harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
f_{p}(T_{f})\approx\frac{D-2+p_{H}+(1-d_{H})/2}{A_{p}C}\left(\frac{m_{s}}{m_{p}}\right)^{\frac{d_{H}-1}{2}}\left(\frac{T_{f}}{m_{s}}\right)^{D-2+p_{H}+(1-d_{H})/2}
S_{M_{j}}=\frac{1}{2}\int\mathrm{d}^{3}x\,\mathrm{d}^{3}yA_{j}{}_{\mu}(x)\left[\frac{\delta^{2}S_{\Lambda}^{\mathrm{sgn}(j)}}{\deltaA_{\mu}(x)\deltaA_{\nu}(y)}-M_{j}\delta^{\mu\nu}\delta(x-y)\right]A_{j}{}_{\nu}(y),
\frac{1}{(k^{2}-\alpham^{2})(k^{2}-\betam^{2})}~=~\frac{1}{(\alpha-\beta)m^{2}}\left[\frac{1}{(k^{2}-\alpham^{2})}~-~\frac{1}{(k^{2}-\betam^{2})}\right]~.
S_{\mathrm{YM/\Phi\,}}^{(\mathrm{q})}={\frac{N\,V}{4g_{0}^{2}}}\mathrm{Tr}\left[\,\mathbf{\Phi}^{-1}\left(\left[\mathbf{A}_{\mu}^{(\mathrm{q})},\mathbf{A}_{\nu}^{(\mathrm{q})}\right]^{2}\right)\,\right]+{\frac{1}{4}}\,\mathrm{Tr}\mathbf{\Phi}\.
\langleE_{\varepsilon_{1}\varepsilon_{1}^{\prime}}^{(1)}\cdotsE_{\varepsilon_{n}\varepsilon_{n}^{\prime}}^{(n)}\rangle_{b}=G_{b}^{(n)}(\overbrace{x^{-1}\zeta,\cdots,x^{-1}\zeta}^{n},\overbrace{\zeta,\cdots,\zeta}^{n})^{-\varepsilon_{n}\cdots-\varepsilon_{1}\varepsilon_{1}^{\prime}\cdots\varepsilon_{n}^{\prime}}.
I_{CS}^{\prime}=-\frac{i\kappa_{r}}{4\pi}\intd^{3}x\varepsilon^{\mu\nu\lambda}\mathrm{Tr}[Z_{A}^{-1}A_{\mu}^{(r)}\partial_{\nu}A_{\lambda}^{(r)}+\frac{2}{3}Z_{g}^{-1}A_{\mu}^{(r)}*A_{\nu}^{(r)}*A_{\lambda}^{(r)}],
\Psi_{\vec{k},n}(\vec{x},\phi)=e^{i\vec{k}.\vec{x}}e^{in\phi}+\int_{o}^{2\pi}d\phi^{\prime}\intd^{3}x^{\prime}G_{o}(K;|\vec{x}-\vec{x}^{\prime}|;(|\phi-\phi^{\prime}|)V(x^{\prime},\phi^{\prime})\Psi_{\vec{k},n}(\vec{x}^{\prime},\phi^{\prime}).
\delta_{[\Sigma_{1},\Sigma_{2}]}\Phi=\left(\varepsilon_{1}^{\alpha}\partial_{\alpha}\Sigma_{2}^{a}\right)\partial_{a}\Phi+\left(\varepsilon_{1}^{\alpha}\partial_{\alpha}\Sigma_{2}^{\beta}\right)\partial_{\beta}\Phi+...-1\leftrightarrow2
\frac{\partial^{2}S_{n}}{\partialt_{+}^{2}}=-\frac{S_{n}}{1-S_{n}^{2}}(\frac{\partialS_{n}}{\partialt_{+}})^{2}-\frac{1}{t_{+}}\frac{\partialS_{n}}{\partialt_{+}}+\frac{(n+1)^{2}}{t_{+}^{2}}\frac{S_{n}}{1-S_{n}^{2}}-4S_{n}(1-S_{n}^{2}),
\begin{array}{l}{d{\hat{s}}^{2}=F\left(dv^{2}-du^{2}\right)-K\,dx_{I}dx_{I}\qquad,\qquade^{2\hat{\phi}}=FK\,,}\\{\hat{H}_{IJK}=-\frac{1}{6}\epsilon_{IJKL}\partial^{L}K\,\quad\hat{H}_{Iuv}=\partial_{I}F\,,\qquad\hat{H}_{IJu}=\hat{H}_{IJv}=0\,.}\\\end{array}
1\simeq\frac{\alpha}{\pi}|eH|\int_{-\infty}^{\infty}dq_{3}\int_{0}^{\infty}d\hat{q}_{\perp}^{2}\frac{\mathrm{e}^{-\hat{q}_{\perp}^{2}}}{\sqrt{Q_{1}Q_{2}}}\frac{1}{\sqrt{Q_{1}}+\sqrt{Q_{2}}}
{\calL}_{kin}=\left[{\frac{m}{2}}\dot{x}^{2}+{\frac{i}{2}}(\chi_{1}\dot{\chi}_{1}+\chi_{2}\dot{\chi}_{2})+{\frac{1}{8}}mx^{2}F^{2}+{\frac{i}{2}}F\chi_{1}\chi_{2}\right]\,.
