harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{l}{{\mathcalD}_{V}f=V^{ab}\nabla_{a}\nabla_{b}f+\frac{n}{n+2}(\nabla_{a}V^{ab})\nabla_{b}f+\frac{n(n-2)}{4(n+2)(n+1)}(\nabla_{a}\nabla_{b}V^{ab})f}\\{\delta_{V}f=V^{ab}\nabla_{a}\nabla_{b}f+\frac{n+4}{n+2}(\nabla_{a}V^{ab})\nabla_{b}f+\frac{n+4}{4(n+1)}(\nabla_{a}\nabla_{b}V^{ab})f.}\\\end{array}
H={\frac{u}{\tilde{L}}}\left[{\frac{P_{\phi}^{2}}{L^{2}}}+{\frac{P_{u}^{2}}{\tilde{L}_{2}/u^{2}}}+{\frac{P_{\rho}^{2}}{L^{2}/(1-\rho^{2})}}+{\frac{(\rhoP_{\phi}-N\rho^{p})^{2}}{L^{2}(1-\rho^{2})}}\right]^{\frac{1}{2}}.
\tilde{\calH}=\frac{1}{2}\,\pi_{a}^{2}+\frac{1}{2}\,{\sigma^{\prime}}_{a}^{2}-(\sigma_{a}\pi^{a})\theta+\frac{1}{2}\lambda\bigl(\sigma_{a}^{2}-1\bigr)+\frac{1}{2}\,\sigma_{a}^{2}\theta^{2}.
\begin{array}{rll}{\bar{X}^{\mu}}&{=\left(\begin{array}{c|c|c|c}{\strutx_{1}^{\mu}{\bf1}}&{{\bf0}}&{\cdots}&{{\bf0}}\\\hline{\strut{\bf0}}&{x_{r}^{\mu}{\bf1}}&{\ddots}&{\vdots}\\\hline{\strut\vdots}&{\ddots}&{\ddots}&{{\bf0}}\\\hline{\strut{\bf0}}&{\cdots}&{{\bf0}}&{x_{M}^{\mu}{\bf1}}\\\end{array}\right)}&{(\mu=1,\ldots,d),}\\{\bar{X}^{m}}&{=0}&{(m=d+1,\ldots,D).}\\\end{array}
e^{AA^{\prime}j}\;{}^{(3)}\nabla_{j}\Bigr({_{e}n_{A}^{\;\;B^{\prime}}}\;\sigma_{B^{\prime}}^{nq}\Bigr)=\Bigr(n+{\frac{3}{2}}\Bigr){_{e}n^{AA^{\prime}}}\Bigr({_{e}n_{A}^{\;\;B^{\prime}}}\;\sigma_{B^{\prime}}^{nq}\Bigr)\;.
\left[g^{\mu\nu}(\vec{\sigma})\right]=\left(\begin{array}{c|c}{\gamma_{0}^{-2}}&{\displaystyle\frac{\vec{\sigma}^{\mathrm{T}}}{c}\gamma_{0}^{-2}}\\\hline{\displaystyle\frac{\vec{\sigma}}{c}\gamma_{0}^{-2}}&{-I}\\\end{array}\right).
\begin{array}{ccc}{(U_{1})_{k,l}}&{=}&{\frac{1}{\sqrt{N}}\omega^{m(l-k)(3l-k-2)}\times\left\{\begin{array}{c}{1}\\{-{\mathrmi}}\\\end{array}\right\}}\\{(U_{2})_{k,l}}&{=}&{\frac{1}{\sqrt{N}}\omega^{m(l-k)(l-3k+2)}\times\left\{\begin{array}{c}{1}\\{-{\mathrmi}}\\\end{array}\right\}}\\{(U_{3})_{k,l}}&{=}&{\delta_{k,l}\omega^{m(k-1)^{2}}}\\\end{array}
A=\int_{M_{D}}\,d^{D}x\,\sqrt{-g}\,\left[2\,R[g]+\frac{1}{2}\partial^{\mu}\,\phi\partial_{\mu}\phi-2\,\Lambda\,e^{-a\,\phi}\right]+\,\mathcal{T}\,\int_{WV_{D-1}}d^{D-1}\xi\,\mathcal{L}_{source}
\left[J^{\alpha\beta}(x),S_{-}^{\alpha^{\prime}\beta^{\prime}}(y)\right]=-\delta(x-y)\left(\delta^{\alpha\alpha^{\prime}}S_{-}^{\beta\beta^{\prime}}(x)+\delta^{\alpha\beta^{\prime}}S_{+}^{\beta\alpha^{\prime}}(x)\right)~~,
\frac{i}2(\sigma_{\mu}{}^{\rho})_{\alpha\beta}L_{\nu\rho}{}^{\alpha\beta}\+\\mathrm{h.c.}=2\mathrm{Re}\{b_{1}^{2}\}\sum_{k=0}^{\infty}\frac{t_{\mathrm{a}i}(k)\,[\mathrm{a~i~}]_{\mu\nu}{}(k)+t_{\mathrm{b}j}(k)\,[\mathrm{b~j~}]_{\mu\nu}{}(k)}{(k!)^{2}}
M={\textstyle{\frac{1}{2}}}({\mathrm{e}}^{-\phi_{0}}|p|+{\mathrm{e}}^{\phi_{0}}|q|)\,\qquad\Sigma={\textstyle{\frac{1}{2}}}({\mathrm{e}}^{-\phi_{0}}|p|-{\mathrm{e}}^{\phi_{0}}|q|)\,\,.
