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\begin{align*} E_2 \leqslant \frac1\alpha \sum_{i=1}^j \int_{t_{i-1}}^{t_i} g^2(t) \big( (t_i-t)^{-2\alpha} + (t-t_{i-1})^{-2\alpha} \big) \, \mathrm{d}t. \end{align*} | |
\begin{align*}{\cal L}^{(\alpha)}=-\frac{1}{\beta}\gamma\theta^{0(\alpha)}\xi,\qquad{\cal L}_{j}^{(\alpha)}=\beta\xi*k_{ij}\theta_{,i}^{0(\alpha)}l.\end{align*} | |
\begin{align*} r = d(1+2n) + \lfloor 4 \| Y \|(d+1) + \frac{3}{2} |{\rm trace}(Y)| + |d-{\rm trace(Y)}| \rfloor~. \end{align*} | |
\begin{align*}\phi(t_{n+1}) = (\tilde{k}\phi(t_n))^{\tilde{d}} \leq (\tilde{k}e^{t_n})^{\tilde{d}} \leq e^{\tilde{k}t_n}=e^{t_{n+1}},\end{align*} | |
\begin{align*}g_{m,b}= ds^2 + u_{m,b}(s)^2 g_*,\end{align*} | |
\begin{align*}-\Delta u+u=(K\ast F(u))F'(u)\mathbb{R}^N,\end{align*} | |
\begin{align*} \theta^m_l(\mathbf{u}) =\begin{cases} u_0 + l u_1 & \colon m=1 \\ (u_0 + l u_m, u_1, u_2, \dots, u_{ m - 1 }) & \colon m > 1\end{cases},\end{align*} | |
\begin{align*}w := a^{ij}v_{ij}, \tilde{w} := a^{ij} v_i v_j.\end{align*} | |
\begin{align*}x_{n-1}^2=x_{n-2}+x_{n-1}\end{align*} | |
\begin{align*}\eth = \sum_i S^{-1}(E_{\xi_i}) \otimes \gamma_{-}(w_i) \in U_q(\mathfrak{g}) \otimes \mathrm{Cl}_q.\end{align*} | |
\begin{align*}h((\mathcal{P}\cap B_S(0)) \cup \{0\} \cup A, 0)=h_{\infty},\end{align*} | |
\begin{align*}\frac{\partial g}{\partial t} = -R\, g\end{align*} | |
\begin{align*}J_k=\partial_{\theta^k}+i\theta^k\partial_{x_k},k=1,2\end{align*} | |
\begin{align*}\lim_{N^\prime}\int_{T_1}^t \left<\varphi, \mathcal{T} (t-\tau) C_2 f_N^{(2)} (\tau) \right> d\tau= 0\end{align*} | |
\begin{align*} (_F(n,k,\Omega))^\bot=_F(n,n-k,\Omega,v) \end{align*} | |
\begin{align*}U(x)\,=\, {\cal P} \exp [-\int_{C_x} dy^{\mu} L_{\mu} (y) ],\end{align*} | |
\begin{align*}v_p(S(n,k))=v_p(S(a_0+\tilde{n}(p-1),k))=v_p(f_{a_0,k}(\tilde{n}))=\beta+lv_p(\tilde{n}-\tilde{x}_0)\end{align*} | |
\begin{align*} A \cdot G = G_{left}^{-1} \cdot G = I_n. \end{align*} | |
\begin{gather*}=\sum_{s=0}^{n}\binom{n}{s}y^{n-s}(1-\rho )^{s}\sum_{t=s}^{n}(-1)^{n-t}\frac{(n-s)!}{(t-s)!t!}\rho ^{t-s}(\beta +t-s)^{(s)} \\=\sum_{s=0}^{n}\binom{n}{s}y^{n-s}(1-\rho )^{s}\sum_{m=0}^{n-s}(-1)^{n-m-s}\binom{n-s}{m}\rho ^{m}(\beta +m)^{(s)}.\end{gather*} | |
