image imagewidth (px) 276 2.95k | latex_formula stringlengths 317 2.94k | category stringclasses 4
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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 & β & TimeLim & 584.771 & TimeLim \\ Mean time/DNL ... | table | |
\[\dot{L}\leq -k_{\mathrm{a}}\|\tilde{\bm{\eta}}\|^{2}-k_{\mathrm{b}}\|\tilde{ \bm{\theta}}^{\dagger}\|^{2}-\left(\frac{\mu}{\tau_{\min}}-\frac{1}{2\tau_{ \min}k_{5}}\right)\|\tilde{\bm{u}}\|^{2}\] \[-\left(\beta-\frac{L\beta\|\bm{B}\|}{\tau_{\min}}-\frac{\beta\| \bm{B}\|}{2\tau_{\min}k_{3}}-\frac{\beta\|\bm{B}\|}{2k_{... | outline | |
\[\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{... | outline | |
\[\sum_{i=1}^{N}\bm{a}(\bm{x_{i}},\bm{w})^{T}\left(\bar{\bm{a}}(\bm {x_{i}})\bar{\bm{a}}(\bm{x_{i}})^{T}\right)\bm{a}(\bm{x_{i}},\bm{w})\] \[\quad=\sum_{i=1}^{N}\bm{a}(\bm{x_{i}},\bm{w})^{T}\left(\bm{a}(\bm {x_{i}},\bm{w})\bm{a}(\bm{x_{i}},\bm{w})^{T}\right)\bm{a}(\bm{x_{i}},\bm{w})\] \[\geq\sum_{i=1}^{N}\hat{\bm{g}}^{... | outline | |
\[\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}(\ma... | outline | |
\[\|\kappa_{r}^{\geqslant j}[f] -\mathbf{E}^{j}\,\kappa_{r}^{\geqslant j+1}[f]\|_{\infty}\] \[\leqslant\sum_{p=2}^{r}\sum_{\{B_{1},\ldots,B_{p}\}\in P_{r}} \big{(}\tfrac{3}{2}\big{)}^{p}(p-1)!\prod_{t=1}^{p}\|\kappa_{|B_{t}|}^{ \geqslant j+1}[f]-\mathbf{E}^{j}\,\kappa_{|B_{t}|}^{\geqslant j+1}[f]\|_{\infty}\] \[\leqsla... | matrix | |
\[\langle Tx,x\rangle = \langle\mbox{Re}(T)x,x\rangle+i\langle\mbox{Im}(T)x,x\rangle\] \[\Rightarrow\langle Tx,x\rangle^{2} = \langle\mbox{Re}(T)x,x\rangle^{2}-\langle\mbox{Im}(T)x,x\rangle^ {2}+2i\langle\mbox{Re}(T)x,x\rangle\langle\mbox{Im}(T)x,x\rangle\] \[\Rightarrow\left|\langle Tx,x\rangle^{2}\right|^{2} = \left|... | outline | |
\[11\cdot 43=473 = 10\cdot 11+11\cdot 33=10\cdot 22+11\cdot 23\] \[= 10\cdot 33+11\cdot 13=10\cdot 44+11\cdot 3,\] \[11\cdot 54=594 = 10\cdot 11+11\cdot 44=10\cdot 22+11\cdot 34\] \[= 10\cdot 33+11\cdot 24=10\cdot 44+11\cdot 14=10\cdot 55+11\cdot 4,\] \[11\cdot 76=836 = 10\cdot 11+11\cdot 66=10\cdot 22+11\cdot 56=10\cd... | outline | |
\[\big{(}\alpha_{1},\!\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},\alpha_{6}, \alpha_{7}\big{)}=\big{(}n-\phi(n)-1,\phi(n)+1+\phi(pq)+\phi(pr),\phi(n)+1+ \phi(pq)+\phi(qr)\big{)},\] \[\phi(n)+1+\phi(pr)+\phi(qr),\phi(n)+1+\phi(p)+\phi(q),\phi(n)+1+ \phi(p)+\phi(r),\phi(n)+1+\phi(q)+\phi(r)\big{)}\] \[\text{and}\quad\bi... | outline | |
\[3S(\Delta_{Ex,Ey}Ez,w)\] \[= S(\left[\!\left[Ex,Ey,w\right]\!\right],Ez)+S(\left[\!\left[Ex,Ey,Ez\right]\!\right],w)+S(\left[\!\left[w,Ez,Ex\right]\!\right],Ey)+2S(\left[\! \left[w,Ez,Ey\right]\!\right],Ex)\] \[= S(E\left[\!\left[x,y,Ew\right]\!\right],Ez)+S(E\left[\!\left[x,y, z\right]\!\right],Ew)+S(E\left[\!\left[... | outline | |
