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\begin{tabular}{|c|c|c|} \hline \multirow{2}{*}{Formulation} & Size of moment matrices (\(d\times d\)) & \(4\) \\ & Penalty parameter (\(\gamma\)) & \(10.0\) \\ & Burer-Monteiro rank & \(5\) (full) \\ & Number product measures (\(L\)) & \(6\) \\ \hline \multirow{3}{*}{Solver (KKT)} & Stopping criterion... | table | |
\[\sup_{\|\theta-\bar{\theta}_{n}\|\leq\bar{\delta}}\left|\sum_{i,j, k=1}^{d}\left(\frac{\partial^{3}}{\partial\theta_{j}\partial\theta_{i} \partial\theta_{k}}\log\pi(\theta)\right)u_{i}v_{j}w_{k}\right|\] \[\leq \sup_{\|\theta-\bar{\theta}_{n}\|\leq\bar{\delta}}\left\{\frac{3( \nu+d)}{\left[\nu+(\theta-\mu)^{T}\Sigma^... | outline | |
\[c\|A\|^{2}\|B\|^{2}-f(A,B;q)\] \[=\sum_{\begin{subarray}{c}i,k\\ i\neq k\end{subarray}}|\sqrt{c-q}\sum_{j}a_{ij}b_{jk}+\varepsilon_{1}\sqrt{c-1 }\sum_{\ell}a_{\ell k}b_{i\ell}|^{2}\] \[+c\sum_{D_{4}}|a_{ij}|^{2}|b_{k\ell}|^{2}+(c-1-q)\sum_{D_{1}}|a_{ ij}|^{2}|b_{k\ell}|^{2}\] \[-c\sum_{\begin{subarray}{c}i,j,k,\ell\\... | outline | |
\[y_{n}(t,x)=\frac{1}{M}\sum_{i=1}^{M}y_{n-1}^{i}(t,x)\] \[+\sum_{j=1}^{Q}\frac{w_{n,j}}{M}\sum_{i=1}^{M}\left[\left(f\left(y_ {n-1},z_{n-1}\right)-f(y_{n-2},z_{n-2})\right)\left(t_{n,j},x+W_{t_{n,j}-t}^{(n,i)}\right)\right]\] \[z_{n}(t,x)=\frac{1}{M}\sum_{i=1}^{M}z_{n-1}^{i}(t,x)\frac{W_{T-t} ^{(n,i)}}{T-t}\] \[+\sum_... | outline | |
\[|\mathcal{F}(\varphi,\sigma)-\mathcal{F}(\varphi_{\rm s},\sigma_ {\rm s})|^{1-\widetilde{\kappa}}\] \[\leq|\mathcal{F}(\varphi,\sigma)-\mathcal{F}(\widetilde{\varphi},\sigma)|^{1-\widetilde{\kappa}}+|\mathcal{F}(\widetilde{\varphi},\sigma)- \mathcal{F}(\varphi_{\rm s},\sigma_{\rm s})|^{1-\widetilde{\kappa}}\] \[\leq ... | outline | |
\[2\gamma\mathbb{E}\left[\mathrm{Gap}(\bar{x}^{N})\right]\leq \frac{3D^{2}}{N}+2\gamma^{2}C_{2}\tau^{2}M^{-2}B^{-2}\sum_{k=0}^ {N-1}\left(\sigma^{2}+\Delta^{2}\mathbb{E}\left[\|x^{k+1/2}-x^{*}\|^{2}\right]\right)\] \[+3\gamma^{2}C_{1}\tau B^{-1}\cdot\frac{1}{N}\sum_{k=0}^{N-1} \left(\sigma^{2}+\Delta^{2}\mathbb{E}\left... | outline | |
\[\text{under}\ \mathcal{M}^{e}_{r+}: d\mathbf{X}/dt=-\mathrm{grad}f(\mathbf{X})=-P_{\mathbf{U}}\nabla f( \mathbf{X})-\nabla f(\mathbf{X})P_{\mathbf{U}}+P_{\mathbf{U}}\nabla f(\mathbf{ X})P_{\mathbf{U}},\] \[\text{under}\ \mathcal{M}^{q_{1}}_{r+}: d\ell([\mathbf{Z}])/dt=-\overline{\mathrm{grad}\,h_{r+}([\mathbf{Y}])} \... | outline | |
\[\begin{split}&\left\|\sqrt{\log\frac{1}{\varepsilon}}.X_{ \varepsilon}^{\tau}(t,x)-\frac{h^{(\tau)}(1)}{\tau!\,s(\tau)}\left(\frac{3 \hat{\lambda}^{2}}{2\pi}\right)^{|\tau|}\hat{\lambda}e^{\mathfrak{m}\,t}P_{t+ \varepsilon^{2}}\eta(x)\right\|_{L^{2}(\mathbb{P})}\\ \leq&\frac{|h^{(\tau)}(1)|}{s(\tau)}\sum_{\kappa\in C... | outline | |
\[\sum_{v_{1},\ldots,v_{n-\eta}}\left[\prod_{i=1}^{n-\eta}\hat{ \phi}\left(\frac{\log v_{i}}{\log R}\right)\left(\frac{\chi_{0}(v_{i})\Lambda( v_{i})}{\sqrt{v_{i}}\log R}\right)\right]J_{k-1}\left(\frac{4\pi m\sqrt{cv_{1} \cdots v_{n-\eta}}}{b\sqrt{N}}\right)\] \[=\ \sum_{\gamma=0}^{a-\eta-1}\sum_{j=\gamma}^{a-\eta-1}(... | outline | |
\[\|\pi_{\mathcal{Q}_{p}}u-\pi_{\mathcal{S}_{p}}u\|_{L^{2}(\hat{ \kappa})}^{2} \leq\|(\pi_{\mathcal{Q}_{p}}u-\pi_{\mathcal{S}_{p}}u)W^{-1}\|_{L^{2 }(\hat{\kappa})}^{2}\] \[=\sum_{\begin{subarray}{c}|i|=p-1\\ p-1\geq i_{k}\geq 1,k=1,2\end{subarray}}^{2(p-1)}|a_{i_{1}i_{2}}|^{2}\prod_{k=1}^{2} \frac{2}{2i_{k}+1}\frac{1}{... | matrix | |
\[\mathbb{E}[\phi_{k+1}(x^{*})-F(x^{*})]\leq(1-\alpha_{k})\mathbb{E }[\phi_{k}(x^{*})-F(x^{*})]+\left(\frac{2}{L}+\frac{1}{\mu}\right)\mathbb{E}[ \|\bar{w}_{k,N_{k}}\|^{2}]\] \[\leq(1-\alpha_{k})(1-\alpha_{k-1})\mathbb{E}[\phi_{k-1}(x^{*})-F( x^{*})]+\alpha_{k}\bigg{(}\frac{2}{L}+\frac{1}{\mu}\bigg{)}\,\mathbb{E}[\|\ba... | outline | |
\[C_{1} =C_{1}\left(\|(1+x^{2}+v^{2})\nabla^{j}\widetilde{W}_{N,s}\|_{L^ {2}_{v}(L^{\infty}_{x})},\,\|\widetilde{W}_{N,s}\|_{H^{j}_{2}}\right)\quad j=0,1;\] \[C_{2} =C_{2}\left(\|v^{2}\nabla^{j}\widetilde{W}_{N,s}\|_{L^{2}_{v}(L^ {\infty}_{x})},\,\|\widetilde{W}_{N,s}\|_{H^{j}_{2}}\right)\quad j\leq 3;\] \[C_{3} =C_{3}... | outline | |
