harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{c}{\omega_{N}=d{\bfp}\wedged{\bfx}+\sum_{i=1}^{N}d{\bfp}_{i}\wedged{\bfe}_{i},}\\{{\calH}={s}\left[{\bfp}{\bfe}_{1}+\sum_{i=1}^{N}{k}_{i-1}({\phi}_{i-1.i}-c_{i-1})+\frac{{k}_{N}}{2c_{N}}({\Phi}_{N.N}-c_{N}^{2})+\sum_{i,j=1}^{N}d_{ij}({\bfe}_{i}{\bfe}_{j}-\delta_{ij})\right],}\\\end{array}
\Theta_{p}(x)=(2\epsilon_{p})^{-1/2}e^{-i\epsilon_{p}t}\varphi_{p}(z),\quad\varphi_{p}(z)=(2\pi)^{-1/2}e^{ipz},\quad\epsilon_{p}=\sqrt{p^{2}+m^{2}},\quad-\infty<p<\infty,
U_{u}(1+re^{2\pii})=\left(\begin{matrix}{1}&{-2}\\{0}&{1}\\\end{matrix}\right)U_{u}(u)\qquad\mathrm{for~}0<r<2\qquad\Rightarrow\T_{(1)}=\left(\begin{matrix}{1}&{-2}\\{0}&{1}\\\end{matrix}\right)\.
2\zeta(3)\lambda^{-3/2}\to2\zeta(3)\lambda^{-3/2}+\frac{1}{24N_{3}^{2}}\lambda^{1/2}+\frac{1}{N_{3}^{3/2}}h(e^{-4\pi^{2}/g_{\mathrm{YM}}^{2}})(1+{\calO}(g_{\mathrm{YM}}^{2})),
S_{\alpha}(z)\,S_{A}(z)=\varepsilon^{\prime}\,\left.\widetildeS_{\alpha}(\bar{z})\,\widetildeS_{A}(\bar{z})\right|_{z=\bar{z}}~~~,~~~~S^{\dot{\alpha}}(z)\,S^{A}(z)=-\varepsilon^{\prime}\,\left.\widetildeS^{\dot{\alpha}}(\bar{z})\,\widetildeS^{A}(\bar{z})\right|_{z=\bar{z}}~~.
\langle\langle\delta^{(1)}j_{1}(x,t)\rangle\rangle=+i\left(\frac{\lambda}{2\pi}\right)^{2}\int_{t_{0}}^{\infty}dt^{\prime}\int_{-\infty}^{\infty}dx^{\prime}A_{0}^{cl}(x^{\prime},t^{\prime})\theta(t-t^{\prime})\frac{\partial}{\partialt}\frac{\partial}{\partialx^{\prime}}\langle\langle[\phi(x,t),\phi(x^{\prime},t^{\prime})]\rangle\rangle.
T_{\mu\nu}^{\alpha\beta}\sim\epsilon_{~\mu}^{\alpha~\rho\sigma}\epsilon_{~\nu}^{\beta~\lambda\gamma}\big(F_{\rho\sigma}^{a}F_{\lambda\gamma}^{a}+\Lambda\tilde{F}_{\rho\sigma}^{a}F_{\lambda\gamma}^{a}+\kappa\tilde{F}_{\rho\sigma}^{a}\tilde{F}_{\lambda\gamma}^{a}\big)\\,
\begin{array}{ll}{ds^{2}=-2dudv-32(|\partial_{k}\mathtt{W}|^{2}+|\phi_{j\bar{k}}z^{j}|^{2})(du)^{2}+dz^{i}d\bar{z}^{i}}\\{\phi_{mn}=\partial_{m}\partial_{n}\mathtt{W},\,\,\,\,\,\,\phi_{\overline{{mn}}}=\partial_{\bar{m}}\partial_{\bar{n}}\overline{{\mathtt{W}}},\,\,\,\,\,\,\phi_{l\bar{m}}=constants}\\\end{array}
\overline{{{\calP}}}_{1}=i\hbar\sqrt{\frac{\gamma}{2}}\left[\left(f^{\ast}+g^{\ast}\right)z_{1}-\left(f+g\right)z_{1}^{\ast}\right]\;,\quad\overline{{{\calP}}}_{2}=-\hbar\sqrt{\frac{\gamma}{2}}\left[\left(f^{\ast}-g^{\ast}\right)z_{1}+\left(f-g\right)z_{1}^{\ast}\right]\;.
Z=\left(\begin{array}{cc}{z^{1}}&{z^{4}}\\{z^{2}}&{z^{3}}\\\end{array}\right)\;,\;Z^{\prime}=MZM^{\dagger}\;,\;D=\left(\begin{array}{cc}{\delta_{1}}&{\delta_{2}}\\{\delta_{4}}&{\delta_{3}}\\\end{array}\right)\;,\;D^{\prime}=(M^{\dagger})^{-1}DM^{-1}\;
\begin{array}{lcr}{\sum_{j=1}^{3}(|\varphi_{j}^{1}|^{2}-|\varphi_{j}^{2}|^{2})}&{=\xi^{3}\qquad}&{(a)}\\{\sum_{j=1}^{3}\varphi_{j}^{1}\overline{{\varphi}}_{j2}}&{=\xi^{1}+i{\xi^{2}}\qquad}&{(b)}\\{\sum_{j=1}^{3}\varphi_{j}^{2}\overline{{\varphi}}_{j1}}&{=\xi^{1}-i{\xi^{2}}\qquad}&{(c).}\\\end{array}
\langlej_{0}(x)\rangle^{(1)}=i4\pi^{2}e^{3}g\intd^{2}y\intdp^{0}\,\mathrm{Tr}\,[\gamma_{0}\,\Sigma(p^{0},{\bfx}-{\bfy})\,\gamma_{0}\,\Sigma(p^{0},{\bf0})\,\gamma_{0}\,\Sigma(p^{0},{\bfy}-{\bfx})]\;.
