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\[\Lambda_{(1,-a)}.\Lambda_{(1,-a)}= -\sum_{\begin{subarray}{c}a^{\prime}\in\mathbb{Z}/p^{r}\mathbb{Z} \\ a^{\prime}\neq a\end{subarray}}\Lambda_{(1,-a^{\prime})}.\Lambda_{(1,-a)}-\sum_ {b\in\mathbb{Z}/p^{r-1}\mathbb{Z}}\Lambda_{(-pb,1)}.\Lambda_{(1,-a)}\] \[= -\deg\mathrm{S}(N)\sum_{\begin{subarray}{c}a^{\prime}\in \mathbb{Z}/p^{r}\mathbb{Z}\\ a^{\prime}\neq a\end{subarray}}p^{2\nu_{p}(a^{\prime}-a)}-\deg\mathrm{S}(N) \sum_{b\in\mathbb{Z}/p^{r-1}\mathbb{Z}}1\] \[= -\deg\mathrm{S}(N)\sum_{\begin{subarray}{c}a^{\prime}\in \mathbb{Z}/p^{r}\mathbb{Z}\\ a^{\prime}\neq 0\end{subarray}}p^{2\nu_{p}(a^{\prime})}-\deg\mathrm{S}(N)\cdot p^{r-1}\]
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\[\frac{\operatorname{d}\!\pi}{\operatorname{d}\!x\!\operatorname{ d}\!y}(x,y) =\exp\left(\frac{f(x)+g(y)-\|x-y\|^{2}}{2\sigma^{2}}\right)\frac{ \operatorname{d}\!\alpha}{\operatorname{d}\!x}(x)\frac{\operatorname{d}\! \beta}{\operatorname{d}\!y}(y)\] \[=\omega\exp\left(\mathcal{Q}(\mathbf{A}^{-1}\mathbf{a}+\mathbf{u},\mathbf{A}^{-1}+\mathbf{U})(x)-\frac{\|x-y\|^{2}}{2\sigma^{2}}+\mathcal{Q}( \mathbf{B}^{-1}\mathbf{b}+\mathbf{v},\mathbf{B}^{-1}+\mathbf{V})(y)\right)\] \[=\omega\exp\left(\mathcal{Q}(\mathbf{U}+\mathbf{A}^{-1})(x)+ \mathcal{Q}(\mathbf{V}+\mathbf{B}^{-1})(y)+\mathcal{Q}(\frac{\frac{14}{\sigma ^{2}}}{-\frac{14}{\sigma^{2}}}\frac{-\frac{14}{\sigma^{2}}}{\sigma^{2}})(x,y)\right)\] \[=\omega\exp\left(\mathcal{Q}\left(\begin{pmatrix}\mathbf{A}^{-1} \mathbf{a}+\mathbf{u}\\ \mathbf{B}^{-1}\mathbf{b}+\mathbf{v}\end{pmatrix},\begin{pmatrix}\mathbf{U}+ \mathbf{A}^{-1}+\frac{14}{\sigma^{2}}&0\\ 0&\mathbf{V}+\mathbf{B}^{-1}+\frac{14}{\sigma^{2}}\end{pmatrix}\right)(x,y)\right)\] \[=\omega\exp\left(\mathcal{Q}\left(\begin{pmatrix}\mathbf{A}^{-1} \mathbf{a}+\mathbf{u}\\ \mathbf{B}^{-1}\mathbf{b}+\mathbf{v}\end{pmatrix},\frac{1}{\sigma^{2}}\begin{pmatrix} \mathbf{F}&-\operatorname{Id}\\ -\operatorname{Id}&\mathbf{G}\end{pmatrix}\right)(x,y)\right)\] \[=\omega\exp\left(\mathcal{Q}(\mu,\Gamma)(x,y)\right)\]
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\[f_{m,p,i,j} =\mathcal{R}_{m,r}f_{m,p,i-1,j}-a_{i}\Psi_{m}\] \[=\operatorname{rem}\left(x^{m-r}f_{m,p,i-1,j},\ x^{m}-1\right)-a_{ i}\Psi_{m}\] \[=\operatorname{rem}\left(x^{m-r}\Theta_{i-1}\Psi_{m},\ x^{m}-1 \right)-a_{i}\Psi_{m}\] \[=\operatorname{rem}\left(x^{m-r}\Theta_{i-1}\Psi_{m},\ \Phi_{m}\Psi_{m}\right)-a_{i}\Psi_{m}\] \[x^{m}-1=\Phi_{m}\Psi_{m}\] \[\Psi_{m}\] \[=\operatorname{rem}\left(x^{m-r}\Theta_{i-1}-a_{i},\ \Phi_{m}\right)\Psi_{m}\] \[\deg\Phi_{m}>\deg a_{i}\] \[=\operatorname{rem}\left(-x^{m-r}\sum_{s=0}^{i-1}a_{s}x^{(m-r)(i -1-s)}-a_{i},\ \Phi_{m}\right)\Psi_{m}\] \[\Theta_{i-1}\] \[=\operatorname{rem}\left(-\sum_{s=0}^{i-1}a_{s}x^{(m-r)(i-s)}-a_ {i},\ \Phi_{m}\right)\Psi_{m}\] \[x^{m-r}\] \[=\operatorname{rem}\left(-\sum_{s=0}^{i}a_{s}x^{(m-r)(i-s)},\ \Phi_{m}\right)\Psi_{m}\] \[a_{i}=a_{i}x^{(m-r)(i-i)}\] \[\Theta_{i}\]
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\[\mathbb{E}\Big{[}\frac{\partial\log f(x_{k}^{0}|\theta_{k})}{ \partial x_{k}}|\theta_{k}\Big{]}=\int_{\mathcal{X}_{k}}\frac{\partial f(x_{k}^ {0}|\theta_{k})}{\partial x_{k}}dx_{k}^{0}:=\Delta f(x_{k}^{0})\big{|}_{x_{k}^ {0}\in\partial\mathcal{X}_{k}}\] \[=\left(\begin{array}{c}\int f(x_{k,1}^{0}=+\infty,x_{k,2}^{0}, \ldots,x_{k,p}^{0})dx_{k,2}^{0}\ldots dx_{k,p}^{0}\\ \int f(x_{k,1}^{0},x_{k,2}^{0}=+\infty,\ldots,x_{k,p}^{0})dx_{k,1}^{0}dx_{k,3}^ {0}\ldots dx_{k,p}^{0}\\ \vdots\\ \int f(x_{k,1}^{0},x_{k,2}^{0},\ldots,x_{k,p}^{0}=+\infty)dx_{k,1}^{0}\ldots x _{k,p-1}^{0}\end{array}\right)\] \[-\left(\begin{array}{c}\int f(x_{k,1}^{0}=-\infty,x_{k,2}^{0}, \ldots,x_{k,p}^{0})dx_{k,2}^{0}\ldots dx_{k,p}^{0}\\ \int f(x_{k,1}^{0},x_{k,2}^{0}=-\infty,\ldots,x_{k,p}^{0})dx_{k,1}^{0}dx_{k,3}^ {0}\ldots dx_{k,p}^{0}\\ \vdots\\ \int f(x_{k,1}^{0},x_{k,2}^{0},\ldots,x_{k,p}^{0}=-\infty)dx_{k,1}^{0}\ldots x _{k,p-1}^{0}\end{array}\right),\]
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\[\begin{split}-(\tilde{\varepsilon}_{2}^{jk_{1}j_{1}}P_{s_{1}}f_{+, \beta}^{j_{1}})(x)=&-\frac{\phi(\beta)\big{(}\frac{1}{2}-\beta \big{)}^{N-j_{1}}}{\big{(}\frac{1}{2}+\alpha\big{)}^{N-j_{1}}\big{(}\frac{1}{2 }-\alpha\big{)}^{N-j_{k}}}\int_{x}^{\infty}dye^{-(y-x)\alpha}e^{-\beta y}\\ &+\frac{\phi(\beta)\big{(}\frac{1}{2}-\beta\big{)}^{N-j_{1}}}{ \big{(}\frac{1}{2}-\alpha\big{)}^{N-j_{1}}\big{(}\frac{1}{2}+\alpha\big{)}^{N-j_ {k}}}\int_{s_{1}}^{\infty}dye^{-(x-y)\alpha}e^{-\beta y}\\ =&-\frac{\phi(\beta)\big{(}\frac{1}{2}-\beta\big{)}^{N-j_{1}}}{ \big{(}\frac{1}{2}+\alpha\big{)}^{N-j_{1}}\big{(}\frac{1}{2}-\alpha\big{)}^{N- j_{k}}}\frac{e^{-\beta x}}{\alpha+\beta}\\ &+\frac{\phi(\beta)\big{(}\frac{1}{2}-\beta\big{)}^{N-j_{1}}e^{- \alpha x}}{\big{(}\frac{1}{2}-\alpha\big{)}^{N-j_{1}}\big{(}\frac{1}{2}+\alpha \big{)}^{N-j_{k}}}\frac{e^{(\alpha-\beta)x}-e^{(\alpha-\beta)s_{1}}}{\alpha-\beta}.\end{split}\]
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\[\frac{d}{dt}\int h_{\theta^{6}}^{2}\,d\theta =2\int(1/k)_{\theta^{6}}k_{\theta^{6}}\,d\theta-2\int h_{\theta^{ 6}}^{2}\,d\theta\] \[\leq-c\int k_{\theta^{6}}^{2}\,d\theta-2\int h_{\theta^{6}}^{2} \,d\theta\] \[\qquad+C\int k_{\theta^{5}}^{2}k_{\theta}^{2}+k_{\theta^{4}}^{2} k_{\theta\theta}^{2}+k_{\theta^{4}}^{2}k_{\theta}^{4}+k_{\theta^{3}}^{4}+k_{ \theta^{3}}^{2}k_{\theta\theta}^{2}k_{\theta}^{2}+k_{\theta^{3}}^{2}k_{\theta} ^{6}+k_{\theta^{2}}^{6}+k_{\theta^{2}}^{4}k_{\theta}^{4}+k_{\theta^{2}}^{2}k_{ \theta}^{8}+k_{\theta}^{12}\,d\theta\] \[\leq-c\int k_{\theta^{6}}^{2}\,d\theta+C\int k_{\theta^{5}}^{2}+k _{\theta^{4}}^{2}+k_{\theta^{3}}^{4}+k_{\theta^{3}}^{2}\,d\theta+C-2\int h_{ \theta^{6}}^{2}\,d\theta\,.\]
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\[\mathbb{E}\left[\max_{\tau\in[0,t]}\left|\frac{\sigma}{\gamma} \int_{0}^{\tau}D(X_{s}^{\alpha}(\rho_{s}^{m_{k}})-\overline{X}_{s}^{m_{k}})dB_ {s}-\sigma\int_{0}^{\tau}D(X_{s}^{\alpha}(\rho)-\widehat{X}_{s})dB_{s}\right| ^{2}\right]\] \[\leq C\mathbb{E}\left[\int_{0}^{t}\left|\frac{1}{\gamma}(X_{s}^{ \alpha}(\rho^{m_{k}})-\overline{X}_{s}^{m_{k}})-(X_{s}^{\alpha}(\rho)-\widehat {X}_{s})\right|^{2}ds\right]\] \[\leq C\mathbb{E}\left[\int_{0}^{t}\frac{1}{\gamma^{2}}\left|(X_{ s}^{\alpha}(\rho^{m_{k}})-\overline{X}_{s}^{m_{k}})-(X_{s}^{\alpha}(\rho)- \widehat{X}_{s})\right|^{2}ds\right]+C\mathbb{E}\left[\int_{0}^{t}\left|\left( \frac{1}{\gamma}-1\right)(X_{s}^{\alpha}(\rho)-\widehat{X}_{s})\right|^{2}ds\right]\] \[\leq C\mathbb{E}\left[\int_{0}^{t}\left|\overline{X}_{s}^{m_{k}}- \widehat{X}_{s}\right|^{2}ds\right]+C|m_{k}|^{2},\quad\forall\,t\in[0,T].\]
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\[\begin{split}&\frac{E_{a}[f(X_{\tau_{y}^{-}}),X_{\tau_{y}^{-}}\in \mathbb{B}_{y}^{(-i)}]}{E_{a+e_{1}}[f(X_{\tau_{y}^{-}}),X_{\tau_{y}^{-}}\in \mathbb{B}_{y}^{(-i)}]}=p_{a,a+e_{1}}\\ &+p_{a,a+e_{2}}\frac{E_{a+e_{2}}[f(X_{\tau_{y}^{-}}),X_{\tau_{y}^{ -}}\in\mathbb{B}_{y}^{(-i)}]}{E_{a+e_{1}+e_{2}}[f(X_{\tau_{y}^{-}}),X_{\tau_{y}^ {-}}\in\mathbb{B}_{y}^{(-i)}]}\bigg{(}\frac{E_{a+e_{1}}[f(X_{\tau_{y}^{-}}),X_{ \tau_{y}^{-}}\in\mathbb{B}_{y}^{(-i)}]}{E_{a+e_{1}+e_{2}}[f(X_{\tau_{y}^{-}}),X _{\tau_{y}^{-}}\in\mathbb{B}_{y}^{(-i)}]}\bigg{)}^{-1}\end{split}\]
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\[\mathbb{E}\left[A_{T_{S}}^{(S)}\left(F(u^{(S)})-F(x^{*})\right)\right]\] \[\leq A_{T_{0}}^{(0)}\left(F(u^{(0)})-F(x^{*})\right)+\mathbb{E}\left[\frac{ \gamma_{0}^{(s)}}{2}\left\|z_{0}^{(s)}-x^{*}\right\|^{2}-\frac{\gamma_{0}^{(S+1) }}{2}\left\|z_{0}^{(S+1)}-x^{*}\right\|^{2}\right]\] \[+\mathbb{E}\left[\sum_{s=1}^{S}\sum_{t=1}^{T_{s}}\frac{\gamma_{t}^ {(s)}-\gamma_{t-1}^{(s)}}{2}\left\|x_{t}^{(s)}-x^{*}\right\|^{2}+\left(\frac{ \left(a^{(s)}\right)^{2}}{2\gamma_{t}^{(s)}}-\frac{\left(a^{(s)}\right)^{2}}{1 6\beta}\right)\left\|g_{t}^{(s)}-g_{t-1}^{(s)}\right\|^{2}\right]\] \[\leq A_{T_{0}}^{(0)}\left(F(u^{(0)})-F(x^{*})\right)+\frac{\gamma_{0} ^{(1)}}{2}\left\|z_{0}^{(s)}-x^{*}\right\|^{2}\] \[+\mathbb{E}\left[\sum_{s=1}^{S}\sum_{t=1}^{T_{s}}\frac{\gamma_{t} ^{(s)}-\gamma_{t-1}^{(s)}}{2}\left\|x_{t}^{(s)}-x^{*}\right\|^{2}+\left(\frac{ \left(a^{(s)}\right)^{2}}{2\gamma_{t}^{(s)}}-\frac{\left(a^{(s)}\right)^{2}}{1 6\beta}\right)\left\|g_{t}^{(s)}-g_{t-1}^{(s)}\right\|^{2}\right]\] \[= A_{T_{0}}^{(0)}\left(F(u^{(0)})-F(x^{*})\right)+\frac{\gamma}{2 }\left\|u^{(0)}-x^{*}\right\|^{2}\] \[+\mathbb{E}\left[\sum_{s=1}^{S}\sum_{t=1}^{T_{s}}\frac{\gamma_{t} ^{(s)}-\gamma_{t-1}^{(s)}}{2}\left\|x_{t}^{(s)}-x^{*}\right\|^{2}+\left(\frac{ \left(a^{(s)}\right)^{2}}{2\gamma_{t}^{(s)}}-\frac{\left(a^{(s)}\right)^{2}}{1 6\beta}\right)\left\|g_{t}^{(s)}-g_{t-1}^{(s)}\right\|^{2}\right],\]
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\[\mathbb{E}_{N_{1,\,\delta,\,\theta}(\mathsf{b}_{n})-1}\tau_{n}\\ =\exp\Big{(}t_{n}\sum_{k=1}^{N_{1,\,\delta,\,\theta}(\mathsf{b}_ {n})-1}\mathsf{b}_{n}^{S_{k-1}}\tilde{\eta}_{k}(\mathsf{b}_{n})-(t_{n}^{2}/2)e ^{4\rho(1+\varepsilon)}\sum_{k=1}^{N_{1,\,\delta,\,\theta}(\mathsf{b}_{n})-1} \mathsf{b}_{n}^{2S_{k-1}}\mathbb{E}_{k-1}(\tilde{\eta}_{k}^{2}(\mathsf{b}_{n}) )\Big{)}\\ \times\mathbb{E}_{N_{1,\,\delta,\,\theta}(\mathsf{b}_{n})-1} \big{(}\exp(t_{n}\mathsf{b}_{n}^{S_{k-1}}\tilde{\eta}_{N_{1,\,\delta,\, \theta}(\mathsf{b}_{n})}(\mathsf{b}_{n}))\big{)}\exp(-(t_{n}^{2}/2)e^{4\rho(1+ \varepsilon)}\mathsf{b}_{n}^{2S_{k-1}}\mathbb{E}_{N_{1,\,\delta,\,\theta}( \mathsf{b}_{n})-1}(\tilde{\eta}_{N_{1,\,\delta,\,\theta}(\mathsf{b}_{n})}^{2} (\mathsf{b}_{n}))\\ \leq\exp\Big{(}t_{n}\sum_{k=1}^{N_{1,\,\delta,\,\theta}(\mathsf{ b}_{n})-1}\mathsf{b}_{n}^{S_{k-1}}\tilde{\eta}_{k}(\mathsf{b}_{n})-(t_{n}^{2}/2)e ^{4\rho(1+\varepsilon)}\sum_{k=1}^{N_{1,\,\delta,\,\theta}(\mathsf{b}_{n})-1} \mathsf{b}_{n}^{2S_{k-1}}\mathbb{E}_{k-1}(\tilde{\eta}_{k}^{2}(\mathsf{b}_{n}) )\Big{)}\quad\text{a.s.}\]
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\[WG\underline{\beta} =\left(\begin{array}{c}\underline{w}^{(1)}G\\ \underline{w}^{(2)}G\\ \vdots\\ \underline{w}^{(l)}G\end{array}\right)\underline{\beta}=\left(\begin{array}[] {cccc}g_{1}(\underline{w}^{(1)})&g_{2}(\underline{w}^{(1)})&\cdots&g_{l}( \underline{w}^{(1)})\\ g_{1}(\underline{w}^{(2)})&g_{2}(\underline{w}^{(2)})&\cdots&g_{l}( \underline{w}^{(2)})\\ \cdots&\cdots&\cdots&\cdots\\ g_{1}(\underline{w}^{(l)})&g_{2}(\underline{w}^{(l)})&\cdots&g_{l}( \underline{w}^{(l)})\end{array}\right)\left(\begin{array}{c}\beta_{1}\\ \beta_{2}\\ \vdots\\ \beta_{l}\end{array}\right)\] \[=\left(\begin{array}{c}\sum_{j=1}^{l}\beta_{j}g_{j}( \underline{w}^{(1)})\\ \sum_{j=1}^{l}\beta_{j}g_{j}(\underline{w}^{(2)})\\ \cdots\\ \sum_{j=1}^{l}\beta_{j}g_{j}(\underline{w}^{(l)})\end{array}\right)= \underline{a}+\underline{\delta},\]
