harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathrm{Solv}\left(\frac{\mathrm{USp}(3,3)}{\mathrm{SU}(3)\times{U}(1)}\right)\not\subset\mathrm{Solv}\left(\frac{\mathrm{SO}(3,6)}{\mathrm{SO}(3)\times\mathrm{SO}(6)}\right)~.
\tilde{S}=\frac{\beta}{2}\sum_{i}(\nabla_{\mu}\theta_{i}-A_{i\mu}-2\pin_{i\mu})^{2}+\frac{1}{2e^{2}}\sum_{i}(\epsilon_{\mu\nu\lambda}\nabla_{\nu}A_{i\lambda}-2\piN_{i\mu})^{2},
V=\left(\begin{matrix}{0}&{\delta}&{0}&{\dots}&{0}\\{0}&{0}&{\delta^{3}}&{\dots}&{0}\\{\dots}&{\dots}&{\dots}&{\dots}&{\dots}\\{0}&{0}&{\dots}&{0}&{\delta^{2n-3}}\\{\delta^{2n-1}}&{0}&{0}&{\dots}&{0}\\\end{matrix}\right)
\rho(E)={\frac{1}{2\pii}}\int_{c-i\infty}^{c+i\infty}Z(\beta)e^{\betaE}d\beta\approxe^{\piQ^{2}}\delta(E)+e^{\piQ^{2}}\sqrt{\frac{2\pi^{2}Q^{3}}{E}}I_{1}\left(\sqrt{8\pi^{2}Q^{3}E}\right)\theta(E),
\Omega_{p=D}=\frac{1}{D!}a_{[\mu_{1}\dots\mu_{D}]}\;\tilde{\epsilon}^{\mu_{1}\,\dots\;\mu_{D}}\,d^{D}x=\frac{1}{D!}a_{[\mu_{1}\,\dots\;\mu_{D}]}\,|e|\epsilon^{\mu_{1}\,\dots\;\mu_{D}}d^{D}x
{\mathcalL}_{{\mathcalN}=1}=-\frac{1}{2}\lambda^{i}\lambda^{j}\frac{\partial^{2}W}{\partial\phi^{i}\partial\phi^{j}}+\frac{i}{\sqrt{2}}\,D\,\frac{\partial\tau}{\partial\phi^{i}}\lambda^{i}\psi-F^{i}\,\frac{\partial\tau}{\partial\phi^{i}}\psi\psi+...
Z[T,(e_{i},\vec{x}_{i})]=\frac{\int[dA_{0}(\vec{x})]\mathrm{~e}^{-S_{\mathrm{eff}}[A_{0}]}~\prod_{j}\mathrm{~e}^{ie_{j}A_{0}(\vec{x}_{j})/T}}{\int[dA_{0}(\vec{x})]\mathrm{~e}^{-S_{\mathrm{eff}}[A_{0}]}}
\square_{4}(x,x^{\prime})_{\mu\lambda\nu\rho}=\int\frac{d^{4}p}{(2\pi)^{4}}\frac{e^{ip\cdot(x-x^{\prime})}}{2p^{2}}\big(g_{\mu\nu}g_{\lambda\rho}+g_{\mu\rho}g_{\lambda\nu}-g_{\mu\lambda}g_{\nu\rho}\big)\,.
F_{min}^{VV}(\beta)=\frac{\Gamma\left(1-\frac{\beta}{2\pii}\right)\Gamma\left(\frac{\beta}{2\pii}+\frac{1}{N}\right)}{\Gamma\left(1+\frac{\beta}{2\pii}\right)\Gamma\left(-\frac{\beta}{2\pii}+\frac{1}{N}\right)}
\left\{A,B\right\}^{\ast}\,=\,\left\{A,B\right\}^{\dagger}\,+\,\frac{p^{2}}{S^{2}(S\cdotp)}p_{\gamma}\epsilon_{\mu\nu\gamma}\left\{A,{\tilde{\varphi}}_{\mu}\right\}^{\dagger}\left\{{\tilde{\varphi}}_{\nu},B\right\}^{\dagger}.
g:(\Gamma_{1}^{1,9},\Gamma_{2}^{1,9},\Gamma_{1}^{1,1},\Gamma_{2}^{1,1},\Gamma^{2,2})\mapsto(\Gamma_{2}^{1,9},\Gamma_{1}^{1,9},\Gamma_{1}^{1,1},-\Gamma_{2}^{1,1},-\Gamma^{2,2}).
\partial_{\mu}^{z}V_{i}^{\mu}(z,u,v)=\frac{ig_{1}}{2}\left[\delta^{4}\left(z-v\right)-\delta^{4}(z-u)\right]\left(Y_{R}^{i}L+Y_{L}^{i}R\right)\Sigma_{1}^{i}(u-v)\,,
{\frac{\xi_{j}-\xi_{j-1}}{1+\xi_{j}^{*}\xi_{j}}}={\frac{-i\epsilonh\xi_{j-1}}{1+\xi_{j}^{*}\xi_{j-1}}}\,\quad{\frac{\xi_{j}^{*}-\xi_{j+1}^{*}}{1+\xi_{j}^{*}\xi_{j}}}={\frac{-i\epsilonh\xi_{j+1}}{1+\xi_{j+1}^{*}\xi_{j}}}\quad\left({1\lej\leN}\right)\.
\begin{array}{l}{\qquadH_{+}=\frac{8}{\pi}\sum_{-\infty}^{\infty}L_{n}e^{i2n(\tau-\sigma)},\qquadL_{n}\equiv\frac12\sum_{-\infty}^{\infty}\alpha_{n-k}^{\mu}\alpha_{k}^{\mu}=0,}\\{\qquadH_{-}=\frac{8}{\pi}\sum_{-\infty}^{\infty}\bar{L}_{n}e^{i2n(\tau+\sigma)},\qquad\bar{L}_{n}\equiv\frac12\sum_{-\infty}^{\infty}\bar{\alpha}_{n-k}^{\mu}\bar{\alpha}_{k}^{\mu}=0,}\\\end{array}
1=\sum_{n,n_{1}}\prod_{i=1}^{n}\sum_{\underline{{m}}_{i}}\prod_{j=1}^{n_{1}}\sum_{\underline{{l}}_{j}}\left|\tilde{\phi}^{(n,n_{1})}(\underline{{m}}_{i},\underline{{l}}_{j},\underline{{K}}-\sum_{i}\underline{{m}}_{i}-\sum_{j}\underline{{l}}_{j})\right|^{2}\,.
