### Nestandardní zápisy čísel

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We study the complexity of the infinite word ${u}_{\beta}$ associated with the Rényi expansion of $1$ in an irrational base $\beta \>1$. When $\beta $ is the golden ratio, this is the well known Fibonacci word, which is sturmian, and of complexity $\u2102\left(n\right)=n+1$. For $\beta $ such that ${d}_{\beta}\left(1\right)={t}_{1}{t}_{2}\cdots {t}_{m}$ is finite we provide a simple description of the structure of special factors of the word ${u}_{\beta}$. When ${t}_{m}=1$ we show that $\u2102\left(n\right)=(m-1)n+1$. In the cases when ${t}_{1}={t}_{2}=\cdots ={t}_{m-1}$ or ${t}_{1}\>max\{{t}_{2},\cdots ,{t}_{m-1}\}$ we show that the first difference of the complexity function $\u2102(n+1)-\u2102\left(n\right)$ takes value in $\{m-1,m\}$ for every $n$, and consequently we determine...

We study the properties of the function $R\left(n\right)$ 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 $R\left(n\right)$ for ${F}_{k}\le n\<{F}_{k+1}$.

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...

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.

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

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...

We study the complexity of the infinite word associated with the Rényi expansion of in an irrational base . When is the golden ratio, this is the well known Fibonacci word, which is Sturmian, and of complexity . For such that is finite we provide a simple description of the structure of special factors of the word . When =1 we show that . In the cases when or max} we show that the first difference of the complexity function takes value in for every , and consequently we determine the complexity...

We study the properties of the function which determines the number of representations of an integer as a sum of distinct Fibonacci numbers . We determine the maximum and mean values of for .

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

We study infinite words over an alphabet $\mathcal{A}$ satisfying the property $\mathcal{P}:\phantom{\rule{3.33333pt}{0ex}}\mathcal{P}\left(n\right)+\mathcal{P}(n+1)=1+\#\mathcal{A}\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}\mathrm{any}\phantom{\rule{4pt}{0ex}}n\in \mathbb{N}$, where $\mathcal{P}\left(n\right)$ denotes the number of palindromic factors of length occurring in the language of . We study also infinite words satisfying a stronger property $\mathrm{\mathcal{P}\mathcal{E}}$: every palindrome of has exactly one palindromic extension in . For binary words, the properties $\mathcal{P}$ and $\mathrm{\mathcal{P}\mathcal{E}}$ coincide and these properties characterize Sturmian words, i.e., words with the complexity + 1 for any $n\in \mathbb{N}$. In this paper, we focus on ternary infinite words with...

We study some arithmetical and combinatorial properties of -integers for being the larger root of the equation . We determine with the accuracy of 1 the maximal number of -fractional positions, which may arise as a result of addition of two -integers. For the infinite word coding distances between the consecutive -integers, we determine precisely also the balance. The word is the only fixed point of the morphism → and → . In the case , the corresponding infinite word is sturmian, and, therefore,...

We consider words coding exchange of three intervals with permutation (3,2,1), here called 3iet words. Recently, a characterization of substitution invariant 3iet words was provided. We study the opposite question: what are the morphisms fixing a 3iet word? We reveal a narrow connection of such morphisms and morphisms fixing Sturmian words using the new notion of amicability.

The aim of this article is to study certain combinatorial properties of infinite binary and ternary words associated to cut-and-project sequences. We consider here the cut-and-project scheme in two dimensions with general orientation of the projecting subspaces. We prove that a cut-and-project sequence arising in such a setting has always either two or three types of distances between adjacent points. A cut-and-project sequence thus determines in a natural way a symbolic sequence (infinite word)...

A simple Parry number is a real number $\beta \>1$ such that the Rényi expansion of $1$ is finite, of the form ${d}_{\beta}\left(1\right)={t}_{1}\cdots {t}_{m}$. We study the palindromic structure of infinite aperiodic words ${u}_{\beta}$ that are the fixed point of a substitution associated with a simple Parry number $\beta $. It is shown that the word ${u}_{\beta}$ contains infinitely many palindromes if and only if ${t}_{1}={t}_{2}=\cdots ={t}_{m-1}\ge {t}_{m}$. Numbers $\beta $ satisfying this condition are the so-called Pisot numbers. If ${t}_{m}=1$ then ${u}_{\beta}$ is an Arnoux-Rauzy word. We show that if $\beta $ is a confluent Pisot number then $\mathcal{P}(n+1)+\mathcal{P}\left(n\right)=\mathcal{C}(n+1)-\mathcal{C}\left(n\right)+2$, where...

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