harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\theta=\left(\begin{array}{c}{-\frac{\sqrt{2\omega_{o}}}{\mu+\lambda+\mu_{o}}d_{4}U(a,x)}\\{d_{2}U(a+1,x)}\\{-\frac{\sqrt{2\omega_{o}}}{\mu+\lambda+\mu_{o}}d_{2}U(a,x)}\\{d_{4}U(a+1,x)}\\\end{array}\right).
\frac12e_{\alpha}^{\;\;\mu}e_{\beta}^{\;\;\nu}\frac{\partial\Gamma^{(2)}}{\partialg^{\mu\nu}}=e_{\alpha}^{\;\;\mu}e_{\beta}^{\;\;\nu}B_{\mu\nu}^{g(2)}+e_{\alpha}^{\;\;\mu}e_{\beta}^{\;\;\nu}K_{\mu\nu}^{g}.
\bar{R}=-{\frac{2(r_{0}^{2}+r_{n}^{2})r_{n}^{2}}{r^{6}+r^{4}r_{5}^{2}+2r^{4}r_{n}^{2}+2r^{2}r_{n}^{2}r_{5}^{2}+r^{2}r_{n}^{4}+r_{n}^{4}r_{5}^{2}}}\simeq0
ds^{2}=\frac{4h^{\prime}(u)g^{\prime}(v)}{\left[\left\{h(u)-g(v)\right\}\right]^{2\left(1-\frac{3b^{2}}{4\kappa^{2}}\right)^{-1}}}\left(-dt^{2}+dy^{2}\right)+\frac{1}{\left[\left\{h(u)-g(v)\right\}\right]^{2\left(1-\frac{3b^{2}}{4\kappa^{2}}\right)^{-1}}}\:\delta_{ij}\:dx^{i}\:dx^{j},
P_{\calT}^{1/2}\left(k\right)=\sqrt{\frac{k^{3}}{2\pi^{2}}}\left|{\frac{v_{k}}{m_{\mathrm{Pl}}a}}\right|={\frac{2^{\nu-3/2}}{2\pi}}{\frac{\Gamma\left(\nu\right)}{\Gamma\left(3/2\right)}}\left({\frac{H}{m_{\mathrm{Pl}}}}\right)\left|C_{\mathrm{+}}\left(k\right)+C_{-}\left(k\right)\right|\left({\frac{y}{1-\epsilon}}\right)^{3/2-\nu},
{F_{\alpha_{1}\alpha_{2}}}{F_{\alpha_{2}\alpha_{3}}}{F_{\alpha_{3}\alpha_{4}}}{F_{\alpha_{4}\alpha_{1}}}\rightarrow{F_{1\alpha_{1}\alpha_{2}}}{F_{2\alpha_{2}\alpha_{3}}}{F_{3\alpha_{3}\alpha_{4}}}{F_{4\alpha_{4}\alpha_{1}}}
F_{D=0}^{\mathrm{TM}}={\frac{1}{2\pia^{2}}}(\hat{Q}_{0}+D\hat{Q}_{1}-\hat{Q}_{2})={\frac{1}{2a^{2}}}\left(1-{\frac{101}{64}}+{\frac{101}{64}}\right)={\frac{1}{2a^{2}}},
\langleT(z,\bar{z})T(0,0)\rangle=\frac{c/2}{z^{4}}\qquad\langle\bar{T}(z,\bar{z})\bar{T}(0,0)\rangle=\frac{c/2}{\bar{z}^{4}}\qquad\langleT\bar{T}\rangle=0
A_{\mu}^{pq}(x)=-i\int_{\tilde{T}^{2}}\psi^{p}{}^{\dagger}D_{x}(\hat{A})\left(D_{x}^{\dagger}(\hat{A})D_{x}(\hat{A})\right)^{-1}\sigma_{\mu}^{\dagger}\psi^{q}+R_{\mu}^{pq}.
\delta^{ab}\,\delta_{xy}\,=\,\langle\,\frac{\deltaS}{\delta\bar{c}_{x}^{a}}\,\bar{c}_{y}^{b}\,\rangle\,=\,\widetildeZ_{3}\,\langle\,i(\partialD_{r}c)_{x}^{a}\,\bar{c}_{y}^{b}\,\rangle\,-\,\frac{Z_{\lambda}\lambda}{Z_{3}}\,\langle\,(s_{r}B_{x}^{a})\,\bar{c}_{y}^{b}\,\rangle\;.
C_{\tau\alpha}U^{a_{1}b_{1}}U^{a_{2}b_{2}}U^{a_{3}b_{3}}\lambda_{\tau_{1}\alpha_{1}}^{b_{1}}\lambda_{\tau_{2}\alpha_{2}}^{b_{2}}\lambda_{\tau_{3}\alpha_{3}}^{b_{3}}=f_{\tau\alpha,\tau_{1}\alpha_{1},\tau_{2}\alpha_{2},\tau_{3}\alpha_{3},a_{1},a_{2},a_{3}}\Psi.
F=\sum_{i=1}^{4}f_{i}V_{i}=8\pi\left[(1-\alpha)^{2}K_{1}P_{1}f_{1}+\alpha(1-\alpha)K_{2}P_{2}f_{2}+\alpha(1-\alpha)K_{3}P_{3}f_{3}+\alpha^{2}K_{4}P_{4}f_{4}\right].
U_{~~~~~\alpha\beta\gamma\delta}^{\mu\rho\nu\sigma}(x,y)=(x\eta_{\alpha\beta}\eta^{\mu\rho}+y\delta_{\alpha}^{\mu}\delta_{\beta}^{\rho})(x\eta_{\gamma\delta}\eta^{\nu\sigma}+y\delta_{\gamma}^{\nu}\delta_{\delta}^{\sigma}).
\left(D_{0}\right)_{\;\;b}^{a}=\delta_{\;\;b}^{a}\partial_{0}+f_{bc}^{a}H_{0}^{c},\;\left(D_{0}\right)_{a}^{\;\;b}=\delta_{a}^{\;\;b}\partial_{0}-f_{ac}^{b}H_{0}^{c}.
