Datasets:
prithivMLmods
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Latex-KIE

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\begin{align*}\log (\log r_{k + j}) = \log (\log n_k) + \frac j{d_k} \cdot \left( \log (\log m_k) - \log (\log n_k) \right) .\end{align*}
\begin{align*}\left| \frac{\varphi(m'_k+1)}{\log (m_k'+1)} - \delta \right| < \frac 1{k+d_k}, & 0 \le \delta < \infty,\\\frac{\varphi(m'_k+1)}{\log (m'_k+1)} > k+d_k, & \delta = \infty.\end{align*}
\begin{align*}\left| \frac{\varphi(m_k)}{\log m_k} - \gamma \right| < \frac 1k 0 \le \gamma < \infty, \frac{\varphi(m_k)}{\log m_k} > k, \gamma = \infty.\end{align*}
\begin{align*}d_k = \left \lfloor \sqrt{\log m_k} - k \right\rfloor.\end{align*}
\begin{align*}\left| \frac{\varphi(m'_k+1)}{\log (m'_k+1)} - \delta \right| < \frac 1{k+d_k}, & 0 \le \delta < \infty,\\\frac{\varphi(m'_k+1)}{\log (m'_k+1)} > k+d_k, & \delta = \infty.\end{align*}
\begin{align*}s=\sum_{i=0}^{\infty}\epsilon_i2^i, \epsilon_i\in\{0,1,t\}.\end{align*}
\begin{align*}\varphi(m_{i+1}) - \varphi(m_{i}) &\ge \varphi(n_{i+1}) - \varphi(n_{i+1}-1) \\&= \varphi(n_{i+1}) -\varphi(n_i)+\varphi(n_i) - \varphi(n_{i+1}-1)> 1. \end{align*}
\begin{align*}\lim_{i \to \infty} \frac{i(n_i+3)}{\ell_i} &\le \lim_{i \to \infty} \frac{i(n_i+3)}{n_i^2 } \\&= \left( \lim_{i \to \infty} \frac{n_i+3}{n_i} \right) \cdot \left( \lim_{i \to \infty} \frac{i}{n_i} \right) = 1 \cdot 0 = 0. \end{align*}
\begin{align*}\lim_{i \to \infty} \frac{i(n_i+3)}{\ell_i} &\le \lim_{i \to \infty} \frac{i(n_i+3)}{n_i \log n_i } \\&= \left( \lim_{i \to \infty} \frac{n_i+3}{n_i} \right) \cdot \left( \lim_{i \to \infty} \frac{i}{\log n_i} \right) = 1 \cdot 0 = 0. \end{align*}
\begin{align*}\tilde \ell_{k+j+1} - \tilde \ell_{k+j} &= \tilde \ell_{k+j}\left( e^{ \frac\alpha2 ( \varphi(m_{k+j+1})- \varphi(m_{k+j})) } -1 \right) \\&> \tilde \ell_{k+j} \frac{\alpha( \varphi(m_{k+j+1})- \varphi(m_{k+j})) }{2} > \frac{\alpha}{2} \tilde \ell_{k+j} \end{align*}
\begin{align*}\tilde \ell_{k+d+1} - \tilde\ell_{k+d} &= \tilde \ell_{k+d} \left( e^{\beta \varphi(m_{k+d+1}) - \alpha\varphi(m_{k+d}) } -1 \right) \\&> \tilde \ell_{k+d} \left( \beta \varphi(m_{k+d+1}) - \alpha\varphi(m_{k+d}) \right) > \beta \tilde \ell_{k+d} > \frac{\alpha}{2} \tilde \ell_{k+d}.\end{align*}
