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\[\langle\phi_{h}(x,a),\psi^{j}_{h+1}\rangle\geq 0\qquad\forall x \in\mathcal{X},a\in\mathcal{A}\] \[\frac{1}{m}\sum_{j=1}^{m}\|\psi^{j}_{h+1}\|_{1}\leq C_{\mathsf{ nrm}}\] \[\mathbb{E}_{(\bar{x},\bar{a})\sim\nu_{h}}\left[\max_{\ell\in[d]} \left|\mathbb{E}_{x^{\prime}\sim\mathbb{P}_{h}(\bar{x},\bar{a})}[\phi^{ \mathsf{avg}}_{h+1}(x^{\prime})_{\ell}]-\frac{1}{m}\sum_{j=1}^{m}\langle\phi_{ h}(\bar{x},\bar{a}),\psi^{j}_{h+1}\rangle\cdot\phi^{\mathsf{avg}}_{h+1}( \tilde{x}^{j}_{h+1})_{\ell}\right|\right]\] \[\leq\frac{9C_{\mathsf{nrm}}^{1/2}\log^{1/4}(36d/\delta^{2})}{m^{ 1/4}}\qquad\forall i\in[n],\ell\in[d].\]
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\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) & 0.17 (0.01) & 0.00 \\ \hline No & MNIST & SVM & 8.60 (0.31) & 8.72 (0.29) & -0.12 \\ \hline No & Semeion & LeNet & 0.18 (0.01) & 0.18 (0.01) & 0.00 \\ \hline No & MNIST & LeNet & 9.00 (0.34) & 9.12 (0.27) & -0.12 \\ \hline No & KMNIST & LeNet & 9.23 (0.33) & 9.04 (0.55) & 0.19 \\ \hline No & Fashion-MNIST & LeNet & 8.56 (0.48) & 9.45 (0.31) & -0.90 \\ \hline No & CIFAR-10 & PreActResNet18 & 45.55 (0.47) & 43.72 (0.93) & 1.82 \\ \hline No & CIFAR-100 & PreActResNet18 & 46.83 (0.90) & 43.95 (1.03) & 2.89 \\ \hline No & SVHN & PreActResNet18 & 71.95 (1.40) & 66.94 (1.67) & 5.01 \\ \hline Yes & Semeion & LeNet & 0.28 (0.02) & 0.28 (0.02) & 0.00 \\ \hline Yes & MNIST & LeNet & 14.44 (0.54) & 14.77 (0.41) & -0.32 \\ \hline Yes & KMNIST & LeNet & 12.17 (0.33) & 11.42 (0.29) & 0.75 \\ \hline Yes & Fashion-MNIST & LeNet & 12.23 (0.40) & 12.38 (0.37) & -0.14 \\ \hline Yes & CIFAR-10 & PreActResNet18 & 48.18 (0.58) & 46.40 (0.97) & 1.78 \\ \hline Yes & CIFAR-100 & PreActResNet18 & 47.37 (0.84) & 44.74 (0.91) & 2.63 \\ \hline Yes & SVHN & PreActResNet18 & 72.29 (1.23) & 67.95 (1.54) & 4.34 \\ \hline \hline \end{tabular}
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\[\|\partial_{\boldsymbol{y}}^{\boldsymbol{\nu}}(u-u_{h})\|_{V}\] \[\leq\,\bigg{[}\frac{a_{\max}}{a_{\min}}\frac{2\overline{\lambda 2_{2}}}{\rho\chi_{1}}\big{(}\overline{u}\overline{h}C_{\lambda}+\overline{ \lambda}C_{\mathcal{P}}+\overline{\lambda}C_{u}\big{)}+\overline{u}\bigg{(} \frac{a_{\max}}{\chi_{1}}C_{u}+\sqrt{\frac{a_{\max}}{\chi_{1}}}C_{\mathcal{P} }\bigg{)}+C_{\mathcal{P}}\bigg{]}h|\boldsymbol{\nu}|!^{1+\epsilon}\overline{ \boldsymbol{\beta}}^{\boldsymbol{\nu}}\] \[+\frac{a_{\max}}{a_{\min}}\frac{2\overline{u}\overline{\lambda }_{2}}{\rho\chi_{1}}(\overline{\lambda}+\overline{u})\sum_{\begin{subarray} {c}\boldsymbol{0}\neq\boldsymbol{m}\leq\boldsymbol{\nu}\\ \boldsymbol{m}\neq\boldsymbol{\nu}\end{subarray}}\binom{\boldsymbol{\nu}}{ \boldsymbol{m}}|\boldsymbol{m}|!^{1+\epsilon}\boldsymbol{\beta}^{\boldsymbol {m}}\Big{[}\|\partial_{\boldsymbol{y}}^{\boldsymbol{\nu}-\boldsymbol{m}}(u- u_{h})\|_{V}+\|\partial_{\boldsymbol{y}}^{\boldsymbol{\nu}-\boldsymbol{m}}( \lambda-\lambda_{h})\|\Big{]}\] \[+\overline{u}^{2}\frac{a_{\max}}{\chi_{1}}\sum_{\begin{subarray} {c}\boldsymbol{0}\neq\boldsymbol{m}\leq\boldsymbol{\nu}\\ \boldsymbol{m}\neq\boldsymbol{\nu}\end{subarray}}\binom{\boldsymbol{\nu}}{ \boldsymbol{m}}|\boldsymbol{m}|!^{1+\epsilon}\boldsymbol{\beta}^{\boldsymbol{ m}}\|\partial_{\boldsymbol{y}}^{\boldsymbol{\nu}-\boldsymbol{m}}(u-u_{h})\|_{V}.\]
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\[\sum_{n=0}^{\infty}l_{n}t^{n} =\frac{(1+x_{1}t)(1+x_{3}t)}{(1-x_{2}t)(1-x_{4}t)}=(1+(x_{1}+x_{3})t +x_{1}x_{3}t^{2})\left(\frac{1}{1-x_{2}t}\right)\left(\frac{1}{1-x_{4}t}\right)\] \[=(1+(x_{1}+x_{3})t+x_{1}x_{3}t^{2})\left(\sum_{n=0}^{\infty}x_{2} ^{n}t^{n}\right)\left(\sum_{n=0}^{\infty}x_{4}^{n}t^{n}\right)\] \[=(1+(x_{1}+x_{3})t+x_{1}x_{3}t^{2})\sum_{n=0}^{\infty}\left(\sum _{\begin{subarray}{c}a+b=n\\ a,b\geq 0\end{subarray}}x_{2}^{a}x_{4}^{b}\right)t^{n}\] \[=\sum_{n=0}^{\infty}\left[\sum_{\begin{subarray}{c}a+b=n\\ a,b\geq 0\end{subarray}}x_{2}^{a}x_{4}^{b}+\left((x_{1}+x_{3})\sum_{ \begin{subarray}{c}a+b=n-1\\ a,b\geq 0\end{subarray}}x_{2}^{a}x_{4}^{b}\right)+\left(x_{1}x_{3}\sum_{ \begin{subarray}{c}a+b=n-2\\ a,b\geq 0\end{subarray}}x_{2}^{a}x_{4}^{b}\right)\right]t^{n}.\]
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\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, 9158, 9159, 9160, \\ & 9161, 9162, 9165, 9166, 9168, 9171, 9172, 9174, 9179, 9180, 9181, \\ & 9182, 9184, 9185, 9186, 9190, 9192, 9193, 9194, 9195, 9198, 9199, \\ & 9202, 9204, 9207, 9208, 9210, 9211, 9212, 9216, 9220, 9224, 9237, \\ & 9242, 9244, 9247, 9258, 9264, 9274, 9280, 9284, 9286, 9306, 9328, \\ & 9330, 9342, 9372, 9374, 9424, 9442, 9458, 9811 \\ \hline \(\Gamma_{72,3}\) & 8502, 8503, 8504, 8505, 8506, 8507, 8518, 8527, 8529, 8534, 8537, \\ & 8543, 8546, 8552, 8555, 8558, 8567, 8571, 8573, 8577, 8583, 8585, \\ & 8594, 8914, 9077, 9083, 9104, 9110, 9113, 9119, 9121, 9122, 9139 \\ \hline \end{tabular}
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\[H_{0,\vec{S},\vec{\Sigma}}(A,\vec{E};\,\phi,\tilde{\pi})= \tfrac{1}{2} \int_{\Sigma}\mathrm{d}^{3}x\,\big{[}\tilde{\epsilon}^{ij}{}_{k} \,N\,\tilde{E}^{a}_{i}\tilde{E}^{b}_{j}F_{ab}{}^{k}\,+\,2S^{a}\tilde{E}^{b}_{k }F_{ab}{}^{k}\big{]}\] \[- \tfrac{1}{2}\,\int_{\Sigma}\mathrm{d}^{3}x\,\big{[}N(\tilde{\pi} ^{2}+V(\phi)q)-N^{-1}(\mathbb{L}_{\bar{N}_{i}}\phi)(\mathbb{L}_{\bar{N}_{j}} \phi)\tilde{q}^{ij}\,+\,2\tilde{\pi}\,{\cal L}_{\bar{S}}\phi\big{]}\] \[- \oint_{\partial\Sigma}\mathrm{d}^{2}S_{a}\big{[}\tilde{\epsilon} ^{ij}{}_{k}\,N\,\tilde{E}^{a}_{i}\tilde{E}^{b}_{j}A^{k}_{b}+2S^{a}\tilde{E}^{b }_{k}\,F_{ab}{}^{k}\big{]}\,.\]
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\[\int_{Q_{T}}|u|^{p}\tilde{\varphi}+CT^{-(\alpha_{1}-\gamma)}\int_ {\mathbb{R}^{N}}u_{1}(x)\varphi_{2}(x)dx\] \[\leqslant\frac{1}{3p}\int_{Q_{T}}|u|^{p}\tilde{\varphi}+\frac{3^ {p^{\prime}-1}}{p^{\prime}}\int_{\Sigma}\varphi_{2}^{-p^{\prime}/p}\varphi_{ 1}^{-p^{\prime}/p}|D^{2+\alpha_{1}-\gamma}_{t\overline{T}}\varphi_{1}|^{p^{ \prime}}\] \[+\frac{1}{3p}\int_{Q_{T}}|u|^{p}\tilde{\varphi}+\frac{3^{p^{\prime }-1}}{p^{\prime}}\int_{\Sigma}\varphi_{2}^{-p^{\prime}/p}\varphi_{1}^{-p^{ \prime}/p}|(-\Delta)^{\delta}\varphi_{2}|^{p^{\prime}}|D^{1+\alpha_{2}-\gamma} _{t\overline{T}}\varphi_{1}|^{p^{\prime}}\] \[+\frac{1}{3p}\int_{Q_{T}}|u|^{p}\tilde{\varphi}+\frac{3^{p^{\prime }-1}}{p^{\prime}}\int_{\Sigma}\varphi_{2}^{-p^{\prime}/p}\varphi_{1}^{-p^{ \prime}/p}|(-\Delta)^{\sigma}\varphi_{2}|^{p^{\prime}}|D^{1-\gamma}_{t \overline{T}}\varphi_{1}|^{p^{\prime}},\]
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\[\left\{\begin{aligned} & f([\bm{\Phi}]_{i_{r}}[x_{0}]_{i_{r}})+\\ &\underset{[\bm{\Phi}]_{i_{r}},[\bm{\Omega}]_{i_{r}},[\bm{\Xi}]_{i_{r} }\geq 0}{\text{argmin}}&\frac{\rho}{2}\left\|[\tilde{\bm{\Phi}}]_{i_{r}}-[ \tilde{H}]_{i_{r}}[\tilde{\bm{\Psi}}]_{i_{r}}^{k}+[\bm{\Lambda}]_{i_{r}}^{k} \right\|_{F}^{2}\\ &\text{s.t.}&[\bm{\Omega}]_{i_{r}}[x_{0}]_{i_{r}}+[\bm{\Xi}]_{i_{r}} [g]_{i_{r}}\leq[h]_{i_{r}}\end{aligned}\right\}\] \[[\bm{\Phi}]_{i_{c}}^{k+1}=\left\{\begin{aligned} &\underset{[\bm{\Psi}]_{i_{c}}}{ \operatorname{argmin}}&\left\|[\tilde{\bm{\Phi}}]_{i_{c}}-[\tilde{H}]_{i _{c}}[\tilde{\bm{\Psi}}]_{i_{c}}+[\bm{\Lambda}]_{i_{c}}^{k}\right\|_{F}^{2}\\ &\text{s.t.}&[Z_{AB}]_{i_{c}}[\bm{\Psi}]_{i_{c}}=[I]_{i_{c}}\end{aligned} \right\}\]
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\[\tilde{K}_{t}^{i}= G_{t}^{i}\Big{[}2cf(r_{t}^{i})+\frac{1}{2}f^{\prime\prime}(r_{t}^ {i})\left(2\sigma_{X}^{2}\varphi_{\text{rc}}\Big{(}|X_{t}^{i,N}-\bar{X}_{t}^{i }|\Big{)}^{2}\right)\] \[+f^{\prime}(r_{t}^{i})\Big{(}(1+\gamma\delta+L_{X}+\delta L_{C})| X_{t}^{i,N}-\bar{X}_{t}^{i}|-|{(X_{t}^{i,N})}^{3}-{(\bar{X}_{t}^{i})}^{3}|\] \[+(1+L_{X}+\delta L_{C}-\delta)|C_{t}^{i,N}-\bar{C}_{t}^{i}|+( \epsilon C_{f,1}+\mathcal{C}_{f,2})\,\sigma_{X}^{2}\varphi_{\text{rc}}\Big{(} |X_{t}^{i,N}-\bar{X}_{t}^{i}|\Big{)}^{2}r_{t}^{i}\Big{)}\Big{]}\] \[+\epsilon f(r_{t}^{i})\left(4\tilde{B}-\frac{\lambda}{16}\tilde{ H}(\bar{Z}_{t}^{i})-\frac{\lambda}{16}\tilde{H}(Z_{t}^{i,N})-\frac{\lambda}{1 6N}\sum_{j=1}^{N}\tilde{H}(\bar{Z}_{t}^{j})-\frac{\lambda}{16N}\sum_{j=1}^{N} \tilde{H}(Z_{t}^{j,N})\right),\]
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\[r_{+}(iD_{t})^{\nu}l_{+}\int_{0}^{\infty}K\biggl{(}\frac{t}{\tau }\biggr{)}r_{+}\varphi(\tau)\,\frac{d\tau}{\tau}\] \[\qquad=r_{+}(iD_{t})^{\nu}l_{+}\int_{0}^{\infty}K(\tau)\,r_{+} \varphi\biggl{(}\frac{t}{\tau}\biggr{)}\,\frac{d\tau}{\tau}\] \[\qquad=\int_{0}^{\infty}K(\tau)\,r_{+}(iD_{t})^{\nu}l_{+}r_{+} \varphi\biggl{(}\frac{t}{\tau}\biggr{)}\,\frac{d\tau}{\tau}\quad\text{by Remark \ref{lem:K-1}}\] \[\qquad=\int_{0}^{\infty}K(\tau)\,r_{+}(iD_{t})^{\nu}\varphi \biggl{(}\frac{t}{\tau}\biggr{)}\,\frac{d\tau}{\tau}\quad\text{by Lemma \ref{lem:K-1}}\] \[\qquad=\int_{0}^{\infty}K(\tau)r_{+}\biggl{[}\biggl{(}\frac{iD_{t }}{\tau}\biggr{)}^{\nu}\varphi\biggr{]}\biggl{(}\frac{t}{\tau}\biggr{)}\,\frac{ d\tau}{\tau}\quad\text{by Lemma \ref{lem:K-1}}\] \[\qquad=\int_{0}^{\infty}\frac{K(\tau)}{\tau^{\nu}}\,r_{+}\bigl{(} (iD_{t})^{\nu}\varphi\bigr{)}\biggl{(}\frac{t}{\tau}\biggr{)}\,\frac{d\tau}{ \tau}.\]
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\[\langle P_{t}(x),v^{\otimes t}\rangle\] \[=\left\langle v^{\otimes t},\sum_{S_{0}\subset[t]}\left(x^{ \otimes S_{0}}\right)\otimes\left(\sum_{\{S_{1},\ldots,S_{t}\}\in Z_{t}([t] \setminus S_{0})}(-1)^{\mathcal{C}\{S_{1},\ldots,S_{t}\}}(\mathcal{C}\{S_{1},\ldots,S_{t}\})!(D_{|S_{1}|})^{(S_{1})}\otimes\cdots\otimes(D_{|S_{t}|})^{(S_ {t})}\right)\,\right\rangle\] \[\geq a^{t}-\sum_{S_{0}\subset[t],S_{0}\neq[t]}a^{|S_{0}|}\left( \sum_{\{S_{1},\ldots,S_{t}\}\in Z_{t}([t]\setminus S_{0})}6^{\mathcal{C}\{S_{1 },\ldots,S_{t}\}}(\mathcal{C}\{S_{1},\ldots,S_{t}\})!|S_{1}|!\cdots|S_{t}!! \right)\] \[\geq a^{t}-\sum_{S_{0}\subset[t],S_{0}\neq[t]}a^{|S_{0}|}6^{t-|S_{ 0}|}\left(\sum_{c=1}^{t-|S_{0}|}\sum_{\begin{subarray}{c}S_{1}\cup\cdots\cup S _{c}=[t]\setminus S_{0}\\ S_{i}\cap S_{j}=\emptyset,S_{i}\neq\emptyset\end{subarray}}|S_{1}|!\cdots|S_{ c}|!\right)\] \[=a^{t}-\sum_{S_{0}\subset[t],S_{0}\neq[t]}a^{|S_{0}|}6^{t-|S_{0}| }\left(\sum_{c=1}^{t-|S_{0}|}\sum_{\begin{subarray}{c}s_{1}+\cdots+s_{s}=t-|S_ {0}|\\ s_{i}>0\end{subarray}}(t-|S_{0}|)!\right)\] \[\geq a^{t}-\sum_{S_{0}\subset[t],S_{0}\neq[t]}a^{|S_{0}|}6^{t-|S_{ 0}|}(t-|S_{0}|)!2^{t-|S_{0}|}\] \[\geq a^{t}-\sum_{c=1}^{t}\binom{t}{c}c!12^{c}a^{t-c}\geq a^{t}- \sum_{c=1}^{t}(12t)^{c}a^{t-c}=a^{t}\left(1-\sum_{c=1}^{t}\left(\frac{12t}{a} \right)^{c}\right)\geq 0.9a^{t}\,.\]
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\[\sum_{k=u+v+1}^{w+v}\sum_{a=1}^{u}\sum_{b=1}^{v}\frac{(-)^{a+v} \binom{k-a-b-1}{u-a,v-b,k-u-v-1}\binom{2w-k-1}{w+v-k}}{(4\pi\tau_{2})^{w-a-b} }\sum_{N=1}^{\infty}\frac{N^{a+b-2}}{\Gamma(a)\Gamma(b)}q^{N}\bar{q}^{N}V_{a,k -a,2w-k}(N,N)\] \[+\sum_{k=u+v+1}^{w+v}\sum_{b=1}^{v}\sum_{c=1}^{k-u-v}\frac{(-)^{ v}\binom{k-b-c-1}{u-1,v-b,k-u-v-c}\binom{2w-k-1}{w+v-k}}{(4\pi\tau_{2})^{w-b-c }}\sum_{N=1}^{\infty}\frac{N^{b+c-2}}{\Gamma(b)\Gamma(c)}q^{N}\bar{q}^{N}\] \[\times\Big{(}4\,\mathfrak{I}_{k-c}\,\sigma_{1-k+c}(N)\sigma_{1-2 w+k-c}(N)-2\sigma_{2-2w}(N)\Big{)}\] \[+(u\leftrightarrow v)\]
