harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\phi_{\mu\nu}(n)=2\xi_{n}\bar{\xi}_{n}(\varepsilon_{\mu}^{n}p_{n,\nu}-\varepsilon_{\nu}^{n}p_{n,\mu})+\frac{4}{T}\xi_{n+1}\bar{\xi}_{n+1}\xi_{n}\bar{\xi}_{n}(\varepsilon_{\mu}^{n+1}\varepsilon_{\nu}^{n}-\varepsilon_{\nu}^{n+1}\varepsilon_{\mu}^{n})\delta(t_{n+1}-t_{n})
I(k^{+},k^{-})=-{\frac{\lambda^{2}}{96\pi^{2}}}\int_{0}^{k^{+}}{\frac{dk_{1}^{+}}{k_{1}^{+}}}\int_{0}^{k_{1}^{+}-k^{+}}{\frac{dk_{2}^{+}}{k_{2}^{+}}}{\frac{f^{2}(k_{1}^{+})f^{2}(k_{2}^{+})}{(k^{+}-k_{1}^{+}-k_{2}^{+})(k^{--}{\frac{\mu^{2}}{k^{+}-k_{1}^{+}-k_{2}^{+}}}-{\frac{\mu^{2}}{k_{1}^{+}}}-{\frac{\mu^{2}}{k_{2}^{+}}})}}.
\left(\lambda_{\alpha\beta}^{g_{1}g_{2},g_{3}}\right)\left(g_{3}^{*}\lambda_{\alpha\beta}^{g_{1},g_{2}}\right)\:=\:\left(\lambda_{\alpha\beta}^{g_{1},g_{2}g_{3}}\right)\left(\lambda_{\alpha\beta}^{g_{2},g_{3}}\right)\left(\gamma_{\alpha}^{g_{1},g_{2},g_{3}}\right)\left(\gamma_{\beta}^{g_{1},g_{2},g_{3}}\right)^{-1}
[-{\frac{\delta_{\rho}}{k^{-}}}+2\delta_{c}]{\frac{v(X)}{X}}=[{\frac{\delta_{a}}{(k^{-})^{2}}}-2{\frac{\delta_{\rho}}{k^{-}}}+\delta_{c}(3-{\frac{k^{4}}{(k^{-})^{4}}})]{\frac{\partialv(X)}{\partialX}},
\omega={\frac{e^{2}}{\theta}}\sum_{a,b=1}^{N}\int_{0}^{T}dt\,{\dot{x}}_{a}^{i}\epsilon_{ij}{\frac{\partial_{a}^{j}}{\nabla_{a}^{2}}}\biggl(\delta({\bfx}_{a}(t)-{\bfx}_{b}(t))-\delta({\bfx}_{a}(t)-{\bfx}_{b}(0))\biggr)\,,
{\calL}_{1}\rightarrow{\calL}_{2}={\calL}_{1}+{\frac{\lambda_{--}}{1-\lambda_{++}\lambda_{--}}}\left[\lambda_{--}A_{+}^{2}+\lambda_{++}A_{-}^{2}+\left(1+\lambda_{++}\lambda_{--}\right)A_{+}A_{-}\right]
\widetilde{\calH}^{\scriptscriptstyleBFV}=\Lambda_{\scriptscriptstyleA}\omega^{\scriptscriptstyleAB}\vec{\partial}_{\scriptscriptstyleB}H+i\bar{\Gamma}_{\scriptscriptstyleA}\omega^{\scriptscriptstyleAC}\stackrel{\rightarrow}{\partial}_{\scriptscriptstyleC}H\stackrel{\leftarrow}{\partial}_{\scriptscriptstyleB}\Gamma^{\scriptscriptstyleB}
\psi\left(m,z\right)=\sqrt{\left|z\right|+\frac{1}{k}}\left(c_{1}J_{2}\left(m\left(\left|z\right|+\frac{1}{k}\right)\right)+c_{2}Y_{2}\left(m\left(\left|z\right|+\frac{1}{k}\right)\right)\right),
{\calL}=-{\frac{1}{4}}e[{\calR}-{\frac{1}{2}}\partial_{\mu}\varphi\partial^{\mu}\varphi+{\frac{1}{4}}\partial_{\mu}g_{ij}\partial^{\mu}g^{ij}-{\frac{1}{4}}e^{\varphi}g_{ij}G_{\mu\nu}^{i}G^{j\,\mu\nu}+{\frac{1}{2}}e^{\varphi}g^{ik}g^{jl}F_{\mu\nu\,ij}^{4}F_{\\\\kl}^{4\,\mu\nu}+\cdots],
{\calL}^{\prime}(x)=\overline{{\psi}}^{1}(x)(i\partial\!\!\!/-m)\psi^{1}(x)+\overline{{\psi}}^{2}(x)(i\partial\!\!\!/)\psi^{2}(x)-\overline{{\psi}}^{3}(x)(i\partial\!\!\!/)\psi^{3}(x).
v^{i}(\kappa_{g})=T_{\,\,j}^{i}(g)\,v^{j}\left(\kappa\right),\quad\lambda^{ia}(\kappa_{g})=T_{\,\,j}^{i}(g)\,\,g_{\,\,\,b}^{a}\,\lambda^{jb}\left(\kappa\right).
\begin{array}{l}{Z_{M_{1}M_{2}}=\Lambda_{M_{1}}^{\,\,\,\,\,N_{1}}Z_{N_{1}N_{2}}^{0}\Lambda_{\,\,\,\,M_{2}}^{N_{2}}}\\{\,Z_{M_{1}M_{2}}^{0}=\left(\begin{array}{lll}{0}&{p_{10}}&{p_{\mu}}\\{-p_{10}}&{0}&{0}\\{-p_{\mu}}&{0}&{0}\\\end{array}\right)}\\\end{array}\,\,\,\,
\mathcal{M}=\left(\begin{array}{cc}{-i(\Phi_{a}\partial_{n}X^{a}+\frac{\alpha^{\prime}}{2}\theta_{ab}\psi^{a}\psi^{b})+u^{2}\Phi_{10}^{2}}&{\sqrt{\alpha^{\prime}}\thetau\psi^{9}}\\{\sqrt{\alpha^{\prime}}\thetau\psi^{9}}&{-i(\Phi_{a}\partial_{n}X^{a}+\frac{\alpha^{\prime}}{2}\theta_{ab}\psi^{a}\psi^{b})+u^{2}\Phi_{10}^{2}}\\\end{array}\right)\.