\frac{F(4,-\frac{i}{2}k^{\prime}+4,-\frac{i}{2}k^{\prime}+5,e^{-4x_{0}})}{k^{\prime}+8i}=\frac{1}{3!}\sum_{n=3}^{\infty}\frac{n(n-1)(n-2)e^{-4(n-3)x_{0}}}{k^{\prime}+i(2+2n)}.
n_{(\alpha)}^{i}({\boldsymbol{p}})M^{ij}({\boldsymbol{p}})n_{(\beta)}^{j*}({\boldsymbol{p}})=\left(\begin{array}{ccc}{1}&{}&{}\\{}&{{\boldsymbol{p}}^{2}\vert\widetilde{\boldsymbol{f}}({\boldsymbol{p}})\vert^{2}}&{}\\{}&{}&{0}\\\end{array}\right)_{\alpha\beta},
\begin{matrix}{\hat{\delta}\psi_{A}=\psi_{A},}&{\hat{\delta}\varepsilon_{A}=\varepsilon_{A},}&{\hat{\delta}\lambda_{i}^{A}=\lambda_{i}^{A},}\\{\hat{\delta}\psi^{A}=-\psi^{A},}&{\hat{\delta}\varepsilon^{A}=-\varepsilon^{A},}&{\hat{\delta}\lambda_{A}^{i}=-\lambda_{A}^{i}.}\\\end{matrix}
\mathrm{r.h.s.}:\begin{array}{l}{R_{23}R_{13}K_{2}R_{12}K_{1}R_{21}R_{31}R_{32}Y_{3}R_{32}^{-1}R_{31}^{-1}}\\{+\eta_{3}{\calP}_{23}R_{13}R_{12}K_{1}R_{21}+\eta_{3}R_{23}K_{2}R_{32}{\calP}_{13}R_{21}\quad.}\\\end{array}
\Phi_{y}=-a_{0}^{-1}\sum_{M\geq0}\frac{x^{M}}{\Gamma^{2}(M+1)}\;\frac1{2\pii}\int_{C}\frac{\Gamma(4s+1)\Gamma(-s)\Gamma(2M-s)}{\Gamma^{2}(s+1)}y^{s}\,ds,
<e^{ip\cdotX_{0m}}>_{o}=\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}e^{i\vec{k}\cdot\vec{Y}_{1}}\int\frac{d^{3}\vec{q}}{(2\pi)^{3}}e^{-i\vec{q}\cdot\vec{Y}_{2}}e^{-\fracl2k_{B}^{2}+\taup\cdotq_{B}}<k_{B}^{\mu}|(p+q_{B})^{\mu}>\;.
J^{(N)}\left(n;1,\ldots,1\right)=2\;\mathrm{i}^{1-2N}\;\pi^{n/2}\;\Gamma\left(N\!-\!{\textstyle{\frac{n}{2}}}\right)\;\left(\prodf_{i}\right)\;\int_{0}^{\infty}\!\ldots\!\int_{0}^{\infty}\frac{\prod\mathrm{d}\alpha_{i}}{\left(\sum\alpha_{i}f_{i}\right)^{n-N}}\;\delta\left(\alpha^{T}\|C\|\alpha-1\right).
(v^{(\kappa)},K)=(1+\tilde{U}_{00})\frac{1-\tilde{\nu}(\kappa)}{\kappa}\quad\mathrm{and}\quad(v^{(\kappa)},\overline{{K}})=(1+\tilde{U}_{00})\frac{1-\overline{{\tilde{\nu}}}(\kappa)}{\kappa}.
\xi\equiv\frac{\rho}{\sqrt{\mu\hbar\omega}}\;,\quad\kappa\equiv\sqrt{\mu\hbar\omega(\beta+\beta^{\prime})}\;,\quad\eta\equiv\frac{\beta}{\beta+\beta^{\prime}}\;,\quad\delta\equiv\frac{\gamma}{\beta+\beta^{\prime}}\;,\quad\varepsilon\equiv\frac{2E}{\hbar\omega}\;,
I=\intdx^{+}dx^{-}(4\Omega\partial_{+}\partial_{-}\rho+\frac{\mu}{2}e^{2\rho}\Omega^{1-\lambda-\gamma/4}+\Omega^{\delta}\partial_{+}f\partial_{-}f-e^{-2\rho}\Omega^{(\gamma-2\epsilon)/4+1}F_{-+}^{2})
\Gamma\sim\frac{v^{4}}{r^{7-p}}+\frac{\theta^{2}v^{3}}{r^{8-p}}+\frac{\theta^{4}v^{2}}{r^{9-p}}+\frac{\theta^{6}v}{r^{10-p}}+\frac{\theta^{8}}{r^{11-p}}
S=\intd^{6}x\left[{\frac{1}{12}}{\epsilon}^{0ijklm}F_{0ij}F_{klm}-{\frac16}F_{klm}F_{klm}+{\frac16}G^{{\mu}{\nu}{\sigma}}(F_{{\mu}{\nu}{\sigma}}-{\partial}_{[{\mu}}A_{{\nu}{\sigma}]})\right],
\omega\midU_{\pm}=\int_{N}\frac{-2iS_{0}}{\left(1+z_{\pm}(\vec{x})\overline{{z}}_{\pm}(\vec{x})\right)^{2}}\delta\overline{{z}}_{\pm}(\vec{x})\wedge\deltaz_{\pm}(\vec{x})d\mu(\vec{x})\,.
