harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
E_{D}=\frac{1}{2}\left(\prod_{i=1}^{p}L_{i}\right)\sum_{n_{p+1},\dots,n_{D-1}}\int\frac{d^{p}k}{(2\pi)^{p}}\left[k^{2}+\sum_{j=p+1}^{D-1}\nu_{n_{j}}^{2}(a_{j})\right]^{1/2},
\frac{1}{N}\left[\dot{\pi}_{\,j}^{i}-\left(N^{m}\pi_{\,j}^{i}\right)_{,m}+N_{\,,m}^{i}\pi_{\,j}^{m}-N_{\,\,,j}^{m}\pi_{\,m}^{i}\right]=\frac{1}{2}\gamma^{-1/2}e^{\phi}\delta_{\,j}^{i}\left[\pi^{mn}\pi^{kl}\gamma_{mk}\gamma_{nl}+\frac{1}{2}\left(\pi^{\phi}\right)^{2}+\pi\pi^{\phi}\right]~~~,
{\calL}_{sfield}=-\frac14\intd^{2}\theta~\mathrm{Tr}\left(~\frac12\overline{{{\bfJ}}}{\bfJ}+i{\bf\Phi}\cdot\tau~(\overline{{D}}\gamma_{p}{\bfJ}+\overline{{{\bfJ}}}\gamma_{p}{\bfJ})~\right)
\frac{\deltaF}{\delta\rho_{a}^{r}}=\frac{\deltaE}{\delta\rho_{a}^{r}}-T\frac{\deltaS}{\delta\rho_{a}^{r}}-T\sum_{b=1}^{l}\frac{\deltaS}{\delta\rho_{b}}\frac{\delta\rho_{b}}{\delta\rho_{a}^{r}}=0\,.
{\calL}_{m}(g)={\frac{1}{2^{m}}}\;\delta_{\lambda_{1}\rho_{1}\cdots\lambda_{m}\rho_{m}}^{\mu_{1}\nu_{1}\cdots\mu_{m}\nu_{m}}\;\;R_{\mu_{1}\nu_{1}}{}^{\lambda_{1}\rho_{1}}\cdotsR_{\mu_{m}\nu_{m}}{}^{\lambda_{m}\rho_{m}}{}.
S(W)=-\frac{1}{2a^{2}}{\frac{2\pia}{e^{2}}}\int_{0}^{\infty}\mathrm{d}r\,\mathrm{e}^{{\frac{1}{2}}B}\left(P^{2}(r)+{\frac{a^{2}e^{2}}{\kappa^{2}}}\mathrm{e}^{B}\right)~\int\mathrm{d}^{4}x[(\partial_{\mu}W_{\nu})^{2}+e^{-A}\partial_{r}W_{\mu}\partial_{r}W_{\mu}].
\begin{array}{rcl}{{\sfs}f}&{=}&{Z_{c}\partial^{\mu}(\partial_{\mu}c+igZ_{g}[c,G_{\mu}])}\\{}&{}&{+Z_{c}Z_{g}(1+\deltav/v)g^{2}v^{2}\xi/2(\hat{\phi}_{0}^{\dagger}ct^{A}\hat{\phi}_{0}+\hat{\phi}_{0}^{\dagger}t^{A}c\hat{\phi}_{0})t^{A}}\\{}&{}&{+Z_{c}Z_{g}g^{2}v\xi/\sqrt{2}(\phi^{\dagger}ct^{A}\hat{\phi}_{0}+\hat{\phi}_{0}^{\dagger}t^{A}c\phi)t^{A}.}\\\end{array}
\left(\deltag\right)_{\alpha\alpha^{\prime}\beta_{1}\cdots\beta_{p+1}}=g_{\alpha^{\prime}\beta_{1}\cdots\beta_{p+1}}g_{\alpha\beta_{1}\cdots\beta_{p+1}}^{-1}g_{\alpha\alpha^{\prime}\beta_{2}\cdots\beta_{p+1}}\cdotsg_{\alpha\alpha^{\prime}\beta_{1}\cdots\beta_{p}}^{(-1)^{p}}=1
{\calL}^{\prime}=p_{-}\frac{(1-h_{+-}-h_{-i}\dot{x}^{i})-\sqrt{(1-h_{+-}-h_{-i}\dot{x}^{i})^{2}-h_{--}(h_{++}+2h_{+i}\dot{x}^{i}+v^{2}+h_{ij}\dot{x}^{i}\dot{x}^{j})}}{h_{--}}\
[(\eta-\eta^{\prime})^{2}]^{\nu}=\sum_{N=0}^{\infty}\sum_{\alpha=0}^{N}\frac{(2a)^{2\nu+n}{\pi}^{\frac{n}{2}}\Gamma(\nu+\frac{n}{2})\Gamma(N-\nu)}{\Gamma(N+n+\nu)\Gamma(-\nu)}Y_{\alpha}^{N}({\eta})Y_{\alpha}^{N}({\eta}^{\prime})
\rho_{p\rightarrow1+2+3}=\frac{(32\pi)^{2-2\ell}}{(\Gamma(\ell-1/2))^{2}(p^{2})^{\ell-1}}\int_{(m_{1}+m_{2})^{2}}^{(\sqrt{p^{2}}-m_{3})^{2}}s^{1-\ell}{\calD}^{\ell-3/2}\,ds,
h(a)=k+{\frac{a^{2}}{l^{2}}}-{\frac{\omega_{D}M}{a^{D-2}}}+{\frac{(D-1)\omega_{D}^{2}Q^{2}}{8(D-2)a^{2D-4}}},~~\omega_{D}={\frac{16\piG_{D+1}}{(D-1)V_{D-1}}}.
\partial_{J}^{I}u_{i}^{K}=\delta_{J}^{K}u_{i}^{I}-\frac{1}{4}\delta_{J}^{I}u_{i}^{K}\;,\qquad\partial_{J}^{I}u_{K}^{i}=-\delta_{K}^{I}u_{J}^{i}+\frac{1}{4}\delta_{J}^{I}u_{K}^{i}\;.
