harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
H^{2}\equiv\left(\frac{\dot{R}}{R}\right)^{2}=-\frac{k}{R^{2}}+\frac{\mu}{R^{4}}-\frac{q^{2}}{R^{6}}+\frac{8\piG_{4}}{3}\,\Lambda_{4}+\left(\frac{4\piG}{3}\right)^{2}\left(\rho^{2}+2\sigma\rho\right)\,.
\sum_{i=1}^{N}\frac{\alpha_{i}}{m_{i}}=\frac{1}{D^{(N)}}\sum_{i=1}^{N}\alpha_{i}\sum_{l=1}^{N}\frac{c_{il}}{m_{l}}F_{l}^{(N)}=\frac{1}{D^{(N)}}\sum_{i=1}^{N}\frac{F_{i}^{(N)}}{m_{i}}\sum_{l=1}^{N}c_{il}\alpha_{l}.
\left(\frac{\dot{R}}{R}\right)^{2}=\frac{\kappa_{5}^{4}\rho_{0}}{18}\rho+\mu\left(\frac{1}{R^{4}}-\frac{1}{r_{0}^{4}}\right)-Q^{2}\left(\frac{1}{R^{6}}-\frac{1}{r_{0}^{6}}\right)+\frac{\kappa_{5}^{4}}{36}\rho^{2},
\mu_{11}=-{\frac{1}{24\pi^{2}}}{\frac{\pi^{D/2}\Gamma(-D/2)}{2^{D}}}\left[(D-3)(D-2)(D-1)\zeta(2-D)+4\pi^{2}(D+1)\zeta(-D)\right]\,{\frac{\hbarL^{D-1}}{a^{D}}},
\partial_{\mu}\vec{n}^{\prime}=i\Gamma_{\mu}^{\prime}\vec{n}^{\prime}=-\vec{\Gamma}_{\mu}^{\prime}\wedge\vec{n}^{\prime}\qquad\mathrm{with}\qquad\Gamma_{\mu}^{\prime}=\Omega(\Gamma_{\mu}+i\partial_{\mu})\Omega^{\dagger}
\begin{array}{c}{\left(D(\beta,L)\right)_{A(\eta),B}=\left(e^{-i\etat}D(\beta,L)e^{i\etat}\right)_{A(0),B}}\\{=\left(\fraci\beta\gamma^{0}\partial_{t}+\fraciL\gamma^{1}\partial_{x}-\frac\eta\beta\gamma^{0}\right)_{A(0),B}\,}\\\end{array}
S_{NJL2}^{F}=-{\frac{G_{2}}{2}}\sum_{x\mu}\bar{\Psi}_{L}^{F}(x)^{i}L_{\mu}^{F}(x)_{i}^{i^{\prime}}U_{\mu}^{F}(x)\Psi_{R}^{F}(x+a_{\mu})_{j^{\prime}}\bar{\Psi}_{R}^{F}(x)^{j}R_{\mu}^{F}(x)_{j}^{j^{\prime}}U_{\mu}^{F}(x)\Psi_{L}^{F}(x+a_{\mu})_{i^{\prime}}.
V(a)=-\frac{\kappa_{(4)}^{2}}{6}\rho_{X0}a^{-3w_{X}+2}-\frac{\kappa_{(4)}^{2}}{12\lambda}\rho_{X0}^{2}a^{-6w_{X}+2}-\frac{\sigma_{0}^{2}}{6}a^{-4}-\frac{\Lambda_{(4)}}{6}a^{2}-\frac{{\calU}_{0}}{\lambda\kappa_{(4)}^{2}}a^{-2}+\frac{k}{2}
\begin{array}{lll}{\gamma^{0}\!=\!\left(\!\!\begin{array}{cc}{\sigma_{3}}&{0}\\{0}&{-\sigma_{3}}\\\end{array}\!\!\right),}&{\!\gamma^{1}\!=\!\left(\!\!\begin{array}{cc}{i\sigma_{1}}&{0}\\{0}&{-i\sigma_{1}}\\\end{array}\!\!\right),}&{\!\gamma^{2}\!=\!\left(\!\!\begin{array}{cc}{i\sigma_{2}}&{0}\\{0}&{-i\sigma_{2}}\\\end{array}\!\!\right).}\\\end{array}
\delta\bar{q}_{\pm\mu\nu}=e^{-\alpha_{\pm}^{(0)}}\deltaq_{\pm\mu\nu}-{\alpha_{\pm}^{\prime}}^{(0)}e^{-\alpha_{\pm}^{(0)}}e^{-2A_{\pm}}\eta_{\pm\mu\nu}\bar{q}_{\pm\mu\nu}^{(0)}\delta\Psi_{\pm},
X^{+Y}(x,\theta^{+},u)=X^{AY}(x)u_{A}^{+}+i\theta_{+}^{+A^{\prime}}\psi_{-A^{\prime}}^{Y}(x)-i(\theta_{+}^{+})^{2}\partial_{--}X^{AY}(x)u_{A}^{-}
T(z)=-\frac{1}{2}:\!\partialX\partialX\!:+\,\frac{Q}{2}\partial^{2}X,\qquad\tilde{T}(\bar{z})=-\frac{1}{2}:\!\bar{\partial}X\bar{\partial}X\!:+\,\frac{Q}{2}\bar{\partial}^{2}X.
