harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left\langle\sigma\right\rangle=-\frac{1}{m_{H}^{2}}\left[2g^{2}\eta\left(2F(m_{W}^{2})+\alphaF(\alpham_{W}^{2})\right)+2\lambda\etaF(\alpham_{W}^{2})+3\lambda\etaF(m_{H}^{2})-2\alphag^{2}\etaF(\alpham_{W}^{2})\right].
S_{k}=\Phi_{B_{m}}^{\dot{A}_{n}}\left(i\partial_{\beta_{m+1}\dot{\alpha}_{n}}\Lambda_{\dot{A}_{n-1}}^{B_{m+1}}+i\partial_{\dot{\alpha}_{1}}^{\beta_{1}}\cdotsi\partial_{\dot{\alpha}_{n}}^{\beta_{n}}\Gamma^{\beta_{n+1}\cdots\beta_{m}}\right)\,
\chi_{S}^{(3)}\left(e_{i},e_{j}\right)={\frac{1}{24\pii}}\oint_{\calC}\left[{\frac{1}{2}}\left(e_{i}e_{j}^{\prime\prime\prime}-e_{i}^{\prime\prime\prime}e_{j}\right)+T^{S}\left(e_{i}e_{j}^{\prime}-e_{i}^{\prime}e_{j}\right)\right].
f(\theta)=-\sqrt{{\frac{2}{\pik}}}e^{-i[\alpha]\theta}\left\{e^{-i\pi\alpha/2}\sum_{n=0}^{\infty}e^{in\theta}A_{n}^{+}(\alpha)+e^{i\pi\alpha/2}\sum_{n=1}^{\infty}e^{-in\theta}A_{n}^{-}(\alpha)\right\},
(\tau,\mathrm{e})>(\tau^{\prime},\mathrm{e}^{\prime})\\\\mathrm{iff}\\\\tau-\tau^{\prime}>2{\mathrm{Arcsin}}\left(\frac{(\mathrm{e}-\mathrm{e}^{\prime})^{2}}{4}\frac{r^{2}}{r^{2}+1}\right)^{\frac{1}{2}}.
\begin{array}{lcl}{{\widehat\Sigma}_{m}^{(r,s)}}&{=}&{\oplus_{k=r}^{s}{\widehat\Sigma}_{m}^{(k,k)}(\theta)}\\{}&{=}&{\oplus_{k=r}^{s}\oplus_{l=0}^{k}{\theta}^{l}{\Sigma}_{m}^{(k-l,k-l)}}\\\end{array}
\begin{array}{l}{\qquad{\tilde{X}}^{\mu}\equivx^{\mu}-\frac12\frac{(P_{y}Y)}{(P_{y}p)}P_{y}^{\mu}={\bfX}^{\mu}-\frac1{\pi}{\bfP}_{\tau}^{\mu}+(oscillators),}\\{\qquad{\tilde{P}}^{\mu}\equivp^{\mu}+(P_{y}p)P_{y}^{\mu}={\bfP}^{\mu}+(oscillators),}\\{\qquad{\bfX}^{\mu}\equivX^{\mu}-\frac12\frac{(P_{y}Y)}{(P_{y}P)}P_{y}^{\mu},\qquad{\bfP}^{\mu}\equivP^{\mu}+(P_{y}P)P_{y}^{\mu},}\\\end{array}
\begin{array}{c}{\left[a(g),a^{*}(f)\right]=\left(g,f\right)1,\;\;\;\mathrm{with~}\left(g,f\right)=\int\bar{g}(\vec{x})f(\vec{x})d^{3}x}\\{\left[a(f),a(g)\right]=0=\left[a^{*}(g),a^{*}(f)\right]}\\\end{array}
\zeta_{-1/2}({\bfp};\alpha,{\bfn})={A}({\bfp};{\bfn})\,\xi^{(0)}({\bfp},\alpha,{\bfn}){\chi}_{-1/2},\quad\chi_{-1/2}=\left(\begin{array}{ccc}{0}\\{1}\\\end{array}\right),
\tilde{\Psi}^{+}=-i\frac1r(2k+1)^{-1/2}\left(\begin{array}{c}{(k+m+1/2)^{1/2}Y_{k}^{m-1/2}(\vartheta,\varphi)B(r)}\\{(k-m+1/2)Y_{k}^{m+1/2}(\vartheta,\varphi)B(r)}\\\end{array}\right)
\intd\deltaZd\delta{\bar{Z}}~e^{-|\deltaZ|^{2}/2}\equiv1=\int[dz_{0}][d{\bar{z}}_{0}]\prod_{n_{1},n_{2}}\intdz_{n_{1}n_{2}}d{\bar{z}}_{n_{1}n_{2}}dz_{n_{1}0}d{\bar{z}}_{n_{1}0}~J~e^{-|\deltaZ|^{2}/2}\quad,
\langle\zeta^{\prime}|p_{\mu}|\zeta^{\prime\prime}\rangle=-i\hbar\frac{\partial}{\partial\eta^{\prime\prime}{}^{\mu}}\langle\zeta^{\prime}|\zeta^{\prime\prime}\rangle+\frac{\partialF(\zeta^{\prime\prime})}{\partial\eta^{\prime\prime}{}^{\mu}}\langle\zeta^{\prime}|\zeta^{\prime\prime}\rangle\,,
\begin{array}{rll}{f_{0}}&{=}&{\tilde{f}_{0}^{-1}\;,}\\{f_{1}f_{0}^{-1/2}}&{=}&{\tilde{f}_{2}\tilde{f}_{0}^{-1/2}\;,}\\{f_{2}f_{0}^{-1/2}}&{=}&{\tilde{f}_{1}\tilde{f}_{0}^{-1/2}\;.}\\\end{array}
