harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tilde{\psi}_{+}=-\partial_{-}^{-1}\frac{\gamma_{-}}{4}\hat{\partial}_{\bot}\tilde{\psi}_{-}\;\;\;\;\bar{\tilde{\psi}}_{+}=-\frac{1}{4}\left(\frac{\partial^{\rho}}{\partial_{-}}\bar{\tilde{\psi}}_{-}\right)\gamma_{\rho}\gamma_{-}
V_{2b}(k,k\,^{\prime};\tau)=\left(\frac{\alpha}{m}\right)^{2b}\,d_{k}^{*}\,d_{k\,^{\prime}}\,\,e^{i\alpha(k^{2}-k\,^{2})\tau}\,\,\int_{-\infty}^{\infty}dx\,K_{ik}(e^{x})\,e^{2bx}\,K_{ik\,^{\prime}}(e^{x}).
u_{11}/u_{12}+\zeta(\nu_{jk})-\zeta(x)-\zeta(\nu_{jk}-x)={\frac{\sigma(\nu_{jk})\sigma(x-\nu_{jm})\sigma(x-\nu_{jk}+\nu_{jm})}{\sigma(\nu_{jm})\sigma(\nu_{jk}-\nu_{jm})\sigma(x-\nu_{jk})\sigma(x)}}
8{\pi}^{2}\left(\frac{\partial{g_{\alpha}^{-2}}}{\partial{\tilde{\Phi}}_{i}}\right)^{(1-loop)}=-i\left(\frac{\partial\Theta_{\alpha}}{\partial{\tilde{\Phi}}_{i}}\right)^{(1-loop)}=\frac{-1}{2}Tr\left(Q_{\alpha}^{2}\frac{\partialM}{\partial{\Phi}_{i}}M^{\dagger}\frac{1}{MM^{\dagger}+O(p^{2})}\right).
\Omega_{\alpha\beta}=\partial_{\beta}\{\frac{1}{\Pi_{0}^{+}\sqrt{W}}[\Pi_{M}\partial_{\alpha}{X}^{M}+\frac{1}{4}V_{\alpha}]\}-\partial_{\alpha}\{\frac{1}{\Pi_{0}^{+}\sqrt{W}}[\Pi_{M}\partial_{\beta}{X}^{M}+\frac{1}{4}V_{\beta}]\}.
Z_{D}(x,x,s)={\frac{1}{(s-1)(s-{\frac{1}{2}})s\Gamma(s)}}\int_{0}^{\infty}dtt^{s}{\frac{\partial}{\partialt}}[t^{\frac{1}{2}}{\frac{\partial}{\partialt}}[t^{\frac{1}{2}}{\frac{\partial}{\partialt}}tK_{D}(x,x,t)]]
P\left(q\right)=\left|a\right|^{2}P_{1}\left(q\right)+\left|b\right|^{2}P_{2}\left(q\right)=\left|a\right|^{2}\left|\Psi_{1}\left(q\right)\right|^{2}+\left|b\right|^{2}\int\left|\Psi_{2}\left(q,y\right)\right|^{2}dy
t=t_{0}+{\frac{1}{5}}\Big({\frac{2^{7}}{m^{5}n^{4}}}\Big)^{1/3}y^{5/2}F\left(\Big[{\frac{5}{6}},{\frac{2}{3}}\Big],\Big[{\frac{11}{6}}\Big],-{\frac{4y^{3}}{m^{2}n^{2}}}\right).
(\;p_{L}\;\vert\;p_{R}\;)=\left(\;{\frac{n-2{\bfq}\cdot{\bfa}}{2r_{11}}}-w({\frac{r_{11}}{\alpha_{h}^{\prime}}}+{\frac{{\bfa}\cdot{\bfa}}{2r_{11}}})\;\Bigg\vert\;{\frac{n-2{\bfq}\cdot{\bfa}}{2r_{11}}}+w({\frac{r_{11}}{\alpha_{h}^{\prime}}}-{\frac{{\bfa}\cdot{\bfa}}{2r_{11}}})\;;\;\sqrt{\frac{2}{\alpha_{h}^{\prime}}}({\bfq}+w{\bfa})\;\right),
\left\{\frac{\left(U_{n}-2+U_{n}^{-1}\right)}{T^{2}}+E^{2}\frac{\left(U_{n}+4+U_{n}^{-1}\right)}6\right\}\tilde{\varphi}_{n}\left(\mathbf{p}\right)={}_{{}_{\!\!\!\!\!\!\!c}}\;\,\tilde{j}_{n}\left(\mathbf{p}\right),
\begin{array}{lll}{}&{}&{0<(\phi-\sqrt{\phi^{2}-1})^{1/4}\leqx_{1}\leq(\phi+\sqrt{\phi^{2}-1})^{1/4}\;}\\{}&{\mathrm{~and~}}&{\phantom{0}-(\phi+\sqrt{\phi^{2}-1})^{1/4}\leqx_{1}\leq-(\phi-\sqrt{\phi^{2}-1})^{1/4}<0\;.}\\\end{array}
\varepsilon_{1}\dot{\calG}_{B12}k_{2}\,\varepsilon_{2}\dot{\calG}_{B23}\varepsilon_{3}\,k_{3}\dot{\calG}_{B31}k_{1}\,-\varepsilon_{1}{\calG}_{F12}k_{2}\,\varepsilon_{2}{\calG}_{F23}\varepsilon_{3}\,k_{3}{\calG}_{F31}k_{1}\quad.
