harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{rcl}{\mathrm{integral}}&{\approx}&{\displaystyle-2(4\pi)^{-4}p^{2}\epsilon\int_{0}^{\infty}dy~y(y+\epsilon)^{-3}\int_{0}^{1}dx~K_{0}\left(\sqrt{(1-x)(xy+\epsilon)y}\right),}\\\end{array}
f=f_{1}f_{2}f_{n}=\left(1+{\frac{r_{1}^{2}}{r^{2}}}\right)\left(1+{\frac{r_{5}^{2}}{r^{2}}}\right)\left(1+{\frac{r_{n}^{2}}{r^{2}}}\right),~~~h=\left(1-{\frac{r_{0}^{2}}{r^{2}}}\right).
\{\phi_{i}(x,t),\Pi_{j}(x^{\prime},t)\}=\delta_{ij}\left[\frac{1}{2\pi}+\frac{1}{\pi}{\Large\sum_{n\geq1}}\mathrm{cos}(nx^{\prime})\mathrm{cos}(nx)\right]=\delta_{ij}\delta(x-x^{\prime}),
eT_{\mu\nu}={\tilde{\overline{\Psi}}_{L}}{\gamma_{\mu}}i\left(D_{\nu}-\frac{1}{2}\Gamma_{\alpha\nu}^{\alpha}\right){\tilde{\Psi}}_{L}-\frac{i}{2}g_{\mu\nu}\partial_{\alpha}({\tilde{\overline{\Psi}}}_{L}\gamma^{\alpha}{\tilde{\Psi}_{L}}).\nonumber\,
Q=+{\frac{\pi}{2}}-\rho\int_{0}^{n^{2}-1}{\frac{du^{\prime\prime}}{u^{\prime\prime}}}{\frac{\sqrt{n^{2}-1-u^{\prime\prime}}\sqrt{u^{\prime\prime}}}{(n^{2}-1-u^{\prime\prime})+\rho^{2}u^{\prime\prime}}}.
\begin{array}{l}{\Lambda(u,\{v_{j}\})=(u(u-d))^{N}\displaystyle{\prod_{k=1}^{M}}S_{1}(u-v_{k}-1/2)S_{-1}(u-v_{k}-1)-}\\{-((u-1)(u-d))^{N}\displaystyle{\prod_{k=1}^{M}}S_{1}(u-v_{k})-(u(u-d+1))^{N}\displaystyle{\prod_{k=1}^{M}}S_{-1}(u-v_{k}-d)\,,}\\\end{array}
|\zeta_{i}\rangle=e^{\zeta_{i}J_{+}}|-j\rangle=\sum_{\mu=-j}^{j}\left[\frac{(2j)!}{(j+\mu)!(j-\mu)!}\right]^{1/2}\zeta^{j+\mu}|\mu\rangle\equiv\sum_{\mu=-j}^{j}c_{\mu}|\mu\rangle.
G=\left(\begin{matrix}{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}\\\end{matrix}\right),\qquad\theta=\sqrt{-1}\left(\begin{matrix}{0}&{a}&{0}&{0}\\{-a}&{0}&{0}&{0}\\{0}&{0}&{0}&{b}\\{0}&{0}&{-b}&{0}\\\end{matrix}\right).
W^{\mu}(x)=\sum_{k}\sum_{r=0}^{3}\frac{1}{(2Vw_{k})^{1/2}}\left[\epsilon_{r}^{\mu}({\bfk})a_{r}({\bfk})e^{-ikx}+\epsilon_{r}^{\mu}({\bfk})b_{r}^{\dagger}({\bfk})e^{ikx}\right]
\pi_{\mu}=\frac{\partial\mathrm{L}}{\partial\dot{n}^{\mu}}=\lambda\dot{n}_{\mu}+m[\dot{y}+\dot{n}^{\nu}S_{\nu}y]^{T}\frac{\partial}{\partial\dot{n}^{\mu}}[\dot{y}+\dot{n}^{\nu}S_{\nu}y]=\lambda\dot{n}_{\mu}-y^{T}S_{\mu}\mathrm{p}
{\mathcalJ}^{0}(x)\;=\;\frac{q}{2\pi}\;\mathrm{div}\,\underline{{\varphi}}(x)~,~~\underline{{{\mathcalJ}}}(x)\;=\;-\,\frac{q}{2\pi}~\frac{\partial}{\partialt}~\underline{{\varphi}}(x)~,
T\left(\begin{array}{cc}{e^{\sigma}}&{0}\\{0}&{e^{-\sigma}}\\\end{array}\right)\left(\begin{array}{cc}{I}&{0}\\{-B}&{I}\\\end{array}\right)\left(\begin{array}{c}{\omega}\\{\mu}\\\end{array}\right)=T\left(\begin{array}{c}{e^{+\sigma}\omega}\\{e^{-\sigma}(\mu-B\omega)}\\\end{array}\right)\;,
\frac{d}{dT}\left[T\left(L_{\mathrm{eff}}(\omega)+E+\frac{\omega}{2}\right)\right]=L_{\mathrm{eff}}(\omega)+E+\frac{\omega}{2}+T\frac{d\omega}{dT}\left(\frac{dL_{\mathrm{eff}}}{d\omega}+\frac{1}{2}\right)=0\,.
ds^{2}=\frac{1}{1-\xi^{0}\xi^{0}}d\xi^{0}d\xi^{0}-(1-\xi^{0}\xi^{0})\left[(1-\rho^{2})^{-2}d\rho^{2}+(1-\rho^{2})^{-1}\rho^{2}d{\bfu}d{\bfu}^{\prime}\right]~,
{\calV}=e^{\varphi_{3}/2}\left[\begin{matrix}{1}&{-\chi_{3}}\\{0}&{e^{-\varphi_{3}}}\\\end{matrix}\right]\otimese^{\varphi_{1}/2}\left[\begin{matrix}{1}&{-\chi_{1}}\\{0}&{e^{-\varphi_{1}}}\\\end{matrix}\right],
\int_{\mathrm{CPV}}dk_{0}{\frac{P(k,p)}{(k^{2}+i\epsilon)[(k+p^{2}+i\epsilon]\zeta^{\alpha}}}={\frac{1}{2}}\biggl(\int_{C}+\int_{C^{\prime}}\biggr)dk_{0}{\frac{P(k,p)}{(k^{2}+i\epsilon)[(k+p^{2}+i\epsilon]\zeta^{\alpha}}}.