{\calL}_{flow}^{1+1QED}=\frac12\left(\partial_{+}{A}_{-}-\partial_{-}{A}_{+}\right)^{2}+\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}-e\gamma^{\mu}{A}_{\mu}-M\right)\psi+{\Lambda}\left(\partial_{+}A_{-}+\alpha\partial_{-}A_{+}\right)\,
\begin{array}{rclcrcl}{(A_{1})_{mn}}&{=}&{0}&{\mathrm{~except~possibly~for}}&{n-m}&{\le}&{-1}\\{(A_{2})_{mn}}&{=}&{0}&{\mathrm{~except~possibly~for}}&{n-m}&{\ge}&{+1}\\{(B_{1})_{mn}}&{=}&{0}&{\mathrm{~except~possibly~for}}&{n-m}&{\le}&{+1}\\{(B_{1})_{mn}}&{=}&{0}&{\mathrm{~except~possibly~for}}&{n-m}&{\ge}&{-1}\\\end{array}
\alpha^{\left(1\right)-1a}=\left(\Gamma_{}^{\left(0\right)0},\Gamma_{}^{\left(1\right)0}\right)^{a}+\widetilde{V}^{a}\Gamma_{}^{\left(1\right)0}+\left(\Gamma_{}^{\left(1\right)-1},\Gamma_{}^{\left(0\right)1}\right)^{a}
D_{T}^{[r,0]}(k^{2})=\frac{k^{2}+u_{r,2}\Lambda^{2}}{(k^{2}+\sigma_{r,1}\Lambda^{2})(k^{2}+\sigma_{r,3}\Lambda^{2})}\prod_{s=1}^{(r-1)/2}\frac{(k^{2}+u_{r,4s}\Lambda^{2})(k^{2}+u_{r,4s+2}\Lambda^{2})}{(k^{2}+\sigma_{r,4s+1}\Lambda^{2})(k^{2}+\sigma_{r,4s+3}\Lambda^{2})}\,,
L=\frac{1}{2}\Gamma_{i}\Lambda^{ij}\dot{\Gamma}_{j}-\rho(\Gamma)=q_{i}\dot{x}_{i}+\frac{B}{2}\epsilon_{ij}x_{i}\dot{x}_{j}-\frac{1}{2}\left(\frac{q_{i}^{2}}{m}+kx_{i}^{2}\right)
(L_{0}-a_{\epsilon})\Psi=\left({\textstyle{\frac{1}{2\alpha}}}P^{\mu}P_{\mu}+{\textstyle{\frac{1}{2\beta}}}(P^{L})^{2}+R+m_{\epsilon}^{2}\right)\Psi\;=\;0\;\;\;,\;\;\;P^{L}\Psi\;=\;p^{L}\Psi
{\calA}_{4}^{1(s)}=\int_{0}^{\infty}dT\oint\frac{dz}{2\pii}\frac{dz}{d\rho}\langle{\calO}_{1}(z_{1})\tilde{{\calO}}_{2}(z_{2})\tilde{{\calO}}_{3}(z_{3}){\calO}_{4}(z_{4})b(z)((\xi(z_{4})-\xi(z_{0}))Y(z_{0})\rangle.
{\frac{cdt}{dx}}={\frac{\hat{c}d\hat{t}}{d\hat{x}}},\\\\\\\{\frac{\hbar}{dEdt}}={\frac{\hat{\hbar}}{d\hat{E}d\hat{t}}},\\\\\\\{\frac{G_{4}dE}{c^{4}dx}}={\frac{\hat{G}_{4}d\hat{E}}{\hat{c}^{4}d\hat{x}}},\\\\\\\{\frac{e^{2}}{dEdx}}={\frac{\hat{e}^{2}}{d\hat{E}d\hat{x}}},
\delta\Gamma^{\sigma}\left(p^{\prime},p\right)=\frac{i36g^{2}}{M^{2}}\int\frac{d^{4}k}{(2\pi)^{4}}\Biggl\{\frac{A^{\sigma}}{\left[(k^{\prime})^{2}-m^{2}\right]\left[k^{2}-m^{2}\right]}-\frac{B^{\sigma}}{\left[k-p\right]^{2}\left[(k^{\prime})^{2}-m^{2}\right]}\Biggr\},
{\calL}_{3}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{\alpha^{2}}{36m_{e}^{4}}(F_{\mu\nu}F^{\mu\nu})^{2}+\frac{7\alpha^{2}}{90m_{e}^{4}}F_{\mu\nu}F_{\sigma\tau}F^{\mu\sigma}F^{\nu\tau},
S^{\prime}={\frac{N^{2}}{16\piG_{10}}}\intd^{10}x\sqrt{-G^{dual}}(Ne^{\phi})^{-{\frac{6}{7}}}\left[{\calR}^{dual}+{\frac{16}{49}}\partial_{M}\phi\partial^{M}\phi-{\frac{1}{2\cdot8!}}{\frac{1}{N^{2}}}F_{M_{1}\cdotsM_{8}}F^{M_{1}\cdotsM_{8}}\right].
\left.\frac{d^{2}V(\rho)}{d\rho^{2}}\right|_{\rho^{(n)}}=\frac{1}{l}\left.\frac{dj}{dM_{0}}\left(\frac{d\rho}{dM_{0}}\right)^{-1}\right|_{\rho^{(n)}}=8\pi^{2}\nu^{2}l^{2}>0.
=\{\varepsilon^{\mu\nu}\}+\left(\begin{array}{cccc}{0}&{-q\varepsilon^{12}-r\varepsilon^{13}}&{p\varepsilon^{12}-r\varepsilon^{23}}&{p\varepsilon^{13}+q\varepsilon^{23}}\\{q\varepsilon^{12}+r\varepsilon^{13}}&{0}&{0}&{0}\\{-p\varepsilon^{12}+r\varepsilon^{23}}&{0}&{0}&{0}\\{-p\varepsilon^{13}-q\varepsilon^{23}}&{0}&{0}&{0}\\\end{array}\right)
(\mathcal{L}_{\eta_{i}^{\nu}}d\Omega)_{\mu}=\frac{1}{2}f_{i}^{2}\left\{3a(u^{\nu}s_{\mu}-s^{\nu}u_{\mu})+\Theta\delta_{\mu}^{\nu}+3(\sigma_{.\mu}^{\nu}+\omega_{.\mu}^{\nu})\right\}dV.
