image imagewidth (px) 276 2.95k | latex_formula stringlengths 317 2.94k | category stringclasses 4
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\[\mathcal{Q}_{\mu}[f_{\xi,\mu}]\] \[=\int_{\mathbb{R}}K(w,t)f_{\xi,\mu}(t)dt\] \[=\int_{\mathbb{R}}\sqrt{\frac{b}{2\pi i}}e^{i(at^{2}+btw+cw^{2}+ dt+ew)}\sqrt{\frac{b}{2\pi i}}e^{i(at^{2}+bt\xi+c\xi^{2}+dt+e\xi)}f(t)dt\] \[=\sqrt{\frac{b}{2\pi i}}\int_{\mathbb{R}}\sqrt{\frac{b}{2\pi i}} e^{i[at^{2}+bt(w+\xi)+c(w+\xi)^... | outline | |
\[\begin{array}{lcl}\sum\limits_{k=1}^{K}\left(\boldsymbol{\rho}_{k}^{n}\right) ^{\top}\widehat{\mathcal{P}}\hat{\boldsymbol{\rho}}_{k}^{n}&&\leq 4b_{1}^{ \max,n}\mathcal{C}^{n-1}+2\sum\limits_{k=1}^{K}\boldsymbol{\rho}_{0}^{\top} \widehat{\mathcal{P}}\hat{\boldsymbol{\rho}}_{0},\\ \sum\limits_{k=1}^{K}\left((\boldsymb... | matrix | |
\[\begin{cases}\partial_{t}\bar{\rho}_{m+1}+\mathrm{div}_{{}_{|}} \bar{u}_{n-1+m,{}^{|}}+\partial_{y}\bar{u}_{n+m,3}=0,\\ \partial_{t}\bar{u}_{m+1,i}+\partial_{i}(\bar{\rho}_{n-1+m}+\bar{ \theta}_{n-1+m})\\ \quad+\sum_{j=1}^{2}\partial_{x_{j}}\langle\mathcal{A}_{ij},( \mathbf{I}-\mathbf{P})\bar{f}_{n-1+m}\rangle+\parti... | outline | |
\[\|e^{it\Delta}u(t)-e^{i\tau\Delta}u(\tau)\|_{L^{2}}\] \[\leq\|\int_{t}^{\tau}e^{is\Delta}V(x)u(s)ds\|_{L^{2}}+\|\int_{t}^ {\tau}e^{is\Delta}F(|u|^{2})u(s)ds\|_{L^{2}}+\|\int_{t}^{\tau}e^{is\Delta}(W*|u |^{2})u(s)ds\|_{L^{2}}\] \[\leq\sum_{j=1}^{2}\|\int_{t}^{\tau}e^{is\Delta}V_{j}(x)u(s)ds\|_ {L^{2}}+\|\int_{t}^{\tau... | outline | |
\[\Big{(}S_{0,q}(S_{\alpha-\beta,p}(\{2^{-\beta i}X_{i}\}))\Big{)} _{j} =\Big{[}S_{0,q}(S_{(\alpha-\beta)p,1}(\{2^{-\beta pi}X_{i}^{p}\}) )\Big{]}_{j}^{\frac{1}{p}}\] \[=\Big{[}S_{0,1}\Big{(}(S_{(\alpha-\beta)p,1}(\{2^{-\beta pi}X_{i} ^{p}\}))^{\frac{q}{p}}\Big{)}\Big{]}_{j}^{\frac{1}{q}}\] \[=\Big{(}S_{0,\frac{q}{p}}(... | outline | |
\[\big{|}\mathfrak{C}[N_{0},N_{1},N_{2},N_{3}](t)\big{|}\] \[\lesssim N_{0}^{-3}N_{1}^{-2}N_{2}^{-2}N_{3}^{-3}\sum_{\begin{subarray}{c} \pm 0,\pm 1,\\ \pm 2,\pm 3\end{subarray}}\sum_{\begin{subarray}{c}n_{0},n_{1},n_{2},n_{3} \in\mathbb{Z}^{3}:\\ n_{0}=n_{123}\end{subarray}}\Big{[}\Big{(}\prod_{j=0}^{3}1_{N_{j}}(n_{j})... | outline | |
\[\big{|}f_{\eta/2^{\ell},\Lambda_{\ell}}(x)-p_{\eta/2^{\ell}}(cx)\,e^{ 2\pi idx}\big{|}\] \[\leq\Big{|}\sum_{j=-M}^{M}a_{j}^{(\eta/2^{\ell})}\Big{(}\exp\big{(} 2\pi is_{\eta/2^{\ell},\Lambda_{\ell}}(j)x\big{)}-e^{2\pi i(cj+d)x}\Big{)}\Big{|} +\Big{|}\sum_{|j|>M}a_{j}^{(\eta/2^{\ell})}\,e^{2\pi i(cj+d)x}\Big{|}\] \[\le... | outline | |
\[J =\frac{(1+w)}{1+\sum_{j\in\mathcal{W}}p_{j}(D_{j}-1)}\left(1+ \sum_{j\in\mathcal{W}}p_{j}(1-e^{\lambda_{0}p_{j}})(e^{-\theta}-1)\right)-1- \log\left(\frac{1+w}{1+\sum_{j\in\mathcal{W}}p_{j}(D_{j}-1)}\right)\] \[\leq w^{2}+0.2\sum_{j\in\mathcal{W}}p_{j}(D_{j}-1)\] \[\leq w^{2}+0.2\sum_{j\in\mathcal{W}}p_{j}\left(D\l... | outline | |
\[\|X-H_{k}^{(j)}\|=\left\|X-X^{(1)}+\sum_{s=1}^{j-1}(X^{(s)}-X^{(s+ 1)})+X^{(j)}-H_{k}^{(j)}\right\|\] \[\leq \|X-X^{(1)}\|+\sum_{s=1}^{j-1}\|X^{(s)}-X^{(s+1)}\|+\|X^{(j)}-H_{k} ^{(j)}\|\] \[\leq \frac{1}{\ell^{(0)}}\|H_{1}^{(1)}-H_{1}\|+\xi^{(0)}\|A_{1}^{(1)}- A_{1}\|+\eta^{(0)}\|G_{1}^{(1)}-G_{1}\|\] \[+\mathcal{O}(... | outline | |
\[a^{n} =u^{n}+\frac{1}{2}\varphi_{1}\left(\frac{1}{2}\mathcal{J}(u^{n}) \Delta t\right)f(u^{n})\Delta t\] \[b^{n} =u^{n}+\varphi_{1}\left(\mathcal{J}(u^{n})\Delta t\right)f(u^{n}) \Delta t+\varphi_{1}\left(\mathcal{J}(u^{n})\Delta t\right)(\mathcal{F}(a^{n} )-\mathcal{F}(u^{n}))\Delta t\] \[u_{3}^{n+1} =u^{n}+\varphi_... | outline | |
\[I_{1}\leq H_{21}\bigg{(}\mathbb{E}\int_{t_{0}}^{t\wedge\theta_{n}}\big{|}e(s) \big{|}^{\bar{q}}ds+\mathbb{E}\int_{t_{0}}^{T}\big{|}\tilde{R}_{\mu}(t,X_{ \Delta}(t),\bar{X}_{\Delta}(t))\big{|}^{\bar{q}}ds+\Delta^{\bar{q}}\bigg{)}\] \[\leq H_{21}\bigg{(}\mathbb{E}\int_{t_{0}}^{t\wedge\theta_{n}}\big{|}e(s )\big{|}^{\ba... | outline | |
\[\left(\iota\circ\mathrm{id}_{\mathsf{Com}^{*}}\right)d_{\kappa} =\left(\iota\circ\mathrm{id}_{\mathsf{Com}^{*}}\right)\left( \mu_{(1)}^{\mathsf{Lie}}\circ\mathrm{id}_{\mathsf{Com}^{*}}\right)\left( \mathrm{id}_{\mathsf{Lie}}\circ^{\prime}\left[\,\left(\kappa\circ\mathrm{id}_{ \mathsf{Com}^{*}}\right)\Delta\right]\rig... | outline | |