{\cal{M}}=\frac{1}{\tau_{2}\rho_{2}}\left(\begin{array}{cccc}{1}&{-\tau_{1}}&{-\tau_{1}\rho_{1}}&{-\rho_{1}}\\{-\tau_{1}}&{|\tau|^{2}}&{\rho_{1}|\tau|^{2}}&{\tau_{1}\rho_{1}}\\{-\tau_{1}\rho_{1}}&{\rho_{1}|\tau|^{2}}&{|\tau|^{2}|\rho|^{2}}&{\tau_{1}|\rho|^{2}}\\{-\rho_{1}}&{\tau_{1}\rho_{1}}&{\tau_{1}|\rho|^{2}}&{|\rho|^{2}}\\\end{array}\right)
G^{ij}=\frac{1}{g_{1}(1-e^{2})}(-\delta_{0}^{i}\delta_{0}^{j}+\delta_{1}^{i}\delta_{1}^{j})+\frac{1}{g_{2}(1+b^{2})}(\delta_{2}^{i}\delta_{2}^{j}+\delta_{3}^{i}\delta_{3}^{j}),
F^{\alpha\beta,\mu\nu}(\nabla)=\sqrt{-g}C^{\alpha\beta,\lambda\sigma}\left\{\Box\delta_{(\lambda}^{\mu}\delta_{\sigma)}^{\nu}-\frac{2\lambda}{1+\lambda}\delta_{(\lambda}^{(\mu}\nabla_{\sigma)}\nabla^{\nu)}+\frac{\lambda}{1+\lambda}g^{\mu\nu}\nabla_{(\lambda}\nabla_{\sigma)}+P_{\lambda\sigma}^{\mu\nu}+M_{\lambda\sigma}^{\mu\nu}\right\},
\bar{\calR}^{\frac{1}{2}\,z\,\omega,\frac{1}{2}\,z\,\omega}\,\sim\,\left(\begin{array}{cccc}{t}&{0}&{0}&{0}\\{0}&{s^{-1}}&{t-t^{-1}}&{0}\\{0}&{0}&{s}&{0}\\{0}&{0}&{0}&{-t^{-1}}\\\end{array}\right)\,.
h_{1}(\epsilon,a)=\int_{0}^{\infty}dt{\frac{t^{-\epsilon}}{\sqrt{t^{2}+a^{2}}}}\left[\sum_{n=-\infty}^{+\infty}{\frac{\sqrt{t^{2}+a^{2}}}{t^{2}+a^{2}+4\pi^{2}n^{2}}}-{\frac{1}{2}}\right]\equivI_{\epsilon}^{(1)}(a^{2})+I_{\epsilon}^{(2)}(a^{2})\;.
\times\,\mathrm{exp}\Bigg\{\sum_{\alpha=0}^{N-1}\left[\frac{i}{2\hbar\epsilon}(r_{\alpha+1}-r_{\alpha})^{2}-\frac{1}{2}i\hbar\epsilon(\ell^{2}-\frac{1}{4})\frac{1}{r_{\alpha}^{2}}-\frac{i\epsilon}{2\hbar}\omega^{2}r_{\alpha}^{2}\right]\Bigg\}\\\.
\begin{aligned}{R(g,A)_{\mu\nu\rho\sigma}}&{{}=R(g)_{\mu\nu\rho\sigma}-\frac{1}{3}D_{[\mu}A^{\phi}\sqrt{g}\,\epsilon_{\nu]\rho\sigma\phi}}\\{}&{{}\quad+\frac{1}{18}\big[g_{\mu[\rho}A_{\sigma]}A_{\nu}-g_{\nu[\rho}A_{\sigma]}A_{\mu}-g_{\mu[\rho}g_{\sigma]\nu}A^{\lambda}A_{\lambda}\big]\,.}\\\end{aligned}
\tilde{\Gamma}_{3\;\mu_{1}\mu_{2}\mu_{3}}^{a_{1}a_{2}a_{3}}\left(k_{1},k_{2},k_{3}\right)=-\displaystyle{\frac{g^{3}}{4\,\pi^{2}}}\displaystyle{\mathrm{ln}\left({T}\right)}\displaystyle{\int\frac{\mathrm{d}\Omega}{4\pi}}\;\;{\calL}_{\mu_{1}\mu_{2}\mu_{3}}^{a_{1}a_{2}a_{3}}\left(k_{1},k_{2},k_{3},\hat{Q}\right)\nonumber,
\frac{1}{(kz)^{3}}\left(-q^{2}-\partial_{z}^{2}+\frac{3}{z}\partial_{z}+\frac{m^{2}}{(kz)^{2}}+\frac{\lambda_{v2}}{z_{v}}\delta(z-z_{v})+\frac{\lambda_{h2}}{z_{h}}\delta(z-z_{h})\right)\chi_{q}(z)=0
\left(a\astb\right)\left(\left\langle\omega\cdot\omega^{\prime}\right\rangle\right)=\int_{\mathbb{S}_{d-1}}d\omega^{\prime\prime}a\left(\left\langle\omega\cdot\omega^{\prime\prime}\right\rangle\right)b\left(\left\langle\omega^{\prime\prime}\cdot\omega^{\prime}\right\rangle\right),\tag{3.17}
\,\,\,Z_{2}\left(\Phi^{\pm}\right)=\pm\Phi^{\pm},\,\,\,\,\,\,\,\,Z_{2}\left(\Phi_{\alpha}^{s}\right)=-\Phi_{\alpha}^{s},\,\,\,\,\,\,\,Z_{2}\left(\Phi_{\alpha}^{c}\right)=+\Phi_{\alpha}^{c}.
g_{\mathrm{mj}}\;\ddot{x}^{\mathrm{j}}+\biggl({\frac{\partialg_{\mathrm{ml}}}{\partialx^{\mathrm{k}}}}-{\frac{1}{2}}{\frac{\partialg_{\mathrm{kl}}}{\partialx^{\mathrm{m}}}}\biggr)\dot{x}^{\mathrm{k}}\dot{x}^{\mathrm{l}}=-{\frac{\partialV}{\partialx^{\mathrm{m}}}}
\bar{u}_{2}^{\prime}(p_{2}^{\prime})\vec{\gamma}u_{2}(p_{2})=c^{2}(\frac{1}{c}){\omega_{2}^{\prime}}^{\ast}\left[(2\vec{p}_{2}-\vec{q})-i((\vec{q})_{x}\hat{j}-(\vec{q})_{y}\hat{i})\right]\omega_{2}~~.
