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Tensors and their geometrical and mechanical applications
Kunz, Daniel ; Kureš, Miroslav (referee) ; Tomáš, Jiří (advisor)
In this thesis I describe construction of tensor algebra and its following usage of this magnificent structure for description of curved surfaces. This structure can be used in geometry or for example in mechanic. The thesis is focused on clear construction and if it is possible than to sustain visual aspect of current problem. Main task of this thesis was got the feel of tensor algebra and its construction and then use it on tasks in physic and mechanic.
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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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Tensors and their applications
Korbel, Filip ; Tomáš, Jiří (referee) ; Doupovec, Miroslav (advisor)
The goal of this thesis is to provide an overview of basic terms and results of tensor calculus. We introduce tensors as multilinear mappings and introduce basic tensor operations. In the next section, we give some examples of tensors, especially from the field of differential geometry.
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Two types of septic trinomials and their use in hyperelliptic cryptography
Felcmanová, Adéla ; Tomáš, Jiří (referee) ; Kureš, Miroslav (advisor)
This thesis deals with two types of septic trinomials and genus three hyperelliptic curves constructed from them. It includes an introduction to the theory of hyperelliptic curves and divisors, as well as terms and algorithms necessary for their implementation in hyperelliptic cryptosystems. The principle of the hyperelliptic curve cryptography is presented along with two examples of cryptosystems. It also contains several exercises, some of which were programmed in Python language.
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Symmetric group, its representation and applications in molecular and quantum chemistry
Krchová, Lenka ; Tomáš, Jiří (referee) ; Kureš, Miroslav (advisor)
The subject of this bachelor thesis is the study of the symmetric groups, their representation and application in molecular chemistry. At first, the particular terms from the algebra are defined, which are which are necessary to define the concept of a group. Many of them are suúpplemented by pictures for clarity and better understanding. Then, the algebraic structures, which are accompanied by clear schemes and concrete examples, are explained. Also, the symmetric groups are demonstrated on the example of the square and triangle. After that, the reader gets into the chapter about the representation of final groups where the structure of the work is similar. First, the relevant terms are defined and then the author focuses on Young's diagrams. These are meticulůously described, few examples are mentioned and so is their working procedure. The last part of the bachelor thesis is dedicated to the operators in quantum chemistry, their principles and functions for two and three particles. This too is accompanied by examples.
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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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