harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
f(\lambda,\omega)=\frac{3}{4}-\frac{3}{8\pi^{2}}\omega^{2}+\frac{69}{64\pi^{4}}\omega^{4}+\frac{\zeta(3)}{64}(2\lambda)^{-3/2}\left(90-\frac{981}{7\pi^{2}}\omega^{2}+\frac{7655}{16\pi^{4}}\omega^{4}\right)+\cdots
H^{\pm}(\lambda)=-\sum_{j=1}^{2N-1}[(e_{2j-1}^{}-{\textstyle\frac{1}{2}})+\lambda(e_{2j}^{}-{\textstyle\frac{1}{2}})]-[(e_{2N-1}^{}-{\textstyle\frac{1}{2}})-\lambda(e_{2N}^{\pm}-{\textstyle\frac{1}{2}})],
H=-\frac{e}{2}\pi^{2}+\imath\pi_{\mu}\psi^{\mu}\chi-\left(\frac{1}{8!}\epsilon_{\nu\mu\rho_{2}\ldots\rho_{9}}\pi^{\mu}\psi^{\rho_{2}}\psi^{\rho_{9}}+\frac{\alpha}{2^{8}}\pi_{\nu}\right)b^{\nu}\;.
\begin{array}{ll}{e^{2\varphi}=}&{e^{\phi_{0}}+\frac{2v\alpha^{\prime}\left(2\rho^{2}+\tau^{2}\right)}{\left(\rho^{2}+\tau^{2}\right)^{2}},}\\{e^{-2\lambda}=}&{e^{-\phi_{0}}+\frac{2\tilde{v}\alpha^{\prime}\left(2\rho^{2}+\tau^{2}\right)}{\left(\rho^{2}+\tau^{2}\right)^{2}}.}\\\end{array}
\sqrt{\frac{25}{4}+4\kappa^{2}\tilde{\omega}_{m,\pm}^{2}}=\frac{\left|8\left(m+\frac12\right)\pm\sqrt{25+4\left(29-4\left(m+\frac12\right)^{2}\right)\kappa^{2}+64(\kappa^{2})^{2}}\right|}{2(1+4\kappa^{2})}.
\delta_{\epsilon,\epsilon^{*}}\langleX\rangle=-\frac{1}{2\pii}\oint_{C}dz\,\epsilon(z)\langleT(z)X^{\prime}\rangle-\frac{1}{2\pii}\oint_{\bar{C}}dz\,\epsilon(z)\langleT(z)X^{\prime}\rangle,
\partial_{y}(e^{2ky}h^{\prime})=-\frac{2}{3k}M^{-3}\bigg(1-\frac{4\alphak^{2}}{M^{2}}\bigg)^{-1}\bigg[T_{55}(y)-e^{2ky}\int_{y}^{y_{m}}dy\partial^{\mu}T_{\mu5}\bigg].
\Lambda=\left(\begin{array}{cc}{\tau}&{0}\\{0}&{-\bar{\tau}}\\\end{array}\right)+i{\frac{\tau_{2}{\frac{g^{2}}{2\pi}}}{2(1+{\frac{g^{2}}{2\pi}})}}\left(\begin{array}{cc}{{\frac{g^{2}}{2\pi}}}&{-2-{\frac{g^{2}}{2\pi}}}\\{-2-{\frac{g^{2}}{2\pi}}}&{{\frac{g^{2}}{2\pi}}}\\\end{array}\right),
M(p)=\frac{{\bar{\alpha}}}{N\pi^{3}}(1\!-\!\xi)\intdq\;\frac{1}{Z(p)}\frac{M(q)}{q^{2}+M^{2}(q)}\left[\frac{2q^{2}p}{p^{2}-q^{2}}\theta(p-q)+\frac{2p^{2}q}{q^{2}-p^{2}}\theta(q-p)\right].
\delta_{D}S_{EH}=\frac\ell{16\piG}\int_{|r|=\bar{r}}Tr[\xi^{r}\{6\,\partial_{[r}(\tilde{A}_{\varphi}A_{t]})-2\,A_{r}(A_{[\varphi}A_{t]}+\tilde{A}_{[\varphi}\tilde{A}_{t]})\}]\,d\varphidt\quad.
\left[T_{\mu\nu}\right]_{P}(x)=\frac{C_{P}}{2}\frac{1}{\sqrt{|g(x)|}}g_{\mu\rho}(x)g_{\nu\lambda}(x)\int_{-\infty}^{+\infty}d\xi\frac{1}{\beta(\xi)}\frac{dy^{\rho}(\xi)}{d\xi}\frac{dy^{\lambda}(\xi)}{d\xi}\delta^{(11)}(x-y(\xi))
\langleT_{\mu}^{\mu}\rangle_{ren}=\frac{1}{(4\pi)^{3}}\left[\frac{135}{16}\Lambda^{3}-\frac{63}{4}\left(\frac{\Lambda^{2}}{r^{2}}\right)+\frac{25}{2}\left(\frac{\Lambda}{r^{4}}\right)-\frac{2249}{840}\left(\frac{1}{r^{6}}\right)\right],
\int\!\!\!\!\!\intd^{4}\hat{x}\Phi_{1}(\hat{x})\left(\hat{\partial}_{A}\Phi_{2}(\hat{x})\right)=\int\!\!\!\!\!\intd^{4}\hat{x}\left(\hat{\partial}_{A}^{+}\Phi_{1}(\hat{x})\right)\Phi_{2}(\hat{x})\,.