\vec{A}(\vec{x},t)=\frac{1}{\sqrt{V}}\sum_{\vec{k},\lambda}{\vec{\varepsilon}}_{\vec{k}}^{(\lambda)}[C_{\vec{k}\lambda}(t)e^{i\vec{k}\cdot\vec{x}}+C_{\vec{k}\lambda}^{\ast}(t)e^{-i\vec{k}\cdot\vec{x}}]
A^{\mu}(\vec{t})=\frac1{2\pi}\sum_{k,n=0,\pm1,\cdots}e^{i(k-n)\theta}\left[a_{\uparrow\downarrow}^{\mu}\bar{\psi}_{-n}^{\uparrow}(t)\psi_{k}^{\downarrow}(t)+a_{\downarrow\uparrow}^{\mu}\bar{\psi}_{-n}^{\downarrow}(t)\psi_{k}^{\uparrow}(t)\right],
V(\Phi)=({\frac{1}{6}}\lambda_{R}-{\frac{1}{24\pi^{2}}})m^{2}\Phi^{2}+{\frac{1}{36}}\lambda_{R}\Phi^{4}+{\frac{1}{144\pi^{2}}}\Phi^{4}(ln{\frac{\Omega^{2}}{-2m^{2}}}-{\frac{3}{2}})+{\calD}_{2}
\Psi(x)\,=\,\sum_{\sigma=+j}^{-j}\int{\frac{d^{3}p}{(2\pi)^{3}}}{\frac{1}{2\,\omega_{\vec{p}}}}\Big[u_{\sigma}(\vec{p}\,)\,a_{\sigma}(\vec{p}\,)\,e^{-ip\cdotx}+v_{\sigma}(\vec{p}\,)\,b_{\sigma}^{\dagger}(\vec{p}\,)\,e^{ip\cdotx}\Bigr]\quad,
X_{\pmz^{a}}=\theta_{a\bar{a}}S^{N_{\pm}}\bar{z}^{\bar{a}}(S^{\dagger})^{N_{\pm}}+\sum_{n=0}^{N_{\pm}-1}\lambda_{a,n}^{\pm}|n\rangle\langlen|\otimesP_{0}^{(2)},
\Gamma_{N}=-ie^{N}\int_{0}^{\infty}\frac{dT}{T}e^{-im^{2}T}\left(\prod_{i=1}^{N}\int_{0}^{T}dt_{i}\right){\calN}\int_{x(T)=x(0)}{\calD}x\:e^{\displaystyle\frac{i}{4}\int_{0}^{T}d\taux\frac{d^{2}}{d\tau^{2}}x}e^{\displaystyle\int_{0}^{T}J(\tau)x(\tau)}.
\begin{array}{llll}{\eta_{z}^{+}\rightarrow\eta_{z},\quad}&{\quad\lambda_{+}\rightarrow\lambda,\quad}&{\quad\beta_{+z}\rightarrow\beta_{z},\quad}&{\quad\gamma^{+}\rightarrow\gamma,}\\{\eta_{z}^{-}\rightarrow\eta,\quad}&{\quad\lambda_{-}\rightarrow\lambda_{z},\quad}&{\quad\beta_{-z}\rightarrow\beta_{zz},\quad}&{\quad\gamma^{-}\rightarrow\gamma^{z}.}\\\end{array}
{\mathbfa}^{\nu}\,\mid0\rangle\:=\:0\;\;,\;\;\langle0\mid{\mathbfa}_{\nu}^{\dagger}\:=\:0\;\;,\;\;{\mathbfb}_{\mu}\,\mid0\rangle\:=\:0\;\;,\;\;\langle0\mid{\mathbfb}^{\dagger\mu}\:=\:0.
{\frac{\delta{\calH}_{2n+1}}{\deltau_{-j}}}=\sum_{k=0}^{\infty}(-1)^{(j+1)k+{\frac{1}{2}}k(k+1)}\left(D^{k}\,{\frac{\partial{\calH}_{2n+1}}{\partialu_{-j}^{[k]}}}\right),
\sigma_{3}(w,b)\sim\Re\int_{b(2+b)}^{w(w-2)}\frac{ds}{s}\int\frac{dt}{t}\,\frac{1}{\sqrt{J(s,t,w^{2},b^{2})}}=\int_{b(2+b)}^{w(w-2)}\frac{ds}{s}\,\frac{\pi}{\sqrt{-J(s,0,w^{2},b^{2})}}
ds^{2}=\left({\frac{{\rho}^{2}-2Mr}{{\rho}^{2}}}\right)dt^{2}-{\frac{{\rho}^{2}}{\Lambda}}dr^{2}-{\rho}^{2}d{\theta}^{2}+{\frac{4MrA{sin}^{2}\theta}{{\rho}^{2}}}dtd\varphi-{\frac{{sin}^{2}\theta}{{\rho}^{2}}}{\Sigma}^{2}d{\varphi}^{2}\,
\frac{1}{(g_{\mathrm{YM}}^{\mathrm{w}})^{2}(\mu)}=\frac{1}{(g_{\mathrm{YM}}^{\mathrm{w}})^{2}}\left(1-\frac{(g_{\mathrm{YM}}^{\mathrm{w}})^{2}N}{4\pi\mu}\right)\,,
\begin{array}{cccl}{p=3}&{D=10}&{AdS_{5}\timesX^{5}}&{\mathrm{~D3~--brane~of~type~IIBtheory}}\\{p=2}&{D=11}&{AdS_{4}\timesX^{7}}&{\mathrm{~M2~--brane~of~M--theory}}\\{p=5}&{D=11}&{AdS_{7}\timesX^{4}}&{\mathrm{~M5~--brane~of~M--theory}\}\\\end{array}
g_{s}^{\prime}=g_{s}\frac{l_{s}^{4}}{R_{6}R_{7}R_{8}R_{9}}=\frac{l_{p}^{9}}{R^{1/2}R_{6}R_{7}R_{8}R_{9}}\,\R_{i}^{\prime}=\frac{l_{s}^{2}}{R_{i}}=\frac{l_{p}^{3}}{R_{i}R}
\mathcal{H}^{\ast}=\intd^{2}x\left[\frac{1}{2}\left(\xi_{5}^{\ast2}+\xi_{6}^{\ast2}\right)+\frac{1}{2}\left(\partial_{1}\xi_{3}^{\ast}-\partial_{2}\xi_{2}^{\ast}\right)^{2}+\frac{1}{2}m^{2}\left(\xi_{2}^{\ast2}+\xi_{3}^{\ast2}\right)+m\left(\xi_{2}^{\ast}\xi_{6}^{\ast}-\xi_{3}^{\ast}\xi_{5}^{\ast}\right)\right].\tag{18}
{\dot{X}}=\left(\begin{array}{cccc}{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}\\{-(L_{R}^{2}-\Omega^{2}-T)}&{0}&{0}&{2\Omega}\\{0}&{-(L_{R}^{2}-\Omega^{2})}&{-2\Omega}&{0}\\\end{array}\right)X
{\sfK}_{g}^{d}({\bfr})=gf^{d\sigmae}\,{\sfTr}\left[V_{\cal{C}}^{-1}({\bf{r}}){\textstyle\frac{\lambda^{e}}{2}}V_{\cal{C}}({\bf{r}}){\textstyle\frac{\lambda^{b}}{2}}\right]A_{{\sfGI}\,i}^{\sigma}({\bfr})\Pi_{i}^{b}({\bfr})\,.
