harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{\partial\Gamma_{\Lambda}[\phi^{c}]}{\partial\Lambda}=\frac{1}{2}tr\left[\frac{\partialC^{-1}}{\partial\Lambda}\;_{.}\left(C^{-1}+\frac{\delta^{2}\Gamma_{\Lambda}}{\delta\phi^{c}\delta\phi^{c}}\right)^{-1}\right]
\displaystyle\begin{array}{rl}{\displaystyleZ_{N}(\vartheta)}&{=N\left[\phi_{1/2}(\vartheta+\Theta)+\phi_{1/2}(\vartheta-\Theta)\right]+\displaystyle\sum_{k=1}^{N_{H}}\phi_{1}(\vartheta-h_{k})}\\{}&{-\displaystyle\sum_{k=1}^{M_{C}+M_{W}}\phi_{1}(\vartheta-\xi_{k})-\displaystyle\sum_{k=1}^{M_{R}+N_{H}}\phi_{1}(\vartheta-x_{k})\:,}\\\end{array}
A_{0i_{2}\ldotsi_{p}}(x_{0},{\bfx})=x_{0}^{\frac{d}{2}}\int{\frac{d^{d}k}{(2\pi)^{d}}}e^{-i{\bfk}\cdot{\bfx}}a_{0i_{2}\ldotsi_{p}}({\bfk})K_{\nu}(|{\bfk}|x_{0}).
\mathcal{J}=2K+12\alpha\delta_{a}^{[l}\delta_{b}^{m}\delta_{c}^{n]}K_{l}^{a}\left[\widehatR^{bc}{}_{mn}-2H_{m}^{b}H_{n}^{c}-\frac{2}{3}K_{m}^{b}K_{n}^{c}\right],
(P_{\mu}P^{\mu})^{\times}|[j,{\mathsfs}_{R}],f^{-}\rangle={\mathsfs}_{R}\,|[j,{\mathsfs}_{R}],f^{-}\rangle\,,\quad\mathrm{for~any}\f(j_{3},b)\in{\calS}({\mathbbR}^{3})\,,
[T_{\alpha_{1}...\alpha_{n}\beta_{1}^{\prime}...\beta_{m}^{\prime}}]_{;\mu}=[T_{\alpha_{1}...\alpha_{n}\beta_{1}^{\prime}...\beta_{m}^{\prime};\mu}]+[T_{\alpha_{1}...\alpha_{n}\beta_{1}^{\prime}...\beta_{m}^{\prime};\mu^{\prime}}].
T={\frac{1}{\pi}}\intd^{2}x\left\{g_{\alpha\beta}(\partial\psi_{\alpha}\bar{\partial}\bar{\psi}_{\beta}-\bar{\partial}\psi_{\alpha}\partial\bar{\psi}_{\beta})+\frac{1}{4}\epsilon_{ij}\partial_{i}\left((v^{a}-Q^{a}(\psi,\bar{\psi}))A_{j}^{a}\right)\right\}.
\tau^{s+d_{j}-1}\,Q_{j}(\vec{\beta},\vec{\lambda})\,e^{\,-\tau\sum_{k,v}A_{kv}^{j}(\vec{\beta},\vec{\lambda})\,p_{k}p_{v}}\,e^{\,-\tau\,m^{2}\sum_{k=1}^{s}\beta_{k}}\biggr]\quad.
\tilde{\rho}_{B}=\frac{1}{12\pi^{2}}\left(D_{i}\xi^{a}D_{j}\xi^{b}D_{k}\xi^{c}\xi^{d}\epsilon^{abcd}-3\xi^{4}F_{ij}^{\alpha}D_{k}\xi^{\alpha}\right)\varepsilon^{ijk}\,
\begin{array}{c}{A_{j}=B_{j}=\left(\begin{array}{cc}{ia_{j}}&{-c_{j}^{*}}\\{c_{j}}&{ib_{j}}\\\end{array}\right),}\\{\Phi_{A}=\left(\begin{array}{cc}{i\alpha_{A}}&{-\gamma_{A}^{*}}\\{\gamma_{A}}&{i\beta_{A}}\\\end{array}\right),\Phi_{B}=\left(\begin{array}{cc}{i\alpha_{B}}&{-\gamma_{B}^{*}}\\{\gamma_{B}}&{i\beta_{B}}\\\end{array}\right).}\\\end{array}
\ddot{\calR}_{g}+\left({\frac{1}{{\calR}_{g}}}-{\frac{3}{2}}{\frac{h^{\prime}}{h}}\right)\dot{\calR}_{g}^{2}+{\frac{1}{2}}h\,h^{\prime}-{\frac{h^{2}}{{\calR}_{g}}}=0\;.
\frac{\Phi_{n+1}^{i}-\Phi_{n}^{i}}{\tau}=\frac{\Phi_{n}^{i+1}-2\Phi_{n}^{i}+\Phi_{n}^{i-1}}{\delta^{2}}-\mu^{2}\Phi_{n}^{i}-\lambda{\Phi_{n}^{i}}^{3}+noise.
\left[\frac{d^{2}}{d\tau^{2}}+q^{2}-1+\int_{0}^{1}dp\left\{\frac{p^{2}}{p^{2}+L^{2}}\left[{\cal{U}}_{p}^{+}(\tau){\cal{U}}_{p}^{-}(\tau)-1\right]\right\}\right]{\cal{U}}_{q}^{\pm}(\tau)=0
K\,=\,\frac{1}{\pi^{2}}\,\frac{g^{2}}{(2\pi)^{4}}\,\frac{m^{2}}{\sqrt{\omega(\vec{p}\,_{1}^{\prime})\omega(\vec{p}\,_{2}^{\prime})\omega(\vec{p}_{1})\omega(\vec{p}_{2})}},
E_{B}=\left(\begin{array}{cc}{E_{1}}&{E_{2}}\\\end{array}\right)\;{\calE}_{B}\;;\;{\calE}_{B}=\left(\begin{array}{c}{1}\\{\nu_{B}^{c}}\\\end{array}\right)\;{\frac{1}{\sqrt{1+\vert\nu_{B}\vert^{2}}}}\;.
