harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\underline{{\Omega}}_{IJ}=\left(\begin{array}{cccc}{0}&{0}&{0}&{1}\\{0}&{0}&{\frac{1}{V}}&{-\underline{{\Omega}}_{C++}}\\{0}&{-\frac{1}{V}}&{0}&{-\underline{{\Omega}}_{C--}}\\{-1}&{\underline{{\Omega}}_{C++}}&{\underline{{\Omega}}_{C--}}&{0}\\\end{array}\right).
S_{q}=\intd^{5}x\left[-\frac{1}{2}\tilde{\varphi}\left\{\partial_{\mu}\partial^{\mu}+(m\kappa)\{a_{q},a_{q}^{\dagger}\}\right\}\tilde{\varphi}-|\kappay|^{5}V\left(|\kappay|^{-\frac{3}{2}}\tilde{\varphi}\right)\right].
\left<\hat{H}_{\alpha}(t)\right>_{n_{\alpha}}=\frac{R^{3}(t)}{2}\left(\dot{\phi}_{\alpha1}(t)\dot{\phi}_{\alpha1}^{*}(t)+\left(m^{2}+\xi{\bfR}+\frac{{\bfk}^{2}}{R^{2}(t)}\right)\phi_{\alpha1}(t)\phi_{\alpha1}^{*}(t)\right)\hbar(2n_{\alpha}+1).
\left(\frac{1}{c^{2}}\left(\frac{\partial}{\partialt}\right)^{2}-\sum_{j=1}^{3}\left(\frac{\partial}{\partialx^{j}}\right)^{2}\right)A_{i}(x;y)=-\frac{4\pi}{c}m_{y}G\frac{dy^{i}}{dt}\delta({\bfx}-{\bfy}(t)).
ds_{B}^{2}=e^{\phi}dzd\bar{z}+e^{\phi}\left(\frac{\partial\xi}{\partialz^{\prime}}dz^{\prime}d\bar{z}+c.c.\right)+\left(e^{\phi}\left|\frac{\partial\xi}{\partialz^{\prime}}\right|^{2}+e^{\psi}\right)dz^{\prime}d\bar{z^{\prime}}.
\sum_{i=1}^{p}\frac{\mu_{1}}{2\pi\lambda\tilde{k}_{i}}\intdtdx_{1}dp_{i}dq_{i}C_{t1,2i,2i+1}^{(4)}=\sum_{i=1}^{p}\mu_{3}\intdtdx_{1}dx_{2i}dx_{2i+1}C_{t1,2i,2i+1}^{(4)}=\sum_{i=1}^{p}\mu_{3}\int_{M_{i}}C^{(4)}\,
e_{\mu}^{a}=e^{\frac{\rho(x)}{2}}\delta_{\mu}^{a},\quade_{a}^{\mu}=e^{-\frac{\rho(x)}{2}}\delta_{a}^{\mu},\quad\omega_{m,ab}=\frac{1}{2}(\eta_{am}\partial_{b}\rho(x)-\eta_{bm}\partial_{a}\rho(x)),\quad\omega_{c,ab}=e_{c}^{m}\omega_{m,ab}=e^{-\frac{\rho(x)}{2}}\omega_{c,ab}.
X=\left(\begin{matrix}{0}&{-p_{8}}&{p_{9}}&{x_{1}}&{x_{3}}\\{}&{0}&{-p_{6}}&{x_{2}}&{x_{5}}\\{}&{}&{0}&{x_{4}}&{x_{7}}\\{}&{a/s}&{}&{0}&{x_{0}}\\{}&{}&{}&{}&{0}\\\end{matrix}\right)
[\hat{S}_{+},\hat{S}_{-}]=\displaystyle\frac{\partialG(S_{3})}{\partialS_{3}}\vert_{S_{3}\to\hat{S}_{3}}\stackrel{def.}{=}[[\hat{S_{3}}]]~,~~~~[\hat{S}_{3},\hat{S}_{\pm}]=\pm\hat{S}_{\pm}~,
H_{int}^{as}=-{\displaystyle\frac{1}{(2\pi)^{\frac{3}{2}}}}\intd^{3}\vec{p}\,\frac{p_{\mu}}{p_{0}}\rho(\vec{p})\int\frac{d^{3}\vec{k}}{\sqrt{2k_{0}}}[a_{\mu}(\vec{k})e^{-i(pk/p_{0})t}+a_{\mu}^{\dagger}(\vec{k})e^{i(pk/p_{0})t}]\,\,,
A(\varepsilon,v)=\sum_{n=1}^{\infty}{\frac{1}{n!}}\int_{0}^{\infty}\prod_{j=1}^{n}{\frac{vd\varepsilon_{j}}{\varepsilon_{j}^{\eta+1}}}\theta(\varepsilon_{j}-m_{1})\theta\left(m_{0}(a)-\varepsilon_{j}\right)\delta\left(\varepsilon-\sum_{j=1}^{n}\varepsilon_{j}\right).
S={\frac{1}{2}}\intd^{4}x\int_{\pi}^{\pi}d\phi\sqrt{g}\left(g^{AB}\partial_{A}\Phi\partial_{B}\Phi-\hat{m}^{2}\Phi^{2}-\xi\hat{R}\Phi^{2}-{\frac{\lambda}{k}}\Phi^{4}\right)\;.
[{\calP}_{0},X_{0}]=i,\quad[{\calP}_{i},X_{j}]=-i\,\delta_{ij}\left(1-\frac{\vec{{\calP}}\,{}^{2}}{\kappa^{2}}\right)\,e^{{\calP}_{0}/\kappa},\quad[{\calP}_{0},X_{i}]=-\frac{2i}\kappa\,{\calP}_{i}\,e^{{\calP}_{0}/\kappa}
(\kappa_{\tau}\frac{\partial}{\partial\kappa_{\tau}}+\gamma_{\phi^{2}(\tau)}^{*}(2+t\frac{\partial}{\partialt}))\Gamma_{R(\tau)}^{(0,2)}=(\kappa_{\tau}^{-2\tau})^{\frac{\epsilon_{L}}{2}}B_{\tau}(u_{\tau}^{*}),
I_{\epsilon}^{(1)}(a^{2})=\sum_{n=-\infty}^{+\infty}{\frac{1}{(a^{2}+4\pi^{2}n^{2})^{\frac{1+\epsilon}{2}}}}\int_{0}^{\infty}dy{\frac{y^{-\epsilon}}{y^{2}+1}}\;.
