harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S_{\mathrm{ct}}=-\frac{1}{8\piG}\intd^{2}x\sqrt{-h}\frac{c_{0}}{l_{\mathrm{eff}}},\\\c_{0}=\sqrt{\frac{2N}{3N-2}},\\\\frac{1}{l_{\mathrm{eff}}}=\sqrt{\Lambda}e^{b\phi/2},
C=\left\langle\sum_{X_{i}\in({\itG}_{1})_{X_{i^{\prime}}^{\prime}}}\sum_{\stackrel{i_{\alpha_{1}}<j_{\alpha_{1}}<k_{\alpha_{1}}}{\left\{i_{\alpha_{1}},j_{\alpha_{1}},k_{\alpha_{1}}\right\}\inJ_{X_{i}}}}t_{i_{\alpha_{1}}}t_{j_{\alpha_{1}}}t_{k_{\alpha_{1}}}{\bf1}_{(w(t_{i_{\alpha_{1}}})=w(t_{j_{\alpha_{1}}})=w(t_{k_{\alpha_{1}}}))}\right.\times
\frac{\langleWW\rangle_{s}}{\langleW\rangle^{2}}\sim\sum_{k,A}{\frac{\alpha_{0}}{B_{k}}}\left({\frac{2k}{(k+1)\alpha^{\prime}}}\right)^{2}Y_{A}^{(k)}(\theta)^{2}{\frac{a^{2k}}{L^{2k}}}=\sum_{k,A}{2^{k}\pik}\,{\frac{g_{s}}{N}}\,Y_{A}^{(k)}(\theta)^{2}{\frac{a^{2k}}{L^{2k}}}.
\left.m(b,b^{\prime})=\sum_{n=0}^{\infty}\frac{1}{n\,!}\,\sum_{i,j=1}^{n}\,\sum_{b\subsetB(X_{i}),b^{\prime}\subsetB(X_{j})}\,\psi_{c}(X_{1}...X_{n})\,\times\left[\prod_{k\neqi,j}\,K(X_{k},sD)\right]\frac{\partial}{\partialD_{b}}K(X_{i},sD)\,\frac{\partial}{\partialD_{b^{\prime}}}K(X_{j},sD)\.\right.
\Omega:\qquad\begin{array}{lcl}{\alpha_{n}^{0}}&{\longrightarrow}&{(-1)^{n}\alpha_{n}^{0}}\\{\alpha_{n}^{1,\ldots,9}}&{\longrightarrow}&{-(-1)^{n}\alpha_{n}^{1,\ldots,9}\,,}\\\end{array}
\left(\delta^{ij}\partial_{i}\partial_{j}+H_{6}\delta^{pq}\partial_{p}\partial_{q}\right)H_{2}+\frac{1}{2}H_{6}\delta^{ij}\partial_{i}b\partial_{j}b\delta(x^{6})\cdots\delta(x^{9})+\kappaT_{2}N\delta(x^{3})\cdots\delta(x^{9})=0\,,
\begin{array}{rcl}{}&{}&{k_{i}X_{j}^{+}k_{i}^{-1}=q_{i}^{1/2}X_{j}^{+}~,}\\{}&{}&{X_{i}^{+}X_{j}^{-}-q_{i}^{a_{ij}/2}X_{j}^{-}X_{i}^{+}=\delta_{ij}\displaystyle\frac{1-k_{i}^{-2}}{1-q_{i}^{-1}}~.}\\\end{array}
{\calL}=-\partial_{\mu}h^{\mu\nu}\partial_{\alpha}h_{\nu}^{\alpha}+\frac{1}{2}\partial_{\alpha}h^{\mu\nu}\partial^{\alpha}h_{\mu\nu}+\partial_{\mu}h^{\mu\nu}\partial_{\nu}h_{\alpha}^{\alpha}-\frac{1}{2}\partial_{\alpha}h_{\mu}^{\mu}\partial^{\alpha}h_{\nu}^{\nu}-\frac{M^{2}}{2}\left(h_{\mu\nu}h^{\mu\nu}-h_{\mu}^{\mu}h_{\nu}^{\nu}\right).
\Pi_{N}^{\mu}\Pi_{N}^{\mu}=-\partial_{\mu}\partial_{\mu}+\frac{3a^{\mu}}{a^{\sigma}a^{\sigma}}\partial_{\mu}+\frac{a^{\mu}a^{\nu}}{a^{\sigma}a^{\sigma}}\partial_{\mu}\partial_{\nu}+\frac{1}{a^{\sigma}a^{\sigma}}(\frac{9}{4}-c^{2})
\int_{0}^{\infty}dx\,\left(x^{2}+u^{2}\right)^{\nu-1}e^{-px}=\frac{\sqrt{\pi}}{2}\left(\frac{2u}{p}\right)^{\nu-\frac{1}{2}}\Gamma\left(\nu\right)\left[H_{\nu-\frac{1}{2}}\left(up\right)-N_{\nu-\frac{1}{2}}\left(up\right)\right],
\langle\vec{r}t\vert\vec{r}_{0}0\rangle=\bigl[{\calK}[H_{0},4u^{2},4u^{2}](\vec{u}t;\vec{u}_{0}0)+{\calK}[H_{0},4u^{2},4u^{2}](\vec{u}t;\vec{u}_{0}0)\bigr]
\dot{\calP}(t)=\frac{d}{dt}{\calP}(t)=\frac{\Gamma_{R}}{\hbar}e^{-i\Gamma_{R}t/\hbar}\,;\,\,\dot{\calP}_{\eta}(t)=\frac{\Gamma_{R\,\eta}}{\hbar}e^{-i\Gamma_{R}t/\hbar}\,;\,\,t\geq0\,,
\phi_{0}(x)={\frac{1}{\sqrt{2\pi}}}\left(q+{\frac{1}{\sqrt{2}}}\sum_{k=1}^{\infty}{\frac{1}{\sqrt{k}}}[(a_{(+)k}+a_{(-)k}^{\dagger})e^{-ikx}+(a_{(+)k}^{\dagger}+a_{(-)k})e^{ikx}]\right),
\begin{array}{cc}{E_{n_{a},n_{b}}^{\mathrm{osc}}=}&{\omega_{+}(n_{a}+{1}/{2})+\omega_{-}(n_{b}+{1}/{2})=}\\{\;}&{=\left[(\sqrt{({\calE}+\kappa)^{2}-4\kappa}(n_{1}+1/2)-(\sqrt{({\calE}+\kappa)^{2}-4\kappa}+\kappa-{\calE})l\right]/\theta\;.}\\\end{array}
R_{\mu\nu}\left(W\right)=\partial_{\rho}W_{\mu\nu}^{\rho}-\partial_{\nu}W_{\mu\rho}^{\rho}+W_{\alpha\rho}^{\rho}W_{\mu\nu}^{\alpha}-W_{\mu\rho}^{\alpha}W_{\alpha\nu}^{\rho}\,\,,
G_{l+\nu}(r^{\prime\prime},r^{\prime};E)=\sum_{n=0}^{\infty}\,\prod_{\alpha=1}^{n}\left[\int_{0}^{\infty}dr_{\alpha}V(r_{\alpha})\right]\,\prod_{\beta=0}^{n}\left[G_{l+\nu}^{(0)}(r_{\beta+1},r_{\beta};E)\right]\;.
