harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(\stackrel{\alpha}{\omega}_{0,p}\!\!\lbrack\beta\rbrack\lhd\stackrel{P}{\calO}{}\!\!_{0,p}^{(\pm)})\stackrel{I}{e}_{a}\!\!(\alpha)=\stackrel{P}{\calF}{}\!\!_{0,p}^{(\pm)}\left(\begin{array}{c}{\!\!\alpha\!\!}\\{\!\!\beta\!\!}\\\end{array}\right)\stackrel{I}{e}_{a}\!\!(\alpha),
R_{\mu\nu}=2\nabla_{\mu}\nabla_{\nu}\phi-{\frac{1}{4}}H_{\mu\xi\rho}H_{nu}^{\xi\rho}+{\frac{1}{4}}e^{2\phi}\sum_{n}{\frac{(-1)^{n}}{(n-1)!}}T^{(n)}{}_{\mu\nu}\,,
\begin{array}{lcr}{\eta_{j}^{+}=u_{\alpha}^{+}\eta_{j}^{\alpha};\qquad\overline{{\eta}}_{j}^{+}=u_{\alpha}^{+}\overline{{\eta}}_{j}^{\alpha}}\\{\eta_{j}^{\alpha}\overline{{\eta}}_{\alphaj}=1;\qquad\eta_{j}^{\alpha}{\eta}_{\alphaj}=\overline{{\eta}}_{j}^{\alpha}\overline{{\eta}}_{\alphaj}=0.}\\\end{array}
\gamma_{1}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\,,\quad\gamma_{2}=\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right)\,,\quad\gamma_{5}=-i\gamma_{1}\gamma_{2}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)\,.
\begin{array}{l}{<1\;,\;T_{1}\,T_{2}\,...\,T_{k}>=I^{\otimesk}=<L_{1}^{\pm}\,L_{2}^{\pm}\,...\,L_{k}^{\pm}\;,\;1>\quad,}\\{<L_{1}^{\pm}\,L_{2}^{\pm}\,...\,L_{k}^{\pm}\;,\;T_{k+1}>=R_{1\,k+1}^{\pm}\,R_{2\,k+1}^{\pm}\,...\,R_{k\,k+1}^{\pm}\quad,}\\{<L_{1}^{\pm}\;,\;T_{2}\,T_{3}\,...\,T_{k+1}>=R_{12}^{\pm}\,R_{13}^{\pm}\,...\,R_{1\,k+1}^{\pm}\quad.}\\\end{array}
\fracd{dt}\left[W_{n}\frac{\overleftarrow{\delta}}{\delta\eta_{n+1}^{\downarrow}(t)}\right]=\left\langle[H_{n}(\bar{\eta},\eta,t),a_{n}^{\uparrow+}]\right\rangle_{t}=\left\langle-\fracnta_{n}^{\uparrow+}+ma_{n+1}^{\downarrow+}-t\bar{\eta}_{-n}^{\uparrow}\right\rangle_{t}.
=-{\frac{1}{2}}\epsilon^{\mu\nu\rho\lambda\sigma}\Lambda_{\rho}(\tilde{H}_{\mu\nu}g_{1}+(\tilde{H}^{3})_{\mu\nu}g_{2})(\tilde{H}_{\lambda\sigma}g_{1}+(\tilde{H}^{3})_{\lambda\sigma}g_{2})+\mathrm{tot.\,deriv.}
({\widehat\varphi}^{AB})_{~v}^{u}=f(x,\rho)\,\Big(\mu^{[Au}\bar{\mu}_{~\,v}^{B]}-\frac{1}{2}\,\mu^{[Ap}\,\bar{\mu}_{~\,p}^{B]}\,\tilde{\delta}_{~v}^{u}\Big)
{\calL}_{A}=-\frac{1}{4}F_{\mu\nu}^{a}F^{a\mu\nu}\\,\\F_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}-gf^{abc}A_{\mu}^{b}A_{\nu}^{c}\.
dx^{\mu_{1}}\wedgedx^{\mu_{2}}\wedge\cdots\wedgedx^{\mu_{r}}=\sum_{\mathrm{{\scriptsize~permutations}}\,P}\mathrm{sign}(P)dx^{\mu_{P(1)}}\otimesdx^{\mu_{P(2)}}\otimes\cdots\otimesdx^{\mu_{P(r)}},
\frac{1}{r}\frac{\partial}{\partialr}(r\frac{\partialh^{1}}{\partialr})+\frac{1}{r^{2}}\frac{\partial^{2}h^{1}}{\partial\theta^{2}}-\frac{4(p-1)ag^{2}}{r(1-g^{2})}\frac{\partialh^{1}}{\partialr}-g^{2}(2(2p-1)\eta^{2}+\frac{4(p-1)a^{2}}{r^{2}(1-g^{2})^{2}})h^{1}=0.
\widetilde{\bfH}_{i}^{(n)}\{g,{\boldsymbol\phi}(u)\}={\bfH}_{i}^{(i)}\{g^{-1},g^{-1}{\boldsymbol\phi}(u)\},\qquad\widetilde{\overline{{\bfH}}}_{i}^{(n)}\{g,{\boldsymbol\phi}(u)\}=\overline{{\bfH}}_{i}^{(n)}\{g^{-1},g^{-1}{\boldsymbol\phi}(u)\}.
{\frac{\calL}{\sqrt{g}}}={\frac{1}{2}}g^{\mu\nu}h_{ij}{\calD}_{\mu}q^{i}{\calD}_{\nu}q^{j}+{\frac{1}{4\beta}}F_{\mu\nu}^{a}F_{\mu\nu}^{a}+{\frac{\beta}{2}}{\calP}_{a}^{u}{\calP}_{a}^{u}.