\begin{align*}\frac{(p^2-4M^2)N}2\int\frac{d^Dk}{(2\pi)^D} \frac{1}{(k^2-M^2)[(k+p)^2-M^2]},\end{align*} | |
\begin{align*}J_{\mathrm{Lie}}^{s}\overline{\eta }_{\mathrm{L}}=\overline{\eta }_{\mathrm{BrL}}^{s}J_{\mathrm{Lie}}^{s}J_{\mathrm{Bialg}}^{s}\overline{\epsilon }_{\mathrm{L}}=\overline{\epsilon }_{\mathrm{BrL}}^{s}J_{\mathrm{Bialg}}^{s}, \end{align*} | |
\begin{align*}T_h({\bf 1}_{\{r\}}\circ\pi_x)(y)&= \int_X {\bf 1}_{\{r\}}(\pi(x,z))\> K_h(y,dz)= K_h(y, \{z\in X:\> \pi(x,z)=r\})\\&=\frac{1}{\omega_h}|\{z\in X:\> \pi(x,z)=r, \> \pi(y,z)=h\}|\\&=\frac{1}{\omega_h}|\{z\in X:\> \pi(z,x)=\bar r, \> \pi(y,z)=h\}|\\&= \frac{1}{\omega_h}p_{h,\bar r}^{\pi(y,x)}=\frac{1}{\omeg... | |
\begin{align*}\rho(t,\vec{x})=\rho(t,r)=\frac{f\left( \frac{r}{a(t)}\right) }{a(t)^{2}},{\normalsize u}_{1}{\normalsize =}\frac{\overset{\cdot}{a}(t)}{a(t)}x-\frac{G(t,r)}{r}yu_{2}=\frac{G(t,r)}{r}x+\frac{\overset{\cdot}{a}(t)}{a(t)}y \end{align*} | |
\begin{align*}U^\ast Uf(x)&=\frac{m_2(\tau^{-1}(x))}{m_1(x)}h(\tau^{-1}(x))^2 f(x)\\U U ^*g(y)&=\frac{m_2(y)}{m_1(\tau(y))}h(y)^2 g(y).\end{align*} | |
\begin{align*}T_1=g^{mn}\nabla _m\nabla _n+m^2\approx 0\qquad T_2={e_a}^m \nabla _mM^a-ms\approx 0\,, \end{align*} | |
\begin{align*} \gamma(0, \cdot) = u^{0}|_{\partial \Omega}, \end{align*} | |
\begin{align*}\Lambda_0:=\sup_{B_{g(0)}\left(x_0,\frac{2\rho}{\sqrt{K}}\right)}|Rm|(x,0)<\infty.\end{align*} | |
\begin{align*} W^!_p=\bigcap_{i+2+j=p} V^{\ast \otimes i}\otimes R^{\perp} \otimes V^{\ast \otimes j}.\end{align*} | |
\begin{align*}\psi (\theta^a) = \sum_{i=0,1,..,3,5,..,d} \quad \sum_{\{ a_1< a_2<...<a_i\}\in \{0,1,..,3,5,..,d\} } \alpha_{a_1, a_2,...,a_i}\theta^{a_1} \theta^{a_2} \cdots \theta^{a_i}. \end{align*} | |
\begin{align*} &\mathbb{P} \bigg[ \sum_{m=1}^j G_m \le a + h(j) \, \forall 1 \le j \le n \bigg] \\ = &\mathbb{P} [ G_1 \le - C - 1 - 3 \log 2] \mathbb{P} \bigg[ \sum_{m=1}^j G_m \le a + h(j) \, \forall 1 \le j \le n \, \big \vert \, G_1 \le -C-1-3 \log 2 \bigg]. \end{align*} | |
\begin{align*}p^{\beta_{m+1}}_{1}(m+1,t)&=-\lambda\left(I_t^{\beta_{m+1}}p^{\beta_{m+1}}_{0}(m+1,t)-I_t^{\beta_{m}}p^{\beta_{m}}_{0}(m,t)\right)=0,\\p^{\beta_{m+1}}_{2}(m+1,t)&=-\lambda\left(I_t^{\beta_{m+1}}p^{\beta_{m+1}}_{1}(m+1,t)-I_t^{\beta_{m}}p^{\beta_{m}}_{1}(m,t)\right)=0.\end{align*} | |
