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\[\mathcal{Q}_{\mu}[f_{\xi,\mu}]\] \[=\int_{\mathbb{R}}K(w,t)f_{\xi,\mu}(t)dt\] \[=\int_{\mathbb{R}}\sqrt{\frac{b}{2\pi i}}e^{i(at^{2}+btw+cw^{2}+ dt+ew)}\sqrt{\frac{b}{2\pi i}}e^{i(at^{2}+bt\xi+c\xi^{2}+dt+e\xi)}f(t)dt\] \[=\sqrt{\frac{b}{2\pi i}}\int_{\mathbb{R}}\sqrt{\frac{b}{2\pi i}} e^{i[at^{2}+bt(w+\xi)+c(w+\xi)^{2}+dt+e(w+\xi)]}\] \[\qquad\qquad\qquad\times e^{i(at^{2}+dt)}f(t)e^{-i(2cw\xi)}dt\] \[=\sqrt{\frac{b}{2\pi i}}e^{-i2cw\xi}\mathcal{Q}_{\mu}[e^{i(at^{2} +dt)}f(t)](w+\xi).\]
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\[\begin{array}{lcl}\sum\limits_{k=1}^{K}\left(\boldsymbol{\rho}_{k}^{n}\right) ^{\top}\widehat{\mathcal{P}}\hat{\boldsymbol{\rho}}_{k}^{n}&&\leq 4b_{1}^{ \max,n}\mathcal{C}^{n-1}+2\sum\limits_{k=1}^{K}\boldsymbol{\rho}_{0}^{\top} \widehat{\mathcal{P}}\hat{\boldsymbol{\rho}}_{0},\\ \sum\limits_{k=1}^{K}\left((\boldsymbol{m}_{i})_{k}^{n}\right)^{\top}\widehat{ \mathcal{P}}(\hat{\boldsymbol{m}}_{i})_{k}^{n}&&\leq 2b_{i+1}^{\max,n} \mathcal{C}^{n-1},\ \ \ \ \ \ \ \ i=1,2,3,\\ \sum\limits_{k=1}^{K}\left(\mathbf{Et}_{k}^{n}\right)^{\top}\widehat{ \mathcal{P}}\widehat{\mathbf{Et}}_{k}^{n}&&\leq 4b_{5}^{\max,n} \mathcal{C}^{n-1}+2\sum\limits_{k=1}^{K}\mathbf{Et}_{0}^{\top}\widehat{ \mathcal{P}}\widehat{\mathbf{Et}}_{0}.\end{array}\]
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\[\begin{cases}\partial_{t}\bar{\rho}_{m+1}+\mathrm{div}_{{}_{|}} \bar{u}_{n-1+m,{}^{|}}+\partial_{y}\bar{u}_{n+m,3}=0,\\ \partial_{t}\bar{u}_{m+1,i}+\partial_{i}(\bar{\rho}_{n-1+m}+\bar{ \theta}_{n-1+m})\\ \quad+\sum_{j=1}^{2}\partial_{x_{j}}\langle\mathcal{A}_{ij},( \mathbf{I}-\mathbf{P})\bar{f}_{n-1+m}\rangle+\partial_{y}\langle\mathcal{A}_{3 {}_{|}},(\mathbf{I}-\mathbf{P})\bar{f}_{n+m}\rangle=0,\quad(i=1,2),\\ \partial_{t}\bar{u}_{m+1,3}+\sum_{j=1}^{2}\partial_{x_{j}}\langle \mathcal{A}_{3j},(\mathbf{I}-\mathbf{P})\bar{f}_{n-1+m}\rangle\\ \quad+\partial_{y}(\bar{\rho}_{n+m}+\bar{\theta}_{n+m})+\partial_{y} \langle\mathcal{A}_{33},(\mathbf{I}-\mathbf{P})\bar{f}_{n+m}\rangle=0,\\ 3\partial_{t}(\bar{\rho}_{m+1}+\bar{\theta}_{m+1})+5\,\mathrm{div}_{{}_{|}} \bar{u}_{n-1+m,{}^{|}}\\ \quad+2\sum_{i=1}^{2}\partial_{x_{i}}\langle\mathcal{B}_{i},( \mathbf{I}-\mathbf{P})\bar{f}_{n-1+m}\rangle+5\partial_{y}\bar{u}_{n+m,3}+2 \partial_{y}\langle\mathcal{B}_{3},(\mathbf{I}-\mathbf{P})\bar{f}_{n+m} \rangle=0.\end{cases}\]
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\[\|e^{it\Delta}u(t)-e^{i\tau\Delta}u(\tau)\|_{L^{2}}\] \[\leq\|\int_{t}^{\tau}e^{is\Delta}V(x)u(s)ds\|_{L^{2}}+\|\int_{t}^ {\tau}e^{is\Delta}F(|u|^{2})u(s)ds\|_{L^{2}}+\|\int_{t}^{\tau}e^{is\Delta}(W*|u |^{2})u(s)ds\|_{L^{2}}\] \[\leq\sum_{j=1}^{2}\|\int_{t}^{\tau}e^{is\Delta}V_{j}(x)u(s)ds\|_ {L^{2}}+\|\int_{t}^{\tau}e^{is\Delta}F(|u|^{2})u(s)ds\|_{L^{2}}\] \[\quad+\sum_{m=1}^{2}\|\int_{t}^{\tau}e^{is\Delta}(W_{m}*|u|^{2})u (s)ds\|_{L^{2}}\] \[\leq C\sum_{j=1}^{2}\left(\int_{t}^{\tau}\left(\int_{\mathbb{R}^ {N}}|V_{j}(x)u|^{r^{\prime}_{j}}dx\right)^{q^{\prime}_{j}\over r^{\prime}_{j}} dt\right)^{1\over q^{\prime}_{j}}+C\left(\int_{t}^{\tau}\left(\int_{\mathbb{R}^{N}} [|F(|u|^{2})||u|]^{r^{\prime}_{3}}dx\right)^{q^{\prime}_{3}\over r^{\prime}_{3} }dt\right)^{1\over q^{\prime}_{3}}\] \[\quad+C\sum_{m=1}^{2}\left(\int_{t}^{\tau}\left(\int_{\mathbb{R}^ {N}}[(|W_{m}|*|u|^{2})|u|]^{r^{\prime}_{m}}dx\right)^{q^{\prime}_{m}\over r^{ \prime}_{m}}dt\right)^{1\over q^{\prime}_{m}}\] \[\coloneqq(I)+(II)+(III).\]
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\[\Big{(}S_{0,q}(S_{\alpha-\beta,p}(\{2^{-\beta i}X_{i}\}))\Big{)} _{j} =\Big{[}S_{0,q}(S_{(\alpha-\beta)p,1}(\{2^{-\beta pi}X_{i}^{p}\}) )\Big{]}_{j}^{\frac{1}{p}}\] \[=\Big{[}S_{0,1}\Big{(}(S_{(\alpha-\beta)p,1}(\{2^{-\beta pi}X_{i} ^{p}\}))^{\frac{q}{p}}\Big{)}\Big{]}_{j}^{\frac{1}{q}}\] \[=\Big{(}S_{0,\frac{q}{p}}(S_{(\alpha-\beta)p,1}(\{2^{-\beta pi}X_ {i}^{p}\}))\Big{)}_{j}^{\frac{1}{p}}\] \[\leq\frac{1}{(1-2^{-(\alpha-\beta)p})^{\frac{1}{p}}}\Big{(}S_{0, \frac{q}{p}}(\{2^{-\beta pi}X_{i}^{p}\})\Big{)}_{j}^{\frac{1}{p}}\] \[\leq\frac{1}{1-2^{-(\alpha-\beta)}}2^{-\beta j}(S_{\beta,q}(X))_{ j},\]
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\[\big{|}\mathfrak{C}[N_{0},N_{1},N_{2},N_{3}](t)\big{|}\] \[\lesssim N_{0}^{-3}N_{1}^{-2}N_{2}^{-2}N_{3}^{-3}\sum_{\begin{subarray}{c} \pm 0,\pm 1,\\ \pm 2,\pm 3\end{subarray}}\sum_{\begin{subarray}{c}n_{0},n_{1},n_{2},n_{3} \in\mathbb{Z}^{3}:\\ n_{0}=n_{123}\end{subarray}}\Big{[}\Big{(}\prod_{j=0}^{3}1_{N_{j}}(n_{j})\Big{)} \Big{(}1+\Big{|}\sum_{j=0}^{3}(\pm_{j})\langle n_{j}\rangle\Big{|}\Big{)}^{-1} \Big{]}\] \[\lesssim N_{\max}^{\epsilon/2}N_{0}^{-3}N_{1}^{-2}N_{2}^{-2}N_{3}^{-3} \sum_{m\in\mathbb{Z}}\sum_{\begin{subarray}{c}\pm 0,\pm 1,\\ \pm 2,\pm 3\end{subarray}}\sum_{\begin{subarray}{c}n_{0},n_{1},n_{2},n_{3} \in\mathbb{Z}^{3}:\\ n_{0}=n_{123}\end{subarray}}\Big{[}\Big{(}\prod_{j=0}^{3}1_{N_{j}}(n_{j})\Big{)} \mathbf{1}\Big{\{}\Big{|}\sum_{j=0}^{3}(\pm_{j})\langle n_{j}\rangle-m\Big{|} \leq 1\Big{\}}\Big{]}.\]
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\[\big{|}f_{\eta/2^{\ell},\Lambda_{\ell}}(x)-p_{\eta/2^{\ell}}(cx)\,e^{ 2\pi idx}\big{|}\] \[\leq\Big{|}\sum_{j=-M}^{M}a_{j}^{(\eta/2^{\ell})}\Big{(}\exp\big{(} 2\pi is_{\eta/2^{\ell},\Lambda_{\ell}}(j)x\big{)}-e^{2\pi i(cj+d)x}\Big{)}\Big{|} +\Big{|}\sum_{|j|>M}a_{j}^{(\eta/2^{\ell})}\,e^{2\pi i(cj+d)x}\Big{|}\] \[\leq\sum_{j=-M}^{M}\big{|}a_{j}^{(\eta/2^{\ell})}\big{|}\cdot \Big{|}\exp\Big{(}2\pi i\big{(}s_{\eta/2^{\ell},\Lambda_{\ell}}(j)-cj-d\big{)} x\Big{)}-1\Big{|}+\sum_{|j|>M}\big{|}a_{j}^{(\eta/2^{\ell})}\big{|}\] \[<\tfrac{\eta}{2^{\ell}}+\tfrac{\eta}{2^{\ell}}=\tfrac{\eta}{2^{ \ell-1}}\;\leq\;\eta.\]
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\[J =\frac{(1+w)}{1+\sum_{j\in\mathcal{W}}p_{j}(D_{j}-1)}\left(1+ \sum_{j\in\mathcal{W}}p_{j}(1-e^{\lambda_{0}p_{j}})(e^{-\theta}-1)\right)-1- \log\left(\frac{1+w}{1+\sum_{j\in\mathcal{W}}p_{j}(D_{j}-1)}\right)\] \[\leq w^{2}+0.2\sum_{j\in\mathcal{W}}p_{j}(D_{j}-1)\] \[\leq w^{2}+0.2\sum_{j\in\mathcal{W}}p_{j}\left(D\left(\frac{8 \lambda_{0}}{m}\right)-1\right)\] \[\leq w^{2}+0.14\sum_{j\in\mathcal{W}}p_{j}\frac{8\lambda_{0}}{m}( e^{-\theta}-e^{\theta})\] \[\leq w^{2}+1.12\beta(p)\frac{n(1+w)}{mD_{*}}(e^{-2\theta}-1).\]
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\[\|X-H_{k}^{(j)}\|=\left\|X-X^{(1)}+\sum_{s=1}^{j-1}(X^{(s)}-X^{(s+ 1)})+X^{(j)}-H_{k}^{(j)}\right\|\] \[\leq \|X-X^{(1)}\|+\sum_{s=1}^{j-1}\|X^{(s)}-X^{(s+1)}\|+\|X^{(j)}-H_{k} ^{(j)}\|\] \[\leq \frac{1}{\ell^{(0)}}\|H_{1}^{(1)}-H_{1}\|+\xi^{(0)}\|A_{1}^{(1)}- A_{1}\|+\eta^{(0)}\|G_{1}^{(1)}-G_{1}\|\] \[+\mathcal{O}(\|(H_{1}^{(1)}-H_{1},A_{1}^{(1)}-A_{1},G_{1}^{(1)}- G_{1})\|^{2})\] \[+\sum_{s=1}^{j-1}\Big{\{}\frac{1}{\ell^{(s)}}\|H_{s+1}^{(s)}-H_{s+ 1}^{(s+1)}\|+\xi^{(s)}\|A_{s+1}^{(s)}-A_{s+1}^{(s+1)}\|+\eta^{(s)}\|G_{s+1}^{( s)}-G_{s+1}^{(s+1)}\|\] \[+\mathcal{O}(\|(H_{s+1}^{(s)}-H_{s+1}^{(s+1)},A_{s+1}^{(s)}-A_{s+ 1}^{(s+1)},G_{s+1}^{(s)}-G_{s+1}^{(s+1)})\|^{2})\Big{\}}\] \[+\left\|[(A_{j}^{(j)})^{\mathsf{T}}(I+X^{(j)}G_{j}^{(j)})^{-1}]^{ 2^{k-j}}X^{(j)}(I+Y^{(j)}X^{(j)})[(I+G_{j}^{(j)}X^{(j)})^{-1}A_{j}^{(j)}]^{2^{ k-j}}\right\|,\]
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\[a^{n} =u^{n}+\frac{1}{2}\varphi_{1}\left(\frac{1}{2}\mathcal{J}(u^{n}) \Delta t\right)f(u^{n})\Delta t\] \[b^{n} =u^{n}+\varphi_{1}\left(\mathcal{J}(u^{n})\Delta t\right)f(u^{n}) \Delta t+\varphi_{1}\left(\mathcal{J}(u^{n})\Delta t\right)(\mathcal{F}(a^{n} )-\mathcal{F}(u^{n}))\Delta t\] \[u_{3}^{n+1} =u^{n}+\varphi_{1}\left(\mathcal{J}(u^{n})\Delta t\right)f(u^{n} )\Delta t+\varphi_{3}(\mathcal{J}(u^{n})\Delta t)(-14\mathcal{F}(u^{n})+16 \mathcal{F}(a^{n})-2\mathcal{F}(b^{n}))\Delta t\] \[u_{4}^{n+1} =u_{3}^{n+1}+\varphi_{4}(\mathcal{J}(u^{n})\Delta t)(36\mathcal{ F}(u^{n})-48\mathcal{F}(a^{n})+12\mathcal{F}(b^{n}))\Delta t\]
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\[I_{1}\leq H_{21}\bigg{(}\mathbb{E}\int_{t_{0}}^{t\wedge\theta_{n}}\big{|}e(s) \big{|}^{\bar{q}}ds+\mathbb{E}\int_{t_{0}}^{T}\big{|}\tilde{R}_{\mu}(t,X_{ \Delta}(t),\bar{X}_{\Delta}(t))\big{|}^{\bar{q}}ds+\Delta^{\bar{q}}\bigg{)}\] \[\leq H_{21}\bigg{(}\mathbb{E}\int_{t_{0}}^{t\wedge\theta_{n}}\big{|}e(s )\big{|}^{\bar{q}}ds+\int_{t_{0}}^{T}\mathbb{E}\big{|}\tilde{R}_{\mu}(t,X_{ \Delta}(t),\bar{X}_{\Delta}(t))\big{|}^{\bar{q}}ds+\Delta^{\bar{q}}\bigg{)}\] \[\leq H_{21}\bigg{(}\mathbb{E}\int_{t_{0}}^{t\wedge\theta_{n}}\big{|}e( s)\big{|}^{\bar{q}}ds+\Delta^{\bar{q}}\big{(}h(\Delta)\big{)}^{2\bar{q}}+ \Delta^{\bar{q}}\bigg{)}.\]
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\[\left(\iota\circ\mathrm{id}_{\mathsf{Com}^{*}}\right)d_{\kappa} =\left(\iota\circ\mathrm{id}_{\mathsf{Com}^{*}}\right)\left( \mu_{(1)}^{\mathsf{Lie}}\circ\mathrm{id}_{\mathsf{Com}^{*}}\right)\left( \mathrm{id}_{\mathsf{Lie}}\circ^{\prime}\left[\,\left(\kappa\circ\mathrm{id}_{ \mathsf{Com}^{*}}\right)\Delta\right]\right)\] \[=\left(\mu_{(1)}^{\mathsf{Graphs}_{M}}\circ\mathrm{id}_{\mathsf{ Com}^{*}}\right)\left(\,\left(\iota\circ_{(1)}\iota\right)\circ\mathrm{id}_{ \mathsf{Com}^{*}}\right)\left(\,\left(\mathrm{id}_{\mathsf{Lie}}\circ_{(1)} \kappa\right)\circ\mathrm{id}_{\mathsf{Com}^{*}}\right)\left(\,\left(\iota \circ_{(1)}\mathrm{id}_{\mathsf{Com}^{*}}\right)\circ\mathrm{id}_{\mathsf{Com }^{*}}\right)\left(\mathrm{id}_{\mathsf{Lie}}\circ^{\prime}\Delta\right)\] \[=\left(\mu_{(1)}^{\mathsf{Graphs}_{M}}\circ\mathrm{id}_{\mathsf{ Com}^{*}}\right)\left(\,\left(\mathrm{id}_{\mathsf{Graphs}_{M}}\circ_{(1)}\iota \kappa\right)\circ\mathrm{id}_{\mathsf{Com}^{*}}\right)\left(\mathrm{id}_{ \mathsf{Graphs}_{M}}\circ^{\prime}\Delta\right)\left(\iota\circ\mathrm{id}_{ \mathsf{Com}^{*}}\right)\] \[=d_{\mathsf{mod}}^{\mathsf{com}}\Big{(}\iota\circ\mathrm{id}_{ \mathsf{Com}^{*}}\Big{)}\]
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\[3\left|A_{3}-\rho A_{2}^{2}\right|\leq\left\{\begin{array}{ll}\left(3-2 \beta\right)\left(3-2\alpha-\beta\right)-3\rho\left(2-\alpha-\beta\right)^{2} &,\ \ \rho\leq\frac{2\left(1-\beta\right)}{3\left(2-\alpha-\beta\right)}\\ \\ 1-2\alpha+\beta\left(3-2\beta\right)+\frac{4}{3\rho}\left(1-\beta\right)^{2}&, \ \frac{2\left(1-\beta\right)}{3\left(2-\alpha-\beta\right)}\leq\rho\leq\frac{2} {3}\\ \\ 3-2\alpha-\beta&,\ \frac{2}{3}\leq\rho\leq\frac{2\left(2-\beta\right) \left(3-2\alpha-\beta\right)}{3\left(2-\alpha-\beta\right)^{2}}\\ \\ \left(2\beta-3\right)\left(3-2\alpha-\beta\right)+3\rho\left(2-\alpha-\beta \right)^{2}&,\ \ \rho\geq\frac{2\left(2-\beta\right)\left(3-2\alpha-\beta\right)}{3\left(2- \alpha-\beta\right)^{2}}\end{array}\right.\]
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\[I_{1} :=\left\langle-\lambda_{1}\boldsymbol{m}_{h,k}^{-}\times \boldsymbol{v}_{h,k}+\lambda_{2}\boldsymbol{v}_{h,k},\boldsymbol{m}_{h,k}^{-} \times\boldsymbol{\psi}-I_{\mathbb{V}_{h}}(\boldsymbol{m}_{h,k}^{-}\times \boldsymbol{\psi})\right\rangle_{\mathbb{L}^{2}(D_{T})},\] \[I_{2} :=\mu\left\langle\nabla(\boldsymbol{m}_{h,k}^{-}+k\theta \boldsymbol{v}_{h,k}),\nabla(\boldsymbol{m}_{h,k}^{-}\times\boldsymbol{\psi}- I_{\mathbb{V}_{h}}(\boldsymbol{m}_{h,k}^{-}\times\boldsymbol{\psi}))\right\rangle_{ \mathbb{L}^{2}(D_{T})},\] \[I_{3} :=\mu\int_{0}^{T}F_{k}(t,\boldsymbol{m}_{h,k}^{-},\boldsymbol{m}_ {h,k}^{-}\times\boldsymbol{\psi})-F_{k}(t,\boldsymbol{m}_{h,k}^{-},I_{ \mathbb{V}_{h}}(\boldsymbol{m}_{h,k}^{-}\times\boldsymbol{\psi}))\,dt.\]
