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\[\langle\phi_{h}(x,a),\psi^{j}_{h+1}\rangle\geq 0\qquad\forall x \in\mathcal{X},a\in\mathcal{A}\] \[\frac{1}{m}\sum_{j=1}^{m}\|\psi^{j}_{h+1}\|_{1}\leq C_{\mathsf{ nrm}}\] \[\mathbb{E}_{(\bar{x},\bar{a})\sim\nu_{h}}\left[\max_{\ell\in[d]} \left|\mathbb{E}_{x^{\prime}\sim\mathbb{P}_{h}(\bar{x},\bar{a})}[\phi^{ \mathsf{... | outline | |
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline Data Aug & Datasets & Model & mini-batch SGD & ordered SGD & difference \\ \hline No & Semeion & Logistic model & 0.15 (0.01) & 0.15 (0.01) & 0.00 \\ \hline No & MNIST & Logistic model & 7.16 (0.27) & 7.32 (0.24) & -0.16 \\ \hline No & Semeion & SVM & 0.17 (0.01)... | table | |
\[\|\partial_{\boldsymbol{y}}^{\boldsymbol{\nu}}(u-u_{h})\|_{V}\] \[\leq\,\bigg{[}\frac{a_{\max}}{a_{\min}}\frac{2\overline{\lambda 2_{2}}}{\rho\chi_{1}}\big{(}\overline{u}\overline{h}C_{\lambda}+\overline{ \lambda}C_{\mathcal{P}}+\overline{\lambda}C_{u}\big{)}+\overline{u}\bigg{(} \frac{a_{\max}}{\chi_{1}}C_{u}+\sqrt{... | matrix | |
\[\sum_{n=0}^{\infty}l_{n}t^{n} =\frac{(1+x_{1}t)(1+x_{3}t)}{(1-x_{2}t)(1-x_{4}t)}=(1+(x_{1}+x_{3})t +x_{1}x_{3}t^{2})\left(\frac{1}{1-x_{2}t}\right)\left(\frac{1}{1-x_{4}t}\right)\] \[=(1+(x_{1}+x_{3})t+x_{1}x_{3}t^{2})\left(\sum_{n=0}^{\infty}x_{2} ^{n}t^{n}\right)\left(\sum_{n=0}^{\infty}x_{4}^{n}t^{n}\right)\] \[=(... | outline | |
\begin{table}
\begin{tabular}{c|r||c|c} \hline \(\Gamma_{72,1}\) & 8244, 8316, 8326, 8366, 8376, 8401, 8403, 8415, 8421, 8426, 8439, \\ & 8445, 8454, 8456, 8458, 8470, 8472, 8475, 8478, 8479, 8481, 8489, \\ & 8490, 8492, 8493, 8494, 8498 \\ \hline \(\Gamma_{72,2}\) & 9144, 9146, 9150, 9151, 9153, 9154, 9156, 9157, 91... | table | |
\[H_{0,\vec{S},\vec{\Sigma}}(A,\vec{E};\,\phi,\tilde{\pi})= \tfrac{1}{2} \int_{\Sigma}\mathrm{d}^{3}x\,\big{[}\tilde{\epsilon}^{ij}{}_{k} \,N\,\tilde{E}^{a}_{i}\tilde{E}^{b}_{j}F_{ab}{}^{k}\,+\,2S^{a}\tilde{E}^{b}_{k }F_{ab}{}^{k}\big{]}\] \[- \tfrac{1}{2}\,\int_{\Sigma}\mathrm{d}^{3}x\,\big{[}N(\tilde{\pi} ^{2}+V(\phi... | outline | |
\[\int_{Q_{T}}|u|^{p}\tilde{\varphi}+CT^{-(\alpha_{1}-\gamma)}\int_ {\mathbb{R}^{N}}u_{1}(x)\varphi_{2}(x)dx\] \[\leqslant\frac{1}{3p}\int_{Q_{T}}|u|^{p}\tilde{\varphi}+\frac{3^ {p^{\prime}-1}}{p^{\prime}}\int_{\Sigma}\varphi_{2}^{-p^{\prime}/p}\varphi_{ 1}^{-p^{\prime}/p}|D^{2+\alpha_{1}-\gamma}_{t\overline{T}}\varphi... | outline | |
\[\left\{\begin{aligned} & f([\bm{\Phi}]_{i_{r}}[x_{0}]_{i_{r}})+\\ &\underset{[\bm{\Phi}]_{i_{r}},[\bm{\Omega}]_{i_{r}},[\bm{\Xi}]_{i_{r} }\geq 0}{\text{argmin}}&\frac{\rho}{2}\left\|[\tilde{\bm{\Phi}}]_{i_{r}}-[ \tilde{H}]_{i_{r}}[\tilde{\bm{\Psi}}]_{i_{r}}^{k}+[\bm{\Lambda}]_{i_{r}}^{k} \right\|_{F}^{2}\\ &\text{s.t... | outline | |
\[\tilde{K}_{t}^{i}= G_{t}^{i}\Big{[}2cf(r_{t}^{i})+\frac{1}{2}f^{\prime\prime}(r_{t}^ {i})\left(2\sigma_{X}^{2}\varphi_{\text{rc}}\Big{(}|X_{t}^{i,N}-\bar{X}_{t}^{i }|\Big{)}^{2}\right)\] \[+f^{\prime}(r_{t}^{i})\Big{(}(1+\gamma\delta+L_{X}+\delta L_{C})| X_{t}^{i,N}-\bar{X}_{t}^{i}|-|{(X_{t}^{i,N})}^{3}-{(\bar{X}_{t}... | outline | |
\[r_{+}(iD_{t})^{\nu}l_{+}\int_{0}^{\infty}K\biggl{(}\frac{t}{\tau }\biggr{)}r_{+}\varphi(\tau)\,\frac{d\tau}{\tau}\] \[\qquad=r_{+}(iD_{t})^{\nu}l_{+}\int_{0}^{\infty}K(\tau)\,r_{+} \varphi\biggl{(}\frac{t}{\tau}\biggr{)}\,\frac{d\tau}{\tau}\] \[\qquad=\int_{0}^{\infty}K(\tau)\,r_{+}(iD_{t})^{\nu}l_{+}r_{+} \varphi\bi... | outline | |
\[\langle P_{t}(x),v^{\otimes t}\rangle\] \[=\left\langle v^{\otimes t},\sum_{S_{0}\subset[t]}\left(x^{ \otimes S_{0}}\right)\otimes\left(\sum_{\{S_{1},\ldots,S_{t}\}\in Z_{t}([t] \setminus S_{0})}(-1)^{\mathcal{C}\{S_{1},\ldots,S_{t}\}}(\mathcal{C}\{S_{1},\ldots,S_{t}\})!(D_{|S_{1}|})^{(S_{1})}\otimes\cdots\otimes(D_{... | matrix | |
\[\sum_{k=u+v+1}^{w+v}\sum_{a=1}^{u}\sum_{b=1}^{v}\frac{(-)^{a+v} \binom{k-a-b-1}{u-a,v-b,k-u-v-1}\binom{2w-k-1}{w+v-k}}{(4\pi\tau_{2})^{w-a-b} }\sum_{N=1}^{\infty}\frac{N^{a+b-2}}{\Gamma(a)\Gamma(b)}q^{N}\bar{q}^{N}V_{a,k -a,2w-k}(N,N)\] \[+\sum_{k=u+v+1}^{w+v}\sum_{b=1}^{v}\sum_{c=1}^{k-u-v}\frac{(-)^{ v}\binom{k-b-c... | outline | |
\begin{table}
