harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\tilde{A}}_{\mu}(x)=4\pi\int\delta\xids\epsilon_{\mu\nu\rho\sigma}\Omega_{\xi}(s,0)L^{\rho\sigma}[\xi|s]\Omega_{\xi}^{-1}(s,0)\dot{\xi}^{\nu}(s)\delta^{4}(x-\xi(s)).
\left.\begin{array}{l}{\left[q_{\mu},q_{\nu}\right]=i(H_{\mu\nu}+T_{\mu\nu}R),}\\{{}}\\{\left[q_{\mu},R\right]=iC_{\mu}S,}\\{{}}\\{\left[q_{\mu},S\right]=-iC_{\mu}R,}\\{{}}\\{\left[R,S\right]=0,}\\\end{array}\right.
\nabla_{\mu}^{x}\;\nabla_{\nu}^{y}\left\{{\frac{1}{\sigma_{\gamma}(x,y)}}\right\}={\frac{1}{\sigma_{\gamma}(x,y)^{2}}}\left\{4\nabla_{\mu}^{x}s_{\gamma}(x,y)\;\nabla_{\nu}^{y}s_{\gamma}(x,y)-\nabla_{\mu}^{x}\;\nabla_{\nu}^{y}\sigma_{\gamma}(x,y)\right\}.
\Psi_{\sigma,N_{s}}=\sum_{N_{r}}M_{N_{s},N_{r}}(\{\sigma_{q}\})\chi_{N_{r}}\quad\mathrm{{where}}\quadM_{N_{s},N_{r}}(\{\sigma_{q}\})=\sum_{p=a,b}\int_{C_{p}^{(r)}}\Psi_{\sigma,N_{s}}(z)\frac{d\thetadz}{2\pii}Y_{p,N_{r}}(t).
\alpha=Au_{0}{\frac{2^{\nu}\Gamma(\nu+1)\Gamma(\nu)\Gamma(c)u_{0}^{-\nu}}{\Gamma(c-a)\Gamma(c-b)}},\beta=Au_{0}{\frac{\Gamma(-\nu+1)\Gamma(-\nu)\Gamma(c)u_{0}^{\nu}}{2^{\nu}\Gamma(a)\Gamma(b)}}.
\begin{array}{ll}{E_{0}\geq\textstyle{\frac{\mu}{12}}(2J_{z}+3J_{u}+2J_{v}+J_{w})~~~}&{~~~\mathrm{for~~}\mu>0\,,}\\{{}}&{{}}\\{E_{0}\geq\textstyle{\frac{|\mu|}{12}}(2J_{z}+J_{u}+2J_{v}+3J_{w})~~~}&{~~~\mathrm{for~~}\mu<0\,.}\\\end{array}
B(\varepsilon)={\frac{1}{\sqrt{1-v^{2}}}}\left(\begin{array}{cc}{1}&{-v}\\{v}&{-1}\\\end{array}\right)={\frac{1}{2\sqrt{\varepsilon}}}\left(\begin{array}{cc}{1+\varepsilon}&{1-\varepsilon}\\{\varepsilon-1}&{-1-\varepsilon}\\\end{array}\right)
Tr_{(s+1)}(e^{-a_{s}^{\dagger}a_{s}/M^{2}})-Tr_{(s)}(e^{-a_{s}a_{s}^{\dagger}/M^{2}})=Tr_{(s+1)}(e^{-a_{s}a_{s}^{\dagger}/M^{2}})-Tr_{(s)}(e^{-a_{s}a_{s}^{\dagger}/M^{2}})=1
\tilde{\delta}_{left}\chi=\tilde{\Lambda}^{(+)}\partial_{(+)}\chi-{\calD}\tilde{\Lambda}^{(+)}\;,\;\;\;\;\;\;\;\tilde{\Lambda}^{(+)}=i(\Lambda^{(-)}{\calD}\chi+\frac{1}{2}{\calD}\Lambda^{(-)}\chi)\;.
{S_{P}^{4}}^{\star}=\intd^{4}x\Bigl[{\frac{1}{2}}T^{\mu\nu}F_{\mu\nu}+{\frac{1}{4}}T^{\mu\nu}T_{\mu\nu}-{\frac{{\mu}^{2}}{2}}A^{\mu}A_{\mu}\Bigl]\;,
\langle\vec{\varphi}(X_{0})\rangle_{\zeta}\simeq\vec{\Phi}e^{\omegaX_{0}}-\frac{\lambda^{\prime}}{48\omega^{2}}\vec{\Phi}e^{3\omegaX_{0}}\left[\vec{\Phi}^{\,2}-\frac{\lambda}{192\pi}(N+2)^{2}\frac{1}{\omega^{2}}\frac{T}{V}\right].
(T_{i}\otimesT_{j})_{{\calG}\{\alpha_{\ell},\alpha_{\ell}^{\prime}\}}=\prod_{\ell\not=i,j}^{r}\delta_{\alpha_{\ell},\alpha_{\ell}^{\prime}}\cdot(T^{\alpha_{i}^{\prime}})_{\alpha_{i},\alpha}(T^{\alpha_{j}})_{\alpha,\alpha_{j}^{\prime}}
\Omega_{j,l,m}=\left(\begin{matrix}{{({\frac{l+{\frac{1}{2}}+m}{2l+1}})^{\frac{1}{2}}}Y_{l,{m-{\frac{1}{2}}}}}\\{{({\frac{l+{\frac{1}{2}}-m}{2l+1}})^{\frac{1}{2}}}Y_{l,{m+{\frac{1}{2}}}}}\\\end{matrix}\right),
D(\omega)=\left[\prod_{v}o(\omega^{m_{\gamma}^{v}+4\theta(m_{y}^{v})-m_{y}^{v}+2\theta(m_{g}^{v})\delta_{0}^{m_{y}^{v}}})\right]\omega^{-2(I_{\gamma}+I_{y}+I_{g})}
\intd{\bfx}{\bfA}_{lmn}^{(\lambda)}({\bfx})\cdot{\bfA}_{l^{\prime}m^{\prime}n^{\prime}}^{(\lambda^{\prime})*}({\bfx})=\delta_{\lambda,\lambda^{\prime}}\delta_{l,l^{\prime}}\delta_{m,m^{\prime}}\delta_{n,n^{\prime}}
\left\langle0\right|A_{n-1}^{+1}(t)A_{-n+1}^{-1}(t^{\prime})\left|0\right\rangle_{cov}=\frac{\alpha+1}{4\left|n-1\right|}\left[\theta(t-t^{\prime})\left(\frac{t^{\prime}}{t}\right)^{\left|n-1\right|}+\theta(t^{\prime}-t)\left(\frac{t}{t^{\prime}}\right)^{\left|n-1\right|}\right].
