An interval doubling is a constructive operation which applies on a poset P and an interval I of P and constructs a new "bigger" poset P = P [I ] by replacing in P the interval I with its direct product with the two-element lattice. The main contribution of this paper is to prove that finite Coxeter lattices are bounded, i.e., that they can be constructed starting with the two-element lattice by a finite series of interval doublings.The boundedness of finite Coxeter lattices strengthens their algebraic property of semidistributivity. It also brings to light a relation between the interval doubling construction and the reflections of Coxeter groups.Our approach to the question is somewhat indirect. We first define a new class HH of lattices and prove that every lattice of HH is bounded. We then show that Coxeter lattices are in HH and the theorem follows. Another result says that, given a Coxeter lattice L W and a parabolic subgroup W H of the finite Coxeter group W , we can construct L W starting from W H by a series of interval doublings. For instance the lattice, associated with A n , of all the permutations on n + 1 elements is obtained from A n−1 by a series of interval doublings. The same holds for the lattices associated with the other infinite families of Coxeter groups B n , D n and I 2 (n).

One fundamental concept in the context of biological systems on which researches have flourished in the past decade is that of the apparent robustness of these systems, i.e., their ability to resist to perturbations or constraints induced by external or boundary elements such as electromagnetic fields acting on neural networks, micro-RNAs acting on genetic networks and even hormone flows acting both on neural and genetic networks. Recent studies have shown the importance of addressing the question of the environmental robustness of biological networks such as neural and genetic networks. In some cases, external regulatory elements can be given a relevant formal representation by assimilating them to or modeling them by boundary conditions. This article presents a generic mathematical approach to understand the influence of boundary elements on the dynamics of regulation networks, considering their attraction basins as gauges of their robustness. The application of this method on a real genetic regulation network will point out a mathematical explanation of a biological phenomenon which has only been observed experimentally until now, namely the necessity of the presence of gibberellin for the flower of the plant Arabidopsis thaliana to develop normally.

In this paper we propose a probabilistic analysis of the fully asynchronous behavior (i.e., two cells are never simultaneously updated, as in a continuous time process) of elementary finite cellular automata (i.e., {0, 1} states, radius 1 and unidimensional) for which both states are quiescent (i.e., (0, 0, 0) → 0 and (1, 1, 1) → 1). It has been experimentally shown in previous works that introducing asynchronism in the global function of a cellular automata was perturbing its behavior, but as far as we know, only few theoretical work exists on the subject. The cellular automata we consider live on a ring of size n and asynchronism is introduced as follows: at each time step one cell is selected uniformly at random and the transition is made on this cell while the others stay in the same state. Among the sixty-four cellular automata belonging to the class we consider, we show that nine of them diverge almost surely on all non-trivial configurations while the fifty-five other converge almost surely to a random fixed point. We show that the exact convergence time of these fifty-five automata can only take the following values: either 0, Θ(n ln n), Θ(n 2), Θ(n 3), or Θ(n2 n). Furthermore, the global behavior of each of these cellular automata is fully determined by reading its code.