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We employ separation of variables to prove weighted resolvent estimates for the semiclassical Schrödinger operator ?h2Δ+V(|x|)?E in dimension n2, where h,E>0, and V:[0,)R is L and compactly supported. The weighted resolvent norm grows no faster than exp?(Ch?1), while an exterior weighted norm grows h?1. We introduce a new method based on the Mellin transform to handle the two-dimensional case.  相似文献   

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Let M be a random m×n rank-r matrix over the binary field F2, and let wt(M) be its Hamming weight, that is, the number of nonzero entries of M.We prove that, as m,n+ with r fixed and m/n tending to a constant, we have thatwt(M)12r2mn2r(12r)4(m+n)mn converges in distribution to a standard normal random variable.  相似文献   

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《Discrete Mathematics》2022,345(1):112640
We show that the lattice point enumerator Gn(?) satisfiesGn(tK+sL+(?1,?t+s?)n)1/ntGn(K)1/n+sGn(L)1/n for any K,L?Rn bounded sets with integer points and all t,s0.We also prove that a certain family of compact sets, extending that of cubes [?m,m]n, with mN, minimizes the functional Gn(K+t[?1,1]n), for any t0, among those bounded sets K?Rn with given positive lattice point enumerator.Finally, we show that these new discrete inequalities imply the corresponding classical Brunn-Minkowski and isoperimetric inequalities for non-empty compact sets.  相似文献   

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《Discrete Mathematics》2022,345(8):112902
For a simple graph G, denote by n, Δ(G), and χ(G) its order, maximum degree, and chromatic index, respectively. A graph G is edge-chromatic critical if χ(G)=Δ(G)+1 and χ(H)<χ(G) for every proper subgraph H of G. Let G be an n-vertex connected regular class 1 graph, and let G? be obtained from G by splitting one vertex of G into two vertices. Hilton and Zhao in 1997 conjectured that G? must be edge-chromatic critical if Δ(G)>n/3, and they verified this when Δ(G)n2(7?1)0.82n. In this paper, we prove it for Δ(G)0.75n.  相似文献   

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Let Q=(0,T)×Ω, where Ω is a bounded open subset of Rd. We consider the parabolic p-capacity on Q naturally associated with the usual p-Laplacian. Droniou, Porretta, and Prignet have shown that if a bounded Radon measure μ on Q is diffuse, i.e. charges no set of zero p-capacity, p>1, then it is of the form μ=f+div(G)+gt for some fL1(Q), G(Lp(Q))d and gLp(0,T;W01,p(Ω)L2(Ω)). We show the converse of this result: if p>1, then each bounded Radon measure μ on Q admitting such a decomposition is diffuse.  相似文献   

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