harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
G_{\mathrm{rad}}^{a}=-\vec{\nabla}\vec{E}^{a}+g\rho_{\mathrm{rad}}^{a}\,\qquad\\\rho_{\mathrm{rad}}^{a}=-g\epsilon^{abc}\vec{A}^{\,b}\vec{E}^{c}\,\qquad\\\rho_{\mathrm{matt}}^{a}=\epsilon^{abc}\phi^{b}\pi^{c}\,
l^{j}-\theta^{ji}\hat{F}_{ii^{\prime}}l^{i^{\prime}}=i\theta^{jj^{\prime}}(ik_{j^{\prime}}+i\partial_{j^{\prime}}\hat{A}_{i^{\prime}}l^{i^{\prime}})+\theta^{jj^{\prime}}l^{i^{\prime}}D_{i^{\prime}}\hat{A}_{j^{\prime}},
G_{H_{3}}(x,x^{\prime})={\frac{1}{4\pi}}\left({\frac{1}{|\Sigma|}}-{\frac{1}{|\hat{\Sigma}|}}\right)={\frac{1}{4\pi}}\left({\frac{1}{|X-X^{\prime}|}}-{\frac{1}{|X+X^{\prime}|}}\right)~~.
S=\frac{1}{16\piG_{D+1}}\intd^{D+1}\hat{x}\sqrt{\hat{g}}\left(\hat{R}+\Upsilon(\hat{R}_{MNPQ}\hat{R}^{MNPQ}-4\hat{R}_{MN}\hat{R}^{MN}+b\hat{R}^{2})\right).
S_{a}=\intd^{4}x\,\widetilde{g}\,\bar{\psi}_{a}{}^{\dagger}\left(\bar{m}^{-1}\widetilde{g}^{\mu\nu}D_{\mu}D_{\nu}+ie_{\alpha}^{\mu}\sigma^{\alpha}D_{\mu}\right)\bar{\psi}_{a}
D_{i}=\Gamma(m-{\frac{D}{2}})\int_{0}^{1}du_{i_{1}}\int_{0}^{u_{i_{1}}}du_{i_{2}}\cdots\int_{0}^{u_{i_{m-2}}}du_{i_{m-1}}{\frac{P_{i}(u_{i_{1}},\ldots,u_{i_{m}})}{{\bigl[-\sum_{r<s}^{m}G_{Bi_{r}i_{s}}K_{i_{r}}\cdotK_{i_{s}}\bigr]}^{m-{\frac{D}{2}}}}}
A=\left|\frac{F_{(\gamma)}^{\mathrm{in}}(r=r_{H})}{F_{(\gamma)}^{\mathrm{in}}(r=\infty)}\right|,~~~~~~R=\left|\frac{F_{(\gamma)}^{\mathrm{out}}(r=\infty)}{F_{(\gamma)}^{\mathrm{in}}(r=\infty)}\right|,
\begin{array}{l}{w_{2i+1}=x_{i}^{-1}x_{i+1},~~i=0,1,2,\cdots,N-1,}\\{w_{2j}=y_{j},~~j=1,2,\cdots,N-1,}\\{w_{2N}=\prod_{l=1}^{N-1}y_{l}^{-1}.}\\\end{array}
{\calM}_{g^{4}(c)}(\theta)=\frac{-2g^{4}}{\Theta^{2}}[\varepsilon_{bac}T^{b}\otimesT^{c}T^{a}]\int\frac{d^{2}{\bfk}}{(2\pi)^{2}}\frac{[{\bfq}\wedge{\bfk}-{\bfp}_{1}\wedge{\bfp}_{3}]}{[({\bfk}-{\bfp}_{1})^{2}][({\bfk}-{\bfp}_{3})^{2}][{\bfq}^{2}]}T_{0}\;,
H=-\nabla_{1}^{2}+\omega^{2}r_{1}^{2}-\nabla_{2}^{2}+\omega^{2}r_{2}^{2}+{\frac{2\beta}{|{\bfr}_{1}-{\bfr}_{2}|}}+\lambda|{\bfr}_{1}-{\bfr}_{2}|
{\calC}={\frac{1}{M}}\left({\calA}(\Phi_{\bfm},\Phi_{-{\bfm}},V_{0})+{\calA}(\Phi_{\bfm}^{\prime},\Phi_{-{\bfm}}^{\prime},V_{0})\right)={\frac{2}{M}}{\calA}(\Phi_{\bfm},\Phi_{-{\bfm}},V_{0})~.
I_{\varepsilon}^{1}\left(a\right)=\sum_{\nu_{1},\nu_{2}\in{\calN}_{\varepsilon}^{+}}\frac{1}{\rho\left(\nu_{1}\right)\rho\left(\nu_{2}\right)}\left[{\LARGEi}\right]+\sum_{\nu_{1},\nu_{2}\in{\calN}_{\varepsilon}^{+}}\frac{1}{\rho\left(\nu_{1}\right)\rho\left(\nu_{2}\right)}\left[{\Largeii}\right],
e^{i\alphaQ^{5}}V_{\eta}^{1/2}(x,\hat{n})V_{\eta}^{*1/2}(x,-\hat{n})e^{-i\alphaQ^{5}}=V_{\eta}^{1/2}(x,\hat{n})V_{\eta}^{*1/2}(x,-\hat{n})e^{i\frac{e}{2\pi}\alpha\phi_{\hat{n}}(x)}
\sum_{i<j}p_{i}\timesp_{j}(\tau_{i}-\tau_{j})=\frac{1}{2}\sum_{i,j}p_{i}\timesp_{j}(\tau_{i}-\tau_{j})=-\sum_{i,j}p_{j}\timesp_{i}~\tau_{i}=\sum_{i}k\timesp_{i}~\tau_{i}
\widetilde{H}_{c}^{\prime}=\frac{1}{4}(B_{ij}-Q_{ij})^{2}-\frac{1}{2}(A_{i}+\partial_{i}\theta)^{2}+\frac{1}{4m^{2}}F_{ij}F^{ij}-\frac{1}{2\cdot3!m^{2}}H_{ijk}H^{ijk}.
