arXiv subjects

Edita Pelantová

Publications and source records attributed to Edita Pelantová.

At least 55 records · Page 3Linked to original sources

arXiv2018

A note on palindromic length of Sturmian sequences

Frid, Puzynina and Zamboni (2013) defined the palindromic length of a finite word $w$ as the minimal number of palindromes whose concatenation is equal to $w$. For an infinite word $u$ we study $PL_{u}$, that is, the function that assigns to each positive integer $n$, the maximal palindromic length of factors of length $n$ in $u$. Recently, Frid (2018) proved that $\limsup_{n\to\infty} PL_{u}(n)=+\infty$ for any Sturmian word $u$. We show that there is a constant $K>0$ such that $PL_{u}(n)\leq K\ln n$ for every Sturmian word $u$, and that for each non-decreasing function $f$ with property $\lim_{n\to\infty}f(n)=+\infty$ there is a Sturmian word $u$ such that $PL_{u}(n)=O(f(n))$.

math.CO↗

arXiv2018

Fixed points of Sturmian morphisms and their derivated words

Any infinite uniformly recurrent word ${\bf u}$ can be written as concatenation of a finite number of return words to a chosen prefix $w$ of ${\bf u}$. Ordering of the return words to $w$ in this concatenation is coded by derivated word $d_{\bf u}(w)$. In 1998, Durand proved that a fixed point ${\bf u}$ of a primitive morphism has only finitely many derivated words $d_{\bf u}(w)$ and each derivated word $d_{\bf u}(w)$ is fixed by a primitive morphism as well. In our article we focus on Sturmian words fixed by a primitive morphism. We provide an algorithm which to a given Sturmian morphism $ψ$ lists the morphisms fixing the derivated words of the Sturmian word ${\bf u} = ψ({\bf u})$. We provide a sharp upper bound on length of the list.

math.CO↗

arXiv2018

Two applications of the spectrum of numbers

Let the base $β$ be a complex number, $|β|>1$, and let $A \subset \C$ be a finite alphabet of digits. The \emph{$A$-spectrum} of $β$ is the set $S_{A}(β) = \{\sum_{k=0}^n a_kβ^k \mid n \in \mathbb{N}, \ a_k \in {A}\}$. We show that the spectrum $S_{A}(β)$ has an accumulation point if and only if $0$ has a particular $(β, A)$-representation, said to be \emph{rigid}. The first application is restricted to the case that $β>1 $ and the alphabet is $A=\{-M, \ldots, M\}$, $M \ge 1$ integer. We show that the set $Z_{β,M}$ of infinite $(β, A)$-representations of $0$ is recognizable by a finite Büchi automaton if and only if the spectrum $S_A(β)$ has no accumulation point. Using a result of Akiyama-Komornik and Feng, this implies that $Z_{β, M}$ is recognizable by a finite Büchi automaton for any positive integer $M \ge \lceil β\rceil -1$ if and only if $β$ is a Pisot number. This improves the previous bound $M \ge \lceil β\rceil $. For the second application the base and the digits are complex. We consider the on-line algorithm for division of Trivedi and Ercegovac generalized to a complex numeration system. In on-line arithmetic the operands and results are processed in a digit serial manner, starting with the most significant digit. The divisor must be far from $0$, which means that no prefix of the $(β,A)$-representation of the divisor can be small. The numeration system $(β,A)$ is said to \emph{allow preprocessing} if there exists a finite list of transformations on the divisor which achieve this task. We show that $(β,A )$ allows preprocessing if and only if the spectrum $S_{A}(β)$ has no accumulation point.

math.NT↗

arXiv2017

On Words with the Zero Palindromic Defect

We study the set of finite words with zero palindromic defect, i.e., words rich in palindromes. This set is factorial, but not recurrent. We focus on description of pairs of rich words which cannot occur simultaneously as factors of a longer rich word.

math.CO↗

arXiv2017

Markov constant and quantum instabilities

For a qualitative analysis of spectra of certain two-dimensional rectangular-well quantum systems several rigorous methods of number theory are shown productive and useful. These methods (and, in particular, a generalization of the concept of Markov constant known in Diophantine approximation theory) are shown to provide a new mathematical insight in the phenomenologically relevant occurrence of anomalies in the spectra. Our results may inspire methodical innovations ranging from the description of the stability properties of metamaterials and of certain hiddenly unitary quantum evolution models up to the clarification of the mechanisms of occurrence of ghosts in quantum cosmology.