\begin{tabular}{c|c|c} Killing vector field & Conserved quantity & Surface integral \\ \hline \(\partial_{0}\) (time translation) & energy \(E\) & \(16\pi E=\int_{S^{2}_{\infty}}\mathbb{U}_{1}(\nu)dS^{2}_{\infty}\) \\ \(\partial_{i}\) (spatial translation) & linear momentum \(P\) & \(8\pi P_{i}=\int_{S^{2... | table | |
\[\bm{H}_{\bm{f}} = -\mathbb{E}\left(\frac{\bm{f}(\bm{X}_{i})}{\pi_{\bm{\beta}_{0}} (\bm{X}_{i})(1-\pi_{\bm{\beta}_{0}}(\bm{X}_{i}))}\left(\frac{\partial\pi_{\bm {\beta}_{0}}(\bm{X}_{i})}{\partial\bm{\beta}}\right)^{\top}\right)\] \[\bm{\Omega} = \mathrm{Var}(\bm{g}_{\bm{\beta}_{0}}(T_{i},\bm{X}_{i}))\] \[\bm{g}_{\bm{\... | outline | |
\[I_{A} \lesssim \int\sum_{N_{1}\gtrsim N}\|P^{x}_{\leq N_{1}}\langle\nabla_{\xi} \rangle^{1+\gamma}\widetilde{g}\|_{L^{6}_{x}L^{2}_{\xi}}\|\langle\nabla_{\xi} \rangle^{r}P^{x}_{N_{1}}P^{\xi}_{M_{1}}\widetilde{f}\|_{\tilde{M}^{2}_{M_{1}}L ^{3}_{x}L^{3}_{\xi}}\|\langle\nabla_{\xi}\rangle^{-r}P^{x}_{N}h\|_{L^{2}_{x}L ^{2... | outline | |
\[\begin{array}{l}\Theta=\sum_{i}\varphi_{\Theta,i}=\sum_{i}\varphi_{\Theta,i}^{ eq},\ \ \ \mathfrak{T}_{\Theta}=\sum_{i}{\bf e}_{i}\varphi_{\Theta,i}^{eq}=\sum_{i}{\bf e}_{i} \varphi_{\Theta,i}\ \ \mbox{(the momentum)}\,\\ \sum_{i}{\bf e}_{i}\otimes{\bf e}_{i}\varphi_{\Theta,i}^{eq}=(\mathfrak{C}_{ \Theta}+d_{\Theta}C... | matrix | |
\[g(p,q) =(2\pi)^{3}(r_{p}r_{q})^{2}\operatorname{Re}\left(\sum_{j=0}^{ \infty}\frac{1}{j+1}\frac{e^{ij(\theta_{p}-\theta_{q})}}{r_{p}^{j+1}r_{q}^{j+1 }}\left(r_{q}^{2j+2}-1-(r_{q}^{2}-1)^{j+1}\right)\right)\] \[=(2\pi)^{3}(r_{p}r_{q})^{2}\operatorname{Re}\left(e^{-i(\theta_{ p}-\theta_{q})}\sum_{j=0}^{\infty}\frac{1}{... | outline | |
\[[\nabla_{\partial_{k}},\nabla_{\partial_{l}}]\,dx^{j} =[\nabla_{\partial_{k}},\nabla_{\partial_{l}}]\,dx^{j}(\partial_{ m})dx^{m}\] \[=\left(\nabla_{\partial_{k}}\left(\nabla_{\partial_{l}}dx^{j} \right)\partial_{m}-\nabla_{\partial_{l}}\left(\nabla_{\partial_{k}}dx^{j} \right)\partial_{m}\right)dx^{m}\] \[=\left(\pa... | outline | |
\[\Big{|}\mathcal{E}-\frac{N^{2}}{I_{N}}\sum_{m_{1}=1}^{M}\sum_{m_{ 2}=1}^{M-m_{1}}\cdots\sum_{m_{M}=0}^{M-m_{1}-m_{2}-\cdots-m_{M-1}}(-1)^{m_{1}+ \cdots+m_{M}}\] \[\quad\times{N-3\choose m_{1}}\cdots{N-3-m_{1}-\cdots-m_{M-1} \choose m_{M}}\,\Big{[}\prod_{j=5}^{M}\chi(m_{1}+\cdots+m_{j}\geq j-2)\Big{]}\] \[\quad\times\... | outline | |
\[\mathbb{P}\bigg{[}\|\widetilde{g}\|_{\infty}>\frac{19L/20}{N\sqrt {n}}\bigg{]}\leq\mathbb{P}[\exists b\in B_{0}:b\in B_{1}]\] \[\leq|B_{0}|\sup_{b\in B_{0}}\mathbb{P}\bigg{[}\exists t:\mathbb{E }_{b^{\prime}}\bigg{[}g\bigg{(}t+\sum_{i=1}^{\lfloor\epsilon n\rfloor}b^{ \prime}_{i}(b_{2i}-b_{2i-1})(Y_{2i-1}-Y_{2i})\bigg... | outline | |
\[\mathbb{E}\|\mathcal{P}_{\mathbf{A}}(\mathbf{g}_{t}^{s})\|^{2} =\mathbb{E}\|\mathcal{P}_{\mathbf{A}}(\nabla F(\mathbf{x}_{t}^{s };\xi_{t}^{s})-\nabla F(\widetilde{\mathbf{x}}_{s};\xi_{t}^{s}))\|^{2}\] \[\leq\mathbb{E}\|\nabla F(\mathbf{x}_{t}^{s};\xi_{t}^{s})-\nabla F (\widetilde{\mathbf{x}}_{s};\xi_{t}^{s})\|^{2}\] ... | outline | |
\[\frac{1}{q+1}\frac{\partial\overline{w}}{\partial y_{n}} =\frac{1}{q+1}\frac{\partial}{\partial y_{n}}\left(\overline{v} ^{q+1}-(q+1)\overline{\phi}(I_{n}-y_{n}\widetilde{\kappa})^{-2}\nabla \overline{u}\cdot\nabla\overline{h}\right)\] \[=\overline{v}^{q}\frac{\partial\overline{v}}{\partial y_{n}}- \frac{\partial\ove... | outline | |
\[\mathcal{C}(\mathcal{F})=(E_{1}|H_{1})\wedge(E_{2}|H_{2})\wedge(E_{3}|H_{3})= \left\{\begin{array}{ll}1,&\mbox{if $E_{1}H_{1}E_{2}H_{2}E_{3}H_{3}$ is true}\\ 0,&\mbox{if $\overline{E}_{1}H_{1}\vee\overline{E}_{2}H_{2}\vee\overline{E}_{3}H_{3}$ is true},\\ x_{1},&\mbox{if $\overline{H}_{1}E_{2}H_{2}E_{3}H_{3}$ is true... | matrix | |
\[\sum_{{\bf k}\neq 0}\sum_{{\bf l}\neq 0}B_{\sigma}({\bf k},{\bf l}) \overline{{\bf z}}^{\bf k}{\bf z}^{\bf l}=\sum_{{\bf k}\neq 0}\sum_{{\bf l} \neq 0}\frac{(\pi i)^{\mathcal{K}({\bf k}+{\bf l})}\overline{{\bf z}}^{\bf k }{\bf z}^{\bf l}}{{\bf k}{\bf l}{\bf l}{\bf l}}\sum_{p}A_{p,\sigma}({\bf k},{ \bf l})\\ -\frac{1}... | outline | |
\[\mathbf{v}(\nu_{m})=\left[\begin{array}{c}G_{\frac{n}{2}}\left(\nu_{m};0,\frac {2\pi}{n+2}\right)-\frac{2}{\nu_{m}-\lambda\frac{n}{2}+1}-\frac{b\,G_{\frac{n} {2}}(\nu_{m};0,0)\bigg{[}G_{\frac{n}{2}}\big{(}\nu_{m};0,\frac{2\pi}{n+2}\big{)} -\frac{2}{\nu_{m}-\lambda\frac{n}{2}+1}\bigg{]}}{b\,G_{\frac{n}{2}}\big{(}\nu_ ... | outline | |