S_{\left\langlep,\phi\right\rangle}^{\left\langleq,\psi\right\rangle}=\frac{1}{|\Omega_{p}||\Omega_{q}|}\sum_{z\in\Omega}\sum_{x,y\in\Omega_{p}\cap\Omega_{zq}}\overline{{\phi}}(x,y)\overline{{\psi}}^{z}(y,x)\prod_{\xi\in\mathcal{O}(x,y)}\Lambda_{p(\xi)}^{zq(\xi)}\left(\frac{\kappa_{\xi}}{\lambda_{\xi}}\right)
\begin{aligned}{\big(M^{(8)},M^{(8)},Z^{(2|1)}\big)}&{{}\rightarrowm=1\,,}&{\big(M^{(5)},M^{(5)},Z^{(2|1)}\big)}&{{}\rightarrown=\frac{1}{2}\,,}\\{\big(M^{(8)},M^{(8)},Z^{(4|2)}\big)}&{{}\rightarrow\mu=2\,,}&{\big(M^{(5)},M^{(5)},Z^{(4|2)}\big)}&{{}\rightarrow\nu=1\,.}\\\end{aligned}
[{\calH}({\bfx}),{\calH}({\bfx^{\prime}})]=(\Lambda({\bfx}){\calH}^{k}({\bfx})+\Lambda({\bfx^{\prime}}){\calH}^{k}({\bfx^{\prime}}))\delta,_{k}({\bfx}-{\bfx^{\prime}})
\bar{M}\to\bar{\Omega}\bar{M}\bar{\Omega}^{T},\\\\bar{{\calA}}_{\bar{\mu}}^{\bar{i}}\to\bar{\Omega}_{\bar{i}\bar{j}}\bar{{\calA}}_{\bar{\mu}}^{\bar{j}},\\\h_{\bar{\mu}\bar{\nu}}\toh_{\bar{\mu}\bar{\nu}},\\\\bar{B}_{\bar{\mu}\bar{\nu}}\to\bar{B}_{\bar{\mu}\bar{\nu}},\\\{\bar{\varphi}}\to{\bar{\varphi}},
(\tilde{\calL}_{c}\omega^{i})_{\mu}=2c^{\tau}F_{[\tau\mu]}^{i},~{}~\tilde{\calL}_{c}\Sigma_{\mu\nu}^{i}=2D_{[\mu}c^{\sigma}\Sigma_{\sigma\nu]}^{i}+3c^{\sigma}D_{[\mu}\Sigma_{\sigma\nu]}^{i}.
E^{\prime}=\frac{1}{-{g^{\prime}}^{tt}}\left\{-{g^{\prime}}^{t\phi}p_{\phi}\pm\left[({g^{\prime}}^{t\phi}p_{\phi})^{2}+(-{g^{\prime}}^{tt})(\mu^{2}+{g^{\prime}}^{\phi\phi}p_{\phi}^{2}+{g^{\prime}}^{rr}p_{r}^{2}+{g^{\prime}}^{\theta\theta}p_{\theta}^{2})\right]^{1/2}\right\}\equiv{E^{\prime}}_{\pm}.
\delta{\calF}=-\frac{i}{24}{\bar{D}}^{4}\left(D^{\alpha{\mathbfj}}L_{{\mathbfi}{\mathbfj}}D_{\alpha}^{\mathbfi}{\calF}\right)-3\sigma{\calA}{\calA}_{D}-\frac{i}{48}{\bar{D}}^{4}\left[(H-{\bar{H}}){\bar{\calA}}{\calA}_{D}\right],
S_{WZ}~=~{\frac{1}{6}}\int_{B}d^{3}x~\epsilon^{\kappa\lambda\mu}\,\omega_{ijk}(\tilde{\varphi})\,\partial_{\kappa}\tilde{\varphi}^{i}\,\partial_{\lambda}\tilde{\varphi}^{j}\,\partial_{\mu}\tilde{\varphi}^{k}~=~\int_{B}\tilde{\varphi}^{\star}\omega~~~.
\langle\bar{\psi}\psi\rangle\;\approx\;\frac{1}{V}\,N_{\mathrm{V}}\intd^{4}x_{0}\;\langle\bar{\psi}\psi\rangle_{(1)}\;=\;\frac{1}{4\pi^{2}}\;m\;\rho_{\mathrm{V}}\;c_{0}\Bigl(m^{2}\,d^{2}\Bigr)\;.
{\calA}(A^{\alpha_{L}\alpha_{R}},B^{\beta_{L}{\bar{\beta}}_{R}},A_{\mu},A_{\nu})_{odd}~=~i\epsilon_{\mu\nu\rho\sigma}p_{3}^{\rho}p_{4}^{\sigma}(\Theta_{AB}^{(1)}-H_{AB,\bar{k}}G^{{\bar{k}}k}\Theta_{k}^{(1)})
\left(\partial_{\alpha}\partial^{\alpha}+M^{2}\right)T^{(\beta\gamma)\nu}=\frac{1}{4}\,J^{(\beta\gamma)\nu}+\frac{1}{4}J^{(\gamma\beta\nu)}+\frac{1}{6M^{2}}\partial_{\alpha}\partial^{\alpha}J^{\left(\gamma\beta\nu\right)},
f_{r_{1}r_{2}}^{-1}=\int{\frac{d^{2}p}{(2\pi)^{2}}}{\frac{1}{p^{2}+\mu^{2}}}e^{ip\cdot(r_{2}-r_{1})}={\frac{1}{2\pi}}K_{0}(\mu|r_{2}-r_{1}|)\;.
\hat{G}_{FF}(p)=\frac{p^{2}}{p^{2}+e^{2}\frac{N}{\pi}}=1-\frac{e^{2}\frac{N}{\pi}}{p^{2}+e^{2}\frac{N}{\pi}}=1-\int_{0}^{\infty}\frac{d\rho(\mu^{2})}{p^{2}+\mu^{2}}\;.
\bar{\eta}_{j_{1}j_{2}j_{3}j_{4}j_{6}I}=\sum_{j_{5}}\eta_{j_{1}j_{2}j_{3}j_{4}j_{5}I}(-)^{2j_{5}}\left\{\begin{array}{lll}{j_{1}}&{j_{2}}&{j_{6}}\\{j_{3}}&{j_{4}}&{j_{5}}\\\end{array}\right\}.