S(D)=\sum_{n=0}^{\infty}{\frac{1}{(n+1)^{2}}}{\frac{1}{n!}}{\frac{\Gamma(n+D-3)}{\Gamma(D-3)}}{\frac{\Gamma(n+4-D/2)}{\Gamma(4-D/2)}}{\frac{\Gamma(D/2)}{\Gamma(n+D/2)}}.
V_{(j,m)}^{(1)}(z,\bar{z})={\large[}-\frac{j}{\alpha_{+}}\partial\phi(z)\gamma^{(j-m)}-(j-m)\partial\gamma\gamma^{j-m-1}(z){\large]}\timesc.c.\timese^{2j\phi(z,\bar{z})/\alpha_{+}}
G_{E}(a_{3},a_{4},\phi)=\mu^{4}\sqrt{a_{3}a_{4}}\pi\sum_{s=0}^{\infty}\frac{(-1)^{s+2}}{2s+2}g^{s+1}\phi^{2s+2}\frac{\Gamma(s)}{\Gamma(s+1)}A_{2}^{c^{2}}(s,a_{3},a_{4})\,.
\frac{d}{dt}\left[\frac{a^{4}r^{2}\dot{r}}{\sqrt{a^{2}r^{2}(1-a^{2}\dot{r}^{2})}}\right]+\frac{a^{2}r(a^{2}\dot{r}^{2}-1)}{\sqrt{a^{2}r^{2}(1-a^{2}\dot{r}^{2})}}=0
\mathrm{Tr}\left[{\mathrm{\small{~\frac12~}}}{\calD}^{2}\,\bar{\calA}\,{\mathrm{\small{~\frac12~}}}{\calD}^{2}\right]=\mathrm{\overline{T}r}\left[\bar{\calA}\,{\mathrm{\small{~\frac12~}}}{\calD}^{2}\,{\mathrm{\small{~\frac12~}}}\bar{\calD}^{2}\right].
\intdt^{\prime}G(0,t;x^{\prime},t^{\prime})=i\intdt^{\prime}\int\int\frac{d\omega^{\prime}}{2\pi}\frac{dk^{\prime}}{2\pi}\left(\frac{e^{-i\omega^{\prime}(t-t^{\prime})}}{\omega^{2}-k^{2}-4+i\rho}\right)\left(\frac{2ik^{\prime}}{ik^{\prime}-2\sigma_{1}}\right)e^{-ik^{\prime}x^{\prime}}.
\frac{e^{4}}{15\pi^{2}}\left(A(\omega_{1},\omega_{2},\omega_{3},\omega_{4})\delta^{ij}\delta^{kl}+A(\omega_{1},\omega_{3},\omega_{2},\omega_{4})\delta^{ik}\delta^{jl}+A(\omega_{1},\omega_{4},\omega_{2},\omega_{3})\delta^{il}\delta^{jk}\right),
{\calZ}_{open,BPS}^{p-p}\int_{0}^{\infty}\frac{ds}{s}\left[\mathrm{Tr}_{\mathrm{NS}}\left(\frac{1+(-1)^{F}}{2}\right)q^{2L_{0}-1}-\mathrm{Tr}_{\mathrm{R}}\left(\frac{1+(-1)^{F}}{2}\right)q^{2L_{0}}\right]~~,
S_{eff}=\int\!d^{4}\!x~[~\pi_{0}\dot{A}^{0}+\pi_{i}\dot{A}^{i}+\pi_{\rho}\dot{\rho}+B_{2}\dot{N}^{2}+\overline{{\calP}}_{i}\dot{\calC}^{i}+\overline{{\calC}}_{2}\dot{\calP}^{2}~]-H_{total}
I=\frac{\bar{\beta}}{\sqrt{\pi}}e^{-\bar{m}^{2}\pi}\sum_{k=1}^{\infty}\frac{\left(\bar{t}_{f}\right)^{2k-1}\left(-1\right)^{k+1}}{\left(2k-1\right)\left(k-1\right)!}\int_{0}^{\infty}\frac{duu^{(2k-5)/2}}{u+\pi}e^{-u\bar{m}^{2}-\frac{\bar{\beta}^{2}}{4u}}
\Psi_{nlm}(r,\theta,\varphi;\delta_{1},\delta_{2})=\sum_{n_{1}=0}^{n-|m|-1}\widetildeW_{nlm}^{n_{1}}(\delta_{1},\delta_{2})\Psi_{n_{1}n_{2}m}(\mu,\nu,\varphi;\delta_{1},\delta_{2}),
f^{kl(1)}(x,y)=\left(\begin{array}{cccc}{0}&{\frac{1}{m^{2}}\partial_{x}^{i}}&{0}&{-\frac{1}{m^{2}}}\\{\frac{1}{m^{2}}\partial_{x}^{i}}&{0}&{\delta_{ij}}&{0}\\{0}&{-\delta_{ij}}&{0}&{0}\\{\frac{1}{m^{2}}}&{0}&{0}&{0}\\\end{array}\right)\delta^{3}(x-y).