x_{r}(t)={\frac{a}{\mathrm{sn}(at,k)}},\quady_{r}(t)=-a^{2}{\frac{\mathrm{cn}(at,k)\,\mathrm{dn}(at,k)}{\mathrm{sn}^{2}(at,k)}},\quadz_{r}(t)=-b^{2}\wp(bt),\quadb=a/\sqrt{e_{1}-e_{3}},
C_{p}^{(i,j)}\phi_{p}\chi_{0}^{(i)}{\bar{\chi}}_{0}^{(j)}\quad,\quadC_{p}^{(i,j)}=|p|e^{-i\pim_{4}({\frac{m_{5}}{2}}-j)}e^{i{\frac{\pi}{2}}({\frac{p_{5}^{2}}{4}}\tau-{\frac{p_{4}^{2}}{\tau}})}\quad;\quadm_{5}+i-j=\mathrm{even}\,
R_{c}^{2}\,\left(n_{m}-n_{\mathrm{loop}}^{\mathrm{unc}}\right)=R_{m}^{2}\,n_{m}-R_{\mathrm{loop}}^{2}\,n_{\mathrm{loop}}^{\mathrm{unc}}\,\,\,\,\,,\,\,\,\,\,
u_{l,m,n}(x)=\frac1{\sqrt{2\piR^{2}L\omega_{l,m,n}}J_{l+1}\left(\alpha_{m}^{(l)}\right)}J_{l}\left(\frac{\alpha_{m}^{(l)}}Rr\right)e^{-i\omega_{l,m,n}t+il\theta+i\frac{n\pi}{2L}z}
{\calL}_{1}=\frac{\kappa}{4}\varepsilon^{\mu\nu\alpha\beta}\varepsilon_{abcd}f_{\mu}{}^{a}f_{\nu}{}^{b}f_{\alpha}{}^{c}\delta_{\beta}{}^{d}\qquad\delta_{1}\Phi_{\mu\nu,c}=\frac{\kappa}{2}f_{[\alpha}{}^{a}\delta_{\beta]}{}^{b}\varepsilon_{abcd}\eta^{d}
N^{AB}-a^{-1}\xi^{AB}-i\epsilon^{(A}\psi^{B)}=0~,\bar{N}^{A^{\prime}B^{\prime}}-a^{-1}\bar{\xi}^{A^{\prime}B^{\prime}}-i\bar{\epsilon}^{(A^{\prime}}\bar{\psi}^{B^{\prime})}=0~,
\begin{array}{cc}{i)\quad}&{I_{ab}^{+}={\calJ}_{a}^{+}{\calJ}_{b}^{+}+f_{+}(z\bar{z}){\bar{z}}^{a}{\bar{z}}^{b},}\\{ii)\quad}&{I_{a\bar{b}}={\calJ}_{a}^{+}{\calJ}_{\bar{b}}^{-}+f_{0}(z\bar{z}){\bar{z}}^{a}{z}^{b}.}\\\end{array}
\tau_{\nu}^{\mu}=\left[\begin{array}{ccc}{-(\rho+\delta\rho)}&{}&{a(\rho+P)(v+B)_{|j}}\\{}&{}&{}\\{-a^{-1}(\rho+P)v^{|i}}&{}&{(P+\deltaP)\delta_{j}^{i}+\delta\pi_{j}^{i}}\\\end{array}\right]\,,
\begin{array}{l}{G(z)=\sqrt{\frac{2}{k+\check{g}}}\{U_{a}\psi^{a}+\frac{1}{2(\check{g}-\check{s})}K_{ab}\psi^{ab}-\frac{1}{2}\psi^{a}\psi^{b}\psi_{ab}\}(z)}\\{\bar{G}(z)=\sqrt{\frac{2}{k+\check{g}}}\{U^{a}\psi_{a}-\frac{1}{2(\check{g}-\check{s})}K^{ab}\psi_{ab}+\frac{1}{2}\psi^{ab}\psi_{a}\psi_{b}\}(z)}\\\end{array}
i\alpha_{N}2\pi\delta(\sum_{i\inin}k_{i}^{0}-\sum_{i\inout}k_{i}^{0})\pi\delta(\sum_{i\inin}^{str}k_{i}^{1}-\sum_{i\inin}^{dsh}k_{i}^{1}-\sum_{i\inout}k_{i}^{1})
{\pi}^{-}=\frac{{\delta}L}{{\delta}{\partial}_{-}A_{-}}=0,\;{\pi}^{+}=\frac{{\delta}L}{{\delta}{\partial}_{-}A_{+}}=F_{-+},\;{\pi}_{B}=\frac{{\delta}L}{{\delta}{\partial}_{+}B}=0,\;{\pi}_{\psi}=\frac{{\delta}L}{{\delta}{\partial}_{-}{\psi}}=i\bar{\psi}{\gamma}^{-}.
\tilde{R}_{~\lambda\mu\nu}^{\sigma}(\Gamma)=\partial_{\mu}\Gamma_{~\lambda\nu}^{\sigma}-\partial_{\nu}\Gamma_{\lambda\mu}^{\sigma}+\Gamma_{~\alpha\mu}^{\sigma}\Gamma_{~\lambda\nu}^{\alpha}-\Gamma_{~\alpha\nu}^{\sigma}\Gamma_{\lambda\mu}^{\alpha}
\widehat{h}=\displaystyle\frac{\widehat{H}}{{\calM}}=\displaystyle\frac{2\pi}{l}\left(\widehat{L_{0}}+\widehat{\overline{{L}}_{0}}-\displaystyle\frac{c}{12}\widehat{I}+\kappa\displaystyle\frac{l^{2-h}}{\left(2\pi\right)^{1-h}}\widehat{B}\right)
{\calV}(\rho,v)=-m_{p}\frac{u_{0}^{7-p}}{\rho^{7-p}}\left\{\frac{9-p}{4(7-p)}v^{2}+\frac{1}{8}\left[\left(\frac{u_{0}}{R}\right)^{7-p}+\left(\frac{R}{u_{0}}\right)^{7-p}v^{4}\right]\right\}
