harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tilde{R}^{ab}=R^{ab}-K_{c}^{a}\wedgeK^{cb}+\frac{1}{2}C_{\;\cdotc}^{ab}\tilde{T}^{c}+\frac{1}{4}\gamma_{J}^{I}\gamma_{L}^{K}\theta_{I}^{a}\theta_{K}^{b}A^{J}\wedgeA^{L}+i\left(DK^{ab}-\frac{1}{4}\gamma_{J}^{I}(\theta_{I}^{a}e^{b}+\theta_{I}^{b}e^{a})\wedgeA^{J}\right).
\rho_{\mathrm{grav}}={\frac{c_{\mathrm{ph}}}{c_{\mathrm{grav}}}}\rho_{\mathrm{ph}},\\\\\\\\\\\\p_{\mathrm{grav}}={\frac{c_{\mathrm{ph}}}{c_{\mathrm{grav}}}}p_{\mathrm{ph}}.
\begin{array}{ll}{\mathrm{Sector}\\}&{\mathrm{Field~content}}\\{\mathrm{even}}&{[1,1],\,[3,3],\,[5,5],\,[7,7],\,[9,9],\,}\\{\mathrm{odd}}&{[2,2],\,[4,4],\,[6,6],\,[8,8],\,}\\{\mathrm{twisted}}&{[1,9],\,[3,7],\,[5,5],\,[7,3],\,[9,1].}\\\end{array}
a\int_{p+1}C\wedged\,\mathrm{Tr}\{T\,\mathrm{e}^{2\pi\alpha^{\prime}{\calF}}\}\wedge\mathrm{e}^{2\pi\alpha^{\prime}{\hat{F}}+B}\wedge\sqrt{\hat{A}(R_{T})/\hat{A}(R_{N})}~,
S^{z}(z,\bar{z},\varphi)=\sum_{i=0}^{\infty}\frac{1}{i!}A_{i}^{z}(z,\bar{z})y^{i}\,,\quadS^{\bar{z}}(z,\bar{z},\varphi)=\sum_{i=0}^{\infty}\frac{1}{i!}A_{i}^{\bar{z}}(z,\bar{z})y^{i}
E^{2}(R)=2{\frac{j(j+1)}{k+2}}-{\frac{m^{2}+\bar{m}^{2}}{k}}+{\frac{1}{2k}}\left[(m-\bar{m}+kM)^{2}R^{2}+{\frac{(m+\bar{m}+kN)^{2}}{R^{2}}}\right]+N^{\prime}+\bar{N}^{\prime}
{\calH}=\frac{1}{2}\left[\pi^{i}\pi^{i}+F_{<ij>}F_{<ij>}+(H_{123})^{2}\right]+\frac{1}{2}m^{2}{\bfA}^{2}+m(A_{1}\pi_{23}+A_{2}\pi_{31}+A_{3}\pi_{12})-A_{0}G-B_{0i}G_{i}
\left(n^{A}({\calR}_{AB}-\frac{1}{2}{\calR}G_{AB}-8\pi{\bfG}{\calT}_{AB})y_{\,,\mu}^{B}\right)^{(1)}+\left(n^{A}({\calR}_{AB}-\frac{1}{2}{\calR}G_{AB}-8\pi{\bfG}{\calT}_{AB})y_{\,,\mu}^{B}\right)^{(2)}=0~.
S=\intd^{3}x\left[\frac{\zeta(-1/2)}{\sqrt{2}\pi}[B(x)]^{3/2}+{\frac{1}{\sqrt{2}(4\pi)^{2}}}{\frac{15}{16\pi}}\zeta(5/2){\frac{[\vec{\nabla}B(x)]^{2}}{[B(x)]^{3/2}}}+\dots\right]
\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{p}}\left[\gamma+\psi(1+kn)\right]=\frac{1}{k}\left(\frac{1}{2^{p}}-1\right)\zeta(p+1)+k^{p-1}\sum_{j=1}^{k}\,S_{p-1,2}\left(\omega^{j+1/2}\right).
\mathrm{Z}\approx{\bfN_{2}}\intd\Omegae^{\gamma\Omega^{00}(1-\frac{1}{2F^{2}}\frac{1}{V}\sum_{n}^{\prime}[1/p_{n}^{2}])}={\bfN_{3}}Y_{N}\left(\gamma\Big[1-\frac{N-1}{2F^{2}}G_{1}\Big]\right)
{\calG}_{{\calM}_{0}}=\left(\mu+\frac{g^{2}\lambda}{8\pir}\right)\,[dr^{2}+r^{2}\sigma_{1}^{2}+r^{2}\sigma_{2}^{2}]+\left(\frac{g^{2}\lambda}{8\pi}\right)^{2}\left(\mu+\frac{g^{2}\lambda}{8\pir}\right)^{-1}\sigma_{3}^{2},
{\calL}={\frac{m}{2}}\,\epsilon^{\mu\nu\lambda}A_{\mu}^{a}(\partial_{\nu}A_{\lambda}^{a}+{\frac{g}{3}}f^{abc}A_{\nu}^{b}A_{\lambda}^{c})-F^{a}(\partial_{\mu}A^{\mua})+\partial^{\mu}\overline{{c}}^{a}(D_{\mu}c^{a})
F(T)=-\frac{\xi_{0}^{2}x_{0}^{4}t}{8\pia}\sum_{l=1}^{\infty}\left[\frac1{\nu(x_{0}^{2}+l^{2})^{2}}+\sum_{n=1}^{\infty}\frac{2\nu^{5}}{(x_{0}^{2}+l^{2}+n^{2}t^{2})^{2}(\nu^{2}+n^{2}t^{2})^{3}}\right],
\left(\dot{m}\right)^{\prime\prime}+\frac{1}{p}\left(2-\frac{1}{N\pi^{2}}(1\!+\!\xi)\right)\left(\dot{m}\right)^{\prime}-\frac{2}{N\pi^{2}}\frac{\xi}{p^{2}}\dot{m}-\frac{1}{N\pi^{2}}\frac{1}{p}m^{\prime}-\frac{2}{N\pi^{2}}m=0;
O=\left(\begin{array}{cc}{O_{11}}&{O_{12}}\\{O_{21}}&{O_{22}}\\\end{array}\right),\quad\begin{array}{ll}{O_{11}=\etaO\eta,}&{O_{12}=\etaO(1-\eta),}\\{O_{21}=(1-\eta)O\eta,}&{O_{22}=(1-\eta)O(1-\eta).}\\\end{array}
F_{\alpha\rho}=f_{\rho,\alpha}^{c}-f_{\alpha,\rho}^{c}=G_{\alpha\rho}-\int\frac{\partialc^{\mu}}{\partialx^{\rho}}\frac{\partialc^{\sigma}}{\partialx^{\alpha}}\frac{\partialc^{\nu}}{\partialt}\cdot[G_{\mu\nu,\sigma}+G_{\nu\sigma,\mu}+G_{\sigma\mu,\nu}]dt
F_{0,0,2}(0,d_{2};0,0,n_{1},n_{2})={\frac{1}{d_{2}}}{\frac{(\lambda_{1}-\lambda_{2})(\lambda_{2}-2\lambda_{1})}{\lambda_{1}^{d_{2}}}}\prod_{j=1}^{2}{\frac{\prod_{l=1}^{n_{j}-1}\left((\lambda_{2}-2\lambda_{1})n_{j}+l\lambda_{1}\right)}{(n_{j}-1)!}}.