\frac{y^{2}}{\Lambda^{2(n_{c}-\tilde{n}_{c})}}=\prod_{a=1}^{\tilde{n}_{c}}(x-\phi_{a})^{2}+\frac{4}{\Lambda^{n_{c}-\tilde{n}_{c}}}\prod_{i=1}^{n_{f}}(x+m_{i}),
H_{c}^{\prime}=\int[\frac{1}{2}\pi_{i}^{2}+\frac{\pi_{\rho}^{2}}{2m^{2}}+\frac{1}{4}F_{ij}^{2}+\frac{m^{2}}{2}(A_{i}+\partial_{i}\rho)^{2}-A_{0}T_{2}^{\prime}]
{\zeta_{P\bar{\calK}}}(s)=\zeta_{\bar{\calV}}(s)+\frac{\Gamma(s+\frac{1}{2})}{\sqrt{\pi}\,\Gamma(s)}\left[\frac{2}{3^{s+\frac{1}{2}}}{}_{2}F_{1}[{\textstyle\frac{1}{2}},{s+\textstyle\frac{1}{2}},{\textstyle\frac{3}{2}},-{\textstyle\frac{1}{3}}]-\frac{1}{4^{s}}\frac{1}{s}\right]
\frac{K_{(\Lambda-1)/2}(k\epsilon)}{K_{(\Lambda+1)/2}(k\epsilon)}=\frac{k\epsilon}{\Lambda-1}\Bigl(1+\sum_{i=1}^{\infty}c_{i}k^{2i}\Bigr)+\frac{\Gamma\bigl((1-\Lambda)/2\bigr)}{\Gamma\bigl((1+\Lambda)/2\bigr)}\biggl(\frac{k\epsilon}{2}\biggr)^{\Lambda}+\cdots
S_{i}S_{j}W_{\alpha}\bar{W}_{\dot{\alpha}},\quad\bar{S}_{i}\bar{S}_{j}W_{\alpha}\bar{W}_{\dot{\alpha}},(S_{i}\bar{S}^{j}-{\frac{1}{3}}\delta_{i}^{j}S_{k}\bar{S}^{k})W_{\alpha}\bar{W}_{\dot{\alpha}}
(x\circy)(a_{1},\dots,a_{n+m-1})=\frac{1}{m!(n-1)!}\sum_{\sigma\inP_{n+m-1}}(-1)^{\sigma}x(a_{\sigma(1)},\dots,a_{\sigma(n-1)},y(a_{\sigma(n)},\dots,a_{\sigma(n+m-1)}))
{\calL}=\sqrt{(1-B_{3}^{2})-(1+B_{1}^{2})\phi_{t}^{2}-2B_{1}B_{2}\phi_{t}\phi_{x}+(1-B_{2}^{2})\phi_{x}^{2}+2B_{1}B_{3}\phi_{t}+2B_{2}B_{3}\phi_{x}},
^{^{\prime}}<\xi|\zeta>^{^{\prime}}=\sum_{n=0}^{p}\frac{\{n\}!}{\{p\}!}\xi^{p-n}(\zeta^{*})^{p-n}=\sum_{n=0}^{p}\frac{\{p-n\}!}{\{p\}!}\xi^{n}(\zeta^{*})^{n}
\begin{array}{lll}{\mathbf{f}_{\mu}(x)\;\longmapsto}&{\;\gamma_{\mu}\otimes\mathbf{1}}&{\inCl(T_{(x,z)}^{h}E\oplusT_{(x,z)}^{v}E)=Cl(T_{(x,z)}^{h}E,g)\otimesCl(T_{(x,z)}^{h}E,-g_{0})}\\{\mathbf{e}_{k}(x)\;\longmapsto}&{\;\omega\otimes\gamma_{k}}&{\inCl(T_{(x,z)}^{h}E\oplusT_{(x,z)}^{v}E)=Cl(T_{(x,z)}^{h}E,g)\otimesCl(T_{(x,z)}^{h}E,-g_{0})\,.}\\\end{array}
\lambdaf(\omega)\;=\;(\omega^{1/2}-\omega^{-1/2})\int_{1}^{\infty}{\frac{d\omega^{\prime}(\omega^{1/2}-\omega^{-1/2})}{(\omega^{\prime}+\omega)(\omega^{\prime}+\omega^{-1})}}{f(\omega^{\prime})}\;,
\sum_{a,\mu}\sum_{b,\nu}\left(\hat{A}_{\mu}^{a}\hat{A}_{\nu}^{b}\right)\left(\hat{A}^{b\mu}\hat{A}^{a\nu}\right)-\sum_{a,\mu}\sum_{b,\nu}\left(\left(\hat{A}_{\mu}^{a}\hat{A}_{\nu}^{b}\right)\hat{A}^{b\mu}\right)\hat{A}^{a\nu}=?
{\mathcalV}(z,\bar{z},q)\,=\,g^{2}\Bigl[\left(g_{ij^{\star}}k_{\Lambda}^{i}k_{\Sigma}^{j^{\star}}+4h_{uv}k_{\Lambda}^{u}k_{\Sigma}^{v}\right)\bar{L}^{\Lambda}L^{\Sigma}
{\tilde{Z}}_{+}(\beta_{1}){\tilde{Z}}_{0}(\beta_{2})={\tilde{S}}_{+,0}^{0,+}(\beta_{12}){\tilde{Z}}_{+}(\beta_{2}){\tilde{Z}}_{0}(\beta_{1})+{\tilde{S}}_{+,0}^{+,0}(\beta_{12}){\tilde{Z}}_{0}(\beta_{2}){\tilde{Z}}_{+}(\beta_{1}).
