也谈广义对角占优阵的特征值的分布

设A=(a_(ij))_(n×n)为n阶复矩阵,记 σ_i=sum from j=1,j≠i to n(|a_(ij)|,i=l,2,…,n)。若|a_(ij)|>σ_i(i=1,2,…n),则称A为(按行)严格对角占优阵,记为A∈D,若|a_(ii)|·|a_(jj)|>σ_iσ_j(i≠j,i,j=1,2,…,n)则称A为严格对角乘积占优阵,记为A∈D_p(在〔1〕中此类矩阵称为广义对角占优阵,并记为GD)。若存在非奇对角阵Q=diag(q_l,…,q_n)使Q~(-1)AQ∈D,则称A为准严格对角占优阵,记为A∈D′(见〔2〕)。若存在非奇对角阵Q=diag(q_1,…,q_n)使Q~(-1)AQ∈D_p,则称A为准严格对角乘积占优阵。记为A∈D′_p。     

广义严格对角占优矩阵的迭代判别方法

正1引言设A=(a_(ij))∈C~(n×n),N={1,2,…,n}.记R_i(A)= sum |a_(ij)| from j≠i (i∈N),又记N_1=N_1(A)={i∈N:0|a_(ii)|≤R_i(A)},N_2=N_2(A)={i∈N:|a_(ii)R_i(A)}.定义1设A=(a_(ij))∈C~(n×n),如果|a_(ii)|R_i(A)(i∈N),则称A为严格对角占优矩阵.严格对角占优矩阵的集合记为D.如果存在n阶正对角矩阵D使得AD∈D,则称A为广义严格对角占优矩阵.广义严格对角占优矩阵的集合记为D.     

广义对角占优矩阵与M—矩阵的判定准则

广义对角占优矩阵与M—矩阵是计算数学中应用极其广泛的矩阵类。作者在文[1]中证明若A=(α_(ij))∈C~(n×n)为具有非零元素链对角占优阵或A满足:|α_(ii)‖α_(kk)|>Λ_iΛ_k,i,k∈N={1,…,n},则A为广义对角占优矩阵,detA≠0,揭示了文[3],[4]中detA≠0的共同本     

非奇异H矩阵的充分条件

1 引言 设A=(a_(ij))∈C~(n,n),R_i(A)=sum from j≠i to(|a_(ij)|,i,j∈N={1,2,…,n}。若|a_(ij)|≥R_i(A),i∈N,则称A为对角占优矩阵,记为A∈D_0;若不等式中每个不等号都是严格的,则称A为严格对角占优矩阵,记为A∈D。若存在正对角矩阵X,使得AX∈D,则称A为广义严格对角占优矩阵,记为A∈D。     

非奇异H-矩阵的新判据

1引言与记号设A=(a_(ij))∈C~(n×n),记N={1,2,…,n},∧_i(?)∧_i(A)=sum from j≠i|a_(ij)|,S_i(?)S_i(A)=sum from j≠i|a_(ij)|,(?)i,j∈N。若|a_(ij)>∧_i(A),(?)i∈N,则称A为严格对角占优矩阵。     

非奇H矩阵的简捷判据

非奇H矩阵在计算数学和矩阵理论的研究中很重要,但简便实用的判定条件较少见。本文给出几个简捷判据。[1,2,3]的主要结果是本文定理1的特例。 记M_n(C)为n阶复阵集合,M_n(R)为n阶实阵集合。设A=(a_(ij))∈M_n(C),记Λ_i(A)=sum from j≠i to |a_(ij)|,i,j∈N≡{1,2,…,n}。若|a_(ii)|>Λ_i(A),i∈N,则称A     

矩阵对角占优性的推广及应用

§1.引言设 A=(a_(ij))_(n×n)为一复矩阵,若有一正向量 d=(d_1,d_2,…,d_n)~T 使得d_i|a_(ij)|≥sum from j≠1 d_j|a_(ij)|,(1)对每一 i∈N={1,2,…,n}都成立,则称 A 为广义对角占优矩阵,记为 A∈D_0~*;如若(1)式中每一不等号都是严格的,则称 A 为广义严格对角占优矩阵,记为 A∈D~*.特别地,当 d=(1,1,…,1)~T 时,A∈D_0~*及 A∈D~*即是通常的对角占优与严格对角占优,分别记作 A∈D_0及 A∈D.利用矩阵的对角占优性质讨论其特征值分布是矩阵论中的重要课题,文献[5]—[10]给出了这方面的重要结果.n 阶实方阵 A 称为 M-矩阵,如果 A具有形式:A=sI-B,s>ρ(B),其中 B 为 n 阶非负方阵,ρ(B)表 B 之谱半径,利用广义严格对角占优的概念,文[1]给出了 M-矩阵的等价表征:若 n 阶实方阵     

