harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{l}{\hat{Z}_{88}^{1}(\Omega)\\stackrel{ZHP}\longrightarrow\hat{Z}_{8}(\tau_{1})\hat{Z}_{8}(\tau_{2})+O(t)}\\{Z_{88}^{1}(\Omega)\\stackrel{NZHP}\longrightarrow<\Phi_{8}(x_{1})\Phi_{8}(x_{2})>(\tau)|t|^{1/2+1/2}+\cdots}\\\end{array}
f_{A}={\frac{\pi(e_{1}+e_{2})Ete^{-\pit\epsilon^{2}/2}}{2sin(\pit\epsilon/2)}}\,prod_{n=1}^{\infty}{\frac{(1-e^{-\pitn})^{2}}{(1-e^{-\pit(n+i\epsilon)})(1-e^{-\pit(n-i\epsilon)})}}\.
B(q(\tau_{i}))=\frac{b_{i}}{(2\pi)^{3/2}\sqrt{\beta}}e^{ik_{i}\cdotq(\tau_{i})}\equiv\frac{b_{i}}{(2\pi)^{3/2}\sqrt{\beta}}e^{i\left[{\bfk}_{i}{\bfq}(\tau_{i})+(\frac{2\pi}{\beta})n_{i}q_{4}(\tau_{i})\right]},
L_{\mathbf{g}}(\psi)^{\mua}=\psi^{\mua}\\\\Rightarrow\\\\psi^{\mua}(\mathbf{x},\mathbf{z}\mathbf{g})=\psi^{\mua}(\mathbf{x},\mathbf{e})\equiv\psi^{\mua}(\mathbf{x})\;.
\frac{1}{4\pi}\int_{\Sigma}d^{3}x\rho=\frac{1}{4\pi}\int_{S_{up}^{2}}dS^{\hat{\imath}}{}^{(3)}\nabla_{\hat{\imath}}\phi+\frac{1}{4\pi}\int_{S_{low}^{2}}dS^{\hat{\imath}}{}^{(3)}\nabla_{\hat{\imath}}\phi\equiv2\Sigma_{p}\,,
\delta_{\mathrm{sp}}^{R}V_{\star}\left(\Phi\star\Phi^{\dagger}\right)=0\quad\mathrm{and}\quad\delta_{\mathrm{sp}}^{L}V_{\star}\left(\Phi\star\Phi^{\dagger}\right)=\frac{1}{2}\omega_{L}^{\alpha\beta}D_{\alpha\beta}V_{\star}\left(\Phi\star\Phi^{\dagger}\right)
\widetilde{(f\starg)}(K)=\frac{1}{(\pi\vartheta)^{2n}}\intdK^{\prime}~e^{\frac{i}{2}\Lambda^{ij}K_{i}K_{j}^{\prime}}\widetilde{f}\Big(\frac{1}{2}K+K^{\prime}\Big)\widetilde{g}\Big(\frac{1}{2}K-K^{\prime}\Big)
Z[\lambda,T]=\frac{\intdA_{\mu}e^{-\int_{0}^{1/T}F_{\mu\nu}^{2}/4+\lambda\intdxe^{ie\int_{0}^{1/T}d\tauA_{0}(\tau,x)+\mathrm{c.c.}}}}{\intdA_{\mu}e^{-\int_{0}^{1/T}F_{\mu\nu}^{2}/4}}
S=\intd^{4}x\sqrt{-g}\left\{e^{-\Phi}\left[R+\left(\nabla\Phi\right)^{2}-6\left(\nabla\gamma\right)^{2}\right]-\frac{1}{48}e^{6\gamma}F_{\mu\nu\lambda\kappa}F^{\mu\nu\lambda\kappa}-\frac{1}{2}m^{2}e^{6\gamma}\right\},
\deltag_{\mu\nu}^{(S)}=\left(\begin{matrix}{2a^{2}\phi}&{-a^{2}B_{i}}&{-abC_{a}}&{}\\{-a^{2}B_{i}}&{2a^{2}\psi\delta_{ij}-2a^{2}E_{ij}}&{-abD_{ia}}&{}\\{-abC_{a}}&{-abD_{ia}}&{2b^{2}\xi\delta_{ab}-2b^{2}G_{ab}}&{}\\\end{matrix}\right)
\frac{p^{2}}{2}(P_{\mu\rho,\nu\sigma}^{(2)}-Q_{\mu\rho,\nu\sigma})=\frac{p^{2}}{2}\left[\begin{array}{cccc}{1}&{}&{}&{}\\{}&{0}&{}&{}\\{}&{}&{-1}&{-\sqrt{2}}\\{}&{}&{-\sqrt{2}}&{-2}\\\end{array}\right],
G_{\mu\nu}^{(4)}=8\piG_{N}\left(T_{\mu\nu}+T_{\mu\nu}^{\mathrm{CFT}}\right)-2l^{2}\left(\alpha-{\frac{1}{4}}\right){\calS}_{\mu\nu}-{\frac{2l^{2}}{3}}\beta{\calK}_{\mu\nu}\.
\psi_{(m_{1},n_{1})}\otimes\psi_{(m_{2},n_{2})}=\sum_{k=|m_{1}-m_{2}|+1}^{m_{1}+m_{2}-1}\;\sum_{l=|n_{1}-n_{2}|+1}^{n_{1}+n_{2}-1}\;\left[\psi_{(k,l)}\right],
\bar{\partial}_{\bar{A}}F_{g}=\frac{1}{2}\bar{C}_{\bar{A}\bar{B}\bar{C}}e^{2K}G^{B\bar{B}}G^{C\bar{C}}\left(D_{B}D_{C}F_{g-1}+\sum_{r}D_{B}F_{r}\cdotD_{C}F_{g-r}\right),
Z(\theta,t)={\frac{(z-1)^{4}}{z}}\,\Big(\sum_{n=0}^{\infty}z^{2n}\Big)\,\Big[1+{\frac{(1-z)^{4}}{z^{2}}}\,q^{2}+{\frac{(1-z)^{4}(1+7z^{2}+z^{4})}{z^{4}}}\,q^{4}+\dots\Big]\,.
