|
|
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
|
|
|
Exceptional Sets in Mathematical Analysis
Rmoutil, Martin ; Kalenda, Ondřej (advisor) ; Holický, Petr (referee) ; Zindulka, Ondřej (referee)
Title: Exceptional Sets in Mathematical Analysis Author: Martin Rmoutil Department: Department of Mathematical Analysis Supervisor: Doc. RNDr. Ondřej Kalenda, Ph.D., DSc., Department of Mathematical Analysis Abstract: The present thesis consists of four research articles. In the first paper we study the notion of σ-lower porous set; our main result is the existence of two closed sets A, B ⊂ R which are not σ-lower porous, but their product in R2 is lower porous. In the second and third article we use a set-theoretical method of el- ementary submodels involving the Lwenheim-Skolem theorem to prove that certain σ-ideals of sets in Banach spaces are separably determined. In the second article we do so for σ-porous sets and σ-lower porous sets. In the next article we refine these methods obtaining separable determination of a wide class of σ-ideals. In both cases we derive interesting corollaries which extend known theorems in separable spaces to the nonseparable setting; for example, we obtain the following theorem. Any continuous convex function on an Asplund space is Frchet differentiable outside a cone small set. In the fourth article we introduce the following notion. A closed set A ⊂ Rd is said to be c-removable if the following is true: Every real function on Rd is convex whenever it is continuous on Rd...
|
|
|
Future predicting and the axiom of choice
Jarosil, Lukáš ; Pyrih, Pavel (advisor) ; Kalenda, Ondřej (referee)
Given arbitrary function f : R → R it seems practically impossible to predict its future values based on our knowledge of its previous values. Nevertheless, axiom of choice surprisingly implies the existence of strategy that from values of the function f on some interval (s, t) correctly predicts its values on interval [t, t+ ) for every t of real line except for countable set. This result of Christopher Hardin and Alan Taylor is presented along with its generalization to mappings from topological space in the context of hat guessing games, mathematical games in which the players are supposed to guess color of their own hat while knowing only colors of other's hats. 1
|
|
|
Isomorphic and isometric classification of spaces of continuous and Baire affine functions
Ludvík, Pavel ; Spurný, Jiří (advisor) ; Kalenda, Ondřej (referee) ; Fabian, Marián (referee)
This thesis consists of five research papers. The first paper: We prove that under certain conditions, the existence of an isomorphism between spaces of continuous affine functions on the compact convex sets imposes home- omorphism between the sets of its extreme points. The second: We investigate a transfer of descriptive properties of elements of biduals of Banach spaces con- strued as functions on dual unit balls. We also prove results on the relation of Baire classes and intrinsic Baire classes of L1-preduals. The third: We identify intrinsic Baire classes of X with the spaces of odd or homogeneous Baire functions on ext BX∗ , provided X is a separable real or complex L1-predual with the set of extreme points of its dual unit ball of type Fσ. We also provide an example of a separable C∗ -algebra such that the second and second intrinsic Baire class of its bidual differ. The fourth: We generalize some of the above mentioned results for real non-separable L1-preduals. The fifth: We compute the distance of a general mapping to the family of mappings of the first resolvable class via the quantity frag and we introduce and investigate a class of mappings of countable oscillation rank.