{\calE_{C}}^{\mathrm{\tiny~bare}}[\phi_{0}]=\sum_{\ell=0}^{\infty}(2\ell+1)\left[\int_{0}^{\infty}\frac{dk}{\pi}E\left(\rho_{\ell}(k)-\rho_{\ell}^{0}(k)\right)+\sum_{j}\mathrm{max}(\omega,E_{j,\ell})\right]\,.
{\calB}\Bigl(\frac{v^{2}}{N_{F}},\tilde{\alpha},\tilde{\lambda},\{\zeta_{i}\}\Bigr)=\frac{v^{2}}{N_{F}}{\calE}_{\mathrm{cl}}\bigl(\tilde{\alpha},\tilde{\lambda},\{\zeta_{i}\}\bigr)+{\calE}^{F}(\{\zeta_{i}\})-1\,.
\epsilon^{\alpha\mu\nu\beta}\gamma_{5}\gamma_{\mu}\partial_{\nu}\Psi_{\beta}^{i}-{\frac{\kappa\epsilon^{ij}}{\sqrt2}}\big[F^{\alpha\beta}+{\scriptstyle{\frac{1}{2}}}\gamma_{5}\epsilon^{\alpha\beta\rho\sigma}F_{\rho\sigma}\big]\Psi_{\beta}^{j}=0\,.
\begin{array}{ll}{\phi_{1}=0}\\{\phi_{2}=\phi=\varphi(r)e^{i\theta}}\\{H_{\mu}=\frac{\sqrt{\beta}}{q}\frac{H(r)}{r}\delta_{\mu}^{\theta}}\\{\tilde{D}=\frac{1}{2\beta}\left(q(\varphi^{2}-\eta^{2})+\alphae\sigma^{2}\right)}\\{G=0,}\\\end{array}
R_{\pm}\tilde{\Theta}_{1}R_{\mp}^{-1}\tilde{\Theta}_{2}+\tilde{\Theta}_{2}R_{\pm}\tilde{\Theta}_{1}R_{\pm}^{-1}=-\frac{1}{\mathrm{tr}D}\left(R_{\pm}[\mathrm{tr}_{q}\Theta,\tilde{\Theta}_{1}]R_{\mp}^{-1}+[\mathrm{tr}_{q}\Theta,\tilde{\Theta}_{2}]\right).
{\calH}=\frac{1}{2}(\vec{E}^{a}\vec{E}^{a}+\vec{B}^{a}\vec{B}^{a})+\frac{1}{2}(\vec{\calE}^{\,2}+\vec{\calB}^{\,2})+\pi^{\dagger}\pi+(\vec{D}\varphi)^{\dagger}(\vec{D}\varphi)+V(\varphi^{\dagger}\varphi)
\mathrm{tr}\left[T_{i}\partial_{\mu}\partial^{\mu}X_{0}T_{i}\right]=\partial_{\mu}\partial^{\mu}m_{i}^{2}\,\quad\mathrm{tr}\left[T_{i}\partial_{\mu}X_{0}T_{j}\partial^{\mu}X_{0}T_{i}\right]=\delta_{ij}\\left(\partial^{\mu}m_{i}^{2}\right)^{2}
\left(\begin{matrix}{\{\chi_{a}^{*}(x),\chi_{b}(y)\}\quad\{\chi_{a}^{*}(x),(\chi^{b})^{*}(y)\}}\\{\{\chi^{a}(x),\chi_{b}(y)\}\quad\{\chi^{a}(x),(\chi^{b})^{*}(y)\}}\\\end{matrix}\right)=M_{ab}(-iD(x-y))
{\frac{m}{(2\pi)^{3}}}\int\left[(g_{1}^{(1)})^{2}+(g_{2}^{(1)})^{2}+(g_{3}^{(1)})^{2}+(g_{4}^{(1)})^{2}\right]{\frac{d^{3}k}{\varepsilon_{k}}}=1.
\bigg(\langle\langleF_{\mu\nu}(z_{j})F_{\rho\sigma}(z_{j}^{\prime})\rangle\rangle-\langle\langleF_{\mu\nu}(z_{j})\rangle\rangle\,\langle\langleF_{\rho\sigma}(z_{j}^{\prime})\rangle\rangle\bigg)^{\mathrm{LR}}=0.
<\bar{0}|:T_{\epsilon}^{\hat{0}\hat{0}}:|_{P_{0}}|\bar{0}>=\frac{1}{24\pi}\frac{\frac{\partial^{3}\bar{f}_{\epsilon}}{\partialx^{3}}|_{P_{0}}}{\frac{\partial\bar{f}_{\epsilon}}{\partialx}|_{P_{0}}}
\chi_{F}=-i(s/\sqrt{2})\left(\begin{array}{c}{C\bar{N}e^{3}}\\{C(\bar{S}e^{3}-\bar{N}e^{1})}\\{A(Ne^{4}+Se^{1})+B(Ne^{2}-Se^{3})}\\{-ANe^{1}+BNe^{3}}\\\end{array}\right),
\psi^{(A,B)}(x)=\frac{1}{\left(2\pi\right)^{\frac{3}{2}}}\int\left(e^{-ipx}\sumu(p,s_{3})a(p,s_{3})+e^{ipx}\sumv(p,s_{3})b^{\ast}(p,s_{3})\right)\frac{d^{3}p}{2p_{0}}
c_{\Lambda}^{(l)}=q^{-l(\lambda,\lambda+2\rho_{0})}\sum_{(\mu,2\kappa,-s)\inD^{+}}\sum_{s=0}^{\infty}(m_{\mu,s}^{+}+(-1)^{l}m_{\mu,s}^{-})q^{l(\mu,\mu+2\rho_{0})/2-sl(2\kappa+g)}\frac{D_{q}[(\mu,2\kappa,-s)]}{D_{q}[(\lambda,\kappa,0)]}\,,~~~~l\in{\bfZ}^{+}
{\mathcalF}_{l}\;=\;e_{N}^{2}\,(\xi-1)\left[\langlex|{\frac{M^{-1}(\sqrt{-\partial^{2}})}{-\partial^{2}}}|x\rangle-\langlex|{\frac{M^{-1}(\sqrt{-\partial^{2}})}{-\partial^{2}}}|y\rangle\right]\;.