m\epsilon_{\mu\alpha\beta}\partial^{\alpha}\hat{A}^{\beta}\approx\epsilon_{\mu\alpha\beta}\partial^{\alpha}(\hat{f}^{\beta}-m\hat{a}^{\beta})-\epsilon_{\nu\alpha\beta}(m\partial^{\alpha}\hat{A}^{\beta}-\partial^{\alpha}\hat{f}^{\beta})\frac{\delta\hat{a}^{\nu}}{\delta\hat{A}^{\mu}}.
\Sigma(Z_{1},Z_{2},T)=\sigma_{c}+{\frac{\left(e^{2h}-1\right)^{2}\left(3+(2+\sqrt{7})e^{2h}\right)^{2}}{20\cdot2^{1/3}e^{4h}\left(1+e^{2h}\right)^{2}}}\left[{\frac{\Phi(Z_{1},T)-\Phi(Z_{2},T)}{Z_{1}-Z_{2}}}\right]\epsilon^{1/3}+{\calO}(\epsilon^{2/3})
V=-\theta^{\alpha}\sigma_{\alpha\dot{\alpha}}^{m}\bar{\theta}^{\dot{\alpha}}V_{m}-\frac{1}{\sqrt{2}}\left(\theta^{\alpha}\theta_{\alpha}\bar{\theta}_{\dot{\alpha}}\bar{\chi}^{\dot{\alpha}}-\bar{\theta}_{\dot{\alpha}}\bar{\theta}^{\dot{\alpha}}\theta^{\alpha}\chi_{\alpha}\right)+\frac{1}{2}\theta^{\alpha}\theta_{\alpha}\bar{\theta}_{\dot{\alpha}}\bar{\theta}^{\dot{\alpha}}D
[\phi_{k}\,,\,\phi_{h}]=-{\frac{i\pi\beta^{2}}{2}}s\left(\frac{k-h}{2N}\right)\quad,\quad[\bar{\phi}_{k}\,,\,\bar{\phi}_{h}]={\frac{i\pi\beta^{2}}{2}}s\left(\frac{k-h}{2N}\right)\quad,\quad[\phi_{k}\,,\,\bar{\phi}_{h}]=0\quad.
I_{Bolt}=\frac{16\pi^{3}(5r_{b}^{8}-(\ell^{2}+14N^{2})r_{b}^{6}+5N^{2}r_{b}^{4}\ell^{2}-5N^{4}(3\ell^{2}-14N^{2})r_{b}^{2}-5N^{6}(\ell^{2}-7N^{2}))}{5(7N^{2}-7r_{b}^{2}-\ell^{2})}
{\calL}_{gK}=\frac{1}{2}\left[i\overline{{\Psi}}_{L}\left(\partial\!\!\!/+\frac{1}{2}(i{\calA}\!\!\!/\,\hat{C}+{\calA}\!\!\!/\:\hat{C}i)\right)\Psi_{L}+\overline{{\Psi}}_{L}\left(\partial\!\!\!/+\frac{1}{2}(i{\calA}\!\!\!/\,\hat{C}+{\calA}\!\!\!/\:\hat{C}i)\right)\Psi_{L}i\right]
[\hat{x}^{\mu},\hat{p}_{\nu}]=i\{x^{\mu},p_{\nu}\}_{D(\Phi_{4}^{(1)})}=i\delta_{\nu}^{\mu}\;,\;\;[\hat{\psi}^{k},\hat{\psi}^{n}]_{+}=i\{\psi^{k},\psi^{n}\}_{D(\Phi_{4}^{(1)})}=-\frac{1}{2}\eta^{kn}\;.
