Browsing by Autor "Alexander Berenstein"

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DIMENSION AND MEASURE IN PSEUDOFINITE <i>H</i>-STRUCTURES
(Cambridge University Press, 2025) Alexander Berenstein; Darío García; Tingxiang Zou
Abstract We study H -structures associated with $SU$ -rank 1 measurable structures. We prove that the $SU$ -rank of the expansion is continuous and that it is uniformly definable in terms of the parameters of the formulas. We also introduce notions of dimension and measure for definable sets in the expansion and prove they are uniformly definable in terms of the parameters of the formulas.
Geometric structures with a dense independent subset
(Birkhäuser, 2015) Alexander Berenstein; Evgueni Vassiliev
Hilbert spaces with generic predicates
(National University of Colombia, 2018) Alexander Berenstein; Tapani Hyttinen; Andrés Villaveces
Estudiamos la teoría de modelos de expansiones de espacios de Hilbert mediante predicados genéricos. Primero demostramos la existencia de modelo-compañeras de expansiones genéricas de espacios de Hilbert mediante una función-distancia a una estructura aleatoria, y luego una distancia a un subconjunto aleatorio. La teoría obtenida con la subestructura aleatoria es ω-estable; la obtenida mediante la distancia a subconjunto aleatorio es TP2 y NSOP1. Este ejemplo es la primera estructura de esta clase de complejidad en lógica continua.
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
The independence property in generalized dense pairs of structures
(Cambridge University Press, 2011) Alexander Berenstein; Alf Dolich; Alf Onshuus
Abstract We provide a general theorem implying that for a (strongly) dependent theory T the theory of sufficiently well-behaved pairs of models of T is again (strongly) dependent. We apply the theorem to the case of lovely pairs of thorn-rank one theories as well as to a setting of dense pairs of first-order topological theories.