harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\left\{{\eta}_{\infty}+\sum_{i=1}^{8}A^{i}(a,\nu)\,\kappa_{i}\right\}}_{\Big\vert{\displaystyle{u=U(t,\,x,\,a,\,\nu)}}}=0\,,\quad\kappa_{i}\equiv{\eta}_{i}-{\xi}_{i}^{1}u_{t}-{\xi}_{i}^{2}u_{x}-{\xi}_{i}^{3}u_{a}-{\xi}_{i}^{4}u_{\nu}\,.
\Theta(Z,\psi,p)=\Theta(p)_{A(l_{1})B(l_{2})...C(l_{r-1})a(l_{r})}Z_{1}^{A(l_{1})}Z_{2}^{B(l_{2})}...Z_{r-1}^{C(l_{r-1})}\widetilde{\psi}^{a(l_{r})}\,
H_{0}(r)=-\frac{2r(r^{d-3}-\omegaM)}{2r^{d-3}-(d-1)\omegaM}H_{1}^{\prime}(r)-\frac{2(d-1)r^{d-3}-(d+1)\omegaM}{2r^{d-3}-(d-1)\omegaM}H_{1}(r).
(2b)=e^{4}\int{\calD}q\int{\calD}k\frac{(2k+q)^{\alpha}\varepsilon_{\alpha\beta\gamma}q^{\gamma}(2k+q-2p)^{\beta}(2k-p)_{\mu}(2k+2q-p)_{\nu}}{k^{2}(k+q)^{2}q^{2}(k-p)^{2}(q+k-p)^{2}}\;,
L=2\sqrt{-\mathrm{det}\Big(G_{\hat{\mu}\hat{\nu}}+iG_{\hat{\mu}\rho}G_{\hat{\nu}\lambda}\tilde{H}^{\rho\lambda}/\sqrt{-G_{5}}\Big)}+{\frac{1}{2}}\tilde{H}^{\mu\nu}\partial_{5}B_{\mu\nu}-{\frac{1}{4}}\epsilon_{\mu\nu\rho\lambda\sigma}{\frac{G^{5\rho}}{G^{55}}}\tilde{H}^{\mu\nu}\tilde{H}^{\lambda\sigma}.
y_{\pm}\simeq\frac{a}{t}\,\sum_{n=-\infty}^{\infty}\left[(n\pmb)^{2}+c^{2}\right]^{1/2},\\b=\frac{\betat}{4\pi}\sqrt{\rho(\lambda^{11}-\lambda^{22})},\\c^{2}=\frac{\rho\beta^{2}t^{2}(\lambda^{11}+\lambda^{22})}{16\pi^{2}}\simeq\frac{\alpha_{0}^{2}}{16\pi^{2}}.
l_{s}^{-2}ds^{2}=\hat{H}^{-1/4}\left[-\hat{f}dt^{2}+\sum_{i=1}^{5}(dx^{i})^{2}+\hat{H}\Big(\hat{f}^{-1}d\hat{r}^{2}+\hat{r}^{2}d\Omega_{3}^{2}\Big)\right]
{\calG}_{k}(U,U^{\prime})=-\frac{g_{YM}^{2}}{N}\frac{1+\gamma}{2\gammak^{2}}\frac{1}{UU^{\prime}}\left(\frac{U_{*}}{U_{>}}\right)^{-\gamma}\left[(\gamma-1)\left(\frac{U_{*}}{U_{<}}\right)^{\gamma}+(\gamma+1)\left(\frac{U_{*}}{U_{<}}\right)^{-\gamma}\right]~,
{\calFF}=\left(\begin{array}{cc}{({\calF}_{ev})^{2}-({\calF}_{od})_{1}({\calF}_{od})_{2}}&{{\calF}_{ev}({\calF}_{od})_{1}+({\calF}_{od})_{1}{\calF}_{ev}}\\{({\calF}_{od})_{2}{\calF}_{ev}+{\calF}_{ev}({\calF}_{od})_{2}}&{({\calF}_{ev})^{2}-({\calF}_{od})_{2}({\calF}_{od})_{1}}\\\end{array}\right).
\omega^{-1}(x)\tilde{F}_{\mu\nu}(x)\omega(x)=-\frac{2}{\bar{N}}\epsilon_{\mu\nu\rho\sigma}\int\!\delta\xids\,E^{\rho}[\xi|s]\dot{\xi}^{\sigma}(s)\dot{\xi}^{-2}(s)\delta(x-\xi(s)).
{\calL}^{3/2}=-\frac{e}{2}\bar{\Psi}_{\mu}\gamma^{\mu\rho\sigma}\nabla_{\rho}\Psi_{\sigma}+\frac{M}{2}\bar{\Psi}_{\mu}\gamma^{\mu\nu}\Psi_{\nu}+\bar{\Psi}^{\mu}J_{\mu}\,,
R_{a}^{i}=\left(\begin{matrix}{\partialT+2T\partial}&{-3/2\partialG-5/2G\partial}&{0}&{0}\\{\partialG+5/2G\partial}&{T^{2}}&{0}&{0}\\{\nabla^{(-1)}}&{-fT}&{0}&{2T^{2}}\\{\partialf-3/2f\partial}&{\nabla^{(-3/2)}}&{9\partialT+4T\partial}&{-15\partialG}\\\end{matrix}\right)\,
{\calA}(H_{i},H_{j},{\bar{H}}_{k},{\bar{H}}_{l})\propto\left({\frac{su}{t}}\,\delta_{il}\delta_{jk}+{\frac{st}{u}}\,\delta_{ik}\delta_{jl}+sK_{H_{j}\bar{H}_{k},H_{i}\bar{H}_{l}}\right).
\begin{array}{rll}{f_{0}}&{\longleftrightarrow}&{\tilde{f}_{0}^{-1}\;,}\\{f_{i}f_{0}^{-1/2}}&{\longleftrightarrow}&{\tilde{f}_{k-i}\tilde{f}_{0}^{-1/2}\;\;\;;i=1,\ldots,k-1\;.}\\\end{array}
{{\calG}^{-1}}_{\mua}^{\nub}\eta_{\nub}^{\mathrm{L,T}}=\frac{1}{2}\left(k^{2}-\Lambda_{l}+\frac{1}{2r^{2}}(N+1)(N-2)-\frac{N+d}{N+d-2}\Lambda\right)\eta_{\mua}^{\mathrm{L,T}},
(F,G)=\frac{F\stackrel{\leftarrow}{\partial}}{\partial\phi^{A}}\,\frac{\stackrel{\rightarrow}{\partial}G}{\partial\phi_{A}^{*}}-\frac{F\stackrel{\leftarrow}{\partial}}{\partial\phi_{A}^{*}}\,\frac{\stackrel{\rightarrow}{\partial}G}{\partial\phi^{A}}\,.
