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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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Packing rectangles
Pavlík, Tomáš ; Šámal, Robert (advisor) ; Mareš, Martin (referee)
This thesis studies the open problem of packing rectangles. Is it possible to pack rectangles with dimensions 1/n x 1/(n+1) into a unit square? The aim of this thesis is analysis of the problem and the related algorithm. Attention will be focused mainly on the implementation of this algorithm and on study of its functioning. Powered by TCPDF (www.tcpdf.org)
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