Q=\frac{1}{32\pi^{2}}\mathrm{tr}\intd^{4}x{\varepsilon}^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}=\frac{1}{\pi}\int_{-\infty}^{0}du\int_{-\infty}^{\infty}dtB^{2}=2\mathrm{Tr}B^{2}=\frac{q}{2}
S_{2,BV}=e^{{\calL}_{\Psi_{7}}}e^{{\calL}_{\Psi_{6}}}e^{{\calL}_{\Psi_{5}}}e^{{\calL}_{\Psi_{4}}}e^{{\calL}_{\Psi_{3}}}e^{{\calL}_{\Psi_{2}}}e^{{\calL}_{\Psi_{1}}}S_{1,BV}
M^{-}=\left(\begin{matrix}{u^{2}}&{uvf^{*}}&{0}&{uvg^{*}}\\{uvf}&{v^{2}}&{-uvg^{*}}&{0}\\{0}&{-uvg}&{u^{2}}&{uvf}\\{uvg}&{0}&{uvf^{*}}&{v^{2}}\\\end{matrix}\right)
[\tau(\lambda)Q(\lambda)\psi]({\vec{x}})=Tr\left[\intd{\vec{t}}d{\vec{y}}\left(\prod_{i=N}^{1}L(\lambda|x_{i},\partial_{x_{i}})R_{\lambda+\beta_{i}-1}(t_{i},x_{i}|t_{i-1},y_{i})\right)\psi({\vec{y}})\right]~.
{\calK}={\textstyle{\frac{1}{4}}}\left\{(Q_{o}+Q_{v})(0;0)\left[P_{1}P_{2}+W_{1}W_{2}\right]+16\times2(Q_{s}+Q_{c})(0;0)\left({\frac{\eta}{\vartheta_{4}(0)}}\right)^{2}\right\}\,
\begin{array}{rl}{\tilde{{\calL}}=}&{-\frac{1}{2}\left(f_{xx}^{\;2}+f_{tx}\,f_{xx}\right)+f_{xxx}\,f_{tt}\,f_{xx}+f_{txx}\,f_{tx}\,f_{tt}}\\{}&{-f_{xxx}\,f_{tt}\,f_{tx}-f_{ttt}\,f_{tx}\,f_{xx}-\frac{1}{2}\,f_{ttt}\,f_{tx}^{\;\;2}}\\\end{array}
\begin{array}{lll}{d_{v}}&{=}&{(2H+2N)k}\\{}&{=}&{2mN+2kH+4mk-m(2m-1)-k(2k+1)~~~when~~m=k,}\\{d_{H}}&{=}&{[(2H-2)2k(2k+1)/2-(H-1)2k(2k+1)-k(2k+1)]}\\{}&{+}&{[(2N+2)4k^{2}/2-N(k(2k+1)+k(2k-1))-k(2k-1)]}\\{}&{+}&{[2k/2+2k/2]}\\{}&{=}&{[-k(2k+1)]+[2k^{2}+k]+[2k]=k+m}\\\end{array}
\left(\begin{array}{c}{\Gamma^{(4)}(q,P)}\\{\Gamma^{(5)}(q,P)}\\\end{array}\right)^{APS}=\gamma_{45}\left(\begin{array}{c}{\Gamma^{(4)}(q,P)}\\{\Gamma^{(5)}(q,P)}\\\end{array}\right)^{AS}.
\vec{p}_{+,t\rightarrowt_{0}}=\left(\begin{array}{c}{+.48}\\{-.45}\\{-.08}\\\end{array}\right)\;,\qquad\vec{p}_{+,t\rightarrow\infty}=\left(\begin{array}{c}{+.06}\\{+.81}\\{+.13}\\\end{array}\right)\;.
\Psi(\theta,\phi)=\left(\begin{array}{lr}{\psi_{n,m-1}(\theta,\phi)}\\{\psi_{n,m}(\theta,\phi)}\\{\psi_{n,m-1}(\theta,\phi)}\\{\psi_{n,m}(\theta,\phi)}\\{\,\,\,\,\,\,\,\,\vdots}\\{\psi_{n,m-1}(\theta,\phi)}\\\end{array}\right)_{(2k+1)\times1}.
\begin{array}{cllcll}{{[a_{m}^{\mu},a_{n}^{\nu}]}}&{=\;m\eta^{\mu\nu}\delta_{m,-n}\;\;\;}&{,\;\;\;}&{[c_{m},c_{n}]}&{=\;m\delta_{m,-n}\;\;\;}&{,}\\{{[a_{0}^{\mu},q_{0}^{\nu}]}}&{=\;-i\eta^{\mu\nu}\;\;\;}&{,\;\;\;}&{[c_{0},q_{0}^{L}]}&{=\;-i\;\;\;}&{,}\\{\{b_{r}^{\mu},b_{s}^{\nu}\}}&{=\;\eta^{\mu\nu}\delta_{r,-s}\;\;\;}&{,\;\;\;}&{\{d_{r},d_{s}\}}&{=\;\delta_{r,-s}\;\;\;}&{,}\\\end{array}
\begin{array}{rll}{f_{0}}&{\longleftrightarrow}&{\tilde{f}_{0}^{-1}\;,}\\{f_{i}f_{0}^{-1/2}}&{\longleftrightarrow}&{\tilde{f}_{k-i}\tilde{f}_{0}^{-1/2}\;\;\;;i=1,\ldots,k-1\;,}\\\end{array}
\sum_{n=0\,(1)}^{\infty}\int_{-\infty}^{\infty}dk\int_{0}^{\infty}\lambda\,d\lambda\,f_{\lambdank}^{*}(r^{\prime},\theta^{\prime},z^{\prime})f_{\lambdank}(r,\theta,z)=\frac{1}{r}\,\delta(r-r^{\prime})\,\delta(z-z^{\prime})\,\delta(\theta-\theta^{\prime})\,{.}
\begin{array}{cc}{\mathrm{y}_{\pm}=\left(\begin{array}{cc}{0}&{-(y_{\pm}{\cdot\sigma})^{t}}\\{y_{\mp}{\cdot\sigma}}&{0}\\\end{array}\right)~~~~}&{~~~~y_{\pm}^{\mu}=y^{\mu}\pmi\bar{\phi}{}^{a}\tilde{\sigma}^{\mu}\phi_{a}=x_{\pm}^{\mu}}\\\end{array}
v_{(2)}{}^{(1)}\otimesv_{(1)}(f^{(2)^{\prime}})\,v_{(2)}{}^{(2)^{\prime}}=v^{(1)}{}_{(2)}\otimesv^{(2)^{\prime}}\,\widehat{v^{(1)}{}_{(1)}}(f^{(2)^{\prime}}),
\Omega(B,m)=-\frac{eB}{2\pi}\sum_{n=1}^{\infty}\sqrt{2neB+m^{2}}+\frac{m^{2}eB}{2\pi}\sum_{n=1}^{\infty}\frac{1}{\sqrt{2neB+m^{2}}}-\frac{\sqrt{2}\zeta(\frac{3}{2})}{8\pi^{2}}(eB)^{\frac{3}{2}}.
