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\[\begin{array}{r l}{\mathrm{d}\chi^{2}}&{=\frac{1}{1-k r^{2}}\mathrm{d}r^{2}}\\ {\hfill}\\ {r^{2}}&{\left(\mathrm{d}\theta^{2}+\sin^{2}(\theta)\mathrm{d}\phi^{2}\right)=S_{k}^{2}(\chi)\left(\mathrm{d}\theta^{2}+\sin^{2}(\theta)\mathrm{d}\phi^{2}\right)}\end{array}\] |
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\[\begin{array}{r l}{\sum_{K\in\mathcal{T}_{h}}(\pmb{\theta}\cdot\pmb{\nu},P_{h}^{n+1})_{\partial K}-\sum_{K\in\mathcal{T}_{h}}(\widehat{\pmb{\theta}}\cdot\pmb{\nu},P_{h}^{n+1})_{\partial K}=}&{\sum_{K\in\mathcal{T}_{h}}(\widehat{P}_{h}^{n+1},\pmb{\nu}\cdot\pmb{\theta})_{\partial K},}\\ {-\sum_{K\in\mathcal{T}_{h}}(\rho,\pmb{\nu}\cdot\pmb{Q}_{h}^{n+1})_{\partial K}+\sum_{K\in\mathcal{T}_{h}}(\widehat{\rho},\pmb{\nu}\cdot\pmb{Q}_{h}^{n+1})_{\partial K}=}&{-\sum_{K\in\mathcal{T}_{h}}(\widehat{\pmb{Q}}_{h}^{n+1}\cdot\pmb{\nu},\rho)_{\partial K}.}\end{array}\] |
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\[\begin{array}{r l}&{I_{1}:=2\sum_{t=\eta_{k}+1}^{\eta_{k}+\widetilde{r}}\vert\vert f_{(s_{k},\eta_{k}]}\ast\mathcal{K}_{{h_{2}}}-f_{(s_{k},\eta_{k}]}+f_{(\eta_{k},e_{k}]}\ast\mathcal{K}_{h_{2}}-f_{(\eta_{k},e_{k}]}\vert\vert_{L_{2}}^{2},\ \mathrm{and,}}\\ &{I_{2}:=2\sum_{t=\eta_{k}+1}^{\eta_{k}+\widetilde{r}}\langle f_{(s_{k},\eta_{k}]}\ast\mathcal{K}_{{h_{2}}}-f_{(\eta_{k},e_{k}]}\ast\mathcal{K}_{h_{2}},F_{t,{h_{2}}}-f_{(\eta_{k},e_{k}]}\ast\mathcal{K}_{h_{2}}\rangle_{L_{2}}.}\end{array}\] |
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\[\begin{array}{r l}{R}&{=\mathbb{E}[(\log_{2}(1+\gamma_{s}))],}\\ &{=\frac{1}{\Gamma(k)\mathrm{{ln}}(2)}H_{3,2}^{1,3}\bigg[p\bigg|\begin{array}{l l l}{(1,1),}&{(1,1),}&{(-k+1,1)}\\ {(1,1),}&{(0,1)}\end{array}\bigg],}\end{array}\] |
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\[\begin{array}{r l}{\frac{\delta}{\delta u}\frac{f_{\bar{Q}}\{F_{\bar{Q}}^{-1}(u)\}}{\phi\{\Phi^{-1}(u)\}}}&{=\frac{f_{\bar{Q}}^{\prime}\{F_{\bar{Q}}^{-1}(u)\}}{f_{\bar{Q}}\{F_{\bar{Q}}^{-1}(u)\}\phi\{\Phi^{-1}(u)\}}-\frac{\Phi^{-1}(u)f_{\bar{Q}}\{F_{\bar{Q}}^{-1}(u)\}}{\phi\{\Phi^{-1}(u)\}^{2}}.}\end{array}\] |
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\[\begin{array}{r l r}&{}&{\mathrm{tr}\left[v_{x}A(\omega;\mathbf{k})v_{x}A(\omega-\Omega;\mathbf{k})\right]}\\ &{}&{=\frac{v_{F}^{2}}{96}\Bigg\{96\left(\hbar\omega-g\right)\left(\hbar\omega-g-\hbar\Omega\right)-(\hbar v_{F}\tilde{k})^{2}+\hbar v_{F}\tilde{k}}\\ &{}&{\times\left[4\sqrt{6}\left(2\hbar\omega-2g-\hbar\Omega\right)\cos{(\varphi)}+5\hbar v_{F}k\cos{(2\varphi)}\right]\Bigg\}}\\ &{}&{\times\sum_{\eta_{1},\eta_{2}=\pm}\frac{\delta(\hbar\omega-\epsilon_{\eta_{1}})}{\epsilon_{\eta_{1}}-\epsilon_{-\eta_{1}}}\frac{\delta(\hbar\omega-\hbar\Omega+\epsilon_{\eta_{2}})}{\epsilon_{\eta_{2}}-\epsilon_{-\eta_{2}}},}\end{array}\] |
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\[\displaystyle\int_{\Omega}\nabla\delta^{n+1}\cdot\nabla z\,\mathrm{dV}=\displaystyle\frac{1}{\Delta t}\int_{\Omega}\mathbf{\widetilde{W}}^{n+1}\cdot\nabla z\,\mathrm{dV}-\displaystyle\frac{1}{\Delta t}\int_{\partial\Omega}G^{n+1}z\,\mathrm{dS}\quad\forall z\in V_{0},\] |
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\[\begin{array}{c c c c c}{\phi_{g}}&{:}&{\ell^{\infty}(\Delta_{d-1})}&{\to}&{\ell^{\infty}(S_{d})}\\ &&{x}&{\mapsto}&{w^{(O_{g})}(t^{(1)},\dots,t^{(G)})x(\textbf{0},t^{(i_{g,1})},\dots,t^{(i_{g,d_{g}})},\textbf{0}),}\end{array}\] |
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\[T_{\mu}^{\nu}=-\delta_{\mu}^{\nu}{\mathcal{L}}+\delta_{\mu}^{\sigma}\partial_{\sigma}\varphi{\frac{\partial{\mathcal{L}}}{\partial\varphi_{,\nu}}}=\left({\frac{\partial{\mathcal{L}}}{\partial\varphi_{,\nu}}}\right)\cdot\varphi_{,\mu}-\delta_{\mu}^{\nu}{\mathcal{L}}\] |
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\[\begin{array}{r l r}&{}&{M(u)=M(u^{(0)})+M^{\prime}(u^{(0)})u^{(1)}\varepsilon+\left(M^{\prime}(u^{(0)})u^{(2)}+\frac{1}{2}M^{\prime\prime}\left(u^{(0)}\right)\left(u^{(1)}\right)^{2}\right)\varepsilon^{2}+\hdots,}\\ &{}&{g(u)=g(u^{(0)})+g^{\prime}(u^{(0)})u^{(1)}\varepsilon+\left(g^{\prime}(u^{(0)})u^{(2)}+\frac{1}{2}g^{\prime\prime}(u^{(0)})\left(u^{(1)}\right)^{2}\right)\varepsilon^{2}+\hdots,}\\ &{}&{q^{\prime}(u)=q^{\prime}(u^{(0)})+q^{\prime\prime}(u^{(0)})u^{(1)}\varepsilon+\left(q^{\prime\prime}(u^{(0)})u^{(2)}+\frac{1}{2}q^{(3)}(u^{(0)})\left(u^{(1)}\right)^{2}\right)\varepsilon^{2}+\hdots,}\end{array}\] |
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\[\begin{array}{r l}&{\|T_{j}f\|_{L^{p}(\mathbb{R}^{n}\setminus Q_{0}^{*})}=\Big\|\int_{Q}K_{j}(x,y)\,f(y)\,\mathrm{d}y\Big\|_{L_{x}^{p}(\mathbb{R}^{n}\setminus Q_{0}^{*})}}\\ &{\leq\Big\|g(x,c_{Q})^{N_{1}}\int_{Q}K_{j}(x,y)\,f(y)\,\mathrm{d}y\Big\|_{L_{x}^{2}(\mathbb{R}^{n}\setminus Q_{0}^{*})}\,\Big\|\frac{1}{g(x,c_{Q})^{N_{1}}}\Big\|_{L^{p^{\prime}}(\mathbb{R}^{n})}}\\ &{\lesssim\int_{Q}\big\|g(x,c_{Q})^{N_{1}}\,K_{j}(x,y)\,f(y)\big\|_{L_{x}^{2}(\mathbb{R}^{n}\setminus Q_{0}^{*})}\,\mathrm{d}y}\\ &{\lesssim\int_{Q}\vert f(y)\vert\,\big\|g(x,y)^{N_{1}}\,K_{j}(x,y)\big\|_{L_{x}^{2}(\mathbb{R}^{n}\setminus Q_{0}^{*})}\!\,\mathrm{d}y}\\ &{\lesssim2^{-k_{Q}(n-n/p)}\,2^{j(n/2+m^{\prime})}=2^{-k_{Q}(n-n/p)}\,2^{j(n-n/p)},}\end{array}\] |
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\[\begin{array}{r l}{r V}&{=r\left(-\frac{m_{1}}{|x_{1}+\frac{x_{0}}{2}|}-\frac{m_{1}}{|x_{1}-\frac{x_{0}}{2}|}-\frac{1}{|x_{0}|}\right)}\\ &{=-\sum_{i=0,1}\frac{m_{1}}{\left(\frac{\sin^{2}\psi}{M_{1}}+\frac{\cos^{2}\psi}{4M_{0}}+(-1)^{i}\frac{\sin\psi\cos\psi}{(M_{1}M_{0})^{1/2}}\cos(\theta_{0}-\theta_{1})\right)^{1/2}}-\frac{M_{0}^{1/2}}{\cos\psi}}\end{array}\] |
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\[\begin{array}{r l}{\lefteqn{\big(\int_{\Omega}|x-y|^{2}d\pi\big)^{\frac{1}{2}}\stackrel{()}{\le}\|(f_{2},f_{3},f_{4},f_{5})\|}}\\ &{\le\|(|x-y|,|z-y|,|x-w|,|z-w|)\|+\sqrt{2}\|(0,\tilde{f}_{3},\tilde{f}_{4},\tilde{f}_{5})\|,}\end{array}\] |
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\[\begin{array}{r l}&{H=-\partial_{t}\;\;,\;\;P_{i}=\partial_{i}\;\;,\;\;J_{i j}=x_{i}\partial_{j}-x_{j}\partial_{i}}\\ &{B_{i}=t\partial_{i}\;\;\;D=-(t\partial_{t}+x^{i}\partial_{i})\;\;,\;\;K_{i}=t^{2}\partial_{i}}\\ &{K=-(t^{2}\partial_{t}+2x_{i}t\partial_{i})}\end{array}\] |
