自反算子代数的环自同构

设Ａ是Ｂａｎａｃｈ空间Ｘ上的自反算子代数，并且Ａ的不变子空间格ＬａｔＡ满足 ０＋≠０和Ｘ_≠Ｘ，ａ：Ａ→Ａ是环自同构．如果Ｘ是实空间，并且ｄｉｍ　Ｘ　＞１；则存在Ｘ上的线性有界可逆算子Ａ，使得ａ（Ｔ）＝ＡＴＡ～（－１）；Ｔ∈Ａ：如果Ｘ是复空间，并且ｄｉｍ Ｘ　＝∞，则ａ（Ｔ）＝ＡＴＡ～（－１），Ｔ∈Ａ．其中Ａ：Ｘ→Ｘ是线性、或者共轭线性有界可逆算子．     

矩阵方程AXB=C的通解

本文给出了矩阵方程 Ａ_（ｍ×ｎ）Ｘ_（ｎ×５）Ｂ_（ｓ×）＝Ｃ_（ｍ×ｔ）有解且有无穷解的通解表达式 Ｘ＝Ｃ~（＊＊）＋［ｋ_（１１）ξ_１~Ｔ＋…＋ｋ_（１（ｎ－ｒ））ξ_（ｎ－ｒ）~Ｔ＋……ｋ_（ｓ１）ξ_１~Ｔ＋…＋ｋ_（ｓ（ｎ－ｒ））ξ_（ｎ－ｒ）~Ｔ］ ＋［Ｐ_（１１）η_１＋…＋Ｐ_（１（ｓ－１））η_（ｓ－１）……Ｐ_（ｎ１）η_１＋…Ｐ_（ｎ（ｓ－１））η_（ｓ－１）］~Ｔ（其中ｋ_（ｉｊ）;Ｐ_（ｉｊ）为任意常数;ξ_１…,ξ_（ｎ－ｒ）;η_１…,η_（ｓ－１）分别为Ａ_（ｍ×ｎ）Ｘ_（ｎ×１）＝０;Ｘ_（１×ｓ）Ｂ_（ｓ×ｔ）＝０的一个基础解系,Ｃ~（＊＊）为ＡＸＢ＝Ｃ的一个特解）及利用矩阵初等变换求其通解的方法．     

On the Matrix Equations of Higher Degrees

１　ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄｎｏｔａｔｉｏｎｓＦｏｌｌｏｗｉｎｇＲｅｆｅｒｅｎｃｅ［１］ｗｅｕｓｅＭｎ（Ｆ）ｆｏｒｎ－ｂｙ－ｎｍａｔｒｉｃｅｓｗｉｔｈｅｎｔｒｉｅｓｆｒｏｍＦ,ｓｐｅｃｉａｌ－ｌｙＭｎｆｏｒＭｎ（Ｃ）,ＡｆｏｒｔｈｅＨｅｒｍｉｔｉａｎａｄｊｏｉｎｔｏｆＡ,ｉ．ｅ．（Ａ）Ｔ,ｗｈｅｒｅＡｉｓｔｈｅｅｎｔｒｙｗｉｓｅｃｏｎｊｕｇａｔｅｏｆＡ,ａｎｄＡＴ,ｔｈｅｔｒａｎｓｐｏｓｅｏｆＡ;ＡＢｆｏｒｔｈｅｄｉｒｅｃｔｓｕｍｏｆｍａｔｒｉｃｅｓＡａｎｄＢ．ＬｅｔＸ∈Ｍｎｂｅｔ…     

构造法简证一类三角不等式

１９９６年,周永良先生在全国第三届初等数学研究学术交流会论文集中提出如下三角不等式在锐角三角形ＡＢＣ中,有ｃｏｓ（Ｂ－Ｃ）ｃｏｓＡ＋ｃｏｓ（Ｃ－Ａ）ｃｏｓＢ＋ｃｏｓ（Ａ－Ｂ）ｃｏｓＣ≥６（１）ｃｏｓＡｃｏｓ（Ｂ－Ｃ）＋ｃｏｓＢｃｏｓ（Ｃ－Ａ）＋ｃｏｓ...     

自反B空间中可测集值随机过程的可选（可料）投影

设Ｘ是可分自反Ｂａｎａｃｈ空间，（Ｆ，Ａ，ｔ∈Ｒ＋）是取值于Ｘ中闭凸集的可测集值随机过程，该文证明了，若任给停时Ｔ，ＦＴＩ（Ｔ〈∞）关于ＡＴ（相应地关于Ａ）是σ一可积，则（Ｆ，ｔ∈Ｒ＋）必存在可选（相应地可料）投影过程。     

A BRANCH－AND－BOUND ALGORITHM IN THE TOURING－PATH PROBLEM AND ITS COMPUTER IMPLEMENTATION

ＡＢＲＡＮＣＨ－ＡＮＤ－ＢＯＵＮＤＡＬＧＯＲＩＴＨＭＩＮＴＨＥＴＯＵＲＩＮＧ－ＰＡＴＨＰＲＯＢＬＥＭＡＮＤＩＴＳＣＯＭＰＵＴＥＲＩＭＰＬＥＭＥＮＴＡＴＩＯＮＸｕＸｕｓｏｎｇ（徐绪松）（ＳｃｈｏｏｌｏｆＭａｎａｇｅｍｅｎｔ，ＷｕｈａｎＵｎｉｖｅｒｓｉｔ...     