\frac{\nu_{B}}{\nu_{A}}=\sqrt{\frac{g_{00}(x_{B})}{g_{00}(x_{A})}}\frac{g_{0\mu}(x_{A})dx_{A}^{\mu}}{g_{0\nu}(x_{B})dx_{B}^{\nu}}=\sqrt{\frac{g_{00}(x_{A})}{g_{00}(x_{B})}}\frac{dx_{A}^{0}}{dx_{B}^{0}}
\frac{1}{8}\mathrm{Tr}^{2}\left[O+\left(\partial\!\!\!/B\!\!\!\!/\right)\right]-\frac{1}{2}\mathrm{Tr}\left[O+\left(\partial\!\!\!/B\!\!\!\!/\right)\right]^{2}=(\partial_{\lbrack\mu}B_{\nu]})(\partial^{\lbrack\mu}B^{\nu]})
\left[M^{2}-4(\vec{k}\,^{2}+m^{2})\right]g_{i}^{(1)}(\vec{k},\hat{n})=\frac{m^{2}}{2\pi^{3}}\int\sum_{j=1}^{4}K_{ij}^{(1)}(\vec{k},\vec{k}\,^{\prime},\hat{n})g_{j}^{(1)}(\vec{k}\,^{\prime},\hat{n})\frac{d^{3}k^{\prime}}{\varepsilon_{k^{\prime}}}
\hat{\mathbf{J}}^{2}\,\Upsilon_{lm}=(\hat{\mathbf{L}}_{C}+\hat{\mathbf{S}}_{C})^{2}\,\Upsilon_{lm}=l(l+1)\,\Upsilon_{lm}\qquad{\mathrm{and}}\qquad\hat{J}_{z}\,\Upsilon_{lm}=m\,\Upsilon_{lm};
{\calI}(A)=c_{1}a_{2}^{ij}\partial_{i}\partial_{j}+(c_{2}a_{1}^{k}+c_{3}\partial_{j}a_{2}^{kj})\partial_{k}+c_{4}\partial_{i}\partial_{j}a_{2}^{ij}+c_{5}\partial_{i}a_{1}^{i}+c_{6}a_{0}
c_{1}({\calN})=n\left({\frac{1}{2}}+\lambda\right)\sigma+\left({\frac{1}{2}}-\lambda\right)\pi^{*}\eta+\left({\frac{1}{2}}+n\lambda\right)\pi^{*}\left(c_{1}(B)\right)+\sum_{i}\kappa_{i}N_{i}
J_{n}^{mixed}=\sumc_{r,s,\vec{i}}\lambda^{m}\chi_{i_{1}}\ldots\chi_{i_{s}}\chi_{i_{r+1}}^{\prime}\ldots\chi_{i_{s}}^{\prime}\chi_{p}^{\prime\prime}\delta_{i_{1}+\cdotsi_{s}+m+p,~n}.
\begin{array}{lcr}{e_{\breve{\beta}}^{--{\breve{\gamma}}}}&{=}&{\partial_{h{\breve{\beta}}}^{+}\partial_{h}^{+{\breve{\gamma}}}e^{-4},}\\{\omega_{{\breve{\beta}}{\breve{\alpha}}}^{-{\breve{\gamma}}}}&{=}&{\partial_{h{\breve{\alpha}}}^{+}\partial_{h{\breve{\beta}}}^{+}\partial_{h}^{+{\breve{\gamma}}}e^{-4}\.}\\\end{array}
\mathrm{Tr}\,\hat{O}\,=\,\int\,d^{m}\eta\,\langle-\eta|\hat{O}|\eta\rangle\qquad\qquad\quad\mathrm{Str}\,\hat{O}\,=\,\int\,d^{m}\eta\,\langle\eta|\hat{O}|\eta\rangle
\left.\begin{array}{llrl}{\gamma_{1,3,(m,n,p)}}&{=}&{{\bf0},}&{(m,n,p=\pm1)}\\{\gamma_{1,{\tilde{7}}_{3},(m)}}&{=}&{-\alpha{\bf1}_{1},}&{(m=\pm1)}\\{\gamma_{1,7_{4},(n)}}&{=}&{\alpha{\bf1}_{1},}&{(n=\pm1)}\\{\gamma_{1,7_{5},(p)}}&{=}&{{\bf0},}&{(p=\pm1)}\\\end{array}\right.
\underbrace{(\rho_{i-\hat{\lambda},\lambda,\mu}\otimes\rho_{i\lambda\mu}^{\ast})\otimes\cdots}_{\lambda\in\{1,\ldots,\mu-1\}}\otimes\underbrace{(\rho_{i-\hat{\nu},\mu,\nu}^{\ast}\otimes\rho_{i\mu\nu})\otimes\cdots}_{\nu\in\{\mu+1,\ldots,d\}}\otimes\underbrace{\tau_{i\mu}}_{\mathrm{new}}
{\calV}_{\mathrm{phys}}^{\mathrm{(II)}}=\{\left|\Phi\right\rangle\in{\calV}^{\mathrm{(II)}};~(Q_{\mathrm{B(KO)}}+Q_{\mathrm{B(Y2)}})\left|\Phi\right\rangle=0\}\subset{\calV}^{\mathrm{(II)}}.
