harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(\hat{X})_{\mathrm{rev}}=\left(\begin{array}{cc}{T}&{0}\\{0}&{T|_{{\calM}_{2}}}\\\end{array}\right)\hat{X}\left(\begin{array}{cc}{T^{-1}}&{0}\\{0}&{(T|_{{\calM}_{1}})^{-1}}\\\end{array}\right),
G(r,r^{\prime},\nu)=\sum_{n=1}^{n_{\mathrm{max}}}\frac{1}{\chi_{n}-\nu^{2}}\frac{s_{n}(r)s_{n}(r^{\prime})}{\int_{-\bar{R}_{1}}^{\bar{R}_{2}}dr^{\prime\prime}\Sigma(r^{\prime\prime})s_{n}^{2}(r^{\prime\prime})}\,,
\bar{\partial}\frac{\deltaZ}{\deltah_{i}(x,\bar{x})}=\frac{1}{\pi}\sum_{j,n}^{}\frac{(-1)^{n-1}}{(n-1)!}\partial^{n-1}\left(h_{j}\left<\left[\Phi^{i}\Phi^{j}\right]_{n}e^{-\frac{1}{\pi}\int{\bfh\cdot\Phi}}\right>\right).
S^{(2)}=\frac{1}{2\pi}\intd^{2}x\sqrt{-g}\left[e^{-2\phi}\left(R^{(2)}+4{(\nabla\phi)}^{2}+4\lambda^{2}\right)-\frac{1}{2}\sum_{i=1}^{N}{(\nablaf_{i})}^{2}\right]
\tilde{B}(p)=2g^{2}\delta_{b}^{a}\delta_{J}^{I}\int\frac{d^{4}k}{(2\pi)^{4}}d^{2}\thetad^{2}{\overline{\theta}}\,\frac{1}{(k^{2}+m^{2})[(p-k)^{2}+m^{2}]}\left\{\Phi_{aI}^{\dagger}(p,\theta,{\overline{\theta}})\Phi^{bJ}(-p,\theta,{\overline{\theta}})\right\}\,.
(\rho^{\prime})^{2}=(\rho^{2}-\rho_{0}^{2})^{2}\left[\frac{mv}{\hbar}\lambda-\left(\frac{mv}{\hbar}\mp\frac{\lambda\sqrt{\xi}}{2}\rho_{0}^{2}\right)^{2}\frac{1}{\rho^{2}}-\frac{\lambda^{2}}{4}\xi(\rho^{2}+2\rho_{0}^{2})\right]
N^{\prime}=n_{2}^{2}+\cdots+n_{k-1}^{2}+n_{i}+\cdots+n_{k-1}+\sum_{j=2}^{k-2}\sum_{p=1}^{n_{j}-n_{j+1}}a_{p}^{(j)}+2\sum_{p=1}^{n_{k-1}}b_{p},
\frac{1}{2\pii}\int_{C}dsx^{-s}\frac{\Gamma(s)}{\Gamma(s+\frac{1}{2})}\;{}_{2}F_{1}(\frac{s}{2},\frac{s+1}{2};s+\frac{1}{2};z^{2})=\frac{1}{\sqrt{\pi}}\frac{\Theta(1-x+\frac{x^{2}z^{2}}{4})}{\sqrt{1-x+\frac{x^{2}z^{2}}{4}}}.
\frac{d^{2}\Phi_{n}}{dr^{2}}+\frac{1}{r}\frac{d\Phi_{n}}{dr}+\frac{\Phi_{n}}{1-\Phi_{n}^{2}}\left[\left(\frac{d\Phi_{n}}{dr}\right)^{2}-\frac{n^{2}}{r^{2}}\right]+\Phi_{n}\left(1-\Phi_{n}^{2}\right)=0.
R_{\mathrm{boundary}}(r)=\frac{1}{r^{\beta}}\left[B_{\mathrm{in}}\left(1+\frac{i}{\pi}\left(\frac{2r_{H}}{r}\right)^{1-2\beta}\right)+B_{\mathrm{out}}\left(1-\frac{i}{\pi}\left(\frac{2r_{H}}{r}\right)^{1-2\beta}\right)\right].
g_{\left(\mathrm{AdS}_{p+2}/\Gamma^{+}\right)\times\mathrm{S}^{D-p-2}}=g_{\mathrm{AdS}_{p+1}}+(z^{+})^{2}(dz)^{2}+(\mathrm{R}_{\mathrm{S}})^{2}\,g_{\mathrm{S}^{D-p-2}}~.
