harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left(\begin{array}{c|c}{\Lambda^{\prime\prime}}&{a^{\prime\prime}}\\\hline{0}&{1}\\\end{array}\right)=\left(\begin{array}{c|c}{\Lambda^{\prime}}&{a^{\prime}}\\\hline{0}&{1}\\\end{array}\right)\left(\begin{array}{c|c}{\Lambda}&{a}\\\hline{0}&{1}\\\end{array}\right).
\begin{array}{l}{(A_{i}^{(1)}F_{jl}^{(2)}+A_{i}^{(2)}F_{jl}^{(1)}+cycl.perm.)\sim}\\{\sim\,\epsilon_{ijl}\left(A_{m}^{(1)}\partial_{m}\chi^{(2)}+A_{m}^{(1)}\partial_{m}\chi^{(2)}\right)=}\\{=\,\epsilon_{ijl}\sum_{st}\frac{p_{s}^{F}p_{t}^{H}+p_{s}^{H}p_{t}^{F}}{r_{s}(z-z_{s}\pmr_{s})\;r_{t}^{3}}\left[(x-x_{s})(y-y_{t})-(y-y_{s})(x-x_{t})\right]\.}\\\end{array}
S_{ij}(\theta)=\prod_{x=1}^{2h}\prod_{y=1}^{2H}\left\{x,y\right\}_{\theta}^{\mu_{ij}(x,y)}=\prod_{x=1}^{h}\prod_{y=1}^{H}\left\{x,y\right\}_{\theta}^{2\mu_{ij}(x,y)}\;.
\left[\begin{array}{c}{\sum_{i}(\alpha_{i}b_{i}+\beta_{i}b_{i}^{*})q}\\{0}\\\end{array}\right]\;\;\;\mathrm{and}\;\;\;\left[\begin{array}{c}{0}\\{\sum_{i}(\alpha_{i}b_{i}+\beta_{i}b_{i}^{*})p}\\\end{array}\right]
\theta^{\mu\nu}=\frac{2\pi\alpha^{\prime}}{\frac{2\pi\alpha^{\prime}}{\theta_{0}}+\biggl(\frac{2\pi\alpha^{\prime}}{\theta_{1}}\biggr)^{2}}\left(\begin{array}{ccc}{0}&{1}&{0}\\{-1}&{0}&{-\frac{2\pi\alpha^{\prime}}{\theta_{1}}}\\{0}&{\frac{2\pi\alpha^{\prime}}{\theta_{1}}}&{0}\\\end{array}\right).
\{\theta_{i}^{~j},t_{k}^{~l}\}=C_{0~ik~s}^{~~~jl~m}\theta_{s}^{~m}+C_{1~ik~i_{1}i_{2}i_{3}}^{~~~jl~j_{1}j_{2}j_{3}}\theta_{j_{1}}^{~i_{1}}\theta_{j_{2}}^{~i_{2}}\theta_{j_{3}}^{~i_{3}}+\ldots~.
D^{+\alpha}D_{\alpha}^{+}K=12ig\\mathrm{Tr}\left(\bar{W}[\tilde{q}^{+},q^{+}]\right)+cg^{2}\\bar{D}_{\dot{\alpha}}^{+}\bar{D}^{+\dot{\alpha}}\mathrm{Tr}\left(\bar{W}\bar{W}\right)\,,
\begin{array}{ll}{{\calL}_{\xi_{I}}\omega_{J}-{\calL}_{\xi_{J}}\omega_{I}=}&{{f_{IJ}}^{K}\omega_{K}+d\rho_{IJ},}\\{{\calL}_{\xi_{I}}\phi_{J}-{\calL}_{\xi_{J}}\phi_{I}=}&{{f_{IJ}}^{K}\phi_{K}-\rho_{IJ},}\\\end{array}
\xi\equiv\zeta\mathrm{e}^{\frac{g_{m}^{2}}{2}D_{m_{H}}(0)}=\frac{m_{W}^{7/2}}{g}\delta\left(\frac{\lambda}{g^{2}}\right)\mathrm{e}^{\frac{2\pim_{W}}{g^{2}}\left(-2\epsilon+\mathrm{e}^{-c}\right)}
d\tilde{s}^{2}=-{\frac{T_{p}^{2}V_{p}^{2}}{E^{2}}}(hr^{2}\rho_{0}^{2})^{p}e^{-2\Phi}g_{tt}^{2}dt^{2}+h^{2}r^{2}\rho_{0}^{2}d\Omega_{p}^{2},
\begin{array}{rcc}{\mathrm{states}}&{J_{0}^{3}}&{L_{0}}\\{|0\rangle}&{j}&{j}\\{Q_{-1/2}^{-\pm}|0\rangle}&{j-1/2}&{j+1/2}\\{Q_{-1/2}^{-+}Q_{-1/2}^{--}|0\rangle}&{j-1}&{j+1.}\\\end{array}
\frac{1}{(4\pi)^{2}}\intd^{8}z(\frac{W^{2}\bar{W}^{2}}{\Phi^{2}\bar{\Phi}^{2}}+\sum_{k=1}^{\infty}(k+1)\cdot\frac{W^{2}\bar{W}^{2}}{\Phi^{2+k}\bar{\Phi}^{2+k}}\cdot(-(\bar{Q}Q+\bar{\tilde{Q}}\tilde{Q}))^{k}).
\hat{H}_{\mathrm{eff}}=\frac{i}{\hbar}\{\hat{\Psi},\hat{Q}_{B}\}=\hat{\lambda}\left[\hbar\omega(a^{\dagger}a+\frac{1}{2})-\Lambda\right]+F(\hat{\lambda})\hat{p}_{\lambda}-i\hbar\hat{c}^{1}\hat{b}_{1}F^{\prime}(\hat{\lambda})-i\hbar\hat{c}^{1}\hat{b}_{2}\,
U_{\mathrm{eff}}(B^{2})={\frac{B^{2}}{g_{R}^{2}(M)}}+{\frac{29}{525\pi^{2}}}{\frac{B^{4}}{M^{4}(T)}}+{\calO}(B^{6}/M^{8}(T))+{\calO}(g_{R}^{2}(M))\.