\[\left[\mathcal{K},Q_{2}^{k}(B_{k})\right] =2\operatorname{Re}\left(\sum_{p\in L_{k}}\left(b_{k}(B_{k}e_{p}) \left[\mathcal{K},b_{-k}\left(e_{-p}\right)\right]+\left[\mathcal{K},b_{k}(B_{ k}e_{p})\right]b_{-k}\left(e_{-p}\right)\right)\right)\] \[=2\operatorname{Re}\left(\sum_{p\in L_{k}}\left(b_{k,p}b_{k}^{*} \left(K... | outline | |
\[\mathbb{E}\left[Y_{t}^{2}\right]= \mathbb{E}\left[\left(e^{-\kappa t}y_{0}+\sigma\int_{0}^{t}e^{- \kappa(t-s)}dW_{s}+\kappa\int_{0}^{t}e^{-\kappa(t-s)}\mu(s)ds+\gamma\sum_{i=1 }^{N_{t}}e^{-\kappa(t-t_{i})}\right)^{2}\right]\] \[= \left(e^{-\kappa t}y_{0}+\kappa\int_{0}^{t}e^{-\kappa(t-s)}\mu(s )ds\right)^{2}+\frac{\s... | outline | |
\[\int_{U^{\prime}_{k,\varepsilon,2}}\sum_{i=1}^{N-1}\varphi_{i}e_{\varepsilon}(u _{\varepsilon,i})\,dx=\int_{U^{\prime}_{k,\varepsilon,2}}\sum_{i\in I_{k}} \varphi_{i}e_{\varepsilon}(u_{\varepsilon,i})\,dx+\int_{U^{\prime}_{k, \varepsilon,2}}\sum_{i\in I^{c}_{k}}\varphi_{i}e_{\varepsilon}(u_{ \varepsilon,i})\,dx.\] \[... | outline | |
\[-\int\epsilon^{2}v_{xx}\beta_{x} =-\int\epsilon^{2}v_{xx}\frac{v_{x}}{u_{s}}-\int\epsilon^{2}v_{xx }v\partial_{x}\frac{1}{u_{s}}\] \[=+\int\frac{\epsilon^{2}}{2}\partial_{x}\frac{1}{u_{s}}v_{x}^{2} -\int_{x=L}\frac{\epsilon^{2}}{2u_{s}}v_{x}^{2}\] \[+\int_{x=0}\frac{\epsilon^{2}}{2u_{s}}v_{x}^{2}+\int\epsilon^{2}v _{... | outline | |
\[\langle\mathbf{d}^{k+1}-\mathbf{d}^{k},\mathbf{x}^{k}-\mathbf{x}^{k -1}\rangle\] \[+\langle\mathbf{d}^{k+1}-\mathbf{d}^{k},2\mathbf{z}^{k}-\mathbf{ z}^{k-1}-\mathbf{z}^{k+1}\rangle\] \[\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}... | outline | |
\[\mathcal{L}\widehat{V}^{(h)}(\psi)\] \[=\int_{\beta,L}d\underline{x}\Big{[}\sum_{e}\psi^{+}_{\underline{ x},e}\psi^{-}_{\underline{x},e}2^{h}\,\frac{Z_{h,e}}{Z_{h-1,e}}n_{h,e}\] \[\quad+\sum_{\underline{e}}\psi^{+}_{\underline{x},e_{1}}\psi^{-} _{\underline{x},e_{2}}\psi^{+}_{\underline{x},e_{3}}\psi^{+}_{\underline{... | outline | |
\[|\langle\Phi,R_{3}\Phi\rangle| =\frac{1}{N-1}\left|\iint w_{N}(x-y)\overline{u(t,x)}\Big{\langle} \Phi,\sqrt{N-\mathcal{N}}a_{y}^{*}a_{y}a_{x}\Phi\Big{\rangle}\,\,\mathrm{d}x\, \mathrm{d}y\right|\] \[\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\... | outline | |
\[e_{ii} =\lambda_{i}+\mathbf{n}_{[1,i-1],i}-\mathbf{n}_{i,[i+1,M+N]}\quad \text{for}\quad i\in\mathfrak{I},\] \[e_{i,i+1} =\sum_{k=1}^{i-1}\mathbf{c}_{ki}^{\dagger}\mathbf{c}_{k,i+1}\] \[\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,[... | outline | |
\[a_{n}\equiv a(u)=-J_{z}+\frac{J_{+}}{2}\frac{\theta_{4}(u)\theta _{1}(u+\eta)}{\theta_{1}(u)\theta_{4}(u+\eta)}-\frac{J_{-}}{2}\frac{\theta_{1 }(u)\theta_{1}(u+\eta)}{\theta_{4}(u)\theta_{4}(u+\eta)}\] \[\stackrel{{(\ref{eq:2})}}{{=}}-e^{-i\pi\eta}\frac{ \theta_{1}(\eta+\frac{1}{2}-\tau)\theta_{1}(\eta+\frac{1}{2})}{... | outline | |
\[\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... | outline | |
\[\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... | outline | |
\begin{table}