f_{\calN}(z)={\frac{P_{\calN}(z)}{{\calA}{\calB}\sum_{i=1}^{\calN}\left({\frac{1}{z-a_{i}}}-{\frac{1}{z-b_{i}}}\right)}}\,\quad\chi_{\calN}(z)={\calB}f_{\calN}(z)\,
S_{0}^{\prime}\left[A_{\;\;\;(\sigma)}^{\alpha\beta},B_{\alpha}^{\;(\sigma)}\right]=\frac{1}{2}\intd^{4}x\left(-B_{\alpha}^{\;(\sigma)}B_{\;\;(\sigma)}^{\alpha}+A_{\;\;\;(\sigma)}^{\alpha\beta}H_{\alpha\beta}^{\;\;\;(\sigma)}\right),
\begin{array}{rcl}{{e^{\mu}}_{a}(y)}&{=}&{\displaystyle{\delta^{\mu}}_{a}+{\frac{1}{6}}R^{(0)\mu}{}_{\!\rho\sigmaa}(y-x)^{\rho}(y-x)^{\sigma},}\\{\omega^{ab}{}_{\!\mu}\sigma_{ab}}&{=}&{\displaystyle{\frac{1}{2}}R^{(0)ab}{}_{\!\mu\lambda}(y-x)^{\lambda}\sigma_{ab}.}\\\end{array}
(-i\epsilon_{~bcd}^{a}e_{a}^{\mu}\omega_{\mu}^{bc}+i\kappae_{a}^{\mu}R_{\mu\nu}^{bc}\epsilon_{~~bc}^{da}(\bar{\beta}^{\nu}-\bar{\alpha}^{\nu\rho}p_{\rho}){\calS}_{1}=\frac{1}{3}\kappa\epsilon_{~~bc}^{da}e_{a}^{\mu}(\nabla_{\rho}R_{\mu\nu}^{bc})\alpha^{\nu\rho}{\calS}_{0}
\left({\frac{\dot{\hat{a}}_{0}}{\hat{a}_{0}}}-{\frac{\dot{\varepsilon}}{\varepsilon}}\right)^{2}={\frac{8\pi\hat{G}_{4}}{3}}\hat{\varrho}+{\frac{\hat{\calC}c_{0}^{2}}{\hat{a}_{0}^{4}}}-{\frac{\hat{k}c_{0}^{2}}{\hat{a}_{0}^{2}}},
{}_{[0]}\langle0\vert\hat{H}\vert0\rangle_{[0]}=\frac{1}{2}\Bigl(\dot{u}^{*}\dot{u}+\omega^{2}u^{*}u+3\lambda\epsilon(u^{*}u)^{2}\Bigr)-\frac{3}{4}\lambda\epsilon(u^{*}u)^{2}.
\begin{array}{c}{H_{X}^{L}\left(g,l_{t}\right)=-\left\langleX,Ad^{*}g\left(l_{t}\right)\right\rangle,}\\{H_{H}^{R}=\left\langleX,l_{t}\right\rangle,}\\{X\in\widehat{{\bfg}}.}\\\end{array}
={\frac{3y^{3}yg_{2}+2y^{2}y^{2}g_{2}G(y)+y^{\prime}yg_{2}(G(y)-G_{0})}{1-y^{3}yg_{2}-y^{2}y^{2}g_{2}G(y)-y^{\prime}yg_{2}(G(y)-G_{0})-g_{2}(G(y)-G_{0}-G_{1}y)-y^{2}G(y)}}\.
{\frac{1}{16\pi^{2}}}\hat{F}^{a}\hat{F}^{a}(x)\rightarrow{\frac{1}{4!}}\delta(x-x_{0})\,dV(x-x_{0})=-{\frac{1}{4!\pi^{2}}}\hat{d}\,\partial_{\mu}{\frac{1}{(x-x_{0})^{2}}}\,d\sigma_{\mu}(x-x_{0}),
[\delta_{1},\delta_{2}]M={\frac{t_{1}\delta(\epsilon_{12},t_{1})-t_{2}\delta(\epsilon_{12},t_{2})}{t_{1}-t_{2}}}M+{\frac{t_{1}t_{2}}{1-t_{1}t_{2}}}\left(\delta(\epsilon_{12}^{\prime},t_{2})-\delta(\epsilon_{21}^{\prime},t_{1})\right)M.
{\calL}\!=\!i\bar{\psi}\gamma^{\mu}\!\left(\!\nabla_{\mu}+i\frac{1+r\gamma_{5}}{2}A_{\mu}\!\right)\psi\!-\!\frac{g^{2}}{4}\bar{\psi}\gamma^{\mu}\psi\bar{\psi}\gamma_{\mu}\psi\!-\!\frac{1}{4e^{2}}F_{\mu\nu}F^{\mu\nu}.
\psi_{(\pm)}^{\dot{b}\alpha}\equiv\xi^{\dot{b}\alpha}\pmi\eta^{\dot{b}\alpha}-\frac{1}{4}\sum_{a=5,\cdots,8}(\Gamma^{\dot{b}}\Gamma_{\dot{a}})_{\;\;\beta}^{\alpha}(\xi^{\dot{a}\beta}\pmi\eta^{\dot{a}\beta}),
A_{n_{1},n_{2},n_{3}}^{(k)}\sim\sum_{k=0}^{l}c_{k}\partial_{a}^{k}\partial_{b}^{(l-k)}\int_{0}^{1}dxx^{n_{2}-1}(1-x)^{n_{3}-1}\partial_{x}^{n_{1}}\frac{1}{[1-ax-(1-x)b]^{B+1}}.