{\calV}=m^{2}g^{2}\int\frac{d^{D}k}{(2\pi)^{D}}\int\frac{d^{D}q}{(2\pi)^{D}}\frac{1}{k^{2}}\frac{1}{q^{2}}\frac{1}{(k-p)^{2}}\frac{1}{(k+q)^{2}}\quad,\quad
\left[-{\frac{d^{2}}{dr^{2}}}-{\frac{d-1}{r}}{\frac{d}{dr}}+\mu^{2}-mE\right]R(r)-{\frac{g^{2}}{8{\pi}\Omega_{d}a^{d-1}(\mu_{2}^{2}-4mE)}}\delta(r-a)R(a)=0
\partial^{\mu}\,\langle0\vert\,T(\,\,N_{3}\,[\,\,J_{_{\mu\,5}}^{^{BPHZ}}\,]\,\,X\,\vert0\rangle\,=\,\,r\,\langle0\,\vert\,T\,(\,N_{4}\,[\,F_{\mu\nu}\,{\tilde{F}}^{\,\mu\nu}\,\,]\,X\vert\,0\,\rangle
T_{\mu\nu}(x)=\sum_{a=1}^{3}F_{\mu\nu}^{a}(x)\widehat{\phi}^{a}(x)-\frac{1}{\lambda}\sum_{a,b,c=1}^{3}\epsilon^{abc}\widehat{\phi}^{a}(x)(D_{\mu}\widehat{\phi}(x))^{b}(D_{\nu}\widehat{\phi}(x))^{c},
W_{n_{1}n_{2}m}^{l}(\delta_{1},\delta_{2})=(-1)^{n_{1}+\frac{m-|m|}{2}}C_{\frac{n+\delta_{2}-1}{2},\frac{|m|+n_{2}-n_{1}+\delta_{2}}{2};\frac{n+\delta_{1}-1}{2},\frac{|m|+n_{1}-n_{2}+\delta_{1}}{2}}^{l+\frac{\delta_{1}+\delta_{2}}{2},|m|+\frac{\delta_{1}+\delta_{2}}{2}}.
2\,\sqrt{\left(\bar{z}_{+}\,\,\bar{z}_{-}\right)\Big(\begin{array}{cc}{\mathrm{e}^{V}}\\\end{array}\Big)\left(\begin{array}{c}{z_{+}}\\{z_{-}}\\\end{array}\right)\left({z}_{-}\,\,-{z}_{+}\right)\Big(\begin{array}{cc}{\mathrm{e}^{-V}}\\\end{array}\Big)\left(\begin{array}{c}{\bar{z}_{-}}\\{-\bar{z}_{+}}\\\end{array}\right)}\,.
S=\frac{1}{4\pi\alpha^{\prime}}\int_{\Sigma}d^{2}z\left(g_{ij}{\partial}_{a}x^{i}{\partial}^{a}x^{j}-2\pii\alpha^{\prime}B_{ij}\epsilon^{ab}{\partial}_{a}x^{i}{\partial}_{b}x^{j}\right).
V_{e}\,=\,-2i\pi\,\hat{\theta}_{e}\,\intdp_{0}^{\prime}dp_{0}\delta(p_{0}^{\prime}\!-\!\mu_{e})\beta_{1}\beta_{2}K_{e}^{T}(p_{0}^{\prime},p_{0})\delta(p_{0}\!-\!\mu_{e})\,\hat{\theta}_{e},
x_{m}={\displaystyle\frac{-3\,m-\sqrt{9\,m^{2}-8\,q^{2}\,b}}{4q^{2}\,A}},\,x_{0}=0,\,x_{p}={\displaystyle\frac{-3\,m+\sqrt{9\,m^{2}-8\,q^{2}\,b}}{4q^{2}\,A}},
Z[j,\eta,\bar{\eta}]=\int[dM][d\psi][d\bar{\psi}]exp[-\sum_{n=0}^{\infty}(j_{n}TrM^{n}+\eta_{n}Tr\bar{\psi}M^{n}+\bar{\eta}_{n}Tr\psiM^{n})]exp[-Tr\bar{\psi}\psi]
A_{n}(k_{i},\epsilon_{i})=\int\mathrm{d}\mu\prod_{i=1}^{n}\mathrm{d}^{2}z_{i}\sum_{s}\eta_{s}Q_{s}\langle\prod_{i=1}^{n}V(k_{i},\epsilon_{i};z_{i},\bar{z}_{i})\rangle_{s}\times(\mathrm{right~part}),
\delta^{\mu\nu}=\sum_{i,j=2}^{N-1}{S_{i}^{\mu}}^{\dagger}[\delta_{ij}+a_{i}M_{ij}a_{j}^{\dagger}]S_{j}^{\nu}\equiv\sum_{i,j=2}^{N-1}{S_{i}^{\mu}}^{\dagger}K_{ij}S_{j}^{\nu}
{\phi}^{*(a-\bar{\varepsilon}_{\mu},a)}(z)=G_{a}^{-1}{\phi}_{(a-\bar{\varepsilon}_{\mu},a)}(-z-\frac{n}{2}),~~~~{\phi}_{(a+\bar{\varepsilon}_{\mu},a)}^{*}(z)=G_{a}{\phi}^{(a+\bar{\varepsilon}_{\mu},a)}(-z-\frac{n}{2}).