\begin{tabular}{|c|c|c|} \hline \(\operatorname{RA}(C)\) & A normal form of \(C\) & A Rosenhain form of \(C\) \\ \hline \(\{1\}\) & β & \(y^{2}=x(x-1)(x-\lambda)(x-\mu)(x-\nu)\) \\ \(\operatorname{C}_{2}\) & \(y^{2}=(x^{2}-1)(x^{2}-a)(x^{2}-b)\) & \(y^{2}=x(x-1)(x-\lambda)(x-\mu)\big{(}x-\frac{\lambda(1-\... | table | |
\[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... | matrix | |
\[=\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)}\... | matrix | |
\[\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})... | outline | |
\[\Pr\left(\lambda_{k}\left(\mathcal{X}\right)\geq\theta\right) =_{1} \Pr\left(e^{\lambda_{k}\left(t\mathcal{X}\right)}\geq e^{t\theta}\right)\] \[\leq_{2} e^{-t\theta}\mathbb{E}e^{\lambda_{k}\left(t\mathcal{X}\right)}\] \[=_{3} e^{-t\theta}\mathbb{E}\exp\left(\min_{\mathcal{U}_{\left( \mathbb{I}_{M}-k+1\right)}}\lambd... | outline | |
\[\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)}\,|\, \bolds... | matrix | |
\[\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... | outline | |
\[\frac{1}{N} \int_{0}^{T}N^{2}e^{-N\left\langle{\Gamma^{N}_{t}},H_{t}\right\rangle }{\cal L}_{\beta}e^{N\left\langle{\Gamma^{N}_{t}},H_{t}\right\rangle}dt\] \[=\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^{... | outline | |
\[\int_{Y_{2d}}\prod_{i\in I}\psi_{m_{i}}\ \mathrm{d}\mu =\int_{Y_{2d}}\prod_{\ell=1}^{k}\Biggl{(}\sum_{K_{\ell}\subset Q_{ \ell}}\bigl{(}-\mu\big{(}\phi\big{)}\bigr{)}^{\#K_{\ell}}g_{\ell,K_{\ell}} \Biggr{)}\circ a^{m_{Q_{\ell}}}\ \mathrm{d}\mu\] \[=\int_{Y_{2d}}\sum_{\begin{subarray}{c}K_{\ell}\subset Q_{\ell}\\ \tex... | outline | |
\[|\xi|^{s}\chi(|\xi|)|\partial_{t}^{j}\widehat{K_{0}^{1}}(t,\xi)| \lesssim e^{-c_{0}t|\xi|^{2(\sigma-\delta_{1})}}|\xi|^{s+2j(\sigma- \delta_{1})},\] \[|\xi|^{s}\chi(|\xi|)|\partial_{t}^{j}\widehat{K_{0}^{2}}(t,\xi)| \lesssim e^{-c_{0}t|\xi|^{2\delta_{1}}}|\xi|^{s+2j\delta_{1}+2( \sigma-2\delta_{1})},\] \[|\xi|^{s}\ch... | outline | |
\[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{... | outline | |
\[\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})... | outline | |
\[\bar{\kappa}(p) :=(M_{1}(p))^{2r}/(4(1+(M_{1}(p))^{2r})),\] \[M_{1}(p) :=\Big{(}2pd(2^{2r+3}K_{G}\mathbb{E}\left[(1+|X_{0}|)^{2\rho} \right]+4b_{F}+6)\] \[\quad+d^{p}{2p\choose p}(2p-1)2^{4p+1}(2+K_{F})^{2p}\mathbb{E}[(1 +|X_{0}|)^{2p\rho}]\Big{)}/\min\left\{1,a_{F}\right\},\] \[\bar{c}_{0}(p) :=c_{4}(p)+2^{2p-2}p(2p... | outline | |
\[\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... | outline | |
\[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}... | outline | |
\[P(\|Y-\phi^{(2)}\|\leq\varepsilon)\] \[=\tilde{P}(\|\tilde{Y}-\phi^{(2)}\|\leq\varepsilon)=E\Big{(} \frac{d\tilde{P}}{dP}\mathbb{I}_{\|B^{H}\|\leq\varepsilon}\Big{)}\] \[=E\Big{(}\exp\Big{(}\int_{0}^{1}\eta_{s}ds-\frac{1}{2}\int_{0}^{1} \eta_{s}^{2}ds\Big{)}\mathbb{I}_{\|B^{H}\|\leq\varepsilon}\Big{)}\] \[=E\Big{(}\e... | outline | |