\rho^{D}(\alpha)=\frac{(-1)^{{\scriptstyle\frac{D}{2}}}\Gamma({\textstyle\frac{D}{2}}-1)}{4(D-2)!\\pi^{{\scriptstyle\frac{D}{2}}}}B_{D-2}(\alpha)\qquad\mathrm{for~0\leq\alpha\leq~1~,}
\begin{array}{l}{D_{\mu}^{ij}[A]\frac{\deltaW}{\deltaA_{\mu}^{j}}+g(f^{bim}\tilde{Y}_{\mu\nu}^{ma}-f^{nia}\tilde{Y}_{\mu\nu}^{bn})\frac{\deltaW}{\delta\tilde{Y}_{\mu\nu}^{ba}}}\\{+gf^{ijk}(J_{\mu}^{j}\frac{\deltaW}{\deltaJ_{\mu}^{k}}+\xi^{j}\frac{\deltaW}{\delta\xi^{k}}+\overline{{\xi}}^{j}\frac{\deltaW}{\delta\overline{{\xi}}^{k}})=0}\\\end{array}
\rho=\sum_{a^{\prime}}\left[\frac{i\omega_{a^{\prime}}}{\sqrt{\Im_{a^{\prime}}}}\lambda_{a^{\prime}}\pi_{a^{\prime}}+\left(\omega_{a^{\prime}}^{2}G_{a^{\prime}}+\frac{i}{2}b_{a^{\prime}}\right)\bar{\eta}_{a^{\prime}}\right],
\frac{V_{i}^{\prime\prime}(0)}{(V_{i}^{\prime}(0))^{2}}=2\frac{\left(z_{i}-z_{i+1}\right)-\ell\left(z_{i+1}-z_{i-1}\right)}{\left(z_{i}-z_{i+1}\right)\left(z_{i}-z_{i-1}\right)}
\sum_{i_{1},...i_{l_{1}};j_{1},...,j_{l_{2}}=1}^{J}\mathrm{tr}((Z\Omega)^{i_{1}}(Z^{\prime}\Omega)(Z\Omega)^{j_{1}-i_{1}}(W\Omega)....(W\Omega)^{j_{l_{2}}}(Z\Omega)^{J-j_{l_{2}}})
{\calL}=\overline{{\psi}}\mathrm{}_{\alpha}^{i}\left(i\gamma_{\alpha\beta}^{\mu}{\calD}_{\mu}-m\delta_{\alpha\beta}\right)\psi_{\beta}^{i}+\chi\overline{{\psi}}\mathrm{}_{\alpha}^{i}\phi^{a}\left(T^{a}\right)_{ij}\psi_{\alpha}^{j}
S_{\xi}=\intd^{4}x\sqrt{-g}\left[-{\frac{1}{2}}|\xi|\hat{\phi}^{2}\tilde{R}-\frac{|\xi|}{\xi}\left({\frac{1}{2}}{\calD}_{\mu}\hat{\phi}{\calD}^{\mu}\hat{\phi}+V(\hat{\phi})\right)-{\frac{1}{4}}H_{\mu\nu}H^{\mu\nu}\right],
[\Lambda_{i\alpha},\Lambda^{j\beta}]=(t_{i}{}^{j}\delta_{\alpha}{}^{\beta}+\delta_{i}{}^{j}t_{\alpha}{}^{\beta}),\qquad[\Lambda_{\alpha\dot{\alpha}},\Lambda^{\beta\dot{\beta}}]=-(t_{\alpha}{}^{\beta}\delta_{\dot{\alpha}}{}^{\dot{\beta}}+\delta_{\alpha}{}^{\beta}t_{\dot{\alpha}}{}^{\dot{\beta}}),\nonumber
D_{T}^{[r,0]}(k^{2})=\frac{\rho_{r,0}}{k^{2}+\sigma_{r,1}\Lambda^{2}}+\frac{\rho_{r,2}}{k^{2}+\sigma_{r,3}\Lambda^{2}}+\sum_{s=1}^{(r-1)/2}\left[\frac{\rho_{r,4s}}{k^{2}+\sigma_{r,4s+1}\Lambda^{2}}+\frac{\rho_{r,4s+2}}{k^{2}+\sigma_{r,4s+3}\Lambda^{2}}\right]\,,
\frac{\partial}{\partial\theta}\Psi_{j}(\bar{x})\left|_{\Psi_{j}(\bar{x})}\equiv0=\intd^{4}x\frac{\partial}{\partial\theta}S_{k}(\bar{y})|_{\Psi_{j}}\frac{{\delta^{\prime}}^{2}\bar{Z}[S]}{\deltaS_{k}(\bar{y})\deltaS_{j}(\bar{x})}\right.
\Pi^{\mu\nu}(q)=\frac{ig^{2}}{4\pi}\mathrm{tr}[T^{a}T^{b}]\left[\left(g^{\mu\nu}-\frac{q^{\mu}q^{\nu}}{q^{2}}\right)\Pi_{1}(q^{2})+im\varepsilon^{\mu\nu\lambda}q_{\lambda}\Pi_{2}(q^{2})\right]\;,
e_{4}(k,N)={\frac{8N^{4}}{C_{(1^{k};0)}}}\left[{\frac{1}{2C_{(1;0)}}}+\sum_{i=1}^{4}{\frac{1}{C_{(1^{k};0)}-3C_{(1;0)}-C_{r_{i}}}}{\frac{d_{r_{i}}}{Nd_{(1^{k};0)}}}\right].
\vec{\calP}\equiv\vec{\calQ}\equiv0\;\;\Rightarrow\;\;\Re^{\prime\,r}(\tau,\vec{\sigma}_{o})\equiv\Re^{r}(\tau,\vec{\sigma}_{o}),\;\;\wp^{\prime\,s}(\tau,\vec{\sigma}_{o})\equiv\wp^{s}(\tau,\vec{\sigma}_{o}).
