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The Helly numbers of systems of sets with bounded algebraic and topological complexity
Sosnovec, Jakub ; Tancer, Martin (advisor) ; Patáková, Zuzana (referee)
Maehara has shown that a family F of at least d+3 spheres in Rd has a nonempty intersection if every d+1 spheres from F have a nonempty intersection. We extend this Helly-type result in two directions. On the one hand, we show an analogous theorem holds for families of pseudospheres, i.e., systems of sets such that the intersection of any nonempty subsystem is homeomorphic to a sphere of some dimension or is empty. On the other hand, a sphere in Rd can be expressed as the zero set of a real polynomial. For a set of polynomials P, the Helly number of the family of zero sets of polynomials from P is bounded by the dimension of the vector space generated by P. For spheres, however, Maehara's result gives a stronger bound. We show some general sufficient assumptions that allow better bounds on the Helly numbers in this context. Powered by TCPDF (www.tcpdf.org)
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The Helly numbers of systems of sets with bounded algebraic and topological complexity
Sosnovec, Jakub ; Tancer, Martin (advisor) ; Patáková, Zuzana (referee)
Maehara has shown that a family F of at least d+3 spheres in Rd has a nonempty intersection if every d+1 spheres from F have a nonempty intersection. We extend this Helly-type result in two directions. On the one hand, we show an analogous theorem holds for families of pseudospheres, i.e., systems of sets such that the intersection of any nonempty subsystem is homeomorphic to a sphere of some dimension or is empty. On the other hand, a sphere in Rd can be expressed as the zero set of a real polynomial. For a set of polynomials P, the Helly number of the family of zero sets of polynomials from P is bounded by the dimension of the vector space generated by P. For spheres, however, Maehara's result gives a stronger bound. We show some general sufficient assumptions that allow better bounds on the Helly numbers in this context. Powered by TCPDF (www.tcpdf.org)
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The Use of Bayesian Methods in Investment Decisions
Sosnovec, Jakub ; Bína, Vladislav (advisor) ; Váchová, Lucie (referee)
My bachelor thesis points at the application of Bayes methods during the decision making and moreover is interested in general types of investments and their advantages and disadvantages. The output of my thesis is the portfolio of materials used for three possible investments, specifically for gold investments, funds and agricultural lands during three economical conditions of the world - during recession, stagnation and growth. The model will be furthemore entered by expert's accurate estimation of possibilities that one or another condition of the world will happen. Apart from this, results will be compared with the variation of decision making during the uncertainty and thanks to this, we will demonstrate the difficulties of this method. My thesis ends with the presentation of methods which develop the process and specify the groundwork for decision making.
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