A_{\gamma\gamma}(s,t,u)=\int\prod_{r=1}^{4}dx_{r}\mu(x)\xi_{1\mu}(k_{1})\xi_{2\nu}(k_{2})\Phi_{3}^{a}\Phi_{4}^{b}\langle0\midV_{k_{1}}^{\mu}(x_{1})V_{k_{2}}^{\nu}(x_{2})S_{k_{3}}^{a}(x_{3})S_{k_{4}}^{b}(x_{4})\mid0\rangle,
\begin{array}{rcl}{S}&{=}&{{\frac{1}{\kappa}}\intd^{11}x\sqrt{|g|}[R]}\\{}&{}&{}\\{}&{}&{-{\frac{T_{KK11}}{2}}\intd^{7}\xi\sqrt{|\gamma|}\left[k^{4/7}\gamma^{ij}\partial_{i}X^{\mu}\partial_{j}X^{\nu}\Pi_{\mu\nu}-5\right]\,,}\\\end{array}
S=\frac{1}{4\pi}\intd^{2}x\left(\frac{1}{2}\partial_{\mu}\phi(x)\partial^{\mu}\phi(x)-4\pi\frac{\tilde{\lambda}}{2}e^{-i\beta\phi(x)/\sqrt{4\pi}}+\sqrt{2}i\alpha_{0}\phi(x)R(x)\right)
\begin{matrix}{\Phi^{\primeA}=\Phi^{A}+{\frac{\partialR^{(n)}}{\partialK_{A}}}+{\calO}(\hbar^{n+1}),}&{K_{A}^{\prime}=K_{A}-{\frac{\partialR^{(n)}}{\partial\Phi^{A}}}+{\calO}(\hbar^{n+1}).}\\\end{matrix}
|B(y,{\bfs})\rangle^{\hat{\mu}}=|B_{X}(y)\rangle_{M^{6}}\otimes|B_{\mathrm{FF}}(y)\rangle\otimes|C\rangle\otimes|B_{\mathrm{gh}}\rangle\otimes|B_{\psi}({\bfs})\rangle^{\hat{\mu}},
\partial_{\alpha}\partial_{\beta}X^{\nu}(\delta_{\nu}^{\mu}g^{\alpha\beta}+3d_{\nu\rho}^{\mu}g^{\alpha\beta\gamma}\partial_{\gamma}X^{\rho}+4d_{\nu\rho\sigma}^{\mu}g^{\alpha\beta\gamma\delta}\partial_{\gamma}X^{\rho}\partial_{\delta}X^{\sigma}+more)
A_{a}^{+}=\frac{1}{2x^{2}}\left[\begin{array}{c}{x_{4}\sigma_{1}+x_{3}\sigma_{2}-x_{2}\sigma_{3}}\\{-x_{3}\sigma_{1}+x_{4}\sigma_{2}+x_{1}\sigma_{3}}\\{x_{2}\sigma_{1}-x_{1}\sigma_{2}+x_{4}\sigma_{3}}\\{-x_{1}\sigma_{1}-x_{2}\sigma_{2}-x_{3}\sigma_{3}}\\{0}\\\end{array}\right]
{\tilde{b}}=\frac{2}{3}\Pi_{\mu\nu}\Pi^{\mu}\Pi^{\nu}+\frac{4}{3}\Pi_{\mu}^{(z)}\Pi^{\mu}\Pi-\frac{2}{15}\Pi_{\alpha\beta}\Pi^{\alpha}\Pi^{\beta}-\frac{3}{5}\Pi_{\mu\alpha}\Pi^{\mu}\Pi^{\alpha}-\frac{3}{5}\Pi_{\alpha}^{(z)}\Pi\Pi^{\alpha}-2\Pi{\calF}\quad.
E={\frac{1}{2}}m_{f}\dot{x}^{2}+V(x);\\\\V(x)=E[1-{\frac{1+{\frac{E}{2m_{f}}}H_{F}}{(1+{\frac{E}{m_{f}}}H_{F})^{2}}}]+{\frac{J_{\parallel}^{2}}{2m_{f}x^{2}}}{\frac{1}{(1+{\frac{E}{m_{f}}}H_{F})^{2}}}.
e^{-1}{\calL}=-{\frac{1}{2}}R-{\frac{1}{4}}G_{IJ}F_{\mu\nu}{}^{I}F^{\mu\nuJ}-{\frac{1}{2}}g_{ij}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}+{\frac{e^{-1}}{8}}\epsilon^{\mu\nu\rho\sigma\lambda}d_{IJK}F_{\mu\nu}^{I}F_{\rho\sigma}^{J}A_{\lambda}^{K}
R_{\alpha\beta\mu\nu}=\frac{1}{\Box}\{\nabla_{\mu}\nabla_{\alpha}R_{\nu\beta}+\nabla_{\nu}\nabla_{\beta}R_{\mu\alpha}-\nabla_{\nu}\nabla_{\alpha}R_{\mu\beta}-\nabla_{\mu}\nabla_{\beta}R_{\nu\alpha}\}+{\calO}({\calR}^{2})\,\,\,,
{\calL}_{jWeyl}^{mass}=\partial_{+}{B}_{i}(\partial_{-}{B}_{i}-\partial_{i}{B}_{-})+\frac12\left(\partial_{+}{B}_{-}\right)^{2}-\frac14\left(\partial_{i}{B}_{j}-\partial_{j}{B}_{i}\right)^{2}-\frac{m^{2}}{2}B_{i}^{2}+{B}_{-}{j}^{-}+{B}_{i}j^{i}
\lambda_{\mathbfi}^{\alpha}(z_{+})=-\frac{i}{8}{\bar{D}}_{{\dot{\alpha}}{\mathbfi}}h^{\alpha{\dot{\alpha}}}\,\qquad\qquad{\bar{\lambda}}^{{\dot{\alpha}}{\mathbfi}}(z_{-})=\frac{i}{8}D_{\alpha}^{\mathbfi}{\bar{h}}^{\alpha{\dot{\alpha}}}\.
\tilde{H}_{n,n+1}^{(\alpha)}=(\phi_{n}^{\alpha}\,\phi_{n+1}^{\alpha})^{-1}\,H_{n,n+1}\,(\phi_{n}^{\alpha}\,\phi_{n+1}^{\alpha})=F_{n,n+1}^{\alpha}\,H_{n,n+1}\,(F_{n,n+1}^{\alpha})^{-1}\,.