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\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 \(P\) & \(8\pi P_{i}=\int_{S^{2}_{\infty}}(\mathbf{i}_{\partial_{i}}\Pi)(\nu)dS^{2}_{\infty}\) \\ \(x_{i}\partial_{0}+x_{0}\partial_{i}\) (boost) & center of mass \(C\) & \(16\pi EC_{i}=\int_{S^{2}_{\infty}}\mathbb{U}_{x_{i}}(\nu)dS^{2}_{\infty}\) \\ \(X_{ij}=x_{i}\partial_{j}-x_{j}\partial_{i}\) (spatial rotation) & angular momentum \(Q\) & \(8\pi Q_{ij}=\int_{S^{2}_{\infty}}(\mathbf{i}_{X_{ij}}\Pi)(\nu)dS^{2}_{\infty}\) \\ \hline \end{tabular}
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\[\bm{H}_{\bm{f}} = -\mathbb{E}\left(\frac{\bm{f}(\bm{X}_{i})}{\pi_{\bm{\beta}_{0}} (\bm{X}_{i})(1-\pi_{\bm{\beta}_{0}}(\bm{X}_{i}))}\left(\frac{\partial\pi_{\bm {\beta}_{0}}(\bm{X}_{i})}{\partial\bm{\beta}}\right)^{\top}\right)\] \[\bm{\Omega} = \mathrm{Var}(\bm{g}_{\bm{\beta}_{0}}(T_{i},\bm{X}_{i}))\] \[\bm{g}_{\bm{\beta}_{0}}(T_{i},\bm{X}_{i}) = \left(\frac{T_{i}}{\pi_{\bm{\beta}_{0}}(\bm{X}_{i})}-\frac{1-T_{i }}{1-\pi_{\bm{\beta}_{0}}(\bm{X}_{i})}\right)\bm{f}(\bm{X}_{i})\] \[\mu_{\bm{\beta}_{0}}(T_{i},Y_{i},\bm{X}_{i}) = \frac{T_{i}Y_{i}}{\pi_{\bm{\beta}_{0}}(\bm{X}_{i})}-\frac{(1-T_{i })Y_{i}}{1-\pi_{\bm{\beta}_{0}}(\bm{X}_{i})}.\]
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\[I_{A} \lesssim \int\sum_{N_{1}\gtrsim N}\|P^{x}_{\leq N_{1}}\langle\nabla_{\xi} \rangle^{1+\gamma}\widetilde{g}\|_{L^{6}_{x}L^{2}_{\xi}}\|\langle\nabla_{\xi} \rangle^{r}P^{x}_{N_{1}}P^{\xi}_{M_{1}}\widetilde{f}\|_{\tilde{M}^{2}_{M_{1}}L ^{3}_{x}L^{3}_{\xi}}\|\langle\nabla_{\xi}\rangle^{-r}P^{x}_{N}h\|_{L^{2}_{x}L ^{2}_{\xi}}dt\] \[\lesssim \int\|\langle\nabla_{\xi}\rangle^{1+\gamma}\widetilde{g}\|_{L^{6 }_{x}L^{2}_{\xi}}\sum_{N_{1}\gtrsim N}\frac{N^{s}}{N_{1}^{s}}\|\langle\nabla_{ x}\rangle^{s}\langle\nabla_{\xi}\rangle^{r}P^{x}_{N_{1}}P^{\xi}_{M_{1}}\widetilde{f} \|_{\tilde{M}^{2}_{M_{1}}L^{3}_{x}L^{3}_{\xi}}\] \[\|\langle\nabla_{x}\rangle^{-s}\langle\nabla_{\xi}\rangle^{-r}P^{ x}_{N}h\|_{L^{2}_{x}L^{2}_{\xi}}dt\]
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\[\begin{array}{l}\Theta=\sum_{i}\varphi_{\Theta,i}=\sum_{i}\varphi_{\Theta,i}^{ eq},\ \ \ \mathfrak{T}_{\Theta}=\sum_{i}{\bf e}_{i}\varphi_{\Theta,i}^{eq}=\sum_{i}{\bf e}_{i} \varphi_{\Theta,i}\ \ \mbox{(the momentum)}\,\\ \sum_{i}{\bf e}_{i}\otimes{\bf e}_{i}\varphi_{\Theta,i}^{eq}=(\mathfrak{C}_{ \Theta}+d_{\Theta}C_{s}^{2}\Theta)\ \ \mbox{(the stress tensor)}\,\\ \Phi_{\Theta}=\sum_{i}\Phi_{\Theta,i},\ \ \sum_{i}e_{i}\Phi_{\Theta,i}=\vec{V}_{ \Theta}\Phi_{\Theta},\ \ \mbox{and}\\ \sum_{i}{\cal S}_{\Theta,i}=0,\ \ \sum_{i}{\bf e}_{i}{\cal S}_{\Theta,i}=A_{ \Theta,1}\nabla\mathfrak{b}_{\Theta,1}+A_{\Theta,2}\nabla\mathfrak{b}_{\Theta,2}. \end{array}\]
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\[g(p,q) =(2\pi)^{3}(r_{p}r_{q})^{2}\operatorname{Re}\left(\sum_{j=0}^{ \infty}\frac{1}{j+1}\frac{e^{ij(\theta_{p}-\theta_{q})}}{r_{p}^{j+1}r_{q}^{j+1 }}\left(r_{q}^{2j+2}-1-(r_{q}^{2}-1)^{j+1}\right)\right)\] \[=(2\pi)^{3}(r_{p}r_{q})^{2}\operatorname{Re}\left(e^{-i(\theta_{ p}-\theta_{q})}\sum_{j=0}^{\infty}\frac{1}{j+1}e^{i(j+1)(\theta_{p}-\theta_{q})} \left(\frac{r_{q}^{j+1}}{r_{p}^{j+1}}-\frac{1}{r_{p}^{j+1}r_{q}^{j+1}}-\frac{( r_{q}^{2}-1)^{j+1}}{r_{p}^{j+1}r_{q}^{j+1}}\right)\right)\] \[=-(2\pi)^{3}(r_{p}r_{q})^{2}\operatorname{Re}\left[e^{-i(\theta_{ p}-\theta_{q})}\bigg{(}\operatorname{Log}(1-e^{i(\theta_{p}-\theta_{q})}r_{q}/r_{p})- \operatorname{Log}(1-e^{i(\theta_{p}-\theta_{q})}1/r_{p}r_{q})\right.\] \[\left.\hskip 113.811024pt-\operatorname{Log}(1-e^{i(\theta_{p}- \theta_{q})}(r_{q}^{2}-1)/r_{p}r_{q})\bigg{)}\right]\] \[=-(2\pi)^{3}|pq|\operatorname{Re}\big{[}\overline{p}q\big{(} \operatorname{Log}(1-\overline{q}/\overline{p})-\operatorname{Log}(1-1/ \overline{p}q)-\operatorname{Log}(1-(|q|^{2}-1)/\overline{p}q)\big{)}\big{]}\,,\]
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\[[\nabla_{\partial_{k}},\nabla_{\partial_{l}}]\,dx^{j} =[\nabla_{\partial_{k}},\nabla_{\partial_{l}}]\,dx^{j}(\partial_{ m})dx^{m}\] \[=\left(\nabla_{\partial_{k}}\left(\nabla_{\partial_{l}}dx^{j} \right)\partial_{m}-\nabla_{\partial_{l}}\left(\nabla_{\partial_{k}}dx^{j} \right)\partial_{m}\right)dx^{m}\] \[=\left(\partial_{k}\left(\nabla_{\partial_{l}}dx^{j}\right) \partial_{m}-\nabla_{\partial_{l}}dx^{j}(\nabla_{\partial_{k}}\partial_{m}) \right)dx^{m}\] \[\quad-\left(\partial_{l}\left(\nabla_{\partial_{k}}dx^{j}\right) \partial_{m}-\nabla_{\partial_{k}}dx^{j}(\nabla_{\partial_{l}}\partial_{m}) \right)dx^{m}\] \[=\partial_{k}\partial_{l}dx^{j}(\partial_{m})dx^{m}-\partial_{k}dx ^{j}(\nabla_{\partial_{l}}\partial_{m})dx^{m}\] \[\quad-\partial_{l}\partial_{k}dx^{j}(\partial_{m})dx^{m}+\partial _{l}dx^{j}(\nabla_{\partial_{k}}\partial_{m})dx^{m}\] \[\quad+\left(-\partial_{l}dx^{j}(\nabla_{\partial_{k}}\partial_{m}) +dx^{j}(\nabla_{\partial_{l}}\nabla_{\partial_{k}}\partial_{m})\right)dx^{m}\] \[\quad+\left(\partial_{k}dx^{j}(\nabla_{\partial_{l}}\partial_{m}) -dx^{j}(\nabla_{\partial_{k}}\nabla_{\partial_{l}}\partial_{m})\right)dx^{m}\] \[=dx^{j}(\operatorname{Rm}_{klm}^{r}\partial_{r})dx^{m}=\operatorname{Rm}_{klm}^{j} dx^{m}\]
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\[\Big{|}\mathcal{E}-\frac{N^{2}}{I_{N}}\sum_{m_{1}=1}^{M}\sum_{m_{ 2}=1}^{M-m_{1}}\cdots\sum_{m_{M}=0}^{M-m_{1}-m_{2}-\cdots-m_{M-1}}(-1)^{m_{1}+ \cdots+m_{M}}\] \[\quad\times{N-3\choose m_{1}}\cdots{N-3-m_{1}-\cdots-m_{M-1} \choose m_{M}}\,\Big{[}\prod_{j=5}^{M}\chi(m_{1}+\cdots+m_{j}\geq j-2)\Big{]}\] \[\quad\times\int\nabla u_{1,3}\cdot\nabla u_{2,3}\prod_{j_{1}=4}^{ m_{1}+3}u_{1,j_{1}}\cdots\prod_{j_{M}=m_{1}+\cdots+m_{M-1}+4}^{m_{1}+\cdots+m_{M}+3}u_{M,j_{M}}\prod_{M+1\leq i<j\leq N}f_{ij}^{2}\Big{|}\] \[\leq C\rho\mathfrak{a}(C\rho\mathfrak{a}\ell^{2})^{M+2}+C\rho \mathfrak{a}(\rho\mathfrak{a}^{2}\ell)\sum_{j=2}^{M}(C\rho\mathfrak{a}\ell^{2} )^{j-2}\,.\]
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\[\mathbb{P}\bigg{[}\|\widetilde{g}\|_{\infty}>\frac{19L/20}{N\sqrt {n}}\bigg{]}\leq\mathbb{P}[\exists b\in B_{0}:b\in B_{1}]\] \[\leq|B_{0}|\sup_{b\in B_{0}}\mathbb{P}\bigg{[}\exists t:\mathbb{E }_{b^{\prime}}\bigg{[}g\bigg{(}t+\sum_{i=1}^{\lfloor\epsilon n\rfloor}b^{ \prime}_{i}(b_{2i}-b_{2i-1})(Y_{2i-1}-Y_{2i})\bigg{)}\bigg{)}\bigg{]}\geq\frac{ 9L/10}{N\sqrt{n}}\bigg{]}\] \[\leq|B_{0}|\sup_{b\in B_{0}}\mathbb{P}\bigg{[}\exists t:\mathbb{E }_{b^{\prime}}\bigg{[}g\bigg{(}t+\sum_{i=1}^{\lfloor\epsilon n\rfloor}b^{ \prime}_{i}(b_{2i}-b_{2i-1})(Y_{2i-1}-Y_{2i})\bigg{)}\bigg{]}\geq\frac{9L/10} {N\sqrt{n}}\bigg{]}.\]
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\[\mathbb{E}\|\mathcal{P}_{\mathbf{A}}(\mathbf{g}_{t}^{s})\|^{2} =\mathbb{E}\|\mathcal{P}_{\mathbf{A}}(\nabla F(\mathbf{x}_{t}^{s };\xi_{t}^{s})-\nabla F(\widetilde{\mathbf{x}}_{s};\xi_{t}^{s}))\|^{2}\] \[\leq\mathbb{E}\|\nabla F(\mathbf{x}_{t}^{s};\xi_{t}^{s})-\nabla F (\widetilde{\mathbf{x}}_{s};\xi_{t}^{s})\|^{2}\] \[\leq 3\mathbb{E}\|\nabla F(\mathbf{x}_{t}^{s};\xi_{t}^{s})-\nabla F (\mathcal{P}_{\mathbf{A}^{\perp}}(\mathbf{x}_{t}^{s});\xi_{t}^{s})\|^{2}+3 \mathbb{E}\|\nabla F(\mathcal{P}_{\mathbf{A}^{\perp}}(\mathbf{x}_{t}^{s}); \xi_{t}^{s})-\nabla F(\widetilde{\mathbf{x}}^{*};\xi_{t}^{s})\|^{2}\] \[\quad+3\mathbb{E}\|\nabla F(\widetilde{\mathbf{x}}_{s};\xi_{t}^{s })-\nabla F(\widetilde{\mathbf{x}}^{*};\xi_{t}^{s})\|^{2}\] \[\leq 3L^{2}\mathbb{E}\|\mathcal{P}_{\mathbf{A}}(\mathbf{x}_{t}^{s})\| ^{2}+6L\mathbb{E}\left[F(\mathcal{P}_{\mathbf{A}^{\perp}}(\mathbf{x}_{t}^{s}))- F(\widetilde{\mathbf{x}}^{*})\right]+6L\mathbb{E}\left[F(\tilde{\mathbf{x}}_{s})-F( \widetilde{\mathbf{x}}^{*})\right]\]
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\[\frac{1}{q+1}\frac{\partial\overline{w}}{\partial y_{n}} =\frac{1}{q+1}\frac{\partial}{\partial y_{n}}\left(\overline{v} ^{q+1}-(q+1)\overline{\phi}(I_{n}-y_{n}\widetilde{\kappa})^{-2}\nabla \overline{u}\cdot\nabla\overline{h}\right)\] \[=\overline{v}^{q}\frac{\partial\overline{v}}{\partial y_{n}}- \frac{\partial\overline{\phi}}{\partial y_{n}}(I_{n}-y_{n}\widetilde{\kappa} )^{-2}\nabla\overline{u}\cdot\nabla\overline{h}-2\overline{\phi}(I_{n}-y_{n} \widetilde{\kappa})^{-3}\widetilde{\kappa}\nabla\overline{u}\cdot\nabla \overline{h}\] \[\qquad-\overline{\phi}(I_{n}-y_{n}\widetilde{\kappa})^{-2}\frac{ \partial}{\partial y_{n}}(\nabla\overline{u})\cdot\nabla\overline{h}-\overline {\phi}(I_{n}-y_{n}\widetilde{\kappa})^{-2}\nabla\overline{u}\cdot\frac{ \partial}{\partial y_{n}}(\nabla\overline{h}).\]
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\[\mathcal{C}(\mathcal{F})=(E_{1}|H_{1})\wedge(E_{2}|H_{2})\wedge(E_{3}|H_{3})= \left\{\begin{array}{ll}1,&\mbox{if $E_{1}H_{1}E_{2}H_{2}E_{3}H_{3}$ is true}\\ 0,&\mbox{if $\overline{E}_{1}H_{1}\vee\overline{E}_{2}H_{2}\vee\overline{E}_{3}H_{3}$ is true},\\ x_{1},&\mbox{if $\overline{H}_{1}E_{2}H_{2}E_{3}H_{3}$ is true},\\ x_{2},&\mbox{if $\overline{H}_{2}E_{1}H_{1}E_{3}H_{3}$ is true},\\ x_{3},&\mbox{if $\overline{H}_{3}E_{1}H_{1}E_{2}H_{2}$ is true},\\ x_{12},&\mbox{if $\overline{H}_{1}\overline{H}_{2}E_{3}H_{3}$ is true},\\ x_{13},&\mbox{if $\overline{H}_{1}\overline{H}_{3}E_{2}H_{2}$ is true},\\ x_{23},&\mbox{if $\overline{H}_{2}\overline{H}_{3}E_{1}H_{1}$ is true},\\ x_{123},&\mbox{if $\overline{H}_{1}\overline{H}_{2}\overline{H}_{3}$ is true}\end{array}\right.\]
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\[\sum_{{\bf k}\neq 0}\sum_{{\bf l}\neq 0}B_{\sigma}({\bf k},{\bf l}) \overline{{\bf z}}^{\bf k}{\bf z}^{\bf l}=\sum_{{\bf k}\neq 0}\sum_{{\bf l} \neq 0}\frac{(\pi i)^{\mathcal{K}({\bf k}+{\bf l})}\overline{{\bf z}}^{\bf k }{\bf z}^{\bf l}}{{\bf k}{\bf l}{\bf l}{\bf l}}\sum_{p}A_{p,\sigma}({\bf k},{ \bf l})\\ -\frac{1}{2}\sum_{{\bf k}_{1},{\bf k}_{2}\neq 0}\sum_{1_{1}, {\bf l}_{2}\neq 0}\frac{(\pi i)^{\mathcal{K}({\bf k}_{1}+{\bf l}_{1}+{\bf k}_{2}+{ \bf l}_{2})}\overline{{\bf z}}^{\bf k}{\bf l}_{1}+{\bf k}_{2}}{{\bf z}^{\bf l} {\bf l}_{1}!{\bf l}_{2}!}\sum_{p}A_{p,\sigma}({\bf k}_{1},{\bf l}_{1})A_{p, \sigma}({\bf k}_{2},{\bf l}_{2}).\]
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\[\mathbf{v}(\nu_{m})=\left[\begin{array}{c}G_{\frac{n}{2}}\left(\nu_{m};0,\frac {2\pi}{n+2}\right)-\frac{2}{\nu_{m}-\lambda\frac{n}{2}+1}-\frac{b\,G_{\frac{n} {2}}(\nu_{m};0,0)\bigg{[}G_{\frac{n}{2}}\big{(}\nu_{m};0,\frac{2\pi}{n+2}\big{)} -\frac{2}{\nu_{m}-\lambda\frac{n}{2}+1}\bigg{]}}{b\,G_{\frac{n}{2}}\big{(}\nu_ {m};0,0)+n+2}\\ G_{\frac{n}{2}}\left(\nu_{m};-\frac{2\pi}{n+2},\frac{2\pi}{n+2}\right)-\frac{b \bigg{[}G_{\frac{n}{2}}\big{(}\nu_{m};-\frac{2\pi}{n+2},0\big{)}-\frac{2}{\nu _{m}-\lambda\frac{n}{2}+1}\bigg{]}\bigg{[}G_{\frac{n}{2}}\big{(}\nu_{m};0, \frac{2\pi}{n+2}\big{)}-\frac{2}{\nu_{m}-\lambda\frac{n}{2}+1}\bigg{]}}{b\,G_{ \frac{n}{2}}\big{(}\nu_{m};0,0\big{)}+n+2}\\ \vdots\\ G_{\frac{n}{2}}\left(\nu_{m};\frac{2(1-k)\pi}{n+2},\frac{2\pi}{n+2}\right)- \frac{1-(-1)^{k}}{\nu_{m}-\lambda\frac{n}{2}+1}-\frac{b\bigg{[}G_{\frac{n}{2} }\big{(}\nu_{m};\frac{2(1-k)\pi}{n+2},0\big{)}-\frac{1+(-1)^{k}}{\nu_{m}- \lambda\frac{n}{2}+1}\bigg{]}\bigg{[}G_{\frac{n}{2}}\big{(}\nu_{m};0,\frac{2 \pi}{n+2}\big{)}-\frac{2}{\nu_{m}-\lambda\frac{n}{2}+1}\bigg{]}}{b\,G_{\frac{n }{2}}\big{(}\nu_{m};0,0\big{)}+n+2}\\ \vdots\\ G_{\frac{n}{2}}\left(\nu_{m};-\frac{2(n+1)\pi}{n+2},\frac{2\pi}{n+2}\right)- \frac{b\bigg{[}G_{\frac{n}{2}}\big{(}\nu_{m};-\frac{2(n+1)\pi}{n+2},0\big{)}- \frac{2}{\nu_{m}-\lambda\frac{n}{2}+1}\bigg{]}\bigg{[}G_{\frac{n}{2}}\big{(} \nu_{m};0,\frac{2\pi}{n+2}\big{)}-\frac{2}{\nu_{m}-\lambda\frac{n}{2}+1}\bigg{]} }{b\,G_{\frac{n}{2}}\big{(}\nu_{m};0,0\big{)}+n+2}\\ \vdots\\ G_{\frac{n}{2}}\left(\nu_{m};-\frac{2(n+1)\pi}{n+2},\frac{2\pi}{n+2}\right)- \frac{b\bigg{[}G_{\frac{n}{2}}\big{(}\nu_{m};-\frac{2(n+1)\pi}{n+2},0\big{)}- \frac{2}{\nu_{m}-\lambda\frac{n}{2}+1}\bigg{]}}{b\,G_{\frac{n}{2}}\big{(}\nu_ {m};0,0\big{)}+n+2}\end{array}\right]\]
