Very simple theories without forking

We prove Vaught's conjecture for minimal trivial simple theories satisfying the generalized independence theorem. Research supported by KBN grant 2 P03A 006 09     

Stable theories

We studyK _T(λ)=sup {|S(A)| : |A|≦λ} and extend some results for totally transcendental theroies to the case of stable theories. We then investigate categoricity of elementary and pseudo-elementary classes. This paper is a part of the author’s doctoral dissertation written at the Hebrew University of Jerusalem, under the kind guidance of Profeossr M. Rabin.     

Measures and forking

Shelah's theory of forking (or stability theory) is generalized in a way which deals with measures instead of complete types. This allows us to extend the method of forking from the class of stable theories to the larger class of theories which do not have the independence property. When restricted to the special case of stable theories, this paper reduces to a reformulation of the classical approach. However, it goes beyond the classical approach in the case of unstable theories. Methods from ordinary forking theory and the Loeb measure construction from nonstandard analysis are used.     

On dividing chains in simple theories

Dividing chains have been used as conditions to isolate adequate subclasses of simple theories. In the first part of this paper we present an introduction to the area. We give an overview on fundamental notions and present proofs of some of the basic and well-known facts related to dividing chains in simple theories. In the second part we discuss various characterizations of the subclass of low theories. Our main theorem generalizes and slightly extends a well-known fact about the connection between dividing chains and Morley sequences (in our case: independent sequences). Moreover, we are able to give a proof that is shorter than the original one. This result motivates us to introduce a special property of formulas concerning independent dividing chains: For any dividing chain there exists an independent dividing chain of the same length. We study this property in the context of low, short and ω -categorical simple theories, outline some examples and define subclasses of low and short theories, which imply this property. The results give rise to further studies of the relationships between some subclasses of simple theories. Research supported by CNPq grant 150309/2003-1. Research supported by CNPq grant 304365/2003-3 (Modelos, Provas e Algoritmos)     

Stable theories and representation over sets

In this paper we give characterizations of the stable and ?₀‐stable theories, in terms of an external property called representation. In the sense of the representation property, the mentioned classes of first‐order theories can be regarded as “not very complicated”.     

The complete finitely axiomatized theories of order are dense

We prove a conjecture of Lauchli and Leonard that every sentence of the theory of linear order which has a model, has a model with a finitely axiomatized theory.     

The stable forking conjecture and generic structures

 We prove that for any simple theory which is constructed via Fr?issé-Hrushovski method, if the forking independence is the same as the d-independence then the stable forking property holds. Received: 22 January 2001 / Published online: 19 December 2002 This article is part of the author's D-Phil thesis, written at the University of Oxford and supported by the Ministry of Higher Education of Iran. The author would like to thank the Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran, for its financial support whilst working on this article. Mathematics Subject Classification (2000): 03C45 Key words or phrases: Generic structures – Fr?issé-Hrushovski method – Predimension – Simple theories – Stable theories – Stable forking conjecture     

Stable theories, pseudoplanes and the number of countable models

We prove that if T is a stable theory with only a finite number (>1) of countable models, then T contains a type-definable pseudoplane. We also show that for any stable theory T either T contains a type-definable pseudoplane or T is weakly normal (in the sense of [9]).     

Almost arcwise connected continua without dense arc components

A continuum M is almost arcwise connected if each pair of nonempty open subsets of M can be joined by an arc in M. An almost arcwise connected plane continuum without a dense arc component can be defined by identifying pairs of endpoints of three copies of the Knaster indecomposable continuum that has two endpoints. In [7] K.R. Kellum gave this example and asked if every almost arcwise connected continuum without a dense arc component has uncountably many arc components. We answer Kellum's question by defining an almost arcwise connected plane continuum with only three arc components none of which are dense. A continuum M is almost Peano if for each finite collection $\text{C}$ of nonempty open subsets of M there is a Peano continuum in M that intersects each element of $\text{C}$. We define a hereditarily unicoherent almost Peano plane continuum that does not have a dense arc component. We prove that every almost arcwise connected planar λ-dendroid has exactly one dense arc component. It follows that every hereditarily unicoherent almost arcwise connected plane continuum without a dense arc component has uncountably many arc components. Using an example of J. Krasinkiewicz and P Minc [8], we define an almost Peano λ-dendroid that do not have a dense arc component. Using a theorem of J.B. Fugate and L. Mohler [3], we prove that every almost arcwise connected λ-dendroid without a dense arc component has uncountably many arc components. In Euclidean 3-space we define an almost Peano continuum with only countably many arc components no one of which is dense. It is not known if the plane contains a continuum with these properties.     

Stable manifolds for nonautonomous equations without exponential dichotomy

We consider nonautonomous ordinary differential equations v^′=A(t)v in Banach spaces and, under fairly general assumptions, we show that for any sufficiently small perturbation f there exists a stable invariant manifold for the perturbed equation v^′=A(t)v+f(t,v), which corresponds to the set of negative Lyapunov exponents of the original linear equation. The main assumption is the existence of a nonuniform exponential dichotomy with a small nonuniformity, i.e., a small deviation from the classical notion of (uniform) exponential dichotomy. In fact, we showed that essentially any linear equation v^′=A(t)v admits a nonuniform exponential dichotomy and thus, the above assumption only concerns the smallness of the nonuniformity of the dichotomy. This smallness is a rather common phenomenon at least from the point of view of ergodic theory: almost all linear variational equations obtained from a measure-preserving flow admit a nonuniform exponential dichotomy with arbitrarily small nonuniformity. We emphasize that we do not need to assume the existence of a uniform exponential dichotomy and that we never require the nonuniformity to be arbitrarily small, only sufficiently small. Our approach is related to the notion of Lyapunov regularity, which goes back to Lyapunov himself although it is apparently somewhat forgotten today in the theory of differential equations.     

Compact groups containing dense pseudocompact subgroups without non-trivial convergent sequences

Let G be compact abelian group such that w(C(G))=w(Cω(G)). We prove that if

|C(G)|?m(G/C(G)),