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Symbolické reprezentace kompaktních prostorů
Kazda, Alexandr
Title: Symbolic representations of compact spaces Author: Alexandr Kazda Department: Department of Algebra Supervisor: Prof. RNDr. Petr Kůrka, CSc. Supervisor's e-mail address: kurka@cts.cuni.cz Abstract: The thesis concerns itself with Möbius number systems. These systems represent points using sequences of Möbius transformations. We are mainly inter- ested in representing the unit circle (which is equivalent to representing R ∪ {∞}). The main aim of the thesis is to improve already known tools for proving that a given subshift-iterative system pair is in fact a Möbius number system. We also study the existence problem: How to describe iterative systems resp. subshifts for which there exists a subshift resp. iterative system such that the resulting pair forms a Möbius number system. While we were unable to provide a complete answer to this question, we present both positive and negative partial results. As Möbius number systems are also subshifts, we can ask when a given Möbius number system is sofic. We give this problem a short treatment at the end of our thesis. Keywords: Möbius transformation, numeral system, subshift
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Quantitative properties of Banach spaces
Krulišová, Hana ; Kalenda, Ondřej (advisor)
The present thesis consists of four research papers. Each article deals with quan- tifications of certain properties of Banach spaces. The first paper is devoted to the Grothendieck property. The main result is that the space ∞ enjoys its quan- titative version. The second paper investigates quantifications of the Banach- Saks and the weak Banach-Saks property. The relationship of compact, weakly compact, Banach-Saks, and weak Banach-Saks sets is quantified, as well as some characterizatons of weak Banach-Saks sets. In the third article we discuss possible quantifications of Pelczy'nski's property (V), their characterizations and relations to quantitative versions of other properties of Banach spaces. The last paper is a continuation of the third one. We prove that C∗ -algebras have a quantita- tive version of the property (V), which generalizes one of the results obtained in the previous paper. Moreover, we establish a relationship between quantita- tive versions of the property (V) and the Grothendieck property in dual Banach spaces. 1
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Stochastic Differential Equations with Gaussian Noise
Janák, Josef ; Maslowski, Bohdan (advisor)
Title: Stochastic Differential Equations with Gaussian Noise Author: Josef Janák Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Bohdan Maslowski, DrSc., Department of Probability and Mathematical Statistics Abstract: Stochastic partial differential equations of second order with two un- known parameters are studied. The strongly continuous semigroup (S(t), t ≥ 0) for the hyperbolic system driven by Brownian motion is found as well as the formula for the covariance operator of the invariant measure Q (a,b) ∞ . Based on ergodicity, two suitable families of minimum contrast estimators are introduced and their strong consistency and asymptotic normality are proved. Moreover, another concept of estimation using "observation window" is studied, which leads to more families of strongly consistent estimators. Their properties and special cases are descibed as well as their asymptotic normality. The results are applied to the stochastic wave equation perturbed by Brownian noise and illustrated by several numerical simula- tions. Keywords: Stochastic hyperbolic equation, Ornstein-Uhlenbeck process, invariant measure, paramater estimation, strong consistency, asymptotic normality.
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Linear volatility modeling in financial time series
Kollárová, Dominika ; Zichová, Jitka (advisor) ; Hendrych, Radek (referee)
The aim of this master thesis is to introduce models belonging to ARCH(∞) representation where a time series volatility is modelled as a linear function of squared residuals. Specifically, the thesis deals with models IGARCH, FIGARCH and HYGARCH that are used to analyse, model and predict a development of financial time series. Definition and graphical illustration of individual models together with their application on real data, is supplemented by a simulation study of first-order FIGARCH model.
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Volumes of unit balls of Lorentz spaces
Doležalová, Anna ; Vybíral, Jan (advisor) ; Lang, Jan (referee)
This thesis studies the volume of the unit ball of finite-dimensional Lorentz sequence spaces p,q n . Lorentz spaces are a generalisation of Lebesgue spaces with a quasinorm described by two parameters 0 < p, q ≤ ∞. The volume of the unit ball Bp,q n of a general finite-dimensional Lorentz space was so far an unknown quantity, even though for the Lebesgue spaces it has been well-known for many years. We present the explicit formula for Vol(Bp,∞ n ) and Vol(Bp,1 n ). We also describe the asymptotic behaviour of the n-th root of Vol(Bp,q n ) with respect to the dimension n and show that [Vol(Bp,q n )]1/n ≈ n−1/p for all 0 < p < ∞, 0 < q ≤ ∞. Furthermore, we study the ratio of Vol(Bp,∞ n ) and Vol(Bp n). We conclude by examining the decay of entropy numbers of embeddings of the Lorentz spaces.
