harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
H_{GS}=\frac{\sqrt{2}\kappa}{4i}\left(\tau\partial_{I}\tau\partial_{I}\bar{\tau}-\left(\frac{\partial_{I}}{\partial_{-}}\tau\right)\partial_{I}\tau\partial_{-}\bar{\tau}-\left(\frac{\partial_{I}}{\partial_{-}}\tau\right)\partial_{I}\bar{\tau}\partial_{-}\tau+\left(\frac{\partial_{I}\partial_{I}}{\partial_{-}^{2}}\tau\right)\partial_{-}\tau\partial_{-}\bar{\tau}\right)+\mathrm{c.c.}~.
|\tilde{\Psi}\rangle_{1}:={}_{2}\langle\Psi|V_{2}\rangle_{12}=e^{\hat{c}_{0}^{(1)}\mu_{0}^{\ast}+\hat{c}^{\dagger(1)}C\mu_{b}^{\ast}+\mu_{c}^{\ast}C\hat{b}^{\dagger(1)}}\hat{c}_{1}^{(1)}|\Omega\rangle_{1}\,.
\epsilon_{l}(H_{G}^{2l}(M))=\;<\left(\begin{array}{ccc}{1}\\{\vdots}\\{1}\\\end{array}\right),\left(\begin{array}{ccc}{\mu_{1}}\\{\vdots}\\{\mu_{k}}\\\end{array}\right),\ldots,\left(\begin{array}{ccc}{\mu_{1}^{l}}\\{\vdots}\\{\mu_{k}^{l}}\\\end{array}\right)>
H=-\frac{g^{2}L}{4}\left(\sum_{i=1}^{N}\frac{\partial^{2}}{\partial\varphi_{i}^{2}}-\frac{1}{N}\left(\sum_{i=1}^{N}\frac{\partial}{\partial\varphi_{i}}\right)^{2}\right)-\frac{Ng^{2}L(N^{2}-1)}{48}\.
U_{\mathrm{w}}^{(c)}\,-\,U_{\mathrm{s}}^{(c)}\;=\;{\frac{1}{4\beta_{c}^{2}}}\int_{a_{c}}^{b_{c}}dz^{\prime}{\frac{\rho_{c}(z^{\prime})r_{c}}{d(d-1)}}\;=\;{\frac{d}{4(d-1)}}a_{c}^{2}\;,
{\frac{1}{M}}\left\{{\calU}({\vec{p}})\overline{{\calU}}({\vec{p}})+{\calV}({\vec{p}})\overline{{\calV}}({\vec{p}})\right\}\,=\,\left(\begin{matrix}{I}&{{S}_{{\vec{p}}}\otimes{S}_{{\vec{p}}}}\\{{S}_{{\vec{p}}}^{-1}\otimes{S}_{{\vec{p}}}^{-1}}&{I}\\\end{matrix}\right)\quad.
\psi_{l}=e^{-\frac{\pi\omega}{L_{1}L{2}}\sum_{\gamma}y_{\gamma}^{2}}e^{vY/L_{1}}\prod_{\gamma<\delta}\mid\theta_{1}\left(z_{\gamma}-z_{\delta}\mid\tau\right)\mid^{\frac{1}{\kappa}}\phi_{l}\left(x,y,v\right)
R(a_{j},a_{j})=\sum_{k\neqj}(-1)^{k}\,{\frac{(q_{j}p_{k}-p_{j}q_{k})^{2}}{a_{j}-a_{k}}}\,+p_{j}^{2}-a_{j}q_{j}^{2}-2p_{j}q_{j}u+2q_{j}^{2}v\,.
\bar{V}(\bar{\phi})=\frac{1}{2}\sum_{l=0}^{M}\sum_{a_{l}=1}^{L_{l}}{\calM}^{2}(\alpha_{l},R)\left(\bar{\phi}_{a_{l}}^{(\alpha_{l})}\right)^{2}+\frac{\lambda}{8}\left(\sum_{l=0}^{M}\sum_{a_{l}=1}^{L_{l}}\left(\bar{\phi}_{a_{l}}^{(\alpha_{l})}\right)^{2}\right)^{2},
\begin{array}{cll}{E_{a}{\circ_{1,Z}}E_{b}}&{=(Ze_{a})(Z^{\dagger}(Ze_{b}))}&{}\\{}&{=(Ze_{a})e_{b}}&{}\\{}&{=-(Ze_{b})e_{a}}&{\mathrm{See~{\bf~[1]}}}\\{}&{=-(Ze_{b})(Z^{\dagger}(Ze_{a}))}&{}\\{}&{=-E_{b}{\circ_{1,Z}}E_{a}.}&{}\\\end{array}
\frac{L^{3}}{z^{3}}(-\partial_{z}^{2}+\frac3z\partial_{z}+\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}+\frac{m^{2}L^{2}}{z^{2}}+\frac{2\lambda_{1}}{z}\delta(z-z_{1})+\frac{2\lambda_{2}}{z}\delta(z-z_{2}))\phi(q,z)=0
J(X_{\tau},\alpha,\beta)=\int_{X_{\tau}}d^{m+n}z\prod_{i=1}^{m}z_{i}^{\alpha}(1-z_{i})^{\beta}\prod_{\hat{i}=\hat{1}}^{\hat{n}}z_{\hat{i}}^{\alpha^{\prime}}(1-z_{\hat{i}})^{\beta^{\prime}}\left[\prodz_{ij}^{2\rho}\prodz_{\hat{i}\hat{j}}^{2\rho^{\prime}}\prodz_{i\hat{j}}^{2\sigma}\right]_{\tau}.
S[q]=S[q_{\alpha}]+\int\frac{\deltaS[q_{\alpha}]}{\deltaq_{\alpha}(\tau)}\varphi_{\alpha}(\tau)d\tau+\frac{1}{2}\int\frac{\delta^{2}S[q_{\alpha}]}{\deltaq_{\alpha}(\tau)\deltaq_{\alpha}(\tau^{\prime})}\varphi_{\alpha}(\tau)\varphi_{\alpha}(\tau^{\prime})d\taud\tau^{\prime}+\cdots.