S=\frac{1}{2}\int\left((e^{-\Phi}\starQ_{0}e^{\Phi})(e^{-\Phi}\star\eta_{0}e^{\Phi})-\int_{0}^{1}dt(e^{-t\Phi}\star\partial_{t}e^{t\Phi})\star\left\{(e^{-t\Phi}\starQ_{0}e^{t\Phi}),(e^{-t\Phi}\star\eta_{0}e^{t\Phi})\right\}\right)\.
\gamma\frac{df}{d\gamma}\;+\;\sum_{n=1}^{\infty}\Big[\lambda_{n}\frac{df}{d\lambda_{n}}+\kappa_{n}\frac{df}{d\kappa_{n}}+\bar{\kappa}_{n}\frac{df}{d\bar{\kappa}_{n}}\Big]\;\;=\;\;f\;.
\Gamma[\phi]=\intd^{D}x\left\{V(\phi^{2},t)+\frac{1}{2}K(\phi^{2},t)\left(\partial_{\mu}\phi^{a}\right)^{2}+\frac{1}{2}Z(\phi^{2},t)\left(\phi^{a}\partial_{\mu}\phi^{a}\right)^{2}+\cdots\right\}
\begin{array}{ll}{[B,~Q_{\alpha(\dot{\alpha})}]=\pm\frac{1}{2}Q_{\alpha(\dot{\alpha})},}&{[B,~Q_{i}]=\frac{1}{2}Q_{i},}\\{[B,~R_{\alpha(\dot{\alpha})}]=\mp\frac{1}{2}R_{\alpha(\dot{\alpha})},}&{[B,~R_{i}]=-\frac{1}{2}R_{i},}\\{[A,~Q_{i}]=-iQ_{i},}&{[A,~R_{i}]=iR_{i}.}\\\end{array}
\begin{array}{rcl}{\delta_{i}x^{\alpha}}&{=0}&{=\displaystyle\frac{1}{i\hbar}[x^{\alpha},L_{i}^{\mu}\circp_{\mu}]}\\{\delta_{i}x^{\mu}}&{=L_{i}^{\mu}}&{=\displaystyle\frac{1}{i\hbar}[x^{\mu},L_{i}^{\mu}\circp_{\mu}]\,.}\\\end{array}
f^{\prime\prime}+\frac{1}{r}f^{\prime}-\left[\left(\frac{m}{r}\right)^{2}-qBm\left(\frac{R}{r}\right)^{2}+\left(\frac{qB}{2}\left(\frac{R^{2}}{r}\right)\right)^{2}-k^{2}\right]f=0
F_{lm}^{0}=\frac{-ig}{\sqrt{L}}({\calA}_{l}{\calA}_{m}^{\dagger}-{\calA}_{m}{\calA}_{l}^{\dagger})+\frac{ig}{\sqrt{L}}({\calB}_{l}{\calB}_{m}^{\dagger}-{\calB}_{m}{\calB}_{l}^{\dagger});
L_{B,int.}\equiv{\frac{m}{d!}}\epsilon^{\mu_{1}\dots\mu_{d}}B_{\mu_{1}\dots\mu_{d-p-2}}\partial_{\,[\,\mu_{d-p-1}}A_{\mu_{d-p}\dots\mu_{d}\,]}\equivm\,B^{\mu_{1}\dots\mu_{d-p-2}}F^{*}{}_{\mu_{1}\dots\mu_{d-p-2}}(A)
\intd^{D}k_{i}\;A(k_{j},\;p_{l})\;f(k_{j}\!\cdot\!n^{*},\;p_{l}\!\cdot\!n^{*})\:h(k_{j}\!\cdot\!n,\;p_{l}\!\cdot\!n)\longrightarrow^{\!\!\!\!\!\!\!\!\!\!\!\!AC}\int\frac{d^{D}k_{i}}{A(k_{j},\;p_{l})}\frac{f(k_{j}\!\cdot\!n^{*},\;p_{l}\!\cdot\!n^{*})}{\:h(k_{j}\!\cdot\!n,\;p_{l}\!\cdot\!n)}
\partial_{x}S(z,x)={\frac{1}{2(z-x)^{2}}}\,{\frac{u(z)+u(x)}{\sqrt{u(z)u(x)}}}-{\frac{S_{1}(x)}{8\,(z-x)}}\,{\frac{u(z)-u(x)}{\sqrt{u(z)u(x)}}}\,.
e^{-2\hat{\phi}}=e^{-2\hat{\phi}_{0}}+{\frac{2\tilde{m}}{r}}\,\qquadK=K_{0}+{\frac{2\hat{m}}{r}}\,\qquadA_{n}=(A_{n})_{0}+{\frac{2q_{n}}{r}}\,\qquadn=4,\dots9.
\frac{g_{ii}}{A^{2}}{\partial_{i}^{({\bfx})}}^{2}\Omega({\bfx},{\bfy},t)\mid_{{\bfx}={\bfy}}=-\frac{g_{ii}}{2d}\int_{{\bfp}}\frac{p^{2}}{\sqrt{p^{2}+\alpha}}\;\;,
S=-{\frac{1}{2}}\intd^{6}x~\partial_{M}\phi~\partial^{M}\phi-a\int_{\Sigma}d^{4}x~{\calO}_{4}-bL^{2}\int_{\Sigma}d^{4}x~\phi~{\calO}_{4}-cL^{-2}\int_{\Sigma}d^{4}x~\phi~.