Q=\frac{\sqrt{R^{2}+D^{2}}}{(1+{R^{\prime}}^{2})^{3/2}},\;\;P=\frac{RR^{\prime}}{\sqrt{(1+{R^{\prime}}^{2})(R^{2}+D^{2})}},\;\;V=D^{2}\frac{\sqrt{1+{R^{\prime}}^{2}}}{(R^{2}+D^{2})^{3/2}}-h
A=\left(\begin{array}{cccc}{-3}&{-1}&{-6}&{4}\\{-3}&{1}&{3}&{3}\\{1}&{0}&{1}&{-1}\\{-1}&{0}&{0}&{1}\\\end{array}\right),\;\left(\begin{array}{cccc}{-3}&{-1}&{-4}&{4}\\{-2}&{1}&{2}&{2}\\{1}&{0}&{1}&{-1}\\{-1}&{0}&{0}&{1}\\\end{array}\right),\;\left(\begin{array}{cccc}{-2}&{-1}&{-1}&{3}\\{0}&{1}&{1}&{0}\\{1}&{0}&{1}&{-1}\\{-1}&{0}&{0}&{1}\\\end{array}\right),
\times\left|H_{n-k}\left(\frac{(T+1)z+\left[\sigma_{pp}-\sigma_{qq}-2i\sigma_{pq}\right]z^{\star}}{\left\{(2T+4d+1)\left[\sigma_{pp}-\sigma_{qq}-2i\sigma_{pq}\right]\right\}^{\frac12}}\right)\right|^{2}.
(-\frac{1}{2}\frac{d^{2}}{{(dx^{1})}^{2}}+\frac{1}{2}{(\Lambda_{1}^{3})}^{2}(x^{1})^{2}+\frac{1}{(x^{1})^{2}}(8(m^{1})^{2}-4m^{1}+\frac{3}{8}))Z_{1}(x^{1})=4E_{IV}^{1}Z_{1}(x^{1}).
\widetildeY\equiv\tilde{e}_{\mu}{}^{r}e_{r}{}^{\rho}g(\tilde{e})_{\rho\nu}\,\mathrm{d}x^{\mu}\wedge\mathrm{d}x^{\nu}=-2m\,\mathrm{d}r\wedge\Sigma^{3}+\frac{1}{m}(r+2m)(r+m)\,\Sigma^{1}\wedge\Sigma^{2}\,,
H=\sum_{k}\omega_{k}A_{1,1}\left(k\right)+\sum_{n,\left\{m\right\},\left\{m^{\prime}\right\}}\sum_{k_{1},\ldotsk_{n}}^{\prime}V_{\left\{m\right\},\left\{m^{\prime}\right\}}\left(k_{1},\ldotsk_{n}\right)\prod_{i=1}^{n}A_{m_{i},m_{i}^{\prime}}\left(k_{i}\right).
\frac{\partial\Sigma^{\check{r}}(\tau,\vec{\sigma}_{o})}{\partial\tau}=-[I^{-1}(\tau,\vec{\Sigma}(\tau,\vec{\sigma}_{o}))]_{\;i}^{\check{r}}\,\frac{\partial\alpha^{i}(\tau,\vec{\Sigma}(\tau,\vec{\sigma}_{o}))}{\partial\tau}=+\frac{J^{\check{r}}(\tau,\vec{\Sigma}(\tau,\vec{\sigma}_{o}))}{J^{\tau}(\tau,\vec{\Sigma}(\tau,\vec{\sigma}_{o}))}.
a_{\alpha}({\bfx})=-\left({\frac{\hbar\theta}{e^{2}\pi}}\right)\intd^{2}x^{\prime}{\frac{\epsilon_{\alpha\beta}\bigl(x^{\beta}-x^{\prime\beta}\bigr)}{\vert{\bfx}-{\bfx}^{\prime}\vert^{2}}}\rho({\bfx}^{\prime}).
\kappa=\frac{\Psi_{1}^{\prime}(r_{0})\Psi_{3}(r_{0})-\Psi_{1}(r_{0})\Psi_{3}^{\prime}(r_{0})}{\Psi_{2}^{\prime}(r_{0})\Psi_{3}(r_{0})-\Psi_{2}(r_{0})\Psi_{3}^{\prime}(r_{0})}\.
F=\left(\begin{array}{cccccc}{0}&{\tilde{\nu}_{1}}&{0}&{0}&{0}&{0}\\{-\tilde{\nu}_{1}}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{\tilde{\nu}_{2}}&{0}&{0}\\{0}&{0}&{-\tilde{\nu}_{2}}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{\tilde{\nu}_{3}}\\{0}&{0}&{0}&{0}&{-\tilde{\nu}_{3}}&{0}\\\end{array}\right)
P_{0}=1,\quadP_{1}=-\tau,\quadP_{2}=\frac{1}{2}\,\kappa^{2},\quadP_{3}=\frac{1}{2}\,\kappa^{2}\tau,\quadP_{4}=\frac{1}{2}\,\kappa^{(1)\,2}+\frac{1}{2}\,\kappa^{2}\tau^{2}-\frac{1}{8}\,\kappa^{4},\quad\ldots,
[X_{j},[X^{j},X^{i}]]+{\frac{i}{2}}\{b^{+j},b^{i}{}_{j}-b^{+-}\delta^{i}{}_{j}\}=0,\qquad[X_{i},[X^{i},X^{-}]]-{\frac{i}{4}}\{b^{ij},b_{ij}\}=0.