\[F\big{(}\widetilde{\mathcal{M}}(\varphi_{RA^{n}}),\mathcal{N}^{ \otimes n}(\varphi_{RA^{n}})\big{)}\] \[= 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 ... | outline | |
\[\|G_{\alpha}^{\lambda}\theta\|_{\dot{H}^{\sigma+\delta_{1}}} \leq\|G_{\alpha}^{\lambda}\theta\|_{\dot{H}^{\sigma-\frac{2\delta_{1}} {2}}}^{-\frac{2\delta_{1}}{2}}\|G_{\alpha}^{\lambda}\theta\|_{\dot{H}^{\sigma+ \frac{\kappa}{2}}}^{\frac{2\delta_{1}}{2}},\] \[\|G_{\alpha}^{\lambda}\theta\|_{\dot{H}^{\sigma+\frac{\kapp... | outline | |
\[\frac{1}{W_{j_{k}}}\sum_{n=J_{k-1}+(i-1)j_{k}+1}^{J_{k-1}+ij_{k}} w_{n} \geqslant\left(\frac{J_{k-1}+(i-1)j_{k}+1}{j_{k}}+1\right)^{1- \theta}-\left(\frac{J_{k-1}+(i-1)j_{k}+1}{j_{k}}\right)^{1-\theta}\] \[=\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}... | outline | |
\[\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})\\ \P... | outline | |
\[\int_{D_{10}\cup D_{8}\cup D_{6}}u_{6}\sqrt{h^{2}+x^{2}+y^{2}}\ \mathrm{d}x \mathrm{d}y =h^{3}\Bigg{(}-\frac{2}{5}\sqrt{5+2\sqrt{5}}\left(I_{00}^{(1)} \left(\frac{1}{2\phi^{2}},\frac{2\pi}{5}\right)-I_{00}^{(1)}\left(\frac{1}{2 \phi^{2}},\frac{\pi}{5}\right)\right)\] \[+2h\left(\!1\!+\!\frac{1}{\sqrt{5}}\!\right)\lef... | outline | |
\[\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}}... | outline | |
\[\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\i... | outline | |
\[\text{Vol}(\delta(S^{\prime})) = \prod_{i,j}|x^{\prime}_{j}-x^{\prime}_{i}|\int_{\delta(S^{\prime})} \prod_{i}dx^{\prime}_{i}*\text{Angular part}\] \[= \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... | outline | |
\[\mathrm{e}^{-\lambda(t\wedge\hat{\tau}_{\kappa+c})} (\kappa-X_{t\wedge\hat{\tau}_{\kappa+c}})=\mathrm{e}^{-\lambda(t \wedge\hat{\tau}_{\kappa+c})}\widehat{V}(X_{t\wedge\hat{\tau}_{\kappa+c}})\] \[=\widehat{V}(X_{0})-\int_{0}^{t\wedge\hat{\tau}_{\kappa+c}} \lambda\mathrm{e}^{-\lambda s}\widehat{V}(X_{s})\,\mathrm{d}s+... | outline | |
\[\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(\fr... | outline | |
\[\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}^{-}... | outline | |
\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 & 1.189... | table | |
\[\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(\under... | matrix | |
\[S_{n_{1}|n_{2}}(f;z)=\frac{1}{n_{1}!n_{2}!}\sum_{k_{1}=1}^{n_{1} }\sum_{k_{2}=1}^{n_{2}}\sum_{m_{1}\geq 0}\sum_{m_{2}\geq 0}\sum_{p\geq 0}\sum_{q=1 }^{p}(-1)^{k_{1}+m_{2}+q}2^{-m_{1}-m_{2}}(k_{1}+k_{2}-1)!\] \[\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},...... | outline | |
\[\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\| \|... | outline | |
\[\Delta_{1,\mathbf{F},\sigma}(t,s,\boldsymbol{\theta}_{s,t}( \boldsymbol{\xi}),\mathbf{y},u) = \left\langle\left(\mathbf{F}_{1}(s,\mathbf{y})-\mathbf{F}_{1}(s, \boldsymbol{\theta}_{s,t}(\boldsymbol{\xi}))\right),D_{\mathbf{y}_{1}}u(s, \mathbf{y})\right\rangle\] \[\quad+\frac{1}{2}\mathrm{Tr}\Big{(}\big{(}a(s,\mathbf{y... | outline | |