S_{YM}=-\,\frac{1}{4g_{\mathrm{YM}}^{2}}\intd^{4}x\,F_{\alpha\beta}^{A}\,F_{A}^{\alpha\beta}+\frac{\theta_{\mathrm{YM}}}{32\pi^{2}}\intd^{4}x\,F_{\alpha\beta}^{A}\,{\widetilde{F}}_{A}^{\alpha\beta}
\iota((\sum_{j=0}^{d}a_{j}^{1}s^{j}t^{d-j}:\cdots:\sum_{j=0}^{d}a_{j}^{N}s^{j}t^{d-j}))=(a_{0}^{1}:\cdots:a_{d}^{1}:a_{0}^{2}:\cdots:a_{d}^{2}:\cdots:a_{0}^{N}:\cdots:a_{d}^{N})
A_{-}=\frac12\left(\begin{array}{cc}{\omega_{-}}&{\frac\sigmar}\\{0}&{-\omega_{-}}\\\end{array}\right)\equivK_{-}\quad,\quad\tilde{A}_{+}=\frac12\left(\begin{array}{cc}{\omega_{+}}&{0}\\{-\frac\sigmar}&{-\omega_{+}}\\\end{array}\right)\equiv\tilde{K}_{+}\qquad,
\mathrm{Im}\sum_{n=0}^{\infty}\frac{(in)^{(r+s)/2}}{\frac{1}{2\sqrt{\alpha^{\prime}}}(2n+1)i-\beta\mu}\sim\mathrm{Re}\left[\beta^{(r+s)/2}(\sqrt{\alpha^{\prime}})^{(r+s)/2+1}\int_{0}^{\infty}dx\frac{(ix)^{(r+s)/2}}{x+i\mu}\right].
R_{\mu\nu}-\frac{1}{2}\,G_{\mu\nu}\,{\calR}-\frac{1}{2}\left(\partial_{\mu}\varphi\,\partial_{\nu}\varphi-\frac{1}{2}\,G_{\mu\nu}\,\partial_{\rho}\varphi\,\partial^{\rho}\varphi\right)=0
\hat{E}_{M}^{~~A}=\left(\begin{matrix}{\delta_{m}^{~~a}}&{0}\\{\frac{1}{2}(\gamma^{a})_{\mu}^{~~\beta}\theta_{\beta}}&{\delta_{\mu}^{~~\alpha}}\\\end{matrix}\right)=\left(\begin{matrix}{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{\theta}&{0}&{1}&{0}\\{0}&{-\bar{\theta}}&{0}&{1}\\\end{matrix}\right)
\tilde{S}_{E}={\frac{1}{2\kappa_{D}^{2}}}\intd^{D}x\sqrt{-G^{E}}\left[{\calR}_{G^{E}}-{\frac{4}{D-2}}(\partial\phi)^{2}-{\frac{1}{2\cdot(D-2)!}}e^{-2a\phi}F_{D-2}^{2}\right],
\left(\begin{array}{c}{\vec{F}^{(1)}}\\{-\vec{F}^{(2)}}\\\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}{1}&{1}\\{-1}&{1}\\\end{array}\right)\left(\begin{array}{c}{\vec{F^{R}}}\\{\vec{F}^{L}}\\\end{array}\right)\,.
[i\hbar\gamma^{\mu}\partial_{\mu}-{\frac{Mc}{\phi(x)}}{\frac{1+\gamma^{5}}{2}}-Mc\phi(x){\frac{1-\gamma^{5}}{2}}+i\hbar\gamma_{-}^{\mu}{\frac{1}{\phi(x)}}\partial_{\mu}\phi(x)]\psi^{D}(x)=0\,.
\delta_{t}{S}=\int_{t_{1}}^{t_{2}}dt\{[{L}_{q^{i}}]\delta_{t}q^{i}+[\frac{d}{dt}{H}+\frac{\partial}{\partialt}{L}]\deltat+\frac{d}{dt}(\frac{\partial{L}}{\partial\dot{q}^{j}}\delta_{t}q^{j}-{H}\deltat)\},
\begin{array}{lcl}{f(x)\starc=f.c+\sum_{\alpha=1}^{\infty}\frac{c}{\alpha!}(\frac{i}{2})^{\alpha}\prod_{i=1}^{\alpha}\theta^{\mu_{i}\nu_{i}}f_{\mu_{1}...\mu_{\alpha}}^{(\alpha)}\prod_{j=1}^{\alpha}\partial_{\nu_{j}}}\\\end{array}
ds^{2}=-{\biggl(}\frac{R^{2}}{l^{2}}-M{\biggr)}dt^{2}-Jdtd\varphi+{\biggl(}\frac{R^{2}}{l^{2}}-M+\frac{J^{2}}{4R^{2}}{\biggr)}^{-1}dR^{2}+R^{2}d\varphi^{2}\,.
\sqrt{-h}e^{-2\varphi}h^{\bar{\mu}\bar{\mu}^{\prime}}h^{\bar{\nu}\bar{\nu}^{\prime}}(ML)_{ij}{\calF}_{\bar{\mu}^{\prime}\bar{\nu}^{\prime}}^{j}=\epsilon^{\bar{\mu}\bar{\nu}\bar{\rho}}\partial_{\bar{\rho}}\psi^{i},
[\tilde{\pi}_{a}(x),\tilde{\theta}_{b}(y)]=i\delta_{ab}\Bigl(\delta(x-y)-\Omega^{-1}\Bigr)\,\qquad[\pi_{a}^{0},\tilde{\theta}_{b}]=[\tilde{\pi}^{a},\theta_{0}^{b}]=0\,\qquad[\pi_{a}^{0},\theta_{0}^{b}]=i\.
