We lay the foundations of first-order model theory Coxeter groups. Firstly, with exception 2-spherical non-affine case (which we leave open), characterize superstable groups finite rank, which show to be essentially affine type. Secondly, rank are domains, a central assumption in algebraic geometry over groups, many respects (e.g. λ -stability) reduces given system its associated irreducible co...

<p style='text-indent:20px;'>We present two different mechanisms for the emergence of strange attractors (observable chaos) in a two-parameter periodically-perturbed family differential equations on plane. The parameters are independent and act ways invariant manifolds consecutive saddles cycle. When both zero, flow exhibits an attracting heteroclinic cycle associated to equilibria. first...

A group is small if it has countably many pure n-types for each integer n. It is shown that in a small group, subgroups which are definable with parameters in a finitely generated algebraic closure satisfy local descending chain conditions. An infinite small group has an infinite abelian subgroup, which may not be definable. A nilpotent small group is the central product of a definable divisibl...

It is shown that a stable division ring with positive characteristic has finite dimension over its centre. This is then extended to simple division rings. Macintyre proved any ω-stable field to be either finite or algebraically closed [7]. This was generalised by Cherlin and Shelah to superstable fields [3]. It follows that a superstable division ring is a field [2]. The result was broadened to...

A group is small if it has countably many complete n-types over the empty set for each natural number n. More generally, a group G is weakly small if it has countably many complete 1-types over every finite subset of G. We show here that in a weakly small group, subgroups which are definable with parameters lying in a finitely generated algebraic closure satisfy the descending chain conditions ...

Assume T is superstable and small. Using the multiplicity rankM we find locally modular types in the same manner as U -rank considerations yield regular types. We define local versions of M-rank, which also yield meager types. 0. Introduction. Throughout, T is superstable, small, and we work in C = C. In [Ne1] we defined the multiplicity rank M and proved that M has additivity properties simila...

Assume p∗ is a meager type in a superstable theory T . We investigate definability properties of p∗-closure. We prove that if T has < 2א0 countable models then the multiplicity rank M of every type p is finite. We improve Saffe’s conjecture. 0. Introduction. Throughout the paper, T is a superstable theory in a countable language L, and we work within a monster model C = C of T . The general ref...

A division ring of positive characteristic with countably many pure types is a field. Wedderburn showed in 1905 that finite fields are commutative. As for infinite fields, we know that superstable [1, Cherlin, Shelah] and supersimple [4, Pillay, Scanlon, Wagner] ones are commutative. In their proof, Cherlin and Shelah use the fact that a superstable field is algebraically closed. Wagner showed ...