harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\zeta\int_{S}^{}dS_{\mu}(x)\int_{S}^{}dS_{\mu}(y)D_{m}(x-y)\simeq\sigma\intd^{3}\xi\sqrt{\hat{g}}+\alpha\intd^{3}\xi\sqrt{\hat{g}}\left(K^{ia}{\,}_{a}\right)^{2}+\beta\intd^{3}\xi\sqrt{\hat{g}}{\calR}.
\mu_{12}=-{\frac{1}{24\pi^{2}}}{\frac{\pi^{D/2}\Gamma(-D/2)}{2^{D}}}\left[-(D-3)(D-2)(D-1)\zeta(2-D)+2\pi^{2}(D+1)\zeta(-D)\right]\,{\frac{\hbarL^{D-1}}{a^{D}}}.
Cf\rightarrow\left(\begin{array}{cc}{0}&{\\\delta_{cd}^{ab}\,\partial_{\mu}}\\{}&{}\\{\partial_{\mu}}&{\\-\eta_{\mub}\,\delta_{cd}^{ab}}\\\end{array}\right)\left(\begin{array}{c}{\Lambda^{a}}\\{\theta^{cd}}\\\end{array}\right).
\mathrm{Tr}\left(e^{-\frac{n\beta}{\sqrt{2}}H_{\mathrm{l.c.}}}\right)=V_{T}\int\frac{d^{D-2}p_{T}}{(2\pi)^{D-2}}~e^{-\frac{n\beta}{\sqrt{2}}\left(\frac{{\vec{p}_{T}}^{~2}+M^{2}}{2p^{-}}\right)}
S_{0}^{(l)}[Q]={\frac{l}{2g^{2}\beta}}\sum_{{\vec{n}}}\sum_{{\hat{e}}}\varphi_{{\vec{n}}}[\varphi_{{\vec{n}}+2{\hat{e}}}-2\varphi_{{\vec{n}}+{\hat{e}}}+\varphi_{\vec{n}}]\;.
\tilde{\calL}=-\frac{1}{4e^{2}}F_{\mu\nu}F^{\mu\nu}+\frac{\kappa}{2}\epsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}A_{\rho}+\frac{1}{2e^{2}}(\partial_{\mu}N)^{2}+|D_{\mu}\phi|^{2}-\tilde{U}-\rho_{e}A_{0},
g_{-}=\left(\begin{array}{cc}{1}&{0}\\{\frac{x_{-}}{\sqrt{2}}}&{1}\\\end{array}\right)\;\;\;\;\;\;g_{0}=\left(\begin{array}{cc}{e^{\frac{x}{2}}}&{0}\\{0}&{e^{-\frac{x}{2}}}\\\end{array}\right)\;\;\;\;\;\;g_{+}=\left(\begin{array}{cc}{1}&{\frac{x_{+}}{\sqrt{2}}}\\{0}&{1}\\\end{array}\right).
\epsilon^{\kappa\lambda\sigma\beta}\left(\partial_{\alpha}h_{\beta}^{\alpha}-\partial_{\beta}h_{\alpha}^{\alpha}\right)-\epsilon^{\kappa\lambda\alpha\beta}\,\partial_{\alpha}h_{\beta}^{\sigma}=\left(\partial^{\kappa}{\tilde{h}}^{\sigma\lambda}-\partial^{\lambda}{\tilde{h}}^{\sigma\kappa}\right)+\left(\partial^{\kappa}\omega^{\sigma\lambda}-\partial^{\lambda}\omega^{\sigma\kappa}\right)\.
T_{A}=\sum_{s=0}^{\infty}T_{A}^{s}\,,\qquadT_{A}^{0}=\Theta_{A}\,,\quadT_{A}^{s}=T_{A\,B_{1},\ldots,B_{s}}^{s}(x)Y^{B_{1}},\ldots,Y^{B_{s}}\,,
Z^{j}{}_{l}(f_{\mu}{}^{l}{}_{i}-f^{\prime}{}_{\mu}{}^{l}{}_{i})=(M^{-1})^{j}{}_{k}M^{\nu}{}_{\mu}M^{h}{}_{i}(f_{\nu}{}^{k}{}_{h}-f^{\prime}{}_{\nu}{}^{k}{}_{h}),
TrD_{\mu\nu}(x-x^{\prime})={\int}d^{4}x\left[3{\int}\frac{d^{4}p}{(2\pi)^{4}}\frac{1}{p^{2}+e^{2}<\rho>^{2}}+\frac{\delta^{4}(0)}{e^{2}<\rho>^{2}}\right].
R_{\lambda_{1}^{C},\lambda_{2}^{C}}(u)=\frac{1}{R_{N^{\prime}-1,N^{\prime}-1}^{N^{\prime}-1,N^{\prime}-1}(\lambda_{1},\lambda_{2};u)}({\calC}\otimes{\calC})R_{\lambda_{1},\lambda_{2}}(u)({\calC}\otimes{\calC})
u_{1(-1)}(z)=\frac{1}{\sqrt{2}}~u_{2(-1)}(z)+2\pi\frac{\Gamma(-\frac{1}{2})}{\Gamma(\frac{1}{8})\Gamma(\frac{3}{8})}~(1+z)^{\frac{1}{2}}~F(\frac{7}{8},\frac{5}{8},\frac{3}{2};{1+z})~.
