harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left\langle{\calO}(k){\calO}(-p)\right\rangle\sim\frac{\alpha_{\mathrm{eff}}^{\prime}\pi}{2G_{o(1)}^{4}}[N(\omega^{2}-\sum_{i=1}^{5}k_{i}^{2})-1]^{1/2}\delta^{6}(k-p)
<\prod_{i=0}^{\infty}\sigma_{i}^{k_{i}}>_{n}^{irr}={\frac{1}{n!}}\left({\frac{1}{2}}{\frac{\partial^{2}}{\partialt_{0}^{2}}}\right)^{n}\left.\prod_{i=0}^{\infty}{\frac{\partial^{k_{i}}}{\partialt_{i}^{k_{i}}}}F(0,t)\right|_{t=0}=<{\frac{1}{n!}}\left({\frac{\sigma_{0}\sigma_{0}}{2}}\right)^{2}\prod_{i=0}^{\infty}\sigma_{i}^{k_{i}}>_{0},
I={\frac{1}{16\pi}}\intd^{4}x\sqrt{-g}\left[-{\calR}+2\partial_{\mu}\varphi\partial^{\mu}\varphi-{\frac{1}{2}}(e^{-2\varphi}F^{\mu\nu}F_{\mu\nu}+e^{2\varphi}\tilde{G}^{\mu\nu}\tilde{G}_{\mu\nu})\right],
U_{(0,\,r-1)}\equiv\frac{1}{2}(Q_{0})_{\alpha_{0}\beta_{0}}(Q_{1})_{\alpha_{1}\beta_{1}}(Q_{2})_{\alpha_{2}\beta_{2}}\cdots(Q_{r-1})_{\alpha_{r-1}\beta_{r-1}}\epsilon^{\beta_{0}\alpha_{1}}\epsilon^{\beta_{1}\alpha_{2}}\cdots\epsilon^{\beta_{r-1}\alpha_{0}}.
K~=~\frac{8\pi^{2\mu}(2\mu-1)^{2}(\mu-2)^{2}}{(2\mu-3)^{2}\Gamma(\mu)\Gamma(\mu+1)}~~,~~P~=~\frac{16\pi^{2\mu}(2\mu-1)(2-\mu)}{(2\mu-3)\Gamma(\mu)\Gamma(\mu+1)}
\langlei|P^{-}|jk\rangle=\left[(p_{i}^{-}-p_{j}^{--}p_{k}^{-})\sqrt{\frac{J_{j}J_{k}}{J_{i}}}C_{ijk}\right]p_{i}^{+}\delta(p_{i}^{+}-p_{j}^{+}-p_{k}^{+})
\begin{array}{rclcrcl}{|u_{L}\rangle}&{=}&{\displaystyle\bigg|\frac{1}{3},\frac{1}{2},\frac{1}{2}\bigg\rangle,}&{\quad}&{|d_{L}\rangle}&{=}&{\displaystyle\bigg|\frac{1}{3},\frac{1}{2},-\frac{1}{2}\bigg\rangle,}\\{|u_{R}\rangle}&{=}&{\displaystyle\bigg|\frac{4}{3},0,0\bigg\rangle,}&{\quad}&{|d_{R}\rangle}&{=}&{\displaystyle\bigg|-\frac{2}{3},0,0\bigg\rangle,}\\\end{array}
a^{3}(z)={\frac{6\piGA}{(z_{-}-z_{+})^{5}}}(C-z)^{2}={{\frac{3}{4}}\pi^{3}\zeta_{R}^{\prime}(-4)G_{5}}{\frac{(z_{0}-z)^{2}}{(z_{-}-z_{+})^{5}}},
S[\bar{\psi},\psi,\varphi]=\intd^{4}x\left(\frac{1}{2}(\partial_{\mu}\varphi_{b})^{2}-\frac{1}{2}m_{0}^{2}\varphi_{b}^{2}+V(\varphi_{b})+\bar{\psi}_{b}(i\not{\partial}-M_{0}-g_{0}\varphi_{b})\psi_{b}\right)
L_{\mathrm{DH}}={P}_{i}^{\prime}\dot{X}_{i}+\frac{\theta}{2}\varepsilon_{ij}({P}_{i}^{\prime}-eA_{i})(\dot{P}_{j}^{\prime}-e\frac{d}{dt}A_{j})+eA_{0}-\frac{1}{2}({P}_{i}^{\prime}-eA_{i})^{2}.
(\Omega_{b})^{2}=\left(\begin{array}{ccccc}{r^{2}}&{-2iv}&{0}&{\cdots}&{0}\\{2iv}&{r^{2}}&{0}&{\ddots}&{0}\\{0}&{0}&{r^{2}}&{\ddots}&{\vdots}\\{\vdots}&{\ddots}&{\ddots}&{\ddots}&{0}\\{0}&{0}&{\cdots}&{0}&{r^{2}}\\\end{array}\right)
\left(\begin{array}{cccc|cc}{\times}&{\times}&{}&{}&{\times}&{}\\{\times}&{\times}&{}&{}&{\times}&{}\\{\times}&{\times}&{\times}&{\times}&{\times}&{\times}\\{\times}&{\times}&{\times}&{\times}&{\times}&{\times}\\\hline{\times}&{\times}&{}&{}&{\times}&{}\\{\times}&{\times}&{\times}&{\times}&{\times}&{\times}\\\end{array}\right)
H^{\prime}\equiv-i\frac{3\hbarc}{2}\gamma^{0}\gamma^{a}(-\sigma^{-1})_{,a}=-i\frac{3\hbarc}{2}\left(\begin{matrix}{\epsilon_{0}}&{\vec{\sigma}\cdot\vec{\epsilon}}\\{\vec{\sigma}\cdot\vec{\epsilon}}&{\epsilon_{0}}\\\end{matrix}\right)\,{,}
\boxed{\frac{T_{c}}{s_{1}}\,\frac{d}{dy}\left(\begin{matrix}{f}\\{g}\\\end{matrix}\right)=\left(\begin{matrix}{s_{1}u^{\prime}}&{\left(y^{2}+\frac{u^{2}}{2}+\frac{Tt}{2s_{1}^{2}}\right)+yu}\\{-\left(y^{2}+\frac{u^{2}}{2}+\frac{Tt}{2s_{1}^{2}}\right)+yu}&{-s_{1}u^{\prime}}\\\end{matrix}\right)\left(\begin{matrix}{f}\\{g}\\\end{matrix}\right),}
\zeta_{0}^{(p)}(\nu)={\frac{2}{(4\pi)^{D/2}\Gamma(\nu)}}\sum_{n=0}^{\infty}\sum_{m=0}^{[n/2]}{\frac{\Gamma(m+1/2)\breve{a}_{2m,n}}{\sqrt{\pi}(2\nu+2(n-m)-D)}}~~.
