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On weakly Gibson $F_{σ}$-measurable mappings

100%
Karlova O., Mykhaylyuk V.
Colloquium Mathematicum
|
2013
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tom 133
|
nr 2
211-219
EN
A function f: X → Y between topological spaces is said to be a weakly Gibson function if $f(Ū) ⊆ \overline{f(U)}$ for any open connected set U ⊆ X. We prove that if X is a locally connected hereditarily Baire space and Y is a T₁-space then an $F_{σ}$-measurable mapping f: X → Y is weakly Gibson if and only if for any connected set C ⊆ X with dense connected interior the image f(C) is connected. Moreover, we show that each weakly Gibson $F_{σ}$-measurable mapping f: ℝⁿ → Y, where Y is a T₁-space, has a connected graph.
2
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On a problem of Mazur from "The Scottish Book" concerning second partial derivatives

100%
Mykhaylyuk V., Plichko A.
Colloquium Mathematicum
|
2015
|
tom 141
|
nr 2
175-181
EN
We comment on a problem of Mazur from ``The Scottish Book" concerning second partial derivatives. We prove that if a function f(x,y) of real variables defined on a rectangle has continuous derivative with respect to y and for almost all y the function $F_{y}(x): = f'_{y}(x,y)$ has finite variation, then almost everywhere on the rectangle the partial derivative $f''_{yx}$ exists. We construct a separately twice differentiable function whose partial derivative $f'_{x}$ is discontinuous with respect to the second variable on a set of positive measure. This solves the Mazur problem in the negative.
3
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On isomorphisms of some Köthe function F-spaces

81%
Kholomenyuk V., Mykhaylyuk V., Popov M.
Open Mathematics
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2011
|
tom 9
|
nr 6
1267-1275
EN
We prove that if Köthe F-spaces X and Y on finite atomless measure spaces (ΩX; ΣX, µX) and (ΩY; ΣY; µY), respectively, with absolute continuous norms are isomorphic and have the property $\mathop {\lim }\limits_{\mu (A) \to 0} \left\| {\mu (A)^{ - 1} 1_A } \right\| = 0$ (for µ = µX and µ = µY, respectively) then the measure spaces (ΩX; ΣX; µX) and (ΩY; ΣY; µY) are isomorphic, up to some positive multiples. This theorem extends a result of A. Plichko and M. Popov concerning isomorphic classification of L p(µ)-spaces for 0 < p < 1. We also provide a new class of F-spaces having no nonzero separable quotient space.
4
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Dividing measures and narrow operators

81%
Mykhaylyuk V., Pliev M., Popov M., Sobchuk O.
Studia Mathematica
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2015
|
tom 231
|
nr 2
97-116
EN
We use a new technique of measures on Boolean algebras to investigate narrow operators on vector lattices. First we prove that, under mild assumptions, every finite rank operator is strictly narrow (before it was known that such operators are narrow). Then we show that every order continuous operator from an atomless vector lattice to a purely atomic one is order narrow. This explains in what sense the vector lattice structure of an atomless vector lattice given by an unconditional basis is far from its original vector lattice structure. Our third main result asserts that every operator such that the density of the range space is less than the density of the domain space, is strictly narrow. This gives a positive answer to Problem 2.17 from "Narrow Operators on Function Spaces and Vector Lattices" by B. Randrianantoanina and the third named author for the case of reals. All the results are obtained for a more general setting of (nonlinear) orthogonally additive operators.
5
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Equiconnected spaces and Baire classification of separately continuous functions and their analogs

81%
Karlova O., Maslyuchenko V., Mykhaylyuk V.
Open Mathematics
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2012
|
tom 10
|
nr 3
1042-1053
EN
We investigate the Baire classification of mappings f: X × Y → Z, where X belongs to a wide class of spaces which includes all metrizable spaces, Y is a topological space, Z is an equiconnected space, which are continuous in the first variable. We show that for a dense set in X these mappings are functions of a Baire class α in the second variable.
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