harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{cc}{X_{p+1}^{(0)\mathrm{diag}}=U^{-1}\left(\begin{array}{cc}{\betax}&{d^{*}}\\{d}&{-\betax}\\\end{array}\right)U}&{=\left(\begin{array}{cc}{\sqrt{(\betax)^{2}+|d|^{2}}}&{0}\\{0}&{-\sqrt{(\betax)^{2}+|d|^{2}}}\\\end{array}\right).}\\\end{array}
\phi(x)=\left(\begin{matrix}{\phi_{1}(x)}\\{\phi_{2}(x)}\\{\phi_{3}(x)}\\{\phi_{4}(x)}\\\end{matrix}\right)=\left(\begin{matrix}{\chi_{1}(x)}\\{\chi_{2}(x)}\\{\bar{\chi}^{\bar{1}}(x)}\\{\bar{\chi}^{\bar{2}}(x)}\\\end{matrix}\right)=\left(\begin{matrix}{\chi_{1}(x)}\\{\chi_{2}(x)}\\{\chi_{2}^{*}(x)}\\{-\chi_{1}^{*}(x)}\\\end{matrix}\right)
\begin{array}{l}{F=u_{0}+v=\alpha+\bar{\beta},}\\{\quadquad\leftrightarrow\quadv=W=\alpha+\bar{\beta}-u_{0},}\\{\bar{F}=u+v_{0}=\bar{\alpha}+\beta,}\\{\quad\quad\leftrightarrow\quadu=\bar{W}=\bar{\alpha}+\beta-v_{0}.}\\\end{array}
\psi_{J}=\sqrt{\frac{\omega^{J+1}}{\piJ!}}z^{J}e^{-{\frac{\omega}{2}}z\bar{z}},\qquad\bar{\psi}_{J}=\sqrt{\frac{\omega^{J+1}}{\piJ!}}\bar{z}^{J}e^{-{\frac{\omega}{2}}z\bar{z}},
\rho_{2}(\lambda)=\frac{1}{\pi}\sqrt{2-\lambda^{2}}\\\\\\\\mathrm{and}\\\\\\\,rho_{4}(\lambda)=\frac{1}{\pi}\left(\sqrt{2}\lambda^{2}+\sqrt{\frac{\sqrt{8}}{3}}\right)\sqrt{\sqrt{\frac{2\sqrt{8}}{3}}-\lambda^{2}}
Z^{\rho}=\left[\begin{array}{c}{Z^{1}}\\{Z^{2}}\\\end{array}\right]=\left[\begin{array}{c}{U(\hbar{\bfu},\mathrm{0)}}\\{U(0,\hbar{\bfu})}\\\end{array}\right]\mathrm{,~i.e.~\rho~=1,2~.~}
\mathrm{S}_{\mathrm{BH}}^{\mathrm{EBTZ}}={\frac{\tilde{A}_{3}}{4\tilde{G}_{3}}}=2\pi\sqrt{Q_{1}Q_{5}N_{K}}={\frac{A_{5}}{4G_{5}}}=\mathrm{S}_{\mathrm{BH}}^{\mathrm{5D}}
{\frac{m}{M_{Pl}}}={\frac{(3\piP_{k}^{S})^{1/2}}{2L}}\left[{\frac{(1+u^{2})^{3/4}}{(1+3u^{2})^{1/4}}}\right]\approx2.6\times10^{-7}\left(10^{10}P_{k}^{S}\right)^{1/2}\left({\frac{60}{L}}\right)\left[{\frac{(1+u^{2})^{3/4}}{(1+3u^{2})^{1/4}}}\right];
\begin{array}{llll}{T^{0}}&{=}&{J^{0}{\omega}^{0}\wedge{\omega}^{3}\nonumber}\\{T^{1}}&{=}&{J^{1}{\omega}^{3}\wedge{\omega}^{1}+J^{2}{\omega}^{0}\wedge{\omega}^{1}\nonumber}\\{T^{2}}&{=}&{J^{1}{\omega}^{3}\wedge{\omega}^{2}+J^{2}{\omega}^{0}\wedge{\omega}^{2}\nonumber}\\{T^{3}}&{=}&{J^{3}{\omega}^{0}\wedge{\omega}^{3}\nonumber}\\\end{array}
\begin{array}{c}{\langleF_{K}^{\ast}F_{K}\rangle~(a_{Q}^{2}+b_{Q}^{2})}\\{\langle\tilde{F}_{\tilde{K}}^{\ast}\tilde{F}_{\tilde{K}}\rangle(\tilde{a}_{\tilde{Q}}^{2}+\tilde{b}_{\tilde{Q}}^{2})}\\{\langleF_{\tau}\rangle\lambda^{\alpha}\lambda_{\alpha}~+~\langleF_{\tau}^{\ast}\rangle\bar{\lambda}_{\dot{\alpha}}\bar{\lambda}^{\dot{\alpha}}}\\\end{array}
y^{2}=x^{3}+\sum_{k=-4}^{4}\sum_{\ell=0}^{8-nk}f_{k\ell}\,z^{\prime}{}^{\ell}z^{4-k}u^{\ell-k}x+\sum_{k=-6}^{6}\sum_{\ell=0}^{12-nk}g_{k\ell}\,z^{\prime}{}^{\ell}z^{6-k}u^{\ell-k}.
0=\left(\frac{\partialU}{\partialu_{\nu}}\right)_{u_{\mu}=e_{\mu}}\rightarrow\frac{1}{g}+\sum_{\mu=1(\neq\nu)}^{N}\frac{2}{e_{\mu}-e_{\nu}}=\sum_{j=1}^{\Omega}\frac{1}{z_{j}-e_{\nu}}\;,
{\calF}_{ij}=\left(\begin{array}{cccc}{0}&{{\calF}_{01}}&{0}&{0}\\{-{\calF}_{01}}&{0}&{0}&{0}\\{0}&{0}&{0}&{{\calF}_{23}}\\{0}&{0}&{-{\calF}_{23}}&{0}\\\end{array}\right).