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\[-\frac{h}{2}\sqrt{\frac{3h^{2}-1}{3-h^{2}}}\Big{(}2{\bf p}_{3}^{\perp}\frac{ \partial^{2}}{\partial p_{2}\partial p_{1}}+3p_{2}p_{1}\frac{\partial}{\partial{\bf p }_{3}^{\perp}}\Big{)}++\underline{h}\Big{(}\tilde{p}_{2}\frac{\partial}{ \partial p_{2}}+p_{2}\frac{\partial}{\partial\tilde{p}_{2}}\Big{)}\left(p_{1}\frac{ \partial}{\partial p_{1}}-1\right)+\frac{6h}{\sqrt{1+h^{2}}}\Big{(}p_{3}\frac{ \partial}{\partial{\bf p}_{3}}+{\bf p}_{3}\frac{\partial}{\partial{\bf p}_{3}} \Big{)}+\] \[+\frac{3}{2}\sqrt{\frac{3(1-h^{2})(3h^{2}-1)}{3-h^{2}}}\Big{(}p_{3} \frac{\partial}{\partial\mathbb{P}_{3}^{\perp}}+\mathbb{P}_{3}^{\perp}\frac{ \partial}{\partial p_{3}}\Big{)}-\frac{\sqrt{3(1-h^{2})}}{2}\Big{(}2p_{3}\frac {\partial^{2}}{\partial\tilde{p}_{2}\partial p_{1}}+3\tilde{p}_{2}p_{1}\frac{ \partial}{\partial p_{3}}\Big{)}\]
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\[p_{\alpha}(J(-\eta;III))=p_{\alpha}(\int\limits_{\delta}^{\eta}f(- it)\cdot(-i)\int\limits_{-\delta}^{\delta}\varphi(a-\delta^{\prime}+is)e^{i(a- \delta^{\prime}+is)(-it)}i\mathrm{d}s\,\mathrm{d}t)\] \[\underset{(\ref{eq:p_alpha}),(\ref{eq:p_alpha})}{\leq}\int \limits_{\delta}^{\eta}|f|_{k,\alpha,K}e^{\frac{1}{k}t-at}\int\limits_{-\delta }^{\delta}\|\varphi\|_{n,K}e^{-\frac{1}{n}|a-\delta^{\prime}|}e^{(a-\delta^{ \prime})t}\mathrm{d}s\,\mathrm{d}t\] \[\leq 2\delta|f|_{k,\alpha,K}\|\varphi\|_{n,K}\int\limits_{\delta} ^{\eta}e^{(\frac{1}{k}-\delta^{\prime})t}\mathrm{d}t\] \[=2\delta|f|_{k,\alpha,K}\|\varphi\|_{n,K}\frac{1}{\frac{1}{k}- \delta^{\prime}}(e^{(\frac{1}{k}-\delta^{\prime})\eta}-e^{(\frac{1}{k}-\delta ^{\prime})\delta})\] \[\underset{(\ref{eq:p_alpha})}{\rightarrow}\frac{2\delta}{ \delta^{\prime}-\frac{1}{k}}e^{(\frac{1}{k}-\delta^{\prime})\delta}|f|_{k, \alpha,K}\|\varphi\|_{n,K},\;\eta\rightarrow\infty.\]
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\[(p)_{\Gamma} = C_{1}+C_{2}\ln R+\delta e^{il\theta}\left(\frac{C_{2}}{R}+D_{1} R^{l}+\frac{D_{2}}{R^{l}}\right)\] \[= \mathcal{G}^{-1}(\kappa)_{\Gamma}+\left(\mathcal{P}-\chi_{\sigma }\right)(\sigma)_{\Gamma}-\mathcal{P}\mathcal{A}\frac{(x\cdot x)_{\Gamma}}{4}\] \[= \mathcal{G}^{-1}\frac{1}{R}-\frac{\mathcal{P}\mathcal{A}}{4}R^{2 }+\left(\mathcal{P}-\chi_{\sigma}\right)(A_{1}I_{0}(R)+A_{2}K_{0}(R))\] \[+\left(\mathcal{G}^{-1}\frac{l^{2}-1}{R^{2}}-\frac{\mathcal{P} \mathcal{A}}{2}R+\left(\mathcal{P}-\chi_{\sigma}\right)(A_{1}I_{1}(R)-A_{2}K _{1}(R)+B_{1}I_{l}(R)+B_{2}K_{l}(R))\right)\delta e^{il\theta}.\]
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\[\begin{split}\big{(}\mathcal{M}\otimes\bar{\mathcal{M}}\big{)}( \Psi_{1},\Psi_{2})&=\big{(}\mathcal{M}\otimes\bar{\mathcal{M}} \big{)}(u_{1}\otimes\bar{u}_{1},u_{2}\otimes\bar{u}_{2})\\ &=(-)^{u_{2}\bar{u}_{1}+|\bar{\mathcal{M}}|(u_{1}+u_{2})} \mathcal{M}(u_{1},u_{2})\otimes\bar{\mathcal{M}}(\bar{u}_{1},\bar{u}_{2})\;, \\ \big{(}\mathcal{T}\otimes\bar{\mathcal{T}}\big{)}(\Psi_{1},\Psi_{ 2},\Psi_{3})&=\big{(}\mathcal{T}\otimes\bar{\mathcal{T}}\big{)}( u_{1}\otimes\bar{u}_{1},u_{2}\otimes\bar{u}_{2},u_{3}\otimes\bar{u}_{3})\\ &=(-)^{u_{2}\bar{u}_{1}+u_{3}(\bar{u}_{1}+\bar{u}_{2})+|\bar{ \mathcal{T}}|(u_{1}+u_{2}+u_{3})}\mathcal{T}(u_{1},u_{2},u_{3})\otimes\bar{ \mathcal{T}}(\bar{u}_{1},\bar{u}_{2},\bar{u}_{3})\;.\end{split}\]
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\[dial_{P}=\{-1,0,2,2\}\] \[p<q\] \[p\] \[zone_{0}\] \[od_{2}\] \[od_{4}\] \[p=\begin{cases}\left(\frac{\Delta-4}{4}\right)^{2}&\text{for }\Delta=4k,k=2m,\ k,m\in\mathbb{N};p=2j+1,\ j\in\mathbb{N}\\ \\ \left(\frac{\Delta-4}{4}\right)\left(\frac{\Delta-4}{4}+1\right)-\left(\frac{ \Delta-4}{4}-1\right)&\text{for }\Delta=4k,k=2m+1,\ k,m\in\mathbb{N};p=2j+1,\ j\in\mathbb{N}\\ \\ \left(2\times\left(\left(\frac{\Delta-6}{4}+1\right)^{2}-1\right)\right)+3& \text{for }\Delta=4k+2,\ k\in\mathbb{N};\ p=2j+1,\ j\in\mathbb{N}\\ \\ \left(2\times\left(\frac{\Delta-4}{4}\right)\left(\frac{\Delta-4}{4}+1\right) \right)+2&\text{for }\Delta=4k,\ k\in\mathbb{N};\ p=2j,\ j\in\mathbb{N}\\ \\ \left(\frac{\Delta-6}{4}\right)\left(\frac{\Delta-6}{4}+1\right)+2&\text{for } \Delta=4k+2,\ k\in\mathbb{N};\ p=2j,\ j\in\mathbb{N}\\ \end{cases}\]
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\[\begin{split}&\left[\begin{array}{c}\mathbb{E}\left[\sum_{i=1}^ {m}\|\bar{x}_{i}^{k+1}-x_{i}^{*}\|_{2}^{2}|\mathcal{F}^{k}\right]\\ \mathbb{E}\left[\sum_{i=1}^{m}\|\mathbf{x}_{i}^{k+1}-\bar{\mathbf{x}}_{i}^{k+1 }\|_{L}^{2}|\mathcal{F}^{k}\right]\end{array}\right]\\ \leq&\left(\left[\begin{array}{cc}1&\kappa_{1}\gamma^{k}\\ 0&1-\kappa_{2}\gamma^{k}\end{array}\right]+a^{k}\mathbf{1}\mathbf{1}^{T} \right)\left[\begin{array}{c}\sum_{i=1}^{m}\|\bar{x}_{i}^{k}-x_{i}^{*}\|_{2} ^{2}\\ \sum_{i=1}^{m}\|\mathbf{x}_{i}^{k}-\bar{\mathbf{x}}_{i}^{k}\|_{L}^{2}\end{array} \right]\\ &+b^{k}\mathbf{1}-c^{k}\left[\begin{array}{c}\left(\phi(\bar{x}^{k})-\phi(x^ {*})\right)^{T}(\bar{x}^{k}-x^{*})\\ 0\end{array}\right],\quad\forall k\geq 0\end{split}\]
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\[\begin{split}&\sum_{\begin{subarray}{c}\|\bm{a}\|_{\infty}\leq A\\ P(\bm{a})=0\end{subarray}}|\mathcal{I}_{\bm{a}}(X)-\mathfrak{S}_{\bm{a}}^{*} \mathfrak{J}_{\bm{a}}^{*}|^{2}\\ &\ll\sum_{\begin{subarray}{c}\|\bm{a}\|_{\infty}\leq A\\ P(\bm{a})=0\end{subarray}}|\mathcal{I}_{\bm{a}}(X,\mathfrak{M})-\mathfrak{S}_{ \bm{a}}^{*}\mathfrak{J}_{\bm{a}}^{*}|^{2}+\sum_{\begin{subarray}{c}\|\bm{a}\|_{ \infty}\leq A\\ P(\bm{a})=0\end{subarray}}|\mathcal{I}_{\bm{a}}(X,\mathfrak{m})|^{2}\\ &\ll\Sigma_{1}+\Sigma_{2}+\sum_{\begin{subarray}{c}\|\bm{a}\|_{ \infty}\leq A\\ P(\bm{a})=0\end{subarray}}|\mathcal{I}_{\bm{a}}(X,\mathfrak{m})|^{2}+O\bigg{(} A^{-2}X^{2n-2d-2}w^{10}\sum_{\begin{subarray}{c}\|\bm{a}\|_{\infty}\leq A\\ P(\bm{a})=0\end{subarray}}1\bigg{)},\end{split}\]
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\[n^{\frac{1}{p}+\frac{1}{q^{\prime}}}\sum_{N_{1}\leq j\leq N_{2}} \|f_{i}\|_{L^{p}(Q_{0},B_{1})}\|g_{i}\|_{L^{q^{\prime}}(3Q_{0},B_{2}^{*})}\] \[\sum_{R\in\mathfrak{R}_{j}}|R|^{-(\frac{1}{p}-\frac{1}{q})}\bigg{(} \sum_{\begin{subarray}{c}P\subset R\\ j-\ell\leq L(P)<j\end{subarray}}|P|\bigg{)}^{\frac{1}{p}}\bigg{(}\sum_{ \begin{subarray}{c}P^{\prime}\subset 3R\\ j-\ell\leq L(P^{\prime})<j\end{subarray}}|P^{\prime}|\bigg{)}^{\frac{1}{q^{ \prime}}}\] \[\leq n^{\frac{1}{p}+\frac{1}{q^{\prime}}}\sum_{N_{1}\leq j\leq N_{ 2}}\|f_{i}\|_{L^{p}(Q_{0},B_{1})}\|g_{i}\|_{L^{q^{\prime}}(3Q_{0},B_{2}^{*})}\] \[\sum_{R\in\mathfrak{R}_{j}}|R|^{-(\frac{1}{p}-\frac{1}{q})}\bigg{(} \sum_{\begin{subarray}{c}P\subset 3R\\ j-\ell\leq L(P)<j\end{subarray}}|P|\bigg{)}^{\frac{1}{p}-\frac{1}{q}}\bigg{(} \sum_{\begin{subarray}{c}P\subset 3R\\ j-\ell\leq L(P)<j\end{subarray}}|P|\bigg{)}.\]
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\[\langle A(t,\bm{v}),\bm{z}\rangle:=\langle A(t,\bm{\varepsilon}( \bm{v})),\bm{\varepsilon}(\bm{z})\rangle_{\mathcal{H}}\ \ \text{for}\ \ \bm{v},\bm{z}\in V,\ \text{a.e.}\ t\in(0,T),\] \[(I\bm{w})(t):=\bm{u}_{0}+\int_{0}^{t}\bm{w}(s)\,ds\ \ \text{for}\ \ \bm{w}\in L^{2}(0,T;V),\ \text{a.e.}\ t\in(0,T),\] \[(S\bm{w})(t):=((I\bm{w})(t))_{\tau}=\bm{u}_{0\tau}+\int_{0}^{t} \bm{w}_{\tau}(s)\,ds\ \ \text{for}\ \ \bm{w}\in L^{2}(0,T;V),\ \text{a.e.}\ t\in(0,T),\] \[\langle(R_{1}\bm{w})(t),\bm{v}\rangle:=\langle\mathcal{B}\bm{ \varepsilon}((I\bm{w})(t))+\int_{0}^{t}\mathcal{C}(t-s)\bm{\varepsilon}(\bm{ w}(s))\,ds,\bm{\varepsilon}(\bm{v})\rangle_{\mathcal{H}}\] \[\qquad\qquad\qquad\qquad\qquad\text{for}\ \ \bm{w}\in L^{2}(0,T;V),\ \bm{v}\in V,\ \text{a.e.}\ t\in(0,T),\] \[(R_{3}\bm{w})(t):=((I\bm{w})(t))_{\nu}=u_{0\nu}+\int_{0}^{t}w_{ \nu}(s)\,ds\ \ \text{for}\ \ \bm{w}\in L^{2}(0,T;V),\ \text{a.e.}\ t\in(0,T),\] \[\varphi(t,\beta,y,\bm{v}):=\int_{\Gamma_{C}}\mu(y)\,h_{1}(\beta )\,h_{2}(\bm{v}_{\tau})\,d\Gamma\ \ \text{for}\ \ \beta\in E,\ y\in Y,\ \bm{v}\in X,\ \text{a.e.}\ t\in(0,T),\] \[F(t,\beta,\bm{w})(\bm{x}):=G(\bm{x},t,\beta(\bm{x},t),\bm{w}_{ \tau}(\bm{x},t))\ \text{for}\ \beta\in E,\ \bm{w}\in U,\ \text{a.e.}\ t\in(0,T),\] \[M\bm{v}:=\bm{v}_{\tau}\ \ \text{for}\ \ \bm{v}\in V,\]
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\[\left|\int_{B_{1}^{\prime}}F\cdot\nabla\bar{v}_{2}|\bar{v}_{2}|^{p-2}\right|\] \[\leq\frac{\varrho}{2}\int_{B_{1}^{\prime}}|x^{\prime}|^{m}|\nabla \bar{v}_{2}|^{2}|\bar{v}_{2}|^{2}+C\int_{B_{1}^{\prime}}|x^{\prime}|^{2\sigma+m }|\bar{v}_{2}|^{p-2}\] \[\leq\frac{\varrho}{2}\int_{B_{1}^{\prime}}|x^{\prime}|^{m}|\nabla \bar{v}_{2}|^{2}|\bar{v}_{2}|^{2}\] \[\quad+C\||\bar{v}_{2}|^{p-2}\|_{L^{\frac{d+m-1+2\mu}{4+m-3+2\mu}}( B_{1}^{\prime},|x^{\prime}|^{m}dx^{\prime})}\left(\int_{B_{1}^{\prime}}|x^{ \prime}|^{\sigma(d+m-1+2\mu)+m}\right)^{\frac{2}{d+m-1+2\mu}}\] \[\leq\frac{\varrho}{2}\int_{B_{1}^{\prime}}|x^{\prime}|^{m}| \nabla\bar{v}_{2}|^{2}|\bar{v}_{2}|^{2}+C\||\bar{v}_{2}|^{p-2}\|_{L^{\frac{d+ m-1+2\mu}{d+m-3+2\mu}}(B_{1}^{\prime},|x^{\prime}|^{m}dx^{\prime})},\]
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\[\frac{1}{x^{2i\lambda}}\,\prod_{k=1}^{L}\int d^{4}x_{k}\frac{1}{x _{k-1k}^{2(2+u_{k})}}\,\frac{1}{x_{k}^{-2u_{k}}}\,\frac{1}{(x_{L}-y)^{2(2+u_{L +1})}}\,\frac{1}{y^{2(-u_{L+1}-i\lambda)}}=\] \[\frac{\pi^{2L-1}}{2}\,\sum_{n\geq 0}\,(n+1)\,\int\limits_{- \infty}^{+\infty}d\nu\,\frac{\hat{x}^{\mu_{1}\cdots\mu_{n}}}{x^{2(1+\frac{i \lambda}{2}+i\nu)}}\,\frac{\hat{y}^{\mu_{1}\cdots\mu_{n}}}{y^{2(1-\frac{i \lambda}{2}-i\nu)}}\,\prod_{k=1}^{L+1}4^{-u_{k}}\,a_{0}(u_{k}+2)\,\tau(u_{k}, \nu,n)\,\frac{\sin\left((n+1)\theta\right)}{\sin\theta}\]
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\[\operatorname{Var}_{\mu}[f] =\operatorname{Var}_{i\sim\mu D_{k\to 1}}\big{[}\mathbb{E}_{\mu}[f \mid i]\big{]}+\mathbb{E}_{i\sim\mu D_{k\to 1}}\big{[}\operatorname{Var}_{\mu}[f \mid i]\big{]}\] \[=\operatorname{Var}_{\mu D_{k\to 1}}[U_{1\to k}f]+\mathbb{E}_{i\sim\mu D_{k\to 1}} \big{[}\operatorname{Var}_{\mu}[f\mid i]\big{]}\] \[=\mathbb{E}[(U_{1\to k}(f-\mathbb{E}[f]))^{2}]+\mathbb{E}_{i\sim\mu D _{k\to 1}}\big{[}\operatorname{Var}_{\mu}[f\mid i]\big{]}\] \[=\mathbb{E}[(f-\mathbb{E}[f])(D_{k\to 1}U_{1\to k}(f-\mathbb{E}[f]))]+ \mathbb{E}_{i\sim\mu D_{k\to 1}}\big{[}\operatorname{Var}_{\mu}[f\mid i]\big{]}\] \[\leq\lambda_{2}(D_{k\to 1}U_{1\to k})\operatorname{Var}_{\mu}[f]+ \mathbb{E}_{i\sim\mu D_{k\to 1}}\big{[}\operatorname{Var}_{\mu}[f\mid i]\big{]}\] \[\leq\frac{C^{\prime}}{k}\operatorname{Var}_{\mu}[f]+\mathbb{E}_ {i\sim\mu D_{k\to 1}}\big{[}\operatorname{Var}_{\mu}[f\mid i]\big{]}\,\]
