In this thesis we aim to build algebraic models in computer using machine learning methods and in particular neural networks. We start with a set of axioms that describe functions, constants and relations and use them to train neural networks approximating them. Every element is represented as a real vector, so that neural networks can operate on them. We also explore and compare different representations. The main focus in this thesis are groups. We train neural representations for cyclic (the simplest) and symmetric (the most complex) groups. Another part of this thesis are experiments with extending such trained models by introducing new "algebraic" elements, not unlike the classic extension of rational numbers Q[ √ 2]. 1

This thesis reports on an evolution of the media image of major athletics competitions, concretely of the World Athletics Championships and the European Athletics Championships. Using grounded theory methodology, the thesis seeks to examine which topics are used by Czech sports journalists publishing on sport.iDnes.cz, iSport.cz, and Sport.cz news websites. This paper aims to uncover the media's agenda and its eventual gradual changes based on articles dedicated to five chosen championships. The qualitative content analysis covers the period from 2009 to 2018. All the detected topics are divided into categories and subcategories, the findings are exemplified by quotations of relevant journalistic texts. The results of this analysis demonstrate the media image of the athletics. Besides that, the thesis also shows how the Czech online media worked with several sporting events. The final part of this paper includes a comparison of observed championships and a description of categories' development. The findings are also represented in a chart.

The dissertation thesis deals with economic consequences of established institutional environment in the post-socialist economies. The institutional environment is associated with political arrangement. The main aim was to investigate whether level of political environment affects economic performance in the post-socialist economies. The level of political arrangement is expressed by two groups of indicators. Firstly, a standard expression using the three indicators is employed (Freedom in World, Polity2, Liberal Democracy Index of the V-Dem Project). Secondly, the level is associated with the individual political regimes (democratic and authoritarian regimes; liberal democracies and non-liberal regimes). Based on received results of linear regression analysis, it can be argued that the higher level of political institution supports economic performance. Simultaneously the individual analyses indicate liberal democracies achieve the highest economic performance within the individual political regimes whereas democracies have similar economic outcomes to authoritarian regimes. The first partial aim was to verify opposite relation, it means whether factors of economic performance representing the Modernization theory have affected formation of political regimes in the post-socialist area. Probit regression analysis was used for attainment of the partial aim. The analysis identified the individual factors of the Modernization theory (level of economic development and schooling) have had impact on establishment of liberal democracies. According to the results of probit analysis the other two theories explaining establishment of political institutions were discussed, namely the Early Developed Institutions and the Natural Resources Curse. In the last part, the second partial aim was achieved by testing reciprocal causal link between level of political system and economic performance with using the test of Granger causality. The results indicate there is not reciprocal causal link in Granger approach between the observed variables. Simultaneously the reciprocal causal relation occurs only within liberal democracies.

The aim of this thesis is to introduce bond portfolio management along with minimization of interest rate risk. The theoretical framework is dedicated to bonds, yield curve, Markowitz portfolio theory and portfolio management which also presents examples of active and passive strategies. The practical part focuses on portfolio bond modelling. The difference between created portfolios is caused by their composition as different combinations of corporate and state bonds are used. To achieve the aim of this thesis a simulation of fictitious market change is implemented, using interest rate decrease and increase while observing its impact on created portfolios. As a conclusion, best portfolio recommended for investors is chosen based on maximizing yield and minimizing interest rate risk.

In this work we study properties of branched polymers in a good solvent. We focus on problematic related to the size exclusion chromatography and predicting elution behavior of randomly branched polymers. We developed a software for generating self-avoiding walks (SAW) of any given non-looping architecture on a cubic lattice using Monte Carlo (MC) simulation and vali- date its reliability by presenting the scaling of different architectures: linear, 3-arm star and 6-arm star and asymmetric star. We calculate distribu- tion coefficients and calibration curves for size exclusion chromatography for various architectures to validate that the hydrodynamic radius is more suitable for predicting elution volume than the radius of gyration. Then we propose a new method for, although approximate, a very fast estimation of radius of gyration and hydrodynamic radius for different architecture using a graph method. It is done by comparing MC results with results obtained from graph theory. Then we introduce a correction to graph-theory results to fit the MC. At the end we present depletion layer calculation from MC and self-consistent field (SCF) method for polymers and their comparison. We show how calculation of depletion layer using SCF can be improved to get significantly better agreement with MC results. v