H=\int{d^{3}}x\left({-\frac{1}{2}\Pi_{i}^{a}\Pi_{a}^{i}+\frac{1}{4}F_{ij}^{a}F^{aij}+c^{a}\left(x\right)\left({\partial^{i}\Pi_{i}^{a}+gf_{abc}A_{b}^{i}\Pi_{i}^{c}-J^{0}}\right)}\right),
\textstyle{\frac{4}{D-2}}\partial_{M}\left[\sqrt{-G}G^{MN}\partial_{N}\phi\right]=\textstyle{\frac{a}{4}}e^{2a\phi}F_{2}^{2}+\kappa_{D}^{2}\sqrt{-g}f^{\prime}(\phi)\delta(z-z_{b}),
\begin{array}{ccc}{\displaystyle{\frac{W(a+1,b+1|u)}{W(a,b|u)}}}&{=}&{\displaystyle{\frac{\theta_{3}(u+(a+b+1)\eta)}{\theta_{3}(u-(a+b+1)\eta)}},}\\{\displaystyle{\frac{W(a+1,b-1|u)}{W(a,b|u)}}}&{=}&{\displaystyle{\frac{\theta_{2}(u+(a-b+1)\eta)}{\theta_{2}(u-(a-b+1)\eta)}}.}\\\end{array}
\begin{aligned}{[m]A}&{{}=H_{1}^{-\frac13}H_{2}^{-\frac13}H_{3}^{-\frac13}}\\{A_{1}}&{{}=H_{1}^{-\frac13}H_{2}^{\frac16}H_{3}^{\frac16}}\\{A_{2}}&{{}=H_{1}^{\frac16}H_{2}^{-\frac13}H_{3}^{\frac16}}\\{A_{3}}&{{}=H_{1}^{\frac16}H_{2}^{\frac16}H_{3}^{-\frac13}}\\{B}&{{}=H_{1}^{\frac16}H_{2}^{\frac16}H_{3}^{\frac16}}\\\end{aligned}
\sum_{\nu_{1},\nu_{2}\in{\calN}_{\varepsilon}^{+}}\frac{1}{\rho\left(\nu_{1}\right)\rho\left(\nu_{2}\right)}\left[{\Largei}\right]\rightarrow\int_{0}^{\infty}\int_{0}^{\infty}d\nu_{1}d\nu_{2}\left[{\Largei}\right]\.
d_{W}\theta^{\alpha}+{\frac{1}{2}}C_{\beta\gamma}^{\alpha}\theta^{\beta}\wedge\theta^{\gamma}=u^{\alpha}\,\{}d_{W}u^{\alpha}=-C_{\beta\gamma}^{\alpha}\theta^{\beta}\wedgeu^{\gamma}\;\{}i_{W\,X_{\alpha}}\theta^{\beta}=\delta_{\alpha}^{\beta}\,\{}i_{W\,X_{\alpha}}u^{\beta}=0\;
\sum_{l=0}^{\infty}f(l)=\int_{0}^{\infty}dl\f(l)+\frac{1}{2}[f(\infty)+f(0)]+\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}[f^{(2k-1)}(\infty)-f^{(2k-1)}(0)]\,
{}^{ee}G_{1234}^{4}=\frac{\delta^{2}W}{\deltaS_{41}^{-1}\deltaS_{32}^{-1}}+\frac{\deltaW}{\deltaS_{41}^{-1}}\frac{\deltaW}{\deltaS_{32}^{-1}}=\frac{\delta\;{}^{e}G_{23}^{2}}{\deltaS_{41}^{-1}}+{}^{e}G_{14}^{2}{}^{e}G_{23}^{2}.
t=\left(\frac{2s_{1}^{4}}{T_{c}}\right)^{1/3}\tilde{t},\quadu=\left(\frac{4T_{c}}{s_{1}}\right)^{1/3}\tilde{u},\quady=\left(\frac{4T_{c}}{s_{1}}\right)^{1/3}\tilde{y}.
\begin{array}{rcl}{\overline{{R}}_{21}^{V_{z_{2}},V_{z_{1}+nw/2}^{*}}}&{=}&{(\overline{{R}}_{12}^{V_{z_{1}},V_{z_{2}}})^{t_{1}}\displaystyle\prod_{p=2}^{n-1}\frac{h(-z_{1}+z_{2}+pw)}{h(w)},}\\{\overline{{R}}_{12}^{V_{z_{1}+nw/2}^{*},V_{z_{2}}}}&{=}&{(\overline{{R}}_{21}^{V_{z_{2}},V_{z_{1}+nw}})^{t_{1}}\displaystyle\prod_{p=1}^{n-2}\frac{h(-z_{1}+z_{2}-pw)}{h(w)},}\\\end{array}
I=\frac{1}{2}\int_{X}\varepsilon^{\mu\nu\rho\sigma}\left(i\Sigma_{\mu\nu}^{\\ab}R_{\rho\sigmaab}+\frac{1}{2}\varepsilon_{abcd}\Sigma_{\mu\nu}^{\\ab}R_{\rho\sigma}^{\\cd}\right)d^{4}x.
\lambda=W(\mu)\sim\mu^{p}A^{(W,Q)}\equivs^{(W,Q)}(\mu){\frac{1}{W^{\prime}(\mu)}}{\frac{\partial}{\partial\mu}}[s^{(W,Q)}(\mu)]^{-1}=={\frac{1}{W^{\prime}(\mu)}}{\frac{\partial}{\partial\mu}}-{\frac{1}{2}}{\frac{W^{\prime\prime}(\mu)}{W^{\prime}(\mu)^{2}}}+Q(\mu)
\int\mathrm{d}^{4}z_{1}\ldots\int\mathrm{d}^{4}z_{n}(z_{1}-x)_{\alpha}\frac{e^{-\frac{(x-z_{1})^{2}}{4\sigma}}}{(4\pi\sigma)^{2}}\frac{e^{-\frac{(z_{1}-z_{2})^{2}}{4\sigma}}}{(4\pi\sigma)^{2}}\ldots\frac{e^{-\frac{(z_{n}-y)^{2}}{4\sigma}}}{(4\pi\sigma)^{2}}=2\sigma\hat{x}_{\alpha}\frac{e^{-\frac{(x-y)^{2}}{4\sigma(n+1)}}}{(4\pi\sigma(n+1))^{2}}
{\widehat\Gamma}_{\Lambda_{1}\dots\Lambda_{n}}^{(n)}(k_{1}\dotsk_{n})=V_{\Lambda_{1}a_{1}}(k_{1}^{0})\dotsV_{\Lambda_{n}a_{n}}(k_{n}^{0})\Gamma_{a_{1}\dotsa_{n}}^{(n)}(k_{1}\dotsk_{n})\,,
\langle\bar{w}(\ell)\rangle_{1}=\langlew(\ell)\rangle_{1}-{\frac{\langleP\rangle_{1}\langlew(\ell)P\rangle_{0}}{\langlePP\rangle_{0}}}-{\frac{1}{2}}\,{\frac{\langlew(\ell)PP\rangle_{0}}{\langlePP\rangle_{0}}}+{\frac{1}{2}}\,{\frac{\langlew(\ell)P\rangle_{0}\langlePPP\rangle_{0}}{(\langlePP\rangle_{0})^{2}}}\.