\mathrm{IIA}_{(10,0)}\stackrel{\mathrm{T}_{st}}{\longleftrightarrow}\mathrm{IIB}_{(9,1)}\stackrel{\mathrm{T}_{st}}{\longleftrightarrow}\mathrm{IIA}_{(8,2)}\stackrel{\mathrm{T}_{st}}{\longleftrightarrow}\mathrm{IIB}_{(7,3)}\stackrel{\mathrm{T}_{st}}{\longleftrightarrow}\mathrm{IIA}_{(6,4)}\stackrel{\mathrm{T}_{st}}{\longleftrightarrow}\mathrm{IIB}_{(5,5)}\,.
M(Q)=(\alpha,\beta)=\left\{\begin{array}{lcc}{(A,\frac{B+\sqrt{\widehat{d}}}{2})}&{\mathrm{if}}&{A>0}\\{(A,\frac{B+\sqrt{\widehat{d}}}{2})\sqrt{\widehat{d}}}&{\mathrm{if}}&{A<0}\\\end{array}\right\},\;\;\;\;\;\widehat{d}=B^{2}-4AC.
{\mathbfL}_{1}=\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{-1}&{0}\\{0}&{0}&{\sqrt2}\\\end{array}\right)\,\qquad{\mathbfL}_{2}=\left(\begin{array}{ccc}{0}&{-1}&{0}\\{1}&{0}&{0}\\{0}&{0}&{0}\\\end{array}\right)
T^{0i}=B\tilde{E}_{i}-(\partial_{j}\Pi_{j})A_{i}+\gamma\partial_{i}(A\Pi)+(\frac{M}{8}+\frac{e^{2}}{8\pi}n\gamma)\partial_{j}(A_{i}\tilde{A}_{j}+A_{j}\tilde{A}_{i})+O(\frac{1}{\Lambda})
\tilde{R}=\tilde{g}\sqrt{\tilde{\alpha}^{\prime}}=\frac{1}{M\prodr_{i}}\,\quad\tilde{l}_{11}=\tilde{g}^{1/3}\sqrt{\tilde{\alpha}^{\prime}}=R_{s}^{1/6}M^{-5/6}\prodr_{i}^{-1/3}\.
\left.\begin{array}{l}{\delta_{a}\bigl([S_{a}]^{-1}[S_{2}]\bigl)=\delta_{a}\bigl((D_{2})(\gamma_{2})(D_{1})^{-1}\bigl)=0,}\\{\delta_{b}\bigl([S_{b}][S_{1}]\bigl)=\delta_{b}\bigl((D_{1})(\gamma_{1})\bigl)=0,}\\\end{array}\right.
q\equiv-\frac{1}{16\pi^{2}}\intd^{4}x\mathrm{~}trG_{\mu\nu}\tilde{G}_{\mu\nu}=\frac{1}{2}\intd\sigma_{1}d\sigma_{2}d\tau_{1}d\tau_{2}\epsilon_{\mu\nu\alpha\beta}\dot{z}_{1\mu}z\prime_{1\nu}\dot{z}_{2\alpha}z\prime_{2\beta}\delta(z_{1}-z_{2})
T_{\mu\nu}={{\overline{\Psi}}_{L_{0}}}{\gamma_{\mu}}iD_{\nu}{\Psi}_{L_{0}}-\frac{i}{2}g_{\mu\nu}\left[\partial_{\alpha}({{\overline{\Psi}}}_{L_{0}}\gamma^{\alpha}{\Psi}_{L_{0}})+\Gamma_{\beta\alpha}^{\beta}{{\overline{\Psi}}}_{L_{0}}\gamma^{\alpha}{\Psi}_{L_{0}})\right].
\begin{aligned}{\eta_{B}(s)}&{{}=\sum_{p>0}\frac{2p}{(p+\lambda^{2}/2+2Z\lambda)^{^{\textstyles}}}}\\{}&{{}\quad+\sum_{p,q>0}\,\left\{\frac{p+q}{\left(\lambda^{2}/2+\sqrt{D(\lambdaS,p,q)}+2Z\lambda\right)^{\textstyles}}-\frac{p+q}{\left(-\lambda^{2}/2+\sqrt{D(\lambdaS,p,q)}-2Z\lambda\right)^{\textstyles}}\,\right\}}\\\end{aligned}
S_{A}=-\frac{1}{4}\intF_{AB}F^{AB}d^{N}X=-\frac{1}{2}\int[\partial_{A}A_{B}(\partial^{A}A^{B}+\partial^{B}A^{A})-2\partial_{A}A^{A}\partial_{B}A^{B}]d^{N}X~,
N^{{\calP}_{0}}~=~\sum_{m_{1}=-1/2}^{1/2}\cdots\sum_{m_{p_{0}}=-1/2}^{1/2}\left[~\delta_{(\sum_{n=1}^{p_{0}}m_{n}),0}~-~\frac12~\delta_{(\sum_{n=1}^{p_{0}}m_{n}),1}~-~\frac12~\delta_{(\sum_{n=1}^{p_{0}}m_{n}),-1}~\right]
\begin{matrix}{\zeta:}&{L}&{\longmapsto}&{\mathcal{E}xt_{\mathcal{O}_{X}}^{1}(L,\mathcal{O}_{X}),}\\{\tilde{\epsilon}:}&{(\mathcal{O}_{C}\toL_{|C})}&{\longmapsto}&{({\mathcal{H}}om_{\mathcal{O}_{C}}(L_{|C},\mathcal{O}_{C})\subset\mathcal{O}_{C}).}\\\end{matrix}
d\Omega+\Omega\wedge\Omega=g\big[F^{\Lambda\Sigma}-\big(\sqrt{2}(u^{\Lambda\Sigma}{}_{AB}+v_{\Lambda\SigmaAB})\bar{\psi}^{A}\wedge\psi^{B}+\mathrm{h.c.}\big)\big]e_{\Lambda\Sigma}^{\alpha}\mathbb{L}^{-1}G_{\alpha}\mathbb{L}\,,
\frac{\partialF}{\partiala_{i}}=\oint_{\beta_{i}}\lambda,\\\,frac{\partialF}{\partialT_{n}}=-2\pii\,mbox{res}\Big(z^{-n}\lambda\Big),\\\,frac{\partialF}{\partialT_{0}}=-2\pii\,\int_{z_{*}=0}^{z=0}\lambda,
\begin{array}{rcl}{\varphi_{1}^{+}\varphi_{2}^{+}\varphi_{4}^{-}}&{=}&{\varphi_{4}^{-}\varphi_{1}^{+}\varphi_{2}^{+}+D_{24}\varphi_{1}^{+}+D_{14}\varphi_{2}^{+},}\\\end{array}
\langle\dot{\phi}_{0}^{2}\rangle=\frac{1}{4\pi}\int_{-\Lambda}^{\Lambda}dkE(k),~~\langle\phi_{0}^{\prime~2}\rangle=\frac{1}{4\pi}\int_{-\Lambda}^{\Lambda}dk\frac{k^{2}}{E(k)},~~\langle\phi_{0}^{2}\rangle=\frac{1}{4\pi}\int_{-\Lambda}^{\Lambda}dk\frac{dk}{E(k)}.