\mathrm{~\frac{1}{2}~}\epsilon_{ab}\mathbf{s}_{m}^{b}\mathbf{s}_{m}^{a}A^{i}=R_{\alpha_{0}}^{i}B^{\alpha_{0}}+\mathrm{~\frac{1}{2}~}\epsilon_{ab}R_{\alpha_{0},j}^{i}R_{\beta_{0}}^{j}C^{\beta_{0}b}C^{\alpha_{0}a},
{\frac{\delta}{\deltaS^{\mu\nu}(z(\tau))}}\int_{a}^{b}d\bar{\tau}\int_{a^{\prime}}^{b^{\prime}}d\bar{\tau}^{\prime}E_{\mathrm{conf}}(\bar{z}-\bar{z}^{\prime},\dot{\bar{z}},\dot{\bar{z}}^{\prime})
\frac{\vert\eta_{\mathrm{i}}\vert^{1-b}}{\gammab}\frac{\mu_{\mathrm{I}}^{\prime}(\eta_{\mathrm{i}})}{\mu_{\mathrm{I}}(\eta_{\mathrm{i}})}=\pm\frac{i}{\sqrt{b_{m}}}\biggl[\frac{\lambda(\eta_{\mathrm{i}})}{l_{\mathrm{C}}}\biggr]^{m}\ll1,
\left[\begin{array}{cccc}{g_{2}+w_{2}^{\2}h_{4}+n_{2}^{\2}h_{5}}&{w_{2}w_{3}h_{4}+n_{2}n_{3}h_{5}}&{w_{2}h_{4}}&{n_{2}h_{5}}\\{w_{2}w_{3}h_{4}+n_{2}n_{3}h_{5}}&{g_{3}+w_{3}^{\2}h_{4}+n_{3}^{\2}h_{5}}&{w_{3}h_{4}}&{n_{3}h_{5}}\\{w_{2}h_{4}}&{w_{3}h_{4}}&{h_{4}}&{0}\\{n_{2}h_{5}}&{n_{3}h_{5}}&{0}&{h_{5}}\\\end{array}\right],
{\calL}_{bosonic}^{{\calN}=8}={\frac{1}{2g_{_{YM}}^{2}}}\left\{-F^{\underline{m}\underline{n}}F_{\underline{m}\underline{n}}+\nabla_{\underline{m}}\overline{{\rho}}_{AB}\nabla^{\underline{m}}\rho^{AB}+{\frac{1}{16}}\left[\overline{{\rho}}_{AB}\rho^{CD}\right]\left[\rho^{AB},\overline{{\rho}}_{CD}\right]\right\}\,.
B_{\ell}={\frac{\Gamma(\ell+1)\Gamma(\ell+1+2ikr_{0})}{\Gamma(\ell+{\frac{3}{2}})\Gamma(2\ell+1)\Gamma(1+2ikr_{0})}}\;{\frac{\sqrtk}{2}}\;\left({\frac{kr_{0}}{2}}\right)^{\ell}
\partial_{u}\hat{\epsilon}(u)-{\frac{1}{2}}H^{-{\frac{1}{2}}}[(\partial_{\hat{a}}S)-{\frac{1}{16}}W^{2}y^{a}]\gamma^{\underline{{u\hat{a}}}}\hat{\epsilon}+{\frac{1}{8}}WH^{-{\frac{1}{2}}}\gamma^{\underline{{y}}}[(\gamma^{\underline{{uv}}}+\eta^{\underline{{uv}}})i\sigma_{2}]\hat{\epsilon}=0\,.
\delta_{\varepsilon}\Phi_{a_{I}}^{I}=\varepsilon^{i}{(C_{i}^{I})_{a_{I}}}^{a_{I+1}}\Phi_{a_{I+1}}^{I+1}+\varepsilon^{i}{(\tilde{C}_{i}^{I})_{a_{I}}}^{a_{I-1}}\frac{d}{d\tau}{\Phi}_{a_{I-1}}^{I-1}.
\hat{H}^{1}=\sum_{i=0}^{N}\frac{\vec{p}_{1i}^{2}}{2m_{i}}=H_{s}^{1}+H_{c}^{1}=\frac{\vec{p}_{1s}^{2}}{2M_{N+1}}+\sum_{j=2}^{N+1}\frac{\vec{\pi}_{1j}^{2}}{2\mu_{1j}}
U(\prod_{i=1}^{g}\left(g_{a_{i}}^{~}g_{b_{i}}^{~}g_{a_{i}}^{-1}g_{b_{i}}^{-1}\right))~\psi~V^{-1}(\prod_{i=1}^{g}\left(g_{a_{i}}^{~}g_{b_{i}}^{~}g_{a_{i}}^{-1}g_{b_{i}}^{-1}\right))=\psi~.
\begin{array}{c}{b^{-}~=~lim_{Q\toq}{\frac{Q^{\frac{\pmkN_{B}}{2}}{(B^{-})}^{k}}{([k]!)^{\frac{1}{2}}}}}\\{b^{+}~=~lim_{Q\toq}{{\frac{{(B^{+})}^{k}Q^{{\frac{\pmkN_{B}}{2}}}}{([k]!)^{\frac{1}{2}}}}}}\\\end{array}
\frac{L^{2}}{M^{2}}\partial_{\alpha}\partial^{\alpha}{\calA}_{\mu}-\ddot{{\calA}}_{\mu}-{\calA}_{\mu}^{\prime\prime}-2{\calH}{\calA}_{\mu}^{\prime}+P^{\prime}\frac{M}{L}[Z_{\mu}^{\prime}-({\calH}-{\calF})Z_{\mu}-\dot{V}_{\mu}]+\alpha{\calA}_{\mu}{\calL}^{2}f^{2}=0,
W_{2}(R)={\frac{1}{\sqrt{2}R}}BJ_{1}B^{-1}=\left(\begin{matrix}{0}&{0}&{0}&{0}\\{0}&{0}&{-i/R^{2}}&{i}\\{0}&{i}&{0}&{0}\\{0}&{i/R^{2}}&{0}&{0}\\\end{matrix}\right),
ds^{2}\,\stackrel{r\to0}{\Longrightarrow}\,ds_{H}^{2}=\underbrace{(k)^{-\frac{\widetilde{d}}{D-2}}\,r^{\frac{\widetilde{d}^{2}}{D-2}}\,dx^{\mu}\otimesdx^{\nu}\,\eta_{\mu\nu}\,+\,(k)^{\frac{d}{D-2}}\,\frac{dr^{2}}{r^{2}}}_{\mathrm{~AdS_{p+2}~metric}}\,+\,(k)^{\frac{d}{D-2}}\,ds_{S^{D-p-2}}^{2}
\Psi=\intd^{26}p\;\left[\phi(p)\;|0_{1};p\rangle+A_{\mu}(p)\;\alpha_{-1}^{\mu}|0_{1};p\rangle+\chi(p)b_{-1}c_{0}|0_{1};p\rangle+B_{\mu\nu}(p)\alpha_{-1}^{\mu}\alpha_{-1}^{\nu}|0_{1};p\rangle+\cdots\right]\,.