\partial_{U}\left(U^{5}f(U)\partial_{U}\phi(U,x_{2})\right)+R^{4}U\partial_{22}\phi(U,x_{2})=\frac{2\kappa_{10}^{2}}{4\pi}\frac{U_{T}^{2}}{R^{2}\Omega_{5}}\delta(U-U_{T})\delta(x_{2})
\begin{aligned}{\hat{S}(P_{i})\Lambda+\partial_{i}\Lambda}&{{}=0,}\\{\hat{S}(D)\Lambda+x^{\mu}\partial_{\mu}\Lambda}&{{}=0,}\\{\hat{S}(K_{i})\Lambda-2x^{\mu}\LambdaS_{i\mu}+\left(2x_{i}x^{\mu}\partial_{\mu}-x^{2}\partial_{i}\right)\Lambda}&{{}=0,}\\{\hat{S}(L_{ij})\Lambda-\LambdaS_{ij}+\left(x_{i}\partial_{j}-x_{j}\partial_{i}\right)\Lambda}&{{}=0.}\\\end{aligned}
\Phi^{\alpha_{0}}=\left(A_{\mu}^{a},H_{\mu}^{a},\varphi_{a},B_{a}^{\mu\nu}\right),\;\Phi_{\alpha_{0}}^{*}=\left(A_{a}^{*\mu},H_{a}^{*\mu},\varphi^{*a},B_{\mu\nu}^{*a}\right),
t-t_{0}=\int_{x_{0}}^{x}dx^{\prime}\frac{1}{\sqrt{2}\sqrt{E+\int_{x_{0}}^{x^{\prime}}f_{0}(x^{\prime\prime})dx^{\prime\prime}+\int_{x_{0}}^{x^{\prime}}f_{1}(x^{\prime\prime})dx^{\prime\prime}}}
\hat{T}_{00}=\frac{1}{2}\left[\dot{\phi}^{2}+\left(\vec{\nabla}\phi\right)^{2}+m^{2}\phi^{2}+\sigma\phi^{2}\right]=\frac{1}{2}\intd^{d}y\phi(y)\left(\hat{T}_{x}^{(0)}+\hat{T}_{x}^{(1)}\right)\phi(y)
J_{1}(\frac\phiv)=\left(\begin{array}{cccc}{\phi_{2}^{1}}&{\phi_{3}^{1}}&{\cdots}&{\phi_{m}^{1}}\\{\phi_{2}^{2}}&{\phi_{3}^{2}}&{\cdots}&{\phi_{m}^{2}}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{\phi_{2}^{m-1}}&{\phi_{3}^{m-1}}&{\cdots}&{\phi_{m}^{m-1}}\\\end{array}\right)
\sum\xi_{I^{2},I_{3},Y}^{(n_{1},n_{2})}\langle(n_{1},n_{2}),I^{2},I_{3},Y|U^{(n_{1},n_{2})}(g)|(n_{1},n_{2}),0,0,-{\textstyle\frac{2}{3}}{\bar{N}}\rangle\nonumber
\mathrm{Tr}(L^{2})=2(p_{1}^{2}+p_{2}^{2})+4m^{2}e^{q_{1}-q_{2}}+4m^{2}e^{-q_{1}-q_{2}}+2m^{2}e^{2q_{2}}=4{\calH}_{A_{3}}^{(2)}.
\frac{\widehat{SU}(n+k)_{1}\times\widehat{SO}(2nk)_{1}}{\widehat{SU}(n)_{k+1}\times\widehat{SU}(k)_{n+1}\times\widehat{U}(1)_{kn(k+n)(k+n+1)}}\cong\frac{\widehat{SU}(n+1)_{k}\times\widehat{SO}(2n)_{1}}{\widehat{SU}(n)_{k+1}\times\widehat{U}(1)_{n(n+1)(k+n+1)}}~,
\sum_{n=1}^{\infty}\frac{1}{\left(2n\atopn\right)}\frac{1}{n^{c}}\prod_{a,b,i,j}\left[\sum_{m=1}^{n-1}\frac{1}{m^{a}}\right]^{i}\left[\sum_{k=1}^{2n-1}\frac{1}{k^{b}}\right]^{j}
E_{c}\sim{\frac{4\pia^{3}}{3}}\int{\frac{(d{\bfk})}{(2\pi)^{3}}}{\frac{1}{2}}k\left(1-{\frac{1}{\sqrt{\epsilon}}}\right)\sim{\frac{a^{3}K^{4}}{12\pi}}\left(1-{\frac{1}{\sqrt{\epsilon}}}\right),
W={\frac{i}{4}}u_{i}^{+}u_{j}^{+}\bar{D}_{\dot{\alpha}}^{i}\bar{D}^{j\dot{\alpha}}\sum_{n=1}^{\infty}\intdu_{1}\ldotsdu_{n}\;{\frac{(-ig)^{n}V^{++}(u_{1})\ldotsV^{++}(u_{n})}{(u^{+}u_{1}^{+})(u_{1}^{+}u_{2}^{+})\ldots(u_{n}^{+}u^{+})}}
\tilde{\rho}_{i}\sim\Omega_{i}^{2}{\frac{k^{2}}{\tilde{a}^{2}}}{\frac{\left.{\calP}_{\delta\sigma_{i}}\right|_{s}}{2\kappa^{2}}}=C^{2}(\mu_{i}){\frac{k^{2}}{\tilde{a}^{2}}}\left({\frac{\tilde{H}}{2\pi}}\right)_{s}^{2}\left({\frac{k}{k_{s}}}\right)^{3-2\mu_{i}}\,,
f_{\scriptscriptstyle{L,L}}(a)=\frac{1}{2\pi\,a^{L}}\int_{0}^{a}dr\,r^{L}\int_{0}^{2\pi}d\theta\,e^{i(1-L)\theta}\,e^{2\phi(r,\theta)}\sum_{m=0}^{\infty}b_{\scriptscriptstyle{m,L}}\,r^{m}\,e^{im\theta}\,.
\int_{0}^{2\pi}d\sigmaX^{i}(\sigma)\hat{P}_{i}(\sigma)=x_{0}^{i}\hat{p}_{0}^{i}+\frac{g_{ij}}{2\sqrt{\alpha^{\prime}}}\sum_{n=1}^{\infty}\left\{\left(\chi_{n}^{i}\alpha_{-n}^{j}+\bar{\chi}_{n}^{i}\tilde{\alpha}_{-n}^{j}\right)+\left(\bar{\chi}_{n}^{i}\alpha_{n}^{j}+\chi_{n}^{i}\tilde{\alpha}_{n}^{j}\right)\right\}~.
\textrm{dim}\,\mathcal{G}(N_{0},N_{1},N_{2})=\left[\frac{N_{0}+2}{2}\right]\textrm{dim}V_{0}+\left[\frac{N_{1}+1}{2}\right]\textrm{dim}V_{1}+\left[\frac{N_{2}}{2}\right]\textrm{dim}V_{2}\quad,
\langle\phi^{2}(x)\rangle_{\beta}=\frac{1}{12\beta^{2}}\eta^{2}+\frac{1}{4\pi^{2}r^{2}}\sum_{n=1}^{\infty}\sum_{l=0}^{\infty}(2l+1)Q_{\nu_{l}-1/2}(z_{n}).
Q_{R}=\frac{1}{\sqrt{2T_{2}U_{2}}}(n_{2}{\bar{U}}\tau+n_{1}\tau-{\bar{U}}l_{1}+l_{2}),\;\;\;{\bar{Q}}_{R}=\frac{1}{\sqrt{2T_{2}U_{2}}}(n_{2}U\tau+n_{1}\tau-Ul_{1}+l_{2}).