C_{\alpha\beta}(\omega)=\pi\sum_{\epsilon,\epsilon^{\prime}}\langle0|{\hat{f}}_{\alpha}^{(0)}(0)|\epsilon,\epsilon^{\prime}\rangle\langle\epsilon,\epsilon^{\prime}|{\hat{f}}_{\beta}^{(0)}(0)|0\rangle\delta(\omega-\omega_{\epsilon}-\omega_{\epsilon^{\prime}}).
p={\frac{1}{2d}}{X_{1}^{2d}}+{\frac{1}{2d}}{X_{2}^{2d}}+{\frac{1}{d\sqrt{c}}}{X_{1}^{d}X_{2}^{d}}+\tilde{W}({\frac{X_{1}X_{2}}{c^{1/d}}},X_{3},X_{4},X_{5}),
{\calL}=\frac{1}{2}\left\{\partial_{m}\phi_{i}\partial^{m}\phi_{i}+\imath\bar{\psi}_{i}\gamma^{m}\partial_{m}\psi_{i}-\frac{\partial{\calW}}{\partial\phi_{i}}\frac{\partial{\calW}}{\partial\phi_{i}}-\frac{\partial^{2}{\calW}}{\partial\phi_{i}\partial\phi_{j}}\bar{\psi}_{i}\psi_{j}\right\}.
\int\delta^{10}u\sqrt{|g_{\alpha\beta}|}e^{-\Phi^{\prime}}\sum_{I=1}^{16}\widehat{F}^{I,\mu\nu}\widehat{F}_{\mu\nu}^{I}=\int\delta^{n^{\prime}+m^{\prime}}u\sqrt{|g_{\alpha^{\prime}\beta^{\prime}}|}e^{-\phi}~\sum_{I=1}^{16}[\widehat{F}^{I,\mu^{\prime}\nu^{\prime}}\widehat{F}_{\mu^{\prime}\nu^{\prime}}^{I}+2
\begin{array}{rcl}{\dot{J}_{1}^{\prime}}&{=}&{[J_{1}^{\prime},H_{q}]_{GPB}=\displaystyle\frac{I-I_{3}}{II_{3}}J_{2}^{\prime}J_{3}^{\prime}~,}\\{\dot{J}_{2}^{\prime}}&{=}&{[J_{2}^{\prime},H_{q}]_{GPB}=\displaystyle\frac{I_{3}-I}{II_{3}}J_{1}^{\prime}J_{3}^{\prime}~,}\\{\dot{J}_{3}^{\prime}}&{=}&{[J_{3}^{\prime},H_{q}]_{GPB}=0~.}\\\end{array}
N=\frac{1}{4\beta\left(\frac{2}{\sqrt{d_{\perp}}}\right)^{d_{\perp}}}~\int_{y_{N}}^{y_{end}}\frac{(1-y)^{d_{\perp}-1}dy}{y^{3}\left(1+(\frac{d_{\perp}}{4}-\frac{3}{2})y\right)}
\begin{aligned}{{}\big[M^{(8)},Z^{(2|1)}\big]}&{{}=m\,Z^{(2|1)}\,,}&{\big[W^{(8)},Z^{(2|1)}\big]}&{{}=\tilde{m}\big[M^{(8)},Z^{(2|1)}\big]\,,}\\{{}\big[M^{(5)},Z^{(2|1)}\big]}&{{}=n\,Z^{(2|1)}\,,}&{\big[W^{(5)},Z^{(2|1)}\big]}&{{}=\tilde{n}\big[M^{(5)},Z^{(2|1)}\big]\,,}\\\end{aligned}
\{f,g\}=\sum_{i=1}^{d}\left(\frac{\partialf}{\partialq^{i}}\frac{\partialg}{\partialp^{i}}-\frac{\partialf}{\partialp^{i}}\frac{\partialg}{\partialq^{i}}\right)=\sum_{i=1}^{d}(f_{2i-1}g_{2i}-f_{2i}g_{2i-1}),
\prod_{i}\left|z-z_{i}\right|^{2\beta_{i}}=\prod_{i}\left|w\right|^{-2\beta_{i}}\left|1-wz_{i}\right|^{2\beta_{i}}=\left|w\right|^{4}\prod_{i}\left|1-wz_{i}\right|^{2\beta_{i}}\mathrm{~.}\tag{5.19}
\left(\begin{array}{c}{x^{+}}\\{x^{-}}\\\end{array}\right)\sim\left(\begin{array}{c}{x^{+}}\\{x^{-}}\\\end{array}\right)+\left(\begin{array}{c}{\frac{R_{0}}{2R}}\\{-R+\frac{R_{0}}{2R}}\\\end{array}\right)
f(y)=Ae^{-y^{2}/4}M\left(\frac{a}{2}+\frac{1}{4},\frac{1}{2},\frac{y^{2}}{2}\right)+Bye^{-y^{2}/4}M\left(\frac{a}{2}+\frac{3}{4},\frac{3}{2},\frac{y^{2}}{2}\right),
dS^{2}={\frac{1}{I_{0}}}\{{\frac{1}{f_{0}}}[e^{2k_{0}}dz\,d{\bar{z}}+\rho^{2}\,d\phi^{2}]-f_{0}(dt+A_{0}d\phi)^{2}\}+I_{0}^{2}dx^{5^{2}},
K_{x\mu\nu}^{\rho\sigma}G_{\rho\sigma,\alpha\beta}(x,y)+K_{x\mu\nu}^{R\rho\sigma}G_{\rho\sigma,\alpha\beta}(x,y)=\frac{1}{2}(\eta_{\mu\alpha}\eta_{\nu\beta}+\eta_{\mu\beta}\eta_{\nu\alpha})\delta^{N}(x-y)
ds^{2}=dUdV+dX^{+}dX^{-}+dX^{i}dX^{i}+R^{d}\Big(\frac{{dX^{+}}^{2}}{U^{d}}+\frac{{X^{+}}^{2}}{U^{d+2}}du^{2}-2\frac{X^{+}}{U^{d+1}}dUdX^{+}\Big)
m^{\mu\nu}(\Sigma)={\frac{1}{2}}\int_{\Sigma}\left\{\tilde{p}^{\mu}X^{\nu}+2\beta\sqrt{h}\eta_{a}\left(K^{i}\gamma^{ab}-K^{ab\,i}\right)n^{\mu}{}_{i}e^{\nu}{}_{b}-\left(\mu\leftrightarrow\nu\right)\right\}\,,
n^{\mathrm{string}}\equiv\frac{\delta\psi}{2\pi}=\frac{1}{2\pi}\int_{C}ds_{j}\,\partial_{j}\psi=\frac{-i}{4\pi}\int_{C}ds_{j}\,\mathrm{tr}\,\tau_{3}\Omega_{\mathrm{D}}\partial_{j}\Omega_{\mathrm{D}}^{\dagger},
\psi_{k_{y}}^{+}=\sqrt{\frac{\pi}{4H_{5}}}e^{{\frac{\pik_{y}}{2H_{5}}}}H_{ik_{y}/H_{5}}^{(2)}(mt)e^{ik_{y}y},\qquad\psi_{k_{y}}^{-}=\sqrt{\frac{\pi}{4H_{5}}}e^{-{\frac{\pik_{y}}{2H_{5}}}}H_{ik_{y}/H_{5}}^{(1)}(mt)e^{ik_{y}y}.