\Theta_{l}=\frac{\partial(\tilde{T}_{a}\Psi^{(0)})}{\partiala^{l}}\left|_{a=0}\right.=\left(\begin{array}{c}{D_{l}^{(0)}\phi^{(0)}}\\{(D_{l}^{(0)}\phi^{(0)})^{\ast}}\\{F_{lk}^{(0)}}\\\end{array}\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].
F_{\mathrm{min}}\left(\theta\right)=\kappa\frac{\left\{\sigma^{2n}I,N_{1}\right\}\left\{\sigma^{2n}\sigma_{0},N_{1}\right\}}{\left\{\sigma^{2n+1}I,N_{1}\right\}\left\{\sigma^{2n}\sigma_{1},N_{1}\right\}}
\stackrel{o}{A_{3}^{+}}=v\;,\quadx^{+}=t\;,\quadw^{2}={\frac{\stackrel{o}{J_{+}^{+}}\stackrel{o}{J_{-}^{+}}}{g^{2}}}\quad\mathrm{{and}}\quad\stackrel{o}{J_{3}^{-}}={\frac{B}{2}}\;.
\sigma({\bfq})=\left\{N_{0}\left(1-e^{-q^{2}\overline{{u_{t}^{2}}}}\right)+e^{-q^{2}\overline{{u_{t}^{2}}}}\left|\sum_{n}e^{i{\bfqr}_{n0}}\right|^{2}\right\}\sigma_{0}({\bfq}),
\begin{array}{rcl}{d\left(\hat{G}_{(5)}+\hat{H}\hat{C}_{(2)}\right)}&{=}&{0\,,}\\{}&{}&{}\\{d\left(\hat{\overline{{G}}}_{(5)}+\hat{H}\hat{\overline{{C}}}_{(2)}\right)}&{=}&{0\,,}\\\end{array}
J_{\mathrm{sym}}=-\intd^{2}x\,\biggl\{\epsilon_{ij}x^{i}\biggl[\dot{f}\partial_{j}f+f^{2}(\dot{\theta}+A_{0})(\partial_{j}\theta+A_{j})\biggr]+\frac{\rho_{e}}{2\kappa}x^{2}\,(\kappaF_{12}-\rho_{e})\biggr\}.
\rho_{\mathrm{rad}}^{3}\to{\calU}^{\dagger}\rho_{\mathrm{rad}}^{3}{\calU}=\frac{1}{g}(\vec{D}\hat{\phi})^{a}\vec{E}^{a}=\frac{1}{g}{\calG}_{\mathrm{rad}}^{a}\hat{\phi}^{a}\.
\varphi_{n_{1}}^{\,\,m_{1}}\star\left(\varphi^{\dagger}\right)_{m_{2}}^{\,\,n_{2}}\star\varphi_{n_{3}}^{\,\,m_{3}}=\delta_{\,\,m_{2}}^{m_{1}}\,\delta_{\,\,n_{3}}^{n_{2}}\,\varphi_{n_{1}}^{\,\,m_{3}},.
\tilde{S}_{\mathrm{IF}}[\phi,\phi^{\prime};\xi_{1},\xi_{2},\xi_{3}]=\mathrm{Re}\left(S_{\mathrm{IF}}[\phi,\phi^{\prime}]\right)+\sum_{N=1}^{3}{\frac{1}{N}}\int_{t_{i}}^{t}d^{4}x\;R_{N}^{-}(x)\xi_{N}(x).
\bar{l}(e_{0})=n^{\kappa}{\calG}_{\kappa\mu}^{(r)}\star\bar{K}_{\quad\nu}^{\mu}\left(\gamma^{\nu}+\frac{ge_{0}}{4}\bar{B}_{\alpha\beta}\gamma^{[\nu}\gamma^{\alpha}\gamma^{\beta]}\right)\bar{\Lambda}(e_{0}).
\begin{array}{ll}{\delta\phi_{1}=\bar{\zeta}\psi_{1}\,,}&{\delta\phi_{2}=\bar{\zeta}\psi_{2}\,,}\\{\delta\psi_{1}=-\imath\partial_{m}\phi_{1}\gamma^{m}\zeta+F_{1}\zeta\,,}&{\delta\psi_{2}=-\imath\partial_{m}\phi_{2}\gamma^{m}\zeta+F_{2}\zeta\,,}\\{\deltaF_{1}=-\imath\bar{\zeta}\gamma^{m}\partial_{m}\psi_{1}\,,}&{\deltaF_{2}=-\imath\bar{\zeta}\gamma^{m}\partial_{m}\psi_{2},}\\\end{array}
ds^{2}=\left({\frac{r^{2}}{b^{2}}}+1-{\frac{w_{p+2}M}{r^{p-1}}}\right)d\tau^{2}+\left({\frac{r^{2}}{b^{2}}}+1-{\frac{w_{p+2}M}{r^{p-1}}}\right)^{-1}dr^{2}+r^{2}d\Omega_{p},
\langlej_{\mu}^{top}(x)j_{\nu}^{top}(y)\rangle~=~(2s)^{2}\langlej_{\mu}^{N}(x)j_{\nu}^{N}(y)\rangle~+~i{\frac{s}{\pi}}\epsilon_{\mu\nu\lambda}\partial_{\lambda}\delta^{3}(x-y)
\xi_{\bfp}^{(1)}(\alpha,{\bfn}){\Omega}_{{\jmath\ell}m}({\bfn}_{p})=D^{\dagger(1)}({\bfn})\,A({\alpha},{\bfn})\,\frac{m}{2}\frac{\xi_{\bfp}^{(0)}({\alpha},{\bfn})}{p_{0}+m}{\Omega}_{{\jmath\ell}m}({\bfn}_{p}),
\psi_{1}=\psi^{s}-\gamma^{5}\psi^{a}\,,\,\,\,\psi_{2}=\psi^{a}+\gamma^{5}\psi^{s}\,,\psi_{3}\equiv\gamma^{5}\psi_{1}\,,\,\,\,\psi_{4}\equiv\gamma^{5}\psi_{2}\,.