\frac{\delta<H_{A}^{\theta}>}{\deltaG(k)}=\frac{1}{4}[k^{2}-G^{-2}(k)+\frac{4\pi^{2}}{g^{2}}(2zk^{-2}G^{-3}(k)-\frac{\deltaz}{\deltaG(k)}4\pi^{2}\int\frac{d^{2}p}{(2\pi)^{2}}p^{-2}G^{-2}(p))]
\lambda_{1}=(\frac{\theta_{1100}\theta_{1000}}{\theta_{0100}\theta_{0000}})^{2},\;\lambda_{2}=(\frac{\theta_{1001}\theta_{1100}}{\theta_{0001}\theta_{0100}})^{2},\;\lambda_{3}=(\frac{\theta_{1001}\theta_{1000}}{\theta_{0001}\theta_{0000}})^{2},
{\mathcalP}^{(N_{\alpha})}\left\{t_{(\alpha)},t_{(\alpha+1)}\right\}:\left\{t_{\alpha,j_{0}}\equivt_{(\alpha)},t_{\alpha,j_{1}},\ldots,t_{\alpha,j_{N_{\alpha}-1}},t_{\alpha,j_{N_{\alpha}}}\equivt_{(\alpha+1)}\right\}\;.
I_{B}=\frac{1}{\widetildeC_{B}}(-1)^{\frac{S-\widetilde{S}}{2}}\:\sum_{m_{j}^{\prime}=0}^{2k_{j}+3}\delta_{\frac{M-\widetilde{M}}{2}+\sum\frac{K^{\prime}(m_{j}^{\prime}+1)}{2k_{j}+4}}^{(K^{\prime})}\prod_{j=1}^{r}\:n_{L_{j}\widetilde{L}_{j}}^{-m_{j}^{\prime}-1}\:,
W[A,\psi,{\bar{\psi}}]\;=\;\intd^{4}x\left(\frac{1}{2}[(\partial_{0}A^{i}-\partial^{i}A_{0})^{2}-B_{i}^{2}]+\bar{\psi}(i{\hat{\partial}}-m)\psi+eJ_{\mu}A^{\mu}\right),
<\tilde{T}_{J,J}\tilde{W}_{J_{1},m_{1}}\prod_{i=2}^{N}\tilde{T}_{J_{i},J_{i}}>=\frac{(J_{1}-m_{1})(J+J_{1}-1)}{J_{1}}<\tilde{W}_{J+J_{1}-1,J+m_{1}}\prod_{i=2}^{N}\tilde{T}_{J_{i},J_{i}}>
\frac{\partial^{2}}{\partialX_{\mu_{0}}\partialX^{\mu_{0}}}B^{\mu_{1}\mu_{2}\ldots\mu_{p}}=-\sum_{i=1}^{p}(-1)^{p\cdoti}\frac{\partial}{\partialX_{\mu_{i}}}\frac{\partial}{\partialX^{\mu_{0}}}B^{\mu_{i+1}\mu_{i+2}\ldots\mu_{p}\mu_{0}\mu_{1}\ldots\mu_{i-1}}-\frac{g}{e}\frac{\partial}{\partialX^{\mu_{0}}}\phi^{\mu_{0}\mu_{1}\ldots\mu_{p}},
S=\intdt\left\{\,\int_{\gamma_{t}}dy^{a}\dot{A}_{a}(y)+{\textstyle{\frac{1}{2}}}\,\intd^{3}xF_{ab}(x)F_{ab}(x)+g^{2}\int_{\gamma_{t}}dy^{a}\int_{\gamma_{t}}dy^{a}f_{\epsilon}(y-y^{\prime})\,\right\}
I^{(m,n)}=\frac{\pi}{\epsilon^{2}}\overline{{\Theta}}_{\Lambda_{24}}\Theta_{(m,n)}-\frac{\pi^{2}}{\tau_{2}}\overline{{\Theta}}_{\Lambda_{24}}\Theta_{(m,n)}-\tau_{2}\overline{{B}}_{14}B_{6}^{(m,n)}+O(\epsilon^{2}).
