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\begin{align*} \mathbb L_{N,M_N} \langle \pi_s^{N,i}, G_{i,s} \rangle = \dfrac{1}{N^{d}}\sum\limits_{x \in \Lambda_{N,M_N}} G_{i,s}(x/N) \tau_x f_i (\zeta_s,\chi_s), \end{align*} |
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\begin{align*}V_{\epsilon N} (\xi,\omega) = \Big| \dfrac{1}{(2\epsilon N+1)^d} \sum\limits_{\|y\| \leq \epsilon N}\tau_y \phi(\xi,\omega) - \widetilde \phi(\widehat \eta^{\epsilon N}(0)) \Big|.\end{align*} |
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\begin{align*}\mu_n(\zeta,\chi)=\mu\big\{ (\xi,\omega)\; :\; (\xi(x),\omega(x))=(\zeta(x),\chi(x)) \; \; {\rm for}\ x \in \Lambda_{N,n} \big\} \, .\end{align*} |
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\begin{align*}s_n(\mu_n | \nu_{\widehat\theta, n}^N) = \sup_{U\in C_b( \widehat \Sigma_{N,n})} \Big\{\int U(\xi,\omega) d\mu_n(\xi,\omega) - \log \int e^{U(\xi,\omega)}d\nu_{\widehat\theta, n}^N (\eta,\xi)\Big\} .\end{align*} |
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\begin{align*} \lim_{r\to\infty}\frac{s_{\mathcal{\tilde{A}}}(r,1)}{3^r}=c_1\end{align*} |
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\begin{align*}s_n(\mu_n | \nu_{\widehat\theta, n}^N)= \int \log \left( f_n(\xi,\omega) \right) d\mu_n (\xi,\omega),\end{align*} |
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\begin{align*} D_n(\mu_n | \nu_{\widehat\theta, n}^N ) =\mathcal D_n^0 (\mu_n | \nu_{\widehat \theta ,n}^N)\,+\, D^{\widehat b}_n (\mu_n | \nu_{\widehat \theta ,n}^N) \, ,\end{align*} |
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\begin{align*}\mathbb D_n (\mu_n | \nu_{\widehat \theta ,n}^N)=\sum_{x\in \Lambda_{N,n}} (\mathbb D_n)^{x} (\mu_n | \nu_{\widehat \theta ,n}^N)\, ,\end{align*} |
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\begin{align*}\Lambda_{N,n}^c=\Lambda_{N,n+1}\setminus\Lambda_{N,n}.\end{align*} |
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\begin{align*} \langle f_{n+1}^t \rangle_{n+1}=f_n^t\, .\end{align*} |
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\begin{align*}\partial_t f_n^t = \langle \mathfrak L_{N,n+1}^* f_{n+1}^t \rangle _{n+1},\end{align*} |
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\begin{align*}\int \eta_i(x) \eta_j(y) f(\xi^{x,y}, \omega^{x,y}) d\nu_{\widehat \theta }^N (\xi,\omega) = \int \eta_j(x) \eta_i(y) (R_{i,j}^{x,y} (\widehat \theta)+1) f(\xi, \omega) d\nu_{\widehat \theta }^N (\xi,\omega)\end{align*} |
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\begin{align*} \int \eta_i(x) f(\sigma^x\xi, \sigma^x\omega) d\nu_{\widehat \theta }^N (\xi,\omega) = \int \eta_j(x) e^{\vartheta_i(x/N) - \vartheta_j(x/N)} f(\xi, \omega) d\nu_{\widehat \theta }^N (\xi,\omega), \end{align*} |
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\begin{align*} \int \eta_i(x) f(\sigma^x\xi, \omega) d\nu_{\widehat \theta }^N (\xi,\omega) = \int \eta_j(x) e^{\vartheta_i(x/N) - \vartheta_j(x/N)} f(\xi, \omega) d\nu_{\widehat \theta }^N (\xi,\omega), \end{align*} |