H_{eff}=\frac{\hbar^{2}}{2m}\intdx\left\{\left\vert\left(\partial_{x}+\nu\intdyK(x-y)(\rho(y)+\eta\tau(y))\right)\psi\right\vert^{2}+\vert\partial\chi\vert^{2}-\etag\tau^{2}\right\}\,,
S=\frac{1}{g^{2}}Tr(-\frac{1}{4}[A_{i},A_{j}][A^{i},A^{j}]+\frac{2}{3}i\alpha\epsilon_{ijk}A^{i}A^{j}A^{k}+\frac{1}{2}\bar{\psi}\sigma^{i}[A_{i},\psi]+\alpha\bar{\psi}\psi).
\mathrm{Im}I_{\mathrm{cl.\,tach.}}(u)=\frac{V_{26-D}}{(\sqrt{16\pi^{2}})^{26-D}}(e^{-a}(Z_{0}(u))^{D})^{2}\frac{1}{(4\pi)^{D/2}}e^{-4/u}\frac{2^{D/2-2}}{\Gamma(D/2)}\,.
\delta^{(1)}e^{ighB_{0}}=\sum_{N=0}^{\infty}{\frac{(igh)^{N}}{N!}}\sum_{l=0}^{N-1}B_{0}^{l}{\frac{ig}{2}}[\delta\omega_{m+1n}+\delta\omega_{mn},B_{0}]B_{0}^{N-l-1},
d_{C}\alpha=\sum_{s=1}^{p}F^{\alpha_{1}}\wedge\dots\wedgeF^{\alpha_{s}}dv_{\alpha_{1}\dots\alpha_{s}}-F^{\beta}i_{\beta}\Omega-\sum_{s=1}^{p}F^{\beta}\wedgeF^{\alpha_{1}}\wedge\dots\wedgeF^{\alpha_{s}}i_{\beta}v_{\alpha_{1}\dots\alpha_{s}}=0\quad,
(\lambda^{i})^{C}\equiv\alpha^{-1}B^{-1}\varepsilon^{ij}(\lambda^{j})^{*}\,,\qquad\bar{\lambda}^{iC}\equiv\overline{{(\lambda^{i})^{C}}}=\alpha^{-1}\left(\bar{\lambda}{}^{k}\right)^{*}B\varepsilon^{ki}\,,
{\calH}=\frac{1}{2}(m_{\alpha}+m_{\beta})\left(\frac{m_{\alpha}Q_{\alpha}+m_{\beta}}{m_{\alpha}+m_{\beta}}\right)^{2}+2\frac{m_{\alpha}m_{\beta}}{m_{\alpha}+m_{\beta}}\left(\frac{Q_{\alpha}-Q_{\beta}}{2}\right)^{2}+{\calH}_{\mathrm{rel}}\left(\frac{Q_{\alpha}-Q_{\beta}}{2};\,r\right).
{\left\{\sum_{Q}(p^{2}-4m_{Q}^{2})m_{Q}^{2}[K_{Q}(p)+H_{Q}(p)]\right\}}^{2}+{\left\{\sum_{Q}(p^{2}-4m_{Q}^{2})m_{Q}^{2}S_{Q}(p)\right\}}^{2}={\left\{\sum_{Q}(p^{2}-4m_{Q}^{2})m_{Q}^{2}R_{Q}(p)\right\}}^{2}.
\left[A_{\kappa\rho},A^{\lambda\sigma}\right]=\delta_{\kappa}^{\lambda}{M^{\sigma}}_{\rho}-\delta_{\kappa}^{\sigma}{M^{\lambda}}_{\rho}-\delta_{\rho}^{\lambda}{M^{\sigma}}_{\kappa}+\delta_{\rho}^{\sigma}{M^{\lambda}}_{\kappa}\,.
A^{g}~=~\int_{M_{g}}\langle\int\prod_{i=1}^{2g}d^{2}x_{i}V_{A}(x_{1})V_{A}(x_{2})\prod_{i=3}^{2g}V_{\lambda}^{\alpha}(x_{i})\prod_{k=1}^{3g-3}|(\mu_{k}b)|^{2}e^{\varphi}G^{-}(z_{k}){\rangle}_{g}~.
\vert{\nu_{1}},{\nu_{2}}\rangle={e^{{{\nu}_{1}}{e_{13}}}}{e^{{{\nu}_{2}}{e_{23}}}}\left(\begin{matrix}{0}\\{0}\\{1}\\\end{matrix}\right)=\left(\begin{matrix}{{{\nu}_{1}}}\\{{{\nu}_{2}}}\\{1}\\\end{matrix}\right)
\rho(t,\vec{x})=\stackrel{0}{\rho}(t)+\stackrel{1}{\rho}(t,\vec{x})\quad,\quadu^{\mu}(t,\vec{x})=\stackrel{0}{{u}\strut^{\mu}}+{\stackrel{1}{u}}\strut^{\mu}(t,\vec{x})\quad,
\phi_{1}\circ\phi_{2}(\alpha_{1},\ldots,\alpha_{n_{1}+n_{2}-1})=\sum_{i}(-1)^{\epsilon_{i}}\phi_{1}(\alpha_{1},\ldots,\alpha_{i},\phi_{2}(\alpha_{i+1},\ldots,\alpha_{i+n_{2}}),\ldots,\alpha_{n_{1}+n_{2}-1}).