\[3\left|A_{3}-\rho A_{2}^{2}\right|\leq\left\{\begin{array}{ll}\left(3-2 \beta\right)\left(3-2\alpha-\beta\right)-3\rho\left(2-\alpha-\beta\right)^{2} &,\ \ \rho\leq\frac{2\left(1-\beta\right)}{3\left(2-\alpha-\beta\right)}\\ \\ 1-2\alpha+\beta\left(3-2\beta\right)+\frac{4}{3\rho}\left(1-\beta\right)^{2}&, \ \frac{2\l... | matrix | |
\[I_{1} :=\left\langle-\lambda_{1}\boldsymbol{m}_{h,k}^{-}\times \boldsymbol{v}_{h,k}+\lambda_{2}\boldsymbol{v}_{h,k},\boldsymbol{m}_{h,k}^{-} \times\boldsymbol{\psi}-I_{\mathbb{V}_{h}}(\boldsymbol{m}_{h,k}^{-}\times \boldsymbol{\psi})\right\rangle_{\mathbb{L}^{2}(D_{T})},\] \[I_{2} :=\mu\left\langle\nabla(\boldsymbol{... | outline | |
\[-\frac{h}{2}\sqrt{\frac{3h^{2}-1}{3-h^{2}}}\Big{(}2{\bf p}_{3}^{\perp}\frac{ \partial^{2}}{\partial p_{2}\partial p_{1}}+3p_{2}p_{1}\frac{\partial}{\partial{\bf p }_{3}^{\perp}}\Big{)}++\underline{h}\Big{(}\tilde{p}_{2}\frac{\partial}{ \partial p_{2}}+p_{2}\frac{\partial}{\partial\tilde{p}_{2}}\Big{)}\left(p_{1}\frac... | outline | |
\[p_{\alpha}(J(-\eta;III))=p_{\alpha}(\int\limits_{\delta}^{\eta}f(- it)\cdot(-i)\int\limits_{-\delta}^{\delta}\varphi(a-\delta^{\prime}+is)e^{i(a- \delta^{\prime}+is)(-it)}i\mathrm{d}s\,\mathrm{d}t)\] \[\underset{(\ref{eq:p_alpha}),(\ref{eq:p_alpha})}{\leq}\int \limits_{\delta}^{\eta}|f|_{k,\alpha,K}e^{\frac{1}{k}t-at... | outline | |
\[(p)_{\Gamma} = C_{1}+C_{2}\ln R+\delta e^{il\theta}\left(\frac{C_{2}}{R}+D_{1} R^{l}+\frac{D_{2}}{R^{l}}\right)\] \[= \mathcal{G}^{-1}(\kappa)_{\Gamma}+\left(\mathcal{P}-\chi_{\sigma }\right)(\sigma)_{\Gamma}-\mathcal{P}\mathcal{A}\frac{(x\cdot x)_{\Gamma}}{4}\] \[= \mathcal{G}^{-1}\frac{1}{R}-\frac{\mathcal{P}\mathc... | outline | |
\[\begin{split}\big{(}\mathcal{M}\otimes\bar{\mathcal{M}}\big{)}( \Psi_{1},\Psi_{2})&=\big{(}\mathcal{M}\otimes\bar{\mathcal{M}} \big{)}(u_{1}\otimes\bar{u}_{1},u_{2}\otimes\bar{u}_{2})\\ &=(-)^{u_{2}\bar{u}_{1}+|\bar{\mathcal{M}}|(u_{1}+u_{2})} \mathcal{M}(u_{1},u_{2})\otimes\bar{\mathcal{M}}(\bar{u}_{1},\bar{u}_{2})\... | outline | |
\[dial_{P}=\{-1,0,2,2\}\] \[p<q\] \[p\] \[zone_{0}\] \[od_{2}\] \[od_{4}\] \[p=\begin{cases}\left(\frac{\Delta-4}{4}\right)^{2}&\text{for }\Delta=4k,k=2m,\ k,m\in\mathbb{N};p=2j+1,\ j\in\mathbb{N}\\ \\ \left(\frac{\Delta-4}{4}\right)\left(\frac{\Delta-4}{4}+1\right)-\left(\frac{ \Delta-4}{4}-1\right)&\text{for }\Delta=... | outline | |
\[\begin{split}&\left[\begin{array}{c}\mathbb{E}\left[\sum_{i=1}^ {m}\|\bar{x}_{i}^{k+1}-x_{i}^{*}\|_{2}^{2}|\mathcal{F}^{k}\right]\\ \mathbb{E}\left[\sum_{i=1}^{m}\|\mathbf{x}_{i}^{k+1}-\bar{\mathbf{x}}_{i}^{k+1 }\|_{L}^{2}|\mathcal{F}^{k}\right]\end{array}\right]\\ \leq&\left(\left[\begin{array}{cc}1&\kappa_{1}\gamma... | matrix | |
\[\begin{split}&\sum_{\begin{subarray}{c}\|\bm{a}\|_{\infty}\leq A\\ P(\bm{a})=0\end{subarray}}|\mathcal{I}_{\bm{a}}(X)-\mathfrak{S}_{\bm{a}}^{*} \mathfrak{J}_{\bm{a}}^{*}|^{2}\\ &\ll\sum_{\begin{subarray}{c}\|\bm{a}\|_{\infty}\leq A\\ P(\bm{a})=0\end{subarray}}|\mathcal{I}_{\bm{a}}(X,\mathfrak{M})-\mathfrak{S}_{ \bm{a... | matrix | |
\[n^{\frac{1}{p}+\frac{1}{q^{\prime}}}\sum_{N_{1}\leq j\leq N_{2}} \|f_{i}\|_{L^{p}(Q_{0},B_{1})}\|g_{i}\|_{L^{q^{\prime}}(3Q_{0},B_{2}^{*})}\] \[\sum_{R\in\mathfrak{R}_{j}}|R|^{-(\frac{1}{p}-\frac{1}{q})}\bigg{(} \sum_{\begin{subarray}{c}P\subset R\\ j-\ell\leq L(P)<j\end{subarray}}|P|\bigg{)}^{\frac{1}{p}}\bigg{(}\su... | outline | |
\[\langle A(t,\bm{v}),\bm{z}\rangle:=\langle A(t,\bm{\varepsilon}( \bm{v})),\bm{\varepsilon}(\bm{z})\rangle_{\mathcal{H}}\ \ \text{for}\ \ \bm{v},\bm{z}\in V,\ \text{a.e.}\ t\in(0,T),\] \[(I\bm{w})(t):=\bm{u}_{0}+\int_{0}^{t}\bm{w}(s)\,ds\ \ \text{for}\ \ \bm{w}\in L^{2}(0,T;V),\ \text{a.e.}\ t\in(0,T),\] \[(S\bm{w})(t... | outline | |
\[\left|\int_{B_{1}^{\prime}}F\cdot\nabla\bar{v}_{2}|\bar{v}_{2}|^{p-2}\right|\] \[\leq\frac{\varrho}{2}\int_{B_{1}^{\prime}}|x^{\prime}|^{m}|\nabla \bar{v}_{2}|^{2}|\bar{v}_{2}|^{2}+C\int_{B_{1}^{\prime}}|x^{\prime}|^{2\sigma+m }|\bar{v}_{2}|^{p-2}\] \[\leq\frac{\varrho}{2}\int_{B_{1}^{\prime}}|x^{\prime}|^{m}|\nabla ... | outline | |
\[\frac{1}{x^{2i\lambda}}\,\prod_{k=1}^{L}\int d^{4}x_{k}\frac{1}{x _{k-1k}^{2(2+u_{k})}}\,\frac{1}{x_{k}^{-2u_{k}}}\,\frac{1}{(x_{L}-y)^{2(2+u_{L +1})}}\,\frac{1}{y^{2(-u_{L+1}-i\lambda)}}=\] \[\frac{\pi^{2L-1}}{2}\,\sum_{n\geq 0}\,(n+1)\,\int\limits_{- \infty}^{+\infty}d\nu\,\frac{\hat{x}^{\mu_{1}\cdots\mu_{n}}}{x^{2... | outline | |
