harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left.{\frac{16\,\pi^{2}}{g_{i,\,\mathrm{eff}}^{2}}}\right\vert_{\mathrm{\field\theory}}=\frac{16\,\pi^{2}}{g_{i,\,\mathrm{bare}}^{2}}+b_{i}\,(4\pi)^{\epsilon}\int_{0}^{\infty}\frac{dt}{t^{1-\epsilon}}\,\Gamma_{FT}\left(\frac{\mu}{\sqrt{\pi}M}\right)\,,
f(h)=\frac{Ah}{2}+\mathrm{ln}{\frac{(h-1/2)(h-b)}{(h-a)(h-c)}}+g(h)\int_{a}^{1/2}\frac{\mathrm{d}s}{(h-s)g(s)}+g(h)\int_{c}^{b}\frac{\mathrm{d}s}{(h-s)g(s)}
\begin{array}{rl}{\Lambda\sqrt{\sigma_{r,3}}=}&{\omega_{r,0}-i\gamma_{r,0}=\Lambda\sqrt{\sigma_{r,1}^{*}}\,,}\\{\varrho_{r,2}i^{-\frac{3}{2}}(\sigma_{r,3})^{\frac{1}{4}}=}&{\chi_{r,0}\,e^{-i\varphi_{r,0}}=\left[\varrho_{r,0}i^{\frac{3}{2}}(\sigma_{r,1})^{\frac{1}{4}}\right]^{*}\,.}\\\end{array}
\begin{array}{l}{(\gammap-m)\Psi(x)-\lambda\left(\bar{\Psi}(x)\Psi(x)\right)\Psi(x)=0,}\\{\bar{\Psi}(x)(\gamma\overleftarrow{p}+m)+\lambda\bar{\Psi}(x)\left(\bar{\Psi}(x)\Psi(x)\right)=0,}\\\end{array}
G_{\mu\nu}(p)=\int_{0}^{\infty}ds{\frac{\rho_{1}(s)}{p^{2}-s+i\epsilon}}\left(\delta_{\mu\nu}-{\frac{p_{\mu}p_{\nu}}{p^{2}}}\right)+{\frac{\theta}{16}}\int_{0}^{\infty}ds{\frac{\rho_{2}(s)}{p^{2}-s+i\epsilon}}\epsilon_{\mu\rho\nu}p^{\rho},
S_{\mathrm{total}}={\frac{3M_{\mathrm{P}}^{2}}{4\pi}}\;2\pi^{2}\;{\frac{1}{2}}\int_{R^{2}(0)}^{\infty}dR^{2}\left(1-\sqrt{1-{\frac{R^{4}(0)}{R^{4}}}}\,\right)={\frac{3\pi^{2}}{8}}M_{\mathrm{P}}^{2}R^{2}(0)\cdot\left(1-{\frac{2}{\pi}}\right)\.
N_{nm}^{rs}=\frac1{n\,m}\oint_{0}\frac{dz}{2\pii}z^{-n}\oint_{0}\frac{d\zeta}{2\pii}\zeta^{-m}\frac{f_{r}^{\prime}(z)f_{s}^{\prime}(\zeta)}{(f_{r}(z)-f_{s}(\zeta))^{2}}
L=-m\sqrt{-\dot{x}^{2}\left(1-\frac{2s}{m}\frac{\dot{\varphi}}{(\dot{x},n)}-\frac{\varrho^{2}}{m^{2}}\frac{\dot{\varphi}^{2}}{(\dot{x},n)^{2}}\right)}+\varrho\frac{(\dot{x},\partial_{\varphi}n)}{(\dot{x},n)}\dot{\varphi}\.
H={\frac{1}{2}}\int_{x,y}\alpha_{ij}^{ab}(x,y)E_{i}^{a}(x)E_{j}^{b}(y)+{\frac{1}{2}}ic_{A}A_{i}^{a}(x)E_{i}^{a}(x)G(x,x)+{\frac{1}{2}}B_{i}^{a}(x)B_{i}^{a}(x)
\begin{array}{l}{X_{n}\equivC^{\alpha_{1}}\dotsC^{\alpha_{n}}X_{n,\alpha_{1}\dots\alpha_{n}},}\\{X_{[n]}\equiv\displaystyle\sum_{k=0}^{n}C^{\alpha_{1}}\dotsC^{\alpha_{k}}X_{k,\alpha_{1}\dots\alpha_{k}},}\\\end{array}
V(M_{1},M_{2})=\frac{1}{N}\mathrm{Tr}\,M_{1}^{2}+\frac{1}{N}\mathrm{Tr}\,M_{2}^{2}+\frac{1}{N^{2}}\mathrm{Tr}\,M_{1}^{2}\,\mathrm{Tr}\,M_{2}^{2}\;.
\zeta\left(\begin{array}{rcr}{s_{1}\;}&{\ldots}&{s_{k}\;}\\{\lambda^{p_{1}}}&{\ldots}&{\lambda^{p_{k}}}\\\end{array}\right)=\sum_{n_{1}>n_{2}>\ldots>n_{k}>0}\;\;\;\prod_{j=1}^{k}\frac{\lambda^{p_{j}n_{j}}}{n_{j}^{s_{j}}}
\left({\frac{\Phi_{\alpha_{1}}\cdots\Phi_{\alpha_{N-2}}{\calB}_{\beta_{1}\dots\beta_{N-4}}}{\Lambda^{3}}}\right)^{\frac{3}{2N-9}}=\left({\frac{\left[\Phi_{\alpha_{1}}\cdots\Phi_{\alpha_{N-2}}{\calB}_{\beta_{1}\dots\beta_{N-4}}\right]^{3}}{\Lambda^{9}}}\right)^{\frac{1}{2N-9}}~.