|a_{n}^{2}(\vec{q},\vec{q})|>\frac{(n-2)!}{2^{n-2}(2(n-1))!}\left(V^{(n-2,0,0)}(\vec{q})\right)^{2}\sim\frac{(n-2)!}{2^{n-2}(2(n-1))!}\left(\frac{(n-2)!}{R_{1}^{n-2}(\vec{q})}\right)^{2}\simn!.
S_{\mathrm{I}}=\frac{1}{16\pi\alpha^{\prime4}}\intd^{10}x\sqrt{|g_{10}|}\left[e^{-\Phi}\left(R_{10}+\left(\nabla\Phi\right)^{2}\right)-\frac{1}{12}H_{3}^{2}-\frac{1}{4}e^{-\Phi/2}F_{2}^{2}\right]
\hat{\varrho}_{\mathrm{fld}}=\hat{\varrho}_{m}\left[1+\zeta\left({\frac{\hat{c}}{c}}\right)^{2}\right]+\hat{\varrho}_{r}\left({\frac{\hat{c}}{c}}\right)^{2},\\\\\\\\hat{\wp}_{\mathrm{fld}}={\frac{1}{3}}\hat{\varrho}_{r}\left({\frac{\hat{c}}{c}}\right)^{2},
\left\{{\frac{Det^{\prime}\left[-(d^{2}/d\tau^{2})+V^{\prime\prime}[x_{c1}(\tau)]\right]}{Det\left[-(d^{2}/d\tau^{2})+\nu^{2}\right]}}\right\}^{-1/2}\,sqrt{{\frac{S_{1}}{2\pi}}}\d\tau_{c}
{\calH}_{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}-A^{0}(\partial^{i}\pi_{i}+m^{2}A^{0}).
\left(\begin{matrix}{\\\\cdot\}&{\cdot}&{0.407663}&{0.405138}&{0.404239}&{}\\{\\\\cdot\}&{0.362383^{*}}&{0.405248}&{0.403736}&{}&{}\\{\\\\cdot\}&{0.395111}&{0.404338}&{}&{}&{}\\{\\\\cdot\}&{0.400335}&{}&{}&{}&{}\\{\\\\cdot\}&{}&{}&{}&{}&{}\\\end{matrix}\right),
\sigma_{1}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\,,\sigma_{2}=\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right)\,,\sigma_{3}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)\,,
c(\gamma_{1},\gamma_{2}):=k\left\{m_{1}\,\mu_{2}-\mu_{1}\,m_{2}+\frac{1}{4\pi}\int_{0}^{2\pi}\left(f_{1}^{\prime}(\theta)f_{2}(\theta)-f_{1}(\theta)f_{2}^{\prime}(\theta)\right)d\theta\right\},
e^{-\frac{1}{2}\betaK^{ij}u_{i}u_{j}}<x|e^{\varphi^{i}v_{i}}|y>=\int\sqrt{g^{\prime}(\varphi^{\prime})}\,d\varphi^{\prime}K_{\beta}(\varphi,\varphi^{\prime})<x|e^{\varphi^{i}v_{i}}|y>
\Gamma=\frac{\pi}{2}A_{h}(\frac{\omega^{2}-m^{2}}{\omega})r_{e}\frac{1}{(e^{\frac{(\omega-m)}{T_{H}}}-1)}\frac{1}{(e^{\frac{1}{2}\pi(\omega+m)r_{e}}-1)}\frac{d^{4}k}{(2\pi)^{4}}.
\frac{\partial^{2}}{\partialx_{2}^{2}}V(x_{2})=\widetilde{\phi}_{1}(x_{2})\widetilde{\phi}_{1}^{\dagger}(x_{2})\frac{e^{2}\;V_{1,1}}{2}+\phi_{1}(x_{2})\phi_{1}^{\dagger}(x_{2})\frac{e^{2}\;V_{\tilde{1},\tilde{1}}}{2}.