1\simeq\frac{2\alpha}{\pi}T|eH|\int_{-\infty}^{\infty}dq_{3}\int_{0}^{\infty}d\hat{q}_{\perp}^{2}~\mathrm{e}^{-\hat{q}_{\perp}^{2}}\sum_{l}\frac{1}{Q_{2}^{2}+4\pi^{2}T^{2}l^{2}}~\frac{1}{Q_{1}^{2}+[\piT(2l-1)-i\mu]^{2}}
{C}_{\gamma_{1}\ldots\gamma_{4}}^{N_{1}\ldotsN_{4}\,N}w_{\gamma_{1}}\ldotsw_{\gamma_{4}}=\sqrt{\mathrm{dim}_{N}}\\left({C}_{\gamma_{1}\gamma_{2}\gamma}^{N_{1}N_{2}N}w_{\gamma_{1}}w_{\gamma_{2}}w_{\gamma}\right)\left({C}_{\gamma_{1}\gamma_{2}\gamma^{\prime}}^{N_{1}N_{2}N}w_{\gamma_{1}}w_{\gamma_{2}}w_{\gamma^{\prime}}\right)={\frac{1}{\sqrt{\mathrm{dim}_{N_{1}}\ldots\mathrm{dim}_{N_{4}}}}}.
\tilde{\rho}_{\mathrm{s}}(x)\;=\;{\frac{1}{8\pi^{2}\beta}}\int_{a^{2}}^{b^{2}}{\frac{dy}{x+y}}\sqrt{{\frac{x(b^{2}-x)(y+a^{2})}{y(b^{2}+y)(x-a^{2})}}}\;.
G(f)_{\beta}^{\mu(n)}=\Sigma_{m=0}^{n}\{\tilde{\Theta}_{\beta}^{\mu(n-m)},\tilde{f}^{(m)}\}_{(q)}+\Sigma_{m=0}^{(n-2)}\{\tilde{\Theta}_{\beta}^{\mu(n-m)},\tilde{f}^{(m+2)}\}_{(\phi)}+\{\tilde{\Theta}_{\beta}^{\mu(n+1)},\tilde{f}^{(1)}\}_{(\phi)}
\begin{array}{ll}{\alpha\gamma=q^{2}\gamma\alpha,\quad}&{\quad[\beta,\delta]=-(\lambda/q)\alpha\beta,}\\{\alpha\beta=q^{-2}\beta\alpha,\quad}&{\quad[\gamma,\delta]=(\lambda/q)\gamma\alpha,}\\{\alpha\delta=\delta\alpha,\quad}&{\quad[\beta,\gamma]=(\lambda/q)(\alpha\delta-\alpha^{2}).}\\\end{array}
H^{2}+\frac{k}{R^{2}}=\frac{8}{(\beta\ell)^{2}}\left\{1+\frac{\beta}{2}\left(1+\frac{\rho}{\sigma}\right)-\sqrt{\left(1+\frac{\beta}{2}\right)^{2}+\beta\left(\frac{\rho}{\sigma}\right)-\frac{(\beta\ell)^{2}}{4}\left(\frac{\mu}{R^{4}}-\frac{q^{2}}{R^{6}}\right)}\right\}\,.
P^{\mu}(\Sigma)+p^{\mu}(\Sigma)=\beta\int_{\Sigma}\sqrt{h}\left\{\eta^{\mu}\left[\kappa_{AB}\kappa^{AB}-\kappa^{2}+\eta^{a}\eta^{b}{\calG}_{ab}\right]+K_{AB}^{i}(h^{AB}\kappa-\kappa^{AB})n^{\mu}{}_{i}\right\}\,.
\begin{matrix}{\tilde{\psi}^{a}={\frac{e}{2}}(\psi_{\alpha})_{\dot{A}}(\bar{\sigma}^{a})^{\dot{A}\alpha},}&{\tilde{\psi}^{ab}=-e{(\bar{\sigma}^{ab})^{\dot{A}}}_{\dot{\alpha}}(\psi^{\dot{\alpha}})_{\dot{A}},}&{\tilde{\psi}=-e(\psi_{\dot{\alpha}})^{\dot{A}}\delta_{\dot{A}}^{\dot{\alpha}},}\\\end{matrix}
d\Gamma=\frac{\left(1-\frac{\lambda^{2}m^{2}}{m_{P}^{2}}\right)^{3}d^{3}p}{2\left[1-\frac{\lambda}{m_{P}c}\sqrt{m^{2}c^{2}+|\vec{p}|^{2}\left(1-\frac{\lambda^{2}m^{2}}{m_{P}^{2}}\right)}\right]^{3}\sqrt{m^{2}c^{2}+|\vec{p}|^{2}\left(1-\frac{\lambda^{2}m^{2}}{m_{P}^{2}}\right)}},
\int_{0}^{1}{\mathrm{d}}x~{\frac{x}{\sqrt{1-x^{2}}}}~K_{0}(\frac{nz~x}{2})={\frac{1}{nz}}\Big[\,{\mathrm{Ei}}\big(\frac{nz}{2}\big){\mathrm{e}}^{-{nz}/{2}}-{\mathrm{Ei}}\big(-\frac{nz}{2}\big){\mathrm{e}}^{{nz}/{2}}\,\Big],
\frac{\deltaS_{m}}{\delta\phi^{A}}\frac{\deltaS_{m}}{\delta\phi_{Aa}^{*}}+V_{m}^{a}S_{m}=0,\qquad\frac{\deltaS_{m}}{\delta\phi^{A}}\frac{\deltaS_{m}}{\delta\eta_{B}}(\sigma_{\alpha})_{B}^{~~\!A}+V_{\alpha}S_{m}=0,\qquad\frac{\deltaS_{m}}{\delta\phi^{A}}\frac{\deltaS_{m}}{\delta\eta_{B}}\gamma_{A}^{B}+V_{m}S_{m}=0,
R(\theta)=\left(\begin{array}{cccc}{1}&{0}&{0}&{0}\\{0}&{\frac{1}{(\theta+1)}}&{\frac{\theta}{(\theta+1)}}&{0}\\{0}&{\frac{\theta}{(\theta+1)}}&{\frac{1}{(\theta+1)}}&{0}\\{0}&{0}&{0}&{1}\\\end{array}\right)\;\;\;.