R_{\mu\kappa\lambda}^{\nu}(x)=\frac{\partialA_{\lambda\mu}^{\nu}(x)}{\partialx^{\kappa}}-\frac{\partialA_{\kappa\mu}^{\nu}(x)}{\partialx^{\lambda}}+\sum_{\sigma=0}^{3}(A_{\lambda\mu}^{\sigma}(x)A_{\kappa\sigma}^{\nu}(x)-A_{\kappa\mu}^{\sigma}(x)A_{\lambda\sigma}^{\nu}(x)).
\mu^{\prime}=P_{\mu}^{GG}+\frac{\alpha^{2}}{2}Nx^{2}\frac{H^{2}H^{2}}{1-H^{2}}\,\\\sigma^{\prime}=P_{\sigma}^{GG}+\alpha^{2}x\sigma\frac{H^{2}H^{2}}{1-H^{2}}
\psi_{+}^{r}(x)=\sum_{\sigma=\pm1/2}\int_{k^{+}>0}{\frac{dk^{+}d^{2}k_{\perp}}{16\pi^{3}k^{+}}}\Bigl[b^{r}(k,\sigma)u_{+}(k,\sigma)e^{-ik\cdotx}+d^{r\dagger}(k,\sigma)v_{+}(k,\sigma)e^{ik\cdotx}\Bigr]\;,
{\calL}={\calL}_{0}+{\calL}_{I}=\left\{\frac{1}{2}\,(\partial\Phi)^{2}-\frac{1}{2}\,m^{2}\Phi^{2}\right\}+\left\{-\frac{\lambda}{4!}\,\Phi^{4}+{\calL}_{\mathrm{ct}}\right\},
m_{Z}=\frac{2^{\frac{-2K}{1-2K}}}{\sqrt{\pi}}\,\mu^{\frac{1}{1-2K}}\,\frac{\Gamma(\frac{K}{1-2K})}{\Gamma(\frac{1}{2-4K})}\,\left(\pi\,\frac{\Gamma(1-2K)}{\Gamma(2K)}\right)^{\frac{1}{2-4K}}\,\lambda_{0}^{\frac{1}{2-4K}}
ds_{S-type}^{2}=d\chi^{2}-dT^{2}+dW^{2}-\frac{1+A^{2}/\tau_{0}^{2p-2}}{(\sqrt{T^{2}-W^{2}}/\tau_{0})^{2p-2}-1}\frac{(WdW-TdT)^{2}}{T^{2}-W^{2}}+W^{2}d\Omega_{p-2}^{2}\.
U^{a_{1}b_{1}}U^{a_{2}b_{2}}U^{a_{3}b_{3}}f_{\tau\alpha,\tau_{1}\alpha_{1},\tau_{2}\alpha_{2},\tau_{3}\alpha_{3},b_{1},b_{2},b_{3}}\Psi=f_{\tau\alpha,\tau_{1}\alpha_{1},\tau_{2}\alpha_{2},\tau_{3}\alpha_{3},a_{1},a_{2},a_{3}}\Psi,
[***]^{2}=[\frac{1}{z-w}k-\bar{\beta}k\frac{1}{z}]^{2}=\frac{k^{2}}{(z-w)^{2}}-2\frac{k\bar{\beta}k}{z-w}\frac{1}{z}+\frac{k\bar{\beta}^{2}k}{z^{2}},
0=[\Phi_{x}^{2}+({\scriptstyle\frac{g(D)}{D}}+1)\Phi_{y}^{2}]\Phi_{xx}+[({\scriptstyle\frac{g(D)}{D}}+1)\Phi_{x}^{2}+\Phi_{y}^{2}]\Phi_{yy}-2{\scriptstyle\frac{g(D)}{D}}\Phi_{x}\Phi_{y}\Phi_{xy}
\int_{1}^{\infty}dx(x^{2}-1)^{\nu/2}e^{-\alphax}J_{\nu}[\beta\sqrt{x^{2}-1}]=\sqrt{\frac{2}{\pi}}\beta^{\nu}(\alpha^{2}+\beta^{2})^{-\nu/2-1/4}K_{\nu+1/2}(\sqrt{\alpha^{2}+\beta^{2}}).
e^{-i(\mu\hat{P}+m\,\hat{\Omega})}\,e^{-i(\nu\hat{P}+n\,\hat{\Omega})}=e^{-\frac12(m\nu-n\mu)[\,\hat{\Omega},\hat{P}\,]}\,e^{-i((\mu+\nu)\hat{P}+(m+n)\hat{\Omega})}.