D(\Omega,\Omega^{\prime})=-\frac{1}{4\pi}\hat{U}^{-1}(n)\,frac{2i(\vec{\sigma}\wedge(\vec{p}-\vec{p}^{~\!\prime}))}{p_{0}}\\frac{(1+n_{0})^{1/2}(1+n_{0}^{\prime})^{\frac{1}{2}}}{(n-n^{\prime})^{2}}\,hat{U}(n^{\prime})
\begin{array}{l}{Y_{i}(u)Y_{j}(v)=Y_{j}(v)Y_{i}(u)~,~~~~|i-j|\geq2~,}\\{Y_{i}(u)Y_{i}(-u)=1~,}\\{Y_{i}(u)Y_{i+1}(u+v)Y_{i}(v)=Y_{i+1}(v)Y_{i}(u+v)Y_{i+1}(u)~.}\\\end{array}
\hat{P}^{\hat{a}\hat{b}\hat{c}}\equiv\hat{\calH}^{\hat{a}\hat{b}\hat{d}}\hat{C}_{\hat{d}}^{\hat{c}}={\frac{1}{K}}\,\left[(1+{\frac{1}{12}}\hat{{\calH}}^{2})\,\hat{\calH}^{\hat{a}\hat{b}\hat{c}}-\,{\frac{1}{4}}(\hat{{\calH}}^{3})^{\hat{a}\hat{b}\hat{c}}\right]\,.
\sum_{i=1}^{4}M_{(i),12}=1+\frac{\lambda_{1}^{3}\lambda_{2}^{3}(\lambda_{3}-1)^{3}-\lambda_{3}^{3}(\lambda_{3}-1)^{3}-\lambda_{3}^{3}(\lambda_{1}-1)^{3}(\lambda_{2}-1)^{3}}{(\lambda_{3}-\lambda_{1})^{3}(\lambda_{3}-\lambda_{2})^{3}}.
\sum_{l_{1}=-1}^{n}\sum_{l_{2}=-1}^{k}\left(\begin{array}{c}{n+1}\\{l_{1}+1}\\\end{array}\right)z^{n-l_{1}}\left(\begin{array}{c}{k+1}\\{l_{2}+1}\\\end{array}\right)z^{k-l_{2}}(l_{1}-l_{2})=(n-k)(z+1)^{n+k+1}.
(dG)_{11\bar{I}\bar{J}\bar{K}\bar{L}}=4\sqrt{2}\pi\left(\frac{\kappa}{4\pi}\right)^{2/3}(J^{(1)}\delta(x^{11})+J^{(2)}\delta(x^{11}-\pi\rho)+\Sigma_{i=1}^{N}\hat{J}^{(i)}\delta(x^{11}-x_{i}))_{\bar{I}\bar{J}\bar{K}\bar{L}}.
\begin{array}{rcl}{T_{a}}&{=}&{\displaystyle+[3D_{m}^{2}(k_{1})D_{m}(k_{2})+D^{2}(k_{1})D(k_{2})](2+K_{12})}\\{}&{}&{\displaystyle+[3D_{m}^{2}(k_{1})D(k_{2})+D^{2}(k_{1})D_{m}(k_{2})](2-K_{12})}\\{}&{}&{\displaystyle+(N-1)[6D_{m}^{2}(k_{1})D(k_{2})+4D^{2}(k_{1})D(k_{2})+2D^{2}(k_{1})D_{m}(k_{2})]}\\{}&{}&{+4(N-1)ND^{2}(k_{1})D(k_{2}),}\\\end{array}
{{\calG}^{-1}}_{\mua}^{\nub}\eta_{\nub}^{\mathrm{T,T}}=\frac{1}{2}\left(k^{2}-\Lambda_{l}+\frac{1}{2r^{2}}(N+1)(N-2)-\frac{N+d}{N+d-2}\Lambda\right)\eta_{\mua}^{\mathrm{T,T}},
\begin{array}{lll}{U_{\pm}^{R}=\left(\begin{array}{cc}{1}&{0}\\{0}&{\mpi}\\\end{array}\right)}&{\mathrm{or}}&{U_{\pm}^{R}=\left(\begin{array}{cc}{0}&{\mpi}\\{1}&{0}\\\end{array}\right)}\\\end{array}
\deltaT_{\mu\nu}=-\,{\textstyle\frac14}\,\bar{\epsilon}\,\gamma_{P}\left(\epsilon_{\mu\lambda}\,\partial^{\lambda}\,S_{\nu}+\epsilon_{\nu\lambda}\,\partial^{\lambda}\,S_{\mu}\right)=-\,{\textstyle\frac12}\,\bar{\epsilon}\,\gamma_{P}\epsilon_{\mu\lambda}\,\partial^{\lambda}\,S_{\nu}~.