G_{R}(x,z;x^{\prime},z^{\prime})=-\int\frac{d^{4}p}{(2\pi)^{4}}e^{ip_{\mu}(x^{\mu}-x^{\prime}{}^{\mu})}\Biggl[\frac{e^{-2k(|z|+|z^{\prime}|)}k}{{\bfp}^{2}-(\omega+i\epsilon)^{2}}+\int_{0}^{\infty}dm\,\frac{v_{m}(z)v_{m}(z^{\prime})}{m^{2}+{\bfp}^{2}-(\omega+i\epsilon)^{2}}\Biggr],
\sum_{n_{1},n_{2}}e^{-\piN(n_{1}^{2}\frac{R_{1}}{R_{2}}+n_{2}^{2}\frac{R_{2}}{R_{1}})}+\sum_{n_{1},n_{2}}\frac{1}{N}e^{-\frac{\pi}{N}(n_{1}^{2}\frac{R_{1}}{R_{2}}+n_{2}^{2}\frac{R_{2}}{R_{1}})}
\frac{c_{2r}^{\nu}}{c_{2r-2}^{\nu}}=\frac{1}{\displaystyleh^{-2}[s^{2}-(\nu+2r)^{2}]-\frac{1}{\displaystyleh^{-2}[s^{2}-(\nu+2r+2)^{2}]-\frac{1}{\displaystyleh^{-2}[s^{2}-(\nu+2r+4)^{2}]\cdot\cdot\cdot}}}
Z[J]=\intD[c,C^{1},C^{2},\chi^{1},\chi^{2},\sigma]\,\delta(\partial_{i}c^{i})\prod_{a=1,2}\delta(\partial_{i}C^{i\,a})e^{i\intd^{4}x{\calL}_{\mathrmeff}+j_{c}c+..}
\begin{array}{c}{(a_{v}+\{\epsilon_{v},.\})R_{v}(\epsilon,a)\equiv(a_{v}+\{\epsilon_{v},.\})(aH_{v}+\{\epsilon,H_{v}\})}\\{=R_{v}(\{\epsilon_{v},\epsilon\},\{\epsilon_{v},a\}-\{\epsilon,a_{v}\}),}\\\end{array}
\langle\overline{{\phi}}^{2}(x)\rangle_{\beta}=\frac{1}{12\beta^{2}}(1+\eta^{2})-\eta^{2}\xi\frac{\sqrt{2}}{8\pi^{2}r^{2}}S_{1}+\eta^{2}\frac{1}{32\pi^{2}r^{2}}S_{2},
L^{+}=\left(\begin{array}{ccccc}{l_{11}^{+}}&{l_{12}^{+}}&{l_{13}^{+}}&{l_{14}^{+}}&{l_{15}^{+}}\\{}&{l_{22}^{+}}&{l_{23}^{+}}&{l_{24}^{+}}&{l_{25}^{+}}\\{}&{}&{l_{33}^{+}}&{l_{34}^{+}}&{l_{35}^{+}}\\{}&{}&{}&{l_{44}^{+}}&{l_{45}^{+}}\\{}&{}&{}&{}&{l_{55}^{+}}\\\end{array}\right)~,
V=\frac{1}{2gNl_{s}(2\pil_{s}^{2})^{2}}\sum_{j=0}^{N}\left(X_{j+1,j+1}^{m}-X_{jj}^{m}\right)^{2}\left(|Z_{j,j+1}^{1}|^{2}+|Z_{j+1,j}^{2}|^{2}\right).
t_{+}^{\ell}(\mu,\nu;a,x)=\frac{N(\mu)}{N(\nu)}\frac{\Gamma(\nu-2\ell)\Gamma(\mu-\nu)}{\Gamma(\mu-2\ell)}\,e^{a(\mu-\ell)}\,x_{+}^{\nu-\mu}\,,\qquad2\ell<\Re(\nu)<\Re(\mu)\,.
\begin{array}{rcl}{{\calL}_{GG\phi}}&{=}&{\displaystyle+\frac{g^{2}v}{2\sqrt{2}}\left(A^{\mu}(\pi_{+}W_{\mu}^{-}+W_{\mu}^{+}\pi_{-})+Z^{\mu}(W_{\mu}^{-}\pi_{+}+\pi_{-}W_{\mu}^{+})\right)}\\{}&{}&{\displaystyle+\frac{g^{2}v}{2}(W_{\mu}^{-}W^{+\mu}+Z_{\mu}Z^{\mu})\sigma,}\\\end{array}
L_{3}(u)\equiv\frac{d}{du}\left(u\frac{d}{du}\right)-\left(u\frac{d}{du}+\frac{5}{6}\right)\,\left(u\frac{d}{du}+\frac{1}{6}\right)=u\,\left(1-u\right)\frac{d^{2}}{du^{2}}+\left(1-2\,u\right)\,\frac{d}{du}-\frac{5}{36}
\Psi(x,t=0)=\sum_{\alpha}\left(\int_{0}^{\infty}b_{\alpha}^{\omega}\psi_{\alpha}^{\omega}(x)d\omega+\int_{-\infty}^{0}d_{\alpha}^{\omega}{}^{\dagger}\psi_{\alpha}^{\omega}(x)d\omega\right)
I_{DivPart}(p)=-i\frac{g_{\lambda\rho}}{2^{d+1}\pi^{\frac{d}{2}}[p^{2}]^{\epsilon}}\frac{\Gamma(\epsilon)}{\Gamma\left(2-\frac{d}{2}\right)}\int_{0}^{1}dz\int_{0}^{z}dtt^{1-\frac{d}{2}}[a^{\prime\prime}-b^{\prime\prime}]^{d-3}\;,
0=\epsilon_{i_{1}i_{2}\dotsi_{n}i_{n+1}}\epsilon_{j_{1}j_{2}\dotsj_{n}j_{n+1}}=\sum_{P\inS_{n+1}}\(-1)^{\pi(P)}\\delta_{i_{1}P(j_{1})}\dots\delta_{i_{n+1}P(j_{n+1})},
S=\frac{1}{2\pi}\intd^{2}x\sqrt{-g}\,e^{-2\phi}\,(R+4{\nabla\phi}^{2}+4\lambda^{2}-(\nablaT)^{2}-V(T))-\frac1{\pi}\intd\Sigma\sqrt{-h}\,e^{-2\phi}\,(K-2{\nabla}_{n}\phi)
\begin{array}{c}{\epsilon_{ab}=\left(\begin{array}{cc}{0}&{q^{\frac{1}{2}}}\\{-q^{-\frac{1}{2}}}&{0}\\\end{array}\right)}\\{\epsilon_{ab}\epsilon_{bc}=-\delta_{ac}}\\{(\epsilon_{\overline{{a}}\overline{{b}}})_{q}=-(\epsilon_{ab})_{q^{-1}}}\\\end{array}
f(p,k)={\frac{2k^{3}(4\alpha^{3}p^{3}+4\alpha^{2}p^{4}+\alpha^{2}p^{2}k^{2}-3p^{4}k^{2}+\alpha^{2}k^{4}+3\alphapk^{4}+3p^{2}k^{4})}{(3\alpha(\alphap+p^{2})^{3})}}\Theta(p-k),
(u^{-1})_{\underline{a}}^{~\underline{c}}R_{\underline{c}}^{~\underline{d}}u_{\underline{d}}^{~\underline{b}}=d\hat{\Omega}_{\underline{a}}^{~\underline{b}}+\hat{\Omega}_{\underline{a}}^{~\underline{c}}\hat{\Omega}_{\underline{c}}^{~\underline{b}}.
