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\begin{align*}{}_{1}^\kappa EllH^{\mu,4;l}_{k,j;n}(t_1, t_2) := {}_{1;0}^{\kappa} EllH^{\mu,4;l}_{k,j;n}(t_1, t_2)+ {}_{1;1}^{\kappa}EllH^{\mu,4;l}_{k,j,n}(t_1, t_2)+ {}_{1;2}^{\kappa}EllH^{\mu,4;l}_{k,j,n}(t_1, t_2),\end{align*} |
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\begin{align*} \clubsuit \widetilde{K_{k_1,j_1;n_1}^{\mu_1,i_1}}(s, \xi,v, V(s)): = \clubsuit {K_{k_1,j_1;n_1}^{\mu_1,i_1}}(s, \xi-\eta, V(s)) + \big[\mathcal{F}_{x\rightarrow \xi}[{}_{}^zT_{k_1,n_1}^{\mu_1}(B)](s,\xi,V(s)) \mathbf{1}_{n_1\geq -2M_t/15 } \end{align*} |
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\begin{align*}\omega f(x) = f(\omega^{-1}x), \omega \in W.\end{align*} |
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\begin{align*}f=\sum_k \phi_k \otimes \psi_k\end{align*} |
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\begin{align*}P_{(-\infty,\lambda]}(H)=\sum_{\mu \leq \lambda} P_{\{\mu\}}(H_1) \otimes P_{(-\infty,\lambda-\mu]}(H_2).\end{align*} |
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\begin{align*}f=P_{(-\infty,\lambda]}(H)f=\sum_{k\colon \mu_k \leq \lambda} \phi_k \otimes \psi_k\end{align*} |
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\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*} |
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\begin{align*}b_i = a_i \end{align*} |
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\begin{align*} \mathcal{HP}_{\mathcal{P}_i}(q)=\frac{\mathcal{HP}_{\textbf{K}[x_j,j\geq 3-i]/<x_j^2,x_jx_{j+1},j\geq 3-i>}(q)}{\prod_{j\geq 3-i}(1-q^j)}.\end{align*} |
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\begin{align*}\frac{ \sum_{\lambda \in \mathcal{N}_i }\delta(\lambda)q^{\mid \lambda \mid}}{\prod_{j\geq 3-i}(1-q^j)^2}=\frac{\mathcal{HP}_{\textbf{K}[x_j,j\geq 3-i]/<x_j^2,x_jx_{j+1},j\geq 3-i>}(q)}{\prod_{j\geq 3-i}(1-q^j)}.\end{align*} |
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\begin{align*} d^2-2n(g-1)=(d')^2-2n'(g-1)\hbox{and} d\equiv d'\mod 2g-2; \end{align*} |
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\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*} |
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\begin{align*} a_2(f)&=216\\ a_{1,1}(f)&=1914,\end{align*} |
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\begin{align*} a_2(h)&=816\\ a_{1,1}(h)&=33480.\end{align*} |
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\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}\end{align*} |
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\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*} |
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\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*} |
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\begin{align*} \sum\limits_{m\geq 0} c(m)q^m=2q^{-1}+20-128q^3+216q^4-1026q^7+1618q^8+\cdots.\end{align*} |
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\begin{align*}\rho = \frac{a_k+a_{\ell}}{2} = \frac{b_k+ b_{\ell}}{2} \end{align*} |
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\begin{align*}\begin{aligned} E_4^{(2)}E_6^{(2)}=~&(1+240\tilde{q}+240q+\cdots+13440\tilde{q}pq+ 30240\tilde{q}q+\cdots) \times \\&(1-504\tilde{q}-504q +\cdots+44352\tilde{q}p q+166320\tilde{q}q+\cdots)\\ =~&1-264\tilde{q}-264q +\cdots+57792 \tilde{q}pq-45360 \tilde{q}q+\cdots . \end{aligned} \end{align*} |
