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\[0= \int_{0}^{T}\int_{\mathbb{R}^{d}}\{-u_{0}(\hat{Y}^{x,n}_{t}( \omega))\rho^{{}^{\prime}}(t)\eta(x)\] \[+b_{n}(t,x+B^{H}_{t}(\omega))\cdot(\nabla u_{0})(\hat{Y}^{x,n}_{ t}(\omega))^{T}\frac{\partial}{\partial x}\hat{Y}^{x,n}_{t}(\omega)\rho(t)\eta(x)\} dxdt\text{, for a.a. }\omega.\] \[\Big{(}\left(t,x\right)\mapsto\hat{Y}^{x,n_{j}}_{t}\Big{)}_{j\geq 1}\] \[\Big{(}\left(t,x\right)\mapsto\frac{\partial}{\partial x}\hat{Y}^{x,n_{j}}_{t} \Big{)}_{j\geq 1}\] \[Y(t,x,\omega)=\hat{Y}^{x}_{t}(\omega)\text{, }Y^{{}^{\prime}}(t,x,\omega)=\frac{ \partial}{\partial x}\hat{Y}^{x}_{t}(\omega)\] \[(\nabla u_{0})(\hat{Y}_{t}^{x}(\omega))^{T}\frac{\partial}{\partial x}\hat{Y }_{t}^{x}(\omega)=\nabla u(\omega,t,x)\quad(t,x)\text{-a.e.}\] \[\int_{0}^{T}\int_{\mathbb{R}^{d}}\{-u(\omega,t,x)\rho^{{}^{\prime}}(t)\eta(x) +b^{*}(t,x)\cdot\nabla u(\omega,t,x)\rho(t)\eta(x)\}dxdt=0\]
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\[\|[e^{it\partial_{x}^{2}}\tilde{w}_{n}^{M}](0)\|_{L^{q}_{\mathbb{R}_{ t}}}\] \[\leq \|[e^{it\partial_{x}^{2}}w_{n}^{M}](0)\|_{L^{q}_{\mathbb{R}_{t}}}+ \sum_{j=1}^{M}\|[e^{it\partial_{x}^{2}}(-\mathrm{NLS}(-t_{n}^{j})\tilde{\phi}^{ j}+e^{-it_{n}^{j}\partial_{x}^{2}}\phi^{j})](0)\|_{L^{q}_{\mathbb{R}_{t}}}\] \[\leq \|[e^{it\partial_{x}^{2}}w_{n}^{M}](0)\|_{L^{q}_{\mathbb{R}_{t}}}+ \sum_{j=1}^{M}\|\mathrm{NLS}(-t_{n}^{j})\tilde{\phi}^{j}-e^{-it_{n}^{j} \partial_{x}^{2}}\phi^{j}\|_{\dot{H}^{\sigma_{c}}_{x}},\] \[\leq \|[e^{it\partial_{x}^{2}}w_{n}^{M}](0)\|_{L^{q}_{\mathbb{R}_{t}}}+ \sum_{j=1}^{M}\|\mathrm{NLS}(-t_{n}^{j})\tilde{\phi}^{j}-e^{-it_{n}^{j} \partial_{x}^{2}}\phi^{j}\|_{H^{1}_{x}}.\]
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\[\frac{1}{2\tau}\left(\|\Phi^{k+1}-\Phi^{*}\|^{2}-\|\Phi^{k}-\Phi^{* }\|^{2}\right)\] \[\leq h\sum_{i=1}^{N-1}\Phi_{i}-h\sum_{i=0}^{N-1}(\Phi_{i})^{2}+ah \frac{\Phi_{m+1}(\Phi_{m+2}-\Phi_{m+1})}{\Phi_{m+2}-\Phi_{m+1}}+ah(1-2\Phi_{m+1})\] \[= h\sum_{i=1}^{N-1}\Phi_{i}+ah-ah\Phi_{m+1}-h\sum_{i=1}^{N-1}(\Phi _{i})^{2}\] \[= -\left[h\sum_{i=1}^{m}(\Phi_{i})^{2}+h(\Phi_{m+1}-a)^{2}+h\sum_{i =m+2}^{N-1}(\Phi_{i}-1)^{2}\right]+2h\sum_{i=1}^{m+1}\Phi_{i}-3ah\Phi_{m+1}+a^ {2}h\] \[\leq -\|\Phi^{k+1}-\Phi^{*}\|^{2}+2h\sum_{i=1}^{m+1}\Phi_{i}+h.\]
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\[\mathbb{E}[H_{s}(\lambda_{t}-\mu)]-\mathbb{E}[H_{s}](\mathbb{E}[ \lambda_{t}]-\mu)\] \[=\mu\int_{0}^{s}\Phi(t-v)\left(1+\int_{v}^{s}\Psi(y-v)dy\right) \left(1+\int_{0}^{v}\Psi(w)dw\right)dv\] \[+\mu\int_{0}^{t}\int_{0}^{s\wedge u}\Phi(t-u)\Psi(u-v)\left(1+ \int_{0}^{v}\Psi(w)dw\right)\left(1+\int_{v}^{s}\Psi(y-v)dy\right)dvdu\] \[=\mu\int_{0}^{s}\Phi(t-v)\left(1+\int_{v}^{s}\Psi(y-v)dy\right) \left(1+\int_{0}^{v}\Psi(w)dw\right)dvd\] \[+\mu\int_{0}^{s}\int_{0}^{u}\Phi(t-u)\Psi(u-v)\left(1+\int_{0}^{v }\Psi(w)dw\right)\left(1+\int_{v}^{s}\Psi(y-v)dy\right)dvdu\] \[+\mu\int_{0}^{s}\Phi(t-v)\Psi(s-v)\left(1+\int_{0}^{v}\Psi(w)dw\right) \int_{v}^{s}\Phi(t-u)\Psi(u-v)dudv\] \[+\mu\int_{0}^{s}\Phi(t-v)\Psi(s-v)\left(1+\int_{0}^{v}\Psi(w)dw \right)dv\] \[+\mu\int_{0}^{t}\int_{0}^{s/u}\Phi(t-u)\Psi(u-v)\Psi(s-v)\left(1+ \int_{0}^{v}\Psi(v-w)dw\right)dv\] \[=\mathbb{E}\left[(\lambda_{s}-\mu)\right]\mathbb{E}\left[(\lambda _{t}-\mu)\right]\] \[+\mu\int_{0}^{s}\Phi(t-v)\Psi(s-v)\left(1+\int_{0}^{v}\Psi(w)dw \right)dv\] \[+\mu\int_{0}^{s}\Psi(s-v)\left(1+\int_{0}^{v}\Psi(v-w)dw\right) \int_{v}^{s}\Phi(t-u)\Psi(u-v)dudv\] \[+\mu\int_{0}^{s}\Psi(s-v)\left(1+\int_{0}^{v}\Psi(v-w)dw\right) \int_{s}^{t}\Phi(t-u)\Psi(u-v)dudv\] \[+\mu\int_{0}^{s}\Psi(s-v)\left(1+\int_{0}^{v}\Psi(v-w)dw\right) \int_{s}^{t}\Phi(t-u)\Psi(u-v)dudv\] \[=\mathbb{E}\left[(\lambda_{s}-\mu)\right]\mathbb{E}\left[(\lambda _{t}-\mu)\right]\] \[+\mu\int_{0}^{s}\Phi(t-v)\Psi(s-v)\left(1+\int_{0}^{v}\Psi(w)dw \right)dv\] \[+\mu\int_{0}^{s}\Psi(s-v)\left(1+\int_{0}^{v}\Psi(v-w)dw\right)\int_{v}^{ t}\Phi(t-u)\Psi(u-v)dudv\] \[=\mathbb{E}\left[(\lambda_{s}-\mu)\right]\mathbb{E}\left[(\lambda_ {t}-\mu)\right]\] \[+\mu\int_{0}^{s}\Phi(t-v)\Psi(s-v)\left(1+\int_{0}^{v}\Psi(w)dw \right)dv\] \[+\mu\int_{0}^{s}\Psi(s-v)\left(1+\int_{0}^{v}\Psi(v-w)dw\right) \int_{0}^{t-v}\Phi(t-v-x)\Psi(x)dxdv\] \[=\mathbb{E}\left[(\lambda_{s}-\mu)\right]\mathbb{E}\left[(\lambda _{t}-\mu)\right]\] \[+\mu\int_{0}^{s}\Psi(s-v)\Psi(t-v)\left(1+\int_{0}^{v}\Psi(v-w)dw \right)dv.\]
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\[z_{r}(\bm{Q}^{\text{RS}}) =\frac{1}{2}\sum_{\sigma^{0},\cdots,\sigma^{r}\in\{\pm 1\}}\exp \Big{[}\beta\lambda b\sum_{a=1}^{r}\sigma^{0}\sigma^{a}+\beta^{2}q\sum_{1\leq a <b\leq r}\sigma^{a}\sigma^{b}\Big{]}\] \[=\sum_{\sigma^{1}\cdots,\sigma^{r}\in\{\pm 1\}}\exp\Big{[}\beta \lambda b\sum_{a=1}^{r}\sigma^{a}+\frac{\beta^{2}q}{2}\Big{(}\sum_{a=1}^{r} \sigma^{a}\Big{)}^{2}-\frac{\beta^{2}qr}{2}\Big{]}\] \[=\mathbb{E}\Big{[}\sum_{\sigma^{1},\cdots,\sigma^{r}}\exp\Big{[} \beta\lambda b\sum_{a=1}^{r}\sigma^{a}+\beta\sqrt{q}g\sum_{a=1}^{r}\sigma^{a} -\frac{\beta^{2}qr}{2}\Big{]}\Big{]}\] \[=\exp\Big{[}-\frac{\beta^{2}qr}{2}\Big{]}\mathbb{E}\Big{[}(2 \text{cosh}\beta(\lambda b+\sqrt{q}g))^{r}\Big{]},\]
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\[g\left(n_{\Sigma_{\tau}},n_{\Sigma_{\tau}}\right) =\begin{cases}-1-\dfrac{2Mr(r^{2}+a^{2})}{\Delta\rho^{2}}+\dfrac {\Delta}{\rho^{2}}\left(\dfrac{df}{dr}\right)^{2}\,,&r\geq 9M/4\\ -1-\dfrac{2Mr}{\rho^{2}}-\dfrac{4Mr}{\rho^{2}}\dfrac{df}{dr}+\dfrac{\Delta}{ \rho^{2}}\left(\dfrac{df}{dr}\right)^{2}\,,&r\leq 15M/8\end{cases}\,,\] \[g\left(n_{\Sigma_{\tau}},L\right) =\begin{cases}1-\dfrac{\Delta}{r^{2}+a^{2}}\dfrac{df}{dr}\,,&r \geq 9M/4\,,\\ 1+\dfrac{2Mr}{r^{2}+a^{2}}-\dfrac{\Delta}{r^{2}+a^{2}}\dfrac{df}{dr}\,,&r \leq 15M/4\,,\end{cases}\,,\] \[g\left(n_{\Sigma_{\tau}},\dfrac{r^{2}+a^{2}}{\Delta}\underline{L}\right) =\begin{cases}\dfrac{r^{2}+a^{2}}{\Delta}+\dfrac{df}{dr}\,,&r \geq 9M/4\,,\\ 1+\dfrac{df}{dr}\,,&r\leq 15M/8\,,\end{cases}\,,\] \[\det g_{\Sigma_{\tau}} =\rho^{2}\sin^{2}\theta\begin{cases}w^{-1}-\Delta\left(\dfrac{ df}{dr}\right)^{2}-a^{2}\sin^{2}\theta\,,&r\geq 9M/4\,,\\ \rho^{2}+2Mr\left(1+2\dfrac{df}{dr}\right)-\Delta\left(\dfrac{df}{dr}\right)^ {2}\,,&r\leq 15M/8\,,\end{cases}\,.\]
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\[\left(\begin{array}{c}Z\\ \Phi\end{array}\right)_{t} =-\mathbf{A}_{kl}\left(\begin{array}{c}Z\\ \Phi\end{array}\right)+2\sum_{i,j=1}^{N_{kl}}\lambda^{kl}_{j}\left\langle \left(\begin{array}{c}\mathcal{L}_{kl}\mathbb{D}^{1}_{\gamma^{kl}_{i}}(\psi^{kl }_{i})\\ \mathbb{D}^{2}_{\gamma^{kl}_{i}}(\psi^{kl}_{i})\end{array}\right),\left( \begin{array}{c}Z^{kls}_{j}\\ \Phi^{kls}_{j}\end{array}\right)\right\rangle\left(\begin{array}{c}Z^{kl}_{j} \\ \Phi^{kl}_{j}\end{array}\right)\] \[+\sum_{i=1}^{N_{kl}}\gamma^{kl}_{i}\left(\begin{array}{c} \mathcal{L}_{kl}\mathbb{D}^{1}_{\gamma^{kl}_{i}}(\psi^{kl}_{i})\\ \mathbb{D}^{2}_{\gamma^{kl}_{i}}(\psi^{kl}_{i})\end{array}\right)-\sum_{i=1}^{N_{ kl}}\left(\begin{array}{c}\mathcal{L}_{kl}\mathbb{D}^{1}_{\gamma^{kl}_{i}}(\psi^{kl }_{i})\\ \mathbb{D}^{2}_{\gamma^{kl}_{i}}(\psi^{kl}_{i})\end{array}\right)_{t}.\]
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\[\frac{\mathrm{d}}{\mathrm{d}t}\|g\|_{\dot{H}^{\frac{1}{2}}}^{2} +C(M)\bigg{[}\log\left(4+||f_{1}||_{2,\frac{1}{3}}\right)^{-\frac{1 }{3}}+\log\left(4+||f_{2}||_{2,\frac{1}{3}}\right)^{-\frac{1}{3}}\bigg{]}\,\|g\|_ {\dot{H}^{1}}^{2}\] \[\lesssim_{M}\log\left(4+\|f_{1}\|_{2,\frac{1}{3}}\right)^{-\frac{ 1}{3}}\|f_{1}\|_{2,\frac{1}{3}}\,\|g\|_{\dot{H}^{\frac{1}{2}}}\,\|g\|_{\dot{H} ^{1}}\] \[+\log\left(4+\|f_{2}\|_{2,\frac{1}{3}}\right)^{-\frac{1}{3}}\|f_{ 2}\|_{2,\frac{1}{3}}^{\frac{1}{2}}\,\|g\|_{\dot{H}^{\frac{1}{2}}}^{\frac{1}{2}} \,\|g\|_{\dot{H}^{1}}^{\frac{3}{2}}\] \[+\log\left(4+\|f_{2}\|_{2,\frac{1}{3}}\right)^{-\frac{1}{3}}\|f_{ 2}\|_{2,\frac{1}{3}}\,\|g\|_{\dot{H}^{\frac{1}{2}}}\,\|g\|_{\dot{H}^{1}}\] \[+\log\left(4+\|f_{1}\|_{2,\frac{1}{3}}\right)^{-\frac{1}{3}}\|f_{ 1}\|_{2,\frac{1}{3}}^{\frac{1}{2}}\,\|g\|_{\dot{H}^{\frac{1}{2}}}^{\frac{1}{2} }\,\|g\|_{\dot{H}^{1}}^{\frac{3}{2}}\,.\]
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\[(A_{H,2})_{i,j} = \int_{\Omega}\zeta(u_{H}+\alpha\widetilde{u}_{h})^{2}\phi_{i,H} \phi_{j,H}d\Omega\] \[= \int_{\Omega}\zeta\big{(}(u_{H})^{2}+2\alpha u_{H}\widetilde{u}_ {h}+\alpha^{2}(\widetilde{u}_{h})^{2}\big{)}\phi_{i,H}\phi_{j,H}d\Omega\] \[= \int_{\Omega}\zeta(u_{H})^{2}\phi_{i,H}\phi_{j,H}d\Omega+2\alpha \int_{\Omega}\zeta\widetilde{u}_{h}u_{H}\phi_{i,H}\phi_{j,H}d\Omega\] \[\quad+\alpha^{2}\int_{\Omega}\zeta(\widetilde{u}_{h})^{2}\phi_{i,H}\phi_{j,H}d\Omega\] \[:= (A_{H,2,1})_{i,j}+2\alpha(A_{H,2,2})_{i,j}+\alpha^{2}(A_{H,2,3})_ {i,j}.\]
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\[\begin{split}&\hat{\mathbb{E}}[(\int_{0}^{T}n|(Y_{s}^{n})^{-}|^{ \alpha-1}(Y_{s}^{m})^{-}ds)^{\alpha^{\prime}}]\\ &\leq\hat{\mathbb{E}}[\sup_{s\in[0,T]}\{|(Y_{s}^{n})^{-}|^{( \alpha-2)\alpha^{\prime}}|(Y_{s}^{m})^{-}|^{\alpha^{\prime}}\}(\int_{0}^{T}n(Y _{s}^{n})^{-}ds)^{\alpha^{\prime}}]\\ &\leq(\hat{\mathbb{E}}[\sup_{s\in[0,T]}|(Y_{s}^{n})^{-}|^{( \alpha-2)\alpha^{\prime}p}])^{\frac{1}{p}}(\hat{\mathbb{E}}[\sup_{s\in[0,T]}|(Y _{s}^{m})^{-}|^{\alpha^{\prime}q}])^{\frac{1}{q}}(\hat{\mathbb{E}}[(\int_{0}^{ T}n(Y_{s}^{n})^{-}ds)^{\alpha^{\prime}r}])^{\frac{1}{r}}\\ &\leq C(\hat{\mathbb{E}}[\sup_{s\in[0,T]}|(Y_{s}^{m})^{-}|^{ \alpha^{\prime}q}])^{\frac{1}{q}},\end{split}\]
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\[\frac{\mathrm{d}}{\mathrm{d}s}\mathcal{Q}_{\bm{\eta}}^{2}\big{(} \dot{\gamma}(s)\big{)} =2\mathrm{tr}\big{\{}\ddot{\gamma}(s)^{\mathsf{T}}\dot{\gamma}(s) \big{\}}+\sum_{j=1}^{P}\eta_{j}\big{[}\mathrm{tr}\big{\{}\ddot{\gamma}(s)^{ \mathsf{T}}\bm{\Pi}_{j1}\dot{\gamma}(s)\bm{\Pi}_{j2}\big{\}}+\mathrm{tr}\big{\{} \dot{\gamma}(s)^{\mathsf{T}}\bm{\Pi}_{j1}\ddot{\gamma}(s)\bm{\Pi}_{j2}\big{\}} \big{]}\] \[\leq 2\big{\{}\|\ddot{\gamma}(s)\|_{F}+\sum_{j=1}^{P}\eta_{j}\| \bm{\Pi}_{j1}^{1/2}\ddot{\gamma}(s)\bm{\Pi}_{j2}^{1/2}\|_{F}\cdot\|\bm{\Pi}_{j 1}^{1/2}\dot{\gamma}(s)\bm{\Pi}_{j2}^{1/2}\|_{F}\] \[\leq 2\big{\{}\|\ddot{\gamma}(s)\|_{F}^{2}+\sum_{j=1}^{P}\eta_{j} \|\bm{\Pi}_{j1}^{1/2}\ddot{\gamma}(s)\bm{\Pi}_{j2}^{1/2}\|_{F}^{2}\Big{\}}^{1/2}\] \[\qquad\times\Big{\{}\|\dot{\gamma}(s)\|_{F}^{2}+\sum_{j=1}^{P}\eta _{j}\|\bm{\Pi}_{j1}^{1/2}\dot{\gamma}(s)\bm{\Pi}_{j2}^{1/2}\|_{F}^{2}\Big{\}}^ {1/2}\] \[\leq 2\mathcal{Q}_{\bm{\eta}}(\ddot{\gamma}(s))\mathcal{Q}_{\bm{ \eta}}(\dot{\gamma}(s))\] \[=2\mathcal{Q}_{\bm{\eta}}(\Pi(\dot{\gamma}(s),\dot{\gamma}(s))) \mathcal{Q}_{\bm{\eta}}(\dot{\gamma}(s))\] \[\leq 2C_{\Pi}\mathcal{Q}_{\bm{\eta}}^{3}(\dot{\gamma}(s)).\]
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\[\mathrm{E}\,U_{1h}(x)^{m} =\mathrm{E}\,[(a+b)^{m}]\] \[=\mathrm{E}\,\Big{[}\sum_{k=0}^{m}\binom{m}{k}a^{m-k}b^{k}\Big{]}\] \[=\mathrm{E}\,\Big{[}a^{m}+b^{m}\Big{]}\qquad\text{because}\qquad a ^{m-k}b^{k}=0\;\forall\;k\in\{1,\ldots,m-1\}\] \[=\mathrm{E}\,\Big{[}t^{m}\,\bigg{(}\frac{1}{q(x)}\frac{1}{h^{2}} \,w^{\prime}\Big{(}\frac{x-V_{i}}{h}\Big{)}-\frac{q^{\prime}(x)}{q^{2}(x)} \frac{1}{h}\,w\Big{(}\frac{x-V_{i}}{h}\Big{)}\bigg{)}^{m}\] \[\quad+(1-t)^{m}(-1)^{m}\left(\frac{1}{q(x)}\frac{1}{h^{2}}\,w^{ \prime}\Big{(}\frac{x+1-V_{i}}{h}\Big{)}-\frac{q^{\prime}(x)}{q^{2}(x)}\frac{ 1}{h}\,w\Big{(}\frac{x+1-V_{i}}{h}\Big{)}\right)^{m}\Big{]}.\]
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\[\int_{\mathbb{R}^{d}}\|\nabla V\|^{4}\,\mathrm{d}\rho,\quad\int_ {\mathbb{R}^{d}}(\Delta V)^{2}\,\mathrm{d}\rho,\quad\int_{\mathbb{R}^{d}}( \Delta\log\rho)^{2}\,\mathrm{d}\rho,\quad\int_{\mathbb{R}^{d}}\|\nabla\log \rho\|^{4}\,\mathrm{d}\rho,\] \[\int_{\mathbb{R}^{d}}\Big{\|}\nabla^{2}\log\rho(y)\nabla\frac{ \delta\mathcal{F}_{\mathrm{KL}}}{\delta\rho}(\rho)(y)+\nabla\mathrm{tr} \Big{(}\nabla^{2}\frac{\delta\mathcal{F}_{\mathrm{KL}}}{\delta\rho}(\rho) \Big{)}(y)\Big{\|}^{2}\,\mathrm{d}\rho,\quad\text{and}\] \[\int_{\mathbb{R}^{d}}\Big{\|}\nabla^{2}\frac{\delta\mathcal{F}_{ \mathrm{KL}}}{\delta\rho}(\rho)(y)\nabla\frac{\delta\mathcal{F}_{\mathrm{KL}}} {\delta\rho}(\rho)(y)\Big{)}\Big{\|}^{2}\,\mathrm{d}\rho,\]
