
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).


Additive families of Borel sets
Hronek, Radek ; Zelený, Miroslav (advisor) ; Spurný, Jiří (referee)
This master thesis focuses on the existence of σdiscrete refinement of point countable Borel additive systems in complete metric spaces. In the first three chapters we deal with the lower Borel classes, namely Gδadditive, Fσadditive and Fσδadditive systems. In all cases we show the existence of σdiscrete refi nement of the systems and even for Gδadditive 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

 

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 HahnMazurkiewicz 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.

 

WiFi and BT monitoring tool
Zelený, Miroslav ; Zamazal, Michal (referee) ; Valach, Soběslav (advisor)
This bachelor thesis deals with possibilities and ways of detection of devices using Bluetooth or WiFi technology, with the possibility of future use for monitoring of human movement. Detection is only in the 2.4 GHz band. It introduces WiFi and Bluetooth technology and their detection. Device detection was performed using Kismet software and Raspberry Pi platform.


Nonabsolute convergence of Newton integral
Konopka, Filip ; Spurný, Jiří (advisor) ; Zelený, Miroslav (referee)
In this thesis we search for sufficient and necessary conditions for non abso lute convergence of Newton integral of function of the form sin φ(x) x . Importantly we analyse how the oscilation of the sine function influences the convergence of the integral. We are dealing with continous nondecreasing functions such that limx→∞ φ(x) = ∞. We proved that bilipschitz of φ is not sufficient. Nevertheless, we proved several theorems about sufficient conditions for the convergence of the integral. 1


Lojasiewicz inequality for various classes of functions
Surma, Martin ; Bárta, Tomáš (advisor) ; Zelený, Miroslav (referee)
Bachelor thesis pursue the Łojasiewicz inequality. The Łojasiewicz inequality is proved here for generalized MorseBott functions and for functions with simple normal crossings. Further on, we study optimality of the Łojasiewicz exponent for those functions. In the last chapter, there are possible applications of the Łojasiewicz inequality to certain gradientlike differential equation stated and proved, such as the theorem on convergence of its solution. There is also shown how one can use the Łojasiewicz exponent to estimate the rate of the convergence. 1


Properties of derivative
Marková, Hana ; Zelený, Miroslav (advisor) ; Hencl, Stanislav (referee)
In the bachelor thesis we relate the concepts of derivative, the Darboux pro perty and the function of the Baire class one. It is shown that each derivative has Darboux property and is of the Baire class one. Furthermore, we characterize the functions of the Baire class one using their associated sets. We introduce the concept of Zahorski classes and put them in connection with the functions of the Baire class one with the Darboux property. At the end of the thesis, we prove the ClarksonDenjoy theorem.


Continuous mappings and fixedpoint theorems
Vondrouš, David ; Holický, Petr (advisor) ; Zelený, Miroslav (referee)
This thesis deals with images of compact convex sets under a continuous mapping. We will show a combinatorial proof of famous Brouwer's fixedpoint theorem based on Sperner's lemma. Later, this theorem will be applied for proving Brouwer's invariance of domain theorem, which asserts that image of an open subset of an euclidean space under a continuous mapping is open too. Then we will compare this proof with another proof using Borsuk's theorems. Their proof is more complicated, nevertheless it turns out that Borsuk's theorems give stronger results. One of them is, for instance, an analogy of the Darboux property for continuous mappings in an multidimensional space. 1
