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Universal quadratic forms over number fields
Svoboda, Josef ; Kala, Vítězslav (advisor) ; Hejda, Tomáš (referee)
The aim of this work is to study universal quadratic forms over biquadratic fields. In the thesis we define biquadratic fields and describe their structure. In particular, we study some distinguished (totally positive and aditively indecomposable) elements, their norms and traces. Then we describe the theory of universal quadratic forms and use special elements to find a lower bound for the number of variables of a universal quadratic form over some biquadratic fields.
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Continued fractions with prescribed period
Kuděj, Martin ; Kala, Vítězslav (advisor) ; Francírek, Pavel (referee)
This thesis concerns continued fractions of quadratic irrationals. Their basic properties are shown, including mentioning necessary theory to do this. Then, this theory is used to find the form of continued fractions of square roots of positive nonsquare integers and their symmetric part (a1, . . . , ak). Next, for a given symmetric sequence of positive integers (a1, . . . , ak), we find all natural numbers N, whose square root has a continued fraction with symmetric part (a1, . . . , ak). These positive N will be described as values of a certain quadratic polynomial, whose properties are studied as well in the thesis. 1
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Universal quadratic forms over number fields
Svoboda, Josef ; Kala, Vítězslav (advisor) ; Hejda, Tomáš (referee)
The aim of this work is to study universal quadratic forms over biquadratic fields. In the thesis we define biquadratic fields and describe their structure. In particular, we study some distinguished (totally positive and aditively indecomposable) elements, their norms and traces. Then we describe the theory of universal quadratic forms and use special elements to find a lower bound for the number of variables of a universal quadratic form over some biquadratic fields.
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A stream cipher based on continued fractions
Krasnayová, Dáša ; Drápal, Aleš (advisor) ; Holub, Štěpán (referee)
This bachelor thesis deals with the theory of continued fractions which is design of a stream cipher in article On the use of continued fractions for stream ciphers based on. Since results about probability for a positive integer number to be a partial quotient of a generalised continued fraction which are necessary for proving the cipher secure, has not been proved yet, there are summarized previous results which could lead to proving them. In particular, basic properties of classical and generalised continued fractions and proof of Kuzmin theorem are presented here. Distribution of probability for a positive integer number to be a partial quotient of a classical continued fraction follows from Kuzmin theorem. The design of the stream cipher from the article is briefly introduced at the end of the thesis. Powered by TCPDF (www.tcpdf.org)
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