harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\gamma^{\mu}=\left(\begin{array}{cc}{0}&{-i\sigma^{\mu\,\alpha\beta^{\prime}}}\\{i\bar{\sigma}_{\alpha^{\prime}\beta}^{\mu}}&{0}\\\end{array}\right)\,\qquad\gamma^{5}=\gamma^{1}\gamma^{2}\gamma^{3}\gamma^{4}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)\.
\lambda_{x}=\frac{\mathrm{cn}(\psi)\pm\mathrm{i}k^{\prime}\mathrm{sn}(\psi)}{1+k\mathrm{sn}(\psi)}\;\;,\;\;\lambda_{y}=\frac{\mathrm{cn}(\psi)\pm\mathrm{i}k^{\prime}\mathrm{sn}(\psi)}{1-k\mathrm{sn}(\psi)}
S_{0}=\intd^{2}xe(g^{\alpha\beta}\partial_{\alpha}\sigma\partial_{\beta}-i\bar{\psi}\gamma^{\beta}\gamma^{\alpha}\partial_{\alpha}\psi+2\bar{\chi}_{\alpha}\gamma^{\beta}\gamma^{\alpha}\psi\partial_{\beta}\sigma+\frac{1}{2}\bar{\psi}\psi\bar{\chi}_{\alpha}\gamma^{\beta}\gamma^{\alpha}\chi_{\beta})
{\calL}_{\mathrm{red}}[J_{\mu}^{\mathrm{cl}}]={\frac{1}{2}}\left(\dot{V}_{i}^{R}R_{ij}^{-1}\dot{V}_{j}^{R}+V_{i}^{R}(\vec{\partial}^{2}-M^{2})R_{ij}^{-1}V_{j}^{R}\right)+J_{i}^{\mathrm{cl}}V_{i}^{R}+{\frac{1}{2}}J_{0}^{\mathrm{cl}}{\frac{1}{\vec{\partial}^{2}-M^{2}}}J_{0}^{\mathrm{cl}}~.
\begin{array}{rcl}{\epsilon(k_{\pm})}&{=}&{1~,~~~~\epsilon(X_{\pm}^{-})=0=\epsilonX_{\pm}^{-}~,}\\{S(k_{\pm})}&{=}&{k_{\pm}^{-1}~,~~~~S(X_{\pm}^{-})=-k_{\pm}X_{\pm}^{-}~,~~~~S(X_{\pm}^{+})=-k_{\pm}X_{\pm}^{+}~.}\\\end{array}
\begin{array}{lcr}{{\nu}={\frac{P_{2}}{P_{2}P_{1}-1}}}\\{Q=-e{\frac{P_{2}\ell_{1}+\ell_{2}}{P_{2}P_{1}-1}}}\\{{\theta}={\frac{\pi}{{P_{2}P_{1}-1}}}[P_{2}{\ell_{1}}^{2}+\ell_{1}{\ell_{2}}^{2}+2\ell_{1}\ell_{2}]}\\\end{array}
{\Pi}^{\underline{{m}}}=dX^{\underline{{m}}}-id{\Theta}^{1\underline{{\mu}}}{\sigma}_{\underline{{\mu}}\underline{{\nu}}}^{\underline{{m}}}{\Theta}^{1\underline{{\nu}}}-id{\Theta}_{\underline{{\mu}}}^{2}\tilde{\sigma}^{\underline{{m}}\underline{{\mu}}\underline{{\nu}}}{\Theta}_{\underline{{\nu}}}^{2}
L_{0}=(2n+1){\frac{|h|}{2}}+|h|d_{0}^{\dagger}d_{0}-{\frac{|h|}{2}}-|h|\sum_{k=1}^{\infty}\left(d_{k}^{\dagger}d_{k}-\tilde{d}_{k}^{\dagger}\tilde{d}_{k}+a_{k}^{\dagger}a_{k}-b_{k}^{\dagger}b_{k}\right)+L_{0}^{free}.
({\calM}_{F}^{(n)})^{2}=\left(\begin{array}{ccc}{(n/R)^{2}+2v^{2}}&{iv\frac{\alpha}{R}}&{-iv\frac{\alpha}{R}}\\{-iv\frac{\alpha}{R}}&{((n+\alpha)/R)^{2}+v^{2}}&{v^{2}}\\{iv\frac{\alpha}{R}}&{v^{2}}&{((n-\alpha)/R)^{2}+v^{2}}\\\end{array}\right).
\begin{array}{c}{[P_{A},K_{B}]=2i(M_{AB}+\eta_{AB}D)}\\{{}}\\{\displaystyle{[P_{A},S^{i}]=\tilde{\gamma}_{A}Q^{i}}~~~~~~~~~[K_{A},Q^{i}]=\gamma_{A}S^{i}}\\{{}}\\{\{Q_{\alpha}^{i},S^{j\beta}\}=i{\calE}^{ij}(2\delta_{\alpha}^{~\beta}D+(\gamma^{[A}\tilde{\gamma}^{B]})_{\alpha}^{~\beta}M_{AB})-4i\delta_{\alpha}^{~\beta}A^{ij}}\\\end{array}
{\Phi}=\left(\begin{array}{c}{l_{I}\cdot{\bfe}_{\mu}^{(R)}}\\{{\bar{k}}_{i}\cdot{\bfe}_{\mu}^{(R)}}\\{k^{j}\cdot{\bfe}_{\mu}^{(R)}}\\\end{array}\right)={\frac{1}{2}}\left(\begin{array}{c}{-AE^{*}}\\{DE^{*}}\\{E^{*}}\\\end{array}\right)\;\inM(28,\;6,\;{\bfR})
S^{m}(y)={\bar{w}}\intd^{4}x\sqrt{-\bar{g}}{\bar{g}}^{nm}D_{n}H^{\dagger}D_{m}H-{\bar{w}}^{2}\intd^{4}x\sqrt{-\bar{g}}\lambda(|H|^{2}-\upsilon_{0}^{2})^{2}.