{\vec{Z}}\rightarrow{\vec{Z}}^{N}\equiv\left(\begin{matrix}{z_{(1,1,1)}^{0}}\\{z_{(1,1,15)}^{1}}\\{z_{(1,1,15)}^{2}}\\{z_{(1,1,15)}^{3}}\\{0}\\{\dots}\\{0}\\{{\bar{z}}_{{\bar{(}1,1,1)}}^{0}}\\{{\bar{z}}_{{\bar{(}1,1,15)}}^{1}}\\{{\bar{z}}_{{\bar{(}1,1,15)}}^{2}}\\{{\bar{z}}_{{\bar{(}1,1,15)}}^{3}}\\{0}\\{\dots}\\{0}\\\end{matrix}\right)
{\calA}=\sum_{e_{1},e_{2}\inQ}VT{\frac{\beta_{1}+\beta_{2}}{2(2\pi)^{26}\epsilon}}\sum_{k=1}^{\infty}(-1)^{k+1}e^{-\pi|\epsilon|k}(|\epsilon|/k)^{13}\eta(ik/|\epsilon|)^{-24}.
\frac{1}{\alphak_{0}^{2}-|\mathbf{k}|^{2}-m^{2}+i\varepsilon}+\frac{1}{\alphak_{0}^{2}-|\mathbf{k}|^{2}-m^{2}+i\varepsilon}\bullet\frac{-\delta\alphak_{0}^{2}}{\alphak_{0}^{2}-|\mathbf{k}|^{2}-m^{2}+i\varepsilon}+......
\tilde{T}(\tilde{u},\tilde{v})=(n(\tilde{u})\;\;n(\tilde{v}))^{-1/2}\;\;\sum_{u\in\tilde{u}}\sum_{v\in\tilde{v}}T(u,v)\;\;=(n(\tilde{u})/n(\tilde{v}))^{1/2}\sum_{v\in\tilde{v}}T(u,v)\;\;.
\begin{array}{rcl}{{}^{\star}F_{tr}^{(n)}}&{\sim}&{i\frac{\tilde{p}^{(n)}}{r^{2}}\,,}\\{}&{}&{}\\{\left(e^{-2\phi}F-ia{}^{\star}F\right)_{tr}^{(n)}}&{\sim}&{\frac{\tilde{q}^{(n)}}{r^{2}}\,,}\\\end{array}
\prod_{i=3}^{N-1}(\intdz_{i}\ointdz_{i}^{\prime})\langle{\calO}^{1}(z_{1}){\calO}^{2}(z_{2})b(z_{3}^{\prime}){\calO}^{3}(z_{3})\cdotsb(z_{N-1}^{\prime}){\calO}^{N-1}(z_{n-1}){\calO}^{N}(z_{N})\rangle,
\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{1}\\\end{array}\right),\quad\left(\begin{array}{ccc}{0}&{0}&{1}\\{1}&{0}&{0}\\{0}&{1}&{0}\\\end{array}\right),\quad\left(\begin{array}{ccc}{0}&{1}&{0}\\{0}&{0}&{1}\\{1}&{0}&{0}\\\end{array}\right)
\begin{array}{cc}{\tilde{t}=\frac{1}{2\lambda}[k_{1}(t+w)+(x^{2}+\lambda^{2})h(t-w)],}\\{\tilde{w}=\frac{1}{2\lambda}[k_{1}(t+w)+(x^{2}-\lambda^{2})h(t-w)],}\\{\tilde{x}^{k}=h(t-w)x^{k},}\\\end{array}
{M(j)_{\rho}}^{\sigma}=4g^{2}\,\left\{\frac{2}{3}\,{j^{a}}_{b}\,{j^{c}}_{d}+2\,{j^{a}}_{d}\,{j^{c}}_{b}\right\}\,{(e_{\rho})_{a}}^{b}\,{(e^{\sigma})_{c}}^{d},
\int\sqrt{g}~\widehat{R}^{\mu\nu\lambda\rho}\widehat{R}_{\mu\nu\lambda\rho},\qquad\int\sqrt{g}~\widehat{R}^{\mu\nu}\widehat{R}_{\mu\nu},\qquad\int\sqrt{g}~\widehat{R}^{2}.
\mathrm{d}\mu_{\,\mathrm{SU}(2)}=\frac{1}{256\pi^{2}}\,e^{-8\pi^{2}/{g^{2}}}\,(M_{\mathrm{PV}})^{6}\,\left(\frac{8\pi^{2}}{g^{2}}\right)^{2}\mathrm{d}^{4}x_{0}\,\mathrm{d}^{2}\theta_{0}\,\mathrm{d}\rho^{2}\,\mathrm{d}^{2}\bar{\beta}\,.
\{x^{\mu},x^{\nu}\}_{DB}=-\frac{S^{\mu\nu}}{M^{2}}~,~\{P^{\mu},x^{\nu}\}_{DB}=g^{\mu\nu}~,~\{P^{\mu},P^{\nu}\}_{DB}=\{P^{\mu},\Lambda^{\nu\lambda}\}_{DB}=\{P^{\mu},S^{\nu\lambda}\}_{DB}=0~,
H=\frac{1}{4\pi^{2}g}\intdzd\theta\bigg[(R^{2}+D_{E}^{2})\sqrt{\frac{(1+{R^{\prime}}^{2})}{(D_{E}^{2}+R^{2}-R^{2}D^{2})}}-\frac{h}{2}R^{2}\bigg].