\[\operatorname{Var}_{\mu}[f] =\operatorname{Var}_{i\sim\mu D_{k\to 1}}\big{[}\mathbb{E}_{\mu}[f \mid i]\big{]}+\mathbb{E}_{i\sim\mu D_{k\to 1}}\big{[}\operatorname{Var}_{\mu}[f \mid i]\big{]}\] \[=\operatorname{Var}_{\mu D_{k\to 1}}[U_{1\to k}f]+\mathbb{E}_{i\sim\mu D_{k\to 1}} \big{[}\operatorname{Var}_{\mu}[f\mid i]... | outline | |
\[\left\|\left\{\sum_{i=1}^{m}\left[\frac{\lambda_{i}}{\|\mathbf{1} _{x_{i}+B_{l_{i}}}\|_{L^{p(i)}(\mathbb{R}^{n})}}\right]^{\!\!p}\mathbf{1}_{x_ {i}+B_{l_{i}}}\right\}^{1/\!p}\right\|_{L^{p(i)}(\mathbb{R}^{n})}^{-1}\sum_{j= 1}^{m}\frac{\lambda_{j}|x_{j}+B_{l_{j}}|^{\frac{1}{2}}}{\|\mathbf{1}_{x_{j}+B_ {l_{j}}}\|_{L^{p... | outline | |
\[f_{1,j}\left(\frac{z_{2}}{z_{3}}\right)f_{i,j}\left(\frac{z_{2}} {z_{1}}\right)f_{1,i}\left(\frac{z_{1}}{z_{3}}\right)T_{1}(z_{3})T_{i}(z_{1}) T_{j}(z_{2})-f_{j,1}\left(\frac{z_{3}}{z_{2}}\right)f_{j,i}\left(\frac{z_{1}}{z_{2}} \right)T_{j}(z_{2})f_{1,i}\left(\frac{z_{1}}{z_{3}}\right)T_{1}(z_{3})T_{i}(z_ {1})\] \[- ... | outline | |
\[\begin{array}{rl}(x_{j}\circ y_{\beta})(x_{i}\circ y_{\alpha})&=q_{ji}q^{ \prime}_{\beta\alpha}(x_{i}x_{j})\circ(y_{\alpha}y_{\beta})=q_{ji}q^{\prime}_{ \beta\alpha}(x_{i}\circ y_{\alpha})(x_{j}\circ y_{\beta}).\\ (x_{j}\circ y_{\alpha})(x_{i}\circ y_{\beta})&=q_{ji}q^{\prime}_{\alpha\beta}( x_{i}x_{j})\circ(y_{\beta... | outline | |
\[\Lambda_{1}^{\mathsf{T}} = (0,0,0,0,0,0,0,0,0);\] \[\Lambda_{2}^{\mathsf{T}} = (0,0,0,0,0,0,(a^{\mathsf{T}}\nabla)_{2}f,-(a^{\mathsf{T}}\nabla) _{1}f,0);\] \[\mathfrak{R}_{ab}(\nabla f,\nabla f)-\Lambda_{1}^{\mathsf{T}}Q^{ \mathsf{T}}Q\Lambda_{1}-\Lambda_{2}^{\mathsf{T}}P^{\mathsf{T}}P\Lambda_{2}+D^ {\mathsf{T}}D+E^{... | outline | |
\begin{table}
\begin{tabular}{l r r r r} \hline \hline Instance & Objective & Time & \#Pen. & \#ADM \\ \hline syn05m04h & 5510.39 & 0 & 3 & 4 \\ syn05m04m & 5499.39 & 1 & 13 & 28 \\ syn10h & 1267.35 & 0 & 3 & 4 \\ syn10m & 560.25 & 0 & 10 & 14 \\ syn10m02h & 2310.30 & 1 & 3 & 4 \\ syn10m02m & 2275.85 & 2 & 13 & 25 \\ s... | table | |
\[\left[\nabla X_{u,t}^{\phi_{s,u}(\mu_{0})}\right](X_{s,u}^{\mu_{1}}(x ))^{\prime}-\left[\nabla X_{u,t}^{\phi_{s,u}(\mu_{0})}\right](X_{s,u}^{\mu_{0}}( x))^{\prime}\] \[=\int_{0}^{1}\left[\nabla^{2}X_{u,t}^{\phi_{s,u}(\mu_{0})}\right]( X_{s,u}^{\mu_{0}}(x)+\epsilon(X_{s,u}^{\mu_{1}}(y)-X_{s,u}^{\mu_{0}}(x)))^{ \prime}... | outline | |
\[\begin{array}{rcl}\nabla\!\!\!/_{4}\left(\frac{\check{\nu}}{r^{4}}\right)&=& \nabla\!\!\!/_{4}\left(\!\not{\!\!div}\,\zeta+2\,^{\left(F\right)}\!\rho\,^{ \left(\check{F}\right)}\!\rho\right)\\ &=&\mathrm{d}\mathrm{f}\mathrm{v}\left(\nabla\!\!\!/_{4}\zeta\right)-\frac{1} {2}\kappa\,\not{\!\!div}\,\zeta+2(\nabla\!\!\!/... | matrix | |
\[\vartheta_{-(N-1)\epsilon^{*},\emptyset}\big{(}\mathbf{u}(z)+ \mathbf{u}(w)-\mathbf{u}(\infty_{-})-\boldsymbol{\tau}(\boldsymbol{\epsilon }^{*})\big{|}\boldsymbol{\tau}\big{)}\vartheta_{-(N+1)\epsilon^{*}, \emptyset}\big{(}-\mathbf{u}(z^{\prime})-\mathbf{u}(w^{\prime})+\mathbf{u}( \infty_{-})+\boldsymbol{\tau}(\bolds... | outline | |
\[I_{t}=\lim_{\varepsilon\to 0+}\int\limits_{0}^{1}\int\limits_{ \mathbb{R}^{N}}\eta(z)\Bigg{(}\int\limits_{K+\tau t\boldsymbol{k}+\varepsilon z }\Bigg{\{}\int\limits_{0}^{1}\nabla_{a}W\Big{(}P_{t}(u_{\varepsilon},y,s, \boldsymbol{k}),u_{\varepsilon}(y)\Big{)}ds\Bigg{\}}\cdot d\Big{[}Du(x)\cdot \boldsymbol{k}\Big{]}\Bi... | outline | |
\[\tau_{i}^{\mathbf{B}}(\mathbf{A}) =\mathbf{a}_{i}^{\top}\left(\left(\begin{matrix}\mathbf{B}\\ \mathbf{a}_{i}^{\top}\end{matrix}\right)^{\top}\left(\begin{matrix}\mathbf{B} \\ \mathbf{a}_{i}^{\top}\end{matrix}\right)\right)^{+}\mathbf{a}_{i}\] \[=\mathbf{a}_{i}^{\top}\left(\left(\mathbf{B}^{\top}\mathbf{B} \right)^{+... | outline | |
\[e^{s_{\ell+1}(\frac{s_{\ell+1}}{2}+z_{m}+\frac{s_{m}}{2})^{2}+ \sum_{i\neq\ell+1}s_{i}z_{i}^{2}}\frac{s_{\ell+1}s_{m}}{s_{\ell+1}+s_{m}}\prod_ {(i,j)\in\tilde{S}_{1}}\frac{z_{j}-\frac{s_{j}}{2}-z_{i}+\frac{s_{j}}{2}\,z_{j }+\frac{s_{j}}{2}-z_{i}-\frac{s_{i}}{2}}{z_{j}+\frac{s_{j}}{2}-z_{i}+\frac{s_{ j}}{2}\,z_{j}-\fr... | outline | |
\[I_{0}^{\pm}=\int_{-\infty}^{\infty}\left(v_{1}^{\pm}\right)^{2} \,dt=\frac{2}{3}\nu\sqrt{\nu}\left(3\phi^{2}-1\right)+4\nu\sqrt{\nu}\,\psi\phi ^{2}\tan^{-1}\left(\psi\mp\phi\right),\] \[I_{1}^{\pm}=\int_{-\infty}^{\infty}u_{1}^{\pm}\,\left(v_{1}^{\pm }\right)^{2}\,dt=-\sqrt{2}\nu^{2}\left\{\psi\left(\frac{5}{2}\phi^{... | outline | |
