\cdot\vec{Z}-\frac{1}{2}\,\omega^{2}{{\xi}_{P}}^{2}\,(Z^{0}+Z^{5})}\\{Z^{i}}&{\to}&{Z^{i}+\omega\,\xi_{P}^{i}\,(Z^{0}+Z^{5})}\\\end{array}
<\vec{x}|N,n>=\sqrt{{\frac{N!}{\pi(N+|n|)!}}}\,\left(m\Omega\right)^{\frac{1+|n|}{2}}r^{|n|}e^{in\theta}e^{-\frac{1}{2}m\Omegar^{2}}L_{N}^{|n|}\left(m\Omegar^{2}\right)
P^{-}=\intdx^{-}\,\mathrm{tr}\Bigg[-\frac{g^{2}}{2}J^{+}\frac{1}{\partial_{-}^{2}}J^{+}+\frac{i}{2}g^{2}[\phi,\psi]\frac{1}{\partial_{-}}[\phi,\psi]\Bigg].
A_{(b)}^{\mu\nu}={\frac{i}{2}}P^{\alpha\beta,\lambda\kappa}iP^{\gamma\delta,\rho\sigma}\tau_{\alpha\beta,\gamma\delta}(p,p^{\prime})\int{\frac{d^{4}\ell}{(2\pi)^{4}}}{\frac{\tau_{\lambda\kappa,\rho\sigma}^{\mu\nu}(\ell,q)}{\ell^{2}(\ell-q)^{2}}}
Z(G)={\frac{1}{\vertG\vert}}\Bigl\{\sum_{a}{\vertG_{a}\vert}~Z(G_{a})-\sum_{a<b}{\vertG_{ab}\vert}~Z(G_{ab})+\sum_{a<b<c}{\vertG_{abc}\vert}~Z(G_{abc})-\cdots\Bigr\}
{\calL}\left(U(\tau),z^{i}(\tau),\bar{z}^{i}(\tau)\right)=\left({\frac{dU}{d\tau}}\right)^{2}+\left|{\frac{dz}{d\tau}}\right|^{2}+e^{2U}\left(|Z(z,p,q)|^{2}+|D_{i}Z(z,p,q)|^{2}\right)\.
\begin{array}{rcl}{{\hat{A}}_{i}^{(1)}}&{=}&{\left(1+e^{2\phi}k^{-2}(i_{k}C^{(1)})^{2}\right)^{-1}\left(A_{i}^{(1)}-e^{2\phi}k^{-2}C_{i}^{(1)}(i_{k}C^{(1)})\right)\,,}\\{}&{}&{}\\{{\hat{A}}_{6}^{(1)}}&{=}&{-\left(1+e^{2\phi}k^{-2}(i_{k}C^{(1)})^{2}\right)^{-1}e^{2\phi}k^{-2}(i_{k}C^{(1)})\,,}\\\end{array}
\frac{\delta<H_{\phi^{3}}^{\theta}>}{\deltaG(k)}=\frac{16\pi^{4}\theta^{2}}{\eta^{2}}\frac{\deltaz}{\deltaG(k)}\int\frac{d^{2}p}{(2\pi)^{2}}[1-K^{2}(p)(p^{2}+\lambda\eta^{2})]
\left(\hat{A}_{\mu}^{a}\hat{A}_{\nu}^{b}\right)\left(\hat{A}^{b\mu}\hat{A}^{a\nu}\right)-\left(\left(\hat{A}_{\mu}^{a}\hat{A}_{\nu}^{b}\right)\hat{A}^{b\mu}\right)\hat{A}^{a\nu}=0.
{\calK}^{\alpha\beta}=-{\frac14}\left[{\frac12}D^{\gamma}D^{\beta}D_{\gamma}+{\frac1\alpha}D^{\alpha}D^{2}D^{\beta}+\muD^{\beta}D^{\alpha}-m_{\lambda}D^{\gamma}D^{\alpha}\theta^{2}D^{\beta}D_{\gamma}\right]~~.