\deltax=\dot{x}\xi\,\,,\,\,\,\,\,\,\deltae=\frac{d}{dt}(e\xi)\,\,,\,\,\,\,\,\,\delta\psi^{n}=\dot{\psi}^{n}\xi\,\,,\,\,\,\,\,\,\delta\chi=\frac{d}{dt}(\chi\xi)\,,
\sum_{l}\left\{\begin{array}{ccc}{d}&{i}&{l}\\{e}&{m}&{c}\\\end{array}\right\}\left\{\begin{array}{ccc}{a}&{b}&{f}\\{e}&{l}&{i}\\\end{array}\right\}\left\{\begin{array}{ccc}{a}&{f}&{k}\\{d}&{d}&{l}\\\end{array}\right\}=\left\{\begin{array}{ccc}{a}&{b}&{k}\\{c}&{d}&{i}\\\end{array}\right\}\left\{\begin{array}{ccc}{k}&{b}&{f}\\{e}&{m}&{c}\\\end{array}\right\}.
K=i{g_{i}}^{j}\nablaz_{j}\wedge\nabla\bar{z}^{i}+{\frac{i}{2}}g(G_{i}k^{i\Lambda}-G^{i}k_{i}^{\Lambda})(dA_{\Lambda}+{f_{\Lambda}}^{\Sigma\Pi}A_{\Sigma}\wedgeA_{\Pi}).
\int_{\partial\Sigma_{t}}\nabla_{\nu}K_{\mu}d\Sigma^{\mu\nu}=4g^{2}\int_{\Sigma_{t}}K_{\nu}d\Sigma^{\nu}+\int_{\Sigma_{t}}(F_{\nu\rho}^{2}-\frac16g_{\nu\rho}F^{2})K^{\rho}d\Sigma^{\nu},
Z^{ij}=\alpha_{ij}^{3}|Z|=i\sigma_{2}\;\left(\begin{matrix}{|Z|}&{0}&{0}&{0}\\{0}&{|Z|}&{0}&{0}\\{0}&{0}&{|Z|}&{0}\\{0}&{0}&{0}&{|Z|}\\\end{matrix}\right)=|Z|\left(\begin{matrix}{0}&{1}&{0}&{0}\\{-1}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}\\{0}&{0}&{-1}&{0}\\\end{matrix}\right)
{\tilde{F}}_{n}={\tilde{F}}^{(0)}(S,M_{i})+{\tilde{F}}^{1}(M^{i})+{\tilde{F}}^{NP}(e^{-8\pi^{2}S},M^{i}),\;\;{\tilde{F}}_{1}=24S,\;\;{\tilde{F}}_{n\geq1}^{o}=const,
R_{j_{1}j_{2}}^{k_{1}k_{2}}R_{k_{1}j_{3}}^{l_{1}k_{3}}R_{k_{2}k_{3}}^{l_{2}l_{3}}=R_{j_{2}j_{3}}^{k_{2}k_{3}}R_{j_{1}k_{3}}^{k_{1}l_{3}}R_{k_{1}k_{2}}^{l_{1}l_{2}}
\langle{F}_{\mu\nu}(p){F}_{\rho\sigma}(-p)\rangle={\frac{1}{p^{2}}}\left(p_{\mu}p_{\rho}\delta_{\nu\sigma}-p_{\mu}p_{\sigma}\delta_{\nu\rho}-p_{\nu}p_{\rho}\delta_{\mu\sigma}+p_{\nu}p_{\sigma}\delta_{\mu\rho}\right),
Q_{\mu_{1}\dots\mu_{p}}^{\mathrm{R}}\equiv{\frac{Q_{p}^{\mathrm{R}}}{v_{p}}}\int_{{\calM}_{p}}dX^{\mu_{1}}\wedge\cdotsdX^{\mu_{p}}\,\quad\mathrm{with}\quadQ_{p}^{\mathrm{R}}={\frac{1}{\mathrm{Vol}\,S^{8-p}}}\int_{S^{8-p}}\!\!{}^{*}G^{(p+2)}\.
q_{1}v_{0}^{(1)}=\partial_{t}\omega_{1}-\frac{\kappaB_{2}}{2q_{1}\mid\phi\mid^{2}}\;\;\;,\;\;\;q_{2}v_{0}^{(2)}=\partial_{t}\omega_{2}-\frac{\kappaB_{1}}{2q_{2}\mid\chi\mid^{2}}\;\;,
\hat{\Phi}=\hat{B}^{-1/2}\hat{\varphi}\hat{B}^{-1/2}=\hat{B}^{-1/2}\left(\begin{matrix}{\hat{\varphi}^{R}}&{0}\\{0}&{\hat{\varphi}^{L}}\\\end{matrix}\right)\hat{B}^{-1/2}
\langleH_{I}\rangle^{\beta}=\frac{\mathrm{Tr}e^{-\betaH_{0}}H_{I}}{\mathrm{Tr}e^{-\betaH_{0}}}=\frac{\sum_{n}e^{-\betaE_{n}}\langlen|H_{I}|n\rangle}{\sum_{n}e^{-\betaE_{n}}}\;.
\partial_{\bar{z}}^{2}\bar{A}=\frac{1}{2}(\partial_{\bar{z}}\bar{A})^{2}e_{0}+e_{4}\partial_{\bar{z}}\bar{A}+e_{5}\,,\quad\partial_{z}^{2}A=\frac{1}{2}(\partial_{z}A)^{2}e_{0}+e_{4}\partial_{z}A+e_{5}\,,
\delta_{F}^{(1)}(k)=\delta_{+}^{(1)}(\omega_{k})+\delta_{+}^{(1)}(-\omega_{k})+\delta_{-}^{(1)}(\omega_{k})+\delta_{-}^{(1)}(-\omega_{k})=-\frac{4G^{2}v}{k}\int_{0}^{\infty}h(x)\,dx.