\begin{align*}\lim_{i \to \infty} \frac{i(n_i+3)}{\ell_i} &\le \lim_{i \to \infty} \frac{i(n_i+3)}{n_i \log n_i } \\&= \left( \lim_{i \to \infty} \frac{n_i+3}{n_i} \right) \cdot \left( \lim_{i \to \infty} \frac{i}{\log n_i} \right) = 1 \cdot 0 = 0. \end{align*}
\begin{align*}\liminf_{n \to \infty} \frac{\log R_{n}(x)}{\varphi(n)} &=\liminf_{i \to \infty} \frac{\log \ell_i}{\varphi(n_i)}= \liminf_{i \to \infty} \left( \frac{\log \ell_i}{\log n_i} \cdot \frac{\log n_i}{\varphi(n_i)} \right) \\&= \lim_{i \to \infty} \frac{\log \ell_i}{\log n_i} \cdot \liminf_{i \to \infty} \frac{\log n_i}{\varphi(n_i)}= A \cdot \frac 1\gamma = \alpha.\end{align*}
\begin{align*}\varphi(m_{i}) \log m_{i} &\ge \varphi(m_{i_0}) \log m_{i_0} + \sum_{j=i_0}^{i-1} \min \left( \varphi(m_1) \log \log m_i , \ \log m_i \right) \\&\ge \varphi(m_{i_0}) \log m_{i_0} + \varphi(m_1) \sum_{j=i_0}^{i-1} \log \log m_j,\end{align*}
\begin{align*}\frac{\tilde \ell_i}{i m_i} &= \frac{ {m_i}^{\rho(\varphi(m_i)/\log m_i)} \varphi(m_i) \log m_i}{i m_i}\ge \frac{\varphi(m_i) \log m_i}{i} \\&> \frac{\varphi(m_1)}{i} \sum_{j=i_0}^{i-1} \log \log m_j \ge \frac{\varphi(m_1)}{i} \sum_{j=i_0}^{i-1} \log \log j \\&\ge \frac{\varphi(m_1)}{i} \int_{i_0}^{i-1} \log \log x \, \mathrm{d}x \to \infty\end{align*}
\begin{align*}\partial f=\partial f'+\partial f'',\end{align*}
\begin{align*}s_\mathcal{A}(r,1)=\sum_{n=2^r}^{2^{r+1}-1}f_\mathcal{A}(n)\approx c(\mathcal{A},1)3^r,\end{align*}
\begin{align*}h(\partial f',\bar\partial f')=0,\end{align*}
\begin{align*}\gamma= f^*g =(u_1^2+u_2^2)|dz|^2,f^*\omega = \frac{i}{2}(u_1^2-u_2^2)dz\wedge d\bar z.\end{align*}
\begin{align*}u_1 =\|\partial f'(Z)\|,\ u_2=\|\partial f''(Z)\|,Z=\partial/\partial z.\end{align*}
\begin{align*}\gamma_1 = h(\partial f',\partial f')=u_1^2dz\overline{dz},\gamma_2=h(\partial f'',\partial f'')= u_2^2dz\overline{dz}\end{align*}
\begin{align*}|dz|^2 = \tfrac12 (dz\overline{dz} + d\bar z\overline{d\bar z}),\end{align*}
\begin{align*}\tau(\rho) = \frac{2}{\pi}\int_\Sigma f^*\omega.\end{align*}
\begin{align*}\tau(\rho) = -\tfrac{2}{3}c_1(\rho)= \tfrac{2}{3} (d_2-d_1).\end{align*}
\begin{align*}R^\nabla = R^{\nabla_E}+[\Phi\wedge\Phi^\dagger] = -i\mu \omega_c I_E.\end{align*}
\begin{align*}\nabla_E = \tfrac12(\nabla + \sigma\nabla\sigma),\Psi = \tfrac12(\nabla - \sigma\nabla\sigma).\end{align*}
\begin{align*}\Phi_2(\sigma_1) = a\sigma_3,\Phi_2(\sigma_2) = b\sigma_3,\Phi_1(\sigma_3) = c\sigma_1.\end{align*}
