harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
G_{ik*}=\sum_{a=0}^{N}\frac{\partial\Psi^{a}}{\partial\Pi^{i}}\frac{\partial\Psi^{*a}}{\partial\Pi^{*k}}=R^{2}\frac{(\sum_{s=1}^{N}|\Pi^{s}|^{2}+R^{2})\delta_{ik}-\frac{3}{4}\Pi^{*i}\Pi^{k}}{(\sum_{s=1}^{N}|\Pi^{s}|^{2}+R^{2})^{2}}.
{\calL}_{BI}=4be^{2\alpha\varphi}\left\{1-\left[1+\frac{e^{-4\alpha\varphi}}{2b}F^{2}-\frac{e^{-8\alpha\varphi}}{16b^{2}}(F\starF)^{2}\right]^{\frac{1}{2}}\right\}.
\varphi(t,x_{1},x_{2},x_{3},r,\theta_{i})=e^{-i\omegat}e^{i(k_{1}x_{1}+k_{2}x_{2}+k_{3}x_{3})}Y_{l}(\theta_{1},\theta_{2},\cdots,\theta_{5})\varphi^{l}(r)
\varphi_{n}(x^{-})=\frac{1}{\sqrt{4\pi}}\sum_{k=1}^{\infty}\frac{\left[a_{n}(k)e^{-ip_{k}^{+}x^{-}}+a_{n}^{\dagger}(k)e^{ip_{k}^{+}x^{-}}\right]}{\sqrt{k-\frac{1}{2}}}
\deltaL(q,\dot{q})=\frac{1}{2}\dot{q}^{\mu}(\deltaq^{\alpha}\partial_{\alpha}g_{\mu\nu}+g_{\mu\alpha}\partial_{\nu}\deltaq^{\alpha}+g\_{\alpha\nu}\partial_{\mu}\deltaq^{\alpha})q^{\nu}++\frac{1}{4}\left[\left[\partial_{\alpha}\deltaq^{\mu},\dot{q}^{\alpha}\right]g_{\mu\nu},\dot{q}^{\nu}\right]-\deltaq^{\alpha}\partial_{\alpha}U_{q}(q)\,.
\begin{array}{rclcrcl}{H_{17}(r)}&{=}&{1+\frac{\vertg^{17}\vert}{r}}&{;}&{H_{24}(r)}&{=}&{1+\frac{\vertg^{24}\vert}{r}}\\{H_{18}(r)}&{=}&{1+\frac{\verte_{18}\vert}{r}}&{;}&{H_{23}(r)}&{=}&{1+\frac{\verte_{23}\vert}{r}}\\\end{array}
\displaystyle{\frac{8}{\kappa}}V_{\mu\nu,\,\alpha\beta}^{^2ghost}(k_{1},k_{2};\,p_{1},p_{2})=-4\,p_{_1\mu}\,p_{_2\nu}\,\eta_{\alpha\beta}+8\,p_{_1\beta}\,p_{_2\nu}\,\eta_{\alpha\mu}-2\,p_{1}\cdotp_{2}\,\eta_{\alpha\mu}\,\eta_{\beta\nu}+p_{1}\cdotp_{2}\,\eta_{\alpha\beta}\,\eta_{\mu\nu}.
\begin{array}{ll}{\Omega_{tor}=dP_{q}\wedgedq+dp_{\mu}\wedgedx^{\mu}+\frac{1}{2}\alpha^{-2}\epsilon_{\mu\nu\lambda}J^{\nu}\wedgedJ^{\lambda},}\\{H_{tor}=\alpha^{-1}q\chi+w\psi,}\\\end{array}
{\sigma}_{R}=\mathrm{exp}[-\frac{1}{2}\int_{0}^{\tilde{\kappa}}\frac{dp_{-}}{p_{-}}]\mathrm{exp}[-\int_{0}^{\tilde{\kappa}}\frac{dp_{-}}{p_{-}}c^{\ast}(p_{-})]\mathrm{exp}[\int_{0}^{\tilde{\kappa}}\frac{dp_{-}}{p_{-}}c(p_{-})].
+\frac12g\sigma^{I}\sigma^{J}\psi^{\tilde{K}}(t_{I}t_{J})_{\tilde{K}}{}^{M}+\frac14g\sigma^{I}\sigma^{\tilde{K}}\psi^{J}\left(t_{I}t_{J}+2t_{J}t_{I}\right)_{\tilde{K}}{}^{M}\,.
\zeta(-x,d)=(-1)^{\lfloorx\rfloor}\frac{2d(x/\pi)^{2x+d/2}\,e^{-2x}}{\Gamma(x-\lfloorx\rfloor+1)\,\Gamma(\lfloorx\rfloor-x)}(1+O(x^{-1}))\quad,\quadx\in\mathbb{R}^{+}\quad.
\frac{1}{\sqrt{-g^{\prime}}}\partial_{\mu}(\sqrt{-g^{\prime}}g^{\prime\mu\nu}\partial_{\nu}\tilde{\varphi})+\frac{e^{\alpha\phi/M_{p}}}{\chi}\frac{dP(\varphi)}{d\varphi}-e^{2}\varphig^{\prime\alpha\beta}A_{\alpha}A_{\beta}=-\frac{h}{\sqrt{2}}\overline{{\Psi}}^{\prime}\Psi^{\prime}\frac{e^{\gamma\phi/M_{p}}}{\chi^{3/2}};
f_{E_{6}}=\frac{e^{-S_{1}-i\frac{x_{0}I_{3}}{y(x_{0}^{2}+y^{2})}}}{(y^{2}+x_{0}^{2})S_{1}}\propto\frac{K_{1/2}(S_{1})e^{-i\frac{x_{0}I_{3}}{y(x_{0}^{2}+y^{2})}}}{(y^{2}+x_{0}^{2})\,\sqrt{S_{1}}}\,.
