H-空间与拟向量变分不等式

本在Hausdorff locally convex H-空间上建立了一个新的不动点定理；利用局部交性质，在Hausdorff locally convex H-空间上建立了相应的拟向量变分不等式．     

The inverses of Toeplitz band matrices

The elements of the inverse of a Toeplitz band matrix are given in terms ofthe solution of a difference equation. The expression for these elements is a quotient of determinants whose orders depend the number of nonzero superdiagonals but not on the order of the matrix. Thus, the formulae are particularly simple for lower triangular and lower Hessenberg Toeplitz matrices. When the number of nonzero superdiagonals is small, sufficient conditions on the solution of the abovementioned difference equation can be given to ensure that the inverse matrix is positive. If the inverse is positive, the row sums can be expressed in terms of the solution of the difference equation.     

Group inverses and Drazin inverses over Banach algebras

For a matrix over a complex commutative unital Banach algebra, necessary and sufficient conditions are given for the existence of its group inverse, and more generally, its Drazin inverses. The conditions are easy to check and explicit formulas for the inverses are provided. Some properties of the inverses and an application to operator theory are discussed. This note is a continuation of an earlier work of the author.     

H-空间中一类广义拟似变分不等式

在H 空间中 ,研究一类抽象形式的广义拟 似变分不等式φ( x , w , y ,x) ≥ 0 ,利用KKM技巧 ,通过作连续选择 ,得到了其解的存在性定理 ,该定理包含了近年来其他作者所做的结果作为特殊情形 .     

On Hermitian generalized inverses and positive semidefinite generalized inverses

The main aim of this paper is to investigate the Hermitian and positive semidefinite generalized inverses of a square matrix. First, we present some conditions for the existence of Hermitian and positive semidefinite generalized inverses. Further, expressions of these generalized inverses are given. Finally, we give two numerical examples to demonstrate our results.     

Planar wiener inverses

Let B(Z₁, z₂) be a power series and C(z₁, z₂) be the least mean-square inverse approximation of B among polynomials of a fixed order. This paper discusses conditions under which $\text{B}\text{?}\text{(z}{}_1\text{, z}{}_2\text{)}$, the least mean-square inverse polynomial approximation of C, does not vanish on the unit bicylinder.     

The Pseudo Drazin inverses in Banach Algebras

Let (A) be a complex Banach algebra and J be the Jacobson radical of(A).(1) We firstly show that a is generalized Drazin invertible in (A) if and only if a+J is generalized Drazin invertible in (A)/J.Then we prove that a is pseudo Drazin invertible in (A) if and only if a + J is Drazin invertible in (A)/J.As its application,the pseudo Drazin invertibility of elements in a Banach algebra is explored.(2) The pseudo Drazin order is introduced in (A).We give the necessary and sufficient conditions under which elements in (A) have pseudo Drazin order,then we prove that the pseudo Drazin order is a pre-order.     

On matrices over an arbitrary semiring and their generalized inverses

In this paper, we consider matrices with entries from a semiring S. We first discuss some generalized inverses of rectangular and square matrices. We establish necessary and sufficient conditions for the existence of the Moore–Penrose inverse of a regular matrix. For an m×n$m\times n$ matrix A  , an n×m$n\times m$ matrix P and a square matrix Q of order m, we present necessary and sufficient conditions for the existence of the group inverse of QAP   with the additional property that P(QAP)^#Q$P{(QAP)}^{\mathrm{\#}}Q$ is a {1,2}$\{1,2\}$ inverse of A  . The matrix product used here is the usual matrix multiplication. The result provides a method for generating elements in the set of {1,2}$\{1,2\}$ inverses of an m×n$m\times n$ matrix A starting from an initial {1} inverse of A  . We also establish a criterion for the existence of the group inverse of a regular square matrix. We then consider a semiring structure (_Mm×n(S),+,°)$({\mathbf{M}}_{m\times n}(\mathbf{S}),+,°)$ made up of m×n$m\times n$ matrices with the addition defined entry-wise and the multiplication defined as in the case of the Hadamard product of complex matrices. In the semiring (_Mm×n(S),+,°)$({\mathbf{M}}_{m\times n}(\mathbf{S}),+,°)$, we present criteria for the existence of the Drazin inverse and the Moore–Penrose inverse of an m×n$m\times n$ matrix. When S is commutative, we show that the Hadamard product preserves the Hermitian property, and provide a Schur-type product theorem for the product A°(CC^?)$A°(C{C}^{?})$ of a positive semidefinite n×n$n\times n$ matrix A   and an n×n$n\times n$ matrix C.     

