In this thesis, the metastable states are studied. The work focuses on a theoretical model of one or two metastable states decaying into a continuum of states which is bounded from below. We examine the time evolution of such systems and how it is affected by the energy of the metastable state(s) and by the position of the poles of the scattering matrix in the complex plane. We also look closely at the spectral line shape. Numerical integration of a system of differential equations is used for solving the problem of the time evolution and spectral line shape while Freshbach-Fano projection operator formalism is used for finding the position of the poles. The results are compared with first order perturbation theory and with semi-analytical formula obtained from adiabatic elimination of the continuum. The last part of the thesis is dedicated to an application of the model on neon-helium-neon cluster. 1

In this thesis we study a deformation of a black-hole spacetime due to another strong sources of gravity. Keeping within static and axially symmetric metrics, we consider a binary of Schwarzschild black holes held apart from each other by a repulsive effect of an Appell ring. After verifying that such a system can rest in static equilibrium (without any supporting struts), we compute its several basic geometric characteristics and we plot simple invariants determined by the metric functions (especially lapse, or, equivalently, potential) and by their first and second derivatives (gravitational acceleration and Kretschmann scalar). Then we extend the analysis below the black-hole horizon and inspect the behaviour of the scalars inside. The geometry turns out to be deformed in a non-trivial way, we even find regions of negative Kretschmann scalar in some cases. In the second part, we present a summary of the perturbative solution describing a slowly rotating system of a black hole surrounded by a thin finite circular disc, and an analysis of equatorial circular geodesics in such a spacetime. 1

Immersion is a notion of graph inclusion related to the notion of graph minors. While the structural theory of graph minors is extensive, there are still numerous open problems in the structural theory of graph immersions. Kuratowski's theorem claims that the class of graphs that do not contain a subdivision of the graphs K3,3 and K5 is exactly the class of planar graphs. The main goal of this thesis is to describe the structure of the graphs that do not contain an immersion of K3,3. Such graphs can be separated by small edge cuts into small graphs or planar 3-regular graphs. 1

The goal of this master thesis is to design and implement an analytical application for Security Information Service by providing a software prototype. The solution proposes an enhancement of existing graph that allows security analytics to analyse, edit and visualize objects and relations that are saved into a relational database. In the thesis, we walk through the process of development step by step. First, we investigate the current version software and the requirements of the customer. Afterwards, we design the architecture to be easily extendable with new modules and reliable libraries. In the next step, we implement the application, present our solution to the customer and conduct excessive testing. The final step is evaluating our solution by comparing it to the current software solution in use.

This thesis is focused on the theoretical bases of contemporary analysis of Turkish nationalism and Turkish identity. The first part of this thesis in two chapters provides a brief review of the historical development of Turkish nationalism and the main sources of its ideas, and then summarizes the evolution of theoretical debates on the nationalism and shows the most influent theories of the end of 20th century. The main part of the thesis analyzes texts of contemporary Turkish experts concerned with Turkish nationalism in the book Milliyetçilik. The analysis is focused in detail on four main topics. The aim of the analysis is to show how today's Turkish experts approach the study of Turkish nationalism and which thoughts and theoretical bases they work with.

of doctoral thesis "Computational Homotopy Theory": We consider several basic problems of algebraic topology, with connections to combinatorial and geometric questions, from the point of view of compu- tational complexity. The extension problem asks, given topological spaces X, Y , a subspace A ⊆ X, and a (continuous) map f : A → Y , whether f can be extended to a map X → Y . For computational purposes, we assume that A, X, Y are represented as finite simplicial complexes and f as a simplicial map. We study the problem under the assumption that, for some d ≥ 1, Y is d- connected, otherwise the problem is undecidable by uncomputability of the fundamental group; We prove that, this problem is still undecidable for dim X = 2d + 2. On the other hand, for every fixed dim X ≤ 2d + 1, we obtain an algorithm that solves the extension problem in polynomial time. We obtain analogous complexity results for the problem of determining the set of homotopy classes of maps X → Y . We also consider the computation of the homotopy groups πk(Y ), k ≥ 2, for a 1-connected Y . Their computability was established by Brown in 1957; we show that πk(Y ) can be computed in polynomial time for every fixed k ≥ 2. On the other hand, we prove that computing πk(Y ) is #P-hard if k is a part of input. It is a strengthening of...

