Michel Morvan

Weak-order extensions of an order

In this paper, at first we describe a digraph representing all the weak-order extensions of a partially ordered set and algorithms for generating them. Then we present a digraph representing all of the minimal weak-order extensions of a partially ordered set. This digraph also implies generation algorithms. Finally, we prove that the number of weak-order extensions of a partially ordered set is a comparability invariant, whereas the number of minimal weak-order extensions of a partially ordered set is not a comparability invariant.

About the Dynamics of Some Systems Based on Integer Partitions and Compositions

In this paper, we study the dynamics of sand grains falling in sand piles. Usually sand piles are characterized by a decreasing integer partition and grain moves are described in terms of transitions between such partitions. We study here four main transition rules. The more classical one, introduced by Brylawski [5] induces a lattice structure L B (n) (called dominance ordering) between decreasing partitions of a given integer n. We prove that a more restrictive transition rule, called SPM rule, induces a natural partition of L B (n) in suborders, each one associated to a fixed point for SPM rule. In the second part, we generalize the SPM rule and obtain other lattice structure parametrized by some θ: L(n, θ), which form for θ ∈ [n, −n + 2, n] a decreasing sequence of lattices. For each θ, we characterize the fixed point of L(n, θ) and give the value of its maximal sized chain’s lenght. We also note that L(n, −n + 2) is the lattice of all compositions of n. In the last section, we extend the SPM rule in another way and obtain a model called Chip Firing Game [8]. We prove that this new model has a structure of lattice.

Simplicial elimination schemes, extremal lattices and maximal antichain lattices

We compared the clinical picture of sarcoidosis in patients diagnosed at Mjölbolsta hospital in Finland in 1955-1987 with those diagnosed in Sapporo in 1964-1988. The female:male ratios showed a slight female predominance. The mean age (SD) at diagnosis was 41.5 (13.0) years at Mjölbolsta and 30.0 (15.4) years in Sapporo. In both series, half of the patients had been detected at mass x-ray surveys. The presenting symptoms varied considerably among the symptomatic patients. At Mjölbolsta hospital, 189 patients (33%) had cough, 21% fever, 21% general malaise, 18% dyspnoea, 18% erythema nodosum, 16% joint pain and only 27 patients (5%) had eye symptoms. In Sapporo, 245 patients (41%) had eye symptoms, 18 (4%) had enlarged peripheral lymph nodes, 14 (3%) had cough, 10 (2%) had fever. Erythema nodosum did not occur as a presenting symptom in Sapporo. The chest radiographs showed bilateral hilar lymphadenopathy (BHL, stage I) in 48% of the Mjölbolsta patients and in 57% of the Sapporo patients. Stage II lesions were seen in 39% and 20%, and stage III lesions in 12% and 5% respectively. Only 1% had a normal chest radiograph at Mjölbolsta hospital as compared with 18% in Sapporo. The Sapporo patients were more obstructive but the proportion of smokers was also higher. No difference in diffusion capacity was seen.

Weak-Order Extensions of an Order

In this paper, at first we describe a graph representing all the weak-order extensions of a partially ordered set and an algorithm generating them. Then we present a graph representing all of the minimal weak-order extensions of a partially ordered set, and implying a generation algorithm. Finally, we prove that the number of weak-order extensions of a partially ordered set is a comparability invariant, whereas the number of minimal weak-order extensions of a partially ordered set is not a comparability invariant.

Parallel interval order recognition and construction of interval representations

Abstract Parallel algorithms for recognizing and representing interval orders are proposed for different models of parallel random access machines (PRAM). The algorithms accept as input a transitively-closed directed graph with N nodes and M edges. They run in time O(log N ) with O( N + M ) processors and O( N + M ) space and in constant time with O( N 2 ) processors and O( N 2 ) space depending on the data structure and the PRAM model used. Optimal probabilistic algorithms for PRAM are also presented as well as algorithms for distributed-memory machines.

TRANSITIVE CLOSURE IN PARALLEL ON A LINEAR NETWORK OF PROCESSORS

In this paper, we propose a linear time parallel algorithm (in the number of edges of the transitive closure) that computes the transitive closure of a directed graph on a linear network of n processors. The underlying architecture is a linear network of processors with neighbouring communications, where the number of processors is equal to the number of vertices of the graph.