{\calR}_{abc}^{0}=\partial_{b}\Omega_{ac}^{0}-\partial_{c}\Omega_{ab}^{0}+\Omega_{0b}^{0}\Omega_{ac}^{0}+\Omega_{db}^{0}\Omega_{ac}^{d}-\Omega_{0c}^{0}\Omega_{ab}^{0}-\Omega_{dc}^{0}\Omega_{ab}^{d},
\left\{\begin{array}{c}{\left(\begin{array}{clc}{\frac{\delta-\sqrt{\delta^{2}+\omega^{\ast}\omega}}{\omega}}&{,}&{1}\\\end{array}\right)}\\{\left(\begin{array}{clc}{\frac{\delta+\sqrt{\delta^{2}+\omega^{\ast}\omega}}{\omega}}&{,}&{1}\\\end{array}\right)}\\\end{array}\right\}.
^{res}_{\lambda\to-n/2-k}P_{+}^{\lambda}=c_{1}\frac{\Box^{k}\delta(P)}{(\lambda+n/2+k)^{2}}+\frac{c_{2}\Box^{k}\delta(x_{1},\dots,x_{n})+c_{2}^{\prime}\delta^{(n/2+k-1)}(P)}{\lambda+n/2+k}
\begin{aligned}{{[}h_{i},e_{n}]}&{{}=e_{n+i},}&{\qquad[h_{i},f_{n}]}&{{}=-f_{n+i},}\\{{[}e_{i},f_{j}]}&{{}=2h_{i+j}+ki\delta_{i+j,0},}&{\qquad[h_{i},h_{j}]}&{{}=\frac12ki\delta_{i+j,0}}\\\end{aligned}
E_{0}\left(\ell,\mu,eB\right)=-2\times\frac{eBL^{2}}{2\pi^{2}\ell}\sum_{p=-1}^{\infty\,\,\,\,\,\,^{\prime}}\,\sum_{k=1}^{\infty}\,\frac{\left(-1\right)^{k+1}}{k}I_{pk}\left(M_{p}\right),
\mathcal{J}_{\alpha}=\frac{R^{2\alpha-1}}{2^{2\alpha-2}}\frac{\Gamma(2\alpha-2)\Gamma(\frac12)\Gamma(\frac{5-2\alpha}2)}{\Gamma(\alpha)\Gamma(\frac{2\alpha+1}2)\Gamma(\frac{2\alpha-1}2)}\;.
\begin{array}{rclcrcl}{\psi_{-}}&{\rightarrow}&{e^{-i\theta}g^{-1}h^{-1}\psi_{-}}&{\quad\quad}&{\psi_{+}}&{\rightarrow}&{\psi_{+}\nonumber}\\{\psi_{-}^{\dagger}}&{\rightarrow}&{\psi_{-}^{\dagger}hge^{i\theta}}&{\quad\quad}&{\psi_{+}^{\dagger}}&{\rightarrow}&{\psi_{+}^{\dagger}}\\\end{array}\quad\,,
S={\frac{1}{2\kappa^{2}}}\intd^{5}x\sqrt{-g}\left({\calR}+{\frac{12}{l^{2}}}\right)-\sum_{i=A,B}\sigma_{i}\intd^{4}x\sqrt{-g^{i\mathrm{brane}}}+\sum_{i=A,B}\intd^{4}x\sqrt{-g^{i\mathrm{brane}}}{\calL}_{\mathrm{matter}}^{i}\,
\left[\eta^{a}Q_{+}^{a},V_{B}^{i}\zeta^{i}\right]=(1/2)V_{F}^{\dot{a}}\left[\tilde{u}^{\dot{a}}-M_{\dot{a}\dot{b}}\tilde{s}^{\dot{b}}\right]\equiv(1/2)V_{F}^{\dot{a}}t^{\dot{a}}
{\calS}_{A}=-\int\mathrm{d}^{4}x~\Big[~\frac14(\partial_{\lambda}A_{\mu\nu})~\partial^{\lambda}A^{\mu\nu}-\frac12(\partial_{\mu}A^{\mu\lambda})~\partial^{\nu}A_{\nu\lambda}\Big],~~~~\deltaA_{\mu\nu}=\partial_{\mu}\Lambda_{\nu}-\partial_{\nu}\Lambda_{\mu}.
{\calF}\left(L,\beta,\nu\right)=\frac{8\pi^{4}}{\sqrt{2\pi\alpha^{\prime}}}\int_{0}^{\infty}\frac{ds}{s^{9/2}}\;e^{s-L^{2}/2s\alpha^{\prime}}\Theta_{2}\left(\nu\left\vert\frac{i\beta^{2}s}{2\pi^{3}\alpha^{\prime}}\right.\right)\prod_{n=1}^{\infty}\left(\frac{1-e^{-(2n+1)s}}{1-e^{-2ns}}\right)^{8},
{\Theta_{I}}^{J}=\left(\begin{array}{cccc}{0}&{1}&{0}&{0}\\{-1}&{-1}&{0}&{0}\\{0}&{0}&{0}&{1}\\{0}&{0}&{-1}&{-1}\\\end{array}\right)~,~~~{\Theta^{I}}_{J}=\left(\begin{array}{cccc}{-1}&{1}&{0}&{0}\\{-1}&{0}&{0}&{0}\\{0}&{0}&{-1}&{1}\\{0}&{0}&{-1}&{0}\\\end{array}\right)~.