W_{\Gamma}=a_{2}x_{3}^{6}x_{4}^{6}+a_{3}x_{3}^{3}x_{4}^{12}+a_{5}x_{0}^{3}+a_{6}x_{1}^{3}+a_{8}x_{3}^{9}+a_{9}x_{4}^{18}=0.
\left.\frac{\{v_{2n-1}-u_{2n-1}-\frac{1}{2}-k\}[u_{2n}-v_{2n-1}-\frac{1}{2}]}{\{k+1\}}\right|_{w_{2n-1}=xz_{2n}}=-\frac{\{u_{2n}-u_{2n-1}-k\}[1]}{\{k+1\}},
S={\frac{1}{2\kappa_{\scriptscriptstyle{D}}^{2}}}\intd^{\scriptscriptstyle{D}}x\sqrt{-g_{\scriptscriptstyle{D}}}\,e^{-\varphi}\left\{R_{\scriptscriptstyle{D}}+(\nabla\varphi)^{2}-{\frac{1}{12}}H^{2}\right\}\,,
V^{(0)}={\frac{m}{2}}\left[p_{i}(t)p^{i}(t)+\omega^{2}q_{i}(t)q^{i}(t)\right]+eA_{0}(t,\vec{q})+2\theta\int\,d^{2}x\,\epsilon_{ij}\partial^{i}A^{j}(t,\vec{x})A_{0}(t,\vec{x})\;.
\bar{q}_{+}^{(0)\mu\nu}k_{\mu}k_{\nu}\bar{f}_{+(T)}=\frac{\kappa_{5}^{2}e^{-2A_{+}-\alpha_{+}^{(0)}}}{\int_{w_{-}}^{w_{+}}dw^{\prime}e^{-3A(w^{\prime})}}\bar{\tau}_{+(T)}
\begin{array}{rcl}{Q_{1}}&{=}&{S_{+}C^{+}+S_{-}C^{-}+\displaystyle\frac{\partialG(S_{3})}{\partialS_{3}}C^{3}+2i\displaystyle\frac{\partial^{2}G(S_{3})}{\partial^{2}S_{3}}(\bar{C}_{-}C^{-}C^{3}-\bar{C}_{+}C^{+}C^{3})}\\{}&{}&{+i\bar{C}_{3}C^{+}C^{-}-4\displaystyle\frac{\partial^{3}G(S_{3})}{\partial^{3}S_{3}}\bar{C}_{+}\bar{C}_{-}C^{+}C^{-}C^{3}~.}\\\end{array}
d\langleE_{AB},w_{{\calM}_{0}}^{(a)}\rangle=\frac{g^{2}\lambda}{8\pi}d\langleE_{AB},w_{\mathrm{flat}}^{(a)}\rangle-\frac{\bar{\mu}}{2}d\left\langleE_{AB},\epsilon^{abc}\left(\sum_{A}dr_{A}^{b}\right)\wedge\left(\sum_{B}dr_{B}^{c}\right)\right\rangle.
\mathcal{S}=\mathcal{S}_{cs}+\intd^{3}x\;\left(b^{a}\partialA^{a}+\partial^{\mu}\overline{{c}}^{a}(D_{\mu}c)^{a}+\;\hat{A}_{\mu}^{*}(D^{\mu}c)^{a}-\frac12\;f_{abc}C^{*a}c^{b}c^{c}\right)\;,
\begin{array}{l}{m_{11}^{2}=\mu^{2}-\frac{15}{4}+\muc_{33}}\\{m_{33}^{2}=\mu^{2}-\frac{15}{4}+\muc_{11}}\\{m_{44}^{2}=\mu^{2}-\frac{15}{4}+\muc_{22}.}\\\end{array}
S=\int\delta^{4}u\sqrt{|g_{\alpha^{\prime}\beta^{\prime}}|}e^{-\varphi}[\widehat{R}^{\prime}+S^{\prime}+(\delta_{\mu^{\prime}}\phi)(\delta^{\mu^{\prime}}\phi)-\frac{1}{2}(\delta_{\mu^{\prime}}\beta)(\delta^{\mu^{\prime}}\beta)-\frac{1}{2}e^{2\varphi}(\delta_{\mu^{\prime}}\sigma)(\delta^{\mu^{\prime}}\sigma)],
\varphi(x)=\epsilon^{1/2}\Big\lbrack\left(\frac{C_{1}}{2C_{2}}\right)^{1/2}\tilde{\varphi}_{0}+\epsilon(-\frac{D_{0}}{D_{1}}\tilde{\varphi}_{0}+\varphi_{1p})\Big\rbrack\;,
g=\pi^{2}\left(\frac{1}{{\beta}_{1}^{2}}-\frac{1}{{\beta}_{2}^{2}}\right)\,~~~~~~~g^{\prime}=\frac{\pi^{2}}{2}\left(\frac{1}{{\beta}_{1}^{2}}+\frac{1}{{\beta}_{2}^{2}}-4\right)\.
\widetilde{W}_{\nu}(\eta,\eta^{\prime})=(rr^{\prime})^{-\frac{d}{2}-\nu}{W}_{\nu}^{\infty}([\tau,\mathrm{e}],[\tau^{\prime},\mathrm{e}^{\prime}])=\frac{1}{2\pi^{\frac{d}{2}}}\frac{\Gamma(\nu+\frac{d}2)}{\Gamma(\nu+1)}\frac{1}{{[-(\eta-\eta^{\prime})^{2}]}^{\frac{d}{2}+\nu}}.