\overline{{\Psi}}_{L}\partial\!\!\!/\Psi_{L}\to\overline{{\Psi}}_{L}e^{-i\beta(\hat{C}_{+}^{\dagger}-\hat{C}_{-}^{\dagger})}e^{i\beta(\hat{C}_{+}-\hat{C}_{-})}\partial\!\!\!/\Psi_{L},
ds^{2}=e^{-2k|y|}\left[-\left(1-\frac{U_{T}^{4}}{k^{4}}e^{4k|y|}\right)dt^{2}+\sum_{i=1}^{3}dx^{i}dx^{i}\right]+\frac{dy^{2}}{1-\frac{U_{T}^{4}}{k^{4}}e^{4k|y|}}
F_{\mu\nu}^{pq}(x)=\int_{\tilde{T}^{2}}d^{2}z\int_{\tilde{T}^{2}}d^{2}z^{\prime}\,\psi^{p}{}^{\dagger}(z;x)\sigma_{\mu}\,f(z,z^{\prime};x)\,\sigma_{\nu}^{\dagger}\psi^{q}(z^{\prime};x)-[\mu\leftrightarrow\nu],
{}^{({\calC}_{H}^{\calR})}\Psi_{Y,X}\circ(\triangleleftu)\circ{}^{({\calC}_{H}^{\calR})}\Psi_{X,Y}={}^{\calC}\Psi_{Y,X}\circ((\triangleleftu)\otimes(\triangleleftu))\circ{}^{\calC}\Psi_{X,Y}
\tau_{\alpha}^{(n+1)}(x)=-\frac{1}{n+2}\int\Phi^{\mu}(z)\omega_{\mu\nu}(z,z^{\prime})X^{\nu\rho}(z^{\prime},z^{\prime\prime})B_{\rho\alpha}^{(n)}(z^{\prime\prime},x)dzdz^{\prime}dz^{\prime\prime}~~(n\geq1)
\epsilon_{i}^{a}(\theta)+\Omega_{k}\astL_{i}^{b}(\theta-\sigma_{ba})=\sum_{j=1}^{k-1}I_{ij}^{su(k)}\,\Omega_{k}\ast(\epsilon_{j}^{a}+L_{j}^{a})(\theta)\;,\quada\neqb\,.
L^{-}=q\left(\begin{array}{ccc}{q^{-h_{2}}}&{0}&{0}\\{(q-q^{-1})\alpha^{-}}&{q^{-h_{1}-h_{2}}}&{0}\\{(q^{-1}-q)b^{-}}&{(q^{-1}-q)\beta^{-}}&{(-1)^{F}q^{-h_{1}-2h_{2}}}\\\end{array}\right)
\frac{1}{|\Omega|}\sum_{z_{1},\ldots,z_{N}\in\Omega}\sum_{x\in\bigcap_{i=1}^{N}\Omega_{z_{i}p_{i}}}\sum_{\theta\in\Omega_{g}(x|z_{1}p_{1},\ldots,z_{N}p_{N})}\prod_{i=1}^{N}\frac{\overline{{\phi_{i}}}\left(c_{i}^{z_{i}},x^{z_{i}}\right)}{|\Omega_{p_{i}}|}\prod_{\xi\in\mathcal{O}(x|\theta)}\mathcal{D}_{\xi}(z_{1}p_{1},\ldots,z_{N}p_{N})
\psi_{1}\left(\rho,\varphi\right)=\sqrt{k_{0}+m}\psi_{n-1,\,\,\,l-1}^{\left(j\right)}\,,\,\psi_{2}\left(\rho,\varphi\right)=\sqrt{k_{0}-m}\psi_{n,l}^{\left(j\right)}\,\,,\;k_{0}^{2}=m^{2}+2\gamma\bar{n}\,\,,
\Gamma_{\mu}^{\mathrm{odd}}={\frac{\theta}{16}}\int_{0}^{\infty}ds\rho_{2}(s)\int{\frac{d^{3}k}{(2\pi)^{3}}}{\frac{(2p-k)_{\lambda}(2p-2q-k)_{\nu}(2p-2k-q)_{\mu}\epsilon_{\lambda\rho\nu}k^{\rho}}{\left[(p-k)^{2}-M^{2}\right]\left[(p-k-q)^{2}-M^{2}\right]\left[k^{2}-s+i\epsilon\right]}}.
S_{massive}=S+\intdx^{+}dx^{-}\,\mathrm{tr}\left[-\frac{1}{2}m_{b}^{2}\phi^{2}+\frac{i}{2}m_{f}^{2}\psi\frac{1}{\partial_{-}}\psi+m_{f}g\psi\frac{1}{\partial_{-}}[\phi,\psi]\right].
\mathcal{L}=\beta^{2}\sqrt{1+\frac{m^{2}}{\beta^{2}}A_{\mu}A^{\mu}}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}-\frac{\chi\theta}{2}\varepsilon^{\mu\nu\rho\lambda}\partial_{\mu}B_{\nu\rho}A_{\lambda}\;.
\theta_{\mu\nu,k\lambda}=\frac{\Box}{2}\\overset{(2)}{P}+\frac{\Box}{2\alpha}\\overset{(1)}{P}_{\!\!m}-\Box\left(\frac{4\alpha-3}{4\alpha}\right)\overset{(0)}{P}_{\!\!s}+\frac{\Box}{4\alpha}\\overset{(0)}{P}_{\!\!w}-\frac{\Box\sqrt{3}}{4\alpha}\\overset{(0)}{P}_{\!\!sw}-\frac{\Box\sqrt{3}}{\alpha}\\overset{(0)}{P}_{\!\!ws}
H^{\bot}=\frac1{4l_{p}^{2}}\int_{\Sigma}^{}d^{3}x\sqrt{g}G^{ijkl}\left[K^{-1\bot}\left(x,x\right)_{ijkl}+\left(\triangle_{2}\right)_{j}^{a}K^{\bot}\left(x,x\right)_{iakl}\right].
T_{\mu\nu}^{eff}(\phi)=\phi_{,\mu}\phi_{,\nu}-\frac{1}{2}\overline{{g}}_{\mu\nu}\phi_{,\alpha}\phi_{,\beta}\overline{{g}}^{\alpha\beta}+\overline{{g}}_{\mu\nu}V_{eff}(\phi),V_{eff}(\phi)=\frac{1}{4U(\phi)}(V+M)^{2}.