{\bf\hat{e}_{\rho}.\hat{e}_{\mu}^{\prime}}={\bf\hat{e}_{\rho}.}L{\bf\hat{e}_{\mu}}=L_{\mu}^{\nu}{\bf\hat{e}_{\rho}.\hat{e}_{\nu}}=L_{\mu}^{\gamma}g_{\rho\gamma}\Rightarrow{L}_{\mu}^{\gamma}=g_{\gamma\rho}^{-1}{\bf\hat{e}_{\rho}.}L{\bf\hat{e}_{\mu}}.
\m^{2}(\beta)=m_{0}^{2}+\frac{\lambda_{0}}{(2\pi)^{D/2}}\sum_{n=1}^{\infty}\biggl(\frac{m(\beta)}{\betan}\biggr)^{\frac{D}{2}-1}K_{\frac{D}{2}-1}\left(m(\beta)n\beta\right).
\left({\frac{A(q,p)}{4}}\right)_{N=8}=\pi\left({|(\sum_{i}|Z_{i}|^{4}-2\sum_{i>j}|Z_{i}|^{2}|Z_{j}|^{2}+8|Z_{1}Z_{2}Z_{3}Z_{4}|})|\right)_{|Z_{2}|=|Z_{3}|=|Z_{4}|=0}^{1/2}=|Z_{1}|_{\mathrm{fix}}^{2}\,
\chi_{1}=\left(\begin{array}{c}{1}\\{0}\\{0}\\\end{array}\right)\;\;;\;\;\chi_{2}=\left(\begin{array}{c}{0}\\{1}\\{0}\\\end{array}\right)\;\;;\;\;\chi_{3}=\left(\begin{array}{c}{0}\\{0}\\{1}\\\end{array}\right)\;.
\frac{1}{2\kappa}\left[\theta(\tau-\tau^{\prime})e^{i\sigma\kappa(\tau-\tau^{\prime})}\theta(\tau^{\prime}-\tau)e^{i\sigma\kappa(\tau^{\prime}-\tau)}\right]=-\frac{i}{2\pi}\intdK\frac{e^{i\sigmaK(\tau-\tau^{\prime})}}{k^{2}+\sigmaK^{2}-i\epsilon}
{\frac{e^{-3T}H}{\kappa^{2}}}[{\frac{\partial}{\partialT}}({\frac{e^{3T}}{H}}){\frac{\partialV}{\partialT}}]-{\frac{\partial^{2}V}{\partial\psi^{2}}}=2\pi{\frac{e^{4}}{N}}\sum_{j}\delta(\psi-\psi_{j}).
{\calR}_{a0c}^{0}=\partial_{0}\Omega_{ac}^{0}-\partial_{c}\Omega_{a0}^{0}+\Omega_{00}^{0}\Omega_{ac}^{0}+\Omega_{d0}^{0}\Omega_{ac}^{d}-\Omega_{0c}^{0}\Omega_{a0}^{0}-\Omega_{dc}^{0}\Omega_{a0}^{d},
\rho(t)=U(t)\rho(0)U(t)^{\dagger}=\sum_{n=0}^{\infty}\sum_{m^{\prime},m\nen}\frac{1}{\Omega}e^{-\betan\omega}{C^{\prime}}_{mn}{C^{\prime}}_{m^{\prime}n}^{*}|{\Psi^{\prime}}_{m}(t)><{\Psi^{\prime}}_{m}^{\prime}(t)|
\frac{p_{r}^{2}}{g_{rr}}+\frac{p_{\theta}^{2}}{g_{\theta\theta}}-\frac{{g^{\prime}}_{tt}}{-\calD}\left(p_{\phi}+\frac{g_{t\phi}+\Omega_{0}g_{\phi\phi}}{{g^{\prime}}_{tt}}E\right)^{2}=-\left(\frac{E^{2}}{{g^{\prime}}_{tt}}+V\right),
{\calL}_{L}^{EPF}=\frac{1}{2}h_{\mu\nu}\left[R_{L}^{\mu\nu}-\frac{1}{2}\eta^{\mu\nu}R_{L}\right]-\frac{\mu^{2}}{2}\left(\left(h_{\mu\nu}+\partial_{\mu}A_{\nu}+\partial_{\nu}A_{\mu}\right)^{2}-\left(h+2\partial\cdotA\right)^{2}\right).
\langle0_{D}|T_{i}^{k}|0_{D}\rangle=\piA_{d}\delta_{i}^{k}\int_{0}^{\infty}dkk^{d}\sum_{n=1}^{\infty}\frac{I_{i\omega}(k\xi_{2})}{I_{i\omega}(k\xi_{1})\frac{\partialD_{i\omega}(k\xi_{1},k\xi_{2})}{\partial\omega}}
R_{1010}=\frac{a^{\prime}}{a(abc)^{2}}\,\quadR_{2020}=\frac{b^{\prime}}{b(abc)^{2}}\,\quadR_{3030}=\frac{c^{\prime}}{c(abc)^{2}}\,\quadR_{i0j0}=0\quad\mathrm{if}\quadi\neqj\.
(B,C)=\sum_{A}\left[\frac{\partialB}{\partial\Phi^{A}}\wedge\frac{\partialC}{\partial\Phi_{A}^{\star}}-(-1)^{\mathrm{deg}(\Phi^{A})}\frac{\partialC}{\partial\Phi_{A}^{\star}}\wedge\frac{\partialC}{\partial\Phi^{A}}\right]\;.
\hat{\calA}_{2}=\theta^{\alpha\beta}\partial_{\beta}[<\partial_{\alpha}\hat{\bar{\psi}}(x_{+})\gamma_{(2n+1)}\gamma_{\mu}\hat{\psi}(x_{-})>\{g(\hat{A}^{\mu}(x_{+})-\hat{A}^{\mu}(x_{-}))-i\partial^{\mu}\hat{U}\}].
