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\begin{table}
\begin{tabular}{l c c c c c} \hline \hline & & \multicolumn{2}{c}{\(b^{max}=6\)} & \multicolumn{2}{c}{\(b^{max}=10\)} \\ \cline{3-6} Variant & A & B & C & B & C \\ \cline{2-6} Total \# walks & 14843 & NF & 27920 & 90 & 68674 \\ Wall clock time & TimeLim & – & TimeLim & 584.771 & TimeLim \\ Mean time/DNL & 85.074 & – & 257.964 & 0.339 & 621.421 \\ Mean time/FP-Update & 8.695 & – & 20.764 & 0.034 & 58.764 \\ \# iterations & 71 & – & 25 & 1457 & 10 \\ \# walks with \(h>0\) & 10 & – & 12 & 11 & 16 \\ \(\Delta h\) & 10.745 & – & 27.236 & 0.100 & 258.073 \\ \(\Delta h\) (relative) & 0.002 & – & 0.005 & 0.000 & 0.045 \\ QoPI (absolute) & 355.168 & – & 364.975 & 40.345 & 1579.097 \\ QoPI & 0.007 & – & 0.003 & 0.001 & 0.027 \\ \hline \hline \end{tabular}
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\[\dot{L}\leq -k_{\mathrm{a}}\|\tilde{\bm{\eta}}\|^{2}-k_{\mathrm{b}}\|\tilde{ \bm{\theta}}^{\dagger}\|^{2}-\left(\frac{\mu}{\tau_{\min}}-\frac{1}{2\tau_{ \min}k_{5}}\right)\|\tilde{\bm{u}}\|^{2}\] \[-\left(\beta-\frac{L\beta\|\bm{B}\|}{\tau_{\min}}-\frac{\beta\| \bm{B}\|}{2\tau_{\min}k_{3}}-\frac{\beta\|\bm{B}\|}{2k_{4}}\right)\|\bm{x}- \pi(\hat{\bm{u}})\|^{2}\] \[-\left(k_{\mathrm{b}}-\frac{k_{5}}{2\tau_{\min}}-\frac{\beta\| \bm{B}\|k_{3}}{2\tau_{\min}}\right)\|\tilde{\bm{\theta}}^{\dagger}\|^{2}+k_{ \mathrm{c}}\|\hat{\bm{\theta}}\|^{2}+\frac{\sigma}{2}\|\bm{\theta}\|^{2}\] \[+\frac{L_{F}\beta\|\bm{B}\|+L}{\tau_{\min}}\|\tilde{\bm{u}}\|\| \bm{x}-\pi(\hat{\bm{u}})\|+\frac{\beta\|\bm{B}\|k_{4}}{2}\|\bm{d}(t)\|^{2}.\]
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\[\begin{split}&\Big{\|}e^{itL}\int_{0}^{t}\widetilde{\mathcal{F}}_{ \xi\to x}^{-1}\big{(}\varphi_{\leqslant\delta_{N}m+5}(\xi)e^{-is\langle\xi \rangle}I(s,\xi)\big{)}\,\tau_{m}(s)ds\Big{\|}_{L^{\infty}_{x}}\\ &\lesssim\langle t\rangle^{3\delta_{N}/8}\Big{\|}\int_{0}^{t}e^{ i(t-s)L}P_{\leqslant\delta_{N}m+5}\widetilde{\mathcal{F}}_{\xi\to x}^{-1} \big{(}I(\xi,s)\big{)}\,\tau_{m}(s)ds\Big{\|}_{L^{8}_{x}}\\ &\lesssim\langle t\rangle^{2\delta_{N}}\int_{0}^{t}\frac{1}{ \langle t-s\rangle^{9/8}}\langle s\rangle^{2\delta_{N}}\big{\|}\widetilde{ \mathcal{F}}_{\xi\to x}^{-1}I(\xi,s)\big{\|}_{L^{8/7}_{x}}\,\tau_{m}(s)ds. \end{split}\]
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\[\sum_{i=1}^{N}\bm{a}(\bm{x_{i}},\bm{w})^{T}\left(\bar{\bm{a}}(\bm {x_{i}})\bar{\bm{a}}(\bm{x_{i}})^{T}\right)\bm{a}(\bm{x_{i}},\bm{w})\] \[\quad=\sum_{i=1}^{N}\bm{a}(\bm{x_{i}},\bm{w})^{T}\left(\bm{a}(\bm {x_{i}},\bm{w})\bm{a}(\bm{x_{i}},\bm{w})^{T}\right)\bm{a}(\bm{x_{i}},\bm{w})\] \[\geq\sum_{i=1}^{N}\hat{\bm{g}}^{T}\left(\bm{a}(\bm{x_{i}},\bm{w}) \bm{a}(\bm{x_{i}},\bm{w})^{T}\right)\hat{\bm{g}}\] \[=\hat{\bm{g}}^{T}\left(\sum_{i=1}^{N}\bm{a}(\bm{x_{i}},\bm{w})\bm{ a}(\bm{x_{i}},\bm{w})^{T}\right)\hat{\bm{g}}\] \[=\lambda_{max}\left(\sum_{i=1}^{N}\bm{a}(\bm{x_{i}},\bm{w})\bm{a}( \bm{x_{i}},\bm{w})^{T}\right).\]
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\[\sum_{n\geq 0}(-1)^{n}\big{[}Q_{L}^{n}\xrightarrow{\mathbf{h} \mathbf{c}_{\mathrm{A}^{3}}}\mathrm{Sym}^{n}(\mathrm{A}^{3})\big{]}_{\mathrm{vir}}\\ ={}^{\circ}\!\mathrm{Exp}_{\cup}\!\left(\sum_{n\geq 1}\!\left( \mathrm{Sym}^{n}(\mathrm{A}^{1})\times\mathrm{G}_{m}\times\mathrm{A}^{1} \hookrightarrow\mathrm{Sym}^{n}(\mathrm{A}^{3})\right)\!\Psi_{n}^{\mathrm{uni}} \mathfrak{B}\big{[}\mathrm{G}_{m}\times\mathrm{A}^{1}\xrightarrow{\mathrm{id} }\mathrm{G}_{m}\times\mathrm{A}^{1}\big{]}\right.\\ +\left.\left(\mathrm{Sym}^{n}(\mathrm{A}^{1})\times\{0\} \times\mathrm{A}^{1}\hookrightarrow\mathrm{Sym}^{n}(\mathrm{A}^{3})\right)\!, \Psi_{n}^{\mathrm{nilp}}\mathfrak{B}\big{[}\{0\}\times\mathrm{A}^{1} \xrightarrow{\mathrm{id}}\{0\}\times\mathrm{A}^{1}\big{]}\right).\]
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\[\|\kappa_{r}^{\geqslant j}[f] -\mathbf{E}^{j}\,\kappa_{r}^{\geqslant j+1}[f]\|_{\infty}\] \[\leqslant\sum_{p=2}^{r}\sum_{\{B_{1},\ldots,B_{p}\}\in P_{r}} \big{(}\tfrac{3}{2}\big{)}^{p}(p-1)!\prod_{t=1}^{p}\|\kappa_{|B_{t}|}^{ \geqslant j+1}[f]-\mathbf{E}^{j}\,\kappa_{|B_{t}|}^{\geqslant j+1}[f]\|_{\infty}\] \[\leqslant\sum_{p=2}^{r}\sum_{\begin{subarray}{c}b_{1},\ldots,b_ {p}\geqslant 1\\ b_{1}+\cdots+b_{p}=r\end{subarray}}\binom{r}{b_{1},\ldots,b_{p}}\Big{(} \tfrac{3}{2}\Big{)}^{p}\frac{1}{p}\prod_{t=1}^{p}80^{b_{t}-1}\tfrac{(b_{t}-1)! }{b_{t}}\alpha^{b_{t}}\] \[=100^{r-1}\frac{(r-1)!}{r}\alpha^{r}\sum_{p=2}^{r}r^{2}\big{(} \tfrac{3}{2}\big{)}^{p}\,80^{1-p}\frac{1}{p}\sum_{\begin{subarray}{c}b_{1}, \ldots,b_{p}\geqslant 1\\ b_{1}+\cdots+b_{p}=r\end{subarray}}\prod_{t=1}^{p}\tfrac{1}{b_{t}^{2}}\] \[\leqslant 100^{r-1}\frac{(r-1)!}{r}\alpha^{r}\sum_{p=2}^{r}\big{(} \tfrac{3}{2}\big{)}^{p}\,80^{1-p}\tfrac{1}{2}5.3^{p-1}\leqslant 0.1\cdot 80^{r-1} \frac{(r-1)!}{r}\alpha^{r}.\]
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\[\langle Tx,x\rangle = \langle\mbox{Re}(T)x,x\rangle+i\langle\mbox{Im}(T)x,x\rangle\] \[\Rightarrow\langle Tx,x\rangle^{2} = \langle\mbox{Re}(T)x,x\rangle^{2}-\langle\mbox{Im}(T)x,x\rangle^ {2}+2i\langle\mbox{Re}(T)x,x\rangle\langle\mbox{Im}(T)x,x\rangle\] \[\Rightarrow\left|\langle Tx,x\rangle^{2}\right|^{2} = \left|\langle\mbox{Re}(T)x,x\rangle^{2}-\langle\mbox{Im}(T)x,x \rangle^{2}\right|^{2}+4\langle\mbox{Re}(T)x,x\rangle^{2}\langle\mbox{Im}(T)x,x\rangle^{2}\] \[\Rightarrow\left|\langle Tx,x\rangle\right|^{4} \leq \max\left\{\left|\|\mbox{Re}(T)\|^{2}-m^{2}(\mbox{Im}(T))\right| ^{2},\left|\|\mbox{Im}(T)\|^{2}-m^{2}(\mbox{Re}(T))\right|^{2}\right\}\] \[+4\|\mbox{Re}(T)\|^{2}\|\mbox{Im}(T)\|^{2}.\]
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\[11\cdot 43=473 = 10\cdot 11+11\cdot 33=10\cdot 22+11\cdot 23\] \[= 10\cdot 33+11\cdot 13=10\cdot 44+11\cdot 3,\] \[11\cdot 54=594 = 10\cdot 11+11\cdot 44=10\cdot 22+11\cdot 34\] \[= 10\cdot 33+11\cdot 24=10\cdot 44+11\cdot 14=10\cdot 55+11\cdot 4,\] \[11\cdot 76=836 = 10\cdot 11+11\cdot 66=10\cdot 22+11\cdot 56=10\cdot 33+11\cdot 46\] \[= 10\cdot 44+11\cdot 36=10\cdot 55+11\cdot 26=10\cdot 66+11\cdot 16\] \[= 10\cdot 77+11\cdot 6,\] \[11\cdot 120=1320 = 10\cdot 11+11\cdot 110=10\cdot 22+11\cdot 100=10\cdot 33+11 \cdot 90\] \[= 10\cdot 44+11\cdot 80=10\cdot 55+11\cdot 70=10\cdot 66+11\cdot 60\] \[= 10\cdot 77+11\cdot 50=10\cdot 88+11\cdot 40=10\cdot 99+11\cdot 30\] \[= 10\cdot 110+11\cdot 20=10\cdot 121+11\cdot 10.\]
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\[\big{(}\alpha_{1},\!\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},\alpha_{6}, \alpha_{7}\big{)}=\big{(}n-\phi(n)-1,\phi(n)+1+\phi(pq)+\phi(pr),\phi(n)+1+ \phi(pq)+\phi(qr)\big{)},\] \[\phi(n)+1+\phi(pr)+\phi(qr),\phi(n)+1+\phi(p)+\phi(q),\phi(n)+1+ \phi(p)+\phi(r),\phi(n)+1+\phi(q)+\phi(r)\big{)}\] \[\text{and}\quad\big{(}\alpha_{1}+r_{1},\alpha_{2}+r_{2},\alpha_{3}+ r_{3},\alpha_{4}+r_{4},\alpha_{5}+r_{5},\alpha_{6}+r_{6},\alpha_{7}+r_{7}\big{)}= \big{(}n-1,\phi(n)+\phi(p)+\phi(pq)\] \[+\phi(pr),\phi(n)+\phi(q)+\phi(pq)+\phi(qr)),\phi(n)+\phi(r)+\phi( pr)+\phi(qr),\phi(n)+\phi(pq)+\phi(p)+\phi(q),\] \[\phi(n)+\phi(pr)+\phi(p)+\phi(r),\phi(n)+\phi(qr)+\phi(q)+\phi(r )\big{)}.\]
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\[3S(\Delta_{Ex,Ey}Ez,w)\] \[= S(\left[\!\left[Ex,Ey,w\right]\!\right],Ez)+S(\left[\!\left[Ex,Ey,Ez\right]\!\right],w)+S(\left[\!\left[w,Ez,Ex\right]\!\right],Ey)+2S(\left[\! \left[w,Ez,Ey\right]\!\right],Ex)\] \[= S(E\left[\!\left[x,y,Ew\right]\!\right],Ez)+S(E\left[\!\left[x,y, z\right]\!\right],Ew)+S(E\left[\!\left[Ew,z,x\right]\!\right],Ey)+2S(E\left[\! \left[Ew,z,y\right]\!\right],Ex)\] \[= -\Big{(}S(\left[\!\left[x,y,Ew\right]\!\right],z)+S(\left[\! \left[x,y,z\right]\!\right],Ew)+S(\left[\!\left[Ew,z,x\right]\!\right],y)+2S( \left[\!\left[Ew,z,y\right]\!\right],x)\Big{)}\] \[= -3S(\Delta_{x,y}z,Ew)\] \[= 3S(E\Delta_{x,y}z,w),\]
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\[\left[\mathcal{K},Q_{2}^{k}(B_{k})\right] =2\operatorname{Re}\left(\sum_{p\in L_{k}}\left(b_{k}(B_{k}e_{p}) \left[\mathcal{K},b_{-k}\left(e_{-p}\right)\right]+\left[\mathcal{K},b_{k}(B_{ k}e_{p})\right]b_{-k}\left(e_{-p}\right)\right)\right)\] \[=2\operatorname{Re}\left(\sum_{p\in L_{k}}\left(b_{k,p}b_{k}^{*} \left(K_{k}B_{k}e_{p}\right)+b_{-k}^{*}\left(K_{-k}B_{-k}e_{-p}\right)b_{-k,-p} \right)\right)\] \[\quad+2\operatorname{Re}\left(\sum_{p\in L_{k}}\left(b_{k}(B_{k}e _{p})\mathcal{E}_{-k}\left(e_{-p}\right)+\mathcal{E}_{k}\left(e_{p}\right)b_{-k }\left(B_{-k}e_{-p}\right)\right)\right)=(\text{I})+(\text{II}).\]
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\[\mathbb{E}\left[Y_{t}^{2}\right]= \mathbb{E}\left[\left(e^{-\kappa t}y_{0}+\sigma\int_{0}^{t}e^{- \kappa(t-s)}dW_{s}+\kappa\int_{0}^{t}e^{-\kappa(t-s)}\mu(s)ds+\gamma\sum_{i=1 }^{N_{t}}e^{-\kappa(t-t_{i})}\right)^{2}\right]\] \[= \left(e^{-\kappa t}y_{0}+\kappa\int_{0}^{t}e^{-\kappa(t-s)}\mu(s )ds\right)^{2}+\frac{\sigma^{2}}{2\kappa}\left(1-e^{-2\kappa t}\right)+ \mathbb{E}\left[\left(\sum_{i=1}^{N_{t}}\gamma_{t_{i}}e^{-\kappa(t-t_{i})} \right)^{2}\right]\] \[+2\left(e^{-\kappa t}y_{0}+\kappa\int_{0}^{t}e^{-\kappa(t-s)}\mu (s)ds\right)\cdot\bar{\gamma}\frac{\nu}{\kappa}\left(1-e^{-\kappa t}\right).\]
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\[\int_{U^{\prime}_{k,\varepsilon,2}}\sum_{i=1}^{N-1}\varphi_{i}e_{\varepsilon}(u _{\varepsilon,i})\,dx=\int_{U^{\prime}_{k,\varepsilon,2}}\sum_{i\in I_{k}} \varphi_{i}e_{\varepsilon}(u_{\varepsilon,i})\,dx+\int_{U^{\prime}_{k, \varepsilon,2}}\sum_{i\in I^{c}_{k}}\varphi_{i}e_{\varepsilon}(u_{ \varepsilon,i})\,dx.\] \[|Du_{i}(x)|=\left|D\frac{x^{\prime\prime}}{|x^{\prime\prime}|}\right|=\frac{ (n-2)^{1/2}}{|x^{\prime\prime}|}.\] \[F_{\varepsilon}(u_{\varepsilon,i},U^{\prime}_{k,\varepsilon,2}) \leq\mathcal{H}^{1}(S_{k})\frac{(n-2)^{(n-1)/2}}{n-1}\int_{B^{n- 1}_{\delta^{\prime}}\setminus B^{n-1}_{\varepsilon}}\frac{dx^{\prime\prime}} {|x^{\prime\prime}|^{n-1}}\]
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\[-\int\epsilon^{2}v_{xx}\beta_{x} =-\int\epsilon^{2}v_{xx}\frac{v_{x}}{u_{s}}-\int\epsilon^{2}v_{xx }v\partial_{x}\frac{1}{u_{s}}\] \[=+\int\frac{\epsilon^{2}}{2}\partial_{x}\frac{1}{u_{s}}v_{x}^{2} -\int_{x=L}\frac{\epsilon^{2}}{2u_{s}}v_{x}^{2}\] \[+\int_{x=0}\frac{\epsilon^{2}}{2u_{s}}v_{x}^{2}+\int\epsilon^{2}v _{x}^{2}\partial_{x}\frac{1}{u_{s}}\] \[+\int\epsilon^{2}vv_{x}\partial_{xx}\{\frac{1}{u_{s}}\}-\int_{x= L}\epsilon^{2}vv_{x}\partial_{x}\{\frac{1}{u_{s}}\}\] \[\gtrsim\int_{x=0}\frac{\epsilon^{2}}{2u_{s}}v_{x}^{2}-\int_{x=L} \frac{\epsilon^{2}}{2u_{s}}v_{x}^{2}-\int_{x=L}\epsilon^{2}vv_{x}\partial_{x} \{\frac{1}{u_{s}}\}\] \[-\epsilon||u_{sx},u_{sxx}||_{L^{\infty}}\mathcal{O}(L)||\sqrt{ \epsilon}v_{x}||_{L^{2}}^{2}.\]
