harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left|2^{js}g_j(x)\right|^r \lesssim\int_{\mathbb{R}^n}\frac{2^{jn}}{(1+2^j|x-z|)^M}\left|2^{js}h_j(z)\right|^r\,dz &=\int_{3P}\frac{2^{jn}}{(1+2^j|x-z|)^M}\left|2^{js}h_j(z)\right|^r\,dz+\sum_{k\in\mathbb{Z}^n,\,\|k\|_\infty\geq2}\int_{P+k\ell(P)}\cdots &=:I_j(x)+J_j(x),
0<\alpha:=\min\Big\{|B_{\frac{r_i}{2}}(x_i)\cap B_{\frac{r_j}{2}}(x_j)\cap\Omega|\,:\,i,j\in\{1,\dots,N\},B_{\frac{r_i}{2}}(x_i)\cap B_{\frac{r_j}{2}}(x_j)\cap\Omega\neq\emptyset\Big\}.
E_{n+1}^\bot(|x|)x_{n+1}s_n= E_{n+1}^\bot(|x|)E_{n}^\bot(|x|x_{n+1}E_{n}(|x|))s_n = E_{n+1}^\bot(|x|)(E_{n}^\bot(|x|)x_{n}E_{n}(|x|))s_n = E_{n+1}^\bot(|x|)(E_{n}^\bot(|x|)x_{n}s_n) = E_{n+1}^\bot(|x|)(E_{n}^\bot(|x|)x_{n}g_n)s_n = 0.
0\leq w_\nu \leq(|\nabla\log u|^2)_{\nu}=2(\log u)_{\nu}(\Delta u-(\log u)_\nu H) = 2\frac{u_\nu}{u}\left(\frac{\Delta u}{u}-\frac{|\nabla u|^2}{u^2}-\frac{u_\nu}{u}H\right) = 2\frac{u_\nu}{u}\left(-h-\frac{|\nabla u|^2}{u^2}-\frac{u_\nu}{u}H\right) \leq 2\frac{u_\nu}{u}\left(-w-\frac{u_\nu}{u}H\right).
\mathbb{P}[u_m x_0 \in S_{x_0}(w_m^{-1}x_0, R_n) ] =\sum_{g\in G}\mathbb{P}[u_m x_0 \in S_{x_0}(w_m^{-1}x_0,R_n) \ | \w_m =g ] \cdot \mu_m(g) &= \sum_{g\in G}\mu_{n-m}(S_{x_0}(g^{-1}x_0,R_n)) \mu_m(g).
h^2 =\frac{\int_M(t-t_0)^2d\mu}{\int_M(s_g-s_0)^2d\mu}&=\frac{-\int_Mt^2d\mu+2\left(\int_Mtd\mu\right)^2/[\omega]^2}{\mathcal{F}(\mathfrak{E},[\omega])}.
\frac{A^\alpha (x) A^\alpha (x')}{4 \pi^{2}a}\int d^2k_{t}\sum_{n=1}^{\infty}\frac{1}{\omega}\sin(k_xv(x))\sin(k_xv(x')) e^{i[k_y(y-y')+k_z(z-z')-\omega (t-t')]}
L^\infty_\sigma\bigl(\Omega_2; B(L^1(\Omega_1),H)\bigr)\, = L^\infty_\sigma\bigl(\Omega_2; (L^1(\Omega_1)\overset{\wedge}{\otimes}H^*)^*\bigr)\, &= \bigl(L^1(\Omega_2)\overset{\wedge}{\otimes}L^1(\Omega_1)\overset{\wedge}{\otimes}H^*\bigr)^* =L^1(\Omega_1\times\Omega_2; H^*)^* = L^\infty_\sigma(\Omega_1\times\Omega_2; H).
E\,=\,\frac{\rho F}{2}\int\limits_{0}^{l}dx\,\left(\dot{y}^2\,+\,r^2\dot{\psi}^2\right)\,+\,\frac{EI}{2}\int\limits_{0}^{l}dx\,\psi'^2\,+\,\frac{kFG}{2}\int\limits_{0}^{l}dx\,\left(y'\,-\,\psi\right)^2.
mi_k(n-\delta-1)+\frac{s(B-1)}{2\binom{k}{2}}mi_k(n-\delta-2)+\frac{(s(\delta-s)+\frac{s(s-B)}{2})}{\binom{k}{2}}mi_k(n-\delta-k)+\frac{\binom{\delta-s}{2}}{\binom{k}{2}}mi_k(n-\delta-B).