\begin{align*} g_{m,n}(\theta)=\sum_{j=0}^n \cos^{2m}{\theta+j\pi \over n+1}\;.\end{align*}
\begin{align*}\sum_{s=1}^{2^n-1}f_\mathcal{A}(s)&=\sum_{j=0}^{n-1}\sum_{s=2^j}^{2^{j+1}-1}f_\mathcal{A}(s)\approx\sum_{j=0}^{n-1}c(\mathcal{A},1)3^j\\&=c(\mathcal{A},1)\left(\frac{3^n-1}{2}\right)\approx \frac{1}{2}c(\mathcal{A},1)3^n.\end{align*}
\begin{align*}\begin{pmatrix} 0&0&c\\ 0&0&0\\ 0&b&0\end{pmatrix}.\end{align*}
\begin{align*}\varphi\circ\bar\partial_E\circ\varphi^{-1}= d\bar z\left[\frac{\partial}{\partial\bar z} +\begin{pmatrix} -\bar Z\log u_1 & -\bar Q/u_1u_2 & 0\\ 0 & \bar Z\log u_2 & 0\\ 0 & 0 & 0\end{pmatrix}\right]\end{align*}
\begin{align*}\varphi\circ\Phi\circ\varphi^{-1} = \begin{pmatrix} 0 & 0 & u_1\\ 0 & 0 & 0\\ 0 & u_2 & 0 \end{pmatrix}dz,\end{align*}
\begin{align*}\sigma_1 = \partial f'(Z)f_0 = \pi_0^\perp Zf_0,\sigma_2 = \bar\partial f'(\bar Z)f_0 = \pi_0^\perp\bar Z f_0.\end{align*}
\begin{align*}Zf_1 & = (Z\log u_1)f_1 + (Q/u_1u_2)f_2,\\Zf_2 & = -(Z\log u_2)f_2 + u_2f_0,\\Zf_0 & = u_1f_1.\end{align*}
\begin{align*}\varphi\circ\nabla^{1,0}\circ\varphi^{-1} =dz\left(\frac{\partial}{\partial z} +\begin{pmatrix} Z\log u_1 & 0 & u_1\\ Q/u_1u_2 & -Z\log u_2 & 0\\ 0 & u_2 & 0\end{pmatrix}\right)\end{align*}
\begin{align*}\tilde f_1 = \frac{z^{-q}\sigma_1}{|z^{-q}\sigma_1|} = (|z|/z)^qf_1,\tilde f_2 = (|z|/\bar z)^pf_2= (z/|z|)^pf_2,\end{align*}
\begin{align*}\tilde\varphi = S(\varphi|V),S = \begin{pmatrix} (z/|z|)^q &0\\0&(|z|/z)^p \end{pmatrix}.\end{align*}
\begin{align*}\xi_{jk} = a_jz^{-(p_j+q_j)} dz_j^{-2} - a_kdz_k^{-2}\ \ U_j\cap U_k,\end{align*}
\begin{align*}\partial a_j/\partial\bar z_j = -\frac{\bar Q_jz_j^{p_j+q_j}}{u_{1j}^2u_{2j}^2},\end{align*}
\begin{align*}\sum_{i=0}^{n-2} \epsilon_i 2^i&\leq t+t\cdot2+t\cdot2^2+\cdots+t\cdot2^{n-k-3}+2^{n-k-2}+2^{n-k-1}+\cdots+2^{n-2}\\&<t\cdot2^{n-k-2}+2^{n-1}-1\\&\leq2^{k+1}\cdot2^{n-k-2}+2^{n-1}-1\\&=2^n-1\\&<2^n.\end{align*}
\begin{align*}\tilde\varphi\circ\bar\partial_E\circ\tilde\varphi^{-1} = d\bar z\left[\frac{\partial}{\partial\bar z} +\begin{pmatrix} -\bar Z\log (u_1/|z|^{q}) & -\bar Qw^{p+q}/u_1u_2 \\ 0 & \bar Z\log(u_2/|z|^{p}) \end{pmatrix} \right].\end{align*}
\begin{align*}c_{jk} = \begin{pmatrix} w_j^{q_j}\frac{dz_j/dz_k}{|dz_j/dz_k|} & 0 \\ 0 & w_j^{-p_j}\frac{|dz_j/dz_k|}{dz_j/dz_k} \end{pmatrix},\end{align*}