\pi_{AB}=\kappa_{n}^{2}\left\{{\frac{1}{2}}\sqrt{-q}\;F(\phi)\;\left(K_{AB}-q_{AB}K\right)+{\frac{1}{2}}\sqrt{-q}\;F_{i}^{\prime}(\phi)\;\left({\frac{\partial\phi^{i}}{\partial\eta}}\right)q_{AB}\right\}.
\begin{array}{rcl}{{\displaystyle\sum_{m=1-h}^{n}\left(\begin{array}{c}{n+h-1}\\{m+h-1}\\\end{array}\right)\zeta^{n-m}m^{2}}}&{=}&{{\displaystyle(\zeta+1)^{n+h-3}\left((1-2\,h+h^{2})\,\zeta^{2}\right.}}\\{}&{}&{{\displaystyle\left.+(h-1+3\,n-2\,h\,n)\,\zeta+n^{2}\right)}}\\\end{array}
\zeta\left(\begin{array}{rcr}{s_{1}}&{\ldots}&{s_{k}}\\{\sigma_{1}}&{\ldots}&{\sigma_{k}}\\\end{array}\right):=\sum_{n_{j}>n_{j+1}>0}\quad\prod_{j=1}^{k}\frac{\sigma_{j}^{n_{j}}}{n_{j}^{s_{j}}}
1=\sum_{n,n_{1}}\prod_{i}^{n}\int\,dq_{i}^{+}d^{2}q_{\perpi}\prod_{j}^{n_{1}}\int\,dr_{j}^{+}d^{2}r_{\perpj}\left|\phi^{(n,n_{1})}(\underline{{q}}_{i},\underline{{r}}_{j};\underline{{P}}-\sum_{i}\underline{{q}}_{i}-\sum_{j}\underline{{r}}_{j})\right|^{2}\,.
\left({\frac{J}{M^{2}}}\right)^{2}\leq\frac{1}{4}\left[2-10\left({\frac{Q}{M}}\right)^{2}-\left({\frac{Q}{M}}\right)^{4}+2\left(1+2\left({\frac{Q}{M}}\right)^{2}\right)^{3/2}\right].
\left[-\frac{1}{N^{2}}\,\lambda_{i}\,\frac{\partial^{2}}{\partial\lambda_{i}^{2}}-\frac{1}{N^{2}}\!\sum_{j\neqi}\lambda_{j}\,\frac{1}{\lambda_{j}-\lambda_{i}}\left(\frac{\partial}{\partial\lambda_{j}}-\frac{\partial}{\partial\lambda_{i}}\right)+\frac{\alpha+\beta-2}{N}\,\frac{\partial}{\partial\lambda_{i}}+(\beta-\alpha)+\lambda_{i}\right]\!Z(\lambda)\!=\!0.
\zeta_{\bar{\calV}}(s)=\frac{mL}{\sqrt{8\pi}}\frac{1}{\Gamma(s)}\int_{0}^{\infty}d\beta\,\beta^{s-\frac{3}{2}}e^{-4\beta}=\frac{mL}{\sqrt{8\pi}}\frac{1}{4^{s-\frac{1}{2}}}\frac{\Gamma(s-\frac{1}{2})}{\Gamma(s)}
W_{2}(k_{1})=\frac{1}{4\pi^{2}R_{c}^{2}}\int_{0}^{2\piR_{c}}dx_{1}dy_{1}<\!\!\frac{1}{N}\mathrm{Tr}\,\left[\,\Omega^{(k_{1})}(x_{1})\,\right]\frac{1}{N}\mathrm{Tr}\,\left[\,\Omega^{(-k_{1})}(y_{1})\,\right]\!\!>,
\sum_{l=1}^{N}{\frac{w_{pR(r)}(n-l)}{w_{qr}(l-m)w_{pq}(k-l)}}=\rho(pqr){\frac{w_{pR(q)}(n-m)w_{R^{-1}(q)r}(k-n)}{w_{pr}(k-m)}}
\frac{d}{dt}\left[\dot{q}_{i}^{a}-\lambda\epsilon^{ab}q_{i}^{b}\right]\,-\,\lambda\epsilon^{ab}\left[\dot{q}_{i}^{b}-\lambda\epsilon^{bc}q_{i}^{c}\right]=-g^{2}\omega^{2}q_{i}^{a}\\,\\a=1,2;\i=1,2,\cdots,d\\,
\begin{array}{rcl}{{\calL}}&{=}&{\partial_{\mu}\phi^{\dagger}\partial^{\mu}\phi-m^{2}\phi^{\dagger}\phi-V(|\phi|)}\\{}&{}&{}\\{}&{=}&{\frac{1}{2}(\partial_{\mu}\phi_{1})^{2}-\frac{1}{2}m^{2}\phi_{1}^{2}+\frac{1}{2}(\partial_{\mu}\phi_{2})^{2}-\frac{1}{2}m^{2}\phi_{2}^{2}-V\left(\sqrt{\phi_{1}^{2}+\phi_{2}^{2}}\right)\,}\\\end{array}
\hat{x}_{12}=x_{1_{A}}-x_{2_{A}}+{\frac{4i}{(12)}}[(1^{-}2)\theta_{1}^{+}\bar{\theta}_{1}^{+}+(2^{-}1)\theta_{2}^{+}\bar{\theta}_{2}^{+}+\theta_{1}^{+}\bar{\theta}_{2}^{+}+\theta_{2}^{+}\bar{\theta}_{1}^{+}]
\hat{H}=\frac{1}{2}\hat{p}_{0}^{2}+\frac{\omega_{0}^{2}}{2}\hat{q}_{0}^{2}+\frac{1}{2}\sum_{k=1}^{N}\left[\hat{p}_{k}^{2}+\omega_{k}^{2}\hat{q}_{k}^{2}+2\lambda_{k}(t)\hat{q}_{0}\hat{q}_{k}\right]\;.