A J-CONVEX SUBSET WHICH IS NOT PSH-CONVEX

ＡＪ－ＣＯＮＶＥＸＳＵＢＳＥＴＷＨＩＣＨＩＳＮＯＴＰＳＨ－ＣＯＮＶＥＸＢｕＳｈａｎｇｑｕａｎ（步尚全）（Ｄｅｐａｒｔｍｅｎｔｏｆａｐｐｌｉｅｄｍａｔｈｅｍａｔｉｃｓ，ＴｓｉｎｇｈｕａＵｎｉｖｅｒｓｉｔｙ，Ｂｅｉｊｉｎｇ１０００８４，Ｃｈｉｎａ）Ａｂｓ...     

A MATHEMATICAL MODEL OF A CLASS OF AUTOCATALYTIC CHEMICAL REACTION

Ｂｙｆｏｌｏｗｉｎｇｍｅｃｈａｎｉｓｍ：ＡａαＸ，ＣｃαＹ，Ｂ＋ＸｂαＹ＋Ｄ，Ｘ＋Ｙｅα３Ｙ，ＹｆαＦ，ＸλαＦ，ａｒｔｉｃｌｅ［１］ｏｂｔａｉｎｅｄｔｈｅｆｏｌｏｗｉｎｇｓｙｓｔｅｍ：ｘ＝ａ－ｂｘ－ｅｘｙ２－λｘ，ｙ＝ｂｘ＋ｅｘｙ２－ｆｙ＋ｃ．（１...     

Hadamard积和酉不变范数不等式

设Ｍｎ，ｍ是ｎ×ｍ复矩阵空间，Ｍｎ≡Ｍｎ，ｎ．对于Ｈｅｒｍｉｔｅ阵Ｇ，Ｈ∈Ｍｎ，ＧＨ表示Ｇ－Ｈ半正定．记Ａ和Ｂ的Ｈａｄａｍａｒｄ积为ＡＢ．本文证明了若Ａ，Ｂ∈Ｍｎ正定，而Ｘ，Ｙ∈Ｍｎ，ｍ任意，则（ＸＡ－１Ｘ）（ＹＢ－１Ｙ）（ＸＹ）（ＡＢ）－１（ＸＹ），ＸＡ－１Ｘ＋ＹＢ－１Ｙ（Ｘ＋Ｙ）（Ａ＋Ｂ）－１（Ｘ＋Ｙ）．这推广和统一了一些现存的结果．设‖·‖为任意酉不变范数，Ｉ是单位矩阵．本文还证明了对于Ｘ∈Ｍｎ，ｍ和Ａ∈Ｍｎ，Ｂ∈Ｍｍ，若ＡＩ，ＢＩ，则函数ｆ（ｐ）＝‖ＡｐＸ＋ＸＢｐ‖在［０，∞）上单调递增．     

体上的矩阵方程AX—XB=C与AX±XB=±C

设Ｆ和Ω分别是一个任意的体和一个具有对合反自同构的有限维中心代数且ｃｈａｒΩ≠２．本研究体上的下列矩阵方程：ＡＸ－ＸＢ＝Ｃ，（１）ＡＸ－ＸＢ＝Ｃ，（２）ＡＸ＋ＸＢ＝－Ｃ（３）分别给出了在Ω上（１）有一般解，（２）自共轭解及（３）有斜自共轭解的充要条件，并将Ｗ．Ｅ．Ｒｏｔｈ的相似定理推广到了任意的体Ｆ上。     

ON HERMITIAN POSITIVE DEFINITE SOLUTION OF NONLINEAR MATRIX EQUATION X＋A^＊X^-2A=Q

Based on the fixed-point theory, we study the existence and the uniqueness of the maximal Hermitian positive definite solution of the nonlinear matrix equation X＋A^＊X^-2A=Q, where Q is a square Hermitian positive definite matrix and A＊ is the conjugate transpose of the matrix A. We also demonstrate some essential properties and analyze the sensitivity of this solution. In addition, we derive computable error bounds about the approximations to the maximal Hermitian positive definite solution of the nonlinear matrix equation X＋A^＊X^-2A=Q. At last, we further generalize these results to the nonlinear matrix equation X＋A^＊X^-nA=Q, where n≥2 is a given positive integer.     

New upper solution bounds of the discrete algebraic Riccati matrix equation

In this note, we present upper matrix bounds for the solution of the discrete algebraic Riccati equation (DARE). Using the matrix bound of Theorem 2.2, we then give several eigenvalue upper bounds for the solution of the DARE and make comparisons with existing results. The advantage of our results over existing upper bounds is that the new upper bounds of Theorem 2.2 and Corollary 2.1 are always calculated if the stabilizing solution of the DARE exists, whilst all existing upper matrix bounds might not be calculated because they have been derived under stronger conditions. Finally, we give numerical examples to demonstrate the effectiveness of the derived results.     

