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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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Algorithms of the interpolation by multivariate polynomials
Doktorová, Alice ; Čermák, Libor (referee) ; Kureš, Miroslav (advisor)
This bachelor's work concerns to algorithms of the multivariate interpolation. The problem of the interpolation over the plane is studied in the first part of this work. In the next section, the multivariate Lagrange interpolation is described and the polynomial degree is discussed. A Mathematica program package was developed, by this, the multivariate interpolation over an arbitrary field can be solved.
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Curves in D^3_1
Navrátil, Dušan ; Doupovec, Miroslav (referee) ; Kureš, Miroslav (advisor)
The Bachelor thesis deals with research on curves in three-dimensional space with Lorentzian inner product. The emphasis is on detailed analysis of dual numbers and dual Lorentzian space properties. Main part of this work is focused on dual functions, their differentiability, both arc length reparametrization and Frenet equations of dual curves and examples of such a curves. Within this work, many statements were derived generalizing from Minkowski space. Properties of dual rectifying curves were described in last section and finally we showed relation between these curves, dual unit spherical curves and ruled surfaces in Minkowski space.
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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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Mathematical principles of Robotics
Pivovarník, Marek ; Kureš, Miroslav (referee) ; Hrdina, Jaroslav (advisor)
Táto diplomová práca sa zaoberá matematickými aparátmi popisujúcimi doprednú a inverznú kinematiku robotického ramena. Pre popis polohy koncového efektoru, teda doprednej kinematiky, je potrebné zaviesť špeciálnu Euklidovskú grupu zobrazení. Táto grupa môže byť reprezentovaná pomocou matíc alebo pomocou duálnych kvaterniónov. Problém inverznej kinematiky, kedy je potrebné z určenej polohy koncového efektoru dopočítať kĺbové parametre robotického ramena, je v tejto práci riešený pomocou exponenciálnych zobrazení a Grobnerovej bázy. Všetky spomenuté popisy doprednej a inverznej kinematiky sú aplikované na robotické rameno s troma rotačnými kĺbami. Odvodené postupy sú následne implementované a vizualizované v prostredí programu Mathematica.
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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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Discretely normed orders of quaternionic algebras
Horníček, Jan ; Skula, Ladislav (referee) ; Kureš, Miroslav (advisor)
Tato práce shrnuje autorův výzkum v oblasti teorie kvaternionových algeber, jejich izomorfismů a maximálních řádů. Nový úhel pohledu na tuto problematiku je umožněn využitím pojmu diskrétní normy. Za hlavní výsledky práce je možná považovat důkaz jednoznačnosti diskrétní normy pro celá čísla, kvadratická rozšíření těles a řády kvaternionových algeber. Dále větu, která umožňuje mezi dvěma kvaternionovými algebrami konstruovat izomorfismy explicitně vyjádřené v maticovém tvaru. A v neposlední řadě důkaz existence nekonečně mnoha různých maximálních řádů kvaternionové algebry. Výsledky uvedené v této diplomové práci budou dále publikovány ve vědeckém článku.
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