2013/2

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    Topological frame extension
    (Vysoké učení technické v Brně, Fakulta strojního inženýrství, Ústav matematiky, 2013) Vaziry, Zohref; Leseberg , Dieter
    The concept of nearness on a set was introduced by H. Herrlich. D. Le- seberg generalized nearness by introducing supernearness, which generalizes also supertopology as de ned by D. Doitchinov. In this paper, our work is based on the representation theorem of M. H. Stone and the de nition of nearness. We de ne proximity and nearness on a Boolean frame and then, by using these, we de ne supertopic frame, supernear frame and paranear frame. We study basic properties of the concepts de ned. We also introduce a topological extension on a Boolean frame and investigate its behavior.
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    From Lie algebras to Lie groups within synthetic differential geometry: Weil sprouts of Lie's third fundamental theorem
    (Vysoké učení technické v Brně, Fakulta strojního inženýrství, Ústav matematiky, 2013) Nishimura, Hirokazu
    Weil prolongations of a Lie group are naturally Lie groups. It is not known in the theory of in nite-dimensional Lie groups how to construct a Lie group with a given Lie algebra as its Lie algebra or whether there exists such a Lie group at all. We will show in this paper how to construct some Weil prolongations of this mythical Lie group from a given Lie algebra. We will do so within our favorite framework of synthetic di erential geometry.
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    Optimal treatment strategies for musculoskeletal diseases
    (Vysoké učení technické v Brně, Fakulta strojního inženýrství, Ústav matematiky, 2013) Haugen, Kjetil K.
    This paper presents a model which aims to find an optimal treatment time for patients with musculoskeletal diseases. The stochastic dynamic programming model presented is based on minimizing expected costs given a stochastic processes for cure with and without treatment as well as costs for treatment and reduced quality of life as a consequence of the disease. The main results derived in the paper include explicit conditions for an interior optimal solution and a simple and intuitive approximate solution. Even though the presented model is specifically aimed at musculoskeletal dis- eases, the model’s generality should make it applicable for a wide range of disease treatment strategies.
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    Mathematical evolution in discrete networks
    (Vysoké učení technické v Brně, Fakulta strojního inženýrství, Ústav matematiky, 2013) Pfaltz, John F.
    This paper provides a mathematical explanation for the phenomenon of \triadic closure" so often seen in social networks. It appears to be a natural consequence when network change is constrained to be continuous. The concept of chordless cycles in the network's \irreducible spine" is used in the analysis of the network's dynamic behavior. A surprising result is that as networks undergo random, but continuous, perturbations they tend to become more structured and less chaotic.
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    Dwindlable R-algebras
    (Vysoké učení technické v Brně, Fakulta strojního inženýrství, Ústav matematiky, 2013) Kureš, Miroslav
    The concept of a dwindlable R-algebra is defined for an arbitrary topo- logical R-algebra A. It is proved that if A is dwindlable, then its subalgebra of fixed points is trivial. It is also demonstrated that for an algebra gradable by radical its dwindlability depends on a dwindlability of its factor.