harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{\partial\bar{z}(w^{-1},t)}{\partialt}=\frac{\partialS(z,t)}{\partialt}+\frac{\partialS(z,t)}{\partialz}\frac{\partialz(w,t)}{\partialt},\;\;\;\;\;\;\frac{\partial\bar{z}(w^{-1},t)}{\partialw}=\frac{\partialS(z,t)}{\partialz}\frac{\partialz(w,t)}{\partialw}
\begin{array}{rcl}{T_{n}(x_{1},\ldots,x_{n})}&{=}&{\sum_{k}:{\mathcal{O}}_{k}(x_{1},\ldots,x_{n}):t_{n}^{k}(x_{1},\ldots,x_{n}),}\\{t_{n}^{k}(x_{1},\ldots,x_{n})}&{=}&{\tilde{t}_{n}^{k}(x_{1},\ldots,x_{n})+\mathrm{sum~of~free~local~normalization~terms}}\\\end{array}
\left[R_{free}^{KG}\right]^{2}=\frac{\int_{-\infty}^{\infty}|\chi|^{2}\mathrm{dz}}{\int_{-\infty}^{\infty}|\varphi|^{2}\mathrm{dz}}=\left[\frac{1-\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\sqrt{1-\frac{v^{2}}{c^{2}}}}\right]^{2}\stackrel{v\rightarrowc}{\longrightarrow}1
\left[A_{\{(n_{1}),...(n_{N})\}}^{\{r,r_{1},...,r_{N}\}},A_{\{(n_{1}),...,(n_{N})\}}^{\{-r,-r_{1},...,-r_{N}\}}\right]=\chi_{\{(n_{1}),(n_{1}),...(n_{N}),(n_{N})\}}^{\{r,-r,r_{1},-r_{1},...,r_{N},-r_{N}\}}r\cdotp
=\int_{F}\frac{d^{2}\tau}{\tau_{2}^{2}}+2\int_{0}^{\infty}\frac{d\tau_{2}}{\tau_{2}^{2}}\sum_{n>0}e^{\frac{-\pix}{2\tau_{2}}n^{2}}=\frac{\pi}{3}+\frac{2\pi}{3x}.
\delta_{1}X_{2}={\frac{t_{2}}{t_{1}-t_{2}}}(\eta_{1}X_{2}-X_{2}\epsilon_{1})+{\frac{t_{1}t_{2}}{1-t_{1}t_{2}}}(M^{-1}\eta_{1}^{\dagger}MX_{2}-X_{2}M_{0}^{-1}\epsilon_{1}^{\dagger}M_{0}),
\deltaf_{3}^{9}\sim2((W_{1}^{9})^{2}+W_{2}^{9}W_{3}^{9})\sum_{i,j}^{3}(2\phi_{i,j,1}-\phi_{i,j,2}-\phi_{i,j,3})-2\sqrt{3}((W_{1}^{9})^{2}+W_{2}^{9}W_{3}^{9})(\rho_{1,1,2}-\rho_{1,1,3})\;.
\psi^{0}(z)=\sum_{r\in{\bfZ}+1/2}b_{r}^{0}z^{-r-\frac{1}{2}}~,\quad\widetilde{\psi}^{0}(\overline{{z}})=\sum_{r\in{\bfZ}+1/2}b_{r}^{0}\overline{{z}}^{-r-\frac{1}{2}}~.
\begin{aligned}{}&{{}U_{1}:\qquaddu_{1}+y_{1}v_{1}dx_{1}+x_{1}v_{1}dy_{1}+x_{1}y_{1}dv_{1}=0}\\{}&{{}U_{2}:\qquadx_{2}du_{2}+u_{2}dx_{2}+y_{2}dv_{2}+v_{2}dy_{2}=0.}\\\end{aligned}
{\calZ}(x)=1+\frac{\alpha}{4\pi}\frac{(4\pi)^{\epsilon}}{\Gamma(2-\epsilon)}\xi\left[{\frac{\epsilon}{2-\epsilon}}\int_{0}^{x}dy{\frac{y^{1-\epsilon}}{x^{2}}}{\calZ}(y)-\int_{x}^{\infty}dyy^{-\epsilon-1}{\calZ}(y)\right]\;\;\;.
{\frac{1}{\omega^{2}}}{\frac{d^{4}{\bfr}}{dt^{4}}}+{\frac{d^{2}{\bfr}}{dt^{2}}}={\frac{e}{m}}\left({\bfE}(t,{\bfr})+{\bfu}\times{\bfB}(t,{\bfr})\right),
E_{\mu\nu\rho\sigma}\;\;:=\;\;G_{\;\;\mu\nu\alpha\rho\sigma}^{\alpha}-\eta_{\rho[\mu}\,G_{\;\;\;\;\;\nu]\alpha\beta\sigma}^{\alpha\beta}+\eta_{\sigma[\mu}\,G_{\;\;\;\;\;\nu]\alpha\beta\rho}^{\alpha\beta}+\frac{1}{3}\,\eta_{\rho[\mu}\,\eta_{\nu]\sigma}\,G_{\;\;\;\;\;\;\alpha\beta\gamma}^{\alpha\beta\gamma}
\frac{1}{2\pi}c^{(f)}:e^{\mpi2\sqrt{\pi}\sqrt{\frac{\pi}{\pi+gN}}U_{1f}\Phi^{(1)}(x)}:_{M^{(1)}}\;\prod_{I=2}^{N}:e^{\mpi2\sqrt{\pi}U_{If}\Phi^{(I)}(x)}:_{M^{(I)}}e^{\pmi\frac{\theta}{N}}\;.
|0\rangle\rightarrow|0\rangle_{\beta}=(1+\mathrm{e}^{-\beta\epsilon})^{1/2}\mathrm{exp}\left\{-\frac{\beta\epsilon}{2}(a^{\dagger}{\tilde{a}}^{\dagger}\otimesb^{\dagger}{\tilde{b}}^{\dagger})\right\}|0\rangle_{a}\otimes|0\rangle_{b}
\begin{aligned}{AB=}&{{}qBA}\\{AC=}&{{}qCA}\\\end{aligned}\qquad\begin{aligned}{DB=}&{{}q^{-1}BD}\\{DC=}&{{}q^{-1}CD}\\\end{aligned}\qquad\begin{aligned}{AD-}&{{}qBC=1}\\{AD-}&{{}DA=(q-q^{-1})BC.}\\\end{aligned}
Z_{a}\equiv\nabla_{a}\,Z=-{\frac{1}{2}}\oint_{S^{2}}g_{ab^{*}}{\calF}^{+\,b^{*}}=(Q_{\Lambda}\nabla_{a}L^{\Lambda}-P^{\Sigma}\nabla_{a}M_{\Sigma})=\langleU_{a}|{\calQ}\rangle.