S_{AB}={\frac{i}{2}}e^{K_{V}/2}~(\sigma_{x})_{A}{}^{C}~\epsilon_{BC}~P_{I}^{x}(q)~X^{I}(z)={\frac{i}{2}}(\sigma_{x})_{A}{}^{C}~\epsilon_{BC}~P_{I}^{x}(q)~L^{I}\,.
{H}_{\theta\theta}={\frac{{r}^{3}}{2}}({\frac{{H}_{rr}}{A}})^{\prime}+{\frac{{r}^{2}{H}_{rr}}{A}}+{\frac{{r}^{3}B^{\prime}{H}_{rr}}{4AB}}+{\frac{{r}^{3}B^{\prime}{H}_{tt}}{4{B}^{2}}}.
{\frac{1}{2\ell}}\sum_{n}\,2(-1)^{n}\,{\frac{1}{k^{2}+(k^{5})^{2}}}=\oint{\frac{dk^{5}}{2\pi}}{\frac{2e^{ik^{5}\ell}}{e^{2ik^{5}\ell}-1}}{\frac{1}{k^{2}+(k^{5})^{2}}}\,
\begin{array}{rcl}{}&{}&{{\calR}=\bar{\calC}{\calK}~,}\\{}&{}&{\bar{\calC}=\displaystyle\sum_{m=0}^{p-1}\frac{q^{-m(m-1)/2}}{[m]!}\left(-(q-q^{-1})\right)^{m}e^{m}\otimesf^{m}~,}\\{}&{}&{{\calK}=\displaystyle\frac{1}{p}\sum_{n,n^{\prime}=0}^{p-1}q^{2nn^{\prime}}t^{n}\otimes\bar{t}^{n^{\prime}}~.}\\\end{array}
H_{\mathrm{eff}}=\frac{1}{2}p_{r}^{2}+\frac{1}{2r^{2}}p_{\theta}^{2}+\frac{1}{2}p_{z}^{2}+\frac{1}{2}\omega^{2}r^{2}+\xi\Big[p_{z}+gp_{\theta}\Big]+F(\xi)p_{\xi}-F^{\prime}(\xi){\calP}_{1}\eta^{1}-{\calP}_{2}\eta^{1}\\\,
i\frac{\partial}{\partialt}\langleq^{\prime},t^{\prime}\midq,t\rangle=-\frac{1}{2}\frac{\partial^{2}}{\partialq^{2}}\langleq^{\prime},t^{\prime}\midq,t\rangle+V(q^{\prime})\langleq^{\prime},t^{\prime}\midq,t\rangle
\widehat{A}_{\mu}=A_{\mu}+\frac1m\varepsilon_{\mu\nu\rho}f(\partial^{2}/m^{2})\partial^{\nu}A^{\rho}-\frac1{m^{2}}h(\partial^{2}/m^{2})(g_{\mu\nu}\partial^{2}-\partial_{\mu}\partial_{\nu})A^{\nu}\;,
{\hat{\gamma}}_{(\lambda,\mu,\alpha)}^{r}\equiv{\hat{O}}_{\lambda}^{r_{1}}\otimes{\hat{O}}_{\mu}^{r_{2}}\otimes{\hat{\sigma}}_{\alpha}^{r_{3}},\quad{\hat{\gamma}}_{r}^{(\lambda,\mu,\alpha)}\equiv{\hat{O}}_{r_{1}}^{\lambda}\otimes{\hat{O}}_{r_{2}}^{\mu}\otimes{\hat{\sigma}}_{r_{3}}^{\alpha}={}^{*}\delta^{\lambda\tau}{}^{*}\delta^{\mu\nu}\delta^{\alpha\beta}{\hat{\gamma}}_{(\tau,\nu,\beta)}^{r},
S=-\frac{1}{2}\sum_{i,j}\mathscr{M}_{ij}\phi^{i}\phi^{j}-\frac{1}{2}\sum_{a,b}\mathscr{F}_{ab}t^{a}t^{b}-\frac{1}{3}\sum_{i,j,k}\mathscr{G}_{ijk}\phi^{i}\phi^{j}\phi^{k}+\sum_{i,a,b}\mathscr{G}_{iab}\phi^{i}t^{a}t^{b},
\langle:\!\!\phi^{2}(0)\!\!:\rangle=\sum_{n,n_{1}}\frac{2Zn}{(2n+2n_{1})!n!n_{1}!}\frac{(g/\mu)^{2n}(g/\mu_{1})^{2n_{1}}}{(16\pi^{2})^{n+n_{1}}}\,.
\mu^{\prime\rho}=-\left(\partial_{\sigma}\eta^{*\alpha(\rho)}\right)B_{\alpha}^{\;(\sigma)}+\eta^{*\alpha(\sigma)}\partial_{\sigma}B_{\alpha}^{\;(\rho)}+\varepsilon^{\alpha\beta\gamma\delta}A_{\alpha\beta}^{*\;\;(\sigma)}\partial_{\sigma}A_{\gamma\delta}^{*\;\;(\rho)},
[{\calD}\mu]={\calD}A^{\mu}{\calD}\lambda{\calD}\pi_{\lambda}{\calD}\rho{\calD}\pi_{\rho}{\calD}\xi\prod_{\beta=0}^{3}\delta(\Gamma_{\beta}[A^{0}+\xi,A^{i},\lambda,\rho])det\mid\{\tilde{\Omega}_{\alpha},\Gamma_{\beta}\}\mid,
\rho_{\mathrm{prod}}(p,k)=\sum_{s}b^{\dagger}(p,s)d^{\dagger}(k-p,r)\xi^{s\dagger}\boldsymbol{\sigma}\cdot\hat{\boldsymbol{q}}\xi^{r}e^{it(E_{p}+E_{p+k}-\omega_{k})}
{\cal{F}}\left\{\left(P+M^{2}\right)^{-1}\right\}(x)=\frac{\pi}{2}M^{\frac{\nu}{2}-1}Q_{-}^{\frac{1}{2}\left(1-\frac{\nu}{2}\right)}{\cal{J}}_{\frac{\nu-3}{2}}\left(MQ_{-}^{\frac{1}{2}}\right)
\partial^{0}J_{05}^{3}\left(x\right)-g\epsilon^{ab3}\frac{1}{2}\left\{J_{05}^{a}\left(x\right),\alpha_{0}^{b}\left(x\right)\right\}+\partial^{1}J_{15}^{3}\left(x\right)=\frac{g}{2\piL}\left(e^{3}+e^{3\,\dagger}\right).
