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Nestandardní zápisy čísel

Edita Pelantová; Štepán Starosta — 2011

Pokroky matematiky, fyziky a astronomie

Complexity of infinite words associated with beta-expansions

Christiane Frougny; Zuzana Masáková; Edita Pelantová — 2004

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We study the complexity of the infinite word $u_{β}$ associated with the Rényi expansion of $1$ in an irrational base $β > 1$ . When $β$ is the golden ratio, this is the well known Fibonacci word, which is sturmian, and of complexity $ℂ (n) = n + 1$ . For $β$ such that $d_{β} (1) = t_{1} t_{2} \dots t_{m}$ is finite we provide a simple description of the structure of special factors of the word $u_{β}$ . When $t_{m} = 1$ we show that $ℂ (n) = (m - 1) n + 1$ . In the cases when $t_{1} = t_{2} = \dots = t_{m - 1}$ or $t_{1} > max {t_{2}, \dots, t_{m - 1}}$ we show that the first difference of the complexity function $ℂ (n + 1) - ℂ (n)$ takes value in ${m - 1, m}$ for every $n$ , and consequently we determine...

Integers with a maximal number of Fibonacci representations

Petra Kocábová; Zuzana Masáková; Edita Pelantová — 2005

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We study the properties of the function $R (n)$ which determines the number of representations of an integer $n$ as a sum of distinct Fibonacci numbers $F_{k}$ . We determine the maximum and mean values of $R (n)$ for $F_{k} \leq n < F_{k + 1}$ .

Sturmian jungle (or garden?) on multiliteral alphabets

L'ubomíra Balková; Edita Pelantová; Štěpán Starosta — 2010

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

The properties characterizing sturmian words are considered for words on multiliteral alphabets. We summarize various generalizations of sturmian words to multiliteral alphabets and enlarge the list of known relationships among these generalizations. We provide a new equivalent definition of rich words and make use of it in the study of generalizations of sturmian words based on palindromes. We also collect many examples of infinite words to illustrate differences in the generalized definitions...

Corrigendum : “Complexity of infinite words associated with beta-expansions”

Christiane Frougny; Zuzana Masáková; Edita Pelantová — 2004

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We add a sufficient condition for validity of Propo- sition 4.10 in the paper Frougny et al. (2004). This condition is not a necessary one, it is nevertheless convenient, since anyway most of the statements in the paper Frougny et al. (2004) use it.

Greedy and lazy representations in negative base systems

Tomáš Hejda; Zuzana Masáková; Edita Pelantová — 2013

Kybernetika

We consider positional numeration systems with negative real base $- β$ , where $β > 1$ , and study the extremal representations in these systems, called here the greedy and lazy representations. We give algorithms for determination of minimal and maximal $(- β)$ -representation with respect to the alternate order. We also show that both extremal representations can be obtained as representations in the positive base $β^{2}$ with a non-integer alphabet. This enables us to characterize digit sequences admissible as greedy...