\frac{1}{a^{2}}\left(\frac{da}{dt}\right)^{2}=\frac{C_{GR}}{a^{3\gamma}}+\frac{C_{\lambda}}{a^{6\gamma}}-\frac{k}{a^{2}}+\frac{\Lambda_{4}}{3}+\frac{C_{\calU}}{a^{4}},
b_{{\bfn}}({\bfx}_{{\bfj}+{\bfe}_{\mu}})-b_{{\bfn}}({\bfx}_{{\bfj}})=\epsilon{\displaystyle\sum_{{\bfl}\in(Z_{N})^{d}}}\,(\nabla_{\!\!\mu})_{{\bfn}}^{{\bfl}}\,b_{{\bfl}}({\bfx}_{{\bfj}})\;.
\mathrm{E}(z)\;=\;\left(\begin{array}{cc}{0}&{+e^{+\frac{z}{2}}}\\{-e^{-\frac{z}{2}}}&{0}\\\end{array}\right),\qquad\mathrm{V}\;=\;\left(\begin{array}{rr}{1}&{1}\\{-1}&{0}\\\end{array}\right).
\frac{F(2,\frac{i}{4}(k+k^{\prime}-2k^{\prime\prime})+1,\frac{i}{4}(k+k^{\prime}-2k^{\prime\prime})+2,e^{-4x_{0}})}{k+k^{\prime}-2k^{\prime\prime}-4i}=\sum_{n=1}^{\infty}\frac{ne^{-4(n-1)x_{0}}}{k+k^{\prime}-2k^{\prime\prime}-4ni}.
\gamma^{1}=-i\sigma^{3},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\gamma^{2}=\sigma^{1},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\gamma^{3}=\sigma^{2},
(\partialX^{j}(z))^{2}\cdot\partial_{n}X^{i}(w)\simG^{ii}\frac{1}{(z-w)^{2}}\partial_{n}X^{i}\sim\frac{t^{-2p}}{(z-w)^{2}}\partial_{n}X^{i},\qquad(\mathrm{no}~~\mathrm{sum}~~\mathrm{over}~~i)
\left[g_{\mu\nu}(\vec{\sigma})\right]=\left(\begin{array}{c|c}{1}&{\displaystyle\frac{\vec{\sigma}^{\mathrm{T}}}{c}\gamma_{0}^{-2}}\\\hline{\displaystyle\frac{\vec{\sigma}}{c}\gamma_{0}^{-2}}&{\displaystyle-I+\frac{\vec{\sigma}\otimes\vec{\sigma}^{\mathrm{T}}}{c^{2}}\gamma_{0}^{-4}}\\\end{array}\right),
{\frac{1}{p^{2}-\alphaM^{2}}}{\frac{1}{(p+k)^{2}-\betaM^{2}}}=\int_{0}^{1}dx{\frac{1}{[p^{2}+2xk\cdotp+\{xk^{2}-(x\beta-x\alpha+\alpha)M^{2}\}]^{2}}},
\delta\tilde{m}^{2}(t,s)_{1}=-\frac{i}{2}\delta(0)\tilde{\lambda}(t)a^{-1}(t)(2\pi)^{-3}\intd^{3}\vec{q}~[\vec{q}^{2}+\tilde{m}^{2}(t)-i\epsilon]^{-1}.
W_{i}^{(n)a}({\bfr};\cdots,{\bfr}_{p},T_{(p)},\cdots,{\bfr}_{q},T_{(q)},\cdots)=W_{i}^{(n)a}({\bfr};\cdots,{\bfr}_{q},T_{(q)},\cdots,{\bfr}_{p},T_{(p)},\cdots)
{\sqrt{g}\mathrm{F}^{\rho\lambda}(\mathrm{D}^{\alpha}{}_{\rho}\partial_{\alpha}\mathrm{A}_{\lambda}+\mathrm{A}_{\alpha}\partial_{\lambda}\mathrm{D}_{\rho}^{\alpha}-\partial_{\rho}(\mathrm{D}^{\alpha}{}_{\lambda}\mathrm{A}_{\alpha}))},
^{c}\omega_{\mu}^{\prime}\,^{a}\,_{b}=U^{a}\,_{d}\left(^{c}\omega_{\mu}\,^{d}\,_{f}\right)\left(U^{-1}\right)^{f}\,_{b}-\left(\partial_{\mu}U\right)^{a}\,_{d}U^{d}\,_{b}.
0\stackrel{0}{\longrightarrow}\Gamma_{\!N}\stackrel{T_{N}}{\longrightarrow}\dots\longrightarrow\Gamma_{\!1}\stackrel{T_{1}}{\longrightarrow}\Gamma\stackrel{T}{\longrightarrow}\Gamma\stackrel{T_{1}^{*}}{\longrightarrow}\Gamma_{\!1}\longrightarrow\dots\stackrel{T_{N}^{*}}{\longrightarrow}\Gamma_{\!N}\stackrel{0}{\longrightarrow}0\,.