\stackrel{(n-1)}{\omega}_{1}=-W_{ab}^{i_{1}\cdotsi_{n-1}}\eta^{a}C_{i_{1}\cdotsi_{n-1}}^{b}-\frac{\left(-\right)^{n}}{2}\left(M_{ab}^{c}\right)^{i_{1}\cdotsi_{n-1}}\eta^{a}\eta^{b}\eta_{ci_{1}\cdotsi_{n-1}},
\begin{array}{ll}{}&{\displaystyle(F_{1},\ldots,F_{M})=(J_{1},\ldots,J_{r+1},W_{1},\ldots,W_{t})}\\{}&{\displaystyle(\rho_{1},\ldots,\rho_{M})=\frac{1}{m}(n_{1},\ldots,n_{r+1},m-d_{1},\ldots,m-d_{t})}\\{}&{\displaystyle(\omega_{1},\ldots,\omega_{N})=\frac{1}{m}(w_{1},\ldots,w_{N})}\\{}&{\displaystyleh=m\,.}\\\end{array}
I_{\mathrm{effective}}=\frac12\intdr\,\biggl\{{\rho^{\prime}}^{2}\mpN^{2}\Bigl(\frac{1}{\rho^{2}}+\rho^{2}\Bigr)+\frac{1}{2r^{2}}(\rho^{2}-1)^{2}+\rho^{2}f^{2}+\frac{r^{2}}{2}f^{\prime2}\biggr\}
\begin{array}{rcl}{U_{a}}&{=}&{\displaystyle+[D_{m}^{2}(k_{1})D_{m}(k_{2})+D^{2}(k_{1})D(k_{2})](2+K_{12})}\\{}&{}&{\displaystyle+[D_{m}^{2}(k_{1})D(k_{2})+D^{2}(k_{1})D_{m}(k_{2})](2-K_{12})}\\{}&{}&{\displaystyle+2(N-1)[D_{m}^{2}(k_{1})+D^{2}(k_{1})]D(k_{2})}\\{}&{}&{\displaystyle+2N[D^{2}(k_{1})D_{m}(k_{2})+D^{2}(k_{1})D(k_{2})]}\\{}&{}&{\displaystyle+4N^{2}D^{2}(k_{1})D(k_{2}),}\\\end{array}
\begin{aligned}{e_{\mu}{}^{r}}&{{}=\phi^{-\alpha}\tilde{e}_{\mu}{}^{r}\,,}&{\quad\lambda}&{{}=\phi^{\alpha-\beta}\tilde{\lambda}\,,}\\{\psi_{\mu}}&{{}=\phi^{-\beta}\tilde{\psi}_{\mu}-\gammae_{\mu}{}^{r}\Gamma_{r}\lambda\,,}&{\quad\epsilon}&{{}=\phi^{-\beta}\tilde{\epsilon}\,,}\\\end{aligned}
\widetilde{W}_{i}^{T}=-{\frac{\beta\beta_{H}}{(4\pi)^{D/2+1}}}\Gamma\left(1-\fracD2\right)n_{i}m_{i}^{D-2}\left[\int_{0}^{\infty}{\frac{dt}{t^{2}}}\sum_{k=1}^{\infty}\eta_{i}^{k+1}e^{-{\frac{\beta^{2}k^{2}}{4t}}}\right]{\calA}^{H}+P_{i}(\epsilon,D)~~~.
F^{g}(u_{L},u_{R})\propto\hat{\sum_{(m,n)}}|n+mT|^{2g-4}\left(\frac{u_{L}^{+}u_{R}^{+}}{n+mT}+\frac{u_{L}^{-}u_{R}^{-}}{n+m\bar{T}}\right)^{4g-4}
\left(\frac{1}{4}(Q+2P)+\mu^{2}\right){\hat{T}}^{(\nu\rho)\sigma}+\mu^{2}\,\left(g^{0\nu}\hat{T}^{(\rho0)\sigma}+g^{0\rho}\hat{T}^{(0\nu)\sigma}\right)+\frac{1}{2}R\,\epsilon^{\nu\rho\alpha\beta}\hat{T}_{(\alpha\beta)}^{\;\;\;\;\sigma}=0
M_{Het}=\left(\begin{array}{rrr}{G^{-1}}&{-G^{-1}C}&{-G^{-1}A^{T}}\\{-C^{T}G^{-1}}&{G+C^{T}G^{-1}C+A^{T}A}&{C^{T}G^{-1}A^{T}+A^{T}}\\{-AG^{-1}}&{AG^{-1}+A}&{\Gamma+AG^{-1}A^{T}}\\\end{array}\right).
L=\frac{1}{2}\left\{-\partial_{\mu}\phi\left(1-\frac{\partial_{x}^{2}}{M^{2}}\right)\partial^{\mu}\phi-\bar{\psi}\left(1-\frac{\partial_{x}^{2}}{M^{2}}\right){\not\partial}\psi+F\left(1-\frac{\partial_{z}^{2}}{M^{2}}\right)F+2U(\phi)F-U^{\prime}(\phi)\bar{\psi}\psi\right\}
\left[T_{i}^{(+)},T_{j}^{(+)}\right]=-\sqrt{2}\epsilon_{ijk}T_{k}^{(+)}\,\qquad\left[T_{i}^{(-)},T_{j}^{(-)}\right]=-\sqrt{2}\epsilon_{ijk}T_{k}^{(-)}\,\qquad\left[T_{i}^{(+)},T_{j}^{(-)}\right]=0\.
W_{n}^{m}=\frac{n!m!}{\left[\left(\frac{n+m}2\right)!\right]^{2}|\xi|}\left|\frac{\eta}{2\xi}\right|^{|n-m|}\left[P_{\mu_{nm}}^{\left(\frac{|n-m|}2,\frac{|n-m|}2\right)}\left(\frac1{|\xi|}\right)\right]^{2},
{\calL}_{bosonic}=-\frac{1}{4l^{2}}\partial_{\mu}l\partial^{\mu}l+\frac{1}{4l^{2}}h^{\mu}h_{\mu}-G_{i{\bar{j}}}\partial_{\mu}z^{i}\partial^{\mu}{\bar{z}}^{\bar{j}}-\frac{i}{2}(G_{ij}\partialz^{j}-G_{i{\bar{j}}}\partial_{\mu}z^{j}-G_{i{\bar{j}}}\partial_{\mu}{\bar{z}}^{\bar{j}})h^{\mu}.