\widehat{\calM}^{(B=0)}(\omega)=\widehat{V}^{(B=0)}(x,t)(\widehat{V}^{(B=0)})^{T}(x,{\frac{1}{t}})=\left(\begin{array}{cc}{{\calM}(\omega)}&{0}\\{0}&{{\calM}^{-1}(\omega)}\\\end{array}\right)
w_{1}={\frac{\partialW}{\partialJ_{1}}}={\frac{\theta}{2\pi}}-{\frac{J_{1}}{2\pi^{2}}}\int{\frac{dr}{r^{2}}}(2mE-{\frac{J_{1}^{2}}{4\pi^{2}r^{2}}}-{\frac{Q^{2}m}{\pi\epsilon_{0}}}ln{\frac{r}{r_{0}}})^{-{\frac{1}{2}}}~,
\lambda_{h2}=\lambda_{h2}^{(0)}+k^{-2}\lambda_{h2}^{(2)}g_{h}^{\mu\nu}\partial_{\mu}\partial_{\nu}+k^{-4}\lambda_{h2}^{(4)}g_{h}^{\mu\nu}g_{h}^{\rho\sigma}\partial_{\mu}\partial_{\nu}\partial_{\rho}\partial_{\sigma}+\ldots
\sigma_{aa}^{2}={\frac{1}{4}}\biggl\{\mu_{1}^{2}-\mu_{4}^{2}+(m_{0}\omega_{0})^{2}\mu_{3}^{2}-{\frac{\mu_{2}^{2}}{(m_{0}\omega_{0})^{2}}}-2i\biggl({\frac{1}{m_{0}\omega_{0}}}\mu_{1}\mu_{2}+m_{0}\omega_{0}\mu_{3}\mu_{4}\biggr)\biggr\}\,,
H_{0}=\frac{1}{2}p_{r}^{2}+\frac{1}{2}\frac{1}{r^{2}}p_{\theta}^{2}+\frac{1}{2}p_{z}^{2}+\frac{1}{2}\omega^{2}r^{2}+\xi\left[p_{z}+gp_{\theta}\right]\\\.
\bar{\Gamma}_{~\mu\nu}^{\sigma}\rightarrow^{\prime}\bar{\Gamma}_{~\mu\nu}^{\sigma}=\bar{\Gamma}_{~\mu\nu}^{\sigma}+\delta_{\mu}^{\sigma}\partial_{\nu}\Phi(x)+g^{\sigma\alpha}\partial_{[\alpha}\lambda_{\mu\nu]}(x)
\tilde{g}_{AB}=e^{\frac{2\kappa}{3}(\pm\phi+a)}\left(\begin{array}{cc}{g_{\mu\nu}}&{0}\\{0}&{e^{-2\kappa(\pm\phi+a)}}\\\end{array}\right)=\left(\begin{array}{cc}{e^{\frac{2\kappa}{3}(\pm\phi+a)}g_{\mu\nu}}&{0}\\{0}&{e^{-\frac{4\kappa}{3}(\pm\phi+a)}}\\\end{array}\right)\,,
c_{n}^{\prime}=\frac{x^{2n}(x^{2};x^{4})_{\infty}^{n}(x^{2};x^{4},x^{4})_{\infty}^{2n}}{(x^{2};x^{2})_{\infty}^{n^{2}}(x^{8};x^{4},x^{4})_{\infty}^{2n}}.
\begin{aligned}{j_{3}=}&{{}j_{21}^{+}-1-n-mb^{-2}}\\{j_{3}=}&{{}j_{21}^{-}-(n+1)-(m+1)b^{-2}}\\{j_{3}=}&{{}-j_{21}^{\pm}-(n+1)-(m+1)b^{-2}}\\\end{aligned}\quad\begin{aligned}{j_{3}=}&{{}j_{21}^{+}+n+(m+1)b^{-2}}\\{j_{3}=}&{{}j_{21}^{-}+n+mb^{-2}}\\{j_{3}=}&{{}-j_{21}^{\pm}+n+mb^{-2}}\\\end{aligned}\quadn,m=0,1,2,\ldots
\left.J^{(N)}(n;\nu_{1},\ldots,\nu_{N})=\int\frac{\mathrm{d}^{n}q}{\prod_{i=1}^{N}\left[q^{2}+2(p_{i}\cdotq)\right]^{\nu_{i}}}\right|_{(p_{j}\cdotp_{l})=m_{j}m_{l}c_{jl}}.
\lambda_{~A}^{a}\rightarrow-{\frac{\partial}{\partial\bar{\lambda}^{(a)A}}}~,~\bar{\psi}_{A}\rightarrow{\frac{\partial}{\partial\psi^{A}}}~,~\pi_{a}\rightarrow{\frac{\partial}{\partiala}},~\pi_{f}\rightarrow-i{\frac{\partial}{\partialf}}~.
Z(\alpha)=\int{\calD}A_{\mu}^{\omega}{\calD}B{\calD}\bar{c}{\calD}ce^{-S_{0}(A^{\omega})-\frac{\alpha}{2}\intB^{2}dx+\int[-iB\frac{1}{\sqrt{-\partial_{\mu}\partial_{\mu}}}\partial_{\mu}A_{\mu}^{\omega}+\bar{c}\sqrt{-\partial_{\mu}\partial_{\mu}}c]dx}
S=-\frac{1}{g_{0}^{2}}\left(\frac{1}{2\alpha^{\prime}}\left<I\circ\Phi(0)Q\Phi(0)\right>+\frac{1}{3}\left<f_{1}\circ\Phi(0)f_{2}\circ\Phi(0)f_{3}\circ\Phi(0)\right>\right)\,
P^{-}=\frac{1}{2p^{+}}(p^{i})^{2}+\frac{1}{2p^{+}}\sum_{n=1}^{\infty}\left(\alpha_{-n}^{i}\alpha_{n}^{i}+\tilde{\alpha}_{-n}^{i}\tilde{\alpha}_{n}^{i}+n\,S_{-n}^{a}S_{n}^{a}+n\,\tilde{S}_{-n}^{\dot{a}}\tilde{S}_{-n}^{\dot{a}}\right)\nonumber
s^{1}:=\frac{2y^{1}}{1-r^{2}},\quads^{2}:=\frac{2y^{2}}{1-r^{2}},\quads^{3}:=\frac{1+r^{2}}{1-r^{2}},\quadr^{2}:=(y^{1})^{2}+(y^{2})^{2}.