\[K(\nabla r;E_{i}) =R^{\nabla r}(E_{i},\nabla r,E_{i},\nabla r)=g_{\nabla r}(R^{ \nabla r}(E_{i},\nabla r)\nabla r,E_{i})\] \[=g_{\nabla r}(D^{\nabla r}_{E_{i}}D^{\nabla r}_{\nabla r}\nabla r -D^{\nabla r}_{\nabla r}D^{\nabla r}_{E_{i}}\nabla r-D^{\nabla r}_{[E_{i}, \nabla r]}\nabla r,E_{i})\] \[=-g_{\nabla r}(D^{\nab... | outline | |
\[\begin{split}\pi_{S}\rho_{S}\{\gamma_{S}(\bar{a})[p]_{\chi}\gamma _{S}(\bar{b})[q]\}=&\sum_{\begin{subarray}{c}-t_{1}-\frac{1}{2} <_{S}(\bar{i},m)_{0}^{\flat}<_{S}0\\ -t_{2}-\frac{1}{2}<_{S}(\bar{j},n)_{0}^{\sharp}<_{S}0\end{subarray}}s(a,\bar{ b})s(a,r_{j_{q}}^{n_{q}})\left(\overline{[b,r_{n_{0}}^{j_{0}}]}^{\sharp S... | matrix | |
\[\widetilde{J_{ij}} = e^{-2\phi}J_{ij}-\frac{e^{-2\phi}}{2}\Bigg{(}g_{ij}\Big{(}2S_{ kl}\nabla^{k}\nabla^{l}\phi-2S_{kl}\nabla^{k}\phi\nabla^{l}\phi-2S\Delta \phi+S\nabla_{k}\phi\nabla^{k}\phi\] \[+ \Delta\phi\Delta\phi-\Delta\phi\nabla_{k}\phi\nabla^{k}\phi- \nabla_{k}\nabla_{l}\phi\nabla^{k}\nabla^{l}\phi+2\nabla_{k... | outline | |
\[\frac{d}{dt}\frac{1}{\det(g_{t})} =-2\langle d\vec{\Phi},d\vec{w}\rangle_{g}e^{-4\lambda}\] \[\frac{d^{2}}{dt^{2}}\frac{1}{\det(g_{t})} =-2\left(|d\vec{w}|_{g}^{2}+\langle d\vec{\Phi},d\vec{w}\rangle_{g }^{2}-16|\partial\vec{\Phi}\,\dot{\otimes}\,\partial\vec{w}|_{\rm WP}^{2}+ \mathscr{R}(\vec{w},\vec{w})\right)e^{-4... | outline | |
\[\left[\begin{array}{cccccccc}A_{1,1}&\mathbf{0}&\mathbf{0}&\mathbf{0}& \cdots&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\cdots&\mathbf{0}&\mathbf{0}& \mathbf{0}\\ \mathbf{0}&\mathbf{0}&-A_{1,1}&\mathbf{0}&\cdots&\mathbf{0}&\mathbf{0}& \mathbf{0}\\ \mathbf{0}&\mathbf{0}&-2A_{1,1}&\... | matrix | |
\[\mathfrak{K}_{2}^{T}\left(D_{z,\alpha}^{2},D_{w,\beta}^{2}\right)\] \[=\sum_{\begin{subarray}{c}p_{1},p_{2},q_{1},q_{2}=0\\ p\neq q\end{subarray}}^{\infty}\frac{1-\overline{\alpha}^{p_{1}T}\beta^{q_{1} T}}{1-\overline{\alpha}^{p_{1}}\beta^{q_{1}}}\cdot\frac{1-\overline{\alpha}^{p_{ 2}T}\beta^{q_{2}T}}{1-\overline{\al... | outline | |
\[ERK(5):\begin{cases}t_{41}:\sum b_{i}=1,\\ t_{42}:\sum b_{i}c_{i}=1/2,\\ t_{43}:\sum b_{i}c_{i}^{2}=1/3,\\ t_{44}:\sum b_{i}c_{i}^{3}=1/4,\\ t_{45}:\sum b_{i}a_{ij}c_{j}=1/6,\\ t_{46}:\sum b_{i}c_{i}a_{ij}c_{j}=1/8,\\ t_{47}:\sum b_{i}a_{ij}c_{j}^{2}=1/12,\\ t_{48}:\sum b_{i}a_{ij}a_{jk}c_{k}=1/24,\end{cases}\begin{c... | outline | |
\[y_{OCT} =z_{1}+z_{2}+z_{3},\] \[1 =z_{1}+z_{2}+z_{3}+z_{4},\] \[x_{2}-\rho||x_{2}||_{q} \geq 0.9319-M(1-z_{1}),\] \[x_{2}+\rho||x_{2}||_{q} \leq 0.9319+M(1-z_{2})-\epsilon,\] \[0.1712x_{1}-0.06246x_{2}+\rho||0.1712x_{1}-0.06246x_{2}||_{q} \leq 0.06421+M(1-z_{2})-\epsilon,\] \[x_{2}+\rho||x_{2}||_{q} \leq 0.9319+M(1-z... | outline | |
\[Z^{(\mathcal{T}_{1}^{\vee})}(\bm{m},\bm{\eta}) =e^{2\pi i\eta N_{1}\delta}\,\int\,\left[d\bm{s}^{1}\right]\left[d \bm{\sigma}^{2}\right]\,\Big{[}\ldots\Big{]}\,Z_{\mathrm{FI}}(\bm{s}^{1},\eta) \,Z_{\mathrm{1-loop}}^{\mathrm{vec}}(\bm{s}^{1})\,Z_{\mathrm{1-loop}}^{ \mathrm{fund}}(\bm{s}^{1},\bm{m}^{1}-\delta)\] \[\qqu... | outline | |
\[y_{*}(t,\epsilon):= \,\eta^{1\!/\!p}\,z_{1}(\eta,\delta)\] \[= \,\eta^{1\!/\!p}\,\delta\big{(}1+\eta\delta^{p-1}+O((\eta\delta^ {p-1})^{2})\big{)}\] \[= \,\gamma M\epsilon^{\alpha}t\big{(}1+{\frac{1}{2}} \gamma^{2}t^{2}+O(t^{4})+|g|K_{p}(\gamma M)^{p-1}\epsilon^{\alpha(p-1)}t^{p}+ \epsilon^{\alpha(p-1)}O(t^{p+2})+\ep... | outline | |
\[\begin{split}\pi_{s}^{*}=&-\frac{2Y_{s-}^{\eta^{*}, \phi^{*}}G(Z_{s},s)}{\Sigma(Z_{s})}\left[\mu\left(Z_{s}\right)-r-\frac{\sigma(Z _{s})b(Z_{s})\rho_{W}G_{z}(Z_{s},s)}{G(Z_{s},s)}\right],\\ \eta_{1,s}^{*}=&-\frac{\sigma(Z_{s})}{\Sigma(Z_{s} )}\left(\mu\left(Z_{s}\right)-r\right)-\rho_{W}b(Z_{s})\frac{\int_{\mathbb{R... | outline | |
\[\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\Big{[}\frac{(PU_{x_{e},\lambda_{e}})^{2^{*}_{\mu}-2}w_{\varepsilon}^{2}(PU_{x_{e},\lambda_{e}}(x))^{ 2^{*}_{\mu}-1}}{|x-\xi|^{\mu}}+\frac{(PU_{x_{e},\lambda_{e}})^{2^{*}_{\mu}-2}w _{\varepsilon}^{2}(PU_{x_{e},\lambda_{e}}(x))^{2^{*}_{\mu}-2}w_{\varepsilon}}{| x-\xi|^{\mu}}\B... | outline | |
\[\sum_{\lambda}\left(\sum_{\mu\geq\lambda/8}w_{\lambda}\|u_{\mu}( \sigma)\|_{L^{2}}\right)d_{\lambda} \lesssim\sum_{\lambda}\left(\sum_{j=-3}^{\infty}a_{j}w_{2^{j} \lambda}\|u_{2^{j}\lambda}(\sigma)\|_{L^{2}}\right)d_{\lambda}=\sum_{j=-3}^{ \infty}a_{j}\left(\sum_{\lambda}w_{2^{j}\lambda}\|u_{2^{j}\lambda}(\sigma)\|_ ... | outline | |
