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Invariance principles
Suchánek, Ondřej ; Staněk, Jakub (advisor) ; Hlubinka, Daniel (referee)
The purpose of this work is to state the Donsker's invariance principle which is about the relation of a random walk and the Wiener process and to make its detailed proof. Then we will deal with the usage of the Donsker's invariance prin- ciple when simulating the trajectory of the Wiener process and we will simulate it with a few of distributions of steps of a random walk. In the next part we will focus on the first passage time of the random walk for which we will derive the distribution and compare to the first passage time of the Wiener process. 1
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Methods for solving differential equations
Zadražil, Tomáš ; Staněk, Jakub (advisor) ; Slavík, Antonín (referee)
The goal of this work is to introduce some elementary methods of solving ordinary differential equations to the reader. These methods are then exemplified by means of using solved samples and particular illustrations from practice. The text intends to be friendly to an inexpert user, therefore it prefers verbal expressions to accumulation of mathematic expressions. The target groups are university students and secondary school teachers who teach optional seminars of mathematic. The knowledge of derivatives, definite and indefinite integrals is required to understand this text. Powered by TCPDF (www.tcpdf.org)
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Computational problems of elementary number theory
Widž, Jiří ; Porubský, Štefan (advisor) ; Staněk, Jakub (referee) ; Šimša, Jaromír (referee)
Title: Computational problems of elementary number theory Author: Mgr. Jiří Widž Department: Department of Mathematics Education Supervisor: Prof. RNDr. Štefan Porubský, DrSc. Institute of Computer Science of the Academy of Sciences of the Czech Republic Abstract: The central notion of this presented thesis is the concept of continued fractions. The origin of this concept as one of the oldest mathematical methods is shown here in its historical connections. The technique of continued fractions belongs to classical parts of mathematics. Although the general theory of continued fractions is manifold and layered considerably, in textbooks it is usually treated according to the intended purpose of its use. In the present text we have summarized the foundations of the general theory of convergence of continued fractions with an emphasis on the theory of simple continued fractions and their most common applications. We show several possibilities how the concept of continued fractions can be generalized to other structures such as the Gaussian integers or polynomial continued fractions. In the chapter devoted to matrix continued fractions we shall demonstrate the possibility how to extend it to non-commutative algebraic structures. We also show how the apparatus of continued fractions can be used to solve...
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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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Program COMSOL Multiphysics jako prostředek pro simulace utužení půdy pojezdem zemědělské techniky
STANĚK, Jakub
This bachelor work is concerned with an applicability of the program COMSOL Multiphysics as a device for a simulation of a solidify of a land. The work is composed of four thematic units. In the first one the land is described shortly and its basic property. The second unit is about problems of a solidify of a land under the influence of a roller of farming machines and its results. The third part is pointed on the software its functions, additionals function and in the end on a short sequence of action, that is essential to do for making a model. On the last part the all process is described for creating of a multiphysical task with concrete numbers, that pretends loading of the land and its deformation due to a effect of weight of a farming machine.
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Elementary mathematics from an advanced standpoint
Kučera, Jiří ; Bečvář, Jindřich (advisor) ; Staněk, Jakub (referee)
Mine thesis is written for self-study of secondary school graduates students which prepare for the mathematically oriented high schools. It clarifies theoretical approach to mathematics. It was chosen topics exponents, square roots, logarithms and equations that contain these objects for this purpose. For this reason students are given the opportunity to gain stance to university mathematics knowledge that should be known from high school. In addition, his work allows him to enlarge knowledge of those topics and learn heavier and unusual examples.
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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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Volume of solids
Tvrdá, Monika ; Halas, Zdeněk (advisor) ; Staněk, Jakub (referee)
This didactic oriented bachelor project helps to approach an origin of relations for the volumes of solids taught at high school. It is focused on high school and university students. At the beginning the project shows historical meaning of the volumes of solids and the processes which were used to enumerate them in the ancient Egypt and Mesopotamia. Further, the project deals with the definition of volume of solids; it is based on Jordan's measure. The relations for volumes of the sorted solids are derived using the integral calculus. In the end the other ways of deriving of these relations are shown. At first, it is the method that Archimedes from Syracuse invented, furthermore by the visual imaginations and the Cavalieri's principle. 1
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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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