\begin{align*}\frac{d\bar z_j/d\bar z_k}{|d\bar z_j/d\bar z_k|} = \overline{\frac{dz_j/dz_k}{|dz_j/dz_k|}}= \frac{|dz_j/dz_k|}{dz_j/dz_k}.\end{align*}
\begin{align*}\chi_j\circ \bar\partial_E\circ\chi_j^{-1} = d\bar z_j\frac{\partial}{\partial\bar z_j},\end{align*}
\begin{align*}R_j[\tilde\varphi\circ\bar\partial_E\circ\tilde\varphi^{-1}]R_j^{-1} = d\bar z_j \frac{\partial}{\partial\bar z_j}.\end{align*}
\begin{align*}R_j = \begin{pmatrix} 1&a_j\\0&1\end{pmatrix}\begin{pmatrix} (u_{1j}/|z_j|^{q_j})^{-1}&0\\0&u_{2j}/|z_j|^{p_j}\end{pmatrix},\end{align*}
\begin{align*}\frac{\partial a_j}{\partial\bar z_j} = -\frac{\bar Q_jz^{p_j+q_j}}{u_{1j}^2u_{2j}^2},\end{align*}
\begin{align*}b_{jk} = R_jc_{jk}R_k^{-1} = \begin{pmatrix} z_j^{q_j}dz_j/dz_k & \lambda_{jk}\\ 0 & z_j^{-p_j}dz_k/dz_j\end{pmatrix},\end{align*}
\begin{align*}\lambda_{jk} = a_jz_j^{-p_j}\frac{dz_k}{dz_j} - a_kz_j^{q_j}\frac{dz_j}{dz_k},\end{align*}
\begin{align*}\xi_{jk} & = (z_j^{q_j}\frac{dz_j}{dz_k})^{-1}\lambda_{jk}dz_k^{-2} \\& = a_jz_j^{-(p_j+q_j)}dz_j^{-2} - a_kdz_k^{-2}.\end{align*}
\begin{align*}\frac{1}{2}c(\mathcal{A},1)3^n\geq 3^{n-k-2}\cdot 2^{k+1}=3^n\cdot \frac{2^{k+1}}{3^{k+2}},\end{align*}
\begin{align*}\bar\partial\eta_j = -\frac{\bar Q_j}{u_{1j}^2u_{2j}^2}dz_j^{-2}\bar dz_j.\end{align*}
\begin{align*}\|\tau\|^2 = B(\tau,\tau) = |z|^{-2d}u_1^2u_2^2,\end{align*}
\begin{align*}Z\bar Z\log u_1^2 & = 2u_1^2 + |Q|^2/u_1^2u_2^2 - u_2^2,\\Z\bar Z\log u_2^2 & = 2u_2^2 + |Q|^2/u_1^2u_2^2 - u_1^2,\end{align*}
\begin{align*}\tan^2(\theta/2) = \frac{u_2^2}{u_1^2},\sin^2(\theta) = \frac{4u_1^2u_2^2}{(u_1^2+u_2^2)^2},\end{align*}
\begin{align*}Z\bar Z\log\sin^2\theta & = Z\bar Zu_1^2 + Z\bar Zu_2^2 - 2Z\bar Z\log(u_1^2+u_2^2)\\ = & u_1^2 + \frac{2|Q|^2}{u_1^2u_2^2} + u_2^2 - \frac{2}{u_1^2+u_2^2}[Z\bar Z\log (u_1^2+u_2^2)](u_1^2+u_2^2),\end{align*}
\begin{align*}\kappa^\perp -\kappa_\gamma = 2\left(1+\frac{2|Q|^2}{u_1^2u_2^2(u_1^2+u_2^2)}\right).\end{align*}
\begin{align*}-1 = \kappa_\gamma + 2\|A_f\|^2.\end{align*}
\begin{align*}f_1 = \frac{1}{s}(f_0)_z,f_2 = \frac{1}{s}(f_0)_{\bar z},s = u_1=u_2.\end{align*}
\begin{align*}\begin{pmatrix} d\Theta_1\\ d\bar\Theta_1\\ d\Theta_2\\ d\bar\Theta_2\end{pmatrix}=\begin{pmatrix} l & k & 0 &-1\\ \bar k & l & -1 & 0 \\ \bar k & l & 1 & 0 \\ l & k & 0 & 1\end{pmatrix}\begin{pmatrix} dz \\ d\bar z \\ dw \\ d\bar w \end{pmatrix},\end{align*}
\begin{align*}l = \tfrac{1}{\sqrt{2}}(1+|w|^2),k = -2Q_0w-\tfrac{1}{\sqrt{2}}\bar w^2.\end{align*}