{\hat{x}}^{i}(\varepsilon)=e^{i{\hat{H}}\varepsilon}{\hat{x}}^{i}(0)e^{-i{\hat{H}}\varepsilon}\simeq{\hat{x}}^{i}(0)+{\frac{{\hat{\pi}}^{i}(0)}{m}}\varepsilon+{\frac{e}{4m^{2}}}(2{\hat{F}}^{ik}{\hat{\pi}}_{k}(0)-i\partial_{k}{\hat{F}}^{ik})\varepsilon^{2}\,,
\Theta^{(D)}(N_{1},N_{2},N_{3})={\frac{\Gamma(g+2p)\Gamma(p+1)}{\Gamma(g+p+1)\Gamma(2p)}}\prod_{i=1}^{3}\left({\frac{(N_{i}+p)\Gamma(g-N_{i}+p)}{\Gamma(p+1)\Gamma(g-N_{i}+1)}}\right).
A_{(0)}=\frac{1}{R_{T}^{2}}\frac{(d-3)^{2}}{k_{d}^{\frac{2}{d-3}}}\frac{F_{d-2}^{2\frac{d-2}{d-3}}}{(\partial_{r}F_{d-2})^{2}+(\partial_{z}F_{d-2})^{2}}
\phi_{n,m}\left(z_{1},\overline{{z}}_{1}\right)\phi_{1,2}\left(z_{2},\overline{{z}}_{2}\right)\sim\frac{C_{nm,12}^{n,m-1}\phi_{n,m-1}\left(z_{2},\overline{{z}}_{2}\right)}{\midz_{1}-z_{2}\mid^{2\left(h_{nm}+h_{12}-h_{n,m-1}\right)}}+\frac{C_{nm,12}^{n,m+1}\phi_{n,m+1}\left(z_{2},\overline{{z}}_{2}\right)}{\midz_{1}-z_{2}\mid^{2\left(h_{nm}+h_{12}-h_{n,m+1}\right)}}
\begin{array}{|c|c|c|c|}\hline{}&{G_{1}}&{G_{2}}&{G_{3}}\\\hline{-}&{\mathrm{isometry}}&{\mathrm{BMN}}&{\mathrm{isometry,isometry}}\\\hline{H_{1}}&{\mathrm{circle}}&{\mathrm{MRV}}&{\mathrm{isometry,circle}}\\\hline{H_{1},H_{2}}&{\mathrm{circle,DLCQ}}&{}&{\mathrm{circle,circle}}\\\hline{H_{2}}&{\mathrm{isometry,DLCQ}}&{\mathrm{IKM,GO}}&{}\\\hline{}\\\end{array}.
{\calH}=\intd^{3}x\left[{\frac{1}{4}}F_{ij}^{a}F^{aij}-{\frac{1}{2}}F_{0i}^{a}F^{a0i}+{\frac{1}{2}}D_{i}\Phi^{a}D^{i}\Phi^{a}-{\frac{1}{2}}D_{0}\Phi^{a}D^{0}\Phi^{a}\right]
\bar{W}_{21}=\frac{2n}{2n+1}R_{n}\left(\mu\right)\mu^{2}\left(1-\mu\right)^{2}\left[\left(\frac{1-\mu}{2-\mu}\right)^{2}+\left(n-1\right)\delta\right]\bar{S^{2}},\;\frac{1}{2}<\delta<1\,,
\dot{\varrho}+3\left(\varrho+{\frac{\wp}{\bar{c}_{4}^{2}}}\right){\frac{\dot{\bar{a}}_{R}}{\bar{a}_{R}}}=-\varrho{\frac{\dot{\bar{G}}_{4}}{\bar{G}_{4}}}+{\frac{3k\bar{c}_{4}\dot{\bar{c}}_{4}}{4\pi\bar{G}_{4}\bar{a}_{R}^{2}}}.
\theta(t_{1}-t_{2})\left[\begin{array}{cc}{0}&{\frac{\theta(-n-\frac12)t_{1}^{n}}{t_{2}^{n+1}}}\\{\frac{\theta(n+\frac12)t_{2}^{n}}{t_{1}^{n+1}}}&{0}\\\end{array}\right]-\theta(t_{2}-t_{1})\left[\begin{array}{cc}{0}&{\frac{\theta(n+\frac12)t_{1}^{n}}{t_{2}^{n+1}}}\\{\frac{\theta(-n-\frac12)t_{2}^{n}}{t_{1}^{n+1}}}&{0}\\\end{array}\right]
\mu\frac{\partial}{\partial\mu}e^{-\Gamma}=\int[d\Phi]e^{-S[\Phi]+i\intd^{4}xJ_{\mu}(x)B_{\mu}(x)}\intd^{4}z\left[-\frac{3N_{c}-N_{f}(1-\gamma)}{32\pi^{2}}{}~(F_{\mu\nu}^{a})^{2}+\frac{1}{4}q(B_{\mu\nu})^{2}\right]
M^{ij}=\frac{1}{L_{x}L_{y}(1+\phi^{2}/T^{2})}\left(\begin{array}{cc}{L_{y}/L_{x}}&{-\phi/T-(1+\phi^{2}/T^{2})T\theta}\\{\phi/T+(1+\phi^{2}/T^{2})T\theta}&{L_{x}/L_{y}}\\\end{array}\right).