\|\partial_{\Lambda}\partial^{p}C_{2m}^{(p,q)}\|\leq{\frac{1}{\Lambda}}P\left(ln{\frac{\Lambda}{\Lambda_{R}}}\right){\frac{1}{\Lambda^{2m-6}}}\left({\frac{\Lambda}{M}}\right)^{2-\epsilon}\left({\frac{M}{\Lambda_{R}}}\right)^{\epsilon^{\prime}},\qquadq>1,M>\Lambda_{R}
=(-1)^{m}\,\left(\frac{\omega}{\pi\hbar}\right)^{1/2}\,\left(\frac{1}{2\pi\hbar}\right)^{1/2}\,\left(\frac{m!}{(m+|\ell|)!}\right)^{1/2}\,e^{i\ell\theta}\,e^{ipz/\hbar}\,u^{|\ell|}\,e^{-u^{2}/2}\,L_{m}^{|\ell|}(u^{2})\\\,
\Gamma^{(3)}=\frac{1}{i}\left\langle\frac{1}{3!}\left(\frac{-i\lambda}{4!}\varphi^{4}\right)^{3}-i\frac{\lambda}{4!}\varphi^{4}\left(-\frac{i}{2}J^{(2)}\varphi^{2}\right)-\frac{i}{2}J^{(3)}\varphi^{2}\right\rangle_{G^{(0)}}^{\mathrm{nself.}}.
V_{Q}^{\prime}\,=\,\frac{{\hbar}^{2}}{8M}\left\{R+\frac{(\partial_{a}f)(\partial^{a}f)}{4M}\left[3g_{\phantom{0},0}^{ij}\,g_{ij,0}-{\left(g^{ij}\,g_{ij,0}\right)}^{2}\right]\right\}.
{\calS}(u)=\left[-\left(\frac{\partialu^{\mu}}{\partialt}\frac{\partialu_{\mu}}{\partialt}\right)\left(\frac{\partialu^{\mu}}{\partials}\frac{\partialu_{\mu}}{\partials}\right)+\left(\frac{\partialu^{\mu}}{\partialt}\frac{\partialu_{\mu}}{\partials}\right)^{2}\right]^{\frac{1}{2}}
\begin{aligned}{[b]}&{{}U_{1}:}&{}&{{}\qquadu_{1}+v_{1}x_{1}y_{1}=\mu}\\{}&{{}U_{2}:}&{}&{{}\qquadu_{2}x_{2}+v_{2}y_{2}=\mu}\\{}&{{}U_{3}:}&{}&{{}\qquadu_{3}x_{3}+v_{3}y_{3}=\mu.}\\\end{aligned}
{\frac{F(x)}{x^{\ell}}}\int_{0}^{x}dyy^{\ell-1}F(y)={\frac{F(x)}{x^{\ell}}}\int_{\Lambda^{2}}^{x}dyy^{\ell-1}F(y)+{\frac{F(x)}{x^{\ell}}}\int_{0}^{\Lambda^{2}}dyy^{\ell-1}F(y),
\mathrm{U}_{\Lambda}\psi_{m_{1}m_{2}}^{j_{1}j_{2}}(x)\mathrm{U}_{\Lambda}^{\dagger}={\calD}_{m_{1}m_{1}^{\prime}}^{j_{1}}(\Lambda^{-1})\bar{{\calD}}_{m_{2}m_{2}^{\prime}}^{j_{2}}(\Lambda^{-1})\psi_{m_{1}^{\prime}m_{2}^{\prime}}^{j_{1}j_{2}}(\Lambdax)
I=\frac{1}{16\piG_{5}}\intd^{5}x\sqrt{-g}\left[R-\frac{1}{2}G^{IJ}H_{I\mu\nu\rho}H_{J}^{\mu\nu\rho}-\mathcal{G}_{ij}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}\right],
\Box\left(\begin{matrix}{\pi^{(lm)}}\\{b^{(lm)}}\\\end{matrix}\right)=\left(\begin{matrix}{l(l+1)+2}&{-4l(l+1)}\\{-1}&{l(l+1)}\\\end{matrix}\right)\left(\begin{matrix}{\pi^{(lm)}}\\{b^{(lm)}}\\\end{matrix}\right),
{\bfV}_{i}^{(int)}(t_{1}){\bfV}_{j}^{(int)}(t_{2})=q^{2({\bfe}_{i}{\bfe}_{j})}{\bfV}_{j}^{(int)}(t_{2}){\bfV}_{i}^{(int)}(t_{1}),\qquadt_{1}>t_{2}
U_{cs}=\frac{2m^{2}}{q^{2}}(q\tilde{k}-m^{2})+2m^{2}\bar{\phi}\phi+\frac{1}{2}\left(\frac{1}{2}q^{2}v^{2}\right)\left(\frac{1}{\sqrt{2}}[\tilde{\phi}+\tilde{\bar{\phi}}]\right)^{2}+...
{}_{2}\phi_{1}\left(\begin{matrix}{a\phantom{c}b}\\{\phantom{a}c\phantom{b}}\\\end{matrix};q,z\right)=\sum_{n=0}^{\infty}\frac{(a;q)_{n}(b;q)_{n}}{(c;q)_{n}(q;q)_{n}}z^{n},\qquad(a;q)_{n}=\prod_{j=0}^{n-1}(1-aq^{j})~.
a\frac{d}{da}\lambda_{3}^{ijk}(-p,p-q,q)=-2\sigma^{ijk}+d\lambda_{3}^{ijk}(-p,p-q,q)+\frac12\lambda_{3}^{ijk}(-p,p-q,q)[\lambda_{2}^{i}(p)+\lambda_{2}^{j}(p-q)+\lambda_{2}^{k}(q)]
\begin{array}{cc}{{\calG}(i)^{-1}=\left(\begin{array}{ccc}{\bar{\epsilon}}&{0}&{0}\\{0}&{\epsilon^{-1}}&{0}\\{0}&{0}&{-\bar{\zeta}_{ab}}\\\end{array}\right)~~~}&{~~~{\calK}(z;i)=\left(\begin{array}{cc}{i(\bar{\epsilon}x_{+}{\cdot\tilde{\sigma}})^{-1}}&{2i(x_{-}{\cdot\tilde{\sigma}})^{-1}\bar{\theta}_{c}\zeta^{cb}}\\{0}&{-v_{~c}^{a}(z)\zeta^{cb}}\\\end{array}\right)}\\\end{array}
V=\left(\begin{array}{c}{V_{R}}\\{V_{L}}\\\end{array}\right)={\frac{1}{\sqrt{2}}}\left(\begin{array}{cc}{E^{-1}}&{E^{T}-E^{-1}B}\\{E^{-1}}&{-E^{T}-E^{-1}B}\\\end{array}\right).