Z^{J}=\sum_{ij}Z_{ij}^{J}{\chi}_{i}\bar{\chi}_{j}=\sum_{i_{0},\mathrm{Rep}}({\chi}_{i_{0}}+{\chi}_{Ji_{0}})(\bar{\chi}_{i_{0}^{c}}+\bar{\chi}_{Ji_{0}^{c}})+2\sum_{f}{{\chi}}_{f}{\bar{\chi}}_{f^{c}}\;\;\;\;,
L_{ij}=\left(\begin{array}{ccccc}{0}&{k_{1}-k_{3}}&{-k_{1}+k_{2}}&{k_{3}-k_{2}}&{0}\\{-k_{1}+k_{3}}&{0}&{k_{1}}&{-k_{3}}&{0}\\{k_{1}-k_{2}}&{-k_{1}}&{0}&{k_{2}}&{0}\\{-k_{3}+k_{2}}&{k_{3}}&{-k_{2}}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}\\\end{array}\right).
\omega^{-1}(\eta(t))\tilde{E}_{\mu}[\eta|t]\omega(\eta(t))=8\pi\epsilon_{\mu\nu\rho\sigma}\dot{\eta}^{\nu}(t)\int\delta\xids\delta_{\alpha}(s)W^{\rho\alpha}[\xi|s]\dot{\xi}^{\sigma}(s)\delta(\xi(s)-\eta(t)),
\begin{array}{rcrcl}{e^{\pmiH_{1}}}&{=}&{\frac1{\sqrt{k}}(\psi^{1}}&{\mp}&{i\psi^{2})~,}\\{e^{\pmi\piN_{1}}e^{\pmiH_{2}}}&{=}&{{\frac{1}{\sqrt{2}}}\psi^{0}}&{\mp}&{{\frac{1}{\sqrt{k}}}\psi^{3}~.}\\\end{array}
\begin{array}{cccc}{\delta_{\mathrm{B}}A_{\mu}=D_{\mu}c,}&{\delta_{\mathrm{B}}b=0,}&{\delta_{\mathrm{B}}c=\{c,c\},}&{\delta_{\mathrm{B}}\overline{c}=b,}\\{\delta_{\mathrm{B}}A_{j}{}_{\mu}=[A_{j}{}_{\mu},c],}&{\delta_{\mathrm{B}}b_{j}=[b_{j},c],}&{\delta_{\mathrm{B}}\overline{c}_{i}=\{\overline{c}_{i},c\},}&{\delta_{\mathrm{B}}c_{i}=\{c_{i},c\}.}\\\end{array}
\begin{array}{c}{\pi_{x}=p_{x}+\frac{\lambda}{2}\left(x\left(xp_{x}+yp_{y}\right)+\left(xp_{x}+yp_{y}\right)x\right),}\\{\pi_{y}=p_{y}+\frac{\lambda}{2}\left(y\left(xp_{x}+yp_{y}\right)+\left(xp_{x}+yp_{y}\right)y\right).}\\\end{array}
\langle0\vert[T_{\mu\nu}^{V}(x),T_{\rho\sigma}^{V}(0)]\vert0\rangle=F_{\mu\nu\rho\sigma}(x)-F_{\rho\sigma\mu\nu}(-x)=\int\!d^{2}p\e^{-ipx}\varepsilon(p^{0})G_{\mu\nu\rho\sigma}(p)\,.
\frac{\partial}{\partial\theta}<A_{\mu,\lambda}^{\prime}>=\frac{\partial}{\partial\theta}\frac{\deltaZ}{\deltaL^{\mu}}=\frac{\delta}{\deltaL^{\mu}}\frac{\partialZ}{\partial\theta}=-K_{,\theta}^{\mu}(x)=\frac{\delta\Gamma}{\deltaA_{\mu}}
\left[T^{k},\psi_{I}\right]=-T_{II^{\prime}}^{k}\psi_{I^{\prime}}\quad\mathrm{with}\quadT_{II^{\prime}}^{k}:=\frac{1}{2}\left(\begin{array}{lr}{\sigma^{k}}&{0}\\{0}&{(-)^{k}\sigma^{k}}\\\end{array}\right)_{\kappa\kappa^{\prime}}\delta_{\alpha\alpha^{\prime}}\,\delta({\bfr}-{\bfr}^{\prime})\quad.
{\frac{1}{A}}{\frac{\partial{\calE}}{\partialm^{2}}}={\frac{i}{2}}\sum_{n=1}^{\infty}\int\int{\frac{dk_{1}dk_{2}}{(2\pi)^{2}}}\int{\frac{dk_{0}}{2\pi}}{\frac{1}{k_{0}^{2}-[k_{1}^{2}+k_{2}^{2}+{\frac{n^{2}\pi^{2}}{a^{2}}}-i\epsilon+m^{2}]}}\,.
A^{+}(x^{+})\rightarrow\frac{aA^{+}(x^{+})+b}{cA^{+}(x^{+})+d},~~~~~A^{-}(x^{-})\rightarrow\frac{aA^{-}(x^{-})+\epsilonb}{\epsiloncA^{-}(x^{-})+d},~~~~{\mathrm{with}}~~~ad-bc=1.
ds^{2}=-dT^{2}+\left[\frac{(D-1)T}{2r_{h}}\right]^{-2(D-3)/(D-1)}d\rho^{2}+r_{h}^{2}\left[\frac{(D-1)T}{2r_{h}}\right]^{4/(D-1)}d\Omega_{D-2}^{2}\.