V_{add}^{\omega}=-\frac{m}{4\pi}g^{2}\frac{3m^{2}}{2M^{2}}(\frac{1}{x}+\frac{1}{x^{2}})\phi{\bfL}\cdot{\bfS}+\frac{m}{4\pi}g^{2}\frac{3m^{4}}{16M^{4}}\frac{1}{x^{2}}\chiQ_{12}
\Gamma_{e.f.}=\int\{Y_{1}(-{e}z_{e}\sqrt{\frac{z_{2}}{z_{1}}}\sqrt{z_{A}}\bar{\varphi}_{2}c+x_{1}q_{1})\;+\;Y_{2}({e}z_{e}\sqrt{\frac{z_{1}}{z_{2}}}\sqrt{z_{A}}(\bar{\varphi}_{1}+\frac{\sqrt{z_{m}}m}{\sqrt{z_{1}}{z_{e}}e})c+x_{2}q_{2})\}
{\calT}_{\mu\nu}^{\alpha\beta}=2{\itg_{2}}^{2}\epsilon_{~\mu}^{\alpha~\rho\sigma}\epsilon_{~\nu}^{\beta~\lambda\gamma}\bigg(F_{\rho\sigma}F_{\lambda\gamma}+\tilde{F}_{\rho\sigma}\tilde{F}_{\lambda\gamma}+{\frac{1}{\sqrt2}}\hat{\theta}_{5}\big(F_{\rho\sigma}\tilde{F}_{\lambda\gamma}+\tilde{F}_{\rho\sigma}F_{\lambda\gamma}\big)\bigg)\,,
\begin{array}{lcr}{{L_{0}}\vert{h,{\bar{h}}}\rangle=h{\vert{h,{\bar{h}}}\rangle},}\\{{L_{n}}{\vert{h,{\bar{h}}}\rangle}=0;\quadn\geq1}\\{{\bar{L}_{0}}{\vert{h,{\bar{h}}}\rangle}=\bar{h}{\vert{h,{\bar{h}}}\rangle},}\\{{{\bar{L}}_{n}}{\vert{h,{\bar{h}}}\rangle}=0;\quadn{\geq}1,}\\{cI\vert{h,{\bar{h}}}\rangle=c\vert{h,{\bar{h}}}\rangle,}\\\end{array}
\left[{\frac{\partial}{\partialt}}+\intdx\;\left(\Pi(x)\;{\frac{\delta}{\delta\Phi(x)}}-\big(m^{2}\Phi(x)-\nabla^{2}\Phi(x)\big){\frac{\delta}{\delta\Pi(x)}}+{\calK}_{I}(x)\right)\right]W[\Phi,\Pi;t]=0\;,
\partial_{[\alpha}\overline{{u}}_{\overline{{\beta}}]a}=H_{\alpha\overline{{\beta}}}^{\,\,\,\,\,\,\gamma}\overline{{\xi}}_{\gammaa}+H_{\alpha\overline{{\beta}}}^{\,\,\,\,\,\,\overline{{\gamma}}}\xi_{\overline{{\gamma}}a},
S=\frac{R_{1}^{2}}{2}\int\partial_{\mu}X_{1}\partial^{\mu}X_{1}+\frac{R_{2}^{2}}{2}\int\partial_{\mu}X_{2}\partial^{\mu}X_{2}+B_{i,\alpha;j,\beta}\intX_{\alpha}^{i}\partial_{t}X_{\beta}^{j}dt,
\mu^{2}(t)=\frac{\lambda_{\mathrm{R}}}{2}\tilde{I}_{1}^{\phi}(t)-\frac{1}{2}a^{2}(t)\hat{m}^{2}\left(1+\frac{\lambda_{\mathrm{R}}}{16\pi^{2}}\bar{c}\right)+O(\lambda_{\mathrm{R}}^{2})\,,
\frac{\partialS}{\partialt}+\frac12\left(\frac{\partialS}{\partialq}\right)^{2}+\frac12V^{2}(q)-\mathrm{i}U(q)\frac{\partialS}{\partial\bar{\psi}}\bar{\psi}+\mathrm{i}U(q)\frac{\partialS}{\partial\psi}\psi=0.
\langle\hat{W}(z)\hat{W}(w)\rangle_{c}={\frac{1}{N^{2}}}{\frac{a^{2}}{2\beta}}{\frac{1}{(z-w)^{2}}}{\frac{{(W(z)-W(w))}^{2}}{(1-{\frac{a^{2}}{4}}W^{2}(z))(1-{\frac{a^{2}}{4}}W^{2}(w))}}.
E^{-\alpha}=\left[E^{\alpha}\right]^{T}\,\leftrightarrow\,\langle{\vec{W}}^{(n)}\vert\,E^{-{\vec{\alpha}}}\,\vert{\vec{W}}^{(m)}\rangle\,=\,\langle{\vec{W}}^{(m)}\vert\,E^{{\vec{\alpha}}}\,\vert{\vec{W}}^{(n)}\rangle
V_{ns}({\tilde{x}})=\left(\frac{{\tilde{m}}N^{2}}{16\pi}\right)Ng^{2ns-1}{\tilde{x}}^{2}\left\{{\tilde{x}}^{2}-\frac{2{\tilde{b}}}{3}{\tilde{x}}+\frac{{\tilde{b}}^{2}}{3}-(-1)^{ns}{\tilde{c}}\right\}\,.
A={\frac{+{r^{\prime}}^{2}}{(r^{\prime}-r_{2})(r^{\prime}-r_{1})}}>0\;,\;\;B={\frac{-{r_{2}}^{2}}{(r^{\prime}-r_{2})(r_{2}-r_{1})}}<0\;,\;\;C={\frac{+{r_{1}}^{2}}{(r^{\prime}-r_{1})(r_{2}-r_{1})}}>0\;.
c(l_{0}+1,l_{1};l_{0},l_{1})=c(l_{0},l_{1};l_{0}+1,l_{1})=c_{0}\sqrt{\frac{(k_{1}-l_{0}-1)(k_{1}+l_{0})}{(l_{1}-l_{0})(l_{1}-l_{0}-1)(l_{1}+l_{0})(l_{1}+l_{0}+1)}},
\frac{1}{4}\gamma_{ab}\psi_{A}\overline{{\psi}}_{B}\gamma^{ab}\psi_{C}\epsilon^{AC}-\frac{1}{2}\gamma_{a}\psi_{A}\overline{{\psi}}_{B}\gamma^{a}\psi_{C}\epsilon^{AC}+3\psi_{C}\overline{{\psi}}_{B}\psi_{A}\epsilon^{BC}=0
\langle\prod_{j=1}^{N}T_{j}(Z_{j})\rangle=\delta^{(d)}\left(\sum_{j=1}^{N}k_{j}\right)\prod_{i\not=j}^{N}\left|Z_{ij}\right|^{k_{i}k_{j}}\left(\frac{\epsilon}{\mu}\right)^{\sum_{j=1}^{N}k_{j}^{2}}\langle\langlee^{-(10-d)S_{sL}}\prod_{j=1}^{N}B_{j}(Z_{j})\rangle\rangle.
