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\[\begin{array}{r l}&{\langle u|u|^{2},b_{j,r+1}\rangle}\\ &{=\sum_{k,l,m}\frac{A_{k}A_{l}A_{m}}{L_{k}L_{l}L_{m}}\left\langle e^{i\Gamma_{k}+i\Gamma_{l}-i\Gamma_{m}}e^{-\frac{|y_{k}|^{2}+|y_{l}|^{2}+|y_{m}|^{2}}{2}},e^{i\Gamma_{j}}e^{-\frac{1}{2}|y_{j}|^{2}}\frac{x_{r}-(X_{j})_{r}}{L_{j}}\right\rangle}\\ &{=\sum_{k,l,m}\frac{C}{L_{j}}\left(\widehat{x_{r}f}-(X_{j})_{r}\widehat{f}\right).}\end{array}\] |
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\[\begin{array}{r l r}{c p_{1}}&{=2m_{0}c^{2}-c\left(\frac{p_{1}-p_{y}}{\cos(\phi)\cos(\psi)}\right)}&{\mathrm{Substitution~of~}\mathrm{~in~}}\\ {c p_{1}+\frac{c p_{1}}{\cos(\phi)\cos(\psi)}}&{=2m_{0}c^{2}+c\left(\frac{p_{y}}{\cos(\phi)\cos(\psi)}\right)}&\\ {c p_{1}\left(\frac{\cos(\phi)\cos(\psi)+1}{\cos(\phi)\cos(\psi)}\right)}&{=2m_{0}c^{2}+c\left(\frac{p_{y}}{\cos(\phi)\cos(\psi)}\right)}&\\ {c p_{1}}&{=\frac{2m_{0}c^{2}\cos(\phi)\cos(\psi)+c p_{y}}{1+\cos(\phi)\cos(\psi)}}&\\ &{\approx m_{0}c^{2}+0.5c p_{y}}&{\mathrm{With~}\cos(\phi)\approx1.0,\cos(\psi)\approx1.0}\end{array}\] |
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\[\underbrace{(w^{h},i\omega\phi^{h})+(w^{h},a\phi_{,x}^{h})+(w_{,x}^{h},\kappa\phi_{,x}^{h})}_{\mathrm{Baseline~Galerkin}}+\underbrace{\sum_{e}\Big(\tau a w_{,x}^{h},r(\phi^{h})\Big)_{\Omega_{e}}}_{\mathrm{The~SUPG~terms}}-\underbrace{\sum_{e}\Big(i\omega\tau w^{h},r(\phi^{h})\Big)_{\Omega_{e}}}_{\mathrm{The~new~VMS~terms}}=0.\] |
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\[\begin{array}{r}{R_{k}^{\mathrm{ub}}=\ln\operatorname*{det}\left(\sum_{i=1}^{K}{\uppercase{\mathbf{P}}}_{i}{\uppercase{\mathbf{M}}}_{i,k}{\uppercase{\mathbf{P}}}_{i}^{H}+{\uppercase{\mathbf{I}}}_{M_{e}}\right)-\ln\operatorname*{det}\left(\sum_{i=1}^{K}{\uppercase{\mathbf{P}}}_{i}{\uppercase{\mathbf{M}}}_{i,k}{\uppercase{\mathbf{P}}}_{i}^{H}+{\uppercase{\mathbf{I}}}_{M_{e}}-{\uppercase{\mathbf{P}}}_{k}{\uppercase{\mathbf{N}}}_{k,k}{\uppercase{\mathbf{P}}}_{k}^{H}\right),}\end{array}\] |
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\[\operatorname*{limsup}_{\varepsilon\downarrow0}\frac{\log N_{U}(\varepsilon,\mathfrak{F}_{\gamma})}{\varepsilon^{-V}}<\infty,\qquad\mathbb{E}[\bar{F}_{\gamma}(\mathbf{Z})^{2}]<\infty,\qquad\operatorname*{limsup}_{\delta\downarrow0}\frac{\mathbb{E}[\bar{D}_{\gamma}^{\delta}(\mathbf{Z})^{2}+\bar{D}_{\gamma}^{\delta}(\mathbf{Z})^{4}]}{\delta}<\infty.\] |
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\[{\begin{array}{r l}{{\hat{\mathbf{r}}}}&{={\frac{\rho{\hat{\boldsymbol{\rho}}}+z{\hat{\mathbf{z}}}}{\sqrt{\rho^{2}+z^{2}}}}}\\ {{\hat{\boldsymbol{\theta}}}}&{={\frac{z{\hat{\boldsymbol{\rho}}}-\rho{\hat{\mathbf{z}}}}{\sqrt{\rho^{2}+z^{2}}}}}\\ {{\hat{\boldsymbol{\varphi}}}}&{={\hat{\boldsymbol{\varphi}}}}\end{array}}\] |
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\[{\mathbb{T L}}_{N_{2}^{k-1}}^{\!{\mathbb Z}_{(p)}}\subseteq{\mathbb{T L}}_{N_{2}^{k-2}}^{\!{\mathbb Z}_{(p)}}\subseteq\cdots\subseteq{\mathbb{T L}}_{N_{2}^{0}}^{\!{\mathbb Z}_{(p)}}={\mathbb{T L}}_{N}^{\!{\mathbb Z}_{(p)}}={\mathbb{T L}}_{n}^{\!{\mathbb Z}_{(p)}}\] |
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\[\left\{\begin{array}{l l l}&{d X_{N,t}^{i,N}=A X_{N,t}^{i,N}\,d t+A^{\prime}\frac1N\sum_{k=1}^{N}X_{N,t}^{k,N}\,d t+B\,d Z_{N,t}^{i},\quad t\in[0,T],\quad i\in\{1,\dots,N\},}\\ &{\overline{{\mu}}_{N,t}^{N}:=\frac{1}{N}\sum_{j=1}^{N}\delta_{X_{N,t}^{j,N}},}\\ &{X_{N,0}^{i,N}=\xi^{i},}\end{array}\right.\] |
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\[\Big|S(N)-\Tilde{S}(N)\Big|\leq\int_{0}^{1}\Big|1-\tilde{I}_{\Phi,\delta}(x)\Big|\Big|\sum_{n\in\mathbf{Z}}\lambda_{f}(n)e(x n)h\Big(\frac{n}{N}\Big)\Big|\Big|\sum_{m\in\mathbf{Z}}\lambda_{g}(m)\chi(m)e(x m)h^{*}\Big(\frac{m}{N}\Big)\Big|d x.\] |
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\[\begin{array}{r l}{\left((\mathbb{R}_{+}^{N_{A}})^{\ast}\otimes(\mathbb{R}_{+}^{N_{B}})^{\ast}\right)^{\ast}=}&{\left(\mathbb{R}_{+}^{N_{A}}\otimes\mathbb{R}_{+}^{N_{B}}\right)^{\ast}=(\mathbb{R}_{+}^{N_{A}N_{B}})^{\ast}=\mathbb{R}_{+}^{N_{A}N_{B}}.}\end{array}\] |
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\[\begin{array}{r l}{C R^{-2}\eta^{-1}\delta^{-1}}&{\geq-C\left(1+\frac{\delta}{|\nabla\tilde{u}_{0}(x_{0})|}\right)\left(|\nabla\xi_{G}|+R^{4-\frac{n+1-2\delta_{0}}{n+1+2\delta_{0}}}\omega\right)+C(\delta^{6}+\delta^{4})}\\ &{\geq C\left[\left(1+\frac{\delta}{2\delta}\right)\left((R\eta)^{-1}+R^{4-\frac{n+1-2\delta_{0}}{n+1+2\delta_{0}}}\omega\right)+\delta^{4}\right].}\end{array}\] |
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\[\begin{array}{r l r}&{}&{\left.\left\langle v_{1}^{\prime\mu}\right\rangle_{\phi^{\prime}}=\left\langle u^{\prime\alpha}v_{1\alpha}^{\prime}u^{\prime\mu}\right\rangle_{\phi^{\prime}}\right.}\\ &{}&{\left.=\left\langle v_{1\alpha}^{\prime}\right\rangle_{\phi^{\prime}}\left\langle u^{\prime\alpha}u^{\prime\mu}\right\rangle_{\phi^{\prime}}+\left\langle u^{\prime\alpha}\left[v_{1\alpha}^{\prime}\right]^{\sim}u^{\prime\mu}\right\rangle_{\phi^{\prime}},\right.}\end{array}\] |
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\[\begin{array}{r}{\|\tilde{u}^{n_{k}}-\tilde{u}\|_{L^{2}([0,T],H)}+\|\tilde{u}_{T}^{n_{k}}-\tilde{u}_{T}\|_{H}+\int_{0}^{T}\|G^{n_{k}}(t,\tilde{u}_{t}^{n_{k}})-\hat{G}(t,\tilde{u}_{t})\|_{H}^{2}d t\rightarrow0,}\end{array}\] |
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\[\begin{array}{r l}{(-1)^{j+1}d x^{1}\wedge\dots\wedge\widehat{d x}^{j}\wedge\dots\wedge d x^{n}}&{=(-1)^{j+2}\frac{b_{j}}{b_{1}}d x^{i}\wedge d x^{2}\wedge\dots\wedge\widehat{d x}^{j}\wedge\dots\wedge d x^{n}}\\ &{=+\frac{b_{j}}{b_{1}}d x^{2}\wedge\dots\wedge d x^{n},}\end{array}\] |