J_{a}\propto\left|\begin{array}{ccccc}{m_{e}}&{m_{d}}&{m_{c}}&{m_{b}}&{m_{a}}\\{\epsilon_{e}-\epsilon_{a}}&{v_{de}}&{v_{ce}}&{0}&{0}\\{0}&{\epsilon_{d}-\epsilon_{a}}&{0}&{0}&{v_{ad}}\\{0}&{0}&{\epsilon_{c}-\epsilon_{a}}&{v_{bc}}&{v_{ac}}\\{0}&{0}&{0}&{\epsilon_{c}-\epsilon_{a}}&{v_{ab}}\\\end{array}\right|\,.
B_{1}(t,q)=\sum_{n=1}^{\infty}t^{n}\sum_{l=1}^{n}\frac{(-1)^{l}}{l!}\frac{l}{2}f^{l-1}(q)f^{(1)}(q)\beta_{nl}\prod_{j=1}^{l}\left(\frac{\lambda(\lambda-1)}{2}-\frac{j(j-1)}{2}\right),
\begin{array}{c}{\Omega=i\partial\bar{\partial}\left(K(z,\bar{z})-ig_{A\bar{B}}\theta^{A}{\bar{\theta}}^{\bar{B}}\right)=}\\{=i(g_{A\bar{B}}+iR_{A\bar{B}C\bar{D}}\theta^{C}\bar{\theta}^{\bar{D}})dz^{A}\wedged{\bar{z}}^{\bar{B}}}\\{+g_{A\bar{B}}D\theta^{A}\wedgeD{\bar{\theta}}^{\bar{B}},}\\\end{array}
V_{q}(z,\bar{z})=\left(\frac{\sqrt{\frac{z}{\bar{z}}}+2+\sqrt{\frac{\bar{z}}{z}}}{4|z-\bar{z}|}\right)^{q^{2}}\,:e^{iq\Phi(z,\bar{z})}:\quad;\quadq=\frac{n}{r}\,
ds_{t\to\infty}^{2}\sim\mathrm{e}^{-\frac{2pf_{1}}{q-1}}\mathrm{e}^{\frac{2p}{q-1}(\alpha+\frac{ac_{1}}{\chi})t}\left(-2^{\frac{2k}{k-1}}\mathrm{e}^{-kt}dt^{2}+2^{\frac2{k-1}}\mathrm{e}^{-t}d\Sigma_{k,-1}^{2}+d\vec{y}_{q-k}^{2}\right)+\mathrm{e}^{2f_{1}}\mathrm{e}^{-2(\alpha+\frac{ac_{1}}{\chi})t}d\vec{x}_{p}^{2},
\Lambda[\xi]^{\mu_{2}\ldots\mu_{s}}:=\intd^{d-1}l\;\xi(x)^{i_{2}\ldotsi_{s}}{\frac{1}{(v\cdotl)^{s+d-2}}}(\hat{\delta}_{i_{2}}^{\mu_{2}}\cdots\hat{\delta}_{i_{s}}^{\mu_{s}}-\mathrm{traces}_{(i)})
\delta{\calL}_{0}=\delta\!\left(F_{\mu\nu}^{k}\,F_{\mu\nu}^{k}\right)=\frac{\delta{\calL}_{0}}{\deltaA_{\alpha}^{l}}\,\deltaA_{\beta}^{l}+\frac{\delta{\calL}_{0}}{\delta\!\left({\partial}_{\beta}A_{\alpha}^{l}\right)}\,\delta\!\left(\partial_{\beta}A_{\alpha}^{l}\right),
\begin{array}{rcl}{(b)_{\sigma}}&{=}&{\displaystyle-\frac{3}{2}\lambdag^{2}v\intO_{k}(k+2p_{+})\cdot(k-2p_{-})D_{k}^{W}D_{k+p_{+}}^{\sigma}D_{k-p_{-}}^{\sigma}.}\\\end{array}
\frac{\delta^{2}\Gamma_{k}[\phi]}{\delta\phi\delta\phi}=\frac{\delta^{2}\widehat{\Gamma}_{k}[\phi,\varphi]}{\delta\phi\delta\phi}{\Big|}_{\varphi=\phi}+2\frac{\delta^{2}\widehat{\Gamma}_{k}[\phi,\varphi]}{\delta\phi\delta\varphi}{\Big|}_{\varphi=\phi}+\frac{\delta^{2}\widehat{\Gamma}_{k}[\phi,\varphi]}{\delta\varphi\delta\varphi}{\Big|}_{\varphi=\phi}.
\begin{array}{lll}{{\widetildeT}_{\alpha{\dot{\alpha}}}^{\phi}}&{=}&{\frac{1}{12}{\nabla}_{\alpha}\Phi{\bar{\nabla}}_{\dot{\alpha}}\bar{\Phi}-\frac{i}{6}\bar{\Phi}\stackrel{\leftrightarrow}{{\nabla}}_{\alpha{\dot{\alpha}}}\Phi\,}\\{{\widetildeJ}^{\phi}}&{=}&{\frac{1}{4}(q-1){\bar{\nabla}}^{2}\left(\bar{\Phi}\Phi\right)\.}\\\end{array}
\widetilde{M}(j_{p},j_{q};\vec{k},\lambda)=-i\langlej_{q},j_{p}|S^{(1)}|\vec{k},\lambda\rangle=-e\int{d^{4}x}\bar{\psi}_{e}(j_{p},x)\;A_{\mu}^{\lambda}(\vec{k},x)\gamma_{\mu}\;\psi_{p}^{c}(j_{q},x)
\mathrm{I=\frac{2\Gamma(4-\omega-\sigma)}{(4\pi)^{2}}\int_{0}^{1}dxx^{-\sigma}\int_{0}^{x}dG\int_{G}^{1-x+G}dy~[~H_{0}^{(1)}~]^{\omega+\sigma-4}+F.T.}
\Lambda_{1}^{M}\,t^{3}+\left(P_{1}(z;u_{n}^{(1)},\Lambda_{1}^{M})+\Lambda_{1}^{M}\right)t^{2}+\left(P_{2}(z;u_{n}^{(2)},\Lambda_{2}^{M})+\Lambda_{2}^{M}\right)t+\Lambda_{2}^{M},
ds^{2}=(1-\frac{r_{+}}{r})(1-\frac{r_{-}}{r})^{\sigma}dt^{2}-(1-\frac{r_{+}}{r})^{-1}(1-\frac{r_{-}}{r})^{-\sigma}dr^{2}-(1-\frac{r_{-}}{r})^{1-\sigma}r^{2}(d\theta^{2}+sin^{2}\thetad\phi^{2})
\{\frac{-l^{2}M^{2}}{2}+N^{(\alpha)}+N^{(d)}+\frac{1}{2}\sum_{i=1}^{D-1}\sum_{m=1,odd}^{\infty}m\delta^{ii}-\frac{1}{2}\sum_{i=1}^{D-1}\sum_{k=1,even}^{\infty}k\delta^{ii}\}\mid\Psi_{phys}\rangle=0
A_{\lambda}^{(1)}(x|z_{1},\cdots,z_{n})=\sum_{\kappa=0}^{\lambda}(-1)^{\kappa}x^{\lambda-\kappa}(\tau^{n(\lambda-\kappa)+\kappa}-\tau^{-n(\lambda-\kappa)-\kappa})\sigma_{\kappa}(z_{1},\cdots,z_{n}).