\pi(x)={\frac{\partial{\calL}_{\sigma}}{\partial\partial_{+}\psi(x)}}=i\bar{\psi}\gamma^{+}=\left(\begin{matrix}{\psi_{R}^{*}}&{\psi_{L}^{*}}\\\end{matrix}\right)\left(\begin{matrix}{i\sqrt{1+s}}&{0}\\{0}&{i\sqrt{1-s}}\\\end{matrix}\right).
\frac{A^{2n+2}}{V^{2n+1}}=\frac{2\pi}{(n+1)}{\mathrm{Vol{(M)}}}\,2^{2n+1}\,(n+1)^{2n+1}\frac{1}{\left(1-\frac{a^{2n+2}}{r^{2n+2}}\right)^{n}}
\left(\frac{1+ix}{1-ix}\right)^{1/3}=\sum_{e\geq0}A_{e}x^{e}+i\sum_{o>0}A_{o}x^{o}\,,~~~~\left(\frac{1+ix}{1-ix}\right)^{2/3}=\sum_{e\geq0}B_{e}x^{e}+i\sum_{o>0}B_{o}x^{o}\,.
\rho_{11}^{[0,2,0]\,\nu\{\mu_{1}\ldots\mu_{s}\}}=(1^{-}2)(12)\,\partial_{1}^{\nu}\left\{\frac{1}{(x_{12}^{2})^{2-\frac{d-s}{2}}(x_{1}^{2}x_{2}^{2})^{\frac{d-s}{2}}}\,V^{\{\mu_{1}}\ldotsV^{\mu_{s}\}}\right\}
\begin{array}{cc}{S^{2}\leftrightarrowM_{0}}\\{V^{2}\leftrightarrow\bar{M_{0}}}\\{S^{2}V^{2}\leftrightarrowM_{2}}\\{S^{2}V^{4}\leftrightarrowM_{4}}\\{S^{4}V^{2}\leftrightarrow\bar{M_{4}}}\\\end{array}
|\kappa,\Omega\rangle(z)=\sqrt{A_{1}(\kappa)}\,\sum_{m=1}^{\infty}V_{m-1}^{(1)}(\kappa)\,\frac{z^{m}}{m}\equiv\sum_{m=1}^{\infty}\langle\kappa,1|m-1,1\rangle\,\frac{z^{m}}{\sqrt{m}}.
\theta_{a_{1}}\theta_{a_{2}}\ldots\theta_{a_{n}}\in\Lambda_{n}\longleftrightarrowF^{c_{1}c_{2}\ldotsc_{n}}=\frac{1}{n!}\varepsilon^{c_{1}c_{2}\ldotsc_{n}}{}_{b_{1}b_{2}\ldotsb_{n}}\varepsilon^{b_{1}b_{2}\ldotsb_{n}}{}_{a_{1}a_{2}\ldotsa_{n}}.\nonumber
{\calL}=\frac{1}{2}\left\{\partial_{\mu}\phi^{a}\,\partial^{\mu}\phi^{a}+i\bar{\psi}^{a}\gamma^{\mu}\partial_{\mu}\psi^{a}-\frac{\partial{\calW}}{\partial\phi^{a}}\,\frac{\partial{\calW}}{\partial\phi^{a}}-\frac{\partial^{2}{\calW}}{\partial\phi^{a}\partial\phi^{b}}\,\bar{\psi}^{a}\psi^{b}\right\}\,.
m^{2}\alpha^{2}=2\sum_{n}\left[\left((n+mH\alpha^{\prime})^{2}+n^{2}\right)A_{1n}A_{1n}^{\dagger}+\left((n-mH\alpha^{\prime})^{2}+n^{2}\right)A_{2n}A_{2n}^{\dagger}\right],
\Pi_{2}~=~\frac{a^{2}(1)a^{2}(2\mu-2)}{(\mu-2)^{2}}\left[\frac{\hat{\Psi}^{2}(\mu)}{2}~+~\frac{\hat{\Phi}(\mu)}{2}~+~\frac{\hat{\Psi}(\mu)}{(\mu-2)}-3\hat{\Theta}(\mu)\right]\eta_{1}^{2}
{\displaystyle\int}{\calD}y\,\mathrm{exp}\Bigl[-\int_{0}^{T}d\tau{\frac{1}{4}}{\dot{y}}^{2}\Bigr]=\mathrm{Det^{\prime}}_{P}^{-{\frac{1}{2}}}\bigl[-{\partial}_{\tau}^{2}\bigr]={\lbrack4\piT\rbrack}^{-{\frac{D}{2}}}\,
\langle\Psi_{k}\,|\,k\,^{\prime}\rangle_{\epsilon}=c_{k}\,\frac{2^{\epsilon+ik\,^{\prime}}}{4\sqrt{2\pi}}\,\Gamma\left(\frac{\epsilon+ik\,^{\prime}-ik}{2}\right)\Gamma\left(\frac{\epsilon+ik\,^{\prime}+ik}{2}\right),
S_{renorm}^{(6)}=-\left(m_{exp}+a^{-\frac32}\frac{e^{2}\sqrt{\pi}}6\right)\intd\tau\sqrt{\dot{x}^{2}}-a^{-\frac12}\frac{e^{2}\sqrt{\pi}}{32}\intd\tau\sqrt{\dot{x}^{2}}(D^{2}x)^{2}
\begin{array}{l}{(T_{1}^{l})^{\dagger}T_{1}^{l}|\Lambda_{L}\rangle=c_{l}(T_{1}^{l-1})^{\dagger}T_{1}^{l-1}|\Lambda_{L}\rangle\,,}\\{{}}\\{(T_{2}^{l}T_{1}^{m})^{\dagger}T_{2}^{l}T_{1}^{m}|\Lambda_{L}\rangle=c_{l+4}(T_{2}^{l-1}T_{1}^{m})^{\dagger}T_{2}^{l-1}T_{1}^{m}|\Lambda_{L}\rangle\,,}\\\end{array}
\eta=\left[{\frac{m+1}{2n}}+{\frac{3m^{2}+7m+6}{8n^{2}}}+O\left({\frac{1}{n^{3}}}\right)\right]\tilde{\epsilon}-\left[{\frac{m+1}{2n}}+{\frac{3(m+1)^{2}}{4n^{2}}}+O\left({\frac{1}{n^{3}}}\right)\right]\tilde{\epsilon}^{2}+O(\tilde{\epsilon}^{3}).