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\begin{align*} \Theta_{\psi_3}=E_4^{(2)}E_6^{(2)}-56160\chi_{10}. \end{align*} |
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\begin{align*} E\exp\biggl\{ i\sum_{j=1}^n \theta_jX_j\biggr\}= \exp\biggl\{ -\int_{S^n} \biggl| \sum_{j=1}^d \theta_js_j\biggr|\,\Gamma(ds_1,\ldots, ds_n)\biggr\}.\end{align*} |
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\begin{align*} \Sigma^{-1} = \left[\begin{matrix}1 & \theta \\\theta & 1\end{matrix}\right] \ \ \end{align*} |
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\begin{align*} \lim_{v\to\infty} v^2g_V(v)=1/\pi,\end{align*} |
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\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\theta x+1\big)^{3/2}}dx,\end{align*} |
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\begin{align*}v^2g_V(v) =v^2\left[ \int_1^\infty +\int_1^\infty\right]+ v^2\left[ \int_0^1 +\int_0^1\right] =I_1(v)+I_2(v).\end{align*} |
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\begin{align*}I_1(v) \sim& (1-\theta^2)w_2^2\int_1^\infty \frac{x^3v^2}{\pi\bigl( w_2^2x^4 +(1-\theta^2)v^2x^2\bigr)^{3/2}}\,dx \\ =&w_2 \int_{w_2/(v(1-\theta^2)^{1/2})}^\infty \frac{1}{\pi(x^2+1)^{3/2}}\,dx \to w_2 \int_0^\infty \frac{1}{\pi (x^2+1)^{3/2}}\,dx = w_2/\pi.\end{align*} |
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\begin{align*} g_V(v) =\int_{-\infty}^\infty \int_{-\infty}^\infty h_{y_1,y_2}(v)f_{Y_1,Y_2}(y_1,y_2)\, dy_1dy_2,\end{align*} |
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\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*} |
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\begin{align*}a_k = \rho + \tau, a_{\ell} = \rho - \tau,\end{align*} |
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\begin{align*}\Delta_{A, p}u:=\textrm{div}_A(|\nabla_A u|^{p-2}\nabla_A u),\end{align*} |
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\begin{align*}\nabla_A u:=\nabla u+i A(x) u; \textrm{div}_{A}F := \textrm{div} F + i A\cdot F, \end{align*} |
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\begin{align*}c(p):=\inf_{\left(s,t\right) \in\mathbb{R}^{2}\setminus\left\{ \left( 0,0\right) \right\}}\frac{\left[ t^{2}+s^{2}+2s+1\right] ^{\frac{p}{2}}-1-ps}{\left[t^{2}+s^{2}\right] ^{\frac{p}{2}}}\in\left( 0,1\right] \end{align*} |
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\begin{align*} \{a \in \mathcal{P}_\kappa\lambda \mid f_q(a) \in G\} \in U &\Leftrightarrow[f_q]_{U_\alpha} = q \in i(G)(\alpha)\\ &\Leftrightarrow k(q) = q \in k \circ i(G) = \overline{G}.\end{align*} |
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\begin{align*} \mu_l^{[m_1m_2]}=\mu_l^{[m_2m_1]},\end{align*} |
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\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,5}, \mu_{l<3}^{[23]}=0.\end{align*} |
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\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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\begin{align*}\mu_l=\mu_{l-1}+\mu_{l-2}+\mu_{l-3}+6\delta_{l,3}+2(\delta_{l,2}+\delta_{l,4}+\delta_{l,5}), \mu_{l<2}=0.\end{align*} |
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\begin{align*}\mu_l=4(T_{l}+T_{l-1})-2\delta_{l,2}.\end{align*} |
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\begin{align*}A_{n}=\delta_{n,0}+A_{n-1}+3A_{n-2}+9A_{n-3}+\sum_{l=4}^n\mu_lA_{n-l},\end{align*} |