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\begin{align*} \int \eta_i(x) f(\xi, \sigma^x\omega) d\nu_{\widehat \theta }^N (\xi,\omega) = \int \eta_j(x) e^{\vartheta_i(x/N) - \vartheta_j(x/N)} f(\xi, \omega) d\nu_{\widehat \theta }^N (\xi,\omega) \, . \end{align*} |
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\begin{align*} \frac{a}{p}+\frac{b}{q} = n\left(1-\frac{1}{p}-\frac{1}{q}\right),\end{align*} |
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\begin{align*} Q_{2m,n,\gamma}(\beta)={1 \over n+1}\int_0^\pi\big |\cos \vartheta\big |^\gamma g_{m,n}(\vartheta-\beta)\; d\vartheta\;,\end{align*} |
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\begin{align*}s_\mathcal{\tilde{A}}(r+2,1)-s_\mathcal{\tilde{A}}(r+1,1)-s_\mathcal{\tilde{A}}(r,1)=c_13^r\cdot 5-c_4-c_5(r-1).\end{align*} |
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\begin{align*}\mathcal L^{x,y} = \sum\limits_{0\le i\not= j\le 3} \mathcal L^{x,y}_{i \leftrightarrow j} \end{align*} |
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\begin{align*}\begin{aligned}\mathcal L^{x,y}_{i \leftrightarrow j} f(\xi,\omega) &= \eta_i(x) \eta_j(y)\Big( f(\xi^{x,y},\omega^{x,y}) - f(\xi,\omega) \Big) +\eta_j(x)\eta_i(y) \Big( f(\xi^{x,y},\omega^{x,y}) - f(\xi,\omega) \Big)\, . \end{aligned}\end{align*} |
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\begin{align*}F_{i,j}^{(1)} (\xi,\omega)= \eta_j(y) f_{n+1}^t (\xi,\omega) \, ,F_{i,j}^{(2)}(\xi,\omega) = \eta_j(y) f_{n+1}^t (\xi^{x,y},\omega^{x,y})\,.\end{align*} |
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\begin{align*}\int L_{\widehat b, N}^x f_n^t d\nu_{\widehat \theta, n}^N=0.\end{align*} |
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\begin{align*}\widetilde \mu^{N,M_N}_{\widehat \theta}=\mu_{N,M_N}\otimes \nu_{\widehat \theta,\L_N\setminus \L_{N,M_N}}^N.\end{align*} |
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\begin{align*}\begin{aligned}\langle \mathfrak L_{N,M_N} \sqrt f, \sqrt f \rangle &\leq - N^2 D_{M_N} ( f) + A_0 |\L_{N,M_N}| \leq - N^2 D_{M_N}^{\widehat b} ( f) + A_0 |\L_{N,M_N}|\end{aligned}\end{align*} |
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\begin{align*} \widetilde\eta_i(x)-b_i(x/N) & = \widetilde\eta_i(x)(1-b_i(x/N)) -b_i(x/N)(1-\widetilde\eta_i(x)) \\& = \sum\limits_{0\le j \neq i\le 3} \Big( \widetilde\eta_i(x) b_j(x/N) - b_i(x/N) \widetilde\eta_j(x/N) \Big) .\end{align*} |
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\begin{align*}{\overline {\mathcal L}}_{N}={\overline {\mathcal L}}_{N}^1+{\overline {\mathcal L}}_{N}^2\, ,\end{align*} |
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\begin{align*} {\overline {\mathbb L}_N}= {\overline {\mathbb L}}^{1,a}_N+{\overline {\mathbb L}}^{1,b}_N + {\overline {\mathbb L}}^2_N ,\end{align*} |
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\begin{align*}\overline s_n(\overline \mu_n | \overline \nu_{\widehat\theta, n}^N)= \int \log \left( \overline f_n((\zeta,\chi),(\xi,\omega)) \right) d\overline \mu_n ((\zeta,\chi),(\xi,\omega)), \end{align*} |
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\begin{align*} \lim_{r\to\infty}\frac{s_{\mathcal{\tilde{A}}}(r,1)}{3^r}=\frac{2}{5}\end{align*} |
