harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\tilde{\beta}}_{\mu\nu}^{G}=R_{\mu\nu}+2\gamma\nabla_{\mu}\partial_{\nu}\Phi,\qquad{\tilde{\beta}}^{\Phi}=-R+4\gamma^{2}(\nabla_{\mu}\Phi)(\nabla_{\nu}\Phi)G^{\mu\nu}-4\gamma\nabla^{2}\Phi+Q^{2}
\bar{\zeta}_{\mu\nu}(s,x|L_{b})=\frac{a^{2s}}2\int_{0}^{\infty}\!\!d\lambda\,\mu(\lambda)\int_{{\mathbbR}^{N-1}}\!\!d^{N-1}k\,(\lambda^{2}+a^{2}b)^{-s}\left(\nabla_{\mu}\phi_{n}^{*}\nabla_{\nu}\phi_{n}+\nabla_{\nu}\phi_{n}^{*}\nabla_{\mu}\phi_{n}\right).
{\calA}=\left(\begin{array}{cccc}{X+\rho^{-2}}&{0}&{0}&{0}\\{0}&{X+1}&{0}&{0}\\{0}&{0}&{X}&{h}\\{0}&{0}&{-h}&{X}\\\end{array}\right)\equiv\left(\begin{array}{cc}{\tilde{X}}&{0}\\{0}&{a}\\\end{array}\right)
\gamma_{5}\equivi\bar{\gamma}^{0}\bar{\gamma}^{1}\bar{\gamma}^{2}\bar{\gamma}^{3}\;\quad\gamma_{5}^{2}={\bf1}_{2^{\omega}}\;\quad\left\{\bar{\gamma}^{\mu},\gamma_{5}\right\}=0=\left[\hat{\gamma}^{\mu},\gamma_{5}\right]\.
d{s^{2}}=-\left(1+{\frac{Q}{R^{2}}}-{\frac{\Lambda}{3}}{R^{2}}\right)d{T^{2}}+{{\left(1+{\frac{Q}{R^{2}}}-{\frac{\Lambda}{3}}{R^{2}}\right)}^{-1}}d{R^{2}}+{R^{2}}d{\Omega^{2}}
\partial_{\hat{x}}[(\hat{x}^{2}-1)(\hat{x}^{2}-\alpha^{2})\partial_{\hat{x}}\hat{n}(\hat{x})]-2m\zeta\partial_{\hat{x}}\hat{n}(\hat{x})+2(\hat{x}^{2}-\beta_{D}^{2})\hat{n}(\hat{x})=0,
b_{\kappa}^{(R)}=\frac{ie^{\pi\kappa/2}}{\pi\sqrt{2}}\,\left\{\int_{z_{0}}^{\infty}F_{R}(z,\kappa)\,dz+\frac12\left(\Gamma(-i\kappa)\vphantom{\frac12}\left(\frac{mz_{0}}{2}\right)^{i\kappa}-\Gamma(i\kappa)\left(\frac{mz_{0}}{2}\right)^{-i\kappa}\right)\phi_{M}(0,z_{0})\right\},
{\calL}^{\prime}=\bar{\psi}\gamma^{\mu}\left[i\partial_{\mu}+e{\calA}_{\mu}\right]\psi-\frac{1}{2G}{\calA}_{\mu}{\calA}^{\mu}-\frac{\kappa}{2}\epsilon_{\mu\nu\rho}{\calA}^{\mu}\partial^{\nu}{\calA}^{\rho},
\pm(q^{\mu_{1}^{1}}+\ldots+q^{\mu_{1}^{l_{1}}})\alpha_{1}\pm\ldots\pm(q^{\mu_{n}^{1}}+\ldots+q^{\mu_{n}^{l_{n}}})\alpha_{n}\rightarrow\pm\mu_{1}^{1},\ldots,\mu_{l_{1}}^{1};\ldots;\mu_{n}^{1},\ldots,\mu_{l_{n}}^{1}\,\,,
S={\frac{1}{\sqrt{M^{2}}}}\varepsilon_{ij}L^{i}\tilde{D}L^{j}={\frac{L^{*}{}^{2}}{\sqrt{M^{2}}}}\varepsilon_{ij}\hat{k}^{i}\tilde{D}\hat{k}^{j}={\frac{L^{*}{}^{2}}{\sqrt{M^{2}}}}\overline{\kappa}_{2}\,.
U=\left(\begin{array}{cccc}{1}&{0}&{\cdots}&{0}\\{0}&{\omega}&{\cdots}&{0}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{0}&{0}&{\cdots}&{\omega^{p-1}}\\\end{array}\right)\,,\quadV=\left(\begin{array}{cccc}{0}&{1}&{\cdots}&{0}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{0}&{0}&{\cdots}&{1}\\{1}&{0}&{\cdots}&{0}\\\end{array}\right)\,,\quad\omega=e^{2\pii\theta}\,.
\frac{\partial^{2}Z_{l}(t,r)}{\partialr_{*}^{2}}-\frac{\partial^{2}Z_{l}(t,r)}{\partialt^{2}}-V(r)Z_{l}(t,r)=\frac{4\piq_{s}m_{0}f}{r^{\frac{d-2}{2}}}(\frac{dt}{d\tau})^{-1}\delta(r-r_{p}(t))Y_{l}(0)\,.