{\calL}_{Skyrme}=\frac{f_{\pi}^{2}}{16}\mathrm{Tr}U^{\dagger}\partial_{\mu}UU^{\dagger}\partial^{\mu}U+\frac{1}{32e^{2}}\mathrm{Tr}[U^{\dagger}\partial_{\mu}U,U^{\dagger}\partial_{\nu}U]^{2}.
\begin{array}{c}{S=-\Sigma_{n}m_{n}\int\sqrt{-g_{\mu\nu}U_{n}^{\mu}U_{n}^{\nu}}d\tau_{n}+\frac{1}{2}\Sigma_{n}\int\lambda_{n}(\tau_{n})[g_{\mu\nu}U_{n}^{\mu}U_{n}^{\nu}+1]d\tau_{n}}\\{+\alpha\Sigma_{n}m_{n}\int\phi[x_{n}(\tau_{n})]d\tau_{n}-\frac{1}{2\beta}\int[\nabla^{\mu}\phi\nabla_{\mu}\phi-\Lambda^{2}e^{2\phi}]\sqrt{g}d^{4}x}\\\end{array}
Z_{T\oplus{\calO}}(\tau,z)=-2\pi\eta(\tau)^{2}\mathop{\mathrm{Res}}_{J=0}\left[\frac{\prod_{i}P(\tau,w_{i}J+z)}{P(\tau,dJ+z)}\frac{P(\tau,-dJ)}{\prod_{i}P(\tau,-w_{i}J)}\right].
{\cal{L}}_{\Lambda}=\frac{1}{2}\left[(\partial\eta)^{2}-\mu^{2}\eta^{2}\right]+\mu^{2}\frac{\gamma}{4!}\eta^{4}-\mu^{2}\frac{\gamma^{2}}{6!}\eta^{6}\dots-\frac{\delta\mu^{2}}{2}\eta^{2}+\dots.
\frac{\Upsilon_{n,0}[\kappa,\lambda,\sigma,\tau]}{\vert\Xi_{n,0}[\kappa,\lambda,\sigma,\tau]\vert^{2}{\calP}(\lambda)^{2}{\calP}(\sigma,\tau)}=\frac{\psi_{n,0}(\kappa,\lambda,\sigma,\tau,;k,\ell,s,t)\Upsilon_{n,0}[\kappa+k,\lambda+\ell,\sigma+s,\tau+t]}{\vert\Xi_{n,0}[\kappa+k,\lambda+\ell,\sigma+s,\tau+t]\vert^{2}{\calP}(\lambda+\ell)^{2}{\calP}(\sigma+s,\tau+t)}.
\begin{array}{lcl}{(1)\:{\calC}_{con}}&{\equiv}&{\{(1-x)^{2}-x^{2}y=0\},}\\{(2)\:{\calC}_{\infty}}&{\equiv}&{\{y=0\},}\\{(3)\:{\calC}_{1}}&{\equiv}&{\{y=1\},}\\{(4)\:{\calC}_{0}}&{\equiv}&{\{x=\infty\}.}\\\end{array}
U_{0}^{n}Str\left(\ldots\underbrace{\phi^{i_{2k}^{\prime}}\phi^{i_{2k-1}^{\prime}}}\ldots\right)=\frac{n}{n+2k}(U_{0}^{n}-U_{0}^{n}|_{i_{n}\leftrightarrowi_{1}^{\prime}}-\ldots-U_{0}^{n}|_{i_{n}\leftrightarrowi_{2k}^{\prime}})Str\left(\ldots\underbrace{\phi^{i_{2k}^{\prime}}\phi^{i_{2k-1}^{\prime}}}\ldots\right).
\deltaS=\frac{i}{2}\int\{D_{\bar{\eta}}({I\!\!L}D_{\eta}f({I\!\!N},{I\!\!R},Z,\bar{Z}))+D_{\eta}({I\!\!L}D_{\bar{\eta}}f({I\!\!N},{I\!\!R},Z,\bar{Z}))\}d\etad\bar{\eta}dt,
\frac{1}{2}(1-(-)^{l_{1}+l_{2}})\chi_{H_{1}^{+}}^{l_{1}}\chi_{H_{2}^{+}}^{l_{2}}=\chi_{H_{1}^{+}}^{l_{1}}\chi_{H_{2}^{+}}^{1-l_{1}}\delta_{l_{2},1-l_{1}}\,
\mathrm{d}s^{2}=\mathrm{e}^{A(r)}\eta_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}+\mathrm{e}^{B(r)}g_{mn}(y)\mathrm{d}y^{m}\mathrm{d}y^{n}+\mathrm{d}r^{2},
\begin{array}{rclcccc}{\mathrm{\underline{{sector}}}}&{{}}&{\mathrm{\underline{{state}}}}&{{}}&{{}}&{\mathrm{\underline{{~R_1~}}}}&{\mathrm{\underline{{~R_2~}}}}\\{\mathrm{NS}}&{{}}&{|2s_{3},2s_{4}\rangle,}&{s_{3}=s_{4}}&{{}}&{+}&{+}\\{\mathrm{R}}&{{}}&{|+,2s_{1},2s_{2}\rangle,}&{s_{1}=-s_{2}}&{{}}&{+}&{-}\\\end{array}
\Re=\frac{1}{12}\frac{q^{4}}{\Lambda_{\mathrm{NC}}^{4}}\frac{{\vec{E}}_{\theta}^{2}}{F_{\nu}^{2}(q^{2})}=\frac{3}{\Lambda_{\mathrm{NC}}^{4}}\frac{{\vec{E}}_{\theta}^{2}}{\langler_{\nu}^{2}\rangle^{2}}.
f(q,p)\starg(q,p)=\mathrm{exp}\Big\{\frac{i\hbar}{2}\Pi_{ij}\Big(\frac{\partial}{\partialq_{i}}\frac{\partial}{\partialp_{j}^{\prime}}-\frac{\partial}{\partialp_{i}}\frac{\partial}{\partialq_{j}^{\prime}}\Big)\Big\}f(q,p)g(q^{\prime},p^{\prime})|_{q=q^{\prime},p=p^{\prime}}.
a_{0}^{i}={\frac{1}{\sqrt{2m}}}(p_{0}^{i}+imx_{0}^{i}),\bar{a}_{0}^{i}={\frac{1}{\sqrt{2m}}}(p_{0}^{i}-imx_{0}^{i}),a_{n}^{I}=\sqrt{\frac{1}{|\omega_{n}|}}\alpha_{n}^{I}.