G_{\nu}^{\mu}(\{\})=8\piG(2\alphag^{\mu\alpha}\partial_{\alpha}\Lambda\partial_{\nu}\Lambda-\alpha\delta_{\nu}^{\mu}g^{\alpha\beta}\partial_{\alpha}\Lambda\partial_{\beta}\Lambda+T_{\nu}^{\mu})=8\piG\tilde{T}_{\nu}^{\mu}
h_{ij}(m)=\!\int\!\mathrm{d}z\,g_{ab}(\phi_{\mathrm{sol}})\,\,\frac{\partial\phi_{\mathrm{sol}}^{a}}{\partialm^{n}}\,\,\frac{\partial\phi_{\mathrm{sol}}^{b}}{\partialm^{k}}\,,\qquadi,j=0,1,2,...,n\,,
[G\frac{\delta}{\deltaa^{\mu}},A_{\nu}^{(0)}]=\sum_{n=1}^{\infty}\sum_{m=0}^{n}\int\widetilde{C}_{\nu\mu}(x,1,\ldots,m;m+1\ldots,n)a(1)\cdotsa(m)\mathrm{Tr\;}[a(m+1)\cdotsa(n)]\,,
<\hat{H}_{0}>_{T}^{(p)}=\,sum_{n=0}^{p}[-{\frac{1}{2}}({\frac{-\lambda}{4}})^{n}\,\,{\bfP}_{n}^{c}\,\,T^{3n-1}\,\,+\,\,{\frac{\lambda}{4}}({\frac{-\lambda}{4}})^{n}\,\,{\bfQ}_{n}^{c}\,\,T^{3n+2}],
\tilde{\psi}_{n}\left(\mathbf{p}\right)=\tilde{\psi}_{n}^{\left(0\right)}\left(\mathbf{p}\right)-T\sum_{m=-\infty}^{\infty}\tilde{S}_{F}^{n-m}\left(\mathbf{p}\right)\tilde{\eta}_{m}\left(\mathbf{p}\right),
\frac{\mathrm{d}}{\mathrm{d}t}\{A,B\}=\{\frac{\mathrm{d}A}{\mathrm{d}t},B\}+\{A,\frac{\mathrm{d}B}{\mathrm{d}t}\}-\{A,B\}\left(\frac{\partial\dot{q}}{\partialq}+\frac{\partial\dot{p}}{\partialp}\right).
g_{ab}^{-1}(s;\tau)=-\left(\begin{array}{l}{2\tau_{2}+s\tau_{2}^{2}+a_{2}w_{3}^{2}s^{2}\tau_{3}^{2},\,3\tau_{3}+a_{1}s\tau_{2}\tau_{3}}\\{3\tau_{3}+a_{1}s\tau_{2}\tau_{3},\,-\frac23w_{3}^{-2}\tau_{2}^{2}+a_{3}s\tau_{3}^{2}}\\\end{array}\right)
\overbrace{\mathrm{SU}(8/1)}^{N=8}\rightarrow\overbrace{\mathrm{SU}(7/1)\otimes\mathrm{U}(1)}^{N=7}\rightarrow\overbrace{\mathrm{SU}(6/1)\otimes[\mathrm{U}(1)]^{2}}^{N=6}\rightarrow\cdots\rightarrow\overbrace{[\mathrm{U}(1)]^{8}}^{N=1}
\Psi^{(A)}=\left(\begin{matrix}{{\bfE}+i{\bfB}}\\{{\bfB}+i{\bfE}}\\\end{matrix}\right)={\frac{1+\gamma^{5}}{2}}\Psi^{(I)}+{\frac{1-\gamma^{5}}{2}}\Psi^{(II)}\,.
\left(f\astg\right)_{\theta}=\displaystyle\intdx\,f(\theta-x)\,g(x)=\displaystyle\frac{1}{p+1}\displaystyle\int\displaystyle\frac{dk}{2\pi}e^{ik\theta\displaystyle\frac{1}{p+1}}\tilde{f}(k)\tilde{g}(k)
\sum_{g}G_{g}W\left(\left.\begin{array}{cc}{g}&{b}\\{d}&{c}\\\end{array}\right|-n-z\right)W\left(\left.\begin{array}{cc}{g}&{d}\\{b}&{a}\\\end{array}\right|z\right)=\delta_{ac}\frac{G_{b}G_{d}}{G_{a}},
\hat{G}(x_{0},\vec{k})=\frac{1}{2}\int_{-\infty}^{+\infty}dk_{0}~\left[(-k^{2}+m^{2}-i0)^{-\alpha}+(-k^{2}+m^{2}+i0)^{-\alpha}\right]e^{ik_{0}x_{0}}
\delta_{F}(\left[v^{2}\theta^{4}\right])+\delta_{B}(\left[v^{3}\theta^{2}\right])+v^{i}\epsilon\left[v\dot{N}^{i(2)}\theta^{2}\right]\theta+\frac{i}{2}\theta\left[v^{2}\dot{M}^{(2)}\theta^{2}\right]\epsilon\simeq0~,
Z_{F}[M^{6};({\bf\beta},{\bf\gamma})]=(-)^{\alpha+\beta}\frac{\theta^{2}(_{\beta}^{\alpha})}{\eta^{2}(\tau)}(-)^{\bar{\alpha}+\bar{\beta}}\frac{\bar{\theta}^{2}(_{\bar{\beta}}^{\bar{\alpha}})}{\bar{\eta}^{2}(\bar{\tau})}\,
\Gamma_{a,s}(l_{1s})=\tilde{\Gamma}_{s}^{-1}\Gamma_{a,s}^{(0)}(l_{1s})\tilde{\Gamma}_{s}\,,\qquad\Gamma_{b,s}(l_{2s})=\widetilde\Gamma_{s}^{-1}\Gamma_{b,s}^{(0)}(l_{2s})\widetilde\Gamma_{s}
\bar{\omega}(q)\,\sim\,n\,\pi\,+\,\frac{2q}{n\pi}\,-\,\left(1\,+\,\frac{2q}{3}\right)\,\frac{(2q)^{2}}{(n\pi)^{3}}\,+\,O(n^{-5})\,{,}\qquadn\to\infty\,{,}
\Phi_{\alpha}\equiv\tilde{\Psi}_{\alpha}+{\frac{1}{2}}\tilde{\Psi}_{{\beta}}D_{\alpha}\tilde{\Psi}_{{\beta}}+{\frac{1}{2}}\tilde{\Psi}_{{i^{\prime}}}D_{\alpha}\tilde{\Psi}_{{i^{\prime}}}\;.