i(-\frac{1}{2}\Gamma^{\mu\nu\rho}\partial_{\rho}^{x}-\frac{1}{i}\Gamma^{\mu}\Gamma^{\rho}\Gamma^{\nu}\partial_{\rho}^{x})\langle\psi_{\nu}(x)\bar{\psi_{\tau}}(y)\rangle=\eta_{\tau}^{\mu}\delta(x-y),
{\calO}_{\pm}^{a}(\{x\}):=\frac{1}{(N!)^{2}}\sum_{\pi,\pi^{\prime}}\prod_{a=1}^{N}\overline{{\psi}}^{(a)}(x^{(\pi(a))})P_{\pm}\psi^{(a)}(x^{(\pi(a))})\;.
h_{12}={\frac{\tau_{1}\tau_{2}}{1-\tau_{1}\tau_{2}}}F_{1}F_{2}\Big[(g^{\dagger})^{-1}\eta_{1}^{\dagger}M\eta_{2}g^{-1}+g\eta_{1}M^{-1}\eta_{2}^{\dagger}g^{\dagger}\Big]-\mathrm{h.c.}.
\begin{array}{rcl}{}&{}&{G_{\mu\nu\rho\sigma}=4\partial_{[\mu}C_{\nu\rho\sigma]}+6B_{m\,[\mu\nu}F^{m}{}_{\rho\sigma]}\,,}\\{}&{}&{}\\{}&{}&{H_{m\,\mu\nu\rho}=3\partial_{[\mu|}B_{m|\,\nu\rho]}+3V_{mn\,[\mu}F^{n}{}_{\nu\rho]}\,,}\\{}&{}&{}\\{}&{}&{F_{mn\,\mu\nu}=2\partial_{[\mu|}V_{mn|\,\nu]}\,,}\\{}&{}&{}\\{}&{}&{F^{m}{}_{\mu\nu}=2\partial_{[\mu}A^{m}{}_{\nu]}\,.}\\\end{array}
\theta_{\mu\nu}~=~g_{ij}(\varphi)\,\partial_{\mu}\varphi^{i}\,\partial_{\nu}\varphi^{j}\,-\,{\textstyle{\frac{1}{2}}}\,h_{\mu\nu}\,h^{\kappa\lambda}\,g_{ij}(\varphi)\,\partial_{\kappa}\varphi^{i}\,\partial_{\lambda}\varphi^{j}~~,
ds^{2}~=~dx^{-}\left[\frac{1-\alpha/2P_{0}|x^{-}|}{1-\alpha/P_{0}|x^{-}|}\right]\left[dx^{+}-\frac{4GP_{0}/|x^{-}|}{1-\alpha/P_{0}|x^{-}|}~dx^{-}\right]-dx_{\perp}^{2}~.
6g^{6}{\frac{p^{4}}{M^{4}}}ln{\frac{m_{0}^{2}+p^{2}}{m_{0}^{2}}}-2g^{6}ln{\frac{\Lambda^{2}}{m_{0}^{2}}}+2g^{6}ln{\frac{m_{0}^{2}+p^{2}}{m_{0}^{2}}}+...
{m_{0}}^{2}=\frac{\kappa^{2}}{4}\,\left(1-e^{-2p_{0}/\kappa}\right)^{2}=\frac{\kappa^{2}}{4}\,\left(1-\left(-\frac{m}{2\kappa}+\sqrt{\frac{m^{2}}{4\kappa^{2}}+1}\right)^{4}\right)^{2},
\int_{10}\hat{\vec{H}}{}^{T}{\calM}^{-1}\{}^{\star}\hat{\vec{H}}\;=\;\int_{9}\left[K^{-\frac{3}{4}}\vec{F}_{(3)}^{\T}{\calM}^{-1}\{}^{\star}\vec{F}_{(3)}-K^{\frac{3}{4}}\vec{F}_{(2)}^{\T}{\calM}^{-1}\{}^{\star}\vec{F}_{(2)}\right]\,.
b_{3}={\frac{1}{6}}b_{\mu\nu\rho}d\sigma^{\mu}d\sigma^{\nu}d\sigma^{\rho}={\frac{1}{2}}\bar{\theta}\Gamma_{MN}d\theta(dX^{M}dX^{N}+dX^{M}\bar{\theta}\Gamma^{N}d\theta+{\frac{1}{3}}\bar{\theta}\Gamma^{M}d\theta\bar{\theta}\Gamma^{N}d\theta).
