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Metric and analytic methods
Kaluža, Vojtěch ; Tancer, Martin (advisor) ; Kleiner, Bruce (referee) ; Fulek, Radoslav (referee)
The thesis deals with two separate problems. In the first part we show that the regular n×n grid of points in Z2 cannot be recovered from an arbitrary n2 -element subset of Z2 using only mappings with prescribed maximum stretch independent of n. This provides a negative answer to a question of Uriel Feige from 2002. The present approach builds on the work of Burago and Kleiner and McMullen from 1998 on bilipschitz non-realisable densities and bilipschitz non-equivalence of separated nets in the plane. We describe a procedure that takes a positive, measurable function and encodes it into a sequence of discrete sets. Then we show that applying this procedure to a typical positive, continuous function on the unit square yields a counter-example to Feige's question. Along the way we provide a new proof of a result on bilipschitz decomposition for Lipschitz regular mappings, which was originally proved by Bonk and Kleiner in 2002. In the second part we provide a constructive proof for the strong Hanani- Tutte theorem on the projective plane. In contrast to the previous proof by Pelsmajer, Schaefer and Stasi from 2009, the presented approach does not rely on characterisation of embeddability into the projective plane via forbidden minors. 1
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Linear codes and a projective plane of order 10
Liška, Ondřej ; Drápal, Aleš (advisor) ; Vojtěchovský, Petr (referee)
Projective plane of order 10 does not exist. Proof of this assertion was finished in 1989 and is based on the nonexistence of a binary code C generated by the incidence vectors of the plane's lines. As part of the proof of the nonexistence of code C, the coefficients of its weight enumerator were studied. It was shown that coefficients A12, A15, A16 and A19 have to be equal to zero, which contradicted other findings about the relationship among the coefficients. Presented diploma thesis elaborately analyses the phases of the proof and, in several places, enhances them with new observations and simplifications. Part of the proof is generalized for projective planes of order 8m + 2. 1
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Metric and analytic methods
Kaluža, Vojtěch ; Tancer, Martin (advisor) ; Kleiner, Bruce (referee) ; Fulek, Radoslav (referee)
The thesis deals with two separate problems. In the first part we show that the regular n×n grid of points in Z2 cannot be recovered from an arbitrary n2 -element subset of Z2 using only mappings with prescribed maximum stretch independent of n. This provides a negative answer to a question of Uriel Feige from 2002. The present approach builds on the work of Burago and Kleiner and McMullen from 1998 on bilipschitz non-realisable densities and bilipschitz non-equivalence of separated nets in the plane. We describe a procedure that takes a positive, measurable function and encodes it into a sequence of discrete sets. Then we show that applying this procedure to a typical positive, continuous function on the unit square yields a counter-example to Feige's question. Along the way we provide a new proof of a result on bilipschitz decomposition for Lipschitz regular mappings, which was originally proved by Bonk and Kleiner in 2002. In the second part we provide a constructive proof for the strong Hanani- Tutte theorem on the projective plane. In contrast to the previous proof by Pelsmajer, Schaefer and Stasi from 2009, the presented approach does not rely on characterisation of embeddability into the projective plane via forbidden minors. 1
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Semifields and planar functions
Hrubešová, Tereza ; Drápal, Aleš (advisor) ; Krýsl, Svatopluk (referee)
The aim of this diploma thesis is to introduce the topic of semifields and to explain its connection with planar functions. From its beginning the thesis leads to the formulation of relation between commutative se- mifields of odd order and planar Dembowski-Ostrom polynomials, which R. S. Coulter and M. Henderson introduce in their article from 2008. At the beginning of the thesis there is a short introduction to projective and affine planes. The thesis further describes coordinatization of projective plane by planar ternary ring. It also aims to investigate properties of ternary ring depending on the number of perspectivities in the projective plane. One of the chapters is dedicated to the isotopy of loops, which can be applied directly on the isotopy of semifields. The thesis mainly focuses on the proof of denoted correspondence between commutative semifields of odd order and planar Dembowski-Ostrom polynomials. Finally, several corrolaries of this relation and the isotopy of semifields are declared. 1
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Linear codes and a projective plane of order 10
Liška, Ondřej ; Drápal, Aleš (advisor) ; Vojtěchovský, Petr (referee)
Projective plane of order 10 does not exist. Proof of this assertion was finished in 1989 and is based on the nonexistence of a binary code C generated by the incidence vectors of the plane's lines. As part of the proof of the nonexistence of code C, the coefficients of its weight enumerator were studied. It was shown that coefficients A12, A15, A16 and A19 have to be equal to zero, which contradicted other findings about the relationship among the coefficients. Presented diploma thesis elaborately analyses the phases of the proof and, in several places, enhances them with new observations and simplifications. Part of the proof is generalized for projective planes of order 8m + 2. 1
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