harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\tilde{b}}=\frac{2}{3}\Pi_{\mu\nu}\Pi^{\mu}\Pi^{\nu}+\frac{4}{3}\Pi_{\mu}^{(z)}\Pi^{\mu}\Pi-\frac{2}{15}\Pi_{\alpha\beta}\Pi^{\alpha}\Pi^{\beta}-\frac{3}{5}\Pi_{\mu\alpha}\Pi^{\mu}\Pi^{\alpha}-\frac{3}{5}\Pi_{\alpha}^{(z)}\Pi\Pi^{\alpha}-2\Pi{\calF}\.
I_{UV}(\rho)=\frac{1}{4\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}d\theta\,e^{4i\theta}e^{\rho^{\prime}e^{2i\theta}}\left(4e^{-2i\theta}K_{1}^{2}(e^{i\theta})+K_{0}^{2}(e^{i\theta})+4e^{-i\theta}K_{0}(e^{i\theta})K_{1}(e^{i\theta})\right).
gf^{ilj}\intd^{4}xA_{\mu}^{l}(x)\langleT[j^{\mu,j}(x)\frac{\deltaj^{\beta,l}}{\deltaA_{\lambda}^{k}}(y)]\rangle+ig(f^{bil}\langle\frac{\deltaj^{\beta,k}}{\deltaA_{\lambda}^{b}}(x)\rangle+f^{kia}\langle\frac{\deltaj^{\beta,a}}{\deltaA_{\lambda}^{l}}(y)\rangle)=0.
{S_{TM}^{4}}^{\star}=\intd^{4}x\Bigl[{\frac{1}{2\mu^{2}}}\partial_{\mu}T^{\mu\nu}\partial_{\lambda}T^{\lambda}{}_{\nu}-{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}-{\frac{1}{2}}T^{\mu\nu}F_{\mu\nu}\Bigl]\;,
\frac{\partialF_{j}}{\partialv^{1}}\mid_{(t^{*},p_{i})}=0,\\quad\quad\frac{\partialF_{j}}{\partialv^{2}}\mid_{(t^{*},p_{i})}=0,\\quad\quad\frac{\partialF_{j}}{\partialt}\mid_{(t^{*},p_{i})}=0,\\quad\quadj=1,2,
{\calM}=\left(\begin{array}{ccccc}{0}&{1}&{0}&{0}&{1}\\{1}&{0}&{1}&{0}&{0}\\{0}&{1}&{0}&{1}&{0}\\{0}&{0}&{1}&{0}&{1}\\{1}&{0}&{0}&{1}&{0}\\\end{array}\right).
{\calF}_{R}={\frac{\kappa_{R}^{2}}{4\pi^{2}R^{2}}}\Big({\frac{5}{128}}-{\frac{c^{3}(c^{2}+5)}{(c-1)^{4}(c+1)^{4}}}\Big)\,\,\,\,\,\,\,0<c<1
0=F_{l}^{\prime\prime}+\frac{2}{r}F_{l}^{\prime}\left[1+rL_{+}^{\prime}(kr)\right]-\frac{s^{\prime}}{s-\frac{\omega}{m}}\left[F_{l}^{\prime}+F_{l}L_{+}^{\prime}(kr)\right]+V(r,-m)F_{l}\.
T:\qquad{\frac{{\tilde{R}}_{9}}{{\tilde{\ell}}_{s}}}={\frac{\ell_{s}}{R_{9}}}\quad,\qquad{\frac{{\tilde{g}}}{\sqrt{{\tilde{R}}_{9}/{\tilde{\ell}}_{s}}}}={\frac{g}{\sqrt{R_{9}/\ell_{s}}}}\quad,\qquad{\tilde{\ell}}_{s}=\ell_{s}\quad.
\begin{array}{ll}{m\dot{u}^{\alpha}+\displaystyle\frac{1}{c^{2}}S^{\alpha\sigma}\ddot{u}_{\sigma}=f^{\alpha}}\\{\dot{S}^{\alpha\beta}-\displaystyle\frac{1}{c^{2}}S^{\alpha\sigma}\dot{u}_{\sigma}u^{\beta}+\displaystyle\frac{1}{c^{2}}S^{\beta\sigma}\dot{u}_{\sigma}u^{\alpha}=0}\\\end{array}
g_{tt}^{-1}=g_{ii}=e^{-2U}=\left({\mathrm{e}}^{-\phi_{0}}+\frac{|q|}{r}\right)\,\left({\mathrm{e}}^{+\phi_{0}}+\frac{|p|}{r}\right)=1+\frac{{\mathrm{e}}^{-\phi_{0}}|p|+{\mathrm{e}}^{\phi_{0}}|q|}{r}+{\frac{|pq|}{r^{2}}}\.
{\calL}_{c}=\deltaZ\,(\frac{i}{2}\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-\frac{i}{2}(\partial_{\mu}\bar{\psi})\gamma^{\mu}\psi)-\deltam\,\,\bar{\psi}\psi+\frac{\delta\theta}{4}\epsilon^{\mu\nu\alpha}F_{\mu\nu}A_{\alpha}+\deltae\,\,\bar{\psi}\gamma^{\mu}A_{\mu}\psi
{\calL}_{0}(\varphi_{1},\varphi_{2})=\frac{1}{2}(\partial\varphi_{1})^{2}+\frac{1}{2}(\partial\varphi_{2})^{2}+\frac{1}{2}m^{2}\varphi_{1}^{2}+\frac{1}{2}m^{2}\varphi_{2}^{2}
\varepsilon^{\mu\nu\rho\sigma}\varepsilon_{abcd}F_{\rho\sigma}^{cd}D_{\nu}\phi^{a}\phi^{b}=2\varepsilon^{\mu\nu\rho\sigma}[2F_{\rho\sigma}^{\alpha}D_{\nu}\phi^{\alpha}\phi^{4}-\partial_{\nu}(F_{\rho\sigma}^{\alpha}\phi^{\alpha}\phi^{4})].