\tilde{Y}_{1}\hat{R}dK_{1}\hat{R}=\hat{R}dK_{1}\hat{R}\tilde{Y}_{1}\quad\Longleftrightarrow\quad\hat{R}dK_{1}^{\dagger}\hat{R}Y_{1}=Y_{1}\hat{R}dK_{1}^{\dagger}\hat{R}\;;
B_{\pm}(\bar{p})=\mu\pmm+\frac{e^{2}}{(2\pi)^{3}}\int_{-\infty}^{\infty}\!\!d\bar{k}\,\frac{B_{\pm}(\bar{k})}{\bar{k}^{2}+B_{\pm}^{2}(\bar{k})}\,\widetilde{D}(\bar{p}-\bar{k}).
A_{n}^{*i}=-\mathrm{e}^{i{\frac{\pi}{4}}}\Omega_{+}^{-{\frac{1}{2}}}\bar{\psi}_{n}^{i\dot{2}},\qquadA_{n}^{i}=\mathrm{e}^{-i{\frac{\pi}{4}}}\Omega_{+}^{{\frac{1}{2}}}\psi_{n1}^{i}.
J=\left(\begin{array}{cc}{\alpha^{t}}&{\tilde{f}_{2}}\\{f_{1}}&{\tilde{A}}\\\end{array}\right)\left(\begin{array}{ll}{0}&{0}\\{0}&{L}\\\end{array}\right)\left(\begin{array}{cc}{\alpha}&{\tilde{f}_{1}}\\{f_{2}}&{A}\\\end{array}\right)=\left(\begin{array}{ll}{\tilde{f}_{2}Lf_{2}}&{\tilde{f}_{2}LA}\\{\tilde{A}Lf_{2}}&{\tilde{A}LA}\\\end{array}\right)
F(x,x^{\prime})=F(x)-\frac{1}{2}F_{;\alpha}(x)\sigma^{;\alpha}+\frac{1}{2}F_{\alpha\beta}(x)\sigma^{;\alpha}\sigma^{;\beta}+\frac{1}{4}\{\frac{1}{6}F_{;\alpha\beta\gamma}(x)-F_{\alpha\beta;\gamma}(x)\}\sigma^{;\alpha}\sigma^{;\beta}\sigma^{;\gamma}+O(\sigma^{2})~,
\phi_{p_{3}n\xir}(x_{\parallel})\,=(2\pi)^{-1/2}\,e^{-ip_{3}x^{3}}\phi_{p_{3}n\xir}(x^{0})\quad,\;\;\phi_{p_{3}n\xir}(x^{0})=\phi_{p_{3}n\xir}(x^{0},p_{z})|_{p_{z}=0}\;
\mathrm{{\it~with}~}\qquad\vert\,\sum_{j\in\tau_{a}}k_{0,j}\,\vert<2m\quad\mathrm{~{\it~for~all}~}\quad\tau_{a}\subseteq\{1,\ldots,n\}\quad,\quadk_{0,n}=-\sum_{j=1}^{n-1}k_{0,j}
S_{p}~=~-2\tau_{p}\intd^{4}\xie^{-\phi}\sqrt{-\hat{G}}-2g_{s}^{-1}\mu_{p}\intd^{4}\xi\left[Q\intd\rhoe^{-2\xi}f^{-1}(T)+P\intd\rhoe^{-2\xi}f(T)\right].
\theta_{\mu\nu}-\theta_{\nu\mu}=-\frac{\mathrm{i}}4\partial_{\alpha}(\psi^{+}\{\gamma_{\alpha},\sigma_{\mu\nu}\}\psi)=\frac12\varepsilon_{\mu\nu\alpha\lambda}\partial_{\alpha}(\psi^{+}\gamma_{\lambda}\gamma_{5}\psi).
\begin{array}{c}{\left(\partial_{X}-\lambdae^{\omegaX}\right)\alpha(X)=i\epsilon\beta(X),\quad\left(\partial_{X}+\lambdae^{\omegaX}\right)\beta(X)=i\epsilon\alpha(X),}\\{\left(\partial_{X}+\lambdae^{\omegaX}\right)\gamma(X)=\epsilon\delta(X),\quad\left(\partial_{X}-\lambdae^{\omegaX}\right)\delta(X)=-\epsilon\gamma(X).}\\\end{array}
G_{m}(D;\alpha_{1},...,\alpha_{m})=[\prod_{i=1}^{m}\Gamma(\alpha_{i})]^{-1}(\prod_{i=1}^{m}\int_{0}^{\infty}dt_{i}\,t_{i}^{\alpha_{i}-1})\,T^{-\frac{D}{2}}e^{-\frac{1}{T}\sum_{i<j}t_{i}t_{j}x_{ij}^{2}}
G_{0}\,\Omega_{0}(p)={\textstyle{\frac{1}{2}}}\left({\frac{p_{\mu}}{\sqrt{\alpha}}}b_{0}^{\mu}+{\frac{p^{L}}{\sqrt{\beta}}}d_{0}+m_{0}g_{0}\right)\,\Omega_{0}(p)\;=\;0\;\;\;.