I(p^{2},m_{2}^{2})=(1-\frac{d}{2})N_{d}(-\frac{1}{\epsilon}+O(\epsilon))\sum_{k=0}^{\infty}C_{\frac{d}{2}-2}^{k}\frac{p^{2k}}{m_{2}^{2k}}\int_{0}^{1}dt(t(1-t))^{k}.
\epsilonL^{(2)}\theta=\tilde{h}_{1}^{(2)}v^{2}\phi^{j}(\epsilon\gamma^{i}\theta)(\theta\gamma^{ij}\theta)+h_{1}^{(2)}v^{i}v^{j}\phi^{k}(\epsilon\gamma^{i}\theta)(\theta\gamma^{jk}\theta),
\frac{m}{\Sigma}+\frac{4g^{2}}{\beta_{c}}\sum_{n=-\infty}^{\infty}\int^{\Lambda}\frac{d^{d}{\bfq}}{(2\pi)^{d}}\frac{\left({\bfq}^{2}+(\rho_{c}-\bar{\omega}_{nc})^{2}\right)\left(\frac{\beta_{c}}{\beta}\right)-\left({\bfq}^{2}+(\rho-\bar{\omega}_{n})^{2}+\Sigma^{2}\right)}{\left({\bfq}^{2}+(\rho_{c}-\bar{\omega}_{nc})^{2}\right)\left({\bfq}^{2}+(\rho-\bar{\omega}_{n})^{2}+\Sigma^{2}\right)}=0.
J_{+}^{q}=J_{+}\sqrt{\frac{[J_{3}+j+1][J_{3}-j]}{(J_{3}+j+1)(J_{3}-j)}}\equivJ_{+}F(J_{3}),\quadJ_{-}^{q}=\bar{J_{+}^{q}},\quadJ_{3}^{q}=J_{3}.
0=F_{a}\,\equiv\,P_{2}\,P_{4}^{(1)}\,\partial_{a}P_{4}^{(0)}+P_{4}^{(0)}\,\left(-4\,P_{4}^{(1)}\,\partial_{a}P_{2}+P_{2}\,\partial_{a}P_{4}^{(1)}\right).
\begin{array}{ll}{\phi_{\omega}^{E}(\sigma,\tau)=[(a/b)_{\omega}exp(\omega\sigma)+exp(-\omega\sigma)]exp(-i\omega\tau)}\\{\phi_{\omega}^{L}(\sigma,\tau)=[(c/b)_{\omega}exp(-i\omega\sigma)+(1/b)_{\omega}exp(i\omega\sigma)]exp(-i\omega\tau)}\\\end{array}
1=\intd^{2}{\bfr}\,|\Psi_{\!\mathrm{(gs)}}({\bfr})|^{2}=A^{2}\,2\pi\kappa^{-2}\,\left\{{\mathcalK}(\kappaa)+\left(\frac{B}{A}\right)^{2}\,{\mathcalI}(\kappaa)\right\}\;,
{\calI}_{\mathrm{1-loop}}^{\mathrm{het}}=\frac{V^{(8)}T_{2}}{2^{8}\pi^{4}}\int\frac{d^{2}\tau}{\tau_{2}^{2}}\sum_{{\bfl},{\bfn}\in{\bfZ}}e^{-\frac{2\pi}{\tau_{2}}(n^{I}\tau-l^{I})(G+B)_{IJ}(n^{J}{\bar{\tau}}-l^{J})}\intDS_{0}^{a}\e^{{-\pi/\tau_{2}}F_{\mu\nu}^{A}\Q^{A}R_{0}^{\mu\nu}}\bar{\calA}(\bar{q})\,,
I=\frac{m_{P}^{2}}{16\pi}\intd^{4}x\sqrt{-g}\Bigl[R-2\Lambda\Bigr]+\frac{m_{P}^{2}}{8\pi}\intd^{3}x\sqrt{h}K+\intd^{4}x\sqrt{-g}\Bigl[\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\Bigr],
H_{CN}=-\sum_{j=1}^{L}\sum_{n=1}^{N^{\prime}-1}\left[t_{n}\left(\Sigma_{j}^{+}\Sigma_{j+1}^{-}\right)^{n}+\bar{t}_{n}\left(\Sigma_{j}^{-}\Sigma_{j+1}^{+}\right)^{n}+h_{n}Z_{j}^{n}\right]
\widehat{V}^{\dagger}(k_{1},k_{2})=e^{i\sqrt{2|\theta^{12}|}\;\left(\kappa\hat{a}+\overline{{\kappa}}\hat{a}^{\dagger}\right)}=e^{-|\theta^{12}|\kappa\overline{{\kappa}}}\,e^{i\sqrt{2|\theta^{12}|}\;\overline{{\kappa}}\hat{a}^{\dagger}}\,e^{i\sqrt{2|\theta^{12}|}\;\kappa\hat{a}}~,
H_{D}\equiv\left(\begin{array}{cc}{h_{11}}&{h_{12}}\\{h_{21}}&{h_{22}}\\\end{array}\right)=\left(\begin{array}{cc}{i\sigma_{j}\partial_{j}+g\sigma_{j}A_{j}}&{m_{f}(s+ip\tau_{j}\hat{x}_{j})}\\{m_{f}(s-ip\tau_{j}\hat{x}_{j})}&{-i\sigma_{j}\partial_{j}}\\\end{array}\right)\,.
{\calL}_{int}=\frac{\kappa}{6}\varepsilon^{\mu\nu\alpha}\varepsilon_{abc}f_{\mu}{}^{a}f_{\nu}{}^{b}f_{\alpha}{}^{c}\qquad\delta_{1}h_{\alphab}=\kappa\varepsilon_{abc}f_{\alpha}{}^{a}\eta^{c}
\left(\begin{array}{cc}{1}&{-i}\\\end{array}\right)\left[e^{\gamma_{5}\Omega_{n}Lx}\left(\begin{array}{c}{1}\\{i}\\\end{array}\right)\otimes\left(\begin{array}{cc}{f(x^{\prime})}&{g(x^{\prime})}\\\end{array}\right)-i\gamma^{1}e^{-\gamma_{5}\Omega_{n}L(1-x\prime)}\right]=0.