\Lambda=1+\sum_{i=1}^{4}\beta_{i}^{2}\vertz_{i}\vert^{2}\quad\mathrm{and}\quadA=\frac{\Lambda^{-1}}{2i}\sum_{i=1}^{4}\beta_{i}\left(\bar{z}_{i}dz_{i}-z_{i}d\bar{z}_{i}\right)~,
\int_{0}^{\infty}dm\\int_{m_{e}}^{\infty}dk\(k^{2}-m_{e}^{2})^{1-s}\partial_{k}\frac{t^{j}}{m^{n}}=-\frac{m_{e}^{2-2s}}{2}\frac{\Gamma(2-s)\Gamma\left(\frac{1+j-n}{2}\right)\Gamma(s+\frac{n-3}{2})}{(Rm_{e})^{n-1}\Gamma(j/2)}\,
M=\left(\begin{array}{cc}{T}&{B}\\{-C}&{U}\\\end{array}\right),\;M\;\rightarrow\;\left(\begin{array}{cc}{a}&{b}\\{c}&{d}\\\end{array}\right),\;\left(\begin{array}{cc}{a}&{b}\\{c}&{d}\\\end{array}\right)\;\in\;Sp(4,Z).
S=\intd^{5}x\left[\frac{1}{2}\left\{\eta^{ij}\partial_{i}\tilde{\varphi}\partial_{j}\tilde{\varphi}-e^{-2\sigma}\left\{m^{2}+\left(\frac{3}{16}-\xi\right)R\right\}\tilde{\varphi}^{2}\right\}-e^{-5\sigma}V\left(e^{\frac{3}{2}\sigma}\tilde{\varphi}\right)\right].
\Gamma^{p_{1}\top_{2}}\equiv\intd^{3}\widetilde{\bfk}_{1}\intd^{3}\widetilde{\bfk}_{2}\frac{d\Gamma^{p_{1}\top_{2}}}{d^{3}\widetilde{\bfk}_{1}d^{3}\widetilde{\bfk}_{2}}\;,
\left({\frac{\dot{a}}{a}}\right)^{2}={\frac{1}{L^{2}}}\left[(\bar{E}^{2}-\bar{P}_{\phi}^{2})\left({\frac{L}{a}}\right)^{2(p+1)}-\bar{E}^{2}\left({\frac{L}{a}}\right)^{2p}+2\bar{P}_{\phi}\left({\frac{L}{a}}\right)^{p+1}-\left({\frac{L}{a}}\right)^{2}\right],
\rho(n,E)=\frac{e^{S(\alpha_{0},\beta_{0})}}{2\pi\sqrt{\frac{\partial^{2}\mathrm{ln}Z}{\partial\alpha^{2}}|_{(\alpha_{0},\beta_{0})}\times\frac{\partial^{2}\mathrm{ln}Z}{\partial\beta^{2}}|_{(\alpha_{0},\beta_{0})}}}.
S_{dual}=\intd^{6}x\left[-{\frac16}G_{{\mu}{\nu}{\sigma}}G^{{\mu}{\nu}{\sigma}}-{\frac13}{\lambda}_{{\mu}{\nu}{\sigma}}{\calG}^{{\mu}{\nu}{\sigma}}+A_{{\nu}{\sigma}}{\partial}_{{\mu}}G^{{\mu}{\nu}{\sigma}}\right].
\Psi^{\alpha_{1}^{1}....\alpha_{1}^{m},\alpha_{2}^{1}....\alpha_{2}^{m}}(X_{1},X_{2})=-\Psi^{\alpha_{2}^{1}....\alpha_{2}^{m},\alpha_{1}^{1}....\alpha_{1}^{m}}(X_{2},X_{1}),
{}\big[\delta_{\epsilon_{1}}^{(\alpha^{\prime})^{0}}+\delta_{\epsilon_{1}}^{(\alpha^{\prime})^{2}},\delta_{\epsilon_{2}}^{(\alpha^{\prime})^{0}}+\delta_{\epsilon_{2}}^{(\alpha^{\prime})^{2}}\big]A_{\mu}={}\big[\delta_{\epsilon_{1}}^{(\alpha^{\prime})^{0}},\delta_{\epsilon_{2}}^{(\alpha^{\prime})^{0}}\big]A_{\mu}+{\calO}\big((\alpha^{\prime})^{4}\big)\,.
Q^{a}={\frac{1}{\sqrt{2p^{+}}}}\int_{0}^{\pi}d\sigmaS^{a}(\sigma),\qquad{Q}^{\dot{a}}={\frac{1}{\pi\sqrt{p^{+}}}}\int_{0}^{\pi}d\sigma\gamma_{\dot{a}b}^{I}\partialX^{I}{S}^{b}(\sigma),
c(n)=\Gamma(-n)e^{-{\frac{n(n+1)T}{2}}}\prod_{i=1}^{n}\intd^{2}z_{i}\,\prod_{i<j}|z_{i}-z_{j}|^{2}\approxe^{-{\frac{n(n+1)T}{2}}}A^{\frac{n(n+1)}{2}}
R_{n\ell}(p)=\sqrt{\frac{2(2n^{\prime}+a+b+1)\,n^{\prime}!\,\Gamma(n^{\prime}+a+b+1)}{\Gamma(n^{\prime}+a+1)\Gamma(n^{\prime}+b+1)}}\,(\beta+\beta^{\prime})^{D/4}\,c^{\lambda+\delta}\,s^{\ell}\,P_{n^{\prime}}^{(a,b)}(z)\;,
\Gamma_{ij}^{\sft}=a\left(\frac{da}{d{\sft}}\right)\gamma_{ij}\;,~~~\Gamma_{{\sft}j}^{i}=\frac{1}{a}\left(\frac{da}{d{\sft}}\right)\delta_{j}^{i}\;,~~~\Gamma_{jk}^{i}=\frac{\gamma^{il}}{2}(\partial_{j}\gamma_{lk}+\partial_{k}\gamma_{lj}-\partial_{l}\gamma_{jk})\;.