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\[\begin{array}{r l}&{K^{(n)}(z_{1},z_{2})=\textstyle\sum_{k=1}^{n}\mathcal{R}Q_{[1,n]}^{-1}A_{k-1}\hat{G}_{0,n}^{(k)}A_{n}^{-1}\mathcal{R}^{-1}(z_{1},z_{2})}\\ &{\quad\textstyle=\sum_{k=1}^{n}\mathcal{R}Q_{[1,n]}^{-1}A_{k-1}Q_{[1,k)}\chi_{y_{k}}\bar{Q}_{[k,n]}^{+}A_{n}^{-1}\mathcal{R}^{-1}(z_{1},z_{2})}\\ &{\qquad\textstyle-\sum_{k=1}^{n}\sum_{\eta^{\prime}\in\mathbb{Z}}\mathcal{R}Q_{[1,n]}^{-1}A_{k-1}Q_{[1,k)}\chi_{y_{k}}Q_{k}^{+}(z_{1},\eta^{\prime})\mathbb{E}_{B_{k}^{+}=\eta^{\prime}}\bigl[\bar{Q}_{({\tau^{+}},n]}^{+}A_{n}^{-1}\mathcal{R}^{-1}(B_{{\tau^{+}}}^{+},z_{2})\mathbf{1}_{{\tau^{+}}<n}\bigr].}\end{array}\] |
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\[\begin{array}{r l}{\|z_{k+1}-y_{k}^{*}\|^{2}}&{=\|z_{k}-y_{k}^{*}\|^{2}+\gamma_{k}^{2}\|\tilde{h}_{z}^{k}\|^{2}-2\gamma_{k}\langle\tilde{h}_{z}^{k},z_{k}-y_{k}^{*}\rangle}\\ &{\le\|z_{k}-y_{k}^{*}\|^{2}+2\gamma_{k}^{2}(\|q_{k}^{z}\|^{2}+\|\tilde{e}_{k}^{z}\|^{2})-2\gamma_{k}\langle q_{k}^{z},z_{k}-y_{k}^{*}\rangle-2\gamma_{k}\langle\tilde{e}_{k}^{z},z_{k}-y_{k}^{*}\rangle.}\end{array}\] |
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\[\begin{array}{r l}&{\frac{1}{2}\frac{d}{d t}\big(\|\partial_{t}^{k}\phi\|^{2}+(\gamma\lambda+)\|\nabla\partial_{t}^{k}\phi\|^{2}+\gamma\lambda\|\Delta\partial_{t}^{k}\phi\|^{2}\big)}\\ &{+\gamma\lambda\|\nabla\partial_{t}^{k}\phi\|^{2}+\|\partial_{t}^{k}\phi_{t}\|^{2}+\gamma\lambda\|\Delta\partial_{t}^{k}\phi\|^{2}+\|\nabla\partial_{t}^{k}\phi_{t}\|^{2}}\\ {=}&{\langle\partial_{t}^{k}K_{\phi},\Delta\partial_{t}^{k}\phi-\partial_{t}^{k}\phi-\partial_{t}^{k}\phi_{t}\rangle-\langle\nabla\partial_{t}^{k}K_{\phi},\nabla\partial_{t}^{k}\phi_{t}\rangle\,,}\end{array}\] |
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\[\begin{array}{r l}&{(t_{\gamma_{i+1}^{l^{\prime}+1}}t_{\gamma_{i+2}^{l^{\prime}+2}}t_{\gamma_{i+1}^{l^{\prime}+2}}\cdots t_{\gamma_{i+2}^{l^{\prime}+k-1}}t_{\gamma_{i+1}^{l^{\prime}+k-1}})t_{\gamma_{i+2}^{l^{\prime}+k}}t_{\gamma_{i+1}^{l^{\prime}+k}}(t_{\gamma_{i+1}^{l^{\prime}+1}}t_{\gamma_{i+2}^{l^{\prime}+2}}t_{\gamma_{i+1}^{l^{\prime}+2}}\cdots t_{\gamma_{i+2}^{l^{\prime}+k-1}}t_{\gamma_{i+1}^{l^{\prime}+k-1}})^{-1}}\\ &{=t_{\gamma_{i+1}^{l^{\prime}}}t_{\gamma_{i+2}^{l^{\prime}+1}}}\end{array}\] |
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\[\begin{array}{r l}{r_{k}}&{=A^{T}A\left[\begin{array}{l l l l}{r_{0}}&{r_{1}}&{\cdots}&{r_{k-1}}\end{array}\right]\left[\begin{array}{l}{y_{0}}\\ {\vdots}\\ {y_{k-1}}\end{array}\right]-A^{T}b}\\ &{=A^{T}A\left(y_{0}r_{0}+\cdots+y_{k-1}r_{k-1}\right)+r_{0}}\\ &{=r_{0}+y_{0}A^{T}A r_{0}+\cdots+y_{k-1}A^{T}A r_{k-1}.}\end{array}\] |
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\[\mathfrak{s}_{G}^{1}(i_{1})_{i l}:=\left\{\begin{array}{l l}{(\emptyset,\{\delta_{i i_{1}}\})}&{\mathrm{if~}l=0,}\\ {(\mathfrak{s}_{G}^{1}(i_{1})_{i,l-1},\{\mathfrak{s}_{G}^{1}(i_{1})_{i^{\prime},l-1}:i^{\prime}\in N_{G}(i)\}^{\#})}&{\mathrm{otherwise}}\end{array}\right.\] |
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\[\begin{array}{r l}{\alpha\,h_{\alpha,\theta}(x\vert\lambda)}&{=x\int_{0}^{\infty}f_{\alpha,\theta}(x\vert t)\,t^{-1}e^{-\lambda t}\,d t}\\ &{=\frac{\Gamma(\theta+1)}{\Gamma(\theta/\alpha+1)}\,x^{1-\theta}\int_{0}^{\infty}f_{\alpha}(x\vert t)\,t^{\theta/\alpha-1}\,e^{-\lambda t}\,d t}\\ {u=x^{-\alpha}t:\quad h_{\alpha,\theta}(x\vert\lambda)}&{=\int_{0}^{\infty}e^{-\lambda x^{\alpha}u}\,d P_{\alpha,\theta}(u)}\\ {\mathrm{where}\quad P_{\alpha,\theta}(t)}&{=\frac{\Gamma(\theta+1)}{\Gamma(\theta/\alpha+1)}\,\frac{1}{\alpha}\int_{0}^{t}f_{\alpha}(u^{-1/\alpha})\,u^{(\theta-1)/\alpha-1}\,d u}\\ {\mathrm{or}\quad d P_{\alpha,\theta}(t)}&{=\frac{\Gamma(\theta+1)}{\Gamma(\theta/\alpha+1)}\,t^{\theta/\alpha}\,d P_{\alpha}(t)}\end{array}\] |
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\[\begin{array}{r l}{\frac{d}{d t}\left(D G_{\delta}(\bar{i}(\omega t))[{A}(t)]\right)}&{=D^{2}G_{\delta}(\bar{i}(\omega t))[(\omega,0,0),A(t)]+D G_{\delta}(\bar{i}(\omega t))[\dot{A}(t)]=\dot{I}(t)=d_{i}X_{H_{\zeta}}(i_{0}(\omega t))[I(t)]}\\ &{=d_{i}X_{H_{\zeta}}(i_{0}(\omega t))\circ D G_{\delta}(\bar{i}(\omega t))[A(t)],}\end{array}\] |
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\[\begin{array}{r l}&{\frac{\partial\mathbf{x}_{1}}{\partial G_{1}}={s i g n}(x_{1})\frac{1}{M_{1}k_{1}}(0,\langle\mathbf{x}_{1},\mathbf{p}_{1}\rangle)={s i g n}(x_{1})\frac{1}{M_{1}k_{1}}(0,r^{1/2}(v\sin^{2}\psi+\frac{1}{2}w\sin2\psi)),}\\ &{\frac{\partial^{2}\mathbf{x}_{1}}{\partial G_{1}^{2}}=\frac{1}{M_{1}k_{1}}(-{s i g n}(x_{1}),0)-\frac{|\mathbf{p}_{1}|^{2}}{M_{1}^{2}k_{1}^{2}}\mathbf{x}_{1}=\frac{1}{M_{1}k_{1}}(-{s i g n}(x_{1})-\frac{1}{M_{1}k_{1}}M_{1}^{1/2}\sin\psi(v\sin\psi+w\cos\psi)^{2},0).}\end{array}\] |
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\[\begin{array}{r l}{\sum_{n=0}^{+\infty}\frac{(a)_{n}}{n!(c)_{n}}\mathcal{Z}_{m,n}^{\kappa,\rho}(z,\overline{{z}})\overline{{\mathcal{Z}_{m,n}^{\kappa,\rho}(w,\overline{{w}})}}}&{=R_{m}^{\kappa,\rho}\left({_1F_{1}}\left(\begin{array}{c}{a}\\ {c}\end{array}\bigg|z\overline{{w}}\right)\right)}\end{array}\] |
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\[\begin{array}{r l}{{\cal H}_{\mathrm{3D}}({\bf k})}&{=M({\bf k})\sigma_{0}\otimes\tau_{z}\!+\!\lambda_{z}\sin(k_{z}a_{z})\sigma_{z}\otimes\tau_{x}\!+\!\lambda_{||}\sin(k_{x}a_{||})\sigma_{x}\otimes\tau_{x}\!+\!\lambda_{||}\sin(k_{y}a_{||})\sigma_{y}\otimes\tau_{x},}\\ {M({\bf k})}&{=m_{0}-4t_{z}\sin^{2}\frac{k_{z}a_{z}}{2}-4t_{||}\left(\sin^{2}\frac{k_{x}a_{||}}{2}+\sin^{2}\frac{k_{y}a_{||}}{2}\right)}\\ &{=m_{0}-4t_{z}\sin^{2}\frac{k_{z}a_{z}}{2}-2t_{||}\left[1-\cos(k_{x}a_{||})+1-\cos(k_{y}a_{||})\right]}\\ &{=(m_{0}-2t_{z}-4t_{||})+2t_{z}\cos(k_{z}a_{z})+2t_{||}\left[\cos(k_{x}a_{||})+\cos(k_{y}a_{||})\right].}\end{array}\] |
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\[\begin{array}{r}{\left(1\wedge\frac{\Phi_{1}^{U_{i}}(v_{i})}{1\wedge s^{1/2}}\right)\left(1\wedge\frac{\Phi_{1}^{U_{i}}(w_{i})}{1\wedge r^{1/2}}\right)e^{-\frac{c_{1}}{6}\frac{|v_{i}-w_{i}|^{2}}{r}}\le C_{0,i}\left(1\wedge\frac{\Phi_{1}^{U_{i}}(w_{i})}{1\wedge s^{1/2}}\right)}\end{array}\] |