S_{F}=\intd^{6}x\left[N_{\varepsilon_{1}}(r)\bar{\psi}_{1}\gamma^{\mu}(\partial_{\mu}+\frac{i}{a}W_{\mu}Q_{1})\psi_{1}+N_{\varepsilon_{2}}(r)\bar{\psi}_{2}\gamma^{\mu}(\partial_{\mu}+\frac{i}{a}W_{2})\psi_{2}\right]~,
30I_{A}^{\prime}[\sigma;g_{\mu\nu}]=\intd^{4}x[8\sigma{\calE}_{4}+\sigmaG^{\mu\nu}D_{\mu}D_{\nu}\sigma+\textstyle{\frac{1}{2}}\Box\sigma(\partial_{\mu}\sigma)^{2}+(\partial_{\mu}\sigma)^{2}(\partial_{\nu}\sigma)^{2}]\;,
\overline{{F}}_{\sigma}^{(n)}(\zeta)^{kk_{1}\cdotsk_{2n-1}k-2l}=\prod_{a\inA^{\prime}}\oint_{C_{a}}\frac{dw_{a}}{2\pi\sqrt{-1}w_{a}}\Psi_{\sigma}^{(n)}(w|\zeta)Q^{(n)}(w|\zeta)^{kk_{1}\cdotsk_{2n-1}k-2l},
\Pi_{\mathrm{gg}}^{\mu\nu}(p,p_{3})=ig^{2}\left(\left[g^{\mu\nu}\frac{p^{\mu}p^{\nu}}{p^{2}}\right]\Pi_{\mathrm{gg}}^{(1)}(p,p_{3})+g^{\mu\nu}\Pi_{\mathrm{gg}}^{(2)}(p,p_{3})\right)\quad(\mu,\nu=0,1,2)\.
(B_{\{k})_{i_{1},...,i_{n-2p}\}}^{m}\equiv(B_{k})_{i_{1},...,i_{n-2p}}^{m}+\sum_{l=1}^{n-2p}(B_{i_{l}})_{i_{1},...,i_{l-1},k,i_{l+1}...,i_{n-2p}}^{m}~~~,
\left<\Psi\right|{\frac{T_{ij}(\vec{y},\tau)}{N}}\left|\Psi\right>={\frac{1}{N}}\left<\Omega\right|e^{-{\frac{i\xi_{mn}}{N}}\int{f(t)T_{mn}(x)}}{T_{ij}(y)}e^{{\frac{i\xi_{mn}}{N}}\int{f(t)T_{mn}(x)}}\left|\Omega\right>.
\tilde{\gamma}_{~\mu\nu}^{\sigma}=\gamma_{~\mu\nu}^{\sigma}+F_{~~~~\mu\nu~\alpha\beta}^{-1\sigma~~\lambda}\left(B_{\lambda~~~\rho\epsilon}^{~\alpha\beta\tau}\nabla_{\tau}-H_{\lambda~~\rho\epsilon}^{~\alpha\beta}\right)h^{\rho\epsilon}
T_{\mu\nu}=\partial_{\mu}\Phi\partial_{\nu}\Phi-g_{\mu\nu}\left(\frac{1}{2}g^{\alpha\beta}\partial_{\alpha}\Phi\partial_{\beta}\Phi+\frac{\mu^{2}}{2}\Phi^{2}+\frac{\lambda}{4!}\Phi^{4}\right)\;\;.
S_{G}=\frac{1}{2(\kappa_{3})^{2}}\sum_{n=1}^{N}\intd^{3}x\frac{1}{2}\epsilon^{ijk}\left(e_{i}^{a}(n)R_{jk}^{a}(n)+\bar{e}_{i}^{a}(n)\bar{R}_{jk}^{a}(n)\right).
R(\lambda)_{0}c_{-{\frac{1}{2}}}\hat{\rho}\equiv2^{-3}\left(3^{2}2^{\lambda^{2}}U(|\lambda\rangle_{+})_{0}-2^{9\lambda^{2}}U(|3\lambda\rangle_{+})_{0}\right)c_{-{\frac{1}{2}}}\hat{\rho}=c_{-{\frac{1}{2}}}\hat{\rho}\,,
\int_{\epsilon}^{\infty}dr~r^{2}~\left({\frac{r_{A}}{r}}\right)^{2(\alpha_{-}-1)}~\left({\frac{r_{B}}{r}}\right)^{3\alpha_{-}}\sim\int_{\epsilon}^{\infty}dr~r^{-1+{\frac{5}{\sqrt{3}}}}~.
A_{0}^{\prime}(y)(4\eta(y,t)+\sigma(y,t))+\frac{\kappa^{2}}{3}\left(\xi_{0}^{\prime\prime}(y)\delta\xi(y,t)-\xi_{0}^{\prime}(y)\delta\xi^{\prime}(y,t)+(\xi_{0}^{\prime}(y))^{2}\frac{\deltab(y,t)}{b_{0}}\right)=0+O(m_{B}^{4}/k^{4}).
H(\lambda)={\frac{1}{2}}\int_{\lambda}^{\infty}d\eta\,\left({\frac{2\,\pi_{0}(\lambda,\eta)^{2}}{3(\eta-\lambda)}}+{\frac{3}{2}}r\,(\eta-\lambda)(\partial_{\eta}\phi_{0}(\lambda,\eta))^{2}\right).
\langle{\calO}_{e^{\alpha^{1}}}{\calO}_{e^{\alpha^{2}}}\cdots{\calO}_{e^{\alpha^{l}}}\rangle_{M_{N}^{k}}=k\cdot\oint_{C_{\infty}}(\frac{f_{\alpha_{1}}^{N,k}(X)\cdotf_{\alpha_{2}}^{N,k}(X)\cdotsf_{\alpha_{l}}^{N,k}(X)}{f_{rel}^{N,k}(X)})dX
\left[z^{\prime}(1-z^{\prime}){\partial_{z^{\prime}}}^{2}+\left(\frac{D+2}{2}-(D+1)z^{\prime}\right)\partial_{z^{\prime}}-\frac{D^{2}}{4}+s^{2}a^{2}\right]h_{HD}(z^{\prime})=0\,.