\ell\,\neq\,5\quad\,\mathrm{SpD}_{56}\left(E^{{\vec{\alpha}}_{\ell}}\right)=\left(\begin{matrix}{A[{{\vec{\alpha}}_{\ell}}]}&{{\bf0}}\\{{\bf0}}&{-A^{T}[{{\vec{\alpha}}_{\ell}}]}\\\end{matrix}\right)
\pi^{i}=\frac{\partial{\calL}}{\partial\dot{A}_{i}}=4\sqrt{2}\epsilon^{ij}\eta^{4}A_{j}-8\sqrt{2}i\epsilon^{ij}(\eta^{2}-\frac{1}{2}{\vert\phi\vert^{2}})(\phi(D_{j}\phi)^{*})
\begin{array}{c}{<A>\left(t\right)=Tr\mathrm{~}\rho\left(0\right)\mathrm{~}U^{-1}\left(t,0\right)\mathrm{~}A\mathrm{}U\left(t,0\right)=}\\{Tr\mathrm{~}\rho\left(0\right)\mathrm{~}U\left(0,T\right)\mathrm{~}U\left(T,t\right)\mathrm{~}A\mathrm{~}U\left(t,0\right)}\\\end{array}
K_{IK\bar{J}\bar{L}}\bigl[\delta\,\Phi^{I}\Phi^{K}\bar{\Phi}^{\bar{J}}\bar{\Phi}^{\bar{L}}\,+\,\tau\,\Phi^{I}\bar{\Phi}^{\bar{J}}\Phi^{K}\bar{\Phi}^{\bar{L}}\bigr],
T_{abcd}^{\kappa\lambda\mu\nu}(p_{1}\ldotsp_{4})=\delta_{ab}\delta_{cd}\frac{\Phi^{\kappa\lambda}(p_{1},p_{2})\Phi^{\mu\nu}(p_{3},p_{4})}{(p_{1}+p_{2})^{2}+c_{B}\Lambda^{2}}
Z_{1}^{*}W_{\pi^{+}}(K)Z=e^{i\frac1930m}e^{i90m}b_{1}z_{1}^{*}z_{1}+e^{i\frac1930m}e^{i90m}b_{2}z_{2}^{*}z_{2}+\frac12e^{i\frac1930m}e^{i90m}b_{3}[z_{1}^{*}(z_{1}+z_{2})+z_{1}^{*}(z_{1}+z_{2})]
{\bfG}(\Box)=\frac{1}{\Boxz_{+}z_{-}}\frac{1}{J_{1}^{+}\,Y_{1}^{-}-J_{1}^{-}\,Y_{1}^{+}}\left[\begin{array}{cc}{\,\sqrt{\Box}z_{-}u_{-}(z_{+})}&{\displaystyle{\frac2\pi}}\\{\displaystyle{\frac2\pi}}&{\sqrt{\Box}z_{+}u_{+}(z_{-})\,}\\\end{array}\right].
\Theta^{\underline{\mu}}(z)=\theta^{\underline{\mu}}(\xi)\,+\,\eta^{-}\lambda_{-}^{\underline{\mu}}(\xi)\,+\,\bar{\eta}^{-}\bar{\lambda}_{-}^{\underline{\mu}}(\xi)\,+\,i\eta^{-}\bar{\eta}^{-}\sigma_{--}^{\underline{\mu}}(\xi)\,,
{\calS}_{J_{H}^{-1}}^{(2k+1)}=(2k+1)\left({J_{H}^{-1}}^{\prime}\right)^{k+1}{L_{-1}^{ch}}^{2k+1}\left({J_{H}^{-1}}^{\prime}\right)^{k}=(2k+1)e^{(k+1)\varphi}L_{-1}^{2k+1}e^{k\varphi}.
\epsilon_{\tau_{0}\cdots\tau_{p}}\Gamma^{\tau_{0}\cdots\tau_{p}}\Gamma^{\mu_{1}\cdots\mu_{q}}=(-1)^{\frac{q(q-1)}{2}}\frac{(p+1)!}{(p+1-q)!}\epsilon_{\tau_{0}\cdots\tau_{p-q}}{^{\mu_{1}\cdots\mu_{q}}}\Gamma^{\tau_{0}\cdots\tau_{p-q}},
\left(\delta^{ij}\partial_{i}\partial_{j}+H_{6}\delta^{pq}\partial_{p}\partial_{q}\right)H_{2}+\frac{1}{2}H_{6}\delta^{ij}\partial_{i}b\partial_{j}b\delta(x^{6})\cdots\delta(x^{9})+\kappaT_{2}N\delta(x^{3})\cdots\delta(x^{9})=0\,.