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\[r(\{1\},\{1\}) =(-1)^{2}T(\{1\}|\{1\})T(\{2,3\}|\{2,3\})=0,\] \[r(\{1\},\{2\}) =(-1)^{3}T(\{1\}|\{2\})T(\{2,3\}|\{1,3\})=-12,\] \[r(\{1\},\{3\}) =(-1)^{4}T(\{1\}|\{3\})T(\{2,3\}|\{1,2\})=-12,\] \[r(\{2\},\{1\}) =(-1)^{3}T(\{2\}|\{1\})T(\{1,3\}|\{2,3\})=-18,\] \[r(\{2\},\{2\}) =(-1)^{4}T(\{2\}|\{2\})T(\{1,3\}|\{1,3\})=-6,\] \[r(\{2\},\{3\}) =(-1)^{5}T(\{2\}|\{3\})T(\{1,3\}|\{1,2\})=0,\] \[r(\{3\},\{1\}) =(-1)^{4}T(\{3\}|\{1\})T(\{1,2\}|\{2,3\})=-6,\] \[r(\{3\},\{2\}) =(-1)^{5}T(\{3\}|\{2\})T(\{1,2\}|\{1,3\})=-6,\] \[r(\{3\},\{3\}) =(-1)^{6}T(\{3\}|\{3\})T(\{1,2\}|\{1,2\})=-12.\]
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\[\begin{array}{l}U_{1}^{1}=u_{0}^{1}+\Delta x\frac{1}{2}\mathcal{W}_{1}^{1}, \quad\mathcal{W}_{1}^{1}=w_{0}^{1}+\Delta x\frac{1}{2}\delta_{x}\mathcal{W}_{ 1}^{1},\\ \overline{u}_{1}^{1}=u_{0}^{1}+\Delta x\mathcal{W}_{1}^{1},\quad\overline{w}_{ 1}^{1}=w_{0}^{1}+\Delta x\delta_{x}\mathcal{W}_{1}^{1},\\ U_{1}^{1}=u_{1}^{0}+\Delta t\frac{1}{2}V_{1}^{1},\quad V_{1}^{1}=v_{1}^{0}+ \Delta t\frac{1}{2}\left(\delta_{x}\mathcal{W}_{1}^{1}-f(U_{1}^{1})\right), \\ \overline{u}_{1}^{1}=u_{1}^{0}+\Delta tV_{1}^{1},\quad\overline{v}_{1}^{1}=v_ {1}^{0}+\Delta t\left(\delta_{x}\mathcal{W}_{1}^{1}-f(U_{1}^{1})\right),\\ u_{1}^{1}=\overline{u}_{1}^{1},\quad w_{1}^{1}=\overline{w}_{1}^{1},\quad v_ {1}^{1}=\overline{v}_{1}^{1}+g(\overline{u}_{1}^{1})\Delta W_{1}^{1}.\end{array}\]
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\[N^{f,\boldsymbol{\alpha}^{\mathbf{x}}}_{t} :=f(\mathbf{X}^{\boldsymbol{\alpha}^{\mathbf{x}}}_{t})-f(\mathbf{X }^{\boldsymbol{\alpha}^{\mathbf{x}}}_{0})-\int_{0}^{t}\sum_{i=1}^{d}\partial_ {i}f(\mathbf{X}^{\boldsymbol{\alpha}^{\mathbf{x}}}_{s-})\,\mathrm{d}B^{j}_{s} -\frac{1}{2}\int_{0}^{t}\sum_{i,j=1}^{d}\partial_{ij}^{2}f(\mathbf{X}^{ \boldsymbol{\alpha}^{\mathbf{x}}}_{s-})\,\mathrm{d}C_{ij}(s)\] \[\quad-\int_{[0,t]\times\mathbb{R}^{n}_{0}}f(\mathbf{X}^{ \boldsymbol{\alpha}^{\mathbf{x}}}_{s-}+\mathbf{y})-f(\mathbf{X}^{\boldsymbol {\alpha}^{\mathbf{x}}}_{s-})-h(\mathbf{y})\cdot\nabla f(\mathbf{X}^{ \boldsymbol{\alpha}^{\mathbf{x}}}_{s-})\eta(\mathrm{d}s,\mathrm{d}\mathbf{y})\]
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\[P_{\theta_{0}}^{(n)}\left[\frac{\alpha}{1-\alpha}\int r_{n}( \theta,\theta_{0})\rho_{n}(d\theta)\geq\frac{\alpha}{1-\alpha}\int\mathrm{E}[ r_{n}(\theta,\theta_{0})]\rho_{n}(d\theta)+\frac{\alpha}{1-\alpha}\sqrt{\frac{ \mathrm{Var}[\int r_{n}(\theta,\theta_{0})\rho_{n}(d\theta)]}{\eta}}+\frac{ \mathcal{K}(\rho_{n},\pi)}{1-\alpha}\right]\] \[=P_{\theta_{0}}^{(n)}\left[\frac{\alpha}{1-\alpha}\int r_{n}( \theta,\theta_{0})\rho_{n}(d\theta)-\frac{\alpha}{1-\alpha}\int\mathrm{E}[r_{ n}(\theta,\theta_{0})]\rho_{n}(d\theta)-\frac{\mathcal{K}(\rho_{n},\pi)}{1- \alpha}\geq\frac{\alpha}{1-\alpha}\sqrt{\frac{\mathrm{Var}[\int r_{n}(\theta, \theta_{0})\rho_{n}(d\theta)]}{\eta}}\right]\] \[\leq\frac{\mathrm{Var}\left[\frac{\alpha}{1-\alpha}\int r_{n}( \theta,\theta_{0})\rho_{n}(d\theta)-\frac{\alpha}{1-\alpha}\int\mathrm{E}[r_{ n}(\theta,\theta_{0})]\rho_{n}(d\theta)-\frac{\mathcal{K}(\rho_{n},\pi)}{1-\alpha} \right]}{\frac{\alpha^{2}}{(1-\alpha)^{2}}\frac{\mathrm{Var}[\int r_{n}( \theta,\theta_{0})\rho_{n}(d\theta)]}{\eta}}.\]
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\[\begin{split}& u^{-5/2}\mathbf{q}_{\text{LO}}^{(1)}(u)=\sum_{n=0}^ {\infty}\frac{((1+\Delta)/2)_{n}}{(2)_{n}}\frac{((1-\Delta)/2)_{n}}{(2)_{n}} \frac{(1+iu)_{n}}{n!}=\,_{3}F_{2}\left(\begin{matrix}1+iu,(1+\Delta)/2,(1- \Delta)/2\\ 2,2\end{matrix};1\right)\,,\\ & u^{-5/2}\mathbf{q}_{\text{NLO}}^{(1)}(u)=\frac{I_{1}}{2}\sum_{n=0}^ {\infty}\frac{((1+\Delta)/2)_{n}}{(2)_{n}}\frac{((1-\Delta)/2)_{n}}{(2)_{n}} \frac{(1+iu)_{n}}{n!}\mathcal{A}_{n}\,,\\ & u^{-5/2}\mathbf{q}_{\text{LO}}^{(2)}(u)=\frac{4i}{u(1-\Delta^{2})}+ \sum_{n=0}^{\infty}\frac{((1+\Delta)/2)_{n}}{(2)_{n}}\frac{((1-\Delta)/2)_{n}} {(2)_{n}}\frac{(1+iu)_{n}}{(1)_{n}}\mathcal{B}_{n}\,,\\ & u^{-5/2}\mathbf{q}_{\text{NLO}}^{(2)}(u)=-\frac{2iI_{1}(\psi(iu)-\psi( 1))}{u(1-\Delta^{2})}+\sum_{n=0}^{\infty}\frac{((1+\Delta)/2)_{n}}{(2)_{n}} \frac{((1-\Delta)/2)_{n}}{(2)_{n}}\frac{(1+iu)_{n}}{n!}\mathcal{C}_{n}\,,\end{split}\]
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\[\zeta^{3}\mu^{4}\phi_{3}^{3}e^{2\zeta\phi_{3}}\int_{-1}^{1}\int_ {\widetilde{\omega}}\Big{(}|U_{1}^{F}|^{2}+|U_{2}^{F}|^{2}\Big{)}dxds\] \[\leq\zeta^{3}\mu^{4}\int_{-b_{0}}^{b_{0}}\int_{\widetilde{\omega} }\theta^{2}\phi^{3}\Big{(}|U_{1}^{F}|^{2}+|U_{2}^{F}|^{2}\Big{)}dxds\] \[\leq\zeta\mu^{2}\int_{-b_{0}}^{b_{0}}\int_{\widetilde{\omega}} \theta^{2}\phi\big{(}|\nabla U_{1}^{F}|^{2}+|\partial_{s}U_{1}^{F}|^{2}+\zeta ^{2}\mu^{2}\phi^{2}|U_{1}^{F}|^{2}+|\nabla U_{2}^{F}|^{2}+|\partial_{s}U_{2}^{ F}|^{2}+\zeta^{2}\mu^{2}\phi^{2}|U_{2}^{F}|^{2}\big{)}dxds\] \[\leq C\Big{[}e^{2\zeta\phi_{2}}\frac{4\lambda^{2}bL}{\pi}e^{ \frac{\lambda^{2}}{2}(b^{2}-(\frac{L}{8})^{2})}\max_{K}\{|\Phi^{\prime\prime} (l)|^{2},|\Phi^{\prime}(l)|^{2}\}\int_{\widetilde{\omega}}\int_{K}\Big{(}|U_{ 1}(x,l)|^{2}+|\partial_{t}U_{1}(x,l)|^{2}\] \[+|U_{2}(x,l)|^{2}+|\partial_{t}U_{2}(x,l)|^{2}\Big{)}dldx\] \[+e^{2\zeta\phi_{1}}\frac{\lambda^{2}bLM_{0}^{2}}{\pi}e^{\frac{ \lambda^{2}}{2}b^{2}}\Big{(}\max\{|\nabla\chi|^{2},|\Delta\chi|^{2}\}\int_{- \frac{L}{2}}^{\frac{L}{2}}\int_{O_{3}\backslash O_{2}}\Big{(}|U_{1}(x,l)|^{2}+ |\nabla U_{1}(x,l)|^{2}\] \[+|U_{2}(x,l)|^{2}+|\nabla U_{2}(x,l)|^{2}\Big{)}dxd\] \[+\frac{\lambda^{2}Lb}{2\pi}\zeta^{5}\mu^{7}\phi_{2}^{5}e^{2 \zeta\phi_{2}}e^{\frac{\lambda^{2}}{2}b^{2}}\int_{\omega_{0}}\int_{-\frac{L}{2 }}^{\frac{L}{2}}|U_{1}(x,l)|^{2}dldx\Big{]}\] \[+Ce^{2\zeta\phi_{4}}\int_{(-b,b_{0})\cup(b_{0},b)}\int_{\widetilde {\omega}}\big{(}|\partial_{s}\widetilde{U}_{1}^{F}|^{2}+|\widetilde{U}_{1}^{F} |^{2}+|\partial_{s}\widetilde{U}_{2}^{F}|^{2}+|\widetilde{U}_{2}^{F}|^{2}\big{)} dxds.\]
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\[\begin{split} d^{-\alpha}&(P,Q)\Big{|}D\delta z_{- }^{\rm(n)}(P)-D\delta z_{-}^{\rm(n)}(Q)\Big{|}\\ &\leq d^{-\alpha}(P,Q)\Big{|}D\delta z_{-}^{\rm(n)}(P)-D\delta z _{-}^{\rm(n)}(Q_{+}(\xi_{P}))\Big{|}\\ &\qquad+d^{-\alpha}(P,Q)\Big{|}D\delta z_{-}^{\rm(n)}(Q_{+}(\xi_{ P}))-D\delta z_{-}^{\rm(n)}(Q)\Big{|}\\ &\leq\mathcal{C}d^{-\alpha}\big{(}P,Q_{+}(\xi_{P})\big{)}\Big{|}D \delta z_{-}^{\rm(n)}(P)-D\delta z_{-}^{\rm(n)}(Q_{+}(\xi_{P}))\Big{|}\\ &\qquad+\mathcal{C}d^{-\alpha}\big{(}Q_{+}(\xi_{P}),Q\Big{)} \Big{|}D\delta z_{-}^{\rm(n)}(Q_{+}(\xi_{P}))-D\delta z_{-}^{\rm(n)}(Q)\Big{|} \\ &=:\mathcal{C}(J_{1}+J_{2}),\end{split}\]
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\[u^{(1)} =\mathrm{e}^{-2F^{\prime}_{D}/\hbar}\frac{\mathrm{i}u^{\prime}}{ 2\pi},\] \[u^{(2)} =\mathrm{e}^{-4F^{\prime}_{D}/\hbar}\,\left(\frac{\mathrm{i}u^{ \prime}}{4\pi}-\frac{u^{\prime\prime}}{8\pi^{2}}+\frac{F^{\prime}_{D}}{2\pi^{ 2}\hbar}\right),\] \[u^{(3)} =\mathrm{e}^{-6F^{\prime}_{D}/\hbar}\,\left(\frac{\mathrm{i}F^{ \prime\prime\prime}_{D}\,u^{\prime}}{8\pi^{3}\hbar}+\frac{\mathrm{i}\,F^{ \prime\prime}_{D}\,u^{\prime\prime}}{4\pi^{3}\hbar}-\frac{3\mathrm{i}\left(F^ {\prime\prime}_{D}\right)^{2}u^{\prime}}{4\pi^{3}\hbar^{2}}+\frac{3F^{\prime \prime}_{D}u^{\prime}}{4\pi^{2}\hbar}-\frac{\mathrm{i}u^{\prime\prime\prime}}{ 48\pi^{3}}-\frac{u^{\prime\prime}}{8\pi^{2}}+\frac{\mathrm{i}u^{\prime}}{6 \pi}\right),\]
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\[\begin{array}{ll}x_{i,t}&=A_{i,1}^{t}x_{i,1}+\sum_{\tau=1}^{t-1}\Big{(}A_{i, \tau+1}^{t}B_{i,\tau}\bm{\zeta}_{{}_{\mathcal{N}_{i}},\tau}+A_{i,\tau+1}^{t}D_{i,\tau}\widehat{\Gamma}_{i,\tau}(\bm{\xi}_{i}^{\tau-1},\bm{s}_{{}_{\mathcal{N}_{i} }}^{\tau})+A_{i,\tau+1}^{t}E_{i,\tau}\xi_{i,\tau}\Big{)}\\ &=A_{i,t-1}^{t}x_{i,1}+\sum_{\tau=1}^{t-1}\Big{(}A_{i,\tau+1}^{t}B_{i,\tau}\bm{ \zeta}_{{}_{\mathcal{N}_{i}}}+A_{i,\tau+1}^{t}D_{i,\tau}\widehat{\Gamma}_{i,\tau }(\bm{\xi}_{i}^{\tau-1},[R_{j}^{t}(\bm{\zeta}_{j}^{\tau})]_{j\in\mathcal{N}_{i}} )+A_{i,\tau+1}^{t}E_{i,\tau}\xi_{i,\tau}\Big{)}\\ &=\widehat{\chi}_{i,t}\left(\bm{\xi}_{i}^{t-1},[R_{j}^{t-1}(\bm{\zeta}_{j}^{t-1 })]_{j\in\mathcal{N}_{i}}\right)\\ &=:\widehat{\chi}_{i,t}\left(\bm{\xi}_{i}^{t-1},\bm{\zeta}_{{}_{\mathcal{N}_{i}}}^{t- 1}\right).\end{array}\]
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\[\|J^{(1+i\eta)}e^{i\eta\beta(|x|+|y|)}f\|_{L^{2}(\mathbb{R}^{2})} \lesssim\|Je^{i\eta\beta(x+y)}f\|_{L^{2}(x\geq 0,y\geq 0)}+\|Je^{i \eta\beta(-x-y)}f\|_{L^{2}(x\geq 0,y<0)}\] \[\quad+\|Je^{i\eta\beta(-x+y)}f\|_{L^{2}(x<0,y\geq 0)}+\|Je^{i \eta\beta(-x-y)}f\|_{L^{2}(x<0,y<0)}\] \[\lesssim\|Je^{i\eta\beta(x+y)}f\|_{L^{2}(\mathbb{R}^{2})}+\|Je^{ i\eta\beta(x-y)}f\|_{L^{2}(\mathbb{R}^{2})}\] \[\quad+\|Je^{i\eta\beta(-x+y)}f\|_{L^{2}(\mathbb{R}^{2})}+\|Je^{ i\eta\beta(-x-y)}f\|_{L^{2}(\mathbb{R}^{2})}\] \[:=r\text{-}h\text{-}s.\]
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\[\begin{split}&(S(f^{(4)})\otimes S(f^{(1)}))_{v_{1},v_{4}}*(\phi_{ 2}^{(1)}\otimes\phi_{3}\otimes\phi_{4}\otimes\phi_{1}^{(2)})_{v_{1},v_{2},v_{3 },v_{4}}=\\ &\langle\phi_{2}^{(1)(1)}\otimes S(f^{(1)})^{(2)},R^{-1}\rangle \langle\phi_{3}^{(1+\tau_{3})}\otimes S(f^{(1)})^{(1)(2)},(S^{\tau_{3}}\otimes S )(R)\rangle\\ &\langle\phi_{4}^{(1+\tau_{4})}\otimes S(f^{(1)})^{(1)(1)(2)},(S ^{\tau_{4}}\otimes S)(R)\rangle\langle\phi_{1}^{(2)(2)}\otimes S(f^{(1)})^{(1 )(1)(1)(2)},R\rangle\\ &(S(f^{(4)})\phi_{2}^{(1)(2)}\otimes\phi_{3}^{(2-\tau_{3})} \otimes\phi_{4}^{(2-\tau_{4})}\otimes S(f^{(1)})^{(1)(1)(1)(1)}\phi_{1}^{(2)(1 )})_{v_{1},v_{2},v_{3},v_{4}}\;,\end{split}\]
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\[\mathbb{P}(\overline{W}\leq z)-\Phi(z) =\mathbb{E}[f_{z}^{\prime}(\overline{W})]-\mathbb{E}[\overline{W}f _{z}(\overline{W})]\] \[=\mathbb{E}[f_{z}^{\prime}(\overline{W})]\left(\sum_{i=1}^{n}\int_ {-\infty}^{1}\bar{K}_{i}(t)dt+\sum_{i=1}^{n}\mathbb{E}[{\xi_{i}}^{2}\mathbb{1} _{\xi_{i}>1}]\right)\] \[=\sum_{i=1}^{n}\mathbb{E}[{\xi_{i}}^{2}\mathbb{1}_{\xi_{i}>1}]E[f _{z}^{\prime}(\overline{W})]\] \[\quad+\sum_{i=1}^{n}\int_{-\infty}^{1}\mathbb{E}[f_{z}^{\prime}( \overline{W}^{(i)}+\bar{x}_{i})-f_{z}^{\prime}(\overline{W}^{(i)}+t)]\bar{K}_ {i}(t)dt\] \[\quad+\sum_{i=1}^{n}\mathbb{E}[\xi_{i}\mathbb{1}_{\xi_{i}>1}]E[f _{z}(\overline{W}^{(i)})]\] \[=R_{1}+R_{2}+R_{3}.\]
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\[\Phi_{\mathsf{D}^{0}}^{\text{joint}}(P_{\mathsf{M}|\mathsf{X}K}) \supseteq\{\mathsf{D}:D_{1}\geq\mathbb{E}[\|X_{1}-\tilde{X}_{1}\|^{2}]+W_{2}^ {2}(P_{\hat{X}_{1}},P_{X_{1}}),\] \[D_{2}\geq\mathbb{E}[\|X_{2}-\tilde{X}_{2}\|^{2}]+\sum_{x_{1}}P_{X _{1}}(x_{1})W_{2}^{2}(P_{\tilde{X}_{2}|\hat{X}_{1}^{*}=x_{1}},P_{X_{2}|X_{1}=x_ {1}}),\] \[D_{3}\geq\mathbb{E}[\|X_{3}-\tilde{X}_{3}\|^{2}]+\sum_{x_{1},x_{2 }}P_{X_{1}X_{2}}(x_{1},x_{2})W_{2}^{2}(P_{\tilde{X}_{3}|\hat{X}_{1}^{*}=x_{1}, \hat{X}_{2}^{*}=x_{2}},P_{X_{3}|X_{1}=x_{1},X_{2}=x_{2}})\}.\]