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\[\begin{array}{r l}{x_{0}^{k+1}}&{=x_{0}^{k}-\tau_{x_{0}}\left(\left(\mathrm{prox}_{\Psi}\left(f(x_{0}^{k},\Theta_{1})\right)-x_{1}^{k}\right)\mathcal{J}_{f}^{x}(x_{0}^{k},\Theta_{1})+\alpha K^{\top}z^{k}\right)\,,}\\ {z^{k+1}}&{=\mathrm{prox}_{\tau_{z}R^{\ast}}\left(z^{k}+\tau_{z}\alpha K\left(2x^{k+1}-x^{k}\right)\right)\,.}\end{array}\] |
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\[\begin{array}{r}{\begin{array}{r}{A_{1}^{*}(z)=\frac{C_{1}\,\sinh(L)}{\cosh(z)(D\operatorname{tanh}(z/D)-\operatorname{tanh}(z))},}\\ {B_{1}^{*}(z)=-\frac{C_{1}\,D\sinh(L/D)}{\cosh(z/D)(D\operatorname{tanh}(z/D)-\operatorname{tanh}(z))},}\end{array}}\end{array}\] |
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\[\begin{array}{r l}{\mu_{S}(A)}&{\geq\frac{n}{k\cdot k^{\prime}}\sum_{i>k:v_{i}\geq k^{\prime}/n}\left(v_{i}-\frac{k^{\prime}}{n}\right)}\\ &{=\frac{n}{k\cdot k^{\prime}}\sum_{i>k+r}\left(v_{i}-\frac{k^{\prime}}{n}\right)}\\ &{=\frac{n}{k\cdot k^{\prime}}\left(k^{\prime}\left(1-\frac{r}{n}\right)-k\cdot\phi_{S}(A)-\frac{k^{\prime}}{n}(k^{\prime}-r)\right)}\\ &{=\frac{n}{k\cdot k^{\prime}}\left(k^{\prime}\left(1-\frac{k^{\prime}}{n}\right)-k\cdot\phi_{S}(A)\right)}\\ &{=\frac{n}{k}\left(1-\frac{k^{\prime}}{n}\right)-\frac{n}{k^{\prime}}\cdot\phi_{S}(A)}\\ &{=1-\frac{n}{k^{\prime}}\cdot\phi_{S}(A)}\\ &{\geq1-2\cdot\phi_{S}(A)}\end{array}\] |
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\[\begin{array}{r l}{T^{\prime}(x;\tau)}&{=\sum_{\sigma\in\mathfrak{S}_{n}}U^{\prime}(x_{\sigma(1)},\dotsc,x_{\sigma(n)};\tau,0)\cdot x_{\sigma(1)}*\cdots*x_{\sigma(n)}\,,}\\ {T(x;\tau)}&{=\sum_{\sigma\in\mathfrak{S}_{n}}U(x_{\sigma(1)},\dotsc,x_{\sigma(n)};\tau,0)\cdot x_{\sigma(1)}*\cdots*x_{\sigma(n)}\,,}\end{array}\] |
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\[\begin{array}{r l}&{\mathbb{E}\left(\frac{1}{n}\sum_{i\in[n]}{\mathbb{I}\left\{{\check{z}_{i}\neq z_{i}^{*}}\right\}}\right)\leq\exp\left(-(1-o(1))\left(J_{n_{1},n_{2},p,q}\wedge J_{n_{2},n_{1},p,q}\right)\right)}\\ {\mathrm{and~}\quad}&{\mathbb{E}\left(\frac{1}{n}\sum_{i\in[n]}{\mathbb{I}\left\{{\check{z}_{i}\neq z_{i}^{*}}\right\}}\right)\geq\exp\left(-(1+o(1))\left(J_{n_{1},n_{2},p,q}\wedge J_{n_{2},n_{1},p,q}\right)\right).}\end{array}\] |
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\[\begin{array}{r}{(\omega_{a b})=\frac{j}{2\vert x\vert^{3}}\left(\begin{array}{l l l}{0}&{x^{3}}&{-x^{2}}\\ {-x^{3}}&{0}&{x^{1}}\\ {x^{2}}&{-x^{1}}&{0}\end{array}\right)=\frac{N-1}{4\vert x\vert^{2}}\left(\sum_{c}\epsilon^{a b c}\frac{x^{c}}{\vert x\vert}\right),}\end{array}\] |
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\[\mathbb{E}_{x}\left[\sum_{v\neq w\in\mathcal{N}_{L^{2}}^{L}}\mathbb{P}_{x_{v}}\left(Z_{t N-L^{2}}^{(v)}>0\right)\mathbb{P}_{x_{w}}\left(Z_{t N-L^{2}}^{(w)}>0\right)\right]=2h(0,x)L^{2}Q_{x}^{2,L^{2}}\left[r(\zeta_{v})h({|v|},\zeta_{v})\prod_{i=1,2}\frac{u(t N-L^{2},\zeta_{v_{i}})}{h(L^{2},{\zeta_{v_{i}}})}\right],\] |
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\[1-\frac{|\boldsymbol{v}_{A}^{\top}\boldsymbol{x}_{2}|}{|\boldsymbol{v}_{A}||\boldsymbol{x}_{2}|}<\varepsilon,\;\left(\boldsymbol{x}_{1}\right)_{V_{A}}^{\top}\boldsymbol{x}_{V_{A}}>0,\;\left(\boldsymbol{x}_{1}\right)_{W_{A}}^{\top}\boldsymbol{x}_{W_{A}}>0.\] |
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\[\left(\begin{array}{l l l l}{a(\theta)a(\theta+\frac{\tau}{2})+b(\theta)-\alpha}&{\beta}&{a(\theta)b(\theta+\frac{\tau}{2})}&{0}\\ {-\beta}&{a(\theta)a(\theta+\frac{\tau}{2})+b(\theta)-\alpha}&{0}&{a(\theta)b(\theta+\frac{\tau}{2})}\\ {a(\theta-\frac{\tau}{2})}&{0}&{b(\theta-\frac{\tau}{2})-\alpha}&{\beta}\\ {0}&{a(\theta-\frac{\tau}{2})}&{-\beta}&{b(\theta-\frac{\tau}{2})-\alpha}\end{array}\right)\] |
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\[v^{t}\cdot\frac{f^{\prime}(x_{i}^{t})\cdot\Big[\sum_{j\in\mathcal{N}^{t},j\ne i}f(x_{j}^{t})\Big]}{\Big[\sum_{j\in\mathcal{N}^{t}}f(x_{j}^{t})\Big]^{2}}\le C^{\prime}\Big(\sum_{t\in\mathcal{T}_{i}}x_{i}^{t}\Big)\mathrm{~with~equality~if~x_i^t~>~0~}.\] |
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\[14,24~\mathrm{{\fontfamily{qpl}\selectfont~v o l u m e-s a r}}~=~(8,0,0)\times(14,24)~\mathrm{{\fontfamily{qpl}\selectfont~s\`{i}l a}\index{s z i l a@s\`{i}l a~(c a p a c i t y~u n i t)}}=1,55,12,0,0~\mathrm{{\fontfamily{qpl}\selectfont~s\`{i}l a}\index{s z i l a@s\`{i}l a~(c a p a c i t y~u n i t)}}\] |
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\[\begin{array}{r l}&{\tau_{2}(u,\lambda,\beta)}\\ &{\!=\lambda\!\int_{0}^{2\pi}\!\!\!\!\int_{0}^{\infty}\!\!\!\!\int_{0}^{\infty}\!\!\!r\,\mathrm{e}^{-\lambda\pi r^{2}-\sigma^{2}w_{u,r,\theta}^{\,\beta}\,z}\,\frac{\rho_{2}(z,r,\theta|u,\lambda,\beta)}{1+z}\,\mathrm{d}z\mathrm{d}r\mathrm{d}\theta,}\end{array}\] |
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\[{\begin{array}{r l}{{\boldsymbol{a}}}&{={\frac{d}{d t}}{\boldsymbol{v}}=\sum_{k=1}^{d}{\dot{v}}_{k}\ {\boldsymbol{e_{k}}}+\sum_{k=1}^{d}v_{k}\ {\dot{\boldsymbol{e_{k}}}}}\\ &{=\sum_{k=1}^{d}\left({\dot{v}}_{k}\ +\sum_{j=1}^{d}\sum_{i=1}^{d}v_{j}{\Gamma^{k}}_{i j}{\dot{q}}_{i}\right){\boldsymbol{e_{k}}}\ .}\end{array}}\] |
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\[\begin{array}{r l}&{d_{\tau}\langle\tilde{b}^{\dagger}b\rangle=-(\gamma_{c}+i\nu)\langle\tilde{b}^{\dagger}b\rangle+N g^{*}\langle\tilde{b}^{\dagger}v^{\dagger}c\rangle,}\\ &{d_{\tau}\langle\tilde{b}^{\dagger}v^{\dagger}c\rangle=-(\gamma+i\nu_{\epsilon})\langle\tilde{b}^{\dagger}v^{\dagger}c\rangle+g\langle\tilde{b}^{\dagger}b\rangle(2\langle c^{\dagger}c\rangle-1).}\end{array}\] |
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\[\operatorname*{min}_{\substack{{x}_{1}\in{\mathcal X}_{1},\left(A,{a}\right)\in{\mathcal A}\left({x}_{1},{\Xi}\right)\,{\mu}\in{\mathcal M}\left(A\right)}}{c}_{1}^{\top}{x}_{1}+{c}_{2}^{\top}\left(A\tilde{\xi}+{a}\right)+{c}_{3,\Xi}^{\top}{\mu}.\] |