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\[\|a\|_{[A_{0},A_{1}]_{r,q}}:=\left\{\begin{array}{l l}{\left(\int_{0}^{\infty}z^{-r q}K(a,z;A_{0},A_{1})^{q}\frac{1}{z}d z\right)^{\frac{1}{q}},}&{\quad q\in[1,\infty),}\\ {\operatorname*{sup}_{z>0}z^{-r}K(a,z;A_{0},A_{1}),}&{\quad q=\infty.}\end{array}\right.\] |
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\[\begin{array}{r l}{J\alpha}&{=\sum_{k<l}\alpha_{k l}J d z^{k}\wedge J d z^{l}=\sum_{i,j}\sum_{k<l}\alpha_{k l}J_{\bar{i}}^{k}J_{\bar{j}}^{l}d\Bar{z}^{i}\wedge d\Bar{z}^{j}}\\ &{=\sum_{i<j}\sum_{k<l}\alpha_{k l}(J_{\bar{i}}^{k}J_{\bar{j}}^{l}-J_{\bar{j}}^{k}J_{\bar{i}}^{l})d\Bar{z}^{i}\wedge d\Bar{z}^{j}.}\end{array}\] |
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\[\begin{array}{r l}{{R_{\alpha}}(P_{\beta_{0}}}&{|P_{\beta_{1}},P_{\beta_{2}})=\exp\Bigg(\log\left(\frac{\theta_{0}^{1-2\alpha}\theta_{1}^{\alpha}\theta_{2}^{\alpha}+(1-\theta_{0})^{1-2\alpha}(1-\theta_{1})^{\alpha}(1-\theta_{2})^{\alpha}}{(1-\theta_{0})^{1-2\alpha}(1-\theta_{1})^{\alpha}(1-\theta_{2})^{\alpha}}\right)}\\ &{\quad-\log\left(\frac{\theta_{0}^{1-\alpha}\theta_{1}^{\alpha}+(1-\theta_{0})^{1-\alpha}(1-\theta_{1})^{\alpha}}{(1-\theta_{0})^{1-\alpha}(1-\theta_{1})^{\alpha}}\right)}\\ &{\quad-\log\left(\frac{\theta_{0}^{1-\alpha}\theta_{2}^{\alpha}+(1-\theta_{0})^{1-\alpha}(1-\theta_{2})^{\alpha}}{(1-\theta_{0})^{1-\alpha}(1-\theta_{2})^{\alpha}}\right)+\log(1-\theta_{0})\Bigg)-1}\\ &{=\frac{\theta_{0}^{1-2\alpha}\theta_{1}^{\alpha}\theta_{2}^{\alpha}+(1-\theta_{0})^{1-2\alpha}(1-\theta_{1})^{\alpha}(1-\theta_{2})^{\alpha}}{\big(\theta_{0}^{1-\alpha}\theta_{1}^{\alpha}+(1-\theta_{0})^{1-\alpha}(1-\theta_{1})^{\alpha}\big)\big(\theta_{0}^{1-\alpha}\theta_{2}^{\alpha}+(1-\theta_{0})^{1-\alpha}(1-\theta_{2})^{\alpha}\big)}-1,}\end{array}\] |
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\[\begin{array}{r l}{{\sf p}_{i}^{\ell}=}&{\;(c_{p}-c_{v})\rho_{i}^{\ell}\theta_{i}^{\ell},}\\ {{\sf e}_{i}^{\ell}=}&{\;c_{v}\theta_{i}^{\ell},}\\ {{\sf s}_{i}^{\ell}=}&{\;c_{v}\log(\theta_{i}^{\ell})-(c_{p}-c_{v})\log(\rho_{i}^{\ell})+c_{v},}\end{array}\] |
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\[\begin{array}{r l}&{\hat{\nu}_{n,m}(\textbf{t})=\frac{1}{k}\sum_{i=1}^{k}\left[\bigvee_{j=1}^{d}\left\{\hat{U}_{n,m,i,j}\right\}^{1/t_{j}}-\frac{1}{d}\sum_{j=1}^{d}\left\{\hat{U}_{n,m,i,j}\right\}^{1/t_{j}}\right],\quad c(\textbf{t})=\frac{1}{d}\sum_{j=1}^{d}\frac{t_{j}}{1+t_{j}},}\end{array}\] |
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\[\begin{array}{r}{\nabla y_{\lambda}^{*}(x)=-\left(\frac{1}{\lambda}\nabla_{y y}^{2}f(x,y_{\lambda}^{*}(x))+\nabla_{y y}^{2}g(x,y_{\lambda}^{*}(x))\right)^{-1}\left(\frac{1}{\lambda}\nabla_{x y}^{2}f(x,y_{\lambda}^{*}(x))+\nabla_{x y}^{2}g(x,y_{\lambda}^{*}(x))\right).}\end{array}\] |
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\[\begin{array}{r l}{\tilde{R}=}&{-2\left(U^{\prime}+\frac{1}{r}\sum_{i=1}^{3}\alpha_{i}\left(1-R_{i}\right)U+\frac{(W+Z)}{r C A}U-\frac{\left(W^{\prime}+Z^{\prime}\right)}{r C}A+24\lambda^{2}\right.}\\ &{\left.-\frac{(W+Z)A}{r^{2}C}\sum_{i=1}^{3}\alpha_{i}\left(1-R_{i}\right)+\frac{\left(1-W Z-\alpha_{3}\left(1+R_{3}\right)+\sum_{i=1}^{2}\alpha_{i}\left(1-R_{i}\right)\cot\theta^{2}\right)}{r^{2}C^{2}}\right).}\end{array}\] |
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\[\begin{array}{r l}{\partial S}&{=\sum_{i}\partial e_{i}-\partial C}\\ &{=\sum_{i}\left(a_{i}+b_{i}+\phi_{[0,T^{s_{i}}(w_{i})]}(w_{i})-\phi_{[0,T^{s_{i}}(v_{i})_{i}]}(v_{i})\right)-c}\\ &{=\sum_{i}\phi_{[0,T^{s_{i}}(v_{i})_{i}]}(v_{i})-\sum_{i}\phi_{[0,T^{s_{i}}(w_{i})]}(w_{i})}\\ &{=\phi_{[0,T^{r}(v_{1})]}(v_{1})-\sum_{i}\phi_{[0,T^{s_{i}}(w_{i})]}(w_{i})}\\ &{=\gamma_{r}-\sum_{i}\gamma_{s_{i}}}\end{array}\] |
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\[\begin{array}{r l}{h_{i,\upsilon}=}&{h-\nabla_{x}F_{i}(x,y_{+};\xi_{\upsilon}^{i})+\nabla_{x}F_{i}(x_{\upsilon}^{i},y_{+};\xi_{\upsilon}^{i})}\\ {=}&{\widetilde h^{D}(x)-\widetilde h^{I}(x)-\nabla_{x}F_{i}(x,y_{+};\xi_{\upsilon}^{i})}\\ &{+\nabla_{x}F_{i}(x_{\upsilon}^{i},y_{+};\xi_{\upsilon}^{i}),}\end{array}\] |
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\[\begin{array}{r l}{(\mathcal{M}^{j}(\nabla\boldsymbol{X})Z)_{k}({\boldsymbol{\theta}})}&{=-\int_{{\mathbb{R}^{2}}}m_{m,k,l}^{j}({\boldsymbol{\theta}},{\boldsymbol{\eta}})\frac{\partial\widehat{X}_{i}}{\partial\eta_{m}}({\boldsymbol{\eta}})Z_{l,i}({\boldsymbol{\eta}})d\eta_{1}d\eta_{2}.}\end{array}\] |
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\[\begin{array}{r l}{r_{P}^{2}=\sum_{i}\alpha_{i}\|p_{i}\|^{2}}&{=\Big\|\sum_{i}\alpha_{i}p_{i}\Big\|^{2}\!+\frac12\sum_{i,j}\alpha_{i}\alpha_{j}\|p_{i}-p_{j}\|^{2}}\\ &{\ge\Big\|\sum_{i}\alpha_{i}q_{i}\Big\|^{2}+\frac12\sum_{i,j}\alpha_{i}\alpha_{j}\|q_{i}-q_{j}\|^{2}=\sum_{i}\alpha_{i}\|q_{i}\|^{2}=r_{Q}^{2}.}\end{array}\] |
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\[\begin{array}{r l}{I+\beta_{k}F^{\prime}(\alpha)}&{=2C_{2}e_{k}+C_{3}e_{k-1}^{2}-2C_{3}e_{k-1}e_{k}-4\left(C_{2}^{2}-C_{3}\right)e_{k}^{2}+O_{3}\left(e_{k},e_{k-1}\right),}\\ {2I-\delta_{k}F^{\prime}(\alpha)}&{=4C_{2}e_{k}+2C_{3}e_{k-1}^{2}-4C_{3}e_{k-1}e_{k}-8\left(C_{2}^{2}-C_{3}\right)e_{k}^{2}+O_{3}\left(e_{k},e_{k-1}\right).}\end{array}\] |
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\[\operatorname*{lim}_{n\rightarrow\infty}\bigg\vert\mathbb{\hat{E}}\bigg[\varphi\left(z+(s/n)^{\frac{1}{\alpha}}S_{n}^{3}\right)-\varphi(z)\bigg]-s\operatorname*{sup}_{F_{\mu}\in\mathcal{L}}\int_{\mathbb{R}^{d}}\delta_{\lambda}\varphi(z)F_{\mu}(d\lambda)\bigg\vert=o(s),\] |
