解广义特征值反问题的同伦方法

1.引言 我们讨论下列广义特征值反问题: (G)已知B是n×n阶对称半正定矩阵,λ=(λ_1,…,λ_(2n-1))~T∈R~(2n-1),且{λ_i}~(n_3),和{λ_i}_(n+1)~(2n-1)严格交错。问题是欲求一个实对称三对角n×n阶矩阵A,使得λ_1…,λ_n是Ax=λBx的特征值,λ_(n+1),…,λ_(2n-1)是A_(n-1)x=λB_(n-1)x的特征值,其中A_(n-1),B_(n-1)分别是矩阵A,B的前n-1阶主子阵。     

加法与乘法逆特征值问题的可解性

1.引言 本文讨论如下代数特征值反问题可解的充分条件: 问题A(加法逆特征值问题)。给定一Hermite矩阵A=(a_(ij))_(n×n)及n个实数λ_1,…,λ_n,求一实对角阵D=diag(c_1…,c_n),使得A+D的特征值为λ_1,…,λ_n。 问题M(乘法逆特征值问题)。给定一正定Hermite矩阵A=(a_(ij))_(n×n)和n个正实数     

标准Jacobi矩阵的混合型特征反问题

0 引言 本文讨论如下标准形式的Jacobi矩阵 其中a_i>0(i=1,2,…,n),b_i>0(i=1,2,…,n-1)。 对于Jacobi矩阵(对称三对角矩阵)的特征反问题,已有的成果[1],基本上集中在由两组频谱或两个特征对(指特征值及相应的特征向量)构造Jacobi矩阵的元素这样两类问题上,习惯上称之为频谱型或特征向量型反问题。本文提出且求解了第三类型——混合型特征反问题。即由一组频谱数据和一个特征向量构造矩阵元素的问题: 问题Ⅰ 给定正数λ~(1),λ~(2),…,λ~(n)和实向量x=(x_1,x_2,…,x_n)~T,其中x_1=1。构造一个标准形式的Jacobi矩阵J,使其第k阶顺序主子阵恰以λ~(k)(k=1,2,…,n)为其特征值。且(λ~(n),x)为其特征对。 问题Ⅱ 给定正数0<λ_1~(n)<λ_1~(n-1)<…<λ_1~(1)和正向量x=(x_1,x_2,…,x_n),其中x_=,x_k>0(k=2,…,n),构造一个标准形式的Jacobi矩阵J,使其第K阶顺序主子阵恰以λ_1~(k)为其最小特征值,而(λ~(n),x)为J的特征对。 问题Ⅲ 给定n个实数0<λ_1)<λ_2<…<λ_n和m个实数λ~(1),λ~(2),…,λ~(m)及m维向量x=(x_1,…,x_m)~T。构造n阶标准形式的Jaeobi矩阵J,使其第K阶顺序主子阵恰以λ~(k)(k=1,2,…,m)为其特征值,而(λ~(m),x)为第m阶顺序主子阵的特征对,且λ_k(k=1,2,…,n)为J的特征值。这里系大于或等     

关于Kahan定理的一个注记

设A∈C~(n×n),B∈C~(k×k)均为Hermite矩阵,它们的特征值分别为{λ_j}_(j=1)~n和{μ_j}_(j=1)~k(k≤n);Q∈~(n×k)为列满秩矩阵.令 (1) 则存在A的k个特征值λ_(j_2),λ_(j_2),…,λ_(j_k),使得 (2) 其中σ_k为Q的最小奇异值,||·||_2表示矩阵的谱范数.这是著名的Kahan定理·1996年曹志浩等在[2]中将(2)加强为 (3) 这是Kahan的猜想.在本文中,我们讨论将Kahan定理中“B为k阶Hermite矩阵”改为B为k阶(任意)方阵后,特征值的扰动估计,有以下结果. 定理 设A∈C~(n×n)为Hermite矩阵,其特征值为{λ_j}_(j=1)~n,B∈C~(k×k)的特征值为{μ_j}_(j=1)~k,而Q∈C~(n×k)为列满秩矩阵.则存在A的k个特征值λ_(j_1),λ_(j_2),…,λ_(j_k),使得     