\begin{array}{ll}{{[}W_{0}^{a},W_{0}^{b}]={f^{ab}}_{c}W_{0}^{c}}&{a=1,2,\dots,n^{2}-1}\\{{[}W_{0}^{a},W_{k}^{b}]={f^{ab}}_{c}W_{k}^{c}}&{k=1,2,\dots,p-1}\\{{[}C_{i},W_{0}^{a}]=[C_{i},W_{k}^{a}]=0\\\\\\}&{i=2,3,\dots,p}\\\end{array}
\hat{p}^{\mu}=\left(\begin{array}{c}{\hat{E}=\frac{p^{0}}{w}=\sqrt{1+\hat{\bboxp}^{2}}}\\{\hat{\bboxp}=\frac{\bboxp}{w}}\\\end{array}\right)\,\,\,\,\,\,\mathrm{and}\,\,\,\,w^{2}={\mathsf{s}}.
L_{eff}=-\Biggl(\frac{1}{2}\gamma_{~b\mu}^{a}F_{a~~m}^{~b\mu~n\nu}\gamma_{~n\nu}^{m}+\frac{1}{2}\lambda_{~\nu}^{b}D_{b~a}^{~\nu~\mu}\lambda_{~\mu}^{a}+\lambda_{~n}^{m}\biggl(G_{m~~a}^{~~nj~bc}\nabla_{j}+T_{m~a}^{~n~bc}\biggr)\gamma_{~bc}^{a}\Biggr)e
\mu_{k}^{\prime\prime}+\biggl\{k^{2}\biggl[\frac{\epsilon_{1}}{1+\mathrm{e}^{x}}+\frac{\epsilon_{3}\mathrm{e}^{x}}{(1+\mathrm{e}^{x})^{2}}\biggr]-(1-6\xi)\frac{\beta(\beta+1)}{\eta^{2}}\biggr\}\mu_{k}=0\,,
\frac{1}{8\piG}(R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu})=T_{\mu\nu}+\langleT_{\mu\nu}\rangle_{\lambda=0}^{G}+\langleT_{\mu\nu}\rangle_{\lambda=0}^{M}+\langleT_{\mu\nu}\rangle_{a(\lambda)}+\langleT_{\mu\nu}\rangle_{b(\lambda)},
(A,B)=\sum_{-\infty}^{+\infty}A_{n}B_{-n}=A_{0}B_{0}+\sum_{1}^{\infty}a_{n}^{\prime}b_{n}^{\prime}+a_{n}^{\prime\prime}b_{n}^{\prime\prime}\,;\;a_{n}^{\prime}=\sqrt{2}\ReA_{n},a_{n}^{\prime\prime}=\sqrt{2}\ImA_{n}
\begin{array}{c}{|\xi\wedge\eta|^{2}=2|P_{1}(\xi\wedge\eta)|^{2}=-{\frac{1}{4}}\left(\xi\wedge\eta,\sum_{a=1}^{3}I_{a}^{2}(\xi\wedge\eta)\right)=}\\{={\frac{3}{2}}|\xi\wedge\eta|^{2}-{\frac{1}{2}}\left(\xi\wedge\eta,\sum_{a=1}^{3}I_{a}\xi\wedgeI_{a}\eta\right)=}\\{={\frac{3}{2}}|\xi\wedge\eta|^{2}+{\frac{1}{2}}(\xi,I_{a}\eta)(\eta,I_{a}\xi)}\\\end{array}
\partial_{({\alpha_{1}\ldots\alpha_{k}}}B_{[\mu_{1}{)}_{1}\ldots\mu_{l}{]}_{2}}^{a(l,p_{a}-l)}\mathrm{\\\and~\\\}\partial_{\alpha_{2}\ldots\alpha_{k}}H_{\alpha_{1}\mu_{1}\ldots\mu_{l}}^{a(l+1,p_{a}-l)},
\begin{array}{l}{R^{(1)}K_{1}R^{(2)}K_{2}=R^{(1)}X_{1}Z_{1}^{\dagger}R^{(2)}X_{2}Z_{2}^{\dagger}=R^{(1)}X_{1}X_{2}Z_{1}^{\dagger}Z_{2}^{\dagger}}\\{\qquad=(\kappa_{1}/\kappa_{2})X_{2}X_{1}Z_{2}^{\dagger}Z_{1}^{\dagger}R^{(4)}=(\kappa_{1}/\kappa_{2})X_{2}Z_{2}^{\dagger}R^{(3)}X_{1}Z_{1}^{\dagger}R^{(4)}\quad.}\\\end{array}
Q_{2}^{1}=r_{0}(-2r^{2}\hat{r}_{2}+6r^{2}\hat{r}_{6}+8r\hat{r}_{2}\hat{r}_{6}+3\hat{r}_{2}^{2}\hat{r}_{6}+\hat{r}_{2}\hat{r}_{6}^{2}),
H^{\prime}=-i\alpha^{j}\frac{\partial}{\partialu^{j}}+\betamR-i\alpha^{3}\Bigl(\left(1+R\vec{\kappa}\cdot\vec{u}\right)R\frac{\partial}{\partials}+\frac{R}{2}\frac{\partial\left(R\vec{\kappa}\cdot\vec{u}\right)}{\partials}\Bigr)+{\calO}({\delta}^{3})\.