\mathrm{tr}\left[Q\,\Gamma^{,\rho\sigma}\right]=2i\delta^{\rho\sigma}t\,\xi^{-1}+i\frac{1}{k^{2}}P^{\rho\sigma}\left[e^{k^{2}(1-\xi^{-1})t}-1\right]-i\frac{1}{k^{2}}P^{,\rho\sigma}\left[e^{k^{2}(\xi^{-1}-1)t}-1\right]
{\calL}^{\prime}=p_{-}\left[\frac{1}{2}v^{2}+\frac{i}{2}\psi^{T}\dot{\psi}+\frac{1}{8}hv^{4}+\frac{i}{16}v^{2}\partial_{i}h(\psi^{T}\gamma^{ij}\psi)v_{j}+\frac{i}{8}v^{2}h\psi^{T}\dot{\psi}+\cdots\right]~.
W\left[f_{\ell}(k,R),\phi_{\ell}(k^{\prime},R)\right]=\left(-k\right)^{\frac{1}{2}-\nu}F_{\ell}(k)+\left(k^{2}-k^{\prime2}\right)\int_{0}^{R}dr\,f_{\ell}(k,r)\phi_{\ell}(k^{\prime},r)\,.
\int_{s_{ij}^{-}}\Omega^{(2,0)}=(-\psi_{s})^{-1/12}\frac{\sqrt{6}}{2\pi}\tilde{\epsilon}^{1/3}(\tilde{x}_{j^{-}}-\tilde{x}_{i^{-}})+{\calO}(\tilde{\epsilon}^{4/3})\.
\frac{d^{2}}{d\lambda^{2}}\left[\frac{1}{\lambda}K_{1}\left(2\pin\mu_{eff}\lambda\right)\right]_{\lambda=1}=-\left(2\pin\mu_{eff}\right)K_{2}\left(2\pin\mu_{eff}\right)+\left(2\pin\mu_{eff}\right)^{2}K_{3}\left(2\pin\mu_{eff}\right),
\int_{C}{d^{4}}{\eta}D_{c}^{(4)}(\eta,{\eta})=\int_{\partialC}d^{3}\etaK-6a\int_{\partialC}d^{3}\etaKK^{\prime}-\frac{2}{3}\frac{1}{a^{2}}\pi^{-\frac{1}{2}}\Gamma(\frac{5}{2})\int_{S_{4}}d^{4}\eta
\omega=2\pi\left(\begin{array}{cccccc}{l}&{}&{}&{}&{}&{}\\{}&{}&{}&{}&{}&{}\\{}&{}&{l}&{}&{}&{}\\{}&{}&{}&{-k}&{}&{}\\{}&{}&{}&{}&{}&{}\\{}&{}&{}&{}&{}&{-k}\\\end{array}\right),
C_{\rho\sigma}=Cg_{\rho\sigma}-2{\frac{\partialC}{c^{2}\partials^{2}}}s_{\rho}s_{\sigma}-4{\frac{\partialA}{c^{2}\partials^{2}}}s_{(\rho}\mu_{\sigma)}-{\frac{\partialA}{c^{2}\partialy^{2}}}\mu_{\rho}\mu_{\sigma}\,
\hat{D}_{i}\theta_{1,2}=\partial_{i}\theta_{1,2}+\frac{1}{4}\omega_{\murs}\Gamma^{rs}\theta_{1,2}\partial_{i}X^{\mu}\pm\frac{1}{2}e_{\mu}{}^{r}I\Gamma^{+}\Gamma_{r}\theta_{2,1}\partial_{i}X^{\mu}\,.
\begin{array}{rcl}{{\calL}_{\mathrm{int}}}&{=}&{\displaystyle-\frac{g_{\varphi}}{2}(\varphi^{\dagger}\star\varphi\star\sigma+\sigma\star\varphi^{\dagger}\star\varphi)}\\{}&{}&{\displaystyle-\frac{g_{\chi}}{2}(\chi^{\dagger}\star\chi\star\sigma+\sigma\star\chi^{\dagger}\star\chi).}\\\end{array}
[-\frac{(d-1)!}{4\,d\,a_{1}}\,(G\,h^{-1})_{p}\,^{t}-2\,(d-2)!\,a_{3}\,(\mathcal{F}^{ti}\,F_{ip})]\,=\,-\frac{a_{2}\,d!}{d-2}\,[\mathrm{det}\,(h^{-1}+\mu\mathcal{F})]^{\alpha}\delta_{p}^{t}
\frac{i}{2}\mathrm{Tr}\,\left(\hat{T}_{x}^{(1)}\frac{1}{\partial^{2}+m^{2}}\sigma\frac{1}{\partial^{2}+m^{2}}\right)-\frac{i}{4}\mathrm{Tr}\,\left(\hat{T}_{x}^{(0)}\frac{1}{\partial^{2}+m^{2}}\sigma\frac{1}{\partial^{2}+m^{2}}\sigma\frac{1}{\partial^{2}+m^{2}}\right)\,.
i\lambda\frac{\partial}{\partialx}\left[\begin{array}{cc}{\Psi_{e}(x)}\\{\Psi_{\mu}(x)}\\\end{array}\right]=H_{\mathrm{flavor}}(x)\left[\begin{array}{cc}{\Psi_{e}(x)}\\{\Psi_{\mu}(x)}\\\end{array}\right]\,,
\sum_{n=1}^{\infty}~e^{-M\sqrt{n^{2}\beta^{2}+\tilde{p}_{2}^{2}}}=-\frac{1}{2}~e^{-M\tilde{p}_{2}}+\sqrt{2\pi}~[~\frac{1}{2}~F(0)+\sum_{n=1}^{\infty}F(2\pin)]~~~,
{\calL}_{H}=\frac{1}{2}H^{\mu\nu}\BoxH_{\mu\nu}-\frac{1}{4}H\BoxH-\frac{1}{2}\H^{\mu\nu}\partial_{\mu}\partial_{\alpha}H_{\nu}^{\alpha}-\frac{1}{2}\H^{\mu\nu}\partial_{\nu}\partial_{\alpha}H_{\mu}^{\alpha}-\frac{1}{2}\\LambdaH^{\mu\nu}H_{\mu\nu}+\frac{1}{4}\\LambdaH^{2}\,
(k-1)\alpha_{k-2}{}^{2}=-\frac{(k+1)(2k+d-2)}{2k+d-4)}\alpha_{k}{}^{2}+\frac{2k(2k+d-5)}{(2k+d-6)}\alpha_{k-1}{}^{2}-2\Omega(2k+d-5),\qquadk\ge2
V^{\tau}(R,l,E)=E-\frac{1}{2}\frac{(f_{+})^{(3k+c_{1}-1)/2}}{(f_{-})^{(3k+c_{1}+1)/2}}\left[E^{2}-\left(\frac{f_{-}}{f_{+}}\right)^{2k}-\frac{l^{2}}{r^{2}}\frac{(f_{-})^{k-1/2}}{(f_{+})^{k+1/2}}\right].