\left({{z_{1}},\{z_{2}}}\right)=\zeta\left({{\frac{\xi}{\sqrt{1-\left|\xi\right|^{2}}}},\1}\right),\quad\left({dz^{*}dz}\right)^{2}=|\zeta|^{2}d\zeta^{*}d\zeta{\frac{d\xi^{*}d\xi}{\left({1-\left|\xi\right|^{2}}\right)^{2}}}\quad,
ds^{2}=\frac{1}{\varsigma^{0}\varsigma^{0}}\left[\frac{d\varsigma^{0}d\varsigma^{0}}{\varsigma^{0}\varsigma^{0}-1}-\left(\varsigma^{0}\varsigma^{0}-1\right)\left[\frac{dr^{2}}{\left(r^{2}-1\right)^{2}}+\frac{d{\bfu}d{\bfu^{\prime}}}{r^{2}-1}\right]\right]~.
\tilde{\sigma}=\int{\frac{d^{2}k}{(2\pi)^{2}}}[{\frac{-i\gamma^{5}}{\not\!k+\lambda^{2}\gamma^{5}\tilde{\sigma}+\lambda^{2}\tilde{\sigma}}}]_{\nu\nu}=\int{\frac{d^{2}k}{(2\pi)^{2}}}[{\frac{4i\lambda^{2}\tilde{\sigma}}{k^{2}-4\lambda^{4}(\sigma^{2}+\tilde{\sigma}^{2})}}]
\frac{8C_{1}C_{2}}{\pi}\int_{0}^{\infty}\int_{0}^{\infty}d{\kappa}_{1}\;d{\kappa}_{2}\left({\kappa}_{1}^{2}+{\sigma}_{1}^{2}\right)^{\frac{\lambda+m}{2}}\left({\kappa}_{2}^{2}+{\sigma}_{2}^{2}\right)^{\frac{\lambda+n}{2}}
\int_{0}^{\infty}dk\,k^{D-2}K_{0}\left(\gamma\sqrt{k^{2}+m^{2}}\right)=2^{(D-3)/2}m^{D-1}\Gamma\left(\frac{D-1}{2}\right)\frac{K_{(D-1)/2}(m\gamma)}{(m\gamma)^{(D-1)/2}},
\begin{array}{c}{\eta(k)\equivW[\varphi,\chi]=\varphi(s;k)\chi^{\prime}(s;k)-\varphi^{\prime}(s;k)\chi(s;k),}\\{\sigma(k)\equivW[\varphi,\rho]=\varphi(s;k)\rho^{\prime}(s;k)-\varphi^{\prime}(s;k)\rho(s;k).}\\\end{array}
\begin{array}{c}{K=h^{\mu\nu}\nabla_{\mu}u_{\nu}=-h^{\mu\nu}\Gamma_{\mu\nu}^{\lambda}u_{\lambda}=-\frac{1}{2\sqrt{f}}h^{\mu\nu}\left(\partial_{\nu}h_{\mu\tau}+\partial_{\mu}h_{\nu\tau}-\partial_{\tau}h_{\mu\nu}\right)}\\{=\frac{1}{2\sqrt{f}}h^{\mu\nu}\partial_{\tau}h_{\mu\nu}}\\\end{array}
\sim{\frac{1}{(4\pi)^{d/2}}}\sum_{n=0}^{\infty}\left[a_{n-\alpha}~{\frac{\lambda^{d/2-n+\alpha-1}}{\Gamma\left(\fracd2-n+\alpha\right)}}+b_{n-\alpha}~{\frac{\lambda^{(d-1)/2-n+\alpha-1}}{\Gamma\left({\frac{d-1}{2}}-n+\alpha\right)}}\right],
G_{rs}=\left(\begin{matrix}{{\frac{H_{0}}{H_{1}}}}&{0}\\{0}&{{\frac{H_{2}}{H_{3}}}}\\\end{matrix}\right)={\frac{H_{2}}{H_{3}}}\left(\begin{matrix}{(\mathrm{Re}\,U)^{2}}&{0}\\{0}&{1}\\\end{matrix}\right)\;,
\sigma_{ik}^{\prime}=\eta\Bigl({\frac{\partialv_{i}}{\partialx_{k}}}+{\frac{\partialv_{k}}{\partialx_{i}}}-{\frac{2}{3}}\delta_{ik}{\frac{\partialv_{l}}{\partialx_{l}}}\Bigr)+\zeta\delta_{ik}{\frac{\partialv_{l}}{\partialx_{l}}}\.
\begin{array}{c}{\left[S_{-},S_{+}\right]=F_{m,n}(H,S_{0}+1)-F_{m,n}(H,S_{0}),}\\{\mathrm{where}\quadF_{m,n}(H,S_{0})=\prod_{k=1}^{m}\left(H/2+S_{0}-\frac{2k-1}{2m}\right)\prod_{\ell=1}^{n}\left(H/2-S_{0}+\frac{2\ell-1}{2n}\right).}\\\end{array}
c_{a,b}^{(u-1)}\left(-\frac{1}{\tau}\right)=\frac{1}{\sqrt{(-i\tau)(u-1)(u+1)}}\,\underset{a^{\prime}\equivb^{\prime}\;\mathrm{mod~}2}{\sum_{a^{\prime}=0}^{u-1}\,\sum_{b^{\prime}=-u+2}^{u-1}}s(a,b,a^{\prime},b^{\prime})\,c_{a^{\prime},b^{\prime}}^{(u-1)}(\tau),
\sqrt{-\gamma}\widetilde{\calJ}^{\nu}=\sqrt{-\gamma}(\eta^{\alpha\nu}\delta\xi_{\alpha}+\delta\eta^{\alpha\nu}\xi_{\alpha})+(\eta^{\alpha\nu}\xi_{\alpha})\delta\sqrt{-\gamma}=\delta(-\sqrt{-\gamma}\eta^{\alpha\nu}\xi_{\alpha}),
H_{1}^{2}=-\frac{2H_{1}(\dot{d}/\ell)}{e^{2d/\ell}-1}+\frac{(\dot{d}/\ell)^{2}}{e^{2d/\ell}-1}+\frac{\rho_{1}+\rho_{2}e^{-4d/\ell}}{3\ell(1-e^{-2d/\ell})}\,.