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\[A =\begin{bmatrix}1&0&-\beta_{1}&0&0&0\\ 0&1&-\gamma_{1}&0&0&0\\ \tau^{2}\lambda&0&\alpha_{1}&0&0&0\\ 0&0&0&1&0&-\beta_{2}\\ 0&0&0&0&1&-\gamma_{2}\\ 0&0&0&\tau^{2}\alpha_{f}\lambda&0&\alpha_{2}\end{bmatrix},\] \[B =\begin{bmatrix}1&1&\frac{1}{2}-\beta_{1}&\frac{1}{6}-\beta_{1}& \frac{1}{24}-\frac{\beta_{1}}{2}&\frac{1}{120}-\frac{\beta_{1}}{6}\\ 0&1&1-\gamma_{1}&\frac{1}{2}-\beta_{1}&\frac{1}{6}-\frac{\beta_{1}}{2}&\frac{1 }{24}-\frac{\beta_{1}}{6}\\ 0&0&\alpha_{1}-1&\alpha_{2}-1&\frac{1}{2}(\alpha_{2}-1)&\frac{1}{6}(\alpha_{2} -1)\\ 0&0&0&1&1&\frac{1}{2}-\beta_{2}\\ 0&0&0&0&1&1-\gamma_{2}\\ 0&0&0&-\tau^{2}(1-\alpha_{f})\lambda&0&\alpha_{2}-1\end{bmatrix},\] \[\mathbf{U_{n}} =\begin{bmatrix}U_{n}\\ \tau V_{n}\\ \tau^{2}A_{n}\\ \tau^{3}\mathcal{L}^{1}(A_{n})\\ \tau^{4}\mathcal{L}^{2}(A_{n})\\ \tau^{5}\mathcal{L}^{3}(A_{n})\end{bmatrix}.\]
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\[\mathbb{E}\big{[}|\varphi(Q_{d,k_{*}}(a,X))-\varphi(Q_{d,k_{*}}(a,X)+\eta Z)|\big{]}\\ =\mathbb{E}_{Z}\mathbb{E}_{\varepsilon,U}\mathbb{E}_{V}\big{[}| (\varphi(Q_{d,k_{*}}(a,X))-\varphi(Q_{d,k_{*}}(a,X)+\eta Z)|\big{]}\\ =\mathbb{E}_{Z}\mathbb{E}_{\varepsilon,U}\mathbb{E}_{V}\Big{[}| (\varphi(Q_{d,k_{*}}(a,X))-\varphi(Q_{d,k_{*}}(a,X)+\eta Z)|(I_{\{\mathbb{D}_{ V}[Q_{d,k_{*}}(a,X)]\geq\theta\}}+I_{\{\mathbb{D}_{V}[Q_{d,k_{*}}(a,X)]<\theta\}}) \Big{]}\\ \leq\mathbb{E}_{Z}\Big{[}\frac{C_{1}(d,k_{*})}{\theta^{\frac{1}{2 dk_{*}}}}|\eta Z|^{\frac{1}{dk_{*}}}\Big{]}+2\|\varphi\|_{\infty}\mathbb{E}_{Z} \big{[}\mathbb{E}_{\varepsilon,U}[I_{\{\mathbb{D}_{V}[Q_{d,k_{*}}(a,X)]<\theta \}}]\big{]}\\ \leq\frac{C_{1}(d,k_{*})}{\theta^{\frac{1}{2dk_{*}}}}|\eta|^{ \frac{1}{dk_{*}}}+2P_{\varepsilon,U}\big{(}\mathbb{D}_{V}\big{[}Q_{d,k_{*}}(a,X )\big{]}<\theta\big{)}.\]
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\[\frac{1}{N}\sum_{i_{3}=1}^{n_{3}}\cdots\sum_{i_{d}=1}^{n_{d}}\sum_{i =1}^{\#\sigma_{>r+1}^{(i_{3},\ldots,i_{d})}(\boldsymbol{\mathcal{P}})}\sigma_{i}( \mathbf{\overline{P}}^{(i_{3},\ldots,i_{d})})-\frac{1}{N}\left(\sum_{i_{3}=1}^{ n_{3}}\cdots\sum_{i_{d}=1}^{n_{d}}\#\sigma_{>r+1}^{(i_{3},\ldots,i_{d})}( \boldsymbol{\mathcal{P}})\right)\sigma_{r+1}^{\max}(\mathbf{\overline{P}})\] \[\geq 1+\frac{\eta}{N}\left(\sigma_{1}(\overline{\nabla f(\boldsymbol {\mathcal{X}}^{*})})-\sigma_{r-\tilde{r}_{\max}+2}^{\max}(\overline{\nabla f( \boldsymbol{\mathcal{X}}^{*})})\right)\sum_{i_{3}=1}^{n_{3}}\cdots\sum_{i_{d} =1}^{n_{d}}\#\sigma_{1}^{(i_{3},\ldots,i_{d})}\] \[\quad-\left(\frac{1}{N\sqrt{\tilde{r}_{\max}}}\sum_{i_{3}=1}^{n_{ 3}}\cdots\sum_{i_{d}=1}^{n_{d}}\#\sigma_{1}^{(i_{3},\ldots,i_{d})}+\frac{ \sqrt{\tilde{r}_{\max}\cdot\mathrm{nnzb}(\overline{\nabla f(\boldsymbol{ \mathcal{X}}^{*})})}}{N}\right)\|\mathbf{\overline{P}}-\mathbf{\overline{P^{*}} }\|_{F}\]
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\[|b_{h}^{E}(\widehat{u}_{h},w_{I})-b^{E} (\widehat{u}_{h},w_{I})|\leq C_{\vartheta}\|\vartheta(\mathbf{x })\cdot\nabla\widehat{u}_{h}-\Pi^{E}(\vartheta(\mathbf{x})\cdot\nabla \widehat{u}_{h})\|_{0,E}\|w_{I}-\Pi^{E}w_{I}\|_{0,E}\] \[\qquad+|\widehat{u}_{h}-\Pi^{\nabla,E}\widehat{u}_{h}|_{1,E}\| \vartheta(\mathbf{x})w_{I}-\vartheta(\mathbf{x})\Pi^{E}w_{I}\|_{0,E}\] \[\qquad+|\widehat{u}_{h}-\Pi^{\nabla,E}\widehat{u}_{h}|_{1,E}\| \vartheta(\mathbf{x})w_{I}-\Pi^{E}(\vartheta(\mathbf{x})w_{I})\|_{0,E}\] \[\qquad\qquad\qquad\lesssim h^{s}\left(\|\widehat{u}-\widehat{u} _{h}\|_{1,E}+|\widehat{u}-\Pi^{\nabla,E}\widehat{u}_{h}|_{1,E}\right.\] \[\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.+\| \vartheta(\mathbf{x})\cdot\nabla\widehat{u}_{h}-\Pi^{E}(\vartheta(\mathbf{x}) \cdot\nabla\widehat{u}_{h})\|_{0,E}\right)|w|_{1+s,E}.\]
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\[\begin{split}&\widehat{\Upsilon}^{0<|y|<|\mathfrak{q}|<|z|<1} _{l,h}\\ &=z^{\frac{-a_{2,h}+a_{2,\bar{h}}+\epsilon}{2\varepsilon_{2}}} \left(\frac{y}{\mathfrak{q}}\right)^{\frac{a_{4,\bar{l}}-a_{4,\bar{l}}+ \epsilon}{2\varepsilon_{1}}}\mathfrak{q}^{-\Delta_{0}-\Delta_{4}+\frac{\epsilon ^{2}-(a_{2,1}-a_{2,2})^{2}}{4\varepsilon_{1}}+\frac{2\varepsilon_{1}+3 \varepsilon_{2}}{4\varepsilon_{1}}}\\ &(1-z)^{\frac{2a_{0}-2a_{2}+\varepsilon_{1}+2\varepsilon_{2}}{2 \varepsilon_{2}}}\left(1-\frac{\mathfrak{q}}{z}\right)^{\frac{2a_{2}-2a_{4}+4 \varepsilon_{1}+5\varepsilon_{2}}{2\varepsilon_{2}}}\left(1-\frac{y}{z} \right)^{-\frac{1}{2}}(1-y)^{\frac{-2a_{0}+2a_{2}+\varepsilon_{1}}{2 \varepsilon_{1}}}\\ &(1-\mathfrak{q})^{\frac{(2a_{0}-2a_{2}-\varepsilon_{1})(2a_{2}- 2a_{4}+4\varepsilon_{1}+5\varepsilon_{2})}{2\varepsilon_{1}\varepsilon_{2}}} \left(1-\frac{y}{\mathfrak{q}}\right)^{\frac{-2a_{2}+2a_{4}-2\varepsilon_{1}-3 \varepsilon_{2}}{2\varepsilon_{1}}}\Upsilon^{0<|y|<|\mathfrak{q}|<|z|<1}_{l,h},\end{split}\]
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\[\begin{split}\langle(p_{1k}\circ\theta_{0}^{\gamma})m_{t}^{1}, \,f\rangle&=\int_{[0,\max\{0,(\tau_{1})^{-1}(t)\}]}f(\Phi_{t}^{e_{1 }}(x,0),t)p_{1k}(\theta_{0}^{\gamma}(x,0))dm_{0}^{1}(x)\\ &+\int_{(\max\{0,(\varsigma_{1})^{-1}(t)\},t]}f(\Phi_{t}^{e_{1}} (0,s),t)p_{1k}(\theta_{0}^{\gamma}(0,s))d\sigma_{0}(s)\\ &=\int_{[0,\max\{0,(\tau_{1})^{-1}(t)\}]}f(\Phi_{t}^{e_{1}}(x,0),t )p_{1k}(\tau_{1}(x))dm_{0}^{1}(x)\\ &+\int_{(\max\{0,(\varsigma_{1})^{-1}(t)\},t]}f(\Phi_{t}^{e_{1}} (0,s),t)p_{1k}(\varsigma_{1}(s))d\sigma_{0}(x).\end{split}\]
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\[(-1)^{|Q_{-1}|}\rho(\beta;\gamma)\pi^{\mathcal{M}_{0,l}(\beta)}{}_ {*}\left(Q_{-1,l}^{\beta}\right)_{*}(d\xi) =\rho(\beta;\gamma)\sum_{\begin{subarray}{c}P\in S_{3}[l]\\ (2:3)=(j)\end{subarray}}(-1)^{n+|\gamma^{(1:3)}|}\pi^{\mathcal{M}_{0,l}(\beta) }{}_{*}\left(Q_{-1,l}^{\beta}\right)_{*}\bar{\gamma}_{P}\] \[=\sum_{\begin{subarray}{c}P\in S_{3}[l]\\ (2:3)=(j)\end{subarray}}(-1)^{|\gamma^{(1:3)}|}\rho(\beta;\widetilde{\gamma}_ {P})\pi^{\mathcal{M}_{0,l}(\beta)}{}_{*}\left(Q_{-1,l}^{\beta}\right)_{*}\bar {\gamma}_{P}\] \[=\sum_{\begin{subarray}{c}P\in S_{3}[l]\\ (2:3)=(j)\end{subarray}}(-1)^{|\gamma^{(1:3)}|}\mathfrak{q}_{-1,l}^{\beta} \left(\gamma^{(1:3)},d\gamma_{j},\gamma^{(3:3)}\right).\]
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\[\int_{G/\Gamma_{\gamma}}f(x\gamma x^{-1})\ {\rm d}\mu_{G/\Gamma_{\gamma}}(\dot{x}) =\ \int_{G/G_{\gamma}}\ \left(\,\int_{G_{\gamma}/\Gamma_{\gamma}}f(x\eta \gamma\eta^{-1}x^{-1})\ {\rm d}\mu_{G_{\gamma}/\Gamma_{\gamma}}(\dot{\eta})\right)\,{\rm d}\mu_{G/G_{ \gamma}}(\dot{x})\] \[=\ \int_{G/G_{\gamma}}\ \left(\,\int_{G_{\gamma}/\Gamma_{\gamma}}f(x \gamma x^{-1})\ {\rm d}\mu_{G_{\gamma}/\Gamma_{\gamma}}(\dot{\eta})\right)\,{\rm d}\mu_{G/G_{ \gamma}}(\dot{x})\] \[=\ \int_{G/G_{\gamma}}\ f(x\gamma x^{-1})\left(\,\int_{G_{\gamma} /\Gamma_{\gamma}}\ {\rm d}\mu_{G_{\gamma}/\Gamma_{\gamma}}\right)\,{\rm d}\mu_{G/G_{ \gamma}}(\dot{x})\] \[=\ {\rm vol}(G_{\gamma}/\Gamma_{\gamma})\int_{G/G_{\gamma}}\ f(x \gamma x^{-1})\ {\rm d}\mu_{G/G_{\gamma}}(\dot{x}).\]
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\[\langle\mathrm{sgn}(\bm{u}_{S}^{*})+\bm{\Lambda}-\bm{v}-2\bm{B}^{ T}\bm{A}\bm{x}^{*}\,,\bm{\delta}_{\bm{2}}\rangle\] \[= \langle\mathrm{sgn}(\bm{u}_{S}^{*})-\bm{v}_{S}\,,(\bm{\delta}_{ \bm{2}})_{S}\rangle-\langle\bm{v}_{S^{\perp}},(\bm{\delta}_{\bm{2}})_{S^{ \perp}}\rangle+||(\bm{\delta}_{\bm{2}})_{S^{\perp}}||_{1}-\langle 2[\bm{B}^{T}\bm{A}\bm{x}^{* }]_{S}\,,(\bm{\delta}_{\bm{2}})_{S}\rangle\] \[- \langle 2[\bm{B}^{T}\bm{A}\bm{x}^{*}]_{S^{\perp}}\,,(\bm{\delta}_{ \bm{2}})_{S^{\perp}}\rangle\] \[\geq -||\,\mathrm{sgn}(\bm{u}_{S}^{*})-\bm{v}_{S}||_{2}||(\bm{\delta} _{\bm{2}})_{S}||_{2}-||\bm{v}_{S^{\perp}}||_{\infty}||(\bm{\delta}_{\bm{2}})_ {S^{\perp}}||_{1}+||(\bm{\delta}_{\bm{2}})_{S^{\perp}}||_{1}\]
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\[a_{1}(x^{2})+x\kappa_{1}(x^{2})+x^{2\iota+1}(a_{1}(x^{2})+x\varphi _{2}(\kappa_{1}(x^{2}))\] \[= a_{1}(x^{2})+x\kappa_{1}(x^{2})+x^{2\iota+1}\varphi_{2}(a_{1}(x^ {2}))+x^{2\iota+1}x^{4n-1}\varphi_{2}(\kappa_{1}(x^{2}))\] \[= a_{1}(x^{2})+x^{2\iota+1}\varphi_{2}(a_{1}(x^{2}))+x\kappa_{1}(x ^{2})+x^{2\iota}\varphi_{2}(\kappa_{1}(x^{2}))\] \[= a_{1}(x^{2})+x^{2\iota+2}\varphi_{1}(a_{1}(x^{2}))+x\kappa_{1}(x ^{2})+x^{2\iota+1}\varphi_{1}(\kappa_{1}(x^{2}))\] \[= (a_{1}(x)+x^{\iota+1}\varphi_{1}(a_{1}(x)))^{2}+x(\kappa_{1}(x)+x ^{\iota}\varphi_{1}(\kappa_{1}(x)))^{2}.\]
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\[\mathbb{E}^{\mathbb{Q}}\left[\int_{0}^{T}\Gamma_{t}\Delta_{t}dt \right]=\frac{A_{0,T}}{\kappa}\Gamma_{0}\Delta_{0}+\int_{0}^{T}\mathbb{E}^{ \mathbb{Q}}\left[\frac{A_{r,T}}{\kappa}Q\Gamma_{r}\partial_{s}P_{r}\phi_{r} \right]dr\] \[+\int_{0}^{T}\frac{A_{r,T}}{\kappa}\mathbb{E}^{\mathbb{Q}}\left[ Q\sigma\partial_{ss}P_{r}\beta_{r}\right]dr+Q\gamma\sigma^{2}\int_{0}^{T} \mathbb{E}^{\mathbb{Q}}\left[\Delta_{r}\partial_{ss}P_{r}\hat{\Gamma}_{r} \right]dr\] \[+Q\gamma\sigma^{2}\int_{0}^{T}\mathbb{E}^{\mathbb{Q}}\left[\Delta_{r} \partial_{ss}P_{r}\left(\check{\Gamma}_{r}+A_{r,T}\int_{0}^{r}e^{-\int_{t}^{r}m( v)dv}\Gamma_{t}A_{t,T}dt-\frac{A_{r,T}\Gamma_{r}}{\kappa}\right)\right]dr\] \[-\frac{\gamma}{\kappa}\int_{0}^{T}\mathbb{E}^{\mathbb{Q}}\left[A_ {r,T}\alpha_{r}\Delta_{r}^{2}\right]dr.\]
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\[\iint_{\mathbb{R}^{2}}\|e^{i\partial^{4}_{x}}\widetilde{v}_{1,y_ {1}}e^{i\partial^{4}_{x}}\widetilde{v}_{2,y_{2}}\|_{L^{2}_{x,x}}dy_{1}dy_{2} \lesssim\iint_{\mathbb{R}^{2}}N_{1}^{-\frac{3}{2}}\|\widetilde{v }_{1,y_{1}}\|_{L^{2}_{x}}\|\widetilde{v}_{2,y_{2}}\|_{L^{2}_{x}}dy_{1}dy_{2}\] \[\lesssim\iint_{\mathbb{R}^{2}}N_{1}^{-\frac{3}{2}}N_{2}^{-\frac{ 3}{2}}\|F_{1}(t,y_{1})\|_{L^{2}_{x}}N_{2}^{-\frac{3}{2}}\|F_{2}(t,y_{2})\|_{L^ {2}_{t}}dy_{1}dy_{2}\] \[=N_{1}^{-3}N_{2}^{-\frac{3}{2}}\|F_{1}\|_{L^{1}_{x}L^{2}_{t}}\|F_ {2}\|_{L^{1}_{x}L^{2}_{t}}.\]
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\[\mathcal{C}_{s}= \mathbb{E}_{v}\left[\mathbb{P}\left(\frac{KP_{s}v^{-\alpha}\chi_{ 0}}{I+B_{s}N_{s}}>\tilde{\gamma}^{\rm rf}\right)\right]\overset{(a)}{=} \mathbb{E}_{v}\left[\mathbb{E}_{I}\left[\frac{\Gamma_{u}(\kappa,\frac{\tilde{ \gamma}^{\rm rf}(I+B_{s}N_{s})}{KP_{s}v^{-\alpha}\Theta})}{\Gamma(\kappa)} \right]\right],\] \[\overset{(b)}{=} \mathbb{E}_{v,I}\left[\exp\left(-\frac{\tilde{\gamma}^{\rm rf}(I+ B_{s}N_{s})}{KP_{s}v^{-\alpha}\Theta}\right)\sum_{n=0}^{\kappa-1}\frac{(\frac{ \tilde{\gamma}^{\rm rf}(I+B_{s}N_{s})}{KP_{s}v^{-\alpha}\Theta})^{n}}{n!}\right],\] \[\overset{(c)}{=} \mathbb{E}_{v}\left[\sum_{n=0}^{\kappa-1}\left.\frac{(-s)^{n}}{n! }\frac{d^{n}}{ds^{n}}\mathcal{L}_{Z}(s)\right]\right|_{s=\frac{\tilde{\gamma }^{\rm rf}v^{\alpha}}{KP_{s}\Theta}},\]
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\[s \geq\sqrt{(1-\rho^{2})\,r^{2}\delta^{-1}-\mathbb{E}[Z^{2};Z<0]},\] \[s \leq\big{(}1+q_{+}(\kappa)\big{)}\cdot\Big{(}\sqrt{(1-\rho^{2}) \,r^{2}\delta^{-1}-\mathbb{E}[Z^{2};Z<0]}\,+\sqrt{1-\rho^{2}}\,\delta^{-1/2} \sqrt{2q_{+}(\kappa)}\Big{)}\] \[\leq\sqrt{(1-\rho^{2})\,r^{2}\delta^{-1}-\mathbb{E}[Z^{2};Z<0]}+ q_{+}(\kappa)\sqrt{1-\rho^{2}}\,\delta^{-1/2}+(1+q_{+}(\kappa))\cdot\sqrt{1- \rho^{2}}\delta^{-1/2}\sqrt{2q_{+}(\kappa)}\] \[\stackrel{{(i)}}{{\leq}}\sqrt{(1-\rho^{2})\,r^{2} \delta^{-1}-\mathbb{E}[Z^{2};Z<0]}+\sqrt{(1-\rho^{2})q_{+}(\kappa)}\,\delta^{ -1/2}+2\sqrt{(1-\rho^{2})q_{+}(\kappa)}\,\delta^{-1/2}\] \[=\sqrt{(1-\rho^{2})\,r^{2}\delta^{-1}-\mathbb{E}[Z^{2};Z<0]}+3 \sqrt{(1-\rho^{2})q_{+}(\kappa)}\,\delta^{-1/2}\]
