harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\omega= \sum_{i=1}^n \frac{du_i^-}{u_i^-}\wedge d\beta_i^- -\sum_{i<j}\frac{d\beta_i^-\wedge d\beta_j^-}{\beta_i^--\beta_j^-}+ \sum_{i=1}^n \frac{du_i^+}{u_i^+}\wedge d\beta_i^+ -\sum_{i<j}\frac{d\beta_i^+\wedge d\beta_j^+}{\beta_i^+-\beta_j^+}.
c_{\sigma}^{+}(M(\chi))\sim(2\pi i)^{-\lceil n/2\rceil w(\chi)}G_{\sigma}(\chi)^{\mathbf{r}_{\sigma}}\delta_{\sigma}(M)a_{\sigma}^{*}(\chi)Q_{\sigma}(\chi)^{\mathbf{r}_{\sigma}-\lceil n/2\rceil}\prod_{j=1}^{\mathbf{s}_{\sigma}}Q_{j,\sigma},
\iint_{Q_T}f^2dxdt\leq C\iint_{Q_T}(u_x^4+\mu^2|\mathbf{w}_x|^4+\nu^2|\mathbf{b}_x|^4+\rho^2u^2\theta_x^2+\rho^2u_x^2\theta^2)dxdt \leq C.
\big|(\partial^{\gamma}_{x_1}g)(x_1)\big| =\Big| \sum_{\sigma\le\gamma}\binom{\gamma}{\sigma}\big(\partial^{\sigma}_{x_1}|x_1+x_j|\big) \big(\partial_{x_1}^{\gamma-\sigma}|x_1|^{-1}\big)\Big| \le\sum_{\sigma\le\gamma}c_{\gamma,\sigma}|x_1+x_j|^{1-|\sigma|}|x_1|^{-1-|\gamma|+|\sigma|}\le \tilde{c}_{\gamma,k}|x_1|^{-|\gamma|}\,.
\big{\langle}M(Z,W) (\nabla_ZJ)_WX,X \big{\rangle}=- \big{\langle}J_WJ_Z (\nabla_ZJ)_WX, (\nabla_ZJ)_WX \big{\rangle}&= \langle Z,W \rangle \| (\nabla_ZJ)_WX \|^2.
c_{4}^{( 1 )}= - \frac{9}{2}\int_{0}^{1}\frac{d x}{x^{2}}\left[ \frac{1 - \frac{1 2 1}{7 2 9}x^{2}}{( 1 - \frac{2 5}{7 2 9}x^{2})^{1 3}}- 1 \right] .
\frac{1}{1-\kappa_{N}'}\left(\frac{2\beta}{\zeta N}+ \kappa_{N}'\right)\le \frac{1}{1-B/N}\left(\frac{2\beta}{\zeta N}+ B/N\right)= \frac{1}{N - B}\left(\frac{2\beta}{\zeta}+ B\right).
\frac{\partial_r[F(\theta)] + \partial_r [w(\theta)]^\top \mathbb{E}_\theta [f(\textbf{X})]}{\mathbb{E}_\theta [h + F(\theta) + w(\theta)^\top f(\textbf{X})]}= \frac{\partial_r[F(\theta)] + \partial_r [w(\theta)]^\top \bar{f}}{h + F(\theta) + w(\theta)^\top\bar{f}}.
\tilde{sec}(U, V) = sec(U, V) + \frac{\lambda^2}{2}\left( Hess\frac{1}{\lambda^2}(U, U) + Hess\frac{1}{\lambda^2}(V, V)\right) & - \frac{\lambda^4}{4}\mid grad \frac{1}{\lambda^2}\mid^2 - \frac{\lambda^4}{4}\left( \left( U\frac{1}{\lambda^2}\right)^2 + \left( V \frac{1}{\lambda^2}\right)^2 \right),
H \psi = \left( \begin{array}{c c}{{P_{-}P_{+}}}&{{0}}\\{{0}}&{{P_{+}P_{-}}}\\ \end{array}\right) \psi = ( E^{2}- m^{2}) \left( \begin{array}{c}{{\psi_{+}}}\\{{\psi_{-}}}\\ \end{array}\right)
P(C_1\cup C_2)-\epsilon \leq P(J_{1,\epsilon}) = \lim_{m\to\infty}\frac{\sum_{i=0}^{m-1}\mathbf{1}_{J_{1,\epsilon}}(g_{n,i}(\mathbf{x}))}{m}\leq \liminf_{m\to\infty}\frac{\sum_{i=0}^{m-1}\mathbf{1}_{C_1\cup C_2}(g_{n,i}(\mathbf{x}))}{m}.
v_t-(\dot{x}-c)w_x =-(v+w)^2v_x-(v^2+2vw)w_x-\dfrac{1}{2}p*\big(v_x^3+3v_x^2w_x+3v_xw_x^2\big) +p_x*\Big(v^3+3v^2w+3vw^2+\dfrac{3}{2}v(v_x+w_x)^2+\dfrac{3}{2}v_x^2w+2v_xww_x\Big).
[a_{(1\,l-1)}, a_{(1\,l)}]_c= -3\xi^2\chi_{(1\,l)}(g_{l-1})\lambda_{l-1l-1l}a_{(1\,l-2)}^2g_{l-1l-1l}&-3\xi^2\lambda_{112}\chi_{(1\,l)}(g_{(2\,l-1)})a_{(3\,l)}a_{(2\,l-1)}+3\lambda_{112}\chi_{1}(g_{(3\,l-1)})a_{(3\,l-1)}a_{(2\,l)}.