60\mathrm{Cl}_{2}(\theta_{3,2})-18\mathrm{Cl}_{2}(2\theta_{3,2})-4\mathrm{Cl}_{2}(3\theta_{3,2})+3\mathrm{Cl}_{2}(4\theta_{3,2})=Z_{24}=24\sum_{k\in\{1,5,7,11\}}\mathrm{Cl}_{2}\left(\frac{2\pik}{24}\right)
G_{r}^{(2)}=\sum_{n_{0},n_{1},\ldots,n_{\nu-3}=0}^{\infty}x^{\frac{r-1}{2}}\frac{1}{(1-T_{1,0})^{1+\tilde{V}_{\nu-2,r}}}\times\prod_{i=0}^{\nu-3}(T_{\nu-1-i,1})^{n_{i}}\times\prod_{i=1}^{\nu-3}\left[\begin{array}{c}{n_{i}+\tilde{n}_{i}+\tilde{V}_{i,r}}\\{n_{i}}\\\end{array}\right],
\bar{\rho}_{\mathrm{w}}(\zeta)\;=\;{\frac{8\beta}{E(k)^{2}}}\left[{\frac{\sqrt{k^{2}-\zeta^{2}}}{\sqrt{1-\zeta^{2}}}}K(k)-\sqrt{k^{2}-\zeta^{2}}\sqrt{1-\zeta^{2}}\Pi(\zeta^{2},k)\right]\;,
U_{c}N_{c}^{\mathrm{gh}}=U_{1}N_{1}^{\mathrm{gh}}GPU_{2}^{\dagger}+U_{1}\left[G,N_{1}^{\mathrm{gh}}\right]PU_{2}^{\dagger}+U_{1}GPU_{2}^{\dagger}N_{2}^{\mathrm{gh}}\.
\begin{array}{c}{\int_{c(r_{2})}\omega^{I\pm}-\int_{c(r_{1})}\omega^{I\pm}=\pm\int_{c\timesI_{[r_{1},r_{2}]}}dx^{\lambda}\wedgedx^{\mu}\wedgedx^{\nu}T_{[\lambda\mu}^{\rho}\omega_{\nu]\rho}^{I\pm}}\\\end{array}
q_{\pm}(r_{+})={\frac{r_{+}^{2}}{2(2+3r_{+}^{2})}}\left(17r_{+}^{2}+4+12r_{+}^{4}+\sqrt{180r_{+}^{8}+444r_{+}^{6}+369r_{+}^{4}+120r_{+}^{2}+16}\right)\,
sB_{\mu\nu}^{a}=\varepsilon_{\mu\nu\lambda\rho}\left(D^{\lambda}\right)_{\;\;b}^{a}\eta^{\rhob},\;sA_{\mu}^{a}=0,\;s\varphi^{(1)a}=\left(D_{\mu}\right)_{\;\;b}^{a}\eta^{\mub},
\bar{W}_{\mathrm{ren}}^{\infty}=\mathrm{}-\frac12\frac{\partial}{\partial\epsilon}\left[\frac{\mu^{2\epsilon}}{\Gamma(\epsilon)}\int_{0}^{\infty}\!\frac{{\mathrm{d}}s}{s^{1-\epsilon}}\,{\mathrm{Tr}}\bar{K}^{\infty}(s)\right]_{\epsilon=0},
F_{\mu\nu}=\left(\begin{array}{cccc}{0}&{E_{x}}&{E_{y}}&{E_{z}}\\{-E_{x}}&{0}&{B_{z}}&{-B_{y}}\\{-E_{y}}&{-B_{z}}&{0}&{B_{x}}\\{-E_{z}}&{B_{y}}&{-B_{x}}&{0}\\\end{array}\right),
\frac{dt}{d\tau}=\mathrm{constant}\doteqg^{tt}K\;\mathrm{~and~}\;\,\frac{d^{2}x^{\mu}}{d\tau^{2}}+\Gamma_{\lambda\rho}^{\mu}\frac{dx^{\lambda}}{d\tau}\frac{dx^{\rho}}{d\tau}=0\,,
K^{A}=\int_{\Sigma}d^{3}xe\Theta^{\mu0}k_{\mu}^{A}=\int_{\Sigma}\Theta^{\mu\nu}k_{\mu}^{A}d\Sigma_{\nu}={\frac{1}{4}}\int_{S=\partial\Sigma}\delta_{\mu\nu\rho}^{\sigma\tau\lambda}\Gamma_{\\b}^{\nua}k^{A\,\rho}e_{a}^{\mu}e_{\lambda}^{b}d\Sigma_{\sigma\tau},
{\frac{\partial^{3}F}{\partialt^{\alpha}\partialt^{\beta}\partialt^{\lambda}}}\eta^{\lambda\mu}{\frac{\partial^{3}F}{\partialt^{\mu}\partialt^{\gamma}\partialt^{\delta}}}={\frac{\partial^{3}F}{\partialt^{\gamma}\partialt^{\beta}\partialt^{\lambda}}}\eta^{\lambda\mu}{\frac{\partial^{3}F}{\partialt^{\mu}\partialt^{\alpha}\partialt^{\delta}}}\tag{2.24b}
\begin{array}{l}{S^{3}=s_{(1)}^{3}\otimes1_{(2)}+1_{(1)}\otimess_{(2)}^{3}~,}\\{S^{\pm}=s_{(1)}^{\pm}\otimesq^{-s_{(2)}^{3}}+q^{s_{(1)}^{3}}\otimess_{(2)}^{\pm}~.}\\\end{array}
A_{ij}=\left(\begin{matrix}{2}&{-1}&{0}&{0}&{\cdots}&{0}&{0}&{0}\\{-1}&{2}&{-1}&{0}&{\cdots}&{0}&{0}&{-1}\\{0}&{-1}&{2}&{-1}&{\cdots}&{0}&{0}&{0}\\{\vdots}&{}&{}&{}&{}&{}&{}&{}\\{0}&{0}&{0}&{0}&{\cdots}&{2}&{-1}&{0}\\{0}&{0}&{0}&{0}&{\cdots}&{-1}&{2}&{-1}\\{0}&{-1}&{0}&{0}&{\cdots}&{0}&{-1}&{2}\\\end{matrix}\right)\,\;\;\;\;\mathrm{for~d~>3~}.