\mathrm{Tr}\,e^{-sQ-s{\calM}^{2}}=\frac{e^{-s{\calM}^{2}}}{16\pi^{2}s^{2}}\sum_{n=0}^{\infty}s^{n}\,\biggl\{\intdx^{4}\sqrt{g}\:{\calL}^{(n)}\biggr\}.
\omega_{g}=U^{-1}dU=\left(\begin{array}{cc}{S^{-1}}&{P}\\{0}&{S}\\\end{array}\right)\left(\begin{array}{cc}{dS}&{0}\\{dP}&{dS^{-1}}\\\end{array}\right)=\left(\begin{array}{cc}{S^{-1}dS+PdP}&{PdS^{-1}}\\{SdP}&{SdS^{-1}}\\\end{array}\right)\,,
{\calV}={\textstyle\frac{1}{4}}e^{-\varphi}\,\left[2{\calM}^{nq}f_{mn}{}^{p}f_{pq}{}^{m}+{\calM}^{mq}{\calM}^{nr}{\calM}_{ps}f_{mn}{}^{p}f_{qr}{}^{s}\right]\,
(\alpha_{0},\beta_{0},C_{0})=\left(\frac{-\sqrt{2K}V_{0}\pm\sqrt{W_{0}}}{2},\frac{2B}{-\sqrt{2K}V_{0}\pm\sqrt{W_{0}}},\frac{W_{0}\sqrt{2K}\mpKV_{0}\sqrt{W_{0}}}{-\sqrt{2K}V_{0}\pm\sqrt{W_{0}}}\right),
\sigma^{Si}(y^{i})=\pm(\sum_{j_{i}=1}^{L-1}(-1)^{j_{i}}k^{y^{i}}|y^{i}-y_{j_{i}}^{i}|+{\frac{1}{2}}(1+(-1)^{L+1})k^{y^{i}}y^{i})~.~\,
\psi_{\mathrm{rot}}(x)=e^{i\gamma_{5}\chi}\psi_{\mathrm{in}}(x)=\left(\begin{matrix}{\nu}&{i\mu}\\{i\mu}&{\nu}\\\end{matrix}\right)\psi_{\mathrm{in}}(x)=\left(\begin{matrix}{\nu+i\mu\lambda}\\{i\mu+\lambda\nu}\\\end{matrix}\right)e^{ikx},
{\frac{\delta}{\delta\Omega}}\sqrt{-g^{\prime}}V(\hat{\phi}^{\prime})=\sqrt{-g^{\prime}}\Omega^{-1}\left[4V(\hat{\phi}^{\prime})-\hat{\phi}^{\prime}{\frac{\deltaV(\hat{\phi}^{\prime})}{\delta\hat{\phi}^{\prime}}}\right]=0,
U(\lambda\mu\rho)={\frac{1}{d(\rho)}}U_{k_{1}\,\,k_{2}}^{\,\,l_{1}\,\,l_{2}}\left(\begin{array}{ccc}{\lambda}&{\mu}&{\rho}\\{k_{1}}&{k_{2}}&{m}\\\end{array}\right)\left(\begin{array}{ccc}{\lambda}&{\mu}&{\rho}\\{l_{1}}&{l_{2}}&{m}\\\end{array}\right)^{\ast}.
S=\intd^{4}x\sqrt{|g_{E}|}\left[\hat{R}(\hat{g}_{E})-{\textstyle\frac{1}{2}}\frac{\partial_{\hat{\mu}}\hat{\lambda}\partial^{\hat{\mu}}\bar{\hat{\lambda}}}{\left(\Im\mathrm{m}\hat{\lambda}\right)^{2}}+{\textstyle\frac{1}{4}}\hat{F}^{I}\{}^{\star}\tilde{\hat{F}^{I}}\right]\,.
\intd^{4}x\left\{\sqrt{g_{1}^{(4)}}\lambda_{1}\left(\Phi^{2}(-l)-v_{1}^{2}\right)^{2}+\sqrt{g_{2}^{(4)}}\lambda_{2}\left(\Phi^{2}(l)-v_{2}^{2}\right)^{2}\right\}\;.
H_{c}^{(n)}=\intdx\,\Bigl[(-1)^{n+1}\theta(n-2)\,\frac{2}{(n-2)!}\,(\eta^{1})^{n-2}\eta^{2}\eta^{2}+\alpha_{n}\,(\eta^{1})^{n}+\beta_{n\alpha}\,\chi_{\alpha}(\eta^{1})^{n-1}\Bigr]
\left\{{\frac{\partial^{2}}{\partialr^{2}}}+{\frac{5}{r}}{\frac{\partial}{\partialr}}+\tilde{\omega}^{2}\left(1+{\frac{\tilde{R}_{\frac{\pi}{2}}^{4}}{r^{4}}}\right)+{\frac{4R_{\frac{\pi}{2}}^{8}}{r^{10}f^{2}}}\right\}\hat{\phi}-{\frac{12R_{\frac{\pi}{2}}^{8}(h_{~2}^{2}+h_{~3}^{3})}{r^{10}f^{2}}}=0.
