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Curves of constant width
Němec, Miroslav ; Slavík, Antonín (advisor) ; Staněk, Jakub (referee)
Title: Curves of constant width Author: Miroslav Němec Department: Department of Mathematics Education Supervisor: RNDr. Antonín Slavík, Ph.D., Department of Mathematics Education Abstract: The aim of this thesis is to familiarize the reader with the notion of the width of a given plane curve, and with curves having the same width in all directions. We focus our attention especially on those curves that are constructible using straightedge and compass. The chapters presenting the definition and measurement of curve width, as well as the construction of curves from circular arcs, are accessible to high-school students. We also briefly mention other related topics, such as the solids of constant width. Keywords: Width of a curve, constant width, construction, properties
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Limits and L'Hospital's rule
Ranšová, Kateřina ; Staněk, Jakub (advisor) ; Halas, Zdeněk (referee)
Title: Limits and l'Hospital's rule Author: Kateřina Ranšová Department: Department of Mathematics Education Supervisor: RNDr. Jakub Staněk, Ph.D., Department of Mathematics Education Abstract: The aim of this BA thesis is to introduce to the reader the concept of the limit of function and the means of its solution. The main impact of the thesis lies in a didactic approach and in the connection of a limit theory with its graphic representation and different methods of algebraic calculation. The text consists of eight chapters which can be divided into two parts according to their content. The first part explores the term "the limit of the function". Individual types of limits are then defined. To facilitate understanding, most of the definitions are accompanied by a particular example and a graphic representation . The first part is concluded by a unified definition which by means of the term "vicinity" summarizes all preceding types of limits. The other part deals with some basic methods of limits' calculation. Other topics include Taylor Series, l'Hospital's Rule and their applications to the limits. The core of the thesis is a comparison of calculation by means of l'Hospital's Rule and Taylor Series. The conclusion of the thesis presents some advantages and disadvantages of applying Taylor Series and...
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Poker and Probability
Jelínek, Roman ; Staněk, Jakub (advisor) ; Slavík, Antonín (referee)
Title: Poker and Probability Author: Roman Jelínek Department: Department of Mathematics Education Supervisor: RNDr. Jakub Staněk, Ph.D., Department of Mathematics Education Abstract: The aim of this Bachelor thesis is to create a historical introduction to the exciting card game and especially proximity of its mathematical aspects to the general public. The work consists of didactic way with many comments to the theory. The text is worded in a way that most people understand the situations, which are described without mathematical basics. For this reason there are no complicated formulas written and used are thoroughly described. The work focuses mainly on amateur poker players, but also on people, who knows anything about this card game. Keywords: history of poker, poker rules, poker hands, the probability of winning
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The beginnings of probability theory
Marcinčín, Martin ; Staněk, Jakub (advisor) ; Halas, Zdeněk (referee)
The purpose of this thesis is to give a summary of historical development and explain fundamentals of the probability theory. Early systematic thoughts, emergence of classical Laplace, geometric and statistical definition of probability with development of theory, independence, conditional probability and Bayes theorem are shown. The thesis describes first mention of random values and the central limit theorem. The alternative, discrete uniform, binomial, Poisson, continuous uniform, normal and exponential distributions are discussed with historical background of their discoveries. The theory is supplemented with illustrative and contemporary examples. The thesis describes development in various fields of probability until publication of the Kolmogorov's probability theory in 1933. Powered by TCPDF (www.tcpdf.org)
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Safety of magnetic resonance imaging for the examinated patient
Staněk, Jakub ; Kašpar, Miroslav (advisor) ; Daníčková, Kateřina (referee)
Magnetic resonance imaging is a modern radiological imaging method. It allows to diagnose pathological changes in the human body without any need to disrupt of its structure. The main advantage of NMR is that it does not use any harmful ionizing radiation. Because NMR is a relatively young technology, any harmful effects have not found yet on the human organism. But from the physical point of view the potential risks arise consequence of biological structures interaction and tissues with the static magnetic field, gradient magnetic field and high frequency electromagnetic field. Due to the development methods it leads to intensification of these fields inside the device. Master's thesis involved research of biophysical effects and potential risks of static magnetic field, gradient magnetic field and high frequency electromagnetic field used in magnetic resonance. Result interactions of these fields with investigated patients' bodies biological structures leads to heat induction inside their bodies. Temperature of the investigated parts of patient body is monitored. Ascertained data are evaluated by the Wilcoxon non-parametric test with simulation Monte Carlo.
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Proofs of the strong law of large numbers
Odintsov, Kirill ; Štěpán, Josef (advisor) ; Staněk, Jakub (referee)
This thesis concentrates on the Strong Law of Large Numbers. It features two proofs of this law. The first is less general, but simpler Borel's proof. The second one is more complex. It uses Kronecker's lemma and Kolmogorov-Khinchin's theorem, which is proven by Kolmogorov's inequality. The text includes all the necessary auxiliary theorems and lemmas along with their proofs. Since all the proofs are explored in a great detail this text is suitable for readers with only basic knowledge of probability theory and measure theory. Furthermore it contains numerous practical and mathematical examples thought out the whole text. Finally to demonstrate the importance of Strong Law of Large Numbers the text features four important applications of the law in mathematics.
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