2\xiJ(x,\xi)\equiv\frac{dI(x,\xi)}{d\xi}=\int\frac{d^{4}p}{(2\pi)^{4}}e^{-ip\cdotx}\frac{2\xi|\vec{p}|^{2}}{-p_{0}^{2}+\xi^{2}|\vec{p}|^{2}-i\epsilon}
S=\frac{1}{2\pi}\int\!\sqrt{-g}d^{2}x\left[e^{-2\phi}(R+\Lambda)+\sum_{k=1}^{N}\bar{\psi}_{k}(i\gamma^{\mu}\nabla_{\mu}+\sigma)\psi_{k}-\frac{N}{2\lambda_{0}}\sigma^{2}\right]\,,
\left(\sigma_{i}F\right)^{(n)}\left(k_{1},k_{2},\ldots,k_{n}\right)=\sum_{l_{i},l_{i+1}\in{\calI}}{\hat{\calR}}\left(l_{i},l_{i+1};k_{i},k_{i+1}\right)F^{(n)}\left(k_{1},k_{2},\ldots,k_{i-1},l_{i},l_{i+1},k_{i+2},\ldots,k_{n}\right)\;,
{\calF}_{IIA}=-\frac{1}{6}C_{ABC}t^{A}t^{B}t^{C}-\frac{\chi\zeta(3)}{2(2\pi)^{3}}\sum_{d_{1}\dots,d_{n}}n_{d_{1},\dots,d_{n}}{\calL}_{i_{3}}e^{2\pii[\sum_{k}d_{k}t^{k}]},
\begin{array}{c}{N_{i}|jm\rangle=N_{i}|n_{1},n_{2}\rangle=n_{i}|n_{1},n_{2}\rangle}\\{N=N_{1}+N_{2}\\\\left[N,N_{i}\right]=0\\\\left[N_{1},N_{2}\right]=0}\\{\left[N_{i},a_{j}^{+}\right]=\delta_{ij}a_{i}^{+}\\\\left[N_{i},a_{j}\right]=-\delta_{ij}a_{i}}\\{\left[N,a_{i}^{+}\right]=a_{i}^{+}\\\\left[N,a_{i}\right]=-a_{i}.}\\\end{array}
m_{\mu\nu}{}^{i}{}_{k}m_{\rho\sigma}{}^{k}{}_{j}-m_{\rho\sigma}{}^{i}{}_{k}m_{\mu\nu}{}^{k}{}_{j}=-\eta_{\mu[\rho}m_{\sigma]\nu}{}^{i}{}_{j}+\eta_{\nu[\rho}m_{\sigma]\mu}{}^{i}{}_{j}\.
H\,=\,\intd^{2}z[\frac{1}{2}\pi_{j}^{T}\pi_{j}^{T}-\frac{\theta}{2}\pi_{j}^{T}\epsilon^{jk}A_{T}^{k}+\frac{1}{4}F^{ij}F^{ij}+\frac{\theta^{2}}{2}A_{T}^{j}A_{T}^{j}],
I=\int\!\!{\frac{d^{^{D}}\!p_{_{E}}}{(2\pi)^{^{D}}}}\Pi(p^{2})(D-1)=\int\!\!{\frac{d^{^{D}}\!p_{_{E}}}{(2\pi)^{^{D}}}}\Pi(p^{2}){\frac{\langle{F}_{\mu\nu}(p){F}_{\mu\nu}(-p)\rangle}{2}}.
\Re\left\{\frac{-x^{1-s}}{\pi}\,e^{\displaystyle{-i\frac{\pi}{2}(s+1)}}\int_{0}^{\infty}(y-\imath)^{-s}\,\displaystyle{\frac{I_{\nu}^{\prime}\left(x(y-\imath)\right)}{I_{\nu}\left(x(y-\imath)\right)}}\,dy\right\}.
\int\frac{\mathrm{d}^{3}{\bfQ}}{(2\pi)^{3}}\int\frac{\mathrm{d}z}{2\pii}N(z)\frac{N^{\mu\nu\lambda\sigma}({\bfk}_{i}=0,{\bfQ},z)}{(z^{2}-q^{2})^{4}},
\bar{F}_{p+1}=\sum_{i_{k},j_{k}}\frac{d}{dx^{+}}\bar{A}_{i_{1}\cdotsi_{p}}(-4x^{+})Q_{j_{1}}^{i_{1}}\cdotsQ_{j_{p}}^{i_{p}}dx^{+}\wedgedx^{j_{1}}\wedge\cdots\wedgedx^{j_{p}}.
h_{2}\simC_{1}e^{-m_{1}r}+C_{2}e^{-m_{2}r},\quad{\frac{1}{i\kappa}}h_{4}\sim{\frac{m_{1}{}^{2}-2e^{2}v^{2}}{\kappa^{2}}}C_{1}e^{-m_{1}r}+{\frac{m_{2}{}^{2}-2e^{2}v^{2}}{\kappa^{2}}}C_{2}e^{-m_{2}r},
{\tilde{\calL}}_{F}=4(D-2)^{2}\kappa^{2-2D}\intdre^{(D-3)A}\left({\frac{1}{W}}{\frac{\partialW}{\partial\phi}}\right)^{2}\bar{\tilde{\eta}}_{-i}\Gamma^{\underline{\mu}}\partial_{\mu}\tilde{\eta}_{-}^{i}+\ldots\,
\intd^{p}x\,\sqrt{g}\,C^{i}\Psi_{\omega}^{*}\partial_{i}\Psi_{\omega}=\Omega_{H}\intd^{p}x\,\sqrt{g}\,\Psi_{\omega}^{*}\partial_{\phi}\Psi_{\omega}\nonumber={\calO}(\omega^{2})
L_{\mathrm{CS}}={\frac{i\kappa\pi^{2}}{Im(\tau)}}\left(\dot{a}a^{*}-\dot{a}^{*}a\right)+i\kappa\int_{\Sigma}\left(\partial_{\bar{z}}\dot{\chi}\partial_{z}\chi^{*}-\partial_{z}\dot{\chi}^{*}\partial_{\bar{z}}\chi\right)
G_{MN}=\left(\begin{array}{cc}{g_{\mu\nu}+h_{ij}V_{\mu}^{i}V_{\nu}^{j}\,\,\,}&{V_{\mu}^{i}h_{ij}}\\{V_{\mu}^{i}h_{ij}}&{h_{ij}}\\\end{array}\right)\,\,.
{\calK}_{I}(x)\equiv-{\frac{i}{\hbar}}{\calL}_{I}\left(\Phi(x)+{\frac{i\hbar}{2}}{\frac{\delta}{\delta\Pi(x)}}\right)+{\frac{i}{\hbar}}{\calL}_{I}\left(\Phi(x)-{\frac{i\hbar}{2}}{\frac{\delta}{\delta\Pi(x)}}\right)\;.
\left\{\mathrm{Tr}\Big[\tau^{(2)}\tau^{(3)}\tau^{(\pi(2))}\tau^{(\pi(3))}...\tau^{(\pi(n))}\Big]-\mathrm{Tr}\Big[\tau^{(2)}\tau^{(\pi(2))}\tau^{(\pi(3))}...\tau^{(\pi(n))}\tau^{(3)}\Big]\right\}\;.
