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Functions in examples and counterexamples
Janda, David ; Pilous, Derek (advisor) ; Zhouf, Jaroslav (referee)
The aim of my Bachelors thesis is to explicate students coming to the uni- versity the key problems in fundamentals of mathematical analysis. I focus on the most notable terms of continuity and limit, which these secondary students were acquainted with. However, majority of them just intuitively and informaly. I am trying to point out the fact, that the knowledge of many students is distortid and uncomplete. As a result it is necessary to practise and clarify this knowledge so that the intuitive imagination of these terms corresponds to the formal definition. I am trying to get this point by brea- king of intuitive imaginations of students by counterexamples. Important is a chapter named The Construction of Functions, which contains instructi- ons leading to the finding functions with specific features. Not only these features, described in this thesis, but also more complex such as derivation, primitive function or uniform convergence. It is a consequence of the fact, that the principle of examples to practise these terms is in many sights similar and repetitious. In chapters named Continuity and Limit, I am interpreting these terms using the special examples, which are in my opinion optimal for rehearsing. My intention is to help illustrate selected problematical sections of mathematical analysis.
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Vlastnosti zobrazení s konečnou distorzí
Campbell, Daniel ; Hencl, Stanislav (advisor) ; Malý, Jan (referee)
We study the continuity of mappings of finite distortion, a set of mappings intended to model elastic deformations in non-linear elasticity. We focus on continuity criteria for the inner-distortion function and prove that certain modulus of continuity estimates are sharp, i.e. cannot be im- proved. We also give a proof of the continuity of mappings of finite distortion under simplified conditions on the integrability of the distortion function. 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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Commuting continuous functions without a common fixed point
Karasová, Klára ; Vejnar, Benjamin (advisor) ; Cúth, Marek (referee)
The topic of the thesis are common fixed points of commuting functions. With the help of the Mountain climbing theorem we will prove the theorem about extending commuting functions, which will allow us to construct commuting self-mappings of the unit interval with no common fixed point. For the next part we prove several versions of the extending commuting functions theorem using different versions of the Mountain climbing theorem. We will also prove that if X is a dendroid, S an abelian semigroup of continuous monotone self-mappings of X and f : X → X commutes with each element of S, then f and S have a common fixed point. 1
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Význam stromořadí pro spojitost systému zeleně sídla
Galíková, Kateřina
This thesis about The importance of alleys for continuity of the green settlement system is dealing with the problematics of urban greenery system, the significance and usage of alleys in cities and the importance of the connection of the urban greenery. This thesis is split into two different parts. The first part, literal outline, is dealing with the problematics of urban greenery system and the significance of the usage of the alleys for the urban greenery. It is taking into consideration its full meaning and functional aspect of the city itself. We will also find in this part the importance of the greenery infrastructure, alley legislation, the possibility of restoration, rating and sorting the alleys. The second part, materials and methodology, is working with the methodology of rating the alleys. The key way to this is to rate the quality of the nature of the alleys and the methodology of rating the spatial links of the alleys. In the first part of the methodology was analysed and rated exactly sixty-five alleys in the area of Breclav with the overall length of 17,6km. The alleys were rated on the scale from 1 (the most significant) to 3 (least significant) depending on the significant of the connection of the urban greenery. The mark 1 scored 39 alleys, with the mark 2 (medium significant) was scored 5 alleys and the remaining 19 were granted the significance of 3. Methodology of rating the spatial links of the alleys in the picture schemes, shows the spatial links of the alleys and combine them with the first part of methodology. From this, we can conclude the following statement about the overall and final rating of the alleys. From the ratings of the alleys, we can take away a recommendation for the improvement of the alleys in Breclav and the overall urban system of the city. In conclusion is the evaluation of what is the meaning of the alleys for the continuity of urban greenery.
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Functions in examples and counterexamples
Janda, David ; Pilous, Derek (advisor) ; Zhouf, Jaroslav (referee)
The aim of my Bachelors thesis is to explicate students coming to the uni- versity the key problems in fundamentals of mathematical analysis. I focus on the most notable terms of continuity and limit, which these secondary students were acquainted with. However, majority of them just intuitively and informaly. I am trying to point out the fact, that the knowledge of many students is distortid and uncomplete. As a result it is necessary to practise and clarify this knowledge so that the intuitive imagination of these terms corresponds to the formal definition. I am trying to get this point by brea- king of intuitive imaginations of students by counterexamples. Important is a chapter named The Construction of Functions, which contains instructi- ons leading to the finding functions with specific features. Not only these features, described in this thesis, but also more complex such as derivation, primitive function or uniform convergence. It is a consequence of the fact, that the principle of examples to practise these terms is in many sights similar and repetitious. In chapters named Continuity and Limit, I am interpreting these terms using the special examples, which are in my opinion optimal for rehearsing. My intention is to help illustrate selected problematical sections of mathematical analysis.
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|
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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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Vlastnosti zobrazení s konečnou distorzí
Campbell, Daniel ; Hencl, Stanislav (advisor) ; Malý, Jan (referee)
We study the continuity of mappings of finite distortion, a set of mappings intended to model elastic deformations in non-linear elasticity. We focus on continuity criteria for the inner-distortion function and prove that certain modulus of continuity estimates are sharp, i.e. cannot be im- proved. We also give a proof of the continuity of mappings of finite distortion under simplified conditions on the integrability of the distortion function. 1
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Použitelnost finančních výkazů
Pohl, Petr ; Tyll, Ladislav (advisor) ; Weber, Radek (referee)
This work is analysing association between stock returns and financial statement information. It studies the usefulness of accounting data for investors and its development in time. In the beginning it summarizes main features of this relationship and derives hypothesis of weakening temporal association. In the analytical part two models for assessment of this phenomenon are introduced and based on various financial indicators available from annual reports it tests this hypothesis year by year. The evidence suggests that the usefulness of financial statement information is not deteriorating over examined period. Nonetheless, the association is very weak despite both increasing investor demand for relevant information and persistent regulator efforts to improve the quality and timeliness of financial information.
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