\Lambda=\left(\begin{array}{cc}{a}&{b}\\{c}&{d}\\\end{array}\right)=\left(\begin{array}{cc}{(1+xp^{-1})\kappa^{\frac{1}{2}}m}&{x\kappa^{-\frac{1}{2}}n}\\{p^{-1}\kappa^{\frac{1}{2}}m}&{\kappa^{-\frac{1}{2}}n}\\\end{array}\right)
\dot{\epsilon}_{\delta\sigma}=\epsilon_{\delta\sigma}\left[3{\frac{\dot{\bar{a}}_{R}}{\bar{a}_{R}}}(1+w)+2{\frac{\dot{\bar{c}}_{4}}{\bar{c}_{4}}}{\frac{1+\epsilon_{\delta\sigma}}{1+\epsilon}}+\left({\frac{\dot{\bar{G}}_{4}}{\bar{G}_{4}}}-4{\frac{\dot{\bar{c}}_{4}}{\bar{c}_{4}}}\right)(1+\epsilon_{\delta\sigma})\right].
\begin{array}{lll}{\beta_{1}=T_{\pm3456}=\pmT_{\pm1278}\,,~}&{\beta_{2}=T_{\pm2475}=\pmT_{\pm1638}\,,~}&{\beta_{3}=T_{\pm1357}=\pmT_{\pm2468}\,,}\\{{}}&{{}}&{{}}\\{\beta_{4}=T_{\pm1256}=\pmT_{\pm3478}\,,~}&{\beta_{5}=T_{\pm1234}=\pmT_{\pm5678}\,,~}&{\beta_{6}=T_{\pm1476}=\pmT_{\pm2538}\,,}\\{{}}&{{}}&{{}}\\{\beta_{7}=T_{\pm2376}=\pmT_{\pm1548}\,.}&{{}}&{{}}\\\end{array}
I=-\intd^{4}x[\frac{1}{2}ve^{-\phi}B_{i}^{1}B_{i}^{1}-\frac{1}{2vdetD}e^{-\phi}E_{i}^{1}E_{i}^{1}-\frac{1}{2vdetD}e^{-2\phi}(E_{i}^{1}B_{i}^{1})^{2}+\frac{1}{2}v+\frac{1}{2v}-\psiE_{i}^{1}B_{i}^{1}-1].
\frac{L}{2}-\sigma=\frac{R^{2}}{U_{0}}\int_{U/U_{0}}^{\infty}\frac{dy}{\sqrt{(y^{4}-1)(y^{4}-1+\epsilon)}}\approx\frac{R^{2}U_{0}^{2}}{3U^{3}}\approx\frac{1}{3by^{3}},
W=\left|\begin{array}{cccc}{0}&{iC_{-}}&{0}&{C_{-}}\\{0}&{-C_{+}}&{0}&{iC_{+}}\\{iC_{-}}&{0}&{C_{-}}&{0}\\{iC_{+}}&{0}&{C_{+}}&{0}\\\end{array}\right|;\quadC_{\mp}\equiv\frac12(C\mp1),\quadC\Psi\equiv\Psi^{*},\quadC\mathcal{E}\equiv\mathcal{E}^{*}.
\displaystyle{0=(1-\kappa\frac{1}{2r^{2}})(1+\kappa\frac{1}{r^{2}})\deltaf^{\prime\prime}+(\frac{1}{r}+\kappa\frac{5}{2r^{3}})\deltaf^{\prime}+(\frac{2}{r^{2}}+\kappa\frac{2}{r^{4}})\deltaf}
A_{k}^{i}\left.\left|0\right\rangle\!\right\rangle=B_{m}^{j}\left.\left|0\right\rangle\!\right\rangle=\tilde{A}_{k}^{i}\left.\left|0\right\rangle\!\right\rangle=\tilde{B}_{m}^{j}\left.\left|0\right\rangle\!\right\rangle=0.
I=\frac{k}{2\pi}\intd^{2}z[\frac{|\partialx|^{2}}{4+x^{2}}+\frac{2}{k}|\partialf|^{2}]+\frac{2c}{\pi}\intd^{2}z(A\overline{{\partial}}f+\overline{{A}}\partialf+2c\overline{{A}}A),
\begin{matrix}{\hat{\delta}F^{0+ab}=4\bar{\calN}^{0\Lambda}F_{\Lambda}^{+ab},}&{\hat{\delta}F_{i}^{+ab}=0,}\\{\hat{\delta}F^{0-ab}=-4{\calN}^{0\Lambda}F_{\Lambda}^{-ab},}&{\hat{\delta}F^{i-ab}=0.}\\\end{matrix}
{\Gamma^{1}}_{a}=\left\{\left(\begin{matrix}{0}&{\sigma_{1}}\\{-\sigma_{1}}&{0}\\\end{matrix}\right),\quad\left(\begin{matrix}{0}&{-I}\\{-I}&{0}\\\end{matrix}\right),\quad\left(\begin{matrix}{I}&{0}\\{0}&{-I}\\\end{matrix}\right)\right\},
\zeta^{(T)}\left(z\right)=\sum_{n=0}^{\infty}\sum_{l=1}^{\infty}D_{n}^{T}(\Lambda_{n,l}^{T})^{-z}+\sum_{n=1}^{\infty}\bar{D}_{n}^{T}(\Lambda_{n,0}^{T})^{-z}~~~
C^{A}\rightarrow\pm\bar{C}^{A},\quad\bar{C}^{A}\rightarrow\mpC^{A},\quadB^{A}\rightarrow-\bar{B}^{A},\quad\bar{B}^{A}\rightarrow-B^{A},\quad{\calA}_{\mu}^{A}\rightarrow{\calA}_{\mu}^{A}.
Q^{1}=J\frac{\psi+\psi^{*}}{1-\vert\psi\vert^{2}},~~Q^{2}=iJ\frac{\psi^{*}-\psi}{1-\vert\psi\vert^{2}},~~Q^{3}=J\frac{1+\vert\psi\vert^{2}}{1-\vert\psi\vert^{2}}.