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\begin{align*} \overline D_n(\overline{\mu}_n | \overline{\nu}_{\widehat\theta, n}^N ) =\overline{ \mathcal D}_n^0 (\overline{\mu}_n | \overline{\nu}_{\widehat \theta ,n}^N)\,+\, \overline D^{\widehat b}_n (\overline{\mu}_n | \overline{\nu}_{\widehat \theta ,n}^N) \, ,\end{align*} |
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\begin{align*}h_{M_N,i}(\widetilde \eta,\eta)\; =\; N^{-d-1} \sum_{n=1}^{M_N-1} e^{-n/N}H_{i,n} (\widetilde \eta,\eta),\quad\mbox{with }H_{i,n} (\widetilde \eta,\eta) \, =\, \sum_{x\in \Lambda_{N,n}}\big|\widetilde\eta_i(x)-\eta_i(x)\big|\, .\end{align*} |
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\begin{align*}\eta_j(y) (1- \widetilde\eta_j(y))+(1-\eta_j(y)) \widetilde\eta_j(y)= |\eta_j(y)- \widetilde\eta_j(y)|\, ,\end{align*} |
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\begin{align*}\overline {\mathbb L}_N |\eta_i(x) - \widetilde \eta_i(x)| & \leq - 2C''_2 |\eta_i(x) - \widetilde \eta_i(x)| + 34 C''_2 \sum\limits_{j =0}^3 (1-\eta_j(x))\widetilde \eta_j(x) \\ & \leq - 2C''_2|\eta_i(x) - \widetilde \eta_i(x)| + 34 C''_2 \sum\limits_{j =0}^3 |\eta_j(x) - \widetilde \eta_j(x)|\end{align*} |
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\begin{align*}& {\overline { L}}_{\widehat b, N}^1|\widetilde \eta_{i}(x)-\eta_{i}(x) |= -\Big(\sum\limits_{j=0}^3 b_j(x/N)\Big) |\widetilde \eta_i(x) - \eta_i(x)|\leq 0. \end{align*} |
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\begin{align*} \left\{ \begin{array}{lll} -\Delta \varphi =\alpha \varphi \,, & \ \\ \varphi \in H^1_0((-1,1)) \, . & \ \end{array} \right. \end{align*} |
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\begin{align*}\left\{ \begin{array}{l}\partial_t \varphi = \partial_{u_1}^2 \varphi\, , \\ \varphi (0 ,\cdot) =\; g (\cdot) \, , \\\varphi (t, \cdot)\in H_0^1((-1,1))\; \; 0< t\le T \;.\end{array}\right. \end{align*} |
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\begin{align*}\Big| \big< \overline{m}_i (\tau,.), f_k \big>\Big|\,\le \, C_1' t\, \Big(\sum_{i=1}^3 \|\rho_i^{(1)}-\rho_i^{(2)} \|_\infty\Big)\, \| f_k\|_1\, .\end{align*} |
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\begin{align*} \|\overline{m}_i \|_\infty \, \le \, C_1' \, t\, \Big(\sum_{i=1}^3 \|\rho_i^{(1)} -\rho_i^{(2)}\|_\infty \Big)\, .\end{align*} |
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\begin{align*}\widetilde Q_t^x (\eta_i) = Q_t^x (\eta_i) - \int_0^t \tau_x f_i (\xi_s,\omega_s) ds \end{align*} |
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\begin{align*}\mathcal{\tilde{A}}:=\{0,a_z-a_{z-1},\ldots,a_z-a_1,a_z\}.\end{align*} |
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\begin{align*} \textrm{vol}(L)=n!\textrm{vol}(\Delta(L)).\end{align*} |
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\begin{align*} \int_{D(L)}\omega_{st}^n=\int_X c_1(L)^n.\end{align*} |
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\begin{align*} |\mathcal{A}(kL)|=h^0(X,kL),\end{align*} |
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\begin{align*} \Delta(L)=[0,deg(L)].\end{align*} |