\gamma({\bfx},t)=\frac{1}{2}\int_{0}^{t}dt^{\prime}\frac{\stackrel{.}{B_{2}}({\bfx},t^{\prime})B_{1}({\bfx},t^{\prime})-B_{2}({\bfx},t^{\prime})\stackrel{.}{B_{1}}({\bfx},t^{\prime})}{|{\bfB}({\bfx},t^{\prime})|^{2}}
F(g,N)=\sum_{k=0}^{\infty}a_{k}(g-g_{*})^{k}+\sum_{k=0}^{\infty}b_{k}(g-g_{*})^{k+\gamma}+O(N^{-2})\\(\gamma\in{\bfR}_{+}\setminus{\bfZ},b_{0}\ne0).
T^{a}{}_{b}=\frac{2}{\sqrt{g_{n+1}}}\,\frac{\deltaS_{\mathrm{br}}}{\deltag_{ac}}\,g_{cb}=\mathrm{diag}(-\rho,p,\ldots,p)=f^{(n+1)/2}(\phi)z(\phi)\,\mathrm{diag}(-\bar{\rho},\bar{p},\ldots,\bar{p})\,,
\begin{array}{l}{\alpha=\frac{M}{A_{8-p}}\left[-1\pm\sqrt{1+\frac{8-p}{(7-p)^{2}}\left(\frac{Q_{0}}{M}\right)^{2}}\right],}\\{r_{H}^{7-p}=\frac{M}{(8-p)A_{8-p}}\left[(9-p)\mp(7-p)\sqrt{1+\frac{8-p}{(7-p)^{2}}\left(\frac{Q_{0}}{M}\right)^{2}}\right].}\\\end{array}
R_{\qquadkl}^{nm}=\delta_{\quadk}^{n}\delta_{\quadl}^{m}+\epsilon^{nm}\epsilon_{kl}+i\eta\epsilon^{nm}\epsilon_{kl}+\frac{1}{2}\epsilon^{nm}b_{kl}+\frac{1}{2}b^{nm}\epsilon_{kl}+o\left(\eta^{2}\right).
S_{si}=\oint_{0}{\frac{dz}{2\pii}}{\frac{1}{z^{s-1}}}\oint_{0}{\frac{dw}{2\pii}}{\frac{1}{w^{i+2}}}{\frac{(f^{\prime}(z))^{2}(f^{\prime}(w))^{-1}}{f(z)-f(w)}}\biggl({\frac{f(w)}{f(z)}}\biggr)^{3}.
<{\frac{\deltaW}{\delta\beta^{a}}}>_{_{\beta^{a}=0}}=\,<(D_{\mu}J_{R}^{\mu}\,)^{a}+{\frac{1}{4\pi}}\epsilon^{\mu\nu}\partial_{\mu}A_{\nu}^{a}>_{_{\beta^{a}=0}}\,=\,0\,\,.
f_{N}\left[\phi(x)\right]={\frac{1}{Z_{0}}}\sum_{n=0}^{N}{\frac{\left(-1\right)^{n}}{n!}}\left({\frac{\lambda}{4}}\int\mathrm{d^{d}}x\phi^{4}\right)^{n}e^{-\int\mathrm{d^{d}}x\left[{\frac{1}{2}}\left(\partial_{\mu}\phi\right)^{2}+{\frac{1}{2}}m^{2}\phi^{2}\right]}
\times\;\prod_{i=1}^{m_{b}^{\prime}}\psi_{2}^{(b)}({y^{\prime}}_{i}^{(b)})\prod_{i=1}^{n_{b}}\overline{{\psi}}_{1}^{(b)}(x_{i}^{(b)}\!+\hat{\tau})\prod_{i=1}^{m_{b}}\psi_{2}^{(b)}(y_{i}^{(b)}\!+\hat{\tau})\prod_{i=1}^{n_{b}^{\prime}}\overline{{\psi}}_{1}^{(b)}({x^{\prime}}_{i}^{(b)})\Big\rangle_{free}\;.
S_{\mathrm{nobound}}={\frac{3\piM_{\mathrm{P}}^{2}}{2}}\int_{0}^{\infty}drR(r)(1-[R^{\prime}(r)]^{2})={\frac{3\pi^{2}}{8}}M_{\mathrm{P}}^{2}R^{2}(0)\.
\int_{M_{4}}\partial_{\mu}j^{\mu}=\int_{M_{4}}\frac{e}{8\pi^{2}}\epsilon^{\mu\nu\lambda\sigma}\partial_{\mu}\rho\partial_{\nu}\theta\partial_{\lambda}A_{\sigma}=\int_{\Sigma_{2}}\frac{e}{4\pi}\epsilon^{ab}\partial_{a}A_{b}=\int_{\Sigma_{2}}\frac{e}{4\pi}F_{03}
\lambda=\pig^{2}C_{D}\lambda_{1}=\frac{\pig^{2}}{2^{D}\pi^{(D+1)/2}\Gamma\left(\frac{D-1}{2}\right)}\lambda_{1}=\frac{g^{2}}{8\pi^{2}}\lambda_{1},\qquad\lambda_{1}=\frac{3}{N_{5}}\left(1-\frac{1}{N_{4}^{2}}\right)=\frac{9}{16^{2}},\,
<BB>=\frac{i}{M_{4}M_{1}^{-1}(M_{1}+4M_{3}^{2}M_{4}^{-1}\Box)}(P_{b}^{1})_{\alpha\beta\gammak}+\frac{i}{(4M_{3}^{-1}M_{1}^{-1}\Box-M_{5})}(P_{e}^{1})_{\alpha\beta\gammak}.