\varrho_{\Phi}=\left({\frac{1}{2}}{\frac{\dot{\Phi}^{2}}{c^{2}}}+V\right){\frac{1}{c^{2}}},\\\\\\wp_{\Phi}={\frac{1}{2}}{\frac{\dot{\Phi}^{2}}{c^{2}}}-V,\\\\\I\equiv1+B{\frac{\dot{\Phi}^{2}}{c^{2}}}.
\begin{array}{l}{\displaystyle{\Upsilon}_{{[p,q]}}={\mathrm{sTr}}(\omegaF_{L}^{p}({g^{-1}F_{R}g})^{q}(\omega^{2})^{m-p-q-1})\quad,}\\{\displaystyle{\Pi}_{{[p,q]}}={\mathrm{sTr}}(F_{L}^{p}({g^{-1}F_{R}g})^{q}(\omega^{2})^{m-p-q})\quad,}\\{\displaystyle{\Gamma}_{{[p,q+1]}}={\mathrm{sTr}}(\omegaF_{L}^{p}({g^{-1}F_{R}g})^{q}[{g^{-1}F_{R}g},\omega](\omega^{2})^{m-p-q-2})\quad.}\\\end{array}
\begin{array}{lcr}{U^{+{\frac{n(n+1)}{2}}}=\prod_{j=0}^{n}(\varphi_{j}^{+})^{n-j}}\\{V^{+{\frac{n(n+1)}{2}}}=\prod_{j=0}^{n}(\varphi_{j}^{+})^{j}}\\{Z^{+{(n+1)}}=\prod_{j=0}^{n}(\varphi_{j}^{+}).}\\\end{array}
R_{MN}-\frac{1}{2}G_{MN}R=\frac{1}{4M^{3}}\left\{\partial_{M}\phi\partial_{N}\phi+\partial_{M}\pi\partial_{N}\pi-G_{MN}\left(\frac{1}{2}(\partial\phi)^{2}+\frac{1}{2}(\partial\pi)^{2}+V\right)\right\}\,,
\phi(x,y;P)=-~ie^{2}\intd^{4}x^{\prime}\intd^{4}y^{\prime}G_{A}(x,x^{\prime})\gamma^{\mu}\phi(x^{\prime},y^{\prime};P)\gamma^{\nu}G_{A}(y^{\prime},y)D_{\mu\nu}(y^{\prime}-x^{\prime}).
\bar{c}_{4}=\sqrt{c^{2}+R_{\tau}^{2}},\\\\\\\\\\\bar{G}_{4}={\frac{m_{0}e^{2m_{0}R_{0}}G_{5}(1+R_{\tau}^{2}/c^{2})^{3/2}}{1-\Omega_{R}^{2}}},
\zeta_{\lambda_{j,N}}=\sum_{j=-\infty}^{\infty}\sum_{N=0}^{\infty}{\frac{(2N+2)^{2}}{\left[\left({\frac{2\pij}{\beta}}+i\mu\right)^{2}+{\frac{(2N+1)(2N+3)}{a^{2}}}+{\frac{6\xi}{a^{2}}}+m^{2}\right]^{s}}}.
r\equiv\intd{\tau}_{1}d{\tau}_{2}d{\tau}_{3}\delta({\tau}_{1})\delta({\tau}_{2})\delta({\tau}_{3})\frac{|{\tau}_{3}-{\tau}_{1}|+|{\tau}_{3}-{\tau}_{2}|}{|{\tau}_{3}-{\tau}_{1}|+|{\tau}_{2}-{\tau}_{1}|+|{\tau}_{3}-{\tau}_{2}|}\;
I=\int_{\cal{M}}d^{4}\!x\sqrt{g}\left(R(g)-{\frac{1}{2}}\left\{\left(\nabla\Phi\right)^{2}-e^{-2\Phi}\left(\nabla\Psi\right)^{2}\right\}-\left(\nablaT\right)^{2}+e^{-\Phi}V(T)\right)\.
{\calA}_{0}\left({t,x}\right)=\int_{0}^{1}{d\alphax^{1}E_{1}^{L}\left({t,\alphax}\right)}=\int_{0}^{1}{d\alphax^{1}\left({1-\frac{{g^{2}}}{\pi}\frac{a}{{a-1}}\frac{1}{{\partial_{1}^{2}}}}\right)}_{\alphax}^{-1}\frac{{\partial_{1}^{\alphax}\left({-J^{0}\left({\alphax}\right)}\right)}}{{\partial_{\alphax}^{2}}},
\delta_{\varepsilon}\Phi_{a_{0}}^{0}=\varepsilon^{i}{(C_{i}^{0})_{a_{0}}}^{a_{1}}\Phi_{a_{1}}^{1},\quad\delta_{\varepsilon}\Phi_{a_{M}}^{M}=\varepsilon^{i}{(\tilde{C}_{i}^{M})_{a_{M}}}^{a_{M-1}}\frac{d}{d\tau}{\Phi}_{a_{M-1}}^{M-1}.
{\frac{1}{(2\pi\tau)^{d/2}\sqrt{\mathrm{det}(\nu)}}}\mathrm{exp}(-\tau[{\frac{1}{2}}({\frac{z_{j}}{\tau}}-\nu_{j})\nu_{jk}^{-1}({\frac{z_{k}}{\tau}}-\nu_{k})+\nu_{o}])
\begin{array}{c}{6\beta^{\prime}v^{2}v^{\prime}+2\beta^{\prime}(D-1)v^{3}+2\beta^{\prime}v^{\prime}+j^{\prime}(v,v^{\prime})=\alpha^{\prime}}\\{v^{\prime}=I_{1}(\alpha^{\prime})/I_{0}(\alpha^{\prime})}\\\end{array}
\sum_{n=0}^{\infty}\frac{n!}{(n+\ell)!}(-1)^{n}L_{n}^{\ell}(x)L_{n}^{\ell}(y)\,t^{n}=\frac{(xyt)^{-\ell/2}}{1+t}e^{t(x+y)/(1+t)}J_{\ell}(2\sqrt{xyt}/(1+t))\,,
S(q)\equiv\frac{\Omega(q)}{\Omega_{0}(q)}=\frac1{N!}\sum_{PP^{\prime}}\frac{\Omega_{0}(\frac{q_{P}+q_{P^{\prime}}}2)}{\Omega_{0}(q_{P})}\hat{w}(\frac{q_{P}+q_{P^{\prime}}}2,q_{P}-q_{P^{\prime}})
{\calD}_{x^{2\xi}}^{2}V_{1,0}(w)={\frac{1}{(x^{\xi}-x^{-\xi})^{2}}}\left(\oint_{w}z^{-3}f_{1,0}\left({\frac{w}{z}}\right)T(z)V_{1,0}(w)dz-(x^{\xi}-x^{-\xi})w^{-2}V_{1,0}(w)\right).