{\frac{1}{g_{1}^{2}}}+{\frac{1}{g_{2}^{2}}}={\frac{1}{4\pig_{s}}},\qquad{\frac{1}{g_{1}^{2}}}-{\frac{1}{g_{2}^{2}}}={\frac{1}{4\pi^{2}g_{s}}}\left(\int_{S^{2}}B-\pi\right).
e^{-S_{B}[A]}\;=\;\frac{{\mathcalN}_{2}}{{\mathcalN}_{1}}\int{\mathcalD}b_{\mu}\,{\mathcalZ}[b_{\mu}]\,e^{i\intd^{3}xb_{\mu}\epsilon_{\mu\nu\lambda}\partial_{\nu}A_{\lambda}}\;,
\eta_{11}^{(2)}=\frac{\pi^{2}B(2-B)}{H^{2}}\,,\quad\eta_{12}^{(2)}=-\eta_{11}^{(2)}\,,\quad\eta_{22}^{(2)}=-2\eta_{11}^{(2)},\quad\eta_{1}^{(2)}=0,\quad\eta_{2}^{(2)}=-3\eta_{11}^{(2)}\,.
\times{D_{2s}}\left[{\frac{{\lambda_{2}}\ointd\hat{s}{e^{\alpha\bar{\phi}/2}}}{2{\mu_{2}}{{\left(\int{d^{2}}\xi\sqrt{\hat{g}}{e^{\alpha\bar{\phi}}}\right)}^{1/2}}}}\right]{e^{-{S_{L}^{0}}[\bar{\phi},\hat{g}]}},
|X_{1}(K)|\leq\frac{1}{4\pi^{2}}|\int_{\pi\beta}^{\piR}duW_{o}(\frac{u}{R};\beta)H(K,u,A)|+(\frac{c_{1}}{\sqrt{\pi^{2}R^{2}+A^{2}}}+c_{2}|K|+\frac{c_{3}}{A})e^{-ImKA}.
V(d)=-G_{\mathrm{N}}{\frac{1}{d}}-\sum_{n=-\infty,n\not=1}^{+\infty}G_{\mathrm{N}}{\frac{1}{d}}e^{+i{\frac{|n|}{L}}d}\sim{\frac{1}{d^{1+1}}}\quadas\quadd\rightarrow0
4\pi\frac{\partialz_{B}^{\prime}}{\partialz_{n}^{\prime}}=-\frac{P_{n}^{\prime}}{(z_{B}^{\prime}-z_{n}^{\prime})^{2}}\frac{{\calP}(z_{B}^{\prime}+z_{1})}{\prod_{C\neqB}(z_{B}^{\prime}-z_{C}^{\prime})}
D_{\mu\nu}=\frac{-i}{p^{2}-\mu^{2}+i0}(g_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}-\frac{i\mu}{p^{2}}\epsilon_{\mu\nu\lambda}p^{\lambda})-i\xi\frac{p_{\nu}}{p^{4}}
c_{m,d}=4^{1-d/2}\frac{\Gamma(h_{+})\Gamma(h_{-})}{\Gamma(\frac{d}{2})\Gamma(\frac{d}{2}-1)}\times\frac{\Gamma(\frac{d}{2})}{2(d-2)\pi^{d/2}}=\frac{\Gamma(h_{+})\Gamma(h_{-})}{(4\pi)^{d/2}\Gamma(\frac{d}{2})}.
\left({\frac{\dot{R}}{R}}\right)^{2}=-{\frac{\xi_{+}^{2}+\xi_{-}^{2}}{2}}R^{-{\frac{2}{3}}k^{2}}+{\frac{\kappa_{5}^{4}}{36}}(\lambda+\rho)^{2}+{\frac{9}{4\kappa_{5}^{4}}}\left(\xi_{+}^{2}-\xi_{-}^{2}\right)^{2}(\lambda+\rho)^{-2}R^{-{\frac{4}{3}}k^{2}}.
\tilde{F}^{\mu}={\frac{1}{2}}[\epsilon_{\mu\nu\lambda}F_{\nu\lambda}^{a}n^{a}-\frac{1}{g}\epsilon^{\mu\nu\lambda}\epsilon^{abc}n_{a}({\calD}_{\nu}n)^{b}({\calD}_{\lambda}n)^{c}]\,,
\tilde{j}^{\mu_{1}\cdots\mu_{D-d}}=(\frac1{\sqrt{g}})J^{\mu_{1}\cdots\mu_{D-d}}(\frac\phix)\sum_{i=1}^{l}\beta_{i}\eta_{i}\int_{N_{i}}\delta^{D}(x-z_{i}(u))\sqrt{g_{u}}d^{(D-d)}u,
(X,Y)=\intd^{n}x\left(\frac{\delta^{R}X}{\delta\Phi^{A}(x)}\frac{\delta^{L}Y}{\delta\Phi_{A}^{*}(x)}-\frac{\delta^{R}X}{\delta\Phi_{A}^{*}(x)}\frac{\delta^{L}Y}{\delta\Phi^{A}(x)}\right)\,,
\deltax:=\overline{{\varepsilon}}\psi+\overline{{\psi}}\varepsilon,\qquad\delta\psi:=-(\mathrm{i}\dot{x}+\Phi(x))\varepsilon,\qquad\delta\overline{{\psi}}=(\mathrm{i}\dot{x}-\Phi(x))\overline{{\varepsilon}},
\begin{array}{llllllccl}{}&{p}&{s}&{t}&{u}&{v}&{x}&{y}&{w}\\{\lambda\}&{0}&{1}&{1}&{n}&{0}&{2(n+2)}&{3(n+2)}&{0}\\{\mu\}&{0}&{0}&{0}&{1}&{1}&{4}&{6}&{0}\\{\nu\}&{0}&{0}&{0}&{0}&{0}&{2}&{3}&{1}\\{\sigma\}&{1}&{1}&{0}&{0}&{q^{\prime}}&{2(q^{\prime}+2)}&{3(q^{\prime}+2)}&{0}\\\end{array}
f({\vec{\theta}}^{1},{\vec{\theta}}^{2})=\theta_{a}^{1}\theta_{b}^{1}\theta_{c}^{1}\theta_{d}^{2}\theta_{e}^{2}\theta_{f}^{2}+C_{a}\theta_{a}^{1}\theta_{b}^{2}\theta_{c}^{2}\theta_{d}^{2}+D_{a}\theta_{a}^{2}\theta_{b}^{1}\theta_{c}^{1}\theta_{d}^{1}\.