math-ph↗

arXiv2016

Constructions of words rich in palindromes and pseudopalindromes

A narrow connection between infinite binary words rich in classical palindromes and infinite binary words rich simultaneously in palindromes and pseudopalindromes (the so-called $H$-rich words) is demonstrated. The correspondence between rich and $H$-rich words is based on the operation $S$ acting over words over the alphabet $\{0,1\}$ and defined by $S(u_0u_1u_2\ldots) = v_1v_2v_3\ldots$, where $v_i= u_{i-1} + u_i \mod 2$. The operation $S$ enables us to construct a new class of rich words and a new class of $H$-rich words. Finally, the operation $S$ is considered on the multiliteral alphabet $\mathbb{Z}_m$ as well and applied to the generalized Thue--Morse words. As a byproduct, new binary rich and $H$-rich words are obtained by application of $S$ on the generalized Thue--Morse words over the alphabet $\mathbb{Z}_4$.

math.CO↗

arXiv2016

On the Zero Defect Conjecture

Brlek et al. conjectured in 2008 that any fixed point of a primitive morphism with finite palindromic defect is either periodic or its palindromic defect is zero. Bucci and Vaslet disproved this conjecture in 2012 by a counterexample over ternary alphabet. We prove that the conjecture is valid on binary alphabet. We also describe a class of morphisms over multiliteral alphabet for which the conjecture still holds. The proof is based on properties of extension graphs.

math.CO↗

arXiv2016

Exchange of three intervals: itineraries, substitutions and palindromicity

Given a symmetric exchange of three intervals, we provide a detailed description of the return times to a subinterval and the corresponding itineraries. We apply our results to morphisms fixing words coding non-degenerate three interval exchange transformation. This allows us to prove that the conjecture stated by Hof, Knill and Simon is valid for such infinite words.

math.CO↗

arXiv2016

On periodic representations in non-Pisot bases

We study periodic expansions in positional number systems with a base $β\in\C,\ |β|>1$, and with coefficients in a finite set of digits $\A\subset\C.$ We are interested in determining those algebraic bases for which there exists $\A\subset \Q(β),$ such that all elements of $\Q(β)$ admit at least one eventually periodic representation with digits in $\A$. In this paper we prove a general result that guarantees the existence of such an $\A$. This result implies the existence of such an $\A$ when $β$ is a rational number or an algebraic integer with no conjugates of modulus $1$. We also consider eventually periodic representations of elements of $\Q(β)$ for which the maximal power of the representation is proportional to the absolute value of the represented number, up to some universal constant. We prove that if every element of $\Q(β)$ admits such a representation then $β$ must be a Pisot number or a Salem number. This result generalises a well known result of Schmidt \cite{Schmidt}.

math.NT↗

arXiv2015

Palindromic sequences generated from marked morphisms

Fixed points ${\bf u}=φ({\bf u})$ of marked and primitive morphisms $φ$ over arbitrary alphabet are considered. We show that if ${\bf u}$ is palindromic, i.e., its language contains infinitely many palindromes, then some power of $φ$ has a conjugate in class ${\mathcal P}$. This class was introduced by Hof, Knill, Simon (1995) in order to study palindromic morphic words. Our definitions of marked and well-marked morphisms are more general than the ones previously used by Frid (1999) or Tan (2007). As any morphism with aperiodic fixed point over binary alphabet is marked, our result generalizes the result of Tan. Labbé (2014) demonstrated that already on a ternary alphabet the property of morphisms to be marked is important for the validity of our theorem. The main tool used in our proof is the description of bispecial factors in fixed points of morphisms provided by Klouda (2012).

math.CO↗

arXiv2014

Spectral properties of cubic complex Pisot units

For a real number $β>1$, Erdős, Joó and Komornik study distances between consecutive points in the set $X^m(β)=\bigl\{\sum_{j=0}^n a_j β^j : n\in\mathbb N,\,a_j\in\{0,1,\dots,m\}\bigr\}$. Pisot numbers play a crucial role for the properties of $X^m(β)$. Following the work of Zaïmi, who considered $X^m(γ)$ with $γ\in\mathbb{C}\setminus\mathbb{R}$ and $|γ|>1$, we show that for any non-real $γ$ and $m < |γ|^2-1$, the set $X^m(γ)$ is not relatively dense in the complex plane. Then we focus on complex Pisot units with a positive real conjugate $γ'$ and $m > |γ|^2-1$. If the number $1/γ'$ satisfies Property (F), we deduce that $X^m(γ)$ is uniformly discrete and relatively dense, i.e., $X^m(γ)$ is a Delone set. Moreover, we present an algorithm for determining two parameters of the Delone set $X^m(γ)$ which are analogous to minimal and maximal distances in the real case $X^m(β)$. For $γ$ satisfying $γ^3 + γ^2 + γ- 1 = 0$, explicit formulas for the two parameters are given.