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\[\langle\mathbf{d}^{k+1}-\mathbf{d}^{k},\mathbf{x}^{k}-\mathbf{x}^{k -1}\rangle\] \[+\langle\mathbf{d}^{k+1}-\mathbf{d}^{k},2\mathbf{z}^{k}-\mathbf{ z}^{k-1}-\mathbf{z}^{k+1}\rangle\] \[\geq \left\|\mathbf{d}^{k+1}-\mathbf{d}^{k}\right\|_{\mathbf{M}}^{2}- \frac{1}{2}\|\mathbf{d}^{k+1}-\mathbf{d}^{k}\|_{\mathbf{M}}^{2}-\frac{1}{2} \|\mathbf{d}^{k}-\mathbf{d}^{k-1}\|_{\mathbf{M}}^{2}\] \[+\langle\mathbf{d}^{k+1}-\mathbf{d}^{k},2\mathbf{z}^{k}-\mathbf{z} ^{k-1}-\mathbf{z}^{k+1}\rangle\] \[= \frac{1}{2}\|\mathbf{d}^{k+1}-\mathbf{d}^{k}\|_{\mathbf{M}}^{2}- \frac{1}{2}\|\mathbf{d}^{k}-\mathbf{d}^{k-1}\|_{\mathbf{M}}^{2}\] \[+\langle\mathbf{d}^{k+1}-\mathbf{d}^{k},2\mathbf{z}^{k}-\mathbf{z} ^{k-1}-\mathbf{z}^{k+1}\rangle,\]
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\[\mathcal{L}\widehat{V}^{(h)}(\psi)\] \[=\int_{\beta,L}d\underline{x}\Big{[}\sum_{e}\psi^{+}_{\underline{ x},e}\psi^{-}_{\underline{x},e}2^{h}\,\frac{Z_{h,e}}{Z_{h-1,e}}n_{h,e}\] \[\quad+\sum_{\underline{e}}\psi^{+}_{\underline{x},e_{1}}\psi^{-} _{\underline{x},e_{2}}\psi^{+}_{\underline{x},e_{3}}\psi^{+}_{\underline{x},e _{4}}\frac{\sqrt{Z_{h,e_{1}}}Z_{h,e_{2}}Z_{h,e_{3}}Z_{h,e_{4}}}{\sqrt{Z_{h-1,e _{1}}Z_{h-1,e_{2}}}Z_{h-1,e_{3}}Z_{h-1,e_{4}}}u_{h,\underline{e}}\Big{]}\] \[\equiv\int_{\beta,L}d\underline{x}\Big{[}\sum_{e}\psi^{+}_{ \underline{x},e}\psi^{-}_{\underline{x},e}2^{h}\nu_{h,e}+\sum_{\underline{e}} \psi^{+}_{\underline{x},e_{1}}\psi^{-}_{\underline{x},e_{2}}\psi^{+}_{ \underline{x},e_{3}}\psi^{+}_{\underline{x},e_{4}}\lambda_{h,\underline{e}} \Big{]}\;,\]
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\[|\langle\Phi,R_{3}\Phi\rangle| =\frac{1}{N-1}\left|\iint w_{N}(x-y)\overline{u(t,x)}\Big{\langle} \Phi,\sqrt{N-\mathcal{N}}a_{y}^{*}a_{y}a_{x}\Phi\Big{\rangle}\,\,\mathrm{d}x\, \mathrm{d}y\right|\] \[\leq\frac{1}{N-1}\iint|w_{N}(x-y)||u(t,x)|\cdot\|a_{y}\sqrt{N- \mathcal{N}}\Phi\|\cdot\|a_{y}a_{x}\Phi\|\,\mathrm{d}x\,\mathrm{d}y\] \[\leq\frac{\|u(t,\cdot)\|_{L^{\infty}}}{N-1}\left(\iint|w_{N}(x-y) |\|a_{x}a_{y}\Phi\|^{2}\,\mathrm{d}x\,\mathrm{d}y\right)^{1/2}\] \[\qquad\qquad\qquad\times\left(\iint|w_{N}(x-y)|\|a_{y}\sqrt{N- \mathcal{N}}\Phi\|^{2}\,\mathrm{d}x\,\mathrm{d}y\right)^{1/2}\] \[\leq\frac{C_{t}}{N}\langle\Phi,C_{\varepsilon}N^{\varepsilon} \mathrm{d}\Gamma(1-\Delta)\mathcal{N}\Phi\rangle^{1/2}\langle\Phi,\mathcal{N}N \Phi\rangle^{1/2}\] \[\leq\frac{C_{t,\varepsilon}N^{\varepsilon}m^{1/2}}{N^{1/2}} \langle\Phi,\mathrm{d}\Gamma(1-\Delta)\Phi\rangle.\]
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\[e_{ii} =\lambda_{i}+\mathbf{n}_{[1,i-1],i}-\mathbf{n}_{i,[i+1,M+N]}\quad \text{for}\quad i\in\mathfrak{I},\] \[e_{i,i+1} =\sum_{k=1}^{i-1}\mathbf{c}_{ki}^{\dagger}\mathbf{c}_{k,i+1}\] \[\times q^{-p_{i}\lambda_{i}+p_{i+1}\lambda_{i+1}-p_{i}\mathbf{n} _{[k+1,i-1],i}+p_{i+1}\mathbf{n}_{[k+1,i],i+1}+p_{i}\mathbf{n}_{i,[i+1,M+N]}- p_{i+1}\mathbf{n}_{i+1,[i+2,M+N]}}\] \[+p_{i}\mathbf{c}_{i,i+1}\left[p_{i}\lambda_{i}-p_{i+1}\lambda_{i +1}-p_{i}\mathbf{n}_{i,[i+1,M+N]}+p_{i+1}\mathbf{n}_{i+1,[i+2,M+N]}+p_{i} \right]_{q}\] \[-p_{i}\sum_{k=i+2}^{M+N}p_{k}\mathbf{c}_{ik}\mathbf{c}_{i+1,k}^{ \dagger}q^{p_{i}\lambda_{i}-p_{i+1}\lambda_{i+1}-p_{i}\mathbf{n}_{i,[k,M+N]}+p _{i+1}\mathbf{n}_{i+1,[k,M+N]}+p_{i}+p_{i+1}},\] \[e_{i+1,i} =\mathbf{c}_{i,i+1}^{\dagger}q^{p_{i}\mathbf{n}_{[1,i-1],i}-p_{i+ 1}\mathbf{n}_{[1,i-1],i+1}}+\sum_{k=1}^{i-1}\mathbf{c}_{k,i+1}^{\dagger} \mathbf{c}_{ki}q^{p_{i}\mathbf{n}_{[1,k-1],i}-p_{i+1}\mathbf{n}_{[1,k-1],i+1}}\] \[\quad\quad\quad\text{for}\quad i\in\mathfrak{I}\setminus\{M+N\}.\]
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\[a_{n}\equiv a(u)=-J_{z}+\frac{J_{+}}{2}\frac{\theta_{4}(u)\theta _{1}(u+\eta)}{\theta_{1}(u)\theta_{4}(u+\eta)}-\frac{J_{-}}{2}\frac{\theta_{1 }(u)\theta_{1}(u+\eta)}{\theta_{4}(u)\theta_{4}(u+\eta)}\] \[\stackrel{{(\ref{eq:2})}}{{=}}-e^{-i\pi\eta}\frac{ \theta_{1}(\eta+\frac{1}{2}-\tau)\theta_{1}(\eta+\frac{1}{2})}{\theta_{1}( \frac{1}{2}+\tau)\theta_{1}(\frac{1}{2})}+e^{-i\pi\eta}\frac{\theta_{1}(u+\eta )\theta_{1}(u-\eta+\tau)}{\theta_{1}(u)\theta_{1}(u+\tau)}\] \[\stackrel{{(\ref{eq:2})}}{{=}}\frac{e^{i\pi\eta} \theta_{1}(\eta+\tau)\theta_{1}(\eta)\theta_{1}(u+\frac{1}{2})\theta_{1}(u+ \frac{1}{2}+\tau)}{\theta_{1}(\frac{1}{2}+\tau)\theta_{1}(\frac{1}{2})\theta_{1} (u)\theta_{1}(u+\tau)}\] \[\stackrel{{(\ref{eq:2})}}{{=}}\frac{\bar{\theta}_{1}( \eta)\,\bar{\theta}_{2}(u)}{\bar{\theta}_{2}(0)\,\bar{\theta}_{1}(u)}.\]
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\[\bigg{|}\int_{(\mathbb{R}^{d})^{k}}\mathcal{W}(t_{\underline{F}} (\boldsymbol{y}_{1},\ldots,\boldsymbol{y}_{k}))\mathrm{e}\bigg{(}\sum_{ \begin{subarray}{c}i=1\\ i\neq l\end{subarray}}^{k}\big{[}\theta_{i}(\tfrac{1}{2}\|\boldsymbol{y}_{i}\|^ {2}-\tfrac{1}{2}\|\boldsymbol{y}_{l}\|^{2})+\mathrm{i}|\theta_{i}|\gamma\big{]} \bigg{)}\\ \times\widehat{\mathcal{A}}_{t,\ell,\underline{F}}(r^{d} \boldsymbol{\theta},\boldsymbol{y}_{1},\ldots,\boldsymbol{y}_{k})\,\mathrm{d} \boldsymbol{y}_{1}\cdots\mathrm{d}\boldsymbol{y}_{k}\bigg{|}\\ \leq\mathrm{e}^{-2\pi\sum_{i\neq l}|\theta_{i}|\gamma}\bigg{|} \int_{(\mathbb{R}^{d})^{k}}\mathcal{W}(t_{\underline{F}}(\boldsymbol{y}_{1}, \ldots,\boldsymbol{y}_{k}))\mathrm{e}\bigg{(}\tfrac{1}{2}\sum_{i=1}^{k}\theta _{i}\|\boldsymbol{y}_{i}\|^{2}\bigg{)}\\ \times\widehat{\mathcal{A}}_{t,\ell,\underline{F}}(r^{d} \boldsymbol{\theta},\boldsymbol{y}_{1},\ldots,\boldsymbol{y}_{k})\,\mathrm{d} \boldsymbol{y}_{1}\cdots\mathrm{d}\boldsymbol{y}_{k}\bigg{|}\]
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\[\left\{\begin{array}{l}\partial_{tt}\phi^{\epsilon}+\mathrm{u}^{\epsilon} \cdot\nabla\partial_{t}\phi^{\epsilon}+\partial_{t}\mathrm{u}^{\epsilon} \cdot\nabla\phi^{\epsilon}+\partial_{t}\phi^{\epsilon}\mathrm{div}\epsilon^{ +}+\phi^{\epsilon}\mathrm{div}\partial_{t}\mathrm{u}^{\epsilon}+\frac{1}{ \epsilon}\mathrm{div}\partial_{t}\mathrm{u}^{\epsilon}=0\,,\\ \partial_{tt}\mathrm{u}^{\epsilon}+\mathrm{u}^{\epsilon}\cdot\nabla\partial_{t} \mathrm{u}^{\epsilon}+\partial_{t}\mathrm{u}^{\epsilon}\cdot\nabla\mathrm{u}^{ \epsilon}+\frac{1}{\epsilon}\partial_{t}(\frac{p^{\epsilon}(\rho^{\epsilon})}{ \rho^{\epsilon}})\nabla\phi^{\epsilon}+\frac{1}{\epsilon}\frac{p^{\prime}(\rho^{ \epsilon})}{\rho^{\epsilon}}\nabla\partial_{t}\phi^{\epsilon}\\ =\partial_{t}(\frac{1}{\rho^{\epsilon}})\mathrm{div}(\Sigma_{1}^{\epsilon}+ \Sigma_{2}^{\epsilon}+\Sigma_{3}^{\epsilon})+\frac{1}{\rho^{\epsilon}}\mathrm{ div}\partial_{t}(\Sigma_{1}^{\epsilon}+\Sigma_{2}^{\epsilon}+\Sigma_{3}^{\epsilon})\,,\end{array}\right.\]
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\begin{table}
\begin{tabular}{|c|c|c|} \hline \(\operatorname{RA}(C)\) & A normal form of \(C\) & A Rosenhain form of \(C\) \\ \hline \(\{1\}\) & — & \(y^{2}=x(x-1)(x-\lambda)(x-\mu)(x-\nu)\) \\ \(\operatorname{C}_{2}\) & \(y^{2}=(x^{2}-1)(x^{2}-a)(x^{2}-b)\) & \(y^{2}=x(x-1)(x-\lambda)(x-\mu)\big{(}x-\frac{\lambda(1-\mu)}{1-\lambda}\big{)}\) \\ \(\operatorname{S}_{3}\) & \(y^{2}=(x^{3}-1)(x^{3}-a)\) & \(y^{2}=x(x-1)(x-\lambda)(x-\frac{\lambda-1}{\lambda})\big{(}x-\frac{1}{1- \lambda}\big{)}\) \\ \((\operatorname{C}_{2})^{2}\) & \(y^{2}=x(x^{2}-1)(x^{2}-a)\) & \(y^{2}=x(x-1)(x+1)(x-\lambda)\big{(}x-\frac{1}{\lambda}\big{)}\) \\ \(\operatorname{D}_{12}\) & \(y^{2}=x^{6}-1\) & \(y^{2}=x(x-1)(x+1)(x-2)\big{(}x-\frac{1}{2}\big{)}\) \\ \(\operatorname{S}_{4}\) & \(y^{2}=x^{5}-x\) & \(y^{2}=x(x-1)(x+1)(x-i)(x+i)\) \\ \(\operatorname{C}_{5}\) & \(y^{2}=x^{5}-1\) & \(y^{2}=x(x-1)(x-1-\zeta)(x-1-\zeta-\zeta^{2})(x-1-\zeta-\zeta^{2}-\zeta^{3})\) \\ \hline \end{tabular}
\end{table}
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\[X_{1}^{[5,3,1]}=\left(\begin{array}{ccc}\frac{1}{q^{6}+q^{8}+q^{10}}&-\frac {1}{q^{5}\sqrt{1+q^{2}+q^{4}}}&\frac{\sqrt{1+q^{2}+q^{4}+q^{6}+q^{8}}}{q^{2}+q^ {4}+q^{6}}\\ \\ \frac{1}{q^{3}\sqrt{1+q^{2}+q^{4}}}&-\frac{1}{q^{2}}+\frac{1}{1+q^{4}}&-\frac {q^{3}\sqrt{\frac{1+q^{2}+q^{4}+q^{6}+q^{8}}{1+q^{2}+q^{4}}}}{1+q^{4}}\\ \frac{q^{4}\sqrt{1+q^{2}+q^{4}+q^{6}+q^{8}}}{1+q^{2}+q^{4}}&\frac{q^{7}\sqrt{ \frac{1+q^{2}+q^{4}+q^{6}+q^{8}}{1+q^{2}+q^{4}}}}{1+q^{2}+2q^{4}+q^{6}+q^{8}} \end{array}\right)\]
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\[=\mathbb{E}\left(\sum_{\mathbf{z}}\frac{\sum\limits_{\begin{subarray} {c}m[\mathcal{K}]\end{subarray}}\sum\limits_{\ell[\mathcal{K}]}W_{Z|X|\mathcal{ K}|}^{\otimes n}\left(\mathbf{z}|\mathbf{X}_{\mathcal{K}}\left(m[ \mathcal{K}],\ell[\mathcal{K}]\right)\right)}{\left(\prod\limits_{k\in \mathcal{K}}M_{k}L_{k}\right)}\log\!\left(\frac{\sum\limits_{\begin{subarray} {c}m[\mathcal{K}]\end{subarray}}\sum\limits_{\ell[\mathcal{K}]}W_{Z|X| \mathcal{K}|}^{\otimes n}\left(\mathbf{z}|\mathbf{X}_{\mathcal{K}}\left( \widetilde{m}[\mathcal{K}],\widetilde{\ell}[\mathcal{K}]\right)\right)}{ \left(\prod\limits_{k\in\mathcal{K}}M_{k}L_{k}\right)Q_{\alpha_{n}}^{\otimes n }\left(\mathbf{z}\right)}\right)\right)\]
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\[\begin{array}{rcl}b_{g_{i}}&=&\langle\delta\operatorname{Trd}_{Q}(z_{p}z_{q} )\rangle\langle 1,z_{p}^{2}z_{q}^{2}(z_{p}z_{q})_{0}^{2}\rangle+\langle-z_{i}^{2}z_{p}^{2} \operatorname{Trd}_{Q}(z_{p}z_{q})\rangle\langle 1,z_{p}^{2}z_{q}^{2}(z_{p}z_{q})_{0}^{2}\rangle \\ &=&\langle\delta\operatorname{Trd}_{Q}(z_{p}z_{q})\rangle\langle 1,-z_{q}^{2} \rangle\langle 1,z_{p}^{2}z_{q}^{2}(z_{p}z_{q})_{0}^{2}\rangle\\ &=&\langle\delta\operatorname{Trd}_{Q}(z_{p}z_{q})\rangle\langle\langle z_{q} ^{2},-z_{p}^{2}z_{q}^{2}(z_{p}z_{q})_{0}^{2}\rangle\rangle.\end{array}\]
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\[\Pr\left(\lambda_{k}\left(\mathcal{X}\right)\geq\theta\right) =_{1} \Pr\left(e^{\lambda_{k}\left(t\mathcal{X}\right)}\geq e^{t\theta}\right)\] \[\leq_{2} e^{-t\theta}\mathbb{E}e^{\lambda_{k}\left(t\mathcal{X}\right)}\] \[=_{3} e^{-t\theta}\mathbb{E}\exp\left(\min_{\mathcal{U}_{\left( \mathbb{I}_{M}-k+1\right)}}\lambda_{\max}\left(t\mathcal{U}_{\left(\mathbb{I} _{M}-k+1\right)}^{H}\star_{M}\mathcal{X}\star_{M}\mathcal{U}_{\left(\mathbb{I} _{M}-k+1\right)}\right)\right)\] \[\leq_{4} \min_{\mathcal{U}_{\left(\mathbb{I}_{M}-k+1\right)}}\mathbb{E} \lambda_{\max}\left(\exp\left(t\mathcal{U}_{\left(\mathbb{I}_{M}-k+1\right)}^ {H}\star_{M}\mathcal{X}\star_{M}\mathcal{U}_{\left(\mathbb{I}_{M}-k+1\right)} \right)\right)\] \[\leq_{5} \min_{\mathcal{U}_{\left(\mathbb{I}_{M}-k+1\right)}}\mathbb{E} \mathrm{Tr}\left(\exp\left(t\mathcal{U}_{\left(\mathbb{I}_{M}-k+1\right)}^{H }\star_{M}\mathcal{X}\star_{M}\mathcal{U}_{\left(\mathbb{I}_{M}-k+1\right)} \right)\right),\]