T_{+2-2}^{+q}=\frac12\gamma_{q\dot{p}}^{i}\left(\Omega_{+2}^{+2i}\psi_{-2\dot{p}}^{1-}-\Omega_{-2}^{+2i}\psi_{+2\dot{p}}^{1-}\right),\T_{+p\pm2}^{+q}=\frac12\gamma_{q\dot{p}}^{i}\Omega_{+p}^{+2i}\psi_{\pm2\dot{p}}^{1-},
\begin{array}{rclrcl}{\hat{\jmath}_{\mu\nu}}&{=}&{\jmath_{\mu\nu}\,,}&{\hat{\jmath}_{mn}}&{=}&{-e^{\bar{G}}\delta_{mn}\,,}\\{\hat{{\calB}}_{\mu\nu}^{(i)}}&{=}&{{\calB}_{\mu\nu}^{(i)}\,,}&{}&{}&{}\\{\hat{D}_{\mu\nu\rho\sigma}}&{=}&{D_{\mu\nu\rho\sigma}\,,}&{\hat{D}_{mnpq}}&{=}&{D_{mnpq}\,.}\\{\hat{\ell}}&{=}&{\ell\,,}&{\hat{\varphi}}&{=}&{\bar{\varphi}+\bar{G}\,.}\\\end{array}
F_{{\hat{M}}{\hat{N}}{\hat{P}}{\hat{Q}}{\hat{R}}}{\hat{\Gamma}}^{{\hat{M}}{\hat{N}}{\hat{P}}{\hat{Q}}{\hat{R}}}=5e^{3a\varphi}F_{[4]}\cdot\Gamma\,\Gamma^{9}(1-\hat{\Gamma}^{11}),
\begin{array}{l}{\,[L^{mn},L^{pk}]=g^{mk}L^{np}+g^{np}L^{mk}-g^{mp}L^{nk}-g^{nk}L^{mp}\quad,}\\{\,}\\{\,[\partial^{k},L^{mn}]=g^{km}\partial^{n}-g^{kn}\partial^{m}\quad,\qquad[\partial^{k},\partial^{l}]=0\quad,}\\\end{array}
E_{1,2}^{2}={\bfp}^{2}+m^{2}+\frac{g^{2}(\lambda_{1}+\lambda_{2})}{4}\pmg\sqrt{\lambda_{1}p_{1}^{2}+\lambda_{2}p_{2}^{2}+\frac{(H_{z}^{3})^{2}}{4}}
Q^{(1)}=\intDx(t)\delta[x(0)-x(s)]\delta[x(s_{1})-x(s)]exp\left(-{\frac{1}{4}}\int_{0}^{s}\,dt_{i}\,\dot{x}^{2}+\sum_{n=1}^{M}\hat{k}(t_{n})\cdotx(t_{n})\right).
B_{\ell}=\overset{(s_{1})}{e}{\!\!}_{A_{1}}\,\dots\,\frac{\partial\,\overset{(s_{\ell})}{e}{\!\!}_{A_{\ell}}}{\partial\,p}\;\dots\,\overset{(s_{N})}{e}{\!\!}_{A_{N}}\,\widetilde{G}_{A_{1}\,\dots\,A_{N}}\;,
[Q(\hat{K}_{L}(\hat{I}),\hat{K}_{L}(\hat{I})]_{ij}=\sum_{k}Q(\hat{K}_{L}(\hat{I}))_{ik}\star\hat{K}_{L}(\hat{I})_{kj}-\hat{K}_{L}(\hat{I})_{ik}\starQ(\hat{K}_{L}(\hat{I}))_{kj}
I(\tau)\;=\;F(\{w_{j}\},\{z_{j}\})\times\Big(\tau^{2}\Big)^{-\frac{1}{N}\sum_{b,b^{\prime}=1}^{N}(n_{b}-m_{b})(n_{b^{\prime}}^{\prime}-m_{b^{\prime}}^{\prime})}\;\left(1+O\Big(\frac{1}{\tau}\Big)\right)\;,
\Gamma={\frac{ge}{(2\pi)^{3}}}\intd^{6}x\,\epsilon^{\mu\nu\rho\sigma\tau\omega}\left(\,e^{2}F_{\rho\sigma}F_{\tau\omega}+g^{2}F_{\rho\sigma}^{5}F_{\tau\omega}^{5}\,\right)\left[\,F_{\mu\nu}{\frac{1}{\Box}}\partialA^{5}-2{\bfa}\,A_{\mu}A_{\nu}^{5}\,\right]
\int_{0}^{\infty}dk\,k\,e^{-yk^{2}}J_{\mu}^{2}(kr)=\frac1{2r^{2}}\mathrm{exp}\left(-\frac{r^{2}}{2y}\right)I_{\mu}\left(\frac{r^{2}}{2y}\right)\,,\,\mu>-1\,,