math.MG↗

arXiv2014

Balances of $m$-bonacci words

The $m$-bonacci word is a generalization of the Fibonacci word to the $m$-letter alphabet $\mathcal{A} = {0,...,m-1}$. It is the unique fixed point of the Pisot--type substitution $ φ_m: 0\to 01, 1\to 02, ..., (m-2)\to0(m-1), and (m-1)\to0$. A result of Adamczewski implies the existence of constants $c^{(m)}$ such that the $m$-bonacci word is $c^{(m)}$-balanced, i.e., numbers of letter $a$ occurring in two factors of the same length differ at most by $c^{(m)}$ for any letter $a\in \mathcal{A}$. The constants $c^{(m)}$ have been already determined for $m=2$ and $m=3$. In this paper we study the bounds $c^{(m)}$ for a general $m\geq2$. We show that the $m$-bonacci word is $(\lfloor κm \rfloor +12)$-balanced, where $κ\approx 0.58$. For $m\leq 12$, we improve the constant $c^{(m)}$ by a computer numerical calculation to the value $\lceil\frac{m+1}{2}\rceil$.

math.CO↗

arXiv2014

Description of spectra of quadratic Pisot units

The spectrum of a real number $β>1$ is the set $X^{m}(β)$ of $p(β)$ where $p$ ranges over all polynomials with coefficients restricted to ${\mathcal A}=\{0,1,\dots,m\}$. For a quadratic Pisot unit $β$, we determine the values of all distances between consecutive points and their corresponding frequencies, by recasting the spectra in the frame of the cut-and-project scheme. We also show that shifting the set ${\mathcal A}$ of digits so that it contains at least one negative element, or considering negative base $-β$ instead of $β$, the gap sequence of the generalized spectrum is a coding of an exchange of three intervals.

math.NT↗

arXiv2013

$k$-block parallel addition versus $1$-block parallel addition in non-standard numeration systems

Parallel addition in integer base is used for speeding up multiplication and division algorithms. $k$-block parallel addition has been introduced by Kornerup in 1999: instead of manipulating single digits, one works with blocks of fixed length $k$. The aim of this paper is to investigate how such notion influences the relationship between the base and the cardinality of the alphabet allowing parallel addition. In this paper, we mainly focus on a certain class of real bases --- the so-called Parry numbers. We give lower bounds on the cardinality of alphabets of non-negative integer digits allowing block parallel addition. By considering quadratic Pisot bases, we are able to show that these bounds cannot be improved in general and we give explicit parallel algorithms for addition in these cases. We also consider the $d$-bonacci base, which satisfies the equation $X^d = X^{d-1} + X^{d-2} + \cdots + X + 1$. If in a base being a $d$-bonacci number $1$-block parallel addition is possible on the alphabet $\mathcal{A}$, then $\#\mathcal{A} \geq d+1$; on the other hand, there exists a $k\in\mathbb{N}$ such that $k$-block parallel addition in this base is possible on the alphabet $\{0,1,2\}$, which cannot be reduced. In particular, addition in the Tribonacci base is $14$-block parallel on alphabet $\{0,1,2\}$.

math.NT↗

arXiv2013

Palindromic closures using multiple antimorphisms

Generalized pseudostandard word $\bf u$, as introduced in 2006 by de Luca and De Luca, is given by a directive sequence of letters from an alphabet ${\cal A}$ and by a directive sequence of involutory antimorphisms acting on ${\cal A}^*$. Prefixes of $\bf u$ with increasing length are constructed using pseudopalindromic closure operator. We show that generalized Thue--Morse words ${\bf t}_{b,m}$, with $b, m \in \N$ and $b, m \geq 2$, are generalized pseudostandard words if and only if ${\bf t}_{b,m}$ is a periodic word or $b \leq m$. This extends the result of de Luca and De Luca obtained for the classical Thue--Morse words.

math.CO↗

arXiv2013

Greedy and lazy representations in negative base systems

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 $β^2$ and a non-integer alphabet. This enables us to characterize digit sequences admissible as greedy and lazy $(-β)$-representation. Such a characterization allows us to study the set of uniquely representable numbers. In case that $β$ is the golden ratio and the Tribonacci constant, we give the characterization of digit sequences admissible as greedy and lazy $(-β)$-representation using a set of forbidden strings.

math.DS↗

arXiv2013

Enumerating Abelian Returns to Prefixes of Sturmian Words

We follow the works of Puzynina and Zamboni, and Rigo et al. on abelian returns in Sturmian words. We determine the cardinality of the set $\mathcal{APR}_u$ of abelian returns of all prefixes of a Sturmian word $u$ in terms of the coefficients of the continued fraction of the slope, dependingly on the intercept. We provide a simple algorithm for finding the set $\mathcal{APR}_u$ and we determine it for the characteristic Sturmian words.

cs.FL↗

arXiv2013

Palindromic richness for languages invariant under more symmetries

For a given finite group $G$ consisting of morphisms and antimorphisms of a free monoid $\mathcal{A}^*$, we study infinite words with language closed under the group $G$. We focus on the notion of $G$-richness which describes words rich in generalized palindromic factors, i.e., in factors $w$ satisfying $Θ(w) = w$ for some antimorphism $Θ\in G$. We give several equivalent descriptions which are generalizations of know characterizations of rich words (in the terms of classical palindromes) and show two examples of $G$-rich words.

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