\[L(x,y)*_{2}M(u,v)= (((a^{\prime}_{2}x+b^{\prime}_{2}y)^{q}(a^{\prime}_{3}u+b^{\prime }_{3}v)+(a^{\prime}_{2}x+b^{\prime}_{2}y)(a^{\prime}_{3}u+b^{\prime}_{3}v)^{q} )^{p^{i}}\] \[+B^{\prime}((c^{\prime}_{2}x+d^{\prime}_{2}y)^{q}(c^{\prime}_{3}u +d^{\prime}_{3}v)+(c^{\prime}_{2}x+d^{\prime}_{2}y)(c^{\prime}_{3}u+d^{\pr... | outline | |
\[\begin{split}|\mathcal{G}_{H(d+1)+i}(\phi)|^{2}&=4 \biggl{(}\int_{[a,b]^{d}}\bigl{[}\mathfrak{R}_{\infty}\bigl{(}\mathfrak{b}_{i }^{\phi}+\sum_{j=1}^{d}\mathfrak{w}_{i,j}^{\phi}x_{j}\bigr{)}\bigr{]}( \mathscr{N}_{\infty}^{\phi}(x)-f(x))\,\mu(\mathrm{d}x)\biggr{)}^{\!\!2}\\ &\leq 4\int_{[a,b]^{d}}\bigl{|}\mathfrak{R}_... | outline | |
\[\int_{\widetilde{\mathbf{F}^{\times}}}\left(\int_{\mathbf{F}^{ \times}}\widehat{\phi}(t)\chi(t)|t|^{\frac{1}{2}}\mathrm{d}^{\times}t\right) \cdot\left(\int_{\mathbf{F}^{\times}}H^{\mathfrak{F}^{3}}(t)\psi(-t)\chi^{-1} (t)|t|^{\frac{1}{2}}\mathrm{d}^{\times}t\right)\mathrm{d}\mu_{\mathrm{PL}}(\chi)\] \[=\int_{\mathbf{... | outline | |
\[\sum_{i}\big{\|}\frac{\sqrt{|\mathbf{A}_{n}|}}{\eta(n)^{2}} \mathbb{E}_{\mu_{n}^{\otimes 2}}\big{[}\bar{h}_{n}^{i}(\bm{\phi}X_{n})\Delta_{in}^{ \bm{\phi}}-\mathbb{E}[\bar{h}_{n}^{i}(\bm{\phi}X_{n})\Delta_{in}^{\bm{\phi}}| \mathbb{G}]\big{|}\mathbf{A}_{n}^{k_{n}}\big{]}\big{\|}_{1}\] \[\leq\sum_{i,j}\big{\|}\frac{|\ma... | outline | |
\[{}_{0}\mathbb{F}_{3}(T):={}_{0}W^{1/2-1/2p}_{p}(J;L_{p}(\Sigma)^{2})\cap L_{p}(J;W^ {1-1/p}_{p}(\Sigma)^{2}),\] \[{}_{0}\mathbb{F}_{4}(T):= {}_{0}W_{p}^{1/2-1/2p}(J;L_{p}(\Sigma))\cap L_{p}(J;W_{p}^{1-1/p}( \Sigma)),\] \[{}_{0}\mathbb{F}_{5}(T):= {}_{0}W_{p}^{1/2-1/2p}(J;L_{p}(S_{1}))\cap L_{p}(J;W_{p}^{1-1/p}(S_{1} ... | outline | |
\[(\delta_{L}\operatorname{Ob}^{\mathfrak{b}}_{(\phi_{\mathfrak{g}},\phi_{\mathfrak{a}})})(x,y,z)\] \[=-\operatorname{Ob}^{\mathfrak{b}}_{(\phi_{\mathfrak{g}},\phi_{ \mathfrak{a}})}([x,y],z)+\operatorname{Ob}^{\mathfrak{b}}_{(\phi_{\mathfrak{g}},\phi_{\mathfrak{a}})}([x,z],y)+\operatorname{Ob}^{\mathfrak{b}}_{(\phi_{ \... | outline | |
\[\left[\begin{array}{cc}L_{n}(F_{m}F_{n}+F_{m+1}F_{n+1})/2&L_{n}(L_{m}F_{n}+ F_{m}L_{n})/4\\ L_{n}(L_{m}F_{n}+F_{m}L_{n})/4&L_{n}(F_{m-1}F_{n-1}+F_{m}F_{n})/2\end{array} \right]+\\ \left[\begin{array}{cc}3F_{n}(F_{m-1}F_{n}+F_{m}F_{n+1})/2&5F_{n}(F_{m-1 }F_{n-1}+2F_{m}F_{n}+F_{m+1}F_{n+1})/4\\ F_{n}(F_{m-1}F_{n-1}+2F_... | matrix | |
\[\begin{split}&\left\|r^{\frac{1}{2}-\frac{3\theta}{2}}(t+2+r)^{ \frac{1-\theta}{2}}(t+2-r)^{\frac{x-1+\theta}{2}+\alpha}\phi\right\|_{L^{ \infty}_{t}L^{\sigma}_{r}L^{\beta}(S^{2})}\\ \leq& C\Big{\|}r^{\frac{1}{2}-\frac{3\theta}{2}}(t+2+r)^{ \frac{1-\theta}{2}}(t+2-r)^{\frac{x-1+\theta}{2}}\\ &\times\int_{0}^{t}\int_{... | outline | |
\[E_{i}(x^{\prime}\xi_{-\lambda}\otimes y\eta_{\lambda+\zeta})\] \[=E_{i}x^{\prime}\xi_{-\lambda}\otimes y\eta_{\lambda+\zeta}+ \tilde{K}_{i}x^{\prime}\xi_{-\lambda}\otimes E_{i}y\eta_{\lambda+\zeta}\] \[=E_{i}x^{\prime}\xi_{-\lambda}\otimes y\eta_{\lambda+\zeta}+ \tilde{K}_{i}x^{\prime}\xi_{-\lambda}\otimes\frac{\tild... | outline | |
\[\left\{\begin{array}{l}d\mathscr{P}_{1}=-\Big{[}\mathscr{P}_{1}(A+B\Theta_{1} )+(A+B\Theta_{1})^{\top}\mathscr{P}_{1}+(C+D\Theta_{1})^{\top}\mathscr{P}_{1}(C +D\Theta_{1})\\ \qquad-\big{[}Q+\Theta_{1}^{\top}S+\Theta_{1}^{\top}R\Theta_{1}+S^{\top}\Theta _{1}\big{]}\Big{]}ds,\\ d\mathscr{P}_{2}=-\Big{\{}\mathscr{P}_{2}... | matrix | |
\[\Pr(\text{adversary guesses assignment wrong}|S=[s_{1},s_{2}])\] \[= \Pr(\text{adversary guesses assignment wrong}|S=[s_{1},s_{2}],D=P_{1}) \Pr(D=P_{1}|S=[s_{1},s_{2}])\] \[+\Pr(\text{adversary guesses assignment wrong}|S=[s_{1},s_{2}],D=P_{2}) \Pr(D=P_{2}|S=[s_{1},s_{2}])\] \[= \frac{\min\{f_{1}(s_{1})f_{2}(s_{2}),f... | outline | |
\[\pi^{*}(D_{\Gamma,d}(\varphi)) =e(\pi,\left[\begin{matrix}1\\ d\end{matrix}\right]_{\Gamma^{\prime}},\left[\begin{matrix}1\\ d\end{matrix}\right]_{\Gamma})\sum_{(db,Al)=d}\varphi(ab)\left[\begin{matrix}a\\ db\end{matrix}\right]_{\Gamma^{\prime}}+\varphi(l)e(\pi,\left[\begin{matrix}1 \\ dl\end{matrix}\right]_{\Gamma^{... | matrix | |
\[\widehat{Q_{1}^{n+1}}(x,y)=(-1)^{x\cdot y}\] \[\times\left(\frac{1}{2}+\frac{1}{2n}\left[(-1)^{y_{1}}+(-1)^{y_{2} }+(-1)^{y_{3}}+\cdots+(-1)^{y_{n}}\right]\right)\] \[\times\left(\frac{1}{2}+\frac{1}{2n}\left[(-1)^{y_{1}}+(-1)^{y_{ 1}+y_{2}}+(-1)^{y_{1}+y_{3}}+\cdots+(-1)^{y_{1}+y_{n}}\right]\right)\] \[\times\left(\... | outline | |