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\[t:=\left\{\begin{array}{ll}3n+2,&i=1\mbox{ and }j=0,\\ 3\lfloor n/3\rfloor+j+2,&i=1\mbox{ and }j>0\mbox{ and }j\equiv 0\mbox{ (mod 3); or }i=m^{\prime}\mbox{ and }j\equiv 1\mbox{ (mod 3);}\\ &\mbox{ or }i=m\mbox{ and }j\equiv 2\mbox{ (mod 3),}\\ 6\lfloor n/3\rfloor+j+4,&i=1\mbox{ and }j\equiv 1\mbox{ (mod 3); or }i=m^{\prime}\mbox{ and }j\equiv 2\mbox{ (mod 3);}\\ &\mbox{ or }i=m\mbox{ and }j>0\mbox{ and }j\equiv 0\mbox{ (mod 3),}\\ j,&i=1\mbox{ and }j\equiv 2\mbox{ (mod 3); or }i=m^{\prime}\mbox{ and }j\equiv 0\mbox{ (mod 3);}\\ &\mbox{ or }i=m\mbox{ and }j\equiv 1\mbox{ (mod 3),}\\ 3n+1,&i=m\mbox{ and }j=0.\end{array}\right.\]
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\[\frac{\partial\log p\left(y_{n}|\hat{x}_{n-1},\mathbf{w}_{n}\right)} {\partial\hat{x}_{n-1}} = 2\rho\operatorname{Re}\left\{\left[y_{n}-\mathbf{w}_{n}^{\text{ H}}\mathbf{a}(\hat{x}_{n-1})\right]^{\text{H}}\cdot\mathbf{w}_{n}^{\text{H}}\frac{ \partial\mathbf{a}(\hat{x}_{n-1})}{\partial\hat{x}_{n-1}}\right\}\] \[= 2\rho\operatorname{Re}\left\{\left(y_{n}-\sqrt{M}\right)^{\text{ H}}\cdot\frac{1}{\sqrt{M}}\left[\sum_{m=1}^{M}-j\frac{2\pi d}{\lambda}(m-1) \right]\right\}\] \[= 2\rho\operatorname{Re}\left\{y_{n}^{\text{H}}\cdot\frac{1}{ \sqrt{M}}\left[\sum_{m=1}^{M}-j\frac{2\pi d}{\lambda}(m-1)\right]\right\}\] \[= \frac{2\sqrt{M}(M-1)\pi d\rho}{\lambda}\cdot\operatorname{Re} \left\{-jy_{n}^{\text{H}}\right\}=-\frac{2\sqrt{M}(M-1)\pi d\rho}{\lambda} \cdot\operatorname{Im}\left\{y_{n}\right\}.\]
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\[\begin{array}{l}\tau^{(1)}_{\mbox{\boldmath$n$}}\tau^{(1)}_{\mbox{\boldmath$ n$}+\mbox{\boldmath$e$}_{1}+\mbox{\boldmath$e$}_{2}+\mbox{\boldmath$e$}_{3}}\;=\; \tau^{(1)}_{\mbox{\boldmath$n$}+\mbox{\boldmath$e$}_{1}}\tau^{(1)}_{\mbox{ \boldmath$n$}+\mbox{\boldmath$e$}_{2}+\mbox{\boldmath$e$}_{3}}+\tau^{(1)}_{ \mbox{\boldmath$n$}+\mbox{\boldmath$e$}_{3}}\tau^{(1)}_{\mbox{\boldmath$n$}+ \mbox{\boldmath$e$}_{1}+\mbox{\boldmath$e$}_{2}}\;,\\ \\ \tau^{(2)}_{\mbox{\boldmath$n$}+\mbox{\boldmath$e$}_{1}+\mbox{\boldmath$e$}_{2}+\mbox{ \boldmath$e$}_{3}}\;=\;\tau^{(2)}_{\mbox{\boldmath$n$}+\mbox{\boldmath$e$}_{1}} \tau^{(2)}_{\mbox{\boldmath$n$}+\mbox{\boldmath$e$}_{2}+\mbox{\boldmath$e$}_{3}}+ \tau^{(2)}_{\mbox{\boldmath$n$}+\mbox{\boldmath$e$}_{3}}\tau^{(2)}_{\mbox{ \boldmath$n$}+\mbox{\boldmath$e$}_{1}+\mbox{\boldmath$e$}_{2}}\;.\end{array}\]
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\[\sum_{k=1}^{n}\frac{(a-\frac{1}{2})_{k}(\frac{2a+2}{3})_{k}(2b-1) _{k}(2c-1)_{k}(2+2a-2b-2c)_{k}(a+n)_{k}(-n)_{k}}{(1)_{k}(\frac{2a-1}{3})_{k}( 1+a-b)_{k}(1+a-c)_{k}(b+c-\frac{1}{2})_{k}(2a+2n)_{k}(-2n)_{k}}\] \[\quad\times\bigg{\{}\big{[}2H_{k}(2c-2)-2H_{k}(1+2a-2b-2c)+H_{k}( a-c)-H_{k}(b+c-\frac{3}{2})\big{]}\] \[\quad\times\bigg{[}4\sum_{i=1}^{k}\frac{1}{(2b-2+i)(1+2a-2b-2c+i )}-\sum_{i=1}^{k}\frac{1}{(a-b+i)(b+c-\frac{3}{2}+i)}\bigg{]}\] \[\quad+\left[8\sum_{i=1}^{k}\frac{1}{(2b-2+i)(1+2a-2b-2c+i)^{2}}+\sum_{i=1}^{k} \frac{1}{(a-b+i)(b+c-\frac{3}{2}+i)^{2}}\right]\biggr{\}}\]
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\[(II)= \sum_{l=1}^{p}\bm{X}_{l}^{T}\dot{\Sigma}^{-1}(\theta_{0})\bm{X}_{ j}(\beta_{l}-\beta_{0l})+\sum_{k=1}^{q}\sum_{l=1}^{p}\bm{X}_{l}^{T}\dot{\Sigma}^{ k}(\theta^{*})\bm{X}_{j}(\beta_{l}-\beta_{0l})(\theta_{k}^{*}-\theta_{0k})\] \[= (7)+(8)\] \[(7)= \sum_{l=1}^{p}\bm{X}_{l}^{T}diag_{n-1}(\Sigma^{-1})(\tilde{I}_{n -1,p}+\tilde{J}_{n-1,p})\bm{X}_{j}(\beta_{l}-\beta_{0l})\] \[= \sum_{l=1}^{p}\left(\sum_{i=1}^{n-1}\bm{X}_{il}^{T}\Sigma^{-1} \bm{X}_{ij}+\sum_{i=1}^{n-1}\bm{X}_{il}^{T}\Sigma^{-1}\sum_{i=1}^{n-1}\bm{X}_{ ij}\right)(\beta_{l}-\beta_{0l})\]
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\[\mathrm{P}\left\{\sup_{\beta\in\mathcal{F}_{p_{n}},\,\|\beta\|_ {L^{2}}\leq 2}\,\frac{\sqrt{n}\|H_{n}(\beta)\|_{K}}{b_{n}\|\beta\|_{L^{2}}^{ \gamma}+1}\geq T_{n}\,\big{|}\mathcal{X}_{n}\right\}\] \[\leq\mathrm{P}\left\{\sup_{\beta\in\mathcal{F}_{p_{n}},\,\|\beta \|_{L^{2}}\leq\mathfrak{g}_{n}^{1/\gamma}}\sqrt{n}\|H_{n}(\beta)\|_{K}\geq T_{ n}\,\big{|}\mathcal{X}_{n}\right\}\] \[\qquad\qquad+\sum_{j=0}^{Q_{n}}\mathrm{P}\left\{\sup_{\beta\in \mathcal{F}_{p_{n}},\,(\theta_{n}2^{j})^{1/\gamma}\leq\|\beta\|_{L^{2}}\leq( \theta_{n}2^{j+1})^{1/\gamma}}\,\frac{\sqrt{n}\|H_{n}(\beta)\|_{K}}{b_{n}\| \beta\|_{L^{2}}^{\gamma}+1}\geq T_{n}\,\big{|}\mathcal{X}_{n}\right\}\] \[\leq 2\exp\big{(}-c_{1}^{-2}W_{n}^{-2}T_{n}^{2}\big{)}+2(Q_{n}+1) \exp\big{(}-c_{1}^{-2}W_{n}^{-2}T_{n}^{2}/4\big{)}\] \[\leq 2(Q_{n}+2)\exp\big{(}-c_{1}^{-2}W_{n}^{-2}T_{n}^{2}/4\big{)}\,.\]
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\begin{table}
\begin{tabular}{c c|c c} \hline \(S\) & \(\theta(S)\) & \(S\) & \(\theta(S)\) \\ \hline \(\emptyset\) & \(\left(1-\sum_{i=1}^{n-1}x_{i}\right)(1-x_{n})\) & \([n]\) & \(x_{1}\left(\sum_{i=2}^{n}x_{i}-(n-2)\right)\) \\ \(\{1\}\) & \(x_{1}(1-x_{n})\) & \([n]\setminus\{2\}\) & \(x_{1}(1-x_{2})\) \\ \(\{2\}\) & \(x_{2}(1-x_{n})\) & \([n]\setminus\{3\}\) & \(x_{1}(1-x_{3})\) \\ \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) \\ \(\{n-1\}\) & \(x_{n-1}(1-x_{n})\) & \([n]\setminus\{n\}\) & \(x_{1}(1-x_{n})\) \\ All other \(S\) with \(n\notin S\) & 0 & All other \(S\) with \(1\in S\) & 0 \\ All \(S\) with \(n\in S\) & \(\theta(S)\) & All \(S\) with \(1\notin S\) & \(\theta(S)\) \\ \hline \end{tabular}
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\[\sum_{0\in S\in\Gamma^{\infty,d}}w_{\beta,h}^{q}(S)e^{a(S)} =\sum_{k=1}^{\infty}e^{-(1+\frac{1}{q-1})hk}\sum_{\begin{subarray} {c}0\in S\in\Gamma^{\infty,d}\\ |V_{S}|=k\end{subarray}}e^{(a(S))}e^{-\beta|E_{S}|}\tilde{Z}_{\beta,0}^{0,q-1 }(S)e^{-(1+\frac{1}{q-1})\beta|\partial_{a}S|}\] \[\leq\sum_{k=1}^{\infty}(q-1)^{k}e^{-(1+\frac{1}{q-1})hk}e^{2dk}| \{S\in\Gamma^{\infty,d}|\;|V_{S}|=k,\;0\in V_{S}\}|\] \[\leq\sum_{k=1}^{\infty}(q-1)^{k}e^{-(1+\frac{1}{q-1})hk}e^{2dk} \left(\frac{(2d+1)^{2d+1}}{(2d)^{2d}}\right)^{k},\]
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\[\big{|}\partial_{i} \phi^{x}_{\rho_{1}}(\xi_{t})-\partial_{i}\phi^{x}_{\rho_{2}}(\xi_{s} )\big{|}=\Big{|}\rho_{1}^{-1}\partial_{i}\phi\Big{(}\frac{\xi_{t}-x}{\rho_{1}} \Big{)}-\rho_{2}^{-1}\partial_{i}\phi\Big{(}\frac{\xi_{s}-x}{\rho_{2}}\Big{)} \Big{|}\] \[\leq\frac{1}{\rho_{1}}\Big{|}\nabla\partial_{i}\phi\Big{(}c_{2} \frac{\xi_{t}-x}{\rho_{1}}+(1-c_{2})\frac{\xi_{s}-x}{\rho_{2}}\Big{)}^{*}\cdot \Big{(}\frac{\xi_{t}-x}{\rho_{1}}-\frac{\xi_{s}-x}{\rho_{2}}\Big{)}\Big{|}\] \[\quad+\Big{|}\partial_{i}\phi^{x}_{\rho_{2}}(\xi_{s})\Big{|}\Big{|} \frac{1}{\rho_{1}}-\frac{1}{\rho_{2}}\Big{|}\] \[\leq\frac{1}{\rho_{1}}\big{(}\mathbf{1}_{B(x,4\rho_{1})}(\xi_{t} )\vee\mathbf{1}_{B(x,4\rho_{2})}(\xi_{s})\big{)}\Big{|}\frac{\xi_{t}-x}{\rho_{ 1}}-\frac{\xi_{s}-x}{\rho_{2}}\Big{|}+\mathbf{1}_{B(x,4\rho_{2})}(\xi_{s}) \Big{|}\frac{1}{\rho_{1}}-\frac{1}{\rho_{2}}\Big{|}.\]
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\[L= \;(2n)^{-1}\Big{\{}\|\mathbf{Y}\|_{F}^{2}-2\bm{u}_{k}^{T}\mathbf{ Y}^{T}\mathbf{X}\bm{v}_{k}+\bm{u}_{k}^{T}\mathbf{X}^{T}\mathbf{X}\bm{u}_{k}\bm{ v}_{k}^{T}\bm{v}_{k}\] \[+2\langle\mathbf{Y},-\mathbf{X}\widehat{\mathbf{C}}^{(1)}\rangle+ 2\langle\mathbf{Y},-\mathbf{X}\mathbf{C}^{(2)}\rangle+2\bm{u}_{k}^{T}\mathbf{ X}^{T}\mathbf{X}\widehat{\mathbf{C}}^{(1)}\bm{v}_{k}\] \[+2\langle\mathbf{X}\widehat{\mathbf{C}}^{(1)},\mathbf{X}\mathbf{ C}^{(2)}\rangle+\|\mathbf{X}\widehat{\mathbf{C}}^{(1)}\|_{F}^{2}+\sum_{j=k+1}^{r^{*}} \bm{u}_{j}^{T}\mathbf{X}^{T}\mathbf{X}\bm{u}_{j}\bm{v}_{j}^{T}\bm{v}_{j} \Big{\}}.\]
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\[\mathcal{B}^{\prime}\equiv\sqrt{a_{\varsigma+s,l,k}}\,\big{[}\det \big{(}-\overline{b}\bm{z}+\overline{a}\big{)}\big{]}^{-\varsigma-s}\ \left((g^{-1}\diamond\bm{z})\cdot(g^{-1}\diamond\bm{z})\right)^{k}\,\mathcal{Y}_{ l-2k,m}\big{(}g^{-1}\diamond\bm{z}\big{)}\] \[=\sum_{l^{\prime\prime\prime}k^{\prime\prime\prime\prime\prime\prime \prime\prime}}\,\mathbf{U}^{(\varsigma+s,0)}_{l,k,m;l^{\prime\prime\prime},k^{ \prime\prime\prime\prime},m^{\prime\prime\prime\prime}}(g)\ \sqrt{a_{\varsigma+s,l^{\prime\prime\prime},k^{\prime\prime\prime\prime \prime}}}\,(\bm{z}\cdot\bm{z})^{k^{\prime\prime\prime\prime}}\,\mathcal{Y}_{l^{ \prime\prime\prime\prime}-2k^{\prime\prime\prime\prime},m^{\prime\prime\prime \prime\prime}}(\bm{z})\,,\]
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\[(-1)^{d-1}s_{(m+2,2^{j-1},1^{d-2j})}(\mathbf{x})+(-1)^{d-1}s_{(1^ {j-1})}(\mathbf{x})s_{(m+1,1^{d-j})}(\mathbf{x})+(-1)^{d}s_{(1^{j})}(\mathbf{x })s_{(m+1,1^{d-j-1})}(\mathbf{x})\] \[= (-1)^{d-1}s_{(m+2,2^{j-1},1^{d-2j})}(\mathbf{x})+(-1)^{d-1}\Big{(} s_{(1^{j-1})}(\mathbf{x})s_{(m+1,1^{d-j})}(\mathbf{x})-s_{(1^{j})}(\mathbf{x})s_{(m+1,1^{d-j-1})}(\mathbf{x})\Big{)}\] \[= (-1)^{d-1}s_{(m+2,2^{j-1},1^{d-2j})}(\mathbf{x})+(-1)^{d-1}\Big{(} -s_{(m+2,2^{j-1},1^{d-2j})}(\mathbf{x})-s_{(m+1,2^{j},1^{d-2j-1})}(\mathbf{x}) \Big{)}\] \[= (-1)^{d}s_{(m+1,2^{j},1^{d-2j-1})}(\mathbf{x}),\]
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\[\frac{1}{2}\log\left(\frac{\lambda}{\lambda+n}\right)+\frac{1}{2} \frac{n^{2}}{n+\lambda}\frac{\bm{\Omega}_{\tilde{\bm{z}}}}{s_{n}^{2}(\bm{Y}_{n}) }(\hat{\delta}^{ols}_{n}-\delta)^{2}<\frac{1}{2}\log(\alpha^{-2})+o_{a.s.}(1)\] \[\Rightarrow \frac{n^{2}}{n+\lambda}\frac{\bm{\Omega}_{\tilde{\bm{z}}}}{s_{n}^ {2}(\bm{Y}_{n})}(\hat{\delta}^{ols}_{n}-\delta)^{2}<\log\left(\frac{\lambda+n} {n\alpha^{2}}\right)+o_{a.s.}(1)\] \[\Rightarrow \frac{\bm{\Omega}_{\tilde{\bm{z}}}}{s_{n}^{2}(\bm{Y}_{n})}(\hat {\delta}^{ols}_{n}-\delta)^{2}<\frac{n+\lambda}{n^{2}}\log\left(\frac{\lambda+ n}{n\alpha^{2}}\right)+o_{a.s.}(n^{-1})\] \[\Rightarrow |\hat{\delta}^{ols}_{n}-\delta|<\sqrt{\frac{s_{n}^{2}(\bm{Y}_{n} )}{\bm{\Omega}_{\tilde{\bm{z}}}}\left(\frac{n+\lambda}{n^{2}}\right)\log \left(\frac{\lambda+n}{n\alpha^{2}}\right)}+o_{a.s.}(n^{-\frac{1}{2}})\]
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\[{}_{3}F_{2}\left[\begin{matrix}a,b,c\\ 3+a-b,a-c\end{matrix}\Big{|}\,1\right]\] \[=\ \frac{2+b^{2}-3b+bc-ac-c}{(b-1)(b-2)}\ \frac{\Gamma(\frac{2+a}{2}) \Gamma(3+a-b)\Gamma(a-c)\Gamma(\frac{4+a}{2}-b-c)}{\Gamma(1+a)\Gamma(\frac{4+a }{2}-b)\Gamma(\frac{9}{2}-c)\Gamma(3+a-b-c)}\] \[+\ \frac{2+b^{2}-3b-bc+ac+c-2c^{2}}{2(b-1)(b-2)}\frac{\Gamma(\frac {1+a}{2})\Gamma(3+a-b)\Gamma(a-c)\Gamma(\frac{3+a}{2}-b-c)}{\Gamma(a)\Gamma( \frac{3+a}{2}-b)\Gamma(\frac{1+a}{2}-c)\Gamma(3+a-b-c)}.\]
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\begin{table}
\begin{tabular}{|l|l||l|l|} \hline Index & Relation & Index & Relation \\ \hline
1 & Adjectival clause modifier & 20 & Fixed multiword expression \\