\[\left\|\nabla_{x}u_{\varepsilon}|m_{p,\varepsilon}^{\frac{1}{p}} \right\|_{\mathrm{L}^{p}((0,T)\times\mathbb{T}^{3})}^{p}\lesssim\varepsilon^ {1-\alpha_{p}}\left\|f_{\varepsilon}^{0}|v|^{p}\right\|_{\mathrm{L}^{1-\alpha_ {p}}_{\mathrm{L}^{1}(\mathbb{R}^{3};\mathrm{L}^{\infty}(\mathbb{R}^{3}))}}^{ \frac{1}{2}}\mathscr... | outline | |
\[\begin{split}\frac{|B_{e}|}{|\{(s,t)\in\mathcal{Q}_{k}^{m}(kN) \times\mathcal{Q}(kN)\colon t\leq s\}|}&=\frac{\sum_{t\in \mathcal{Q}(kN)}|\{s\in\mathcal{Q}_{k}^{m}(kN)\colon(s,t)\in B_{e}\}|}{\sum_{t \in\mathcal{Q}(kN)}|\{s\in\mathcal{Q}_{k}^{m}(kN)\colon t\leq s\}}\\ &=\frac{\sum_{t\in\mathcal{Q}(kN)}\ |\{s\in\mathc... | outline | |
\begin{table}
\begin{tabular}{c|c c c|c c c|c c c} \hline \(u_{0}\) & \(H\) & \(h\) & \(\|e^{h}\|_{0,\infty,h}\) & \(H\) & \(h\) & \(\|e^{h}\|_{0,\infty,h}\) & \(H\) & \(h\) & \(\|e^{h}\|_{0,\infty,h}\) \\ \hline
1 & \(2^{-4}\) & \(2^{-8}\) & 1.95e+0 & \(2^{-5}\) & \(2^{-10}\) & 1.95e+0 & \(2^{-6}\) & \(2^{-12}\) & 1.9... | table | |
\[L_{0} \coloneqq\begin{bmatrix}l_{0}&\overline{l_{0}}\end{bmatrix}, \qquad\text{with }l_{0}\coloneqq e^{3}\stackrel{{ 0,0}}{{\odot}}\frac{2 \kappa_{1}{}^{2}\bar{R}_{1^{\prime}}}{3\bar{\Psi}_{2}}\mathcal{K}^{2}\mathcal{ K}^{2},\] \[L_{1} \coloneqq\begin{bmatrix}l_{1}&\overline{l_{1}}\end{bmatrix}, \qquad\text{with }l_{... | matrix | |
\[\nabla_{X_{a}}d[\omega_{1},\omega_{2}]_{SN} = -\frac{p+q}{(p+1)(q+1)}\bigg{(}-c(p+1)i_{X^{b}}(e_{a}\wedge\omega_ {1})\wedge i_{X_{b}}d\omega_{2}\] \[-c(q+1)i_{X^{b}}d\omega_{1}\wedge i_{X_{b}}(e_{a}\wedge\omega_{2 })\bigg{)}\] \[-c(p+q)\bigg{(}\frac{1}{p+1}i_{X_{a}}d\omega_{1}\wedge\omega_{2}+ \frac{1}{q+1}\omega_{1}... | outline | |
\[\omega(\psi_{n})\|(\psi_{n})_{+}\|_{L^{2}}^{2}+o(1)=\frac{1}{2}d \mathcal{I}^{(m)}(\psi_{n})[(\psi_{n})_{+}]\\ = \|(\psi_{n})_{+}\|_{H^{1/2}}^{2}-m\mathrm{e}^{2}\int_{\mathbb{R}^ {3}\times\mathbb{R}^{3}}\frac{\rho_{\psi_{n}}(x)\operatorname{Re}(\psi_{n},( \psi_{n})_{+})(y)}{|x-y|}dxdy\\ +m\mathrm{e}^{2}\int_{\mathbb{... | outline | |
\[\sup_{\begin{subarray}{c}P_{X^{n}\gamma_{n}\in\mathcal{C}_{s}(P_{X^ {n}}):}\\ \rho_{m}(X^{n};Y^{n})\leq\rho\end{subarray}}H(Z^{n})\] \[\geq\sup_{\begin{subarray}{c}P_{X^{n}-1\gamma_{n-1}\in\mathcal{C}_ {s}(P_{X^{n-1}}):}\\ \rho_{m}(X^{n-1};Y^{n-1})\leq\rho\end{subarray}}H(Z^{n-1})\] \[\qquad+\sup_{\begin{subarray}{c}... | matrix | |
\begin{table}
\begin{tabular}{|c|l|c|c|c|c|c|c|} \hline Pattern & Source & \(I_{MAX}(X)\) & \(I_{S}(X)\) & \(I_{SSM}(X)\) & \(I_{ZIP}(X)\) & \(I_{7Z}(X)\) & \(I_{ZPAQ}(X)\) \\ \hline \hline \(\mathrm{X}_{A}\) & Random binary pattern. & 48 & 46 & 40 & & & \\ \hline \(\mathrm{X}_{B}\) & Repeating binary pattern. & 48 & 4... | table | |
\[u^{\top}\frac{1}{T}\sum_{t=0}^{T-1}\Theta^{(t)\top}B\Theta^{(t)}v= \frac{1}{T}\sum_{t=0}^{T-1}\sum_{i,j=1}^{t+m+1}u^{(i)\top}\Psi_{t +m+1-i}^{\top}B\Psi_{t+m+1-j}v^{(j)}\] \[= \frac{1}{T}\sum_{i,j=1}^{T+m}u^{(i)\top}\left[\sum_{t=(i\lor j-m- 1)\lor 0}^{T-1}\Psi_{t+m+1-i}^{\top}B\Psi_{t+m+1-j}\right]v^{(j)}\] \[\leq \... | outline | |
\[r_{\pi}\mathcal{R}r_{\pi}^{-1} =r_{\pi}(t_{\alpha_{1}}^{2}t_{\alpha_{2}}t_{\alpha_{3}}\cdots t_{ \alpha_{2g-1}})^{g-1}t_{\alpha_{2g+1}}^{-1}r_{\pi}^{-1}\] \[=((r_{\pi}t_{\alpha_{1}}r_{\pi}^{-1})^{2}\cdot r_{\pi}t_{\alpha_ {2}}r_{\pi}^{-1}\cdot r_{\pi}t_{\alpha_{3}}r_{\pi}^{-1}\cdots r_{\pi}t_{\alpha _{2g-1}}r_{\pi}^{... | outline | |
\[\begin{split}& 2\eta_{j}B_{j}(1-\lambda)(1-(1-\lambda)L\eta_{j}- \dfrac{(1-\lambda)b_{j}}{B_{j}})\mathbb{E}\parallel\nabla f(\tilde{x}_{j}) \parallel^{2}\\ &+\dfrac{b_{j}^{3}-(1-\lambda)^{2}\eta_{j}^{2}L^{2}b_{j}B_{j}-(1- \lambda)^{2}\eta_{j}^{3}L^{3}B_{j}^{2}}{b_{j}\eta_{j}B_{j}}\mathbb{E}\parallel \tilde{x}_{j}-\ti... | outline | |
\[\tilde{\mathbb{E}}_{7}= ({\mathbb{F}^{1}}_{8}+{\mathbb{F}^{8}}_{1})+({\mathbb{F}^{2}}_{4} +{\mathbb{F}^{4}}_{2})+({\mathbb{F}^{3}}_{7}+{\mathbb{F}^{7}}_{3})+({\mathbb{ F}^{5}}_{6}+{\mathbb{F}^{6}}_{5})+\] \[({\mathbb{F}^{9}}_{16}+{\mathbb{F}^{16}}_{9})+({\mathbb{F}^{10}}_ {12}+{\mathbb{F}^{12}}_{10})+({\mathbb{F}^{11... | outline | |
\[2C|\mathcal{B}(\boldsymbol{0},\gamma)|^{2}\limsup_{n\to\infty} \sum_{k<z\leq n}\Big{\{}\frac{z^{w-1}}{\varepsilon}\Big{(}1-\Phi\Big{(}\Big{(} \frac{1}{2}\inf_{\boldsymbol{v}\in L_{\gamma}(\boldsymbol{\ell}_{\mathcal{F}}, \boldsymbol{\ell}_{\mathcal{F}}):\boldsymbol{\ell}_{\mathcal{F}}\in\mathbb{Z}^{w },\|\boldsymbol{... | outline | |