S_{\mathrm{sugra}}=\frac{\pi\alpha_{\mathrm{eff}}^{\prime}}{G_{o(1)}^{4}}\intd^{6}k[div(\Lambda,k)-\frac{1}{2\pi}e^{-i(\nu+\frac{1}{2})\pi}+....]\varphi_{0}(k)\varphi_{0}(-k)
\Pi_{\prime\prime\,tad}^{\mu\nu,\alpha\beta}\equiv\frac\kappa4\left(\displaystyle\frac{a+4c+2d}{a+4b}\right)\left(\eta^{\mu\alpha}\Gamma^{\nu\beta}+\eta^{\mu\beta}\Gamma^{\nu\alpha}+\eta^{\nu\alpha}\Gamma^{\mu\beta}+\eta^{\nu\beta}\Gamma^{\mu\alpha}\right).
\begin{array}{rcl}{a_{i,\kappa}}&{=}&{\frac{\kappa}{n-s}+\frac{n-i}{n}-\frac{\gamma}{n(n-s)}\,\,\,(i=1,2,\dots,n),}\\{b_{i,\kappa}}&{=}&{\frac{\kappa}{n-s}+\frac{s-i+1}{s}-\frac{\gamma}{s(n-s)}\,\,\,(i=1,2,\dots,s),}\\{b_{i,\kappa}}&{=}&{\frac{\kappa}{n-s}+\frac{i-s}{n-s}+\frac{\delta_{i}}{n-s}\,\,\,(i=s+1,\dots,n-1).}\\\end{array}
\begin{array}{l}{\langle:\Phi\Phi^{*}:(x):\Phi\Phi^{*}:(y)\rangle}\\{=\intd\thetadrrr^{-d-(1-\gamma)(D-d)}F(\theta,r^{-\frac{1}{2}}({\bfx}_{F}-{\bfy}_{F}),r^{-\frac{1}{2}+\frac{\gamma}{2}}({\bfx}_{G}-{\bfy}_{G}))}\\\end{array}
D(k^{2})=\frac{1}{\hbar^{2}k^{2}-\hbar^{4}k^{4}/m_{p}^{2}-m^{2}}\simeq\left(\frac{1}{1-\hbar^{2}k^{2}/m_{p}^{2}}\right)\left(\frac{1}{\hbar^{2}k^{2}-m^{2}}\right)\,.
T_{\frac{1}{2},s_{q}}(\lambda)=(\lambda+is_{q})^{N}\prod_{j=1}^{n}\frac{\lambda-\lambda_{j}-i}{\lambda-\lambda_{j}}+(\lambda-is_{q})^{N}\prod_{j=1}^{n}\frac{\lambda-\lambda_{j}+i}{\lambda-\lambda_{j}},
\Big[\begin{array}{cc}{(\alpha_{1}+b_{3})(f)}&{\alpha_{2}(f)}\\{(\beta_{1}+b_{4})(f)}&{\beta_{2}(f)}\\\end{array}\Big]=\Big[\begin{array}{cc}{(\alpha_{1}+b_{3})(f)}&{1}\\{(\beta_{1}+b_{4})(f)}&{1}\\\end{array}\Big].
\frac{1}{\overline{{\eta}}^{24}}\left[{\overline{{Q}}}_{\alpha}^{2}-\frac{1}{4\pi\Im\tau}\right]\overline{{\Omega}}=\frac{b_{\alpha}}{3}+2\left[{\overline{{\Omega}}}_{2}^{\phantom{2}}-\frac{1}{8\pi\Im\tau}\right]\frac{\overline{{\Omega}}}{\overline{{\eta}}^{24}}+\frac{\overline{{j}}}{24}-42\,
\frac{p_{r}^{2}}{g_{rr}}+\frac{p_{\theta}^{2}}{g_{\theta\theta}}=\frac{p_{\phi}-\left(\frac{g_{\phi\phi}E^{2}}{\calD}+\mu^{2}\right)/\left(\frac{2g_{t\phi}}{\calD}E\right)}{\frac{-{\calD}}{2g_{t\phi}E}},
\tilde{H}=\int\mathrm{d}^{2}x\left[\tilde{\Pi}_{\alpha}^{*}\tilde{\Pi}_{\alpha}+(\partial_{i}\tilde{Z}_{\alpha}^{*})(\partial_{i}\tilde{Z}_{\alpha})-(\tilde{Z}_{\alpha}^{*}\partial_{i}\tilde{Z}_{\alpha})(\tilde{Z}_{\beta}\partial_{i}\tilde{Z}_{\beta}^{*})\right].
\begin{array}{l}{E_{2}{\circ_{1,Z}}E_{4}=E_{5},\;\;\;\;E_{3}{\circ_{1,Z}}E_{4}=E_{1},}\\{E_{2}{\circ_{1,Z}}E_{6}=E_{1},\;\;\;\;E_{5}{\circ_{1,Z}}E_{6}=E_{3}.}\\\end{array}
{\widetildeS}={\widehatM}_{P}^{D-d-2}\int_{\widetilde\Sigma}\d^{D}x~f^{(d)}(x^{i}-x_{*}^{i})~\sqrt{-\widehatG}~{\widehatR}+M_{P}^{D-2}\int\d^{D}x~\sqrt{-G}~R~.
\begin{array}{l}{\psi_{\tau}(x)=E\Big[exp\Big(\frac{i}{c\hbar}\int_{0}^{\tau}dsA_{\mu}(x+\sigmavb_{s})\sigmav_{a}^{\mu}db_{s}^{a}\Big)\psi\big(x+\sigmavb_{\tau})\Big]}\\\end{array}
(R_{p_{ij},q_{ij}})_{kl}^{ij}=\delta_{k}^{i}\delta_{l}^{j}[\delta^{ij}+{\frac{\theta(j-i)}{q_{ij}}}+\theta(i-j){\frac{1}{p_{ij}}}]+(1-p_{ij}^{-1}q_{ij}^{-1})\delta_{l}^{i}\delta_{k}^{j}\theta(i-j).