\Lambda^{(j)2F}=\Lambda^{2kF-F}\frac{(hk(v\omega^{j-1})^{k-1})^{F}}{(k(v\omega^{j-1})^{(k-1)})^{2F}}={\frac{\Lambda^{2kF-F}h^{F}}{k^{F}(v\omega^{j-1})^{(k-1)F}}}
\frac{1}{(1+|u|^{2})^{s+m+1}}=\frac{1}{(1+|u|^{2})^{j+1}}\sum_{t=0}^{j-s-m}\frac{(j-s-m)!}{t!(j-s-m-t)!}|u|^{2j-2s-2m-2t}.
C_{4}=4L^{4}\frac{q}{3!}\left(\psi\,\omega^{2}-\frac{1}{3}\,A\wedge\eta\wedge\omega\right)-\frac{r^{4}}{L^{4}}d^{4}x+\frac{q}{9}L^{3}(\star_{5}dA)\wedge\eta\.
\rho_{n}^{\left(+\right)}=\rho^{\left(-\right)}{}_{n}+\frac{iT}6\nabla\cdot\left[\varphi_{n}^{*}\overleftrightarrow{\nabla}\varphi_{n+1}+\varphi_{n+1}^{*}\overleftrightarrow{\nabla}\varphi_{n}\right]
F^{a{\mu}_{1}{\cdots}{\mu}_{p+1}}=\frac{(-1)^{p}}{(p+1)!}{\Gamma}^{ab}{\epsilon}^{{\mu}_{1}{\cdots}{\mu}_{p+1}{\nu}_{1}{\cdots}{\nu}_{p+1}}G_{{\nu}_{1}{\cdots}{\nu}_{p+1}}^{b}.
{\calV}^{(0)}(\Phi)={\frac{1}{2}}I_{0}(\mu^{2})-{\frac{1}{2}}I_{0}({\calM}^{2})+{\frac{{\calM}^{2}}{4}}I_{1}({\calM}^{2})+{\frac{m^{2}}{\beta^{2}}}\;.
\frac{d^{2}\eta_{0}}{d\zeta^{2}}+(-\frac{1}{\zeta+1}+\frac{1}{\zeta+2}+\frac{1}{\zeta})\frac{d\eta_{0}}{d\zeta}-\frac{m_{0}^{2}}{8b^{2}}(-\frac{2}{\zeta+1}+\frac{1}{\zeta+2}+\frac{1}{\zeta})\eta_{0}=0
{\hat{M}}=-\frac{1}{2\pigP}\frac{1}{R^{\prime}}\frac{d}{dz}\frac{R{R^{\prime}}^{2}(1+D^{2}{R^{\prime}}^{2})}{(1+{R^{\prime}}^{2})^{3/2}}\frac{d}{dz}\frac{1}{R^{\prime}}.
U=\left(\begin{array}{cc}{\mathrm{e}^{i\alpha}}&{0}\\{0}&{\mathrm{e}^{i\beta}}\\\end{array}\right),\qquadV=-\sigma_{2}\left(\begin{array}{cc}{\mathrm{e}^{-i\alpha}}&{0}\\{0}&{\mathrm{e}^{-i\beta}}\\\end{array}\right).
\tilde{N}=\left(\begin{array}{cccc}{\hat{N}^{11}(x,x)}&{0}&{0}&{0}\\{0}&{\hat{N}^{22}(x,x)}&{\hat{N}^{23}(x,y)}&{0}\\{0}&{\hat{N}^{32}(y,x)}&{\hat{N}^{33}(y,y)}&{0}\\{0}&{0}&{0}&{\hat{N}^{11}(y,y)}\\\end{array}\right),
\Gamma_{\alpha\betaa}\psi^{*\alpha}\psi^{\beta}\sim\Gamma_{b\bar{c}a}\lambda^{\bar{c}}\lambda^{b}+k_{\bar{I}ab}\lambda^{\bar{I}}\lambda^{a}+a_{I\bar{J}a}\lambda^{\bar{J}}\lambda^{I}+a^{\prime}{}_{\bar{I}^{\prime}J^{\prime}a}\nu^{J^{\prime}}\nu^{\bar{I}^{\prime}}
\mathrm{Re}\,\chi_{T}^{\mathrm{(TM)}}(\omega)\approx\frac{\hbarA}{4\pi^{2}}\left(\frac{k_{B}T}{\hbar}\right)^{5}\Omega^{5}e^{-\Omega/2}\int_{-1}^{1}ds\frac{e^{-\Omegas^{2}/2}}{\sqrt{1-s^{2}}}\int_{-\infty}^{\infty}dt\frac{1+t^{2}-s^{2}}{\sqrt{1+t^{2}}}\left[(1+t^{2})(s^{2}-1/2)-s^{2}/2\right]^{2}e^{-\Omegat^{2}/2}.
{\calD}_{C}^{\;\;A}\;\nu_{(\lambda)}^{C}=-{\frac{1}{2}}{\frac{\partial}{\partial\tau}}\nu_{(\lambda)}^{A}-{_{e}n_{C^{\prime}}^{\;\;\;(C}}\;e^{A)C^{\prime}i}\;{}^{(4)}\nabla_{i}{\nu_{C}}_{(\lambda)}\;,
S^{reg}[{\mathcalA}]\;=\;\frac{\kappa}{2}\langleA_{\mu}|u(\frac{\partial^{2}}{M^{2}})R_{\mu\nu}|A_{\nu}\rangle\,+\,\frac{1}{4}\langleF_{\mu\nu}|v(\frac{\partial^{2}}{M^{2}})|F_{\mu\nu}\rangle\;,
E_{\\nu}^{\mu}=K_{\\nu,y}^{\mu}-\delta_{\nu}^{\mu}K_{,y}-K_{\\lambda}^{\mu}K_{\\nu}^{\lambda}+\delta_{\nu}^{\mu}K_{\\beta}^{\alpha}K_{\\alpha}^{\beta}-{\frac{3}{l^{2}}}\delta_{\nu}^{\mu}\,
\frac{dI_{\epsilon}}{dR}(R)=-\frac{4\sqrt{\pi}}{R_{0}}\left(\frac{R^{2}+R_{0}^{2}}{R^{2}}\right)sig(R-R_{0})\Gamma\left(\frac{1}{2},\pi\epsilon\left(\frac{R^{2}-R_{0}^{2}}{RR_{0}}\right)^{2}\right)
S=\frac{1}{\kappa}\int_{\Sigma}\rho^{\mu}d\chi_{\mu}+Q\int_{\Sigma}\Bigl(tg_{\mu\bar{\mu}}\rho^{\mu}*d\bar{z}^{\bar{\mu}}-\frac{1}{2\kappa}\Gamma_{\mu\nu}^{\lambda}\rho^{\mu}\rho^{\nu}\chi_{\lambda}\Bigr).