\[S_{6.2} \leq c\tau^{\delta}\varrho^{\delta+\vartheta\gamma(\theta\delta+n)/p} \left(\int_{\tau\varrho}^{\varrho}\frac{\mathrm{d}\nu}{\nu^{1+s+\vartheta n/p}} \right)^{\gamma}[\mathsf{gl}_{\vartheta,\delta}^{+}(\varrho)]^{\vartheta\gamma}\] \[\leq c\tau^{\delta-s\gamma-n\vartheta\gamma/p}\varrho^{\theta\delta \varthet... | outline | |
\[\gamma_{-1}(A_{s,m}) \lesssim\int_{B_{s,m}}\int_{0}^{r_{s,m}(x^{\prime})}r^{n-1}e^{r^ {2}}\,dr\,dx^{\prime}\] \[\lesssim\int_{B_{s,m}}r_{s,m}(x^{\prime})^{n-2}e^{r_{s,m}(x^{ \prime})^{2}}\,dx^{\prime}\] \[=\frac{1}{s}\,e^{-c2^{2m}}\int_{B_{s,m}}I_{m}(r_{s,m}(x^{\prime}),x^{\prime})\,dx^{\prime}\] \[\leqslant\frac{1}{... | outline | |
\[c+o(1)\|Dv_{n}\|_{p}=F_{\lambda}(v_{n})-\frac{1}{\alpha p^{*}}F_ {\lambda}^{\prime}(v_{n})(G^{-1}(v_{n})g(G^{-1}(v_{n})))\psi_{\varepsilon}\] \[\quad-\frac{1}{\alpha p^{*}}\int_{\mathbb{R}^{N}}G^{-1}(v_{n})g(G ^{-1}(v_{n}))|Dv_{n}|^{p-2}Dv_{n}\cdot D\psi_{\varepsilon}dx+\frac{\lambda}{ \alpha p^{*}}\int_{\mathbb{R}^{... | outline | |
\[u_{1}=v_{1,1}^{\varepsilon(1)}\overset{h_{i(l)}^{\varepsilon(1)}}{\mapsto}v_{ 1,1}^{-\varepsilon(1)}\overset{c_{1}}{\mapsto}v_{1,2}^{\varepsilon(2)} \overset{h_{i(2)}^{\varepsilon(2)}}{\mapsto}\cdots\overset{h_{i(l-1)}^{ \varepsilon(l-1)}}{\mapsto}v_{1,l-1}^{-\varepsilon(l-1)}\overset{c_{l-1}}{ \mapsto}v_{1,l}^{\vare... | outline | |
\[\boldsymbol{M}_{jk}^{(2)}=\langle\phi_{k},\phi_{j}^{*}\rangle_{N}=\begin{cases} \frac{2(j-1)}{j(2j+1)}d_{j}c_{j-2},&k=j-2,\ \ 2\leq j\leq N-1,\\ \frac{4}{(j+1)(j+2)}d_{j}c_{j-1},&k=j-1,\ \ 1\leq j\leq N-1,\\ \Big{(}\frac{2}{2j+1}-\frac{2(2j+3)}{(j+2)^{2}}+\Big{(}\frac{j+1}{j+2} \Big{)}^{2}\frac{2}{2j+5}\Big{)}d_{j}c_... | outline | |
\[\begin{split}\sum_{r=1}^{R}&\int_{\Omega}\frac{1}{1+ \varepsilon|g(v^{\varepsilon})|}\left[\frac{(v^{\varepsilon})^{y_{r}}}{v_{ \infty}^{y_{r}}}-\frac{(v^{\varepsilon})^{y_{r}^{\prime}}}{v_{\infty}^{y_{r}^ {\prime}}}\right]^{2}dx\\ &\geq\sum_{r=1}^{R}\int_{\Omega_{1}}\frac{1}{1+\varepsilon|g(v^{ \varepsilon})|}\left[... | outline | |
\[\begin{split} I&=\int_{0}^{\infty}x\exp\left(- \frac{p^{2}x^{2}}{2}\right)\!I_{0}\left(cx\right)\\ &\times\Bigg{(}1+\exp\left(-\frac{\alpha^{2}x^{2}+\beta^{2}}{2} \right)\!I_{0}\left(axb\right)-Q\left(\alpha x,\beta\right)\Bigg{)}\,dx\\ &=\int_{0}^{\infty}x\exp\left(-\frac{p^{2}x^{2}}{2}\right)\!I_{0 }\left(cx\right)... | outline | |
\[\mathcal{W}_{3}(z_{1})\mathcal{W}_{3}(z_{2}) \sim \,\frac{\frac{c}{3}}{z_{12}^{6}}\,+\,\frac{2\,\mathcal{L}}{z_{12 }^{4}}\,+\,\frac{\mathcal{L}^{\prime}}{z_{12}^{3}}\,+\,\frac{1}{z_{12}^{2}} \Big{(}\frac{32\mathcal{L}^{2}}{5c}+\frac{32\mathcal{W}_{4}}{\sqrt{105}}+ \frac{3\mathcal{L}^{\prime\prime}}{10}\Big{)}\,+\,\fr... | outline | |
\[(A.23)\] \[\leq C_{E}\left(\left(\frac{\epsilon\theta_{i}}{2}+\frac{1}{4 \epsilon}\right)\left\|\partial^{\nu}\sigma\right\|_{V}^{2}+\frac{\theta_{i}}{ \epsilon}\left\|(1-\Pi)\partial^{\nu}h\right\|_{V,\omega}^{2}\right)\] \[+\frac{\theta_{i}}{2}\sum_{\nu_{j}\neq 0}C_{j}\left(\epsilon\nu_{j}^{2} \left\|\partial^{\nu-... | outline | |
\[(R6)\ w_{\beta}(u)h_{\gamma}(s)w_{\beta}(u)^{-1}=\begin{cases}h_{\gamma}(s)& \text{if }\ (\beta,\gamma)=0,\\ h_{\mp\beta}(u^{\mp 1}su^{\mp 1})h_{\mp\beta}(u^{\pm 2})&\text{if }\ \gamma=\pm\beta,\\ h_{\sigma_{\beta}(\gamma)}(u^{-1}s)h_{\sigma_{\beta}(\gamma)}(u)&\text{if }\ \beta\pm\gamma\neq 0\text{ and }i=k,\\ h_{\s... | outline | |
\[\frac{1}{\delta}|(\widetilde{\rho}\partial^{\alpha}\partial_{x}u _{1},\frac{2\bar{\theta}}{3\bar{\rho}^{2}}\partial^{\alpha}\widetilde{\rho})| \leq C\frac{1}{\delta}(\|\widetilde{\rho}\|_{L^{\infty}}\|\partial^{ \alpha}\partial_{x}\widetilde{u}_{1}\|\|\partial^{\alpha}\widetilde{\rho}\|+\| \partial^{\alpha}\partial_{... | outline | |
\[\begin{array}{|c|c|c|c|}\hline\mathbf{a}&c_{\mathbf{a},\lambda_{2}}&d_{ \lambda_{3}}&e_{\mathbf{a}}\\ \hline(1^{3},3^{2})&\frac{4}{5}\left(\frac{9\times 2^{3\lambda_{2}+3}+5}{2 ^{3}-1}\right)&\left(\frac{2\times 3^{3\lambda_{3}+2}-5}{3^{3}-1}\right)&\frac{8} {5}\\ \hline(1,3^{2},6^{2})&\frac{2}{5}\left(\frac{13\times... | matrix | |