{\calL}_{1/m^{2}}^{(3+1)scal}=\frac{\alpha}{720\pim^{2}}\left[6F^{\nu\lambda}F_{\nu\lambda,\mu}^{~~~~\mu}+4F^{\nu\lambda,\mu}F_{\nu\lambda,\mu}+F_{~~~\lambda}^{\nu\lambda,}F_{\nu\mu,}^{~~~\mu}\right],
\mathcal{H}^{\ast}=\intd^{2}\,x\,\left[\frac{1}{2}\,\left(\pi_{i}^{\perp}\,\pi_{i}^{\perp}+4\,m^{2}A\,_{i}^{\perp}\,A\,_{i}^{\perp}\right)+\frac{1}{2}\left(\,\epsilon^{\,i\,j}\,\partial_{\,i}\,A\,_{j}^{\perp}\right)^{2}+2\,m\,\left(A_{1}^{\perp}\,\pi\,_{2}^{\perp}-A\,_{2}^{\perp}\,\pi_{1}^{\perp}\right)\right].
\begin{array}{lll}{f_{0}}&{=}&{\tilde{f}_{0}^{-1}\;,}\\{f_{i}f_{0}^{-1/2}}&{=}&{\tilde{f}_{k-i}\tilde{f}_{0}^{-1/2}\;\;\;;i=1,\ldots,k-1\;.}\\\end{array}
\begin{aligned}{\sum_{i}\left(H_{1}^{i}T_{G}^{a}(H_{1}^{i})^{\dagger}-(H_{2}^{i})^{\dagger}T_{G}^{a}H_{2}^{i}\right)}&{{}=0,}\\{\sum_{i}\left((H_{1}^{i})^{\dagger}T_{G}^{b}H_{1}^{i}-H_{2}^{i}T_{G}^{b}(H_{2}^{i})^{\dagger}\right)}&{{}=0.}\\\end{aligned}
\Phi_{1}^{(2)}=\pi^{2}\;,\;\;\Phi_{2}^{(2)}=\pi_{\mu}\psi^{\mu}\;,\;\;\Phi_{3\mu}^{(2)}=\frac{1}{8!}\epsilon_{\mu\nu\rho_{2}\ldots\rho_{9}}\pi^{\nu}\psi^{\rho_{2}}\ldots\psi^{\rho_{9}}+\frac{\alpha}{2^{8}}\pi_{\mu}\;.
G_{{\calQ}^{4}}(x_{p})=\frac{3^{3}\,\Gamma(11)}{2^{5}\left(\pi^{3}\Gamma(4)\right)^{4}}\:g^{8}e^{-\frac{8\pi^{2}}{g^{2}}+i\theta}\,x_{12}^{4}x_{34}^{4}\,Q(x_{pq})\,.
L_{0}={\frac{1}{k+2}}\sum_{s=0}^{2|k|-1}\sum_{n\in\mathbb{Z}}:\left[J_{-n}^{(s)+}J_{n}^{(s)-}+J_{-n}^{(s)-}J_{n}^{(s)+}+J_{-n}^{(s)0}J_{n}^{(s)0}\right]:.
x\frac{K_{|l+\alpha|}^{\prime}(x)}{K_{|l+\alpha|}(x)}=-\alpha+|l|+\alpha\frac{|l|+l+1+(x^{2}/2\alpha)}{|l|+1}\frac{{}_{1}F_{1}\left(\frac{|l|+l+3}{2}+(x^{2}/4\alpha),|l|+2,\alpha\right)}{{}_{1}F_{1}\left(\frac{|l|+l+1}{2}+(x^{2}/4\alpha),|l|+1,\alpha\right)},
\delta\overline{{\bfA}}_{a}^{M}=-\bbox\nabla\overline{{\delta\Lambda}}_{a}^{M}-gf_{ab}^{c}\left(\overline{{\bfA}}_{c}^{M}\overline{{\delta\Lambda}}_{b}^{M}+{\frac{(\deltat)^{2}}{4}}\dot{\bfA}_{c}^{M}\delta\dot{\Lambda}_{b}^{M}\right),
S=T\,e^{-i\int_{t_{0}}^{\infty}\,dt\hat{H}_{\mathrm{int}}^{(I)}(t)}\,T\,e^{-i\int_{-\infty}^{t_{0}}\,dt\hat{H}_{\mathrm{int}}^{(I)}(t)}=T\,e^{-i\int_{-\infty}^{\infty}dt\int_{(\infty)}d^{3}\vec{x}\,\hat{\calH}_{\mathrm{int}}^{(I)}}\,
\tilde{B}(p)=g^{2}N(N^{2}-1)\delta_{b}^{a}\delta_{J}^{I}\int\frac{d^{4}k}{(2\pi)^{4}}\,d^{4}\theta\,\left\{\frac{\Phi_{aI}^{\dagger}(p,\theta,{\overline{\theta}})\Phi^{bJ}(-p,\theta,{\overline{\theta}})}{(k^{2}+m^{2})[(p-k)^{2}+m^{2}]}\right\}\,.
\deltaS[\theta,\bar{A}_{+},\bar{\Pi}^{\perp-}]=\frac{k}{8\pi}\int_{\partial\calM}\partial_{+}\theta\,\partial_{-}\theta+\partial_{-}\theta\left(\bar{A}_{+}-\mathrm{~\frac{8\pi}{k}~}\bar{\Pi}^{\perp-}\right)\,.
H=\intd\sigma^{1}\left[n^{\mu}\partial_{1}A_{0}^{\mu}+\frac1\phi(n^{2}+1)+\lambda_{\phi}\pi_{\phi}+\lambda_{n}^{\mu}p_{n}^{\mu}+\lambda_{0}^{\mu}p_{0}^{\mu}+\lambda_{1}^{\mu}(p_{1}^{\mu}-n^{\mu})\right],
[{\calL}_{\scriptscriptstyleK},{\calL}_{\scriptscriptstyleL}]\omega=({\calL}_{\scriptscriptstyleK}{\calL}_{\scriptscriptstyleL})\omega-(-1)^{[\widetilde{\calH}_{\scriptscriptstyleL}][\widetilde{\calH}_{\scriptscriptstyleK}]}({\calL}_{\scriptscriptstyleL}{\calL}_{\scriptscriptstyleK})\omega
\gamma_{\calD}^{\mathrm{1-inst}}={\frac{C_{N}^{K=\pm1}}{N^{4}}}{\frac{\Gamma(N-{\frac{1}{2}})}{\Gamma(N-{1})}}e^{-{\frac{8\pi^{2}}{g^{2}}}}(e^{i\theta}+e^{-i\theta})\;,
\overline{{\Omega}}^{\alpha}\equiv(-1)^{m}\frac{1}{(2m-2)!}{k_{\betaj_{1}\dotsj_{m-2}}}^{\alpha}{C_{i_{1}i_{2}}}^{j_{1}}\ldots{C_{i_{2m-5}i_{2m-4}}}^{j_{m-2}}\pi^{\beta}\pi^{i_{1}}\ldots\pi^{i_{2m-4}}\quad.