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\[\sum_{k=1}^{m}\left(\Sigma\left(\frac{\partial}{\partial Z(t)_{k} }\begin{pmatrix}0&\mathbf{GC}_{XY}\\ \mathbf{C}_{XY}{}^{\prime}\mathbf{G}&0\end{pmatrix}\right)Z(t)\right)_{k}\] \[= \frac{1}{T}\sum_{k=1}^{n}\mathbf{e}^{\prime}_{k}\mathcal{A}\left( z^{2}\mathbf{Ge}_{k}Y(t)^{\prime}\widetilde{\mathbf{G}}Y(t)+\mathbf{GC}_{XY}Y(t) \mathbf{e}^{\prime}_{k}\mathbf{GC}_{XY}Y(t)\right)+\] \[\frac{1}{T}\sum_{k=1}^{n}\mathbf{e}^{\prime}_{k}\mathcal{C}\left( z^{2}\widetilde{\mathbf{G}}Y(t)\mathbf{e}^{\prime}_{k}\mathbf{G}X(t)+\mathbf{C}_{XY} {}^{\prime}\mathbf{Ge}_{k}Y(t)^{\prime}\mathbf{C}_{XY}{}^{\prime}\mathbf{G}X(t )\right)+\] \[\frac{1}{T}\sum_{k=1}^{p}\mathbf{e}^{\prime}_{k}\mathcal{C}^{ \prime}\left(z^{2}\mathbf{G}X(t)\mathbf{e}^{\prime}_{k}\widetilde{\mathbf{G}}Y (t)+\mathbf{GC}_{XY}\mathbf{e}_{k}X(t)^{\prime}\mathbf{GC}_{XY}Y(t)\right)+\] \[\frac{1}{T}\sum_{k=1}^{p}\mathbf{e}^{\prime}_{k}\mathcal{B}\left( z^{2}\widetilde{\mathbf{G}}\mathbf{e}_{k}X(t)^{\prime}\mathbf{G}X(t)+\mathbf{C}_{XY} {}^{\prime}\mathbf{G}X(t)\mathbf{e}^{\prime}_{k}\mathbf{C}_{XY}{}^{\prime} \mathbf{G}X(t)\right)\] \[= \frac{1}{T}\left(z^{2}(\operatorname{Tr}\mathcal{A}\mathbf{G})Y( t)^{\prime}\widetilde{\mathbf{G}}Y(t)+Y(t)^{\prime}\mathbf{C}_{XY}{}^{ \prime}\mathbf{G}\mathcal{A}\mathbf{GC}_{XY}Y(t)\right)+\] \[\frac{2}{T}\left(z^{2}X(t)^{\prime}\mathbf{GC}\widetilde{\mathbf{ G}}Y(t)+(\operatorname{Tr}\mathcal{C}\mathbf{C}_{XY}{}^{\prime}\mathbf{G})Y(t)^{ \prime}\mathbf{C}_{XY}{}^{\prime}\mathbf{G}X(t)\right)+\] \[\frac{1}{T}\left(z^{2}(\operatorname{Tr}\mathcal{B}\widetilde{ \mathbf{G}})X(t)^{\prime}\mathbf{G}X(t)+X(t)^{\prime}\mathbf{GC}_{XY}\mathcal{B }\mathbf{C}_{XY}{}^{\prime}\mathbf{G}X(t)\right)\]
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\[\sum_{\begin{subarray}{c}p_{1},p_{2}\in[P_{1},P_{2}]\\ p_{1}\equiv p_{2}\equiv 1\ (\text{mod}\ D_{f})\end{subarray}}\frac{1}{p_{1}p_{2}} \sum_{\begin{subarray}{c}\mathbf{b}_{1}\in\mathbb{Z}^{4}\cap\mathcal{C}_{1}, \mathbf{b}_{2}\in\mathbb{Z}^{4}\cap\mathcal{C}_{2}\\ \mathbf{b}_{1}\equiv\mathbf{b}_{2}\equiv\mathbf{b}_{0}\ (\text{mod}\ m)\\ \wedge(\mathbf{b}_{1},\mathbf{b}_{2})\neq 0\end{subarray}}\frac{|g_{ \mathbf{b}_{1}}g_{\mathbf{b}_{2}}|D_{\mathbf{b}_{1},\mathbf{b}_{2}}}{\wedge( \mathbf{b}_{1},\mathbf{b}_{2})}\] \[\ll \sum_{\begin{subarray}{c}p_{1},p_{2}\in[P_{1},P_{2}]\\ p_{1}\equiv p_{2}\equiv 1\ (\text{mod}\ D_{f})\end{subarray}}\frac{1}{p_{1}p_{2}} \sum_{\begin{subarray}{c}\mathbf{b}_{1},\mathbf{c}_{2}\ (\text{mod}\ p_{1})\\ p_{1}|N(\mathbf{c}_{1})\end{subarray}}\sum_{\begin{subarray}{c}\mathbf{b}_{1} \in\mathbb{Z}^{4}\cap\mathcal{C}_{1},\mathbf{b}_{2}\in\mathbb{Z}^{4}\cap \mathcal{C}_{2}\\ \wedge(\mathbf{b}_{1},\mathbf{b}_{2})\neq 0\end{subarray}}\frac{|g_{ \mathbf{b}_{1}}g_{\mathbf{b}_{2}}|D_{\mathbf{b}_{1},\mathbf{b}_{2}}}{\wedge( \mathbf{b}_{1},\mathbf{b}_{2})}\] \[\ll \sum_{\begin{subarray}{c}p_{1},p_{2}\in[P_{1},P_{2}]\\ p_{1}\equiv p_{2}\equiv 1\ (\text{mod}\ D_{f})\end{subarray}}\frac{1}{p_{1}p_{2}} p_{1}^{7}\Big{(}\frac{B^{6}}{p_{1}^{8}}+B^{17/3}\Big{)}\] \[\ll \frac{B^{6}}{P_{1}}+P_{2}^{7}B^{17/3}.\]
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\[\begin{array}{lcl}&\alpha^{n}\oint_{\infty}\frac{dz}{2i\pi}z^{-n-1}e^{-\sum _{k>0}(\gamma^{k}t_{k}-t_{k}^{\prime})\frac{1-t^{k}}{1-q^{k}}z^{k}}\tau_{u}( \boldsymbol{t}+[\gamma^{-1}z^{-1}],\bar{\boldsymbol{t}}|G)\tau_{uq^{-n}t^{-1} }(\boldsymbol{t}^{\prime}-[\gamma^{-2}z^{-1}],\bar{\boldsymbol{t}}^{\prime}|G) \\ &+\oint_{0}\frac{dz}{2i\pi}z^{-n-1}e^{\sum_{k>0}(\gamma^{k}t_{-k}-t_{-k}^{ \prime})p^{k}z^{-k}}\tau_{ut^{-1}}(\boldsymbol{t},\bar{\boldsymbol{t}}-[ \gamma z]_{q,t}|G)\tau_{uq^{-n}}(\boldsymbol{t}^{\prime},\bar{\boldsymbol{t}}^ {\prime}+[z]_{q,t}|G)=0,\\ &\alpha^{m}\oint_{\infty}\frac{dz}{2i\pi}z^{-m-1}e^{\sum_{k>0}( \gamma^{k}t_{k}-t_{k}^{\prime})z^{k}}\tau_{u}(\boldsymbol{t}-[\gamma z^{-1}] _{t,q},\bar{\boldsymbol{t}}|G)\tau_{uqt^{m}}(\boldsymbol{t}^{\prime}+[z^{-1}] _{t,q},\bar{\boldsymbol{t}}^{\prime}|G)\\ &+\oint_{0}\frac{dz}{2i\pi}z^{-m-1}e^{-\sum_{k>0}(\gamma^{k}t_{-k}-t_{-k}^{ \prime})\frac{1-q^{k}}{1-t^{k}}z^{-k}}\tau_{uq}(\boldsymbol{t},\bar{ \boldsymbol{t}}+[\gamma z]|G)\tau_{ut^{m}}(\boldsymbol{t}^{\prime},\bar{ \boldsymbol{t}}^{\prime}-[z]|G)=0,\end{array}\]
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\[\mathcal{L}_{c}(p^{k+1},q^{k},\nu^{k+1})-\mathcal{L}_{c}(p^{k+1}, q^{k+1},\nu^{k+1})\] \[= G(q^{k})-G(q^{k+1})+\langle\nu^{k+1},Bq^{k+1}-Bq^{k}\rangle+\frac {c}{2}\|Ap^{k+1}-Bq^{k}\|^{2}-\frac{c}{2}\|Ap^{k+1}-Bq^{k+1}\|^{2}\] \[\geq \langle\nabla G(q^{k+1})-B^{T}\nu^{k+1},q^{k}-q^{k+1}\rangle- \frac{\omega_{G}}{2}\|Bq^{k}-Bq^{k+1}\|^{2}+\frac{c}{2}\|Ap^{k+1}-Bq^{k}\|^{2} -\frac{c}{2}\|Ap^{k+1}-Bq^{k+1}\|^{2}\] \[= c(Ap^{k+1}-Bq^{k+1},Bq^{k}-Bq^{k+1})-\frac{\omega_{G}}{2}\|Bq^{k }-Bq^{k+1}\|^{2}+\frac{c}{2}\|Ap^{k+1}-Bq^{k}\|^{2}-\frac{c}{2}\|Ap^{k+1}-Bq^{ k+1}\|^{2}\] \[= \frac{c-\omega_{G}}{2}\|Bq^{k}-Bq^{k+1}\|^{2}.\]
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\[(\mathrm{D}) =\int_{B_{R}^{d}}\left|1-\frac{\pi_{\mathrm{loc}}(y)\overline{\pi }(x)}{\pi_{\mathrm{loc}}(x)\overline{\pi}(y)}\right|\frac{\overline{\pi}(y) \widetilde{Q}(y,x)}{\overline{\pi}(x)}\,\mathrm{d}y\] \[\leq(\exp(2\widetilde{\varepsilon})-1)\int_{B_{R}^{d}}\frac{ \overline{\pi}(y)\widetilde{Q}(y,x)}{\overline{\pi}(x)}\,\mathrm{d}y\] \[\leq(\exp(2\widetilde{\varepsilon})-1)\left(1+\int\left|Q^{\Delta }(y,x)-\frac{\overline{\pi}(y)Q^{\Delta}(y,x)}{\overline{\pi}(x)}\,\mathrm{d }y\right|\,\mathrm{d}y+\int_{B_{R}^{d}}\left|Q^{\Delta}(y,x)-\widetilde{Q}(y,x )\right|\frac{\overline{\pi}(y)}{\overline{\pi}(x)}\,\mathrm{d}y\right)\] \[\leq\frac{19}{18}(\exp(2\widetilde{\varepsilon})-1)\] \[\leq\frac{9}{100}.\]
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\begin{table}
\begin{tabular}{l|l l l} & **Performance** & **Offline time** & **Online time** \\ & **score** & (seconds) & (seconds) \\ \hline
**MPC** & 0.487 & - & 9.82 \(10^{-4}\) \\ \hline
**OLFC-10** & 0.506 & - & 1.14 \(10^{-2}\) \\ \hline
**OLFC-50** & 0.513 & - & 8.62 \(10^{-2}\) \\ \hline
**OLFC-100** & 0.510 & - & 1.87 \(10^{-1}\) \\ \hline \hline
**SDP** & 0.691 & 2.67 & 3.09 \(10^{-4}\) \\ \hline
**SDP-AR(1)** & 0.794 & 38.1 & 4.44 \(10^{-4}\) \\ \hline
**SDP-AR(2)** & 0.795 & 468 & 5.55 \(10^{-4}\) \\ \hline
**Upper bound** & **1.0** & - & - \\ \end{tabular}
\end{table}
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\[\|N(\varphi)e^{i\mu\psi}\|_{\dot{H}^{\delta}}\] \[\lesssim\||D|^{\delta-1}((\nabla\varphi)\psi e^{i\mu\psi})\|_{L^{2} }+\||D|^{\delta-1}(\varphi(\nabla\psi)e^{i\mu\psi})\|_{L^{2}}+|\mu|\||D|^{ \delta-1}(\varphi\psi(\nabla\psi)e^{i\mu\psi})\|_{L^{2}}\] \[\lesssim\||D|^{\delta-1}\nabla\varphi\|_{L^{2}}\|\psi e^{i\mu \psi}\|_{L^{\infty}}+\|\nabla\varphi\|_{L^{\frac{2d}{d-2(\delta-1)}}}\||D|^{ \delta-1}(\psi e^{i\mu\psi})\|_{L^{\frac{d}{\delta-1}}}\] \[\qquad+\||D|^{\delta-1}(\varphi\nabla\psi)\|_{L^{2}}\|e^{i\mu \psi}\|_{L^{\infty}}+\|\varphi\nabla\psi\|_{L^{\frac{2d}{d-2(\delta-1)}}}\||D |^{\delta-1}e^{i\mu\psi}\|_{L^{\frac{d}{\delta-1}}}\] \[\qquad+|\mu|\||D|^{\delta-1}(\varphi\nabla\psi)\|_{L^{2}}\|\psi e ^{i\mu\psi}\|_{L^{\infty}}+|\mu|\|\varphi\nabla\psi\|_{L^{\frac{2d}{d-2( \delta-1)}}}\||D|^{\delta-1}(\psi e^{i\mu\psi})\|_{L^{\frac{d}{\delta-1}}}\] \[\lesssim\|\varphi\|_{H^{\delta}}\Big{(}\|\psi\|_{L^{\infty}}+\|| D|^{\delta-1}(\psi e^{i\mu\psi})\|_{L^{\frac{d}{\delta-1}}}\Big{)}\] \[\qquad+\||D|^{\delta-1}(\varphi\nabla\psi)\|_{L^{2}}\Big{(}1+\|| D|^{\delta-1}(\psi e^{i\mu\psi})\|_{L^{\frac{d}{\delta-1}}}\Big{)}\] \[\qquad+|\mu|\||D|^{\delta-1}(\varphi\nabla\psi)\|_{L^{2}}\Big{(} \|\psi\|_{L^{\infty}}+\||D|^{\delta-1}(\psi e^{i\mu\psi})\|_{L^{\frac{d}{ \delta-1}}}\Big{)}.\]
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\[\prod_{x\in X/S}F\left(1+\sum_{\underline{i}\in I}X_{\underline{ i}}\underline{t}^{\underline{i}}\right)_{x} = \prod_{x\in X/S}\left(1+\sum_{\underline{i}\in I}X_{\underline{ i},x}\prod_{k=1}^{n}s_{1}^{a_{k,1}i_{k}}\ldots s_{m}^{a_{k,m}i_{k}}\right)\] \[= \prod_{x\in X/S}\left(1+\sum_{\underline{i}\in I}X_{\underline{i},x}\prod_{j=1}^{m}s_{j}^{\sum_{k=1}^{n}a_{k,j}i_{k}}\right)\] \[= \prod_{x\in X/S}\left(1+\sum_{\underline{i}\in J}\left(\sum_{ \begin{subarray}{c}\underline{i}\\ \underline{i}\cdot\underline{A}=\underline{j}\end{subarray}}X_{\underline{i},x} \right)\underline{s}^{\underline{j}}\right)\] \[= \sum_{\nu=(n_{\underline{j}})_{\underline{i}\in J}}\left(\prod_{ \begin{subarray}{c}\underline{i}\\ \underline{i}\end{subarray}}\operatorname{Sym}_{X/S}^{n_{\underline{j}}}\left( \sum_{\underline{i}\cdot\underline{A}=\underline{j}}X_{\underline{i}}\right) \right)_{*}\prod_{\underline{i}}\underline{s}^{n_{\underline{j}}\underline{j}}\] \[= \sum_{\underline{k}=(k_{1},\ldots,k_{m})}\underline{s}^{\underline {k}}\left(\sum_{\begin{subarray}{c}(n_{\underline{j}})_{\underline{i}}\\ \sum_{\underline{i}}n_{\underline{j}}\underline{j}=\underline{k}\end{subarray}} \left(\prod_{\underline{i}}\operatorname{Sym}_{X/S}^{n_{\underline{j}}}\left( \sum_{\underline{i}\cdot\underline{A}=\underline{j}}X_{\underline{i}} \right)\right)_{*}\right)\] \[= \sum_{\underline{k}=(k_{1},\ldots,k_{m})}\tfrac{\underline{s}^{ \underline{k}}}{\left(\sum_{\begin{subarray}{c}(n_{\underline{j}})_{\underline{j }}\\ \sum_{\underline{i}}n_{\underline{j}}\underline{j}=\underline{k}\end{subarray}} \left(\prod_{\begin{subarray}{c}\underline{j}\\ \underline{i}\end{subarray}}\sum_{\begin{subarray}{c}(n_{\underline{j}})_{ \underline{i}}\\ \sum_{\underline{i}}n_{\underline{j}}\underline{j}=\underline{k}\end{subarray}} \prod_{\begin{subarray}{c}\underline{i}\in I_{\underline{i}}\\ \sum n_{\underline{i}}=n_{\underline{j}}\end{subarray}}\right)_{*}\right)}.\]
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\begin{table}
\begin{tabular}{|l||l|l|l|l|l|l|l||l|} \hline Objects & A & B & C & D & E & F & G & Results \\ \hline
1 & 2 & 2 & 2 & 3 & 2 & 4 & 3 & 1 \\ \hline
2 & 3 & 3 & 2 & 3 & 2 & 5 & 4 & 1 \\ \hline
3 & 3 & 3 & 1 & 4 & 2 & 4 & 4 & 1 \\ \hline
4 & 3 & 3 & 1 & 4 & 2 & 4 & 4 & 1 \\ \hline
5 & 2 & 2 & 1 & 3 & 2 & 5 & 3 & 1 \\ \hline οΏ½ & οΏ½ & οΏ½ & οΏ½ & οΏ½ & οΏ½ & οΏ½ & οΏ½ & οΏ½ \\ \hline
71 & 2 & 1 & 3 & 2 & 2 & 3 & 1 & 2 \\ \hline
72 & 2 & 2 & 3 & 2 & 2 & 3 & 2 & 2 \\ \hline
73 & 2 & 2 & 2 & 2 & 2 & 3 & 2 & 2 \\ \hline
74 & 1 & 1 & 1 & 2 & 1 & 4 & 1 & 2 \\ \hline
75 & 2 & 2 & 2 & 2 & 2 & 3 & 2 & 2 \\ \hline οΏ½ & οΏ½ & οΏ½ & οΏ½ & οΏ½ & οΏ½ & οΏ½ & οΏ½ & οΏ½ \\ \hline
141 & 3 & 3 & 3 & 3 & 3 & 2 & 3 & 3 \\ \hline
142 & 3 & 3 & 3 & 3 & 3 & 1 & 3 & 3 \\ \hline
143 & 3 & 3 & 3 & 3 & 3 & 2 & 3 & 3 \\ \hline
144 & 3 & 3 & 3 & 4 & 3 & 2 & 3 & 3 \\ \hline
145 & 4 & 3 & 4 & 4 & 4 & 3 & 3 & 3 \\ \hline οΏ½ & οΏ½ & οΏ½ & οΏ½ & οΏ½ & οΏ½ & οΏ½ & οΏ½ & οΏ½ \\ \hline \end{tabular}
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\[\Pi_{\bar{\mathcal{B}}}(\mathbf{t}_{N\backslash\{i\}}^{*},\mathbf{ s}_{N\backslash\{i\}}^{*}) =\mathbb{E}[V(D,\mathbf{s}_{N\backslash\{i\}}^{*}(D))-\sum_{j\in N ^{*}(\mathcal{C})}(\bar{P}_{j}(\mathbf{s}_{N\backslash\{i\}}^{*}(D))+\bar{R}_{ j}(\mathbf{t}_{N\backslash\{i\}}^{*}))\] \[\quad-\sum_{j\notin N^{*}(\mathcal{C})}(C_{j}(\mathbf{s}_{N \backslash\{i\}}^{*}(D))+E_{j}(\mathbf{t}_{N\backslash\{i\}}^{*}))]\] \[=\mathbb{E}[V(D,\mathbf{s}_{N\backslash\{i\}}^{*}(D))-\sum_{j\in N }(C_{j}(\mathbf{s}_{N\backslash\{i\}}^{*}(D))+E_{j}(\mathbf{t}_{N\backslash\{ i\}}^{*}))]\] \[\quad-\sum_{j\in I(\mathbf{t}_{N\backslash\{i\}}^{*})\cap N^{*} (\mathcal{C})}\left(\Pi_{\mathcal{C}}^{*}(N)-\Pi_{\mathcal{C}}^{*}(N \backslash\{j\})\right)\] \[=\Pi_{\mathcal{C}}^{*}(N\backslash\{i\})-\sum_{j\in N^{*}( \mathcal{C})\backslash\{i\}}\left(\Pi_{\mathcal{C}}^{*}(N)-\Pi_{\mathcal{C}}^{* }(N\backslash\{j\})\right),\]