Z^{\prime}=\prod_{n_{1},n_{2}=0}^{N^{\prime}-1}\left|\frac{\vartheta\left[^{(\alpha_{1}+n_{1})/N^{\prime}-1/2}_{(\alpha_{2}+n_{2})/N^{\prime}-1/2}\right](0,\tau)}{\eta(\tau)}\right|^{2}
H^{1}({\calS},{\calO}_{{\calS}}(-1)\oplus\bigoplus_{i=3}^{7}{\calO}_{{\calS}}(-i))=H^{0}({\calS},{\calO}_{{\calS}}(-1)\oplus\bigoplus_{i=3}^{7}{\calO}_{{\calS}}(-2+i))^{*}.
\langle{\calO}_{1/2}^{[m,0,0]}(1){\calO}_{1/2}^{[0,0,n]}(2){\calO}^{(m+n;\0,0;\m,0,n)}(3)\rangle=\langleW^{1}(1)\bar{W}_{4}(3)\rangle^{m}\\langleW^{1}(3)\bar{W}_{4}(2)\rangle^{n}\,.
G^{\mu\nu}=\left(\begin{array}{cccc}{-\frac{1}{g(1-e^{2})}}&{0}&{0}&{0}\\{0}&{\frac{1}{g(1-e^{2})}}&{0}&{0}\\{0}&{0}&{\frac{1}{1+g^{2}b^{2}}}&{0}\\{0}&{0}&{0}&{\frac{1}{1+g^{2}b^{2}}}\\\end{array}\right),
N_{i}^{in}=(|\alpha_{i}|^{2}+|\beta_{i}|^{2})[N_{i}^{out}+{\frac{(2\omega+1)}{2}}]+\alpha_{i}\beta_{i}b_{i}b_{i}+\alpha_{i}^{*}\beta_{i}^{*}b_{i}^{\dagger}b_{i}^{\dagger}-{\frac{(2\nu+1)}{2}}.
ds^{2}=-\frac{r^{2}}{\gamma^{4}}\left(1-\frac{2{\sqrt2}\gamma^{2}M}{Qr}\right)dt^{2}+\left(1-\frac{2{\sqrt2}\gamma^{2}M}{Qr}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}
\begin{array}{lclclcl}{f_{\underline{{[IJ]}}\underline{{[KL]}}}{}^{\underline{{[MN]}}}}&{=}&{-8\delta_{{}^{[}I{}_{[}K}\delta_{L{}_{]}J{}^{]}}^{MN}}&{\quad}&{f_{\underline{{[IJ]}}A}{}^{B}}&{=}&{\frac{1}{2}\Gamma_{AB}^{IJ}}\\{f_{B\underline{{[IJ]}}}{}^{A}}&{=}&{\frac{1}{2}\Gamma_{AB}^{IJ}}&{\quad}&{f_{AB}{}^{\underline{{[IJ]}}}}&{=}&{-\frac{1}{2}\Gamma_{AB}^{IJ}.}\\\end{array}
\nabla_{\mathrm{{(AdS)}}}^{2}a=\left(\partial_{\bar{r}}\partial_{\bar{r}}+\partial_{\bar{z}}\partial_{\bar{z}}+\frac{4}{\bar{r}}\,\partial_{\bar{r}}-\frac{3}{\bar{z}}\,\partial_{\bar{z}}\right)a=0
\textbf{1}=\sum_{n=0}^{\infty}a_{m_{1}}^{\dagger}\ldotsa_{m_{n}}^{\dagger}\mid0\rangle\langle0\mida_{m_{n}}\ldotsa_{m_{1}}=\sum_{n=0}^{\infty}\tilde{a}_{m_{1}}^{\dagger}\ldots\tilde{a}_{m_{n}}^{\dagger}\mid0\rangle\langle0\mid\tilde{a}_{m_{n}}\ldots\tilde{a}_{m_{1}}
S=\displaystyle\intd^{8}zE^{-1}\left[\bar{Q}Q+\bar{Q}^{c}Q^{c}+\lambda_{0}(\bar{Q}^{c}Q)(\bar{Q}Q^{c})+\bar{\xi}_{1}(\bar{Q}Q)+\bar{\xi}_{2}(\bar{Q}^{c}Q^{c})\right],
{\calU}_{N}=(-1)^{N-1}\,\mathrm{det}\,\left(\begin{array}{cccccc}{f_{N}}&{g_{-1}}&{g_{0}}&{g_{1}}&{\ldots}&{g_{N-5}}\\{f_{N-1}}&{-1}&{g_{-1}}&{g_{0}}&{\ldots}&{g_{N-6}}\\{f_{N-2}}&{0}&{-1}&{g_{-1}}&{\ldots}&{g_{N-7}}\\{\vdots}&{{}}&{\ddots}&{\ddots}&{\ddots}&{\vdots}\\{f_{4}}&{0}&{\ldots}&{0}&{-1}&{g_{-1}}\\{f_{3}}&{0}&{\ldots}&{\ldots}&{0}&{-1}\\\end{array}\right),
P\psi=\left(\begin{array}{c}{0_{4\times1}}\\{\sqrt{m}\varphi}\\\end{array}\right),\P^{\mu}\psi=\frac{i}{\sqrt{m}}\left(\begin{array}{c}{0_{4\times1}}\\{D^{\mu}\varphi}\\\end{array}\right),\D^{\mu}D_{\mu}\varphi+m^{2}\varphi=0.