S_{fi}^{(1)}=-i\intdt\,d{\bfx}\,\Psi_{{\bfp}^{^{\prime}}}^{\dagger}({\bfx})\,\tau_{3}\,H_{\mathrm{int}}({\bfx,t)\,}\Psi_{{\bfp}}({\bfx})=-i\,2\pi\,\delta(w_{{\bfp}^{^{\prime}}}-w_{{\bfp}})\,{\calA}_{fi}^{(1)}\;\;
{\calC}={\calC}^{-1}=\left(\begin{matrix}{1}&{0}&{0}&{0}\\{0}&{-1}&{0}&{0}\\{0}&{0}&{-1}&{0}\\{0}&{0}&{0}&{-1}\\\end{matrix}\right)\quad,\quad{\calC}\alpha^{\mu}{\calC}^{-1}=\overline{{\alpha}}^{\mu\,\,\ast}
\ddot{\deltas}+3H\,\dot{\deltas}+\left(\frac{k^{2}}{a^{2}}+3\dot{\theta}^{2}+V_{ss}+\Gamma_{\alpha}\right)\deltas=-\frac{V_{s}}{\dot{\sigma}^{2}}\frac{\mp^{2}\,k^{2}}{2\pia^{2}}\,\Phi\,,
a_{D}=\partial_{a}{\calF}={\frac{\sqrt2}{\pi}}\int_{1}^{u}{\frac{dx\sqrt{x-u}}{\sqrt{x^{2}-1}}},\qquada={\frac{\sqrt2}{\pi}}\int_{-1}^{1}{\frac{dx\sqrt{x-u}}{\sqrt{x^{2}-1}}}.
P_{n}^{~n^{\prime}}=\frac{1}{\sqrt{k+2}}e^{\piinn^{\prime}/2(k+2)}E_{n+n^{\prime}+k+2}~,~~P_{s}^{~s^{\prime}}=\frac{1}{\sqrt{2}}e^{-\piiss^{\prime}/4}E_{s+s^{\prime}+2}~.
{\calE}_{nl}(t)=\frac18\int_{0}^{1}d\rho\left(e^{\frac{2-\rho}{8}{\partial}^{2}}(-q^{2}{\partial}^{2}+1)\Psi\right)\overleftrightarrow{\partial}\left(\partiale^{\frac{\rho}{8}{\partial}^{2}}\Psi\right).
\frac{2\,J(r)^{2}K(r)^{2}}{\mathcal{L}(r)^{2}}=\frac{\left[M(r)^{4}\mathcal{L}(r)^{2}J^{\prime}(r)\right]^{\prime}}{M(r)^{4}\mathcal{L}(r)^{2}}J(r)-\alphaJ(r)^{2}\left(J(r)^{2}-1\right)\,.
\begin{array}{c}{\frac{1}{e^{2\omegaX}+e^{-2\omegaT}}\left[\left(\partial_{X}S\right)^{2}-\left(\partial_{T}S\right)^{2}\right]+\left(\omega_{o}z-\partial_{y}S\right)^{2}+\left(\partial_{z}S\right)^{2}+m^{2}=0}\\\end{array}
a_{(\pm)k}[T]={\frac{1}{2\pi\sqrt{k}}}\int_{0}^{2\pi}e^{\pmikx}T_{,x}^{\pm}\left[\pm{\frac{ip}{\sqrt{2}}}\pm\sum_{n=1}^{\infty}\sqrt{n}(a_{(\pm)n}e^{-inT^{\pm}(x)}-a_{(\pm)n}^{\dagger}e^{inT^{\pm}(x)})\right]\,dx,
\frac{d^{2}f}{dx^{2}}-\frac{(x^{3}-2)}{x(1-x^{3})}\frac{df}{dx}-\frac{l(l+3)}{x^{2}(1-x^{3})}f+\frac{\lambda^{2}}{x(1-x^{3})^{2}}\left(1+\frac{\rho_{0}^{3}}{(\omegaR)^{3}x^{3}}\right)f=0\,,
\begin{array}{llll}{|1>}&{=\sum_{x}e^{\imath\thetax}[\gamma(-\theta)|x_{\,-}>+\;|x_{\,0}>],\;\;}&{E_{1}}&{=1+\gamma(\theta)\gamma(-\theta)}\\{|2>}&{=\sum_{x}e^{\imath\thetax}[|x_{\,-}>-\;\gamma(\theta)|x_{\,0}>],\;\;}&{E_{2}}&{=0,}\\\end{array}
\bar{T}_{E\lambda}^{\rho}=T_{0\lambda}^{\rho}+\frac{\delta\bar{S}}{\delta\Phi}\bar{s}_{\lambda}^{\rho}+\partial_{\mu}\Sigma_{\lambda}^{\mu\rho}+\partial_{\sigma}R_{\lambda}^{[\sigma\rho]}+\delta_{\lambda}^{\rho}k_{\sigma}^{\sigma},
\begin{array}{lcl}{a_{2}}&{=}&{\frac{-1}{6}u_{2}}\\{a_{3}}&{=}&{\frac{1}{3}\thetau_{2}^{\prime}}\\{a_{4}}&{=}&{-\frac{4}{9}{\theta}^{2}u_{2}^{\prime\prime}+\frac{7}{72}u_{2}^{2}}\\{a_{5}^{\prime}}&{=}&{\frac{4}{9}{\theta}^{3}u_{2}^{(4)}-\theta(\frac{17}{18}u_{2}^{\prime\prime}u_{2}+\frac{7}{9}{u_{2}^{\prime}}^{2})+\frac{1}{9}u_{3}^{\prime}u_{2}}\\\end{array}
\nabla^{2}\vec{B}-\frac{\partial^{2}\vec{B}}{\partialt^{2}}=\vec{\nabla}\times\left(g\\frac{\partialf}{\partialt}\\vec{E}-g\vec{\nabla}f\times\vec{B}+\Gamma_{\mu\beta}^{i}F^{\mu\beta}\right)\.
