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Jordan Curve Theorem
Dudák, Jan ; Vejnar, Benjamin (advisor) ; Kurka, Ondřej (referee)
Title: The Jordan Curve Theorem Author: Jan Dudák Department: Department of Mathematical Analysis Supervisor: Mgr. Benjamin Vejnar, Ph.D., Department of Mathematical Analysis Abstract: The crucial part of this work is the proof of the Jordan curve theorem. To this end, the work starts by introducing necessary notions (e.g. a curve or an arc) and showing their basic properties. Further on, the Brouwer fixed point theorem is proved (in the 2-dimensional case) as well as some of its corollaries which are then used (together with several other proven assertions) in the proof of the Jordan curve theorem. The last chapter of this work briefly informs about possible generalisations of the Jordan curve theorem, referencing to appropriate bibliography. Keywords: Jordan curve, arc, plane, connected component 1
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Big families of incomparable continua
Doležalová, Anna ; Vejnar, Benjamin (advisor) ; Kurka, Ondřej (referee)
The goal of the thesis is to define the basic concepts of continuum theory and explore properties of some special continuous mappings between them. These are used for the construction of infinite families of continua which are incomparable by homeomorphic, open or monotone mappings. Special concern is given to families of dendrites. In particular, we describe an uncountable family of homeomorphically incomparable dendrites, an uncountable family of open incomparable dendrites and a countable family of monotone incomparable local dendrites. Existence of an uncountable family of monotone incomparable dendrites is open problem, in this thesis we describe a family of such dendrites of arbitrary finite cardinality. Powered by TCPDF (www.tcpdf.org)
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The James theorem and the boundary problem
Lechner, Jindřich ; Spurný, Jiří (advisor) ; Kurka, Ondřej (referee)
Let G be a subset of the dual of a real Banach space X and F ⊂ G. Then F is a James boundary of G if each w∗ -continuous linear functional on X attains its supremum over G on an element of the set F. We ask whether a norm bounded subset of X which is countably compact for the topology generated by F is ne- cessary sequentially compact for the topology generated by G. The main content of our work is a positive solution to this problem. As a corollary we obtain James characterization of weakly compact subsets of a real Banach space. Due to the Eberlein-Šmuljan theorem a positive solution to the so called boundary problem is shown as a special case of the affirmative answer to the question raised above. The question is further discussed for a case of Banach spaces defined over the complex field. In this case we cannot use the old definition of the James boun- dary but by a "natural" way it is possible to redefine the term James boundary and then we are able to answer our question positively again. 1
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Absolutely continuous function and functions of bounded variation
Hladký, Filip ; Hencl, Stanislav (advisor) ; Kurka, Ondřej (referee)
In this thesis we will study relationship between space of absolutely continuous func- tions and space of functions with bounded variation. In first three chapters we will study properties of absolutely continuous functions and functions with bounded variation and we will show nessesary and sufficient condition for functions with bounded variation to be absolutely continuous. Moreover we will show one part of fundamental theorem of calculus for Lebesgue's integral. In the last chapter we will study relationship between absolutely continuous mappings and mappings with bounded variation from Rn to Rm. 1
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Analysis in Banach spaces
Novotný, Matěj ; Hájek, Petr (advisor) ; Kurka, Ondřej (referee)
Univerzita Karlova Abstract of the diploma thesis Analysis in Banach spaces Matěj Novotný, Praha 2013 In the thesis, connection between two certain types of equivalence on Ba- nach spaces is studied: Between Lipschitz and linear one. In general, linear equivalence of two Banach spaces implies their Lipschitz equivalence, but the converse need not be true, which is shown by some nonseparable examples. There are summarized several examples to this question in the thesis, both positive and negative ones. Moreover, it is shown that James' quasi-reflexive space and its dual space have unique Lipschitz structure. To prove this, theory of linearization of Lipschitz mappings and at the same time linear structure of the two mentioned spaces is used. 1
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Properties of Poulsen simplices
Jaroň, Zdeněk ; Spurný, Jiří (advisor) ; Kurka, Ondřej (referee)
Title: Properties of Poulsen simplices Author: Zdeněk Jaroň Department: Department of Mathematical Analysis Supervisor: Doc. RNDr. Jiří Spurný, Ph.D. Abstract: In the present thesis, we study a generalisation of concept of the Poulsen simplex in general, non-metrizable case. First, for any given simplex F we con- struct a new one S, containing F as a face, having dense set of extreme points and preserving some important properties of F. In the next part, we employ this con- struction to build up, for any given infinite cardinal κ, two simplices S1, S2 with dense extreme boundary, with density character equal to κ and with spaces of affine functions Ac (S1) and Ac (S2) having the same density character, but which are not affinely homeomorphic. Keywords: Poulsen simplex, projective limit, Helly space
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Convergence of probability measures
Starý, Ladislav ; Spurný, Jiří (advisor) ; Kurka, Ondřej (referee)
In this thesis we define two most common types of convergence of probability measures and show relations between them. Using examples, we show that our result is sharp. After that, we discuss weak convergence of probability measures and convergence of random variables in distribution defined by weak convergence of probability distributions. Above all, we provide a survey among various types of convergence of random variables and relations among them.