R=\frac{4({r_{H}}^{2}+2\zeta(3)\omega^{2}m^{2}\ell^{6})+\pi^{2}\omega^{2}\ell^{4}(1+4m^{2}\ell^{2})-4\pir_{H}\omega\ell^{2}(1+2m^{2}\ell^{2})}{4({r_{H}}^{2}+2\zeta(3)\omega^{2}m^{2}\ell^{6})+\pi^{2}\omega^{2}\ell^{4}(1+4m^{2}\ell^{2})+4\pir_{H}\omega\ell^{2}(1+2m^{2}\ell^{2})}.
\begin{array}{ll}{\Pi_{J_{0}}^{e}\equiv\Omega_{J_{0}}^{e}}&{\Pi_{\gamma_{0}}^{s}\equiv\Omega^{s_{\gamma_{0}}}\oplus\bigoplus_{1\lep<\frac{1}{2}(x-1)}\Omega^{sr^{p}}}\\{\Pi_{J_{1}}^{e}\equiv\Omega_{J_{1}}^{e}}&{\Pi_{\gamma_{1}}^{s}\equiv\Omega^{s_{\gamma_{1}}}\oplus\bigoplus_{1\lep<\frac{1}{2}(x-1)}\Omega^{sr^{p}}}\\{\Pi_{\beta_{l}}^{r^{k}}\equiv\Omega_{\beta_{l}}^{r^{k}}}\\\end{array}
det[\delta_{b}^{a}\partial_{t}-\omega^{ac}\partial_{c}\partial_{b}H]=\int~{\calD}c^{a}{\calD}{\bar{c}}_{a}~exp~-\int{\bar{c}}_{a}[\delta_{b}^{a}\partial_{t}-\omega^{ac}\partial_{c}\partial_{b}H]c^{b}~dt
E^{i}\equiv\Pi^{\underline{{m}}}u_{\underline{{m}}}^{i}=0\qquad\Leftrightarrow\qquad\Pi_{+q}^{\underline{{m}}}\equiv{\calD}_{+q}X^{\underline{{m}}}-i{\calD}_{+q}\Theta\Gamma^{\underline{{m}}}\Theta,
|\tilde{Z}_{1,2}|^{2}={\frac{1}{(4G)^{2}}}\left[\tilde{\calQ}^{2}+\tilde{\calP}^{2}\pm2\left(\tilde{\calQ}^{2}\tilde{\calP}^{2}-(\tilde{\calQ}\tilde{\calP})^{2}\right)^{\frac{1}{2}}\right]\,
y^{2}=x^{3}+x^{2}(\frac{\Lambda_{1}^{4}\Lambda_{3}^{4}}{M_{3}^{2}}+\Lambda_{2}^{4}-M_{1}M_{2}+\frac{T^{2}}{M_{3}})+x\frac{\Lambda_{1}^{4}\Lambda_{2}^{4}\Lambda_{3}^{4}}{M_{3}^{2}}.\nonumber
\Phi^{\ell}(a,x)=\frac{1}{2\pii}\int_{c-i\infty}^{c+i\infty}d\nu\,\left[b_{+}(\nu)t_{+}^{\ell}(0,\nu;a,x)+b_{-}(\nu)t_{-}^{\ell}(0,\nu;a,x)\right]\equive^{-\ella}\varphi(x)\,.
S_{\mathrm{AC}}(\alpha)=\int_{M}d^{4}x\,\Bigr\{(D^{a}\varphi_{ac})(D_{b}\varphi^{bc})^{\dagger}+\alpha(\varphi_{ac}\varphi^{\daggerbc})(\varphi^{ad}\varphi_{bd}^{\dagger})\Bigr\}.
ds^{2}=\rho^{2}\left[\left(e^{t}+e^{-3t}\frac{r^{2}}{1-r^{2}}\right)dr^{2}+e^{t}r^{2}d\Omega_{5}^{2}+e^{-3t}(1-r^{2})d\theta^{2}\right],
\begin{array}{lcr}{\nu={\sum_{ij}}t^{i}{(K^{-1})_{i}}^{j}t_{j}}\\{Q=-e{\sum}_{ij}{t}^{i}(K^{-1})_{i}^{j}{\ell}_{j}}\\{\theta=\pi{\sum}_{ij}{\ell}^{i}(K^{-1})_{i}^{j}{\ell}_{j}}\\\end{array}
\begin{array}{|c|c|c|c|}\hline{}&{e^{\ell\rho}}&{e^{\ell\phi}}&{e^{\ell\chi}}\\\hline{\mathrm{in~the~original~theory}}&{(\frac{1}{2}\ell(\ell-3),0)}&{(-\frac{1}{2}\ell(\ell+2),0)}&{(\frac{1}{2}\ell(\ell-1),0)}\\\hline{\mathrm{in~the~twisted~theory}}&{(\frac{1}{2}\ell(\ell-1),0)}&{(-\ell^{2}/2,0)}&{(\frac{1}{2}\ell(\ell+1),0)}\\\hline{}\\\end{array}
Z[J,K]=e^{\frac{1}{\hbar}W[J,K]}=\int\!D\Phi\,e^{\frac{1}{\hbar}\left[-S[\Phi]+\frac{1}{2}\int\!\!J(x)\Phi^{2}(x)\,d^{n}x+\frac{1}{24}\int\!\!K(x)\Phi^{4}(x)\,d^{n}x\right]}.
G_{n}^{(N)}(L,r,l)=\widetilde{\sum}_{\tilde{m}\in{\bfZ}^{N+1}}q^{mC^{-1}m}(q)_{L-1}\prod_{i=0}^{N}\frac{1}{(q)_{m_{i}}}\left\{\sum_{k=n}^{N}q^{-e_{n-l}C^{-1}m}B^{(N)}(L,r,l|k)\right\}
B=\frac{\pi|\beta|^{2}}{2}\int_{0}^{\infty}dr\left(\frac{r}{(1+|\beta|^{2}r^{2}+|\lambda|^{2}r^{4})^{2}}+\frac{|\lambda|^{2}r^{5}}{(1+|\beta|^{2}r^{2}+|\lambda|^{2}r^{4})^{3}}\right).