\[\begin{array}{l}\mathfrak{B}^{\rm LL}=\frac{h^{\rm L}_{2}(u)-h^{\rm L}_{1}(v) }{h^{\rm L}_{2}(u)-h^{\rm L}_{1}(u)}\frac{h^{\rm L}_{2}(v)-h^{\rm L}_{1}(u)}{h^ {\rm L}_{2}(v)-h^{\rm L}_{1}(v)}\sigma^{\rm LL}(u,v)\sigma^{\rm LL}(v,u)\\ \mathfrak{B}^{\rm RR}=\frac{h^{\rm R}_{2}(u)-h^{\rm R}_{1}(v)}{h^{\rm R}_{2}(u)- h^{\... | matrix | |
\begin{table}
\begin{tabular}{c c c|c c|c c|c c} \(N_{S}\) & \(n\) & \(n/N_{S}\) & \(n^{\Gamma}\) & \(n_{C}\) & its. & \(t_{set-up}\) & \(t_{PCG}\) \\ \hline
16/4 & 1.2\(\cdot\)10\({}^{8}\) & 7.5\(\cdot\)10\({}^{6}\) & 1.6\(\cdot\)10\({}^{4}\)/25 & 72/10 & 25 & 265 & 173 \\
32/6 & 1.2\(\cdot\)10\({}^{8}\) & 3.7\(\cdot\... | table | |
\[R^{d+1}_{\boldsymbol{\gamma}}((x^{w_{1}}g_{1},\ldots,x^{w_{d}}g _{d},x^{w_{d+1}}g^{*}),x^{m})=-(1+\gamma_{d+1})\prod_{i=1}^{d}(1+\gamma_{i})\] \[+ \frac{1}{p^{m}}\sum_{v\in G_{p,m}}\prod_{i=1}^{d}\left(1+\gamma_{ i}+\gamma_{i}\sum_{h\in G_{p,m}\setminus\{0\}}r_{p}(h)X_{p}\left(\frac{v}{x^{m}} hx^{w_{i}}g_{i}\right)\r... | outline | |
\[B_{i}(\theta) :=\operatorname{vtf}_{n_{u}}(\theta_{B_{i}}) \text{for }i\in\{1,\ldots,\kappa_{B}\},\] \[C_{i}(\theta) :=\operatorname{vtf}_{n_{y}}(\theta_{C_{i}}) \text{for }i\in\{1,\ldots,\kappa_{C}\},\] \[D_{i}(\theta) :=\operatorname{vtf}_{n_{y}}(\theta_{D_{i}}) \text{for }i\in\{1,\ldots,\kappa_{D}\},\] \[J_{i}(\th... | outline | |
\[\begin{array}{ll}\mathrm{d}X_{1}(t)&=[X_{1}(t)(1-X_{1}(t)(2Y_{1}(t)-1)\\ &+X_{1}(t)X_{2}(t)(2Y_{1}(t)-1)+2X_{1}(t)X_{3}(t)]\mathrm{d}t\\ &+\sigma_{1}X_{1}(t)(X_{1}(t)-1)\,\mathrm{d}W_{1}(t)\\ &+\sigma_{2}X_{1}(t)X_{2}(t)\,\mathrm{d}W_{2}(t)+\sigma_{3}X_{1}(t)X_{3}(t)\, \mathrm{d}W_{3}(t)\,,\\ \mathrm{d}X_{2}(t)&=[X_{... | outline | |
\[\begin{split}&\frac{d}{dt}\int_{\mathbb{T}^{3}}\frac{n(1+|v|^{2})} {2}\log(1+|v|^{2})dx+\int_{\mathbb{T}^{3}}n\log(1+|v|^{2})(\eta|\mathbb{D}(v)| ^{2}+\sqrt{\varepsilon}|\nabla v|^{2})dx\\ &\quad+\varepsilon\int_{\mathbb{T}^{3}}n|v|^{5}(1+\log(1+|v|^{2} ))dx+\frac{\varepsilon}{2}\int_{\mathbb{T}^{3}}(1+|v|^{2})\log(1... | outline | |
\[\text{W1}: \quad\text{d}s_{1}^{2}=2\text{d}u(\text{d}v+V\text{d}u)+2\text{d}U (\text{d}V+av^{4}\text{d}U)\] \[\text{W2}: \quad\text{d}s_{2}^{2}=2\text{d}u(\text{d}v+(av^{2}+bV^{2})\text{ d}u)+2\text{d}U(\text{d}V+(cv^{2}+dV^{2})\text{d}U)\] \[\text{W3}: \quad\text{d}s_{3}^{2}=2\text{d}u(\text{d}v+(av^{2}+bV^{2})\text... | outline | |
\[\begin{split}&\|X_{s_{1},t_{1}}^{x_{1}}-X_{s_{1},t_{2}}^{x_{2}}\|_{ L^{p}(\mathbb{P};H)}\\ &\quad\leq\sqrt{|t_{1}-t_{2}|}ce^{\alpha_{0}\gamma(T-s_{1})}\Big{|}p \gamma+e^{-\alpha_{0}s_{1}}V_{0}(x_{1})+\int_{0}^{T-s_{1}}\frac{e^{-\alpha_{0}s _{1}}\beta_{0}}{e^{\alpha_{0}u}}\,du\Big{|}^{\gamma}\Big{(}\sqrt{T-s_{1}}+p \B... | outline | |
\[\|w_{t}-\hat{w}_{t}\|^{2} \leq \sum_{j=t-\tau}^{t-1}\|\eta_{j}d_{\xi_{j}}(I-\Sigma_{t,j})S_{u_{j}} ^{\xi_{j}}\nabla f(\hat{w}_{j};\xi_{j})\|^{2}\] \[+\sum_{i\neq j\in\{t-\tau,\ldots,t-1\}}[\|\eta_{j}d_{\xi_{j}}(I- \Sigma_{t,j})S_{u_{j}}^{\xi_{j}}\nabla f(\hat{w}_{j};\xi_{j})\|^{2}\] \[+\|\eta_{i}d_{\xi_{i}}(I-\Sigma_... | outline | |
\begin{table}
\begin{tabular}{c c} \hline \([I]\) & Branch-node incidence matrix \([I]^{\prime}\) is the transpose) \\ \(r^{s}\) & Bus load curtailment for scenario \(s\) \\ \(d^{s}\) & Bus load, scenario \(s\) \\ \(g^{s}\) & Bus generation, scenario \(s\) \\ \(f_{e}^{s}\) & Flow for existing circuits, scenario \(s\) \... | table | |
\[q^{s}\begin{bmatrix}h-1\\ s\end{bmatrix}_{q}\begin{bmatrix}h+k-s-1\\ h\end{bmatrix}_{q}+\begin{bmatrix}h-1\\ s-1\end{bmatrix}_{q}\begin{bmatrix}h+k-s\\ h\end{bmatrix}_{q}=\] \[=q^{s}\begin{bmatrix}h-1\\ s\end{bmatrix}_{q}\begin{bmatrix}h+k-s-1\\ h\end{bmatrix}_{q}\] \[\quad+\begin{bmatrix}h-1\\ s-1\end{bmatrix}_{q}\b... | matrix | |
\[\tilde{b}_{1}=\frac{1-\alpha(1+\pi)^{\theta-1}}{(1+\pi)^{\theta-2}} \bigg{/}\bigg{(}\nu^{\prime}\bigg{(}\frac{\Delta Y}{A}\bigg{)}\frac{Y}{A}+\alpha \beta\frac{\mathbb{E}(1+\pi)^{\theta}\nu^{\prime}(\Delta Y/A)Y/A}{1-\alpha\beta \mathbb{E}(1+\pi)^{\theta}}\bigg{)}\] \[\times\bigg{\{}\beta\bigg{[}(1+\eta)\nu^{\prime}(... | outline | |
