We provide moment bounds for expressions of the type
Stochastic networks for the clock were identified by ensemble methods using genetic algorithms that captured the amplitude and period variation in single cell oscillators of
 Award ID(s):
 1713746
 NSFPAR ID:
 10192589
 Publisher / Repository:
 Nature Publishing Group
 Date Published:
 Journal Name:
 Scientific Reports
 Volume:
 10
 Issue:
 1
 ISSN:
 20452322
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract where$$(X^{(1)} \otimes \cdots \otimes X^{(d)})^T A (X^{(1)} \otimes \cdots \otimes X^{(d)})$$ ${({X}^{\left(1\right)}\otimes \cdots \otimes {X}^{\left(d\right)})}^{T}A({X}^{\left(1\right)}\otimes \cdots \otimes {X}^{\left(d\right)})$ denotes the Kronecker product and$$\otimes $$ $\otimes $ are random vectors with independent, mean 0, variance 1, subgaussian entries. The bounds are tight up to constants depending on$$X^{(1)}, \ldots , X^{(d)}$$ ${X}^{\left(1\right)},\dots ,{X}^{\left(d\right)}$d for the case of Gaussian random vectors. Our proof also provides a decoupling inequality for expressions of this type. Using these bounds, we obtain new, improved concentration inequalities for expressions of the form .$$\Vert B (X^{(1)} \otimes \cdots \otimes X^{(d)})\Vert _2$$ $\Vert B({X}^{\left(1\right)}\otimes \cdots \otimes {X}^{\left(d\right)}){\Vert}_{2}$ 
Abstract The elliptic flow
of$$(v_2)$$ $\left({v}_{2}\right)$ mesons from beautyhadron decays (nonprompt$${\textrm{D}}^{0}$$ ${\text{D}}^{0}$ was measured in midcentral (30–50%) Pb–Pb collisions at a centreofmass energy per nucleon pair$${\textrm{D}}^{0})$$ ${\text{D}}^{0})$ TeV with the ALICE detector at the LHC. The$$\sqrt{s_{\textrm{NN}}} = 5.02$$ $\sqrt{{s}_{\text{NN}}}=5.02$ mesons were reconstructed at midrapidity$${\textrm{D}}^{0}$$ ${\text{D}}^{0}$ from their hadronic decay$$(y<0.8)$$ $\left(\righty<0.8)$ , in the transverse momentum interval$$\mathrm {D^0 \rightarrow K^\uppi ^+}$$ ${D}^{0}\to {K}^{}{\pi}^{+}$ GeV/$$2< p_{\textrm{T}} < 12$$ $2<{p}_{\text{T}}<12$c . The result indicates a positive for nonprompt$$v_2$$ ${v}_{2}$ mesons with a significance of 2.7$${{\textrm{D}}^{0}}$$ ${\text{D}}^{0}$ . The nonprompt$$\sigma $$ $\sigma $ meson$${{\textrm{D}}^{0}}$$ ${\text{D}}^{0}$ is lower than that of prompt nonstrange D mesons with 3.2$$v_2$$ ${v}_{2}$ significance in$$\sigma $$ $\sigma $ , and compatible with the$$2< p_\textrm{T} < 8~\textrm{GeV}/c$$ $2<{p}_{\text{T}}<8\phantom{\rule{0ex}{0ex}}\text{GeV}/c$ of beautydecay electrons. Theoretical calculations of beautyquark transport in a hydrodynamically expanding medium describe the measurement within uncertainties.$$v_2$$ ${v}_{2}$ 
Abstract Let
denote the standard Haar system on [0, 1], indexed by$$(h_I)$$ $\left({h}_{I}\right)$ , the set of dyadic intervals and$$I\in \mathcal {D}$$ $I\in D$ denote the tensor product$$h_I\otimes h_J$$ ${h}_{I}\otimes {h}_{J}$ ,$$(s,t)\mapsto h_I(s) h_J(t)$$ $(s,t)\mapsto {h}_{I}\left(s\right){h}_{J}\left(t\right)$ . We consider a class of twoparameter function spaces which are completions of the linear span$$I,J\in \mathcal {D}$$ $I,J\in D$ of$$\mathcal {V}(\delta ^2)$$ $V\left({\delta}^{2}\right)$ ,$$h_I\otimes h_J$$ ${h}_{I}\otimes {h}_{J}$ . This class contains all the spaces of the form$$I,J\in \mathcal {D}$$ $I,J\in D$X (Y ), whereX andY are either the Lebesgue spaces or the Hardy spaces$$L^p[0,1]$$ ${L}^{p}[0,1]$ ,$$H^p[0,1]$$ ${H}^{p}[0,1]$ . We say that$$1\le p < \infty $$ $1\le p<\infty $ is a Haar multiplier if$$D:X(Y)\rightarrow X(Y)$$ $D:X\left(Y\right)\to X\left(Y\right)$ , where$$D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J$$ $D({h}_{I}\otimes {h}_{J})={d}_{I,J}{h}_{I}\otimes {h}_{J}$ , and ask which more elementary operators factor through$$d_{I,J}\in \mathbb {R}$$ ${d}_{I,J}\in R$D . A decisive role is played by theCapon projection given by$$\mathcal {C}:\mathcal {V}(\delta ^2)\rightarrow \mathcal {V}(\delta ^2)$$ $C:V\left({\delta}^{2}\right)\to V\left({\delta}^{2}\right)$ if$$\mathcal {C} h_I\otimes h_J = h_I\otimes h_J$$ $C{h}_{I}\otimes {h}_{J}={h}_{I}\otimes {h}_{J}$ , and$$I\le J$$ $\leftI\right\le \leftJ\right$ if$$\mathcal {C} h_I\otimes h_J = 0$$ $C{h}_{I}\otimes {h}_{J}=0$ , as our main result highlights: Given any bounded Haar multiplier$$I > J$$ $\leftI\right>\leftJ\right$ , there exist$$D:X(Y)\rightarrow X(Y)$$ $D:X\left(Y\right)\to X\left(Y\right)$ such that$$\lambda ,\mu \in \mathbb {R}$$ $\lambda ,\mu \in R$ i.e., for all$$\begin{aligned} \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}\mathcal {C})\text { approximately 1projectionally factors through }D, \end{aligned}$$ $\begin{array}{c}\lambda C+\mu (\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}C)\phantom{\rule{0ex}{0ex}}\text{approximately 1projectionally factors through}\phantom{\rule{0ex}{0ex}}D,\end{array}$ , there exist bounded operators$$\eta > 0$$ $\eta >0$A ,B so thatAB is the identity operator ,$${{\,\textrm{Id}\,}}$$ $\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}$ and$$\Vert A\Vert \cdot \Vert B\Vert = 1$$ $\Vert A\Vert \xb7\Vert B\Vert =1$ . Additionally, if$$\Vert \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}\mathcal {C})  ADB\Vert < \eta $$ $\Vert \lambda C+\mu (\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}C)ADB\Vert <\eta $ is unbounded on$$\mathcal {C}$$ $C$X (Y ), then and then$$\lambda = \mu $$ $\lambda =\mu $ either factors through$${{\,\textrm{Id}\,}}$$ $\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}$D or .$${{\,\textrm{Id}\,}}D$$ $\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}D$ 
Abstract Consider two halfspaces
and$$H_1^+$$ ${H}_{1}^{+}$ in$$H_2^+$$ ${H}_{2}^{+}$ whose bounding hyperplanes$${\mathbb {R}}^{d+1}$$ ${R}^{d+1}$ and$$H_1$$ ${H}_{1}$ are orthogonal and pass through the origin. The intersection$$H_2$$ ${H}_{2}$ is a spherical convex subset of the$${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$ ${S}_{2,+}^{d}:={S}^{d}\cap {H}_{1}^{+}\cap {H}_{2}^{+}$d dimensional unit sphere , which contains a great subsphere of dimension$${\mathbb {S}}^d$$ ${S}^{d}$ and is called a spherical wedge. Choose$$d2$$ $d2$n independent random points uniformly at random on and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of$${\mathbb {S}}_{2,+}^d$$ ${S}_{2,+}^{d}$ . A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on$$\log n$$ $logn$ . The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a halfsphere.$${\mathbb {S}}_{2,+}^d$$ ${S}_{2,+}^{d}$ 
Abstract It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.
arXiv:2010.09793 ) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators associated to a domain$$L_{\beta ,\gamma } = {\text {div}}D^{d+1+\gamma n} \nabla $$ ${L}_{\beta ,\gamma}=\text{div}{D}^{d+1+\gamma n}\nabla $ with a uniformly rectifiable boundary$$\Omega \subset {\mathbb {R}}^n$$ $\Omega \subset {R}^{n}$ of dimension$$\Gamma $$ $\Gamma $ , the now usual distance to the boundary$$d < n1$$ $d<n1$ given by$$D = D_\beta $$ $D={D}_{\beta}$ for$$D_\beta (X)^{\beta } = \int _{\Gamma } Xy^{d\beta } d\sigma (y)$$ ${D}_{\beta}{\left(X\right)}^{\beta}={\int}_{\Gamma}{Xy}^{d\beta}d\sigma \left(y\right)$ , where$$X \in \Omega $$ $X\in \Omega $ and$$\beta >0$$ $\beta >0$ . In this paper we show that the Green function$$\gamma \in (1,1)$$ $\gamma \in (1,1)$G for , with pole at infinity, is well approximated by multiples of$$L_{\beta ,\gamma }$$ ${L}_{\beta ,\gamma}$ , in the sense that the function$$D^{1\gamma }$$ ${D}^{1\gamma}$ satisfies a Carleson measure estimate on$$\big  D\nabla \big (\ln \big ( \frac{G}{D^{1\gamma }} \big )\big )\big ^2$$ $D\nabla (ln(\frac{G}{{D}^{1\gamma}})){}^{2}$ . We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical distance function from David et al. (Duke Math J, to appear).$$\Omega $$ $\Omega $