|
|
|
Pologrupy operátorů a jejich orbity
Vršovský, Jan ; Müller, Vladimír (advisor) ; Kalenda, Ondřej (referee) ; Fašangová, Eva (referee)
Title: Semigroups of operators and its orbits Author: Jan Vršovský Department: Institute of Mathematics of the Academy of Sciences of the Czech Republic Supervisor: prof. RNDr. Vladimír Müller, DrSc., Institute of Mathematics of the AS CR Abstract: The orbit of a bounded linear operator T on a Banach space is a se- quence T n x, n = 0, 1, 2, . . ., where x is a fixed vector. The orbits are closely connected to the dynamics of operator semigroups and to the invariant sub- spaces and subsets. The thesis studies the relation between the operator and its orbits. The subject of the first part is the relation between sequences T n x and T n , stability and orbits tending to infinity. The second part deals with dense orbits - hypercyclicity and related notions. In the third part, an ana- logue of reflexive algebras of operators, orbit reflexive operators are defined and studied. Apart from "normal" orbits of a single operator, the weak orbits and orbits of C0-semigroups are also touched. Keywords: operator, semigroup, orbit, hypercyclic, orbit reflexive
|
|
|
Creditor coordination model - discrete case and limit behaviour
Šedek, Jan ; Kalenda, Ondřej (advisor) ; Polyák, Oliver (referee)
This thesis concerns with the theory of global games, in particular its ap- plication to creditor's coordination models. Several variants of these models are introduced throughout the text. All of the variants are based on the as- sumption of the finite number of players. This is rather unusual, because the literature on creditor coordination models mainly builds on the assumption of a continuum of players. We firstly describe the model for two players then we generalize it to the symmetric n-player model. After that, we analyze the limit behavior of n-player symmetric model and show the convergence of the model to the model, which is based on the assumption of a continuum of players. The thesis then deals with the asymmetric models of one big and other small players. An analysis of existence and uniqueness of equilibrium is attached to the solution of the models, where the analysis turns out to be feasible. 1
|
|
|
Composition operators on function spaces
Novotný, Matěj ; Spurný, Jiří (advisor) ; Kalenda, Ondřej (referee)
Univerzita Karlova Abstract of the bachelor thesis Composition operators on function spaces Matěj Novotný, Praha 2011 In the thesis we define what is an composition operator on the space of continuous or measurable functions of one complex variable so that we may proceed to study its properties depending on properties of the mapping the operator is induced by. We search for conditions under which the operator is continuous, compact and an isomorphism. We roughly estimate the spectrum of an operator defined on a space of continuous functions. 1
|
|
|
New measures of weak non-compactness
Bendová, Hana ; Kalenda, Ondřej (advisor) ; Holický, Petr (referee)
The main topic of this thesis is the measures of weak non-compactness, which, in different ways, measure weak non-compactness of bounded sets in Banach spa- ces. Besides some known measures of weak non-compactness, we introduce new measures, that are more natural in some sense, and we show the relationships be- tween them. We prove quantitative versions of Eberlein-Grothendieck, Eberlein- Šmulian, and James' theorems. Afterwards, we deal with measures of weak non-compactness of the unit ball and measures of weak non-compactness of sets in Banach spaces with w∗ -angelic dual unit ball. We prove that in these cases some of the defined measures coincide. Finally, we focus on the behaviour of the defined measures while passing to convex and absolute convex hull. We prove quantitative version of Krein's theorem and we also prove that most of the mea- sures do not change when passing to convex and absolute convex hull in Banach spaces with w∗ -angelic dual unit ball.
|
|
|
Some results in convexity and in Banach space theory
Kraus, Michal ; Lukeš, Jaroslav (advisor) ; Kalenda, Ondřej (referee) ; Smith, Richard (referee)
This thesis consists of four research papers. In the first paper we construct nonmetrizable compact convex sets with pathological sets of simpliciality, show- ing that the properties of the set of simpliciality known in the metrizable case do not hold without the assumption of metrizability. In the second paper we construct an example concerning remotal sets, answering thus a question of Martín and Rao, and present a new proof of the fact that in every infinite- dimensional Banach space there exists a closed convex bounded set which is not remotal. The third paper is a study of the relations between polynomials on Banach spaces and linear identities. We investigate under which conditions a linear identity is satisfied only by polynomials, and describe the space of poly- nomials satisfying such linear identity. In the last paper we study the coarse and uniform embeddability between Orlicz sequence spaces. We show that the embeddability between two Orlicz sequence spaces is in most cases determined only by the values of their upper Matuszewska-Orlicz indices. 1
|
|
|
Topologies defined using ideals
Dvořáková, Karolína ; Kalenda, Ondřej (advisor) ; Murtinová, Eva (referee)
In this thesis we study the topologies formed by a modification of some given topology using ideals - we focus on localizable and strongly localizable ideals. In the first chapter we use a certain set mapping to define ideal topology, then we show its relation to the initial topology. Next we investigate what properties the elements of ideal obtains in the new topology, for example on certain conditions the ideal becomes exactly the set of all nowhere dense sets in the ideal topology. Finally, we show when the new topology is regular and formulate necessary and sufficient conditions for a set with ideal topology to be a Baire space. In the second chapter we apply the results on concrete examples of ideals and topologies defined by them.
|
guest ::
RSS feed.