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\begin{align*} \Delta(L)\cap \{x_1=0\}=\Delta(L_{|Y_1}),\end{align*} |
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\begin{align*} \Delta(L)\cap\{x_1\geq r\}=\Delta(L-rL_1)+re_1.\end{align*} |
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\begin{align*} s_{\alpha}(\tau^{\gamma}z)=\tau^{\alpha\cdot \gamma}(z^{\alpha}+o(|\tau|))\end{align*} |
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\begin{align*} \phi(z)=\lambda\ln|z-w|^2+O(1)\end{align*} |
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\begin{align*}S = \{ (\xi_1, \xi_2, \xi_1 \xi_2) \in \mathbb R^3:~ (\xi_1, \xi_2) \in D(1) \},\end{align*} |
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\begin{align*}E f(x) = \int_S e^{2 \pi i x \cdot \xi} f(\xi) d\sigma(\xi)\sim \int_{D(1)} e^{2 \pi i(x_1\xi_1+x_2\xi_2+x_3 \xi_1\xi_2)} \tilde f(\xi_1,\xi_2) d \xi_1 d\xi_2\end{align*} |
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\begin{align*}\left(\begin{array}{c}f_{\mathcal{\tilde{A}}}(2n)\\f_{\mathcal{\tilde{A}}}(2n-1)\\\vdots\\f_{\mathcal{\tilde{A}}}(2n-a_z)\end{array}\right)=M_{\mathcal{\tilde{A}}}\left(\begin{array}{c}f_{\mathcal{\tilde{A}}}(n)\\f_{\mathcal{\tilde{A}}}(n-1)\\\vdots\\f_{\mathcal{\tilde{A}}}(n-a_z)\end{array}\right).\end{align*} |
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\begin{align*}\max_{\tau} |Ef_\tau(x)| + \max_{L = L_{\parallel e_1} L_{\parallel e_2 }} \Big| \sum_{\tau: \bar\tau \subset L }Ef_\tau(x) \Big| \le \alpha |Ef(x)|.\end{align*} |
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\begin{align*} \int_{B(R)} |E f(x)|^{p_0} &\lesssim \int_{B(R)} |\mathbf{B}_{\alpha} [Ef](x)|^{p_0} \\&+ \alpha^{-p_0} \Big( \sum_{\tau} \int_{B(R)} |Ef_{\tau}(x)|^{p_0} + \sum_{L = L_{\parallel e_1} L_{\parallel e_2 }} \int_{B(R)} |\sum_{\tau: \bar\tau \subset L } Ef_{\tau}(x)|^{p_0} \Big).\end{align*} |
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\begin{align*}\| [Ef_\tau^K](K^{-1}\cdot,K^{-1}\cdot, K^{-2}\cdot) \|_{L^{p_0}(B(R))} = K^{4/p_0}\| Ef_\tau^K \|_{L^{p_0}(T)} \end{align*} |
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\begin{align*}\| [Ef_L^K](\cdot,K^{-1}\cdot, K^{-1}\cdot) \|_{L^{p_0}(B(R))} = K^{2/p_0}\| Ef_L^K \|_{L^{p_0}(L^*)} \end{align*} |
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\begin{align*} \delta = \epsilon^2 ,~ \delta_{1} =\epsilon^4 ~ ~ \delta_{2} = \epsilon^6.\end{align*} |
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\begin{align*} M = R^{\delta_1} \end{align*} |
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\begin{align*}\int_{B(R)} |\mathbf{B}_\alpha [Ef]|^{p_0} = \sum_i \int_{B(R) \cap O_i'} |\mathbf{B}_\alpha [Ef]|^{p_0} + \int_{B(R) \cap W} |\mathbf{B}_\alpha [Ef]|^{p_0}.\end{align*} |
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\begin{align*} N A \le C_2 \sum_{i=1}^N X_i.\end{align*} |
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\begin{align*} f = \sum_{T \in \mathbb T} f_T.\end{align*} |
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\begin{align*}f_\tau = \sum_{T \in \mathbb T : \omega(T) \in \tau}f_T.\end{align*} |