\begin{gather*}T_{i}(E_{j,j}(s)) = \begin{cases}E_{i+1,i+1}(s) & j=i, \\E_{i,i}(s) & j=i+1,\\E_{j,j}(s) \ &,\end{cases}i \neq 0.\end{gather*} | |
\begin{align*}X_A := \left\{ x \in ((\mathbb{R}/\mathbb{Z})^k)^\Gamma~:~ (xA^*)_g = \mathbb{Z}^k \mbox{ for every } g \in \Gamma\right\}\end{align*} | |
\begin{align*}P_G(K) = P_{G/N}(K/N) \geq \mathrm{tp}(D_{n/k}) = \left(\frac{1}{2}\right)^{\frac{\frac{n}{k}-i}{2}},i = \left.\begin{cases} 1, & n/k \equiv 1\,(2)\\2,& n/k \equiv 0\,(2)\end{cases}\right\}.\end{align*} | |
\begin{align*}\begin{aligned}\mathcal H(k,d,l)= &\left\{F\in\binom{[n]}{k}:[d-1]\subseteq F,\, F\cap[d,d+l-1]\neq\emptyset\right\}\\&\cup\left\{F\in\binom{[n]}{k}:|[d-1]\cap F|=d-2,\, [d,d+l-1]\subseteq F\right\}.\end{aligned}\end{align*} | |
\begin{align*}\left[ D_I{}^J,D_K{}^L \right] = \delta_K{}^J D_I{}^L - \delta_I{}^LD_K{}^J\end{align*} | |
\begin{align*}(\delta f)^{-1}(0)= D_M:=\{(u,u): u\in M\}.\end{align*} | |
\begin{align*}\int_{a\exp\{-\int_0^tI(y_2,s)ds\}}^{b\exp\{-\int_0^tI(y_2,s)ds\}}\frac{1}{((x_2-y_2)^2+y_1^2)^2}dy_1\sim_{a,b}\frac{e^{-G}}{((x_2-y_2)^2+e^{-2G})^2}.\end{align*} | |
\begin{align*}U = \bigcup_{j=0}^{p^n-1} \xi^j (1+px_j)R_n^* \mbox{and} V = \bigcup_{j=0}^{p^n-1} (1+px_j)R_n^*.\end{align*} | |
\begin{align*}M_{ir}^{2} = \frac{1}{2} \hbar n ~~,~~ n = 1,2,3,...\end{align*} | |
\begin{align*}I \coloneqq \langle \varepsilon_1^{c_1},\ \varepsilon_2^{c_2},\ \varepsilon_2^{b/g}\alpha_k - \alpha_k \varepsilon_1^{c/g}\mid k=1,2,\dots, g \rangle.\end{align*} | |
\begin{align*}\square_\gg \Phi^A + {}^{(h)} \Gamma_{BC}^A\circ\Phi\, \partial_\mu \Phi^B \partial_\nu \Phi^C \gg^{\mu\nu} = 0\end{align*} | |
\begin{align*} \sum_{i=1}^k\|xb_i-b_i x\|_2\geq C \|x-E_A(x)\|_2.\end{align*} | |
\begin{align*}{\rm Prob}(a\leq x_1<b,x\in S^2) &\ = \ \frac{\int_a^b\frac{1}{\sqrt{1-x_1^2}}S_{1}(\sqrt{1-x_1^2}) dx_1}{S_2(1)}\\&\ = \ \frac{\int_a^b2\pi dx_1}{4\pi }\\&\ = \ \frac{b-a}{2},\end{align*} | |
\begin{align*} \sum_{k=0}^n \binom{n}{k}^2 \binom{k}{c}\binom{n-k}{d} = \binom{2n-c-d}{n}\binom{n}{c}\binom{n}{d}\end{align*} | |