\phi_{k}(z)=\frac{1}{2}q_{k}+\frac{z}{4\pi}p_{k}+\frac{i}{\sqrt{4\pi}}\sum_{n\neq0}\frac{a_{n}^{(k)}}{n}e^{-inz},~~~\bar{\phi}_{k}(\bar{z})=\frac{1}{2}q_{k}+\frac{\bar{z}}{4\pi}p_{k}+\frac{i}{\sqrt{4\pi}}\sum_{n\neq0}\frac{\bar{a}_{n}^{(k)}}{n}e^{-in\bar{z}}.
T_{QED}^{\mu\nu}=F^{\mu\alpha}F_{\alpha}^{~~\nu}+(\partial_{\alpha}F^{\mu\alpha})A^{\nu}+i\overline{{\Psi}}\gamma^{\mu}\partial^{\nu}\Psi+\frac{1}{4}g^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}-g^{\mu\nu}\overline{{\Psi}}(i\partial\!\!\!/-eA\!\!\!/-m)\Psi\.
{\hat{T}}_{\\nu}^{\mu}[C]={\frac{1}{2(p+1)!}}e^{2\Phi}G^{\mu\lambda\ldots\sigma}G_{\nu\lambda\ldots\sigma}-{\frac{(p+1)}{16}}e^{2\Phi}{\frac{G^{2}}{(p+2)!}}g_{\,nu}^{\mu}\,.
\begin{array}{ll}{\alpha_{k}=\frac{1}{2}}&{k=1,2,\cdots,g}\\{\beta_{k}=\frac{1}{2}(1-\delta_{m_{k-1},m_{k}})}&{k=1,2,\cdots,g}\\{\mathrm{and}\\\{\bfm}=(1,m_{1},\cdots,m_{g-1},1).}&{}\\\end{array}
S_{\mathrm{wv}}=-\frac{T_{3}}{\kappa}\left\{\int_{D3}\sqrt{-\mathrm{det}G}\,{m^{J}b_{J}}+\int_{D3}C_{(4)}\,{m^{J}b_{J}}+\int_{D3}{m^{J}{\mathcalA}_{\,4,J}}\right\}~.
\bigl\{A,\,B\bigr\}={\frac{\partialA}{\partialy^{\mu}}}f^{\mu\nu}{\frac{\partialB}{\partialy^{\nu}}}+{\frac{\partialA}{\partial\psi^{\alpha}}}\omega^{\alpha\beta}{\frac{\partialB}{\partial\psi^{\beta}}}\,,
\Gamma_{(1)}(M)=\intdx{\frac{1}{2}}(4\pi)^{-d/2}{\frac{\pi(-1)^{(d-1)/2}}{\Gamma(d/2+1)}}\Biggl\{r(d-2)M^{d}+2\sum_{{\alpha}>0}J_{d/2,\,q}\left({\frac{M^{2}}{H}}\right)H^{d/2}\Biggr\}
{\calS}=\frac18(\tau^{+}-\tau^{-})\intd^{4}x{}\epsilon^{\mu\nu\alpha\beta}{}{\calF}_{\mu\nu}^{\tau\lambda}{}{\calF}_{\alpha\beta}^{\sigma\rho}{}\epsilon_{\tau\lambda\sigma\rho}+\fraci8(\tau^{+}+\tau^{-})\intd^{4}x{}\epsilon^{\mu\nu\alpha\beta}{}{\calF}_{\mu\nu}^{\tau\lambda}{}{\calF}_{\alpha\beta}^{\sigma\rho}{}\delta_{\tau\lambda\sigma\rho}.
A_{3,\zeta}=Q\rho\tau_{,\zeta}\\,\\\\A_{3,\bar{\zeta}}=-Q\rho\tau_{,\bar{\zeta}}\\,\\\\\kappa^{2}=I^{3}={\frac{h^{3}e^{\tau_{0}\tau}}{(1-{\frac{2m}{r}})g_{22}^{\beta}}}
E^{a}{}_{\alpha}=\delta^{a\beta}\left[\delta_{\beta\alpha}-\frac{1}{|\vec{\chi}^{R}|^{2}+(\vec{\chi}^{R}\vec{\chi}^{L})}\left((1-\sqrt{1-\frac{|\vec{\chi}^{L}|^{2}}{|\vec{\chi}^{R}|^{2}}})\chi_{\beta}^{R}\chi_{\alpha}^{R}\+\\chi_{\beta}^{L}\chi_{\alpha}^{R}\right)\right]
\prod_{a<b}^{4}{z_{ab}}^{b^{2}\mu_{ab}}{x_{ab}}^{\lambda_{ab}}F(z,x)\sim\prod_{a<b}^{4}{z_{ab}}^{-2b^{2}j_{a}j_{b}}\ointdz_{0}dx_{0}\prod_{a}z_{0a}^{2b^{2}j_{a}}x_{0a}^{2j_{a}}\cdot\sum_{a}\frac{j_{a}}{z_{0a}x_{0a}}~.
G_{o}(K;\vec{x},\vec{x^{\prime}};\phi,\phi^{\prime})=-\frac{1}{8\pi^{2}}\frac{e^{iK|\vec{x}-\vec{x^{\prime}}|}}{|\vec{x}-\vec{x}^{\prime}|}+X(K;\vec{x},\vec{x^{\prime}};\phi,\phi^{\prime});
\frac{\partial}{\partial\beta}\int_{\Lambda_{2}}^{\Lambda_{1}}d\lambda\frac{1}{\sqrt{2(E_{F}^{(0)}-V)}}=\int_{\Lambda_{2}}^{\Lambda_{1}}d\lambda\frac{\partial}{\partial\beta}\left\{\frac{1}{\sqrt{2(E_{F}^{(0)}-V)}}\right\}
f_{5}=\frac{5}{6}(\frac{1}{2}+h_{2}-\frac{q^{2}}{m_{H}^{2}}a_{0}^{2})f_{3}+\frac{5}{2}(\frac{1}{6}h_{4}+\frac{1}{2}h_{2}^{2}+f_{1}^{2}-\frac{2q^{2}}{m_{H}^{2}}a_{0}a_{2})f_{1}.
