关于组合的两类问题

设有自然数集合A={1、2、…,n},从中任意取出k个来(k相似文献   

关于Erds和Nathanson的一个组合问题

Erds和Nathanson研究2阶渐近基中存在2阶极小渐近基的条件时研究了一族元数≤2的互不相交集合之满足某种条件的代表元系的个数问题,给出了其最佳上界估计,并且指出对于集合元数≤h(h≥3)的一般情形乃是一个尚未解决的组合问题。本文给出了一般情形时的最佳上界估计,从而彻底解决了Erds和Nathanson的这一组合问题。     

有限集合的子集族问题分类解析

将有限集合中符合某一特性的所有子集合,称之为有限集合的子集族.在各类集合问题中,与子集族相关的问题是其中极为重要的一类.这类问题题型新颖,解答灵活,给同学们的学习造成了一定的困难.本文拟对这类问题分类进行解析.1.求有限定条件的子集个数例1(03希望杯高一竞赛题)集合S={1,2,3,4,5,6},A是S的一个子集,当x∈A时,若有x-1A,且x 1A,则称x为A的一个“孤立元素”,那么S中无“孤立元素”的4元子集族中子集的个数是.解4个元素为连续自然数的子集有{1,2,3,4},{2,3,4,5},{3,4,5,6},共3个,不都连续的子集有{1,2,4,5},{1,2,5,6},{2,3,5,6},共…     

关于度集合的极小盖

度a称为度集合A的极小盖指a为A中所有元素的极小盖。如果a为A的极小盖并且满足?c相似文献   

A（x）－C（x）的渐近公式

设Ａ、Ｃ是一些自然数的集合。对于Ａ中任一自然数ｍ，每一ｍ阶群都是Ａｂｅｌ群；对于Ｃ中任一自然数ｎ，每一ｎ阶群都是循环群。本文的目的是证明下面的渐近公式：此处γ是Ｅｕｌｅｒ常数，ｌｏｇｒｘ＝ｌｏｇ（ｌｏｇ＿（ｒ－１）ｘ），ｌｏｇ＿１ｘ＝ｌｏｇｘ。     

谈一类极值问题的解法

本文将利用集合的子集类的思想去解决一类较为复杂的极值问题 .为此我们引入以下的概念和定理 .设Ａ是一个非空有限集 ,集合Ａ的元素个数称为集合Ａ的阶 ,记作｜Ａ｜ .当｜Ａ｜=ｎ ,称Ａ是一个ｎ阶集 .对于一个集合Ａ的一个子集类 {Ａ1,Ａ2 ,… ,Ａｋ},若对任何二个子集Ａｉ,Ａｊ(ｉ≠ｊ)都有Ａｉ Ａｊ,Ａｊ Ａｉ,则称这个类是互不包含的子集类 .对这种子集类我们有定理　有一个ｎ(ｎ≥ 1 )阶集合Ａ的一切互不包含的子集类中 ,子集个数最多的类含有Ｃｎ2ｎ个子集 ,其中 ｎ2 表示不超过 ｎ2 的最大整数 .证明　记 {Ａ1,Ａ2 ,… ,Ａｋ}为Ａ的…     

例谈“正难则反”策略的应用

<正>数学中的很多问题,若从正面入手,则较为繁琐或困难较大,往往从其反面进行思考,即所谓"正难则反".下面谈谈"正难则反"的一些策略.例1设集合A、B是非空集合M的两个不同的子集,满足:A不是B的子集,B不是A的子集.(1)若M={a1,a2,a3,a4},直接写出所有不同的有序集合对(A,B)的个数;(2)若M={a1,a2,a3,…,an},求所有不     

一类图中的最大团

本文中的图均指有限阶的简单图,未加说明的术语和记号均见[2].设 G 是连通图,若对于 S(?)V(G),G\S 是不连通的(指 G\S 至少包含两个连通分支),则称 S 是 G 的一个割集.若 S 是 G 的割集,但 S 的任何真子集不是 G 的割集,则称 S 是 G 的一个极小割集.设 h 是一个正整数.若 S 是一个极小割集且|S|≤h,则称 S 是 G 的一个下 h-割集.若对于每个 v∈V(G),存在下 h-割集 S 使得 v∈S,则称 G 是一个处处 h-可断图.     