\gamma_{m}^{N,k,d}=\sum_{l=1}^{d}\sum_{\sum_{i=1}^{l}d_{i}=d}(-1)^{l-1}\sum_{j_{l}=m}^{N-1-(N-k)d}\cdots\sum_{j_{2}=m}^{j_{3}}\sum_{j_{1}=m}^{j_{2}}(\prod_{i=1}^{l}L_{j_{i}+(\sum_{n=1}^{i-1}d_{n})(N-k)}^{N,k,d_{i}})
\langleE_{r}(r)E_{r}(r^{\prime})\rangle_{eff}=-\frac{1}{\pir^{4}}\int_{0}^{x_{0}}\frac{dx}{n^{3}x}\sum_{l=1}^{\infty}\frac{2l+1}{4\pi}l(l+1)A_{G}(x)s_{l}^{2}(nx\tilde{r}),
\begin{array}{rcl}{R_{12}K_{1}R_{21}dK_{2}}&{=}&{dK_{2}R_{12}K_{1}R_{12}^{-1}\;,}\\{}&{}&{}\\{R_{12}dK_{1}R_{21}dK_{2}}&{=}&{-dK_{2}R_{12}dK_{1}R_{12}^{-1}\;,}\\{}&{}&{}\\{Y_{2}R_{21}^{-1}dK_{1}R_{21}}&{=}&{R_{12}dK_{1}R_{21}Y_{2}\quad.}\\\end{array}
\chi=a^{A}P_{A}+\bar{\varepsilon}_{i}Q^{i}+\lambdaD+\textstyle{\frac{1}{2}}\omega^{AB}M_{AB}+b^{A}K_{A}+\bar{\rho}_{i}S^{i}+\textstyle{\frac{1}{2}}T_{i}^{~j}A_{j}^{~i}
\tilde{T}=\left(\begin{matrix}{\sum_{i=1}^{n-1}\bar{z}^{m_{i}}}\\{-1}\\{\vdots}\\{-1}\\\end{matrix}\right)+\left(\begin{matrix}{1}\\{z^{m_{1}}}\\{\vdots}\\{z^{m_{n-1}}}\\\end{matrix}\right)\,\frac{-2i}{1+\sum_{i=1}^{n-1}\bar{z}^{m_{i}}z^{m_{i}}}\,\biggl(t\sum_{j=1}^{n-1}m_{j}\bar{z}^{m_{j}-1}+h(\bar{z})\biggr).
F(r)=\frac{(r-n)(5r^{4}+20nr^{3}+(\ell^{2}+22n^{2})r^{2}+(4n\ell^{2}-12n^{3})r-35n^{4}+5\ell^{2}n^{2}}{5(r+n)^{3}\ell^{2}}
\lambda_{l_{1},l_{2}}^{L}=(L+1)(-1)^{l_{1}+l_{2}+L}\left\{\begin{array}{ccc}{l_{1}}&{{\frac{L}{2}}}&{{\frac{L}{2}}}\\{{}}&{{}}&{}\\{l_{2}}&{{\frac{L}{2}}}&{{\frac{L}{2}}}\\\end{array}\right\}.
{\Gamma_{\phi}^{ij}}^{-1}(\nu)=\delta_{ij}m_{i}\bar{\gamma}_{i}^{3}\bar{R}_{i}^{2}\nu^{2}\frac{\nu^{2}-\left(2\bar{\gamma}_{i}^{2}+1\right)\omega^{2}}{\nu^{2}+\left(2\bar{\gamma}_{i}^{2}-1\right)\omega^{2}}-\sigmaG_{\phi}^{ij}(\nu)\,.
q^{2(\lambda,\lambda+2\rho_{0})}=\sum_{(\mu,2\kappa,-s)\inD^{+}}\sum_{s=0}^{\infty}(m_{\mu,s}^{+}-m_{\mu,s}^{-})q^{(\mu,\mu+2\rho_{0})/2-s(2\kappa+g)}\frac{D_{q}[(\mu,2\kappa,-s)]}{D_{q}[(\lambda,\kappa,0)]}
D_{[a}B_{bc]}^{I\alpha}=-{\frac{1}{12}}g\mathbb{L}_{~~AB}^{I\alpha}\mathcal{H}^{AB~|de}\varepsilon_{abcde}-\mathrm{i}\sqrt{2}\mathbb{L}_{AB}^{-1~I\alpha}\bar{\chi}^{ABC}\gamma_{[a}\rho_{bc]C}\,.
V_{eff}(\phi_{c})=\frac{\mu_{0}^{2}}{2}\phi_{c}^{2}+\frac{\lambda}{4!N}(\phi_{c}^{2})^{2}+\frac{N}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\mathrm{ln}(1+\frac{m^{2}}{k^{2}})-\frac{\lambda}{4!N}I^{2}.
{\calF}_{\mu\nu}=\partial_{\mu}(\hat{\phi}^{a}W_{\nu}^{a})-\partial_{\nu}(\hat{\phi}^{a}W_{\mu}^{a})-{\frac{1}{g}}\epsilon^{abc}\hat{\phi}^{a}(\partial_{\mu}\hat{\phi}^{b})(\partial_{\nu}\hat{\phi}^{c})
\mathrm{det}\Big(1+\frac{1}{2}(\theta)^{2}\Big)=\Big(1+\frac{1}{6}(\theta)^{2}\Big)^{3}\,\quad\mathrm{det}(1-(\theta_{\mathrm{s}})^{2})=\Big(1-\frac{1}{2}(\theta_{\mathrm{s}})^{2}\Big)^{2}\,
N=\int_{t_{\mathrm{i}}}^{t_{\mathrm{f}}}dtH\simeq\frac{-1}{M_{Pl}^{2}\sqrt{1+4\alpha\Lambda_{b}/3M^{5}}}\int_{\phi_{\mathrm{i}}}^{\phi_{\mathrm{f}}}d\phi{\frac{V}{V^{\prime}}}\left[1+{\frac{V}{2\Lambda_{2}^{+}}}\right]\,.
[D^{F\nu},[D^{F\mu},Q_{\pm}^{\taulm}]]=\sum_{\tau^{\prime},l^{\prime},m^{\prime}}Q_{\pm}^{\tau^{\prime}l^{\prime}m^{\prime}}\left(D^{B\nu}D^{B\mu}\right)_{\tau^{\prime}l^{\prime}m^{\prime},\taulm}.
