关于完全正矩阵的几点注记(英文)

本文给出了一个 n×n非负、对称、弱对角占优矩阵 A为完全正的一个充分条件 .我们还给出了较好的算法 ,用以获得关于矩阵 A(当 A为完全正时 )的分解指数的一个上界 .     

中心化子的刻画

令X为实或复域F上的Banach空间,■为X上的标准算子代数,I是■的单位元.设Φ:■→■是可加映射.本文证明了,如果有正整数m,n,使得Φ满足条件Φ(A~(m+n+1))-A~mΦ(A)A~n∈FI对任意A成立,则存在λ∈F,使得对所有的A∈■,都有Φ(A)=λA.同样的结果对于自伴算子空间上的可加映射也成立.此外,本文还给出了中心素代数上满足条件(m+n)Φ(AB)-mAΦ(B)-nΦ(A)B∈FI的可加映射Φ的完全刻画.     

一类模糊双向联想记忆网络极限环的最大长度

讨论了一类模糊双向联想记忆网络极限环的最大长度.首先,A是强连通布尔矩阵时,极限环的最大长度是A的周期指数per(A);其次,A具有形式(☆)时,若gcd(per(A1),per(A2))=1,则极限环的最大长度为max(per(A1),per(A2));若gcd(per(A1),per(A2))≥2,则其最大长度为lcm(per(A1),per(A2));最后对其进行推广.     

符号空间有限型子转移混沌集的Hausdorff测度与Parry测度

设A是一个每列至少有二个元素为1的不可约0，1方阵，(∑4，σA)为由A所决定的符号空间有限型子转移．在∑A上定义一个与其拓扑相容的度量d使得(∑A，d)的Hausdorff维数为1．若G是H^-可测的σA的Li—Yorke混沌集，则H^1(C)＝0；若A是本原的，则存在一个σA的有限型混沌集S使得H^1(S)＝1，其中H^1为1-维的Hausdorff测度．     

Strong conjugation actions induced by flags of exact Borel subalgebras

Let A be a quasi-hereditary algebra with a strong exact Borel subalgebra. It is proved that for any standard semisimple subalgebra T there exists an exact Borel subalgebra B of A such that T is a maximal semisimple subalgebra of B. It is shown that the maximal length of flags of exact Borel subalgebras of A is the difference of the radium and the rank of Grothendic group of A plus 2. The number of conjugation-classes of exact Borel subalgebras is 1 if and only if A is basic; the number is 2 if and only if A is semisimple. For all other cases, this number is 0 or no less than 3. Furthermore, it is shown that all the exact Borel subalgebras are idempotent-conjugate to each other, that is, for any exact Borel subalgebras B and C of A, there exists an idempotent e of A, and an invertible element u of A, such that eBe = u^-1eCeu.     

Inconsistencies and S.O.R. Convergence for the Discrete Neumann Problem

The behaviour of S.O.R. iterations for linear equations AU =Bis described for the case when A is singular. Small inconsistenciesmay arise in practical application, for example in systems derivedfrom Neumann problems. In that case the S.O.R. iterations donot converge. A simple transformation is presented under whichthe iterations converge to an approximate solution of AU = B,provided that the singularity of A is of a simple type. A practicalway of measuring the appropriate convergence rate is also described.For problems with property (A) and consistent ordering the optimumacceleration parameter is unaffected by the simple singularityof A. The behaviour of the iterations when A has singularitiesof general type is also described.     

自伴标准算子代数上强保持k-斜交换性映射的刻画

令H是维数大于2的复Hilbert空间,A是H上自伴标准算子代数.对于给定的正整数k≥1,H上算子A与B的k-斜交换子递推地定义为_*[A,B]_k=_*[A,_*[A,B]_k-1],其中_*[A,B]₀=B,_*[A,B]₁=AB-BA^*.设k≥4,φ是A上的值域包含所有一秩投影的映射.本文证明了φ满足_*[φ（A）,φ（B）]_k=_*[A,B]_k对任意A,B∈A都成立的充分必要条件是φ（A）=A对任意A∈A都成立,或φ（A）=-A对任意A∈A都成立.当k是偶数时后一情形不出现.     

ON HERMITIAN POSITIVE DEFINITE SOLUTIONS OF MATRIX EQUATION X-A^＊X^-2 A=I

The Hermitian positive definite solutions of the matrix equation X-A^＊X^-2 A=I are studied. A theorem for existence of solutions is given for every complex matrix A. A solution in case A is normal is given. The basic fixed point iterations for the equation are discussed in detail. Some convergence conditions of the basic fixed point iterations to approximate the solutions to the equation are given.     

A stochastic optimal control approach to a class of production and inventory problems

A comprehensible and unified system control approach is presented to solve a class of production/inventory smoothing problems. A nonstationary, non-Gaussian, finite-time linear optimal solution with an attractive computation scheme is obtained for a general quadratic and linear cost structure. A complete solution to a classical production/inventory control problem is given as an example. A general solution to the discrete-time optimal regulator with arbitrary but known disturbance is provided and discussed in detail. A computationally attractive closed-loop suboptimal scheme is presented for problems with constraints or nonquadratic costs. Implementation and interpretation of the results are discussed.     

关于无穷维算子的有限维最佳逼近

A method of approaching to the infinite-dimensional linear operators by the finite-dimensional operators is discussed. It is shown that,for every infinite-dimensional operator A and every natural number n,there exists an n-dimensional optimal approximation to A. The norm error is found and the necessary and sufficient condition for such n-dimensional optimal approximations to be unique is obtained.     

