Freeness for 13 lines arrangements is combinatorial

We prove Terao conjecture saying that the freeness is determined by the combinatorics for arrangements of 13 lines in the complex projective plane.     

Stable intersection of affine Cantor sets defined by two expanding maps

It is shown that in the context of affine Cantor sets with two increasing maps, the arithmetic sum of both of its elements is a Cantor set otherwise, it is a closure of countable union of nontrivial intervals. Also, a new family of pairs of affine Cantor sets is introduced such that each element of it has stable intersection. At the end, pairs of affine Cantor sets are characterized such that the sum of elements of each pair is a closed interval.     

二部图形式的Erd\H{O}s-S\'{o}s猜想

二部图形式的Erd\H{O}s-S\''{o}s猜想     

Optimality conditions at junctions of singular and nonsingular controls

Some necessary conditions for the optimality of junctions between singular and nonsingular subarcs arising in optimal control problems are well established. The second part of McDanell's conjecture has been proved false. The first part still awaits verification or otherwise. This paper briefly summarizes previous work and presents some evidence to suggest that the first part of the conjecture is also false.     

On spectral properties of periodic polyharmonic matrix operators

We consider a matrix operatorH = (-Δ)^l +V inR ⁿ, wheren ≥ 2,l ≥ 1, 4l > n + 1, andV is the operator of multiplication by a periodic inx matrixV(x). We study spectral properties ofH in the high energy region. Asymptotic formulae for Bloch eigenvalues and the corresponding spectral projections are constructed. The Bethe-Sommerfeld conjecture, stating that the spectrum ofH can have only a finite number of gaps, is proved.     

The weight twoK-theory of fields

We verify the weight-two portion of the Quillen-Lichtenbaum conjectures, relating the modn weight-twoK-groups of a field with the appropriate étale cohomology groups. This extends the work of Merkurjev and Suslin onK ₂ of a field, and the works of Suslin and the author on the indecomposableK ₃ of a field.Partially supported by the NSF.     

完全r部图乘积上的Graham猜想

图G的Pebbling数f(G)是最小的正整数n,使得不论n个Pebble如何放置在G的顶点上,总可以通过一系列的Pebbling移动把1个Pebble移到任意一点上,其中Pebbling移动是从一个顶点处移走两个Pebble而把其中一个移到与其相邻的一个顶点上。Graham猜测对于任意的连通图G和H有f(G×H)≤f(G)f(H)。本文证明对于一个完全r部图和一个具有2-Pebbleing性质的图来说,Graham猜想成立。作为一个推论,当G和H均为完全r部图时,Graham猜想成立。     

A companion matrix approach to the study of zeros and critical points of a polynomial

In this paper, we introduce a new type of companion matrices, namely, D-companion matrices. By using these D-companion matrices, we are able to apply matrix theory directly to study the geometrical relation between the zeros and critical points of a polynomial. In fact, this new approach will allow us to prove quite a number of new as well as known results on this topic. For example, we prove some results on the majorization of the critical points of a polynomial by its zeros. In particular, we give a different proof of a recent result of Gerhard Schmeisser on this topic. The same method allows us to prove a higher order Schoenberg-type conjecture proposed by M.G. de Bruin and A. Sharma.     

随机配置问题的概率分析

假设(ti,j)1≤i,j≤n是一个n×n的具有独立同分布,参数为1的指数费用矩阵.考虑最优配置费用Ane=:minπsumfromi=1ton=1ti,π(i),其中π=(π(1),…,π(n))为1,2,…,n的排列.本文目的是对最近关于平均最优配置费用EAne的研究进展作些评论,特别关注Aldous的目标方法和局部弱收敛性,以及著名的Parisi猜想和证明.文章结尾包含了一些尚未解决的问题,值得进一步研究.     

Particle systems with variable length

We present a new class of one-dimensional particle systems, in which the number of components may change in the process of interaction. We suggest that some of these systems display spontaneous symmetry breaking. Received: 16 August 2002