\gamma_{q\dot{q}}^{i}h_{+p\dot{q}}^{1-}+\gamma_{p\dot{q}}^{i}h_{+q\dot{q}}^{1-}=0,\\tilde{\gamma}_{\dot{q}p}^{i}h_{-p\dot{p}}^{2+}+\tilde{\gamma}_{\dot{p}p}^{i}h_{-p\dot{q}}^{2+}=0,\\gamma_{q\dot{q}}^{i}\chi_{\dot{q}\dot{p}}^{1}+\tilde{\gamma}_{\dot{p}q}^{i}\chi_{pq}^{2}=0.
\sigma_{x}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right);\qquad\sigma_{y}=\left(\begin{array}{rr}{0}&{-i}\\{i}&{0}\\\end{array}\right);\qquad\sigma_{z}=\left(\begin{array}{rr}{1}&{0}\\{0}&{-1}\\\end{array}\right).
\mu_{\bar{u}}{}^{u}=\frac{\mu_{\bar{z}}{}^{z}\cdot\partial_{z}u-\partial_{\bar{z}}u}{\partial_{\bar{z}}\bar{u}-\mu_{\bar{z}}{}^{z}\cdot\partial_{z}\bar{u}}=\frac{\partial_{\bar{u}}z+\mu_{\bar{z}}{}^{z}\cdot\partial_{\bar{u}}\bar{z}}{\partial_{u}z+\mu_{\bar{z}}{}^{z}\cdot\partial_{u}\bar{z}}
{\calL}_{skyrme}=\frac{|W_{t}|^{2}-2|W_{z}|^{2}}{(1+|W|^{2})^{2}}+8\theta_{1}\frac{|W_{z}|^{2}}{(1+|W|^{2})^{4}}(|W_{t}|^{2}-|W_{z}|^{2}).
\delta\lambda=\left[\frac{10}{6}\frac{\lambda^{2}}{64\pi^{2}}{\bfI_{4}}+\frac{{\tilde{g}^{4}}}{16\pi^{2}}\left(\begin{array}{ll}{1}&{-3}\\{-3}&{1}\\\end{array}\right)\right]\left(-\frac{1}{\varepsilon}+\frac{\gamma}{2}+ln{\frac{2\beta\mu}{\sqrt{\pi}}}\right),
\hat{V}_{\mu\nu\rho}^{(1\,z)\,AB}=2\cdot2\,\mathrm{Tr}\,T^{A}T^{B}\,(Y_{R}^{(d)}-Y_{L}^{(d)})\,V_{\mu\nu\rho}^{(1)}=-4\,\mathrm{Tr}\,T^{A}T^{B}\,V_{\mu\nu\rho}^{(1)}\,.
\left|\left.a,1/2\right|n\right\rangle=\frac{1}{g_{\left|\left.a,1/2\right|n\right\rangle}^{\left|1/2\right\rangle}}K_{a\frac{1}{2}}\left(i\,\nu_{n}\right)\left|\frac{1}{2}\right\rangle\;,\quad\mathrm{where}\;\left|\frac{1}{2}\right\rangle\equiv\left|B_{\frac{1}{2}}\right\rangle\;,
a_{11}=\frac{(1^{24}2^{12})}{(1^{12}2^{12})},\;\;\;a_{12}=\frac{(1^{14}2^{12})}{(1^{12}2^{12})}=-\frac{(1^{23}2^{12})}{(1^{12}2^{12})},\;\;\;a_{22}=-\frac{(1^{13}2^{12})}{(1^{12}2^{12})}
z=C\Bigg(\frac{8\pi^{2}}{e^{2}}\Bigg)^{N_{c}}e^{-8\pi^{2}/e^{2}}M^{3N_{c}-N_{f}}\Bigg(\frac{m_{\lambda}^{2/3}}{B^{11/12}}\Bigg)^{N_{c}}\Big(\mathrm{det}(m_{q})_{i}^{j}\Big)^{2/3}\Big(\mathrm{det}(m_{sq})_{i}^{j}\Big)^{1/3}
\int_{C^{\prime}}\frac{\mathrm{d}\;\gamma}{2\pi}\frac{i\pi(\mu-\beta)(\mu+\gamma+\frac{i\pi}{2})\gamma}{(\gamma-\beta-\frac{i\pi}{2})(\gamma-\beta+\frac{i\pi}{2})(\gamma+\beta+\frac{i\pi}{2})}\;\mathrm{e}^{-i{\bar{\phi}^{\prime}}{}_{-}(\gamma)-i{\bar{\phi}^{\prime}}{}_{-}(-\gamma)}\;|B\rangle=(\beta\leftrightarrow-\beta),
\left(-\partial_{r}^{2}-\frac{3}{r}\partial_{r}+2\frac{(j(j+1)+l(l+1)-1)}{r^{2}}-\frac{8R^{2}}{(r^{2}+R^{2})^{2}}\right)\varphi=\lambda\frac{4R^{4}}{(r^{2}+R^{2})^{2}}\varphi\,.
P^{\prime}(w^{\prime})=L^{\varphik}\;\sum_{\left[n_{i^{\prime}}\right]_{i^{\prime}=0}^{k}}\;\sum_{\left[X_{i}\right]_{i=0}^{n}}\int\prod_{i=0}^{n}dt_{i}\;\prod_{j^{\prime}=0}^{k}\;\delta(\sum_{{i}=m_{j^{\prime}-1}+1}^{m_{j^{\prime}}}\;\;t_{i}-L^{\varphi}t_{j^{\prime}}^{\prime})\times
F^{S}=\sqrt{3}m_{3/2}{\tilde{C}}(S+{\bar{S}})sin\theta,\;\;F^{T}=m_{3/2}{\tilde{C}}(T+{\bar{T}})cos\theta,\;\;{\tilde{C}}=1+\frac{V_{o}}{3m_{3/2}^{3}},
S_{1D}=-\intd^{5}\sigma\left(\sqrt{-G}\sqrt{1+\frac{\tilde{{\calH}}^{2}}{2(-G)(1+C_{1}^{2})}+\frac{(\tilde{{\calH}}^{2})^{2}-2\tilde{{\calH}}^{4}}{8(-G)^{2}(1+C_{1}^{2})^{2}}}+\frac{\epsilon_{\mu\nu\lambda\sigma\tau}C^{\mu}\tilde{{\calH}}^{\nu\lambda}\tilde{{\calH}}^{\sigma\tau}}{8(1+C_{1}^{2})}\right),
z=\frac{\Lambda^{3N_{c}-N_{f}}S^{N_{f}-N_{c}}}{\mathrm{det}M-(\tilde{B}^{A_{1}A_{2}\ldotsA_{N_{f}-N_{c}}}M_{A_{1}}{}^{B_{1}}M_{A_{2}}{}^{B_{2}}\ldotsM_{A_{N_{f}-N_{c}}}{}^{B_{N_{f}-N_{c}}}B_{B_{1}B_{2}\ldotsB_{N_{f}-N_{c}}})}.