A_{{\bfk}}^{(1)\sigma}=-i{\frac{\pi}{\omega^{(0)}}}E_{{\bfk}0}^{\sigma}[e^{-i\phi_{\bfk}^{\sigma}}\delta(\omega^{(0)}-\omega_{\bfk}^{\sigma})+e^{i\phi_{\bfk}^{\sigma}}\delta(\omega^{(0)}+\omega_{\bfk}^{\sigma})],
\frac{1}{u^{2}}\phi(u,v)=log\,u\,[-\frac{48}{(1-Y)}\sum_{m=0}^{\infty}\frac{(1-V)^{m}}{m!}\frac{(m+1)!(m+2)!}{(m+5)!}]+\,\mathrm{terms~continuous~at}\,u=0,
\frac{k_{\mu}k_{\lambda}l_{\nu}^{\,\,\nu\mu\lambda}}{4k^{2}}-\frac{V(x)}{2\sqrt{k^{2}}}-\frac{k_{\mu}k^{\lambda}l_{\,\,\,\,\nu\lambda}^{\mu\nu}}{4(k^{2})^{3/2}}-\frac{k_{\mu}k_{\lambda}l_{\nu}^{\,\,\,\,\nu\mu\lambda}}{4(k^{2})^{3/2}}).
\phi^{\prime\prime}+{\frac{5r^{4}-r_{0}^{4}}{r(r^{4}-r_{0}^{4})}}\phi^{\prime}+\omega^{2}{\frac{r^{4}(r^{4}+R^{4})}{(r^{4}-r_{0}^{4})^{2}}}\phi=0\,.
\left\{\frac{d^{2}}{d\varrho^{2}}+\alpha^{2}\varrho^{2(\alpha-1)}\left[E-V(\varrho^{\alpha})\right]-\frac{\alpha^{2}(l+\nu)^{2}-1/4}{\varrho^{2}}\right\}\widetilde{u}(\varrho)=0\;,
q_{\mu}\Gamma_{A}^{\mu}(p_{1},p_{2})=S_{F}^{-1}(p_{1})\gamma_{5}+\gamma_{5}S_{F}^{-1}(p_{2})+2m\Gamma_{5}(p_{1},p_{2})+i\frac{g^{2}}{16\pi^{2}}F(p_{1},p_{2}),
{\calD}^{\alpha_{i}\widehat{e}}\widetilde{x}_{i}^{\mu}\circ\gamma[\widetilde{\varphi}_{i}]^{-1}=\mathrm{e}^{\frac{d}{16}(\widetilde{\varphi}_{i}(0)+\widetilde{\varphi}_{i}(1))}{\calD}^{\widehat{e}}\widetilde{x}_{i}^{\mu}\;\;\;,\;\;\;(i\rightarrowf)\;\;\;,
m^{2}\alpha^{\prime}=2\sum_{n>0}\frac{2n^{2}+m^{2}H^{2}\alpha^{2}}{\sqrt{n^{2}+m^{2}H^{2}\alpha^{2}}}+\sum_{n>0}\frac{2n^{2}+m^{2}H^{2}\alpha^{2}}{\sqrt{n^{2}+m^{2}H^{2}\alpha^{2}}}\left[(a_{n}^{R})_{n}^{R}+(\tilde{a}_{n}^{R})^{\dagger}\tilde{a}_{n}^{R}\right].
(X)_{ll^{\prime}}=\frac{2\alpha}{2n+\delta_{1}+\delta_{2}}\biggl(A_{nm}^{l+1}\delta_{l^{\prime},l+1}+A_{nm}^{l}\delta_{l^{\prime},l-1}\biggr)-\frac{\alpha(2|m|+\delta_{1}+\delta_{2})(\delta_{1}-\delta_{2})}{(2l+\delta_{1}+\delta_{2})(2l+\delta_{1}+\delta_{2}+2)}\delta_{l^{\prime}l},
\omega=\frac{1}{\gamma^{2}}\,\int_{\tau}^{\tau+2\pi}\,\,\frac{1}{\psi^{2}(z)}\,\delta\psi^{\prime}(z)\wedge\,\delta\psi(z)\,dz+\frac{1}{\gamma^{2}}\,\delta\lambda\wedge\,\frac{\delta\psi(\tau)}{\psi(\tau)}.
m_{l}=(b-a)\sqrt{m^{2}+(l/b\sigma)^{2}},\quadc_{\alpha}(y)=\frac{\tilde{A}_{\alpha}+y\tilde{B}_{\alpha}/(b-a)}{\tilde{A}_{\alpha}-y\tilde{B}_{\alpha}/(b-a)},\quad\alpha=a,b,
\begin{array}{rcl}{\langleM|j(x)j(0)|M\rangle}&{=}&{\displaystyle\sum_{\alpha}\int\frac{d^{3}\vec{p}}{(2\pi)^{3}2E_{\vec{p},\alpha}}\langleM|j(x)|\vec{p},\alpha\rangle\langle\vec{p},\alpha|j(0)|M\rangle}\\{}&{=}&{\displaystyle\sum_{\alpha}\int\frac{d^{3}\vec{p}}{(2\pi)^{3}2E_{\vec{p},\alpha}}|\langleM|j(0)|\vec{p},\alpha\rangle|^{2}e^{iMx_{0}}e^{-ip_{+}\cdotx}.}\\\end{array}
j_{m}\bar{Z}(j)-G_{m}(\bar{Z}^{-1}(j)\frac{\partial}{\partialj}\bar{Z}(j))+E_{mn}(\bar{Z}^{-1}(j)\frac{\partial}{\partialj}\bar{Z}(j))\bar{Z}^{-1}(j)\frac{\partial}{\partialj_{n}}=0.