\left(1-\frac{w\kappa}{4}e^{2\phi}\right)\left(\delta^{2}\phi-2\delta\phi\,\delta\rho\right)-(\delta\phi)^{2}=\frac{\kappa}{8}e^{2\phi}\left[(\deltaZ)^{2}+2\delta^{2}Z-4\deltaZ\delta\rho\right];
I(p)=1+\sum_{n=1}^{\infty}{\frac{1}{n!}}{\frac{{\hat{\lambda}}^{n}}{\epsilon^{n}}}{\frac{\Gamma(1+n\epsilon)\Gamma(1-\epsilon)^{n+1}}{\Gamma(2-(n+1)\epsilon)\prod_{k=2}^{n}(1-k\epsilon)}}{\frac{1}{[p_{\phi}^{2}]^{n\epsilon}}}
H^{(1)}(q,{\bfn})-m{\quad}\rightarrow{\quad}-\frac{1}{{2m}{q}^{2}}\frac{\partial}{{\partial}{q}}{q}^{2}\frac{\partial}{{\partial}{q}}+\frac{\widetilde{\bfJ^{2}}+2[\widetilde{\tau}-(\widetilde{\bfs}\cdot\widetilde{\bfJ})\widetilde\tau-\widetilde\tau(\widetilde{\bfs}\cdot\widetilde{\bfJ})]}{{2mq^{2}}}.
(G_{OM}^{-1})_{ab}={\frac{\left((2K^{2}-1)-2K^{2}\sqrt{1-K^{-2}}\right)^{\frac{1}{6}}}{K}}\,\left[\hat{\eta}_{\hat{a}\hat{b}}+\,{\frac{1}{4}}(\hat{{\calH}}^{2})_{\hat{a}\hat{b}}\right]\,.
S^{(2)}=-g^{2}N_{p}(2\pi)^{3}\delta^{3}(p_{1}^{\prime}+p_{2}^{\prime}-p_{1}-p_{2})\{(p_{1}^{\prime}+p_{1})_{\mu}(p_{2}^{\prime}+p_{2})_{\nu}\frac{1}{m^{2}}D^{\mu\nu}(k)+p_{1}\leftrightarrowp_{2}\}
\frac{\partial^{2}\varphi_{E}(\tau)}{\partial\tau^{2}}+3\Bigl(\frac{\partiala(\tau)/\partial\tau}{a(\tau)}\Bigr)\frac{\partial\varphi_{E}(\tau)}{\partial\tau}-\frac{\delta^{2}V(\hat{\phi})}{\delta\hat{\phi}^{2}}\varphi_{E}(\tau)=0.
\langle\varphi^{2}(x)\rangle_{b}\approx-\frac{\nu_{0}\Gamma(2\nu_{0}+1/2)\Gamma(\nu_{0}+1/2)}{2^{2\nu_{0}+1}(a\sigma)^{D-1}S_{D}\Gamma^{3}(\nu_{0}+1)}\frac{A+B\nu_{0}}{A-B\nu_{0}}\left(\frac{a}{r}\right)^{2\nu_{0}+D-1}.
\psi_{-}(\vec{x})=\frac{1}{\sqrt{2}}\left(\frac{1}{i\partial_{-}-eV_{-}}\ast\xi\right)(\vec{x})+\frac{e^{2}}{2\sqrt{2}L}\left({\calG}_{(\perp)}[{\calM}^{2}]\ast\Gamma^{i}\right)(x_{\perp})\left(\frac{1}{i\partial_{-}-eV_{-}}\ast\psi_{+}\right)(\vec{x})\;,
S_{boundary}=-\frac{\sqrt{2}}{\kappa_{5}^{2}}\int_{M_{4}^{(1)}}{\sqrt{-g}V^{-1}\alpha_{i}^{(1)}b^{i}}-\frac{\sqrt{2}}{\kappa_{5}^{2}}\int_{M_{4}^{(2)}}{\sqrt{-g}V^{-1}\alpha_{i}^{(2)}b^{i}}.
i\pi_{rs}^{m}\psi_{0}(\phi)=\frac{\partial}{\partial\phi_{sr}^{m}}\;\psi_{0}(\phi)=-F_{rs}^{m}(\phi)\;\psi_{0}(\phi),\;\;\;\;\;\;(F_{rs}^{m}(\phi))^{\dagger}=F_{sr}^{m}(\phi)
\mathrm{d}\mu_{G}=\frac{e^{-8\pi^{2}/g^{2}}}{2^{\,4T_{G}+3}\,\pi^{4}}\,\frac{\Omega_{G}}{\Omega_{H}}\,(M_{\mathrm{PV}})^{\,3T_{G}}\left(\frac{8\pi^{2}}{g^{2}}\right)^{T_{G}}\!\mathrm{d}\rho^{2}\,\mathrm{d}^{4}x_{0}\,\mathrm{d}^{2}\theta_{0}\,\mathrm{d}^{2}\bar{\beta}\,\prod_{i=1}^{2T_{G}-4}\!\rho\,\mathrm{d}\bar{\xi}_{i}\,.
{\calS}_{g}=\intdt\,\sum_{j=1}^{n}\left[\Psi_{j}^{\dagger}(i\partial_{t}-A_{0})\Psi_{j}\right]+\sum_{j=n+1}^{N}\left[\Psi_{j}^{\dagger}(-i\partial_{t}+A_{0})\Psi_{j}\right]~.