\chi^{\mu}=-{\frac{C}{2\gamma\Phi}}\nabla^{\mu}\varphi+{\frac{\Theta}{2\gamma\Phi}}(\nabla^{\mu}\Phi)\varphi+{\frac{C}{4\gamma\Phi}}(\nabla^{\mu}\Phi)h-\nabla^{\nu}\bar{h}_{\mu\nu}-{\frac{1}{\Phi}}(\nabla^{\nu}\Phi)\bar{h}_{\nu}^{\mu}.
K_{0}=24\lambda+\frac{\Lambda}{\int\frac{d^{n}pd^{n}qd^{n}r}{(2\pi)^{3n}(p^{2}+\Omega^{2})(q^{2}+\Omega^{2})(r^{2}+\Omega^{2})((p+q+r)^{2}+\Omega^{2})}}.
\begin{array}{rclcrcl}{\displaystylet_{\underline{{a}}}t_{\underline{{a}}}}&{=}&{C_{F},}&{}&{C_{F}}&{=}&{\displaystyle\frac{N^{2}-1}{2N}}\\{\displaystylet_{\underline{{a}}}t_{\underline{{b}}}t_{\underline{{a}}}}&{=}&{(C_{F}-C_{A}/2)t_{\underline{{b}}},}&{}&{C_{A}}&{=}&{N}\\\end{array}
\biggl(\frac{x_{14}\partial_{4}}{1+2\epsilon}-1\biggr)\,K_{a2}\,=\,(x_{14}\partial_{4}-1)\int\frac{d^{4}x_{5}}{x_{15}^{2}\,x_{45}^{2}\,(x_{35}^{2})^{\beta}}+O(\epsilon)\,,\quad\;
K_{\alpha}^{-}~=~{\textstyle\frac{1}{2}}\,\epsilon\,(H_{\alpha}^{})_{\mathfrak{b}}^{}~~,~~M_{\alpha}^{-}~=~{\textstyle\frac{1}{2}}\,\epsilon\,(H_{\alpha}^{})_{\mathfrak{a}}^{}~.
D_{--}(x)=i\langle0|T^{+}A_{-}({x})A_{-}(0)|0\rangle=-\int_{-\infty}^{\infty}\frac{d^{2}{k}}{(2\pi)^{2}}e^{-i{k}\cdotx}\frac{1}{[k_{+}+i\epsilon\\mathrm{sgn}\(k_{-})]^{2}}.
\Bigl[-\partial_{4}+m(x_{4})\Bigr]\xi_{M}^{-}(x_{4})=M\xi_{M}^{+}(x_{4})\,,\quad\Bigl[\partial_{4}+m(x_{4})\Bigr]\xi_{M}^{+}(x_{4})=M\xi_{M}^{-}(x_{4})
e^{-s}=\frac{1}{2\pii}\int_{c-i\infty}^{c+i\infty}\,d\alpha\,\Gamma\left(\alpha\right)\,s^{-\alpha},\,\,\,\,\,\,\,\,\,\,\,\,s,c\,\,\mathrm{real},\,\,\left|s\right|<1,\,\,c>1.
A_{bh}=4\pi\prod_{i=1}^{n}(q_{i}+r_{0})^{2/n}=4\pi\prod_{i=1}^{n}\left[\left(\hat{q}_{i}^{2}+{\frac{r_{0}^{2}}{4}}\right)^{1/2}+{\frac{r_{0}}{2}}\right]^{2/n},
\frac{D_{1}-2}{2}A^{\prime}h_{rm}+h_{m\underline{l};}^{\quad\underline{l}}-\kappa^{2}\left(\phi^{\dagger}D_{m}\Phi+(D_{m}\Phi)^{\dagger}\phi+V_{\underline{l}}F_{n}^{\quad\underline{l}}\right)~,
\sum_{j=0}^{i}q^{jr}\chi_{1,r(k+1)+k-i+2j}^{(p,kp+p-1)}(q)=\displaystyle\sum_{n_{1}\geq\cdots\geqn_{k}\geq0}\frac{q^{n_{1}^{2}+\cdots+n_{k}^{2}+(r-1)(n_{1}+\cdots+n_{i})+r(n_{i+1}+\cdots+n_{k})}}{(q)_{n_{1}-n_{2}}\cdots(q)_{n_{k-1}-n_{k}}(q)_{2n_{k}+r}}F_{p-1;r}^{(2n_{k}+r)}(q).
\begin{array}{ll}{\delta_{k}\theta=\tilde{\Gamma}^{m}p_{m}k,}&{\qquad\delta_{k}x^{m}=i\theta\Gamma^{m}\delta\theta,}\\{\delta_{k}e=4i\dot{\theta}k,}&{\qquad\delta_{k}A^{m}=(i\theta\Gamma^{m}\delta\theta)^{\cdot}.}\\\end{array}
\hat{g}=\frac{1}{\tilde{g}}=(R_{s}M)^{1/2}\prodr_{i}\,\quad\hat{\alpha}^{\prime}=\tilde{g}\tilde{\alpha}^{\prime}=\frac{\prodr_{i}}{M^{2}}\,\quad\hat{R}_{i}=\tilde{R}_{i}\.