Conjunctural Bratislava is complex regional study which emphasises the topic of identity and social features of the city and its impact on the physical infrastucture and organisation of the city. First theoretical part introduces the Theory of conjunctural city based on connection between hierarchy and dynamic development in broad sense. Second part provides the systematic view of Bratislava, explaining some aspects of the theory (population, economy, urbanism, transportation, culture). Short chapter is dedicated to current conceptual ideas of Bratislava in terms of a city and a region, too. The last part of the study brings a brief prognostic view of main prospects of the city and some possible problems which may occure soon or later. Empirical study in appendix is aimed to Bratislava's hiphop culture and its musical production. I have chosen four songs for the analysis (three local ones and one from Prague hip hop group PSH). Bratislava has been chosen for its uniqueness, being one of the very few remaining conjunctural cities in Europe. Keywords: Bratislava, city, identity, hierarchy, theory, regional development, hip hop

Conjunctural Bratislava is regional study which emphasises the topic of identity and social features of the city and its impact on the physical infrastucture and organisation of the city. First theoretical part introduces the Theory of conjunctural city based on connection between hierarchy and dynamic development in broad sense. Second part provides the systematic view of Bratislava, explaining some aspects of the theory (population, economy, urbanism, transportation, culture). Short chapter is dedicated to current conceptual ideas of Bratislava in terms of a city and a region, too. The last part of the study brings a brief prognostic view of main prospects of the city and some possible problems which may occure soon or later. Empirical study in appendix is aimed to Bratislava's hiphop culture and its musical production. I analysed three local hip hop songs. Bratislava has been chosen for its uniqueness, being one of the very few remaining conjunctural cities in Europe.

The need to understand the structure of complex networks increases as both their complexity and the dependency of human society on them grows. Network centralities help to recognize the key elements of these networks. Betweenness centrality is a network centrality measure based on shortest paths. More precisely, the contribution of a pair of vertices u, v to a vertex w ̸= u, v is the fraction of the shortest uv-paths which lead through w. Betweenness centrality is then given by the sum of contributions of all pairs of vertices u, v ̸= w to w. In this work, we have summarized known results regarding both exact values and bounds on betweenness. Additionally, we have improved an existing bound and obtained more exact formulation for r-regular graphs. We have made two major contributions about betweenness uniform graphs, whose vertices have uniform betweenness value. The first is that all betweenness uniform graphs of order n with maximal degree n − k have diameter at most k, by which we have solved a conjecture posed in the literature. The second major result is that betweenness uniform graphs nonisomorphic to a cycle that are either vertex- or edge-transitive are 3-connected, by which we have partially solved another conjecture. 1

This diploma thesis deals with the problematics of revolutions. Namely with various interpretations of Velvet revolution in Czechoslovakia. First part tries to outline historical development of theories of revolution. The Research of revolutions undergone several generations and shifts of interests. This work tries to map the evolution from the oldest modern authors to the current ones and describe their points of view and how they approach revolutions. In second part, there are presented various views on Velvet revolution in Czechoslovakia. First, they are presented schematically and put in context, then they are described in detail. These interpretations are based on various assumptions and see Velvet revolution from various angles. In Third part the various interpretations are juxtaposed and are described, respectively their view of three aspects of Velvet revolution - causes, process and outcomes of revolution. These various views are then commented by using authors of general theories of revolutions. The goal of this approach is to enable better understanding of Velvet revolution, by sharpening the contures of the whole proces and also show the possibilities of use of general theories of revolution. Key words: revolution, theory of revolution, Velvet revolution, causes, process, outcomes

In this thesis, we study the discrete logarithm problem in the context of TFNP - the complexity class of search problems with a syntactically guaranteed existence of a solution for all instances. Our main results show that suitable variants of the discrete logarithm problem, which we call Index and DLog, are complete for the classes PPP and PWPP, respectively. Additionally, our reductions provide new structural insights into PWPP by establishing two new PWPP-complete problems. First, the problem Dove, a relaxation of the PPP-complete problem Pigeon. Dove is the first PWPP-complete problem not defined in terms of an explicitly shrinking function. Second, the problem Claw, a total search problem capturing the computational complexity of breaking claw-free permuta- tions. In the context of TFNP, the PWPP-completeness of Claw matches the known intrinsic relationship between collision-resistant hash functions and claw-free permuta- tions established in the cryptographic literature. 1