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\[\left\|\nabla_{x}u_{\varepsilon}|m_{p,\varepsilon}^{\frac{1}{p}} \right\|_{\mathrm{L}^{p}((0,T)\times\mathbb{T}^{3})}^{p}\lesssim\varepsilon^ {1-\alpha_{p}}\left\|f_{\varepsilon}^{0}|v|^{p}\right\|_{\mathrm{L}^{1-\alpha_ {p}}_{\mathrm{L}^{1}(\mathbb{R}^{3};\mathrm{L}^{\infty}(\mathbb{R}^{3}))}}^{ \frac{1}{2}}\mathscr{E}_{\varepsilon}(0)^{\frac{p}{2}}\] \[+\left\|f_{\varepsilon}^{0}\right\|_{\mathrm{L}^{1}(\mathbb{R}^ {3};\mathrm{L}^{\infty}(\mathbb{T}^{3}))}\mathscr{E}_{\varepsilon}(0)^{\frac {(1-\beta_{p})p}{2}}\Psi_{\varepsilon,0}^{\frac{p}{2(1-\beta_{p})}}\] \[+\left(\varepsilon^{1-\alpha_{p}}+\left\|f_{\varepsilon}^{0} \right\|_{\mathrm{L}^{1}(\mathbb{R}^{3};\mathrm{L}^{\infty}(\mathbb{T}^{3}))} \mathscr{E}_{\varepsilon}(0)^{\frac{(1-\beta_{p})p}{2}}\right)\|\Delta_{x}u_{ \varepsilon}\|_{\mathrm{L}^{p}((0,T)\times\mathbb{T}^{3})}^{p}.\]
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\[\begin{split}\frac{|B_{e}|}{|\{(s,t)\in\mathcal{Q}_{k}^{m}(kN) \times\mathcal{Q}(kN)\colon t\leq s\}|}&=\frac{\sum_{t\in \mathcal{Q}(kN)}|\{s\in\mathcal{Q}_{k}^{m}(kN)\colon(s,t)\in B_{e}\}|}{\sum_{t \in\mathcal{Q}(kN)}|\{s\in\mathcal{Q}_{k}^{m}(kN)\colon t\leq s\}}\\ &=\frac{\sum_{t\in\mathcal{Q}(kN)}\ |\{s\in\mathcal{Q}_{k}^{m}(kN) \colon h_{t}^{e}\leq s\}|}{\sum_{t\in\mathcal{Q}(kN)}|\{s\in\mathcal{Q}_{k}^{ m}(kN)\colon t\leq s\}}\\ &\leq\frac{\sum_{t\in\mathcal{Q}(kN)}\ |\{s\in\mathcal{Q}_{k}^{m}(kN) \colon h_{t}^{e}\leq s\}|}{\sum_{t\in\mathcal{Q}(kN)}\ |\{s\in\mathcal{Q}_{k}^{m}(kN)\colon t\leq s\}}\\ &\leq\frac{1}{|\mathcal{Q}(k)|}\end{split}\]
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\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 & \(2^{-6}\) & \(2^{-12}\) & 1.95e+0 \\ -1 & \(2^{-4}\) & \(2^{-8}\) & 2.25e+1 & \(2^{-5}\) & \(2^{-10}\) & 2.74e+2 & \(2^{-6}\) & \(2^{-12}\) & 2.99e+2 \\
0 & \(2^{-4}\) & \(2^{-8}\) & 1.95e+0 & \(2^{-5}\) & \(2^{-10}\) & 1.95e+0 & \(2^{-6}\) & \(2^{-12}\) & 1.95e+0 \\ \(\omega\) & \(2^{-4}\) & \(2^{-8}\) & 3.06e+0 & \(2^{-5}\) & \(2^{-10}\) & 3.05e+0 & \(2^{-6}\) & \(2^{-12}\) & 3.05e+0 \\ \(u^{DL}\) & \(2^{-4}\) & \(2^{-8}\) & 3.27e-3 & \(2^{-5}\) & \(2^{-10}\) & 1.98e-4 & \(2^{-6}\) & \(2^{-12}\) & 1.23e-5 \\ \hline \end{tabular}
\end{table}
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\[L_{0} \coloneqq\begin{bmatrix}l_{0}&\overline{l_{0}}\end{bmatrix}, \qquad\text{with }l_{0}\coloneqq e^{3}\stackrel{{ 0,0}}{{\odot}}\frac{2 \kappa_{1}{}^{2}\bar{R}_{1^{\prime}}}{3\bar{\Psi}_{2}}\mathcal{K}^{2}\mathcal{ K}^{2},\] \[L_{1} \coloneqq\begin{bmatrix}l_{1}&\overline{l_{1}}\end{bmatrix}, \qquad\text{with }l_{1}\coloneqq e_{1}\stackrel{{ 1,1}}{{\odot}}e_{2}\stackrel{{ 1,0}}{{\odot}}\frac{2}{3\bar{ \Psi}_{2}}\mathcal{K}^{1}\mathcal{K}^{2},\] \[\mathcal{W} \coloneqq\begin{bmatrix}\vartheta\Psi\end{bmatrix}, \qquad\text{with }\vartheta\Psi\coloneqq\begin{bmatrix}\frac{1}{2} \mathscr{C}\mathscr{C}&-\frac{3}{32}\bar{\Psi}_{2}\mathcal{K}^{0}\mathcal{K}^{0 }\end{bmatrix},\] \[\mathcal{W}^{\mathcal{A}} \coloneqq\begin{bmatrix}\vartheta\Psi^{A}\\ \vartheta\Psi^{A}\end{bmatrix}, \qquad\text{with }\vartheta\Psi^{A}\coloneqq\begin{bmatrix}-\frac{3}{4}\Psi_{2} \mathcal{K}^{0}\mathcal{K}^{1}e^{1}\stackrel{{ 0,1}}{{\odot}}\bullet&-\frac{3}{4}\Psi_{2} \mathcal{K}^{0}\mathcal{K}^{1}e^{2}\stackrel{{ 0,1}}{{\odot}}\bullet \end{bmatrix},\] \[\mathcal{W}^{D} \coloneqq\begin{bmatrix}\vartheta\Psi^{D}\\ \vartheta\Psi^{D}\end{bmatrix}, \qquad\text{with}\] \[\vartheta\Psi^{D} \coloneqq\begin{bmatrix}0&p^{1}&0&p^{2}\end{bmatrix},\quad p^{i} \coloneqq-\frac{1}{2}e^{i}\stackrel{{ 0,1}}{{\odot}}\mathcal{T} \bullet-\frac{1}{4}e^{1}\stackrel{{ 0,1}}{{\odot}}\mathcal{T}^{i} \stackrel{{ 0,1}}{{\odot}}\bullet-\frac{1}{4}e^{2}\stackrel{{ 0,1}}{{ \odot}}\mathcal{T}^{i}\stackrel{{ 0,0}}{{\odot}}\bullet,\]
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\[\nabla_{X_{a}}d[\omega_{1},\omega_{2}]_{SN} = -\frac{p+q}{(p+1)(q+1)}\bigg{(}-c(p+1)i_{X^{b}}(e_{a}\wedge\omega_ {1})\wedge i_{X_{b}}d\omega_{2}\] \[-c(q+1)i_{X^{b}}d\omega_{1}\wedge i_{X_{b}}(e_{a}\wedge\omega_{2 })\bigg{)}\] \[-c(p+q)\bigg{(}\frac{1}{p+1}i_{X_{a}}d\omega_{1}\wedge\omega_{2}+ \frac{1}{q+1}\omega_{1}\wedge i_{X_{a}}d\omega_{2}\bigg{)}\] \[= -c(p+q)\bigg{(}\frac{1}{q+1}e_{a}\wedge i_{X^{b}}\omega_{1}\wedge i _{X_{b}}d\omega_{2}\] \[+\frac{(-1)^{p}}{p+1}e_{a}\wedge i_{X^{b}}d\omega_{1}\wedge i_{X _{b}}\omega_{2}\bigg{)}\]
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\[\omega(\psi_{n})\|(\psi_{n})_{+}\|_{L^{2}}^{2}+o(1)=\frac{1}{2}d \mathcal{I}^{(m)}(\psi_{n})[(\psi_{n})_{+}]\\ = \|(\psi_{n})_{+}\|_{H^{1/2}}^{2}-m\mathrm{e}^{2}\int_{\mathbb{R}^ {3}\times\mathbb{R}^{3}}\frac{\rho_{\psi_{n}}(x)\operatorname{Re}(\psi_{n},( \psi_{n})_{+})(y)}{|x-y|}dxdy\\ +m\mathrm{e}^{2}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\frac{J_ {\psi_{n}}(x)\cdot\operatorname{Re}(\psi_{n},\boldsymbol{\alpha}(\psi_{n})_{+ })(y)}{|x-y|}dxdy\\ \geq \|(\psi_{n})_{+}\|_{H^{1/2}}^{2}-2\mathrm{e}^{2}\gamma_{K}\|\psi_ {n}\|_{H^{1/2}}\|(\psi_{n})_{+}\|_{H^{1/2}}\\ \geq \|(\psi_{n})_{+}\|_{H^{1/2}}^{2}-4\mathrm{e}^{2}\gamma_{K}\|(\psi _{n})_{+}\|_{H^{1/2}}^{2}\geq(1-4\mathrm{e}^{2}\gamma_{K})\|(\psi_{n})_{+}\|_{ H^{1/2}}^{2}.\]
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\[\sup_{\begin{subarray}{c}P_{X^{n}\gamma_{n}\in\mathcal{C}_{s}(P_{X^ {n}}):}\\ \rho_{m}(X^{n};Y^{n})\leq\rho\end{subarray}}H(Z^{n})\] \[\geq\sup_{\begin{subarray}{c}P_{X^{n}-1\gamma_{n-1}\in\mathcal{C}_ {s}(P_{X^{n-1}}):}\\ \rho_{m}(X^{n-1};Y^{n-1})\leq\rho\end{subarray}}H(Z^{n-1})\] \[\qquad+\sup_{\begin{subarray}{c}P_{X^{n}\gamma_{n}|X^{n-1}Y^{n-1 }\in\mathcal{C}_{s}(P_{X^{n}|X^{n-1}}):}\\ \rho_{m}(X_{n};Y_{n}|X^{n-1},Y^{n-1})\leq\rho\end{subarray}}H(Z_{n}|Z^{n-1})\] \[\geq\sup_{\begin{subarray}{c}P_{X^{n}-1\gamma_{n-1}\in\mathcal{C}_ {s}(P_{X^{n-1}}):}\\ \rho_{m}(X^{n-1};Y^{n-1})\leq\rho\end{subarray}}H(Z^{n-1})\] \[\qquad+\inf_{\begin{subarray}{c}P_{X^{n}-1\gamma_{n-1}\in \mathcal{C}_{s}(P_{X^{n-1}}):}\\ \rho_{m}(X^{n-1};Y^{n-1})\leq\rho\end{subarray}}\sup_{\begin{subarray}{c}P_{X^ {n}\gamma_{n}|X^{n-1}Y^{n-1}\in\mathcal{C}_{s}(P_{X^{n}|X^{n-1}}):}\\ \rho_{m}(X_{n};Y_{n}|X^{n-1},Y^{n-1})\leq\rho\end{subarray}}H(Z_{n}|Z^{n-1})\] \[\geq\cdots\cdots.\] \[\geq\sum_{i=1}^{n}\inf_{\begin{subarray}{c}P_{X^{i-1}Y^{i-1}\in \mathcal{C}_{s}(P_{X^{i-1}}):}\\ \rho_{m}(X^{i-1};Y^{i-1})\leq\rho\end{subarray}}\sup_{\begin{subarray}{c}P_{X^ {i}\gamma_{i}|X^{i-1}Y^{i-1}\in\mathcal{C}_{s}(P_{X^{i}|X^{i-1}}):}\\ \rho_{m}(X_{i};Y_{i}|X^{i-1},Y^{i-1})\leq\rho\end{subarray}}H(Z_{i}|Z^{i-1}),\]
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\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 & 48 & 2 & & & \\ \hline \(\mathrm{X}_{C}\) & Repeating binary pattern. & 48 & 48 & 13 & & & \\ \hline \(\mathrm{X}_{D}\) & Repeating text. & 362 & 343 & 58 & & & \\ \hline \(\mathrm{X}_{E}\) & Duplicate text with one character error. & 374 & 347 & 116 & & & \\ \hline \(\mathrm{X}_{F}\) & Random DNA pattern. & 471 & 422 & 409 & & & \\ \hline \(\mathrm{X}_{G}\) & DNA segment of COVID virus. & 471 & 405 & 388 & & & \\ \hline \(\mathrm{X}_{H}\) & Random string (0-9, x-z, A-Z). & 1209 & 1174 & 1174 & & & \\ \hline \(\mathrm{X}_{I}\) & English text (James Herriot’s Cat Stories). & 1104 & 971 & 971 & & & \\ \hline \(\mathrm{X}_{J}\) & Solar activity between 1700-2021 (A-Z). & 1495 & 1349 & 1295 & & & \\ \hline \(\mathrm{X}_{K}\) & Isaac Asimov: True love. & 50901 & 37266 & 32649 & 30904 & 29968 & 25248 \\ \hline \(\mathrm{X}_{L}\) & Binary ECG signal. & 80000 & 79491 & 47646 & 52320 & 41032 & 36968 \\ \hline \(\mathrm{X}_{M}\) & Binary seismic data. & 313664 & 312320 & 171546 & 83920 & 66064 & 45824 \\ \hline \(\mathrm{X}_{N}\) & Speech recording. & 325472 & 325342 & 277489 & 286760 & 257856 & 251408 \\ \hline \(\mathrm{X}_{O}\) & Lena. & 524288 & 524216 & 422085 & 443096 & 371360 & 337408 \\ \hline \end{tabular}
\end{table}
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\[u^{\top}\frac{1}{T}\sum_{t=0}^{T-1}\Theta^{(t)\top}B\Theta^{(t)}v= \frac{1}{T}\sum_{t=0}^{T-1}\sum_{i,j=1}^{t+m+1}u^{(i)\top}\Psi_{t +m+1-i}^{\top}B\Psi_{t+m+1-j}v^{(j)}\] \[= \frac{1}{T}\sum_{i,j=1}^{T+m}u^{(i)\top}\left[\sum_{t=(i\lor j-m- 1)\lor 0}^{T-1}\Psi_{t+m+1-i}^{\top}B\Psi_{t+m+1-j}\right]v^{(j)}\] \[\leq \frac{1}{T}\sum_{i,j=1}^{T+m}\|u^{(i)}\|_{2}\|v^{(j)}\|_{2}\|B\|_ {2}\sum_{l=0}^{\infty}\left\|\Psi_{|i-j|+l}\right\|_{2}\left\|\Psi_{l}\right\| _{2},\]
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\[r_{\pi}\mathcal{R}r_{\pi}^{-1} =r_{\pi}(t_{\alpha_{1}}^{2}t_{\alpha_{2}}t_{\alpha_{3}}\cdots t_{ \alpha_{2g-1}})^{g-1}t_{\alpha_{2g+1}}^{-1}r_{\pi}^{-1}\] \[=((r_{\pi}t_{\alpha_{1}}r_{\pi}^{-1})^{2}\cdot r_{\pi}t_{\alpha_ {2}}r_{\pi}^{-1}\cdot r_{\pi}t_{\alpha_{3}}r_{\pi}^{-1}\cdots r_{\pi}t_{\alpha _{2g-1}}r_{\pi}^{-1})^{g-1}r_{\pi}t_{\alpha_{2g+1}}^{-1}r_{\pi}^{-1}\] \[=(t_{r_{\pi}(\alpha_{1})}^{2}t_{r_{\pi}(\alpha_{2})}t_{\pi(\alpha _{3})}\cdots t_{r_{\pi}(\alpha_{2g-1})})^{g-1}t_{r_{\pi}(\alpha_{2g+1})}^{-1}\] \[=(t_{\alpha_{2g+1}}^{2}t_{\alpha_{2g}}t_{\alpha_{2g-1}}\cdots t_{ \alpha_{3}})^{g-1}t_{\alpha_{1}}^{-1}.\]
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\[\begin{split}& 2\eta_{j}B_{j}(1-\lambda)(1-(1-\lambda)L\eta_{j}- \dfrac{(1-\lambda)b_{j}}{B_{j}})\mathbb{E}\parallel\nabla f(\tilde{x}_{j}) \parallel^{2}\\ &+\dfrac{b_{j}^{3}-(1-\lambda)^{2}\eta_{j}^{2}L^{2}b_{j}B_{j}-(1- \lambda)^{2}\eta_{j}^{3}L^{3}B_{j}^{2}}{b_{j}\eta_{j}B_{j}}\mathbb{E}\parallel \tilde{x}_{j}-\tilde{x}_{j-1}\parallel^{2}\\ &+2\eta_{j}B_{j}\mathbb{E}<e_{j},\nabla f(\tilde{x}_{j})>+2b_{j} \mathbb{E}<e_{j},(\tilde{x}_{j}-\tilde{x}_{j-1})>\\ &=2\eta_{j}B_{j}(1-\lambda)(1-(1-\lambda)L\eta_{j}-\dfrac{(1- \lambda)b_{j}}{B_{j}}+\dfrac{(2\lambda-1)^{2}}{2\eta_{j}B_{j}(1-\lambda)} \mathbb{E}\parallel\nabla f(\tilde{x}_{j})\parallel^{2}\\ &+\dfrac{b_{j}^{3}-(1-\lambda)^{2}\eta_{j}^{2}L^{2}b_{j}B_{j}-(1- \lambda)^{2}\eta_{j}^{3}L^{3}B_{j}^{2}}{b_{j}\eta_{j}B_{j}}\mathbb{E} \parallel\tilde{x}_{j}-\tilde{x}_{j-1}\parallel^{2}-2\eta_{j}B_{j}\mathbb{E} \parallel\tilde{e_{j}}\parallel^{2}\text{ ( Lemma B.10)}\\ &\leq-2(1-\lambda)b_{j}\mathbb{E}<\nabla f(\tilde{x}_{j}),(\tilde {x}_{j}-\tilde{x}_{j-1})>+2b_{j}\mathbb{E}(f(\tilde{x}_{j-1})-f(\tilde{x}_{j}) )+(2L\eta_{j}^{2}B_{j}+2\eta_{j}b_{j})\mathbb{E}\parallel\tilde{e_{j}} \parallel^{2}\end{split}\]