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\[\sum_{1\leq i\leq n}\sum_{\begin{subarray}{c}|\boldsymbol{u}_{ \mathrm{III}}^{(r)}|=(2,0)\\ i\leq r<n\end{subarray}}\sum_{|\boldsymbol{u}_{\mathrm{II}}^{(n)}|=1}\prod_{i<k \leq n}\Gamma_{k}(\boldsymbol{u}_{\mathrm{II}}^{(k-1)};\boldsymbol{u}_{ \mathrm{I}}^{(1...n)})\,K(\boldsymbol{u}_{\mathrm{II}}^{(k-1)}\,|\, \boldsymbol{u}_{\mathrm{II},\mathrm{III}}^{(k)})\] \[\times\frac{\Gamma_{\hat{n}}(\boldsymbol{u}_{\mathrm{II}}^{(n)} ;\boldsymbol{u}_{\mathrm{I}}^{(1...n)})}{\boldsymbol{u}_{\mathrm{II}}^{(n)}- \boldsymbol{u}_{\mathrm{III}}^{(n)}}\,E_{i}^{(\hat{n})}\otimes E_{i}^{(\hat{n} )}\otimes\Psi(\boldsymbol{u}_{\mathrm{I}}^{(1...n)})\] \[+\sum_{1\leq i<n}\sum_{\begin{subarray}{c}|\boldsymbol{u}_{ \mathrm{II}}^{(r)}|=1\\ i\leq r<n\end{subarray}}\prod_{i<k<n}\frac{\Gamma_{k}(\boldsymbol{u}_{ \mathrm{II}}^{(k-1)};\boldsymbol{u}_{\mathrm{I}}^{(1...n)})}{\boldsymbol{u}_{ \mathrm{II}}^{(k-1)}-\boldsymbol{u}_{\mathrm{II}}^{(k)}}\cdot\frac{\Gamma_{n} (\boldsymbol{u}_{\mathrm{II}}^{(n-1)};\boldsymbol{u}_{\mathrm{I}}^{(1...n)}) \Gamma_{\hat{n}}(\boldsymbol{u}_{\mathrm{II}}^{(n)};\boldsymbol{u}_{\mathrm{ I}}^{(1...n)})}{\boldsymbol{u}_{\mathrm{II}}^{(n)}-\boldsymbol{u}_{\mathrm{III}}^{(n)}}\]
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\[\begin{split}& A_{1}+A_{2}=\sum_{k=0}^{\rho-1}\frac{(-1)^{k}}{k! }z^{-\nu-k}\frac{\Gamma(\rho)\Gamma\left(\frac{3}{2}\right)}{\Gamma(\rho-k)} \left(\frac{\Gamma(\nu+k)}{\Gamma\left(\frac{1}{2}-\nu-k\right)}+\frac{\Gamma \left(\nu+\frac{1}{2}+k\right)}{\Gamma(1-\nu-k)}\right)+\\ &+O\big{(}z^{-\nu-\rho+1}\big{)}\sum_{k=0}^{\rho-1}\frac{(-1)^{k}} {k!}z^{-\nu-k}\frac{\Gamma(\rho)\Gamma\left(\frac{3}{2}\right)}{\Gamma(\rho-k) }\left(\frac{\pi}{\sin\left(\frac{3\pi}{4}+\pi k\right)}+\frac{\pi}{\sin\left( \frac{3\pi}{4}+\frac{\pi}{2}+\pi k\right)}\right)\\ &+O\big{(}z^{-\nu-\rho+1}\big{)}=O\big{(}x^{-2m_{0}-1}\big{)}. \end{split}\]
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\[\frac{1}{N} \int_{0}^{T}N^{2}e^{-N\left\langle{\Gamma^{N}_{t}},H_{t}\right\rangle }{\cal L}_{\beta}e^{N\left\langle{\Gamma^{N}_{t}},H_{t}\right\rangle}dt\] \[=\frac{1}{4}\int_{0}^{T}dt\int_{\gamma^{N}_{t}(\varepsilon)} \frac{({\sf v}^{\varepsilon})^{2}}{{\sf v}}\big{[}{\bf T}^{\varepsilon}\cdot{ \bf m}\big{(}\gamma^{N}_{t}(s)\big{)}\big{]}\;{\bf T}^{\varepsilon}\cdot \nabla H\big{(}t,\gamma^{N}_{t}(s)\big{)}ds\] \[\quad+\frac{1}{2}\int_{0}^{T}dt\int_{\gamma^{N}_{t}(\varepsilon)} \frac{({\sf v}^{\varepsilon})^{2}}{{\sf v}}|{\bf T}^{\varepsilon}\cdot{\bf b} _{1}||{\bf T}^{\varepsilon}\cdot{\bf b}_{2}|H\big{(}t,\gamma^{N}_{t}(s)\big{)} ^{2}ds+\int_{0}^{T}\omega(H_{t},\delta,\varepsilon,A,\gamma^{N}_{t})dt\] \[\quad-\frac{1}{2}\int_{0}^{T}\sum_{k=1}^{4}(1/2-e^{-\beta})\big{[} H(t,L_{k}(\gamma^{N}_{t}))+H(t,R_{k}(\gamma^{N}_{t}))\big{]}dt,\]
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\[\int_{Y_{2d}}\prod_{i\in I}\psi_{m_{i}}\ \mathrm{d}\mu =\int_{Y_{2d}}\prod_{\ell=1}^{k}\Biggl{(}\sum_{K_{\ell}\subset Q_{ \ell}}\bigl{(}-\mu\big{(}\phi\big{)}\bigr{)}^{\#K_{\ell}}g_{\ell,K_{\ell}} \Biggr{)}\circ a^{m_{Q_{\ell}}}\ \mathrm{d}\mu\] \[=\int_{Y_{2d}}\sum_{\begin{subarray}{c}K_{\ell}\subset Q_{\ell}\\ \text{for all $\ell$}\end{subarray}}\bigl{(}-\mu\big{(}\phi\big{)}\bigr{)}^{\#K_{1}+ \cdots+\#K_{k}}\prod_{\ell=1}^{k}\ g_{\ell,K_{\ell}}\circ a^{m_{Q_{\ell}}}\ \mathrm{d}\mu\] \[=\sum_{\begin{subarray}{c}K_{\ell}\subset Q_{\ell}\\ \text{for all $\ell$}\end{subarray}}\bigl{(}-\mu\big{(}\phi\big{)}\bigr{)}^{\#K_{1}+ \cdots+\#K_{k}}\int_{Y_{2d}}\prod_{\ell=1}^{k}\ g_{\ell,K_{\ell}}\circ a^{m_{Q_ {\ell}}}\ \mathrm{d}\mu.\]
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\[|\xi|^{s}\chi(|\xi|)|\partial_{t}^{j}\widehat{K_{0}^{1}}(t,\xi)| \lesssim e^{-c_{0}t|\xi|^{2(\sigma-\delta_{1})}}|\xi|^{s+2j(\sigma- \delta_{1})},\] \[|\xi|^{s}\chi(|\xi|)|\partial_{t}^{j}\widehat{K_{0}^{2}}(t,\xi)| \lesssim e^{-c_{0}t|\xi|^{2\delta_{1}}}|\xi|^{s+2j\delta_{1}+2( \sigma-2\delta_{1})},\] \[|\xi|^{s}\chi(|\xi|)|\partial_{t}^{j}\widehat{K_{1}^{1}}(t,\xi)| \lesssim e^{-c_{0}t|\xi|^{2(\sigma-\delta_{1})}}|\xi|^{s+2j(\sigma- \delta_{1})-2\delta_{1}},\] \[|\xi|^{s}\chi(|\xi|)|\partial_{t}^{j}\widehat{K_{1}^{2}}(t,\xi)| \lesssim e^{-c_{0}t|\xi|^{2\delta_{1}}}|\xi|^{s+2(j-1)\delta_{1}},\]
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\[p_{k}^{\alpha_{k}}\bigg{(}\frac{m}{p_{k}^{\alpha_{k}}}\psi( \mathbb{Z}_{n})+\sum_{J\subsetneq I^{\prime}}\prod_{i\in J}p_{i}^{\alpha_{i}} \prod_{i\in I^{\prime}\setminus J}(\psi(\mathbb{Z}_{p_{i}^{\alpha_{i}}})-p_{i} ^{\alpha_{i}})\psi(\langle b^{\beta_{I^{\prime}\setminus J}\cdot o_{M_{I^{ \prime}J}}}\rangle))\bigg{)}\] \[+(\psi(\mathbb{Z}_{p_{k}^{\alpha_{k}}})-p_{k}^{\alpha_{k}})\bigg{(} \frac{m}{p_{k}^{\alpha_{k}}}\psi(\langle b^{o^{\emptyset}_{k}}\rangle)+\sum_{J \subsetneq I^{\prime}}\prod_{i\in J}p_{i}^{\alpha_{i}}\prod_{i\in I^{\prime} \setminus J}(\psi(\mathbb{Z}_{p_{i}^{\alpha_{i}}})-p_{i}^{\alpha_{i}})\psi( \langle b^{\beta_{I^{\prime}\setminus J}\cdot o_{M_{I^{\prime}J}}^{I^{\prime} \setminus(J\cup\{M_{I^{\prime}J}\}}}\rangle))\bigg{)}\] \[=m\psi(\mathbb{Z}_{n})+\frac{m}{p_{k}^{\alpha_{k}}}(\psi(\mathbb{ Z}_{p_{k}^{\alpha_{k}}})-p_{k}^{\alpha_{k}})\psi(\langle b^{o^{\emptyset}_{k}}\rangle)\] \[+\sum_{J\subsetneq I^{\prime}}\bigg{(}\prod_{i\in J\cup\{k\}}p_{i }^{\alpha_{i}}\prod_{i\in I^{\prime}\setminus J}(\psi(\mathbb{Z}_{p_{i}^{ \alpha_{i}}})-p_{i}^{\alpha_{i}})\psi(\langle b^{\beta_{I^{\prime}\setminus J }\cdot o_{M_{I^{\prime}J}}^{I^{\prime}\setminus(J\cup\{M_{I^{\prime}J}\})}} \rangle)\] \[+\prod_{i\in J}p_{i}^{\alpha_{i}}\prod_{i\in(I^{\prime}\cup\{k\}) \setminus J}(\psi(\mathbb{Z}_{p_{i}^{\alpha_{i}}})-p_{i}^{\alpha_{i}})\psi( \langle b^{\beta_{(I^{\prime}\cup\{k\})\setminus J}\cdot o_{M_{(I^{\prime} \cup\{k\})J}}}\rangle)\bigg{)}\] \[=m\psi(\mathbb{Z}_{n})+\frac{m}{p_{k}^{\alpha_{k}}}(\psi(\mathbb{ Z}_{p_{k}^{\alpha_{k}}})-p_{k}^{\alpha_{k}})\psi(\langle b^{o^{\emptyset}_{k}} \rangle)\] \[+\sum_{J\subsetneq I^{\prime}}\bigg{(}\prod_{i\in J\cup\{k\}}p_{i }^{\alpha_{i}}\prod_{i\in I^{\prime}\setminus J}(\psi(\mathbb{Z}_{p_{i}^{ \alpha_{i}}})-p_{i}^{\alpha_{i}})\psi(\langle b^{\beta_{I^{\prime}\setminus J }\cdot o_{M_{I^{\prime}J}}^{I^{\prime}\setminus(J\cup\{M_{I^{\prime}J}\})}} \rangle)\] \[+\prod_{i\in J}p_{i}^{\alpha_{i}}\prod_{i\in I\setminus J}(\psi( \mathbb{Z}_{p_{i}^{\alpha_{i}}})-p_{i}^{\alpha_{i}})\psi(\langle b^{\beta_{I \setminus J}\cdot o_{M_{IJ}}^{I\setminus(J\cup\{M_{IJ}\})}}\rangle)\bigg{)}\] \[=m\psi(\mathbb{Z}_{n})+\sum_{J\subsetneq I}\prod_{i\in J}p_{i}^{ \alpha_{i}}\prod_{i\in I\setminus J}(\psi(\mathbb{Z}_{p_{i}^{\alpha_{i}}})-p_{ i}^{\alpha_{i}})\psi(\langle b^{\beta_{I\setminus J}\cdot o_{M_{IJ}}^{I \setminus(J\cup\{M_{IJ}\})}}\rangle),\]
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\[\widetilde{E}_{l}(t,\mathfrak{f},\partial_{t}\mathfrak{f})\coloneqq \int_{\Gamma_{t}}\Biggl{|}\biggl{(}-\mathcal{N}_{+}^{\frac{1}{2}} \bigtriangleup_{\Gamma_{t}}\mathcal{N}_{+}^{\frac{1}{2}}\biggr{)}^{\frac{l}{ 2}}\mathcal{N}_{+}^{\frac{1}{2}}\bigl{[}(\partial_{t}\mathfrak{f}+\mathrm{D}_ {\mathbf{v}_{*}}\mathfrak{f})\circ\Phi_{\Gamma_{t}}^{-1}\bigr{]}\Biggr{|}^{2}\] \[\qquad\qquad+\Biggl{|}\biggl{(}-\mathcal{N}_{+}^{\frac{1}{2}} \bigtriangleup_{\Gamma_{t}}\mathcal{N}_{+}^{\frac{1}{2}}\biggr{)}^{\frac{l}{ 2}}\mathcal{N}_{+}^{\frac{1}{2}}\bigl{[}(\mathrm{D}_{\mathbf{h}_{*}}\mathfrak{ f})\circ\Phi_{\Gamma_{t}}^{-1}\bigr{]}\Biggr{|}^{2}\] \[\qquad\qquad+\Biggl{|}\biggl{(}-\mathcal{N}_{+}^{\frac{1}{2}} \bigtriangleup_{\Gamma_{t}}\mathcal{N}_{+}^{\frac{1}{2}}\biggr{)}^{\frac{l}{ 2}}\mathcal{N}_{+}^{\frac{1}{2}}\Bigl{[}(\mathrm{D}_{\mathbf{\hat{h}}_{*}} \mathfrak{f})\circ\Phi_{\Gamma_{t}}^{-1}\Bigr{]}\Biggr{|}^{2}\,\mathrm{d}S_{t}\,.\]
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\[\bar{\kappa}(p) :=(M_{1}(p))^{2r}/(4(1+(M_{1}(p))^{2r})),\] \[M_{1}(p) :=\Big{(}2pd(2^{2r+3}K_{G}\mathbb{E}\left[(1+|X_{0}|)^{2\rho} \right]+4b_{F}+6)\] \[\quad+d^{p}{2p\choose p}(2p-1)2^{4p+1}(2+K_{F})^{2p}\mathbb{E}[(1 +|X_{0}|)^{2p\rho}]\Big{)}/\min\left\{1,a_{F}\right\},\] \[\bar{c}_{0}(p) :=c_{4}(p)+2^{2p-2}p(2p-1)d\beta^{-1}((M_{2}(p))^{2(p-1)}+c_{4}(p-1 ))+2^{2p-4}(2p(2p-1))^{p+1}(d\beta^{-1})^{p},\] \[c_{4}(p) :=pd\mathbb{E}\left[(1+|X_{0}|)^{2\rho}\right](2+4b_{F}+2^{2r+2}K_ {G})+2a_{F}\bar{\kappa}(p)(M_{1}(p))^{2p}\] \[\quad+pd\mathbb{E}\left[(1+|X_{0}|)^{2\rho}\right](2^{2r+3}K_{G}+4 b_{F}+6)(M_{1}(p))^{2r+2p-1}\] \[\quad+d^{p}{2p\choose p}(2p-1)2^{4p}(2+K_{F})^{2p}\mathbb{E}[(1+|X_{0}|)^{2p \rho}](1+(M_{1}(p))^{2r+2p-1}),\] \[M_{2}(p) :=\left(2^{2p-2}p(2p-1)d\beta^{-1}/(a_{F}\bar{\kappa}(p))\right)^{1/ 2}.\]
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\[\begin{split} d_{3,t}&\big{\|}P\partial_{t}f_{R}\big{\|} _{L^{2}_{t}L^{p}_{x}}\\ \lesssim&\Big{\{}\frac{1}{\kappa}\left(\|\partial_{t} \mathcal{I}\|_{L^{\infty}_{t,x,v}}+\varepsilon\|(2.49)\|_{L^{\infty}_{t,x,v}} \right)\Big{\}}\Big{\{}\|Pf_{R}\|_{L^{2}_{t,x}}+\|\sqrt{\nu}(\mathbf{I}- \mathbf{P})f_{R}\|_{L^{2}_{t,x,v}}\Big{\}}\\ &+\Big{\{}\frac{1}{\kappa\varepsilon}+\frac{\varepsilon^{1/2}}{ \kappa}\|\mathfrak{w}f_{R}\|_{L^{\infty}_{t,x,v}}+\frac{1}{\kappa}\left(\| \mathcal{I}\|_{L^{\infty}_{t,x}}+\varepsilon\|(2.36)\|_{L^{\infty}_{t,x}} \right)\Big{\}}\|\sqrt{\nu}(\mathbf{I}-\mathbf{P})\partial_{t}f_{R}\|_{L^{2}_ {t,x,v}}\\ &+(\kappa\varepsilon)^{\frac{2}{p-2}}\|\mathfrak{w}_{\varrho^{ \prime}}\partial_{t}f_{R}\|_{L^{2}_{t}L^{\infty}_{x,v}}+\|\partial_{t}f_{R}\| _{L^{\infty}_{t}L^{2}_{v,v}}+\|\partial_{t}f_{R}(0)\|_{L^{2}_{\tau}}+\frac{1} {\kappa}\|\mathcal{I}\|_{L^{\infty}_{t,x}}\|P\partial_{t}f_{R}\|_{L^{2}_{t}L^ {2}_{x}}\\ &+\varepsilon^{1/2}\|(2.65)\|_{L^{2}_{t,x,v}}+\frac{\varepsilon^{ 1/2}}{\kappa}\|\kappa(2.66)\|_{L^{2}_{t,x,v}}+\varepsilon^{3/2}\|(2.68)\|^{2} _{L^{2}_{t,x,v}}+\frac{\varepsilon^{3/2}}{\kappa}\|\kappa(2.69)\|^{2}_{L^{2}_ {t,x,v}},\end{split}\]
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\[J_{5}^{(2)}: \begin{pmatrix}(-2,a_{112},\ldots)&(-2,a_{122},\ldots)&(0,a_{132}, \ldots)\\ (1,a_{212},\ldots)&(-2,a_{222},\ldots)&(0,a_{232},\ldots)\\ (a_{311},a_{312},\ldots)&(a_{321},a_{322},\ldots)&(1,a_{332},\ldots)\\ \end{pmatrix}_{\mathbb{Z}_{5}}\overset{K_{5}^{\prime}\circ\pi_{1}}{\longmapsto }C=\begin{pmatrix}-2&-2\\ 1&-2\\ \end{pmatrix}_{\mathbb{Z}/5}\overset{\rho_{2}\circ\varphi}{\longmapsto} \begin{pmatrix}-\frac{1}{2}&-\frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}&-\frac{1}{2}\\ \end{pmatrix}_{\mathbb{C}}\] \[\begin{pmatrix}(1,a_{112},\ldots)&(0,a_{122},\ldots)&(0,a_{132}, \ldots)\\ (0,a_{212},\ldots)&(-1,a_{222},\ldots)&(0,a_{232},\ldots)\\ (a_{311},a_{312},\ldots)&(a_{321},a_{322},\ldots)&(-1,a_{332},\ldots)\\ \end{pmatrix}_{\mathbb{Z}_{5}}\overset{K_{5}^{\prime}\circ\pi_{1}}{\longmapsto }Z=\begin{pmatrix}1&0\\ 0&-1\\ \end{pmatrix}_{\mathbb{Z}/5}\overset{\rho_{2}\circ\varphi}{\longmapsto} \begin{pmatrix}1&0\\ 0&-1\\ \end{pmatrix}_{\mathbb{C}}.\]
