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Gromov-Hausdorff metric
Horský, Miroslav ; Vejnar, Benjamin (advisor) ; Vlasák, Václav (referee)
In this bachelor's thesis we define the notion of Hausdorff metric and Gromov-Hausdorff metric. We will define both metrics in several different ways and then we will show that these definitions are equivalent. We will also proof some fundamental properties of those metrics. At last we will explore Gromov-Hausdorff convergence and proof some strong properties of this convergence. 1
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Mountain Pass Theorem and its applications
Příhoda, Vojtěch ; Černý, Robert (advisor) ; Vlasák, Václav (referee)
Cílem této práce je formulovat a dokázat Větu o horském sedle a ukázat příklady její aplikace tak, aby student třetího ročníku matematiky nebo fy- ziky, jenž absolvoval alespoň úvodní kurs funkcionální analýzy, byl schopen práci porozumět. D·kaz Věty o horském sedle provedeme s pomocí Ekelan- dova variačního principu, který si rovněž formulujeme a dokážeme. Následně si ukážeme dva příklady její aplikace na d·kaz existence netriviálního sla- bého řešení eliptické parciální diferenciální rovnice obsahující nelinearitu se subkritickým r·stem. 1
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Collections of compact sets in descriptive set theory
Vlasák, Václav ; Zelený, Miroslav (advisor) ; Holický, Petr (referee) ; Tišer, Jaroslav (referee)
1 Title: Collections of compact sets in descriptive set theory Author: Václav Vlasák Department: Department of Mathematical Analysis Supervisor: Doc. RNDr. Miroslav Zelený, Ph.D. Author's e-mail address: vlasakmm@volny.cz Abstract: This work consists of three articles. In Chapter 2, we dissert on the connections between complexity of a function f from a Polish space X to a Polish space Y and complexity of the set C(f) = {K ∈ K(X); f K is continuous}, where K(X) denotes the space of all compact subsets of X equipped with the Vietoris topology. We prove that if C(f) is analytic, then f is Borel; and assuming ∆1 2-Determinacy we show that f is Borel if and only if C(f) is coanalytic. Similar results for projective classes are also presented. In Chapter 3, we continue in our investigation of collection C(f) and also study its restriction on convergent sequences (C(f)). We prove that C(f) is Borel if and only if f is Borel. Similar results for projective classes are also presented. The Chapter 4 disserts on HN -sets, which form an important subclass of the class of sets of uniqueness for trigonometric series. We investigate the size of these classes which is reflected by the family of measures called polar which annihilate all the sets belonging to the given class. The main aim of this chapter is to answer in...
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Nonabsolute convergence of Newton integral
Konopka, Filip ; Spurný, Jiří (advisor) ; Vlasák, Václav (referee)
In this thesis, we look for sufficient conditions for non-absolute convergence of Newton integral. Importantly we analyse how the oscillation of the sine function influences the con- vergence of the integral. We are dealing with the convergence of integrals based on the limits of the antiderivative, which does not need to convergence in the Lebesgue sense. 1
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Collections of compact sets in descriptive set theory
Vlasák, Václav ; Zelený, Miroslav (advisor)
1 Title: Collections of compact sets in descriptive set theory Author: Václav Vlasák Department: Department of Mathematical Analysis Supervisor: Doc. RNDr. Miroslav Zelený, Ph.D. Author's e-mail address: vlasakmm@volny.cz Abstract: This work consists of three articles. In Chapter 2, we dissert on the connections between complexity of a function f from a Polish space X to a Polish space Y and complexity of the set C(f) = {K ∈ K(X); f K is continuous}, where K(X) denotes the space of all compact subsets of X equipped with the Vietoris topology. We prove that if C(f) is analytic, then f is Borel; and assuming ∆1 2-Determinacy we show that f is Borel if and only if C(f) is coanalytic. Similar results for projective classes are also presented. In Chapter 3, we continue in our investigation of collection C(f) and also study its restriction on convergent sequences (C(f)). We prove that C(f) is Borel if and only if f is Borel. Similar results for projective classes are also presented. The Chapter 4 disserts on HN -sets, which form an important subclass of the class of sets of uniqueness for trigonometric series. We investigate the size of these classes which is reflected by the family of measures called polar which annihilate all the sets belonging to the given class. The main aim of this chapter is to answer in...
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Nonabsolute convergence of Newton integral
Konopka, Filip ; Spurný, Jiří (advisor) ; Vlasák, Václav (referee)
In this thesis, we look for sufficient conditions for non-absolute convergence of Newton integral. Importantly we analyse how the oscillation of the sine function influences the con- vergence of the integral. We are dealing with the convergence of integrals based on the limits of the antiderivative, which does not need to convergence in the Lebesgue sense. 1
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Mountain climbing theorem
Šmídová, Kristýna ; Vejnar, Benjamin (advisor) ; Vlasák, Václav (referee)
Title: Mountain climbing theorem Author: Kristýna Šmídová Department: Department of Mathematical Analysis Supervisor: Mgr. Benjamin Vejnar, Ph.D., Department of Mathematical Ana- lysis Abstract: The subject of this theses is the so-called Muntain Climbers' Pro- blem. We ask for which pairs of continuous functions f, g : [0,1] → [0,1] such that f(0) = g(0) = 0 and f(1) = g(1) = 1 there exist some functions k, h with the same properties such that f (k(x)) = g (h(x)) for all x in the inter- val of [0,1]. For piecewise injective functions we prove the existence using a convenient graph model and handshaking lemma. For locally non-constant functions we provide a constructive proof using uniform convergence. There is also an example of pair of continuons functions for which there exists no suitable pair of functions that solve the problem. The aim is to provide a clear and visual explanation of all the mathematical constructions included. Keywords: continuous function, mountain climber, uniform convergence, hand- shaking lemma
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Descriptive set properties of collections of exceptional sets in Harmonic analysis
Kovařík, Vojtěch ; Vlasák, Václav (advisor) ; Zelený, Miroslav (referee)
We study families of small sets which appear in Harmonic analysis. We focus on the systems H(N) , N ∈ N, U and U0. In particular we compare their sizes via comparing the polars of these classes, i.e. the families of measures annihilating all sets from given class. Lyons showed that in this sense, the family N∈N H(N) is smaller than U0. The main goal of this thesis is the study of the question whether this also holds when the system U0 is replaced by the much smaller system U. To this end we define a new system H(∞) and systems of sets of type N where N ∈ N∪{∞}. We then prove some of their properties, which might be useful in solving the studied question. 1
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Construction of a von Koch snowflake
Bouchala, Ondřej ; Hencl, Stanislav (advisor) ; Vlasák, Václav (referee)
A Mapping from C onto C is quasiconformal, if it maps "infinitesimally small circles" onto "infinitesimally small ellipses". In other words, its real derivative in almost every point (which is for each point linear mapping from plane to plane) maps circles to ellipses with bounded ratio of axes. Koch snowflake is well-known inductively defined fractal: Using Beurling-Ahlfors extension we will prove, that there exists quasi- conformal mapping from the plane onto the plane, which maps unit disk onto Koch snowflake. 1
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