{\calL}_{YM}(x,y)\;=\;\gamma^{2}\theta(y)^{2}F_{\mu\nu}F^{\mu\nu}+\alpha\gamma\theta(y){\calF}_{\mu}^{\dagger}{\calF}^{\mu}+\alpha^{2}\left(\Psi^{\dagger}\Psi-1\right)^{2}
X_{l,m}^{(L)}\equiv(\partial_{A_{1}}\cdots\partial_{A_{l+2m}}\omega^{L})\,K_{[l,m]}^{A_{1}\dotsA_{l+2m},B_{1}\dotsB_{l+2m}}\,(\partial_{B_{1}}\cdots\partial_{B_{l+2m}}\omega^{\primeL})\;.
<\,\,X\,\,>_{_{\Psi\,,J\,}}\,=\,\int\prodD\phi^{A}\,X\,exp\left({\frac{i}{\hbar}}W[\phi^{A},\phi_{A}^{\ast}={\frac{\partial\Psi}{\partial\phi^{A}}}]+J_{A}\,\phi^{A}\right)
\Phi^{m}=\left(\begin{matrix}{0}&{\phi_{12}^{m}}&{}&{}&{}&{}&{}\\{}&{0}&{\phi_{23}^{m}}&{}&{}&{}&{}\\{}&{}&{0}&{\quad\cdot}&{}&{}&{}\\{}&{}&{}&{\quad\cdot}&{\quad\cdot}&{}&{}\\{}&{}&{}&{}&{\quad\cdot}&{\quad\cdot}&{}\\{\phi_{k1}^{m}}&{}&{}&{}&{}&{\quad\cdot}&{\quad\cdot}\\\end{matrix}\right)
\int_{0}^{1}dt\delta_{\vec{u}}(t)<\mathrm{Tr}(H_{12}(\gamma)H_{34}(\gamma))>=-\alpha[{\frac{i\pi}{k}}\mathrm{Tr}(H_{12}(\gamma)H_{34}(\gamma))-{\frac{2i\pi}{k}}\mathrm{Tr}(H_{12}(\gamma))\mathrm{Tr}(H_{34}(\gamma))]
\bar{D}^{2}{\calJ}^{f}=\bar{D}^{2}\,(\bar{Q}_{f}e^{V}Q_{f})=4\,Q_{f}\frac{\partial{\calW}}{\partialQ_{f}}+\frac{T(R_{f})}{2\pi^{2}}\,\mathrm{Tr}\,W^{2}\,.
M=\varepsilonM^{\prime}:=\left(\begin{array}{cc}{0}&{M^{\prime}}\\{-M^{\prime}}&{0}\\\end{array}\right)\,,~~M^{\prime}=\left(\begin{array}{cc}{A}&{0}\\{0}&{B}\\\end{array}\right)\,,~~~\bar{A}=A\,,~~~\bar{B}=B\,,~~~~~\lambda=0
D_{1}\eta_{1}^{\alpha_{1}}=\frac12C_{\;\;\beta_{1}\gamma_{1}}^{\alpha_{1}}\eta_{1}^{\beta_{1}}\eta_{1}^{\gamma_{1}}+C_{\;\;\beta_{1}\gamma_{3}}^{\alpha_{1}}\eta_{1}^{\beta_{1}}\eta_{1}^{\gamma_{3}},
ds^{2}~=~-\left(1-\left(\frac{r_{0}}{r}\right)^{2}\right)^{2}~dt^{2}+\left(1-\left(\frac{r_{0}}{r}\right)^{2}\right)^{-2}~dr^{2}+r^{2}d\Omega_{3}^{2}~.
0=\intd^{5}x\left(ie\frac{\delta\Gamma}{\delta\Phi_{c}^{*}}\Phi_{c}^{*}w_{c}-ie\frac{\delta\Gamma}{\delta\Phi_{c}}w_{c}\Phi_{c}+D_{\alpha}\frac{\delta\Gamma}{\delta\Gamma_{\alpha}^{c}}w_{c}+\frac{1}{\xi}\frac{\delta\Gamma}{\deltau_{c}}D^{\alpha}\Gamma_{\alpha}^{c}\right).
\left(\begin{array}{c}{\hat{D}}\\{\hat{Y}}\\\end{array}\right)=\left(\begin{array}{c}{A^{-1}\,D-A^{-1}CB^{-1}\,Y}\\{B^{-1}\,Y}\\\end{array}\right)\quad,\quad\quad\quad(\begin{array}{cc}{\hat{H}\,,}&{\hat{\Theta}}\\\end{array})=(\begin{array}{cc}{H\,A\,,}&{H\,C+\Theta\,B}\\\end{array})\quad.
\frac{8\pi}{\Phi}T_{\mu\nu}+\frac{\omega}{\Phi^{2}}\left(\nabla_{\mu}\Phi\nabla_{\nu}\Phi-\frac{1}{2}g_{\mu\nu}\nabla_{\sigma}\Phi\nabla^{\sigma}\Phi\right)+\frac{1}{\Phi}\left(\nabla_{\mu}\nabla_{\nu}\Phi-g_{\mu\nu}\Box\Phi\right)=\Lambdag_{\mu\nu}\;.