共轭对角占优矩阵的特征值分布

<正> 设 A=(a_(rs)_(n×n)为 n 阶复矩阵.记μ_r=sum from s≠r |a_(rs)|,N={1,2,…,n},J(A)={r∈N||a_(rr)>μ_r}.我们引入下述定义:定义1 若对r=1,2,…,n 皆有|a_(rr)|>μ_r,则称 A 为按行严格对角占优矩阵,记为 A∈D.若对 r=1,2,…,n 皆有|a_(rr)|≥μ_r,J(A)非空集,且对任一 k(?)J(A),有a_(ks_1)a_(s_1s_2)…a_(s_m)l≠0,l∈J(A),则称 A 为按行准严格对角占优矩阵,记为 A∈SC.若 A为此二类矩阵之一,则记为 A∈D∪SC.     

两类对角占优矩阵的特征值分布

§1.引言 由于矩阵特征值分布的重要性,迄今已有许多人对其进行研究,国内这方面的主要工作参见[1]—[5]。本文将进一步研究以下两类矩阵的特征值分布。 定义1 设A=(a_ij)_n×n为n阶复矩阵,记,若对任意都成立,称A∈DD_0(R). 定义2 若2|Rea_(ij)|+|Rea_(ij)|>以Λ_i+Λ_j对任意i≠j,i,j∈N均成立,称A∈SD(R).若|Rea_(ij)|+|Rea_(ij)|≥Λ_i+Λ_j对任意i≠j,i,j∈N均成立,称A∈SD_0(R).     

局部双对角占优矩阵的注记

1引言非奇异H矩阵是计算数学、数学物理、控制论等领域中具有广泛应用的重要矩阵类,研究其充分条件自然引起人们的兴趣．文[1]中定义了一类局部双对角占优矩阵,并由此得到了非奇异H矩阵的判别方法．我们指出,文[1]所获充分条件中所给出的四个不等式条件,其中第四个不等式条件可蕴涵其余三个,进而定义了另一类局部双对角占优矩阵,并由此获得了非奇异H矩阵新的判别方法．设A=(a_(ij))∈C~(n×n),R_i(A)=sum from j≠i|a_(ij)|,i∈N={1,2,…,n}．若|a_(ii)|≥R_i(A),(?)i∈N,则称A为对角占优矩阵,记为A∈D_o;若不等式中每个不等号都是严格的,则称A为     

Some non-interior path-following methods based on a scaled central path for linear complementarity problems

In this paper we present some non-interior path-following methods for linear complementarity problems. Instead of using the standard central path we use a scaled central path. Based on this new central path, we first give a feasible non-interior path-following method for linear complementarity problems. And then we extend it to an infeasible method. After proving the boundedness of the neighborhood, we prove the convergence of our method. Another point we should present is that we prove the local quadratic convergence of feasible method without the assumption of strict complementarity at the solution.     

Optimal control of nonlinear evolution inclusions

In this paper, we study the optimal control of nonlinear evolution inclusions. First, we prove the existence of admissible trajectories and then we show that the set that they form is relatively sequentially compact and in certain cases sequentially compact in an appropriate function space. Then, with the help of a convexity hypothesis and using Cesari's approach, we solve a general Lagrange optimal control problem. After that, we drop the convexity hypothesis and pass to the relaxed system, for which we prove the existence of optimal controls, we show that it has a value equal to that of the original one, and also we prove that the original trajectories are dense in an appropriate topology to the relaxed ones. Finally, we present an example of a nonlinear parabolic optimal control that illustrates the applicability of our results.This research was supported by NSF Grant No. DMS-88-02688.     

Well-posedness and blow-up for a modified two-component Camassa–Holm equation

In this article, we consider a newly modified two-component Camassa–Holm equation. First, we establish the local well-posedness result, then we present a precise blow-up scenario. Afterwards, we derive a new conservation law, by which and the precise blow-up scenario we prove three blow-up results and a blow-up rate estimate result.     