\begin{align*}& B \dot{u}(t)+ A u(t) =f(t) \ae, \\& u(0)=u_0. \end{align*} | |
\begin{align*}w^{(a)}=w^{(a)}_{\rm flat}\equiv-\sum_A \left(\frac{1}{2r_A}\epsilon^{abc}dr_A^b\wedge dr_A^c+dr_A^a\wedge (d\psi_A+\cos\theta_A d\phi_A)\right),\end{align*} | |
\begin{align*}\min\,\,& \varphi_0(x(0),x(T)),\\&\dot{x}=F(x,u,v),{\rm a.e.}\ {\rm on}\ [0,T],\\& \eta_j(x(0),x(T))=0,\mathrm{for}\j=1\hdots,d_{\eta},\\& \varphi_i(x(0),x(T))\leq 0,\mathrm{for}\i=1,\hdots,d_{\varphi},\\& u(t)\in U ,\,\, v(t)\in V,{\rm a.e.}\ {\rm on}\ [0,T],\end{align*} | |
\begin{align*}\sum_{\mathbf{t}\in\{0,\dots,9\}^k}\prod_{i=1}^k G(t_i,\dots,t_{i+4})\ll 10^{27k/77}.\end{align*} | |
\begin{align*}\mathsf{G}=\left[\begin{array}{lll}a\; -\bar b\;\; p\\b\; \;\;\;\;\bar a\;\; q\\0\; \;\;\;\;0\;\; 1\end{array}\right].\end{align*} | |
\begin{align*}I_2^{(5)}=&r^{2-n}\Big[\frac{(n-6)(n-4)}{12(n-1)}|W(p)|^2r^2-\frac{32}{9(n-2)}\sum_{k,l,s}\big((W_{ikls}(p)+W_{ilks}(p))x^i\big)^2 \\& +\frac{4}{3(n-2)(n-1)}|W(p)|^2r^2-\frac{16(7n-8)}{n-2}\sigma_1(A)_{,ij}(p)x^ix^j\Big]+O(r^{5-n}).\end{align*} | |
\begin{align*}S_{N,r+1}^{(4,b)}&=\sum_{k=0}^{r+1}N_{4k+b}=S_{N,r}^{(4,b)}+N_{4(r+1)+b} \\&=(5S_{N,r-1}^{(4,b)}-2S_{N,r-2}^{(4,b)}+S_{N,r-3}^{(4,b)}+1)+(5N_{4r+b}-2N_{4(r-1)+b}+N_{4(r-2)+b}) \\&=5S_{N,r}^{(4,b)}-2S_{N,r-1}^{(4,b)}+S_{N,r-2}^{(4,b)}+1,\end{align*} | |
\begin{align*}=\left\langle GA(s),t\right\rangle =0,\end{align*} | |
\begin{align*}\left[ T_{i}^{\alpha },T_{j}^{\alpha }\right] =i\epsilon _{ijk}T_{k}^{\alpha }\, .\end{align*} | |
\begin{align*}\iint F(u,u_*) = \iint F(u(x),u(x_*)) \d x_* \d x.\end{align*} | |
\begin{align*}H_t(x,y) = \sum_{m=1}^d g_m \sum_{\ell=0}^m {m \choose \ell}^2 x^{2 \ell} y^{2m-2 \ell} - t \log x - t \log y. \end{align*} | |
\begin{align*}1 = R(x)R(x) = 2R(R(x)) - R(x^2) = 2 - R(x^2),\end{align*} | |
\begin{align*}K_{m+\frac 12}(kx_0)-i\frac{\hat k}{k}K_{m-\frac12}(kx_0)\end{align*} | |
\begin{align*}f(\tau ,x)=\left( e^{(v-\tau )s\Delta _{K}/2}\phi \right) \left( x\right)=\left\langle h,\pi (x)e^{(v-\tau )s\pi \left( \Delta _{K}\right)/2}w\right\rangle , \end{align*} | |
\begin{align*} \nabla k ( \nabla h (x) ) = x/ h(x) \mbox{ while } k ( \nabla h ) = 1.\end{align*} | |