\tilde{\psi}_{B}(r,\nu)=\sum_{i=1}^{2}(-1)^{i+1}\Sigma((-1)^{i}\bar{R}_{i})\bar{\gamma}_{i}^{-2}\tilde{\psi}_{b,i}(\nu)\frac{\partial}{\partialr^{\prime}}G(r,r^{\prime},\nu)\Bigg|_{r^{\prime}=(-1)^{i}\bar{R}_{i}}\,.
G_{ab}=\left[\begin{array}{cc}{\gamma_{ik}}&{0}\\{0}&{-\delta_{\alpha\beta}}\\\end{array}\right]+O(\rho)\rightarrowG_{ab}^{eff}=\left[\begin{array}{cc}{\gamma_{ik}}&{0}\\{0}&{-h_{\alpha\beta}(\xi)}\\\end{array}\right]+O(\xi)
\hat{\calL}_{CSM}^{0}={\frac{1}{2}}\epsilon^{\alpha\mu\nu}B_{\alpha}^{*}G_{\mu\nu}-{\frac{1}{2M}}G_{\mu\nu}^{*}G^{\mu\nu}+{\frac{1}{\hat{\alpha}}}(\partial_{\mu}B^{*\mu})(\partial_{\nu}B^{\nu})\;,
\mathcal{S(}\Sigma\mathcal{)=}\intd^{4}x\frac{\delta\Sigma}{\delta\varphi^{i}}\frac{\delta\Sigma}{\delta\varphi^{*i}}\;+\;\frac12C_{\;AB}^{C}\varepsilon^{A}\varepsilon^{B}\frac{\partial\Sigma}{\partial\varepsilon^{C}}\;,
P=\frac{2\pi\lambda}{[\nu!(\nu-1)!]^{2}}\frac{|\Gamma\left(x_{1}+i\lambda/2\right)\Gamma\left(x_{2}+i\lambda/2\right)|^{2}}{|\Gamma(1+i\lambda)|^{2}}\left(\frac{\omegar_{0}}{2}\right)^{2\nu}\,
\tilde{k}^{u\Lambda}=i{\nu}_{m}^{(1)\Lambda}\left(\frac{\partial}{\partialC_{0}}-\frac{\partial}{\partial\bar{C}_{0}}\right)^{u}+i\nu_{m}^{(2)\Lambda}\left(\frac{\partial}{\partialS}-\frac{\partial}{\partial\bar{S}}\right)^{u}.
S=6\intd^{4}x\big[-{\frac{RR^{\prime2}}{N}}-NR+{\frac{1}{12}}{\frac{R^{3}}{N}}D^{\prime2}+{\frac{Q^{2}}{48\pi^{4}}}{\frac{N}{R^{3}}}e^{{\frac{2}{\sqrt3}}D}\big],
\langle\langle\prod_{i=1}^{L}\intV_{n_{i}}\rangle\rangle=\mu^{S}\Gamma(-S)\langlec{\bar{c}}V_{n_{1}}(0)c{\bar{c}}V_{n_{2}}(1)c{\bar{c}}V_{n_{3}}(\infty)\prod_{i=4}^{L}\intV_{n_{i}}(\intV_{0})^{S}{\frac{1}{R!}}(Q_{-})^{R}\rangle,
|\mu_{k}(\eta_{\mathrm{i}})|=\frac{1}{\sqrt{2\omega(k,\eta_{\mathrm{i}})}},\qquad\mu_{k}^{\prime}(\eta_{\mathrm{i}})=[{\calH}+i\omega(k,\eta_{\mathrm{i}})]\mu_{k}(\eta_{\mathrm{i}}).
Z[A_{+}]=\int\!Dge^{-S[g]-\frac{i}{4\pi}\int\!d^{2}x\,[\mathrm{Tr}(A_{+}(\partial_{-}g)g^{-1})+\mathrm{Tr}(A_{-}g^{-1}\partial_{+}g)+A_{+}gA_{-}g^{-1}+A_{+}A_{-}]}
\frac{\epsilon_{\rho}(q)}{Z_{3}^{1/2}}q^{2}{\calD}_{\mu\rho}(q)=\frac{\epsilon_{\mu}(q)}{Z_{3}^{1/2}}q^{2}\frac{Z_{3}}{q^{2}}=Z_{3}^{1/2}\epsilon_{\mu}(q),
\textstyle{\frac{9}{2}}D_{j\delta}D_{i\alpha}\tilde{\mathrm{A}}^{\beta\gamma}=D_{i\delta}D_{j\epsilon}\tilde{\mathrm{A}}^{\epsilon[\beta}\delta_{\alpha}^{~\gamma]}+\delta_{\delta}^{~[\beta}\delta_{\alpha}^{~\gamma]}D_{i\epsilon}D_{j\eta}\tilde{\mathrm{A}}^{\epsilon\eta}
X_{\pm}(s)=\frac{\beta}{2}\sum_{n}\left[E_{n}^{2}+\frac{\lambda\phi^{2}}{3}+\mu^{2}\pm2\muE_{n}\pm\frac{\mu\lambda\phi^{2}}{3E_{n}}\pm\frac{\lambda^{2}\phi^{4}}{144\muE_{n}}\mp\frac{\mu\lambda^{2}\phi^{4}}{36E_{n}^{3}}\right]^{-\frac{s}{2}}\;.
