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A comparison of two notions of porosity

100%
Strobin F.
Commentationes Mathematicae
|
2008
|
tom 48
|
nr 2
EN
In the paper we compare two notions of porosity: the R-ball porosity \((R \rt 0)\) defined by Preiss and Zajı́ček, and the porosity which was introduced by Olevskii (here it will be called the O-porosity). We find this comparison interesting since in the literature there are two similar results concerning these two notions. We restrict our discussion to normed linear spaces since the R-ball porosity was originally defined in such spaces.
2
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Descriptive properties of density preserving autohomeomorphisms of the unit interval

63%
Głąb S., Strobin F.
Open Mathematics
|
2010
|
tom 8
|
nr 5
928-936
EN
We prove that density preserving homeomorphisms form a Π11-complete subset in the Polish space ℍ of all increasing autohomeomorphisms of unit interval.
3
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Large free subgroups of automorphism groups of ultrahomogeneous spaces

63%
Głąb S., Strobin F.
Colloquium Mathematicum
|
2015
|
tom 140
|
nr 2
279-295
EN
We consider the following notion of largeness for subgroups of $S_{∞}$. A group G is large if it contains a free subgroup on 𝔠 generators. We give a necessary condition for a countable structure A to have a large group Aut(A) of automorphisms. It turns out that any countable free subgroup of $S_{∞}$ can be extended to a large free subgroup of $S_{∞}$, and, under Martin's Axiom, any free subgroup of $S_{∞}$ of cardinality less than 𝔠 can also be extended to a large free subgroup of $S_{∞}$. Finally, if Gₙ are countable groups, then either $∏_{n∈ℕ} Gₙ$ is large, or it does not contain any free subgroup on uncountably many generators.
4
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Dichotomies for Lorentz spaces

51%
Głąb S., Strobin F., Yang C.
Open Mathematics
|
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.
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