\mathcal{I}_{D^I}=\langle x_1^{m_1}, x_2^{m_2}, x_3(-x_1+x_2+x_3)(x_2+x_3)^{m_3-2}\rangle, \mathcal{I}_{D^{I\!I}}=\langle x_1^{m_1}, x_2(x_1+x_2)^{m_2-1}, x_3(x_1+x_2+x_3)^{m_3-1}\rangle.
|\mathbf{w}_x|^2\leq \Big|\frac{\mathbf{w}(b,t)-\mathbf{w}(a,t)}{b-a}\Big|^2+2\int_\Omega|\mathbf{w}_{x}||\mathbf{w}_{xx}|dx \leq C+C\left(\int_\Omega|\mathbf{w}_{x}|^2dx\right)^{1/2}\left(\int_\Omega|\mathbf{w}_{xx}|^2dx\right)^{1/2},
\big(f, g \big)_{\mathfrak{J}^{L,C}}= (\mathfrak{J}^{L,C}f, g) = \int \limits_{C}\, (\, \mathfrak{J}_L (f_x ), g_x \,) \, d\mu_{\mathfrak{G}/G_1}, \,\,\, f,g \in{\mathcal{H}^{L}_{C}}'.
J_{gf}=\left( \begin{array}{cc}0 J_{+}1 0\end{array}\right) \quad ;\quad \tilde{J}_{gf}=\left( \begin{array}{cc}0 1 J_{+}0\end{array}\right) \quad ;\quad g=\left( \begin{array}{cc}-J_{+}g_{22}-g_{21}g_{21}g_{22}\end{array}\right) .
d=[\begin{matrix}(\lambda+\alpha)(C_{10}^{j+1}+C_{10}^{j})\\ 0\\ 0\\ 0\\ (\lambda+\alpha)(C_{20}^{j+1}+C_{20}^{j})\\ 0\\ 0\\ 0\\ (\lambda+\alpha)(C_{30}^{j+1}+C_{30}^{j})\\ 0\\ 0\\ 0\end{matrix}].
||x||_J \geq\frac{1}{\sqrt{2}}\sqrt{\sum_{i<k}(x_{p_i}-x_{p_{i+1}})^2}&=\frac{1}{\sqrt{2}}\sqrt{\sum_{i<k}(e^*_{p_i}(x)-e^*_{p_{i+1}}(x))^2}&>\frac{1}{\sqrt{2}}\sqrt{k\lceil 1/\epsilon\rceil^2}&\geq1.
d \left( \mathcal{S}(\mathbf{p}),\widetilde{\mathcal{S}}(\mathbf{p}) \right) \le c \left( \delta(\mathbf{p}) \right)^2 , \delta(\mathbf{p}) = \sup_{i \in \mathbb{Z}}d(p_i,p_{i+1}) , c \in \mathbb{R}_+ .
F(\overline{x}) =\left(\hat p(0), (d\hat pv_1)(0), \ldots, (d\hat pv_{n-1})(0)\right) &=(p_{d, 0, \ldots, 0}, p_{d-1, 0,1 0, \ldots, 0}, \ldots, p_{d-1, 0, \ldots, 0,1, 0, \ldots 0})
A_{\nu}( q ) = \frac{1}{2}\left( \begin{array}{c c c}{0}&{\overline{\psi}( q ) \gamma_{\mu}\gamma_{\nu}}&{- ( \gamma_{\nu}\gamma_{\mu}\psi ( q ) )^{T}}\\{0}&{0}&{0}\\{0}&{0}&{0}\\ \end{array}\right)_{( \mu , \lambda )},
\begin{array}{l l l}{{l_{q}}}&{{=}}&{{q^{2}[ ( x^{0})^{2}- q^{- 1}x^{+}x^{-}- q x^{-}x^{+}- ( x^{3})^{2}]}}\\{{\,}}&{{=}}&{{q^{2}[ ( x_{0})^{2}- q x_{+}x_{-}- q^{- 1}x_{-}x_{+}- ( x_{3})^{2}] \; ;}}\\{{\,}}&{{\,}}&{{\,}}\\{{l_{q}}}&{{=}}&{{C D - q^{2}A B \quad .}}\\ \end{array}
\mathbf{R}_{A_{dB}}^{i,j}=\frac{\mathbf{R}_{A_{dB}}^{i,i}\mathbf{R}_{A_{dB}}^{j,j}/N_f+\mathbf{R}_{A_{dB}}^{i,i}\circ\mathbf{R}_{A_{dB}}^{j,j}}{2},
v_{m}(\lambda, r) = e^{-\kappa r}\, (2\kappa\, r)^{|m|}\, M\Big(\frac 12+|m|+\frac{m\alpha}{\kappa}, 1+2 |m|, 2\kappa r\Big) u_m(\lambda, r) = e^{-\kappa r}\, (2\kappa\, r)^{|m|}\, U\Big(\frac 12+|m|+\frac{m\alpha}{\kappa}, 1+2 |m|, 2\kappa r\Big) \ ,
B(X)=\, X^3 (1+b+a_1 z)+X^2 (1+b+a_1 k+z+a_1 z+b z)\cr +X (b+k+a_1 k+b k+z+a_1 z+b z+a_1 k z)\cr +b+a_1 k+a_1 k^2+a_1 z+b z+k z+b k z.