\begin{array}{rcl}{A_{(321)}(Q)}&{=}&{Y_{1}(k_{2}-k_{3})Y_{2}(k_{1}-k_{3})Y_{1}(k_{1}-k_{2})A_{(123)}(Q)}\\{}&{=}&{Y_{2}(k_{1}-k_{2})Y_{1}(k_{1}-k_{3})Y_{2}(k_{2}-k_{3})A_{(123)}(Q)~.}\\\end{array}
=4\pi\frac{\partial\beta_{n}}{\partialM}-P_{nz}\sum_{B}\frac{\partial\beta_{B}}{\partialM}\frac{{\calP}(z_{B})}{(z_{n}-z_{B})^{2}\prod_{C\neqB}(z_{B}-z_{C})}+P_{nz}~p_{1}^{\prime}(z_{n}).
ds^{2}=-4dx^{+}dx^{-}-\mu^{2}\left(4x_{2}^{2}+\sum_{I=5}^{8}x_{I}^{2}\right){dx^{+}}^{2}+\sum_{i=1}^{8}{dx_{i}}^{2}+4\mux_{4}dx_{3}dx^{+},
S_{\mathrm{T}}=\intd^{4}z\,d^{4}\bar{z}\,\mathrm{tr}\Bigr\{\mathrm{~\frac{1}{12}~}\Bigr(\epsilon^{ab}F_{\!~~~~ab}^{AB}\epsilon^{cd}F_{ABcd}-\epsilon_{AB}F_{\!~~~~ab}^{AB}\epsilon_{CD}F^{CDab}-F_{\!~~~~ab}^{AB}F_{AB}^{\!~~~~ba}\Bigr)\Bigr\},
S_{A}=-\frac{2}{\kappa_{11}^{2}}\intd^{11}x\left\{\frac{\sqrt{|g|}}{2\cdot2\cdot4!}F^{\mu\nu\lambda\xi}F_{\mu\nu\lambda\xi}+\frac{1}{12}\frac{1}{3!(4!)^{2}}\epsilon^{\mu_{1}\ldots\mu_{11}}A_{\mu_{1}\mu_{2}\mu_{3}}F_{\mu_{4}\ldots\mu_{7}}F_{\mu_{8}\ldots\mu_{11}}\right\}
[D_{a\alpha},{\calL}]=-D_{a\alpha}\lambda^{b\beta}D_{b\beta}=(\textstyle{\frac{1}{2}}\delta_{a}{}^{b}\delta_{\alpha}{}^{\beta}\hat{\lambda}(z)+\delta_{a}{}^{b}\hat{w}^{\beta}{}_{\alpha}(z)-\delta_{\alpha}{}^{\beta}\hat{t}{}_{a}{}^{b}(z))D_{b\beta}\,.
{\calG}_{\mathrm{rel}}\simeq\frac{g^{2}}{16\pi}\sum_{V}\lambda_{V}\left(\frac{1}{r_{V}}d{\bfr}_{V}^{2}+r_{V}\,(d\psi_{V}+{\bfw}({\bfr}_{V})\cdotd{\bfr}_{V})^{2}\right)+\cdots.
\begin{array}{c}{\gamma_{0}=\sigma_{1}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right),\quad\gamma_{1}=\sigma_{2}=\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right),}\\{\gamma_{5}=-i\gamma_{0}\gamma_{1}=\sigma_{3}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right),}\\\end{array}
A_{123}=-\left(\frac{z^{*}}{|z|}\right)^{3}e^{\hat{K}/2}\left[\hat{K}^{S}F_{S}+\hat{K}^{T}F_{T}\left(1+\sum_{i=1}^{3}\frac{n_{i}}{3}-\frac{T+T^{*}}{3}\frac{y_{123}^{\prime}(T)}{y_{123}(T)}\right)\right]
\langle\dot{b}_{n}^{2}\rangle_{\mathrm{sol}}=\frac{\omega_{n}}{2}\,,\quad\langleb_{n}^{2}\rangle_{\mathrm{sol}}=\frac{1}{2\omega_{n}}\,,\quad\langle\eta_{n}\xi_{n}\rangle_{\mathrm{sol}}=\frac{i}{2}\,.
\hat{t}=\frac{2}{2+\lambdam}t^{1+\frac{\lambdam}{2}}\;,\;\;\hat{a}=At^{n+\frac{\lambdam}{2}}\;,\;\;\hat{R}=\frac{2}{3}\;(1-3\lambda^{2})\;t^{-(2+\lambdam)}\;.
{\mathrm{\bf~M}}(v)=\left(\begin{array}{cc}{\frac{1+v^{2}}{1-v^{2}}}&{\frac{+2v}{1-v^{2}}}\\{\frac{+2v}{1-v^{2}}}&{\frac{1+v^{2}}{1-v^{2}}}\\\end{array}\right)=\frac{1}{\gamma^{2}}\left(\begin{array}{cc}{1+v^{2}}&{+2v}\\{+2v}&{1+v^{2}}\\\end{array}\right)\equiv\frac{1}{\gamma^{2}}{\mathrm{\bf~L}}(v)\,.
\Lambda_{\cdot\,B}^{A\,\cdot}(\omega)=\delta_{B}^{A}+\omega_{\cdot\,B}^{A\,\cdot}+\frac{1}{2}\,\omega_{\cdot\,C}^{A\,\cdot}\omega_{\cdot\,B}^{C\,\cdot}+...+\frac{1}{n!}\,\underbrace{\omega_{\cdot\,C}^{A\,\cdot}\omega_{\cdot\,C^{\prime}}^{C\,\cdot}...\omega_{\cdot\,B}^{D\,\cdot}}_{n}+...\,.