A_{\alpha,\,\gamma}^{lepton}=\textstyle{\frac{1}{2}}\sqrt{(\eta\,{R_{1}^{(1)}})^{2}\,(\eta\,{R_{2}^{(1)}})^{2}+(\eta\,{R_{1}^{(1)}})^{2}\,{\vec{\calA}_{\alpha,\,\gamma}}^{2}+(\eta{R_{2}^{(1)}})^{2}\,{\vec{\calB}_{\alpha,\,\gamma}}^{2}+4{A_{\alpha,\,\gamma}^{quark}}^{2}},
H_{SUSY}=[Q_{-},Q_{+}]_{+}=\left(\begin{array}{cc}{{\calA}^{+}{\calA}^{-}}&{0}\\{0}&{{\calA}^{-}{\calA}^{+}}\\\end{array}\right)_{4\mathrm{x}4}=\left(\begin{array}{cc}{{\calH}_{-}}&{0}\\{0}&{{\calH}_{+}}\\\end{array}\right),
R^{2}=l^{2}\frac{r^{2}-r_{-}^{2}}{r_{+}^{2}-r_{-}^{2}},\qquad\left(\begin{array}{c}{\theta_{1}}\\{\theta_{2}}\\\end{array}\right)=\left(\begin{array}{cc}{r_{+}/l^{2}}&{-r_{-}/l^{2}}\\{-r_{-}/l^{2}}&{r_{+}/l^{2}}\\\end{array}\right)\left(\begin{array}{c}{y}\\{t}\\\end{array}\right)
\left\{Q_{\alpha}\,,\bar{Q}_{\dot{\alpha}}\right\}=2P_{\alpha\dot{\alpha}}\,,\quad\left\{Q_{\alpha}\,,Q_{\beta}\right\}=0\,,\quad\left\{\bar{Q}_{\dot{\alpha}}\,,\bar{Q}_{\dot{\beta}}\right\}=0\,,\quad\left[Q_{\alpha}\,,P_{\beta\dot{\beta}}\right]=0\,,
\begin{array}{cl}{}&{\overline{{G}}_{\sigma}^{(n)}(\cdots,\zeta_{j+1},\zeta_{j},\cdots)^{\cdots+-\cdots}}\\{=}&{b(\zeta_{j}/\zeta_{j+1})\overline{{G}}_{\sigma}^{(n)}(\cdots,\zeta_{j},\zeta_{j+1},\cdots)^{\cdots-+\cdots}+c(\zeta_{j}/\zeta_{j+1})\overline{{G}}_{\sigma}^{(n)}(\cdots,\zeta_{j},\zeta_{j+1},\cdots)^{\cdots+-\cdots}.}\\\end{array}
+\left.\frac{1}{48}(2\pi\alpha^{\prime})^{2}\mathrm{Tr}\left(F_{ab}F_{cd}F^{ab}F^{cd}+2F_{ab}F^{ab}F_{cd}F^{cd}-4F_{ab}F^{bc}F_{cd}F^{da}-8F_{ab}F^{bc}F_{ad}F_{c}^{d}\right)\right].
\begin{array}{lcl}{\{p^{n},f\}_{\theta}}&{=}&{\sum_{k=0}{\theta}^{2k}f^{(2k+1)}p^{n-2k-1}}\\{\{p^{-n},f\}_{\theta}}&{=}&{-\sum_{k=0}{\theta}^{2k}c_{2k+n}^{2k+1}f^{(2k+1)}p^{-2k-n-1}}\\\end{array}
I_{\varepsilon}\left(a,g\right)=\frac{{\itTr}_{{\calH}}\left[e^{-2\pi{\bfK}_{0}+\frac{2\piia}{\beta}{\bfQ}}T\left(e^{-\int{\bfK}_{I}\left(\eta,\theta\right)d\etad\theta}\right)\right]}{{\itTr}_{{\calH}}\left[e^{-2\pi{\bfK}_{0}}T\left(e^{-\int{\bfK}_{I}\left(\eta,\theta\right)d\etad\theta}\right)\right]}{\it\}.
\begin{aligned}{{\calL}_{W}}&{{}=\partial_{x}+\sum_{i=1}^{m}\Phi_{i}\partial_{x}^{-1}\Psi_{i},\quad\mathrm{with}}\\{\Phi_{i}(t)}&{{}=(\psi_{W}(t,\lambda),\lambdah_{i}(-\lambda)),}\\{\Psi_{i}(t)}&{{}=(\psi_{W}^{*}(t,\lambda),\lambdah_{i}(\lambda))}\\{\}&{{}=(\psi_{W}((-)^{j+1}t_{j},-\lambda),\lambdah_{i}(\lambda))}\\{\}&{{}=(\psi_{W}((-)^{j+1}t_{j},\lambda),\lambdah_{i}(-\lambda)),}\\\end{aligned}
\int_{0}^{1}\prod_{i=1}^{n}du_{i}\left(\prod_{j=1}^{n-1}\theta(u_{j}-u_{j+1})\right)S^{+}(u_{n})\dotsS^{+}(u_{1})=\frac{1}{\left\{n\right\}_{q}!}\left(\int_{0}^{1}{du}S^{+}(u)\right)^{n}
\begin{array}{l}{\varphi_{1E}\left(r,\theta\right)=\sum_{n=-\infty}^{\infty}f_{n}\left(r\right)e^{i\left(n+\frac{1}{2}\right)\theta}}\\{\chi_{1E}\left(r,\theta\right)=\sum_{n=-\infty}^{\infty}g_{n}\left(r\right)e^{i\left(n+\frac{1}{2}\right)\theta}}\\\end{array}
\left(\begin{array}{c}{p_{\beta,f}}\\{p_{\varphi,f}}\\\end{array}\right)=P\left(\begin{array}{c}{p_{\beta,i}}\\{p_{\varphi,i}}\\\end{array}\right)\;,\qquadP=\frac{1}{2}\left(\begin{array}{rr}{1}&{1}\\{3}&{-1}\\\end{array}\right)\;.
H_{n}={}-\frac12\left(\partial_{r}^{2}+r^{-1}\partial_{r}-r^{-2}\left(A_{\varphi}^{2}(r)-2iA_{\varphi}(r)\partial_{\varphi}-\partial_{\varphi}^{2}\right)\right)+\fracn2B(r)\sigma_{3}.