\begin{align*}\sum_{i=0}^{\infty}\epsilon_i2^i\geq t\cdot2^{i_0}\geq t\cdot2^{n-k}>2^k2^{n-k}=2^n.\end{align*}
\begin{align*}\theta_1 = \frac{1}{(1-2|w|^2)^2}\left(2|\phi|^2 -(w\phi+\bar w\bar\phi)^2\right),\theta_2 = \frac{1}{(1-2|w|^2)^2}\left(2|dw|^2+ (wd\bar w-\bar w dw)^2\right).\end{align*}
\begin{align*}\theta_1 = \frac{1}{(1-2r^2)^2}\begin{pmatrix} \phi_1 & \phi_2\end{pmatrix}\begin{pmatrix} 2-4w_1^2 & 4w_1w_2\\4w_1w_2 & 2-4w_2^2\end{pmatrix} \begin{pmatrix} \phi_1 \\ \phi_2\end{pmatrix}.\end{align*}
\begin{align*}|\phi|^2\geq\varepsilon_2\gamma = \varepsilon_2(dx^2+dy^2),\end{align*}
\begin{align*}\begin{pmatrix} \phi_1\\ \phi_2 \end{pmatrix} = B \begin{pmatrix} dx \\ dy\end{pmatrix}B=\begin{pmatrix} l+k_1& k_2\\k_2 & l-k_1\end{pmatrix}.\end{align*}
\begin{align*}\lambda^2 - 2(l^2+|k|^2)\lambda +(l^2-|k|^2)^2=0,\end{align*}
\begin{align*}l-|k| = \frac{l^2-|k|^2}{l+|k|},\end{align*}
\begin{align*}l^2-|k|^2 & = \tfrac12(1+r^2)^2 - (2Q_0w+\tfrac1{\sqrt{2}}\bar w^2)(2Q_0\bar w+\tfrac1{\sqrt{2}}w^2) \\& = r^2(1 -4Q_0^2) +\tfrac12 - Q_0(\sqrt{2}r)^3\cos(3\alpha) ,\end{align*}
\begin{align*}\varphi_r^*g = \frac{1}{(1-2r^2)^2}\begin{pmatrix} dx & dy \end{pmatrix} B^t\begin{pmatrix} 2-4w_1^2 & 4w_1w_2\\ 4w_1w_2 & 2-4w_2^2\end{pmatrix}B\begin{pmatrix} dx\\ dy\end{pmatrix},\end{align*}
\begin{align*}a(r) &= \frac{1}{(1-2r^2)^2} 2\sqrt{1-2r^2}(l^2-|k|^2)\\&= \frac{1}{(1-2r^2)^{3/2}}\left( 1+2r^2(1-4Q_0^2)-4\sqrt{2}Q_0r^3\cos(3\alpha)\right),\end{align*}
\begin{align*}\frac{da}{dr} = \frac{4r}{(1-2r^2)^{5/2}}\left[ (1+r^2)(1-4Q_0^2) + \tfrac32 (1-2\sqrt{2}Q_0r\cos(3\alpha))\right].\end{align*}
\begin{align*}|G/N_n|=|G/L_{k_n}|.|L_{k_n}/N_n|\leq A.k_n^{d+\epsilon}.\end{align*}
\begin{align*}\deg(V_1)<\tfrac12\deg(V_1\oplus 1) = \tfrac12 \deg(V_1),\ \deg(V_1)<0,\end{align*}
\begin{align*}b<2(g-1),\deg(V_2)=\tfrac12 b- g+1.\end{align*}
\begin{align*}\sigma_1 = \pi_0^\perp Zf_0 = Zf_0,\sigma_2 = \pi_1^\perp Z\sigma_1=\pi_1^\perp ZZf_0.\end{align*}
\begin{align*}(\nabla_Z Z)f_0 = \pi_0^\perp Z\pi_0^\perp Z f_0- \pi_0^\perp Z \pi_0 Zf_0=\pi_0^\perp ZZf_0,\end{align*}
\begin{align*}Zf_1& = (Z\log u_1) f_1 + u_1^{-1}s_2f_2,\\Zf_2& = (Z\log s_2) f_2, \\Zf_0& = u_1f_1.\end{align*}
\begin{align*}\varphi\circ\bar\partial_E\circ\varphi^{-1} =d\bar z\left[ \frac{\partial}{\partial\bar z} - \begin{pmatrix} \bar Z\log u_1 &s_2/u_1\\ 0 &\bar Z\log s_2 \end{pmatrix}\right].\end{align*}