U_{g}^{a}=\left(\begin{array}{cccc}{\mathrm{e}^{i\lambda_{a,1}}}&{}&{}&{}\\{}&{\ddots}&{}&{}\\{}&{}&{\ddots}&{}\\{}&{}&{}&{\mathrm{e}^{i\lambda_{a,N}}}\\\end{array}\right),\quad\sum_{i=1}^{N}\lambda_{a,i}=0,\quad(a=1,\cdots,n),
I_{a,b,c}=\int\frac{d^{d}q_{1}}{N_{d}\pi^{d/2}}\int\frac{d^{d}q_{2}}{N_{d}\pi^{d/2}}\frac{1}{(q_{1}^{2}+1)^{a}(q_{2}^{2}+1)^{b}((q_{1}+q_{2})^{2}+1)^{c}}
F(r)=\frac{5r^{8}+(\ell^{2}-28N^{2})r^{6}+5N^{2}(14N^{2}-\ell^{2})r^{4}+5N^{4}(3\ell^{2}-28N^{2})r^{2}-10mr\ell^{2}+5N^{6}(\ell^{2}-7N^{2})}{5\ell^{2}(r^{2}-N^{2})^{3}}
\nabla^{2}=\frac{4}{d^{2}(\xi^{2}-\eta^{2})}\left\{\frac{\partial}{\partial\xi}(\xi^{2}-1)\frac{\partial}{\partial\xi}+\frac{\partial}{\partial\eta}(1-\eta^{2})\frac{\partial}{\partial\eta}+\frac{\xi^{2}-\eta^{2}}{(\xi^{2}-1)(1-\eta^{2})}\frac{\partial^{2}}{\partial\phi^{2}}\right\}.
\Pi_{\mu\nu}^{M}(k)=\frac{ie^{2}}{(2\pi)^{3}}\sum_{i=0}^{n_{f}}c_{i}\intd^{3}p\frac{P(M_{i})}{({M_{i}}^{2}-{p_{1}}^{2})({M_{i}}^{2}-{p_{2}}^{2})}\,\,,
\alpha^{\prime}M^{2}=-4+2(N_{R}+N_{L})+\alpha^{\prime}\,m_{I}\,G^{IJ}m_{J}+\frac{1}{\alpha^{\prime}}\,n^{I}(G-BG^{-1}B)_{IJ}\,n^{J}+2n^{I}B_{IK}G^{KJ}m_{J}\,,
\rho_{\mu}(x)=\rho_{\mu^{\prime}}(x)\frac{q^{C(\mu)/2}+\epsilon(\mu)\epsilon(\mu^{\prime})xq^{C(\mu^{\prime})/2}}{xq^{C(\mu)/2}+\epsilon(\mu)\epsilon(\mu^{\prime})q^{C(\mu^{\prime})/2}},~~~~\forall\mu\neq\mu^{\prime}.
\widetilde{\Gamma}_{i}^{(9)}=\left(\begin{matrix}{0}&{t_{i}}\\{-t_{i}}&{0}\\\end{matrix}\right)\,,\qquad\widetilde{\Gamma}_{i}^{(0)}=\left(\begin{matrix}{0}&{1}\\{1}&{0}\\\end{matrix}\right)\,,\qquad\widetilde{\Gamma}_{i}^{(8)}=\left(\begin{matrix}{1}&{0}\\{0}&{-1}\\\end{matrix}\right)\,,
i\frac{\partial}{\partialt}\rho({\bfk},{\bfk}^{\prime})=(\omega_{\bfk}-\omega_{{\bfk}^{\prime}})\rho({\bfk},{\bfk}^{\prime})+ig^{2}\left[F({\bfk},t)+F^{*}({\bfk}^{\prime},t)\right]\rho({\bfk},{\bfk}^{\prime})
z_{j}\;\longmapsto\;\frac{a\,z_{j}+b}{c\,z_{j}+d}\,\\quad\bar{z}_{j}\;\longmapsto\;\frac{a^{*}\,\bar{z}_{j}+b^{*}}{c^{*}\,\bar{z}_{j}+d^{*}}\,\quad\j=1,\ldots,n\,
\int_{\calM}\partial_{\mu}(\sqrt{-G}G^{\mu\nu}\partial_{\nu}\phi)=\frac{p-3}{4}\sum_{i}q_{i}\sqrt{-g^{nc}(x,y_{i})}e^{{\frac{(p-3)}{4}}\phi(x,y_{i})}+\frac{3-p}{4}\int_{\calM}e^{{\frac{(3-p)}{2}}\phi}\sqrt{-{g}}\frac{|F_{p+2}|^{2}}{(p+2)!}\,.
0=\mathrm{~n~-th~order~of~}\left\langle\frac{1}{2}\varphi^{2}e^{-\frac{i\lambda}{4!}\varphi^{4}+\frac{i}{2}(J^{(1)}+J^{(2)}+\cdots)\varphi^{2}}\right\rangle_{G^{(0)}}
S=\intd\tau\{p_{m}(\dot{x}^{m}+i\theta\sigma^{m}\dot{\bar{\theta}}-i\dot{\theta}\sigma^{m}\bar{\theta}+i\psi\sigma^{m}\bar{\rho}-i\rho\sigma^{m}\bar{\psi})-\frac{ep^{2}}{2}-{\rho^{\alpha}}{\dot{\theta}}_{\alpha}-{\bar{\rho}}_{\dot{\alpha}}{\dot{\bar{\theta}}}^{\dot{\alpha}}\},
=g_{\mu\nu}+2g_{\mu\nu}\phi_{\lambda}\xi^{\lambda}+{\frac{1}{2}}\left[4g_{\mu\nu}\phi_{\lambda}\phi_{\sigma}+g_{\mu\nu}\left(\phi_{\lambda;\sigma}+\phi_{\sigma;\lambda}\right)+{\frac{1}{3}}\left(R_{~\lambda\mu\sigma}^{\kappa}g_{\kappa\nu}+R_{~\lambda\nu\sigma}^{\kappa}g_{\kappa\mu}\right)\right]\xi^{\lambda}\xi^{\sigma}+...