\sum_{j=2}^{3}\phi_{j}^{a}f_{1t}^{j}=-\sum_{j=2}^{3}[\phi_{jt}^{a}f_{1}^{j}+\phi_{j1}^{a}f_{t}^{j}+\sum_{k=2}^{3}(\phi_{jk}^{a}f_{t}^{k})f_{1}^{j}]-\phi_{1t}^{a}\quad\quada=1,2,3
S_{4*}=\intd^{4}x\sqrt{-g}\left[R+\frac{1}{4}\mathrm{Tr}\left[\nablaP\nablaP^{-1}\right]-\frac{1}{2}\left(\nablax\right)^{2}-\frac{1}{2}e^{\sqrt{3}x}\nabla\tau^{T}P\nabla\tau\right].
\left(\begin{array}{c}{\vec{x}}\\{\vec{p}}\\\end{array}\right)=\left(\begin{array}{cc}{\widetilde{\mathbfa}}&{\widetilde{\mathbfb}}\\{\widetilde{\mathbfd}}&{\widetilde{\mathbfc}}\\\end{array}\right)\left(\begin{array}{c}{\vec{Q}}\\{\vec{P}}\\\end{array}\right)
K^{\mu}\;:=\;\mathrm{diag}\left(\frac{\pi}{\pi+gN}\frac{1}{-\triangle+\frac{e^{2}N}{\pi+gN}}\;,\;\frac{1}{-\triangle+\mu}\;,\;.\;.\;.\;.\;,\;\frac{1}{-\triangle+\mu}\right)\;,
\langle\varphi^{2}(x_{1}){\varphi}^{2}(x_{2}){\overline{{\varphi}}}^{2}(x_{3}){\overline{{\varphi}}}^{2}(x_{4})\rangle_{_{\mathrm{\tiny{free}}}}=\frac{1}{(4\pi^{2})^{4}}\left({\frac{4N^{2}}{x_{13}^{4}x_{24}^{4}}}+{\frac{4N^{2}}{x_{14}^{4}x_{23}^{4}}}+{\frac{16N}{x_{41}^{2}x_{13}^{2}x_{32}^{2}x_{24}^{2}}}\right)\,.
c_{\nabla\phi\nabla\phi}^{(0)}(\vec{\nabla}\phi(\vec{t}))^{2}+c_{\nabla\phi\nabla\phi}^{\hat{t}\hat{t},(0)}(\hat{t}\cdot\vec{\nabla}\phi(\vec{t}))^{2}+(M^{2}c_{\phi^{2}}^{(0)}+\frac{1}{t^{2}}c_{\phi^{2}}^{(2)})\phi^{2}(\vec{t})+c_{\phi^{4}}^{(0)}\phi^{4}(\vec{t}).
r_{0}^{6}={\frac{1}{6^{2}}}\{(t^{I}q_{I})_{\mathrm{hor}}\}^{3}\qquad\Longrightarrow\qquad{\frac{1}{2^{11/2}}}(-q_{0})^{2}q_{1}={\frac{8(Q_{H}Q_{F})^{2}}{\pi^{2}}}\,
G_{B}(\tau,\tau^{\prime})=\frac{|\tau-\tau^{\prime}|}{T}(T-|\tau-\tau^{\prime}|),\quad\frac{\partial}{\partial\tau}G_{B}(\tau,\tau^{\prime})=sign(\tau-\tau^{\prime})-{\frac{2(\tau-\tau^{\prime})}{T}}.
V_{\alpha}=\bar{\phi}_{B}(\sigma_{\alpha})_{~A}^{B}\frac{\delta}{\delta\bar{\phi}_{A}}+\bigr(\phi_{Ab}^{*}(\sigma_{\alpha})_{~a}^{b}+\phi_{Ba}^{*}(\sigma_{\alpha})_{~A}^{B}\bigr)\frac{\delta}{\delta\phi_{Aa}^{*}}+\eta_{B}(\sigma_{\alpha})_{~A}^{B}\frac{\delta}{\delta\eta_{A}}.
\frac{L}{2}=\frac{R^{2}}{U_{\mathrm{\tiny{min}}}}\biggl[F_{0}(l_{\psi},l_{\phi_{1}})+\left(\frac{r_{0}}{U_{\mathrm{\tiny{min}}}}\right)F_{1}(l_{\psi},l_{\phi_{1}})+{\calO}\left(\left(\frac{r_{0}}{U_{\mathrm{\tiny{min}}}}\right)^{2}\right)\biggr],
\eta^{\dagger}n_{s}^{1/2}\left[\left({\frac{1}{2}}mv^{M}v_{M}+V-{\frac{1}{2m}}\partial^{M}\partial_{M}-\mu\right)-i\left({\frac{1}{2}}\partial^{M}v_{M}+v^{M}\partial_{M}\right)\right]n_{s}^{1/2}\eta=0.
B_{\underline{i}}^{(n)}={\frac{e^{2U}}{\overline{{\calH}}_{2}}}\biggl[\partial_{\underline{i}}{\overline{{\calH}}_{2}}\bigl(k^{(n)}{\calH}_{1}+{\overline{k}}^{(n)}{\overline{{\calH}}_{1}}\bigr)-\partial_{\underline{i}}{\overline{{\calH}}_{1}}\bigl(k^{(n)}{\calH}_{2}+{\overline{k}}^{(n)}{\overline{{\calH}}_{2}}\bigr)\biggr]\,.