2 & Adverbial clause modifier & 21 & Flat multiword expression \\
3 & Adverbial modifier & 22 & Goes with \\
4 & Adjectival modifier & 23 & Indirect object \\
5 & Appositional modifier & 24 & List \\
6 & Auxiliary & 25 & Marker \\
7 & Case marking & 26 & Nominal modifier \\
8 & Coordinating conjunction & 27 & Nominal subject \\
9 & Clausal complement & 28 & Numeric modifier \\
10 & Classifier & 29 & Object \\
11 & Compound & 30 & Oblique nominal \\
12 & Conjunct & 31 & Orphan \\
13 & Copula & 32 & Parataxis \\
14 & Clausal subject & 33 & Punctuation \\
15 & Unspecified dependency & 34 & Overridden disfluency \\
16 & Determiner & 35 & Root \\
17 & Discourse element & 36 & Vocative \\
18 & Dislocated elements & 37 & Open clausal complement \\
19 & Expletive & & \\ \hline \end{tabular}
\end{table}
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\[\sum_{k=1}^{K}\sum_{x,a}\bm{\omega}_{k}(x,a)f_{x,a}(n_{k}(x,a)) =\sum_{x,a}\sum_{k=1}^{\tau(x,a)-1}\bm{\omega}_{k}(x,a)f_{x,a}(n_{ k}(x,a))+\sum_{x,a}\sum_{k=\bm{\tau}(x,a)}f_{x,a}(n_{k})\] \[\leq\sum_{x,a}\sum_{k=1}^{\tau(x,a)-1}\bm{\omega}_{k}(x,a)f_{ \max}+\sum_{x,a}\sum_{k=\bm{\tau}(x,a)}f_{x,a}(n_{k})\] \[\leq SAH_{\text{sample}}f_{\max}+\sum_{x,a}\sum_{k=\bm{\tau}(x,a) }f_{x,a}(n_{k}(x,a))\] \[\leq SAH_{\text{sample}}f_{\max}+\sum_{x,a}\sum_{k=\bm{\tau}(x,a) }f_{x,a}(\overline{n}_{k}(x,a)/4),\]
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\[S(z) =\left[\big{(}\Sigma_{\alpha\triangle m}^{{}_{(\mathcal{D})}} \big{)}^{(1,1)}(z)\widetilde{F}(z)+\big{(}\Sigma_{\alpha\triangle m}^{{}_{( \mathcal{D})}}\big{)}^{(1,2)}(z)\right]\] \[\cdot\left[\big{(}\Sigma_{\alpha\triangle m}^{{}_{( \mathcal{D})}}\big{)}^{(2,1)}(z)\widetilde{F}(z)+\big{(}\Sigma_{\alpha \triangle m}^{{}_{(\mathcal{D})}}\big{)}^{(2,2)}(z)\right]^{-1}\] \[=\left[\big{(}\Sigma_{-\alpha\triangleright m}^{{}_{(\mathcal{D})}} \big{)}^{(1,1)}(-z)\big{(}-\widetilde{G}\big{)}(-z)-\big{(}\Sigma_{-\alpha \triangleright m}^{{}_{(\mathcal{D})}}\big{)}^{(1,2)}(-z)\right]\] \[\cdot\left[-\big{(}\Sigma_{-\alpha\triangleright m}^{{}_{(\mathcal{ D})}}\big{)}^{(2,1)}(-z)\big{(}-\widetilde{G}\big{)}(-z)+\big{(}\Sigma_{-\alpha \triangleright m}^{{}_{(\mathcal{D})}}\big{)}^{(2,2)}(-z)\right]^{-1}\] \[=-\left[\big{(}\Sigma_{-\alpha\triangleright m}^{{}_{(\mathcal{D})} }\big{)}^{(1,1)}(-z)\widetilde{G}(-z)+\big{(}\Sigma_{-\alpha\triangleright m}^{ {}_{(\mathcal{D})}}\big{)}^{(1,2)}(-z)\right]\] \[\cdot\left[\big{(}\Sigma_{\alpha\triangleright m}^{{}_{(\mathcal{ D})}}\big{)}^{(2,1)}(-z)\widetilde{G}(-z)+\big{(}\Sigma_{-\alpha\triangleright m}^{{}_{( \mathcal{D})}}\big{)}^{(2,2)}(-z)\right]^{-1}=-T(-z)\]
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\[||(\frac{\partial^{m_{1}+m_{2}}v_{r,\gamma}(w)}{\partial x_{1}^{m_{1}} \partial x_{2}^{m_{2}}}-v_{r,\gamma}(\frac{\partial^{m_{1}+m_{2}}w}{\partial x_{ 1}^{m_{1}}\partial x_{2}^{m_{2}}}))1_{|x|\geq\frac{1}{2}}||_{L^{\infty}}\] \[\leq C(||v_{1,\gamma}(w)1_{|x|\geq\frac{1}{2}}||_{C^{m_{1}+m_{2}- 1}}+||v_{2,\gamma}(w)1_{|x|\geq\frac{1}{2}}||_{C^{m_{1}+m_{2}-1}}),\] \[||(\frac{\partial^{m_{1}+m_{2}}v_{r,\gamma}(w)}{\partial x_{1}^{m _{1}}\partial x_{2}^{m_{2}}}-v_{r,\gamma}(\frac{\partial^{m_{1}+m_{2}}w}{ \partial x_{1}^{m_{1}}\partial x_{2}^{m_{2}}}))1_{|x|\geq\frac{1}{2}}||_{L^{2}}\] \[\leq C(||v_{1,\gamma}(w)1_{|x|\geq\frac{1}{2}}||_{H^{m_{1}+m_{2}- 1}}+||v_{2,\gamma}(w)1_{|x|\geq\frac{1}{2}}||_{H^{m_{1}+m_{2}-1}})\]
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\[\frac{d^{2}}{ds^{2}}\mathcal{J}(\mathcal{H}(u,s))\Big{|}_{s=s_{u}}= 8e^{4s_{u}}\int_{\mathbb{R}^{4}}|\Delta u|^{2}dx+2\beta e^{2s_{u} }\int_{\mathbb{R}^{4}}|\nabla u|^{2}dx\] \[-\frac{(8-\mu)^{2}}{2}e^{(\mu-8)s_{u}}\int_{\mathbb{R}^{4}}(I_{ \mu}*F(e^{2s_{u}}u))F(e^{2s_{u}}u)dx\] \[+(28-4\mu)e^{(\mu-8)s_{u}}\int_{\mathbb{R}^{4}}(I_{\mu}*F(e^{2s_{ u}}u))f(e^{2s_{u}}u)e^{2s_{u}}udx\] \[-4e^{(\mu-8)s_{u}}\int_{\mathbb{R}^{4}}(I_{\mu}*F(e^{2s_{u}}u))f^{ \prime}(e^{2s_{u}}u)e^{4s_{u}}u^{2}dx\] \[= -2\beta\int_{\mathbb{R}^{4}}|\nabla\mathcal{H}(u,s_{u})|^{2}dx-( 8-\mu)(6-\frac{\mu}{2})\int_{\mathbb{R}^{4}}(I_{\mu}*F(\mathcal{H}(u,s_{u}))) F(\mathcal{H}(u,s_{u}))dx\] \[+(36-4\mu)\int_{\mathbb{R}^{4}}(I_{\mu}*F(\mathcal{H}(u,s_{u}))) f(\mathcal{H}(u,s_{u}))\mathcal{H}(u,s_{u})dx\] \[-4\int_{\mathbb{R}^{4}}(I_{\mu}*f(\mathcal{H}(u,s_{u}))\mathcal{H }(u,s_{u}))f(\mathcal{H}(u,s_{u}))\mathcal{H}(u,s_{u})dx\] \[-4\int_{\mathbb{R}^{4}}(I_{\mu}*F(\mathcal{H}(u,s_{u})))f^{\prime }(\mathcal{H}(u,s_{u}))\mathcal{H}^{2}(u,s_{u})dx\] \[= -2\beta\int_{\mathbb{R}^{4}}|\nabla\mathcal{H}(u,s_{u})|^{2}dx+4 \int_{\mathbb{R}^{4}}\int_{\mathbb{R}^{4}}\frac{A}{|x-y|^{\mu}}dxdy<0,\]
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\[\mathcal{S}^{(Rt)}_{4,8} = \alpha_{1,1}\log\left(\frac{p_{1678}p_{2345}}{p_{1345}p_{2678}} \right)+\alpha_{1,2}\log\left(\frac{p_{1245}p_{1678}p_{3456}}{p_{1345}p_{1456} p_{2678}}\right)+\alpha_{1,3}\log\left(\frac{p_{1256}p_{1678}p_{4567}}{p_{1456}p_{1567 }p_{2678}}\right)\] \[+ \alpha_{1,4}\log\left(\frac{p_{1267}p_{5678}}{p_{1567}p_{2678}} \right)+\alpha_{2,1}\log\left(\frac{p_{1278}p_{1345}}{p_{1245}p_{1378}}\right) +\alpha_{2,2}\log\left(\frac{p_{1235}p_{1278}p_{1456}}{p_{1245}p_{1256}p_{1378 }}\right)\] \[+ \alpha_{2,3}\log\left(\frac{p_{1236}p_{1278}p_{1567}}{p_{1256}p_{ 1267}p_{1378}}\right)+\alpha_{2,4}\log\left(\frac{p_{1237}p_{1678}}{p_{1267}p_{ 1378}}\right)+\alpha_{3,1}\log\left(\frac{p_{1238}p_{1245}}{p_{1235}p_{1248}}\right)\] \[+ \alpha_{3,2}\log\left(\frac{p_{1234}p_{1238}p_{1256}}{p_{1235}p_{ 1236}p_{1248}}\right)+\alpha_{3,3}\log\left(\frac{p_{1234}p_{1238}p_{1267}}{p_{ 1236}p_{1237}p_{1248}}\right)+\alpha_{3,4}\log\left(\frac{p_{1234}p_{1278}}{p_ {1237}p_{1248}}\right)\]
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\[J_{21} =\sum_{k_{j}}2^{k_{3}n/2}\left\|P_{k_{3}}\left[\sum_{i=1}^{n}v_{i} \partial_{x_{i}}P_{\geq k_{1}+k_{2}-9}Q_{\geq k_{1}+k_{2}-10}w(P_{k_{1}}fP_{k_{ 2}}g)\right]\right\|_{L_{t}^{1}L_{x}^{2}}\] \[\lesssim\sum_{k_{j}}2^{k_{3}n/2}\|v\|_{L_{t,x}^{\infty}}\sum_{i=1 }^{n}\sum_{k_{1}+k_{2}\in\mathbb{Z}}\left\|P_{k_{1}+k_{2}}Q_{\geq k_{1}+k_{2}-1 0}\partial_{x_{i}}w\right\|_{L_{t}^{2}L_{x}^{2}}\left\|P_{k_{1}}f\right\|_{L_ {x_{j}}^{2,\infty}}\left\|P_{k_{2}}g\right\|_{L_{x_{j}}^{\infty,2}}\] \[\lesssim\sum_{k_{j}}2^{k_{3}n/2}\|v\|_{L_{t,x}^{\infty}}2^{(k_{1} +k_{2})n/2}2^{k_{1}+k_{2}}\sum_{j\geq k_{1}+k_{2}-10}\left\|Q_{j}(P_{k_{1}+k_{ 2}}w)\right\|_{L_{t,x}^{2}}\left\|P_{k_{1}}f\right\|_{L_{x_{j}}^{2,\infty}} \left\|P_{k_{2}}g\right\|_{L_{x_{j}}^{\infty,2}}\] \[\lesssim\sum_{k_{j}}2^{k_{3}n/2+(k_{1}+k_{2})n/2+k_{1}+k_{2}}\|v \|_{L_{t,x}^{\infty}}\left[\sum_{j\geq k_{1}+k_{2}-10}\left(2^{j}\|Q_{j}(P_{k_{ 1}+k_{2}}w)\|_{L_{t,x}^{2}}\right)^{2}\right]^{\frac{1}{2}}\] \[\quad\left[\sum_{j\geq k_{1}+k_{2}-10}2^{-2j}\right]^{\frac{1}{2} }\left\|P_{k_{1}}f\right\|_{L_{x_{j}}^{2,\infty}}\left\|P_{k_{2}}g\right\|_{L_ {x_{j}}^{\infty,2}}\] \[\lesssim\sum_{k_{j}}2^{k_{3}n/2+(k_{1}+k_{2})n/2+k_{1}+k_{2}}\|v \|_{L_{t,x}^{\infty}}2^{-(k_{1}+k_{2})}\|P_{k_{1}+k_{2}}w\|_{X^{0,1,2}}\|P_{k _{1}}f\|_{L_{x_{j}}^{2,\infty}}\|P_{k_{2}}g\|_{L_{x_{j}}^{\infty,2}}\] \[\lesssim\sum_{k_{j}}2^{(k_{3}-k_{2})n/2+(k_{1}+k_{2})/2}\|v\|_{L_ {t,x}^{\infty}}\|P_{k_{1}+k_{2}}w\|_{Y^{n/2}}\|P_{k_{1}}f\|_{F^{n/2}}\|P_{k_{2} }g\|_{F^{n/2}}.\]
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\[\begin{split} 0=&\int_{\mathbb{R}^{N}}\partial_{t}^{2} \mathbf{u}_{\varepsilon}\partial_{t}\mathbf{u}_{\varepsilon}\,d\mathbf{x}- \int_{\mathbb{R}^{N}}(K\mathbf{u}_{\varepsilon}(\cdot,t))(\mathbf{x}) \partial_{t}\mathbf{u}_{\varepsilon}\,d\mathbf{x}+\varepsilon\int_{\mathbb{R} ^{N}}\Delta^{2}\mathbf{u}_{\varepsilon}\partial_{t}\mathbf{u}_{\varepsilon}\,d \mathbf{x}\\ =&\int_{\mathbb{R}^{N}}\partial_{t}^{2}\mathbf{u}_{ \varepsilon}\partial_{t}\mathbf{u}_{\varepsilon}\,d\mathbf{x}-\int_{\mathbb{R }^{N}}(K\mathbf{u}_{\varepsilon}(\cdot,t))(\mathbf{x})\partial_{t}\mathbf{u}_{ \varepsilon}\,d\mathbf{x}+\varepsilon\int_{\mathbb{R}^{N}}\Delta\mathbf{u}_{ \varepsilon}\partial_{t}\Delta\mathbf{u}_{\varepsilon}\,d\mathbf{x}\\ =&\frac{d}{dt}\int_{\mathbb{R}^{N}}\frac{|\partial_ {t}\mathbf{u}_{\varepsilon}|^{2}+\varepsilon|\Delta\mathbf{u}_{\varepsilon}|^ {2}}{2}\,d\mathbf{x}\\ &+\int_{\mathbb{R}^{N}}\int_{B_{\delta}(\mathbf{0})}\mathbf{f}( \mathbf{y},\mathbf{u}_{\varepsilon}(\mathbf{x},t)-\mathbf{u}_{\varepsilon}( \mathbf{x}-\mathbf{y},t))\partial_{t}\mathbf{u}_{\varepsilon}(\mathbf{x},t) \,d\mathbf{x}\,d\mathbf{y}.\end{split}\]
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\begin{table}
\begin{tabular}{l l|l l} \hline parameter & value & parameter & value \\ \hline hidden layers & depth=3, width=30 & \(N_{e}\) & 3 \\
1st hidden layer & \(C^{\infty}\) periodic layer \(\mathcal{L}_{p}(11,30)\), & \(Q\) & 30 (\(C^{\infty}\) periodic DNN), \\ & or \(C^{k}\) periodic layer \(\mathcal{L}_{C^{k}}(11,30)\) & & or 40 (\(C^{k}\) periodic DNN) \\ activation & tanh & optimizer & Adam \\ maximum epochs & 10000 & learning rate & \(1e-3\) \\ input data & \(x_{i}^{e}\) (\(0\leqslant e\leqslant N_{e}-1\), \(0\leqslant i\leqslant Q-1\)) & label data & \(u_{1}(x_{i}^{e})\), or \(u_{2}(x_{i}^{e})\), or \(u_{3}(x_{i}^{e})\) \\ \(x_{i}^{e}\) & Gauss-Lobatto-Legendre quadrature & \(\omega\) & \(2\pi/V_{\Omega}\) \\ \hline \end{tabular}
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\[\mathbb{E}\left[\prod_{j=1}^{p}u(t,x_{j})\right] =\mathbb{E}\left[\prod_{j=1}^{p}\sum_{n_{j}=0}^{\infty}I_{n_{j}}( f_{n_{j}}(\cdot,t,x_{j}))\right]\] \[=\sum_{n_{1}=0}^{\infty}\cdots\sum_{n_{p}=0}^{\infty}\mathbb{E} \Big{[}I_{n_{1}}(f_{n_{1}}(\cdot,t,x_{1}))\cdots I_{n_{p}}(f_{n_{p}}(\cdot,t,x_ {p}))\Big{]}\] \[=\sum_{m=0}^{\infty}\sum_{\begin{subarray}{c}\vec{n}\in\mathbb{N }^{p}\\ |\vec{n}|=2m\end{subarray}}\sum_{\mathcal{D}\in\mathbb{D}_{\vec{n}}}F_{\mathcal{ D}}\left(g_{n_{1}}\left(\cdot;t,x_{1}\right),\ldots,g_{n_{p}}\left(\cdot;t,x_{p} \right)\right),\]
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\[X_{1}(t) \geq X_{1m}^{(0,t_{1}]}(t)\] \[\geq\int_{0}^{t}\mu\bigg{(}N-\frac{5N\mu}{s}e^{su}\bigg{)}e^{s(t -u)-7e^{-C_{1}}}du-\sqrt{\frac{48}{\epsilon}\cdot\frac{N\mu}{s^{2}}}e^{st}\] \[=\bigg{(}e^{-7e^{-C_{1}}}\int_{0}^{t}(se^{-su}-5\mu)du-\sqrt{ \frac{48}{\epsilon}\cdot\frac{1}{N\mu}}\bigg{)}\bigg{(}\frac{N\mu}{s}e^{st} \bigg{)}\] \[\geq\bigg{(}(1-7e^{-C_{1}})(1-e^{-st}-5\mu t)-\sqrt{\frac{48}{ \epsilon}\cdot\frac{1}{N\mu}}\bigg{)}\bigg{(}\frac{N\mu}{s}e^{st}\bigg{)}\] \[\geq\bigg{(}(1-7e^{-C_{1}})(1-e^{-st_{0,r}}-5\mu t_{1})-\sqrt{ \frac{48}{\epsilon}\cdot\frac{1}{N\mu}}\bigg{)}\bigg{(}\frac{N\mu}{s}e^{st} \bigg{)}.\]
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\[\begin{split} I_{21}&\triangleq c_{v}[(u+U\phi^{\prime}) \partial_{x}+(v-U_{x}\phi)\partial_{y}]D^{\alpha}\theta,\\ I_{22}&\triangleq c_{v}[D^{\alpha},(u+U\phi^{\prime}) \partial_{x}+(v-U_{x}\phi)\partial_{y}]\theta,\\ I_{23}&\triangleq D^{\alpha}(c_{v}\Theta_{x}\phi^{ \prime}u+c_{v}\Theta\phi^{\prime\prime}v-\mu\theta(u_{y})^{2}-\mu(U\phi^{ \prime\prime})^{2}\theta-2\mu U\phi^{\prime\prime}\theta u_{y}-\mu\Theta\phi^ {\prime}(u_{y})^{2})\\ &\quad-D^{\alpha}(\mu\theta U\phi^{\prime}\phi^{\prime\prime}u_{y }+2\mu U\phi^{\prime\prime}u_{y}+\mu(u_{y})^{2}+\nu(h_{y})^{2}+2\nu H\phi^{ \prime}h_{y}).\end{split}\]