\Lambda_{n}=\left[\begin{matrix}{0}&{1}&{0}&{\cdots}&{0}\\{\vdots}&{0}&{1}&{\ddots}&{\vdots}\\{\vdots}&{{}}&{\ddots}&{\ddots}&{0}\\{0}&{{}}&{{}}&{\ddots}&{1}\\{\lambda}&{0}&{\cdots}&{\cdots}&{0}\\\end{matrix}\right]
\int\frac{d^{3}k}{(2\pi)^{3}}\prod_{j=1}^{m}\frac{1}{(k-p_{j})^{2}-m^{2}+i\epsilon}=\frac{1}{2}\sum_{p}\sum_{s=\pm}\left[{\calT}_{p}^{s}\int\frac{1}{(2\pi)^{3}}\prod_{i\inP}\frac{1}{(k-p_{i})^{2}-m^{2}+i\epsilon}\right]
{\cal{Z}}^{PF}(X_{1},X_{2};p)=\frac{1}{(1-X_{1})}\frac{1}{(1-X_{2})}\left[\frac{(1-(X_{1}X_{2})^{p+2})}{(1-X_{1}X_{2})}-\frac{({X_{1}}^{p+2}-{X_{2}}^{p+2})}{(X_{1}-X_{2})}\right]~~~~.
{\hat{Y}}_{n}^{K}=\sum_{IlJm}(-)^{N+I+J}\sqrt{(2I+1)(2J+1)}\Bigl\{\begin{array}{ccc}{I}&{J}&{K}\\{{\frac{N}{2}}}&{{\frac{N}{2}}}&{{\frac{N}{2}}\}\\\end{array}\Bigr\}C_{lmn}^{IJK}{\hat{Y}}_{l}^{I}{\hat{Y}}_{m}^{J}.
ds^{2}=\sqrt{a_{1}a_{2}b}c_{1}c_{2}\left(dr^{2}+\frac{1}{a_{1}^{2}}\sigma_{1}^{2}+\frac{1}{a_{2}^{2}}\sigma_{2}^{2}+\frac{1}{b^{2}}\sigma_{3}^{2}+\frac{1}{c_{1}^{2}}(\sigma_{4}^{2}+\sigma_{5}^{2})+\frac{1}{c_{2}^{2}}(\sigma_{6}^{2}+\sigma_{7}^{2})\right).
\int_{x_{1}}^{x_{2}}dv\langleT_{vv}^{\bar{M}}(\Sigma)\rangle=\frac{1}{12}\left.\left[\left(\partial_{v}\bar{v}\right)^{1/2}\partial_{v}\left(\partial_{v}\bar{v}\right)^{-1/2}\right]\right|_{x_{1}}^{x_{2}}+\frac{1}{48}\int_{x_{1}}^{x_{2}}dv\left(\frac{\partial_{v}^{2}\bar{v}}{\partial_{v}\bar{v}}\right)^{2}
A\,=\,\frac{1}{6}\left(\begin{array}{cccc}{3\sqrt{2}}&{-3\sqrt{2}}&{0}&{0}\\{\sqrt{6}}&{\sqrt{6}}&{-2\sqrt{6}}&{0}\\{\sqrt{3}}&{\sqrt{3}}&{\sqrt{3}}&{-3\sqrt{3}}\\{3}&{3}&{3}&{3}\\\end{array}\right)\,.
{\frac{d{\psi}}{ds}}\approx\tilde{\lambda}^{\alpha}\left\{\psi,\\phi_{\alpha}\right\}_{p,q,t,\pi_{t}}={\tilde{\lambda}}^{\alpha}\left\{\pi_{t}+\H_{0},\phi_{\alpha}\right\}={\frac{1}{\tilde{\mu}}}{\tilde{\lambda}}^{\alpha}C_{\alpha\beta}{\tilde{\lambda}}^{\beta},
S_{0}=-l^{3}\inta^{3}(t)dt[(\sum_{i}{\frac{\dot{\psi}_{i}^{2}}{2}}-\sum_{i,j}V_{i,j})+({\frac{R}{12g}})+(\lambda({\frac{n_{f}^{2}\dot{\theta}_{f}^{2}}{2(\rho_{f}+p_{f})}}-{\frac{\rho_{f}+p_{f}}{2}})+p_{f})],
\left.\begin{array}{llr}{\sum_{r=3}^{5}(\delta_{2,0+b_{r}}-\delta_{2,0-b_{r}})}&{=}&{-3}\\{\sum_{r=3}^{5}(\delta_{2,3+b_{r}}-\delta_{2,3-b_{r}})}&{=}&{-3}\\{\sum_{r=3}^{5}(\delta_{3,1+b_{r}}-\delta_{3,1-b_{r}})}&{=}&{-3}\\{\sum_{r=3}^{5}(\delta_{3,2+b_{r}}-\delta_{3,2-b_{r}})}&{=}&{3}\\\end{array}\right.
\hat{\phi}_{(I)}(x)=\int_{(\infty)}\frac{d^{3}\vec{k}}{(2\pi)^{3}2\omega(\vec{k}\,)}\left[a(\vec{k}\,)e^{-ik\cdotx}\,+\,a^{\dagger}(\vec{k}\,)e^{ik\cdotx}\right]\,
{\calA}\,=\,\frac{c-1}{2}M^{2}\,\int_{}^{}\,d^{2}\xi\sqrt{g}\,g^{ij}\,D_{i}x^{\lambda}\,\frac{1}{c\,-\,e^{D^{2}/\mu^{2}}}\,D_{j}x_{\lambda}\,{,}
S=\intd^{D}x\left(K_{+-}-\frac{1}{2}\gamma^{ij}K_{ij}+\left(\frac{\partial_{j}}{\partial_{-}}\gamma^{ij}\right)K_{i-}-\frac{1}{2}\left(\frac{\partial_{i}\partial_{j}}{\partial_{-}^{2}}\gamma^{ij}\right)K_{--}\right)~.
+\frac{1}{(z-w)^{\frac{D}{8}-\frac{1}{2}}}\gamma_{\mu\alpha\beta}\psi_{\mu}(w)+\frac{1}{(z-w)^{\frac{D}{8}-1}}\frac{1}{2}\gamma_{\mu\alpha}^{~~\gamma}\gamma_{\nu\gamma\beta}\psi_{\mu}\psi_{\nu}(w)~+~...~.