\begin{array}{l}{{\calK}={\calA}+VS_{m}^{\dagger}+S_{m}W^{\dagger}+S_{m}\tilde{{\calK}}S_{m}^{\dagger}=\left(\begin{array}{cc}{{\calA}}&{V}\\{W^{\dagger}}&{\tilde{{\calK}}}\\\end{array}\right)\,,}\\{\varphi=XS_{m}^{\dagger}+S_{m}\tilde{h}S_{m}^{\dagger}=\left(\begin{array}{cc}{0}&{X}\\{0}&{\tilde{h}}\\\end{array}\right)\,.}\\\end{array}
\begin{array}{c}{\displaystyle{\sum_{\nu=\nu_{<}+1}^{\nu_{>}}\left(f(\nu)-g(\nu)\right)=\int_{\nu_{<}+1}^{\nu_{>}}\left(f(\nu)-g(\nu)\right)\,d\nu+}}\\{\displaystyle{+\left(f(\nu_{<}+1)-g(\nu_{<}+1)\right)+\int_{\nu_{<}+1}^{\nu_{>}}\left(x-[x]\right)\left(f^{\prime}(x)-g^{\prime}(x)\right)\,dx}.}\\\end{array}
[{\calT}_{\beta}^{\dagger},{\calT}_{\alpha}^{\dagger}]={\calT}_{\gamma}^{\dagger}U_{\alpha\beta}^{\dagger\gamma}+\bar{C}_{\gamma}^{\dagger}[U_{\alpha\beta}^{\dagger\gamma},\Omega](-1)^{\varepsilon_{\alpha}+\varepsilon_{\beta}},
\Gamma_{\mathrm{r}}^{(4)}(q_{m})_{i,j,k,l}=-6\mu^{\varepsilon}\left[F_{1}(\delta_{\mathrm{r}},g_{\mathrm{r}})\left(S_{ijkl}-\delta_{ijkl}\right)+F_{2}(\delta_{\mathrm{r}},g_{\mathrm{r}})\delta_{ijkl}\right]
A_{\J}^{I\mu}(k_{1},k_{2},k_{3})=iK\sqrt{g_{s}}(\alpha^{\prime})^{3/4}(2\pi)^{4}\delta^{4}(\sum_{i=1}^{3}k_{i})\bar{u}_{1}\gamma^{\mu}u_{2}\left(\tilde{\lambda}_{1}\lambda_{2}\right)^{I}{}_{J}\.
\begin{array}{ccccc}{}&{SO(2n)}&{\subset}&{SO(2n+1)}&{R_{M}={\bf(2n+1)}}\\{}&{SU(3)}&{\subset}&{G_{2}}&{R_{M}={\bf7}}\\{\tilde{R}_{M}={\bf9}}&{SO(9)}&{\subset}&{F_{4}}&{R_{M}={\bf26}}\\{\ast}&{SU(8)}&{\subset}&{E_{7}}&{\ast}\\{\ast}&{SO(16)}&{\subset}&{E_{8}}&{\ast}\\{\ast}&{SU(9)}&{\subset}&{E_{8}}&{\ast}\\\end{array}\quad.
\Gamma_{\mu}^{(\mathrm{f)}}=\int{\frac{d^{3}k}{(2\pi)^{3}}}G_{\alpha\beta}(k){\frac{(2p-k)_{\alpha}}{(p-k)^{2}+M^{2}}}\Sigma(q-k)\Gamma_{\beta\mu}(k,q-k,-q),
g_{\alpha\beta\gamma}:=\int_{\sum_{2}}d^{2}\varphiY_{\alpha}(\frac{\partialY_{\beta}}{\partial\varphi^{1}}\frac{\partialY_{\gamma}}{\partial\varphi^{2}}-\frac{\partialY_{\gamma}}{\partial\varphi^{1}}\frac{\partialY_{\beta}}{\partial\varphi^{2}})
\frac{1}{8\pi^{2}}\left[R^{ab}\mathrm{\tiny~\wedge~}R_{ab}+\frac{2}{l^{2}}(T^{a}{\mathrm{\tiny~\wedge~}}T_{a}-R_{ab}{\mathrm{\tiny~\wedge~}}e^{a}{\mathrm{\tiny~\wedge~}}e^{b})\right].
\bar{Q}_{a}=\bar{\psi}_{a}^{i}R_{i}-2i\bar{\psi}_{a}^{i}m^{j}(\partial_{ij}^{2}A)+2i\psi_{a}^{i}n^{j}(\partial_{ij}^{2}A)-\frac{1}{2}i\lambda_{ai}^{c}\bar{\psi}_{c}^{i},
t_{K\widetildeI}{}^{\widetildeL}t_{\left(\widetildeJ\widetildeL\right)}{}^{M}+t_{K\widetildeJ}{}^{\widetildeL}t_{\left(\widetildeI\widetildeL\right)}{}^{M}-t_{\left(\widetildeI\widetildeJ\right)}{}^{\widetildeL}t_{K\widetildeL}{}^{M}=0\,,
\langle\zeta^{\prime}|Q|\zeta^{\prime\prime}\rangle=-i\hbar\frac{\vec{\partial}}{\vec{\partial}\overline{{\psi}}^{\prime\prime}}\langle\zeta^{\prime}|\zeta^{\prime\prime}\rangle+\frac{\vec{\partial}\Omega(\zeta^{\prime\prime})}{\vec{\partial}\overline{{\psi}}^{\prime\prime}}\,,
{\frac{\dot{\varrho}_{m}}{\varrho_{m}}}=-3{\frac{\dot{a}}{a}}(1+w)-2{\frac{\dot{c}}{c}}{\frac{\varrho_{c}}{\varrho_{m}}}+2{\frac{\dot{c}}{c}}{\frac{\varrho+\varrho_{\delta\sigma}}{\varrho_{m}}}-{\frac{\varrho}{\varrho_{m}}}{\frac{\dot{G}_{4}}{G_{4}}}.
\biggl<S_{4}(\phi_{s},\phi_{f})\biggr>_{0f}=-\frac{1}{4!}\biggl<\int(\phi_{s}+\phi_{f})_{4}(\phi_{s}+\phi_{f})_{3}(\phi_{s}+\phi_{f})_{2}(\phi_{s}+\phi_{f})_{1}u(4321)\biggr>_{0f}.
