最大无关组判定定理及其它

本文给出了一个判定最大无关组的充要性定理，同时对用矩阵的行变换求最大无关组这一问题进行了点滴分析并介绍了一个解齐次线性方程组的简便方法。     

用初等变换求最大无关组的几个问题

本文讨论了求行最大无关组的一些问题 ,并给出了用行初等变换求行最大无关组的方法 .     

用初等变换求最大无关组的几个问题

本讨论了求行最大无关组的一些问题,并给出了用行初等变换求行最大无关组的方法。     

巧妙设计线性代数习题

通过具体实例的设计与求解,深化对向量组的最大线性无关组、最大线性无关组的不唯一性、向量组的线性表示等知识的理解,同时展示这部分知识在实际中的一种应用。增强学生应用所学知识解决实际问题的能力。     

枢轴运算与线性关系

本文给出求一个向量组的极大无关组并同时求出向量组中任何一个向量由该极大无关组线性表出系数的简易方法。     

最大乘积型Fuzzy上下无关文法与下推自动机

在文献[1]的基础上,讨论了最大乘积型Fuzzy上下文无关文法与最大乘积型Fuzzy下推自动机的关系,即:由给定的最大乘积型Fuzzy上下文无关文法可构造一个最大乘积型Fuzzy下推自动机使得二者接受的语言集相同,反之亦然。从而达到自动识别语言的目的。     

矩阵初等变换方法的几何原理

用向量的线性运算解释了矩阵初等行变换的本质,完善了用初等行变换求最大无关组的方法.     

可加函数的结构

本文讨论了可加函数的结构.在§1中定义了无理数的有理相关和有理无关,引入了无理数的最大有理无关集的概念,证明了最大有理无关集的存在,并由此得出,任一无理数均可表为某一最大有理无关集的有限个元素的有理组合,且这种表示法是唯一的.在§2中     

关于向量组的秩与最大无关组的教学改革研究

对向量的秩与最大无关组的教学过程进行了分析研究，从教学内容组织与教学手段等方面提出了改革的方案．对有的教材的不足提出了改进办法.     

求最大线性无关组应注意的问题

参考文献[1]中介绍的求最大线性无关组的方法是不确切的.指出产生错误的根源及避免产生错误的方法.     

Random Laplacian Matrices and Convex Relaxations

The largest eigenvalue of a matrix is always larger or equal than its largest diagonal entry. We show that for a class of random Laplacian matrices with independent off-diagonal entries, this bound is essentially tight: the largest eigenvalue is, up to lower order terms, often the size of the largest diagonal. entry. Besides being a simple tool to obtain precise estimates on the largest eigenvalue of a class of random Laplacian matrices, our main result settles a number of open problems related to the tightness of certain convex relaxation-based algorithms. It easily implies the optimality of the semidefinite relaxation approaches to problems such as \({{\mathbb {Z}}}_2\) Synchronization and stochastic block model recovery. Interestingly, this result readily implies the connectivity threshold for Erd?s–Rényi graphs and suggests that these three phenomena are manifestations of the same underlying principle. The main tool is a recent estimate on the spectral norm of matrices with independent entries by van Handel and the author.     

Trees with the second largest number of maximal independent sets

A maximal independent set is an independent set that is not a proper subset of any other independent set. In this paper, we determine the second largest number of maximal independent sets among all trees and forests of order n≥4. We also characterize those extremal graphs achieving these values.     

Subcritical, Critical and Supercritical Size Distributions in Random Coagulation-Fragmentation Processes

We consider the asymptotic probability distribution coagula-tion-fragmentation process in the thermodynamic limit of the size of a reversible random We prove that the distributions of small, medium and the largest clusters converge to Gaussian, Poisson and 0-1 distributions in the supercritical stage （post-gelation）, respectively. We show also that the mutually dependent distributions of clusters will become independent after the occurrence of a gelation transition. Furthermore, it is proved that all the number distributions of clusters are mutually independent at the critical stage （gelation）, but the distributions of medium and the largest clusters are mutually dependent with positive correlation coefficient in the supercritical stage. When the fragmentation strength goes to zero, there will exist only two types of clusters in the process, one type consists of the smallest clusters, the other is the largest one which has a size nearly equal to the volume （total number of units）.     

Reinsurance under the LCR and ECOMOR treaties with emphasis on light-tailed claims

Suppose that, over a fixed time interval of interest, an insurance portfolio generates a random number of independent and identically distributed claims. Under the LCR treaty the reinsurance covers the first l largest claims, while under the ECOMOR treaty it covers the first l−1 largest claims in excess of the lth largest one. Assuming that the claim sizes follow an exponential distribution or a distribution with a convolution-equivalent tail, we derive some precise asymptotic estimates for the tail probabilities of the reinsured amounts under both treaties.     

Trees with extremal numbers of maximal independent sets including the set of leaves

A subset S of vertices of a graph G is independent if no two vertices in S are adjacent. In this paper we study maximal (with respect to set inclusion) independent sets in trees including the set of leaves. In particular the smallest and the largest number of these sets among n-vertex trees are determined characterizing corresponding trees. We define a local augmentation of trees that preserves the number of maximal independent sets including the set of leaves.     

Polynomial-time solvability of the independent set problem in a certain class of subcubic planar graphs

The independent set problem for a given simple graph consists in computing the size of a largest subset of its pairwise nonadjacent vertices. In this article, we prove the polynomial solvability of the problem for the subcubic planar graphs with no induced tree obtained by identifying the ends of three paths of lengths 3, 3, and 2 respectively.     

On Domination and Independence Numbers of Graphs

A set S of vertices of a graph G is dominating if each vertex x not in S is adjacent to some vertex in S, and is independent if no two vertices in S are adjacent. The domination number, γ(G), is the order of the smallest dominating set in G. The independence number, α(G), is the order of the largest independent set in G. In this paper we characterize bipartite graphs and block graphs G for which γ(G) = α(G).     

Gompertz分布顺序统计量的一些随机比较

考虑两组相互独立的来自非齐次总体Gompertz分布的样本,给出了最小顺序统计量的反向失效率序、散度序以及凸变换序之间的比较和最大顺序统计量的普通随机序的比较.     

Constraints on the number of maximal independent sets in graphs

A maximal independent set of a graph G is an independent set that is not contained properly in any other independent set of G. Let i(G) denote the number of maximal independent sets of G. Here, we prove two conjectures, suggested by P. Erdös, that the maximum number of maximal independent sets among all graphs of order n in a family Φ is o(3^n/3) if Φ is either a family of connected graphs such that the largest value of maximum degrees among all graphs of order n in Φ is o(n) or a family of graphs such that the approaches infinity as n → ∞.