We call a subset S of a topological vector space V linearly Borel, if for every finite number n, the set of all linear combinations of S of length n is a Borel subset of V . It will be shown that a Hamel base of an infinite dimensional Banach space can never be linearly Borel. This answers a question of Anatolij Plichko. In the sequel, let X be any infinite dimensional Banach space. A subset S ...

The foundations of regular variation for Borel measures on a complete separable space S, that is closed under multiplication by nonnegative real numbers, is reviewed. For such measures an appropriate notion of convergence is presented and the basic results such as a Portmanteau theorem, a mapping theorem and a characterization of relative compactness are derived. Regular variation is defined in...

The paper provides a speci…cation of belief systems for models of large economies with anonymity in which aggregate states depend only on crosssection distributions of types. For belief systems satisfying certain conditions of mutual absolute continuity, the paper gives a necessary and su¢ cient condition for the existence of a common prior. Under the given conditions, the common prior is uniqu...

In Section 1, we give a review on Stewart Gough (SG) platforms with self-motions and the related Borel Bricard problem. Moreover, in Section 2, we report about recent results achieved by the author on this topic (SG platforms with type II DM self-motions). In context of these results, we also present two new theorems in Section 3, which open the way for addressed future work. In Section 4, we g...

We handle divergent {\epsilon} expansions in different universality classes derived from modified Landau-Wilson Hamiltonian. Hamiltonian can cater for describing critical phenomena on a wide range of physical systems which differ symmetry conditions and the associated class. Numerically parameters are most interesting quantities characterize singular behaviour around point. More precise estimat...

1 Ergodic theory References for this section: CFS]. 1. The basic setting of ergodic theory: a measure-preserving transformation T of a probability space (X; B; m). Usually we assume T is invertible. (More generally, measure-preserving means R f T = R f; equivalently, m(T ?1 (A)) = mA.) How many measure spaces are there? Standard Borel spaces: any Borel subset of a complete, separable metric spa...

(A) A Polish metric space is a complete separable metric space (X, d). Our first goal in this paper is to determine the exact complexity of the classification problem of Polish metric spaces up to isometry. Our work was motivated by a recent paper of Vershik [1998], where the author remarks (in the beginning of Section 2): “The classification of Polish spaces up to isometry is an enormous task....

The continuum is here presented as a formal space by means of a finitary inductive definition. In this setting a constructive proof of the Heine-Borel covering theorem is given. 1 I n t r o d u c t i o n It is well known that the usual classical proofs of the Heine-Borel covering theorem are not acceptable from a constructive point of view (cf. [vS, F]). An intuitionistic alternative proof that...

We discuss the method of conformal mappings applied to perturbative QCD. The approach is based on Borel-Laplace integral regulated with principal value prescription and expansion Borel transform in powers variable which performs mapping cut plane onto unit disk. write down expression for most general location singularities review properties corresponding expansions correlators. Unlike standard ...