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\[\left\|\left\{\sum_{i=1}^{m}\left[\frac{\lambda_{i}}{\|\mathbf{1} _{x_{i}+B_{l_{i}}}\|_{L^{p(i)}(\mathbb{R}^{n})}}\right]^{\!\!p}\mathbf{1}_{x_ {i}+B_{l_{i}}}\right\}^{1/\!p}\right\|_{L^{p(i)}(\mathbb{R}^{n})}^{-1}\sum_{j= 1}^{m}\frac{\lambda_{j}|x_{j}+B_{l_{j}}|^{\frac{1}{2}}}{\|\mathbf{1}_{x_{j}+B_ {l_{j}}}\|_{L^{p(i)}(\mathbb{R}^{n})}}\] \[\qquad\qquad\times\left[\sum_{\ell\in\mathbb{Z}}\int_{\{x\in \mathbb{R}^{n}:\ (x,\ell)\in x_{j}+\widetilde{B}_{l_{j}}\}}\left|\phi_{-\ell}*b_{j}^{(2)}(x) \right|^{2}\ dx\right]^{\!\!\frac{1}{2}}\] \[\quad\quad\quad\times\left\{\left[\int_{x_{j}+B_{l_{j}+\omega}} \left|b(x)-P_{x_{j}+B_{l_{j}+\omega}}^{s}b(x)\right|^{2}\ dx\right]^{\!\!\frac{1 }{2}}\right.\] \[\quad\quad\quad+\left.\frac{1}{\left|x_{j}+B_{l_{j}}\right|^{\frac {1}{2}}}\int_{x_{j}+B_{l_{j}+\omega}}\left|b(x)-P_{x_{j}+B_{l_{j}+\omega}}^{s} b(x)\right|\ dx\right\}.\]
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\[f_{1,j}\left(\frac{z_{2}}{z_{3}}\right)f_{i,j}\left(\frac{z_{2}} {z_{1}}\right)f_{1,i}\left(\frac{z_{1}}{z_{3}}\right)T_{1}(z_{3})T_{i}(z_{1}) T_{j}(z_{2})-f_{j,1}\left(\frac{z_{3}}{z_{2}}\right)f_{j,i}\left(\frac{z_{1}}{z_{2}} \right)T_{j}(z_{2})f_{1,i}\left(\frac{z_{1}}{z_{3}}\right)T_{1}(z_{3})T_{i}(z_ {1})\] \[- c(r,x)\delta\left(\frac{x^{-j-1}z_{2}}{z_{3}}\right)\Delta_{1} \left(\frac{x^{-i}z_{1}}{z_{3}}\right)f_{j+1,i}\left(\frac{x^{-j}z_{1}}{z_{3}} \right)T_{j+1}(x^{j}z_{3})T_{i}(z_{1})\] \[+ c(r,x)\delta\left(\frac{x^{j+1}z_{2}}{z_{3}}\right)\Delta_{1} \left(\frac{x^{i}z_{1}}{z_{3}}\right)f_{j+1,i}\left(\frac{x^{j}z_{1}}{z_{3}} \right)T_{j+1}(x^{-j}z_{3})T_{i}(z_{1}).\]
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\[\begin{array}{rl}(x_{j}\circ y_{\beta})(x_{i}\circ y_{\alpha})&=q_{ji}q^{ \prime}_{\beta\alpha}(x_{i}x_{j})\circ(y_{\alpha}y_{\beta})=q_{ji}q^{\prime}_{ \beta\alpha}(x_{i}\circ y_{\alpha})(x_{j}\circ y_{\beta}).\\ (x_{j}\circ y_{\alpha})(x_{i}\circ y_{\beta})&=q_{ji}q^{\prime}_{\alpha\beta}( x_{i}x_{j})\circ(y_{\beta}y_{\alpha})=q_{ji}q^{\prime}_{\alpha\beta}(x_{i} \circ y_{\beta})(x_{j}\circ y_{\alpha}).\\ (x_{i}\circ y_{\beta})(x_{j}\circ y_{\alpha})=x_{i}x_{j}\circ y_{\beta}y_{ \alpha}=q^{\prime}_{\beta\alpha}(x_{i}x_{j})\circ(y_{\alpha}y_{\beta})=q^{ \prime}_{\beta\alpha}(x_{i}\circ y_{\alpha})(x_{j}\circ y_{\beta})\end{array}\]
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\[\Lambda_{1}^{\mathsf{T}} = (0,0,0,0,0,0,0,0,0);\] \[\Lambda_{2}^{\mathsf{T}} = (0,0,0,0,0,0,(a^{\mathsf{T}}\nabla)_{2}f,-(a^{\mathsf{T}}\nabla) _{1}f,0);\] \[\mathfrak{R}_{ab}(\nabla f,\nabla f)-\Lambda_{1}^{\mathsf{T}}Q^{ \mathsf{T}}Q\Lambda_{1}-\Lambda_{2}^{\mathsf{T}}P^{\mathsf{T}}P\Lambda_{2}+D^ {\mathsf{T}}D+E^{\mathsf{T}}E\] \[= -\Gamma_{1}(f,f)+\frac{1}{2}\Gamma_{1}^{z}(f,f)-(a^{\mathsf{T}} \nabla)_{1}V\partial_{z}f(a^{\mathsf{T}}\nabla)_{2}f+(a^{\mathsf{T}}\nabla)_{ 2}V\partial_{z}f(a^{\mathsf{T}}\nabla)_{1}f\] \[+\Big{[}\frac{\partial^{2}V}{\partial x\partial x}+\frac{y^{2}}{ 4}\frac{\partial^{2}V}{\partial z\partial z}-y\frac{\partial^{2}V}{\partial x \partial z}\Big{]}|(a^{\mathsf{T}}\nabla)_{1}f|^{2}\] \[+\Big{[}\frac{\partial^{2}V}{\partial y\partial y}+\frac{x^{2}}{ 4}\frac{\partial^{2}V}{\partial z\partial z}+x\frac{\partial^{2}V}{\partial y \partial z}\Big{]}|(a^{\mathsf{T}}\nabla)_{2}f|^{2}\] \[+2\Big{[}\frac{\partial^{2}V}{\partial x\partial y}+\frac{x}{2} \frac{\partial^{2}V}{\partial x\partial z}-\frac{y}{2}\frac{\partial^{2}V}{ \partial y\partial z}-\frac{xy}{4}\frac{\partial^{2}V}{\partial z\partial z} \Big{]}(a^{\mathsf{T}}\nabla)_{1}f(a^{\mathsf{T}}\nabla)_{2}f;\] \[\mathfrak{R}_{zb}(\nabla f,\nabla f) = \Big{(}\frac{\partial^{2}V}{\partial x\partial z}-\frac{y}{2} \frac{\partial^{2}V}{\partial z\partial z}\Big{)}(a^{\mathsf{T}}\nabla)_{1}f(z ^{\mathsf{T}}\nabla)_{1}f+\Big{(}\frac{\partial^{2}V}{\partial y\partial z}+ \frac{x}{2}\frac{\partial^{2}V}{\partial z\partial z}\Big{)}(z^{\mathsf{T}} \nabla)_{1}f(a^{\mathsf{T}}\nabla)_{2}f;\] \[\mathfrak{R}_{\pi}(\nabla f,\nabla f) = 0.\]
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\begin{table}
\begin{tabular}{l r r r r} \hline \hline Instance & Objective & Time & \#Pen. & \#ADM \\ \hline syn05m04h & 5510.39 & 0 & 3 & 4 \\ syn05m04m & 5499.39 & 1 & 13 & 28 \\ syn10h & 1267.35 & 0 & 3 & 4 \\ syn10m & 560.25 & 0 & 10 & 14 \\ syn10m02h & 2310.30 & 1 & 3 & 4 \\ syn10m02m & 2275.85 & 2 & 13 & 25 \\ syn10m03h & 3354.68 & 1 & 3 & 4 \\ syn10m03m & 2773.64 & 2 & 25 & 38 \\ syn10m04h & 4557.06 & 0 & 3 & 4 \\ syn10m04m & 3814.77 & 3 & 26 & 41 \\ syn15h & 853.28 & 0 & 3 & 4 \\ syn15m & 503.63 & 1 & 10 & 25 \\ syn15m02h & 2832.75 & 0 & 3 & 4 \\ syn15m02m & 2777.75 & 1 & 14 & 25 \\ syn15m03h & 3850.18 & 0 & 3 & 4 \\ syn15m03m & 3795.18 & 3 & 17 & 34 \\ syn15m04h & 4937.48 & 1 & 3 & 4 \\ syn15m04m & 4882.48 & 2 & 18 & 31 \\ syn20h & 924.26 & 1 & 3 & 4 \\ syn20m & 722.64 & 2 & 8 & 19 \\ syn20m02h & 1752.13 & 1 & 3 & 4 \\ syn20m02m & 1693.13 & 2 & 23 & 38 \\ syn20m03h & 2646.95 & 1 & 3 & 4 \\ syn20m03m & 2574.01 & 3 & 18 & 41 \\ syn20m04h & 3532.74 & 1 & 3 & 4 \\ syn20m04m & 3475.74 & 3 & 18 & 35 \\ syn30h & 134.03 & 0 & 3 & 5 \\ syn30m & -38.75 & 2 & 15 & 28 \\ syn30m02h & 393.25 & 1 & 3 & 5 \\ syn30m02m & -13.34 & 2 & 16 & 29 \\ syn30m03h & 646.05 & 2 & 3 & 5 \\ syn30m03m & 31.70 & 3 & 15 & 34 \\ syn30m04h & 859.05 & 3 & 3 & 5 \\ syn30m04m & 111.41 & 5 & 13 & 33 \\ syn40h & 58.66 & 1 & 3 & 5 \\ syn40m & -25.08 & 2 & 27 & 46 \\ syn40m02h & 379.76 & 1 & 3 & 4 \\ syn40m02m & 149.17 & 3 & 11 & 30 \\ syn40m03h & 390.15 & 4 & 4 & 6 \\ syn40m03m & -12.17 & 6 & 28 & 51 \\ syn40m04h & 896.96 & 4 & 3 & 4 \\ syn40m04m & 609.95 & 6 & 12 & 32 \\ synheat & 219858.00 & 2 & 18 & 43 \\ synthe1 & 7.09 & 1 & 8 & 14 \\ \hline \hline \end{tabular}
\end{table}
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\[\left[\nabla X_{u,t}^{\phi_{s,u}(\mu_{0})}\right](X_{s,u}^{\mu_{1}}(x ))^{\prime}-\left[\nabla X_{u,t}^{\phi_{s,u}(\mu_{0})}\right](X_{s,u}^{\mu_{0}}( x))^{\prime}\] \[=\int_{0}^{1}\left[\nabla^{2}X_{u,t}^{\phi_{s,u}(\mu_{0})}\right]( X_{s,u}^{\mu_{0}}(x)+\epsilon(X_{s,u}^{\mu_{1}}(y)-X_{s,u}^{\mu_{0}}(x)))^{ \prime}\;\left[X_{s,u}^{\mu_{1}}(x)-X_{s,u}^{\mu_{0}}(x)\right]d\epsilon\] \[=\left[\nabla^{2}X_{u,t}^{\phi_{s,u}(\mu_{0})}\right](X_{s,u}^{\mu _{0}}(x))^{\prime}\;\left[X_{s,u}^{\mu_{1}}(x)-X_{s,u}^{\mu_{0}}(x)\right]\] \[+\int_{0}^{1}(1-\epsilon)\;\left[\nabla^{3}X_{u,t}^{\phi_{s,u}( \mu_{0})}\right]\left(X_{s,u}^{\mu_{0}}(x)+\epsilon(X_{s,u}^{\mu_{1}}(y)-X_{s,u}^{\mu_{0}}(x))\right)^{\prime}\;\left[X_{s,u}^{\mu_{1}}(x)-X_{s,u}^{\mu_{0 }}(x)\right]^{\otimes 2}\;d\epsilon\]
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\[\begin{array}{rcl}\nabla\!\!\!/_{4}\left(\frac{\check{\nu}}{r^{4}}\right)&=& \nabla\!\!\!/_{4}\left(\!\not{\!\!div}\,\zeta+2\,^{\left(F\right)}\!\rho\,^{ \left(\check{F}\right)}\!\rho\right)\\ &=&\mathrm{d}\mathrm{f}\mathrm{v}\left(\nabla\!\!\!/_{4}\zeta\right)-\frac{1} {2}\kappa\,\not{\!\!div}\,\zeta+2(\nabla\!\!\!/_{4}\,^{\left(F\right)}\!\rho) \,^{\left(\check{F}\right)}\!\rho+2\,^{\left(F\right)}\!\rho\nabla\!\!\!/_{4} \,^{\left(\check{F}\right)}\!\rho\\ &=&\mathrm{d}\mathrm{f}\mathrm{v}\left(-\kappa\zeta-\beta-\,^{\left(F \right)}\!\rho\,^{\left(F\right)}\!\beta\right)-\frac{1}{2}\kappa\,\not{\! \!div}\,\zeta-2\kappa\,^{\left(F\right)}\!\rho\,^{\left(\check{F}\right)}\! \rho+2\,^{\left(F\right)}\!\rho(-\kappa\,^{\left(\check{F}\right)}\!\rho-\, ^{\left(F\right)}\!\rho\check{\kappa}+\,\not{\!\!div}\,^{\left(F\right)}\! \beta)\end{array}\]
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\[\vartheta_{-(N-1)\epsilon^{*},\emptyset}\big{(}\mathbf{u}(z)+ \mathbf{u}(w)-\mathbf{u}(\infty_{-})-\boldsymbol{\tau}(\boldsymbol{\epsilon }^{*})\big{|}\boldsymbol{\tau}\big{)}\vartheta_{-(N+1)\epsilon^{*}, \emptyset}\big{(}-\mathbf{u}(z^{\prime})-\mathbf{u}(w^{\prime})+\mathbf{u}( \infty_{-})+\boldsymbol{\tau}(\boldsymbol{\epsilon}^{*})\big{|}\boldsymbol{ \tau}\big{)}\] \[=e^{2i\pi\boldsymbol{\epsilon}^{*}\cdot\boldsymbol{\tau}( \boldsymbol{\epsilon}^{*})+2i\pi\boldsymbol{\epsilon}^{*}\cdot(\mathbf{u}(z)+ \mathbf{u}(w)+\mathbf{u}(w^{\prime})-2\mathbf{u}(\infty_{-})-2\boldsymbol{ \tau}(\boldsymbol{\epsilon}^{*}))}\] \[\quad\times\vartheta_{-N\epsilon^{*},\emptyset}\big{(} \mathbf{u}(z)+\mathbf{u}(w)-\mathbf{u}(\infty_{-})\big{|}\boldsymbol{\tau} \big{)}\vartheta_{-N\epsilon^{*},\emptyset}\big{(}-\mathbf{u}(z^{\prime})- \mathbf{u}(w^{\prime})+\mathbf{u}(\infty_{-})\big{|}\boldsymbol{\tau}\big{)}.\]
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\[I_{t}=\lim_{\varepsilon\to 0+}\int\limits_{0}^{1}\int\limits_{ \mathbb{R}^{N}}\eta(z)\Bigg{(}\int\limits_{K+\tau t\boldsymbol{k}+\varepsilon z }\Bigg{\{}\int\limits_{0}^{1}\nabla_{a}W\Big{(}P_{t}(u_{\varepsilon},y,s, \boldsymbol{k}),u_{\varepsilon}(y)\Big{)}ds\Bigg{\}}\cdot d\Big{[}Du(x)\cdot \boldsymbol{k}\Big{]}\Bigg{)}dzd\tau\\ =O\bigg{(}\|Du\|\Big{(}\cup_{\tau\in[0,1]}\big{(}\partial K+\tau t \boldsymbol{k}\big{)}\Big{)}\bigg{)}+\\ \lim_{\varepsilon\to 0+}\int\limits_{0}^{1}\int\limits_{ \mathbb{R}^{N}}\eta(z)\Bigg{(}\int\limits_{K+\tau t\boldsymbol{k}}\Bigg{\{} \int\limits_{0}^{1}\nabla_{a}W\Big{(}P_{t}(u_{\varepsilon},y,s,\boldsymbol{k} ),u_{\varepsilon}(y)\Big{)}ds\Bigg{\}}\cdot d\Big{[}Du(x)\cdot\boldsymbol{k} \Big{]}\Bigg{)}dzd\tau\\ =O\bigg{(}\|Du\|\Big{(}\cup_{\tau\in[0,1]}\big{(}\partial K+\tau t \boldsymbol{k}\big{)}\Big{)}\bigg{)}+\\ \lim_{\varepsilon\to 0+}\int\limits_{0}^{1}\Bigg{(}\int\limits_{K+ \tau t\boldsymbol{k}}\Bigg{\{}\int\limits_{0}^{1}\int\limits_{\mathbb{R}^{N}} \eta(z)\nabla_{a}W\Big{(}P_{t}(u_{\varepsilon},y,s,\boldsymbol{k}),u_{ \varepsilon}(y)\Big{)}dzds\Bigg{\}}\cdot d\Big{[}Du(x)\cdot\boldsymbol{k} \Big{]}\Bigg{)}d\tau.\]
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\[\tau_{i}^{\mathbf{B}}(\mathbf{A}) =\mathbf{a}_{i}^{\top}\left(\left(\begin{matrix}\mathbf{B}\\ \mathbf{a}_{i}^{\top}\end{matrix}\right)^{\top}\left(\begin{matrix}\mathbf{B} \\ \mathbf{a}_{i}^{\top}\end{matrix}\right)\right)^{+}\mathbf{a}_{i}\] \[=\mathbf{a}_{i}^{\top}\left(\left(\mathbf{B}^{\top}\mathbf{B} \right)^{+}-\frac{\left(\mathbf{B}^{\top}\mathbf{B}\right)^{+}\mathbf{a}_{i} \mathbf{a}_{i}^{\top}\left(\mathbf{B}^{\top}\mathbf{B}\right)^{+}}{1+\mathbf{ a}_{i}^{\top}\left(\mathbf{B}^{\top}\mathbf{B}\right)^{+}\mathbf{a}_{i}}\right) \mathbf{a}_{i}\] \[=\frac{\mathbf{a}_{i}^{\top}\left(\mathbf{B}^{\top}\mathbf{B} \right)^{+}\mathbf{a}_{i}}{\mathbf{a}_{i}^{\top}\left(\mathbf{B}^{\top} \mathbf{B}\right)^{+}\mathbf{a}_{i}+1}.\]
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\[e^{s_{\ell+1}(\frac{s_{\ell+1}}{2}+z_{m}+\frac{s_{m}}{2})^{2}+ \sum_{i\neq\ell+1}s_{i}z_{i}^{2}}\frac{s_{\ell+1}s_{m}}{s_{\ell+1}+s_{m}}\prod_ {(i,j)\in\tilde{S}_{1}}\frac{z_{j}-\frac{s_{j}}{2}-z_{i}+\frac{s_{j}}{2}\,z_{j }+\frac{s_{j}}{2}-z_{i}-\frac{s_{i}}{2}}{z_{j}+\frac{s_{j}}{2}-z_{i}+\frac{s_{ j}}{2}\,z_{j}-\frac{s_{j}}{2}-z_{i}-\frac{s_{i}}{2}}\\ \prod_{j\in S_{2}}\frac{z_{j}-\frac{s_{j}}{2}-z_{m}+\frac{s_{m}}{ 2}}{z_{j}+\frac{s_{j}}{2}-z_{m}+\frac{s_{m}}{2}}\frac{z_{j}+\frac{s_{j}}{2}-z_{ m}-s_{\ell+1}-\frac{s_{m}}{2}}{z_{j}-\frac{s_{j}}{2}-z_{m}-s_{\ell+1}-\frac{s_{m}} {2}}.\]