\chi^{\mathrm{D}}(\omega)={\frac{i\hbarA}{\pi^{2}}}\intd\omega_{\mathrm{in}}\int_{0}^{|\omega_{\mathrm{in}}|}dk_{\|}\,k_{\|}\,\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],
\frac{\partial}{\partialu}\:{\bfC}=\frac{\partial}{\partialu}\:{\bfC}_{source}+\frac{\partial}{\partialu}\:{\bfC}_{vac}(2)+\frac{\partial}{\partialu}\:{\bfC}_{vac}(3)+O(\hbar^{2})+O[R_{..}^{3}]
{\mathsfL}_{++}^{(210)}=\frac{1}{4!}g_{ab}^{^{(210)}}<\Psi_{(+)a}|\gamma^{0}\gamma^{A}\Gamma_{[\mu}\Gamma_{\nu}\Gamma_{\rho}\Gamma_{\lambda]}|\Psi_{(+)b}>\Phi_{A\mu\nu\rho\lambda}.
\chi_{\Lambda}(C_{m}^{\Lambda_{0}})=\sum_{(\lambda+\lambda_{0}^{\prime},\kappa+\kappa_{0},-s)\inD^{+}}\sum_{s=0}^{\infty}\,m_{\lambda_{0}^{\prime},s}\,\alpha_{\lambda_{0}^{\prime},s}^{m}(\Lambda)\frac{D_{q}[(\lambda+\lambda_{0}^{\prime},\kappa+\kappa_{0},-s)]}{D_{q}[(\lambda,\kappa,0)]}\,,~~~m\in{\bfZ}^{+}
\frac{1}{1+H^{2}r^{2}}\left(\frac{dr}{d\xi^{1}}\right)^{2}-(1+H^{2}r^{2})\left(\left(\frac{dt}{d\xi^{1}}\right)^{2}+c_{1}^{2}\right)+r^{2}\left(\left(\frac{d\phi}{d\xi^{1}}\right)^{2}+c_{2}^{2}\right)=0.
R^{pqrs}R_{pq}{}^{tu}R_{r}{}^{v}{}_{t}{}^{w}R_{svuw}=A_{5},\quadR^{pqrs}R_{p}{}^{t}{}_{r}{}^{u}R_{t}{}^{v}{}_{q}{}^{w}R_{uvsw}=A_{7}\,.
\left(\begin{array}{cc}{v_{2j-1}\cdotv_{2j-1}}&{v_{2j-1}\cdotv_{2j}}\\{v_{2j}\cdotv_{2j-1}}&{v_{2j}\cdotv_{2j}}\\\end{array}\right)=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right).
f_{\lambda}^{(nl)}(z_{1}\tau^{3}|z^{\prime}|z^{\prime\prime},z_{1}\tau^{4})=(-\tau)^{l-n+\lambda-2}(1-\tau^{2})\varphi_{\lambda~l-n+\lambda-2}^{(n~l-2)}(z^{\prime}|z^{\prime\prime})
i\hbar\partial_{\tau}\psi=\frac{1}{2M}g^{\mu\nu}(-i\hbar\partial_{\mu}+\frac{1}{c}A_{\mu})(-i\hbar\partial_{\nu}+\frac{1}{c}A_{\nu})\psi\equiv-\frac{\hbar^{2}}{2M}\Box_{A}\psi
{\calL}(x)=\overline{{\Psi}}(x)\left(i\gamma^{\mu}\partial_{\mu}-eA^{\mu}(x)\gamma_{\mu}\right)\Psi(x)-\frac{1}{4}F^{\mu\nu}(x)F_{\mu\nu}(x)-\frac{\lambda}{2}\left(\partial_{\mu}A^{\mu}(x)\right)^{2}\;,
\rho(w)=2N\,{[}\theta_{2}\theta_{3}{]}\Bigl(0|N\tau\Bigr)\,\frac{\theta_{3}\Bigl(2\sqrt{-1}Nw|N\tau\Bigr)}{\theta_{2}\Bigl(2\sqrt{-1}Nw|N\tau\Bigr)},\quad-\frac{\kappa}{4}\leqw<\frac{\kappa}{4},
\langle\prod_{j}e^{ie_{j}\int_{0}^{1/T}d\tauA_{0}(\tau,x_{j})}\rangle\equiv{\frac{\intdA_{\mu}(x)e^{-\int_{0}^{1/T}dxd\tauF_{01}^{2}/2}\prod_{j}e^{ie_{j}\int_{0}^{1/T}d\tauA_{0}(\tau,x_{j})}}{\intDA_{\mu}(x)e^{-\int_{0}^{1/T}dxd\tauF_{01}^{2}/2}}}
{\calL}_{\mathrm{rel}}^{(2)}=\frac{1}{2}\biggl(\mu+\frac{g^{2}\lambda}{8\pir}\biggr)\,\dot{{\bfr}}^{2}+\frac{1}{2}\left(\frac{g^{2}\lambda}{8\pi}\right)^{2}\biggl(\mu+\frac{g^{2}\lambda}{8\pir}\biggr)^{-1}(\dot{\psi}+{\bfw}({\bfr})\cdot\dot{{\bfr}})^{2}.