\pi_{i}=\frac{\partialL}{\partialv^{i}}=\frac{{m}\left[v^{i}-u^{i}u^{0}(1+u^{0}\vec{u}\vec{v})\right]}{\sqrt{(1+u^{0}\vec{u}\vec{v})^{2}-(\vec{v})^{2}}}=-m\omega_{i}
\left[\left(hr^{3}\partial_{r}\right)^{2}+\omega^{2}r^{6}f-{\frac{8hr^{4}r_{\pm}^{4}}{(r^{2}+r_{\pm}^{2})^{2}}}\left(1+{\frac{r_{0}^{2}}{r_{\pm}^{2}}}\right)\right]\phi_{\pm}=0,
K_{+}^{+}(\zeta;r)=\frac{\mathrm{snh}(\alpha+u)}{\mathrm{snh}(\alpha-u)}\frac{\kappa(\zeta^{2};r)}{\kappa(\zeta^{-2};r)},\qquadK_{-}^{-}(\zeta;r)=\frac{\kappa(\zeta^{2};r)}{\kappa(\zeta^{-2};r)},
\begin{array}{rcl}{{\calL}_{\mathrm{g.f.}}}&{=}&{\displaystyle-\frac{1}{2\xi_{1}}f_{1}f_{1}-\frac{1}{2\xi_{2}}f_{2}f_{2},}\\{f_{1}}&{=}&{\partial^{\mu}G_{1\mu}+ig_{1}\xi_{1}(\phi^{\dagger}\phi_{0}-\phi_{0}^{\dagger}\phi),}\\{f_{2}}&{=}&{\partial^{\mu}G_{2\mu}-ig_{2}\xi_{2}(\phi^{\dagger}\phi_{0}-\phi_{0}^{\dagger}\phi).}\\\end{array}
\begin{array}{l}{P^{+}(z)\,\times\,:X^{-}(w)\,P^{+}(w):}\\{\quad=:P^{+}(z)\,X^{-}(w)\,P^{+}(w):}\\{\quad\quad+<P^{+}(z)\,X^{-}(w)>P^{+}(w)}\\{\quad\rightarrow\frac{-\,i}{z-w}P^{+}(w)+\cdots}\\\end{array}
\begin{array}{rcl}{{\calL}_{\phi}}&{=}&{{\calL}\left(\phi,D_{\mu}\phi\right)=|(\partial_{\mu}+igA_{\mu})\phi|^{2}-m^{2}|\phi|^{2}-V(|\phi|)}\\{}&{}&{}\\{}&{=}&{|\partial_{\mu}\phi|^{2}-m^{2}|\phi|^{2}-V(|\phi|)-igA_{\mu}\left[\phi^{\dagger}\partial^{\mu}\phi-\partial^{\mu}\phi^{\dagger}\phi\right]+g^{2}A_{\mu}A^{\mu}\.}\\\end{array}
e_{j}\;=\;-\frac12\,\Bigl(\sigma_{j}^{x}\sigma_{j+1}^{x}\;+\;\sigma_{j}^{y}\sigma_{j+1}^{y}\;+\;\frac{q+q^{-1}}2\,(\sigma_{j}^{z}\sigma_{j+1}^{z}-1)\;+\;\frac{q-q^{-1}}2\,(\sigma_{j}^{z}-\sigma_{j+1}^{z})\Bigr)\,.
Z_{AB}^{(N)}=\left(\begin{matrix}{Z_{1}\epsilon}&{0}&{0}&{0}\\{0}&{Z_{2}\epsilon}&{0}&{0}\\{0}&{0}&{Z_{3}\epsilon}&{0}\\{0}&{0}&{0}&{Z_{4}\epsilon}\\\end{matrix}\right)\,;\quadZ_{i}\epsilon=\left(\begin{matrix}{0}&{Z_{i}}\\{-Z_{i}}&{0}\\\end{matrix}\right)
\left\{\alpha,x_{0}^{\mu},x_{k}^{-},x_{k}^{i}\right\}_{k=1,...}^{\mu=0,...,d;i=1,...,d-1}\leftarrow\!\!\rightarrow\left\{\varphi_{0},x_{0}^{\mu},\varphi_{k},x_{k}^{i}\right\}_{k=1,...}^{\mu=0,...,d;i=1,...,d-1}\;\;\;.
ds_{\mathrm{asym}}^{2}=\frac{1}{2}M_{ij}d{\bfx}_{i}\cdotd{\bfx}_{j}+\frac{g^{4}}{2(4\pi)^{2}}(M^{-1})_{ij}(d\xi_{i}+{\bfW}_{ik}\cdotd{\bfx}_{k})(d\xi_{j}+{\bfW}_{jl}\cdotd{\bfx}_{l})\,\,.
{\calV}_{\sigma}({\bfp})={\frac{m}{\sqrt{2}}}\left(\begin{matrix}{D^{J}\left(\alpha({\bfp})\Theta_{[1/2]}\right)\xi_{\sigma}^{*}}\\{D^{J}\left(\alpha^{-1\,\dagger}({\bfp})\Theta_{[1/2]}\right)(-1)^{2J}\xi_{\sigma}^{*}}\\\end{matrix}\right)\quad,
\begin{array}{l}{{\calX}^{-k}\vertn,z\rangle_{-}=q^{2n}z\vertn,z\rangle_{-k}~,}\\{{\calD}_{-k}\vertn,z\rangle_{-}=-q^{-1-2n}\lambda^{-1}z^{-1}\vertn+1,z\rangle_{-k}~,}\\{\mu_{-k}\vertn,z\rangle_{-}=\vertn-1,z\rangle_{-k}~.}\\\end{array}
U_{(0,\,r-1)}^{2}+\Lambda_{1}^{4}\Omega_{(2,\,r-1)}+\Lambda_{r}^{4}\Omega_{(1,\,r-2)}-M_{0}\Omega_{(1,\,r-1)}\pm2(\prod_{i=1}^{r}\Lambda_{i}^{4})^{1/2}=0.