\langle\,S_{(\lambda,\mu)},S_{(\lambda,\mu)}\,\rangle_{\beta}=\frac{m!(N-m)!}{f_{\mu}}\;\frac{d_{(\lambda^{R},\mu)}^{\prime}d_{(\lambda^{R},\mu^{R})}}{d_{(\lambda^{R},\mu^{R})}^{\prime}d_{(\lambda,\mu^{R})}}\langle\,E_{(\lambda^{R},\mu^{R})},E_{(\lambda^{R},\mu^{R})}\,\rangle_{\beta}\,,
\Pi_{-}^{AB}|0\rangle_{D}=2\delta^{AB}|0\rangle_{D},\quad\Pi_{-}^{\alpha\beta}|0\rangle_{D}=0,\quad\Pi_{-}^{A\beta}|0\rangle_{D}\neq0,\quad\Pi_{-}^{\alphaB}|0\rangle_{D}\neq0.
g\,\langle\langleF_{\mu\nu}(z_{1})\rangle\rangle^{\mathrm{SR}}=\frac{4}{3}g^{2}\int_{t_{\mathrm{i}}}^{t_{\mathrm{f}}}dt_{2}\,i[\partial_{\nu}D_{\mu\rho}(z_{1}-z_{2})-\partial_{\mu}D_{\nu\rho}(z_{1}-z_{2})]\dot{z}_{2}^{\rho}
\partial_{\mu}\partial^{\mu}f^{a}-\frac{f^{a}f_{b}^{*}\partial_{\mu}\partial^{\mu}f^{b}}{|f|^{2}}=\frac{2(\partial_{\mu}f^{a})f_{b}^{*}\partial^{\mu}f^{b}}{|f|^{2}}-\frac{2f^{a}f_{b}^{*}(\partial_{\mu}f^{b})f_{c}^{*}\partial^{\mu}f^{c}}{|f|^{4}}.
\left.\begin{array}{lll}{\frac{1}{g_{Y}^{2}}}&{=}&{\frac{6}{g_{3}^{2}}(\frac{2}{3})^{2}+\frac{4}{g_{2}^{2}}(\frac{1}{2})^{2}+\frac{2}{g_{0}^{2}}}\\{}&{}&{}\\{}&{=}&{\frac{8}{3g_{3}^{2}}+\frac{1}{g_{2}^{2}}+\frac{2}{g_{0}^{2}}}\\\end{array}\right.
S=-\frac{T}{2}\intd^{p+1}d\xi\left(\frac{\varphi^{\frac{p-1}{p+1}}}{N}\left[-{\dot{X}}^{2}+2N^{a}\dot{X}^{\mu}\partial_{a}X_{\mu}+\left(N^{2}\bar{\gamma}\bar{\gamma}^{ab}-N^{a}N^{b}\right)g_{ab}\right]-(p-1)\varphiN\bar{\gamma}\right).
{\calL}={\frac{1}{2}}m\dot{x}^{2}+{\frac{i}{2}}(\chi_{1}\dot{\chi}_{1}+\chi_{2}\dot{\chi}_{2})+{\frac{1}{8}}mx^{2}F^{2}+F(\ell+{\frac{i}{2}}\chi_{1}\chi_{2})\quad.
<1>\stackrel{O(h^{2})}{=}\frac{1}{8\pi^{2}}\intd^{2}z\,d^{2}z^{\prime}\,h_{\bar{z}\bar{z}}(z,\bar{z})h_{\bar{z}\bar{z}}(z^{\prime},\bar{z}^{\prime})<T_{zz}(z)T_{zz}(z^{\prime})>\+\\mathrm{local}.
I_{1}={\frac{\pic^{2-s}}{4(s-1)(s-2)a\alpha^{\frac{3}{2}}}}+{\frac{(2c)^{2-s}\pi}{a\alpha^{\frac{3}{2}}\Gamma(s)}}\sum_{n=1}^{\infty}\frac{1}{X_{n}^{2-s}}\{K_{-s+2}(X_{n})-X_{n}K_{-s+3}(X_{n})\}\,
\left\langle\Theta_{22}(\vec{x})\right\rangle=-\frac{1}{16\pi^{2}}D_{2}(\vec{x})+\frac{1}{48\pi^{2}}D_{1}(\vec{x})-\frac{1}{24\pi^{2}}D_{22}(\vec{x})+\frac{1}{64\pi^{2}}F_{2}(\vec{x})-\frac{1}{192\pi^{2}}F_{1}(\vec{x})
N_{\xi}=\frac{1}{\pii}\int_{\frac{3}{2}+iT_{o}}^{\frac{3}{2}+iT}d\nu\frac{\xi^{\prime}}{\xi}+\frac{1}{\pii}\int_{\frac{3}{2}+iT}^{iT}d\nu\frac{\xi^{\prime}}{\xi}+\frac{1}{\pii}\int_{iT_{o}}^{\frac{3}{2}+iT_{o}}d\nu\frac{\xi^{\prime}}{\xi},=\frac{1}{\pi}\{\bigtriangleuparg\xi(\nu+1/2)\};
L_{+}=\left(\begin{array}{cc}{q^{S^{3}}}&{(q-q^{-1})S^{-}}\\{0}&{q^{-S^{3}}}\\\end{array}\right);\quadL_{-}=\left(\begin{array}{cc}{q^{-S^{3}}}&{0}\\{-(q-q^{-1})S^{+}}&{q^{S^{3}}}\\\end{array}\right)\quad;
\begin{array}{|l|c|c|c|}\hline{}&{\gamma=1}&{\gamma=0}&{\gamma=-1}\\\hline{}&{}&{}&{}\\{\mathrm{odd~}k\mathrm{~}(\alpha=1)}&{\mu>0}&{\mu>0}&{\begin{array}{ll}{\mu\geq\mu_{c}}&{:\mu_{c}<0}\\\end{array}}\\{}&{}&{}&{}\\\hline{\mathrm{even~}k\mathrm{~}}&{\mu>0}&{\mu>0}&{\begin{array}{ll}{\alpha=1}&{:\mu\geq0}\\{}&{}\\{\alpha=-1}&{:\mu_{c}\geq\mu\geq0}\\\end{array}}\\\hline{}\\\end{array}
l^{4}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=40+72\frac{r_{0}^{8}}{r^{8}}+72f(r/r_{0},\alpha_{1},\alpha_{2})\frac{r_{0}^{16}}{r^{8}(r^{4}-r_{0}^{4})^{2}},