\[\mathcal{U}_{13,4,5} = \Big{\{}7936,7360,6816,3008,3488,3680,5776,6496,6544,6736,7216,\] \[3720,5456,5512,5704,3400,1840,1924,5668,7180,3604,4904,4932,\] \[4994,1858,2840,2852,5410,6666,3346,1264,1698,4516,4801,5281,\] \[6406,6409,6661,7171,1380,1473,2706,3217,4328,4706,4748,4881,\] \[740,929,1801,2264,2388,2444,2636... | outline | |
\[I_{2}+II= -\langle w_{i}\partial_{i}\partial_{t}T_{\beta}T_{q}\bar{f},( \partial_{t}+w_{i}\partial_{i})T_{\beta}T_{q}\bar{f}\rangle+R_{2},\] \[= -\langle w_{i}\partial_{i}\partial_{t}T_{\beta}T_{q}\bar{f},w_{i} \partial_{i}T_{\beta}T_{q}\bar{f}\rangle-\langle w_{i}\partial_{i}\partial_{t }T_{\beta}T_{q}\bar{f},\parti... | outline | |
\[u(x_{1},x_{2},t)=e^{-a_{0}x_{1}}\left\{\kappa_{1}\left(E^{1}_{\alpha,1}(\gamma_{1}t^{\alpha})+\sum_{m=1}^{n}\mu^{m}(t-m\tau)^{\alpha m}E^{m+1}_{ \alpha,\alpha m+1}(\gamma_{1}(t-m\tau)^{\alpha})\right)\right.\] \[\left.+\hat{\kappa}_{1}\left(tE^{1}_{\alpha,2}(\gamma_{1}t^{ \alpha})+\sum_{m=1}^{n}\mu^{m}(t-m\tau)^{\alp... | outline | |
\[A_{\alpha,1} = \frac{\alpha}{ABC(\alpha)\Gamma(\alpha)}\int_{0}^{t_{n+1}}(t_{n+1 }-t)^{\alpha-1}\left\{\frac{t-t_{n-1}}{h}f(t_{n},u_{n})-\frac{t-t_{n}}{h}f(t_{ n},u_{n})\right\}\] \[= \frac{\alpha f(t_{n},u_{n})}{ABC(\alpha)\Gamma(\alpha)h}\left\{ \int_{0}^{t_{n+1}}(t_{n+1}-t)^{\alpha-1}f(t-t_{n-1})\right\}dt\] \[-\f... | outline | |
\[s_{a_{k}}^{*}as_{a_{k}}p_{r(A_{l},a_{k})} =s_{a_{k}}^{*}\left(\sum_{j=1}^{n}\mu_{j}s_{a_{j}}p_{B_{j}}s_{a_{j }}^{*}\right)s_{a_{k}}p_{r(A_{l},a_{k})}\] \[=\sum_{j=1}^{n}\mu_{j}s_{a_{k}}^{*}s_{a_{j}}p_{B_{j}}s_{a_{j}}^{* }s_{a_{k}}p_{r(A_{l},a_{k})}\] \[=\sum_{j\,:\,a_{j}=a_{k}}\mu_{j}p_{r(a_{k})}p_{B_{j}}p_{r(a_{k})}... | outline | |
\[\phi^{\prime}_{z}(z,\omega,t) =2\pi i(z\omega-t)\omega,\] \[\phi^{\prime\prime}_{z\zeta}(z,\omega,t) =2\pi i\begin{bmatrix}\omega^{\prime}\\ \omega_{n}\end{bmatrix}\begin{bmatrix}{}^{t}z^{\prime}-{z_{n}}^{t}\omega^{ \prime}/\omega_{n}&-1\end{bmatrix}+2\pi i(z\omega-t)\begin{bmatrix}E_{n-1}&0\\ {}^{-t}\omega^{\prime}/... | matrix | |
\[V_{0} =\frac{(N+1)}{2N}\int dxdy\,N^{2}V(N(x-y))\] \[\qquad\times\big{[}b^{*}(\gamma_{x})b^{*}(\gamma_{y})+b^{*}( \gamma_{x})b(\sigma_{x})+b^{*}(\gamma_{y})b(\sigma_{y})+b(\sigma_{x})b(\sigma _{y})+\langle\sigma_{x},\gamma_{y}\rangle\] \[\qquad-N^{-1}\langle\sigma_{x},\gamma_{y}\rangle\,\mathcal{N}-N^ {-1}a^{*}(\gamm... | outline | |
\[-\frac{(k-1)}{k}\sum_{j_{1}=0}^{k-1}\sum_{j_{2}=0}^{k-1}\cos^{k-2 }\left(\frac{2\pi(j_{1}-j_{2})}{k}\right)\sin^{2}\left(\frac{2\pi(j_{1}-j_{2}) }{k}\right)+\frac{1}{k}\sum_{j_{1}=0}^{k-1}\sum_{j_{2}=0}^{k-1}\cos^{k-1} \left(\frac{2\pi(j_{1}-j_{2})}{k}\right)\cos\left(\frac{2\pi(j_{1}-j_{2})}{k }\right)\] \[=-(k-1)\s... | outline | |
\[\alpha_{n,j} = \frac{\sin\left((2r_{n}+1)j\pi/n\right)}{2r_{n}\sin(j\pi/n)}-\frac {1}{2r_{n}}\] \[= \frac{(2r_{n}+1)j\pi/n}{2r_{n}j\pi/n}\Bigg{[}1-\frac{(2r_{n}+1)^{ 2}j^{2}\pi^{2}}{6n^{2}}+O\left(\frac{r_{n}^{4}j^{4}}{n^{4}}\right)\Bigg{]} \times\Bigg{[}1+O\left(\frac{j^{2}}{n^{2}}\right)\Bigg{]}-\frac{1}{2r_{n}}\] ... | outline | |
\[c(\mathbf{w})=\prod_{v\in S}\|(\mathbf{w}^{(v)})\|_{v,2} =\left(\prod_{v\in S_{r}}\|\mathbf{w}^{(v)}\|_{v,2}\right)\left( \prod_{v\in S_{e}}\|\mathbf{w}^{(v)}\|_{v,2}\right)\left(\prod_{v\in S_{f}}\| \mathbf{w}^{(v)}\|_{v,2}\right)\] \[<\left(\prod_{v\in S_{r}}\sqrt{(m+n)\varepsilon^{2}}\right)\left( \prod_{v\in S_{e... | outline | |
\[\mathcal{T}^{n}v(x)=\mathcal{T}\left(\mathcal{T}^{n-1}v\right)(x)\] \[\leq\inf_{\mathbb{P}_{0}\in\mathbf{P}_{a_{\mathrm{loc}}^{*}}} \int_{\Omega_{\mathrm{loc}}}\bigg{[}r\left(x,{a_{\mathrm{loc}}}^{*}(x), \omega_{1}\right)\] \[\qquad\qquad+\alpha\inf_{\mathbb{P}\in\mathfrak{P}_{\omega_{1}, \mathbf{a}^{*}}}\int_{\Omega... | matrix | |
\[f^{\prime\prime}(0) =\sum_{i=1}^{n}\sum_{j=1}^{m_{i}-1}(b_{i,j}-b_{i,j+1})^{2}/4+ \sum_{j=1}^{n-1}((b_{j,m_{j}}-b_{j+1,1})^{2}/4-(g_{j})^{2}/4)\] \[\leq\sum_{i=1}^{n}\sum_{j=1}^{m_{i}-1}\delta(b_{i,j+1}-b_{i,j})/2 +\sum_{j=1}^{n-1}(-b_{j,m_{j}}+b_{j+1,1}-g_{j})(-b_{j,m_{j}}+b_{j+1,1}+g_{j})/4\] \[\leq\delta/2+\sum_{j... | outline | |
\[\parallel\ ^{(n-1)}\tilde{\underline{\tilde{\chi}}}\parallel_{L^{2}( {\cal K}^{\tau_{1}})}^{2} =\int_{0}^{\tau_{1}}\parallel\ ^{(n-1)}\tilde{\underline{\chi}} \parallel_{L^{2}(S_{\tau,\tau})}^{2}d\tau\] \[\leq C\int_{0}^{\tau_{1}}\tau^{-2}\|r\ ^{(n-1)}\tilde{ \underline{\tilde{\chi}}}\|_{L^{2}(S_{\tau,\tau})}^{2}d\ta... | outline | |
