Certain basic concepts of geometrical stability theory are generalized to a class of closure operators containing algebraic closure. A specific case of a generalized closure operator is developed which is relevant to Vaught’s conjecture. As an application of the methods, we prove Theorem A. Let G be a superstable group of U− rank ω such that the generics of G are locally modular and Th(G) has f...

We study the dynamics of the travelling interface arising from a bistable piece-wise linear one-way coupled map lattice. We show how the dynamics of the interfacial sites, separating the two superstable phases of the local map, is finite dimensional and equivalent to a toral map. The velocity of the travelling interface corresponds to the rotation vector of the toral map. As a consequence, a ra...

By a classifiable theory we shall mean a theory which is superstable, without the dimensional order property, which has prime models over pairs. In order to define what we mean by unique decomposition, we remind the reader of several definitions and results. We adopt the usual conventions of stability theory and work inside a large saturated model of a fixed classifiable theory T ; for instance...

A structure M is pregeometric if the algebraic closure is a pregeometry in all M ′ elementarily equivalent to M . We define a generalisation: structures with an existential matroid. The main examples are superstable groups of U-rank a power of ω and d-minimal expansion of fields. Ultraproducts of pregeometric structures expanding a field, while not pregeometric in general, do have an unique exi...

Dans un article récent [1], Abderezak Ould Houcine montre qu’un groupe superstable résiduellement C pour une pseudo-variété C admet une série normale définissable G = G0⊲G1⊲· · ·⊲Gn telle que Gi/Gi+1 ∈ C pour i < n, et Gn est résoluble ; si G est ω-stable, on peut choisir Gn nilpotent. Nous étendons son résultat aux groupes supersimples, et montrons en même temps qu’on peut toujours prendre Gn ...

For a smooth dynamical system xn+1 = f (C, xn) (depending on a parameter C), there may be infinitely many periodic windows, that is, intervals inC having a region of stable periodic behaviour. However, the smaller of these windows are easily destroyed with tiny perturbations, so that only finitely many of the windows can be detected for a given level of noise. For a fixed perturbation size , we...

The Dirichlet form given by the intrinsic gradient on Poisson space is associated with a Markov process consisting of a countable family of interacting diffusions. By considering each diffusion as a particle with unit mass, the randomly evolving configuration can be thought of as a Radon measure valued diffusion. The quasi-sure analysis of Dirichlet forms is used to find exceptional sets for th...

We continue [Sh:h].II, studying stability theory for abstract elementary classes. In [Sh E46], Shelah obtained a non-forking relation for an AEC, (K, ), with LST -number at most λ, which is categorical in λ and λ and has less than 2 + models of cardinality λ, but at least one. This non-forking relation satisfies the main properties of the non-forking relation on stable first order theories. Her...