\psi_{+2\dot{q}}^{2+}=-\frac{i}{2}\nabla_{+}\chi_{\dot{q}}^{2},\\psi_{-2\dot{q}}^{2+}=-\frac{i}{4}\Omega_{-}^{+2i}\tilde{\gamma}_{\dot{q}q}^{i}\chi_{q}^{2},
{\calE}_{PE}[\rho_{\pm},\theta,\phi=0;R]=\frac{2\piR\Lambda^{4}}{g^{2}}-\frac{2\piR}{g^{2}}\bigg(\Lambda^{2}-\bigg(\frac{\alpha}{R}\bigg)^{2}\bigg)^{2}\qquad\mathrm{for}\quadR\geq\frac{\alpha}{\Lambda}.
ds_{5}^{2}=-f^{2}(R)\left(1-\frac{2m}{R^{2}}\right)dt^{2}+f^{-1}(R)\left[\left(1-\frac{2m}{R^{2}}\right)^{-1}dR^{2}+R^{2}d\Omega_{3}^{2}\right]
\begin{array}{rl}{\mathrm{bosons:}}&{(3,3)\!+\!5(3,1)\!+\!5(1,3)\!+\!16(2,2)\!+\!25(1,1)\nonumber}\\{\mathrm{fermions:}}&{4(3,2)+4(2,3)+10(2,1)+10(1,2).}\\\end{array}
I_{D}[g]=\intd^{D}x\sqrt{g}\left[\alpha_{o}^{\prime}+\alpha_{1}^{\prime}R+\alpha_{2}^{\prime}(R^{\alpha\beta\gamma\delta}R_{\alpha\beta\gamma\delta}-4R^{\alpha\beta}R_{\alpha\beta}+R^{2})+\cdot\cdot\cdot\right].
Z_{1}(R)=2\intdx^{2}\intdx^{1}\partial_{1}{\calW}^{*}(A^{*})=2\int_{x^{2-}}^{x^{2+}}dx^{2}\left[{\calW}^{*}\left(x^{1+}\left(x^{2}\right),x^{2}\right)-{\calW}^{*}\left(x^{1-}\left(x^{2}\right),x^{2}\right)\right],
\sigma_{1}(\tau)=\left[(1+a_{1}^{2})c_{1}^{2}\mid\tau\mid^{8/3}+\frac{6a_{1}K_{1}^{1/2}\mid\tau\mid}{5}+\frac{9K_{1}}{25c_{1}^{2}\mid\tau\mid^{2/3}}\right]^{1/2},
2D(c_{1}+6c_{2}+42c_{3})H_{0}^{4}-\{4D-[3c_{2}+4(2D-1)c_{3}]\Lambda\}H_{0}^{2}-{\frac{1}{12}}\left[(D-2)\Lambda-(D-4)\left({\frac{2c_{1}}{D(D-1)}}+{\frac{c_{2}}{D}}+c_{3}\right)\Lambda^{2}\right]=0.
E=\intd^{3}x\,\Theta_{00}=\intd^{3}x\,\left\{\left[\frac{1}{2}(B_{i}^{a})^{2}+(E_{i}^{a})^{2}+(\Pi^{a})^{2}+[(D_{i}\Phi)^{a}]^{2}\right]+V(\Phi)\right\}\,,
S_{\mathrm{SMB}}=\intd^{3}\xi\biggl[-\frac{1}{2}\sqrt{-\gamma}\gamma^{ij}\partial_{i}\rho^{M}\partial_{j}\rho^{N}g_{MN}(\rho)+\frac{1}{2}\sqrt{-\gamma}+\frac{1}{3!}\epsilon^{ijk}\partial_{i}\rho^{M}\partial_{j}\rho^{N}\partial_{k}\rho^{P}A_{MNP}(\rho)\biggr].
f=\left(\begin{array}{cc}{(Z_{1}^{\prime\prime}+Z_{2}^{\prime}+\zeta)(Z^{\prime\prime}~^{\prime})^{-1}}&{-\sqrt{\zeta}(Z^{\prime\prime}~^{\prime})^{-1}\bar{z}_{1}}\\{-\sqrt{\zeta}z_{1}(Z^{\prime\prime}~^{\prime})^{-1}}&{(Z_{1}^{\prime}+Z_{2}^{\prime}-\theta_{1})(Z^{\prime}~^{\prime})^{-1}}\\\end{array}\right).
A(\vec{u})=\left(\begin{array}{cccc}{u^{3}}&{u^{4}}&{u^{1}}&{u^{2}}\\{u^{4}}&{-u^{3}}&{-u^{2}}&{u^{1}}\\{u^{1}}&{u^{2}}&{-u^{3}}&{-u^{4}}\\{u^{2}}&{-u^{1}}&{u^{4}}&{-u^{3}}\\\end{array}\right).
T^{-1}\,=\,\left(\begin{array}{cc}{a^{-1}+a^{-1}bd^{-1}ca^{-1}}&{-a^{-1}bd^{-1}}\\{-d^{-1}ca^{-1}}&{d^{-1}+d^{-1}ca^{-1}bd^{-1}}\\\end{array}\right)
Z_{TM}^{red}=\int{\calD}p{\calD}qdet{\bigl\{\Theta_{a},\Upsilon_{b}\bigr\}}\delta(\theta)\delta({\theta}_{i}^{T})\delta(\Upsilon)\delta({\Upsilon}_{i}^{T})e^{i\int(p\dot{q}-{\calH}_{TM})}\;,
{\calA}(r,v)=V_{26}\int_{0}^{\infty}{\frac{dl}{l}}(8\pi^{2}\alpha^{\prime}l)^{-p/2}e^{-r^{2}l/2\pi\alpha^{\prime}}[l^{-1/2}\eta(i/l)]^{-22}[{\frac{\eta(i/l)}{-\Theta_{11}(-iu/\pi,i/l)}}],
\frac{\delta^{2}}{\deltaS_{12}^{-1}\deltaS_{34}^{-1}}=\int_{5678}\,S_{51}S_{26}S_{73}S_{48}\,\frac{\delta^{2}}{\deltaS_{56}\deltaS_{78}}+\int_{56}\left[S_{53}S_{41}S_{26}+S_{23}S_{46}S_{51}\right]\,\frac{\delta}{\deltaS_{56}}\,.