\begin{array}{rl}{\mathrm{Bekenstein-Verlinde~bound}:}&{S_{\mathrm{BV}}=\frac{2\pi}{n}ER,}\\{\mathrm{Bekenstein-Hawking~bound}:}&{S_{\mathrm{BH}}=(n-1)\frac{V}{4G_{n}R},}\\{\mathrm{Hubble~bound}:}&{S_{\mathrm{H}}=(n-1)\frac{HV}{4G_{n}}.}\\\end{array}
\left[l_{\dot{J}}^{\dot{I}},f_{\dot{L}}^{\dot{K}}\right]=\sum_{\dot{K}_{1}\dot{K}_{2}=\dot{K}}\delta_{\dot{J}}^{\dot{K}_{1}}f_{\dot{L}}^{\dot{I}\dot{K}_{2}}-\sum_{\dot{L}_{1}\dot{L}_{2}=\dot{L}}\delta_{\dot{L}_{1}}^{\dot{I}}f_{\dot{J}\dot{L}_{2}}^{\dot{K}}.
\phi_{ij}(\theta)\underset{k\rightarrow\infty}{\longrightarrow}2\pi\,\delta(\theta)\,\left(\delta_{ij}-2\left(A_{ij}^{su(k)}\right)^{-1}\right)\quad\mathrm{and\quad~}\psi_{ij}(\theta)\underset{k\rightarrow\infty}{\longrightarrow}2\pi\,\delta(\theta)\,\left(A_{ij}^{su(k)}\right)^{-1},\;
{\mathcal{L}}_{I}=iw_{2}(\mu){\overline{{\Psi}}}\gamma^{\mu}\partial_{\mu}\Psi-w_{0}(\mu)m{\overline{{\Psi}}}\Psi-w_{3}(\mu)\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+iw_{1}(\mu)e\overline{{\Psi}}{\gamma^{\mu}}{A_{\mu}}{\Psi}~.
R_{vv}^{\ast}-{\sigma}^{ij}\frac{\partial\beta}{\partialy^{i}}\frac{\partial\beta}{\partialy^{j}}-\frac{2\Lambda}{(n-1)}\beta^{2}=\frac{1}{\alpha^{3}}\frac{d^{2}\alpha}{du^{2}}-\frac{1}{\alpha^{4}}\left(\frac{d\alpha}{du}\right)^{2}=\frac{1}{\alpha(v)}\frac{d^{2}\alpha}{dv^{2}}\quad,
<n>=\frac{1}{e^{\hbar\omega/kT}-1}-\lambda^{2}\frac{\hbar\omega}{kT}\frac{e^{3\hbar\omega/kT}+4e^{2\hbar\omega/kT}+e^{\hbar\omega/kT}}{(e^{\hbar\omega/kT}-1)^{4}}.
{\frac{1}{C_{A}}}=\frac{2^{\frac{r}{2}}\prod(k_{j}+2)}{K\kappa_{\alpha}^{A}\kappa_{\widetilde{\alpha}}^{A}},\quad{\frac{1}{C_{B}}}=\frac{2^{\frac{r}{2}}}{\kappa_{\alpha}^{B}\kappa_{\widetilde{\alpha}}^{B}},
B=\left(\begin{array}{cc}{{\frac{R^{2}}{4}}}&{1-{\frac{R^{2}}{4}}}\\{1-{\frac{R^{2}}{4}}}&{{\frac{R^{2}}{4}}}\\\end{array}\right),A={\frac{Q^{\prime}}{p}}(1,-1),~~m_{1}-m_{2}\equivQ(\mathrm{mod}2p^{\prime})
A_{Q}(-i\Omega_{m},\stackrel{\rightharpoonup}{p},\stackrel{\rightharpoonup}{l})=T\sum_{n}\frac{1}{(\omega_{n}+i\mu_{Q})^{2}+\omega_{Ql}^{2}}\frac{1}{(\omega_{n}+\Omega_{m}+i\mu_{Q})^{2}+\omega_{Ql+p}^{2}},
R_{n}^{\ell}\left(r\right)=-\frac{2}{n^{2}}\sqrt{\frac{\left(n-\ell-1\right)!}{\left(\left(n+\ell\right)!\right)^{3}}}e^{-r/n}\left(\frac{2r}{n}\right)^{\ell}L_{n+\ell}^{2\ell+1}\left(\frac{2r}{n}\right).