A_{0}^{U}(t,{\bfx})=\frac{1}{\beta}\int_{0}^{\beta}dt^{\prime}A_{0}(t^{\prime},{\bfx})={\calA}_{0}({\bfx}),\,\A_{i}^{U}(t,{\bfx})=A_{i}(0,{\bfx})+{\tilde{E}}_{i}(t,{\bfx}).
R_{r_{g}}^{(y)}(r)=D_{\mathrm{in}}\left(\Omega_{\beta}^{(+)}\frac{1}{r^{1-\beta}}+i\Omega_{\beta}^{(-)}\frac{1}{r^{\beta}}\right)+D_{\mathrm{out}}\left(\Omega_{\beta}^{(+)}\frac{1}{r^{1-\beta}}-i\Omega_{\beta}^{(-)}\frac{1}{r^{\beta}}\right).
\mu_{\alpha\beta}=\pi^{4}{\frac{\hbarL^{D-1}}{a^{6}}}\sum_{l_{1},l_{2}}s_{\alpha\beta}(l_{1},l_{2})(l_{1}l_{2})^{2}\int{\frac{d^{D-1}k_{\|}}{(2\pi)^{D-1}}}\frac{1}{\omega_{l_{1}}\omega_{l_{2}}(\omega_{l_{1}}+\omega_{l_{2}})^{3}}.
\hat{T}_{k}=e^{\lambda_{-1}^{(k)}(\hat{L}_{-1}+\bar{\hat{L}}_{-1})}e^{\lambda_{0}^{(k)}(\hat{L}_{0}+\bar{\hat{L}}_{0})}e^{\lambda_{1}^{(k)}(\hat{L}_{1}+\bar{\hat{L}}_{1})},
\Lambda={\left(\begin{array}{cccc}{\frac{1}{2}}&{0}&{0}&{\frac{c}{eB}}\\{0}&{\frac{1}{2}}&{-\frac{c}{eB}}&{0}\\{0}&{-\frac{eB}{4c}}&{\frac{1}{2}}&{0}\\{\frac{eB}{4c}}&{0}&{0}&{\frac{1}{2}}\\\end{array}\right)}\;.
\langle{\calT}(a_{1}){\calT}(a_{2})\rangle=(-1)^{2}{\frac{\delta^{2}\Gamma_{SC}}{\delta\hat{\mu}(a_{1})\delta\hat{\mu}(a_{2})}}\bigg\vert_{\hat{\mu}=0}=\kappa\partial_{1}^{2}D_{1}\left({\frac{\theta_{1}-\theta_{2}}{z_{1}-z_{2}-\theta_{1}\theta_{2}}}\right)
\begin{array}{rcl}{}&{\tilde{L}_{i}=\frac{(2\pi)^{2}{l_{p}}^{3}}{L_{i}R}}\\{}&{g^{2}=\frac{(2\pi)^{6}{l_{p}}^{6}}{L_{1}L_{2}L_{3}L_{4}R}}\\\end{array}
\hat{T}_{~0}^{0}=T_{~0}^{0}-\displaystyle{{\frac{\hbar^{2}}{8m}}}\partial_{x}^{2}\rho^{2},\\\\\hat{T}_{~0}^{x}=T_{0}^{x}-\displaystyle{{\frac{\hbar^{2}}{8m}}}\partial_{x}^{2}J,\\\\,hat{T}_{~x}^{0}=T_{~x}^{0},\\\\,hat{T}_{~x}^{x}=T_{~x}^{x},
\{b_{r}^{\mu,j},b_{s}^{\nu,j^{\prime}}\}=\eta^{\mu\nu}\delta^{j,j^{\prime}}\delta_{r,-s};~~~~~~~~~~~~~~~\{b_{r}^{\mu,k},b_{s}^{\mu,k^{\prime}}\}=\eta^{\mu\nu}\delta^{k,k^{\prime}}\delta_{r,-s}~~~:~~~~~~~~NS
\phi_{1}={\frac{\phi_{0}\;(\eta_{ab}\;\zeta^{a}\;\zeta^{b})\;\left\{\eta_{ab}\;[x^{a}-x_{0}^{a}]\;[x^{b}-x_{0}^{b}]-\eta_{ab}\;\zeta^{a}\;\zeta^{b}\right\}}{(\eta_{ab}\;[x^{a}-x_{0}^{a}]\;[x^{b}-x_{0}^{b}]-\eta_{ab}\;\zeta^{a}\;\zeta^{b})^{2}+4(\eta_{ab}\;[x^{a}-x_{0}^{a}]\;\zeta^{b})^{2}}};
S(A,\bar{A})=\frac{k}{4\pi}\intd^{2}z\left\{\frac{1}{2}\partialx\bar{\partial}x+e^{x}\partialx_{+}\bar{\partial}x_{-}+\bar{A}e^{x}\partialx_{+}+\lambda\left(\partial\bar{A}-\bar{\partial}A\right)\right\}.
\varepsilon^{\prime\alpha}\psi_{\alpha}=\varepsilon_{\beta}^{\prime}C^{\beta\alpha}\psi_{\alpha}=\psi_{\alpha}C^{\alpha\beta}\varepsilon_{\beta}^{\prime}=\psi^{\beta}\varepsilon_{\beta}^{\prime}=-\psi_{\alpha}\varepsilon^{\prime\alpha}.
V_{1}=(0(0~0)^{3}\vert\vert0^{2}~{\tilde{e}}^{1}\vert({\frac{1}{3}})_{r}~({\frac{1}{3}})_{r}~({\frac{2}{3}})_{r}~0_{r}^{5}\vert(-{\frac{1}{3}})_{r}~(-{\frac{1}{3}})_{r}~(-{\frac{2}{3}})_{r}~0_{r}^{5})\nonumber
A_{+}^{a}=-\frac{1}{\partial_{-}}\left(\partial_{\rho}A^{a\rho}\right)-g\frac{1}{\partial_{-}^{2}}\left[iA_{\rho}T^{a}(\partial_{-}A^{\rho})-\frac{1}{2}{\bar{\psi}}_{-}\gamma_{-}t^{a}\psi_{-}\right]
\omega^{\alpha}=-\phi^{j}\,(A+\frac{1}{3!}A^{3}+\cdots)_{j}^{\,\,\,\alpha}+d\xi^{\beta}\,(\mathrm{1}+\frac{1}{3!}A^{2}+\frac{1}{5!}A^{4}+\cdots)_{\beta}^{\,\,\,\,\alpha}.