\hat{H}=\hbar\Biggl\{\frac{\Omega_{[1]}}{2}\Bigl(\hat{a}^{\dagger}\hat{a}+\hat{a}\hat{a}^{\dagger}\Bigr)+\frac{3\lambda\hbar}{8\Omega_{G}^{2}}\hat{a}^{\dagger}\hat{a}+\frac{\lambda\hbar}{16\Omega_{G}^{2}}\sum_{k=0}^{4}{\binom{4}{k}}\hat{a}^{\daggerk}\hat{a}^{4-k}\Biggr\},
\begin{array}{ccccc}{p_{j}=-p_{i},}&{}&{\alpha_{n}^{(j)}=-\bar{\alpha}_{-n}^{(i)}=-\bar{\alpha}_{n}^{(i)\dagger},}&{}&{\bar{\alpha}_{n}^{(j)}=-\alpha_{-n}^{(i)}=-\alpha_{n}^{(i)\dagger}.}\\\end{array}
S_{\mathrm{eff}}[\chi]=\int\!d^{2}x\,\Bigl({\frac{1}{2}}\,\partial_{\alpha}\chi\partial_{\alpha}\chi-{\frac{\deltav}{2}}\,\chi^{2}+{\frac{\piV}{N}}\,\chi^{4}+{\calO}\bigl(\chi^{6}/N^{2}\bigr)\Bigr),
\left({\frac{\dot{a}}{a}}\right)^{2}=\left[{\frac{\dot{r}}{r}}\left({\frac{1}{2}}{\frac{h^{\prime}}{h}}r+1\right)+{\frac{\dot{\rho}}{\rho}}\right]^{2},\\\\\\\\\{\frac{\ddot{a}}{a}}=\left[1+{\frac{1}{2}}a{\frac{\partial}{\partiala}}\right]\left({\frac{\dot{a}}{a}}\right)^{2},
T_{\mu\nu}^{T}=T_{3}\frac{V(T)}{\sqrt{1+e^{-2\Phi}\nabla_{\alpha}T\nabla^{\alpha}T}}\left[e^{-2\Phi}\nabla_{\mu}T\nabla_{\nu}T-g_{\mu\nu}(1+e^{-2\Phi}\nabla_{\alpha}T\nabla^{\alpha}T)\right].
e^{i\Gamma_{f}[A_{+},A_{-}]}=\intd\psi^{\dagger}d\psie^{i\intd^{2}x(\psi_{L}^{\dagger}(i\partial_{+}+A_{+})\psi_{L}+\psi_{R}^{\dagger}(i\partial_{-}+A_{-})\psi_{R})}\,,
\tilde{\Lambda}(x)=\frac{4\sqrt{6}}{\bar{N}}\int\delta\xids\omega(\xi(s_{+}))\tilde{\Lambda}_{\nu}[\xi|s]\omega^{-1}(\xi(s_{+}))\dot{\xi}^{\nu}(s)\dot{\xi}(s)^{-2}\delta(x-\xi(s)).
{\frac{dz(u)}{du}}=e^{\lambda_{0}}\prod_{u_{\mu}\in\,\partial{\calD}}(u-u_{\mu})^{-\beta_{\mu}}\prod_{u_{i}\in\,\mathrm{int}\,{\calD}}(u-u_{i})^{-\beta_{i}}(\bar{u}_{i}u-1)^{-\beta_{i}}
\tilde{\calL}^{(0)}=\pi_{a}\dot{\sigma}^{a}+\Psi\dot{\theta}-\frac{1}{2}\,{\sigma^{\prime}}_{a}^{2}-\frac{1}{2}\,\lambda\,\bigl(\sigma_{a}^{2}-1\bigr)-\frac{1}{2}\,\pi_{a}^{2}+\sigma_{a}\pi^{a}\theta-\frac{1}{2}\sigma_{a}^{2}\theta^{2}.
e_{i}^{(k)}=\frac{e_{i}^{k}}{[k]_{q}!},~~~f_{i}^{(k)}=\frac{f_{i}^{k}}{[k]_{q}!}\,,~~~[k]_{q}=\frac{q^{k}-q^{-k}}{q-q^{-1}}\,,~~~[k]_{q}!=[k]_{q}[k-1]_{q}\cdots[1]_{q}
a_{j}=\frac{1}{\left(j+1\right)!}a_{a_{1}\cdotsa_{j}}\left(\left[F_{\mu\nu}^{a}\right],\left[y^{i}\right],\left[A_{a}^{*\mu}\right],\left[y_{i}^{*}\right],\left[\eta_{a}^{*}\right]\right)\eta^{a_{1}}\cdots\eta^{a_{j}}.
F^{1111}=\frac{(12)(34)}{x_{12}^{2}\,x_{34}^{2}}\;\alpha(x_{0},\ldots,x_{4})+\frac{(14)(23)}{x_{14}^{2}\,x_{23}^{2}}\;\beta(x_{0},\ldots,x_{4})+\frac{(13)(24)}{x_{13}^{2}\,x_{24}^{2}}\;\gamma(x_{0},\ldots,x_{4})\;.
\sqrt{-G}\frac{8}{3}G^{55}\partial_{5}\Phi\Big|_{y=0}=-\sqrt{-g^{0}}b_{0}V_{0}e^{b_{0}\Phi}\Big|_{y=0}\,\quad\sqrt{-G}\frac{8}{3}G^{55}\partial_{5}\Phi\Big|_{y=L}=\sqrt{-g^{L}}b_{L}V_{L}e^{b_{L}\Phi}\Big|_{y=L}\.
L_{\mathrm{SUSY;int}}^{(0)}=-\frac{1}{4k}\epsilon_{ij}Q_{i}\dot{Q}_{j}+\frac{1}{8k^{2}}\overrightarrow{Q}^{2}-2ik^{2}\rho_{k}\dot{\rho}_{k}-ik\epsilon_{ij}\rho_{i}\rho_{k}\,,
S_{\mathrm{eff.}}\left(\Sigma_{\mathrm{min.}}\right)=2\int_{\Sigma_{\mathrm{min.}}}^{}d\sigma_{\mu\nu}(x)\int_{\Sigma_{\mathrm{min.}}}^{}d\sigma_{\mu\nu}(x^{\prime})D\left(\frac{(x-x^{\prime})^{2}}{T_{g}^{2}}\right)
S_{\alpha\beta}^{\gamma\delta}(\theta)=\left(\frac{\rho_{\gamma}\rho_{\delta}}{\rho_{\alpha}\rho_{\beta}}\right)^{-\frac{\theta}{2\pii}}R(\theta)\,\,W\!\!\left(\left.\begin{array}{cc}{\alpha}&{\delta}\\{\gamma}&{\beta}\\\end{array}\right|\lambda\theta\right),\;\;\;\;\alpha,\beta,\gamma,\delta\in\{0,1\}
\biggl(\sum_{K=0}^{\infty}a_{K}g^{K}\biggr)\biggl(\frac{\partialF}{\partialg}\biggr)=-g\biggl(\sum_{K=0}^{\infty}b_{K}g^{K}\biggr)\biggl(\frac{\partial^{2}F}{\partialg^{2}}\biggr)+\biggl(\sum_{K=0}^{\infty}c_{K}g^{K}\biggr)
\left[\mathrm{Tr}e^{-\bar{H}_{d}^{2}t}\right]_{\mathrm{div}}=r_{d}{\frac{\Gamma\left(1-\fracD2\right)}{(4\pi)^{(D-1)/2}}}{\frac{m^{D-4}}{2\kappat^{3/2}}}\int_{\Sigma}\left[\left(m^{2}+{\frac{R}{12}}-{\frac{R_{ii}}{8}}\right)t-{\frac{P}{4\kappa^{2}}}\right]~~~.