\varepsilon_{0}^{\beta\rightarrow0}=-\frac{\verte\vertH}{(2\pi)^{2}}\frac{1}{\Gamma(1+\delta)}\sum_{n\sigma}\int_{0^{+}}^{\infty}d\eta\;\eta^{\delta}\mathrm{e}^{-(4+\beta^{2}\alpha_{n\sigma})\eta}\left\{2\sqrt{\pi}\alpha_{n\sigma}\;\eta^{-1}-\frac{\partial}{\partial\beta}\left(\sqrt{\frac{\pi}{\beta\eta}}\right)\right\},
\epsilon(t,z,r,\theta,\phi)=e^{{\frac{i}{2}}\Gamma_{4}\theta}e^{{\frac{1}{2}}\Gamma_{3}\Gamma_{4}\phi}\Big((1+\Gamma_{0}\Gamma_{1})r^{\frac{-1}{4}}-{\frac{(t-z)}{2(Z_{m})_{\mathrm{cr}}^{\frac{3}{2}}}}r^{\frac{1}{4}}(\Gamma_{0}-\Gamma_{1})\Gamma_{2}\Big)\kappa_{0}.
\Pi_{33}(0)=2g^{2}\frac{1}{L}\sum_{n=-\infty}^{\infty}\int\frac{d^{3}p}{\left(2\pi\right)^{3}}\left[m_{\mathrm{tp}}(p)+m_{\mathrm{gh}}(p)+m_{\mathrm{gg}}(p)\right]\,
{\calM}^{2}=\left(n^{I}+\frac{1}{6}n^{IJKL}{\calC}_{JKL}^{(11)}\right)g_{IM}\left(n^{M}+\frac{1}{6}n^{MNPQ}{\calC}_{NPQ}^{(11)}\right)+\frac{1}{24}n^{IJKL}g_{IM}~g_{JN}~g_{KP}~g_{LQ}~n^{MNPQ}\.
\phi_{<}^{cl}(\vec{x},s)=[\phi_{<f}{\frac{sin(k_{0}s)}{sin(k_{0}t)}}+\phi_{<i}{\frac{sin[k_{0}(t-s)]}{sin(k_{0}t)}}]cos(\vec{k}_{0}.\vec{x})\equiv\phi_{<}^{cl}(s)cos(\vec{k}_{0}.\vec{x}),
\begin{array}{c}{\omega=d{\bfp}\wedged{\bfx}+d{\bfp}_{+}\wedged{\bfe_{+}}+d{\bfp}_{1}\wedged{\bfe}_{1}}\\{{\calH}={\tilde{\sigma}}\left[\phi_{0}+\lambda\phi_{1}+\sum_{i,j=+,1}d_{ij}u_{ij}\right]}\\\end{array}
G_{R}(p_{+},p_{-})={\frac{\etap_{+}p_{-}}{\pi^{2}lT_{L}T_{R}}}\left[\psi\left(1-{\frac{ip_{+}}{2\piT_{L}}}\right)+\psi\left(1-{\frac{ip_{-}}{2\piT_{R}}}\right)\right]\,.
\left(-\frac{1}{2m}\nabla^{2}-\mu^{2}\right)^{2}-\frac{\partial^{2}}{\partialt^{2}}=\left(-\frac{1}{2m}\nabla^{2}-\mu-\frac{\partial}{\partialt}\right)\left(-\frac{1}{2m}\nabla^{2}-\mu+\frac{\partial}{\partialt}\right)
L_{i_{2}}^{(2)}(x):=\intd^{2}y\left[W_{i_{2}j}^{(2)}(x,y)\ddot{\varphi}^{j}(y)+\alpha_{i_{2}}^{(2)}(y)\delta^{2}(x-y)\right]=0,~~~i_{2}=1,\cdots,13,
T=\left(\begin{array}{ccccccc}{e^{i\theta_{1}}}&{}&{}&{}&{}&{}&{}\\{}&{e^{-i\theta_{2}}}&{}&{}&{}&{}&{}\\{}&{}&{e^{i\theta_{2}}}&{}&{}&{}&{}\\{}&{}&{}&{e^{-i\theta_{1}}}&{}&{}&{}\\{}&{}&{}&{}&{e^{i(\theta_{1}+\theta_{2})}}&{}&{}\\{}&{}&{}&{}&{}&{1}&{}\\{}&{}&{}&{}&{}&{}&{e^{-i(\theta_{1}+\theta_{2})}}\\\end{array}\right)
21\frac{d\bar{x}_{2}}{dt}+\left(2\beta_{0}\bar{x}_{0}(t)\right)\bar{x}_{2}=-\beta_{2}\bar{x}_{0}^{4}(t)-3\beta_{1}\bar{x}_{0}^{2}(t)\bar{x}_{1}(t)-\beta_{0}\bar{x}_{1}^{2}(t),\;\;\;\bar{x}_{2}(0)=0,
G_{M}^{+}(x^{\prime},y^{\prime})=\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}dq\int_{-\infty}^{\infty}dkN_{1}^{2}e^{-i\omega(t_{1}^{\prime}-t_{2}^{\prime})+im(\theta_{1}^{\prime}-\theta_{2}^{\prime})}J_{m}^{2}(qR_{0}).