{\widehatRes}({\mathcalL}_{3}\star{\tilde{\mathcalL}_{-3}})=3v_{0}^{\prime}+u_{-2}^{\prime}v_{-2}+u_{-2}v_{-2}^{\prime}+4\theta^{2}v_{-2}^{\prime\prime\prime}+6\thetav_{-1}^{\prime\prime}
d\hat{s}(\xi)\oint{d}\hat{s}(\xi^{\prime\prime}){\hat{K}_{D}}(\xi,\xi^{\prime\prime}){\delta_{\rho}}{\hat{G}_{K}}(\xi^{\prime\prime},\xi^{\prime})=-{\frac{\rho(\xi)d\hat{s}(\xi)}{2\oint{d}\hat{s}(\eta)}}+{\frac{d\hat{s}(\xi)\oint{d}\hat{s}(\zeta)\rho(\zeta)}{2{{\left[\oint{d}\hat{s}(\eta)\right]}^{2}}}}.
q^{(j+m)(m-n)}\left[\begin{array}{l}{j-m}\\{n-m}\\\end{array}\right]_{q^{2}}^{1/2}\left[\begin{array}{l}{j+n}\\{n-m}\\\end{array}\right]_{q^{2}}^{1/2}a^{-m-n}(c^{*})^{n-m}P_{j+n}^{n-m,-m-n}(c^{*}c;q^{2}),
A={\frac{1}{M_{P}^{2}M_{s}^{2}}}\sum_{n=0}^{\infty}c_{n}\int_{0}^{\infty}dl\sum_{n_{1},\dotsn_{6}}e^{-\pil/2[-s/M_{s}^{2}+(n_{1}^{2}+\dots+n_{6}^{2})(RM_{s})^{2}+4n]}\,
\begin{array}{rcl}{dA+A^{2}+\hat{\psi}^{2}-\psi^{2}+\hat{A}}&{=}&{0,}\\{d\hat{\psi}+[A,\hat{\psi}]-\{\psi,\hat{A}\}}&{=}&{0,}\\{d\psi+\{A,\psi\}+[\hat{A},\hat{\psi}]}&{=}&{0,}\\{d\hat{A}+[A,\hat{A}]+\{\hat{\psi},\psi\}}&{=}&{0.}\\\end{array}
S=\frac{1}{4\pi}\intd^{3}y\epsilon^{\mu\nu\rho}\left(K_{ab}\partial_{\mu}A_{\nu}^{a}A_{\rho}^{b}+C_{abc}A_{\mu}^{a}\{A_{\nu}^{b},A_{\rho}^{c}\}\right)+{\calO}(2),
K_{3/2}(\pin_{1}|\vec{N}|(1+T))=\frac{1+\pin_{1}|\vec{N}|(1+T)}{\pin_{1}|\vec{N}|(1+T)}\,(2\pin_{1}|\vec{N}|(1+T)))^{\frac{-1}{2}}\,e^{-\pin_{1}|\vec{N}|(1+T)}.\,[5pt]
\begin{array}{ll}{i)}&{p_{s}^{F}p_{t}^{H}=-p_{t}^{F}p_{s}^{H}}\\{ii)}&{\mathrm{all~centers~line-up,~i.e.~}\x_{s}=x_{t}\,,\,y_{s}=y_{t}\;\forall\s,t\.}\\\end{array}
Z_{\mu\nu}\equiv-(k^{2}+i\epsilon)\biggl[g_{\mu\rho}+\biggl({\frac{1}{\lambda}}(1-\kappa)^{2}-1\biggr){\frac{k_{\mu}k_{\rho}}{k^{2}+i\epsilon}}-i{\frac{\kappa(1-\kappa)}{\lambda}}{\frac{k_{[\mu}\eta_{\rho]}}{k^{2}+i\epsilon}}+{\frac{\kappa^{2}}{\lambda}}{\frac{\eta_{\mu}\eta_{\rho}}{k^{2}+i\epsilon}}\biggr],
\langleM\rangle={\frac{\langle\mathrm{Tr}(e^{h}U+e^{-h}V)^{k}\mathrm{Tr}(U-V)\rangle_{n}}{\langle\mathrm{Tr}(e^{h}U+e^{-h}V)^{k}\mathrm{Tr}(U+V)\rangle_{n}}}\.
{\calA}={\frac{V}{2\pi}}\,(8\pi^{2}\alpha^{\prime})^{-1/2}\int_{0}^{\infty}dt\left({\frac{\pi}{t}}\right)^{\frac{9}{2}}\left[{\frac{f_{3}^{8}(q)}{f_{1}^{8}(q)}}-{\frac{f_{4}^{8}(q)}{f_{1}^{8}(q)}}\right]
\begin{aligned}{q^{\alpha}\mathcal{O}}&{{}=\chi^{\alpha},}\\{q^{\alpha}\chi^{\beta}}&{{}=(\sigma_{i}c^{-1})^{\alpha\beta}\partial_{i}\mathcal{O}+(c^{-1})^{\alpha\beta}\mathcal{F},}\\{q^{\alpha}\mathcal{F}}&{{}=-(\sigma_{i}\partial_{i}\chi)^{\alpha}.}\\\end{aligned}
F_{\mu\nu}(x^{\mu})=\int{\frac{d^{3}{\bfp}}{(2\pi)^{3}2E_{p}}}\sum_{\sigma}\left[(F_{\mu\nu})_{\sigma}^{(+)}({\bfp})a_{\sigma}({\bfp})e^{-ip\cdotx}+(F_{\mu\nu})_{\sigma}^{(-)}({\bfp})b_{\sigma}^{\dagger}({\bfp})e^{+ip\cdotx}\right]\,,
U=\frac{1-\sigma^{3}}{2}\rho^{3}+\frac{1+\sigma^{3}}{2}\rho^{2},\quad\gamma_{\mathrm{Weyl}}^{\mu}=U^{\dagger}\gamma^{\mu}U,\quadB_{\mathrm{Weyl}}=U^{\dagger}BU^{*}=\sigma^{2}\rho^{2}.