\sigma_{1}=\left(\begin{array}{cc}{{0}}&{{1}}\\{{1}}&{{0}}\\\end{array}\right)\quad,\quad\sigma_{2}=\left(\begin{array}{cc}{{0}}&{{-i}}\\{{i}}&{{0}}\\\end{array}\right)\quad,\quad\sigma_{3}=\left(\begin{array}{cc}{{1}}&{{0}}\\{{0}}&{{-1}}\\\end{array}\right)\quad.
\psi_{,TT}^{0}-\psi_{,\sigma\sigma}^{0}+\frac{2\alpha}{1+\alpha}(T+\tilde{\Lambda})^{-1}\psi_{,T}^{0}-\frac{\alpha}{(1+\alpha)^{2}}(T+\tilde{\Lambda})^{-2}\psi^{0}=0,
\begin{array}{c}{h^{m(s)}=\varphi^{m(s)}+\eta^{m(2)}\,\chi^{m(s-2)}\;;\;{\varphi_{l}}^{lm(s-2)}=0,}\\{\epsilon^{m(s-1)}=\varepsilon^{m(s)}+\eta^{m(2)}\,\zeta^{m(s-3)}\;;\;{\varepsilon_{l}}^{lm(s-3)}=0,}\\\end{array}
v_{i}^{CM}=\dot{x}_{i}^{CM}(t)=\frac{\displaystyle{\int_{-\infty}^{\infty}dx~J_{i}(x,t)}}{N_{i}},\\\\\\mathrm{with}\\N_{i}\equiv\int_{-\infty}^{\infty}dx~\rho_{i}^{2}(x,t).
\frac{d^{2}f}{dr^{2}}+\frac{1}{r^{2}\Lambda}\frac{d\left(r^{2}\lambda\right)}{dr}\frac{df}{dr}-\frac{1}{\Lambda^{2}}\left[\frac{l(l+1)}{r^{2}}-E^{2}\right]~f~=~0~.
\frac{\bar{F}_{a}(a,b)}{A}=\frac{\bar{F}(a)}{A}-\bar{P}_{a}(a,b,a)=\frac{\eta^{2}}{24\pia^{4}}(c_{1}\Lambda+c_{2}\Lambda^{\prime})-\frac{\eta^{2}\Lambda^{\prime}}{3}P_{a}(a,b,a),
\int_{0}^{\infty}dx\,J_{1}(\mux)\Bigl(q\frac{\partial}{\partialq}F\Bigr)\biggl|_{2q=-x^{2}}=\frac{1}{\mu}\Bigl(q\frac{\partial}{\partialq}F\Bigr)\biggl|_{q=0}+\frac{1}{2}\mathrm{~Re~}\int_{-\infty}^{\infty}dx\,H_{1}^{(1)}(\mux+\mathrm{i}\epsilon)\Bigl(q\frac{\partial}{\partialq}F\Bigr)\biggl|_{2q=-x^{2}}
{\calI}_{8}=-\frac{1}{16}\left[\mathrm{tr}R^{2}-2(\mathrm{tr}F_{N_{+}}^{2}-\mathrm{tr}F_{N_{-}}^{2})\right]\left[\mathrm{tr}R^{2}-2(\mathrm{tr}F_{D_{+}}^{2}-\mathrm{tr}F_{D_{-}}^{2})\right]
\Psi_{_{(2)}}^{-1}=\left(\begin{array}{cc}{(\Psi_{_{(2)}}^{-1})_{_{11}}}&{(\Psi_{_{(2)}}^{-1})_{_{12}}}\\{-(\Psi_{_{(2)}}^{-1})_{_{12}}^{^{T}}}&{(\Psi_{_{(2)}}^{-1})_{_{22}}}\\\end{array}\right)
\left\langle0\right|F_{n}^{12}(t)F_{n^{\prime}}^{12}(t^{\prime})\left|0\right\rangle=\frac{1}{2\pi}\intd\thetad\theta^{\prime}e^{-in\theta}e^{-in^{\prime}\theta^{\prime}}\left\langle0\right|F^{12}(\vec{t}
\langle\hat{\phi}(x)\hat{\phi}(x^{\prime})\rangle_{\beta}=\int_{0}^{\infty}d\omega\intd^{2}k\left[n_{\omega}(\beta)~U_{\omega,k}^{*}(x)U_{\omega,k}(x^{\prime})+(n_{\omega}(\beta)+1)U_{\omega,k}(x)U_{\omega,k}^{*}(x^{\prime})\right]~,
\langleT_{zz}(sz,s\bar{z})T_{zz}(0,0)\rangle\rightarrow\frac{1}{2s^{4}z^{4}}\int_{0}^{\infty}d\mu^{2}\,\tilde{c}(\mu^{2})=\frac{c_{UV}}{2s^{4}z^{4}}\\\mathrm{for}\s\rightarrow0.
t_{x}(e^{{\mathrmi}k_{x}}\psi_{n+1}+e^{-{\mathrmi}k_{x}}\psi_{n-1})+t_{y}(e^{{\mathrmi}(k_{y}+n\phi)}+e^{-{\mathrmi}(k_{y}+n\phi)})\psi_{n}=E\psi_{n}
A=\left(\begin{array}{cc}{\frac{d\tau}{2\tau}}&{{\calO}(1/\tau)}\\{}&{}\\{\frac{i\tau}{l}dz}&{-\frac{d\tau}{2\tau}}\\\end{array}\right)\,,\qquad\bar{A}=\left(\begin{array}{cc}{-\frac{d\tau}{2\tau}}&{-\frac{i\tau}{l}d\bar{z}}\\{}&{}\\{{\calO}(1/\tau)}&{\frac{d\tau}{2\tau}}\\\end{array}\right)\,.