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\[\tilde{\mathbb{E}}_{7}= ({\mathbb{F}^{1}}_{8}+{\mathbb{F}^{8}}_{1})+({\mathbb{F}^{2}}_{4} +{\mathbb{F}^{4}}_{2})+({\mathbb{F}^{3}}_{7}+{\mathbb{F}^{7}}_{3})+({\mathbb{ F}^{5}}_{6}+{\mathbb{F}^{6}}_{5})+\] \[({\mathbb{F}^{9}}_{16}+{\mathbb{F}^{16}}_{9})+({\mathbb{F}^{10}}_ {12}+{\mathbb{F}^{12}}_{10})+({\mathbb{F}^{11}}_{15}+{\mathbb{F}^{15}}_{11})+ ({\mathbb{F}^{13}}_{14}+{\mathbb{F}^{14}}_{13}),\] \[\tilde{\mathbb{E}}_{8}= -({\mathbb{F}^{1}}_{5}-{\mathbb{F}^{5}}_{1})-({\mathbb{F}^{2}}_{3 }-{\mathbb{F}^{3}}_{2})-({\mathbb{F}^{4}}_{7}-{\mathbb{F}^{7}}_{4})+({\mathbb{ F}^{6}}_{8}-{\mathbb{F}^{8}}_{6})-\] \[({\mathbb{F}^{9}}_{13}-{\mathbb{F}^{13}}_{9})-({\mathbb{F}^{10}}_ {11}-{\mathbb{F}^{11}}_{10})-({\mathbb{F}^{12}}_{15}-{\mathbb{F}^{15}}_{12})+ ({\mathbb{F}^{14}}_{16}-{\mathbb{F}^{16}}_{14}),\] \[\tilde{\mathbb{E}}_{9}= ({\mathbb{F}^{1}}_{16}-{\mathbb{F}^{16}}_{1})+({\mathbb{F}^{2}}_{12 }-{\mathbb{F}^{12}}_{2})-({\mathbb{F}^{3}}_{15}-{\mathbb{F}^{15}}_{3})+({\mathbb{ F}^{4}}_{10}-{\mathbb{F}^{10}}_{4})-\] \[({\mathbb{F}^{5}}_{14}-{\mathbb{F}^{14}}_{5})-({\mathbb{F}^{6}}_{1 3}-{\mathbb{F}^{13}}_{6})-({\mathbb{F}^{7}}_{11}-{\mathbb{F}^{11}}_{7})+({ \mathbb{F}^{8}}_{9}-{\mathbb{F}^{9}}_{8}),\]
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\[2C|\mathcal{B}(\boldsymbol{0},\gamma)|^{2}\limsup_{n\to\infty} \sum_{k<z\leq n}\Big{\{}\frac{z^{w-1}}{\varepsilon}\Big{(}1-\Phi\Big{(}\Big{(} \frac{1}{2}\inf_{\boldsymbol{v}\in L_{\gamma}(\boldsymbol{\ell}_{\mathcal{F}}, \boldsymbol{\ell}_{\mathcal{F}}):\boldsymbol{\ell}_{\mathcal{F}}\in\mathbb{Z}^{w },\|\boldsymbol{\ell}_{\mathcal{F}}\|=z}\delta(\boldsymbol{v})\Big{)}^{1/2} \Big{)}\Big{)}\] \[\quad+\mathcal{O}\Big{(}\frac{z^{w-1}}{m_{n}^{d}}\Big{)}\Big{\}}\] \[\leq\frac{2C|\mathcal{B}(\boldsymbol{0},\gamma)|^{2}}{\varepsilon} \limsup_{n\to\infty}\sum_{k<z<\infty}\Big{\{}z^{w-1}\Big{(}\exp\Big{\{}-\frac{1} {4}\inf_{\boldsymbol{v}\in L_{\gamma}(\boldsymbol{\ell}_{\mathcal{F}}, \boldsymbol{\ell}_{\mathcal{F}}):\boldsymbol{\ell}_{\mathcal{F}}\in\mathbb{Z}^{w },\|\boldsymbol{\ell}_{\mathcal{F}}\|=z-\gamma}\delta(\boldsymbol{v})\Big{\}} \Big{)}\Big{\}}\] \[\quad+\mathcal{O}\Big{(}\frac{r_{n}^{w}}{m_{n}^{d}}\Big{)}\] \[\leq\frac{2C|\mathcal{B}(\boldsymbol{0},\gamma)|^{2}}{\varepsilon} \limsup_{n\to\infty}\sum_{k<z<\infty}\Big{\{}z^{w-1}\Big{(}\exp\Big{\{}-\frac{1} {4}\inf_{\boldsymbol{v}\in L_{\gamma}^{(1)}\times\mathbb{Z}^{w}:\|\boldsymbol{v} \|\geq z-\gamma}\delta(\boldsymbol{v})\Big{\}}\Big{)}\Big{\}}\Big{\}}\] \[+\mathcal{O}\Big{(}\frac{r_{n}^{w}}{m_{n}^{d}}\Big{)},\]
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\[\begin{array}{l}\mathfrak{B}^{\rm LL}=\frac{h^{\rm L}_{2}(u)-h^{\rm L}_{1}(v) }{h^{\rm L}_{2}(u)-h^{\rm L}_{1}(u)}\frac{h^{\rm L}_{2}(v)-h^{\rm L}_{1}(u)}{h^ {\rm L}_{2}(v)-h^{\rm L}_{1}(v)}\sigma^{\rm LL}(u,v)\sigma^{\rm LL}(v,u)\\ \mathfrak{B}^{\rm RR}=\frac{h^{\rm R}_{2}(u)-h^{\rm R}_{1}(v)}{h^{\rm R}_{2}(u)- h^{\rm R}_{1}(u)}\frac{h^{\rm R}_{2}(v)-h^{\rm R}_{1}(v)}{h^{\rm R}_{2}(v)-h^{ \rm R}_{1}(v)}\sigma^{\rm RR}(u,v)\sigma^{\rm RR}(v,u)\\ \mathfrak{B}^{\rm LR}=\frac{1+h^{\rm L}_{2}(u)h^{\rm R}_{2}(v)}{1+h^{\rm L}_{1}( v)h^{\rm L}_{2}(v)}\frac{1+h^{\rm L}_{1}(v)h^{\rm L}_{1}(u)}{1+h^{\rm L}_{1}(v)h^{\rm L }_{2}(u)}\sigma^{\rm LR}(u,v)\sigma^{\rm RL}(v,u)\\ \mathfrak{B}^{\rm RL}=\frac{1+h^{\rm L}_{2}(v)h^{\rm R}_{2}(u)}{1+h^{\rm L}_{1}( v)h^{\rm R}_{2}(u)}\frac{1+h^{\rm R}_{1}(v)h^{\rm L}_{1}(u)}{1+h^{\rm L}_{1}(v)h^{\rm L }_{2}(v)}\sigma^{\rm RL}(u,v)\sigma^{\rm LR}(v,u)\end{array}\]
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\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\(\cdot\)10\({}^{6}\) & 2.5\(\cdot\)10\({}^{4}\)/46 & 151/20 & 28 & 150 & 116 \\
64/8 & 1.2\(\cdot\)10\({}^{8}\) & 1.9\(\cdot\)10\({}^{6}\) & 4.4\(\cdot\)10\({}^{4}\)/81 & 341/26 & 34 & 76 & 80 \\
128/12 & 1.2\(\cdot\)10\({}^{8}\) & 9.3\(\cdot\)10\({}^{5}\) & 7.0\(\cdot\)10\({}^{4}\)/202 & 707/40 & 49 & 39 & 73 \\
256/16 & 1.2\(\cdot\)10\({}^{8}\) & 4.7\(\cdot\)10\({}^{5}\) & 1.2\(\cdot\)10\({}^{5}\)/368 & 1862/66 & 37 & 23 & 40 \\
512/22 & 1.2\(\cdot\)10\({}^{8}\) & 2.3\(\cdot\)10\({}^{5}\) & 1.9\(\cdot\)10\({}^{5}\)/814 & 3713/121 & 62 & 15 & 114 \\ \end{tabular}
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\[R^{d+1}_{\boldsymbol{\gamma}}((x^{w_{1}}g_{1},\ldots,x^{w_{d}}g _{d},x^{w_{d+1}}g^{*}),x^{m})=-(1+\gamma_{d+1})\prod_{i=1}^{d}(1+\gamma_{i})\] \[+ \frac{1}{p^{m}}\sum_{v\in G_{p,m}}\prod_{i=1}^{d}\left(1+\gamma_{ i}+\gamma_{i}\sum_{h\in G_{p,m}\setminus\{0\}}r_{p}(h)X_{p}\left(\frac{v}{x^{m}} hx^{w_{i}}g_{i}\right)\right)\] \[\times\left(1+\gamma_{d+1}+\gamma_{d+1}\sum_{h\in G_{p,m} \setminus\{0\}}r_{p}(h)X_{p}\left(\frac{v}{x^{m}}hx^{w_{d+1}}g^{*}\right)\right)\] \[= (1+\gamma_{d+1})R^{d}_{\boldsymbol{\gamma}}((x^{w_{1}}g_{1},\ldots,x^{w_{d}}g_{d}),x^{m})+L(g^{*}),\]
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\[B_{i}(\theta) :=\operatorname{vtf}_{n_{u}}(\theta_{B_{i}}) \text{for }i\in\{1,\ldots,\kappa_{B}\},\] \[C_{i}(\theta) :=\operatorname{vtf}_{n_{y}}(\theta_{C_{i}}) \text{for }i\in\{1,\ldots,\kappa_{C}\},\] \[D_{i}(\theta) :=\operatorname{vtf}_{n_{y}}(\theta_{D_{i}}) \text{for }i\in\{1,\ldots,\kappa_{D}\},\] \[J_{i}(\theta) :=\operatorname{vtsu}(\theta_{J_{i}})-\operatorname{vtsu}\left( \theta_{J_{i}}\right)^{\mathsf{T}} \text{for }i\in\{1,\ldots,\kappa_{J}\},\] \[R_{i}(\theta) :=\operatorname{vtu}(\theta_{R_{i}})\operatorname{vtu}\left( \theta_{R_{i}}\right)^{\mathsf{T}} \text{for }i\in\{1,\ldots,\kappa_{R}\},\] \[Q_{i}(\theta) :=\operatorname{vtu}(\theta_{Q_{i}})\operatorname{vtu}\left( \theta_{Q_{i}}\right)^{\mathsf{T}} \text{for }i\in\{1,\ldots,\kappa_{Q}\}.\]
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\[\begin{array}{ll}\mathrm{d}X_{1}(t)&=[X_{1}(t)(1-X_{1}(t)(2Y_{1}(t)-1)\\ &+X_{1}(t)X_{2}(t)(2Y_{1}(t)-1)+2X_{1}(t)X_{3}(t)]\mathrm{d}t\\ &+\sigma_{1}X_{1}(t)(X_{1}(t)-1)\,\mathrm{d}W_{1}(t)\\ &+\sigma_{2}X_{1}(t)X_{2}(t)\,\mathrm{d}W_{2}(t)+\sigma_{3}X_{1}(t)X_{3}(t)\, \mathrm{d}W_{3}(t)\,,\\ \mathrm{d}X_{2}(t)&=[X_{2}(t)(1-X_{2}(t)(1-2Y_{1}(t))\\ &+X_{1}(t)X_{2}(t)(1-2Y_{1}(t))+2X_{2}(t)X_{3}(t)]\mathrm{d}t\\ &+\sigma_{2}X_{2}(t)(X_{2}(t)-1)\,\mathrm{d}W_{2}(t)+\\ &\sigma_{1}X_{2}(t)X_{1}(t)\,\mathrm{d}W_{1}(t)+\sigma_{3}X_{2}(t)X_{3}(t)\, \mathrm{d}W_{3}(t)\,,\\ \mathrm{d}X_{3}(t)&=[-2X_{3}(t)(1-X_{3}(t))-X_{3}(t)X_{2}(t)(1-2Y_{1}(t))\\ &-X_{3}(t)X_{1}(t)(2Y_{1}(t)-1)]\mathrm{d}t\\ &+\sigma_{3}X_{3}(t)(X_{3}(t)-1)\,\mathrm{d}W_{3}(t)\\ &+\sigma_{1}X_{3}(t)X_{1}(t)\,\mathrm{d}W_{1}(t)+\sigma_{2}X_{2}(t)X_{3}(t)\, \mathrm{d}W_{2}(t)\,,\\ \mathrm{d}Y_{1}(t)&=[Y_{1}(t)(1-Y_{1}(t)(2X_{2}(t)-2X_{1}(t)]\,\mathrm{d}t\\ &+\tilde{\eta}Y_{1}(t)(1-Y_{1}(t))\,\mathrm{d}\tilde{W}(t)\,.\end{array}\]
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\[\begin{split}&\frac{d}{dt}\int_{\mathbb{T}^{3}}\frac{n(1+|v|^{2})} {2}\log(1+|v|^{2})dx+\int_{\mathbb{T}^{3}}n\log(1+|v|^{2})(\eta|\mathbb{D}(v)| ^{2}+\sqrt{\varepsilon}|\nabla v|^{2})dx\\ &\quad+\varepsilon\int_{\mathbb{T}^{3}}n|v|^{5}(1+\log(1+|v|^{2} ))dx+\frac{\varepsilon}{2}\int_{\mathbb{T}^{3}}(1+|v|^{2})\log(1+|v|^{2})| \nabla\sqrt{n}|^{4}dx\\ &\quad+\kappa\int_{\mathbb{T}^{3}}n|v|^{2}(1+\log(1+|v|^{2}))dx \\ &=\frac{\varepsilon}{2}\int_{\mathbb{T}^{3}}(1+|v|^{2})\sqrt{n} \Delta\sqrt{n}\log(1+|v|^{2})dx+\frac{\varepsilon}{2}\int_{\mathbb{T}^{3}}n^ {-12}(1+v^{2})\log(1+|v|^{2})dx\\ &\quad-\varepsilon\int_{\mathbb{T}^{3}}n^{-12}|v|^{2}(1+\log(1+| v|^{2}))dx+\int_{\mathbb{T}^{3}}v\cdot\nabla n(1+\log(1+|v|^{2}))dx\\ &\quad+\kappa\int_{\mathbb{T}^{3}}nu\cdot v(1+\log(1+|v|^{2}))dx.\end{split}\]
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\[\text{W1}: \quad\text{d}s_{1}^{2}=2\text{d}u(\text{d}v+V\text{d}u)+2\text{d}U (\text{d}V+av^{4}\text{d}U)\] \[\text{W2}: \quad\text{d}s_{2}^{2}=2\text{d}u(\text{d}v+(av^{2}+bV^{2})\text{ d}u)+2\text{d}U(\text{d}V+(cv^{2}+dV^{2})\text{d}U)\] \[\text{W3}: \quad\text{d}s_{3}^{2}=2\text{d}u(\text{d}v+(av^{2}+bV^{2})\text {d}u)+2\text{d}U(\text{d}V+cV^{2}\text{d}U)\] \[\text{W4}: \quad\text{d}s_{4}^{2}=2\text{d}u(\text{d}v+av^{2}\text{d}u)+2 \text{d}U(\text{d}V+bV^{2}\text{d}U)\]
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\[\begin{split}&\|X_{s_{1},t_{1}}^{x_{1}}-X_{s_{1},t_{2}}^{x_{2}}\|_{ L^{p}(\mathbb{P};H)}\\ &\quad\leq\sqrt{|t_{1}-t_{2}|}ce^{\alpha_{0}\gamma(T-s_{1})}\Big{|}p \gamma+e^{-\alpha_{0}s_{1}}V_{0}(x_{1})+\int_{0}^{T-s_{1}}\frac{e^{-\alpha_{0}s _{1}}\beta_{0}}{e^{\alpha_{0}u}}\,du\Big{|}^{\gamma}\Big{(}\sqrt{T-s_{1}}+p \Big{)}\\ &\quad+\|x_{1}-x_{2}\|_{H}\exp\!\left(\int_{0}^{T-s_{1}}\Big{(}\phi(s _{1}+r)+\frac{e^{-\alpha_{0}s_{1}}\beta_{0}}{q_{0}e^{\alpha_{0}r}}+\frac{e^{- \alpha_{1}s_{1}}\beta_{1}}{q_{1}e^{\alpha_{1}r}}\Big{)}\,dr+\sum_{i=0}^{1} \frac{V_{i}(x_{1})+V_{i}(x_{2})}{2q_{i}e^{\alpha_{i}s_{1}}}\right).\end{split}\]
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\[\|w_{t}-\hat{w}_{t}\|^{2} \leq \sum_{j=t-\tau}^{t-1}\|\eta_{j}d_{\xi_{j}}(I-\Sigma_{t,j})S_{u_{j}} ^{\xi_{j}}\nabla f(\hat{w}_{j};\xi_{j})\|^{2}\] \[+\sum_{i\neq j\in\{t-\tau,\ldots,t-1\}}[\|\eta_{j}d_{\xi_{j}}(I- \Sigma_{t,j})S_{u_{j}}^{\xi_{j}}\nabla f(\hat{w}_{j};\xi_{j})\|^{2}\] \[+\|\eta_{i}d_{\xi_{i}}(I-\Sigma_{t,j})S_{u_{i}}^{\xi_{i}}\nabla f (\hat{w}_{i};\xi_{i})\|^{2}]\sqrt{\Delta}/2\] \[= (1+\sqrt{\Delta}\tau)\sum_{j=t-\tau}^{t-1}\|\eta_{j}d_{\xi_{j}}(I -\Sigma_{t,j})S_{u_{j}}^{\xi_{j}}\nabla f(\hat{w}_{j};\xi_{j})\|^{2}\] \[\leq (1+\sqrt{\Delta}\tau)\sum_{j=t-\tau}^{t-1}\eta_{j}^{2}\|d_{\xi_{ j}}S_{u_{j}}^{\xi_{j}}\nabla f(\hat{w}_{j};\xi_{j})\|^{2}.\]
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\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 \(s\) \\ \(f_{c}^{s}\) & Flow for candidate circuits, scenario \(s\) \\ \(\Gamma_{e}\) & Suceptance matrix for existing circuits \\ \(\Gamma_{c}\) & Suceptance matrix for candidate circuits \\ \(\Theta^{s}\) & Bus voltage angle, scenario \(s\) \\ \(\bar{f}_{e}\) & Flow limit for existing circuits \\ \(\bar{f}_{c}\) & Flow limit for candidate circuits \\ \(M_{c}\) & big-M coefficient for candidates \\ \hline \end{tabular}
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\[q^{s}\begin{bmatrix}h-1\\ s\end{bmatrix}_{q}\begin{bmatrix}h+k-s-1\\ h\end{bmatrix}_{q}+\begin{bmatrix}h-1\\ s-1\end{bmatrix}_{q}\begin{bmatrix}h+k-s\\ h\end{bmatrix}_{q}=\] \[=q^{s}\begin{bmatrix}h-1\\ s\end{bmatrix}_{q}\begin{bmatrix}h+k-s-1\\ h\end{bmatrix}_{q}\] \[\quad+\begin{bmatrix}h-1\\ s-1\end{bmatrix}_{q}\begin{pmatrix}\begin{bmatrix}h+k-s-1\\ k-s-1\end{bmatrix}_{q}+q^{k-s}\begin{bmatrix}h+k-s-1\\ k-s\end{bmatrix}_{q}\end{pmatrix}\] \[=\begin{bmatrix}h\\ s\end{bmatrix}_{q}\begin{bmatrix}h+k-s-1\\ k-s-1\end{bmatrix}_{q}+q^{k-s}\begin{bmatrix}h-1\\ s-1\end{bmatrix}_{q}\begin{bmatrix}h+k-s-1\\ k-s\end{bmatrix}_{q}\] \[=\frac{[h]_{q}!}{[s]_{q}![h-s]_{q}!}\frac{[h+k-s-1]_{q}!}{[h]_{q}![ k-s-1]_{q}!}+q^{k-s}\frac{[h-1]_{q}!}{[s-1]_{q}![h-s]_{q}!}\frac{[h+k-s-1]_{q}!}{[h- 1]_{q}![k-s]_{q}!}\] \[=\frac{[k]_{q}!}{[s]_{q}![k-s]_{q}!}\frac{[h+k-s-1]_{q}!}{[k-1]_{q}! [h-s]_{q}!}\frac{1}{[k]_{q}}\left([k-s]_{q}+q^{k-s}[s]_{q}\right)\] \[=\begin{bmatrix}k\\ s\end{bmatrix}_{q}\begin{bmatrix}h+k-s-1\\ h-s\end{bmatrix}_{q}.\]