(i)^{n}\frac{\delta^{n}W[f(t);x_{f},t_{f};x_{i},t_{i}]}{\deltaf(t_{1})\\deltaf(t_{2})\...\\deltaf(t_{n})}\left|_{f(t)=0}\right.=\langlex_{f},t_{f}|{\calT}[\hat{q}(t_{1})\\hat{q}(t_{2})\...\\hat{q}(t_{n})]|x_{i},t_{i}\rangle,
\begin{aligned}{}&{{}\alpha_{n}+\alpha_{-n+2d}^{\prime}=a_{n},}&{}&{{}\qquad\beta_{n}-\beta_{-n+d}^{\prime}=b_{n}}\\{}&{{}}&{}&{{}\qquad\delta_{n}-\delta_{-n-2d}^{\prime}=d_{n}}\\\end{aligned}
B_{ij}^{\bar{g}(1)}={\bar{R}}_{ij}-\frac{1}{4}a_{i}a_{j}-\frac{1}{2}(aF_{ik}F_{j}^{\;\;k}+\frac{1}{a}G_{ik}G_{j}^{\;\;k})-\frac{1}{4}{\hat{H}}_{ikm}{\hat{H}}_{j}^{\;\;km}+2{\bar{\nabla}}_{i}{\bar{\nabla}}_{j}\Phi,
\Box\phi-\frac{\alpha}{M_{p}(\zeta+b)}\left[sM^{4}e^{-2\alpha\phi/M_{p}}-\frac{(\zeta-b)V_{1}+2V_{2}}{\zeta+b}\right]=-\frac{\alpha}{M_{p}}[F_{E}\overline{{E^{\prime}}}E^{\prime}+F_{U}\overline{{U^{\prime}}}U^{\prime}+F_{D}\overline{{D^{\prime}}}D^{\prime}],
\breve{M}\to\breve{\Omega}\breve{M}\breve{\Omega}^{T},\\,breve{\calA}_{\breve{\mu}}^{\breve{i}}\to\breve{\Omega}_{\breve{i}\breve{j}}\breve{\calA}_{\breve{\mu}}^{\breve{j}},\\\breve{\varphi}\to\breve{\varphi},\\,breve{g}_{\breve{\mu}\breve{\nu}}\to\breve{g}_{\breve{\mu}\breve{\nu}},\\\breve{B}_{\breve{\mu}\breve{\nu}}\to\breve{B}_{\breve{\mu}\breve{\nu}},
\epsilon_{\lambda}=\sum_{i=1}^{M}\Bigl(\lambda_{i}^{2}+\beta(N+1-2i)\lambda_{i}\Bigr)=\sum_{i=1}^{M^{\prime}}\Bigl(-\beta\lambda_{i}^{\prime2}+(\betaN+2i-1)\lambda_{i}^{\prime}\Bigr),
\begin{matrix}{D3:}&{1}&{2}&{3}&{-}&{-}&{-}&{-}&{-}&{-}\\{D1:}&{-}&{-}&{-}&{4}&{-}&{-}&{-}&{-}&{-}\\{F1:}&{-}&{-}&{-}&{-}&{5}&{-}&{-}&{-}&{-}\\\end{matrix}
\intd^{3}\vec{x}\,\theta^{\mu\nu}\theta^{\alpha\beta}\partial_{\mu}\partial_{\alpha}\dot{\phi}\,\partial_{\nu}\partial_{\beta}\delta\phi=\intd^{3}\vec{x}\,\bigl(\bar{\partial}^{2}\!\stackrel{...}{\phi}\!\delta\phi+2\,\bar{\partial}^{2}\ddot{\phi}\,\delta\dot{\phi}+\bar{\partial}^{2}\dot{\phi}\,\delta\ddot{\phi}\bigr)
V_{G}(\phi_{0})={\frac{1}{V}}\sum_{n=0}^{\infty}\frac{(-i)^{n}}{n!}\int_{-\infty}^{0}dt_{n}...\int_{-\infty}^{0}dt_{n}\langle0|T(H_{I}(0)H_{I}(t_{1})...H_{I}(t_{n}))|0{\rangle}_{conn}
\Gamma_{+}^{-1}A^{(r)}C_{r}-\alpha_{r}C_{3}\Gamma_{+}^{-1}A^{(r)}=-\frac{1}{2}\frac{\alpha}{1+\mu\alphak}CU_{3}^{-1}YY^{\mathrm{T}}A^{(r)}U_{r}^{-1}C,\qquadr\in\{1,2\}.
(\lambda-2)v^{\prime}(0)=\frac{(N+2)v^{\prime\prime}(0)}{\{2V^{\prime}(0)+2\}^{3/2}}-3\frac{(N+2)v^{\prime}(0)V^{\prime\prime}(0)}{\{2V^{\prime}(0)+2\}^{5/2}}\quad.
