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Random measurable sets
Fojtík, Vít ; Rataj, Jan (advisor) ; Pawlas, Zbyněk (referee)
The aim of this thesis is to compare two major models of random sets, the well established random closed sets (RACS) and the more recent and more general random measurable sets (RAMS). First, we study the topologies underlying the models, showing they are very different. Thereafter, we introduce RAMS and RACS and reformulate prior findings about their relationship. The main result of this thesis is a characterization of those RAMS that do not induce a corresponding RACS. We conclude by some examples of such RAMS, including a construction of a translation invariant RAMS. 1
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Gradient polyconvexity and its application to problems of mathematical elasticity and plasticity
Zeman, Jiří ; Kružík, Martin (advisor) ; Zeman, Jan (referee)
Polyconvexity is a standard assumption on hyperelastic stored energy densities which, together with some growth conditions, ensures the weak lower semicontinuity of the respective energy functional. The present work first reviews known results about gradient polyconvexity, introduced by Benešová, Kružík and Schlömerkemper in 2017. It is an alternative property to polyconvexity, better-suited e.g. for the modelling of shape-memory alloys. The principal result of this thesis is the extension of an elastic material model with gradient polyconvex energy functional to an elastoplastic body and proving the existence of an energetic solution to an associated rate- independent evolution problem, proceeding from previous work of Mielke, Francfort and Mainik. 1
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Probability distributions on metric groups.
Ondřej, Josef ; Štěpán, Josef (advisor) ; Dostál, Petr (referee)
Title: Probability distributions on metric groups Author: Josef Ondřej Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Josef Štěpán, DrSc., Department of Probability and Mathematical Statistics Abstract: In this thesis we deal with the space of Borel probability measures at first on a metric space and later on a metric group. We define the notion of a weak convergence of Borel probability measures and in a special case we show this convergence is metrizable. Further we introduce operation of convolution of Borel probability measures on a metric group and we show that together with this operation the space of measures becomes a topological semigroup. We use the notion of convolution to define idempotent and Haar measure and we show a relation between them. Finally we use the mentioned results to describe all solutions of Choquet problem. At the end we demonstrate how the theory that we have developed applies to a group of complex units. Keywords: Metric group, weak convergence, Prokhorov's theorem, Choquet's theorem.
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