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\[\begin{array}{r l}{F\left(R_{1},\ldots,R_{N}\right)\approx}&{\int_{0}^{R_{1}}\ldots\int_{0}^{R_{N}}f_{\left|\boldsymbol{h}\right|}\left(\left|h_{1}\right|,\ldots,\left|h_{N}\right|\right)d\left|h_{1}\right|\cdots d\left|h_{N}\right|}\\ {=}&{\stackrel[s_{1}=0]{s_{0}}{\sum}\sum_{s_{2}=0}^{s_{1}}\ldots\sum_{s_{T}=0}^{s_{T-1}}\frac{g\left(\boldsymbol{s}^{*}\right)}{\pi^{N}\mathrm{{det}\ensuremath{\left(\boldsymbol{J}\right)}}}\prod_{t=1}^{T}\frac{\left(-2K_{m,n}\right)^{s_{t}^{*}}}{s_{t}^{*}!\mathrm{{det}\ensuremath{\left(\boldsymbol{J}\right)}}^{s_{t}^{*}}}\int_{0}^{R_{1}}\ldots\int_{0}^{R_{N}}\times}\\ &{\stackrel[n=1]{N}{\prod}\left|h_{n}\right|\prod_{n=1}^{N}\prod_{m<n}^{N}\left|h_{n}\right|^{s_{n}^{*}}\left|h_{m}\right|^{s_{m}^{*}}\exp\left\{-\frac{\sum_{n=1}^{N}\left|h_{n}\right|^{2}K_{n,n}}{\mathrm{{det}\ensuremath{\left(\boldsymbol{J}\right)}}}\right\}d\left|h_{1}\right|\cdots d\left|h_{N}\right|}\\ {=}&{\stackrel[s_{1}=0]{s_{0}}{\sum}\sum_{s_{2}=0}^{s_{1}}\ldots\sum_{s_{T}=0}^{s_{T-1}}\frac{g\left(\boldsymbol{s}^{*}\right)}{\pi^{N}\mathrm{{det}\ensuremath{\left(\boldsymbol{J}\right)}}}\prod_{t=1}^{T}\frac{\left(-2K_{m,n}\right)^{s_{t}^{*}}}{s_{t}^{*}!\mathrm{{det}\ensuremath{\left(\boldsymbol{J}\right)}}^{s_{t}^{*}}}\times}\\ &{\stackrel[n=1]{N}{\prod}\int_{0}^{R_{n}}\left|h_{n}\right|^{\bar{s}_{n}+1}\exp\left\{-\frac{\left|h_{n}\right|^{2}K_{n,n}}{\mathrm{{det}\ensuremath{\left(\boldsymbol{J}\right)}}}\right\}d\left|h_{n}\right|}\\ {=}&{\stackrel[s_{1}=0]{s_{0}}{\sum}\sum_{s_{2}=0}^{s_{1}}\ldots\sum_{s_{T}=0}^{s_{T-1}}\frac{g\left(\boldsymbol{s}^{*}\right)}{\pi^{N}\mathrm{{det}\ensuremath{\left(\boldsymbol{J}\right)}}}\prod_{t=1}^{T}\frac{\left(-K_{m,n}\right)^{s_{t}^{*}}}{s_{t}^{*}!\mathrm{{det}\ensuremath{\left(\boldsymbol{J}\right)}}^{s_{t}^{*}}}\times}\\ &{\stackrel[n=1]{N}{\prod}\frac{1}{2}\left(\frac{\mathrm{\ensuremath{K_{n,n}}}}{\mathrm{{det}\ensuremath{\left(\boldsymbol{J}\right)}}}\right)^{-\frac{\bar{s}_{n}}{2}-\frac{1}{2}}\left[\Gamma\left(\frac{1+\bar{s}_{n}}{2}\right)-\Gamma\left(\frac{1+\bar{s}_{n}}{2},\frac{\mathrm{\ensuremath{K_{n,n}}}R_{n}^{2}}{\mathrm{{det}\ensuremath{\left(\boldsymbol{J}\right)}}}\right)\right],}\end{array}\] |
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\[\begin{array}{r l r}{\tau(1(\rho_{i}\rho_{j})^{2}1)}&{=}&{(\gamma(1,\rho_{i})\gamma(\rho_{1},\rho_{j}))^{2}}\\ &{=}&{\left\{\begin{array}{l l}{\alpha_{j}^{-2}~}&{\quad\textrm{f o r}\quad j\geq3,}\\ {(\alpha_{i}\alpha_{j}^{-1})^{2}}&{\quad\textrm{f o r}\quad i\geq2\quad\mathrm{and}\quad i+2\leq j\leq n-1,}\end{array}\right.}\\ &{}&\\ {\tau(y_{1}(\rho_{i}\rho_{j})^{2}y_{1})}&{=}&{(\gamma(y_{1},\rho_{i})\gamma(y_{1}\rho_{1},\rho_{j}))^{2}}\\ &{=}&{\left\{\begin{array}{l l}{\beta_{j}^{-2}}&{\quad\textrm{f o r}\quad3\leq j\leq n-1,}\\ {(\beta_{i}\beta_{j}^{-1})^{2}}&{\quad\textrm{f o r}\quad2\leq i\leq n-2\quad\mathrm{and}\quad j\geq i+2,}\end{array}\right.}\\ &{}&\\ {\tau(\rho_{1}(\rho_{i}\rho_{j})^{2}\rho_{1})}&{=}&{(\gamma(\rho_{1},\rho_{i})\gamma(1,\rho_{j}))^{2}}\\ &{=}&{\left\{\begin{array}{l l}{\alpha_{j}^{2}}&{\quad\textrm{f o r}\quad3\leq j\leq n-1,}\\ {(\alpha_{i}^{-1}\alpha_{j})^{2}}&{\quad\textrm{f o r}\quad i\geq2\quad\mathrm{and}\quad i+2\leq j\leq n-1,}\end{array}\right.}\\ &{}&\\ {\tau(y_{1}\rho_{1}(\rho_{i}\rho_{j})^{2}\rho_{1}y_{1})}&{=}&{(\gamma(y_{1}\rho_{1},\rho_{i})\gamma(y_{1},\rho_{j}))^{2}}\\ &{=}&{\left\{\begin{array}{l l}{\beta_{j}^{2}}&{\quad\textrm{f o r}\quad j\geq3,}\\ {(\beta_{i}^{-1}\beta_{j})^{2}}&{\quad\textrm{f o r}\quad i\geq2\quad\mathrm{and}\quad i+2\leq j\leq n-1.}\end{array}\right.}\end{array}\] |
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\[\begin{array}{r l}{\Big\Vert\mathrm{K}_{G}^{\mathrm{free}}[H]\Big\Vert_{\mathrm{L}^{2}(0,T;\mathrm{L}^{2}(\mathbb T^{d}))}+\Big\Vert\mathrm{K}_{G}^{\mathrm{fric}}[H]\Big\Vert_{\mathrm{L}^{2}(0,T;\mathrm{L}^{2}(\mathbb T^{d}))}}&{\leq C(1+T)\underset{0\leq s,t\leq T}{\operatorname*{sup}}\Vert\partial_{s}G(t,s)\Vert_{\mathcal{H}_{\sigma}^{p}}\Vert H\Vert_{\mathrm{L}^{2}(0,T;\dot{\mathrm{H}}^{-1}(\mathbb T^{d}))}.}\end{array}\] |
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\[\begin{array}{r l}{\Vert S-\hat{S}_{j}\Vert_{F}^{2}}&{=\Vert X\tilde{R}_{j}X^{T}\Vert_{F}^{2}=\frac{\textnormal{t r}\big(Y\tilde{R}_{j}Y^{T}(Y\tilde{R}_{j}Y^{T})^{H}\big)}{|x_{1}^{T}J x_{2}|^{2}}}\\ &{=\frac{\textnormal{t r}\big(Y\tilde{R}_{j}Y^{T}\overline{{Y}}\tilde{R}_{j}^{H}Y^{H}\big)}{|x_{1}^{T}J x_{2}|^{2}}=\frac{\textnormal{t r}\big(Y^{H}Y\tilde{R}_{j}\overline{{(Y^{H}Y)}}\tilde{R}_{j}^{H}\big)}{|x_{1}^{T}J x_{2}|^{2}}.}\end{array}\] |
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\[\begin{array}{r l r}{E1_{P V}}&{=}&{\sum_{I\ne i}\frac{\langle\Phi_{f}|D|\Phi_{I}\rangle\langle\Phi_{I}|H_{W}|\Phi_{i}\rangle}{{\cal E}_{i}-{\cal E}_{I}}}\\ &{}&{+\sum_{I\ne f}\frac{\langle\Phi_{f}|H_{W}|\Phi_{I}\rangle\langle\Phi_{I}|D|\Phi_{i}\rangle}{{\cal E}_{f}-{\cal E}_{I}}.}\end{array}\] |
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\[\begin{array}{r l}{\mathrm{d}\alpha^{0}=}&{-2\psi_{0}^{0}{\,{\wedge}\;}\alpha^{0}+\alpha^{1}{\,{\wedge}\;}\alpha^{2},}\\ {\mathrm{d}\alpha^{1}=}&{(-\psi_{0}^{0}-\psi_{1}^{1}){\,{\wedge}\;}\alpha^{1}-\beta^{1}{\,{\wedge}\;}\alpha^{0},}\\ {\mathrm{d}\alpha^{2}=}&{(-\psi_{0}^{0}+\psi_{1}^{1}){\,{\wedge}\;}\alpha^{2}-\beta^{2}{\,{\wedge}\;}\alpha^{0}.}\end{array}\] |
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\[\begin{array}{r l}{I(1,\lambda)}&{=2\int_{0}^{1}d\xi\int_{0}^{\infty}d u(\xi(1-\xi))^{\frac{1}{2}}e^{-(1+\lambda)u\xi(1-\xi)}}\\ &{=2\int_{0}^{1}d\xi\frac{1}{(1+\lambda)\sqrt{\xi(1-\xi)}}}\\ &{=\frac{2\pi}{1+\lambda}.}\end{array}\] |
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\[{\displaystyle\mathrm{d}^{(i)}(\mu_{j_{1}}\wedge\dots\wedge\mu_{j_{i}}\otimes\frac{\partial}{\partial x_{j}})=\sum_{l=1}^{i}(-1)^{l+1}f_{j_{l}}\mu_{j_{1}}\wedge\cdots\wedge{\widehat{\mu_{j_{l}}}}\wedge\cdots\wedge\mu_{j_{i}}}\otimes\frac{\partial}{\partial x_{j}},\] |
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\[\begin{array}{r l}&{\frac{1}{T V^{a,b*}(P)}\sum_{t=1}^{T}\Bigg(T V^{a,b*}(P)\mathbb{E}_{P}\big[(\xi_{t}^{a,b}(P))^{2}\big]-\mathbb{E}_{P}\Bigg[\sum_{k\in\{a,b\}}\frac{\big(Y_{t}^{k}-\mu^{k}(P)(X_{t})\big)^{2}}{w^{*}(k|X_{t})}+\Big(\mu^{a}(P)(X_{t})-\mu^{b}(P)(X_{t})-\Delta^{a,b}(P)\Big)^{2}\Bigg]\Bigg)}\\ &{\leq\frac{\widetilde{t}}{T V^{a,b*}(P)}+\epsilon.}\end{array}\] |