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\[P(\|Y-\phi^{(2)}\|\leq\varepsilon)\] \[=\tilde{P}(\|\tilde{Y}-\phi^{(2)}\|\leq\varepsilon)=E\Big{(} \frac{d\tilde{P}}{dP}\mathbb{I}_{\|B^{H}\|\leq\varepsilon}\Big{)}\] \[=E\Big{(}\exp\Big{(}\int_{0}^{1}\eta_{s}ds-\frac{1}{2}\int_{0}^{1} \eta_{s}^{2}ds\Big{)}\mathbb{I}_{\|B^{H}\|\leq\varepsilon}\Big{)}\] \[=E\Big{(}\exp\Big{(}\int_{0}^{1}\Big{(}(K^{H})^{-1}\Big{(}b( \tilde{X}_{u},\tilde{Y}_{u})\Big{)}(s)-\dot{\phi}_{s}^{(2)}\Big{)}ds-\frac{1} {2}\int_{0}^{1}\Big{(}(K^{H})^{-1}\Big{(}b(\tilde{X}_{u},\tilde{Y}_{u})\Big{)} (s)-\dot{\phi}_{s}^{(2)}\Big{)}^{2}ds\Big{)}\mathbb{I}_{\|B^{H}\|\leq \varepsilon}\Big{)}\] \[=E\Big{(}\exp\Big{(}\int_{0}^{1}\big{(}(K^{H})^{-1}b(\tilde{X}_{u}, \phi_{u}^{(2)}+B_{u}^{H})\big{)}(s)dW_{s}+\int_{0}^{1}(-\dot{\phi}_{s}^{(2)})dW_{ s}+\frac{1}{2}\int_{0}^{1}\big{|}\dot{\phi}_{s}^{(2)}-\big{(}(K^{H})^{-1}b( \phi_{u})\big{)}(s)\big{|}^{2}ds\] \[-\frac{1}{2}\int_{0}^{1}\big{|}\dot{\phi}_{s}^{(2)}-\big{(}(K^{H}) ^{-1}b(\phi_{u})\big{)}(s)\big{|}^{2}ds-\frac{1}{2}\int_{0}^{1}\Big{(}(K^{H})^ {-1}\Big{(}b(\tilde{X}_{u},\tilde{Y}_{u})\Big{)}(s)-\dot{\phi}_{u}^{(2)}\Big{)} ^{2}ds\Big{)}\mathbb{I}_{\|B^{H}\|\leq\varepsilon}\Big{)}\] \[=E\Big{(}\exp(I_{1}+I_{2}+I_{3}+I_{4})\mathbb{I}_{\|B^{H}\|\leq \varepsilon}\Big{)}\times\exp\Big{(}-\frac{1}{2}\int_{0}^{1}\big{|}\dot{\phi }_{s}^{(2)}-\big{(}(K^{H})^{-1}b(\phi_{u})\big{)}(s)\big{|}^{2}ds\Big{)},\]
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\[F\big{(}\widetilde{\mathcal{M}}(\varphi_{RA^{n}}),\mathcal{N}^{ \otimes n}(\varphi_{RA^{n}})\big{)}\] \[= F\Big{(}\sum_{\pi\in S_{n}}\frac{1}{|S_{n}|}W_{B^{n}}^{\pi^{-1}} \circ\mathcal{M}\circ W_{A^{n}}^{\pi}(\varphi_{RA^{n}}),\sum_{\pi\in S_{n}} \frac{1}{|S_{n}|}W_{B^{n}}^{\pi^{-1}}\circ\mathcal{N}^{\otimes n}\circ W_{A^{n }}^{\pi}(\varphi_{RA^{n}})\Big{)}\] \[\geq \sum_{\pi\in S_{n}}\frac{1}{|S_{n}|}F\big{(}W_{B^{n}}^{\pi^{-1}} \circ\mathcal{M}\circ W_{A^{n}}^{\pi}(\varphi_{RA^{n}}),W_{B^{n}}^{\pi^{-1}} \circ\mathcal{N}^{\otimes n}\circ W_{A^{n}}^{\pi}(\varphi_{RA^{n}})\big{)}\] \[= \sum_{\pi\in S_{n}}\frac{1}{|S_{n}|}F\big{(}\mathcal{M}(W_{A^{n}} ^{\pi}(\varphi_{RA^{n}})),\mathcal{N}^{\otimes n}(W_{A^{n}}^{\pi}(\varphi_{RA^ {n}}))\big{)}\] \[\geq \sum_{\pi\in S_{n}}\frac{1}{|S_{n}|}\min_{\varphi_{RA^{n}}^{\pi}}F \big{(}\mathcal{M}(\varphi_{RA^{n}}^{\prime}),\mathcal{N}^{\otimes n}(\varphi_ {RA^{n}}^{\prime})\big{)}\] \[= \min_{\varphi_{RA^{n}}^{\prime}}F\big{(}\mathcal{M}(\varphi_{RA^ {n}}^{\prime}),\mathcal{N}^{\otimes n}(\varphi_{RA^{n}}^{\prime})\big{)},\]
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\[\|G_{\alpha}^{\lambda}\theta\|_{\dot{H}^{\sigma+\delta_{1}}} \leq\|G_{\alpha}^{\lambda}\theta\|_{\dot{H}^{\sigma-\frac{2\delta_{1}} {2}}}^{-\frac{2\delta_{1}}{2}}\|G_{\alpha}^{\lambda}\theta\|_{\dot{H}^{\sigma+ \frac{\kappa}{2}}}^{\frac{2\delta_{1}}{2}},\] \[\|G_{\alpha}^{\lambda}\theta\|_{\dot{H}^{\sigma+\frac{\kappa}{2}}} \leq\|G_{\alpha}^{\lambda}\theta\|_{\dot{H}^{\sigma-\frac{3-\kappa}{2}}}^{- \frac{3-\kappa}{2}}\|G_{\alpha}^{\lambda}\theta\|_{\dot{H}^{\sigma+\frac{\kappa}{2}}}^ {\frac{3-\kappa}{2}},\] \[\|G_{\alpha}^{\lambda}\theta\|_{\dot{H}^{\sigma+\frac{\kappa}{2}} \frac{\delta_{1}}{2}} \leq\|G_{\alpha}^{\lambda}\theta\|_{\dot{H}^{\sigma}}^{\frac{\delta_{1}}{ 2}}\|G_{\alpha}^{\lambda}\theta\|_{\dot{H}^{\sigma+\frac{\kappa}{2}}}^{-\frac{ \delta_{1}}{2}}.\]
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\[\frac{1}{W_{j_{k}}}\sum_{n=J_{k-1}+(i-1)j_{k}+1}^{J_{k-1}+ij_{k}} w_{n} \geqslant\left(\frac{J_{k-1}+(i-1)j_{k}+1}{j_{k}}+1\right)^{1- \theta}-\left(\frac{J_{k-1}+(i-1)j_{k}+1}{j_{k}}\right)^{1-\theta}\] \[=\left(\frac{J_{k-1}}{j_{k}}+i+\frac{1}{j_{k}}\right)^{1-\theta} -\left(\frac{J_{k-1}}{j_{k}}+i-1+\frac{1}{j_{k}}\right)^{1-\theta}\] \[\geqslant\left(\frac{J_{k-1}}{j_{k}}+i\right)^{1-\theta}-\left( \frac{J_{k-1}}{j_{k}}+i-1+\frac{1}{2}\right)^{1-\theta}\] \[=i^{\theta}\left[\left(\frac{J_{k-1}}{j_{k}}+i\right)^{1-\theta} -\left(\frac{J_{k-1}}{j_{k}}+i-\frac{1}{2}\right)^{1-\theta}\right]w_{i}\]
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\[\left\{\begin{array}{l}\Phi_{q_{x}}=\rho\,u\,\xi_{q}\,,\,\,\Phi_{q_{y}}=\rho \,v\,\xi_{q}\,,\,\,\Phi_{q_{z}}=\rho\,w\,\xi_{q}\,,\,\,\xi_{q}\equiv|{\bf u}|^{ 2}-3\,\lambda^{2}+5\,c_{s}^{2}\\ \Phi_{x_{yz}}=\rho\,u\,(v^{2}-w^{2})\,,\,\,\Phi_{y_{zx}}=\rho\,v\,(w^{2}-u^{2} )\,,\,\,\Phi_{z_{xy}}=\rho\,w\,(u^{2}-v^{2})\\ \Phi_{xyz}=\rho\,u\,v\,w\\ \Phi_{r_{x}}=\rho\,u\,\lambda^{2}\,\big{(}5\,\lambda^{2}-9\,c_{s}^{2}-(u^{2}+3 \,v^{2}+3\,w^{2})\big{)}\\ \Phi_{r_{y}}=\rho\,v\,\lambda^{2}\,\big{(}5\,\lambda^{2}-9\,c_{s}^{2}-(v^{2}+3 \,w^{2}+3\,u^{2})\big{)}\\ \Phi_{r_{z}}=\rho\,w\,\lambda^{2}\,\big{(}5\,\lambda^{2}-9\,c_{s}^{2}-(w^{2}+3 \,u^{2}+3\,v^{2})\big{)}\,.\end{array}\right.\]
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\[\int_{D_{10}\cup D_{8}\cup D_{6}}u_{6}\sqrt{h^{2}+x^{2}+y^{2}}\ \mathrm{d}x \mathrm{d}y =h^{3}\Bigg{(}-\frac{2}{5}\sqrt{5+2\sqrt{5}}\left(I_{00}^{(1)} \left(\frac{1}{2\phi^{2}},\frac{2\pi}{5}\right)-I_{00}^{(1)}\left(\frac{1}{2 \phi^{2}},\frac{\pi}{5}\right)\right)\] \[+2h\left(\!1\!+\!\frac{1}{\sqrt{5}}\!\right)\left(I_{10}^{(1)} \left(\frac{1}{2\phi^{2}},\frac{2\pi}{5}\right)\!-\!I_{10}^{(1)}\left(\frac{1 }{2\phi^{2}},\frac{\pi}{5}\right)\right)\!-\!h^{2}\sqrt{2\!+\!\frac{2}{\sqrt {5}}}\left(I_{20}^{(1)}\left(\frac{1}{2\phi^{2}},\frac{2\pi}{5}\right)\!-\!I_{ 20}^{(1)}\left(\frac{1}{2\phi^{2}},\frac{\pi}{5}\right)\right)\Bigg{)}.\]
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\[\mathbf{D}_{\mathbf{a}_{1}}\cdots\mathbf{D}_{\mathbf{a}_{k}}\left( \tau q\mathfrak{F}^{a}\mathbf{B}_{a}f\right) =\tau q\Big{(}\theta_{\mathbf{a}_{1}}\mathbf{D}_{\mathbf{a}_{2}} \cdots\mathbf{D}_{\mathbf{a}_{k}}-\sum_{2\leq j\leq k}\mathbf{\Gamma}^{ \mathbf{c}}_{\mathbf{a}_{j}\mathbf{a}_{1}}\mathbf{D}_{\mathbf{a}_{2}}\cdots \mathbf{D}_{\mathbf{a}_{c}}\cdots\mathbf{D}_{\mathbf{a}_{k}}\Big{)}\left( \mathfrak{F}^{a}\mathbf{B}_{a}f\right)\] \[=\tau q\theta_{\mathbf{a}_{1}}\cdots\theta_{\mathbf{a}_{k}} \left(\mathfrak{F}^{a}\mathbf{B}_{a}f\right)+\ldots+\tau q(-1)^{k-1}\mathbf{ \Gamma}^{\mathbf{c}_{1}}_{\mathbf{a}_{k}\mathbf{a}_{1}}\mathbf{\Gamma}^{ \mathbf{c}_{2}}_{\mathbf{c}_{1}\mathbf{a}_{2}}\cdots\mathbf{\Gamma}^{ \mathbf{c}_{k-1}}_{\mathbf{c}_{k-2}\mathbf{a}_{k-1}}\left(\mathfrak{F}^{a} \mathbf{B}_{a}f\right)\,,\]
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\[\sum_{l\in\mathbb{L}_{small}^{k}}\frac{\eta_{k}\partial_{f}(\bm{w }_{k,0})^{2}}{2\max_{i\in[n]}|\partial_{f_{i}}(\bm{w}_{k,0})|+\varepsilon}\leq \sum_{l\in\mathbb{L}_{small}^{k}}\frac{\eta_{k}\partial_{f}(\bm{w}_{k,0})^{2}}{2 \max_{i\in[n]}|\partial_{f_{i}}(\bm{w}_{k,0})|+\varepsilon}\leq\frac{n}{2}\eta_{ k}\sum_{l\in[n]}\max_{l\in[n]}|\partial_{f_{i}}(\bm{w}_{k,0})|\] \[\leq \frac{n\eta_{k}}{2}\left(C_{3}\eta_{k}+C_{4}\sum_{r=1}^{k-1}\sqrt {\beta_{2}}^{(r-1)n}\eta_{k-r}\sum_{j=0}^{n-1}\|\nabla f(\bm{w}_{k-r,j})\|+C_{ 4}n\sum_{r=1}^{k-1}\sqrt{\beta_{2}}^{(r-1)n}\eta_{k-r}+\eta_{k}C_{4}\sum_{j=0 }^{n-1}\|\nabla f(\bm{w}_{k,j})\|\right).\]
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\[\text{Vol}(\delta(S^{\prime})) = \prod_{i,j}|x^{\prime}_{j}-x^{\prime}_{i}|\int_{\delta(S^{\prime})} \prod_{i}dx^{\prime}_{i}*\text{Angular part}\] \[= \beta(\frac{\epsilon^{2}n^{\epsilon}}{\ln^{8}n})^{p_{1}n^{\frac{1} {2}-\frac{\epsilon}{2}}}\prod_{i,j}|x_{j}-x_{i}|\int_{\delta(S)}\prod_{i}dx_{i}* \text{Angular part}*\frac{\int_{\delta(S^{\prime})}\prod_{i}dx^{\prime}_{i}}{ \int_{\delta(S)}\prod_{i}dx_{i}}=\] \[= \beta(\frac{\epsilon^{2}n^{\epsilon}}{\ln^{8}n})^{p_{1}n^{\frac{1 }{2}-\frac{\epsilon}{2}}}\text{Vol}(\delta(S))*\left(\frac{\int_{\delta(S^{ \prime})}\prod_{i}dx^{\prime}_{i}}{\int_{\delta(S)}\prod_{i}dx_{i}}\right).\]
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\[\mathrm{e}^{-\lambda(t\wedge\hat{\tau}_{\kappa+c})} (\kappa-X_{t\wedge\hat{\tau}_{\kappa+c}})=\mathrm{e}^{-\lambda(t \wedge\hat{\tau}_{\kappa+c})}\widehat{V}(X_{t\wedge\hat{\tau}_{\kappa+c}})\] \[=\widehat{V}(X_{0})-\int_{0}^{t\wedge\hat{\tau}_{\kappa+c}} \lambda\mathrm{e}^{-\lambda s}\widehat{V}(X_{s})\,\mathrm{d}s+\int_{0}^{t \wedge\hat{\tau}_{\kappa+c}}\mathrm{e}^{-\lambda s}\widehat{V}^{\prime}(X_{s} )\,\mathrm{d}X_{s}\] \[+\int_{0}^{t\wedge\hat{\tau}_{\kappa+c}} \mathrm{e}^{-\lambda s}\frac{1}{2}\widehat{V}^{\prime\prime}(X_{s})\,\mathrm{ d}\langle X\rangle_{s}\] \[=\widehat{V}(X_{0})-\int_{0}^{t\wedge\hat{\tau}_{\kappa+c}} \mathrm{e}^{-\lambda s}\big{(}\mathcal{L}_{X}\widehat{V}(X_{s})-\lambda \widehat{V}(X_{s})\big{)}\,\mathrm{d}s-\int_{0}^{t\wedge\hat{\tau}_{\kappa+c} }\mathrm{e}^{-\lambda s}\sigma X_{s}\,\mathrm{d}B_{s}\] \[=\widehat{V}(X_{0})-\int_{0}^{t\wedge\hat{\tau}_{\kappa+c}} \lambda\mathrm{e}^{-\lambda s}(\kappa-X_{s})\,\mathrm{d}s-\int_{0}^{t\wedge \hat{\tau}_{\kappa+c}}\mathrm{e}^{-\lambda s}\sigma X_{s}\,\mathrm{d}B_{s},\]
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\[\frac{H_{nk+r}\left(e^{-(nk+r)V_{0,k}}\right)}{H_{n}\left(e^{-2 \widetilde{\sigma}nx^{2}}\right)^{k-r}H_{n+1}\left(e^{-2\widetilde{\sigma}(n +1)x^{2}}\right)^{r}}=\frac{D_{1}(1)^{-k+1}}{2^{n^{2}k(k-1)+2nr(k-1)+(r-1)^{2 }}2^{k-1}}\\ \times\exp\left[\frac{1}{2}\left(1-\frac{r}{k}\right)\log\sigma+ (1-r)\log 2+\left(\frac{r}{k}-1\right)\log\left(\sqrt{\sigma}+\sqrt{\sigma-1} \right)\right]\\ \times\frac{\theta((nk+r)\Omega|\tau)}{\theta((nk+r)\Omega+ \Upsilon|\tau)}\left(\frac{2}{1+\sqrt{1-\sigma^{-1}}}\right)^{\frac{k-1}{2}} \left[\gamma_{1}(1)+\gamma_{1}(1)^{-1}\right]^{k-1}\\ \times e^{-\mathcal{Q}_{J_{0,k}}(f)+k\mathcal{Q}_{J_{0,1}}(f)}(1+ \mathcal{O}(1/n)),\]
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\[\sup_{\eta\in B([s,t]\times\Sigma^{h})}I^{[s,t]}(\eta) =\left\|\left[\frac{\mathrm{d}J}{\mathrm{d}\Theta_{\rho}^{+}} \right]^{+}\right\|_{L^{2}([s,t]\times\Sigma^{h},\Theta_{\rho}^{+})}^{2}+\left\| \left[\frac{\mathrm{d}J}{\mathrm{d}\Theta_{\rho}^{-}}\right]^{-}\right\|_{L^{2} ([s,t]\times\Sigma^{h},\Theta_{\rho}^{-})}^{2}\] \[=\int_{s}^{t}\sum_{(K,L)\in\Sigma^{h}}\tau_{K|L}^{h}\Bigg{(}u_{K}( r)\Bigg{|}\frac{j_{K|L}^{+}(r)}{\tau_{K|L}^{h}u_{K}(r)}\Bigg{|}^{2}+u_{L}(r) \Bigg{|}\frac{j_{K|L}^{-}(r)}{\tau_{K|L}^{h}u_{L}(r)}\Bigg{|}^{2}\Bigg{)}\, \mathrm{d}r\] \[=\int_{s}^{t}\mathcal{R}_{\mathrm{up},h}(\rho_{r},j_{r})\, \mathrm{d}r,\]
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\begin{table}
\begin{tabular}{c|c|c|c} \hline \hline \(\varepsilon\) & **IE** & **TRI** & **RE** \\ \hline
0.1 & 1.602E-06 & 7.107E-04 & 2.25E-03 \\ & & **2.09** & \\
0.05 & 1.602E-06 & 1.671E-04 & 9.58E-03 \\ & & **1.90** & \\
0.025 & 1.602E-06 & 4.4716E-05 & 3.58E-02 \\ & & **1.91** & \\
0.0125 & 2.002E-07 & 1.189E-05 & 1.68E-02 \\ & & **1.95** & \\
0.00625 & 2.002E-07 & 3.073E-06 & 6.51E-02 \\ & & **1.92** & \\
0.00313 & 2.002E-07 & 8.117E-07 & 2.46E-01 \\ \hline \hline \end{tabular}
\end{table}
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\[\left(\begin{array}{c}k_{x}^{(0)}\\ k_{\phi}^{(0)}\\ \boldsymbol{q}^{(0)}\\ \boldsymbol{m}^{(0)}\end{array}\right):=\widetilde{\chi}(\mathrm{i}^{-1} \,\partial_{x})\left(\begin{array}{c}\underline{k}_{x}\partial_{x}\varphi_{ x}^{(0)}\\ \underline{k}_{x}\partial_{x}\varphi_{\phi}^{(0)}\\ \delta\mathfrak{G}\left(\underline{\mathcal{U}},\underline{k}_{x}\partial_{ x}\underline{\mathcal{U}}+\underline{k}_{\phi}\mathcal{J}\underline{\mathcal{U}} \right)\widetilde{\mathbf{V}}_{0}-(\mathcal{U}[\underline{\mathcal{U}}]- \underline{m})\,\left(\underline{k}_{\phi}\partial_{x}\varphi_{x}^{(0)}- \underline{k}_{x}\partial_{x}\varphi_{\phi}^{(0)}\right)\\ \delta\mathcal{M}\left[\underline{\mathcal{U}}\right]\widetilde{\mathbf{V}}_{0 }-(\mathcal{M}[\underline{\mathcal{U}}]-\underline{m})\,\partial_{x}\varphi_ {x}^{(0)}\end{array}\right).\]