\begin{array}{lcl}{{\displaystyle\left(-\partial_{\tau}^{2}+\partial_{\sigma}^{2}+\frac{\alpha}{\tau^{2}}\right)X}}&{=}&{0}\\{}&{}&{}\\{{\displaystyle\left(-\partial_{\tau}^{2}+\partial_{\sigma}^{2}-\frac{\alpha}{\tau^{2}}\right)Y}}&{=}&{0}\\\end{array}
\partial_{\mu}^{x}\,U(x,x^{\prime})=ig\left[A_{\mu}(x)-\!\!\int_{0}^{1}\!\!ds\,s\,{\frac{d\omega^{\nu}}{ds}}\U(x,\omega)\,F_{\nu\mu}(\omega)\,U(\omega,x)\right]U(x,x^{\prime})
\begin{array}{c}{{\bfx}^{\prime}={\bfe}_{+},\quad{\bfe}_{+}^{\prime}={\bfe}_{1},}\\{{\bfe}_{1}^{\prime}=K{\bfe}_{+}+{\bfe}_{-},\quad{\bfe}_{-}^{\prime}=K{\bfe}_{1},}\\\end{array}
T_{J_{1}}^{R_{1}}\starT_{J_{1}}^{R_{2}}=\sum_{R,J}\frac{1}{N}\left(\begin{array}{ccc}{R}&{R_{1}}&{R_{2}}\\{J}&{J_{1}}&{J_{2}}\\\end{array}\right)\left\{\begin{array}{ccc}{R}&{R_{1}}&{R_{2}}\\{\frac{p}{2}}&{\frac{p}{2}}&{\frac{p}{2}}\\\end{array}\right\}T_{J}^{R},
J^{(3)}(n;\nu_{1},\nu_{2},\nu_{3})\equiv\int\frac{\mathrm{d}^{n}q}{\big[(p_{2}-q)^{2}-m_{1}^{2}\big]^{\nu_{1}}\big[(p_{1}+q)^{2}-m_{2}^{2}\big]^{\nu_{2}}\big[q^{2}-m_{3}^{2}\big]^{\nu_{3}}}\;.
H_{0}=\intdpf(p)A_{f}^{\dagger}(p)A_{f}(p)=\int_{x}[{\frac{1}{2}}\Pi_{x}^{2}+{\frac{1}{2}}(\partial_{x}\phi_{x})^{2}+{\frac{1}{2}}\mu^{2}(\phi_{x}-\varphi)^{2}-{\frac{1}{2}}I_{0}(\mu)]\;.
p^{-}=\frac{1}{N}P^{-}=\alphac_{ij}^{\dagger}c_{ij}+\frac{1}{N}\betac_{ij}^{\dagger}(ccc)_{ij}+\frac{1}{N}\beta(c^{\dagger}c^{\dagger})_{ij}(cc)_{ij}+\frac{1}{N}\beta(c^{\dagger}c^{\dagger}c^{\dagger})_{ij}(c)_{ij}.
nB_{2}+lA_{2}=b_{l}^{n}-(k-1)U_{l}^{n}Str\left(\ldots\phi^{i_{2k}^{\prime}}\ldots\underbrace{\phi^{i_{2j}^{\prime}}\phi^{i_{2j-1}^{\prime}}\phi^{i_{2k-1}^{\prime}}}\ldots\right)
\begin{array}{l}{G_{\mathrm{nn}}^{1}(0,0)=G_{\mathrm{nn}}^{1}(n,0)=G_{\mathrm{nn}}^{1}(0,n^{\prime})=0,}\\{G_{\mathrm{nn}}^{1}(n,n)=2(F_{(0,0,4)}-F_{(0,2,2)}-F_{(2,0,2)}+F_{(0,1,3)}+F_{(1,0,3)}-F_{(1,1,2)}),}\\{G_{\mathrm{nn}}^{1}(n,-n)=F_{(1,0,3)}+F_{(3,0,1)}+2F_{(1,1,2)}+2F_{(2,1,1)}-6F_{(1,2,1)}.}\\\end{array}
x_{jk}^{\gamma\delta}=\left(\begin{array}{cc}{{\frac{p_{0}(p.S)}{2(p.\Lambda)}}(\epsilon_{jk}-{\frac{\epsilon_{jl}p_{l}\Lambda_{0k}}{p_{0}\Lambda_{00}}})}&{g_{jk}}\\{-{\frac{1}{(p.\Lambda)}}(g_{jk}+{\frac{p_{j}\Lambda_{0k}}{p_{0}\Lambda_{00}}})}&{0}\\\end{array}\right).
\varepsilon^{jkl}V_{a}^{\;k}V_{b}^{\;l}=\frac{-2}{1+4\Phi^{2}}\,\left(\Phi^{a}\delta_{bj}-\Phi^{b}\delta_{aj}\right)+\frac1{1+4\Phi^{2}}\,\varepsilon^{abj}.
^t(q^{H_{i}})=q^{H_{i}}\,,\,\,^{t}E_{i}=F_{i}q^{H_{i}-1}\,,\,\,^{t}F_{i}={\bar{q}}^{H_{i}-1}E_{i}\,\,\Leftrightarrow\,\,^{t}M_{+}=M_{-}^{-1}\,,
{\frac{d\tilde{m}_{0}}{d\tilde{\lambda}}}=-{\frac{\tilde{m}_{0}}{2}}\left[{\calW}^{-1}{\calW}^{\prime}-{\calW}^{-1}\partial_{y}({\calW}g_{\mu\nu}){\frac{dx^{\mu}}{d\tilde{\lambda}}}{\frac{dx^{\nu}}{d\tilde{\lambda}}}\right]{\frac{dy}{d\tilde{\lambda}}}.
\widehat{V}^{(B=0)}(x,t)=\mathrm{diag}({\scriptstyle\sqrt{-\frac{t_{3}}{t_{1}}}\frac{t-t_{1}}{t-t_{3}},\sqrt{-\frac{t_{4}}{t_{2}}}\frac{t-t_{2}}{t-t_{4}},\sqrt{-\frac{t_{1}}{t_{3}}}\frac{t-t_{3}}{t-t_{1}},\sqrt{-\frac{t_{2}}{t_{4}}}\frac{t-t_{4}}{t-t_{2}}})
a_{D}=2^{2-D/2}\int_{0}^{1}{dx_{3}}\int_{0}^{x_{3}}{dx_{2}}x_{2}\left[{x_{2}(1-x_{2})+(x_{3}-x_{2})(1-x_{3}+x_{2})+x_{3}(1-x_{3})}\right]^{D/2-2}
W[\Phi_{0},a,M]=-{\frac{1}{64\pi^{2}}}\tau({\frac{4}{3}}\pia^{3})\Phi_{0}^{4}{\frac{1}{35}}Log{\frac{M^{2}}{\Phi_{0}^{2}}}(1+{\frac{70}{63}}{\frac{1}{\Phi_{0}^{2}a^{2}}}+{\frac{35}{2}}{\frac{m^{2}}{\Phi_{0}^{2}}})+...