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\[\begin{array}{r l r}&{}&{H=\Omega/\sqrt{2}(|g g\rangle\langle r_{0}g^{+}|+h.c.)+\Omega/\sqrt{2}(|r_{0}g^{+}\rangle\langle r_{0}r_{0}|+h.c.)}\\ &{}&{+\sum_{i,j,k,l}\frac{C_{3_{i j,k l}}(\theta)}{R^{3}}(|r_{l}r_{k}\rangle\langle r_{i}r_{j}|+h.c.)+\sum_{i,j}\delta_{i j}|r_{i}r_{j}\rangle\langle r_{i}r_{j}|}\end{array}\] |
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\[\begin{array}{r}{\mathrm{i}\left(\overline{{\omega}}\cdot l-\left(\left(-\frac{1}{2}\lambda_{\alpha}(k+j)-(k+j)\frac{T_{\alpha}}4\right)-\left(-\frac{1}{2}\lambda_{\alpha}(j)-j\frac{T_{\alpha}}4\right)\right)\right)\widehat{\rho}_{1,s}^{\varphi,x}(\omega,l,k,j)=\widehat{\mathfrak{b}}_{1,s}^{\varphi,x}(\omega,l,k,j),}\end{array}\] |
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\[\partial_{2}F(x,\phi(x),\phi^{\prime}(x),\phi^{\prime\prime}(x))-\frac{\mathrm{d}}{\mathrm{d}x}(\partial_{3}F(x,\phi(x),\phi^{\prime}(x),\phi^{\prime\prime}(x)))+\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}(\partial_{4}F(x,\phi(x),\phi^{\prime}(x),\phi^{\prime\prime}(x)))=0.\] |
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\[\begin{array}{r l r l}{F_{1}^{(l)}(\zeta,t)}&{=-i(\omega k_{4})^{l}Z_{1}(\zeta,t)+O(t^{-1})}&&{\mathrm{as~}t\to\infty,}\\ {F_{2}^{(l)}(\zeta,t)}&{=-i(\omega^{2}k_{2})^{l}Z_{2}(\zeta,t)+O(t^{-1})}&&{\mathrm{as~}t\to\infty,}\end{array}\] |
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\[\begin{array}{r l}&{\mathbb{E}[\mathrm{env}_{\psi}^{\eta}(x^{k+1})\vert\mathcal{F}_{k}]\leq\mathrm{env}_{\psi}^{\eta}(x^{k})+\frac{\eta}{2}\big[\alpha_{k}(\rho-\eta^{-1}-\lambda)\big]\|\nabla\mathrm{env}_{\psi}^{\eta}(x^{k})\|^{2}+\frac{\alpha_{k}^{2}}{2\eta}\theta\beta.}\end{array}\] |
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\[\begin{array}{r l}{m_{2}}&{(m_{2}(x_{1},x_{2}),x_{3})-m_{2}(x_{1},m_{2}(x_{2},x_{3}))=}\\ &{d m_{3}(x_{1},x_{2},x_{3})+m_{3}(d x_{1},x_{2},x_{3})+m_{3}(x_{1},d x_{2},x_{3})+m_{3}(x_{1},x_{2},d x_{3}).}\end{array}\] |
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\[\begin{array}{r l r}{\mathrm{Q}_{j}^{(1)}}&{=}&{(L/2-1)/2-(j-1),~~j=1,\dots,L/2}\\ {\mathrm{Q}_{j}^{(2)}}&{=}&{(L-1)/2-(j-1),~~j=1,\dots,L}\\ {\mathrm{Q}_{j}^{(3)}}&{=}&{(L/2-1)/2-(j-1),~~j=1,\dots,L/2}\end{array}\] |
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\[\begin{array}{r l}{\mathcal{G}_{\bar{t}_{s}}-\mathcal{G}_{\bar{t}_{s-1}}}&{\leq-\frac{\eta}{2}\sum_{t=\bar{t}_{s-1}}^{\bar{t}_{s}-1}\mathbb{E}\|\nabla h(\bar{x}_{t})\|^{2}-\frac{5\eta\hat{L}^{2}}{2}\sum_{t=\bar{t}_{s-1}}^{\bar{t}_{s}-1}B_{t}-\frac{\eta}{8}\sum_{t=\bar{t}_{s-1}}^{\bar{t}_{s}-1}E_{t}}\\ &{\qquad+\left(\frac{360}{\mu^{2}\gamma^{2}}+4I\right)\eta^{3}\kappa^{2}\hat{L}^{2}I\sum_{t=\bar{t}_{s-1}}^{\bar{t}_{s}-1}D_{t}+\frac{27I\hat{L}^{2}\gamma\eta\sigma^{2}}{b_{y}\mu}}\\ &{\qquad+\frac{90I\kappa^{2}\hat{L}^{2}\eta^{3}G_{2}^{2}}{\mu^{2}\gamma^{2}b_{x}M}+\frac{I\eta^{2}\bar{L}G_{2}^{2}}{2b_{x}M}+I\eta G_{1}^{2}}\end{array}\] |
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\[\begin{array}{r l}{H:\,}&{\mathbb{R}^{D}\to\operatorname{End}(\mathcal{H})}\\ &{\left(x^{a}\right)\mapsto H_{x}:=\frac{1}{2}\sum_{a}\left(X^{a}-x^{a}\mathbb{1}\right)^{2}=\frac{1}{2}\sum_{a,b}\delta_{a b}\left(X^{a}-x^{a}\mathbb{1}\right)\left(X^{b}-x^{b}\mathbb{1}\right),}\end{array}\] |
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\[\begin{array}{r l r}{I_{3,i}}&{\leq}&{\int_{B}|\partial_{i}u|f_{i}^{\frac{1}{2}}(\partial_{i}u)\Gamma^{\frac{\gamma}{2}}(|\partial_{i}u|)\eta^{k}f_{i}^{\frac{1}{2}}(\partial_{i}u)\Gamma^{\frac{\gamma}{2}}(|\partial_{i}u|)\eta^{k-1}|\nabla\eta|\,\mathrm{d}x}\\ &{\leq}&{\varepsilon\int_{B}|\partial_{i}u|^{2}f_{i}(\partial_{i}u)\Gamma^{\gamma}(|\partial_{i}u|)\eta^{2k}d x}\\ &{}&{+c(\varepsilon,r)\int_{B}f_{i}(\partial_{i}u)\Gamma^{\gamma}(|\partial_{i}u|)\eta^{2k-2}\,\mathrm{d}x\,.}\end{array}\] |
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\[\begin{array}{r l r}{2^{\rho-1}\sum_{k=1}^{\mu-\rho-1}2^{\nu-k-1}k}&{=}&{2^{\nu+2\rho-\mu-1}\sum_{k=1}^{\mu-\rho-1}2^{\mu-\rho-k-1}k}\\ &{=}&{2^{\nu+2\rho-\mu-1}\sum_{i=0}^{\mu-\rho-2}2^{i}(\mu-\rho-i-1)}\\ &{=}&{2^{\nu+2\rho-\mu-1}\left((\mu-\rho-1)\sum_{i=0}^{\mu-\rho-2}2^{i}-\sum_{i=0}^{\mu-\rho-2}2^{i}i\right)}\\ &{=}&{2^{\nu+2\rho-\mu-1}\left((\mu-\rho-1)(2^{\mu-\rho-1}-1)-((\mu-\rho-3)2^{\mu-\rho-1}+2)\right)}\\ &{=}&{2^{\nu+2\rho-\mu-1}(2^{\mu-\rho}-\mu+\rho-1).}\end{array}\] |
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\[\begin{array}{r l}{S(s)}&{=\left(\begin{array}{c c}{\vert\frac s\eta\vert^{\gamma_{1}}}&{\vert\frac s\eta\vert^{\gamma_{2}}}\\ {-\frac{\gamma_{1}}{c}\frac s\eta\vert\frac s\eta\vert^{\gamma_{1}-2}}&{-\frac{\gamma_{2}}{c}\frac s\eta\vert\frac s\eta\vert^{\gamma_{2}-2}}\end{array}\right),}\end{array}\] |
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\[\begin{array}{r l r}{{\rho_{f}}}&{{=}}&{{\rho_{e}\,\mathrm{,}}}\\ {{u_{f}}}&{{=}}&{{u_{e}\,\mathrm{,}}}\\ {{v_{f}}}&{{=}}&{{v_{e}\,\mathrm{,}}}\\ {{w_{f}}}&{{=}}&{{w_{e}\,\mathrm{,}}}\\ {{e_{f}}}&{{=}}&{{e_{e}\,\mathrm{,}}}\end{array}\] |
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\[\left\{\begin{array}{r l}&{{{{\bar{a}}}_{1}}=-\frac{2{{{\bar{Q}}}_{i-2}}+{{{\bar{Q}}}_{i-1}}-{{{\bar{Q}}}_{i+1}}-2{{{\bar{Q}}}_{i+2}}}{10\Delta x}}\\ &{{{{\bar{a}}}_{2}}=\frac{4{{{\bar{Q}}}_{i-2}}+{{{\bar{Q}}}_{i-1}}-10{{{\bar{Q}}}_{i}}+{{{\bar{Q}}}_{i+1}}+4{{{\bar{Q}}}_{i+2}}}{34\Delta{{x}^{2}}}}\end{array}\right..\] |
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\[\begin{array}{r l}&{\chi_{L[n]}=(-1)^{n}\chi_{L},\ \chi_{L(\frac{n}{2})}=\sqrt{q}^{-n}\chi_{L},\ \chi_{L\boxtimes L^{\prime}}=\chi_{L}\otimes\chi_{L^{\prime}},}\\ &{\chi_{f_{!}(L)}=f_{!}(\chi_{L}),\ \chi_{f^{*}(L^{\prime})}=f^{*}(\chi_{L^{\prime}}).}\end{array}\] |