Z=\frac{1}{\sqrt{g_{1}^{2}+g_{2}^{2}}}\left(g_{1}A_{1}^{0}+g_{2}A_{2}^{0}\right)\quad\mathrm{and}\quad\gamma=\frac{1}{\sqrt{g_{1}^{2}+g_{2}^{2}}}\left(g_{2}A_{1}^{0}-g_{1}A_{2}^{0}\right)
{\calO}_{1}^{i}{}_{k}{\calO}_{2}^{k}{}_{j}=\left[\begin{matrix}{0}&{\mathrm{e}^{i{\frac{\pi}{4}}}\Omega_{+}^{-{\frac{1}{2}}}}\\{-\mathrm{e}^{-i{\frac{\pi}{4}}}\Omega_{+}^{\frac{1}{2}}}&{0}\\\end{matrix}\right]{\calO}_{B}^{i}{}_{j}\left[\begin{matrix}{0}&{-\mathrm{e}^{i{\frac{\pi}{4}}}\Omega_{+}^{-{\frac{1}{2}}}}\\{\mathrm{e}^{-i{\frac{\pi}{4}}}\Omega_{+}^{\frac{1}{2}}}&{0}\\\end{matrix}\right].
G(x,0)=\frac{{\displaystyle\int}{\calD}A_{\mu}{\calD}{\bar{\psi}}{\calD}\psi~{\bar{\psi}}(x)~\psi(x)~{\bar{\psi}}(0)~\psi(0)\,\,\,{\displaystylee}^{iS_{SM}}}{{\displaystyle\int}{\calD}A_{\mu}{\calD}{\bar{\psi}}{\calD}\psi\,\,\,{\displaystylee}^{iS_{SM}}},
B_{L,N^{\prime},N^{\prime\prime}}^{(c/2\mu)}=\frac{\betac}{2\mu}\int_{-1}^{+1}dy\frac{(1-y^{2})^{L+\frac12}C_{n^{\prime}}^{L+1}(y)C_{n^{\prime\prime}}^{L+1}(y)}{1-y}
P^{\mu}=\intd^{3}x\left[\dot{\Phi}({\vec{x}},t)\partial^{\mu}\Phi({\vec{x}},t)-{\frac{1}{2}}\eta^{\mu0}(\partial_{\nu}\Phi({\vec{x}},t)\partial^{\nu}\Phi({\vec{x}},t)-m^{2}\Phi^{2}({\vec{x}},t))\right],
\Phi=\sum_{\sigma=\pm}\int_{0}^{\infty}d\Omega\sum_{\{T,\lambda\}}\sum_{A}\left(u_{\Omega\{T,\lambda\}A}^{(\sigma)}b_{\Omega\{T,\lambda\}}^{(\sigma)A}+b_{\Omega\{T,\lambda\}}^{(\sigma)A\dagger}u_{\Omega\{T,\lambda\}A}^{(\sigma)*}\right)\.
G_{\\nu}^{\mu}=\frac{\kappa^{2}}{l\Psi_{0}}\left(\stackrel{A}{T_{\\nu}^{\mu}}+(1-\Psi_{0})\stackrel{B}{T_{\\nu}^{\mu}}\right)+\frac{1}{\Psi_{0}}\left(\delta\Psi_{\|\nu}^{|\mu}-\delta_{\nu}^{\mu}\Box\delta\Psi\right)\.
\bar{H}(z)=\left(\begin{array}{cc}{\hat{w}(z)+\textstyle{\frac{1}{2}}\hat{\lambda}(z)+i\textstyle{\frac{1}{2}}\hat{\psi}(z)}&{0}\\{2\bar{\hat{\rho}}{}^{a}(z)}&{\hat{t}{}^{a}{}_{b}(z)++i\textstyle{\frac{2}{{\calN}}}(\hat{\psi}(z)+\delta^{4}{}_{{\calN}}\,\Omega)\delta^{a}{}_{b}}\\\end{array}\right)\,.
\frac{a_{\nu}a_{\mu}\nabla^{\nu}R^{1/2}\nabla^{\mu}R^{1/2}}{2R}+\frac{a_{\mu}b\nabla^{\mu}R^{1/2}}{R^{1/2}}+\frac{a_{\nu}\nabla^{\nu}a_{\mu}\nabla^{\mu}R^{1/2}}{2R^{1/2}})-\frac{a_{\mu}a_{\nu}\nabla^{\mu}\nabla^{\nu}R^{1/2}}{2R},
\deltax_{i}^{\mu}=\varepsilon_{ij}e^{jk}\,\partial_{\tau}x_{k}^{\mu},\quad\deltae_{ij}=\partial_{\tau}\varepsilon_{ij},\quad\deltap_{\mu}^{i}=e^{ij}\varepsilon_{jk}\,\partial_{\tau}p_{\mu}^{k}\,,
\int_{-\infty}^{\infty}{\frac{dp_{3}}{2\pi}}\left(m^{2}+p_{3}^{2}\right)^{3/2-2k-j}={\frac{(m^{2})^{2-2k-j}}{2\sqrt{\pi}}}{\frac{\Gamma(2k+j-2)}{\Gamma(2k+j-3/2)}}
H_{k_{n}}=\frac{L^{3}}{2}\pi_{\chi}({\bfk}_{n},t)\pi_{\chi}(-{\bfk}_{n},t)+\frac{1}{2L^{3}}({\bfk}_{n}^{2}+m^{2})\chi({\bfk}_{n},t)\chi(-{\bfk}_{n},t).
{\calB}_{a}(\tau,\bar{\tau})=\frac{2}{|\eta(\tau)|^{4}}\sum_{{even}\;s}\frac{(-1)^{s_{1}+s_{2}}}{2\pii}\frac{dZ_{\psi}(\bfs,\bar{\tau})}{d\tau}tr_{s_{1}}\left[(Q_{a}^{2}-\frac{k_{a}}{8\pi\tau_{2}})(-1)^{s_{2}F}q^{H}\bar{q}^{\bar{H}}\right]_{int}.