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\[S_{n_{1}|n_{2}}(f;z)=\frac{1}{n_{1}!n_{2}!}\sum_{k_{1}=1}^{n_{1} }\sum_{k_{2}=1}^{n_{2}}\sum_{m_{1}\geq 0}\sum_{m_{2}\geq 0}\sum_{p\geq 0}\sum_{q=1 }^{p}(-1)^{k_{1}+m_{2}+q}2^{-m_{1}-m_{2}}(k_{1}+k_{2}-1)!\] \[\times\frac{\partial^{m_{1}}B_{n_{1}|k_{1}}(f(z);z)\partial^{m_{2 }}B_{n_{2}|k_{2}}(f(z);z)B_{p|q}(g_{1},...,g_{p-q+1})}{m_{1}!m_{2}!p!(f^{ \prime}(z))^{k_{1}+k_{2}}}\] \[g_{s}=2^{-s-1}(1+(-1)^{s})\frac{\frac{d^{s+1}f}{dz^{s+1}}}{(s+1) f^{\prime}(z)};s=1,...,p-q+1\]
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\[\frac{1}{a!(k-a)!}\frac{|\langle l,m^{(a)}m^{-1}m^{(k-a)}\rangle| }{|\beta\langle l,b\rangle|} \leq\frac{|\beta_{a}|}{a!}\frac{|\beta_{k-a}|}{(k-a)!}\frac{| \langle l,bm^{-1}b\rangle|}{|\beta\langle l,b\rangle|}+\frac{C_{1}^{2}C_{2}^{ k-2}}{a^{\alpha}(k-a)^{\alpha}\rho^{2k-1}(\rho+|\sigma|)^{k}}\frac{\rho^{2}\|l\| \|m^{-1}\|}{|\beta\langle l,b\rangle|}(2\|b\|+\rho)\] \[\leq\frac{C_{1}^{2}C_{2}^{k-2}}{a^{\alpha}(k-a)^{\alpha}\rho^{2k -1}(\rho+|\sigma|)^{k}}\frac{\rho(\rho+|\sigma|)}{|\beta\langle l,b\rangle|} \Big{(}\frac{|\langle l,bm^{-1}b\rangle|}{\rho+|\sigma|}+\|l\|m^{-1}\|(2\|b\|+ \rho)\Big{)}\]
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\[\Delta_{1,\mathbf{F},\sigma}(t,s,\boldsymbol{\theta}_{s,t}( \boldsymbol{\xi}),\mathbf{y},u) = \left\langle\left(\mathbf{F}_{1}(s,\mathbf{y})-\mathbf{F}_{1}(s, \boldsymbol{\theta}_{s,t}(\boldsymbol{\xi}))\right),D_{\mathbf{y}_{1}}u(s, \mathbf{y})\right\rangle\] \[\quad+\frac{1}{2}\mathrm{Tr}\Big{(}\big{(}a(s,\mathbf{y})-a(s, \boldsymbol{\theta}_{s,t}(\boldsymbol{\xi}))\big{)}D_{\mathbf{y}_{1}}^{2}u(s, \mathbf{y})\Big{)},\] \[\Delta_{i,\mathbf{F}}(t,s,\boldsymbol{\theta}_{s,t}(\boldsymbol{ \xi}),\mathbf{y}) = \mathbf{F}_{i}(s,\mathbf{y})-\mathbf{F}_{i}(s,\boldsymbol{\theta}_{s,t}( \boldsymbol{\xi}))-D_{\mathbf{x}_{i-1}}\mathbf{F}_{i}(s,\boldsymbol{\theta}_{s,t}(\boldsymbol{\xi}))(\mathbf{y}-\boldsymbol{\theta}_{s,t}(\boldsymbol{\xi}))_{ i-1}.\]
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\[\mathbb{E}\left(\sup_{t\in[0,T]}|X_{t}^{\varepsilon}-X_{t}|^{2}\right) \leqslant \mathrm{e}^{2\|\nabla b\|_{\infty}T}\mathbb{E}\left(\sup_{t\in[0,T]}\left|\int_{0}^{t}\varepsilon b(s,X_{s-}^{\varepsilon})\mathrm{d} \widetilde{\mathcal{N}}_{s}^{\varepsilon}\right|^{2}\right)\] \[\leqslant 4\mathrm{e}^{2\|\nabla b\|_{\infty}T}\mathbb{E}\left| \int_{0}^{T}\varepsilon b(s,X_{s-}^{\varepsilon})\mathrm{d}\widetilde{ \mathcal{N}}_{s}^{\varepsilon}\right|^{2}\] \[= 4\mathrm{e}^{2\|\nabla b\|_{\infty}T}\mathbb{E}\left(\int_{0}^{ T}|\varepsilon b(s,X_{s}^{\varepsilon})|^{2}\mathrm{d}\left(\frac{\varepsilon}{ \varepsilon}\right)\right)\] \[\leqslant 4\mathrm{e}^{2\|\nabla b\|_{\infty}T}\|b\|_{\infty}^{2}T\varepsilon.\]
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\[\begin{vmatrix}\partial_{1}f&\partial_{2}f&\partial_{3}f\\ \partial_{1}g&\partial_{2}g&\partial_{3}g\\ \partial_{1}h&\partial_{2}h&\partial_{3}h\end{vmatrix}=\partial_{1}f\begin{vmatrix} \partial_{2}g&\partial_{3}g\\ \partial_{2}h&\partial_{3}h\end{vmatrix}-\partial_{2}f\begin{vmatrix}\partial_{ 1}g&\partial_{3}g\\ \partial_{1}h&\partial_{3}h\end{vmatrix}+\partial_{3}f\begin{vmatrix}\partial_{ 1}g&\partial_{2}g\\ \partial_{1}h&\partial_{2}h\end{vmatrix}\] \[=\nabla f\cdot\left(\begin{vmatrix}\partial_{2}g&\partial_{3}g\\ \partial_{2}h&\partial_{3}h\end{vmatrix},-\begin{vmatrix}\partial_{1}g&\partial _{3}g\\ \partial_{1}h&\partial_{3}h\end{vmatrix},\begin{vmatrix}\partial_{1}g&\partial _{2}g\\ \partial_{1}h&\partial_{2}h\end{vmatrix}\right).\]
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\[\kappa_{m}(\{\beta_{q}\}\times A_{p})-\nu_{m}(\{\beta_{q}\}\times A _{p})=\frac{1}{4M}\sum_{k=1}^{M}h_{\mathsf{c}}^{a}(J_{p,q,1,1}^{r})-h_{\mathsf{ c}}^{a}(J_{p,q,1,1}^{l})\] \[-h_{\mathsf{c}}^{a}(J_{p,q,k,1}^{r})-h_{l}^{a}(J_{p,q,k,1}^{r})+ h_{\mathsf{c}}^{a}(J_{p,q,k,1}^{l})+h_{l}^{a}(J_{p,q,k,1}^{l})\] \[=\frac{1}{4M}\sum_{k=1}^{M}h_{l}^{a}(J_{p,q,k,1}^{l})-h_{l}^{a}(J _{p,q,k,1}^{r})+\frac{1}{4M}\sum_{k=2}^{M}h_{\mathsf{c}}^{a}(J_{p,q,1,1}^{r}) -h_{\mathsf{c}}^{a}(J_{p,q,k,1}^{r})\] \[-\frac{1}{4M}\sum_{k=2}^{M}h_{\mathsf{c}}^{a}(J_{p,q,1,1}^{l})-h_ {\mathsf{c}}^{a}(J_{p,q,k,1}^{l})\]
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\[\|\mathcal{Z}(s)\|^{2}= \|\mathcal{Z}(\tau^{\prime})\|^{2}+\int_{\tau^{\prime}}^{s}[ \epsilon^{2}\|\mathcal{Z}\|^{2}-2\|\Delta\mathcal{Z}\|^{2}-4(\Delta\mathcal{Z},\mathcal{Z})-2a\|\mathcal{Z}\|^{2}-2(\mathcal{Z}^{2},G\ast u_{2}^{2})] \mathrm{d}s\] \[-2\int_{\tau^{\prime}}^{s}(\mathcal{Z}u_{1},G\ast(u_{2}^{2}-u_{1 }^{2}))\mathrm{d}\varsigma+2\epsilon\int_{\tau^{\prime}}^{t}\|\mathcal{Z}( \varsigma)\|^{2}\mathrm{d}W_{\varsigma}\] \[\leqslant \|\mathcal{Z}(\tau^{\prime})\|^{2}+\!\!\int_{\tau^{\prime}}^{s}[( \epsilon^{2}+2-2a)\|\mathcal{Z}(\varsigma)\|^{2}\!+\!2\beta\|\mathcal{Z}( \varsigma)\|\|u_{1}\|\|u_{2}^{2}-u_{1}^{2}\|]\mathrm{d}\varsigma+2\epsilon\!\! \int_{\tau^{\prime}}^{s}\|\mathcal{Z}(\varsigma)\|^{2}\mathrm{d}W_{\varsigma}\] \[\leqslant \|\mathcal{Z}(\tau^{\prime})\|^{2}+c\!\!\int_{\tau^{\prime}}^{s}[ \|\mathcal{Z}\|^{2}+\|\mathcal{Z}\|^{2}(\|u_{1}(\varsigma)\|^{2}+\|u_{2}( \varsigma)\|^{2})]\mathrm{d}\varsigma+2\epsilon\!\int_{\tau^{\prime}}^{s}\| \mathcal{Z}(\varsigma)\|^{2}\mathrm{d}W_{\varsigma},\]
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\[r =\frac{1}{k}\left[\omega_{L}\|h[T]\|_{2,1}+(1-\omega_{1})\|h[T \cup\cup_{i=1}^{L}\widetilde{T}_{i}\backslash\cup_{i=1}^{L}(\widetilde{T}_{i} \cap T)]\|_{2,1}\right.\] \[\quad+\sum_{i=2}^{L}(\omega_{i-1}-\omega_{i})\|h[T\cup\cup_{j=i}^ {L}\widetilde{T}_{j}\backslash\cup_{j=i}^{L}(\widetilde{T}_{j}\cap T)]\|_{2,1 }+2\bigg{(}\sum_{i=1}^{L}\omega_{i}\|x[T^{c}]\|_{2,1}\] \[\quad+(1-\sum_{i=1}^{L}\omega_{i})\|x[\widetilde{T}^{c}\cap T^{c }]\|_{2,1}-\sum_{i=1}^{L}(\sum_{j=1}^{L}\omega_{j}-\omega_{i})\|x[\widetilde{ T}_{i}\cap T^{c}]\|_{2,1}\bigg{)}\bigg{]},\]
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\[\left\{\begin{aligned} \dot{\bar{x}}^{1}(t)&=\left(s_{1} \bar{u}^{1}\cos\theta_{1}(0),\ s_{1}\bar{u}^{1}\sin\theta_{1}(0)\right),\\ \dot{\bar{x}}^{2}(t)&=\left(s_{2}\bar{u}^{2}\cos \theta_{1}(0),\ s_{2}\bar{u}^{2}\sin\theta_{1}(0)\right)\ \text{ for }\ t\in[0,t_{1}),\end{aligned}\right.\] \[\left\{\begin{array}{l}\dot{\bar{x}}^{1}(t)=\left(s_{1}\bar{u}^{1}\cos\theta_{1}( t_{1})-\eta^{1}(t),\;s_{1}\bar{u}^{1}\sin\theta_{1}(t_{1})-\eta^{1}(t)\right),\\ \dot{\bar{x}}^{2}(t)=\left(s_{2}\bar{u}^{2}\cos\theta_{1}(t_{1})+\eta^{1}(t), \;s_{2}\bar{u}^{2}\sin\theta_{1}(t_{1})+\eta^{1}(t)\right)\;\;\mbox{for}\;\;t \in[t_{1},T],\end{array}\right.\]
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\[\left\|g-G_{hh}^{-1}(\theta_{0},h_{0})\left[Z_{h}(\theta_{0}+ \zeta,h_{0}+g)-U_{\theta h}(\theta_{0},h_{0})U_{\theta}^{-1}(\theta_{0},h_{0})Z _{\theta}(\theta_{0}+\zeta,h_{0}+g)\right]-(\bar{h}_{\lambda}-h_{0})\right\|_{ \mathcal{H}}\] \[\quad=\left\|G_{hh}^{-1}(\theta_{0},h_{0})\left[R_{h}(\theta_{0}, h_{0})\zeta g-U_{\theta h}(\theta_{0},h_{0})U_{\theta}^{-1}(\theta_{0},h_{0})R_{ \theta}(\theta_{0},h_{0})\zeta g\right]\right\|_{\mathcal{H}}\] \[\qquad\leq\frac{1}{2}\left(K_{h}^{1}\left\|g\right\|_{\mathcal{H} }+K_{h}^{2}\left\|\zeta\right\|_{\mathbb{R}^{p}}\right)\left\|g\right\|_{ \mathcal{H}}+\frac{1}{2}\left(K_{h}^{3}\left\|g\right\|_{\mathcal{H}}+K_{h}^{4 }\left\|\zeta\right\|_{\mathbb{R}^{p}}\right)\left\|\zeta\right\|_{\mathbb{R}^ {p}}.\]
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\[AX^{\circ}_{\infty}+BU^{\circ}_{\infty}+EW=\tilde{A}X^{\circ}_{ \infty}+BF\mathbb{E}[W]+EW\] \[\stackrel{{\mathcal{H}^{\circ}}}{{=}}\tilde{A} \Big{(}(I-\tilde{A})^{-1}\tilde{F}\mathbb{E}[W]\phi^{-2}+\sum_{j=0}^{\infty} \tilde{A}^{j}E\mathbb{w}^{0}\phi^{j}\Big{)}\] \[\qquad\qquad\qquad\qquad+BF\mathbb{E}[W]+E(\mathbb{E}[W]\phi^{-2} +\mathbb{w}^{0}\phi^{w})\] \[\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad+BF \mathbb{E}[W]+E(\mathbb{E}[W]\phi^{-2}+\mathbb{w}^{0}\phi^{w})\] \[\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad=\Big{(}\tilde{A}(I- \tilde{A})^{-1}+I\Big{)}\tilde{F}\mathbb{E}[W]\phi^{-2}+EW^{0}\phi^{w}+ \sum_{j=1}^{\infty}\tilde{A}^{j}E\mathbb{w}^{0}\phi^{j},\] \[\stackrel{{\mathcal{H}^{\circ}}}{{=}}(I-\tilde{A})^{-1}F \mathbb{E}[W]\phi^{-2}+\sum_{j=0}^{\infty}\tilde{A}^{j}E\mathbb{w}^{0}\phi^{j} \stackrel{{\mathcal{H}^{\circ}}}{{=}}X^{\circ}_{\infty},\]
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\begin{table}
\begin{tabular}{|c|c|c|c|c|} \hline \# & \(L_{2}\) & \(\theta(O^{+}(L_{2}))\) & subcase & \(\lambda\) \\ \hline \hline A1 & \(\langle 1,1,2^{4}\rangle\) & \(\{1,2,5,10\}\bar{\mathbb{Q}}_{2}^{2}\) & (b)(iii) & 1 \\ \hline A2 & \(\langle 1,1,2^{5}\rangle\) & \(\{1,2,5,10\}\bar{\mathbb{Q}}_{2}^{2}\) & (b)(iii) & 2 \\ \hline A3 & \(\langle 1,1,2^{6}\rangle\) & \(\{1,2,5,10\}\bar{\mathbb{Q}}_{2}^{2}\) & (b)(iii) & 3 \\ \hline A4 & \(\langle 1,2^{2},2^{4}\rangle\) & \(\{1,5\}\bar{\mathbb{Q}}_{2}^{2}\) & (b)(iii) & 1 \\ \hline A5 & \(\langle 1,2^{4},2^{4}\rangle\) & \(\{1,2,5,10\}\bar{\mathbb{Q}}_{2}^{2}\) & (b)(i) & 1 \\ \hline A6 & \(\langle 5,2^{2},5\cdot 2^{6}\rangle\) & \(\{1,5\}\bar{\mathbb{Q}}_{2}^{2}\) & (b)(ii) & 1 \\ \hline A7 & \(\langle 1,2^{2},2^{6}\rangle\) & \(\{1,5\}\bar{\mathbb{Q}}_{2}^{2}\) & (b)(iii) & 3 \\ \hline A8 & \(\langle 1,2^{3},2^{6}\rangle\) & \(\{1,2,3,6\}\bar{\mathbb{Q}}_{2}^{2}\) & (c)(iii) & 1 \\ \hline A9 & \(\langle 1,2^{4},2^{6}\rangle\) & \(\{1,5\}\bar{\mathbb{Q}}_{2}^{2}\) & (b)(iii) & 3 \\ \hline A10 & \(\langle 1,2^{5},2^{5}\rangle\) & \(\{1,2,5,10\}\bar{\mathbb{Q}}_{2}^{2}\) & (b)(iv) & 1 \\ \hline A11 & \(\langle 5,2^{4},5\cdot 2^{6}\rangle\) & \(\{1,5\}\bar{\mathbb{Q}}_{2}^{2}\) & (b)(ii) & 3 \\ \hline A12 & \(\langle 1,2^{4},2^{8}\rangle\) & \(\{1,5\}\bar{\mathbb{Q}}_{2}^{2}\) & (b)(iii) & 5 \\ \hline A13 & \(\langle 1,2^{6},2^{6}\rangle\) & \(\{1,2,5,10\}\bar{\mathbb{Q}}_{2}^{2}\) & (b)(i) & 3 \\ \hline \end{tabular}
\end{table}
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\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Data** & \multicolumn{2}{c}{**FT**} & \multicolumn{2}{c}{**MACE**} & \multicolumn{2}{c}{**OAE**} & \multicolumn{2}{c}{**OCEAN**} \\ & T(s) & R & T(s) & R & T(s) & R & T(s) & R \\ \hline
**AD** & 3.03 & 15.9 & 20.60 & 1.1 & 28.37 & 1.0 & 1.22 & 1.0 \\
**CC** & 29.44 & 10.2 & 41.25 & 1.2 & 5.52 & 1.0 & 1.34 & 1.0 \\
**CP** & 22.68 & 4.5 & 15.82 & 1.0 & 0.38 & 1.0 & 0.52 & 1.0 \\
**GC** & 16.26 & 4.8 & 19.03 & 1.0 & 5.08 & 1.0 & 1.16 & 1.0 \\
**ON** & 10.05 & 31.7 & \(>\)900 & — & \(>\)900 & — & 2.97 & 1.0 \\
**PH** & 10.95 & 1.4 & \(>\)900 & — & 0.94 & 1.0 & 0.52 & 1.0 \\
**SP** & NA & — & \(>\)900 & — & \(>\)900 & — & 2.73 & 1.0 \\
**ST** & NA & — & \(>\)900 & — & 69.64 & 1.0 & 1.10 & 1.0 \\ \hline \hline \end{tabular}
\end{table}
