Browsing by Autor "Alexander Usvyatsov"

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Additivity of the dp-rank
(American Mathematical Society, 2013) Itay Kaplan; Alf Onshuus; Alexander Usvyatsov
The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank cannot be reduced to the study of its dp-minimal types, and we discuss the possible relations between dp-rank and VC-density.
Generic stability, forking, and þ-forking
(American Mathematical Society, 2012) Darío García; Alf Onshuus; Alexander Usvyatsov
Abstract notions of “smallness” are among the most important tools that model theory offers for the analysis of arbitrary structures. The two most useful notions of this kind are forking (which is closely related to certain measure zero ideals) and thorn-forking (which generalizes the usual topological dimension). Under certain mild assumptions, forking is the finest notion of smallness, whereas thorn-forking is the coarsest. In this paper we study forking and thorn-forking, restricting ourselves to the class of generically stable types. Our main conclusion is that in this context these two notions coincide. We explore some applications of this equivalence.
Model theory for metric structures
(Cambridge University Press, 2008) Itaï Ben Yaacov; Alexander Berenstein; C. Ward Henson; Alexander Usvyatsov
A metric structure is a many-sorted structure with each sort a metric space, which for convenience is assumed to have finite diameter. Additionally there are functions (of several variables) between sorts, assumed to be uniformly continuous. Examples include metric spaces themselves, measure algebras (with the metric d(A, B) = µ(A∆B), where ∆ is symmetric difference), and structures based on Banach spaces (where one interprets the sorts as balls), including Banach lattices, C*algebras, etc. The usual first-order logic does not work very well for such structures, and several good alternatives have been developed. One is the logic of positive bounded formulas with an approximate semantics (see [3]). Another is the setting of compact abstract theories (cats) (see [1]). A recent development, which will be emphasized in this course, is the convergence of these two points of view and the realization that they are also connected to the [0,1]-valued continuous logic that was studied extensively in the 1960s and then dropped (see [2]). This leads to a new formalism for the logic of metric structures in which the operations of sup and inf play a central role. The analogy between this logic for metric space structures and the usual first order logic for ordinary structures is far reaching. This new logic satisfies the compactness theorem, Löwenheim-Skolem