\[\mathbb{E}\left(\sup_{t\in[0,T]}|X_{t}^{\varepsilon}-X_{t}|^{2}\right) \leqslant \mathrm{e}^{2\|\nabla b\|_{\infty}T}\mathbb{E}\left(\sup_{t\in[0,T]}\left|\int_{0}^{t}\varepsilon b(s,X_{s-}^{\varepsilon})\mathrm{d} \widetilde{\mathcal{N}}_{s}^{\varepsilon}\right|^{2}\right)\] \[\leqslant 4\mathrm{e}^{2\|\nabla b\|_{\... | outline | |
\[\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\\ \... | outline | |
\[\kappa_{m}(\{\beta_{q}\}\times A_{p})-\nu_{m}(\{\beta_{q}\}\times A _{p})=\frac{1}{4M}\sum_{k=1}^{M}h_{\mathsf{c}}^{a}(J_{p,q,1,1}^{r})-h_{\mathsf{ c}}^{a}(J_{p,q,1,1}^{l})\] \[-h_{\mathsf{c}}^{a}(J_{p,q,k,1}^{r})-h_{l}^{a}(J_{p,q,k,1}^{r})+ h_{\mathsf{c}}^{a}(J_{p,q,k,1}^{l})+h_{l}^{a}(J_{p,q,k,1}^{l})\] \[=\frac{1}... | outline | |
\[\|\mathcal{Z}(s)\|^{2}= \|\mathcal{Z}(\tau^{\prime})\|^{2}+\int_{\tau^{\prime}}^{s}[ \epsilon^{2}\|\mathcal{Z}\|^{2}-2\|\Delta\mathcal{Z}\|^{2}-4(\Delta\mathcal{Z},\mathcal{Z})-2a\|\mathcal{Z}\|^{2}-2(\mathcal{Z}^{2},G\ast u_{2}^{2})] \mathrm{d}s\] \[-2\int_{\tau^{\prime}}^{s}(\mathcal{Z}u_{1},G\ast(u_{2}^{2}-u_{1 }^... | outline | |
\[r =\frac{1}{k}\left[\omega_{L}\|h[T]\|_{2,1}+(1-\omega_{1})\|h[T \cup\cup_{i=1}^{L}\widetilde{T}_{i}\backslash\cup_{i=1}^{L}(\widetilde{T}_{i} \cap T)]\|_{2,1}\right.\] \[\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... | outline | |
\[\left\{\begin{aligned} \dot{\bar{x}}^{1}(t)&=\left(s_{1} \bar{u}^{1}\cos\theta_{1}(0),\ s_{1}\bar{u}^{1}\sin\theta_{1}(0)\right),\\ \dot{\bar{x}}^{2}(t)&=\left(s_{2}\bar{u}^{2}\cos \theta_{1}(0),\ s_{2}\bar{u}^{2}\sin\theta_{1}(0)\right)\ \text{ for }\ t\in[0,t_{1}),\end{aligned}\right.\] \[\left\{\begin{array}{l}\do... | outline | |
\[\left\|g-G_{hh}^{-1}(\theta_{0},h_{0})\left[Z_{h}(\theta_{0}+ \zeta,h_{0}+g)-U_{\theta h}(\theta_{0},h_{0})U_{\theta}^{-1}(\theta_{0},h_{0})Z _{\theta}(\theta_{0}+\zeta,h_{0}+g)\right]-(\bar{h}_{\lambda}-h_{0})\right\|_{ \mathcal{H}}\] \[\quad=\left\|G_{hh}^{-1}(\theta_{0},h_{0})\left[R_{h}(\theta_{0}, h_{0})\zeta g-... | outline | |
\[AX^{\circ}_{\infty}+BU^{\circ}_{\infty}+EW=\tilde{A}X^{\circ}_{ \infty}+BF\mathbb{E}[W]+EW\] \[\stackrel{{\mathcal{H}^{\circ}}}{{=}}\tilde{A} \Big{(}(I-\tilde{A})^{-1}\tilde{F}\mathbb{E}[W]\phi^{-2}+\sum_{j=0}^{\infty} \tilde{A}^{j}E\mathbb{w}^{0}\phi^{j}\Big{)}\] \[\qquad\qquad\qquad\qquad+BF\mathbb{E}[W]+E(\mathbb{... | outline | |
\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)(ii... | table | |
\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 \\
**C... | table | |
\[\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}\ri... | outline | |
\[f(\bm{U}^{\prime})=\frac{1}{m}\left\|\mathcal{A}(\bm{U}^{\prime }\bm{U}^{\prime\mathrm{T}}-\bm{X}^{\star})-\bm{s}^{\star}\right\|_{1}\] \[=\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}^... | outline | |