\int\frac{dk_{\tau}}{2\pi}\rightarrow\frac{1}{\beta}\sum_{n_{1}=-\infty}^{+\infty},\;\;\int\frac{dk_{x}}{2\pi}\rightarrow\frac{1}{L}\sum_{n_{2}=-\infty}^{+\infty}\;\;\;\;;\;\;\;k_{\tau}\rightarrow\frac{2n_{1}\pi}{\beta}\;,\;\;k_{x}\rightarrow\frac{2n_{2}\pi}{L}.
14-\intd^{4}x\left\{Ng^{-1/2}g_{ij}\left[(eE_{T}^{i}+\muB_{T}^{i})V^{j}+\textstyle{\frac{1}{2}}(e^{2}+\mu^{2})V^{i}V^{j}\right]-\epsilon_{ijk}N^{i}V^{j}(eB_{T}^{k}-\muE_{T}^{k})\right\}\;.
\int_{0}^{\infty}dw\,(wz)^{1/2}\,w^{\sigma-1}\,K_{\mu}\left(a/w\right)\,K_{\nu}(wz)=2^{-\sigma-5/2}a^{\sigma}\,G_{04}^{40}\left(\left.\frac{a^{2}z^{2}}{16}\right|\frac{\mu-\sigma}{2}\,,\frac{-\mu-\sigma}{2},\frac{1}{4}+\frac{\nu}{2}\,,\frac{1}{4}-\frac{\nu}{2}\right).
(f\starg)(x)=\int\frac{d^{4}k}{(2\pi)^{4}}\int\frac{d^{4}p}{(2\pi)^{4}}\,\mathrm{e}^{-\mathrm{i}(k_{\mu}+p_{\mu})x^{\mu}}\,\mathrm{e}^{-\frac{\mathrm{i}}{2}\theta^{\mu\nu}k_{\mu}p_{\nu}}\tilde{f}(k)\,\tilde{g}(p)\;,
D(\Phi^{\prime\prime}+{\frac{2r-r_{0}}{r^{2}K}}\Phi^{\prime}+{\frac{Z_{2}Z_{6}}{K^{2}}}\omega^{2}\Phi)+P_{2}^{1}(\Psi_{2}^{\prime}+2{\calZ}_{6}^{\prime})+Q_{1}^{1}\Phi+Q_{2}^{1}\Psi_{2}+Q_{3}^{1}{\calZ}_{6}=0,
R_{\\\mu\nu\kappa}^{\lambda}=\frac{1}{2}\left[\partial_{\mu}\left(\partial_{\kappa}h_{\nu}^{\lambda}-\partial_{\nu}h_{\kappa}^{\lambda}\right)-\partial^{\lambda}\left(\partial_{\kappa}h_{\mu\nu}-\partial_{\nu}h_{\mu\kappa}\right)\right]\,
({\calV}^{(n)}(u){\calV}^{(n)}(u+v))_{\alpha,\alpha^{\prime}}=\mathrm{Tr}X^{\prime}(\alpha_{1},\alpha_{1}^{\prime})X^{\prime}(\alpha_{2},\alpha_{2}^{\prime})\cdotsX^{\prime}(\alpha_{n},\alpha_{n}^{\prime})~.
[{{P^{a}}_{i}}^{j}\,\Q_{\alpha}]_{DB}=-i\sqrt{R}\left(2{[X_{d}\,,\,\theta^{\alpha^{\prime}}]_{i}}^{j}(\gamma^{ab})_{\alpha\alpha^{\prime}}+\mathrm{Tr}[\{D({{}^{j}}_{i})\,,\,\theta^{\alpha^{\prime}}\}\,,\,X_{d}](\gamma^{ad})_{\alpha\alpha^{\prime}}\right),
I_{s}={\frac{1}{4\epsilon^{2}}}{\frac{\Gamma(1+2\epsilon)\Gamma(1-\epsilon)^{3}}{(1-3\epsilon)(1-2\epsilon)\Gamma(1-3\epsilon)}}={\frac{1}{2}}\left[{\frac{1}{2\epsilon^{2}}}+{\frac{1}{2\epsilon}}+{\frac{1}{\epsilon}}(2-C)\right]
\Gamma_{\mu_{1}\mu_{2}\mu_{3}}^{\mathrm{ln}}=-\left(\frac{17}{6}-\frac{3\xi}{2}\right)\frac{g^{2}C_{G}}{\left(4\pi\right)^{2}}\mathrm{ln}\left(T\right)i\left(\eta_{\mu_{1}\mu_{2}}\left(k_{1}-k_{2}\right)_{\mu_{3}}+\eta_{\mu_{2}\mu_{3}}\left(k_{2}-k_{3}\right)_{\mu_{1}}+\eta_{\mu_{3}\mu_{1}}\left(k_{3}-k_{1}\right)_{\mu_{2}}\right).
S_{M}=-M\intd\tau\left[1-\frac{\kappa}{2}s_{\mu\nu}{\dot{x}}^{\mu}{\dot{x}}^{\nu}-\frac{\kappa^{2}}{8}s_{\mu\nu}s_{\rho\sigma}{\dot{x}}^{\mu}{\dot{x}}^{\nu}{\dot{x}}^{\rho}{\dot{x}}^{\sigma}+\ldots\right],
\left(\begin{array}{c}{{\calQ}_{d}^{\prime}}\\{{\calQ}_{a}^{\prime}}\\\end{array}\right)=\left(\begin{array}{cc}{\frac{d^{2}-c^{2}}{\sqrt{c^{2}+d^{2}}}}&{\frac{2cd}{\sqrt{c^{2}+d^{2}}}}\\{}&{}\\{\frac{-2cd}{\sqrt{c^{2}+d^{2}}}}&{\frac{d^{2}-c^{2}}{\sqrt{c^{2}+d^{2}}}}\\\end{array}\right)\left(\begin{array}{c}{{\calQ}_{d}}\\{{\calQ}_{a}}\\\end{array}\right)\,,
x=\left(\begin{array}{cc}{\bar{u}_{2}}&{u_{1}}\\{-\bar{u}_{1}}&{u_{2}}\\\end{array}\right),~~~~~~\bar{x}=\left(\begin{array}{cc}{u_{2}}&{-u_{1}}\\{\bar{u}_{1}}&{\bar{u}_{2}}\\\end{array}\right)
\mathrm{P}{\int}_{0}^{\infty}dk_{+}\left(\frac{1}{(k_{+}-a)^{2}}+\frac{1}{(k_{+}+a)^{2}}\right)=\mathrm{P}{\int}_{-\infty}^{\infty}dk_{+}\frac{1}{(k_{+}-a)^{2}}.