N=n_{1}^{2}+\cdots+n_{k-1}^{2}+n_{i}+\cdots+n_{k-1}+\sum_{j=1}^{k-2}\sum_{p=1}^{n_{j}-n_{j+1}}a_{p}^{(j)}+2\sum_{p=1}^{n_{k-1}}b_{p},
4\pi^{2}\rho_{m_{0}\rightarrowm_{1}+m_{2}+m_{3}}^{2\ell}=\int\!\!\!\int\!\!\!\int\!d\mu_{1}\,d\mu_{2}\,d\mu_{3}\,\delta(\mu_{1}+\mu_{2}+\mu_{3})\rho_{m_{0}\rightarrowm_{1\mu_{1}}+m_{2\mu_{2}}+m_{3\mu_{3}}}^{2\ell-1}
V(\phi)\,=\,\left(\,-\,m^{2}\,+\,{\frac{F^{2}}{M_{p}^{2}}}\,+...\right)\,|\phi|^{2}\,+\,\left(\,1\,+\,{\frac{F^{2}}{M_{p}^{4}}}\,+..\right)|\phi|^{4}\,+...
\partial_{\mu}\partial_{\nu}(\eta_{A}q^{A})-\partial_{\nu}(\eta_{A}q^{A})\partial_{\mu}\sigma-\partial_{\mu}(\eta_{A}q^{A})\partial_{\nu}\sigma-\eta_{\mu\nu}\partial^{\lambda}(\eta_{A}q^{A})\partial_{\lambda}\sigma=[\frac{\pi\kappa}{2}\hat{V}_{Q}(\eta_{A}q^{A})-\lambda]e^{2\sigma}\eta_{\mu\nu}.
\Omega_{D}^{2}(a,b,{\bfk}_{n},{\bfk}_{n^{\prime}})\equiv\Omega_{r}^{ab}({\bfk}_{n})\Omega_{r}^{ba}(-{\bfk}_{n^{\prime}})-\Omega_{i}^{ab}({\bfk}_{n})\Omega_{i}^{ba}(-{\bfk}_{n^{\prime}}),
\mathrm{Res}_{s=1/2-k}\\zeta_{G}(s)=\frac12\,\mathrm{res}\G^{k-1/2}=\frac{(2k-1)!!\,q^{k}}{k!\,2^{k}},\,mbox{Res}_{s=-k}\\zeta_{G}(s)=\frac12\,\mathrm{res}\G^{k}=0,\k=0,1,2,\ldots,
i[{\calH}_{C\omega},L_{\mu\nu}^{\pm}]=\sum_{\lambda\in{\calR}}\left(L_{\mu\lambda}^{\pm}\widetilde{M}_{\lambda\nu}-\widetilde{M}_{\mu\lambda}L_{\lambda\nu}^{\pm}\right)\pmi\omegaL_{\mu\nu}^{\pm},\qquad\mu,\nu\in{\calR}.
\tilde{\calA}_{n}=\left[(-1)^{A}({{\cal{\varepsilon}}}^{2})_{A}{}^{B}\,J_{B}{}^{C}\,\left({{\calO}}_{C}{}^{D}I_{D}{}^{A}\right)^{n}\right]_{\mathrm{anti}},\quad\quad\mathrm{for~n=1,3~},
S=\frac14{^+\tau}\int_{M}\varepsilon^{\mu\nu\alpha\beta}\,{^+{\calR}}_{\mu\nu}^{ab}\,{^+{\calR}}_{\alpha\beta}^{cd}\epsilon_{abcd}-\frac14{^-\tau}\int_{M}\varepsilon^{\mu\nu\alpha\beta}\,{^-{\calR}}_{\mu\nu}^{ab}\,{^-{\calR}}_{\alpha\beta}^{cd}\epsilon_{abcd}\,,
g_{20}(g,g_{f},\phi_{0},q_{1},q_{2})={\frac{g}{4}}{\frac{q_{1}q_{2}}{(q_{1}^{2}+q_{2}^{2}+q_{1}q_{2})}}[C_{0}(g_{f},\phi_{0},q_{1})+C_{0}(g_{f},\phi_{0},q_{2})+2\phi_{0}].
W[Y]=\frac{1}{2}\intd^{4}x\int\frac{d^{4}p}{(2\pi)^{4}}\int_{0}^{\infty}\frac{dt}{t^{3}}\rho(t,\Lambda^{2})e^{-p^{2}}\mathrm{tr}\left(e^{-t({\calM}^{2}+A)}\right)\cdot1.