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\[\begin{array}{r l}{\mathbb{P}\left(\left|\frac{1}{s}\left\langle{\boldsymbol Y}_{*j}^{(k)},\tilde{{\boldsymbol u}}^{(k)}\right\rangle-\|\tilde{{\boldsymbol u}}^{(k)}\|_{2}\|{\boldsymbol p}\|_{2}r_{j}^{(k)}\cos\theta\right|>\frac{\delta_{1}\|{\boldsymbol p}\|_{2}}{2}\right)}&{\leq2\exp\left(-\frac{\Theta(\delta_{1}^{2}\|{\boldsymbol p}\|_{2}^{2})}{\Theta\left(\frac{1}{s}\right)+\Theta\left(\delta_{1}\|{\boldsymbol p}\|_{2}/\sqrt{n}\right)}\right)}\\ &{\leq\exp\left(-\Theta(s\delta_{1}^{2}\|{\boldsymbol p}\|_{2}^{2})\right)}\end{array}\] |
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\[\begin{array}{r l}{\rho_{\ell-1}(0)}&{=\frac{t_{1}^{\ell}-t_{0}^{\ell}}{t_{2}^{\ell}-t_{0}^{\ell}}=\frac{\sin^{2}\left(\frac{\pi}{2}\cdot\frac{1}{2^{\ell}}\right)-\sin^{2}(0)}{\sin^{2}\left(\frac{\pi}{2}\cdot\frac{2}{2^{\ell}}\right)-\sin^{2}(0)}=\frac{\sin^{2}(\frac{1}{x})}{\sin^{2}(\frac{2}{x})}}\end{array}\] |
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\[\begin{array}{r}{\mathbb{E}\big[V_{\epsilon}^{t-1}\big]-\mathbb{E}\big[U_{\epsilon}^{t-1}\big]\geq\frac{t}{q_{\epsilon}^{t}-1}+\frac{p_{\epsilon}^{t-1}(p_{\epsilon}-1)-(p_{\epsilon}+1)}{2(p_{\epsilon}+1)(p_{\epsilon}^{t}-p_{\epsilon}^{t-1})}>0.}\end{array}\] |
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\[\begin{array}{r l}{\left\|{\mathcal I}^{k,\beta}[p](.,.;t)\right\|_{L^{1}({\mathbb R}^{d+1},d m d x)}\leq}&{D\|B\|_{\infty}2^{(d+1)/2}\int_{0}^{t}\int_{{\mathbb R}^{d+1}}\frac{1}{\sqrt{t-s}}|p(m,a;s)|d m d a d s}\\ {\leq}&{D\|B\|_{\infty}2^{(d+1)/2}\int_{0}^{t}\frac{1}{\sqrt{t-s}}\operatorname*{sup}_{u\leq s}\|p(.,.;u)\|_{L^{1}(\mathbb{R}^{d+1},d m d a)}d s,}\end{array}\] |
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\[\mathcal{S}_{2}:=\left\lbrace\left\lVert\frac{1}{T}\sum_{t=1}^{T}\boldsymbol{\epsilon}_{t}\boldsymbol{\epsilon}_{t}^{\prime}-\frac{1}{T}\sum_{t=1}^{T}\mathbb{E}\boldsymbol{\epsilon}_{t}\boldsymbol{\epsilon}_{t}^{\prime}\right\rVert_{\operatorname*{max}}\leq\frac{N^{4/m}}{T^{(m-2)/m}}\eta_{T}^{-1}\right\rbrace\] |
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\[{\begin{array}{r l}{\underbrace{X_{1/T}\left({\frac{k}{N T}}\right)}_{X_{k}}}&{=\sum_{n=-\infty}^{\infty}x[n]\cdot e^{-i2\pi{\frac{k}{N}}n}\quad\quad k=0,\dots,N-1}\\ &{=\underbrace{\sum_{N}x_{_{N}}[n]\cdot e^{-i2\pi{\frac{k}{N}}n},}_{\mathrm{DFT}}\quad\scriptstyle{{\mathrm{(sum~over~any~}}n{\mathrm{-sequence~of~length~}}N)}}\end{array}}\] |
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\[\begin{array}{r l}&{\mathbb{E}[|\xi_{b,i}|e^{W_{b}^{(i)}/2}(|\bar{\Delta}_{1n,x}-\bar{\Delta}_{1n,x}^{(i)}|+|\bar{\Delta}_{2n,x}-\bar{\Delta}_{2n,x}^{(i)}|)]}\\ &{\leq C\mathbb{E}[|\xi_{b,i}|e^{W_{b}^{(i)}/2}(|\bar{D}_{1n}-\bar{D}_{1n}^{(i)}|+x|\bar{D}_{2n}-\bar{D}_{2n}^{(i)}|)]}\\ &{\leq C\Big\{\|\xi_{b,i}e^{W_{b}^{(i)}/2}\|_{p_{1}}\|\bar{D}_{1n}-\bar{D}_{1n}^{(i)}\|_{q_{1}}+x\|\xi_{b,i}e^{W_{b}^{(i)}/2}\|_{p_{2}}\|\bar{D}_{2n}-\bar{D}_{2n}^{(i)}\|_{q_{2}}\Big\}}\\ &{\leq C\Big\{\|\xi_{i}\|_{p_{1}}\|\bar{D}_{1n}-\bar{D}_{1n}^{(i)}\|_{q_{1}}+x\|\xi_{i}\|_{p_{2}}\|\bar{D}_{2n}-\bar{D}_{2n}^{(i)}\|_{q_{2}}\Big\},}\end{array}\] |
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\[\tilde{w}_{\mathrm{min}}=-\frac{\tilde{F}}{2}\times\left\{\begin{array}{l l}{\sqrt{\frac{1}{2}+\tilde{a}(1+\tilde{a})}e^{\arctan(1+2\tilde{a})-\pi}}&{\mathrm{Case~A}}\\ {\frac{\tilde{L}^{2}-2\tilde{a}^{2}+\tilde{L}\tilde{a}}{(\tilde{L}-\tilde{a})^{3}}}&{\mathrm{Case~C.}}\end{array}\right.\] |
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\[\begin{array}{r l}&{{\bf M}_{\epsilon}^{1}\partial_{t}\mathbf{e}=\widetilde{\bf K}_{2}\widetilde{\mathbf{D}}^{1}\mathbf{h}-\widetilde{\bf K}_{2}\mathbf{j},}\\ &{\widetilde{{\bf M}}_{\mu}^{1}\partial_{t}\mathbf{h}=-\mathbf{K}_{2}\mathbf{D}^{1}\mathbf{e}.}\end{array}\] |
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\[\begin{array}{r l}{\left\vert\sum_{\substack{\pi\in\mathcal{S}_{4}:(\pi,G_{0})\in\mathcal{L}_{1}^{3}}}\Gamma_{\pi,G_{0}}^{3}\right\vert}&{\leq\int|g_{14}||g_{24}|\left\vert\rho_{\mathrm{t}}^{(\{1,2,4\},\{3\})}\right\vert\,\textnormal{d}x_{4}\leq C a b^{2}\rho^{8}|x_{1}-x_{2}|^{2}}\\ {\left\vert\sum_{\substack{\pi\in\mathcal{S}_{4}:(\pi,G_{1})\in\mathcal{L}_{1}^{3}}}\Gamma_{\pi,G_{1}}^{3}\right\vert}&{\leq\int|g_{14}||g_{24}||g_{34}|\rho^{(4)}\,\textnormal{d}x_{4}\leq C a b^{2}\rho^{8}|x_{1}-x_{2}|^{2}.}\end{array}\] |
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\[\begin{array}{r l r}{\sum_{j_{1}=1,\cdots,j_{N}=1}^{J_{1},\cdots,J_{N}}\mathbb{E}f_{\mathcal{X}}(\theta_{j_{1},\cdots,j_{N}})}&{=}&{\frac{\prod_{j=1}^{N}J_{j}}{\prod_{i=1}^{M}I_{i}}\sum_{j_{1}=1,\cdots,j_{N}=1}^{J_{1},\cdots,J_{N}}\mathbb{E}(\theta_{j_{1},\cdots,j_{N}})\left\vert\langle\mathcal{U}_{j_{1},\cdots,j_{N}},\mathcal{X}\rangle\right\vert^{2}}\\ &{=}&{\sum_{j_{1}=1,\cdots,j_{N}=1}^{J_{1},\cdots,J_{N}}\left\vert\langle\mathcal{U}_{j_{1},\cdots,j_{N}},\mathcal{X}\rangle\right\vert^{2}}\\ &{=}&{\left\Vert\mathcal{U}\star_{N}\mathcal{X}\right\Vert_{2}^{2}=1.}\end{array}\] |
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\[\begin{array}{r l}{\|E\|}&{\;\leq\;\kappa\,\|[H,D_{0}]\|\;+\;2\,\|H^{s}\|\,\operatorname*{max}\big\{\|[H,G_{\rho}(|D_{0}|)]\|,\|H,G_{\rho}(|D_{0}^{*}|)]\|\big\}}\\ &{\;\leq\;\kappa\,\|[H,D_{0}]\|\;+\;\frac{16}{\rho}\,\|H^{s}\|\,\operatorname*{max}\big\{\|[H,|D_{0}|]\|,\|H,|D_{0}^{*}|]\|\big\}\;.}\end{array}\] |
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\[\begin{array}{r l}{S_{0,3}^{\ast}(\xi_{i};\alpha_{i})}&{=(\alpha_{1}^{2}+\alpha_{2}^{2}-\alpha_{3}^{2})\log|\xi_{1}-\xi_{2}|+(\alpha_{1}^{2}+\alpha_{3}^{2}-\alpha_{2}^{2})\log|\xi_{1}-\xi_{3}|}\\ &{\quad\quad\quad+(\alpha_{2}^{2}+\alpha_{3}^{2}-\alpha_{1}^{2})\log|\xi_{2}-\xi_{3}|+s(\alpha_{1},\alpha_{2},\alpha_{3})\,.}\end{array}\] |