\hat{D}_{3}=D_{3}-\displaystyle{{\frac{\hbar^{2}}{8m}}}\partial_{x}^{2}\rho^{2},\\\\\hat{F}_{3}=F_{3}-\displaystyle{{\frac{\hbar^{2}}{8m}}}\partial_{x}^{2}(\sum_{k=1}^{N}J_{k}),\\\\,hat{D}_{2}=D_{2},\\\\,hat{F}_{2}=F_{2},
{\calN}_{Q}[{\calH}_{x}]={\frac{1}{2}}\Pi_{x}^{2}+{\frac{1}{2}}(\nabla\phi_{x})^{2}-{\frac{1}{2}}I_{0}[Q^{2}]+{\frac{1}{4}}Q^{2}I_{1}[Q^{2}]+{\calN}_{Q}[V(\phi_{x})]\;\;,
\mathrm{Im}\,\Big\langle\bar{D}^{2}\Big(\phi^{*}e^{2V}\phi+\tilde{\phi}^{*}e^{-2V}\tilde{\phi}\Big)\Big\rangle=\frac{1}{2\pi^{2}}\mathrm{Im}\Big(W_{a}C^{ab}W_{b}\Big).
\left.\frac{d}{d\tau}\right|_{\tau=0}x^{\alpha}(g_{A}(\tau)g)=\left.\frac{\partialx^{\alpha}(g_{A}(\tau)g)}{\partialx^{M}(g_{A}(\tau))}\right|_{g_{A}=e}\cdot\left.\frac{dx^{M}(g_{A}(\tau))}{d\tau}\right|_{\tau=0}=R_{M}^{\alpha}(g)\delta_{A}^{M}=R_{A}^{\alpha}(g).
M-M_{0}=\frac{\tilde{z}h}{2p_{0}Iw_{n=1}}\{1+(\frac{\tilde{z}\hat{J}_{-}(0)h}{4ip_{0}I\hat{J}_{-}(iw_{n=1})\tilde{a}_{n=1}})^{\frac{2w_{j=1}}{w_{n=1}}}g\}+\dots\
\langle{k_{y}},\vec{k},{k_{y}^{\prime}},\vec{k^{\prime}},\mathrm{out}|0,\mathrm{in}\rangle={\frac{i\mu^{2}}{2m^{2}H_{5}V}}(2\pi)^{3}\delta^{3}(\vec{k}+\vec{k}^{\prime})\delta_{k_{y}+k_{y}^{\prime},0}\,{\frac{(k_{y}/H_{5})}{m^{2}\mathrm{sinh}(k_{y}\pi/H_{5})}}.
m^{2}\left(\left(\dot{x}_{1}\right)^{2}+\left(\dot{x}_{2}\right)^{2}\right)=k_{1}^{2},\quadm^{2}\left(\left(\dot{x}_{0}\right)^{2}-\left(\dot{x}_{3}\right)^{2}\right)=m^{2}+\left(k_{1}\right)^{2},
\int_{a}^{b}{f(x)dx}=R[f(z),g(z)]-\frac{1}{2}\int_{-i\infty}^{+i\infty}{\left[g(u)+{\mathrm{sgn}}({{\mathrm{Im}}}z)f(u)\right]_{u=a+z}^{u=b+z}dz,}
\begin{aligned}{a}&{{}=a_{1}\left(\frac{1}{2}a_{3}-\frac{1}{48}a_{1}^{3}\right),}\\{b}&{{}=\frac{1}{4}a_{3}^{2}+\frac{1}{864}a_{1}^{6}-\frac{1}{24}a_{1}^{3}a_{3},}\\{\delta}&{{}=\frac{1}{16}a_{3}^{3}(27a_{3}-a_{1}^{3}).}\\\end{aligned}
\tilde{L_{1}}(t)(\alpha^{*},\alpha)=\frac{g^{2}}{2}\left(\frac{1}{3!}V_{abc}W^{abc}+\alpha_{a}^{*}(V_{\;\;cd}^{a}W_{\;\;\;b}^{cd}+V_{bcd}W^{cda})\alpha^{b}\right)
Q_{\calM}=\left(\begin{array}{ccccccccc}{1}&{1}&{-3}&{1}&{0}&{0}&{0}&{0}\\{0}&{0}&{1}&{-2}&{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}&{-2}&{1}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}&{-2}&{1}&{0}\\{0}&{0}&{0}&{0}&{0}&{1}&{-2}&{1}\\\end{array}\right).
F_{m}^{\dagger}(\phi)=F_{m}(\phi),\;\;\;\;\;\;\tilde{F}_{m}^{\dagger}(\tilde{\phi})=\tilde{F}_{m}(\tilde{\phi}),\;\;\;\;\;\;\tilde{F}_{m}(\tilde{\phi})\mid0\rangle=F_{m}(\phi)\mid0\rangle
\left\{\sum_{n=0}^{\infty}\frac{(-1)\\rho_{n}^{(2)}(s)}{(t-n-\frac{1}{2})^{2}}-\sum_{n=0}^{\infty}\frac{(-1)^{2}\2!\r_{n}^{(1)}(s)}{(s-n-\frac{1}{2})^{3}}\right\}_{s=0,\t=0}=0.