\tilde{W}=M_{ij}q_{i}\bar{q}_{j}+\tilde{h}(q_{i_{1}}q_{i_{2}}\dotsq_{i_{N_{f}-N_{c}}}+\bar{q}_{i_{1}}\bar{q}_{i_{2}}\dots\bar{q}_{i_{N_{f}-N_{c}}}+\dots).
S_{\mathrm{ct}}=-\frac{1}{8\piG_{p+2}}\intd^{p+1}x\sqrt{-h}\frac{c_{0}}{l_{\mathrm{eff}}},\\\c_{0}=\sqrt{p(p+1)},\\\\frac{1}{l_{\mathrm{eff}}}=be^{-a\phi/2}\sqrt{\frac{1}{2p(p+1)}},
\Gamma[F,V]=\int_{0}^{\infty}\!{\frac{dT}{T}}\;\frac{\mathrm{e}^{-m^{2}T}}{[4\piT]^{d/2}}\;\mathrm{tr}\;\int\!d^{d}x_{0}\;\sum_{n=1}^{N}\;\frac{(-T)^{n}}{n!}\;O_{n}[F,V]\,,
Z_{2}(p)\!=\!1\!+\!{\frac{(\tilde{g}_{2})^{2}}{N_{f}}}\!\int_{k,q}\!\left(\!4w(p+q)w(k\!-\!{\frac{q}{2}})\right)^{2}\!S_{RR}(p+q)S_{LL}(p)R(k,\!q).
d^{D}\mathbf{x}^{\prime}\,d^{D}\mathbf{p}^{\prime}=\left|\frac{\partial(x_{1}^{\prime},\cdots,x_{D}^{\prime},p_{1}^{\prime},\cdots,p_{D}^{\prime})}{\partial(x_{1},\cdots,x_{D},p_{1},\cdots,p_{D})}\right|d^{D}\mathbf{x}\,d^{D}\mathbf{p}\;.
I=\int_{X}d^{4}x\\varepsilon^{\mu\nu\rho\sigma}\left[2{R_{\mu\nu}}^{0i}(\omega^{+})R_{\rho\sigma0i}(\omega^{+})+{R_{\mu\nu}}^{ij}(\omega^{+})R_{\rho\sigmaij}(\omega^{+})\right].
\begin{array}{rl}{\displaystyle\int_{\infty}^{0}d(1/x)G(1/x)(1/x)^{s^{\prime}/2-1}}&{=}\\{\displaystyle\int_{0}^{\infty}dxG(x)x^{(1-s^{\prime})/2-1}}&{=2Z(1-s^{\prime}).}\\\end{array}
\tilde{H}=\int_{\Omega}:\rho^{-2}g_{ab}\tilde{\pi}^{a}\tilde{\pi}_{b}+\rho^{2}\nabla\tilde{\theta}^{a}\cdot\nabla\tilde{\theta}^{b}:\,=\int(\rho^{-2}\tilde{\pi}^{a}+i\sqrt{\nabla}\tilde{\theta}_{a})\rho^{2}g_{ab}(\rho^{-2}\tilde{\pi}^{b}-i\sqrt{\nabla}\tilde{\theta}^{b})\.
\hat{Y}W:=\left[m+e_{a}^{\mu}\gamma^{a}(p_{\mu}+\frac{1}{2}i\nabla_{\mu})+\frac{1}{2}i\kappae_{a}^{\mu}\gamma^{a}R_{\mu\nu}^{bc}\left\{\frac{\partial}{\partialp_{\nu}}\cdot,\sigma_{bc}\right\}\right]W
\Gamma_{t}[\phi]=\int\!\!d^{D}x\,\left\{V(\phi^{2},t)+\frac{1}{2}K(\phi^{2},t)\left(\partial_{\mu}\phi^{a}\right)^{2}+\frac{1}{2}Z(\phi^{2},t)\left(\phi^{a}\partial_{\mu}\phi^{a}\right)^{2}+\cdots\right\}
S=\intd^{5}x\sqrt{-G}\left[{\frac{c^{4}}{16\piG_{5}}}{\calR}-\Lambda\right]+\intd^{4}x\sqrt{-g}\left[{\calL}_{\mathrm{mat}}+{\calL}_{\psi}+e^{-2\psi}{\calL}_{\mathrm{em}}-\sigma\right],
+[(pq+p\cdotq)\epsilon_{\nu\sigma\eta}(\frac{q_{\sigma}}{q}-\frac{p_{\sigma}}{p})+(\frac{q_{\nu}}{q}+\frac{p_{\nu}}{p})\epsilon_{\sigma\tau\eta}q_{\sigma}p_{\tau}]\gamma_{\eta}\frac{1}{(p+q+|p-q|)^{2}}
\frac{1}{\pi}\frac{d\delta_{\ell}}{dk}=\frac{2k}{\pi}\mathsf{Im}\int_{0}^{\infty}\left(G_{\ell}(r,r,k+i\epsilon)-G_{\ell}^{(0)}(r,r,k+i\epsilon)\right)dr=\rho_{\ell}(k)-\rho_{\ell}^{(0)}(k)\,.
{\calA}=\intd^{3}x\langle\langle\widehat\chi\rangle\rangle=\intd^{3}x\left({\calF}_{+}^{\mu}{\calL}_{\mu\nu}^{+}{\calF}_{+}^{\nu}+{\calF}_{-}^{\mu}{\calL}_{\mu\nu}^{-}{\calF}_{-}^{\nu}\right)
\frac{1}{L}\sum_{n=-\infty}^{\infty}F\left(\frac{2n+1}{L}\right)=\frac{1}{L}\sum_{n=-\infty}^{\infty}F\left(\frac{n}{L}\right)-\frac{1}{L}\sum_{n=-\infty}^{\infty}F\left(\frac{2n}{L}\right)\,,
Z_{\mathrm{log}}[p]=\sum_{\lambda=0}^{2p-1}|\Theta_{\lambda,p}|^{2}+\alpha\sum_{{\mu=1\atop\mu\neqp}}^{2p-1}\frac{1}{2}\left((\partial\Theta)_{\mu,p}(\nabla\Theta)_{\mu,p}^{*}+(\nabla\Theta)_{\mu,p}(\partial\Theta)_{\mu,p}^{*}\right)\,,
S_{\mathrm{hyp}}(\epsilon_{\kappa})\stackrel{0\leftarrow\tau/\sigma}{\longleftarrow}S_{\mathrm{catenary}}(\epsilon_{\sigma})\stackrel{\tau/\sigma\rightarrow\infty}{\longrightarrow}S_{\mathrm{3/2-parab}}(\epsilon_{\kappa})~.