\hat{H}_{I}^{(q)}(\hat{x},\hat{p})=f^{2}\sum_{k=1}^{\infty}\frac{V^{(k)}(0)}{k!}\biggl(\sum_{i=0}^{k-1}\hat{x}^{(k-1)-i}g(\hat{x},\hat{p})\hat{x}^{i}\biggr),
\left(\begin{array}{c}{m^{I}}\\{e_{I}}\\\end{array}\right)\to\left(\begin{array}{cc}{U}&{Z}\\{W}&{V}\\\end{array}\right)\left(\begin{array}{c}{m^{I}}\\{e_{I}}\\\end{array}\right)\,\qquad\left(\begin{array}{c}{\check{F}^{I}}\\{\check{G}_{I}}\\\end{array}\right)\to\left(\begin{array}{cc}{U}&{Z}\\{W}&{V}\\\end{array}\right)\left(\begin{array}{c}{\check{F}^{I}}\\{\check{G}_{I}}\\\end{array}\right)\,
\rho_{m_{0}\rightarrowm_{1}+m_{2}}^{2\ell}=\frac{\pi^{1-\ell}\Gamma(\ell-1)(\lambda(m_{0}^{2},m_{1}^{2},m_{2}^{2}))^{2\ell-3}}{2^{2\ell-1}m_{0}^{2\ell-2}\Gamma(2\ell-2)},
p^{2}\xi_{20}(p,q)-p.q\xi_{11}(p,q)={\frac{1}{2}}\left\{-\left[{\frac{1}{2}}+m^{2}\xi_{00}(p,q)\right]+{\frac{q^{2}}{2}}\xi_{01}(p,q)+{\frac{3p^{2}}{2}}\xi_{10}(p,q)\right\}
\begin{array}{rclcc}{F}&{=}&{-2\,\frac{k}{\sqrt{a^{2}+1}}\,\frac{1}{r^{3}}\left(1+\frac{k}{r}\right)^{-2}dt\,\wedge\,{\vec{x}}\,\cdot\,{\vec{x}}}&{;}&{\mathrm{electric~0~-brane}}\\{{F}}&{=}&{-2\,\frac{k}{\sqrt{a^{2}+1}}\,\frac{1}{r^{3}}\,\epsilon_{abc}\,x^{a}\,dx^{b}\,\wedge\,dx^{c}}&{;}&{\mathrm{magnetic~solitonic~0~-brane}}\\\end{array}
S^{\mathrm{HS}}[g_{(0)},\epsilon]=\intd^{d}x\left[\,d\int_{\epsilon}d\rho\,\sqrt{g}\,\rho^{-d/2-1}+\left(4\rho^{-d/2+1}\sqrt{g}^{\,\prime}-2\,d\,\rho^{-d/2}\sqrt{g}\right)\Bigg|_{\rho=\epsilon}\right].
{\calD}_{\alpha}\perp_{\mu\nu}=0,\quadu^{\mu_{i}}{\calD}_{\mu_{1}\ldots\mu_{i}\dots\mu_{n}}^{\nu_{1}\ldots\nu_{p}}=u_{\nu_{j}}{\calD}_{\mu_{1}\ldots\mu_{n}}^{\nu_{1}\ldots\nu_{j}\ldots\nu_{p}}=u^{\alpha}{\calD}_{\mu_{1}\ldots\mu_{n}}^{\nu_{1}\ldots\nu_{p}}=0.
S_{FT}=\int{\calL}dx\;,\;\;{\calL}=\sqrt{-g}\left[\left(P_{\mu}\varphi\right)^{\ast}g^{\mu\nu}P_{\nu}\varphi-m^{2}\varphi^{\ast}\varphi\right],\;\;P_{\mu}=i\partial_{\mu}-qA_{\mu}\;.
\hat{h}_{\mu\nu}^{-}=a^{2}\xi_{\mu\nu}+2a^{2}\left(\int\frac{dz}{a^{4}}\right)\partial_{\mu}\partial_{\nu}\xi-2\frac{a^{2}}{a_{-}^{2}}\left(\int\frac{dz}{a^{2}}\right)\partial_{\mu}\partial_{\nu}\xi-\frac{U_{B}}{2}\frac{a^{2}}{a_{-}^{2}}\xi\eta_{\mu\nu}
{\frac{\partial^{2}\psi}{\partialk_{y}^{2}}}\;+\;{\frac{\partial^{2}\psi}{\partial\kappa_{y}^{2}}}\;=\;0\qquad\qquad\left({\frac{\partial\;}{\partialp_{y}}}{\frac{\partial\psi}{\partial\overline{{p_{y}}}}}\;=\;0\right)
{\mathcalP}^{(N)}\left\{t^{\prime},t^{\prime\prime}\right\}=\left\{{\mathcalP}^{(N_{0})}\left\{t^{\prime},t_{1}\right\};{\mathcalP}^{(N_{1})}\left\{t_{1},t_{2}\right\};\ldots;{\mathcalP}^{(N_{n})}\left\{t_{n},t^{\prime\prime}\right\}\right\}\;,
L\mathrm{e}^{-{\beta_{H}\varepsilon}}={m_{1}}^{l}\intdx_{0}\cdotsdx_{l}\mathrm{exp}\left[\sum_{j=0}^{l}(h(x_{j}-w/r_{j})+w/r_{j}-{\beta_{H}m_{1}}(r_{j}-1)x_{j})\right]\delta\left(x-\sum_{j=0}^{l}r_{j}x_{j}\right).