V(0)={\frac{1}{4\alpha}}[n_{1}n_{1}^{\prime}\mathrm{Tr}(Y^{2})\mu_{0}+n_{3}n_{3}^{\prime}\mathrm{Tr}(\lambda_{i}\lambda_{j})\mu_{3}+n_{8}n_{8}^{\prime}\mathrm{Tr}(T_{a}T_{b})\mu_{8}]\<\0\.
i\partial_{t}\Psi[\xi,t)=\intd^{2}x\biggl[i\biggl(A_{0}+{\frac{1}{\theta}}J_{i}\epsilon_{ij}{\frac{\partial_{j}}{\nabla^{2}}}\biggr){\frac{\delta}{\delta\xi}}+A_{0}\rho-\xi{\dot{\rho}}\biggr]\Psi[\xi,t)\,,
E[\sigma]=\sum_{\ell}N_{\ell}\left[\int_{0}^{\infty}\frac{dk}{2\pi}\sqrt{k^{2}+m^{2}}\frac{d}{dk}[\delta_{\ell}(k)]_{N}+\frac{1}{2}\sum_{j}\omega_{\ellj}\right]+\sum_{i=1}^{N}E_{\mathrm{FD}}^{(i)}+E_{\mathrm{CT}}
D_{\mu\nu}({x}-{y})=\int\frac{d^{d+2}k}{(2\pi)^{d+2}}{e^{-i{k}\cdot({x}-{y})}}\left\{\frac{1}{k^{2}+i\epsilon}\left[g_{\mu\nu}-\frac{(k_{\mu}N_{\nu}+k_{\nu}N_{\mu})}{k_{+}+i\\epsilon^{\prime}\\mathrm{sgn}(k_{-})}\right]-\alpha\frac{k_{\mu}k_{\nu}}{[k_{+}+i\\epsilon^{\prime}\\mathrm{sgn}(k_{-})]^{2}}\right\},
I=-{\frac{\lambda_{0}^{2}}{2\pi}}\sqrt{\frac{4m_{r}^{2}-k^{2}}{-k^{2}}}log\big[{\frac{\sqrt{4m_{r}^{2}-k^{2}}-\sqrt{-k^{2}}}{\sqrt{4m_{r}^{2}-k^{2}}+\sqrt{-k^{2}}}}\big]-1.
\begin{array}{lcl}{a_{2}}&{=}&{-\frac{1}{4}u_{2}}\\{a_{3}^{\prime}}&{=}&{\frac{1}{800\theta}\{9u_{2}^{2}+100\thetau_{3}^{\prime}-280{\theta}^{2}u_{2}\}}\\{a_{4}}&{=}&{\frac{49}{400}u_{2}^{2}-\frac{3}{8}\thetau_{3}^{\prime}+\frac{41}{20}{\theta}^{2}u_{2}^{\prime\prime}}\\{\vdots}\\\end{array}
\partial_{\mu}^{2}u=\frac{1}{{\delta^{\prime}}^{2}}(2a_{\mu}a_{\mu}^{\prime}-\frac{\delta^{\prime\prime}}{\delta^{\prime}}a_{\mu}^{2})=\frac{1}{{\delta^{\prime}}^{2}}\left(\frac{\partial}{\partialu}-\frac{\delta^{\prime\prime}}{\delta^{\prime}}\right)(a_{\mu}^{2}).
V^{\lambda\sigma,\mu^{\prime}}\bigtriangleup_{\lambda\sigma,\lambda^{\prime}\sigma^{\prime}}^{B}V^{\lambda^{\prime}\sigma^{\prime},\nu^{\prime}}=-m^{2}(g^{\mu^{\prime}\nu^{\prime}}-p^{\mu^{\prime}}p^{\nu^{\prime}}/p^{2})\equiv\theta^{\mu^{\prime}\nu^{\prime}},
\Gamma_{R}^{(0)}[\phi]=\intd^{4}x\Big(\frac12\big(\partial_{\mu}\phi(x)\partial_{\mu}\phi(x)+m^{2}\phi^{2}(x)-\phi(x)\frac{\tilde{c}}{\tilde{\partial}^{2}}\phi(x)+\phi(x)\frac{\tilde{c}}{\tilde{\partial}^{2}}\phi(x)\big)+\frac{g^{2}}{4!}\phi\star\phi\star\phi\star\phi(x)\Big),
{\frac{\partialt_{1}}{\partial\phi^{1}}}=-{\frac{1}{\sqrt3}}\left({\frac{4}{3}}\right)^{2/3}e^{{\frac{\phi}{3}}}\,\qquad{\frac{\partialt_{0}}{\partial\phi^{1}}}=-\sqrt{\frac{2}{3}}\left({\frac{4}{3}}\right)^{2/3}e^{-{\frac{2\phi}{3}}}\.
Z=\intd\mu\,\mathrm{exp}\left\{-s\Psi+\sum_{k=0}^{4}\sum_{p=4-k}^{4n}\sum_{\beta_{k}=1}^{b^{k}({\calM})}\sum_{\alpha_{p}=1}^{b^{p}({\calN})}s_{\alpha_{p},\beta_{k}}^{p,k}{\calD}_{p,k}^{\alpha_{p},\beta_{k}}\right\},
\psi_{k_{x},+}(x)=\frac{1}{\sqrt{2}}\left(\begin{array}{c}{1}\\{1}\\{\cdot}\\{\cdot}\\\end{array}\right)e^{-ik^{0}(t-x_{1})}\;\;,\;\;\psi_{k_{x},-}(x)=\frac{1}{\sqrt{2}}\left(\begin{array}{c}{\cdot}\\{\cdot}\\{1}\\{1}\\\end{array}\right)e^{-ik^{0}(t-x_{1})}
{\bar{\eta}^{a}}_{\mu\nu}=-{\bar{\eta}^{a}}_{\nu\mu},\qquad{\bar{\eta}^{a}}_{\mu\nu}{\bar{\eta}^{b}}_{\mu\sigma}={\delta}_{ab}{\delta}_{\nu\sigma}+{\epsilon}_{abc}{\bar{\eta}^{c}}_{\nu\sigma}.
