In 1937, Baer [5] introduced the notion of the type of an element in a torsion-free abelian group and showed that this notion provided a complete invariant for the classification problem for torsion-free abelian groups of rank 1. Since then, despite the efforts of such mathematicians as Kurosh [23] and Malcev [25], no satisfactory system of complete invariants has been found for the torsion-fre...

The literature provides dichotomies involving homomorphisms (like the G0 dichotomy) or reductions (like the characterization of sets potentially in a Wadge class of Borel sets, which holds on a subset of a product). However, part of the motivation behind the latter result was to get reductions on the whole product, like in the classical notion of Borel reducibility considered in the study of an...

An internal characterization of metric spaces which are absolute Borel sets of multiplicative classes is given. This characterization uses complete sequences of covers, a notion introduced by Froĺık for characterizing Čechcomplete spaces. We also show that the absolute Borel class of X is determined by the uniform structure of the space of continuous functions Cp(X); however the case of absolut...

The infrared freezing of observables is known to hold at fixed orders of perturbative QCD if the Minkowskian quantities are defined through the analytic continuation from the Euclidean region. In a recent paper [1] it is claimed that infrared freezing can be proved also for Borel resummed allorders quantities in perturbative QCD. In the present paper we obtain the Minkowskian quantities by the ...

This paper is a continuation of the study of topological properties of omega context free languages (ω-CFL). We proved in [Topological Properties of Omega Context Free Languages, Theoretical Computer Science, Volume 262 (1-2), 2001, p. 669697] that the class of ω-CFL exhausts the finite ranks of the Borel hierarchy, and in [Borel Hierarchy and Omega Context Free Languages, Theoretical Computer ...

In this paper we prove some Tauberian theorems for the product of Borel and Hölder summability methods which improve the classical Tauberian theorems for the Borel summability method. Mathematics Subject Classification 2010: 40E05, 40G05, 40G10.

We demonstrate that a reproducing kernel Hilbert space of functions on a separable absolute Borel space or an analytic subset of a Polish space is separable if it possesses a Borel measurable feature map.

Let G be a locally compact Abelian topological group having a countable basis of its topology. We also suppose that G has the T0–property, that is, ⋂ U∈Ne U = {e}, where e denotes the identity element of G and Ne is the collection of all Borel neighbourhoods of e. (By a Borel neighbourhood U of e we mean a Borel subset of G for which there exists an open subset Ũ of G such that e ∈ Ũ ⊂ U.) Let ...

We study the Borel summation method. We obtain a general sufficient condition for a given matrix A to have the Borel property. We deduce as corollaries, earlier results obtained by G. Müller and J.D. Hill. Our result is expressed in terms belonging to the theory of Gaussian processes. We show that this result cannot be extended to the study of the Borel summation method on arbitrary dynamical s...

We show that it is relatively consistent with ZF that the Borel hierarchy on the reals has length ω2. This implies that ω1 has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument produces models of ZF in which the Borel hierarchy has length any given limit ordinal less than ω2, e.g., ω or ω1 + ω1.