D_{0}(y)=P_{0}\,\,\,y^{-\frac{b}{2}}F\left[\frac{1}{2}\left(a+b\right),1-b,y\right]+Q_{0}\,\,\,y^{\frac{b}{2}}F\left[\frac{1}{2}\left(a-b\right),1+b,y\right],
\delta=\epsilon^{0}\,\underline{\epsilon}_{0}+\epsilon^{+}\,\underline{\epsilon}_{+}+\epsilon^{-}\,\underline{\epsilon}_{-}+\alpha^{I}\,\underline{\alpha}_{I}+\beta_{I}\,\underline{\beta}^{I}+\hat{\alpha}{}^{I}\,\underline{{\hat{\alpha}}}_{I}+\hat{\beta}_{I}\,\underline{{\hat{\beta}}}{}^{I}+\omega^{i}\,\underline{\omega}_{i}\.
\alpha_{ab}\;:=\;\frac{\pi}{\pi\!+\!gN}U_{1a}U_{1b}+\sum_{I=2}^{N}U_{Ia}U_{Ib}\;=\;\frac{\pi}{\pi\!+\!gN}\frac{1}{N}\;+\;\delta_{ab}-\frac{1}{N}\;=\;\delta_{ab}-\frac{g}{\pi\!+\!gN}\;.
\dot{S}=\intdV\Bigl[{\frac{\sigma_{ik}^{\prime}}{2T}}\Bigl({\frac{\partialv_{i}}{\partialx_{k}}}+{\frac{\partialv_{k}}{\partialx_{i}}}\Bigr)-{\frac{{\bfq}\cdot\mathrm{grad}\,T}{T^{2}}}\Bigr]\.
\left[\Gamma_{FT{\overline{{F}}}}^{\nu}(p_{1},k,-p_{2})\right]^{[r,0]}=t^{\nu\mu}(k)\left\{C_{0,r}^{\mu}(p_{1},-p_{2})+\sum_{t=1}^{r}\,C_{t,r}^{\mu}(p_{1},-p_{2})\left(\frac{\Lambda^{2}}{k^{2}+{\overline{{u}}}_{r,2t}^{\prime}\Lambda^{2}}\right)\right\}\,,
\begin{array}{c}{lim}\\{\varepsilon\rightarrow0}\\\end{array}\left\{\begin{array}{c}{lim}\\{\theta\rightarrow\theta_{0}^{+}}\\\end{array}\frac{\partial}{\partial\theta}ImI^{\prime}-\begin{array}{c}{lim}\\{\theta\rightarrow\theta_{0}^{-}}\\\end{array}\frac{\partial}{\partial\theta}ImI^{\prime}\right\}=0
\begin{array}{c}{\chi(a,b)=\chi(a)+\chi(b)+n_{2}(a)n_{1}(b)}\\{\chi(c,d)=\chi(c)+\chi(d)+n_{2}(c)n_{1}(d)}\\{\chi(a,d)=\chi(a)+\chi(d)+n_{2}(a)n_{1}(d)}\\{\chi(b)=\chi(c)}\\{(\epsilon_{(b,c)})_{n}=(-)^{n_{2}(b)}q^{\frac{1}{2}(n_{1}(b)-n_{2}(b))}}\\\end{array}
\tilde{G}(p,y;s)=\frac{1}{p^{2}-s^{2}}-\frac{1}{12}\frac{R^{0}}{(p^{2}-s^{2})^{2}}+\frac{2}{3}\frac{{R^{0}}^{\mu\nu}p_{\mu}p_{\nu}}{(p^{2}-s^{2})^{3}}+\mathrm{O}(R_{;\mu}^{0},(R^{0})^{2})\,.
g_{\mu\nu}g^{\alpha\beta}Q_{\alpha\rho}^{\kappa}Q_{\kappa\beta}^{\rho}(-2+\frac{a(a-2)(5a^{2}+45a+26)}{30(1-a)^{2}})+g^{\alpha\beta}Q_{\mu\alpha\kappa}Q_{\beta\nu}^{\kappa}(\frac{-a(a-2)}{2}),
\Gamma[A,\langleQ\rangle,\langleC\rangle,\langle\overline{{C}}\rangle]=W[A,J,\xi,\overline{{\xi}}]-\intd^{4}x(J_{\mu}^{i}\langleQ^{\mu,i}\rangle+\overline{{\xi}}^{i}\langleC^{i}\rangle+\langle\overline{{C}}^{i}\rangle\xi^{i})
J_{b}^{\mu_{1}\mu_{2}\ldots\mu_{p}}\equive_{b}\sum_{i=1}^{p}\intd\xi_{b}^{0}d\xi_{b}^{1}\ldots\hat{d\xi_{b}^{i}}\ldotsd\xi_{b}^{p}\left[K_{bi}^{\mu_{1}\mu_{2}\ldots\mu_{p}}\delta^{(D)}(X-X_{b})\right]_{\xi_{b}^{i}=0}^{\xi_{b}^{i}=l_{b}^{i}},
\begin{array}{cccc}{}&{U(1)_{k}}&{{}}&{U(1)_{R}}\\{\Phi_{i}}&{\delta_{ik}}&{}&{1}\\{\lambda_{i}}&{-\delta_{ik}}&{}&{1}\\{m_{ij}}&{-(\delta_{ik}+\delta_{jk})}&{}&{0}\\{g_{ijl}}&{-(\delta_{ik}+\delta_{jk}+\delta_{lk})}&{}&{-1}\\\end{array}
\|C_{2m}^{(p,q)}\|\leq\left({\frac{\Lambda}{M}}\right)^{2-\epsilon}{\frac{1}{\Lambda^{2m-6}}}P\left(ln{\frac{\Lambda}{\Lambda_{R}}}\right)\left({\frac{M}{\Lambda_{R}}}\right)^{\epsilon^{\prime}},q\geq1,M>\Lambda_{R},\epsilon,\epsilon^{\prime}\in(0,1)
{\hat{\calT}}_{J}^{R}=\sum_{J^{1},J^{2}}\left\langle\begin{array}{ll}{R^{1}}&{R^{2}}\\{J^{1}}&{J^{2}}\\\end{array}\right.\left|\begin{array}{l}{R}\\{J}\\\end{array}\right\rangle{\hat{\calT}}_{J^{1}}^{R^{1}}(1){\hat{\calT}}_{J^{2}}^{R^{2}}(2),
a_{D}(u,\Lambda)=\partial_{a}{\calF}={\frac{\sqrt2}{\pi}}\int_{\Lambda^{2}}^{u}{\frac{dx\sqrt{x-u}}{\sqrt{x^{2}-\Lambda^{4}}}},\qquada(u,\Lambda)={\frac{\sqrt2}{\pi}}\int_{-\Lambda^{2}}^{\Lambda^{2}}{\frac{dx\sqrt{x-u}}{\sqrt{x^{2}-\Lambda^{4}}}},
\widetildeJ^{0}(k,x_{1},x_{2},x_{1}^{\prime},x_{2}^{\prime})\=\i\frac{1}{k_{0}}\big[g_{1}(e^{ik.x_{1}}-e^{ik.x_{1}^{\prime}})-g_{2}(e^{ik.x_{2}}-e^{ik.x_{2}^{\prime}})\big]\.