R_{\alpha{\bar{\beta}}{\nu}{\bar{\mu}}}=G_{\alpha{\bar{\beta}}}G_{{\nu}{\bar{\mu}}}-W_{\alpha\nu}G^{s{\bar{s}}}(e^{G}{\bar{W}}_{{\bar{\beta}}{\bar{\mu}}{\bar{s}}}-T_{{\bar{\beta}}{\bar{\mu}}{\bar{s}}}),
C^{\mu\nu}=\bar{\eta}\left(\bar{\eta}^{\mu\nu}+\bar{F}^{\mu\nu}+\bar{\eta}^{\mu\alpha}\bar{\eta}^{\beta\gamma}\bar{\eta}^{\delta\nu}\bar{F}_{\alpha\beta}^{*}\bar{F}_{\gamma\delta}^{*}+\bar{\eta}^{\mu\alpha}\bar{\eta}^{\beta\gamma}\bar{F}_{\alpha\beta}^{*}\bar{F}_{\gamma\delta}^{*}\bar{F}^{\delta\nu}\right),
\rho_{2}^{2}=-\frac{G}{4G_{5}^{2}}\big(\frac{1}{2}\mathrm{tr}\,(\tilde{\calH}^{2})^{2}-\mathrm{tr}\,(\tilde{\calH}^{4})\big)=-G\big(\frac{1}{2}z_{1}^{2}-z_{2}\big).
\intd^{2}\mathrm{\bf~x}\,\varrho(\mathrm{\bf~x})=\intd^{2}\mathrm{\bf~x}\,\Big(\sqrt{\frac{\kappa}{2}}f_{ij}f^{ij}+\sqrt{\frac{\kappa}{8}}(V+\sqrt{2\kappa}Af)_{ij}{}^{k}(V+\sqrt{2\kappa}Af)^{ij}{}_{k}
\tilde{\Gamma}_{t}^{(4)}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4})=\tilde{\Gamma}_{t,s=0}^{(4)}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4})\;,
\tilde{A}_{1}(p)=-g^{2}N(N^{2}-1)\delta_{b}^{a}\delta_{J}^{I}\int\frac{d^{4}k}{(2\pi)^{4}}d^{4}\theta\,\frac{1}{k^{2}[(p-k)^{2}+m^{2}]}\left\{\Phi_{aI}^{\dagger}(p,\theta,{\overline{\theta}})\Phi^{bJ}(-p,\theta,{\overline{\theta}})\right\}.
v~=~{\frac{r_{0}^{2}}{r^{2}}};~~~A~=~{\frac{\omega^{2}}{4}}\left({\frac{r_{1}r_{5}r_{n}}{r_{0}}}\right)^{2};~~B~=~{\frac{\omega^{2}}{4}}\left({\frac{r_{1}r_{5}}{r_{0}}}\right)^{2}.
T_{k,l}:e^{2\pi{i}t}\rightarrow\left\{\left[\begin{array}{ccc}{e^{2\pi{i}kt}}&{0}&{0}\\{0}&{e^{2\pi{i}lt}}&{0}\\{0}&{0}&{e^{-2\pi{i}(k+l)t}}\\\end{array}\right]\mid{t}\in\mathbb{R}/\mathbb{Z}\right\}
\begin{array}{lcr}{\sum_{i=0}^{r}q_{i}^{a}V_{i}+\sum_{l=r+1}^{r+4}q_{l}^{a}V_{l}=0}\\{\sum_{i=0}^{r}q_{i}^{a}+\sum_{l=r+1}^{r+4}q_{l}^{a}=0.}\\\end{array}
S(\alpha,\psi)=S_{0}(\Psi(\alpha,\psi))=\bar{\psi}(i\gamma^{\mu}\partial_{\mu}+\gamma^{\mu}V_{\mu}+\gamma^{\mu}\gamma^{5}A_{\mu}+\alpha{\frac{f_{\pi}}{m}}\gamma^{\mu}\gamma^{5}\partial_{\mu}\varphi)\psi,
[\Sigma_{\lambda\alpha}]_{ab}=i\int\frac{d^{4}S}{(2\pi)^{4}}(-ie\gamma_{\beta\alpha}^{\mu})(-ig_{\mu\nu}[\tau_{3}D(S)\tau_{3}]_{ba})(-ie\gamma_{\lambda\gamma}^{\nu})i(P+S)_{\tau}\gamma_{\gamma\beta}^{\tau}[D(P+S)]_{ab}.
F={\frac{1}{Z_{0}}}\e^{-\int\mathrm{d^{d}}x\left[{\frac{1}{2}}\left(\partial_{\mu}\phi\right)^{2}+{\frac{1}{2}}m^{2}\phi^{2}\right]-{\frac{\lambda}{4}}\int\mathrm{d^{d}}x\phi^{4}}
\left(\tilde{M}_{\xi}^{2}(X)\right)_{\mathrm{\scriptsize{I}}}+\left(\tilde{M}_{\xi}^{2}(X)\right)_{\mathrm{\scriptsize{II}}}=M_{\xi}^{2}(X)-\mu^{2}\;\;\;\;\;\;(\xi=\pi,\,\sigma).
\Biggl(\phi_{1}(x),{\binom{\phi_{2}(x)}{\phi_{3}(x)}},{\binom{\phi_{4}(x)}{\phi_{5}(x)}}\Biggr)\rightarrow\Biggl(\phi_{1}(x),f(x){\binom{\phi_{2}(x)}{\phi_{3}(x)}},f(x){\binom{\phi_{4}(x)}{\phi_{5}(x)}}\Biggr).
\left[a_{\left[k\right]}^{\mu_{m}}\left(p\right),a_{\left[l\right]}^{\mu_{n}}\left(q\right)\right]=\left[a_{\left[k\right]}^{+\mu_{m}}\left(p\right),a_{\left[l\right]}^{+\mu_{n}}\left(q\right)\right]=0\;\,.
\left[{\frac{m^{2}}{a^{2}}}+\partial_{z}^{2}+{\frac{a^{\prime}}{a}}\partial_{z}-{\frac{a^{\prime}}{a}}(g\chi\tau_{1}+\mu\tau_{2})-g\chi^{\prime}\tau_{1}-(g^{2}\chi^{2}+\mu^{2})\right]\Psi_{L}=0.
\sum_{l_{2}=2,4,..}^{*}{\frac{l_{2}^{2}}{(l_{1}^{2}-l_{2}^{2})^{3}}}={\frac{1}{16}}\sum_{k=1}^{*}{\frac{k^{2}}{((l_{1}/2)^{2}-k^{2})^{3}}}.