\frac{\dot{a}^{2}+k}{a^{2}}=\frac{8\piG}{3}\left[\frac{1}{2}\frac{\dot{\phi}^{2}}{1+2\etaf}+\frac{\Lambda}{8\piG}\frac{1}{(1+4\etaf)}+\frac{\rho_{0}}{a^{4}}\right]
S=t\int_{\Sigma}\Bigl(g_{\mu\bar{\mu}}dz^{\mu}*d\bar{z}^{\bar{\mu}}-g_{\mu\bar{\mu}}\rho^{\mu}*D\bar{\eta}^{\bar{\mu}}\Bigr)+\frac{1}{\kappa}\int_{\Sigma}\Bigl(\rho^{\mu}D\chi_{\mu}-\frac{1}{2}R_{\mu\bar{\mu}\nu}^{\lambda}\rho^{\mu}\rho^{\nu}\bar{\eta}^{\bar{\mu}}\chi_{\lambda}\Bigr),
x^{\mu}-y^{\mu}-\frac{2i}{(1^{12}2^{12})}[\xi^{2/12}\sigma^{\mu}\bar{\theta}^{1}-\xi^{1/12}\sigma^{\mu}\bar{\theta}^{2}]\,,\qquad\xi^{I/12}\equiv[(1^{I3}2^{12})\theta_{3}+(1^{I4}2^{12})\theta_{4}]\,.
S_{N}(x)={\frac{1}{\sqrt{1-\beta}}}\sum_{l=0}^{N}\beta^{l}\sum_{n=0}^{l}c_{n}{\frac{\Gamma(n/2+l-1/2)}{\Gamma(3n/2-1/2)\Gamma(l-n+1)}}\left[{\frac{g}{x(x^{2}-1)}}\right]^{n}.
\tau_{k}^{(j)}=\frac{1}{W_{k}}\mathrm{det}\left(\begin{matrix}{y_{k+1},}&{y_{k+2},}&{\cdots,}&{y_{k},}&{\cdots,}&{y_{k+n+1}}\\{\vdots}&{}&{}&{}&{}&{\vdots}\\{\partial^{n}y_{k+1},}&{\partial^{n}y_{k+2},}&{\cdots,}&{\partial^{n}y_{k},}&{\cdots,}&{\partial^{n}y_{k+n+1}}\\\end{matrix}\right).
q=\frac1{32\pi^{2}}\int_{0}^{\infty}dr\int_{0}^{\pi}d\chi\,\int_{0}^{\pi}d\theta\,\int_{0}^{2\pi}d\phi\,\epsilon_{\alpha\beta\gamma\delta}\,\mathrm{tr}\,\hat{F}^{\alpha\beta}\,\hat{F}^{\gamma\delta}.
\frac{1}{\sqrt{-i}}\left(\frac{x^{-}}{x^{+}}\right)^{2}F(x^{+},x^{-},0)=\sum_{\alpha}\frac{1}{2(2\pi)^{5/2}}\frac{M_{\alpha}^{9/2}}{\sqrt{r}}K_{9/2}(M_{\alpha}r)\frac{|\langleu|\alpha\rangle|^{2}}{lK^{3}|N_{u}|^{2}}
{\calT}_{stan}\,=\,\left[\begin{array}{cc}{1}&{\qquad0}\\{}&{}\\{1}&{\qquad1}\\\end{array}\right]\qquad\qquad{\calS}_{stan}\,=\,\left[\begin{array}{cc}{0}&{\qquad-1}\\{}&{}\\{1}&{\qquad0}\\\end{array}\right]
\nabla_{\mu}\nabla^{\mu}\Phi=\frac{\kappa^{2}}{2}T_{3}\;e^{3\Phi}V(T)\frac{3+2e^{-2\Phi}\nabla_{\alpha}T\nabla^{\alpha}T}{\sqrt{1+e^{-2\Phi}\nabla_{\alpha}T\nabla^{\alpha}T}},
d\Pi_{\mu_{1}\dots\mu_{p-1}\alpha_{1}}=a_{1}\left((C\Gamma_{\nu\mu_{1}\dots\mu_{p-1}})_{\beta\alpha_{1}}\Pi^{\nu}\Pi^{\beta}+{\frac{a_{s}}{a_{0}}}(C\Gamma^{\nu})_{\beta\alpha_{1}}\Pi_{\nu\mu_{1}\dots\mu_{p-1}}\Pi^{\beta})\right)
E=\frac{4\pi}{\kappag}\int_{0}^{\infty}\left[\frac{1}{2}\varphi_{\infty}^{2}+e^{2\varphi_{\infty}}\frac{n^{2}}{2x^{4}}\right]x^{2}dx=\frac{4\pi}{\kappag}n\,
\nabla_{\sigma_{o}}^{2}\,\Gamma_{\Sigma}(\vec{\sigma}_{o},\vec{\sigma}_{o}^{\prime})=\nabla_{\sigma_{o}}^{2}\,c(\vec{\sigma}_{o}-\vec{\sigma}_{o}^{\prime})=\delta^{3}(\vec{\sigma}_{o}-\vec{\sigma}_{o}^{\prime}).