奇异M—矩阵和广义对角占成阵的实用判定准则

1　引言和符号首先对本文所采用的符号和术语作一约定和说明,而不再重申.N表示前面n个自然数的集合,而分别用Mn(C)和Mn(R)表示所有n阶复方阵和n阶实方阵的集合,Rn表示n维实列向量.Zn={A|A=(aij)∈Mn(R),aij≤0,i≠j,i,j∈N}.若A∈Zn则称A为Z-矩阵,有时也简记为A∈Z.I恒表示适当阶的单位矩阵.设α和β为N的非空子集,对于A∈Mn(C),把由A中行标属于α而列标属于β的元素按照原来相对位置所构成的子矩阵记为A(α,β),特别地,把主子阵A(α,α)简记为A(α)、当A(α)可逆时,其逆阵记为A(α)-1,此时称矩阵A/A(α)=A(α)-A(α,α).…     

谈集合间的关系

本刊争鸣栏问题 5 8是 :集合间的关系有几种 ?要回答这个问题 ,我们从数学中的“关系”谈起 .在抽象代数中 ,规定集合A的元素间的一种关系R是A×A ={ (x ,y) |x∈A ,y∈A}的一个子集R .即A×A的任一个子集均确定集合A的元素间的一种关系 .判断R是否成为集合A的元素间的一种关系常按如下方法进行 :若对于任意的a ,b∈A ,要么a与b满足关系R ,要么a与b不满足关系R ,二者必居其一 ,这时我们就说R是集合A的元素间的一种关系 .否则R就不是A的元素间的一种关系 .依据上面关于“关系”的判定方法 ,我们说集合间关系在高中教材中只介绍了两种…     

Minimal bases and maximal nonbases in additive number theory

An asymptotic basis $\text{A}$ of order h is minimal if no proper subset of $\text{A}$ is an asymptotic basis of order h. Examples are constructed of minimal asymptotic bases, and also of an asymptotic basis of order two no subset of which is minimal.If $\text{B}$ is a set of nonnegative integers which is not a basis (resp. asymptotic basis) of order h, but such that every proper superset of $\text{B}$ is a basis (resp. asymptotic basis) of order h, then $\text{B}$ is a maximal nonbasis (resp. maximal asymptotic nonbasis) of order h. Examples of such sets are constructed, and it is proved that every set not a basis of order h is a subset of a maximal nonbasis of order h.     

Dense minimal asymptotic bases of order two

We call a set A of positive integers an asymptotic basis of order h if every sufficiently large integer n can be written as a sum of h elements of A. If no proper subset of A is an asymptotic basis of order h, then A is a minimal asymptotic basis of that order. Erd?s and Nathanson showed that for every h?2 there exists a minimal asymptotic basis A of order h with d(A)=1/h, where d(A) denotes the density of A. Erd?s and Nathanson asked whether it is possible to strengthen their result by deciding on the existence of a minimal asymptotic bases of order h?2 such that A(k)=k/h+O(1). Moreover, they asked if there exists a minimal asymptotic basis with lim sup(ai+1−ai)=3. In this paper we answer these questions in the affirmative by constructing a minimal asymptotic basis A of order 2 fulfilling a very restrictive condition

     

Minimal asymptotic bases for the natural numbers

The sequence A of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer can be written as the sum of h elements of A. Let $\text{M}{}_{\mathrm{h}}{}^{\mathrm{A}}$ denote the set of elements that have more than one representation as a sum of h elements of A. It is proved that there exists an asymptotic basis A such that $\text{M}{}_{\mathrm{h}}{}^{\mathrm{A}}\text{(x) = O(x}{}^{\mathrm{<mtext>1?1</mtext><mtext>h+?</mtext>}}\text{)}$ for every ? > 0. An asymptotic basis A of order h is minimal if no proper subset of A is an asymptotic basis of order h. It is proved that there does not exist a sequence A that is simultaneously a minimal basis of orders 2, 3, and 4. Several open problems concerning minimal bases are also discussed.     