\[-4\pi\nabla_{\tau}\beta^{\rm sv}\Big{[}\begin{smallmatrix}j\\ k_{1}\\ z_{1}\end{smallmatrix};\tau\Big{]} =(k{-}j{-}2)\beta^{\rm sv}\Big{[}\begin{smallmatrix}j+1\\ k_{2}\\ z_{1}\end{smallmatrix};\tau\Big{]}+\delta_{j,k-2}(\tau{-}\bar{\tau})^{k}f^{(k)}(z |\tau)\] \[-4\pi\nabla_{\tau}\beta^{\rm sv}\Big{[}\begin{smallmat... | matrix | |
\[-\psi(z_{N_{k}|k}^{0},d_{N_{k}-1}^{0})-\sum_{i=N_{k+1}+1}^{N_{k}- 1}\ell(\hat{z}_{i|k}^{0},\hat{\nu}_{i|k}^{0},d_{i}^{0})\] \[=-\ell(x_{k},\kappa(x_{k}),d_{0}^{0})+\psi(\hat{z}_{N_{k+1}|k+1}^{ 0},d_{N_{k+1}-1}^{0})\] \[-\psi(z_{N_{k}|k}^{0},d_{N_{k}-1}^{0})-\sum_{i=N_{k+1}+1}^{N_{k}- 1}\ell(\hat{z}_{i|k}^{0},\hat{\nu... | outline | |
\[e^{-rt}\left\|(x,y)^{\delta}(t)\right\|_{\mathcal{H}}^{2}+2\int _{0}^{t}e^{-rs}\left\|\nabla x^{\delta}(s)\right\|_{H}^{2}\,ds+2\varepsilon \int_{0}^{t}e^{-rs}\left\|\nabla_{\Gamma}y^{\delta}(s)\right\|_{H_{\Gamma}}^{2 }\,ds\] \[\qquad+2\int_{0}^{t}e^{-rs}\left((\xi,\xi_{\Gamma})^{\delta}(s),(x,y)^{\delta}(s)\right)_... | outline | |
\[\frac{1}{t}\iint\limits_{[0,t]\times\Omega}\phi^{N}(s,x)\bar{\rho }_{0}(x)\ \mathrm{d}s\ \mathrm{d}x =\frac{1}{t}\iint\limits_{[0,t]\times\Omega}\left(\phi^{N}(0^{+},x)+\int_{0}^{s}\partial_{t}\phi^{N}(r,x)\ \mathrm{d}r\right)\bar{\rho}_{0}(x)\ \mathrm{d}s\ \mathrm{d}x\] \[\geqslant\int_{\Omega}\phi^{N}(0^{+},\cdot)\... | outline | |
\[(J_{\theta})(s)=\frac{\|f(s)\|_{X}^{p-2}J_{X}(f(s))}{\|f\|_{L_{p} (S;X)}^{p-2}}\] \[\qquad=\sum_{n\in M}\frac{\|f_{A_{n}}(s)\|_{X}^{p-2}J_{X}(f_{A_{n} }(s))}{\|f\|_{L_{p}(S;X)}^{p-2}}\] \[\qquad=\sum_{n\in M}\frac{\|f_{A_{n}}(s)\|_{X}^{p-2}J_{X}(f_{A_{n} }(s))}{\|f\|_{L_{p}(S;X)}^{p-2}}\] \[\qquad=\frac{1}{\|f\|_{L_{... | outline | |
\[A =-t^{j+1}\left\{\mathbf{c}\left(\left(\begin{array}{cc}1&0\\ 0&1\end{array}\right)\left(\begin{array}{cc}1&0\\ 0&t\end{array}\right)\right)+\sum\limits_{v\in\mathbb{F}_{q}^{*}}\mathbf{c} \left(\left(\begin{array}{cc}1&0\\ vt&1\end{array}\right)\left(\begin{array}{cc}1&0\\ 0&t\end{array}\right)\right)\right\}\left(X... | matrix | |
\[\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{TV}(\bar{\rho}(t)) =\sum_{i=0}^{N}\sigma_{i}(\rho_{i+1}^{\prime}-\rho_{i}^{\prime})=- \sum_{i=1}^{N}\mu_{i}\rho_{i}^{\prime}=\sum_{i=1}^{N}\mu_{i}N\rho_{i}^{2}(x_{i}^ {\prime}-x_{i-1}^{\prime})\] \[=\sum_{i=1}^{N}\mu_{i}N\rho_{i}^{2}(v_{i}U_{i}-v_{i-1}U_{i-1})\] \[=\sum_{i=... | outline | |
\[\big{\|}S_{\alpha}(t)u\mid\mathcal{N}^{\sigma}_{p,q,r}\big{\|} =\bigg{\|}\alpha\int_{0}^{\infty}\theta\Phi_{\alpha}(\theta)e^{t^ {\alpha}\theta\Delta}u(x)d\theta\mid\mathcal{N}^{\sigma}_{p,q,r}\bigg{\|}\] \[\leqslant\alpha\int_{0}^{\infty}\theta\Phi_{\alpha}(\theta) \big{\|}e^{t^{\alpha}\theta\Delta}u\mid\mathcal{N}^... | outline | |
\begin{table}
\begin{tabular}{|c|c|c|c|} \hline Rogue wave & Solution & \(M_{1}\) in (4.12) & \(M_{2}\) in (4.13) \\ \hline \(\lambda_{1}=\sqrt{z_{1}}\) & (1.2) with \(k=0.9\) & 1.45 & 3.96 \\ \(\lambda_{2}=\sqrt{z_{2}}\) & same & 1.71 & 4.68 \\ \(\lambda_{3}=\sqrt{z_{3}}\) & same & 1.84 & 5.03 \\ \(\lambda_{1}=\sqrt{z... | table | |
\[\sum_{\omega\in M_{L},\omega|\nu}\frac{[L_{\omega}:\mathbb{Q}_{ \nu}]}{[L:\mathbb{Q}]} \sum_{\phi^{n}(\alpha)=\beta}\log\max\{|\alpha|_{\omega},1\}\] \[=\sum_{\omega\in M_{L},\omega|\nu}\frac{[L_{\omega}:\mathbb{Q}_{ \nu}]}{[L:\mathbb{Q}]}\left(\sum_{\phi^{n}(\alpha)=\beta}\log|\alpha|_{\omega }-\sum_{\phi^{n}(\alpha... | outline | |
\[\mathrm{D}_{\Xi}(\Phi) =\inf_{\begin{subarray}{c}u_{i}\in C_{\mathrm{lin}}(\mathbb{R}_{ +},\mathbb{R}_{+})\\ H_{i},H_{i,j,k}\in C_{b}(\mathbb{R}^{n}):\\ \Psi^{V}_{(H_{i}),(H_{i,j,k}),(u_{i})}\geq\Phi\end{subarray}}\sum_{i=1}^{n}\int u _{i}(s_{i})\,\mathrm{d}\mu_{i}(s_{i})\] \[\geq\inf_{\begin{subarray}{c}u_{i}\in C_{... | outline | |
\[\|\Pi^{2}(v_{1})(t,\cdot)-\Pi^{2}(v_{2})(t,\cdot)\|_{L^{1}( \mathbb{R}^{d})}\] \[\leq C^{2}\int\limits_{r}^{t}\int\limits_{r}^{s}\frac{1}{\sqrt{t-s}} \frac{1}{\sqrt{s-\theta}}\|v_{1}(\theta,\cdot)-v_{2}(\theta,\cdot)\|_{L^{1}( \mathbb{R}^{d})}\,\mathrm{d}\theta\,\mathrm{d}s\] \[= C^{2}\int\limits_{r}^{t}\int\limits_{... | outline | |