\begin{array}{ll}{g_{i}^{2}=(q-1)g_{i}+q}&{\;\;\;i=1,\,2,\,\cdots,\,n-1}\\{g_{i}g_{i+1}g_{i}=g_{i+1}g_{i}g_{i+1}}&{\;\;\;i=1,\,2,\,\cdots,\,n-2}\\{g_{i}g_{j}=g_{j}g_{i}}&{\;\;\;{\mathrm{if~}}|i-j|\geq2}\\\end{array}
\begin{array}{llll}{\varrho=x^{1}+ix^{2},}&{\varrho^{*}=x^{1}-ix^{2},}&{\partial=\frac{\partial}{\partial\varrho},}&{\partial^{*}=\frac{\partial}{\partial\varrho^{*}}}\\{x^{\pm}=x^{0}\pmx^{3}}&{\partial_{\pm}=\frac{\partial}{\partialz^{\pm}}}&{}&{}\\\end{array}
f_{k}^{(D)}(\Omega^{({D})})=-\frac{1}{4\pi}\,\frac{2M}{\hbar^{2}}\,\left(\frac{k}{2\pi}\right)^{({D}-3)/2}\left.\intd^{{D}}r^{\prime\prime}\intd^{{D}}r^{\prime}e^{i\left({\bfk}^{\prime}\cdot{\bfr}^{\prime}-{\bfk}^{\prime\prime}\cdot{\bfr}^{\prime\prime}\right)}T_{D}({\bfr^{\prime\prime}},{\bfr^{\prime}};E_{k})\right|_{|{\bfk}^{\prime}|=|{\bfk}^{\prime\prime}|\equiv|{\bfk}|}\;,
{\hat{v}}(\eta,{\bfx})\,=\,\frac{1}{(2\pi)^{3/2}}\int\mathrm{d}^{3}{\bfk}[{\hat{c}}_{\bfk}(\eta)e^{i{\bfk}\cdot{\bfx}}+{\hat{c}}_{\bfk}^{\dagger}(\eta)e^{-i{\bfk}\cdot{\bfx}}]\,,
{\frac{1}{2m_{p+1}V_{p}}}\,\int_{\Sigma_{p}}{\frac{d^{p}s}{\sqrth}}\,\left(\,{\frac{\deltaS^{eff}}{\deltay_{\mu}(\,s\,)}}-N^{m}(\,s\,)\,\partial_{m}\,\phi_{\mu}\,\right)\,\left(\,{\frac{\deltaS^{eff}}{\deltay^{\mu}(\,s\,)}}-N^{j}(\,s\,)\,\partial_{j}\,\phi^{\mu}\,\right)-\frac{m_{p+1}}{2}\,p={\frac{\partialS^{eff}}{\partial\Omega_{p+1}}}\.
\Theta\left[^a_{b}\right]\left(z|\Omega\right)=\sum_{k\in{\bfZ}^{h}}e^{\pii(k+a)\cdot\Omega\cdot(k+a)+2\pii(k+a)\cdot(z+b)},\qquad\Theta\left(z|\Omega\right)\equiv\Theta\left[^0_{0}\right]\left(z|\Omega\right),
S_{\mathrm{grav}}=\frac{1}{2}\intd^{4}x[{\bfg}^{\mu\nu}R_{\mu\nu}+\mathrm{compl.conj.}]=\intd^{4}x[{\bfs}^{\mu\nu}A_{\mu\nu}+{\bfa}^{\mu\nu}B_{\mu\nu}],
\partial_{\alpha_{j}}r(\alpha_{j})\rightarrow\partial_{\alpha_{j}}r(\alpha_{j})-\partial_{\alpha_{2}}r(\alpha_{2})\rightarrow\frac{\partial_{\alpha_{j}}r(\alpha_{j})-\partial_{\alpha_{2}}r(\alpha_{2})}{\alpha_{j}-\alpha_{2}}\xrightarrow{\alpha_{2}\rightarrow\alpha_{j}}\partial_{\alpha_{j}}^{2}r(\alpha_{j})
k=-\frac{1}{24\pi^{2}}\oint_{S_{x\to0}^{3}(\mathrm{space})}d{\mit\Omega}_{\mu}\epsilon_{\mu\nu\rho\sigma}\mathrm{tr}\,\left\{\left(U^{-1}\partial_{\nu}U\right)\left(U^{-1}\partial_{\rho}U\right)\left(U^{-1}\partial_{\sigma}U\right)\right\}\.
\psi_{0}=\prod_{j<k}{{\left(\frac{2}{\pi}\right)}^{\frac{d-1}{2}}(\lambda_{j}-\lambda_{k})}^{\frac{d-1}{2}}\cdote^{-(\lambda_{j}-\lambda_{k})z_{\mu,jk}^{\ast}z_{\mu,jk}}\cdot\prod_{\alpha=1}^{d-1}{\Theta^{\prime}}_{\alpha,jk}^{\dagger}\mid0\rangle\,
K_{m+a}((m+a)z^{\prime})\\sim\\sqrt{\frac{\pi}{2(m+a)}}\frac{e^{-(m+a)\eta^{\prime}}}{(1+z^{2})^{1/4}}\left\{1+\sum_{j=1}^{\infty}(-1)^{j}\frac{u_{j}([1+z^{2}]^{-1/2})}{(m+a)^{j}}\right\}\,
\left(\begin{array}{cc}{s}&{1}\\{u-\lambda+sS}&{S}\\\end{array}\right)\left(\begin{array}{cc}{x}&{1}\\{u+\beta+xX}&{X}\\\end{array}\right)=\left(\begin{array}{cc}{y}&{1}\\{u+\beta+yY}&{Y}\\\end{array}\right)\left(\begin{array}{cc}{t}&{1}\\{u-\lambda+tT}&{T}\\\end{array}\right)\;\;.