S=\intdt\int_{0}^{L}dx\left\{\partial_{0}A^{1}\pi_{1}+\frac{1}{2}i\psi^{\dagger}\partial_{0}\psi-\frac{1}{2}i\partial_{0}\psi^{\dagger}\psi-{\calH}-A^{0}\phi+\partial_{1}(A^{0}\pi_{1})\right\}\,
\alpha_{1}(t)=\frac{m}{\sqrt{\lambda}}\left\{b_{1}e^{i\omega_{1}t}+\frac{3\pi}{8}\left(-3+\frac{1}{2}\omega_{1}^{2}t^{2}-\frac{1}{8}\omega_{1}^{4}t^{4}\right)\right\},
\left(\begin{matrix}{h^{T}}&{\tilde{h}^{T}}\\\end{matrix}\right)\left(\begin{matrix}{(\bar{\Sigma}-\Sigma)^{-1}}&{(\Sigma-\bar{\Sigma})^{-1}\Sigma}\\{\bar{\Sigma}(\Sigma-\bar{\Sigma})^{-1}}&{\bar{\Sigma}(\bar{\Sigma}-\Sigma)^{-1}\Sigma}\\\end{matrix}\right)\left(\begin{matrix}{h}\\{\tilde{h}}\\\end{matrix}\right)=-i.
\Lambda_{vac}(t)=\Lambda_{0}+\frac{2\,m_{F}^{4}}{(4\pi)^{2}}\;\;\int_{0}^{t}U^{2}(t)\,dt-\frac{2}{(4\pi)^{2}}\sum_{i}\,N_{i}\int_{o}^{t}m_{i}^{4}(t)\,dt\,,
p_{M}\to\overline{{a}}_{M}^{N}\circp_{N}=p_{N}\overline{{a}}_{M}^{N}-\frac{1}{2}[p_{N},\overline{{a}}_{M}^{N}]=p_{N}\overline{{a}}_{M}^{N}+\frac{i\hbar}{2}(\partial_{N}\overline{{a}}_{M}^{N}+b_{M}^{P}\partial_{P}\overline{{a}}_{M}^{N})\,,
{\calA}_{\mathrm{QED}}[\bar{\psi},\psi,A]=\intd^{4}x\,\left[\bar{\psi}_{\alpha}(i\gamma_{\alpha\beta}^{\mu}\partial_{\mu}+m)\psi_{\beta}+\frac{1}{2}A_{\mu}(-\partial^{2})A^{\mu}-e\bar{\psi}_{\alpha}\gamma_{\alpha\beta}^{\mu}A_{\mu}\psi_{\beta}\right]
V^{\prime\prime}(A_{0})=\frac{2}{\ell^{2}}-\frac{2\,A_{0}^{2}/\ell^{4}}{k+A_{0}^{2}/\ell^{2}}-\frac{2\,k}{A_{0}^{2}}[(d-2)+(d-1)\,v_{s_{0}}^{2}].
\begin{array}{rcl}{F^{(n)}(u_{0}|\{u_{j}\}_{1\leqslantj\leqslant2n}|\{v_{a}\}_{1\leqslanta\leqslant(r+1)n})^{\varepsilon}}&{=}&{\displaystyle\sum_{k=0}^{2r-1}(\sqrt{-1})^{k\sum_{j=0}^{2n}\varepsilon_{j}}\displaystyle\prod_{j=1}^{2n}h_{\varepsilon_{j}}(k+j+u_{0}+u_{j}-2v_{j})}\\{}&{\times}&{\displaystyle\prod_{i=1}^{(r-1)n}\{v_{2n+i}-u_{0}+\frac{1}{2}-k-2(n+i)\}.}\\\end{array}
\tilde{S}_{\frac{SU(3)\otimesSU(3)}{SU(2)}}=\left(\begin{array}{rrrrr}{-1}&{-(\sigma_{12},2)}&{1}&{-(\sigma_{12}/2,3)}&{1}\\{(\sigma_{21},2)}&{-1}&{-(\sigma_{32},2)}&{-(\sigma_{21}/2,1)}&{-(\sigma_{23}/2,1)}\\{1}&{(\sigma_{23},2)}&{-1}&{1}&{-(\sigma_{32}/2,3)}\\{-(\sigma_{21}/2,1)}&{-(\sigma_{12}/2,3)}&{1}&{-1}&{1}\\{1}&{-(\sigma_{32}/2,3)}&{-(\sigma_{23}/2,1)}&{1}&{-1}\\\end{array}\right).
{\hat{x}}^{n}=x^{n}\quad,\qquad{\hat{p}}_{n}=-i\frac{\partial}{\partialx^{n}}\quad,\qquad{\hat{\psi}}_{n}={\textstyle{\frac12}}e^{\frac{3\pii}{4}}\gamma_{n}\quad,\qquad\hat{\xi}=\xi\quad,\qquad{\hat{p}}_{\xi}=i\frac{\partial}{\partial\xi}\quad,
{\cal{P}}_{L}=\frac{{\cal{F}}_{0}}{{\cal{F}}_{\infty}}=\frac{\piLk^{+2}N^{L}\omega^{2L+1}}{l^{2}\rho_{+}2^{2L}(\rho_{+}-\rho_{-})^{2}(\Gamma(L+2)\Gamma(L+1))^{2}|E_{1}|^{2}}
Z(s)=\sum_{n_{1},n_{2}}\left(\tilde{n}_{1}^{2}+\frac{1}{|\tau|^{2}}\tilde{n}_{2}^{2}\right)^{-s}=\frac{1}{\Gamma(s)}\sum_{n_{1},n_{2}}\int_{0}^{\infty}\frac{dt}{t}t^{s}e^{-(\tilde{n}_{1}^{2}+\frac{1}{|\tau|^{2}}\tilde{n}_{2}^{2})t}\equiv\frac{1}{\Gamma(s)}J(s),
u_{1}^{\prime\prime}+\left(\delta^{2}+\frac{\tilde{\beta}(2\delta+1)-1}{2\,\xi}+\frac{(\tilde{\beta}+2\delta)^{2}-1}{4\,\xi^{2}}+\frac{1-\tilde{\beta}}{1-\xi}+\frac{(3/4}{(\xi-1)^{2}}\right)u_{1}=0.