\partial^{\alpha}\partial_{\alpha}h_{\mu\nu}+\partial_{\mu}\partial_{\nu}h_{\alpha}^{\alpha}-\left(\partial_{\mu}\partial_{\alpha}h_{\nu}^{\alpha}+\partial_{\nu}\partial_{\alpha}h_{\mu}^{\alpha}\right)-\eta_{\mu\nu}\left(\partial^{\alpha}\partial_{\alpha}h_{\beta}^{\beta}-\partial_{\alpha}\partial_{\beta}h^{\alpha\beta}\right)=\Theta_{\mu\nu}\,
{\frac{\partial}{\partialT_{2}}}(e^{-H}-e^{-H_{0}})=-e^{-H}{\frac{H-T_{1}lP_{1}^{*}}{T_{2}}}+e^{-H_{0}}{\frac{H_{0}-T_{1}lP_{1}^{*}}{T_{2}}}+e^{-H}{\frac{K^{2}}{M_{1}}}\left[{\frac{(M_{1}-l)T_{1}T_{2}}{(MT_{2}+M_{1}T_{1})^{2}}}\right].
\lambda(\beta)=\lambda_{0}-\frac{3}{2}\frac{\lambda_{0}^{2}}{(2\pi)^{D/2}}\sum_{n=1}^{\infty}\biggl(\frac{m_{0}}{\betan}\biggr)^{\frac{D}{2}-2}K_{\frac{D}{2}-2}(m_{0}n\beta),
U=\left(\begin{array}{ccc}{q_{1}}&{q_{1}p_{1}}&{q_{1}p_{3}}\\{q_{1}p_{1}}&{q_{2}+q_{1}p_{1}^{2}}&{q_{2}p_{2}+q_{1}p_{1}p_{3}}\\{q_{1}p_{3}}&{q_{2}p_{2}+q_{1}p_{1}p_{3}}&{q_{1}p_{3}^{2}+q_{2}p_{2}^{2}+(q_{1}q_{2})^{-1}}\\\end{array}\right).
\begin{array}{ccccccc}{x}&{y}&{z}&{s}&{t}&{u}&{v,}\\{2}&{3}&{1}&{0}&{0}&{0}&{0,}\\{4}&{6}&{0}&{1}&{1}&{0}&{0,}\\{2n+4}&{3n+6}&{0}&{n}&{0}&{1}&{1.}\\\end{array}
\begin{array}{rclrcl}{A^{(1)m\prime}}&{=}&{-A^{(2)}{}_{m}\,,}&{\rho_{m}^{\prime}}&{=}&{-\rho_{m}\,,}\\{}&{}&{}&{}&{}&{}\\{A^{(2)}{}_{m}^{\prime}}&{=}&{-A^{(1)m}\,.}&{}&{}&{}\\\end{array}
H(t)=\frac{1}{2}\mathrm{p}^{2}+\frac{1}{2}M^{2}\mathrm{x}^{2}+\frac{g(t)}{\lambda}\left[\frac{m^{2}-M^{2}}{2}\mathrm{x}^{2}+\frac{1}{4}\lambda\mathrm{x}^{4}\right]\;,
a^{2}(\tau,y)=\frac{D(\tau)}{2k}e^{2kby}+\frac{1}{2kD(\tau)}\left[\left(\frac{\lambda^{2}(\tau)+K}{2k}\right)^{2}-C(\tau)\right]e^{-2kby}-\frac{\lambda^{2}(\tau)+K}{2k^{2}},
{{\calA}}({\calT})\wedge{{\calA}}({\calN})\;=\;{{\calA}}({\widehat{\calS}})\;=\;1+{\frac{(4\pi^{2}\alpha^{\prime})^{2}}{192\pi^{2}}}\;\mathrm{tr}\;({\widehat{\calR}}\wedge{\widehat{\calR}})+\cdots\,
\sumq^{\frac{1}{2}p^{2}}=\sum_{n}q^{\frac{30}{4}\left(2n+1+\frac{5m_{1}+3m_{3}}{30}\right)^{2}}=\sum_{n}q^{30\left(n+\frac{5m_{1}+3m_{3}+30}{60}\right)^{2}}=\theta_{5m_{1}+3m_{3}+30,30}\:(\tau).
J_{\alpha}^{\mu{\mathbfi}}=192\left[i\sigma_{\alpha\dot{\alpha}}^{\mu}{\bar{D}}_{\mathbfj}^{\dot{\alpha}}D^{{\mathbfi}{\mathbfj}}T-3i{\bar{\sigma}}^{\mu\dot{\alpha}\beta}{\bar{D}}_{\dot{\alpha}}^{\mathbfi}D_{\alpha\beta}T-12\partial^{\mu}D_{\alpha}^{\mathbfi}T+a\sigma_{\alpha}^{\mu\nu\,\beta}\partial_{\nu}D_{\beta}^{\mathbfi}T\right]|\,
\begin{array}{l}{\partial_{0}x^{i}=-p^{i},\qquad\partial_{0}p^{i}=-\partial_{1}\partial_{1}x^{i},}\\{(\partial_{0}+\partial_{1})\theta_{a}=0,\qquad(\partial_{0}-\partial_{1})\bar{\theta}_{\dot{a}}=0,}\\\end{array}
\tilde{S}=\intd^{2}{x}\left\{g^{2}(1+\varphi_{1}^{2})^{2}(\partial\tilde{\varphi_{0}})^{2}+8\thetag^{2}\partial_{a}\tilde{\varphi_{0}}\partial^{a}\varphi_{1}+\frac{(1+16\theta^{2}g^{4})}{g^{2}(1+\varphi_{1}^{2})^{2}}(\partial\varphi_{1})^{2}\right\}
\Bigl\{b^{r}(k,\sigma),b^{s\dagger}(p,\sigma^{\prime})\Bigr\}=\Bigl\{d^{r}(k,\sigma),d^{s\dagger}(p,\sigma^{\prime})\Bigr\}=16\pi^{3}k^{+}\delta_{rs}\delta_{\sigma\sigma^{\prime}}\delta^{3}(k-p)\;.