{\bfJ}\cdot\widehat{\bfp}\,\,\Theta_{[j]}\,\left[\phi_{_{L,R}}^{h^{\prime}}({p}^{\mu})\right]^{\ast}\,=\,-\,h^{\prime}\,\,\Theta_{[j]}\,\left[\phi_{_{L,R}}^{h^{\prime}}({p}^{\mu})\right]^{\ast}\quad.
\vec{r}_{1}=(\frac{1}{2},\frac{\sqrt{3}}{2}),\\,vecr_{2}=(-\frac{1}{2},\frac{\sqrt{3}}{2}),\\,vecr_{3}=(1,0),\\\vec{r}_{4}=-\vec{r}_{1},\\,vecr_{5}=-\vec{r}_{2},\\\vec{r}_{6}=-\vec{r}_{3}
E_{tot}^{\bar{M}}(\Sigma)={\frac{\lambda^{2}}{48\pi}}\left[\int_{-\infty}^{v_{1}}\left(1-\partial_{v}\bar{v}\right)^{2}+\int_{v_{1}}^{v_{2}}\left(1-\partial_{v}\bar{v}\right)^{2}+\int_{v_{2}}^{\infty}\left(1-\partial_{v}\bar{v}\right)^{2}\right]
\frac{F}{V}=-\sum_{n,m,p,q}\int_{\calF}d\tau_{1}\frac{d\tau_{2}}{2\tau_{2}}\frac{\left|\eta\left(\tau\right)\right|^{-48}}{(4\pi^{2}\alpha^{\prime}\tau_{2})^{13}}e^{-S_{w.m}(m,n,p,q)}
\begin{array}{rcl}{}&{}&{e=\displaystyle\frac{1}{q-q^{-1}}x_{1}(a_{1}Z-a_{1}^{-1}Z^{-1})X~,}\\{}&{}&{f=\displaystyle\frac{1}{q-q^{-1}}x_{1}^{-1}(a_{2}Z^{-1}-a_{2}^{-1}Z)X^{-1}~,}\\{}&{}&{t=\displaystyle\frac{a_{1}}{a_{2}}Z^{2}~,}\\\end{array}
D=i\gamma^{\mu}\partial_{\mu}=\left(\begin{array}{cc}{0}&{i(\partial_{t}+\vec{\sigma}\cdot\vec{\partial})}\\{i(\partial_{t}-\vec{\sigma}\cdot\vec{\partial})}&{0}\\\end{array}\right)=\left(\begin{array}{cc}{0}&{D^{(+)}}\\{D^{(-)}}&{0}\\\end{array}\right)
S_{ij}=g^{AB}{\calR}_{\mu\nu}\eta_{A}^{\mu}\eta_{B}^{\nu}g_{ij}-\frac{1}{2}g^{AB}{\calR}_{\mu\nu\rho\sigma}\eta_{A}^{\mu}\eta_{B}^{\sigma}Z_{,i}^{\nu}Z_{,j}^{\rho}
||J_{-}^{l-m}v||^{2}=(-1)^{l-m}\langlev|J_{+}^{l-m}J_{-}^{l-m}|v\rangle=(-1)^{l-m}{\frac{1}{2^{l-m}}}{\frac{(l-m)!}{(l+m)!}}(2l)!||v||^{2}.
P^{\mu}=\frac{\partial{\calL}}{\partial\dot{X}_{\mu}}=\frac{1}{{\calA}}[(X^{\prime}N^{\prime})\dot{N}^{\mu}-(X^{\prime}\dot{N})N^{\mu}]~;~P_{N}^{\mu}=\frac{\partial{\calL}}{\partial\dot{N}_{\mu}}=\frac{1}{{\calA}}[(X^{\prime}N^{\prime})\dot{X}^{\mu}-(\dot{X}N^{\prime})X^{\mu}],
W_{i_{1}j}^{(1)}(x,y)=\left(\begin{array}{cc}{\widetilde{W}_{ij}^{(0)}}\\\hline{}\\{\begin{array}{ccccc}{0}&{\partial_{1}^{x}}&{-e}&{-(a-1)e}\\\end{array}}\\\end{array}\right)\delta(x-y),
L(h,V)=V^{\underline{n}}\left(D_{\underline{m}}h_{\underline{l}\underline{n}}F^{\underline{l}\underline{m}}-h_{\underline{l}}^{\underline{m}}D_{\underline{m}}F_{\quad\underline{n}}^{\underline{l}}\right)-\frac{1}{2}A^{\prime}F^{r\underline{m}}V_{\underline{m}}h_{i}^{i}+\frac{D_{1}-2}{2}A^{\prime}F^{\underline{l}\underline{m}}V_{\underline{m}}h_{r\underline{l}}~,
\xi=\left(\begin{array}{cc}{0}&{1}\\{0}&{0}\\\end{array}\right)=\frac{1}{2}(\sigma_{1}+i\sigma_{2})\,\overline{{\xi}}=\left(\begin{array}{cc}{0}&{0}\\{1}&{0}\\\end{array}\right)=\frac{1}{2}(\sigma_{1}-i\sigma_{2})\.