k_{ij}=\left(\begin{array}{cccc}{k_{00}}&{0}&{0}&{k_{30}}\\{{\frac{1}{2}}}&{{\frac{1}{2}}(v^{I})^{2}}&{-k_{21}+v^{I}A^{I}}&{0}\\{0}&{k_{21}}&{{\frac{1}{2}}(A^{I})^{2}+{\frac{1}{3}}}&{0}\\{k_{30}}&{0}&{0}&{k_{30}+{\frac{1}{2}}({\tilde{A}}^{I})^{2}+{\frac{1}{2}}}\\\end{array}\right)~.
\frac{1}{4!}\int_{{\Lambda}}\frac{d^{4}\vec{p}_{1}}{(2{\pi})^{4}}\frac{d^{4}\vec{p}_{2}}{(2{\pi})^{4}}\frac{d^{4}\vec{p}_{3}}{(2{\pi})^{4}}\frac{d^{4}\vec{p}_{5}}{(2{\pi})^{4}}{\delta}{\lambda}_{4}(1235){\delta}^{4}(\vec{p}_{1}+\vec{p}_{2}+\vec{p}_{3}+\vec{p}_{5})\hat{\phi}_{2}(\vec{p}_{1})\hat{\phi}_{2}(\vec{p}_{2})\hat{\phi}_{2}(\vec{p}_{3})\hat{\phi}_{2}(\vec{p}_{5}).
i_{\{\alpha_{p}}\Omega_{(p)\,\alpha_{1}\dots\alpha_{p-1}\}}^{\prime}={\frac{2m-p}{p}}\Pi_{(p)\,\alpha_{1}\dots\alpha_{p}}^{\prime}={\frac{2m-p}{p}}d\Omega_{(p+1)\,\alpha_{1}\dots\alpha_{p}}^{\prime}
\begin{array}{rl}{}&{<\chi\chi|S|\pi\pi>}\\{=}&{\displaystyle(-ig_{\pi})(-ig_{\chi})\intd^{4}x_{5}\intd^{4}x_{6}}\\{}&{\displaystyle\times<\chi(k_{3+})\chi(k_{4+})|T\left((\pi\star\sigma\star\pi)_{5}(\chi\star\sigma\star\chi)_{6}\right)|\pi(k_{1+})\pi(k_{2+})>}\\{\displaystyle}&{+\mathrm{~higher~orders},}\\\end{array}
\int_{C}d^{n}{\eta}Y_{\alpha}^{N}(\eta)Y_{\alpha^{\prime}}^{N^{\prime}}(\eta)=Aa^{2}\oint_{\partialC}d^{n-1}{\eta}[KY_{\alpha}^{N}(\eta)Y_{\alpha^{\prime}}^{N^{\prime}}(\eta)+2n_{p}Y_{\alpha^{\prime}}^{N^{\prime}}(\eta)D_{p}Y_{\alpha}^{N}(\eta)]
\langleT_{i}^{k}\rangle_{{\mathrm{sub}}}^{(R)}=-\frac{\delta_{i}^{k}\xi^{-d-1}}{2^{d-1}\pi^{d/2}\Gamma(d/2)}\int_{0}^{\infty}\frac{\omega^{d}g^{(i)}(\omega)d\omega}{e^{2\pi\omega}+(-1)^{d}}
\intdt\left\{-\Lambda\frac{dV_{\Lambda}}{d\Lambda}+\frac{1}{2}\Lambda\frac{dZ_{\Lambda}}{d\Lambda}{\dot{\phi}}^{2}\right\}=\frac{\Lambda^{3}}{4\pi^{2}}\intdt\left\{\sqrt{A_{\Lambda}}+\frac{1}{2}\dot{\phi}^{2}\left[\frac{B_{\Lambda}}{\sqrt{A_{\Lambda}}}-\left(\frac{C_{\Lambda}}{\sqrt{A_{\Lambda}}}\right)^{\prime}\right]\right\}
\begin{aligned}{F_{0,2,0}(d_{1},0;m_{1},m_{2},0,0)={\frac{1}{d_{1}}}{\frac{\lambda_{1}(\lambda_{2}-2\lambda_{1})}{(\lambda_{1}-\lambda_{2})^{d_{1}}}}\prod_{i=1}^{2}{\frac{\prod_{k=1}^{m_{i}-1}\left((\lambda_{2}-2\lambda_{1})m_{i}+k(\lambda_{1}-\lambda_{2})\right)}{(m_{i}-1)}}.}\\\end{aligned}
\begin{array}{c}{\mathcal{J}_{\nu}(w)=w\j_{l}(w)=\displaystyle{\sqrt{\frac{\piw}{2}}}J_{\nu}(w)}\\{\mathcal{Y}_{\nu}(w)=w\y_{l}(w)=\displaystyle{\sqrt{\frac{\piw}{2}}}Y_{\nu}(w)}\\\end{array}
\begin{array}{rcl}{}&{}&{[X_{+}^{+},X_{-}^{+}]=0~,}\\{}&{}&{k_{\pm}X_{\pm}^{+}k_{\pm}^{-1}=q_{\pm}X_{\pm}^{+}~,~~~~k_{\pm}X_{\mp}^{+}k_{\pm}^{-1}=-X_{\mp}^{+}~,}\\{}&{}&{X_{\pm}^{+}X_{\pm}^{-}-q_{\pm}X_{\pm}^{-}X_{\pm}^{+}=\displaystyle\frac{1-k_{\pm}^{-2}}{1-q_{\pm}^{-1}}~,}\\{}&{}&{X_{\pm}^{+}X_{\mp}^{-}-q_{\pm}X_{\mp}^{-}X_{\pm}^{+}=0~.}\\\end{array}
A^{(i)}={\frac{12g_{YM}^{4}}{\pi^{4}}}\sum_{k_{1},\dotsk_{6}=-\infty}^{+\infty}\sum_{n_{1},\dotsn_{4}\geq0}p_{n_{1},\dots,n_{4}}^{(i)}({\frac{s}{M_{s}^{2}}},{\frac{t}{M_{s}^{2}}})B(s,t,\{n_{i}M_{s}^{2}\!+\!(k_{1}^{2}\!+\!\dots\!+\!k_{6}^{2})/R^{2}\})\.