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\[\left\{\begin{array}{ll}\overline{\phi}_{1}(z)=\begin{cases}(a-1)Bze^{- \lambda_{1}^{*}z},&z>z_{1},\\ a-1,&z\leq z_{1},\end{cases}\ \ \ \ \ \overline{\phi}_{2}(z)=\begin{cases}(a-1)Bze^{- \lambda_{2}^{*}z},&z>z_{2},\\ a-1,&z\leq z_{2},\end{cases}\\ \underline{\phi}_{1}(z)=\begin{cases}(a-1)Bze^{-\lambda_{1}^{*}z}-p_{3}z^{1/2} e^{-\lambda_{1}^{*}z},&z>z_{3},\\ 0,&z\leq z_{3},\end{cases}\\ \underline{\phi}_{2}(z)=\begin{cases}(a-1)Bze^{-\lambda_{2}^{*}z}-p_{4}z^{1/2} e^{-\lambda_{2}^{*}z},&z>z_{4},\\ 0,&z\leq z_{4},\end{cases}\\ \overline{\phi}_{3}(z)=1,\ \ \underline{\phi}_{3}(z)=\max\{1-e^{- \lambda_{0}(z-z_{0})},0\},\end{array}\right.\]
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\[\mathsf{U}_{q,t}= \frac{t^{1-N}}{1-t}A_{2}q\sum_{s=1}^{\infty}s\tau_{s}\mathsf{Schur }_{\{s\}}\left(p_{k}=\frac{1-t^{k}}{q^{k}}\partial_{k}\right)+\] \[+\frac{t^{1-N}}{1-t}A_{1}\sum_{s=1}^{\infty}s\tau_{s}\mathsf{Schur }_{\{s-1\}}\left(p_{k}=\frac{1-t^{k}}{q^{k}}\partial_{k}\right)-\frac{t}{1-t} A_{1}\tau_{1}+\] \[+\frac{t^{1-N}}{1-t}q^{-1}\sum_{s=1}^{\infty}s\tau_{s}\mathsf{Schur }_{\{s-2\}}\left(p_{k}=\frac{1-t^{k}}{q^{k}}\partial_{k}\right)-\frac{q^{-1}t} {1-t}2\tau_{2}+\] \[+\frac{t^{N}}{1-t}\sum_{\ell=0}^{\infty}\sum_{s=1}^{\infty}s\tau_{ s}\mathsf{Schur}_{\{\ell\}}\left(p_{k}=-k(1-q^{k})\tau_{k}\right)\mathsf{Schur }_{\{\ell+s-2\}}\left(p_{k}=(1-t^{-k})\partial_{k}\right)+\] \[-\frac{1}{1-t}2\tau_{2}+\frac{1-q}{1-t}\tau_{1}^{2}\]
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\[\begin{split}\begin{array}{|c|cccccccc|}\hline Q\backslash n&-3&-2&-1&0& 1&2&3\\ \hline 0&0&0&0&1&144&10008&446304\\ 1&0&0&0&0&1053&149526&10238319\\ 2&0&0&0&0&52812&8053182&591031890\\ 3&0&0&0&3402&6914214&1001912544&71961634872\\ 4&0&0&0&5520393&1937967282&225717793668&14749020131814\\ 5&0&0&5520393&5626721862&1006811225253&88682916004956&4943255069504250\\ 6&10206&3838378&24521163804&666284688372&768849614982540&52757172850696986&2484705136566 6066336\\ \hline\end{array}\end{split}\]
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\[\sum_{r=1}^{m}\mathbb{P}(\sup_{d_{r}+g_{n}\leq t\leq\tilde{a}_{r} }(2II_{r,2}(t)\mu_{r}(t)+\frac{1}{6}(\mu_{r}^{2}(t)-\mu_{r}^{2}(d_{r})))\geq 0 |\mathcal{E}_{n},u_{r},l_{r})\] \[=\sum_{r=1}^{m}\mathbb{P}(\sup_{d_{r}+g_{n}\leq t\leq\tilde{a}_{ r}}\left|\frac{2\mu_{r}(t)\lambda([d_{r},t])}{\lambda([l_{r},u_{r}])} \right|(-\tilde{S}_{[l_{r},u_{r}]}\mathrm{sgn}(\mu_{r}(t))\] \[\qquad\qquad\qquad\qquad\qquad+\frac{(\mu_{r}^{2}(t)-\mu_{r}^{2} (d_{r}))\lambda([l_{r},u_{r}])}{12|\mu_{r}(t)\lambda([d_{r},t])|}\geq 0\big{|} \mathcal{E}_{n},u_{r},l_{r})\] \[\leq\sum_{r=1}^{m}\mathbb{P}\left(|\tilde{S}_{[l_{r},u_{r}]}| \geq C(\tilde{\alpha})n|\Delta_{r,n}|z_{n}||\mathcal{E}_{n},u_{r},l_{r}\right)\]
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\[x_{3}x_{4}x_{6}+x_{3}x_{4}x_{7}+x_{3}x_{4}x_{8}+x_{3}x_{5}x_{6}+x_{3}x_{5}x_{7}+ x_{3}x_{5}x_{8}+x_{3}x_{6}x_{7}+x_{3}x_{7}x_{8}+x_{4}x_{6}x_{8}+x_{4}x_{7}x_{8}+x _{5}x_{6}x_{8}+x_{5}x_{7}x_{8}+x_{6}x_{7}x_{8})(sx_{2}x_{3}x_{4}x_{5}x_{6}+sx_{ 2}x_{3}x_{4}x_{5}x_{7}+sx_{2}x_{3}x_{4}x_{5}x_{8}+sx_{2}x_{3}x_{3}x_{5}x_{7}x_{8}+ sx_{3}x_{4}x_{5}x_{7}x_{8})-(sx_{3}x_{5}x_{7}x_{8}+ux_{2}x_{4}x_{6}x_{7}+ux_{2}x_{4}x_{6 }x_{8}+ux_{2}x_{5}x_{6}x_{8}+ux_{2}x_{6}x_{7}x_{8}+ux_{3}x_{4}x_{6}x_{7}+ux_{4}x_{ 4}x_{6}x_{7}x_{8})(x_{2}x_{3}x_{4}x_{6}+x_{2}x_{3}x_{4}x_{7}+x_{2}x_{3}x_{4}x_{8 }+x_{2}x_{3}x_{5}x_{6}+x_{2}x_{3}x_{5}x_{7}+x_{2}x_{3}x_{5}x_{8}+x_{2}x_{3}x_{6}x_{7 }+x_{2}x_{3}x_{7}x_{8}+x_{2}x_{4}x_{5}x_{6}+x_{2}x_{4}x_{5}x_{7}+x_{2}x_{4}x_{5}x _{8}+x_{2}x_{4}x_{6}x_{7}+x_{2}x_{4}x_{6}x_{8}+x_{2}x_{5}x_{6}x_{8}+x_{2}x_{5}x_{7 }x_{8}+x_{2}x_{6}x_{7}x_{8}+x_{3}x_{4}x_{5}x_{6}+x_{3}x_{4}x_{5}x_{7}+x_{3}x_{4}x_{5 }x_{8}+x_{3}x_{4}x_{6}x_{7}+x_{3}x_{4}x_{7}x_{8}+x_{4}x_{5}x_{6}x_{8}+x_{4}x_{5}x _{7}x_{8}+x_{4}x_{6}x_{7}x_{8})\}.\]
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\[g^{c} =\frac{\rho+c}{\rho}g_{\mathbb{C}\mathrm{H}^{n-1}}+\frac{1}{4\rho ^{2}}\frac{\rho+2c}{\rho+c}\mathrm{d}\rho^{2}\] \[\quad+\frac{1}{4\rho^{2}}\frac{\rho+c}{\rho+2c}\bigg{(}\mathrm{d} \tilde{\phi}-4\operatorname{Im}\left(\bar{w}^{0}\mathrm{d}w^{0}-\sum_{a=1}^{n- 1}\bar{w}^{a}\mathrm{d}w^{a}\right)+\frac{2c}{1-\|X\|^{2}}\operatorname{Im} \bigg{(}\sum_{a=1}^{n-1}\bar{X}^{a}\mathrm{d}X^{a}\bigg{)}\bigg{)}^{2}\] \[\quad-\frac{2}{\rho}\bigg{(}\mathrm{d}w^{0}\mathrm{d}\bar{w}^{0}- \sum_{a=1}^{n-1}\mathrm{d}w^{a}\mathrm{d}\bar{w}^{a}\bigg{)}+\frac{\rho+c}{\rho ^{2}}\frac{4}{1-\|X\|^{2}}\Big{|}\mathrm{d}w^{0}+\sum_{a=1}^{n-1}X^{a}\mathrm{d }w^{a}\Big{|}^{2}\]
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\[\mathbf{y}(t)=\int_{0}^{t}\mathbf{C}\Phi(\sigma_{1})\mathbf{Bu}(t _{\sigma_{1}})d\sigma_{1}\\ +\sum_{\xi=2}^{d}\int_{0}^{t}\underbrace{\int_{0}^{t_{\sigma_{1} }}\cdots\int_{0}^{t_{\sigma_{1}}}}_{\xi-\text{times}}\mathbf{C}\Phi(\sigma_{1} )\mathbf{H}_{\xi}\left(\Phi(\sigma_{2})\mathbf{B}\otimes\cdots\otimes\Phi( \sigma_{\xi+1})\mathbf{B}\right)\\ \times\left(\mathbf{u}(t_{\sigma_{1}}-\sigma_{2})\cdots\mathbf{u} (t_{\sigma_{1}}-\sigma_{\xi+1})\right)d\sigma_{1}d\sigma_{2}\cdots d\sigma_{ \xi+1}\\ +\sum_{\eta=1}^{d-1}\int_{0}^{t}\underbrace{\int_{0}^{t_{\sigma_{1 }}}\cdots\int_{0}^{t_{\sigma_{1}}}}_{\eta-\text{times}}\mathbf{C}\Phi(\sigma_ {1})\mathbf{N}_{\eta}\left(\Phi(\sigma_{2})\mathbf{B}\otimes\cdots\otimes\Phi( \sigma_{\eta+1})\mathbf{B}\right)\\ \times\left(\mathbf{u}(t_{\sigma_{1}})\mathbf{u}(t_{\sigma_{1}}- \sigma_{2})\cdots\mathbf{u}(t_{\sigma_{1}}-\sigma_{\eta+1})\right)d\sigma_{1}d \sigma_{2}\cdots d\sigma_{\eta+1}+\cdots,\]
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\[\begin{split}& I\left(Z;W\right)=\mathrm{H}\left(Z\right)- \mathrm{H}\left(Z\middle|W\right)\\ &=\mathrm{H}\left(Z\right)-\mathrm{H}\left(Z\middle|W=0\right) \frac{\alpha}{1+\alpha}-\mathrm{H}\left(Z\middle|W=1\right)\frac{1}{1+\alpha} \\ &=-\sum_{x\in\mathcal{X}}\left(\frac{\alpha}{1+\alpha}P\left(x \right)+\frac{1}{1+\alpha}Q\left(x\right)\right)\\ &\qquad\qquad\times\log\left(\frac{\alpha}{1+\alpha}P\left(x \right)+\frac{1}{1+\alpha}Q\left(x\right)\right)\\ &+\frac{\alpha}{1+\alpha}\sum_{x\in\mathcal{X}}P\left(x\right)\log P \left(x\right)+\frac{1}{1+\alpha}\sum_{x\in\mathcal{X}}Q\left(x\right)\log Q \left(x\right)\\ &=\frac{1}{1+\alpha}\mathrm{GJS}\left(P,Q,\alpha\right)\end{split}\]
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\[c_{p}^{r}\] \[= \frac{1}{(MN)^{r}}\int_{T^{r}}\sum_{i_{1}^{1}\ldots i_{p}^{r}} \sum_{b_{1}^{1}\ldots b_{p}^{r}}\frac{Q^{1}_{i_{1}b_{1}^{1}}Q^{1}_{i_{1}b_{1}^ {1}}}{Q^{1}_{i_{1}b_{2}^{1}}Q^{1}_{i_{1}b_{1}^{1}}}\cdots\frac{Q^{1}_{i_{p}^{1 }b_{1}^{1}}Q^{1}_{i_{p}^{2}b_{1}^{1}}}{Q^{1}_{i_{p}^{1}b_{1}^{1}}Q^{1}_{i_{p}^ {2}b_{p}^{1}}}\cdots\cdots\frac{Q^{r}_{i_{1}^{r}b_{1}^{r}}Q^{r}_{i_{1}^{1}b_{2}^ {r}}}{Q^{r}_{i_{1}^{r}b_{2}^{r}}Q^{r}_{i_{1}^{1}b_{1}^{r}}}\cdots\frac{Q^{r}_{ i_{p}^{r}b_{p}^{r}}Q^{r}_{i_{1}^{1}b_{1}^{r}}}{Q^{r}_{i_{p}^{r}b_{1}^{r}}Q^{r}_{i_{1}^{1 }b_{p}^{r}}}\] \[\frac{1}{M^{pr}}\sum_{j_{1}^{1}\ldots j_{p}^{r}}\frac{K_{i_{1}^{1 }j_{1}^{1}}K_{i_{1}^{2}j_{2}^{1}}}{K_{i_{1}^{1}j_{2}^{1}}K_{i_{1}^{2}j_{1}^{1}} }\cdots\frac{K_{i_{p}^{1}j_{p}^{1}}K_{i_{p}^{2}j_{1}^{1}}}{K_{i_{p}^{1}j_{1}} K_{i_{p}^{2}j_{p}^{1}}}\cdots\frac{K_{i_{7}^{r}j_{1}^{r}}K_{i_{1}^{1}j_{2}^{r}}}{K_{i_{ 7}^{r}j_{1}^{r}}K_{i_{1}^{1}j_{1}^{r}}}\cdots\frac{K_{i_{7}^{r}j_{p}^{r}}K_{i_{ 7}^{1}j_{1}^{r}}}{K_{i_{7}^{r}j_{1}^{r}}K_{i_{7}^{1}j_{p}^{r}}}\] \[\frac{1}{N^{pr}}\sum_{a_{1}^{1}\ldots a_{p}^{r}}\frac{L_{a_{1}^{1 }b_{1}^{1}}L_{a_{1}^{2}b_{1}^{1}}}{L_{a_{1}^{1}b_{2}^{1}}L_{a_{1}^{2}b_{1}^{1} }}\cdots\frac{L_{a_{p}^{1}b_{1}^{1}}L_{a_{p}^{2}b_{1}^{1}}}{L_{a_{p}^{1}b_{1}^ {1}}L_{a_{p}^{2}b_{p}^{1}}}\cdots\cdots\frac{L_{a_{1}^{r}b_{1}^{r}}L_{a_{1}^{1 }b_{2}^{r}}}{L_{a_{1}^{r}b_{2}^{r}}L_{a_{1}^{1}b_{1}^{r}}}\cdots\frac{L_{a_{p} ^{r}b_{p}^{r}}L_{a_{p}^{1}b_{1}^{r}}}{L_{a_{p}^{r}b_{1}^{r}}L_{a_{p}^{1}b_{p}^ {r}}}\,dQ\]
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\[[\partial_{t^{*cr}_{n,m}},\partial_{t^{*dj}_{l,k}}]\mathcal{L}_{i}\] \[= [\sum_{p,s=0}^{\infty}\dfrac{n^{p}(m\log q)^{s}}{p!s!}\partial_{ t^{cr}_{p,s}},\sum_{a,b=0}^{\infty}\dfrac{l^{a}(k\log q)^{b}}{a!b!}\partial_{ t^{dj}_{a,b}}]\mathcal{L}_{i}\] \[= \sum_{p,s=0}^{\infty}\sum_{a,b=0}^{\infty}\dfrac{n^{p}(m\log q)^{ s}}{p!s!}\dfrac{l^{a}(k\log q)^{b}}{a!b!}[\partial_{t^{cr}_{p,s}},\partial_{ t^{dj}_{a,b}}]\mathcal{L}_{i}\] \[= \sum_{p,s=0}^{\infty}\sum_{a,b=0}^{\infty}\dfrac{n^{p}(m\log q)^{ s}}{p!s!}\dfrac{l^{a}(k\log q)^{b}}{a!b!}\sum_{\alpha\beta}C^{(ps)(ab)}_{ \alpha\beta}\delta_{rj}\partial_{\bar{t}^{cr}_{\alpha,\beta}}\mathcal{L}_{i}\] \[= (q^{ml}-q^{nk})\sum_{\alpha,\beta=0}^{\infty}\dfrac{(n+l)^{\alpha }((m+k)\log q)^{\beta}}{\alpha!\beta!}\delta_{cd}\delta_{rj}\partial_{\bar{t} ^{cr}_{\alpha,\beta}}\mathcal{L}_{i}\] \[= (q^{ml}-q^{nk})\delta_{cd}\delta_{rj}\partial_{t^{*cr}_{n+l,m+k}} \mathcal{L}_{i}.\]
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\[d_{\{t_{1},\ldots,t_{j+1-m}\}}(x,r)d_{\{t_{j+3-m},\ldots,t_{j+1}\}}(x, r)\prod_{p=1}^{j+1-m}\prod_{\begin{subarray}{c}q=j+3-m\\ t_{q}=t_{p}\end{subarray}}^{j+1}\Delta\big{(}x^{2(q-p+t_{p}-N-2)}\big{)}\] \[\qquad\times\prod_{\begin{subarray}{c}p=1\\ t_{j+2-m}=\overline{t}_{p}\end{subarray}}^{j+1-m}\Delta\big{(}x^{2(j+t_{p}-m- p-N)}\big{)}\prod_{\begin{subarray}{c}q=j+3-m\\ t_{q}=\overline{t}_{p}\end{subarray}}^{j+1}\Delta\big{(}x^{2(q+m+t_{j+2-m}-j- N-3)}\big{)}\] \[=d_{\{t_{1},\ldots,t_{j-m}\}}(x,r)d_{\{t_{j-m+2},\ldots,t_{j+1}\}} (x,r)\prod_{p=1}^{j-m}\prod_{\begin{subarray}{c}q=j+2-m\\ t_{q}=\overline{t}_{p}\end{subarray}}^{j+1}\Delta\big{(}x^{2(q-p+t_{p}-N-2)} \big{)}\] \[\qquad\times\prod_{\begin{subarray}{c}p=1\\ t_{j+1-m}=\overline{t}_{p}\end{subarray}}^{j-m}\Delta\big{(}x^{2(j+t_{p}-m-p- N)}\big{)}\prod_{\begin{subarray}{c}q=j+2-m\\ t_{q}=\overline{t}_{j+1-m}\end{subarray}}^{j+1}\Delta\big{(}x^{2(q+m+t_{j+1-m}- j-N-3)}\big{)}\]
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\[\sum_{i=1}^{t}\left((1-p)^{\binom{s_{i}}{k-1}}\left(n-\sum_{j=1} ^{i}s_{j}\right)\right)\] \[\geq\sum_{i=1}^{t^{\prime}}\left((1-p)^{\binom{s_{i}}{k-1}} \left(n-\sum_{j=1}^{i}s_{j}\right)\right)\] \[\geq\frac{n}{\log^{2}d}\sum_{i=1}^{t^{\prime}}(1-p)^{\binom{s_{i} }{k-1}}\geq\frac{nt^{\prime}}{\log^{2}d}(1-p)^{\binom{s^{\prime}}{k-1}}\geq \frac{nt}{2\log^{2}d}(1-p)^{\binom{m_{0}\left(1+\frac{2}{\log d}\right)}{k-1}}\] \[\geq\frac{nt}{2\log^{2}d}\exp\left\{-(p+p^{2})\binom{\left(\frac{ \left(\log d-5(k-1)\log\log d\right)}{(k-1)d}\right)^{1/(k-1)}\left(1+\frac{2}{ \log d}\right)n}{k-1}\right\}\] \[\geq\frac{nt}{2\log^{2}d}\exp\left\{-\frac{\log d-5(k-1)\log\log d}{k-1} \cdot\left(1+\frac{3(k-1)}{\log d}\right)\right\}\] \[\geq\frac{nt\log^{2}d}{d^{1/(k-1)}}.\]