\tilde{S}=\int\,d\tau\,d\eta\,(iE^{-1}D\Theta\gamma_{m}{D\Theta}D\hat{X}^{m}+\hat{\Omega}^{m}{\calA}_{m}+{\frac{1}{2}}l\hat{\Omega}^{\alpha\beta}{(\sigma^{mn})}_{\alpha\beta}{\calF}_{mn})
\begin{aligned}{R_{\mu\nu}-\frac{1}{4}H_{\mu\rho\sigma}H_{\nu}{}^{\rho\sigma}+2\nabla_{\mu}\nabla_{\nu}\Phi}&{{}=0,}\\{\nabla^{2}\Phi-2(\nabla\Phi)^{2}+\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}}&{{}=0,}\\{\nabla^{\mu}\left(e^{-2\Phi}H_{\mu\nu\rho}\right)}&{{}=0.}\\\end{aligned}
\frac{1}{2{\pi}}{\int}_{-\infty}^{\infty}dq_{-}\left({\delta}(q_{-}){\int}_{-\infty}^{\infty}dq_{-}\frac{1}{q_{-}^{2}}-\frac{1}{q_{-}^{2}}\right)e^{-q_{-}x^{-}}=\frac{1}{{\pi}}{\int}_{0}^{\infty}dq_{-}\frac{1-\mathrm{cos}q_{-}x^{-}}{q_{-}^{2}}=\frac{|x^{-}|}{2},
\Psi_{w}^{\prime\prime}+\left(m^{2}-{\frac{\xi_{w}^{\prime\prime}}{\xi_{w}}}\right)\Psi_{w}=0,~~~~~~~~~~~~\Psi_{i}^{\prime\prime}+\left(m^{2}-{\frac{\xi_{i}^{\prime\prime}}{\xi_{i}}}\right)\Psi_{i}=0.
\ddot{\hat{\alpha}}+2\dot{\hat{\alpha}}^{2}+\frac{1}{6}\dot{\hat{\phi}}^{2}=\frac{\sqrt{2}}{36}\kappa^{2}\sigma((1-3w)\rho+4U+3\sqrt{2}bU^{\prime})e^{-\sqrt{2}b\hat{\phi}}-\frac{1}{6}V(\phi^{(0)}),
\hat{{\calG}}_{0}^{Box}[a,b;E]=\hat{G}_{0}^{Box}[a,b;E]+\frac{\hat{G}_{0}^{Box}[a,0;E]\hat{G}_{0}^{Box}[0,b;E]}{\frac{1}{v_{1}}-\hat{G}_{0}^{Box}[0^{+},0;E]}.
G_{l,s}^{\,\alpha}(\vec{\alpha},\vec{\lambda},p)=\Bigl[-\tilde{G}_{l,s}(\vec{\alpha},\vec{\lambda}^{\prime},0)-\frac{1}{2}\sum_{\nu,\,\mu=0}^{3}\,p_{\mu}\,p_{\nu}\,\partial_{\mu}\partial_{\nu}\tilde{G}_{l,s}(\vec{\alpha},\vec{\lambda}^{\prime},0)\Bigr]\,\Theta(\alpha_{s}-\alpha)
\frac12{\bar{e}}_{\alpha}^{\;\;i}{\bar{e}}_{\beta}^{\;\;j}\frac{\partial\delta\Gamma^{(1)}}{\partial\bar{g}^{ij}}={\bar{e}}_{\alpha}^{\;\;i}{\bar{e}}_{\beta}^{\;\;j}\deltaB_{ij}^{\bar{g}(2)}+e_{\alpha}^{\;\;\mu}e_{\beta}^{\;\;\nu}X_{\mu\nu}^{g}.
\mathcal{W}^{\mathrm{U(1)}}=\partial_{\mu}\frac\delta{\deltaA_{\mu}}+\varepsilon^{ab}\left(A_{\mu}^{a}\frac\delta{\deltaA_{\mu}^{b}}+c^{a}\frac\delta{\deltac^{b}}+b^{a}\frac\delta{\deltab^{b}}+\overline{{c}}^{a}\frac\delta{\delta\overline{{c}}^{b}}+A_{\mu}^{a*}\frac\delta{\deltaA_{\mu}^{b*}}+c^{a*}\frac\delta{\deltac^{b*}}\right)\;.
Z_{cs}={\calN}\int{\calD}a_{\mu}{\calD}c{\calD}\bar{c}\mathrm{exp}\left(i\intd^{3}x(\frac{1}{2}\epsilon_{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}+a_{\mu}j_{\mu}+i\bar{c}\partial^{2}c+\frac{1}{2\rho}(\partial_{\mu}a_{\mu})^{2})\right)\;.