{\frac{1}{A}}{\frac{\partial{\calE}}{\partialm^{2}}}={\frac{1}{2}}\sum_{n=1}^{\infty}\int{\frac{dk_{1}dk_{2}dk_{0}}{(2\pi)^{3}}}\int_{0}^{\infty}dse^{-is(-k_{0}^{2}+k_{1}^{2}+k_{2}^{2}+{\frac{n^{2}\pi^{2}}{a^{2}}}-i\epsilon+m^{2})}\,.
H=\frac{P^{2}}{2\Omega}+\OmegaV(\varphi)+\int_{\Omega}d^{3}x\left\{\frac{1}{2}\pi^{2}(x)+\frac{1}{2}(\vec{\nabla}\psi(x))^{2}+\frac{1}{2}V^{\prime\prime}(\varphi)\psi^{2}(x)+\cdots\right\}
I_{3}=-{\frac{\lambda_{0}^{2}}{2\pi}}\sqrt{\frac{4m_{r}^{2}-k^{2}}{-k^{2}}}log\big[{\frac{\sqrt{4m_{r}^{2}-k^{2}}-\sqrt{-k^{2}}}{\sqrt{4m_{r}^{2}-k^{2}}+\sqrt{-k^{2}}}}\big]-1+{\frac{\lambda_{0}^{2}}{\pi}}
\begin{array}{l}{e^{+2}\nabla\psi_{+2\dot{q}}^{1-}+e^{-2}\nabla\psi_{-2\dot{q}}^{1-}+e_{p}^{+}\nablah_{+p\dot{q}}^{1-}+e_{\dot{p}}^{-}\nabla\chi_{\dot{p}\dot{q}}^{1}+T^{+2}\psi_{+2\dot{q}}^{1-}+T^{-2}\psi_{-2\dot{q}}^{1-}+T_{p}^{+}h_{+p\dot{q}}^{1-}+T_{\dot{p}}^{-}\chi_{\dot{p}\dot{q}}^{1}}\\{=-\frac12\tilde{\gamma}_{\dot{q}q}^{i}\Omega^{-2i}e_{q}^{+},}\\\end{array}
(r_{b}|r_{a})_{E,l}=\frac{\hbar}{Mc}\sum_{n=l+1}^{\infty}\left(\frac{E}{Mc^{2}}\right)^{2}\frac{2Mc^{2}i\hbar}{E^{2}-E_{nl}^{2}}R_{nl}(r_{b})R_{nl}(r_{a})+\dots~.
\begin{aligned}{F_{inst}(g_{s},t_{1},t_{2},t_{c},}&{{}U_{1},U_{2},V_{1},V_{2})}\\{}&{{}=\sum_{n=1}^{\infty}\left[ia_{n}e^{-nt_{1}}+ib_{n}e^{-nt_{2}}-c_{n}e^{-nt_{c}}+2d_{n}e^{-n(t_{1}+t_{2})}\right]}\\\end{aligned}
T_{\rho\mu}(\phi(\hat{x})):={\frac{1}{2}}\big(\hat{\partial}_{\rho}\phi(\hat{x})\hat{\partial}_{\mu}\phi(\hat{x})+\hat{\partial}_{\mu}\phi(\hat{x})\hat{\partial}_{\rho}\phi(\hat{x})\big)-\delta_{\rho\mu}\bar{\mathcal{L}}^{(0)}.
\left(\begin{array}{ccc}{A_{3}+A_{8}/\sqrt{3}}&{R\bar{G}}&{R\bar{B}}\\{G\bar{R}}&{-A_{3}+A_{8}/\sqrt{3}}&{G\bar{B}}\\{B\bar{R}}&{B\bar{G}}&{-2A_{8}/\sqrt{3}}\\\end{array}\right)
{\calT}_{\mu\nu}^{\alpha\beta}\sim\epsilon_{~\mu}^{\alpha~\rho\sigma}\epsilon_{~\nu}^{\beta~\lambda\gamma}\big(AF_{\rho\sigma}F_{\lambda\gamma}+BF_{\rho\sigma}\tilde{F}_{\lambda\gamma}+C\tilde{F}_{\rho\sigma}\tilde{F}_{\lambda\gamma}\big)\,,
\overline{{\Theta}}_{\mu\nu}^{ab}=-g_{\mu\nu}\left[\left(\nabla_{\lambda}-ig(n^{c}T^{c})\overline{{A}}_{\lambda}\right)^{2}\right]^{ab}+ig(n^{c}T^{c})^{ab}\overline{{F}}_{\mu\nu}-\delta^{ab}R_{\mu\nu}.