S_{E}\,=\,M_{0}^{2}\,\int_{0}^{T}\!\!dt\int_{0}^{R}\!\!dr\left[1\,+\,\frac{1}{2}{\bfu}\,\partial^{2}{\bfu}\right]\,+\,\sum_{a=1}^{2}\,\frac{m_{a}}{2}\,\int_{t_{1}}^{t_{2}}\!\!dt\,{\dot{\bfu}}^{2}(t,r_{a})\,{,}
dv_{\underline{{\mu}}}^{~\underline{{\alpha}}}=1/4v_{\underline{{\mu}}}^{~\underline{{\beta}}}(\Gamma_{\underline{{a}}\underline{{b}}})_{\underline{{\beta}}}^{~\underline{{\alpha}}}\Omega^{~\underline{{a}}\underline{{b}}}(d)\qquad
S=\frac{1}{g^{2}}Tr(-\frac{1}{4}[A_{i},A_{j}][A^{i},A^{j}]+\frac{2}{3}i\alpha\epsilon_{ijk}A^{i}A^{j}A^{k}+\frac{1}{2}\bar{\psi}\sigma^{i}[A_{i},\psi]).
\textrm{Tr}\;e^{-\frac{i}{\hbar}{\cal{H}}T}=\intdq_{0}K(q_{0},T\midq_{0})=\underset{n}{\sum}e^{-\frac{i}{\hbar}E_{n}T}\intdq_{0}\midu_{n}(q_{0})\mid^{2}.
\sigma_{1}=\left(\begin{matrix}{1}&{0}\\{0}&{-1}\\\end{matrix}\right),\quad\sigma_{2}=\left(\begin{matrix}{0}&{1}\\{1}&{0}\\\end{matrix}\right),\quad\sigma_{3}=\left(\begin{matrix}{0}&{-i}\\{i}&{0}\\\end{matrix}\right).
W_{\alpha\beta}(x-y)=\left(\begin{array}{cccc}{0}&{0}&{\delta(x-y)}&{0}\\{0}&{0}&{0}&{\delta(x-y)}\\{-\delta(x-y)}&{0}&{0}&{0}\\{0}&{-\delta(x-y)}&{0}&{0}\\\end{array}\right).
[\gamma_{l},\bar{\gamma}_{m}]=\frac{1}{4\pik^{\prime}}G_{lm}\\\\\mathrm{or}\\\\,bar{\gamma}_{l}=-\frac{1}{4\pik^{\prime}}G_{lm}\frac{\partial}{\partial\gamma_{m}}=-\frac{1}{4\pik^{\prime}}\frac{\partial}{\partial\gamma^{l}}
I=\intd\mu{\calL}=\intd^{4}x\widetilde{D}_{k_{1}l_{1},\ldotsk_{q}l_{q}}^{i_{1}j_{1},\ldots,i_{p}j_{p}}L_{i_{1}j_{1},\ldots,i_{p}j_{p}}^{k_{1}l_{1},\ldots,k_{q}l_{q}}
{I_{D}}^{\mathrmr}\equiv-VT{\frac{{\mathrm{tr}}{\bf1}}{2(4\pi)^{D/2}}}\int_{1/\Lambda^{2}}^{\infty}\!d\tau\,\tau^{-{\frac{D}{2}}-1}\left\{{\mathrme}^{-\tau(m^{2}+\epsilon)}G_{D}(\tauF)-1\right\}\.
{\frac{\ddot{a}}{a}}=-{\frac{1}{\tilde{L}^{2}}}\left[p(\tilde{E}^{2}-\tilde{\ell}^{2})\left({\frac{\tilde{L}}{a}}\right)^{2(p+1)}-(p-1)\tilde{\ell}^{2}\left({\frac{\tilde{L}}{a}}\right)^{2p}+(p-1)\tilde{E}\left({\frac{\tilde{L}}{a}}\right)^{p+1}\right],
\begin{aligned}{\mathrm{brane:}\qquad}&{{}(n_{1},m_{1})\times(N_{2},M_{2})\times(n_{3},M_{3})}\\{c_{1}:\qquad}&{{}(1,0)\times(2,0)\times(4^{b},0),}\\{c_{2}:\qquad}&{{}(1,0)\times(0,-2)\times(0,-1),}\\{c_{3}:\qquad}&{{}(1,1)\times(1,1)\times(4^{b},0),}\\{c_{4}:\qquad}&{{}(1,1)\times(1,-1)\times(0,-1).}\\\end{aligned}
S~=~\int\,d^{d}x\,\left[\,\intd^{4}\theta\,\left(\bar{\Phi}^{i}\Phi^{i}+\frac{\bar{\sigma}\sigma}{g^{2}}\right)~-~\frac{1}{2}\intd^{2}\theta\,\sigma{\Phi}^{2}~-~\frac{1}{2}\intd^{2}\bar{\theta}\,\bar{\sigma}\bar{\Phi}^{2}\right]
\vert\phi_{1}\rangle=L_{-1}G_{-{\frac{1}{3}}}^{-}\vert{h;0}\rangle{}~,{}~\vert\phi_{2}\rangle=G_{-1}^{+}G_{-{\frac{1}{3}}}^{+}\vert{h;0}\rangle{}~,{}~\vert\phi_{3}\rangle=G_{-{\frac{4}{3}}}^{-}\vert{h;0}\rangle{}~,
S[\psi,{\overline{\psi}},A]=S_{\mathrm{YM}}[A]+S_{A}[\psi,{\overline{\psi}}]=S_{\mathrm{YM}}[A]+\intd^{4}x\,{\overline{\psi}}\left[iD_{\mu}({\bfr})\right]{\overline{\sigma}}^{\mu}\psi\,,
\frac{\partial{Z_{k}(\rho)}}{\partial{t}}\bigg\vert_{\rho}=\frac{\partial{Z_{k}(\rho)^{(a)}}}{\partial{t}}\bigg\vert_{\rho}+\frac{\partial{Z_{k}(\rho)^{(b)}}}{\partial{t}}\bigg\vert_{\rho}
\begin{array}{rcll}{J_{r}(A_{1},\ldots,A_{r})J_{r^{\prime}}(A_{1}^{\prime},\ldots,A_{r^{\prime}}^{\prime})}&{=}&{J_{r}(A_{1}A_{1}^{\prime},\ldots,A_{r}A_{r}^{\prime})}&{\quad{\mathrm{if}}\;\;r=r^{\prime}}\\{}&{=}&{0}&{\quad{\mathrm{if}}\;\;r\neqr^{\prime}\;.}\\\end{array}
a_{\mathrm{UV}}-a_{\mathrm{IR}}=\int_{-\infty}^{+\infty}\mathrm{d}t\\beta^{2}(t)f(t)=-\int_{\lambda_{\mathrm{UV}}}^{\lambda_{\mathrm{IR}}}\mathrm{d}\lambda\\beta(\lambda)f(\lambda)\geq0.