{\calI}=\int_{{\calF}}\frac{d^{2}\tau}{Im\tau}\sum_{m_{1},m_{2}\atopn_{1},n_{2}}\left(\sum_{b=even}q^{\frac{1}{2}p_{L}^{2}}\bar{q}^{\frac{1}{2}p_{R}^{2}}h_{0}(\tau)+\sum_{b=odd}q^{\frac{1}{2}p_{L}^{2}}\bar{q}^{\frac{1}{2}p_{R}^{2}}h_{1}(\tau)-c(0)\right)
S=\frac{1}{\pi}\int{d^{2}x\left(\frac{\tilde{e}}{2}\tilde{\nabla}_{_{\!\!+\!\!\!+}}X\tilde{\nabla}_{_{\!\!=}}X+\tilde{\chi}_{_{-=}}b_{1}+\tilde{\chi}_{_{++\!\!\!+}}\bar{b}_{1}\right)}.
\left\langle\Phi_{B}(\zeta^{\prime})\Phi_{A}(\zeta)\right\rangle_{0}+\left\langle\Phi_{A}(\zeta^{\prime})\Phi_{A}(\zeta)\right\rangle_{0}=\frac{1}{N^{2}}a^{-2}\frac{10c}{s^{2}}[2w(y,y^{\prime})+u(y,y^{\prime})]+O(\frac{1}{N^{2}}a^{-5/3}),
\langle\langle\prod_{i=1}^{L}\intV_{n_{i}}^{\alpha_{i}}\rangle\rangle=\rho\mu^{S}{\frac{\Gamma({\frac{\sum_{i}n_{i}}{2}}+{\frac{(\sum_{i}\alpha_{i})+2}{6}})}{\Gamma{(S+1)}}}\quad.
{x}=y_{(2)}={\frac{-i}{\pi_{C^{\prime}}{\bar{\pi}}_{C}t^{C{C^{\prime}}}}}[{\frac{1}{\sqrt{2}}}({\bar{\beta}}_{A}{\bar{\beta}}^{B}-{\bar{\alpha}}_{A}{\bar{\alpha}}^{B}){\bar{\pi}}_{B}\omega^{A}-{\frac{1}{\sqrt{2}}}(\beta_{A^{\prime}}\beta^{B^{\prime}}-\alpha_{A^{\prime}}\alpha^{B^{\prime}})\pi_{B^{\prime}}{\bar{\omega}}^{A^{\prime}}],
D_{2m+3}=\frac{\kappa^{2}\mathrm{dn}^{2m}[u,\kappa]\mathrm{sn}[u,\kappa]\mathrm{cn}[u,\kappa]-2m\kappa^{\prime2}D_{2m-1}+(2m+1)(2-\kappa^{2})D_{2m+1}}{2(m+1)},
f=\,-i\epsilon^{1}\left(\psi^{\ast^{\prime}}\psi\,-\,{\bar{\psi}}{\bar{\psi}}^{\ast^{\prime}}\right)\,-\,i\epsilon^{2}\left(\psi^{\ast^{\prime}}\gamma_{5}\psi\,+\,{\bar{\psi}}\gamma_{5}{\bar{\psi}}^{\ast^{\prime}}\right)\,,
\left({\frac{\dot{a}}{a}}\right)^{2}={\frac{1}{r^{2}}}\left({\frac{dt}{d\eta}}\right)^{2}\left({\frac{dr}{dt}}\right)^{2},\\\\\\\\\{\frac{\ddot{a}}{a}}=\left[1+{\frac{1}{2}}a{\frac{\partial}{\partiala}}\right]\left({\frac{\dot{a}}{a}}\right)^{2}.
S_{1}(g,\phi)=\intd^{4}x~g^{\mu\nu}\sqrt{g}\left(\frac{1}{2}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{{\itk}^{2}}R_{\mu\nu}(g)+\frac{1}{12}\phi^{2}R_{\mu\nu}(g)\right)
=\frac{\left\langle\Psi\left|H^{Schw.}-H^{Flat}\right|\Psi\right\rangle}{\left\langle\Psi|\Psi\right\rangle}+\frac{\left\langle\Psi\left|H_{quasilocal}\right|\Psi\right\rangle}{\left\langle\Psi|\Psi\right\rangle}.
M_{\alpha\beta}=\left(\begin{array}{ccccc}{-\frac{1+f^{2}}{1-f^{2}}}&{\frac{2f}{1-f^{2}}}&{}&{}&{}\\{-\frac{2f}{1-f^{2}}}&{\frac{1+f^{2}}{1-f^{2}}}&{}&{}&{}\\{}&{}&{1}&{}&{}\\{}&{}&{}&{\ddots}&{}\\{}&{}&{}&{}&{1}\\\end{array}\right)
\prod_{i}\frac{1}{[\epsilon_{i}+2q_{i}\cdotl+(s-s_{0})l^{2}]}=\prod_{i}\frac{1}{[\epsilon_{i}+2q_{i}\cdotl]}\times\big\lbrace1-\sum_{j}{\frac{(s-s_{0})l^{2}}{[\epsilon_{j}+2q_{j}\cdotl]}}+\ldots\big\rbrace.
S^{1}=\left|\begin{array}{ccc}{0}&{0}&{0}\\{0}&{0}&{-i}\\{0}&{i}&{0}\\\end{array}\right|,\quadS^{2}=\left|\begin{array}{ccc}{0}&{0}&{i}\\{0}&{0}&{0}\\{-i}&{0}&{0}\\\end{array}\right|,\quadS^{3}=\left|\begin{array}{ccc}{0}&{-i}&{0}\\{i}&{0}&{0}\\{0}&{0}&{0}\\\end{array}\right|,\quad\overrightarrow{S}^{2}=1(1+1)I.
{\mathrm{}^{*}\mathcal{L}}_{eff}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{q-2p}{p}(\beta^{2})\left(-\frac{H^{2}}{24\beta^{3}}\right)^{\frac{p}{2p-q}}-\frac{\chi}{2\theta}\varepsilon^{\mu\nu\rho\lambda}B_{\nu\rho}\partial_{\mu}A_{\lambda}\;.