\[M_{2,p}(\mathcal{F}_{r_{1},X,\delta_{1},q_{1}}\times\mathcal{F}_{r_{2},X,\delta_{2 },q_{2}})\text{ to be}\] \[\left(\sum_{\begin{subarray}{c}\xi_{1}<X^{\delta_{1}}\end{subarray} }\dim H^{*}_{k_{1},q_{1}}(\chi_{0})\sum_{\begin{subarray}{c}k_{2}<X^{\delta_{ 2}}\end{subarray}}\dim H^{*}_{k_{2},q_{2}}(\chi_{0})\right)^{-... | outline | |
\[(S_{TR})_{0,3;i_{1}i_{2}i_{3}} =2\sum_{\begin{subarray}{c}\alpha_{1},\alpha_{2},\alpha_{3}\in Ram,j_{1},j_{2},j_{3}>0\\ j_{1}\ odd\end{subarray}}Res_{z_{\alpha}=0}\left(-\frac{1}{4j_{1}}f_{j_{1}, \alpha_{1}}(z_{\alpha})f_{j_{2},\alpha_{2}}(z_{\alpha})f_{j_{3},\alpha_{3}}(- z_{\alpha})\right)c_{i_{1}}^{j_{1},\alpha_{1... | matrix | |
\[jN_{m+3}^{(3)}-3JN_{m+3}^{(3)} =j_{m+3}^{(3)}+j_{m+4}^{(3)}\mathbf{i}+j_{m+5}^{(3)}\mathbf{j}+j_{ m+6}^{(3)}\mathbf{k}\] \[\quad-3(J_{m+3}^{(3)}+J_{m+4}^{(3)}\mathbf{i}+J_{m+5}^{(3)} \mathbf{j}+J_{m+6}^{(3)}\mathbf{k})\] \[=(j_{m+3}^{(3)}-3J_{m+3}^{(3)})+(j_{m+4}^{(3)}-3J_{m+4}^{(3)}) \mathbf{i}\] \[\quad+(j_{m+5}^{(... | outline | |
\[v_{2}= \bar{J}_{p,p-d}^{\top}\mathbb{E}[(X-t(x))f(X)\chi_{B_{\varepsilon}^{ \mathbb{R}^{p}}(t(x))}(X)]\] \[= \frac{|S^{d-1}|}{d+2}\frac{f(x)P(x)\bar{J}_{p,p-d}^{\top}\mathfrak{ M}_{0}(x)}{2}\varepsilon^{d+2}+\frac{|S^{d-1}|}{24}\frac{f(x)P(x)\bar{J}_{p,p-d}^{ \top}\mathfrak{N}_{1}(x)}{2}\varepsilon^{d+4}\] \[+\frac{|... | outline | |
\[= \frac{\rho+\mu}{8}\!\!\left\|\begin{bmatrix}\mathbf{v}^{k}\!-\! \mathbf{v}^{k+1}\\ \mathbf{e}_{1}^{k}\!-\!\mathbf{e}_{1}^{k+1}\\ \mathbf{e}_{2}^{k}\!-\!\mathbf{e}_{2}^{k+1}\end{bmatrix}\right\|_{2}^{2}+\rho \omega\beta V^{2}-\rho\omega\beta\|\mathcal{X}^{k}\|_{2}^{2}.\] \[\frac{\rho\!+\!\mu}{4}\!\!\left\|\begin{bma... | outline | |
\[|e_{2}| \leq h\|\bm{\phi}\|_{C^{1}(\overline{\Omega})}\sum_{\sigma\in\Sigma _{\text{int}}}\int_{\sigma}\left\|(\{\bm{F}_{h}\}-\bm{F}(\bm{U}_{\sigma}^{RP})) \cdot\bm{n}\right\|\ dS_{\bm{x}}\] \[\lesssim h\|\phi\|_{C^{1}(\overline{\Omega})}\sum_{\sigma:=L|R\in \Sigma_{\text{int}}}\int_{\sigma}\left\|(\bm{F}(\bm{U}_{L})... | outline | |
\[S_{n} \leq\lambda_{max}D\sum_{j\in[d]}\Big{(}\Big{|}\hat{M}_{1}[j]-M_{1 }(\theta^{*})[j]\Big{|}\Big{)}\Big{(}\big{|}\hat{M}_{1}[j]\big{|}+\big{|}M_{1 }(\theta^{*})[j]\big{|}+2\sup_{\theta\in\Theta}\big{|}M_{1}(\theta)[j]\big{|} \Big{)}\] \[+\lambda_{max}D\sum_{j,k\in[d]}\Big{(}\Big{|}\hat{M}_{2}[j,k]-M_{ 2}(\theta^{*... | outline | |
\[\int_{0}^{1}\!\int_{\Omega}\left(\frac{\|\tilde{m}\|^{2}}{\rho}+ \tau^{2}\beta^{-2}\frac{\|\nabla\rho\|^{2}}{\rho}+2\tau\beta^{-1}\frac{\tilde{ m}\cdot\nabla\rho}{\rho}\right)\mathrm{d}t\mathrm{d}x+2\tau\mathcal{E}_{2}( \rho(1,\cdot))\] \[= \int_{0}^{1}\!\int_{\Omega}\left(\frac{\|\tilde{m}\|^{2}}{\rho}+ \tau^{2}\bet... | outline | |
\begin{table}
\begin{tabular}{c c c c} Nodes & \(n\) & \(\omega\) & parameters \\ \hline \(\{(0,0,0)\}\) & 4 & \(\frac{41-9\sqrt{2}}{41160}\) & - \\ \(\{(a,0,0)\}\) & 12 & \(\frac{8+9\sqrt{2}}{13720}\) & \(\frac{3-\sqrt{3(\sqrt{2}-1)}}{6}\) \\ \(\{(b,b,0)\}\) & 12 & \(\frac{10-\sqrt{2}}{1715}\) & \(\frac{4-\sqrt{2}}{12... | table | |
\[\begin{array}{rl}P_{k|k}&=\mathbf{E}[(\mathcal{X}_{k}-\hat{\mathcal{X}}_{k| k})(\mathcal{X}_{k}-\hat{\mathcal{X}}_{k|k})^{T}]\\ &=\mathbf{E}[((\mathcal{X}_{k}-\hat{\mathcal{X}}_{k|k-1})-K_{k}(H_{k}\mathcal{ X}_{k}+r_{k}-H_{k}\hat{\mathcal{X}}_{k|k-1}))((\mathcal{X}_{k}-\hat{\mathcal{X}}_{ k|k-1})-K_{k}(H_{k}\mathcal{... | outline | |
\[B_{s}(i,j) =\lambda_{i}\,d_{sKL}(\text{PG}_{i},\text{PG}_{ij})+\lambda_{j}\, d_{sKL}(\text{PG}_{j},\text{PG}_{ij})\] \[=\frac{1}{2}\lambda_{i}\,\text{tr}\left(\Sigma_{ij}^{-1}\Sigma_{i }+\Sigma_{i}^{-1}\Sigma_{ij}+(\Sigma_{i}^{-1}+\Sigma_{ij}^{-1})(\mu_{i}-\mu_{ ij})(\mu_{i}-\mu_{ij})^{\top}\right)-d\,\lambda_{i}\] \... | outline | |
\[\left(\begin{array}{c}0\\ y_{0}\end{array}\right)=\left(\begin{array}{cc}A-i\omega I&B\\ C&D\end{array}\right)\underbrace{\left(\begin{array}{c}x_{0}\\ u_{0}\end{array}\right)}_{\neq 0},\ \omega\in\mathbb{R}\ \ \mbox{or}\ \ \ y_{0}=Du_{0},\ u_{0}\neq 0\ \Longrightarrow\ \ \left(\begin{array}{c}y_{0}\\ u_{0}\end{array... | matrix | |