阶极小渐近基

Let N denote the set of all nonnegative integers and A be a subset of N.Let W be a nonempty subset of N.Denote by F~*(W) the set of all finite,nonempty subsets of W.Fix integer g≥2,let A_g(W) be the set of all numbers of the form sum f∈Fa_fg~f where F∈F~*(W)and 1≤a_f≤g-1.For i=0,1,2,3,let W_i = {n∈N|n≡ i(mod 4)}.In this paper,we show that the set A = U_i~3=0 A_g(W_i) is a minimal asymptotic basis of order four.     

3阶不可约符号模式的精细惯量极小临界集

Let S be a nonempty, proper subset of all possible refined inertias of real matrices of order n. The set S is a critical set of refined inertias for irreducible sign patterns of order n,if for each n × n irreducible sign pattern A, the condition S ? ri(A) is sufficient for A to be refined inertially arbitrary. If no proper subset of S is a critical set of refined inertias, then S is a minimal critical set of refined inertias for irreducible sign patterns of order n.All minimal critical sets of refined inertias for full sign patterns of order 3 have been identified in [Wei GAO, Zhongshan LI, Lihua ZHANG, The minimal critical sets of refined inertias for 3×3 full sign patterns, Linear Algebra Appl. 458(2014), 183–196]. In this paper, the minimal critical sets of refined inertias for irreducible sign patterns of order 3 are identified.     

Domain Structure of Bulk Ferromagnetic Crystals in Applied Fields Near Saturation

We investigate the ground state of a uniaxial ferromagnetic plate with perpendicular easy axis and subject to an applied magnetic field normal to the plate. Our interest is in the asymptotic behavior of the energy in macroscopically large samples near the saturation field. We establish the scaling of the critical value of the applied field strength below saturation at which the ground state changes from the uniform to a multidomain magnetization pattern and the leading order scaling behavior of the minimal energy. Furthermore, we derive a reduced sharp interface energy, giving the precise asymptotic behavior of the minimal energy in macroscopically large plates under a physically reasonable assumption of small deviations of the magnetization from the easy axis away from domain walls. On the basis of the reduced energy and by a formal asymptotic analysis near the transition, we derive the precise asymptotic values of the critical field strength at which non-trivial minimizers (either local or global) emerge. The non-trivial minimal energy scaling is achieved by magnetization patterns consisting of long slender needle-like domains of magnetization opposing the applied field.     

Answers to two questions posed by Farhi concerning additive bases

Let A be an asymptotic basis for N and X a finite subset of A such that A?X is still an asymptotic basis. Farhi recently proved a new batch of upper bounds for the order of A?X in terms of the order of A and a variety of parameters related to the set X. He posed two questions concerning possible improvements to his bounds. In this note, we answer both questions.     

Irreducible function bases of isotropic invariants of a third order three-dimensional symmetric and traceless tensor

Third order three-dimensional symmetric and traceless tensors play an important role in physics and tensor representation theory. A minimal integrity basis of a third order three-dimensional symmetric and traceless tensor has four invariants with degrees two, four, six, and ten, respectively. In this paper, we show that any minimal integrity basis of a third order three-dimensional symmetric and traceless tensor is also an irreducible function basis of that tensor, and there is no syzygy relation among the four invariants of that basis, i.e., these four invariants are algebraically independent.     

On the Restricted Order of Asymptotic Bases of Order Two

The restricted order of an asymptotic basis A is the least integer h, if it exists, such that every sufficiently large integer is the sum of h or fewer distinct elements of A. We show that any asymptotic basis of order 2 has a restricted order at most equal to 4. We also provide an example of an additive basis of order 2 whose restricted order is 4.To Jean-Louis Nicolas2000 Mathematics Subject Classification: Primary—11B13     

Bases and nonbases of square-free integers

A basis is a set A of nonnegative integers such that every sufficiently large integer n can be represented in the form n = a_i + a_j with a_i, a_i ∈ A. If A is a basis, but no proper subset of A is a basis, then A is a minimal basis. A nonbasis is a set of nonnegative integers that is not a basis, and a nonbasis A is maximal if every proper superset of A is a basis. In this paper, minimal bases consisting of square-free numbers are constructed, and also bases of square-free numbers no subset of which is minimal. Maximal nonbases of square-free numbers do not exist. However, nonbases of square-free numbers that are maximal with respect to the set of square-free numbers are constructed, and also nonbases of square-free numbers that are not contained in any nonbasis of square-free numbers maximal with respect to the square-free numbers.