\delta_{\epsilon}\lambda=\left(\not\!\partial\phi+{\textstyle\frac{1}{2\cdot3!}}\Gamma_{11}\not\!\!H\right)\epsilon+{\textstyle\frac{5i}{4}}me^{\phi}\epsilon-{\textstyle\frac{3i}{8}}e^{\phi}\not\!G^{(2)}\Gamma_{11}\epsilon\,,
\sigma=\sigma_{0}f(\omega,T_{L},T_{R})=\pi^{3}r_{1}^{2}r_{5}^{2}\omega\frac{e^{\frac{\omega}{T_{H}}}-1}{\left(e^{\frac{\omega}{2T_{L}}}-1\right)\left(e^{\frac{\omega}{2T_{R}}}-1\right)}\,,
\overline{{{\cal{B}}_{(\eta)i}^{\beta}}}({\bf{r}})=\{a_{i}^{\beta}({\bf{r}})+\,[\delta_{ij}-{\textstyle\frac{\eta}{\eta+1}}{\textstyle\frac{\partial_{i}\partial_{j}}{\partial^{2}}}]\overline{{{\cal{A}}_{j}^{\beta}}}({\bf{r}})\,\}\;,
\frac{\partial}{\partialt}\tilde{m}_{A}^{2}=(-2+\tilde{\eta}_{F})\tilde{m}_{A}^{2}-\frac{N_{c}}{8\pi^{2}}g_{k}^{2}\left(\frac{9}{4}l_{A,1}^{4}-\frac{5}{4}l_{A,2}^{6}\right)
^{+}{\calS}_{2}=i{^+\tau}\int_{M}eF^{\mu\nui}F^{\alpha\betaj}g_{ij}\,R_{\mu\nu}^{ab}\,\Sigma_{\alpha\beta}^{cd}\,\epsilon_{abcd}-{^+\tau}\int_{M}eF^{\mu\nui}F^{\alpha\betaj}g_{ij}\,R_{\mu\nu}^{ab}\,\Sigma_{\alpha\beta}^{cd}\,\delta_{ab,cd}\,,
2T_{0}^{0}=w\Phi^{2}+\frac{{F}}{r^{2}}e^{-2\Phi}+\frac{4\beta}{r^{2}}\left\{2w(w-1)(\Phi^{\prime\prime}-2\Phi^{2})+\Phi^{\prime}w^{\prime}(3w-1)\right\}e^{-2\Phi}\;.
\left\{i\frac{U_{n}^{-1}-U_{n}}{2T}+\frac{_1}{^6}\overrightarrow{H}\left(U_{n}+4+U_{n}^{-1}\right)\right\}S_{F}^{n}\left(\mathbf{x}\right)=-\frac{\delta_{n}}T\gamma^{0}\delta^{3}\left(\mathbf{x}\right).
\begin{array}{cl}{(w+ie^{-\phi},M_{s}^{-1}R)}&{\mathrm{for~IIA}[S^{1}]}\\{(\widetilde{\varphi}+ie^{-\phi},2M_{s}R^{-1})}&{\mathrm{for~IIB}[S^{1}]}\\{(\tau_{\mathrm{cplx.}},1/\tilde{g}^{2})}&{\mathrm{for~M(atrix)}[\widetilde{T}^{2}],}\\\end{array}
\gamma_{2}=\frac{1}{2}\left\langle\sum_{X_{i}\in({\itG}_{1})_{X_{i^{\prime}}^{\prime}}}\sum_{\stackrel{i_{\alpha_{1}}<j_{\alpha_{1}}<i_{\alpha_{2}}<j_{\alpha_{2}}}{\left\{i_{\alpha_{1}},j_{\alpha_{1}},i_{\alpha_{2}},j_{\alpha_{2}}\right\}\inJ_{X_{i}}}}t_{i_{\alpha_{1}}}t_{j_{\alpha_{1}}}{\bf1}_{(w(t_{i_{\alpha_{1}}})=w(t_{j_{\alpha_{1}}}))}\times\right.
<{\calO}_{1}\cdots{\calO}_{k}\,\mathrm{exp}\left\{\sum_{k=0}^{4}\sum_{p=4-k}^{4n}\sum_{\beta_{k}=1}^{b^{k}({\calM})}\sum_{\alpha_{p}=1}^{b^{p}({\calN})}s_{\alpha_{p},\beta_{k}}^{p,k}{\calD}_{p,k}^{\alpha_{p},\beta_{k}}\right\}>.
{\calL}_{\mathrm{matter}}=\frac{1}{4}\sum_{i}\int\!\mathrm{d}^{2}\theta\mathrm{d}^{2}\bar{\theta}\,\bar{Q}_{i}e^{V}Q_{i}+\frac{1}{2}\left\{\int\!\mathrm{d}^{2}\theta\,{\calW}(Q_{i})+\mathrm{H.c.}\right\}\,.
L_{R}=\sum_{i}\int_{V}d^{3}{\bfx}e^{3Ht}\frac{1}{2}\left[(\partial_{0}\eta_{i}({\bfx},t))^{2}-(e^{-Ht}{\bf\nabla}\eta_{i}({\bfx},t))^{2}-\mu_{i}^{2}\eta_{i}^{2}({\bfx},t)\right].
\begin{aligned}{H_{Bosonic}=}&{{}\frac{1}{2}P^{0m}P_{m}^{0}+\frac{1}{2}(P^{Am}P_{m}^{-A})+\frac{1}{4}[X^{m},X^{n}]^{2}+\frac{1}{2}(\widetilde{\mathcal{D}}_{r}X^{mA})^{2}+}\\{}&{{}\frac{1}{8}n^{2}+\frac{1}{2}\Pi^{r0}\Pi_{r}^{-0}+\frac{1}{2}\Pi^{rA}\Pi_{r}^{-A}+\frac{1}{4}(\mathcal{F}_{rs}^{A})^{2}}\\{}&{{}+\Lambda^{A}([X^{m},P_{m}]+\widetilde{\mathcal{D}}_{r}\Pi_{r})^{-A}}\\\end{aligned}
m^{2}\sim{\frac{1}{r_{*d}^{2}}}\sim\left({\frac{V_{*}}{V}}g_{s}N\right)^{-{2/3}}{\frac{1}{\alpha^{\prime}}}=\left(\left({\frac{V_{*}}{V}}\right)^{3/4}\mu\right)^{-{2/3}}{\frac{1}{\alpha^{\prime}}}.