\begin{array}{lcr}{P(W_{2})=(y^{2}+x^{3}+z^{6}+\muxyz)+w(x^{{\frac{3}{2}}}z^{{\frac{3}{2}}})+w^{2}(y+xz)+w^{3}x^{{\frac{1}{2}}}z^{{\frac{1}{2}}}+w^{4}.}\\\end{array}
\left[T_{\mu\nu}\right]_{A}=\frac{1}{12\kappa_{11}^{2}}\left(F_{\mu\lambda\xi\rho}F_{\nu}^{\lambda\xi\rho}-\frac{1}{8}g_{\mu\nu}F^{\rho\sigma\lambda\xi}F_{\rho\sigma\lambda\xi}\right)
\begin{array}{c}{\frac{da_{m}(t)}{dt}=-\fraci\hbar{\calE}_{m}a_{m}(t)-\fraci\hbar\sum_{n}<\varphi_{m}|H_{1}|\varphi_{n}>a_{n}(t)}\\{-\fraci\hbar\int_{c}<\varphi_{m}|H_{1}|\varphi^{(+)}(k)>b(k;t)dk}\\\end{array}
D(\vec{V},\vec{\sigma})=\left(\begin{array}{c|c}{\gamma}&{0}\\\hline{\displaystyle-\frac{\vec{V}}{c}\gamma^{-1}}&{I+\displaystyle\frac{\vec{V}\otimes\vec{V}^{\mathrm{T}}}{c^{2}\gamma\left[\gamma+\sqrt{\gamma^{2}+\displaystyle\frac{\vec{V}}{c}^{2}}\right]}-\frac{\vec{V}\otimes\vec{\sigma}^{\mathrm{T}}}{c^{2}\gamma\gamma_{0}}}\\\end{array}\right)
\delta\phi(t,r^{\ast})=\frac{-ce^{-2\lambdar^{\ast}}+\alpha^{-}~e^{\lambda(t-r^{\ast})}+\alpha^{+}~e^{-\lambda(t+r^{\ast})}}{(1+\frac{m}{\lambda}e^{-2\lambdar^{\ast}})}.
g_{p}=\int_{0}^{\infty}xdx\\frac{x^{p}}{2}f(x)+\sum_{m=1}^{\infty}\int_{m}^{\infty}xdx\x^{p}f(x)-\int_{0}^{\infty}dm\int_{m}^{\infty}xdx\x^{p}f(x).
H^{\prime}=\Big(-{\frac{\partial}{\partial\bar{\phi}_{C}}}\tilde{\Omega}(C,C^{\prime}){\frac{\partial}{\partial{\phi}_{C^{\prime}}}}+{\frac{1}{4}}\omega(C)\tilde{\Omega}^{-1}(C,C^{\prime})\bar{\omega}(C^{\prime})\Big)+V
\partial_{t}^{2}\left(\begin{array}{c}{\varphi}\\{{\pi_{\chi}/\kappa}}\\\end{array}\right)=\left(\begin{array}{cc}{\nabla^{2}-\mu^{2}-{\frac{\kappa^{2}}{2}}}&{\kappa^{2}}\\{\mu^{2}+{\frac{\kappa^{2}}{4}}}&{\nabla^{2}-\mu^{2}-{\frac{\kappa^{2}}{2}}}\\\end{array}\right)\left(\begin{array}{c}{\varphi}\\{{\pi_{\chi}/\kappa}}\\\end{array}\right)
S_{\mathrm{total}}={\frac{\sqrt{3\pi}}{8}}\,\Bigl(1-{\frac{2}{\pi}}\Bigr){\frac{M_{\mathrm{P}}}{f_{0}}}={\frac{3\pi^{2}}{8}}\left(1-{\frac{2}{\pi}}\right)M_{\mathrm{P}}^{2}R^{2}(0)=1.34M_{\mathrm{P}}^{2}R^{2}(0)\.
{\bfX_{L}}:={\frac{\partial}{\partialt}}+\dot{q}^{s}{\frac{\partial}{\partialq^{s}}}+a^{s}(q,\dot{q}){\frac{\partial}{\partial\dot{q}^{s}}}+\eta^{\mu}{\bf\Gamma}_{\mu}=:{\bfX}_{0}+\eta^{\mu}{\bf\Gamma}_{\mu},
{\calL}=\partial_{+}\sigma\partial_{-}\sigma+\partial_{+}\pi\partial_{-}\pi-\frac{1}{2}(\partial_{\bot}\sigma)^{2}-\frac{1}{2}(\partial_{\bot}\pi)^{2}-\frac{1}{2}\mu^{2}(\sigma^{2}+\pi^{2})-\frac{\lambda}{4}(\sigma^{2}+\pi^{2})^{2},
\beta^{2}=\left(\begin{array}{ccccc}{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}\\{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{-1}&{0}&{0}\\\end{array}\right);\\beta^{3}=\left(\begin{array}{ccccc}{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}\\{0}&{0}&{0}&{-1}&{0}\\\end{array}\right).
[i\gamma_{\mu}\partial^{\mu},M^{kl}]=i\gamma_{\rho}g^{\rhol}\partial^{k}-i\gamma_{\rho}g^{\rhok}\partial^{l}+i\gamma_{\rho}Z_{\nu}^{l\rho}\partial^{\nu}\partial^{k}-i\gamma_{\rho}Z_{\nu}^{k\rho}\partial^{\nu}\partial^{l}
s_{2}[f;m^{2}]=-{\frac{1}{16\pim^{2}}}\,\int\!d^{2}x\,\phi^{2}(x)-{\frac{1}{96\pim^{4}}}\,\int\!d^{2}x\,\phi(x)\partial^{2}\phi(x)+{\calO}(\partial^{4}).
\begin{aligned}{\rho^{1}(J_{A^{\prime}})^{B}{}_{C^{\prime}}\,\rho^{(1,0)}\bigl((L_{\mathrm{I}+}^{\Lambda})^{A^{\prime}}{}_{A}\bigr)^{C^{\prime}}{}_{C}}&{{}=-q^{2}\varepsilon_{C}{}^{B}{}_{A}}\\{\rho^{1}(J_{A^{\prime}})^{B}{}_{C^{\prime}}\,\rho^{(0,1)}\bigl((L_{\mathrm{I}+}^{\Lambda})^{A^{\prime}}{}_{A}\bigr)^{C^{\prime}}{}_{C}}&{{}=-q^{-2}\varepsilon_{C}{}^{B}{}_{A}\,,}\\\end{aligned}
S={\frac{1}{2\kappa_{2}^{2}}}\intd^{2}x\sqrt{-\tilde{g}}\left[\Phi{\calR}_{\tilde{g}}+{\frac{1}{2}}\partial_{\mu}\Phi\partial^{\mu}\Phi+\Lambda\Phi^{1-{\frac{\gamma}{\delta^{2}}}}e^{\frac{\Phi}{2}}\right].