J_{i}\equiv\bar{\psi}\gamma_{i}\psi=-e\kappa\LambdaA_{i}+\frac{1}{e\kappa\Lambda}\left[2\tilde{E}_{i}(A\tilde{E})-E^{2}A_{i}+4(p_{1}+p_{2})m\Lambda\tilde{E}_{i}\right]
A~=~\mathrm{diag}~\left(~a_{1}~\sigma_{2},~a_{2}~\sigma_{2},\ldots,~a_{N/2}~\sigma_{2}~\right)~,~\mathrm{with}~|v_{i}|^{2}~=~2~|a_{i}|^{2}~+~c~.
\hat{V}=\frac{1}{3}\left(\begin{array}{ccc}{1}&{\alpha}&{\alpha^{*}}\\{1}&{\alpha^{*}}&{\alpha}\\{1}&{1}&{1}\\\end{array}\right)\left(\begin{array}{ccc}{\hat{V}_{D}^{11}}&{0}&{0}\\{0}&{\hat{V}_{D}^{22}}&{0}\\{0}&{0}&{\hat{V}_{D}^{33}}\\\end{array}\right)\left(\begin{array}{ccc}{1}&{1}&{1}\\{\alpha^{*}}&{\alpha}&{1}\\{\alpha}&{\alpha^{*}}&{1}\\\end{array}\right)
\left(Q_{2}^{\dagger}\right)^{31}Q_{2}^{13}\u_{A}({r})={\frac{1}{m}}\left[-{\frac{\partial^{2}}{\partialr^{2}}}-{\frac{1}{r}}{\frac{\partial}{\partialr}}+{\frac{\left(2\ell+1-\lambda\right)^{2}}{r^{2}}}+\lambda{\frac{\delta(r)}{r}}\right]u_{A}=Eu_{A}({r})
h_{3}=\frac{1}{12\kappa^{2}}\frac{\alpha_{1}^{4}+\alpha_{2}^{4}+\alpha_{3}^{4}}{\alpha_{1}^{2}\alpha_{2}^{2}\alpha_{3}^{2}}\left(P_{\perp}^{2}-P_{\parallel}^{2}\right)~,
\vec{n}_{\pm}=\left(\frac{1}{\xi^{2}}\right){\frac{\vec{\partial}}{\partial\tau}}\mp\left(\frac{\alpha}{Y_{0}}\right)\left[{\frac{\vec{\partial}}{\partialy}}\pm\left(\frac{\alpha}{Y_{0}}\right){\frac{\vec{\partial}}{\partial\tau}}\right]\!.
\deltaG_{\mu\nu}^{[NS1]}\simg^{2}|Z|{\frac{1}{r^{7-p}}}\rightarrow0\quad,\qquad\deltaG_{\mu\nu}^{[Rp]}\simg^{2}\frac{|Z|}{g}{\frac{1}{r^{7-p}}}\rightarrow0\quad.
\delta_{\ell}\simeq-\frac{1}{4}\intd^{2}x\chi_{\ell}^{\dagger}U\chi_{\ell}\,,\qquad\delta_{-\ell}\simeq-\frac{1}{4}\intd^{2}x\chi_{\ell}U\chi_{\ell}^{\dagger}\qquad\mathrm{for~\ell\ge0}\,.
^{+}{\calS}_{1}=\frac{i}{2}{^+\tau}\int_{M}eF^{\mu\nui}F^{\alpha\betaj}g_{ij}\,R_{\mu\nu}^{ab}\,R_{\alpha\beta}^{cd}\epsilon_{abcd}-\frac{1}{2}{^+\tau}\int_{M}eF^{\mu\nui}F^{\alpha\betaj}g_{ij}\,R_{\mu\nu}^{ab}\,R_{\alpha\beta}^{cd}\delta_{ab,cd}\,,
\Phi_{n}=\left(\begin{matrix}{0}&{h}&{0}&{\ldots}&{0}\\{{\frac{a_{1}}{h}}}&{\ddots}&{\ddots}&{\ddots}&{\vdots}\\{{\frac{a_{2}}{h^{2}}}}&{\ddots}&{\ddots}&{h}&{0}\\{\vdots}&{\ddots}&{{\frac{a_{1}}{h}}}&{0}&{h}\\{{\frac{a_{n-1}}{h^{(n-1)}}}}&{\ldots}&{{\frac{a_{2}}{h^{2}}}}&{{\frac{a_{1}}{h}}}&{0}\\\end{matrix}\right)~dz\quad,
H\equivi\partial_{+}=P^{-}=\frac1{2p^{+}}\left((p^{I})^{2}-m_{1}^{2}\sum_{i=1}^{4}\partial_{p^{i}}^{2}-m_{2}^{2}\sum_{i^{\prime}=5}^{8}\partial_{p^{i^{\prime}}}^{2}\right)-\frac{\alpha}{2},
\begin{array}{l}{\displaystyley^{0}=\frac{\xi^{n+1}}{\xi^{0}}~,}\\{\displaystyley^{i}=\frac{\xi^{i}}{\xi^{0}}~,~~~~~~(i=1,~2,~\cdots,~n;~~\xi^{0}\not=0)~.}\\\end{array}
\chi_{12}(p_{30})\,=\,\beta_{1}\beta_{2}\,T_{12}^{0}(p_{30})\,\intd\,p_{120}\,G_{01}G_{02}\,\beta_{1}\beta_{2}\,[\,\,\chi_{23}(p_{10})\,+\,\chi_{31}(p_{20})\,]
F(t,V)=\sum_{g=0}^{\infty}\sum_{h=0}^{\infty}\sum_{d_{1},\ldots,d_{h}}^{\infty}\lambda^{2g-2+h}F_{g;d_{1},\ldots,d_{h}}(t)\prod_{i=1}^{h}\mathrm{tr}V^{d_{i}},
R_{2}^{1}=-2i\frac{dg_{1}\wedged\overline{{g}}_{1}}{(1-g_{1}\overline{{g}}_{1})^{2}},~~~R_{4}^{3}=+2i\frac{dg_{2}\wedged\overline{{g}}_{2}}{(1+g_{2}\overline{{g}}_{2})^{2}},
\beta={\frac{1}{T_{H}}}={\frac{2\pir_{+}^{2}\sqrt{\prod_{i=1}^{3}(r_{+}^{2}+q_{i})}}{2r_{+}^{6}+r_{+}^{4}(1+\sum_{i=1}^{3}q_{i})-\prod_{i=1}^{3}q_{i}}}\.