\[\Sigma(t) = E[(D_{t}-\mathbb{E}[D_{t}])(D_{t}-\mathbb{E}[D_{t}])^{\top}]\] \[= e^{\int_{0}^{t}\mathcal{A}(s)ds}E[(D_{0}-\mathbb{E}[D_{0}])(D_{0 }-\mathbb{E}[D_{0}])^{\top}]\left(e^{\int_{0}^{t}\mathcal{A}(s)ds}\right)^{\top}\] \[+ \left(\int_{0}^{t}e^{\int_{s}^{t}\mathcal{A}(u)du}dM_{s}\right) \left(\int_{0}^{t}e^{\i... | outline | |
\[G_{1}^{(+)}=\begin{pmatrix}\mathsf{G}_{11}(x,t)&\mathsf{G}_{12}(x,t)&\mathsf{ G}_{12}(-x,-t)&\overset{\ast}{\mathsf{G}}_{12}(x,t)&\overset{\ast}{\mathsf{G}}_{1 2}(-x,-t)\\ \overset{\ast}{\mathsf{G}}_{12}(x,t)&\mathsf{G}_{22}(x,t)&\overset{\ast}{ \mathsf{G}}_{32}(x,t)&\overset{\ast}{\mathsf{G}}_{42}(x,t)&\overset{\ast... | outline | |
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Keywords:** & & \\ \hline Biometrics; & Filtering; & Publishing; \\ Categorization; & Geospatial; & Resolving terms; \\ City; & Government; & Safety; \\ Clustering; & Information management; & Searching; \\ Context management; & Information technology; & Secure; \\ C... | table | |
\[|G_{12}|\leq \sum_{p>Q_{b,h}}\|b_{p}\|_{\infty}\sum_{-1\leq p^{\prime}\leq p-2 }\lambda_{p^{\prime}}\|w_{p^{\prime}}\|_{2}\sum_{|q-p|\leq 2}\lambda_{q}^{2s}\|m_{q }\|_{2}\] \[\leq c_{r}\kappa\sum_{p>Q_{b,h}}\sum_{-1\leq p^{\prime}\leq p-2} \lambda_{p^{\prime}}^{s+1}\|w_{p^{\prime}}\|_{2}\lambda_{p^{\prime}}^{-s}\sum ... | outline | |
\[\left|e^{(\tau-\bar{\tau})\mathscr{L}_{\infty}}\hat{\varepsilon}_ {-}(\bar{\tau})(y,\tau)\right| \lesssim \sum_{j=2}^{\infty}e^{(\frac{\alpha}{2}-j)(\tau-\bar{\tau})} \left|\langle\hat{\varepsilon}_{-}(\bar{\tau})\phi_{j,\infty}\rangle_{L^{2}_{ \rho}}\right|\left|\phi_{j,\infty}(y)\right|\] \[\lesssim \sum_{j=2}^{\in... | outline | |
\[\lim_{s\to 0}\left\|\frac{\nabla^{n}_{H_{R}}\varphi(x+sv_{n})- \nabla^{n}_{H_{R}}\varphi(x)}{s}-T_{n-1}\left(\nabla^{n+1}_{R}\varphi(x)( \cdot,\ldots,\cdot,h_{n})\right)\right\|_{\mathscr{L}^{(n-1)}(H_{R})}\] \[\quad=\lim_{s\to 0}\left\|T_{n-1}\left(\frac{\nabla^{n}_{R} \varphi(x+sv_{n})-\nabla^{n}_{R}\varphi(x)}{s}-... | outline | |
\[W_{F_{a}}(\omega) =\sum_{x\in\mathbb{F}_{2^{n}}}(-1)^{\operatorname{Tr}_{2^{n}/2}( ax^{2^{e}}h(\operatorname{Tr}_{2^{n}/2^{m}}(x)))+\operatorname{Tr}_{2^{n}/2}( \omega x)}\] \[=\sum_{u\in\mathbf{U}}\sum_{y\in\mathbb{F}_{2^{m}}}(-1)^{ \operatorname{Tr}_{2^{n}/2}(a(y+u)^{2^{e}}h(\operatorname{Tr}_{2^{n}/2^{m}}(u )))+\o... | outline | |
\[\|\Delta(U_{k,\varepsilon}^{+}\psi_{k,\varepsilon}^{+})\|_{ \mathsf{L}^{2}(B_{k,\varepsilon}^{+})}^{2} \leq C\varrho_{k,\varepsilon}^{\pm}\|2(\nabla U_{k,\varepsilon}^{ +},\nabla\psi_{k,\varepsilon}^{+})+U_{k,\varepsilon}^{+}\Delta\psi_{k, \varepsilon}^{+}\|_{\mathsf{L}^{\infty}(B_{k,\varepsilon}^{+}\setminus \mathca... | outline | |
\begin{table}
\begin{tabular}{|l|c|c|c|c|} \hline Design variable & Best known & \(x_{\text{lb}}\) & \(x_{\text{ub}}\) & Starting guess \\ \hline \hline Wing span \(x_{1}\) & 44.19 & 30.0 & 45.0 & 37.5 \\ Root cord \(x_{2}\) & 6.75 & 6.0 & 12.0 & 9.0 \\ Taper ratio \(x_{3}\) & 0.28 & 0.28 & 0.50 & 0.39 \\ Angle of atta... | table | |
\[\left\|\bm{x}^{*}-\widetilde{\bm{x}}\right\|_{\mathbf{W}_{k}^{-1}}\] \[= \left\|\sum_{i\in[k]}\mathbf{W}_{i}\mathbf{B}\left[\begin{array}[ ]{c}-\mathbf{L}_{FF}^{-1}\mathbf{L}_{FC}\\ \mathbf{I}\end{array}\right](\mathbf{SC}(\mathbf{L}_{i},C)^{\dagger}-\widetilde {\mathbf{SC}}_{i}^{\dagger})\left[\begin{array}{c}-\math... | matrix | |
\[\sin(\zeta)k(z,y)\geq\eta(z,y):=\] \[\begin{array}{ll}\cos([\max\{|\zeta a+\frac{\pi}{4}|,|\zeta b+\frac{\pi}{4}|\} )\cos[\zeta(y-1)-\frac{\pi}{4}],&|z|<y,\ y\in[b,1],\\ \cos([\max\{|\zeta a+\frac{\pi}{4}|,|\zeta y+\frac{\pi}{4}|\})\cos[\zeta(y-1)- \frac{\pi}{4}],&|z|<y,\ y\in[a,b),\\ \cos([\max\{|\zeta(1-y)-\frac{\p... | outline | |
\[\frac{(k-4)C-4s^{2}c_{0}|p-2|}{4k^{2}C}\int_{\Omega}\left|\frac{ \eta\nabla f^{k}}{(\phi\circ u)^{k/2}}\right|^{2}+\frac{2s(k+2-2\alpha)}{k^{2} }\int_{\Omega}\left|f^{k}\eta\nabla\left(\frac{1}{(\phi\circ u)^{k/2}}\right) \right|^{2}\] \[+\left(s\beta-\frac{c_{0}|p-2|}{C}-A(\phi)\right)\int_{\Omega} \left(\frac{f^{2}... | outline | |
\[[\mathcal{O}_{\mathbf{Q}_{G}(s_{1}s_{2}\cdots s_{k}s_{k+1})}] =-\sum_{0\leq l\leq k}(-1)^{l}\mathbf{e}^{(l+1)\varpi_{1}}\sum_{ \begin{subarray}{c}J\subset[k]\\ |J|=l\end{subarray}}\left(\prod_{j\notin J,\,j+1\in J}(1-\mathfrak{t}_{j}) \right)[\mathcal{O}_{\mathbf{Q}_{G}}(w_{\circ}\epsilon_{J\cup\{k+1\}})]\] \[\quad+\... | outline | |
