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Perfect functions of the first Baire class
Skovajsa, Břetislav ; Spurný, Jiří (advisor) ; Zajíček, Luděk (referee)
A wide class of problems in mathematical analysis can be described as searching for properties P such that for each F from a given system of mappings F between spaces K and L an arbitrary real valued function on L has the property P if and only if its composition with F also has this property. The inspiration for this text comes from [1], where the mentioned problem is examined in the form of stability of Baire classes of functions towards composition with a continuous mapping between compact topological spaces. The goal of this text will be to get acquainted with the original result, to slightly improve it on compact metric spaces, then to take a closer look at the finer structure of B1 functions and to try to find a similar kind of stability in this environment. [1] J. Lukeš, J. Malý, I. Netuka, J. Spurný, Integral representation theory: ap- plications to convexity, Banach spaces and potential theory, Walter de Gruyter (2010).
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Limit behavior of the Nash equlibrium
Kovařík, Vojtěch ; Spurný, Jiří (advisor) ; Bárta, Tomáš (referee)
The subject of study of game theory - games - serves as mathematical models for real-life problems. In every game there are two or more players who aim to maximize their own profit by choosing their actions. A situation where no player can benefit from changing his own action alone has got particular importance in the study of games - it is called Nash equilibrium. Games with a finite number of players have certain advantages over those with an infinite number of players. For one, problems whose model is a game with a finite number of players are quite common. Moreover, one of the classical results of game theory is that (with certain additional assumptions) in every game with a finite number of players there exists a Nash equilibrium. On the other hand, when trying to describe the properties of a game with an infinite number of players we might be able to use calculus instead of going trough all possibilities (as is common for games with a finite number of players), which tends to be computationally demanding. However, if we want to use these advantages of games with an infinite number of players, it is important first to know whether there is any relationship between games with a finite and infinite number of players at all. The goal of this thesis is to define terms and to introduce tools which would allow...
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Skorokompaktní vnoření prostorů funkcí
Křepela, Martin ; Pick, Luboš (advisor) ; Spurný, Jiří (referee)
This work is dealing with almost-compact embeddings of function spaces, in particular, the class of classical and weak Lorentz spaces with a norm given by a general weight fuction is studied. These spaces are not Banach function spaces in general, thus the almost-compact em- bedding is defined for more general sturctures of rearrangement-invariant lattices. A general characterization of when an r.i. lattice is almost-compactly embedded into a Lorentz space, involving an optimal constant of a certain continuous embedding, is proved. Based on this the- orem and appropriate known results about continuous embeddings, explicit characterizations of mutual almost-compact embeddings of all subtypes of Lorentz spaces are obtained. 1
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Quantitative weak compactness
Rolínek, Michal ; Spurný, Jiří (advisor) ; Kalenda, Ondřej (referee)
In this thesis we study quantitative weak compactness in spaces (C(K), τp) and later in Banach spaces. In the first chapter we introduce several quantities, which in different manners measure τp-noncompactness of a given uniformly bounded set H ⊂ RK . We apply the results in Banach spaces in chapter 2, where we prove (among others) a quantitative version of the Eberlein-Smulyan theorem. In the third chapter we focus on convex closures and how they affect measures of noncompactness. We prove a quantitative version of the Krein-Smulyan theorem. The first three chapters show that measuring noncompactness is intimately related to measuring distances from function spaces. We follow this idea in chapters 4 and 5, where we measure distances from Baire one functions first in RK and later also in Banach spaces. 1
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Radon-Nikodým compact spaces
Cepák, Jiří ; Spurný, Jiří (advisor) ; Holický, Petr (referee)
In the present work we study Radon-Nikodým compact spaces (RN compacta for short) their topological characterizations and properties with emphasis on those related to the problem of continuous image of RN compact. First chapter consists of auxiliary results. In second chapter we give eight characterizations of RN compacta as well as several examples. In third chapter we introduce three notions weaker than that of RN compact and stable under continuous images and we show that they are equivalent. Last chapter is devoted to partial positive solutions to the problem of continuous image.
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Properties of Poulsen simplices
Jaroň, Zdeněk ; Spurný, Jiří (advisor) ; Kurka, Ondřej (referee)
Title: Properties of Poulsen simplices Author: Zdeněk Jaroň Department: Department of Mathematical Analysis Supervisor: Doc. RNDr. Jiří Spurný, Ph.D. Abstract: In the present thesis, we study a generalisation of concept of the Poulsen simplex in general, non-metrizable case. First, for any given simplex F we con- struct a new one S, containing F as a face, having dense set of extreme points and preserving some important properties of F. In the next part, we employ this con- struction to build up, for any given infinite cardinal κ, two simplices S1, S2 with dense extreme boundary, with density character equal to κ and with spaces of affine functions Ac (S1) and Ac (S2) having the same density character, but which are not affinely homeomorphic. Keywords: Poulsen simplex, projective limit, Helly space
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Separable reduction theorems, systems of projections and retractions
Cúth, Marek ; Kalenda, Ondřej (advisor) ; Kubiš, Wieslaw (referee) ; Spurný, Jiří (referee)
This thesis consists of four research papers. In the first paper we study whether certain properties of sets (functions) are separably determined. In our results we use the "method of elementary submodels". In the second paper we generalize some results concerning Valdivia compacta (equivalently spaces with a commutative retractional skeleton) to the context of spaces with a retractional skeleton (not necessarily commutative). The third paper further studies the structure of spaces with a projectional (resp. retractional) skeleton. Under certain conditions we prove the existence of a "simultaneous projectional skeleton" and we use this result to prove other statements concerning the structure of spaces with a projectional (resp. retractional) skeleton. In the last paper we study the method of elementary submodels in a greater detail and we compare it with the "method of rich families". 1
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Open problems in Continuum thory
Seifrt, Jan ; Pyrih, Pavel (advisor) ; Spurný, Jiří (referee)
ISYixev prace: Otevfene problemy teorie konlinm Autor: Jan Seifrt Katcdra (ustav): Katedra matematicke analyzy Vedouci ba.kaliirske prace: Doc. RNDr. Pavel Pyrih, CSc. e-mail vedouciho: poliodai6.gniail.com Anstrakt: PiYdlo/ena pracc sc /a))yva v/tahrin iiia^iicli('l\ycb a koiH-rnr pn- riodickych bodu v jislycli koiiipaktiiu-li .souvislych mno/iuach. Teziste prace s])ociva v in)di-()l)in''in ro/boru dvou ])ul)liku\>uiycli vyslcdkii (motor a null- comb). Fungovam Irclito ])fikladu JL- /achycono na fade pomocnydi obraxkii. Pia(.:c obsabnjc polrubnc definicr a /aktadni 1 vr/cm' Inv, dukaxu. V praci JHOU dale doka'/ana i dalsi tvr/oni / dam'1 problcmatiky. vii shiva: dcndril., ill]I' vla.st.imsr, ina^iiutickr body a mill-comb Tillc: Open problems in Continuum thmry Author: Jan Scifrt Department.; Department of Matheinal ical Analysis Supervisor: Doc. HNDr. Pavel Pyrili, CSc. Supervisor's e-mail address: Abstract: In t lit1 present work \ve study the relation between non-wandering ami eventually-periodic- points in certain compact conned ed sets. The goal of the work consists of detailed study of two published results (engine and null-comb). How these examples work is demonstrated by a. sequence of fi- gures. The work contain all needed definitions a.nd lacts wit.lumt proofs. In the work are proved some other...
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