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\[\tilde{b}_{1}=\frac{1-\alpha(1+\pi)^{\theta-1}}{(1+\pi)^{\theta-2}} \bigg{/}\bigg{(}\nu^{\prime}\bigg{(}\frac{\Delta Y}{A}\bigg{)}\frac{Y}{A}+\alpha \beta\frac{\mathbb{E}(1+\pi)^{\theta}\nu^{\prime}(\Delta Y/A)Y/A}{1-\alpha\beta \mathbb{E}(1+\pi)^{\theta}}\bigg{)}\] \[\times\bigg{\{}\beta\bigg{[}(1+\eta)\nu^{\prime}(\Delta Y/A)\frac {Y}{A}\mathbb{E}(1+\pi)^{\theta-1}\] \[-(1-\sigma)\psi u^{\prime}(Y)\frac{\theta-1}{\theta}\frac{(1- \alpha)^{1/(\theta-1)}}{(1-\alpha(1+\pi)^{\theta-1})^{1/(\theta-1)}}\mathbb{ E}(1+\pi)^{\theta}\bigg{]}\] \[-\frac{1}{\alpha}\frac{u^{\prime}(Y)}{u^{\prime\prime}(Y)}\frac{ \mathbb{E}\psi u^{\prime\prime}(Y)/(1+\pi)}{\mathbb{E}\psi u^{\prime}(Y)/(1+ \pi)}\sigma\bigg{/}\bigg{(}\sigma+\frac{a_{y}\beta}{\psi u^{\prime}(Y)} \mathbb{E}\psi u^{\prime}(Y)/(1+\pi)\bigg{)}\] \[\times\bigg{(}(1+\eta)\nu^{\prime}\bigg{(}\frac{\Delta Y}{A} \bigg{)}\frac{Y}{A}-\] \[(1-\sigma)\psi u^{\prime}(Y)\frac{(\theta-1)}{\theta}\frac{(1- \alpha)^{1/(\theta-1)}}{(1-\alpha(1+\pi)^{\theta-1})^{1/(\theta-1)}}\bigg{)} \bigg{\}}\]
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\[\tilde{J}_{A,1}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}] \equiv\left(-\frac{1}{2}I+W^{*}_{\partial\Omega^{o}}\right)[\mu^{o }]+\nu_{\Omega^{o}}\cdot\nabla v^{-}_{\Omega^{i}}[\mu^{i}]_{|\partial\Omega^{o}} \qquad\text{on }\partial\Omega^{o},\] \[\tilde{J}_{A,2}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}] \equiv\left(\frac{1}{2}I+W^{*}_{\partial\Omega^{i}}\right)[\mu^{i}]+ \nu_{\Omega^{i}}\cdot\nabla v^{+}_{\Omega^{i}}[\mu^{o}]_{|\partial\Omega^{i}}\] \[\qquad-(A_{11},A_{12})\cdot(v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial \Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}],V_{\partial\Omega^{i}}[\eta^{i}]) \qquad\text{on }\partial\Omega^{i},\] \[\tilde{J}_{A,3}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}] \equiv\left(-\frac{1}{2}I+W^{*}_{\partial\Omega^{i}}\right)[\eta^{ i}]\] \[\qquad-(A_{21},A_{22})\cdot(v^{+}_{\Omega^{i}}[\mu^{o}]_{| \partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}],V_{\partial\Omega^{i}}[\eta^{ i}]) \qquad\text{on }\partial\Omega^{i},\] \[\tilde{J}_{A,4}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}] \equiv\rho^{o},\] \[\tilde{J}_{A,5}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}] \equiv\rho^{i},\]
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\[[m+n]_{a,b;q,p} =\frac{\theta\big{(}bq,\tfrac{a}{b}q;p\big{)}}{\theta\big{(}q,aq, bq^{m+n},\tfrac{a}{b}q^{m+n};p\big{)}}\cdot\frac{\theta\big{(}\tfrac{a}{b}q^{m}, bq^{m},aq^{m+n},q^{m+n};p\big{)}}{\theta\big{(}\tfrac{a}{b}q^{m},bq^{m};p\big{)}}\] \[=\frac{\theta\big{(}bq,\tfrac{a}{b}q;p\big{)}}{\theta\big{(}q,aq, bq^{m+n},\tfrac{a}{b}q^{m+n};p\big{)}}\] \[\quad\times\frac{\theta\big{(}\tfrac{a}{b}q^{m+n},bq^{m+n},aq^{m},q^{m};p\big{)}+q^{m}\theta\big{(}\tfrac{a}{b},b,aq^{2m+n},q^{n};p\big{)}}{ \theta\big{(}\tfrac{a}{b}q^{m},bq^{m};p\big{)}}\] \[=[m]_{a,b;q,p}+\frac{\theta\big{(}aq^{2m+n},b,bq^{n},\tfrac{a}{b} q^{n},\tfrac{a}{b};p\big{)}}{\theta\big{(}aq^{n},bq^{m+n},\tfrac{a}{b}q^{m}, \tfrac{a}{b}q^{m+n};p\big{)}}q^{m}[n]_{a,b;q,p}.\]
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\[V_{K_{a,b}} = t^{\frac{a^{2}+b^{2}-4ab-a-b}{2}}\left[\frac{1-t^{n+2}-t^{(b+1)( a+1)}(1-t^{a-b})}{1-t^{2}}\right.\] \[\left.-t\frac{1-t^{n+2}-t^{b(a+2)}(1-t^{a-b+2})}{1-t^{2}}\right]\] \[V_{K^{\prime}_{a,b}} = t^{\frac{a^{2}+b^{2}-4ab-3a-3b}{2}}\left[t\frac{1-t^{n}-t^{(b+1)( a+1)}(1-t^{a-b-2})}{1-t^{2}}\right.\] \[\left.-\frac{1-t^{n}-t^{b(a+2)}(1-t^{a-b})}{1-t^{2}}\right]\] \[V_{K^{\prime\prime}_{a,b}} = t^{b}t^{\frac{(n-1)(n-2)}{2}}\left[-t^{n+2}\frac{1-t^{n-2}-t^{ba }(1-t^{a-b-2})}{1-t^{2}}\right.\] \[\left.+\frac{1-t^{n}-t^{(b+1)(a+1)}(1-t^{a-b-2})}{1-t^{2}}\right]\]
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\[\varpi_{n}^{\sharp}:=\varpi_{n}^{\sharp}(\gamma):=\frac{1}{\gamma^{3 /2}}\Bigg{\{}\frac{\{(b_{\mathfrak{g}}\vee\sigma_{\mathfrak{h}})\overline{ \sigma}_{\mathfrak{g}}K_{n}^{3/2}\}^{1/2}}{n^{1/4}}+\frac{b_{\mathfrak{g}}K_{n }}{n^{1/2-1/q}}+\frac{(\overline{\sigma}_{\mathfrak{g}}\nu_{\mathfrak{h}})^{1/ 2}K_{n}}{n^{3/8-1/(2q)}}\] \[\qquad\qquad\qquad\qquad\qquad\qquad+\frac{\overline{\sigma}_{ \mathfrak{g}}^{1/2}(\sigma_{\mathfrak{h}}b_{\mathfrak{h}})^{1/4}K_{n}^{7/8}}{ n^{3/8}}+\frac{(\overline{\sigma}_{\mathfrak{g}}b_{\mathfrak{h}})^{1/2}K_{n}^{5/4}}{ n^{1/2-1/(2q)}}+\overline{\sigma}_{\mathfrak{g}}^{1/2}\chi_{n}^{1/2}K_{n}^{1/2} \Bigg{\}}.\]
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\[\tfrac{\hat{h}(k^{2},t)+ik\tilde{g}(k^{2},t)}{2}=\tfrac{1}{\Upsilon (k)}\Big{\{} \hat{f}_{0}(k)[(1+r)\exp(ik)-2r]\] \[+\hat{f}_{0}(-k)(1-r)\exp(-ik)\] \[-\exp(k^{2}t/2)\hat{f}(k,t)[(1+r)\exp(ik)-2r]\] \[-\exp(k^{2}t/2)\hat{f}(-k,t)(1-r)\exp(-ik)\Big{\}},\] \[\tfrac{\hat{h}(k^{2}/2,t)+ik\tilde{g}(k^{2}/2,t)}{2}=\frac{1}{ \Upsilon(k)}\Big{\{} \hat{f}_{0}(k)[2\exp(ik)-(1+r)]+\hat{f}_{0}(-k)(1-r)\] \[-\exp(k^{2}t/2)\hat{f}(k,t)[2\exp(ik1)-(1+r)]\] \[-\exp(k^{2}t/2)\hat{f}(-k,t)(1-r)\Big{\}}.\]
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\[\langle\mathfrak{d}\otimes\mathbf{X}^{m},\mathbf{\tilde{\Upsilon}}_{ \mathfrak{t}}^{\hat{M}F}\rangle =\langle\mathfrak{d}\otimes\uparrow^{m}\mathbf{1},\mathbf{\tilde{ \Upsilon}}_{\mathfrak{t}}^{\hat{M}F}\rangle=\langle\mathfrak{d}\otimes \mathbf{1},\partial^{m}\mathbf{\tilde{\Upsilon}}_{\mathfrak{t}}^{\hat{M}F}\rangle\] \[=\langle\mathfrak{d}\otimes\mathbf{1},\partial^{m}(M\mathbf{ \Upsilon}^{F})_{\mathfrak{t}}\rangle=\langle\mathfrak{d}\otimes M^{*} \mathbf{1},\partial^{m}\mathbf{\tilde{\Upsilon}}_{\mathfrak{t}}^{F}\rangle\] \[=\langle\mathfrak{d}\otimes\uparrow^{m}M^{*}\mathbf{1},\mathbf{ \tilde{\Upsilon}}_{\mathfrak{t}}^{F}\rangle=\langle\mathfrak{d}\otimes M^{*} \uparrow^{m}\mathbf{1},\mathbf{\tilde{\Upsilon}}_{\mathfrak{t}}^{F}\rangle\] \[=\langle\mathfrak{d}\otimes X^{m},M\mathbf{\tilde{\Upsilon}}_{ \mathfrak{t}}^{F}\rangle\,\]
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\[\begin{bmatrix}0&0&0&0&0\\ 0&K_{\mathrm{u}}&0&0&0\\ 0&0&M_{\mathrm{p}}&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\end{bmatrix}\begin{bmatrix}\dot{w}_{h}\\ u_{h}\\ \dot{p}_{h}\\ \dot{\lambda}_{\mathrm{u},h}\\ \dot{\lambda}_{\mathrm{p},h}\end{bmatrix}=\begin{bmatrix}0&-K_{\mathrm{u}}&D^{ T}&B_{\mathrm{u}}^{T}&0\\ K_{\mathrm{u}}&0&0&0&0\\ -D&0&-K_{\mathrm{p}}(u_{h})&0&B_{\mathrm{p}}^{T}\\ -B_{\mathrm{u}}&0&0&0&0\\ 0&0&-B_{\mathrm{p}}&0&0\end{bmatrix}\begin{bmatrix}w_{h}\\ u_{h}\\ p_{h}\\ \lambda_{\mathrm{u},h}\\ \lambda_{\mathrm{p},h}\end{bmatrix}+\begin{bmatrix}f_{h}\\ 0\\ g_{h}\\ \dot{w}_{\mathrm{b},h}\\ p_{\mathrm{b},h}\end{bmatrix},\]
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\[\frac{J_{\beta,\mathbb{H}^{n}}^{\prime}(1)}{J_{\beta,\mathbb{H}^{n}}(1)}=\frac{ \int_{\partial B_{1}^{\mathbb{H}^{n}}(0)}\frac{|\nabla_{\mathbb{H}^{n}}u_{1}( \kappa)|^{2}}{\sqrt{|x|^{2}+|y|^{2}}}dP_{\mathbb{H}^{n}}^{B_{1}^{\mathbb{H}^{n }}(0)}(\kappa)}{\int_{B_{1}^{\mathbb{H}^{n}}(0)}\frac{|\nabla_{\mathbb{H}^{n} }u_{1}(\kappa)|^{2}}{|\kappa|_{\mathbb{H}^{n}}^{\mathcal{G}^{-2}}}d\kappa}+ \frac{\int_{\partial B_{1}^{\mathbb{H}^{n}}(0)}\frac{|\nabla_{\mathbb{H}^{n}}u _{2}(\kappa)|^{2}}{\sqrt{|x|^{2}+|y|^{2}}}dP_{\mathbb{H}^{n}}^{B_{1}^{\mathbb{ H}^{n}}(0)}(\kappa)}{\int_{B_{1}^{\mathbb{H}^{n}}(0)}\frac{|\nabla_{\mathbb{H}^{n}}u _{2}(\kappa)|^{2}}{|\kappa|_{\mathbb{H}^{n}}^{\mathcal{G}^{-2}}}d\kappa}-\beta.\]
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\[\sum_{m=3}^{\infty}\mathcal{P}_{\ell,m,0} \leq C\left\|u\right\|_{H^{r}_{\ell}}^{2}+C\tau\left\|u\right\|_{H^ {r}}\left\|u\right\|_{Y_{\tau,\ell}},\] \[\sum_{m=3}^{\infty}\mathcal{P}_{\ell,m,1} \leq C(1+\tau)\left\|u\right\|_{H^{r}_{\ell}}^{2}+C\tau^{2}\left\| u\right\|_{H^{r}}\left\|u\right\|_{Y_{\tau,\ell}},\] \[\sum_{m=5}^{\infty}\mathcal{P}_{\ell,m,2} \leq C\tau^{2}\left\|u\right\|_{H^{r}_{\ell}}^{2}+C\tau^{3}\left\| u\right\|_{H^{r}}\left\|u\right\|_{Y_{\tau,\ell}},\] \[\sum_{m=8}^{\infty}\sum_{k=3}^{[m/2]-1}\mathcal{P}_{\ell,m,k} \leq C\tau^{3/2}\left\|u\right\|_{X_{\tau}}\left\|u\right\|_{Y_{ \tau,\ell}},\] \[\sum_{m=6}^{\infty}\sum_{j=[m/2]}^{m-3}\mathcal{P}_{\ell,m,j} \leq C\tau^{3/2}\left\|u\right\|_{X_{\tau}}\left\|u\right\|_{Y_{ \tau,\ell}},\] \[\sum_{m=4}^{\infty}\mathcal{P}_{\ell,m,m-2} \leq C\tau\left\|u\right\|_{H^{r}_{\ell}}^{2}+C\tau^{2}\left\|u \right\|_{H^{r}}\left\|u\right\|_{Y_{\tau,\ell}},\] \[\sum_{m=3}^{\infty}\mathcal{P}_{\ell,m,m-1} \leq C\left\|u\right\|_{H^{r}_{\ell}}^{2}+C\tau\left\|u\right\|_{H ^{r}}\left\|u\right\|_{Y_{\tau,\ell}}.\]
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\[\langle S_{i}\eta,\eta\rangle =a_{i}(F_{i}\eta,F_{i}\eta)+a_{i}(F_{i}\eta,R_{i}\eta-F_{i}\eta)-(f _{i},R_{i}\eta)_{L^{2}(\Omega_{i})}\] \[=a_{i}(F_{i}\eta,F_{i}\eta)+(f_{i},R_{i}\eta-F_{i}\eta)_{L^{2}( \Omega_{i})}-(f_{i},R_{i}\eta)_{L^{2}(\Omega_{i})}\] \[\geq c_{i}(\|\nabla F_{i}\eta\|_{L^{p}(\Omega_{i})^{d}}^{p}+\|F_{ i}\eta\|_{L^{r}(\Omega_{i})}^{r})-(f_{i},F_{i}\eta)_{L^{2}(\Omega_{i})}\] \[\geq c_{i}(\|\nabla F_{i}\eta\|_{L^{p}(\Omega_{i})^{d}},\|F_{i} \eta\|_{L^{r}(\Omega_{i})})\|F_{i}\eta\|_{V_{i}}-\|f_{i}\|_{L^{2}(\Omega_{i})} \|F_{i}\eta\|_{V_{i}}\] \[\geq c_{i}\big{(}P(\|\nabla F_{i}\eta\|_{L^{p}(\Omega_{i})^{d}},\|F _{i}\eta\|_{L^{r}(\Omega_{i})})-\|f_{i}\|_{L^{2}(\Omega_{i})}\big{)}\|\eta\|_{ \Lambda_{i}},\]
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\[w^{4}_{t}\left(\begin{array}{cc}X&Y\\ Z&W\end{array}\right) \leq 128(1-t)^{4}\max\{w^{4}(X),w^{4}(W)\}\] \[+16\alpha(1-t)^{4}\max\{\left\||Z|^{4}+|Y^{*}|^{4}\right\|,\left\|| Y|^{4}+|Z^{*}|^{4}\right\|\}\] \[+32\alpha(1-t)^{4}\max\{w(YZ),w(ZY)\}\] \[\times\max\{\left\||Z|^{2}+|Y^{*}|^{2}\right\|,\left\||Y|^{2}+|Z^ {*}|^{2}\right\|\}\] \[+32\alpha(1-t)^{4}\max\{w^{2}(YZ),w^{2}(ZY)\}\] \[+32(1-\alpha)(1-t)^{4}\max\left\{\left\||Z|^{2}+|Y^{*}|^{2} \right\|,\left\||Y|^{2}+|Z^{*}|^{2}\right\|\right\}\] \[\times w^{2}\left(\begin{array}{cc}O&Y\\ Z&O\end{array}\right)\] \[+64(1-\alpha)(1-t)^{4}\max\{w(YZ),w(ZY)\}w^{2}\left(\begin{array}[] {cc}O&Y\\ Z&O\end{array}\right).\]
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\[\{\delta(k^{2}+m^{2})a_{k},\delta(k^{\prime 2}+m^{2})a_{k^{\prime}}\}\] \[= \delta(k^{2}+m^{2})\delta(k^{\prime 2}+m^{2})\{a_{k},a_{k^{ \prime}}\}\] \[= \delta(k^{2}+m^{2})\delta(k^{\prime 2}+m^{2})\frac{1}{2\pi^{2}} \int\int\{k^{I}n_{I}(x)\phi(x)+\mathrm{i}\Pi(x),k^{\prime J}n_{J}(y)\phi(y)+ \mathrm{i}\Pi(y)\}\mathrm{e}^{-\mathrm{i}\vec{k}\cdot\int^{x}\vec{e}- \mathrm{i}\vec{k}^{\prime}\cdot\int^{y}\vec{e}}\mathrm{d}^{2}x\mathrm{d}^{2}y\] \[= \delta(k^{2}+m^{2})\delta(k^{\prime 2}+m^{2})\frac{\mathrm{i}}{2\pi^{2}} \int\int\left(-k^{I}n_{I}(x)\delta(x-y)+k^{\prime J}n_{J}(y)\delta(x-y)\right) \mathrm{e}^{-\mathrm{i}\vec{k}\cdot\int^{x}\vec{e}-\mathrm{i}\vec{k}^{\prime }\cdot\int^{y}\vec{e}}\mathrm{d}^{2}x\mathrm{d}^{2}y\] \[= \delta(k^{2}+m^{2})\delta(k^{\prime 2}+m^{2})\frac{\mathrm{i}}{2\pi^{2}} \int(k^{\prime}-k)^{I}n_{I}\mathrm{e}^{-\mathrm{i}(\vec{k}+\vec{e})\cdot\int^{ x}\vec{e}}\mathrm{d}^{2}x.\]
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\[|g_{k}^{T}(\bar{d}_{k}-\tilde{d}_{k})| =|(g_{k}+J_{k}^{T}(y_{k}+\delta_{k})-J_{k}^{T}(y_{k}+\delta_{k}) )^{T}(\bar{d}_{k}-\tilde{d}_{k})|\] \[\leq|(g_{k}+J_{k}^{T}(y_{k}+\delta_{k}))^{T}(\bar{d}_{k}-\tilde{d }_{k})|+|(J_{k}^{T}(y_{k}+\delta_{k}))^{T}(\bar{d}_{k}-\tilde{d}_{k})|\] \[\leq|(H_{k}d_{k})^{T}(\bar{d}_{k}-\tilde{d}_{k})|+\|y_{k}+\delta _{k}\|_{\infty}\|\bar{r}_{k}\|_{1}\] \[\leq\frac{\kappa_{H}\sqrt{\theta_{2}}}{\sqrt{\kappa_{\Delta l,d}} }\beta^{\sigma}\Delta l(x_{k},\tau_{k},g_{k},d_{k})+\kappa_{y\delta}\omega_{a} \beta^{\sigma}\Delta l(x_{k},\bar{\tau}_{k},\bar{g}_{k},\bar{d}_{k}).\]
