harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\delta_{t o t}| \phi \rangle = \Bigl ( \xi \partial + \frac{d - 2}{2 z}\xi^{z}+ \frac{1}{2}\partial^{I}\xi^{J}M^{I J}+ M^{I J}\partial^{+}\xi^{I}\frac{\partial^{J}}{\partial^{+}}- \frac{d - 4}{2 z \partial^{+}}\partial^{+}\xi^{I}R^{z I}- \frac{\partial^{+}\xi^{z}}{z \partial^{+}}\Bigr ) | \phi \rangle
\Vert g^n\Vert_{W_2^1[0,T]}^2 \leq \Vert[g]_n\Vert_{w_2^1}^2 +C\sqrt T \sqrt{\tau}\sqrt{\sum\limits_{k=1}^n\tau g_k^2}+ \frac13\tau^2\sum\limits_{k=1}^n\tau g_{k\bar t}^2 \leq R^2+O(\sqrt{\tau}) .
\prod_{i=1}^n \left( uX_i + vX_i^{-1}+ w \right) \prod_{1 \leq p < q \leq n}\left( u E_{k_p}+ v E_{k_q}^{-1}+ w E_{k_p}E_{k_q}^{-1}\right) \tilde{s}_{(k_n,k_{n-1},\ldots,k_1)}(X_1,X_2,\ldots,X_n).
\tau^- = \begin{array}{c}\left( \begin{array}{ccc}0 0 0 1 0 0 0 0 0 \end{array}\right),\qquad\varepsilon^0 = \left( \begin{array}{ccc}0 0 0 0 1 0 0 0 1 \end{array}\right),\qquad\varepsilon^z = \left( \begin{array}{ccc}0 0 0 0 1 0 0 0 -1 \end{array}\right),\end{array}
\|y\|_{C^{\alpha/2,\alpha}}\leq K_1\big(\|u_0\|_{C^{\alpha}(\overline{D})}&\quad+\|B(u)+(A(u)-)\nabla z\|_{L^{2r_0}(D_T)}+\|F(u)\|_{L^{r_0}(D_T)}\big)
\int_{\mathbb{R}}\Phi(\alpha (|u|)^{\frac{1}{1-s}})\, dx = \int_{I^c}\Phi(\alpha (|u|)^{\frac{1}{1-s}})\, dx+\int_{I}\Phi(\alpha (|u|)^{\frac{1}{1-s}})\, dx = :(I)+(II),
\widetilde{U}(t)=U^{j-1}+\frac{t-t_{j-1}}{t_j-t_{j-1}}\Big(U^{j}-U^{j-1}\Big)=\frac{t-t_{j-1}}{\Delta t_j}U^j+\frac{t_j-t}{\Delta t_j}U^{j-1},
\frac{\partial}{\partial \phi}\Re{f\Big(\tau+ \frac{R}{\eta_{-}}e^{i\phi}\Big)}=\Big(-x+\frac{1}{1+R^2+ 2R \cos\phi}\Big) R \sin\phi < -\Big(x- \frac{1}{(R-1)^2}\Big) R \sin\phi <0,
v(\theta ,\phi )=\left[ \begin{matrix}{{g}_{1}}(\theta ,\phi )\exp (j2\pi{{f}_{0}}{{\tau}_{1}}){{g}_{2}}(\theta ,\phi )\exp (j2\pi{{f}_{0}}{{\tau}_{2}}) \vdots{{g}_{N}}(\theta ,\phi )\exp (j2\pi{{f}_{0}}{{\tau}_{N}}) \end{matrix}\right],
\vec{a}^{\prime}( 0 ) = \left( \begin{array}{c}{c o s \alpha}\\{s i n \alpha}\\{0}\\ \end{array}\right) ; \vec{a}^{\prime}( 2 \pi ) = \left( \begin{array}{c}{c o s \alpha}\\{- s i n \alpha}\\{0}\\ \end{array}\right) ,
\omega^{A_{\Delta}}(V) \leq \sum_{j\in J}\omega^{A_{\Delta}}(5\Delta_{j}) \le C\sum_{j\in J}\omega^{A_{\Delta}}(\Delta_{j}) \le C\sum_{j\in J}\frac{\omega(\Delta_{j})}{\omega(\Delta)}= C\frac{\omega\left( \bigcup_{j\in J}\Delta_{j}\right)}{\omega(\Delta)}\le C \frac{\omega(V)}{\omega(\Delta)}.