S_{D}=-\intd^{4}\sigma\sqrt{-\mathrm{det}\,(G_{\mu\nu}+\frac{1}{\sqrt{1+e^{2\phi}C_{0}^{2}}}(e^{\frac{\phi}{2}}\tilde{F}_{\mu\nu}+C_{\mu\nu}+e^{\phi}C_{0}b_{\mu\nu})}+\int\Omega_{D},
\Pi_{1}^{M}(k^{2})\rightarrow\Pi^{(1)}(k^{2})=-\frac{e^{2}k^{2}}{(2\pi)}\int_{0}^{1}d\xi\,\,\xi\,(1-\xi)\frac{1}{{(M^{2})}^{\frac{1}{2}}}\,\,,
_0\left<\right|T\left(\Psi(x_{2})\bar{\Psi}(x_{1})\right)\left|\right>_{0}=\frac{1}{(2\pi)^{3}}\int{\mathrmd}^{3}p\frac{-i\hat{p}+m}{2E}e^{ip(x_{2}-x_{1})}
\frac{L_{eff}}{m}=-R\approx\frac{1}{2}\Bigl[(\delta\dot{r})^{2}+r_{0}^{2}\,(\delta\dot{\theta})^{2}+(\delta\dot{y}^{i})^{2}\Bigr]+\frac{1}{2}\left[(\eta-\hat{q})\frac{k\tilde{d}(\tilde{d}+1)}{r_{0}^{\tilde{d}+2}}-3\left(\frac{\ell}{r_{0}^{2}}\right)^{2}\right](\deltar)^{2}\,.
U_{j}(\underline{{p}},\underline{{q}})\equiv\frac{gm_{j}}{\sqrt{16\pi^{3}}}\frac{1}{p^{+}\sqrt{q^{+}}}\,,\;\;V_{2s}(\underline{{p}},\underline{{q}})\equiv\frac{g}{\sqrt{8\pi^{3}}}\frac{\vec{\epsilon}_{2s}^{\,*}\cdot\vec{p}_{\perp}}{p^{+}\sqrt{q^{+}}}\,,
T_{2}(x,y)^{\mathrm{corr}}=i\,\frac{\kappa^{2}}{4}\,:\!T_{m_{1}}^{\gamma\delta}(x)\,T_{m_{2}}^{\rho\sigma}(y)\!:\,\Omega(x-y)_{\gamma\delta\rho\sigma}^{\mathrm{corr}}\,,
\begin{array}{l}{3(3,2)_{\frac{1}{6}}+3(\overline{{3}},1)_{\frac{1}{3}}+3(\overline{{3}},1)_{-\frac{2}{3}}+6(1,2)_{-\frac{1}{2}}+3(1,2)_{\frac{1}{2}}}\\{\\\+6(1,1)_{1}+9(1,1)_{0}+3(1,1)_{-1}.}\\\end{array}
\begin{array}{ccc}{\deltax^{A}=a^{A}-i\bar{\theta}_{i}\gamma^{A}\varepsilon^{i}-i\bar{\tilde{\theta}}_{\tilde{\imath}}\tilde{\gamma}^{A}\tilde{\varepsilon}^{\tilde{\imath}}~~~}&{~~~\delta\theta^{i}=\varepsilon^{i}~~~}&{~~~\delta\tilde{\theta}^{\tilde{\imath}}=\tilde{\varepsilon}^{\tilde{\imath}}}\\\end{array}
ds_{t\to0}^{2}\sim\mathrm{e}^{-\frac{2pf_{0}}{q-1}}(-dt^{2}+d\Sigma_{k,0}^{2}+d\vec{y}_{q-k}^{2})+\mathrm{e}^{2f_{0}}d\vec{x}_{p}^{2}.
\{P_{i},P_{j}\}_{\mathrm{D}}=-\frac{\partial{f}_{s}}{\partial{q}^{i}}(D^{-1})^{st}\left.\left[\frac{\partial}{\partial{q}^{j}}\left(g^{kl}\frac{\partial{f}_{t}}{\partial{q}^{k}}\right)P_{l}-\frac{\partial}{\partial{q}^{j}}\left(g^{kl}\frac{\partial{f}_{t}}{\partial{q}^{k}}A_{l}\right)\right]\right|_{f=0}-(i\leftrightarrowj)\;.
\sigma_{1}=\left(\begin{array}{rcl}{0}&{1}\\{1}&{0}\\\end{array}\right)~,~\sigma_{2}=\left(\begin{array}{rcl}{0}&{-i}\\{i}&{0}\\\end{array}\right)~,~\sigma_{3}=\left(\begin{array}{rcl}{1}&{0}\\{0}&{-1}\\\end{array}\right)~,
L_{B}=L\,-\,\frac{1}{2}\chi^{\alpha}\gamma_{\alpha\beta}\chi^{\beta}\,-\,\bar{c}_{\alpha}s(q^{r})\chi_{;r}^{\alpha}\,+\,\dot{\bar{c}}_{\alpha}s(q^{r})\frac{\partial\chi^{\alpha}}{\partial\dot{q}^{r}},
\begin{array}{lll}{h_{b}(z,T,x^{10})=\tilde{h}_{b+N^{\prime}/2}(\bar{z},-1/\bar{T},-x^{10})}&{,}&{\tilde{h}_{b}(z,T,x^{10})=h_{b+N^{\prime}/2}(\bar{z},-1/\bar{T},-x^{10})}\\{h_{c}(z,T,x^{10})=\tilde{h}_{c}(\bar{z},-1/\bar{T},-x^{10})}&{,}&{}\\\end{array}
\renewcommand{\arraystretch}{1}\sum_{\begin{array}{c}{\scriptstyle\sigma_{1},\ldots,\sigma_{L-1}=0,1}\\{\scriptstyle\sigma_{j}\sigma_{j+1}=0}\\\end{array}}q^{\;\;\displaystyle\sum_{k=1}^{L-1}k\sigma_{k}}\qquad\qquad\sigma_{0}=a,\;\sigma_{L}=0,
\left(\begin{array}{c}{{\frac{\partial\Phi_{0}}{\partialt}}}\\{{\frac{\partial\Phi_{1}}{\partialt}}}\\\end{array}\right)={\calD}_{1}\left(\begin{array}{c}{{\frac{\deltaH_{3}}{\delta\Phi_{0}}}}\\{{\frac{\deltaH_{3}}{\delta\Phi_{1}}}}\\\end{array}\right)={\calD}_{2}\left(\begin{array}{c}{{\frac{\deltaH_{2}}{\delta\Phi_{0}}}}\\{{\frac{\deltaH_{2}}{\delta\Phi_{1}}}}\\\end{array}\right)
I_{div}=\frac{1}{4\pi^{2}}\int_{m_{f}}^{\infty}dq\left(\frac{q^{2}}{\sqrt{q^{2}+m_{i}^{2}}}\right)\left\{1+\frac{1}{2}\left(\frac{m_{i}^{2}+m_{f}^{2}}{q^{2}-m_{f}^{2}}\right)\right\}
{\psi}_{m}\propto\quad<\mathrm{source}_{after}\vert<0\vert\frac{(a_{i}(\vec{k}))^{m+1}}{\sqrt{[m+1]_{q}!}}\vert\quada_{i}^{\dagger}g_{i}\quad\vert\frac{(a_{i}^{\dagger}(\vec{k}))^{m}}{\sqrt{[m]_{q}!}}\vert0>\vert\mathrm{source}_{before}>
(-4U+2S)_{[i}{}^{k}L_{j]k}^{M}-V_{[ij]}^{a}L_{a}^{M}+2(S+U)_{[i}{}^{k}L_{j]k}^{M}=\Sigma^{2}\frac{1}{\sqrt{2}}\Lambda^{M}{}_{N}L_{ij}^{N},
\left\langle\mathrm{Tr}(\tilde{q}q\tilde{q}q)-\mathrm{det}\tilde{q}\,\mathrm{det}q\right\rangle=\langle\mathrm{Tr}(\tilde{q}q\tilde{q}q)\rangle-\langle\tilde{q}_{1}q_{1}\tilde{q}_{2}q_{2}\rangle+\Lambda^{4}\.
