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Baire and Harmonic Functions
Pošta, Petr ; Lukeš, Jaroslav (advisor)
Title: Baire and Harmonic Functions Author: Petr Pošta Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Jaroslav Lukeš, DrSc., Department of Mathematical Analysis Abstract: The present thesis consists of six research papers. The first four articles deal with topics related to potential theory, Baire-one functions and its important subclasses, in particular differences of semicontinuous functions. The first paper is devoted to the stability of the Dirichlet problem for which a new criterion in terms of Poisson equation is provided. The second paper improves the recent result obtained by Lukeš et al. It shows that the classical Dirichlet solution belongs to the B1/2 subclass of Baire-one functions. A generalization of this result to the abstract context of the Choquet theory on functions spaces is provided. Finally, an abstract Dirichlet problem for the boundary condition belonging to the class of differences of semincontinuous functions is discussed. The third paper concentrates on the Lusin-Menshov property and the approximation of Baire- one and finely continuous functions by differences of semicontinuous and finely continuous functions. It provides an exposition of topologies (various density topologies as well as the fine topologies in both linear and non-linear potential...
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Baire and Harmonic Functions
Pošta, Petr ; Lukeš, Jaroslav (advisor)
Title: Baire and Harmonic Functions Author: Petr Pošta Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Jaroslav Lukeš, DrSc., Department of Mathematical Analysis Abstract: The present thesis consists of six research papers. The first four articles deal with topics related to potential theory, Baire-one functions and its important subclasses, in particular differences of semicontinuous functions. The first paper is devoted to the stability of the Dirichlet problem for which a new criterion in terms of Poisson equation is provided. The second paper improves the recent result obtained by Lukeš et al. It shows that the classical Dirichlet solution belongs to the B1/2 subclass of Baire-one functions. A generalization of this result to the abstract context of the Choquet theory on functions spaces is provided. Finally, an abstract Dirichlet problem for the boundary condition belonging to the class of differences of semincontinuous functions is discussed. The third paper concentrates on the Lusin-Menshov property and the approximation of Baire- one and finely continuous functions by differences of semicontinuous and finely continuous functions. It provides an exposition of topologies (various density topologies as well as the fine topologies in both linear and non-linear potential...
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Baire and Harmonic Functions
Pošta, Petr ; Lukeš, Jaroslav (advisor) ; Benyaiche, Allami (referee) ; Netuka, Ivan (referee)
Title: Baire and Harmonic Functions Author: Petr Pošta Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Jaroslav Lukeš, DrSc., Department of Mathematical Analysis Abstract: The present thesis consists of six research papers. The first four articles deal with topics related to potential theory, Baire-one functions and its important subclasses, in particular differences of semicontinuous functions. The first paper is devoted to the stability of the Dirichlet problem for which a new criterion in terms of Poisson equation is provided. The second paper improves the recent result obtained by Lukeš et al. It shows that the classical Dirichlet solution belongs to the B1/2 subclass of Baire-one functions. A generalization of this result to the abstract context of the Choquet theory on functions spaces is provided. Finally, an abstract Dirichlet problem for the boundary condition belonging to the class of differences of semincontinuous functions is discussed. The third paper concentrates on the Lusin-Menshov property and the approximation of Baire- one and finely continuous functions by differences of semicontinuous and finely continuous functions. It provides an exposition of topologies (various density topologies as well as the fine topologies in both linear and non-linear potential...
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Construction of dendroids and their properties
Marciňa, Radek ; Pyrih, Pavel (advisor) ; Hušek, Miroslav (referee)
This thesis is about dendroids and their properties. Another example of a dendroid with two intersecting shore continua whose union is not a shore continuum is constructed. Moreover, a simplification of a proof that the union of finitely many pairwise disjoint shore continua is again a shore continuum has been made. But the main result is an affirmative answer to the question whether the union of finitely many closed shore sets is again a closed shore set in the case dendroids with only finitely many branch points. Powered by TCPDF (www.tcpdf.org)
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