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\[\Lambda_{(1,-a)}.\Lambda_{(1,-a)}= -\sum_{\begin{subarray}{c}a^{\prime}\in\mathbb{Z}/p^{r}\mathbb{Z} \\ a^{\prime}\neq a\end{subarray}}\Lambda_{(1,-a^{\prime})}.\Lambda_{(1,-a)}-\sum_ {b\in\mathbb{Z}/p^{r-1}\mathbb{Z}}\Lambda_{(-pb,1)}.\Lambda_{(1,-a)}\] \[= -\deg\mathrm{S}(N)\sum_{\begin{subarray}{c}a^{\prime}\in \m... | outline | |
\[\frac{\operatorname{d}\!\pi}{\operatorname{d}\!x\!\operatorname{ d}\!y}(x,y) =\exp\left(\frac{f(x)+g(y)-\|x-y\|^{2}}{2\sigma^{2}}\right)\frac{ \operatorname{d}\!\alpha}{\operatorname{d}\!x}(x)\frac{\operatorname{d}\! \beta}{\operatorname{d}\!y}(y)\] \[=\omega\exp\left(\mathcal{Q}(\mathbf{A}^{-1}\mathbf{a}+\mathbf{u},... | matrix | |
\[f_{m,p,i,j} =\mathcal{R}_{m,r}f_{m,p,i-1,j}-a_{i}\Psi_{m}\] \[=\operatorname{rem}\left(x^{m-r}f_{m,p,i-1,j},\ x^{m}-1\right)-a_{ i}\Psi_{m}\] \[=\operatorname{rem}\left(x^{m-r}\Theta_{i-1}\Psi_{m},\ x^{m}-1 \right)-a_{i}\Psi_{m}\] \[=\operatorname{rem}\left(x^{m-r}\Theta_{i-1}\Psi_{m},\ \Phi_{m}\Psi_{m}\right)-a_{i}\... | outline | |
\[\mathbb{E}\Big{[}\frac{\partial\log f(x_{k}^{0}|\theta_{k})}{ \partial x_{k}}|\theta_{k}\Big{]}=\int_{\mathcal{X}_{k}}\frac{\partial f(x_{k}^ {0}|\theta_{k})}{\partial x_{k}}dx_{k}^{0}:=\Delta f(x_{k}^{0})\big{|}_{x_{k}^ {0}\in\partial\mathcal{X}_{k}}\] \[=\left(\begin{array}{c}\int f(x_{k,1}^{0}=+\infty,x_{k,2}^{0},... | matrix | |
\[\begin{split}-(\tilde{\varepsilon}_{2}^{jk_{1}j_{1}}P_{s_{1}}f_{+, \beta}^{j_{1}})(x)=&-\frac{\phi(\beta)\big{(}\frac{1}{2}-\beta \big{)}^{N-j_{1}}}{\big{(}\frac{1}{2}+\alpha\big{)}^{N-j_{1}}\big{(}\frac{1}{2 }-\alpha\big{)}^{N-j_{k}}}\int_{x}^{\infty}dye^{-(y-x)\alpha}e^{-\beta y}\\ &+\frac{\phi(\beta)\big{(}\frac{1... | outline | |
\[\frac{d}{dt}\int h_{\theta^{6}}^{2}\,d\theta =2\int(1/k)_{\theta^{6}}k_{\theta^{6}}\,d\theta-2\int h_{\theta^{ 6}}^{2}\,d\theta\] \[\leq-c\int k_{\theta^{6}}^{2}\,d\theta-2\int h_{\theta^{6}}^{2} \,d\theta\] \[\qquad+C\int k_{\theta^{5}}^{2}k_{\theta}^{2}+k_{\theta^{4}}^{2} k_{\theta\theta}^{2}+k_{\theta^{4}}^{2}k_{\... | outline | |
\[\mathbb{E}\left[\max_{\tau\in[0,t]}\left|\frac{\sigma}{\gamma} \int_{0}^{\tau}D(X_{s}^{\alpha}(\rho_{s}^{m_{k}})-\overline{X}_{s}^{m_{k}})dB_ {s}-\sigma\int_{0}^{\tau}D(X_{s}^{\alpha}(\rho)-\widehat{X}_{s})dB_{s}\right| ^{2}\right]\] \[\leq C\mathbb{E}\left[\int_{0}^{t}\left|\frac{1}{\gamma}(X_{s}^{ \alpha}(\rho^{m_{... | outline | |
\[\begin{split}&\frac{E_{a}[f(X_{\tau_{y}^{-}}),X_{\tau_{y}^{-}}\in \mathbb{B}_{y}^{(-i)}]}{E_{a+e_{1}}[f(X_{\tau_{y}^{-}}),X_{\tau_{y}^{-}}\in \mathbb{B}_{y}^{(-i)}]}=p_{a,a+e_{1}}\\ &+p_{a,a+e_{2}}\frac{E_{a+e_{2}}[f(X_{\tau_{y}^{-}}),X_{\tau_{y}^{ -}}\in\mathbb{B}_{y}^{(-i)}]}{E_{a+e_{1}+e_{2}}[f(X_{\tau_{y}^{-}}),X... | outline | |
\[\mathbb{E}\left[A_{T_{S}}^{(S)}\left(F(u^{(S)})-F(x^{*})\right)\right]\] \[\leq A_{T_{0}}^{(0)}\left(F(u^{(0)})-F(x^{*})\right)+\mathbb{E}\left[\frac{ \gamma_{0}^{(s)}}{2}\left\|z_{0}^{(s)}-x^{*}\right\|^{2}-\frac{\gamma_{0}^{(S+1) }}{2}\left\|z_{0}^{(S+1)}-x^{*}\right\|^{2}\right]\] \[+\mathbb{E}\left[\sum_{s=1}^{S}... | outline | |
\[\mathbb{E}_{N_{1,\,\delta,\,\theta}(\mathsf{b}_{n})-1}\tau_{n}\\ =\exp\Big{(}t_{n}\sum_{k=1}^{N_{1,\,\delta,\,\theta}(\mathsf{b}_ {n})-1}\mathsf{b}_{n}^{S_{k-1}}\tilde{\eta}_{k}(\mathsf{b}_{n})-(t_{n}^{2}/2)e ^{4\rho(1+\varepsilon)}\sum_{k=1}^{N_{1,\,\delta,\,\theta}(\mathsf{b}_{n})-1} \mathsf{b}_{n}^{2S_{k-1}}\mathb... | outline | |
\[WG\underline{\beta} =\left(\begin{array}{c}\underline{w}^{(1)}G\\ \underline{w}^{(2)}G\\ \vdots\\ \underline{w}^{(l)}G\end{array}\right)\underline{\beta}=\left(\begin{array}[] {cccc}g_{1}(\underline{w}^{(1)})&g_{2}(\underline{w}^{(1)})&\cdots&g_{l}( \underline{w}^{(1)})\\ g_{1}(\underline{w}^{(2)})&g_{2}(\underline{w... | outline | |
\[r(\{1\},\{1\}) =(-1)^{2}T(\{1\}|\{1\})T(\{2,3\}|\{2,3\})=0,\] \[r(\{1\},\{2\}) =(-1)^{3}T(\{1\}|\{2\})T(\{2,3\}|\{1,3\})=-12,\] \[r(\{1\},\{3\}) =(-1)^{4}T(\{1\}|\{3\})T(\{2,3\}|\{1,2\})=-12,\] \[r(\{2\},\{1\}) =(-1)^{3}T(\{2\}|\{1\})T(\{1,3\}|\{2,3\})=-18,\] \[r(\{2\},\{2\}) =(-1)^{4}T(\{2\}|\{2\})T(\{1,3\}|\{1,3\})... | outline | |