ds^{2}=H^{-\frac{d-2}{D-2}b}\Big(-fdt^{2}+\sum_{i=1}^{p}(dy^{i})^{2}\Big)+H^{\frac{p+1}{D-2}b}\Big(f^{-1}dr^{2}+r^{2}d\Omega_{d-1}^{2}\Big)
\frac{1}{\sqrt{-\overline{{g}}}}\partial_{\mu}(\sqrt{-\overline{{g}}}\enspace\overline{{g}}^{\mu\nu}\partial_{\nu}\varphi)+\frac{dV_{eff}^{(p)}}{d\varphi}+e^{2}\varphig^{\alpha\beta}A_{\alpha}A_{\beta}=0
\sum_{s=1,2,5}g(s)(\varepsilon_{s})^{\mu}(\varepsilon_{s})^{\nu}=\sum_{s=1,2,5}(\varepsilon_{s})^{\mu}(\varepsilon_{s})^{\nu}=g^{\mu\nu}+\frac{1}{\kappa^{2}}k^{\mu}k^{\nu}=g^{\mu\nu}-\frac{1}{k^{2}}k^{\mu}k^{\nu}
\left({\frac{\dot{\hat{a}}_{0}}{\hat{a}_{0}}}+{\frac{\dot{\hat{c}}}{\hat{c}}}\right)^{2}={\frac{8\pi\hat{G}_{4}}{3}}\hat{\varrho}_{\mathrm{fld}}+{\frac{4\pi\omega\hat{G}_{4}}{3c^{4}}}{\frac{\dot{\hat{c}}^{2}}{\hat{c}^{2}}}+{\frac{\hat{c}^{2}}{3}}\hat{\Lambda}_{\mathrm{eff}}+{\frac{\hat{\calC}\hat{c}^{2}}{\hat{a}_{0}^{4}}}-{\frac{\hat{k}\hat{c}^{2}}{\hat{a}_{0}^{2}}},
\nu_{\alpha\beta\gamma}^{g_{1}g_{2}}\:=\:\left(\nu_{\alpha\beta\gamma}^{g_{2}}\right)\left(g_{2}^{*}\nu_{\alpha\beta\gamma}^{g_{1}}\right)\lambda_{\alpha\beta}^{g_{1},g_{2}}\,\lambda_{\beta\gamma}^{g_{1},g_{2}}\,\lambda_{\gamma\alpha}^{g_{1},g_{2}}
\tilde{C}(p)=-\gamma\,g^{2}N(N^{2}-1)\delta_{a}^{b}\delta_{J}^{I}\int\frac{d^{4}k}{(2\pi)^{4}}d^{4}\theta\frac{1}{k^{4}}\left\{\Phi_{I}^{a\dagger}(p,\theta,{\overline{\theta}}){\Phi}_{b}^{J}(-p,\theta,{\overline{\theta}})\right\}\,,
H=\intd^{3}x\left(\frac12\pi_{ia}\pi_{i}^{a}+\frac14F_{ij}^{a}F_{a}^{ij}-A_{0}^{a}D_{i}\pi_{a}^{i}+\frac12M^{2}A_{\mu}^{a}A_{a}^{\mu}\right),
\Gamma_{\nu}=\varepsilon^{\mu,[\mu\nu]}+\varepsilon^{[\mu\nu],\mu}+\varepsilon^{\nu,0}+\varepsilon^{o,\nu}+\varepsilon^{\widetilde{\nu},\widetilde{0}}+\varepsilon^{\widetilde{0},\widetilde{\nu}}+\frac12e_{\mu\nu\rho\omega}\left(\varepsilon^{\widetilde{\mu},[\rho\omega]}+\varepsilon^{[\rho\omega],\widetilde{\mu}}\right).
\begin{array}{rcl}{U_{g}/m^{4}}&{=}&{\displaystyle+6D_{m}(k_{1})D(k_{1}+p)D_{m}(k_{2})D(k_{2}+p)D_{m}(k_{1}-k_{2})(1+K_{12})}\\{}&{}&{\displaystyle+4D_{m}(k_{1})D(k_{1}+p)D(k_{2}+p)D_{m}(k_{2})D(k_{1}+k_{2}+p)K_{12},}\\\end{array}
{\calL}=\bar{\psi}(i\!\not{\!\partial}+J)\psi+\frac12\sigmaiG_{\sigma}^{(0)-1}\sigma+\frac12A^{\mu}iD_{\mu\nu}^{(0)-1}A^{\nu}+\frac{1}{\sqrt{N}}\bar{\psi}\not{\!\!A}\psi-\frac{1}{\sqrt{N}}\sigma\bar{\psi}\psi\,
\begin{array}{c}{C_{0}^{\prime}=0\,,}\\{C_{1}^{\prime}+G_{1}(r_{1})C_{0}=0\,,}\\{\dots}\\{C_{k+1}^{\prime}+C_{k}^{\prime\prime}+G_{1}(r_{1})C_{k}=0\,,}\\{\dots}\\\end{array}
{\calL}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{{e^{2}}}{{4\pi}}F_{\mu\nu}\frac{1}{{\partial^{2}+4\pim\Sigma}}F^{\mu\nu}+em\Sigma\frac{1}{{\partial^{2}+4\pim\Sigma}}\theta\varepsilon_{\mu\nu}F^{\mu\nu}-A_{0}J^{0}.
S=\intd^{4}x\sqrt{-g}[\frac{\mathcal{R}}{16\piG}-\frac{1}{4\pi}((D_{\mu}\Phi)^{\dagger}(D^{\mu}\Phi)+V(\Phi)+\xi\Phi^{\dagger}\Phi\mathcal{R})-\frac{1}{4\pi}\frac{1}{4}\midF\mid^{2}].
m^{2}(t)=\frac{2a}{(4\pi)^{2}}y_{\Lambda}^{2}(t)\left(exp\left\{\frac{(\mu_{0}/\Lambda)^{\epsilon}-(\mu_{0}/\mu)^{\epsilon}}{\epsilon}\right\}\right)^{\alpha/\alpha_{c}}\Lambda^{2}\left[\frac{1}{g_{4}(\Lambda)}-\frac{1}{w}\right]
\Gamma_{\,\,\,\alpha\beta}^{\sigma}=\frac{q^{2}}{2A^{2}m^{2}}\tilde{g}^{\sigma\nu}[F_{\,\,\,\nu}^{\lambda}(F_{\lambda\alpha,\beta}+F_{\lambda\beta,\alpha})+F_{\,\,\,\alpha}^{\lambda}F_{\beta\nu,\lambda}+F_{\,\,\,\beta}^{\lambda}F_{\alpha\nu,\lambda}]\,
\begin{array}{ll}{a(\partiald)=(\partiald)a+\mu(\partialb)c+\mub(\partialc),}&{c(\partiald)=q(\partiald)c+\mud(\partialc),}\\{b(\partiald)=q(\partiald)b+\muq(\partialb)d,}&{d(\partiald)=q^{2}(\partiald)d.}\\\end{array}
{\calC}_{k}^{\prime\prime}+\biggl[k^{2}-V_{k}(\eta)\biggr]{\calC}_{k}=0,~~~V_{k}(\eta)=\frac{k^{2}r^{2}}{1+r^{2}}\biggl(\frac{b^{2}-a^{2}}{b^{2}}\biggr)+\frac{{\calG}^{\prime\prime}}{{\calG}}
\left(G_{4}\varrho_{m}a^{3(w+1)}\right)^{.}\approx{\frac{8\pi\sigma_{0}\delta\sigmaG_{5}^{2}a^{3(w+1)}}{3b_{0}c^{6}}}\left(3{\frac{\dot{c}}{c}}-{\frac{\dot{G}_{5}}{G_{5}}}\right)+{\frac{3kc\dot{c}a^{3w+1}}{4\pi}}.