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\[M \lesssim\sum_{k=0}^{d-2}\frac{\rho^{k+2}}{\varepsilon}\int_{ \tilde{H}_{\epsilon_{1}}}...\int_{\tilde{H}_{\epsilon_{d-k}}}\ 1\wedge\left(\frac{\rho}{|\sum_{i=1}^{d-k}(x_{i}-y_{i})|} \right)^{d-k-\frac{3}{4}}d\nu_{k,\sum_{i=d-k+1}^{d}x_{i}}^{1}(y_{d-k},...,y_{ 1})\] \[+\sum_{k=0}^{d-2}\rho^{k+1}\mathbf{1}_{|\sum_{i=1}^{d}(y_{0i}-x_{ i})|\leq 2\varepsilon}\int_{\tilde{H}_{\epsilon_{1}}}...\int_{\tilde{H}_{ \epsilon_{d-k}}}1\wedge\left(\frac{\rho}{|\sum_{i=1}^{d-k}(x_{i}-y_{i})|} \right)^{d-k+\frac{1}{4}}d\nu_{k,\sum_{i=d-k+1}^{d}x_{i}}^{2}(y_{d-k},..,y_{1}).\]
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\[\begin{split}\mathrm{(i)}&\sum\nolimits_{j}\int_{ \widetilde{\Omega}_{j}^{\eta,\gamma}}\big{|}\mathrm{sym}\big{(}(R_{j}^{\eta, \gamma})^{T}\nabla y-\mathrm{Id}\big{)}\big{|}^{2}\leq C_{0}\big{(}1+C_{\eta} \gamma^{-5d/q}\varepsilon\big{)}\int_{\Omega\setminus\overline{E}} \operatorname{dist}^{2}(\nabla y,SO(d))\,,\\ \mathrm{(ii)}&\sum\nolimits_{j}\int_{\widetilde{ \Omega}_{j}^{\eta,\gamma}}\big{|}(R_{j}^{\eta,\gamma})^{T}\nabla y-\mathrm{Id }\big{|}^{2}\leq C_{\eta}\gamma^{-2d/q}\int_{\Omega\setminus\overline{E}} \operatorname{dist}^{2}(\nabla y,SO(d))\,,\\ \mathrm{(iii)}&\sum\nolimits_{j}\int_{\widetilde{ \Omega}_{j}^{\eta,\gamma}}\big{|}y-(R_{j}^{\eta,\gamma}x+b_{j}^{\eta,\gamma}) \big{|}^{2}\leq C_{\eta}\gamma^{(2-4d)/q}\int_{\Omega\setminus\overline{E}} \operatorname{dist}^{2}(\nabla y,SO(d))\,,\end{split}\]
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\[\mathfrak{X}_{[4]}\mathfrak{X}_{[3]} \mathfrak{X}_{[2]2}r(\mathfrak{X}^{[1]})\] \[\overset{(\ref{eq:2.2})}{=}(q-q^{-1})(q^{2}c_{2})\mathfrak{X}_{[4 ]}\mathfrak{X}_{[3]}\mathfrak{X}_{[2]}E_{2}\mathfrak{X}^{[1]}\] \[\overset{(\ref{eq:2.3})}{=}(q-q^{-1})(q^{2}c_{2})\mathfrak{X}_{[4 ]}\mathfrak{X}_{[3]}E_{2}\mathfrak{X}^{[2]}\mathfrak{X}^{[1]}\] \[\overset{(\ref{eq:2.3})}{=}(q-q^{-1})(q^{2}c_{2})\mathfrak{X}_{[4 ]}E_{2}\mathfrak{X}^{[3]}\mathfrak{X}^{[2]}\mathfrak{X}^{[1]}\] \[\overset{(\ref{eq:2.3})}{-}(q-q^{-1})(q^{2}c_{2})\mathfrak{X}_{[4 ]}\mathfrak{X}_{[3]}T_{213}(E_{2})T_{2}(E_{1})T_{2}(E_{3})\mathfrak{X}^{[2]} \mathfrak{X}^{[1]}\] \[\overset{(\ref{eq:2.3})}{=}(q-q^{-1})(q^{2}c_{2})\Big{(}E_{2} \mathfrak{X}^{[4]}-q^{-1}(q-q^{-1})c_{1}\mathfrak{X}_{[4]}\big{(}E_{1}T_{2}(E _{3})+E_{3}T_{2}(E_{1})\big{)}\] \[\qquad-(q-q^{-1})c_{1}\big{(}\mathfrak{X}_{[4]}+(q-q^{-1})c_{1} \mathfrak{X}_{[4]}E_{1}E_{3}\big{)}T_{213}(E_{2})\Big{)}\mathfrak{X}^{[3]} \mathfrak{X}^{[2]}\mathfrak{X}^{[1]}\] \[\qquad-(q-q^{-1})^{3}(q^{6}c_{1}^{2}c_{2}^{2})\mathfrak{X}_{[4 ]}\mathfrak{X}_{[3]}T_{213}(E_{2})T_{2}(E_{1})T_{2}(E_{3})\mathfrak{X}^{[2]} \mathfrak{X}^{[1]}.\]
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\[\Big{|}\mathcal{F}_{\alpha}(f)(m)\Big{|}= \Big{|}\frac{A_{\alpha}^{n}e_{\alpha}(m)}{(2\pi im_{j}\csc\alpha )^{s}}\int_{\mathbf{T}_{\alpha}^{n}}\partial_{j}^{s}(e_{\alpha}f)(x)e^{-2\pi im \cdot x\csc\alpha}dx\Big{|}\] \[\leq \Big{(}\frac{\sqrt{n}}{2\pi|\csc\alpha|m|}\Big{)}^{s}\frac{|A_{ \alpha}^{n}|}{2|\csc\alpha|^{n}}\frac{\|\partial_{j}^{s}(e_{\alpha}f)\|_{\dot{ \Lambda}_{\gamma}}}{(2|m_{j}\csc\alpha|)^{\gamma}}\] \[\leq \Big{(}\frac{\sqrt{n}}{2\pi|\csc\alpha|m|}\Big{)}^{s}\frac{1}{2| \csc\alpha|^{n/2}}\Big{(}\frac{\sqrt{n}}{2|\csc\alpha|m|}\Big{)}^{\gamma}\| \partial_{j}^{s}(e_{\alpha}f)\|_{\dot{\Lambda}_{\gamma}}\] \[\leq \frac{(\sqrt{n})^{s+\gamma}}{(2\pi)^{s}|\csc\alpha|^{s+\gamma+n/2 }2^{\gamma+1}}\frac{\max_{|\partial|=s}\|\partial_{j}^{s}(e_{\alpha}f)\|_{ \dot{\Lambda}_{\gamma}}}{|m|^{s+\gamma}}.\]
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\[a_{5}(15n+6,10,12,13)\equiv 0\pmod{3},\] \[a_{7}(21n+3,8,11,15,17,18)\equiv 0\pmod{3},\] \[a_{11}(33n+3,11,12,20,24,26,27,29,30,32)\equiv 0\pmod{3},\] \[a_{13}(39n+3,7,9,10,15,16,18,22,28,31,33,36)\equiv 0\pmod{ 3},\] \[a_{17}(51n+10,14,16,19,20,23,25,26,28,34,35,38,41,46,47,49)\equiv 0 \pmod{3},\] \[a_{19}(57n+7,14,16,17,19,22,25,26,31,35,37,38,41,44,50,52,55,56) \equiv 0\pmod{3},\] \[a_{23}(69n+3,9,16,22,27,30,31,33,34,36,42,43,46,48,51,52,58,60,61, 63,64,64,67)\] \[\equiv 0\pmod{3}.\]
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\begin{table}
\begin{tabular}{|c|c|c|c|c|} \hline \(i\) & \(d_{i}\) & \([r_{1}d_{i}]\) & \((r_{1},d_{i})\) & \((a_{r_{1}},u_{d_{i}})_{V}\) \\ \hline \(1\) & \((6,7,8)(9,10,11)\) & \(2^{2}\cdot 3^{2}\) & \(3\times 3\) & \(0\) \\ \hline \(2\) & \((4,5,6)(9,10,11)\) & \(2\cdot 3\cdot 4\) & \(3\times S_{4}\) & \(\frac{1}{36}\) \\ \hline \(3\) & \((3,4,5)(9,10,11)\) & \(2^{2}\cdot 3\) & \(3\times S_{3}\) & \(\frac{1}{20}\) \\ \hline \(4\) & \((3,5,7)(4,6,8)\) & \(2\cdot 6\) & \(2\times A_{4}\) & \(\frac{1}{45}\) \\ \hline \(5\) & \((2,3,4)(9,10,11)\) & \(3^{2}\) & \(A_{4}\) & \(\frac{1}{9}\) \\ \hline \(6\) & \((2,3,5)(9,10,11)\) & \(3\cdot 5\) & \(3\times A_{5}\) & \(\frac{11}{360}\) \\ \hline \(7\) & \((2,3,5)(4,6,7)\) & \(7\) & \(L_{3}(2)\) & \(\frac{1}{24}\) \\ \hline \(8\) & \((2,5,7)(4,6,8)\) & \(4^{2}\) & \(S_{4}\) & \(\frac{13}{180}\) \\ \hline \(9\) & \((2,5,7)(3,4,6)\) & \(2\cdot 4\) & \(S_{4}\) & \(\frac{1}{36}\) \\ \hline \(10\) & \((1,2,3)(4,5,6)\) & \(5\) & \(A_{5}\) & \(\frac{1}{18}\) \\ \hline \(11\) & \((1,2,5)(3,4,6)\) & \(2^{2}\) & \(S_{3}\) & \(\frac{1}{4}\) \\ \hline \(12\) & \((1,3,5)(2,4,6)\) & \(3^{2}\) & \(A_{4}\) & \(\frac{1}{9}\) \\ \hline \(13\) & \((1,3,6)(2,5,4)\) & \(2\cdot 4\) & \(S_{4}\) & \(\frac{1}{36}\) \\ \hline \end{tabular}
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\[\sum_{j_{1},j_{2},j_{3},j_{4}} \delta_{i_{1}j_{1}}\delta_{i_{2}j_{2}}\delta_{i_{3}j_{3}}\delta_ {i_{4}j_{4}}\mathbb{E}[Y_{j_{1}}Y_{j_{2}}Y_{j_{3}}Y_{j_{4}}]\] \[=\sum_{j_{1},j_{2},j_{3}}\delta_{i_{1}j_{1}}\delta_{i_{2}j_{2}} \delta_{i_{3}j_{3}}\left(\sum_{j_{4}\in\{j_{1},j_{2},j_{3}\}}\delta_{i_{4}j_{4 }}\mathbb{E}[Y_{j_{1}}Y_{j_{2}}Y_{j_{3}}Y_{j_{4}}]\right)\] \[=\sum_{j_{1},j_{2},j_{3}}p_{3}(j_{1},j_{2},j_{3})\delta_{i_{1}j_{ 1}}\delta_{i_{2}j_{2}}\delta_{i_{3}j_{3}}\left(\sum_{j_{4}\in\{j_{1},j_{2},j_{ 3}\}}\delta_{i_{4}j_{4}}+p_{4}(j_{1},j_{2},j_{3})\sum_{j_{4}\notin\{j_{1},j_{2 },j_{3}\}}\delta_{i_{4}j_{4}}\right)\] \[=\sum_{j_{1},j_{2},j_{3}}p_{3}(j_{1},j_{2},j_{3})q_{4}(j_{1},j_{2},j_{3})\delta_{i_{1}j_{1}}\delta_{i_{2}j_{2}}\delta_{i_{3}j_{3}}\sum_{j_{4}\in \{j_{1},j_{2},j_{3}\}}\delta_{i_{4}j_{4}},\]
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\[\sum_{\begin{subarray}{c}q\leqslant Q\\ q(q,a_{1}a_{2})=1\\ q\equiv c_{0}\,(\text{mod}\ c)\end{subarray}}\sum_{\begin{subarray}{c}n\in \underline{\times}\\ n\equiv a_{1}\overline{a_{2}}\,(\text{mod}\ d)\\ n\equiv d_{0}\,(\text{mod}\ d)\end{subarray}}\Lambda(n)=\sum_{\begin{subarray}{ c}q\leqslant Q\\ (q,a_{1}a_{2})=1\\ q\equiv c_{0}\,(\text{mod}\ c)\end{subarray}}\frac{1}{\phi(q)\phi(d)}\sum_{ \begin{subarray}{c}\psi\,(\text{mod}\ d)\\ \psi\,(\text{mod}\ d)\\ \chi\,(\text{mod}\ q)\end{subarray}}\sum_{n\in\underline{\times}}\Lambda(n) \chi(n\overline{a_{1}}a_{2})\psi(n\overline{d_{0}})\] \[=\sum_{\begin{subarray}{c}q\leqslant Q\\ (q,a_{1}a_{2})=1\\ q\equiv c_{0}\,(\text{mod}\ c)\end{subarray}}\frac{1}{\phi(q)\phi(d)}\sum_{ \begin{subarray}{c}\psi\,(\text{mod}\ d)\\ \chi\,(\text{mod}\ q)\\ \text{cond}(\chi)\leqslant R\end{subarray}}\sum_{n\in\underline{\times}} \Lambda(n)\chi(n\overline{a_{1}}a_{2})\psi(n\overline{d_{0}})\] \[\quad+\sum_{\begin{subarray}{c}q\leqslant Q\\ q(q,a_{1}a_{2})=1\\ q\equiv c_{0}\,(\text{mod}\ c)\end{subarray}}\frac{1}{\phi(d)}\sum_{\psi\,( \text{mod}\ d)}\sum_{n\in\underline{\times}}\Lambda(n)u_{R}(n\overline{a_{1}}a _{2};q)\psi(n\overline{d_{0}}):=\mathcal{S}_{1}+\mathcal{S}_{2},\]
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\[||\alpha_{\delta}^{1,0}-\alpha_{\delta^{*}}^{1,0}||\leq||\Delta_{ \delta}^{0,0}-\Delta_{\delta^{*}}^{0,0}||\] \[+\beta||\sum_{s^{\prime}}P(s^{\prime}|s,\delta^{*}(\zeta))\alpha_{ \delta^{*}}^{1,0}(\bar{\zeta}(s^{\prime},s_{2}^{L^{\prime}},s_{1}^{L^{\prime}}, \zeta,\delta^{*}(\zeta)))-\sum_{s^{\prime}}P(s^{\prime}|s,\delta(\zeta))\alpha_ {\delta}^{1,0}(\bar{\zeta}(s^{\prime},s_{2}^{L^{\prime}},s_{1}^{L^{\prime}}, \zeta,\delta(\zeta)))||\] \[\leq||\Delta_{\delta}^{0,0}-\Delta_{\delta^{*}}^{0,0}||+\beta \left(d(\delta,\delta^{*})||\alpha_{\delta^{*}}^{1,0}||+||\alpha_{\delta}^{1,0 }(\bar{\zeta}(s^{\prime},s_{2}^{L^{\prime}},s_{1}^{L^{\prime}},\zeta,\delta( \zeta)))-\alpha_{\delta^{*}}^{1,0}(\bar{\zeta}(s^{\prime},s_{2}^{L^{\prime}},s _{1}^{L^{\prime}},\zeta,\delta(\zeta)))||\right)\] \[\leq||\Delta_{\delta}^{0,0}-\Delta_{\delta^{*}}^{0,0}||+\beta \left(d(\delta,\delta^{*})||\alpha_{\delta^{*}}^{1,0}||+||\alpha_{\delta}^{1,0 }-\alpha_{\delta^{*}}^{1,0}||\right).\] \[||\alpha_{\delta}^{1,0}-\alpha_{\delta^{*}}^{1,0}||\leq\frac{1}{1 -\beta}||\Delta_{\delta}^{0,0}-\Delta_{\delta^{*}}^{0,0}||+\frac{\beta}{1- \beta}\left(d(\delta,\delta^{*})||\alpha_{\delta^{*}}^{1,0}||\right)\] \[\leq\frac{\beta}{1-\beta}d(\delta,\delta^{*})\eta^{1}+\frac{\beta }{1-\beta}\left(||\alpha_{\delta^{*}}^{0,0}(\bar{\zeta}(s^{\prime},s_{2}^{L^{ \prime}},s_{1}^{L^{\prime}},\zeta,\delta(\zeta)))-\alpha_{\delta}^{0,0}(\bar{ \zeta}(s^{\prime},s_{2}^{L^{\prime}},s_{1}^{L^{\prime}},\zeta,\delta(\zeta))) ||\right)\] \[+\frac{\beta}{1-\beta}\left(||\sum_{z^{F^{\prime}}_{1}\neq s_{2}^{ L^{\prime}}}\sigma_{s_{2}^{L^{\prime}},z^{F_{1}^{\prime}}}\left[\alpha_{ \delta^{*}}^{0,0}(\bar{\zeta}(s^{\prime},z^{F_{1}^{\prime}},s_{1}^{L^{\prime} },\zeta,\delta(\zeta)))-\alpha_{\delta}^{0,0}(\bar{\zeta}(s^{\prime},z^{F_{1} ^{\prime}},s_{1}^{L^{\prime}},\zeta,\delta(\zeta)))\right]||\right)\]
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\[\left|t_{P,H}(z)-\widetilde{t}_{P,H}(z)\right| =\left|\sum_{i=1}^{\infty}\alpha_{P,i}z^{M_{i}}-\sum_{i=1}^{ \infty}\sum_{H^{\prime}\in\mathcal{H}^{\prime}}\alpha_{P,H^{\prime},i}z^{M_{i }}\right|\] \[\leq\sum_{i=1}^{\infty}\left|\alpha_{P,i}-\sum_{H^{\prime}\in \mathcal{H}^{\prime}}\alpha_{P,H^{\prime},i}\right|\cdot\left|z^{M_{i}}\right|\] \[\leq\sum_{i=1}^{\infty}\left|\alpha_{P,i}-\sum_{H^{\prime}\in \mathcal{H}^{\prime}}\alpha_{P,H^{\prime},i}\right|\] \[\leq\frac{\varepsilon}{8}+\sum_{i=i_{0}+1}^{\infty}\left|\alpha_ {P,i}-\sum_{H^{\prime}\in\mathcal{H}^{\prime}}\alpha_{P,H^{\prime},i}\right|\] \[\leq\frac{\varepsilon}{8}+\sum_{i=i_{0}+1}^{\infty}\left|\alpha_ {P,i}\right|+\sum_{i=i_{0}+1}^{\infty}\left|\sum_{H^{\prime}\in\mathcal{H}^{ \prime}}\alpha_{P,H^{\prime},i}\right|\] \[\leq\frac{\varepsilon}{8}+2\sum_{i=i_{0}+1}^{\infty}\sum_{H^{ \prime}\in\mathcal{H}}\left|\alpha_{P,H^{\prime},i}\right|\leq\frac{\varepsilon }{4}\,.\]
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\[\beta_{0}^{l} =\frac{1}{4}(W_{i-2}-4W_{i-1}+3W_{i})^{2}+\frac{13}{12}(W_{i-2}-2 W_{i-1}+W_{i})^{2},\] \[\beta_{1}^{l} =\frac{1}{36}(W_{i-2}-6W_{i-1}+3W_{i}+2W_{i+1})^{2}+\frac{13}{12} (W_{i-1}-2W_{i}+W_{i+1})^{2}\] \[+\frac{1043}{960}(-W_{i-2}+3W_{i-1}-3W_{i}+W_{i+1})^{2}\] \[+\frac{1}{432}(W_{i-2}-6W_{i-1}+3W_{i}+2W_{i+1})(-W_{i-2}+3W_{i-1 }-3W_{i}+W_{i+1}),\] \[\beta_{2}^{l} =\frac{1}{36}(-2W_{i-1}-3W_{i}+6W_{i+1}-W_{i+2})^{2}+\frac{13}{1 2}(W_{i-1}-2W_{i}+W_{i+1})^{2}\] \[+\frac{1043}{960}(-W_{i-1}+3W_{i}-3W_{i+1}+W_{i+2})^{2}\] \[+\frac{1}{432}(-2W_{i-1}-3W_{i}+6W_{i+1}-W_{i+2})(-W_{i-1}+3W_{i} -3W_{i+1}+W_{i+2}),\] \[\beta_{3}^{l} =\frac{1}{36}(-11W_{i}+18W_{i+1}-9W_{i+2}+2W_{i+3})^{2}+\frac{13 }{12}(2W_{i}-5W_{i+1}+4W_{i+2}+W_{i+3})^{2}\] \[+\frac{1043}{960}(-W_{i}+3W_{i+1}-3W_{i+2}+W_{i+3})^{2}\] \[+\frac{1}{432}(-11W_{i}+18W_{i+1}-9W_{i+2}+2W_{i+3})(-W_{i}+3W_{i +1}-3W_{i+2}+W_{i+3}).\]