(\delta_{1}X_{2})_{\mathrm{extra}}={\frac{\tau_{1}\tau_{2}}{1-\tau_{1}\tau_{2}}}F_{1}M^{-1}\eta_{1}^{\dagger}MX_{2}+{\frac{t_{1}t_{2}}{t_{1}t_{2}-1}}X_{2}M_{0}^{-1}\epsilon_{1}^{\dagger}M_{0}.
{\frac{c_{n+1}}{c_{n}}}={\frac{\tilde{c}}{2(n+1)}}\Gamma(-\epsilon)\Gamma(3-\epsilon){\frac{\Gamma(1+\epsilonn+\epsilon)\Gamma(2-\epsilon-\epsilonn)}{\Gamma(2-2\epsilon-\epsilonn)\Gamma(1+\epsilonn)}}\;\;\;.
I_{0}[\beta]=-2e^{-2\phi_{h}}(1-{\frac{\beta}{\beta_{H}}})-{\frac{1}{2\pi}}\int_{M}^{}d^{2}x\sqrt{-g}e^{-2\phi}[R+4(\nabla\phi)^{2}+4\lambda^{2}]-{\frac{1}{\pi}}\int_{\partialM}^{}e^{-2\phi}kds
<0|T\left(\hat{\phi}_{(I)}(x)\hat{\phi}_{(I)}(y)\right)|0>=\int_{(\infty)}\frac{d^{4}k^{\mu}}{(2\pi)^{4}}\,\frac{i}{k^{2}-m^{2}+i\epsilon}\,e^{-ik\cdot(x-y)}\,
\begin{array}{cr}{the\,\,\,nonperturbative\,\,\,extension\,\,\,of\,\,\,\Gamma^{\mathrm{{(pert)}}}}&{\mathrm{}}\\{should\,\,\,continue\,\,\,to\,\,\,be\,\,\,perturbatively\,\,\,renormalizable.}&{\mathrm{}}\\\end{array}
\tilde{\calL}_{\mathrm{Bose}}=g_{A\bar{B}}\left(-\partial_{\mu}\lambda^{A}\partial^{\mu}\bar{\lambda}^{B}+E^{A}\bar{E}^{B}\right)+\left(E^{A}\frac{\partial{\calW}}{\partial\lambda^{A}}+\bar{E}^{A}\frac{\partial\bar{{\calW}}}{\partial\bar{\lambda}^{A}}\right)
<\Psi_{M}\Psi_{N}\Biggl[1-\frac{1}{16}[\varphi(V)-\varphi(U)]\biggl(P^{2}\Phi+i(P\Psi)\biggl)\Biggl]-\frac{i}{4}\Psi_{M}\Psi_{N}(\Psi\tilde{p})\Phi>=0\,,
L_{n}^{k,k,d}=\sum_{m=0}^{d-1}\sum_{\sigma_{m}\inP_{m}}(C_{m}^{d}(\sigma_{m})\cdot\prod_{i=1}^{l(\sigma_{m})}(\tilde{L}_{1}^{k,k,d_{i}}))\cdot(\tilde{L}_{n}^{k,k,d-m}-\tilde{L}_{1}^{k,k,d-m}),
\frac{1}{(q)_{\infty}}\sum_{r=-\infty}^{\infty}(-1)^{r}q^{r(Lr+L-2i)/2}=\displaystyle\sum_{n_{1}\geq\cdots\geqn_{k-1}\geq0}\frac{q^{n_{1}^{2}+\cdots+n_{k-1}^{2}+n_{i}+\cdots+n_{k-1}}}{(q)_{n_{1}-n_{2}}\cdots(q)_{n_{k-2}-n_{k-1}}(q^{a};q^{a})_{n_{k-1}}},
\left({h^{(\pm)},\h^{(0)}}\right)\equiv\left({\pmh,\h}\right);\quad\left({J^{(\pm)},\J^{(0)}}\right)\equiv\left({J,\K}\right);\quad\left({\kappa^{(\pm)},\\kappa^{(0)}}\right)\equiv\left({\kappa,\-\kappa}\right)\.
\mathcal{T}_{1,\kappa}^{\lambda}F_{n}^{\mathcal{O}}(\theta_{1},\ldots,\theta_{n})\simF_{\kappa}^{\mathcal{O}^{\prime}}(\theta_{1},\ldots,\theta_{\kappa})F_{n-\kappa}^{\mathcal{O}^{\prime\prime}}(\theta_{\kappa+1},\ldots,\theta_{n})\,\,.
R_{0}=\frac{1}{4}\left(\begin{array}{cc}{-\sigma_{3}+1}&{0}\\{0}&{\sigma_{3}-1}\\\end{array}\right)\;,\;\;\;R_{+}=\frac{1}{2}\left(\begin{array}{cc}{0}&{-\sigma_{3}+1}\\{0}&{0}\\\end{array}\right)\;,\;\;\;R_{-}=\frac{1}{2}\left(\begin{array}{cc}{0}&{0}\\{-\sigma_{3}+1}&{0}\\\end{array}\right),
\langle({\calO}_{j_{1}m_{1}}^{r_{1}~+}{\calO}_{j_{2}m_{2}}^{r_{2}~+}){\calO}_{j_{3}~{-m_{3}}}^{r_{3}~-}\rangle=\langle{\calO}_{j_{1}m_{1}}^{r_{1}~+}({\calO}_{j_{2}m_{2}}^{r_{2}~+}{\calO}_{j_{3}~{-m_{3}}}^{r_{3}~-})\rangle\nonumber
X_{1}={\frac{x_{14}}{x_{14}^{2}}}-{\frac{x_{24}}{x_{24}^{2}}}\;,\quadX_{2}={\frac{x_{14}}{x_{14}^{2}}}-{\frac{x_{34}}{x_{34}^{2}}}\;,\quadX_{3}={\frac{x_{34}}{x_{34}^{2}}}-{\frac{x_{24}}{x_{24}^{2}}}
\left[M_{\alphaA}^{\beta},M_{\alpha^{\prime}A^{\prime}}^{\beta^{\prime}\dagger}\right]+\left[D_{\alphaA}^{\beta^{\prime}\dagger},D_{\alpha^{\prime}A^{\prime}}^{\beta}\right]=-2ix_{jE}f_{EAA^{\prime}}(\delta_{\beta\beta^{\prime}}\Gamma_{\alpha\alpha^{\prime}}^{j}+\delta_{\alpha\alpha^{\prime}}\Gamma_{\beta\beta^{\prime}}^{j})\.