\begin{array}{rcl}{M_{M2_{6}}}&{=}&{{\displaystyle\frac{R_{3}R_{4}(R_{5}\ldotsR_{10})^{2}}{({}^{-}\!\!\!\!\ell_{\mathrm{Planck}}^{(11)})^{15}}}\,,}\\{}&{}&{}\\{M_{M5_{3}}}&{=}&{{\displaystyle\frac{R_{3}\ldotsR_{6}(R_{7}R_{8}R_{9})^{2}R_{10}}{({}^{-}\!\!\!\!\ell_{\mathrm{Planck}}^{(11)})^{12}}}\,.}\\\end{array}
\begin{array}{ccccc}{\alpha+\beta+\gamma}&{=}&{\bar{\alpha}+\bar{\beta}+\bar{\gamma}}&{=}&{kn-m-n-1,}\\{\alpha^{\prime}+\beta^{\prime}+\gamma^{\prime}}&{=}&{\bar{\alpha}^{\prime}+\bar{\beta}^{\prime}+\bar{\gamma}^{\prime}}&{=}&{km-m-n-1,}\\\end{array}
S={\frac{1}{16\piG_{3}}}\int_{\calM}d^{3}x\,\sqrt{-g}\left(R+{\frac{2}{\ell^{2}}}\right)-{\frac{1}{8\piG_{3}}}\int_{\partial{\calM}}\sqrt{-h}\left(\theta+{\frac{1}{\ell}}\right),
{\frac{1}{\omega}}\left(-{\frac{9}{20}}-{\frac{z_{0}^{2}}{l^{2}}}\right)<{\frac{1}{32\omega^{2}l^{2}}}\left(-32\omegaz_{0}^{2}+9l^{2}+3\sqrt{-64\omegaz_{0}^{2}l^{2}+9l^{4}-64l^{2}z_{0}^{2}\omega^{2}}\right)\,,
\bar{\gamma}^{(4)}\equiv\left\langle\gamma^{(4)}(\hat{\bf{q}},-\hat{\bf{q}},0,0)\right\rangle_{\hat{\bf{q}}}=\int_{0}^{1}d\lambda\lambda^{3-D}\left\langleI^{(4)}(\lambda\hat{\bf{q}},-\lambda\hat{\bf{q}},0,0)\right\rangle_{\hat{\bf{q}}}\;,
P(q,q^{\prime})=\sum_{k=0,k^{\prime}=0}^{\infty}\frac{1}{k!k^{\prime}!}q^{n_{1}}...q^{n_{k^{\prime}}}q^{\prime}{}^{m_{1}}...q^{\prime}{}^{m_{k}}P_{\{m_{1}...m_{k}|n_{1}...n_{k^{\prime}}\}}(\partial_{l})
|f^{\prime}|\leq{\frac{1}{|w_{2}|^{N}}}{\frac{|1-w_{2}|^{1/2}}{|w_{2}-\beta|}}\left({\frac{N}{|w_{2}|}}+{\frac{1}{|w_{2}-\beta|}}+{\frac{1/2}{|w_{2}-1|}}\right)<{\frac{\sqrt{2}s^{N}}{\beta}}\left(Ns+{\frac{1}{\beta}}+{\frac{1}{2}}\right),
\mathcal{A}_{bound}=\int_{H}d^{2}x\left[\frac{1}{8\pi}(\partial_{\mu}\varphi)^{2}+\sum_{i=0}^{r}\mu_{i}e^{be_{i}\cdot\varphi}\right]+\intdx\sum_{0}^{r}\nu_{i}e^{be_{i}\cdot\varphi/2}.
I_{S}=\lambda^{2}B_{n}^{4}\sum_{N=0}^{\infty}\sum_{\alpha=0}^{N}\frac{(2a)^{8-3n}{\pi}^{\frac{n}{2}}\Gamma(\frac{8-3n}{2})\Gamma(N-\frac{8-4n}{2})}{\Gamma(N+\frac{8-2n}{2})\Gamma(\frac{4n-8}{2})}{\int}_{S}d^{n}{\eta}d^{n}{\eta^{\prime}}Y_{\alpha}^{N}({\eta})Y_{\alpha}^{N}({\eta}^{\prime})
\begin{array}{rcl}{\frac{A^{2n+2}}{V^{2n+1}}}&{=}&{4\pi(n+1)\,\lambda^{n}\left(\frac{n+1}{2^{n}}\right)^{n}\mathrm{Vol(M)}\,\rho^{n+1}(\rho+1)^{n(n+1)}}\\{}&{\times}&{\frac{\left(\mathrm{{Appell}}F_{1}[n+1,2,-n,n+2,1-\rho,\frac{1-\rho}{2}]\right)^{n+1}}{\left(_2F_{1}[1+n,-n,n+2,\frac{1-\rho}{2}]\right)^{2n+1}}.}\\\end{array}
\left(\begin{array}{c}{\dot{\Phi}}\\{\Phi^{\prime}}\\\end{array}\right)\;\;\rightarrow\;\;\left(\begin{array}{cc}{cosh\,\theta}&{sinh\,\theta}\\{sinh\,\theta}&{cosh\,\theta}\\\end{array}\right)\;\;\left(\begin{array}{c}{\tilde{\dot{\Phi}}}\\{\tilde{\Phi}^{\prime}}\\\end{array}\right)\;\;.
\frac{\tau_{i}^{\prime}}{\tau_{i}}+\varepsilon\,\eta_{i}\,\frac{\alpha_{i}^{2}}{2}\prod_{j=0}^{n}\tau_{j}^{-\frac{a_{ij}}2}=\eta_{i}\,C~~~~\mathrm{at~}x=0,~~~i=0,\dots,n,
{\Gamma}_{t}^{(6)}({\bf{q}}_{1},\ldots,{\bf{q}}_{6})=\int_{0}^{t}d\taug_{t-\tau}^{3}e^{-(2D-6)\tau}I^{(6)}(e^{-\tau}{\bf{q}}_{1},\ldots,e^{-\tau}{\bf{q}}_{6})\;,
\deltaK_{\mu\nu}=\frac{1}{2}e_{\mu}^{(0)M}e_{\nu}^{(0)N}({\calL}_{\deltan}g_{MN}^{(0)}+{\calL}_{\deltaZ}{\calL}_{n^{(0)}}g_{MN}^{(0)}-2n^{(0)L}\delta\Gamma_{LMN}).
\widetilde{\Gamma}_{kp}\=\\frac{1}{\mu_{k}{-}\bar{\mu}_{p}}\,T_{k}^{\dagger}\,T_{p}\quad,\qquad\textrm{i.e.}\qquad\sum_{k=1}^{m}\,\Gamma^{\ellk}\,\widetilde{\Gamma}_{kp}\=\\delta_{\p}^{\ell}\quad.
T(a,b,d)=\frac{4}{(2\pi)^{d-2}ab}\sum_{n,n^{\prime}=1}^{\infty}\intd^{d-2}p\frac{1}{\left(\vec{p}^{\,2}+(\frac{n\pi}{a})^{2}+(\frac{n^{\prime}\pi}{b})^{2}+m^{2}\right)}.