\pi_{a}(x,t)={\frac{1}{\Omega}}\pi_{a}^{0}(t)+\tilde{\pi}_{a}(x,t)\,\qquad\int_{\Omega}\tilde{\pi}_{a}=0\,\qquad\pi_{a}^{0}=\int_{\Omega}\pi_{a}(x,t)\.
\begin{array}{ccccccc}{}&{}&{U(3)}&{}&{}&{}&{}\\{}&{\diagup}&{}&{\diagdown}&{}&{}&{}\\{SU(4)}&{}&{}&{}&{U(2)\timesU(1)}&{\;\;-\;\;}&{U(1)\timesU(1)\timesU(1).}\\{}&{\diagdown}&{}&{\diagup}&{}&{}&{}\\{}&{}&{S(U(2)\timesU(2))}&{}&{}&{}&{}\\\end{array}
U^{2,\nu}{\frac{H\rightarrow0,\nu\rightarrow\infty}{H\nu=m}}\rightarrowP^{>}(m,2)\qquad\mathrm{or}\qquadU^{2,-\nu}{\frac{H\rightarrow0,\nu\rightarrow\infty}{H\nu=m}}\rightarrowP^{<}(m,2).
\frac{N^{\prime}}{\mu}(2-\frac2{x_{i}^{2}})+2\sum_{j\neqi}\left(\frac1{x_{i}-x_{j}}-\frac1{x_{i}+x_{j}}\right)+\sum_{a=0,1}\frac2{x_{i}-\lambda_{a}}-\frac2{x_{i}+\lambda_{a}}
\tilde{\phi}^{\prime\prime}+\left({\frac{2}{r}}+{\frac{1}{(r-M)}}\right)\tilde{\phi}^{\prime}+{\frac{1}{8}}{\frac{k}{r-M}}\left({\frac{\omega^{2}}{r-M}}-{\frac{\mu^{2}}{r}}\right)\tilde{\phi}=0.
\left\langle\left\langleg^{4}F_{\mu_{1}\nu_{1}}^{a_{1}}F_{\mu_{1}\nu_{1}}^{a_{1}}F_{\mu_{2}\nu_{2}}^{a_{2}}F_{\mu_{2}\nu_{2}}^{a_{2}}\right\rangle\right\rangle=\mathcal{N}\left(\left\langleg^{2}FF\right\rangle\right)^{2},
R_{\nkl}^{m}=\left[\frac{\epsilon}{K^{2}}-\frac{1}{a^{2}}\left(\frac{\dot{K}}{K}\right)^{2}\right]\left(G_{\k}^{m}G_{nl}-G_{\l}^{m}G_{nk}\right)\\\,\\\R_{\n0l}^{0}=-\frac{1}{a^{2}}\frac{\ddot{K}}{K}G_{nl}\.
(\partial_{\lambda}g^{\mu\nu}){\frac{\deltaS_{DG}}{\deltag^{\mu\nu}}}+2\partial_{\alpha}\left(g^{\alpha\nu}{\frac{\deltaS_{DG}}{\deltag^{\lambda\nu}}}\right)+\left(\partial_{\lambda}\phi\right){\frac{\deltaS_{DG}}{\delta\phi}}=0.
\left(\intd^{2}zd^{2}w\,{\frac{1}{z-w}}\,\psi^{\dagger}(z)U^{q}(z)\psi^{\dagger}(w)U^{q}(w)\mathrm{e}^{-{\frac{1}{4}}(|z|^{2}+|w|^{2})}\right)^{N/2}|0\rangle.
<\Phi(Z_{1}){\bar{\Phi}}(Z_{2})>=\mathrm{ln}(Z_{12})-{\frac{\theta_{12}\bar{\theta}_{12}}{2Z_{12}}}\;,\;\;<{\calV}(Z_{1}){\bar{\calV}}(Z_{2})>=\mathrm{ln}(Z_{12})-{\frac{\theta_{12}\bar{\theta}_{12}}{2Z_{12}}}\;.
{\calL}=-\frac{i}{2}\epsilon_{\mu\nu\lambda}B_{\mu}^{*}(\partial_{\nu}-i2ga_{\nu}+\delta_{\lambda0}\mu)B_{\lambda}+\frac{M}{2}B_{\mu}^{*}B_{\mu}-\frac{i}{2}\epsilon_{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}+(\partial_{\mu}\bar{c})(\partial_{\mu}c)+\frac{1}{2\rho}(\partial_{\mu}a_{\mu})^{2}\;,
+H_{1}^{\frac{1}{4}}H_{2}^{-\frac{1}{4}}(dz_{1}^{2}+dz_{2}^{2}+dz_{3}^{2}+dz_{4}^{2})+H_{1}^{\frac{1}{4}}H_{2}^{\frac{3}{4}}(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}+dx_{4}^{2}).
\varrho={\frac{p(p-1)}{256\piG}}\left[(p-3)^{2}\bar{\calE}^{2}\bar{\calP}_{\phi}^{-2{\frac{p-1}{p-3}}}a^{2{\frac{p+1}{p-3}}}-(p+1)(p-7){\frac{1}{a^{2}}}\right],
A_{i}^{\mathrm{gh}}(s)={\frac{2}{\Gamma(s)}}\Bigr[r_{+}^{2s}+(-1)^{i}r_{-}^{2s}\Bigr]\zeta_{R}(2s+i-2)\sum_{l=0}^{i}x_{i,l}{\frac{\Gamma(s+l+{\frac{i}{2}})}{\Gamma(l+{\frac{i}{2}})}},
Z[J,K]=e^{\frac{1}{\hbar}W[J,K]}=\int\!D\Phi\,e^{\frac{1}{\hbar}\left[-S[\Phi]+\frac{1}{2}\int\!\!J(x)\Phi^{2}(x)\,d^{n}x+\frac{1}{24}\int\!\!K(x)\Phi^{4}(x)\,d^{n}x\right]}
S_{1}\,[\psi\,,\,{\overline{\psi}}\,,\,A_{\mu}\,,\,g\,]\,=\,\intd^{k}x\Big(-{\frac{1}{4}}Tr(F^{\mu\nu}\,F_{\mu\nu})+i\overline{\psi}\gamma^{\mu}\big(\partial_{\mu}-i{\tilde{A}}_{\mu}\big)\psi\,\Big)
{\calR}\Psi=\left[\begin{array}{ll}{R}&{{\bf0}}\\{{\bf0}}&{R^{\prime}}\\\end{array}\right]\left[\begin{array}{l}{Q_{1}^{0}-iQ_{1}^{3}}\\{Q_{1}^{2}-iQ_{1}^{1}}\\{Q_{2}^{0}-iQ_{2}^{3}}\\{Q_{2}^{2}-iQ_{2}^{1}}\\\end{array}\right].