f^{(2)}=\left(\begin{array}{cccc}{0}&{-\delta_{ij}}&{2a_{i}}&{\pi_{i}}\\{\delta_{ij}}&{0}&{0}&{a_{i}}\\{-2a_{i}}&{0}&{0}&{0}\\{-\pi_{i}}&{-a_{i}}&{0}&{0}\\\end{array}\right),
<\lambda|\tilde{\Phi}_{\mu}^{V_{2}^{(1)}\lambda}(z_{2})\tilde{\Phi}_{\lambda}^{V_{1}^{(1)}\mu}(z_{1})|\lambda>=\sum_{m_{1},m_{2}=0,1\atopm_{1}+m_{2}=1}v_{m_{1}}^{(1)}\otimesv_{m_{2}}^{(1)}<\lambda|\tilde{\Phi}_{\mu,m_{2}}^{V_{2}^{(1)}\lambda}(z_{2})\tilde{\Phi}_{\lambda,m_{1}}^{V_{1}^{(1)}\mu}(z_{1})|\lambda>.
{\calL}={\frac{3}{16}}e^{4Y}[16(Y^{\prime})^{2}-2(h^{\prime})^{2}-\frac{1}{2}e^{-2h}|a^{\prime}|^{2}+2e^{-2h}-\frac{1}{4}e^{-4h}(|a|^{2}-1)^{2}+4],
\begin{array}{rcl}{\hat{J}_{\pm}D_{MK}^{J}}&{=}&{\sqrt{(J\mpM)(J\pmM-1)}D_{M\pm1,K}^{J}~,}\\{\hat{J}_{3}D_{MK}^{J}}&{=}&{KD_{MK}^{J}~,~~~~\hat{J}_{z}D_{MK}^{J}=MD_{MK}^{J}~,}\\\end{array}
\psi_{\scriptscriptstyle{l}}(k,r)=\frac{e^{-i\piW/2}\,e^{i\pi|l|}}{2\sqrt2}\,\left(H_{\scriptscriptstyle{W}}^{(-)}(kr)+e^{i\pi(W-|l|)}\,e^{2i\delta_{l}(k)}\,H_{\scriptscriptstyle{W}}^{(+)}(kr)\right)\,,
\begin{array}{cccccc}{\phi_{1}^{1}}&{\phi_{1}^{2}}&{\ldots}&{\phi_{1}^{a}}&{\ldots}&{\phi_{1}^{m}}\\{\vdots}&{\vdots}&{}&{\vdots}&{}&{\vdots}\\{}&{}&{}&{}&{}&{\phi_{N_{m}}^{m}}\\{\vdots}&{\phi_{N_{2}}^{2}}&{}&{\vdots}\\{\phi_{N_{1}}^{1}}\\{\}&{}&{}&{\phi_{N_{a}}^{a}}\\\end{array}
I(s)\sim\int_{1}^{t_{0}}{\frac{(t-1)^{-s}(t+1)^{-s}dt}{\mathrm{e}^{2\piat}-1}}\sim{\frac{1}{{2(\mathrm{e}^{2\pia}-1})}}\int_{1}^{t_{0}}(t-1)^{-s}dt\,.
\Omega_{s}^{\mathrm{SUSY}}=\mathrm{d}\Theta^{\mathrm{SUSY}}=-\mathrm{d}x^{a}\wedge\mathrm{d}p_{a}+\Omega_{{\calL}^{1|2}}\qquad\qquad\Omega_{{\calL}^{1|2}}=\mathrm{d}\Sigma_{{\calL}^{1|2}}
ds^{2}=\sum_{i=1}^{3}dz_{i}d\bar{z}_{i}=\sum_{i=1}^{3}dr_{i}^{2}+R^{2}d\theta_{1}^{2}+R^{2}d\theta_{2}^{2}+\frac{a^{2}R^{2}}{\Gamma^{2}}d\phi^{2}
w=w_{0}+\frac{b_{0}\alpha}{6(k+b_{0}^{2})}t^{2}+\left(\frac{g^{2}b_{0}^{2}w_{0}(k-w_{0}^{2})}{9(k+b_{0}^{2})^{2}}+\frac{b_{0}\alpha(\lambda-2g)}{216(k+b_{0}^{2})}\right)t^{4}+O(t^{6}),\quadt\to0,
R_{MN}-{\frac{1}{2}}G_{MN}R+{\frac{1}{2}}{\frac{\sqrt{-{\widehatG}}}{\sqrt{-G}}}~{\delta_{M}}^{\alpha}{\delta_{N}}^{\beta}{\widehatG}_{\alpha\beta}~{\widetildef}_{D-2}~\delta^{(2)}(x^{i})=0~,
D_{j\delta}D_{i\alpha}W_{~\gamma}^{\beta}=-\delta_{\alpha}^{~\beta}D_{i\gamma}D_{j\delta}W_{~\epsilon}^{\epsilon}=-D_{i\gamma}D_{j\alpha}W_{~\delta}^{\beta}=\delta_{~\gamma}^{\beta}D_{j\alpha}D_{i\delta}W_{~\epsilon}^{\epsilon}
\langleV_{\mu_{1}}(\theta_{1})\cdotsV_{\mu_{n}}(\theta_{n})|V_{\mu_{1}^{\prime}}(\theta_{1}^{\prime})\cdotsV_{\mu_{n}^{\prime}}(\theta_{n}^{\prime})\rangle=\prod_{i=1}^{n}2\pi\delta_{\mu_{i}\mu_{i}^{\prime}}\delta(\theta_{i}-\theta_{i}^{\prime}),
(x\alpha)^{m}=x^{m}+\alpha^{m}-{\frac{1}{2}}\Gamma_{nr,s}^{m}(e)x^{n}x^{r}\alpha^{s}-{\frac{1}{2}}\Gamma_{n(r,s)}^{m}(e)x^{n}\alpha^{r}\alpha^{s}+\ldots\,.