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\begin{align*}\mathcal{A}:=\{0,2b_2,\ldots,2b_s,2c_1+1,\ldots,2c_t+1\}, \end{align*} |
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\begin{align*}f_{\tau,i} = \sum_{T \in \mathbb T_i : \omega(T) \in \tau} f_{T} f_i = \sum_{\tau} f_{\tau,i}\end{align*} |
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\begin{align*}\mathbb T_i(\Omega) = \{ T \in \mathbb T(\Omega) : T \cap O_i' \neq \emptyset \},\end{align*} |
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\begin{align*}Ef =\sum_{T \in \mathbb T} Ef_T = \sum_{T \in \mathbb T_i} Ef_T + \sum_{T \in \mathbb T \setminus \mathbb T_i} Ef_T.\end{align*} |
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\begin{align*} Ef(x) = Ef_{i}(x) + O\big(R^{-990}\sum_{\tau}\|f_{\tau}\|_{L^2(S)} \big).\end{align*} |
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\begin{align*}|Ef(x)| &\le \Big|\sum_{\tau \in \acute I} Ef_{\tau}(x) \Big| + \Big|\sum_{\tau \in \acute I^c} Ef_{\tau}(x) \Big| \\&\le \Big|\sum_{\tau \in \acute I} Ef_{\tau}(x) \Big| + 16\max_{L=L_{\parallel e_1} L_{\parallel e_2}} \bigg| \sum_{\tau:\bar\tau \subset L} Ef_{\tau}(x) \bigg| \\&\le |Ef_{\acute I}(x)| + 16 \alpha |Ef(x)|.\end{align*} |
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\begin{align*}\Big| \sum_{T :\omega(T) \in \tau,~ T \cap B_j \cap W= \emptyset}Ef_{T}(x) \Big| \lesssim R^{-990}\|f_\tau\|_{L^2(S)}.\end{align*} |
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\begin{align*} Ef_\tau(x) = Ef_{\tau,j}^{\sharp}(x) + Ef_{\tau,j}^{\flat}(x) + O(R^{-990}\|f_\tau\|_{L^2(S)}).\end{align*} |
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\begin{align*} \Big|\sum_{\tau \in \acute I: \bar\tau \subset \varLambda} Ef_{\tau,j}^{\sharp}(x) \Big| &\le \Big|\sum_{\tau: \bar\tau \subset \varLambda} Ef_\tau(x) \Big| + 16\max_{\tau} |Ef_\tau(x)| \\&+ \sum_{\tau \in \acute I: \bar\tau \subset \varLambda} |Ef_{\tau,j}^{\flat}(x)| + O(R^{-990}\sum_{\tau}\|f_\tau\|_{L^2(S)}) .\end{align*} |
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\begin{align*}\mathbb T_{1,Q}^{\flat} &=\{ T \in \mathbb T_{j}^{\flat}: T \cap Q \neq \emptyset,~ \omega(T) \in \tau_1 \}, \\\mathbb T_{2,Q}^{\flat} &=\{ T \in \mathbb T_{j}^{\flat}: T \cap Q \neq \emptyset,~ \omega(T) \in \tau_2 \}. \end{align*} |
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\begin{align*}\xi_1 + \xi_2 &= \xi_1' + \xi_2' + O(R^{-1/2}), \\\zeta_1 + \zeta_2 &= \zeta_1' + \zeta_2' + O(R^{-1/2}), \\\xi_1 \zeta_1 + \xi_2 \zeta_2 &= \xi_1' \zeta_1' + \xi_2' \zeta_2' + O(R^{-1/2}). \end{align*} |
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\begin{align*}f_{\mathcal{A}}(2n-2j)=f_{\mathcal{A}}(n-j)+f_{\mathcal{A}}(n-j-b_2)+\cdots+f_{\mathcal{A}}(n-j-b_s)\end{align*} |
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\begin{align*} (\xi_2',\zeta_2')=(\xi_1+\xi_2-\xi_1',\zeta_1+\zeta_2-\zeta_1')+O(R^{-1/2}).\end{align*} |
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\begin{align*} (\xi_1 - \xi_1')(\zeta_2 - \zeta_1') + (\zeta_1-\zeta_1')(\xi_2 - \xi_1') = O(R^{-1/2}).\end{align*} |