F_{k}=i\sum_{l=1\atopl\neqk}^{3}\left(\alpha_{kl}[B_{k},B_{l}]+\widehat{\alpha}_{kl}([B_{k},\widehat{B}_{l}]+[\widehat{B}_{k},B_{l}])\right)+\beta_{1,k}\chi_{1}+\beta_{2,k}\chi_{2}.
S_{\alpha\beta}(x,x^{\prime})=\left[\begin{array}{cc}{S_{\mathrm{AA}}(x,x^{\prime})}&{S_{\mathrm{AB}}(x,x^{\prime})}\\{S_{\mathrm{BA}}(x,x^{\prime})}&{S_{\mathrm{BB}}(x,x^{\prime})}\\\end{array}\right]
S_{2}={\frac{1}{2}}\intdp_{1}dp_{2}dp_{3}dp_{4}\big[{\frac{-g^{2}}{4(2\pi)^{d}}}\delta(p_{1}-p_{2}+p_{3}-p_{4})+i\delta(p_{1}-p_{2})\delta(p_{3}-p_{4}){\sigma^{0}}^{-1}(p_{1}){\sigma^{0}}^{-1}(p_{3})\big]\eta(p_{4},p_{1})\eta(p_{2},p_{3})
\sqrt{\theta}=\left(\frac{1}{8\pi\rho_{\mathrm{V}}}\right)^{1/4}\simeq\left(\frac{1}{8\pi\times10^{-47}\,\,\mathrm{GeV}^{4}}\right)^{1/4}\simeq5.0\times10^{-3}\,\,\mathrm{cm}.
\frac{\sqrt{3}}{2}(U_{m\,\,0}+{\delta}_{m\,\,0})+\frac{2i}{\pi}\sum_{n=0}^{\infty}\frac{(-)^{n}}{(2n+1)^{3/2}}(U_{m\,\,2n+1}+{\delta}_{m\,\,2n+1})\,\,.
\left.+\frac{e^{2}\vec{p}^{2}}{m^{2}c^{2}}\frac{1}{\vec{q}^{2}}-\frac{e^{2}}{m^{2}c^{2}}\frac{(\vec{q}\cdot\vec{p})^{2}}{\vec{q}^{4}}\right]\omega_{1}\omega_{2}~~.
\tilde{f}(q,\phi)\equivf(\tilde{q})=f(q)+\Sigma_{n=1}^{\infty}\tilde{f}(q,\phi)^{(n)}~~;~~\tilde{f}^{(1)}=-\phi^{\beta}\omega_{\beta\gamma}X^{\gamma\delta}\{\Theta_{\delta},f\}_{(q)}
D_{q}[\lambda]=\frac{q^{-2(\rho,\rho)}}{P_{q}(2\rho)}\sum_{t\in{\calT}}q^{2(t(\lambda+\rho),\rho)}\prod_{\alpha\in\Phi_{0}^{+}}\left(1-q^{-2(t(\lambda+\rho),\alpha)}\right)
D_{k}({\vec{x}},{\vec{x}^{\prime}})=\sum_{n=-\infty}^{\infty}\int_{0}^{\infty}{\frac{qdq}{2\pi\alpha}}{\frac{e^{in(\theta-\theta^{\prime})/\alpha}}{q^{2}+k_{4}^{2}}}J_{\left|n/\alpha\right|}(qr)J_{\left|n/\alpha\right|}(qr^{\prime}),
\left(\begin{matrix}{a}&{b}\\{c}&{d}\\\end{matrix}\right)=\left(\begin{matrix}{I-e_{i}}&{e_{i}}\\{e_{i}}&{I-e_{i}}\\\end{matrix}\right)\\\mathrm{s.t.}\\e_{i}=\mathrm{diag}(\underbrace{\overbrace{0,...,0}^{i-1},1,\overbrace{0,...,0}^{d-i-2}}_{d}).
A(z_{A})\starB(z_{A})=e^{\kappa\left[\sum_{i=1}^{n}(\partial_{x_{i}}\partial_{\tilde{p}_{i}}-\partial_{p_{i}}\partial_{\tilde{x}_{i}})+\sum_{\alpha=1}^{N}(\partial_{\theta_{\alpha}}\partial_{\tilde{p}_{\theta_{\alpha}}}+\partial_{p_{\theta_{\alpha}}}\partial_{\tilde{\theta}_{\alpha}})\right]}\left.A(z_{A})B(\tilde{z}_{A})\right|_{\tilde{z}_{A}=z_{A}}
K\sim\frac{\tau_{3}}{2}L^{2}\frac{3}{2^{5/3}}\left(\frac{u^{2}{\alpha^{\prime}}^{2}}{L^{2}}\right)^{4/3}=\frac{1}{8\pi^{2}g_{YM}^{2}}\frac{3}{2^{5/3}}\left(\frac{{\alpha^{\prime}}^{2}}{L^{2}}\right)^{1/3}(u^{2})^{4/3}\,
2\sum_{ij}\left(\partial_{i}\bbox{\phi}\cdot\partial_{j}\bbox{\phi}\right)^{2}\geq\sum_{ij}\left(\partial_{i}\bbox{\phi}\right)^{2}\left(\partial_{j}\bbox{\phi}\right)^{2}\geq\sum_{ij}\left(\partial_{i}\bbox{\phi}\cdot\partial_{j}\bbox{\phi}\right)^{2}\,,
\langlet_{1}\cdot\cdot\cdott_{N}|\Psi\rangle=e^{-B\sum_{i}t_{i}^{2}/2}e^{-{\frac{1}{4B}}\sum_{i}\left({\frac{\partial}{\partialt_{i}}}\right)^{2}}\prod_{i<j}(t_{i}-t_{j})^{m}
\begin{array}{lcr}{(|X_{0}|^{2}+|-\overline{{Y}}_{1}|^{2})-(|Y_{0}|^{2}+|X_{1}|^{2})=R_{1}}\\{X_{0}\overline{{Y}}_{0}+X_{1}(-\overline{{Y}}_{2})=\overline{{X}}_{0}Y_{0}+\overline{{X}}_{1}(-Y_{2})=0.}\\\end{array}
F_{j_{1},m_{1},j_{2},m_{2}}^{j_{1}+j_{2}-1,m_{1}+m_{2}}=\frac{N(j_{3},m_{3})(j_{3}-1)!\sqrt{j_{3}}}{N(j_{1},m_{1})(j_{1}-1/2)!\sqrt{j_{1}}N(j_{2},m_{2})(j_{2}-1/2)!\sqrt{j_{2}}}\frac{1}{\sqrt{2}}(j_{2}m_{1}-j_{1}m_{2}).