\[\left\|\mathbf{P}_{n}\left(T_{u^{k-1}}u-\sum_{j}\mathbf{P}_{j}u(S _{j+1}u)^{k-1}\right)\right\|_{L_{x}^{1}}\] \[= \left\|\mathbf{P}_{n}\sum_{j\geq n-3}\mathbf{P}_{j}u\left(S_{j-1} u^{k-1}-(S_{j+1}u)^{k-1}\right)\right\|_{L_{x}^{1}}\] \[\lesssim \sum_{j\geq n-3}\left\|\mathbf{P}_{j}u\right\|_{L_{x}^{p}}\left\| S_{j-1}... | outline | |
\[C_{2}k^{2}\|u\|_{L^{k+1-m}(\Omega)}^{k+1-m}\] \[= 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}}(... | outline | |
\[\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}... | outline | |
\[\frac{1}{|V|}\left(\int_{V}u(\cdot,t)d\mu\right)^{2} = \frac{1}{|V|}\left(\int_{A_{t}}u(\cdot,t)d\mu+\int_{V\setminus A_ {t}}u(\cdot,t)d\mu\right)^{2}\] \[= \frac{1}{|V|}\left(\int_{A_{t}}u(\cdot,t)d\mu\right)^{2}+\frac{1} {|V|}\left(\int_{V\setminus A_{t}}u(\cdot,t)d\mu\right)^{2}\] \[+\frac{2}{|V|}\int_{A_{t}}u(\cd... | outline | |
\[\frac{1-m}{p}C_{\psi}=\frac{2m(1-m)}{p-1}\left[\int_{A_{R_{0},R_{ 1}}}\frac{|\nabla\psi|^{\frac{2p}{1-m}}}{\psi^{\frac{2(p+m-1)}{1-m}}}|x|^{ \gamma\frac{p+m-1}{1-m}-\beta\frac{p}{1-m}}\,\mathrm{d}x\right]^{\frac{1-m}{p}}\] \[\leq\left[\frac{C_{2}^{\frac{p}{1-m}}\,c_{0,p}^{\frac{p}{1-m}}}{ \left(R_{0}-R_{1}\right)^{\f... | outline | |
\[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_{... | outline | |
\[2VD_{s}\Psi^{*} =2b_{0}^{*}D_{s}\Psi^{*}+2b_{1}^{*}(z-z_{*})^{*}(b_{1t}-b_{0}^{*} D_{s}^{2}U^{*})+\mathcal{L}^{1}_{2,\beta-1}(m)\] \[\Psi D_{s}V^{*} =-c_{1}^{*}b_{1}-c_{1}^{*}D_{s}U^{*}-b_{0}b_{1}(D_{s}V^{*})^{*}-b_ {0}|D_{s}V^{*}|^{2}+b_{1}b_{1t}^{*}(z-z_{*})^{*}+\mathcal{L}^{1}_{2,\beta-1}(m)\] \[V^{*}|D_{s}V^{*}|^... | outline | |
\[\mathbb{E}_{\lambda,\mu}\Big{[}\prod_{j=1}^{r_{1}}Z_{n,k_{j},l_{j} }^{m_{j}}\Big{]}\mathbb{E}_{\lambda,\mu}\Big{[}\prod_{j=r_{1}+1}^{r}(Z_{n,k_{j },l_{j}})^{m_{j}}\Big{]}=\frac{1}{n^{\sum_{j=r_{1}+1}^{r_{1}}l_{j}m_{j}}}\times\] \[\mathbb{E}_{\sigma}\Big{[}\sum_{j\in[r],1\leq q_{j}\leq m_{j}, \omega_{j,q_{j}}}\mathbb{... | outline | |
\[\int_{Q}\Biggl{(}\frac{1}{s^{2}}\left(|\partial_{t}u|^{2}+|\Delta u |^{2}\right)+\frac{1}{s}\left(|\partial_{t}H|^{2}+|\Delta H|^{2}\right)+| \nabla u|^{2}+s|\nabla H|^{2}\] \[+s^{2}|u|^{2}+s^{3}|H|^{2}+\frac{1}{s}|\nabla p|^{2}+s|p|^{2} \Biggr{)}e^{2s\varphi}dxdt\] \[\leq C\int_{Q}\left(|F|^{2}+|G|^{2}+|\nabla_{x,t}... | outline | |
\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_... | table | |
\[b_{n-1,n-2}^{(n-2)} =\left|\begin{smallmatrix}0&-b_{n-1,n-3}^{{}^{\prime}(n-3)}&b_{n-2,n-3}^{{}^{\prime}(n-3)}\\ b_{n-2,n-3}^{{}^{\prime}(n-3)}&b_{n-2,n-2}^{(n-3)}&b_{n-2,n-1}^{(n-3)}\\ \rho_{n-1,n-3}^{{}^{\prime}(n-3)}&b_{n-1,n-2}^{(n-3)}&b_{n-1,n-1}^{(n-3)}\\ \end{smallmatrix}\right|\] \[=\frac{1}{e_{n-3}^{2}}\left... | outline | |
\[\mathbb{E}\Big{[} \operatorname{DTV}\left(f(x_{1:n});\,\tilde{w}\right)\Big{]}\] \[\leq p_{\max}^{2}n^{2}\int_{\Omega}\int_{\Omega}|f(y)-f(x)|\Bigg{(} c_{0}n^{-(d-1)/d}\mathbb{P}_{3:n}\{\mathcal{H}(\bar{V}_{x}\cap\bar{V}_{y})>0\}\] \[\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+C_{1} \frac{1\{\|x-y\|\leq C_... | outline | |