T\xi_{\omega}=\left(\frac{2}{\pi}\right)^{1/2}\frac{k}{(p(k^{2})^{2}+k^{2}q(k^{2})^{2})^{1/2}}\frac{p(k^{2})-\omegaq(k^{2})}{\omega^{2}+k^{2}},
{\calL}={\frac{1}{2g\sqrt{\alpha^{\prime}}}}~\mathrm{Tr}~\left(\left({\dot{\Phi}}_{i}-{\frac{i\lambda}{2(2\pi\alpha^{\prime})}}\epsilon_{ijk}\left[\Phi_{i},\Phi_{j}\right]\right)^{2}+{\frac{1+\lambda^{2}}{2(2\pi\alpha^{\prime})^{2}}}\left[\Phi_{i},\Phi_{j}\right]^{2}\right)~.
\begin{array}{ll}{{\calL}_{\chi}}&{=(2\piR){\calL}_{5},_{\chi}=\tilde{g}^{MN}(\partial_{M}\chi)^{*}(\partial_{N}\chi)-V}\\{}&{=e^{\sqrt{\frac23}\kappa\varphi}g^{\mu\nu}(\partial_{\mu}\chi)^{*}(\partial_{\nu}\chi)-V}\\\end{array}
{\frac{1}{2}}\Big(\mathrm{Tr}_{A}e^{iF_{1}}+\mathrm{Tr}_{S}e^{iF_{2}}-\mathrm{tr}e^{iF_{1}}\mathrm{tr}e^{iF_{2}}\Big)\hat{A}(R)+{\frac{1}{16}}L(R).
\begin{array}{lll}{d_{v}}&{=}&{4\sum_{t=1}^{k}t+k(2n-4k-1)=2nk-k(2k-1),}\\{d_{H}}&{=}&{[2\sum_{t=1}^{k-1}((2t)^{2}+2t(2t+2))+2(2k)^{2}+(2n-4k)2k(2k+1)+2(2k)]/2}\\{}&{-}&{[2\sum_{t=1}^{k-1}(t(2t-1)+t(2t+1))+2k(2k-1)+(2n-4k+1)k(2k+1)]}\\{}&{=}&{k}\\\end{array}
\gamma_{g,9}:|D_{\mathrm{inv}}\rangle\rightarrow{\frac{1}{\sqrt{D}}}\sum_{\alpha}\sum_{\beta,\gamma}c_{\alpha\beta}{\overline{c}}_{\gamma\alpha}|d_{\beta}\otimes{\overline{d}}_{\gamma}\rangle={\frac{1}{\sqrt{D}}}\sum_{\alpha}|d_{\alpha}\otimes{\overline{d}}_{\alpha}\rangle=|D_{\mathrm{inv}}\rangle~,
l_{p}^{-2}ds^{2}=\hat{H}^{-1/3}\left[-dt^{2}+\sum_{i=1}^{5}(dx^{i})^{2}+\hat{H}\Big(d\hat{z}^{2}+d\hat{r}^{2}+\hat{r}^{2}d\Omega_{3}^{2}\Big)\right]
g\,\langle\langleF_{hk}(z_{1})\rangle\rangle=(\frac{4}{3}\frac{\alpha_{s}}{m_{2}}\frac{1}{r^{3}}+\frac{\sigma}{m_{2}}\frac{1}{r})(r^{h}p_{2}^{k}-r^{k}p_{2}^{h})+O(v^{3}),
\hat{\pi}_{(I)}(x)=\int_{(\infty)}\frac{d^{3}\vec{k}}{(2\pi)^{3}2\omega(\vec{k}\,)}\left(-i\omega(\vec{k}\,)\right)\left[a(\vec{k}\,)e^{-ik\cdotx}\,-\,a^{\dagger}(\vec{k}\,)e^{ik\cdotx}\right]\,
V(u)=E\left\{1-\frac{m_{p-2}\tilde{f}^{2}}{2E}\left(\frac{u}{R}\right)^{7-p}\left[1-\frac{{\calC}\tilde{f}}{{\calB}^{2}}\right]\right\}+\frac{L^{2}\tilde{f}^{3}}{2m_{p-2}u^{2}{\calB}^{2}},
\begin{aligned}{\delta_{\xi}\phi\equiv\left(\xiQ+\bar{\xi}\bar{Q}\right),\qquad\left(\xiQ+\bar{\xi}\bar{Q}\right)^{*}=\xiQ+\bar{\xi}\bar{Q}}\\{\delta_{\xi}\phi^{*}=\left(\delta_{\xi}\phi\right)^{*}=\left(\xiQ+\bar{\xi}\bar{Q}\right)\phi^{*}\quad,}\\\end{aligned}
\hat{\Theta}^{\hat{a}\hat{b}\hat{c}}\equiv-\,\ell_{p}^{3}\,\hat{W}^{\hat{a}\hat{b}\hat{c}}=-\,\ell_{p}^{3}\,z^{-2}\,\hat{g}^{\hat{a}\hat{k}}\hat{\calH}_{\hat{k}\hat{l}\hat{m}}\,(\hat{G}_{OM})^{\hat{b}\hat{l}}\,(\hat{G}_{OM})^{\hat{c}\hat{m}}\,.