H_{D}^{fer}=\frac{1}{\sqrt{2}}\intd^{3}x\xi^{\dagger}(\vec{x})\left({\calG}_{(-)}[V_{-}]\ast\xi\right)(\vec{x})+\frac12\frac{e^{2}}{2L}\intd^{2}x_{\perp}\Gamma^{i}(x_{\perp})\left({\calG}_{(\perp)}[{\calM}^{2}]\ast\Gamma^{i}\right)(x_{\perp})
\gamma^{0}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)\,,\qquad\gamma^{1}=\left(\begin{array}{cc}{0}&{1}\\{-1}&{0}\\\end{array}\right)\,,\qquad\gamma^{2}=\left(\begin{array}{cc}{0}&{i}\\{i}&{0}\\\end{array}\right)
\mathrm{(a1)}+\mathrm{(a2)}+\mathrm{(a3)}-i\delta_{\mathrm{T}}(q^{2}g_{\mu\nu}-q_{\mu}q_{\nu})\delta^{AB}-i\frac{\delta_{\mathrm{L}}}\lambdaq_{\mu}q_{\nu}\delta^{AB}\equiv0,
\left[\Bigr({\frac{1}{2}}({\mathrm{Tr}}K)\delta_{\nu}^{\mu}+i\gamma_{5}{\widehatK}^{\mu}n_{\nu}+i\gamma_{5}n^{\mu}{\widehatK}_{\nu}+n_{\alpha}\nabla^{\alpha}\delta_{\nu}^{\mu}\Bigr){\calP}_{+}\kappa_{\mu}\right]_{\partialM}=0\;,
q_{c}=\frac{1}{8\pi^{2}a^{4}}\sum_{m=-\infty}^{+\infty}\int_{-\infty}^{+\infty}{dk\sum_{\lambda=0}^{1}\sum_{n=1}^{\infty}\frac{j_{m,n}^{(\lambda)3}\psi_{\mu}(j_{m,n}^{(\lambda)}/a)}{\sqrt{k^{2}a^{2}+j_{m,n}^{(\lambda)2}}}T_{m}(j_{m,n}^{(\lambda)})f_{m}^{(q)}(j_{m,n}^{(\lambda)}x)},\quadq=\varepsilon,p_{1},p_{2}
\partial_{t}\left(\frac{1}{n}ba^{3}\dot{\Phi}\right)-\partial_{y}\left(\frac{1}{b}a^{3}n\Phi^{\prime}\right)+ba^{3}n\left[V^{\prime}+V_{0}^{\prime}\delta(by)+V_{1}^{\prime}\delta(b(y-1))\right]=0.
F(\tilde{p}\sqrt{\alpha^{\prime}})=F(\alpha^{\prime}\tilde{s},\alpha^{\prime}\tilde{t})\approx(\alpha^{\prime}\tilde{s})^{2+\frac{1}{2}\alpha^{\prime}\tilde{t}}\frac{\Gamma(-\frac{1}{4}\alpha^{\prime}\tilde{t})}{\Gamma(1+\frac{1}{4}\alpha^{\prime}\tilde{t})}\.
H_{D}^{F}=\tilde{\lambda}^{\mu}(\tau)\tilde{H}_{\mu}(\tau)-\frac12\tilde{\lambda}^{\mu\nu}(\tau)\tilde{H}_{\mu\nu}(\tau)+\int{d^{3}\sigma\Big[-A_{\tau}(\tau,\vec{\sigma})\Gamma(\tau,\vec{\sigma})+\mu_{\tau}(\tau,\vec{\sigma})\pi^{\tau}(\tau,\vec{\sigma})\Big]}
\int{\calD}[A]{\calD}[\Lambda]{\calD}\lambda\quade^{\int{\calD}A_{h}F(A+A_{h},\psi,\overline{\psi},\phi)}e^{(A,d\lambda+i\astd\Lambda)}=e^{\int{\calD}A_{h}F(A_{h},\psi,\overline{\psi},\phi)}.
\left(\begin{array}{c}{\frac{\Gamma(\frac{1}{2}+M)\Gamma(\frac{1}{2}+M)}{\Gamma(-\frac{1}{2}+M)\Gamma(\frac{3}{2}+M)}z^{\frac{M}{2}}(1-z)^{-\frac{M}{2}}F(1,-1,\frac{1}{2}-M;1-z)}\\{\pm\frac{i}{\frac{1}{2}+M}z^{-\frac{M}{2}}(1-z)^{\frac{1-M}{2}}F(\frac{3}{2}-M,-\frac{1}{2}-M,\frac{3}{2}-M;1-z)}\\\end{array}\right),
{W^{(n)}}_{c}^{\pm}=\frac{\scriptstyle{V^{(n)}}_{c}^{\pm}-{\displaystyle\sum_{k=3}^{n-1}}2\binom{n-1}{k-1}{W^{(k)}}_{c}^{\pm}\left({W^{(n-k+2)}}_{c}^{\pm}-{W^{(n-k)}}_{c}^{\pm}\right)+2(n-1){W^{(2)}}_{c}^{\pm}{W^{(n-2)}}_{c}^{\pm}}{\scriptstyle2\left(n{W^{(2)}}_{c}^{\pm}-W_{c}^{\pm}\right)}\.
\hat{\calW}_{\mu}\toU_{\alpha}U_{\beta}\hat{\calW}_{\mu}U_{\beta}^{-1}U_{\alpha}^{-1}+i(\partial_{\mu}U_{\alpha})U_{\alpha}^{-1}+iU_{\alpha}(\partial_{\mu}U_{\beta})U_{\beta}^{-1}U_{\alpha}^{-1}.