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\[\begin{array}{r l}{\mathcal{Z}_{m,n}^{\kappa,\rho}(z,\overline{{z}})}&{=\frac{m!\Gamma(\rho+1)}{(\kappa+m+1)_{n}}\sum_{j=0}^{m\wedge^{*}\rho}\frac{(-1)^{j}}{j!(m-j)!\Gamma(\rho-j+1)}\frac{\left(1-|z|^{2}\right)^{j}}{z^{j}}\mathcal{Z}_{m-j,n}^{\kappa+j}(z,\overline{{z}}).}\end{array}\] |
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\[\begin{array}{r l r}{\mathsf{T}^{2}(A)=A[\epsilon][\epsilon^{\prime}]=\lbrace a+b\epsilon+c\epsilon^{\prime}+d\epsilon\epsilon^{\prime}\vert~\forall a,b,c,d\in A\rbrace}&{}&{\epsilon^{2}={\epsilon^{\prime}}^{2}=0}\\ {\mathsf{T}_{n}(A)=A[\epsilon_{1},\hdots,\epsilon_{n}]=\lbrace a+b_{1}\epsilon_{1}+\hdots+b_{n}\epsilon_{n}\vert~\forall a,b_{j}\in A\rbrace}&{}&{\epsilon_{i}\epsilon_{j}=0}\end{array}\] |
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\[\begin{array}{r l}{\mu(\mathcal{F})}&{=\frac{\deg(\mathcal{Q})-\deg(\mathcal{R})}{\operatorname{rk}(\mathcal{Q})-\operatorname{rk}(\mathcal{R})}}\\ &{\leq\frac{\deg(\mathcal{Q})}{\operatorname{rk}(\mathcal{Q})-q}}\\ &{\leq(q+1)\mu(\mathcal{Q})}\end{array}\] |
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\[\frac{\mathrm{LF}_{\mathbb H,\varepsilon}^{(\alpha_{2},i),(\beta,0)}(\ell)}{\mathrm{LF}_{\mathbb H}^{(\alpha_{1},i),(\beta,0)}(\ell)}(\phi):=\varepsilon^{\frac{1}{2}(\alpha_{2}^{2}-\alpha_{1}^{2})}e^{(\alpha_{2}-\alpha_{1})\phi_{\varepsilon}(i)}.\] |
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\[\begin{array}{r}{\partial_{t}\varrho+u_{1}\cdot\nabla_{x}\varrho+\frac{\varrho_{1}}{1-\rho_{f_{1}}}\mathrm{div}_{x}\Big[\mathrm{K}_{1,G_{1}}^{\mathrm{free}}(\varrho)-\mathrm{K}_{G_{1}}^{\mathrm{free}}(\varrho)u_{1}\Big]=S[\varrho],}\end{array}\] |
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\[\begin{array}{r l}{\frac{1}{|x-y|}}&{=\frac{1}{\sqrt{r^{2}+s^{2}-2r s(\cos\theta\cos\theta^{\prime}+\sin\theta\sin\theta^{\prime}\cos(\varphi-\varphi^{\prime}))}}}\\ &{=\sum_{\ell\geq0}\sum_{|m|\leq\ell}\frac{4\pi}{2\ell+1}\frac{(r\wedge s)^{\ell}}{(r\vee s)^{\ell+1}}Y_{\ell,m}(\theta,\varphi)Y_{\ell,m}(\theta^{\prime},\varphi^{\prime})^{*},}\end{array}\] |
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\[\begin{array}{r l r}{\textstyle g_{\xi,\eta}^{+}[(1+\theta_{\gamma})e_{\gamma}]}&{\leq}&{(1-\lambda_{\xi,\eta}^{+})(1+\theta_{\gamma})+\lambda_{\xi,\eta}^{+}(1+\theta_{\gamma})\mu_{\gamma}}\\ &{=}&{1+\theta_{\gamma}+\lambda_{\xi,\eta}^{+}(\mu_{\gamma}+\theta_{\gamma}\mu_{\gamma}-1-\theta_{\gamma})}\\ &{\leq}&{\textstyle\frac{1+\theta_{\gamma}+\mu_{\gamma}+\theta_{\gamma}\mu_{\gamma}}{2},}\end{array}\] |
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\[\begin{array}{r l r l}{{2}d}&{\in\mathbb{N},\ \ }&&{\theta_{l e x}(h)\leq2\Bigg(V_{d}(c)\frac{d!V a r(\Lambda^{\prime})\beta^{\alpha}}{2^{d+1}}\sum_{k=0}^{d}\frac{\left(\frac{2\psi(h)}{c}\right)^{k}}{k!\left(\frac{2\psi(h)}{c}+\beta\right)^{\alpha-d-1+k}}\frac{\Gamma(\alpha-d-1+k)}{\Gamma(\alpha)}\Bigg)^{\frac{1}{2}}.}\end{array}\] |
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\[\begin{array}{r l}&{\mathbb{E}_{\xi}\|\nabla f_{\xi}(w_{\theta})-\nabla f_{\xi}(w)\|^{2}}\\ &{=\mathbb{E}_{\xi}\Big\|\sum_{k=0}^{n-1}\big(\nabla f_{\xi}(w_{\theta(k+1)/n})-\nabla f_{\xi}(w_{\theta k/n})\big)\Big\|^{2}}\\ &{\overset{(i)}{\le}n\sum_{k=0}^{n-1}\mathbb{E}_{\xi}\big\|\nabla f_{\xi}(w_{\theta(k+1)/n})-\nabla f_{\xi}(w_{\theta k/n})\big\|^{2}}\\ &{\overset{(i i)}{\le}n\sum_{k=0}^{n-1}\|w_{\theta(k+1)/n}-w_{\theta k/n}\|^{2}\mathbb{E}_{\xi}\Big(L_{0}+L_{1}\operatorname*{max}_{\theta^{\prime}\in[0,1]}\|\nabla f_{\xi}\big(\theta^{\prime}w_{\theta(k+1)/n}+(1-\theta^{\prime})w_{\theta k/n}\big)\|^{\alpha}\Big)^{2}}\\ &{=\theta^{2}\|w^{\prime}-w\|^{2}\mathbb{E}_{\xi}\sum_{k=0}^{n-1}\frac{1}{n}\operatorname*{max}_{\theta^{\prime}\in[0,1]}\big(L_{0}+L_{1}\|\nabla f_{\xi}\big(w_{\theta^{\prime}\theta(k+1)/n+(1-\theta^{\prime})\theta k/n}\big)\|^{\alpha}\big)^{2}}\\ &{\overset{(i i i)}{=}\theta^{2}\|w^{\prime}-w\|^{2}\mathbb{E}_{\xi}\sum_{k=0}^{n-1}\frac{1}{n}\operatorname*{max}_{u\in[k/n,(k+1)/n]}h(u)}\end{array}\] |
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\[\begin{array}{r}{U^{-1}B_{1}=\left(\begin{array}{l l}{\frac{1}{n}\mathbf{1}_{n}^{\intercal}}&{\mathbf{0}}\\ {\mathbf{0}}&{\frac{1}{n}\mathbf{1}_{n}^{\intercal}}\\ {U_{L,l}}&{U_{L,r}}\end{array}\right)\left(\begin{array}{l l}{\tilde{W}}&{-\tilde{W}\tilde{V}}\\ {\tilde{V}}&{\tilde{W}}\end{array}\right)=\left(\begin{array}{l l}{\frac{1}{n}\mathbf{1}_{n}^{\intercal}}&{\mathbf{0}}\\ {\mathbf{0}}&{\frac{1}{n}\mathbf{1}_{n}^{\intercal}}\\ {U_{L,l}\tilde{W}+U_{L,r}\tilde{V}}&{U_{L,r}\tilde{W}-U_{L,l}\tilde{W}\tilde{V}}\end{array}\right)=\left(\begin{array}{l l}{\frac{1}{n}\mathbf{1}_{n}^{\intercal}}&{\mathbf{0}}\\ {\mathbf{0}}&{\frac{1}{n}\mathbf{1}_{n}^{\intercal}}\\ {P_{1}U_{L,l}}&{P_{1}U_{L,r}}\end{array}\right).}\end{array}\] |
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\[\begin{array}{r l r l r}{{3}}&{s_{0}=0,\qquad}&&{{\mathcal S}=A H-\frac{\big(\alpha^{2}\big)^{\prime}}{2}(\partial_{y}A)P_{x}P_{y}+\frac12\big(\alpha^{2}\partial_{y}^{2}A-\big(\alpha^{2}\big)^{\prime\prime}A\big)P_{y}^{2},}&\\ &{s_{0}=\pm1,\qquad}&&{{\mathcal S}=A H-\frac{\big(\alpha^{2}\big)^{\prime}}{2}(\partial_{y}A)P_{x}P_{y}-\frac{s_{0}}{2}\big(\alpha^{2}-a_{0}\big)A P_{y}^{2}.}&\end{array}\] |
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\[\begin{array}{r l}{\tilde{R}^{-n}}&{\left(\mu_{\varepsilon}(\cup_{x\in X\cap X_{t_{1}}^{t_{3}}}B_{\tilde{R}}(x)\cap S_{t_{1}}^{t_{3}})+\mu_{\varepsilon}(\cup_{x\in X\cap X_{t_{3}}^{t_{2}}}B_{\tilde{R}}(x)\cap S_{t_{3}}^{t_{2}})\right)\leq}\\ &{\left(1+\frac{1}{\Gamma}\right)^{n}R^{-n}\mu_{\varepsilon}\left(\cup_{x\in X}B_{R}(x)\cap S_{t_{1}}^{t_{2}}\right)+C R^{\gamma_{0}}+2\omega.}\end{array}\] |