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\[\alpha_{\xi(n)}\beta_{\xi(n)} = 1+\frac{bc}{a}\left(\frac{a}{b}\right)^{2\xi(n)}-\frac{bc^{3}}{a }\left(\frac{a}{b}\right)^{2\xi(n)}-\left(\frac{a}{b}\right)^{\xi(n)}bc\] \[+\left(b\left(\frac{a}{b}\right)^{\xi(n)}+\left(\frac{a}{b}\right) ^{2\xi(n)}\frac{bc}{a}\Delta\right)i\] \[+\left(\left(ab+2c\right)+\frac{bc}{a}\left(\frac{a}{b}\right)^{2 \xi(n)}\Delta+\frac{c^{2}}{a}\left(\frac{a}{b}\right)^{\xi(n)}\Delta\right)\epsilon\] \[+\left(\left(\frac{a}{b}\right)^{\xi(n)}\left(ab^{2}+3bc\right)- \frac{c}{a}\left(\frac{a}{b}\right)^{\xi(n)}\Delta\right)h\] \[= 1+\left(\frac{a}{b}\right)^{\xi(n)}bc\left(\frac{1}{a}\left(\frac{a}{ b}\right)^{\xi(n)}-\frac{c^{2}}{a}\left(\frac{a}{b}\right)^{\xi(n)}-1\right)\] \[+\left(\frac{a}{b}\right)^{\xi(n)}\left(\left(\frac{b}{a}\right)^{ \xi(n)}2+bi+\left(\frac{b}{a}\right)^{\xi(n)}\left(ab+2c\right)\epsilon+\left( b\left(ab+3c\right)\right)h\right)-2\] \[+\Delta\left(\frac{a}{b}\right)^{\xi(n)}\frac{c}{a}\left(\left( \frac{a}{b}\right)^{\xi(n)}bi+\left(b\left(\frac{a}{b}\right)^{\xi(n)}+c \right)\epsilon-h\right)\]
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\begin{table}
\begin{tabular}{c c c c c c c} \hline n & r & CMILP+MTZ & CMILP+MTZ+2Clq & CMILP+DL & CMILP+SC & CMILP+2C \\ \hline
3 & 5 & 0.43 & 0.78 & 0.46 & 0.48 & 0.50 \\
3 & 10 & 0.71 & 0.78 & 0.72 & 0.88 & 1.10 \\
3 & 15 & 2.13 & 1.88 & 1.88 & 2.32 & 2.63 \\
3 & 20 & 25.96 & 27.39 & 22.46 & 4.82 & 5.89 \\
3 & 25 & 227.72 & 255.11 & 385.28 & 9.27 & 11.65 \\
3 & 30 & 37.27 & 46.44 & 46.52 & 17.82 & 21.97 \\
3 & 35 & 3495.71 & 1363.08 & 1108.21 & 31.65 & 36.34 \\
3 & 40 & 120.75 & 127.70 & 154.93 & 53.32 & 59.40 \\
3 & 45 & 98.77 & 83.54 & 70.79 & 86.12 & 101.78 \\
3 & 50 & 1673.92 & 816.66 & 1425.62 & 131.76 & 160.49 \\ \hline
6 & 5 & 0.52 & 0.52 & 0.49 & 0.62 & 0.67 \\
6 & 10 & 0.97 & 1.06 & 0.97 & 1.62 & 2.07 \\
6 & 15 & 3.06 & 3.26 & 2.94 & 5.81 & 6.18 \\
6 & 20 & 53.23 & 48.17 & 43.15 & 14.95 & 15.38 \\
6 & 25 & 519.55 & 762.68 & 972.94 & 32.48 & 36.12 \\
6 & 30 & 77.10 & 72.25 & 74.25 & 66.70 & 72.07 \\ \hline \end{tabular}
\end{table}
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\[\Theta_{(0,0,0)} = \sum_{n=-\infty}^{\infty}(-1)^{n}Q_{\tau}^{\tfrac{3}{2}(n-\tfrac {1}{6})^{2}}Q_{1}^{n}Q_{2}^{-n+\tfrac{1}{3}}={\rm e}^{\tfrac{\pi{\rm i}}{6}}Q_ {1}^{\tfrac{1}{6}}Q_{2}^{\tfrac{1}{6}}\theta_{4}^{[-\tfrac{1}{6}]}(3\tau,t_{1} -t_{2})\,\] \[\Theta_{(-1,0,0)} = \sum_{n=-\infty}^{\infty}(-1)^{n}Q_{\tau}^{\tfrac{3}{2}(n-\tfrac {1}{6})^{2}}Q_{1}^{-2n+1}Q_{2}^{-n+\tfrac{1}{3}}={\rm e}^{\tfrac{\pi{\rm i}}{6} }Q_{1}^{\tfrac{2}{3}}Q_{2}^{\tfrac{1}{6}}\theta_{4}^{[-\tfrac{1}{6}]}(3\tau,- 2t_{1}-t_{2})\,\] \[\Theta_{(0,1,0)} = \sum_{n=-\infty}^{\infty}(-1)^{n}Q_{\tau}^{\tfrac{3}{2}(n-\tfrac {1}{6})^{2}}Q_{1}^{n}Q_{2}^{2n+\tfrac{1}{3}}={\rm e}^{\tfrac{\pi{\rm i}}{6}}Q _{1}^{\tfrac{1}{6}}Q_{2}^{\tfrac{3}{2}}\theta_{4}^{[-\tfrac{1}{6}]}(3\tau,t_{ 1}+2t_{2})\,\]
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\[\begin{array}{ll}\Gamma_{m}^{+}(1_{1})\cap\Gamma_{m}^{+}(1_{0})=\emptyset,& \Gamma_{m}^{-}(1_{1})\cap\Gamma_{m}^{+}(1_{0})=\emptyset,\\ \Gamma_{m}^{+}(x_{m-1})\cap\Gamma_{m}^{+}(1_{0})=\{x_{0}\},&\Gamma_{m}^{-}(x_{ m-1})\cap\Gamma_{m}^{+}(1_{0})=\emptyset,\\ \Gamma_{m}^{+}(x_{0}^{-1})\cap\Gamma_{m}^{+}(1_{1})=\emptyset,&\Gamma_{m}^{-}( x_{0}^{-1})\cap\Gamma_{m}^{+}(1_{1})=\emptyset,\\ \Gamma_{m}^{+}(x_{1+1})\cap\Gamma_{m}^{+}(1_{i})=\{x_{i}\}\ (i\neq 0),&\Gamma_{m}^{-}(x_{ 1+1})\cap\Gamma_{m}^{+}(1_{i})=\emptyset\ (i\neq 0),\\ \Gamma_{m}^{+}(x_{i-1})\cap\Gamma_{m}^{+}(1_{i})=\{x_{i}\}\ (i\neq 0),1&\Gamma_{m}^{-}( x_{i-1})\cap\Gamma_{m}^{+}(1_{i})=\emptyset\ (i\neq 0,1).\end{array}\]
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\[\hat{\mathcal{L}}_{l} \coloneqq\frac{1}{2}\|\hat{v}_{\mathbb{E}_{l}}\|_{2}^{2}+\frac{1}{2} \|\hat{v}_{\mathbb{T}_{l}}\|_{2}^{2}-\sigma_{\mathbb{T}_{l}}\mu_{\mathbb{T}_{l} }\sum\log(\hat{w}_{\mathbb{T}_{l}})-\sigma_{\mathcal{I}_{\mathcal{L},l-1}}\mu_{ \mathcal{I}_{\mathcal{L},l-1}}\sum\log(\hat{w}_{\mathcal{I}_{\mathcal{L},l-1}})\] \[+\lambda_{\mathbb{E}_{l}}^{T}(A_{\mathbb{E}_{l}}\Delta x-b_{ \mathbb{E}_{l}}-\hat{v}_{\mathbb{E}_{l}})+\lambda_{\mathbb{I}_{l}}^{T}(A_{ \mathbb{I}_{l}}\Delta x-b_{\mathbb{I}_{l}}-\hat{v}_{\mathbb{T}_{l}}-\hat{w}_{ \mathbb{T}_{l}})\] \[+\lambda_{\mathcal{A}_{\mathcal{U},l-1}}^{T}(A_{\mathcal{A}_{ \mathcal{U},l-1}}\Delta x-b_{\mathcal{A}_{\mathcal{U},l-1}}-\hat{v}_{ \mathcal{A}_{\mathcal{U},l-1}}^{*})+\lambda_{\mathcal{I}_{\mathcal{U},l-1}}^{ T}(A_{\mathcal{L}_{\mathcal{U},l-1}}\Delta x-b_{\mathcal{I}_{\mathcal{U},l-1}}- \hat{w}_{\mathcal{I}_{\mathcal{L},l-1}})\]
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\[I_{q,t}^{2}\lesssim \sum_{k}\theta(2^{-q}k)^{2}\sum_{k_{1},k_{2},k_{3},k_{1}^{\prime}, k_{2}^{\prime},k_{3}^{\prime}\neq 0:k_{123}=k_{123}^{\prime}=k}\] \[\times(1_{k_{1}=k_{1}^{\prime},k_{2}=k_{2}^{\prime},k_{3}=k_{3}^{ \prime}}+1_{k_{1}=k_{1}^{\prime},k_{2}=k_{3}^{\prime},k_{3}=k_{2}^{\prime}}) \int_{[0,t]^{2}}e^{-|k_{12}|^{2}(t-s)-|k_{12}^{\prime}|^{2}(t-\bar{s})}\] \[\qquad\times|k_{12}||k_{12}^{\prime}|\left(\frac{\sum_{i=1}^{3}| \epsilon k_{i}|^{\eta}}{\prod_{i=1}^{3}|k_{i}|^{2}}+\frac{\sum_{i=1}^{3}| \epsilon k_{i}^{\prime}|^{\eta}}{\prod_{i=1}^{3}|k_{i}^{\prime}|^{2}}\right)dsd \bar{s}\] \[\lesssim \epsilon^{\eta}t^{\eta}\sum_{k}\theta(2^{-q}k)^{2}\sum_{k_{1},k_{2},k_ {3}\neq 0:k_{123}=k}\] \[\times[|k_{12}|^{-2+2\eta}\frac{\sum_{i=1}^{3}|k_{i}|^{\eta}}{\prod_ {i=1}^{3}|k_{i}|^{2}}+|k_{12}|^{-1+\eta}|k_{13}|^{-1+\eta}\frac{\sum_{i=1}^{3} |k_{i}|^{\eta}}{\prod_{i=1}^{3}|k_{i}|^{2}}]\lesssim\epsilon^{\eta}t^{\eta}2^{ q(1+3\eta)}\]
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\[M_{2} =\int_{\Omega}[\boldsymbol{u}^{m}\cdot\nabla(\boldsymbol{v}^{n}- \boldsymbol{v}^{m})]\cdot(\boldsymbol{u}^{n}-\boldsymbol{u}^{m})\] \[=-\int_{\Omega}[\boldsymbol{u}^{m}\cdot\nabla(\boldsymbol{u}^{n} -\boldsymbol{u}^{m})]\cdot(\boldsymbol{v}^{n}-\boldsymbol{v}^{m})\] \[=-\int_{\Omega}[\boldsymbol{u}^{m}\cdot\nabla(\boldsymbol{u}^{n} -\boldsymbol{u}^{m})]\cdot(\boldsymbol{u}^{n}-\boldsymbol{u}^{m})+\int_{ \Omega}[\boldsymbol{u}^{m}\cdot\nabla(\boldsymbol{u}^{n}-\boldsymbol{u}^{m})] \cdot(\Delta\boldsymbol{u}^{n}-\Delta\boldsymbol{u}^{m})\] \[=-\int_{\Omega}[\boldsymbol{u}^{m}\cdot\nabla(\boldsymbol{u}^{n} -\boldsymbol{u}^{m})]\cdot(\boldsymbol{u}^{n}-\boldsymbol{u}^{m})-\int_{ \Omega}[\partial_{i}\boldsymbol{u}_{j}^{m}\cdot\partial_{j}(\boldsymbol{u} _{k}^{n}-\boldsymbol{u}_{k}^{m})]\cdot\partial_{i}(\boldsymbol{u}_{k}^{n}- \boldsymbol{u}_{k}^{m}),\]
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\[Z(z_{\text{out}},x_{\text{in}})=\int\mathcal{D}z_{\text{fl}} \;\mathcal{D}x_{\text{fl}}\;e^{\frac{i}{\hbar}S^{f}\left(z_{\text{out}}^{-}+z _{\text{fl}},\widetilde{x_{\text{in}}}+x_{\text{fl}}\right)}\\ =\int\mathcal{D}z_{\text{fl}}\;\mathcal{D}x_{\text{fl}}\;e^{\frac {i}{\hbar}\left(i\int_{I}z_{\text{fl}}\;dx_{\text{fl}}+\left(-ix_{\text{fl}}(1) z_{\text{out}}+\frac{i}{2}z_{\text{out}}^{2}\right)+\left(-iz_{\text{fl}}(0)\;x_{ \text{in}}+\frac{i}{2}x_{\text{in}}^{2}\right)\right)}\\ =e^{-\frac{1}{\hbar}\left(\frac{z_{\text{out}}^{2}}{4}-z_{\text {out}}x_{\text{in}}+\frac{x_{\text{in}}^{2}}{2}\right)}.\]
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\[\bar{I}_{h}^{n}(t) =\bar{I}_{h}^{n}(0)+\bar{N}_{1}^{n}\left(\beta_{h}^{n}\int_{0}^{t} \bar{I}_{v}^{n}(u)\bar{S}_{h}^{n}(u)\mathrm{d}u\right)-\bar{N}_{2}^{n}\left( \gamma_{h}^{n}\int_{0}^{t}\bar{I}_{h}^{n}(u)\mathrm{d}u\right),\] \[\bar{S}_{v}^{n}(t) =\bar{S}_{v}^{n}(0)+\bar{N}_{3}^{n}\left(\gamma_{v}^{n}\int_{0}^{t }\bar{S}_{v}^{n}(u)+\bar{I}_{v}^{n}(u)\mathrm{d}u\right)-\bar{N}_{4}^{n}\left( \beta_{v}^{n}\int_{0}^{t}\bar{I}_{h}^{n}(u)\bar{S}_{v}^{n}(u)\mathrm{d}u\right)\] \[\quad-\bar{N}_{5}^{n}\left(\gamma_{v}^{n}\int_{0}^{t}\bar{S}_{v}^ {n}(u)\mathrm{d}u\right),\] \[\bar{I}_{v}^{n}(t) =\bar{I}_{v}^{n}(0)+\bar{N}_{4}^{n}\left(\beta_{v}^{n}\int_{0}^{t }\bar{I}_{h}^{n}(u)\bar{S}_{v}^{n}(u)\mathrm{d}u\right)-\bar{N}_{6}^{n}\left( \gamma_{v}^{n}\int_{0}^{t}\bar{I}_{v}^{n}(u)\mathrm{d}u\right).\]
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\[-\int_{0}^{T^{\prime}}\int_{\Omega}\mathrm{n}_{\varepsilon}\cdot \partial_{t}\xi\left|\nabla\psi_{\varepsilon}\right|\mathrm{d}x\,\mathrm{d}t =\int_{0}^{T^{\prime}}\int_{\Omega}(\nabla\cdot\xi)(B\cdot\mathrm{ n}_{\varepsilon})\left|\nabla\psi_{\varepsilon}\right|\mathrm{d}x\,\mathrm{d}t\] \[\quad-\int_{0}^{T^{\prime}}\int_{\Omega}(\mathrm{Id}-\mathrm{n}_ {\varepsilon}\otimes\mathrm{n}_{\varepsilon}):\nabla B\left|\nabla\psi_{ \varepsilon}\right|\mathrm{d}x\,\mathrm{d}t\] \[\quad-\int_{0}^{T^{\prime}}\int_{\partial\Omega}\psi_{\varepsilon }(\mathrm{n}_{\partial\Omega}\cdot\xi)(\nabla^{\tan}\cdot B)\,\mathrm{d} \mathcal{H}^{d-1}\,\mathrm{d}t\] \[\quad-\int_{0}^{T^{\prime}}\int_{\partial\Omega}\psi_{\varepsilon }(B\cdot\nabla)(\xi\cdot\mathrm{n}_{\partial\Omega})\,\mathrm{d}\mathcal{H}^{ d-1}\,\mathrm{d}t\] \[\quad-\int_{0}^{T^{\prime}}\int_{\Omega}(\mathrm{n}_{\varepsilon} -\xi)\cdot\left(\partial_{t}\xi+(B\cdot\nabla)\xi+(\nabla B)^{\mathsf{T}}\xi \right)\left|\nabla\psi_{\varepsilon}\right|\mathrm{d}x\,\mathrm{d}t\] \[\quad-\int_{0}^{T^{\prime}}\int_{\Omega}\xi\cdot\left(\partial_{ t}\xi+(B\cdot\nabla)\xi\right)\left|\nabla\psi_{\varepsilon}\right|\mathrm{d}x\, \mathrm{d}t\] \[\quad-\int_{0}^{T^{\prime}}\int_{\Omega}(\mathrm{n}_{\varepsilon} -\xi)\otimes(\mathrm{n}_{\varepsilon}-\xi):\nabla B\left|\nabla\psi_{ \varepsilon}\right|\mathrm{d}x\,\mathrm{d}t\] \[\quad-\int_{0}^{T^{\prime}}\int_{\Omega}(\mathrm{n}_{\varepsilon} \cdot\xi-1)(\nabla\cdot B)\left|\nabla\psi_{\varepsilon}\right|\mathrm{d}x\, \mathrm{d}t.\]
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\[2\mathcal{D}_{\text{H}}^{2}\big{(}Q_{X,Y|\widetilde{Z}=m}, \widetilde{Q}_{X,Y|\widetilde{Z}=m}\big{)}\ =\ \int\int\big{(}\sqrt{q_{XY|\widetilde{Z}}}-\sqrt{q_{X \cdot|\widetilde{Z}}q_{\cdot Y|\widetilde{Z}}}\big{)}^{2}d\mu_{X}d\mu_{Y}\] \[= \int\int\frac{\big{(}q_{XY|\widetilde{Z}}-q_{X\cdot|\widetilde{Z}}q_{ Y|\widetilde{Z}}\big{)}^{2}}{(\sqrt{q_{XY|\widetilde{Z}}}+\sqrt{q_{X\cdot| \widetilde{Z}}q_{Y|\widetilde{Z}}})^{2}}d\mu_{X}d\mu_{Y}\] \[\leq \int\int\frac{\big{(}q_{XY|\widetilde{Z}}-q_{X\cdot|\widetilde{Z}} q_{Y|\widetilde{Z}}\big{)}^{2}}{q_{XY|\widetilde{Z}}+q_{X\cdot|\widetilde{Z}}q_{Y| \widetilde{Z}}}d\mu_{X}d\mu_{Y}\] \[\leq 3\int\int\frac{\big{(}q_{XY|\widetilde{Z}}-q_{X\cdot|\widetilde{ Z}}q_{Y|\widetilde{Z}}\big{)}^{2}}{q_{XY|\widetilde{Z}}+3q_{X\cdot|\widetilde{Z}}q_{Y| \widetilde{Z}}}d\mu_{X}d\mu_{Y}.\]
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\[\log T\int_{\theta_{0}}^{\theta_{-\eta}}1-\frac{\sigma(1+2\eta)} {2\eta}+\frac{\sigma+\eta}{2\eta}d\theta+\log\log T\int_{\theta_{0}}^{\theta_ {-\eta}}\frac{\sigma+\eta}{\eta}c_{2}d\theta\] \[+\int_{\theta_{0}}^{\theta_{-\eta}}-\frac{\sigma}{\eta}\log\frac{ 1+\eta}{c_{1}(2\pi)^{\eta}}+\log\frac{c_{1}}{\sqrt{2\pi}}d\theta+\frac{1}{2T} \int_{\theta_{0}}^{\theta_{-\eta}}L_{-1}^{\star}(\theta)d\theta\] \[+\frac{1}{T}\int_{\theta_{0}}^{\theta_{-\eta}}\left(-\frac{ \sigma(1+2\eta)}{4\eta}+\frac{\sigma+\eta}{4\eta}\right)L_{Q_{4}}^{\star}( \theta)d\theta+\frac{1}{2T\log T}\int_{\theta_{0}}^{\theta_{-\eta}}\frac{ \sigma+\eta}{\eta}c_{2}L_{Q_{4}}^{\star}(\theta)d\theta.\]
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\[w_{1}^{\left(2\right)}(z)=\frac{d}{dz}\ln\left(\frac{\tau_{0}^{ \left(2\right)}}{\tau_{1}^{\left(2\right)}}\right)=-\frac{1}{z},\quad\alpha_{2 }=1;\quad w_{2}^{\left(2\right)}(z)=\frac{d}{dz}\ln\left(\frac{\tau_{1}^{\left( 2\right)}}{\tau_{2}^{\left(2\right)}}\right)=\frac{4t_{1}-2z^{3}}{4t_{1}z+z^{ 4}},\quad\alpha_{2}=2;\] \[w_{3}^{\left(2\right)}(z)=\frac{d}{dz}\ln\left(\frac{\tau_{2}^{ \left(2\right)}}{\tau_{3}^{\left(2\right)}}\right)=\frac{3\left(160t_{1}^{2}z ^{2}+8t_{1}\left(z^{5}-24\right)+\left(z^{5}+96\right)z^{3}\right)}{\left(4t_ {1}+z^{3}\right)\left(80t_{1}^{2}-20t_{1}z^{3}-z^{6}+144z\right)},\quad\alpha_{ 2}=3.\]
