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A characterization of sets with positive reach
Fryš, Filip ; Rataj, Jan (advisor) ; Pokorný, Dušan (referee)
The main source of this thesis is the article Sets with positive reach by the German mathematician Prof. Dr. Victor Bangert from 1982. In this paper, Victor Bangert gives a characterization of sets with positive reach as subsets of connected Riemannian manifolds using weakly regular sublevel sets of functions, the class of which he introduces in his earlier article Analytische Eigenschaften konvexer Funtionen auf Riemannschen Mannigfaltigkeiten from 1979. The aim of this thesis is to study the above mentioned article from 1982 from Bangert and to give a detailed proof for the special case of the Riemannian manifold Rn . After the introductory chapter, where we shall get acquainted with Bangert's article and the aim of the thesis, the first chapter follows, in which we will introduce the basic notation and introduce some necessary knowledge and definitions. In the second chapter we shall deal with the sets with positive reach themselves, give some examples and their basic properties. In the third chapter we will take a closer look at the Bangert's class of functions, and in the fourth chapter we will characterize the sets with positive reach in Rn . 1
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Sets with positive reach and their intersections
Komárek, Daniel ; Rataj, Jan (advisor) ; Pokorný, Dušan (referee)
The goal of this thesis is to collect various properties of sets with positive reach and to describe generalization of the directional curvatures in R3 as the intersection of a plane and a set with positive reach. Firstly, we define sets with positive reach, their Tangent and Normal cones, show basic properties accompanied by some characterizations of sets with positive reach. Then, we generalize principal curvatures for sets with positive reach and describe generalization of Euler's identity about normal curvature in R3 . 1
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A guide to fractal geometry
Hajmová, Kateřina ; Pokorný, Dušan (advisor) ; Boček, Leo (referee)
This text is intended for the general public. The aim of this work is acquaint readers with foundations of the fractal geometry. The thesis explains important terminology, such as the coastline paradox or the fractal dimension. A great emphasis is placed on explaining the concept of the box-counting dimension. The thesis includes the construction methods of the L-systems, IFS, TEA and random fractals. In addition, it shows the use of fractal geometry in practice. The text is completed with illuminating figures drawn in most cases in Geogebra software and Wolfram Mathematica.
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A guide to fractal geometry
Hajmová, Kateřina ; Pokorný, Dušan (advisor) ; Boček, Leo (referee)
This text is intended for the general public. The aim of this work is acquaint readers with foundations of the fractal geometry. The thesis explains important terminology, such as the coastline paradox or the fractal dimension. A great emphasis is placed on explaining the concept of the box-counting dimension. The thesis includes the construction methods of the L-systems, IFS, TEA and random fractals. In addition, it shows the use of fractal geometry in practice. The text is completed with illuminating figures drawn in most cases in Geogebra software and Wolfram Mathematica.
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