V_{ang}=\left[\left(\frac{1}{m_{1}}+\frac{1}{m_{2}}\right)^{2}\frac{V_{+}}{4r^{2}}+\left(\frac{1}{m_{1}}-\frac{1}{m_{2}}\right)^{2}\frac{V_{-}}{4r^{2}}\right]\bf(r\timesp)\cdot(r\timesp),
\left\langlet_{\Lambda},\phi\right\rangle=\left\langlet,\phi_{\Lambda^{-1}}\right\rangle=\left\langle{^0t},\phi-w_{\Lambda}\sum_{|\alpha|\leq\omega}\frac{x^{\alpha}}{\alpha!}D^{\alpha}\phi(0)\right\rangle+\sum_{|\alpha|\leq\omega}\frac{\left(D(\Lambda)c\right)^{\alpha}}{\alpha!}D^{\alpha}\phi(0).
\bar{S}_{int}=\int_{0}^{\beta}dt\int_{\Sigma}d\sigma_{x}\lbrace\mu^{j}(\Phi^{\dagger}T_{j}\dot{\Phi}-\dot{\Phi}^{\dagger}T_{j}\Phi)-\mu^{j}\mu^{k}\Phi^{\dagger}T_{j}T_{k}\Phi\rbrace
{\vec{\theta}}=f(\mu)\,\left(\begin{matrix}{\frac{1}{2\sqrt{a}}\left(1+{\vec{Y}}^{2}\right)}\\{\frac{\mathrm{i}}{2\sqrt{b}}(1-{\vec{Y}}^{2})}\\{Y_{i}}\\\end{matrix}\right)\qquad;\qquad\begin{array}{l}{i=1,\dots,n_{\mathrm{tr}}}\\{{\vec{Y}}^{2}\,\equiv\,Y_{i}Y_{j}\,h^{ij}}\\\end{array}\.
{W_{\nu}^{d}}(Z,Z^{\prime})=\frac{1}{2(2\pi)^{\frac{d}{2}}}\Gamma\left(\frac{d-1}{2}+i\nu\right)\Gamma\left(\frac{d-1}{2}-i\nu\right)\,\,((Z\cdotZ^{\prime})^{2}-1)^{-\frac{d-2}{4}}\,P_{-\frac{1}{2}+i\nu}^{-\frac{d-2}{2}}\left({Z\cdotZ^{\prime}}\right),
T_{\alpha\beta}[\phi]=(\frac{2}{3}\nabla_{\alpha}\phi\nabla_{\beta}\phi-\frac{1}{6}g_{\alpha\beta}\nabla_{\gamma}\phi\nabla^{\gamma}\phi)-\frac{1}{3}(\phi\nabla_{\alpha}\nabla_{\beta}\phi-g_{\alpha\beta}\phi\Box\phi)+\frac{1}{6}\phi^{2}G_{\alpha\beta}
C=i_{\hat{z}}\hat{C}+(dy+i_{\hat{z}}\hat{B}_{2})\wedge\left(\hat{C}^{(-)}-\,{\frac{\hat{E}^{{\#}(-)}}{i_{\hat{z}}\hat{E}^{{\#}}}}\,\wedgei_{\hat{z}}\hat{C}\right)\,.
\Phi^{\pm}(z,\bar{z})=i\sum_{m\in{\mathbbZ}}\left(\frac{\psi_{m\mp\lambda}^{\pm}}{m\mp\lambda}z^{-m\pm\lambda}+\frac{\bar{\psi}_{m\pm\lambda}^{\pm}}{m\pm\lambda}\bar{z}^{-m\mp\lambda}\right).
\delta\sigma_{i}=\pmi\kappa\sqrt{\frac{\pi}{k}}\;e^{i(2|p_{i}|+1)\pi/4}\;\left(\frac{-\Gamma(|p_{i}|)}{\pi\bar{a}_{i}}\right)\left(\frac{2}{-k\eta}\right)^{|p_{i}|-(1/2)}\,.
\delta^{B}g_{ij}=\{g_{ij},T_{\mu}\}C^{\mu},\:\delta^{B}\pi^{ij}=\{\pi^{ij},T_{\mu}\}C^{\mu},\:\delta^{B}N^{\mu}=\dot{C}^{\mu}-U_{\alpha\beta}^{\mu}N^{\alpha}C^{\beta}.
\langleW[L]\rangle\=\exp\left\{-{\frac{2\pii}{k}}\Bigl[\sum_{\ell}^{s}n_{\ell}^{2}~{\calSL}(K_{\ell})+\sum_{\ell\nem}^{s}n_{\ell}n_{m}{\calL}(K_{\ell},K_{m})\Bigr]\right\}
{\calZ}_{B}=Tr\;e^{-\beta\varepsilon_{\kappa}(\overline{{\Phi}}_{1,\kappa}\Phi_{1,\kappa}+\overline{{\Phi}}_{2,\kappa}\Phi_{2,\kappa})}e^{-\beta\mu(N_{1,\kappa}+N_{2,\kappa})},
{\calM}^{odd}=\frac18b_{ij}\left\{\epsilon^{ij}<p\cdot\psip\cdot\bar{\psi}>^{odd}+2ip_{k}\epsilon^{kj}\left(k^{i}+i<\psi^{i}p\cdot\bar{\psi}>_{o}^{odd}-<\psi^{i}p\cdot\psi>_{o}^{odd}\right)\right\}\;,
\Psi_{\mathrm{RN}}[M,Q;\psi_{1},\chi;k]=\Psi_{\mathrm{RN}}^{0}e^{i\left[\frac{1}{4}\left(-\frac{Q}{M}+2M\right)+k^{2}\frac{M}{Q^{2}-M^{2}}\pm2\sqrt{\pi}k(\psi_{1}+\chi)\right]}\zeta(A_{1}),
\begin{array}{rclc}{\left(0,1,0,1,e^{\phi}\right)u_{+}}&{=}&{e^{\phi}\cdotu_{+}}&{.}\\{\left(V_{1},1,0,1,1\right)u_{+}}&{=}&{u_{+}}&{.}\\{\left(0,1,V_{2},1,1\right)u_{+}}&{=}&{u_{+}}&{.}\\{\left(0,C_{1},0,1,1\right)u_{+}}&{=}&{u_{+}}&{.}\\{\left(0,1,0,C_{2},1\right)u_{+}}&{=}&{u_{+}}&{.}\\\end{array}
\Theta(D_{e};D_{\bar{e}})=-\int_{0}^{\infty}dl^{\prime}\frac{df(D_{e}^{\kappa}l^{\prime})}{dl^{\prime}}f(D_{\bar{e}}^{\kappa}l^{\prime})=-\int_{0}^{\infty}dz\frac{df(z)}{dz}f\left(\left(\frac{D_{\bar{e}}}{D_{e}}\right)^{\kappa}z\right).