\frac{1}{u}(\lambda_{0}-\mu)=(v^{2}-c^{2})+i\int\frac{d^{d}k}{(2\pi)^{d}}[\frac{1}{k^{2}-\lambda_{0}}-\frac{1}{k^{2}-\mu}-\frac{\lambda_{0}-\mu}{(k^{2}-\mu)^{2}}]
\begin{array}{rcl}{\Phi}&{=}&{\displaystyle\frac{1}{N}\sum_{n=0}^{\infty}\left\langle\mathrm{Tr}\;(uU+vV)^{n}\right\rangle}\\{}&{=}&{\displaystyle\frac{1}{N}\sum_{w(u,v)}w(u,v)\left\langle\mathrm{Tr}\;w(U,V)\right\rangle\.}\\\end{array}
m=\left(\begin{matrix}{x}&{y}&{y}&{z}&{y}&{z}&{z}\\{z}&{x}&{y}&{y}&{z}&{y}&{z}\\{z}&{z}&{x}&{y}&{y}&{z}&{y}\\{y}&{z}&{z}&{x}&{y}&{y}&{z}\\{z}&{y}&{z}&{z}&{x}&{y}&{y}\\{y}&{z}&{y}&{z}&{z}&{x}&{y}\\{y}&{y}&{z}&{y}&{z}&{z}&{x}\\\end{matrix}\right)
k_{1}^{\mu}T_{\rho\sigma\mu\nu}^{abcd}(p_{1},p_{2};k_{1},k_{2})=-ig^{2}e^{\rho}(p_{1})e^{\sigma}(p_{2})[k_{2\nu}S_{\rho\sigma}^{abcd}+G_{\rho\sigma\nu}^{abcd}]
\Psi_{E}(r,\theta)=\sum_{n=-\infty}^{\infty}e^{in\theta}\left(\begin{array}{c}{A_{n}\,J_{n-\kappa}\left(|E|r\right)+B_{n}\,J_{\kappa-n}\left(|E|r\right)}\\{-i\frac{|E|}{E}\left(A_{n}\,J_{n+1-\kappa}\left(|E|r\right)-B_{n}\,J_{\kappa-n-1}\left(|E|r\right)\right)e^{i\theta}}\\\end{array}\right)
{\calW}_{\mu\nu\rho\sigma}={\calW}_{\mu\nu\rho\sigma}^{+}+{\calW}_{\mu\nu\rho\sigma}^{-},{\calW}_{\mu\nu\rho\sigma}^{\mp}:=\frac{1}{2}\left({\calW}_{\mu\nu\rho\sigma}\pm\frac{i}{2}\varepsilon_{\mu\nu}^{\\\\lambda\tau}{\calW}_{\lambda\tau\rho\sigma}\right)
\begin{array}{c}{if\space\space{\psi}({\bfx})={\mu}^{\frac{3}{2}}f({\mu}{\bfx};\{\alpha_{i}\})\space\space\spacethen\space\space{\tilde{\psi}}({\bfx})={\tilde{\mu}}^{\frac{3}{2}}f({\tilde{\mu}}{\bfx};\{\alpha_{i}\}).}\\\end{array}
{\calL}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{\kappa}{2}\epsilon^{\mu\nu\lambda}A_{\mu}\partial_{\nu}A_{\lambda}+(D_{\mu}\Phi)^{*}(D^{\mu}\Phi)-\frac{\lambda}{4}(\Phi^{*}\Phi-v^{2})^{2}
\omega=-{\frac{e^{2}}{2\theta}}\sum_{a,b=1}^{N}\int_{0}^{T}dt\,\int_{0}^{t}d\tau\,{\dot{x}}_{a}^{i}(t)\epsilon_{ij}{\dot{x}}_{b}^{j}(\tau)\,\delta({\bfx}_{a}(t)-{\bfx}_{b}(\tau))\,.
Z(\lambda,\beta_{I})=\sum_{N_{2}}\mathrm{e}^{-\lambdaN_{2}}\sum_{\mathrm{causal}\atopT\in{\calT}_{N_{2}}}\sum_{\{\sigma_{i}=\pm1\}}\mathrm{e}^{\frac{\beta_{I}}{2}\sum_{<ij>}\sigma_{i}\sigma_{j}},
F_{\sigma}^{(n)}(\zeta)^{kk_{1}\cdotsk_{2n-1}k}=c_{n}\prod_{1\leqslantj<k\leqslant2n}\zeta_{j}^{\frac{r-1}{r}}g(z_{j}/z_{k})\times\overline{{F}}_{\sigma}^{(n)}(\zeta)^{kk_{1}\cdotsk_{2n-1}k}.
i\frac{\partial}{\partialt}\phi_{v}(\vec{r^{\prime}},t)=\left[-\frac{{\nabla}_{r^{\prime}}^{2}}{2m}-\frac{m}{2}\left(\frac{\ddot{a}}{a}\right){\vec{r^{\prime}}}^{2}\right]\\phi_{v}(\vec{r^{\prime}},t).
F_{A}\equiv\frac{\partialW}{\partial\Phi^{A}}+\left(\frac{\partialK}{\partial\Phi^{A}}\right)W,\qquad\left(K_{A}^{B}\right)^{-1}\equiv\left(\frac{\partial^{2}K}{\partial\Phi^{A}\partial\Phi_{B}^{\dagger}}\right)^{-1}.
<\frac{1}{2},m_{s}|{\psi}_{-}^{(\frac{1}{2})}(g)>\equivD_{m_{s}-\frac{1}{2}}^{(\frac{1}{2})}(g)=<g,l+\frac{1}{2}|T_{m_{s}-}^{\frac{1}{2}}|g,l>.
V(\phi)=m^{2}\frac{1}{2}\sum_{i=0}^{r}n_{i}\alpha_{i}^{a}\alpha_{i}^{b}\phi^{a}\phi^{b}+m^{2}\beta\frac{1}{6}\sum_{i=0}^{r}n_{i}\alpha_{i}^{a}\alpha_{i}^{b}\alpha_{i}^{c}\phi^{a}\phi^{b}\phi^{c}+...