\left(\begin{array}{cccc}{\thetaV^{\prime\prime}(c_{0})+2}&{-2}&{}&{}\\{}&{\thetaV^{\prime\prime}(c_{1})+6-\frac{2^{2}}{\thetaV^{\prime\prime}(c_{0})+2}}&{-4}&{}\\{}&{}&{\thetaV^{\prime\prime}(c_{2})+10-\frac{4^{2}}{\thetaV^{\prime\prime}(c_{1})+6-\frac{2^{2}}{\thetaV^{\prime\prime}(c_{0})+2}}}&{-6}\\{}&{}&{}&{\ddots}\\\end{array}\right)
\Psi(x)\,=\,\sum_{\sigma=0,\pm1}\int{\frac{d^{3}p}{(2\pi)^{3}}}{\frac{1}{2\,\omega_{\vec{p}}}}\Big[u_{\sigma}(\vec{p}\,)\,a_{\sigma}(\vec{p}\,)\,e^{-ip\cdotx}+v_{\sigma}(\vec{p}\,)\,b_{\sigma}^{\dagger}(\vec{p}\,)\,e^{ip\cdotx}\Bigr]\quad,
\hat{G}_{c}(t,x)={\frac{|t|}{2L}}\Big[\theta(t)\sum_{n=1}^{\infty}exp\big({\frac{-i\pin}{L}}(t+x)\big)+\theta(-t)\sum_{n=1}^{\infty}exp\big({\frac{i\pin}{L}}(t+x)\big)\Big].
T(z)=\frac12J_{\alpha\beta}\mathopen:\psi^{\alpha}(z)\psi^{\beta}(z)\mathclose:,\qquad\bar{T}(\bar{z})=\frac12J_{\alpha\beta}\mathopen:\bar{\psi}^{\alpha}(\bar{z})\bar{\psi}^{\beta}(\bar{z})\mathclose:
\varphi_{0}\frac{f^{\prime}(\varphi_{0})-\frac{2-\eta}{2}}{f(\varphi_{0})-\frac{2-\eta}{2}\,\varphi_{0}}=\frac{d-2+\eta}{d+2-\eta},\qquadZ^{\prime}(\varphi_{0})=0.
\left[-\frac{d^{2}}{dz^{2}}+\frac{l(l+1)}{(z+R)^{2}}+m^{2}+\frac{\lambda}{2}\phi_{\mathrm{wall}}^{2}(z)+\frac{\sigma}{4!}\phi_{\mathrm{wall}}^{4}(z)\right]\chi_{n,l}(z)=E_{\mathrm{wall}}^{2}(n,l)\chi_{n,l}(z)\;.
\gamma_{\mu}=\left(\!\!\begin{array}{cc}{\sigma_{\mu}\!\!}&{\!\!0}\\{0\!\!}&{\!\!-\sigma_{\mu}}\\\end{array}\!\!\right)\;\\mu=1\sim3\,\\\gamma_{4}=\left(\!\!\begin{array}{cc}{0\!\!}&{\!\!{\bf1}}\\{{\bf1}\!\!}&{\!\!0}\\\end{array}\!\!\right)\,\\\gamma_{5}=\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}=\left(\!\!\begin{array}{cc}{0\!\!}&{\!\!i{\bf1}}\\{-i{\bf1}\!\!}&{\!\!0}\\\end{array}\!\!\right)\.
J^{(N)}(N-1;1,\ldots,1)=-\frac{1}{2\;D^{(N)}}\sum_{i=1}^{N}\frac{1}{m_{i}^{2}}\;F_{i}^{(N)}\;\left.J^{(N-1)}(N-1;1,\ldots,1)\right|_{\mathrm{\scriptsize~without}\;i}.
A_{d}(p^{2}+\omega_{n}^{2})^{\frac{1}{2}d-2}\left[\left(1+v^{2}\right)^{\frac{1}{2}d-\frac{3}{2}}+\frac{\Gamma(\frac{1}{2}d-\frac{1}{2})}{\sqrt{\pi}\Gamma(d)}\frac{(v^{2})^{\frac{d}{2}-1}}{1+v^{2}}{}_{2}F_{1}\left(\textstyle{{\frac{1}{2}d-\frac{1}{2},1;\frac{1}{2}d;\frac{v^{2}}{1+v^{2}}}}\right)\right]\,\\cdot
\frac{1}{2}{\calH}^{\mu}{\calV}_{\mu}=\frac{i}{2}(V_{j}z^{j}-V_{\,\bar{\!\jmath}}z^{\,\bar{\!\jmath}})\,\partial_{\mu}{\calH}^{\mu}=-\frac{i}{8}(V_{j}z^{j}-V_{\,\bar{\!\jmath}}z^{\,\bar{\!\jmath}})\,[F_{\mu\nu}^{a}\widetilde{F}_{a}^{\mu\nu}-2\partial_{\mu}(\lambda^{a}\sigma^{\mu}\bar{\lambda}_{a})]+\dots\,,
S_{R^{2}bulk}^{(1)}=\frac{1}{2k^{2}}\intd^{D}x\frac{\alpha^{\prime}}{4}e^{\gamma\Phi}\bigl(R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}-4R^{\mu\nu}R_{\mu\nu}+R^{2}\bigr)
[N_{i},p_{j}]=i\,\delta_{ij}\left({\frac{\kappa}{2}}\left(1-e^{-2{p_{0}/\kappa}}\right)+{\frac{1}{2\kappa}}\vec{p}\,{}^{2}\,\right)-i\,{\frac{1}{\kappa}}p_{i}p_{j},\quad[N_{i},p_{0}]=ip_{i},