i\frac{g\tilde{g}}{\bar{N}}\omega(x_{0})\left[\int\delta\xiU_{\xi}(s_{0})X^{\alpha}\tilde{X}_{\alpha}U_{\xi}^{-1}(s_{0})\dot{\xi}^{-2}(s_{0})\delta(\xi(s_{0})-x_{0})\right]\omega^{-1}(x_{0}).
W_{\Lambda}\left(\phi,\phi^{*},\overline{{\phi}},\pi\right)\equiv\overline{{W}}\left(\phi,\phi^{*},\overline{{\phi}},\pi,\varphi_{q}\left(\phi,\phi^{*},\overline{{\phi}},\pi\right),\varphi^{*}=0,\overline{{\varphi}}=0,\pi_{\varphi}=0\right)
\mathbb{J}_{SU}^{x}=\left(\begin{array}{c}{\mathrm{i}{\bar{\psi}}^{\star}\,\wedge\Gamma_{\underline{{a}}}\psi\,\wedge\,V^{\underline{{a}}}}\\{\\mathrm{i}{\bar{\psi}}\,\wedge\Gamma_{\underline{{a}}}\psi^{\star}\,\wedge\,V^{\underline{{a}}}\}\\\end{array}\right)\quad\quad(x=\pm)
S_{R}(\phi,g)={\frac{1}{2}}Z_{\phi}^{2}\partial_{i}{\phi_{\alpha}}\partial_{i}{\phi_{\alpha}}+{\frac{1}{2}}Z_{\phi}^{2}Z_{\tau}\tau{\bf\phi}^{2}+{\frac{1}{4!}}Z_{\phi}^{4}Z_{g}g\mu^{\epsilon}({\bf\phi}^{2})^{2}
\langleG\vertJ_{GB}^{i}\vertG\rangle=-\epsilon^{ilm}\intd^{3}x\langleG\vertx^{l}(i\partial_{(x)}^{m}[\delta^{3}({\bfx-y})])\vertG\rangle=-\epsilon^{ilm}\intd^{3}x\left(i\partial_{(x)}^{m}[\delta^{3}({\bfx-y})]\right)x^{l}
\varepsilon_{\mu\beta}\partial^{\mu}\left(\partial_{\alpha}\partial_{\nu}h^{\alpha\beta}-\partial_{\alpha}\partial^{\alpha}h_{\nu}^{\\beta}\right)=\varepsilon_{\mu\nu}\partial^{\mu}\left(\partial_{\alpha}\partial_{\beta}h^{\alpha\beta}-\partial_{\alpha}\partial^{\alpha}h_{\beta}^{\\beta}\right).
\frac{e}{4m}F_{\alpha\beta}[y^{T}{\calL}{\calM}^{\alpha\beta}{\calL}^{T}\mathrm{p}+\frac{mb^{2}}{\lambda}n_{\mu}({\calM}^{\alpha\beta})^{\mu\nu}(\pi_{\nu}+y^{T}S_{\nu}\mathrm{p})]
\intd\mu_{B}=\int\frac{d^{3}x}{(2\pi)^{\frac{3}{2}}}({\calJ}_{X})^{\frac{3}{2}}\int_{0}^{\frac{2\pi}{k}}\frac{d\theta}{(2\pi)^{\frac{1}{2}}}({\calJ}_{\theta})^{\frac{1}{2}}
\Omega=\left(\begin{array}{cccc}{J_{11}}&{\cdots}&{J_{1N}}&{Q_{1}}\\{\vdots}&{\ddots}&{\vdots}&{\vdots}\\{J_{N1}}&{\cdots}&{J_{NN}}&{Q_{N}}\\{-{\overline{{Q}}}_{1}}&{\cdots}&{-{\overline{{Q}}}_{N}}&{\sum_{\mu=1}^{N}J_{\mu\mu}}\\\end{array}\right)
a_{kb}^{\prime}={\calK}_{kn}^{-1}\Bigl(G^{11}({H^{n}}_{b}+e^{\rho}\alpha_{n}\gamma_{b})-G^{n1}({H^{b}}_{J}b^{J}+{\frac{1}{2}}e^{\rho}\gamma^{b}b^{I}d_{IJ}b^{J})\Bigr),
{\calP}_{\calR}^{\eta_{k}}=\frac{k^{3}}{2\pi^{2}}\,\left\vert\frac{\varphi_{k}^{\eta_{k}}}{z}\right\vert_{k=aH}^{2}\,\,,\,\,z=-m_{\mathrm{Planck}}\,a\sqrt{\epsilon}\,\,,
\intd^{4}z_{1}d^{4}z_{2}\langle\Phi(x_{1})\Phi(x_{2})\Phi^{*}(y_{1})\Phi^{*}(y_{2}):(\Phi\Phi^{*})^{2}:(z_{1}):(\Phi\Phi^{*})^{2}:(z_{2})\rangle
A_{\mu}(x;y)=\eta_{\mu\mu}m_{y}G\left(\sum_{\alpha=0}^{3}\eta_{\alpha\alpha}(x^{\alpha}-y^{\alpha}(t^{\prime}))\frac{dy^{\alpha}(t^{\prime})}{dt^{\prime}}\right)^{-1}\frac{dy^{\mu}(t^{\prime})}{dt^{\prime}}
\langleJ^{\alpha}(\bar{z})J^{\beta}(0)\rangle=e(\alpha,\beta)c_{\alpha+\beta}\langlee^{i\alpha\cdot\phi_{q}(\bar{z})}e^{i\beta\cdot\phi_{q}(0)}\rangle\sum_{P_{L}\in\Lambda_{24}}e^{-i\pi\bar{\tau}-2\pii\,z\,\alpha\cdotP_{L}}.