D_{mj}^{j}(s_{\pm}(\theta,\phi))=\sum_{m^{\prime},\,q^{\prime}}D_{mm^{\prime}}^{j}(s_{\mp}(\pi-\theta,\pi+\phi))D_{m^{\prime}q^{\prime}}^{j}(e^{\pmi\sigma_{3}\pi/2})D_{q^{\prime}j}^{j}(e^{-i\sigma_{1}\pi/2}).
w(l,\mu)\:=\;\int_{-\infty}^{+\infty}\!\!dp\int_{\mu}^{\infty}\!\!dx\;\mathrm{e}^{\textstyle-lp^{2}-\sqrt{x}\,l}\;=\;\frac{2\sqrt{\pi}}{l^{5/2}}\;\mathrm{e}^{\textstyle-\sqrt{\mu}\,l}\;\left[1+\sqrt{\mu}\,l\right],
g_{\mathrm{B}}=\frac{l_{\mathrm{H}}^{2}}{V_{\mathrm{H}}}\,\quadl_{\mathrm{B}}=g_{\mathrm{H}}l_{\mathrm{H}}^{2}\,\quadV_{\mathrm{B}}=\frac{g_{\mathrm{H}}^{2}l_{\mathrm{H}}^{4}}{V_{\mathrm{H}}}
\mathrm{d}\mu_{\,\mathrm{SU}(N)}=\frac{\pi^{2N-6}\,e^{-8\pi^{2}/g^{2}}}{2^{8}\,(N-1)!\,(N-2)!}\,(M_{\mathrm{PV}})^{3N}\left(\frac{8\pi^{2}}{g^{2}}\right)^{N}\!\!\mathrm{d}\rho^{2}\,\mathrm{d}^{4}x_{0}\,\mathrm{d}^{2}\theta_{0}\,\mathrm{d}^{2}\bar{\beta}\,\prod_{i=1}^{2N-4}\rho\,\mathrm{d}\bar{\xi}_{i}\,.
{\bfB}\equiv\left[\begin{array}{l}{B^{1}}\\{B^{2}}\\{B^{3}}\\\end{array}\right]=\left[\begin{array}{l}{X^{7}dX^{1}-X^{1}dX^{7}-X^{2}dX^{4}+X^{4}dX^{2}}\\{X^{1}dX^{4}-X^{4}dX^{1}-X^{2}dX^{7}+X^{7}dX^{2}}\\{X^{4}dX^{7}-X^{7}dX^{4}-X^{2}dX^{1}+X^{1}dX^{2}}\\\end{array}\right]\;\;.
I=\intd^{n}x~{\calL}=-\intd^{n}x\sqrt{-g}\left(\frac{1}{2}g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi+\frac{\mu^{2}}{2}\Phi^{2}+\frac{\lambda}{4!}\Phi^{4}\right)\;\;,
S=\intd^{4}x\mathrm{Tr}\bigg(-{\frac{1}{12}}{\widetildeH}_{\mu\nu\lambda}{\widetildeH}^{\mu\nu\lambda}-{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}+{\frac{m}{4}}\epsilon^{\mu\nu\rho\lambda}B_{\mu\nu}F_{\rho\lambda}\bigg).
(M_{\pm})_{\beta}^{\alpha}a_{\gamma}^{\varepsilon}=a_{\tau}^{\varepsilon}(R^{\pm})_{\sigma\gamma}^{\alpha\tau}(M_{\pm})_{\beta}^{\sigma}\,,\quad({\bar{M}}_{\pm}^{-1})_{\sigma}^{\alpha}(R^{\pm})_{\beta\tau}^{\sigma\gamma}{\bar{a}}_{\varepsilon}^{\tau}={\bar{a}}_{\varepsilon}^{\gamma}({\bar{M}}_{\pm}^{-1})_{\beta}^{\alpha}\,,\quad\varepsilon=+\,-\,.
\left({\frac{\dot{a}}{a}}\right)^{2}+{\frac{kc^{2}}{a^{2}}}={\frac{8\pim_{0}G_{5}}{3(1-\Omega_{R}^{2})}}\left[{\frac{\tilde{\varrho}_{1}\Omega_{R}+\varrho_{2}\Omega_{R}^{5}}{\sqrt{\Omega_{R}^{2}-\dot{R}^{2}/c^{2}}}}+V_{r}/c^{2}\right].
I_{2}\equiv{\frac{1}{2}}\left\{\int_{0}^{\Lambda}{\frac{kdk}{\sqrt{k^{2}+m_{F}^{2}}}}\left({\calP}\int_{-\infty}^{\infty}dq\,{\frac{P_{+}(q)}{k-q}}+2\mathrm{arctan}\left({\frac{K_{0}^{+}}{k}}\right)+[(+)\leftrightarrow(-)]\right)\right\}
S_{gauge}=-\,\frac{1}{g_{\mathrm{YM}}^{2}(\mu)}\intd^{4}x\left\{\frac{1}{2}\partial_{\alpha}\Phi^{i}\partial^{\alpha}\Phi^{i}+\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}\right\}+\frac{\theta_{\mathrm{YM}}}{32\pi^{2}}\intd^{4}xF_{\alpha\beta}{\tilde{F}}^{\alpha\beta}
(\gamma^{a})_{\alpha}^{~\beta}E_{a}^{\underline{\beta}}v_{\underline{\beta},\betaq^{\prime}}=E_{a}^{\underline{\beta}}(\eta^{ab}E_{b}^{\underline{a}}\Gamma_{\underline{a}})_{\underline{\beta}\underline{\gamma}}v_{\alphaq^{\prime}}^{~~~\underline{\gamma}}=0
\eta_{2}(\beta\sqrt{-\triangle})=\int_{0}^{\infty}{\frac{{\mathrm{d}}s}{s}}\left[\theta_{3}\Big(0,{\mathrme}^{-{\frac{\beta^{2}}{4s}}}\Big)-1\right]\frac{f(-s\triangle)-1-\frac16s\triangle}{(s\triangle)^{2}}.
S_{\mathrm{YM}}=S_{\mathrm{T}}+\intd^{4}z\,d^{4}\bar{z}\,\mathrm{tr}\Bigr\{\mathrm{~\frac{1}{6}~}\Omega_{ABCD}F_{\!~~~~ab}^{AB}F^{CDab}+\mathrm{~\frac{1}{6}~}\epsilon_{AB}F_{\!~~~~ab}^{AB}\epsilon_{CD}F^{CDab}\Bigr\}.