Q^{\mu}\equiv-N\xi_{1}\hat{p}^{\mu}+N_{1}\xi_{1}\Pi_{1}^{\mu}-\xi_{0}\partial_{1}x^{\mu}+\frac12[(\bar{\theta}\Gamma_{\nu}\partial_{1}\theta)\bar{\theta}\Gamma^{\mu\nu}+(\bar{\theta}\Gamma^{\mu\nu}\partial_{1}\theta)\bar{\theta}\Gamma_{\nu}]\lambda_{\theta}.
\begin{array}{ll}{\phi_{mn}=\frac{1}{3!}(\varphi_{[4]})_{m\overline{{ijk}}}\epsilon^{\overline{{ijkn}}}g_{n\bar{n}}}\\{\phi_{\overline{{mn}}}=\frac{1}{3!}(\varphi_{[4]})_{\overline{{m}}ijk}\epsilon^{ijkn}g_{n\bar{n}}}\\\end{array}
{\calZ}_{-}^{(1)}(t)\equive^{{\frac{4\pii}{5}}}{\calZ}_{+}\left(e^{{\frac{2\pii}{5}}}t\right),\qquad{\calZ}_{-}^{(2)}(t)\equive^{-{\frac{4\pii}{5}}}{\calZ}_{+}\left(e^{-{\frac{2\pii}{5}}}t\right).
I=\int\mathrm{d}^{4}x\mathrm{d}^{4}yh_{\mu\nu}(x)S_{sub}^{\mu\nu\,\alpha\beta}(x-y)h_{\alpha\beta}(y)+\int\mathrm{d}^{4}x\mathrm{d}^{4}yJ^{\mu\nu}(x)X_{\mu\nu\,\lambda}(x-y)\eta^{\lambda}(y)+\cdots.
{\tilde{\nu}}_{\tilde{\alpha}}^{(1)}\frac{\partialV_{(n=1)}^{(1)}(a_{i},p_{i},\theta)}{\partial{\tilde{\xi}}_{\tilde{\alpha}}^{(1)}}-\frac{\partial{\calG}^{(n=2)}(a_{i},p_{i},\theta)}{\partial\theta}=0.
\begin{array}{ccl}{(0~0~-\frac{7}{2})}&{\rightarrow}&{(0~-\frac{7}{2})(+\frac{7}{2})\oplus(0~-\frac{5}{2})(+\frac{3}{2}),}\\{(0~0~+\frac{3}{2})}&{\rightarrow}&{(0~+\frac{5}{2})(-\frac{7}{2})\oplus(0~+\frac{3}{2})(-\frac{3}{2}).}\\\end{array}
\Gamma\Bigl({\frac{D}{2}}-1\Bigr){{\frac{\dot{x}_{a}\cdot\dot{x}_{b}}{{\Bigl[{(x_{a}-x_{b})}^{2}\Bigr]}^{{\frac{D}{2}}-1}}}\quad}\nonumber\,-{\frac{1-\alpha}{4}}\Gamma\Bigl({{\frac{D}{2}}-2}\Bigr){\frac{\partial}{\partial\tau_{a}}}{\frac{\partial}{\partial\tau_{b}}}{\Bigl[{(x_{a}-x_{b})}^{2}\Bigr]}^{2-{\frac{D}{2}}}
{\frac{1}{4!}}\Omega_{i\,i_{1}i_{2}i_{3}i_{4}}\epsilon_{i_{1}i_{2}i_{3}i_{4}j_{1}j_{2}j_{3}j_{4}}={\frac{2}{\sqrt{3}}}\delta_{i[j_{1}}f_{j_{2}j_{3}j_{4}]}
F_{n,k_{1},{\vec{c}}}=\sum_{{\calP}\inS_{N_{A}}}\sum_{K=0}^{M-1}D_{{\vec{c}}+Kp^{2}r\omega,n}\theta\left[\begin{array}{c}{{\vec{0}}}\\{\frac{{\calP}{\vec{c}}}{\ell}+\frac{K}{N}{\hat{e}}}\\\end{array}\right]\left({\vec{z}}-\frac{\tau}{\omega\kappa}\left(n+\frac{\beta}{2\pi}\right){\hat{e}}+\frac{\alpha}{2\pi\omega\kappa}{\hat{e}},\Omega\right)
{\bfE}(k)=\int_{0}^{\frac{\pi}{2}}da\;\sqrt{1-k^{2}\mathrm{sin}^{2}a}\\,\\\\\mathrm{and}\\\\{\bfK}(k)=\int_{0}^{\frac{\pi}{2}}\frac{da}{\sqrt{1-k^{2}\mathrm{sin}^{2}a}}\\.
I=-\intd^{5}x\left({\frac{1}{4g}}{\calF}_{\bar{\mu}\bar{\nu}}{\calF}^{\bar{\mu}\bar{\nu}}+{\frac{i}{4}}\epsilon^{\bar{\mu}\bar{\nu}\bar{\rho}\bar{\alpha}\bar{\beta}}C_{\bar{\mu}}{\calF}_{\bar{\nu}\bar{\rho}}{\calF}_{\bar{\alpha}\bar{\beta}}+i\epsilon^{\bar{\mu}\bar{\nu}\bar{\rho}\bar{\alpha}\bar{\beta}}C_{\bar{\mu}\bar{\nu}\bar{\rho}}{\calF}_{\bar{\alpha}\bar{\beta}}\right),
\pm(q^{\mu_{1}^{1}}+\ldots+q^{\mu_{1}^{l_{1}}})\alpha_{1}\pm\ldots\pm(q^{\mu_{r}^{1}}+\ldots+q^{\mu_{r}^{l_{r}}})\alpha_{r}\rightarrow\pm\mu_{1}^{1},\ldots,\mu_{l_{1}}^{1};\ldots;\mu_{r}^{1},\ldots,\mu_{l_{r}}^{1}\,\,,
N_{i_{e}j_{e}}^{\;\;\;k_{e}}=N_{i_{0}j_{1}}^{\;\;\;k_{1}}+N_{Ji_{0},j_{1}}^{\;\;\;k_{1}}\;\;\;,\;\;\;N_{i_{e}j_{e}}^{\;\;\;f_{e,\mu}}=N_{i_{0}j_{1}}^{\;\;\;f}
H^{c}=\frac{1}{2}\left[p_{a}^{\prime}p_{a}^{\prime}+\partial_{a}^{\prime}W^{\prime}(x^{\prime})\partial_{a}^{\prime}W^{\prime}(x^{\prime})+\hbar\partial_{a}^{\prime}\partial_{b}^{\prime}W^{\prime}(x^{\prime})\chi_{ab}^{(i)}\right]
D^{\mu}\varphi_{a}=\partial^{\mu}\varphi_{a}+gW_{ab}A^{b\mu}\varphi_{a},\;\bar{F}_{\mu\nu}^{a}=\partial_{\left[\mu\right.}A_{\left.\nu\right]}^{a}+g\frac{\deltaW_{bc}}{\delta\varphi_{a}}A_{\mu}^{b}A_{\nu}^{c}.