2C_{S}=\left(\begin{matrix}{2}&{-2}&{1}&{-2}&{4}&{-2}&{1}&{-2}\\{-2}&{2}&{-2}&{1}&{-2}&{4}&{-2}&{1}\\{1}&{-2}&{2}&{-2}&{1}&{-2}&{4}&{-2}\\{-2}&{1}&{-2}&{2}&{-2}&{1}&{-2}&{4}\\{4}&{-2}&{1}&{-2}&{2}&{-2}&{1}&{-2}\\{-2}&{4}&{-2}&{1}&{-2}&{2}&{-2}&{1}\\{1}&{-2}&{4}&{-2}&{1}&{-2}&{2}&{-2}\\{-2}&{1}&{-2}&{4}&{-2}&{1}&{-2}&{2}\\\end{matrix}\right),
I_{2}=\frac{\pi^{3/2}}{12}\biggl(\frac{\thetap}{2}\biggr)^{1/2}[15\nu^{-1/2}K_{1/2}(\nu\thetap)-20\biggl(\frac{\thetap}{2}\biggr)^{3/2}\nu^{1/2}K_{1/2}(\nu\thetap)+4\biggl(\frac{\thetap}{2}\biggr)^{5/2}\nu^{3/2}K_{3/2}(\nu\thetap)],
\left(\begin{matrix}{{Q_{E}}^{U}}\\{{Q_{M}}^{U}}\\\end{matrix}\right)={\frac{e}{4\pi}}\intdS_{k}\,\sqrt{-g}\,U(x)\left(\begin{matrix}{F^{k0}}\\{\tilde{F}^{k0}}\\\end{matrix}\right)U^{-1}(x)
V^{q}(\psi\otimes\bar{w},z)\;(\chi\otimes\bar{v}):=\sum_{\bar{w}^{\prime},\bar{v}^{\prime}}\left(\widetilde{F}^{-1}\right)_{\bar{w}^{\prime},\bar{v}^{\prime}}^{\bar{w},\bar{v}}V(\psi\otimes\bar{w}^{\prime},z)\;(\chi\otimes\bar{v}^{\prime}),
\begin{array}{rcl}{[T_{a},T_{b}]}&{=}&{iF_{abc}T_{c}\;,}\\{{}~[T_{a},-T_{b}^{\ast}]}&{=}&{0\;,}\\{{}~[-T_{a}^{\ast},-T_{b}^{\ast}]}&{=}&{iF_{abc}(-T_{c}^{\ast})\;,}\\\end{array}
F_{a}^{\pm}[A^{\pm}]=\frac{1}{2}\eta_{ab}\epsilon^{ij}(\partial_{i}A_{j}^{\pm\;b}-\partial_{j}A_{i}^{\pm\;b}+\epsilon_{\;cd}^{b}A_{i}^{\pm\;c}A_{j}^{\pm\;d})=0
\frac{\partial^{2}\tilde{U}}{\partialR^{2}}\mid_{crit.}=\frac{1}{r^{2}(r^{2}-2mr)^{3/2}}\left((N^{2}-m^{2})r^{2}-6mN^{2}r+\frac{15}{2}m^{2}N^{2}\right)\mid_{crit.}
\begin{array}{c}{(0~0~0~0~-\frac{5}{2})\rightarrow(0~0~0~-\frac{5}{2})(\frac{5}{2})\oplus(0~0~0~-\frac{3}{2})(-\frac{3}{2}),}\\{(0~0~0~0~-\frac{3}{2})\rightarrow(0~0~0~-\frac{3}{2})(\frac{3}{2})\oplus(0~0~0~-\frac{1}{2})(-\frac{5}{2}).}\\\end{array}
\begin{array}{l}{K_{\tau}(x,y)=E[\delta(y-q_{\tau}(x))]=}\\{=E[\delta({\bfy}_{F}-{\bfx}_{F}-{\bfb}_{F}(\tau))\prod_{\mu}\delta\left(y_{\mu}-q_{\mu}\left(\tau,x\right)\right)]}\\\end{array}
\left(\begin{array}{c}{{\calH}_{1}^{\prime}}\\{{\calH}_{2}^{\prime}}\\\end{array}\right)=\left(\begin{array}{cc}{a}&{\,\,\,b}\\{}&{}\\{c}&{\,\,\,d}\\\end{array}\right)\left(\begin{array}{c}{{\calH}_{1}}\\{{\calH}_{2}}\\\end{array}\right)\,,
Z_{orb}=\langleB_{orb}|\tilde{q}^{H^{(P)}}|B_{orb}\rangle=\sum_{h\inH}|a_{h}|^{2}\sum_{k,k^{\prime}\inG/H}\langleKB_{h}|\tilde{q}^{H^{(P)}}|K^{\prime}B_{h}\rangle\,
e^{i{\bfk}\cdot({\bfx}-{\bfx_{0}})}=\sum_{m,n=-\infty}^{\infty}J_{m}(k\rho)J_{n}(k\rho_{0})(-1)^{n}e^{im\varphi}e^{in\varphi_{0}}e^{i(m+n)\theta},
R_{\mu\nu\rho\lambda}R^{\mu\nu\rho\lambda}={\frac{8\,\left({M^{4}}+4\,{M^{2}}\,{{r_{0}}^{2}}+12\,M\,{{r_{0}}^{3}}+7\,{{r_{0}}^{4}}-2\,{M^{2}}\,{|\Upsilon|^{2}}+2\,{{r_{0}}^{2}}\,{|\Upsilon|^{2}}+{|\Upsilon|^{4}}\right)}{(r_{\pm}^{2}-|\Upsilon|^{2})^{4}}}
y=\varphi-\varphi_{0}\left(\!z=\alpha_{0}(\sigma,\bar{\sigma})+\!\!\sum_{n=1}^{\infty}\frac{{\hat{y}}^{n}}{n!}\alpha_{n}(\sigma,\bar{\sigma}),\,\bar{z}=\bar{\alpha}_{0}(\sigma,\bar{\sigma})+\!\!\sum_{m=1}^{\infty}\frac{{\hat{y}}^{m}}{m!}\bar{\alpha}_{m}(\sigma,\bar{\sigma})\right)\,.
(\hat{f}\ast\hat{g})(x)=\int_{\mathbbQ_{p}^{D}}\int_{\mathbbQ_{p}^{D}}dkdk^{\prime}\\chi_{p}(-(x^{i}k_{i}+x^{j}k_{j}^{\prime})+\frac{1}{2}k_{i}k_{j}^{\prime}\theta^{ij})\tilde{f}(k)\tilde{g}(k^{\prime}).
\begin{array}{ll}{Z=\frac{1}{2}(}&{\mathrm{Tr\thinspace~}e^{-\beta(H^{(+)}-\mu\hat{N})}+\mathrm{Tr\thinspace~}e^{-\beta(H^{(-)}-\mu\hat{N})}+}\\{}&{\mathrm{Tr\thinspace~}\taue^{-\beta(H^{(+)}-\mu\hat{N})}-\mathrm{Tr\thinspace}\taue^{-\beta(H^{(-)}-\mu\hat{N})})}\\\end{array}
\left(\begin{matrix}{\tilde{\xi}^{+}}\\{\tilde{\xi}^{-}}\\\end{matrix}\right)=\left(\begin{matrix}{e^{\alpha/2}}&{0}\\{0}&{e^{-\alpha/2}}\\\end{matrix}\right)\left(\begin{matrix}{\xi^{+}}\\{\xi^{-}}\\\end{matrix}\right)\,\,,
\Bigl(X^{aA,bB}(x,y)\Bigr)=\left(\begin{array}{cc}{\frac{g}{2m^{2}}\,f^{abc}\bigl(D_{i}\pi^{i}\bigr)^{c}}&{\delta^{ab}}\\{-\,\frac{1}{m^{2}}\,\delta^{ab}}&{0}\\\end{array}\right)\,\delta(x-y)\,.
r_{d}=\frac{(2q_{1}^{2}+q_{2}^{2})(r_{0}-r_{1})r_{1}+q_{2}^{2}(r_{2}-r_{0})r_{2}}{2q_{1}^{2}(r_{0}-r_{1})+q_{2}^{2}(r_{2}-r_{1})}\,.