\[-e^{2(-F+K)} \Big{(}\mathsf{c}^{2}\rho\frac{1+\frac{1}{\mathsf{c}^{2}} \mathfrak{Q}_{1}}{1-\frac{1}{\mathsf{c}^{2}}\mathfrak{Q}_{1}}+P\frac{3-\frac {2}{\mathsf{c}^{2}}\mathfrak{Q}_{1}}{1-\frac{1}{\mathsf{c}^{2}}\mathfrak{Q}_ {1}}\Big{)}+\mathsf{c}^{2}\rho_{\mathsf{N}}=\] \[=-\mathsf{c}^{2}[Q1]+[Q2]-3P-P\Big{(}e^{2(-F... | outline | |
\begin{table}
\begin{tabular}{c c c c} \hline \(\tau\) & \(1/20\) & \(1/40\) & \(1/80\) \\ \hline \(e_{B,2}(t_{n}=4)\) & 4.346e-10 & 6.010e-12 & 9.207e-14 \\ rate & - & 6.176 & 6.028 \\ \hline \(e_{\rho,2}(t_{n}=4)\) & 5.571e-10 & 8.602e-12 & 1.345e-13 \\ rate & - & 6.017 & 5.999 \\ \hline \(e_{u,2}(t_{n}=4)\) & 1.066e... | table | |
\[|\Pi_{f_{0},6}| \lesssim\sum_{k=1}^{3}\Big{(}t^{k}\|\partial_{v}^{k+1}f_{0}\|_{ \mathcal{G}^{\lambda,\beta;s}}\|g\|_{\mathcal{G}^{\lambda,\beta;s}}\|\partial_ {v}^{k}f_{0}\|_{\mathcal{G}^{\lambda,\beta;s}}+t^{k}\|\partial_{v}^{k+1}f_{0} \|_{\mathcal{G}^{\lambda,\beta;s}}\|\partial_{v}^{2}g\|_{\mathcal{G}^{\lambda,\be... | outline | |
\[\mathcal{K}^{l}_{m,k}=\left\{\begin{array}{ll}-2\Delta t^{n+1} \frac{h_{j}^{y}}{h_{i-\frac{1}{2}}^{x}}\mathcal{E}_{x}^{-}S_{i,j}^{l,n}\mu_{m +N_{y}}^{l,n}-2\Delta t^{n+1}\frac{h_{j}^{y}}{h_{i-\frac{1}{2}}^{x}}\mathcal{E }_{x}^{+}S_{i-1,j}^{l,n}\mu_{m}^{l,n},\quad m=[i-1,j],\\ \\ -2\Delta t^{n+1}\frac{h_{i}^{y}}{h_{j-... | outline | |
\[\mathop{\mathbb{E}}_{G\sim\text{Gin}(d_{1},d_{2})}\Biggl{[}\exp \left(\sum_{i=1}^{n}\frac{8\varepsilon^{2}}{d_{2}^{2}}\left(\frac{x_{i}^{ \dagger}Gy_{i}}{x_{i}^{\dagger}Ax_{i}+y_{i}^{\dagger}By_{i}}\right)^{2} \right)\Biggr{]} =\mathop{\mathbb{E}}_{G\sim\text{Gin}(d_{1},d_{2})}\Biggl{[}\exp \left(\frac{8\varepsilon^{... | outline | |
\[\sum_{j=1}^{n} \xi_{i}x_{j}^{2}\otimes 1-2\xi_{i}x_{j}\otimes x_{j}+\xi_{i}\otimes x_{j} ^{2}-x_{j}^{2}\otimes\xi_{i}+2x_{j}\otimes x_{j}\xi_{i}-1\otimes x_{j}^{2}\xi_{i}\] \[=2r_{i}\cdot D(X)\] \[=2\lim_{k\to\infty}p_{k}\otimes 1-1\otimes p_{k}\] \[=[(1\otimes\tau^{\circ})(2r_{i}\cdot D(X))]\otimes 1+1\otimes[(\tau ... | outline | |
\[\|\text{proj}_{t}h\|_{2}^{2} =a_{I}^{\top}\Gamma_{m}a_{I}+2\sum_{i\leq m<j}a_{i}a_{j}\mathbb{E }\left\{\mathbb{E}[e_{i}(W_{t},S_{t},A_{t})\mid Z_{t},S_{t},A_{t}]\mathbb{E}[ e_{j}(W_{t},S_{t},A_{t})\mid Z_{t},S_{t},A_{t}]\right\}\] \[\quad+\mathbb{E}\left(\sum_{j>m}a_{j}\mathbb{E}[e_{j}(W_{t},S_{t },A_{t})\mid Z_{t},S... | outline | |
\[\frac{C_{s+1}}{(2s+1)L}\frac{\partial}{\partial A_{s}}\mathcal{F} _{g,m}^{[\mathbb{C}^{n}/\mathbb{Z}_{n}]}[b_{0},\ldots,b_{n-1}]\] \[= \frac{1}{2}\mathcal{F}_{g-1,m+2}^{[\mathbb{C}^{n}/\mathbb{Z}_{n}] }[b_{0},\ldots,b_{s-1},b_{s}+2,b_{s+1},\ldots,b_{n-1}]\] \[+\frac{1}{2}\sum_{\begin{subarray}{c}g_{1}+g_{2}=g\\ m_{1}... | outline | |
\[\left(\begin{matrix}x\\ y\\ z\end{matrix}\right)(\epsilon_{1},\epsilon_{2})\ =\ r\ \left(\begin{matrix}\sin(\theta{+}{\rm i}\epsilon_{1})\cos(\phi{+}{\rm i} \epsilon_{2})\\ \sin(\theta{+}{\rm i}\epsilon_{1})\sin(\phi{+}{\rm i}\epsilon_{2})\\ \cos(\theta{+}{\rm i}\epsilon_{1})\end{matrix}\right)\ =\ r\ \left(\begin{ma... | matrix | |
\begin{table}
\begin{tabular}{|l|l|} \hline \(s\) & \(\Lambda\) \\ \hline \hline
0 & \(\{1\}\) \\ \hline
1 & \(\{-2\sqrt{2}-5,2\sqrt{2}-5\}\) \\ \hline
2 & \(\{\frac{2^{\sqrt{2}-9}+i\sqrt{1455}}{3^{2/3}}+\frac{16}{\sqrt[3]{3}\left(-9+i \sqrt{1455}\right)}-7,-\frac{\left(1+i\sqrt{3}\right)\sqrt[3]{-9}+i\sqrt{1455} }{3^{... | table | |
\[\begin{array}{rl}P_{\mathrm{b}}(2)=&P_{\mathrm{boa}}(2)+(1-P_{ \mathrm{boa}}(2))P_{\mathrm{pre}}(2)\\ =&\sum_{i=0}^{k}\pi_{i,k-i}+\frac{\lambda_{1}}{\lambda_{2}} \times\left(\sum_{i=0}^{k}\pi_{i,k-i}-\pi_{k,0}\right)\\ =&\pi_{k,0}+(1+\frac{\lambda_{1}}{\lambda_{2}})\times\left( \sum_{i=0}^{k}\pi_{i,k-i}-\pi_{k,0}\rig... | matrix | |
\[f^{*}=\sum_{\alpha\in\operatorname{Irr}(\mathbb{G})}\sum_{i,j=1}^{n_{\alpha}}d_{ \alpha}(\widehat{f}(\alpha)Q_{\alpha})_{i,j}(u_{j,i}^{\alpha})^{*}\] \[\langle\mu,f^{*}\rangle_{M(\mathbb{G}),C_{r}(\mathbb{G})} =\sum_{\alpha\in\operatorname{Irr}(\mathbb{G})}\sum_{i,j=1}^{n_{ \alpha}}d_{\alpha}\overline{(\widehat{f}(\a... | outline |
Subsets and Splits
Complex LaTeX Formulas Query
Retrieves a large sample of LaTeX formulas from a benchmark dataset excluding table categories, providing basic data access but offering limited analytical value for understanding patterns or relationships.