h_{2}=-8A^{ka}A_{ja}\pi_{ik}\pi^{ij}+\frac{1}{3}\left(f_{abc}A_{i}^{a}A_{j}^{b}A_{k}^{c}+A_{\left[i\right.}^{a}F_{a\left.jk\right]}\right)^{2}+8A_{a}^{j}\pi_{ij}\eta_{2}^{a}{\calP}_{2}^{i},
{\calS}_{MTM}=\intd^{2}x\left\{i\overline{{\Psi}}\gamma^{\nu}\partial_{\nu}\Psi-M\\overline{{\Psi}}\Psi-\frac{g}{2}\left(\overline{{\Psi}}\gamma^{\nu}\Psi\right)\left(\overline{{\Psi}}\gamma_{\nu}\Psi\right)\right\},
C_{\beta_{0}\gamma_{0}}^{\alpha_{0}}C_{\rho_{0}\sigma_{0}}^{\gamma_{0}}+R_{\beta_{0}}^{j}C_{\rho_{0}\sigma_{0},j}^{\alpha_{0}}+Z_{\alpha_{1}}^{\alpha_{0}}F_{\beta_{0}\rho_{0}\sigma_{0}}^{\alpha_{1}}+(\beta_{0}\rho_{0}\sigma_{0}-cycle)
\begin{array}{rcl}{{\calA}}&{=}&{\displaystylei\intd^{4}x~e^{ik\cdotx}\tau(x_{0})\sum_{\alpha}f_{\alpha}((i\tilde{\partial})^{2})\left[e^{iMx_{0}}D(x,m_{\alpha}^{2})-e^{-iMx_{0}}D(-x,m_{\alpha}^{2})\right],}\\\end{array}
\lambda={\frac{\eta_{P}e^{3\varphi}{\bfP}_{ij\,\infty}+\eta_{Q}e^{-3\varphi}{\bfQ}_{ik\,\infty}}{\eta_{P}{\bfP}_{ij\,\infty}+\eta_{Q}{\bfQ}_{ik\,\infty}}},
\pi:={\frac{\partialK(\phi,\bar{\phi})}{\partial\bar{\phi}}}-{\frac{\partialK(X,\bar{\phi})}{\partial\bar{\phi}}},\qquad\bar{\pi}:={\frac{\partialK(\phi,\bar{\phi})}{\partial\phi}}-{\frac{\partialK(\phi,\bar{X})}{\partial\phi}}.
\phi_{dis}(x;\mu^{2})=\frac{1}{(2\pi)^{3/2}}\int\frac{d^{3}k}{2E_{\mu}({\bfk})}[A({\bfk})e^{-ik\cdotx}+A^{\dagger}({\bfk})e^{ik\cdotx}],~~k^{0}=E_{\mu}({\bfk}),
\left.K_{{}_{\bar{S}}}^{\pm}\right\vert_{q}=\left.K_{{}_{X^{a}}}^{\pm}\right\vert_{q}-\left.k_{a}^{\pm}\gamma_{{}_{\bar{S}}}^{AB}\frac{\partial^{2}\Phi^{a}}{\partial\mu^{A}\partial\mu^{B}}\right\vert_{q}
\int_{-\pi/2}^{+\pi/2}~\frac{d\xi^{1}}{\pi}~O_{M}^{cl}(\{\sigma_{+}+\xi^{1}\};\{\sigma_{-}-\xi^{1}\})\equiv\int_{-\pi/2}^{+\pi/2}~\frac{d\xi^{1}}{\pi}~O_{M}^{cl}(\{\sigma^{0}\};\{\sigma^{1}+\xi^{1}\})
{\tt\frac{1}{z}}\equiv\frac{1}{\sqrt{2\theta}}\sum_{n=0}^{\infty}\frac{1}{\sqrt{n+1}}|n+1\rangle\langlen|\,,\quad{\tt\frac{1}{\bar{z}}}\equiv\frac{1}{\sqrt{2\theta}}\sum_{n=0}^{\infty}\frac{1}{\sqrt{n+1}}|n\rangle\langlen+1|\,.
S_{H}=-\intd^{D}x\sqrt{|g_{D}|}e^{\gamma}\left[\frac{1}{2n!}H_{\mu_{1}\ldots\mu_{n}}^{(n)}H^{(n)\mu_{1}\ldots\mu_{n}}+\frac{1}{2(n-1)!}e^{-2\gamma}H_{\mu_{1}\ldots\mu_{n-1}}^{(n-1)}H^{(n-1)\mu_{1}\ldots\mu_{n-1}}\right]
a_{0}z_{1}z_{2}z_{3}z_{4}z_{5}+a_{1}z_{1}^{4}+a_{2}z_{2}^{4}+a_{3}z_{3}^{4}+a_{4}z_{4}^{8}+a_{5}z_{5}^{8}+a_{6}z_{4}^{4}z_{4}^{4}.
[\delta_{1},\delta_{2}]A_{\mu}^{a}=D_{\mu}(f_{bc}^{a}\epsilon_{1}^{(0)b}\epsilon_{2}^{(0)c})+f_{bc}^{a}B_{\mu}^{(n)b}f_{de}^{c}\epsilon_{1}^{(0)d}\epsilon_{2}^{(0)e},
[{\calZ}_{A}^{I},\bar{\calZ}_{B}^{J}]_{P.B.}=i\delta^{IJ}\delta_{AB}\,,\quad[{\calZ}_{A}^{I},{\calZ}_{B}^{J}]_{P.B.}=[\bar{\calZ}_{A}^{I},\bar{\calZ}_{B}^{J}]_{P.B.}=0\,,
{\calL}=-\frac14F_{\mu\nu}^{2}+\frac18\theta_{\alpha\beta}F_{\alpha\beta}F_{\mu\nu}^{2}-\frac12\theta_{\alpha\beta}F_{\mu\alpha}F_{\nu\beta}F_{\mu\nu}+{\calO}(\theta^{2})+A_{\mu}J_{\mu},
E^{\underline{\alpha}}(X^{\underline{m}}(z),\Theta^{\underline{\mu}}(z))={\frac{1}{2}}(1-\Gamma_{0})_{~\underline{\beta}}^{\underline{\alpha}}\,e^{a}E_{a}^{\underline{\beta}}+e^{\alphaq}(v_{\alphaq}^{~\underline{\alpha}}(z)+h_{\alpha\beta}C_{qr}v^{\betar,\underline{\beta}}),
{\calA}_{v}=-{\frac{\mathrm{e}^{-\kappa\pi/2}}{1+\mathrm{e}^{-\kappa\pi/2}}},\,\,\,\,\,\,\,\,\,{\calB}_{v}={\frac{1}{1+\mathrm{e}^{-\kappa\pi/2}}}.