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\[\|Qx\|_{\mathbb{A}}^{2}=(Qx,Qx)_{\mathbb{A}}=(Q^{\#_{\mathbb{A}} }Qx,x)_{\mathbb{A}} =\begin{pmatrix}\begin{bmatrix}P_{\overline{\mathcal{R}(A)}}&O\\ O&P_{\overline{\mathcal{R}(A)}}\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix},\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix}\end{pmatrix}_{\mathbb{A}}\] \[=\begin{pmatrix}\begin{bmatrix}AP_{\overline{\mathcal{R}(A)}}&O \\ O&AP_{\overline{\mathcal{R}(A)}}\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix},\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix}\end{pmatrix}\] \[=\begin{pmatrix}\begin{bmatrix}AA^{\dagger}A&O\\ O&AA^{\dagger}A\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix},\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix}\end{pmatrix}\] \[=\begin{pmatrix}\begin{bmatrix}A&O\\ O&A\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix},\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix}\end{pmatrix}\] \[=\|x\|_{\mathbb{A}}^{2}.\]
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\[M_{2}(x_{k},x_{k+n}) = \max\left\{\begin{array}{c}d(x_{k},x_{k+n}),d(Tx_{k},x_{k}),d( Tx_{k+n},x_{k+n}),\\ d(Tx_{k},x_{k+n}),d(Tx_{k+n},x_{k})\end{array}\right\}\] \[= \max\left\{\begin{array}{c}d(x_{k},x_{k+n}),d(x_{k+1},x_{k}),d( x_{k+n+1},x_{k+n}),\\ d(x_{k+1},x_{k+n}),d(x_{k+n+1},x_{k})\end{array}\right\}\] \[\leq \max\left\{\begin{array}{c}d(x_{k},x_{k+n}),d(x_{k},x_{k+1}),d( x_{k+n},x_{k+n+1}),\\ d(x_{k},x_{k+1})+d(x_{k},x_{k+n}),d(x_{k+n},x_{k+n+1})+d(x_{k},x_{k+n})\end{array} \right\}.\]
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\[\sum_{j=1}^{n}\frac{1}{(aj+b)^{2k}}\sin\frac{2\pi(aj+b)}{m}=- \frac{1}{2b^{2k}}\left(\sin\frac{2\pi b}{m}-\sum_{j=0}^{k-1}\frac{(-1)^{j}}{(2 j+1)!}\left(\frac{2\pi b}{m}\right)^{2j+1}\right)\\ +\frac{1}{2(an+b)^{2k}}\left(\sin\frac{2\pi(an+b)}{m}-\sum_{j=0}^ {k-1}\frac{(-1)^{j}}{(2j+1)!}\left(\frac{2\pi(an+b)}{m}\right)^{2j+1}\right)- \sum_{j=0}^{k-1}\frac{(-1)^{k-j}}{(2k-1-2j)!}\left(\frac{2\pi}{m}\right)^{2k-1 -2j}HP_{2j+1}(n)\\ -\frac{(-1)^{k}}{2(2k-1)!}\left(\frac{2\pi}{m}\right)^{2k}\int_{ 0}^{1}(1-u)^{2k-1}\left(\cos\frac{2\pi(an+b)u}{m}-\cos\frac{2\pi bu}{m}\right) \cot\frac{\pi au}{m}\,du\]
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\[\varphi_{n}(x)=\sqrt{2}\cos\Big{(}\nu_{n}x+\frac{3-\alpha}{8}\pi- \arctan\Delta_{\alpha}\Big{)}\] \[\quad-\frac{\sqrt{5-\alpha}}{\pi}\int_{0}^{\infty}\rho_{0}(u) \left(Q_{0}(u)e^{-u\nu_{n}x}-(-1)^{n}Q_{1}(u)e^{-u\nu_{n}(1-x)}\right)du\] \[\quad+C_{0}e^{-c\nu_{n}x}\cos\Big{(}s\nu_{n}x+\varkappa_{0} \Big{)}+C_{1}e^{-c\nu_{n}(1-x)}\cos\Big{(}s\nu_{n}(1-x)+\varkappa_{1}\Big{)} +n^{-1}r_{n}(x)\] \[r_{n}(x)\] \[n\in\mathbb{N}\] \[x\in[0,1]\] \[\rho_{0}(u)\] \[Q_{0}(u)\] \[Q_{1}(u)\] \[c:=\cos\frac{\pi}{2}\frac{1-\alpha}{5-\alpha}\quad\text{and}\quad s:=\sin\frac{ \pi}{2}\frac{1-\alpha}{5-\alpha},\]
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\[\begin{array}{ll}\mathrm{s.t.}&\mathrm{model}\left(\ref{eq:model}\right) -\left(\ref{eq:model}\right)\\ &\delta x_{\mathrm{L}}[h]=\delta\hat{x}_{\mathrm{L}}[h|h]\\ \delta q_{\mathrm{f}}\in&[0-q_{\mathrm{r}}^{\mathrm{r}},q_{\mathrm{f}, \mathrm{max}}-q_{\mathrm{r}}^{\mathrm{s}}]\\ \delta q_{\mathrm{g}}^{\mathrm{B}}\in&[q_{\mathrm{min}}^{\mathrm{B}}-q_{ \mathrm{g}}^{\mathrm{B,as}},q_{\mathrm{max}}^{\mathrm{B}}-q_{\mathrm{g}}^{ \mathrm{B,as}}]\\ \delta T_{\mathrm{w}}\in&[T_{\mathrm{min}}-T_{\mathrm{w}}^{\mathrm{s}},T_{ \mathrm{max}}-T_{\mathrm{w}}^{\mathrm{s}}]\\ \delta l_{\mathrm{w}}\in&[l_{\mathrm{min}}-l_{\mathrm{w}}^{\mathrm{s}},l_{ \mathrm{max}}-l_{\mathrm{w}}^{\mathrm{s}}]\end{array}\]
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\[\left.\frac{dl}{d\epsilon}\right|_{\epsilon=0}=\int_{x_{1}}^{x_{ 2}}\frac{d}{d\epsilon}\left(\frac{\sqrt{1+(y^{\prime}+\epsilon h^{\prime})^{2 }}}{y+\epsilon h}\right)|_{\epsilon=0}dx=\] \[=\int_{x_{1}}^{x_{2}}\left(-\frac{h\sqrt{1+(y^{\prime}+\epsilon h ^{\prime})^{2}}}{(y+\epsilon h)^{2}}+\frac{h^{\prime}(y^{\prime}+\epsilon h ^{\prime})}{(y+\epsilon h)\sqrt{1+(y^{\prime}+\epsilon h^{\prime})^{2}}} \right)|_{\epsilon=0}dx=\] \[=\int_{x_{1}}^{x_{2}}\left(-\frac{h\sqrt{1+{y^{\prime}}^{2}}}{y^{2}}+\frac{h^{ \prime}y^{\prime}}{y\sqrt{1+{y^{\prime}}^{2}}}\right)dx=\int_{x_{1}}^{x_{2}}- \frac{h\sqrt{1+{y^{\prime}}^{2}}}{y^{2}}dx+\left[\frac{hy^{\prime}}{y\sqrt{1+{y ^{\prime}}^{2}}}\right]_{x_{1}}^{x_{2}}\] \[-\int_{x_{1}}^{x_{2}}h\frac{d}{dx}\frac{y^{\prime}}{y\sqrt{1+{y^{ \prime}}^{2}}}dx\] \[=\int_{x_{1}}^{x_{2}}\left(-\frac{\sqrt{1+{y^{\prime}}^{2}}}{y^{2}}-\frac{d }{dx}\frac{y^{\prime}}{y\sqrt{1+{y^{\prime}}^{2}}}\right)hdx\]
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\[w\in K_{g}(\bar{z},\bar{\lambda}) \iff\,(w,0)\in\mathrm{gph}\,N_{K_{g}(\bar{z},\bar{\lambda})}\] \[\iff\exists\,r^{\prime}>0:t(w,0)\in\big{(}\mathrm{gph}\,N_{K_{g}( \bar{z},\bar{\lambda})}\big{)}\cap\mathbb{B}_{r}(0,0)\quad\text{for all}\ \ t\in[0,r^{\prime})\] \[\iff\exists\,r^{\prime}>0:(\bar{z}+tw,\bar{\lambda})\in\big{(} \mathrm{gph}\,\partial g\big{)}\cap\mathbb{B}_{r}(\bar{z},\bar{\lambda})\quad \text{for all}\ \ t\in[0,r^{\prime})\] \[\iff\exists\,r^{\prime}>0:\bar{\lambda}\in\partial g(\bar{z}+tw) \quad\text{for all}\ \ t\in[0,r^{\prime})\] \[\iff\exists\,r^{\prime}>0:\bar{z}+tw\in\partial g^{*}(\bar{ \lambda})\quad\text{for all}\ \ t\in[0,r^{\prime})\] \[\iff w\in T_{\partial g^{*}(\bar{\lambda})}(\bar{z})=\big{(}N_{ \partial g^{*}(\bar{\lambda})}(\bar{z})\big{)}^{*}=K_{g^{*}}(\bar{\lambda}, \bar{z})^{*},\]
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\[\sum_{a}\,\hat{\cal A}_{a}^{(z_{1})}|0\rangle^{1}\otimes\hat{\cal A} _{(z_{2}/q)}^{a}|0\rangle^{2}=\] \[\qquad=\frac{\alpha^{\prime}g_{D}^{2}}{8\pi i}\int\frac{{\rm d}^{D} k}{(2\pi)^{D}}e^{ik\cdot(x_{0}^{(z_{1})}-x_{0}^{(z_{2})})}\bar{q}^{\frac{\alpha^{ \prime}}{4}k^{2}-1}q^{\frac{\alpha^{\prime}}{4}k^{2}-1/2}\] \[\qquad\times\exp\Big{[}\sum_{n=1}^{\infty}\bar{q}^{n}\Big{(}- \frac{1}{n}\tilde{\alpha}_{-n}^{(z_{1})}\cdot\tilde{\alpha}_{-n}^{(z_{2})}+ \tilde{c}_{-n}^{(z_{1})}\tilde{b}_{-n}^{(z_{2})}-\tilde{b}_{-n}^{(z_{1})} \tilde{c}_{-n}^{(z_{2})}\Big{)}\Big{]}\] \[\qquad\times\exp\Big{[}\sum_{n=1}^{\infty}i\eta\bar{q}^{\,n-1/2} \Big{(}\lambda_{-n+1/2}^{(z_{1})}\cdot\lambda_{-n+1/2}^{(z_{2})}\Big{)}\Big{]}\] \[\qquad\times\exp\Big{[}\sum_{n=1}^{\infty}q^{n}\Big{(}-\frac{1}{ n}\alpha_{-n}^{(z_{1})}\cdot\alpha_{-n}^{(z_{2})}+c_{-n}^{(z_{1})}b_{-n}^{(z_{2} )}-b_{-n}^{(z_{1})}c_{-n}^{(z_{2})}\Big{)}\Big{]}\] \[\qquad\times\exp\Big{[}\sum_{n=1}^{\infty}i\eta q^{n-1/2}\Big{(} \beta_{-n+1/2}^{(z_{1})}\gamma_{-n+1/2}^{(z_{2})}-\gamma_{-n+1/2}^{(z_{1})} \beta_{-n+1/2}^{(z_{2})}+\psi_{-n+1/2}^{(z_{1})}\cdot\psi_{-n+1/2}^{(z_{2})} \Big{)}\Big{]}\] \[\qquad\times\eta\big{[}1+(\tilde{c}_{0}^{(z_{1})}+\tilde{c}_{0}^ {(z_{2})})(c_{0}^{(z_{1})}+c_{0}^{(z_{2})})\big{]}\tilde{c}_{1}^{(z_{1})}c_{1 }^{(z_{1})}\tilde{c}_{1}^{(z_{2})}c_{1}^{(z_{2})}\delta(\gamma_{1/2}^{(z_{1}) })\delta(\gamma_{1/2}^{(z_{2})})|0\rangle^{1}\otimes|0\rangle^{2}\]
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\[II\leq\left\langle u_{(Q\cup R)^{c},t}\right|J_{(Q\cup R)^{c}}\left|u_{(Q\cup R )^{c},t}\right\rangle\sqrt{\overline{N}_{Q^{c}}}\sqrt{\overline{N}_{R^{c}}} \frac{\bm{d}_{\bm{N}}}{\overline{N}}\] \[\times\left\|\left(\mathcal{N}_{\mathrm{total}}+p\mathcal{I}\right)^{- 1/2}\mathcal{W}^{*}(\mathbf{N}^{\odot 1/2}\odot\mathbf{u})\frac{\left( \boldsymbol{a}^{*}(\mathbf{u})\right)^{\boldsymbol{N}}}{\sqrt{N!}}\omega\right\| _{\mathcal{F}^{\otimes p}}\] \[\times\left\|\left(\mathcal{N}_{\mathrm{total}}+p\mathcal{I} \right)^{1/2}\left(d\boldsymbol{\Gamma}^{(Q\cap R)}(J_{Q\cap R})\cdot\mathcal{ R}^{(Q\Delta R)}(J_{Q\Delta R}\odot u_{Q\Delta R,t})\right)\omega\right\|_{\mathcal{F}^{ \otimes p}}.\]
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\[M_{n}(\xi,u)\] \[= \frac{1}{n}\sum_{i=1}^{n}\max_{\Delta:X_{i}+n^{-1/2}\Delta\in \Omega}\left[n^{1/2}\xi^{\mathrm{ T}}\left\{h(X_{i}+n^{-1/2}\Delta,\beta_{*}+n^{-1/2}u)-h(X_{i},\beta_{*}) \right\}-\left\|\Delta\right\|_{q}^{2}\right]\] \[= \frac{1}{n}\sum_{i=1}^{n}\max_{\Delta:X_{i}+n^{-1/2}\Delta\in \Omega}\left\{\xi^{\mathrm{ T}}\int_{0}^{1}D_{x}h\left(X_{i}+n^{-1/2}t\Delta,\beta_{*}+n^{-1/2}tu \right)\Delta\mathrm{d}t\right.\] \[\left.+\xi^{\mathrm{ T}}\int_{0}^{1}D_{\beta}h\left(X_{i}+n^{-1/2}t \Delta,\beta_{*}+n^{-1/2}tu\right)u\mathrm{d}t-\left\|\Delta\right\|_{q}^{2} \right\},\]
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\[\frac{\mathrm{d}^{3}}{\mathrm{d}t^{3}}\mathbf{C}(t)= \frac{3!}{3!0!0!}f_{1}^{(3)}f_{2}^{(0)}f_{3}^{(0)}+\frac{3!}{1!1!1! 1!}f_{1}^{(1)}f_{2}^{(1)}f_{3}^{(1)}+\frac{3!}{2!1!0!}f_{1}^{(2)}f_{2}^{(1)}f_{3 }^{(0)}+\frac{3!}{2!0!1!1}f_{1}^{(2)}f_{2}^{(0)}f_{3}^{(1)}\] \[= f_{1}^{(3)}f_{2}^{(0)}f_{3}^{(0)}+6f_{1}^{(1)}f_{2}^{(1)}f_{3}^{ (1)}+3f_{1}^{(2)}f_{2}^{(1)}f_{3}^{(0)}+3f_{1}^{(2)}f_{2}^{(0)}f_{3}^{(1)}\] \[= \Bigg{(}-\mathrm{tr}(\partial\mathbf{V}\partial\mathbf{G}[t]^{-1 })\cdot\frac{1}{\det(\partial\mathbf{G}[t])}\left[\mathrm{tr}(\partial\mathbf{ V}\partial\mathbf{G}[t]^{-1})^{2}+\mathrm{tr}\left[(\partial\mathbf{V} \partial\mathbf{G}[t]^{-1})(\partial\mathbf{V}\partial\mathbf{G}[t]^{-1}) \right]\right]\] \[-2\cdot\frac{1}{\det(\partial\mathbf{G}[t])}\cdot\mathrm{tr}( \partial\mathbf{V}\partial\mathbf{G}[t]^{-1})\cdot\mathrm{tr}(\partial \mathbf{V}\partial\mathbf{G}[t]^{-1}\partial\mathbf{V}\partial\mathbf{G}[t]^{ -1})\] \[-\frac{1}{\det(\partial\mathbf{G}[t])}\cdot\mathrm{tr}\Big{[}( \partial\mathbf{V}\partial\mathbf{G}[t]^{-1}\partial\mathbf{V}\partial \mathbf{G}[t]^{-1})(\partial\mathbf{V}\partial\mathbf{G}[t]^{-1})+\partial \mathbf{V}\partial\mathbf{G}[t]^{-1}\partial\mathbf{V}\partial\mathbf{G}[t]^ {-1}\partial\mathbf{V}\partial\mathbf{G}[t]^{-1}\Big{]}\Bigg{)}\partial \mathbf{G}[t]^{\top}\partial\mathbf{G}[t]^{-1}\partial\mathbf{V}\] \[+3\cdot\frac{1}{\det(\partial\mathbf{G}[t])}\left[\mathrm{tr}( \partial\mathbf{V}\partial\mathbf{G}[t]^{-1})^{2}+\mathrm{tr}\left[(\partial \mathbf{V}\partial\mathbf{G}[t]^{-1})(\partial\mathbf{V}\partial\mathbf{G}[t ]^{-1})\right]\right]\partial\mathbf{V}^{\top}\partial\mathbf{G}[t]\] \[+3\cdot\frac{1}{\det(\partial\mathbf{G}[t])}\left[\mathrm{tr}( \partial\mathbf{V}\partial\mathbf{G}[t]^{-1})^{2}+\mathrm{tr}\left[(\partial \mathbf{V}\partial\mathbf{G}[t]^{-1})(\partial\mathbf{V}\partial\mathbf{G}[t ]^{-1})\right]\right]\partial\mathbf{G}[t]^{\top}\partial\mathbf{V}.\]