\begin{array}{ll}{\begin{array}{ll}{\frac{d}{dr}\sigma(r)=0,}&{r=0,}\\{\sigma(r)=0,}&{r\rightarrow\infty,}\\\end{array}}&{\begin{array}{ll}{A(r)\neq0,}&{r\rightarrow\infty,}\\{A(r)=1,}&{r=0.}\\\end{array}}\\\end{array}
\begin{array}{l}{h_{1}=2\displaystyle{\frac{x_{0}x_{+}}{q-q^{-1}}}(s(k^{2}+k^{-2})+q^{-1}s^{-1}w)\,,}\\{h_{2}=-2\displaystyle{\frac{x_{0}x_{+}}{q-q^{-1}}}(s^{-1}(k^{2}+k^{-2})+qsw)\,,}\\{h_{3}=\displaystyle{\frac{1}{q-q^{-1}}}(4x_{0}^{2}-x_{+}x_{-}(k^{2}+k^{-2})w)\,.}\\\end{array}
{\frac{M_{Pl}}{m}}=\left({\frac{3n^{3}P_{k}^{S}}{128\pi}}\right)^{{\frac{1}{n-4}}}\left({\frac{\phi}{M_{Pl}}}\right)^{-{\frac{n+2}{n-4}}}=\left({\frac{3n^{3}P_{k}^{S}}{128\pi}}\right)^{{\frac{1}{n-4}}}\left({\frac{4\pi}{nL}}\right)^{{\frac{n+2}{2(n-4)}}},
Ref_{-}(s,t)=\frac{s^{2n-1}}{\pi}P\!\int_{s_{0}}^{\infty}ds^{\prime}Im\left(\frac{f_{-}(s^{\prime},t)}{s^{2n-1}}\right)\left[\frac{2s^{2}}{s^{\prime}(s^{2}-s^{2})}\right].
\frac{1}{r}\frac{\partial}{\partialr}\left(r\frac{\partial\varphi}{\partialr}\right)-h^{2}\varphi^{3}e^{-2gM}+2hmM\varphi-\frac{{\calH}^{2}\varphi}{r^{2}}-2gM^{\prime}\varphi^{\prime}+g^{2}M^{2}\varphi=0,
H^{++{\breve{\beta}}-}=x_{h}^{{\breve{\alpha}}-}\partial_{h{\breve{\alpha}}}^{-}H^{++{\breve{\beta}}+}=x_{h}^{{\breve{\alpha}}-}\partial_{h{\breve{\alpha}}}^{-}\partial_{h}^{-{\breve{\beta}}}{\calL}^{+4},
L=-\frac{\kappa_{0}}{2}\dot{x}^{2}+i\zeta\cdot\dot{\zeta}+\,\frac{1}{T}\,\dot{x}\cdot\zeta\,\chi-e\,\dot{x}\cdotA(x)-\frac{ie}{\kappa_{0}}F_{\mu\nu}(x)\,\zeta^{\mu}\zeta^{\nu}.
\chi_{\Lambda}(C^{\Lambda_{0}})=\sum_{(\lambda_{0}^{\prime},\kappa_{0},-s)\inD^{+}}\,q^{2(\lambda_{0}^{\prime},\lambda+\rho_{0})}\,\sum_{s=0}^{\infty}n_{\lambda_{0}^{\prime},s}q^{-2s(\kappa+g)}
\int_{C}d^{n}{\eta}Y_{\alpha}^{N}({\eta})Y_{\alpha^{\prime}}^{N^{\prime}}({\eta})\int_{C}d^{n}{\eta^{\prime}}Y_{\alpha}^{N}({\eta^{\prime}})Y_{\alpha^{\prime}}^{N^{\prime}}({\eta^{\prime}})
\mu=\left(\begin{array}{cc}{-\omega^{ab}\frac{1}{4}\gamma_{ab}+eP^{a}\frac{i}{2}\gamma_{a}}&{\sqrt{e}\,\psi_{M}}\\{\sqrt{e}\,\overline{{\psi}}^{N}}&{A_{.M}^{N}}\\\end{array}\right)+B\:1_{8}
\deltaX^{a}=\frac{i\tilde{u}_{0}v}{2u_{0}}(\chi\gamma^{a}P_{\pm}\epsilon),\qquad\delta\chi^{\alpha}=i\tilde{u}_{0}X^{b}(\gamma_{b}P_{\pm}\epsilon)^{\alpha}+u_{0}(\gamma^{3}\epsilon)^{\alpha},
\{{\calQ},\omega_{d}\}=(-1)^{g}i_{v_{1}}\cdotsi_{v_{g}}d\Omega_{d+g}+\sum_{k=1}^{g}(-1)^{k+1}{\calL}_{v_{k}}i_{v_{1}}\cdotsi_{v_{k-1}}i_{v_{k+1}}\cdotsi_{v_{g}}\Omega_{d+g}=d\omega_{d}.