\overline{{\gamma}}_{5}\:\kappa^{q}=\rho\:\kappa^{q}\;\;\;\;\kappa_{c\,q}\equiv\overline{{C}}\overline{{\kappa}}_{q}^{T}=\epsilon_{qp}\:\kappa^{p}\;\;\;\;\overline{{\kappa}}_{p}\kappa^{q}=\delta_{p}^{q}
A_{\ell}=\frac{1}{2\,Z_{\ell}}\;\frac{\partialZ_{\ell}}{\partialp}\,\overset{(s_{1})}{e}{\!\!}_{A_{1}}\,\dots\,\overset{(s_{N})}{e}{\!\!}_{A_{N}}\,\widetilde{G}_{A_{1}\,\dots\,A_{N}}\;,
\tilde{P}^{2}=\tilde{g}^{{\mu^{\prime}}{\nu^{\prime}}}P_{\mu^{\prime}}P_{\nu^{\prime}},\qquad\tilde{g}^{{\mu^{\prime}}{\nu^{\prime}}}=\eta^{{a^{\prime}}{b^{\prime}}}\tilde{e}_{a^{\prime}}^{\mu^{\prime}}\tilde{e}_{b^{\prime}}^{\nu^{\prime}}.
\left(\deltaF_{p}\right)\left(\phi_{0},\cdots,\phi_{p+1}\right)=\sum_{i=0}^{p+1}(-1)^{i}\,F_{p}\left(\phi_{0},\cdots,\phi_{i-1},\widehat{\phi}_{i},\phi_{i+1},\cdots,\phi_{p+1}\right)
u_{l}(r)\stackrel{\scriptscriptstyle(r\rightarrow0)}{\sim}\frac{\sqrt{r}}{\pii}\left\{\left[A_{l}^{(+)}e^{-i\pi(l+\nu)}-A_{l}^{(-)}e^{i\pi(l+\nu)}\right]\Gamma(-\nu)\left(\frac{kr}{2}\right)^{l+\nu}+\left[A_{l}^{(+)}-A_{l}^{(-)}\right]\Gamma(\nu)\left(\frac{kr}{2}\right)^{-(l+\nu)}\right\}\;.
f\bulletg=f\g+\sum_{m=1}^{\infty}(-1)^{m}\frac{1}{m!}\\frac{(2j+1)!}{(2j+1+m)!}\J^{a_{1}b_{1}}J^{a_{2}b_{2}}\cdotsJ^{a_{m}b_{m}}\\partial_{a_{1}}\cdots\partial_{a_{m}}f\,partial_{b_{1}}\cdots\partial_{b_{m}}g.
S=\int\left(\frac{R}{16\piG}-\frac{1}{2}\mathrm{Tr}(F_{\mu\nu}F^{\mu\nu})-\frac{1}{4}\mathrm{Tr}(D_{\mu}\PhiD^{\mu}\Phi)-\frac{1}{8}\lambda\mathrm{Tr}(\Phi^{2}-\eta^{2})^{2}\right)\sqrt{-g}d^{4}x,\
\frac{d}{dt}\left(\begin{matrix}{a}\\{b}\\{c}\\{d}\\\end{matrix}\right)=\left(\begin{matrix}{0}&{\hat{H}}&{0}&{0}\\{-\hat{H}}&{0}&{0}&{0}\\{0}&{0}&{0}&{-\hat{H}}\\{0}&{0}&{\hat{H}}&{0}\\\end{matrix}\right)\left(\begin{matrix}{a}\\{b}\\{c}\\{d}\\\end{matrix}\right)
\tilde{m}_{0}^{\ast}:=K^{\ast-1}m_{0}K^{\ast}=\left(\begin{array}{cc}{0}&{-\kappa_{e}^{\frac{1}{2}}T\kappa_{o}^{-{\frac{1}{2}}}}\\{-\kappa_{o}^{-{\frac{1}{2}}}\bar{T}\kappa_{e}^{\frac{1}{2}}}&{0}\\\end{array}\right)=-\tilde{m}_{0}\,.
\begin{array}{c}{A_{x}=\left(\begin{array}{cc}{ia}&{0}\\{0}&{-ia}\\\end{array}\right),B_{x}=\left(\begin{array}{cc}{0}&{A_{1}+A_{-1}}\\{-\left(\overline{{A}}_{1}+\overline{{A}}_{-1}\right)}&{0}\\\end{array}\right),}\\{B_{t}=\left(\begin{array}{cc}{0}&{A_{1}-A_{-1}}\\{-\left(\overline{{A}}_{1}-\overline{{A}}_{-1}\right)}&{0}\\\end{array}\right),}\\\end{array}
<u_{0}\midf[u_{0}]>=2\sum_{n}\left(\frac{1}{h_{n}}+\frac{1}{2}\frac{1}{h_{n}-l(2\omega_{sph})}\right)\mid<G_{2}[u_{0}]\midu_{n}>\mid^{2}-\frac{3}{2}<u_{0}\midG_{3}[u_{0}]>.