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\[\frac{1}{2}\frac{d}{dt}||\xi_{u}||^{2}+||\xi_{p}||^{2} =\sum_{i=1}^{N}\int_{I_{i}}(\eta_{p}-\xi_{p})(\xi_{u})_{x}dx+\sum _{i=2}^{N}\left(\eta_{p_{i-\frac{1}{2}}}-\xi_{p_{i-\frac{1}{2}}}+\alpha_{i- \frac{1}{2}}\frac{\left[\eta_{u}-\xi_{u}\right]_{i-\frac{1}{2}}}{\Delta\tilde {x}_{i-\frac{1}{2}}}\right)\left[\xi_{u}\right]_{i-\frac{1}{2}}\] \[\quad-\sum_{i=1}^{N}\int_{I_{i}}(\eta_{u}-\xi_{u})(\xi_{p})_{x}dx -\sum_{i=2}^{N}[\eta_{u}-\xi_{u}]_{i-\frac{1}{2}}\xi_{p}(x_{i-\frac{1}{2}})\] \[=\sum_{i=1}^{N}\int_{I_{i}}\eta_{p}(\xi_{u})_{x}dx+\sum_{i=2}^{N} \left(\eta_{p_{i-\frac{1}{2}}}+\alpha_{i-\frac{1}{2}}\frac{\left[\eta_{u} \right]_{i-\frac{1}{2}}}{\Delta\tilde{x}_{i-\frac{1}{2}}}\right)\left[\xi_{u} \right]_{i-\frac{1}{2}}\] \[\quad-\sum_{i=1}^{N}\int_{I_{i}}\eta_{u}(\xi_{p})_{x}dx-\sum_{i=2 }^{N}[\eta_{u}]_{i-\frac{1}{2}}\xi_{p}(x_{i-\frac{1}{2}})+H_{u}^{N}(\xi_{u}, \xi_{p},\xi_{u})+H_{p}^{N}(\xi_{u},\xi_{p})\] \[=R_{1}+R_{2}+R_{3},\]
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\[f(\bm{U}^{\prime})=\frac{1}{m}\left\|\mathcal{A}(\bm{U}^{\prime }\bm{U}^{\prime\mathrm{T}}-\bm{X}^{\star})-\bm{s}^{\star}\right\|_{1}\] \[=\frac{1}{m}\left\|\mathcal{A}(\bm{U}\bm{U}^{\mathrm{T}}-\bm{X}^{ \star}+\bm{U}\bm{\Delta}^{\mathrm{T}}+\bm{\Delta}\bm{U}^{\mathrm{T}}+\bm{ \Delta}\bm{\Delta}^{\mathrm{T}})-\bm{s}^{\star}\right\|_{1}\] \[\geq\frac{1}{m}\left\|\mathcal{A}(\bm{U}\bm{U}^{\mathrm{T}}-\bm{X }^{\star}+\bm{U}\bm{\Delta}^{\mathrm{T}}+\bm{\Delta}\bm{U}^{\mathrm{T}})-\bm{s }^{\star}\right\|_{1}-\frac{1}{m}\left\|\mathcal{A}(\bm{\Delta}\bm{\Delta}^{ \mathrm{T}})\right\|_{1}\] \[\geq\frac{1}{m}\left\|\mathcal{A}(\bm{U}\bm{U}^{\mathrm{T}}-\bm{X }^{\star}+\bm{U}\bm{\Delta}^{\mathrm{T}}+\bm{\Delta}\bm{U}^{\mathrm{T}})-\bm{s }^{\star}\right\|_{1}-\left(\sqrt{\frac{2}{\pi}}+\delta\right)\left\|\bm{ \Delta}\bm{\Delta}^{\mathrm{T}}\right\|_{F}\] \[\geq f(\bm{U})+\frac{1}{m}\left\langle\bm{d},\mathcal{A}(\bm{U} \bm{\Delta}^{\mathrm{T}}+\bm{\Delta}\bm{U}^{\mathrm{T}})\right\rangle-\frac{ \tau}{2}\|\bm{\Delta}\|_{F}^{2}\]
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\[\left\|\mathbf{P}_{n}\left(T_{u^{k-1}}u-\sum_{j}\mathbf{P}_{j}u(S _{j+1}u)^{k-1}\right)\right\|_{L_{x}^{1}}\] \[= \left\|\mathbf{P}_{n}\sum_{j\geq n-3}\mathbf{P}_{j}u\left(S_{j-1} u^{k-1}-(S_{j+1}u)^{k-1}\right)\right\|_{L_{x}^{1}}\] \[\lesssim \sum_{j\geq n-3}\left\|\mathbf{P}_{j}u\right\|_{L_{x}^{p}}\left\| S_{j-1}(u^{k-1}-(S_{j-k}u)^{k-1})+\left((S_{j-k}u)^{k-1}-(S_{j+1}u)^{k-1} \right)\right\|_{L_{x}^{q}}\] \[\lesssim \sum_{j\geq n-3}\left\|\mathbf{P}_{j}u\right\|_{L_{x}^{p}}\left( \left\|u^{k-1}-(S_{j-k}u)^{k-1}\right\|_{L_{x}^{q}}+\left\|(S_{j-k}u)^{k-1}-(S _{j+1}u)^{k-1}\right\|_{L_{x}^{q}}\right).\]
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\[C_{2}k^{2}\|u\|_{L^{k+1-m}(\Omega)}^{k+1-m}\] \[= C_{2}k^{2}\|u^{\frac{k+m-1}{2}}\|_{L^{\frac{2(k+1-m)}{k+m-1}}( \Omega)}^{\frac{2Nk(2-2m-2)}{k+m-1}}\|u^{\frac{k+m-1}{2}}\|_{L^{\frac{k}{k+m-1 }}(\Omega)}^{\frac{2(k+1-m)}{k+m-1}-\frac{2N(k-2m-2)}{(N+2k+2N(m-1)}}+C_{5}k^ {2}\|u^{\frac{k+m-1}{2}}\|_{L^{\frac{k}{k+m-1}}(\Omega)}^{\frac{2(k-1+m)}{k+m- 1}}\] \[\leq \delta\|\nabla u^{\frac{k+m-1}{2}}\|_{L^{2}(\Omega)}^{2}+C_{6}k^ {\frac{(N+2)k+2N(m-1)}{k+2N(m-1)}}\|u^{\frac{k+m-1}{2}}\|_{L^{\frac{k}{k+m-1}} (\Omega)}^{\frac{k(k+2(m-1))}{(k+m-1)(\frac{k}{2}+N(m-1))}}+C_{5}k^{2}\|u^{ \frac{k+m-1}{2}}\|_{L^{\frac{k+m-1}{k+m-1}}(\Omega)}^{\frac{2(k+1+m)}{k+m-1}}\] \[\leq \delta\|\nabla u^{\frac{k+m-1}{2}}\|_{L^{2}(\Omega)}^{2}+C_{6}k^ {N+2}\|u\|_{L^{\frac{k}{2}}(\Omega)}^{\frac{k(k+2(m-1))}{k+2N(m-1)}}+C_{5}k^{2} \|u\|_{L^{\frac{k}{k}}(\Omega)}^{k+1-m}.\]
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\[\begin{split} F_{k}\stackrel{{\text{def}}}{{=}}& -\partial_{t}A_{k}-\sum_{\ell=0}^{k}F_{1,\ell}-\sum_{\ell=2}^{k+1}F_{2, \ell}-\sum_{\ell=1}^{k}F_{3,\ell}-\sum_{\ell=1}^{k+1}F_{4,\ell}+\sum_{\ell_{1 }+\ell_{2}=k-1}F_{5,\ell_{1},\ell_{2}}\\ &-\sum_{\begin{subarray}{c}\ell_{1}+\ell_{2}+j=k\\ 2\leq j\leq k\end{subarray}}F_{6,\ell_{1},\ell_{2},j}-\sum_{\begin{subarray}{ c}\ell_{1}+\ell_{2}+j=k+1\\ 2\leq j\leq k+1\end{subarray}}F_{7,\ell_{1},\ell_{2},j}-\sum_{\begin{subarray}{ c}\ell_{1}+\ell_{2}+j=k+2\\ 2\leq j\leq k+2\end{subarray}}F_{8,\ell_{1},\ell_{2},j}\end{split}\]
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\[\frac{1}{|V|}\left(\int_{V}u(\cdot,t)d\mu\right)^{2} = \frac{1}{|V|}\left(\int_{A_{t}}u(\cdot,t)d\mu+\int_{V\setminus A_ {t}}u(\cdot,t)d\mu\right)^{2}\] \[= \frac{1}{|V|}\left(\int_{A_{t}}u(\cdot,t)d\mu\right)^{2}+\frac{1} {|V|}\left(\int_{V\setminus A_{t}}u(\cdot,t)d\mu\right)^{2}\] \[+\frac{2}{|V|}\int_{A_{t}}u(\cdot,t)d\mu\int_{V\setminus A_{t}}u (\cdot,t)d\mu\] \[\leq \frac{C^{2}|A_{t}|^{2}}{|V|}+\frac{1}{|V|}\left(\int_{V\setminus A _{t}}u(\cdot,t)d\mu\right)^{2}\] \[+\frac{C^{2}|A_{t}|^{2}}{\epsilon|V|}+\frac{\epsilon}{|V|}\left( \int_{V\setminus A_{t}}u(\cdot,t)d\mu\right)^{2},\]
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\[\frac{1-m}{p}C_{\psi}=\frac{2m(1-m)}{p-1}\left[\int_{A_{R_{0},R_{ 1}}}\frac{|\nabla\psi|^{\frac{2p}{1-m}}}{\psi^{\frac{2(p+m-1)}{1-m}}}|x|^{ \gamma\frac{p+m-1}{1-m}-\beta\frac{p}{1-m}}\,\mathrm{d}x\right]^{\frac{1-m}{p}}\] \[\leq\left[\frac{C_{2}^{\frac{p}{1-m}}\,c_{0,p}^{\frac{p}{1-m}}}{ \left(R_{0}-R_{1}\right)^{\frac{2p}{1-m}}}\int_{A_{R_{0},R_{1}}}|x|^{(\gamma- \beta)\frac{p}{(1-m)}}\frac{\mathrm{d}x}{|x|^{\gamma}}\right]^{\frac{1-m}{p}} \leq c_{p}\frac{h_{\sigma}(R_{0},R_{1},x_{0})}{\left(R_{0}-R_{1}\right)^{\sigma }}\mu_{\gamma}\left(A_{R_{0},R_{1}}\right)^{\frac{1-m}{p}}:=K_{R_{0},R,p, \sigma,x_{0}}\,,\]
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\[W(g_{t,l,v})=\begin{cases}-\zeta_{F}(2)^{-1}\zeta_{F}(1)q^{-1-\frac{a(\chi)}{2 }-t}\chi(v^{-1})\epsilon(\frac{1}{2},\chi^{-1})&\text{ if }t>-2,\\ q\zeta_{F}(1)^{-2}K(\chi\circ N_{\mathrm{E/F}},(\varpi^{-1},\varpi^{-1}),v \varpi^{-l})&\text{ if }t=-2\text{ and }l=1,\\ \chi(v^{-1})\epsilon(\frac{1}{2},\chi^{-1})\zeta_{F}(1)^{-1}q^{1-\frac{a(\chi )}{2}}S(1,-b_{\chi}v^{-1},1)&\text{ if }t=-2\text{ and }l>1,\\ q^{-\frac{l}{2}}\zeta_{F}(1)^{-2}K(\chi\circ N_{\mathrm{E/F}},(\varpi^{\frac{l }{2}},\varpi^{\frac{l}{2}}),v\varpi^{-l})&\text{ if }-2l\leq t<-2\text{ even},\\ 0&\text{ else.}\end{cases}\]
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\[2VD_{s}\Psi^{*} =2b_{0}^{*}D_{s}\Psi^{*}+2b_{1}^{*}(z-z_{*})^{*}(b_{1t}-b_{0}^{*} D_{s}^{2}U^{*})+\mathcal{L}^{1}_{2,\beta-1}(m)\] \[\Psi D_{s}V^{*} =-c_{1}^{*}b_{1}-c_{1}^{*}D_{s}U^{*}-b_{0}b_{1}(D_{s}V^{*})^{*}-b_ {0}|D_{s}V^{*}|^{2}+b_{1}b_{1t}^{*}(z-z_{*})^{*}+\mathcal{L}^{1}_{2,\beta-1}(m)\] \[V^{*}|D_{s}V^{*}|^{2} =b_{0}|D_{s}V^{*}|^{2}+b_{1}|b_{1}|^{2}(z-z_{*})+\mathcal{L}^{1}_ {2,\beta-1}(m)\] \[V^{2}D_{s}^{2}V^{*} =(b_{0}^{*})^{2}D_{s}^{2}U^{*}+2b_{0}^{*}b_{1t}^{*}(z-z_{*})^{*}D_ {s}^{2}U^{*}+\mathcal{L}^{1}_{2,\beta-1}(m).\]
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\[\mathbb{E}_{\lambda,\mu}\Big{[}\prod_{j=1}^{r_{1}}Z_{n,k_{j},l_{j} }^{m_{j}}\Big{]}\mathbb{E}_{\lambda,\mu}\Big{[}\prod_{j=r_{1}+1}^{r}(Z_{n,k_{j },l_{j}})^{m_{j}}\Big{]}=\frac{1}{n^{\sum_{j=r_{1}+1}^{r_{1}}l_{j}m_{j}}}\times\] \[\mathbb{E}_{\sigma}\Big{[}\sum_{j\in[r],1\leq q_{j}\leq m_{j}, \omega_{j,q_{j}}}\mathbb{E}_{\lambda,\mu}\Big{[}\Big{(}\prod_{j=1}^{r_{1}} \prod_{q_{j}\leq m_{j}}\prod_{e\in E_{\omega_{j,q_{j}},A}}A_{e}\Big{)}\mid \sigma\Big{]}\mathbb{E}_{\lambda,\mu}\Big{[}\Big{(}\prod_{j=r_{1}+1}^{r}\prod_ {q_{j}\leq m_{j}}\prod_{e\in E_{\omega_{j,q_{j}},A}}A_{e}\Big{)}\mid\sigma\Big{]}\times\] \[\mathbb{E}_{\lambda,\mu}\Big{[}\prod_{j=r_{1}+1}^{r}\prod_{q_{j} \leq m_{j}}\prod_{e\in E_{\omega_{j,q_{j}},B}}B_{e}\mid\sigma\Big{]}\Big{]}.\]
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\[\int_{Q}\Biggl{(}\frac{1}{s^{2}}\left(|\partial_{t}u|^{2}+|\Delta u |^{2}\right)+\frac{1}{s}\left(|\partial_{t}H|^{2}+|\Delta H|^{2}\right)+| \nabla u|^{2}+s|\nabla H|^{2}\] \[+s^{2}|u|^{2}+s^{3}|H|^{2}+\frac{1}{s}|\nabla p|^{2}+s|p|^{2} \Biggr{)}e^{2s\varphi}dxdt\] \[\leq C\int_{Q}\left(|F|^{2}+|G|^{2}+|\nabla_{x,t}h|^{2}\right)e^{2 s\varphi}dxdt\] \[+Cs^{3}\int_{\partial\Omega\times I}(|u|^{2}+|H|^{2}+|\nabla_{x,t }u|^{2}+|\nabla_{x,t}H|^{2})e^{2s\varphi}dSdt\] \[+Cs^{3}\int_{\Omega}(|u|^{2}+|H|^{2}+|\nabla u|^{2}+|\nabla H|^{2 })e^{2s\varphi}dx\Big{|}_{t=t_{0}\pm\delta}\]
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\begin{table}
\begin{tabular}{c l} \hline \(x^{t},\hat{x}^{t}\) & server, shared hidden state at time \(t\) \\ \(L\) & L-smoothness constant of the loss function \\ \(P,p\) & number, index of local steps at client \\ \(K,k\) & number, index of clients at the buffer \\ \(N,n\) & number, index of total clients \\ \(\eta_{g},\eta_{\ell}^{(p)}\) & server, client (at step \(p\)) learning rates \\ \(Q_{s},Q_{c}\) & server, client quantizers \\ \(\overline{\Delta}^{t},\Delta_{k}^{t}\) & server, client \(k\)’s update at time \(t\) \\ \(\mathcal{S}^{t}\) & set of client indices at the buffer at time \(t\) \\ \(y_{k,p}^{t}\) & local state at client \(k\), during local step \(p\) at time \(t\) \\ \(\pm\) & plus and minus the same quantity, i.e., \(a\pm b=a+b-b=a\) \\ \hline \end{tabular}
\end{table}
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\[b_{n-1,n-2}^{(n-2)} =\left|\begin{smallmatrix}0&-b_{n-1,n-3}^{{}^{\prime}(n-3)}&b_{n-2,n-3}^{{}^{\prime}(n-3)}\\ b_{n-2,n-3}^{{}^{\prime}(n-3)}&b_{n-2,n-2}^{(n-3)}&b_{n-2,n-1}^{(n-3)}\\ \rho_{n-1,n-3}^{{}^{\prime}(n-3)}&b_{n-1,n-2}^{(n-3)}&b_{n-1,n-1}^{(n-3)}\\ \end{smallmatrix}\right|\] \[=\frac{1}{e_{n-3}^{2}}\left|\begin{smallmatrix}0&-b_{n-1,n-3}^{{} ^{\prime}(n-3)}&b_{n-2,n-3}^{{}^{\prime}(n-3)}\\ b_{n-2,n-3}^{{}^{\prime}(n-3)}&b_{n-2,n-2}^{(n-3)}&b_{n-2,n-1}^{(n-3)}\\ b_{n-1,n-3}^{{}^{\prime}(n-3)}&b_{n-1,n-2}^{(n-3)}&b_{n-1,n-1}^{(n-3)}\end{smallmatrix}\right|.\]
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\[\mathbb{E}\Big{[} \operatorname{DTV}\left(f(x_{1:n});\,\tilde{w}\right)\Big{]}\] \[\leq p_{\max}^{2}n^{2}\int_{\Omega}\int_{\Omega}|f(y)-f(x)|\Bigg{(} c_{0}n^{-(d-1)/d}\mathbb{P}_{3:n}\{\mathcal{H}(\bar{V}_{x}\cap\bar{V}_{y})>0\}\] \[\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+C_{1} \frac{1\{\|x-y\|\leq C_{2}(\log n/n)^{1/d}\}}{n^{(d-1)/d}}+\frac{C_{1}}{n^{2} }\Bigg{)}\;dy\;dx\] \[\leq p_{\max}^{2}c_{0}n^{-(d-1)/d}\mathbb{E}\left[ \operatorname{DTV}(f(x_{1:n});\tilde{w}^{\mathrm{V}})\right]+p_{\max}^{2}C_{1 }n^{-(d-1)/d}\mathbb{E}\left[\operatorname{DTV}(f(x_{1:n});w^{e\gets C_{2} (\log n/n)^{1/d}})\right]\] \[\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad+p_{\max}^{2}C_{1}\int_{\Omega}\int_{\Omega}|f(y)-f(x)|\;dy \;dx\] \[=T_{1}+T_{2}+T_{3}.\]