\alpha={\frac{M^{2}}{g^{2}\left(\lambda_{1}^{2}-\lambda_{2}^{2}\right)}}\left({\frac{M^{2}}{g^{2}}}F\left(\lambda_{1}^{2},\lambda_{2}^{2},\Lambda^{2}\right)+2ln{\frac{\lambda_{1}^{2}+m_{0}^{2}}{\lambda_{2}^{2}+m_{0}^{2}}}\right)
{\calS}\,=\,\int\,dt\,\left[\frac{2\,\lambda\,\dot{a}^{i}\,\dot{a}^{i}}{a^{i}\,a^{i}}\,-\,\frac{2\,\lambda\,({a^{i}}\dot{a}^{i})^{2}}{(\,a^{i}\,a^{i}\,)^{2}}\right]\;\;,
ds^{2}=\left({\frac{{\rho}^{2}-2Mr+q^{2}}{{\rho}^{2}}}\right)dt^{2}-{\frac{{\rho}^{2}}{\Lambda}}dr^{2}-{\rho}^{2}d{\theta}^{2}+{\frac{2A{sin}^{2}\theta(2Mr-q^{2})}{{\rho}^{2}}}dtd\varphi-{\frac{{sin}^{2}\theta}{{\rho}^{2}}}{\Sigma}^{2}d{\varphi}^{2}
U(|\vec{x}|)\simG_{4}\frac{m^{*}}{|\vec{x}|}\left[1+\frac12\left(\frac{r_{H}}{|\vec{x}|}\right)^{2}+\sum_{\ell\neq0}\left(\frac{r_{H}}{|\vec{x}|}\right)^{\delta+2+2\ell}\right].
\Phi({\bfr}_{1}^{(1)},\ldots,{\bfr}_{n_{M}}^{(M)},t)=e^{-\frac{i}{\hbarc}\Lambda}\bar{\Phi}({\bfr}_{1}^{(1)},\ldots,{\bfr}_{n_{M}}^{(M)},t)\,,
\frac{F}{V}=-\sum_{\stackrel{n=1}{n~\mathrm{odd}}}^{\infty}\sum_{k=1}^{\infty}\sum_{s=0}^{k-1}\frac{1}{k^{2}}\left(\frac{1}{4\pi^{2}\alpha^{\prime}\tau_{2}}\right)^{5}2^{9}\left|\theta_{4}\left(0,2\tau\right)\right|^{-16}e^{-n^{2}\beta^{2}/4\pi\alpha^{\prime}\tau_{2}}
\zeta_{m}\left(s\right)=\sum_{N=1}^{\infty}\int_{-\infty}^{+\infty}\frac{\Omega}{\left(2\pi\right)^{3}a}d^{3}k\left[k^{2}+\frac{\pi^{2}N^{2}}{a^{2}}+\frac{\lambda}{2}\phi_{c}^{2}\right]^{-s}.
\{T_{a},T_{b}\}=\varepsilon_{abc}((1-\frac{me\gamma}{s})F^{c}-(\frac{m^{2}}{s}+\frac{2me\gamma}{s}(Fn))n^{c}+\frac{me\delta}{s}F^{ck}n_{k})\equiv\varepsilon_{abc}N^{c}
L_{eff}=L_{eff}^{0}+\gamma^{a\mu\nu}(x)(\hat{\Theta}_{\mu\nu0}^{a}(x)+\theta\hat{\Theta}_{\mu\nu1}^{a}(x))+\zeta^{a\rho}(x)\hat{\Psi}_{\rho0}^{a}(x)
i\left<\bar{A}_{\mu}^{a}(p)\tilde{A}_{\rho}^{b}(q)\tilde{A}_{\sigma}^{c}(r)\right>_{\mathrm{bare}}=gf^{abc}[(q-r)_{\mu}g_{\rho\sigma}+(r-p)_{\rho}g_{\sigma\mu}+(p-q)_{\sigma}g_{\mu\rho}].
\partial_{f}^{(\tau\beta)}\partial_{(\tau\beta)}^{f}a^{(\lambda\alpha)}-(1-\zeta_{0}^{-1})\partial_{f}^{(\lambda\alpha)}\partial_{(\tau\beta)}^{f}a^{(\tau\beta)}=-\frac{1}{2}\sqrt{-g}\frac{\partialg^{(\tau\beta)(\mu\gamma)}}{\partiala_{(\lambda\alpha)}}T_{(\tau\beta)(\mu\gamma)}.
F(x^{-},x^{+})=\sum_{i}\left|{\frac{L}{\pi}}\langle0|T^{++}(K)|i\rangle\right|^{2}\left({\frac{x^{+}}{x^{-}}}\right)^{2}{\frac{M_{i}^{4}}{8\pi^{2}K^{3}}}K_{4}\left(M_{i}\sqrt{2x^{+}x^{-}}\right)\,,
\xi_{IJ}R^{IJ}(\midN\rangle{\mid\bar{N}\rangle}+i\mid\dot{a}\rangle\mid\bar{\dot{a}}\rangle)=\xi^{IJ}\midI\rangle\mid\bar{J}>+i\xi^{IJ}\gamma_{\dot{a}\dot{b}}^{IJ}\mid\dot{a}\rangle\mid\bar{\dot{a}}\rangle
\frac{E_{D}(B)}{V}\,=\,\frac{(gB)^{\frac{3}{2}}}{2\pi}\,\left[\frac{g(\frac{\lambda}{2})}{\sqrt{2}}\,+\,\frac{1}{2\sqrt{\lambda}}\right]\;,\;\lambda\,=\,\frac{gB}{m^{2}}
Z_{1}=a\cdot\frac{\prod_{i=1}^{k}(z-b_{i})}{\prod_{i=1}^{k}(z-c_{i})},\;\;\;\;\;Z_{2}=\bar{a}\cdot\frac{\prod_{i=1}^{k}(z-\bar{b}_{i})}{\prod_{i=1}^{k}(z-\bar{c}_{i})}.