\left\{DX_t=0,\ \forall t\in [l(\omega), u(\omega)] \right\}\Rightarrow \left\{\exists t\in \mathbb{Q}\cap [0,1],\DX_t=0 \right\}&\Rightarrow \bigcup_{t\in \mathbb{Q}\cap[0,1]}\left\{DX_t=0 \right\}.
Z ({\bf p}) = \frac{2 m M \left[ \sqrt{{\bf p}^{2}+ m^{2}}+ \sqrt{{\bf p}^{2}+ M^{2}}\right]}{\sqrt{{\bf p}^{2}+ m^{2}}\sqrt{{\bf p}^{2}+ M^{2}}\left[ \sqrt{{\bf p}^{2}+ m^{2}}+ \sqrt{{\bf p}^{2}+ M^{2}}+ E \right]}
\left[\begin{array}{cc}\ast 0 0 \ast\end{array}\right], \mbox{for $A(t)$, $B(t)$, $A'(t)$, $B'(t)$, $F(t)$ and $G(t)$}, I(t)=\left[\begin{array}{c}0 \ast\end{array}\right],J(t)=\left[\begin{array}{cc}0 \ast\end{array}\right]
\mathfrak{M}\left[{}^{M\!L\!\!}{B}_{p}(x,y) : s\right]=\int_{0}^{\infty}p^{s-1}\left[\int_{0}^{1}t^{x-1}(1-t)^{y-1}E_{\alpha, \beta}^{\gamma}\left(-\frac{p}{t(1-t)}\right)dt\right]dp.
\sigma_{\Lambda}( k^{2}) = \left( 1 + \frac{( 1 + \alpha ) m^{2}M^{2}+ \alpha m^{4}}{\Lambda^{6}}k^{2}\right) e^{- \frac{( k^{2}+ m^{2}) ( k^{2}+ \alpha m^{2}) ( k^{2}+ M^{2})}{\Lambda^{6}}}.
S =\{s_1\}\cup\{|s_1|, 2|s_1|,\ldots, (l-1)|s_1|\}&\cup\{s_j, s_j+|s_1|,\ldots, s_j+(l_2-1)|s_1|\}D(|s_1|) =\{2|s_1|\}\cup\{|s_1|,2|s_1|,\ldots,(l-2)|s_1|\}&\cup\{|s_j-|s_1||,s_j, s_j+|s_1|,\ldots,(l_2-2)|s_1|\}
- \int_{0}^{\sigma}d s \int_{- \infty}^{\infty}\frac{d \omega}{2 \pi}\frac{s}{( 4 \pi )^{\frac{D - 1}{2}}}( \omega^{2}+ s^{2})^{\frac{D - 3}{2}}\Gamma ( - \frac{D - 3}{2}) \mathrm{t r}\mathrm{\boldmath ~ 1 ~}.
\begin{aligned}\lambda_{|I|}(C(\widehat{A},\lambda)_I^TC(\widehat{A},\lambda)_I) \geq \lambda_{|I|}(C(A,\lambda)_I^TC(A,\lambda)_I) - \frac{1}{\epsilon H}\geq \left(1 - \frac{1}{\epsilon}\right) \lambda_{|I|}(C(A,\lambda)_I^TC(A,\lambda)_I), \end{aligned}
f(X_u) (v-u) + \sigma(X_u)B^H_{u,v}+ D\sigma(X_u)\sigma(X_u)\mathbb{B}^H_{u,v}&= \int_u^v f(X_r)dr + \int_u^v \sigma(X_r)d\mathbf{B}^H_r + o(|v-u|) &= X_v - X_u + o(|v-u|).
\sqrt{- g}R ( g_{a b}) = \partial_{t}\left[ \frac{1}{G}\left( \dot{C}- 2 B^{\prime}+ B \frac{C^{\prime}}{C}\right) \right] + \partial_{x}\left[ \frac{1}{{G}}\left( A^{\prime}- B \frac{\dot{C}}{C}\right) \right] .
b(\tau) = \Psi_\infty \exp{\left( \left( 1 -\frac{2}{\alpha}\right)\left( \int_{\tau_0}^\tau 2\beta(\zeta) d\zeta + \tau_0 \right) \right)}\left[1 + O\left(I^\eta(\tau)\right) \right] \tau \to +\infty.