\sum_{\sigma}[\frac{(-)^{i}}{|[\sigma]|}\(\sum_{R}\frac{d_{R}|[\sigma]|}{n!^{2}}(\frac{{\tilde{C}}(R)}{2})^{i}\\chi_{R}(\sigma)\Upsilon_{\sigma}(U)\N^{n-i})+O(N^{n-i-1})]
{\overline{{\mathcalR}}}(\overline{{y}}_{j},{x}_{i}^{\prime}+{x}_{i}^{\prime\prime})={\overline{{\mathcalR}}}(\overline{{y}}_{j},{x}_{i}^{\prime})\,{\overline{{\mathcalR}}}(\overline{{y}}_{j},{x}_{i}^{\prime\prime})
\Phi({\bfr})=U_{3}^{-1}({\bfr})\left(\begin{matrix}{t_{4}-{\frac{1}{2y_{R}}}}&{0}&{0}&{0}\\{0}&{t_{2}-{\frac{1}{2y_{L}}}+{\frac{1}{2y_{R}}}}&{0}&{0}\\{0}&{0}&{t_{2}}&{0}\\{0}&{0}&{0}&{t_{1}+{\frac{1}{2y_{L}}}}\\\end{matrix}\right)U_{3}({\bfr})
K_{N}(p_{1j}^{2};M_{N}^{2},\epsilon)=\frac{-i}{32\pi^{4}}\frac{1}{M_{N}^{2\epsilon}}\left[I(\frac{p_{12}^{2}}{M_{N}^{2}},\epsilon)+I(\frac{p_{13}^{2}}{M_{N}^{2}},\epsilon)+I(\frac{p_{14}^{2}}{M_{N}^{2}},\epsilon)\right].
S_{\|}\left[g,h\right]~=~-{\frac{1}{G}}\int\sqrtg\left(\sqrth~R_{h}+{\frac{1}{4}}\sqrthh^{ij}\partial_{i}g_{\alpha\beta}\partial_{j}g_{\gamma\delta}{\epsilon}^{\alpha\gamma}{\epsilon}^{\beta\delta}\right),
\begin{array}{rcl}{Y}&{=}&{\frac{1}{2}(x+y)=\frac{1}{2}(z_{1}^{2n-4}+z_{2}^{2n-4})}\\{X}&{=}&{\frac{1}{2}z(x-y)=\frac{1}{2}z_{1}z_{2}(z_{1}^{2n-4}-z_{2}^{2n-4})}\\{Z}&{=}&{z^{2}=z_{1}^{2}z_{2}^{2}}\\\end{array}
T_{F}^{-1}=\frac{i}{2\sqrt{2}}\sum_{k}e^{-iH_{k}}\biggl[(1-i)e^{iH_{k^{\prime}}}e^{iH_{\bar{k}^{\prime}}}+(1+i)e^{iH_{k^{\prime}}}e^{-iH_{\bar{k}^{\prime}}}+(1+i)e^{-iH_{k^{\prime}}}e^{iH_{\bar{k}^{\prime}}}+(1-i)e^{-iH_{k^{\prime}}}e^{-iH_{\bar{k}^{\prime}}}\biggr]
\left.\left(\begin{array}{ccc}{\partial\beta_{g}/\partialg}&{\partial\beta_{g}/\partialh}&{\partial\beta_{g}/\partialf}\\{\partial\beta_{h}/\partialg}&{\partial\beta_{h}/\partialh}&{\partial\beta_{h}/\partialf}\\{\partial\beta_{f}/\partialg}&{\partial\beta_{f}/\partialh}&{\partial\beta_{f}/\partialf}\\\end{array}\right)\right|_{g=g_{\ast},h=h_{\ast},f=f_{\ast}}
ds^{2}=-(\frac{\rho^{2}}{\ell^{2}}-M)dt^{2}+\frac{d\rho^{2}}{\frac{\rho^{2}}{\ell^{2}}-M+\frac{J^{2}}{4\rho^{2}}}+\rho^{2}d\varphi^{2}+Jd\varphidt\qquad,
T^{\prime\prime}(t,x)=\left.{\frac{\partialT^{\prime}(t,x)}{\partialq}}\right|_{t}\left.{\frac{\partialq}{\partialx}}\right|_{t}={\frac{T_{i,qq}}{1-\frac{T_{i,qq}}{(1+T_{i,q})^{3/2}}\,t}}\,
T=\left(\begin{array}{rrrrrrrr}{x}&{y}&{0}&{a}&{0}&{\cdots}&{0}&{d}\\{z}&{-x}&{0}&{b}&{0}&{\cdots}&{0}&{e}\\{\frac{-v_{2}}{v_{1}}b}&{\frac{v_{2}}{v_{1}}a}&{0}&{c}&{0}&{\cdots}&{0}&{f}\\{0}&{0}&{0}&{0}&{0}&{\cdots}&{0}&{0}\\{\vdots}&{\vdots}&{\vdots}&{\vdots}&{\vdots}&{}&{\vdots}&{\vdots}\\{0}&{0}&{0}&{0}&{0}&{\cdots}&{0}&{0}\\\end{array}\right).
\frac{\partial^{2}F}{\partialx^{1}\partialt}=\phi_{1t}^{3}+\sum_{j=2}^{3}[\phi_{1j}^{3}f_{t}^{j}+\phi_{tj}^{3}f_{1}^{j}+\phi_{j}^{3}f_{1t}^{j}+\sum_{k=2}^{3}(\phi_{jk}^{3}f_{t}^{k})f_{1}^{j}]
J^{\mu_{2}...\mu_{p+1}}(x)={\frac{-i}{(D-p-1)!}}\int_{W_{q}}\epsilon^{\mu_{1}\mu_{2}...\mu_{p+1}\mu_{p+2}...\mu_{D}}{\frac{\partial}{\partialy^{\mu_{1}}}}\delta^{(D)}(x-y)dy^{\mu_{p+2}...\mu_{D}},
\begin{array}{rcl}{G_{_{++\!\!\!+}}}&{=}&{b_{1}(1+c_{1}\nabla_{_{\!\!+\!\!\!+}}c_{1})-\frac{1}{2}c_{1}(\nabla_{_{\!\!+\!\!\!+}}X)^{2},}\\{G_{_{-=}}}&{=}&{\bar{b}_{1}(1+\bar{c}_{1}\nabla_{_{\!\!=}}\bar{c}_{1})-\frac{1}{2}\bar{c}_{1}(\nabla_{_{\!\!=}}X)^{2},\nonumber}\\\end{array}
(u_{\omega{\bfk}_{\perp}},u_{\omega{\bfk}_{\perp}})=(2\pi)^{3}\,\delta^{(3)}(0)\,\left[\omega\left(1+R_{\omega{\bfk}_{\perp}}^{2}\right)+(\omega+qV)\,T_{\omega{\bfk}_{\perp}}^{2}\right].