\[\begin{array}{l}U_{1}^{1}=u_{0}^{1}+\Delta x\frac{1}{2}\mathcal{W}_{1}^{1}, \quad\mathcal{W}_{1}^{1}=w_{0}^{1}+\Delta x\frac{1}{2}\delta_{x}\mathcal{W}_{ 1}^{1},\\ \overline{u}_{1}^{1}=u_{0}^{1}+\Delta x\mathcal{W}_{1}^{1},\quad\overline{w}_{ 1}^{1}=w_{0}^{1}+\Delta x\delta_{x}\mathcal{W}_{1}^{1},\\ U_{1}^{1}=u_{1}^{... | outline | |
\[N^{f,\boldsymbol{\alpha}^{\mathbf{x}}}_{t} :=f(\mathbf{X}^{\boldsymbol{\alpha}^{\mathbf{x}}}_{t})-f(\mathbf{X }^{\boldsymbol{\alpha}^{\mathbf{x}}}_{0})-\int_{0}^{t}\sum_{i=1}^{d}\partial_ {i}f(\mathbf{X}^{\boldsymbol{\alpha}^{\mathbf{x}}}_{s-})\,\mathrm{d}B^{j}_{s} -\frac{1}{2}\int_{0}^{t}\sum_{i,j=1}^{d}\partial_{ij... | outline | |
\[P_{\theta_{0}}^{(n)}\left[\frac{\alpha}{1-\alpha}\int r_{n}( \theta,\theta_{0})\rho_{n}(d\theta)\geq\frac{\alpha}{1-\alpha}\int\mathrm{E}[ r_{n}(\theta,\theta_{0})]\rho_{n}(d\theta)+\frac{\alpha}{1-\alpha}\sqrt{\frac{ \mathrm{Var}[\int r_{n}(\theta,\theta_{0})\rho_{n}(d\theta)]}{\eta}}+\frac{ \mathcal{K}(\rho_{n},\pi... | outline | |
\[\begin{split}& u^{-5/2}\mathbf{q}_{\text{LO}}^{(1)}(u)=\sum_{n=0}^ {\infty}\frac{((1+\Delta)/2)_{n}}{(2)_{n}}\frac{((1-\Delta)/2)_{n}}{(2)_{n}} \frac{(1+iu)_{n}}{n!}=\,_{3}F_{2}\left(\begin{matrix}1+iu,(1+\Delta)/2,(1- \Delta)/2\\ 2,2\end{matrix};1\right)\,,\\ & u^{-5/2}\mathbf{q}_{\text{NLO}}^{(1)}(u)=\frac{I_{1}}{2... | outline | |
\[\zeta^{3}\mu^{4}\phi_{3}^{3}e^{2\zeta\phi_{3}}\int_{-1}^{1}\int_ {\widetilde{\omega}}\Big{(}|U_{1}^{F}|^{2}+|U_{2}^{F}|^{2}\Big{)}dxds\] \[\leq\zeta^{3}\mu^{4}\int_{-b_{0}}^{b_{0}}\int_{\widetilde{\omega} }\theta^{2}\phi^{3}\Big{(}|U_{1}^{F}|^{2}+|U_{2}^{F}|^{2}\Big{)}dxds\] \[\leq\zeta\mu^{2}\int_{-b_{0}}^{b_{0}}\in... | outline | |
\[\begin{split} d^{-\alpha}&(P,Q)\Big{|}D\delta z_{- }^{\rm(n)}(P)-D\delta z_{-}^{\rm(n)}(Q)\Big{|}\\ &\leq d^{-\alpha}(P,Q)\Big{|}D\delta z_{-}^{\rm(n)}(P)-D\delta z _{-}^{\rm(n)}(Q_{+}(\xi_{P}))\Big{|}\\ &\qquad+d^{-\alpha}(P,Q)\Big{|}D\delta z_{-}^{\rm(n)}(Q_{+}(\xi_{ P}))-D\delta z_{-}^{\rm(n)}(Q)\Big{|}\\ &\leq\ma... | outline | |
\[u^{(1)} =\mathrm{e}^{-2F^{\prime}_{D}/\hbar}\frac{\mathrm{i}u^{\prime}}{ 2\pi},\] \[u^{(2)} =\mathrm{e}^{-4F^{\prime}_{D}/\hbar}\,\left(\frac{\mathrm{i}u^{ \prime}}{4\pi}-\frac{u^{\prime\prime}}{8\pi^{2}}+\frac{F^{\prime}_{D}}{2\pi^{ 2}\hbar}\right),\] \[u^{(3)} =\mathrm{e}^{-6F^{\prime}_{D}/\hbar}\,\left(\frac{\math... | outline | |
\[\begin{array}{ll}x_{i,t}&=A_{i,1}^{t}x_{i,1}+\sum_{\tau=1}^{t-1}\Big{(}A_{i, \tau+1}^{t}B_{i,\tau}\bm{\zeta}_{{}_{\mathcal{N}_{i}},\tau}+A_{i,\tau+1}^{t}D_{i,\tau}\widehat{\Gamma}_{i,\tau}(\bm{\xi}_{i}^{\tau-1},\bm{s}_{{}_{\mathcal{N}_{i} }}^{\tau})+A_{i,\tau+1}^{t}E_{i,\tau}\xi_{i,\tau}\Big{)}\\ &=A_{i,t-1}^{t}x_{i,... | outline | |
\[\|J^{(1+i\eta)}e^{i\eta\beta(|x|+|y|)}f\|_{L^{2}(\mathbb{R}^{2})} \lesssim\|Je^{i\eta\beta(x+y)}f\|_{L^{2}(x\geq 0,y\geq 0)}+\|Je^{i \eta\beta(-x-y)}f\|_{L^{2}(x\geq 0,y<0)}\] \[\quad+\|Je^{i\eta\beta(-x+y)}f\|_{L^{2}(x<0,y\geq 0)}+\|Je^{i \eta\beta(-x-y)}f\|_{L^{2}(x<0,y<0)}\] \[\lesssim\|Je^{i\eta\beta(x+y)}f\|_{L^... | outline | |
\[\begin{split}&(S(f^{(4)})\otimes S(f^{(1)}))_{v_{1},v_{4}}*(\phi_{ 2}^{(1)}\otimes\phi_{3}\otimes\phi_{4}\otimes\phi_{1}^{(2)})_{v_{1},v_{2},v_{3 },v_{4}}=\\ &\langle\phi_{2}^{(1)(1)}\otimes S(f^{(1)})^{(2)},R^{-1}\rangle \langle\phi_{3}^{(1+\tau_{3})}\otimes S(f^{(1)})^{(1)(2)},(S^{\tau_{3}}\otimes S )(R)\rangle\\ &... | outline | |
\[\mathbb{P}(\overline{W}\leq z)-\Phi(z) =\mathbb{E}[f_{z}^{\prime}(\overline{W})]-\mathbb{E}[\overline{W}f _{z}(\overline{W})]\] \[=\mathbb{E}[f_{z}^{\prime}(\overline{W})]\left(\sum_{i=1}^{n}\int_ {-\infty}^{1}\bar{K}_{i}(t)dt+\sum_{i=1}^{n}\mathbb{E}[{\xi_{i}}^{2}\mathbb{1} _{\xi_{i}>1}]\right)\] \[=\sum_{i=1}^{n}\m... | outline | |
\[\Phi_{\mathsf{D}^{0}}^{\text{joint}}(P_{\mathsf{M}|\mathsf{X}K}) \supseteq\{\mathsf{D}:D_{1}\geq\mathbb{E}[\|X_{1}-\tilde{X}_{1}\|^{2}]+W_{2}^ {2}(P_{\hat{X}_{1}},P_{X_{1}}),\] \[D_{2}\geq\mathbb{E}[\|X_{2}-\tilde{X}_{2}\|^{2}]+\sum_{x_{1}}P_{X _{1}}(x_{1})W_{2}^{2}(P_{\tilde{X}_{2}|\hat{X}_{1}^{*}=x_{1}},P_{X_{2}|X_... | outline | |
\[A =\begin{bmatrix}1&0&-\beta_{1}&0&0&0\\ 0&1&-\gamma_{1}&0&0&0\\ \tau^{2}\lambda&0&\alpha_{1}&0&0&0\\ 0&0&0&1&0&-\beta_{2}\\ 0&0&0&0&1&-\gamma_{2}\\ 0&0&0&\tau^{2}\alpha_{f}\lambda&0&\alpha_{2}\end{bmatrix},\] \[B =\begin{bmatrix}1&1&\frac{1}{2}-\beta_{1}&\frac{1}{6}-\beta_{1}& \frac{1}{24}-\frac{\beta_{1}}{2}&\frac{... | matrix | |