\{\chi_{\mu\nu}\}=\left(\begin{array}{cccc}{-2i\mu\varepsilon_{0}}&{-i\mu\varepsilon_{1}}&{-i\mu\varepsilon_{2}}&{-i\mu\varepsilon_{3}}\\{-i\mu\varepsilon_{1}}&{\chi_{11}}&{\chi_{12}}&{\chi_{13}}\\{-i\mu\varepsilon_{2}}&{\chi_{12}}&{\chi_{22}}&{\chi_{23}}\\{-i\mu\varepsilon_{3}}&{\chi_{13}}&{\chi_{23}}&{\chi_{33}}\\\end{array}\right)
S=\int_{X}d^{4}x\varepsilon^{\mu\nu\alpha\beta}\varepsilon_{abcd}\bigg({^+}\tau{^+}F_{\mu\nu}^{ab}{^+}F_{\alpha\beta}^{cd}-{^-}\tau{^-}F_{\mu\nu}^{ab}{^-}F_{\alpha\beta}^{cd}\bigg),
{\calA}(x)=\frac{1}{8\pi^{2}}\left[R^{ab}\mathrm{\tiny~\wedge~}R_{ab}+\frac{2}{l^{2}}(T^{a}{\mathrm{\tiny~\wedge~}}T_{a}-R_{ab}{\mathrm{\tiny~\wedge~}}e^{a}{\mathrm{\tiny~\wedge~}}e^{b})\right].
\omega_{F}\left(\begin{array}{c}{\psi}\\{\chi}\\\end{array}\right):=i\partial_{0}\left(\begin{array}{c}{\psi}\\{\chi}\\\end{array}\right)=-i\gamma^{0}\left(\begin{array}{cc}{\gamma^{j}D_{j},}&{-\sqrt{2}e\phi}\\{\sqrt{2}e\phi^{*},}&{\gamma^{k}\partial_{k}}\\\end{array}\right)\left(\begin{array}{c}{\psi}\\{\chi}\\\end{array}\right)\,.
{\hat{\calY}}_{J}^{R}=\frac{D(r_{1}=I,r_{2}=I)}{A(\Omega)}\intd\Omega\sum_{r_{2z}}\langler_{2},r_{2z}|{\calD}_{JJ^{(0)}}^{(R)}(\Omega)|\Omega,\frac{P}{2}\rangle\langle\Omega,\frac{P}{2}|.
f^{1}=\frac{1}{m}(q\varepsilon^{12}+r\varepsilon^{13}),~~~f^{2}=\frac{1}{m}(-p\varepsilon^{12}+r\varepsilon^{23}),~~~f^{3}=\frac{-1}{m}(p\varepsilon^{13}+q\varepsilon^{23})
L_{2}=\dot{r}p_{r}+\dot{\theta}p_{\theta}+\dot{z}p_{z}-\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]\\\.
Z=L^{I}q_{I}^{(e)}-M_{I}q_{(m)}^{I}=\left({\frac{q_{0}^{(e)}q_{1}^{(e)}+q_{(m)}^{0}q_{(m)}^{1}}{(q_{1}^{(e)})^{2}+(q_{(m)}^{0})^{2}}}\right)^{1/2}(q_{1}^{(e)}+iq_{(m)}^{0}),
\Sigma_{f}^{1},\Sigma_{f}^{2},\Sigma_{f}^{3}\in\Omega^{0,2}\oplus\Lambda^{2}\rightarrow\Sigma_{f}^{1},\Sigma_{f}^{2},\Sigma_{f}^{3}\in({\bfR}S\oplus2K)^{\ast}\otimes\Lambda^{2}.
{\frac{1}{2\pi}}\int_{0}^{2\pi}d\varphi\,\left[R(r,\varphi)\,K_{1}(R(r,\varphi))\right]^{2}=\sum_{n=-\infty}^{\infty}\left[r(I_{n}\,K_{n})^{\prime}(r)\right]^{2}\equivF(r),
L^{2}({\calA}_{\gamma}\times{\calF}_{\gamma})=\bigotimes_{e\inE}\bigl(\bigoplus_{\rho\in\mathrm{Rep}(G)}\rho\otimes\rho^{*}\bigr)\;\otimes\;\bigotimes_{v\inV}\bigl(\bigoplus_{\rho\inS}\rho\bigr).
\langle\overline{{\phi}}^{2}(x)\rangle_{\beta}=\frac{1}{4\pi^{2}r^{2}}\sum_{n=1}^{\infty}\sum_{l=0}^{\infty}(2l+1)Q_{\nu_{l}-1/2}\left(1+\alpha^{2}\frac{n^{2}\beta^{2}}{2r^{2}}\right)\.