{\bf1}\frac{d}{d\eta}\chi(\eta)=\left(\begin{array}{llll}{0}&{1}\\{1}&{\frac{1}{\eta}}\\\end{array}\right)\chi^{(0)}(\eta),\quad\chi^{(0)}(\eta)=\left(\begin{array}{l}{\chi_{1}^{(0)}}\\{\chi_{2}^{(0)}}\\\end{array}\right),
\begin{array}{rcl}{\left({\calV}_{1}\otimes{\calV}_{1}\right)_{\mathrm{f}}}&{=}&{{\calV}_{0}\,,}\\{\left({\calV}_{-1/8}\otimes{\calV}_{-1/8}\right)_{\mathrm{f}}}&{=}&{{\calR}_{0}\,,}\\{\left({\calV}_{-1/8}\otimes{\calV}_{3/8}\right)_{\mathrm{f}}}&{=}&{{\calR}_{1}\,,}\\{\left({\calV}_{3/8}\otimes{\calV}_{3/8}\right)_{\mathrm{f}}}&{=}&{{\calR}_{0}\,,}\\\end{array}
M_{I{\bar{J}}}^{2}=-\frac{4g^{2}}{16\pi^{2}}R_{I}R_{J}\sum_{p_{5},\,p_{6}}\int_{0}^{\infty}dl\,\,\,\,\left(G^{IJ}-\frac{2\,G^{IK}G^{JL}{p}_{K}{p}_{L}}{l}\right)\,\,e^{-(p_{M}G^{MN}p_{N})/2l}
F(t;q^{\prime},q)=1+\sum_{n=1}^{\lambda-1}\frac{(-1)^{n}t^{n}}{q^{n}q^{n}}\frac{1}{n!}\prod_{j=1}^{n}\left(\frac{\lambda(\lambda-1)}{2}-\frac{j(j-1)}{2}\right).
J^{\mu\nu}\,=\,\intd^{3}x\,J^{\,0,\mu\nu}(x)\quad,\quad(\hat{W}\cdotn)=S^{\,12}\quad,\quad\mathrm{if}\quad{\bfn}\,\,\vert\vert\,\,{\bfp}\,\,\vert\vert\,\,OZ\quad
{\cal{F}}\left\{\left(P+m^{2}\right)^{-1}\right\}(x)=\frac{\pi}{2}m^{\frac{\nu}{2}-1}Q_{-}^{\frac{1}{2}\left(1-\frac{\nu}{2}\right)}{\cal{J}}_{1-\frac{\nu}{2}}\left(mQ_{-}^{\frac{1}{2}}\right)
g=-4dx^{+}dx^{-}+\left(\sum_{i=1}^{3}\left(\frac{\mu}{3}\right)^{2}x^{i}x^{i}+\sum_{i=4}^{9}\left(\frac{\mu}{6}\right)^{2}x^{i}x^{i}\right)dx^{+}dx^{+}+\sum_{i=1}^{9}dx^{i}dx^{i}~.
F_{\mathrm{sing}}(N,g_{3},g_{4})\simb\cdot\left[\deltag_{3}+\left(\frac{3^{1/4}}{4}-\frac{3^{7/4}}{2}\right)\deltag_{4}\right]^{5/2}+b^{\prime}\cdot[\deltag_{4}]^{-6}+\cdots.
M_{n_{1},n_{2}}^{d}(w_{1},w_{2})=k^{2(n_{1}+n_{2}-1)-d}\int_{0}^{\infty}dxx^{\frac{d}{2}}\frac{\partial}{\partial{t}}\left\{\dot{P}^{2}(P+w_{1})^{-n_{1}}(\tilde{P}+w_{2})^{-n_{2}}\right\}
S=\int\frac{1}{16\piG}R\sqrt{-g}d^{4}x-\int\left[\frac{1}{4}F_{\mu\nu}^{a}F^{a\mu\nu}+\frac{1}{2}D_{\mu}\phi^{a}D^{\mu}\phi^{a}+\frac{1}{4}\lambda(\phi^{a}\phi^{a}-v^{2})^{2}\right]\sqrt{-g}d^{4}x\
\sqrt{\pi}\left(\frac{1}{2}\frac{d^{2}}{dT^{2}}\left[T\left(L_{\mathrm{eff}}(\omega)+E+\frac{\omega}{2}\right)\right]\right)^{-1/2}=\sqrt{2\piT}\left(\omega^{2}\frac{dL_{\mathrm{eff}}}{d\omega^{2}}\right)^{-1/2}\,,
S(\delta_{ji,A}^{k})=-\delta_{ji,A}^{k}-\sum_{a=1}^{|A|!}\sum_{\mathcal{C}_{a}}(-1)^{|\mathcal{C}_{a}|}\;P^{\mathcal{C}_{a}}(t_{a}^{|A|})\,R^{\mathcal{C}_{a}}(t_{a}^{|A|})~,
\phi_{i}(x,y)=\frac{\phi_{0}(x-y)}{v_{1}}\left[v_{1}+\frac{\sigma(X)+i\varphi^{0}(X)}{\sqrt{2}}\right]\delta_{i}{}^{5}+\sqrt{2}i\frac{\phi_{0}(x-y)}{v_{2}}\varphi^{\rho}(X)\left(S^{\rho}\right)_{i}{}^{5}+\ldots,
H_{\mu\nu}\rightarrowH_{\mu\nu}+\frac{1}{m^{2}}\bigl(\lambda_{1}R_{\mu}{}^{\alpha}{}_{\nu}{}^{\beta}H_{\alpha\beta}+\lambda_{2}R^{\alpha}{}_{(\mu}H_{\nu)\alpha}+\lambda_{3}RH_{\mu\nu}+\lambda_{4}G_{\mu\nu}R^{\alpha\beta}H_{\alpha\beta}\bigr)+O\left(\frac{1}{m^{4}}\right)