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Characterization of functions with zero traces via the distance function
Turčinová, Hana ; Nekvinda, Aleš (advisor) ; Edmunds, David Eric (referee)
Consider a domain Ω ⊂ RN with Lipschitz boundary and let d(x) = dist(x, ∂Ω). It is well known for p ∈ (1, ∞) that u ∈ W1,p 0 (Ω) if and only if u/d ∈ Lp (Ω) and ∇u ∈ Lp (Ω). Recently a new characterization appeared: it was proved that u ∈ W1,p 0 (Ω) if and only if u/d ∈ L1 (Ω) and ∇u ∈ Lp (Ω). In the author's bachelor thesis the condition u/d ∈ L1 (Ω) was weakened to the condition u/d ∈ L1,p (Ω), but only in the case N = 1. In this master thesis we prove that for N ≥ 1, p ∈ (1, ∞) and q ∈ [1, ∞) we have u ∈ W1,p 0 (Ω) if and only if u/d ∈ L1,q (Ω) and ∇u ∈ Lp (Ω). Moreover, we present a counterexample to this equivalence in the case q = ∞. 1
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Nonabsolute convergence of Newton integral
Konopka, Filip ; Spurný, Jiří (advisor) ; Zelený, Miroslav (referee)
In this thesis we search for sufficient and necessary conditions for non abso- lute convergence of Newton integral of function of the form sin φ(x) x . Importantly we analyse how the oscilation of the sine function influences the convergence of the integral. We are dealing with continous non-decreasing functions such that limx→∞ φ(x) = ∞. We proved that bilipschitz of φ is not sufficient. Nevertheless, we proved several theorems about sufficient conditions for the convergence of the integral. 1
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Stochastic Differential Equations with Gaussian Noise
Janák, Josef ; Maslowski, Bohdan (advisor)
Title: Stochastic Differential Equations with Gaussian Noise Author: Josef Janák Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Bohdan Maslowski, DrSc., Department of Probability and Mathematical Statistics Abstract: Stochastic partial differential equations of second order with two un- known parameters are studied. The strongly continuous semigroup (S(t), t ≥ 0) for the hyperbolic system driven by Brownian motion is found as well as the formula for the covariance operator of the invariant measure Q (a,b) ∞ . Based on ergodicity, two suitable families of minimum contrast estimators are introduced and their strong consistency and asymptotic normality are proved. Moreover, another concept of estimation using "observation window" is studied, which leads to more families of strongly consistent estimators. Their properties and special cases are descibed as well as their asymptotic normality. The results are applied to the stochastic wave equation perturbed by Brownian noise and illustrated by several numerical simula- tions. Keywords: Stochastic hyperbolic equation, Ornstein-Uhlenbeck process, invariant measure, paramater estimation, strong consistency, asymptotic normality.
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Stochastic Differential Equations with Gaussian Noise
Janák, Josef ; Maslowski, Bohdan (advisor) ; Duncan, Tyrone E. (referee) ; Pawlas, Zbyněk (referee)
Title: Stochastic Differential Equations with Gaussian Noise Author: Josef Janák Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Bohdan Maslowski, DrSc., Department of Probability and Mathematical Statistics Abstract: Stochastic partial differential equations of second order with two un- known parameters are studied. The strongly continuous semigroup (S(t), t ≥ 0) for the hyperbolic system driven by Brownian motion is found as well as the formula for the covariance operator of the invariant measure Q (a,b) ∞ . Based on ergodicity, two suitable families of minimum contrast estimators are introduced and their strong consistency and asymptotic normality are proved. Moreover, another concept of estimation using "observation window" is studied, which leads to more families of strongly consistent estimators. Their properties and special cases are descibed as well as their asymptotic normality. The results are applied to the stochastic wave equation perturbed by Brownian noise and illustrated by several numerical simula- tions. Keywords: Stochastic hyperbolic equation, Ornstein-Uhlenbeck process, invariant measure, paramater estimation, strong consistency, asymptotic normality.
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Quantum phase transitions in systems with a finite number of degrees of freedom
Kloc, Michal ; Cejnar, Pavel (advisor) ; Schaller, Gernot (referee) ; Šindelka, Milan (referee)
In the thesis we investigate and classify critical phenomena in the extended Dicke model (EDM) which describes the interaction between two-level atoms and a single-mode bosonic field (schematic model for cavity quantum electrodynamics). The model belongs to the class of so-called finite models, which keep the number of degrees of freedom f constant independently on the size of the system N . The important property of these systems is that the thermodynamic limit N → ∞ coincides with the classical limit ħ → 0. This allows us to study various quantum critical phenomena, in particular the ground-state quantum phase transitions (QPTs) and the excited-state quantum phase transitions (ESQPTs), by means of semiclassical methods. Using the semiclassical approach we identify and classify the QPTs and ESQPTs in various settings of the EDM and make a link to thermal phase transitions. We study the entanglement properties of both the ground state and the excited states as a function of the atom-field interaction strength. In the integrable version of the EDM we make a link between the ESQPT and monodromy, and discuss its effect on classical dynamics. The fate of monodromy under a non-integrable perturbation is observed. The dynamical consequences of the ESQPTs are examined using quantum quenches. The influence of the...
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