\[D\Big{(}\frac{p}{q}\Big{)}(T) =\Big{(}\frac{r}{t}\Big{)}(\overline{T})^{-1}\Big{(}D\Big{(}\frac {rp}{tq}\Big{)}(T)-\Big{(}\frac{p}{q}\Big{)}(T)D\Big{(}\frac{r}{t}\Big{)}(T) \Big{)}\] \[=\big{(}r[\overline{T}]t[\overline{T}]^{-1}\big{)}^{-1}\Big{(}Dr[ T]p[T]t[T]q[T]+r[\overline{T}]Dp[T]t[T]q[T]\] \[\qquad\qquad\qquad\... | outline | |
\[\mathcal{D}_{1}=\mathcal{A}_{1}=\mathcal{R}_{1}=\{\mathcal{B}_{1},\mathcal{B}_{5},\mathcal{B}_{7}\},\] \[\mathcal{D}_{2}=\mathcal{A}_{2}=\mathcal{R}_{2}=\{\mathcal{B}_{1},\mathcal{B}_{2},\mathcal{B}_{6}\},\] \[\mathcal{D}_{3}=\mathcal{A}_{3}=\mathcal{R}_{3}=\{\mathcal{B}_{2},\mathcal{B}_{3},\mathcal{B}_{7}\},\] \[\ma... | outline | |
\[e^{x}H_{1,2}^{1,1}\left[-x^{3}\Bigg{|}\begin{array}{c}(0,1)\\ (0,1),(0,3)\end{array}\right] = e^{x}{}_{0}F_{2}\left[\begin{array}{c}-\\ \frac{1}{3},\frac{2}{3}\end{array};\frac{x^{3}}{27}\right]\] \[= \frac{2}{3}\sum_{m=0}^{\infty}\frac{x^{m}}{m!}\left[2^{m-1}+cos \left(\frac{m\pi}{3}\right)\right]\] \[= \frac{e^{2x}... | outline | |
\[I_{T,t}^{*(i_{1}i_{2}i_{3}i_{4})}=\underset{p\to\infty}{\text{l.i.m.}}\ \int\limits_{t}^{*\,T}\ \int\limits_{t}^{*\,t_{4}}\ \int\limits_{t}^{*\,t_{3}}\ \int\limits_{t}^{*\,t_{2}}d\mathbf{w}_{t_{1}}^{(i_{1})}d\mathbf{w}_{t_{2}}^{ (i_{2})}d\mathbf{w}_{t_{3}}^{(i_{3})}d\mathbf{w}_{t_{4}}^{(i_{4})}\] \[I_{T,t}^{*(i_{1}i_... | outline | |
\[\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}+9... | outline | |
\[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}... | outline | |
\[\bigg{|}\int_{0}^{\infty}\int_{t}^{\infty}(O(\alpha t)x-y)_{1} \big{(}(\cos\alpha t)A_{1}+(\sin\alpha t)A_{2}\big{)}G(O(\alpha t)x-y,s) \frac{\mathrm{d}s}{8s^{3}}\,\mathrm{d}t-\partial_{y_{1}}L^{112}(x,y)\] \[=\,\bigg{|}\int_{0}^{\infty}\int_{t}^{\infty}\bigg{(}(\cos 2\alpha t )D_{1}+(\sin 2\alpha t)D_{2}+D_{3}\bigg{... | outline | |
\[C_{j^{{}^{\prime\prime}}}(z)\rightarrow(A\times B)_{\psi(j^{{}^{ \prime\prime}})}(\phi(z)) = C_{j^{{}^{\prime\prime}}}(z)\rightarrow(A\times B)_{(\psi_{1}(j^{{ }^{\prime\prime}}),\psi_{2}(j^{{}^{\prime\prime}}))}(\phi_{1}(z),\phi_{2}(z))\] \[= C_{j^{{}^{\prime\prime}}}(z)\rightarrow(A_{\psi_{1}(j^{{}^{ \prime\prime}}... | outline | |
\[\lambda_{m,0}\leq H_{m,0}\Psi(y,m,\gamma,\lambda_{m,\gamma})\] \[=H_{m,\gamma}\Psi(y,m,\gamma,\lambda_{m,\gamma})-\int_{-1}^{1} \Bigg{(}\frac{U^{\prime\prime}(y)}{U_{m,\gamma}(y)}-\frac{U^{\prime\prime}(y) }{U(y)}\Bigg{)}|\Psi(y,m,\gamma,\lambda_{m,\gamma})|^{2}\;dy\] \[\quad-\int_{-1}^{1}\frac{m}{\gamma}\sigma(y/\ga... | outline | |
\[h_{k}\left(g\cdot x^{I}y^{J}\right) =h_{k}\left(a_{1}^{i_{1}+j_{1}}\cdots a_{n}^{i_{n}+j_{n}}x_{\sigma \left(1\right)}^{i_{1}}\cdots x_{\sigma\left(n\right)}^{i_{n}}y_{\sigma\left( 1\right)}^{j_{1}}\cdots y_{\sigma\left(n\right)}^{j_{n}}\right)\] \[=\left(-1\right)^{i_{k}+j_{k}}a_{1}^{i_{1}+j_{1}}\cdots a_{n}^{i_ {n}... | outline | |