\mathrm{Var}_0\left[\sum_{l=3}^m\sum_{1\le p<q<l}C^{(pq)}\right] = \sum_{1 \leq p < q \leq m-1}(m - q)^2 \mathrm{Var}_0\left[C^{(pq)}\right] &= \frac{1}{12}m(m-2)(m-1)^2 \mathrm{Var}_0 \left[ C^{(12)}\right].
\mathcal{I}_{\mathrm{D}}(\delta_1, \delta_2)- \mathcal{I}_{\mathrm{D}}(0,0) = \sup_{\widetilde{\gamma}\in \Sigma_{\mathrm{D}}(\delta)}\int_{\mathcal{V}}g \, d \widetilde{\gamma}- \mathcal{I}_{\mathrm{D}}(0,0) \leq \Psi( \delta_1 + \epsilon , \delta_2 + \epsilon).
(16 T^2+44 T-1)Z_{TTT}+ (48 T^2+66 T)Z_{TT}+ (44 T^2+34 T) Z_{T}+ (12 T^2+6 T)Z = 0, Z_{T}= T \frac{d}{dT}Z, T = \frac{R^{5}(1 - 11R^{5}- R^{10})}{(1 + R^{10})^{2}}.
F(\xi,\tau) =\left[ \int_{0}^{\infty}g_{1/2}(\eta)e^{-\eta\tau^{2}(1-\partial_{\xi}^{2})}d\eta\right] f(\xi), & = \int_{0}^{\infty}g_{1/2}(\eta)e^{-\eta\tau^{2}}D(\xi, \eta\tau^{2}) d\eta.
|S_{out}(x, t)|\lesssim \mu_0^{\frac{n-2s}{2}}(\mu_0^{2s}+\mu_0^{n-4s})\lesssim \mu_0^{\min(n-4s,2s)-(a-2s)-\sigma}(t_0)\sum_{j=1}^k\frac{\mu_j^{-2s}\mu_0^{\frac{n-2s}{2}+\sigma}}{1+|y_j|^{a}}.
C_{p_l}^k =\left(\frac{(k/(k-1))^{(k-1)/k}}{k^{1/k}}\right)^{k/2}&=\frac{1}{k^{1/2}}\left(1+\frac{1}{k-1}\right)^{(k-1)/2}&\leq\frac{\sqrt{e}}{k^{1/2}}.
I_{22}\lesssim\sum_{i=1}^\infty \frac{\ell(Q)^{2s}}{\ell(2^iQ)^{1+2s}}\,\Big(\ell(2^iQ)^{\frac Nq+ 2s}\,\ell(Q)^{\frac N{q'}}+ i\,\ell(Q)^N\,\ell(2^iQ)^{2s}\Big) &=\frac{\ell(Q)^{2s}\ell(Q)^{N+2s}}{\ell(Q)^{1+2s}}\sum_{i=1}^\infty \,\left(2^{i(\frac Nq-1)}+\frac i{2^i}\right).
x\Psi_{B}^{\bf f}({\bf t}) = x + \int_0^{t_1}\cdots\int_0^{t_m}x\Psi_{B}^{\bf f}\left(\underset{m}{\bf t},s_m\right){B}({\bf f})^{\left(\underset{{{\{m_1,m\}}}}{\bf t}{{, s_m}},{\bf{s}}\right)}ds_1\dots ds_m{{.}}\,\,\,
S_{- 1}^{E}= \int d^{2}\sigma \, \left( - i Y^{*}C^{- 1}+ \overline{{{C}}}^{* \, - 1}B + \overline{{{C}}}^{* \, 0}\overline{{{\lambda}}}_{0}+ \overline{{{C}}}^{* \, r}\overline{{{\omega}}}_{r}\right) \, .