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\[D\Big{(}\frac{p}{q}\Big{)}(T) =\Big{(}\frac{r}{t}\Big{)}(\overline{T})^{-1}\Big{(}D\Big{(}\frac {rp}{tq}\Big{)}(T)-\Big{(}\frac{p}{q}\Big{)}(T)D\Big{(}\frac{r}{t}\Big{)}(T) \Big{)}\] \[=\big{(}r[\overline{T}]t[\overline{T}]^{-1}\big{)}^{-1}\Big{(}Dr[ T]p[T]t[T]q[T]+r[\overline{T}]Dp[T]t[T]q[T]\] \[\qquad\qquad\qquad\qquad\qquad\qquad-r[T]p[T]t[T]Dq[T]-Dr[T]p[T]t[T]q[ \overline{T}]\Big{)}(tq)[T]^{-1}(tq)[\overline{T}]^{-1}\] \[=r[\overline{T}]^{-1}\Big{(}Dr[T]p[T]q[T]+r[\overline{T}]Dp[T]q[T]\] \[\qquad\qquad\qquad\qquad\qquad\qquad-r[T]p[T]Dq[T]-Dr[T]p[T]q[ \overline{T}]\Big{)}q[T]^{-1}q[\overline{T}]^{-1}\] \[=\big{(}Dp[T]q[T]-Dq[T]p[T]\big{)}q[T]^{-1}q[\overline{T}]^{-1},\]
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\[\mathcal{D}_{1}=\mathcal{A}_{1}=\mathcal{R}_{1}=\{\mathcal{B}_{1},\mathcal{B}_{5},\mathcal{B}_{7}\},\] \[\mathcal{D}_{2}=\mathcal{A}_{2}=\mathcal{R}_{2}=\{\mathcal{B}_{1},\mathcal{B}_{2},\mathcal{B}_{6}\},\] \[\mathcal{D}_{3}=\mathcal{A}_{3}=\mathcal{R}_{3}=\{\mathcal{B}_{2},\mathcal{B}_{3},\mathcal{B}_{7}\},\] \[\mathcal{D}_{4}=\mathcal{A}_{4}=\mathcal{R}_{4}=\{\mathcal{B}_{1},\mathcal{B}_{3},\mathcal{B}_{4}\},\] \[\mathcal{D}_{5}=\mathcal{A}_{5}=\mathcal{R}_{5}=\{\mathcal{B}_{2},\mathcal{B}_{4},\mathcal{B}_{5}\},\] \[\mathcal{D}_{6}=\mathcal{A}_{6}=\mathcal{R}_{6}=\{\mathcal{B}_{3},\mathcal{B}_{5},\mathcal{B}_{6}\},\] \[\mathcal{D}_{7}=\mathcal{A}_{7}=\mathcal{R}_{7}=\{\mathcal{B}_{4},\mathcal{B}_{6},\mathcal{B}_{7}\}.\]
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\[e^{x}H_{1,2}^{1,1}\left[-x^{3}\Bigg{|}\begin{array}{c}(0,1)\\ (0,1),(0,3)\end{array}\right] = e^{x}{}_{0}F_{2}\left[\begin{array}{c}-\\ \frac{1}{3},\frac{2}{3}\end{array};\frac{x^{3}}{27}\right]\] \[= \frac{2}{3}\sum_{m=0}^{\infty}\frac{x^{m}}{m!}\left[2^{m-1}+cos \left(\frac{m\pi}{3}\right)\right]\] \[= \frac{e^{2x}}{3}+\frac{2\pi}{3}H_{1,2}^{1,0}\left[-x\Bigg{|} \begin{array}{c}(\frac{1}{2},\frac{1}{3})\\ (0,1),(\frac{1}{2},\frac{1}{3})\end{array}\right]\] \[= \frac{e^{2x}}{3}+\frac{2}{3}e^{\frac{\pi}{2}}cos\left(\frac{\sqrt{3 }x}{2}\right)\]
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\[I_{T,t}^{*(i_{1}i_{2}i_{3}i_{4})}=\underset{p\to\infty}{\text{l.i.m.}}\ \int\limits_{t}^{*\,T}\ \int\limits_{t}^{*\,t_{4}}\ \int\limits_{t}^{*\,t_{3}}\ \int\limits_{t}^{*\,t_{2}}d\mathbf{w}_{t_{1}}^{(i_{1})}d\mathbf{w}_{t_{2}}^{ (i_{2})}d\mathbf{w}_{t_{3}}^{(i_{3})}d\mathbf{w}_{t_{4}}^{(i_{4})}\] \[I_{T,t}^{*(i_{1}i_{2}i_{3}i_{4})}=\underset{p\rightarrow\infty}{\text{l.i.m.}}\ \int\limits_{t}^{T}\int\limits_{t}^{t_{4}}\int\limits_{t}^{t_{3}}\int\limits_{t}^{t _{2}}d{\bf w}_{t_{1}}^{(i_{1})p}d{\bf w}_{t_{2}}^{(i_{2})p}d{\bf w}_{t_{3}}^{(i _{3})p}d{\bf w}_{t_{4}}^{(i_{4})p}\]
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\[\begin{cases}9b_{1}^{2}c_{0}^{2}+6b_{1}^{2}c_{0}+9b_{1}^{2}c_{1}^{2}+b_{1}^{2 }+9b_{1}c_{0}^{2}c_{1}+6b_{1}c_{0}c_{1}+9b_{1}c_{1}^{3}+b_{1}c_{1}+9b_{2}^{2}c _{1}^{2}+9b_{3}^{2}c_{1}^{2}\\ +9c_{0}^{2}c_{1}^{2}+6c_{0}c_{1}^{2}+9c_{1}^{2}+7c_{1}^{2}=0\\ 2b_{1}+7c_{1}+3b_{0}b_{1}+9b_{1}c_{0}+6c_{0}c_{1}+9b_{1}c_{0}^{2}+9b_{1}c_{1} ^{2}+9b_{1}^{2}c_{1}+9b_{2}^{2}c_{1}+9b_{3}^{2}c_{1}\\ +9c_{0}^{2}c_{1}+9c_{1}^{3}+9b_{0}b_{1}c_{0}=0\\ 9b_{2}^{3}+9b_{2}b_{3}^{2}+9b_{2}c_{0}^{2}+6b_{2}c_{0}+9b_{2}c_{1}^{2}+7b_{2} =0\\ 9b_{2}^{2}b_{3}+9b_{3}^{3}+9b_{3}c_{0}^{2}+6b_{3}c_{0}+9b_{3}c_{1}^{2}+7b_{3} =0\\ b_{0}^{2}+b_{0}c_{0}+b_{0}+b_{1}^{2}+b_{1}c_{1}+b_{2}^{2}+b_{3}^{2}+c_{0}^{2}+ c_{0}+c_{1}^{2}+1=0\\ 2b_{2}+3b_{0}b_{2}+3b_{2}c_{0}=0\\ 2b_{3}+3b_{0}b_{3}+3b_{3}c_{0}=0\\ c_{1}-b_{1}+3b_{0}c_{1}-3b_{1}c_{0}=0\\ b_{1}b_{2}+b_{2}c_{1}=0\\ b_{1}b_{3}+b_{3}c_{1}=0\\ a_{0}+b_{0}+c_{0}+1=0\\ a_{1}+b_{1}+c_{1}=0\end{cases}\]
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\[I_{3,\varepsilon}= \int_{\partial B_{\varepsilon}}\frac{y}{\varepsilon}\left(\frac{ \partial\big{(}H_{B_{\varepsilon}^{c}}(x,y)-H_{\Omega_{\varepsilon}}(x,y) \big{)}}{\partial\nu_{y}}\right)^{2}d\sigma_{y}=\int_{\partial B_{\varepsilon }}\frac{y}{\varepsilon}\left(\phi_{\varepsilon}(x,y)\cdot\nu_{y}+O\Big{(} \frac{1}{|\ln\varepsilon|}\Big{)}\right)^{2}d\sigma_{y}\] \[= \int_{\partial B_{\varepsilon}}\frac{y}{\varepsilon}\left(\phi_{ \varepsilon}(x,y)\cdot\nu_{y}\right)^{2}d\sigma_{y}+O\Big{(}\frac{1}{|\ln \varepsilon|}\Big{)}\underbrace{\int_{\partial B_{\varepsilon}}|\phi_{ \varepsilon}(x,y)|d\sigma_{y}}_{=O(1)}+O\Big{(}\frac{\varepsilon}{(\ln \varepsilon)^{2}}\Big{)}\] \[= \int_{\partial B_{\varepsilon}}\frac{y}{\varepsilon}\left(\phi_{ \varepsilon}(x,y)\cdot\nu_{y}\right)^{2}d\sigma_{y}+O\Big{(}\frac{1}{|\ln \varepsilon|}\Big{)}=\frac{1}{4\pi^{2}\varepsilon^{2}}\left(\frac{G_{\Omega} (x,0)}{-\frac{1}{2\pi}\ln\varepsilon-H_{\Omega}(0,0)}-1\right)^{2} \underbrace{\int_{\partial B_{\varepsilon}}\frac{y}{\varepsilon}d\sigma_{y}} _{=0}\] \[+\frac{2}{\pi\varepsilon^{2}}\left(1-\frac{\ln|x|}{\ln\varepsilon} +O\Big{(}\frac{1}{|\ln\varepsilon|}\Big{)}\right)\int_{\partial B_{ \varepsilon}}\frac{y}{\varepsilon}\left[\nabla_{y}H_{\Omega}(x,0)\cdot y- \nabla_{y}H_{\Omega}(0,0)\cdot y\frac{\ln|x|}{\ln\varepsilon}\right]d\sigma_ {y}\] \[+\underbrace{\int_{\partial B_{\varepsilon}}\frac{|y|}{\varepsilon }O(1)d\sigma_{y}}_{=O(\varepsilon)}+O\Big{(}\frac{1}{|\ln\varepsilon|}\Big{)}\] \[= \frac{2}{\pi\varepsilon^{3}}\left(1-\frac{\ln|x|}{\ln\varepsilon} +O\Big{(}\frac{1}{|\ln\varepsilon|}\Big{)}\right)\sum_{j=1}^{2}\left(\frac{ \partial H_{\Omega}(x,0)}{\partial y_{j}}-\frac{\partial H_{\Omega}(0,0)}{ \partial y_{j}}\frac{\ln|x|}{\ln\varepsilon}\right)\underbrace{\int_{ \partial B_{\varepsilon}}yy_{j}d\sigma_{y}}_{=\pi\varepsilon^{3}\delta_{j}^{ i}}+O\Big{(}\frac{1}{|\ln\varepsilon|}\Big{)}\] \[= 2\left(1-\frac{\ln|x|}{\ln\varepsilon}\right)\left(\nabla_{y}H_{ \Omega}(x,0)-\nabla_{y}H_{\Omega}(0,0)\frac{\ln|x|}{\ln\varepsilon}\right)+O \Big{(}\frac{1}{|\ln\varepsilon|}\Big{)}\] \[= 2\nabla_{y}H_{\Omega}(x,0)+O\left(\left|\frac{\ln|x|}{\ln\varepsilon }\right|\right)+O\Big{(}\frac{1}{|\ln\varepsilon|}\Big{)}.\]
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\[\bigg{|}\int_{0}^{\infty}\int_{t}^{\infty}(O(\alpha t)x-y)_{1} \big{(}(\cos\alpha t)A_{1}+(\sin\alpha t)A_{2}\big{)}G(O(\alpha t)x-y,s) \frac{\mathrm{d}s}{8s^{3}}\,\mathrm{d}t-\partial_{y_{1}}L^{112}(x,y)\] \[=\,\bigg{|}\int_{0}^{\infty}\int_{t}^{\infty}\bigg{(}(\cos 2\alpha t )D_{1}+(\sin 2\alpha t)D_{2}+D_{3}\bigg{)}\,G(O(\alpha t)x-y,s)\frac{\mathrm{d}s }{8s^{3}}\,\mathrm{d}t\bigg{|}\] \[\leq\bigg{|}\int_{0}^{\infty}\int_{t}^{\infty}\bigg{(}(\cos 2 \alpha t)D_{1}+(\sin 2\alpha t)D_{2}+D_{3}\bigg{)}\,\bigg{(}G(O(\alpha t)x-y,s)-G(x,s) \bigg{)}\frac{\mathrm{d}s}{8s^{3}}\,\mathrm{d}t\bigg{|}\] \[\quad+\bigg{|}\int_{0}^{\infty}\int_{t}^{\infty}\bigg{(}(\cos 2 \alpha t)D_{1}+(\sin 2\alpha t)D_{2}+D_{3}\bigg{)}\,G(x,s)\frac{\mathrm{d}s}{8s^{3}} \,\mathrm{d}t\bigg{|}\] \[\leq C\frac{|y|}{|x|^{2}}+C\min\left\{\frac{1}{|\alpha||x|^{3}}, \frac{1}{|x|}\right\},\qquad|x|>2|y|\,.\]
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\[C_{j^{{}^{\prime\prime}}}(z)\rightarrow(A\times B)_{\psi(j^{{}^{ \prime\prime}})}(\phi(z)) = C_{j^{{}^{\prime\prime}}}(z)\rightarrow(A\times B)_{(\psi_{1}(j^{{ }^{\prime\prime}}),\psi_{2}(j^{{}^{\prime\prime}}))}(\phi_{1}(z),\phi_{2}(z))\] \[= C_{j^{{}^{\prime\prime}}}(z)\rightarrow(A_{\psi_{1}(j^{{}^{ \prime\prime}})}\times B_{\psi_{2}(j^{{}^{\prime\prime}})})(\phi_{1}(z),\phi_{2} (z))\] \[= C_{j^{{}^{\prime\prime}}}(z)\to A_{\psi_{1}(j^{{}^{\prime\prime}} )}(\phi_{1}(z))\wedge B_{\psi_{2}(j^{{}^{\prime\prime}})}(\phi_{2}(z))\] \[\geq C_{j^{{}^{\prime\prime}}}(z)\to C_{j^{{}^{\prime\prime}}}(z) \otimes l_{1}\wedge C_{j^{{}^{\prime\prime}}}(z)\otimes l_{2}\] \[= (C_{j^{{}^{\prime\prime}}}(z)\to C_{j^{{}^{\prime\prime}}}(z) \otimes l_{1})\] \[\wedge(C_{j^{{}^{\prime\prime}}}(z)\to C_{j^{{}^{\prime\prime}} }(z)\otimes l_{2})\] \[\geq l_{1}\otimes l_{2}.\]
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\[\lambda_{m,0}\leq H_{m,0}\Psi(y,m,\gamma,\lambda_{m,\gamma})\] \[=H_{m,\gamma}\Psi(y,m,\gamma,\lambda_{m,\gamma})-\int_{-1}^{1} \Bigg{(}\frac{U^{\prime\prime}(y)}{U_{m,\gamma}(y)}-\frac{U^{\prime\prime}(y) }{U(y)}\Bigg{)}|\Psi(y,m,\gamma,\lambda_{m,\gamma})|^{2}\;dy\] \[\quad-\int_{-1}^{1}\frac{m}{\gamma}\sigma(y/\gamma)\frac{|\Psi(y, m,\gamma,\lambda_{m,\gamma})|^{2}}{\frac{U(y)}{y}+\gamma m\widetilde{\Gamma}(y/ \gamma)/(y/\gamma)}\;dy-\frac{m|\Psi(0,m,\gamma,\lambda_{m,\gamma})|^{2}}{U^ {\prime}(0)}\] \[=\lambda_{m,\gamma}-\int_{-1}^{1}\Bigg{(}\frac{U^{\prime\prime}(y )}{U_{m,\gamma}(y)}-\frac{U^{\prime\prime}(y)}{U(y)}\Bigg{)}|\Psi(y,m,\gamma, \lambda_{m,\gamma})|^{2}\;dy\] \[\quad-m\int_{-1/\gamma}^{1/\gamma}\sigma(y)\bigg{(}\frac{\gamma y }{U(\gamma y)}|\Psi(\gamma y,m,\gamma,\lambda_{m,\gamma})|^{2}-\frac{|\Psi(0, m,\gamma,\lambda_{m,\gamma})|^{2}}{U^{\prime}(0)}\bigg{)}\;dy\] \[\quad-m\int_{|y|\geq 1/\gamma}\sigma(y)\frac{|\Psi(0,m, \gamma,\lambda_{m,\gamma})|^{2}}{U^{\prime}(0)}\;dy+m\int_{-1}^{1}\frac{ \sigma(y/\gamma)y^{2}}{\gamma U^{2}(y)}\frac{\gamma m\widetilde{\Gamma}(y/ \gamma)/(y/\gamma)|\Psi(y,m,\gamma,\lambda_{m,\gamma})|^{2}}{1+\frac{y}{U(y) }\gamma m\widetilde{\Gamma}(y/\gamma)/(y/\gamma)}\;dy\] \[=:\lambda_{m,\gamma}+S_{1}+S_{2}+S_{3}+S_{4},\]
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\[h_{k}\left(g\cdot x^{I}y^{J}\right) =h_{k}\left(a_{1}^{i_{1}+j_{1}}\cdots a_{n}^{i_{n}+j_{n}}x_{\sigma \left(1\right)}^{i_{1}}\cdots x_{\sigma\left(n\right)}^{i_{n}}y_{\sigma\left( 1\right)}^{j_{1}}\cdots y_{\sigma\left(n\right)}^{j_{n}}\right)\] \[=\left(-1\right)^{i_{k}+j_{k}}a_{1}^{i_{1}+j_{1}}\cdots a_{n}^{i_ {n}+j_{n}}x_{\sigma\left(1\right)}^{i_{1}}\cdots x_{\sigma\left(n\right)}^{i_ {n}}y_{\sigma\left(1\right)}^{j_{1}}\cdots y_{\sigma\left(n\right)}^{j_{n}}\] \[=-a_{1}^{i_{1}+j_{1}}\cdots a_{n}^{i_{n}+j_{n}}x_{\sigma\left(1 \right)}^{i_{1}}\cdots x_{\sigma\left(n\right)}^{i_{n}}y_{\sigma\left(1\right)} ^{j_{1}}\cdots y_{\sigma\left(n\right)}^{j_{n}}\] \[=-g\cdot x^{I}y^{J}.\]
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\[\begin{split}\widetilde{\mathbb{L}}^{\mbox{\tiny dTac}}& (\tau_{2},y_{i};\tau_{1},x_{j})\\ &\stackrel{{**}}{{=}}\frac{1}{\sqrt{\pi}}\left[ \begin{array}{l}+\mathbb{1}_{n_{2}>n_{1}}\sum_{\alpha=0}^{n_{1}-1}e^{-\frac{ y_{i}^{2}}{2}}\widetilde{H}_{n_{2}-n_{1}+\alpha}(y_{i})\sqrt{\pi}e^{\frac{x_{j}^{2}} {2}}g_{n_{2}-n_{1}+\alpha}(\vec{y},x_{j})\\ +\mathbb{1}_{n_{2}>n_{1}}\sum_{\alpha=0}^{n_{2}-n_{1}-1}e^{-\frac{ y_{i}^{2}}{2}}\widetilde{H}_{\alpha}(y_{i})\sqrt{\pi}e^{\frac{x_{j}^{2}}{2}}( \Phi_{n_{2}-n_{1}-\alpha-1}(-x_{j})+g_{\alpha}(\vec{y},x_{j}))\\ +\sum_{\alpha=0}^{n_{1}-r-1}e^{-\frac{y_{i}^{2}}{2}}\widetilde{H}_{n_{2}-n_{1 }+\alpha}(y_{i})e^{-\frac{x_{j}^{2}}{2}}\widehat{H}_{\alpha}(x_{j})\\ +\sum_{\alpha=n_{1}-r}^{n_{1}-1}e^{-\frac{y_{i}^{2}}{2}}G_{n_{2}-n_{1}+ \alpha}(y_{i})e^{-\frac{x_{j}^{2}}{2}}\widehat{H}_{\alpha}(x_{j})\end{array}\right] \\ =\left\{\begin{array}{l}A_{2}^{\top}(y_{i})B_{1}(x_{j}),\mbox{ after setting }r<n_{1}<n_{2}\mbox{ in }\stackrel{{**}}{{=}},\\ A_{2}^{\top}(y_{i})B_{2}(y_{j}),\mbox{ after setting }\left\{\begin{array}{l}\rho<\tau_{1}=\tau_{2} \\ r<n_{1}=n_{2}\end{array}\right\}\mbox{ and }x_{j}=y_{j},\mbox{ in }\stackrel{{**}}{{=}}. \end{array}\right.\end{split}\]
