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On the statistical and σ-cores

Hüsamettın Çoşkun, Celal Çakan, Mursaleen (2003)

Studia Mathematica

In [11] and [7], the concepts of σ-core and statistical core of a bounded number sequence x have been introduced and also some inequalities which are analogues of Knopp’s core theorem have been proved. In this paper, we characterize the matrices of the class ( S ∩ m , V σ ) r e g and determine necessary and sufficient conditions for a matrix A to satisfy σ-core(Ax) ⊆ st-core(x) for all x ∈ m.

The Hurewicz covering property and slaloms in the Baire space

Boaz Tsaban (2004)

Fundamenta Mathematicae

According to a result of Kočinac and Scheepers, the Hurewicz covering property is equivalent to a somewhat simpler selection property: For each sequence of large open covers of the space one can choose finitely many elements from each cover to obtain a groupable cover of the space. We simplify the characterization further by omitting the need to consider sequences of covers: A set of reals X has the Hurewicz property if, and only if, each large open cover of X contains a groupable subcover. This...

Uncountable γ-sets under axiom C P A c u b e g a m e

Krzysztof Ciesielski, Andrés Millán, Janusz Pawlikowski (2003)

Fundamenta Mathematicae

We formulate a Covering Property Axiom C P A c u b e g a m e , which holds in the iterated perfect set model, and show that it implies the existence of uncountable strong γ-sets in ℝ (which are strongly meager) as well as uncountable γ-sets in ℝ which are not strongly meager. These sets must be of cardinality ω₁ < , since every γ-set is universally null, while C P A c u b e g a m e implies that every universally null has cardinality less than = ω₂. We also show that C P A c u b e g a m e implies the existence of a partition of ℝ into ω₁ null compact sets....

Very slowly varying functions. II

N. H. Bingham, A. J. Ostaszewski (2009)

Colloquium Mathematicae

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.

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