M=\frac{V_{p}\Omega_{d-2}2\piR_{T}}{16\piG}r_{0}^{d-3}\frac{d-1}{2}+\sqrt{\frac{1}{4}\left(\frac{V_{p}\Omega_{d-2}2\piR_{T}}{16\piG}(d-3)r_{0}^{d-3}\right)^{2}+Q^{2}}
G_{SM}(x-y,\gamma)=\frac{1}{\calN}\int[d\psi][d\bar{\psi}][dA]\,\psi(x)e^{ie\int_{\gamma}dx_{\mu}^{\prime}\,A_{\mu}}\bar{\psi}(y)e^{-\intd^{2}x^{\prime}\,{\calL}_{SM}},
\Lambda_{-1,1}=\lambda^{-1}(e_{1,2p}-e_{2,2p+1})+(e_{2,1}-e_{2p+1,2p})+2\sum_{k=1}^{p-1}e_{2+k,1+k}-2\sum_{k=1}^{p-1}e_{p+1+k,p+k},
({\calH}_{V})_{cl}=\sum_{{\vec{n}},{\vec{m}}}\sum_{s,s^{\prime}}\kappa_{\vec{m}}\left[(a_{\vec{n}}^{s\dagger}\bar{u}_{\vec{n}}^{s}+b_{-{\vec{n}}}^{s}\bar{v}_{-{\vec{n}}}^{s})(a_{\vec{m}}^{s^{\prime}}\gamma^{0}u_{\vec{m}}^{s^{\prime}}-b_{-{\vec{m}}}^{s^{\prime}\dagger}\gamma^{0}v_{-{\vec{m}}}^{s^{\prime}})\right]_{cl}F({\vec{n}}-{\vec{m}},X),
\begin{array}{llll}{S}&{=}&{{\displaystyle{\frac{T^{-\epsilon}}{\left(2\pi\right)^{1+\epsilon}}}}}&{\Biggl[\;2\Psi\left(|l+1/2|\right)-2\Psi\left(|l+1/2|-|l|\right)+{\displaystyle{\frac{1}{|l+1/2|}}}+\Biggr.}\\{}&{}&{}&{\;\;\;\zeta\left(1+\epsilon,l+1/2\right)+\zeta\left(1+\epsilon,-l-1/2\right)\Biggl.\;\Biggr]}\\\end{array}
A_{\tilde{\varphi}}\rightarrowL^{-1}e^{\phi_{0}}\frac{\hat{B}\rho^{2}}{2\tilde{\Lambda}},\,B_{\tilde{\varphi}}\rightarrowLe^{-\phi_{0}}\frac{\hat{E}\rho^{2}}{2\tilde{\Psi}},
j_{+}{}^{n}=\sum_{m=-\infty}^{\infty}j_{++}^{m,m+n},\quadj_{+}{}^{0}=Q_{+},\qquadj_{-}{}^{n}=\sum_{m=-\infty}^{\infty}j_{--}^{m+n,m},\quadj_{-}{}^{0}=Q_{-}.
\prod_{\sigma}[\alpha_{q}^{(\sigma)},\hat{\alpha}_{q}^{(\sigma)}][\beta_{q^{\prime}}^{(\sigma)},\hat{\beta}_{q^{\prime}}^{(\sigma)}]=e^{\frac{\epsilon}{2}\sum_{\sigma}[(\alpha^{(\sigma)2}-\alpha^{(\sigma)\dagger2})-(\beta^{(\sigma)2}-\beta^{(\sigma)\dagger2})]}\,{.}
g_{mn}(z)\equive_{m}^{~a}e_{n}^{~b}\eta_{ab}=\partial_{m}Z^{\underline{M}}E_{\underline{M}}^{\underline{a}}\partial_{n}Z^{\underline{N}}E_{\underline{N}}^{\underline{b}}\eta_{\underline{{ab}}}=E_{m}^{\underline{a}}E_{n\underline{a}}
\frac{1}{2\pi}\frac{1}{(x-y)^{2}}\left(\begin{array}{cc}{0}&{e^{-e[\varphi(x)-\varphi(y)]}\;\overline{{(\tilde{x}-\tilde{y})}}}\\{e^{+e[\varphi(x)-\varphi(y)]}\;(\tilde{x}-\tilde{y})}&{0}\\\end{array}\right)\;,
{}~~~~~~~\times\Bigg\{\sum_{\sigma=0}^{N-1}{\frac{w(v_{2},b-f+\sigma)w(v_{3},d-h-\sigma)s(\sigma,a)s(\sigma,h)}{w(v_{1},g-a+\sigma)w(v_{4},e-c-\sigma)s(\sigma,c)s(\sigma,f)}}\Bigg\}_{0},
\hat{C}\;=\;C_{(3)}-\textstyle{\frac{1}{2}}\vec{A}_{(1)}^{T}\varepsilon\vec{A}_{(2)}+\vec{A}_{(2)}^{T}\vec{\theta}+\textstyle{\frac{1}{2}}A_{(1)}\vec{\theta}^{T}\varepsilon\vec{\theta}\;,
ds^{2}=2dX^{+}dX^{-}-\mu^{2}\sum_{i=1}^{8}(X^{i})^{2}(dX^{+})^{2}+\sum_{i=1}^{8}(dX^{i})^{2}+(dX^{9}+\muX^{8}dX^{+})^{2}.
{\calZ}={\frac{\Gamma((N+1)/2)}{\sqrt{\pi}(Nr)^{N/2}}}\left[\sum_{k=1}^{K}{\frac{\Gamma(N/2+k)\Gamma(k+1/2)}{\sqrt{\pi}\,\Gamma(N/2)k!}}\Bigl({\frac{2}{rN}}\Bigr)^{k}+{\calO}(1/r^{K+1})\right]\,,
d(x,y;e^{2})=d(t,y;e^{2})d\left(\frac{x}{t},\frac{y}{t};e^{2}d(t,y;e^{2})\right),\;s(x,y;e^{2})=s(t,y;e^{2})s\left(\frac{x}{t},\frac{y}{t};e^{2}d(t,y;e^{2})\right)
\nabla^{\mu}\left(e^{-2\phi}\partial_{\mu}\mathcal{T}\right)+V^{\prime}(\hat{\calT})-h^{\prime}(\mathcal{T})\left[\sum_{n}\textstyle{\frac{(-)^{n}}{2\cdotn!}}F_{(n)}^{+\2}\,-\,\sum_{n}\textstyle{\frac{(-)^{n}}{2\cdotn!}}F_{(n)}^{-\2}\right]=0\,,
\left({\frac{\calZ}{|Z|}}\right)_{mag}=\left(\begin{matrix}{i\sigma_{2}\;\left(\begin{matrix}{-1}&{0}&{0}&{0}\\{0}&{-1}&{0}&{0}\\{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}\\\end{matrix}\right)}&{\mathrm{I}}\\{-\mathrm{I}}&{i\sigma_{2}\;\left(\begin{matrix}{-1}&{0}&{0}&{0}\\{0}&{-1}&{0}&{0}\\{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}\\\end{matrix}\right)}\\\end{matrix}\right)\.
