点可迁图的限制边连通性

3限制边割是连通图的一个边割, 它将此图分离成阶不小于3的连通分支. 图G的最小3限制边割所含的边数称为此图的3限制边连通度, 记作λ\-3(G). 它以图G的3阶连通点导出 子图的余边界的最小基数ξ_3(G)为上界. 如果λ_3(G)=ξ_3(G), 则称图G是极大3限制边连通的 . 已知在某种程度上,3限制边连通度较大的网络有较好的可靠性. 作者在文中证明: 如果k正则连通点可迁图的 围长至少是5, 那么它是是极大3限制边连通的.     

正则图的限制性边连通度

将连通图分离成阶至少为二的分支之并的边割称为限制性边割,最小限制性边割的阶称为限制性边连通度. 用λ′(G)表示限制性连通度,则λ′(G)≤ξ(G),其中ξ(G)表示最小边度. 如果上式等号成立,则称G是极大限制性边连通的. 本文证明了当k＞|G|/2时,k正则图G是极大限制性边连通的,其中k≥2, |G|≥4; k的下界在某种程度上是不可改进的.     

具有割点的标号Euler图的计数

本文讲座了具有k(k≥2）个割点,并且所有割点均分布在一个2－连能Euler图的标号Euler图的计数,在这里给出了有含有n个2－连能Euler图和k(k≥2）个割点,并且所有割点均分布在其中一2－连能Euler图的标号Euler图的指数型生成函数。     

对称本原图的集指数与本原简单图的广义上指数的极图

一个有向图D称为本原的，如果存在某个正整数k，使得对于D中的任一点x到任一点y都有长为k的途径，这样的正整数k中的最小者称为D的本原指数，作为本原指数概念的推广，R．A．Brualdi和柳柏濂于1990年引入了本原有向图的广义本原指数的新概念，本文给出了对称本原图的集指数的一些性质，并对本原简单图的广义上指数的极图进行了完全刻划。     

极小Cayley图的限制性边连通度

一个连通图X的边集的一个子集C称为一个限制性边割,如果它是一个边割,且X／C不含孤立点。X的限制性边连通度λ′（X）定义为所有限制性边割的最小基数。本文完全决定了极小Cayley图的限制性边连通度。     

点可迁图的限制边连通度

设S是连通图G的边子集.如果G-S不连通而且不含孤立点,那么称S是G的一个限制边割.G中所有限制边割中最小边数称为G的限制边连通度,记为′(G).限制边连通度是对传统边连通度的推广,而且是计算机互连网络容错性的一个重要度量.点可迁图是一类重要的网络模型.本文证明了如下结论 设G是连通的点可迁图.如果G的点数n4,而且点度k2,那么或者′(G)=2k-2,或者n是偶数,G含三角形且存在整数m2,使得k′(G)=n/m2k-3.     

本原对称有向图上广义指数的极图

设D是n阶有向图(允许有环但不允许有重复弧),X C V(D),集指数expD(X)是这样的最小正整数P,使得对D中每个点v,存在从X的至少一个点到V的长为P的途径.若这样的正整数P不存在,则定义expD(X)=∞.D的第k重上广义指数F(D,k):=max{expD(X)| X C V(D),|X|=k},1≤k≤n.如果F(D,k)<∞,则称D是k-上本原的.本文完全刻划了k-上本原对称有向图的第k重上广义指数的极图.     

极大3限制边连通二部图的充分条件

设G=(V,E)是一个连通图.称一个边集合S■E是一个k限制边割,如果G-S的每个连通分支至少有k个顶点.称G的所有k限制边割中所含边数最少的边割的基数为G的k限制边连通度,记为λ_k(G).定义ξ_k(G)=min{[X,■]:|X|=k,G[X]连通,■=V(G)\X}.称图G是极大k限制边连通的,如果λ_k(G)=ξ_k(G).本文给出了围长为g>6的极大3限制边连通二部图的充分条件.     

退化图中的子图计数问题研究

令G表示n个顶点的图,如果G的每个子图中都包含一个度至多为k的顶点,则称G为k-退化图.令N(G,F)表示G中F子图的个数.主要研究了k-退化图中完全子图和完全二部子图的计数问题,给出了计数的上界以及相应的极图.首先,证明了Ν(G,K_t)≤(n-k)(k t-1)+(k t).其次,如果s,t≥1,n≥k+1且s+t≤k,我们证明了Ν(G,K_s,t)≤{(k s)(n-s s)-1/2(k s)(k-s s),t=s,(k s)(n-s t)+(k t)(n-t s)-(k t)(k-t s),t≠s.此外,还研究了在最大匹配和最小点覆盖为给定值的情况下,图G中的最大边数.记v(G),K(G)分别为图G的最大匹配数和最小点覆盖.证明了当v(G)≤k,K(G)=k+r且n≥2k+2r²+r+1时,有e(G)≤(k+r+1 2)+(k-r)(n-k-r-1).     