M_{k,l}^{\calW}=M_{k,l}^{{\calW}^{\dagger}}=\left(\begin{array}{ccccc}{-4}&{2}\\{2}&{-4}&{2}\\{}&{2}&{\ddots}&{\ddots}\\{}&{}&{\ddots}&{-4}&{2}\\{}&{}&{}&{2}&{-4}\\\end{array}\right).\quad(k,l=1,2,\cdots,J)
L^{0}\,=\,\eta_{ij}\Big(\epsilonP_{0}^{i}\partial_{\tau}X_{0}^{j},+\,P_{1}^{i}\partial_{\tau}X_{1}^{j}\,+...+P_{n}^{i}\partial_{\tau}X_{n}^{j}\,+\,...+P_{N-1}^{i}\partial_{\tau}X_{N-1}^{j}\,+\,\epsilonP_{N}^{i}\partial_{\tau}X_{N}^{j}\,\Big)\,+V
\Gamma_{r}^{\,\alpha,\,\alpha_{0}}(\Phi_{c})=\sum_{n\geq2}\,\int\,\prod_{k=1}^{n-1}\,\frac{d^{\,4}p_{k}}{(2\pi)^{\,4}}\\Phi_{c}(p_{k})\,\Phi_{c}\bigl(-\sum_{j=1}^{n-1}p_{j}\bigr)\\Gamma_{r,n}^{\,\alpha,\,\alpha_{0}}(p_{1},\ldots,p_{n-1})\quad.
\partial_{0}^{2}h_{j}^{i}+\Boxh_{j}^{i}+\frac{1-d}{x_{0}}\partial_{0}h_{j}^{i}-\frac{1}{x_{0}}\partial_{0}h\delta_{j}^{i}+\partial^{i}\partial_{j}h-\partial^{i}\partial_{l}h_{j}^{l}-\partial_{j}\partial^{l}h_{l}^{i}=0.
T_{\hat{M}}^{~\hat{N}}=\mathrm{diag}\left(\frac{n(n+1)}{2\ell^{2}}-\frac{{\calQ}^{2}}{a^{2n}},\frac{n(n+1)}{2\ell^{2}}+\frac{{\calQ}^{2}}{a^{2n}},\cdots,\frac{n(n+1)}{2\ell^{2}}+\frac{{\calQ}^{2}}{a^{2n}},\frac{n(n+1)}{2\ell^{2}}-\frac{{\calQ}^{2}}{a^{2n}}\right).
\left[a_{\left[k\right]}^{\mu_{m}}\left[\varphi\right],a_{\left[l\right]}^{\mu_{n}}\left[\chi\right]\right]=\left[a_{\left[k\right]}^{+\mu_{m}}\left[\varphi\right],a_{\left[l\right]}^{+\mu_{n}}\left[\chi\right]\right]=0\;\,,
\zeta(\Omega^{2}\omega_{3}\omega_{5})=\sum_{r,s>0}\frac{\lambda^{3r+5s}}{(r+s)^{3}s}=\sum_{n_{1}>n_{2}>0}\frac{\lambda^{3n_{1}}}{n_{1}^{3}}\,\frac{\lambda^{2n_{2}}}{n_{2}}=\zeta\left(\begin{array}{rl}{3,}&{1}\\{\lambda^{3},}&{\lambda^{2}}\\\end{array}\right)
\vec{T}_{0}^{1}\vec{T}_{1}^{1}\;=\;s_{-1}^{r-1}s_{1}^{r-1}f_{-1}^{1}f_{1}^{1}\,\vec{I}+s_{0}^{r-1}f_{0}^{1}\,\vec{T}_{0}^{2}
\vert\nu\rangle=\vert{\nu_{1}},{\nu_{2}},{\nu_{3}}\rangle={e^{{\nu_{1}}{e_{14}}}}{e^{{\nu_{2}}{e_{24}}}}{e^{{\nu_{3}}{e_{34}}}}\left(\begin{matrix}{0}\\{0}\\{0}\\{1}\\\end{matrix}\right)=\left(\begin{matrix}{{\nu_{1}}}\\{{\nu_{2}}}\\{{\nu_{3}}}\\{1}\\\end{matrix}\right)
\deltaI[g,\phi]=\int_{\calM}\sqrt{-g}d^{4}x\left[E^{\mu\nu}\deltag_{\mu\nu}+E_{\phi}\delta\phi+\nabla_{\mu}\theta^{\mu}(\deltag,\delta\phi)\right]-\delta\left[\int_{\partial{\calM}}\sqrt{-h}d^{3}xB[g,\phi]\right]~~~.
\widetilde{W}_{i}^{T}=-\int_{0}^{\infty}{\frac{dt}{t}}{\frac{\beta}{\sqrt{4\pit}}}\sum_{k=1}^{\infty}\eta_{i}^{k+1}e^{-{\frac{\beta^{2}k^{2}}{4t}}}\mathrm{Tr}~e^{-(L_{i}+m_{i}^{2}g_{\tau\tau})t}~~~.
(\rho^{+}\oplus\rho^{-})\otimes(\rho^{+}\oplus\rho^{-})\cong\bigoplus_{p=0}^{8}{\wedge}^{p}\rho_{v}\cong2\left(1\oplus\rho_{v}\oplus{\wedge}^{2}\rho_{v}\oplus{\wedge}^{3}\rho_{v}\right)\oplus{\wedge}^{4}\rho_{v}.