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\[I_{0}^{\pm}=\int_{-\infty}^{\infty}\left(v_{1}^{\pm}\right)^{2} \,dt=\frac{2}{3}\nu\sqrt{\nu}\left(3\phi^{2}-1\right)+4\nu\sqrt{\nu}\,\psi\phi ^{2}\tan^{-1}\left(\psi\mp\phi\right),\] \[I_{1}^{\pm}=\int_{-\infty}^{\infty}u_{1}^{\pm}\,\left(v_{1}^{\pm }\right)^{2}\,dt=-\sqrt{2}\nu^{2}\left\{\psi\left(\frac{5}{2}\phi^{2}-\frac{1} {3}\right)+\left(5\psi^{4}+6\psi^{2}+\frac{1}{3}\right)\tan^{-1}(\psi\mp\phi) \right\},\] \[I_{2}^{\pm}=\int_{-\infty}^{\infty}\left(u_{1}^{\pm}\,v_{1}^{\pm }\right)^{2}\,dt=\nu^{2}\sqrt{\nu}\left\{\frac{16}{15}+\frac{23}{3}\psi^{2}+7 \psi^{4}+2\psi\phi^{2}\left(3\phi^{2}+4\psi^{2}\right)\tan^{-1}\left(\psi\mp \phi\right)\right\},\]
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\[S_{6.2} \leq c\tau^{\delta}\varrho^{\delta+\vartheta\gamma(\theta\delta+n)/p} \left(\int_{\tau\varrho}^{\varrho}\frac{\mathrm{d}\nu}{\nu^{1+s+\vartheta n/p}} \right)^{\gamma}[\mathsf{gl}_{\vartheta,\delta}^{+}(\varrho)]^{\vartheta\gamma}\] \[\leq c\tau^{\delta-s\gamma-n\vartheta\gamma/p}\varrho^{\theta\delta \vartheta\gamma/p}\varrho^{\delta-s\gamma}[\mathsf{gl}_{\vartheta,\delta}^{+}( \varrho)]^{\vartheta\gamma}\] \[\leq\varrho^{\theta\vartheta\vartheta\gamma/p}\tau^{-n\vartheta \gamma/p}[\mathsf{gl}_{\vartheta,\delta}(\varrho)]^{p}+c(\mathsf{A}_{\gamma}+ \mathsf{B}_{\gamma})\varrho^{\theta\vartheta\vartheta\gamma/p}\tau^{-n\vartheta \gamma/p}\varrho^{\frac{p(\delta-s\gamma)}{p-\vartheta\gamma}}\] \[\leq c\varrho^{\theta\vartheta\vartheta\gamma/p}\tau^{-n\vartheta \gamma/p}[\mathsf{gl}_{\vartheta,\delta}(\varrho)]^{p}\] \[\leq c\tau^{\delta(a-n/\chi)}[\mathfrak{gl}^{+}_{\theta,\delta}(\varrho)]^{p}\,.\]
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\[\gamma_{-1}(A_{s,m}) \lesssim\int_{B_{s,m}}\int_{0}^{r_{s,m}(x^{\prime})}r^{n-1}e^{r^ {2}}\,dr\,dx^{\prime}\] \[\lesssim\int_{B_{s,m}}r_{s,m}(x^{\prime})^{n-2}e^{r_{s,m}(x^{ \prime})^{2}}\,dx^{\prime}\] \[=\frac{1}{s}\,e^{-c2^{2m}}\int_{B_{s,m}}I_{m}(r_{s,m}(x^{\prime}),x^{\prime})\,dx^{\prime}\] \[\leqslant\frac{1}{s}\,e^{-c2^{2m}}\int_{0}^{\infty}\rho^{2n-2} \int_{\mathbb{S}^{n-1}}f(\rho y^{\prime})\left(\int_{\{|y^{\prime}-x^{\prime}| \,\wedge\,|y^{\prime}+x^{\prime}|<2^{m+1}/\rho\}}\,dx^{\prime}\right)\,dy^{ \prime}e^{\rho^{2}}\,d\rho\] \[\lesssim\frac{1}{s}\,e^{-c2^{2m}}\int_{0}^{\infty}\rho^{2n-2} \left(\frac{2^{m}}{\rho}\right)^{n-1}\int_{\mathbb{S}^{n-1}}f(\rho y^{\prime} )\,dy^{\prime}e^{\rho^{2}}\,d\rho\] \[\approx\frac{1}{s}\,2^{m(n-1)}e^{-c2^{2m}}\|f\|_{L^{1}(\gamma_{-1 })}.\]
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\[c+o(1)\|Dv_{n}\|_{p}=F_{\lambda}(v_{n})-\frac{1}{\alpha p^{*}}F_ {\lambda}^{\prime}(v_{n})(G^{-1}(v_{n})g(G^{-1}(v_{n})))\psi_{\varepsilon}\] \[\quad-\frac{1}{\alpha p^{*}}\int_{\mathbb{R}^{N}}G^{-1}(v_{n})g(G ^{-1}(v_{n}))|Dv_{n}|^{p-2}Dv_{n}\cdot D\psi_{\varepsilon}dx+\frac{\lambda}{ \alpha p^{*}}\int_{\mathbb{R}^{N}}V|G^{-1}(v_{n})|^{k}\psi_{\varepsilon}dx\] \[\geq\frac{1}{N}\int_{\mathbb{R}^{N}}|Dv_{n}|^{p}\psi_{\varepsilon }dx-\frac{\lambda}{k}\int_{\mathbb{R}^{N}}V|G^{-1}(v_{n})|^{k}dx\] \[\quad-\frac{1}{\alpha p^{*}}\int_{B_{\epsilon}(x_{j})\setminus B_ {\epsilon/2}(x_{j})}G^{-1}(v_{n})g(G^{-1}(v_{n}))|Dv_{n}|^{p-1}|D\psi_{ \varepsilon}|dx.\]
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\[u_{1}=v_{1,1}^{\varepsilon(1)}\overset{h_{i(l)}^{\varepsilon(1)}}{\mapsto}v_{ 1,1}^{-\varepsilon(1)}\overset{c_{1}}{\mapsto}v_{1,2}^{\varepsilon(2)} \overset{h_{i(2)}^{\varepsilon(2)}}{\mapsto}\cdots\overset{h_{i(l-1)}^{ \varepsilon(l-1)}}{\mapsto}v_{1,l-1}^{-\varepsilon(l-1)}\overset{c_{l-1}}{ \mapsto}v_{1,l}^{\varepsilon(l)}\overset{h_{i(l)}^{\varepsilon(l)}}{\mapsto}v_ {1,l}^{-\varepsilon(l)}=w_{1}\\ \vdots\\ u_{d}=v_{d,1}^{\varepsilon(1)}\overset{h_{i(l)}^{\varepsilon(1)}}{\mapsto }v_{d,1}^{-\varepsilon(1)}\overset{c_{1}}{\mapsto}v_{d,2}^{\varepsilon(2)} \overset{h_{i(2)}^{\varepsilon(2)}}{\mapsto}\cdots\overset{h_{i(l-1)}^{ \varepsilon(l-1)}}{\mapsto}v_{d,l-1}^{-\varepsilon(l-1)}\overset{c_{l-1}}{ \mapsto}v_{d,l}^{\varepsilon(l)}\overset{h_{i(l)}^{\varepsilon(l)}}{\mapsto}v _{d,l}^{-\varepsilon(l)}=w_{d}.\]
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\[\boldsymbol{M}_{jk}^{(2)}=\langle\phi_{k},\phi_{j}^{*}\rangle_{N}=\begin{cases} \frac{2(j-1)}{j(2j+1)}d_{j}c_{j-2},&k=j-2,\ \ 2\leq j\leq N-1,\\ \frac{4}{(j+1)(j+2)}d_{j}c_{j-1},&k=j-1,\ \ 1\leq j\leq N-1,\\ \Big{(}\frac{2}{2j+1}-\frac{2(2j+3)}{(j+2)^{2}}+\Big{(}\frac{j+1}{j+2} \Big{)}^{2}\frac{2}{2j+5}\Big{)}d_{j}c_{j},&k=j,\ \ 0\leq j\leq N-2,\\ \frac{-N^{3}-2N^{2}+4N+2}{N(N+1)^{2}(2N-1)(2N+1)},&k=j=N-1,\\ -\frac{4}{(j+2)(j+3)}d_{j}c_{j+1},&k=j+1,\ \ 0\leq j\leq N-2,\\ \frac{2(j+1)}{(j+2)(2j+5)}d_{j}c_{j+2},&k=j+2,\ \ 0\leq j\leq N-3.\end{cases}\]
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\[\begin{split}\sum_{r=1}^{R}&\int_{\Omega}\frac{1}{1+ \varepsilon|g(v^{\varepsilon})|}\left[\frac{(v^{\varepsilon})^{y_{r}}}{v_{ \infty}^{y_{r}}}-\frac{(v^{\varepsilon})^{y_{r}^{\prime}}}{v_{\infty}^{y_{r}^ {\prime}}}\right]^{2}dx\\ &\geq\sum_{r=1}^{R}\int_{\Omega_{1}}\frac{1}{1+\varepsilon|g(v^{ \varepsilon})|}\left[\frac{(v^{\varepsilon})^{y_{r}}}{v_{\infty}^{y_{r}}}- \frac{(v^{\varepsilon})^{y_{r}^{\prime}}}{v_{\infty}^{y_{r}^{\prime}}}\right] ^{2}dx\\ &\geq K_{8}\sum_{r=1}^{R}\int_{\Omega_{1}}\left(\frac{1}{2}\left[ \frac{\overline{v}^{\varepsilon y_{r}}}{v_{\infty}^{y_{r}}}-\frac{\overline{v }^{\varepsilon y_{r}^{\prime}}}{v_{\infty}^{y_{r}^{\prime}}}\right]^{2}-K_{9} \sum_{i=1}^{I}|\delta_{i}(x)|^{2}\right)dx\\ &\geq\frac{K_{8}}{2}|\Omega_{1}|\sum_{r=1}^{R}\left[\frac{ \overline{v}^{\varepsilon y_{r}}}{v_{\infty}^{y_{r}}}-\frac{\overline{v}^{ \varepsilon y_{r}^{\prime}}}{v_{\infty}^{y_{r}^{\prime}}}\right]^{2}-K_{8}K_{9 }R\sum_{i=1}^{I}\|\delta_{i}\|_{L^{2}(\Omega)}^{2}.\end{split}\]
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\[\begin{split} I&=\int_{0}^{\infty}x\exp\left(- \frac{p^{2}x^{2}}{2}\right)\!I_{0}\left(cx\right)\\ &\times\Bigg{(}1+\exp\left(-\frac{\alpha^{2}x^{2}+\beta^{2}}{2} \right)\!I_{0}\left(axb\right)-Q\left(\alpha x,\beta\right)\Bigg{)}\,dx\\ &=\int_{0}^{\infty}x\exp\left(-\frac{p^{2}x^{2}}{2}\right)\!I_{0 }\left(cx\right)\\ &+\int_{0}^{\infty}x\exp\Bigg{(}\frac{-\left(\alpha^{2}+p^{2} \right)x^{2}+\beta^{2}}{2}\Bigg{)}\!I_{0}\left(cx\right)\!I_{0}\left(axb \right)\\ &-\int_{0}^{\infty}x\exp\Bigg{(}-\frac{p^{2}x^{2}}{2}\Bigg{)}\!I _{0}\left(cx\right)\!Q\left(\alpha x,\beta\right)dx.\end{split}\]
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\[\mathcal{W}_{3}(z_{1})\mathcal{W}_{3}(z_{2}) \sim \,\frac{\frac{c}{3}}{z_{12}^{6}}\,+\,\frac{2\,\mathcal{L}}{z_{12 }^{4}}\,+\,\frac{\mathcal{L}^{\prime}}{z_{12}^{3}}\,+\,\frac{1}{z_{12}^{2}} \Big{(}\frac{32\mathcal{L}^{2}}{5c}+\frac{32\mathcal{W}_{4}}{\sqrt{105}}+ \frac{3\mathcal{L}^{\prime\prime}}{10}\Big{)}\,+\,\frac{1}{z_{12}}\Big{(} \frac{32\mathcal{L}\mathcal{L}^{\prime}}{5c}+\frac{16\mathcal{W}_{4}^{ \prime}}{\sqrt{105}}+\frac{\mathcal{L}^{(3)}}{15}\Big{)}\] \[\mathcal{W}_{3}(z_{1})\mathcal{W}_{4}(z_{2}) \sim \,\frac{\sqrt{105}}{\sqrt{105}}\left(\frac{3\mathcal{W}_{3}}{z_{12 }^{4}}+\frac{\mathcal{W}_{3}^{\prime}}{z_{12}^{3}}\right)\,+\,\frac{1}{z_{12} ^{2}}\Big{(}\frac{416}{7c}\sqrt{\frac{3}{35}}\mathcal{L}\mathcal{W}_{3}+ \frac{4}{7}\sqrt{\frac{3}{35}}\mathcal{W}_{3}^{\prime\prime}+\frac{5}{7}\sqrt {15}\mathcal{W}_{5}\Big{)}\] \[+\,\frac{1}{z_{12}}\Big{(}\frac{40}{7c}\sqrt{\frac{15}{7}} \mathcal{W}_{3}\mathcal{L}^{\prime}+\frac{144}{7c}\sqrt{\frac{3}{35}}\mathcal{ L}\mathcal{W}_{3}^{\prime}+\frac{2}{7}\sqrt{15}\mathcal{W}_{5}^{\prime}+\frac{2 \mathcal{W}_{3}(3)}{7\sqrt{105}}\Big{)}\]
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\[(A.23)\] \[\leq C_{E}\left(\left(\frac{\epsilon\theta_{i}}{2}+\frac{1}{4 \epsilon}\right)\left\|\partial^{\nu}\sigma\right\|_{V}^{2}+\frac{\theta_{i}}{ \epsilon}\left\|(1-\Pi)\partial^{\nu}h\right\|_{V,\omega}^{2}\right)\] \[+\frac{\theta_{i}}{2}\sum_{\nu_{j}\neq 0}C_{j}\left(\epsilon\nu_{j}^{2} \left\|\partial^{\nu-e_{j}}h\right\|_{V}^{2}+\frac{1}{\epsilon}\left\|(1-\Pi) \partial^{\nu}h\right\|_{V,\omega}^{2}\right)+\frac{1}{4\epsilon}\sum_{\nu_{j} \neq 0}C_{j}\left(\nu_{j}^{2}\left\|\partial^{\nu-e_{j}}\sigma\right\|^{2}+ \left\|\partial^{\nu}\partial_{x}\sigma\right\|^{2}\right)\] \[\leq C_{E}\left(\frac{1}{2}\left(\epsilon\theta_{i}+\frac{1}{ \epsilon}\right)\left\|\partial^{\nu}\sigma\right\|_{V}^{2}+\frac{3\theta_{i}}{ 2\epsilon}\left\|(1-\Pi)\partial^{\nu}h\right\|_{V,\omega}^{2}\right)+\sum_{\nu _{j}\neq 0}\nu_{j}^{2}C_{j}\left(\frac{\epsilon\theta_{i}}{2}+\frac{1}{4\epsilon} \right)\left\|\partial^{\nu-e_{j}}h\right\|_{V}^{2},\]
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\[(R6)\ w_{\beta}(u)h_{\gamma}(s)w_{\beta}(u)^{-1}=\begin{cases}h_{\gamma}(s)& \text{if }\ (\beta,\gamma)=0,\\ h_{\mp\beta}(u^{\mp 1}su^{\mp 1})h_{\mp\beta}(u^{\pm 2})&\text{if }\ \gamma=\pm\beta,\\ h_{\sigma_{\beta}(\gamma)}(u^{-1}s)h_{\sigma_{\beta}(\gamma)}(u)&\text{if }\ \beta\pm\gamma\neq 0\text{ and }i=k,\\ h_{\sigma_{\beta}(\gamma)}(su)h_{\sigma_{\beta}(\gamma)}(u^{-1})&\text{if }\ \beta\pm\gamma\neq 0\text{ and }i=l,\\ h_{\sigma_{\beta}(\gamma)}(us)h_{\sigma_{\beta}(\gamma)}(u^{-1})&\text{if }\ \beta\pm\gamma\neq 0\text{ and }j=k,\\ h_{\sigma_{\beta}(\gamma)}(su^{-1})h_{\sigma_{\beta}(\gamma)}(u)&\text{if }\ \beta\pm\gamma\neq 0\text{ and }j=l,\\ \end{cases}\]
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\[\frac{1}{\delta}|(\widetilde{\rho}\partial^{\alpha}\partial_{x}u _{1},\frac{2\bar{\theta}}{3\bar{\rho}^{2}}\partial^{\alpha}\widetilde{\rho})| \leq C\frac{1}{\delta}(\|\widetilde{\rho}\|_{L^{\infty}}\|\partial^{ \alpha}\partial_{x}\widetilde{u}_{1}\|\|\partial^{\alpha}\widetilde{\rho}\|+\| \partial^{\alpha}\partial_{x}\bar{u}_{1}\|_{L^{\infty}}\|\widetilde{\rho}\|\| \partial^{\alpha}\widetilde{\rho}\|)\] \[\leq\widetilde{C}\delta\|\partial^{\alpha}\partial_{x}\widetilde{u }_{1}\|\|\partial^{\alpha}\widetilde{\rho}\|+C\|\widetilde{\rho}\|\|\partial^{ \alpha}\widetilde{\rho}\|\] \[\leq\eta\frac{1}{\delta^{1/2}}\|\partial^{\alpha}\partial_{x} \widetilde{u}_{1}\|^{2}+C_{\eta}\widetilde{C}\frac{1}{\varepsilon}\delta^{1/2} \delta^{2}\|\partial^{\alpha}\widetilde{\rho}\|^{2}+C\mathcal{E}_{2}(t)\] \[\leq\eta\frac{\varepsilon}{\delta^{1/2}}\|\partial^{\alpha}\partial_{x} \widetilde{u}_{1}\|^{2}+C_{\eta}\mathcal{E}_{2}(t).\]
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\[\begin{array}{|c|c|c|c|}\hline\mathbf{a}&c_{\mathbf{a},\lambda_{2}}&d_{ \lambda_{3}}&e_{\mathbf{a}}\\ \hline(1^{3},3^{2})&\frac{4}{5}\left(\frac{9\times 2^{3\lambda_{2}+3}+5}{2 ^{3}-1}\right)&\left(\frac{2\times 3^{3\lambda_{3}+2}-5}{3^{3}-1}\right)&\frac{8} {5}\\ \hline(1,3^{2},6^{2})&\frac{2}{5}\left(\frac{13\times 2^{3\lambda_{2}+1}-5}{ 2^{3}-1}\right)&\left(\frac{16\times 3^{3\lambda_{3}}+10}{3^{3}-1}\right)& \frac{4}{5}\\ \hline(1^{3},6^{2})&2\left(\frac{3\times 2^{3\lambda_{2}+1}+1}{2^{3}-1} \right)&\left(\frac{4\times 3^{3\lambda_{3}+2}-10}{3^{3}-1}\right)&4\\ \hline(1,6^{4})&\frac{2}{5}\left(\frac{3\times 2^{3\lambda_{2}+2}-5}{ 2^{3}-1}\right)&\left(\frac{16\times 3^{3\lambda_{3}}+10}{3^{3}-1}\right)& \frac{8}{5}\\ \hline(1,2^{2},6^{2})&\frac{2}{5}\left(\frac{2^{3\lambda_{2}+4}+5}{ 2^{3}-1}\right)&\left(\frac{4\times 3^{3\lambda_{3}+2}-10}{3^{3}-1} \right)&\frac{4}{5}\\ \hline(2^{3},3,6)&\frac{4}{5}\left(\frac{3\times 2^{3\lambda_{2}+2}-5}{ 2^{3}-1}\right)&\left(\frac{7\times 3^{3\lambda_{3}+1}+5}{3^{3}-1}\right)& \frac{-4}{5}\\ \hline(2,3^{3},6)&20\left(\frac{3\times 2^{3\lambda_{2}+1}+1}{2^{3}-1} \right)&\sigma_{3}(3^{\lambda_{3}-1})&0\\ \hline(2,3,6^{3})&4\left(\frac{2^{3\lambda_{2}+4}+5}{ 2^{3}-1}\right)&\sigma_{3}(3^{\lambda_{3}-1})&0\\ \hline(1^{2},2,3,6)&\frac{4}{5}\left(\frac{13\times 2^{3\lambda_{2}+1}-5}{ 2^{3}-1}\right)&\left(\frac{11\times 3^{3\lambda_{3}+1}-7}{3^{3}-1}\right)& \frac{8}{5}\\ \hline\end{array}\]