E[V]_{a=0}\;=\;\lambda\int_{0}^{\infty}\,2\pidrr\,(v^{4}-\rho^{4})\;=\;\frac{\pi^{3}}{12}\,\frac{\eta}{\lambda}\;=\;\frac{\pi^{3}}{12}\,(\mu^{2}-\frac{\xiu}{\lambda})\;.
R_{\;bcd}^{a}={\frac4{(1+4\Phi^{2})^{3}}\;}\left(\begin{array}{c}{\left(3+12\Phi^{2}+16\Phi^{4}\right)\;\left(\delta^{ac}\delta^{bd}-\delta^{ad}\delta^{bc}\right)}\\{+\left(16\Phi^{2}\right)\;\Phi^{b}\left(\Phi^{c}\delta^{ad}-\Phi^{d}\delta^{ac}\right)}\\{+\left(12+16\Phi^{2}\right)\;\Phi^{a}\left(\Phi^{d}\delta^{bc}-\Phi^{c}\delta^{bd}\right)}\\\end{array}\right),
y^{2}=x^{3}+\sum_{k=-4}^{4}\sum_{l=0}^{8-nk}f_{kl}z^{\primel}z^{4+k}p^{8-l+q^{\prime}k}x+\sum_{k=-6}^{6}\sum_{l=0}^{12-nk}g_{kl}z^{\primel}z^{6+k}p^{12-l+q^{\prime}k}
\hat{\calA}=-{\frac{g^{2}}{16\pi^{2}}}\epsilon_{\mu\nu\rho\lambda}[\hat{F}^{\mu\nu}\hat{F}^{\rho\lambda}-g\theta_{\alpha\beta}\partial^{\beta}(\hat{A}^{\alpha}\hat{F}^{\mu\nu}\hat{F}^{\rho\lambda})].
\mathrm{d}s^{2}=\alpha^{\prime}\left[\left({\frac{u}{\calR}}\right)^{2}\mathrm{d}x_{d}^{2}+\left({\frac{\calR}{u}}\right)^{2}\mathrm{d}u^{2}+\mathrm{d}\Omega_{\tilde{d}+1}^{2}\right],\quade^{\phi}=g_{s}{r_{0}}^{{\frac{-(D-2)a}{8}}\left({\frac{\beta+1}{\beta}}\right)}\left({\frac{u}{\calR}}\right)^{{\frac{-(D-2)a}{8}}\left({\frac{\beta+1}{\beta}}\right)}\,.
\left.\begin{array}{lcl}{G_{\;\;z}^{z}}&{=}&{3\,\left(\beta^{2}-{{a^{\prime}}^{2}}\right)e^{-2\,a}}\\{G_{\;\,i}^{i}}&{=}&{\left(\beta^{2}-{{a^{\prime}}^{2}}-2\,a^{\prime\prime}\right)e^{-2\,a}}\\\end{array}\right.
\stackrel{(0)}{\omega}_{1}=W_{ab}\left(\eta^{a}H_{0}^{b}-A^{ai}C_{i}^{b}\right)+M_{ab}^{c}\left(A_{i}^{a}\eta^{b}B_{c}^{0i}-A_{i}^{a}A_{j}^{b}\eta_{c}^{ij}\right).
\frac12\frac{\epsilon^{\alpha\beta\gamma\delta}}{\sqrt{g}}\frac{\partialx^{\mu}}{\partialq^{\alpha}}\frac{\partialx^{\nu}}{\partialq^{\beta}}e_{\gamma}^{c}\,e_{\delta}^{d}\,\hat{\xi}_{cd}^{+}=\frac12\frac{\partialx^{\mu}}{\partialq^{\alpha}}e_{a}^{\alpha}\,\frac{\partialx^{\nu}}{\partialq^{\beta}}e_{b}^{\beta}\,\epsilon^{abcd}\,\hat{\xi}_{cd}^{+}=e^{a\,\mu}\,e^{b\,\nu}\,\hat{\xi}_{ab}^{+}.
K_{\bar{c}}(\theta+i\pi)=K_{\bar{a}}(\theta+i\pi+i{\bar{\theta}_{ac}^{b}})S_{ab}(2\theta+i{\bar{\theta}_{ac}^{b}}-i{\bar{\theta}_{bc}^{a}})K_{\bar{b}}(\theta+i\pi-i{\bar{\theta}_{bc}^{a}}).
{V\equiv\left(\begin{matrix}{V_{-}^{1}}&{V_{+}^{1}}\\{V_{-}^{2}}&{V_{+}^{2}}\\\end{matrix}\right)={\frac{1}{\sqrt{-2i\tau_{2}}}}\left(\begin{matrix}{\bar{\tau}e^{-i\phi}}&{\taue^{i\phi}}\\{e^{-i\phi}}&{e^{i\phi}}\\\end{matrix}\right),}
\frac{\partial^{n}V}{\partial\Phi^{n}}=\sum_{k=1}^{n}2{\binom{n-1}{k-1}}\frac{\partial^{k}W}{\partial\Phi^{k}}\left(\frac{27}{4}\frac{\partial^{n-k+2}W}{\partial\Phi^{n-k+2}}-12\frac{\partial^{n-k}W}{\partial\Phi^{n-k}}\right).