\[\mathbb{P}(K,z,1,B\times\{\hat{p}_{i}\})\] \[\geq d_{1}\cdot\mathbb{P}\bigg{\{}\begin{array}{l}(\sigma_{0}, \sigma_{\bar{p}_{1}}-\bar{t}_{1},\ldots,\sigma_{\bar{p}_{j}}-\bar{t}_{j}, \sigma_{\hat{p}_{4}}-\hat{t}_{4},\ldots,\sigma_{\hat{p}_{i-1}}-\hat{t}_{i-1})=w \in W,\\ (\sigma_{\hat{p}_{1}}-\hat{t}_{1},\sigma_{\hat{... | outline | |
\[\mathcal{L}^{1}([0,1]\setminus A_{*}) =\mathcal{L}^{1}\left([0,1]\setminus\bigcap_{k=2}^{\infty}A_{k} \right)=\mathcal{L}^{1}\left(\bigcup_{k=2}^{\infty}\big{(}[0,1]\setminus A_{k }\big{)}\right)\] \[=\mathcal{L}^{1}\left(\bigcup_{k=2}^{\infty}\left(\big{(}[0,1] \setminus A_{k}\big{)}\setminus\left(\bigcup_{l=1}^{k-1... | outline | |
\[\mathrm{Per}_{s}(E,B_{R+1})-\mathrm{Per}_{s}(F,B_{R+1})\] \[\leq \int_{(\Omega^{(\delta,\rho)})^{+}}t^{1-s}\big{(}|\nabla\mathbf{ E}_{E}(X)|^{2}-|\nabla V^{(\delta)}(X)|^{2}\big{)}\,dX+\delta\] \[= \mathcal{F}_{s,\sigma}(\mathbf{E}_{E},\Omega^{(\delta,\rho)})- \mathcal{F}_{s,\sigma}(V^{(\delta)},\Omega^{(\delta,\rho)... | outline | |
\[\mathbb{E}\left[|\xi_{i}-\langle\mathbf{X}_{i},\mathbf{M}_{1}- \mathbf{M}^{*}\rangle|-|\xi_{i}-\langle\mathbf{X}_{i},\mathbf{M}-\mathbf{M}^{ *}\rangle|\;\middle|\mathbf{X}_{i}\right]\] \[= 2\int_{(\mathbf{X}_{i},\mathbf{M}-\mathbf{M}^{*})}^{(\mathbf{X}_ {i},\mathbf{M}-\mathbf{M}^{*})+\langle\mathbf{X}_{i},\mathbf{M}_... | outline | |
\begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{Sample Sizes} & \multicolumn{3}{c}{Shapiro-Wilks Test} & \multicolumn{3}{c}{Kolmogorov-Smirnov Test} \\ \cline{2-7} & NN & TRI & ND & NN & TRI & ND \\ \hline
50 & 0.878 & 0.884 & 0.881 & 0.584 & 0.597 & 0.595 \\
100 & 0.098 & 0.095 & 0.095 & 0.472 & 0... | table | |
\[\tilde{l}:=\frac{1}{2}\left(\frac{r}{\tau}-1+\sqrt{(1-\rho)^{2}-( \frac{r}{\tau})^{2}}-\rho\right),\qquad\rho_{\tilde{l}}:=1-\sqrt{(1-\rho)^{2} -(\frac{r}{\tau})^{2}}+\tilde{l},\] \[\tilde{r}_{\delta,c}^{2}:=\min\left\{2\rho,\frac{1}{2}((\frac{r}{ \tau})^{2}-\tilde{l}^{2}+\rho(2-\rho)+\rho_{\tilde{l}}(2-\rho_{\tilde{... | outline | |
\[\begin{split}&\dot{\bm{x}}(t)=\left(A_{0}+B_{0}K\right)\bm{x}(t)+ \left[\!\left[\left(A_{i}+B_{i}K\right)\right]_{i=1}^{\nu}\!\bm{\chi}(t,-1) \right.\\ &+\sum_{i=1}^{\nu}\int_{\mathcal{I}_{i}}\!\!\left(\widehat{A}_{ i}+\widehat{B}_{i}\left(I_{\kappa_{i}}\otimes K\right)\right)G_{i}(\tau)\bm{x}(t+ \tau)\mathrm{d}\tau+... | outline | |
\[\Phi_{N}^{1}(x,y) = \widetilde{P}_{0}^{(\alpha)}(x)\widetilde{P}_{2}^{(\alpha)}(y)+ \widetilde{P}_{2}^{(\alpha)}(x)\widetilde{P}_{0}^{(\alpha)}(y)\] \[-(4x^{2}+4y^{2}-2)\bigl{(}\widetilde{P}_{0}^{(\alpha)}(x) \widetilde{P}_{0}^{(\alpha)}(y)+\widetilde{P}_{1}^{(\alpha)}(x)\widetilde{P}_{ 1}^{(\alpha)}(y)\bigr{)}\] \[\... | outline | |
\[Q\mathit{Ell}_{E_{8}}(S^{4})= K_{\Lambda_{E_{8}}(1)}((S^{4})^{1})\times K_{\Lambda_{E_{8}}(-1)}((S ^{4})^{-1})\times K_{\Lambda_{E_{8}}(y_{3})}((S^{4})^{y_{3}})\] \[\times K_{\Lambda_{E_{8}}(y_{4})}((S^{4})^{y_{4}})\times K_{ \Lambda_{E_{8}}(y_{5})}((S^{4})^{y_{5}})\times K_{\Lambda_{E_{8}}(y_{6})}((S^ {4})^{y_{6}})\... | outline | |
\[\int_{\mathcal{N}_{\varepsilon}(\delta_{\varepsilon})}\mu_{ \varepsilon}\mathbf{A}_{\varepsilon}\mathbf{D}_{\varepsilon}\nabla u\cdot \mathbf{D}_{\varepsilon}\nabla\boldsymbol{v} =\int_{\mathcal{N}_{\varepsilon}(\delta_{\varepsilon})}\mu_{ \varepsilon}\mathbf{A}_{\varepsilon}\mathbf{D}_{\varepsilon}\nabla u\cdot \mat... | outline | |
\[\int_{\mathcal{D}_{i}}Q(\partial^{x}D_{t}^{k}\mathrm{div}\;u, \partial^{x}D_{t}^{k}P)dx\] \[= -\int_{\mathcal{D}_{i}}Q(\partial^{x}D_{t}^{k}(\frac{\rho^{\prime}( p)}{\rho}),\partial^{x}D_{t}^{k}(\frac{1}{2}|B|^{2}))-\int_{\mathcal{D}_{i}}Q( \partial^{x}D_{t}^{k}(\frac{\rho^{\prime}(p)}{\rho}),\partial^{x}D_{t}^{k}p)d... | outline | |
\[\sum_{j\in[a:b]}g(i(j),i(j+1))\] \[=g(i(a),i(a+1))+g(i(a+1),i(a+2))\] \[\quad+g(i(a+2),i(a+3))+\cdots+g(i(b),i(b+1))\] \[\geq R_{i(a+1)}+g(i(a),i(a+1),i(a+2))\] \[\quad+g(i(a+2),i(a+3))+\cdots+g(i(b),i(b+1))\] \[\geq R_{i(a+1)}+R_{i(a+2)}+g(i(a),i(a+1),i(a+2),i(a+3))\] \[\quad+\cdots+g(i(b),i(b+1))\] \[\ldots\] \[\ge... | outline | |
\[1-P_{D}=P(\hat{V}\neq V^{*}|V=V^{*})\leq\sum_{i,j}P(\hat{v}_{ij} \neq v_{ij}^{*}|V=V^{*})\] \[=\sum_{\begin{subarray}{c}i,j\\ v_{ij}=1\end{subarray}}P(\hat{v}_{ij}\neq v_{ij}^{*}|V=V^{*})+\sum_{ \begin{subarray}{c}i,j\\ v_{ij}=0\end{subarray}}P(\hat{v}_{ij}\neq v_{ij}^{*}|V=V^{*})\] \[=\sum_{\begin{subarray}{c}i,j\\ ... | matrix | |