\delta\simeq-{\frac{V_{(p)}}{(2\pi\sqrt{\alpha^{\prime}})^{p}}}\;\Gamma\left(3-{\frac{p}{2}}\right)\left({\frac{4\pi\alpha^{\prime}}{b^{2}}}\right)^{3-p/2}{\frac{\mathrm{sinh}^{4}(\pi\epsilon/2)}{\mathrm{sinh}(\pi\epsilon)}}+\;\cdots\,
\frac{2}{(4\pis)^{d/2}}\intdx\,\sum_{n=2}^{\infty}\sum_{m=2}^{n}V\underbrace{\frac{1}{\Box}V...\frac{1}{\Box}V}_{n-m}\frac{1}{\Box^{2}}\underbrace{V\frac{1}{\Box}...V\frac{1}{\Box}}_{m-2}\,V(x).
\hat{a}{\calP}=\frac{1}{q^{2}}{\calP}\hat{a}-\frac{\mu}{q^{3}}\hat{a}~~,~~\hat{b}{\calP}=\frac{1}{q^{2}}{\calP}\hat{b}-\frac{\mu}{q^{3}}\hat{d}\hat{c}^{-1}\hat{a},
f(r)=1-\frac{3\sigma_{0}^{2}}{\beta+3\lambda^{2}\sigma_{0}^{2}}\frac{1}{r^{2}}-\frac{9[2\beta\epsilon^{2}\sigma_{0}^{4}+3(2\epsilon^{2}-3)\lambda^{2}\sigma_{0}^{6}]}{2(2\beta-3\lambda^{2}\sigma_{0}^{2})(\beta+3\lambda^{2}\sigma_{0}^{2})^{2}}\frac{1}{r^{4}}+O(\frac{1}{r^{6}})
\left(\begin{matrix}{A}&{B}\\{C}&{D}\\\end{matrix}\right)^{-1}=\left(\begin{matrix}{(A-BD^{-1}C)^{-1}}&{(C-DB^{-1}A)^{-1}}\\{(B-AC^{-1}D)^{-1}}&{(D-CA^{-1}B)^{-1}}\\\end{matrix}\right),
H(z_{2}^{\prime},\dotsz_{\calN}^{\prime},P_{2}^{\prime},\dotsP_{\calN})=-\int_{\{z_{0}\}}^{\{z_{B}^{\prime}\}}\sum_{B}\frac{\partial\beta_{B}}{\partial\mu}(z_{2}^{\prime},\dotsz_{\calN}^{\prime},z_{A}^{\prime\prime},\dots,\mu)~dz_{B}^{\prime\prime}+f(z_{2}^{\prime},\dotsz_{\calN}^{\prime}).
\Gamma^{n}(|{\bfp}|,|{\bfq}|,E)=K^{n}(|{\bfp}|,|{\bfq}|)+\int\frac{d|{\bfk}|^{2}}{4\pi}K^{n}(|{\bfp}|,|{\bfk}|)\frac{i}{E-\frac{{\bfk}^{2}}{2m}+i\epsilon}\Gamma^{n}(|{\bfk}|,|{\bfq}|,E).
{\cal{G}}_{k_{\perp}}^{\pm(\eta)}(t,t^{\prime})=-{\frac{\Gamma(-\nu)}{2\sqrt{\pi}}}e^{-{\frac{i\pi}{4}}}{\frac{1}{S^{\prime}}}D_{\nu}\left({\frac{1+i}{\sqrt{\hbar}}}S(t)\right)D_{\nu}\left(-{\frac{1+i}{\sqrt{\hbar}}}S(t^{\prime})\right)
a_{5}=-\;4{\frac{b_{5}}{(b_{1})^{5}}}\;+{\frac{32b_{2}b_{4}+18(b_{3})^{3}}{(b_{1})^{6}}}\;-144{\frac{(b_{2})^{2}b_{3}}{(b_{1})^{7}}}\;+112{\frac{(b_{2})^{4}}{(b_{1})^{8}}}\;;
J_{3}={\alpha}_{ijk}\,\partial\varphi_{i}\,\partial\varphi_{j}\,\partial\varphi_{k}+{\beta}_{ij}\partial\varphi_{i}\,\partial^{2}\varphi_{j},\\\{\overline{J}}_{3}=v_{r}\,B_{ri}\,\partial\varphi_{i}
G_{\lambda}^{(4)}(p_{1},p_{2},\dot{-}p_{3},\dot{-}p_{4})=\frac{\delta^{4}\bar{Z}^{(1)}[J(k)]}{\deltaJ(\dot{-}p_{1})\deltaJ(\dot{-}p_{2})\deltaJ(p_{3})\deltaJ(p_{4})}\Bigg|_{J=0}~,
S^{\pm}=\left(\begin{array}{lll}{\partial_{1}X^{0}}&{...}&{\partial_{1}X^{p}}\\{\partial_{2}X^{0}}&{...}&{\partial_{2}X^{p}}\\{...}&{...}&{...}\\{G^{0\nu}(P_{\nu}\pmTn_{\nu})}&{...}&{G^{p\nu}(P_{\nu}\pmTn_{\nu})}\\\end{array}\right)\,
{p_{L}}=\frac{1}{\sqrt{2ImTImU}}(m_{1}+m_{2}{\bar{U}}+n_{1}{\bar{T}}+n_{2}{\bar{U}}{\bar{T}}),\;\;p_{R}=\frac{1}{\sqrt{2ImTImU}}(m_{1}+m_{2}{\bar{U}}+n_{1}T+n_{2}T{\bar{U}})
{\calL}={\calL}[\varphi_{\mathrm{cl}},\alpha_{\mathrm{cl}}]-\frac{r}{4}\left(\vec{u}\cdot{\calM}_{\mathrm{B}}\cdot\vec{u}+i\bar{\Psi}_{\alpha}\hat{\calM}_{\mathrm{F}}^{\alpha\beta}\Psi_{\beta}\right)
\bar{D}^{2}e^{-2V}D^{2}e^{2V}\rightarrow\bar{D}^{2}e^{i\Lambda}e^{-2V}e^{-i\bar{\Lambda}}D^{2}e^{i\bar{\Lambda}}e^{2V}e^{-i\Lambda}=e^{i\Lambda}\left(\bar{D}^{2}e^{-2V}D^{2}e^{2V}\right)e^{-i\Lambda}.