\psi_{\mu}^{M}=\left(\begin{array}{c}{\psi_{1}}\\{\psi_{2}}\\\end{array}\right)\,,\quad\epsilon^{M}=\left(\begin{array}{c}{\epsilon_{1}}\\{\epsilon_{2}}\\\end{array}\right)\,,\quad\lambda^{M}=\left(\begin{array}{c}{\lambda_{1}}\\{\lambda_{2}}\\\end{array}\right)\,,\quad\tilde{\lambda}^{M}=\left(\begin{array}{c}{\tilde{\lambda}_{1}}\\{\tilde{\lambda}_{2}}\\\end{array}\right)
\begin{array}{l}{(3\pm\beta_{k})(\beta_{i}\mp\beta_{j})=(\beta_{l}\mp\beta_{s})(\beta_{m}\mp\beta_{r})\,,}\\{{}}\\{(3\pm\beta_{k})(\beta_{m}\mp\beta_{r})=(\beta_{l}\mp\beta_{s})(\beta_{i}\mp\beta_{j})\,,}\\\end{array}
ds^{2}=R^{2}\left[(1+b^{2}-{\frac{2MG_{5}}{R^{2}b^{2}}})dt^{2}-(1+b^{2}-{\frac{2MG_{5}}{R^{2}b^{2}}})^{-1}db^{2}-b^{2}d\Omega^{2}\right]
S=-\frac{1}{2\pi}\intd^{2}\sigma~~e~~\left[h^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X_{\mu}-i\bar{\Psi}^{\mu}\rho^{\alpha}\partial_{\alpha}\Psi_{\mu}+2\bar{\chi}_{\alpha}\rho^{\beta}\rho^{\alpha}\Psi^{\mu}\partial_{\beta}X_{\mu}+\frac{1}{2}\bar{\Psi}^{\mu}\Psi_{\mu}\bar{\chi}_{\beta}\rho^{\beta}\rho^{\alpha}\chi_{\alpha}\right].
\sum_{n=-n_{0}}^{\infty}\frac{a_{n}^{(1)}e^{-n\pii/2}}{\left(x-x_{0}\right)^{n}}=\sum_{n=-n_{0}}^{\infty}\frac{(-1)^{n}a_{n}^{(2)}e^{-n\pii/2}}{\left(x-x_{0}\right)^{n}},
{\calL}_{\mu_{1}\mu_{2}\mu_{3}}^{tadpole}\left(k_{1},k_{2},k_{3},\hat{Q}\right)=-{\frac{9\,{k_{3}\cdotk_{3}}\,k_{{3\mu_{1}}}\eta_{{\mu_{2}\mu_{3}}}}{8\,{\left(\hat{Q}\cdotk_{3}\right)}^{2}}}+{\frac{9\,{k_{3}\cdotk_{3}}\eta_{{\mu_{1}\mu_{3}}}k_{{3\mu_{2}}}}{8\,{\left(\hat{Q}\cdotk_{3}\right)}^{2}}}
{\calL}=-\frac{1}{4}F_{\mu\nu}\left({1-\frac{{e^{2}}}{\pi}\frac{1}{{\partial_{1}^{2}-4\pim\Sigma}}}\right)F^{\mu\nu}-em\Sigma\frac{1}{{\partial_{1}^{2}-4\pim\Sigma}}\theta-A_{0}J^{0}.
\sum_{\alpha_{n}=0}^{m_{n}-1}(-1)^{\alpha_{n}}\left(\frac{d}{dt}\right)^{\alpha_{n}}\frac{\partialL_{0}}{\partialq_{n,\alpha_{n}}}+(-1)^{m_{n}}\left(\frac{d}{dt}\right)^{m_{n}}\frac{\partialL_{0}}{\partial\dot{q}_{n,m_{n}-1}}=0\.
\int{\calD}A{\calD}c{\calD}{\bar{c}}O^{GI}[A]e^{iS_{\mathrm{eff}}^{L}[A,c,{\bar{c}}]}=\int{\calD}A^{\prime}{\calD}c^{\prime}{\calD}{\bar{c}}^{GI}[A^{\prime}]e^{iS_{\mathrm{eff}}^{A}[A^{\prime},c^{\prime},{\bar{c}}^{\prime}]}.
\begin{array}{c}{{\calR}_{1}(\pi)=\left[\begin{array}{ccc}{1}&{}&{0}\\{}&{-1}&{}\\{0}&{}&{-1}\\\end{array}\right],\\quad{\calR}_{2}(\pi)=\left[\begin{array}{ccc}{-1}&{}&{0}\\{}&{1}&{}\\{0}&{}&{-1}\\\end{array}\right]\,}\\{{\calR}_{3}(\pi)=\left[\begin{array}{ccc}{-1}&{}&{0}\\{}&{-1}&{}\\{0}&{}&{1}\\\end{array}\right]\,}\\\end{array}
\begin{array}{ll}{\left[A,B\right]=-q^{-1}\lambdaCD+q\lambdaD^{2}\quad,}&{\qquad\left[B,C\right]=-q^{-1}\lambdaBD\;,}\\{\left[A,C\right]=q\lambdaAD\quad,}&{\qquadBD=q^{2}DB\;,}\\{AD=q^{-2}DA\quad,}&{\qquad\left[D,C\right]=0\;.}\\\end{array}
{\calE}={\frac{2\pi\mu^{2}}{\lambda}}\int_{0}^{\infty}dr\;r{\frac{1}{2}}\left[\left({\frac{1}{r}}{\frac{df}{dr}}\right)^{2}+\left({\frac{d\varphi}{dr}}\right)^{2}+{\frac{1}{r^{2}}}f^{2}\varphi^{2}+{\frac{\lambda}{2}}\left(\varphi^{2}-1\right)^{2}\right].