T\mathrm{}_{k}^{(+)}(\alpha|u)=\prod_{j=1}^{\alpha}\displaystyle{\frac{\theta_{k}(u+(2j-1)\eta)}{\theta_{k}(u-(2j-1)\eta)}}\quad\mathrm{and}\quadT\mathrm{}_{k}^{(-)}(\alpha|u)=T\mathrm{}_{k}^{(+)}(\alpha|\lambda-u),
\begin{array}{lll}{A(k_{1},k_{2},k_{3})}&{=}&{A_{123}\xi_{1}^{k_{1}}\xi_{2}^{k_{2}}\xi_{3}^{k_{3}}+A_{132}\xi_{1}^{k_{1}}\xi_{2}^{k_{3}}\xi_{3}^{k_{2}}+A_{213}\xi_{1}^{k_{2}}\xi_{2}^{k_{1}}\xi_{3}^{k_{3}}}\\{}&{}&{}\\{}&{}&{+A_{231}\xi_{1}^{k_{2}}\xi_{2}^{k_{3}}\xi_{3}^{k_{1}}+A_{312}\xi_{1}^{k_{3}}\xi_{2}^{k_{1}}\xi_{3}^{k_{2}}+A_{321}\xi_{1}^{k_{3}}\xi_{2}^{k_{2}}\xi_{3}^{k_{1}}}\\\end{array}
\intd^{2}x\sqrt{-\eta}\left[\vec{f}\left(A+2\gamma^{5}\epsilon^{\mu\nu}e^{2(\psi-\varphi)}\varphi_{,\mu}\psi_{,\nu}\right)\vec{f}+\epsilon^{\mu\nu}e^{2(\psi-\varphi)}\varphi_{,\mu}\partial_{\nu}(\vec{f}\gamma^{5}\vec{f})\right]
d_{H}{\frac{4\beta_{3}-\beta_{0}}{\beta_{0}}}=4\left(1+\sqrt{1+{\frac{8}{Q^{2}}}}\right)\left[{\frac{1}{1+\sqrt{1-{\frac{2}{Q^{2}}}}}}-{\frac{1}{1+\sqrt{1-{\frac{8}{Q^{2}}}}}}\right]<0
D_{R\mu\alpha}(k)=\frac{1}{k^{2}-m^{2}+i\epsilonk_{0}}\left[g_{\mu\alpha}-\frac{k_{\mu}k_{\alpha}}{k^{2}+i\epsilonk_{0}}-im\varepsilon_{\mu\alpha\rho}\frac{k^{\rho}}{k^{2}+i\epsilonk_{0}}\right]+\mathrm{gauge~fixing~term}
H_{3}=\hat{H}_{3}-\left(d\hat{z}+{\frac{\hat{E}^{{\#}(-)}}{i_{\hat{z}}\hat{E}^{{\#}}}}\right)\wedgei_{\hat{z}}\hat{H}_{3}\;+\left(dy+{\frac{{E}^{{*}(-)}}{i_{y}\hat{E}^{{*}}}}\,\right)\wedgei_{y}H_{3}\;,
{\tilde{\calL}}_{F}=4(D-2)^{2}\intdre^{(D-3)A}\left(\left({\frac{\partialW}{\partial\phi}}\right)^{2}-\kappa^{D-2}\left({\frac{D-3}{D-2}}\right)W^{2}\right)\bar{\tilde{\eta}}_{-i}\Gamma^{\underline{\mu}}\partial_{\mu}\tilde{\eta}_{-}^{i}\.
\stackrel{(-1,0)}{\Phi}_{\alpha_{L-4}}^{*(1)}Z_{\;\;\alpha_{L-3}}^{\alpha_{L-4}}=\delta_{1}\left(\stackrel{(-2,0)}{\eta}_{\alpha_{L-3}}^{*(11)}-\stackrel{(-2,0)}{\gamma}_{\beta_{L-2}}A_{\alpha_{L-3}}^{\;\;\alpha_{L-2}}\right).