\Pi^{(1)}(k^{2},m)=\frac{g^{2}}{2\pi}\left[\frac{\sqrt{m^{2}}}{2}-\left(\frac{m^{2}}{2}+\frac{k^{2}}{8}\right)\frac{1}{\sqrt{k^{2}}}ln\left(\frac{2\sqrt{m^{2}}+\sqrt{k^{2}}}{2\sqrt{m^{2}}-\sqrt{k^{2}}}\right)\right]\,
\begin{array}{lll}{E_{\pm}^{\dagger}=E_{\mp},}&{\quad\quadH^{\dagger}=H,}&{\quad\quad\quad\mathrm{for}\quadsu(2),}\\{E_{\pm}^{\dagger}=-E_{\mp},}&{\quad\quadH^{\dagger}=H,}&{\quad\quad\quad\mathrm{for}\quadsu(1,1).}\\\end{array}
Z_{ab}^{V}={z_{aa^{\prime}}^{V}}^{T}\tilde{I}_{a^{\prime}b^{\prime}}z_{b^{\prime}b}^{V}=\bigl(O(\theta){z^{V}}\bigr)_{aa^{\prime}}^{T}\tilde{I}_{a^{\prime}b^{\prime}}\bigl(O(\theta)z^{V}\bigr)_{b^{\prime}b}
\begin{array}{ll}{\delta\phi_{1}=-\bar{\eta}\psi_{2}\,,}&{\delta\phi_{2}=\bar{\eta}\psi_{1}\,,}\\{\delta\psi_{1}=-\imath\partial_{\mu}\phi_{2}\gamma^{\mu}\eta-F_{2}\eta\,,}&{\delta\psi_{2}=\imath\partial_{\mu}\phi_{1}\gamma^{\mu}\eta+F_{1}\eta\,,}\\{\deltaF_{1}=-\imath\bar{\eta}\gamma^{\mu}\partial_{\mu}\psi_{2}\,,}&{\deltaF_{2}=\imath\bar{\eta}\gamma^{\mu}\partial_{\mu}\psi_{1}.}\\\end{array}
D_{++}^{(ML)cd}(x)=D_{++}^{ML}(x)\delta^{cd}={\frac{i\delta^{cd}}{\pi^{2}}}\intd^{2}k\,e^{ikx}{\frac{k_{+}^{2}}{(k^{2}+i\epsilon)^{2}}}={\frac{\delta^{cd}}{\pi}}{\frac{(x^{-})^{2}}{(-x^{2}+i\epsilon)}}\
F_{min}^{SS}(\beta)=\frac{\beta-2\pii/N}{\beta+2\pii/N}\frac{\Gamma\left(1-\frac{\beta}{2\pii}\right)\Gamma\left(\frac{\beta}{2\pii}+2\frac{1}{N}\right)}{\Gamma\left(1+\frac{\beta}{2\pii}\right)\Gamma\left(-\frac{\beta}{2\pii}+2\frac{1}{N}\right)}.
\frac{\mathrm{d}P_{n}(\lambda)}{\mathrm{d}\lambda}=\int_{-\infty}^{\infty}\mathrm{d}\mu(t)\frac{\mathrm{d}P_{n}(t)}{\mathrm{d}t}\sum_{k=0}^{n-1}P_{k}(\lambda)P_{k}(t)\,
s_{\psi}B_{\mu\nu}^{a}=\varepsilon_{\mu\nu\lambda\rho}\left(D^{\lambda}\right)_{\;\;b}^{a}C^{\rhob}+f_{\;\;bc}^{a}\partial_{\left[\mu\right.}\bar{C}_{\left.\nu\right]}^{c}{\calC}^{b},\;s_{\psi}C_{\mu}^{a}=\left(D_{\mu}\right)_{\;\;b}^{a}{\calC}^{b},s_{\psi}{\calC}^{a}=0,
\begin{array}{ll}{\tilde{\omega}=}&{\left[\,(r_{x}+q_{xx})\,dp_{x}+(r_{x}-p_{xx})\,dq_{x}-(p_{xx}+q_{xx})\,dr\,\right]\wedge(dp+dq)}\\{}&{-\frac{1}{2}dp\wedgedp_{x}}\\\end{array}
F[L,\beta,\nu]=\frac{8\pi^{4}}{\sqrt{2\pi\alpha^{\prime}}}\int_{0}^{\infty}\frac{ds}{s^{9/2}}\;e^{s}\,e^{-L^{2}/2s\alpha^{\prime}}\Theta_{2}\left(\nu\left\vert\frac{i\beta^{2}s}{2\pi^{3}\alpha^{\prime}}\right.\right)\prod_{n=1}^{\infty}\left(\frac{1-e^{-(2n+1)s}}{1-e^{-2ns}}\right)^{8}.
\langleU^{(\nu)}(C,x_{0})\rangle={\frac{1}{Z}}\sum_{V_{1}}\sum_{V_{2}}\sum_{V_{n}}D^{(\nu)}(V_{1}V_{2}\cdotsV_{n})\times\prod_{i=1}^{n}\sum_{\mu^{*}}C^{\mu^{*}}(\beta)\chi^{\mu^{*}}(V_{i}),
=4\pi^{2}\zeta\int_{S}^{}dS_{\mu}(x)\int_{S}^{}dS_{\mu}(y)D_{m}(x-y)+\frac{g^{2}}{4}\int_{\Sigma}^{}d\sigma_{\mu\nu}(x)\int_{\Sigma}^{}d\sigma_{\mu\nu}(y)\left[9D_{m}(x-y)-5D_{0}(x-y)\right].
