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Mixing of Markov chains - lower bounds for mixing
Ditrich, Jakub ; Prokešová, Michaela (advisor) ; Swart, Jan (referee)
The focus of the thesis is the convergence of irreducible aperiodic homoge- neous Markov chains with a finite and discrete set of states. Specifically, lower bounds on the time needed for the chain's marginal probability distribution to be sufficiently close to the stationary distribution, so called mixing time. Multiple methods are introdu- ced, properly motivated and proven. Finally, each method is demonstrated on a suitable example. The result is an overview of three methods that can be used to derive lower bounds for mixing time. 1
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Random walks on networks and mixing of Markov chains
Gemrotová, Kateřina ; Prokešová, Michaela (advisor) ; Pawlas, Zbyněk (referee)
The thesis presents the study of deriving upper bounds of the speed of convergence of reversible Markov chains with discrete time and discrete finite space state to their stationary distributions. We express the derived upper bound in terms of several variables and we make use of the theory of electrical networks, which will help us to represent random walks on a graph. The result of this thesis will be simply obtainable upper bound of mixing time of random walks on connected graphs with an arbitrary number of vertices and edges. Partial results will be demonstrated on simple examples and counterexamples. 1
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Speed of convergence of damped oscilations
Fara, Jakub ; Bárta, Tomáš (advisor) ; Pražák, Dalibor (referee)
We study solutions convergence of ordinary differential second order equation u′′(t)+ f(u′(t), t)u′(t) + |u|βu = 0, where β is a positive constant and f is a positive function. Physical meaning of this equation is one-dimensional damped oscilation with time va- riable environment resistance. We convert this studied function to the system of two equations of the first order. It enables us to proof the existence of some positively in- variant sets, hence we derive trajectory behaviour of solutions of this system. Thanks to that we will be able to do speed estimates of energy decrease for non-oscillation solution. Then in many cases we will be able to establish when the system solution for each time will oscillate or on the contrary when the oscillations will stop. 1
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Ekonomická konvergence v EU založená na doplněnem Solowovem modelu
Ryban, Ivan ; Klosová, Anna (advisor) ; Taušer, Josef (referee)
The topic of convergence in real GDP per capita has become a very sensitive issue, its results often depending on how the sample group, time period, estimation approach and theoretical concept are chosen. This dissertation presents a study and a convenient explanation of the Mankiw, Romer and Weil's (1992) augmentation of the Solow's (1956) neoclassical growth model and its subsequent empirical application to the EU27 over the period 1970-2010. The application is based on the convergence models designed by the Augmented Solow's model and studies convergence speed and patterns among the EU27 countries. The evidence indicates that the pace of convergence within the EU27 is much slower than what the model predicts. Nevertheless, the analysis shows that an increase in human capital has a stronger impact on per capita GDP and, by extension, on convergence than a similar increase in physical capital.
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Vztah mezi globalizací a reálnou konvergencí: ovplyvňuje konvergence v globalizaci konvergenci reálného HDP na hlavu?
Rybanová, Soňa ; Klosová, Anna (advisor) ; Nin, Lin Lin (referee)
This dissertation poses the question of whether there is a relationship between the speed of convergence of globalisation and the speed of convergence of GDP per capita. Firstly, the concepts of globalisation and real convergence and their relationship are thoroughly explained from both the theoretical and empirical point of view. And secondly, the answer to the question comes in the form of beta and sigma convergence analysis of this relationship. Thirdly, the analysis splits the countries into two groups (developed and developing countries) and finds interesting but ambiguous results in their comparison. Finally, in order to correctly interpret the results of absolute and conditional beta and sigma convergence, their theoretical and empirical overview is discussed in depth. The dissertation concludes by providing some answers to the initial question for every particular analysis. Namely, it shows that this relationship is indeed very ambiguous.
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