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\[\mathbb{E}_{0}Y_{n}1_{H} = \frac{1}{k^{n}}\sum_{\sigma^{1}\in S^{|\mathcal{V}(H)|}}\sum_{ \sigma^{2}\in S^{[n]/|\mathcal{V}(H)|}}\mathbb{E}_{0}1_{H}\prod_{(i_{1},\ldots,i _{m})\in\mathcal{E}(H)}\left(\frac{p_{i_{1}:i_{m}}(\sigma)}{p_{0}}\right)^{A_{i _{1}:i_{m}}}\left(\frac{q_{i_{1}:i_{m}}(\sigma)}{q_{0}}\right)^{1-A_{i_{1}:i_{m }}}\] \[= k^{-|\mathcal{V}(H)|}\sum_{\sigma^{1}\in S^{|\mathcal{V}(H)|}} \mathbb{E}_{0}1_{H}\prod_{(i_{1},\ldots,i_{m})\in\mathcal{E}(H)}\left(\frac{p_ {i_{1}:i_{m}}(\sigma^{1})}{p_{0}}\right)^{A_{i_{1}:i_{m}}}\left(\frac{q_{i_{1}: i_{m}}(\sigma^{1})}{q_{0}}\right)^{1-A_{i_{1}:i_{m}}}.\]
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\[J_{1,1}=-\int\Delta^{2}\varrho^{\kappa}\Delta\left(v^{\kappa} \cdot\nabla\varrho^{\kappa}\right)-\int\Delta^{2}\varsigma^{\kappa}\Delta \left(v^{\kappa}\cdot\nabla\varsigma^{\kappa}\right),\] \[J_{1,2}=-D\int\nabla\Delta\varrho^{\kappa}\cdot\nabla\Delta\left( \nabla\varsigma^{\kappa}\cdot\nabla\Psi^{\kappa}+\varsigma\Delta\Psi^{\kappa} \right)-D\int\nabla\Delta\varsigma^{\kappa}\cdot\nabla\Delta\left(\nabla \varrho^{\kappa}\cdot\nabla\Psi^{\kappa}+\varrho\Delta\Psi^{\kappa}\right),\] \[J_{1,3}=-\int\nabla\Delta\varrho^{\kappa}\cdot\nabla\Delta(u^{ \kappa}\cdot\nabla\varrho^{\kappa})-\int\nabla\Delta\varsigma^{\kappa}\cdot \nabla\Delta(u^{\kappa}\cdot\nabla\varsigma^{\kappa})\] \[\qquad\qquad-\int\nabla\Delta\varrho^{\kappa}\cdot\nabla\Delta(v ^{\kappa}\cdot\nabla\rho^{\kappa})-\int\nabla\Delta\varsigma^{\kappa}\cdot \nabla\Delta(v^{\kappa}\cdot\nabla\sigma^{\kappa}),\] \[J_{1,4}=-D\int\Delta^{2}\varrho^{\kappa}\Delta(\nabla\sigma^{ \kappa}\cdot\nabla\Psi^{\kappa}+\sigma^{\kappa}\Delta\Psi^{\kappa})-D\int \Delta^{2}\varrho^{\kappa}\Delta(\nabla\varsigma^{\kappa}\cdot\nabla\Phi+ \varsigma^{\kappa}\Delta\Phi^{\kappa})\] \[\qquad\qquad-D\int\Delta^{2}\varsigma^{\kappa}\Delta(\nabla\rho^{ \kappa}\cdot\nabla\Psi^{\kappa}+\rho^{\kappa}\Delta\Psi^{\kappa})-D\int \Delta^{2}\varsigma^{\kappa}\Delta(\nabla\varrho^{\kappa}\cdot\nabla\Phi+ \varrho^{\kappa}\Delta\Phi^{\kappa}).\]
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\[2\begin{bmatrix}0\\ \xi_{2}\end{bmatrix}^{T}(S(x,y)-\mu I)\begin{bmatrix}0\\ \xi_{2}\end{bmatrix} =2\begin{bmatrix}0\\ \xi_{2}\end{bmatrix}^{T}\begin{bmatrix}u^{T}/u_{0}\\ -I\end{bmatrix}(S_{2}-\mu I)\begin{bmatrix}u^{T}/u_{0}\\ -I\end{bmatrix}^{T}\begin{bmatrix}0\\ \xi_{2}\end{bmatrix}\] \[=2\begin{bmatrix}\xi_{0}\\ \xi_{1}\end{bmatrix}^{T}\begin{bmatrix}u^{T}/u_{0}\\ -I\end{bmatrix}(S_{2}-\mu I)\begin{bmatrix}u^{T}/u_{0}\\ -I\end{bmatrix}^{T}\begin{bmatrix}\xi_{0}\\ \xi_{1}\end{bmatrix}\] \[=2\begin{bmatrix}\xi_{0}\\ \xi_{1}\end{bmatrix}^{T}(S(x,y)-\mu I)\begin{bmatrix}\xi_{0}\\ \xi_{1}\end{bmatrix}\] \[=-\gamma.\]
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\[\left\|\mathcal{D}_{k+1}^{n}F^{n}(t_{m+1},X_{t_{m}}^{n})-\mathcal{ D}_{\ell}^{n}F^{n}(t_{m+1},X_{t_{m}}^{n})\frac{\sigma(t_{k+1},X_{t_{k}}^{n}) \nabla X_{t_{\ell-1}}^{n,t_{k},X_{t_{k}}^{n}}}{\sigma(t_{\ell},X_{t_{\ell-1}}^ {n})}\right\|\] \[\leq \|(\mathcal{D}_{k+1}^{n}X_{t_{m}}^{n})(F_{x}^{n,(k+1,m+1)}-F_{x}^ {n,(\ell,m+1)})\|\] \[+\left\|F_{x}^{n,(\ell,m+1)}\!\left((\mathcal{D}_{k+1}^{n}X_{t_{m }}^{n})-(\mathcal{D}_{\ell}^{n}X_{t_{m}}^{n})\frac{\sigma(t_{k+1},X_{t_{k}}^{n })\nabla X_{t_{\ell-1}}^{n,t_{k},X_{t_{k}}^{n}}}{\sigma(t_{\ell},X_{t_{\ell-1} }^{n})}\right)\right\|\] \[=: A_{1}+A_{2}.\]
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\[\sum_{\ell=1}^{m}\ \sum_{\sigma\in\operatorname{unsh}(\iota,- \iota)}\hskip-14.226378pt(-)^{|\mu_{k}|(m-\ell)}\chi(\sigma)f_{m-\ell+1}(\mu_{ \ell}(x_{\sigma_{1}},\ldots,x_{\sigma_{\ell}}),\ x_{\sigma_{\ell+1}},\ldots x_{ \sigma_{\ell}})=\] \[=\hskip-14.226378pt\sum_{\begin{subarray}{c}1\leq\ell\leq m\\ k_{1}\leq\cdots\leq k_{\ell}\end{subarray}}\hskip-14.226378pt(-)^{\sum_{i=1}^{ \ell-1}(|f_{k_{i}}|)(\ell-i)}\hskip-14.226378pt\sum_{\sigma\in\operatorname{ unsh}(\zeta_{1},\ldots,k_{\ell})}\hskip-14.226378pt\chi(\sigma)(-)^{\beta}\ f_{k_{1}}(x_{\sigma_{1}},\ldots,x_{\sigma_{k_{1}}}) \otimes\cdots\otimes f_{k_{\ell}}(x_{\sigma_{(m+1-k_{\ell})}},\ldots,x_{\sigma _{m}})\]
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\[\|a_{i}^{2}W_{i}^{p}\otimes W_{i}^{c}+a_{i}^{2}W_{i}^{c}\otimes W _{i}^{p}+a_{i}^{2}W_{i}^{c}\otimes W_{i}^{c}\|_{L^{1}}\leq\|a_{i}\|_{L^{\infty} }(2\|W_{i}^{p}\|_{L^{2}}\|W_{i}^{c}\|_{L^{2}}+\|W_{i}^{c}\|_{L^{2}}^{2})\leq \varepsilon C(t_{0},\|R_{0}\|_{L^{\infty}})\,,\] \[\|(u_{1}^{(c)}+u_{1}^{(t)})\otimes(u_{1}-u_{0})+(u_{1}-u_{0}) \otimes(u_{1}^{(c)}+u_{1}^{(t)})\|_{L^{1}}\leq 2\|u_{1}^{(c)}+u_{1}^{(t)}\|_{L^{ 2}}\|u_{1}-u_{0}\|_{L^{2}}\leq\varepsilon C(t_{0},\|R_{0}\|_{C_{2}}),\] \[\|\mathcal{R}(\nabla a_{i}^{2}\cdot(W_{i}^{p}\otimes W_{i}^{p}- \int_{\mathbb{T}^{2}}W_{i}^{p}\otimes W_{i}^{p}))\|_{L^{1}}\leq C\varepsilon\| \nabla a_{1}\|_{C^{1}}\|W_{i}^{p}\otimes W_{i}^{p}\|_{L^{1}}\leq\varepsilon C( t_{0},\|R_{0}\|_{C^{2}})\,.\]
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\[\begin{split}\|\chi^{\prime}(\frac{x-x_{0}}{R})\int_{0}^{t}\int \int_{|y-z|>|x-y|}e^{-i\frac{(x-y)^{2}}{4(t-\tau)}}\phi^{(2)}(y-z)(1-\chi( \frac{z-x_{0}}{4R}))f(z)d\tau dydz\|_{L^{2}}\\ \lesssim\|\chi^{\prime}(\frac{x-x_{0}}{R})\int_{0}^{t}\int\int_{| y-z|>|x-y|}\frac{1}{1+|x-y|^{2}}\cdot\frac{1}{1+|x-z|^{2}}\\ \times|\phi^{(2)}(y-z)(y-z)^{4}|(1-\chi(\frac{z-x_{0}}{4R}))|f(z) |dzdyd\tau\|_{L^{2}}\\ \lesssim_{\phi}\sum_{j\neq 0}\frac{1}{1+j^{2}R^{2}}\int_{0}^{t}\| \chi(\frac{x-x_{0}-jR}{R})f\|_{L^{4/3}}d\tau\lesssim\frac{1}{R^{2}}\int_{0}^{ t}(\sup_{x_{0}}\|\chi(\frac{x-x_{0}}{R})f\|_{L^{4/3}}).\end{split}\]
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\[w_{n+1}^{-}(0,1)-w_{n+1}^{-}(0,0)\] \[=\bigg{[}\frac{\lambda}{\alpha}\cdot w_{n}^{-}(1,1)+\frac{\nu}{ \alpha}\cdot w_{n}^{-}(0,2)+\frac{\gamma}{\alpha}\cdot w_{n}^{-}(0,0)+\frac{ \mu+\gamma\cdot(b-1)}{\alpha}\cdot w_{n}^{-}(0,1)\bigg{]}\] \[\quad-\bigg{[}\frac{\nu}{\alpha}\cdot w_{n}^{-}(0,1)+\frac{ \lambda+\mu+\gamma\cdot b}{\alpha}\cdot w_{n}^{-}(0,0)\bigg{]}\] \[=\frac{\lambda}{\alpha}\cdot\underbrace{(w_{n}^{-}(1,1)-w_{n}^{- }(1,0))}_{\geq 0,\text{ by \eqref{eq:13}}}+\frac{\lambda}{\alpha}\cdot \underbrace{(w_{n}^{-}(1,0)-w_{n}^{-}(0,0))}_{\geq 0,\text{ by \eqref{eq:14}}}+\frac{\nu}{\alpha}\cdot \underbrace{(w_{n}^{-}(0,2)-w_{n}^{-}(0,1))}_{\geq 0,\text{ by \eqref{eq:13}}}\] \[\quad+\frac{\mu+\gamma\cdot(b-1)}{\alpha}\cdot\underbrace{(w_{n}^{ -}(0,1)-w_{n}^{-}(0,0))}_{\geq 0,\text{ by \eqref{eq:13}}}\geq 0.\]
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\[A_{j,1}(u) \coloneqq\operatorname{Re}\overline{e^{-i\theta_{j}}\chi_{j}(u)} \sum_{k=1}^{r}\alpha_{k}e^{-i\theta_{k}}\chi_{k}(u),\] \[A_{j,2}(u) \coloneqq\sum_{\ell=2}^{r}\sum_{k=1}^{r}\alpha_{k}\operatorname{ Re}\left(e^{i(\theta_{1}-\theta_{\ell})}\overline{\chi_{1}}\chi_{\ell}(u) \right)\operatorname{Re}\left(e^{i(\theta_{j}-\theta_{k})}\overline{\chi_{j}} \chi_{k}(u)\right),\] \[A_{j,3}(u) \coloneqq\operatorname{Re}\left(e^{-i(\theta_{1}-\theta_{j})} \chi_{1}\overline{\chi_{j}}(u)\right)\operatorname{Re}\left(\sum_{\ell=2}^{r} e^{i(\theta_{1}-\theta_{\ell})}\overline{\chi_{1}}\chi_{\ell}(u)\right)^{2},\] \[A_{j,4}(u) \coloneqq\operatorname{Re}\left(e^{-i(\theta_{1}-\theta_{j})} \chi_{1}\overline{\chi_{j}}(u)\right)\left(\operatorname{Re}\sum_{\ell=2}^{r} e^{i(\theta_{1}-\theta_{\ell})}\overline{\chi_{1}}\chi_{\ell}(u)\right)^{2}.\]
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\[\frac{d}{dt} \left(\frac{1}{2}\int_{\Omega}\rho^{(\varepsilon)}|u^{(\varepsilon )}-u|^{2}\,dx+\frac{c_{P}}{\varepsilon}\int_{\Omega}\mathcal{U}(\rho^{( \varepsilon)}|\rho)\,dx-\frac{c_{K}}{2\varepsilon}\int_{\Omega}(\rho^{( \varepsilon)}-\rho)\Lambda^{\alpha-d}(\rho^{(\varepsilon)}-\rho)\,dx\right)\] \[+\frac{1}{2\varepsilon}\int_{\Omega}\rho^{(\varepsilon)}|u^{( \varepsilon)}-u|^{2}\,dx\] \[\leq\frac{c_{P}C}{\varepsilon}(\gamma-1)\int_{\Omega}\mathcal{U} (\rho^{(\varepsilon)}|\rho)\,dx+\frac{C}{\varepsilon}\int_{\Omega}(\rho^{( \varepsilon)}-\rho)\Lambda^{\alpha-d}(\rho^{(\varepsilon)}-\rho)\,dx\] \[\quad+C\varepsilon^{2}\left(\int_{\Omega}\rho_{0}^{(\varepsilon)}|u _{0}^{(\varepsilon)}|^{2}\,dx+\frac{1}{\varepsilon}\int_{\Omega}\rho_{0}^{( \varepsilon)}\Lambda^{\alpha-d}\rho_{0}^{(\varepsilon)}\,dx\right)+C\varepsilon,\]
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\[\|f-T_{n_{k+1}}\|\geq \|f-T_{n_{k+1}}\|_{[-b_{k},b_{k}]}=\big{\|}P_{r}+\sum_{j=k}^{ \infty}f_{n_{j+1},b_{j}}-T_{n_{k+1}}\big{\|}_{[-b_{k},b_{k}]}\] \[= \big{\|}\big{(}P_{r}+f_{n_{k+1},b_{k}}-T_{n_{k+1}}\big{)}+\sum_{j =k+1}^{\infty}f_{n_{j+1},b_{j}}\big{\|}_{[-b_{k},b_{k}]}\] \[\geq \big{\|}P_{r}+f_{n_{k+1},b_{k}}-T_{n_{k+1}}\big{\|}_{[-b_{k},b_{k }]}-\big{\|}\sum_{j=k+1}^{\infty}f_{n_{j+1},b_{j}}\big{\|}\] \[\geq \frac{c_{10}b_{k}^{r(m+1)}}{n_{k+1}^{m+1}}-\frac{c_{10}b_{k}^{r(m +1)}}{5n_{k+1}^{m+1}}=\frac{4c_{10}b_{k}^{r(m+1)}}{5n_{k+1}^{m+1}}.\]
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\[\mathcal{F}R^{\varepsilon,\ell}_{n,k,J,I_{1},\ldots,I_{k}}(\cdot,t,x)\big{(}(\xi_{j})_{\begin{subarray}{c}j\in J\\ j\neq\ell\end{subarray}}\big{)}=e^{-i\big{(}\sum_{\begin{subarray}{c}j\in J \,\xi_{j}\\ j\neq\ell\end{subarray}}\xi_{j}\big{)}\cdot x}\int_{T_{n+1}(t)}\int_{(\mathbb{R} ^{d})^{k+1}}e^{-\frac{\varepsilon}{2}|\xi_{\ell}|^{2}}\Big{(}1-e^{-\frac{ \varepsilon}{2}\sum_{j\in J}|\xi_{j}|^{2}}\Big{)}\] \[\mathcal{F}G(t-t_{n+1},\cdot)\big{(}\sum_{\begin{subarray}{c}j \in J\\ j\neq\ell\end{subarray}}\xi_{j}\big{)}\prod_{j=1}^{n}\mathcal{F}G(t_{j+1}-t_{j}, \cdot)\big{(}\sum_{s=1}^{j}\xi_{s}\big{)}\prod_{i=1}^{k}1_{\{\xi_{\ell_{i}}=- \xi_{m_{i}}=\eta_{i}\}}\prod_{i=1}^{k}\mu_{\varepsilon}(d\eta_{i})\mu(d\xi_{ \ell})d\mathbf{t}.\]
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\[\int_{\varkappa_{0}}^{\varkappa_{0}+2\mathbf{K}}\frac{\, \widetilde{\mathsf{Q}}^{\alpha}_{(0);2\ell+1}(\varkappa)}{(\sigma+\mathfrak{U} _{2}\,\mathrm{cn}^{2}\varkappa)^{\frac{2\ell+1}{2}}}d\varkappa= 2\int_{0}^{\frac{\pi}{2}}\frac{\,\widetilde{\mathsf{Q}}^{ \alpha}_{(0);2\ell+1}(\varphi)}{(\sigma+\mathfrak{U}_{2}-\mathfrak{U}_{2}\sin^ {2}\varphi)^{\frac{2\ell+1}{2}}}\frac{d\varphi}{\sqrt{1-q\sin^{2}\varphi}}\] \[= 2\int_{0}^{\frac{\pi}{2}}\frac{\,\widetilde{\mathsf{Q}}^{\alpha} _{(0);2\ell+1}(\varphi)(1+\frac{1}{2}q\sin^{2}\varphi+\frac{3}{8}q^{2}\sin^{ 4}\varphi+\cdots)}{(\sigma+\mathfrak{U}_{2})^{\frac{2\ell+1}{2}}(1-\mathbf{ x}^{2}\sin^{2}\varphi)^{\frac{2\ell+1}{2}}}d\varphi,\]
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\[\begin{array}{rcl}-{\cal E}_{4_{f}}&=&x_{1}^{3}x_{2}+x_{0}x_{1}x_{2}^{2}+x_{ 1}^{2}x_{2}^{2}+x_{0}x_{2}^{3}+x_{1}x_{2}^{3}+x_{2}^{4}+x_{1}^{3}x_{3}\\ &&+x_{0}x_{1}x_{2}x_{3}+x_{1}^{2}x_{2}x_{3}+x_{0}x_{2}^{2}x_{3}+x_{1}x_{2}^{2}x_ {3}+x_{2}^{3}x_{3}\\ &&+x_{0}x_{1}x_{3}^{3}+x_{1}^{2}x_{3}^{2}+x_{0}x_{2}x_{3}^{2}+x_{1}x_{2}x_{3}^{2} +x_{2}^{2}x_{3}^{2}+x_{0}x_{3}^{3}\\ &&+x_{1}x_{3}^{3}+x_{2}x_{3}^{3}+x_{3}^{4}.\end{array}\]