\inti|{\bar{a}}\rangle=\inti(M_{\bar{a}}^{l}|\sigma_{2}(l)\rangle)=(\partial_{i}M_{\bar{a}}^{l}+M_{\bar{a}}^{n}A_{in}^{l})|\sigma_{2}(l)\rangle+C_{in}^{l}M_{\bar{a}}^{n}|\sigma_{1}(l)\rangle
\chi_{ij}^{a}=\frac{N^{2}}{2(4\pi^{2})}L_{ij}^{(4)}\quad,\quad\chi_{ij}^{g}=\frac{N^{2}}{(4\pi)^{2}}L_{ij}^{(4)}\quad,\quadc=a=\frac{N^{2}}{4(4\pi)^{2}}\quad.
\begin{array}{c}{Z_{+}(-\infty)=\alpha+g_{+}(-\infty)+\displaystyle\frac{\chi_{\infty}}{\pi}{\calQ}_{+}(-\infty)}\\{Z_{-}(+\infty)=\alpha+g_{-}(+\infty)+\displaystyle\frac{\chi_{\infty}}{\pi}{\calQ}_{-}(+\infty)\,.}\\\end{array}
S=\intd\tau\left[{\frac{1}{2e}}(\dot{x}^{\underline{m}}\dot{x}_{\underline{m}}+i\dot{\chi}^{\underline{m}}\chi_{\underline{m}})-{\frac{i}{e^{2}}}\psi\chi_{\underline{m}}\dot{x}^{\underline{m}}\right].
{\cal{L}}_{\psi}=\frac{1}{2}\,\overline{{\Psi}}^{a}\left[\sqrt{-\left(D^{2}\right)^{ac}+\left(F_{\rho\sigma}^{b}S^{\rho\sigma}\right)\left(t^{b}\right)^{ac}}\right]\Psi^{c}
V^{(0)}=\frac{1}{4}\,\pi_{ij}\pi^{ij}+\partial_{i}A_{0j}\pi^{ij}-\frac{1}{4}\,A_{ij}\nabla^{2}A^{ij}+\frac{1}{2}\,A_{ij}\,\partial^{j}\partial_{k}A^{ik}
T^{\underline{\alpha}}={\frac{1}{288}}F_{\underline{b}_{1}...\underline{b}_{4}}E^{\underline{a}}\left(\Gamma_{\underline{a}}^{~\underline{b}_{1}...\underline{b}_{4}}-8\delta_{\underline{a}}^{[\underline{b}_{1}}\Gamma^{\underline{b}_{2}...\underline{b}_{4}]}\right)_{~\underline{\beta}}^{\underline{\alpha}}E^{\underline{\beta}},
i\hbar\partial_{\tau}\tilde{\psi_{\tau}}=-i\hbar\partial_{t}\tilde{\psi_{\tau}}+\frac{1}{2M}(-i\hbar\nabla+\frac{1}{c}{\bfA})^{2}\tilde{\psi_{\tau}}+V({\bfx},t)\tilde{\psi_{\tau}}\equivK\tilde{\psi_{\tau}}
\bigl<n;t|H(t)|n;t\bigr>=(n+\frac{1}{2})\hbar\omega~\Bigl(\frac{\omega}{\tilde{\omega}}\Bigr),~~~\Bigl<n;t|i\hbar\frac{\partial}{\partialt}|n;t\bigr>=(n+\frac{1}{2})\hbar\omega~\Bigl(\frac{\mu^{2}}{\omega\tilde{\omega}}\Bigr).
U(z)={\frac{B}{A}}\left[{\frac{3}{4}}{\frac{B^{\prime\prime}}{B}}+{\frac{3}{16}}\left({\frac{B^{\prime}}{B}}\right)^{2}-{\frac{3}{8}}{\frac{A^{\prime}B^{\prime}}{AB}}+2{\frac{B^{\prime}}{Br}}-{\frac{1}{2}}{\frac{A^{\prime}}{Ar}}\right]\,.
\sum_{\nu\ne\Omega(\mu)}w_{\nu}I_{\nu\mu}^{(o)}+w_{\mu}I_{\mu\Omega(\mu)}^{(o)}-4I_{\mu\Omega(\mu)}^{(o)}=\sum_{\nu}w_{\nu}I_{\nu\mu}^{(o)}-4I_{\mu\Omega(\mu)}^{(o)}.
f_{dc}^{a}\tilde{f}_{a}^{rs}=\tilde{f}_{c}^{as}f_{da}^{r}+\tilde{f}_{c}^{ra}f_{da}^{s}-\tilde{f}_{d}^{as}f_{ca}^{r}-\tilde{f}_{d}^{ra}f_{ca}^{s}
\xi_{1}^{(i)}(x)\equiv\langle\lambda_{max}^{(i)}|\mathrm{exp(}\frac{1}{2}\Phi)T(x),{}~~~~\xi_{2}^{(i)}(x)\equiv\langle\lambda_{max}^{(i)}|E_{i}\mathrm{exp(}\Psi_{+})\mathrm{exp(}\frac{1}{2}\Phi)T(x)
S\;=\;\intd^{3}x\,\left[-\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi\,-\,\frac{1}{2}{\bar{\psi}}\not\!\partial\psi\,-\,\frac{1}{2}(\frac{\kappa}{2}\phi+g\phi^{2})^{2}\,-\,\frac{1}{2}(\frac{\kappa}{2}+2g\phi){\bar{\psi}}\psi\right]\;.
