A perturbation formalism is presented which shows how the stationary distribution and fundamental matrix of a Markov chain containing a single irreducible set of states change as the transition probabilities vary. Expressions are given for the partial derivatives of the stationary distribution and fundamental matrix with respect to the transition probabilities. Semi-group properties of the generators of transformations from one Markov chain to another are investigated. It is shown that a perturbation formalism exists in the multiple subchain case if and only if the change in the transition probabilities does not alter the number of, or intermix the various subchains. The formalism is presented when this condition is satisfied.
We call a quasi-adequate semi-group whose set of idempotents forms a left [right] quasi-normal band a left [right] semi-perfect abundant semi-group. After obtaining some characterization theorems of such quasi-adequate semi-groups, we establish a structure for left [right] semi-perfect abundant semi-groups of type W. Our results generalize and strengthen the results of El-Qallali and Fountain on quasi-adequate semi-groups.
<pre><!--StartFragment-->In this paper, we present how to use an interval arithmetic framework based on free algebra construction, in order to build better defined inclusion function for interval semi-group and for its associated vector space. One introduces the <span>psi</span>-algorithm, which performs set inversion of functions and exhibits some numerical examples developed with the python programming langage<!--EndFragment--></pre>.
Probabilist Set Inversion using Pseudo-Intervals Arithmetic