Z^{(1)}=i\sqrt{-g}\{(\bar{\tilde{\calH}}^{i}v^{j}-\bar{\tilde{\calH}}^{j}v^{i})\kappa\gamma_{i}E_{j}-(\tilde{\calH}^{i}v^{j}-\tilde{\calH}^{j}v^{i})\bar{\kappa}\gamma_{i}\bar{E}_{j}\},
\partial_{-}V_{+}=-\partial_{i}V_{i}-e^{A}\left(\frac{D_{1}-2}{2}A^{\prime}V_{r}+D_{\underline{m}}V^{\underline{m}}\right)-ie^{A}\left(\phi^{\dagger}T\Phi-\Phi^{\dagger}T\phi\right)e~,
\Theta\frac{\partial}{\partial\theta}Z[J,\theta,\varepsilon]=\begin{array}{c}{lim}\\{\varepsilon\rightarrow0}\\\end{array}\begin{array}{c}{lim}\\{p_{i}^{2}\rightarrowm_{i}^{2}-i\varepsilon}\\\end{array}\Pi_{i}(p_{i}^{2}-m_{i}^{2}+i\varepsilon)\frac{\partial}{\partial\theta}\frac{\phi_{i}(p_{i})}{\sqrt{Z_{i}}}F.T.[\frac{\delta^{n}}{\deltaJ^{n}(x)}]Z[J,\theta,\varepsilon]
\delta^{L}\Phi(X)=\frac{i}{2}\omega^{\alpha\beta}(X)\star\left(\mathcal{D}_{\alpha\beta}\Phi\right)(X)=\frac{i}{2}\left(\omega^{\alpha\beta}(X)\star\mathbf{Q}_{\alpha\beta}\right)\star\Phi(X)\equivi\omega_{L}(X)\star\Phi(X).
\left(\Omega\left[\begin{matrix}{h}\\{0}\\\end{matrix}\right],\Phi^{a}\left[\begin{matrix}{h+1/2}\\{1/2}\\\end{matrix}\right],\bar{\Phi}^{a}\left[\begin{matrix}{h+1/2}\\{1/2}\\\end{matrix}\right],\Gamma\left[\begin{matrix}{h+1}\\{0}\\\end{matrix}\right],\bar{\Gamma}\left[\begin{matrix}{h+1}\\{0}\\\end{matrix}\right],\Sigma\left[\begin{matrix}{h+1}\\{0}\\\end{matrix}\right]\right),
M=|z|=|q_{\Lambda}^{(e)}X^{\Lambda}-{\frac{i}{2}}q_{(m)}^{\Lambda}F_{\Lambda}|=e^{K/2}|q_{0}^{(e)}+q_{i}^{(e)}Z^{i}+{\frac{i}{2}}(q_{(m)}^{0}Z^{i}-q_{(m)}^{i})f_{i}-iq_{(m)}^{0}f|\,
\left(\begin{array}{c}{{\calQ}_{e}^{I\\prime}}\\{{\calQ}_{m}^{I\\prime}}\\\end{array}\right)=\left(\begin{array}{cc}{\frac{d}{\sqrt{c^{2}+d^{2}}}}&{\frac{c}{\sqrt{c^{2}+d^{2}}}}\\{}&{}\\{\frac{-c}{\sqrt{c^{2}+d^{2}}}}&{\frac{d}{\sqrt{c^{2}+d^{2}}}}\\\end{array}\right)\left(\begin{array}{c}{{\calQ}_{e}^{I}}\\{{\calQ}_{m}^{I}}\\\end{array}\right)\,,
M=\frac{q_{0}^{-1}}{6H}\left(\begin{matrix}{{(\frac{{\partial^{2}}V}{\partial{T_{1}}^{2}})}_{0}}&{{(\frac{{\partial^{2}}V}{\partial{T_{1}}\partial{T_{2}}})}_{0}}\\{{(\frac{{\partial^{2}}V}{\partial{T_{1}}\partial{T_{2}}})}_{0}}&{{(\frac{{\partial^{2}}V}{\partial{T_{2}}^{2}})}_{0}}\\\end{matrix}\right)
t(0)=0,\;\;t(\tau_{\mathrm{hor}})=\infty,\;\;t(\frac{\omega_{2}}{2})=\frac{Elr_{m}}{2\sqrt{MH_{2}}\sqrt{r_{m}^{2}-Ml^{2}}}\frac{Z[\epsilon,k]}{\sqrt{r_{m}^{2}(1-k^{2})+k^{2}Ml^{2}}},
\hat{t}=\frac{2}{2-\lambdam}t^{1-\frac{\lambdam}{2}}\;,\;\;\hat{a}=At^{n-\frac{\lambdam}{2}}\;,\;\;\hat{R}=\frac{2}{3}\;(1-3\lambda^{2})\;t^{-(2-\lambdam)}\;.
