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1.
Let Sd denote the unit sphere in the Euclidean space Rd+1(d1). We develop LeVeque type inequalities for the discrepancy between the rotationally invariant probability measure and the normalized counting measures on Sd. We obtain both upper bound and lower bound estimates. We then use these inequalities to estimate the discrepancy of the normalized counting measures associated with minimal energy configurations on Sd.  相似文献   

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In this article, we obtain the sharp bounds from LP(Gn) to the space wLP(Gn) for Hardy operators on product spaces. More generally, the precise norms of Hardy operators on product spaces from LP(Gn) to the space LPI(Gn) are obtained.  相似文献   

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Consider a branching random walk, where the underlying branching mechanism is governed by a Galton–Watson process and the migration of particles by a simple random walk in Zd. Denote by Zn(z) the number of particles of generation n located at site zZd. We give the second order asymptotic expansion for Zn(z). The higher order expansion can be derived by using our method here. As a by-product, we give the second order expansion for a simple random walk on Zd, which is used in the proof of the main theorem and is of independent interest.  相似文献   

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We investigate the regularity of random attractors for the non-autonomous non-local fractional stochastic reaction–diffusion equations in Hs(Rn) with s(0,1). We prove the existence and uniqueness of the tempered random attractor that is compact in Hs(Rn) and attracts all tempered random subsets of L2(Rn) with respect to the norm of Hs(Rn). The main difficulty is to show the pullback asymptotic compactness of solutions in Hs(Rn) due to the noncompactness of Sobolev embeddings on unbounded domains and the almost sure nondifferentiability of the sample paths of the Wiener process. We establish such compactness by the ideas of uniform tail-estimates and the spectral decomposition of solutions in bounded domains.  相似文献   

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Let Xn be a hypersurface in Pn+1 with n1 defined over a finite field Fq of q elements. In this note, we classify, up to projective equivalence, hypersurfaces Xn as above which reach two elementary upper bounds for the number of Fq-points on Xn which involve a Thas’ invariant.  相似文献   

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We consider the nonlinear Schrödinger equations (NLS) on Rd with random and rough initial data. By working in the framework of Lp(Rd) spaces, p>2, we prove almost sure local well-posedness for rougher initial data than those considered in the existing literature. The main ingredient of the proof is the dispersive estimate.  相似文献   

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For a supercritical catalytic branching random walk on Zd, dN, with an arbitrary finite catalysts set we study the spread of particles population as time grows to infinity. It is shown that in the result of the proper normalization of the particles positions in the limit there are a.s. no particles outside the closed convex surface in Rd which we call the propagation front and, under condition of infinite number of visits of the catalysts set, a.s. there exist particles on the propagation front. We also demonstrate that the propagation front is asymptotically densely populated and derive its alternative representation.  相似文献   

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Equivariant Ham Sandwich Theorems are obtained for the classical algebras F=R,C, and H and the finite subgroups G of their unit spheres. Given any n F-valued Borel measures on Fn and any n-dimensional free F-unitary representation of G, it is shown that there exists a Voronoi partition of Fn naturally determined by G which “G-balances” each measure, as realized by the simultaneous vanishing of each “G-average” of the measures of the partition?s isometric fundamental domains. Applications for real measures follow, among them that any n signed mass distributions on C(p?1)n/2 can be equipartitioned by a single complex regular p-fan if p is an odd prime.  相似文献   

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In this paper, we consider a uniformly ergodic Markov process (Xn)n0 valued in a measurable subset E of Rd with the unique invariant measure μ(dx)=f(x)dx, where the density f is unknown. We establish the large deviation estimations for the nonparametric kernel density estimator fn* in L1(Rd,dx) and for 6fn*-f6L1(Rd,dx), and the asymptotic optimality fn* in the Bahadur sense. These generalize the known results in the i.i.d. case.  相似文献   

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A sharp version of the Balian–Low theorem is proven for the generators of finitely generated shift-invariant spaces. If generators {fk}k=1K?L2(Rd) are translated along a lattice to form a frame or Riesz basis for a shift-invariant space V, and if V has extra invariance by a suitable finer lattice, then one of the generators fk must satisfy Rd|x||fk(x)|2dx=, namely, fk??H1/2(Rd). Similar results are proven for frames of translates that are not Riesz bases without the assumption of extra lattice invariance. The best previously existing results in the literature give a notably weaker conclusion using the Sobolev space Hd/2+?(Rd); our results provide an absolutely sharp improvement with H1/2(Rd). Our results are sharp in the sense that H1/2(Rd) cannot be replaced by Hs(Rd) for any s<1/2.  相似文献   

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The boxicity box(H) of a graph H is the smallest integer d such that H is the intersection of d interval graphs, or equivalently, that H is the intersection graph of axis-aligned boxes in Rd. These intersection representations can be interpreted as covering representations of the complement Hc of H with co-interval graphs, that is, complements of interval graphs. We follow the recent framework of global, local and folded covering numbers (Knauer and Ueckerdt, 2016) to define two new parameters: the local boxicity box?(H) and the union boxicity box¯(H) of H. The union boxicity of H is the smallest d such that Hc can be covered with d vertex–disjoint unions of co-interval graphs, while the local boxicity of H is the smallest d such that Hc can be covered with co-interval graphs, at most d at every vertex.We show that for every graph H we have box?(H)box¯(H)box(H) and that each of these inequalities can be arbitrarily far apart. Moreover, we show that local and union boxicity are also characterized by intersection representations of appropriate axis-aligned boxes in Rd. We demonstrate with a few striking examples, that in a sense, the local boxicity is a better indication for the complexity of a graph, than the classical boxicity.  相似文献   

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