\frac{d^{2}z_{a}^{\pm}}{d\lambda_{a}^{2}}+2\frac{\partial\rho(z_{a})}{\partialz_{a}^{\pm}}\frac{dz_{a}^{\pm}}{d\lambda_{a}}\frac{dz_{a}^{\pm}}{d\lambda_{a}}+2e^{-4\rho(z_{a})}\frac{\partial\rho}{\partialz_{a}^{\mp}}=0,
T_{\\mu\rho\sigma}^{\mu}=-p^{2}T_{1}\Biggl[\left(p_{\rho}p_{\sigma}-p^{2}g_{\rho\sigma}\right)\mp\frac{1}{4}\left(\varepsilon_{\rho\lambda}p^{\lambda}p_{\sigma}+\varepsilon_{\sigma\lambda}p^{\lambda}p_{\rho}\right)\Biggl]\,.
{}^{\gamma\gamma}G_{1234}^{4}=4\left\{\frac{\delta^{2}W}{\deltaD_{12}^{-1}\deltaD_{34}^{-1}}+\frac{\deltaW}{\deltaD_{12}^{-1}}\frac{\deltaW}{\deltaD_{34}^{-1}}\right\}=-2\frac{\delta\;{}^{\gamma}G_{12}^{2}}{\deltaD_{34}^{-1}}+{}^{\gamma}G_{12}^{2}{}^{\gamma}G_{34}^{2}.
\alpha(\rho,\theta)=\left(\begin{array}{ll}{e^{i\frac{1}{2}\theta}}&{\rho}\\{0}&{e^{-i\frac{1}{2}\theta}}\\\end{array}\right),\,\,p_{R}\sim\left(\begin{array}{ll}{2}&{0}\\{0}&{0}\\\end{array}\right)
{\mathcal{L}}={\frac{1}{4}}(\partial^{\hat{\mu}}{\hat{h}}^{{\hat{\nu}}{\hat{\rho}}}\partial_{\hat{\mu}}{\hat{h}}_{{\hat{\nu}}{\hat{\rho}}}-\partial^{\hat{\mu}}\hat{h}\partial_{\hat{\mu}}\hat{h}-2{\hat{h}}^{\hat{\mu}}{\hat{h}}_{hat\mu}+2{\hat{h}}^{\hat{\mu}}\partial_{\hat{\mu}}\hat{h})
\hat{j}_{q}^{(2)}(\hat{x},t)=i\frac{f^{2}}{6}\Bigl\{\psi^{*}(\hat{x},t)\hat{x}^{3}V^{\prime}(\hat{x})\partial_{\hat{x}}\psi(\hat{x},t)-\bigl[\partial_{\hat{x}}\psi^{*}(\hat{x},t)\bigr]\hat{x}^{3}V^{\prime}(\hat{x})\psi(\hat{x},t)\Bigr\}.
gl(3n)\downarrowgl(3)+gl(n)\quad{\{}\overline{{1^{K}}}{\}}\cdot{\{}1^{L}{\}}\downarrow\sum_{\rho\vdashK,\sigma\vdashL}{\{}\overline{{\rho^{\prime}}}{\}}\cdot{\{}\sigma^{\prime}{\}}\times{\{}\overline{{\rho}}{\}}\cdot{\{}\sigma{\}},
\nabla_{a}{\calP}^{a\mu}=\left(\nabla_{a}{\calP}^{ab}+K^{b}{}_{ai}{\calP}^{ai}\right)e^{\mu}{}_{b}+\left(\widetilde{\nabla}_{a}{\calP}^{ai}-K_{ab}{}^{i}{\calP}^{ab}\right)n^{\mu}{}_{i}\,,
\Lambda\frac{d}{d\Lambda}Z(\Lambda)=\langle(-\frac{1}{2}\int\frac{d^{2}p}{(2\pi)^{2}}\phi(p)\phi(-p)p^{2}\Lambda\frac{\partialK^{-1}(p^{2}/\Lambda^{2})}{\partial\Lambda}-\Lambda\frac{\partialV(\phi,\Lambda)}{\partial\Lambda})\rangle=0.
{\calL}_{bosonic}=-\frac{1}{4l^{2}}\partial_{\mu}l\partial^{\mu}l+\frac{1}{4l^{2}}h^{\mu}h_{\mu}-G_{i{\bar{j}}}\partial_{\mu}z^{i}\partial^{\mu}{\bar{z}}^{\bar{j}}-\frac{i}{2}(G_{ij}\partialz^{j}-G_{i{\bar{j}}}\partial_{\mu}{\bar{z}}^{\bar{j}})h^{\mu}.
\tilde{\vartheta}_{2}^{2}=\tilde{\vartheta}_{3}^{2}{\frac{\vartheta_{4}^{2}}{\vartheta_{3}^{2}}}\left(1-4{\frac{\vartheta_{2}^{4}}{\vartheta_{3}^{4}}}\right),\;\tilde{\vartheta}_{4}^{2}=\tilde{\vartheta}_{3}^{2}{\frac{\vartheta_{2}^{2}}{\vartheta_{3}^{2}}}\left(1-4{\frac{\vartheta_{4}^{4}}{\vartheta_{3}^{4}}}\right).
\frac16\left(\frac{\Thetag_{m}}{2\pi^{2}}\right)^{2}\intd^{4}xd^{4}yj_{\mu}(x)\left[\frac{1}{(x-y)^{2}}-4\pi^{2}{\calD}_{m}^{(4)}(x-y)\right]j_{\mu}(y).
I_{\mu}^{\\nu}\equiv\left[\delta_{\mu}^{\nu}+\phi^{-1}\phi_{\mu\alpha}h^{\alpha\nu}\right]\;,\quadI^{-1}\_{\mu}^{\\alpha}I_{\alpha}^{\\nu}=\delta_{\mu}^{\nu}\;,\quadI\equivdetI_{\mu}^{\\nu}\;.