E_{\mathrm{PP}}(\lambdaa\gg1)\to\frac{1}{2(4\pi)^{2}}\,\frac{\pi^{4}}{45}\,\frac{1}{a^{3}}\simeq\frac{1}{2(4\pi)^{2}}\,\times\,2.16\dots\,\times\,\frac{1}{a^{3}},
R_{1/2,1/2}(x,y)=f_{q}(x,y)\cdot\left(e_{11}+e_{44}+\frac{q^{-1}(y-x)}{y-q^{-2}x}(e_{22}+e_{33})+\frac{1-q^{-2}}{y-q^{-2}x}(ye_{23}+xe_{32})\right)
\overline{W}_{B\dot{\alpha}}^{\prime}=e^{-V_{B}}\overline{W}_{B\dot{\alpha}}e^{V_{B}}={\frac{1}{4}}[\nabla^{\alpha},\{\nabla_{\alpha},\overline{D}_{\dot{\alpha}}\}]\quad\mathrm{and}\quad\overline{W}_{B\dot{\alpha}}={\frac{1}{4}}D^{2}(e^{V_{B}}\overline{D}_{\dot{\alpha}}e^{-V_{B}}).
Q_{g}(z)^{2}\deltag(z)=\sum_{j=1}^{q}Q_{g(>j)}^{2}(z)[Y_{k_{s}}(z_{j})\deltak_{s}+\frac{(1-k_{s}^{n_{j}})k_{s}}{k_{s}^{n_{j}}((1-k_{s})}Y_{u_{s}}(z_{j})\deltau_{s}+\frac{(1-k_{s}^{n_{j}})}{((1-k_{s})}Y_{v_{s}}(z_{j})\deltav_{s}]
\Pi_{00}^{\mathrm{thermal}}(k_{0},\vec{k})={\frac{4e^{2}}{(2\pi)^{4}i}}\intd^{3}p\int_{-i\infty+\epsilon}^{i\infty+\epsilon}dp_{0}\,n_{F}(p_{0})\left[{\frac{m^{2}+p^{2}+p\cdotk-2p_{0}k_{0}-2p_{0}^{2}}{(p^{2}+m^{2})((p+k)^{2}+m^{2})}}+(k_{0}\leftrightarrow-k_{0})\right]
\langleS_{\lambda}(\beta)\rangle=(-1)^{\frac{A(A-1)}{2}}{\frac{\prod_{f\,\mathrm{odd}}f!!\prod_{f^{\prime}\,\mathrm{even}}f^{\prime}!!}{\prod_{f\,\mathrm{odd},f^{\prime}\,\mathrm{even}}(f-f^{\prime})}}\mathrm{dim}\,\lambda,
\int_{D}d^{3}x~V^{-2}(\partial_{\underline{{i}}}V)^{2}~=~-\int_{D}d^{3}x~\partial_{\underline{{i}}}(V^{-1}\partial_{\underline{{i}}}V)~=~-\int_{\partialD}dS^{\underline{{i}}}\V^{-1}\partial_{\underline{{i}}}V\,,
Z=\left(\begin{matrix}{Z_{\mathrm{BB}}}&{Z_{\mathrm{BF}}}\\{Z_{\mathrm{FB}}}&{Z_{\mathrm{FF}}}\\\end{matrix}\right)\;,\qquad\tilde{Z}=\left(\begin{matrix}{\tilde{Z}_{\mathrm{BB}}}&{\tilde{Z}_{\mathrm{BF}}}\\{\tilde{Z}_{\mathrm{FB}}}&{\tilde{Z}_{\mathrm{FF}}}\\\end{matrix}\right)
\vert\varphi_{k}^{\eta_{k}}(\eta)\vert^{2}=\frac{\pi}{4}(-\eta)\left[(1+2\vert\beta_{k}^{\eta_{k}}\vert^{2})\vert{\calH}_{1/2-q}\vert^{2}+2\Re\left\{\alpha_{k}^{\eta_{k}}(\beta_{k}^{\eta_{k}})^{\ast}\left({\calH}_{1/2-q}\right)^{2}\right\}\right]\,.
