Ing. Petr Ambrož (Ph.D. alumnus)
| advisor: | Prof. Ing. Edita Pelantová, Csc. |
| study start date: | 01.10.2003 |
| study form: | prezenční |
| state exam: | 10.06.2005 |
| title of dissertation thesis: | Algebraic and Combinatorial Properties
of Nonstandard Numeration Systems |
| POPIS_DISERTACNI_PRACE: | This thesis studies the so-called beta-numeration systems, that is positional numeration systems with an arbitrary real base. We define two subsets of R: Zb, the set of real numbers having no fractional part in their beta-expansion and Fin, the set of numbers with a finite fractional part. We derive one necessary and two sufficient conditions for Fin to be ring. Then we inspect the values of Lp and Lt, that is, the maximal length of the fractional part of sum and product of two beta-integers. We compute upper bounds of Lp and Lt for some classes of Pisot numbers. We study also the alpha-adic representation - a representation in a system whose base is an algebraic conjugate of a Pisot number. We prove that a number belong to Q(alpha) if and only if it has an eventually periodic alpha-adic expansion. Then we consider alpha-adic expansions of elements of the extension ring Z[alpha]. |
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