\Omega_{r}^{(2)}={\frac{1}{4\pi^{2}}}\int_{C_{r}}{\frac{3\rho^{2}+(x_{0}^{1})^{2}+(x_{0}^{2})^{2}+({\bfx}-{\bfx}_{0})^{2}}{[\rho^{2}+(x_{0}^{1})^{2}+(x_{0}^{2})^{2}+({\bfx}-{\bfx}_{0})^{2}]^{3}}}(x-x_{0})^{\mu}dx^{\nu}\varepsilon_{\mu\nu\rho\sigma}dx_{0}^{\rho}dx_{0}^{\sigma},
\psi_{a_{1}\cdotsa_{2k-2}}\;=\;F_{\beta}^{-1}\:(N^{\dagger}\:N)^{-1}\:N_{a}^{(\beta)\:\dagger}\:D_{b}^{(\beta)}\:F_{\beta}\:\psi_{a_{1}\cdotsa_{2k-2}ab}\:+\:\psi_{a_{1}\cdotsa_{2k-2}}^{[h]}
\bigg|\frac{\pi_{a}}{\pi_{1}}\bigg|^{2}=1+\frac{(M^{2}/\Lambda^{2})(\big|\pi_{1}\big|^{2}-\big|\pi_{a}\big|^{2})}{G_{S}W_{\mathrm{eff};\lambda}^{\ast}\left(1-\alpha_{B}\right)+(M^{2}/\Lambda^{2})\big|\pi_{a}\big|^{2}+\beta_{B}X_{B}},
V_{[N]\times\underline{{[N]}}}\bar{V}_{\underline{{[N]}}\times[N]}=1_{[N]\times[N]}-u_{[N]\times\underline{{[k]}}\times\underline{{[2]}}}f_{\underline{{[k]}}\times\underline{{[k]}}}\bar{u}_{\underline{{[2]}}\times\underline{{[k]}}\times[N]}.
W_{q}^{2}=q^{2}+V^{\prime\prime}+{\frac{5}{16}}\left[\left({\frac{V^{\prime\prime\prime}}{q^{2}+V^{\prime\prime}}}\right)^{2}-{\frac{V^{\prime\prime\prime\prime}}{4(q^{2}+V^{\prime\prime})}}\right]\dot{\phi}^{2}-{\frac{V^{\prime\prime\prime}}{4(q^{2}+V^{\prime\prime})}}\ddot{\phi}+\ldots,
S^{+}=\left(\begin{array}{cccc}{0}&{a_{1}}&{0}&{0}\\{0}&{0}&{a_{2}}&{0}\\{0}&{0}&{0}&{a_{1}}\\{0}&{0}&{0}&{0}\\\end{array}\right),\quadS^{-}=(S^{+})^{t}\,,\quadS^{3}=\left(\begin{array}{cccc}{\frac32}&{0}&{0}&{0}\\{0}&{\frac12}&{0}&{0}\\{0}&{0}&{-\frac12}&{0}\\{0}&{0}&{0}&{-\frac32}\\\end{array}\right),
\alpha_{3}-i\beta_{3}=-\frac{1}{2}\gamma_{2}^{2};\;\alpha_{4}-i\beta_{4}=-\gamma_{2}\gamma_{3};\;\alpha_{5}-i\beta_{5}=-\gamma_{2}\gamma_{4}-\frac{1}{2}(\alpha_{3}^{2}+\beta_{3}^{2}+\gamma_{3}^{2}),\dots
L_{x}=\left(\begin{matrix}{x_{0}}&{-x_{1}}&{-x_{2}}&{-x_{3}}\\{x_{1}}&{x_{0}}&{-x_{3}}&{x_{2}}\\{x_{2}}&{x_{3}}&{x_{0}}&{-x_{1}}\\{x_{3}}&{-x_{2}}&{x_{1}}&{x_{0}}\\\end{matrix}\right)\qquadR_{x}=\left(\begin{matrix}{x_{0}}&{-x_{1}}&{-x_{2}}&{-x_{3}}\\{x_{1}}&{x_{0}}&{x_{3}}&{-x_{2}}\\{x_{2}}&{-x_{3}}&{x_{0}}&{x_{1}}\\{x_{3}}&{x_{2}}&{-x_{1}}&{x_{0}}\\\end{matrix}\right)
W=\left(\begin{array}{cc}{<\phi^{*}\phi>}&{<\phi*\chi>}\\{-<\chi*\phi>}&{-<\chi*\chi>}\\\end{array}\right)=\left(\begin{array}{cc}{W_{\phi}}&{W_{mix}}\\{-W_{mix}^{*}}&{-W_{\chi}}\\\end{array}\right)\,
(\stackrel{(1,0)}{\varphi^{\alpha_{0}}}+\stackrel{(0,1)}{\varphi^{\alpha_{0}}})=\phi^{\alpha_{0}}\qquad\mathrm{and}\qquad(\stackrel{(1,0)}{\varphi^{\alpha_{0}}}-\stackrel{(0,1)}{\varphi^{\alpha_{0}}})=\bar{\phi}^{\alpha_{0}},
S=\sum_{n_{1}=0}^{M-N_{4}}\,\sum_{n_{4}=0}^{N_{4}}\,(-1)^{n_{1}+n_{3}}\frac{M!}{n_{1}!\,n_{2}!\,n_{3}!\,n_{4}!}\,(1-Y)^{n_{2}}(\delta_{1})_{n}\,(\delta_{3})_{n+m}\,,
S_{2}^{(0)}={\frac{1}{2}}x^{i}\zeta_{i}x^{i}+{\frac{1}{2}}x_{i}^{*}B^{ij}x_{j}^{*}+x_{i}^{*}C^{i\alpha}y_{\alpha}^{*}+{\frac{1}{2}}y_{\alpha}^{*}D^{\alpha\beta}y_{\beta}^{*},
\ddot{\phi}({\bfk}_{n},t)+3H{\dot{\phi}}({\bfk}_{n},t)+[e^{-2Ht}{\bfk}_{n}^{2}+m^{2}]\phi({\bfk}_{n},t)+\frac{\deltaV_{F}}{\delta\phi(-{\bfk}_{n},t)}+\sum_{i}\alpha_{i}\eta_{i}({\bfk}_{n},t)=0