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\[H(S|A)_{\{\Pi^{A}_{j^{\prime}},\Pi^{S}_{i^{\prime}}\}} = -\sum_{i^{\prime},j^{\prime}}P_{i^{\prime}j^{\prime}}log(\frac{P _{i^{\prime}j^{\prime}}}{P_{j^{\prime}}})\] \[= -\sum_{i^{\prime},j^{\prime}}(\sum_{i,j}P_{i}P^{i}_{j}P(n_{ij},n_ {i^{\prime}})P(k_{ij},k_{j^{\prime}}))log(\frac{\sum_{i,j}P_{i}P^{i}_{j}P(n_{ ij},n_{i^{\prime}})P(k_{ij},k_{j^{\prime}})}{\sum_{i,j}P_{i}P^{i}_{j}P(k_{ij},k_{j^{ \prime}})})\] \[= -\sum_{i^{\prime},j^{\prime}}(\sum_{i,j,j^{\prime\prime}}P_{i}P^ {i}_{j}P(n_{ij},n_{i^{\prime}})P(k_{ij},\bar{k}_{j^{\prime\prime}})P(\bar{k}_ {j^{\prime\prime}},k_{j^{\prime}}))log(\frac{\sum_{i,j,j^{\prime\prime}}P_{i}P ^{i}_{j}P(n_{ij},n_{i^{\prime}})P(k_{ij},\bar{k}_{j^{\prime\prime}})P(\bar{k}_ {j^{\prime\prime}},k_{j^{\prime}})}{\sum_{i,j,j^{\prime\prime}}P^{i}_{j}P(k_{ ij},\bar{k}_{j^{\prime\prime}})P(\bar{k}_{j^{\prime\prime}},k_{j^{\prime}})})\] \[\geq -\sum_{i^{\prime},j^{\prime},j^{\prime\prime}}(\sum_{i,j}P_{i}P^ {i}_{j}P(n_{ij},n_{i^{\prime}})P(k_{ij},\bar{k}_{j^{\prime\prime}})P(\bar{k}_ {j^{\prime\prime}},k_{j^{\prime}}))log(\frac{\sum_{i,j}P_{i}P^{i}_{j}P(n_{ij}, n_{i^{\prime}})P(k_{ij},\bar{k}_{j^{\prime\prime}})P(\bar{k}_{j^{\prime\prime}},k_{j^{ \prime}})}{\sum_{i,j}P_{i}P^{i}_{j}P(k_{ij},\bar{k}_{j^{\prime\prime}})P(\bar{k }_{j^{\prime\prime}},k_{j^{\prime}})})\] \[\geq -\sum_{i^{\prime},j^{\prime\prime}}(\sum_{i,j}P_{i}P^{i}_{j}P(n_{ ij},n_{i^{\prime}})P(k_{ij},\bar{k}_{j^{\prime\prime}}))log(\frac{\sum_{i,j}P_{i}P ^{i}_{j}P(n_{ij},n_{i^{\prime}})P(k_{ij},\bar{k}_{j^{\prime\prime}})}{\sum_{i, j}P_{i}P^{i}_{j}P(k_{ij},\bar{k}_{j^{\prime\prime}})})\] \[\geq H(S|A)_{\{\bar{\Pi}^{A}_{j^{\prime\prime}},\Pi^{S}_{i^{ \prime}}\}}\]
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\[\begin{split}&\int_{\Omega}z_{\varepsilon}(\cdot,T)+c_{1}\int_{0}^{T }\int_{\Omega}z_{\varepsilon}|\nabla w_{\varepsilon}|^{2}\\ &\quad+\frac{4(p+1)}{p}\int_{0}^{T}\int_{\Omega}\left|\nabla z_{ \varepsilon}^{\frac{1}{2}}+\frac{2k+p(p+1)\frac{u_{\varepsilon}}{u_{ \varepsilon}+1}}{2\sqrt{p(p+1)}}z_{\varepsilon}^{\frac{1}{2}}\nabla w_{ \varepsilon}\right|^{2}\\ &\leq\int_{\Omega}(u_{0\varepsilon}+1)^{-p}e^{-kw_{0\varepsilon }}-p\int_{0}^{T}\int_{\Omega}\frac{u_{\varepsilon}z_{\varepsilon}}{u_{ \varepsilon}+1}(1-u_{\varepsilon}^{\theta-1}-v_{\varepsilon})\\ &\quad+k\int_{0}^{T}\int_{\Omega}w_{\varepsilon}(u_{\varepsilon} +1)^{-p}e^{-kw_{\varepsilon}}-k\int_{0}^{T}\int_{\Omega}\frac{(u_{\varepsilon }+v_{\varepsilon})z_{\varepsilon}}{1+\varepsilon(u_{\varepsilon}+v_{ \varepsilon})}\\ &\leq\left(1+\frac{T}{e}\right)|\Omega|+p\int_{0}^{T}\int_{ \Omega}u_{\varepsilon}\left|1-u_{\varepsilon}^{\theta-1}-v_{\varepsilon} \right|\\ &\leq c_{2}(1+T),\end{split}\]
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\[Ec_{2}^{*}s_{2}E =\frac{1}{4}E^{1/2}(D-E)E^{1/2}+\frac{1}{4}E^{1/2}(D-E)D^{-1}E^{3/2}\] \[=\frac{1}{\pi}\int_{0}^{\infty}\frac{E^{1/2}}{E^{2}+t^{2}}D^{1/2} KD^{1/2}\frac{t^{2}}{D^{2}+t^{2}}E^{1/2}\mathrm{d}t\] \[\quad+\frac{1}{\pi}\int_{0}^{\infty}\frac{E^{1/2}}{E^{2}+t^{2}}D^{ 1/2}KD^{1/2}D^{-1}\frac{t^{2}}{D^{2}+t^{2}}E^{3/2}\mathrm{d}t\] \[=\frac{1}{\pi}\int_{0}^{\infty}\frac{1}{E^{2}+t^{2}}(E^{1/2}D^{-1/ 2})(DKD)\frac{t^{2}}{D^{2}+t^{2}}(D^{-1/2}E^{1/2})\mathrm{d}t\] \[\quad+\frac{1}{\pi}\int_{0}^{\infty}\frac{t}{E^{2}+t^{2}}(E^{1/2}D ^{-1/2})(DKD)D^{-3/2}E\frac{E^{1/2}t}{E^{2}+t^{2}}\mathrm{d}t\] \[\quad+\frac{2}{\pi}\int_{0}^{\infty}\frac{1}{E^{2}+t^{2}}(E^{1/2}D ^{-1/2})(DKD)D^{-3/2}\frac{t^{2}}{D^{2}+t^{2}}D^{1/2}KD^{1/2}\frac{E^{3/2}}{E ^{2}+t^{2}}\mathrm{d}t.\]
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\[s_{i}^{k+1}+e_{i}^{k+1}+x_{i}^{k+1}+r_{i}^{k+1}=s_{i}^{k}+e_{i}^ {k}+x_{i}^{k}+r_{i}^{k}+h\left(-\beta_{i}^{k}x_{i}^{k}s_{i}+\beta_{i}^{k}x_{i }^{k}s_{i}-\sigma_{i}^{k}e_{i}^{k}+\sigma_{i}^{k}e_{i}^{k}-\delta_{i}^{k}x_{i }^{k}+\delta_{i}^{k}x_{i}^{k}\right)\] \[+\frac{h}{N_{i}}\sum_{j\neq i}\left(F_{ij}^{k}(P(s_{i}|T_{i})+P(e _{i}|T_{i})+P(x_{i}|T_{i})-P_{ji}^{k}(P(s_{j}|T_{j})+P(e_{j}|T_{j})+P(x_{j}|T_{ j})+P(r_{j}|T_{j}))\right)\] \[=s_{i}^{k}+e_{i}^{k}+x_{i}^{k}+r_{i}^{k}+\frac{h}{N_{i}}\left(F^{ +}-F^{-}\right)=1,\]
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\[f_{deg}=\frac{2(1-36\beta)^{2}}{3}(52542675\beta^{3}+178185258 \beta^{2}-9896841\beta-47632)\beta^{4}\] \[-11(397050199920\beta^{5}-40790893923\beta^{4}+4055047758\beta^{3} -243771759\beta^{2}+6417616\beta\] \[-59392)\beta^{3}m_{1}+(5465578392450\beta^{6}+19309935720393\beta ^{5}-3995019640449\beta^{4}\] \[+327340481715\beta^{3}-13039336341\beta^{2}+250520816\beta-185753 6)\beta^{2}m_{1}^{2}+(2408448\] \[-15298708984020\beta^{6}-29436067209393\beta^{5}+7048034089254 \beta^{4}-562788423405\beta^{3}\] \[+20645100208\beta^{2}-359200768\beta)\beta m_{1}^{3}+3[(182185946 4150\beta^{6}+4980794507091\beta^{5}\] \[-1182106602432\beta^{4}+94244985459\beta^{3}-3452615664\beta^{2}+ 59975680\beta-401408)\] \[m_{1}^{4}(2\beta+m_{1}^{2}-2m_{1}+1)].\]
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\begin{table}
\begin{tabular}{|r|r r r r r r|} \hline \(r\)\(\sigma\) & \(0.1\) & \(0.3\) & \(1\) & \(3\) & \(10\) & \(30\) \\ \hline \(0.5\) & \(0.0863\) & \(0.0798\) & \(0.0502\) & \(0.0195\) & \(0.00582\) & \(0.00192\) \\ \(1\) & \(0.454\) & \(0.444\) & \(0.380\) & \(0.276\) & \(0.198\) & \(0.157\) \\ \(2.5\) & \(1.872\) & \(1.861\) & \(1.775\) & \(1.621\) & \(1.531\) & \(1.507\) \\ \(5\) & \(4.350\) & \(4.338\) & \(4.246\) & \(4.084\) & \(4.012\) & \(4.001\) \\ \(10\) & \(9.340\) & \(9.328\) & \(9.233\) & \(9.071\) & \(9.009\) & \(9.001\) \\ \hline \end{tabular}
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\[\left(\mathchoice{{\vbox{\hbox{$-$}}\kern-7.499886pt}}{{ \vbox{\hbox{$-$}}\kern-6.374903pt}}{{\vbox{\hbox{$-$}} \kern-4.499931pt}}{{\vbox{\hbox{$-$}}\kern-3.749943pt}}\! \int_{2B^{\prime}}|\nabla w_{\varepsilon}|^{q}\right)^{1/q} \leq C\left(\mathchoice{{\vbox{\hbox{$-$}}\kern-7.499886pt}}{{ \vbox{\hbox{$-$}}\kern-6.374903pt}}{{\vbox{\hbox{$-$}} \kern-4.499931pt}}{{\vbox{\hbox{$-$}}\kern-3.749943pt}}\! \int_{4B^{\prime}}|\nabla w_{\varepsilon}|^{2}\right)^{1/2}\] \[\leq C\left(\mathchoice{{\vbox{\hbox{$-$}}\kern-7.499886pt}}{{ \vbox{\hbox{$-$}}\kern-6.374903pt}}{{\vbox{\hbox{$-$}} \kern-4.499931pt}}{{\vbox{\hbox{$-$}}\kern-3.749943pt}}\! \int_{4B^{\prime}}|\nabla u_{\varepsilon}|^{2}\right)^{1/2}+C\left(\mathchoice{{ \vbox{\hbox{$-$}}\kern-7.499886pt}}{{\vbox{\hbox{$-$}} \kern-6.374903pt}}{{\vbox{\hbox{$-$}}\kern-4.499931pt}}{{ \vbox{\hbox{$-$}}\kern-3.749943pt}}\! \int_{4B^{\prime}}|\nabla v_{\varepsilon}|^{2}\right)^{1/2}\] \[\leq C\left(\mathchoice{{\vbox{\hbox{$-$}}\kern-7.499886pt}}{{ \vbox{\hbox{$-$}}\kern-6.374903pt}}{{\vbox{\hbox{$-$}} \kern-4.499931pt}}{{\vbox{\hbox{$-$}}\kern-3.749943pt}}\! \int_{4B^{\prime}}|F|^{2}\right)^{1/2}+C\left(\mathchoice{{\vbox{\hbox{$-$}} \kern-7.499886pt}}{{\vbox{\hbox{$-$}}\kern-6.374903pt}}{{\vbox{\hbox{$-$}} \kern-4.499931pt}}{{\vbox{\hbox{$-$}}\kern-3.749943pt}}\! \int_{4B^{\prime}}|f|^{2}\right)^{1/2}.\]
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\[\bigg{(}\frac{1}{P(t)}\bigg{)}^{(l)}=\bigg{(}\frac{1}{(1+\beta_{1 }t)(1+\beta_{2}t)\cdots(1+\beta_{n}t)}\bigg{)}^{(l)}\] \[=\sum_{\begin{subarray}{c}j_{1}+j_{2}+\cdots+j_{n}=l;\\ j_{1},j_{2},\ldots,j_{n}\geq 0\end{subarray}}\binom{l}{j_{1};j_{2};\ldots ;j_{n}}\bigg{(}\frac{1}{1+\beta_{1}t}\bigg{)}^{(j_{1})}\bigg{(}\frac{1}{1+ \beta_{2}t}\bigg{)}^{(j_{2})}\cdots\bigg{(}\frac{1}{1+\beta_{n}t}\bigg{)}^{(j _{n})}\] \[=\sum_{\begin{subarray}{c}j_{1}+j_{2}+\cdots+j_{n}=l;\\ j_{1},j_{2},\ldots,j_{n}\geq 0\end{subarray}}\binom{l}{j_{1};j_{2};\ldots ;j_{n}}\frac{(-1)^{j_{1}}j_{1}!\beta_{1}^{j_{1}}}{(1+\beta_{1}t)^{j_{1}+1}} \frac{(-1)^{j_{2}}j_{2}!\beta_{2}^{j_{2}}}{(1+\beta_{2}t)^{j_{2}+1}}\cdots \frac{(-1)^{j_{n}}j_{n}!\beta_{n}^{j_{n}}}{(1+\beta_{n}t)^{j_{n}+1}}\] \[=\frac{(-1)^{l}l!}{P(1)}\sum_{\begin{subarray}{c}j_{1}+j_{2}+ \cdots+j_{n}=l;\\ j_{1},j_{2},\ldots,j_{n}\geq 0\end{subarray}}\frac{\beta_{1}^{j_{1}}\beta_{2}^ {j_{2}}\cdots\beta_{n}^{j_{n}}}{(1+\beta_{1}t)^{j_{1}}(1+\beta_{2}t)^{j_{2}} \cdots(1+\beta_{n}t)^{j_{n}}},\]
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\[\mathcal{S}_{11}:= -2[D_{t}^{2}\zeta,\mathcal{H}_{\zeta}\frac{1}{\zeta_{\alpha}}+ \bar{\mathcal{H}}_{\zeta}\frac{1}{\bar{\zeta}_{\alpha}}]\partial_{\alpha}D_{t} \zeta+2[\tilde{D}_{t}^{2}\tilde{\zeta},\mathcal{H}_{\zeta}\frac{1}{\bar{\zeta }_{\alpha}}+\bar{\mathcal{H}}_{\tilde{\zeta}}\frac{1}{\bar{\zeta}_{\alpha}}] \partial_{\alpha}\tilde{D}_{t}\tilde{\zeta}\] \[+2[(D_{t}^{0})^{2}\omega,\mathcal{H}_{\omega}\frac{1}{\omega_{ \alpha}}+\bar{\mathcal{H}}_{\omega}\frac{1}{\bar{\omega}_{\alpha}}]\partial_{ \alpha}D_{t}^{0}\omega-2[(\tilde{D}_{t}^{0})^{2}\tilde{\omega},\mathcal{H}_{ \tilde{\omega}}\frac{1}{\bar{\omega}_{\alpha}}+\bar{\mathcal{H}}_{\tilde{ \omega}}\frac{1}{\bar{\overline{\omega}}_{\alpha}}]\partial_{\alpha}\tilde{D}_{t }^{0}\tilde{\omega}\]
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\begin{table}
\begin{tabular}{|l|l|l|} \hline degree & types & toric system \\ \hline \(9\) & \(\mathbb{P}^{2}\) & \(L,L,L\) \\ \hline \(8\) & \(\mathbb{F}_{0}\) & \(H_{1},H_{2},H_{1},H_{2}\) \\ \hline \(8\) & \(\mathbb{F}_{1}\) & \(L_{1},E_{1},L_{1},L\) \\ \hline \(8\) & \(\mathbb{F}_{2}\) & \(F,S-F,F,S-F\) (where \(F^{2}=0,S^{2}=2,FS=1\)) \\ \hline \(7\) & any & \(L_{1},E_{1},L_{12},E_{2},L_{2}\) \\ \hline \(6\) & any & \(L_{13},E_{1},L_{12},E_{2},L_{23},E_{3}\) \\ \hline \(5\) & \((\emptyset)\); \((A_{1})\); \((2A_{1})\); \((A_{2})\); \((A_{1}\)+\(A_{2})\) & \(L_{134},E_{4},E_{1}\)\(-\)\(E_{4},L_{12},E_{2},L_{23},E_{3}\) \\ \hline \(4\) & \((\emptyset)\); \((A_{1})\); \((2A_{1},9)\); \((2A_{1},8)\); \((A_{2})\); \((3A_{1})\); \((3A_{1})\); \((A_{1}\)+\(A_{2})\); \((3A_{3})\); \((4A_{1})\); \((2A_{1}\)+\(A_{3})\) & \\ \hline \(3\) & \((\emptyset)\); \((A_{1})\); \((2A_{1})\); \((A_{2})\); \((3A_{1})\); \((A_{1}\)+\(A_{2})\); \((4A_{1})\); \((2A_{1}\)+\(A_{2})\); \((2A_{1}\)+\(A_{2})\); \((3A_{2})\) & \(E_{2}\)\(-\)\(E_{4},L_{125},E_{5},E_{1}\)\(-\)\(E_{5},L_{136},E_{6},E_{3}\)\(-\)\(E_{6},L_{234},E_{4}\) \\ \((A_{1}\)+\(A_{2})\); \((4A_{1})\); \((2A_{1}\)+\(A_{2})\); \((2A_{1}\)+\(A_{2})\); \((3A_{2})\) & \\ \hline \end{tabular}
\end{table}
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\[E_{\mathcal{X}^{c}}\left(\sum_{j=1}^{M}(I-z_{j}A_{c})^{-1}B_{c}u _{j}\right)\] \[=\sum_{j=1}^{M}\sum_{k=1}^{M}\left\langle B_{c}^{*}(I-\overline{ z_{k}}A_{c}^{*})^{-1}(I-z_{j}A_{c})^{-1}B_{c}u_{j},u_{k}\right\rangle_{\mathcal{U}}\] \[=\sum_{j=1}^{M}\sum_{k=1}^{M}\left\langle B_{0}^{*}(I-\overline{ z_{k}}A_{0}^{*})^{-1}(I-z_{j}A_{0})^{-1}B_{0}u_{j},u_{k}\right\rangle_{\mathcal{U}}.\] \[=E_{\mathcal{X}_{0}}\left(\sum_{j=1}^{M}(I-z_{j}A_{0})^{-1}B_{0} u_{j}\right)\] \[=E_{\mathcal{X}_{0}}\left(R\left(\sum_{j=1}^{M}(I-z_{j}A_{c})^{-1 }B_{c}u_{j}\right)\right).\]
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\[\mathbb{E}\Big{[}\big{|}H\big{(}\widetilde{\pi}_{n}^{0},\ldots, \widetilde{\pi}_{n}^{(m-1)},\widehat{\pi}_{n}^{m}\big{)}-H\big{(}\widetilde{ \pi}_{n}^{0},\ldots,\widetilde{\pi}_{n}^{(m-1)},\overline{\pi}_{n}^{m}\big{)} \big{|}\Big{]}\] \[=\mathbb{E}\Big{[}\big{|}H\big{(}\widetilde{\pi}_{n}^{0},\ldots, \widetilde{\pi}_{n}^{(m-1)},\widehat{\pi}_{n}^{m}\big{)}-H\big{(}\widetilde{ \pi}_{n}^{0},\ldots,\widetilde{\pi}_{n}^{(m-1)},\overline{\pi}_{n}^{m}\big{)} \big{|}\mathbb{1}_{\mathcal{A}_{n,M}^{c}}\big{]}\] \[\leq\mathbb{E}\Big{[}\big{|}H\big{(}\pi_{n}^{0},\ldots,\pi_{n}^{( m-1)},\pi_{n}^{m}\big{)}-H\big{(}\pi_{n}^{0},\ldots,\pi_{n}^{(m-1)},\overline{ \pi}_{n}^{m}\big{)}\big{|}\mathbb{1}_{\mathcal{A}_{n,M}^{c}}\mathbb{1}_{ \mathcal{B}_{n,M}}\Big{]}+2||H||_{\infty}\mathbb{P}\big{[}\mathcal{B}_{n,M}^{c} \big{]}.\]