P_{Y}^{(6)...(i_{1}i_{2})...}=\frac{1}{2}(\delta_{i_{1}i_{1}^{\prime}}\delta_{i_{2}i_{2}^{\prime}}+\delta_{i_{1}i_{2}^{\prime}}\delta_{i_{2}i_{1}^{\prime}})P_{Y}^{(6)...i_{1}^{\prime}i_{2}^{\prime}...}
S_{\mathrm{gf}}+S_{\mathrm{ghost}}=\intd^{n}x\,\left(-{\frac{1}{2\xi}}(\partial_{\mu}A^{\mu})\star(\partial_{\nu}A^{\nu})+\partial^{\mu}\overline{{c}}\star(\partial_{\mu}c-ie[A_{\mu},c]_{\mathrm{MB}})\right)
\left({\frac{\dot{a}}{a}}\right)^{2}={\frac{8\pi}{3}}{\frac{B}{a^{3w+3}}}+{\frac{32(w+1)\pi^{2}}{3(3w+2{\sfq}-6{\sfn}+3)}}{\frac{\tilde{G}_{5}^{2}\sigma_{0}\delta\sigma}{b_{0}c_{0}^{6}}}a^{2{\sfq}-6{\sfn}}-{\frac{3w+2{\sfn}-1}{3w+2{\sfn}+1}}kc_{0}^{2}a^{2{\sfn}-2},
B^{[A_{\mu}\,]\,1}\,,\,\,B^{[c\,]\,1}\,,\,\,B^{[\overline{c}\,]\,1}\,,\,\,B^{[G\,]\,1}\,\,;\,\,\lambda^{[A_{\mu}\,]\,1}\,,\,\,\lambda^{[c\,]\,1}\,,\,\,\lambda^{[\overline{c}\,]\,1}\,,\,\,\lambda^{[G\,]\,1}\,\,.\nonumber
S=\frac{1}{4g_{c}^{2}}\intdx^{2}\mathrm{Tr}\left[\left(F_{\mu\nu}-\frac{m}{2\piR_{c}^{2}N}\epsilon_{\mu\nu}I\!\!I\right)\left(F^{\mu\nu}-\frac{m}{2\piR_{c}^{2}N}\epsilon^{\mu\nu}I\!\!I\right)\right],
{\cal{C}}={\bf\mathrm{1}}\Big(\eta^{(0)}+\eta^{(2)}\Big)+{\bf\mathrm{i}}\eta^{(1)}+{\bf\mathrm{j}}\Big(\widetilde{C}^{(0)}+\widetilde{C}^{(2)}\Big)+{\bf\mathrm{k}}\widetilde{C}^{(1)}\qquad\in\Lambda_{+}.
\zeta_{en}(-1/\tau)=\intde^{\prime}dn^{\prime}\\frac{i}{k}\\mathrm{exp}-\frac{2i\pi}{k}[e^{\prime}(\tilde{n}-1/2)+e(\widetilde{n^{\prime}}-1/2)]\\zeta_{e^{\prime}n^{\prime}}(\tau)
M=4\pi\int_{0}^{\infty}dr~r^{2}\{\frac{F_{\pi}^{2}}{8}[F^{2}+2\frac{sin^{2}F}{r^{2}}]+\frac{1}{2e^{2}}~\frac{sin^{2}F}{r^{2}}~[\frac{sin^{2}F}{r^{2}}+2F^{2}]\}
p_{2}=\int_{0}^{\infty}\,\frac{d\omega}{2\pi}\int_{0}^{\infty}\,\frac{d\omega^{\prime\prime}}{2\pi}\vert\langleout\,\omega\omega^{\prime\prime}\vertin\rangle\vert^{2}=\int_{0}^{\infty}\,\frac{d\omega}{2\pi}p_{2\omega}.
G_{NP}^{\prime}(\sigma_{i},\sigma_{j})=-{\frac{2\alpha^{\prime}}{1+b^{2}}}\mathrm{ln}|{\frac{\Theta_{2}({\frac{il\sigma_{ij}}{2}},{\frac{il}{2}})}{\eta({\frac{il}{2}})}}|^{2}+{\frac{2\pi\alpha^{\prime}l}{1+b^{2}}}\left[(\sigma_{ij})^{2}-{\frac{b^{2}}{3l^{2}}}\right]\quad\quad\sigma_{+}^{2}=1\quad.