\phi^{\mu_{0}\mu_{1}\ldots \mu_{p}}( X ) \rightarrow \phi^{\mu_{0}\mu_{1}\ldots \mu_{p}}( X ) + \sum_{i = 0}^{p}( - 1 )^{p \cdot i}\frac{\partial}{\partial X_{\mu_{i}}}\Lambda^{\mu_{i + 1}\mu_{i + 2}\ldots \mu_{p}\mu_{0}\mu_{1}\ldots \mu_{i - 1}},
\lim_{M\to\infty}N_1^{1/3}c_3(\ell,k) =\frac{\alpha}{(2\pi i)^2}\int_{\Gamma_{d_1}}dz\int_{\Gamma_{-d_2}}d\zeta\frac{e^{\frac 13z^3- \nu_1 z^2- (\lambda_1-x)z}}{(z-\zeta)e^{\frac 13\zeta^3- \nu_1 \zeta^2- (\lambda_1-y)\zeta}}&=\phi_3(x,y).
\sum_{j = 1}^k{( - 1)^{j - 1}G_{2j - 1}^3}= - \frac{3}{{50}}G_0^3 + \frac{{37}}{{50}}G_1^3 + \frac{9}{{50}}G_2^3 - \frac{2}{{25}}G_3^3 - \frac{{23}}{{25}}( - 1)^k G_{2k + 1}^3 &\qquad+ \frac{9}{{50}}( - 1)^k G_{2k + 2}^3 + \frac{{13}}{{50}}( - 1)^k G_{2k + 3}^3 - \frac{3}{{50}}( - 1)^k G_{2k + 4}^3\,.
u = u \cdot \omega \omega +\sum_{i=1,2,3}(u\cdot (\omega\times \mathbf{e}_i))(\omega\times \mathbf{e}_i)\Longrightarrow B_3= B\cdot \omega(\omega\cdot \mathbf{e}_3) -\sum_{i=1,2,3}(B\cdot(\omega \times \mathbf{e}_i))( \omega \times \mathbf{e}_3)\cdot \mathbf{e}_i.
[ x_{k}^{( \alpha )}, x_{l}^{( \beta )}] \; = \; \frac{i}{2 \pi \rho_{0}q^{( \alpha )}}\, \delta_{\alpha \beta}\, \epsilon_{k l}\; \equiv \; \frac{i}{q^{\alpha}}\, \theta \, \delta_{\alpha \beta}\, \epsilon_{k l}
\log P(r,k) =\sum_{i=0}^{r-1}\log \Bigl(1+\dfrac{2i}{k}\Bigr)<\sum_{i=0}^{r-1}\dfrac{2i}{k}=\dfrac{r(r-1)}{k}\ \implies P(r,k) < e^{\frac{r(r-1)}{k}}.
\vert\partial_{p}^{\omega}\,G^{\,\alpha}_{l,s}(\vec\alpha,\vec\lambda,\vec p)\vert\ \leq\e^{\,-m^{2}\sum_{j=1}^{s}\alpha_{j}}\,P(\vert\vec p\vert)\,Q(\sqrt\alpha_{1},\ldots,\sqrt\alpha_{s})\quad,
0= \left\{\lambda_j\mu_j-\frac{\alpha}2(\lambda_j+\mu_j)-2c\right\}\{g(X_j,\xi_a)^2+g(X_j,\phi\xi_a)^2\}, a\in\{2,3\}0= \left\{\lambda_j\mu_j-\frac{\alpha}2(\lambda_j+\mu_j)-2c+2c\eta_1(\xi)^2\right\}\{g(X_j,\phi\xi_1)^2+g(X_j,\xi_1)^2\}.
V(x)=\exp\left[ \frac{q-(p-1)}{p-1}\int_{r}^{1}(-u_{2}'(\tau))^{q-(p-1)}d\tau \right] -1= \left(\frac{1}{r}\right)^{\frac{(N-1)q-(p-1)N}{q-(p-1)}}-1,r=H^{o}(x)
\operatorname{u}_p(m_p) =-\sin_p(\pi_p m_p)^{p-1}\cos_p(\pi_p m_p)^{2-p}=-(1-\cos_p(\pi_p m_p)^p)^\frac{p-1}{p}\cos_p(\pi_p m_p)^{2-p}=-c_p.
Z_{4}= \frac{( m L )^{4}}{4 !}I_{1}^{4}- ( m L )^{3}I_{1}^{2}I_{2}+ \frac{( M L )^{2}}{2}\left[ \frac{2}{3}I_{1}I_{3}+ \left( \frac{I_{2}}{2}\right)^{2}\right] - \frac{( m L )}{4}I_{4}.
