The circular chromatic numbers of planar digraphs

The circular chromatic number is a refinement of the chromatic number of a graph. It has been established in [3,6,7] that there exists planar graphs with circular chromatic number r if and only if r is a rational in the set {1} ∪ [2,4]. Recently, Mohar, in [1,2] has extended the concept of the circular chromatic number to digraphs and it is interesting to ask what the corresponding result is for digraphs. In this article, we shall prove the new result that there exist planar digraphs with circular chromatic number r if and only if r is a rational in the interval [1,4]. © 2006 Wiley Periodicals, Inc. J Graph Theory 55: 14–26, 2007     

NA序列广义Jamison型加权和的几乎处处收敛性

本文从随机变量序列广义Ｊａｍｉｓｏｎ型加权和的一个系数指标函数自身性质出发，讨论了广义Ｊａｍｉｓｏｎ型加权和的强稳定性，避免了控制函数的引入， 进一步还得到了更一般的ＮＡ序列广义Ｊａｍｉｓｏｎ型加权和的几乎处处收敛的条件，并将前人的若干结论包含为特例．     

不同分布NA序列加权和的完全收敛性

本文讨论了不同分布NA随机变量序列加权和的完全收敛性，获得了较[7]中的定理1及定理A更为一般的安全收敛性，并得到了完全收敛速度与矩条件之间的等价关系。     

一类非平稳时间序列的周期性

本文研究了一类非平稳时间序列的稳定性及周期性，得到了它为周期相关序列的充分必要条件，推广了文献[3]的结论．     

Nontrivial periodicity in discrete delay models of population growth

In this paper, we will study the periodicity in discrete model An+1=λAn+F(An−m) of population growth, where the delay m is large enough and the nonlinearity F is unimodal function. Actually, we prove that there is a slowly oscillated periodic solution. Our method is Browder nonextremal fixed point theorem.     

Extensions of partially ordered partial abelian monoids

The notion of a partially ordered partial abelian monoid is introduced and extensions of partially ordered abelian monoids by partially ordered abelian groups are studied. Conditions for the extensions to exist are found. The cases when both the above mentioned structures have the Riesz decomposition property, or are lattice ordered, are treated. Some applications to effect algebras and MV-algebras are shown.     

两两独立同分布序列Cesàro强大数定律的收敛速度

设{X,Xn,n≥0}是两两独立同分布的随机变量序列,11．为了证明这一结论而获得到的两两负相关随机变量序列的Cesaro强大数定律收敛速度的结果本身也是有意义的．此结果对于同分布的两两NQD序列也是对的．     

On the structure of cluster point sets of iteratively generated sequences

LetX be a closed subset of a topological spaceF; leta(·) be a continuous map fromX intoX; let {x _i} be a sequence generated iteratively bya(·) fromx ₀ inX, i.e.,x _i+1 =a(x _i),i=0, 1, 2, ...; and letQ(x ₀) be the cluster point set of {x _i}. In this paper, we prove that, if there exists a pointz inQ(x ₀) such that (i)z is isolated with respect toQ(x ₀), (ii)z is a periodic point ofa(·) of periodp, and (iii)z possesses a sequentially compact neighborhood, then (iv)Q(x ₀) containsp points, (v) the sequence {x _i} is contained in a sequentially compact set, and (vi) every point inQ(x ₀) possesses properties (i) and (ii). The application of the preceding results to the caseF=E ⁿ leads to the following: (vii) ifQ(x ₀) contains one and only one point, then {x _i} converges; (viii) ifQ(x ₀) contains a finite number of points, then {x _i} is bounded; and (ix) ifQ(x ₀) containsp points, then every point inQ(x ₀) is a periodic point ofa(·) of periodp.     

Subordination,rank, and determinism of multivariate stationary sequences

Necessary and sufficient conditions for an arbitrary q-variate stationary sequence x_t, t ∈ $\text{Z}$, to be deterministic are presented. A characterization of the rank r(x) of x_t, t ∈ $\text{Z}$, and a method to construct the Wold-Cramér decomposition for x_t, t ∈ $\text{Z}$, are given. Subordination of q-variate bounded orthogonally scattered vector measures is considered.