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\[W_{a}(x) :=V_{a}(x)-c_{a}(x)(\pi\,p)^{\prime}(x)B(x),\] \[\left[\begin{matrix}\mathscr{P}_{1,1}(x,\,y)\,\,\mathscr{P}_{1,2} (x,\,y)\\ \mathscr{P}_{2,1}(x,\,y)\,\,\mathscr{P}_{2,2}(x,\,y)\end{matrix}\right] :=\left[\begin{matrix}\mathscr{V}_{1,1}^{\tilde{P}_{\rm e}}(x,\,y)+ \mathscr{V}_{1,2}^{\tilde{P}_{\rm o}}(x,\,y)\,\,\mathscr{V}_{1,1}^{\chi\, \tilde{P}_{\rm o}}(x,\,y)+\mathscr{V}_{1,2}^{\tilde{P}_{\rm e}}(x,\,y)\\ \mathscr{V}_{2,1}^{\tilde{P}_{\rm e}}(x,\,y)+\mathscr{V}_{2,2}^{\tilde{P}_{ \rm o}}(x,\,y)\,\,\mathscr{V}_{2,1}^{\chi\,\tilde{P}_{\rm o}}(x,\,y)+\mathscr{ V}_{2,2}^{\tilde{P}_{\rm e}}(x,\,y)\end{matrix}\right]\] \[\left[\begin{matrix}P_{\rm e}(x)&-x\,P_{\rm o}(x)\\ -P_{\rm o}(x)&P_{\rm e}(x)\end{matrix}\right],\] \[\left[\begin{matrix}\kappa_{1,1}(x,\,y)\,\,\kappa_{1,2}(x,\,y)\\ \kappa_{2,1}(x,\,y)\,\,\kappa_{2,2}(x,\,y)\end{matrix}\right] :=\left[\begin{matrix}\tilde{P}_{\rm e}(y)&y\,\tilde{P}_{\rm o}(y) \\ \tilde{P}_{\rm o}(y)&\tilde{P}_{\rm e}(y)\end{matrix}\right]\left[\begin{matrix}( A_{1}(y))_{[n]}^{\top}\\ (A_{2}(y))_{[n]}^{\top}\end{matrix}\right]\left[(W_{1}(x))_{[n]}\,\,(W_{2}(x))_{[n]}\right]\] \[\quad+\left[\begin{matrix}\mathscr{P}_{1,1}(x,\,y)\,\,\mathscr{P}_ {1,2}(x,\,y)\\ \mathscr{P}_{2,1}(x,\,y)\,\,\mathscr{P}_{2,2}(x,\,y)\end{matrix}\right].\]
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\[c\left\|\left|x\right|^{-\frac{b}{2\sigma+2}}u_{2}(t)\right\|_{L^ {2\sigma+2}}^{2\sigma+2} =c\left\|\left|x\right|^{-\frac{b}{2\sigma+2}}\left(1-\phi\left( \frac{x}{R(t)}\right)\right)u(t)\right\|_{L^{2\sigma+2}}^{2\sigma+2}\] \[\leq c\int_{|x|\leq R(t)}|x|^{-b}|u(x,t)|^{2\sigma+2}\,dx\leq \frac{1}{8}\|\nabla u(t)\|_{L^{2}}^{2}+\widetilde{c}\frac{\left\|u_{0}\right\| _{L^{2}}^{2\left(\sigma+2\right)}}{R(t)^{\frac{2\left(\sigma(N-1)+b\right)}{2 -\sigma}}}\] \[\leq\frac{\tilde{c}\,c_{1}}{4}\|\nabla u\|_{L^{2}}^{2}\leq\frac{ 1}{4}\|\nabla u\|_{L^{2}}^{2}.\]
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\[\sum_{k=1}^{K}\mathbb{E}_{\pi^{k}\sim p^{k}}\big{[}f^{M^{\star}}(\pi _{M^{\star}})-f^{M^{\star}}(\pi^{k})\big{]}\] \[=\sum_{k=1}^{K}\mathbb{E}_{\pi^{k}\sim p^{k}}\,\mathbb{E}_{\widehat {\phi}^{k}\sim\mu^{k}}\Big{[}f^{\widehat{\psi}^{k}}(\pi_{\widehat{\psi}^{k}})- f^{M^{\star}}(\pi^{k})\Big{]}+\mathbb{E}_{\widehat{\phi}^{k}\sim\mu^{k}}\Big{[}(f^{M^{ \star}}(\pi_{M^{\star}})-f^{\widehat{\psi}^{k}}(\pi_{\widehat{\phi}^{k}}))\Big{]}\] \[=\sum_{k=1}^{K}\mathbb{E}_{\pi^{k}\sim p^{k}}\,\mathbb{E}_{\widehat {\phi}^{k}\sim\mu^{k}}\Big{[}f^{\widehat{\psi}^{k}}(\pi_{\widehat{\psi}^{k}})- f^{M^{\star}}(\pi^{k})-\gamma\cdot D^{\pi^{k}}\Big{(}\widehat{\psi}^{k}\parallel M ^{\star}\Big{)}\Big{]}\] \[\qquad\qquad+\mathbb{E}_{\widehat{\phi}^{k}\sim\mu^{k}}\Big{[} \gamma\cdot D^{\pi^{k}}\Big{(}\widehat{\psi}^{k}\parallel M^{\star}\Big{)}+(f ^{M^{\star}}(\pi_{M^{\star}})-f^{\widehat{\psi}^{k}}(\pi_{\widehat{\psi}^{k}})) \Big{]}\] \[=\sum_{k=1}^{K}\mathbb{E}_{\pi^{k}\sim p^{k}}\,\mathbb{E}_{\widehat {\phi}^{k}\sim\mu^{k}}\Big{[}f^{\widehat{\psi}^{k}}(\pi_{\widehat{\psi}^{k}})- f^{M^{\star}}(\pi^{k})-\gamma\cdot D^{\pi^{k}}\Big{(}\widehat{\psi}^{k} \parallel M^{\star}\Big{)}\Big{]}+\gamma\cdot\mathbf{OptEst}_{\gamma}^{D}.\]
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\[\frac{d}{dt}\int_{\mathbb{R}^{3}}(n^{\varepsilon,\tau}+1)\ln(n^{ \varepsilon,\tau}+1)(\cdot,t)dx+\int_{\mathbb{R}^{3}}\frac{1}{n^{\varepsilon, \tau}+1}|\nabla(n^{\varepsilon,\tau}+1)|^{2}\] \[+\ \frac{2}{\Theta_{0}}\frac{d}{dt}||\nabla\sqrt{c^{\varepsilon, \tau}(t)}||_{L^{2}}^{2}+\frac{4}{3\Theta_{0}}||\Delta\sqrt{c^{\varepsilon, \tau}(t)}||_{L^{2}}^{2}\] \[+\ \frac{4}{3\Theta_{0}}\sum_{i=j}\int_{\mathbb{R}^{3}}(\sqrt{c^{ \varepsilon,\tau}})^{-2}(\partial_{j}\sqrt{c^{\varepsilon,\tau}})^{2}( \partial_{i}\sqrt{c^{\varepsilon,\tau}})^{2}\] \[\leq \int_{\mathbb{R}^{3}}\nabla\cdot\left(\frac{1}{1+\tau n^{ \varepsilon,\tau}}\nabla c^{\varepsilon,\tau}\chi(c^{\varepsilon,\tau})\right) \ln(n^{\varepsilon,\tau}+1)dx\] \[+\ \frac{4}{\Theta_{0}}\int_{\mathbb{R}^{3}}u^{\varepsilon, \tau}\cdot\nabla\sqrt{c^{\varepsilon,\tau}}\Delta\sqrt{c^{\varepsilon,\tau}}: =J_{1}+J_{2}.\]
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\[\max_{T_{1},T_{2}} r_{\text{A}}^{\text{c}}\] \[10\log_{10}\mu_{\text{sb}}=-10\alpha\log_{10}\left(r_{\text{A} }^{\text{c}}\right)+\delta_{\text{sb}}+\mu_{0},\] \[10\log_{10}\mu_{\text{br}}=-10\alpha\log_{10}\left(r_{\text{A}} ^{\text{c}}\right)+\delta_{\text{br}}+\mu_{0},\] \[\int_{\delta_{\text{br}}^{\text{lb}}}^{+\infty}\int_{F_{\delta}^{ \text{c}}(r_{\text{A}}^{\text{c}},\delta_{\text{br}})}^{+\infty}f_{\text{p}} \left(\delta_{\text{sb}},\delta_{\text{br}}\right)\mathrm{d}\delta_{\text{sb}} \mathrm{d}\delta_{\text{br}}\geq P_{\text{A}},\] \[T_{1}+T_{2}\leq D_{\text{t}},\] \[T_{1}=k_{1}T_{\text{f}},T_{2}=k_{2}T_{\text{f}},k_{1},k_{2}\in \mathbb{Z},\]
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\[\begin{array}{ll}\|x_{1}-x_{3}-\lambda(f(x_{1},p_{1},q_{1})-f(x_{3},p_{2},q_ {2}))\|\\ \leq\|x_{1}-x_{3}-\lambda(f(x_{1},p_{1},q_{1})-f(x_{3},p_{1},q_{1}))\|+\lambda \|f(x_{3},p_{2},q_{2})-f(x_{3},p_{1},q_{1})\|\\ =\left[\|x_{1}-x_{3}\|^{2}-2\lambda\langle f(x_{1},p_{1},q_{1})-f(x_{3},p_{1}, q_{1}),x_{1}-x_{3}\rangle+\lambda^{2}\|f(x_{1},p_{1},q_{1})-f(x_{3},p_{1},q_{1})\|^{2 }\right]^{\frac{1}{2}}\\ \qquad\qquad\qquad\qquad+\lambda\|f(x_{3},p_{2},q_{2})-f(x_{3},p_{1},q_{1})\| \\ \leq\left[\|x_{1}-x_{3}\|^{2}-2\lambda\sigma\|x_{1}-x_{3}\|^{2}+\lambda^{2}L^ {2}\|x_{1}-x_{3}\|^{2}\right]^{\frac{1}{2}}+\lambda L\big{(}d_{1}(p_{1},p_{2}) +d_{2}(q_{1},q_{2})\big{)}\\ \leq(1+\lambda^{2}L^{2}-2\lambda\sigma)^{\frac{1}{2}}\|x_{1}-x_{3}\|+\lambda L \sqrt{2\eta d_{1}(p_{1},p_{2})^{\frac{1}{2}}}+\lambda Ld_{2}(q_{1},q_{2}), \end{array}\]
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\[\mathcal{M}^{(3)}_{f,g}(s,w;it)=\sum_{j\geq 1}\frac{\sqrt{\pi}(4\pi)^ {-w+\frac{1}{2}}\Gamma\left(w-\frac{1}{2}+ir_{j}\right)\Gamma\left(w-\frac{1} {2}-ir_{j}\right)}{\Gamma(w)}\mathcal{L}(s^{\prime},it;\overline{u_{j}}) \left\langle u_{j},V_{f,g}\right\rangle\\ +\frac{1}{4\pi}\int_{-\infty}^{\infty}\frac{\sqrt{\pi}(4\pi)^{-w+ \frac{1}{2}}\Gamma\left(w-\frac{1}{2}+ir\right)\Gamma\left(w-\frac{1}{2}-ir \right)}{\Gamma(w)}\sum_{\mathfrak{a}}\mathcal{L}_{\mathfrak{a}}(s^{\prime}, it;ir)\left\langle E_{\mathfrak{a}}(*,1/2+ir),V_{f,g}\right\rangle\ dr\\ +\delta_{1/2\leq\Re(s^{\prime})<1}\zeta(-1+2s^{\prime})\frac{ \sqrt{\pi}(4\pi)^{-w+\frac{1}{2}}}{\Gamma\left(w\right)}\bigg{\{}\frac{\Gamma \left(-s^{\prime}+\frac{1}{2}+it+w\right)\Gamma\left(s^{\prime}-\frac{3}{2}- it+w\right)}{\Gamma\left(s^{\prime}-\frac{1}{2}-it\right)}\\ \times\pi^{s^{\prime}-\frac{1}{2}-it}\zeta(1+2it)\sum_{\mathfrak{a }}\frac{\mathcal{P}_{\mathfrak{a}}(s^{\prime},it;1-s+it)}{\prod_{p\mid N}(1-p ^{1-2s^{\prime}+2it})}\left\langle E_{\mathfrak{a}}(*,3/2-s^{\prime}+it),V_{f, g}\right\rangle\\ +\frac{\Gamma\left(-s^{\prime}+\frac{1}{2}-it+w\right)\Gamma\left(s ^{\prime}-\frac{3}{2}+it+w\right)}{\Gamma\left(s^{\prime}-\frac{1}{2}+it\right) }\\ \times\pi^{s^{\prime}-\frac{1}{2}+it}\zeta(1-2it)\sum_{\mathfrak{a }}\frac{\mathcal{P}_{\mathfrak{a}}(s^{\prime},it;1-s-it)}{\prod_{p\mid N}(1-p^{1- 2s^{\prime}-2it})}\left\langle E_{\mathfrak{a}}(*,3/2-s^{\prime}-it),V_{f,g} \right\rangle\bigg{\}}.\]
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\[\mathbb{A}_{11} =1-c^{2}+\frac{c\alpha_{1}\lambda_{1}}{2},\mathbb{A}_{22}=-c+(1+ \frac{c+\alpha}{2})\frac{\alpha_{2}\lambda_{2}}{2},\] \[\mathbb{A}_{33} =-c+(1+\frac{c+\alpha}{16})\frac{\alpha_{3}\lambda_{3}}{2}, \mathbb{A}_{44}=\frac{c\alpha_{1}}{2\lambda_{1}}-\beta_{1}^{2},\] \[\mathbb{A}_{55} =(1+\frac{c+\alpha}{2})\frac{\alpha_{2}}{2\lambda_{2}}-\beta_{1 }^{2},\mathbb{A}_{66}=(1+\frac{c+\alpha}{16})\frac{\alpha_{3}}{2\lambda_{3}} -\beta_{1}^{2},\] \[\mathbb{A}_{14} =\mathbb{A}_{41}=\frac{c(1-\alpha_{1})}{2},\mathbb{A}_{25}= \mathbb{A}_{52}=\frac{(1+\frac{c+\alpha}{2})(1-\alpha_{2})}{2},\] \[\mathbb{A}_{36} =\mathbb{A}_{63}=\frac{(1+\frac{c+\alpha}{16})(1-\alpha_{3})}{2},\]
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\[\|\omega\|_{L^{p(\cdot)}(B)}^{-1}\left|\left\|T_{\prod\bar{b}}(f_{1 },f_{2})\right\|_{\mathscr{B}}\right\|_{L^{p(\cdot)}(B,\omega)}\] \[\leq \|\omega\|_{L^{p(\cdot)}(B)}^{-1}\left|\left\|T_{\prod\bar{b}}(f_ {1}^{0},f_{2}^{0})\right\|_{\mathscr{B}}\right\|_{L^{p(\cdot)}(B,\omega)}+ \|\omega\|_{L^{p(\cdot)}(B)}^{-1}\left|\left\|T_{\prod\bar{b}}(f_{1}^{0},f_{2} ^{\infty})\right\|_{\mathscr{B}}\right\|_{L^{p(\cdot)}(B,\omega)}\] \[+ \|\omega\|_{L^{p(\cdot)}(B)}^{-1}\left|\left\|T_{\prod\bar{b}}(f_ {1}^{\infty},f_{2}^{0})\right\|_{\mathscr{B}}\right\|_{L^{p(\cdot)}(B,\omega) }+\|\omega\|_{L^{p(\cdot)}(B)}^{-1}\left|\left\|T_{\prod\bar{b}}(f_{1}^{\infty },f_{2}^{\infty})\right\|_{\mathscr{B}}\right\|_{L^{p(\cdot)}(B,\omega)}\] \[:= L_{1}(x_{0},r)+L_{2}(x_{0},r)+L_{3}(x_{0},r)+L_{4}(x_{0},r).\]
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\[\mathbb{E}\bigg{[}L_{\rho}^{r+1}-L_{\rho}^{r}\bigg{]}\stackrel{{ \mbox{(i)}}}{{\leq}}\frac{9\tilde{\sigma}_{g}^{2}}{\rho J\sigma_{\min}}+ \frac{6L_{\mu}^{2}}{\rho\sigma_{\min}}\mathbb{E}\|z^{r}-z^{r-1}\|^{2}\] \[+\frac{3\rho\|L^{+}\|}{\sigma_{\min}}\mathbb{E}\|w^{r}\|_{L^{+}}^ {2}+\frac{\hat{L}^{2}-2\rho+\hat{L}}{2}\mathbb{E}\|z^{r+1}-z^{r}\|^{2}+\frac{1 }{2\hat{L}^{2}}\mathbb{E}\|\nabla g(z^{r+1})-G_{\mu}^{J,r}\|^{2}\] \[\stackrel{{\mbox{(ii)}}}{{\leq}}\bigg{(}\frac{9}{ \rho\sigma_{\min}}+\frac{3}{2\hat{L}^{2}}\bigg{)}\frac{\tilde{\sigma}_{g}^{2}} {J}+\frac{3\mu^{2}(Q+3)^{3}}{8}+\frac{6L_{\mu}^{2}}{\rho\sigma_{\min}}\mathbb{ E}\|z^{r}-z^{r-1}\|^{2}\] \[+\frac{3\rho\|L^{+}\|}{\sigma_{\min}}\mathbb{E}\|w^{r}\|_{L^{+}}^ {2}+\frac{\hat{L}^{2}-2\rho+\hat{L}+3}{2}\mathbb{E}\|z^{r+1}-z^{r}\|^{2},\]
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\[\begin{array}{ccl}\binom{c+3}{\lfloor\frac{c-2}{4}\rfloor+1}&=&\frac{c(c+1) (c+2)(c+3)}{(c+2-\lfloor\frac{c-2}{4}\rfloor)(c+1-\lfloor\frac{c-2}{4}\rfloor )(c-\lfloor\frac{c-2}{4}\rfloor)(\lfloor\frac{c-2}{4}\rfloor+1)}\binom{c-1} {\lfloor\frac{c-2}{4}\rfloor}\\ &\leq&\frac{4^{4}}{3^{3}}\frac{3c(3c+3)}{(3c+13)(3c+5)}\binom{c-1}{\lfloor \frac{c-1}{4}\rfloor}\\ &\leq&\frac{4^{4}}{3^{3}}\frac{3c(3c+3)}{(3c+13)(3c+5)}\left(\frac{9}{7\sqrt{c }}\right)\left(\frac{4}{3^{\frac{3}{4}}}\right)^{c-1}\left(\frac{2c-4}{3c-2} \right)\\ &=&\left(\frac{4}{3^{\frac{3}{4}}}\right)^{c+3}\left(\frac{9}{7\sqrt{c}} \right)\frac{3c(3c+3)(2c-4)}{(3c+13)(3c+5)(3c-2)}\\ \end{array}\]
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\[\underline{\bm{I}}^{k}_{\text{dev\,grad},T}\bm{\upsilon}\coloneqq \bigg{(}\bm{\pi}^{k-1}_{\mathcal{P},T}\bm{\upsilon},\big{(}\bm{ \pi}^{k}_{\mathcal{P},F}(\bm{\upsilon}_{\bm{n}_{F}}),\bm{\pi}^{k-1}_{ \mathcal{P},F}(\bm{\upsilon}_{\bm{t},F}),\bm{\pi}^{k-1}_{\mathcal{P},F}( \text{div}\,\bm{\upsilon})\big{)}_{F\in\mathcal{T}_{T}},\] \[\big{(}\bm{\pi}^{k-1}_{\mathcal{P},E}(\bm{\upsilon}\cdot\bm{t}_{ E}),\bm{\pi}^{k}_{\mathcal{P},E}(\bm{\upsilon}_{\bm{n},E}),\bm{\pi}^{k}_{ \mathcal{P},E}(\text{grad}\,\bm{\upsilon})_{\bm{nn},E}\big{)}_{E\in\mathcal{E} _{T}},\] \[\big{(}\bm{\upsilon}(\bm{\chi}_{V}),\text{grad}\,\bm{\upsilon}(\bm {\chi}_{V})\big{)}_{V\in\mathcal{V}_{T}}\Big{)},\]