\tilde{I}_{v}[A_{k}]=-\frac12\intdV\left[\nabla^{\mu}A_{k}^{\nu}\nabla_{\mu}A_{k\nu}+R_{\mu\nu}A_{k}^{\mu}A_{k}^{\nu}+m_{v,k}^{2}A_{k}^{\mu}A_{k\mu}\right]~~~.
\Psi_{s,l}^{\gamma}(z,\bar{z})={\textstyle{\sqrt{\frac{(2\gamma+2s+1)\Gamma(s+1)\Gamma(2\gamma+s+1)}{4\pi\Gamma(l+s+1)\Gamma(2\gamma-l+s+1)}}}}\frac{z^{l}}{(1\!+\!z\bar{z})^{\scriptscriptstyle{\gamma}}}P_{s}^{(l,2\gamma-l)}\!\left(\frac{1\!-\!z\bar{z}}{1\!+\!z\bar{z}}\right).
\tilde{S}^{\prime}\[X^{\mu},\hat{P}_{(0)}{}^{\hat{\imath}},\hat{b}_{\hat{\imath}},\hat{\gamma}_{\hat{\imath}\hat{\jmath}}]=\intd^{3}\hat{\xi}\left[-{\hat{P}}_{(0)}{}^{\hat{\imath}}\partial_{\hat{\imath}}\hat{c}^{(0)}+\hat{\calL}\right]\,,
\phi(t^{\prime},r^{\prime},\theta^{\prime},z^{\prime})=\sum_{m^{\prime}}\intdq^{\prime}\,dk^{\prime}\,\left[b_{q^{\prime}m^{\prime}k^{\prime}}v_{q^{\prime}m^{\prime}k^{\prime}}(t^{\prime},r^{\prime},\theta^{\prime},z^{\prime})+b_{q^{\prime}m^{\prime}k^{\prime}}^{\dagger}v_{q^{\prime}m^{\prime}k^{\prime}}^{*}(t^{\prime},r^{\prime},\theta^{\prime},z^{\prime})\right],
\tilde{\Psi}_{y}=\tilde{\Psi}_{xx}+2(\alpha-1)(\tilde{\Psi}\tilde{\Psi}^{*})\Psi_{x}+2\alpha\tilde{\Psi}^{2}\tilde{\Psi}_{x}^{*}-\alpha(1+\alpha)(\tilde{\Psi}\tilde{\Psi}^{*})^{2}\tilde{\Psi}.
\left(\begin{array}{cc}{R^{\alpha\beta}}&{0}\\{0}&{R^{\alpha\beta}}\\\end{array}\right)-\frac{1}{2}\left(\begin{array}{cc}{g^{\alpha\beta}}&{0}\\{0}&{g^{\alpha\beta}}\\\end{array}\right)R=\left(\begin{array}{cc}{T^{\alpha\beta}}&{0}\\{0}&{T^{\alpha\beta}}\\\end{array}\right)
\mu=-\frac{a^{2}}{\beta}\frac{W(\zeta)}{\left(1+W(\zeta)\right)^{2}}~,\quad\frac{\nu^{\prime}}{\nu}=\frac{a^{\prime}}{a}\frac{W(\zeta)\left(5+3W(z\right))}{(1+W(\zeta))^{2}}~.
\left(\begin{array}{cc}{s}&{-r}\\{-q}&{p}\\\end{array}\right)\left(\begin{array}{cc}{a}&{b}\\{c}&{d}\\\end{array}\right)\left(\begin{array}{cc}{p}&{r}\\{q}&{s}\\\end{array}\right)=\left(\begin{array}{cc}{1}&{b_{\rho}}\\{0}&{1}\\\end{array}\right),
\Omega^{(1)}=\frac{\eta^{k-2}d\zeta}{\partialP/\partial\eta},\;\;\Omega^{(2)}=\frac{\eta^{k-3}d\zeta}{\partialP/\partial\eta},\;\;\Omega^{(3)}=\frac{\eta^{k-3}\zetad\zeta}{\partialP/\partial\eta},\;\;\ldots,\;\;\Omega^{(g)}=\frac{\zeta^{2k-4}d\zeta}{\partialP/\partial\eta}.
\Psi_{p}(x)=\int\frac{d^{4}q}{(2\pi)^{4}}\tilde{\psi}(q-p)e^{iq\cdotx}=e^{ip\cdotx}\int\frac{d^{4}\rho}{(2\pi)^{4}}\tilde{\psi}(\rho)e^{i\rho\cdotx}=e^{ip\cdotx}G(x)
D_{\mu(\alpha,\phi)}^{\nu^{\prime}}=(1-2\xi)D_{\mu}{\overline{{D}}}^{\nu^{\prime}}-\xi(D_{\mu}D^{\nu}+\overline{{D}}_{\mu^{\prime}}\overline{{D}}^{\nu^{\prime}})+(2\xi-\frac12)\delta_{\mu}^{\nu}D_{\sigma}\overline{{D}}^{\sigma^{\prime}}\,
\left\langleT_{ii}(x)\right\rangle=\frac{1}{4}\sum_{n}\omega_{n}\left|\phi_{n}\right|^{2}-\frac{1}{4}\sum_{n}\frac{1}{\omega_{n}}\left|\vec{\nabla}\phi_{n}\right|^{2}+\frac{1}{2}\sum_{n}\frac{1}{\omega_{n}}\left|\partial_{i}\phi_{n}\right|^{2}
\rho_{2}={\frac{1}{2}}\tau_{1}\psi^{2}+i\tau_{2}{\calF},\quad\rho_{4}={\frac{1}{24}}i\tau_{2}\psi^{4}+{\frac{1}{2}}\tau_{1}{\calF}\psi^{2}+{\frac{1}{2}}i\tau_{2}{\calF}^{2},\quad\mathrm{etc.},
h_{0}=\frac{l_{p}^{\prime9}}{R_{6}^{\prime}R_{7}^{\prime}R_{8}^{\prime}R_{9}^{\prime}R_{11}^{\prime}}\left(\frac{Q_{0}}{R_{11}^{\prime}}\right)\frac{1}{r^{3}}=\frac{Q_{0}l_{p}^{3}}{r^{3}}