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\[\mathcal{U}_{13,4,5} = \Big{\{}7936,7360,6816,3008,3488,3680,5776,6496,6544,6736,7216,\] \[3720,5456,5512,5704,3400,1840,1924,5668,7180,3604,4904,4932,\] \[4994,1858,2840,2852,5410,6666,3346,1264,1698,4516,4801,5281,\] \[6406,6409,6661,7171,1380,1473,2706,3217,4328,4706,4748,4881,\] \[740,929,1801,2264,2388,2444,2636,3333,5254,696,1617,2274,\] \[3114,3142,4308,1228,2228,2609,2819,4426,5146,5189,376,466,\] \[714,1308,4274,4630,426,1550,2217,2245,4209,409,617,4156,\] \[661,355,1078,1081,309,333,2131,4235,391,1099,2078,583,\] \[110,539,2087\Big{\}}\]
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\[I_{2}+II= -\langle w_{i}\partial_{i}\partial_{t}T_{\beta}T_{q}\bar{f},( \partial_{t}+w_{i}\partial_{i})T_{\beta}T_{q}\bar{f}\rangle+R_{2},\] \[= -\langle w_{i}\partial_{i}\partial_{t}T_{\beta}T_{q}\bar{f},w_{i} \partial_{i}T_{\beta}T_{q}\bar{f}\rangle-\langle w_{i}\partial_{i}\partial_{t }T_{\beta}T_{q}\bar{f},\partial_{t}T_{\beta}T_{q}\bar{f}\rangle+R_{2}\] \[= -\langle\partial_{t}(w_{i}\partial_{i}T_{\beta}T_{q}\bar{f}),w_{i }\partial_{i}T_{\beta}T_{q}\bar{f}\rangle+\langle\frac{1}{2}(\partial_{i}w_{i })\partial_{t}T_{\beta}T_{q}\bar{f}+(\partial_{t}w_{i})\partial_{i}\partial_{t }T_{\beta}T_{q}\bar{f},\partial_{t}T_{\beta}T_{q}\bar{f}\rangle+R_{2}\] \[= -\frac{1}{2}\frac{d}{dt}\|w_{i}\partial_{i}T_{\beta}T_{q}\bar{f} \|^{2}_{L^{2}}+R_{2}.\]
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\[u(x_{1},x_{2},t)=e^{-a_{0}x_{1}}\left\{\kappa_{1}\left(E^{1}_{\alpha,1}(\gamma_{1}t^{\alpha})+\sum_{m=1}^{n}\mu^{m}(t-m\tau)^{\alpha m}E^{m+1}_{ \alpha,\alpha m+1}(\gamma_{1}(t-m\tau)^{\alpha})\right)\right.\] \[\left.+\hat{\kappa}_{1}\left(tE^{1}_{\alpha,2}(\gamma_{1}t^{ \alpha})+\sum_{m=1}^{n}\mu^{m}(t-m\tau)^{\alpha m+1}E^{m+1}_{\alpha,\alpha m+2 }(\gamma_{1}(t-m\tau)^{\alpha})\right)\right.\] \[\left.+\int_{0}^{t}\sum_{m=0}^{n}\mu^{m+1}(\xi-m\tau)^{\alpha(m+1 )-1}E^{m+1}_{\alpha,\alpha(m+1)}(\gamma_{1}(\xi-m\tau)^{\alpha})\phi_{1}(t- \tau-\xi)f(t-\tau-\xi)d\xi\right\}\] \[\left.+e^{-(a_{0}x_{1}+b_{1}x_{2})}\left\{\kappa_{2}\left(E^{1}_{ \alpha,1}(\gamma_{2}t^{\alpha})+\sum_{m=1}^{n}\mu^{m}(t-m\tau)^{\alpha m}E^{m+ 1}_{\alpha,\alpha m+1}(\gamma_{2}(t-m\tau)^{\alpha})\right)\right.\right.\] \[\left.+\hat{\kappa_{2}}\left(tE^{1}_{\alpha,2}(\gamma_{2}t^{ \alpha})+\sum_{m=1}^{n}\mu^{m}(t-m\tau)^{\alpha m+1}E^{m+1}_{\alpha,\alpha m+2 }(\gamma_{2}(t-m\tau)^{\alpha})\right)\right.\] \[\left.+\int_{0}^{t}\sum_{m=0}^{n}\mu^{m+1}(\xi-m\tau)^{\alpha(m+1 )-1}E^{m+1}_{\alpha,\alpha(m+1)}(\gamma_{2}(\xi-m\tau)^{\alpha})\phi_{2}(t- \tau-\xi)f(t-\tau-\xi)d\xi\right\},\]
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\[A_{\alpha,1} = \frac{\alpha}{ABC(\alpha)\Gamma(\alpha)}\int_{0}^{t_{n+1}}(t_{n+1 }-t)^{\alpha-1}\left\{\frac{t-t_{n-1}}{h}f(t_{n},u_{n})-\frac{t-t_{n}}{h}f(t_{ n},u_{n})\right\}\] \[= \frac{\alpha f(t_{n},u_{n})}{ABC(\alpha)\Gamma(\alpha)h}\left\{ \int_{0}^{t_{n+1}}(t_{n+1}-t)^{\alpha-1}f(t-t_{n-1})\right\}dt\] \[-\frac{\alpha f(t_{n-1},u_{n-1})}{ABC(\alpha)\Gamma(\alpha)h} \left\{\int_{0}^{t_{n+1}}(t_{n+1}-t)^{\alpha-1}f(t-t_{n-1})\right\}dt\] \[= \frac{\alpha f(t_{n},u_{n})}{ABC(\alpha)\Gamma(\alpha)h}\left\{ \frac{2ht_{n+1}^{\alpha}}{\alpha}-\frac{t_{n+1}^{\alpha+1}}{\alpha+1}\right\} -\frac{\alpha f(t_{n-1},u_{n-1})}{ABC(\alpha)\Gamma(\alpha)h}\left\{\frac{ht_ {n+1}^{\alpha}}{\alpha}-\frac{t_{n+1}^{\alpha+1}}{\alpha+1}\right\}.\]
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\[s_{a_{k}}^{*}as_{a_{k}}p_{r(A_{l},a_{k})} =s_{a_{k}}^{*}\left(\sum_{j=1}^{n}\mu_{j}s_{a_{j}}p_{B_{j}}s_{a_{j }}^{*}\right)s_{a_{k}}p_{r(A_{l},a_{k})}\] \[=\sum_{j=1}^{n}\mu_{j}s_{a_{k}}^{*}s_{a_{j}}p_{B_{j}}s_{a_{j}}^{* }s_{a_{k}}p_{r(A_{l},a_{k})}\] \[=\sum_{j\,:\,a_{j}=a_{k}}\mu_{j}p_{r(a_{k})}p_{B_{j}}p_{r(a_{k})} p_{r(A_{l},a_{k})}\] \[=\sum_{j\,:\,a_{j}=a_{k}}\mu_{j}p_{B_{j}}p_{r(A_{l},a_{k})}\] \[=\sum_{\begin{subarray}{c}j\,:\,a_{j}=a_{k}\text{ and }\\ r(A_{l},a_{k})\cap B_{j}\neq\emptyset\end{subarray}}\mu_{j}p_{r(A_{l},a_{k}) \cap B_{j}},\]
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\[\phi^{\prime}_{z}(z,\omega,t) =2\pi i(z\omega-t)\omega,\] \[\phi^{\prime\prime}_{z\zeta}(z,\omega,t) =2\pi i\begin{bmatrix}\omega^{\prime}\\ \omega_{n}\end{bmatrix}\begin{bmatrix}{}^{t}z^{\prime}-{z_{n}}^{t}\omega^{ \prime}/\omega_{n}&-1\end{bmatrix}+2\pi i(z\omega-t)\begin{bmatrix}E_{n-1}&0\\ {}^{-t}\omega^{\prime}/\omega_{n}&0\end{bmatrix}\] \[=2\pi i\begin{bmatrix}\omega^{\prime\prime}z^{\prime}-{z_{n}} \omega^{\prime\prime}{}^{t}\omega^{\prime}/\omega_{n}&-\omega^{\prime}\\ \omega_{n}{}^{t}z^{\prime}-{z_{n}}^{t}\omega^{\prime}&-\omega_{n}\end{bmatrix}+2 \pi i(z\omega-t)\begin{bmatrix}E_{n-1}&0\\ {}^{-t}\omega^{\prime}/\omega_{n}&0\end{bmatrix}.\]
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\[V_{0} =\frac{(N+1)}{2N}\int dxdy\,N^{2}V(N(x-y))\] \[\qquad\times\big{[}b^{*}(\gamma_{x})b^{*}(\gamma_{y})+b^{*}( \gamma_{x})b(\sigma_{x})+b^{*}(\gamma_{y})b(\sigma_{y})+b(\sigma_{x})b(\sigma _{y})+\langle\sigma_{x},\gamma_{y}\rangle\] \[\qquad-N^{-1}\langle\sigma_{x},\gamma_{y}\rangle\,\mathcal{N}-N^ {-1}a^{*}(\gamma_{y})a(\sigma_{x})\big{]}\] \[\qquad\times\big{[}b(\gamma_{y})b(\gamma_{x})+b^{*}(\sigma_{y})b (\gamma_{x})+b^{*}(\sigma_{x})b(\gamma_{y})+b^{*}(\sigma_{y})b^{*}(\sigma_{x}) +\langle\sigma_{x},\gamma_{y}\rangle\] \[\qquad-N^{-1}\langle\sigma_{x},\gamma_{y}\rangle\,\mathcal{N}-N^ {-1}a^{*}(\sigma_{x})a(\gamma_{y})\big{]},\]
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\[-\frac{(k-1)}{k}\sum_{j_{1}=0}^{k-1}\sum_{j_{2}=0}^{k-1}\cos^{k-2 }\left(\frac{2\pi(j_{1}-j_{2})}{k}\right)\sin^{2}\left(\frac{2\pi(j_{1}-j_{2}) }{k}\right)+\frac{1}{k}\sum_{j_{1}=0}^{k-1}\sum_{j_{2}=0}^{k-1}\cos^{k-1} \left(\frac{2\pi(j_{1}-j_{2})}{k}\right)\cos\left(\frac{2\pi(j_{1}-j_{2})}{k }\right)\] \[=-(k-1)\sum_{j=0}^{k-1}\cos^{k-2}\left(\frac{2\pi j}{k}\right) \sin^{2}\left(\frac{2\pi j}{k}\right)+\sum_{j=0}^{k-1}\cos^{k}\left(\frac{2 \pi j}{k}\right)\] \[=-(k-1)\sum_{j=0}^{k-1}\cos^{k-2}\left(\frac{2\pi j}{k}\right)+k \sum_{j=0}^{k-1}\cos^{k}\left(\frac{2\pi j}{k}\right)\] \[=\begin{cases}\frac{k^{2}}{2^{k-1}}&\text{if $k$ odd}\\ -\frac{k(k-1)}{2^{k-2}}\left(\frac{k-2}{\frac{k}{2}-1}\right)+\frac{k^{2}}{2^ {k}}\left(\binom{k}{\frac{k}{2}}+2\right)&\text{if $k$ even}\end{cases}.\]
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\[\alpha_{n,j} = \frac{\sin\left((2r_{n}+1)j\pi/n\right)}{2r_{n}\sin(j\pi/n)}-\frac {1}{2r_{n}}\] \[= \frac{(2r_{n}+1)j\pi/n}{2r_{n}j\pi/n}\Bigg{[}1-\frac{(2r_{n}+1)^{ 2}j^{2}\pi^{2}}{6n^{2}}+O\left(\frac{r_{n}^{4}j^{4}}{n^{4}}\right)\Bigg{]} \times\Bigg{[}1+O\left(\frac{j^{2}}{n^{2}}\right)\Bigg{]}-\frac{1}{2r_{n}}\] \[= \left(1+\frac{1}{2r_{n}}\right)\Bigg{[}1-\frac{(2r_{n}+1)^{2}j^ {2}\pi^{2}}{6n^{2}}+O\left(\frac{j^{2}}{n^{2}}\right)+O\left(\frac{r_{n}^{4}j ^{4}}{n^{4}}\right)\Bigg{]}-\frac{1}{2r_{n}}\] \[= 1-\frac{(2r_{n}+1)^{2}j^{2}\pi^{2}}{6n^{2}}+O\left(\frac{r_{n}j ^{2}}{n^{2}}\right)+O\left(\frac{r_{n}^{4}j^{4}}{n^{4}}\right)\] \[= 1-\frac{2r_{n}^{2}j^{2}\pi^{2}}{3n^{2}}+O\left(\frac{r_{n}j^{2}} {n^{2}}\right)+O\left(\frac{r_{n}^{4}j^{4}}{n^{4}}\right)\,,\]
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\[c(\mathbf{w})=\prod_{v\in S}\|(\mathbf{w}^{(v)})\|_{v,2} =\left(\prod_{v\in S_{r}}\|\mathbf{w}^{(v)}\|_{v,2}\right)\left( \prod_{v\in S_{e}}\|\mathbf{w}^{(v)}\|_{v,2}\right)\left(\prod_{v\in S_{f}}\| \mathbf{w}^{(v)}\|_{v,2}\right)\] \[<\left(\prod_{v\in S_{r}}\sqrt{(m+n)\varepsilon^{2}}\right)\left( \prod_{v\in S_{e}}(m+n)\varepsilon^{2}\right)\left(\prod_{v\in S_{f}}\varepsilon\right)\] \[=\left((m+n)^{|S_{r}|/2}\,\varepsilon^{|S_{r}|}\right)\left((m+n )^{|S_{e}|}\,\varepsilon^{2|S_{e}|}\right)\left(\varepsilon^{|S_{f}|}\right)\] \[=(m+n)^{|S_{r}|/2+|S_{e}|}\,\varepsilon^{|S_{r}|+2|S_{e}|+|S_{f}|}\] \[<(m+n)^{|S_{r}|/2+|S_{e}|}\,\varepsilon,\]
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\[\mathcal{T}^{n}v(x)=\mathcal{T}\left(\mathcal{T}^{n-1}v\right)(x)\] \[\leq\inf_{\mathbb{P}_{0}\in\mathbf{P}_{a_{\mathrm{loc}}^{*}}} \int_{\Omega_{\mathrm{loc}}}\bigg{[}r\left(x,{a_{\mathrm{loc}}}^{*}(x), \omega_{1}\right)\] \[\qquad\qquad+\alpha\inf_{\mathbb{P}\in\mathfrak{P}_{\omega_{1}, \mathbf{a}^{*}}}\int_{\Omega}\bigg{\{}\sum_{t=1}^{n-1}\alpha^{t-1}r(\omega_{t},{a_{\mathrm{loc}}}^{*}(\omega_{t}),\omega_{t+1})+\alpha^{n-1}v\left(\omega_{n }\right)\bigg{\}}\mathbb{P}(\mathrm{d}\omega)\bigg{]}\mathbb{P}_{0}(x; \mathrm{d}\omega_{1})\] \[=\inf_{\mathbb{P}_{0}\in\mathbf{P}_{a_{\mathrm{loc}}^{*}}}\int_{ \Omega_{\mathrm{loc}}}\bigg{[}r\left(x,{a_{\mathrm{loc}}}^{*}(x),\omega_{1}\right)\] \[\qquad\qquad+\alpha\inf_{\begin{subarray}{c}\tau_{t}\in\mathbf{P }_{a_{\mathrm{loc}}^{*}},\\ t=1,\ldots,n-1\end{subarray}}\int_{\Omega_{\mathrm{loc}}}\cdots\int_{\Omega_{ \mathrm{loc}}}\bigg{\{}\sum_{t=1}^{n-1}\alpha^{t-1}r(\omega_{t},{a_{\mathrm{ loc}}}^{*}(\omega_{t}),\omega_{t+1})\] \[\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\alpha^{ n-1}v\left(\omega_{n}\right)\bigg{\}}\mathbb{P}_{n-1}(\omega_{n-1};\mathrm{d} \omega_{n})\cdots\mathbb{P}_{1}(\omega_{1};\mathrm{d}\omega_{2})\mathbb{P}_{0} (x;\mathrm{d}\omega_{1})\] \[\leq\inf_{\begin{subarray}{c}\tau_{t}\in\mathbf{P}_{a_{\mathrm{ loc}}^{*}},\\ t=0,\ldots,n-1\end{subarray}}\int_{\Omega_{\mathrm{loc}}}\int_{\Omega_{\mathrm{ loc}}}\cdots\int_{\Omega_{\mathrm{loc}}}\bigg{\{}\sum_{t=0}^{n-1}\alpha^{t}r( \omega_{t},{a_{\mathrm{loc}}}^{*}(\omega_{t}),\omega_{t+1})\] \[\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\alpha^{ n}v\left(\omega_{n}\right)\bigg{\}}\mathbb{P}_{n-1}(\omega_{n-1};\mathrm{d} \omega_{n})\cdots\mathbb{P}_{1}(\omega_{1};\mathrm{d}\omega_{2})\mathbb{P}_{0} (x;\mathrm{d}\omega_{1})\] \[=\inf_{\mathbb{P}\in\mathfrak{P}_{\mathbf{a},\mathbf{a}^{*}}} \mathbb{E}_{\mathbb{P}}\bigg{[}\sum_{t=0}^{n-1}\alpha^{t}r(X_{t},{a_{\mathrm{ loc}}}^{*}(X_{t}),X_{t+1})+\alpha^{n}v(X_{n})\bigg{]}.\]
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\[f^{\prime\prime}(0) =\sum_{i=1}^{n}\sum_{j=1}^{m_{i}-1}(b_{i,j}-b_{i,j+1})^{2}/4+ \sum_{j=1}^{n-1}((b_{j,m_{j}}-b_{j+1,1})^{2}/4-(g_{j})^{2}/4)\] \[\leq\sum_{i=1}^{n}\sum_{j=1}^{m_{i}-1}\delta(b_{i,j+1}-b_{i,j})/2 +\sum_{j=1}^{n-1}(-b_{j,m_{j}}+b_{j+1,1}-g_{j})(-b_{j,m_{j}}+b_{j+1,1}+g_{j})/4\] \[\leq\delta/2+\sum_{j=1}^{n-1}\delta(-b_{j,m_{j}}+b_{j+1,1}+g_{j})/2\] \[\leq\delta/2+\delta(2(n-1)\delta+1)/2=\delta+(n-1)\delta^{2}\leq n\delta.\]