M_{4}^{2}=M_{10}^{8}e^{-2\varphi}\left(\intd^{6}y\sqrt{g^{\cal{B}}}-\sum_{n\neq0}{\frac{1}{\lambda^{2}}}\intd^{6}y^{\prime}\sqrt{g^{\cal{B}}}\rho(y^{\prime})f_{\lambda}(y^{\prime})\intd^{6}y\sqrt{g^{\cal{B}}}\bar{f}_{\lambda}(y)\right)~.
ds_{D}^{2}=H^{-\frac{d-2}{D-2}}\left[-fdt^{2}+\sum_{i=1}^{p}(dx^{i})^{2}+HR_{T}^{2}\left(f^{-1}dR^{2}+dv^{2}+R^{2}d\Omega_{d-2}^{2}\right)\right]
D_{q^{2}}^{+}\left[D_{q^{2}}^{+}f+D_{q^{2}}^{-}f\right]\left(q^{2}x^{+}\right)=D_{q^{2}}^{+}\left(\left[D_{q^{2}}^{+}f+D_{q^{2}}^{-}f\right]\left(q^{2}x^{+}\right)\right).
\int_{\xi_{1}}^{\xi_{2}}\frac{d\xi}{\xi}\phi_{\omega}^{(1)}(\xi)\phi_{v}^{(2)}(\xi)=\frac{\xi}{\omega^{2}-\upsilon^{2}}\left[\phi_{\omega}^{(1)}(\xi)\frac{d\phi_{\upsilon}^{(2)}(\xi)}{d\xi}-\phi_{\nu}^{(2)}(\xi)\frac{d\phi_{\omega}^{(1)}(\xi)}{d\xi}\right]_{\xi_{1}}^{\xi_{2}}.
\mathcal{M}=\left(\begin{array}{cc}{-i(\Phi_{a}\partial_{n}X^{a}+\frac{\alpha^{\prime}}{2}\theta_{ab}\psi^{a}\psi^{b})+u^{*a}u^{b}\Phi_{a}\Phi_{b}}&{\sqrt{\alpha^{\prime}}\theta_{ab}u^{a*}\psi^{b}}\\{\sqrt{\alpha^{\prime}}\theta_{ab}u^{a}\psi^{b}}&{-i(\Phi_{a}\partial_{n}X^{a}+\frac{\alpha^{\prime}}{2}\theta_{ab}\psi^{a}\psi^{b})+u^{a*}u^{b}\Phi_{a}\Phi_{b}}\\\end{array}\right)\.
{\calL}(\frac{1}{2},\alpha,g)=\bar{\psi}[\sigma_{\mu}(\partial_{\mu}+ia_{\mu}+iC_{\mu})+M]\psi+\frac{g}{4}(\bar{\psi}\psi)^{2}-\frac{i}{8\pi\alpha}\epsilon_{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}\;,
2\mu{\frac{\sqrt{J_{1}J_{2}J_{3}}}{N}}\big(\bar{O}^{\primeJ_{3}}\Pi^{\primeJ_{1}}\bar{\Pi}^{\primeJ_{2}}\delta_{J_{3}+J_{1},J_{2}}+O^{J_{3}}\Pi^{\primeJ_{1}}\bar{\Pi}^{\primeJ_{2}}\delta_{J_{3}+J_{2},J_{1}}\big).
\begin{array}{lcl}{X_{n}^{(J)}(\tau)}&{=}&{{\displaystyleC_{n}^{(J)}\sqrt{-\tau}~J_{-\nu}\left(-\frac{2\pi}{\lambda_{J}}|n|\tau\right)+D_{n}^{(J)}\sqrt{-\tau}~J_{\nu}\left(-\frac{2\pi}{\lambda_{J}}|n|\tau\right)}~,~(n\neq0)}\\{}&{}&{}\\{X_{0}^{(J)}(\tau)}&{=}&{{\displaystyleC_{0}^{(J)}(-\tau)^{\frac{1}{2}-\nu}+D_{0}^{(J)}(-\tau)^{\frac{1}{2}+\nu}}}\\\end{array}
\delta\mathrm{D}=2\xi_{\mathrm{B}}^{^{\prime}}\mathrm{D}+\mathrm{D}^{^{\prime}}\xi_{\mathrm{B}}+\frac{c\mu}{24\pi}\xi_{\mathrm{B}}^{\prime\prime\prime}-\mathrm{Tr}\left(\mathrm{A}\Lambda_{\mathrm{B}}^{\prime}\right)=0
\widehat{R}_{\mu\nu}=\mathrm{E}_{\mu\nu}-\frac{1}{d-2}g_{\mu\nu}g^{\alpha\beta}\mathrm{E}_{\alpha\beta}+{\calO}\left(\widehat{R}^{2}\right),\qquad\widehat{R}=-\frac{2}{d-2}g^{\mu\nu}\mathrm{E}_{\mu\nu}+{\calO}\left(\widehat{R}^{2}\right).
Q_{\ell}|V_{A_{1}}(\theta_{1})V_{A_{2}}(\theta_{2})\cdotsV_{A_{n}}(\theta_{n})\rangle=\left[\sum_{i=1}^{n}\xi_{i}^{\ell}(M_{i}e^{\theta_{i}})^{\ell}\right]|V_{A_{1}}(\theta_{1})V_{A_{2}}(\theta_{2})\cdotsV_{A_{n}}(\theta_{n})\rangle
\Gamma_{\4V}^{(0)pert}\=\\frac{3}{2}\,C^{(4)}\,\left(\,L_{(2)}-L_{(3)}\,\right)\+\\frac{1}{2}\,\left(\,C^{(5)}-C^{(6)}\,\right)\left(-2L_{(1)}+L_{(2)}+L_{(3)}\,\right).
A^{a}=U_{b}^{a}\,\psi^{\dagger}\sigma^{\alpha}\psi=e_{\mu}^{a}(x)\,\frac{\partialx^{\mu}}{\partialq^{\alpha}}\,e_{b}^{\alpha}(q)\,\psi^{\dagger}(q)\,VV^{\dagger}\,\sigma^{b}\,VV^{\dagger}\,\psi(q).