M=2m={\frac{1}{2}}m+{\frac{1}{2}}m+{\frac{1}{2}}m+{\frac{1}{2}}m,\\\\p_{1}^{\mathrm{elec}}=p_{2}^{\mathrm{elec}}=p_{1}^{\mathrm{mag}}=p_{4}^{\mathrm{mag}}=ma.
{\calM}=\left(\begin{array}{cc}{\frac{1}{\lambda}+{\calG}_{0}(0,0)}&{{\calG}_{0}(0,\ell)}\\{}&{}\\{{\calG}_{0}(\ell,0)}&{\frac{1}{\lambda}+{\calG}_{0}(\ell,\ell)}\\\end{array}\right)=\left(\begin{array}{cc}{\frac{1}{\lambda}+\frac{1}{2\sigma}}&{\frac{e^{-\sigma\ell}}{2\sigma}}\\{}&{}\\{\frac{e^{-\sigma\ell}}{2\sigma}}&{\frac{1}{\lambda}+\frac{1}{2\sigma}}\\\end{array}\right).
\Phi_{c}=\left(\begin{matrix}{r_{1}^{1/2}e^{i\alpha}}&{0}\\{0}&{r_{2}^{1/2}}\\\end{matrix}\right)\,\qquadr_{1}+r_{2}=\mu^{2},\qquadr_{4}+ir_{5}=(r_{1}r_{2})^{1/2}e^{i\alpha}.
\tilde{\mathcalL}_{m}^{R}(q_{\perp})=\frac{\pi}{2}\int\frac{d^{2}x_{\perp}}{(2\pi)^{2}}e^{-iq_{\perp}x_{\perp}}(({\mathcalA}_{1}^{R})_{m}(x_{\perp})+({\mathcalA}_{2}^{R})_{m}(x_{\perp})+4({\mathcalA}_{3}^{R})_{m}(x_{\perp})-4({\mathcalA}_{4}^{R})_{m}(x_{\perp})).
2\frac{J^{2}}{A^{2}}=-\left(1+\frac{1}{x^{2}}-\frac{A^{\prime}}{AB}\right)\pm\sqrt{\left(1-\frac{1}{x^{2}}+\frac{A^{\prime}}{AB}\right)^{2}+\frac{2}{xB^{2}}\left(\frac{A^{\prime}}{A}+\frac{B^{\prime}}{B}\right)}.
\langleA_{\mu}^{a}\rangle=\intA_{\mu}^{a}e^{iS\left[A_{\mu}^{a}\right]}DA_{\mu}^{a}\approx\int\left(\tilde{A}_{\mu}^{a}\right)_{\phi}e^{iS\left[\left(\tilde{A}_{\mu}^{a}\right)_{\phi}\right]}D\left(\tilde{A}_{\mu}^{a}\right)_{\phi}=\sum_{\substack{\mathrm{over~all}\,\mathrm{classical~solutions}}}\left(\tilde{A_{\mu}^{a}}\right)_{\phi}p_{\phi}
T_{\mu\nu}^{PVV}=\frac{i}{2m}\left\{\left(k_{3}-k_{2}\right)_{\lambda}T_{\lambda\mu\nu}^{AVV}+T_{\nu\mu}^{AV}\left(k_{3},k_{1};m\right)-T_{\mu\nu}^{AV}\left(k_{1},k_{2};m\right)\right\}.
C_{2}=\frac{1}{P^{\prime\prime}(y)}\left|P^{\prime}(y)-\frac{\omega^{\prime}(S)}{\omega(S)}\right|^{2}+\frac{1}{3}\left|3+(T+\bar{T})\frac{h^{\prime}(T)}{h(T)}\right|^{2}-3,
I_{\mathrm{BI}}=T_{(9)}^{I}(\pi\alpha^{\prime})^{2}\;\intd^{10}x\;e^{-\Phi}\left[\;\mathrm{tr}\;(F_{\mu\nu}F^{\mu\nu})-{\frac{(\pi\alpha^{\prime})^{2}}{12}}\;\mathrm{tr}\;(t_{8}F^{4})+\cdots\;\right]
{\calT}_{mod}\equiv\frac{2}{3}C_{i,jk}\nabla\phi^{i}\nabla\phi^{j}{F}^{k}+\frac{2}{3}C_{[i,j]k}d(\phi^{i}A^{j}{F}^{k}-\phi^{i}d\phi^{j}{F}^{k}),
\Phi_{\mu_{N-1}}^{\mu_{N}V_{N}}(z_{N})\circ\cdots\circ\Phi_{\mu_{0}}^{\mu_{1}V_{1}}(z_{1}):V(\mu_{0})\rightarrowV(\mu_{N})\otimesV_{z_{N}}^{(l_{N})}\otimes\cdots\otimesV_{z_{1}}^{(l_{1})}.