S_{kin}=\intd^{4}x\left[\frac{1}{2}(\partial_{\mu}\phi_{+})^{2}+\frac{i\epsilon}{2}\phi_{+}^{2}\right]-\intd^{4}x\left[\frac{1}{2}(\partial_{\mu}\phi_{-})^{2}-\frac{i\epsilon}{2}\phi_{-}^{2}\right]
\begin{array}{ll}{{[J_{01},J_{02}]}=-J_{12}}&{{[J_{02},J_{03}]}=-J_{23}}\\{{[J_{01},J_{03}]}=-J_{13}}&{{[J_{02},J_{12}]}=J_{01}}\\{{[J_{01},J_{12}]}=J_{02}}&{{[J_{02},J_{23}]}=-J_{03}}\\{{[J_{01},J_{13}]}=J_{03}}&{{[J_{03},J_{13}]}=J_{01}}\\{{[J_{03},J_{23}]}=J_{02}}&{{[J_{12},J_{13}]}=-J_{23}}\\{{[J_{12},J_{23}]}=-J_{13}}&{{[J_{13},J_{23}]}=J_{12}}\\\end{array}
\langlen|n\rangle=\langle0|(\gamma^{*})^{n}\gamma^{n}|0\rangle=\langle0|\beta^{n}\gamma^{n}|0\rangle=(\hat{\lambda})^{n}[n;q^{2}]!\Pi_{k=1}^{n}(l-q^{2(k-1)}\alpha_{0}^{2})\;,
\displaystylef(\hat{x})=\oint_{\Gamma_{\hat{x}}}\frac{d\mu}{2\pii}\frac{f(\mu)}{\mu-\hat{x}}=\oint_{\Gamma_{\hat{x}}}\frac{d\mu}{2\pii}f(\mu)\frac{(-1)^{\delta}e^{iZ_{N}(\mu)}iZ_{N}^{\prime}(\mu)}{1+(-1)^{\delta}e^{iZ_{N}(\mu)}}
F=-{\frac{1}{\pi}}\int_{0}^{\infty}{\frac{dE}{e^{\betaE}-1}}\int_{x_{+}+h}^{L}{\frac{dx}{g(x)}}\left(E-2(E^{2}-\mu^{2}g(x))^{1/2}+(E^{2}-2\mu^{2}g(x))^{1/2}\right)~~.
\frac{\partialV_{eff}}{\partial\sigma}=0\Longrightarrow-m_{B}^{2}+\lambda_{B}v^{2}=2g_{B}^{2}\int^{\Lambda}\frac{d^{d}q}{(2\pi)^{d}}\frac{1}{q^{2}+g_{B}^{2}v^{2}}.
P^{+}={\frac{\pi}{L}}\left\{\sum_{n\geN}(n+{\frac{1}{2}})a_{n+{\frac{1}{2}}}^{\dagger}a_{n+{\frac{1}{2}}}-\sum_{n<N}(n+{\frac{1}{2}})a_{n+{\frac{1}{2}}}a_{n+{\frac{1}{2}}}^{\dagger}-{\frac{1}{2}}(N-M(\hat{q}))^{2}-M(\hat{q})Q\right\}
L_{i}\ast_{M}^{}F=L_{i}F+\nu\sum_{1\leqj,k\leq3}\varepsilon_{ijk}L_{k}{\frac{\partialF}{\partialL_{j}}}+\nu^{2}\Bigl(2{\frac{\partialF}{\partialL_{i}}}+\sum_{1\leqj\leq3}L_{j}{\frac{\partial^{2}F}{\partialL_{i}\partialL_{j}}}\Bigr),\quad1\leqi\leq3.
A=-\frac{\kappa^{2}}{4}\,\frac{p_{11}^{(1)}p_{11}^{(2)}}{\left(p_{\perp}^{(1)}-p_{\perp}^{(4)}\right)^{2}}\left(\frac{p_{\perp}^{(1)}}{p_{11}^{(1)}}-\frac{p_{\perp}^{(2)}}{p_{11}^{(2)}}\right)^{4}
\dot{V}\left(\frac{\partial}{\partialL}\right)\langleW(L)\rangle-\int_{0}^{L}dJ\langleW(J)W(L-J)\rangle=-\frac{2}{N}\sum_{n=0}^{\infty}\frac{L^{n}}{n!}\left(\frac{\partialE}{\partial\beta_{n+1}}\right)_{\beta_{n+1}=0}
E_{AB}=e^{-i(\psi_{A}-\psi_{B})/2}\left[f_{AB}^{(a)}\left(\frac{\partial}{\partialr_{A}^{a}}-\frac{\partial}{\partialr_{B}^{a}}\right)+g_{AB}\frac{\partial}{\partial\psi_{A}}+g_{BA}^{*}\frac{\partial}{\partial\psi_{B}}\right].
S_{B}:\begin{array}{llllll}{\frac{\partial\tilde{m}_{s}}{\partialB_{ij}}}&{=}&{\frac{1}{2}\tilde{m}^{ij}\,\quad}&{\frac{\partial\tilde{m}^{ij}}{\partialB_{kl}}}&{=}&{\frac{1}{2}\tilde{m}^{sijkl}\,\quad}\\{\frac{\partial\tilde{m}^{sijkl}}{\partialB_{mn}}}&{=}&{\frac{1}{2}\tilde{m}^{s;sijklmn}\,\quad}&{\frac{\partial\tilde{m}^{s;sijklmn}}{\partialB_{pq}}}&{=}&{0\,\quad}\\\end{array}
L_{\hphantom{\mu}\nu}^{\mu}=\left(\begin{array}{cc}{\frac{p^{0}}{m}}&{-\frac{p_{n}}{m}}\\{\frac{p^{m}}{m}}&{\delta_{n}^{m}\!-\!\frac{\frac{p^{m}}{m}\frac{p_{n}}{m}}{1+\frac{p^{0}}{m}}}\\\end{array}\right)\,,
\betaF=-\left({\frac{2^{21}3^{2}5^{7}\pi^{14}}{7^{19}}}\right)^{1/5}\!\!N^{2}\left({\frac{T}{(g_{YM}^{2}N)^{1/3}}}\right)^{9/5}=-4.115~N^{2}\left({\frac{T}{(g_{YM}^{2}N)^{1/3}}}\right)^{1.8}
\sum_{m=1,2}\int_{0}^{\infty}{z^{3}\psi_{\mu}(z/a)\Omega_{\lambdal}^{(m)}(z)D_{l}^{(mq)}(zx)dz}=2\int_{0}^{\infty}{z^{3}{\mathrm{Re}}\left[\psi_{\mu}(iz/a)\Omega_{\lambdal}^{(1)}(iz)D_{l}^{(1q)}(izx)\right]dz}.