\Phi\left(p^{\mu}\right)=\Lambda\left(\dot{p}^{\mu}\top^{\mu}\right)\Phi\left(\dot{p}^{\mu}\right)=e^{i(\vec{J}\cdot\vec{\vartheta}+\vec{K}\cdot\vec{\varphi})}~\Phi\left(\dot{p}^{\mu}\right)\,,
\langleN\rangle_{\mathrm{T}}=\frac{1}{2\pi}\int_{-\infty}^{\infty}dx\,\theta^{\prime}-\sqrt{\frac{2mT}{\pi}}\,e^{-m/T}\,\int_{-\infty}^{\infty}dx\,\mathrm{sinh}\left(\frac{\theta^{\prime}}{2T}\right)+\dots
\partial_{\bar{a}}C_{i_{1}...i_{s}}^{g}=fact_{g}({\bar{C}}_{\bar{a}})-\sum_{i=1}^{s}{\bar{C}}_{{\bar{a}}i}^{l}C_{\sigma_{1}(l)i_{1}..{\hat{i}}..i_{s}}^{g}+{\bar{C}}_{{\bar{a}}0}^{l}C_{\sigma_{2}(l)i_{1}...i_{s}}^{g}
S={\frac{a_{H}^{n}\mathrm{Vol}(M_{n})}{4G_{n+2}}},\\\\\\{\calT}={\frac{h^{\prime}(a_{H})}{4\pi}}={\frac{(n+1)a_{H}^{2}+(n-1)kL^{2}}{4\piL^{2}a_{H}}},
u([(g_{\mathrm{reg}},w)][(h_{\mathrm{reg}},1)])=u([(g_{\mathrm{reg}}h_{\mathrm{reg}}g_{\mathrm{reg}}^{-1},1)]\,\hat{g})e^{k\Gamma(g_{\mathrm{reg}},h_{\mathrm{reg}})},
C^{\mu}={\frac{1}{6}}\varepsilon^{\mu\nu\rho\sigma}\left(A_{\nu}^{a}\partial_{\rho}A_{\sigma}^{a}+{\frac{1}{3}}\varepsilon_{abc}A_{\nu}^{a}A_{\rho}^{b}A_{\sigma}^{c}\right)={\frac{4}{3D^{3}}}(x-x_{0})^{\mu}[3\rho^{2}+(x-x_{0})^{2}].
\hat{U}(\theta_{0})\,\hat{A}_{i}^{\pm\pm}(n_{1},n_{2})\,\hat{U}^{-1}(\theta_{0})=\hat{A}_{i}^{\pm\pm}(n_{1},n_{2})\,+\,\Big[\frac{i}{\hbar}\int_{T_{2}}d^{2}\vec{x}\\theta_{0}(\vec{x})\hat{\phi}(\vec{x})\,\\hat{A}_{i}^{\pm\pm}(n_{1},n_{2})\Big]\\\.
R_{\mu\nu}\,^{ab}=\partial_{\mu}\left.^{c}\omega_{\nu}\,^{ab}\right.-\partial_{\nu}\left.^{c}\omega_{\mu}\,^{ab}\right.+\left.^{c}\omega_{\mu}\,^{a}\,_{d}\right.\left.^{c}\omega_{\nu}\,^{db}\right.-\left.^{c}\omega_{\nu}\,^{a}\,_{d}\right.\left.^{c}\omega_{\mu}\,^{db}\right..
\partial_{\mu}f^{\mu\beta}=8g^{2}\epsilon^{\mu\nu\alpha\beta}\langleF_{\mu\nu}\rangle\frac{1}{\Box+m_{\mathrm{A}}^{2}}\epsilon^{\rho\sigma\gamma\delta}\langleF_{\rho\sigma}\rangle\partial_{\alpha}f_{\gamma\delta}.
e_{1}\equiv\frac{M}{12}+i\frac{r_{m}}{2l^{2}}\sqrt{r_{m}^{2}-Ml^{2}},\;\;\;e_{3}\equiv\frac{M}{12}-i\frac{r_{m}}{2l^{2}}\sqrt{r_{m}^{2}-Ml^{2}}\;\;\;e_{2}\equiv-\frac{M}{6},
\begin{array}{ll}{i_{1}=k_{4}-k_{3},}&{j_{1}=k_{1},}\\{i_{2}=k_{1}+k_{2}+k_{3}-k_{4},}&{j_{2}=k_{2},}\\{i_{3}=k_{4}-k_{1},}&{j_{3}=k_{3}.}\\\end{array}
Z^{(2)}=\frac{Q-1}{2}\left\{\sum_{h\mathrm{\tiny{even}}\neq0}2Z\left(\begin{array}{c}{x=-1}\\{h}\\\end{array}\right)-Z\left(\begin{array}{c}{x=1}\\{h}\\\end{array}\right)-\sum_{h\mathrm{\tiny{odd}}}Z\left(\begin{array}{c}{x=1}\\{h}\\\end{array}\right)+Z\left(\begin{array}{c}{x=-1}\\{0}\\\end{array}\right)\right\}
\sum_{n_{1},\,n_{2}}\,\!\!\!^{\prime}\,f(n_{1},\,n_{2})=\sum_{n_{1}=1}^{\infty}\\sum_{n_{2}=-\infty}^{\infty}\f(n_{1},\,n_{2})+\sum_{n_{2}=0}^{\infty}\f(0,\,n_{2})
{\sfM}^{ab}[A^{\omega}]=\left.\frac{\delta\chi^{a}[(A^{\omega})^{\Omega}]}{\delta\varphi^{b}}\right|_{\varphi=0}=\left.\frac{\delta\chi^{a}[(A^{\tilde{\Omega}})^{\omega}]}{\delta\varphi^{b}}\right|_{\varphi=0}
H(q,\eta)\;=\;\frac{i}{2}\sum_{j=1}^{L-1}\left(\eta\,\tau_{j}^{2}\tau_{j+1}^{1}\-\\eta^{-1}\,\tau_{j}^{1}\tau_{j+1}^{2}\+\q\,\tau_{j}^{1}\tau_{j}^{2}\+\q^{-1}\,\tau_{j+1}^{1}\tau_{j+1}^{2}\,\right).