\[\mathbb{E}\big{[}|\varphi(Q_{d,k_{*}}(a,X))-\varphi(Q_{d,k_{*}}(a,X)+\eta Z)|\big{]}\\ =\mathbb{E}_{Z}\mathbb{E}_{\varepsilon,U}\mathbb{E}_{V}\big{[}| (\varphi(Q_{d,k_{*}}(a,X))-\varphi(Q_{d,k_{*}}(a,X)+\eta Z)|\big{]}\\ =\mathbb{E}_{Z}\mathbb{E}_{\varepsilon,U}\mathbb{E}_{V}\Big{[}| (\varphi(Q_{d,k_{*}}(a,X))-\varph... | outline | |
\[\frac{1}{N}\sum_{i_{3}=1}^{n_{3}}\cdots\sum_{i_{d}=1}^{n_{d}}\sum_{i =1}^{\#\sigma_{>r+1}^{(i_{3},\ldots,i_{d})}(\boldsymbol{\mathcal{P}})}\sigma_{i}( \mathbf{\overline{P}}^{(i_{3},\ldots,i_{d})})-\frac{1}{N}\left(\sum_{i_{3}=1}^{ n_{3}}\cdots\sum_{i_{d}=1}^{n_{d}}\#\sigma_{>r+1}^{(i_{3},\ldots,i_{d})}( \boldsymbol{\... | outline | |
\[|b_{h}^{E}(\widehat{u}_{h},w_{I})-b^{E} (\widehat{u}_{h},w_{I})|\leq C_{\vartheta}\|\vartheta(\mathbf{x })\cdot\nabla\widehat{u}_{h}-\Pi^{E}(\vartheta(\mathbf{x})\cdot\nabla \widehat{u}_{h})\|_{0,E}\|w_{I}-\Pi^{E}w_{I}\|_{0,E}\] \[\qquad+|\widehat{u}_{h}-\Pi^{\nabla,E}\widehat{u}_{h}|_{1,E}\| \vartheta(\mathbf{x})w_{... | outline | |
\[\begin{split}&\widehat{\Upsilon}^{0<|y|<|\mathfrak{q}|<|z|<1} _{l,h}\\ &=z^{\frac{-a_{2,h}+a_{2,\bar{h}}+\epsilon}{2\varepsilon_{2}}} \left(\frac{y}{\mathfrak{q}}\right)^{\frac{a_{4,\bar{l}}-a_{4,\bar{l}}+ \epsilon}{2\varepsilon_{1}}}\mathfrak{q}^{-\Delta_{0}-\Delta_{4}+\frac{\epsilon ^{2}-(a_{2,1}-a_{2,2})^{2}}{4\va... | outline | |
\[\begin{split}\langle(p_{1k}\circ\theta_{0}^{\gamma})m_{t}^{1}, \,f\rangle&=\int_{[0,\max\{0,(\tau_{1})^{-1}(t)\}]}f(\Phi_{t}^{e_{1 }}(x,0),t)p_{1k}(\theta_{0}^{\gamma}(x,0))dm_{0}^{1}(x)\\ &+\int_{(\max\{0,(\varsigma_{1})^{-1}(t)\},t]}f(\Phi_{t}^{e_{1}} (0,s),t)p_{1k}(\theta_{0}^{\gamma}(0,s))d\sigma_{0}(s)\\ &=\int_... | outline | |
\[(-1)^{|Q_{-1}|}\rho(\beta;\gamma)\pi^{\mathcal{M}_{0,l}(\beta)}{}_ {*}\left(Q_{-1,l}^{\beta}\right)_{*}(d\xi) =\rho(\beta;\gamma)\sum_{\begin{subarray}{c}P\in S_{3}[l]\\ (2:3)=(j)\end{subarray}}(-1)^{n+|\gamma^{(1:3)}|}\pi^{\mathcal{M}_{0,l}(\beta) }{}_{*}\left(Q_{-1,l}^{\beta}\right)_{*}\bar{\gamma}_{P}\] \[=\sum_{\... | outline | |
\[\int_{G/\Gamma_{\gamma}}f(x\gamma x^{-1})\ {\rm d}\mu_{G/\Gamma_{\gamma}}(\dot{x}) =\ \int_{G/G_{\gamma}}\ \left(\,\int_{G_{\gamma}/\Gamma_{\gamma}}f(x\eta \gamma\eta^{-1}x^{-1})\ {\rm d}\mu_{G_{\gamma}/\Gamma_{\gamma}}(\dot{\eta})\right)\,{\rm d}\mu_{G/G_{ \gamma}}(\dot{x})\] \[=\ \int_{G/G_{\gamma}}\ \left(\,\int_{... | outline | |
\[\langle\mathrm{sgn}(\bm{u}_{S}^{*})+\bm{\Lambda}-\bm{v}-2\bm{B}^{ T}\bm{A}\bm{x}^{*}\,,\bm{\delta}_{\bm{2}}\rangle\] \[= \langle\mathrm{sgn}(\bm{u}_{S}^{*})-\bm{v}_{S}\,,(\bm{\delta}_{ \bm{2}})_{S}\rangle-\langle\bm{v}_{S^{\perp}},(\bm{\delta}_{\bm{2}})_{S^{ \perp}}\rangle+||(\bm{\delta}_{\bm{2}})_{S^{\perp}}||_{1}-\... | outline | |
\[a_{1}(x^{2})+x\kappa_{1}(x^{2})+x^{2\iota+1}(a_{1}(x^{2})+x\varphi _{2}(\kappa_{1}(x^{2}))\] \[= a_{1}(x^{2})+x\kappa_{1}(x^{2})+x^{2\iota+1}\varphi_{2}(a_{1}(x^ {2}))+x^{2\iota+1}x^{4n-1}\varphi_{2}(\kappa_{1}(x^{2}))\] \[= a_{1}(x^{2})+x^{2\iota+1}\varphi_{2}(a_{1}(x^{2}))+x\kappa_{1}(x ^{2})+x^{2\iota}\varphi_{2}(... | outline | |
\[\mathbb{E}^{\mathbb{Q}}\left[\int_{0}^{T}\Gamma_{t}\Delta_{t}dt \right]=\frac{A_{0,T}}{\kappa}\Gamma_{0}\Delta_{0}+\int_{0}^{T}\mathbb{E}^{ \mathbb{Q}}\left[\frac{A_{r,T}}{\kappa}Q\Gamma_{r}\partial_{s}P_{r}\phi_{r} \right]dr\] \[+\int_{0}^{T}\frac{A_{r,T}}{\kappa}\mathbb{E}^{\mathbb{Q}}\left[ Q\sigma\partial_{ss}P_{... | outline | |
\[\iint_{\mathbb{R}^{2}}\|e^{i\partial^{4}_{x}}\widetilde{v}_{1,y_ {1}}e^{i\partial^{4}_{x}}\widetilde{v}_{2,y_{2}}\|_{L^{2}_{x,x}}dy_{1}dy_{2} \lesssim\iint_{\mathbb{R}^{2}}N_{1}^{-\frac{3}{2}}\|\widetilde{v }_{1,y_{1}}\|_{L^{2}_{x}}\|\widetilde{v}_{2,y_{2}}\|_{L^{2}_{x}}dy_{1}dy_{2}\] \[\lesssim\iint_{\mathbb{R}^{2}}... | outline | |
\[\mathcal{C}_{s}= \mathbb{E}_{v}\left[\mathbb{P}\left(\frac{KP_{s}v^{-\alpha}\chi_{ 0}}{I+B_{s}N_{s}}>\tilde{\gamma}^{\rm rf}\right)\right]\overset{(a)}{=} \mathbb{E}_{v}\left[\mathbb{E}_{I}\left[\frac{\Gamma_{u}(\kappa,\frac{\tilde{ \gamma}^{\rm rf}(I+B_{s}N_{s})}{KP_{s}v^{-\alpha}\Theta})}{\Gamma(\kappa)} \right]\ri... | outline | |