O^{\prime}=\left[\begin{array}{cccccccc}{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}&{1}&{0}&{1}&{1}\\{0}&{1}&{1}&{0}&{0}&{1}&{0}&{1}\\{0}&{1}&{1}&{1}&{0}&{0}&{1}&{0}\\{0}&{0}&{1}&{1}&{1}&{0}&{0}&{1}\\{0}&{1}&{0}&{1}&{1}&{1}&{0}&{0}\\{0}&{0}&{1}&{0}&{1}&{1}&{1}&{0}\\{0}&{0}&{0}&{1}&{0}&{1}&{1}&{1}\\\end{array}\right].
\renewcommand{\arraystretch}{1.3}\begin{array}{rrc}{T^{k}T^{k}\left|a\right>=t(t+1)\left|a\right>,}&{\,\,T^{3}\left|a\right>=t^{z}\left|a\right>,}&{\,\,F\left|a\right>=f\left|a\right>,}\\{Q\left|a\right>=q\left|a\right>,}&{}&{}\\\end{array}
{\calL}_{B}=\frac12\frac{(\partial_{\mu}\Phi)^{2}}{\Phi^{2}}+\frac12\Phi^{2}((\partial_{\mu}X)^{2}+2(\vec{L}\partial_{\mu}X)^{2})+\frac12D_{\mu}\vec{L}D_{\mu}\vec{L},
\left(g_{ab}\right)=\left(\begin{matrix}{\hat{g}_{\mu\nu}}&{n_{\mu}}\\{n_{\nu}}&{n_{\lambda}n^{\lambda}+n^{2}}\\\end{matrix}\right),\qquad\left(g^{ab}\right)=\frac1{n^{2}}\left(\begin{matrix}{n^{2}\hat{g}^{\mu\nu}+n^{\mu}n^{\nu}}&{-n^{\mu}}\\{-n^{\nu}}&{1}\\\end{matrix}\right),
\mathrm{Im}\,\Pi^{\mathrm{(non\mathrm{-}planar)}}=\frac{\theta(p^{2})\pi^{\frac{n+3}{2}}}{(2\pi)^{n}}\,(p^{2})^{\frac{n-4}{2}}\left(|\bar{\theta}|(p^{2})^{\frac{1}{2}}\right)^{\frac{3-n}{2}}J_{\frac{n-3}{2}}(\frac{|\bar{\theta}|(p^{2})^{\frac{1}{2}}}{2})
\Sigma_{ext}=\int_{\cal{M}}d^{4}x\left\{\Omega^{a\mu}(D_{\mu}c)^{a}+\tau^{a\mu}(s\psi_{\mu}^{a})+\frac{1}{2}L^{a}f^{abc}c^{b}c^{c}+D^{a}(s\varphi^{a})\right\},
h_{rel}=\sum_{j,k=2}^{N}A_{jk}\frac{\partial^{2}}{\partial\tau_{j}\partial\tau_{k}}-2\sum_{i=2}^{N}i\tau_{i}{\frac{\partial}{\partial\tau_{i}}}-\left(\frac{1}{N}+\nu^{\star}\right)\sum_{i=2}^{N}(N-i+2)(N-i+1)\tau_{i-2}{\frac{\partial}{\partial\tau_{i}}}
{\Bigl({\frac{d^{2}}{{d\tau}^{2}}}-{\frac{B_{ab}}{\bar{T}}}\Bigl)}^{-1}={\Bigl({\frac{d}{d\tau}}\Bigr)}^{-2}+{\Bigl({\frac{d}{d\tau}}\Bigr)}^{-2}{\frac{B_{ab}}{\bar{T}}}{\Bigl({\frac{d}{d\tau}}\Bigr)}^{-2}+{\Bigl({\frac{d}{d\tau}}\Bigr)}^{-2}{\frac{B_{ab}}{\bar{T}}}{\Bigl({\frac{d}{d\tau}}\Bigr)}^{-2}{\frac{B_{ab}}{\bar{T}}}{\Bigl({\frac{d}{d\tau}}\Bigr)}^{-2}+\cdots,
\stackrel{\mathrm{G}}{\Sigma}_{\alpha}{}^{\!i}=\delta_{\alpha}^{i}\stackrel{\mathrm{G}}{{\mathcalL}}-\frac{1}{2m}\left(D^{i}\overline{{\Psi}}D_{\alpha}\Psi+D_{\alpha}\overline{{\Psi}}D^{i}\Psi\right)
\frac{1}{2(p+1)!}\sqrt{-\hat{g}}\hat{g}^{\mu_{1}\nu_{1}}\cdots\hat{g}^{\mu_{p+1}\nu_{p+1}}\hat{F}_{\mu_{1}\cdots\mu_{p+1}}\hat{F}_{\nu_{1}\cdots\nu_{p+1}}=\frac{1}{2}\hat{F}_{I}A_{IJ}\hat{F}_{J}~.
R=\frac{R\,{\frac{\Gamma(-{\frac{1}{2}})}{\Gamma({\frac{1}{2}}-{\frac{\epsilon}{2}})}}+{\frac{\Gamma({\frac{1}{2}})}{\Gamma(1-{\frac{\epsilon}{2}})}}}{{\frac{\Gamma(-{\frac{1}{2}})}{\Gamma({\frac{1}{2}}-{\frac{\epsilon}{2}})}}-(2\epsilon-1)R\,{\frac{\Gamma({\frac{1}{2}})}{\Gamma(1-{\frac{\epsilon}{2}})}}}
N(1,2)\equiv\frac{2}{2k+Q_{g}}\sum_{I}\left(\begin{array}{cc}{T_{I}^{(2)}}&{0}\\{0}&{-T_{I}^{(2)}}\\\end{array}\right)\otimes\left(\begin{array}{cc}{T_{I}^{(1)}}&{0}\\{0}&{-T_{I}^{(1)}}\\\end{array}\right)\.