\widetilde{\calH}_{\scriptscriptstyleP}=\{Q,\widehat{X}_{(1)}\cdots\widehat{X}_{(p)}\}=\sum_{i=1}^{p}(-1)^{i+1}\widehat{X}_{(1)}\cdots{\widehat{\widehatX}}_{(i)}\cdots\widehat{X}_{(p)}\widetilde{\calH}_{\scriptscriptstyleX_{(i)}}
\begin{array}{c}{\psi=\psi_{0}\{th[\frac\gamma2(r_{+}-R)]-\epsilon/\lambda\},z<0,}\\{\psi=\psi_{0}\{th[\frac\gamma2(r_{-}-R)]-\epsilon/\lambda\},z>0,}\\{r_{\pm}=\sqrt{x^{2}+y^{2}+\left(z\pmb\right)^{2}},b>R.}\\\end{array}
<\overline{{\psi}}_{L}^{({\calN})}\ldots\overline{{\psi}}_{L}^{(1)}\psi_{R}^{(1)}\ldots\psi_{R}^{({\calN})}>\rightarrow(-1)^{{\calN}}<\overline{{\psi}}_{L}^{({\calN})}\ldots\overline{{\psi}}_{L}^{(1)}\psi_{R}^{(1)}\ldots\psi_{R}^{({\calN})}>
[a_{m},\tilde{a}_{n}^{\dagger}]=[\tilde{a}_{m},a_{n}^{\dagger}]=\delta_{mn}\mid0\rangle\langle0\mid,\;\;\;\;\;\;[a_{m},\tilde{a}_{n}]=[a_{m}^{\dagger},\tilde{a}_{n}^{\dagger}]=0
\left(P^{\mu_{1}\nu_{1}\,\mu_{2}\nu_{2}}\right)^{-1}=d_{A}{\calT}_{A\;{\mu_{1}\nu_{1}\,\mu_{2}\nu_{2}}}+d_{B}{\calT}_{A\;{\mu_{1}\nu_{1}\,\mu_{2}\nu_{2}}}+d_{C}{\calT}_{A\;{\mu_{1}\nu_{1}\,\mu_{2}\nu_{2}}}+\sum_{i=4}^{14}d_{i}T_{i\;{\mu_{1}\nu_{1}\,\mu_{2}\nu_{2}}}
\begin{array}{c}{\lbracke_{i},f_{j}\rbrack=\delta_{ij}\frac{k_{i}-k_{i}^{-1}}{Q_{i}-Q_{i}^{-1}}}\\{k_{i}e_{j}k_{i}^{-1}=Q_{i}^{a_{ij}}e_{j},~~k_{i}f_{j}k_{i}^{-1}=Q_{i}^{-a_{ij}}f_{j}}\\{k_{i}k_{i}^{-1}=k_{i}^{-1}k_{i}=1,~~k_{i}k_{j}=k_{j}k_{i}}\\{\lbracke_{i},e_{i}\rbrack=\lbrackf_{i},f_{i}\rbrack=0,\mathrm{if}~~~a_{ii}=0}\\\end{array}
S_{q}(y,M)=\sum_{\alpha,\beta\gamma,Z,KK}g_{\alpha}^{2}(M)g_{\beta}^{2}(M){\frac{(v_{e}^{\alpha}v_{e}^{\beta}+a_{e}^{\alpha}a_{e}^{\beta})(v_{l}^{\alpha}v_{l}^{\beta}+a_{l}^{\alpha}a_{l}^{\beta})}{(s-m_{\alpha}^{2}+i\Gamma{_\alpha}m_{\alpha})(s-m_{\beta}^{2}-i\Gamma_{\beta}m_{\beta})}}\.
\gamma_{p}\equiv\left.\frac{\Omega}{T}\right|_{p}=\left.\frac{2\pilr_{+}}{2r_{+}^{2}+l^{2}}\right|_{2m=u^{2}(u^{2}+l^{2})}=\frac{\sqrt{2}\pil\sqrt{\sqrt{l^{4}+4u^{2}(u^{2}+l^{2})}-l^{2}}}{\sqrt{l^{4}+4u^{2}(u^{2}+l^{2})}},
{G_{SP}=\left(\begin{array}{ccc}{1}&{\theta}&{iX+{\frac{1}{2}}\theta\bar{\theta}}\\{0}&{1}&{\bar{\theta}}\\{0}&{0}&{1}\\\end{array}\right)\times\left(\begin{array}{ccc}{L}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{(L^{+})^{-1}}\\\end{array}\right)}.
S=-\tau_{p}V_{p}-\frac{1}{4g_{\mathrm{YM}}^{2}}\intd^{p+1}\xi\left(F_{\alpha\beta}F^{\alpha\beta}+\frac{2}{(2\pi\alpha^{\prime})^{2}}\partial_{\alpha}X^{a}\partial^{\alpha}X^{a}\right)+{\calO}(F^{4})
{\frac{\sqrt{{\boldsymbol{\nabla}}^{2}({\boldsymbol{\nabla}}^{2}\vert\widetilde{\boldsymbolf}(-i{\boldsymbol{\nabla}})\vert^{2}+1)}}{{\boldsymbol{\nabla}}^{2}\vert\widetilde{\boldsymbolf}(-i{\boldsymbol{\nabla}})\vert^{2}}}{\mit\Pi}^{(2)}({\boldsymbolx})+{\frac{1}{\vert\widetilde{\boldsymbolf}(-i{\boldsymbol{\nabla}})\vert^{2}}}A^{0}({\boldsymbolx};\phi)+J_{0}(t,{\boldsymbolx})=0.