\[s \geq\sqrt{(1-\rho^{2})\,r^{2}\delta^{-1}-\mathbb{E}[Z^{2};Z<0]},\] \[s \leq\big{(}1+q_{+}(\kappa)\big{)}\cdot\Big{(}\sqrt{(1-\rho^{2}) \,r^{2}\delta^{-1}-\mathbb{E}[Z^{2};Z<0]}\,+\sqrt{1-\rho^{2}}\,\delta^{-1/2} \sqrt{2q_{+}(\kappa)}\Big{)}\] \[\leq\sqrt{(1-\rho^{2})\,r^{2}\delta^{-1}-\mathbb{E}[Z^{2};Z<0]}+ q_{+}(... | outline | |
\[t:=\left\{\begin{array}{ll}3n+2,&i=1\mbox{ and }j=0,\\ 3\lfloor n/3\rfloor+j+2,&i=1\mbox{ and }j>0\mbox{ and }j\equiv 0\mbox{ (mod 3); or }i=m^{\prime}\mbox{ and }j\equiv 1\mbox{ (mod 3);}\\ &\mbox{ or }i=m\mbox{ and }j\equiv 2\mbox{ (mod 3),}\\ 6\lfloor n/3\rfloor+j+4,&i=1\mbox{ and }j\equiv 1\mbox{ (mod 3); or }i=m... | matrix | |
\[\frac{\partial\log p\left(y_{n}|\hat{x}_{n-1},\mathbf{w}_{n}\right)} {\partial\hat{x}_{n-1}} = 2\rho\operatorname{Re}\left\{\left[y_{n}-\mathbf{w}_{n}^{\text{ H}}\mathbf{a}(\hat{x}_{n-1})\right]^{\text{H}}\cdot\mathbf{w}_{n}^{\text{H}}\frac{ \partial\mathbf{a}(\hat{x}_{n-1})}{\partial\hat{x}_{n-1}}\right\}\] \[= 2\rh... | outline | |
\[\begin{array}{l}\tau^{(1)}_{\mbox{\boldmath$n$}}\tau^{(1)}_{\mbox{\boldmath$ n$}+\mbox{\boldmath$e$}_{1}+\mbox{\boldmath$e$}_{2}+\mbox{\boldmath$e$}_{3}}\;=\; \tau^{(1)}_{\mbox{\boldmath$n$}+\mbox{\boldmath$e$}_{1}}\tau^{(1)}_{\mbox{ \boldmath$n$}+\mbox{\boldmath$e$}_{2}+\mbox{\boldmath$e$}_{3}}+\tau^{(1)}_{ \mbox{\b... | outline | |
\[\sum_{k=1}^{n}\frac{(a-\frac{1}{2})_{k}(\frac{2a+2}{3})_{k}(2b-1) _{k}(2c-1)_{k}(2+2a-2b-2c)_{k}(a+n)_{k}(-n)_{k}}{(1)_{k}(\frac{2a-1}{3})_{k}( 1+a-b)_{k}(1+a-c)_{k}(b+c-\frac{1}{2})_{k}(2a+2n)_{k}(-2n)_{k}}\] \[\quad\times\bigg{\{}\big{[}2H_{k}(2c-2)-2H_{k}(1+2a-2b-2c)+H_{k}( a-c)-H_{k}(b+c-\frac{3}{2})\big{]}\] \[\... | outline | |
\[(II)= \sum_{l=1}^{p}\bm{X}_{l}^{T}\dot{\Sigma}^{-1}(\theta_{0})\bm{X}_{ j}(\beta_{l}-\beta_{0l})+\sum_{k=1}^{q}\sum_{l=1}^{p}\bm{X}_{l}^{T}\dot{\Sigma}^{ k}(\theta^{*})\bm{X}_{j}(\beta_{l}-\beta_{0l})(\theta_{k}^{*}-\theta_{0k})\] \[= (7)+(8)\] \[(7)= \sum_{l=1}^{p}\bm{X}_{l}^{T}diag_{n-1}(\Sigma^{-1})(\tilde{I}_{n -... | outline | |
\[\mathrm{P}\left\{\sup_{\beta\in\mathcal{F}_{p_{n}},\,\|\beta\|_ {L^{2}}\leq 2}\,\frac{\sqrt{n}\|H_{n}(\beta)\|_{K}}{b_{n}\|\beta\|_{L^{2}}^{ \gamma}+1}\geq T_{n}\,\big{|}\mathcal{X}_{n}\right\}\] \[\leq\mathrm{P}\left\{\sup_{\beta\in\mathcal{F}_{p_{n}},\,\|\beta \|_{L^{2}}\leq\mathfrak{g}_{n}^{1/\gamma}}\sqrt{n}\|H_{... | outline | |
\begin{table}
\begin{tabular}{c c|c c} \hline \(S\) & \(\theta(S)\) & \(S\) & \(\theta(S)\) \\ \hline \(\emptyset\) & \(\left(1-\sum_{i=1}^{n-1}x_{i}\right)(1-x_{n})\) & \([n]\) & \(x_{1}\left(\sum_{i=2}^{n}x_{i}-(n-2)\right)\) \\ \(\{1\}\) & \(x_{1}(1-x_{n})\) & \([n]\setminus\{2\}\) & \(x_{1}(1-x_{2})\) \\ \(\{2\}\) ... | table | |
\[\sum_{0\in S\in\Gamma^{\infty,d}}w_{\beta,h}^{q}(S)e^{a(S)} =\sum_{k=1}^{\infty}e^{-(1+\frac{1}{q-1})hk}\sum_{\begin{subarray} {c}0\in S\in\Gamma^{\infty,d}\\ |V_{S}|=k\end{subarray}}e^{(a(S))}e^{-\beta|E_{S}|}\tilde{Z}_{\beta,0}^{0,q-1 }(S)e^{-(1+\frac{1}{q-1})\beta|\partial_{a}S|}\] \[\leq\sum_{k=1}^{\infty}(q-1)^{... | matrix | |
\[\big{|}\partial_{i} \phi^{x}_{\rho_{1}}(\xi_{t})-\partial_{i}\phi^{x}_{\rho_{2}}(\xi_{s} )\big{|}=\Big{|}\rho_{1}^{-1}\partial_{i}\phi\Big{(}\frac{\xi_{t}-x}{\rho_{1}} \Big{)}-\rho_{2}^{-1}\partial_{i}\phi\Big{(}\frac{\xi_{s}-x}{\rho_{2}}\Big{)} \Big{|}\] \[\leq\frac{1}{\rho_{1}}\Big{|}\nabla\partial_{i}\phi\Big{(}c_... | outline | |
\[L= \;(2n)^{-1}\Big{\{}\|\mathbf{Y}\|_{F}^{2}-2\bm{u}_{k}^{T}\mathbf{ Y}^{T}\mathbf{X}\bm{v}_{k}+\bm{u}_{k}^{T}\mathbf{X}^{T}\mathbf{X}\bm{u}_{k}\bm{ v}_{k}^{T}\bm{v}_{k}\] \[+2\langle\mathbf{Y},-\mathbf{X}\widehat{\mathbf{C}}^{(1)}\rangle+ 2\langle\mathbf{Y},-\mathbf{X}\mathbf{C}^{(2)}\rangle+2\bm{u}_{k}^{T}\mathbf{ ... | outline | |
\[\mathcal{B}^{\prime}\equiv\sqrt{a_{\varsigma+s,l,k}}\,\big{[}\det \big{(}-\overline{b}\bm{z}+\overline{a}\big{)}\big{]}^{-\varsigma-s}\ \left((g^{-1}\diamond\bm{z})\cdot(g^{-1}\diamond\bm{z})\right)^{k}\,\mathcal{Y}_{ l-2k,m}\big{(}g^{-1}\diamond\bm{z}\big{)}\] \[=\sum_{l^{\prime\prime\prime}k^{\prime\prime\prime\pri... | outline | |
\[(-1)^{d-1}s_{(m+2,2^{j-1},1^{d-2j})}(\mathbf{x})+(-1)^{d-1}s_{(1^ {j-1})}(\mathbf{x})s_{(m+1,1^{d-j})}(\mathbf{x})+(-1)^{d}s_{(1^{j})}(\mathbf{x })s_{(m+1,1^{d-j-1})}(\mathbf{x})\] \[= (-1)^{d-1}s_{(m+2,2^{j-1},1^{d-2j})}(\mathbf{x})+(-1)^{d-1}\Big{(} s_{(1^{j-1})}(\mathbf{x})s_{(m+1,1^{d-j})}(\mathbf{x})-s_{(1^{j})}... | outline | |