\delta_{\epsilon}\mathrm{{\bf~A}}=\left[\begin{array}{cc}{\epsilon^{k}\bar{\psi}_{k}-\psi^{k}\bar{\epsilon}_{k}}&{D\epsilon_{j}}\\{-D\bar{\epsilon}^{i}}&{\bar{\epsilon}^{i}\psi_{j}-\bar{\psi}^{i}\epsilon_{j}}\\\end{array}\right],
i\,\hat{\Gamma}_{r,n+2}^{\,\alpha,\,\alpha_{0}}(p,-p,p_{1},\ldots,p_{n-1}):=(n+1)(n+2)\,i\,\Gamma_{r,n+2}^{\,\alpha,\,\alpha_{0}}(p,-p,p_{1},\ldots,p_{n-1})
\int_{0}^{\infty}\frac{dk}{\pi}k^{4}\frac{d}{dk}\left(\delta_{m}^{\ell}(k)-\delta_{m}^{(1)\ell}(k)-\delta_{m}^{(2)\ell}(k)\right)+\sum_{j}(\kappa_{j,m}^{\ell})^{4}=0
\Xi_{\nu}^{-2}=\intd^{4}x\;\left(C^{*a}A_{\nu}^{a}+\;\frac12\varepsilon_{\sigma\tau\mu\nu}A^{*a\sigma}B^{*a\tau\mu}-\phi^{*a}\xi_{\nu}^{a}-\xi^{*a\mu}B_{\mu\nu}^{a}\right)\;.
\sum_{l_{1}}\frac{\bar{g}_{1l_{1}}}{2I+1}C_{\sigma_{1}\sigma_{1}^{\prime}}C_{\sigma_{2}\sigma_{2}^{\prime}}\delta_{km_{I}}\left\{\begin{array}{lll}{j_{1}}&{j_{2}}&{I}\\{j_{1}}&{j_{2}}&{l_{1}}\\\end{array}\right\}\left(\begin{array}{lll}{j_{1}}&{j_{2}}&{I}\\{\sigma_{1}^{\prime}}&{\sigma_{2}^{\prime}}&{m}\\\end{array}\right).
\triangle_{1}h(r)=\bar{\gamma}G^{2}[G^{2}h(r)-(G^{2}-1)f(r)]\;\;,\;\;\triangle_{k}\equiv(\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\frac{k^{2}}{r^{2}})\;\;.
S={\frac{1}{4\pi\alpha^{\prime}}}\intd^{2}\xi\left[\sqrt{-h}h^{\alpha\beta}g_{\mu\nu}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}-\epsilon^{\alpha\beta}{\calF}_{ij}\partial_{\alpha}X^{i}\partial_{\beta}X^{j}\right],
\tilde{H}=\int\mathrm{d}x~\left[|\tilde{\Pi}_{\alpha}+\frac{\Theta}{8\pi^{2}}\tilde{\Pi}_{\alpha}^{H}|^{2}+|\partial_{i}\tilde{Z}_{\alpha}|^{2}-(\tilde{Z}_{\alpha}^{*}\partial_{i}\tilde{Z}_{\alpha})(\tilde{Z}_{\beta}\partial_{i}\tilde{Z}_{\beta}^{*})\right].
\displaystyle\frac{1}{\lambda-x}=\displaystyle\frac{1}{\epsilon}=\displaystyle\frac{(-1)^{\delta}e^{iZ_{N}(x+\epsilon)}iZ_{_{N}}^{\prime}(x+\epsilon)}{1+(-1)^{\delta}e^{iZ_{N}(x+\epsilon)}}+...={\calR}_{\delta}(x+\epsilon)+...
\vertt^{n}\partial_{\xi}^{\alpha}\tilde{d}_{-1-j}(x,t,\xi,s;\lambda)\vert\leqCe^{-c(t+s)(\vert\xi\vert+\vert\lambda\vert)}(\vert\xi\vert+\vert\lambda\vert)^{-j-n-\vert\alpha\vert},
\frac{e}{m}B_{i}\longrightarrow-\,\frac{1}{2}\varepsilon_{ikl}\gamma_{klc}u^{c};\;\;\;\frac{e}{m}E_{i}\longrightarrow\gamma_{0ic}u^{c},\;\;\;\gamma=\,{\frac{1}{\sqrt{1-v^{2}}}}\,\longrightarrowu_{w}^{0}.
\omega^{-1}(\eta(t))\tilde{E}_{\mu}[\eta|t]\omega(\eta(t))=-8\pi\delta_{\mu}(t)\int\delta\xids\dot{\eta}^{\nu}(t)\mathrm{\mathrm{}~^*\!~}W_{\nu\sigma}[\xi|s]\dot{\xi}^{\sigma}(s)\delta(\xi(s)-\eta(t)),
\begin{array}{l}{Z=\intDX~e^{-S}~~,}\\{S=\frac{1}{4\pi\alpha^{\prime}}\int_{M}d^{2}z\sqrt{g}\left(g^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}G_{\mu\nu}+i\frac{\epsilon^{ab}}{\sqrt{g}}\partial_{a}X^{\mu}\partial_{b}X^{\nu}B_{\mu\nu}+\alpha^{\prime}R^{(2)}\phi+\alpha^{\prime}T\right)~.}\\\end{array}
\Gamma_{n+1}^{\mu_{0}....\mu_{n}}(k^{0},...,k^{n})=\sum_{P}\sum_{l=0}^{m}\sum_{j_{0},...j_{2l-1}=0}^{n}A_{i_{0}...i_{n}}^{l;~j_{0}...j_{2l-1}}(k^{j_{0}})^{\mu_{i_{0}}}...(k^{j_{2l-1}})^{\mu_{i_{2l-1}}}E^{\mu_{i_{2l}}\mu_{i_{2l+1}}}....E^{\mu_{i_{n-1}}\mu_{i_{n}}}
V_{AB}^{CP}=-\frac{\hbarc}{(4\pi)^{3}r^{7}}[23(\alpha_{A}^{E}\alpha_{B}^{E}+\alpha_{A}^{M}\alpha_{B}^{M})-7(\alpha_{A}^{E}\alpha_{B}^{M}+\alpha_{B}^{E}\alpha_{A}^{M})]
D^{\{m_{1},m_{2},\ldots,m_{n}\}}=\left|\begin{array}{cccc}{z_{1}^{m_{1}}}&{z_{2}^{m_{1}}}&{\ldots}&{z_{n}^{m_{1}}}\\{z_{1}^{1+m_{2}}}&{z_{2}^{1+m_{2}}}&{\ldots}&{z_{n}^{1+m_{2}}}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{z_{1}^{n-1+m_{n}}}&{z_{2}^{n-1+m_{n}}}&{\ldots}&{z_{n}^{n-1+m_{n}}}\\\end{array}\right|\,,
\sum_{\ell=0}^{\infty}C_{\ell}^{(-1/3)}\left\{\epsilon_{m-\ell+{\frac{1\pmq}{3}}}^{\mu\pm}\epsilon_{n+\ell+{\frac{2\mpq}{3}}}^{\nu\mp}+\epsilon_{n-\ell+{\frac{1\mpq}{3}}}^{\nu\mp}\epsilon_{m+\ell+{\frac{2\pmq}{3}}}^{\mu\pm}\right\}=g^{\mu\nu}\delta_{m+n+1}.