\lambda_{SW}=-{\frac{1}{2\pii}}{\frac{1}{\sqrt{-4\mu^{2}}}}\sum_{n=0}^{\infty}{\frac{(2n)!}{2^{2n}(n!)^{2}}}{\frac{W_{G}(x,t)^{2n+1}dx}{(2n+1)(4\mu^{2})^{n}}}.
\begin{array}{lcllcl}{X_{j}^{12}}&{=}&{\theta_{j}^{\dagger}}&{X_{j}^{21}}&{=}&{\theta_{j}}\\{X_{j}^{11}}&{=}&{\theta_{j}^{\dagger}\theta_{j}}&{X_{j}^{22}}&{=}&{\theta_{j}\theta_{j}^{\dagger}\;.}\\\end{array}
A|_{\nu\leqx_{0}}=\frac{(n-1)^{2}}{4\pi^{2}a^{4}}\frac{C}{8}(\frac{1}{3}x_{0}^{3}+\frac{1}{2}x_{0}^{2}+\frac{1}{6}x_{0}-\frac{1}{4}).
G^{(N=2)}|_{\theta=0}=a_{1}(s,t){\frac{(12)^{2}(34)^{2}}{x_{12}^{4}x_{34}^{4}}}+a_{3}(s,t){\frac{(14)^{2}(23)^{2}}{x_{14}^{4}x_{23}^{4}}}+b_{2}(s,t){\frac{(12)(23)(34)(41)}{x_{12}^{2}x_{23}^{2}x_{34}^{2}x_{41}^{2}}}
\Gamma_{(a)}^{\mu}(p,p^{\prime})=\frac{-g^{3}}{\Theta}\int\frac{d^{3}k}{(2\pi)^{3}}\left[\frac{(2p-k)^{\sigma}\varepsilon_{\sigma\rho\lambda}\bar{k}^{\lambda}(2p^{\prime}-k)^{\rho}(p+p^{\prime}-k)^{\mu}}{{\bfk}^{2}[(p-k)^{2}-m^{2}+i\epsilon][(p^{\prime}-k)^{2}-m^{2}+i\epsilon]}\right]\;.
\begin{array}{c}{\partial_{t}B_{t}+\left[A_{t},B_{t}\right]=\partial_{x}B_{x}+\left[A_{x},B_{x}\right],}\\{\partial_{x}A_{t}-\partial_{t}A_{x}+\left[A_{x},A_{t}\right]+\left[B_{x},B_{t}\right]=0,}\\{\partial_{x}B_{t}-\partial_{t}B_{x}+\left[A_{x},B_{t}\right]-\left[A_{t},B_{x}\right]=0.}\\\end{array}
\frac{1}{\sqrt{-G}}\frac{\partial}{\partialu^{a}}(\sqrt{-G}\;G^{ab}\frac{\partial\Phi}{\partialu^{b}})+\frac{1}{h}\partial_{\xi}(h\;\partial_{\xi}\Phi)-2\lambda(\Phi^{2}-\frac{M^{2}}{4\lambda})\Phi=0,
\tau_{j}=\sum_{\mu_{1}=0}^{1}\cdots\sum_{\mu_{N}=0}^{1}\prod_{p=1}^{N}\left(\left(\prod_{q=1}^{p-1}\left(X_{pq}\right)^{\mu_{q}}\right)e^{\Omega_{p}}\omega^{ja_{p}}\right)^{\mu_{p}}.
\left[\,\partial_{\mu}\,\partial^{\mu}+{\frac{V_{2}}{3!}}\,{\frac{\partial^{2}}{\partial\sigma_{\mu_{1}\mu_{2}\mu_{3}}\,\partial\sigma^{\mu_{1}\mu_{2}\mu_{3}}}}-(p+1)\,M_{0}^{2}\,\right]\,\Psi\left(\,x\,\sigma\,\right)=0\.
\sum_{n=0}^{\infty}\mu_{3}(n)q^{n}=\prod_{i=1}^{\infty}(1-q^{i})^{-(i^{2}+i)/2}=1+q+4q^{2}+10q^{3}+26q^{4}+59q^{5}+141q^{6}+\cdots,
\bar{h}_{00}=\frac{4G_{4}m_{*}}{r}\Big[1+\frac{5(1-\gamma)}{12(1+\gamma)}\,\frac{\ell^{2}}{r^{2}}\Big],\,\,\,\bar{h}_{ij}=\frac{4G_{4}m_{*}}{r}\Big[{\calO}(0)+\frac{1-\gamma}{12(1+\gamma)}\,\frac{\ell^{2}}{r^{2}}\Big]\,\delta_{ij}\,.