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\[f(v_{b}^{\prime})= f(v_{b})-f(v_{b+e_{0}-e_{1}}^{\prime})-\cdots-f(v_{b+e_{0}-e_{d}} ^{\prime})\] \[= w_{b}+w_{b+e_{0}-e_{1}}+w_{b+e_{0}-e_{2}}+\cdots+w_{b+e_{0}-e_{d}}\] \[-w_{b+e_{0}-e_{1}}+(-1)^{n-k-b_{0}-1}\sum_{b^{\prime}\in D}{n-k-b_ {0}-1\choose b_{1}-1-b_{1}^{\prime},b_{2}-b_{2}^{\prime},\cdots,b_{d}-b_{d}^{ \prime}}u_{b^{\prime}}\] \[-w_{b+e_{0}-e_{2}}+(-1)^{n-k-b_{0}-1}\sum_{b^{\prime}\in D}{n-k-b _{0}-1\choose b_{1}-b_{1}^{\prime},b_{2}-1-b_{2}^{\prime},\cdots,b_{d}-b_{d}^ {\prime}}u_{b^{\prime}}\] \[-\cdots\] \[-w_{b+e_{0}-e_{d}}+(-1)^{n-k-b_{0}-1}\sum_{b^{\prime}\in D}{n-k-b _{0}-1\choose b_{1}-b_{1}^{\prime},b_{2}-b_{2}^{\prime},\cdots,b_{d}-1-b_{d}^ {\prime}}u_{b^{\prime}}\] \[= w_{b}-(-1)^{n-k-b_{0}}\sum_{b^{\prime}\in D}{n-k-b_{0}-1\choose b _{1}-b_{1}^{\prime},b_{2}-b_{2}^{\prime},\cdots,b_{d}-1-b_{d}^{\prime}}u_{b^{ \prime}}\]
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\[|S_{\leq Y}^{\prime}|\lesssim\sum_{r_{k}|k^{\infty}}r_{k}^{- \varepsilon}\max_{\mathrm{Re}(s)=\varepsilon}\sum_{\mathbf{g}}\int_{-2T}^{2T} \sum_{\theta\,(\mathrm{mod}\ r_{k}k)}\sum_{\begin{subarray}{c}(q^{\prime},kg_{1 }g_{2})=1\\ q^{\prime}\leq Y/r_{k}\end{subarray}}\sum_{\chi^{\prime}\,(\mathrm{mod}\ q^{ \prime})}^{*}\\ \Big{|}\sum_{\begin{subarray}{c}(a_{1}b_{1},a_{2}b_{2})=1\\ (\mathbf{a},\mathbf{g})=1\end{subarray}}\delta_{a_{1}b_{1}}(s)\alpha_{a_{1},b_{1 }}^{(1,\mathbf{g})}\delta_{a_{2}b_{2}}(s)\overline{\alpha}_{a_{2},b_{2}}^{(2, \mathbf{g})}\chi^{\prime}\theta\Big{(}\frac{\gamma_{1}a_{1}b_{2}}{\gamma_{2}b_{ 1}a_{2}}\Big{)}\Big{(}\frac{\gamma_{1}a_{1}b_{2}}{\gamma_{2}b_{1}a_{2}}\Big{)} ^{it}\Big{|}dt.\]
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\[\begin{split}\mu(x^{\lambda}y^{1-\lambda},\beta,\alpha)& =\int_{0}^{\infty}\frac{x^{\lambda(t+\alpha)}y^{(1-\lambda)(t+ \alpha)}t^{\beta}}{\Gamma(t+\alpha+1)\Gamma(\beta+1)}dt\\ &=\int_{0}^{\infty}\left[\frac{x^{t+\alpha}t^{\beta}}{\Gamma(t+ \alpha+1)\Gamma(\beta+1)}\right]^{\lambda}\left[\frac{y^{t+\alpha}t^{\beta}}{ \Gamma(t+\alpha+1)\Gamma(\beta+1)}\right]^{1-\lambda}dt\\ &\leq\left[\int_{0}^{\infty}\frac{x^{t+\alpha}t^{\beta}}{\Gamma(t+ \alpha+1)\Gamma(\beta+1)}dt\right]^{\lambda}\left[\int_{0}^{\infty}\frac{y^{t +\alpha}t^{\beta}}{\Gamma(t+\alpha+1)\Gamma(\beta+1)}dt\right]^{1-\lambda}\\ &=\left[\mu(x,\beta,\alpha)\right]^{\lambda}\left[\mu(y,\beta, \alpha)\right]^{1-\lambda}.\end{split}\]
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\[\operatorname{\mathbb{E}}\bigl{[}q_{0}(\theta)-q_{0}(\theta^{*}) \ |\ \mathcal{F}_{0}\bigr{]} = \operatorname{\mathbb{E}}\Bigl{[}\frac{(X_{0}-f_{\theta}^{0})^{2 }}{H_{\theta}^{0}}+\log(H_{\theta}^{0})-\frac{(X_{0}-f_{\theta^{*}}^{0})^{2}}{ H_{\theta^{*}}^{0}}-\log(H_{\theta^{*}}^{0})\ |\ \mathcal{F}_{0}\Bigr{]}\] \[= \log\Big{(}\frac{H_{\theta}^{0}}{H_{\theta^{*}}^{0}}\Big{)}+ \frac{\operatorname{\mathbb{E}}\bigl{[}(X_{0}-f_{\theta^{*}}^{0})^{2}\ |\ \mathcal{F}_{0}\bigr{]}}{H_{\theta}^{0}}-\frac{ \operatorname{\mathbb{E}}\bigl{[}(X_{0}-f_{\theta^{*}}^{0})^{2}\ |\ \mathcal{F}_{0}\bigr{]}}{H_{\theta}^{0}}\] \[= \log\Big{(}\frac{H_{\theta}^{0}}{H_{\theta^{*}}^{0}}\Big{)}-1+ \frac{\operatorname{\mathbb{E}}\bigl{[}(X_{0}-f_{\theta^{*}}^{0}+f_{\theta^{*} }^{0}-f_{\theta}^{0})^{2}\ |\ \mathcal{F}_{0}\bigr{]}}{H_{\theta}^{0}}\] \[= \frac{H_{\theta^{*}}^{0}}{H_{\theta}^{0}}-\log\Big{(}\frac{H_{ \theta^{*}}^{0}}{H_{\theta}^{0}}\Big{)}-1+\frac{(f_{\theta^{*}}^{0}-f_{\theta }^{0})^{2}}{H_{\theta}^{0}}\]
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\[[W^{(2)}_{p,p}t,W^{(2)}_{i,i}t]\] \[=(W^{(2)}_{p,i})_{(-1)}W^{(1)}_{i,p}t^{2}-(W^{(1)}_{p,i})_{(-1)}W^ {(2)}_{i,p}t^{2}\] \[\quad+\alpha_{1}(\partial W^{(1)}_{p,i})_{(-1)}W^{(1)}_{i,p}t^{2} +(\partial W^{(1)}_{p,p})_{(-1)}W^{(1)}_{i,i}t^{2}\] \[\quad-\delta_{p,i}\sum_{w\leq m-n}(W^{(1)}_{w,i})_{(-1)}W^{(2)}_{p,w}t^{2}-\delta_{p,i}\sum_{x,w\leq m-n}(\partial W^{(1)}_{w,x})_{(-1)}W^{(1)}_{ x,w}t^{2}-\delta_{p,i}\alpha_{2}\partial W^{(2)}_{p,i}t^{2}\] \[\quad+\delta_{p,i}\sum_{w\leq m-n}(\partial W^{(1)}_{p,i})_{(-1)} W^{(1)}_{w,w}t^{2}-\frac{1}{2}\delta_{p,i}(\alpha_{1}+2\alpha_{2})\sum_{x\leq m-n} \partial^{2}W^{(1)}_{x,x}t^{2}\] \[\quad-\delta_{i,p}\frac{\alpha_{1}(\alpha_{1}+\alpha_{2})+1}{2} \partial^{2}W^{(1)}_{p,i}t^{2}\] \[\quad+\delta_{p,i}\sum_{x\leq m-n}(W^{(1)}_{x,p})_{(-1)}W^{(2)}_{ i,x}t^{2}+\delta_{p,i}\sum_{x\leq m-n}(\partial W^{(1)}_{x,x})_{(-1)}W^{(1)}_{i,p}t^{2}\] \[\quad-\partial W^{(2)}_{p,p}t^{2}-\frac{1}{2}(\alpha_{1}+2\alpha_ {2})\partial^{2}W^{(1)}_{p,p}t^{2}-\frac{1}{2}\partial^{2}\sum_{w\leq m-n}W^ {(1)}_{w,w}t^{2}\] \[\quad+\alpha_{1}(W^{(1)}_{p,i})_{(-1)}W^{(1)}_{i,p}t+(W^{(1)}_{p, p})_{(-1)}W^{(1)}_{i,i}t\] \[\quad-\delta_{p,i}\alpha_{2}W^{(2)}_{p,i}t+\delta_{p,i}\sum_{w\leq m -n}(W^{(1)}_{p,i})_{(-1)}W^{(1)}_{w,w}t-\delta_{p,i}\alpha_{1}(\alpha_{1}+ \alpha_{2})\partial W^{(1)}_{p,i}t\] \[\quad-\delta_{p,i}\alpha_{2}W^{(2)}_{i,p}t+\delta_{p,i}\sum_{x \leq m-n}(W^{(1)}_{x,x})_{(-1)}W^{(1)}_{i,p}t\] \[\quad-W^{(2)}_{p,p}t-2\alpha_{2}\partial W^{(1)}_{p,p}t-W^{(2)}_ {i,i}t\] \[\quad-\delta_{p,i}\sum_{x,w\leq m-n}(W^{(1)}_{w,x})_{(-1)}W^{(1)}_ {x,w}t-\delta_{p,i}(\alpha_{1}+2\alpha_{2})\sum_{x\leq m-n}\partial W^{(1)}_{ x,x}t.\]
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\[J^{-}(r) :=\int_{-R}^{r}e^{z(x-r)}\sin(b_{k}x)\mathrm{d}x\] \[=\frac{1}{b_{k}}\left[-\cos(b_{k}x)e^{z(x-r)}\right]_{-R}^{r}+ \frac{z}{b_{k}}\int_{-R}^{r}e^{z(x-r)}\cos(b_{k}x)\mathrm{d}x\] \[=-\frac{\cos(b_{k}r)}{b_{k}}+\frac{\cos(b_{k}R)}{b_{k}}e^{-z(R+r) }+\frac{z}{b_{k}^{2}}\left[\sin(b_{k}x)e^{z(x-r)}\right]_{-R}^{r}-\frac{z^{2}} {b_{k}^{2}}J^{-}(r)\] \[=-\frac{\cos(b_{k}r)}{b_{k}}+\frac{z}{b_{k}^{2}}\sin(b_{k}r)+ \frac{b_{k}\cos(b_{k}R)+z\sin(b_{k}R)}{b_{k}^{2}}e^{-z(R+r)}-\frac{z^{2}}{b_{k }^{2}}J^{-}(r)\] \[=\frac{-\cos(b_{k}r)}{b_{k}}+\frac{z}{b_{k}^{2}}\sin(b_{k}r)- \frac{z^{2}}{b_{k}^{2}}J^{-}(r),\]
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\[\begin{split}& u_{1}^{2}+u_{2}^{2}\\ &=(t+C_{2})^{2}+\frac{C_{1}^{2}}{(t+C_{2})^{2}+C_{1}^{2}}\left[C_{ 1}\left[\ln\left(\frac{t+C_{2}+\sqrt{(t+C_{2})^{2}+C_{1}^{2}}}{C_{1}}\right) \right]+C_{3}\right]^{2}\\ &+\frac{(t+C_{2})^{2}}{(t+C_{2})^{2}+C_{1}^{2}}\left[C_{1}\left[ \ln\left(\frac{t+C_{2}+\sqrt{(t+C_{2})^{2}+C_{1}^{2}}}{C_{1}}\right)\right]+C_ {3}\right]^{2}+C_{1}^{2}\\ &=(t+C_{2})^{2}+C_{1}^{2}+\left[C_{1}\left[\ln\left(\frac{t+C_{2 }+\sqrt{(t+C_{2})^{2}+C_{1}^{2}}}{C_{1}}\right)\right]+C_{3}\right]^{2}\\ &=q_{1}^{2}+q_{2}^{2}\\ &=1\;\;\text{[by equation \ref{eq:C1}]}\end{split}\]
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\[B(t)=\sin(2\mathcal{E}t)\int\limits_{-1}^{1}\frac{dE}{2\pi}\rho T (1-\rho T)+\frac{\left(\varepsilon^{2}-1\right)^{2}}{2\left(\varepsilon^{2}+1 \right)}\sin(4\mathcal{E}t)\left(\int\limits_{-1}^{1}\frac{dE}{2\pi}\rho T \right)^{2}\] \[+2\left(\int\limits_{-1}^{1}\frac{dE}{2\pi}\rho T\right)\left(2 \mathcal{E}\cos(2\mathcal{E}t)\int\limits_{-1}^{1}\frac{dE}{2\pi}\rho\frac{ \rho T^{2}}{v^{2}}-\frac{1}{2}\frac{\varepsilon^{2}-1}{\varepsilon^{2}+1}\sin (2\mathcal{E}t)\int\limits_{-1}^{1}\frac{dE}{2\pi}\rho v\frac{E^{2}( \mathcal{E}^{2}-E^{2})+2\mathcal{E}^{2}}{(\mathcal{E}^{2}-E^{2})^{2}}\right).\]
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\[\operatorname{Res}_{\boldsymbol{\zeta}_{0}}\boldsymbol{\zeta}_{0}^{m} (\boldsymbol{\zeta}_{0}+\boldsymbol{x})^{2+i}L(\boldsymbol{\zeta}_{0}+ \boldsymbol{x})\left(\mathcal{Y}_{t}(w_{1},\boldsymbol{x})\otimes\bar{w}_{c,h_{ 0}}\right)\] \[=\operatorname{Res}_{\boldsymbol{\zeta}_{1}}\operatorname{Res}_{ \boldsymbol{\zeta}_{0}}(\boldsymbol{\zeta}_{1}-\boldsymbol{x})^{m}\boldsymbol {\zeta}_{1}^{2+i}\boldsymbol{\zeta}_{1}^{-1}\delta\left(\frac{\boldsymbol{x}+ \boldsymbol{\zeta}_{0}}{\boldsymbol{\zeta}_{1}}\right)\mathcal{Y}_{t}\left(L( \boldsymbol{\zeta}_{0})w_{1},\boldsymbol{x}\right)\otimes\bar{w}_{c,h_{0}}\] \[=\operatorname{Res}_{\boldsymbol{\zeta}_{0}}\boldsymbol{\zeta}_{0 }^{m}(\boldsymbol{\zeta}_{0}+\boldsymbol{x})^{2+i}\mathcal{Y}_{t}\left(L( \boldsymbol{\zeta}_{0})w_{1},\boldsymbol{x}\right)\otimes\bar{w}_{c,h_{0}}.\]
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\[\int_{\mathbb{R}^{2}}\int_{\mathbb{T}^{2}}(F_{t}^{N,R})^{3} \mathrm{div}_{v}(\theta^{\varepsilon_{N}}v*S_{t}^{N,R})\,dxdv=2\int_{\mathbb{ R}^{2}}\int_{\mathbb{T}^{2}}(F_{t}^{N,R})^{4}\,dxdv\\ \int_{\mathbb{R}^{2}}\int_{\mathbb{T}^{2}}(F^{N,R})^{3}\int_{ \mathbb{R}^{2}}\int_{\mathbb{T}^{2}}\theta^{0,\varepsilon_{N}}(x-x^{\prime}) \nabla_{v}\theta^{1,\varepsilon_{N}}(v-v^{\prime})\cdot(v-v^{\prime})\,S_{t}^ {N,R}(dx^{\prime},dv^{\prime})dxdvdt\\ \leq 2\left|\left|F_{t}^{N,R}\right|\right|_{L^{4}(\mathbb{T}^{2} \times\mathbb{R}^{2})}^{4}+\\ \int_{\mathbb{R}^{2}}\!\!\int_{\mathbb{T}^{2}}\!\!\!\left|\nabla( F^{N,R})^{3}\right|\int_{\mathbb{R}^{2}}\!\!\int_{\mathbb{T}^{2}}\!\!\!\theta^{0, \varepsilon_{N}}(x-x^{\prime})\theta^{1,\varepsilon_{N}}(v-v^{\prime})\,|v-v^ {\prime}|S_{t}^{N,R}(dx^{\prime},dv^{\prime})dxdvdt.\]
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\[K=(F\cdot G)\cdot H\] \[E_{K}=E_{F}+E_{G}+E_{H}\qquad\Lambda_{K}=\Lambda_{F}+\Lambda_{G} +\Lambda_{H}\] \[S_{K}=S_{F}-S_{F}[\bullet]+S_{G}-S_{G}[\bullet]+S_{H}\qquad T_{ K}=T_{F}+T_{G}-T_{G}[\bullet]+T_{H}-T_{H}[\bullet]\] \[\kappa_{K}(v)=\begin{cases}\kappa_{H}(\pi_{j}(S_{H}[\bullet])& \text{if }\kappa_{F}(v)=\pi_{i}(S_{F}[\bullet])\wedge\kappa_{G}(\pi_{i}(T_{G}[ \bullet]))=\pi_{j}(S_{G}[\bullet])\\ \kappa_{H}(\pi_{i}(S_{H}[\bullet]))&\text{if }\kappa_{G}(v)=\pi_{i}(S_{G}[ \bullet])\\ \kappa_{G}(\pi_{i}(S_{G}[\bullet]))&\text{if }\kappa_{F}(v)=\pi_{i}(S_{F}[ \bullet])\\ \kappa_{F}+\kappa_{G}+\kappa_{H}&\text{otherwise}\end{cases}\]
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\[T_{11} \geq\frac{1}{n^{2}}\sum_{i=2}^{N/3}\min_{2\frac{k-1}{n}\leq y_{1} \leq y_{2}\leq\ldots\leq y_{N}\leq 2\frac{k}{n}}\sum_{\ell=1}^{i}\sum_{m=1}^{ \lfloor\frac{N-\ell}{i}\rfloor}(V-V_{n})(y_{mi+\ell}-y_{(m-1)i+\ell})\] \[\geq\frac{1}{n^{2}}\sum_{i=2}^{N/3}\sum_{\ell=1}^{i}\min_{2\frac{ k-1}{n}\leq y_{1}\leq y_{2}\leq\ldots\leq y_{N}\leq 2\frac{k}{n}}\sum_{m=1}^{ \lfloor\frac{N-\ell}{i}\rfloor}(V-V_{n})(y_{mi+\ell}-y_{(m-1)i+\ell})\] \[=\frac{1}{n^{2}}\sum_{i=2}^{N/3}\sum_{\ell=1}^{i}\min_{2\frac{k-1 }{n}\leq y_{\ell}\leq y_{i+\ell}\leq y_{2i+\ell}\leq\ldots\leq y_{\lfloor\frac {N-\ell}{i}\rfloor+\ell}\leq 2\frac{k}{n}}\sum_{m=1}^{\lfloor\frac{N-\ell}{i} \rfloor}(V-V_{n})(y_{mi+\ell}-y_{(m-1)i+\ell})\] \[=\frac{1}{n^{2}}\sum_{i=2}^{N/3}\sum_{\ell=1}^{i}\min_{2\frac{k-1 }{n}\leq z_{0}\leq z_{1}\leq\ldots\leq z_{\lfloor\frac{N-\ell}{i}\rfloor}\leq 2 \frac{k}{n}}\sum_{m=1}^{\lfloor\frac{N-\ell}{i}\rfloor}(V-V_{n})(z_{m}-z_{m-1}),\]
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\[\begin{split}\mathbb{E}\mathfrak{B}_{n,N}&=\frac{1 }{1+\frac{n}{\theta}}+\sum_{j=n}^{N-1}\frac{1}{1+\frac{1}{\theta}(j+1)}\left( \frac{1}{1+\frac{j-n+1}{\theta}}+\sum_{k=j-n+1}^{j-1}\frac{\mathbb{E} \mathfrak{B}_{k-(j-n+1)+1,k}}{1+\frac{1}{\theta}(k+1)}\right)\\ &=\frac{1}{1+\frac{n}{\theta}}+\sum_{j=n}^{N-1}\frac{1}{(1+\frac{ 1}{\theta}(j+1))(1+\frac{j-n+1}{\theta})}\\ &+\sum_{j=n}^{N-1}\sum_{k=1}^{n-1}\frac{1}{1+\frac{1}{\theta}(j+ 1)}\frac{\mathbb{E}\mathfrak{B}_{k,k+j-n}}{1+\frac{1}{\theta}(k+j-n+1)}.\end{split}\]
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