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\[f(x_{t}) \leq f(x_{sm})-\frac{\eta}{2}\sum_{j=sm+1}^{t}\|\nabla f(x_{j-1}) \|^{2}-\big{(}\frac{1}{2\eta}-\frac{L}{2}\big{)}\sum_{j=sm+1}^{t}\|x_{j}-x_{j -1}\|^{2}\] \[\qquad+\frac{\eta}{2}\sum_{k=sm+1}^{t-1}\frac{C^{\prime 2}L^{2} \sum_{j=sm+1}^{k}\|x_{j}-x_{j-1}\|^{2}}{b}\] \[\leq f(x_{sm})-\frac{\eta}{2}\sum_{j=sm+1}^{t}\|\nabla f(x_{j-1}) \|^{2}-\big{(}\frac{1}{2\eta}-\frac{L}{2}\big{)}\sum_{j=sm+1}^{t}\|x_{j}-x_{j -1}\|^{2}\] \[\qquad+\frac{\eta C^{\prime 2}L^{2}}{2b}\sum_{k=sm+1}^{t-1}\sum_{j =sm+1}^{k}\|x_{j}-x_{j-1}\|^{2}\] \[\leq f(x_{sm})-\frac{\eta}{2}\sum_{j=sm+1}^{t}\|\nabla f(x_{j-1}) \|^{2}-\big{(}\frac{1}{2\eta}-\frac{L}{2}\big{)}\sum_{j=sm+1}^{t}\|x_{j}-x_{j -1}\|^{2}\] \[\qquad+\frac{\eta C^{\prime 2}L^{2}(t-1-sm)}{2b}\sum_{j=sm+1}^{t} \|x_{j}-x_{j-1}\|^{2}\] \[\leq f(x_{sm})-\frac{\eta}{2}\sum_{j=sm+1}^{t}\|\nabla f(x_{j-1}) \|^{2}-\big{(}\frac{1}{2\eta}-\frac{L}{2}-\frac{\eta C^{\prime 2}L^{2}}{2} \big{)}\sum_{j=sm+1}^{t}\|x_{j}-x_{j-1}\|^{2}\]
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\[\left\{\begin{array}{l}x_{-\frac{1}{2}}^{0}<\ x_{\frac{1}{2}}^{0}<x_{\frac{ 2}{2}}^{0}<\cdots<x_{J-\frac{1}{2}}^{0},\\ 0<\underline{c^{0}}=\min\limits_{j\in\overline{0,J-1}}c_{j}^{0}\leq \overline{c^{0}}=\max\limits_{j\in\overline{0,J-1}}c_{j}^{0}<\infty,\\ 0<\underline{\rho^{0}}=\min\limits_{j\in\overline{0,J-1}}\rho_{j}^{0}\leq \overline{\rho^{0}}=\max\limits_{j\in\overline{0,J-1}}\rho_{j}^{0}<\infty,\\ \left\|\left(u_{j+\frac{1}{2}}\right)_{j\in\overline{0,J-1}}\right\|_{\dot{B }_{1}^{1}}^{2}=\sum\limits_{j=0}^{J-1}\left|u_{j+\frac{1}{2}}^{0}\right|^{2} \Delta x_{j}^{0}+\sum\limits_{j=0}^{J-1}\left|\frac{u_{j+\frac{1}{2}}^{0}-u_{ j-\frac{1}{2}}^{0}}{\Delta x_{j}}\right|^{2}\Delta x_{j}^{0}<\infty.\end{array}\right.\]
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\[\begin{cases}\frac{\partial x_{1}}{\partial t}=D_{h}\Delta_{y}x_{1}+b_{1}-\mu x _{1}-\beta_{2}x_{1}x_{4}-\beta_{1}x_{1}x_{2}+\phi x_{2},&y\in\mathbb{R},t>0,\\ \frac{\partial x_{2}}{\partial t}=D_{h}\Delta_{y}x_{2}+\beta_{1}x_{1}x_{2}+ \beta_{2}x_{1}x_{4}-(\mu+\phi)x_{2},&y\in\mathbb{R},\ t>0,\\ \frac{\partial x_{3}}{\partial t}=D_{v}\Delta_{y}x_{3}+b_{2}-\eta x_{3}- \beta x_{3}x_{2},&y\in\mathbb{R},\ t>0,\\ \frac{\partial x_{4}}{\partial t}=D_{v}\Delta_{y}x_{4}+\beta x_{3}x_{2}-\eta x _{4},&y\in\mathbb{R},\ t>0,\end{cases}\]
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\[B_{1} =\{1_{2},1_{3}\ ;\ 2_{2},2_{3}\ ;\ 3_{3},3_{1}\}, B_{2} =\{1_{1},1_{3}\ ;\ 2_{1},2_{3}\ ;\ 3_{2},3_{1}\}, B_{3} =\{1_{1},1_{2}\ ;\ 2_{1},2_{2}\ ;\ 3_{2},3_{3}\},\] \[B_{4} =\{1_{1},1_{3}\ ;\ 2_{1},2_{2}\ ;\ 3_{3},3_{1}\}, B_{5} =\{1_{2},1_{3}\ ;\ 2_{1},2_{3}\ ;\ 3_{2},3_{3}\}, B_{6} =\{1_{1},1_{2}\ ;\ 2_{2},2_{3}\ ;\ 3_{2},3_{1}\},\] \[B_{7} =\{1_{1},1_{2}\ ;\ 2_{1},2_{3}\ ;\ 3_{3},3_{1}\}, B_{8} =\{1_{2},1_{3}\ ;\ 2_{1},2_{2}\ ;\ 3_{2},3_{1}\}, B_{9} =\{1_{1},1_{3}\ ;\ 2_{2},2_{3}\ ;\ 3_{2},3_{3}\}.\]
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\[\begin{pmatrix}g_{1}(\mathbf{x}_{1})&g_{1}(\mathbf{x}_{2})&\ldots&g_{1}(\mathbf{ x}_{m})\\ g_{2}(\mathbf{x}_{1})&g_{2}(\mathbf{x}_{2})&\ldots&g_{2}(\mathbf{x}_{m})\\ \vdots&\vdots&\ldots&\vdots\\ g_{\ell}(\mathbf{x}_{1})&g_{\ell}(\mathbf{x}_{2})&\ldots&g_{\ell}(\mathbf{x}_{m} )\end{pmatrix}\approx\begin{pmatrix}\eta_{1}&\ldots&\eta_{r}\end{pmatrix} \begin{pmatrix}\zeta_{1}(\mathbf{x}_{1})&\zeta_{1}(\mathbf{x}_{2})&\ldots&\zeta_ {1}(\mathbf{x}_{m})\\ \zeta_{2}(\mathbf{x}_{1})&\zeta_{2}(\mathbf{x}_{2})&\ldots&\zeta_{2}(\mathbf{ x}_{m})\\ \vdots&\vdots&\ldots&\vdots\\ \zeta_{r}(\mathbf{x}_{1})&\zeta_{r}(\mathbf{x}_{2})&\ldots&\zeta_{r}(\mathbf{ x}_{m})\end{pmatrix}.\]
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\[\begin{split} 6\mathscr{F}_{\mathrm{IMS}}=&-(n_{\mathbf{2,1} }-4n_{\mathbf{2,4}}+8n_{\mathbf{3,1}}-8)\psi_{1}^{3}-8(n_{\mathbf{1,10}}+n_{ \mathbf{1,5}}-1)\phi_{1}^{3}-(8n_{\mathbf{1,10}}+n_{\mathbf{1,4}}-8)\phi_{2}^ {3}\\ &-3\phi_{1}^{2}\phi_{2}(4n_{\mathbf{1,10}}+n_{\mathbf{1,4}}-4n_{ \mathbf{1,5}}-4)+3(6n_{\mathbf{1,10}}+n_{\mathbf{1,4}}-2n_{\mathbf{1,5}}-6) \phi_{1}\phi_{2}^{2}\\ &+\psi_{1}\left(-12n_{\mathbf{2,4}}\phi_{1}^{2}+12n_{\mathbf{2,4 }}\phi_{1}\phi_{2}-6n_{\mathbf{2,4}}\phi_{2}^{2}\right)\end{split}\]
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\[\biggl{[}\frac{1-z_{1}/z_{3}}{1-z_{1}/(z_{3}t)}\cdot\frac{1-z_{2} /z_{3}}{1-z_{2}/(z_{3}t)}\biggr{]}_{3}\] \[= \biggl{[}(1+(t^{-1}-1)\sum_{k=1}^{\infty}t^{-(k-1)}z_{1}^{k}z_{3} ^{-k})\cdot(1+(t^{-1}-1)\sum_{l=1}^{\infty}t^{-(l-1)}z_{2}^{l}z_{3}^{-l}) \biggr{]}_{3}\] \[= [1]_{3}+(t^{-1}-1)\sum_{k=1}^{\infty}t^{-(k-1)}z_{1}^{k}\biggl{[} z_{3}^{-k}\biggr{]}_{3}+(t^{-1}-1)\sum_{l=1}^{\infty}t^{-(l-1)}z_{2}^{l}\biggl{[} z_{3}^{-l}\biggr{]}_{3}\] \[+ (t^{-1}-1)^{2}\sum_{k,l=1}^{\infty}t^{-(k-1)}z_{1}^{k}\cdot t^{-( l-1)}z_{2}^{l}\biggl{[}z_{3}^{-k-l}\biggr{]}_{3}\] \[= \{1\}_{1}+(t^{-1}-1)\sum_{k=1}^{\infty}t^{-(k-1)}z_{1}^{k}(-uQ)^{ k-1}\cdot Q\frac{(1-u)(1-uvQ)}{1-uQ}\] \[+ (t^{-1}-1)\sum_{l=1}^{\infty}t^{-(l-1)}z_{2}^{l}(-uQ)^{l-1}\cdot Q \frac{(1-u)(1-uvQ)}{1-uQ}\] \[+ (t^{-1}-1)^{2}\sum_{k,l=1}^{\infty}t^{-(k-1)}z_{1}^{k}\cdot t^{-( l-1)}z_{2}^{l}(-uQ)^{k+l-1}\cdot Q\frac{(1-u)(1-uvQ)}{1-uQ}.\]
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\[\int_{B_{R}\setminus G}|\nabla\tilde{n}|^{2}-\int_{B_{R}\setminus G }|\nabla n|^{2}\] \[=\int_{B_{R}\setminus G}\left(2\nabla n\cdot\nabla(\tilde{n}-n)+| \nabla(\tilde{n}-n)|^{2}\right)\] \[=2\int_{\partial B_{R}}\partial_{r}n\cdot(\tilde{n}-n)+\int_{B_{R }\setminus G}\left(-2\Delta n\cdot(\tilde{n}-n)+|\nabla(\tilde{n}-n)|^{2}\right)\] \[=2\int_{\partial B_{R}}\partial_{r}n\cdot(\tilde{n}-n)+\int_{B_{ R}\setminus G}2|\nabla n|^{2}n\cdot(\tilde{n}-n)+\int_{B_{R}\setminus G}| \nabla(\tilde{n}-n)|^{2}\] \[=2\int_{\partial B_{R}}\partial_{r}n\cdot(\tilde{n}-n)-\int_{B_{ R}\setminus G}|\nabla n|^{2}|\tilde{n}-n|^{2}+\int_{B_{R}\setminus G}|\nabla( \tilde{n}-n)|^{2}\]
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\[F_{3} \equiv \gamma_{14}e^{\frac{-(\eta_{1}-\eta_{4})+(\eta_{2}-\eta_{5})+( \eta_{3}-\eta_{6})}{2}}+\gamma_{25}e^{\frac{(\eta_{1}-\eta_{4})-(\eta_{2}-\eta _{5})+(\eta_{3}-\eta_{6})}{2}}+\gamma_{36}e^{\frac{(\eta_{1}-\eta_{4})+(\eta_{2 }-\eta_{5})-(\eta_{3}-\eta_{6})}{2}}+\] \[R_{14}e^{\frac{\eta_{1}-\eta_{4}-\theta_{14}}{2}}\cosh\left[\frac{ (\eta_{2}+\eta_{5})+(\eta_{3}+\eta_{6})-i\alpha_{14}}{2}\right]+\] \[R_{25}e^{\frac{\eta_{2}-\eta_{5}-\theta_{25}}{2}}\cosh\left[\frac{ (\eta_{1}+\eta_{4})+(\eta_{3}+\eta_{6})-i\alpha_{25}}{2}\right]+\] \[R_{36}e^{\frac{\eta_{3}-\eta_{6}-\theta_{36}}{2}}\cosh\left[\frac{ (\eta_{1}+\eta_{4})+(\eta_{2}+\eta_{5})-i\alpha_{36}}{2}\right]+\] \[A_{1}\,r_{14}e^{\frac{\eta_{1}-\eta_{4}}{2}}\cosh\left[\frac{( \eta_{2}+\eta_{5})-(\eta_{3}+\eta_{6})+i\beta_{14}}{2}\right]+\] \[A_{2}\,r_{25}e^{\frac{n_{2}-n_{5}}{2}}\cosh\big{[}\frac{(\eta_{1}+ \eta_{4})-(\eta_{3}+\eta_{6})+i\beta_{25}}{2}\big{]}+\] \[A_{3}\,r_{36}e^{\frac{n_{3}-n_{6}}{2}}\cosh\big{[}\frac{(\eta_{1} +\eta_{4})-(\eta_{2}+\eta_{5})+i\beta_{36}}{2}\big{]}.\]
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\[\rho_{n,x}\left(X_{T}^{u}-xe^{r(T-n)}\right)=-\int_{n}^{T}m_{s}(u)\mathrm{d}s +\begin{cases}\varrho^{\tilde{L}_{1}}\left\|u_{s}^{T}\sigma_{s}ve^{r(T-s)} \right\|_{\mathcal{L}^{\alpha}((n,T]\times\mathbb{S}^{d},\mathrm{d}s\times \tilde{\sigma}(\mathrm{d}v))}&\text{if $\alpha<2\ \&\ 3a$}\\ \varrho^{\tilde{L}_{1}}\left\|u_{s}^{1}\sigma_{s}^{11}e^{r(T-s)}\right\|_{ \mathcal{L}^{\alpha}((n,T],\mathrm{d}s)}&\text{if $\alpha<2\ \&\ 3b$}\\ \varrho^{\tilde{W}_{1}^{1}}\left\|u_{s}^{T}\sigma_{s}e^{r(T-s)}\right\|_{ \mathcal{L}^{2}((n,T],\mathrm{d}s;H_{s})}&\text{if $\alpha=2$}\end{cases}.\]
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\[(f\circ g)(x_{1},\ldots,x_{m+1})= (-1)^{(0)(m+1)}\frac{f(x_{1},\ldots,x_{m})}{x_{1}^{2}-x_{m}^{2}}\] \[\times\left(g(\bar{x}_{1},x_{m+1})+(-1)^{m+1}\frac{x_{m}}{x_{1}}g (\bar{x}_{1},x_{m+1})\right.\] \[\left.-g(x_{m},x_{m+1})+(-1)^{m+1}\frac{x_{m}}{x_{m+1}}g(x_{m},x_{ m+1})\right)\] \[+(-1)^{m+1}\frac{f(x_{2},\ldots,x_{m+1})}{x_{2}^{2}-x_{m+1}^{2}}\] \[\times\left(g(\bar{x}_{1},\bar{x}_{2})-(-1)^{m+1}\frac{x_{m+1}}{x _{2}}g(\bar{x}_{1},\bar{x}_{2})\right.\] \[\left.-g(\bar{x}_{1},x_{m+1})+(-1)^{m+1}\frac{x_{2}}{x_{m+1}}g( \bar{x}_{1},x_{m+1})\right)\]
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\[\Delta_{0}(x,s) =u_{\gamma}^{0}\int_{x}^{x+i\delta}\phi(z)(\mathrm{Im}\,z)^{s-1/2} \,\mathrm{d}s(z)-|j(\gamma,x)|^{2s-1}\int_{\gamma x}^{\gamma x+i\eta(x)}\phi(z )(\mathrm{Im}\,z)^{s-1/2}\,\mathrm{d}s(z)\] \[=u_{\gamma}^{0}\int_{x}^{x+i\delta}\phi(z)(\mathrm{Im}\,z)^{s-1/2} \,\mathrm{d}s(z)-|j(\gamma,x)|^{2s-1}\int_{x}^{\gamma^{-1}(\gamma x+i\eta(x))} \phi(\gamma z)(\mathrm{Im}\,\gamma z)^{s-1/2}\,\mathrm{d}s(z)\] \[=\int_{x}^{x+i\delta}u_{\gamma}(x)\phi(z)(\mathrm{Im}\,z)^{s-1/2 }\,\mathrm{d}s(z)-\int_{x}^{\gamma^{-1}(\gamma x+i\eta(x))}u_{\gamma}(z)\phi( z)(\mathrm{Im}\,z)^{s-1/2}\left|\frac{j(\gamma,x)}{j(\gamma,z)}\right|^{2s-1}\, \mathrm{d}s(z).\]
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\[\begin{array}{l}X_{13}\,+\,X_{24}\,-\,X_{14}\,=\,c_{13}\\ X_{24}\,+\,X_{3\bar{1}}\,-\,X_{\bar{1}1}\,=\,c_{24}\\ X_{2\bar{1}}\,+\,X_{14}\,-\,X_{24}\,-\,X_{1\bar{1}}\,=\,c_{13}\\ X_{2\bar{4}}\,+\,X_{1\bar{3}}\,-\,X_{2\bar{3}}\,=\,c_{2\bar{4}}\\ X_{2\bar{3}}\,+\,X_{3\bar{4}}\,-\,X_{2\bar{4}}\,-\,X_{3\bar{3}}\,=\,c_{2\bar{4 }}\\ X_{1\bar{1}}\,+\,X_{2\bar{2}}\,-\,X_{1\bar{2}}\,-\,X_{2\bar{1}}\,=\,c_{1\bar{ 1}}\\ X_{2\bar{2}}\,+\,X_{3\bar{3}}\,-\,X_{2\bar{3}}\,-\,X_{3\bar{2}}\,=\,c_{2\bar{2 }}\\ X_{4\bar{4}}\,+\,X_{3\bar{3}}\,-\,X_{3\bar{4}}\,-\,X_{4\bar{3}}\,=\,c_{3\bar{ 3}}\\ X_{1\bar{2}}\,+\,X_{2\bar{3}}\,-\,X_{1\bar{3}}\,-\,X_{2\bar{2}}\,=\,c_{1\bar{ 2}}\end{array}\]
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\[\begin{split} Nat_{\text{pro-$\mathcal{O}$}}(E_{m}(-),E_{n}(-))& =[\{\operatorname{Free}_{\mathcal{O}}(S^{n}\otimes DE_{\alpha})\}, \{\operatorname{Free}_{\mathcal{O}}(S^{m}\otimes DE_{\beta})\}]_{\text{pro-$ \mathcal{O}$}}\\ &=\pi_{0}\lim_{\beta}\operatorname{colim}_{\alpha}\operatorname{ Map}_{\mathcal{O}}(\operatorname{Free}_{\mathcal{O}}(S^{n}\otimes DE_{\alpha}), \operatorname{Free}_{\mathcal{O}}(S^{m}\otimes DE_{\beta}))\\ &=\pi_{0}\lim_{\beta}\operatorname{colim}_{\alpha}\operatorname{ Map}_{\mathcal{Sp}}(S^{n}\otimes DE_{\alpha},\operatorname{Free}_{\mathcal{O}}(S^{m} \otimes DE_{\beta}))\\ &=\pi_{0}\lim_{\beta}\operatorname{Map}_{\mathcal{Sp}}(S^{n},E \otimes\operatorname{Free}_{\mathcal{O}}(S^{m}\otimes DE_{\beta}))\\ &=\pi_{n}\lim_{\beta}E\otimes(\operatorname{Free}_{\mathcal{O}}( S^{m}\otimes DE_{\beta})).\end{split}\]
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\[-\frac{1}{4}\sum_{i=1}^{m}\left(\sum_{\alpha=1}^{n}\bar{\gamma} \left(\hat{A}^{\alpha}(e_{i})\cdot\gamma_{\alpha}\right)\right)^{2}|_{\Sigma^{ \pm}}\] \[= \frac{1}{4}\sum_{i=1}^{2}\sum_{\alpha,\beta=1}^{2}\bar{\gamma} \left(\hat{A}^{\alpha}(e_{i})\cdot\hat{A}^{\beta}(e_{i})\cdot\gamma_{\alpha} \cdot\gamma_{\beta}\right)\] \[= \frac{1}{4}\left|\hat{A}\right|^{2}+\frac{1}{4}\sum_{i=1}^{2}\sum _{\alpha\neq\beta}\bar{\gamma}\left(\hat{A}^{\alpha}(e_{i})\cdot\hat{A}^{ \beta}(e_{i})\cdot\gamma_{\alpha}\cdot\gamma_{\beta}\right)\] \[= \frac{1}{4}\left|\hat{A}\right|^{2}+\frac{1}{4}\sum_{i=1}^{2}\sum _{\alpha\neq\beta}\bar{\gamma}\left(A^{\alpha}(e_{i})\cdot A^{\beta}(e_{i}) \cdot\gamma_{\alpha}\cdot\gamma_{\beta}\right)\] \[= \frac{1}{4}\left|\hat{A}\right|^{2}+\frac{1}{4}\sum_{i=1}^{2}\sum _{\gamma\neq k}\sum_{\alpha\neq\beta}\left\langle A^{\alpha}(e_{i}),e_{j} \right\rangle\left\langle A^{\beta}(e_{i}),e_{k}\right\rangle\bar{\gamma} \left(e_{j}\cdot e_{k}\cdot\nu_{\alpha}\cdot\nu_{\beta}\right)\] \[= \frac{1}{4}\left|\hat{A}\right|^{2}+\frac{1}{2}\sum_{i=1}^{2} \left(\left\langle A^{1}(e_{i}),e_{1}\right\rangle\left\langle A^{2}(e_{i}),e_ {2}\right\rangle-\left\langle A^{1}(e_{i}),e_{2}\right\rangle\left\langle A^{ 2}(e_{i}),e_{1}\right\rangle\right)\bar{\gamma}\left(e_{1}\cdot e_{2}\cdot\nu _{1}\cdot\nu_{2}\right)\] \[= \frac{1}{4}\left|\hat{A}\right|^{2}\pm\frac{1}{2}\left(\kappa_{N} -\tilde{R}(e_{1},e_{2},\nu_{1},\nu_{2})\right).\]