{\calL}_{Y.M.}^{(5)}=-\frac{1}{2g_{5}^{2}}F_{MN}^{2}+\frac{1}{g_{5}^{2}}\left(\left(D_{M}\Sigma\right)^{2}+\overline{{\lambda}}i\Gamma^{M}D_{M}\lambda+\left(X^{a}\right)^{2}-\overline{{\lambda}}\left[\Sigma,\lambda\right]\right)
J_{i}^{(N)}(N-1;1,\ldots,1)=\frac{\mathrm{i}^{1-2N}\;\pi^{(N-1)/2}}{m_{0}^{2}\prod_{l\neqi}^{}m_{l}}\Gamma\left(\frac{N+1}{2}\right)\int_{0}^{\infty}\ldots\int_{0}^{\infty}\prod_{l\neqi}^{}\mathrm{d}\alpha_{l}\;\theta\left(1-\left.\left(\alpha^{T}\|c\|\alpha\right)\right|_{\alpha_{i}=0}\right).
{\frac{\langle:\dot{p}_{i}\dot{p}^{i}:\rangle}{r_{1}^{2}}}\sim{\frac{\kappa^{2}N^{2}}{L^{4}(Q_{1}Q_{5})^{2}Q_{5}}};\\{\frac{\langle:\dot{\gamma}^{2}:\rangle}{r_{1}^{2}r_{5}^{2}}}\sim{\frac{\kappa^{2}N^{2}}{L^{4}(Q_{1}Q_{5})^{2}}}
S_{++}(qt,p_{1}t_{1})=p_{1i}\tilde{\tilde{q}}_{1}^{~i}+\int_{\gamma_{2}}(\tilde{\tilde{p}}_{~i}d\tilde{\tilde{q}}^{i}(\tau)-H(\tilde{\tilde{q}}(\tau),\tilde{\tilde{p}}(\tau)))d\tau
\bar{H}_{\mathrm{Ds}}(q^{r},p^{r})=H_{\mathrm{D}}(q^{r},p^{r})+\lambda_{a}^{(1)s_{a}}\pi_{a}^{s_{a}}+\lambda_{a}^{(2)s_{a}}\psi_{a}^{s_{a}}+\lambda_{A}\phi_{A},
\Pi_{\mu_{1}\mu_{2}}^{1}(k,u)=\frac{1}{2\pii}\int\frac{\mathrm{d}^{3}Q}{\left(2\pi\right)^{3}}\int_{\cal{C}}\mathrm{dQ_{0}}N\left(Q_{0}\right)t_{\mu_{1}\mu_{2}}^{\alpha\beta\rho\sigma}\left(Q_{0},{\bfQ},Q_{0}+k_{0},{\bfQ}+{\bfk}\right)D_{\alpha\beta}(Q)D_{\rho\sigma}(Q+k),
{\calS}(\Sigma)\,\,=\,\,\displaystyle{\int}d^{3}x\,\displaystyle{\sum_{\Phi=A_{\mu}^{a},\,c^{a},\,\Psi_{_A},\,\varphi_{i}}^{}}{\displaystyle{\frac{\delta\Sigma}{\delta\Phi^{*}}}}{\displaystyle{\frac{\delta\Sigma}{\delta\Phi}}}\,+b\,\Sigma=0\,,\quad\mathrm{with~}\b=\displaystyle{\int}d^{3}x\,b^{a}{{\displaystyle{\frac{\delta}{\delta\bar{c}^{a}}}}}\,.