Q_{\mu}^{\nu}=\frac{1}{4\pi}\biggl[\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}\delta_{\mu}^{\nu}-F_{\mu\sigma}F^{\nu\sigma}\biggr]=\frac{1}{8\pi}\biggl[-F_{\mu\sigma}F^{\nu\sigma}-(*F)_{\mu\sigma}(*F)^{\nu\sigma}\biggr],
\Theta^{0}=\ell\,\frac{d\rho}{\rho}\,,\quad\Theta^{+}=\frac\rho\ell\,dx^{+}+\frac\ell\rho\,L_{-}(x^{-})\,dx^{-}\,,\quad\Theta^{-}=\frac\rho\ell\,dx^{-}+\frac\ell\rho\,L_{+}(x^{+})\,dx^{+}\,.
{\Gamma^{0}}_{a}=\left\{\left(\begin{matrix}{\sigma_{3}}&{0}\\{0}&{\sigma_{3}}\\\end{matrix}\right),\quad\left(\begin{matrix}{\sigma_{2}}&{0}\\{0}&{-\sigma_{2}}\\\end{matrix}\right),\quad\left(\begin{matrix}{0}&{\sigma_{2}}\\{\sigma_{2}}&{0}\\\end{matrix}\right)\right\}.
\Psi^{\sigma_{1}\sigma_{2}}=\sum_{\sigma_{1}^{\prime}\sigma_{2}^{\prime}}\beta_{\sigma_{1}\sigma_{1}^{\prime}}^{*}(k_{1})\beta_{\sigma_{2}\sigma_{2}^{\prime}}^{*}(k_{2})\psi_{\sigma_{2}^{\prime}\sigma_{1}^{\prime}}
\hat{\eta}_{\alpha}^{\prime}=\eta_{\alpha}^{\prime}+\Gamma_{\alpha},\\\\\\,hat{\bar{\eta}}_{\dot{\alpha}}^{\prime}=\bar{\eta}_{\dot{\alpha}}^{\prime}+\bar{\Gamma}_{\dot{\alpha}}\\\\\\,hatp_{\mu}=p_{\mu}+\Gamma_{\mu}
[b_{Q}^{a}({\bf{k}}),\,\sum_{n=2}^{\infty}{\calA}_{n}\,]\,-\,g\,f^{a\beta\gamma}\intd{\bf{r}}\,e^{-i{\bf{k\cdotr}}}\,[a_{i}^{\beta}({\bf{r}})+x_{i}^{\beta}({\bf{r}})]\,[\sum_{n=1}^{\infty}{\calA}_{n},\,\Pi_{i}^{\gamma}({\bfr})]\approx0\;,
L_{n_{1},n_{2}}^{d}(w_{1},w_{2})=k^{2(n_{1}+n_{2})-d}\int_{0}^{\infty}dxx^{\frac{d}{2}-1}\frac{\partial}{\partial{t}}\left\{(P+w_{1})^{-n_{1}}(\tilde{P}+w_{2})^{-n_{2}}\right\}
K\mathrm{}_{\\,N-b}^{j\,N-a}(u)=(-1)^{j(n-1)+1}K\mathrm{}_{\b}^{ja}(u),\quadK\mathrm{}_{j\,N-b}^{\\,N-a}(u)=(-1)^{j(n-1)+1}K\mathrm{}_{jb}^{\a}(u).
\dot{S}=\intdV{\frac{\eta}{2T}}\Bigl[\Bigl({\frac{\partialv_{i}}{\partialx_{k}}}-{\frac{\partialv_{k}}{\partialx_{i}}}\Bigr)^{2}+p{\frac{\partialv_{l}}{\partialx_{l}}}+{\frac{\partial}{\partialx_{i}}}\Bigl(v_{k}{\frac{\partialv_{i}}{\partialx_{k}}}\Bigr)\Bigr]\.
v_{\mu}={\frac{i\left(\tilde{\phi}^{\dagger}T\overset{\leftrightarrow}\partial_{\mu}\tilde{\phi}+\tilde{\chi}T\overset{\leftrightarrow}\partial_{\mu}\tilde{\chi}^{\dagger}\right)}{\tilde{\phi}^{\dagger}\tilde{\phi}+\tilde{\chi}\tilde{\chi}^{\dagger}}}.
\bar{A}_{\beta,1}^{(2)}=\left.A_{\beta,1}^{(0)}\right|_{{\calC}_{\beta}}N^{(2)}(d-2)\int_{\Sigma}+\left.A_{\beta,1}^{(2)}\right|_{{\calC}_{\beta}}\int_{\Sigma}+\left.A_{\beta,1}^{(1)}\right|_{{\calC}_{\beta}}(d-2)\int_{\Sigma}~~~.
R_{1p+1}(z)R_{2p+1}(z\tau^{2})\cdotsR_{pp+1}(z\tau^{2p-2})v^{(i_{1}\cdotsi_{p})}\otimesv_{i}=\prod_{j=1}^{p-1}b(z\tau^{2j})v^{(i_{1}\cdotsi_{p})}\otimesv_{i}.
{\frac{\partial^{R}{\calH}}{\partialW_{\lambda}^{i}}}-\partial_{t\mu}{\frac{\partial^{R}{\calH}}{\partial\left[{\partial_{t\mu}W_{\lambda}^{i}}\right]}}+\partial_{n}\Pi_{W}^{i\lambda}=0,\{\frac{\partial^{R}{\calH}}{\partial\Pi_{W\lambda}^{i}}}-\partial_{t\mu}{\frac{\partial^{R}{\calH}}{\partial\left[{\partial_{t\mu}\Pi_{W\lambda}^{i}}\right]}}-\partial_{n}W^{i\lambda}=0,
(\Psi_{\pm})_{\mu}^{A}=\frac{1}{2}(\delta\pm\gamma_{5}\sigma_{3})_{\;B}^{A}\Psi_{\mu}^{B}\,\quad(\eta_{\pm})^{A}=\frac{1}{2}(\delta\pm\gamma_{5}\sigma_{3})_{\;B}^{A}\eta^{B}\.
\chi_{\bar{k}}=\left({\frac{1-\Gamma}{2}}\right)=\left({\frac{1-\gamma^{0}\gamma^{9}}{2}}\right)\,\qquad\chi_{k}=\left({\frac{1+\Gamma}{2}}\right)=\left({\frac{1+\gamma^{0}\gamma^{9}}{2}}\right)\.