{\calL}_{eff}^{nm}=\partial_{+}A_{i}\partial_{-}A_{i}-\frac12(\partial_{i}A_{i}+\frac{1}{\partial_{-}}\astJ_{nm}^{+})^{2}-\frac14\left(\partial_{i}{A}_{j}-\partial_{j}{A}_{i}\right)^{2}+A_{i}J_{nm}^{i}\.
(\bar{S}\bar{R})_{oo^{\prime}}={\frac{16\,i^{o+o^{\prime}+2}}{\pi^{2}o^{\prime}}}\sum_{e=2}^{\infty}{\frac{e^{3}}{(e^{2}-(o^{\prime})^{2})(e^{2}-o^{2})}}=\pm\infty\,.
(\bar{\epsilon}\Gamma^{\mu}\epsilon^{\prime}){\frac{\partial}{\partialx^{\mu}}}=(\bar{\epsilon}\Gamma^{0}\epsilon^{\prime})e_{0}{}^{t}\;{\frac{\partial}{\partialt}}=(\bar{\epsilon}_{o}\Gamma^{0}\epsilon_{o}^{\prime})K^{2}e_{0}{}^{t}\;{\frac{\partial}{\partialt}}={\frac{\partial}{\partialt}}
\left[4\intH(\sigma^{\prime},\sigma^{\prime\prime})X_{\mu}(\sigma^{\prime\prime})d\sigma^{\prime\prime}\intH(\sigma,\sigma^{\prime\prime})X_{\nu}(\sigma^{\prime\prime})d\sigma^{\prime\prime}+2H(\sigma,\sigma^{\prime})\eta_{\mu\nu}\right]\langleX|0\rangle\,\,.
\sigma_{T}\left[(m^{2}+\xiR)-4\kappa\sigma_{T}^{2}+\frac{3\kappam}{\piT}+\frac{15\lambda^{2}T}{4\pim}-\frac{15\lambda^{2}\sigma_{T}^{2}m}{2\piT}+3\lambda^{2}\sigma_{T}^{4}\right]=0
{\frac{[n^{\prime}]_{0}}{n_{0}b_{0}}}=-{\frac{8\piG_{5}}{3c^{4}}}\left[\sigma-2\varrho_{m}c^{2}(1+\zetae^{-2\psi})-3\varrho_{r}c^{2}e^{-2\psi}-5\varrho_{\psi}c^{2}\right],
l={\frac{2\pil_{p}^{3}(n_{1}n_{2}n_{3})^{\frac{1}{3}}}{V_{6}^{\frac{1}{3}}}}~~~;~~G_{3}={\frac{\pil_{p}^{3}}{2(n_{1}n_{2}n_{3})^{\frac{2}{3}}V_{6}^{\frac{1}{3}}}}
\left\langle\frac{\partial^{2}V}{\partial\left(\mathrm{Re}F\right)^{2}}\right\rangle=-\left\langle\frac{\partial^{2}V}{\partial\left(\mathrm{ImF}\right)^{2}}\right\rangle=\frac{2}{\Lambda_{YM}^{4}}\left(\frac{11N_{c}}{12}\right)^{2}\.
\begin{array}{l}{\dot{J}_{1}=[J_{1},H]_{GPB}=\displaystyle\frac{I-I_{3}}{II_{3}}J_{2}J_{3}~,}\\{\dot{J}_{2}=[J_{2},H]_{GPB}=\displaystyle\frac{I_{3}-I}{II_{3}}J_{1}J_{3}~,}\\{\dot{J}_{3}=[J_{3},H]_{GPB}=0~.}\\\end{array}
\mathrm{A}_{\mu}^{(n)}\longrightarrow\mathrm{A}_{\mu}^{(n)}+\frac{\partial}{\partialy^{\mu}}\Theta\qquad\chi_{\mu}\longrightarrow\chi_{\mu}+(\frac{\partial}{\partialn^{\mu}}+y^{T}S_{\mu}\nabla_{\bfy})\Theta.
\tilde{H}^{a}=e^{a}+ih^{a},\;\;\;\;\;\;\tilde{H}^{aa^{\prime}}=\frac{1}{2}\left(\stackrel{2}{\omega}{^{ab}}+i\left(\frac{1}{2}C_{\;\cdotc}^{ab}h^{c}+K^{ab}+S^{ab}\right)\right),
\sum_{ks}\delta\left(\frac{n\tau_{1}}{\nu\tau_{2}}+\frac{2\pi\alpha^{\prime}B^{E}n}{\nu}+s\right)\delta\left(\frac{n}{\nu\tau_{2}}-k\right)=\sum_{ks}\frac{\tau_{2}}{k^{2}}\delta^{2}\left(\tau-\frac{s+\tau_{0}n}{k}\right)
\hat{H}_{free}=\int\left\{\pi^{+}(\stackrel{\rightharpoonup}{x})\pi(x)+(\bigtriangledown\varphi^{+}(\stackrel{\rightharpoonup}{x}))\cdot(\bigtriangledown\varphi(\stackrel{\rightharpoonup}{x}))\right\}d^{3}\stackrel{\rightharpoonup}{x}
\begin{array}{c}{A:\qquadu=p_{A}^{+}\sigma_{per}^{+}+q_{A}^{+},\quadv=p_{A}^{-}\sigma_{per}^{-}+q_{A}^{-}}\\{B:\qquadu=p_{B}^{+}\sigma_{per}^{-}+q_{B}^{+},\quadv=p_{B}^{-}\sigma_{per}^{+}+q_{B}^{-}.}\\\end{array}