S(x)={\calS}_{0}(x)e^{-ie^{2}\beta(x)}+B_{0}e^{ie^{2}\beta(x)}+C_{0}\gamma^{5}e^{ie^{2}\beta(x)}+D_{0}\frac{\gamma^{5}\not\!x}{x^{2}}e^{-ie^{2}\beta(x)}\;,
e_{\mu}^{a}=\left(\begin{array}{cccc}{N}&{0}&{0}&{0}\\{0}&{f^{-1}}&{0}&{0}\\{0}&{0}&{g}&{0}\\{RN^{\phi}}&{0}&{0}&{R}\\\end{array}\right),~~~e_{a}^{\mu}=\left(\begin{array}{cccc}{N^{-1}}&{0}&{0}&{0}\\{0}&{f}&{0}&{0}\\{0}&{0}&{g^{-1}}&{0}\\{-N^{-1}N^{\phi}}&{0}&{0}&{R^{-1}}\\\end{array}\right)
\epsilon=e^{-\frac{1}{2}V}\epsilon_{+}^{0}-e^{\frac{1}{2}V}\left(l\gamma_{0}t+l\sum\gamma_{\alpha_{i}}z^{\alpha_{i}}+\gamma_{2}\theta+\gamma_{3}\phi\right)\epsilon_{+}^{0}+e^{\frac{1}{2}V}\epsilon_{-}^{0}
H^{ij}=\frac{17}{60}\theta^{i0}\theta^{j0}+\sum_{k\neq0}\frac{1}{10}\theta^{k0}\theta^{k0}\delta^{ij}+\frac{1}{4}\Big(\sum_{l>k\neq0}\theta^{kl}\theta^{kl}\delta^{ij}-\sum_{k\neq0}\theta^{ki}\theta^{kj}\Big)>0~,
C=\Gamma_{2}\Gamma_{4}=\left(\begin{array}{cc}{I}&{0}\\{0}&{\widetilde{I}}\\\end{array}\right)\,\quadI=\widetilde{I}=\left(\begin{array}{ccc}{\begin{array}{cc}{0}&{1}\\{-1}&{0}\\\end{array}}&{|}&{0}\\{---}&{|}&{---}\\{0}&{|}&{\begin{array}{cc}{0}&{1}\\{-1}&{0}\\\end{array}}\\\end{array}\right)
e^{-S_{b}\left[X;P^{(B)}\right]}=\left.\left\langleP_{\mu}^{(B)}(\sigma)=-\sum_{r=1}^{M}\left(k_{\mu}^{(r)}-a_{r}A_{\mu}(k^{(r)})\partial_{\sigma_{r}}\right)\delta(\sigma-\sigma_{r})\right|X_{N}\right\rangle~,
\psi_{a}^{L}(x)=\sum_{j=1}^{N}\sum_{\sigma^{\prime}\in{\calW}^{\prime}}A_{a}^{L}(\sigma^{\prime}q_{j}^{\prime},id)e^{\kappa_{j}x_{1}}e^{i(\sigma^{\prime}q_{j}^{\prime}k|x^{\prime})}
\Omega_{AB}=\left(\begin{array}{ll}{\Omega_{\alpha\beta}}&{\Omega_{\alpha}^{\;\;\beta}}\\{\Omega_{\;\;\beta}^{\alpha}}&{\Omega^{\alpha\beta}}\\\end{array}\right)=\left(\begin{array}{cc}{0}&{-\delta_{\alpha}^{\beta}}\\{\delta_{\beta}^{\alpha}}&{0}\\\end{array}\right).
H=\mathrm{Tr}\frac{g_{YM}^{2}R_{0}^{2}}{2V}P^{i}g_{ij}P^{j}+\mathrm{Tr}\frac{V}{4g_{YM}^{2}}(F_{ij}+\phi_{ij}\cdot{\bf1})g^{ik}g^{il}(F_{kl}+\phi_{kl}\cdot{\bf1})\,.
ds^{2}=(\frac{r^{2}}{b^{2}}-\frac{b^{2}}{r^{2}})d\tau^{2}+\frac{dr^{2}}{(\frac{r^{2}}{b^{2}}-\frac{b^{2}}{r^{2}})}+r^{2}\sum_{i=1}^{3}dx_{i}^{2}+b^{2}d\Omega_{5}^{2}
J^{(N)}(n;1,\ldots,1)=\mathrm{i}^{1-2N}\pi^{n/2}\;\Gamma\left(N-{\textstyle{\frac{n}{2}}}\right)\;\frac{m_{0}^{n-N}}{\sqrt{D^{(N)}}\;\prodm_{i}}\;\Omega^{(N;n)},
\mathcal{L}_{0}=\intd^{2}\vartheta(\Phi\cdotJ_{0}+\mathrm{h.c.}),\qquadJ_{0}=\frac{1}{2}\left(\frac{1}{g_{0}^{2}}-i\frac{\theta_{0}}{8\pi^{2}}\right)=\mathrm{const.}
(0~0~0~0~0~0~0~-\frac{7}{2})=[{\bf1}_{B}\oplus{\bf\overline{{8}}}_{F}\oplus{\bf\overline{{28}}}_{B}\oplus{\bf\overline{{56}}}_{F}\oplus{\bf70}_{B}\oplus{\bf56}_{F}\oplus{\bf28}_{B}\oplus{\bf8}_{F}\oplus{\bf1}_{B}].