T_{\alpha\beta}=-\frac{4\pi\alpha^{\prime}}{\sqrt{-\gamma}}\frac{\deltaS}{\delta\gamma^{\alpha\beta}}=\partial_{\alpha}X^{\mu}\partial_{\beta}X_{\mu}-\frac{1}{2}\gamma_{\alpha\beta}\gamma^{\delta\gamma}\partial_{\delta}X^{\mu}\partial_{\gamma}X_{\mu},
\curvearrowrightF_{\alpha_{1}....\alpha_{n}}^{A}(s_{ij}+i\varepsilon)=F_{\alpha_{1}^{\prime}....\alpha_{n}^{\prime}}^{A}(s_{ij}-i\varepsilon)S_{\alpha_{1}....\alpha_{n}}^{\alpha_{1}^{\prime}....\alpha_{n}^{\prime}}(s_{ij})
\frac{1}{t^{2}}\left[\sum_{p\leqq}(2-\delta_{pq})\frac{p(d-q)}{d}\left(\frac{(m^{[p]})^{2}}{V_{d}~p!}\right)\left(\frac{(m^{[q]})^{2}}{V_{d}~q!}\right)\right]
\chi_{\alpha}^{12\ldotsk\;k+1}(x,u)=D_{\alpha}^{k+1}W^{12\ldotsk}\vert_{0}\;,\qquad\bar{\psi}_{\dot{\alpha}}^{23\ldotsk}(x,u)=\bar{D}_{1\dot{\alpha}}W^{12\ldotsk}\vert_{0}\;.
\begin{array}{c}{DZ_{i}^{A}=dZ_{i}^{A}+dx^{\mu}\omega_{\mu}{}^{A}{}_{B}Z_{i}^{B}\,}\\{D\psi_{a}=d\psi_{a}+dx^{\mu}\omega_{\muAB}\Sigma_{a}^{ABb}\psi_{b},}\\\end{array}
\epsilon^{(1)}=e^{{\frac{g\etat}{\sqrt{2}}}\epsilon_{ij}}{\calH}^{-1/8}\left[\sqrt{f^{1/2}+1}-\sqrt{f^{1/2}-1}\gamma_{\overline{{r}}}\right]e^{{\frac{\eta}{2}}\gamma_{\overline{{0\thetar}}}\epsilon^{ij}}e^{-{\frac{1}{2}}\gamma_{\overline{{\phi\theta}}}}(1-\eta\gamma_{\overline{{0}}}\epsilon^{ij})\epsilon_{0}^{(1)}.
V_{(\mu)}=\frac{1}{2}\left(\begin{matrix}{V_{(\mu)}^{c}+V_{(\mu)}^{s}}&{V_{(\mu)}^{c}-V_{(\mu)}^{s}}\\{V_{(\mu)}^{c}-V_{(\mu)}^{s}}&{V_{(\mu)}^{c}+V_{(\mu)}^{s}}\\\end{matrix}\right),
\Psi_{\mathrm{T}}(\alpha)=\frac{1}{2e^{2}}\intd^{4}x\,\sqrt{g}\,\Bigr\{\xi_{-}^{\nu}(\nabla^{\mu}T_{\mu\nu}^{+}-Y_{\nu}^{+})+(\nabla_{\mu}T_{-}^{\mu\nu}-Y_{-}^{\nu})\xi_{\nu}^{+}+\alpha\zeta_{\!~~\nu}^{\mu}(T_{\mu\rho}^{-}T_{+}^{\nu\rho}-G_{\mu}^{\!\!~~\nu})\Bigr\}.
\left[\begin{array}{cc}{k}&{[m]_{n}}\\{s}&{[m]_{n-1}}\\\end{array}\right]=\sqrt{\frac{\prod_{s^{\prime}\neqs}^{n-1}(p_{kn}-p_{s^{\prime},n-1})\prod_{k^{\prime}\neqk}^{n}(p_{s,n-1}-p_{k^{\prime}n}+1)}{\prod_{k^{\prime}\neqk}^{n}(p_{kn}-p_{k^{\prime}n})\prod_{s^{\prime}\neqs}^{n-1}(p_{s,n-1}-p_{s^{\prime},n-1}+1)}}
S=\overline{{n}}_{s}\mu\intd^{4}x\,f^{\dagger}\left[-\frac{1}{\rho^{3}}\frac{d}{d\rho}\left(\rho^{3}\frac{df}{d\rho}\right)-\frac{\overline{{a}}^{6}}{\rho^{6}(f^{\dagger}f)^{2}}f-f+\frac{1}{2}(f^{\dagger}f)f\right].
{\calS}_{1}=\int\sqrt{-G}[-\partial_{\xi}\phi_{(1)}\partial_{\xi}\phi_{(0)}-\frac{\partialV}{\partial\phi}\mid_{(0)}\phi_{(1)}+K_{\xi}(-\frac{1}{2}\partial_{\xi}\phi_{(0)}\partial_{\xi}\phi_{(0)}-V(\phi_{(0)}))\xi]d^{3}\sigmad\xi.