平面图 4-割问题的 O(|V|~2)算法

给定一个简单图 G=(V,E).V 是顶点集,E■V×V 是边集.所谓 k-割乃是E 的一个子集 E_1,它使图 G_1=(V,E—E_1)恰包含 k 个分支.寻找一个图的最小 k-割问题,无论在理论上和实践中都有重要的意义.Hochbaum 和 shmoys 在文献[1]中给出了平面图最小3-割的 O(|V|~2)算法.本文将给出一个平面图最小4-割的O(|V|~2)算法.本文用到的概念及符号记法均与文献[1]一致.     

Estimation of the number of components of finite mixtures of multivariate distributions

An estimator of the number of components of a finite mixture ofk-dimensional distributions is given on the basis of a one-dimensional independent random sample obtained by a transformation of ak-dimensional independent random sample. A consistency of the estimator is shown. Some simulation results are given in a case of finite mixtures of two-dimensional normal distributions.     

Eigenvector p-Bases of Rings of Constants of Derivations

Let A be a UFD of characteristic p > 0, let 𝒵 be a set of some eigenvectors of a derivation of A. We prove, under some additional assumptions, a necessary and sufficient condition for 𝒵 to be a p-basis of the minimal ring of constants containing 𝒵. The main preparatory result is the unique decomposition theorem with respect to a factor from a given subalgebra containing A^p.     

Galois Structure of K-Groups of Rings of Integers

N/Kbe a Galois extension of number fields with finite Galois group G.We describe a new approach for constructing invariants of the G-module structure of the K groups of the ring of integers of N in the Grothendieck group of finitely generated projective Z[G]modules. In various cases we can relate these classes, and their function field counterparts, to the root number class of Fröhlich and Cassou-Noguès.     

有资格限制的指派问题的求解方法

在实际的指派工作中，常会遇到某个人有没有资格去承担某项工作的问题，因此，本建立了有资格限制的指派问题的数学模型。在此数学模型中，将效益矩阵转化为判定矩阵，由此给出了判定此种指派问题是否有解的方法；在有解的情况下，进一步将效益矩阵转化为求解矩阵，从而将有资格限制的指派问题化为传统的指派问题来求解。最后给出了一个数值例子来说明这样的处理方法是有效的。     

On Sums of Coefficients of Products of Polynomials

We study the nilpotency of the sums of all coefficients of some sorts of products of polynomials over reversible, IFP, and NI rings, and introduce an SCN ring as a generalization. We characterize SCN rings in relation with related ring properties, and also provide several useful properties and ring extensions of SCN rings.     

Convergence of sequences of sets of associated primes

It is a well-known result of M. Brodmann that if is an ideal of a commutative Noetherian ring , then the set of associated primes of the -th power of is constant for all large . This paper is concerned with the following question: given a prime ideal of which is known to be in for all large integers , can one identify a term of the sequence beyond which will subsequently be an ever-present? This paper presents some results about convergence of sequences of sets of associated primes of graded components of finitely generated graded modules over a standard positively graded commutative Noetherian ring; those results are then applied to the above question.

     

Rates of Convergence of Means of Euclidean Functionals

Let L be the Euclidean functional with p-th power-weighted edges. Examples include the sum of the p-th power-weighted lengths of the edges in minimal spanning trees, traveling salesman tours, and minimal matchings. Motivated by the works of Steele, Redmond and Yukich (Ann. Appl. Probab. 4, 1057–1073, 1994, Stoch. Process. Appl. 61, 289–304, 1996) have shown that for n i.i.d. sample points {X ₁,…,X _n } from [0,1] ^d , L({X ₁,…,X _n })/n ^(d−p)/d converges a.s. to a finite constant. Here we bound the rate of convergence of EL({X ₁,…,X _n })/n ^(d−p)/d . Y. Koo supported by the BK21 project of the Department of Mathematics, Sungkyunkwan University. S. Lee supported by the BK21 project of the Department of Mathematics, Yonsei University.     

Number of connected components of groups of real points of adjoint groups

We present a unified approach to compute the number of connected components in the group of real points of adjoint almost simple real algebraic groups.     

Enumeration of perfect matchings of a type of Cartesian products of graphs

Let G be a graph and let Pm(G) denote the number of perfect matchings of G.We denote the path with m vertices by P_m and the Cartesian product of graphs G and H by G×H. In this paper, as the continuance of our paper [W. Yan, F. Zhang, Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians, Adv. Appl. Math. 32 (2004) 175-188], we enumerate perfect matchings in a type of Cartesian products of graphs by the Pfaffian method, which was discovered by Kasteleyn. Here are some of our results:1. Let T be a tree and let C_n denote the cycle with n vertices. Then Pm(C₄×T)=∏(2+α²), where the product ranges over all eigenvalues α of T. Moreover, we prove that Pm(C₄×T) is always a square or double a square.2. Let T be a tree. Then Pm(P₄×T)=∏(1+3α²+α⁴), where the product ranges over all non-negative eigenvalues α of T.3. Let T be a tree with a perfect matching. Then Pm(P₃×T)=∏(2+α²), where the product ranges over all positive eigenvalues α of T. Moreover, we prove that Pm(C₄×T)=[Pm(P₃×T)]².