\[\begin{split}\mathcal{I}_{21}&\leq\|\tilde{f}^{n}\| _{L^{\infty}_{q}}\int_{0}^{\frac{q}{2}}\int_{0}^{\pi}\int_{0}^{2\pi}\int_{R+ \Delta v+\Delta I}^{\infty}\frac{\delta\left(3+\delta\right)^{\frac{3}{2}} \left(\frac{3+\delta}{2}\right)^{\frac{\delta}{2}}r^{\delta+2}|\sin\varphi \cos^{\delta-1}k\sin^{2}k|}{r^{q-2}}drd... | outline | |
\[\text{minimize} \mathrm{Tr}\left(P^{\xi^{\top}}R_{x}P^{\xi}M+P^{w\top}R_{x}P^{w}M\right)\] \[+\mathrm{Tr}\left(Q^{w\top}R_{u}Q^{w}M+Q^{\xi^{\top}}R_{u}Q^{\xi}M\right)\] \[+\overline{x}^{\top}R_{x}\overline{x}+\overline{u}^{\top}R_{u} \overline{u}\] \[\text{subject to} Q^{w}\in\mathcal{Q}_{N},Q^{\xi}\in\mathcal{Q}_{C}... | outline | |
\[D_{2} (2k,2l,2m,2n+1)\] \[=D_{1}(2k+2m,2l+2n+1)D_{1}(2k-2m,2l-2n-1)\] \[=\left\{4(k+m)^{2}-4(l+n)^{2}-4(l+n)-1\right\}\left\{4(k-m)^{2}-4 (l-n)^{2}+4(l-n)-1\right\}\] \[\equiv-4(k+m)^{2}+4(l+n)^{2}+4(l+n)-4(k-m)^{2}+4(l-n)^{2}-4(l-n)+1\] \[\equiv-8k^{2}-8m^{2}+8l^{2}+8n(n+1)+1\] \[\equiv 8(k+l+m)+1\pmod{16},\] \[D_{2... | outline | |
\[\Big{\|}(\bm{L}_{t+1}\bm{Q}_{t}-\bm{L}_{\star})\bm{\Sigma}_{\star} ^{1/2}\Big{\|}_{\mathrm{F}}^{2} =\underbrace{\Big{\|}(1-\eta)\bm{\Delta}_{L}\bm{\Sigma}_{\star}^{1/2}- \eta\bm{L}_{\star}\bm{\Delta}_{R}^{\top}\bm{R}(\bm{R}^{\top}\bm{R})^{-1}\bm{ \Sigma}_{\star}^{1/2}\Big{\|}_{\mathrm{F}}^{2}}_{\mathrm{F}}\] \[\quad-... | outline | |
\[a(t)^{3} =\left(\int_{D\times V}\tilde{\varphi}(r,\upsilon)u_{t}(r, \upsilon)\mathrm{d}r\mathrm{d}\upsilon\right)^{3}\leq C_{3}\int_{D}\left(\int_ {V}\tilde{\varphi}(r,\upsilon)u_{t}(r,\upsilon)\mathrm{d}\upsilon\right)^{3} \mathrm{d}r\] \[=C_{3}\int_{D}\mathrm{d}r\int_{V\times V\times V}\tilde{\varphi}( r,\upsilon)\... | outline | |
\[(t-s)^{\frac{\vartheta}{\alpha}}\frac{|D_{x}w(s,x)-D_{x}w(s,x^{ \prime})|}{|x-x^{\prime}|^{\vartheta}} \leq |Dw(s,x)-Dw(s,x^{\prime})|\leq|Dw(s,x)|+|Dw(s,x^{\prime})|\] \[\leq C\bigg{(}\mathbf{d}_{\eta,s,t}(\mathbf{P}_{1},\mathbf{P}_{2})(t-s )^{\zeta}\] \[+\int_{s}^{t}dv(v-s)^{-\frac{1}{\alpha}+\frac{2\eta}{\alpha}} ... | outline | |
\[(J\circ m_{\mathcal{P}})^{\prime}(u)(v)=\lim_{t\to 0}\frac{J(m_{ \mathcal{P}}(u+tv))-J(m_{\mathcal{P}}(u))}{t}\] \[=\lim_{t\to 0}\frac{(r(u+tv)^{2-N}-r(u)^{2-N})\int_{\mathbb{R}^{N}}| \nabla u|^{2}\,dx+r(u+tv)^{2-N}t\int_{\mathbb{R}^{N}}\langle\nabla(2u+tv), \nabla v\rangle\,dx}{2t}\] \[\qquad-\lim_{t\to 0}\frac{(r(u... | outline | |
\[5\left|\int_{I}\int_{\mathbb{R}^{3}}Fv^{4}\partial_{t}v\mathrm{ d}x\mathrm{d}t\right|\] \[= \left|\int_{I}\int_{\mathbb{R}^{3}}F\partial_{t}(v^{5})\mathrm{d }x\mathrm{d}t\right|\] \[\leqslant \left|\int_{I}\int_{\mathbb{R}^{3}}\partial_{t}(F)\ v^{5}\mathrm{d} x\mathrm{d}t\right|+\int_{\mathbb{R}^{3}}|F|(b,x)|v|^{5}(b... | outline | |
\[\frac{1}{2}(\omega^{X}-2\epsilon_{1}g)H_{0}\cdot\nabla_{n_{\Sigma _{\tau}}}H_{0}\] \[= \frac{1}{2}\left(1-\frac{2M}{r}\right)^{-1/2}(\omega^{X}-2 \epsilon_{1}g)H_{0}\cdot\nabla_{t}H_{0}\] \[= \frac{1}{2}\left(1-\frac{2M}{r}\right)^{-1/2}(\omega^{X}-2 \epsilon_{1}g)H_{0}\cdot\nabla_{L}H_{0}-\frac{1}{2}\left(1-\frac{2M... | outline | |
\[\frac{V(\bm{Z}+\delta_{j}\bm{e}_{j},\bm{\theta})-V(\bm{Z},\bm{ \theta})}{a_{j}(\bm{\theta})h\sqrt{\nu_{j}}}\] \[\quad=\frac{1}{a_{j}(\bm{\theta})h\sqrt{\nu_{j}}}\Big{(}\underbrace {\bm{r}(\bm{Z},\bm{\theta})^{\top}\left(\bm{x}(\bm{Z}+\delta_{j}\bm{e}_{j},\bm {\theta})-\bm{x}(\bm{Z},\bm{\theta})\right)}_{\leq 0\text{ ... | outline | |
\[\det T = \det\left[\begin{array}{cccc}\mathbf{s}_{3}&\mathbf{t}_{3} \end{array}\right]\det\left[\begin{array}{cccc}\bar{\mathbf{u}}_{7}&\bar{ \mathbf{v}}_{7}\end{array}\right]\] \[= \det\left[\begin{array}{cccc}-y\mathbf{e}_{d-2}&\mathbf{0}_{(d -2)\times 1}\\ \mathbf{0}_{1\times(d-2)}&-y^{2}\end{array}\right]\det\lef... | matrix | |
\[\widetilde{K}_{n}(a) =\sum_{k=1}^{\lfloor nt_{1}\rfloor-1}a^{k}({\rm e}^{{\rm i}k \theta_{1}}-1-{\rm i}k\theta_{1})+\frac{a^{\lfloor nt_{1}\rfloor}}{1-a}\big{(} {\rm e}^{{\rm i}\lfloor nt_{1}\rfloor\theta_{1}}-1-{\rm i}\lfloor nt_{1} \rfloor\theta_{1}\big{)}\] \[\quad+(1-a)\sum_{\ell=1}^{\lfloor nt_{1}\rfloor}\sum_{k... | outline | |
