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The Frölicher-Nijenhuis bracket and its applications in geometry and calculus of variations
Šramková, Kristína ; Tomáš, Jiří (referee) ; Kureš, Miroslav (advisor)
This Master's thesis clarifies the significance of Frölicher-Nijenhuis bracket and its applications in problems of physics. The basic apparatus for these applications is differential geometry on manifolds, tensor calculus and differential forms, which are contained in the first part of the thesis. The second part summarizes the basic theory of calculus of variations on manifolds and its selected applications in the field of physics. The last part of the thesis is devoted to the applications of Frölicher-Nijenhuis bracket in the derivation of Maxwell's equations and to the description of the geometry of ordinary differential equations.
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Invariants of jet groups and applications in continuum mechanics
Buriánek, Martin ; Doupovec, Miroslav (referee) ; Kureš, Miroslav (advisor)
This thesis is focused on jet groups and their matrix representations. The opening section deals with group representations, group actions on sets and invariants of actions. Another section explains terms such as smooth manifolds, Lie group and Lie algebra. The following part clarifies terms jet and jet group as a special example of Lie group. First of all, groups $G_1^r$ and $G_n^1$ are described, then description of group $G_n^2$ and its subgroups ensues. Representations of these jet groups are proposed. Finally, applications of jet groups in continuum mechanics are mentioned. The thesis is complemented with algorithm of chosen problems in program Wolfram Mathematica.
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Lie groups from the point of view of kinematics and applications in robotics
Kalenský, Jan ; Kureš, Miroslav (referee) ; Tomáš, Jiří (advisor)
This diploma thesis deals with the kinematic and robotic implications of Lie theory. In the introductory section, a manifold is defined as a basic element of configuration space. The main body of the thesis deals with the definition of a structure in the configuration space - Lie group. Tangent space with vector field including a structure of Lie algebra is defined to represent velocity. These two structures are connected using exponential mapping. The conclusion of the thesis focuses on fibre space, especially considering principal bundle and principal connection. Throughout the thesis, numerous examples are presented to illustrate the terms used.
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Lie groups and their physical applications
Kunz, Daniel ; Kureš, Miroslav (referee) ; Tomáš, Jiří (advisor)
In this thesis I describe construction of Lie group and Lie algebra and its following usage for physical problems. To be able to construct Lie groups and Lie algebras we need define basic terms such as topological manifold, tensor algebra and differential geometry. First part of my thesis is aimed on this topic. In second part I am dealing with construction of Lie groups and algebras. Furthermore, I am showing different properties of given structures. Next I am trying to show, that there exists some connection among Lie groups and Lie algebras. In last part of this thesis is used just for showing how this apparat can be used on physical problems. Best known usage is to find physical symmetries to establish conservation laws, all thanks to famous Noether theorem.
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Alternative Ontology: topological Imagination and Topological Materialism
Mrva, Jozef ; Csefalvay,, András (referee) ; Kořínek,, David (referee) ; Cenek, Filip (advisor)
The dissertation Alternative Ontology, subtitled Topological Imagination and Topological Materialism, focuses on the analysis of spatial phenomena and space in the intentions of the mathematical discipline of topology, which is interested in spaces from the point of view of set theory. My goal is to present topology as a tool not only for contemporary philosophy, but also for artistic creation. For the purpose of the dissertation, I formulate two concepts: Topological imagination and Topological materialism. Topological imagination is a tool and method for creating and thinking with the consciousness of space as a dynamic structure, which is not bound only by fixed laws of geometry. This method originated as the name of my long-term artistic practice, which is largely based on the study of space, topology, knot theory and the search for ways of their application in artistic and theoretical work. I propose Topological materialism as a concept that combines the thinking of networks and multi-dimensional spaces with the philosophical currents of the materialist tradition, especially the New Materialism. My basic thesis is that these cannot be perceived separately. Materialism cannot be thought without its spatial dimension, and topology without anchoring in the material world becomes a mere abstraction. The second part of the dissertation is devoted to the analysis of specific spaces: the space we inhabit, which I call phenomenological, infrastructure, logistics space, information space and the space of capital. In addition to individual analyzes, I also focus on their intersections, connections and joint operation.