J^{\alpha\beta}(x)={\frac{1}{2}}\left[\psi_{A}^{\alpha\dagger}(x),\psi_{A}^{\beta}(x)\right]=\psi_{A}^{\alpha\dagger}(x)\psi_{A}^{\beta}(x)-\delta^{\alpha\beta}{\frac{{\calN}_{C}}{2}}=(J^{\beta\alpha}(x))^{\dagger}
{\calH}^{(1)}(\xi)\mid_{\Omega^{(1)}=0}~=~\frac{1}{2}(\pi_{i})^{2}+\frac{1}{4}F_{ij}F^{ij}+\frac{1}{2}m^{2}(A^{0})^{2}+\frac{1}{2}m^{2}(A^{i})^{2}.
\phi_{\nu}^{\mu}(x^{+}=0)=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}\frac{dk^{+}}{\sqrt{2k^{+}}}\left[a_{\nu}^{\mu}(k^{+})\mathrm{e}^{-\mathrm{i}k^{+}x^{-}}+a_{\nu}^{\dagger\mu}(k^{+})\mathrm{e}^{\mathrm{i}k^{+}x^{-}}\right]
A_{i_{1}\ldotsi_{p}}^{(p)}=-i2x_{0}^{\frac{d}{2}}\int{\frac{d^{d}k}{(2\pi)^{d}}}e^{-i{\bfk}\cdot{\bfx}}\left(k_{i_{1}}a_{0i_{2}\ldotsi_{p}}+(-)^{p-1}k_{i_{2}}a_{0i_{3}\ldotsi_{p}i_{1}}+\cdots\right){\frac{1}{|{\bfk}|^{2}}}H(|{\bfk}|x_{0}).
T_{\mu}^{\nu}=\left(\begin{array}{cccc}{-\frac{\alpha\beta}{\kappa}\frac{e^{-\betar}}{r^{2}}}&{0}&{0}&{0}\\{0}&{-\frac{\alpha\beta}{\kappa}\frac{e^{-\betar}}{r^{2}}}&{0}&{0}\\{0}&{0}&{\frac{\alpha\beta^{2}}{2\kappa}\frac{e^{-\betar}}{r}}&{0}\\{0}&{0}&{0}&{\frac{\alpha\beta^{2}}{2\kappa}\frac{e^{-\betar}}{r}}\\\end{array}\right)
\frac{d\sigma_{c}}{dE_{+}}\approx\frac{4\pi^{2}Z^{2}e^{4}}{\omega^{2}a_{2}a_{3}}\sum_{m,g_{y}}\frac{2g_{y}^{2}+R^{-2}}{g_{m\parallel}^{2}(g_{y}^{2}+R^{-2})^{3/2}}\left[\frac{\omega^{2}}{2E_{+}E_{-}}-1+2\frac{\delta}{g_{m\parallel}}\left(1-\frac{\delta}{g_{m\parallel}}\right)\right]J_{m}^{2}(g_{y}u_{0}),
{\overline{{V_{0}+V_{1}}}}=(-{\frac{1}{2}}({\frac{1}{6}}~{\frac{1}{3}})^{3}\vert\vert0~({\frac{1}{3}})^{2}\vert({\frac{1}{3}})^{5}~0_{r}^{5}~({\frac{2}{3}})_{r})\,
\hat{\nabla}_{-q}\psi^{-i}\equiv\left(\delta_{j}^{i}D_{-q}-\omega_{-q,-j}^{~~~~-i}-\delta_{j}^{i}D_{-q}Z^{\underline{M}}A_{\underline{M}}\right)\psi^{-j}=\gamma_{qq^{\prime}}^{i}P^{q^{\prime}},
{\calL}^{F}=2iv_{0}:\!\psi_{+}^{\dagger}\partial_{+}\psi_{+}+\psi_{-}^{\dagger}\partial_{-}\psi_{-}\!:+v_{1}J_{+}J_{-}+iv_{2}\left(\psi_{+}\psi_{--}\psi_{-}^{\dagger}\psi_{+}^{\dagger}\right).