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\[L(\underline{{\theta}},\underline{{a}})=p_{1}W\left(\frac{a_{1}}{\theta_{1}}-1\right)+p_{2}W\left(\frac{a_{2}}{\theta_{2}}-1\right),\;\underline{{\theta}}=(\theta_{1},\theta_{2})\in\Theta_{0},\;\underline{{a}}=(a_{1},a_{2})\in\mathcal{A}=\Re_{++}^{2},\] |
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\[\begin{array}{r}{S(X,t)=\frac{\int_{U}(X-y(z)e^{-\frac{t}{2}})e^{-\frac{\|X-y(z)e^{-\frac{t}{2}}\|^{2}}{2(1-e^{-t})}}\hat{\rho}(z)d z}{(1-e^{-t})\int_{U}e^{-\frac{\|X-y(z)e^{-\frac{t}{2}}\|^{2}}{2(1-e^{-t})}}\hat{\rho}(z)d z}.}\end{array}\] |
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\[\begin{array}{r l}{\mathcal{H}^{d}(Z(u_{\lambda})\cap\Omega_{B}\times[-1,1])}&{\le\sum_{k}\mathcal{H}^{d}(Z(u_{\lambda})\cap Q_{k})}\\ &{\le\sum_{k}C N^{**}(Q_{k})^{\beta}s(Q_{k})^{d}}\\ &{\le C(1+\lambda)^{\beta/2}}\\ &{\le C\lambda^{\beta/2}.}\end{array}\] |
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\[\begin{array}{r}{N_{0}=\frac{1}{2}\cdot\frac{\log\big(\frac{\|Q_{N}-P^{*}\|_{*}\cdot\kappa_{P^{*}}\cdot\|A_{K}^{*}\|\cdot\|B\|}{\epsilon\cdot\lambda_{\operatorname*{min}}(R)}\big)}{\log\big(\frac{1}{\|A_{K}^{*}\|_{*}}\big)}+1.}\end{array}\] |
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\[\begin{array}{r l}{u_{\lambda}(x)\geq}&{w(x)-\int_{-T}^{0}w(\gamma_{\lambda}^{x}(t))\frac{\mathrm{d}}{\mathrm{d}t}\Big(e^{\lambda\int_{t}^{0}\frac{\partial L}{\partial u}(\gamma_{\lambda}^{x}(s),\dot{\gamma}_{\lambda}^{x}(s),0)\,\mathrm{d}s}\Big)\,\mathrm{d}t-(\|u_{\lambda}\|_{\infty}+\|w\|_{\infty})\,e^{-\epsilon T}}\\ &{\qquad\qquad\qquad\qquad+\int_{-T}^{0}\Omega_{\lambda,x}(t)\,e^{\lambda\int_{t}^{0}\frac{\partial L}{\partial u}(\gamma_{\lambda}^{x}(s),\dot{\gamma}_{\lambda}^{x}(s),0)\,\mathrm{d}s}\,\mathrm{d}t}\end{array}\] |
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\[\begin{array}{r l}{u(x,t)}&{=u_{\mathrm{sol}}(x,t)-i\sqrt{3}\frac{\partial}{\partial x}(1,1,1)Z(\zeta,t)\left(\begin{array}{l}{\omega^{2}i k_{1}^{-1}-\omega i k_{1}}\\ {\omega i k_{1}^{-1}-\omega^{2}i k_{1}}\\ {i k_{1}^{-1}-i k_{1}}\end{array}\right)+O(t^{-1}\ln t).}\end{array}\] |
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\[\begin{array}{r l}{\partial_{z}\partial_{w}f_{j,N}(z_{0},w_{0})}&{=\int_{\partial\Omega}\partial_{z}^{2}f_{j,N}(z,w)\operatorname{d}\mu_{1}+\int_{\partial\Omega}\partial_{w}^{2}f_{j,N}(z,w)\operatorname{d}\mu_{2}}\\ &{\;+\int_{\partial\Omega}\partial_{z}f_{j,N}(z,w)\operatorname{d}\nu_{1}+\int_{\partial\Omega}\partial_{w}f_{j,N}(z,w)\operatorname{d}\nu_{2}+\int_{\partial\Omega}f_{j,N}(z,w)\operatorname{d}\nu_{3}.}\end{array}\] |
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\[\begin{array}{r l}{\frac{\mathrm{d}\rho}{\mathrm{d}t}=}&{i[\rho,H]}\\ {+}&{\gamma_{\mathrm{GX}}^{\mathrm{A}}\mathcal{L}(\tau_{\mathrm{AA}}\sigma_{\mathrm{GX}})+\gamma_{\mathrm{GX}}^{\mathrm{B}}\mathcal{L}(\tau_{\mathrm{BB}}\sigma_{\mathrm{GX}})+\gamma_{\mathrm{GY}}^{\mathrm{A}}\mathcal{L}(\tau_{\mathrm{AA}}\sigma_{\mathrm{GY}})+\gamma_{\mathrm{GY}}^{\mathrm{B}}\mathcal{L}(\tau_{\mathrm{BB}}\sigma_{\mathrm{GY}})}\\ {+}&{\sum_{m}\gamma_{\mathrm{SS}}\mathcal{L}(b_{m})}\\ {+}&{\sum_{m}\gamma_{\mathrm{GX}_{m}}^{\mathrm{A}}\mathcal{L}(\tau_{\mathrm{AA}}\sigma_{\mathrm{GX}}b_{m})+\gamma_{\mathrm{GX}_{m}}^{\mathrm{B}}\mathcal{L}(\tau_{\mathrm{BB}}\sigma_{\mathrm{GX}}b_{m})+\gamma_{\mathrm{GY}_{m}}^{\mathrm{A}}\mathcal{L}(\tau_{\mathrm{AA}}\sigma_{\mathrm{GY}}b_{m})+\gamma_{\mathrm{GY}_{m}}^{\mathrm{B}}\mathcal{L}(\tau_{\mathrm{BB}}\sigma_{\mathrm{GY}}b_{m})}\\ {+}&{\sum_{m}\gamma_{\mathrm{GX}_{m}}^{\mathrm{A}}\mathcal{L}(\tau_{\mathrm{AA}}\sigma_{\mathrm{GX}}b_{m}^{\dagger})+\gamma_{\mathrm{GX}_{m}}^{\mathrm{B}}\mathcal{L}(\tau_{\mathrm{BB}}\sigma_{\mathrm{GX}}b_{m}^{\dagger})+\gamma_{\mathrm{GY}_{m}}^{\mathrm{A}}\mathcal{L}(\tau_{\mathrm{AA}}\sigma_{\mathrm{GY}}b_{m}^{\dagger})+\gamma_{\mathrm{GY}_{m}}^{\mathrm{B}}\mathcal{L}(\tau_{\mathrm{BB}}\sigma_{\mathrm{GY}}b_{m}^{\dagger})}\\ {+}&{\gamma_{\mathrm{AB}}^{\mathrm{XA}}\mathcal{L}(\sigma_{\mathrm{XX}}\tau_{\mathrm{BA}})+\gamma_{\mathrm{AB}}^{\mathrm{YA}}\mathcal{L}(\sigma_{\mathrm{YY}}\tau_{\mathrm{BA}})+\gamma_{\mathrm{AB}}^{\mathrm{XB}}\mathcal{L}(\sigma_{\mathrm{XX}}\tau_{\mathrm{AB}})+\gamma_{\mathrm{AB}}^{\mathrm{YB}}\mathcal{L}(\sigma_{\mathrm{YY}}\tau_{\mathrm{AB}})}\\ {+}&{\gamma_{x y}[\mathcal{L}(\tau_{\mathrm{AA}}\sigma_{\mathrm{XY}})+\mathcal{L}(\tau_{\mathrm{BB}}\sigma_{\mathrm{YX}})]}\\ {+}&{\gamma_{\phi}[\mathcal{L}(\sigma_{\mathrm{XX}})+\mathcal{L}(\sigma_{\mathrm{YY}})].}\end{array}\] |
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\[\begin{array}{r l}&{\left\langle U_{j}H-U_{j}^{*},R-I_{d}\right\rangle+\left\langle U_{j}\left(M-\hat{P}H^{\top}\right)^{\top}\hat{P},R-I_{d}\right\rangle}\\ &{\leq\left(1+\frac{16C_{0}(1+\sigma\sqrt{d})}{3\sqrt{n p}}\right)\left(\left\|{B_{j}}\right\|_{\mathrm{op}}+\left\|{F_{j1}}\right\|_{\mathrm{op}}+\left\|{F_{j2}}\right\|_{\mathrm{op}}+\left\|{G_{j1}}\right\|_{\mathrm{op}}\right)\sqrt{2d h_{R}}}\\ &{\quad+\frac{16C_{0}(1+\sigma\sqrt{d})}{3\sqrt{n p}}\sqrt{2d h_{R}}\left\|{G_{j2}}\right\|_{\mathrm{op}}+\left\langle G_{j2},R-I_{d}\right\rangle+\frac{16C_{0}(1+\sigma\sqrt{d})}{3\sqrt{n p}}\frac{\sqrt{2d h_{R}}}{\sqrt{n}}}\\ &{\leq2\left(1+\frac{16C_{0}(1+\sigma\sqrt{d})}{3\sqrt{n p}}\right)\left(\left\|{F_{j1}}\right\|_{\mathrm{op}}+\left\|{F_{j2}}\right\|_{\mathrm{op}}+\left\|{G_{j1}}\right\|_{\mathrm{op}}\right)\sqrt{2d h_{R}}}\\ &{\quad+\left(\frac{41}{33}+\frac{16C_{0}(1+\sigma\sqrt{d})}{3\sqrt{n p}}\right)22C_{0}\left(\frac{1+\sigma\sqrt{d}}{\sqrt{n p}}\right)\left\|{G_{j2}}\right\|_{\mathrm{op}}\sqrt{2d h_{R}}+\left\langle G_{j2},R-I_{d}\right\rangle}\\ &{\quad+\left(\frac{41}{33}+\frac{16C_{0}(1+\sigma\sqrt{d})}{3\sqrt{n p}}\right)22C_{0}\left(\frac{1+\sigma\sqrt{d}}{\sqrt{n p}}\right)\frac{\sqrt{2d h_{R}}}{\sqrt{n}}}\\ &{\leq2\left(1+\frac{16C_{0}(1+\sigma\sqrt{d})}{3\sqrt{n p}}\right)\left(\left\|{F_{j1}}\right\|_{\mathrm{op}}+\left\|{F_{j2}}\right\|_{\mathrm{op}}\right)\sqrt{2d h_{R}}}\\ &{\quad+\left(\frac{41}{33}+\frac{16C_{0}(1+\sigma\sqrt{d})}{3\sqrt{n p}}\right)22C_{0}\left(\frac{1+\sigma\sqrt{d}}{\sqrt{n p}}\right)\left\|{G_{j2}}\right\|_{\mathrm{op}}\sqrt{2d h_{R}}+\left\langle G_{j2},R-I_{d}\right\rangle}\\ &{\quad+\left(\frac{41}{33}+\frac{16C_{0}(1+\sigma\sqrt{d})}{3\sqrt{n p}}\right)22C_{0}\left(\frac{1+\sigma\sqrt{d}}{\sqrt{n p}}\right)\frac{\sqrt{2d h_{R}}}{\sqrt{n}}}\\ &{\quad+4\left(1+\frac{16C_{0}(1+\sigma\sqrt{d})}{3\sqrt{n p}}\right)C_{0}\left(\frac{\sqrt{\log n}+\sigma\sqrt{d}}{\sqrt{n p}}\right)\frac{\sqrt{2d h_{R}}}{\sqrt{n}}}\\ &{\leq2\left(1+\frac{16C_{0}(1+\sigma\sqrt{d})}{3\sqrt{n p}}\right)\left(\left\|{F_{j1}}\right\|_{\mathrm{op}}+\left\|{F_{j2}}\right\|_{\mathrm{op}}\right)\sqrt{2d h_{R}}}\\ &{\quad+\left(\frac{41}{33}+\frac{16C_{0}(1+\sigma\sqrt{d})}{3\sqrt{n p}}\right)22C_{0}\left(\frac{1+\sigma\sqrt{d}}{\sqrt{n p}}\right)\left\|{G_{j2}}\right\|_{\mathrm{op}}\sqrt{2d h_{R}}+\left\langle G_{j2},R-I_{d}\right\rangle}\\ &{\quad+\left(\frac{41}{33}+\frac{16C_{0}(1+\sigma\sqrt{d})}{3\sqrt{n p}}\right)26C_{0}\left(\frac{\sqrt{\log n}+\sigma\sqrt{d}}{\sqrt{n p}}\right)\frac{\sqrt{2d h_{R}}}{\sqrt{n}}\,,}\end{array}\] |