e^{i(\alpha-\frac{1}{4})\pi}f(\theta)-e^{-i(\alpha-\frac{1}{4})\pi}f^{*}(-\theta)=i\sqrt{\frac{k}{2\pi}}\int_{-\pi}^{\pi}d\theta^{\prime}f^{*}(\theta^{\prime})f(\theta^{\prime}+\theta+\pi)
Z_{{\itlog}}[p]=\frac{1}{\eta\bar{\eta}}\left(|\chi_{0}|^{2}+|\chi_{p}|^{2}+\sum_{1\leqs\leqp-1}\left(|\chi_{s}^{0}|^{2}+\chi_{s}^{+}(\chi_{s}^{-})^{*}+\chi_{s}^{-}(\chi_{s}^{+})^{*}\right)\right)\,.
L=\sqrt{-g}\left[R-\frac{1}{48}\left({\calA}_{\mu}^{ijk\ell}\right)^{2}-\frac{1}{4}F_{\mu\nu,IJ}^{+}\left[2S^{IJ,KL}-\delta_{KL}^{IJ}\right]F_{KL}^{+\mu\nu}-g^{2}\left(\frac{3}{2}|A^{ij}|^{2}-\frac{1}{12}|A_{i}^{\;jk\ell}|^{2}\right)\right],\nonumber\,
{\cal\phi}(1){\cal\psi}(2)=\sum_{(\sigma)}\int_{3,3^{\prime}}\langle{\cal\phi}(1){\cal\psi}(2){\calO}_{(\sigma)}(3)\rangle\\langle{\calO}_{(\sigma)}(3){\calO}_{(\sigma)}(3^{\prime})\rangle^{-1}\{\calO}_{(\sigma)}(3^{\prime})\,.
\Big[{\bfa}_{p}(\vec{k})\,,\,{\bfa}_{p^{\prime}}^{\dagger}({\vec{k}}^{\prime})\Big]\,=\,2\,(2\pi)^{3}w_{p}(\vec{k})\delta_{p\,p^{\prime}}\,\delta^{3}(\vec{k}-{\vec{k}}^{\prime})\,.
\Gamma[\phi]=\sum_{n}\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\cdots\frac{d^{4}p_{n}}{(2\pi)^{4}}(2\pi)^{4}\delta^{4}\left(\sum_{i}p_{i}\right)\Gamma^{(n)}(\{p_{i}\})\hat{\phi}(p_{1})\cdots\hat{\phi}(p_{n})
ds^{2}/\alpha^{\prime}=\frac{U}{R_{5}f_{1}^{1/2}}(-dt^{2}+dx_{1}^{2})+\frac{Uf_{1}^{1/2}}{R_{5}}(dx_{2}^{2}+\cdots+dx_{5}^{2})+UR_{5}f_{1}^{1/2}\left(\frac{dU^{2}}{U^{2}}+d\Omega_{3}^{2}\right).
M^{2}=\frac{1}{9}(NT_{0})^{2}\left(\omega_{1}^{2}R_{1}^{4}+\omega_{2}^{2}R_{2}^{4}+\omega_{3}^{2}R_{3}^{4}\right),~~~~E=\frac{NT_{0}}{4}\left(\omega_{1}^{2}R_{1}^{2}+\omega_{2}^{2}R_{2}^{2}+\omega_{3}^{2}R_{3}^{2}\right).
\tilde{f}_{(5)}=d\tilde{c}_{(4)}-\Bigl(\hat{F}_{(3)}+\mathrm{~\frac12~}f_{(3)}\Bigr)\wedgeb_{(2)}+c_{(2)}\wedge\Bigl(\hat{H}_{(3)}+\mathrm{~\frac12~}h_{(3)}\Bigr)\.
{J_{5}^{\mu}}=-\sum_{r=0,2,...}{\tilde{\overline{\Psi}}}_{L_{r}}\gamma^{\mu}\tilde{\Psi}_{L_{r}}+\sum_{s=1,3,...}{\tilde{\overline{\Phi}}}_{L_{s}}\sigma^{3}\gamma^{\mu}\tilde{\Phi}_{L_{s}}.