{\frac{dn(\omega)}{d\omega}}\simeq{\frac{2\omega^{D-2}}{(4\pi)^{(D-1)/2}}}\sum_{n=0}^{\infty}\left[{\frac{c_{n}}{\Gamma\left({\frac{D-1}{2}}-n\right)}}\omega^{-2n}+{\frac{d_{n}}{\Gamma\left({\frac{D-2}{2}}-n\right)}}\omega^{-(2n+1)}\right]~~~,
\begin{array}{rcl}{}&{}&{T_{i}(e_{i})=-f_{i}t_{i}~,}\\{}&{}&{T_{i}(e_{j})=\displaystyle\sum_{m=0}^{-a_{ij}}(-1)^{m-a_{ij}}q_{i}^{-m}\frac{e_{i}^{-a_{ij}-m}}{[-a_{ij}-m]_{q_{i}}!}e_{j}\frac{e_{i}^{m}}{[m]_{q_{i}}!}~,~~~\mathrm{if~}i\not=j~;}\\\end{array}
\bar{\phi}^{2}(y)=\intdk_{1}^{\prime}\dk_{2}^{\prime}\e^{ik_{1}^{\prime}\wedgek_{2}^{\prime}}\\bar{\phi}(k_{1}^{\prime})\bar{\phi}(k_{2}^{\prime})\e^{i(k_{1}^{\prime}+k_{2}^{\prime})\cdoty}
L_{ij}=\delta_{ij}\xi_{i}+(1-\delta_{ij})\lambda_{i\,x}\Phi(\lambda_{i}-\lambda_{j},z),\qquad\mathrm{where}\qquad\xi_{i}=\frac{\lambda_{i\,y}-\lambda_{i\,xx}-b_{i}}{2\lambda_{i\,x}},
2\pi\Theta^{\prime}=\frac{C+D\!\cdot\!2\pi\Theta}{A+B\!\cdot\!2\pi\Theta},\quad\frac{\Phi^{\prime}}{2\pi}=(A+B\!\cdot\!2\pi\Theta)^{2}\frac{\Phi}{2\pi}+B(A+B\!\cdot\!2\pi\Theta).
S_{int}=\Phi(k)\int{\frac{[d\xidx]}{(2\pi)^{n}}}(\mathrm{Pf}B)\int\prod_{\alpha=1}^{n}d\tau_{\alpha}e^{ik_{\mu}\xi^{\mu}}~\mathrm{tr}P_{\star}[\prod_{\alpha=1}^{n}O_{\alpha}(x^{i}+\theta^{ji}k_{j}\tau_{\alpha})W_{t}]\stare^{ik_{i}x^{i}}
\chi^{(p)}(\Omega_{1},\Omega_{2})=[\prod_{i=1}^{\infty}(1-q^{i})]^{-r}\sum_{s\inw}\sum_{\lambda\in\Lambda_{R}}det(s)q^{[ps(\Omega_{1})-(p+1)\Omega_{2}+p(p+1)\lambda]^{2}/2p(p+1)}
R_{\:2}^{1}=-2i\frac{dg_{1}\wedged\overline{{g}}_{1}}{(1-g_{1}\overline{{g}}_{1})^{2}};~R_{\:4}^{3}=-2i\frac{dg_{2}\wedged\overline{{g}}_{2}}{(1\mpg_{2}\overline{{g}}_{2})^{2}},
\frac{\ddot{a}}{a}+2\frac{\dot{a}^{2}}{a^{2}}+D\frac{\dot{a}}{a}\frac{\dot{b}}{b}+\frac{2}{a^{2}}=\frac{8\pi\overline{{G}}}{D+2}\left[\rho+(D-1)p_{a}-Dp_{b}\right]
ds^{2}\approx\left[1-\left(\frac{4\piG_{D}-2|\Lambda|}{d(p+d-1)}-\frac{\hat{R}}{d(p+1)}\right)r^{2}\right]\hat{g}_{\mu\nu}(x)dx^{\mu}dx^{\nu}+\frac{dr^{2}}{1+\bar{B}_{2}r^{2}}+r^{2}d\Omega_{d-1}^{2},
\Omega_{l}(a)=\frac{C_{l}^{(+)}}{C_{l}^{(-)}}=-\frac{\sqrt{\lambda_{l}^{(\ast)}}+\sqrt{\lambda_{l}^{(\ast)}-\lambda}}{\sqrt{\lambda_{l}^{(\ast)}}-\sqrt{\lambda_{l}^{(\ast)}-\lambda}}\;a^{-2\sqrt{\lambda_{l}^{(\ast)}-\lambda}}\;.
