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Baire category results for quasi–copulas

100%
Durante F., Fernández-Sánchez J., Trutschnig W.
Dependence Modeling
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2016
|
tom 4
|
nr 1
EN
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.
2
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Dichotomies for Lorentz spaces

100%
Głąb S., Strobin F., Yang C.
Open Mathematics
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2013
|
tom 11
|
nr 7
1228-1242
EN
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.
3
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The Dugundji extension property can fail in ωµ -metrizable spaces

88%
Stares I. S., Vaughan J. E.
Fundamenta Mathematicae
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1996
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tom 150
|
nr 1
11-16
EN
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.
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