Narrow Results

The derivation of mean first passage times in Markov chains involves the solution of a family of linear equations. By exploring the solution of a related set of equations, using suitable generalized inverses of the Markovian kernel I - P, where P is the transition matrix of a finite irreducible Markov chain, we are able to derive elegant new results for finding the mean first passage times. As a by-product we derive the stationary distribution of the Markov chain without the necessity of any further computational procedures. Standard techniques in the literature, using for example Kemeny and Snell's fundamental matrix Z, require the initial derivation of the stationary distribution followed by the computation of Z, the inverse of I - P + eπ^T where e^T = (1, 1, …, 1) and π^T is the stationary probability vector. The procedures of this paper involve only the derivation of the inverse of a matrix of simple structure, based upon known characteristics of the Markov chain together with simple elementary vectors. No prior computations are required. Various possible families of matrices are explored leading to different related procedures.

In a semiprime ring, von Neumann regular elements are determined by their inner inverses. In particular, for elements $a,b$ of a von Neumann regular ring $R$, $a=b$ if and only if $I(a)=I(b)$, where $I(x)$ denotes the set of inner inverses of $x\in R$. We also prove that, in a semiprime ring, the same is true for reflexive inverses.

In this paper, we investigate inclusion properties in a ∗-ring. It is proven that for two $\{1,3\}$-invertible elements $a$ and $b$ in a ∗-ring, $a=b$ if and only if $a\{1,3\}=b\{1,3\}$. For two $\{1,2,3\}$-invertible elements $a$ and $b$, $a\{1,2,3\}=b\{1,2,3\}$ if and only if $(a-b)Ra=(a-b)Rb=0$. Moreover, we give a characterization of $\{3\}$-inverses, and the absorption law for $\{3\}$-inverses is studied.

We establish the formulas of the maximal and minimal ranks of the common solution of certain linear matrix equations A₁X = C₁, XB₂ = C₂, A₃XB₃ = C₃ and A₄XB₄ = C₄ over an arbitrary division ring. Corresponding results in some special cases are given. As an application, necessary and sufficient conditions for the invariance of the rank of the common solution mentioned above are presented. Some previously known results can be regarded as special cases of our results.

In this paper, we consider the set of all generalized inverses of an $m\times n$ matrix $A$, denoted by $A\{1\}$ and define two binary operations, ⊕ and ⋆ with some properties which offer the structure of a semi ring to $A\{1\}$. The compatibility of some partial orders like Sussman’s order, Conrad’s order, etc. is also checked which make it an ordered matrix semiring.

This survey is articulated around two major axis. The first one concerns the Kaplansky problem; the history of the problem and several results are presented. The second one concerns some new preserver problems (concerning the generalized inverse, Fredholm or semi-Fredholm operators). The common point of these results is that they are interesting only in the infinite dimensional situation. Several open questions are mentioned over all the paper.