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\[\parallel\ ^{(n-1)}\tilde{\underline{\tilde{\chi}}}\parallel_{L^{2}( {\cal K}^{\tau_{1}})}^{2} =\int_{0}^{\tau_{1}}\parallel\ ^{(n-1)}\tilde{\underline{\chi}} \parallel_{L^{2}(S_{\tau,\tau})}^{2}d\tau\] \[\leq C\int_{0}^{\tau_{1}}\tau^{-2}\|r\ ^{(n-1)}\tilde{ \underline{\tilde{\chi}}}\|_{L^{2}(S_{\tau,\tau})}^{2}d\tau\] \[=C\int_{0}^{\tau_{1}}\tau^{-2}\frac{\partial}{\partial\tau}\left( \|r\ ^{(n-1)}\tilde{\underline{\tilde{\chi}}}\|_{L^{2}({\cal K}^{\tau})}^{2} \right)d\tau\] \[=C\left\{\tau_{1}^{-2}\|r^{(n-1)}\tilde{\underline{\tilde{\chi}}} \|_{L^{2}({\cal K}^{\tau_{1}})}^{2}+2\int_{0}^{\tau_{1}}\tau^{-3}\|r^{(n-1)} \tilde{\underline{\tilde{\chi}}}\|_{L^{2}({\cal K}^{\tau})}^{2}\right\}\] \[\qquad\leq C\tau_{1}^{2c_{0}-2}\overline{\underline{X}}^{2}(\tau_ {1})\]
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\[-4\pi\nabla_{\tau}\beta^{\rm sv}\Big{[}\begin{smallmatrix}j\\ k_{1}\\ z_{1}\end{smallmatrix};\tau\Big{]} =(k{-}j{-}2)\beta^{\rm sv}\Big{[}\begin{smallmatrix}j+1\\ k_{2}\\ z_{1}\end{smallmatrix};\tau\Big{]}+\delta_{j,k-2}(\tau{-}\bar{\tau})^{k}f^{(k)}(z |\tau)\] \[-4\pi\nabla_{\tau}\beta^{\rm sv}\Big{[}\begin{smallmatrix}j_{1} \atop k_{1}&k_{2}\\ z_{1}&z_{2}\end{smallmatrix};\tau\Big{]} =(k_{1}{-}j_{1}{-}2)\beta^{\rm sv}\Big{[}\begin{smallmatrix}j_{1 }+1\\ k_{1}\\ z_{1}\end{smallmatrix};\begin{smallmatrix}j_{2}\\ k_{2}\\ z_{2}\end{smallmatrix};\tau\Big{]}+(k_{2}{-}j_{2}{-}2)\beta^{\rm sv}\Big{[} \begin{smallmatrix}j_{1}&j_{2}+1\\ k_{1}&k_{2}\\ z_{1}&z_{2}\end{smallmatrix};\tau\Big{]}\] \[\qquad+\delta_{j_{2},k_{2}-2}(\tau{-}\bar{\tau})^{k_{2}}f^{(k_{2 })}(z_{2}|\tau)\beta^{\rm sv}\Big{[}\begin{smallmatrix}j_{1}\\ k_{1}\\ z_{1}\end{smallmatrix};\tau\Big{]}\]
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\[-\psi(z_{N_{k}|k}^{0},d_{N_{k}-1}^{0})-\sum_{i=N_{k+1}+1}^{N_{k}- 1}\ell(\hat{z}_{i|k}^{0},\hat{\nu}_{i|k}^{0},d_{i}^{0})\] \[=-\ell(x_{k},\kappa(x_{k}),d_{0}^{0})+\psi(\hat{z}_{N_{k+1}|k+1}^{ 0},d_{N_{k+1}-1}^{0})\] \[-\psi(z_{N_{k}|k}^{0},d_{N_{k}-1}^{0})-\sum_{i=N_{k+1}+1}^{N_{k}- 1}\ell(\hat{z}_{i|k}^{0},\hat{\nu}_{i|k}^{0},d_{i}^{0})\] \[=-\ell(x_{k},\kappa(x_{k}),d_{0}^{0})\] \[+\psi(\hat{z}_{N_{k+1}|k+1}^{0},d_{N_{k+1}-1}^{0})-V_{N_{k}-N_{k+1 }-1}^{\text{ahm}}(z_{N_{k+1}+1|k}^{0},\mathbf{d}^{0})\]
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\[e^{-rt}\left\|(x,y)^{\delta}(t)\right\|_{\mathcal{H}}^{2}+2\int _{0}^{t}e^{-rs}\left\|\nabla x^{\delta}(s)\right\|_{H}^{2}\,ds+2\varepsilon \int_{0}^{t}e^{-rs}\left\|\nabla_{\Gamma}y^{\delta}(s)\right\|_{H_{\Gamma}}^{2 }\,ds\] \[\qquad+2\int_{0}^{t}e^{-rs}\left((\xi,\xi_{\Gamma})^{\delta}(s),(x,y)^{\delta}(s)\right)_{\mathcal{H}}^{2}\,ds\] \[\qquad+2\int_{0}^{t}e^{-rs}\left((\xi,\xi_{\Gamma})^{\delta}(s), (x,y)^{\delta}(s)\right)_{\mathcal{H}}\,ds\] \[\qquad=\left\|(x_{0},y_{0})^{\delta}\right\|_{\mathcal{H}}^{2}+ \int_{0}^{t}e^{-rs}\left\|\mathcal{B}^{\delta}(s)\right\|_{\mathscr{L}_{2}( \mathcal{U},\mathcal{H})}^{2}\,ds+2\int_{0}^{t}e^{-rs}(x,y)^{\delta}(s) \mathcal{B}^{\delta}(s)\,d\mathcal{W}_{s}\,.\]
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\[\frac{1}{t}\iint\limits_{[0,t]\times\Omega}\phi^{N}(s,x)\bar{\rho }_{0}(x)\ \mathrm{d}s\ \mathrm{d}x =\frac{1}{t}\iint\limits_{[0,t]\times\Omega}\left(\phi^{N}(0^{+},x)+\int_{0}^{s}\partial_{t}\phi^{N}(r,x)\ \mathrm{d}r\right)\bar{\rho}_{0}(x)\ \mathrm{d}s\ \mathrm{d}x\] \[\geqslant\int_{\Omega}\phi^{N}(0^{+},\cdot)\bar{\rho}_{0}-\frac{1 }{t}\iint\limits_{[0,t]\times\Omega}s\alpha^{N}(s,x)\bar{\rho}_{0}(x)\ \mathrm{d}s\ \mathrm{d}x\] \[\geqslant\int_{\Omega}\phi^{N}(0^{+},\cdot)\bar{\rho}_{0}(x)- \iint\limits_{[0,t]\times\Omega}\alpha^{N}(s,x)\bar{\rho}_{0}(x)\ \mathrm{d}s\ \mathrm{d}x.\]
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\[(J_{\theta})(s)=\frac{\|f(s)\|_{X}^{p-2}J_{X}(f(s))}{\|f\|_{L_{p} (S;X)}^{p-2}}\] \[\qquad=\sum_{n\in M}\frac{\|f_{A_{n}}(s)\|_{X}^{p-2}J_{X}(f_{A_{n} }(s))}{\|f\|_{L_{p}(S;X)}^{p-2}}\] \[\qquad=\sum_{n\in M}\frac{\|f_{A_{n}}(s)\|_{X}^{p-2}J_{X}(f_{A_{n} }(s))}{\|f\|_{L_{p}(S;X)}^{p-2}}\] \[\qquad=\frac{1}{\|f\|_{L_{p}(S;X)}^{p-2}}\sum_{n\in M_{f}}\frac{ \|f_{A_{n}}\|_{L_{p}(S;X)}^{p-2}\|f(s)\|_{X}^{p-2}J_{X}(f_{A_{n}}(s))}{\|f_{A_{n }}\|_{L_{p}(S;X)}^{p-2}}\] \[\qquad=\frac{1}{\|f\|_{L_{p}(S;X)}^{p-2}}\sum_{n\in M_{f}}\left\| f_{A_{n}}\right\|_{L_{p}(S;X)}^{p-2}\frac{\|f_{A_{n}}(s)\|_{X}^{p-2}J_{X}(f_{A_{n }}(s))}{\|f_{A_{n}}\|_{L_{p}(S;X)}^{p-2}}\] \[= \frac{1}{\|f\|_{L_{p}(S;X)}^{p-2}}\sum_{n\in M}\|f_{A_{n}}\|_{L_{p}(S; X)}^{p-2}(J_{p}f_{A_{n}})(s)\] \[= \frac{1}{\|f\|_{L_{p}(S;X)}^{p-2}}\sum_{n\in M}\|f_{A_{n}}\|_{L_{p} (S;X)}^{p-2}(J_{p}f_{A_{n}})(s),\text{ for all }s\in S.\]
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\[A =-t^{j+1}\left\{\mathbf{c}\left(\left(\begin{array}{cc}1&0\\ 0&1\end{array}\right)\left(\begin{array}{cc}1&0\\ 0&t\end{array}\right)\right)+\sum\limits_{v\in\mathbb{F}_{q}^{*}}\mathbf{c} \left(\left(\begin{array}{cc}1&0\\ vt&1\end{array}\right)\left(\begin{array}{cc}1&0\\ 0&t\end{array}\right)\right)\right\}\left(X^{j}Y^{k-2-j}\right)\] \[=-t^{j+1}\left\{\mathbf{c}\left(\left(\begin{array}{cc}1&0\\ 0&t\end{array}\right)\right)+\sum\limits_{v\in\mathbb{F}_{q}^{*}}\left( \begin{array}{cc}1&0\\ vt&1\end{array}\right)\mathbf{c}\left(\left(\begin{array}{cc}1&0\\ 0&t\end{array}\right)\right)\right\}\left(X^{j}Y^{k-2-j}\right)\] \[=-t^{j+1}\mathbf{c}\left(\left(\begin{array}{cc}1&0\\ 0&t\end{array}\right)\right)\left(X^{j}Y^{k-2-j}\right)-t^{j+1}\sum\limits_{v \in\mathbb{F}_{q}^{*}}\mathbf{c}\left(\left(\begin{array}{cc}1&0\\ 0&t\end{array}\right)\right)\left(X^{j}(vtX+Y)^{k-2-j}\right)\] \[=-t^{j+1}\mathbf{c}\left(e\right)\left(X^{j}Y^{k-2-j}\right)-t^{j +1}\sum\limits_{v\in\mathbb{F}_{q}^{*}}\mathbf{c}\left(e\right)\left(\sum \limits_{n=0}^{k-2-j}\binom{k-2-j}{n}v^{n}t^{n}X^{j+n}Y^{k-2-j-n}\right)\] \[=-t^{j+1}\mathbf{c}\left(e\right)\left(X^{j}Y^{k-2-j}\right)+t^{j +1}\mathbf{c}\left(e\right)\left(\sum\limits_{n\geq 0}\ \binom{(\bmod\ q-1)}{n}\binom{k-2-j}{n}t^{n}X^{j+n}Y^{k-2-j-n}\right)\] \[=t^{j+1}\mathbf{c}\left(e\right)\left(\sum\limits_{n\equiv 0} \ \binom{(\bmod\ q-1)}{n}\binom{k-2-j}{n}t^{n}X^{j+n}Y^{k-2-j-n}\right)\]
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\[\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{TV}(\bar{\rho}(t)) =\sum_{i=0}^{N}\sigma_{i}(\rho_{i+1}^{\prime}-\rho_{i}^{\prime})=- \sum_{i=1}^{N}\mu_{i}\rho_{i}^{\prime}=\sum_{i=1}^{N}\mu_{i}N\rho_{i}^{2}(x_{i}^ {\prime}-x_{i-1}^{\prime})\] \[=\sum_{i=1}^{N}\mu_{i}N\rho_{i}^{2}(v_{i}U_{i}-v_{i-1}U_{i-1})\] \[=\sum_{i=1}^{N}\mu_{i}N\rho_{i}^{2}[v(\rho_{i})(U_{i}-U_{i-1})+(v _{i}-v(\rho_{i}))U_{i}-\big{(}v_{i-1}-v(\rho_{i})\big{)}U_{i-1}]\] \[\leq\sum_{i=1}^{N}\mu_{i}N\rho_{i}^{2}v(\rho_{i})(U_{i}-U_{i-1})= \sum_{i=1}^{N}\mu_{i}\rho_{i}I_{i},\]
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\[\big{\|}S_{\alpha}(t)u\mid\mathcal{N}^{\sigma}_{p,q,r}\big{\|} =\bigg{\|}\alpha\int_{0}^{\infty}\theta\Phi_{\alpha}(\theta)e^{t^ {\alpha}\theta\Delta}u(x)d\theta\mid\mathcal{N}^{\sigma}_{p,q,r}\bigg{\|}\] \[\leqslant\alpha\int_{0}^{\infty}\theta\Phi_{\alpha}(\theta) \big{\|}e^{t^{\alpha}\theta\Delta}u\mid\mathcal{N}^{\sigma}_{p,q,r}\big{\|}d\theta\] \[\leqslant C\alpha\int_{0}^{\infty}\theta\Phi_{\alpha}(\theta)(t^ {\alpha}\theta)^{\frac{s-\sigma}{2}}\big{\|}u\mid\mathcal{N}^{s}_{p,q,r}\big{\|}d\theta\] \[=C\big{\|}u\mid\mathcal{N}^{s}_{p,q,r}\big{\|}t^{\frac{(s-\sigma) \alpha}{2}}\int_{0}^{\infty}\theta^{1+\frac{s-\sigma}{2}}\Phi_{\alpha}(\theta )d\theta\] \[\leqslant C\big{\|}u\mid\mathcal{N}^{s}_{p,q,r}\big{\|}t^{\frac{(s -\sigma)\alpha}{2}}.\]
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\begin{table}
\begin{tabular}{|c|c|c|c|} \hline Rogue wave & Solution & \(M_{1}\) in (4.12) & \(M_{2}\) in (4.13) \\ \hline \(\lambda_{1}=\sqrt{z_{1}}\) & (1.2) with \(k=0.9\) & 1.45 & 3.96 \\ \(\lambda_{2}=\sqrt{z_{2}}\) & same & 1.71 & 4.68 \\ \(\lambda_{3}=\sqrt{z_{3}}\) & same & 1.84 & 5.03 \\ \(\lambda_{1}=\sqrt{z_{1}}\) & (1.3) with \(k=0.8\) & 1.80 & 4.67 \\ \(\lambda_{2}=\sqrt{\xi+i\eta}\) & same & 1.60 & 4.15 \\ \(\lambda_{1}=\sqrt{z_{1}}\) & (1.3) with \(k=0.2\) & 1.58 & 3.55 \\ \(\lambda_{2}=\sqrt{\xi+i\eta}\) & same & 1.71 & 3.84 \\ \hline \end{tabular}
\end{table}
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\[\sum_{\omega\in M_{L},\omega|\nu}\frac{[L_{\omega}:\mathbb{Q}_{ \nu}]}{[L:\mathbb{Q}]} \sum_{\phi^{n}(\alpha)=\beta}\log\max\{|\alpha|_{\omega},1\}\] \[=\sum_{\omega\in M_{L},\omega|\nu}\frac{[L_{\omega}:\mathbb{Q}_{ \nu}]}{[L:\mathbb{Q}]}\left(\sum_{\phi^{n}(\alpha)=\beta}\log|\alpha|_{\omega }-\sum_{\phi^{n}(\alpha)=\beta,|\alpha|_{\omega}<1}\log|\alpha|_{\omega}\right)\] \[=\sum_{\omega\in M_{L},\omega|\nu}\frac{[L_{\omega}:\mathbb{Q}_{ \nu}]}{[L:\mathbb{Q}]}\left(\log|\phi^{n}(0)-\beta|_{\omega}-\sum_{\phi^{n}( \alpha)=\beta,|\alpha|_{\omega}<1}\log|\alpha|_{\omega}\right)\] \[=r_{\nu}\log|\phi^{n}(0)-\beta|_{\nu}-\sum_{\omega\in M_{L}, \omega|\nu}\frac{[L_{\omega}:\mathbb{Q}_{\nu}]}{[L:\mathbb{Q}]}\sum_{\phi^{n }(\alpha)=\beta,|\alpha|_{\omega}<1}\log|\alpha|_{\omega},\]
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\[\mathrm{D}_{\Xi}(\Phi) =\inf_{\begin{subarray}{c}u_{i}\in C_{\mathrm{lin}}(\mathbb{R}_{ +},\mathbb{R}_{+})\\ H_{i},H_{i,j,k}\in C_{b}(\mathbb{R}^{n}):\\ \Psi^{V}_{(H_{i}),(H_{i,j,k}),(u_{i})}\geq\Phi\end{subarray}}\sum_{i=1}^{n}\int u _{i}(s_{i})\,\mathrm{d}\mu_{i}(s_{i})\] \[\geq\inf_{\begin{subarray}{c}u_{i}\in C_{\mathrm{lin}}(\mathbb{R} _{+},\mathbb{R}_{+})\\ H_{i}\in C_{b}(\mathbb{R}^{n}):\\ \Psi_{(H_{i}),(u_{i})}\geq\Phi\end{subarray}}\sum_{i=1}^{n}\int u_{i}(s_{i})\, \mathrm{d}\mu_{i}(s_{i})\] \[=\sup_{\mathbb{Q}\in\mathcal{M}(\mu_{1},\ldots,\mu_{n})}\int_{ \mathbb{R}^{n}_{+}}\Phi(s)\,\mathrm{d}\mathbb{Q}(s)\]
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\[\|\Pi^{2}(v_{1})(t,\cdot)-\Pi^{2}(v_{2})(t,\cdot)\|_{L^{1}( \mathbb{R}^{d})}\] \[\leq C^{2}\int\limits_{r}^{t}\int\limits_{r}^{s}\frac{1}{\sqrt{t-s}} \frac{1}{\sqrt{s-\theta}}\|v_{1}(\theta,\cdot)-v_{2}(\theta,\cdot)\|_{L^{1}( \mathbb{R}^{d})}\,\mathrm{d}\theta\,\mathrm{d}s\] \[= C^{2}\int\limits_{r}^{t}\int\limits_{\theta}^{t}\frac{1}{\sqrt{t -s}}\frac{1}{\sqrt{s-\theta}}\|v_{1}(\theta,\cdot)-v_{2}(\theta,\cdot)\|_{L^{ 1}(\mathbb{R}^{d})}\,\mathrm{d}s\,\mathrm{d}\theta\] \[= C^{2}\int\limits_{r}^{t}\Big{(}\int\limits_{\theta}^{t}\frac{1}{ \sqrt{t-s}}\frac{1}{\sqrt{s-\theta}}\,\mathrm{d}s\Big{)}\|v_{1}(\theta,\cdot )-v_{2}(\theta,\cdot)\|_{L^{1}(\mathbb{R}^{d})}\,\mathrm{d}\theta\] \[= C^{2}\int\limits_{r}^{t}\Big{(}\int\limits_{0}^{t-\theta}\frac{1 }{\sqrt{(t-\theta)-\omega}}\frac{1}{\sqrt{\omega}}\,\mathrm{d}\omega\Big{)} \|v_{1}(\theta,\cdot)-v_{2}(\theta,\cdot)\|_{L^{1}(\mathbb{R}^{d})}\,\mathrm{ d}\theta\] \[= \pi C^{2}\int\limits_{r}^{t}\|v_{1}(\theta,\cdot)-v_{2}(\theta, \cdot)\|_{L^{1}(\mathbb{R}^{d})}\,\mathrm{d}\theta.\]