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\[\begin{array}{r l r}{u(\phi)\!=\!\left\{\begin{array}{l l}{u_{m}}&{\!\!\!\!\mathrm{towards~the~antenna~direction}}\\ {u_{s}}&{\!\!\!\!\mathrm{otherwise}}\end{array}\right.,}&{}&{v(\phi)\!=\!\left\{\begin{array}{l l}{v_{m}}&{\!\!\!\mathrm{towards~the~antenna~direction.}}\\ {v_{s}}&{\!\!\!\mathrm{otherwise}.}\end{array}\right.}\end{array}\] |
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\[\begin{array}{r l}{\check{h}_{k+1}}&{=P_{1}\check{h}_{k}+P_{1}J_{L,l}Q_{1}^{\intercal}\eta\left(\nabla F(\mathbf{1}\bar{x}_{k}^{\intercal})-\nabla F(\mathbf{x}_{k})\right)+(J_{L,l}\tilde{\Lambda}_{1}+J_{L,r}\sqrt{I-\tilde{\Lambda}_{1}})Q_{1}^{\intercal}\eta\hat{\mathbf{g}}_{k}}\\ &{\quad+\left(J_{L,r}-J_{L,l}\sqrt{I-\tilde{\Lambda}_{1}}\right)\sqrt{I-\tilde{\Lambda}_{1}}Q_{1}^{\intercal}E_{k}+J_{L,r}(I-\tilde{\Lambda}_{1})^{-\frac{1}{2}}\eta Q_{1}^{\intercal}\left(\nabla F(\mathbf{1}\bar{x}_{k+1}^{\intercal})-\nabla F(\mathbf{1}\bar{x}_{k}^{\intercal})\right),}\end{array}\] |
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\[\begin{array}{r l}{\langle~{\boldsymbol w}_{k}^{(t+1)}~,~{\boldsymbol p}_{+}~\rangle-\langle~{\boldsymbol w}_{k}^{(t)}~,~{\boldsymbol p}_{+}~\rangle\ge}&{~c_{\eta}(I_{1}-|I_{2}|)}\\ {\ge}&{~c_{\eta}\cdot(\alpha-\sigma\sqrt{\frac{(1+r^{2})\log q}{|\mathcal{D}|}}).}\end{array}\] |
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\[\begin{array}{r l r}{\varepsilon_{n,\nu}^{(1)}}&{=}&{\frac{\hbar^{3}c^{3}\mathcal{N}^{2}}{8\lambda^{3}}\int_{0}^{\rho_{\nu}}W_{\nu}\bigg(\frac{\hbar c\rho}{2\lambda}\bigg)\Big\{m_{e}c^{2}\,\big[Q_{1}^{2}(\rho)+Q_{2}^{2}(\rho)\big]}\\ &{}&{+2\varepsilon\,Q_{1}(\rho)Q_{2}(\rho)\Big\}\rho^{2\gamma}e^{-\rho}d\rho,}\end{array}\] |
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\[\begin{array}{l}{i\partial_{t}\tilde{u}+\displaystyle\frac{\Delta_{x}\tilde{u}}{2}+i k\cdot\nabla_{x}\tilde{u}=\gamma\sigma_{1}\star\displaystyle\int_{\mathbb R^{n}}\sigma_{2}\tilde{\Psi}\,{\mathrm{d}}z+\left(\gamma\sigma_{1}\star\displaystyle\int_{\mathbb R^{n}}\sigma_{2}\tilde{\Psi}\,{\mathrm{d}}z\right)\tilde{u},}\\ {\displaystyle\frac{1}{c^{2}}\partial_{t t}^{2}\tilde{\Psi}-\Delta_{z}\tilde{\Psi}=-2\gamma\sigma_{2}\sigma_{1}\star\mathrm{Re}(\tilde{u})-\gamma\sigma_{2}\sigma_{1}\star|\tilde{u}|^{2}.}\end{array}\] |
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\[{\mathcal{J}}^{2}={\mathcal{P}}^{2}=-{\frac{1}{\sin^{2}\beta}}\left({\frac{\partial^{2}}{\partial\alpha^{2}}}+{\frac{\partial^{2}}{\partial\gamma^{2}}}-2\cos\beta{\frac{\partial^{2}}{\partial\alpha\partial\gamma}}\right)-{\frac{\partial^{2}}{\partial\beta^{2}}}-\cot\beta{\frac{\partial}{\partial\beta}}.\] |
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\[\hat{\mathbf{G}}_{v}=J^{-1}\left\{\begin{array}{c}{0}\\ {\zeta_{x}{\tau}_{x x}+\zeta_{y}{\tau}_{x y}+\zeta_{z}{\tau}_{x z}}\\ {\zeta_{x}{\tau}_{x y}+\zeta_{y}{\tau}_{y y}+\zeta_{z}{\tau}_{y z}}\\ {\zeta_{x}{\tau}_{x z}+\zeta_{y}{\tau}_{y z}+\zeta_{z}{\tau}_{z z}}\\ {\zeta_{x}{\beta}_{x}+\zeta_{y}{\beta}_{y}+\zeta_{z}{\beta}_{z}}\end{array}\right\}\,\mathrm{~.}\] |
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\[\operatorname*{min}\left\lbrace\|u\|_{L^{\mathcal{H}_{x}^{*}}({\mathbb R}^{N})}^{h_{1}^{*}},\|u\|_{L^{\mathcal{H}_{x}^{*}}({\mathbb R}^{N})}^{h_{2}^{*}}\right\rbrace\leq\int_{{\mathbb R}^{N}}\mathcal{H}_{x}^{*}(x,|u|)d x\leq\operatorname*{max}\left\lbrace\|u\|_{L^{\mathcal{H}_{x}^{*}}({\mathbb R}^{N})}^{h_{1}^{*}},\|u\|_{L^{\mathcal{H}_{x}^{*}}({\mathbb R}^{N})}^{h_{2}^{*}}\right\rbrace,\] |
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\[\begin{array}{r l}&{\quad\langle\nabla{\xi}_{k-1}^{*}\left(v(\alpha^{k-1})\right),y^{k-1}-y^{k-2}\rangle-\langle x^{k-1},y^{k-1}-y^{k-2}\rangle}\\ &{=\langle\nabla{\xi}_{k-1}^{*}\left({\tilde{u}_{(k-1)}}^{T_{k-1}}\right)-\nabla{\tilde{\xi}}_{k-1}^{*}(u_{(k-1)}^{T_{k-1}}),y^{k-1}-y^{k-2}\rangle}\\ &{\leq\frac{1}{2}\|\nabla{\xi}_{k-1}^{*}\left({\tilde{u}_{(k-1)}}^{T_{k-1}}\right)-\nabla{\xi}_{k-1}^{*}(u_{(k-1)}^{T_{k-1}})\|^{2}+\frac{1}{2}\|y^{k-1}-y^{k-2}\|^{2}}\\ &{\leq\frac{1}{2}\|e_{(k-1)}^{T_{k-1}}\|^{2}+\frac{1}{2}\|y^{k-1}-y^{k-2}\|^{2},}\end{array}\] |
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\[\begin{array}{r l}{\partial_{\rho}\bigg(\nu(\theta_{0})\big(\partial_{\rho}\hat{\mathbf{v}}_{\frac{1}{2}}-(\mathbf{u}_{\frac{1}{2}}d_{\Gamma}+\mathbf{u}_{0}d_{\frac{1}{2}})\eta^{\prime}\big)\bigg)}&{=\left(2\theta_{0}^{\prime}\theta_{0}^{\prime\prime}+\partial_{\rho}\hat{p}_{-1}\right)\nabla d_{\frac{1}{2}}+\partial_{\rho}\hat{p}_{-\frac{1}{2}}\nabla d_{\Gamma},}\\ {\big(\partial_{\rho}\hat{\mathbf{v}}_{\frac{1}{2}}-(\mathbf{u}_{\frac{1}{2}}d_{\Gamma}+\mathbf{u}_{0}d_{\frac{1}{2}})\eta^{\prime}\big)\cdot\nabla d_{\Gamma}}&{=\left(-\partial_{\rho}\hat{\mathbf{v}}_{0}+\mathbf{u}_{0}d_{\Gamma}\eta^{\prime}\right)\cdot\nabla d_{\frac{1}{2}}.}\end{array}\] |
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\[\begin{array}{r l r}{D_{\mathrm{JS}}^{G}(p:q)}&{=}&{\frac{1}{2}\left(D_{\mathrm{KL}}(p:(p q)^{G})+D_{\mathrm{KL}}(q:(p q)^{G})\right)\geq0,}\\ &{=}&{\frac{1}{2}\left(H^{\times}(p:(p q)^{G})-H(p)+H^{\times}(q:(p q)^{G})-H(q)\right)\geq0,}\\ &{=}&{H^{\times}((p q)^{A}:(p q)^{G})-\frac{H(p)+H(q)}{2}\geq0.}\end{array}\] |