\vert\{\lambda\},\{\rho\},\chi\rangle{}~\equiv{}~(\alpha_{-{n_{1}}}^{0})^{\lambda_{1}}...(\alpha_{-{n_{p}}}^{0})^{\lambda_{p}}(\epsilon_{-r_{1}}^{(\pm)0})^{\rho_{1}}...(\epsilon_{-r_{s}}^{(\pm)0})^{\rho_{s}}\vert\chi;q\rangle{}~,
{\calL}=\frac{f_{\pi}^{2}}{4}\mathrm{Tr}\partial_{\mu}U^{\dagger}\partial^{\mu}U+\frac{1}{32e^{2}}\mathrm{Tr}\left(\left[U^{\dagger}\partial_{\mu}U,U^{\dagger}\partial_{\nu}U\right]^{2}\right),
\partial_{M}J^{M}(x,y)=\frac{1}{96\pi^{2}}\left[\mathrm{Tr}\left(Q_{L}(y){\calF}_{L}(x)\tilde{\calF}_{L}(x)\right)-\mathrm{Tr}\left(Q_{R}(y){\calF}_{R}(x)\tilde{\calF}_{R}(x)\right)\right]\;,
\begin{array}{rl}{}&{<0|T\left(\varphi_{1}\varphi_{2}\varphi_{3}(\varphi\star\varphi\star\varphi)(x_{4})\right)|0>}\\{=}&{\sum_{\lambda_{1}\lambda_{2}\lambda_{3}}\tau_{14}^{\lambda_{1}}\tau_{24}^{\lambda_{2}}\tau_{34}^{\lambda_{3}}\{D_{14}^{\lambda_{1}},D_{24}^{\lambda_{2}},D_{34}^{\lambda_{3}}\}_{\star},}\\\end{array}
\int_{(\infty)}d^{4}x^{\mu}\,\left\{-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}\,+\,\mid(\nabla_{\mu}+iqA_{\mu})\psi\mid^{2}-\frac{1}{4}\lambda_{0}\left(\mid\psi\mid^{2}-a^{2}\right)^{2}\right\},
\left[\cdots\right]={\frac{1}{\eta{\bar{\eta}}}}\left[(2\tau_{2})^{-1/2}\sum_{n,w\inZ}e^{-{\frac{\pix^{2}}{2\tau_{2}}}|n-w\tau|^{2}}e^{\piinwx^{2}}+\left(|\Theta_{3}\Theta_{4}|+|\Theta_{2}\Theta_{3}|+|\Theta_{2}\Theta_{4}|\right)\right]\quad.
{\frac{[a^{\prime}]_{0}}{a_{0}b_{0}}}=-{\frac{8\piG_{5}}{3c^{4}}}\left[\sigma+\varrho_{m}c^{2}(1+\zetae^{-2\psi})+\varrho_{r}c^{2}e^{-2\psi}+\varrho_{\psi}c^{2}\right],
\langleP[\overline{{\psi}},\psi,A,h]\rangle:=\frac{1}{Z}\int{\calD}h{\calD}A{\calD}\overline{{\psi}}{\calD}\psi\;P[\overline{{\psi}},\psi,A,h]\;e^{-S[\overline{{\psi}},\psi,A,h]}\;.
\hat{R}dK_{1}^{\dagger}\hat{R}K_{1}=K_{1}\hat{R}dK_{1}^{\dagger}\hat{R}^{-1}\;,\quad\hat{R}dK_{1}^{\dagger}\hat{R}dK_{1}^{\dagger}=-dK_{1}^{\dagger}\hat{R}dK_{1}^{\dagger}\hat{R}^{-1}\;.
\lambda_{3}^{ijk}(-p,p-q,q)=2\sigma^{ijk}a^{d}\int_{a}^{\infty}dz\frac{1}{z^{d+1}}\Big(\frac{z}{a}\Big)^{\frac{3d}{2}}\frac{K_{\nu_{i}}(-pz)K_{\nu_{j}}((p-q)z)K_{\nu_{k}}(qz)}{K_{\nu_{i}}(-pa)K_{\nu_{j}}((p-q)a)K_{\nu_{k}}(qa)}
\begin{array}{c}{\delta_{1}^{^{\prime}2}=s_{2}\left(s_{2}+1\right)\,\quad\delta_{2}^{^{\prime}2}=m^{2}s_{1}\left(s_{1}+3\right)}\\{s_{1}\geqs_{2}\qquad,\qquads_{1},s_{2}=0,1,2,...}\\\end{array}
\hat{H}=-\frac{g^{2}}{2}\int_{0}^{L}\mathrm{d}x\:\frac{\delta}{\mathrm{i}\,\deltaA_{1}^{a}(x)}\frac{\delta}{\mathrm{i}\,\deltaA_{1}^{a}(x)}=-\frac{g^{2}}{2}\bigtriangleup_{{\mathcalA}_{0}}\;,
v(\beta_{j}):=\frac{1}{N}A_{N-1}^{-1}\left(\begin{array}{c}{\beta_{j}^{(1)}}\\{\beta_{j}^{(2)}}\\{\vdots}\\{\beta_{j}^{(N-1)}}\\\end{array}\right),\;\;\beta_{j}^{(n)}\in{\bfZ}.
G^{\pm}={\frac{1}{\sqrt6}}\left(e^{\pm\pii/4}U^{\mu}\otimes{\bf1}\otimes\epsilon_{\mu}^{\pm}+e^{\mp\pii/4}{\bf1}\otimesU^{\mu}\otimes\epsilon_{\mu}^{\pm}+{\frac{3}{\sqrt2}}{\bf1}\otimes{\bf1}\otimess^{\pm}\right).
u^{\mu}={\frac{dx^{\mu}}{d\tau}}~~,~~\sigma^{\mu\nu}=\Lambda_{\rho}^{~\mu}{\frac{d\Lambda^{\rho\nu}}{d\tau}}=-\sigma^{\nu\mu}~~,~~\Lambda_{\rho}^{~\mu}\Lambda^{\rho\nu}=\Lambda_{~\rho}^{\mu}\Lambda^{\nu\rho}=g^{\mu\nu}~~,~~g^{00}=-g^{ii}=1.