\[\frac{1}{2}\log\left(\frac{\lambda}{\lambda+n}\right)+\frac{1}{2} \frac{n^{2}}{n+\lambda}\frac{\bm{\Omega}_{\tilde{\bm{z}}}}{s_{n}^{2}(\bm{Y}_{n}) }(\hat{\delta}^{ols}_{n}-\delta)^{2}<\frac{1}{2}\log(\alpha^{-2})+o_{a.s.}(1)\] \[\Rightarrow \frac{n^{2}}{n+\lambda}\frac{\bm{\Omega}_{\tilde{\bm{z}}}}{s_{n}^ {2}(\bm{Y}_... | outline | |
\[{}_{3}F_{2}\left[\begin{matrix}a,b,c\\ 3+a-b,a-c\end{matrix}\Big{|}\,1\right]\] \[=\ \frac{2+b^{2}-3b+bc-ac-c}{(b-1)(b-2)}\ \frac{\Gamma(\frac{2+a}{2}) \Gamma(3+a-b)\Gamma(a-c)\Gamma(\frac{4+a}{2}-b-c)}{\Gamma(1+a)\Gamma(\frac{4+a }{2}-b)\Gamma(\frac{9}{2}-c)\Gamma(3+a-b-c)}\] \[+\ \frac{2+b^{2}-3b-bc+ac+c-2c^{2}}{2(... | matrix | |
\begin{table}
\begin{tabular}{|l|l||l|l|} \hline Index & Relation & Index & Relation \\ \hline
1 & Adjectival clause modifier & 20 & Fixed multiword expression \\
2 & Adverbial clause modifier & 21 & Flat multiword expression \\
3 & Adverbial modifier & 22 & Goes with \\
4 & Adjectival modifier & 23 & Indirect object \... | table | |
\[\sum_{k=1}^{K}\sum_{x,a}\bm{\omega}_{k}(x,a)f_{x,a}(n_{k}(x,a)) =\sum_{x,a}\sum_{k=1}^{\tau(x,a)-1}\bm{\omega}_{k}(x,a)f_{x,a}(n_{ k}(x,a))+\sum_{x,a}\sum_{k=\bm{\tau}(x,a)}f_{x,a}(n_{k})\] \[\leq\sum_{x,a}\sum_{k=1}^{\tau(x,a)-1}\bm{\omega}_{k}(x,a)f_{ \max}+\sum_{x,a}\sum_{k=\bm{\tau}(x,a)}f_{x,a}(n_{k})\] \[\leq S... | outline | |
\[S(z) =\left[\big{(}\Sigma_{\alpha\triangle m}^{{}_{(\mathcal{D})}} \big{)}^{(1,1)}(z)\widetilde{F}(z)+\big{(}\Sigma_{\alpha\triangle m}^{{}_{( \mathcal{D})}}\big{)}^{(1,2)}(z)\right]\] \[\cdot\left[\big{(}\Sigma_{\alpha\triangle m}^{{}_{( \mathcal{D})}}\big{)}^{(2,1)}(z)\widetilde{F}(z)+\big{(}\Sigma_{\alpha \triangl... | outline | |
\[||(\frac{\partial^{m_{1}+m_{2}}v_{r,\gamma}(w)}{\partial x_{1}^{m_{1}} \partial x_{2}^{m_{2}}}-v_{r,\gamma}(\frac{\partial^{m_{1}+m_{2}}w}{\partial x_{ 1}^{m_{1}}\partial x_{2}^{m_{2}}}))1_{|x|\geq\frac{1}{2}}||_{L^{\infty}}\] \[\leq C(||v_{1,\gamma}(w)1_{|x|\geq\frac{1}{2}}||_{C^{m_{1}+m_{2}- 1}}+||v_{2,\gamma}(w)1_... | outline | |
\[\frac{d^{2}}{ds^{2}}\mathcal{J}(\mathcal{H}(u,s))\Big{|}_{s=s_{u}}= 8e^{4s_{u}}\int_{\mathbb{R}^{4}}|\Delta u|^{2}dx+2\beta e^{2s_{u} }\int_{\mathbb{R}^{4}}|\nabla u|^{2}dx\] \[-\frac{(8-\mu)^{2}}{2}e^{(\mu-8)s_{u}}\int_{\mathbb{R}^{4}}(I_{ \mu}*F(e^{2s_{u}}u))F(e^{2s_{u}}u)dx\] \[+(28-4\mu)e^{(\mu-8)s_{u}}\int_{\mat... | outline | |
\[\mathcal{S}^{(Rt)}_{4,8} = \alpha_{1,1}\log\left(\frac{p_{1678}p_{2345}}{p_{1345}p_{2678}} \right)+\alpha_{1,2}\log\left(\frac{p_{1245}p_{1678}p_{3456}}{p_{1345}p_{1456} p_{2678}}\right)+\alpha_{1,3}\log\left(\frac{p_{1256}p_{1678}p_{4567}}{p_{1456}p_{1567 }p_{2678}}\right)\] \[+ \alpha_{1,4}\log\left(\frac{p_{1267}p... | outline | |
\[J_{21} =\sum_{k_{j}}2^{k_{3}n/2}\left\|P_{k_{3}}\left[\sum_{i=1}^{n}v_{i} \partial_{x_{i}}P_{\geq k_{1}+k_{2}-9}Q_{\geq k_{1}+k_{2}-10}w(P_{k_{1}}fP_{k_{ 2}}g)\right]\right\|_{L_{t}^{1}L_{x}^{2}}\] \[\lesssim\sum_{k_{j}}2^{k_{3}n/2}\|v\|_{L_{t,x}^{\infty}}\sum_{i=1 }^{n}\sum_{k_{1}+k_{2}\in\mathbb{Z}}\left\|P_{k_{1}+... | outline | |
\[\begin{split} 0=&\int_{\mathbb{R}^{N}}\partial_{t}^{2} \mathbf{u}_{\varepsilon}\partial_{t}\mathbf{u}_{\varepsilon}\,d\mathbf{x}- \int_{\mathbb{R}^{N}}(K\mathbf{u}_{\varepsilon}(\cdot,t))(\mathbf{x}) \partial_{t}\mathbf{u}_{\varepsilon}\,d\mathbf{x}+\varepsilon\int_{\mathbb{R} ^{N}}\Delta^{2}\mathbf{u}_{\varepsilon}\... | outline | |
\begin{table}
\begin{tabular}{l l|l l} \hline parameter & value & parameter & value \\ \hline hidden layers & depth=3, width=30 & \(N_{e}\) & 3 \\
1st hidden layer & \(C^{\infty}\) periodic layer \(\mathcal{L}_{p}(11,30)\), & \(Q\) & 30 (\(C^{\infty}\) periodic DNN), \\ & or \(C^{k}\) periodic layer \(\mathcal{L}_{C^{... | table | |
\[\mathbb{E}\left[\prod_{j=1}^{p}u(t,x_{j})\right] =\mathbb{E}\left[\prod_{j=1}^{p}\sum_{n_{j}=0}^{\infty}I_{n_{j}}( f_{n_{j}}(\cdot,t,x_{j}))\right]\] \[=\sum_{n_{1}=0}^{\infty}\cdots\sum_{n_{p}=0}^{\infty}\mathbb{E} \Big{[}I_{n_{1}}(f_{n_{1}}(\cdot,t,x_{1}))\cdots I_{n_{p}}(f_{n_{p}}(\cdot,t,x_ {p}))\Big{]}\] \[=\sum... | matrix | |
\[X_{1}(t) \geq X_{1m}^{(0,t_{1}]}(t)\] \[\geq\int_{0}^{t}\mu\bigg{(}N-\frac{5N\mu}{s}e^{su}\bigg{)}e^{s(t -u)-7e^{-C_{1}}}du-\sqrt{\frac{48}{\epsilon}\cdot\frac{N\mu}{s^{2}}}e^{st}\] \[=\bigg{(}e^{-7e^{-C_{1}}}\int_{0}^{t}(se^{-su}-5\mu)du-\sqrt{ \frac{48}{\epsilon}\cdot\frac{1}{N\mu}}\bigg{)}\bigg{(}\frac{N\mu}{s}e^{... | outline | |