\frac{R(1+{R^{\prime}}^{2})^{-3/2}}{P}=\frac{h^{3}R^{4}}{8}P^{2},\;\;(1+D^{2}{R^{\prime}}^{2})=\frac{1}{P^{2}}+\frac{D^{2}R_{+}^{2}}{R^{2}}
f_{TTT}=\frac{96i}{\pi}\frac{(\phi_{T}(T))^{2}}{\phi(T)(\phi(T)-\phi({i\sqrt{2}}))}\frac{\phi^{\frac{1}{2}}(U)}{\phi_{U}(U)}\frac{{(\phi(U)-\phi({i\sqrt{2}}))}^{\frac{1}{2}}}{(\phi(U)-\phi(T))},
\begin{array}{c}{q_{j}\mapstoq_{j}-\frac{n_{j}}{dimE}}\\{p^{j}\mapstop^{j}}\\{a({\bfk})\mapstoexp(i\tilde{\vartheta}(n_{2}k_{1}-n_{1}k_{2}))a({\bfk})}\\{a^{*}({\bfk})\mapstoexp(-i\tilde{\vartheta}(n_{2}k_{1}-n_{1}k_{2}))a^{*}({\bfk})}\\\end{array}
{\calA}_{\mathrm{eff}}^{\mathrm{mf}}\,=\,\frac{R\betaM^{2}}{2}\,\sqrt{\rho_{0}\rho_{1}}\,\left[\left(\frac{\lambda_{0}\,+\,1}{\rho_{0}}\,+\,\frac{\lambda_{1}\,+\,1}{\rho_{1}}\right)\,-\,\left(\lambda_{0}\,+\lambda_{1}\right)\right],
(\Psi\star\Phi)[\{l_{2k+1}\};\{r_{2k+1}\}]=\int\;\prod_{k=0}^{\infty}dy_{2j+1}\;\Psi[\{l_{2k+1}\};\{y_{2j+1}\}]\;\Phi[\{y_{2j+1}\};\{r_{2k+1}\}]\,.
R_{\vec{n}}=\left(\begin{matrix}{1}&{0}&{0}&{0}\\{0}&{1+\frac{(n^{1})^{2}}{n^{3}-1}}&{\frac{n^{1}n^{2}}{n^{3}-1}}&{n^{1}}\\{0}&{\frac{n^{1}n^{2}}{n^{3}-1}}&{1+\frac{(n^{2})^{2}}{n^{3}-1}}&{n^{2}}\\{0}&{n^{1}}&{n^{2}}&{n^{3}}\\\end{matrix}\right)
\begin{array}{c}{D_{TE}^{(-1)}(t)=\displaystyle{\frac{{n_{0}}^{2}\,R^{2}\,t\,\left(1-\frac{{n_{0}}^{2}\,R^{2}\,t^{2}}{4\,a^{2}}\right)}{2\,a^{2}\,{\left(1+\frac{{n_{0}}^{2}\,R^{2}\,t^{2}}{a^{2}}\right)}^{\frac{5}{2}}}-\frac{\left({n_{0}}^{2}+\frac{2}{t^{2}}\right)\,Z^{2}}{2\,t\,{\sqrt{1+{n_{0}}^{2}\,t^{2}}}}}}\\\end{array},
G_{l}(r,r^{\prime};t-t^{\prime})=i\sum_{n}{\frac{\omega_{ln}f_{l}^{<}(\omega_{ln}r)f_{l}^{<}(\omega_{ln}r^{\prime})}{2[{\frac{1}{2}}[r^{3}{\frac{d}{dr}}f_{l}^{<}(\omega_{ln}r)]^{2}|_{a}^{R}]}}e^{-i\omega|t-t^{\prime}|}
\Omega^{\prime}=\frac{\ell\tau}{\omega\kappaN_{\Phi}}\left(\begin{array}{c}{\left(\omega\kappa-1\right)\mathrm{~}-1\cdots-1}\\{\ddots}\\{-1\cdots-1\mathrm{~}\left(\omega\kappa-1\right)}\\\end{array}\right)
\left[\stackrel{(0)}{h}_{B},\stackrel{(1)}{\omega}\right]=-2C_{\;\;ab}^{c}k_{cd}\varepsilon^{0ij}A_{j}^{b}\left(\eta_{1}^{d}A_{i}^{a}+\eta_{2}^{d}\partial_{i}A_{0}^{a}\right)-2C_{\;\;ab}^{c}
T^{\mathrm{red}}=\pii\gamma\sum_{r=1}^{N}\left[\left(2R^{2}\frac{\bar{z}_{r}}{1+|z_{r}|^{2}}+\tilde{b}_{r}\right)\dot{z}_{r}-\left(2R^{2}\frac{{z}_{r}}{1+|z_{r}|^{2}}+\bar{\tilde{b}}_{r}\right)\dot{\bar{z}}_{r}\right].
{\hat{F}[L,~f]}~=~\alpha^{-\sigma[L,~f]}~\sum_{\{R_{i}\}}~~\mu_{R_{1}}~\mu_{R_{2}}~....~\mu_{R_{r}}~V[L;~n_{1},~n_{2},~...~n_{r};~R_{1},~R_{2},~....~R_{r}]
\cdot{\frac{\left|\left[\sum_{k=0}^{\infty}{\calG}_{k}\left({\frac{A}{\Lambda}}\right)^{-4k}\right]^{2}-\left({\frac{A}{\Lambda}}\right)^{-4}\right|}{\left|{\frac{2i}{\pi}}+\sum_{k=0}^{\infty}{\calF}_{k}4k(4k-2)\left({\frac{A}{\Lambda}}\right)^{-4k}\right|^{4}}}.
A^{\alpha}=\frac{c}{\epsilon^{3/2}}\partial^{\alpha}\phi^{\xi}(x^{\beta})~exp\left(-\frac{\xi^{2}}{\epsilon^{2}}\right)~,~~~~~A^{\xi}=-\frac{2c}{\epsilon^{7/2}}~\xi\phi^{\xi}(x^{\beta})~exp\left(-\frac{\xi^{2}}{\epsilon^{2}}\right)~,