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\[\begin{array}{r l}{\frac{d}{d\varepsilon}\mathcal{B}_{p}[\gamma_{\varepsilon}]}&{=\int_{0}^{L}|\kappa_{\varepsilon}|^{p}\frac{(\gamma_{\varepsilon}^{\prime},\eta^{\prime})}{|\gamma_{\varepsilon}^{\prime}|}\,d s+\int_{0}^{L}p|\kappa_{\varepsilon}|^{p-2}\big(\kappa_{\varepsilon},\partial_{\varepsilon}\kappa_{\varepsilon}\big)|\gamma_{\varepsilon}^{\prime}|\,d s,}\\ {\partial_{\varepsilon}\kappa_{\varepsilon}}&{=\frac{1}{|\gamma_{\varepsilon}^{\prime}|^{2}}\eta^{\prime\prime}-2\frac{(\gamma_{\varepsilon}^{\prime},\eta^{\prime})}{|\gamma_{\varepsilon}^{\prime}|^{4}}\gamma_{\varepsilon}^{\prime\prime}}\\ &{\quad-\bigg(\frac{(\gamma_{\varepsilon}^{\prime},\gamma_{\varepsilon}^{\prime\prime})}{|\gamma_{\varepsilon}^{\prime}|^{4}}\eta^{\prime}+\Big[\frac{(\gamma_{\varepsilon}^{\prime},\eta^{\prime\prime})+(\gamma_{\varepsilon}^{\prime\prime},\eta^{\prime})}{|\gamma_{\varepsilon}^{\prime}|^{4}}-4\frac{(\gamma_{\varepsilon}^{\prime},\gamma_{\varepsilon}^{\prime\prime})(\gamma_{\varepsilon}^{\prime},\eta^{\prime})}{|\gamma_{\varepsilon}^{\prime}|^{6}}\Big]\gamma_{\varepsilon}^{\prime}\bigg).}\end{array}\] |
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\[\begin{array}{r l}{{\mathrm{dist}}({\mathrm{span}}\{\left[\begin{array}{l}{G S}\\ {W C}\end{array}\right]\},{\mathrm{span}}\{\left[\begin{array}{l}{Y}\\ {Z}\end{array}\right]\})}&{\geq\underset{E\in\mathbb{R}^{p\times p}}{\operatorname*{min}}\left\|G-F R E S^{-1}\right\|\sigma_{\operatorname*{min}}(S)}\\ &{=\underset{E\in\mathbb{R}^{p\times p}}{\operatorname*{min}}\left\|G-F E\right\|\sigma_{\operatorname*{min}}(S)}\\ &{={\mathrm{dist}}({\mathrm{span}}\{G\},{\mathrm{span}}\{F\})\beta_{1},}\\ &{={\mathrm{dist}}({\mathrm{span}}\{\tilde{Y}\},{\mathrm{span}}\{Y\})\beta_{1}.}\end{array}\] |
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\[\begin{array}{r l}{\left(\begin{array}{l}{\frac{\partial^{2}u}{\partial m_{j}\partial m_{k}}}\\ {\frac{\partial^{2}u}{\partial m_{j}\partial m_{k}}\vert_{\partial\Omega}}\end{array}\right)}&{=\mathscr{S}_{\boldsymbol{m},\omega}\,\left[\mathcal{G}_{\boldsymbol{m},\omega}\left(\begin{array}{l}{u_{j,k}}\\ {u_{j,k}\vert_{\partial\Omega}}\end{array}\right)-\left(\begin{array}{l}{0}\\ {\left(\frac{\mathrm{i}\omega}{4m^{3/2}}\right)\beta_{j}\beta_{k}u\vert_{\partial\Omega}}\end{array}\right)\right],}\end{array}\] |
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\[\begin{array}{r l r}&{}&{\kappa+\gamma B_{1}=\kappa+\gamma y_{0}(\Gamma)<\sigma(i_{1,1})<\cdots<\sigma(i_{1,t_{1}})\leq\kappa+\gamma y_{1}(\Gamma)<\sigma(i_{2,1})<\cdots<\sigma(i_{2,t_{2}})}\\ &{}&{\leq\cdots<\sigma(i_{T_{1}+T_{2}-1,1})<\cdots<\sigma(i_{T_{1}+T_{2}-1,t_{T_{1}+T_{2}-1}})\leq\kappa+\gamma y_{T_{1}+T_{2}-1}(\Gamma)=\kappa+\gamma B_{2}.}\end{array}\] |
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\[\begin{array}{r l}{N_{1}}&{=2}\\ {N_{2}}&{=3}\\ {N_{3}}&{=7}\\ {N_{4}}&{=43}\\ {N_{5}}&{=1807=13\cdot139}\\ {N_{6}}&{=3263443}\\ {N_{7}}&{=10650056950807=547\cdot607\cdot1033\cdot31051}\\ {N_{8}}&{=113423713055421844361000443=29881\cdot67003\cdot9119521\cdot6212157481}\end{array}\] |
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\[\begin{array}{r l}{\operatorname*{min}_{\mathbf{x}\in\mathcal{X}}\quad}&{\frac{1}\!\left[2\right]\mathbf{x}^{\top}\mathbf{Q}\mathbf{x}+\mathbf{c}^{\top}\mathbf{x}+\mathbf{d}+\mathbb{E}_{\mathbf{U}^{\top}\mathbf{P}^{\top}\mathbf{P}\mathbf{U}}}\\ {\textrm{s.t.}\quad}&{\mathbf{A}\mathbf{x}\preccurlyeq\mathbf{b}+\mathbf{q}_{\varepsilon}}\end{array}\] |
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\[\begin{array}{r l}{|I_{\delta}|}&{\leq\int_{0}^{T}\int_{\Omega}|\eta_{\delta}^{\prime}||u|^{2}|\chi_{\delta}|\,\mathrm{d}y\,\mathrm{d}\tau\leq\int_{0}^{T}|\eta_{\delta}^{\prime}(\tau)|\|u(\tau)\|_{L^{r}(B_{2\delta}(x))}^{2}\|\chi_{\delta}\|_{L^{\frac{r}{r-2}}}\,\mathrm{d}\tau}\\ &{\lesssim\delta^{d\frac{r-2}{r}}\int_{0}^{T}|\eta_{\delta}^{\prime}(\tau)|\|u(\tau)\|_{L^{r}(B_{2\delta}(x))}^{2}\,\mathrm{d}\tau\leq\delta^{d\frac{r-2}{r}}\|u\|_{L^{q}L^{r}(\mathcal{C}_{2\delta}^{\alpha}(x,t))}^{2}\|\eta_{\delta}^{\prime}\|_{L^{\frac{q}{q-2}}}}\\ &{\lesssim\|u\|_{L^{q}L^{r}(\mathcal{C}_{2\delta}^{\alpha}(x,t))}^{2}\delta^{d\frac{r-2}{r}-\alpha\frac{2}{q}}.}\end{array}\] |
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\[\begin{array}{r l r l}{A_{j}^{l}(x,t)}&{=\frac{\partial}{\partial x_{j}}u^{l}(x,t),}&{F^{1}(x,t)}&{=\frac{\partial}{\partial x_{2}}\theta(x,t)-\frac{\partial}{\partial x_{2}}\theta_{0}(x_{1},x_{2}),}\\ {F^{2}(x,t)}&{=-\frac{\partial}{\partial x_{1}}\theta(x,t)+\frac{\partial}{\partial x_{1}}\theta_{0}(x_{1},x_{2}),}&{F^{3}(x,t)}&{=0,}\end{array}\] |
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\[\begin{array}{r l}{\sum_{(t,t^{\prime})\in\mathcal{E}_{\leq}^{(n)}}\log\ell_{t,t^{\prime}}^{(n)}}&{\leq(m^{(n)}-m_{\star}^{(n)})\log n+\frac{n}{2}\Bigg(\sum_{c\in\mathcal{C}^{(\delta,h)}}\mathbb{E}_{\mu}\left[D_{c}([T,o])\right]\log\mathbb{E}_{\mu}\left[D_{c}([T,o])\right]}\\ &{\qquad-\mathbb{E}_{\mu}\left[D_{c}([T,o])\right]-2\mathbb{E}_{\mu}\left[\log D_{c}([T,o])!\right]\Bigg)+o(n).}\end{array}\] |
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\[\Bigl(\mathbb{E}\bigl|\iota_{\varepsilon}\bigl(\Pi_{z}^{\gamma,2}\tau\bigr)(\phi_{z}^{\lambda})\bigr|^{p}\Bigr)^{\frac{1}{p}}\lesssim\mathfrak{e}^{2(a-3)}(\lambda\vee\mathfrak{e})^{\frac{5}{2}-2a}\Bigl(\varepsilon^{\frac94}+\varepsilon^{\frac92-\bar{\kappa}}\mathfrak{e}^{-\frac{5}{2}}\Bigr).\] |