\phi_{A}^{\ast\,a}\,{\overline{\delta}}_{a}\phi^{A}+{\frac{1}{2}}{\bar{\phi}}_{A}{\overline{\delta}}_{2}{\overline{\delta}}_{1}\phi^{A}\,+\,{\frac{1}{2}}\epsilon_{ab}\phi_{A}^{\ast\,a}\,\pi_{A}^{b}
G(x,x^{\prime})=i\int\frac{d^{3}q}{(2\pi)^{3}}e^{i{\bfq}.({\bfx}-{\bfx}^{\prime})}\left(\begin{array}{ll}{G^{++}({\bfq},t-t^{\prime})}&{\:\:G^{+-}({\bfq},t-t^{\prime})}\\{G^{-+}({\bfq},t-t^{\prime})}&{\:\:G^{--}({\bfq},t-t^{\prime})}\\\end{array}\right)\:,
2\left(p^{2}\frac{\partial}{\partialp^{2}}+m^{2}\frac{\partial}{\partialm^{2}}\right)t^{(2)}(p;m^{2})=\beta_{m}^{(1)}u^{(1)}(p;m^{2})\frac{m^{2}}{p^{2}+m^{2}}
\frac{dM_{susy}}{dm}=\frac{d(\deltaM_{susy})}{dm}+\frac{\hbar}{2}\left(\underset{N_{B_{-}},\neq\nu_{B}^{\prime}+1,...\nu_{B}-1}{\overset{N_{B_{+}}}{\sum}}\frac{d\omega_{B}^{K}}{dm}-\underset{N_{F_{-}},\neq\nu_{F}^{\prime}+1,...\nu_{F}-1}{\overset{N_{F_{+}}}{\sum}}\frac{d\omega_{F}^{K}}{dm}\right)
\partial_{J}^{\,I}=u_{i}^{I}{\frac{\partial}{\partialu_{i}^{J}}}-u_{J}^{i}{\frac{\partial}{\partialu_{I}^{i}}}-{\frac{1}{N}}\sum_{K=1}^{N}\delta_{J}^{\,I}\left(u_{i}^{K}{\frac{\partial}{\partialu_{i}^{K}}}-u_{K}^{i}{\frac{\partial}{\partialu_{K}^{i}}}\right)\;.
{\bfG}_{1}=-{\frac{i}{\sqrt{2}}}\left(\begin{array}{ccc}{0}&{1}&{0}\\{1}&{0}&{0}\\{0}&{0}&{0}\\\end{array}\right),\quad{\bfG}_{2}=-{\frac{i}{\sqrt{2}}}\left(\begin{array}{ccc}{0}&{-i}&{0}\\{i}&{0}&{0}\\{0}&{0}&{0}\\\end{array}\right),\quad{\bfG}_{3}=-{\frac{i}{\sqrt{2}}}\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{-1}&{0}\\{0}&{0}&{0}\\\end{array}\right),
<\frac{1}{2},m_{s}|{\psi}_{F}^{\pm}>=\Big[\sum_{j,m}{\xi}_{m}^{j\pm}T_{m{\mp}}^{j}\Big]T_{m_{s}{\pm}}^{\frac{1}{2}},\\,{\xi}_{m}^{j\pm}{\in}{\bfC},m_{s}=+\frac{1}{2},-\frac{1}{2}.
\psi^{\prime\prime}+[3{\calH}-2\frac{\varphi^{\prime\prime}}{\varphi^{\prime}}]\psi^{\prime}+[4{\calH}^{\prime}-4{\calH}\frac{\varphi^{\prime\prime}}{\varphi^{\prime}}]\psi-\partial_{\alpha}\partial^{\alpha}\psi=0.
\begin{array}{l}{\mathrm{S}_{\mathrm{2}}=\int{(dx)(c(h_{\mu\nu}\partial_{\nu\lambda}^{4}h_{\lambda\mu}}-\frac{1}{2}h_{\mu\lambda}\partial^{2}\partial_{\mu\nu}^{2}h_{\nu\lambda}))+}\\{h_{\mu\sigma}\partial_{\mu\nu}\partial_{\lambda\sigma}^{3}h_{\nu\lambda}-(h_{\mu\sigma}\partial^{2}\partial_{\mu\nu}\partial_{\lambda\sigma}h_{\nu\lambda})+(1+c)hG_{4}h)}\\\end{array}
\mathrm{g}(\Omega_{1}+i\Omega_{2})=\left(\begin{array}{cc}{\Omega_{1}\Omega_{2}^{-1}\Omega_{1}+\Omega_{2}}&{-\Omega_{1}\Omega_{2}^{-1}}\\{-\Omega_{2}^{-1}\Omega_{1}}&{\Omega_{2}^{-1}}\\\end{array}\right).
\lambda_{\mathrm{R}}\xi_{\mathrm{R}}(\xi_{\mathrm{R}}-1)\left(\xi_{\mathrm{R}}-{\frac{1}{2}}\right)=0,\quad\lambda_{\mathrm{R}}\left[\frac{13}6-\frac{\lambda_{\mathrm{R}}}{2}+\lambda_{\mathrm{R}}\xi_{\mathrm{R}}(1-\xi_{\mathrm{R}})\right]=0
\sum_{n=0}^{\infty}\frac{n!}{(n+\ell)!}L_{n}^{\ell}(x)L_{n+1}^{\ell-1}(x)=\frac{e^{x}}{2x^{\ell}}\left(1-2\frac{\gamma(\ell,x)}{\Gamma(\ell)}\right)\,,
\left(\begin{array}{cc}{u}&{i\alpha}\\{i\beta}&{\bar{u}}\\\end{array}\right)=\left(\begin{array}{cc}{e^{i\psi}\sqrt{1+t^{2}}}&{ie^{r_{-}\phi}t}\\{ie^{-r_{-}\phi}t}&{e^{-i\psi}\sqrt{1+t^{2}}}\\\end{array}\right)\,.