m_{6}(x)=-4\intd^{4}r\;e^{-ip\,r}\phi_{0}(r)=6\lambda\,\intdz\frac{zm_{6}(z)}{z+m_{6}(z)^{2}}\left[\frac{\theta(z-x)}{z}+\frac{\theta(x-z)}{x}\right],
{\calL}_{I}=\frac{2}{9}\left[\left\{\phantom{|}_{ab}^{\mu\nu}\right\}\Omega_{\mu}{}^{acde}\Omega_{\nu}{}^{bcde}-\frac{1}{48}\left\{\phantom{|}_{abcde}^{\mu\nu\alpha\beta\gamma}\right\}\Omega_{\mu}{}^{abcd}E_{\nu\alpha\beta\gamma}{}^{e}\right]
\frac{y^{2}}{(n_{c}-n_{f}/2)^{2}\Lambda^{2n_{c}-n_{f}-2}}=\prod_{a=1}^{n_{f}/2+1}(x-\phi_{a})^{2}-4\frac{\Lambda^{2}}{(n_{c}-n_{f}/2)^{2}}\prod_{i=1}(x+m_{i}).
J^{\mu~j}=-N_{jb}^{\mu\nu}\bigl(\varepsilon_{\nu\lambda}\partial^{\lambda}\Phi^{b}-\fraci2\varepsilon^{bcd}\overline{{\chi}}^{c}\gamma_{\nu}\chi^{d}\bigr)={\calJ}^{\muj}+N_{jk}^{\mu\nu}K_{\nu}^{k},
{\mathbf{P}}_{\alpha_{ab}}={diag(1,\alpha_{ab},}\alpha_{ab}^{2},...,\alpha_{ab}^{d+1});\quad\mathbf{Q}=\left(\begin{array}{ccccccc}{0}&{0}&{0}&{.}&{.}&{.}&{1}\\{1}&{0}&{0}&{.}&{.}&{.}&{0}\\{0}&{1}&{0}&{.}&{.}&{.}&{0}\\{.}&{.}&{.}&{.}&{.}&{.}&{.}\\{.}&{.}&{.}&{.}&{.}&{.}&{.}\\{0}&{0}&{0}&{.}&{1}&{0}&{0}\\{0}&{0}&{0}&{.}&{.}&{1}&{0}\\\end{array}\right)
\frac{Z_{\lambda}\lambda}{Z_{3}}\,\langle\,B_{x}^{a}\,B_{y}^{b}\,\rangle\,=\,\delta^{ab}\,\delta_{xy}\,-\,\frac{Z_{3}}{Z_{\lambda}\lambda}\,\langle\,\partialA_{x}^{a}\,\partialA_{y}^{b}\,\rangle\;.
W_{0}^{1}=\prod_{l>l^{\prime}}(z_{l}-z_{l^{\prime}})^{r^{2}}\sum_{q}z_{q}\frac{\partial}{\partialz_{q}}\prod_{l>l^{\prime}}(z_{l}-z_{l^{\prime}})^{-r^{2}}=\sum_{q}z_{q}\frac{\partial}{\partialz_{q}}-\frac{r^{2}}{2}N(N-1)
\sigma_{1}^{2}=2-\frac{1}{\sigma_{2}^{2}}-\frac{4\sigma_{2}^{4}}{1+\sigma_{2}^{4}-2\sigma_{2}^{2}-\sqrt{1-4\sigma_{2}^{2}+6\sigma_{2}^{4}+4\sigma_{2}^{6}+\sigma_{2}^{8}}}\quad,
(\alpha_{opt}^{2})^{(ev)}=\sqrt{\frac{2\mid\beta\mid^{2}tanh\mid\beta\mid^{2}+2\mid\beta\mid^{2}cos~2\theta+1}{2\mid\beta\mid^{2}tanh\mid\beta\mid^{2}-2\mid\beta\mid^{2}cos~2\theta+1}}(\alpha_{opt}^{2})^{(o)}\nonumber\,