G_{\mu\nu}^{0}=\frac{\epsilon_{\mu\nu\lambda}p_{\lambda}+\delta_{\mu\nu}M+p_{\mu}p_{\nu}/M}{p^{2}+M^{2}}\;,~~D_{\mu\nu}^{0}(p)=\frac{\epsilon_{\mu\nu\lambda}p_{\lambda}}{p^{2}}\;,~~\mathrm{and}~~\Gamma_{\mu\nu\lambda}^{0}=g\epsilon_{\mu\nu\lambda}
\int\frac{dw_{2}dw_{3}dw_{4}dw}{{(2\pi)}^{4}}\frac{-(w_{2}+w_{3}+w_{4})X_{i}^{8}[-(w_{2}+w_{3}+w_{4})]w_{2}X_{j}^{8}(w_{2})w_{3}X_{k}^{8}(w_{3})w_{4}X_{l}^{8}(w_{4})}{[(w+w_{2})^{2}-R^{2}][(w+w_{2}+w_{3})^{2}-R^{2}][(w+w_{2}+w_{3}+w_{4})^{2}-R^{2}][w^{2}-R^{2}]}
p_{\alpha}A^{\alpha}\rightarrow\frac{1}{2}\left[\hat{p}_{\alpha},A^{\alpha}(\hat{x})\right]_{+}\,,\,\,\,\,\,\,\,\,p_{\alpha}F^{\alpha\beta}\rightarrow\frac{1}{2}\left[\hat{p}_{\alpha},F^{\alpha\beta}(\hat{x})\right]_{+}\,,
\times\Bigg\{\sum_{\sigma\inZ_{N}}\frac{\omega^{\sigmak_{2}}w(x_{3},x_{13},x_{1}|\sigma)w(x_{4},x_{24},x_{2}|k_{4}+\sigma)}{w(x_{4},x_{14},x_{1}|k_{3}+\sigma)w(x_{3}/\omega,x_{23},x_{2}|k_{1}+\sigma)}\Bigg\}_{0}
S[u,t]=\frac{1}{g_{o}^{2}\alpha^{\,\prime\frac{(p+1)}{2}}}\int\frac{d^{p+1}k}{(2\pi)^{p+1}}\left[u(-k)u(k)-\frac12t(-k)t(k)\,\left[\alpha^{\,\prime}k_{\alpha}^{2}-1/2\right]\right.
\begin{array}{ccc}{{\calE}=\int_{M_{3}}<(H_{[\mu\nu}\otimesI_{E}-I_{E}\otimesK_{[\mu\nu})}&{\wedge}&{(D_{\rho]}^{A}\Phi_{A}\otimesI_{E}-I_{E}\otimesD_{\rho]}^{B}\Phi_{B})>}\\{-\int_{M_{3}}<(D_{[\mu}^{A}\Phi_{A}\otimesI_{E})\wedge(H_{\nu\rho]}\otimesI_{E})>}&{+}&{(A\leftrightarrowB,\Phi_{A}\leftrightarrow\Phi_{B}).}\\\end{array}
i\Pi_{\mathrm{ghost+33}}^{\mu\nu}(q)\simi\frac{g^{2}}{16\pi^{2}}\left\{g^{\mu\nu}\frac{24}{-\tilde{q}^{2}}+\frac{10}{3}\left(g^{\mu\nu}q^{2}-q^{\mu}q^{\nu}\right)\mathrm{ln}(q^{2}\tilde{q}^{2})+32\frac{\tilde{q}^{\mu}\tilde{q}^{\nu}}{\tilde{q}^{4}}-\frac{4}{3}\frac{q^{2}}{\tilde{q}^{2}}\tilde{q}^{\mu}\tilde{q}^{\nu}\right\}\,.
H=\intd^{2}{\bfr}\left[\frac{1}{2}{\bf\Pi}^{\dagger}\cdot{\bf\Pi}-\frac{\lambda}{2}(\psi^{\dagger}\psi)^{2}+A^{0}(\frac{\kappa}{2}\epsilon_{ij}F_{ij}+\vert\psi\vert^{2})\right],
SL(2,{\bfR})_{S}:\\\left(\begin{matrix}{\vec{\hat{\alpha}}}\\{\vec{\hat{\beta}}}\\\end{matrix}\right)\to\left(\begin{matrix}{a}&{c}\\{b}&{d}\\\end{matrix}\right)\left(\begin{matrix}{\vec{\hat{\alpha}}}\\{\vec{\hat{\beta}}}\\\end{matrix}\right).
\stackrel{\mathrm{HW}}{S^{\alpha\beta}}=\int\Psi^{\dagger}S_{\mathrm{op}}^{\alpha\beta}\Psid^{3}x,\quadS_{\mathrm{op}}^{\alpha\beta}=\frac{1}{2}\,\sigma^{\alpha\beta}+\frac{i}{2m}\left(p^{\alpha}\gamma^{\beta}-p^{\beta}\gamma^{\alpha}\right)\,.
V_{rad}=\left[\left(\frac{1}{m_{1}}-\frac{1}{m_{2}}\right)^{2}\frac{V_{\|}}{4r^{2}}+\left(\frac{1}{m_{1}}+\frac{1}{m_{2}}\right)^{2}\frac{V_{L}}{4r^{2}}\right]\bf(r\cdotp)(r\cdotp).
\begin{array}{cc}{\partial_{0}x^{i}=-p^{i},\qquad}&{\partial_{0}p^{i}=-\partial_{1}\partial_{1}x^{i};}\\{(\partial_{0}+\partial_{1})\theta_{a}=0,}&{(\partial_{0}+\partial_{1})\bar{\theta}_{\dot{a}}=0}\\\end{array}
\varphi_{p}^{o}(x)={\frac{2a}{3(1+\beta^{2})}}\left[-\varphi_{1}^{h}(x)\int_{0}^{x}dy\left(h(y)\varphi_{2}^{h}(y)\right)+\varphi_{2}^{h}(x)\int_{0}^{x}dy\left(h(y)\varphi_{1}^{h}(y)\right)\right],
\intd^{2}\xi\sqrt{g}h^{\alpha\beta\gamma}(\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\partial_{\gamma}X^{\rho}G_{\mu\nu\rho}(X(\xi))+\partial_{\alpha}\partial_{\beta}X^{\mu}\partial_{\gamma}X^{\rho}\tilde{G}_{\mu\rho}).