\intd^{10}x\sqrt{-g}\frac{{\tilde{F}}_{5}^{2}}{4\cdot5!}\rightarrow\intd^{4}x\sqrt{-g_{4}}\intd^{6}y\sqrt{g_{6}}\,{\frac{e^{-4A}}{2}}\,(\partial_{m}\alpha)^{2}\,.
\Phi(z,x)=\intdqd^{d-1}\vec{k}z^{\frac{d}{2}}J_{\nu}(qz)\left(\frac{q}{2(2\pi)^{d-1}\omega}\right)^{\frac{1}{2}}(e^{-i\omegat+i\vec{k}\cdot\vec{x}}\hat{b}_{q\vec{k}}+\mathrm{h}.\mathrm{c}.)
\frac{V^{2}}{H_{0}^{2}V_{0}^{2}}=\frac{\Omega_{B}}{2}\left(\frac{V}{V_{0}}\right)+\frac{1}{4}\left[\Omega_{M}\left(\frac{V}{V_{0}}\right)^{\frac{\omega+3}{2(1+\omega)}}+\Omega_{\phi}\left(\frac{V}{V_{0}}\right)^{\frac{3}{2}}\right]^{2}.
i\frac{\partialF_{l}^{(\pm)}}{\partialt}=-\frac{1}{2}\frac{\partial^{2}F_{l}^{(\pm)}}{\partialq^{2}}+\frac{q^{\prime}\pmq}{it}\frac{\partialF_{l}^{(\pm)}}{\partialq^{\prime}}+U_{l}(q^{\prime})F_{l}^{(\pm)}.
a_{l/2}(f,R,\Omega,K,E,\Gamma,S)=\sum_{r=0}^{l-2}f^{(r)}\sum_{i=1}^{i_{l-r-1}}{\calU}_{i}^{(l/2,(l-r)/2)}[\Gamma^{2}]I_{i}^{(l-r)/2}[R,\Omega,K,E,\Gamma,S],
\phi(x)\,\star\,\chi(x)=\phi(y)\,\mathrm{exp}\,ix_{\mu}{\widetilde{\gamma}}\^{\mu}\left({\frac{\overleftarrow{\partial}}{\partialy}},{\frac{\overrightarrow{\partial}}{\partialz}}\right)\,\chi(z)\big|_{y=z=x}\,,
E_{\pmN}=\Big\lbrace\mu^{2}+M_{N}^{2}\pm\Big\lbrack4\mu^{2}M_{N}^{2}+|\bar{\Phi}|^{4}\Big(V_{tree}^{\prime\prime}\Big)^{2}\Big\rbrack^{1/2}\Big\rbrace^{1/2}\;,
S_{\sigma}[X]=-\frac{1}{4\pi\alpha^{\prime}}\int_{\partial\calM}d^{2}z~\left(\sqrt{h}~h^{\alpha\beta}g_{IJ}\,\partial_{\alpha}X^{I}\partial_{\beta}X^{J}+\epsilon^{\alpha\beta}B_{IJ}\,\partial_{\alpha}X^{I}\partial_{\beta}X^{J}\right)\,,
\biggl[{\frac{\bigl(\bar{u}(\bar{p},\sigma)\gamma^{+}(\notp+m)\gamma^{\nu}u(p,s)\bigr)\bigl(-f_{\nu}(k,p)+{\frac{k_{\nu}}{k^{+}}}f^{+}(k,p)+{\frac{\delta_{\nu+}}{k^{+}}}k\cdotf\bigr)}{p^{-}-k^{-}-(p-k)^{-}}}\biggr]\;.
\frac{\phi(\nu/2+1)}{\phi(\nu/2)}=\frac{h^{2}}{4(\nu+1)}+\frac{h^{6}(\nu^{2}+4\nu+7)}{128(\nu-1)(\nu+1)^{3}(\nu+2)}+O(h^{10})=-\frac{c_{2}^{\nu}}{c_{0}^{\nu}}
f+h.c=-4\pi(T-{\bar{T}})(U-{\bar{U}})\int\frac{d^{2}\tau}{\tau_{2}^{2}}{\bar{F}}({\bar{\tau}})\partial_{{\bar{\tau}}}[\tau_{2}^{2}\partial_{{\bar{\tau}}}(\tau_{2}\sum_{P_{L},P_{R}}{\bar{P}}_{L}{\bar{P}}_{L}e^{i\pi\tau|P_{L}|^{2}}e^{-i\pi{\bar{\tau}}|P_{R}|^{2}})],
S=-\frac{1}{2\sqrt{\alpha^{\prime}}}\int\!{d\tau}[\frac{m^{2}R^{2}}{\alpha^{\prime}}-R^{2}\lambda^{2}-2kR\lambda{A_{\mu}}{\dot{x}}^{\mu}-(g_{\mu\nu}+k^{2}A_{\mu}A_{\nu}){\dot{x}}^{\mu}{\dot{x}}^{\nu}].
S_{bulk}\,=\,{\frac{1}{2\kappa^{2}}}\intd^{10}x\sqrt{-G}e^{-2\phi}\bigl[R+4G^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi-{\frac{1}{12}}H_{\mu\nu\alpha}H^{\mu\nu\alpha}\bigr]\,,
{s_{_\mathrm{S}}}=s\Big(1-{\frac{{v}^{2}}{c^{2}}}\Big)^{-1/2}={\frac{y^{2}}{\mu}}\,\qquad{\mu_{_\mathrm{N}}}=\mu\Big(1-{\frac{{v}^{2}}{c^{2}}}\Big)^{-1/2}={\frac{y^{2}}{s}}\,
\begin{array}{lcl}{{a^{\prime}}_{1}}&{=}&{\displaystyle{-\frac{16(4\pi)^{3/2}}{7\mu^{3}}\biggl[\Lambda-\frac{9\mu^{2}}{5(31+36~\xi)}\biggr]}\,\nonumber}\\{}&{}&{\nonumber}\\{{a^{\prime}}_{2}}&{=}&{\displaystyle{-\frac{72(4\pi)^{3/2}}{(31+36~\xi)\mu^{3}}}\.}\\\end{array}
F_{g}=\epsilon_{ij}^{1}\epsilon_{ij}^{2}k_{0}^{2}{\frac{\Gamma(1-2k_{0}^{2})\Gamma(\vec{k}\cdot\vec{p}+k_{0}^{2})}{\Gamma(\vec{k}\cdot\vec{p}-k_{0}^{2}+1)}}\.