T=\frac{(d-3)}{2^{\frac{1}{d-3}}4\pi}\frac{r_{0}^{2-\frac{1}{d-3}}}{\prod_{i=1}^{i=n}\left[r_{0}+\sqrt{r_{0}^{2}+q_{i}^{2}}\right]^{2r_{i}}}\,,
\left({\frac{\dot{a}_{0}}{a_{0}}}\right)^{2}={\frac{16\pi^{2}G_{5}^{2}}{9c^{6}}}(\varrho_{\mathrm{tot}}^{2}c^{4}+2\sigma\varrho_{\mathrm{tot}}c^{2})+c^{2}{\frac{a_{R\,0}^{\prime\prime}}{a_{0}}}+{\frac{8\piG_{5}}{3c^{2}}}\left(\Lambda+{\frac{2\piG_{5}}{3c^{4}}}\sigma^{2}\right)-{\frac{kc^{2}}{a_{0}^{2}}},
m^{2}=\left(\begin{array}{cccc}{\mu^{2}-\frac{15}{4}+\muc_{33}}&{0}&{-\muc_{13}}&{\muc_{23}}\\{0}&{\mu^{2}+\mu-\frac{15}{4}}&{0}&{0}\\{-\muc_{13}}&{0}&{\mu^{2}-\frac{15}{4}+\muc_{11}}&{-\muc_{12}}\\{\muc_{23}}&{0}&{-\muc_{12}}&{\mu^{2}-\frac{15}{4}+\muc_{22}}\\\end{array}\right)
\begin{array}{lcl}{{\calW}_{+}}&{\stackrel{CP}{\longrightarrow}}&{-{\calW}_{-}}\\{{\calW}_{-}}&{\stackrel{CP}{\longrightarrow}}&{-{\calW}_{+}}\\\end{array}\qquad\begin{array}{lcl}{{\calW}_{3}}&{\stackrel{CP}{\longrightarrow}}&{-{\calW}_{3}}\\{{\calW}_{4}}&{\stackrel{CP}{\longrightarrow}}&{-{\calW}_{4}}\\\end{array}
T_{n}(L,r)=\sum_{m,\frac{L}{2}-\frac{r}{2}-\frac{m}{2}\in{\bfZ}_{\geq0}}q^{\frac{m^{2}}{2}-\frac{nm}{2}}\frac{(q)_{L}}{(q)_{\frac{L}{2}-\frac{r}{2}-\frac{m}{2}}(q)_{\frac{L}{2}+\frac{r}{2}-\frac{m}{2}}(q)_{m}}.
\begin{aligned}{\tilde{\Gamma}^{i}}&{{}=\gamma^{i}\,,}&{\tilde{\Gamma}^{a}}&{{}=\gamma^{*}\gamma^{a}\,,}\\{\gamma^{*}}&{{}=\gamma^{7\ldots10}\,,}&{C}&{{}={\calC}_{(11)}\gamma^{*}\,,}\\\end{aligned}
B_{\mu\nu}|_{(k^{2}=0)}=\frac{1}{s^{2}}\left[\begin{array}{ccc}{0}&{\begin{array}{c}{-isk_{0}}\\\end{array}}&{0}\\{isk_{0}}&{0}&{-isk_{0}}\\{0}&{\begin{array}{c}{isk_{0}}\\\end{array}}&{0}\\\end{array}\right],
\begin{array}{ll}{\vec{\xi}_{1}=2\varphi{\partial}_{\varphi}+2\sigma{\partial}_{\sigma},}\\{\vec{\xi}_{2}={\partial}_{\sigma},}\\{\vec{\xi}_{3}=-4\varphi\sigma{\partial}_{\varphi}+2(\varphi^{2}-{\sigma}^{2}){\partial}_{\sigma}.}\\\end{array}
\vertj\;I\rangle=\;a_{j}^{\dagger}\;\chi_{1}^{\dagger}\chi_{I}^{\dagger}\;\vert0\rangle\,\\\\mathrm{and}\\\,verta\;I\rangle=\;\theta_{a}^{\dagger}\chi_{1}^{\dagger}\chi_{I}^{\dagger}\;\vert0\rangle\,
T\left(\vartheta|\vartheta_{1},\vartheta_{2},\ldots,\vartheta_{N}\right)_{a_{1}a_{2}\ldotsa_{N}}^{b_{1}b_{2}\ldotsb_{N}}=\prod_{j=1}^{N}S_{b_{j}a_{j}}^{a_{j+1}b_{j+1}}\left(\vartheta-\vartheta_{j}\right)
N_{i}=\sum_{k=0}^{\infty}\sum_{j=0}^{k}c_{k,j}\sum_{p_{1}....p_{j}}\theta_{ip_{1}}.....\theta_{ip_{j}}a_{p_{j}}^{+}.....a_{p_{1}}^{+}(a_{i}^{+})^{k-j+1}(a_{i})^{k-j+1}a_{p_{1}}......a_{p_{j}}
{\calH}(p)=-i\int{\calD}q{\calD}k\;\varepsilon^{\mu\nu\rho}\frac{(p+q)_{\rho}}{(p+q)^{2}}\;\varepsilon^{\alpha\beta\lambda}\frac{q_{\lambda}}{q^{2}}\;\frac{g_{\nu\alpha}(2k+p-q)_{\mu}(2k-q)_{\beta}}{(k+p)^{2}(k-q)^{2}k^{2}}\;.