\[\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... | outline | |
\[(D_{A}S)_{B}C-(D_{B}S)_{A}C+[S_{A},S_{B}]C-\frac{1}{f_{\rm H}} \omega_{\rm H}(A,B)(D_{C}Z+S_{Z}C)\] \[=\frac{1}{4f_{\rm H}^{2}}\Bigg{(}2\sum_{\mu}\Big{[}\big{(}\omega_{ \rm H}(Z,A)\omega_{\mu}(I_{\rm H}B,C)-\omega_{\rm H}(Z,B)\omega_{\mu}(I_{\rm H }A,C)\big{)}I_{\mu}Z\] \[-\omega_{\rm H}(A,B)g(I_{\mu}I_{\rm H}Z,C)I_{... | outline | |
\[(n+1)^{2}\left(k+n^{2}+2\right)\left(3kn^{2}-4k^{2}-5kn-12k+2n^{ 3}+2n^{2}-8n-8\right)F(n,k+1)\] \[\qquad\qquad+(n+1)^{2}\left(k+n^{2}+3\right)\left(2k^{2}-2kn^{2} +2kn+6k-n^{3}-n^{2}+4n+4\right)F(n,k+2)\] \[\qquad\qquad\qquad+(n+1)^{2}(k+n+1)\left(2k-n^{2}+n+4\right) \left(k+n^{2}+1\right)F(n,k)\] \[\qquad\qquad\qqu... | outline | |
\[\langle\psi_{1}|U_{\left(\begin{smallmatrix}N&0\\ 0&1\end{smallmatrix}\right)},\psi_{2}\rangle\] \[= 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{smallmat... | outline | |
\[\lim\limits_{p\to\infty}\sum\limits_{j_{1},j_{2}=0}^{p}C_{j_{2}j_{2}j_{1}j_{1}}=\] \[=\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_... | outline | |
\[\left(\sum_{\begin{subarray}{c}\sigma\in S(A,B)\\ \sigma(a_{j})=a_{j}\end{subarray}}(-1)^{j-1}\operatorname{sgn}(\sigma)e(( \sigma T)_{j})+\sum_{\begin{subarray}{c}\sigma\in S(A,B)\\ \sigma(a_{j+1})=a_{j}\end{subarray}}(-1)^{j}\operatorname{sgn}(\sigma)e(( \sigma T)_{j+1}))\right)\otimes x_{a_{j}}\] \[= \left(\sum_{\... | outline | |
\[\left(\mathcal{A}^{-}(h_{i_{1}},l_{1})\mathcal{A}^{+}(h_{i_{1}+1 })\mathcal{A}^{+}(h_{2n})\Omega\right)\left(x_{i_{1}},\ldots,x_{i_{2n}} \setminus x_{i_{1}},x_{j_{1}}\right)\] \[\quad=\int_{X^{2}}\sigma^{(2)}(dx_{i_{1}}\,dx_{j_{1}})\,\mathrm{ Tr}_{1}\,Q_{2}(x_{i_{1}},x_{j_{1}^{(1)}})Q_{3}(x_{i_{1}},x_{j_{2}^{(1)}})\]... | outline | |
\[\nabla u(t,x)=\frac{1}{\tau(t)^{d/2}}\nabla_{x}\left(v\left(t, \frac{x}{\tau(t)}\right)e^{i\frac{\dot{\tau}(t)}{\tau(t)}\frac{|x|^{2}}{2}}\right)\] \[=\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\rig... | outline | |
\[\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 \epsil... | outline | |
\[\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}... | outline | |
\[d\big{\langle}P^{(i+1)}(X^{(i)}-\mathbb{E}_{t}[X^{(i)}]),X^{(i)}- \mathbb{E}_{t}[X^{(i)}]\big{\rangle}\] \[=\bigg{\{}\Big{\langle}\big{(}(A+BK^{(i)})^{\top}P^{(i+1)}+P^{(i+1) }(A+BK^{(i)})\] \[\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{\... | outline | |
\[\rho(\mathbf{c};\frac{M}{2})+\rho(\mathbf{d};\frac{M}{2}) =\sum_{i=0}^{\frac{M}{2}-1}\left(c_{i}c_{i+\frac{M}{2}}+d_{i}d_{i +\frac{M}{2}}\right)\] \[=\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... | outline | |
\[\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}^{\varepsi... | outline |
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