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\[(D_{A}S)_{B}C-(D_{B}S)_{A}C+[S_{A},S_{B}]C-\frac{1}{f_{\rm H}} \omega_{\rm H}(A,B)(D_{C}Z+S_{Z}C)\] \[=\frac{1}{4f_{\rm H}^{2}}\Bigg{(}2\sum_{\mu}\Big{[}\big{(}\omega_{ \rm H}(Z,A)\omega_{\mu}(I_{\rm H}B,C)-\omega_{\rm H}(Z,B)\omega_{\mu}(I_{\rm H }A,C)\big{)}I_{\mu}Z\] \[-\omega_{\rm H}(A,B)g(I_{\mu}I_{\rm H}Z,C)I_{\mu}Z\Big{]}\] \[+\sum_{\mu,\lambda}\big{(}g(I_{\mu}I_{\rm H}A,Z)g(I_{\lambda}I_{ \rm H}B,C)-g(I_{\mu}I_{\rm H}B,Z)g(I_{\lambda}I_{\rm H}A,C)\big{)}I_{\lambda}I _{\mu}Z\Bigg{)}\] \[+\frac{1}{4f_{Z}^{2}}\Bigg{(}2\sum_{\mu}\Big{[}\alpha_{1}(A)\big{(}g(I_{ \mu}I_{1}Z,B)I_{\mu}C+g(I_{\mu}I_{1}Z,C)I_{\mu}B\big{)}\] \[\qquad\qquad\qquad-\alpha_{1}(B)\big{(}g(I_{\mu}I_{1}Z,A)I_{\mu}C +g(I_{\mu}I_{1}Z,C)I_{\mu}A\big{)}\Big{]}\] \[\qquad\qquad+\sum_{\mu,\lambda}\Big{[}\big{(}g(I_{\mu}I_{1}Z,A)g( I_{\lambda}I_{1}Z,B)-g(I_{\mu}I_{1}Z,B)g(I_{\lambda}I_{1}Z,A)\big{)}I_{\mu}I_{ \lambda}C\] \[\qquad\qquad\qquad+g(I_{\mu}I_{1}Z,B)g(I_{\lambda}I_{1}Z,C)I_{ \lambda}I_{\mu}A-g(I_{\mu}I_{1}Z,A)g(I_{\lambda}I_{1}Z,C)I_{\lambda}I_{\mu}B \Big{]}\Bigg{)}\] \[+\frac{1}{4f_{\rm H}}\Bigg{(}\sum_{\mu}\Big{[}\omega_{\mu}(I_{ \rm H}B,C)I_{\mu}I_{\rm H}A-\omega_{\mu}(I_{\rm H}A,C)I_{\mu}I_{\rm H}B+4 \omega_{\mu}(R(A,B)Z,C)I_{\mu}Z\Big{]}\] \[\qquad\qquad\qquad-2\omega_{\rm H}(A,B)I_{\rm H}C\Bigg{)}\] \[+\frac{1}{4f_{Z}}\Bigg{(}\sum_{\mu}\Big{[}g(I_{\mu}A,C)I_{\mu}B-g (I_{\mu}B,C)I_{\mu}A+\big{(}\omega_{\mu}(A,B)-\omega_{\mu}(B,A)\big{)}I_{\mu} C\Big{]}\Bigg{)}\]
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\[(n+1)^{2}\left(k+n^{2}+2\right)\left(3kn^{2}-4k^{2}-5kn-12k+2n^{ 3}+2n^{2}-8n-8\right)F(n,k+1)\] \[\qquad\qquad+(n+1)^{2}\left(k+n^{2}+3\right)\left(2k^{2}-2kn^{2} +2kn+6k-n^{3}-n^{2}+4n+4\right)F(n,k+2)\] \[\qquad\qquad\qquad+(n+1)^{2}(k+n+1)\left(2k-n^{2}+n+4\right) \left(k+n^{2}+1\right)F(n,k)\] \[\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-(k+1)n^{ 2}(n+2)^{2}\left(k+n^{2}+2n+2\right)F(n+1,k)\] \[\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad+kn^{2}(n+2)^{2}\left(k+n^{2}+2n+3\right)F(n+1,k+1)=0\]
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\[\langle\psi_{1}|U_{\left(\begin{smallmatrix}N&0\\ 0&1\end{smallmatrix}\right)},\psi_{2}\rangle\] \[= N^{-2n^{\prime}}\sum_{\lambda_{1},\mu_{1}\in\mathbb{Z}^{(n^{ \prime},1)}}\int_{\mathcal{F}_{n^{\prime},2}(N)}N^{-2n^{\prime}}\psi_{1}(\tau,z )\overline{(\psi_{2}(*,*\left(\begin{smallmatrix}N^{-1}&0\\ 0&1\end{smallmatrix}\right))|X)(\tau,z)}\] \[\times\det(v)^{k-n^{\prime}-3}e(\frac{-4\pi}{2\pi i}\mathcal{M}v ^{-1}[y])\,du\,dv\,dx\,dy.\] \[= N^{-2n^{\prime}}\int_{\mathcal{F}_{n^{\prime},2}(N)}\psi_{1}(\tau,z)\overline{(\psi_{2}|U^{*}_{\left(\begin{smallmatrix}N&0\\ 0&1\end{smallmatrix}\right)})(\tau,z)}\det(v)^{k-n^{\prime}-3}e(\frac{-4\pi}{2 \pi i}\mathcal{M}v^{-1}[y])\,du\,dv\,dx\,dy\] \[= \int_{\mathcal{F}_{n^{\prime},2}}\psi_{1}(\tau,z)\overline{( \psi_{2}|U^{*}_{\left(\begin{smallmatrix}N&0\\ 0&1\end{smallmatrix}\right)})(\tau,z)}\det(v)^{k-n^{\prime}-3}e(\frac{-4\pi}{2 \pi i}\mathcal{M}v^{-1}[y])\,du\,dv\,dx\,dy\] \[= \langle\psi_{1},\psi_{2}|U^{*}_{\left(\begin{smallmatrix}N&0\\ 0&1\end{smallmatrix}\right)}\rangle.\]
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\[\lim\limits_{p\to\infty}\sum\limits_{j_{1},j_{2}=0}^{p}C_{j_{2}j_{2}j_{1}j_{1}}=\] \[=\sum_{j_{1},j_{2}=0}^{\infty}\int\limits_{t}^{T}\psi_{4}(t_{4})\phi_{j_{2}}(t_{4} )\int\limits_{t}^{t_{4}}\psi_{3}(t_{3})\phi_{j_{2}}(t_{3})\int\limits_{t}^{t_{3} }\psi_{2}(t_{2})\phi_{j_{1}}(t_{2})\int\limits_{t}^{t_{2}}\psi_{1}(t_{1})\phi_{ j_{1}}(t_{1})dt_{1}dt_{2}dt_{3}dt_{4}=\] \[=\frac{1}{4}\int\limits_{t}^{T}\psi_{4}(t_{4})\psi_{3}(t_{4})\int \limits_{t}^{t_{2}}\psi_{2}(t_{2})\psi_{1}(t_{2})dt_{2}dt_{4},\]
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\[\left(\sum_{\begin{subarray}{c}\sigma\in S(A,B)\\ \sigma(a_{j})=a_{j}\end{subarray}}(-1)^{j-1}\operatorname{sgn}(\sigma)e(( \sigma T)_{j})+\sum_{\begin{subarray}{c}\sigma\in S(A,B)\\ \sigma(a_{j+1})=a_{j}\end{subarray}}(-1)^{j}\operatorname{sgn}(\sigma)e(( \sigma T)_{j+1}))\right)\otimes x_{a_{j}}\] \[= \left(\sum_{\begin{subarray}{c}\sigma\in S(A_{j},B)\\ \sigma(b_{1})\leq a_{j-1}\end{subarray}}(-1)^{j-1}\operatorname{sgn}(\sigma)e (\sigma(T_{j}))+\sum_{\begin{subarray}{c}\tau\in S(A_{j},B)\\ \tau(b_{1})\geq a_{j+1}\end{subarray}}(-(-1)^{j}\operatorname{sgn}(\tau)e( \tau(T_{j})))\right)\otimes x_{a_{j}}\] \[= (-1)^{j-1}g_{A_{j},B}e(T_{j})\otimes x_{a_{j}}\] \[= 0.\]
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\[\left(\mathcal{A}^{-}(h_{i_{1}},l_{1})\mathcal{A}^{+}(h_{i_{1}+1 })\mathcal{A}^{+}(h_{2n})\Omega\right)\left(x_{i_{1}},\ldots,x_{i_{2n}} \setminus x_{i_{1}},x_{j_{1}}\right)\] \[\quad=\int_{X^{2}}\sigma^{(2)}(dx_{i_{1}}\,dx_{j_{1}})\,\mathrm{ Tr}_{1}\,Q_{2}(x_{i_{1}},x_{j_{1}^{(1)}})Q_{3}(x_{i_{1}},x_{j_{2}^{(1)}})\] \[\quad\cdots Q_{l_{1}}(x_{i_{1}},x_{j_{l_{1}-1}^{(1)}})h_{i_{1}}(x _{i_{1}})\otimes h_{i_{2}}(x_{i_{2}})\otimes\cdots\otimes h_{2n}(x_{2n})\] \[\quad=\int_{X^{2}}\sigma^{(2)}(dx_{i_{1}}\,dx_{j_{1}})\,\mathbb{T }^{(2n-i_{1}+1)}(i_{1};\,i_{1},\mathbf{j}_{1}^{(1)})\] \[\quad\quad Q(x_{i_{1}},x_{j_{1}^{(1)}})[i_{1};\,\mathbf{j}_{1}^{ (1)},\mathbf{j}_{2}^{(1)}]Q(x_{i_{1}},x_{j_{2}^{(1)}})[i_{1};\,\mathbf{j}_{2}^ {(1)},\mathbf{j}_{3}^{(1)}]\cdots Q(x_{i_{1}},x_{j_{l_{1}-1}^{(1)}})[i_{1};\, \mathbf{j}_{l_{1}-1}^{(1)},\mathbf{j}_{l_{1}}^{(1)}]\] \[h_{i_{1}}(x_{i_{1}})\otimes h_{i_{2}}(x_{i_{2}})\otimes\cdots\otimes h_{2n}(x_{2n}).\]
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\[\nabla u(t,x)=\frac{1}{\tau(t)^{d/2}}\nabla_{x}\left(v\left(t, \frac{x}{\tau(t)}\right)e^{i\frac{\dot{\tau}(t)}{\tau(t)}\frac{|x|^{2}}{2}}\right)\] \[=\underbrace{\frac{1}{\tau(t)}\frac{1}{\tau(t)^{d/2}}\nabla_{y}v \left(t,\frac{x}{\tau(t)}\right)e^{i\frac{\dot{\tau}(t)}{\tau(t)}\frac{|x|^{2} }{2}}}_{\left\|\cdot\right\|_{L^{2}}=\frac{\dot{\tau}}{\left\|\nabla v\right\| _{L^{2}}=\mathcal{O}(1).}}+\underbrace{i\dot{\tau}\frac{1}{\tau(t)^{d/2}}\frac {x}{\tau}v\left(t,\frac{x}{\tau(t)}\right)e^{i\frac{\dot{\tau}(t)}{\tau(t)} \frac{|x|^{2}}{2}}}_{\left\|\cdot\right\|_{L^{2}}=\dot{\tau}\left\|yv\right\| _{L^{2}}\sim\dot{\tau}\left\|y\gamma\right\|_{L^{2}}\approx\sqrt{\ln t}},\]
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\[\begin{split}\chi^{e}_{x}-\chi(x\epsilon)&=\mathscr{K}^{e \,-1}\bigg{(}(f^{e}_{x}-\mathscr{R}_{e}f_{x})+(\mathscr{R}_{e}(\xi^{ns}*(\chi- \mathscr{P}_{e}\mathscr{R}_{e}\chi)))_{x}\\ &+(\mathscr{R}_{e}\mathscr{I})_{x}+\sum_{j=-\infty}^{-1}( \epsilon\int_{-\frac{1}{2}}^{+\frac{1}{2}}\xi^{ns}(x\epsilon-j\epsilon-t \epsilon)(\mathscr{P}_{e}\mathscr{R}_{e}\chi(t\epsilon))dt-\overline{\mathbb{ K}}^{ns:e}_{x-j}\,\chi(j\epsilon))\\ &+\sum_{j=-\infty}^{-1}(\overline{\mathbb{K}}^{ns:e}_{x-j}- \widetilde{\mathbb{K}}^{e}_{x-j})\chi(j\epsilon)+\sum_{j\in S^{e}}(\beta \overline{\mathbb{K}}^{sing:e}_{x-j}-\mathbb{K}^{e}_{x-j})\chi(j\epsilon)\\ &+\beta\xi^{sing}*(\chi-\mathscr{R}_{e}\chi)(x\epsilon)+(1-\beta )(\xi^{sing}*\chi)(x\epsilon)\bigg{)}.\end{split}\]
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\[\mathcal{J}_{81} \lesssim\sum_{\begin{subarray}{c}0\leq|I_{1}|,|I_{2}|\leq N-6\\ 0\leq|I_{3}|,|I_{4}|\leq N-6\end{subarray}}\left\|\langle s+r\rangle\Gamma^{I _{1}}v\right\|_{L^{\infty}_{x}}\left\|\Gamma^{I_{2}}v\right\|_{L^{\infty}_{x} }\left\|\Gamma^{I_{3}}w\right\|_{L^{1}_{x}}\left\|\Gamma^{I_{4}}v\right\|_{L^ {2}_{x}}\] \[\lesssim C_{0}^{4}\varepsilon^{4}\langle s\rangle^{-1}\langle s \rangle^{\delta}\langle s\rangle^{\delta}\lesssim C_{0}^{4}\varepsilon^{4} \langle s\rangle^{-1+2\delta},\] \[\mathcal{J}_{82} \lesssim\sum_{\begin{subarray}{c}0\leq|I_{1}|,|I_{2}|\leq N-6\\ 0\leq|I_{3}|,|I_{4}|\leq N-6\end{subarray}}\left\|\langle s+r\rangle\Gamma^{I _{1}}v\right\|_{L^{\infty}_{x}}\left\|\Gamma^{I_{2}}v\right\|_{L^{\infty}_{x} }\left\|S\Gamma^{I_{3}}w\right\|_{L^{2}_{x}}\left\|\Gamma^{I_{4}}v\right\|_{L ^{2}_{x}}\] \[\lesssim C_{0}^{4}\varepsilon^{4}\langle s\rangle^{-1}\langle s \rangle^{\delta}\langle s\rangle^{\delta}\lesssim C_{0}^{4}\varepsilon^{4} \langle s\rangle^{-1+2\delta},\]
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\[d\big{\langle}P^{(i+1)}(X^{(i)}-\mathbb{E}_{t}[X^{(i)}]),X^{(i)}- \mathbb{E}_{t}[X^{(i)}]\big{\rangle}\] \[=\bigg{\{}\Big{\langle}\big{(}(A+BK^{(i)})^{\top}P^{(i+1)}+P^{(i+1) }(A+BK^{(i)})\] \[\quad+(C+DK^{(i)})^{\top}P^{(i+1)}(C+DK^{(i)})\big{)}(X^{(i)}- \mathbb{E}_{t}[X^{(i)}]),X^{(i)}-\mathbb{E}_{t}[X^{(i)}]\Big{\rangle}\] \[\quad+\Big{\langle}(\widehat{C}+\widehat{D}\widehat{K}^{(i)})^{ \top}P^{(i+1)}(\widehat{C}+\widehat{D}\widehat{K}^{(i)})\mathbb{E}_{t}[X^{(i) }],\mathbb{E}_{t}[X^{(i)}]\Big{\rangle}\] \[\quad+\Big{\langle}(\widehat{C}+\widehat{D}\widehat{K}^{(i)})^{ \top}P^{(i+1)}(C+DK^{(i)})(X^{(i)}-\mathbb{E}_{t}[X^{(i)}]),\mathbb{E}_{t}[X^ {(i)}]\Big{\rangle}\bigg{\}}ds+\{...\}\,dW(s).\]
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\[\rho(\mathbf{c};\frac{M}{2})+\rho(\mathbf{d};\frac{M}{2}) =\sum_{i=0}^{\frac{M}{2}-1}\left(c_{i}c_{i+\frac{M}{2}}+d_{i}d_{i +\frac{M}{2}}\right)\] \[=\sum_{i=1}^{\frac{M}{2}-2}\left(c_{i}c_{i+\frac{M}{2}}+d_{i}d_{ i+\frac{M}{2}}\right)+\left(c_{0}c_{\frac{M}{2}}+d_{0}d_{\frac{M}{2}}+c_{ \frac{M}{2}-1}c_{M-1}+d_{\frac{M}{2}-1}d_{M-1}\right)\] \[=c_{0}c_{\frac{M}{2}}+d_{0}d_{\frac{M}{2}}+c_{\frac{M}{2}-1}c_{M -1}+d_{\frac{M}{2}-1}d_{M-1}\] \[=c_{0}\left(c_{\frac{M}{2}}+\frac{d_{0}}{c_{0}}d_{\frac{M}{2}} \right)+c_{M-1}\left(c_{\frac{M}{2}-1}-\frac{d_{0}}{c_{0}}d_{\frac{M}{2}-1} \right),\]
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\[\frac{d}{dt}\mathbb{E}\|Y_{t}^{\varepsilon}\|_{H_{2}}^{2p}= \ \frac{2p}{\varepsilon}\mathbb{E}\left[\|Y_{t}^{\varepsilon}\|_{H_{2}}^{ 2p-2}{}_{V_{2}^{*}}\langle B(X_{t}^{\varepsilon},\tilde{Y}_{t}^{\varepsilon}), \tilde{Y}_{t}^{\varepsilon}\rangle_{V_{2}}\right]+\frac{p}{\varepsilon} \mathbb{E}\left[\|Y_{t}^{\varepsilon}\|_{H_{2}}^{2p-2}\|G_{2}(X_{t}^{ \varepsilon},\tilde{Y}_{t}^{\varepsilon})\|_{L_{2}(U_{2},H_{2})}^{2}\right]\] \[+\frac{2p(p-1)}{\varepsilon}\mathbb{E}\left[\|Y_{t}^{\varepsilon }\|_{H_{2}}^{2p-4}\|G_{2}(X_{t}^{\varepsilon},\tilde{Y}_{t}^{\varepsilon})^{* }Y_{t}^{\varepsilon}\|_{U_{2}}^{2}\right]\] \[\leqslant \ \frac{2p}{\varepsilon}\mathbb{E}\left[\|Y_{t}^{\varepsilon}\|_{H _{2}}^{2p-2}(-\hat{\gamma}\|Y_{t}^{\varepsilon}\|_{H_{2}}^{2}+C\|X_{t}^{ \varepsilon}\|_{H_{2}}^{2}+C)\right]\] \[+\frac{C_{p}}{\varepsilon}\mathbb{E}\left[\|Y_{t}^{\varepsilon }\|_{H_{2}}^{2p-2}(1+\|X_{t}^{\varepsilon}\|_{H_{1}}^{2}+\|Y_{t}^{ \varepsilon}\|_{H_{2}}^{2\zeta})\right]\] \[\leqslant \ -\frac{C_{p,\gamma}}{\varepsilon}\mathbb{E}\|Y_{t}^{\varepsilon}\|_ {H_{2}}^{2p}+\frac{C_{p}}{\varepsilon}\mathbb{E}\|X_{t}^{\varepsilon}\|_{H_{1}}^ {2p}+\frac{C_{p}}{\varepsilon}.\]
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