{\calJ}_{i}^{\mu\nu}[\xi|s]=4\pi\intd^{4}x\,\epsilon^{\mu\nu\rho\sigma}\left[\Omega_{\xi}(s,0)T_{i}\Omega_{\xi}^{-1}(s,0)\right]^{j}\dot{\xi}_{\sigma}(s)\widetilde{j}_{\rho}^{j}(x)\delta^{4}(x-\xi(s)).
ds^{2}=H_{1}^{\frac{2}{D-2}}H_{2}^{\frac{2(D-q-2)}{q(D-2)}}[(H_{1}H_{2})^{-\frac{2}{q}}\eta_{\mu\nu}dy^{\mu}dy^{\nu}+H_{2}^{-\frac{2}{q}}dz^{m}dz^{m}+dx^{\alpha}dx^{\alpha}],
\kappa^{+q}\equivi_{\delta}E^{+q1}=\delta{\Theta}^{1\underline{{\mu}}}v_{\underline{{\mu}}q}^{~+},\qquad\kappa^{-{q}}\equivi_{\delta}E_{{q}}^{-2}=\delta{\Theta}_{\underline{{\mu}}}^{2}v_{q}^{-\underline{{\mu}}}.
{\calL}=2{\sqrt{-1}}\intd\theta^{+}d{\overline{{\theta}}}^{+}\frac{\partialK}{\partial\Phi^{i}}\partial_{-}\Phi^{i}-\intd\theta^{+}d{\overline{{\theta}}}^{+}h_{a\overline{{b}}}\Lambda^{a}\Lambda^{\overline{{b}}}\,,
\tilde{\mu}_{\mathrm{ph}}=\tilde{\mu}-\frac{\tilde{e}^{2}\tilde{\mu}}{8\pi}\int_{2m}^{\infty}\frac{1+(4m/a^{2})(m-\tilde{\mu})}{a^{2}-\tilde{\mu}^{2}}da+O(\tilde{e}^{4}).
\partial_{\tau}({\frac{1}{e(\tau)}}{\calE}_{\tau}^{\underline{a}})=\partial_{\tau}p^{\underline{a}}=0,\quad({\calE}_{\tau}^{\underline{a}}\Gamma_{\underline{a}})_{~~\underline{\beta}}^{\underline{\alpha}}\partial_{\tau}\theta^{\underline{\beta}}=(p^{\underline{a}}\Gamma_{\underline{a}})_{~~\underline{\beta}}^{\underline{\alpha}}\partial_{\tau}\theta^{\underline{\beta}}=0,
\begin{array}{c}{\delta_{i}^{+}z^{a}=\epsilon\eta_{i}^{a},}\\{\delta_{i}^{+}\eta_{j}^{a}=\epsilon\left(i\epsilon_{ij}{\bar{U}}_{\bar{b}}g^{\bar{b}a}+\Gamma_{bc}^{a}\eta_{i}^{b}\eta_{j}^{c}\right),}\\{\delta_{i}^{-}z^{a}=0,}\\{\delta_{i}^{-}\eta_{j}^{a}=\epsilon(\delta_{ij}{\dot{z}}^{a}-\epsilon_{ij}G^{a}).}\\\end{array}
\int_{R_{+}+h_{+}}^{R_{c}-h_{c}}\frac{r^{2}}{\Big(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{r^{2}}{l^{2}}\Big)^{2}}dr=\int_{R_{+}+h_{+}}^{L}\frac{r^{6}}{f^{2}(r)}dr+\int_{L}^{R_{c}-h_{c}}\frac{r^{6}}{f^{2}(r)}dr
B_{\mathrm{vN}}(R,E)={\frac{\sigma_{\mathrm{N}}(x)}{x}}\approx\left({\frac{2^{7}a_{r}}{3^{5}\pi^{2}x}}\right)^{\frac{1}{4}}=\left[{\frac{2^{6}}{3^{6}5x}}(n_{b}+{\frac{7}{8}}n_{f})\right]^{\frac{1}{4}},
\widetildeW_{nlm}^{n_{1}}(\delta_{1},\delta_{2})=(-1)^{n_{1}+\frac{m-|m|}{2}}C_{\frac{n+\delta_{2}-1}{2},\frac{n+\delta_{2}-1}{2}-n_{1};\frac{n+\delta_{1}-1}{2},n_{1}+|m|-\frac{n-\delta_{1}-1}{2}}^{l+\frac{\delta_{1}+\delta_{2}}{2},|m|+\frac{\delta_{1}+\delta_{2}}{2}}
T=\left(\begin{array}{cccccc}{e^{i\theta_{1}}}&{}&{}&{}&{}&{}\\{}&{\ddots}&{}&{}&{}&{}\\{}&{}&{e^{i\theta_{N}}}&{}&{}&{}\\{}&{}&{}&{e^{-i\theta_{1}}}&{}&{}\\{}&{}&{}&{}&{\ddots}&{}\\{}&{}&{}&{}&{}&{e^{-i\theta_{N}}}\\\end{array}\right)
\sqrt{g}\Box(R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta})=-{\frac{2}{3}}g_{\mu\nu}{\frac{\delta}{\deltag_{\mu\nu}}}\intd^{6}x\sqrt{g}R_{\mu\nu}^{~~~\alpha\beta}R_{\alpha\beta}^{~~~\rho\sigma}R_{\rho\sigma}^{~~~\mu\nu}
\varphi=\mu_{-}\nu_{+}\gamma_{-}=\mu_{+}\nu_{-}\eta\gamma_{-},\qquad\varphi^{\prime}=\mu_{-}^{\prime}\nu_{+}^{\prime}\gamma_{-}^{\prime}=\mu_{+}^{\prime}\nu_{-}^{\prime}\eta^{\prime}\gamma_{-}^{\prime}.