S_{3+1}\,=\,\int\,d^{4}x\,F^{2}\,+\,\int\,\phi^{N}\,d\sigma_{\mu\nu\gamma}\,\Pi^{\mu\mu^{\prime}}\Pi^{\nu\nu^{\prime}}\Pi^{\gamma\gamma^{\prime}}\,A_{\mu^{\prime}\nu^{\prime}\gamma^{\prime}}
B_{ij}\equiv(\varepsilon\psi_{i},\psi_{j})=(\int_{\mathcal{I}}dy\varepsilon(x-y)\psi_{i}(y)\,\\psi_{j})=\int_{\mathcal{I}}\int_{\mathcal{I}}dx\dy\\varepsilon(x-y)\psi_{i}(y)\psi_{j}(x),
\left(\begin{array}{cccc}{(E-m)}&{0}&{-P_{z}}&{-\pi_{-}}\\{0}&{(E-m)}&{-\pi_{+}}&{P_{z}}\\{-P_{z}}&{-\pi_{-}}&{(E+m)}&{0}\\{-\pi_{+}}&{P_{z}}&{0}&{(E+m)}\\\end{array}\right)\left(\begin{array}{c}{\varphi_{+}}\\{\varphi_{-}}\\{\chi_{+}}\\{\chi_{-}}\\\end{array}\right)=0
{}^{\pm}E_{j}{}^{i}=\pm\frac{4\varrho^{2}}{q(r^{2}+\varrho^{2})^{2}}\delta_{j}{}^{i},\qquad{}^{\pm}B_{j}{}^{i}=-\frac{4\varrho^{2}}{q(r^{2}+\varrho^{2})^{2}}\delta_{j}{}^{i},
\tau_{1}=\left(\begin{array}{ll}{i}&{0}\\{0}&{\mathrm{~-i~}}\\\end{array}\right),\\\\\tau_{2}=\left(\begin{array}{ll}{0}&{1}\\{-1}&{0}\\\end{array}\right),\\\\\tau_{3}=\left(\begin{array}{ll}{0}&{i}\\{i}&{0}\\\end{array}\right).
\frac12\int_{s_{0}}^{s_{1}}[\frac12\left(\frac{dA_{1}}{ds}-\frac{dA_{2}}{ds}\right)^{*}\left(\frac{dA_{1}}{ds}-\frac{dA_{2}}{ds}\right)+\sum_{i=1}^{2}\left(\frac{dz^{*}}{ds}+ieA_{i}z^{*}\right)\left(\frac{dz}{ds}-ieA_{i}z\right)]ds
S=\frac{1}{2}\intd^{4}x\varepsilon^{\mu\nu\alpha\beta}\varepsilon_{abcd}\bigg[({^+}\tau-{^-}\tau)F_{\mu\nu}^{ab}F_{\alpha\beta}^{cd}+({^+}\tau+{^-}\tau)F_{\mu\nu}^{ab}\tilde{F}_{\alpha\beta}^{cd}\bigg].
E_{0}={\frac{1}{2}}\,\sum_{n=-\infty}^{\infty}\sqrt{\left({\frac{2\pi\,n}{L}}\right)^{2}+m^{2}}={\frac{m}{2}}+\sum_{n=1}^{\infty}\sqrt{\left({\frac{2\pi\,n}{L}}\right)^{2}+m^{2}}.
\left.\hat{F}_{\alpha\beta}^{I}\right|_{\mathrm{reg}}=-\frac{2\,e_{\alpha}^{a}\,\hat{\xi}_{ab}^{+}\,e_{\beta}^{b}}{\left(r^{2}+\rho^{2}\right)^{2}}\qquad\mathrm{or}\qquad\left.\hat{F}_{ab}^{I}\right|_{\mathrm{reg}}=-\frac{2\,\hat{\xi}_{ab}^{+}}{\left(r^{2}+\rho^{2}\right)^{2}}
\int_{0}^{\epsilon}dr~r^{2}~\left({\frac{r_{A}}{r}}\right)^{2(\alpha_{+}-1)}~\left({\frac{r_{B}}{r}}\right)^{3\alpha_{+}}\sim\int_{0}^{\epsilon}dr~r^{-1-{\frac{5}{\sqrt{3}}}}~.
\lambda^{\prime}=\frac{a_{1}\hat{u}-a_{2}^{2}+\sqrt{(a_{1}^{2}-a_{2}^{2})(\hat{u}^{2}-a_{2}^{2})}}{a_{1}\hat{u}-a_{2}^{2}-\sqrt{(a_{1}^{2}-a_{2}^{2})(\hat{u}^{2}-a_{2}^{2})}}.
\Gamma^{(0)}=\widehat{\Gamma}^{(0)}+\intd^{4}x\,\,\left\{b_{a}\left(\partial_{\mu}^{a}+\rho^{ai}{\phi}_{i}\right)+\frac12\alphab_{a}b^{a}+\frac12\chi\bar{c}_{a}b^{a}+\eta^{ai}\bar{c}_{a}{\phi}_{i\,\,}\right\}~~,
S_{E}={\frac{4VNT^{2}}{2\pi}}\left({\frac{3\Sigma(T)}{T}}+3+{\frac{\Sigma^{2}(T)}{T^{2}}}-{\frac{\Sigma}{T}}{\frac{d\Sigma}{dT}}\right)\e^{-\frac{\Sigma(T)}{T}}.
\left(\begin{array}{c}{a_{D}}\\{a}\\\end{array}\right)\rightarrow\left(\begin{array}{cc}{1}&{1}\\{0}&{1}\\\end{array}\right)\left(\begin{array}{c}{a_{D}}\\{a}\\\end{array}\right)=\left(\begin{array}{c}{a+a_{D}}\\{a}\\\end{array}\right).