{\calA}^{\mu}\Sigma_{\mu}=\left(\begin{matrix}{\Sigma_{y}}&{-\Sigma_{0}}&{0}&{0}\\{-\Sigma_{0}}&{\Sigma_{y}}&{0}&{0}\\{0}&{0}&{-\Sigma_{y}}&{\Sigma_{0}}\\{0}&{0}&{\Sigma_{0}}&{-\Sigma_{y}}\\\end{matrix}\right)\,.
\begin{array}{l}{(m,\tilde{m},n=0,p)=(m+\frac{k}{(k,2k^{\prime})},\tilde{m}-\frac{k}{(k,2k^{\prime})},n=0,p-\frac{2k^{\prime}}{(k,2k^{\prime})}),}\\{(m,\tilde{m},n=1,p)=(m+\frac{k}{(k,2k^{\prime})},\tilde{m}-\frac{k}{(k,2k^{\prime})},n=1,p+\frac{2k^{\prime}}{(k,2k^{\prime})})}\\\end{array}
\begin{array}{l}{\ell_{3}^{+}=-\ell^{+};\;\ell^{-}=0}\\{\hat{h}=\frac{k^{-}}{k}\ell^{+}=-\frac{\hat{q}}{2}}\\{\hat{h}=\hat{h}^{\prime};\;\hat{q}=\hat{q}^{\prime}}\\\end{array}
i\hbar{\frac{\partial\Phi(N)}{\partialt}}={\frac{\hbar^{2}}{2m}}\sum_{p=1}^{N}{\frac{1}{\sqrt{g(x_{p})}}}\left(-{\calD}\bar{\calD}(x_{p})+{\frac{k}{2}}\sqrt{g(x_{p})}\right)\Phi(N)\,.
\{f(z),g(z)\}_{\alpha}=\nabla_{i}f(z)(\omega_{(\alpha)ij}+R_{ijkl}\theta^{k}\theta^{l})^{-1}\nabla_{j}g(z)+\frac12\frac{\partial_{r}f(z)}{\partial\theta^{i}}g^{ij}\frac{\partial_{l}g(z)}{\partial\theta^{j}},
\left(\begin{array}{ccc}{D^{2}+(\phi^{3})^{2}}&{0}&{\pm2iD_{i}\phi^{3}}\\{0}&{D^{2}+(\phi^{3})^{2}}&{\mp2iG_{0i}^{3}}\\{\mp2iD_{i}\phi^{3}}&{\mp2iG_{0i}^{3}}&{D^{2}+(\phi^{3})^{2}}\\\end{array}\right)\left(\begin{array}{c}{\phi^{\pm}}\\{W_{0}^{\pm}}\\{W_{i}^{\pm}}\\\end{array}\right)=0
Q_{\xi}=\int\,X(y)\left(2\xi^{\prime}(y)\,D(y)+D(y)^{\prime}\,\xi+{\frac{ic\mu}{12}}\,\xi^{\prime\prime\prime}\right)+\int\,E(y)\left(\xi^{\prime}(y)A(y)+A^{\prime}(y)\xi(y)\right).
\frac{e}{4m}F_{\alpha\beta}[y^{T}{\calL}{\calM}^{\alpha\beta}{\calL}^{T}\mathrm{p}+n_{\mu}({\calM}^{\alpha\beta})^{\mu\nu}(\pi_{\nu}+y^{T}S_{\nu}\mathrm{p})]=\frac{e}{4m}F_{\alpha\beta}X^{\alpha\beta}.
u_{1}={\frac{w_{2}}{w_{1}}},\quadu_{2}={\frac{1}{w_{1}}},\quadv_{1}={\frac{1}{u_{1}}},\quadv_{2}={\frac{u_{2}}{u_{1}}},\quadw_{1}={\frac{v_{1}}{v_{2}}},\quadw_{2}={\frac{1}{v_{2}}}.
\mu^{2}\frac{\partial}{\partial\mu^{2}}\Gamma_{m}(A|\phi,\phi_{a}^{*},\bar{\phi},\eta)=\bigr(\beta_{g}g\frac{\partial}{\partialg}+\gamma_{Q}\mathbf{\tilde{N}}_{Q}+\gamma_{B}\mathbf{\tilde{N}}_{B}\bigr)\Gamma_{m}(A|\phi,\phi_{a}^{*},\bar{\phi},\eta),
\lambda^{i}=2(D-2){\frac{\partialW}{\partial\phi}}e^{-{\frac{1}{2}}A}\tilde{\eta}_{-}^{i}\,\qquad\psi_{r}^{i}=0\,\qquad\psi_{\mu}^{i}=2\kappa^{D-2}W\Gamma_{\underline{\mu}}e^{{\frac{1}{2}}A}\tilde{\eta}_{-}^{i}\.
{\calL}_{g,3A}=-\frac{1}{4}\sum_{n,m,l\neq(0,0,0)}(\sum_{j=0}^{N}a_{jn}a_{jm}a_{jl})g_{L}f^{abc}\partial_{[\mu}\tilde{A}_{\nu]}^{n}\tilde{A}^{m~\mu}\tilde{A}^{l~\nu}.
\left\{{A_{i}^{a}\left(x\right),\Pi^{bj}\left(y\right)}\right\}^{*}=\delta^{ab}\delta_{i}^{j}\delta^{(3)}\left({x-y}\right)-\int_{0}^{1}{d\lambda}\left({\delta^{ab}\frac{\partial}{{\partialx^{i}}}-gf^{abc}A_{i}^{c}\left(x\right)}\right)x^{j}\delta^{(3)}\left({\lambdax-y}\right).
\deltaZ_{A\phi}=-\frac{g^{2}}{M_{W}(h)+M_{A\phi}(h)}\frac{1}{\pi}+\left(\frac{g^{4}h^{2}}{M_{W}(h)^{3}}+\frac{(\frac{\lambda}{6}+g^{2})^{2}h^{2}}{M_{A\phi}(h)^{3}}\right)\frac{1}{48\pi}.
e^{iS_{\Lambda-\delta\Lambda}(\phi_{+},\phi_{-})}\equiv\int_{\mathrm{CTC}}\prod_{\Lambda_{0}>|{\bfq}|>\Lambda-\delta\Lambda}{\calD}[\phi_{+}({\bfq},t)]{\calD}[\phi_{-}({\bfq},t)]\;e^{iS_{\mathrm{cl}}[\phi_{+},\phi_{-}]}.