\mathfrak g=\mathfrak g^{(1)}\supset\mathfrak g^{(2)}\supset\dots \supset \mathfrak g^{(k)}\supset \mathfrak g^{(k+1)}=\{0\}, \mathfrak g^{(i+1)}=[\mathfrak g,\mathfrak g^{(i)}], \i=1,\dots, k,
N_3 = O_{F,k}\left(Z^{\frac{2}{d}}\sum_{\frac{1}{2kd}\log Z < p \leq (\log Z)^9}\frac{1}{p^k}\right) & = O_{F,k}\left(Z^{\frac{2}{d}}\sum_{\frac{1}{2kd}\log Z < p}\frac{1}{p^2}\right),
U_{\varphi_0}(s,t) =\int_{-\infty}^{\infty}K(s-\xi,t)\varphi_0(\xi)d\xi, ~~~~~~~~~~~~~~~~~~~~ (s,t)\in\mathbb R\times[0,\infty), H^{\psi}_0 (s,t) =\int_0^t\int_{-\infty}^{\infty}K(s-\xi,t-\tau)h(\psi)(\xi,\tau) d\xi d\tau, ~ (s,t)\in\mathbb R\times[0,\infty),
\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\theta \psi{\mathrm{d}x}- \int_{\Omega}\theta(\boldsymbol u \cdot \nabla \psi){\mathrm{d}x}+ \kappa \int_{\Omega}(\Lambda^{\beta}\theta)(\Lambda^{\beta}\psi){\mathrm{d}x}= \int f \psi{\mathrm{d}x}, \forall \psi \in C^{\infty}(\Omega).
|\langle h, D^2\Psi(x^m) h\rangle|=\frac{\|h^m\|_H^2}{\sqrt{M+\|x^m\|^2}}-\frac{|\langle h^m, x^m\rangle |^2}{(M+\|x^m\|_H^2)^{\frac32}}\le\frac{\|h\|_H^2}{\sqrt{M+\|x^m\|_H^2}}.
P_0 =2\cdot{}_2F_3\left(\frac{1}{2},\frac{1}{2};1,1,\frac{3}{2};-\frac{4\pi^2 D^2}{\lambda^2}\right)-J_0^2\left(\frac{2\pi D}{\lambda}\right)-J_1^2\left(\frac{2\pi D}{\lambda}\right) &\approx 2\cdot{}_2F_3\left(\frac{1}{2},\frac{1}{2};1,1,\frac{3}{2};-\frac{4\pi^2 D^2}{\lambda^2}\right),
V_1^{\mu}\equiv p^\mu - \left [\frac{(i+\omega)p^-}{(1+i+l+\omega)(n\!\cdot\!n^*)}\right ]\;n^\mu - \left [\frac{(i+\omega)p^+}{(1+i+j+\omega)(n\!\cdot\!n^*)}\right ]\;n^{*\mu}\:.
\delta_{2^{s}}^{i_k}\!=\!\delta_{2^{s}}\![i_1,\!\cdots,\!i_{2^n}\!]\textrm{Col}_q(L\!_j\!)\!=\!G_{\!1\!j}\delta_{2^{s}}^{i_{q}}\!=\!G_{\!1\!j}\delta_{2^{s}}^{l}\!\!=\!\delta_{2^{s}}^{\alpha^j_l}.
G=\frac{\bar{\mathbf{w}}^T(\mathbf{H}+\bar{\mathbf{H}})\mathbf{w}}{4\cosh^2\pi\epsilon}|K|^2+\frac{\mathbf{w}_3^T(\mathbf{H}+\bar{\mathbf{H}})\mathbf{w}_4}{4\cos^2\pi\kappa}K_3 K_4.