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\[(n!)^{3}\cdot H_{n}^{(3)}=\left[{\begin{array}{l}{n+1}\\ {2}\end{array}}\right]^{3}-3\left[{\begin{array}{l}{n+1}\\ {1}\end{array}}\right]\left[{\begin{array}{l}{n+1}\\ {2}\end{array}}\right]\left[{\begin{array}{l}{n+1}\\ {3}\end{array}}\right]+3\left[{\begin{array}{l}{n+1}\\ {1}\end{array}}\right]^{2}\left[{\begin{array}{l}{n+1}\\ {4}\end{array}}\right].\] |
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\[\Lambda(n)={\left\{\begin{array}{l l}{\log p}&{{\mathrm{if~}}n=2,3,4,5,7,8,9,11,13,16,\ldots=p^{k}{\mathrm{~is~a~prime~power}}}\\ {0}&{{\mathrm{if~}}n=1,6,10,12,14,15,18,20,21,\dots\;\;\;\;{\mathrm{~is~not~a~prime~power}}.}\end{array}\right.}\] |
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\[\begin{array}{r l}{\tilde{\varphi}\colon\mathcal{C}\cup(\mathcal{D}+\{3R\})}&{\to\mathbb R^{m\times n}}\\ {{\boldsymbol{x}}}&{\mapsto\left\{\begin{array}{l l}{\varphi({\boldsymbol{x}}),}&{{\boldsymbol{x}}\in\mathcal{C}}\\ {\psi({\boldsymbol{x}}-3R),}&{{\boldsymbol{x}}\in\mathcal{D}+\{3R\}}\end{array}\right..}\end{array}\] |
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\[U_{t_{L}}\times U_{t_{R}}=f\left(\begin{array}{l}{\frac{x_{L}+x_{R}}{2}\sin({\theta_{R}-\theta_{L}})-\frac{x_{R}-x_{L}}{2}\sin(\theta_{L}+\theta_{R})}\\ {y\sin(\theta_{R}-\theta_{L})-(x_{R}-x_{L})(\cos(\theta_{R}-\theta_{L})-\cos(\theta_{L}+\theta_{R}))}\\ {f\sin(\theta_{R}-\theta_{L})}\end{array}\right)\] |
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\[\begin{array}{r}{R=\left(\begin{array}{l l l l}{\cos^{2}\theta}&{\cos\theta\sin\theta}&{\cos\theta\sin\theta}&{\sin^{2}\theta}\\ {-\sin\theta\cos\theta}&{\cos^{2}\theta}&{-\sin^{2}\theta}&{\sin\theta\cos\theta}\\ {-\sin\theta\cos\theta}&{-\sin^{2}\theta}&{\cos^{2}\theta}&{\sin\theta\cos\theta}\\ {\sin^{2}\theta}&{-\sin\theta\cos\theta}&{-\sin\theta\cos\theta}&{\cos^{2}\theta}\end{array}\right).}\end{array}\] |
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\[\begin{array}{r l}{v(T n/N,\mathbf u)}&{\approx\sum_{m=0}^{n}\binom n m(1-\lambda)^{n-m}\lambda^{m}v(0,(A_{M}^{s i g})^{\circ m}(\mathbf u))}\\ &{=\sum_{m=0}^{n}\binom n m(1-\lambda)^{n-m}\lambda^{m}\exp((A_{M}^{s i g})^{\circ m}(\mathbf u)_{\emptyset}),}\end{array}\] |
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\[\begin{array}{r l}&{\geq\frac{1}{C}\|\mathcal P_{\gamma}^{\perp}g_{\alpha}^{(s)}\|_{L_{x}^{2}(\mathcal H_{\sigma})_{\xi}}^{2}-\kappa\|g_{\alpha}^{(s)}\|_{L_{x}^{2}(\mathcal H_{\sigma})_{\xi}}^{2}}\\ &{\quad-C_{\kappa}\|g_{\alpha+1}\|_{L_{x}^{2}(\mathcal H_{\sigma})_{\xi}}^{2}}\end{array}\] |
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\[\begin{array}{r l}{\sum_{j=1}^{n-1}w_{j}^{2}\sum_{k=j+1}^{n}(a_{k-j-1}^{(k)}-a_{k-j}^{(k)})=}&{\sum_{j=1}^{n-1}w_{j}^{2}\Big[a_{0}^{(j+1)}+\sum_{k=j+1}^{n-1}(a_{k-j}^{(k+1)}-a_{k-j}^{(k)})-a_{n-j}^{(n)}\Big]}\\ {\le}&{\sum_{j=1}^{n-1}a_{0}^{(j+1)}w_{j}^{2}.}\end{array}\] |
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\[\left\{\begin{array}{r l r}{\frac{\partial c}{\partial t}}&{=D\nabla^{2}c-\gamma c}&{\quad\forall x\mathrm{~in~}[0,\infty)}\\ {\left.\frac{\partial c}{\partial x}\right|_{0}}&{=-\frac{\omega}{2D}}\\ {\left.\frac{\partial c}{\partial x}\right|_{x\rightarrow\infty}}&{=0}\\ {c(x,0)}&{=0}&{\forall x\mathrm{~in~}[0,\infty)}\end{array}\right.\] |
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\[{\begin{array}{r l}{s_{\mathrm{a}}(t)}&{\triangleq{\mathcal{F}}^{-1}[S_{\mathrm{a}}(f)]}\\ &{={\mathcal{F}}^{-1}[S(f)+\operatorname{sgn}(f)\cdot S(f)]}\\ &{=\underbrace{{\mathcal{F}}^{-1}\{S(f)\}}_{s(t)}+\overbrace{\underbrace{{\mathcal{F}}^{-1}\{\operatorname{sgn}(f)\}}_{j{\frac{1}{\pi t}}}*\underbrace{{\mathcal{F}}^{-1}\{S(f)\}}_{s(t)}}^{\mathrm{convolution}}}\\ &{=s(t)+j\underbrace{\left[{\frac{1}{\pi t}}*s(t)\right]}_{\operatorname{\mathcal{H}}[s(t)]}}\\ &{=s(t)+j{\hat{s}}(t),}\end{array}}\] |
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\[p(y|x)=\frac{1}{\sqrt{4\pi\sigma^{2}}}e^{-\frac{(y-x)^{2}}{4\sigma^{2}}}\frac{\left[Q\left(\frac{0-(x+y)/2}{\sigma/\sqrt{2}}\right)-Q\left(\frac{1-(x+y)/2}{\sigma/\sqrt{2}}\right)\right]}{\int_{0}^{1}\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(x-u)^{2}}{2\sigma^{2}}}\left[Q\left(\frac{-u}{\sigma}\right)-Q\left(\frac{1-u}{\sigma}\right)\right]\,d u}.\] |
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\[{\begin{array}{r l}&{\nabla^{2}\left({\mathcal{M}}-{\frac{\mathcal{B}}{1+\nu}}\,q\right)=-q}\\ &{\kappa G h\left(\nabla^{2}w+{\frac{\mathcal{M}}{D}}\right)=-\left(1-{\cfrac{{\mathcal{B}}c^{2}}{1+\nu}}\right)q}\\ &{\nabla^{2}\left({\frac{\partial\varphi_{1}}{\partial x_{2}}}-{\frac{\partial\varphi_{2}}{\partial x_{1}}}\right)=c^{2}\left({\frac{\partial\varphi_{1}}{\partial x_{2}}}-{\frac{\partial\varphi_{2}}{\partial x_{1}}}\right)}\end{array}}\] |
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\[\int_{{\mathbb R}^{N}}\int_{{\mathbb R}^{N}}\mathcal{H}\left(x,y,\frac{|u_{n}(x)-u_{n}(y)|}{|x-y|^{s}}\right)d\mu=\int_{{\mathbb R}^{N}}\int_{{\mathbb R}^{N}}\mathcal{H}\left(x,y,\frac{|u_{n}(x)-u_{n}(y)|}{|x-y|^{s}}\right)\frac{d x\ d y}{|x-y|^{N}}<\infty\] |
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\[\Delta_{j}=\frac{1-\sum_{j^{\prime}=1}^{j-1}\Delta_{j^{\prime}}\cdot(\sum_{i^{\prime}=1,i^{\prime}\neq i}^{n}\frac{v_{i^{\prime}}(j)}{1-\sum_{\tilde{j}=1}^{j^{\prime}}v_{i^{\prime}}(\tilde{j})})}{1+\sum_{i^{\prime}\neq i}\frac{v_{i^{\prime}}(j)}{1-\sum_{\tilde{j}=1}^{j-1}v_{i^{\prime}}(\tilde{j})}}.\] |
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\[\begin{array}{r l}{|\mathcal{D}_{1}(\omega k_{4})|=}&{\;\frac{e^{\frac{1}{2\pi}\int_{\Gamma_{2}^{(2)}}\frac{\arg{s}}{2}d\ln(1+r_{1}(s)r_{2}(s))-\nu_{1}\frac{\arg(\omega k_{4})}{2}}e^{2\nu_{3}\frac{\arg(\omega k_{4})}{2}-2\nu_{4}\frac{\arg(\omega^{2}k_{2})}{2}-2\frac{1}{2\pi}\int_{\Gamma_{5}^{(2)}}\frac{\arg{s}}{2}d\ln(f(s))}}{e^{\nu_{1}\frac{\arg(\omega k_{4})}{2}-\nu_{2}\frac{\arg(\omega^{2}k_{2})}{2}-\frac{1}{2\pi}\int_{\Gamma_{5}^{(2)}}\frac{\arg{s}}{2}d\ln(1+r_{1}(s)r_{2}(s))}}.}\end{array}\] |
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\[\begin{array}{r l}&{Q_{k}^{\prime}:=\{q\in Q:q\left(V_{k}(s_{k},\lambda)-r_{k}(s_{k},h_{k},\lambda,\bar{x})-V_{k+1}(s_{k+1},\lambda)\right))=0\},k=1,...,N,}\\ &{Q_{0}^{\prime}:=\{q\in Q:q\left(\nu-V_{1}(s_{1},\lambda)\right))=0\}}\end{array}\] |