\begin{align*}\aligned0&\,=\,-\,\tfrac{1}{10}\,F_{5,0}+\tfrac{1}{120}\,F_{6,0}+\tfrac{1}{36}\,F_{5,1},\\0&\,=\,-\tfrac{7}{120}\,F_{5,0}+\tfrac{1}{120}\,F_{5,1}+\tfrac{2}{27},\endaligned\end{align*} | |
\begin{align*} \beta_n=\sum\limits_{l=0}^{n-1}\omega^l\wedge\omega^{n+l},\beta_n^n=\wedge^n\beta_n=n!~\Omega_{2n},\end{align*} | |
\begin{align*} |x|^\frac{2p}{1-q} &= |x|^\frac{2(p-q+2)}{1-q} \eta^{-\frac{4}{1-q}} |\xi|^{-2\gamma} \cdot |x|^\frac{2(q-2)}{1-q} \eta^\frac{4}{1-q} |\xi|^{2\gamma} \\ &= \eta^\frac{2(p-q)}{1-q} |\xi|^{\frac{2(p-q+2)}{1-q}-2\gamma} \cdot |x|^\frac{2(q-2)}{1-q} \eta^\frac{4}{1-q} |\xi|^{2\gamma}, \end{align*} | |
\begin{align*}S_i={{w}_i} \rho c_v\frac{\Delta t}{2} \partial_t^2T,\end{align*} | |
\begin{align*}\aligned 1/p^*-1/p&=1/p_1=(n-k p)/np,\\ s_2&>\gamma_2,\\ s_2&=r+1,\\ kp&<n.\endaligned\end{align*} | |
\begin{align*}m<0 \; \;\;\;\psi(x)= \sum^{\infty}_{n=1} \int^{+\infty}_{-\infty} d p \; a_{np} \psi^{(+)}_{np} (x) + \sum^{\infty}_{n=0} \int^{+\infty}_{-\infty} d p \; b^{\dag}_{np} \psi^{(-)}_{np} \; ,\end{align*} | |
\begin{align*}\partial_\theta D(p||p_{\theta,\varphi}) &= -\kappa( \cos(\theta)\cos(\varphi) \mathbb{E}_p[x_1] + \cos(\theta) \sin(\varphi) \mathbb{E}_p[x_2] - \sin(\theta)\mathbb{E}_p[x_3]),\\\partial_\varphi D(p||p_{\theta,\varphi}) &= -\kappa( -\sin(\theta)\sin(\varphi) \mathbb{E}_p[x_1] + \sin(\theta)\cos(\varphi) ... | |
\begin{align*}Z= \frac{1}{2} \left ( (\alpha+\beta)X +2Y +2\alpha\beta \pm(\alpha -\beta) \sqrt{X^2 -4Y} \right )\end{align*} | |
\begin{align*} \frac{\nu_{N,t}(\eta^{x-e_i,x})}{\nu_{N,t}(\eta)}=\frac{\theta_{x}}{\theta_{x-e_i}}\frac{\eta_{x-e_i}(K-\eta_x)}{(K-\eta_{x-e_i}+1)(\eta_x+1)},\end{align*} | |
\begin{align*}u(x, t) = \inf_{a \in B_{V t}}\left\{ u_0(a)+t L\left({d(a, x) \over t}\right) \right\} \end{align*} | |
\begin{align*}\Lambda_{t}=\left\{ \left( A\left( t-t_{0}\right) ^{\beta},A\beta\left(t-t_{0}\right) ^{\beta-1}\right) \in\mathbb{R}^{2}:t_{0}\in\left[0,t\right] \right\} .\end{align*} | |
\begin{align*}V_{\pm}=a(a\pm \frac{p}{\sqrt{2}})f^2_2+b(b\mp\frac{q}{\sqrt{2}})f^{-2}_2 + 2a b\pm\frac{1}{\sqrt{2}}(a q-p b),\end{align*} | |
\begin{align*}\iint (x-K)^+ \; q(ds,dx) = C(K) \qquad\forall K\end{align*} | |