\[\begin{split} I_{21}&\triangleq c_{v}[(u+U\phi^{\prime}) \partial_{x}+(v-U_{x}\phi)\partial_{y}]D^{\alpha}\theta,\\ I_{22}&\triangleq c_{v}[D^{\alpha},(u+U\phi^{\prime}) \partial_{x}+(v-U_{x}\phi)\partial_{y}]\theta,\\ I_{23}&\triangleq D^{\alpha}(c_{v}\Theta_{x}\phi^{ \prime}u+c_{v}\Theta\phi^{\prime\prime}v-\mu\the... | outline | |
\[\left\{\begin{array}{ll}\overline{\phi}_{1}(z)=\begin{cases}(a-1)Bze^{- \lambda_{1}^{*}z},&z>z_{1},\\ a-1,&z\leq z_{1},\end{cases}\ \ \ \ \ \overline{\phi}_{2}(z)=\begin{cases}(a-1)Bze^{- \lambda_{2}^{*}z},&z>z_{2},\\ a-1,&z\leq z_{2},\end{cases}\\ \underline{\phi}_{1}(z)=\begin{cases}(a-1)Bze^{-\lambda_{1}^{*}z}-p_{... | outline | |
\[\mathsf{U}_{q,t}= \frac{t^{1-N}}{1-t}A_{2}q\sum_{s=1}^{\infty}s\tau_{s}\mathsf{Schur }_{\{s\}}\left(p_{k}=\frac{1-t^{k}}{q^{k}}\partial_{k}\right)+\] \[+\frac{t^{1-N}}{1-t}A_{1}\sum_{s=1}^{\infty}s\tau_{s}\mathsf{Schur }_{\{s-1\}}\left(p_{k}=\frac{1-t^{k}}{q^{k}}\partial_{k}\right)-\frac{t}{1-t} A_{1}\tau_{1}+\] \[+\... | outline | |
\[\begin{split}\begin{array}{|c|cccccccc|}\hline Q\backslash n&-3&-2&-1&0& 1&2&3\\ \hline 0&0&0&0&1&144&10008&446304\\ 1&0&0&0&0&1053&149526&10238319\\ 2&0&0&0&0&52812&8053182&591031890\\ 3&0&0&0&3402&6914214&1001912544&71961634872\\ 4&0&0&0&5520393&1937967282&225717793668&14749020131814\\ 5&0&0&5520393&5626721862&1006... | outline | |
\[\sum_{r=1}^{m}\mathbb{P}(\sup_{d_{r}+g_{n}\leq t\leq\tilde{a}_{r} }(2II_{r,2}(t)\mu_{r}(t)+\frac{1}{6}(\mu_{r}^{2}(t)-\mu_{r}^{2}(d_{r})))\geq 0 |\mathcal{E}_{n},u_{r},l_{r})\] \[=\sum_{r=1}^{m}\mathbb{P}(\sup_{d_{r}+g_{n}\leq t\leq\tilde{a}_{ r}}\left|\frac{2\mu_{r}(t)\lambda([d_{r},t])}{\lambda([l_{r},u_{r}])} \rig... | outline | |
\[x_{3}x_{4}x_{6}+x_{3}x_{4}x_{7}+x_{3}x_{4}x_{8}+x_{3}x_{5}x_{6}+x_{3}x_{5}x_{7}+ x_{3}x_{5}x_{8}+x_{3}x_{6}x_{7}+x_{3}x_{7}x_{8}+x_{4}x_{6}x_{8}+x_{4}x_{7}x_{8}+x _{5}x_{6}x_{8}+x_{5}x_{7}x_{8}+x_{6}x_{7}x_{8})(sx_{2}x_{3}x_{4}x_{5}x_{6}+sx_{ 2}x_{3}x_{4}x_{5}x_{7}+sx_{2}x_{3}x_{4}x_{5}x_{8}+sx_{2}x_{3}x_{3}x_{5}x_{7... | inline | |
\[g^{c} =\frac{\rho+c}{\rho}g_{\mathbb{C}\mathrm{H}^{n-1}}+\frac{1}{4\rho ^{2}}\frac{\rho+2c}{\rho+c}\mathrm{d}\rho^{2}\] \[\quad+\frac{1}{4\rho^{2}}\frac{\rho+c}{\rho+2c}\bigg{(}\mathrm{d} \tilde{\phi}-4\operatorname{Im}\left(\bar{w}^{0}\mathrm{d}w^{0}-\sum_{a=1}^{n- 1}\bar{w}^{a}\mathrm{d}w^{a}\right)+\frac{2c}{1-\|X... | outline | |
\[\mathbf{y}(t)=\int_{0}^{t}\mathbf{C}\Phi(\sigma_{1})\mathbf{Bu}(t _{\sigma_{1}})d\sigma_{1}\\ +\sum_{\xi=2}^{d}\int_{0}^{t}\underbrace{\int_{0}^{t_{\sigma_{1} }}\cdots\int_{0}^{t_{\sigma_{1}}}}_{\xi-\text{times}}\mathbf{C}\Phi(\sigma_{1} )\mathbf{H}_{\xi}\left(\Phi(\sigma_{2})\mathbf{B}\otimes\cdots\otimes\Phi( \sigm... | outline | |
\[\begin{split}& I\left(Z;W\right)=\mathrm{H}\left(Z\right)- \mathrm{H}\left(Z\middle|W\right)\\ &=\mathrm{H}\left(Z\right)-\mathrm{H}\left(Z\middle|W=0\right) \frac{\alpha}{1+\alpha}-\mathrm{H}\left(Z\middle|W=1\right)\frac{1}{1+\alpha} \\ &=-\sum_{x\in\mathcal{X}}\left(\frac{\alpha}{1+\alpha}P\left(x \right)+\frac{1}... | outline | |
\[c_{p}^{r}\] \[= \frac{1}{(MN)^{r}}\int_{T^{r}}\sum_{i_{1}^{1}\ldots i_{p}^{r}} \sum_{b_{1}^{1}\ldots b_{p}^{r}}\frac{Q^{1}_{i_{1}b_{1}^{1}}Q^{1}_{i_{1}b_{1}^ {1}}}{Q^{1}_{i_{1}b_{2}^{1}}Q^{1}_{i_{1}b_{1}^{1}}}\cdots\frac{Q^{1}_{i_{p}^{1 }b_{1}^{1}}Q^{1}_{i_{p}^{2}b_{1}^{1}}}{Q^{1}_{i_{p}^{1}b_{1}^{1}}Q^{1}_{i_{p}^ {2... | outline | |
\[[\partial_{t^{*cr}_{n,m}},\partial_{t^{*dj}_{l,k}}]\mathcal{L}_{i}\] \[= [\sum_{p,s=0}^{\infty}\dfrac{n^{p}(m\log q)^{s}}{p!s!}\partial_{ t^{cr}_{p,s}},\sum_{a,b=0}^{\infty}\dfrac{l^{a}(k\log q)^{b}}{a!b!}\partial_{ t^{dj}_{a,b}}]\mathcal{L}_{i}\] \[= \sum_{p,s=0}^{\infty}\sum_{a,b=0}^{\infty}\dfrac{n^{p}(m\log q)^{ ... | outline | |
\[d_{\{t_{1},\ldots,t_{j+1-m}\}}(x,r)d_{\{t_{j+3-m},\ldots,t_{j+1}\}}(x, r)\prod_{p=1}^{j+1-m}\prod_{\begin{subarray}{c}q=j+3-m\\ t_{q}=t_{p}\end{subarray}}^{j+1}\Delta\big{(}x^{2(q-p+t_{p}-N-2)}\big{)}\] \[\qquad\times\prod_{\begin{subarray}{c}p=1\\ t_{j+2-m}=\overline{t}_{p}\end{subarray}}^{j+1-m}\Delta\big{(}x^{2(j+... | matrix | |