\stackrel{P}{K}{}\!\!_{0,p}^{(\pm)}\left(\begin{array}{c}{\alpha}\\{\beta,s}\\\end{array}\right)=\stackrel{P^{l}}{K}{}\!\!_{0,p}^{l(\pm)}\left(\begin{array}{c}{\alpha^{l}}\\{\beta^{l},s^{l}}\\\end{array}\right)\stackrel{P^{r}}{K}{}\!\!_{0,p}^{r(\pm)}\left(\begin{array}{c}{\alpha^{r}}\\{\beta^{r},s^{r}}\\\end{array}\right).
S=-\frac{1}{g_{0}^{2}}\left(\frac{1}{2\alpha^{\prime}}\left<I\circ\Phi(0)Q_{B}\Phi(0)\right>+\frac{1}{3}\left<f_{1}\circ\Phi(0)f_{2}\circ\Phi(0)f_{3}\circ\Phi(0)\right>\right)\,
M(a_{1},a_{2},a_{3},a_{4})=x^{4}\int\frac{\mathrm{d}^{4}y}{\pi^{2}}\prod_{i}P_{a_{i}}(x,y)=\sum_{n_{i}}A\left(\{n_{i}\}\right)\,R\left(\{a_{i}\};\{n_{i}\}\right)\,,
<\dot{x}_{\mu}(t_{1})\dot{x}_{\nu}(t_{2})>_{S_{1}}={\calN}(\alpha)\left[\dot{x}_{\mu}^{cl}(t_{1})\dot{x}_{\nu}^{cl}(t_{2})-2\delta_{\mu\nu}\frac{\partial^{2}}{\partialt_{1}\partialt_{2}}K^{(N)}(t_{1},t_{2})\right]e^{-S_{1}[x^{cl}]}
j_{\mu}^{\mathrm{mix}}=\frac{1}{2}\sum_{k,k^{\prime}}(\bar{v}_{k}\gamma_{\mu}u_{k^{\prime}}d_{k}b_{k^{\prime}}+\bar{u}_{k^{\prime}}\gamma_{\mu}v_{k}b_{k^{\prime}}^{\dagger}d_{k}^{\dagger}).
\kappa^{2}g^{2}\alpha^{\prime}\left(\frac{1}{2}(\frac{11\cdot2^{4}}{3^{5}})^{2}-24\cdot\frac{1}{2}(\frac{5\cdot2^{7/2}}{3^{5}})^{2}\right)\left(2A_{i}^{2}A_{j}^{2}\right)=-\kappa^{2}g^{2}\alpha^{\prime}\left(\frac{22912}{59049}\right)\left(2A_{i}A_{i}A_{j}A_{j}\right).
\varphi(\omega,y)=\frac{1}{16\pi}\frac{1}{y}\int_{m}^{\infty}dt\Bigg(\frac{1}{2t-y}\frac{e^{\betat}+\omega}{e^{\betat}-\omega}-\frac{1}{2t+y}\frac{e^{\beta(t+y)}+\omega}{e^{\beta(t+y)}-\omega}\Bigg)+\mathrm{p.p.c.}
{\tilde{Z}}_{m}^{[3]}\=\\partial_{\theta}{\tilde{V}}_{m\\mathrm{sing}}^{\mathrm{cl}}\+\{\frac{1}{M_{W}}}\{\calD}_{m}^{\mathrm{cl}}\\Bigl({\tilde{\phi}}_{\mathrm{sing}}^{\mathrm{cl}}-M_{W}\tau^{3}/2\Bigr)\=\{\frac{1}{M_{W}}}\{\tilde{v}}_{m3\\mathrm{sing}}^{\mathrm{cl}}\,
\gamma_{1}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\;\;,\;\;\gamma_{2}=\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right)\;\;,\;\;\gamma_{5}=i\gamma_{2}\gamma_{1}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)\;.
g=\frac{1}{4\pi}\intd^{3}x\partial_{i}{\calB}^{i}=\frac{1}{8\pigv^{3}}\int_{s_{\infty}^{2}}d\sigma_{i}\varepsilon^{ijk}\varepsilon^{abc}\phi^{a}\partial_{j}\phi^{b}\partial_{k}\phi^{c}=\frac{n}{g}
\begin{aligned}{M_{ii}}&{{}=\lambda_{i\,x}\biggl(\sum_{k=1}^{N}\lambda_{k\,x}\biggr)\wp(z)+m_{i}^{0},}\\{M_{ij}}&{{}=-\lambda_{i\,x}\biggl(\sum_{k=1}^{N}\lambda_{k\,x}\biggr)\Phi^{\prime}(\lambda_{i}-\lambda_{j},z)+m_{ij}\Phi(\lambda_{i}-\lambda_{j},z),\qquadi\nej}\\\end{aligned}
\left(\begin{array}{c}{\vec{x}}\\{\vec{p}}\\\end{array}\right)=\left(\begin{array}{cc}{a\,{\mathbfR}^{T}}&{b\,{\mathbfK}{\mathbfR}^{T}}\\{d\,{\mathbfK}{\mathbfR}^{T}}&{c\,{\mathbfR}^{T}}\\\end{array}\right)\left(\begin{array}{c}{\vec{\alpha}}\\{\vec{\beta}}\\\end{array}\right)
ds^{2}~=~-f(r)~\left(1-\left(\frac{r_{0}}{r}\right)^{2}\right)~dt^{2}+f(r)^{-1}\left(1-\left(\frac{r_{0}}{r}\right)^{2}\right)^{-1}~dr^{2}+r^{2}d\Omega_{3}^{2}~,
R_{N}(\beta)=(1-\beta)^{1/2}\beta^{N+1}\oint_{\Gamma_{A}^{\prime}+\Gamma_{B}^{\prime}}{\frac{d\beta^{\prime}}{2\pii}}{\frac{\Psi(\beta^{\prime})}{(\beta^{\prime})^{N+1}(\beta^{\prime}-\beta)}},
\phi_{f}({\calK})={\frac{1}{4\pi}}\oint_{\calK}dx^{\mu}\oint_{{\calK}^{f}}dy^{\nu}\epsilon_{\mu\nu\rho}{\frac{(x-y)^{\rho}}{|x-y|^{3}}}=\mathrm{lk}({\calK},{\calK}^{f}).