{\mathcalU}(\Lambda)|\hat{\bboxp}j_{3}({\mathsf{s}}_{R}j_{R})^{-}\rangle=\sum_{j_{3}^{\prime}}D_{j_{3}^{\prime}j_{3}}^{j_{R}}({\mathcalR}(\Lambda,\hat{p}))|\bbox{\Lambda}\hat{\bbox{p}}j_{3}^{\prime}({\mathsf{s}}_{R}j_{R})^{-}\rangle,
\dot{\calG}_{B}(\tau_{i},\tau_{i})\rightarrow\dot{\calG}_{B}(\tau_{i},\tau_{i})-{\calG}_{F}(\tau_{i},\tau_{i})={\frac{i}{\mathrm{sin}(eFT)\mathrm{cos}(eFT)}}-{\frac{i}{eFT}}\quad.
i{\calG}_{4}(p)=-[8ie^{4}B^{4}]\left[{\frac{2p_{\bot}^{2}p_{L}^{2}-3(p_{\bot}^{2})^{2}}{(p^{2}-m^{2})^{7}}}\right]D-8e^{5}B^{5}\left[{\frac{15(p_{\bot}^{2})^{2}-16p_{\bot}^{2}p_{L}^{2}+2p_{L}^{2}}{(p^{2}-m^{2})^{8}}}\right]\bar{D}.
O=\left[\begin{array}{cccccccc}{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}&{1}&{0}&{1}&{1}\\{0}&{1}&{1}&{0}&{0}&{1}&{0}&{1}\\{0}&{1}&{1}&{1}&{0}&{0}&{1}&{0}\\{0}&{0}&{1}&{1}&{1}&{0}&{0}&{1}\\{0}&{1}&{0}&{1}&{1}&{1}&{0}&{0}\\{0}&{0}&{1}&{0}&{1}&{1}&{1}&{0}\\{0}&{0}&{0}&{1}&{0}&{1}&{1}&{1}\\\end{array}\right].
ds^{2}=\sqrt{H_{A}}\left[{\frac{1}{\sqrt{H_{B}(y_{a},y_{\perp})}}}\left(-dT^{2}+d{{y}}_{I}^{2}+d{{y}}_{b}^{2}\right)+{\sqrt{H_{B}(y_{a},y_{\perp})}}\left(d{{y}}_{a}^{2}+d{{y}}_{\perp}^{2}\right)\right]\,.
M(p;q_{1},q_{2})=\int\frac{dx\;dy}{(1+x+y)^{3}}exp[\frac{xy}{1+x+y}\frac{p^{2}}{\Lambda^{2}}+\frac{x}{1+x+y}\frac{q_{1}^{2}}{\Lambda^{2}}+\frac{y}{1+x+y}\frac{q_{2}^{2}}{\Lambda^{2}}],
S=\int\Theta\,,\qquad\Theta=p_{a}\mathrm{d}x^{a}+is\frac{\bar{z}\mathrm{d}z-z\mathrm{d}\bar{z}}{1-z\bar{z}}\equivp_{a}\mathrm{d}x^{a}+\Sigma_{\calL}\,,\qquad\mathrm{d}\Sigma_{\calL}=\Omega_{\calL}\,.
V={x^{p-2}}\,{y^{p-2}}-{x^{p-4}}\,{y^{p-4}}\,\left({x^{3}}+{y^{3}}\right)+{x^{p-6}}\,{y^{p-6}}\,\left({x^{6}}+{y^{6}}\right)+\cdots+x\,y\,\left({x^{3\,{\frac{p-3}{2}}}}+{y^{3\,{\frac{p-3}{2}}}}\right)
\Gamma_{\mu\nu}^{AB}(p)=\delta^{AB}\left(im\epsilon_{\mu\nu\lambda}p^{\lambda}(1+\Pi_{2}(p^{2}))+(p^{\mu}-{\frac{p^{2}n^{\mu}}{n\cdotp}})(p^{\nu}-{\frac{p^{2}n^{\nu}}{n\cdotp}})\Pi_{3}(p^{2})\right)
\frac{\partial\tilde{F}}{\partialt}=\frac{1}{2}\frac{\partial^{2}\tilde{F}}{\partialq^{2}}+\frac{1}{\sqrt{t}\Phi}\frac{d\Phi}{dz}\frac{\partial\tilde{F}}{\partialq^{\prime}}-V(q^{\prime})\tilde{F}.
\mathcal{R}_{\mu\nu}=\overline{{\mathcal{R}}}_{\mu\nu}-\overline{{\nabla}}_{\mu}{C^{\alpha}}_{\alpha\nu}+\overline{{\nabla}}_{\alpha}{C^{\alpha}}_{\mu\nu}+{C^{\alpha}}_{\mu\nu}{C^{\beta}}_{\alpha\beta}-{C^{\alpha}}_{\mu\beta}{C^{\beta}}_{\nu\alpha},
\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}}\\\end{array}\quad.
\Theta^{ij}=2\pi\alpha^{\prime}\left(\begin{array}{cccc}{0}&{-\frac{cy^{+}}{R\sqrt{c^{2}+g_{s}^{2}d^{2}}}}&{0}&{0}\\{\frac{cy^{+}}{R\sqrt{c^{2}+g_{s}^{2}d^{2}}}}&{0}&{0}&{0}\\{0}&{0}&{0}&{-\frac{g_{s}d}{R\sqrt{c^{2}+g_{s}^{2}d^{2}}}}\\{0}&{0}&{\frac{g_{s}d}{R\sqrt{c^{2}+g_{s}^{2}d^{2}}}}&{0}\\\end{array}\right),