\langle\Psi|\overbrace{\phi^{2}(x)\cdots\phi^{2}(x)}\pi^{2}(x)|\Psi\rangle\:\mathrm{and}\:\langle\Psi|\overbrace{\phi^{2}(x)\cdots\phi^{2}(x)}{\phi^{\prime}}^{2}(x)|\Psi\rangle,
Q^{(k-1)}=(2-Q^{(k)})\left\{1-\frac{2(a_{k}+b_{k}-1)Q^{(k)}\left[rQ_{r}^{(k)}-2a_{k}+(a_{k}+b_{k})Q^{(k)}\right]}{\left[rQ_{r}^{(k)}-2a_{k}+(a_{k}+b_{k})Q^{(k)}\right]^{2}+r^{2}Q^{(k)2}(2-Q^{(k)})^{2}}\right\},
(2\pi)^{3}{\hat{G}}({\hat{\xi}}{\hat{\eta}};{\hat{\xi}}^{\prime}{\hat{\eta}}^{\prime})=\sum_{123}D^{-1}({\hat{q}}_{12})\intd{{\hat{q}}_{12}}^{\prime\prime}K({\hat{q}}_{12},{{\hat{q}}_{12}}^{\prime\prime}){\hat{G}}({\hat{\xi}}^{\prime\prime}{\hat{\eta}}^{\prime\prime};{\hat{\xi}}^{\prime}{\hat{\eta}}^{\prime})
\langle\Sigma_{f},J_{\phi}^{f}\vert\Sigma_{i},J_{\phi}^{i}\rangle=\delta(Q_{f}-Q_{i})\intD[g,\phi]{e^{-S+i\int_{t_{i}}^{t_{f}}d^{3}x\sqrt{g}J_{\phi}^{\mu}\partial_{\mu}\phi}}
Q^{(1)}=(2\pi)^{4}\delta\left(\sum_{n=1}^{M}p_{n}\right)\frac{1}{(4\pi)^{D}[s(s-s_{1})]^{D/2}}exp\left[\sum_{m>n}\hat{k}(t_{n})\cdot\hat{k}(t_{m})G^{(1)}(t_{n},t_{m})\right]
E_{vac}\,=\,\frac{1}{2}\sum_{\alpha}\langle0\vertQ_{\alpha}^{\daggeri}Q_{\alpha}^{i}\vertO\rangle\,=\,\frac{1}{2}\sum_{\alpha}\vert\vertQ_{\alpha}^{i}\vertO\rangle\vert\vert\,\,\,\,(\mathrm{no~summation~over~i})
\tau(x,t,w)\,=\,\left(\begin{matrix}{1}&{-t}&{0}&{t}&{0}\\{t}&{{\frac{2-{t^{2}}+{w^{2}}+{x^{2}}}{2}}}&{w}&{{\frac{{t^{2}}-{w^{2}}-{x^{2}}}{2}}}&{x}\\{0}&{w}&{1}&{-w}&{0}\\{t}&{{\frac{-{t^{2}}+{w^{2}}+{x^{2}}}{2}}}&{w}&{{\frac{2+{t^{2}}-{w^{2}}-{x^{2}}}{2}}}&{x}\\{0}&{x}&{0}&{-x}&{1}\\\end{matrix}\right)
h=-\frac{1}{k}\int_{\Sigma_{\tau}}d\bar{\eta}d\eta~~{\calE}(-\eta)\sum_{i}q_{i}E_{ii}+\frac{1}{2k}\sum_{i\neqj}\int_{\Sigma_{\tau}}d\bar{\eta}d\eta~~\left(\frac{u_{j}}{u_{i}}w_{ij}v_{ij}+\frac{u_{i}}{u_{j}}w_{ji}v_{ji}\right)\left(E_{ii}-\frac{1}{n}I\right).
i\intdt^{\prime}\int\int\frac{d\omega^{\prime\prime}}{2\pi}\frac{dk^{\prime\prime}}{2\pi}\frac{e^{-i\omega^{\prime\prime}(t-t^{\prime})}}{\omega^{2}-k^{2}-4+i\rho}\left(1+\frac{ik^{\prime\prime}+2\sigma_{1}}{ik^{\prime\prime}-2\sigma_{1}}\right).
A_{4}(k_{1},k_{2},k_{3},k_{4})\rightarrow-i\int{\frac{d^{7}k}{(2\pi)^{7}}}{\frac{\sum_{IJ\mu}A_{\J}^{I\mu}(k_{1},k_{2},k)A_{\I\mu}^{J}(k_{3},k_{4},-k)}{k^{2}-i\epsilon}}\,
{\left(\frac{\rho_{\phi}}{\rho_{BI}}\right)_{end}}={\frac{3\pi}{64}}\left({\frac{1}{h_{GW}^{2}\left(F_{1}(\alpha)/F_{2}(\alpha)\right)^{2}}}\right)\left({\frac{\rho_{g}}{\rho_{BI}}}\right)_{eq}
\hat{\xi}_{\alpha\beta}^{+}=e_{\alpha}^{a}\,e_{\beta}^{b}\,\hat{\xi}_{ab}^{+}\qquad\mathrm{and}\qquad\hat{\xi}_{\alpha\beta}^{-}=e_{\alpha}^{a}\,e_{\beta}^{b}\,\hat{\xi}_{ab}^{-}.
\begin{array}{lll}{|\vec{r}|=e^{\eta}r_{1}}&{\quad\mathrm{and}\quad}&{r_{1}=\frac{\bar{r}}{1+\frac{1}{4}\bar{r}^{2}}}\\{x^{0}=e^{\eta}\sqrt{1-r_{1}^{2}}}&{}&{\eta=\int\frac{dt}{K}~.}\\\end{array}
T_{L}\equiv4\pi\left({\frac{\deltaS}{\deltah}}\right)_{h,\bar{h}=0}=L_{ab}J_{L}^{a}J_{L}^{b},\;\;\;\;\;\;T_{R}\equiv4\pi\left({\frac{\deltaS}{\delta\bar{h}}}\right)_{h,\bar{h}=0}=L_{ab}J_{R}^{a}J_{R}^{b},
C^{m}(k_{1},l_{1};\ldots;k_{m},l_{m})=\left(\prod_{j=2}^{m}\;\frac{1}{l_{j}!}\;\prod_{q_{j}=1}^{l_{j}}\Bigl(K_{j-1}+(q_{j}-2)k_{j}\Bigr)\right)C^{1}(k_{1},l_{1}).