\frac{\lambda_{i}-(\beta-1)i}{\betaN}=\frac{w_{i}-1}{2},\\\frac{\beta\lambda_{i}^{\prime}-i}{\betaN}=\frac{1-v_{i}}{2},\\\frac{r}{\betaN}=-\frac{w_{0}+1}{2},\\\frac{\nu_{i}-(\beta-1)i}{\betaN}=\frac{\xi_{i}-1}{2}
{\hat{\gamma}}_{(\lambda,\mu,\alpha)}^{r}\Rightarrow{\hat{\gamma}}_{(\lambda,\mu,\alpha)}^{r}\eta_{r}^{\lambda\mu},\quad{\hat{\gamma}}_{r}^{(\lambda,\mu,\alpha)}\Rightarrow\eta_{r}^{\lambda\mu}{\hat{\gamma}}_{r}^{(\lambda,\mu,\alpha)}={\hat{\gamma}}_{r}^{(\lambda,\mu,\alpha)}\eta_{r}^{\lambda\mu},\quad{(\eta_{r}^{\lambda\mu})}^{+}=\eta_{r}^{\lambda\mu},
\partial_{z}\left(\theta^{n}A_{n}\right)+\partial_{\bar{z}}\left(\theta^{n}\bar{A}_{n}\right)=-\theta^{n}\sum_{p=2}^{n-2}\left(\!\!\begin{array}{c}{n}\\{p}\\\end{array}\!\!\right)\!\bar{A}_{p+1}A_{n-p+1}-\theta^{n}\left.\frac{d^{n}}{d\varphi^{n}}V\left(\varphi\right)\right|_{\varphi=\varphi_{0}}
T_{0}^{\mu\nu}=-\left(\frac{g_{str}\alpha^{\prime}}{2}\right)\prod_{u}\frac{\delta(\sqrt{\alpha^{\prime}}k_{u})}{\sqrt{\tau_{2}}}\frac{\eta^{\mu\nu}}{2\pi\alpha^{\prime}}\left(\prod_{o}\frac{\delta(\sqrt{\alpha^{\prime}}k_{o})}{\sqrt{\tau_{2}}}X_{1,1}+2^{-(d+1)}X_{1,0}\right)
{\calB}_{\Sigma}\,\,=\,\,\displaystyle{\int}{d}^{3}x\,\displaystyle{\,sum_{\Phi=A_{\mu}^{a},\,c^{a},\,\Psi_{_A},\,\varphi_{i}}^{}}\left({\displaystyle{\\frac{\delta\Sigma}{\delta\Phi^{*}}}}{\displaystyle{\frac{\delta}{\delta\Phi}}}+{\displaystyle{\frac{\delta\Sigma}{\delta\Phi}}}{\displaystyle{\\frac{\delta}{\delta\Phi^{*}}}}\right)\,+b\,.
\left(\begin{array}{l}{\zeta_{2}}\\{\zeta_{3}}\\{\zeta_{0}}\\{\zeta_{1}}\\\end{array}\right)\sim[(\zeta_{0}+\zeta_{1}e_{1})e_{5}e_{6}+(\zeta_{2}-\zeta_{3}e_{1})e_{6}]P_{+3}P_{-\alpha}P_{+\beta},
\hat{U}(k_{1},k_{2})=C(k_{1},k_{2})\,e^{\frac{2i\pi}{\hbar}\sqrt{\frac{\hbarN_{k}}{\tau_{2}}}\Big\{\left[\bar{\tau}k_{1}-k_{2}\right]\alpha^{++}(0,0)+\left[\tauk_{1}-k_{2}\right]{\alpha^{++}}^{\dagger}(0,0)\Big\}}\\\,
h=H[{\bfv}]=\intd^{3}r\intd^{3}r^{\prime}\,\frac{(\nabla\times{\bfv})\times(\nabla\times{\bfv})^{\prime}\cdot({\bfr}-{\bfr}^{\prime})}{4\pi|{\bfr}-{\bfr}^{\prime}|^{3}}.
G_{\,+}^{\ellk}(q^{0})=i\int_{-\infty}^{\infty}dx^{0}e^{iq^{0}x^{0}}\intd\vec{x}\left(\chi_{p},\left(U(\infty,-\infty)j^{\ell}(\vec{x},x_{0})j^{k}(\vec{y},0)\right)_{+}\chi_{p}\right)~~.
\begin{array}{ll}{F_{\sigma\tau}(C)=\partial_{\sigma}C_{\tau}-\partial_{\tau}C_{\sigma}}&{=\frac{1}{2}\frac{1}{\sqrt{g}}\,e^{-\frac{8}{3}\phi}\,g_{\sigma\alpha}\,g_{\tau\beta}\,\epsilon^{\alpha\beta\mu\nu\rho}H_{\mu\nu\rho}}\\{}&{=\frac{1}{2}\delta_{\sigmaI}\,\delta_{\tauJ}\,\epsilon^{IJKL}\partial_{K}\omega_{L},}\\\end{array}
\delta_{B}(f^{(2)}u^{3}\left[\psi^{2}\right])+\delta_{F}(f^{(4)}u^{2}\left[\psi^{4}\right])+u^{i}u\epsilon\dot{N}^{i(2)}\left[\psi^{2}\right]\psi+\frac{i}{2}u^{2}\psi\dot{M}^{(2)}\left[\psi^{2}\right]\epsilon=0~,
H(q,\eta)\;=\;-\frac{1}{2}\sum_{j=1}^{L-1}\:\left(\,\eta\:\sigma_{j}^{x}\sigma_{j+1}^{x}\;+\;\eta^{-1}\sigma_{j}^{y}\sigma_{j+1}^{y}\;+\;q\,\sigma_{j}^{z}\;+\;q^{-1}\sigma_{j+1}^{z}\right)\;,
\begin{array}{lcll}{\mathrm{floor}}&{\mathrm{SU(3/1)}}&{\mathrm{field}}&{\mathrm{helicity}}\\{\mid\mathrm{gnd}>}&{(0~0~-\frac{1}{2})}&{\chi}&{+\frac{1}{2}}\\{\mid\mathrm{1st}>}&{(0~1~-\frac{1}{2})}&{\overline{3}\phi}&{~0}\\{\mid\mathrm{2nd}>}&{(1~0~+\frac{1}{2})}&{3\lambda}&{-\frac{1}{2}}\\{\mid\mathrm{3rd}>}&{(0~0~+\frac{3}{2})}&{A_{\mu}}&{-1.}\\\end{array}
T_{i}^{\vec{\alpha}}(\theta)=\left(\prod_{k=1}^{n}\mathcal{M}_{\alpha_{k}}^{i}(\theta)\right)_{11}^{-1},\,\,\,\quadR_{i}^{\vec{\alpha}}(\theta)=-\left(\prod_{k=1}^{n}\mathcal{M}_{\alpha_{k}}^{i}(\theta)\right)_{12}\left(\prod_{k=1}^{n}\mathcal{M}_{\alpha_{k}}^{i}(\theta)\right)_{11}^{-1}.