\[\tilde{J}_{A,1}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}] \equiv\left(-\frac{1}{2}I+W^{*}_{\partial\Omega^{o}}\right)[\mu^{o }]+\nu_{\Omega^{o}}\cdot\nabla v^{-}_{\Omega^{i}}[\mu^{i}]_{|\partial\Omega^{o}} \qquad\text{on }\partial\Omega^{o},\] \[\tilde{J}_{A,2}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}] \equiv\lef... | outline | |
\[[m+n]_{a,b;q,p} =\frac{\theta\big{(}bq,\tfrac{a}{b}q;p\big{)}}{\theta\big{(}q,aq, bq^{m+n},\tfrac{a}{b}q^{m+n};p\big{)}}\cdot\frac{\theta\big{(}\tfrac{a}{b}q^{m}, bq^{m},aq^{m+n},q^{m+n};p\big{)}}{\theta\big{(}\tfrac{a}{b}q^{m},bq^{m};p\big{)}}\] \[=\frac{\theta\big{(}bq,\tfrac{a}{b}q;p\big{)}}{\theta\big{(}q,aq, bq^... | outline | |
\[V_{K_{a,b}} = t^{\frac{a^{2}+b^{2}-4ab-a-b}{2}}\left[\frac{1-t^{n+2}-t^{(b+1)( a+1)}(1-t^{a-b})}{1-t^{2}}\right.\] \[\left.-t\frac{1-t^{n+2}-t^{b(a+2)}(1-t^{a-b+2})}{1-t^{2}}\right]\] \[V_{K^{\prime}_{a,b}} = t^{\frac{a^{2}+b^{2}-4ab-3a-3b}{2}}\left[t\frac{1-t^{n}-t^{(b+1)( a+1)}(1-t^{a-b-2})}{1-t^{2}}\right.\] \[\le... | outline | |
\[\varpi_{n}^{\sharp}:=\varpi_{n}^{\sharp}(\gamma):=\frac{1}{\gamma^{3 /2}}\Bigg{\{}\frac{\{(b_{\mathfrak{g}}\vee\sigma_{\mathfrak{h}})\overline{ \sigma}_{\mathfrak{g}}K_{n}^{3/2}\}^{1/2}}{n^{1/4}}+\frac{b_{\mathfrak{g}}K_{n }}{n^{1/2-1/q}}+\frac{(\overline{\sigma}_{\mathfrak{g}}\nu_{\mathfrak{h}})^{1/ 2}K_{n}}{n^{3/8-... | outline | |
\[\tfrac{\hat{h}(k^{2},t)+ik\tilde{g}(k^{2},t)}{2}=\tfrac{1}{\Upsilon (k)}\Big{\{} \hat{f}_{0}(k)[(1+r)\exp(ik)-2r]\] \[+\hat{f}_{0}(-k)(1-r)\exp(-ik)\] \[-\exp(k^{2}t/2)\hat{f}(k,t)[(1+r)\exp(ik)-2r]\] \[-\exp(k^{2}t/2)\hat{f}(-k,t)(1-r)\exp(-ik)\Big{\}},\] \[\tfrac{\hat{h}(k^{2}/2,t)+ik\tilde{g}(k^{2}/2,t)}{2}=\frac{... | outline | |
\[\langle\mathfrak{d}\otimes\mathbf{X}^{m},\mathbf{\tilde{\Upsilon}}_{ \mathfrak{t}}^{\hat{M}F}\rangle =\langle\mathfrak{d}\otimes\uparrow^{m}\mathbf{1},\mathbf{\tilde{ \Upsilon}}_{\mathfrak{t}}^{\hat{M}F}\rangle=\langle\mathfrak{d}\otimes \mathbf{1},\partial^{m}\mathbf{\tilde{\Upsilon}}_{\mathfrak{t}}^{\hat{M}F}\rangl... | outline | |
\[\begin{bmatrix}0&0&0&0&0\\ 0&K_{\mathrm{u}}&0&0&0\\ 0&0&M_{\mathrm{p}}&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\end{bmatrix}\begin{bmatrix}\dot{w}_{h}\\ u_{h}\\ \dot{p}_{h}\\ \dot{\lambda}_{\mathrm{u},h}\\ \dot{\lambda}_{\mathrm{p},h}\end{bmatrix}=\begin{bmatrix}0&-K_{\mathrm{u}}&D^{ T}&B_{\mathrm{u}}^{T}&0\\ K_{\mathrm{u}}&0&0&0... | matrix | |
\[\frac{J_{\beta,\mathbb{H}^{n}}^{\prime}(1)}{J_{\beta,\mathbb{H}^{n}}(1)}=\frac{ \int_{\partial B_{1}^{\mathbb{H}^{n}}(0)}\frac{|\nabla_{\mathbb{H}^{n}}u_{1}( \kappa)|^{2}}{\sqrt{|x|^{2}+|y|^{2}}}dP_{\mathbb{H}^{n}}^{B_{1}^{\mathbb{H}^{n }}(0)}(\kappa)}{\int_{B_{1}^{\mathbb{H}^{n}}(0)}\frac{|\nabla_{\mathbb{H}^{n} }u_... | outline | |
\[\sum_{m=3}^{\infty}\mathcal{P}_{\ell,m,0} \leq C\left\|u\right\|_{H^{r}_{\ell}}^{2}+C\tau\left\|u\right\|_{H^ {r}}\left\|u\right\|_{Y_{\tau,\ell}},\] \[\sum_{m=3}^{\infty}\mathcal{P}_{\ell,m,1} \leq C(1+\tau)\left\|u\right\|_{H^{r}_{\ell}}^{2}+C\tau^{2}\left\| u\right\|_{H^{r}}\left\|u\right\|_{Y_{\tau,\ell}},\] \[\s... | outline | |
\[\langle S_{i}\eta,\eta\rangle =a_{i}(F_{i}\eta,F_{i}\eta)+a_{i}(F_{i}\eta,R_{i}\eta-F_{i}\eta)-(f _{i},R_{i}\eta)_{L^{2}(\Omega_{i})}\] \[=a_{i}(F_{i}\eta,F_{i}\eta)+(f_{i},R_{i}\eta-F_{i}\eta)_{L^{2}( \Omega_{i})}-(f_{i},R_{i}\eta)_{L^{2}(\Omega_{i})}\] \[\geq c_{i}(\|\nabla F_{i}\eta\|_{L^{p}(\Omega_{i})^{d}}^{p}+\... | outline | |
\[w^{4}_{t}\left(\begin{array}{cc}X&Y\\ Z&W\end{array}\right) \leq 128(1-t)^{4}\max\{w^{4}(X),w^{4}(W)\}\] \[+16\alpha(1-t)^{4}\max\{\left\||Z|^{4}+|Y^{*}|^{4}\right\|,\left\|| Y|^{4}+|Z^{*}|^{4}\right\|\}\] \[+32\alpha(1-t)^{4}\max\{w(YZ),w(ZY)\}\] \[\times\max\{\left\||Z|^{2}+|Y^{*}|^{2}\right\|,\left\||Y|^{2}+|Z^ {*... | matrix | |
\[\{\delta(k^{2}+m^{2})a_{k},\delta(k^{\prime 2}+m^{2})a_{k^{\prime}}\}\] \[= \delta(k^{2}+m^{2})\delta(k^{\prime 2}+m^{2})\{a_{k},a_{k^{ \prime}}\}\] \[= \delta(k^{2}+m^{2})\delta(k^{\prime 2}+m^{2})\frac{1}{2\pi^{2}} \int\int\{k^{I}n_{I}(x)\phi(x)+\mathrm{i}\Pi(x),k^{\prime J}n_{J}(y)\phi(y)+ \mathrm{i}\Pi(y)\}\mathr... | outline | |