{\calL}_{HE}^{(1)\prime}(a,B)=\sum_{n=1}^{\infty}{\frac{(-1)^{n-1}}{4\pi^{2}}}\int_{0}^{\infty}{\frac{ds}{s^{3}}}e^{-sm^{2}-(an)^{2}/s}\,[seB\,L(seB)-{\frac{1}{3}}(seB)^{2}]\,.
H_{\sfGI}=\intd{\bfr}\left[\{\textstyle\frac{1}{2}}\Pi_{{\sfGI}\,i}^{a\,{\dagger}}({\bfr})\Pi_{{\sfGI}\,i}^{a}({\bfr})+{\textstyle\frac{1}{4}}F_{{\sfGI}\,ij}^{a}({\bfr})F_{{\sfGI}\,ij}^{a}({\bfr})+{\psi^{\dagger}}({\bfr})\left(\betam-i\alpha_{i}\partial_{i}\right)\psi({\bfr})\right]+\tilde{H}^{\prime}\,,
\left(\frac{T_{k}}{\dot{T}}\right)^{.}=\left(1-\frac{1}{4\piG}\,\,\frac{k^{2}}{a^{2}}\,\,\frac{(1-\dot{T}^{2})^{3/2}}{V\dot{T}^{2}}\right)\Phi_{k}\,
\int\frac{d\taud\overline{{\tau}}}{\tau_{2}^{2}}\int\frac{d^{8}p}{(2\pi)^{8}}e^{-2\pip^{2}\tau_{2}}\left|\frac1{2\eta(\tau)^{4}}\left(\theta_{3}^{4}(0|\tau)-\theta_{4}^{4}(0|\tau)^{4}-\theta_{2}^{4}(0|\tau)^{4}-\theta_{1}^{4}(0|\tau)\right)\right|^{2}
\langleqQ\,|\,H_{Q-Q}\,|\,Qq\rangle={\int}d{\bfr}d{\bfx}\,{\bar{\calF}}^{ba}({\bfr},{\bfx}){\langle}q_{k}{\cdots}q_{1}|j_{0}^{b}({\bfr})|q_{1}{\cdots}q_{k}{\rangle}{\langle}Q_{n}{\cdots}Q_{1}|j_{0}^{a}({\bfx})|Q_{1}{\cdots}Q_{n}{\rangle}\,,
W=2\,i\,\omega\,\frac{2^{\frac{i\omega}{M^{1/2}}}\Gamma(2)\Gamma(-\frac{i\omega}{M^{1/2}})}{M^{3/4}\Gamma(1+i\frac{m-\omega}{2\sqrt{M}})\Gamma(1-i\frac{m+\omega}{2\sqrt{M}})}.
D^{2}\\rightarrow\\partial_{t}^{2}-\left(\frac1r\partial_{r}r\partial_{r}+\frac{1}{r^{2}}\partial_{\varphi}^{2}+\partial_{z}^{2}\right)\+\\frac{2i\beta\Theta(r-R)}{r^{2}}\\partial_{\varphi}\+\\frac{\beta^{2}\Theta^{2}(r-R)}{r^{2}}
\int\mathrm{d}^{2}a\,{\frac{\partial}{\partial\bar{a}}}f(a,\bar{a})\times{\frac{1}{a^{n}}}=\int\mathrm{d}^{2}a\,{\frac{\partial}{\partial\bar{a}}}\left[f(a,\bar{a})\times{\frac{1}{a^{n}}}\right],
[\beta_{q^{\prime}}^{(\sigma)},\hat{\beta}_{q^{\prime}}^{(\sigma)}]=\sqrt{q^{\prime}}\,e^{-\frac{\epsilon}{2}(\beta^{(\sigma)2}-\beta^{(\sigma)\dagger2})}\equiv\sqrt{q^{\prime}}{\calS}_{2}(\epsilon),\quadq^{\prime}\equive^{-\epsilon}=1/q\,{,}
\frac{p_{r}^{2}}{g_{rr}}+\frac{p_{\theta}^{2}}{g_{\theta\theta}}+\frac{-{g^{\prime}}_{tt}}{-\calD}\left(p_{\phi}+\frac{g_{t\phi}+\Omega_{0}g_{\phi\phi}}{{g^{\prime}}_{tt}}E\right)^{2}=\left(\frac{E^{2}}{-{g^{\prime}}_{tt}}-\mu^{2}\right),
4\pi{\calD}{}^{*}\!J=\frac{1}{4!}\left(\partial_{\lambda}\Gamma_{\rho\mu}^{\rho}\right){}^{*}\!F_{\nu\omega}\delta_{\alpha\beta\gamma\delta}^{\lambda\mu\nu\omega}\,dx^{\alpha}\wedgedx^{\beta}\wedgedx^{\gamma}\wedgedx^{\delta},
G_{1}(x_{1},x_{1}^{\prime},\bf{x}_{2})=e^{-ix_{1}^{0}h}\frac{1}{2}\Big(\epsilon(x_{1}^{0}-x_{1}^{\prime0})+\frac{h}{\sqrt{h^{2}}}\Big)\delta^{3}(\bf{x}_{1}-\bf{x}_{1}^{\prime})\gamma_{0}e^{ix_{1}^{\prime0}h^{\prime}}.