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\begin{align*}v(T) = \frac{1}{\sqrt{\xi^2+\zeta^2+1}}(\zeta,\xi,-1).\end{align*} |
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\begin{align*}Ef_{\tau_k,j}^{\flat} = \sum_{T \in \mathbb T_{k,Q}^{\flat}} Ef_T + O(R^{-990}\|f\|_{L^2(S)}).\end{align*} |
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\begin{align*} \sum_{\tau} \int_S |f_{\tau,j}^{\flat}|^2 \lesssim R^{1/2+C\delta} R^{-1} = R^{-1/2+C\delta}.\end{align*} |
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\begin{align*}\Big( \sum_{\tau} \|f_{\tau,j}^{\flat}\|_{L^2(S)}^2 \Big)^{p/2} &= \Big( \sum_{\tau} \|f_{\tau,j}^{\flat}\|_{L^2(S)}^2 \Big)^{(p-3)/2} \Big( \sum_{\tau} \|f_{\tau,j}^{\flat}\|_{L^2(S)}^2 \Big)^{3/2} \\&\lesssim R^{C\delta} R^{\frac{3}{4} - \frac{p}{4}} \Big( \sum_{\tau} \|f_{\tau,j}^{\flat}\|_{L^2(S)}^2 \Big)^{3/2}.\end{align*} |
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\begin{align*}\tilde{f}_T := \psi_{\bar \Omega}(\widehat\phi_{D_T}*\tilde{f}_{\bar \Omega})\end{align*} |
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\begin{align*} T = \{ (x',x_3) : | x'- x'_T + x_3 (\omega_2,\omega_1) | \lesssim R^{1/2+ \delta} \}\end{align*} |
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\begin{align*}\tilde f = \sum_{\Omega} \sum_{T \in \widetilde{\mathbb T}(\Omega)} \tilde f_T.\end{align*} |
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\begin{align*}Ef(x) = \sum_{\Omega}\sum_{T \in \widetilde{\mathbb T}(\Omega)} Ef_{T}(x).\end{align*} |
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\begin{align*}f_{\mathcal{A}}(2n-2j-1)=f_{\mathcal{A}}(n-j-c_1-1)+\cdots+f_{\mathcal{A}}(n-j-c_t-1)\end{align*} |
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\begin{align*}[d\pi(v), d\pi(v^{\prime})]_{Sch}=d\pi[v, v^{\prime}]_{Sch}\end{align*} |
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\begin{align*}d\pi(v)=\sum_{i,j,k\neq r,l\neq r}(a_{ijkl}\zeta_{i}\zeta_{j} - a_{ijkr}\zeta_{i}\zeta_{j}\zeta_{k} - a_{ijrl}\zeta_{i}\zeta_{j}\zeta_{l}) \frac{\partial }{\partial \zeta_{k}}\wedge \frac{\partial }{\partial \zeta_{l}}\end{align*} |
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\begin{align*}a_{ij}=(-1)^{i+j}\bar{a}_{\varphi(i)\varphi(j)}.\end{align*} |
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\begin{align*}a_{ijkl}=(-1)^{i+j+k+l}\bar{a}_{\varphi(i)\varphi(j)\varphi(k)\varphi(l)}. \end{align*} |
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\begin{align*}a_{ijklmn}=(-1)^{i+j+k+l+m+n}\bar{a}_{\varphi(i)\varphi(j)\varphi(k)\varphi(l)\varphi(m)\varphi(n)}. \end{align*} |
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\begin{align*}\alpha=i_{\boldsymbol{l}\wedge w}(dz_0\wedge dz_1\wedge dz_2\wedge dz_3)\end{align*} |
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\begin{align*}w=\sum_{i,j=0}^{3}\frac{dz_i \wedge dz_j \wedge d\alpha}{dz_0\wedge dz_1\wedge dz_2\wedge dz_3}\frac{\partial }{\partial z_i}\wedge \frac{\partial }{\partial z_j}\end{align*} |