E={\frac{1}{2}}\int_{\infty}^{+\infty}dre^{4A}\left\{[g^{1/2}\phi^{\prime}\mp3Z_{,\phi}g^{-1/2}]^{2}-12[A^{\prime}\pmZ]^{2}\pm3[e^{4}AZ]_{-\infty}^{+\infty}\right\}.
V_{y}=2\,{x^{8}}\,y-4\,{x^{7}}\,{y^{3}}+6\,{x^{6}}\,{y^{5}}-7\,{x^{4}}\,{y^{6}}+8\,{x^{2}}\,{y^{7}}+3\,{y^{8}}=0
|\vec{p}_{f}|=\frac{\rho}{2}\sqrt{1+\left[\frac{m_{1}^{2}-m_{2}^{2}}{\sqrt{s}\rho}\right]^{2}}=\frac{1}{2\sqrt{s}}\sqrt{s^{2}-2s(m_{1}^{2}+m_{2}^{2})+(m_{1}^{2}-m_{2}^{2})^{2}}=\frac{\lambda^{\frac{1}{2}}(s,m_{1}^{2},m_{2}^{2})}{2\sqrt{s}}
L_{\textrm{SR}}\equiv\dot{x}^{\mu}\dot{x}_{\mu}\longrightarrowL_{\textrm{ESR}}\equiv\dot{X}^{\mu}\dot{X}_{\mu}=(\dot{x}^{\mu}\dot{x}_{\mu}+\dot{u}^{\mu}\dot{u}_{\mu})+2I\dot{x}^{\mu}\dot{u}_{\mu},
k\frac{\partial}{\partialk}U_{k}(\Phi)=-\frac{\Omega_{d}k^{d}}{2(2\pi)^{d}}\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{-U_{k}^{\prime\prime}(\Phi)}{k^{2}+U_{k}^{\prime\prime}(\Phi)}\right)^{n},
\gamma_{5}\left(\begin{array}{c}{\lambda_{A}^{i^{\star}}}\\{\zeta^{\alpha}}\\{\psi^{A}}\\\end{array}\right)\,=\,-\,\left(\begin{array}{c}{\lambda_{A}^{i^{\star}}}\\{\zeta^{\alpha}}\\{\psi^{A}}\\\end{array}\right)
D(j_{3},j_{2},j_{1})=\frac{G(j_{1}+j_{2}+j_{3}+1)G(j_{1}+j_{2}-j_{3})G(j_{1}+j_{3}-j_{2})G(j_{2}+j_{3}-j_{1})}{(\nu(b))^{-j_{1}-j_{2}-j_{3}-1}\;G_{0}\;G(2j_{1}+1)G(2j_{2}+1)G(2j_{3}+1)}.
A_{0}\left(x\right)=\frac1{\xi_{1}}\left(-C_{1}e^{-x\xi_{1}}+C_{2}e^{x\xi_{1}}\right)+\frac1{\xi_{2}}\left(-C_{3}e^{-x\xi_{2}}+C_{4}e^{x\xi_{2}}\right)+C_{7}
\tilde{F}_{L}^{\prime}(\tilde{\xi}^{++\prime})=\omega_{L}^{-3/2}(\tilde{\xi}^{++})\tilde{F}_{L}(\tilde{\xi}^{++}),~~~\tilde{F}_{R}^{\prime}(\tilde{\xi}^{--\prime})=\omega_{R}^{-3/2}(\tilde{\xi}^{--})\tilde{F}_{R}(\tilde{\xi}^{--}),
{\frac{\deltaM_{1}}{\delta\beta^{a}}}\vert_{_{\beta=0}}\,=\,-\,\,\alpha\epsilon^{\mu\nu\rho\sigma}Tr\,\lbrace\,\partial_{\mu}\Big(A_{\nu}\partial_{\rho}A_{\sigma}\,-\,{\frac{i}{2}}A_{\nu}A_{\rho}A_{\sigma}\Big)T^{a}\rbrace\,\,.
{}^{(4)}G_{\mu\nu}={\frac{2\kappa_{5}^{2}}{3}}\left[{}^{(5)}T_{AB}g_{\mu}^{~A}g_{\nu}^{~B}+\left({}^{(5)}T_{AB}n^{A}n^{B}-{\frac{1}{4}}{}^{(5)}T_{~C}^{C}\right)g_{\mu\nu}\right]+KK_{\mu\nu}-K_{\mu}^{~\sigma}K_{\nu\sigma}-{\frac{1}{2}}g_{\mu\nu}\left(K^{2}-K^{\alpha\beta}K_{\alpha\beta}\right)-E_{\mu\nu},
\mathcal{O}:={\frac{1}{\Omega^{(D)}}}\intd^{D}x\\mathrm{tr}\left[{\frac{1}{2}}\mathscr{A}_{\mu}(x)\cdot\mathscr{A}_{\mu}(x)+\alphai\bar{\mathscr{C}}(x)\cdot\mathscr{C}(x)\right],
{\calL}_{-1}=\left(\phi_{x}^{\;2}+\frac{1}{2}\,\phi_{xxx}\right)\phi_{t}+\left(\frac{5}{2}\,\phi_{x}^{\;4}-5\,\phi_{x}\,\phi_{xx}^{\;\;2}+\frac{1}{2}\,\phi_{x}^{\,3}\right)
A_{T}^{i(\pm)}(x^{0},\vec{x})\,=\,\frac{1}{2\pi}\int\frac{d^{2}k}{\sqrt{2\omega_{\vec{k}}}}e^{\pmi(\omega_{\vec{k}}x^{0}-\vec{k}\cdot\vec{x})}\,A_{T}^{i(\pm)}(\vec{k}),
E(r)=T_{2}\left(\sqrt{-g_{00}\left({\frac{\alpha^{2}N^{2}}{4}}+r^{4}\right)}-Fr^{3}\right)\approxT_{2}\left({\frac{N\alpha^{\prime}}{2}}+N\alpha^{\prime}F^{2}r^{2}-Fr^{3}+\ldots\right)
S_{\mathrm{E}}[\phi]=\int\!\mathrm{d}^{4}x\,\left\{\frac{1}{2}\,\partial_{\mu}\phi(x)\,\partial_{\mu}\phi(x)+{\displaystyle\frac{m^{2}}{2}}\phi^{2}(x)+V\left(\phi(x)\right)\right\}\,.
