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On convergence of a series
Procházka, Antonín ; Zelený, Miroslav (advisor) ; Kaplický, Petr (referee)
in English This text is devoted to the series, whose n-th term is defined by (−1)n |sin n| /n. The goal of this work is to prove convergence of this series. The solution uses standard convergence tests, the theory of Fourier Series and findings about approximation of number π. 1
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Banach-Tarski paradox
Klůjová, Jana ; Zelený, Miroslav (advisor) ; Kaplický, Petr (referee)
In the present work we study the Banach-Tarski Paradox and other paradoxical decompositions of sets, groups and semigroups. These decompositions are described especially using free groups and semigroups. We can construct such groups using words made of letters. We study both finite and denumerable paradoxical decomposition. Further we deal with equidecomposability, which we need for a proof of Banach-Tarski Paradox. We present a proof of Banach-Schröder-Berstein Theorem.
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Automated object tracking using robotic manipulator
Zelený, Miroslav ; Ligocki, Adam (referee) ; Chromý, Adam (advisor)
This diploma thesis deals with the tracking of objects using a robotic manipulator Epson C3 and a color camera. The work describes the basic qualities of the device to be used. The OpenCV library and its wrapper EmguCV are used as software tools for computer vision. It discusses the basic issues and principles of tracking objects in the image and introduces some methods of tracking. These methods have been tested and therefore their strengths and weaknesses, which appeared during testing, are listed here. Furthermore, there is a procedure for calculating the new coordinates of the camera and the manipulator effector using homogeneous transformations. The work contains the results of testing the algorithms and their evaluation. The output of the work is a test application for the Epson C3 robot.
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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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Complexity of classification problems in ergodic theory
Vaněček, Ondřej ; Zelený, Miroslav (advisor) ; Doucha, Michal (referee)
In the thesis we acquaint ourselves with the terms from ergodic theory and re- presentation theory of topological groups. We pay attention particularly to terms unitary representation, realizability by an action, dual group, unitary equivalence and Kazhdan's property (T). We achieve a result regarding unitary representati- ons realizable by an action on finite abelian groups according to article [5] and show that it is possible to generalize it to all finite groups at the end of the thesis according to article [6]. A large part of the text subsequently deals with proper- ties of unitary representations and their relations. We connect the terms compact topological group and Kazhdan's property (T).
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Additive families of Borel sets
Hronek, Radek ; Zelený, Miroslav (advisor) ; Spurný, Jiří (referee)
This master thesis focuses on the existence of σ-discrete refinement of point countable Borel additive systems in complete metric spaces. In the first three chapters we deal with the lower Borel classes, namely Gδ-additive, Fσ-additive and Fσδ-additive systems. In all cases we show the existence of σ-discrete refi- nement of the systems and even for Gδ-additive systems we don't need point countability. In the fourth chapter we deal with general Borel additive systems, but we place a limiting condition on the weight of space. In the fifth chapter we present an overview of the results that can be obtained by assuming certain additional axioms. 45
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Complexity of compact metrizable spaces
Dudák, Jan ; Vejnar, Benjamin (advisor) ; Zelený, Miroslav (referee)
We study the complexity of the homeomorphism relation on the classes of metrizable compacta and Peano continua using the notion of Borel reducibil- ity. For each of these two classes we consider two different codings. Metrizable compacta can be naturally coded by the space of compact subsets of the Hilbert cube with the Vietoris topology. Alternatively, we can use the space of continuous functions from the Cantor space to the Hilbert cube with the topology of uniform convergence, where two functions are considered as equivalent iff their images are homeomorphic. Similarly, Peano continua can be coded either by the space of Peano subcontinua of the Hilbert cube, or (due to the Hahn-Mazurkiewicz theo- rem) by the space of continuous functions from r0, 1s to the Hilbert cube. We show that for both classes the two codings have the same complexity (the complexity of the universal orbit equivalence relation). Among other results, we also prove that the homeomorphism relation on the space of nonempty compact subsets of any given Polish space is Borel bireducible with the above mentioned equivalence relation on the space of continuous functions from the Cantor space to the Polish space.
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