\frac{e^{-4\pi\tau}}{1-e^{-4\pi\tau}}\sim-\frac14\frac{q^{2}}{p^{(2)2}}\;,\;\;\frac{e^{-4\pil^{\prime}}}{1-e^{-4\pil^{\prime}}}\sim-\frac14\frac{k^{2}}{p^{(1)2}}\;.
S[{\bfA}]=\frac{\ell}{16\piG}\int_{\calV}{Tr(2\A_{t}F_{r\varphi}+A_{\varphi}\dot{A}_{r}-A_{r}\dot{A}_{\varphi})}\,dr\,d\varphi\,dt\,+\,{\calB}^{(r)}[{\bfA}]\qquad,
\begin{array}{cccc}{q_{2}^{\prime}=}&{q_{1}^{\prime}+p_{1}}&{}&{}\\{\vdots}&{\vdots}&{}&{}\\{q_{j}^{\prime}=}&{q_{1}^{\prime}+p_{1}+}&{\ldots}&{+p_{j-1}}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{q_{n}^{\prime}=}&{q_{1}^{\prime}+p_{1}+}&{\dots}&{+p_{n-1}}\\\end{array}
\sum_{m}(X_{m_{1}}^{J_{1}})_{lm}(X_{m_{2}}^{J_{2}})_{mn}=\sum_{J,M}\left(\begin{array}{ccc}{J_{1}}&{J_{2}}&{J}\\{m_{1}}&{m_{2}}&{M}\\\end{array}\right)\left\{\begin{array}{ccc}{J_{1}}&{J_{2}}&{J}\\{S}&{S}&{S}\\\end{array}\right\}(X_{M}^{J})_{ln},
J^{M_{1}\cdotsM_{p+1}}(x)=T_{p}\intd^{p+1}\xi\,\varepsilon^{i_{1}\cdotsi_{p+1}}\partial_{i_{1}}X^{M_{1}}\cdots\partial_{i_{p+1}}X^{M_{p+1}}{\frac{\delta^{D}(x-X)}{\sqrt{-g}}}.
w_{1}\oplusw_{2}=\left(\begin{array}{cc}{\alpha_{2}\otimes\alpha_{1}-q\gamma_{2}^{*}\otimes\gamma_{1}}&{-q\left(\gamma_{2}^{*}\otimes\alpha_{1}^{*}+\alpha_{2}\otimes\gamma_{1}^{*}\right)}\\{\gamma_{2}\otimes\alpha_{1}+\alpha_{2}^{*}\otimes\gamma_{1}}&{\alpha_{2}^{*}\otimes\alpha_{1}^{*}-q\gamma_{2}\otimes\gamma_{1}^{*}}\\\end{array}\right)\inM_{2}\left(R^{\prime}\otimesR^{\prime}\right),\tag{2.2}
I_{ab}=\widehat{\chi}(R_{a},R_{b})=\left(\begin{matrix}{1}&{4}&{11}&{24}&{46}&{80}\\{0}&{1}&{4}&{11}&{24}&{46}\\{0}&{0}&{1}&{4}&{11}&{24}\\{0}&{0}&{0}&{1}&{4}&{11}\\{0}&{0}&{0}&{0}&{1}&{4}\\{0}&{0}&{0}&{0}&{0}&{1}\\\end{matrix}\right)
(2\pi)^{-n}\int_{V^{*}}e^{i\;s(z^{j}\bar{\psi_{j}}-i\sum_{j}b_{j}\bar{\psi}_{j})}=(\sqrt{\pi})^{-n}e^{-\sum_{i}(z^{j})^{2}}\mathrm{d}z^{1}\wedge\ldots\wedge\mathrm{d}z^{n}
P_{\\nu}^{\mu}=-{\frac{1}{4}}R_{\\lambda}^{\mu}R_{\\nu}^{\lambda}+{\frac{1}{6}}RR_{\\nu}^{\mu}+{\frac{1}{8}}\delta_{\nu}^{\mu}R_{\\beta}^{\alpha}R_{\\alpha}^{\beta}-{\frac{1}{16}}\delta_{\\nu}^{\mu}R^{2}\.
\ddot{\Phi}-3\dot{\Phi}^{2}+6\lambda^{\prime}\alpha\big\{H^{4}+2H^{2}\dot{H}-4\dot{\Phi}H^{3}-4H\dot{H}\dot{\Phi}-2H^{2}(\ddot{\Phi}-2\dot{\Phi}^{2})\big\}=0
{\Gamma}_{\mu}=-\frac{n_{\mu}}{{\partial}_{\bot}^{2}+n^{2}{\partial}_{+}^{2}}B-\frac{{\partial}_{\mu}}{{\partial}_{\bot}^{2}+n^{2}{\partial}_{+}^{2}}(C-n^{2}x^{-}B-\frac{n^{2}\mathrm{sin}2{\theta}{\partial}_{+}}{{\partial}_{\bot}^{2}+n^{2}{\partial}_{+}^{2}}B).
\sigma_{j}^{a}=\mathbf{1}_{V}^{\otimes(j-1)}\otimes\sigma_{}^{a}\otimes\mathbf{1}_{V}^{\otimes(N-j)},\qquad{\calV}=V^{\otimesN}=\underbrace{V\otimesV\otimes\ldots\otimesV}_{\mathrm{~N~factors}},
\langle\sigma^{z}\rangle^{(i)}=\frac{{}_{\;+}^{(i)}\langle1,1|1,1\rangle_{+}^{(i)}-{}_{\;-}^{(i)}\langle1,1|1,1\rangle_{-}^{(i)}}{{}_{\;+}^{(i)}\langle1,1|1,1\rangle_{+}^{(i)}+{}_{\;-}^{(i)}\langle1,1|1,1\rangle_{-}^{(i)}}.
