
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...

 
 
 

Applications of descriptive set theory in mathematical analysis
Doležal, Martin ; Zelený, Miroslav (advisor) ; Holický, Petr (referee) ; Zapletal, Jindřich (referee)
We characterize various types of σporosity via an infinite game in terms of winning strategies. We use a modification of the game to prove and reprove some new and older in scribing theorems for σideals of σporous type in locally compact metric spaces. We show that there exists a closed set which is σ(1 − ε)symmetrically porous for every 0 < ε < 1 but which is not σ1symmetrically porous. Next, we prove that the realizable by an action unitary representations of a finite abelian group Γ on an infinitedimensional complex Hilbert space H form a comeager set in Rep(Γ, H). 1


Properties of sigmaporous sets
Rmoutil, Martin ; Zajíček, Luděk (advisor) ; Zelený, Miroslav (referee)
In the present thesis we prove several new results concerning porous sets. In the first two chapters we examine some properties of related sets in the space R while in the third chapter we concentrate on an entirely different problem formulated in the setting of topologically complete metric spaces. To be more specific, in the first chapter we prove non porosity of the set Ad of all real numbers x (0, 1) with decimal expansion containing the number 9 with density d. In spite of being relatively difficult, this new result has little importance in itself. It merely answers a natural question which arises from an article of L. Zajíček [8]. The main result presented in the second chapter is a significant improve ment of the following result of R.J. Najáres and L. Zajíček from the article [5]: There exists a closed set F R which is right porous, but is not left porous. Thus for any kind of "upper" porosity (i.e. a porosity defined using limsup) it is now even more unlikely for any connection between "left" and "right" to be discovered. From another work [10] of L. Zajíček arises the following question: If A X and B Y are two non lower porous G subsets of topologically complete metric spaces X and Y , is it necessarily true that the Cartesian product A × B is also non lower porous? The article [10]...

 
 