\hat{\zeta}_{m_{i},m_{i}^{\prime}}=\sum_{m_{i}^{\prime\prime}=1}^{M}\mathrm{sgn}\,(m_{i}^{\prime\prime}-m_{i})\,\mathrm{sgn}\,(m_{i}^{\prime\prime}-m_{i}^{\prime})\zeta_{m_{i}^{\prime\prime}}.
e\sim\left(\begin{matrix}{0}&{U_{R_{0},R_{1}}}&{0}&{\dots}&{0}\\{0}&{0}&{U_{R_{1},R_{2}}}&{\dots}&{0}\\{\vdots}&{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{U_{R_{k^{\prime}-1},R_{0}}}&{0}&{0}&{\dots}&{0}\\\end{matrix}\right).
\tilde{V}_{\nu}(r)=-\omega^{2}(f-1)+h\left\{{\frac{3}{4r^{2}}}\left(1+{\frac{3r_{0}^{2}}{r^{2}}}\right)+{\frac{8R^{4}}{r^{2}(r^{2}+R^{2})^{2}}}\right\},
F_{(c)k}^{[r,0]}(p_{1}^{2},p_{2}^{2},p_{3}^{2})=\frac{N_{3V(c)k}^{(r)}(p_{1}^{2},p_{2}^{2},p_{3}^{2})}{\left[\prod_{s=1}^{r}\left(p_{1}^{2}+u_{r,2s}^{\prime}\Lambda^{2}\right)\right]\left[\prod_{s=1}^{r}\left(p_{2}^{2}+u_{r,2s}^{\prime}\Lambda^{2}\right)\right]\left[\prod_{s=1}^{r}\left(p_{3}^{2}+u_{r,2s}^{\prime}\Lambda^{2}\right)\right]}
P(\sigma_{1},\sigma_{2},\cdots,\sigma_{N})=\sum_{m_{N}=0}^{n_{p}}\cdots\sum_{m_{2}=0}^{n_{p}}\sum_{m_{1}=0}^{n_{p}}\left(a_{m_{1}m_{2}\cdotsm_{N}}^{(1)}\sigma_{N}^{m_{N}}\cdots\sigma_{2}^{m_{2}}\right)\sigma_{1}^{m_{1}}\quad.
f\starg=\frac{1}{(2\pi)^{2}}\sum_{m,n}\int_{-\pi}^{+\pi}d\theta^{\prime}d\theta^{\prime\prime}\int_{-\infty}^{+\infty}dP^{\prime}dP^{\prime\prime}\delta(P_{\theta}-P^{\prime}+m)\delta(P_{\theta}-P^{\prime\prime}-n)e^{[-{2i}A(\tilde{\bfr},\tilde{\bfr}^{\prime},\tilde{\bfr}^{\prime\prime})]}f(\tilde{\bfr}^{\prime})g(\tilde{\bfr^{\prime\prime}}).
\hat{M}^{(8)}:=\frac{1}{2\tilde{m}-1}\Big((\tilde{m}-1)\,M^{(8)}+W^{(8)}\Big)\quad\mathrm{and}\quad\hat{M}^{(5)}:=\frac{1}{2\tilde{n}-1}\Big((\tilde{n}-1)\,M^{(5)}+W^{(5)}\Big)
V={x^{6}}\,{y^{6}}-{x^{4}}\,{y^{4}}\,\left({x^{3}}+{y^{3}}\right)+{x^{2}}\,{y^{2}}\,\left({x^{6}}+{y^{6}}\right)+{\frac{{x^{9}}+{y^{9}}}{3}}
\sqrtg\;L_{GB}=-{\frac{2\,r_{+}\,{{\left(1-{\frac{r_{+}}{r}}\right)}^{-1+{\frac{2}{1+{a^{2}}}}}}}{\left(1+{a^{2}}\right)\,{r^{2}}}}\left(1-\left(\left(1-{\frac{r_{+}}{r}}\right)+{\frac{{a^{2}}\,r_{+}}{\left(1+{a^{2}}\right)\,r}}\right)^{2}\right)\.
{\calH}=-K\cdot\left({\calK}\frac{\delta}{\deltaK}\right)-K\cdot\left(\frac{\delta}{\deltaK}\vee\frac{\delta}{\deltaK}\right)-\frac{1}{N^{2}}K\cdot\left(K\cdot\left(\wedge\frac{\delta}{\deltaK}\right)\right)-K\cdotT.
{\frac{\partialH^{\prime}}{\partialg}}=U{\frac{\partialH}{\partialg}}U^{-1}+\left[{\frac{\partialU}{\partialg}}U^{-1},UHU^{-1}\right]=\widetilde{{\frac{\partialH}{\partialg}}}+\left[{\frac{\partialU}{\partialg}}U^{-1},H^{\prime}\right].
f_{\mathrm{out}}^{\mathrm{TM}}=-{\frac{i}{\pi^{(D+1)/2}2^{D}a^{D+1}\Gamma({\frac{D-1}{2}})}}\int_{0}^{\infty}{\frac{dx}{x}}\sum_{n=0}^{\infty}w(n,D)(x^{2}-n(n+D-2)){\frac{e_{n}(x)}{e_{n}^{\prime}(x)}},