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\[\Sigma(t) = E[(D_{t}-\mathbb{E}[D_{t}])(D_{t}-\mathbb{E}[D_{t}])^{\top}]\] \[= e^{\int_{0}^{t}\mathcal{A}(s)ds}E[(D_{0}-\mathbb{E}[D_{0}])(D_{0 }-\mathbb{E}[D_{0}])^{\top}]\left(e^{\int_{0}^{t}\mathcal{A}(s)ds}\right)^{\top}\] \[+ \left(\int_{0}^{t}e^{\int_{s}^{t}\mathcal{A}(u)du}dM_{s}\right) \left(\int_{0}^{t}e^{\int_{s}^{t}\mathcal{A}(u)du}dM_{s}\right)^{\top}\] \[= e^{\int_{0}^{t}\mathcal{A}(s)ds}\Sigma(0)e^{\int_{0}^{t} \mathcal{A}^{\top}(s)ds}+\int_{0}^{t}e^{\int_{s}^{t}\mathcal{A}(u)du} \mathcal{B}(s)e^{\int_{s}^{t}\mathcal{A}^{\top}(u)du}ds.\]
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\[G_{1}^{(+)}=\begin{pmatrix}\mathsf{G}_{11}(x,t)&\mathsf{G}_{12}(x,t)&\mathsf{ G}_{12}(-x,-t)&\overset{\ast}{\mathsf{G}}_{12}(x,t)&\overset{\ast}{\mathsf{G}}_{1 2}(-x,-t)\\ \overset{\ast}{\mathsf{G}}_{12}(x,t)&\mathsf{G}_{22}(x,t)&\overset{\ast}{ \mathsf{G}}_{32}(x,t)&\overset{\ast}{\mathsf{G}}_{42}(x,t)&\overset{\ast}{ \mathsf{G}}_{52}(-x,-t)\\ \overset{\ast}{\mathsf{G}}_{12}(-x,-t)&\mathsf{G}_{32}(x,t)&\overset{\ast}{ \mathsf{G}}_{22}(-x,-t)&\overset{\ast}{\mathsf{G}}_{52}(-x,-t)&\overset{\ast} {\mathsf{G}}_{42}(-x,-t)\\ \mathsf{G}_{12}(x,t)&\mathsf{G}_{42}(x,t)&\mathsf{G}_{52}(-x,-t)&\mathsf{G}_{ 22}(x,t)&\overset{\ast}{\mathsf{G}}_{32}(-x,-t)\\ \mathsf{G}_{12}(-x,-t)&\mathsf{G}_{52}(x,t)&\mathsf{G}_{42}(-x,-t)&\mathsf{G}_ {32}(-x,-t)&\overset{\ast}{\mathsf{G}}_{22}(-x,-t)\end{pmatrix}\]
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\begin{table}
\begin{tabular}{l l l} \hline \hline
**Keywords:** & & \\ \hline Biometrics; & Filtering; & Publishing; \\ Categorization; & Geospatial; & Resolving terms; \\ City; & Government; & Safety; \\ Clustering; & Information management; & Searching; \\ Context management; & Information technology; & Secure; \\ Counter terrorism; & Infrastructure; & Semantic consistency; \\ Crime; & Knowledge management; & Terrorism; \\ Data; & Machine-Learning; & Urban; \\ Database processing; & Predictive modelling; & Video processing; \\ Event detection; & Prevention; & Visualization; \\ Event notification; & Public space; & Workflow management. \\ \hline \hline \end{tabular}
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\[|G_{12}|\leq \sum_{p>Q_{b,h}}\|b_{p}\|_{\infty}\sum_{-1\leq p^{\prime}\leq p-2 }\lambda_{p^{\prime}}\|w_{p^{\prime}}\|_{2}\sum_{|q-p|\leq 2}\lambda_{q}^{2s}\|m_{q }\|_{2}\] \[\leq c_{r}\kappa\sum_{p>Q_{b,h}}\sum_{-1\leq p^{\prime}\leq p-2} \lambda_{p^{\prime}}^{s+1}\|w_{p^{\prime}}\|_{2}\lambda_{p^{\prime}}^{-s}\sum _{|q-p|\leq 2}\lambda_{q}^{s+1}\|m_{q}\|_{2}\lambda_{q}^{s-1}\] \[\lesssim c_{r}\kappa\sum_{p>Q_{b,h}}\lambda_{p}^{s+1}\|m_{p}\|_{2}\sum_{-1 \leq p^{\prime}\leq p-2}\lambda_{p^{\prime}}^{s+1}\|w_{p^{\prime}}\|_{2} \lambda_{p^{\prime}}^{-s}\lambda_{p}^{s-1}\] \[\lesssim c_{r}\kappa\sum_{q\geq-1}(\lambda_{q}^{2s+2}\|w_{q}\|_{2}^{2}+ \lambda_{q}^{2s+2}\|m_{q}\|_{2}^{2}).\]
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\[\left|e^{(\tau-\bar{\tau})\mathscr{L}_{\infty}}\hat{\varepsilon}_ {-}(\bar{\tau})(y,\tau)\right| \lesssim \sum_{j=2}^{\infty}e^{(\frac{\alpha}{2}-j)(\tau-\bar{\tau})} \left|\langle\hat{\varepsilon}_{-}(\bar{\tau})\phi_{j,\infty}\rangle_{L^{2}_{ \rho}}\right|\left|\phi_{j,\infty}(y)\right|\] \[\lesssim \sum_{j=2}^{\infty}4^{j}j!e^{(\frac{\alpha}{2}-j)(\tau-\bar{ \tau})}b^{\frac{\alpha}{2}+\eta}(\bar{\tau})\left|\phi_{j,\infty}(y)\right|\] \[\lesssim \langle y\rangle^{4}y^{-\gamma}b^{\frac{\alpha}{2}+\tilde{\eta}} (\tau)\sum_{j=2}^{\infty}|\beta_{j}|\,e^{(\frac{\alpha}{2}-j)(\tau-\bar{\tau })}b^{-\frac{\alpha}{2}-\tilde{\eta}}(\tau)b^{\frac{\alpha}{2}+\eta}(\bar{\tau })y^{2(j-1)}.\]
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\[\lim_{s\to 0}\left\|\frac{\nabla^{n}_{H_{R}}\varphi(x+sv_{n})- \nabla^{n}_{H_{R}}\varphi(x)}{s}-T_{n-1}\left(\nabla^{n+1}_{R}\varphi(x)( \cdot,\ldots,\cdot,h_{n})\right)\right\|_{\mathscr{L}^{(n-1)}(H_{R})}\] \[\quad=\lim_{s\to 0}\left\|T_{n-1}\left(\frac{\nabla^{n}_{R} \varphi(x+sv_{n})-\nabla^{n}_{R}\varphi(x)}{s}-\nabla^{n+1}_{R}\varphi(x)( \cdot,\ldots,\cdot,h_{n})\right)\right\|_{\mathscr{L}^{(n-1)}(H_{R})}\] \[\quad=\lim_{s\to 0}\left\|\frac{\nabla^{n}_{R}\varphi(x+sRh_{n})- \nabla^{n}_{R}\varphi(x)}{s}-\nabla^{n+1}_{R}\varphi(x)(\cdot,\ldots,\cdot,h_ {n})\right\|_{\mathscr{L}^{(n-1)}((\ker R)^{\perp})}=0.\]
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\[W_{F_{a}}(\omega) =\sum_{x\in\mathbb{F}_{2^{n}}}(-1)^{\operatorname{Tr}_{2^{n}/2}( ax^{2^{e}}h(\operatorname{Tr}_{2^{n}/2^{m}}(x)))+\operatorname{Tr}_{2^{n}/2}( \omega x)}\] \[=\sum_{u\in\mathbf{U}}\sum_{y\in\mathbb{F}_{2^{m}}}(-1)^{ \operatorname{Tr}_{2^{n}/2}(a(y+u)^{2^{e}}h(\operatorname{Tr}_{2^{n}/2^{m}}(u )))+\operatorname{Tr}_{2^{n}/2}(\omega(y+u))}\] \[=\sum_{u\in\mathbf{U}}(-1)^{\operatorname{Tr}_{2^{n}/2}(au^{2^{e} }h(\operatorname{Tr}_{2^{n}/2^{m}}(u))+\omega u)}\sum_{y\in\mathbb{F}_{2^{m}} }(-1)^{\operatorname{Tr}_{2^{n}/2}(ay^{2^{e}}h(\operatorname{Tr}_{2^{n}/2^{m} }(u))+\omega y)}.\]
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\[\|\Delta(U_{k,\varepsilon}^{+}\psi_{k,\varepsilon}^{+})\|_{ \mathsf{L}^{2}(B_{k,\varepsilon}^{+})}^{2} \leq C\varrho_{k,\varepsilon}^{\pm}\|2(\nabla U_{k,\varepsilon}^{ +},\nabla\psi_{k,\varepsilon}^{+})+U_{k,\varepsilon}^{+}\Delta\psi_{k, \varepsilon}^{+}\|_{\mathsf{L}^{\infty}(B_{k,\varepsilon}^{+}\setminus \mathcal{B}(\frac{\varrho_{k,\varepsilon}}{4},x_{k,\varepsilon}))}^{2}\] \[\leq C_{1}\varrho_{k,\varepsilon}^{n}\left(\varrho_{k,\varepsilon }^{-2}\|\nabla U_{k,\varepsilon}^{+}\|_{\mathsf{L}^{\infty}(B_{k,\varepsilon} ^{+}\setminus\mathcal{B}(\frac{\varrho_{k,\varepsilon}}{4},x_{k,\varepsilon }))}^{2}+\varrho_{k,\varepsilon}^{-4}\|U_{k,\varepsilon}^{+}\|_{\mathsf{L}^{ \infty}(B_{k,\varepsilon}^{+}\setminus\mathcal{B}(\frac{\varrho_{k, \varepsilon}}{4},x_{k,\varepsilon}))}^{2}\right)\] \[\leq C_{2}\varrho_{k,\varepsilon}^{n-2}\gamma_{k,\varepsilon}^{2} \varkappa_{k,\varepsilon}^{2}.\]
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\begin{table}
\begin{tabular}{|l|c|c|c|c|} \hline Design variable & Best known & \(x_{\text{lb}}\) & \(x_{\text{ub}}\) & Starting guess \\ \hline \hline Wing span \(x_{1}\) & 44.19 & 30.0 & 45.0 & 37.5 \\ Root cord \(x_{2}\) & 6.75 & 6.0 & 12.0 & 9.0 \\ Taper ratio \(x_{3}\) & 0.28 & 0.28 & 0.50 & 0.39 \\ Angle of attack at root \(x_{4}\) & 3.0 & -1.0 & 3.0 & 1.1 \\ Angle of attack at tip and at rest \(x_{5}\) & 0.72 & -1.0 & 3.0 & 1.0 \\ Tube external diameter \(x_{6}\) & 4.03 & 1.6 & 5.0 & 3.3 \\ Tube thickness \(x_{7}\) & 0.3 & 0.3 & 0.79 & 0.545 \\ \hline \hline Objective function value & \(-16.61011\) & \(10^{20}\) & \(-8.0157\) & \(-10.93552\) \\ \hline Constraint violation & 0 & \(3\times 10^{40}\) & 0 & \(2.01\times 10^{7}\) \\ \hline \end{tabular}
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\[\left\|\bm{x}^{*}-\widetilde{\bm{x}}\right\|_{\mathbf{W}_{k}^{-1}}\] \[= \left\|\sum_{i\in[k]}\mathbf{W}_{i}\mathbf{B}\left[\begin{array}[ ]{c}-\mathbf{L}_{FF}^{-1}\mathbf{L}_{FC}\\ \mathbf{I}\end{array}\right](\mathbf{SC}(\mathbf{L}_{i},C)^{\dagger}-\widetilde {\mathbf{SC}}_{i}^{\dagger})\left[\begin{array}{c}-\mathbf{L}_{FF}^{-1} \mathbf{L}_{FC}\\ \mathbf{I}\end{array}\right]^{\top}\mathbf{B}^{\top}\mathbf{W}_{i}^{1/2}(h^{( i)}\bm{v}^{(i)})\right\|_{\mathbf{W}_{k}^{-1}}\] \[\leq r\sum_{i\in[k]}\left\|\mathbf{W}_{i}\mathbf{B}\left[\begin{array} []{c}-\mathbf{L}_{FF}^{-1}\mathbf{L}_{FC}\\ \mathbf{I}\end{array}\right](\mathbf{SC}(\mathbf{L}_{i},C)^{\dagger}-\widetilde {\mathbf{SC}}_{i}^{\dagger})\left[\begin{array}{c}-\mathbf{L}_{FF}^{-1} \mathbf{L}_{FC}\\ \mathbf{I}\end{array}\right]^{\top}\mathbf{B}^{\top}\mathbf{W}_{i}^{1/2}(h^{( i)}\bm{v}^{(i)})\right\|_{\mathbf{W}_{i}^{-1}}\] \[\leq r\sum_{i\in[k]}\left\|\mathbf{SC}(\mathbf{L}_{i},C)^{1/2}( \mathbf{SC}(\mathbf{L}_{i},C)^{\dagger}-\widetilde{\mathbf{SC}}_{i}^{\dagger })\mathbf{SC}(\mathbf{L}_{i},C)^{1/2}\right\|_{2}=O(rk\delta)\]
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\[\sin(\zeta)k(z,y)\geq\eta(z,y):=\] \[\begin{array}{ll}\cos([\max\{|\zeta a+\frac{\pi}{4}|,|\zeta b+\frac{\pi}{4}|\} )\cos[\zeta(y-1)-\frac{\pi}{4}],&|z|<y,\ y\in[b,1],\\ \cos([\max\{|\zeta a+\frac{\pi}{4}|,|\zeta y+\frac{\pi}{4}|\})\cos[\zeta(y-1)- \frac{\pi}{4}],&|z|<y,\ y\in[a,b),\\ \cos([\max\{|\zeta(1-y)-\frac{\pi}{4}|,|\zeta(1-b)-\frac{\pi}{4}|\}\cos( \zeta y-\frac{\pi}{4}),&z>|y|,\ y\in[a,b),\\ \cos[\max\{|\zeta(1-a)-\frac{\pi}{4}|,|\zeta(1-b)-\frac{\pi}{4}|\}\cos( \zeta y-\frac{\pi}{4}),&z>|y|,\ y\in[-a,a),\\ \cos[\max\{|\zeta(1-y)-\frac{\pi}{4}|,|\zeta(1-b)-\frac{\pi}{4}|\}\cos( \zeta y-\frac{\pi}{4}),&z>|y|,\ y\in[-b,-a),\\ \cos([\max\{|\zeta a+\frac{\pi}{4}|,|\zeta y+\frac{\pi}{4}|\})\cos[\zeta(1+y)- \frac{\pi}{4}],&-|z|>y,\ y\in[-b,-a),\\ \cos([\max\{|\zeta a+\frac{\pi}{4}|,|\zeta b+\frac{\pi}{4}|\})\cos[\zeta(1+y)- \frac{\pi}{4}],&-|z|>y,\ y\in[-1,-b).\end{array}\]
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\[\frac{(k-4)C-4s^{2}c_{0}|p-2|}{4k^{2}C}\int_{\Omega}\left|\frac{ \eta\nabla f^{k}}{(\phi\circ u)^{k/2}}\right|^{2}+\frac{2s(k+2-2\alpha)}{k^{2} }\int_{\Omega}\left|f^{k}\eta\nabla\left(\frac{1}{(\phi\circ u)^{k/2}}\right) \right|^{2}\] \[+\left(s\beta-\frac{c_{0}|p-2|}{C}-A(\phi)\right)\int_{\Omega} \left(\frac{f^{2}}{\phi\circ u}\right)^{k+1}\eta^{2}\] \[\leq \frac{2kc_{0}^{2}(2s+p-1+|p-2|)^{2}}{3C^{2}}\int_{\Omega}\frac{f^ {2k+2}\eta^{2}}{(\phi\circ u)^{k+1}}+\frac{2s+4(p-1+|p-2|)^{2}}{k}\int_{\Omega} \left|\frac{f^{k}\nabla\eta}{(\phi\circ u)^{k/2}}\right|^{2}\] \[+K\int_{\Omega}\left(\frac{f^{2}}{\phi\circ u}\right)^{k}\eta^{2}.\]
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\[[\mathcal{O}_{\mathbf{Q}_{G}(s_{1}s_{2}\cdots s_{k}s_{k+1})}] =-\sum_{0\leq l\leq k}(-1)^{l}\mathbf{e}^{(l+1)\varpi_{1}}\sum_{ \begin{subarray}{c}J\subset[k]\\ |J|=l\end{subarray}}\left(\prod_{j\notin J,\,j+1\in J}(1-\mathfrak{t}_{j}) \right)[\mathcal{O}_{\mathbf{Q}_{G}}(w_{\circ}\epsilon_{J\cup\{k+1\}})]\] \[\quad+\sum_{0\leq l\leq k}(-1)^{l}\mathbf{e}^{l\varpi_{1}}\sum_{ \begin{subarray}{c}J\subset[k]\\ |J|=l\end{subarray}}\left(\prod_{j\notin J,\,j+1\in J}(1-\mathfrak{t}_{j}) \right)[\mathcal{O}_{\mathbf{Q}_{G}}(w_{\circ}\epsilon_{J})]\] \[\quad+\sum_{1\leq m\leq k}\sum_{0\leq l\leq m-1}(-1)^{l}\mathbf{e }^{l\varpi_{1}}\times\] \[\qquad\qquad\sum_{J\subset[m-1]}\mathfrak{t}_{\alpha_{m,k}^{ \vee}}\left(\prod_{j\notin J,\,j+1\in J}(1-\mathfrak{t}_{j})\right)[ \mathcal{O}_{\mathbf{Q}_{G}}(w_{\circ}\epsilon_{J}+w_{\circ}\epsilon_{k+1}-w_ {\circ}\epsilon_{m})]\] \[\quad+\sum_{1\leq m\leq k}\sum_{0\leq l\leq m}(-1)^{l}\mathbf{e} ^{l\varpi_{1}}\times\] \[\sum_{\begin{subarray}{c}J\subset[m]\\ |J|=l\end{subarray}}\mathsf{t}_{\alpha_{m,k}^{\vee}}\left(\prod_{j\notin J,\,j+ 1\in J}(1-\mathsf{t}_{j})\right)[\mathcal{O}_{\mathbf{Q}_{G}}(w_{\circ} \epsilon_{J}+w_{\circ}\epsilon_{k+1}-w_{\circ}\epsilon_{m})].\]