R\;\mathrm{Vol}=\frac{1}{2}d\Phi\wedge{*}d\Phi+\frac{g_{s}e^{-\Phi}}{4}H_{3}\wedge{*}H_{3}+\frac{3g_{s}^{1/2}e^{3\Phi/2}}{8}F_{2}\wedge{*}F_{2}+\frac{g_{s}^{3/2}e^{\Phi/2}}{8}\tilde{F}_{4}\wedge{*}\tilde{F}_{4}\.
\begin{array}{rcl}{}&{}&{T(z)\Phi_{i}(w)=\displaystyle\frac{h}{(z-w)^{2}}\Phi_{i}(w)+\frac{1}{z-w}\partial_{w}\Phi_{i}(w)+\cdots~,}\\{}&{}&{T(z)J^{a}(w)=\displaystyle\frac{1}{(z-w)^{2}}J^{a}(w)+\frac{1}{z-w}\partial_{w}J^{a}(w)+\cdots~.}\\\end{array}
\frac{16\pi^{2}}{g_{\alpha,{\scriptstyle\mathrm{bare}}}^{2}}=\frac{16\pi^{2}}{g_{\alpha}^{2}(\mu)}-b_{\alpha}(4\pi)^{\epsilon}\int_{0}^{\infty}\frac{dt}{t^{1-\epsilon}}e^{-t\frac{\mu^{2}}{M^{2}}}.
\begin{array}{rl}{\tau^{\prime}-\tau}&{=\delta\tau=\Lambda(\tau,\eta)-{\frac{1}{2}}\etaD\Lambda,}\\{\eta^{\prime}-\eta}&{=\delta\eta=-{\frac{i}{2}}D\Lambda,}\\{D^{\prime}-D}&{=\deltaD=-{\frac{1}{2}}\dot{\Lambda}D,}\\\end{array}
\delta\hat{\psi}_{M}=\partial_{M}\hat{\epsilon}-{\frac{1}{4}}\hat{\omega}_{M}^{AB}\hat{\gamma}_{AB}\hat{\epsilon}-{\frac{1}{288}}\left(\hat{\gamma}_{M}^{BCDE}-8\hat{e}_{M}^{B}\hat{\gamma}^{CDE}\right)\hat{F}_{BCDE},
S[x^{\hat{i}},x^{\alpha}]=\intd^{2}\xi\left[\partialx^{\hat{i}}f_{\hat{i}\hat{j}}\bar{\partial}x^{\hat{j}}+\partialx^{\hat{i}}f_{\hat{i}\beta}\bar{\partial}x^{\beta}+\partialx^{\alpha}f_{\alpha\hat{j}}\bar{\partial}x^{\hat{j}}+\partialx^{\alpha}f_{\alpha\beta}\bar{\partial}x^{\beta}\right].
\lambda\{\frac{\partial(S_{N+1}-S_{N})}{\partial\lambda}}|_{\lambda_{0}}=\sum_{n=1}^{N}{\frac{n}{p(N)^{n}}}\left({\frac{A_{n}(N+1)}{c(N)^{n}}}-A_{n}(N)\right)+{\frac{(N+1)A_{N+1}(N+1)}{c(N)^{N+1}p(N)^{N+1}}}\,.
\left(\begin{matrix}{\alpha}&{a}\\{b}&{\beta}\\\end{matrix}\right)\left(\begin{matrix}{\gamma}&{c}\\{d}&{\delta}\\\end{matrix}\right)=\left(\begin{matrix}{\alpha\gamma+a\cdotd}&{\alphac+\deltaa-b\timesd}\\{\gammab+\betad+a\timesc}&{\beta\delta+b\cdotc}\\\end{matrix}\right)
\langleT_{\mu\nu}(x)T_{\rho\sigma}(0)\rangle=-{\frac{1}{48\pi^{4}}}\Pi_{\mu\nu\rho\sigma}^{(2)}\left[{\frac{c_{H}(x)}{x^{4}}}\right]+\pi_{\mu\nu}\pi_{\rho\sigma}\left[{\frac{f(x)}{x^{4}}}\right],
\Phi_{n+1}^{i}=(1-2\frac{\tau}{\delta^{2}}-\mu^{2}\tau)\Phi_{n}^{i}-\lambda\tau\Phi_{n}^{i^{3}}+\frac{\tau}{\delta^{2}}(\Phi_{n}^{i+1}+\Phi_{n}^{i-1})+\tau\cdotnoise
{\calW}(q_{0}{\calH}^{n})\equiv{\frac{1}{2^{n}}}\left[(_{0}^{n})q_{0}{\calH}^{n}+(_{1}^{n}){\calH}q_{0}{\calH}^{n-1}+(_{2}^{n}){\calH}^{2}q_{0}{\calH}^{n-2}+...+(_{n}^{n}){\calH}^{n}q_{0}\right].
\{F,G\}_{\alpha}=(\sigma_{\alpha})_{B}^{~~\!A}\bigr(\frac{\deltaF}{\delta\phi^{A}}\frac{\deltaG}{\delta\eta_{B}}+(-1)^{\epsilon(F)\epsilon(G)}\frac{\deltaG}{\delta\phi^{A}}\frac{\deltaF}{\delta\eta_{B}}\bigr),
\left[a_{ij}(p^{+},n_{\perp}),a_{lk}^{\dagger}(q^{+},m_{\perp})\right]=\left\{b_{ij}(p^{+},n_{\perp}),b_{lk}^{\dagger}(q^{+},m_{\perp})\right\}=\delta(p^{+}-q^{+})\delta_{n_{\perp},m_{\perp}}\delta_{il}\delta_{jk}.