[Q_{i},\;a_{m}^{\dagger}]=\psi_{\dot{\alpha}}^{\dagger}(\gamma_{i}^{\dagger})_{\dot{\alpha}m},\;\;\;\;\;\;[Q_{i},\;\psi_{\dot{\alpha}}^{\dagger}]_{+}=a_{m}^{\dagger}(\gamma_{i})_{m\dot{\alpha}}
\sum_{s,t\inS_{n}}^{}D_{R}(sts^{-1}t^{-1})=\sum_{s,t}^{}D_{R}(sts^{-1})D_{R}(t^{-1})=\sum_{t\inS_{n}}^{}\frac{n!}{d_{R}}\chi_{R}(t)D_{R}(t^{-1})I_{R}=(\frac{n!}{d_{R}})^{2}I_{R}.
H=\intdx\left\{\frac{1}{2}\left[\xi_{5}^{2}+\xi_{6}^{2}+(\partial_{1}\xi_{3})^{2}-e^{2}\xi_{2}^{2}\right]+e\xi_{6}\xi_{2}+\xi_{1}(\partial_{1}\xi_{5}+e\partial_{1}\xi_{3})+\lambda\xi_{4}\right\}.
{e^{\ast}}_{\mu}^{(\lambda)}\gamma_{\mu}=\left(\begin{matrix}{0}&{\alpha_{\lambda}}\\{-\alpha_{\lambda}}&{0}\\\end{matrix}\right),\;\alpha_{\sigma}=\left(\begin{matrix}{0}&{-ie^{-i\varphi_{k}}}\\{ie^{i\varphi_{k}}}&{0}\\\end{matrix}\right),\;\alpha_{\pi}=\left(\begin{matrix}{k_{\perp}}&{-k_{3}e^{-i\varphi_{k}}}\\{-k_{3}e^{i\varphi_{k}}}&{-k_{\perp}}\\\end{matrix}\right){\frac{1}{\omega_{k}}}.
Z=\int_{0}^{\infty}\frac{ds}{s}\mathrm{Tr}_{\mathrm{NS-R}}\left[P_{\mathrm{GSO}}\left(\frac{\mathbf{1}+h_{1}+h_{2}+h_{3}}{4}\right)\mathrm{e}^{-2\pis(L_{0}-a)}\right]\,
A_{\ell}=\frac{1}{2Z_{\ell}}\;\frac{\partialZ_{\ell}}{\partialp}\;\widetilde{G}_{A_{1}\,\dots\,A_{N}}^{\,amp}\;\overset{(s_{1})}{e}{\!\!}_{A_{1}}\,\dots\,\overset{(s_{N})}{e}{\!\!}_{A_{N}}\;\prod_{i=1}^{N}\;\frac{1}{\mu_{i}Z_{i}}\;,
\left(\begin{array}{ccc}{1}&{-\bar{t}_{a}R}&{-\bar{t}_{a}R}\\{-\bar{t}_{b}R}&{1}&{-\bar{t}_{b}R}\\{-\bar{t}_{c}R}&{-\bar{t}_{c}R}&{1}\\\end{array}\right)\left(\begin{array}{c}{r_{aa}(s_{a}^{\prime})}\\{r_{ab}(s_{a}^{\prime})}\\{r_{ac}(s_{a}^{\prime})}\\\end{array}\right)=\left(\begin{array}{c}{0}\\{t_{a}(s_{a}^{\prime})R\bar{t}_{b}}\\{t_{a}(s_{a}^{\prime})R\bar{t}_{c}}\\\end{array}\right)
D_{0}=\frac{1}{2}\left(\begin{array}{cc}{\sigma_{3}}&{0}\\{0}&{\sigma_{3}}\\\end{array}\right)\;,\;\;\;D_{+}=\frac{1}{2}\left(\begin{array}{cc}{0}&{\sigma_{+}}\\{\sigma_{+}}&{0}\\\end{array}\right)\;,\;\;\;D_{-}=\frac{1}{2}\left(\begin{array}{cc}{0}&{\sigma_{-}}\\{\sigma_{-}}&{0}\\\end{array}\right),
\Gamma_{\mu}^{\mathrm{H}}(p,q)=\frac{p^{2}A(p)-q^{2}A(q)}{p^{2}-q^{2}}\gamma_{\mu}+\frac{A(p)-A(q)}{p^{2}-q^{2}}\not\!p\gamma_{\mu}\not\!q-i\,\frac{B(p)-B(q)}{p^{2}-q^{2}}(\gamma\cdotp\gamma_{\mu}+\gamma_{\mu}\gamma\cdotq).
\inte^{i\vec{q}\cdot\vec{\varrho}}\frac{\vec{q}}{\vec{q}^{2}(\vec{q}^{2}+\mu^{2})}\frac{d^{2}\vec{q}}{(2\pi)^{2}}=-i\frac{1}{2\pi|\mu|}K_{1}(|\mu|\varrho)\hat{\varrho}+i\frac{1}{2\pi\mu^{2}}\frac{1}{\varrho^{2}}\vec{\varrho}~~,
ds_{4}^{2}=-\frac{1}{\sqrt{-{\frac{1}{6}}H^{0}\,C_{ABC}H^{A}H^{B}H^{C}}}\,dt^{2}+\sqrt{-{\textstyle{\frac{1}{6}}}H_{0}\,C_{ABC}H^{A}H^{B}H^{C}}\,d\vec{x}^{2}\;.