F_{\mu\nu}=\partial_{\mu}(\hat{\phi}^{a}W_{\nu}^{a})-\partial_{\nu}(\hat{\phi}^{a}W_{\mu}^{a})-{\frac{1}{g}}\epsilon^{abc}\hat{\phi}^{a}(\partial_{\mu}\hat{\phi}^{b})(\partial_{\nu}\hat{\phi}^{c})
\frac{1+g_{p+1}}{1-g_{p+1}}=-\Biggl(\frac{1+a_{p}}{2}\Bigl(\frac{1+g_{p}}{1-g_{p}}\Bigr)+\frac{1-a_{p}}{2}\Bigl(\frac{1-g_{p}}{1+g_{p}}\Bigr)\Biggr)
R(u)=\left(\begin{array}{ccc|ccc|ccc}{a}&{}&{}&{}&{}&{}&{}&{}&{}\\{}&{b}&{}&{c}&{}&{}&{}&{}&{}\\{}&{}&{b}&{}&{}&{}&{c}&{}&{}\\\hline{}&{d}&{}&{b}&{}&{}&{}&{}&{}\\{}&{}&{}&{}&{a}&{}&{}&{}&{}\\{}&{}&{}&{}&{}&{b}&{}&{c}&{}\\\hline{}&{}&{d}&{}&{}&{}&{b}&{}&{}\\{}&{}&{}&{}&{}&{d}&{}&{b}&{}\\{}&{}&{}&{}&{}&{}&{}&{}&{a}\\\end{array}\right),
\mathcal{L}_{E}=-\frac{3}{2\kappa^{2}}\int{d^{4}{\theta}E[K]}+\frac{1}{8}\int{d^{4}\theta\frac{E}{R}f(\Phi_{i})_{ab}W^{{\alpha}a}{W_{\alpha}}^{b}}+\mathrm{h.c.},
\psi_{+2\dot{q}}^{1-}=\frac{i}{32}\tilde{\gamma}_{\dot{q}p}^{i}\Omega_{+p}^{-2i},\\Omega_{+2}^{-2i}=-\frac14\nabla_{+q}\gamma_{q\dot{q}}^{i}\psi_{+2\dot{q}}^{1-},
\int_{0}^{T}dt\intd\thetak\cdotDX(t,\theta)\;e^{-ik\cdotX(t,\theta)}=i\int_{0}^{T}dt\intd\thetaD\,e^{-ik\cdotX(t,\theta)}=i\left(1-e^{-ik\cdotx}\right)\;\;\;.
f(\theta)=\left\{\frac{\vartheta_{4}\left[\frac{1}{2i}\left(\theta-i\frac{\pi}{2}a\right)\right]\vartheta_{4}\left[\frac{1}{2i}\left(\theta+i\frac{\pi}{2}(1+a)\right)\right]}{\vartheta_{4}\left[\frac{1}{2i}\left(\theta+i\frac{\pi}{2}a\right)\right]\vartheta_{4}\left[\frac{1}{2i}\left(\theta-i\frac{\pi}{2}(1+a)\right)\right]}\right\}^{2}\,\,\,,
A^{\mu}=-i\partial_{4}^{--}\partial_{4}^{\mu}\left(\Pi\cdotf\right)-4i\sum_{a=1}^{3}{\frac{(a4^{-})}{(a4)}}x_{a4}^{\mu}{\frac{\partial\Pi}{\partialx_{a4}^{2}}}\cdotf+\Pi\cdot(f_{s}S_{44}^{\mu}+f_{t}T_{44}^{\mu})\;.
S=S_{\mathrm{p}}-\sum_{I=1}^{K}\,\intd\tau_{I}\,\mathrm{tr}\,\lambda_{I}^{-1}Z_{I}{\dot{\lambda}}_{I}-\intd^{D}x\,\mathrm{tr}\left(j_{\mu}\,A^{\mu}+\frac{1}{4\,\Omega_{D-2}}\,F_{\mu\nu}\,F^{\mu\nu}\right),
g^{-1}(x,y)=i\left[\Box+m_{0}^{2}+\frac{\lambda_{0}}{6}(\frac{\phi^{2}}{N}+g(x,x))+\frac{\eta_{0}}{5!}(\frac{\phi^{2}}{N}+g(x,x))^{2}\right]\delta^{D}(x-y)+O(\frac{1}{N}).
\gammaB_{a}^{0i}=\partial_{j}\eta_{a}^{(2)ij},\;\gammaB_{a}^{ij}=\eta_{a}^{(1)ij},\;\gammaH_{i}^{a}=C_{i}^{(1)a},\;\gammaH_{0}^{a}=-\partial^{i}C_{i}^{(2)a},
{\calT}|{\bfb}+\frac{1}{2};n_{1},n_{2},\ldots,n_{b}\rangle=\sum_{i=0}^{2^{b-1}}(-)^{p_{i}+1}|{\bf{p_{i}+1/2}};\tau_{i}(n_{b},n_{b-1},\ldots,n_{1}),\frac{1}{2}\rangle,
\widehat{L}_{\mathrm{part}}={\calP}_{i}\dot{\eta}_{i}-\frac{1}{2}{\calP}_{i}^{2}+e(\widehat{A}_{i}\dot{X}_{i}+\widehat{A}_{0}+\frac{e\theta}{2}\varepsilon_{ij}\widehat{A}_{i}\frac{d}{dt}\widehat{A}_{j})-\frac{1}{2}\frac{d}{dt}(e\theta\varepsilon_{ij}{\calP}_{i}\widehat{A}_{j})\,.
\left(\sum_{l\neqi}^{}\frac{\alpha_{l}}{m_{l}}+\frac{\alpha_{i}}{m_{0}}\right)\delta\left(\left.\left(\alpha^{T}\|c\|\alpha\right)\right|_{\alpha_{i}=0}+2m_{0}\alpha_{i}\sum_{l\neqi}^{}\frac{\alpha_{l}}{m_{l}}+\alpha_{i}^{2}\!-\!1\right).