\frac{\partial}{\partialq^{2}}\,\mathrm{Re}\,\Gamma_{\mathrm{b}}^{(2)}(q)=1+q^{-2\varepsilon}\left(\frac{4}{3}-\frac{2}{3}\delta+\frac{1}{3}{\delta}^{2}\right)g^{2}\left[\frac{3}{4(8\pi^{2})^{2}\varepsilon}+\mathrm{P}+{\calO}(\varepsilon)\right]\,,
S_{D0}=m\int_{{\calM}^{1}}{\tilde{\calL}}_{1}=\int_{{\calM}^{1}}\left(\tilde{\Pi}^{\underline{{m}}}u_{\underline{{m}}}^{(0)}(\tau)+i\left(d\tilde{\Theta}^{1\underline{{\mu}}}\tilde{\Theta}_{\underline{{\mu}}}^{2}-\tilde{\Theta}^{1\underline{{\mu}}}d\tilde{\Theta}_{\underline{{\mu}}}^{2}\right)\right),
B_{0}(t,q)=1+\sum_{n=1}^{\infty}t^{n}\sum_{l=1}^{n}\frac{(-1)^{l}}{l!}f^{l}(q)\beta_{nl}\prod_{j=1}^{l}\left(\frac{\lambda(\lambda-1)}{2}-\frac{j(j-1)}{2}\right),
[det(\delta_{\mu\nu}+L_{\mu\nu})]^{\frac{1}{2}}=\frac{[1+\frac{1}{8}f_{ij}^{2}+\frac{1}{16}\frac{1}{4!}\epsilon^{ijkl}\epsilon^{mnpq}f_{mi}f_{nj}f_{pk}f_{ql}]^{2}}{[-det(\eta_{ab}+\frac{1}{2}F_{ab})]^{\frac{1}{2}}}
\frac{1}{8}\mathrm{Tr}\left[\left(M^{-1}\dot{M}\right)^{2}\right]=\frac{1}{4d}\left[\mathrm{Tr}\left(G^{-1}\dot{G}\right)\right]^{2}+\frac{1}{4}\left\{\mathrm{Tr}\left[\left(G^{-1}\dot{G}\right)_{t}^{2}\right]-\mathrm{Tr}\left[\left(G^{-1}\dot{B}\right)^{2}\right]\right\},
\Sigma_{\mathrm{gf}}=-\,s\,\displaystyle{\int}d^{3}x\,\,e\,g^{\mu\nu}\partial_{\mu}\bar{c}_{a}A_{\nu}^{a}\,=\,-\,\displaystyle{\int}d^{3}x\,\,e\,g^{\mu\nu}\left(\partial_{\mu}b_{a}A_{\nu}^{a}+\partial_{\mu}\bar{c}_{a}D_{\nu}c^{a}\right)\,,
{\mathcal{S}}\left(\begin{matrix}{w_{\alpha}}\\{w_{\dot{\beta}}^{\prime}}\\\end{matrix}\right)=\left(\begin{matrix}{\nabla_{\alpha}}\\{\widetilde{\nabla}_{\dot{\beta}}}\\\end{matrix}\right)c,\quad\mathcal{S}c=c^{2},\quad\mathcal{S}^{2}=0
\widetilde{U}_{1}(\phi)=\widetilde{\mu}_{1}e^{-\widetilde{b}\phi_{1}};\quad\widetilde{U}_{2}(\phi)=\widetilde{\mu}_{2}e^{-2\widetilde{b}\phi_{1}};\quad\widetilde{U}_{2}(\phi)=\widetilde{\mu}_{3}e^{-\widetilde{b}(\phi_{1}+\phi_{2})}.
Q_{\alphaA}=T_{A}^{\,\,\,\,a}Q_{\alphaa},\quadQ_{\dot{\alpha}\bar{A}}=T_{\bar{A}}^{\,\,\,\,\dot{a}}Q_{\dot{\alpha}\dot{a}},\quadT_{\bar{A}}^{\,\,\,\,\dot{a}}=\left(T_{A}^{\,\,\,\,a}\right)^{*}
R_{\nu\lambda\delta}^{\mu}=\Gamma_{\nu\delta,\lambda}^{\mu}-\Gamma_{\nu\lambda,\delta}^{\mu}+\Gamma_{\sigma\lambda}^{\mu}\Gamma_{\nu\delta}^{\sigma}+\Gamma_{\sigma\delta}^{\mu}\Gamma_{\nu\lambda}^{\sigma},\quad[\nabla_{\mu},\nabla_{\nu}]V^{\lambda}=R_{\delta\mu\nu}^{\lambda}V^{\delta}\quad.