关于两个正定矩阵之积的特征值估计的一个注记

令A>0及B>0记两个n×n(n≥2)厄尔米特正定矩阵;μ_1≥μ_2≥…μ_n及ν_1≥ν_2≥…≥ν_n记A和B的特征值;设λ为AB的任意特征值.ShaHu-yun证得2/nμ_n~2ν_n~2/μ_n~2 ν_n~2<λ相似文献   

Hermite矩阵特征值反问题的几乎处处不可解性

§1.引言 Hermite矩阵的特征值反问题是Downing和Householder在[2]中提出的,其形式如下: 问题A. 给定Hermite矩阵A,k个非零实数λ_1…,λ_k,以及满足r_+r_1+…+r_k=n的k+1个非负整数r_1,r_1,…,r_k,求一实对角矩阵D=diag(d_1,…,d_n),使得A+D的特征值为0,λ_1,…,λ_k,并且相应的重数为 r_0,r_1,…,r_k.     

由谱数据构造周期Jacobi矩阵

本文研究如下周期Jacobi矩阵特征值问题的反问题: 问题PJP 给定实数列{λ_i}_(i=1)~n和{u_i}_(i=1)~(n-1)及正实数β且满足     

矩阵特征值分离度的下界

这里λ_n<λ_(n-1)<…<λ_1。 分离度是矩阵特征值计算中的一个有用的概念,它与矩阵特征值计算的难易程度关系极为密切。估计分离度的界限,能够预测用特定方法计算特征值的运算次数,这个课题是M.Newman提出的。 众所周知,求矩阵特征值等价于高次方程求根。有关多项式根的分离度(其定义与(1)类似),在[1],[2]中有了一些结果。但是,它们都含有多项式的系数,对于矩阵,使用     

关于图的规范拉普拉斯特征值和的若干结果（英文）

对任意一个连通图G,记L(G)和L(G)分别为G的拉普拉斯矩阵和规范拉普拉斯矩阵.令μ_1≥μ_2≥…≥μ_n=0和λ_1≥λ_2≥…≥λ_n=0分别为G的拉普拉斯特征值和规范拉普拉斯特征值.本文给出了λ_1的三个新的下界.前两个下界优于Das等在[Ars Cormbin.,2015,118:143-154]中给出的下界,第三个下界优于张晓东在[Ars Combin.,2004,72:191-198]中给出的下界.另一方面讨论了规范拉普拉斯特征值与G的度序列之间的关系.同时也讨论了图的拉普拉斯特征值和规范拉普拉斯特征值之间的关系.     

一个矩阵不等式及其应用

本文给出正定Hermitè矩阵的一个不等式,然后用它推导出算一几不等式. 命题:设A_1,A_2,…,A_n是,n(≥2)个正定(同级)Hermite矩阵,λ_1,λ_2,…,λ_n是n个正实数,且λ_1+λ_2+…+λ_n=1,则     

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.     

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.     

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.     

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.     

图表示范畴的两个子范畴

引进图表示范畴的两个子范畴,研究它们的同调性质。     

Asymptotics of families of solutions of nonlinear difference equations

One method to determine the asymptotics of particular solutions of a difference equation is by solving an associated asymptotic functional equation. Here we study the behaviour of the solutions in an asymptotic neighbourhood of such individual solutions. We identify several types of attraction and repulsion, which range from almost orthogonality to almost parallelness. Necessary and sufficient conditions for these types of behaviour are given.     

L－fuzzy集网的R－收敛性质

本文在ＬＦ拓扑空间中建立了Ｌ－ｆｕｚｚｙ集网的弱收敛（Ｒ－收敛）概念，应用文［４］中的Ｒ－闭包，系统讨论了它们的性质，证明了等式ＲｌｉｍＡ＿ｎ＝∧（∨Ａ＿ｍ）＿Ｒ和ＲｌｉｍＡ＿ｎ＝Ａ＿ｎ＝∧（∨Ａ＿ｍ）＿Ｒ并且给出了Ｌ－ｆｕｚｚｙ集网与其子网之间的关系。