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Infinite products
Kudrnáč, Vojtěch ; Rokyta, Mirko (advisor) ; Černý, Robert (referee)
This paper offers a brief insight into the basic theory of convergence of the infinite products of real or complex sequences. Then it focuses mainly on the possibilities of developing some selected functions into the form of infinite product and on the corollaries and utilizations of being familiar with these. Purpose of the paper is not to prove the existence of infinite products for functions with certain characteristics in general, but rather to derive specific formulas and prove their validity. The attention is paid to those elementary functions which are derived from the exponential function, especially the sinus function, the nonelementary functions mentioned are the gamma and the zeta function. The text should be understandable even for a person, who has never came upon infinite products before.
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Method of Green function for boundary value problems for ODR
Héda, Ivan ; Rokyta, Mirko (advisor) ; Pražák, Dalibor (referee)
The main aim of this work is to summarize the basic knowledge of method which is using Green's functions for solving boundary value problems for linear diffe- rential equations. These functions will be defined and, with some not very strong presumptions, uniquely constructed. This method is primarily derived for solving problems with homogenous boundary conditions. However it will be shown that there is no more presumtions needed to use this method to solve problems with non-homogenous linear boundary conditions. As a main consequence of preceding existence and uniqueness of solution for relatively wide class of linear boundary problems will be provided. 1
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Riemann zeta function
Čoupek, Petr ; Rokyta, Mirko (advisor) ; Zahradník, Miloš (referee)
Riemann zeta function represents an important tool in analytical number theory with various applications in quantum mechanics, probability theory and statistics. First introduced by Bernhard Riemann in 1859, zeta function is a central object of many outstanding problems. From previous results follows the importance of zeta function for further development in the field of number theory. This thesis provides basic properties of the Riemann zeta function. In particular, we prove theorems concerning the distribution of its roots outside and inside the critical strip which leads to the formulation of the Riemann hypothesis and theorems concerning the irrationality of selected values of the Riemann zeta function including the proof of the irrationality of ζ(3). 1
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Conformal mappings and Laplace equation
Kincl, Ondřej ; Rokyta, Mirko (advisor) ; Pražák, Dalibor (referee)
in English Ondřej Kincl 30 May 2018 This paper studies conformal maps over the field of complex numbers with an emphasis on physical applications. In the first two parts (which are mostly theoretic) we shall introduce mathematical terms and theorems which will allow us to solve the Laplace partial differential equation in various regions with an interesting geometry in R2 (parts 3 and 4). We will show how methods of the complex analysis can be applied to decsribe the induced charge on a conductor. In the field of aerodynamics, we will use these methods to explain the underlying principle behind the function of wings.
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Riemann zeta function
Čoupek, Petr ; Rokyta, Mirko (advisor) ; Zahradník, Miloš (referee)
Riemann zeta function represents an important tool in analytical number theory with various applications in quantum mechanics, probability theory and statistics. First introduced by Bernhard Riemann in 1859, zeta function is a central object of many outstanding problems. From previous results follows the importance of zeta function for further development in the field of number theory. This thesis provides basic properties of the Riemann zeta function. In particular, we prove theorems concerning the distribution of its roots outside and inside the critical strip which leads to the formulation of the Riemann hypothesis and theorems concerning the irrationality of selected values of the Riemann zeta function including the proof of the irrationality of ζ(3). 1
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Method of Green function for boundary value problems for ODR
Héda, Ivan ; Rokyta, Mirko (advisor) ; Pražák, Dalibor (referee)
The main aim of this work is to summarize the basic knowledge of method which is using Green's functions for solving boundary value problems for linear diffe- rential equations. These functions will be defined and, with some not very strong presumptions, uniquely constructed. This method is primarily derived for solving problems with homogenous boundary conditions. However it will be shown that there is no more presumtions needed to use this method to solve problems with non-homogenous linear boundary conditions. As a main consequence of preceding existence and uniqueness of solution for relatively wide class of linear boundary problems will be provided. 1
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