{\calL}_{\mathrm{ct}}=a\,\mathrm{tr}\,\left(\partial_{\mu}\Phi\partial^{\mu}\Phi^{\dagger}\right)-b\,\mathrm{tr}\,\left(\Phi\Phi^{\dagger}-v^{2}\right)-c\,\mathrm{tr}\,\left(\Phi\Phi^{\dagger}-v^{2}\right)^{2}
{\calL}_{a}=\frac{1}{2}\widetilde{g}\left(-\bar{m}^{-1}\widetilde{g}^{\mu\nu}D_{\mu}\psi_{a}^{\dagger}D_{\nu}\psi_{a}+i\psi_{a}^{\dagger}e_{\alpha}^{\mu}\sigma^{\alpha}D_{\mu}\psi_{a}\right)+h.c.
|\Phi|^{2}(t,\vec{x})\simf_{n}^{2}(x_{1}^{2}+x_{2}^{2})^{n}+2ntf_{n}^{2}k_{-n}\sum_{p=0}^{[n/2]}(-1)^{n+p}\left(\begin{array}{c}{n}\\{2p}\\\end{array}\right)x_{1}^{n-2p}x_{2}^{2p}+Kt^{2},
S_{BH}=2\pi\left(\sqrt{N_{2}^{(1)}}+\sqrt{\bar{N}_{2}^{(1)}}\right)\left(\sqrt{N_{2}^{(2)}}+\sqrt{\bar{N}_{2}^{(2)}}\right)\left(\sqrt{N_{2}^{(3)}}+\sqrt{\bar{N}_{2}^{(3)}}\right)
Q_{N}=\frac{1}{a^{p}}\sum_{I}(-1)^{|I|}\prod_{i\inI}\left[\prod_{k=0}^{q}\frac{w_{k}-v_{i}-(\beta-1)a}{w_{k}-v_{i}}\prod_{k=1}^{q-1}\frac{\xi_{k}-v_{i}-a}{\xi_{k}-v_{i}}\prod_{j\not\inI}\frac{v_{j}-v_{i}+a}{v_{j}-v_{i}}\right].
Z_{h}=\intD\PhiD\varphie^{-\int\frac{1}{8\pi}(\partial\Phi)^{2}+\mue^{2a\Phi}+Q\hat{R}^{h=0}\Phi+\frac{1}{8\pi}(\partial\varphi)^{2}}\prod_{i=1}^{2h+2}e^{2\gamma_{i}\Phi(a_{i},\bar{a}_{i})}\sigma_{0,\epsilon_{i}}(a_{i},\bar{a}_{i})
\psi_{1}^{\dagger}(x)\psi_{1}(x)=\frac{1}{8\pi}\intd\hat{\epsilon}\left\{\left[\psi^{\dagger}-1(x+\epsilon),\psi-1(x-\epsilon)\right],e^{ie\int_{x-\epsilon}^{x+\epsilon}dx_{i}A_{i}^{T}}\right\}_{|\eta|,|\xi|,|\epsilon|\propto1/\Lambda}
\left\langleU_{k\pml}(x)U_{k}^{-1}(x)\right\rangle={M\beta^{2}\overwithdelims()2}^{2\beta^{2}}\Gamma^{2}(u)\frac{\Gamma(2+(l-1)u)\Gamma(1+(l-1)u)}{\Gamma(1+lu)\Gamma(2+(l-2)u)}.
Tr_{L}\Phi^{\left[1\right]}=D+\frac{{32}}{{T^{2}}}\bar{\xi}_{4}\xi_{4}\bar{\xi}_{3}\xi_{3}\bar{\xi}_{2}\xi_{2}\bar{\xi}_{1}\xi_{1}(\varepsilon^{1}\cdot\varepsilon^{4}\varepsilon^{2}\cdot\varepsilon^{3}-\varepsilon^{1}\cdot\varepsilon^{3}\varepsilon^{2}\cdot\varepsilon^{4})\delta(u_{4}-u_{3})\delta(u_{2}-u_{1})+\,f.t.