E_{n}^{m}\vert_{\mathrm{exchange}}=\sum_{\{n_{j},m_{j}\}}\prod_{i=1}^{n-m-1}E_{n_{i}}^{m_{i}}\qquad\mathrm{with}\qquad\sum_{i=1}^{n-m-1}n_{i}=2(n-1)-m
\mathrm{S}=\intd^{4}xd\tau\left\{\psi^{*}(i\partial_{\tau}+e_{0}a_{5})\psi-\frac{1}{2M}\psi^{*}(-i\partial_{\mu}-e_{0}a_{\mu})(-i\partial^{\mu}-e_{0}a^{\mu})\psi-\frac{\lambda}{4}f_{\alpha\beta}f^{\alpha\beta}\right\}\,
G^{{\mu}{\nu}{\sigma}}=-F^{{\mu}{\nu}{\sigma}}-\frac{1}{({\partial}_{\lambda}a)({\partial}^{\lambda}a)}\left[{\partial}^{[{\mu}}a{\calF}^{{\nu}{\sigma}]{\rho}}{\partial}_{\rho}a+\frac{1}{3!}{\epsilon}^{{\mu}{\nu}{\sigma}{\rho}{\eta}{\delta}}{\partial}_{[{\rho}}a{\calF}_{{\eta}{\delta}]{\theta}}{\partial}^{\theta}a\right].
{\calL}=-\frac{1}{4}F^{a\mu\nu}F_{\mu\nu}^{a}+\frac{1}{2}i\bar{\lambda}^{a}\gamma_{\mu}{\calD}^{\mu}(A)\lambda^{a}+\frac{\thetag^{2}}{32\pi^{2}}F^{a\mu\nu}F_{\mu\nu}^{a}.
<\intd^{4}z~\left[\Theta(z)-\left(1-{\frac{3N_{c}}{N_{f}}}-\gamma(g)\right)L_{matter}(z)\right]>=u_{0}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{4}B(k)^{\mu\nu}{B}_{\mu\nu}(-k).
L/\sqrt{-\gamma}=-\frac{1}{4}\partial_{\rho}b_{\mu\nu}\partial^{\rho}b^{\mu\nu}-\frac{1}{4}(\partial_{4}b_{\mu\nu}-2\partial_{4}\sigmab_{\mu\nu})(\partial_{4}b^{\mu\nu}+2\partial_{4}\sigmab^{\mu\nu})-\frac{3}{4}c\,e^{-2\sigma}\partial_{\mu}\phi\partial^{\mu}\phi.
k_{(i)\mu\nu}=-{\frac{\mu}{4}}(f_{~\mu\lambda}^{Y}f_{~~\nu}^{(i)\lambda}+f_{~\nu\lambda}^{Y}f_{~~\mu}^{(i)\lambda})+{\frac{1}{2\mu}}(k_{(5)\mu}k_{(i)\nu}+k_{(5)\nu}k_{(i)\mu}).
\frac{1}{2}\int_{\epsilon^{2}R^{2}}^{\infty}\frac{dt}{t^{3}}\sum_{k=-\infty}^{\infty}e^{-k^{2}\pit/R^{2}}=R^{-4}{\calI}(\epsilon^{2},\alpha=-2)=\frac{1}{R^{4}}\left\{\frac{3}{4\pi^{2}}\zeta(5)+\frac{1}{5}\epsilon^{-5}\right\}
e^{\overline{{\phi}}/d}\left(e^{-\overline{{\phi}}/d}\right)^{\prime\prime}=-\frac{1}{d^{2}}\phi^{2}-\frac{1}{4d}\left\{\mathrm{{Tr}}\left[\left(G^{-1}G^{\prime}\right)_{t}^{2}\right]-{\mathrm{Tr}}\left[\left(G^{-1}B^{\prime}\right)^{2}\right]\right\},
U({\bfr}^{\prime},{\bfr}_{2})=-\int_{-\infty}^{+\infty}d\omega\,4\pi\alpha_{1}({\bfr}^{\prime},i|\omega|)\alpha_{2}({\bfr}_{2},i|\omega|)\,(D_{pj}^{(0)}({\bfr}^{\prime},{\bfr}_{2},i\omega))^{2}.
\Big[{\frac{1}{4\,v^{2}}}-{\frac{f^{\prime}}{vf}}+{\frac{b}{v}}-{\frac{f^{\prime\prime}}{f}}+{\frac{2bf^{\prime}}{f}}-b^{2}-{\frac{1}{4\,v^{2}}}-m^{2}\Big]\varphi^{(0)}=0~.
i\hbar\partial_{t}\psi(t;x_{1},x_{2})={\frac{\hbar^{2}}{2m}}\left\{\Bigl({\frac{1}{i}}\partial_{x_{1}}+{\frac{i\nu}{x_{1}-x_{2}}}R\Bigr)^{2}+\Bigl({\frac{1}{i}}\partial_{x_{2}}+{\frac{i\nu}{x_{2}-x_{1}}}R\Bigr)^{2}\right\}\psi(t;x_{1},x_{2})
I=-A.\intd^{3}x\left[{\frac{1}{4g}}F_{\mu\nu}^{2}+{\frac{1}{2g}}g^{mn}\partial_{\mu}\phi_{m}\partial^{\mu}\phi_{n}+i\epsilon^{mn}\epsilon^{\mu\nu\rho}C_{m}\partial_{\mu}\phi_{n}F_{\nu\rho}\right]
\Xi=\left(\begin{array}{cccc}{d_{1}^{3}}&{-u_{1}d_{1}^{2}}&{-u_{1}^{2}d_{1}}&{u_{1}^{3}}\\{d_{1}^{2}d_{2}}&{-u_{1}d_{1}d_{2}+u_{2}d_{1}^{2}}&{-u_{1}u_{2}d_{1}+u_{1}^{2}d_{2}}&{u_{1}^{2}u_{2}}\\{-d_{1}d_{2}^{2}}&{u_{2}d_{1}d_{2}-u_{1}d_{2}^{2}}&{u_{1}u_{2}d_{2}-u_{2}^{2}d_{1}}&{-u_{1}u_{2}^{2}}\\{-d_{2}^{3}}&{u_{2}d_{2}^{2}}&{u_{2}^{2}d_{2}}&{-u_{2}^{3}}\\\end{array}\right)
\frac{R^{\prime\prime}}{R}=\frac{1}{R}\frac{\partial}{\partial\tilde{U}}\left(\frac{\partial}{\partial\tilde{U}}R\right)=\frac{G^{2}}{RW^{2}}\frac{\partial}{\partialu}\left(\frac{G^{2}}{W^{2}}\frac{\partial}{\partialu}R\right).
