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Pell's equation, continued fractions and Diophantine approximations of irrational numbers
Kodýtek, Jakub ; Beran, Filip (advisor) ; Jančařík, Antonín (referee)
This bachelor's thesis deals with Pell's equation, while clearly presenting structured information from studied domestic and foreign books, articles, and other sources. The goal of this thesis is to create study material primarily for university students but also for inquisitive high school students, and thus explain as intuitively as possible what Pell's equation is, how to find its solutions, and how it is related, for example, to continued fractions, approximations of irrational numbers, and invertible elements in Z[√n ]. The main motivation for solving Pell's equation throughout the work is specifically that its solutions give best approximations of irrational square roots. Pell's equation is presented in a brief historical context. Further, it is proved that there is a non-trivial integer solution for every Pell equation, and the theory of continued fractions is used to find it. To make the creation of continued fractions easier, the so-called Tenner's algorithm is introduced. Specifically, the search for a solution to Pell's equation is derived using convergents and the periodicity of continued fractions of irrational roots. Subsequently, the structure of the solution is described: it is proved that there is a so-called minimal solution that generates all positive solutions, and a set of...
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Continued fractions in coding theory
Fišer, Jan ; Šťovíček, Jan (advisor) ; Holub, Štěpán (referee)
The first part of the thesis acquaints us with the Reed-Solomon codes, methods of their construction and encoding. At the same time we provide the evi- dence of their most important properties including the relevant theoretical basis. In the second chapter we introduce the theory of continued fractions over a field and examine their structure. Applying the executed general ob- servations on the specific case of the formal Laurent series we get to efficient Reed-Solomon decoding algorithm. Without complete proofs we also men- tion other two decoding algorithms that are based on solving the key equation as well, namely Berlekamp-Massey and Euclidean algorithm. In the end we show the equivalence of these three algorithms.
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The number π and continued fractions
Švejdová, Aneta ; Halas, Zdeněk (advisor) ; Slavík, Antonín (referee)
This bachelor thesis deals with one of the well-known mathematical constants, the number π. The form is understandable to higher-year students of secondary schools interested in mathematics. At first, it presents the best known ways people in history tried to approximate the number π. It includes the methods of Egyptians, the people of ancient Mesopotamia and the method of Archimedes. It also presents expressing π in the form of infinite product according to F. Viète and J. Wallis. The second part of the thesis focuses on expressing the number π by continued fractions, which are at first generally defined. We introduce essential relations among them. Then the thesis presents expressing the number π in the form of continued fractions according to J. H. Lambert, L. Euler and W. Brouncker. Finally, proofs of the irrationality of π using continued fractions are presented together with a simple proof of its transcendence. The aim of the thesis is to extend information about π stated in popular books, to explain and clarify basic ideas leading to these claims.
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Continued fractions in coding theory
Fišer, Jan ; Šťovíček, Jan (advisor) ; Holub, Štěpán (referee)
The first part of the thesis acquaints us with the Reed-Solomon codes, methods of their construction and encoding. At the same time we provide the evi- dence of their most important properties including the relevant theoretical basis. In the second chapter we introduce the theory of continued fractions over a field and examine their structure. Applying the executed general ob- servations on the specific case of the formal Laurent series we get to efficient Reed-Solomon decoding algorithm. Without complete proofs we also men- tion other two decoding algorithms that are based on solving the key equation as well, namely Berlekamp-Massey and Euclidean algorithm. In the end we show the equivalence of these three algorithms.
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