\deltaS=\int_{\mathcalM}\delta\overline{\psi}\left(i\gamma^{a}\nabla_{a}-m\right)\psi-\int_{\mathcalM}\left[i\overline{{(\nabla_{a}\psi)}}\gamma^{a}+m\overline{\psi}\right]\delta\psi+i\sum_{s}\int_{\mathcalB}\left(\overline{\psi}_{s}\gamma^{a}\delta\psi_{s}-\delta\overline{\psi}_{s}\gamma^{a}\psi_{s}\right)n_{a}^{s}\,,
\psi_{F}(x)=\sum_{\vec{k},s}\frac{1}{\sqrt{V}}e^{i[\vec{k}\cdot\vec{x}-\hbar^{-1}E(\vec{k})t]}u_{s}{(\vec{k})}c_{s}(\vec{k}),\;\;\psi_{A}(x)=\sum_{\vec{k},s}\frac{1}{\sqrt{V}}e^{i[\vec{k}\cdot\vec{x}-\hbar^{-1}E(\vec{k})t]}v_{s}{(\vec{k})}d_{s}(\vec{k}).
\rho=|\dot{\Phi}|^{2}+V\,,~~~~p=|\dot{\Phi}|^{2}-V\,,~~~~V(\Phi)=e^{\calK}\,\left(\left|\partial_{\Phi}W+{\frac{\Phi^{*}}{M_{P}^{2}}}W\right|^{2}-{\frac{3}{M_{P}^{2}}}{\left|W\right|^{2}}\right)\,.
\epsilon^{\mu\nu\lambda\rho\sigma}\hat{G}_{\mu}^{(n)}\hat{G}_{\nu}^{(n)}\hat{G}_{\lambda}^{(n)}\hat{G}_{\rho}^{(n)}=\epsilon^{\mu\nu\lambda\rho\sigma}\hat{G}_{\mu\nu}^{(n)}\hat{G}_{\lambda\rho}^{(n)}=(8n+16)\hat{G}_{\sigma}^{(n)},
\sigma_{0}\;\equiv\;\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)\;\;\sigma_{1}\;\equiv\;\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\;\;\sigma_{2}\;\equiv\;\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right)\;,
\int_{x}e^{-ipx}\langle\Psi_{R}^{i}(0)\bar{\psi}_{L}^{j}(x)\rangle=\int_{x}e^{-ipx}\left({\frac{1}{a}}\right)\langle\psi_{R}(0)\bar{\psi}_{L}^{j}(x)\rangle\langle\psi_{L}^{i}(0)\bar{\psi}_{R}(0)\rangle+O\left({\frac{\tilde{g}_{2}}{N_{f}}}\right),
u=\left(\begin{matrix}{1}\\{p}\\\end{matrix}\right),\;\;v=\left(\begin{matrix}{1}\\{q}\\\end{matrix}\right),\;\;w=\left(\begin{matrix}{1}\\{r}\\\end{matrix}\right),\;\;u^{\prime}=\left(\begin{matrix}{1}\\{p^{\prime}}\\\end{matrix}\right),\;\;v^{\prime}=\left(\begin{matrix}{1}\\{q^{\prime}}\\\end{matrix}\right),\;\;w^{\prime}=\left(\begin{matrix}{1}\\{r^{\prime}}\\\end{matrix}\right).
{\calL}_{f}=\sqrt{2}i(\alpha^{\prime}p^{+})\left\{\theta_{+}^{\dagger}(\partial_{\tau}+\partial_{\sigma})\theta_{+}+\theta_{-}^{\dagger}(\partial_{\tau}-\partial_{\sigma})\theta_{-}+2\theta_{+}^{\dagger}\tilde{\Omega}_{-}\theta_{-}+2\theta_{-}^{\dagger}\tilde{\Omega}_{+}\theta_{+}\right\}
c_{\alpha\beta\gamma}(x(u))=\frac{\partial^{3}F(x(u))}{\partialx^{\alpha}\partialx^{\beta}\partialx^{\gamma}}=\sum_{i=1}^{n}\frac{a_{i\alpha}(u)a_{i\beta}(u)a_{i\gamma}(u)}{a_{i1}(u)}
\chi_{k}^{\pi}=\frac{\Gamma\left(-n_{s}^{\pi}\right)}{2\sqrt{\pi}}k\sqrt{\eta}\left(\frac{k\eta}2\right)^{n_{s}^{\pi}}\left[1-\frac{(k\eta)^{2}}{2(2+2n_{s}^{\pi})}\right]-i\frac{\Gamma\left(n_{s}^{\pi}\right)}{2\sqrt{\pi}}k\sqrt{\eta}\left(\frac{k\eta}2\right)^{-n_{s}^{\pi}}\left[1-\frac{(k\eta)^{2}}{2(2-2n_{s}^{\pi})}\right],
{\frac{\ddot{a}_{0}}{a_{0}}}=-{\frac{16\pi^{2}G_{5}^{2}}{9c^{6}}}(2\varrho_{\mathrm{tot}}^{2}c^{4}+\sigma\varrho_{\mathrm{tot}}c^{2}+3\sigma\wp_{\mathrm{tot}}+3\wp_{\mathrm{tot}}\varrho_{\mathrm{tot}}c^{2})-c^{2}{\frac{a_{R\,0}^{\prime\prime}}{a_{0}}}+{\frac{16\pi^{2}G_{5}^{2}}{9c^{6}}}\sigma^{2},
S_{F^{2}}=\int\!\mathrm{d}^{10}x\,e\big[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-8\,\bar{\chi}\not\!\!D(\omega)\chi+2\,\bar{\chi}\Gamma^{\mu}\Gamma^{\nu\rho}\psi_{\mu}\,F_{\nu\rho}\,\big]
\frac{1}{\mu(q)}=\frac{1}{\mu}+\frac{1}{4\mu^{2}c^{2}}\left\{2\left(1+\frac{\mu}{m}+4A-4B\right)U(q)-\left(1-\frac{3\mu}{m}\right)q\frac{dU}{dq}\right\}
p(u)={\frac{\kappa^{2}\pi}{\tilde{L}^{2}}}\sum_{k=1}^{Q_{1}Q_{5}}\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}e^{-2\pii(n+m)(u+kL)/\tilde{L}}:\alpha_{m}\alpha_{n}: