ANNOUNCEMENT ON“SHARP ERROR ESTIMATE OF BDF2 SCHEME WITH VARIABLE TIME STEPS FOR LINEAR REACTION-DIFFUSION EQUATIONS”

In this note we announce the sharp error estimate of BDF2 scheme for linear diffusion reaction problem with variable time steps.Our analysis shows that the optimal second-order convergence does not require the high-order methods or the very small time stepsτ1=O(τ2)for the first level solution u1.This is,the first-order consistence of the first level solution u1 like BDF1(i.e.Euler scheme)as a starting point does not cause the loss of global temporal accuracy,and the ratios are updated to rk≤4.8645.     

球面上的次临界最佳Sobolev不等式

本文在球面S^N上建立了一类最佳Sobolev不等式:||∫||²LqS^N≤(q-2)Γ(N-d/2+1/dΓ(N+d/2)(∫_S^Nf(§d§-Γ(N+d/2)/Γ(N-d)/2∫_S^(N|∫|²d§),其中Ad(0N的高阶保形算子,d§S^N的归一化曲面测度,2≤q<2N/N-d.     

Weak sharp solutions for variational inequalities in Banach spaces

In this paper, we introduce the notion of a weak sharp set of solutions to a variational inequality problem (VIP) in a reflexive, strictly convex and smooth Banach space, and present its several equivalent conditions. We also prove, under some continuity and monotonicity assumptions, that if any sequence generated by an algorithm for solving (VIP) converges to a weak sharp solution, then we can obtain solutions for (VIP) by solving a finite number of convex optimization subproblems with linear objective. Moreover, in order to characterize finite convergence of an iterative algorithm, we introduce the notion of a weak subsharp set of solutions to a variational inequality problem (VIP), which is more general than that of weak sharp solutions in Hilbert spaces. We establish a sufficient and necessary condition for the finite convergence of an algorithm for solving (VIP) which satisfies that the sequence generated by which converges to a weak subsharp solution of (VIP), and show that the proximal point algorithm satisfies this condition. As a consequence, we prove that the proximal point algorithm possesses finite convergence whenever the sequence generated by which converges to a weak subsharp solution of (VIP).     

基于左截尾g-h分布的船舶溢油损失分布拟合研究

由于船舶溢油损失对保险公司的影响非常大,保险费率厘定过程中对损失分布拟合的研究非常重视.以全国船舶溢油事故损失数据为样本,对船舶溢油损失序列的分布特征进行研究.通过实证研究发现,船舶溢油损失数据不服从正态分布,具有"右偏、尖峰、厚尾"的特性.考虑船舶溢油损失数据的左截尾特性,将左截尾g-h分布引入船舶溢油损失序列的分布拟合中,结果表明,这种分布对船舶溢油损失数据的拟合效果与传统分布相比更好.     

Moving boundary i.b.v.p.s for Schrödinger equations: Solution bounds and related results

The paper is concerned with i.b.v.p.s for Schrödinger equations, linear and nonlinear, in a straight line region with prescribed, moving boundaries, upon which (time-dependent) Dirichlet conditions are specified. Bounds, in terms of data, are obtained for the L₂ norm of the spatial derivative of the solutions, or for a measure related thereto: in the context of expanding boundaries, pointwise bounds for the solution may be inferred both in the linear case and in some nonlinear cases (e.g. the defocusing case). Asymptotic properties of the bounds for the aforementioned norm are discussed in the linear case. The methodology of the paper is based on a particular compact formula for the aforementioned norm of an arbitrary, complex-valued function whose values are assigned, as functions of time, on the assigned, moving boundaries of a straight line region. The application of the methodology to i.b.v.p.s for other p.d.e.s is discussed briefly.     

Vanishing of cohomology groups of random simplicial complexes

We consider k‐dimensional random simplicial complexes generated from the binomial random (k + 1)‐uniform hypergraph by taking the downward‐closure. For 1 ≤ j ≤ k ? 1, we determine when all cohomology groups with coefficients in from dimension one up to j vanish and the zero‐th cohomology group is isomorphic to . This property is not deterministically monotone for this model, but nevertheless we show that it has a single sharp threshold. Moreover we prove a hitting time result, relating the vanishing of these cohomology groups to the disappearance of the last minimal obstruction. We also study the asymptotic distribution of the dimension of the j‐th cohomology group inside the critical window. As a corollary, we deduce a hitting time result for a different model of random simplicial complexes introduced by Linial and Meshulam, previously only known for dimension two.     

用于波分复用的缺陷态复周期结构光子晶体滤波器的研究

利用传输矩阵法，研究了一种用于波分复用的光子晶体滤波器。该结构是中间包含一个缺陷层的一维三层念质的光子晶体，叫做缺陷态复周期结构光子晶体。通过数值计算得出它的色散关系和滤波特性。当改变缺陷层的厚度并保持其折射率不变时，带隙中将出现个数不同的局域模。缺陷层厚度增加时，局域模数也随着增加。适当调节缺陷层厚度，使带隙中出现八个窄带滤波窗口，能很好的用作八通道波分复用一解复用滤波器。由于很高的传输率和低的传输损耗，故在高速，长距离光通信中将有很好的应用。     

New Uniform Parametric Error Bounds

A uniform parametric error bound is a uniform error estimate for feasible solutions of a family of parametric mathematical programming problems. It has been proven useful in exact penalty formulation for bilevel programming problems. In this paper, we derive new sufficient conditions for the existence of uniform parametric error bounds.     

Equivalence constants for certain matrix norms II

We give an almost complete solution of a problem posed by Klaus and Li [A.-L. Klaus, C.-K. Li, Isometries for the vector (p, q) norm and the induced (p, q) norm, Linear and Multilinear Algebra 38 (1995) 315–332]. Klaus and Li’s problem, which arose during their investigations of isometries, was to relate the Frobenius (or Hilbert–Schmidt) norm of a matrix to various operator norms of that matrix. Our methods are based on earlier work of Feng [B.Q. Feng, Equivalence constants for certain matrix norms, Linear Algebra Appl. 374 (2003) 247–253] and Tonge [A. Tonge, Equivalence constants for matrix norms: a problem of Goldberg, Linear Algebra Appl. 306 (2000) 1–13], but introduce as a new ingredient some techniques developed by Hardy and Littlewood [G.H. Hardy, J.E. Littlewood, Bilinear forms bounded in space [p, q], Quart. J. Math. (Oxford) 5 (1934) 241–254].     

Smoothness spaces of higher order on lower dimensional subsets of the Euclidean space

We study Sobolev type spaces defined in terms of sharp maximal functions on Ahlfors regular subsets of and the relation between these spaces and traces of classical Sobolev spaces.