M_{\ellK}=U^{T}M_{K}U=\left(\begin{array}{cccc}{\frac{r_{0}(1-f)}{2Mf}}&{-\frac{1}{f}}&{0}&{\frac{r_{0}\chi}{2Mf}}\\{-\frac{1}{f}}&{\frac{2M}{r_{0}f}}&{0}&{-\frac{\chi}{f}}\\{0}&{0}&{-\frac{2M}{r_{0}}}&{-1}\\{\frac{r_{0}\chi}{2Mf}}&{-\frac{\chi}{f}}&{-1}&{\frac{{r_{0}}^{2}\chi}{4M^{2}f}(\chi^{2}+f^{2}-f)}\\\end{array}\right)\,.
{\frac{1}{N}}\left(\begin{array}{ccccc}{N-1}&{-1}&{\cdots}&{\cdots}&{-1}\\{-1}&{N-1}&{}&{}&{\vdots}\\{\vdots}&{}&{\ddots}&{}&{\vdots}\\{\vdots}&{}&{}&{\ddots}&{\vdots}\\{-1}&{\cdots}&{\cdots}&{\cdots}&{N-1}\\\end{array}\right).
\Phi^{+}(E_{7})=\mathbb{D}_{1}^{+}\oplus\mathbb{D}_{2}^{+}\oplus\mathbb{D}_{3}^{+}\oplus\mathbb{D}_{4}^{+}\oplus\mathbb{D}_{5}^{+}\oplus\mathbb{D}_{6}^{+}
{\frac{\alpha^{2}}{2m}}\sum_{i=1}^{N}\left\{2\sum_{i<j}^{N}{\frac{1}{r_{ij}^{2}}}+\sum_{i\nek,l(k\nel)}^{N}{\frac{(\mathrm{\bfk}\times\mathrm{\bfr}_{ik})\cdot(\mathrm{\bfk}\times\mathrm{\bfr}_{il})}{r_{ik}^{2}r_{il}^{2}}}\right\}
H_{\mathrm{LC}}\;\psi=\left(\begin{array}{cc}{H_{NN}}&{H_{NP}}\\{H_{PN}}&{H_{PP}}\\\end{array}\right)\left(\begin{array}{c}{\psi_{\gamma}}\\{\psi_{e\bar{e}}}\\\end{array}\right)=\omega\left(\begin{array}{c}{\psi_{\gamma}}\\{\psi_{e\bar{e}}}\\\end{array}\right).
Z_{2}^{\alpha}\left(\frac{\mu}{M_{s}}\right)=\int_{\calF}\frac{d^{2}\tau}{\Im\tau}\Gamma\left(\frac{\mu}{M_{s}}\right)\sum_{i=1}^{3}\frac{\Gamma_{2,2}(T_{i},U_{i})}{\overline{{\eta}}^{24}}\left[{\overline{{Q}}}_{\alpha}^{2}-\frac{k_{\alpha}}{4\pi\Im\tau}\right]\overline{{\Omega}}\.
A_{\mu}=A_{i\mu}m_{i}^{a}+\frac1gf^{abc}\partial_{\mu}m_{i}^{b}m_{i}^{c}+\frac1g\rho^{ij}f^{abc}\partial_{\mu}m_{i}^{b}m_{j}^{c}+\frac1g\sigma^{ij}d^{abc}\partial_{\mu}m_{i}^{b}m_{j}^{c}
(\hat{r}^{2}-1)\frac{d^{2}T_{EN}}{d\hat{r}^{2}}+(3\hat{r}-\frac{1}{\hat{r}})\frac{dT_{EN}}{d\hat{r}}+(\frac{E^{2}}{\hat{r}^{2}-1}-\frac{N^{2}}{\hat{r}^{2}}-4\lambda^{2}-1)T_{EN}=0.
F({\bfu},{\bfv})=\frac{ieB}{2}\,\epsilon_{ij}u^{i}v^{j}+\frac{1}{4s}\,({\bfu}+{\bfv})^{2}+\frac{eB}{2}\,({\bfu}^{2}+{\bfu}\cdot{\bfv}+{\bfv}^{2})\;,
\chi_{\alpha}(x)=\sum_{\pms}\frac{1}{\sqrt{L}}\sum_{k_{3}}\int\frac{d^{2}k}{2\pi}\sqrt{\frac{m}{E_{k}}}\left[b(k,s)u(k,s)e^{-ikx}+d^{\dagger}(k,s)v(k,s)e^{ikx}\right].
{\mathcalD}^{(1/2,1/2)}\otimes{\mathcalD}^{(1/2,1/2)}={\mathcalD}^{(1,1)}\oplus{\mathcalD}^{(1,0)}\oplus{\mathcalD}^{(0,1)}\oplus{\mathcalD}^{(0,0)}\,.
h_{E}=\frac{p_{\xi}^{2}}{2m}+\frac{p_{\eta}^{2}}{2m}+\frac{p_{\phi}^{2}}{2m}\left(\frac{1}{\xi^{2}}+\frac{1}{\eta^{2}}\right)-E(\xi^{2}+\eta^{2})-2e^{2}-\frac{\hbar^{2}}{8m}\left(\frac{1}{\xi^{2}}+\frac{1}{\eta^{2}}\right)
\frac{d}{dt}S_{WZ}(t)=i^{[\frac{p+1}{2}]}\,\frac{2\,i}{p!}\int\phi^{*}\,\Pi^{a_{1}}(t)\wedge\cdots\wedge\Pi^{a_{p}}(t)\wedge\left(\bar{\nu}(t)\Gamma_{a_{1}}\cdots\Gamma_{a_{p}}\,t\,d\theta\right)
a=\frac{y-q^{2}x}{y-x}\,,~~~a^{\prime}=\frac{y-q^{-2}x}{y-x}\,,~~~b=\frac{y-q^{4}x}{y-q^{2}x}\,,~~~b^{\prime}=\frac{y-q^{-4}x}{y-q^{-2}x}
\int_{-\infty}^{\infty}d\theta\;\Omega_{i}(\theta)\,e^{it\theta}=2\pi\sum_{m=0}^{\infty}(-i)^{m}\nu_{i}^{(m)}t^{m}=\pi\sum_{m=0}^{\infty}\frac{E_{2m}}{(2m)!}(\pi\omega_{i})^{2m}\,t^{2m}\,\,.