V_{z\llb,\,x\ll\sqrt{Q_{p}b}}\toE-{\frac{m_{f}}{2Q_{p}Q_{F}}}z(x^{2}+4Q_{p}z)^{\frac{n}{2}}+{\frac{m_{f}J_{\perp}^{2}}{2E^{2}Q_{p}^{2}Q_{F}^{2}}}(x^{2}+4Q_{p}z)^{n},
D_{\mu\nu}=\frac{1}{(p^{2}+M_{+}^{2})(p^{2}+M_{-}^{2})}\,\left[P_{\mu\nu}\,(p^{2}+\Pi_{2})+Q_{\mu\nu}\,(p^{2}+\Pi_{1})-(\kappa+\Pi_{\mathrm{odd}})\,\epsilon_{\mu\nu\lambda}\,p_{\lambda}\right]+\frac{\xi\,p_{\mu}p_{\nu}}{(p^{2})^{2}}\,.
{\tilde{Z}}_{m}^{T}~=~\int\prod_{\mu}{\calD}P^{\mu}e^{Tr\left\{\sum_{\mu}(P^{\mu})^{2}~+~ln[\sum_{\mu}(P^{\mu})^{2}]\right\}}\prod_{\mu;i\neqj}\delta\left((1-D_{ij})P_{ij}^{\mu}\right)\prod_{\mu;i}\delta(\sum_{j}P_{ij}^{\mu})~~.
U_{DQRS}={\frac{N}{g}}tr\left({\frac{1}{2}}S^{2}+{\frac{1}{2}}D^{2}+{\frac{g_{1}}{4}}S^{4}+{\frac{g_{2}}{4}}D^{4}+{\frac{g_{3}}{2}}(SDSD+2S^{2}D^{2})\right)
D_{++}^{+}(q)=\frac{q_{\bot}^{2}}{q_{-}^{2}}\cdot\frac{i}{q^{2}+i{\epsilon}}-\frac{2{\delta}(q_{-})}{q_{\bot}^{2}}\int_{0}^{\infty}dk_{+}(\frac{ik_{+}}{q_{+}-k_{+}+i{\epsilon}}-\frac{ik_{+}}{q_{+}+k_{+}-i{\epsilon}}).
{\hat{\calH}}\,\psi({\bfX})=\intd^{3}x\left[\,\frac{g^{2}}{2}{\hat{X}}^{ax}{\hat{X}}^{ax}+\,\frac{1}{4g^{2}}\,(\partial_{[a}{\hat{P}}_{b]x})^{2}\right]\,\psi({\bfX})
H=\sum_{\alpha,\beta}h_{\alpha\beta}\sum_{\sigma=1}^{n}a_{\alpha\sigma}^{\dagger}a_{\beta\sigma}+\sum_{\alpha,\beta,\gamma,\delta}v_{\alpha\beta\gamma\delta}\sum_{\sigma_{1},\sigma_{2}=1}^{n}a_{\alpha\sigma_{1}}^{\dagger}a_{\beta\sigma_{2}}^{\dagger}a_{\delta\sigma_{2}}a_{\gamma\sigma_{1}}\,.
\begin{array}{l}{(H^{n})_{W}=H^{n}+\frac{1}{24}(2n-1)n(n-1)H^{n-2}}\\{+\frac{1}{5760}(20n^{2}-48n+7)\prod_{k=0}^{3}(n-k)H^{n-4}}\\{+\frac{1}{2903040}(280n^{3}-1596n^{2}+1874n-93)\prod_{k=0}^{5}(n-k)H^{n-6}+...}\\\end{array}
R=\frac{1}{r^{3/2}}[e^{i\omegar}(\frac{\alpha}{2}e^{-i3\pi/4}-\frac{\beta}{2}e^{-i\pi/4})+e^{-i\omegar}(\frac{\alpha}{2}e^{i3\pi/4}-\frac{\beta}{2}e^{i\pi/4})].
\left\langleAB_{1}B_{2}\right\rangle-\left\langleAB_{1}\right\rangle\left\langle\tilde{B}_{2}\right\rangle-\left\langleAB_{2}\right\rangle\left\langle\tilde{B}_{1}\right\rangle+\left\langleA\right\rangle\left\langle\tilde{B}_{1}\right\rangle\left\langle\tilde{B}_{2}\right\rangle
\sqrt{-g}=\gamma^{-1}-r^{2}\omega\gamma\dot{\phi}-\frac{1}{2}r^{2}\gamma\left(\dot{\theta}^{2}-\omega^{2}\theta^{2}-\gamma^{-2}\theta^{2}\right)-\frac{1}{2}r^{2}\gamma^{3}\left(\dot{\phi}^{2}-\gamma^{-2}\phi^{2}\right)\,,
L_{eff}^{ii}=\frac{1}{2\pi}\left(\hat{\Phi}^{i}~\hat{\eta}^{i}\right)\left(\begin{array}{cc}{A^{ii}}&{C^{ii}/2}\\{C^{ii}/2}&{B^{ii}}\\\end{array}\right)~\left(\begin{array}{c}{\hat{\Phi}^{i}}\\{\hat{\eta}^{i}}\\\end{array}\right),
h_{\mu\nui}^{(1)}=B_{\mu\nui}^{\prime}+V_{\mu\nu}^{k}b_{ki}\,\,\,,\,\,\,h_{\mu\nui}^{(2)}=[V_{\mu}^{k}B_{i\nuk}^{\prime}-(\mu\leftrightarrow\nu)]-\frac{1}{2}\partial_{i}U_{\mu\nu}-F_{\mu\nu}^{k}b_{ki}\,\,.
[\bar{\delta}(\epsilon_{1}),\bar{\delta}(\epsilon_{2})]=gg_{0}^{-1}([\tilde{\epsilon}_{2}^{\dagger},\tilde{\epsilon}_{1}^{\dagger}]+[\tilde{\epsilon}_{2}^{\dagger},\tilde{\epsilon}_{1}]+[\tilde{\epsilon}_{2},\tilde{\epsilon}_{1}^{\dagger}])g.
