
Mathematical plays and game strategies
Michalik, Jindřich ; Slavík, Antonín (advisor) ; Křížek, Michal (referee) ; Stehlík, Petr (referee)
This work focuses on selected results in the field of combinatorial game theory. Standalone chapters are dedicated to the descriptions of the optimal strategies for the games of Nim and Checkers. However, a major part of the work consists of new results. They include optimal strategies for two games by Sidney Sackson, namely Hold That Line and Cutting Corners, and the game Prisoners, created by the author of this work, including strategies for various modifications of these games. Some of these results were obtained using computer programs created by the author. The thesis also contains a chapter dedicated to the game of Master Mind, a game without complete information. 1


Trigonometric and Fourier series and their applications
Sirotková, Veronika ; Slavík, Antonín (advisor) ; Veselý, Jiří (referee)
Fourier series are an important tool of mathematical analysis with many applicati ons. This thesis focuses on their use in several specific mathematical problems. The first application is the proof of the isoperimetric inequality, according to which the circle has the greatest area among closed curves of a given length. Next topic is the sequence of fractional parts of numbers of the form nγ, where n runs through natural numbers and γ is an irrational number. The socalled equidistribution theorem holds for this sequence describing how this sequence fills the interval (0, 1). Fourier series are then also used to obtain a formula for calculating the sum of even powers of the reciprocals of natu ral numbers. The last chapter is devoted to the Gauss circle problem, which investigates the estimation of the number of lattice points inside a circle of a given radius. 1

 

KurzweilStieltjes integral and its generalizations
Konopka, Filip ; Slavík, Antonín (advisor) ; Tvrdý, Milan (referee)
The thesis deals with the HKSp α integral, which is generalization of the HKS integral, its properties and the concepts of ordinary oscillation and poscillation, which are needed for the construction of the integral. This integral is nonabsolutely convergent and more general than the Lebesgue integral. This thesis is based on recent results in the theory of integrals and its goal is to introduce this integral to a wide readership interested in mathematical analysis. 1


Definition of vector product
Holý, David ; Halas, Zdeněk (advisor) ; Slavík, Antonín (referee)
Title: Definition of vector product Author: David Holý Department: Department of Mathematics Education Supervisor: Mgr. Zdeněk Halas, DiS., Ph.D., Department of Mathematics Edu cation Abstract: The main goal of this thesis is to present a compelling and well motivated definition of the vector product and to explain its properties. Torque serves as a medium through which "the rotating effect of force"is studied on simple physical examples. Elaboration leads to revealing essential properties that define the vector product. The thesis contains the derivation of Cartesian coor dinates of the vector product. It also contains a list of its basic mathematical properties and applications. Lorentz force is presented as a concrete example of its application and is thoroughly analyzed. In the closing section, the term curl of a vector field is introduced and conceptually explained. The thesis was focused on bringing a good didactic presentation of a vector product, its concrete appli cations in practice, and its connection to more advanced fields of mathematical inquiry. Keywords: vector product, cross product, torque, Lorentz force, vector field, curl 1


Mathematical models of synchronization
Adámek, Libor ; Slavík, Antonín (advisor) ; Snětinová, Marie (referee)
The thesis aims to discuss spontaneous synchronization phenomenon in dynamic sys tems, which are noticable all around us (beating heart, synchronization of clapping crowd, flashing of fireflies, synchronized ticking of metronomes). The goal is to describe different approaches in the construction of mathematical models of different systems. Concrete systems studied in the thesis are fireflies under the influence of stimuli and synchroni zation of two coupled metronomes. Representative of a general model, Kuramoto model, is analyzed and discussed as well. Several numerical solutions to all those models are presented, dependence on initial values is studied and analyzed both quantitatively and qualitatively. Additionaly, interactive applications/animations were created in Wolfram Mathematica to provide visual support of solutions and even some insight into these solutions. 1


Duels and truels in mathematics
Matis, Anton ; Slavík, Antonín (advisor) ; Staněk, Jakub (referee)
The bachelor thesis describes and analyzes a game in which two or three players compete against each other with the task of eliminating opponents. The game has several variants, which differ in the specified rules. The aim of the work is to review the optimal strategy and calculate the probabilities of success for individual players. 1


Mathematics in the game of SET
Koblížková, Iva ; Slavík, Antonín (advisor) ; Halas, Zdeněk (referee)
This bachelor thesis provides a mathematical description of the card game of SET. The reader is introduced to the history and the rules of this game. Further off, some combinatorial aspects of the game are investigated. In the following chapter, affine geometry is used to describe the game. Thanks to this generalization, the socalled maximum cap can be calculated. The last chapter shows how linear algebra can be used to characterize the game and some of the previously introduced ideas. 1


Selected problems in differential geometry
Paclt, Jan ; Slavík, Antonín (advisor) ; Šír, Zbyněk (referee)
This thesis covers an overview and solution to selected problems in the differential geometry of plane curves. It focuses mainly on calculating areas of regions bounded by plane curves and also on evolutes, involutes and related trochoids and their properties. The work also provides a selfcontained theoretical introduction to the differential geome try of plane curves. Some ideas and mathematical derivations obtained from the original publications were further expanded and generalized by the author. All derivations men tioned in the work are given in a uniform convention, which should make it easier for the reader to orientate and find relations between the topics discussed. The thesis can be used as a study support for students of bachelor's courses in geometry or specifically for students with a focus on descriptive geometry. 1


History of Kurzweil integral
Berková, Andrea ; Halas, Zdeněk (advisor) ; Slavík, Antonín (referee)
The presented work deals with history of Kurzweil integral. It focuses primarily on its comparison with other important integrals, namely Newton, Rie mann, Lebesgue, Perron and McShane integral. Each of them is discussed in a separate chapter which acquaints with their authors and theories. Attention is also oriented to Jaroslav Kurzweil and Ralph Henstock. There are also mentioned the circumstances of the discovery of the Kurzweil integral. The aim is to high light the theory of integration, which has its origins in Bohemia and despite its elementary definition, which is very general and usable in many applications.