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\begin{align*}\left\{\begin{array}{ll}f=cz_0z_1,& |a_0|\neq |a_1|, \\f=f_2+f_1z_0+\frac{\overline{a}_1}{a_0}\Phi(f_1)z_1+\frac{\sqrt{-1}\,r}{a_0-a_1},& |a_0|=|a_1|,\ a_0\neq a_1, \\ f=f_2+f_1z_0+\frac{\overline{a}_1}{a_0}\Phi(f_1)z_1+c',& a_0= a_1 \end{array}\right.\end{align*} |
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\begin{align*}R_n (x) &=\inf\{j \geq 1 : x_{j+1}x_{j+2}\cdots x_{j+n}=x_1x_2\cdots x_n\},\\R'_n (x) &=\inf\{j \geq n: x_{j+1}x_{j+2}\cdots x_{j+n}=x_1x_2\cdots x_n\}.\end{align*} |
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\begin{align*}\liminf_{n\to\infty}d(T^nx,x)=0,\end{align*} |
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\begin{align*}S=\left(\begin{array}{c c c c c}0 &0 &\cdots &0 &1\\0 &0 &\cdots &1 &0\\\vdots &\vdots & &\vdots &\vdots\\0 &1 &\cdots &0 &0\\1 &0 &\cdots &0 &0\end{array}\right),\end{align*} |
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\begin{align*}\underline{R}(x)=\underline{d}_\mu(x),\ \ \overline{R}(x)=\overline{d}_\mu(x), \ \ \ \mu-\end{align*} |
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\begin{align*}\dim_{\rm H}\left\{x\in\Sigma: d(\sigma^n(x),x)<\psi(n)\ \ \right\}=\frac{1}{1+b},\end{align*} |
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\begin{align*}\dim_{\rm H}\left\{x\in\Sigma: \liminf_{n\to\infty}\frac{\log R_n(x)}{\log n}\leq\alpha\right\}=0.\end{align*} |
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\begin{align*}\liminf_{n\to\infty}\frac{\log R_n(x)}{\varphi(n)}&=\liminf_{i\to\infty}\frac{\log\ell_i}{\varphi(n_i)},\\ \limsup_{n\to\infty}\frac{\log R_n(x)}{\varphi(n)}&=\limsup_{i\to\infty}\frac{\log\ell_i}{\varphi(n_{i-1}+1)}.\end{align*} |
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\begin{align*}d_k = \left \lfloor \frac{\log (\log m_k) - \log (\log n_k)}{\log C} \right\rfloor \ge k\end{align*} |
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Latex-KIE Dataset
The Latex-KIE dataset is a large-scale collection of paired LaTeX formula images and their corresponding LaTeX code. It is specifically designed for training and evaluating models for Image-to-LaTeX, Key Information Extraction (KIE), and Optical Character Recognition (OCR) tasks in scientific domains.
📊 Dataset Summary
- Images: Rendered LaTeX math formulas (black text on white background)
- Text: Corresponding raw LaTeX code for each image
- Split:
train - Total Samples: 92,057
- Format: Parquet (
.parquet) - Size: ~439 MB
🧾 Data Fields
Each data sample consists of:
| Column | Type | Description |
|---|---|---|
image |
Image | Rendered image of the LaTeX formula |
latex_formula |
string |
Corresponding LaTeX string representation |
📂 Example
{
"image": "<Rendered Image of LaTeX>",
"latex_formula": "\\begin{align*} L_{N,M,N} = \\frac{1}{N^d} \\sum ... \\end{align*}"
}
🧠 Use Cases
This dataset is intended for:
- Training models for Image-to-LaTeX generation
- Key Information Extraction (KIE) from scientific formulas
- Benchmarking OCR models on scientific/math notation
- Pretraining/fine-tuning Transformer or CNN-based encoders for math-to-text generation
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