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Descriptive and topological aspects of Banach space theory
Kurka, Ondřej
of doctoral thesis Descriptive and topological aspects in Banach space theory Deskriptivní a topologické aspekty v teorii Banachových prostorů Ondřej Kurka The thesis consists of three papers of the author. In the first paper, it is shown that the sets of Fréchet subdifferentiability of Lipschitz functions on a Banach space X are Borel if and only if X is reflexive. This answers a ques- tion of L. Zajíček. In the second paper, a problem of G. Debs, G. Godefroy and J. Saint Raymond is solved. On every separable non-reflexive Banach space, equivalent strictly convex norms with the set of norm-attaining func- tionals of arbitrarily high Borel class are constructed. In the last paper, binormality, a separation property of the norm and weak topologies of a Ba- nach space, is studied. A result of P. Holický is generalized. It is shown that every Banach space which belongs to a P-class is binormal. It is also shown that the asplundness of a Banach space is equivalent to a related separation property of its dual space. 1
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Legal regulation of building savings
Kurka, Ondřej ; Karfíková, Marie (advisor) ; Bakeš, Milan (referee)
The main purpose of the thesis is to analyse contemporary legal regulations governing the field of building savings in the Czech Republic and recent changes of these legal regulations. The thesis is composed of six chapters, each of them dealing with different aspects of building savings' legal regulation, the introduction, the closing part, lists and annexes. The matters already mentioned above are described in chapters two, three and five. Divided into remaining chapters and within the capacity limits, the thesis also includes secondary issues, such as the evolution of building savings and its legal regulation in Europe and in the Czech Republic in connection with the description of changes made to the legal regulations rather long time ago and followed by research of the practical application of these legal regulations. The aim of the thesis is to complete pieces of knowledge gathered from the legal regulations, from practical research and from my two years expirience as a building savings commercial broker and create an integrated thesis that is transparent and complete from different angles of view. Conclusions are drawn particularly in the closing part of the thesis where are briefly summarized the author's opinions on problems described in each chapter.
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Descriptive and topological aspects of Banach space theory
Kurka, Ondřej ; Holický, Petr (advisor) ; Fabian, Marián (referee) ; Hájek, Petr (referee)
The thesis consists of three papers of the author. In the first paper, it is shown that the sets of Fréchet subdifferentiability of Lipschitz functions on a Banach space X are Borel if and only if X is reflexive. This answers a ques- tion of L. Zajíček. In the second paper, a problem of G. Debs, G. Godefroy and J. Saint Raymond is solved. On every separable non-reflexive Banach space, equivalent strictly convex norms with the set of norm-attaining func- tionals of arbitrarily high Borel class are constructed. In the last paper, binormality, a separation property of the norm and weak topologies of a Ba- nach space, is studied. A result of P. Holický is generalized. It is shown that every Banach space which belongs to a P-class is binormal. It is also shown that the asplundness of a Banach space is equivalent to a related separation property of its dual space. 1
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