嵌入定理与代数数域上的大筛法

<正> 本文把数理方程研究中常用的嵌入定理稍作推广,应用到代数数域上来,并把[4]中第四章的定理4.2和[1,5]中的均值定理推广到代数数域上. 为此,先介绍一些符号与约定,基本上采自[2]. 设K为-n次代数数域,按通常的记号,记作n=r_1+2r_2.以Z_k表K中的整数环. 1.设为一理想,如α,β∈Z_k,|(α-β),则记α≡β(mod ).按此可把K中的整数分类,其类数为N.Z_k中与互素的整数在上述分类中占住类数为     

Nonlinear elliptic differential equations with multivalued nonlinearities

In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is defined on all ℝ. Assuming the existence of an upper and of a lower solution, we prove the existence of a solution between them. Also for a special version of the problem, we prove the existence of extremal solutions in the order interval formed by the upper and lower solutions. Then we drop the requirement that the monotone nonlinearity is defined on all of ℝ. This case is important because it covers variational inequalities. Using the theory of operators of monotone type we show that the problem has a solution. Finally in the last part we consider an eigenvalue problem with a nonmonotone multivalued nonlinearity. Using the critical point theory for nonsmooth locally Lipschitz functionals we prove the existence of at least two nontrivial solutions (multiplicity theorem).     

Curves as measured foliation on noncompact surfaces

In the present work, that regards the Thurston's theory, we prove that, if we choose a closed curve, how we wish, on a noncompact surface, it is always possible to construct a particular masured foliation that has the choosed curve like a leaf; we also prove this foliation has a remarkable property that makes very easy to mesure all homotopy classes of closed curves of our surface. To prove this statement we need some Propositions and some Lemma that we also demonstre.     

Proportionally Modular Diophantine Inequalities and Full Semigroups

A proportionally modular numerical semigroup is the set of nonnegative integer solutions to a Diophantine inequality of the type ax mod b ≤ cx. We give a new presentation for these semigroups and we relate them with a type of affine full semigroups. Next, we describe explicitly the minimal generating system for the affine full semigroups we are considering. As a consequence, we obtain generating systems for proportionally modular numerical semigroups and we exhibit several families of these semigroups in terms of their generators. Finally, we use the concept of fundamental gap to study when a proportionally modular numerical semigroup is symmetric and we propose some open problems.     

On intermediate inquisitive and dependence logics: An algebraic study

This article provides an algebraic study of intermediate inquisitive and dependence logics. While these logics are usually investigated using team semantics, here we introduce an alternative algebraic semantics and we prove it is complete for all intermediate inquisitive and dependence logics. To this end, we define inquisitive and dependence algebras and we investigate their model-theoretic properties. We then focus on finite, core-generated, well-connected inquisitive and dependence algebras: we show they witness the validity of formulas true in inquisitive algebras, and of formulas true in well-connected dependence algebras. Finally, we obtain representation theorems for finite, core-generated, well-connected, inquisitive and dependence algebras and we prove some results connecting team and algebraic semantics.     

Controlling uncertain Lü system using backstepping design

In this paper, we discussed how to control Lü system with unknown parameters. Firstly we designed an observer to identify the unknown parameter of Lü system, then we used backstepping design method to control the system, and track any desired trajectory by the same way. At the same time we gave the numerical simulation for the results we had gained.     

Hereditary atomicity in integral domains

If every subring of an integral domain is atomic, we say that the latter is hereditarily atomic. In this paper, we study hereditarily atomic domains. First, we characterize when certain direct limits of Dedekind domains are Dedekind domains in terms of atomic overrings. Then we use this characterization to determine the fields that are hereditarily atomic. On the other hand, we investigate hereditary atomicity in the context of rings of polynomials and rings of Laurent polynomials, characterizing the fields and rings whose rings of polynomials and rings of Laurent polynomials, respectively, are hereditarily atomic. As a result, we obtain two classes of hereditarily atomic domains that cannot be embedded into any hereditarily atomic field. By contrast, we show that rings of power series are never hereditarily atomic. Finally, we make some progress on the still open question of whether every subring of a hereditarily atomic domain satisfies ACCP.