F_{\mu\nu}=\left(\begin{array}{ccccc}{0}&{-\omega_{1}}&{}&{}&{}\\{\omega_{1}}&{0}&{}&{}&{}\\{}&{}&{\ddots}&{}&{}\\{}&{}&{}&{0}&{-\omega_{5}}\\{}&{}&{}&{\omega_{5}}&{0}\\\end{array}\right),
\begin{aligned}{}&{{}(K_{ij}f)(x_{1},\dots,x_{i},\dots,x_{j},\dots,x_{N})=f(x_{1},\dots,x_{j},\dots,x_{i},\dots,x_{N})\,,}\\{}&{{}(K_{i}f)(x_{1},\dots,x_{i},\dots,x_{N})=f(x_{1},\dots,-x_{i},\dots,x_{N})\,.}\\\end{aligned}
\prod_{\nu=0\atop\nu\neq\mu}^{n-1}\frac{[a_{\mu\nu}+1]}{[a_{\mu\nu}]}+\sum_{\lambda=0\atop\lambda\neq\mu}^{n-1}\frac{[z+a_{\lambda\mu}+1][1]}{[z+1][a_{\lambda\mu}]}\prod_{\nu=0\atop\nu\neq\lambda,\mu}^{n-1}\frac{[a_{\lambda\nu}+1]}{[a_{\lambda\nu}]}=\frac{[z+n]}{[z+1]}.
{\calP}({\calE})=\sum_{\vert\Psi\rangle}\vert{\calA}({\calE})\vert^{2}=\int_{-\infty}^{\infty}d\tau\,\int_{-\infty}^{\infty}d\tau^{\prime}\,e^{-i{\calE}(\tau-\tau^{\prime})}\,{\tilde{G}}\left[{\tilde{x}}(\tau),{\tilde{x}}(\tau^{\prime})\right],
Q^{b}=L_{i}\psi^{bi}+2i\psi^{bi}m^{j}\frac{\partial^{2}A}{\partial\phi^{i}\partial\phi^{j}}+2i\bar{\psi}^{bi}\bar{n}^{j}\frac{\partial^{2}A}{\partial\phi^{i}\partial\phi^{j}}+\frac{1}{2}i\lambda_{di}^{b}\psi^{di},
L^{IJ}\,=\,\frac{1}{2a}\left(\begin{array}{ccccccc}{0}&{1}&{}&{}&{}&{}&{}\\{1}&{0}&{}&{}&{}&{}&{}\\{}&{}&{\ddots}&{}&{}&{}&{}\\{}&{}&{}&{0}&{1}&{}&{}\\{}&{}&{}&{1}&{0}&{}&{}\\{}&{}&{}&{}&{}&{0}&{-1}\\{}&{}&{}&{}&{}&{-1}&{-\frac{b+r}{a}}\\\end{array}\right)\,.
Z_{\;\;\alpha_{1}}^{\alpha_{0}}\frac{\deltaA_{\beta_{2k-1}}^{\;\;\alpha_{2k}}}{\delta\Phi^{\alpha_{0}}}\approxC_{\;\;\alpha_{1}\beta_{2k-1}}^{\gamma_{2k-1}}A_{\gamma_{2k-1}}^{\;\;\alpha_{2k}}+C_{\;\;\alpha_{1}\beta_{2k-1}}^{\gamma_{2k+1}}Z_{\;\;\gamma_{2k+1}}^{\alpha_{2k}},\;k=2,\cdots,a,
P_{+}-5P_{-}=4\frac{\sqrt{\lambda}}{2\pi}\frac{\rho_{0}^{2}\mathrm{csh}(\rho_{0})\mathrm{sech}(\rho_{0})}{\sqrt{1-\rho_{0}\mathrm{csh}(\rho_{0})\mathrm{sech}(\rho_{0})}}\underset{\rho_{0}\gg}{\rightarrow}8\frac{\sqrt{\lambda}}{\pi}\rho_{0}^{2}e^{-2\rho_{0}}\,.
\Gamma_{\mu\nu}^{\lambda}=\{_{\mu\nu}^{\lambda}\}|_{\tilde{g}_{\mu\nu}}=\frac{1}{2}\tilde{g}^{\lambda\alpha}(\partial_{\nu}\tilde{g}_{\alpha\mu}+\partial_{\mu}\tilde{g}_{\alpha\nu}-\partial_{\alpha}\tilde{g}_{\mu\nu}).
S_{IIB}={8\,V_{4}}\,\int\,d^{6}\!x\,\sqrt{-g}\,\left[e^{-2\phi}\left(R+4\,(\partial\phi)^{2}-(\partial\sigma)^{2}-{\frac{1}{12}}\,H^{2}\right)-{\frac{e^{2\sigma}}{12}}\,F_{(3)}^{2}\right].
t_{\pm}=-{\frac{1}{\kappa}}\partial_{\pm}\chi\partial_{\pm}\chi+\partial_{\pm}^{2}\chi+{\frac{1}{\kappa}}\partial_{\pm}\Omega\partial_{\pm}\Omega+{\frac{1}{2}}\sum_{i=1}^{N}\partial_{\pm}f_{i}\partial_{\pm}f_{i}\.
I=\int\left[-\frac{k\epsilon^{ij}}{4\pi}\left(\frac{\eta_{ab}}{2}A_{i}^{a}\dot{A}_{j}^{b}+i\bar{\psi}_{i}\dot{\psi}_{j}\right)-A_{0}^{a}{\calG}_{a}-\bar{\psi}_{0}{\calS}\right]
h_{ab}=\delta_{ab}\frac{\mu^{2}\kappa_{11}^{2}p^{+}}{\pi^{4}}\frac{1}{96}\left[\frac{15}{2}\frac{\sum_{k=1}^{3}(x^{k})^{2}}{|\vec{x}|^{7}}-\frac{1}{|\vec{x}|^{5}}\right]
\frac{1}{\tilde{\kappa}^{2}}\frac{1}{CN^{4}[\tilde{t}\mid\frac{1}{2}]}\frac{d^{2}A}{d\tilde{t}^{2}}=\frac{-3}{\tilde{\kappa}^{2}}\frac{A}{CN^{6}[\tilde{t}\mid\frac{1}{2}]}+A-A^{3},
{\bfz}_{1}(t_{1})={\bfz}_{1}(t)+{\frac{1}{2}}\tau\dot{\bfz}_{1}(t)+{\frac{1}{8}}\tau^{2}\ddot{\bfz}_{1}(t)+\dots,\quad\qquad{\bfz}_{2}(t_{2})={\bfz}_{2}(t)-{\frac{1}{2}}\dot{\bfz}_{2}(t)+{\frac{1}{8}}\tau^{2}\ddot{\bfz}_{2}(t)-\dots
\delta_{h}I_{h}=-{\frac{1}{4\pi}}\int\mathrm{Tr}\bigg[h^{-1}\deltah\bigg(\partial_{t}\partial_{\phi}\alpha-k\partial_{\phi}(h^{-1}\partial_{t}h)+[h^{-1}\partial_{t}h,\partial_{\phi}\alpha]\bigg)\bigg]dtd\phi=0,
c=\frac{v_{2}e^{-(1-\gamma)k_{1}l}-v_{1}e^{-(1-\gamma)k_{2}l}}{e^{2\gammak_{1}l}-e^{2\gammak_{2}l}}\;,\qquadd=\frac{v_{2}e^{-(1+\gamma)k_{1}l}-v_{1}e^{-(1+\gamma)k_{2}l}}{e^{-2\gammak_{1}l}-e^{-2\gammak_{2}l}}\;.