w ( \mu , t , r ) = - \frac{1}{\mu}\mathrm{~ R e ~}\int_{\textstyle{\cal C}}d z \, H_{1}^{( 1 )}( \mu z ) \Bigl [ q \frac{\partial}{\partial q}F ( q , t , r ) \Bigr ] \biggl |_{2 q = - z^{2}}
\langle{\bf p}\, \vert{\bf 0}, 0 \rangle = \langle{\bf p}\, \vert U ( B ( p ) ) \vert{\bf 0}, 0 \rangle = \sqrt{{\frac{m}{\omega_{m}({\bf p}\, )}}}\langle{\bf p}={\bf 0}\vert{\bf x}={\bf 0}, t = 0 \rangle = N \sqrt{{\frac{m}{\omega_{m}({\bf p}\, )}}}
\rho_{\mu}\left( x,\xi\right) :=\left( \left[ \sigma_{\mathbb{L}}\left(x,\frac{1}{i}\partial_{x}+\xi\right) -\sigma_{\mathbb{L}}\left(x,\xi\right) \right] \frac{1}{\sigma_{\mathbb{L}}\left( x,\xi\right)+i\mu}\right) ,
(2n+\alpha+\beta) P_{n}^{(\alpha,\beta-1)}(x) =(n+\alpha+\beta)P_n^{(\alpha,\beta)}(x)+(n+\alpha)P_{n-1}^{(\alpha,\beta)}(x) , (2n+\alpha+\beta) P_{n}^{(\alpha-1,\beta)}(x) =(n+\alpha+\beta)P_{n}^{(\alpha,\beta)}(x)-(n+\beta)P_{n-1}^{(\alpha,\beta)}(x) ,
\begin{aligned}\textup{\textbf{w}}p \otimes \mathcal{R}(k^{2}) \textup{\textbf{w}}^{\ast}= \frac{i}{k}\times \frac{1}{2}\textup{\textbf{w}}\big( p \otimes \tau \big) \textup{\textbf{w}}^{\ast}+ \textup{\textbf{w}}\big( p \otimes s(k) \big) \textup{\textbf{w}}^{\ast},\end{aligned}
\phi^{A}\psi^{B}-\phi^{B}\psi^{A}=\frac{1}{8}\left(\frac{1}{2!}(\phi^{T}\Gamma^{\mu\nu}\psi){\Gamma_{\mu\nu}}^{AB}+\frac{1}{6!}(\phi^{T}\Gamma^{\mu\nu\alpha\beta\lambda\rho}\psi){\Gamma_{\mu\nu\alpha\beta\lambda\rho}}^{AB}\right).
G ( z \vert \tau ) = \operatorname{l o g}\chi \ , \ \ \chi = 2 \pi e^{- \pi y^{2}/ \tau_{2}}\Bigl \vert{\frac{\theta_{1}( z \vert \tau )}{\theta_{1}^{\prime}( 0 \vert \tau )}}\Bigr \vert \ \ ,
q^{(n-1)^2-{n\choose 2}}=q^{1-n}q^{\frac{n(n-1)}{2}}&\equiv q^{1-n}\left(1-\frac{(n-1)(1-q^n)}{2}\right) &\equiv q^{1-n}+\frac{(1-n)q(1-q^n)}{2}\pmod{\Phi_n(q)^2}.
\Gamma_{\Lambda}(\bar{\psi}^{c}_{\phantom{j}},\psi^{c}_{\phantom{j}})\ + bar{\psi}^{c\alpha}_{\phantom{c}a}\ \Delta^{-1\phantom{\alpha}\beta}_{IR\alpha}\ \psi^{c}_{\phantom{c}\beta a}\ =\ -W(\bar{\zeta},\zeta)\ + bar{\psi}^{c\alpha}_{\phantom{c}a}\ \zeta^{\phantom{\rho}}_{\alpha a}\ + bar{\zeta}^{\alpha}_{a}\ \psi^{c}_{\phantom{c}\alpha a}
\mathrm{Tr}({}^t z d) = \sum_{y \in \mathcal{F}}\mathrm{Tr}(({}^t z\,y e_{B^\sim})({}^t z\,y e_{B^\sim})^*) = \sum_{y \in \mathcal{F}}\mathrm{Tr}({}^t z\, e_{B^\sim}y^* y e_{B^\sim}) = \sum_{y \in \mathcal{F}}\varphi(zE_B(y^* y)) < +\infty.
\frac{M+N}{2}=\frac{\frac{1}{\kappa_{1}}+\frac{1}{\kappa_{2}}}{2}=\frac{M}{2}\cdot(1+\frac{a^{4}}{(bN)^{2}})=\frac{N}{2}\cdot(\frac{(bN)^{2}}{a^{4}}+1);