\begin{array}{lll}{J(z)}&{=}&{\frac{1}{k+\check{g}}\{S_{a}^{a}-\frac{1}{(\check{g}-\check{s})}\Omega_{ca}^{cb}S_{b}^{a}+k\psi^{a}\psi_{a}}\\{}&{}&{+\Omega_{ab}^{bc}\psi^{a}\psi_{c}+\frac{(k-\check{g}+\check{s})}{2(\check{g}-\check{s})}\psi^{ab}\psi_{ab}\}(z)}\\\end{array}
\begin{array}{l}{ds_{11}^{2}={\frac{1}{(H^{1}H^{2}H^{3})^{\frac{1}{3}}}}\Big[du\,dv+H_{0}\,du^{2}+H^{1}H^{2}H^{3}\,d\vec{x}^{2}+}\\{\qquad\quad+H^{1}(dy_{1}^{2}+dy_{2}^{2})+H^{2}(dy_{3}^{2}+dy_{4}^{2})+H^{3}(dy_{5}^{2}+dy_{6}^{2})\Big]\;.}\\\end{array}
x^{A\dot{B}}=\left\{\left(\begin{array}{c}{c^{A}}\\{0}\\\end{array}\right),\left(\begin{array}{c}{0}\\{c^{\ast\dot{B}}}\\\end{array}\right)\right\},\\\\\left\{\left(\begin{array}{c}{c^{A}}\\{0}\\\end{array}\right),\left(\begin{array}{c}{c^{B}}\\{o}\\\end{array}\right)\right\}=0
A_{rs}^{m}\equiv\frac{1}{\sqrt2}(F_{rs}^{m}(\phi)\;+\;i\pi_{rs}^{m}),\;\;\;\;\;\;(A^{m\dagger})_{rs}\equiv\frac{1}{\sqrt2}(F_{rs}^{m}(\phi)\;-\;i\pi_{rs}^{m})
\bar{\Gamma}\equiv{\frac{1}{(p+1)!\sqrt{-\gamma}}}\varepsilon^{i_{0}i_{1}\cdotsi_{p}}\partial_{i_{0}}X^{M_{0}}\partial_{i_{1}}X^{M_{1}}\cdots\partial_{i_{p}}X^{M_{p}}\Gamma_{M_{0}M_{1}\cdotsM_{p}},
w_{\alpha\beta}=D_{(\alpha}^{i}\Phi_{a}(\gamma^{i}\gamma^{j})_{ab}D_{\beta)}^{j}\Phi_{b}^{\prime}+32i(\Phi_{a}\;\partial_{\alpha\beta}\Phi_{a}^{\prime}-\partial_{\alpha\beta}\Phi_{a}\;\Phi_{a}^{\prime})\;.
\Gamma(s)=\frac{\left(-i\lambda\right)^{2}}{2}\int\int\frac{d^{4}l}{(2\pi)^{4}}\frac{i}{(l-p)^{2}-\mu_{0}^{2}-i\epsilon}\frac{i}{l^{2}-\mu_{0}^{2}+i\epsilon}
M_{\mathrm{non-ext}}=\sqrt{(\breve{P}_{1}^{(1)})^{2}+\beta^{2}}+\sqrt{(\breve{P}_{1}^{(2)})^{2}+\beta^{2}}+\sqrt{(\breve{Q}_{2}^{(1)})^{2}+\beta^{2}}+\sqrt{(\breve{Q}_{2}^{(2)})^{2}+\beta^{2}},
\Pi^{ij}(x)=a\left(\begin{matrix}{0}&{-x_{1}x_{3}}&{-x_{2}x_{3}}&{x_{1}^{2}+x_{2}^{2}}\\{x_{1}x_{3}}&{0}&{0}&{-x_{0}x_{1}}\\{x_{2}x_{3}}&{0}&{0}&{-x_{0}x_{2}}\\{-(x_{1}^{2}+x_{2}^{2})}&{x_{0}x_{1}}&{x_{0}x_{2}}&{0}\\\end{matrix}\right)
m_{el}^{2}\;=\;-ie^{2}\frac{1}{\beta}\,\sum_{n=-\infty}^{+\infty}\int\frac{d^{2}p}{(2\pi)^{2}}\,\frac{\partial}{\partialp_{j}}\mathrm{tr}\left[\gamma_{j}S_{F}(p,n)\right]
\begin{array}{rlrlrl}{\hat{O}}&{}&{\stackrel{\calW}{\longleftrightarrow}}&{~}&{\tilde{o}(z)\nonumber}\\{}&{\searrow\nwarrow}&{}&{~~W\swarrow\nearrow}&{\nonumber}\\{}&{}&{~~~~~~~o(y,x)}&{}&{}\\\end{array}
Q^{Y}=\frac{r}{\mu}\left\{H\,,\left(\begin{array}{cc}{\sigma_{r}}&{0}\\{0}&{-\sigma_{r}V^{-1}}\\\end{array}\right)\right\}=i\frac{r}{\mu}\left[Q_{0}\,,\left(\begin{array}{cc}{\sigma_{r}}&{0}\\{0}&{\sigma_{r}V^{-1}}\\\end{array}\right)\right]\,.
\frac{T_{m,m-1}}{T_{m+1,m}}=\frac{T_{m,m}}{T_{m+1,m}}=\frac{T_{m+1,m}}{T_{m+2,m}}=\frac{T_{m+1,m}}{T_{m+2,m+1}(1-T_{m+1,m})^{2}}.
L_{BCL}=\frac{1}{2e}(\dot{x}_{\mu}+im^{-1}\dot{\xi}_{*}\xi_{\mu})^{2}-\frac{e}{2}m^{2}-\frac{i}{2}\xi_{\mu}\dot{\xi}{}^{\mu}+\frac{i}{2}\xi_{*}\dot{\xi}_{*},
S_{\mathrm{eff}}^{\mathrm{CS}}=-i\frac{N_{f}}{2}\,\frac{1}{4\pi}\,\frac{m}{|m|}\intd^{3}x\epsilon^{\mu\nu\rho}tr\left(A_{\mu}\partial_{\nu}A_{\rho}+{\frac{2}{3}}A_{\nu}A_{\rho}\right)
c_{lm}^{j}(r)=0\quad\mathrm{unless}\quadD_{m0}^{l}(a_{R}^{-1}s_{R})=D_{mm^{\prime}}^{l}(a_{R}^{-1})D_{m^{\prime}0}^{l}(s_{R})\quad\mathrm{is}\quadD_{m0}^{l}(s_{R}).
||h_{mn}||=\left(\begin{array}{ccc}{(A_{1},A_{1})}&{Re(A_{1},A_{2})}&{Re(A_{1},A_{3})}\\{Re(A_{2},A_{1})}&{(A_{2},A_{2})}&{Re(A_{2},A_{3})}\\{Re(A_{3},A_{1})}&{Re(A_{3},A_{2})}&{(A_{3},A_{3})+1}\\\end{array}\right)
L_{q}^{QCD}=g\,\hbarc^{\prime}\intd^{3}x^{\prime}\,\bar{q}_{i}(x^{\prime})\gamma^{\mu^{\prime}}\frac{\lambda_{ij}^{a}}{2}q_{j}(x^{\prime})A_{\mu^{\prime}}^{a}\cdot\,W_{A},