Markov two-components processes

We propose Markov two-components processes (M2CP) as a probabilistic model of asynchronous systems based on the trace semantics for concurrency. Considering an asynchronous system distributed over two sites, we introduce concepts and tools to manipulate random trajectories in an asynchronous framework: stopping times, an Asynchronous Strong Markov property, recurrent and transient states and irreducible components of asynchronous probabilistic processes. The asynchrony assumption implies that there is no global totally ordered clock ruling the system. Instead, time appears as partially ordered and random. We construct and characterize M2CP through a finite family of transition matrices. M2CP have a local independence property that guarantees that local components are independent in the probabilistic sense, conditionally to their synchronization constraints. A synchronization product of two Markov chains is introduced, as a natural example of M2CP.Comment: 34 page

Faster algorithms for minimum path cover by graph decomposition

Minimum-cost minimum path cover is a graph-theoretic problem with an application in gene sequencing problems in bioinformatics. This thesis studies decomposing graphs as a preprocessing step for solving the minimum-cost minimum path cover problem. By decomposing graphs, we mean splitting graphs into smaller pieces. When the graph is split along the maximum anti-chains of the graph, the solution for the minimum-cost minimum path cover problem can be computed independently in the small pieces. In the end all the partial solutions are joined together to form the solution for the original graph. As a part of our decomposition pipeline, we will introduce a novel way to solve the unweighted minimum path cover problem and with that algorithm, we will also obtain a new time/space tradeoff for reachability queries in directed acyclic graphs. This thesis also includes an experimental section, where an example implementation of the decomposition is tested on randomly generated graphs. On the test graphs we do not really get a speedup with the decomposition compared to solving the same instances without the decomposition. However, from the experiments we get some insight on the parameters that affect the decomposition's performance and how the implementation could be improved

M\"obius Functions and Semigroup Representation Theory II: Character formulas and multiplicities

We generalize the character formulas for multiplicities of irreducible constituents from group theory to semigroup theory using Rota's theory of M\"obius inversion. The technique works for a large class of semigroups including: inverse semigroups, semigroups with commuting idempotents, idempotent semigroups and semigroups with basic algebras. Using these tools we are able to give a complete description of the spectra of random walks on finite semigroups admitting a faithful representation by upper triangular matrices over the complex numbers. These include the random walks on chambers of hyperplane arrangements studied by Bidigare, Hanlon, Rockmere, Brown and Diaconis. Applications are also given to decomposing tensor powers and exterior products of rook matrix representations of inverse semigroups, generalizing and simplifying earlier results of Solomon for the rook monoid.Comment: Some minor typos corrected and references update

Computing Equilibria of Semi-algebraic Economies Using Triangular Decomposition and Real Solution Classification

In this paper, we are concerned with the problem of determining the existence of multiple equilibria in economic models. We propose a general and complete approach for identifying multiplicities of equilibria in semi-algebraic economies, which may be expressed as semi-algebraic systems. The approach is based on triangular decomposition and real solution classification, two powerful tools of algebraic computation. Its effectiveness is illustrated by two examples of application.Comment: 24 pages, 5 figure