The representation and approximations of outer generalized inverses

We present a unified representation theorem for the class of all outer generalized inverses of a bounded linear operator. Using this representation we develop a few specific expressions and computational procedures for the set of outer generalized inverses. The obtained result is a generalization of the well-known representation theorem of the Moore--Penrose inverse as well as a generalization of the well-known results for the Drazin inverse and the generalized inverse A_T,S ⁽²⁾. Also, as corollaries we get corresponding results for reflexive generalized inverses.     

Involutory functions and Moore-Penrose inverses of matrices in an arbitrary field

Let F be a field, and M be the set of all matrices over F. A function ? from M into M, which we write ?(A) = A^s for A∈M, is involutory if (1) (AB)^s = B^sA^s for all A, B in M whenever the product AB is defined, and (2) (A^s)^s = A for all A∈M. If ? is an involutory function on M, then A^s is n×m if A is m×n; furthermore, Rank A = Rank A^s, the restriction of ? to F is an involutory automorphism of F, and (aA + bB)^s = a^sA^s + b^sB^s for all m×n matrices A and B and all scalars a and b. For an A∈M, an Ã∈M is called a Moore-Penrose inverse of A relative to ? if (i) AÃA = A, ÃAÃ = Ã and (ii) (AÃ)^s = AÃ, (ÃA)^s = ÃA. A necessary and sufficient condition for A to have a Moore-Penrose inverse relative to ? is that Rank A = Rank AA^s = Rank A^sA. Furthermore, if an involutory function ? preserves circulant matrices, then the Moore-Penrose inverse of any circulant matrix relative to ? is also circulant, if it exists.     

Re-nnd generalized inverses

In this paper, we give some necessary and sufficient conditions for the existence of Re-nnd and nonnegative definite {1,3}$\{1,3\}$- and {1,4}$\{1,4\}$-inverses of a matrix A∈C^n×n$A\in {\mathbb{C}}^{n\times n}$ and completely described these sets. Moreover, we prove that the existence of nonnegative definite {1,3}$\{1,3\}$-inverse of a matrix A   is equivalent with the existence of its nonnegative definite {1,2,3}$\{1,2,3\}$-inverse and present the necessary and sufficient conditions for the existence of Re-nnd {1,3,4}$\{1,3,4\}$-inverse of A.     

Weak Drazin inverses

A new type of generalized inverse is defined which is a weakened form of the Drazin inverse. These new inverses are called (d)-inverses. Basic properties of (d)-inverses are developed. It is shown that (d)-inverses are often easier to compute than Drazin inverses and can frequently be used in place of the Drazin inverse when studying systems of differential equations with singular coefficients or when studying Marcov chains.     

The distribution of patterns of inverses modulo a prime

We investigate the distribution of the numbers x∈[1,p] for which all lie in a subset of the set of multiplicative inverses modulo a prime p. Here the a_i are integers coprime to p and the numbers are distinct .     

The Drazin inverses of sum and difference of idempotents

In this note, the Drazin inverses of sum and difference of idempotents on a Hilbert space are established under some conditions.     

Nonnegative Drazin inverses

A characterization of nonnegative matrices which have a nonnegative Drazin inverse is given. A necessary and sufficient condition for a real matrix to have a nonnegative Drazin inverse is also presented.