{\calH}_{\theta}\Psi=\Big\{P_{0}\psi_{0}^{\prime}+P_{1}\psi_{1}^{\prime}+P_{2}\psi_{2}^{\prime}-\sqrt{\kappa}(P_{0}^{\prime}+P_{1}^{\prime})+\sum_{i=1}^{N}\pi_{i}f_{i}^{\prime}\Big\}\Psi=0
={\frac{\alpha}{\pi}}\sum_{k=0}^{\infty}\sum_{n=0}^{\infty}{\frac{\Gamma(z+k)\Gamma(2z+2k+2n-1)}{2^{2k}k!~\Gamma(z)\Gamma(2z+2k)}}~\zeta_{R}(2z+2k+2n-1,1/2)~{\frac{B_{2n}}{(2n)!}}\left({\frac{2\pi}{\alpha}}\right)^{2n}
\sum_{p_{1},\ldots,p_{i}\inT_{2}}\sum_{s_{1},t_{1},\ldots,s_{G},t_{G}\inS_{n}}N^{n(2-2G)-i}\sum_{R\inY_{n}^{N}}\left[\frac{d_{R}}{n!^{2}}\chi_{R}(p_{1}\cdotsp_{i}\Omega_{n}^{2-2G}\prod_{j}s_{j}t_{j}s_{j}^{-1}t_{j}^{-1})\right].
a(X)=\frac{1}{{(2\pi)}^{3/2}}{\int}_{X}\sqrt{g}d^{3}{\varphi}_{0}^{i}{\varepsilon}^{I_{1}J_{1}}{\varepsilon}^{I_{2}J_{2}}{\varepsilon}^{I_{3}J_{3}}{\varepsilon}^{I_{4}J_{4}}{\Omega}_{I_{1}I_{2}I_{3}I_{4}}{\Omega}_{J_{1}J_{2}J_{3}J_{4}}.
M_{\mathrm{ext}}=-\frac{(d-2)(d-3)\Sigma_{k}l^{2}r_{\mathrm{min}}^{d-5}}{16\piG(d-1)^{2}}\left(1-\frac{d-1}{d-3}\frac{4\tilde{\alpha}}{l^{2}}+\sqrt{1-\frac{(d-1)(d-5)}{(d-3)^{2}}\frac{4\tilde{\alpha}}{l^{2}}}\right).
2\tilde{\nabla}_{[a}\Sigma_{b]}^{ij}=-2\left(\Lambda_{a[}^{ik}\Lambda_{b]k}^{j}+\Sigma_{[a}^{ik}\Sigma_{b]k}^{j}\right)^{str}+4\Lambda_{[a}^{k(i}\Lambda_{b]k}^{j)}
{_1F}_{1}\left(-n+l+1,2l+\delta_{1}+\delta_{2}+2;\varepsilonr\right)=\sum_{s=0}^{n-l-1}\frac{(-n+l+1)_{s}}{(2l+\delta_{1}+\delta_{2}+2)_{s}}\frac{(\varepsilonr)^{s}}{s!}
{\frac{1}{\mathrm{Order}(D)}}\int\frac{[d\deltaX][d\deltag]}{Vol[\mathrm{Diff}]}~\to~\int{\frac{[d\deltae_{I}(\lambda,{\hat{e}})]}{Vol[\mathrm{Diff}_{\partialM}]}}\int_{[{\hat{e}}]}\frac{[d\deltag]}{Vol[\mathrm{Diff}_{M}]}\int_{[{\hat{g}};{\hat{e}}]}[d\deltaX]\quad,
\mathcal{L}_{\mathrm{spin~1}}=\frac12\left(\mathbf{E}^{2}-\mathbf{H}^{2}\right)+\frac{\alpha^{2}}{10m^{4}}\left[\frac{29}4\left(\mathbf{E}^{2}-\mathbf{H}^{2}\right)^{2}-53(\mathbf{EH})^{2}\right]
\omega^{\bot}\;=\;\left(\begin{array}{cc}{\gamma^{\mu\nu}\,\omega_{1\mu\nu}}&{k\,\gamma^{\mu}\gamma^{5}\,{\binom{\omega_{1\mu}}{\omega_{2\mu}}}}\\{k^{*}\gamma^{\mu}\gamma^{5}\,(\tilde{\omega}_{1\mu},\tilde{\omega}_{2\mu}}&{\gamma^{\mu\nu}\omega_{2\mu\nu}+(k^{*}k-1)\,\omega}\\\end{array}\right)
{\calI}=\int\frac{d^{2}\tau}{\tau_{2}^{\;2}}\bar{F}(\bar{\tau})\,\partial_{\bar{\tau}}(\tau_{2}\sum_{p_{L},p_{R}}e^{\pii\tau\,|p_{L}|^{2}}\,e^{-\pii\bar{\tau\,}|p_{R}|^{2}})
{\calL}_{GN}^{(k)}=\frac{f_{1}^{\mu\nu}}{2}(\partial_{\mu}\rho_{j}\partial_{\nu}\rho_{j})+\frac{f_{2}^{\mu\nu}}{\rho^{2}}(\rho_{j}\partial_{\mu}\rho_{j})(\rho_{i}\partial_{\nu}\rho_{i})
S^{\mathrm{grav.}}=-{\frac{T_{KK11}}{2}}\intd^{7}\hat{\xi}\sqrt{|\hat{\gamma}|}\\left[\hat{\hat{k}}^{4/7}\hat{\gamma}^{\hat{\imath}\hat{\jmath}}\hat{D}_{\hat{\imath}}\hat{\hat{X}}^{\hat{\hat{\mu}}}\hat{D}_{\hat{\jmath}}\hat{\hat{X}}^{\hat{\hat{\nu}}}\hat{\hat{g}}_{\hat{\hat{\mu}}\hat{\hat{\nu}}}-5\right]\,,