\[\int_{-\infty}^{\infty}\int_{0}^{2\pi}\int_{0}^{\pi}|\psi(t,r, \theta,\phi)|^{2}\sin\theta d\theta d\phi dt=\int_{-\infty}^{\infty}\sum_{m, \ell}\left|\psi_{m\ell}^{(a\omega)}(r)\right|^{2}d\omega,\] \[\int_{-\infty}^{\infty}\int_{0}^{2\pi}\int_{0}^{\pi}\left| \partial_{r}\psi(t,r,\theta,\phi)\right|^{2}\sin\theta d\... | outline | |
\[f_{0,2} = 1,\] \[f_{0,3} = 1-2q+46q^{2}-1010q^{3}+21550q^{4}-463502q^{5}+{\cal O}(q^{6}),\] \[f_{2,1} = \frac{5}{2}-10q+50q^{2}-1090q^{3}+18770q^{4}-360310q^{5}+{\cal O} (q^{6}),\] \[f_{2,2} = \frac{5}{2}-32q+616q^{2}-14720q^{3}+338440q^{4}-7750832q^{5}+{ \cal O}(q^{6}),\] \[f_{2,3} = 5-46q+1058q^{2}-27910q^{3}+70397... | outline | |
\[K_{n}(\vec{y},\vec{x})=\sum_{I}(\sum_{I_{a_{1}}=0}^{C_{1}}... \text{no k}=\frac{|\mathcal{A}_{n}|-1}{2}...\sum_{I_{a_{|\mathcal{A}_{n}|}}=0}^ {C_{|\mathcal{A}_{n}|}}\begin{pmatrix}\sum_{\vec{a}\in\mathcal{A}_{n}}I_{a}\\ \Pi_{a\in\mathcal{A}_{n}}(I_{\vec{a}})\end{pmatrix})e^{mI}\\ C_{j}=min(\lfloor\frac{proj_{t}(\vec{... | matrix | |
\[\sum_{p_{1},p_{2},\ldots,p_{J}}\sum_{\rho\subseteq\{1,2,\ldots,J \}}\sum_{p^{\prime}_{j},j\in\rho}\sum_{n\sim M/(p_{1}\cdots p_{J}\prod_{j\in \rho}p^{\prime}_{j})}\tilde{g}(n)^{2}\] \[\lesssim M\prod_{p\leq x}\left(1+\frac{\tilde{g}(p)^{2}-1}{p} \right)\sum_{p_{1},p_{2},\ldots,p_{J}}\sum_{\rho\subseteq\{1,2,\ldots,J\... | outline | |
\[C=\left(\begin{array}{cc|cc}\frac{\partial^{2}f}{\partial k^{2}}\Big{|}_{A} &\frac{\partial^{2}f}{\partial k\partial\ell}\Big{|}_{A}&\frac{\partial^{2}f} {\partial k\partial m}\Big{|}_{A}\\ \frac{\partial^{2}f}{\partial k\partial\ell}\Big{|}_{A}&\frac{\partial^{2}f} {\partial\ell^{2}}\Big{|}_{A}&\frac{\partial^{2}f}{... | matrix | |
\[\mathbf{R}_{\alpha}(\tau,\mathbf{L}(\lambda,\mathbf{p}), \mathbf{p})\] \[=\left(\begin{array}{c}\mathbf{S}\left(\frac{\tau}{2}\left\{f( \mathbf{L}(\boldsymbol{\pi}_{\mathrm{T}}^{|\alpha|-1}\mathbf{p}_{1}), \boldsymbol{\pi}_{\mathrm{T}}^{|\alpha|-1}\mathbf{p}_{m})-f(0,0)-[Df(0,0)] \begin{pmatrix}\mathbf{L}(\boldsymbol... | matrix | |
\[|I_{1}| \leq C\|\partial_{x}^{2}\psi^{\varepsilon}\|_{L^{2}(0,t;L^{2}(\Omega _{\eta}))}\|\nabla\bm{v}^{\varepsilon}\|_{L^{2}(0,t;L^{2}(\Omega_{\eta}))}\] \[\leq C\varepsilon^{3/2}\left(\int_{0}^{t}\left(\varepsilon\| \partial_{x}^{3}\eta^{\varepsilon}(s)\|_{L^{2}(\omega)}^{2}+\varepsilon^{3/2} \|\partial_{x}^{3}\eta^... | outline | |
\[\begin{split}\frac{1}{2}\int_{0}^{T}\int_{G}&\big{(} \bm{\varepsilon}+(\mathbf{m}\cdot\nabla)\bm{\varepsilon}\big{)}\mathbf{E} \cdot\mathbf{E}+\big{(}\bm{\mu}+(\mathbf{m}\cdot\nabla)\bm{\mu}\big{)}\mathbf{ H}\cdot\mathbf{H}\mathrm{d}\mathbf{x}\mathrm{d}t\\ &=-\frac{1}{2}\int_{0}^{T}\int_{\Gamma}\bm{\nu}\cdot\mathbf{m... | outline | |
\[\sum_{\begin{subarray}{c}k=0\\ p\nmid k\end{subarray}}^{n}\frac{a_{n-k}a_{k}}{k^{2}}\stackrel{{ k\rightharpoonup}}{{=}} \stackrel{{ k\rightharpoonup}}{{=}} \stackrel{{ m-1}}{{=}}\sum_{\begin{subarray}{c}\ell=0 \\ p\nmid\ell\end{subarray}}^{m-1}\sum_{\begin{subarray}{c}\ell=0\\ p\nmid\ell\end{subarray}}^{p^{r}}\frac{a... | outline | |
\[\left\|\bm{X}_{k,l+1}-\bm{X}_{k}\right\|_{F} \leq\left\|\bm{X}_{k,l+1}-\mathcal{P}_{\bm{Q}_{k,l}}(\bm{X}_{k,l} +\alpha_{k,l}\bm{G}_{k,l})\right\|_{F}+\left\|\bm{X}_{k}-\mathcal{P}_{\bm{Q}_ {k,l}}(\bm{X}_{k,l}+\alpha_{k,l}\bm{G}_{k,l})\right\|_{F}\] \[\leq 2\left\|\bm{X}_{k}-\mathcal{P}_{\bm{Q}_{k,l}}(\bm{X}_{k,l}+ \a... | outline | |
\[\mathbb{K}_{a,0,0}^{-1}(\mathbf{w}_{0},\mathbf{b}_{1}+2ue_{1}+2 ve_{2})=\mathbb{K}_{a,0,0}^{-1}(\mathbf{w}_{0},\mathbf{b}_{1}+2ue_{1}-2ve_{2})\\ =\mathbb{K}_{a,0,0}^{-1}(\mathbf{w}_{1},\mathbf{b}_{0}-2ue_{1}+2 ve_{2})=\mathbb{K}_{a,0,0}^{-1}(\mathbf{w}_{1},\mathbf{b}_{0}-2ue_{1}-2ve_{2})\\ =i\mathbb{K}_{a,0,0}^{-1}(\... | outline | |
\[\begin{array}{ll}0\leq&||v||^{2}+\int_{\Omega}b(x)(g(v))^{22^{*}-1}g^{ \prime}(v)v-\int_{\Omega}h(x)(g(v))^{-\gamma}g^{\prime}(v)v\\ &+\epsilon\left[\int_{\Omega}\nabla v\nabla\varphi+b(x)(g(v))^{22^{*}-1}g^{ \prime}(v)\varphi-h(x)(g(v))^{-\gamma}g^{\prime}(v)\varphi\right]\\ &-\int_{[v+\epsilon\varphi<0]}\hskip-14.2... | outline | |
\[\begin{array}{rcl}\omega_{r}&=&(a_{1,1},a_{1,2},\ldots,a_{1,7})(a_{2,2},a_{2,3}\ldots,a_{2,8})\cdots(a_{n,n},a_{n,1}\ldots,a_{n,6}),\\ \omega_{c}&=&(a_{n-5,1},a_{n-4,1},\ldots,a_{n,1},a_{1,1})(a_{n-4,2},a_{n-3,2},\ldots,a_{n,2},a_{1,2},a_{2,2})\cdots\\ &&(a_{n,6},a_{1,6},a_{2,6},\ldots,a_{6,6})(a_{1,7},a_{2,7},\ldots... | matrix | |