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\[\begin{array}{r l}&{1.\;f_{1}\left(d_{0}^{\star},d_{1}^{\star}\right)\leq0;\quad2.\;\nu^{\star}\geq0;\quad3.\;\nu^{\star}f_{1}\left(d_{0}^{\star},d_{1}^{\star}\right)=0;}\\ &{4.\;-2a_{0}b_{0}d_{0}^{\star}e^{-b_{0}d_{0}^{\star^{2}}}-2a_{1}b_{1}d_{0}^{\star}e^{-b_{1}d_{0}^{\star^{2}}}+\nu^{\star}\left(4d_{0}^{\star}+2M^{\prime}d_{1}^{\star}\right)=0;}\\ &{5.\;-\frac{1}{6}M^{\prime}a_{2}b_{2}d_{1}^{\star}e^{-b_{2}d_{1}^{\star^{2}}}-\frac{2}{3}M^{\prime}a_{2}b_{2}d_{1}^{\star}e^{-\frac{4}{3}b_{2}d_{1}^{\star^{2}}}+\nu^{\star}\left(\frac{2M^{\prime}\left(2M^{\prime}+1\right)}{3}d_{1}^{\star}+2M^{\prime}d_{0}^{\star}\right)=0;}\end{array}\] |
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\[\begin{array}{r l}&{2\mu\int_{a}^{b}\left[\rho Y^{\prime}(\rho)H^{\prime}(\rho)+\frac{Y(\rho)H(\rho)}{\rho}\right]d\rho+\lambda\int_{a}^{b}\rho\left[Y^{\prime}(\rho)+\frac{Y(\rho)}{\rho}+\frac{\pi k}{h}\,Z(\rho)\right]\left[H^{\prime}(\rho)+\frac{H(\rho)}{\rho}+\frac{\pi k}{h}\,K(\rho)\right]d\rho}\\ &{\quad+\mu\int_{a}^{b}\rho\left[Z^{\prime}(\rho)-\frac{\pi k}{h}\,Y(\rho)\right]\left[K^{\prime}(\rho)-\frac{\pi k}{h}\,H(\rho)\right]d\rho+2\mu\frac{\pi^{2}k^{2}}{h^{2}}\int_{a}^{b}\rho Z(\rho)K(\rho)\,d\rho=\int_{a}^{b}\rho\Psi_{k}(\rho)K(\rho)\,d\rho}\end{array}\] |
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\[\begin{array}{r l}{\hat{\rho}(t+\Delta t)}&{=\left(1-\sum_{i}p_{i}\Delta t\right)\left(\hat{\rho}(t)\mathrm{-i}\Delta t[\hat{H},\hat{\rho}(t)]\right)+\sum_{i}p_{i}\Delta t\,\hat{\tau}_{i}\otimes\mathrm{tr}_{i}[\hat{\rho}(t)]+\mathcal{O}(\Delta t^{2})}\\ &{=\hat{\rho}(t)\mathrm{-i}\Delta t[\hat{H},\hat{\rho}(t)]+\sum_{i}p_{i}\Delta t\,\left(\hat{\tau}_{i}\otimes\mathrm{tr}_{i}[\hat{\rho}(t)]-\hat{\rho}(t)\right)+\mathcal{O}(\Delta t^{2}),}\end{array}\] |
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\[\begin{array}{r l}{\operatorname*{lim}_{\rho\to\infty}\bar{P}_{r}=}&{\frac{1}{24}\int_{0}^{\infty}\exp\left(-\frac{\rho\bar{\eta}}{4}\gamma\right)d\gamma+\frac{1}{8}\int_{0}^{\infty}\exp\left(-\frac{\rho\bar{\eta}}{3}\gamma\right)d\gamma}\\ {=}&{\frac{13}{24\rho\bar{\eta}}.}\end{array}\] |
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\[\begin{array}{r}{\hat{\mathsf{Z}}=\mathsf{Z}\cup\mathsf{Z}^{*}\cup\omega\mathsf{Z}\cup\omega\mathsf{Z}^{*}\cup\omega^{2}\mathsf{Z}\cup\omega^{2}\mathsf{Z}^{*}\cup\mathsf{Z}^{-1}\cup\mathsf{Z}^{-*}\cup\omega\mathsf{Z}^{-1}\cup\omega\mathsf{Z}^{-*}\cup\omega^{2}\mathsf{Z}^{-1}\cup\omega^{2}\mathsf{Z}^{-*},}\end{array}\] |
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\[\sum_{t=i}^{j-1}(L_{T_{S_{k}}}(u_{t})+L_{T_{S_{k}}}(u_{t+1}))-(j-i)(d+1)+(d+1)\leq\sum_{t=i}^{j-1}(L_{T}(u_{t})+L_{T}(u_{t+1})+2)-(j-i)(d_{0}+3)+(d_{0}+3)=\sum_{t=i}^{j-1}(L_{T}(u_{t})+L_{T}(u_{t+1}))-(j-i)(d_{0}+1)+(d_{0}+1)+2\leq d_{T}(u_{i},u_{j})+2=d_{T_{S_{k}}}(u_{i},u_{j})\] |
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\[\begin{array}{r l r l}&{|d_{0}(\zeta,t)|=e^{2\pi\nu},}&&{\zeta\in\mathcal{I},\ t\geq2,}\\ &{|d_{1}(\zeta,k)-\frac{\mathcal{P}(\zeta,\omega k_{1})}{\mathcal{P}(\zeta,\omega^{2}k_{1})}|\leq C|k-k_{1}|(1+|\ln|k-k_{1}||),}&&{\zeta\in\mathcal{I},\ k\in\mathcal{X}^{\epsilon}.}\end{array}\] |
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\[\mathbf{G}:=\left(\begin{array}{l l l l l l l l l}{K_{M+1,M+1}}&{\cdots}&{K_{M+1,2M+1}}&{0}&{\cdots}&{0}&{K_{M+1,1}}&{\cdots}&{K_{M+1,M}}\\ {\vdots}&{\ddots}&{\vdots}&{\vdots}&{\ddots}&{\vdots}&{\vdots}&{\ddots}&{\vdots}\\ {K_{2M+1,M+1}}&{\cdots}&{K_{2M+1,2M+1}}&{0}&{\ddots}&{0}&{K_{2M+1,1}}&{\cdots}&{K_{2M+1,M}}\\ {0}&{\cdots}&{0}&{0}&{\cdots}&{0}&{0}&{\cdots}&{0}\\ {\vdots}&{\ddots}&{\ddots}&{\ddots}&{\ddots}&{\ddots}&{\ddots}&{\ddots}&{\vdots}\\ {0}&{\cdots}&{0}&{0}&{\cdots}&{0}&{0}&{\cdots}&{0}\\ {K_{1,M+1}}&{\cdots}&{K_{1,2M+1}}&{0}&{\ddots}&{0}&{K_{1,1}}&{\cdots}&{K_{2M+1,M}}\\ {\vdots}&{\ddots}&{\vdots}&{\vdots}&{\ddots}&{\vdots}&{\vdots}&{\ddots}&{\vdots}\\ {K_{M,M+1}}&{\cdots}&{K_{M,2M+1}}&{0}&{\cdots}&{0}&{K_{M,1}}&{\cdots}&{K_{M,M}}\end{array}\right)\] |
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\[\begin{array}{r l}{\mu_{S}(A)}&{=\frac{1}{2}\|A1_{S}^{\prime}-A1_{\overline{{S}}}^{\prime}\|_{1}}\\ &{=\frac{1}{2}\sum_{i}\left|\frac{1-v_{i}}{k}-\frac{v_{i}}{k^{\prime}}\right|}\\ &{=\frac{1}{2}\sum_{i}\left|\frac{1}{k}-\frac{n\cdot v_{i}}{k\cdot k^{\prime}}\right|}\\ &{=\frac{1}{2}\frac{n}{k\cdot k^{\prime}}\sum_{i}\left|\frac{k^{\prime}}{n}-v_{i}\right|}\\ &{\geq\frac{1}{2}\frac{n}{k\cdot k^{\prime}}\sum_{i=1}^{k}\left(\frac{k^{\prime}}{n}-v_{i}\right)}\\ &{\mathrm{\\\[since~\ensuremath{A}~i s~\ensuremath{1/2}-l a z y]}}\\ &{=\frac{1}{2}\left(1-\frac{n}{k^{\prime}}\phi_{S}(A)\right)}\\ &{\geq\frac{1}{2}-\phi_{S}(A)}\end{array}\] |
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\[\begin{array}{r l}{T}&{=\sum_{i_{1},\ldots,i_{m}\in\mathcal{R}}c_{i_{1},\ldots,i_{m}}G_{i_{1},\ldots,i_{m}},}\\ {T^{*}}&{=\sum_{i_{1},\ldots,i_{m}\in\mathcal{R}}c_{i_{1},\ldots,i_{m}}^{*}G_{i_{1},\ldots,i_{m}},}\end{array}\] |
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\[\left\{\begin{array}{l l}{Q_{A}=\mathrm{{div}}(\kappa_{A}\nabla\theta_{A})}&{\mathrm{~in~}\Omega_{A}(t),}\\ {Q_{B}={\mathrm{div}}(\kappa_{B}\nabla\theta_{B})}&{\mathrm{~in~}\Omega_{B}(t),}\\ {Q_{S}=\mathrm{{div}}_{\Gamma}(\kappa_{S}\nabla_{\Gamma}\theta_{S})+\kappa_{B}(n_{\Gamma}\cdot\nabla)\theta_{B}-\kappa_{A}(n_{\Gamma}\cdot\nabla)\theta_{A}}&{\mathrm{~on~}\Gamma(t).}\end{array}\right.\] |