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\[\mathcal{G}f\left(x\right) \approx\sum_{i=1}^{d}\left(-\frac{\mu^{i}\left(x\right)}{2h}+ \frac{A^{i,i}\left(x\right)}{2h^{2}}-\sum_{j=1,j\neq i}^{d}\frac{\left|A^{i,j} \left(x\right)\right|}{2h^{2}}\right)f\left(x-he_{i}\right)\] \[\qquad+\sum_{i=1}^{d}\left(-\frac{A^{i,i}\left(x\right)}{h^{2}}+ \sum_{j=1,j\neq i}^{d}\frac{\left|A^{i,j}\left(x\right)\right|}{h^{2}}\right) f\left(x\right)\] \[\qquad+\sum_{i=1}^{d}\left(\frac{\mu^{i}\left(x\right)}{2h}+ \frac{A^{i,i}\left(x\right)}{2h^{2}}-\sum_{j=1,j\neq i}^{d}\frac{\left|A^{i,j} \left(x\right)\right|}{2h^{2}}\right)f\left(x+he_{i}\right)\] \[\qquad+\sum_{i,j=1,j\neq i}^{d}\frac{\left(A^{i,j}\left(x\right) \right)^{+}}{4h^{2}}\left(f\left(x+he_{i}+he_{j}\right)+f\left(x-he_{i}-he_{j} \right)\right)\] \[\qquad+\sum_{i,j=1,j\neq i}^{d}\frac{\left(A^{i,j}\left(x\right) \right)^{-}}{4h^{2}}\left(f\left(x+he_{i}-he_{j}\right)+f\left(x-he_{i}+he_{j} \right)\right).\]
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\[\left(\frac{1}{N}\sum_{i,j}s_{ij}^{6}(\hat{\alpha}^{0}_{N})\right) \sqrt{N}(\hat{\rho}^{0}_{N}-\rho_{0})\] \[=\frac{1}{\sqrt{N}}\sum_{i,j}s_{ij}^{3}(\hat{\alpha}^{0}_{N})\left( \hat{c}^{3}_{ij}-3s_{ij}(\hat{\alpha}^{0}_{N})\sigma_{ij}^{2}(\hat{\tau}^{0}_{N} )\hat{c}^{0}_{N}-s_{ij}^{3}(\hat{\alpha}^{0}_{N})\rho_{0}\right)\] \[=O_{p}(1)+\frac{1}{\sqrt{N}}\sum_{i,j}s_{ij}^{3}(\hat{\alpha}^{0 }_{N})\left\{\left(\overline{e}_{ij}-\frac{1}{\sqrt{N}}\hat{H}_{ij}\right)^{3}-3s_ {ij}(\alpha_{0})\sigma_{ij}^{2}(\tau_{0})c_{0}-s_{ij}^{3}(\alpha_{0})\rho_{0}\right\}\] \[=O_{p}(1)+\frac{1}{\sqrt{N}}\sum_{i,j}s^{3}_{ij}(\alpha_{0})\left(\tilde{e}^{3}_{ ij}-3s_{ij}(\alpha_{0})\sigma^{2}_{ij}(\tau_{0})c_{0}-s^{3}_{ij}(\alpha_{0})\rho_{0} \right)=O_{p}(1).\]
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\[\frac{1}{n\tau}\log a_{n\tau}(f,K,D) \leq\frac{1}{\tau}\sum\limits_{j:\rho_{j}\geq 0}d_{j}\log( \left\lfloor e^{(\rho_{j}+\xi)\tau}\right\rfloor+1)+\frac{1}{n\tau}\sum_{i=0}^ {n\tau-1}f(\varphi(i,x^{0},u^{0}),u^{0}_{i})+\varepsilon\] \[\leq\frac{1}{\tau}\sum\limits_{j:\rho_{j}\geq 0}d_{j}\log(2e^{( \rho_{j}+\xi)\tau})+\frac{1}{\tau^{0}}\sum_{i=0}^{\tau^{0}-1}f(\varphi(i,x^{0},u^{0}),u^{0}_{i})+\varepsilon\] \[\leq\frac{d}{\tau}\log 2+\frac{1}{\tau}\sum\limits_{j:\rho_{j} \geq 0}d_{j}(\rho_{j}+\xi)\tau+\frac{1}{\tau^{0}}\sum_{i=0}^{\tau^{0}-1}f( \varphi(i,x^{0},u^{0}),u^{0}_{i})+\varepsilon\] \[\leq\varepsilon+d\xi+\sum\limits_{j:\rho_{j}\geq 0}d_{j}\rho_{j}+ \frac{1}{\tau^{0}}\sum_{i=0}^{\tau^{0}-1}f(\varphi(i,x^{0},u^{0}),u^{0}_{i})+\varepsilon\] \[<S_{0}+\frac{1}{\tau^{0}}\sum_{i=0}^{\tau^{0}-1}f(\varphi(i,x^{0},u^{0}),u^{0}_{i})+2\varepsilon.\]
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\[\hat{\Psi}_{1}=\begin{bmatrix}\ell_{0}(x_{0})\ell_{0}(y_{0})&\ell_{0}(x_{0})\ell _{1}(y_{0})&\ldots&\ell_{0}(x_{0})\ell_{n}(y_{0})\\ \ell_{1}(x_{0})\ell_{0}(y_{0})&\ell_{1}(x_{0})\ell_{1}(y_{0})&\ldots&\ell_{1}(x _{0})\ell_{n}(y_{0})\\ \vdots&&\vdots&\\ \ell_{m}(x_{0})\ell_{0}(y_{0})&\ell_{m}(x_{0})\ell_{1}(y_{0})&\ldots&\ell_{m}( x_{0})\ell_{n}(y_{0})\\ \vdots&&\vdots&\\ \ell_{0}(x_{N-1})\ell_{0}(y_{N-1})&\ell_{0}(x_{N-1})\ell_{1}(y_{N-1})&\ldots &\ell_{0}(x_{N-1})\ell_{n}(y_{N-1})\\ \ell_{1}(x_{N-1})\ell_{0}(y_{N-1})&\ell_{1}(x_{N-1})\ell_{1}(y_{N-1})&\ldots &\ell_{1}(x_{N-1})\ell_{n}(y_{N-1})\\ \vdots&&\vdots&\\ \ell_{m}(x_{N-1})\ell_{0}(y_{N-1})&\ell_{m}(x_{N-1})\ell_{1}(y_{N-1})&\ldots &\ell_{m}(x_{N-1})\ell_{n}(y_{N-1})\end{bmatrix}.\]
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\[\int_{0}^{t}\|\eta^{\mathfrak{h},\mathfrak{l}}(s)\|^{6}_{L^{6}_{ x}}\,\mathrm{d}s+\int_{0}^{t}\|\eta^{\mathfrak{h},\mathfrak{g}}(s)\|^{6}_{L^{6}_{ x}}\,\mathrm{d}s+\int_{0}^{t}\|\eta^{\mathfrak{h},\mathfrak{t}}(s)\|^{6}_{L^{6}_{ x}}\,\mathrm{d}s,\] \[\int_{0}^{t}\|u(s)\|^{6}_{L^{6}_{x}}\,\mathrm{d}s+\int_{0}^{t}\| \eta^{\mathfrak{h},\mathfrak{l}}(s)\|^{6}_{L^{6}_{x}}\,\mathrm{d}s+\int_{0}^{t }\|\zeta^{\mathfrak{h},\mathfrak{g},\mathfrak{l}}(s)\|^{6}_{L^{2}_{x}}\, \mathrm{d}s,\] \[\int_{0}^{t}\|u(s)\|^{6}_{L^{6}_{x}}\,\mathrm{d}s+\int_{0}^{t}\| \eta^{\mathfrak{h},\mathfrak{g}}(s)\|^{6}_{L^{6}_{x}}\,\mathrm{d}s+\int_{0}^{t }\|\zeta^{\mathfrak{h},\mathfrak{h},\mathfrak{l}}(s)\|^{6}_{L^{2}_{x}}\, \mathrm{d}s,\] \[\int_{0}^{t}\|u(s)\|^{6}_{L^{2}_{x}}\,\mathrm{d}s+\int_{0}^{t}\| \eta^{\mathfrak{h},\mathfrak{l}}(s)\|^{6}_{L^{2}_{x}}\,\mathrm{d}s+\int_{0}^{t }\|\zeta^{\mathfrak{h},\mathfrak{g},\mathfrak{l}}(s)\|^{6}_{L^{2}_{x}}\, \mathrm{d}s.\]
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\[\begin{split}&\mathcal{J}_{N}(\pi_{1:N-1|0},p_{1:N-1|0})+\mathbb{E}_{ \pi_{1:N-1|0}}\left[\mathcal{D}_{\text{KL}}\left(\pi_{N|N-1}||p_{N|N-1}\right) \right]-\mathbb{E}_{\pi_{N-1:N-1}}\left[\bar{r}_{N}(\mathbf{X}_{N-1})\right]\\ &=\mathcal{J}_{N}(\pi_{1:N-1|0},p_{1:N-1|0})+\mathbb{E}_{\pi_{N-1: N-1}}\left[\mathcal{D}_{\text{KL}}\left(\pi_{N|N-1}||p_{N|N-1}\right)\right]- \mathbb{E}_{\pi_{N-1:N-1}}\left[\bar{r}_{N}(\mathbf{X}_{N-1})\right]\\ &=\mathcal{J}_{N}(\pi_{1:N-1|0},p_{1:N-1|0})+\mathbb{E}_{\pi_{N-1: N-1}}\left[\mathcal{D}_{\text{KL}}\left(\pi_{N|N-1}||p_{N|N-1}\right)-\tilde{r}_{N}( \mathbf{X}_{N-1})\right].\end{split}\]
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\[\lim_{s\to 0}\frac{\mathcal{F}[\bm{n}+s\delta\bm{n}]-\mathcal{F}[\bm{n}] }{s}\] \[=\int_{\Omega}k_{1}(\nabla\cdot\bm{n})(\nabla\cdot\delta\bm{n})+k_ {2}(\bm{n}\cdot\nabla\times\bm{n})(\delta\bm{n}\cdot\nabla\times\bm{n}+\bm{n} \cdot\nabla\times\delta\bm{n})\] \[\qquad+k_{3}(\bm{n}\times\nabla\times\bm{n})\cdot(\delta\bm{n} \times\nabla\times\bm{n}+\bm{n}\times\nabla\times\delta\bm{n})dV\] \[=\int_{\Omega}\delta\bm{n}\cdot\bigg{[}-k_{1}\nabla(\nabla\cdot \bm{n})+k_{2}\Big{(}(\bm{n}\cdot\nabla\times\bm{n})(\nabla\times\bm{n})+ \nabla\times\big{(}(\bm{n}\cdot\nabla\times\bm{n})\big{)}\Big{)}\] \[\qquad+k_{3}\Big{(}(\nabla\times\bm{n})\times(\bm{n}\times\nabla \times\bm{n})+\nabla\times\big{(}(\bm{n}\times\nabla\times\bm{n})\times\bm{n }\big{)}\Big{)}\bigg{]}dV\] \[\qquad+k_{3}(\bm{n}\times\nabla\times\bm{n})\cdot\big{(}\bm{n} \times(\bm{\nu}\times\delta\bm{n})\big{)}dS,\]
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\[x\rhd(y\rhd z) = x\rhd(y\rhd z)+\sum_{j=1}^{k}\langle z,b_{j}\rangle f_{j}(\langle y,a_{j}\rangle)x\rhd z_{j}\] \[= x\rhd(y\rhd z)+\sum_{j=1}^{k}\langle y\rhd z,b_{j}\rangle f_{j}( \langle x,a_{j}\rangle)z_{j}+\sum_{j=1}^{k}\langle z,b_{j}\rangle f_{j}(\langle y,a_{j}\rangle)z_{j},\] \[= x\rhd(y\rhd z)+\sum_{j=1}^{k}\langle z,b_{j}\rangle f_{j}( \langle x,a_{j}\rangle)z_{j}+\sum_{j=1}^{k}\langle z,b_{j}\rangle f_{j}( \langle y,a_{j}\rangle)z_{j},\] \[(x\rhd y)\rhd(x\rhd z) = (x\rhd y)\rhd(x\rhd z)+\sum_{j=1}^{k}\langle z,b_{j}\rangle f_{j} (\langle x,a_{j}\rangle)z_{j}\] \[= (x\rhd y)\rhd(x\rhd z)+\sum_{j=1}^{k}\langle x\rhd z,b_{j}\rangle f _{j}(\langle x>y,a_{j}\rangle)z_{j}+\sum_{j=1}^{k}\langle z,b_{j}\rangle f_{j} (\langle x,a_{j}\rangle)z_{j}\] \[= (x\rhd y)\rhd(x\rhd z)+\sum_{j=1}^{k}\langle z,b_{j}\rangle f_{j} (\langle y,a_{j}\rangle)z_{j}+\sum_{j=1}^{k}\langle z,b_{j}\rangle f_{j}( \langle x,a_{j}\rangle)z_{j}.\]
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\[\operatorname{Tr}(P_{g}L_{X}L_{Y}P_{g})=\sum_{i<j}\left<e_{i}e_{j} ^{T}-e_{j}e_{i}^{T},g^{-1}Xg(e_{i}e_{j}^{T}-e_{j}e_{i}^{T})Y\right>+\sum_{i<j} \left<e_{i}e_{j}^{T}+e_{j}e_{i}^{T},g^{-1}Xg(e_{i}e_{j}^{T}+e_{j}e_{i}^{T})Y\right>\] \[+\sum_{i}2\left<e_{i}e_{i}^{T},g^{-1}Xge_{i}e_{i}^{T}\right>\] \[=\sum_{i<j}2\bigl{(}\left<e_{i}e_{j}^{T},g^{-1}Xge_{i}e_{j}^{T}Y \right>+\left<e_{j}e_{i}^{T},g^{-1}Xge_{j}e_{i}^{T}\right>\bigr{)}+\sum_{i}2 \left<e_{i}e_{i}^{T},g^{-1}Xge_{i}e_{i}^{T}\right>\] \[=\sum_{ij}2\left<e_{i}e_{j}^{T},g^{-1}Xge_{i}e_{j}^{T}\right>-2 \sum_{i}\left<e_{i}e_{i}^{T},g^{-1}Xge_{i}e_{i}^{T}\right>+2\sum_{i}\left<e_{ i}e_{i},g^{-1}Xge_{i}e_{i}^{T}\right>=\sum_{ij}\operatorname{Re}(g^{-1}Xg)_{ii}Y_{jj}\] \[=\operatorname{Re}\operatorname{Tr}X\operatorname{Tr}Y\]
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\[\mathbf{C}^{-1}=\begin{pmatrix}\frac{\tau}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2} }&0&\frac{-\tau_{1}}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}&\frac{-\tau_{2}}{ \tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}\\ 0&\frac{\tau}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}&\frac{\tau_{2}}{\tau^{2}-\tau_ {1}^{2}-\tau_{2}^{2}}&\frac{-\tau_{1}}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}\\ \frac{-\tau_{1}}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}&\frac{\tau_{2}}{\tau^{2}- \tau_{1}^{2}-\tau_{2}^{2}}&\frac{\tau}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}&0\\ \frac{-\tau_{2}}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}&\frac{-\tau_{1}}{\tau^{2}- \tau_{1}^{2}-\tau_{2}^{2}}&0&\frac{\tau}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}} \end{pmatrix}.\]
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\[H^{1,\,0}_{\bar{\partial}}(X,\mathbb{C}) = \bigg{\langle}[\alpha]_{\bar{\partial}},\,[\beta]_{\bar{\partial}},\,[\gamma]_{\bar{\partial}}\bigg{\rangle},\ \ \ H^{0,\,1}_{\bar{\partial}}(X,\mathbb{C})=\bigg{\langle}[\overline{\alpha}]_{ \bar{\partial}},\,[\overline{\beta}]_{\bar{\partial}}\bigg{\rangle}=\pi^{\star }H^{0,\,1}_{\bar{\partial}}(B,\mathbb{C}),\] \[H^{1,\,1}_{\bar{\partial}}(X,\,\mathbb{C}) = \bigg{\langle}[\alpha\wedge\overline{\alpha}]_{\bar{\partial}},\,[ \alpha\wedge\overline{\beta}]_{\bar{\partial}},\,[\beta\wedge\overline{\alpha}] _{\bar{\partial}},\,[\beta\wedge\overline{\beta}]_{\bar{\partial}},\,[\gamma \wedge\overline{\alpha}]_{\bar{\partial}},\,[\gamma\wedge\overline{\beta}]_{ \bar{\partial}}\bigg{\rangle},\] \[H^{3,\,0}_{\bar{\partial}}(X,\mathbb{C}) = \bigg{\langle}[\alpha\wedge\beta\wedge\gamma]_{\bar{\partial}} \bigg{\rangle},\ \ H^{0,\,3}_{\bar{\partial}}(X,\mathbb{C})=\bigg{\langle}[\overline{\alpha} \wedge\overline{\beta}\wedge\overline{\gamma}]_{\bar{\partial}}\bigg{\rangle},\] \[H^{2,\,1}_{\bar{\partial}}(X,\,\mathbb{C}) = \bigg{\langle}[\alpha\wedge\gamma\wedge\overline{\alpha}]_{\bar{ \partial}},\,[\alpha\wedge\gamma\wedge\overline{\beta}]_{\bar{\partial}},\,[ \beta\wedge\gamma\wedge\overline{\alpha}]_{\bar{\partial}},\,[\beta\wedge \gamma\wedge\overline{\beta}]_{\bar{\partial}}\bigg{\rangle}\oplus\bigg{\langle} [\alpha\wedge\beta\wedge\overline{\alpha}]_{\bar{\partial}},\,[\alpha\wedge \beta\wedge\overline{\beta}]_{\bar{\partial}}\bigg{\rangle}\] \[= [\gamma\wedge\pi^{\star}H^{1,\,1}_{\bar{\partial}}(B,\,\mathbb{C} )]_{\bar{\partial}}\oplus\pi^{\star}H^{2,\,1}_{\bar{\partial}}(B,\,\mathbb{C}),\] \[H^{1,\,2}_{\bar{\partial}}(X,\,\mathbb{C}) = \bigg{\langle}[\alpha\wedge\overline{\alpha}\wedge\overline{\gamma }]_{\bar{\partial}},\,[\beta\wedge\overline{\alpha}\wedge\overline{\gamma}]_{ \bar{\partial}},\,[\alpha\wedge\overline{\beta}\wedge\overline{\gamma}]_{\bar {\partial}},\,[\beta\wedge\overline{\beta}\wedge\overline{\gamma}]_{\bar{ \partial}}\bigg{\rangle}\oplus\bigg{\langle}[\gamma\wedge\overline{\alpha} \wedge\overline{\gamma}]_{\bar{\partial}},\,[\gamma\wedge\overline{\beta} \wedge\overline{\gamma}]_{\bar{\partial}}\bigg{\rangle}.\]