P^{(N_{f})}(\lambda_{\mathrm{min}},m)=2\lambda_{\mathrm{min}}P^{(N_{f})}(x_{\mathrm{min}},m)=-2\lambda_{\mathrm{min}}\left.\frac{dE(0,s)}{ds}\right|_{s=\lambda_{\mathrm{min}}^{2}}\:.
\left[~r^{2}\left(1+\frac{2GM}{r}\right)E^{2}+\partial_{r}\left\{r^{2}\left(1-\frac{2GM}{r}\right)\partial_{r}\right\}-\hat{L}^{2}-\frac{4GJE}{r}~\hat{L}_{z}~\right]\psi~=~0~,
{\frac{1}{2}}\left(1-{\frac{e^{2}}{\kappa^{2}}}|\Phi|^{2}\right)B^{2}+{\frac{1}{2}}|\nabla_{0}\Phi|^{2}={\frac{1}{2}}{\frac{\kappa^{2}}{e^{2}|\Phi|^{2}}}\left(1-{\frac{e^{2}}{\kappa^{2}}}|\Phi|^{2}\right)\,B^{2}\,.
\int_{-\infty}^{\infty}dq^{\prime}U^{-1}(q,q^{\prime})\phi(q^{\prime})=\frac{(1-i)}{2}\sqrt{\pi}\mathrm{e}^{\frac{i\pia}{2}}\mathrm{sgn}(q)\left|kq\right|^{\frac{1}{2}}J_{a}(k\left|q\right|).
A_{3}={\frac{qA_{03}}{q^{2}-8s^{2}f_{0}\kappa_{0}^{-2}}}\qquad\mathrm{and}\qquadA_{4}={\frac{-qs(1-8s^{2}f_{0}\kappa_{0}^{-2})}{2\sqrt2(q^{2}-8s^{2}f_{0}\kappa_{0}^{-2})}},
\begin{array}{rclcrclcrclcrcl}{\alpha_{6}}&{=}&{\frac{i}{6!}}&{;}&{\alpha_{4}}&{=}&{\frac{1}{2\,4!}}&{;}&{\alpha_{0}}&{=}&{\frac{1}{48}}&{;}&{\alpha_{2}}&{=}&{\frac{i}{16}}\\\end{array}
\sum_{s_{j}}\sum_{s_{k}}\frac{(N_{i}-2s_{i})!N_{j}!N_{k}!}{2^{s_{j}+s_{k}}s_{j}!s_{k}!t_{ij}!t_{ik}!t_{jk}!}=\frac{(N_{j}+N_{k})!}{2^{(N_{j}+N_{k}-N_{i}+2s_{i})/2}(\frac{N_{j}+N_{k}-N_{i}+2s_{i}}{2})!}
T_{MN}^{4-brane,l}=\Lambda_{l}A(z_{1},z_{2})\delta(\vec{n}\cdot\vec{z})\left(\begin{array}{rrrrrr}{1}&{}&{}&{}&{}&{}\\{}&{-1}&{}&{}&{}&{}\\{}&{}&{-1}&{}&{}&{}\\{}&{}&{}&{-1}&{}&{}\\{}&{}&{}&{}&{-\mathrm{cos}^{2}\varphi_{l}}&{-\mathrm{sin}\varphi_{l}\mathrm{cos}\varphi_{l}}\\{}&{}&{}&{}&{-\mathrm{sin}\varphi_{l}\mathrm{cos}\varphi_{l}}&{-\mathrm{sin}^{2}\varphi_{l}}\\\end{array}\right),
A_{i}^{\gamma}\,^{L}({\bfr})=\frac{1}{g}\left[{\delta}_{{i}\,{\gamma}}\,\frac{{\calN}(r)}{r}+\frac{r_{i}\,r_{\gamma}}{r}\,\left(\frac{{\calN}(r)}{r}\right)^{\prime}\,\right]
\hat{\delta}S_{eff}=\int{d^{5}\sigmad\tau\hat{\delta}[\mu\dot{C}+\mu_{\rho}\dot{C}^{\rho}+P^{\mu\nu}\dot{B}_{\mu\nu}+\Pi_{a}\dot{X}^{a}+\hat{\mu}^{i}\dot{\hat{C}}_{i}+\widehat{\mu}\dot{\widehat{C}}]}=\int{d\tau\dot{Q}}=0
V_{0}[\zeta]=\sum_{n={\frac{1}{2}}}^{\Lambda}\sum_{m={\frac{1}{2}}}^{\Lambda}\Bigl({\frac{u_{m}}{v_{n}}}-{\frac{v_{n}}{u_{m}}}\Bigr)^{2}w_{m+n}^{4}+\sum_{l=1}^{\Lambda+\frac{1}{2}}\sum_{m={\frac{1}{2}}}^{\Lambda}\Bigl[\Bigl({\frac{w_{l}}{v_{m}}}-{\frac{v_{m}}{w_{l}}}\Bigr)^{2}v_{m+l}^{4}+\Bigl({\frac{w_{l}}{u_{m}}}-{\frac{u_{m}}{w_{l}}}\Bigr)^{2}u_{m+l}^{4}\Bigr]
\triangleS_{\mathrm{NR}}^{d=2}|_{T=0;\mu\not=0}=\frac{B}{2\pi}\left\{2\sum_{n=1}^{\left[\frac{m\mu_{\mathrm{NR}}}{B}\right]}\left(\mu_{\mathrm{NR}}-\frac{B}{m}n\right)+\mu_{\mathrm{NR}}\right\}
\begin{array}{ccc}{K_{ab}^{-1}=}&{p\left(\begin{array}{cc}{\sqrt{g}}&{{\frac{1}{\sqrt{g}}}}\\{{\frac{1}{\sqrt{g}}}}&{\sqrt{g}}\\\end{array}\right)}&{+(1-p)\left(\begin{array}{cc}{{\frac{1}{\sqrt{g}}}}&{\sqrt{g}}\\{\sqrt{g}}&{{\frac{1}{\sqrt{g}}}}\\\end{array}\right)}\\\end{array}