\frac{F}{V}=\frac{1}{4}\left(\frac{1}{4\pi^{2}\alpha^{\prime}}\right)^{5}\int_{-1/2}^{1/2}d\tau_{1}\int_{0}^{\infty}\frac{d\tau_{2}}{\tau_{2}^{6}}\left|\frac{1}{\eta(\tau)}\right|^{24}\sum_{n=-\infty}^{+\infty}e^{-\beta^{2}n^{2}/(4\pi\alpha^{\prime}\tau_{2})}C(n,\tau)
\frac{8\pi}{3}G_{N}\rho_{eff}=(\frac{\dot{\alpha}}{\alpha})^{2}=\frac{(C+E)^{2}g_{S}e^{2\Phi}-|g_{00}|(g_{S}g^{3}+\ell^{2}e^{2\Phi})}{4|g_{00}|g_{rr}g_{S}g^{3}}(\frac{g^{\prime}}{g})^{2}
W_{1}^{VC}(\beta,r_{B},r_{+},\epsilon)=W_{1}^{VC}[C_{\alpha}]-{\frac{1}{24\pi}}\left(\int_{M_{B,\epsilon}}(R\sigma-(\nabla\sigma)^{2})+\int_{\Sigma_{B}}(2K\sigma+3\sigma_{,\mu}n^{\mu})\right).
\nabla\Phi_{\gamma_{1}\ldots\gamma_{k}\dot{\gamma}_{1}\ldots\dot{\gamma}_{k}}=ie^{\alpha\dot{\alpha}}\Phi_{\gamma_{1}\ldots\gamma_{k}\alpha\dot{\gamma}_{1}\ldots\dot{\gamma}_{k}\dot{\alpha}}-ik^{2}e_{(\gamma_{k}|(\dot{\gamma}_{k}|}\Phi_{|\gamma_{1}\ldots\gamma_{k-1})|\dot{\gamma}_{1}\ldots\dot{\gamma}_{k-1})}.
J(M,T)=\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{p^{2}+M^{2}}+2\int_{-\infty+i\epsilon}^{\infty+i\epsilon}\frac{dz}{2\pi}\int\frac{d^{2}p}{(2\pi)^{2}}\frac{1}{z^{2}+{\bfp}^{2}+M^{2}}\frac{1}{e^{-i\betaz}-1}\;.
e^{\alpha\left({\frac{\delta}{\deltaiJ_{w}^{i}}}{\frac{\delta}{\deltaiJ_{w}^{i}}}\right)}\cdotP(J)R_{(j)}(e^{J})|_{J=0}=e^{\alpha(j+1/2)^{2}}=e^{\alpha(c(j)/2+1/4)}
ig^{\mu\sigma}\partial_{\mu}\widetilde{\psi}(x,\gamma^{\sigma})+i\gamma^{\mu}\partial_{\mu}\frac{\partial\widetilde{\psi}(x,\gamma^{\sigma})}{\partial\gamma_{\sigma}}-m\frac{\partial\widetilde{\psi}(x,\gamma^{\sigma})}{\partial\gamma_{\sigma}}=0
S[\Psi,\Psi^{\dagger}]=\int_{0}^{\beta}dt\int_{\Sigma}d\sigma_{x}\left(\frac{1}{2}(\Psi^{\dagger}\frac{\partial\Psi}{\partialt}-\frac{\partial\Psi^{\dagger}}{\partialt}\Psi)+\frac{1}{2m}|\nabla\Psi|^{2}-\mu|\Psi|^{2}\right)\;.
{\frac{1}{2}}\sigma_{\mu\nu}F_{\mu\nu}={\frac{1}{2}}F_{\mu\nu}\epsilon_{\mu\nu\rho}\left(\begin{array}{cc}{\sigma_{\rho}}&{0}\\{0}&{\sigma_{\rho}}\\\end{array}\right)=\left(\begin{array}{cc}{E_{2}\sigma_{1}-E_{1}\sigma_{2}+B\sigma_{3}}&{0}\\{0}&{E_{2}\sigma_{1}-E_{1}\sigma_{2}+B\sigma_{3}}\\\end{array}\right)\,
H^{I_{A}(\mu_{1},\mu_{2})J_{B}(\nu_{2},\nu_{1})}=\mathrm{Tr}\eta^{i_{A(\mu_{1}+1)}}\cdots\eta^{i_{A(\mu_{2}-1)}}\eta^{j_{B(\nu_{2}+1)}}\cdots\eta^{j_{B(\nu_{1}-1)}}
{^{(2)}\!\phi^{AB}}_{u}{}^{v}=-\left\{\frac{8\sqrt{2}}{(x^{2}+\rho^{2})^{2}}+\frac{C_{1}}{x^{2}(x^{2}+\rho^{2})}+\frac{C_{2}}{\rho^{4}}\left(1+\frac{x^{4}}{3\rho^{2}(x^{2}+\rho^{2})}\right)\right\}\xi_{u}^{[A}\xi^{B],v}\,