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\[(\bm{I}-\bm{R})^{\top}(\bm{I}-\bm{R}) =2\bm{I}-\bm{R}-\bm{R}^{\top}=\begin{pmatrix}2\bm{I}-\bm{R}_{11}- \bm{R}_{11}^{\top}&-\bm{R}_{12}-\bm{R}_{21}^{\top}\\ -\bm{R}_{21}-\bm{R}_{12}^{\top}&2\bm{I}-\bm{R}_{22}-\bm{R}_{22}^{\top}\end{pmatrix}\] \[=\begin{pmatrix}(\bm{I}-\bm{R}_{11}^{\top})(\bm{I}-\bm{R}_{11})+ \bm{I}-\bm{R}_{11}^{\top}\bm{R}_{11}&-\bm{R}_{12}-\bm{R}_{21}^{\top}\\ -\bm{R}_{21}-\bm{R}_{12}^{\top}&(\bm{I}-\bm{R}_{22}^{\top})(\bm{I}-\bm{R}_{22} )+\bm{I}-\bm{R}_{22}^{\top}\bm{R}_{22}\end{pmatrix}\] \[=\begin{pmatrix}(\bm{I}-\bm{R}_{11}^{\top})(\bm{I}-\bm{R}_{11})+ \bm{R}_{21}^{\top}\bm{R}_{21}&-\bm{R}_{12}-\bm{R}_{21}^{\top}\\ -\bm{R}_{21}-\bm{R}_{12}^{\top}&(\bm{I}-\bm{R}_{22}^{\top})(\bm{I}-\bm{R}_{22 })+\bm{R}_{12}^{\top}\bm{R}_{12}\end{pmatrix}.\]
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\[\sum_{k>\ell}|\mathbb{E}[\partial_{ij}^{2}\omega(x,X_{i})\partial _{ij}^{2}\omega(x,X_{j})]|\] \[=\sum_{k>\ell}|\mathbb{E}[\partial_{ij}^{2}\omega(x,X_{\ell}) \partial_{ij}^{2}\omega(x,X_{k})]|^{\frac{1}{\log(n)}}|\mathbb{E}[\partial_{ ij}^{2}\omega(x,X_{\ell})\partial_{ij}^{2}\omega(x,X_{k})]|^{1-\frac{1}{\log(n)}}\] \[\lesssim\sum_{k>\ell}\alpha_{k-\ell}^{\frac{1}{\log(n)}}(\sup_{y \in\mathcal{M}}|\partial_{ij}^{2}\omega(x,y)|)^{\frac{2}{\log(n)}}(\mathbb{E} [|\partial_{ij}^{2}\omega(x,X_{1})|^{2}])^{1-\frac{1}{\log(n)}}\] \[\stackrel{{\eqref{eq:1.14},\eqref{eq:1.15}}}{{ \lesssim}}\delta^{-4(1-\frac{1}{\log(n)})}t^{-1-\frac{1}{\log(n)}}\sum_{k=0}^{ \infty}\exp(-b\frac{k^{\eta}}{\log(n)})\] \[\lesssim\delta^{\frac{4}{\log(n)}}t^{-\frac{1}{\log(n)}}\log^{ \frac{1}{\eta}}(n)\delta^{-4}t^{-1},\]
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\[\sup_{\gamma<xh_{0}}J_{1}(K,\gamma)=\begin{cases}-(K-xh_{0})^{2},&\text{ if }K>xh_{0};\\ +\infty,&\text{ if }K<xh_{0}\text{ and }P_{1,0}=h_{0}^{2};\\ \frac{P_{1,0}(K-xh_{0})^{2}}{h_{0}^{2}-P_{1,0}},&\text{ if }K<xh_{0}\text{ and }P_{1,0}<h_{0}^{2};\\ 0,&\text{ if }K=xh_{0},\end{cases}\] \[\sup_{\gamma\geqslant xh_{0}}J_{2}(K,\gamma)=\begin{cases}+\infty,&\text{ if }K>xh_{0}\text{ and }P_{2,0}=h_{0}^{2};\\ \frac{P_{2,0}(K-xh_{0})^{2}}{h_{0}^{2}-P_{2,0}},&\text{ if }K>xh_{0}\text{ and }P_{2,0}<h_{0}^{2};\\ -(K-xh_{0})^{2},&\text{ if }K<xh_{0};\\ 0,&\text{ if }K=xh_{0}.\end{cases}\]
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\[-2y_{0}+x_{0}(x_{0}+s_{6}),\ -2y_{2}+x_{2}(s_{4}+x_{1}s_{7}),\] \[-y_{1}-y_{2}+(2x_{0}+s_{6})y_{0}+x_{0}s_{2},\ x_{3},...,\ x_{n}, \ x_{0}(x_{1}+s_{5}),\ x_{0}x_{2},\] \[(x_{1}+s_{5})^{2}+x_{2}^{2},\ 3x_{1}^{2}x_{2}-(x_{1}+s_{5})(s_{4}+x_ {1}s_{7})+x_{2}s_{3}+x_{2}^{2}s_{7},\] \[-\frac{1}{2}x_{0}(x_{0}+s_{6})(3x_{0}+s_{6})-2x_{0}s_{2}+3x_{1}^{ 2}(x_{1}+s_{5})\] \[\ \ \ \ \ \ \ +x_{2}(s_{4}+x_{1}s_{7})+(x_{1}+s_{5})s_{3}+(x_{1}+s_{5})x_ {2}s_{7}.\]
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\[\int_{t+1}^{\infty}|\lambda^{\prime\prime}(s)||\partial_{2}K_{1}( s-t,\lambda(t))-\frac{\lambda(t)}{2(1+s-t)}|ds|\lambda^{\prime}(t)|\] \[\leq\frac{C}{t^{3}\log^{2b+2}(t)}\int_{t+1}^{\infty}|\partial_{2 }K_{1}(s-t,\lambda(t))-\frac{\lambda(t)}{2(1+s-t)}|ds\] \[\leq\frac{C}{t^{3}\log^{2b+2}(t)}\int_{1}^{\infty}dw\left(\frac{ \lambda(t)}{2w}-\frac{\lambda(t)}{2(1+w)}\right)\] \[+\frac{C}{t^{3}\log^{2b+2}(t)}\int_{0}^{\infty}\frac{RdR}{(1+R^{ 2})^{3}}\int_{1}^{\infty}\rho\log(\rho)\frac{4\lambda(t)R^{2}(1+\rho^{2}+R^{2 }\lambda(t)^{2})}{(4R^{2}\lambda(t)^{2}+(1+\rho^{2}-R^{2}\lambda(t)^{2})^{2}) ^{3/2}}d\rho\] \[\leq\frac{C}{t^{3}\log^{3b+2}(t)}+\frac{C}{t^{3}\log^{3b+2}(t)} \int_{0}^{\infty}\frac{R^{3}dR}{(1+R^{2})^{3}}\log(2+R\lambda(t))\] \[\leq\frac{C}{t^{3}\log^{3b+2}(t)}\]
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\[\big{|}I_{k,\ell}^{1}\big{|}\lesssim\sum_{\begin{subarray}{c}|k^{ \prime}-k|\leq 4\\ |\ell^{\prime}-\ell|\leq 4\end{subarray}}\Big{(}2^{-k}\big{\|}S_{k^{\prime}-1}^{ \mathrm{h}}S_{\ell^{\prime}-1}^{\mathrm{v}}\nabla_{\mathrm{h}}v^{\mathrm{h}} \big{\|}_{L_{\frac{6p}{p+4}}^{\frac{6p}{p+4}}}\big{\|}\Delta_{k^{\prime}}^{ \mathrm{h}}\Delta_{\ell^{\prime}}^{\mathrm{v}}\nabla_{\mathrm{h}}\partial_{3}v^ {3}\big{\|}_{L^{2}}\big{\|}\Delta_{k}^{\mathrm{h}}\Delta_{\ell}^{\mathrm{v}} \partial_{3}v^{3}\big{\|}_{L^{\frac{3p}{p-2}}}\] \[\qquad\qquad\qquad+2^{-\ell}\big{\|}S_{k^{\prime}-1}^{\mathrm{h}}S _{\ell^{\prime}-1}^{\mathrm{v}}\partial_{3}v^{\mathrm{h}}\big{\|}_{L_{\mathrm{h} }^{\infty}L_{\mathrm{v}}^{2}}\big{\|}\Delta_{k^{\prime}}^{\mathrm{h}}\Delta_{ \ell^{\prime}}^{\mathrm{v}}\nabla_{\mathrm{h}}\partial_{3}v^{3}\big{\|}_{L^{2} _{\mathrm{h}}L_{\infty}^{\infty}}\big{\|}\Delta_{k}^{\mathrm{h}}\Delta_{\ell}^ {\mathrm{v}}\partial_{3}v^{3}\big{\|}_{L^{2}}\Big{)}\] \[\stackrel{{\mathrm{def}}}{{=}}I_{k,\ell}^{1,1}+I_{k, \ell}^{1,2}.\]
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\[\alpha^{P}_{s}\leq\min_{\begin{subarray}{c}\mathbf{u}_{\mathcal{A}} \in\mathbb{R}^{\mathcal{X}}_{+},\|\mathbf{u}_{\mathcal{A}}\|=1\\ \mathbf{v}_{\mathcal{S}}\in\mathbb{R}^{\mathcal{S}}_{+},\|\mathbf{v}_{s}\|=1 \end{subarray}}\mathbf{u}^{\top}_{\mathcal{A}}P_{0}(\cdot|s,\cdot)\mathbf{v}_{ \mathcal{S}} =\min_{\begin{subarray}{c}\mathbf{u}_{\mathcal{A}}\in\mathbb{R}^{ \mathcal{X}}_{+},\|\mathbf{u}_{\mathcal{A}}\|=1\\ \mathbf{v}_{\mathcal{S}}\in\mathbb{R}^{\mathcal{S}}_{+},\|\mathbf{v}_{\mathcal{S}} \|=1\end{subarray}}\langle\mathbf{u}_{\mathcal{A}},P_{0}(\cdot|s,\cdot) \mathbf{v}_{\mathcal{S}}\rangle\leq\left\langle\frac{\pi_{s}}{\|\pi_{s}\|},P_{ 0}(\cdot|s,\cdot)\frac{(v_{2}-v_{1})}{\|v_{2}-v_{1}\|}\right\rangle\!,\]
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\[\left(\begin{array}{ccccccccc}4\mathbb{A}_{1}&2\mathbb{A}_{3}&8\mathbb{A}_{1} &6\mathbb{A}_{1}&5\mathbb{A}_{1}&6\mathbb{A}_{1}&6\mathbb{A}_{1}\\ &2\mathbb{A}_{1}&6\mathbb{A}_{1}&(4\mathbb{A}_{1})_{I}&3\mathbb{A}_{1}&(4 \mathbb{A}_{1})_{I}&(4\mathbb{A}_{1})_{I}\\ &&4\mathbb{A}_{1}&2\mathbb{A}_{3}&5\mathbb{A}_{1}&6\mathbb{A}_{1}&6\mathbb{A} _{1}\\ &&&&2\mathbb{A}_{1}&3\mathbb{A}_{1}&(4\mathbb{A}_{1})_{I}&(4\mathbb{A}_{1})_{I} \\ &&&&&\mathbb{A}_{1}&\mathbb{A}_{3}&3\mathbb{A}_{1}\\ &&&&&2\mathbb{A}_{1}&(4\mathbb{A}_{1})_{II}\\ &&&&&2\mathbb{A}_{1}\end{array}\right)\]
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\[I_{4,n}(t) =\frac{1}{2}\int_{0}^{t}\langle\varphi_{n}^{\prime\prime}(\rho_{1}(s )-\rho_{2}(s))\sum_{i=1}^{\infty}\left|\sigma_{i}(\cdot,g^{-1}(\rho_{1}(s)))- \sigma_{i}(\cdot,g^{-1}(\rho_{2}(s)))\right|^{2},\psi\rangle_{H}\,ds\] \[\leq\frac{c}{2}\int_{0}^{t}\langle\varphi_{n}^{\prime\prime}( \rho_{1}(s)-\rho_{2}(s))\left|\rho_{1}(s)-\rho_{2}(s)\right|^{2},\psi\rangle_ {H}\,ds\] \[\leq\frac{c\|\psi\|_{H}}{n}\int_{0}^{t}(\|\rho_{1}(s)\|_{H}+\|\rho _{2}(s)\|_{H})ds\] \[\leq\frac{c_{T}\|\psi\|_{H^{1}}}{n}\int_{0}^{t}(\|\rho_{1}(s)\|_{ H^{1}}^{2}+\|\rho_{2}(s)\|_{H^{1}}^{2})ds.\]
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\[\mathcal{R}(\mathcal{X}^{(k+1)})=\mathcal{R}(\mathcal{X}^{(k)}+ \xi_{k}\mathcal{S}^{(k)})\] \[=(1-\xi_{k})\mathcal{U}^{(k)}\mathcal{D}\left(\mathcal{U}^{(k)} \right)^{T}+\xi_{k}\widetilde{\mathcal{W}}_{\ell}\widetilde{\mathcal{W}}_{ \ell}^{T}-\xi_{k}^{2}\Delta\widetilde{\mathcal{K}}^{(k+1)}\left(\Delta \widetilde{\mathcal{K}}^{(k+1)}\right)^{T}\] \[=(1-\xi_{k})\left(\mathcal{W}^{(k)}\left(\mathcal{W}^{(k)} \right)^{T}-\Delta\mathcal{K}^{(k)}\left(\Delta\mathcal{K}^{(k)}\right)^{T} \right)+\xi_{k}\widetilde{\mathcal{W}}_{\ell}\widetilde{\mathcal{W}}_{\ell}^ {T}-\xi_{k}^{2}\Delta\widetilde{\mathcal{K}}^{(k+1)}\left(\Delta\widetilde{ \mathcal{K}}^{(k+1)}\right)^{T}\] \[=\begin{bmatrix}\left[\sqrt{(1-\xi_{k})}\,\mathcal{W}^{(k)}\,\, \sqrt{\xi_{k}}\,\widetilde{\mathcal{W}}_{\ell}\right]\,\left[\sqrt{(1-\xi_{k} )}\,\Delta\mathcal{K}^{(k)}\,\,\xi_{k}\Delta\widetilde{\mathcal{K}}^{(k+1)} \right]\end{bmatrix}\times\begin{bmatrix}I&0\\ 0&-I\end{bmatrix}\] \[\quad\times\left[\left[\sqrt{(1-\xi_{k})}\,\mathcal{W}^{(k)}\,\, \sqrt{\xi_{k}}\,\widetilde{\mathcal{W}}_{\ell}\right]\,\left[\sqrt{(1-\xi_{k} )}\,\Delta\mathcal{K}^{(k)}\,\,\xi_{k}\Delta\widetilde{\mathcal{K}}^{(k+1)} \right]\right]^{T}\!\!\!,\]
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\[E_{C_{\psi_{2}}}(g) =\frac{1}{2\pi}\int_{\alpha=0}^{2\pi}\int_{r=0}^{\infty}\exp\left[ \frac{r^{2}f^{2}(\alpha)}{C_{\psi_{2}}^{2}}-\frac{r^{2}}{2}\right]\,r\,d\alpha dr\] \[=\frac{1}{4\pi}\int_{\alpha=0}^{2\pi}\frac{2C_{\psi_{2}}^{2}}{2f^ {2}(\alpha)-C_{\psi_{2}}^{2}}\exp\left[\left(\frac{f^{2}(\alpha)}{C_{\psi_{2}} ^{2}}-\frac{1}{2}\right)r^{2}\right]_{r=0}^{r\rightarrow\infty}\,d\alpha\] \[=\frac{1}{2\pi}\int_{\alpha=0}^{2\pi}\frac{C_{\psi_{2}}^{2}}{C_{ \psi_{2}}^{2}-2f^{2}(\alpha)}d\alpha\leq\frac{1}{2\pi}\int_{\alpha=0}^{2\pi} \frac{4f^{2}(\alpha)}{4f^{2}(\alpha)-2f^{2}(\alpha)}d\alpha=2\,.\]
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\[\|(h_{t*}-\phi_{t*}^{X})V(p)\| = \|Dh_{t}(h_{t}^{-1}(p))V(h_{t}^{-1}(p))-D\phi_{t}^{X}(\phi_{-t}^{ X}(p))V(\phi_{-t}^{X}(p))\|\] \[\leq \|Dh_{t}(h_{t}^{-1}(p))[V(h_{t}^{-1}(p))-V(\phi_{-t}^{X}(p))]\|\] \[+\|[Dh_{t}(h_{t}^{-1}(p))-Dh_{t}(\phi_{-t}^{X}(p))]V(\phi_{-t}^{ X}(p))\|\] \[+\|[Dh_{t}(\phi_{-t}^{X}(p))-D\phi_{t}^{X}(\phi_{-t}^{X}(p))]V( \phi_{-t}^{X}(p))\|\] \[\leq Cd(h_{t}^{-1}(p)-\phi_{-t}^{X}(p))^{\beta}+Cd(h_{t}^{-1}(p),\phi_ {-t}^{X}(p)))+C\|Dh_{t}-D\phi_{t}\|\] \[\leq Ct^{2\beta}+Ct^{2}+Ct^{r-1}=O(t^{\min\{2\beta,r-1\}}),\]
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\[f_{3}(n+1) = 64f^{4}+384f^{3}g+192f^{3}h+32f^{3}t+960f^{2}g^{2}+312f^{2}h^{2}+ 12f^{2}t^{2}+1056f^{2}gh\] \[+192f^{2}gt+120f^{2}ht+1152fg^{3}+304fh^{3}+4ft^{3}+2064fg^{2}h+ 408fg^{2}t\] \[+1320fgh^{2}+204fh^{2}t+60fgt^{2}+48fht^{2}+552fght+552g^{4}+138h ^{4}+t^{4}\] \[+1416g^{3}h+304g^{3}t+708gh^{3}+144h^{3}t+12gt^{3}+12ht^{3}+660g^ {2}ht+516gh^{2}t\] \[+132ght^{2}+1452g^{2}h^{2}+78g^{2}t^{2}+60h^{2}t^{2}\,\]
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\[\begin{split} J_{n}&=\prod_{k=1}^{K\!-\!1}\!\!\left( \!1\!-\!\frac{p_{k}}{c}\!\!\int_{0}^{c}\!\!\!\frac{\xi s\,\mathrm{d}x}{s\!+\! \left(x\!+\!a_{k}\right)^{\!\eta}}\!\right)\!\!\approx\!\prod_{k=1}^{K\!-\!1} \!\!\left(\!1\!-\!\int_{0}^{c}\!\!\!\frac{\lambda\xi s\,\mathrm{d}x}{s\!+\! \left(x\!+\!a_{k}\right)^{\!\eta}}\!\right)\\ &\stackrel{{(b)}}{{\approx}}1\!-\!\sum_{k=1}^{K\!- \!1}\!\!\int_{0}^{c}\!\!\!\frac{\lambda\xi s\,\mathrm{d}x}{s\!+\!\left(x\!+\!a _{k}\right)^{\!\eta}}\stackrel{{(c)}}{{\approx}}\exp\!\left(\!- \!\sum_{k=1}^{K\!-\!1}\!\!\int_{0}^{c}\!\!\!\frac{\lambda\xi s\,\mathrm{d}x}{s \!+\!\left(x\!+\!a_{k}\right)^{\!\eta}}\!\right)\\ &\stackrel{{(d)}}{{=}}\exp\!\left(\!-\!\lambda\xi\! \!\int_{c+d}^{R\!+\!d\,\frac{s\,\mathrm{d}x}{s\!+\!x^{\eta}}}\!\right)\!\!. \end{split}\]
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\[\begin{cases}(\rho^{(n+m+1)}-\rho^{(n+1)})_{t}+u^{(n+m)}v^{(n+m)}(\rho^{(n+m+1 )}-\rho^{(n+1)})_{x}=(u^{(n)}v^{(n)}-u^{(n+m)}v^{(n+m)})\rho^{(n+1)}_{x}+R^{ 1}_{n,m},\\ (m^{(n+m+1)}-m^{(n+1)})_{t}+u^{(n+m)}v^{(n+m)}(m^{(n+m+1)}-m^{(n+1)})_{x}=(u^{ (n)}v^{(n)}-u^{(n+m)}v^{(n+m)})m^{(n+1)}_{x}+R^{2}_{n,m},\\ (n^{(n+m+1)}-n^{(n+1)})_{t}+u^{(n+m)}v^{(n+m)}(n^{(n+m+1)}-n^{(n+1)})_{x}=(u^{ (n)}v^{(n)}-u^{(n+m)}v^{(n+m)})n^{(n+1)}_{x}+R^{3}_{n,m},\\ (\rho^{(n+m+1)}-\rho^{(n+1)},m^{(n+m+1)}-m^{(n+1)},n^{(n+m+1)}-n^{(n+1)})=((S_{ n+m+1}-S_{n+1})(\rho_{0},m_{0},n_{0})),\end{cases}\]
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For more details, please refer to the πππ±πππ₯π₯ππ« GitHub repository.
- IMPORTANT NOTE!!! The handwritten subset of this dataset was collected entirely from existing open source work, which includes all test sets. If you want to use this subset for your experimental ablation, please filter it yourself based on the latex label of the test set
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