G^{(\psi)}(z_{1},z_{2})=\frac{1}{16}\left[\begin{array}{cc}{0}&{-\frac{\bar{D}_{1}^{2}D_{2}^{2}}{\Box}\delta^{8}(z_{1}-z_{2})}\\{-\frac{D_{1}^{2}\bar{D}_{2}^{2}}{\Box}\delta^{8}(z_{1}-z_{2})}&{-\frac{D_{1}^{2}D_{2}^{2}}{\Box}\psi(z_{1})\frac{\bar{D}_{1}^{2}}{4\Box_{1}}\delta^{8}(z_{1}-z_{2})}\\\end{array}\right]
{\bar{\psi}}\gamma^{\mu}\stackrel{\leftrightarrow}{\nabla}_{\mu}\psi={\bar{\psi}_{R}}\gamma^{\mu}\stackrel{\leftrightarrow}{\nabla}_{\mu}\psi_{R}+{\bar{\psi}_{L}}\gamma^{\mu}\stackrel{\leftrightarrow}{\nabla}_{\mu}\psi_{L}
X_{2}=\int_{\sqrt{2(e-d)+e^{2}}}^{\sqrt{2(e+d)+e^{2}}}{dx{\frac{1}{(\lambda^{2}-x^{2})^{1/2}\left[(\lambda^{2}-x^{2})^{1/2}+1\right]}}\left[{\frac{1}{x^{2}}}+{\frac{1}{\left[(\lambda^{2}-x^{2})^{1/2}+1\right]}}\right]},
\hat{{\calH}}=\frac{1}{2}g^{\mu\nu}(P_{\mu}-\OmegaA_{\mu})(P_{\nu}-\OmegaA_{\nu})+\frac{1}{2}g_{\phi\phi}(L(\omega)+\frac{\Omega^{2}}{2g_{\phi\phi}dL/d\omega})^{2}-\frac{\Omega^{2}}{2}\frac{(n+A_{\phi})^{2}}{g_{\phi\phi}}=0,
H\,=\,\intdx\left[\,{\frac{1}{2}}\,\bar{g}^{ab}\pi_{a}\pi_{b}+\,{\frac{1}{2}}\,\partial_{x}\phi^{a}\partial_{x}\phi^{a}\,-\,\lambda(|\phi|^{2}-1)+v_{\lambda}\pi_{\lambda}\,\right]\,,
S=\frac{1}{2\pi\alpha^{\prime}}\intd\sigma^{+}d\sigma^{-}\,\left(\partial_{-}X^{\mu}\partial_{+}X_{\mu}+\frac{i}{2}\psi_{+}^{\mu}\partial_{-}\psi_{+\mu}+\frac{i}{2}\psi_{-}^{\mu}\partial_{+}\psi_{-\mu}\right).
(1-s^{2})f^{\prime\prime}-(2\lambda+1)\,s\,f^{\prime}+\left[\left\{\frac{\varepsilon}{\kappa^{2}}-\lambda\right\}+\left\{\lambda(\lambda-1)-\frac{1}{\kappa^{4}}\right\}\frac{s^{2}}{c^{2}}\right]f=0\;.
{\calH}_{_0}^{\prime}=-i\overline{{\psi}}^{1}\gamma^{1}\partial_{_1}\psi^{1}-i\overline{{\psi}}^{2}\gamma^{1}\partial_{_1}\psi^{2}+i\overline{{\psi}}^{3}\gamma^{1}\partial_{_1}\psi^{3}+m\overline{{\psi}}^{1}\psi^{1}.
h_{ij}^{\prime\prime}+(\frac{f^{\prime}}{f}-\frac{g^{\prime}}{g})h_{ij}^{\prime}-2[(\frac{g^{\prime}}{g})^{\prime}+\frac{g^{\prime}}{g}(2\frac{g^{\prime}}{g}+\frac{f^{\prime}}{f})-k^{2}]h_{ij}=0
\hat{\calM}=\left(\hat{\calM}_{ij}\right)=\frac{1}{\Im\mathrm{m}\lambda}\left(\begin{array}{rr}{|\hat{\lambda}|^{2}}&{-\Re\mathrm{e}\hat{\lambda}}\\{-\Re\mathrm{e}\hat{\lambda}}&{1}\\\end{array}\right)\,,
\sum_{n=0}^{\infty}x^{2n}\,\frac{e^{-|z|^{2}}}{(n!)^{2}}\,\left\{\sum_{\mu,\nu=1}^{M}\left(e^{i\omega_{\mu}}|z_{\mu+}^{+}|^{2}+e^{-i\omega_{\mu}}|z_{\mu-}^{+}|^{2}\right)\left(e^{i\omega_{\nu}}|z_{\nu+}^{-}|^{2}+e^{-i\omega_{\nu}}|z_{\nu-}^{-}|^{2}\right)\right\}^{n}\\\.
\tilde{S}\equivTr(\frac{1}{2}a_{\mu}(P_{\lambda}^{2}\delta_{\mu\nu}-2iF_{\mu\nu})a_{\nu}-\frac{1}{2}\bar{\varphi}\Gamma^{\mu}P_{\mu}\varphi+bP_{\lambda}^{2}c+\bar{\varphi}\Gamma^{\mu}\Thetaa_{\mu}).
g=\left(\begin{array}{cc}{e^{i\psi/2}}&{0}\\{0}&{e^{-i\psi/2}}\\\end{array}\right)\,\frac{1}{\sqrt{1-|t|^{2}}}\,\left(\begin{array}{rr}{1}&{\bar{t}}\\{t}&{1}\\\end{array}\right)\,,\qquad|t|<1\,,
\langle\bar{w}(\ell_{1})\bar{w}(\ell_{2})\rangle_{0}(\bar{t}\,)=\left\{\langlew(\ell_{1})w(\ell_{2})\rangle_{0}-{\frac{\langlew(\ell_{1})P\rangle_{0}\langlePw(\ell_{2})\rangle_{0}}{\langlePP\rangle_{0}}}\right\}\Bigg\vert_{t=t(\bar{t}\,)}\,