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Shapley Value in Economics
Maruniaková, Zuzana ; Návrat,, Aleš (referee) ; Hrdina, Jaroslav (advisor)
The subject of this bachleor thesis is to introduce cooperative games, Shapley value and its application to economics and other human interests, using mathematical methods. This thesis is devoted to the important concepts and characteristics , which are illustrated by examples. The thesis also focused on describing selected classes of coalitional games and applications of Shapley´s value. These resulting findings are used in a model of election.
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Mayers Value in Economics
Karmazin, Alexandr ; Vašík,, Petr (referee) ; Hrdina, Jaroslav (advisor)
Bachelor thesis is focused on the coalition games in game theory. At the beginning, important terms that relate to these games are defined. Next part of the work is also the application of this knowledge to the real situation, namely the determination of the bargaining power of political parties in the Czech Republic using Myerson value. The work also includes custom application developed in mathematical software Matlab to calculate this value.
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Representations and transformations of color spaces via quaternions
Tichý, Radek ; Druckmüller, Miloslav (referee) ; Hrdina, Jaroslav (advisor)
This thesis deals with quaternions for edge detection, particularly on image processing represented in various color spaces. First, the key concepts and the quaternion algebra properties are mentioned, then some of the commonly used color spaces are described and afterwards the thesis dedicates to the basic filters for edge detection in grayscale image. After that there are presented the color filters that use quaternions for color edge detection in RGB color space. Towards the end, these filters are used for color edge detection in HSV color space.
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Game Theory in Waste Management
Osička, Ondřej ; Vašík, Petr (referee) ; Hrdina, Jaroslav (advisor)
V této práci je vytvořen model rozhodovací situace v odpadovém hospodářství využívající metody teorie her. Model tvoří nekooperativní hra pro reprezentaci konfliktu zpracovatelů odpadu a kooperativní hra pro reprezentaci konfliktu producentů odpadu. Pro konflikt zpracovatelů odpadu je k nalezení strategií při volbě cen na bráně využit koncept Nashovy rovnováhy, takto nalezené stabilní strategie mohou sloužit jako předpověď budoucí situace. Pro zpřesnění množin strategií jsou určeny dolní a horní meze. Pro konflikt producentů odpadu se uvažuje spolupráce všech producentů a určuje se pro ni přerozdělení nákladů pomocí Shapleyho hodnoty a nucleolu. Pro konflikt více producentů jsou vyvinuty aproximační algoritmy pro Shapleyho hodnotu i nucleolus. Tyto algoritmy jsou založeny na předpokladu, že se vzdálení hráči vzájemně neovlivňují. Model je aplikován na situaci v České republice. Pro konflikt zpracovatelů odpadu je nalezen jeden bod Nashovy rovnováhy. Pro konflikt producentů odpadu jsou určeni někteří producenti s vysokým kooperativním potenciálem.
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Algebra of dual quaternions in image analysis
Hrubý, Jan ; Návrat, Aleš (referee) ; Hrdina, Jaroslav (advisor)
This work has two goals. Firstly it is to acquaint the reader with the classical use of quaternions and dual quaternions in geometry. Secondly the generalization of the Fourier transform into the set of dual quaternions. At first it goes into algebraic properties and structure of quaternions and ways of their inscriptions. Later dual numbers are introduced and consecutively with their help dual quaternions. Then the work deals with description of rotations and translations using quaternions and dual quaternions, that enable their easy description. Finally the discreet dual quaternion Fourier transform is defined, and for its effective calculation the algorithm is derived, which is then brought into effect as a code in program environment MATLAB.
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Geometric optimal control of a snake robot
Vechetová, Jana ; Hrdina, Jaroslav (referee) ; Vašík, Petr (advisor)
This thesis deals with the description of robotic snake the trident snake robot. From a viewpoint of control theory the robot is classified as a nonholonomic system whose controllability is determined by vector fields. We use the operation Lie bracket to create other necessary control vector fields to ensure local controllability of this system. Then we propose the motion planning algorithm. Finally some of the motions caused by the control vector fields are verified in a simulation environment called V-rep.
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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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Geometric structures based on quaternions.
Floderová, Hana ; Vašík, Petr (referee) ; Hrdina, Jaroslav (advisor)
A pair (V, G) is called geometric structure, where V is a vector space and G is a subgroup GL(V), which is a set of transmission matrices. In this thesis we classify structures, which are based on properties of quaternions. Geometric structures based on quaternions are called triple structures. Triple structures are four structures with similar properties as quaternions. Quaternions are generated from real numbers and three complex units. We write quaternions in this shape a+bi+cj+dk.
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Control Theory of robotic snakes with more than three links
Tejkal, Martin ; Návrat, Aleš (referee) ; Hrdina, Jaroslav (advisor)
The subject of this Bachelor's thesis is control theory of mechanism that simulates snake's movement. From a viewpoint of control theory the robot is classified as nonholonomic system, controllability of which is determined by vector fields. Based on nonholomic constrain a set of input vector fields is obtained from a system of nonholonomic equations. The other vector fields that are necessary for controllability of the system are derived from the set of input vector fields by application of Lie bracket operation on two input fields. This set of vector fields is further analysed in particular points of the configuration space. Finally we discuss changes that need to be done in order to describe a mechanism created by adding one, or more new links.
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Control Theory of robotic snakes with missing wheels
Reichmanová, Barbora ; Vašík, Petr (referee) ; Hrdina, Jaroslav (advisor)
This thesis looks into the mathematical description of a three-sectional robot. The thesis deals with cases of wheels missing either on the middle or the last section or solely on the middle section. At first theoretical basis is mentioned including the terms such as vector and affinne space, Lie algebra, distribution or controllable system. Subsequently, there is presented formulation of equations describing a snake robot with missing wheels, solutions of equations, calculation of Lie brackets and discussion of controllability. The calculations are demonstrated on examples of various configurations of the robot.
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