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Semilattices and indecomposable elements
Kuděj, Martin ; Kala, Vítězslav (advisor) ; Korbelář, Miroslav (referee)
This thesis concerns the theory of semilattices, which are non-trivial discrete additive submonoids of Rn , which are contained in a cone. Special emphasis is on their indecomposable elements. The most important example of semilattices is derived from real quadratic number fields, which involves the most parts of the thesis and all indecomposable elements of such semilattices are characterised in two ways. That includes using various tools from number theory, mainly con- tinued fractions, their corresponding semiconvergents and their approximation properties, Farey pairs, but also some tools from algebraic number theory. The final part of the thesis concerns the upper bound of the norm of indecompos- able elements in a semilattice, derived from the Minkowski embedding of the corresponding number field. 1
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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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