\[|g_{k}^{T}(\bar{d}_{k}-\tilde{d}_{k})| =|(g_{k}+J_{k}^{T}(y_{k}+\delta_{k})-J_{k}^{T}(y_{k}+\delta_{k}) )^{T}(\bar{d}_{k}-\tilde{d}_{k})|\] \[\leq|(g_{k}+J_{k}^{T}(y_{k}+\delta_{k}))^{T}(\bar{d}_{k}-\tilde{d }_{k})|+|(J_{k}^{T}(y_{k}+\delta_{k}))^{T}(\bar{d}_{k}-\tilde{d}_{k})|\] \[\leq|(H_{k}d_{k})^{T}(\bar{d}_{k}-\... | outline | |
\[\mathbb{E}_{0}Y_{n}1_{H} = \frac{1}{k^{n}}\sum_{\sigma^{1}\in S^{|\mathcal{V}(H)|}}\sum_{ \sigma^{2}\in S^{[n]/|\mathcal{V}(H)|}}\mathbb{E}_{0}1_{H}\prod_{(i_{1},\ldots,i _{m})\in\mathcal{E}(H)}\left(\frac{p_{i_{1}:i_{m}}(\sigma)}{p_{0}}\right)^{A_{i _{1}:i_{m}}}\left(\frac{q_{i_{1}:i_{m}}(\sigma)}{q_{0}}\right)^{1-A... | outline | |
\[J_{1,1}=-\int\Delta^{2}\varrho^{\kappa}\Delta\left(v^{\kappa} \cdot\nabla\varrho^{\kappa}\right)-\int\Delta^{2}\varsigma^{\kappa}\Delta \left(v^{\kappa}\cdot\nabla\varsigma^{\kappa}\right),\] \[J_{1,2}=-D\int\nabla\Delta\varrho^{\kappa}\cdot\nabla\Delta\left( \nabla\varsigma^{\kappa}\cdot\nabla\Psi^{\kappa}+\varsigma... | outline | |
\[2\begin{bmatrix}0\\ \xi_{2}\end{bmatrix}^{T}(S(x,y)-\mu I)\begin{bmatrix}0\\ \xi_{2}\end{bmatrix} =2\begin{bmatrix}0\\ \xi_{2}\end{bmatrix}^{T}\begin{bmatrix}u^{T}/u_{0}\\ -I\end{bmatrix}(S_{2}-\mu I)\begin{bmatrix}u^{T}/u_{0}\\ -I\end{bmatrix}^{T}\begin{bmatrix}0\\ \xi_{2}\end{bmatrix}\] \[=2\begin{bmatrix}\xi_{0}\\... | outline | |
\[\left\|\mathcal{D}_{k+1}^{n}F^{n}(t_{m+1},X_{t_{m}}^{n})-\mathcal{ D}_{\ell}^{n}F^{n}(t_{m+1},X_{t_{m}}^{n})\frac{\sigma(t_{k+1},X_{t_{k}}^{n}) \nabla X_{t_{\ell-1}}^{n,t_{k},X_{t_{k}}^{n}}}{\sigma(t_{\ell},X_{t_{\ell-1}}^ {n})}\right\|\] \[\leq \|(\mathcal{D}_{k+1}^{n}X_{t_{m}}^{n})(F_{x}^{n,(k+1,m+1)}-F_{x}^ {n,(\e... | outline | |
\[\sum_{\ell=1}^{m}\ \sum_{\sigma\in\operatorname{unsh}(\iota,- \iota)}\hskip-14.226378pt(-)^{|\mu_{k}|(m-\ell)}\chi(\sigma)f_{m-\ell+1}(\mu_{ \ell}(x_{\sigma_{1}},\ldots,x_{\sigma_{\ell}}),\ x_{\sigma_{\ell+1}},\ldots x_{ \sigma_{\ell}})=\] \[=\hskip-14.226378pt\sum_{\begin{subarray}{c}1\leq\ell\leq m\\ k_{1}\leq\cdot... | matrix | |
\[\|a_{i}^{2}W_{i}^{p}\otimes W_{i}^{c}+a_{i}^{2}W_{i}^{c}\otimes W _{i}^{p}+a_{i}^{2}W_{i}^{c}\otimes W_{i}^{c}\|_{L^{1}}\leq\|a_{i}\|_{L^{\infty} }(2\|W_{i}^{p}\|_{L^{2}}\|W_{i}^{c}\|_{L^{2}}+\|W_{i}^{c}\|_{L^{2}}^{2})\leq \varepsilon C(t_{0},\|R_{0}\|_{L^{\infty}})\,,\] \[\|(u_{1}^{(c)}+u_{1}^{(t)})\otimes(u_{1}-u_{... | outline | |
\[\begin{split}\|\chi^{\prime}(\frac{x-x_{0}}{R})\int_{0}^{t}\int \int_{|y-z|>|x-y|}e^{-i\frac{(x-y)^{2}}{4(t-\tau)}}\phi^{(2)}(y-z)(1-\chi( \frac{z-x_{0}}{4R}))f(z)d\tau dydz\|_{L^{2}}\\ \lesssim\|\chi^{\prime}(\frac{x-x_{0}}{R})\int_{0}^{t}\int\int_{| y-z|>|x-y|}\frac{1}{1+|x-y|^{2}}\cdot\frac{1}{1+|x-z|^{2}}\\ \time... | outline | |
\[w_{n+1}^{-}(0,1)-w_{n+1}^{-}(0,0)\] \[=\bigg{[}\frac{\lambda}{\alpha}\cdot w_{n}^{-}(1,1)+\frac{\nu}{ \alpha}\cdot w_{n}^{-}(0,2)+\frac{\gamma}{\alpha}\cdot w_{n}^{-}(0,0)+\frac{ \mu+\gamma\cdot(b-1)}{\alpha}\cdot w_{n}^{-}(0,1)\bigg{]}\] \[\quad-\bigg{[}\frac{\nu}{\alpha}\cdot w_{n}^{-}(0,1)+\frac{ \lambda+\mu+\gamm... | outline | |
\[A_{j,1}(u) \coloneqq\operatorname{Re}\overline{e^{-i\theta_{j}}\chi_{j}(u)} \sum_{k=1}^{r}\alpha_{k}e^{-i\theta_{k}}\chi_{k}(u),\] \[A_{j,2}(u) \coloneqq\sum_{\ell=2}^{r}\sum_{k=1}^{r}\alpha_{k}\operatorname{ Re}\left(e^{i(\theta_{1}-\theta_{\ell})}\overline{\chi_{1}}\chi_{\ell}(u) \right)\operatorname{Re}\left(e^{i(... | outline | |
\[\frac{d}{dt} \left(\frac{1}{2}\int_{\Omega}\rho^{(\varepsilon)}|u^{(\varepsilon )}-u|^{2}\,dx+\frac{c_{P}}{\varepsilon}\int_{\Omega}\mathcal{U}(\rho^{( \varepsilon)}|\rho)\,dx-\frac{c_{K}}{2\varepsilon}\int_{\Omega}(\rho^{( \varepsilon)}-\rho)\Lambda^{\alpha-d}(\rho^{(\varepsilon)}-\rho)\,dx\right)\] \[+\frac{1}{2\va... | outline | |
\[\|f-T_{n_{k+1}}\|\geq \|f-T_{n_{k+1}}\|_{[-b_{k},b_{k}]}=\big{\|}P_{r}+\sum_{j=k}^{ \infty}f_{n_{j+1},b_{j}}-T_{n_{k+1}}\big{\|}_{[-b_{k},b_{k}]}\] \[= \big{\|}\big{(}P_{r}+f_{n_{k+1},b_{k}}-T_{n_{k+1}}\big{)}+\sum_{j =k+1}^{\infty}f_{n_{j+1},b_{j}}\big{\|}_{[-b_{k},b_{k}]}\] \[\geq \big{\|}P_{r}+f_{n_{k+1},b_{k}}-T_... | outline | |
\[\mathcal{F}R^{\varepsilon,\ell}_{n,k,J,I_{1},\ldots,I_{k}}(\cdot,t,x)\big{(}(\xi_{j})_{\begin{subarray}{c}j\in J\\ j\neq\ell\end{subarray}}\big{)}=e^{-i\big{(}\sum_{\begin{subarray}{c}j\in J \,\xi_{j}\\ j\neq\ell\end{subarray}}\xi_{j}\big{)}\cdot x}\int_{T_{n+1}(t)}\int_{(\mathbb{R} ^{d})^{k+1}}e^{-\frac{\varepsilon}... | outline | |