\left\|\varphi_{\Xi}^{(n)}(\zeta)(\zeta_{1}..,\zeta_{n})\right\|_{\Lambda}\leq\sqrt{n!}2^{n}c^{n}\left\|\Xi\right\|_{(\gamma)}\left\|\zeta_{1}\right\|\cdot\cdot\cdot\left\|\zeta_{n}\right\|f_{\gamma}(\left\|\zeta\right\|)
A^{2}=\left(\begin{matrix}{-\omega_{1}^{2}}&{0}&{0}&{0}\\{0}&{-\omega_{1}^{2}}&{0}&{0}&{0}\\{0}&{0}&{-\omega_{2}^{2}}&{0}\\{0}&{0}&{0}&{-\omega_{2}^{2}}\\\end{matrix}\right),
\hat{T}_{jk}([\vec{r}^{N}])=\^{*}\sum_{l_{1}}\cdot\cdot\cdot\^{*}\sum_{l_{N}}\bar{A}_{j}(N_{r_{1}}+l_{1},...,N_{r_{N}}+l_{N})A_{k}(N_{r_{1}}+l_{1},...,N_{r_{N}}+l_{N}).
H_{\mathrm{vis,\pm}}^{2}=\frac{\rho_{\mathrm{vis}}(\rho_{\mathrm{vis}}+2\Lambda_{2}^{\pm})}{36M^{6}(1+4\alpha\Lambda_{b}/3M^{5})}=\frac{\pm\rho_{\mathrm{vis}}}{3M_{Pl}^{2}\sqrt{1+4\alpha\Lambda_{b}/3M^{5}}}\left[1+\frac{\rho_{\mathrm{vis}}}{2\Lambda_{2}^{\pm}}\right]~~.
G_{0}(\vec{p})\equiv\intd^{25-p}\!k\\varphi_{0}(\vec{k})\varphi_{0}(\vec{p}-\vec{k})=\int\frac{d^{25-p}\!x}{(2\pi)^{25-p}}\varphi_{0}^{2}(\vec{x})e^{-i\vec{p}\cdot\vec{x}}.
L=i\overline{{\psi}}_{0}^{\prime}{\gamma}^{\mu}{\partial}_{\mu}{\psi}_{0}^{\prime}+i\overline{{\psi}}_{0}^{\prime}e^{i{t}^{a}{\omega}_{a}}e^{-i{t}^{b}{\theta}_{b}}{\gamma}^{\mu}\partial_{\mu}\left(e^{i{t}^{c}{\theta}_{c}}e^{-i{t}^{l}{\omega}_{l}}\right){\psi}_{0}^{\prime}-m\overline{{\psi}}_{0}^{\prime}{\psi}_{0}^{\prime}.
\left[z^{2}\left(4\partial_{+}\partial_{-}+\partial_{z}^{2}-{\frac{1}{z}}\partial_{z}\right)-{\frac{4s^{2}z^{2}}{z^{4}-2(1-\omega^{+}\omega^{-})z^{2}+(1+{\omega^{+}}^{2})(1+{\omega^{-}}^{2})}}-m^{2}l^{2}\right]\Phi_{s}=0.
\sum_{l=1}^{N}\bar{w}_{qr}^{CP}(m-l)w_{pr}^{CP}(n-l)\bar{w}_{pq}^{CP}(l-k)=R_{pqr}w_{pq}^{CP}(n-m)\bar{w}_{pr}^{CP}(m-k)w_{qr}^{CP}(n-k)
\hat{a}_{n}(x^{0})=\sqrt{\frac{\pi}{2\omega_{n}}}\,\,\hat{\Lambda}_{n}e^{-i\omega_{n}x^{0}}+\sqrt{\frac{\pi}{2\omega_{n}}}\,\,\hat{\Lambda}_{n}^{\dagger}e^{i\omega_{n}x^{0}},
\left(\begin{array}{ccc}{r(\alpha-1)s(\alpha-1)}&{z}&{C_{2}(R)y}\\{4z}&{\frac{p(\beta)q(\beta)}{NM}+z^{2}\Pi^{\prime}}&{-\,C_{2}(R)yz\Pi^{\prime}}\\{4y}&{-\,yz\Pi^{\prime}}&{\frac{p(\gamma)q(\gamma)}{16T(R)N}+\frac{y^{2}\Pi^{\prime}(2C_{2}(R)-C_{2}(G))}{2}}\\\end{array}\right)
\langle\mathrm{T}\Big({\calV}_{\calT}(x)\,{\calV}_{\calT}(y)\,{\calV}_{R}(z)\Big)\rangle_{A=0}=-\langle\mathrm{T}\Big({\calV}_{\calT}(x)\,{\calV}_{\calT}(y)\,{\calV}_{R}(z)\Big)\rangle_{A=1}~.
{\calL}=\frac{1}{g^{2}}\left[(D_{\mu}z)^{\dagger}(D^{\mu}z)+ib\,\epsilon^{\mu\nu}(\partial_{\mu}z)^{\dagger}(\partial_{\nu}z)-\frac{M^{2}}{2}\,b^{2}-\lambda\,(z^{\dagger}z-1)\right],
g_{\theta}\equiv\bar{\zeta}\tilde{p}d_{\theta}+\bar{d}_{\theta}\tilde{p}\zeta=-i(\bar{\zeta}\tilde{p}p_{\theta}+\bar{p}_{\theta}\tilde{p}\zeta)-4k^{2}m^{2}(\theta\zeta+\bar{\zeta}\bar{\theta})\approx0\,,
{\calS}_{\pm}^{(0)}=\intd^{4}k\left[\pm\omega({\bfk})\dot{\varphi}_{a}^{(\pm)}\epsilon_{ab}\varphi_{b}^{(\pm)}\,-\,\omega^{2}({\bfk})\varphi_{a}^{(\pm)}\varphi_{a}^{(\pm)}\,\right].