2g^{2}\Gamma(ap)[\int_{-\pi}^{\pi}\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{H^{2}(k+ap)+(am\Gamma)^{2}(k+ap)}\frac{1}{H^{2}(k)+(am\Gamma)^{2}(k)}]\Gamma(ap).
v_u\left(\mathrm{det}(\phi_{\mathfrak{M}(\mathcal{H})})\right) \le M(k)\left(\frac{p-1}{p^h-1}\right)\left(\frac{1-p^{-n}}{1-p^{-1}}\right)e & = \left(\frac{p-1}{p^{\delta}-1}\right)\left(\frac{1-p^{-k}}{1-p^{-1}}\right)\left(\frac{1-p^{-n}}{1-p^{-1}}\right)e.
Z [ J ( x ) ] = \int{\mathcal D}[ \phi ] \, e^{i \int d^{4}x \, [ \frac{1}{2}\partial^{\mu}\phi ( x ) \partial_{\mu}\phi ( x ) - \frac{m^{2}}{2}\phi^{2}( x ) - \frac{\lambda}{2 4}\phi^{4}( x ) + \frac{1}{2}J ( x ) \phi ( x ) + \frac{1}{2}\phi ( x ) J ( x ) ]}~ .
T_{2}= c_{+ +}J_{n}^{+}J_{n}^{+}+ c_{+ 0}J_{n}^{+}J_{n}^{0}+ c_{0 0}J_{n}^{0}J_{n}^{0}+ c_{0 -}J_{n}^{0}J_{n}^{-}+ c_{--}J_{n}^{-}J_{n}^{-}+
\begin{aligned}\|(\rho-1, u, \nabla d)\|_{H^N}^2+\int_0^t (\|\nabla \rho\|_{H^{N-1}}^2+\|(\nabla u, \nabla^2 d)\|_{H^N}^2 )d\tau\le C \|(\rho_0-1, u_0, \nabla d_0)\|_{H^N}^2.\end{aligned}
\int_{\mathbb{R}_{+}}\frac{tr_{0}}{(1-tr_{0})^{2}}\,d\mu_{1}(t)=\int_{\mathbb{R}_{+}}\frac{1}{(1-tr_{0})^{2}}\,d\mu_{1}(t)-\int_{\mathbb{R}_{+}}\frac{1}{1-tr_{0}}\,d\mu_{1}(t)
M^{\rm OPT}_{nm}:=b_{nm}\times\underset{\eqref{eq:OTS_NP_Flow_S}-\eqref{eq:OTS_NP_slack}\,\cap\,\mathcal{X}_{nm}}{\max}(\theta_n-\theta_m) & M^{\rm OPT}_{mn}:=b_{nm}\times\underset{\eqref{eq:OTS_NP_Flow_S}-\eqref{eq:OTS_NP_slack}\,\cap\,\mathcal{X}_{nm}}{\max}(\theta_m-\theta_n)
M_n^{(\alpha,\beta)}= \sum_{k=0}^{n+1}\bigg(\frac{\alpha+\sqrt{\alpha^2-4\beta}}2\bigg)^{n-2k}\cdot\frac{\beta^k}{n+1}\binom{n+1}{k}\binom{n+1}{k+1}= \frac{(\alpha^2-4\beta)^{\frac{n+1}{2}}}{\beta}\sum_{k=0}^{n+1}\bigg(\frac{\alpha}{\sqrt{\alpha^2-4\beta}}-1\bigg)^{k+1}\cdot\frac{C_k}{2^{k+1}}\binom{n+k}{2k}.
\Sigma=\Sigma(\eta) =\left(\begin{array}{cccc}1 \frac{1}{2}\frac{1}{2}0 \frac{1}{2}\frac{3+\eta}{8}\frac{1-\eta}{8}0 \frac{1}{2}\frac{1-\eta}{8}\frac{3+\eta}{8}0 0 0 0 \frac{1-\eta}{8}\end{array}\right).
\prod_{i=0}^{r-1}\prod_{h=0}^{m-1}\Gamma_p\left(\langle(\frac{x+h}{m})p^i\rangle\right)=\omega \left(m^{(1-x)(1-q)}\right)\prod_{i=0}^{r-1}\Gamma_p(\langle xp^i\rangle)\prod_{h=1}^{m-1}\Gamma_p\left(\langle\frac{hp^i}{m}\rangle\right).
\Vert u\Vert_{W_{G}^{1,2;p}}^{p}=\mathbb{E}[\sup_{s\in \lbrack0,T]}|u_{s}|^{p}+\int_{0}^{T}(|\mathcal{D}_{s}u_{s}|^{p}+|\mathcal{D}_{x}^{2}u_{s}|^{p})ds+( \int_{0}^{T}|\mathcal{D}_{x}u_{s}|^{2}ds)^{p/2}].
\int_{L_{p_{1}}}^{L_{p_{2}}}\overset{\infty}{\underset{\mathop{k=1}}{\mathop{\sum}}}(c_{k}e^{i\omega_{k}\vartheta t}+c_{-k}e^{-i\omega_{k}\vartheta t})\sin(\omega_{k}z)\mathrm{d}z=0\,.