\bar{Q}=\left(\begin{array}{cccccc}{\rho}&{0}&{0}&{0}&{0}&{0}\\{0}&{\rho}&{0}&{0}&{0}&{0}\\\end{array}\right)~,~X=\left(\begin{array}{cccccc}{0}&{\lambda_{1}}&{0}&{0}&{0}&{0}\\{-\lambda_{1}}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{\lambda_{2}}&{0}&{0}\\{0}&{0}&{-\lambda_{2}}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{\lambda_{2}}\\{0}&{0}&{0}&{0}&{-\lambda_{2}}&{0}\\\end{array}\right)~,
\Big\langle\overline{{\psi}}_{\alpha_{0}}^{(a_{0})}(x_{0})\psi_{\beta_{0}}^{(b_{0})}(x_{0})\overline{{\psi}}_{\alpha_{1}}^{(a_{1})}(x_{1})\psi_{\beta_{1}}^{(b_{1})}(x_{1}).....\overline{{\psi}}_{\alpha_{n}}^{(a_{n})}(x_{n})\psi_{\beta_{n}}^{(b_{n})}(x_{n})\Big\rangle_{0\;c}^{\theta}\;,
\begin{array}{rcl}{}&{}&{\deltaC=3\partial\Lambda-6A^{1\,m}\partial\Lambda_{m}+3\epsilon_{mnp}A^{1\,m}A^{1\,n}\partial\Lambda^{2\,p}\,,}\\{}&{}&{}\\{}&{}&{\deltaB_{m}=2\partial\Lambda_{m}-2\epsilon_{mnp}A^{1\,n}\partial\Lambda^{2\,p}\,,}\\{}&{}&{}\\{}&{}&{\deltaA^{2\,m}=\partial\Lambda^{2\,m}\,.}\\\end{array}
G(x)=2\pix\int_{0}^{1}{\frac{\pi^{2}B_{2}(t)}{(x^{2}+(2\pit)^{2})^{{\frac{3}{2}}}}}dt+2\pix\int_{1}^{\infty}{\frac{\pi^{2}B_{2}(t-[t])}{(x^{2}+(2\pit)^{2})^{{\frac{3}{2}}}}}dt\.
\langles_{1}\cdot\cdot\cdots_{N}|\Psi\rangle=e^{-B\sum_{i}s_{i}^{2}/2}e^{-{\frac{1}{4B}}\sum_{i}\left({\frac{\partial}{\partials_{i}}}\right)^{2}}\prod_{i<j}(s_{i}-s_{j})^{m}.
\chi^{\mathrm{(TM)}}={\frac{i\hbarA}{\pi^{2}}}\intd\omega_{\mathrm{in}}\int_{0}^{|\omega_{\mathrm{in}}|}dk_{\|}\,k_{\|}\,\frac{\left[k_{\|}^{2}-\omega_{\mathrm{in}}(\omega+\omega_{\mathrm{in}})\right]^{2}}{\biggl[(\omega+\omega_{\mathrm{in}}+i\varepsilon)^{2}-k_{\|}^{2}\biggr]^{1/2}\,\sqrt{{\omega_{\mathrm{in}}}^{2}-k_{\|}^{2}}}\,\left[{\frac{1}{2}}+{\overline{n}}(|\omega_{\mathrm{in}}|)\right].
f^{1}(u)\equivf^{(+1,0)}(u)=f^{i}u_{i}^{1}+g^{(ij)k}u_{i}^{1}u_{j}^{1}u_{k}^{4}+h^{ijk}u_{1}^{1}u_{j}^{2}u_{k}^{3}+\ldots\;.
\deltaK_{\mu\nu}=\frac{\epsilon}{2}\deltag_{\perp\perp}K_{\mu\nu}^{(0)}-\delta\Gamma_{\perp\mu\nu}-\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\deltaZ_{\perp}+\epsilon\deltaZ_{\perp}(K_{\mu}^{(0)\rho}K_{\rho\nu}^{(0)}-R_{\perp\mu\perp\nu\perp}^{(0)})+\bar{\calL}_{\deltaZ_{\parallel}}K_{\mu\nu}^{(0)},
\begin{array}{ccc}{S}&{=}&{-\frac1{16\piG_{p+1}}\intd^{p+1}y\sqrt{-\tilde{g}}\Pia_{s}e^{-\phi}\{-2a^{i}D^{A}\partial_{A}a_{i}-a^{i}a^{j}\partial^{A}a_{i}\partial_{A}a_{j}}\\{}&{}&{}\\{}&{}&{+a^{i}a^{j}\partial^{A}a_{k}\partial_{A}a_{l}\eta_{ij}^{kl}+\partial^{A}\phi\partial_{A}\phi+\tilde{R}+2\Lambda\},}\\\end{array}
\begin{array}{ccccccc}{F_{A}}&{\equiv}&{<H^{A}\Phi_{A}>}&{=}&{-iF\mathrm{tr}(\Phi_{A})}&{=}&{0,}\\{F_{B}}&{\equiv}&{<K^{B}\Phi_{B}>}&{=}&{-iF\mathrm{tr}(\Phi_{B})}&{=}&{F(\alpha_{B}+\beta_{B})/2,}\\\end{array}
L_{d+2}^{\Psi}=\bar{\Psi}\gamma\cdotX\,\,\gamma\cdot\left(\partial-iA\right)\Psi-h\Phi^{2/\left(d-2\right)}\bar{\Psi}\gamma\cdotX\,\Psi+\bar{\Psi}\gamma\cdotX\,\xi.
\langle\cdots\rangle_{\midA}=\frac{1}{Z(A)}\int\mathrm{~\cal{D}~}_{g}\phi\mathrm{~\cal{D}~}_{g}\sigma\,d\Lambda\,\delta\left(1-\frac{1}{A}\intd^{2}z\sqrt{g}\right)\,\delta\left(\LambdaA+4\pi\right)\,e^{-S_{g}}\left(\cdots\right).
p_{0}=-\frac{L_{p}^{(3-d)}}{4\pi}\left\{\frac{(d-2)}{A_{0}}\left(k+\frac{A_{0}^{2}}{\ell^{2}}\right)^{1/2}+\frac{A_{0}/\ell^{2}}{\sqrt{k+A_{0}^{2}/\ell^{2}}}\right\}.