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Analytic Baire spaces

100%
Ostaszewski A.
Fundamenta Mathematicae
|
2012
|
tom 217
|
nr 3
189-210
EN
We generalize to the non-separable context a theorem of Levi characterizing Baire analytic spaces. This allows us to prove a joint-continuity result for non-separable normed groups, previously known only in the separable context.
2
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Normed versus topological groups: Dichotomy and duality

68%
Bingham N., Ostaszewski A.
Instytut Matematyczny Polskiej Akademii Nauk
EN
The key vehicle of the recent development of a topological theory of regular variation based on topological dynamics \cite{BO-TI}, and embracing its classical univariate counterpart (cf. \cite{BGT}) as well as fragmentary multivariate (mostly Euclidean) theories (eg \cite{MeSh}, \cite{Res}, \cite{Ya}), are groups with a right-invariant metric carrying flows. Following the vector paradigm, they are best seen as normed groups That concept only occasionally appears explicitly in the literature despite its frequent disguised presence, and despite a respectable lineage traceable back to the Pettis closed-graph theorem, to the Birkhoff-Kakutani metrization theorem and further back still to Banach's Théorie des opérations linéaires Its most recent noteworthy appearance has been in connection with the Effros Open Mapping Principle. We collect together known salient features and develop their theory including Steinhaus theory unified by the Category Embedding Theorem \cite{BO-LBII}, the associated themes of subadditivity and convexity, and a topological duality inherent to topological dynamics. We study the latter both for its independent interest and as a foundation for topological regular variation.
3
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Very slowly varying functions. II

64%
Bingham N., Ostaszewski A.
Colloquium Mathematicum
|
2009
|
tom 116
|
nr 1
105-117
EN
This paper is a sequel to papers by Ash, Erdős and Rubel, on very slowly varying functions, and by Bingham and Ostaszewski, on foundations of regular variation. We show that generalizations of the Ash-Erdős-Rubel approach-imposing growth restrictions on the function h, rather than regularity conditions such as measurability or the Baire property-lead naturally to the main result of regular variation, the Uniform Convergence Theorem.
4
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Beyond Lebesgue and Baire II: Bitopology and measure-category duality

64%
Bingham N., Ostaszewski A.
Colloquium Mathematicum
|
2010
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tom 121
|
nr 2
225-238
EN
We re-examine measure-category duality by a bitopological approach, using both the Euclidean and the density topologies of the line. We give a topological result (on convergence of homeomorphisms to the identity) obtaining as a corollary results on infinitary combinatorics due to Kestelman and to Borwein and Ditor. We hence give a unified proof of the measure and category cases of the Uniform Convergence Theorem for slowly varying functions. We also extend results on very slowly varying functions of Ash, Erdős and Rubel.
5
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Beyond Lebesgue and Baire: generic regular variation

64%
Bingham N., Ostaszewski A.
Colloquium Mathematicum
|
2009
|
tom 116
|
nr 1
119-138
EN
We show that the No Trumps combinatorial property (NT), introduced for the study of the foundations of regular variation by the authors, permits a natural extension of the definition of the class of functions of regular variation, including the measurable/Baire functions to which the classical theory restricts itself. The "generic functions of regular variation" defined here characterize the maximal class of functions to which the three fundamental theorems of regular variation (Uniform Convergence, Representation and Characterization Theorems) apply. The proof uses combinatorial variants of the Steinhaus and Ostrowski Theorems deduced from NT in an earlier paper of the authors.
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