\[\sum_{i=1}^{t}\left((1-p)^{\binom{s_{i}}{k-1}}\left(n-\sum_{j=1} ^{i}s_{j}\right)\right)\] \[\geq\sum_{i=1}^{t^{\prime}}\left((1-p)^{\binom{s_{i}}{k-1}} \left(n-\sum_{j=1}^{i}s_{j}\right)\right)\] \[\geq\frac{n}{\log^{2}d}\sum_{i=1}^{t^{\prime}}(1-p)^{\binom{s_{i} }{k-1}}\geq\frac{nt^{\prime}}{\log^{2}d}(1-p)^{\binom... | outline | |
\[\|Qx\|_{\mathbb{A}}^{2}=(Qx,Qx)_{\mathbb{A}}=(Q^{\#_{\mathbb{A}} }Qx,x)_{\mathbb{A}} =\begin{pmatrix}\begin{bmatrix}P_{\overline{\mathcal{R}(A)}}&O\\ O&P_{\overline{\mathcal{R}(A)}}\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix},\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix}\end{pmatrix}_{\mathbb{A}}\] \[=\begin{pm... | matrix | |
\[M_{2}(x_{k},x_{k+n}) = \max\left\{\begin{array}{c}d(x_{k},x_{k+n}),d(Tx_{k},x_{k}),d( Tx_{k+n},x_{k+n}),\\ d(Tx_{k},x_{k+n}),d(Tx_{k+n},x_{k})\end{array}\right\}\] \[= \max\left\{\begin{array}{c}d(x_{k},x_{k+n}),d(x_{k+1},x_{k}),d( x_{k+n+1},x_{k+n}),\\ d(x_{k+1},x_{k+n}),d(x_{k+n+1},x_{k})\end{array}\right\}\] \[\le... | matrix | |
\[\sum_{j=1}^{n}\frac{1}{(aj+b)^{2k}}\sin\frac{2\pi(aj+b)}{m}=- \frac{1}{2b^{2k}}\left(\sin\frac{2\pi b}{m}-\sum_{j=0}^{k-1}\frac{(-1)^{j}}{(2 j+1)!}\left(\frac{2\pi b}{m}\right)^{2j+1}\right)\\ +\frac{1}{2(an+b)^{2k}}\left(\sin\frac{2\pi(an+b)}{m}-\sum_{j=0}^ {k-1}\frac{(-1)^{j}}{(2j+1)!}\left(\frac{2\pi(an+b)}{m}\rig... | outline | |
\[\varphi_{n}(x)=\sqrt{2}\cos\Big{(}\nu_{n}x+\frac{3-\alpha}{8}\pi- \arctan\Delta_{\alpha}\Big{)}\] \[\quad-\frac{\sqrt{5-\alpha}}{\pi}\int_{0}^{\infty}\rho_{0}(u) \left(Q_{0}(u)e^{-u\nu_{n}x}-(-1)^{n}Q_{1}(u)e^{-u\nu_{n}(1-x)}\right)du\] \[\quad+C_{0}e^{-c\nu_{n}x}\cos\Big{(}s\nu_{n}x+\varkappa_{0} \Big{)}+C_{1}e^{-c\... | outline | |
\[\begin{array}{ll}\mathrm{s.t.}&\mathrm{model}\left(\ref{eq:model}\right) -\left(\ref{eq:model}\right)\\ &\delta x_{\mathrm{L}}[h]=\delta\hat{x}_{\mathrm{L}}[h|h]\\ \delta q_{\mathrm{f}}\in&[0-q_{\mathrm{r}}^{\mathrm{r}},q_{\mathrm{f}, \mathrm{max}}-q_{\mathrm{r}}^{\mathrm{s}}]\\ \delta q_{\mathrm{g}}^{\mathrm{B}}\in&... | matrix | |
\[\left.\frac{dl}{d\epsilon}\right|_{\epsilon=0}=\int_{x_{1}}^{x_{ 2}}\frac{d}{d\epsilon}\left(\frac{\sqrt{1+(y^{\prime}+\epsilon h^{\prime})^{2 }}}{y+\epsilon h}\right)|_{\epsilon=0}dx=\] \[=\int_{x_{1}}^{x_{2}}\left(-\frac{h\sqrt{1+(y^{\prime}+\epsilon h ^{\prime})^{2}}}{(y+\epsilon h)^{2}}+\frac{h^{\prime}(y^{\prime... | outline | |
\[w\in K_{g}(\bar{z},\bar{\lambda}) \iff\,(w,0)\in\mathrm{gph}\,N_{K_{g}(\bar{z},\bar{\lambda})}\] \[\iff\exists\,r^{\prime}>0:t(w,0)\in\big{(}\mathrm{gph}\,N_{K_{g}( \bar{z},\bar{\lambda})}\big{)}\cap\mathbb{B}_{r}(0,0)\quad\text{for all}\ \ t\in[0,r^{\prime})\] \[\iff\exists\,r^{\prime}>0:(\bar{z}+tw,\bar{\lambda})\i... | outline | |
\[\sum_{a}\,\hat{\cal A}_{a}^{(z_{1})}|0\rangle^{1}\otimes\hat{\cal A} _{(z_{2}/q)}^{a}|0\rangle^{2}=\] \[\qquad=\frac{\alpha^{\prime}g_{D}^{2}}{8\pi i}\int\frac{{\rm d}^{D} k}{(2\pi)^{D}}e^{ik\cdot(x_{0}^{(z_{1})}-x_{0}^{(z_{2})})}\bar{q}^{\frac{\alpha^{ \prime}}{4}k^{2}-1}q^{\frac{\alpha^{\prime}}{4}k^{2}-1/2}\] \[\q... | outline | |
\[II\leq\left\langle u_{(Q\cup R)^{c},t}\right|J_{(Q\cup R)^{c}}\left|u_{(Q\cup R )^{c},t}\right\rangle\sqrt{\overline{N}_{Q^{c}}}\sqrt{\overline{N}_{R^{c}}} \frac{\bm{d}_{\bm{N}}}{\overline{N}}\] \[\times\left\|\left(\mathcal{N}_{\mathrm{total}}+p\mathcal{I}\right)^{- 1/2}\mathcal{W}^{*}(\mathbf{N}^{\odot 1/2}\odot\ma... | outline | |
\[M_{n}(\xi,u)\] \[= \frac{1}{n}\sum_{i=1}^{n}\max_{\Delta:X_{i}+n^{-1/2}\Delta\in \Omega}\left[n^{1/2}\xi^{\mathrm{ T}}\left\{h(X_{i}+n^{-1/2}\Delta,\beta_{*}+n^{-1/2}u)-h(X_{i},\beta_{*}) \right\}-\left\|\Delta\right\|_{q}^{2}\right]\] \[= \frac{1}{n}\sum_{i=1}^{n}\max_{\Delta:X_{i}+n^{-1/2}\Delta\in \Omega}\left\{\x... | outline | |
\[\frac{\mathrm{d}^{3}}{\mathrm{d}t^{3}}\mathbf{C}(t)= \frac{3!}{3!0!0!}f_{1}^{(3)}f_{2}^{(0)}f_{3}^{(0)}+\frac{3!}{1!1!1! 1!}f_{1}^{(1)}f_{2}^{(1)}f_{3}^{(1)}+\frac{3!}{2!1!0!}f_{1}^{(2)}f_{2}^{(1)}f_{3 }^{(0)}+\frac{3!}{2!0!1!1}f_{1}^{(2)}f_{2}^{(0)}f_{3}^{(1)}\] \[= f_{1}^{(3)}f_{2}^{(0)}f_{3}^{(0)}+6f_{1}^{(1)}f_{2... | outline | |
\[f(v_{b}^{\prime})= f(v_{b})-f(v_{b+e_{0}-e_{1}}^{\prime})-\cdots-f(v_{b+e_{0}-e_{d}} ^{\prime})\] \[= w_{b}+w_{b+e_{0}-e_{1}}+w_{b+e_{0}-e_{2}}+\cdots+w_{b+e_{0}-e_{d}}\] \[-w_{b+e_{0}-e_{1}}+(-1)^{n-k-b_{0}-1}\sum_{b^{\prime}\in D}{n-k-b_ {0}-1\choose b_{1}-1-b_{1}^{\prime},b_{2}-b_{2}^{\prime},\cdots,b_{d}-b_{d}^{ ... | outline | |