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Lie groups from the point of view of kinematics and applications in robotics
Kalenský, Jan ; Kureš, Miroslav (referee) ; Tomáš, Jiří (advisor)
This diploma thesis deals with the kinematic and robotic implications of Lie theory. In the introductory section, a manifold is defined as a basic element of configuration space. The main body of the thesis deals with the definition of a structure in the configuration space - Lie group. Tangent space with vector field including a structure of Lie algebra is defined to represent velocity. These two structures are connected using exponential mapping. The conclusion of the thesis focuses on fibre space, especially considering principal bundle and principal connection. Throughout the thesis, numerous examples are presented to illustrate the terms used.
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Singular points of algebraic varieties
Vančura, Jiří ; Příhoda, Pavel (advisor) ; Šťovíček, Jan (referee)
This thesis is an introduction to exploring singularities of algebraic varieties. In the first chapter, we state basic definitions and theorems necessary for exploring singularities. Firstly, we define algebraic varieties and their corresponding ideals and explain the term of Krull dimension. We also focus on the local properties of varieties. In the second chapter, we begin by examining the term of singularity in detail and introducing methods for searching for singularities. We prove two theorems about the shape and the dimension of singularities. In the second part, we prove theorems about the zero divisors, which enable us to define Cohen-Macaulay and Gorenstein rings. We use them to roughly classify singularities of algebraic varieties.
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Alternative Ontology: topological Imagination and Topological Materialism
Mrva, Jozef ; Csefalvay,, András (referee) ; Kořínek,, David (referee) ; Cenek, Filip (advisor)
The dissertation Alternative Ontology, subtitled Topological Imagination and Topological Materialism, focuses on the analysis of spatial phenomena and space in the intentions of the mathematical discipline of topology, which is interested in spaces from the point of view of set theory. My goal is to present topology as a tool not only for contemporary philosophy, but also for artistic creation. For the purpose of the dissertation, I formulate two concepts: Topological imagination and Topological materialism. Topological imagination is a tool and method for creating and thinking with the consciousness of space as a dynamic structure, which is not bound only by fixed laws of geometry. This method originated as the name of my long-term artistic practice, which is largely based on the study of space, topology, knot theory and the search for ways of their application in artistic and theoretical work. I propose Topological materialism as a concept that combines the thinking of networks and multi-dimensional spaces with the philosophical currents of the materialist tradition, especially the New Materialism. My basic thesis is that these cannot be perceived separately. Materialism cannot be thought without its spatial dimension, and topology without anchoring in the material world becomes a mere abstraction. The second part of the dissertation is devoted to the analysis of specific spaces: the space we inhabit, which I call phenomenological, infrastructure, logistics space, information space and the space of capital. In addition to individual analyzes, I also focus on their intersections, connections and joint operation.
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Invariants of jet groups and applications in continuum mechanics
Buriánek, Martin ; Doupovec, Miroslav (referee) ; Kureš, Miroslav (advisor)
This thesis is focused on jet groups and their matrix representations. The opening section deals with group representations, group actions on sets and invariants of actions. Another section explains terms such as smooth manifolds, Lie group and Lie algebra. The following part clarifies terms jet and jet group as a special example of Lie group. First of all, groups $G_1^r$ and $G_n^1$ are described, then description of group $G_n^2$ and its subgroups ensues. Representations of these jet groups are proposed. Finally, applications of jet groups in continuum mechanics are mentioned. The thesis is complemented with algorithm of chosen problems in program Wolfram Mathematica.
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Variety potkanů a jejich genetická determinace
Tichá, Petra
The work describes variations of chat stroctures and colour of fancy rat (Rattus norvegicus var. alba) bred by hobby breeders. The document contains list of various coat types, colours and also frequently occuring coat colour patterns. The document is not limited to listing specific variations but describes their genetics origin. The work deals with specific mutations leading to differentiations of coat colour patterns. The source information used was obtained from subject articles about genetics of laboratory rats and complemented by authors own experience in fancy rat breeding. The document includes an example of genetics determination of selected types of rat varieties. The point of this work is to create a comprehensive notion of mutations affecting the fancy rat appearance and their heredity. Hobby breeders will find various breeding wals leading to occurence of standardized coat types, specigic colour types and pattern types, broadening their awareness of the problematic.
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