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\[\begin{array}{r l}&{\frac{\hat{Q}_{\mathrm{W}j}}{\sum_{j=\pm1}^{\pm j_{\operatorname*{max}}}\hat{Q}_{\mathrm{W}j}+\frac{3}{e_{2}^{2}}(1-e_{2})^{j_{\operatorname*{max}}}}\le q_{\mathrm{W}j}\le\frac{\hat{Q}_{\mathrm{W}j}+\frac{1}{e_{2}}(1-e_{2})^{j_{\operatorname*{max}}}}{\sum_{j=\pm1}^{\pm j_{\operatorname*{max}}}\hat{Q}_{\mathrm{W}j}}.}\\ &{\frac{\sum_{j=\pm1}^{\pm j_{\operatorname*{max}}}\hat{Q}_{\mathrm{W}j}\mu_{j,A}}{\sum_{j=\pm1}^{\pm j_{\operatorname*{max}}}\hat{Q}_{\mathrm{W}j}+\frac{3}{e_{2}^{2}}(1-e_{2})^{j_{\operatorname*{max}}}}\le\bar{p}_{\mathrm{W}A}\le\frac{\sum_{j=\pm1}^{\pm j_{\operatorname*{max}}}\hat{Q}_{\mathrm{W}j}\mu_{j,A}+\frac{3}{e_{2}^{2}}(1-e_{2})^{j_{\operatorname*{max}}}}{\sum_{j=\pm1}^{\pm j_{\operatorname*{max}}}\hat{Q}_{\mathrm{W}j}}.}\end{array}\] |
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\[\begin{array}{r l}{\mathbb{\widetilde{R}}_{\mu\nu}}&{=\Big(\mathbb{\Gamma}_{\beta\alpha}^{\alpha}\mathbb{\Gamma}_{\mu\nu}^{\beta}-\mathbb{\Gamma}_{\beta\nu}^{\alpha}\mathbb{\Gamma}_{\mu\alpha}^{\beta}\Big)-\frac{1}{2}\mathbb{g}_{\mu\nu}\mathbb{g}^{\kappa\lambda}\Big(\mathbb{\Gamma}_{\beta\alpha}^{\alpha}\mathbb{\Gamma}_{\kappa\lambda}^{\beta}-\mathbb{\Gamma}_{\beta\kappa}^{\alpha}\mathbb{\Gamma}_{\lambda\alpha}^{\beta}\Big)}\\ &{=\frac{1}{4}\mathbb{g}^{\alpha\kappa}\mathbb{g}^{\beta\lambda}\Big[\langle\partial_{\alpha}\varrho\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle\langle\partial_{\mu}\varrho\partial_{\nu}\varrho\partial_{\lambda}\varrho\rangle-\langle\partial_{\nu}\varrho\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle\langle\partial_{\mu}\varrho\partial_{\alpha}\varrho\partial_{\lambda}\varrho\rangle\Big].}\end{array}\] |
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\[{\begin{array}{l c l}{f{\bigl(}^{1}\!\!/\!_{3}{\bigr)}=f(0.0{\overline{{2}}}_{3})=0.0{\overline{{1}}}_{2}=\!\!}&{\!\!0.1_{2}\!\!}&{\!\!=0.1{\overline{{0}}}_{2}=f(0.2{\overline{{0}}}_{3})=f{\bigl(}^{2}\!\!/\!_{3}{\bigr)}.}\\ &{\parallel}\\ &{^{1}\!\!/\!_{2}}\end{array}}\] |
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\[\begin{array}{r}{\mathbb{E}\|\nabla\mathrm{env}_{\psi}^{\eta}(x_{\sim}^{K})\|^{2}\leq\frac{2(\mathrm{env}_{\psi}^{\eta}(x^{0})-\operatorname*{inf}\psi)}{\alpha(1-\eta(\rho-\lambda))\sqrt{K}}+\frac{\beta\theta}{\eta(1-\eta(\rho-\lambda))}\frac{\alpha}{\sqrt{K}},}\end{array}\] |
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\[\begin{array}{r l}{d_{\phi_{1}}}&{=\left(\frac{\omega_{z}}{\omega_{0}}+\frac{\tau\omega_{*i}}{\omega_{0}}\right)\left[\Gamma_{0}-\Gamma_{z}-\frac{\omega_{*i}}{\omega_{z}}\left(F_{1}-\Gamma_{z}\right)-\frac{\left(1-\Gamma_{0}\right)\left(b_{z}-b_{0}\right)}{b_{0}}\right]}\\ &{-\frac{\tau\omega_{*i}b_{+}\left(1-\Gamma_{0}\right)F_{1}}{\omega_{0}b_{0}\sigma_{0}\left(1-\frac{\omega_{z}^{2}}{\omega_{0}^{2}}\right)},}\end{array}\] |
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\[\begin{array}{r l}{A_{T}}&{=\left\{z\in\bigtimes_{i\in\overline{{T}}}\mathbb{Z}_{m_{i}}:d\leq\mathrm{outdeg}\left(z,G^{[\kappa]}(T)\right)\leq2d\mathrm{~and~}\rho(z)\mathrm{~is~}(t+1)\mathrm{-contributing}\right\}}\\ {B_{T}}&{=\left\{w\in\bigtimes_{i\in\overline{{T}}}\mathbb{Z}_{m_{i}}:\rho(w)\mathrm{~is~}t\mathrm{-contributing}\right\}.}\end{array}\] |
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\[\begin{array}{r l}{\left(\frac{d n}{d t}\right)_{\mathrm{MW}}}&{=\frac{1}{4}n\left(\frac{g_{\mathrm{NV}}\mu_{\mathrm{B}}}{\hbar}\right)^{2}B_{1}^{2}g\left(\omega-\omega_{\mathrm{ESR}}\right),}\\ {\left(\frac{d n}{d t}\right)_{\mathrm{T1}}}&{=\frac{n-n_{0}}{T_{1}}.}\end{array}\] |
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\[\begin{array}{r l}{\left(\mathbb{C}_{n_{1},m_{1}},\mathbb{C}_{b_{1},m_{(1)}}^{(I_{1})},\ldots,\mathbb{C}_{b_{1},m_{(1)}}^{(I_{H})}\right)}&{\rightsquigarrow\left(\mathbb{C},\mathbb{C}^{(1)},\ldots,\mathbb{C}^{(H)}\right),}\\ {\left(\mathbb{D}_{n_{2},m_{2}},\mathbb{D}_{b_{2},m_{(2)}}^{(I_{1})},\ldots,\mathbb{D}_{b_{2},m_{(2)}}^{(I_{H})}\right)}&{\rightsquigarrow\left(\mathbb{D},\mathbb{D}^{(1)},\ldots,\mathbb{D}^{(H)}\right).}\end{array}\] |
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\[\begin{array}{r l}{\widehat{L_{4\tau,2h}}}&{:\mathcal{F}_{4\tau,2h}(4\theta_{t},2\theta_{x})\to\mathcal{F}_{4\tau,2h}(4\theta_{t},2\theta_{x})\;,}\\ {\widehat{L_{4\tau,2h}}}&{:=\left(\hat{L}_{4\tau,2h}(4\theta_{t},2\theta_{x})\right),}\end{array}\] |
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\[\begin{array}{r l r}{\Omega}&{=}&{\hbar\sum_{j=1}^{g}d q_{j}\wedge d p_{j}-\sum_{s=1}^{n}\sum_{i=1}^{2}\sum_{k=1}^{r_{s}-1}d t_{X_{s}^{(i)},k}\wedge d\mathrm{Ham}^{(\mathbf{e}_{X_{s}^{(i)},k})}-\sum_{i=1}^{2}\sum_{k=1}^{r_{\infty}-1}d t_{\infty^{(i)},k}\wedge d\mathrm{Ham}^{(\mathbf{e}_{\infty^{(i)},k})}}\\ &{}&{-\sum_{s=1}^{n}d X_{s}\wedge d\,\mathrm{Ham}^{(\mathbf{e}_{X_{s}})}}\\ &{=}&{\hbar\sum_{j=1}^{g}d\check{q}_{j}\wedge d\check{p}_{j}-\sum_{\tau\in\mathcal{T}_{\mathrm{iso}}}^{g}d\tau\wedge d\mathrm{Ham}^{(\boldsymbol{\alpha}_{\tau})}}\end{array}\] |
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\[\begin{array}{r l r}&{}&{|L I S(\sigma|_{\mathcal{R}_{s}})-\sqrt{2}L\beta_{n}^{-1\slash2}|\leq20L^{1\slash2}e^{-L}\beta_{n}^{-1\slash2}+5000L^{2}e^{2L}}\\ &{}&{+\sqrt{2}L\beta_{n}^{-1\slash2}\operatorname*{max}\Big\{e^{4L^{-1}}(1+C_{L}r_{s}^{-1\slash10})^{1\slash2}(1+\operatorname*{max}\{c_{L}\beta_{n}^{-1}\Psi_{s},1\}^{-1\slash6})-1,}\\ &{}&{\quad\quad\quad\quad\quad\quad\quad\quad\quad1-e^{-6L^{-1}}\Psi_{s}^{1\slash2}(1-\operatorname*{max}\{c_{L}\beta_{n}^{-1}\Psi_{s},1\}^{-1\slash6})\Big\}.}\end{array}\] |
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\[\begin{array}{r l}{\Phi(x,t)}&{=\int_{0}^{t}f(x,r)\,\mathrm{d}r+\int_{0}^{t}F(x,r)\,\mathrm{d}W_{r}}\\ &{+\int_{0}^{t}\int_{\|z\|\leq1}\varphi(x,r,z)\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)+\int_{0}^{t}\int_{\|z\|>1}\varphi(x,r,z)\,N(\mathrm{d}z,\mathrm{d}r).}\end{array}\] |