\theta_{\alpha}^{i}=u^{+i}\theta_{\alpha}^{--}u^{-i}\theta_{\alpha}^{+}\,\qquad\theta_{\alpha}^{+}=\theta_{\alpha}^{i}u_{i}^{+}\,\qquad\theta_{\alpha}^{-}=\theta_{\alpha}^{i}u_{i}^{-}
\{x^{i},x^{j}\}_{1}=0,\quad\{x^{i},\theta^{j}\}_{1}=-\{\theta^{j},x^{i}\}_{1}=\omega^{ij},\quad\{\theta^{i},\theta^{j}\}_{1}=-\{\theta^{j},\theta^{i}\}_{1}=\frac{\partial\omega^{ij}}{\partialx^{k}}\theta^{k}
L_{0}~=~\sum_{i}^{N}(K_{i}^{\dagger}K_{i}+1)\Phi_{i}^{\dagger}\Phi_{i}\vert_{D}~+~\lambda_{i}\Phi_{i}\vert_{F}+m_{ij}\Phi_{i}\Phi_{j}\vert_{F}+g_{ijk}\Phi_{i}\Phi_{j}\Phi_{k}\vert_{F},
\left(Y^{\mu}\right)(\tau,\sigma^{1},\sigma^{2})=\left(\begin{array}{c}{\tau}\\{Y^{1}(\tau,\sigma^{1},\sigma^{2})}\\{Y^{2}(\tau,\sigma^{1},\sigma^{2})}\\{Y^{3}(\tau,\sigma^{1},\sigma^{2})}\\\end{array}\right),
F_{\mathrm{div}}^{E}[g,\beta,D]=-\eta{\frac{\Gamma\left(1-\fracD2\right)}{(4\pi)^{D/2}}}{\frac{\pi^{2}m^{D-4}}{3\kappa\beta^{2}}}\int_{\Sigma}\left[f_{1}m^{2}-\left(p_{1}{\frac{4\pi^{2}}{\kappa^{2}\beta^{2}}}P+p_{2}R+p_{3}R_{ii}\right)\right]~~~,
R_{\mathrm{massive}}^{\mathrm{boundary}}(r)=A_{\mathrm{out}}\frac{1}{r^{\beta}}\left(1-\frac{i}{\pi}\left(\frac{2r_{H}}{r}\right)^{1-2\beta}\right)+A_{\mathrm{in}}\frac{1}{r^{\beta}}\left(1+\frac{i}{\pi}\left(\frac{2r_{H}}{r}\right)^{1-2\beta}\right),
ds^{2}=-(\frac{r^{2}}{l^{2}}-M+\frac{J^{2}}{4r^{2}})dt^{2}+\frac{dr^{2}}{\frac{r^{2}}{l^{2}}-M+\frac{J^{2}}{4r^{2}}}+r^{2}(d\varphi-\frac{J}{2r^{2}}dt)^{2}\,,
\tilde{T}_{i}=z_{i}{\frac{\partial}{\partialz_{i}}}+{a}\sum_{j<i}\,{\frac{z_{i}}{z_{i}-z_{j}}}\,(1-K_{ij})+{a}\sum_{j>i}\,{\frac{z_{j}}{z_{i}-z_{j}}}\,(1-K_{ij})+1-i,
\left[\mathcal{S}^{\mu\nu},\mathcal{S}^{\rho\sigma}\right]=g^{\nu\rho}\mathcal{S}^{\mu\sigma}-g^{\nu\sigma}\mathcal{S}^{\mu\rho}+g^{\mu\sigma}\mathcal{S}^{\nu\rho}-g^{\mu\rho}\mathcal{S}^{\nu\sigma}.
S_{\mathrm{loc};\,0}[e^{\sigma(x)}G(x),\phi(x)]-S_{\mathrm{loc};\,0}[G(x),\phi(x)]=\intd^{d}x\,\sigma(x)\,G_{\mu\nu}\,{\frac{\deltaS_{\mathrm{loc};0}}{\deltaG_{\mu\nu}}}.
S=\int\sqrt{\mathfrak{a}^{2}-g^{2}}d\tau,\qquad\textrm{where}d\tau^{2}=dx^{\mu}dx^{\mu},\quadg^{2}=-\frac{d^{2}x^{\mu}}{d\tau^{2}}\frac{d^{2}x_{\mu}}{d\tau^{2}}
{\calY}(T,U)\stackrel{{\Gamma^{o}(3)}_{T}}{\rightarrow}(i\gammaT+\delta)\;{\calY}(T,U)-i\gamma\{(\partial_{\frac{U}{3}}\eta^{-2}(\frac{U}{3})\;\eta^{-2}(T)+(\partial_{U}\eta^{-2}(U))\;\eta^{-2}(\frac{T}{3}))\}.
\omega^{2}-\omega_{0}^{2}=-\rho^{2}\frac{4\pi\tilde{G}}{3\omega_{0}^{2}}V^{(3)}(\phi_{0})g_{1,\xi}-\rho^{4}2\pi\tilde{G}\left[\frac{V^{(4)}(\phi_{0})}{3\omega_{0}^{2}}+2\omega_{0}^{2}\right]g_{1,\xi}^{2}
\zeta_{\epsilon}(\nu)=\mathrm{Tr}\left({\frac{P_{\epsilon}}{\mu^{2}}}\right)^{-\nu}={\frac{2\mu^{2\nu}}{\Gamma(\nu)}}\int_{0}^{\infty}{\frac{d\xi}{\xi}}\xi^{2\nu}\mathrm{Tr}\left[e^{-\xi^{2}P_{\epsilon}}\right].
\Gamma_{\beta\gamma}^{\alpha}=-{\frac{1}{2}}g^{\alpha\lambda}(g_{\lambda\beta,\gamma}+g_{\lambda\gamma,\beta}-g_{\beta\gamma,\lambda})+\delta_{\beta}^{\alpha}\phi_{\gamma}+\delta_{\gamma}^{\alpha}\phi_{\beta}-g_{\beta\gamma}\phi^{\alpha}.
(\Pi_{0}^{(N)})^{2}-(\Pi^{(N)})^{2}=(\frac{\pi_{0}^{(N)}}{1-\frac{\pi_{0}^{(N)}}{N}})^{2}-(\frac{\pi_{i}^{(N)}}{1-\frac{\pi_{0}^{(N)}}{N}})^{2}=(\mu^{(N)})^{2}
\mu(k\eta,q)\rightarrow\sqrt{\frac{\eta_{c}}{q(\eta+2\eta_{c})}}(c_{-}e^{\frac{i}{2}q\frac{(\eta+2\eta_{c})^{2}}{\eta_{c}}}+c_{+}e^{-\frac{i}{2}q\frac{(\eta+2\eta_{c})^{2}}{\eta_{c}}})
\langle0|\theta_{\mu}^{\mu}|0\rangle=4\epsilon_{V}=-\Lambda^{4}\sum_{i}\frac{\partialf}{\partialg_{i}^{(0)}}\beta_{i}(\{g_{i}^{(0)}\})=-\sum_{i}\frac{\partial\epsilon_{V}}{\partialg_{i}^{(0)}}\beta_{i}(\{g_{i}^{(0)}\}).
