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It is well known that nice conditions on the canonical module of a local ring have a strong impact in the study of strong F-regularity and F -purity. In this note, we prove that if (R,m) is an equidimensional and S2 local ring that admits a canonical ideal I≅ωR such that R/I is F-pure, then R is F-pure. This greatly generalizes one of the main theorems in [2]. We also provide examples to show that not all Cohen–Macaulay F-pure local rings satisfy the above property. 相似文献
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Let I=[0,1] and let P be a partition of I into a finite number of intervals. Let τ1, τ2; I→I be two piecewise expanding maps on P . Let G⊂I×I be the region between the boundaries of the graphs of τ1 and τ2. Any map τ:I→I that takes values in G is called a selection of the multivalued map defined by G . There are many results devoted to the study of the existence of selections with specified topological properties. However, there are no results concerning the existence of selection with measure-theoretic properties. In this paper we prove the existence of selections which have absolutely continuous invariant measures (acim). By our assumptions we know that τ1 and τ2 possess acims preserving the distribution functions F(1) and F(2). The main result shows that for any convex combination F of F(1) and F(2) we can find a map η with values between the graphs of τ1 and τ2 (that is, a selection) such that F is the η-invariant distribution function. Examples are presented. We also study the relationship of the dynamics of our multivalued maps to random maps. 相似文献
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We study the problem (−Δ)su=λeu in a bounded domain Ω⊂Rn, where λ is a positive parameter. More precisely, we study the regularity of the extremal solution to this problem. Our main result yields the boundedness of the extremal solution in dimensions n≤7 for all s∈(0,1) whenever Ω is, for every i=1,...,n, convex in the xi-direction and symmetric with respect to {xi=0}. The same holds if n=8 and s?0.28206..., or if n=9 and s?0.63237.... These results are new even in the unit ball Ω=B1. 相似文献
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Let K be a closed convex subset of a q-uniformly smooth separable Banach space, T:K→K a strictly pseudocontractive mapping, and f:K→K an L-Lispschitzian strongly pseudocontractive mapping. For any t∈(0,1), let xt be the unique fixed point of tf+(1-t)T. We prove that if T has a fixed point, then {xt} converges to a fixed point of T as t approaches to 0. 相似文献
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Given k pairs of vertices (si,ti)(1≤i≤k) of a digraph G, how can we test whether there exist k vertex-disjoint directed paths from si to ti for 1≤i≤k? This is NP-complete in general digraphs, even for k=2 [2], but for k=2 there is a polynomial-time algorithm when G is a tournament (or more generally, a semicomplete digraph), due to Bang-Jensen and Thomassen [1]. Here we prove that for all fixed k there is a polynomial-time algorithm to solve the problem when G is semicomplete. 相似文献
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We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if u is a solution of (−Δ)su=g in Ω , u≡0 in Rn\Ω, for some s∈(0,1) and g∈L∞(Ω), then u is Cs(Rn) and u/δs|Ω is Cα up to the boundary ∂Ω for some α∈(0,1), where δ(x)=dist(x,∂Ω). For this, we develop a fractional analog of the Krylov boundary Harnack method. 相似文献
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Given n independent standard normal random variables, it is well known that their maxima Mn can be normalized such that their distribution converges to the Gumbel law. In a remarkable study, Hall proved that the Kolmogorov distance dn between the normalized Mn and its associated limit distribution is less than 3/log?n. In the present study, we propose a different set of norming constants that allow this upper bound to be decreased with dn≤C(m)/log?n for n≥m≥5. Furthermore, the function C(m) is computed explicitly, which satisfies C(m)≤1 and limm→∞?C(m)=1/3. As a consequence, some new and effective norming constants are provided using the asymptotic expansion of a Lambert W type function. 相似文献
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Jean-Pierre Kahane 《Comptes Rendus Mathematique》2014,352(5):383-385
For almost all x>1, (xn)(n=1,2,…) is equidistributed modulo 1, a classical result. What can be said on the exceptional set? It has Hausdorff dimension one. Much more: given an (bn) in [0,1[ and ε>0, the x -set such that |xn−bn|<ε modulo 1 for n large enough has dimension 1. However, its intersection with an interval [1,X] has a dimension <1, depending on ε and X. Some results are given and a question is proposed. 相似文献
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Given two positive integers e and s we consider Gorenstein Artinian local rings R whose maximal ideal m satisfies ms≠0=ms+1 and rankR/m(m/m2)=e. We say that R is a compressed Gorenstein local ring when it has maximal length among such rings. It is known that generic Gorenstein Artinian algebras are compressed. If s≠3, we prove that the Poincaré series of all finitely generated modules over a compressed Gorenstein local ring are rational, sharing a common denominator. A formula for the denominator is given. When s is even this formula depends only on the integers e and s . Note that for s=3 examples of compressed Gorenstein local rings with transcendental Poincaré series exist, due to Bøgvad. 相似文献
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Consider in a real Hilbert space H the Cauchy problem (P0): u′(t)+Au(t)+Bu(t)=f(t), 0≤t≤T; u(0)=u0, where −A is the infinitesimal generator of a C0-semigroup of contractions, B is a nonlinear monotone operator, and f is a given H-valued function. Inspired by the excellent book on singular perturbations by J.L. Lions, we associate with problem (P0) the following regularization (Pε): −εu″(t)+u′(t)+Au(t)+Bu(t)=f(t), 0≤t≤T; u(0)=u0, u′(T)=uT, where ε>0 is a small parameter. We investigate existence, uniqueness and higher regularity for problem (Pε). Then we establish asymptotic expansions of order zero, and of order one, for the solution of (Pε). Problem (Pε) turns out to be regularly perturbed of order zero, and singularly perturbed of order one, with respect to the norm of C([0,T];H). However, the boundary layer of order one is not visible through the norm of L2(0,T;H). 相似文献
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Let f(t) be an operator monotone function. Then A?B implies f(A)?f(B), but the converse implication is not true. Let A?B be the geometric mean of A,B?0. If A?B, then B−1?A?I; the converse implication is not true either. We will show that if f(λB+I)−1?f(λA+I)?I for all sufficiently small λ>0, then f(λA+I)?f(λB+I) and A?B. Moreover, we extend it to multi-variable matrices means. 相似文献