Jean Mairesse

Bio: Jean Mairesse is an academic researcher from University of Paris. The author has contributed to research in topics: Matching (graph theory) & Petri net. The author has an hindex of 22, co-authored 90 publications receiving 1491 citations. Previous affiliations of Jean Mairesse include Centre national de la recherche scientifique & Paris Diderot University.

Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture

Abstract: Given an Iterated Function System (IFS) of topical maps verifying some conditions, we prove that the asymptotic height optimization problems are equivalent to finding the extrema of a continuous functional, the average height, on some compact space of measures. We give general results to determine these extrema, and then apply them to two concrete problems. First, we give a new proof of the theorem that the densest heaps of two Tetris pieces are sturmian. Second, we construct an explicit counterexample to the Lagarias-Wang finiteness conjecture.

Modeling and analysis of timed Petri nets using heaps of pieces

Abstract: The authors show that safe timed Petri nets can be represented by special automata over the (max, +) semiring, which compute the height of heaps of pieces. This extends to the timed case the classical representation a la Mazurkiewicz of the behavior of safe Petri nets by trace monoids and trace languages. For a subclass including all safe free-choice Petri nets, we obtain reduced heap realizations using structural properties of the net (covering by safe state machine components). The authors illustrate the heap-based modeling by the typical case of safe jobshops. For a periodic schedule, the authors obtain a heap-based throughput formula, which is simpler to compute than its traditional timed event graph version, particularly if one is interested in the successive evaluation of a large number of possible schedules.

Deciding unambiguity and sequentiality from a finitely ambiguous max-plus automaton

Abstract: Finite automata with weights in the max-plus semiring are considered. The main result is: it is decidable whether a series that is recognized by a finitely ambiguous max-plus automaton is unambiguous, or is sequential. Furthermore, the proof is constructive. A collection of examples is given to illustrate the hierarchy of max-plus series with respect to ambiguity.

Reversibility and Further Properties of FCFS Infinite Bipartite Matching

Abstract: The model of FCFS infinite bipartite matching was introduced in [10]. In this model there is a sequence of items that are chosen i.i.d. from C = {c1,. .. , cI } and an independent sequence of items that are chosen i.i.d. from S = {s1,. .. , sJ }, and a bipartite compatibility graph G between C and S. Items of the two sequences are matched according to the compatibility graph, and the matching is FCFS, each item in the one sequence is matched to the earliest compatible unmatched item in the other sequence. In [4] a Markov chain associated with the matching was analyzed, a condition for stability was verified, a product form stationary distribution was derived and and the rates rc i ,s j of matches between compatible types ci and sj were calculated. In the current paper we present further results on this model. We use a Loynes type scheme to show that if the system is stable there is a unique matching of the sequence over all the integers. We show dynamic reversibility: if in every matched pair we exchange the positions of the items the resulting permuted sequences are again independent and i.i.d., and the matching between them is FCFS in reversed time. We use these results to derive stationary distributions of various Markov chains associated with the model. We also calculate the distribution of the link lengths between matched items.

Stability of the stochastic matching model

Abstract: We introduce and study a new model that we call the matching model. Items arrive one by one in a buffer and depart from it as soon as possible but by pairs. The items of a departing pair are said to be matched. There is a finite set of classes 𝒱 for the items, and the allowed matchings depend on the classes, according to a matching graph on 𝒱. Upon arrival, an item may find several possible matches in the buffer. This indeterminacy is resolved by a matching policy. When the sequence of classes of the arriving items is independent and identically distributed, the sequence of buffer-content is a Markov chain, whose stability is investigated. In particular, we prove that the model may be stable if and only if the matching graph is nonbipartite.

Convergence of Probability Measures

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Applied Probability and Queues

Abstract: (1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

Stability Criteria for Switched and Hybrid Systems

Abstract: The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving them in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability and also represent problems in which significant progress has been made. We also comment on the inherent difficulty in determining stability of switched systems in general, which is exemplified by NP-hardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell-time requirements and state-dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems are reviewed. We briefly comment on the classical Lur'e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.

Flows in Networks

Abstract: In this chapter, you will see how the simplex method simplifies when it is applied to a class of optimization problems that are known as “network flow models.” You will also see that if a network flow model has “integer-valued data,” the simplex method finds an optimal solution that is integer-valued.

Probability on Trees and Networks

Abstract: Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.