P^{(2)}(\lambda,m)=\frac{\lambda}{2}e^{-\lambda^{2}/4}\frac{1}{I_{0}^{2}(m)-I_{1}^{2}(m)}\left[I_{2}^{2}(t)-I_{1}(t)I_{3}(t)\right]_{t=\sqrt{\lambda^{2}+m^{2}}}\:.
{\calS}\,=\,\int\,dt\,\left[\frac{2\,\lambda\,\dot{a}^{i}\,\dot{a}^{i}}{a^{i}\,a^{i}}\,-\,\frac{2\,\lambda\,\dot{\phi}^{i}\,\dot{\phi}^{i}}{(\,a^{i}\,a^{i}\,)^{2}}\right]\;\;,
{\calL}=\bar{\psi}^{a}i\not\!\partial\psi^{a}+{\frac{\lambda^{2}}{2N}}(\bar{\psi}^{a}\psi^{a})^{2}-{\frac{\lambda_{5}^{2}}{2N}}(\bar{\psi}^{a}\gamma^{5}\psi^{a})^{2}
\\\left.\left.+\frac{3}{2}\gamma\varphi_{0}^{2}\right)\dot{\phi}\left(\frac{\dot{a}}{a}\right)\right]+e^{-2k\phi}\left[\frac{\gamma}{2}\varphi_{0}^{2}k^{2}\dot{\phi}^{2}-V_{0}\varphi_{0}^{\alpha}-e^{\psi(2-\alpha)}\right]\.
\{f(z),g(z)\}_{\alpha}=\nabla_{i}f(z)(\omega_{(\alpha)ij}+R_{ijkl}\theta^{k}\theta^{l})^{-1}\nabla_{j}g(z)+\frac{\partial_{r}f(z)}{\partial\theta^{i}}g^{ij}\frac{\partial_{l}g(z)}{\partial\theta^{j}},
\left[1\,+\,{\frac{M_{*}^{N-2}}{c_{1}\,M_{*}^{N-2}+c_{2}\,\nabla^{N-2}}}\,{\frac{1}{(r_{c}\,\nabla)^{2}}}\right]\,R\,=\,8\piG_{N}\,T_{\mu}^{\mu}\,.
\begin{array}{c}{D(\beta,L)-i\mu\gamma^{0}=e^{\mu\betat}D(\beta,L)\e^{-\mu\betat},}\\{K_{bc}(t,x;t^{\prime},x^{\prime})=e^{\mu\betat}k(t,x;t^{\prime},x^{\prime})\e^{-\mu\betat\prime},}\\\end{array}
\Gamma^{Gauge}=\intd^{4}x(\frac{1}{2}G_{\mu}^{a}(x)G^{a\mu}(x)-\bar{c}_{\mu}^{a}(x)(D^{\mu}D^{\nu}c_{\nu}(x))^{a}+\bar{c}^{a}(x)(D^{\nu}D_{\nu}c(x))^{a})
\int_{-m/2}^{m/2}{s\,g_{+}^{2}\,ds}=\int_{-m/2}^{m/2}{s\left(s^{2}-\frac{m^{2}}{4}\right)^{k-1}e^{2rs}\,ds}\approxe^{mr}\int_{-\infty}^{0}e^{2ru}u^{k-1}(m+u)^{k-1}\left({\frac{m}{2}}+u\right)\,du\,.
{\calL}=-\frac{T}{2}\partial_{\alpha}X^{\mu}(\sigma,\tau)\partial^{\alpha}X_{\mu}(\sigma,\tau)+\sqrt{-1}\sum_{j=1}^{N}k_{\mu}^{(j)}X^{\mu}(\sigma,\tau)\delta(\sigma-\sigma_{j})\delta(\tau-\tau_{j}).
\begin{array}{c}{g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi_{c}+[(m_{r}^{2}+\deltam^{2})+(\xi_{r}+\delta\xi)R]\phi_{c}-4(\kappa_{r}+\delta\kappa)\phi_{c}^{3}-12(\kappa_{r}+\delta\kappa)\phi_{c}<\phi_{q}^{2}>}\\{+\3(\lambda_{r}^{2}+\delta\lambda^{2})\phi_{c}^{5}+30(\lambda_{r}^{2}+\delta\lambda^{2})\phi_{c}^{3}<\phi_{q}^{2}>+15(\lambda_{r}^{2}+\delta\lambda^{2})\phi_{c}<\phi_{q}^{4}>\=0}\\\end{array}
\hat{\cal{L}}_{MCS}=-\frac{1}{2}[\partial_{\alpha}(\hat{A}_{\beta}+\hat{a}_{\beta})-\partial_{\beta}(\hat{A}_{\alpha}+\hat{a}_{\alpha})][\partial^{\alpha}(\hat{A}^{\beta}+\hat{a}_{\beta})]+\frac{m}{2}\epsilon^{\alpha\beta\gamma}(\hat{A}_{\alpha}+\hat{a}^{\alpha})\partial_{\beta}(\hat{A}_{\gamma}+\hat{a}_{\gamma}).