
Gamma null sets
Zelko, Matúš ; Zelený, Miroslav (advisor) ; Kaplický, Petr (referee)
The thesis is devoted to Gammanull sets, which is a σideal closely related to the differentiability of Lipschitz functions on Banach spaces. However, apart from the in troduction, where we quickly summarize some known results on the differentiability of Lipschitz functions, the work does not focus on this aspect. The aim of the thesis is to show that the Gammanull sets are well defined and to supplement the proofs of some known properties. The main contribution is a detailed treatment and completion of the omitted steps of the proof that Gammanull and Lebesguenull sets in Rn coincide. The main steps of the proof, as well as the concept of Gammanullness, come from the paper by Joram Lindenstrauss and David Preiss, On Fréchet differentiability of Lipschitz maps between Banach spaces (2003), which the thesis builds upon. Furthermore, the thesis demonstrates that Gammanull sets form a nontrivial σideal, the proof is not directly taken from the literature. 1


Complexity of classification problems in topology
Dudák, Jan ; Vejnar, Benjamin (advisor) ; Krupski, Pawel (referee) ; Zelený, Miroslav (referee)
This thesis consists of three articles. The first article focuses on compact metrizable spaces homeomorphic to their respective squares, the main result being that there ex ists a family of size continuum of pairwise nonhomeomorphic compact metrizable zero dimensional spaces homeomorphic to their respective squares. This result answers a question of W. J. Charatonik. In the second article we prove that there exists a Borel measurable mapping assigning to each Peano continuum X a continuous function from [0, 1] onto X. We also show that there exists a Borel measurable mapping assigning to each triple (X, x, y), where X is a Peano continuum and x, y are distinct points in X, an arc in X with endpoints x, y. In the third article we prove that the homeomorphism relation for absolute retracts in R2 is Borel bireducible with the isomorphism relation for countable graphs. Moreover, we prove that neither the homeomorphism relation for Peano continua in R2 nor the homeomorphism relation for absolute retracts in R3 is clas sifiable by countable structures. We also show that the homeomorphism relation (as well as the ambient homeomorphism relation) for compacta in [0, 1]n is Borel reducible to the homeomorphism relation for continua in [0, 1]n+1 . 1


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.

 
 

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.


Operation of a distribution system with resistance welders
Zelený, Miroslav ; Šlezingr, Jan (referee) ; Drápela, Jiří (advisor)
This Diploma thesis deals with assessment of the influence of two fundamental types of resistance welders operation on chosen power quality parameters at the point of common coupling (PCC) of the power network. The assessed parameters of power quality are the total harmonic distortion of the supply voltage, asymmetry of the supply voltage and the level of short term flicker at the point of common coupling. The assessment is based on the comparison of the results of computer simulations done in PSCad 4.2.0 with the requirements of technical standards. The outcome of this thesis is the determination of allowable limits for physical and operation parameters for the general arrangement of a power distribution network and a resistance welder that should guarantee the power quality compliance.


Gradient mapping of functions of several variables
Jechumtál Skálová, Alena ; Zelený, Miroslav (advisor)
Title: Gradient mapping of functions of several variables Author: Alena Skálová Department: Department of Mathematical Analysis Supervisor: doc. RNDr. Miroslav Zelený, Ph.D., Department of Mathematical Analysis Abstract: In the thesis we prove that the following statement holds true. For each d ≥ 2, for each open bounded set U ⊂ Rd and for each set F ⊂ Rd of the Borel class Fσ there exists an everywhere differentiable function u: Rd → R such that ∇u(x) ∈ U for all x ∈ Rd , ∇u(x) ∈ U for all x ∈ F, ∇u(x) ∈ ∂U for λdalmost all x ∈ Rd \ F.


Mathematical paradoxes
Wintrová, Lucie ; Pick, Luboš (advisor) ; Zelený, Miroslav (referee)
In the presented bachelor thesis we will focus on mathematical paradoxes, especially the BanachTarski paradox. We will show several paradoxes concerning decompositions of sets, such as the SierpińskiMazurkiewicz paradox. Next, we perform a constructive proof of the BanachTarski theorem in R3 using a special group of rotations. Finally, we generalize the notion of equidecomposability to continuous equidecomposability and prove that the BanachTarski pardox holds even under the stricter condition of continuous equidecomposability. This will answer de Groot's question. 1


Sigmaporous sets and the differentiation theory
Koc, Martin ; Zajíček, Luděk (advisor) ; Zelený, Miroslav (referee) ; Kolář, Jan (referee)
of the dissertation thesis Title: Sigmaporous sets and the differentiation theory Author: Martin Koc Department: Department of mathematical analysis Supervisor: Prof. RNDr. Luděk Zajíček, DrSc., Department of mathematical analysis Abstract: The thesis consists of five research articles. In the first one, it is shown that there exists a closed upper porous (in a strong sense) subset of a nonempty, topolo gically complete metric space without isolated points that is not σlower porous (in a weak sense). In the second article, a new notion of porosity with respect to a measure, that generalizes the upper porosity of a measure, is introduced. Several natural definitions of this notion are investigated. The main result of this chapter is a decomposition theorem for sets that are σporous with respect to a measure. The third article deals with sets of points at which arbitrary real functions are Lipschitz from one side and not Lipschitz from another side. A full characterization of the system generated by sets of this type is proved. In the fourth article, several results on relations among metric derived numbers for functions with values in metric spaces are shown. The last chapter deals with existence of differentiable extensions for functions defined on closed subsets of Rn . Its main result...