G_{d}(x;-m^{2})=\frac{1}{(2\pi)^{d-1}}\biggl[\int_{p_{\perp}^{2}>m^{2}}d^{d-1}p\,\,G_{1}(x;p_{\perp}^{2}-m^{2})+\int_{p_{\perp}^{2}<m^{2}}d^{d-1}p\,\,G_{1}(x;-m^{2}+p_{\perp}^{2})\biggr]\quad,
{\frac{\ddot{\hat{c}}}{\hat{c}}}+3{\frac{\dot{\hat{a}}_{0}}{\hat{a}_{0}}}{\frac{\dot{\hat{c}}}{\hat{c}}}=-{\frac{2}{\omega}}\zeta\left({\frac{\hat{c}}{c}}\right)^{2}\hat{\varrho}_{m}.
\left(\begin{matrix}{P^{I}}\\{iQ_{I}}\\\end{matrix}\right)\rightarrow\Gamma\left(\begin{matrix}{P^{I}}\\{iQ_{I}}\\\end{matrix}\right)=\left(\begin{matrix}{U}&{Z}\\{W}&{V}\\\end{matrix}\right)\left(\begin{matrix}{P^{I}}\\{iQ_{I}}\\\end{matrix}\right),
J_{H}^{-1}\longrightarrow\gamma\cdotJ_{H}^{-1}={\frac{aJ_{H}^{-1}+b}{cJ_{H}^{-1}+d}},\quad\left(\begin{array}{c}{a}\\{c}\\\end{array}\begin{array}{cc}{b}\\{d}\\\end{array}\right)\in\Gamma.
\sigma_{\lambda}={\frac{e^{2}}{2\pi^{2}}}\;{\frac{1}{\omega_{k}^{3}}}\;\int_{M}^{\frac{\omega_{k}}{2}}dx\;{\frac{1}{x^{3}\sqrt{x^{2}-M^{2}}}}\int_{0}^{\omega_{k}-2x}d\varepsilon{\frac{(\omega_{k}^{2}-\varepsilon^{2}+4x^{2})\;C_{\lambda}(\varepsilon,x)}{\sqrt{(\omega_{k}^{2}-\varepsilon^{2})^{2}-8x^{2}(\omega_{k}^{2}+x^{2})+16x^{4}}}}.
M_{x}^{2}=\sum_{i=1}^{N_{c}}\mu_{Li(Ri)}^{2}\big|\frac{\pi_{x}}{\Lambda^{N_{c}-1}}\big|^{2}+G_{B}^{\prime}\big|\pi_{x}\big|^{2}\sum_{y=b,{\bar{b}}}\big|W_{\mathrm{eff};y}\big|^{2}.
{\calL}=-\,\frac{1}{4}\,F_{\mu\nu}\,F^{\mu\nu}+\,\frac{1}{12}\,H_{\mu\nu\rho}\,H^{\mu\nu\rho}\,+{\frac{m}{2}}\,\epsilon^{\mu\nu\rho\lambda}\,B_{\mu\nu}F_{\rho\lambda}\,,
\begin{array}{rrl}{{\calE}(A,B)=\int_{M_{3}}<(I_{E}\otimesK^{B})\wedge(I_{E}\otimesD^{B}\Phi_{B})>}&{-}&{<(I_{E}\otimesK^{B})\wedge(D^{A}\Phi_{A}\otimesI_{E})>}\\{-<(H^{A}\otimesI_{E})}&{\wedge}&{(I_{E}\otimesD^{B}\Phi_{B})>,}\\\end{array}
[Q_{+{\frac{1}{2}}}\,\{Q_{+{\frac{1}{2}}}\,Q_{+{\frac{1}{2}}}\}]=0\qquad\Longrightarrow\qquad\left[\left({\calM}_{nearhor}^{2}\right)_{\beta_{+}}^{\alpha_{+}}\right]_{g.f.}=-\theta^{\gamma_{+}}(f_{\gamma_{+}A}^{\alpha})_{sub}\;\theta^{\delta_{+}}(f_{\delta_{+}\beta_{+}}^{A})_{sub}=0.
P_{N}^{+}={\frac{\pi}{L}}\sum_{n}n(a{_n^{\dagger}}a_{n}+b{_n^{\dagger}}b_{n}+d{_n^{\dagger}}d_{n})+{\frac{\pi}{L}}\int_{-L}^{L}dx^{+}Z_{R}(x^{+})\tilde{J}_{3}^{L}(x^{+}).
\begin{matrix}{r^{p-1}Y_{p-1}(\hat{\phi})F,}&{r^{p-2}Y_{p-2}(\hat{\phi})F^{2},}&{r^{p-2}Y_{p-2}(\hat{\phi})F\bar{F},}\\{r^{p-3}Y_{p-3}(\hat{\phi})F^{2}\bar{F},}&{r^{p-4}Y_{p-4}(\hat{\phi})F^{2}\bar{F}^{2}}&{}\\\end{matrix}
T^{AB}(\tau,\vec{\sigma})=-[{\frac{2}{\sqrt{g}}}{\frac{\deltaS}{\deltag_{AB}}}](\tau,\vec{\sigma})=-\sum_{i=1}^{N}\delta^{3}(\vec{\sigma}-{\vec{\eta}}_{i}(\tau)){\frac{m_{i}{\dot{\eta}}_{i}^{A}(\tau){\dot{\eta}}_{i}^{B}(\tau)}{\sqrt{g_{\tau\tau}+2g_{\tauu}{\dot{\eta}}_{i}^{u}+g_{uv}{\dot{\eta}}_{i}^{u}{\dot{\eta}}_{i}^{v}}}}(\tau,\vec{\sigma}).