\sum_{n=0}^{\infty}h(r_{n})=\frac{V(F)}{2}\int_{-\infty}^{\infty}h(r)\psi_{2}(r)\dr+\sum_{\wp}\sum_{k=1}^{\infty}\frac{\chi(P(\gamma)^{k})}{S_{2}(k;l_{\gamma})}\hat{h}(kl_{\gamma}),
\Bigg|\prod_{i=1}^{n_{a}}\big(\hat{x}_{i}^{(a)}-\tilde{x}_{i}^{(a)}\big)\big(\tilde{y}_{i}^{(a)}-\hat{y}_{i}^{(a)}\big)\Bigg|^{[1-\frac{g}{\pi+gN}]}\;\leq\;\left(8^{[1-\frac{g}{\pi+gN}]}\right)^{2n_{a}}\;,
\mathrm{[}\hat{\Lambda}_{n},\hat{\Lambda}_{m}]=[\hat{\Lambda}_{n}^{\dagger},\hat{\Lambda}_{m}^{\dagger}]=0;\,\,\,[\hat{\Lambda}_{n}\,\,,\,\,\hat{\Lambda}_{m}^{\dagger}]=\frac{R}{\pi}\delta_{n,m}.
\begin{array}{c}{\Omega=\omega_{0}-i\partial{\bar{\partial}}{\bfg}=}\\{=d\pi_{a}\wedgedz^{a}+d{\bar{\pi}}_{a}\wedged{\bar{z}}^{a}+}\\{+R_{a{\bar{b}}c\bar{d}}\eta_{i}^{a}\bar{\eta}_{i}^{b}dz^{a}\wedged{\bar{z}}^{b}+g_{a\bar{b}}D\eta_{i}^{a}\wedge{D{\bar{\eta}}_{i}^{b}}}\\\end{array}
\begin{aligned}{\partial_{k}\beta_{ij}(u)=\beta_{ik}(u)\beta_{kj}(u),\quad}&{{}i\nek\nej,}\\{\frac{\partial\beta_{ij}(u)}{\partialu_{i}}+\frac{\partial\beta_{ji}(u)}{\partialu_{j}}+\sum_{k\nei,j}\beta_{ki}\beta_{kj}=0,\quad}&{{}i\nej,}\\\end{aligned}
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\piG\left(\frac{V}{\sqrt{1+\nabla_{\alpha}T\nabla^{\alpha}T}}\,\,\nabla_{\mu}T\,\,\nabla_{\nu}T-g_{\mu\nu}V\,\sqrt{1+\nabla_{\alpha}T\nabla^{\alpha}T}\right)\,
[\vec{\nabla^{2}}+\frac{1}{R^{2}}\frac{\partial}{\partial\phi^{2}}+K^{2}]G_{o}(K;\vec{x},\phi;\vec{x}^{\prime},\phi^{\prime})=\delta^{3}(\vec{x}-\vec{x}^{\prime})\delta(\phi-\phi^{\prime}).
\begin{array}{lccc}{}&{(1~-1)}\\{\mid\mathrm{ground}>}&{(1~-1)}&{=}&{Q_{1/2}^{+}}\\{\mid\mathrm{1st}>}&{(2~-1)}&{=}&{\mathrm{SU(2)}}\\{}&{(0~0)}&{=}&{\mathrm{U(1)}}\\{\mid\mathrm{2nd}>}&{(1~0)}&{=}&{Q_{1/2}^{-}.}\\\end{array}
\partial_{\tau}\rho_{jk}=i[{\calA},\rho]_{jk}-\frac{1}{2}\overline{{R}}_{lrj}R_{lrm}\rho_{mk}-\frac{1}{2}\overline{{R}}_{lrm}R_{lrk}\rho_{jm}+\overline{{R}}_{lkm}R_{ljr}\rho_{rm}
M=\frac{\Omega_{d-2}}{8\piG_{d+1}}R_{T}^{d-2}R_{m}^{d-2}\int_{0}^{2\pi}dv\left[-g^{\prime}-u\left(\frac{1}{2}\frac{A_{(0)}^{\prime}}{A_{(0)}}+\frac{d-2}{R_{m}}\right)\right]\.
d{\calL}_{4}^{WZ}=R_{5}+{\calF}\wedgeR_{3}+{\frac{1}{2}}{\calF}\wedge{\calF}\wedgeR_{1}={\frac{i}{2}}(\hat{E}\gamma^{(3)}\hat{E})+(\hat{E}\tilde{\gamma}^{(4)}\hat{\Lambda}),
{\mathbbE}\,=\,\left(\begin{array}{c}{E_{1}}\\{E_{2}}\\{E_{3}}\\\end{array}\right)\:\:\:\:\:\mathrm{and}\:\:\:\:\:{\mathbbB}\,=\,\left(\begin{array}{c}{B_{1}}\\{B_{2}}\\{B_{3}}\\\end{array}\right)\;\;.
ds_{\mathrm{target}}^{2}=\frac{1}{2}\mathrm{Tr}\left[(d{\cal{V}}{\cal{V}}^{-1})(d{\cal{V}}{\cal{V}}^{-1})+(d{\cal{V}}{\cal{V}}^{-1})(d{\cal{V}}{\cal{V}}^{-1})^{\mathrm{T}}\right]