\left(\begin{array}{cc}{\,\,\,u}&{a}\\{-b}&{v}\\\end{array}\right)=\left(\begin{array}{cc}{\,\,\,\,u_{L}}&{a_{L}}\\{-b_{L}}&{v_{L}}\\\end{array}\right)\left(\begin{array}{cc}{v_{R}}&{-a_{R}}\\{b_{R}}&{\,\,u_{R}}\\\end{array}\right)
{\calA}=N\int_{z_{-}}^{z_{1}}\!dz\,{\frac{1-z}{z(1+z/N)^{2}}}{\frac{\bigl[1-1/N-z/N^{2}\bigr]\bigl[1+z/(N(1-z))\bigr]}{\displaystyle\sqrt{1-N^{2}(1+z/N)^{2N}/(z^{2}r^{2N})}}}\,\cdotp
u{\otimes}_{\bfA}u{\otimes}_{\bfA}\ldots{\otimes}_{\bfA}u={\tau}_{{\mu}_{1}}^{(1)}a_{{\mu}_{1}}{\tau}_{{\mu}_{2}}^{(2)}a_{{\mu}_{2}}\ldots{\tau}_{{\mu}_{N}}^{(N)}a_{{\mu}_{N}}.
S_{B}=\intd^{2}x\left[{\frac12}({\partial}_{\mu}\phi)({\partial}^{\mu}\phi)+eA^{\mu}({\epsilon}_{{\mu}{\nu}}-rg_{{\mu}{\nu}}){\partial}^{\nu}{\phi}+{\frac12}e^{2}aA_{\mu}A^{\mu}-{\frac14}F_{{\mu}{\nu}}F^{{\mu}{\nu}}\right],
f(x)=-i\frac{x^{2}}{\pi}\int_{0}^{\pi}g(-ix\mathrm{cos}\,\theta)\,\mathrm{cos}\,\theta\,d\theta=-i\frac{x^{2}}{\pi}\int_{0}^{\pi/2}\left[g(-ix\mathrm{cos}\,\theta)\,-g(ix\mathrm{cos}\,\theta)\right]\,\mathrm{cos}\,\theta\,d\theta.
S_{\mathrm{quadr.}}\sim\int\!\mathrm{d}^{2}\sigma\,\left(-\delta\dot{\rho}^{2}+\delta{\rho^{\prime}}^{2}+\delta\rho^{2}-\delta\dot{\theta}^{2}+\delta{\theta^{\prime}}^{2}-\delta\theta^{2}\right)\,.
G\left(z^{\prime}\right)=G\left(z\right)+\sum_{b\,\in\,z-\left(z\capz^{\prime}\right)}\left[\#\sigma\left(z^{\prime},b\right)-1\right]\cdote_{{\calB}\left(z\right)}\left(b\right)+\sum_{b\,\in\,z-\left(z\capz^{\prime}\right)}G_{{\calB}\left(z^{\prime},b\right)}\quad.
S_{int}(\phi_{\pm})=\intd^{4}x[-V_{\Lambda}(\phi_{\pm})+\frac{1}{2}Z_{\Lambda}(\phi_{\pm})\dot{\phi}_{\pm}^{2}-\frac{1}{2}Y_{\Lambda}(\phi_{\pm})({\vec{\nabla}}\phi_{\pm})^{2}+\ldots]
\frac{1}{\alpha^{\prime}}\biggl(\frac{1}{g^{4}}+\frac{s^{2}}{g^{2}}\biggr)^{1/2}\ll\frac{N^{1/2}}{g^{3/2}\alpha^{\prime}}\quad\Rightarrow\quad1+g^{2}s^{2}\llgN\.
S_{eff}^{2+1}=-2\pi\intd\tau\sqrt{g_{\tau\tau}}\;\;\;,\;\;\;g_{\tau\tau}=(\frac{d\xi^{0}}{d\tau})^{2}-h_{\alpha\beta}(\xi)\frac{d\xi^{\alpha}}{d\tau}\frac{d\xi^{\beta}}{d\tau}\;\;,
\begin{array}{c}{Unknown}\\{(6,2,1)}\\\end{array}\,\,\,d\hat{s}_{IIB}^{2}=H\left[H^{-1/2}\eta_{ij}dy^{i}dy^{j}-H^{1/2}dx^{2}\right]-H^{-1/2}dy^{2}-H^{-3/2}dz^{2}\,.
{\cal{R}}_{(\eta)}^{\vec{\alpha}}({\bf{r}})=\prod_{m=1}^{\eta}{\cal{X}}^{\alpha[m]}({\bf{r}})\;\;\;\mathrm{with}\;\;\;{\cal{X}}^{\alpha}({\bf{r}})=[\,{\textstyle\frac{\partial_{i}}{\partial^{2}}}A_{i}^{\alpha}({\bf{r}})\,]\;.
S_{d}=\intd^{d}x[\sqrt{\mathrm{det}(\delta_{\mu\nu}+2\pi\alpha^{\prime}\tilde{F}_{\mu\nu})}+\frac{1}{2}\midD_{\mu}\varphi\mid^{2}+\lambda(\mid\varphi\mid^{2})^{2}+\frac{1}{2}i\Lambda^{\mu\nu}(\tilde{F}_{\mu\nu}-2\partial_{\mu}A_{\nu})],
\ddot{a}=\left(\frac{\left(2+r\right)r}{\left(1+r\right)^{2}}\right)^{2}a^{\prime\prime}+\left(2\frac{\left(2+r\right)^{2}}{\left(1+r\right)^{3}}-\frac{4+r}{\left(1+r\right)^{2}}\right)a^{\prime}-\kappa^{2}r^{2}f^{2}a.
\omega_{\mu}{}^{ab}(e)=-e^{\nu[a}\left(\partial_{\mu}e_{\nu}{}^{b]}-\partial_{\nu}e_{\mu}{}^{b]}\right)-e^{\rho[a}e^{\sigmab]}\left(\partial_{\sigma}e_{c\rho}\right)e_{\mu}{}^{c}\.
\frac{C_{\mu4}-B_{\mu7}}{\sqrt{2}}\,\\,frac{B_{\mu4}+C_{\mu7}}{\sqrt{2}}\,\\,frac{C_{\mu5}-B_{\mu8}}{\sqrt{2}}\,\\,frac{B_{\mu5}+C_{\mu8}}{\sqrt{2}}\,\\,frac{C_{\mu6}-B_{\mu9}}{\sqrt{2}}\,\\,frac{B_{\mu6}+C_{\mu9}}{\sqrt{2}}\.