W_{n}^{m}=\frac{\mu_{nm}!}{\nu_{nm}!}\left[\frac{|n-m|!}{\left(\frac{|n-m|}2\right)!}\right]^{2}\frac1{|\xi|}\left|\frac{\eta}{2\xi}\right|^{|n-m|}\left[C_{\mu_{nm}}^{\frac{|n-m|+1}2}\left(\frac1{|\xi|}\right)\right]^{2}.
Z_{K}^{red}(V,\rho_{K})=\frac{C^{K}}{K!}\int\prod_{i=1}^{K}d{\bfa}_{i}\prod_{j=1}^{K-1}d\rho_{j}\prod_{k=1}^{K}\rho_{k}^{b-d-1}\Theta(\{{\bfa}_{l}\},\{\rho_{l}\})\,,
\{a(\vec{p},\lambda),a^{\dagger}({\vec{p}}^{\,\prime},\lambda^{\prime})\}=\{b(\vec{p},\lambda),b^{\dagger}({\vec{p}}^{\,\prime},\lambda^{\prime})\}=\delta_{\lambda,\lambda^{\prime}}\delta^{3}(\vec{p}-{\vec{p}}^{\,\prime})\,,
ds_{RB}^{2}=\frac{{\rho}^{2}}{M^{2}}\,dt^{2}-\frac{M^{2}}{\rho^{2}}\,d\rho^{2}-M^{2}\,d\Omega^{2}\,\qquadF_{RB}=\pm{\frac{1}{M}}d\rho\wedgedt\.
\int_{E_{\alpha}}R^{\mu\nu\lambda\rho}R_{\mu\nu\lambda\rho}=\alpha\int_{E}\bar{R}^{\mu\nu\lambda\rho}\bar{R}_{\mu\nu\lambda\rho}+8\pi(1-\alpha)\int_{\Sigma}\bar{R}_{abab}+O((1-\alpha)^{2})~~~,
\langleT_{\nu}^{\mu}\left[g_{\alpha\beta}\right]\rangle_{\mathrm{ren}}^{(b)}=-\left(\frac{k_{D}z}{z_{2}-z_{1}}\right)^{D+1}\frac{J_{D}(B_{1},B_{2})}{2^{D}\pi^{D/2}\Gamma(D/2+1)}\mathrm{diag}(1,1,\ldots,1,-D),
\widehat{W}={\widehatM}_{\infty}=\left(\begin{array}{cccccc}{-1}&{0}&{0}&{2}&{0}&{0}\\{0}&{1}&{0}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{-1}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{0}&{0}&{1}\\\end{array}\right)
\begin{array}{rcl}{\delta\Psi^{\mu}}&{\propto}&{\left(1+\frac{1}{r^{3}}\right)^{-3/2}\cdot\frac{1}{r^{3}}}\\{\delta\Psi^{m}}&{\propto}&{\left(1+\frac{1}{r^{3}}\right)^{-7/6}\cdot\frac{1}{r^{3}}}\\\end{array}
ds_{D}^{2}=H^{-\frac{d-2}{D-2}}\left[-fdt^{2}+\sum_{i=1}^{p}(dx^{i})^{2}+HR_{T}^{2}\left(f^{-1}AdR^{2}+\frac{A}{K^{d-2}}dv^{2}+KR^{2}d\Omega_{d-2}^{2}\right)\right]
\tilde{\varphi}^{\overline{{\alpha}}}=\bar{\varphi}^{\overline{{\alpha}}}+qV^{a}\overline{{\xi}}_{a}^{\overline{{\alpha}}}+\frac{q^{2}}{2}V^{a}V^{b}\xi_{b}^{\overline{{\beta}}}\partial_{\overline{{\beta}}}\xi_{a}^{\overline{{\alpha}}}+O(q^{3}).
\left(\begin{array}{c}{{J^{\prime}}_{\mu}^{1}}\\{{J^{\prime}}_{\mu}^{2}}\\\end{array}\right)=\left(\begin{array}{cc}{a}&{c}\\{b}&{d}\\\end{array}\right)\left(\begin{array}{c}{J_{\mu}^{1}}\\{J_{\mu}^{2}}\\\end{array}\right)
V\overrightarrow{K}V^{-1}=-ich(lm)P_{0}\overrightarrow{\bigtriangledown}-ish(lm)(m\overrightarrow{\bigtriangledown}+m^{-1}\overrightarrow{P}(\overrightarrow{P}.\overrightarrow{\bigtriangledown}))-e^{-lm}\frac{\overrightarrow{P}\times\overrightarrow{S}}{P_{0}+m}
\psi\,=-2i\pi\,g^{0}\,\tau\,\intdp_{0}^{\prime}dp_{0}\delta(p_{0}^{\prime}\!-\!s_{1})\beta_{1}\beta_{2}K^{T}(p_{0}^{\prime},p_{0})\delta(p_{0}\!-\!s_{1})\,\psi.
C_{Q}=\left({\frac{\partialM}{\partialT_{H}}}\right)_{Q}=2\pir_{+}^{2}\left({\frac{3r_{+}^{4}+b^{2}r_{+}^{2}-b^{2}Q^{2}}{3r_{+}^{4}-b^{2}r_{+}^{2}+3b^{2}Q^{2}}}\right).
\hat{U}(k_{1},k_{2})=C(k_{1},k_{2})\mathrm{exp}\left\{\frac{2i\pi}{\hbar}\sqrt{\frac{\hbarN_{k}}{\tau_{2}}}\left[(\bar{\tau}k_{1}-k_{2})\alpha+(\tauk_{1}-k_{2})\alpha^{\dagger}\right]\right\}\\\.