R_{\rho\mu\nu}^{\lambda}=\partial_{\mu}\Gamma_{\rho\nu}^{\lambda}-\partial_{\nu}\Gamma_{\rho\mu}^{\lambda}+\Gamma_{\sigma\mu}^{\lambda}\Gamma_{\rho\nu}^{\sigma}-\Gamma_{\sigma\nu}^{\lambda}\Gamma_{\rho\mu}^{\sigma}.
R=\left(\begin{array}{cccccccc}{a}&{a}&{0}&{0}&{0}&{0}&{0}&{0}\\{b}&{b}&{0}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{x}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{x}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{y}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{y}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}&{c}&{c}\\{0}&{0}&{0}&{0}&{0}&{0}&{d}&{d}\\\end{array}\right),
\begin{array}{l}{\mu(R)=p+\frac{1}{2}a_{ij}\,\alpha^{i}\wedge\alpha^{j}+q\,\,\alpha^{1}\wedge\alpha^{2}\wedge\alpha^{3}\wedge\alpha^{4},}\\{\mu(\hat{R})=q-\frac{1}{4}\epsilon^{ijkl}a_{ij}\,\beta_{k}\wedge\beta_{l}+p\,\,\beta_{1}\wedge\beta_{2}\wedge\beta_{3}\wedge\beta_{4}}\\\end{array}
G({\bfx},{\bfy},E)=\frac{m}{\pi\hbar^{2}}\int_{o}^{\infty}\frac{kdk}{q^{2}-k^{2}}\sum_{l=-\infty}^{il(\phi_{x}-\phi_{y})}J_{|l+\alpha|}(kr_{x})J_{|l+\alpha|}(kr_{y}).
P^{-}=\int_{-L}^{+L}\!\!dx^{-}\mathrm{Tr}\\bigl(\partial_{+}{\bfV}-{\bfD}_{-}{\bfA}\bigr)^{2}=\int_{-L}^{+L}\!\!dx^{-}\mathrm{Tr}\\bigl(\\partial_{+}{\bfV}\partial_{+}{\bfV}-g^{2}{\bfJ}^{+}{\frac{1}{{\bfD}_{-}^{2}}}{\bfJ}^{+}\\bigr)\.
I_{1}(0)={\frac{(23)^{2}}{x_{23}^{4}}}\intdu_{5}\left[{\frac{(12)^{2}(35)}{(25)}}\partial_{2}\cdot\partial_{2}+{\frac{(13)^{2}(25)}{(35)}}\partial_{3}\cdot\partial_{3}+(12)(13)\;2\partial_{2}\cdot\partial_{3}\right]f(1,2,1,3)\.
S_{T}=\int_{-\infty}^{\infty}[\;{\frac{1}{2}}(1+{\frac{x^{2}}{T^{2}}})\left({\partial_{x}\phi}\right)^{2}\;+\;{\frac{\lambda}{4}}{\frac{1}{(1+{\frac{x^{2}}{T^{2}}})}}\phi^{4}\;]\;dx.
Q^{ij}{}_{\mu\nu}=(\delta^{ij}\delta_{\mu\nu}-(\underline{{F}}^{2}\pmi\overline{{F}}^{2})^{-1}{}_{\mu\sigma}(F_{\sigma\lambda}^{i}F_{\lambda\nu}^{j}\pmi{*}F_{\sigma\lambda}^{i}F_{\lambda\nu}^{j}))
\rho(x)=\frac{1}{x^{2}}+\sum_{\{q\}\neq0}\{\frac{1}{(x-q)^{2}}-\frac{1}{q^{2}}-\frac{2q\cdotx}{q^{4}}-\frac{1}{q^{6}}(4(q\cdotx)^{2}-q^{2}x^{2})\}
M(x)=x^{-1/2}\left[C_{1}J_{\lambda}\left(\frac{\sqrt{4\alphac_{\epsilon}}}{\epsilonx^{\frac{\epsilon}{2}}}\right)+C_{2}J_{-{\lambda}}\left(\frac{\sqrt{4\alphac_{\epsilon}}}{\epsilonx^{\frac{\epsilon}{2}}}\right)\right]\;\;\;,
{\calO}_{\Sigma}^{bare}(\Phi^{bare},g^{bare})=\sum_{B}\,Z_{\Sigma}{}^{\Xi}{\calO}_{\Xi}^{ren}(Z_{3}^{\Phi}\Phi^{ren},{\frac{Z_{1}}{Z_{3}^{\Phi}{}^{3/2}}}g^{ren})
D^{a}=\bigg(1-\frac{g^{2}\epsilon(q)}{2Lp_{a}}V_{b}\bigg)\partial_{k}{\calA}_{k}^{\dagger}+\bigg(1+\frac{\epsilon(q)}{2}\bigg)\frac{g^{2}}{Lp_{a}}(\partial_{k}V_{b}){\calA}_{k}^{\dagger}+\frac{-ig}{\sqrt{L}}\frac{p_{c}+p_{b}}{p_{a}}{\calC}_{k}^{\dagger}{\calB}_{k};
ds^{2}=-\left(\frac{m-\rho^{2}}{m+\rho^{2}}\right)^{2}e^{2A}dt^{2}+\left(1+\frac{m}{\rho^{2}}\right)^{2}\left(e^{2B}(dr^{2}+dz^{2})+e^{2C}r^{2}d\Omega^{2}\right)