矩阵方程X+A~*X~(-n)A=P的Hermite正定解及其扰动分析

考虑非线性矩阵方程X A~*X~(-n)A=P,其中A是m阶非奇异复矩阵,P是m阶Hermite正定矩阵.本文利用不动点理论讨论了该方程Hermite正定解的存在性及包含区间,给出了极大解的性质及求极大,极小解的迭代算法.研究了极大解的扰动问题,利用微分等方法获得了两个新的一阶扰动界,并给出数值例子对所得结果进行了比较说明.     

Approximate Solutions and Error Bounds for Solving Matrix Differential Equations Without Increasing the Dimension of the Problem

In this paper we present a method for solving initial value problems related to second order matrix differential equations. This method is based on the existence of a solution of a certain algebraic matrix equation related to the problem, and it avoids the increase of the dimension of the problem for its resolution. Approximate solutions, and their error bounds in terms of error bounds for the approximate solutions of the algebraic problem, are given.     

Solution Bounds of the Continuous and Discrete Lyapunov Matrix Equations

A unified approach is proposed to solve the estimation problem for the solution of continuous and discrete Lyapunov equations. Upper and lower matrix bounds and corresponding eigenvalue bounds of the solution of the so-called unified algebraic Lyapunov equation are presented in this paper. From the obtained results, the bounds for the solutions of continuous and discrete Lyapunov equations can be obtained as limiting cases. It is shown that the eigenvalue bounds of the unified Lyapunov equation are tighter than some parallel results and that the lower matrix bounds of the continuous Lyapunov equation are more general than the majority of those which have appeared in the literature.     

Sensitivity analysis of the algebraic Riccati equations

Summary. In this paper, some sharp perturbation bounds for the Hermitian positive semi-definite solution to an algebraic Riccati equation are developed. A further analysis for these bounds is done. This analysis shows that there is, presumably, some intrinsic relation between the sensitivity of the solution to the algebraic Riccati equation and the distance of the spectrum of the closed-loop matrix from the imaginary axis. Received December 16, 1994     

On a relaxed SOR-method applied to nonsymmetric linear systems

Nonsymmetric linear systems are by far not as common as syemmtric ones but nevertheless systems with nonsymmetric matrices appear, e. g., in the numerical solution of the biharmonic equation, the computation of splines or the solution of some special integral equations. The SOR-method applied to linear systems X = BX + C with skew-symmetric matrix B is studied. Described is a region in the complex plane which contains the eigenvalues of the SOR-operator. Using this information a relaxed SOR-method is proposed; bounds for the spectral radius of the iteration operator are derived. The advantage is that the values of the corresponding iteration parameters can be directly calculated from the norm of the given matrix.     

Realistic Eigenvalue Bounds for the Galerkin Mass Matrix

In time-dependent finite-element calculations, a mass matrixnaturally arises. To avoid the solution of the correspondingalgebraic equation system at each time step, ‘mass lumping’is widely used, even though this pragmatic diagonalization ofthe mass matrix often reduces accuracy. We show how the unassembled form of finite-element equationscan be used to establish (in an element-by-element manner) realisticupper and lower bounds on the eigenvalues of the fully consistentmass matrix when preconditioned by its diagonal entries. Weuse this technique to give specific results for a number ofdifferent types of finite elements in one, two, and three dimensions.The bounds are found by independent calculations on the elements,and, for certain element types, are independent of mesh irregularity.We give examples of when some of the bounds are attained. These results indicate that the preconditioned conjugate-gradientmethod is appropriate and very rapid for the solution of Galerkinmass-matrix equations.     

Estimates for the inverse elements of tridiagonal matrices

In this work, new upper and lower bounds for the inverse entries of the tridiagonal matrices are presented. The bounds improve the bounds in D. Kershaw [Inequalities on the elements of the inverse of a certain tridiagonal matrix, Math. Comput. 24 (1970) 155–158], P.N. Shivakumar, C.X. Ji [Upper and lower bounds for inverse elements of finite and infinite tridiagonal matrices, Linear Algebr. Appl. 247 (1996) 297–316], R. Nabben [Two-sided bounds on the inverse of diagonally dominant tridiagonal matrices, Linear Algebr. Appl. 287 (1999) 289–305] and R. Peluso, T. Politi [Some improvements for two-sided bounds on the inverse of diagonally dominant tridiagonal matrices, Linear. Algebr. Appl. 330 (2001) 1–14].     

求矩阵方程AXB+CXD=F的中心对称最小二乘解的迭代算法

该文建立了求矩阵方程AXB+CXD=F的中心对称最小二乘解的迭代算法.使用该算法不仅可以判断该矩阵方程的中心对称解的存在性,而且无论中心对称解是否存在,都能够在有限步迭代计算之后得到中心对称最小二乘解.选取特殊的初始矩阵时,可求得极小范数中心对称最小二乘解.同时,也能给出指定矩阵的最佳逼近中心对称矩阵.