{\frac{d^{2}x^{\rho}}{d\lambda^{2}}}+\Gamma_{\mu\nu}^{\rho}{\frac{dx^{\mu}}{d\lambda}}{\frac{dx^{\nu}}{d\lambda}}+{\calW}^{-1}g^{\rho\sigma}\partial_{y}({\calW}g_{\sigma\mu}){\frac{dx^{\mu}}{d\lambda}}{\frac{dy}{d\lambda}}=0,
A_{D}=\frac{{\calN}_{D}}{2\pi^{2}\pi^{3}}\int_{0}^{1}\frac{dz\,z}{\sqrt{z}\,\sqrt{1-z}}\int_{0}^{\infty}\frac{dt}{t^{2}}\left(\frac{1}{2\pi^{2}t}\right)^{{\textstyle\frac{k+1-D}{2}}}e^{-t(Y^{2}+2lz)}
{\Pi^{\mu\nu,\alpha\beta}(k)\equiv\sum_{\pm}{\epsilon_{\pm}^{\mu}(k)\epsilon_{\pm}^{\nu}(k)\overline{{\epsilon_{\pm}^{\alpha}(k)}}\overline{{\epsilon_{\pm}^{\beta}(k)}}}=\frac{1}{2}\left(\Pi^{\mu\alpha}\Pi^{\nu\beta}+\Pi^{\mu\beta}\Pi^{\nu\alpha}-\Pi^{\mu\nu}\Pi^{\alpha\beta}\right)}
[\hat{H}_{A}^{(r)}(m+\frac{r}{\lambda};d),\hat{E}_{\alpha}^{(s)}(n+\frac{s-d\cdot\alpha}{\lambda};d)]=\alpha_{A}\hat{E}_{\alpha}^{(r+s)}(m+n+\frac{r+s-d\cdot\alpha}{\lambda};d)
H=\sum_{i=2}^{M}(A_{i,i-1}c_{i}^{+}c_{i-1}+1/2B_{i}c_{i}^{+}c_{i})+A_{1M}q^{\hat{N}}c_{1}^{+}c_{M}+1/2B_{1}c_{1}^{+}c_{1}+H.c.
Z(i{\lambda}\,;R(S_{\nabla}),\langle\cdot\,,\cdot\rangle_{H},\langle\cdot\,,\cdot\rangle)=\pi^{\zeta/2}\,e^{\mp\frac{i\pi}{4}\eta}\,k^{-\zeta/2}\,\tau(M,\chi,\langle\cdot\,,\cdot\rangle_{H})^{1/2}\,.
{\calL}\,=\,\pi\,\dot{\phi}\,-\,\frac{\left[\pi\,+\,2(u^{0}+u^{1})^{2}\phi^{\prime}\right]^{2}}{4\left[1-(u^{0}+u^{1})^{2}\right]}\,-\,[1\,+\,(u^{0}+u^{1})^{2}]\,{\phi^{\prime}}
\begin{array}{ll}{\Gamma^{i}=\sigma^{i}\otimes\gamma^{(7)}}&{~~~\mathrm{for~}~i=1,2,3\,,}\\{{}}&{{}}\\{\Gamma^{a}=1\otimes\gamma^{a}}&{~~~\mathrm{for~}~a=4,5,6,7,8,9\,.}\\\end{array}
F_{o}(z)=\sum_{N=0}^{\infty}\rho_{o}(N)e^{-zN}=\prod_{\mathrm{SW};\;{\bfn}\ne0}\left(1-e^{-z\omega_{{\bfn}}}\right)^{-d}=\prod_{{\bfn}\ne0}\left(1-e^{-z\omega_{{\bfn}}}\right)^{-\frac{d}{2}}
f(p_{1},p_{1}^{\prime},p_{2},p_{2}^{\prime})=\frac{m_{e}(-ie)^{2}(2\pi)^{3}}{\sqrt{w(p_{1})w(p_{2})w(p_{1}^{\prime})w(p_{2}^{\prime})}}\delta^{(3)}(p_{1}^{\prime}+p_{2}^{\prime}-p_{1}-p_{2}).
\intd^{2}x\left(\frac{e^{2}}{2}\sum_{i,a}\left(\frac{1}{i}\frac{\delta}{\deltaA_{i}^{a}(\vec{x})}-\frac{k}{8\pi}\varepsilon_{ij}A_{j}^{a}(\vec{x})\right)^{2}+\frac{1}{2e^{2}}\sum_{a}B^{a}(\vec{x})^{2}\right)\Psi[A]~=~{\calE}~\Psi[A]
S_{\mathrm{q}}=\int_{\Sigma_{g}}B^{0}\partial_{z}\partial_{\overline{{z}}}\Phi+(b^{z\overline{{z}}}\partial_{\overline{{z}}}c_{z}+\mathrm{c.c.})+\chi^{0}\partial_{z}\partial_{\overline{{z}}}\psi+(B^{z\overline{{z}}}\partial_{\overline{{z}}}\phi_{z}+\mathrm{c.c.}),
{\frac{e(B_{5}^{m})}{e(B_{2}^{m})}}={\frac{d_{1}^{d_{1}-1}H^{-1}}{(d_{1}-1)!(\lambda_{1}-\lambda_{2})^{d_{1}}}}\prod_{k=1}^{d_{1}-1}\left(\lambda_{2}-2\lambda_{1}+{\frac{k}{d_{1}}}(\lambda_{1}-\lambda_{2})\right).
{\calL}={\frac{1}{2}}\left[\tilde{\varphi}^{\prime}\dot{\varphi}+\dot{\tilde{\varphi}}\varphi^{\prime}-{\frac{1}{2\sqrt{-\eta}\eta^{00}}}\left({{\tilde{\varphi}}^{\prime}}{{\tilde{\varphi}}^{\prime}}+{{\varphi}^{\prime}}{{\varphi}^{\prime}}\right)+{\frac{2\eta^{01}}{\eta^{00}}}{\tilde{\varphi}}^{\prime}\varphi^{\prime}\right]
K_{M}=\left[\frac{2q^{2}+3}{3}-\frac{r_{-}}{r_{+}}\right]\left\{\frac{1}{r_{+}}\left[\frac{2q^{2}+3}{3}+\frac{r_{-}}{r_{+}}\right]+\frac{2q^{2}+6}{2q^{2}+3}\frac{r_{-}^{2}}{r_{+}^{3}}\left[1-\frac{r_{-}}{r_{+}}\right]^{-1}\right\}^{-1}.