image imagewidth (px) 42 746 | latex_formula stringlengths 36 422 |
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\begin{align*}&\left< \frac{i}h [-h^2\Delta+V(x)-E(h), \chi_\alpha(x_n)hD_n] u_h,\, u_h \right>_{L^2(\Omega_\Gamma)} \\=& \int_{\Sigma_{\Omega_\Gamma}} \left( 2\chi_\alpha'(x_n) \xi_n^2-\chi_\alpha(x_n)\partial_{x_n}(R+V) \right) \, d \mu+o(1) \\=& I_1-I_2+o(1)\end{align*} | |
\begin{align*}\big| det\big(\begin{bmatrix}\sin \theta \cos\phi - \sin \tilde{\theta} \cos\tilde{\phi} & \sin \theta \sin\phi - \sin \tilde{\theta}\sin \tilde{\phi} & \cos \theta - \cos \tilde{\theta}\\ -s \sin \theta \sin\phi & s \sin \theta \cos\phi & 0\\ s\cos\theta\cos\phi & s\cos\theta \sin \phi & -s\sin\theta\\ \... | |
\begin{align*}\lim_n\langle B_nRA\xi, C\xi\rangle&=\lim_n\langle RA\xi, B_nC\xi\rangle=\lim_n\langle RA\xi, CB_n\xi\rangle\\&=\langle RA\xi, CT\xi\rangle=\langle RA\xi, TC\xi\rangle=\langle TRA\xi, C\xi\rangle\,,\end{align*} | |
\begin{align*} a_2(f)&=2g-d+2(e-1)\\ a_{1,1}(f)&=d-3g-e(e-1)+\frac{3}{2}(e-1)(e-2)\end{align*} | |
\begin{align*} a_2(f)&=216\\ a_{1,1}(f)&=1914,\end{align*} | |
\begin{align*} a_2(h)&=816\\ a_{1,1}(h)&=33480.\end{align*} | |
\begin{align*} \begin{aligned} a_2(h)=&~4h_*(\alpha_1)\beta_1^2+2h_*(\alpha_1^2+\alpha_2)\beta_1-2h_*(\alpha_1)\beta_2+2h_*(\alpha_1\alpha_2)\\ 2(a_2(h)+a_{1,1}(h))=&~\delta^2-\beta_1\delta+(-2h_*(\alpha_1\alpha_2)-h_*(\alpha_3) -2h_*(\alpha_1^2+\alpha_2)\beta_1\\& +h_*(\alpha_1)(-4\beta_1^2+3\beta_2)) \end{aligned}\en... | |
\begin{align*}\begin{aligned} \Theta_\psi(\tau): &=\sum\limits_{\beta\geq 0} (\deg(\psi^\ast Z(\beta))) e^{2\pi i \mathrm{tr} (\beta \tau )} \\ &=\sum\limits_{k,l,m\geq 0} N_{k,l,m}\tilde{q}^kp^lq^m\end{aligned}\end{align*} | |
\begin{align*} \begin{aligned}\chi_{10}(\tau)& =\tilde{q} p q\prod\limits_{(r,s,t)> 0} (1-\tilde{q}^rp^s q^t)^{c(4rt-s^2)} \\&=\tilde{q}pq-2\tilde{q}q-16 \tilde{q}pq^2+\cdots \end{aligned}\end{align*} | |
\begin{align*} g_V(0) =&\int_{0}^{\infty}\frac{(1-\theta^2)(w_2^2x^2-2\theta w_1w_2x+w_1^2)x}{2\pi\big(w_2^2x^2-2\theta w_1w_2x+w_1^2\big)^{3/2}\bigl(x^2+2\theta x+1\big)^{3/2}}dx \\&+\int_{0}^{\infty}\frac{(1-\theta^2)(w_2^2x^2-2\theta w_1w_2x+w_1^2)x}{2\pi\big(w_2^2 x^2-2\theta w_1w_2x+w_1^2\bigr)^{3/2}\bigl(x^2-2\th... | |
\begin{align*}I_1(v) \sim& (1-\theta^2)w_1^2\int_0^1 \frac{xv^2}{2\pi(w_1^2 +(1-\theta^2)v^2x^2)^{3/2}}\,dx \\ =& w_1 \int_0^{v(1-\theta^2)^{1/2}/w_1}\frac{x}{2\pi(x^2+1)^{3/2}}\,dx \to w_1 \int_0^\infty\frac{x}{2\pi(x^2+1)^{3/2}}\,dx=w_1/(2\pi).\end{align*} | |
\begin{align*} \mu_l^{[12]}&=\mu_{l-1}^{[12]}+\mu_{l-2}^{[12]}+\mu_{l-3}^{[12]}+\delta_{l,2}+\delta_{l,4},\mu_{l<2}^{[12]}=0, \\ \mu_l^{[13]}&=\mu_{l-1}^{[13]}+\mu_{l-2}^{[13]}+\mu_{l-3}^{[13]}+2\delta_{l,3}, \mu_{l<3}^{[13]}=0, \\ \mu_l^{[23]}&=\mu_{l-1}^{[23]}+\mu_{l-2}^{[23]}+\mu_{l-3}^{[23]}+\delta_{l,3}+\delta_{l,... | |
\begin{align*} \mu_l^{[12]}&=\mu_{l-1}^{[12]}+\mu_{l-1}^{[13]}+\delta_{l,2},\\ \mu_l^{[13]}&=\mu_{l-1}^{[12]}+\mu_{l-1}^{[23]}+\delta_{l,3},\\ \mu_l^{[23]}&=\mu_{l-1}^{[12]},\end{align*} |
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