Title: Model Problems of the Theory of Gravitation Author: Marián Pilc Department: Institute of Theoretical Physics Faculty of Mathematics and Physics Supervisor: prof. RNDr. Jiří Bičák, DrSc., dr. h. c., Institute of Theoretical Physics Faculty of Mathematics and Physics Abstract: Equations of motion for general gravitational connection and orthonormal coframe from the Einstein-Hilbert type action of the Einstein-Cartan theory are derived. Ad- ditional gauge freedom is geometrically interpreted. Our formulation does not fix coframe to be tangential to spatial section hence Lorentz group is still present as part of gauge freedom. 3+1 decomposition introduces tangent Minkowski structures hence Hamilton-Dirac approach to dynamics works with Lorentz connection over spatial sec- tion. The second class constraints are analyzed and Dirac bracket is defined.Reduction of phase space is performed and canonical coordinates are introduced. The second part of this thesis is dedicated to quantum formulation of Einstein-Cartan theory. Point version of Einstein-Cartan phase space is introduced. Basic variables, crucial for quan- tization, are derived via groups acting on the phase space and their selfadjoint represen- tation is found. Representation of basic variables of Einstein-Cartan theory is derived via infinite...

Název práce: Studium efektivních Lagrangianů a jejich aplikace Autor: Jaroslav Trnka Katedra: Ústav částicové a jaderné fyziky Vedoucí disertační práce: RNDr. Jiří Novotný, CSc., ÚČJF Abstrakt: V této práci studujeme různé aspekty efektivních teorií pole pro kvan- tovou chromodynamiku (QCD). V prvních dvou kapitolách se zaměříme na efek- tivní teorii pro resonance, která interpoluje mezi nízkoenergetickou efektivní teorií (Chirální poruchová teorie) a vysokoenergetickou QCD. V rámci této teorie studu- jeme jednosmyčkovou renormalizaci, jak z pohledu výpočetního pomocí SS-PP korelátoru, tak i čistě koncepčního studiem dynamicky generovaných stupňů vol- nosti. Ve čtvrté kapitole studujeme amplitudy rozptylu v rámci nelineárního sig- ma modelu, který představuje vedoucí člen nízkoenergetické efektivní teorie pro QCD. V návaznosti na nedávné objevy v rámci Yang-Mills teorie se nám podaří v rámci tohoto modelu zkonstruovat rekurzivní relace pro stromové amplitudy. Kromě čistě teoretické důležitosti tohoto faktu představuje tato metoda efektivní výpočetní nástroj nezávislý na formulaci amplitud pomocí Feynmanovských dia- gramů. Klíčová slova: efektivní teorie pole, kvantová chromodynamika, nelineární sigma model...

Title: Extremal combinatorics of matrices, sequences and sets of permutations Author: Josef Cibulka Department: Department of Applied Mathematics Supervisor: Doc. RNDr. Pavel Valtr, Dr., Department of Applied Mathematics Abstract: This thesis studies questions from the areas of the extremal theory of {0, 1}-matrices, sequences and sets of permutations, which found many ap- plications in combinatorial and computational geometry. The VC-dimension of a set P of n-element permutations is the largest integer k such that the set of restrictions of the permutations in P on some k-tuple of positions is the set of all k! permutation patterns. We show lower and upper bounds quasiexponential in n on the maximum size of a set of n-element permutations with VC-dimension bounded by a constant. This is used in a paper of Jan Kynčl to considerably improve the upper bound on the number of weak isomorphism classes of com- plete topological graphs on n vertices. For some, mostly permutation, matrices M, we give new bounds on the number of 1-entries an n × n M-avoiding matrix can have. For example, for every even k, we give a construction of a matrix with k2 n/2 1-entries that avoids one specific k-permutation matrix. We also give almost tight bounds on the maximum number of 1-entries in matrices avoiding a fixed layered...

of doctoral thesis Structural graph theory Jan Hladký In the thesis we make progress on the Loebl-Komlós-Sós Conjecture which is a classic problem in the field of Extremal Graph Theory. We prove the following weaker version of the Conjecture: For every α > 0 there exists a number k0 such that for every k > k0 we have that every n-vertex graph G with at least (1 2 +α)n vertices of degrees at least (1+α)k contains each tree T of order k as a subgraph. The proof of our result follows a strategy common to approaches which employ the Szemerédi Regularity Lemma: the graph G is decomposed, a suitable combinatorial structure inside the decomposition is found, and then the tree T is embedded into G using this structure. However the decomposition given by the Regularity Lemma is not of help when G sparse. To surmount this shortcoming we develop a decomposition technique that applies also to sparse graphs: each graph can be decomposed into vertices of huge degrees, regular pairs (in the sense of the Regularity Lemma), and two other components each exhibiting certain expander-like properties. The results were achieved in a joint work with János Komlós, Diana Piguet, Miklós Simonovits, Maya Jakobine Stein and Endre Szemerédi. 1