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Lebesgue density theorem for Haar measure
Sterzik, Marek ; Simon, Petr (advisor) ; Zahradník, Miloš (referee)
In this work, we study Lebesgue theorem analogy in the space 2k with Haar measure and a related theorem about -k-linkedness of the measure algebra of this space. The whole text is divided in three chapters. In the first chapter we explain some important definitions and basic properties of the measure space. The Lebesgue theorem is studied in the second chapter. After the essential definition of the point of density, the major part of the chapter is dedicated to the proof of the theorem. The theorem states, that the symmetric difference between any measurable set and the set of its points of density has measure zero. In the third chapter we study the -k-linkedness theorem; a theorem which states that the measure algebra of the space 2 is -k-linked, if 2 .
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3D-surface smoothing
Mácha, Radek ; Pergel, Martin (advisor) ; Sterzik, Marek (referee)
The present work deals with 3D surface subdivision methods. The key areas are the quality of subdivision sequence outputs and the development of methods that measure it. The Catmull-Clark and Butterfly (8-point stencil) outputs are being measured for edge length, face surface and angles between adjacent faces. Furthermore, combined sequences of subdivision algorithms are being discussed, as well as the selection of one that meets custom criteria.
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Covering All Lines Intersecting a Convex Domain
Sterzik, Marek ; Valtr, Pavel (advisor)
For a given covnex body we try to find the shortest possible set (optionally admitting some prescribed properties) meeting all lines meeting the given body. The size of the covering set is measured by the Hausdorff 1-dimensional measure 1. In the first chapter there is given an introduction to the problem. In the second chapter we discuss the upper bound for the minimal covering set. In the third chapter we discuss the existence and properties of the minimal covering. In the fourth chapter we show some lower bounds for the size of a covering. In the fifth chapter we study some related topics and a generalization of the problem.
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Pokrývání sečen konvexní oblasti
Sterzik, Marek ; Matoušek, Jiří (referee) ; Valtr, Pavel (advisor)
For a given covnex body we try to find the shortest possible set (optionally admitting some prescribed properties) meeting all lines meeting the given body. The size of the covering set is measured by the Hausdorff 1-dimensional measure 1. In the first chapter there is given an introduction to the problem. In the second chapter we discuss the upper bound for the minimal covering set. In the third chapter we discuss the existence and properties of the minimal covering. In the fourth chapter we show some lower bounds for the size of a covering. In the fifth chapter we study some related topics and a generalization of the problem.
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3D-surface smoothing
Mácha, Radek ; Sterzik, Marek (referee) ; Pergel, Martin (advisor)
The present work deals with 3D surface subdivision methods. The key areas are the quality of subdivision sequence outputs and the development of methods that measure it. The Catmull-Clark and Butterfly (8-point stencil) outputs are being measured for edge length, face surface and angles between adjacent faces. Furthermore, combined sequences of subdivision algorithms are being discussed, as well as the selection of one that meets custom criteria.
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|
|
Covering All Lines Intersecting a Convex Domain
Sterzik, Marek ; Valtr, Pavel (advisor)
For a given covnex body we try to find the shortest possible set (optionally admitting some prescribed properties) meeting all lines meeting the given body. The size of the covering set is measured by the Hausdorff 1-dimensional measure 1. In the first chapter there is given an introduction to the problem. In the second chapter we discuss the upper bound for the minimal covering set. In the third chapter we discuss the existence and properties of the minimal covering. In the fourth chapter we show some lower bounds for the size of a covering. In the fifth chapter we study some related topics and a generalization of the problem.
|
|
|
Lebesgue density theorem for Haar measure
Sterzik, Marek ; Zahradník, Miloš (referee) ; Simon, Petr (advisor)
In this work, we study Lebesgue theorem analogy in the space 2k with Haar measure and a related theorem about -k-linkedness of the measure algebra of this space. The whole text is divided in three chapters. In the first chapter we explain some important definitions and basic properties of the measure space. The Lebesgue theorem is studied in the second chapter. After the essential definition of the point of density, the major part of the chapter is dedicated to the proof of the theorem. The theorem states, that the symmetric difference between any measurable set and the set of its points of density has measure zero. In the third chapter we study the -k-linkedness theorem; a theorem which states that the measure algebra of the space 2 is -k-linked, if 2 .
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