\Phi_{E}^{(p)}={\frac{1}{\sqrt{n_{1}!n_{2}!...n_{D}!}}}(z^{1})^{n_{1}}(z^{2})^{n_{2}}...(z^{D})^{n_{D}}(dz^{1})^{p_{1}}(dz^{2})^{p_{2}}...(dz^{D})^{p_{D}};
A(z,z^{\prime})=\frac{W_{+}(z)W_{-}(z^{\prime})}{e^{-i\hbar\gamma^{2}h^{+}(z-z^{\prime})}}\delta^{+}(z-z^{\prime})+\frac{W_{-}(z^{\prime})W_{+}(z)}{e^{i\hbar\gamma^{2}h^{-}(z-z^{\prime})}}\delta^{-}(z-z^{\prime}).
\frac{1}{2M}\left(-i\hbar\frac{\partial}{\partialx_{j}}-\hbarA_{j}^{a}\hat{T}_{a}\right)^{2}\psi+\frac{\hbar^{2}}{2Mr^{2}}{\hat{T}}^{2}\psi-\frac{e^{2}}{r}\psi=\epsilon\psi
{\calL}^{(4)}=\frac12f_{\pi}^{2}\mathrm{tr}\left(K_{1}\partial_{\mu}U\partial_{\rho}U^{-1}\partial_{\mu}U\partial_{\rho}U^{-1}+K_{2}\partial_{\mu}U\partial_{\mu}U^{-1}\partial_{\rho}U\partial_{\rho}U^{-1}+h.c.\right).
\begin{array}{ccc}{}&{}&{U(1)_{R}}\\{W}&{}&{1}\\{\tau}&{}&{0}\\{Q}&{}&{(N_{f}-N_{c})/N_{f}}\\{\tilde{Q}}&{}&{(N_{f}-N_{c})/N_{f}}\\{m}&{}&{2N_{c}/N_{f}}\\{K}&{}&{\mathrm{arbitrary}}\\\end{array}
\mathcal{S}_{\mathrm{eff}}=\frac12\intd^{3}x\varepsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}A_{\rho}+\frac\tau{4!}\intd^{3}x\widetilde{F}^{\mu}\widetilde{F}_{\mu}\widetilde{F}^{\nu}\widetilde{F}_{\nu}\;,
\delta^{(1)}(A_{j})_{\overline{{m_{j}^{\prime}-1}},{\bfm}_{\perp},n}=ig{\frac{1}{2}}[\delta\omega_{m_{j},{\bfm}_{\perp},n}+\delta\omega_{m_{j}-1,{\bfm}_{\perp},n},(A_{j})_{\overline{{m_{j}^{\prime}-1}},{\bfm}_{\perp},n}],
<A_{2}|A_{1}>=\frac{\sqrt{k}}{2\pi}\,e^{-\frac{ik}{2\pi}A_{1}A_{2}}\\\,\\\<A_{1}|A_{2}>=\frac{\sqrt{k}}{2\pi}\,e^{+\frac{ik}{2\pi}A_{1}A_{2}}\\\.
\intdx^{4}\phi_{[\omega,l,m_{1},m_{2}]}^{\ast}(x)\phi_{[\omega^{\prime},l^{\prime},m_{1}^{\prime},m_{2}^{\prime}]}(x)=(2\pi)\delta(\omega-\omega^{\prime})\delta_{ll^{\prime}}\delta_{m_{1}m_{1}^{\prime}}\delta_{m_{2}m_{2}^{\prime}}
\begin{array}{c}{\dot{t}_{0}+[T_{0},t_{0}]+[T_{1},t_{1}]+[T_{2},t_{2}]+[T_{3},t_{3}]=0,}\\{\dot{t}_{1}+[T_{0},t_{1}]-[T_{1},t_{0}]+[T_{2},t_{3}]-[T_{3},t_{2}]=0,}\\{\dot{t}_{2}+[T_{0},t_{2}]-[T_{1},t_{3}]-[T_{2},t_{0}]+[T_{3},t_{1}]=0,}\\{\dot{t}_{3}+[T_{0},t_{3}]+[T_{1},t_{2}]-[T_{2},t_{1}]-[T_{3},t_{0}]=0.}\\\end{array}
\tilde{t}^{2}-\tilde{w}^{2}-\tilde{x}^{2}-\tilde{y}^{2}-\tilde{z}^{2}=\left(\frac{\tilde{t}-\tilde{w}}{\lambda}\right)k_{1}\left[h^{-1}\left(\frac{\tilde{t}-\tilde{w}}{\lambda}\right)\right].
2(D_{i\alpha}W_{~\gamma}^{\beta}-\delta_{\alpha}^{~\beta}D_{i\gamma}W_{~\delta}^{\delta})=D_{i\delta}W_{~\{\alpha}^{\delta}\delta_{~\gamma\}}^{\beta}-D_{i\{\alpha}W_{~\gamma\}}^{\beta}-3D_{i\{\alpha}\delta_{~\gamma\}}^{\beta}W_{~\delta}^{\delta}
{\calL}_{1}\sim{\frac{v^{4}}{r^{7}}}+{\frac{v^{3}\psi^{2}}{r^{8}}}+{\frac{v^{2}\psi^{4}}{r^{9}}}+{\frac{v\psi^{6}}{r^{10}}}+{\frac{\psi^{8}}{r^{11}}}
J^{0}=-\frac{1}{2}i\left[\psi^{*}(\partial_{\tau}\psi-i\varphi\psi)-(\partial_{\tau}\psi^{*}+i\varphi\psi^{*})\psi\right]\,\\vec{J}=\frac{1}{2}i\left[\psi^{*}(\vec{\partial}\psi+i\vec{a}\psi)-(\vec{\partial}\psi^{*}-i\vec{a}\psi^{*})\psi\right],
v_{+23}=\frac{u_{3}^{2}}{u_{1}}\,\qquadv_{+24}=\frac{u_{3}u_{4}}{u_{1}}+u_{2}\,\qquadv_{-23}=-\frac{u_{3}u_{4}}{u_{1}}+u_{2}\,\qquadv_{-24}=-\frac{u_{4}^{2}}{u_{1}}\.