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\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 \\ \(g_{\lambda},g_{\lambda,\mu}\) & Biased estimates (9.3, 9.4) \\ \(\hat{g}\) & Empirical estimate (9.5) \\ \(\hat{g}_{p}\) & Empirical estimate with low-rank approximation (Algo. 1) \\ \(\mathcal{H}\) & Reproducing kernel Hilbert space \\ \(k\) & Reproducing kernel \\ \(S\) & Embedding of \(\mathcal{H}\) in \(L^{2}\) \\ \(S^{\star}\) & Adjoint of \(S\), operating from \(L^{2}\) to \(\mathcal{H}\) \\ \(\Sigma=S^{\star}S\) & Covariance operator on \(\mathcal{H}\) \\ \(K=SS^{\star}\) & Equivalent of \(\Sigma\) on \(L^{2}\) \\ \(\mathcal{L}\) & Diffusion operator (a.k.a. Laplacian) \\ \(L=S^{\star}\mathcal{L}S\) & Restriction of the diffusion operator to \(\mathcal{H}\) \\ \(g\) & Generic element in \(L^{2}\) \\ \(\theta\) & Generic element in \(\mathcal{H}\) \\ \(\lambda_{i}\) & Generic eigenvalue \\ \(e_{i}\) & Generic eigenvector in \(L^{2}\) \\ \hline \hline \end{tabular}
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\[\frac{H_{nk+r}\left(e^{-(nk+r)V_{0,k}}\right)}{H_{n}\left(e^{-2 \widetilde{\sigma}nx^{2}}\right)^{k-r}H_{n+1}\left(e^{-2\widehat{\sigma}(n+1) x^{2}}\right)^{r}}=\frac{1}{2^{n(k-1)(nk-1)+2nr(k-1)+(r-1)^{2}}}\\ \times\frac{\theta((nk+r)\Omega|\tau)}{\theta((nk+r)\Omega+ \Upsilon|\tau)}\left(\frac{2}{1+\sqrt{1-\sigma^{-1}}}\right)^{\frac{k-1}{2}} \\ \times\left[\gamma_{1}(1)+\gamma_{1}(1)^{-1}\right]^{k-1}\frac{1} {2^{k-1}}D_{1}(1)^{-k+1}e^{-(nk+r)\int_{J_{0,k}}f(x)d\mu_{V_{0,k}}(x)}e^{- \mathcal{Q}_{J_{0,k}}(f)}\\ \times\exp\left(-\frac{\widehat{\sigma}(n+1)}{\pi}\int_{- \widehat{\sigma}^{-1/2}}^{\widehat{\sigma}^{-1/2}}\sqrt{1/\widehat{\sigma}-x ^{2}}\log(1-x^{2})dx+k\mathcal{Q}_{J_{0,1}}(f)\right)(1+\mathcal{O}(1/n)),\]
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\[\left(\int_{\mathbb{R}^{n}}\left[\left|\widehat{a}(x)\right|\min \left\{\left[\rho_{*}(x)\right]^{1-\frac{1}{\rho_{-}}-\frac{1}{\rho_{+}}},\, \left[\rho_{*}(x)\right]^{1-\frac{2}{\rho_{+}}}\right\}\right]^{p_{+}}\,dx \right)^{1/p_{+}}\] \[\qquad\lesssim\left(\int_{(A^{*})^{-i_{0}+1}B_{0}^{*}}\left[ \left|\widehat{a}(x)\right|\min\left\{\left[\rho_{*}(x)\right]^{1-\frac{1}{ \rho_{-}}-\frac{1}{\rho_{+}}},\,\left[\rho_{*}(x)\right]^{1-\frac{2}{\rho_{+}} }\right\}\right]^{p_{+}}\,dx\right)^{1/p_{+}}\] \[\qquad+\left(\int_{((A^{*})^{-i_{0}+1}B_{0}^{*})\hat{C}}\left[ \left|\widehat{a}(x)\right|\min\left\{\left[\rho_{*}(x)\right]^{1-\frac{1}{ \rho_{-}}-\frac{1}{\rho_{+}}},\,\left[\rho_{*}(x)\right]^{1-\frac{2}{\rho_{+}} }\right\}\right]^{p_{+}}\,dx\right)^{1/p_{+}}\] \[\qquad=:\mathrm{I}_{1}+\mathrm{I}_{2},\]
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\[\int_{0}^{\infty}t\|\nabla u\|_{L_{\infty}(\mathbb{R}^{3})}^{2}\,dt \lesssim\int_{0}^{\infty}t^{-1/3}\|t\nabla u\|_{\dot{W}^{1}_{10/3 }(\mathbb{R}^{3})}^{4/3}\|\nabla u\|_{\dot{W}^{1}_{5/2}(\mathbb{R}^{3})}^{2/3 }\,dt\] \[\lesssim\|t^{-1/3}\|_{L_{3,\infty}(\mathbb{R}_{+})}\|\|t\nabla u \|_{\dot{W}^{1}_{10/3}(\mathbb{R}^{3})}^{4/3}\|_{L_{5/2,1}(\mathbb{R}_{+})}\| \|\nabla u\|_{\dot{W}^{1}_{5/2}(\mathbb{R}^{3})}^{2/3}\|_{L_{15/4}(\mathbb{R}_ {+})}\] \[\lesssim\|t\nabla^{2}u\|_{L_{10/3,1}(\mathbb{R}_{+};L_{10/3}( \mathbb{R}^{3}))}^{4/3}\|\nabla^{2}u\|_{L_{5/2}(\mathbb{R}_{+}\times\mathbb{R }^{3})}^{2/3}.\]
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\[n[L_{n}(G)-L_{n}(1/2)]\] \[\geq\sum_{t=1}^{n}\big{[}y_{t}\Big{(}2g_{\lambda,z_{1:N},\varepsilon _{1:N}}(x_{t})-2.2g_{\lambda,z_{1:N},\varepsilon_{1:N}}^{2}(x_{t})\Big{)}\] \[\qquad\qquad+(1-y_{t})\Big{(}-2g_{\lambda,z_{1:N},\varepsilon_{1: N}}(x_{t})-2.2g_{\lambda,z_{1:N},\varepsilon_{1:N}}^{2}(x_{t})\Big{)}\big{]}\] \[=\sum_{t=1}^{n}\big{[}2(2y_{t}-1)g_{\lambda,z_{1:N},\varepsilon _{1:N}}(x_{t})-2.2g_{\lambda,z_{1:N},\varepsilon_{1:N}}^{2}(x_{t})\big{]}\] \[\geq\lambda\sum_{t=1}^{n}(2y_{t}-1)f(x_{t})+2\lambda^{2}\sum_{t= 1}^{n}\sum_{i=1}^{N}(2y_{t}-1)\varepsilon_{i}\omega_{\eta}(z_{i}-x_{t})-2.2 \lambda^{2}\sum_{t=1}^{n}(f(x_{t}))^{2},\]
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\[\left\{\begin{array}{l}n_{t}+u\cdot\nabla n=\Delta n-\nabla\cdot(nS(x,n,c) \cdot\nabla c)-nm,\quad x\in\Omega,t>0,\\ c_{t}+u\cdot\nabla c=\Delta c-c+m,\quad x\in\Omega,t>0,\\ m_{t}+u\cdot\nabla m=\Delta m-nm,\quad x\in\Omega,t>0,\\ u_{t}+\kappa(u\cdot\nabla)u+\nabla P=\Delta u+(n+m)\nabla\phi,\quad x\in \Omega,t>0,\\ \nabla\cdot u=0,\quad x\in\Omega,t>0,\\ (\nabla n-nS(x,n,c))\cdot\nu=\nabla c\cdot\nu=\nabla m\cdot\nu=0,u=0,\ \ x\in\partial\Omega,t>0,\\ n(x,0)=n_{0}(x),c(x,0)=c_{0}(x),m(x,0)=m_{0}(x),u(x,0)=u_{0}(x),\ \ x\in\Omega\end{array}\right.\]
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\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 \\ \hline \hline Greedy-M & 78.80 & 143.25 & 98.82 & 221.75 & 60.88 & 287.52 & 83.60 & 449.95 \\ Random-M & 74.96 & 136.66 & 85.74 & 206.81 & 65.50 & 301.77 & 87.73 & 469.44 \\ MILP-M & 67.26 & 247.00 & **81.92** & 482.13 & 53.40 & 438.38 & 75.22 & 788.65 \\ Diverse-M & 70.44 & 130.84 & 83.74 & 196.92 & 57.80 & 259.46 & 83.07 & 449.74 \\ Ours & **66.94** & **118.89** & 82.04 & **183.76** & **52.96** & **246.63** & **74.80** & **398.37** \\ \hline \hline \end{tabular}
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\[2\epsilon\geq P^{\mathrm{rand}}_{e,\text{\small{mal A}}}+P^{ \mathrm{rand}}_{e,\text{\small{mal B}}}\geq\frac{1}{N_{\mathsf{A}}\cdot N_{ \mathsf{B}}}\sum_{m_{\mathsf{A}},m_{\mathsf{B}}}\sum_{f_{\mathsf{A}},f_{\mathsf{B}}}p _{F_{\mathsf{A}}}(f_{\mathsf{A}})p_{F_{\mathsf{B}}}(f_{\mathsf{B}})\Bigg{(}\sum_{ \bm{x}}Q_{\mathbf{X}^{\prime}|\bm{X}}(\bm{x}|f_{\mathsf{A}}(m_{\mathsf{A}}))W ^{n}\left(\mathcal{E}^{\mathsf{B},m_{\mathsf{B}}}_{f_{\mathsf{A}},f_{\mathsf{B}}} \Big{|}\bm{x},f_{\mathsf{B}}(m_{\mathsf{B}})\right)\] \[\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\sum_{\bm{y}}Q_{ \mathbf{Y}^{\prime}|\bm{Y}}(\bm{y}|f_{\mathsf{B}}(m_{\mathsf{B}}))W^{n}\left( \mathcal{E}^{\mathsf{A},m_{\mathsf{A}}}_{f_{\mathsf{A}},f_{\mathsf{B}}}\Big{|}f_{ \mathsf{A}}(m_{\mathsf{A}}),\bm{y}\right)\Bigg{)}\] \[=\frac{1}{N_{\mathsf{A}}\cdot N_{\mathsf{B}}}\sum_{m_{\mathsf{A}},m_{ \mathsf{B}}}\sum_{f_{\mathsf{A}},f_{\mathsf{B}}}p_{F_{\mathsf{A}}}(f_{\mathsf{A}})p_{F_{ \mathsf{B}}}(f_{\mathsf{B}})\Bigg{(}\tilde{W}^{n}\left(\mathcal{E}^{\mathsf{B},m _{\mathsf{B}}}_{f_{\mathsf{A}},f_{\mathsf{B}}}\Big{|}f_{\mathsf{A}}(m_{\mathsf{A}}),f_{ \mathsf{B}}(m_{\mathsf{B}})\right)\] \[\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\tilde{W} ^{n}\left(\mathcal{E}^{\mathsf{A},m_{\mathsf{A}}}_{f_{\mathsf{A}},f_{\mathsf{B}}} \Big{|}f_{\mathsf{A}}(m_{\mathsf{A}}),f_{\mathsf{B}}(m_{\mathsf{B}})\right) \Bigg{)}\] \[\geq\frac{1}{N_{\mathsf{A}}\cdot N_{\mathsf{B}}}\sum_{m_{\mathsf{A}},m_{ \mathsf{B}}}\sum_{f_{\mathsf{A}},f_{\mathsf{B}}}p_{F_{\mathsf{A}}}(f_{\mathsf{A}})p_{F_{ \mathsf{B}}}(f_{\mathsf{B}})\tilde{W}^{(n)}\left(\mathcal{E}^{\mathsf{A},m_{\mathsf{A}},m_{\mathsf{A}}}_{f_{\mathsf{A}},f_{\mathsf{B}}}\cup\mathcal{E}^{\mathsf{B},m_{ \mathsf{B}}}_{f_{\mathsf{A}},f_{\mathsf{B}}}\Big{|}f_{\mathsf{A}}(m_{\mathsf{A}}),f (m_{\mathsf{B}})\right),\]
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\[\check{X}^{5}_{m_{1}m_{2}n_{1}n_{2}} =\tfrac{1}{3}R_{m_{1}m_{2}p_{1}p_{2}}R_{n_{1}p_{1}q_{1}q_{2}}R_{n_{ 2}p_{2}q_{1}q_{2}}+\tfrac{1}{3}R_{n_{1}n_{2}p_{1}p_{2}}R_{m_{1}p_{1}q_{1}q_{2}} R_{m_{2}p_{2}q_{1}q_{2}}\] \[\quad-\tfrac{2}{3}R_{m_{1}n_{1}p_{1}p_{2}}R_{n_{2}p_{1}q_{1}q_{2} }R_{m_{2}p_{2}q_{1}q_{2}}\] \[\quad+\tfrac{2}{5}g_{m_{2}n_{1}}R_{n_{2}r_{1}q_{1}q_{2}}R_{q_{1}q_ {2}p_{1}p_{2}}R_{p_{1}p_{2}m_{1}r_{1}}\] \[\check{X}^{6}_{m_{1}m_{2}n_{1}n_{2}} =\tfrac{2}{3}R_{m_{1}p_{1}n_{1}p_{2}}R_{m_{2}q_{1}p_{1}q_{2}}R_{n_ {2}q_{1}p_{2}q_{2}}+\tfrac{1}{3}R_{m_{1}p_{2}n_{1}p_{1}}R_{m_{2}q_{1}p_{1}q_{2} }R_{n_{2}q_{1}p_{2}q_{2}}\] \[\quad+\tfrac{1}{12}R_{m_{1}m_{2}p_{1}p_{2}}R_{n_{1}q_{1}p_{1}q_{2} }R_{n_{2}q_{1}p_{2}q_{2}}+\tfrac{1}{12}R_{n_{1}n_{2}p_{1}p_{2}}R_{m_{1}q_{1}p_ {1}q_{2}}R_{m_{2}q_{1}p_{2}q_{2}}\] \[\quad-\tfrac{1}{10}g_{m_{2}n_{1}}R_{n_{2}r_{1}q_{1}q_{2}}R_{q_{1} q_{2}p_{1}p_{2}}R_{p_{1}p_{2}m_{1}r_{1}}+\tfrac{2}{5}g_{m_{2}n_{1}}R_{n_{2}q_{1}r_{1 }q_{2}}R_{q_{1}p_{1}q_{2}p_{2}}R_{p_{1}m_{1}p_{2}r_{1}}\] \[\quad+\tfrac{1}{5}g_{m_{2}n_{1}}R_{n_{2}q_{1}m_{2}}R_{q_{1}p_{1} p_{2}r_{1}}R_{q_{2}p_{1}p_{2}r_{1}}\] \[\quad+\tfrac{1}{120}g_{m_{2}n_{1}}g_{m_{1}n_{2}}R_{q_{1}q_{2}r_{1 }r_{2}}R_{r_{1}r_{2}p_{1}p_{2}}R_{p_{1}p_{2}q_{1}q_{2}}-\tfrac{1}{30}g_{m_{2}n _{1}}g_{m_{1}n_{2}}R_{q_{1}r_{1}q_{2}r_{2}}R_{r_{1}p_{1}r_{2}p_{2}}R_{p_{1}q_ {1}p_{2}q_{2}}.\]
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\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) \\ \(E_{0},E_{\infty}\) & Largest and smallest energies of \(p\)-spin Hamiltonian local maxima & (1.14) and (3.7) \\ \(\beta_{2}(q)\) & Quantity appearing in Plefka’s condition & (1.5) \\ \(\xi(x)=x^{p}\) & Covariance function of \(p\)-spin Hamiltonian & (1.11) \\ On\((q)\) & Onsager correction & (1.4) \\ \(f(E,q)\) & \(h_{\rm TAP}(q^{p/2}E,q)\) & (4.3) \\ \(D_{\beta},A(q,\beta)\) & \(D_{\beta}=\{q:A(q,\beta)\leq 0\}\) set of possible squared radii of relevant TAP solutions & (1.6), (1.7) \\ \(U_{\rm max},U_{\rm min}\) & Largest and smallest TAP energies of nonzero relevant TAP solutions & (4.39), (6.3) \\ \(V_{*},U_{*},q_{*}\) & Energy, TAP energy and squared radius with largest contribution to relevant TAP free energy & (1.31) \\ \(E_{q},q_{E}\) & Energy of nonzero relevant TAP solution of given squared radius, and squared radius of nonzero relevant TAP solution of given energy & (4.31), (4.33) \\ \(E_{U},q_{U}\) & Energy, squared radius of nonzero relevant TAP solution for given TAP energy & (4.42), (6.7) \\ \(I_{\rm TAP}(U,V,q)\) & Entropy of relevant TAP local maxima of TAP energy & (1.25) \\ \(E_{\rm min},q_{\rm min}\) & Minimal energy on unit sphere of nonzero relevant TAP solutions and corresponding squared radius & (4.25), (4.27) \\ \(r_{\pm}\left(\frac{E}{E_{\infty}}\right)\) & Solutions of \(\frac{E}{E_{\infty}}=\frac{1}{2}(\frac{1}{x}+x)\) with \(r_{-}\left(\frac{E}{E_{\infty}}\right)\leq 1\leq r_{+}\left(\frac{E}{E_{ \infty}}\right)\) & (4.6) \\ \(\bar{r}\) & \(r_{-}(\frac{E_{0}}{E_{\infty}})\) & (1.17) \\ \({\cal N}_{N}\) & Number of relevant TAP solutions with TAP energy, Hamiltonian energy and squared radius in given sets & (1.24) \\ \hline \end{tabular}
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\[\frac{1}{2}\sum_{t=T/2}^{T-1}E\left[||\omega_{t}-\omega_{\theta_ {t}}||\right]^{2} \leq \frac{1}{\alpha_{T}}\frac{4}{\mu}||\omega_{T/2}-\omega_{\theta_{T/ 2}}||^{2}+c_{\alpha}^{2}\alpha_{T}\frac{T}{2}\frac{4}{\mu}\sigma_{c}^{2}\] \[+2L_{\omega}^{2}\frac{c_{\beta}^{2}\beta_{T}^{2}}{\alpha_{T}}\frac {T}{2}\frac{4}{\mu}\sigma_{a}^{2}+6L_{\omega}^{2}\frac{c_{\beta}^{2}\beta_{T}^ {2}}{\alpha_{T}}\frac{T}{2}\frac{4}{\mu}\vec{\delta}^{2}\] \[+4\frac{c_{\beta}^{2}\beta_{T}^{2}\sigma_{a^{\prime}}^{2}(\lambda/ 2)+c_{\beta}\beta_{T}L_{\omega}\bar{\delta}}{\alpha_{T}\mu}\sqrt{T/2}\sqrt{E \sum_{t=T/2}^{T-1}||\Delta_{t}||^{2}}\] \[+\frac{c_{\beta}\beta_{T}}{\alpha_{T}}\frac{4L_{\omega}}{\mu} \sum_{t=T/2}^{T-1}E||\nabla_{t}||^{2},\]
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