\begin{align*}\tilde\varphi\circ\nabla^{0,1}\circ\tilde\varphi^{-1} =d\bar z\left[ \frac{\partial}{\partial\bar z} - \begin{pmatrix} \bar Z\log (u_1/|z|^p) &-s_2w^{p-q}/u_1\\ 0 &\bar Z\log (s_2/|z|^q) \end{pmatrix}\right].\end{align*}
\begin{align*}R = \begin{pmatrix} 1& a\\ 0&1\end{pmatrix}\begin{pmatrix} |z|^p/u_1 & 0 \\ 0 & |z|^q/s_2\end{pmatrix},\end{align*}
\begin{align*}\partial a/\partial\bar z =- s_2^2z^{p-q}/u_1^2.\end{align*}
\begin{align*}\tilde c_{jk} = \begin{pmatrix} \frac{dz_j/dz_k}{|dz_j/dz_k|}& 0 \\ 0 & \frac{(dz_j/dz_k)^2}{|dz_j/dz_k|^2}\end{pmatrix}\begin{pmatrix} w_j^{p_j}w_k^{-p_k} & 0 \\ 0 & w_j^{q_j}w_k^{-q_k}\end{pmatrix}.\end{align*}
\begin{align*}|\Gamma/\Gamma(5^k)|=5^{2k}.\delta(5^k)\leq 6.5^{3k}\end{align*}
\begin{align*}b_{jk} = R_j\tilde c_{jk} R_k^{-1} = \begin{pmatrix} z_j^{p_j}z_k^{-p_k}dz_j/dz_k & \lambda_{jk}\\ 0 & z_j^{q_j}z_k^{-q_k} (dz_j/dz_k)^2 \end{pmatrix},\end{align*}
\begin{align*}\lambda_{jk} = a_jz_j^{q_j}z_k^{-q_k} (dz_j/dz_k)^2 - a_k z_j^{p_j}z_k^{-p_k}dz_j/dz_k,\end{align*}
\begin{align*}u_{1k}/u_{1j} = |dz_j/dz_k|,s_{2k}/s_{2j} = |dz_j/dz_k|^2.\end{align*}
\begin{align*}\eta_{jk}& = z_j^{-p_j}z_k^{p_k} (dz_k/dz_j)\lambda_{jk} dz_k\\& = a_jz_j^{q_j-p_j}dz_j - a_kz^{q_k-p_k}dz_k.\end{align*}
\begin{align*}\bar\partial\tau_j = -\frac{s_{2j}^2}{u_{1j}^2}dz_jd\bar z_j,\end{align*}
\begin{align*}E_1\oplus E_2\oplus E_3= K(-D_2) \oplus 1\oplus K^{-1}(D_1),\end{align*}
\begin{align*}\bar\partial\varphi_2'(\bar Z) = \pi_2^\perp \bar Z.\end{align*}
\begin{align*}\begin{pmatrix} & 0 & \\ Q/u_1u_2 & & 0\\ & u_2 & \end{pmatrix} dz +\begin{pmatrix} &-\bar Q/u_1u_2 & \\ 0 & & u_2\\ & 0 & \end{pmatrix} d\bar z. \end{align*}
\begin{align*}\deg(K^{-2}(D_1+D_2)) = d_1+d_2-4(g-1)<0,\end{align*}
\begin{align*}h^1(K^{-2}(D_1+D_2)) = 5g-5 -d_1-d_2.\end{align*}
\begin{align*}\delta(N_k)^{1-\alpha}=o(N_k^{2\alpha}).\end{align*}
\begin{align*} e(a_1b_1\dots b_{n-1}a_n) = \begin{cases} 2e(a_1b_1\dots b_{n-1}) & \\0 & \end{cases} \end{align*}
\begin{align*}\mathrm{Lyr}_n(G):=\lbrace G^\prime\le H\le G\mid (G:H)=p^n\rbrace\end{align*}
\begin{align*}\tau_n(G):=(H/H^\prime)_{H\in\mathrm{Lyr}_n(G)}\end{align*}
\begin{align*}\tau(G):=\lbrack\tau_0(G);\ldots;\tau_v(G)\rbrack.\end{align*}
\begin{align*}\varkappa_n(G):=(\ker(T_{G,H}))_{H\in\mathrm{Lyr}_n(G)}\end{align*}