\[\int_{\varkappa_{0}}^{\varkappa_{0}+2\mathbf{K}}\frac{\, \widetilde{\mathsf{Q}}^{\alpha}_{(0);2\ell+1}(\varkappa)}{(\sigma+\mathfrak{U} _{2}\,\mathrm{cn}^{2}\varkappa)^{\frac{2\ell+1}{2}}}d\varkappa= 2\int_{0}^{\frac{\pi}{2}}\frac{\,\widetilde{\mathsf{Q}}^{ \alpha}_{(0);2\ell+1}(\varphi)}{(\sigma+\mathfrak{U}_{2}-\ma... | outline | |
\[\begin{array}{rcl}-{\cal E}_{4_{f}}&=&x_{1}^{3}x_{2}+x_{0}x_{1}x_{2}^{2}+x_{ 1}^{2}x_{2}^{2}+x_{0}x_{2}^{3}+x_{1}x_{2}^{3}+x_{2}^{4}+x_{1}^{3}x_{3}\\ &&+x_{0}x_{1}x_{2}x_{3}+x_{1}^{2}x_{2}x_{3}+x_{0}x_{2}^{2}x_{3}+x_{1}x_{2}^{2}x_ {3}+x_{2}^{3}x_{3}\\ &&+x_{0}x_{1}x_{3}^{3}+x_{1}^{2}x_{3}^{2}+x_{0}x_{2}x_{3}^{2}+x_{1... | matrix | |
\[W_{a}(x) :=V_{a}(x)-c_{a}(x)(\pi\,p)^{\prime}(x)B(x),\] \[\left[\begin{matrix}\mathscr{P}_{1,1}(x,\,y)\,\,\mathscr{P}_{1,2} (x,\,y)\\ \mathscr{P}_{2,1}(x,\,y)\,\,\mathscr{P}_{2,2}(x,\,y)\end{matrix}\right] :=\left[\begin{matrix}\mathscr{V}_{1,1}^{\tilde{P}_{\rm e}}(x,\,y)+ \mathscr{V}_{1,2}^{\tilde{P}_{\rm o}}(x,\,y)... | matrix | |
\[c\left\|\left|x\right|^{-\frac{b}{2\sigma+2}}u_{2}(t)\right\|_{L^ {2\sigma+2}}^{2\sigma+2} =c\left\|\left|x\right|^{-\frac{b}{2\sigma+2}}\left(1-\phi\left( \frac{x}{R(t)}\right)\right)u(t)\right\|_{L^{2\sigma+2}}^{2\sigma+2}\] \[\leq c\int_{|x|\leq R(t)}|x|^{-b}|u(x,t)|^{2\sigma+2}\,dx\leq \frac{1}{8}\|\nabla u(t)\|_... | outline | |
\[\sum_{k=1}^{K}\mathbb{E}_{\pi^{k}\sim p^{k}}\big{[}f^{M^{\star}}(\pi _{M^{\star}})-f^{M^{\star}}(\pi^{k})\big{]}\] \[=\sum_{k=1}^{K}\mathbb{E}_{\pi^{k}\sim p^{k}}\,\mathbb{E}_{\widehat {\phi}^{k}\sim\mu^{k}}\Big{[}f^{\widehat{\psi}^{k}}(\pi_{\widehat{\psi}^{k}})- f^{M^{\star}}(\pi^{k})\Big{]}+\mathbb{E}_{\widehat{\ph... | outline | |
\[\frac{d}{dt}\int_{\mathbb{R}^{3}}(n^{\varepsilon,\tau}+1)\ln(n^{ \varepsilon,\tau}+1)(\cdot,t)dx+\int_{\mathbb{R}^{3}}\frac{1}{n^{\varepsilon, \tau}+1}|\nabla(n^{\varepsilon,\tau}+1)|^{2}\] \[+\ \frac{2}{\Theta_{0}}\frac{d}{dt}||\nabla\sqrt{c^{\varepsilon, \tau}(t)}||_{L^{2}}^{2}+\frac{4}{3\Theta_{0}}||\Delta\sqrt{c^... | outline | |
\[\max_{T_{1},T_{2}} r_{\text{A}}^{\text{c}}\] \[10\log_{10}\mu_{\text{sb}}=-10\alpha\log_{10}\left(r_{\text{A} }^{\text{c}}\right)+\delta_{\text{sb}}+\mu_{0},\] \[10\log_{10}\mu_{\text{br}}=-10\alpha\log_{10}\left(r_{\text{A}} ^{\text{c}}\right)+\delta_{\text{br}}+\mu_{0},\] \[\int_{\delta_{\text{br}}^{\text{lb}}}^{+\... | outline | |
\[\begin{array}{ll}\|x_{1}-x_{3}-\lambda(f(x_{1},p_{1},q_{1})-f(x_{3},p_{2},q_ {2}))\|\\ \leq\|x_{1}-x_{3}-\lambda(f(x_{1},p_{1},q_{1})-f(x_{3},p_{1},q_{1}))\|+\lambda \|f(x_{3},p_{2},q_{2})-f(x_{3},p_{1},q_{1})\|\\ =\left[\|x_{1}-x_{3}\|^{2}-2\lambda\langle f(x_{1},p_{1},q_{1})-f(x_{3},p_{1}, q_{1}),x_{1}-x_{3}\rangle... | matrix | |
\[\mathcal{M}^{(3)}_{f,g}(s,w;it)=\sum_{j\geq 1}\frac{\sqrt{\pi}(4\pi)^ {-w+\frac{1}{2}}\Gamma\left(w-\frac{1}{2}+ir_{j}\right)\Gamma\left(w-\frac{1} {2}-ir_{j}\right)}{\Gamma(w)}\mathcal{L}(s^{\prime},it;\overline{u_{j}}) \left\langle u_{j},V_{f,g}\right\rangle\\ +\frac{1}{4\pi}\int_{-\infty}^{\infty}\frac{\sqrt{\pi}(... | outline | |
\[\mathbb{A}_{11} =1-c^{2}+\frac{c\alpha_{1}\lambda_{1}}{2},\mathbb{A}_{22}=-c+(1+ \frac{c+\alpha}{2})\frac{\alpha_{2}\lambda_{2}}{2},\] \[\mathbb{A}_{33} =-c+(1+\frac{c+\alpha}{16})\frac{\alpha_{3}\lambda_{3}}{2}, \mathbb{A}_{44}=\frac{c\alpha_{1}}{2\lambda_{1}}-\beta_{1}^{2},\] \[\mathbb{A}_{55} =(1+\frac{c+\alpha}{2... | outline | |
\[\|\omega\|_{L^{p(\cdot)}(B)}^{-1}\left|\left\|T_{\prod\bar{b}}(f_{1 },f_{2})\right\|_{\mathscr{B}}\right\|_{L^{p(\cdot)}(B,\omega)}\] \[\leq \|\omega\|_{L^{p(\cdot)}(B)}^{-1}\left|\left\|T_{\prod\bar{b}}(f_ {1}^{0},f_{2}^{0})\right\|_{\mathscr{B}}\right\|_{L^{p(\cdot)}(B,\omega)}+ \|\omega\|_{L^{p(\cdot)}(B)}^{-1}\le... | outline | |
\[\mathbb{E}\bigg{[}L_{\rho}^{r+1}-L_{\rho}^{r}\bigg{]}\stackrel{{ \mbox{(i)}}}{{\leq}}\frac{9\tilde{\sigma}_{g}^{2}}{\rho J\sigma_{\min}}+ \frac{6L_{\mu}^{2}}{\rho\sigma_{\min}}\mathbb{E}\|z^{r}-z^{r-1}\|^{2}\] \[+\frac{3\rho\|L^{+}\|}{\sigma_{\min}}\mathbb{E}\|w^{r}\|_{L^{+}}^ {2}+\frac{\hat{L}^{2}-2\rho+\hat{L}}{2}\... | outline | |
\[\begin{array}{ccl}\binom{c+3}{\lfloor\frac{c-2}{4}\rfloor+1}&=&\frac{c(c+1) (c+2)(c+3)}{(c+2-\lfloor\frac{c-2}{4}\rfloor)(c+1-\lfloor\frac{c-2}{4}\rfloor )(c-\lfloor\frac{c-2}{4}\rfloor)(\lfloor\frac{c-2}{4}\rfloor+1)}\binom{c-1} {\lfloor\frac{c-2}{4}\rfloor}\\ &\leq&\frac{4^{4}}{3^{3}}\frac{3c(3c+3)}{(3c+13)(3c+5)}\... | outline | |