\begin{array}{rcl}{}&{}&{tv_{k}=\pmq^{l-2k}v_{k}~,}\\{}&{}&{ev_{k}=[e,f^{(k)}]v_{0}=\pm[l-k+1]v_{k-1}~,}\\{}&{}&{fv_{k}=\displaystyle\frac{f^{k+1}}{[k]!}v_{0}=[k+1]v_{k+1}~,}\\\end{array}
[R^{1},R^{2},R^{3}]_{D({\delta_{1}},{\delta_{2}},{\delta_{3}})}=\sum_{\alpha,\beta,\gamma}\,C_{\alpha\beta\gamma}\;\rho_{A({\alpha_{1}},{\alpha_{2}},{\alpha_{3}})}^{1}\rho_{B({\beta_{1}},{\beta_{2}},{\beta_{3}})}^{2}\rho_{C({\gamma_{1}},{\gamma_{2}},{\gamma_{3}})}^{3},
S=\frac{1}{\pi}\intd^{2}x\left[e^{-2\phi}(2\partial_{+}\partial_{-}\rho-4\partial_{+}\phi\partial_{-}\phi+\lambda^{2}e^{2\rho})+\frac{1}{2}\partial_{+}f\partial_{-}f-\frac{\beta}{16}(f^{2}-\frac{\mu^{2}}{\beta})^{2}e^{2\rho-2\phi}\right],
{\mathcalG}_{l+\nu}^{(n)}(r^{\prime\prime},r^{\prime};E)=\lambda^{n}\,\prod_{\alpha=1}^{n}\left[\int_{0}^{\infty}dr_{\alpha}\,\epsilon_{r_{\alpha}}\right]\,\prod_{\beta=0}^{n}\left[{\mathcalG}_{l+\nu}^{(0)}(r_{\beta+1},r_{\beta};E)\right]\;.
K^{\prime}(z,u_{a},D)\equiv\int_{0}^{\infty}d\hat{T}\,I(z,u_{a},\hat{T},D)=K_{02}(z,u_{a},D)+f(z,D)G_{Bab}^{1-{\frac{D}{2}}}+O\bigl(z^{4},G_{Bab}^{2-{\frac{D}{2}}}\bigr)\nonumber\,
{\bfS}=C\,e^{S_{+-}S_{--}^{-1}b^{\dagger}d^{\dagger}}:e^{(S_{++}^{\dagger-1}-1)b^{\dagger}b}::e^{(1-S_{--}^{-1})dd^{\dagger}}:e^{S_{--}^{-1}S_{-+}db}\,\,,
H_{T}=\intd^{3}x\left[\frac{1}{2}\left(\Pi_{ai}-\frac{\theta}{8\pi^{2}}B_{ai}\right)^{2}+\frac{1}{2}\,B_{ai}^{2}-A_{a0}\left(D_{i}(A)\right)_{ac}\Pi_{ci}+\lambda_{a}\,\Pi_{a}\right]\,.
\omega^{i}(g)=\left[\delta_{j}^{i}+\frac{1}{2!}c_{jk}^{i}g^{k}+\sum_{n=2}^{\infty}\frac{1}{(n+1)!}c_{jk_{1}}^{h_{1}}c_{h_{1}k_{2}}^{h_{2}}\ldotsc_{h_{n-1}k_{n-1}}^{h_{n-1}}c_{h_{n-1}k_{n}}^{i}g^{k_{1}}g^{k_{2}}\ldotsg^{k_{n-1}}g^{k_{n}}\right]dg^{j}\quad.
\left(\sqrt{-g}\,g^{{\mu}{\nu}}\right)^{-1}=\eta_{{\mu}{\nu}}-\kappa\,h_{{\mu}{\nu}}+\kappa^{2}\,h_{{\mu}{\alpha}}\,h_{{\alpha}{\nu}}-\kappa^{3}\,h_{{\mu}{\alpha}}\,h_{{\alpha}{\beta}}\,h_{{\beta}{\nu}}+O(\kappa^{4})
\hat{\Psi}_{i}\;=\;K^{1/14}\chi_{i}\;,\,\hat{\Psi}_{a}\;=\;\K^{1/14}\left\{\Psi_{a}\,-\,\textstyle{\frac{1}{7}}\Gamma_{a}\Gamma^{i}\chi_{i}\right\}\;,\;\hat{\epsilon}\;=\;K^{-1/14}\epsilon\;,
\int_{{\calF}_{2}^{-}}\frac{d^{2}\tau}{\tau_{2}^{2}}\Phi(\tau)=\int_{{\calF}}\frac{d^{2}\tau}{\tau_{2}^{2}}\left[\Phi(\tau)+\Phi\left(-\frac{1}{\tau}\right)+\Phi\left(-\frac{1}{\tau+1}\right)\right]
G_{\\nu}^{\mu}(h)={\frac{\kappa^{2}}{l\Psi}}T_{\quad\\nu}^{A\mu}+{\frac{\kappa^{2}(1-\Psi)}{l\Psi}}T_{\quad\\nu}^{B\mu}+{\frac{1}{\Psi}}\left(\Psi_{\|\nu}^{|\mu}-\delta_{\nu}^{\mu}\Psi_{\|\alpha}^{|\alpha}\right)+{\frac{\omega(\Psi)}{\Psi^{2}}}\left(\Psi^{|\mu}\Psi_{|\nu}-{\frac{1}{2}}\delta_{\nu}^{\mu}\Psi^{|\alpha}\Psi_{|\alpha}\right)\,
\{{\calF}_{\rho\sigma}(x,y),u(z)\}\;=\;-\;ic_{\mu\nu\lambda}u(z)T_{\lambda}\;\Big({\calF}_{\nu\rho}(z,x)\;{\calF}_{\mu\sigma}(z,y)\;+\;\alpha\;\delta_{\mu\rho}\delta_{\nu\sigma}\Bigr)\quad.