\left(\begin{array}{c}{x_{2}^{\prime}}\\{x_{3}^{\prime}}\\\end{array}\right)=\frac{1}{(w^{2})^{2}+(w^{3})^{2}}\left(\begin{array}{cc}{w^{3}}&{-w^{2}}\\{w^{2}}&{w^{3}}\\\end{array}\right)\left(\begin{array}{c}{x_{2}}\\{x_{3}}\\\end{array}\right).
\mathrm{tr}\left[P\,\Gamma^{,\rho\sigma}\right]=2i\delta^{\rho\sigma}t\,(\mathrm{tr}P)-i\frac{1}{k^{2}}P^{\rho\sigma}\left[e^{k^{2}(1-\xi^{-1})t}-1\right]+i\frac{1}{k^{2}}P^{\rho\sigma}\left[e^{k^{2}(-1+\xi^{-1})t}-1\right]
a^{2}(|y|)=\left[H_{U}\frac{2}{\sqrt{3}}\sqrt{H_{U}-\sqrt{H_{U}^{2}-\frac{9}{4}H_{T}^{2}}}+H_{T}\sqrt{3}\sqrt{H_{U}+\sqrt{H_{U}^{2}-\frac{9}{4}H_{T}^{2}}}\right]^{2/3}\,.
{\calL}_{\pm}^{MWY}=\sum_{k=0}^{\infty}\left[\mp\dot{\phi}_{k}\phi_{k}^{\prime}-{\calG_{\pm}}\phi_{k}^{2}-2\phi_{k}^{\prime}\sum_{m=1}^{k-1}\left(\dot{\phi}_{m}+{\calG_{\pm}}\phi_{m}^{\prime}\right)\right]
\Pi_{i}(p_{i}^{2}-m_{i}^{2})FT\frac{\delta^{4}}{\deltaJ^{4}}\begin{array}{c}{lim}\\{\varepsilon\rightarrow0}\\\end{array}<<[i\delta\alpha\intd^{4}x\phi\partial_{0}^{2}\phi]>>\mid_{_{J=0;\alpha\rightarrow\alpha_{0}^{-}}}
2A={\frac{i}{2}}\int_{\Sigma}\omega_{n,m}\wedge\overline{\omega}_{n,m}=-\mathrm{Im}\sum_{j=1}^{h}\oint_{\alpha_{j}}\omega_{n,m}\oint_{\beta_{j}}\overline{\omega}_{n,m}=\sum_{l,k=1}^{h}(a_{k}a_{l}+b_{k}b_{l})\Omega_{lk}^{(2)},
D_{\mu\nu}(p,\xi)=\frac{1}{p^{2}}\left(\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right)+\xi\frac{p_{\mu}p_{\nu}}{p^{4}}=D_{\mu\nu}^{T}(p)+\xi\frac{p_{\mu}p_{\nu}}{p^{4}},
\begin{aligned}{\bigg(R_{mn}-\frac{1}{4}H_{mpq}H_{n}{}^{pq}}&{{}+2\nabla_{m}\nabla_{n}\Phi\bigg)\gamma^{n}\epsilon}\\{}&{{}=\frac{1}{12}dH_{mnpq}\gamma^{npq}\epsilon+\frac{1}{2}e^{2\Phi}\nabla^{p}\left(e^{-2\Phi}H_{pmn}\right)\gamma^{n}\epsilon}\\\end{aligned}
\Gamma=\frac{\varepsilon^{i_{1}i_{2}}}{\sqrt{-det(G_{ij}+F_{ij}-b_{2ij})}}\left(\frac{1}{2}\tau_{1}\hat{L}_{i_{1}}\hat{L}_{i_{2}}+\frac{i\tau_{2}}{2}(F_{i_{1}i_{2}}-b_{2i_{1}i_{2}})\right)\,.
{H}_{int}=\bar{\psi}*e\gamma^{\mu}\bar{B}_{\mu}\left[1+\frac{e}{2}\gamma^{+}\frac{1}{i\partial_{-}-e\bar{B}_{-}}e\gamma^{\nu}\bar{B}_{\nu}\right]\ast\psi=\bar{\psi}\ast{\calV}[\bar{B}_{\mu}]\ast\psi.
\left\{\psi_{L}\left(\theta\right),\psi_{L}^{\dagger}\left(\theta^{\prime}\right)\right\}=\delta\left(\theta-\theta^{\prime}\right),\quad\left\{\psi_{R}\left(\theta\right),\psi_{R}^{\dagger}\left(\theta^{\prime}\right)\right\}=\delta\left(\theta-\theta^{\prime}\right).
{\calL}_{i}=2^{-i}\delta_{c_{1}d_{1}\cdotsc_{i}d_{i}}^{a_{1}b_{1}\cdotsa_{i}b_{i}}R_{\\a_{1}b_{1}}^{c_{1}d_{1}}\cdotsR_{\\a_{i}b_{i}}^{c_{i}d_{i}}.
E^{L}=-{\frac{3}{4}}+{\frac{1}{2}}({\frac{e_{1}}{2}}+p_{L})^{2}+{\frac{1}{2}}(n_{1}^{5}+{\overline{n}}_{1}^{5}+n_{1}^{6}+{\overline{n}}_{1}^{6})~+~...
I=\frac1{16\pi}\intd^{4}x\sqrt{{-\tilde{g}}}\left[{\tilde{\phi}}{\tilde{R}}-\frac{\omega(\tilde{\phi})}{\tilde{\phi}}\partial_{\mu}\tilde{\phi}\partial^{\mu}\tilde{\phi}\right]+I_{m}(\tilde{g}_{\mu\nu},\Psi)\;.
\left(\hat{f}^{*}\hat{\bf\omega}^{n}\right)\left.\left(\hat{\bfA}_{1},\ldots,\hat{\bfA}_{n}\right)\right|_{\Phi_{p}}=\hat{f}_{*}\hat{\bf\omega}^{n}\left.\left(\hat{f}_{*}\hat{\bfA}_{1},\ldots,\hat{f}_{*}\hat{\bfA}_{n}\right)\right|_{f\left(\Phi_{p}\right)}.