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\[-e^{2(-F+K)} \Big{(}\mathsf{c}^{2}\rho\frac{1+\frac{1}{\mathsf{c}^{2}} \mathfrak{Q}_{1}}{1-\frac{1}{\mathsf{c}^{2}}\mathfrak{Q}_{1}}+P\frac{3-\frac {2}{\mathsf{c}^{2}}\mathfrak{Q}_{1}}{1-\frac{1}{\mathsf{c}^{2}}\mathfrak{Q}_ {1}}\Big{)}+\mathsf{c}^{2}\rho_{\mathsf{N}}=\] \[=-\mathsf{c}^{2}[Q1]+[Q2]-3P-P\Big{(}e^{2(-F+K)}-\frac{\frac{1}{ \mathsf{c}^{2}}\mathfrak{Q}_{1}}{1-\frac{1}{\mathsf{c}^{2}}\mathfrak{Q}_{1}} \Big{)}=\] \[=-(Df^{\rho}_{\mathsf{N}}(u_{\mathsf{N}})w+\Upsilon_{1}\rho_{ \mathsf{N}}u_{\mathsf{N}})+2\rho_{\mathsf{N}}(\Phi_{N}+2\Omega^{2}\varpi^{2})+\] \[-3P_{\mathsf{N}}-\mathsf{c}^{2}H_{\rho}(w)+\frac{1}{\mathsf{c}^ {2}}\mathfrak{Q}_{5};\]
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\begin{table}
\begin{tabular}{c c c c} \hline \(\tau\) & \(1/20\) & \(1/40\) & \(1/80\) \\ \hline \(e_{B,2}(t_{n}=4)\) & 4.346e-10 & 6.010e-12 & 9.207e-14 \\ rate & - & 6.176 & 6.028 \\ \hline \(e_{\rho,2}(t_{n}=4)\) & 5.571e-10 & 8.602e-12 & 1.345e-13 \\ rate & - & 6.017 & 5.999 \\ \hline \(e_{u,2}(t_{n}=4)\) & 1.066e-09 & 1.677e-11 & 2.634e-13 \\ rate & - & 5.990 & 5.992 \\ \hline \hline \(e_{B,\infty}(t_{n}=4)\) & 3.522e-10 & 5.346e-12 & 7.997e-14 \\ rate & - & 6.042 & 6.063 \\ \hline \(e_{\rho,\infty}(t_{n}=4)\) & 6.493e-10 & 9.865e-12 & 1.545e-13 \\ rate & - & 6.041 & 5.996 \\ \hline \(e_{u,\infty}(t_{n}=4)\) & 9.499e-10 & 1.489e-11 & 2.334e-13 \\ rate & - & 5.996 & 5.995 \\ \hline \end{tabular}
\end{table}
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\[|\Pi_{f_{0},6}| \lesssim\sum_{k=1}^{3}\Big{(}t^{k}\|\partial_{v}^{k+1}f_{0}\|_{ \mathcal{G}^{\lambda,\beta;s}}\|g\|_{\mathcal{G}^{\lambda,\beta;s}}\|\partial_ {v}^{k}f_{0}\|_{\mathcal{G}^{\lambda,\beta;s}}+t^{k}\|\partial_{v}^{k+1}f_{0} \|_{\mathcal{G}^{\lambda,\beta;s}}\|\partial_{v}^{2}g\|_{\mathcal{G}^{\lambda,\beta;s}}\|\partial_{v}f_{0}\|_{\mathcal{G}^{\lambda,\beta;s}}\Big{)}\] \[\lesssim\mathrm{D}_{lo,f_{0}}^{\frac{1}{2}}\mathrm{E}_{lo,f_{0} }^{\frac{1}{2}}\left(\frac{\epsilon}{\langle t\rangle^{\frac{7}{4}}}+\frac{1} {\langle t\rangle^{2}}\mathrm{E}_{lo,g}^{\frac{1}{2}}\right)+\mathrm{D}_{lo,f_ {0}}^{\frac{1}{2}}\mathrm{E}_{lo,f_{0}}^{\frac{1}{2}}\left(\frac{\epsilon}{ \langle t\rangle}+\frac{\mathrm{E}_{lo,g}^{\frac{1}{2}}}{\langle t\rangle} \right),\] \[|\Pi_{f_{0},7}| \lesssim\sum_{k=1}^{3}\Big{(}\epsilon t^{k}\|\partial_{v}^{k+1}f_ {0}\|_{\mathcal{G}^{\lambda,\beta;s}}^{2}+t^{\frac{k}{2}}\|\partial_{v}^{k+1}f _{0}\|_{\mathcal{G}^{\lambda,\beta;s}}t\|h\|_{\mathcal{G}^{\lambda,\beta+2;s}} t^{\frac{1}{2}}\|\partial_{vv}f_{0}\|_{\mathcal{G}^{\lambda,\beta;s}}\Big{)}\] \[\lesssim\epsilon\mathrm{D}_{lo,f_{0}}+\mathrm{E}_{lo,h}^{\frac{1 }{2}}\mathrm{D}_{lo,f_{0}}.\]
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\[\mathcal{K}^{l}_{m,k}=\left\{\begin{array}{ll}-2\Delta t^{n+1} \frac{h_{j}^{y}}{h_{i-\frac{1}{2}}^{x}}\mathcal{E}_{x}^{-}S_{i,j}^{l,n}\mu_{m +N_{y}}^{l,n}-2\Delta t^{n+1}\frac{h_{j}^{y}}{h_{i-\frac{1}{2}}^{x}}\mathcal{E }_{x}^{+}S_{i-1,j}^{l,n}\mu_{m}^{l,n},\quad m=[i-1,j],\\ \\ -2\Delta t^{n+1}\frac{h_{i}^{y}}{h_{j-\frac{1}{2}}^{y}}\mathcal{E}_{y}^{+}S_{i,j-1}^{l,n}\mu_{m}^{l,n}-2\Delta t^{n+1}\frac{h_{i}^{x}}{h_{j-\frac{1}{2}}^{y} }\mathcal{E}_{y}^{-}S_{i,j}^{l,n}\mu_{m+1}^{l,n},\quad m=[i,j-1],\\ \\ 2\Delta t^{n+1}\left(\frac{h_{j}^{y}}{h_{i+\frac{1}{2}}^{x}}\mathcal{E}_{x}^{+ }S_{i,j}^{l,n}+\frac{h_{j}^{y}}{h_{i-\frac{1}{2}}^{x}}\mathcal{E}_{x}^{-}S_{i,j}^{l,n}+\frac{h_{i}^{x}}{h_{j+\frac{1}{2}}^{y}}\mathcal{E}_{y}^{+}S_{i,j}^{l, n}+\frac{h_{i}^{x}}{h_{j-\frac{1}{2}}^{y}}\mathcal{E}_{y}^{-}S_{i,j}^{l,n} \right)\mu_{m}^{l,n}\\ \\ \quad+2\Delta t^{n+1}\frac{h_{j}^{y}}{h_{i-\frac{1}{2}}^{x}}\mathcal{E}_{x}^{+ }S_{i-1,j}^{l,n}\mu_{m-N_{y}}^{l,n}+2\Delta t^{n+1}\frac{h_{j}^{y}}{h_{i+\frac {1}{2}}^{x}}\mathcal{E}_{x}^{-}S_{i+1,j}^{l,n}\mu_{m+N_{y}}^{l,n}\\ \\ \quad+2\Delta t^{n+1}\frac{h_{j}^{x}}{h_{j+\frac{1}{2}}^{y}}\mathcal{E}_{y}^{- }S_{i,j+1}^{l,n}\mu_{m+1}^{l,n}+2\Delta t^{n+1}\frac{h_{i}^{x}}{h_{j-\frac{1}{ 2}}^{y}}\mathcal{E}_{y}^{+}S_{i,j-1}^{l,n}\mu_{m-1}^{l,n},\ m=k,\\ \\ -2\Delta t^{n+1}\frac{h_{i}^{x}}{h_{j+\frac{1}{2}}^{y}}\mathcal{E}_{y}^{-}S_{i,j+1}^{l,n}\mu_{m+1}^{l,n}-2\Delta t^{n+1}\frac{h_{i}^{x}}{h_{j+\frac{1}{2}}^ {y}}\mathcal{E}_{y}^{+}S_{i,j}^{l,n}\mu_{m}^{l,n},\quad m=[i,j+1],\\ \\ -2\Delta t^{n+1}\frac{h_{j}^{y}}{h_{i+\frac{1}{2}}^{x}}\mathcal{E}_{x}^{-}S_{i +1,j}^{l,n}\mu_{m+N_{y}}^{l,n}-2\Delta t^{n+1}\frac{h_{j}^{y}}{h_{i+\frac{1}{2} }^{x}}\mathcal{E}_{x}^{+}S_{i,j}^{l,n}\mu_{m}^{l,n},\quad m=[i+1,j].\end{array}\right.\]
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\[\mathop{\mathbb{E}}_{G\sim\text{Gin}(d_{1},d_{2})}\Biggl{[}\exp \left(\sum_{i=1}^{n}\frac{8\varepsilon^{2}}{d_{2}^{2}}\left(\frac{x_{i}^{ \dagger}Gy_{i}}{x_{i}^{\dagger}Ax_{i}+y_{i}^{\dagger}By_{i}}\right)^{2} \right)\Biggr{]} =\mathop{\mathbb{E}}_{G\sim\text{Gin}(d_{1},d_{2})}\Biggl{[}\exp \left(\frac{8\varepsilon^{2}}{d_{2}^{2}}\mathsf{vec}(G)^{\dagger}Q\mathsf{vec }(G)\right)\Biggr{]}\] \[=\mathop{\mathbb{E}}_{v\sim N(0,I_{4}d_{2})}\Biggl{[}\exp\left( \frac{8\varepsilon^{2}}{d_{1}d_{2}^{2}}v^{\dagger}Qv\right)\Biggr{]}\] \[=\prod_{j=1}^{d_{1}d_{2}}\mathop{\mathbb{E}}_{g\sim N(0,1)} \Biggl{[}\exp\left(\frac{8\varepsilon^{2}g^{2}}{d_{1}d_{2}^{2}}\lambda_{j} \right)\Biggr{]}\] \[\leq e^{10\varepsilon^{2}(\lambda_{1}+\cdots+\lambda_{d_{1}d_{2}}) /(d_{1}d_{2}^{2})}\] \[\leq e^{\sqrt{d_{1}d_{2}}/70}\,,\]
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\[\sum_{j=1}^{n} \xi_{i}x_{j}^{2}\otimes 1-2\xi_{i}x_{j}\otimes x_{j}+\xi_{i}\otimes x_{j} ^{2}-x_{j}^{2}\otimes\xi_{i}+2x_{j}\otimes x_{j}\xi_{i}-1\otimes x_{j}^{2}\xi_{i}\] \[=2r_{i}\cdot D(X)\] \[=2\lim_{k\to\infty}p_{k}\otimes 1-1\otimes p_{k}\] \[=[(1\otimes\tau^{\circ})(2r_{i}\cdot D(X))]\otimes 1+1\otimes[(\tau \otimes 1)(2r_{i}\cdot D(X))]\] \[=\left(\sum_{j=1}^{n}\xi_{i}(x_{j}^{2}+\tau(x_{j}^{2}))+2x_{j} \tau(x_{j}\xi_{i})-\tau(x_{j}^{2}\xi_{i})\right)\otimes 1+1\otimes\left(\sum_{j=1}^ {n}\tau(\xi_{i}x_{j}^{2})-2\tau(\xi_{i}x_{j})x_{j}-(\tau(x_{j}^{2})+x_{j}^{2}) \xi_{i}\right)\] \[=\sum_{j=1}^{n}\xi_{i}x_{j}^{2}\otimes 1+\tau(x_{j}^{2})d(\xi_{i}) +2\tau(x_{j}\xi_{i})d(x_{j})-1\otimes x_{j}^{2}\xi_{i}.\]
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\[\|\text{proj}_{t}h\|_{2}^{2} =a_{I}^{\top}\Gamma_{m}a_{I}+2\sum_{i\leq m<j}a_{i}a_{j}\mathbb{E }\left\{\mathbb{E}[e_{i}(W_{t},S_{t},A_{t})\mid Z_{t},S_{t},A_{t}]\mathbb{E}[ e_{j}(W_{t},S_{t},A_{t})\mid Z_{t},S_{t},A_{t}]\right\}\] \[\quad+\mathbb{E}\left(\sum_{j>m}a_{j}\mathbb{E}[e_{j}(W_{t},S_{t },A_{t})\mid Z_{t},S_{t},A_{t}]\right)\] \[\geq a_{I}^{\top}\Gamma_{m}a_{I}-2\sum_{i\leq m<j}|a_{i}a_{j}| \mathbb{E}\left\{\mathbb{E}[e_{i}(W_{t},S_{t},A_{t})\mid Z_{t},S_{t},A_{t}] \mathbb{E}[e_{j}(W_{t},S_{t},A_{t})\mid Z_{t},S_{t},A_{t}]\right\}\] \[\geq a_{I}^{\top}\Gamma_{m}a_{I}-2\sum_{i\leq m<j}|a_{i}a_{j}|c \nu_{m}\] \[\geq\nu_{m}\|a_{I}\|_{2}^{2}-2c\nu_{m}\sum_{i\leq m}|a_{i}|\sum_{ j>m}|a_{j}|\] \[\geq\nu_{m}\|a_{I}\|_{2}^{2}-2c\nu_{m}\sqrt{\sum_{i\leq m}\lambda _{i}}\sqrt{\sum_{i\leq m}\frac{|a_{i}|^{2}}{\lambda_{i}}}\sqrt{\sum_{j>m} \lambda_{j}}\sqrt{\sum_{j>m}\frac{|a_{j}|^{2}}{\lambda_{j}}}\] \[\geq\nu_{m}\|a_{I}\|_{2}^{2}-2c\nu_{m}B\sqrt{\sum_{i=1}^{\infty} \lambda_{i}}\sqrt{\sum_{j>m}\lambda_{j}},\quad\text{since }\sum_{j=1}^{\infty} \frac{|a_{j}|^{2}}{\lambda_{j}}\leq B.\]
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\[\frac{C_{s+1}}{(2s+1)L}\frac{\partial}{\partial A_{s}}\mathcal{F} _{g,m}^{[\mathbb{C}^{n}/\mathbb{Z}_{n}]}[b_{0},\ldots,b_{n-1}]\] \[= \frac{1}{2}\mathcal{F}_{g-1,m+2}^{[\mathbb{C}^{n}/\mathbb{Z}_{n}] }[b_{0},\ldots,b_{s-1},b_{s}+2,b_{s+1},\ldots,b_{n-1}]\] \[+\frac{1}{2}\sum_{\begin{subarray}{c}g_{1}+g_{2}=g\\ m_{1}^{2}+m_{2}=m\\ b_{i}+b_{i}=b_{i}\end{subarray}}\mathcal{F}_{g_{1},m_{1}+1}^{[\mathbb{C}^{n}/ \mathbb{Z}_{n}]}[\bar{b}_{0},\ldots,\bar{b}_{s-1},\bar{b}_{s}+1,\bar{b}_{s+1}, \ldots,\bar{b}_{n-1}]\mathcal{F}_{g_{2},m_{2}+1}^{[\mathbb{C}^{n}/\mathbb{Z}_ {n}]}[\tilde{b}_{0},\ldots,\tilde{b}_{s-1},\tilde{b}_{s}+1,\tilde{b}_{s+1}, \ldots,\tilde{b}_{n-1}]\] \[-\frac{b_{s+1}}{2s+1}\mathcal{F}_{g,m}^{[\mathbb{C}^{n}/\mathbb{Z }_{n}]}[b_{0},\ldots,b_{s},b_{s+1}-1,b_{s+2},\ldots,b_{n-1};1].\]
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\[\left(\begin{matrix}x\\ y\\ z\end{matrix}\right)(\epsilon_{1},\epsilon_{2})\ =\ r\ \left(\begin{matrix}\sin(\theta{+}{\rm i}\epsilon_{1})\cos(\phi{+}{\rm i} \epsilon_{2})\\ \sin(\theta{+}{\rm i}\epsilon_{1})\sin(\phi{+}{\rm i}\epsilon_{2})\\ \cos(\theta{+}{\rm i}\epsilon_{1})\end{matrix}\right)\ =\ r\ \left(\begin{matrix}c_{1}c_{2}\,x-{\rm i}c_{1}s_{2}\,y+s_{1}s_{2}\,\frac{z\,y}{ \rho}+{\rm i}s_{1}c_{2}\,\frac{z\,x}{\rho}\\ c_{1}c_{2}\,y+{\rm i}c_{1}s_{2}\,x-s_{1}s_{2}\,\frac{z\,x}{\rho}+{\rm i}s_{1}c_{2} \,\frac{z\,y}{\rho}\\ c_{1}\,z\ -\ {\rm i}s_{1}\,\rho\end{matrix}\right)\]
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\begin{table}
\begin{tabular}{|l|l|} \hline \(s\) & \(\Lambda\) \\ \hline \hline
0 & \(\{1\}\) \\ \hline
1 & \(\{-2\sqrt{2}-5,2\sqrt{2}-5\}\) \\ \hline
2 & \(\{\frac{2^{\sqrt{2}-9}+i\sqrt{1455}}{3^{2/3}}+\frac{16}{\sqrt[3]{3}\left(-9+i \sqrt{1455}\right)}-7,-\frac{\left(1+i\sqrt{3}\right)\sqrt[3]{-9}+i\sqrt{1455} }{3^{2/3}}-\frac{8\left(1-i\sqrt{3}\right)}{\sqrt[3]{3}\left(-9+i\sqrt{1455} \right)}-7,\) \\ & \(-\frac{\left(1-i\sqrt{3}\right)\sqrt[3]{-9}+i\sqrt{1455}}{3^{2/3}}-\frac{8\left( 1+i\sqrt{3}\right)}{\sqrt[3]{3}\left(-9+i\sqrt{1455}\right)}-7\}\) \\ \hline \end{tabular}
\end{table}
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\[\begin{array}{rl}P_{\mathrm{b}}(2)=&P_{\mathrm{boa}}(2)+(1-P_{ \mathrm{boa}}(2))P_{\mathrm{pre}}(2)\\ =&\sum_{i=0}^{k}\pi_{i,k-i}+\frac{\lambda_{1}}{\lambda_{2}} \times\left(\sum_{i=0}^{k}\pi_{i,k-i}-\pi_{k,0}\right)\\ =&\pi_{k,0}+(1+\frac{\lambda_{1}}{\lambda_{2}})\times\left( \sum_{i=0}^{k}\pi_{i,k-i}-\pi_{k,0}\right)\\ =&\frac{A_{1}^{k}/k!}{\sum_{i=0}^{k}\frac{A_{1}^{i}}{i!}}+ \left(\frac{\lambda_{1}+\lambda_{2}}{\lambda_{2}}\right)\\ &\times\left(\frac{(A_{1}+A_{2})^{k}/k!}{\sum_{i=0}^{k}\frac{(A_{1 }+A_{2})^{i}}{i!}}-\frac{A_{1}^{k}/k!}{\sum_{i=0}^{k}\frac{A_{1}^{i}}{i!}}\right) \\ =&\frac{(A_{1}+A_{2})E_{k}(A_{1}+A_{2})-A_{1}E_{k}(A_{1})}{A_{2}}.\end{array}\]
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\[f^{*}=\sum_{\alpha\in\operatorname{Irr}(\mathbb{G})}\sum_{i,j=1}^{n_{\alpha}}d_{ \alpha}(\widehat{f}(\alpha)Q_{\alpha})_{i,j}(u_{j,i}^{\alpha})^{*}\] \[\langle\mu,f^{*}\rangle_{M(\mathbb{G}),C_{r}(\mathbb{G})} =\sum_{\alpha\in\operatorname{Irr}(\mathbb{G})}\sum_{i,j=1}^{n_{ \alpha}}d_{\alpha}\overline{(\widehat{f}(\alpha)Q_{\alpha})_{i,j}}\langle\mu,( u_{j,i}^{\alpha})^{*}\rangle_{M(\mathbb{G}),C_{r}(\mathbb{G})}\] \[=\sum_{\alpha\in\operatorname{Irr}(\mathbb{G})}\sum_{i,j=1}^{n_{ \alpha}}d_{\alpha}\overline{(\widehat{f}(\alpha)Q_{\alpha})_{i,j}}[\widehat{ \mu}(\alpha)]_{i,j}\] \[=\sum_{\alpha\in\operatorname{Irr}(\mathbb{G})}\sum_{i,j=1}^{n_{ \alpha}}d_{\alpha}\overline{(\widehat{f}(\alpha)Q_{\alpha})_{i,j}}[\widehat{ \mu}(\alpha)^{*}]_{j,i}\] \[=\sum_{\alpha\in\operatorname{Irr}(\mathbb{G})}d_{\alpha} \overline{\operatorname{tr}(\widehat{f}(\alpha)Q_{\alpha}\widehat{\mu}( \alpha)^{*})}\] \[=\sum_{\alpha\in\operatorname{Irr}(\mathbb{G})}d_{\alpha} \operatorname{tr}(\widehat{\mu}(\alpha)Q_{\alpha}\widehat{f}(\alpha)^{*}).\]
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