\begin{align*}\frac 25 \sum_x \pi(x)(\mathbb{P}_x[T_A \le t])^{2} & \le \sum_x \pi(x)(\mathbb{P}_x[T_A \le \tau])^{2} \le \sum_x \pi(x)\left(\sum_{a \in A} \mathbb{P}_x[T_a \le \tau]\right)^{2} \\ & \le k^{2}\sum_x \pi(x) \mathbb{(P}_x[T_o \le \tau])^{2},\end{align*} | |
\begin{align*}\frac{d^{2}\xi^{0}}{dt^{2}}=-\frac{1}{2}h_{ab,00}\frac{d\xi^{a}}{dt}S^{zb},\end{align*} | |
\begin{align*}V^{+,+} &= \left\{ (A^+,\underline{A}^+) :\ A\in \binom{[n]}{k} \right\},\\V^{+,-} &= \left\{ (A^+,\underline{A}^-) :\ A\in \binom{[n]}{k} \right\},\\V^{-,+} &= \left\{ (A^-,\underline{A}^+) :\ A\in \binom{[n]}{k} \right\},\\V^{-,-} &= \left\{ (A^-,\underline{A}^-) :\ A\in \binom{[n]}{k} \right\}.\end{ali... | |
\begin{align*}u_t&=\Delta u -\chi\nabla v\cdot\nabla u +u(a(x,t)-u(b(x,t)-\chi\mu)-\chi\lambda v)\\&\ge \Delta u -\chi\nabla v\cdot\nabla u +u(a_{\inf}-\|u(\cdot,t;t_0,u_0)\|_\infty(b_{\sup}-\chi\mu)-\chi\lambda \frac{\mu}{\lambda}\|u(\cdot,t;t_0,u_0)\|_\infty)\\&=\Delta u -\chi\nabla v\cdot\nabla u +u(a_{\inf}-\|u(\cd... | |
\begin{align*}p^u_i&:=\sup\{t>0: Q_\epsilon(f^{i-n}(x))\geq e^{-{\epsilon}n}t\textrm{ for all }n\geq 0\},\\p^s_i&:=\sup\{t>0: Q_\epsilon(f^{i+n}(x))\geq e^{-{\epsilon}n}t \textrm{ for all }n\geq 0\}.\end{align*} | |
\begin{align*}\overline{\textrm{OPT}}_{\eta}(T) \leq 3 + 3 \times T \times \bigg( \sum_{k = t_{\eta}(T)}^T P(\tau = k) \times c^2_k \bigg)^{\frac{1}{2}} + 15 \times T^2 \times \sum_{k = t_{\eta}(T)}^T P(\tau = k) \times c^2_k.\end{align*} | |
\begin{align*}\sum_{g \in G} p^{*n}(g) \frac{dg\nu}{d\nu}(y) = 1, \textrm{for $\nu$-a.e. $y$}.\end{align*} | |
\begin{align*} A_S=(1-\delta)I+\delta S,\end{align*} | |
\begin{align*}w_\alpha(0)=1\quad\lim_{\alpha\to\infty}\left( \left\vert\frac{\nabla w_\alpha(0)}{w_\alpha(0)}\right\vert^n+\left\vert\frac{\nabla^2 w_\alpha(0)}{w_\alpha(0)} \right\vert^{\frac{n}{2}}\right) \leq 1\end{align*} | |
\begin{align*} x_Q:=\frac{1}{\sqrt{|Q|}}\sum_{j\in Q}x_j\,. \end{align*} | |
\begin{align*}\nabla X=dX+i\left[{\cal A},X\right]\,.\end{align*} | |
\begin{align*}0\in S(0)~~\operatorname{and}~~S(\lambda x)=\lambda S(x)~~\operatorname{for~all}~\lambda>0~\operatorname{and}~x \in X\end{align*} | |
