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Baire category results for quasi–copulas

100%

Durante F. , Fernández-Sánchez J. , Trutschnig W.

Dependence Modeling

2016

tom 4

nr 1

The aim of this manuscript is to determine the relative size of several functions (copulas, quasi– copulas) that are commonly used in stochastic modeling. It is shown that the class of all quasi–copulas that are (locally) associated to a doubly stochastic signed measure is a set of first category in the class of all quasi– copulas. Moreover, it is proved that copulas are nowhere dense in the class of quasi-copulas. The results are obtained via a checkerboard approximation of quasi–copulas.

Dichotomies for Lorentz spaces

100%

Głąb S. , Strobin F. , Yang C.

Open Mathematics

2013

tom 11

nr 7

1228-1242

Assume that L p,q, $L^{p_1 ,q_1 } ,...,L^{p_n ,q_n } $ are Lorentz spaces. This article studies the question: what is the size of the set $E = \{ (f_1 ,...,f_n ) \in L^{p_{1,} q_1 } \times \cdots \times L^{p_n ,q_n } :f_1 \cdots f_n \in L^{p,q} \} $. We prove the following dichotomy: either $E = L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ or E is σ-porous in $L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $, provided 1/p ≠ 1/p 1 + … + 1/p n. In general case we obtain that either $E = L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ or E is meager. This is a generalization of the results for classical L p spaces.

A result related to the problem cn of Fremlin

88%

Kolman O. , Shelah S.

Journal of Applied Analysis

1998

tom Vol. 4, nr 2

161-165

We show that the set of injective functions from any uncountable cardinal less than the continuum into the real numbers is of the second category in the box product topology.

The Dugundji extension property can fail in ωµ -metrizable spaces

88%

Stares I. S. , Vaughan J. E.

Fundamenta Mathematicae

1996

tom 150

nr 1

11-16

We show that there exist $ω_μ$-metrizable spaces which do not have the Dugundji extension property ($2^{ω_1}$ with the countable box topology is such a space). This answers a question posed by the second author in 1972, and shows that certain results of van Douwen and Borges are false.

Fractal sets satisfying the strong open set condition in complete metric spaces

75%

Goodman G. S.

Opuscula Mathematica

2008

tom Vol. 28, no. 4

463-470

Let K be a Hutchinson fractal in a complete metric space X, invariant under the action S of the union of a finite number of Lipschitz contractions. The Open Set Condition states that X has a non-empty subinvariant bounded open subset V, whose images under the maps are disjoint. It is said to be strong if V meets K. We show by a category argument that when K ⊄ V and the restrictions of the contractions to V are open, the strong condition implies that [formula] termed the core of V, is non-empty. In this case, it is an invariant, proper, dense, subset of K, made up of points whose addresses are unique. Conversely, [formula] implies the SOSC, without any openness assumption.

On non-injectivity of Borel functions selecting points from compact sets. 2

75%

Balcerzak M. , Peredko J. , Pol R.

Bulletin of the Polish Academy of Sciences. Mathematics

2001

tom Vol. 49, no 2

103--105

The note is a supplement to (1]. We refine a result from [1] on the non-injectivity of Borel selections on the hyperspaces and we discuss the relations of the results in [1] with some results obtained by Lecomte (4].

Finite partitions of topological groups into congruent thick subsets

63%

Weber H. , Zoli E.

Journal of Applied Analysis

2008

tom Vol. 14, nr 1

1-12

We establish fairly general sufficient conditions for a locally compact group (a Baire topological group) to admit partitions into finitely many congruent μ-thick (everywhere of second category) subset.