g\,t_{i_{1}\ldotsi_{n}\Phi^{2}}+U\,t_{i_{1}\ldotsi_{n}\Phi}+c\,(t_{(i_{1}+1)i_{2}\ldots(i_{n}-1)\Phi}+t_{(i_{1}-1)i_{2}\ldots(i_{n}+1)\Phi})-\sum_{j=1}^{n-1}t_{i_{1}\ldotsi_{j}}\,t_{i_{j+1}\ldotsi_{n}}=0
{\calD}\bar{\calM}\equiv{\calD}\bar{B}_{\mu\nu}{\calD}\bar{B}_{\mu5}{\calD}\bar{B}_{55}{\calD}\bar{\zeta}_{\mu}{\calD}\bar{\zeta}_{4}{\calD}\bar{\zeta}_{\mu,\theta}{\calD}\bar{\zeta}_{4,\theta}{\calD}\bar{\eta}{\calD}\bar{\eta}_{,\theta}\,,
\partial_{\mu}(\sqrt{g}R^{\mu})=-\frac{1}{3g^{3}}\widetilde{\beta}(g)(F\widetilde{F})-\frac{\tilde{b}(g)}{48\pi^{2}}(B\widetilde{B})+\frac{\tilde{c}(g)-a(g)}{24\pi^{2}}R\widetilde{R}+\frac{5a(g)-3\tilde{c}(g)}{9\pi^{2}}(V\widetilde{V})
\frac{1}{\rho}\partial_{\rho}\rho\partial_{\rho}\,H+\frac{1}{r^{3}}\partial_{r}\left(r^{3}\,g(r)^{2}\,\partial_{r}H\right)=-\frac{2\,a^{4}}{\pi^{2}\,r^{8}}\,|\partial\gamma(z)|^{2},
\prod_{x\in\Sigma}~\delta\left[\phi(t,{\bfx}({\bfu}))\right]=\int{\calD}\psi(t,{\bfu})~\mathrm{exp}~i\int\,dt\,\int_{\Sigma}d^{2}u\sqrt{g}\psi(t,{\bfu})\,\phi(t,{\bfx}({\bfu}))\,.
\begin{array}{rcl}{{\calA}(r,\omega;\omega_{\bot})}&{=}&{\displaystyle\frac{1}{\pi}\int_{-\infty}^{\infty}d\omega^{\prime}\frac{\mathrm{Im}~{\calA}(r,\omega^{\prime};\omega_{\bot})}{\omega^{\prime}-\omega-i\epsilon}.}\\\end{array}
H_{F}=1+\sum_{i}{\frac{Q_{i}}{[|\vec{x}-\vec{x}_{0\,i}|^{2}+{\frac{4Q}{(p-4)^{2}}}|\vec{z}-\vec{z}_{0}|^{p-4}]^{\frac{p^{2}-6p+12}{2(p-4)}}}},\\\H_{p}={\frac{Q}{|\vec{z}-\vec{z}_{0}|^{6-p}}}.
L_{1}^{*}=\intd^{4}x\bigg[\dots+A_{\mu}^{i}[{\bfZ}]_{ij}^{\mu\nu}A_{\nu}^{j}+i\varepsilon^{\mu\nu\rho\sigma}\bigg(f_{cdj}^{i}A_{\mu}^{cd}{\calH}_{\rho\sigmai}-2\partial_{\mu}{\calH}_{\rho\sigmaj}\bigg)
J^{\alpha}=J^{(\alpha\beta)}{}_{\beta}=\frac{2}{e}\left(\frac{1}{3}\epsilon_{\,\,\,\,\,\,\,\,\,\,\beta}^{\alpha\beta\rho}\partial_{\rho}\Theta_{\alpha}^{\alpha}-\epsilon^{\alpha\beta\sigma\nu}\partial_{\sigma}\Theta_{\beta\nu}\right)=0,
S[V^{++},q^{+},\breve{q}^{+}]=\frac{1}{2g^{2}}\intd^{4}xd^{4}\theta\mathrm{tr}W^{2}-\frac{1}{2g^{2}}\intd\zeta^{(-4)}\mathrm{tr}q^{+i}\nabla^{++}q_{i}
{\frac{1}{m}}{\frac{dm}{d\tilde{\lambda}}}=g_{\mu\nu}A^{\mu}{\frac{dx^{\nu}}{d\tilde{\lambda}}}=\sqrt{h}\left[2-h\left({\frac{dt}{d\tilde{\lambda}}}\right)^{2}+y^{2}\sum_{i}\left({\frac{dx^{i}}{d\tilde{\lambda}}}\right)^{2}\right]\left[{\frac{h^{\prime}}{2}}\left({\frac{dt}{d\tilde{\lambda}}}\right)^{2}-y\sum_{i}\left({\frac{dx^{i}}{d\tilde{\lambda}}}\right)^{2}\right].
\lambda_{n,1}^{(2)}=\frac{\partial\overline{{H}}_{0}}{\partialq_{n,0}}\,\\,lambda_{n,j_{n}}^{(2)}=\frac{\partial\overline{{H}}_{0}}{\partialq_{n,j_{n}-1}}-\mu_{n,j_{n}-1}\,\\j_{n}=2,3,\cdots,m_{n}-1\.
\begin{aligned}{\{q^{\alpha},q^{\beta}\}}&{{}=2\left(\sigma_{i}c^{-1}\right)^{\alpha\beta}P_{i},}\\{\{s^{\alpha},s^{\beta}\}}&{{}=-2\left(\sigma_{i}c^{-1}\right)^{\alpha\beta}K_{i},}\\{\{q^{\alpha},s^{\beta}\}}&{{}=\left(2c^{-1}D-\sigma_{ij}L_{ij}\right)^{\alpha\beta}.}\\\end{aligned}
\delta\hat{\tilde{N}}=16a\partial\hat{\Sigma}+{\textstyle\frac{8!}{3\cdot4!}}\partial\hat{\chi}\hat{C}\left(i_{\hat{k}}\hat{C}\right)+3{\textstyle\frac{8!}{6!}}\partial\hat{\tilde{\chi}}\left(i_{\hat{k}}\hat{C}\right)-8m\hat{\lambda}\left(i_{\hat{k}}\hat{\tilde{N}}\right)\,,
S^{\prime}={\frac{(\Psi\mathrm{cos}\gamma-\mathrm{sin}\gamma)+ie^{-\varphi}\mathrm{cos}\gamma}{(\Psi\mathrm{sin}\gamma+\mathrm{cos}\gamma)+ie^{-\varphi}\mathrm{sin}\gamma}},\\\\\\\M^{\prime}={\calU}^{T}\left(\begin{matrix}{U_{6}}&{0}\\{0}&{U_{22}}\\\end{matrix}\right){\calU}M{\calU}^{T}\left(\begin{matrix}{U_{6}^{T}}&{0}\\{0}&{U_{22}^{T}}\\\end{matrix}\right){\calU},