\phi_{h,\vec{\jmath}}(g_{w})=tr_{\otimes_{w\inh}V_{j_{w}}}\left({\frac{1}{\sqrt{2\pix}}}e^{-\frac{1}{2x^{2}}\Omega^{ij}\Omega_{ij}}\otimes_{w\inhV_{j_{w}}}(2j_{w}+1)R_{j_{w}}(g_{w})\right).
{\calL}=\bar{\psi}(i\!\not{\!\partial}+J)\psi+\frac{1}{\sqrt{N}}\bar{\psi}\not{\!\!A}\psi-\frac{1}{\sqrt{N}}\sigma\bar{\psi}\psi-\frac{1}{2G_{S}}\sigma^{2}+\frac{1}{2G_{V}}A_{\mu}^{2}-\frac{1}{2\xi}(\partial^{\mu}A_{\mu})^{2}\,
\sigma_{j}^{BW}(E)\sim\left|a_{j}^{BW}(E)\right|^{2}=\left|\frac{r_{\eta}}{E-(E_{R}-i\frac{\Gamma}{2})}\right|^{2}\sim\frac{1}{(E-E_{R})^{2}+(\frac{\Gamma}{2})^{2}}\,,
r\widetilde{E}\left(\beta\right)\approx\frac{\pi^{4}}{15}\left(\frac{r}{\beta}\right)^{4}-\frac{1}{240}-\frac{\mu_{eff}^{2}}{2}\left(\frac{r}{\beta}\right)^{2}+\frac{\mu_{eff}^{2}}{48},
L(q_{i}^{a},\dot{q}_{i}^{a})=\frac{1}{2g^{2}}\left[\dot{q}_{i}^{a}+f^{abc}\lambda^{b}q_{i}^{c}\right]^{2}-V(q_{i}^{a})\\,\\V(q_{i}^{a})=\frac{1}{2}\omega^{2}\left(q_{i}^{a}\right)^{2}\,
\gamma_{V}^{2}\equivu_{3}-\left(\frac{u_{1}-u_{2}}{2}\right)^{2}>0,\qquad\gamma_{F}^{2}\equivw_{3}-\left(\frac{w_{1}-w_{2}}{2}\right)^{2}>0,\qquad\left|w_{1}+w_{2}\right|>0,
P=T\frac{1}{T^{\dagger}T}T^{\dagger}=\left(\begin{matrix}{K}&{K\bar{z}^{m_{1}}}&{\cdots}&{K\bar{z}^{m_{n-1}}}\\{z^{m_{1}}K}&{z^{m_{1}}K\bar{z}^{m_{1}}}&{\cdots}&{z^{m_{1}}K\bar{z}^{m_{n-1}}}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{z^{m_{n-1}}K}&{z^{m_{n-1}}K\bar{z}^{m_{1}}}&{\cdots}&{z^{m_{n-1}}K\bar{z}^{m_{n-1}}}\\\end{matrix}\right),
\!\!\!\!{\calL}_{DSM}=\frac1{1+4{\Phi}^{2}}\left(\phantom{\fracII}\!{\textstyle\frac12}\left(\delta^{ij}+4\Phi^{i}\Phi^{j}\right)\partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}-\varepsilon^{\mu\nu}\varepsilon^{ijk}\Phi^{i}\partial_{\mu}\Phi^{j}\partial_{\nu}\Phi^{k}\right)~.
\mathrm{RHS}=\displaystyle\frac{(1-\tau^{2})\zeta_{N-1}}{\prod_{j=1}^{n}(z_{N-1}-z_{j}\tau^{2})}\sum_{k=1}^{n}\zeta_{k}\displaystyle\prod_{j=1\atopj\neqk}^{n}\frac{(z_{N-1}-z_{j})(z_{k}-z_{j}\tau^{2})}{z_{k}-z_{j}}\displaystyle\overline{{G}}_{\varepsilon,k}^{(n-1,l-1)}(\zeta^{\prime}).
\tilde{\Phi}^{J}(x,t,\theta)=\Phi^{J}\Bigl\vert_{\theta^{+}=\theta^{-}=\theta}\qquad\mathrm{and}\qquad\tilde{\Sigma}_{J}(x,t,\theta)=\Sigma_{J}\Bigl\vert_{\theta^{+}=\theta^{-}=\theta}\.