\beta=\left(\frac{\beta_{1}-\beta_{0}e^{k_{2}y_{1}}}{1-e^{(k_{2}-k_{1})y_{1}}}\right)\;e^{k_{1}(y-y_{1})}\;+\;\left(\frac{\beta_{0}-\beta_{1}e^{-k_{1}y_{1}}}{1-e^{(k_{2}-k_{1})y_{1}}}\right)\;e^{k_{2}y}\;,
\left\{\phi_{a}^{(1)},\phi^{(1)}\right\}=\left\{\phi_{a}^{(1)},T\right\}=\left.\left\{T,\phi_{3,n}^{(1)}\right\}\right|_{\phi=T=0}=\left.\left\{T,T\right\}\right|_{\phi=T=0}=0\;,\;\;a=1,2\,.
\psi_{m}(z)=c_{1}{\frac{\sqrt{{\calZ}-z}}{z}}J_{{\frac{1}{2}}\sqrt{1+8K}}[\sqrt{2}m({\calZ}-z)]+c_{2}{\frac{\sqrt{{\calZ}-z}}{z}}Y_{{\frac{1}{2}}\sqrt{1+8K}}[\sqrt{2}m({\calZ}-z)]\,,
\dot{\epsilon}_{\delta\hat{\sigma}}=\epsilon_{\delta\hat{\sigma}}\left[(1+\epsilon_{\delta\hat{\sigma}}){\frac{\dot{\hat{G}}_{4}}{\hat{G}_{4}}}+\left({\frac{3\hat{\calC}c_{0}^{2}}{4\pi\hat{G}_{4}\hat{a}_{0}^{4}}}-5-3w-2\epsilon_{\delta\hat{\sigma}}\right){\frac{\dot{\varepsilon}}{\varepsilon}}\right].
ds^{2}=\sqrt{a_{1}a_{2}b}c^{2}\left(dr^{2}+\frac{1}{a_{1}^{2}}\sigma_{1}^{2}+\frac{1}{a_{2}^{2}}\sigma_{2}^{2}+\frac{1}{b^{2}}\sigma_{3}^{2}+\frac{1}{c^{2}}(\sigma_{4}^{2}+\sigma_{5}^{2}+\sigma_{6}^{2}+\sigma_{7}^{2})\right).
\Gamma_{\eta}=2\pi\sum_{b=b_{\eta}}\int_{0}^{\infty}d{E}\,|\langleb,E|V|\psi^{G}\rangle|^{2}\,\frac{\frac{\Gamma_{R}}{2\pi}}{(E-E_{R})^{2}+(\frac{\Gamma_{R}}{2})^{2}}
\begin{array}{ccccccc}{124+}&{125+}&{126-}&{127+}&{134-}&{135+}&{136-}\\{137-}&{145+}&{146-}&{157-}&{167+}&{234-}&{235+}\\{236+}&{237+}&{245+}&{247+}&{256+}&{267-}&{346-}\\{347-}&{356+}&{357+}&{456+}&{457+}&{467-}&{567+.}\\\end{array}
P_{L}^{\prime}=({\rho}\frac{\breve{m}}{2}+(G_{\bot}-B_{\bot}-\frac{1}{4}A_{\bot}^{t}C_{\bot}A_{\bot}){\breve{n}}-\frac{1}{2}A_{\bot}^{t}C_{\bot}{\breve{l}},\;{\breve{l}}+A_{\bot}{\breve{n}})=(p_{L}^{\prime},\;p_{R}^{\prime})
V_{3,i}=(2\pi)^{3}r_{m_{i}}r_{m_{i}^{\prime}}r_{m_{i}^{\prime\prime}}\,\quad\mathbf{\omega}_{{\mathbf3},i}=\frac{1}{V_{3,i}}\mathbf{dy}^{m_{i}}\wedge\mathbf{dy}^{m_{i}^{\prime}}\wedge\mathbf{dy}^{m_{i}^{\prime\prime}}\.
\kappa_{\mathrm{ind}}^{(T)}=\frac{N_{f}}{2}\,T\,\sum_{n=-\infty}^{\infty}\;\int{\frac{d^{2}k}{(2\pi)^{2}}}{\frac{2m}{[((2n+1)\piT)^{2}+\vec{k}^{2}+m^{2}]^{2}}}
\frac{a^{\prime}}{a}(x^{\perp}\to\pm0,r)=-\frac{n^{\prime}}{n}(x^{\perp}\to\pm0,r)=\pm\frac{m_{3}^{2}}{2m_{4}^{3}}\left[\frac{1}{a^{2}r}\left(\frac{\check{a}}{a}+\frac{\check{n}}{n}\right)\right]_{x^{\perp}=0}.
{\calU}_{k}(x)={\calP}\left\{U_{k}(x)U_{k}(x+\hat{k})+\alpha_{f}\sum_{\pm(j\neqk)}U_{j}(x)U_{k}(x+\hat{j})U_{k}(x+\hat{j}+\hat{k})U_{j}^{\dagger}(x+2\hat{k})\right\}.
e^{i\Gamma[V^{++}]}=e^{iS_{SYM}[V^{++}]}\int{\calD}v^{++}{\calD}b{\calD}c{\calD}\phi{\calD}\stackrel{\smile}{q}^{+}{\calD}q^{+}{\calD}\omega\mathrm{Det}^{1/2}(\stackrel{\frown}{\Box})e^{iS_{total}}
E_{\varepsilon}(t)=\intd^{3}\vec{x}\,\left\{\frac{\hbarc}{2\lambda^{2}}\left[\frac{1}{c^{2}}\dot{\varepsilon}^{2}(x)+\left(\vec{\partial}\,\varepsilon(x)\right)^{2}-\frac{1}{\lambda^{2}}{\varepsilon}^{2}(x)\right]+g_{\varepsilon}\hbar\,\partial_{\mu}j^{\mu}(x)\varepsilon(x)\right\}\,,
\langle\langle\prod_{i=1}^{L}\intV_{n_{i}}\rangle\rangle=\mu^{S}\Gamma(-S)\langlec{\bar{c}}V_{0}(0)c{\bar{c}}V_{0}(1)c{\bar{c}}V_{k-1}(\infty)(\intV_{k})^{L+S-3}{\frac{1}{R!}}(Q_{-})^{R}\rangle