\[||\mathcal{T}\mathbb{E}\kappa_{1}(x)-\mathcal{T}\mathbb{E}\kappa _{2}(x)||\] \[\leq \mathbb{E}\int_{\mathbb{R}}M_{\epsilon}(\omega)e^{(\lambda_{1}+ \epsilon)|s|}||\gamma(\Phi(-s,\omega)x+\kappa_{1}(\Phi(-s,\omega)x))-\gamma( \Phi(-s,\omega)x+\kappa_{2}(\Phi(-s,\omega)x))||ds\] \[\leq \mathbb{E}M_{\epsilon}(\omega)c_{... | outline | |
\[\langle V_{\sigma},z_{\sigma}\rangle=\langle V_{\sigma},z_{ \sigma}\rangle =\prod_{q=1}^{d+1}\left(-\frac{U_{q,\,1}}{x_{q,\,1}}+\frac{U_{q,\, \ell_{q}}}{x_{q,\,\ell_{q}}}\right)-\prod_{q=1}^{d+1}\frac{D_{q,\,\ell_{q}}}{x _{q,\,\ell_{q}}}\] \[=\prod_{q=1}^{d+1}\left(\frac{-x_{q,\,\ell_{q}}U_{q,\,1}+x_{q,\, 1}U_{q,\,\e... | outline | |
\[\left(uu_{xx}+u_{x}^{2}-2u^{2}\right)_{x}= (1-\partial_{x}^{2})u_{t}+2\kappa u_{x}\] \[= (1-\partial_{x}^{2})\left(\epsilon_{t}+\frac{d}{dt}\sum R_{j} \right)+2\kappa\left(\epsilon_{x}+\sum R_{j,x}\right)\] \[= (1-\partial_{x}^{2})\epsilon_{t}+(1-\partial_{x}^{2})\sum\left( \dot{c}_{j}(t)\frac{d}{dc}R_{j}-\dot{x}_{j}... | outline | |
\[\ll(\tilde{y}_{\max}^{(J,p_{1},m)})(\tilde{y}_{\max}^{(J,p_{2},m)} )\bigg{(}\sum_{\begin{subarray}{c}u\leq R\\ (u,\tilde{W})=1\end{subarray}}\frac{\mu^{2}(u)}{f^{*}(u)}\bigg{)}^{|I|}\bigg{(} \sum_{u\leq R}\frac{\mu^{2}(u)}{g^{*}(u)}\bigg{)}^{|J|-1}\sum_{ \begin{subarray}{c}p>D_{0}\\ p\equiv 3\ (\text{mod 4})\end{suba... | outline | |
\[\left\|f_{0}^{k}-f_{0}^{l}\right\|_{L^{2}(\Omega)}^{2}+\left\|f_{1 }^{k}-f_{1}^{l}\right\|_{L^{2}(\Omega)}^{2}+\left\|f_{2}^{k}-f_{2}^{l}\right\|_ {L^{2}(\Omega)}^{2}+\left\|\mathbf{f}^{k}-\mathbf{f}^{l}\right\|_{\mathbf{L}^ {2}(\Omega)}^{2}\] \[\leq C\sum_{i=1}^{m+2}\left\|\frac{\partial u_{tt}^{(i)}(f_{0}^{k},f_ {1... | outline | |
\[F(u) =|f_{M}(u)-f_{N}(u)|^{2}\] \[=\left(\frac{3}{4}\right)^{2}|S_{M}^{1}(u)-S_{N}^{1}(u)+S_{M}^{2}( u)-S_{N}^{2}(u)+S_{M}^{3}(u)-S_{N}^{3}(u)|^{2}\] \[=\frac{9}{16}|\sum_{N<|n_{1}|\leq M}\frac{2n_{1}|g_{n_{1}}|^{4}}{ \langle n_{1}\rangle^{4}}+2\sum_{N<|n_{1}|\leq M}\sum_{|n_{2}|\leq M,|n_{2}|<|n _{1}|}\frac{(n_{1}+n... | outline | |
\[\triangle_{h}\partial_{\theta}F_{1}^{1} =\int_{\mathbb{S}^{1}}\mathcal{T}_{h}\left(\partial_{\theta}\left( \frac{|\Delta\bm{Y}|^{2}\Delta\bm{W}\cdot\partial_{\theta^{\prime}}\bm{Y}^{ \prime}-\Delta\bm{Y}\cdot\partial_{\theta^{\prime}}\bm{Y}^{\prime}\Delta\bm{Y} \cdot\Delta\bm{W}}{|\Delta\bm{Y}|^{4}}\right)\right)(\pa... | outline | |
\[-\int_{0}^{T}\int_{\Omega\times D}M\,\widehat{\psi}_{\rm OB}\, \frac{\partial\widehat{\varphi}}{\partial t}\,\underset{\widetilde{\phantom{ \omega}}}{\mathrm{d}q}\,\underset{\widetilde{\phantom{\omega}}}{\mathrm{d}x}\, \underset{\widetilde{\phantom{\omega}}}{\mathrm{d}t}+\int_{0}^{T}\int_{ \Omega\times D}M\,\left[\va... | outline | |
\[(\nabla r^{z},\nabla v)_{\omega_{z}}=-(\psi_{z}\mathcal{G}(u_{hp} ),\nabla v)_{\omega_{z}}+\sum_{T\in\mathcal{T}(z)}(\Pi_{T}^{p_{z}}f^{z},v)_{T }-\sum_{E\in\mathcal{E}(z)\cap\mathcal{E}(\partial\Omega)}(\Pi_{E}^{p_{z}}g^{z },v)_{E}\] \[\quad=\sum_{T\in\mathcal{T}(z)}\int_{T}(\operatorname{div}(\psi_{z }\mathcal{G}(u_... | outline | |
\[\frac{\mathrm{d}}{\mathrm{d}t}E[u_{\varepsilon}|\xi]\] \[=-\sum_{\begin{subarray}{c}i,j=1\\ i<j\end{subarray}}^{N}\int_{\mathbb{R}^{d}}\frac{1}{2\varepsilon}\big{|} \operatorname{H}_{\varepsilon}-\varepsilon(B\cdot\xi_{i,j})\xi_{i,j}|\nabla u _{\varepsilon}|\big{|}^{2}\chi_{\mathcal{T}_{i,j}}(u_{\varepsilon})\,\mathr... | outline | |
\[\mathbb{E}\left[S_{N}\right] =\!\mathbb{E}\left[\sum\limits_{n_{1},n_{2}}\rho^{|n_{1}-n_{2}|} \,x[n_{1}]x_{m}^{*}[n_{2}]\right]\!=\!\sum\limits_{n}\mathbb{E}\left[|x[n]|^{2 }\right]\!+\!\!\sum\limits_{n_{1}\neq n_{2}}\rho^{|n_{1}-n_{2}|}\,\mathbb{E} \left[x[n_{1}]\right]\mathbb{E}\left[x_{m}^{*}[n_{2}]\right]\] \[\ov... | outline | |
\[x_{1}[(\alpha_{1}-\alpha_{3})y_{1}^{2}+(\alpha_{4}-\alpha_{3})y_ {2}^{2}]+y_{1}[(\alpha_{1}-\alpha_{4})x_{2}y_{2}+(\alpha_{3}+\alpha_{5})x_{3}y _{3}] = (\theta-\alpha_{3})x_{1},\] \[x_{2}[(\alpha_{1}-\alpha_{3})y_{2}^{2}+(\alpha_{4}-\alpha_{3})y_ {1}^{2}]+y_{2}[(\alpha_{1}-\alpha_{4})x_{1}y_{1}+(\alpha_{3}+\alpha_{5}... | outline |
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