{\calA}={\frac{3^{4}}{4^{4}}}\left(\int_{-\infty}^{\infty}{\frac{dy}{(1+y^{2})^{\frac{5}{2}}}}\right)^{4}{\frac{1}{2}}(1+H(r-\bar{x}))={\frac{1}{2}}(1+H(r-\bar{x})),
S=\frac{1}{16\piG_{p+2}}\intd^{p+2}x\sqrt{-g}\left[R-\frac{1}{2}(\partial\phi)^{2}-\frac{1}{4}e^{a\phi}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}b^{2}e^{-a\phi}\right],
b_{2k}^{0}=\frac{(-1)^{k+1}}{k+2}\left[\frac{(k+1)(k+2)}{2(k+3)}\frac{\alpha}{q^{k+3}}-\sum_{m=0}^{k}\frac{1}{m+1}\frac{\alpha^{2}}{q^{k+2}}\right],
\hat{H}(\tau)=\mathrm{bdiag}\left(\hat{h}(\tau),\,-\sigma_{3}\hat{h}(-\tau)\sigma_{3}\right)\,,\;\;\hat{h}(\tau)=\left.\hat{h}(x^{0})\right|_{x^{0}=\tau},\;\;\hat{h}(x^{0})=qA_{0}I+\hat{\omega}\;.
\left\{-\frac{\gamma}{8N^{2}}\intd\theta\rho(\theta)\left\{\left(\frac{\partial}{\partial\theta}\frac{\delta}{\delta\rho(\theta)}\right)^{2}-{\calV}^{2}(\theta)\right\}+V[\rho]\right\}~\tilde{\psi}[\rho]~=~E~\tilde{\psi}[\rho]\;.
S_{(n)}=\intd^{4}x\Bigl[\frac{1}{6}\,H_{(n)}^{\mu\nu\rho}H_{(n)\mu\nu\rho}^{\ast}-\frac{2n^{2}\pi^{2}}{R^{2}}\Bigl(iB_{(n)}^{\mu\nu}-\frac{R}{2n\pi}\,F_{(n)}^{\mu\nu}\Bigr)\Bigl(-\,iB_{(n)\mu\nu}^{\ast}-\frac{R}{2n\pi}\,F_{(n)\mu\nu}^{\ast}\Bigr)\Bigr]
\begin{array}{lll}{A)}&{J_{1},J_{2},J_{3}\mathrm{~unrestricted,}\qquad}&{l\geq6+\frac{1}{2}(J_{1}+2J_{2}+3J_{3})+2(a_{1}+a_{2})}\\{B)}&{J_{3}=0,\qquad}&{l=4+\frac{1}{2}(J_{1}+2J_{2})+2(a_{1}+a_{2})}\\{C)}&{J_{2}=J_{3}=0,\qquad}&{l=2+\frac{1}{2}J_{1}+2(a_{1}+a_{2})}\\{D)}&{J_{1}=J_{2}=J_{3}=0,\qquad}&{l=2(a_{1}+a_{2})}\\\end{array}
E_{n}(R)=8i\pi^{2}G_{\mathrm{N}}L\int_{0}^{R}dr\int_{0}^{r}d\ell\\rho(r)\rho(\ell){\frac{r\ell}{|n|}}\Bigl[e^{i{\frac{|n|}{L}}(r+\ell)}-e^{i{\frac{|n|}{L}}(r-\ell)}\Bigr],
G_{\mu\nu}(x)=\sum_{a=1}^{N^{2}-1}F_{\mu\nu}^{a}(x)M^{a}(x)-\frac{1}{\lambda}\sum_{a,b,c=1}^{N^{2}-1}f^{abc}M^{a}(g)(D_{\mu}M(g))^{b}(D_{\nu}M(g))^{c},
S_{2}=-\int{\mathrm{Htr}}\Bigl\{\phi(d\omega+\omega^{2}+\{\chi_{0},\chi_{2}\}-\chi_{1}^{2})+B(\phi^{2}+\chi_{0}^{2})+\chi_{1}(d\chi_{0}+[\omega,\chi_{0}])\Bigr\}.
\langleT_{01}\rangle=\langleT_{10}\rangle={\frac{1}{2}}\int_{0}^{\infty}d\omega\Biggl\{(\partial_{0}\phi_{\omega})(\partial_{1}\phi_{\omega}^{\ast})+(\partial_{1}\phi_{\omega})(\partial_{0}\phi_{\omega}^{\ast})\Biggr\}.
\psi_{g}:\left(\begin{array}{l}{f_{\alpha}^{1/2}}\\{(f_{\alpha}^{-1/2})^{*}}\\\end{array}\right)(p)\mapstoe^{i\omega_{\alpha\beta}(g;p)/2}\,g\left(\begin{array}{l}{f_{\beta}^{1/2}}\\{(f_{\beta}^{-1/2})^{*}}\\\end{array}\right)(g^{-1}p),\qquadg\inSU(2).
F^{\mu}(k)=G^{\mu}(k)+\frac{1}{2\pi}\intd^{2}k^{\prime}(g^{{\mu}{\nu}}+{\epsilon}^{{\mu}{\nu}}){\lambda}_{{\nu}{\sigma}}(k-k^{\prime})\left[G^{\sigma}(k^{\prime})-{\epsilon}^{{\sigma}{\rho}}G_{\rho}(k^{\prime})\right].
{\calA}_{\beta}=\frac{1}{8{\pi}^{2}}[-2(\beta/l^{2})l^{-2}e\cdotR\cdote+(\beta/l^{2})^{2}R^{ab}{\mathrm{\tiny~\wedge~}}R_{ab}+2(\beta/l^{2})^{2}l^{-2}T\cdotT]+O[(\beta/l)^{-2})].
\left(\begin{matrix}{\phi}\\{\psi_{1}}\\{\psi_{2}}\\\end{matrix}\right)\rightarrow\left(\begin{matrix}{e^{-2i\omega_{1}}\phi}\\{e^{-i(\omega_{1}+\omega_{2})}\psi_{1}}\\{e^{-i(\omega_{1}-\omega_{2})}\psi_{2}}\\\end{matrix}\right)\.