\varepsilon=\langle{\bfx}\cdot\nablaV({\bfx})\rangle+\langleV({\bfx})\rangle+\langle{\frac{m_{1}^{2}}{\sqrt{{\bfp}^{2}+m_{1}^{2}}}}+{\frac{m_{2}^{2}}{\sqrt{{\bfp}^{2}+m_{2}^{2}}}}\rangle
g_{q}(y,M)={\frac{M}{18\pi}}x_{a}x_{b}\[f_{q}^{(a)}(x_{a},M^{2})f_{\bar{q}}^{(b)}(x_{b},M^{2})+f_{\bar{q}}^{(a)}(x_{a},M^{2})f_{q}^{(b)}(x_{b},M^{2})]\,
\begin{array}{c}{U_{0}(z)=\frac12(u_{0}+u_{-1})+\frac1{\sqrt{2}}(u_{0}-u_{-1})\,z_{per}}\\{V_{0}(z)=\frac12(v_{0}+v_{-1})+\frac1{\sqrt{2}}(v_{0}-v_{-1})\,z_{per},}\\\end{array}
\displaystyle\begin{array}{c}{\displaystyleE=\displaystyle\frac{1}{a}\displaystyle\sum_{j=1}^{M}\left(\phi(\Theta-\vartheta_{j},1/2)+\phi(\Theta+\vartheta_{j},1/2)-2\pi\right)}\\{P=\displaystyle\frac{1}{a}\displaystyle\sum_{j=1}^{M}\left(\phi(\Theta-\vartheta_{j},1/2)-\phi(\Theta+\vartheta_{j},1/2)\right)+2\omega}\\\end{array}
{\calL}_{\mathrm{Yuk}}=\frac{h}{2}\left(i\overline{{\Psi}}_{L}\frac{1}{2}(\Phi\hat{C}_{\ast}-i\Phi\hat{C}_{\ast}i)\Psi_{R}+\overline{{\Psi}}_{L}\frac{1}{2}(\Phi\hat{C}_{\ast}-i\Phi\hat{C}_{\ast}i)\Psi_{R}i\right),
d^{D-1}\sigma_{\alpha_{1}\ldots\alpha_{D-1}}=\epsilon_{k_{1}\ldotsk_{D-1}}\frac{\partial\bar{x}_{\alpha_{1}}}{\partial\sigma_{k_{1}}}\cdots\frac{\partial\bar{x}_{\alpha_{D-1}}}{\partial\sigma_{k_{D-1}}}d\sigma_{1}\ldotsd\sigma_{D-1}\.
\mathrm{-\nabla_{0}\vec{H}=rot\vec{E},\quad\mathrm{div\vec{H}=0,\quad\mathrm{rot\vec{H}=\nabla_{0}\vec{E}+\frac{4{\pi}a}{c}{\vec{j}},\quad\mathrm{div\vec{E}=4{\pi}a{\rho}.}}}}
\left({\frac{x^{-}}{x^{+}}}\right)^{2}F(x^{-},x^{+})={\frac{N}{k}}\sum_{n}{\frac{3(k-2n)^{2}}{2\pi^{2}k^{2}r^{4}}}\rightarrow{\frac{N}{2\pi^{2}r^{4}}}\.
L_{m}=P^{+}(m)\alpha_{m}^{-}+P^{-}(m)\alpha_{m}^{+}+\sum_{k\ne0,m}\alpha_{m-k}^{+}\alpha_{k}^{-}+\sum_{s}(s+\frac{1}{2})\psi_{m-s}^{+}\psi_{s}^{-}
\begin{array}{rcl}{x_{1}+ix_{2}}&{=}&{y(\partial_{\sigma}\bar{y})^{1/3}/\mathrm{Re}\;(\partial_{\sigma}y)^{1/3}}\\{x_{-}}&{=}&{-\mathrm{Im}\;(\partial_{\sigma}y)^{1/3}/\mathrm{Re}\;(\partial_{\sigma}y)^{1/3}}\\{x_{+}}&{=}&{x_{-}|y|^{2}+i\intd\sigma\;(y\partial_{\sigma}\bar{y}-\bar{y}\partial_{\sigma}y)+C}\\\end{array}
A_{2}(s)=-{\frac{1}{\Gamma(s)}}\Bigr(r_{+}^{2s}+r_{-}^{2s}\Bigr)\zeta_{H}(2s;2)\sum_{l=0}^{2}(x_{2,l}+z_{2,l}){\frac{\Gamma(s+l+1)}{\Gamma(l+1)}}.
\gamma_{i}\gamma_{i_{1}\cdotsi_{p}\bar{j}_{1}\cdots\bar{j}_{q}}=\gamma_{ii_{1}\cdotsi_{p}\bar{j}_{1}\cdots\bar{j}_{q}}+\sum_{k=1}^{q}(-1)^{k+p}g_{i\bar{j}_{k}}\gamma_{i_{1}\cdotsi_{p}\bar{j}_{1}\cdots\widehat{\bar{j}_{k}}\cdots\bar{j}_{q}}
\langle\hat{H}_{L}\rangle=\frac{\langle\Phi_{L}\vert\hat{H}_{L}\vert\Phi_{L}\rangle}{\langle\Phi_{L}\vert\Phi_{L}\rangle},~~\langle\bar{D}^{2}\rangle=\frac{\langle\Phi_{L}\vert\hat{H}_{L}\vert\Phi_{L}\rangle}{\langle\Phi_{L}\vert\Phi_{L}\rangle}.
\int_{\mu_{1}}^{\infty}dx\,\left(x^{2}-\mu_{1}^{2}\right)^{\nu-1}e^{-\mu_{2}x}=\frac{1}{\sqrt{\pi}}\left(\frac{2\mu_{1}}{\mu_{2}}\right)^{\nu-1/2}\Gamma\left(\nu\right)K_{\nu-\frac{1}{2}}\left(\mu_{1}\mu_{2}\right),
F\equiv(F_{\mu\nu})=\left(\begin{array}{cccc}{0}&{E_{1}}&{E_{2}}&{E_{3}}\\{-E_{1}}&{0}&{B_{3}}&{-B_{2}}\\{-E_{2}}&{-B_{3}}&{0}&{B_{1}}\\{-E_{3}}&{B_{2}}&{-B_{1}}&{0}\\\end{array}\right).
\begin{array}{lcl}{K_{m+a}((m+a)z^{\prime})\}&{\sim\}&{\sqrt{\frac{\pi}{2(m+a)}}\frac{e^{-(m+a)\eta}}{(1+\left(z\frac{m}{m+a}\right)^{2})^{1/4}}}\\{}&{}&{\left\{1+\sum_{j=1}^{\infty}(-1)^{j}\frac{u_{j}([1+\left(z\frac{m}{m+a}\right)^{2}]^{-1/2})}{m^{j}\left(1+\frac{a}{m}\right)^{j}}\right\}}\\\end{array}
\varepsilon(a,0)\approx-\frac{D}{D-1}p(a,0)\approx\frac{(4\xi-1)(am)^{D+1/2}e^{-2am}}{2^{D}\pi^{(D-1)/2}\Gamma(D/2)a^{D+1}}\left(2\delta_{B0}-1\right).