\begin{array}{l}{<T_{i}^{n},T_{j}^{m}>=2(h_{i},h_{j})\delta_{n,m},~~~n,m>0,}\\{<W_{\alpha}^{n},W_{\beta}^{m}>=2(e_{\alpha},e_{-\alpha})\delta_{\alpha\beta}\delta_{n,m},~~~n,m>0,}\\\end{array}
\int\frac{d^{4}p^{\prime}}{\left(2\pi\right)^{4}}\left[{D}\left(p,p^{\prime},P\right)+{K}\left(p,p^{\prime},P\right)\right]{G}^{\left(4\right)}\left(p^{\prime},p^{\prime\prime},P\right)=\delta^{\left(4\right)}\left(p-p^{\prime\prime}\right),
{\vartheta}\equiv\int\Omega^{(2,0)}\equiv\left(\begin{matrix}{\vartheta_{12}}\\{\vartheta_{21}}\\{\vartheta_{11}}\\{\vartheta_{22}}\\\end{matrix}\right)=\frac{\sqrt{3}}{\psi_{0}}\left(\begin{matrix}{\xi_{1}(r)\,\xi_{2}(s)}\\{\xi_{2}(r)\,\xi_{1}(s)}\\{\xi_{1}(r)\,\xi_{1}(s)}\\{\xi_{2}(r)\,\xi_{2}(s)}\\\end{matrix}\right)\,
\Omega^{ij}\;=\;\left(\begin{array}{cccc}{k_{1}}&{0}&{0}&{k_{2}}\\{0}&{k_{2}}&{0}&{0}\\{0}&{0}&{k_{2}}&{0}\\{k_{2}}&{0}&{0}&{0}\\\end{array}\right)\,,\,\Omega_{ij}\;=\;\left(\begin{array}{cccc}{0}&{0}&{0}&{q_{2}}\\{0}&{q_{2}}&{0}&{0}\\{0}&{0}&{q_{2}}&{0}\\{q_{2}}&{0}&{0}&{q_{1}}\\\end{array}\right)\,,
{\bar{y}}=-(\frac{3\sqrt{2\pi}\sigma}{2k_{5}^{2}})^{\frac{1}{2}}\frac{(T_{44}-\frac{V}{4k^{2}}+\sqrt{\frac{\pi}{2}}\frac{\sigma}{k^{2}}\Lambda)^{\frac{1}{2}}}{T_{44}-\frac{1}{16k^{2}}T^{4}},
\frac{1}{N}\left[\dot{\pi}^{\phi}-(N^{i}\phi^{\phi})_{,i}\right]=-\gamma^{-1/2}e^{\phi}\left[\pi^{ij}\pi^{kl}\gamma_{ik}\gamma_{jl}+\frac{1}{2}\left(\pi^{\phi}\right)^{2}+\pi\pi^{\phi}\right]~~~.
U(\nu;x)=4\sum_{n=0}^{\infty}(\frac{1}{\pi})^{n+1}\int_{1}^{\infty}d\alpha_{o}...\int_{1}^{\infty}d\alpha_{n}\frac{(\prod_{j=0}^{n}D(\nu;\alpha_{j})e^{-2\alpha_{j}x})}{(\prod_{j=0}^{n-1}(\alpha_{j}+\alpha_{j+1}))}(\sum_{j=0}^{n}\alpha_{j});~~Rex>x_{o}.
{\calG}^{a{\mu}_{1}{\cdots}{\mu}_{p+1}}\equiv{\Gamma}^{ab}G^{b{\mu}_{1}{\cdots}{\mu}_{p+1}}-\frac{1}{(p+1)!}{\epsilon}^{{\mu}_{1}{\cdots}{\mu}_{p+1}{\nu}_{1}{\cdots}{\nu}_{p+1}}G_{{\nu}_{1}{\cdots}{\nu}_{p+1}}^{a},
H_{1}=A_{1}^{+}A_{1}^{-}+E_{1}^{(0)},\quadA_{1}^{(-)}=\psi_{1}^{(0)}\left(-\frac{1}{\sqrt{2}}\frac{d}{dx}\right)\frac{1}{\psi_{1}^{(0)}}=(A_{1}^{+})^{\dagger},\quadE_{1}^{(0)}=0,
\phi_{1}=\left(\begin{array}{cc}{s-ib_{12}}&{b_{13}-ib_{23}}\\{-b_{13}-ib_{23}}&{s+ib_{12}}\\\end{array}\right),\quad\phi_{2}=\left(\begin{array}{cc}{-b_{03}+ip}&{-b_{01}+ib_{02}}\\{-b_{01}-ib_{02}}&{b_{03}+ip}\\\end{array}\right).
\Bigl[-\,\frac{1}{N^{2}}(\lambda_{i}+\xi)\partial_{i}^{2}-\frac{1}{N^{2}}\sum_{j\neqi}\frac{\lambda_{j}+\xi}{\lambda_{j}-\lambda_{i}}(\partial_{j}-\partial_{i})+\frac{\alpha+\beta-2}{N}\,\partial_{i}+\beta-\alpha+\lambda_{i}+\xi\Bigr]Z=0.
e(\varphi_{1}-\varphi_{2})=\frac{e(\varphi_{1})(dn(\varphi_{1})+1)(dn(\varphi_{2})+1)-k^{2}e(\varphi_{2})sn(\varphi_{1})sn(\varphi_{2})}{e(\varphi_{2})(dn(\varphi_{1})+1)(dn(\varphi_{2})+1)-k^{2}e(\varphi_{1})sn(\varphi_{1})sn(\varphi_{2})}
\begin{aligned}{\nabla_{M}^{\pm}\epsilon^{\pm}\equiv\left(\nabla_{M}\pm\frac{1}{8}H_{MNP}\Gamma^{NP}\right)\epsilon^{\pm}}&{{}=0,}\\{\left(\Gamma^{M}\partial_{M}\Phi\pm\frac{1}{12}H_{MNP}\Gamma^{MNP}\right)\epsilon^{\pm}}&{{}=0,}\\\end{aligned}
\chi_{\alpha}\equiv\Pi_{\alpha}-\frac{i}{2}\left(\gamma^{0}\right)_{\;\;\alpha}^{\beta}\bar{\psi}_{\beta}\approx0,\;\bar{\chi}^{\alpha}\equiv\bar{\Pi}^{\alpha}-\frac{i}{2}\left(\gamma^{0}\right)_{\;\;\beta}^{\alpha}\psi^{\beta}\approx0,