\left({\frac{\dot{\hat{a}_{0}}}{\hat{a}_{0}}}+{\frac{\dot{\hat{c}}}{\hat{c}}}\right)^{2}={\frac{8\pi\hat{G}_{4}}{3}}\hat{\varrho}+{\frac{\hat{c}^{2}}{3}}\hat{\Lambda}_{\mathrm{eff}}+{\frac{\hat{\calC}\hat{c}^{2}}{\hat{a}_{0}^{4}}}-{\frac{\hat{k}\hat{c}^{2}}{\hat{a}_{0}^{2}}},
\begin{array}{rcl}{\{\Gamma_{0}^{A},\Gamma_{0}^{B}\}}&{=}&{0}\\{\{\Gamma_{0}^{A},\Gamma_{1}^{B}\}}&{=}&{E^{AB}(\Gamma_{0})}\\{\{\Gamma_{1}^{A},\Gamma_{1}^{B}\}}&{=}&{\Gamma_{1}^{C}\partial_{C}E^{AB}(\Gamma_{0})\,.}\\\end{array}
Z_{m}(A|J)=\mathrm{exp}\Bigr\{\!-(i/\hbar)z_{Q}^{-1}\bigr(J_{i}A^{i}-\mathrm{~\frac{1}{2}~}m^{2}g_{ij}A^{i}A^{j}\bigr)\Bigr\}\,\mathrm{exp}\Bigr\{-m^{2}g_{ij}A^{i}\frac{\delta}{\deltaJ_{j}}\Bigr\}Z_{m}(0|J).
{\calL}=-\frac{1}{4}\left\{\phantom{|}_{abcd}^{\mu\nu\alpha\beta}\right\}[3\Omega_{\mu}{}^{aef}\Omega_{\nu}{}^{bef}\delta_{\alpha}{}^{c}\delta_{\beta}{}^{d}-\frac{1}{3}\Omega_{\mu}{}^{abc}T_{\nu\alpha\beta}{}^{d}]
F(x^{+},x^{-},0)_{F}=\frac{i}{4(2\pi)^{3}}m^{5}\left(\frac{x^{+}}{x^{-}}\right)^{2}\frac{1}{x}\left[K_{7/2}(mx)K_{3/2}(mx)-K_{5/2}^{2}(mx)\right]\,.
\bar{D}^{\dot{\alpha}}J_{\alpha\dot{\alpha}}^{0}=\frac{1}{3}D_{\alpha}\left\{\left[3{\calW}-\sum_{i}S_{i}\,\frac{\partial{\calW}}{\partialS_{i}}\right]-\left[\frac{b}{16\pi^{2}}\mathrm{Tr}\,W^{2}+\frac{1}{8}\sum_{i}\gamma_{i}Z_{i}\bar{D}^{2}\left({S_{i}}^{\dagger}e^{V}S_{i}\right)\right]\right\}
W^{(1)}=-ig^{2}\,t^{a}t^{a}\int\frac{d^{d}q}{(2\pi)^{d}}\frac{v_{\mu}{\mit\Pi}_{R}^{\mu\nu}(q)v_{\nu}^{\prime}}{q^{4}(q\cdotv)(q\cdotv^{\prime})}\,.
{J^{\prime}}_{1}=\left(\begin{array}{ccc}{0}&{0}&{D^{3}}\\{0}&{D^{3}}&{-D^{2}a\,D}\\{D^{3}}&{-D\,a\,D^{2}}&{{\begin{array}{c}{D^{2}b\,D+D\,b\,D^{2}}\\{+D\,a\,D\,a\,D}\\\end{array}}}\\\end{array}\right)\,,
\int_{Y_{4}}F_{4}\wedge\starF_{4}=\int_{Y_{4}}F_{4}\wedgeF_{4}-4\int_{Y_{4}}F_{3,1}\wedgeF_{1,3}-2\int_{Y_{4}}J\wedgeF_{1,1}^{(0)}\wedgeJ\wedgeF_{1,1}^{(0)}\,
H_{\mathrm{eff}}=-K\sum_{[x,i]}\psi_{A}^{\dagger}(x+\hat{i})\psi_{B}(x)\psi_{B}^{\dagger}(x)\psi_{A}(x+\hat{i})=K\sum_{[x,i]}\rho(x+\hat{i})\rho(x)
f_{\nu}(s)=\sum_{l=1}^{\infty}\\left[(\j_{\nu,l})^{-s}-(l\pi)^{-s}+s\left(\frac{2\nu-1}4\right)\pi(l\pi)^{-s-1}\right]+\\pi^{-s}\left[\zeta(s)-s\left(\frac{2\nu-1}4\right)\zeta(s+1)\right],
\begin{aligned}{{}\big[M^{(8)},Z^{(4|2)}\big]=\mu\,Z^{(4|2)}\,,\quad\quad}&{{}\big[W^{(8)},Z^{(4|2)}\big]=\tilde{\mu}\big[M^{(8)},Z^{(4|2)}\big]\,,}\\{{}\big[M^{(5)},Z^{(4|2)}\big]=\nu\,Z^{(4|2)}\,,\quad\quad}&{{}\big[W^{(5)},Z^{(4|2)}\big]=\tilde{\nu}\big[M^{(5)},Z^{(4|2)}\big]\,.}\\\end{aligned}
({\widehatA}_{\mu}(x))_{~v}^{u}=w_{\dot{\alpha}}^{~u}\,(\bar{\sigma}_{\nu\mu})_{~\dot{\beta}}^{\dot{\alpha}}\,{\bar{w}}_{~v}^{\dot{\beta}}\,\frac{(x-x_{0})^{\nu}}{(x-x_{0})^{2}\Big[(x-x_{0})^{2}+\rho^{2}\Big]}~~.
\begin{array}{l}{|A_{1}|^{2}+|A_{2}|^{2}=|B_{1}|^{2}+|B_{2}|^{2}}\\{|B_{1}|^{2}+|B_{2}|^{2}=|C_{1}|^{2}+|C_{2}|^{2}}\\\end{array}
\mu_{\mathbf{R}}^{1}=\mu_{\mathbf{C}}^{1}=\mu_{\mathbf{C}}^{2}=\mu_{\mathbf{C}}^{2\,\,\,\prime}=\mu_{\mathbf{R}}^{3}=\mu_{\mathbf{C}}^{3}=0\,.