\[\underline{\bm{I}}^{k}_{\text{dev\,grad},T}\bm{\upsilon}\coloneqq \bigg{(}\bm{\pi}^{k-1}_{\mathcal{P},T}\bm{\upsilon},\big{(}\bm{ \pi}^{k}_{\mathcal{P},F}(\bm{\upsilon}_{\bm{n}_{F}}),\bm{\pi}^{k-1}_{ \mathcal{P},F}(\bm{\upsilon}_{\bm{t},F}),\bm{\pi}^{k-1}_{\mathcal{P},F}( \text{div}\,\bm{\upsilon})\big{)}_{F\in\mathc... | outline | |
\begin{table}
\begin{tabular}{c c} \hline \hline Symbol & Description \\ \hline \((X_{i})_{i\leq n}\) & \(n\) samples of input data \\ \((Y_{i})_{i\leq n_{i}}\) & \(n_{l}\) labels \\ \(\rho\) & Distribution of \((X,Y)\) \\ \(g_{\rho}\) & Function to learn (9.1) \\ \(\lambda,\mu\) & Regularization parameters \\ \(g_{\la... | table | |
\[\frac{H_{nk+r}\left(e^{-(nk+r)V_{0,k}}\right)}{H_{n}\left(e^{-2 \widetilde{\sigma}nx^{2}}\right)^{k-r}H_{n+1}\left(e^{-2\widehat{\sigma}(n+1) x^{2}}\right)^{r}}=\frac{1}{2^{n(k-1)(nk-1)+2nr(k-1)+(r-1)^{2}}}\\ \times\frac{\theta((nk+r)\Omega|\tau)}{\theta((nk+r)\Omega+ \Upsilon|\tau)}\left(\frac{2}{1+\sqrt{1-\sigma^{-... | outline | |
\[\left(\int_{\mathbb{R}^{n}}\left[\left|\widehat{a}(x)\right|\min \left\{\left[\rho_{*}(x)\right]^{1-\frac{1}{\rho_{-}}-\frac{1}{\rho_{+}}},\, \left[\rho_{*}(x)\right]^{1-\frac{2}{\rho_{+}}}\right\}\right]^{p_{+}}\,dx \right)^{1/p_{+}}\] \[\qquad\lesssim\left(\int_{(A^{*})^{-i_{0}+1}B_{0}^{*}}\left[ \left|\widehat{a}(... | outline | |
\[\int_{0}^{\infty}t\|\nabla u\|_{L_{\infty}(\mathbb{R}^{3})}^{2}\,dt \lesssim\int_{0}^{\infty}t^{-1/3}\|t\nabla u\|_{\dot{W}^{1}_{10/3 }(\mathbb{R}^{3})}^{4/3}\|\nabla u\|_{\dot{W}^{1}_{5/2}(\mathbb{R}^{3})}^{2/3 }\,dt\] \[\lesssim\|t^{-1/3}\|_{L_{3,\infty}(\mathbb{R}_{+})}\|\|t\nabla u \|_{\dot{W}^{1}_{10/3}(\mathbb{... | outline | |
\[n[L_{n}(G)-L_{n}(1/2)]\] \[\geq\sum_{t=1}^{n}\big{[}y_{t}\Big{(}2g_{\lambda,z_{1:N},\varepsilon _{1:N}}(x_{t})-2.2g_{\lambda,z_{1:N},\varepsilon_{1:N}}^{2}(x_{t})\Big{)}\] \[\qquad\qquad+(1-y_{t})\Big{(}-2g_{\lambda,z_{1:N},\varepsilon_{1: N}}(x_{t})-2.2g_{\lambda,z_{1:N},\varepsilon_{1:N}}^{2}(x_{t})\Big{)}\big{]}\]... | outline | |
\[\left\{\begin{array}{l}n_{t}+u\cdot\nabla n=\Delta n-\nabla\cdot(nS(x,n,c) \cdot\nabla c)-nm,\quad x\in\Omega,t>0,\\ c_{t}+u\cdot\nabla c=\Delta c-c+m,\quad x\in\Omega,t>0,\\ m_{t}+u\cdot\nabla m=\Delta m-nm,\quad x\in\Omega,t>0,\\ u_{t}+\kappa(u\cdot\nabla)u+\nabla P=\Delta u+(n+m)\nabla\phi,\quad x\in \Omega,t>0,\\... | outline | |
\begin{table}
\begin{tabular}{l|c c|c c|c c|c c} \hline \hline \multirow{2}{*}{Strategy} & \multicolumn{2}{c}{CSP (\(L\)=500)} & \multicolumn{2}{c}{CSP (\(L\)=1000)} & \multicolumn{2}{c}{GCP (\(N\)=75)} & \multicolumn{2}{c}{GCP (\(N\)=100)} \\ \cline{2-9} & \# Itr & Time & \# Itr & Time & \# Itr & Time & \# Itr & Time... | table | |
\[2\epsilon\geq P^{\mathrm{rand}}_{e,\text{\small{mal A}}}+P^{ \mathrm{rand}}_{e,\text{\small{mal B}}}\geq\frac{1}{N_{\mathsf{A}}\cdot N_{ \mathsf{B}}}\sum_{m_{\mathsf{A}},m_{\mathsf{B}}}\sum_{f_{\mathsf{A}},f_{\mathsf{B}}}p _{F_{\mathsf{A}}}(f_{\mathsf{A}})p_{F_{\mathsf{B}}}(f_{\mathsf{B}})\Bigg{(}\sum_{ \bm{x}}Q_{\ma... | outline | |
\[\check{X}^{5}_{m_{1}m_{2}n_{1}n_{2}} =\tfrac{1}{3}R_{m_{1}m_{2}p_{1}p_{2}}R_{n_{1}p_{1}q_{1}q_{2}}R_{n_{ 2}p_{2}q_{1}q_{2}}+\tfrac{1}{3}R_{n_{1}n_{2}p_{1}p_{2}}R_{m_{1}p_{1}q_{1}q_{2}} R_{m_{2}p_{2}q_{1}q_{2}}\] \[\quad-\tfrac{2}{3}R_{m_{1}n_{1}p_{1}p_{2}}R_{n_{2}p_{1}q_{1}q_{2} }R_{m_{2}p_{2}q_{1}q_{2}}\] \[\quad+\t... | outline | |
\begin{table}
\begin{tabular}{|l|l|l|} \hline Notation & Description & Definition \\ \hline \hline \(H_{N}(m)\) & Hamiltonian energy & (1.1) \\ \(H_{\rm TAP}(m)\) & TAP energy as function of \(m\) & (1.3) \\ \(h_{\rm TAP}(E,|m|^{2})\) & TAP energy as function of Hamiltonian energy \(E=H_{N}(m)\) and \(|m|^{2}\) & (1.34... | table | |
\[\frac{1}{2}\sum_{t=T/2}^{T-1}E\left[||\omega_{t}-\omega_{\theta_ {t}}||\right]^{2} \leq \frac{1}{\alpha_{T}}\frac{4}{\mu}||\omega_{T/2}-\omega_{\theta_{T/ 2}}||^{2}+c_{\alpha}^{2}\alpha_{T}\frac{T}{2}\frac{4}{\mu}\sigma_{c}^{2}\] \[+2L_{\omega}^{2}\frac{c_{\beta}^{2}\beta_{T}^{2}}{\alpha_{T}}\frac {T}{2}\frac{4}{\mu}... | outline |
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