A Class of Constrained Inverse Eigenproblem and Associated Approximation Problem for Symmetric Reflexive Matrices

Let S∈Rn×n be a symmetric and nontrival involution matrix. We say that A∈E R n×n is a symmetric reflexive matrix if AT = A and SAS = A. Let S R r n×n(S)={A|A= AT,A = SAS, A∈Rn×n}. This paper discusses the following two problems. The first one is as follows. Given Z∈Rn×m (m < n),∧= diag(λ1,...,λm)∈Rm×m, andα,β∈R withα<β. Find a subset (?)(Z,∧,α,β) of SRrn×n(S) such that AZ = Z∧holds for any A∈(?)(Z,∧,α,β) and the remaining eigenvaluesλm 1 ,...,λn of A are located in the interval [α,β], Moreover, for a given B∈Rn×n, the second problem is to find AB∈(?)(Z,∧,α,β) such that where ||.|| is the Frobenius norm. Using the properties of symmetric reflexive matrices, the two problems are essentially decomposed into the same kind of subproblems for two real symmetric matrices with smaller dimensions, and then the expressions of the general solution for the two problems are derived.     

Closed Prime Ideals in Algebras with Semiprime Multiplication Algebra

Let A be an algebra whose multiplication algebra M(A) is semiprime. We prove that, except in an exceptional case, the proper closed prime ideals of A are the maximal closed ideals of A, for the closure operations π and ?. In fact, these sets agree for both closures. The same can be said in M(A) for the closure operations π and ?′. Moreover, we establish the relationships between the proper closed prime ideals of A and the ones of the algebras M(A),U and A/U, for a given ideal U of A.     

算子代数上的Lie可导映射

设A为有单位且包含一非平凡幂等元的环,M为A双模.称δ:A→M为Lie可导映射(无可加或连续假设),若δ([A,B])=[δ(A),B]+[A,δ(B)],(?)A,B∈A.在一定条件下该文证明了Lie可导映射δ具有形式δ(A)=τ(A)+f(A),其中r:A→M是可加导子,f是从A到M的中心且满足f([A,B])=0,(?)A,B∈A的映射.由此刻画了因子von Neuamnn代数和套代数上的Lie可导映射.     

有限维张量积空间上的强可分算子

设F是一个域,a∈F~nF~m.若存在h∈F~m,k∈F~m,使得a=hk,则称a是可分的.空间F~nF~m上的线性算子A称为是强可分的,是指x∈F~nF~m,x可分Ax可分.本文证明了F~nF~n上的线性算子A是强可分的当且仅当存在F~n上的线性双射A_1与A_2,使得A=A_1A_2或A=A_1~T A_2;证明了F~nF~m(n≠m)上线性算子A是强可分的当且仅当存在F~n与F~m上的线性双射A_1与A_2,使得A=A_1A_2.最后,给出了可分算子、强可分算子和秩1保持映射之间的关系.     

亚正定矩阵的Kronecker积

研究亚正定矩阵kronecker积的亚正定性,得到了一个充要条件,同时得到Hadamard积亚正定性的一个充要条件.     

实矩阵的Hadamard积的一些性质

对于任意的n阶实矩阵A,给出了A(A*)T与A的奇异性间的关系,指出了A(A*)T的行和与列和为矩阵A的行列式|A|,最后给出了矩阵类A(A*)T与n阶方阵的一个等价类的一一对应关系.     

A formal Frobenius theorem and argument shift

A formal Frobenius theorem, which is an analog of the classical integrability theorem for smooth distributions, is proved and applied to generalize the argument shift method of A. S. Mishchenko and A. T. Fomenko to finite-dimensional Lie algebras over any field of characteristic zero. A completeness criterion for a commutative set of polynomials constructed by the formal argument shift method is obtained.     

Digraphs and Inclusion Intervals of Brualdi-type for Singular Values

A complex matrix A is said to be a matrix realization of the digraph D if D is the associated digraph of A, and A is said to have the property B if every singular value of A is contained in the union of Brualdi-type intervals. A digraph D is said to be a forcible B-digraph if every matrix realization of D has the property B. In this paper, we give a sufficient condition for a matrix to have the property B and characterize the forcible B-digraphs.     

谱约束下的矩阵最佳逼近问题

在对电学、光学或自动控制的线性系统进行测试或复原时,由于原有资料不全或者要对已有资料进行核验等原因,曾提出如下的数学问题. 根据对线性系统的某种矩阵A(例如传递函数矩阵)的n~2个元素     

p-亚正常算子的本质谱与拟相似

本文证明了每一个ｐ－亚正常算子Ａ,都相应存在一个亚正常算子 ,使得Ａ与 有相同的闭值域点、相同的本质谱和谱．由此推出如果Ａ是ｐ－亚正常算子,Ｂ是任一有界线性算子,若存在有界线性算子Ｘ有稠值域,使得ＸＢ＝ＡＸ,则σ（Ａ）（Ｂ）此外还证明了,如果Ａ是ｐ－亚正常算子且 Ｒ（Ａ）闭或ＫｅｒＡ＝ＫｅｒＡ*, Ｂ是任一有界线性算子,Ａ与Ｂ拟相似,则ｅ（Ａ）（Ｂ）．