\left[A,R_{1},R_{2},R_{3},L_{1},L_{2}\right]=\frac{3}{2}\left(i\hbar\right)^{2}\left[\left\{A,L_{3}\right\},I_{R}\right]+\frac{1}{2}\left(i\hbar\right)^{2}\sum_{i}\left[\,\left[\left[A,R_{i}\right],R_{i}\right]\,,\,L_{3}\,\right]\;.
T=\left(\begin{array}{cccc}{0}&{x_{3}}&{-x_{2}}&{-x_{1}}\\{-x_{3}}&{0}&{x_{1}}&{-x_{2}}\\{x_{2}}&{-x_{1}}&{0}&{-x_{3}}\\{x_{1}}&{x_{2}}&{x_{3}}&{0}\\\end{array}\right)
X^{m}(\sigma^{\pm2},\eta^{\pm})=X^{m}(\sigma^{\pm2})+\frac{i}{\sqrt{8}}\eta^{+}\psi_{+}^{m}(\sigma^{\pm2})+\frac{i}{\sqrt{8}}\eta^{-}\psi_{-}^{m}(\sigma^{\pm2})+i\eta^{+}\eta^{-}F^{m}(\sigma^{\pm2}),
T^{+2}=-2ie_{q}^{+}e_{q}^{+}-2i\pi_{q}^{2+}\pi_{q}^{2+},\quadT^{-2}=-2i\pi_{\dot{q}}^{1-}\pi_{\dot{q}}^{1-}-2ie_{\dot{q}}^{-}e_{\dot{q}}^{-}
\hat{\calA}_{2}=-g{\frac{(-1)^{n+1}}{2^{2n-1}n!}}({\frac{g}{\pi}})^{n}\epsilon_{\mu_{1}\nu_{1}...\mu_{n}\nu_{n}}\theta^{\alpha\beta}\partial_{\beta}(\hat{A}_{\alpha}\hat{F}_{\mu_{1}\nu_{1}}...\hat{F}_{\mu_{n}\nu_{n}}).
i\hbar\frac{\partial}{\partialt}\bigl<q|\phi(t)\bigr>=\Bigl[-\hbar^{2}X(t)\frac{\partial^{2}}{\partialq^{2}}-i\hbarY(t)\bigl(q\frac{\partial}{\partialq}+\frac{\partial}{\partialq}q\Bigr)+Z(t)q^{2}\Bigr]\bigl<q|\phi(t)\bigr>.
\begin{array}{rcl}{r}&{=}&{r_{-}\equiv-n-\sqrt{4n^{2}-1}\,,}\\{x}&{=}&{x_{\pm}\equiv\alpha_{1}\displaystyle\frac{\alpha_{3}\pm||\alpha||}{(\alpha_{1}^{2}+\alpha_{2}{}^{2})}\,,}\\{y}&{=}&{y_{\pm}\equiv\displaystyle-\alpha_{2}\frac{\alpha_{3}\pm||\alpha||}{(\alpha_{1}^{2}+\alpha_{2}^{2})}\,.}\\\end{array}
S=\pi\left\{(m^{2}-|{\calZ}_{1}|^{2})+(m^{2}-|{\calZ}_{2}|^{2})+2\sqrt{(m^{2}-|{\calZ}_{1}|^{2})(m^{2}-|{\calZ}_{2}|^{2})-J^{2}}\,.\right\}\,.
p_{bj}(z_{1},z_{2})=-\frac{\eta^{4}J_{3}(B_{1},B_{2})}{4\alpha_{betw}^{4}\pi^{3/2}a^{4}\Gamma(3/2)}=-\frac{\eta^{4}\Lambda_{betw}^{2}J_{3}(B_{1},B_{2})}{36\pi^{3/2}a^{4}\Gamma(3/2)}.
S_{F}(p)\Gamma_{\nu}(p,k)S_{F}(p+k)D^{\mu\nu}(k)=\intd^{D}x\intd^{D}y\,\,e^{ik\cdotx+ip\cdoty}\langleO\vertT\left[\psi(0)A_{\mu}(x)\bar{\psi}(y)\right]\vertO\rangle
{\eta_{\beta}^{A}\to{\frac{1}{\sqrt{\rho_{0}}}}\left(\rho_{0}\eta_{\beta}^{A}+(x-x_{0})_{\mu}\sigma_{\beta\dot{\beta}}^{\mu}\bar{\xi}^{\dot{\beta}A}\right)\equiv\left({\frac{1-\gamma^{5}}{2}}\right){\frac{1}{\sqrt{\rho_{0}}}}\gamma_{M}z^{M}\zeta^{(0)},}
V_{D}^{\prime}=\left(\begin{array}{ccc}{C}&{0}&{0}\\{0}&{U^{\prime}}&{0}\\{0}&{0}&{{\bar{U}^{\prime}}}\\\end{array}\right),~~~~~{\calO}=\frac{1}{\sqrt3}\left(\begin{array}{ccc}{1}&{1}&{1}\\{\alpha^{*}}&{\alpha}&{1}\\{\alpha}&{\alpha^{*}}&{1}\\\end{array}\right).
lA_{1}+nB_{1}=b_{l}^{n}-(k-1)U_{l}^{n}Str\left(\ldots\phi^{i_{2k-1}^{\prime}}\ldots\underbrace{\phi^{i_{2j}^{\prime}}\phi^{i_{2j-1}^{\prime}}\phi^{i_{2k}^{\prime}}}\ldots\right).