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\[{\begin{array}{r l}{\operatorname*{lim}_{p\to\infty}M_{p}(x_{1},\dots,x_{n})}&{=\operatorname*{lim}_{p\to\infty}\left(\sum_{i=1}^{n}w_{i}x_{i}^{p}\right)^{1/p}}\\ &{=x_{1}\operatorname*{lim}_{p\to\infty}\left(\sum_{i=1}^{n}w_{i}\left({\frac{x_{i}}{x_{1}}}\right)^{p}\right)^{1/p}}\\ &{=x_{1}=M_{\infty}(x_{1},\dots,x_{n}).}\end{array}}\] |
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\[0\longrightarrow\widetilde{H}_{\mathrm{f}}^{1}(G_{\mathbb{Q},\Sigma},T_{2}^{\dagger},\Delta_{\mathrm{bal}})\longrightarrow\widetilde{H}_{\mathrm{f}}^{1}(G_{\mathbb{Q},\Sigma},T_{2}^{\dagger},\Delta_{+})\xrightarrow{\textup{r e s}_{\mathrm{bal+}}^{-}}H^{1}(G_{p},F_{\mathrm{bal+}}^{+}T_{2}^{\dagger}/F_{\mathrm{bal}}^{+}T_{2}^{\dagger})\] |
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\[\begin{array}{r l}{\mathcal{W}_{11}^{\flat}}&{=\gamma^{-1}\langle\bigl(\widehat{\mathcal{W}_{0,\gamma}}(0)-\widehat{\mathcal{P}_{0}}(0)\bigr)\omega_{s_{0}},\omega_{s_{0}}^{*}\rangle}\\ &{=\gamma^{-1}\langle\bigl(F_{g}G_{g}\delta_{g}^{*}\bigr)\omega_{s_{0}},\omega_{s_{0}}^{*}\rangle=\gamma^{-1}\langle\delta_{g}^{*}\omega_{s_{0}},G_{g}B_{g}\omega_{s_{0}}^{*}\rangle}\\ &{=\langle\delta_{g}^{*}\omega_{s_{0}},\big(2\mathfrak{c}\otimes_{s}\omega_{s_{0}}^{*}-\frac12g^{-1}(\mathfrak{c},\omega_{s_{0}}^{*})g\big)\rangle}\\ &{=4\pi(\mathfrak{b}-1)}\end{array}\] |
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\[\begin{array}{r l}{|f_{i}(t,x,y)-f_{i}(t,x,y^{\prime})|}&{\lesssim(1+|y|^{h-1}+|y^{\prime}|^{h-1})|y-y^{\prime}|,}\\ {|F_{i}(t,x,y)-F_{i}(t,x,y^{\prime})|}&{\lesssim(1+|y|^{\frac{h-1}{2}}+|y^{\prime}|^{\frac{h-1}{2}})|y-y^{\prime}|.}\\ {\|g_{i}(t,x,y)-g_{i}(t,x,y^{\prime})\|_{\ell^{2}}}&{\lesssim(1+|y|^{\frac{h-1}{2}}+|y^{\prime}|^{\frac{h-1}{2}})|y-y^{\prime}|.}\end{array}\] |
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\[\check{X}_{t_{n+1}}^{i,\theta}=\check{X}_{t_{n}}^{i,\theta}+b(t_{n},\check{X}_{t_{n}}^{i,\theta},\check{\mu}_{t_{n}}^{N,\theta},\alpha_{t_{n}}(\check{X}_{t_{n}}^{i,\theta};\theta_{n}))\Delta t+\sigma(t_{n},\check{X}_{t_{n}}^{i,\theta},\check{\mu}_{t_{n}}^{N,\theta},\alpha_{t_{n}}(\check{X}_{t_{n}}^{i,\theta};\theta_{n}))\Delta\check{W}_{n}^{i},\quad n=0,\dots,N_{T}-1,\] |
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\[\begin{array}{r}{\frac{\alpha_{k}}{\beta_{k}}=\frac{\frac{\partial f(\mu,\kappa,\varepsilon,\zeta,\xi)}{\partial\xi}\frac{\partial f(\mu,\kappa,\varepsilon,\zeta,\xi)}{\partial\zeta}-\frac{\partial f(\mu,\kappa,\varepsilon,\zeta,\xi)}{\partial\varepsilon}\frac{\partial f(\mu,\kappa,\varepsilon,\zeta,\xi)}{\partial\kappa}}{\frac{\partial f(\mu,\kappa,\varepsilon,\zeta,\xi)}{\partial\xi}\frac{\partial f(\mu,\kappa,\varepsilon,\zeta,\xi)}{\partial\varepsilon}-\frac{\partial f(\mu,\kappa,\varepsilon,\zeta,\xi)}{\partial\mu}\frac{\partial f(\mu,\kappa,\varepsilon,\zeta,\xi)}{\partial\zeta}}.}\end{array}\] |
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\[L_{i,j}^{\mathrm{sym}}:={\left\{\begin{array}{l l}{1}&{{\mathrm{if~}}i=j{\mathrm{~and~}}\deg(v_{i})\neq0}\\ {-{\frac{1}{\sqrt{\deg(v_{i})\deg(v_{j})}}}}&{{\mathrm{if~}}i\neq j{\mathrm{~and~}}v_{i}{\mathrm{~is~adjacent~to~}}v_{j}}\\ {0}&{{\mathrm{otherwise}}.}\end{array}\right.}\] |
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\[\begin{array}{r}{\frac{\gamma_{n+1}}{2}\left\Vert s_{n+1}\right\Vert^{2}-\sum_{k=0}^{n}\frac{\gamma_{k}}{2}\lambda_{k}^{2}\left\Vert g_{k}\right\Vert^{2}=\hat{d}_{n+1}\left\Vert s_{n+1}\right\Vert\le\widetilde{d}_{n+1}\left\Vert s_{n+1}\right\Vert\le d_{n+1}\left\Vert s_{n+1}\right\Vert.}\end{array}\] |
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\[\begin{array}{r l}{\int_{0}^{1}f_{a}\left(x\right)d x}&{=2\int_{0}^{1}g_{a}\left(x\right)d x-1=1-\frac{a}{1+a}\left(1-\frac{\pi}{a+1}\cot\frac{\pi}{a+1}\right)}\\ &{=\frac{1}{a+1}-\frac{a\pi}{\left(a+1\right)^{2}}\cot\frac{a\pi}{a+1}.}\end{array}\] |
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\[\operatorname*{sup}_{i\leq N}\mathbb{E}\operatorname*{sup}_{t\leq T}|X_{t}^{i,N}-X_{t}^{i,\infty}|\leq C\left\{\begin{array}{l l}{(\ln(N))^{\frac{1}{\alpha}}N^{\frac{1}{\alpha}-1},\,}&{\mathrm{if}\quad d=1,2\quad\mathrm{or}\quad d\geq3\,\,\mathrm{and}\,\,\alpha<\frac{d}{d-1},}\\ {N^{-\frac1d},\,}&{\mathrm{if}\quad d\geq3\,\,\mathrm{and}\,\,\alpha>\frac{d}{d-1},}\end{array}\right.\] |
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\[\begin{array}{r l}{\partial_{t}\eta_{(0)}+\nabla\cdot\left(\left(\eta_{0}-b\right)\,\mathbf{v}_{0}\right)}&{=0,}\\ {\partial_{t}\mathbf{q}_{(0)}+\nabla\cdot\left(\mathbf{v}_{(0)}\otimes\mathbf{q}_{(0)}\right)+\eta_{(2)}\nabla\eta_{(0)}+\eta_{(1)}\nabla\eta_{(1)}+\left(\eta_{(0)}-b\right)\nabla\eta_{(2)}}&{=\mathbf{0}.}\end{array}\] |
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\[\begin{array}{r l}&{\hat{\mathbf{E}}\left(\mathbf{r},\omega_{f},t\right)=i\frac{\hbar}{\pi\epsilon_{0}}\sum_{s}\frac{\omega_{f}^{2}}{c^{2}}\mathrm{Im}\overleftrightarrow{G}\left(\mathbf{r},\mathbf{r}_{s};\omega_{f}\right)\cdot\mathbf{d}_{s}^{*}}\\ &{\times\hat{\sigma}_{s}^{12}\left(t\right)\left(\pi\delta\left(\omega_{f}-\omega_{s}\right)+i\mathcal{P}\frac{1}{\omega_{s}-\omega_{f}}\right).}\end{array}\] |
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\[\begin{array}{r l r l}{\Lambda_{j}}&{=\frac{m_{0}m_{j}}{m_{0}+m_{j}}\sqrt{\mathcal{G}(m_{0}+m_{j})a_{j}}\ ,\quad}&{\lambda_{j}}&{=M_{j}+\omega_{j}\ ,}\\ {\xi_{j}}&{=\sqrt{2\Lambda_{j}}\sqrt{1\!-\!\sqrt{1\!-\!e_{j}^{2}}}\cos(\omega_{j})\ ,}&{\eta_{j}}&{=-\sqrt{2\Lambda_{j}}\sqrt{1\!-\!\sqrt{1\!-\!e_{j}^{2}}}\sin(\omega_{j})\ ,}\end{array}\] |
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\[{\begin{array}{r l}{y[m]}&{=\sum_{k=-\infty}^{\infty}x_{L}\left[{\bigl\lfloor}{\frac{m}{L}}{\bigr\rfloor}L-k L\right]\cdot h{\Bigl[}\overbrace{m-{\bigl\lfloor}{\frac{m}{L}}{\bigr\rfloor}L+k L}^{r}{\Bigr]}}\\ &{=\sum_{k=-\infty}^{\infty}x\left[{\bigl\lfloor}{\frac{m}{L}}{\bigr\rfloor}-k\right]\cdot h\left[m-{\bigl\lfloor}{\frac{m}{L}}{\bigr\rfloor}L+k L\right]\quad{\stackrel{m\ \triangleq\ j+n L}{\longrightarrow}}\quad y[j+n L]=\sum_{k=0}^{K}x[n-k]\cdot h[j+k L],\ \ j=0,1,\ldots,L-1\quad{\mathsf{(E q.1)}}}\end{array}}\] |