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Rainbow arithmetic progressions and extremal subsets of lattices
Voborník, Jan ; Šámal, Robert (advisor) ; Pangrác, Ondřej (referee)
When numbers $1,\ldots,tn$ are colored with $t$ colors (each color is used $n$ times), there exists a rainbow arithmetic progression of length $k$ (rainbow progression is a progression whose terms are colored with pairwise distinct colors). This holds true for $t>k^3$. Let $T_k$ denote the smallest $t$ for which it applies. Jungic et al. conjectured $T_k=O(k^2)$. Problem relates to extremal problems in discrete hypercubes. We present a method which uses lattices (discrete hypercubes which can contain indistinguishable elements) which can lead to improving the upper bound of $T_k$ down to $O(k^2\log k)$. In this thesis, we solve several extremal problems in lattices which have corollaries in various branches of mathematics. For example, using lattices we solve edge isoperimetric inequality in Hammilton cube, we find a graph with maximal sum of squares of degrees and convex set $M\subseteq [0,b]\times[0,a]$ which maximizes function $G(M)=\int_{x=0}^a \lambda_1(M_x)^2+\int_{y=0}^b \lambda_1(M_y)^2$. Powered by TCPDF (www.tcpdf.org)
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Algoritmy pro L-omezené toky
Voborník, Jan ; Kolman, Petr (advisor) ; Kučera, Luděk (referee)
We study the problem of maximum $L$-bounded flow, a flow decomposable to flow paths of length bounded by $L$. We review the basic results and related problems. Maximum $L$-bounded flow can be computed in polynomial time in networks with unit edge lengths but combinatorial algorithm is not known. We study combinatorial approach to this question. In networks with general edge lengths, the problem is \cNP-hard; for this problem we describe a fully polynomial approximation scheme (FPTAS) based on an algorithm for maximum multicommodity flow. This approach is practically more efficient than the previous FPTAS which was based on the ellipsoid method. Powered by TCPDF (www.tcpdf.org)
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Rainbow arithmetic progressions and extremal subsets of lattices
Voborník, Jan ; Šámal, Robert (advisor) ; Pangrác, Ondřej (referee)
When numbers $1,\ldots,tn$ are colored with $t$ colors (each color is used $n$ times), there exists a rainbow arithmetic progression of length $k$ (rainbow progression is a progression whose terms are colored with pairwise distinct colors). This holds true for $t>k^3$. Let $T_k$ denote the smallest $t$ for which it applies. Jungic et al. conjectured $T_k=O(k^2)$. Problem relates to extremal problems in discrete hypercubes. We present a method which uses lattices (discrete hypercubes which can contain indistinguishable elements) which can lead to improving the upper bound of $T_k$ down to $O(k^2\log k)$. In this thesis, we solve several extremal problems in lattices which have corollaries in various branches of mathematics. For example, using lattices we solve edge isoperimetric inequality in Hammilton cube, we find a graph with maximal sum of squares of degrees and convex set $M\subseteq [0,b]\times[0,a]$ which maximizes function $G(M)=\int_{x=0}^a \lambda_1(M_x)^2+\int_{y=0}^b \lambda_1(M_y)^2$. Powered by TCPDF (www.tcpdf.org)
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Analýza zemědělského pojištění na trhu ČR
Voborník, Jan ; Ducháčková, Eva (advisor)
Zemědělské pojištění je v dnešní době nedílnou a velmi důležitou součástí eliminace rizik zemědělského pojištění. Zemědělská výroba je stále více ohrožována nenadálými projevy klimatu, které s sebou mohou přinést katastrofické škody. Jednou z mála možností jak se těmto škodám bránit je zemědělské pojištění. Ve své práci se věnuji zemědělskému pojištění jak obecně tak z pohledu nabídky konkrétních pojistných produktů na našem trhu.
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