\sigma_{0}=\left(\begin{array}{cc}{1}&{0}\\{0}&{1}\\\end{array}\right),\sigma_{1}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right),i\sigma_{2}=\left(\begin{array}{cc}{0}&{1}\\{-1}&{0}\\\end{array}\right),\sigma_{3}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right).
g_{\mu\nu}^{j}-G_{\mu\nu}^{j+1}=h_{\mu\nu}^{j}-h_{\mu\nu}^{j+1}+\pi_{\mu,\nu}^{j}+\pi_{\nu,\mu}^{j}+\pi_{,\mu}^{j\alpha}\pi_{\alpha,\nu}^{j}+\cdots
S={\frac{1}{2\kappa_{D}^{2}}}\intd^{D}x\sqrt{-\hat{G}}e^{-2\hat{\Phi}}[{\calR}_{\hat{G}}+4\partial_{M}\hat{\Phi}\partial^{M}\hat{\Phi}-{\frac{1}{12}}\hat{H}_{MNP}\hat{H}^{MNP}],
\pi^{\mu\nu}=\frac{\partial{\calL}}{\partial\left(\partial_{0}h_{\mu\nu}\right)}-2\partial_{\rho}\frac{\partial{\calL}}{\partial\left(\partial_{\rho}\partial_{0}h_{\mu\nu}\right)}+\partial_{0}\frac{\partial{\calL}}{\partial\left(\partial_{0}\partial_{0}h_{\mu\nu}\right)}\;\,,
F^{ab}={\frac{1}{2}}({F^{+}}{}^{ab}+{F^{-}}{}^{ab})={\frac{1}{2}}({f_{\beta}^{+}}{}^{\alpha}{(\sigma^{ab})_{\alpha}}{}^{\beta}+{{f^{-}}^{\dot{\beta}}}_{\dot{\alpha}}{(\bar{\sigma}^{ab}){}^{\dot{\alpha}}}_{\dot{\beta}}),
\{\hat{X}^{\mu},\hat{S}^{\nu\lambda}\}^{\prime}=2\frac{\bar{P}^{\nu}\hat{S}^{\lambda\mu}-\bar{P}^{\lambda}\hat{S}^{\nu\mu}}{(P+\bar{P})^{2}}-\frac{4P^{\mu}}{(P+\bar{P})^{2}}(\bar{P}^{\lambda}D^{\nu\rho}A_{\rho}-\bar{P}^{\nu}D^{\lambda\rho}A_{\rho}).\nonumber\,
H^{B}=\frac{m^{2}R_{1}^{2}}{2l_{s}^{4}p^{+}}+2\muN_{0}+\sum_{n\neq0}(\omega_{n}+\mu)N_{n}+\sum_{n\neq0}(\omega_{n}-\mu)\tilde{N}_{n}+\sum_{I=3}^{8}\sum_{n=-\infty}^{\infty}\omega_{n}N_{n}^{I},
\xi_{1}=(TrX_{\varphi,1}+{\frac{1}{2}}TrX_{+}){\frac{1}{192\pi^{2}}}M_{5}^{2}\,\,\xi_{2}=(TrX_{\varphi,2}+{\frac{1}{2}}TrX_{+}){\frac{1}{192\pi^{2}}}M_{5}^{2}\
\begin{array}{rcl}{}&{}&{\deltaA_{m}=\sigma_{mn}A_{n}+m\lambda_{m}\,,}\\{}&{}&{}\\{}&{}&{\deltaV_{mn}={\calD}\sigma_{mn}\,,}\\{}&{}&{}\\{}&{}&{\deltaB_{m}=2\partial\lambda_{m}+2\partial\sigma_{mn}A_{n}+m\sigma_{mn}B_{n}\,,}\\\end{array}
\nabla_{\alpha}v\nabla_{\mu}v\:m_{\beta}m_{\nu}R^{\alpha\beta\mu\nu}\biggl|_{{\calI}^{+}}=-\:\frac{8}{r}\:\Bigl(\frac{\partial^{2}}{{\partialu}^{2}}C_{1}+i\frac{\partial^{2}}{{\partialu}^{2}}C_{2}\Bigr)+O\left(\frac{1}{r^{2}}\right)
S_{\mathrm{gf}}+S_{\mathrm{FP}}=\int_{M}g^{-2}(\varphi_{\mathrm{o}})\left(\frac{1}{2}\nabla^{\mu}A_{\mu}\nabla^{\nu}A_{\nu}+\partial_{\mu}\bar{\theta}\partial^{\mu}\theta\right)\sqrt{h}\,d^{3}x,
\theta^{\mu\nu}=\frac{2\pi\alpha^{\prime}}{g_{2}g(1-e^{2})+g^{2}b^{2}}\left(\begin{array}{ccc}{0}&{g_{2}e}&{0}\\{-g_{2}e}&{0}&{-gb}\\{0}&{gb}&{0}\\\end{array}\right).
f(r)\equiv\frac12\left\{\fracn{n+1}+{\frac{(r^{2}/2)^{n-1}}{(n-1)!}}{\frac{e^{-r^{2}/2}}{Q(r^{2}/2)}}\left(n-{\frac{r^{2}}{2}}-{\frac{(r^{2}/2)^{n}}{(n-1)!}}{\frac{e^{-r^{2}/2}}{Q(r^{2}/2)}}\right)\right\}.
\begin{array}{ll}{(0~0~0~0~0~0~0~-\frac{7}{2})\longrightarrow}&{~(0~0~0)(0~0~0~-\frac{7}{2})(2)\oplus(0~0~1)(0~0~0~-\frac{5}{2})(1)}\\{}&{\oplus~(0~1~0)(0~0~0~-\frac{3}{2})(0)\oplus(1~0~0)(0~0~0~-\frac{1}{2})(-1)}\\{}&{\oplus~(0~0~0)(0~0~0~\frac{1}{2})(-2),}\\\end{array}
H=\frac{1}{2m}g^{\mu\nu}(x)\left[p_{\mu}\!-\!A_{\mu}(x)\!+\!\omega_{\mu}(x)S(\psi)\right]\left[p_{\nu}\!-\!A_{\nu}(x)\!+\!\omega_{\nu}(x)S(\psi)\right]-\frac{1}{m}B(x)S(\psi)