{\calL}=2\sqrt{2}\epsilon^{\mu\nu\rho}[\eta^{4}A_{\rho}-2i(\eta^{2}-\frac{1}{2}\vert\phi\vert^{2})\phi(D_{\rho}\phi)^{*}]F_{\mu\nu}+4(\eta^{2}-\vert\phi\vert^{2})^{2}\vertD_{\mu}\phi\vert^{2}-V
{\calL}_{\mathrm{dP}}={\calL}_{\mathrm{orb}}={\calL}_{\mathrm{ell}}\circ\varTheta_{z}=\left\{\varTheta_{z}^{2}-z(\varTheta_{z}+\alpha_{1})(\varTheta_{z}+\alpha_{2})\right\}\circ\varTheta_{z},
\vec{\gamma}=\left(\begin{array}{cc}{0}&{\vec{\sigma}}\\{-\vec{\sigma}}&{0}\\\end{array}\right)\;,\;\gamma_{5}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)\;,\;\gamma_{0}=\left(\begin{array}{cc}{0}&{-1}\\{-1}&{0}\\\end{array}\right)
S=\frac14{^+\tau}\int_{M}E^{\mu\nu\alpha\beta}\,\,{^+{\calR}}_{\mu\nu}^{ab}\,\,{^+{\calR}}_{\alpha\beta}^{cd}\,\epsilon_{abcd}-\frac{1}{4}{^-\tau}\int_{M}E^{\mu\nu\alpha\beta}\,\,{^-{\calR}}_{\mu\nu}^{ab}\,\,{^-{\calR}}_{\alpha\beta}^{cd}\,\epsilon_{abcd},
\begin{array}{rl}{E_{8}^{+}=}&{\{\pme_{a}\}}\\{\cup}&{\{(\pme_{a}\pme_{b}\pme_{c}\pme_{d})/2:a,b,c,d\mathrm{~distinct},\;e_{a}(e_{b}(e_{c}e_{d}))=\pm1\},}\\{}&{a,b,c,d\in\{0,...,7\}.}\\\end{array}
g^{-1}=\left(\begin{array}{ccc}{-2\tau_{2}}&{-3\tau_{3}}&{-4\tau_{4}}\\{-3\tau_{3}}&{-4\tau_{4}+\tau_{2}^{2}}&{+\frac12\tau_{2}\tau_{3}}\\{-4\tau_{4}}&{+\frac12\tau_{2}\tau_{3}}&{-2\tau_{2}\tau_{4}+\frac34\tau_{3}^{2}}\\\end{array}\right)
i\widetilde{\Gamma}_{\mathrm{o}}[{\calD}^{\prime},{\calD}]\equiv-\frac{1}{2}\int_{\xi}^{\infty}\frac{1}{\tau}\,d\tau\,\mathrm{Tr}\,P(\,e^{-\tau{\calD}^{\prime}{\calD}}-\,e^{-\tau{\calD}{\calD}^{\prime}}).
{\calH}=\frac{1}{m}\left(-\frac{d^{2}}{dr^{2}}-\frac{1}{r}\frac{d}{dr}+\frac{[L-\alpha\varepsilon(r)]^{2}}{r^{2}}-\frac{2\sigma\alpha\varepsilon^{\prime}(r)}{r}\right)+\frac{1}{4}m\omega^{2}r^{2};
[C_{3}^{-2},\Gamma_{+}^{-1}]B=-\frac{1}{2}\frac{\alpha}{1+\mu\alphak}[(k-\muk_{1})(C_{3}^{-1}-\muC_{3}^{-2})-(k_{1}-\muk_{2})(1-\muC_{3}^{-1})]Y
\frac{1}{g^{2}}F=ET+N\theta-\frac{16\pi^{2}}{g^{2}}-2\pi^{2}v^{2}\rho^{2}+\frac{16\pi^{2}\rho^{4}}{g^{2}}\int_{0}^{\infty}\frac{dk\;k^{3}}{1-\mathrm{e}^{-kT-\theta}}\left[\mathrm{e}^{-2kT_{1}}+\mathrm{e}^{2kT_{1}-kT-\theta}\right]
S_{ij}=\frac{2}{\sqrt{5}}\left(\begin{array}{cc}{\frac{\sqrt{10-2\sqrt{5}}}{4}}&{\frac{\sqrt{10+2\sqrt{5}}}{4}}\\{\frac{\sqrt{10+2\sqrt{5}}}{4}}&{-\frac{\sqrt{10-2\sqrt{5}}}{4}}\\\end{array}\right),\qquadi,j=0,1.
\begin{array}{rcl}{{\calA}(\omega;\omega_{\bot})}&{=}&{\displaystyle\intdx_{0}dx_{3}~e^{i(\omegax_{0}-\sqrt{\omega^{2}-\omega_{\bot}^{2}}x_{3})}\tau(x_{0})g(x_{0},x_{3}^{2};\omega_{\bot}).}\\\end{array}
{\cal{R}}_{GB}^{2}={\frac{48M^{2}(r^{2}-A^{2}{cos}^{2}\theta)[(r^{2}+A^{2}{cos}^{2}\theta)^{2}-16r^{2}A^{2}{cos}^{2}\theta]}{(r^{2}+A^{2}{cos}^{2}\theta)^{6}}}\,
\phi_{m_{1},n_{1}}\times\phi_{m_{2},n_{2}}=\sum_{m_{3}=|m_{1}-m_{2}|+1}^{m_{1}+m_{2}-1}\sum_{n_{3}=|n_{1}-n_{2}|+1}^{n_{1}+n_{2}-1}\phi_{m_{3},n_{3}}\,,
D_{1}^{++}\langle\tilde{q}^{+}(1)q^{+}(2)\rangle=\delta^{4}(x_{12})\;\delta^{2}(\theta_{1}^{+}-\theta_{2}^{+})\;\delta^{2}(\bar{\theta}_{1}^{+}-\bar{\theta}_{2}^{+})\;\delta(u_{1},u_{2})\;.