\begin{align*} m_{0}&=0 \\ m_{2k} &= m_{2k-1}-1 \\ m_{2k+1} &= m_{2k}-2(\alpha_{2k}-\alpha_{2k+1})+1. \end{align*} | |
\begin{align*}S^T J S = J\end{align*} | |
\begin{align*}\sum_{\ell=0}^{M-1}c_{\ell}w^{\ell}\leq\delta w^{M}+C_{\delta}~\forall~w\geq0.\end{align*} | |
\begin{align*}i)\ X|_{x_{0}} \not=0,ii)\ g(X|_{x_{0}},\nabla d(x_{0}))= 0,\end{align*} | |
\begin{align*} \int_a^b e^{i\lambda \phi(x)}\psi(x)dx = \int_a^b e^{i\lambda \phi(x)} (D^t)^N(\psi)(x) dx.\end{align*} | |
\begin{align*}K^*=\begin{bmatrix}-0.3575 & 0.0228 & 0.1682 & -0.1102 & -0.1300\\0.2188 & 0.6072 & 0.6793 & -0.7718 & -0.3158\end{bmatrix},\end{align*} | |
\begin{align*} {{\bf{\bar x}}_{\rm t}}\left[m, n \right] {\rm =} \frac{1}{\sqrt K }\sum\limits_{k = 0}^{K-1} {{{\bf{u}}_k}s\left[m, k \right]{e^{j\frac{{2\pi }}{K}kn}}},\ n {\rm =} - {{N}_{\rm CP }}. \cdots ,K - 1. \end{align*} | |
\begin{align*}\|(w_n)_+\|_{H^1(\Omega)}^2=\|(w_n)_+\|_{L^2(\Omega)}^2+\|\nabla (w_n)_+\|_{L^2(\Omega)}^2\le\|w_n\|_{L^2(\Omega)}^2+\|\nabla w_n\|_{L^2(\Omega)}^2=\|w_n\|_{H^1(\Omega)}^2.\end{align*} | |
\begin{align*}&\mathbf{q}_p^{\mathrm{H}}(\mathbf{H}_{\mathrm{e}}^{\mathrm{H}}\mathbf{H}_{\mathrm{e}})^{-1}\mathbf{q}_p\leq\lambda_p^{-1}\cos^2\theta_{p,p}+\sum_{i\neq p}\lambda_i^{-1}\sin^2\theta_{p,p}\\&=\lambda_p^{-1}+\{\mathrm{Tr}[(\mathbf{H}_{\mathrm{e}}^{\mathrm{H}}\mathbf{H}_{\mathrm{e}})^{-1}]-2\lambda_p^{-1}\}\... | |
\begin{align*}\sum_{k=0}^{n}\bigg[\genfrac{}{}{0pt}{}{n}{k}\bigg]_qX(k)=Y(n),\end{align*} | |
\begin{align*}\Delta&=\{\alpha_1= e_1-e_2, \ldots, \alpha_{n-1}= e_{n-1} -e_n, \alpha_n= e_{n-1}+e_n\}, \\\Delta^\vee &=\{ \alpha_1^{\vee}= e_1^*-e_2^*, \ldots, \alpha_{n-1}^\vee= e_{n-1}^* -e_n^*, \alpha_n^\vee= e_{n-1}^* + e_n^* - e_0^*\},\end{align*} | |
\begin{align*} [e_A, z]_{\underset{K}{\frown}}=-b_K(z).\end{align*} | |
\begin{align*}(B(u)-B(v) ) u = B(u)u-B(v) v - B(v)(u-v).\end{align*} | |
\begin{align*}y^q+y=\frac{x^q}{x^{q-1}+1}\end{align*} | |
\begin{align*}\mathcal{I}_{\nu}(t) &= \left(\frac{t}{2}\right)^{\nu}\sum_{k=0}^{\infty}\frac{(t^2/4)^{k}}{k! \Gamma(\nu + k +1)},\\K_{\nu}(t) &= \frac{\pi}{2}\frac{\mathcal{I}_{-\nu}(t) - \mathcal{I}_{\nu}(t)}{\sin(\pi \nu)}, \nu \notin \mathbb{Z},\end{align*} |
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