{\calL}=\sqrt{-g}\left[{\calR}_{g}-{\textstyle{\frac{1}{2}}}\partial_{\mu}\varphi\partial^{\mu}\varphi-{\textstyle{\frac{1}{2}}}\mathrm{e}^{2\varphi}\partial_{\mu}\Psi\partial^{\mu}\Psi-{\textstyle{\frac{1}{4}}}\mathrm{e}^{-\varphi}{\calF}_{\mu\nu}^{i}(LML)_{ij}{\calF}^{j\,\mu\nu}+{\textstyle{\frac{1}{8}}}\mathrm{Tr}(\partial_{\mu}ML\partial^{\mu}ML)\right],
\left[-\frac{d^{2}}{dr^{2}}-\frac{2}{r}\frac{d}{dr}+\frac{l(l+1)}{r^{2}}+m^{2}+\frac{\lambda}{2}\phi_{\mathrm{wall}}^{2}(r-R)+\frac{\sigma}{4!}\phi_{\mathrm{wall}}^{4}(r-R)\right]\psi_{i}(r)=E_{\mathrm{wall}}^{2}(i)\psi_{i}(r)\;,
\left|\alpha_{k}\right|=\left|\beta_{k}\right|=2^{n_{s}-n_{1}}\left|\frac{\Gamma\left(n_{s}\right)}{\Gamma\left(n_{1}\right)}\right|z_{S}^{1/2+m-n_{1}}\left(\frac{g_{S}}{g_{1}}\right)^{-l}\left(\frack{k_{s}}\right)^{n_{1}-n_{s}}\mathrm{~for~}n_{1}<1
{\calL}=\frac{1}{4}\left(F_{\mu\nu\alpha}\right)^{2}+\frac{1}{\Lambda^{4}}\left\{a\left[\left(F_{\mu\nu\alpha}\right)^{2}\right]^{2}+bF_{\mu\nu\alpha}F_{\nu\rho\alpha}F_{\rho\sigma\beta}F_{\sigma\mu\beta}+cF_{\mu\nu\alpha}F_{\nu\rho\beta}F_{\rho\sigma\alpha}F_{\sigma\mu\beta}\right\}+\cdots
\begin{array}{lcl}{\frac{\partial}{\partialt_{2}}{u_{2}}}&{=}&{2{v_{3}^{\prime}}}\\{\frac{\partial}{\partialt_{2}}{v_{3}}}&{=}&{-\frac{2}{3}\{u_{2}u_{2}^{\prime}+\theta^{2}u_{2}^{\prime\prime\prime}\}}\\\end{array}
\begin{array}{clr}{\sum_{i=0}^{r+4}q_{i}^{a}s_{i}}&{=}&{0}\\{\sum_{i=0}^{r+4}q_{i}^{a}n_{i}}&{=}&{0}\\{\sum_{i=0}^{r+4}q_{i}^{a}m_{i}}&{=}&{0.}\\\end{array}
\Pi_{\pmq}^{m}={\calD}_{\pmq}X^{m}-i\left({\calD}_{\pmq}\Theta^{\alpha1}\sigma_{\alpha\beta}^{m}\Theta^{\beta1}+{\calD}_{\pmq}\Theta_{\alpha}^{2}\tilde{\sigma}^{m\alpha\beta}\Theta_{\beta}^{2}\right)=0,
{\frac{1}{z\,K}}\sqrt{\frac{A_{T^{2}}^{\mathrm{(OM)}}}{A_{T^{2}}^{\mathrm{(bulk)}}}}={\frac{\left(1+{\frac{1}{2}}g_{s}({\calF}^{4})^{2}\right)^{\frac{3}{4}}}{1+{\frac{1}{4}}g_{s}({\calF}^{4})^{2}}}\,.
W_{cr}=-({\frac{g^{2}}{4\pi}})^{2}\,4\,C_{F}\,(C_{F}-{\frac{C_{A}}{2}})({\calA})^{2}\int_{0}^{1}d\,\xi\int_{0}^{1}d\,\eta\,log{\frac{|sin\rho(\xi-\eta)|}{|sin\rho\xi|}}log{\frac{|sin\rho(\xi-\eta)|}{|sin\rho\eta|}},
g^{\prime}{}_{y^{\prime}\bar{y}^{\prime}}^{(2)}=g_{y^{\prime}\bar{y}^{\prime}}^{(2)}-g_{y^{\prime}y^{\prime}}^{(2)}{\frac{|y^{\prime}|^{2}}{\bar{y}^{\prime}{}^{2}}}-g_{y^{\prime}\bar{y}^{\prime}}^{(0)}H(|y^{\prime}|^{2}).
H_{c}^{r}\,=\,{\frac{1}{2}}\int\left[\left(\pi_{1}\phi_{2}-\pi_{2}\phi_{1}\right)^{2}\,-\,{\frac{1}{\phi_{1}^{2}+\phi_{2}^{2}}}\left(\phi_{1}\partial_{i}\partial_{i}\phi_{1}+\phi_{2}\partial_{i}\partial_{i}\phi_{2}\right)\right]dx\,.
\bar{G}_{n}^{(N)}(L,r,l)=q^{\frac{L}{2}+\frac{r}{N}}\widetilde{\sum_{\tilde{m}\in{\bfZ}^{N+1}}}q^{mC^{-1}m}(q)_{L-1}\prod_{i=0}^{N}\frac{1}{(q)_{m_{i}}}\left\{\sum_{k=0}^{n-1}q^{-e_{N+n-l-1}C^{-1}m}\bar{B}^{(N)}(L,r,l|k)\right\}
T_{r}^{r}=-\varphi^{2}e^{-2gM}+\frac{A^{2}\varphi^{2}}{r^{2}}e^{-2gM}-{\calP}M^{2}-\left(\frac{A^{\prime}}{rh}\right)^{2}+{\calR}\left(\frac{G}{gr}\right)^{2}+U;
\sigma_{a^{\dagger}a^{\dagger}}^{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\}\\.
\rho^{\prime\prime}+\frac{\mu}{8}(1-\lambda-\frac{\gamma}{4})e^{2\rho}\Omega^{-\lambda-\gamma/4}+\frac{1}{4}f^{\prime2}-\frac{1}{4}(1+\frac{\gamma}{4}-\frac{\epsilon}{2})e^{\chif-2\rho}\Omega^{\gamma/4-\epsilon/2}A^{\prime2}=0,
\tilde{\calH}\mid_{free}={\frac{1}{2}}(\tilde{\Pi}_{\theta}\tilde{v_{i}}\tilde{v_{i}})={\frac{1}{2}}\Pi_{\theta}[\partial_{i}(\theta+\alpha\phi_{2})+(\alpha+{\frac{\phi_{1}}{\Pi_{\theta}}})\partial_{i}(\beta-\phi_{2})]^{2}.
22S_{k}[w]=\frac{-\frac{1}{2b_{3}}\int_{1}^{w}dr\,r^{1/2}(r-1)^{k}\left[\sum_{n=1}^{k}\frac{b_{2n+3}}{r}\left[2(r-1)\frac{d}{dr}+2(k-n)+1\right]S_{k-n}[r]\right]}{w^{1/2}(w-1)^{k}}.
{\calZ}_{ENJL}=\intD\Phie^{-{\frac{i}{2}}\intd^{4}x(\Phi-m_{0})Q^{-1}(\Phi-m_{0})}\intDqD\bar{q}e^{-i\intd^{4}x\bar{q}(i\gamma^{\mu}\partial_{\mu}-\Phi)q}\,.