\psi(p)\Bigr|_{U_{+}}=\left(\begin{matrix}{\omega(p)}\\{1}\\\end{matrix}\right)\varphi_{+}(p)\delta(p^{2}),\qquad\qquad\psi(p)\Bigr|_{U_{-}}=\left(\begin{matrix}{1}\\{\tilde{\omega}(p)}\\\end{matrix}\right)\varphi_{-}(p)\delta(p^{2}),
\int\Pi_{\mu\nu\lambda\sigma}A_{\mu}A_{\nu}A_{\lambda}A_{\sigma}=\frac{1}{2880m^{4}}\intd^{4}x\,[14F_{\mu\nu}F_{\nu\alpha}F_{\alpha\beta}F_{\beta\mu}-5(F_{\mu\nu}F_{\mu\nu})^{2}]+\cdots.
S_{\alpha}(z)\,S_{A}(z)=\left.\widetildeS_{\alpha}(\bar{z})\,\widetildeS_{A}(\bar{z})\right|_{z=\bar{z}}~~~,~~~~S^{\dot{\alpha}}(z)\,S^{A}(z)=\left.\widetildeS^{\dot{\alpha}}(\bar{z})\,\widetildeS^{A}(\bar{z})\right|_{z=\bar{z}}~~,
W=H_{23}+H_{45}+\ldots+H_{n1}+Hl^{2}+\bar{a}x^{2}+\tilde{M}p+\frac{\tilde{\Lambda}_{n}^{n+1}}{(\tilde{M}\tilde{X}^{(n-4)/2}\mathrm{Pf}\,A)^{1/2}},
{L}(q^{i}(t),\frac{d}{dt}q^{i}(t);t)\rightarrow{L}(q^{i}(t^{\prime}),\frac{d}{dt^{\prime}}q^{i}(t^{\prime});t^{\prime})={L}(q^{i}(t),\frac{d}{dt}q^{i}(t);t)+\delta_{t}{L},
\left(\begin{matrix}{\hat{H}^{(1)}}\\{\hat{H}^{(2)}}\\\end{matrix}\right)\to(\omega^{\mathrm{T}})^{-1}\left(\begin{matrix}{\hat{H}^{(1)}}\\{\hat{H}^{(2)}}\\\end{matrix}\right),\\\\hat{\calM}\to\omega\hat{\calM}\omega^{\mathrm{T}},\\\\\omega\inSL(2,{\bfR}).
X=\hat{q}-2i\hat{p}\tau+\sum_{1}^{\infty}\frac{1}{\sqrt{m}}\Bigl[a_{m}^{\dagger}e^{m\tau+im\sigma}+\tilde{a}_{m}^{\dagger}e^{m\tau-im\sigma}+a_{m}e^{-m\tau-im\sigma}+\tilde{a}_{m}e^{-m\tau+im\sigma}\Bigr]
z_{\mu+}^{\pm}=\frac{1}{\sqrt{2}}\left[z_{2\mu-1}^{\pm}-iz_{2\mu}^{\pm}\right]\\\,\\\z_{\mu-}^{\pm}=\frac{1}{\sqrt{2}}\left[z_{2\mu-1}^{\pm}+iz_{2\mu}^{\pm}\right]\\\.
C_{\alpha\beta}^{\mu\nu\rho\sigma}=\eta_{\alpha}^{\nu}\eta_{\beta}^{\rho}\eta^{\mu\sigma}-\eta_{\alpha}^{\mu}\eta_{\beta}^{\rho}\eta^{\nu\sigma}-\eta_{\alpha}^{\nu}\eta_{\beta}^{\sigma}\eta^{\mu\rho}+\eta_{\alpha}^{\mu}\eta_{\beta}^{\sigma}\eta^{\nu\rho}.
\hat{h}_{\mu\nu}(x,0)=\frac{2G_{5}km_{0}}{r}\left[\textrm{diag}(1,1,1,1)+\frac{1}{3k^{2}r^{2}}\textrm{diag}(2,1,1,1)+\mathcal{O}(1/r^{3})\right]
E_{1}(r,a)\Bigl|_{r\gga}=N_{mol}\Bigl(\frac{4\pia^{3}}{3}\Bigr)\frac{-23\alpha\alpha_{ball}}{4\pir^{7}}=N_{mol}\Bigl(\frac{4\pia^{3}}{3}\Bigr)E_{Cas-Pol}\,.
\langleA_{\mu}(x)\psi(x){\bar{\psi}}(y)\rangle\;=\;\int{\mathcalD}A\,A_{\mu}(x)\,\langlex|(\not\!\partial+ie\not\!\!A+M)^{-1}|y\rangle\,e^{-S_{G}[A]-\Gamma_{F}[A]}
\bar{R}_{\;\;\;(\sigma)}^{\alpha\beta\;\;\;\delta(\rho)}=\varepsilon^{\alpha\beta\gamma\delta}\left(\left(D_{\gamma}\right)_{\;\;\sigma}^{\rho}+g\delta_{\sigma}^{\rho}\left(\partial_{\tau}B_{\gamma}^{\;(\tau)}+B_{\gamma}^{\;(\tau)}\partial_{\tau}\right)\right).
S_{Bulk}=\frac{1}{2}\sum_{n=1}^{\infty}nX_{-n}^{\mu}X_{n}^{\nu}\delta_{\mu\nu}+i\sum_{r=1/2}^{\infty}r\psi_{-r}^{\mu}\psi_{r}^{\nu}\delta_{\mu\nu},\\mu,\nu=1,\dots,10\.
\phi_{x}^{i}(y)=x^{i}+y^{i}-\frac12\,\Gamma_{rs}^{i}(x)\,y^{r}y^{s}+\frac1{3!}\left(2\Gamma_{jr}^{i}(x)\Gamma_{st}^{j}(x)-\partial_{r}\Gamma_{st}^{i}(x)\right)\,y^{r}y^{s}y^{t}+\cdots.
\langle\langle\prod_{i=1}^{L}V_{n_{i}}^{\alpha_{i}}\rangle\rangle=\rho\mu^{S}{\frac{\Gamma({\frac{\sum_{i}n_{i}}{2}}+{\frac{(\sum_{i}\alpha_{i})+2}{2p}})}{\Gamma{(S+1)}}}\quad,