g_{ik}^{(1)}=\eta_{mn}\frac{\partial\xi^{m}}{\partial\tau^{i}}\frac{\partial\xi^{n}}{\partial\tau^{k}}-h_{\alpha\beta}(\xi)\frac{\partial\xi^{\alpha}}{\partial\tau^{i}}\frac{\partial\xi^{\beta}}{\partial\tau^{k}}=g_{ik}^{(2)}\equivg_{ik}\;\;,
S=\frac{1}{2\pi\alpha^{\prime}}\intd\sigma^{+}d\sigma^{-}\left(\partial_{-}X^{\mu}\partial_{+}X_{\mu}+\frac{i}{2}\psi_{-}^{\mu}\partial_{+}\psi_{-\mu}+\frac{i}{2}\sum_{A=1}^{N}\lambda_{+}^{A}\partial_{-}\lambda_{+A}\right).
\begin{aligned}{\mathcal{L}_{\mathrm{IIB}}}&{{}=\frac{1}{2}\sqrt{-\hat{g}}\,[-\hat{R}-\frac{1}{2}(\partial\hat{\phi})^{2}-\frac{1}{2}e^{2\hat{\phi}}(\partial\hat{\chi})^{2}]}\\{}&{{}=\frac{1}{2}\sqrt{-\hat{g}}\,[-\hat{R}+\frac{1}{4}\mathrm{Tr}(\partial\hat{\mathcal{M}}\partial\hat{\mathcal{M}}^{-1})]\,.}\\\end{aligned}
3\mathrm{Tr}(\phi_{SS/SS}(\phi_{M/SS}\phi_{SS/N}+\phi_{SS/M}\phi_{N/SS}))=-\frac{3ig}{8\sqrt{2}\pi^{2}}\frac{\mathrm{Tr}(\phi_{SS/SS}^{2})}{y^{2}+\rho^{2}}\sum_{k=3}^{N_{c}}(\xi_{Mk}\zeta_{Nk}+\xi_{Nk}\zeta_{Mk}).
\intd^{d}x\sqrt{|\hat{g}|}\,{\textstyle\frac{1}{2}}\left(\partial\phi\right)^{2}\,=\,\intd^{d-1}x\sqrt{|g|}\,k\left[{\textstyle\frac{1}{2}}\left({\calD}\phi\right)^{2}\,-\,{\textstyle\frac{1}{2}}k^{-2}m^{2}\right]\,.
H_{G/H}=\frac{1}{2}\left(\pi_{\alpha}-\frac{i\hbar}{2}\Gamma_{\alpha}\right)g^{\alpha\beta}\left(\pi_{\beta}+\frac{i\hbar}{2}\Gamma_{\beta}\right)=\frac{1}{2}\pi_{\alpha}g^{\alpha\beta}\pi_{\beta}+V_{G/H}\,,
\begin{array}{rcl}{Q_{2}}&{=}&{R_{+}C^{+}+R_{-}C^{-}+L_{3}C^{3}+2i\bar{C}_{-}C^{-}C^{3}-2i\bar{C}_{+}C^{+}C^{3}}\\{}&{}&{+i(\displaystyle\frac{1}{2}(\lambda-8E)+4\lambdaL^{2})\bar{C}_{3}C^{+}C^{-}~,}\\\end{array}
ds_{10}^{2}=e^{\Phi_{D}}\left[dx_{0,3}^{2}+N\alpha^{\prime}g_{s}\left(d\rho^{2}+e^{2g(\rho)}d\Omega_{2}^{2}+\frac{1}{4}(\omega^{a}-A^{a})^{2}\right)\right]\,,
{\calF}(k^{2})=-\frac{N^{2}k^{4}}{64\pi^{2}}\left[\Psi\left(\frac{1}{2}+\frac{1}{2}\sqrt{1-e^{2\lambda}k^{2}}\right)+\Psi\left(\frac{1}{2}-\frac{1}{2}\sqrt{1-e^{2\lambda}k^{2}}\right)\right]\,.
S^{c}={\calC}{\calK}=e^{i\vartheta_{[1]}^{c}}\left(\begin{matrix}{0}&{\Theta_{[1]}}\\{-\Theta_{[1]}}&{0}\\\end{matrix}\right){\calK}\,\,,\quad\Theta_{[1]}=\left(\begin{matrix}{0}&{0}&{1}\\{0}&{-1}&{0}\\{1}&{0}&{0}\\\end{matrix}\right)
L=\intd^{4}\theta\left(\sum_{i=1}^{N+1}\overline{{\Phi}}_{i}e^{2\sum_{a=1}^{N-1}Q_{ai}V_{a}}\Phi_{i}-\sum_{a=1}^{N-1}\frac{1}{2e_{a}^{2}}\overline{{\Sigma}}_{a}\Sigma_{a}\right)-\frac{1}{2}\left(\intd^{2}\widetilde{\theta}\sum_{a=1}^{N-1}t_{a}\Sigma_{a}+\mathrm{c.c.}\right),
g_{\mu\nu}=\kappa\phiI_{\mu}^{\\beta}h_{\beta\nu}\:,\quadg^{\mu\nu}=\kappa^{-1}\phi^{-1}h^{\mu\alpha}I^{-1}\_{\alpha}^{\\nu}\;,\quadg=\kappa^{4}\phi^{4}Ih\;\;,
\frac{\ddot{R}}{R}=-\frac{4{\pi}}{3{\phi}}({\rho}+3p-\frac{8{\pi}{\sigma}^{2}}{\phi})-\frac{1}{3}({\omega}+\frac{3}{2})\frac{{\dot{\phi}}^{2}}{{\phi}^{2}}+\frac{{\ddot{\phi}}}{2{\phi}}
\Theta_{\mu\nu}^{\lambda}\equiv\zeta_{\mu\nu\rho}\omega^{\rho\lambda}+\frac{1}{3}\frac{\partial\omega_{\nu\rho}}{\partialx^{\mu}}\omega^{\rho\lambda}+\frac{1}{3}\frac{\partial\omega_{\mu\rho}}{\partialx^{\nu}}\omega^{\rho\lambda}
S^{p}=\int\sqrt{g}\delta_{[\alpha_{1}\cdots\alpha_{2p}]}^{[\beta_{1}\cdots\beta_{2p}]}R_{\;\;\;\beta_{1}\beta_{2}}^{\alpha_{1}\alpha_{2}}\cdotsR_{\;\;\;\beta_{2p-1}\beta_{2p}}^{\alpha_{2p-1}\alpha_{2p}}d^{d-2}x\,