\[|S_{\leq Y}^{\prime}|\lesssim\sum_{r_{k}|k^{\infty}}r_{k}^{- \varepsilon}\max_{\mathrm{Re}(s)=\varepsilon}\sum_{\mathbf{g}}\int_{-2T}^{2T} \sum_{\theta\,(\mathrm{mod}\ r_{k}k)}\sum_{\begin{subarray}{c}(q^{\prime},kg_{1 }g_{2})=1\\ q^{\prime}\leq Y/r_{k}\end{subarray}}\sum_{\chi^{\prime}\,(\mathrm{mod}\ q^{ \prime})}^... | matrix | |
\[\begin{split}\mu(x^{\lambda}y^{1-\lambda},\beta,\alpha)& =\int_{0}^{\infty}\frac{x^{\lambda(t+\alpha)}y^{(1-\lambda)(t+ \alpha)}t^{\beta}}{\Gamma(t+\alpha+1)\Gamma(\beta+1)}dt\\ &=\int_{0}^{\infty}\left[\frac{x^{t+\alpha}t^{\beta}}{\Gamma(t+ \alpha+1)\Gamma(\beta+1)}\right]^{\lambda}\left[\frac{y^{t+\alpha}t^{\beta}}... | outline | |
\[\operatorname{\mathbb{E}}\bigl{[}q_{0}(\theta)-q_{0}(\theta^{*}) \ |\ \mathcal{F}_{0}\bigr{]} = \operatorname{\mathbb{E}}\Bigl{[}\frac{(X_{0}-f_{\theta}^{0})^{2 }}{H_{\theta}^{0}}+\log(H_{\theta}^{0})-\frac{(X_{0}-f_{\theta^{*}}^{0})^{2}}{ H_{\theta^{*}}^{0}}-\log(H_{\theta^{*}}^{0})\ |\ \mathcal{F}_{0}\Bigr{]}\] \[=... | outline | |
\[[W^{(2)}_{p,p}t,W^{(2)}_{i,i}t]\] \[=(W^{(2)}_{p,i})_{(-1)}W^{(1)}_{i,p}t^{2}-(W^{(1)}_{p,i})_{(-1)}W^ {(2)}_{i,p}t^{2}\] \[\quad+\alpha_{1}(\partial W^{(1)}_{p,i})_{(-1)}W^{(1)}_{i,p}t^{2} +(\partial W^{(1)}_{p,p})_{(-1)}W^{(1)}_{i,i}t^{2}\] \[\quad-\delta_{p,i}\sum_{w\leq m-n}(W^{(1)}_{w,i})_{(-1)}W^{(2)}_{p,w}t^{2... | outline | |
\[J^{-}(r) :=\int_{-R}^{r}e^{z(x-r)}\sin(b_{k}x)\mathrm{d}x\] \[=\frac{1}{b_{k}}\left[-\cos(b_{k}x)e^{z(x-r)}\right]_{-R}^{r}+ \frac{z}{b_{k}}\int_{-R}^{r}e^{z(x-r)}\cos(b_{k}x)\mathrm{d}x\] \[=-\frac{\cos(b_{k}r)}{b_{k}}+\frac{\cos(b_{k}R)}{b_{k}}e^{-z(R+r) }+\frac{z}{b_{k}^{2}}\left[\sin(b_{k}x)e^{z(x-r)}\right]_{-R}... | outline | |
\[\begin{split}& u_{1}^{2}+u_{2}^{2}\\ &=(t+C_{2})^{2}+\frac{C_{1}^{2}}{(t+C_{2})^{2}+C_{1}^{2}}\left[C_{ 1}\left[\ln\left(\frac{t+C_{2}+\sqrt{(t+C_{2})^{2}+C_{1}^{2}}}{C_{1}}\right) \right]+C_{3}\right]^{2}\\ &+\frac{(t+C_{2})^{2}}{(t+C_{2})^{2}+C_{1}^{2}}\left[C_{1}\left[ \ln\left(\frac{t+C_{2}+\sqrt{(t+C_{2})^{2}+C_... | outline | |
\[B(t)=\sin(2\mathcal{E}t)\int\limits_{-1}^{1}\frac{dE}{2\pi}\rho T (1-\rho T)+\frac{\left(\varepsilon^{2}-1\right)^{2}}{2\left(\varepsilon^{2}+1 \right)}\sin(4\mathcal{E}t)\left(\int\limits_{-1}^{1}\frac{dE}{2\pi}\rho T \right)^{2}\] \[+2\left(\int\limits_{-1}^{1}\frac{dE}{2\pi}\rho T\right)\left(2 \mathcal{E}\cos(2\m... | outline | |
\[\operatorname{Res}_{\boldsymbol{\zeta}_{0}}\boldsymbol{\zeta}_{0}^{m} (\boldsymbol{\zeta}_{0}+\boldsymbol{x})^{2+i}L(\boldsymbol{\zeta}_{0}+ \boldsymbol{x})\left(\mathcal{Y}_{t}(w_{1},\boldsymbol{x})\otimes\bar{w}_{c,h_{ 0}}\right)\] \[=\operatorname{Res}_{\boldsymbol{\zeta}_{1}}\operatorname{Res}_{ \boldsymbol{\zeta... | outline | |
\[\int_{\mathbb{R}^{2}}\int_{\mathbb{T}^{2}}(F_{t}^{N,R})^{3} \mathrm{div}_{v}(\theta^{\varepsilon_{N}}v*S_{t}^{N,R})\,dxdv=2\int_{\mathbb{ R}^{2}}\int_{\mathbb{T}^{2}}(F_{t}^{N,R})^{4}\,dxdv\\ \int_{\mathbb{R}^{2}}\int_{\mathbb{T}^{2}}(F^{N,R})^{3}\int_{ \mathbb{R}^{2}}\int_{\mathbb{T}^{2}}\theta^{0,\varepsilon_{N}}(x... | outline | |
\[K=(F\cdot G)\cdot H\] \[E_{K}=E_{F}+E_{G}+E_{H}\qquad\Lambda_{K}=\Lambda_{F}+\Lambda_{G} +\Lambda_{H}\] \[S_{K}=S_{F}-S_{F}[\bullet]+S_{G}-S_{G}[\bullet]+S_{H}\qquad T_{ K}=T_{F}+T_{G}-T_{G}[\bullet]+T_{H}-T_{H}[\bullet]\] \[\kappa_{K}(v)=\begin{cases}\kappa_{H}(\pi_{j}(S_{H}[\bullet])& \text{if }\kappa_{F}(v)=\pi_{i... | outline | |
\[T_{11} \geq\frac{1}{n^{2}}\sum_{i=2}^{N/3}\min_{2\frac{k-1}{n}\leq y_{1} \leq y_{2}\leq\ldots\leq y_{N}\leq 2\frac{k}{n}}\sum_{\ell=1}^{i}\sum_{m=1}^{ \lfloor\frac{N-\ell}{i}\rfloor}(V-V_{n})(y_{mi+\ell}-y_{(m-1)i+\ell})\] \[\geq\frac{1}{n^{2}}\sum_{i=2}^{N/3}\sum_{\ell=1}^{i}\min_{2\frac{ k-1}{n}\leq y_{1}\leq y_{2}... | outline | |
\[\begin{split}\mathbb{E}\mathfrak{B}_{n,N}&=\frac{1 }{1+\frac{n}{\theta}}+\sum_{j=n}^{N-1}\frac{1}{1+\frac{1}{\theta}(j+1)}\left( \frac{1}{1+\frac{j-n+1}{\theta}}+\sum_{k=j-n+1}^{j-1}\frac{\mathbb{E} \mathfrak{B}_{k-(j-n+1)+1,k}}{1+\frac{1}{\theta}(k+1)}\right)\\ &=\frac{1}{1+\frac{n}{\theta}}+\sum_{j=n}^{N-1}\frac{1}... | outline |
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