\intdp^{+}\intd^{2}p_{\perp}f(p^{+},\bbox{p}_{\perp})\simeq\frac{2\pi}{L}\left(\frac{\pi}{L_{\perp}}\right)^{2}\sum_{n}\sum_{n_{x},n_{y}=-N_{\perp}}^{N_{\perp}}f(n\pi/L,\bbox{n}_{\perp}\pi/L_{\perp})\,.
u_{--}=\frac{1}{2}\left(\begin{array}{r}{0}\\{1}\\{0}\\{-1}\\\end{array}\right),u_{-+}=\frac{(-1)}{2}\left(\begin{array}{l}{1}\\{0}\\{1}\\{0}\\\end{array}\right),u_{+-}=\frac{1}{2}\left(\begin{array}{l}{0}\\{1}\\{0}\\{1}\\\end{array}\right),u_{++}=\frac{1}{2}\left(\begin{array}{r}{1}\\{0}\\{-1}\\{0}\\\end{array}\right),
\begin{array}{rcl}{\epsilon}&{\rightarrow}&{\epsilon^{\Gamma}=D\epsilon-C\epsilon^{\prime}+{\frac{1}{2}}\mathrm{diag}(CD^{t})~,}\\{\epsilon^{\prime}}&{\rightarrow}&{\epsilon^{\prime\,\Gamma}=-B\epsilon+A\epsilon^{\prime}+{\frac{1}{2}}\mathrm{diag}(AB^{t})~.}\\\end{array}
\tilde{\calL}={\frac{1}{\kappa^{D-2}}}\intdr\e^{(D-3)A}\tilde{e}\left\{\tilde{R}+\bar{\tilde{\psi}}_{\mu+i}\tilde{\Gamma}^{\mu\nu\rho}\tilde{\nabla}_{\nu}\tilde{\psi}_{\rho+}^{i}\right\}\.
\mathcal{D}_{P+1}(N,G)=\sum_{\begin{array}{l}{{g_{1},g_{2},....,g_{P}\geq0}}\\{\sum_{j=1}^{P}j.g_{j}=N}\\\end{array}}G![(G-\sum_{k=1}^{P}G_{P})!\prod_{l=1}^{P}(g_{l}!)]^{-1}
\nonumber\left(C_{\gamma_{1}\gamma_{2}\gamma}^{N_{1}N_{2}N}w_{\gamma_{1}}w_{\gamma_{2}}w_{\gamma}\right)^{2}=\intdg\,D_{00}^{(N_{1})}D_{00}^{(N_{2})}\overline{{D_{00}^{(N)}}}.
I_{m}=\intd^{4}x\sqrt{{\tilde{g}}}[-\frac{1}{2}D_{\mu}\Phi(D^{\mu}\Phi)^{*}-\frac{1}{2}D_{\mu}\Sigma(D^{\mu}\Sigma)^{*}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}H_{\mu\nu}H^{\mu\nu}-V(|\Phi|,|\Sigma|)],
\intd^{25-p}q_{\bot}\,(2)^{q_{\bot}^{2}}\left(\frac{kW^{2}}{1+W^{2}}\right)^{q_{\bot}^{2}}\frac{2}{W^{2}}\,c\partialX(i)\,\tilde{c}\bar{\partial}X(i)e^{iq_{\bot}\cdotX(i)}.
B=\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
\lambda_{\mathrm{os}}^{4/3}=(\hat{G}_{\mathrm{OM}}^{-1})_{yy}=z^{-1}(\hat{C}^{-1})_{yy}=\left({\frac{{\frac{1}{2}}({\calD}_{0}+1)-\sqrt{{\frac{1}{4}}({\calD}_{0}-1)^{2}+({\calD}_{0}-{\calD})}}{\sqrt{\calD}}}\right)^{1/3}\sqrt{\calD}\,,
\delta\alpha_{ab}=\partial_{\mu}k_{\;\;ab}^{\mu},\;\delta\alpha_{\;\;ab}^{\mu\nu}=\partial_{\beta}k_{\;\;ab}^{\beta\mu\nu},\;\delta\alpha_{\;\;ab}^{\mu\nu\;\lambda\rho}=\partial_{\beta}k_{\;\;ab}^{\beta\;\mu\nu\;\lambda\rho}.
{\calC}=\left(\begin{matrix}{0}&{2}&{-1}&{1}&{-2}\\{-2}&{0}&{2}&{-1}&{1}\\{1}&{-2}&{0}&{2}&{-1}\\{-1}&{1}&{-2}&{0}&{2}\\{2}&{-1}&{1}&{-2}&{0}\\\end{matrix}\right),
S_{\mathrm{fermion}}=i\intd^{4}x\sqrt{-g}{\bar{\Psi}}e_{\ell}^{\mu}\gamma^{\ell}\left(\partial_{\mu}-\frac{i}{4}\omega_{\mu}^{mn}\sigma_{mn}-\frac{i}{2}eA_{\mu}^{a}\tau^{a}\right)\frac{1-\gamma_{5}}{2}\Psi.
{\tilde{\gamma}_{5}}={\frac{\epsilon^{cd}}{4\sqrt{g}}}{\hat{\epsilon}^{EF}}\partial_{c}X^{\mu}\partial_{d}X^{\nu}N_{E}^{\lambda}N_{F}^{\rho}\frac14[\gamma_{\mu},\gamma_{\nu}]\,[\gamma_{\lambda},\gamma_{\rho}].
\eta^{n}\cdot\tau^{n}{\cal{G}}\;\;:=\;\;{\cal{Y}}_{[p_{1},...,p_{N}]}\circ\left({\frac{1}{2^{n}}}\sum_{i_{1},...,i_{2n}=1}^{N}\left(\sum_{\pi}\eta^{(\pi(i_{1})\pi(i_{2}))}...\eta^{(\pi(i_{2n-1})\pi(i_{2n}))}\right)\tau^{(i_{1}i_{2})}...\tau^{(i_{2n-1}i_{2n})}\right){\cal{G}}
\left[\nabla^{\mu}\nabla_{\mu}+4\left(\begin{matrix}{-k(k+3)}&{9}\\{-4k(k+3)}&{-k(k+3)-18}\\\end{matrix}\right)\right]\left(\begin{matrix}{48\sqrt{2}b^{I}}\\{h_{2}^{I}}\\\end{matrix}\right)Y^{I}=0.