arXiv subjects

Lubomira Dvorakova

Publications and source records attributed to Lubomira Dvorakova.

6 recordsLinked to original sources

arXiv2022

On balanced sequences and their critical exponent

We study aperiodic balanced sequences over finite alphabets. A sequence vv of this type is fully characterised by a Sturmian sequence u and two constant gap sequences y and y'. We show that the language of v is eventually dendric and we focus on return words to its factors. We develop a method for computing the critical exponent and asymptotic critical exponent of balanced sequences, provided the associated Sturmian sequence u has a quadratic slope. The method is based on looking for the shortest return words to bispecial factors in v. We illustrate our method on several examples; in particular we confirm a conjecture of Rampersad, Shallit and Vandomme that two specific sequences have the least critical exponent among all balanced sequences over 9-letter (resp., $0-letter) alphabets.

cs.FL↗

arXiv2020

Antipalindromic numbers

Everybody has certainly heard about palindromes: words that stay the same when read backwards. For instance kayak, radar, or rotor. Mathematicians are interested in palindromic numbers: positive integers whose expansion in a certain integer base is a palindrome. The following problems are studied: palindromic primes, palindromic squares and higher powers, multibased palindromic numbers, etc. In this paper, we define and study antipalindromic numbers: positive integers whose expansion in a certain integer base is an antipalindrome. We present new results concerning divisibility and antipalindromic primes, antipalindromic squares and higher powers, and multibased antipalindromic numbers. We provide a user-friendly application for all studied questions.

math.CO↗

arXiv2018

Normalization of ternary generalized pseudostandard words

This paper focuses on generalized pseudostandard words, defined by de Luca and De Luca in 2006. In every step of the construction, the involutory antimorphism to be applied for the pseudopalindromic closure changes and is given by a so called directive bi-sequence. The concept of a normalized form of directive bi-sequences was introduced by Blondin-Massé et al. in 2013 and an algorithm for finding the normalized directive bi-sequence over a binary alphabet was provided. In this paper, we present an algorithm to find the normalized form of any directive bi-sequence over a ternary alphabet. Moreover, the algorithm was implemented in Python language and carefully tested, and is now publicly available in a module for working with ternary generalized pseudostandard words.

math.CO↗

arXiv2017

Fixed points of morphisms among binary generalized pseudostandard words

We introduce a class of fixed points of primitive morphisms among aperiodic binary generalized pseudostandard words. We conjecture that this class contains all fixed points of primitive morphisms among aperiodic binary generalized pseudostandard words that are not standard Sturmian words.

math.CO↗

arXiv2016

A new estimate on complexity of binary generalized pseudostandard words

Generalized pseudostandard words were introduced by de Luca and De Luca in 2006. In comparison to the palindromic and pseudopalindromic closure, only little is known about the generalized pseudopalindromic closure and the associated generalized pseudostandard words. We present a counterexample to Conjecture 43 from a paper by Blondin Massé et al. that estimated the complexity of binary generalized pseudostandard words as ${\mathcal C}(n) \leq 4n$ for all sufficiently large $n$. We conjecture that ${\mathcal C}(n)<6n$ for all $n \in \mathbb N$.

math.CO↗

arXiv2015

On periodicity of generalized pseudostandard words

Generalized pseudostandard words were introduced by de Luca and De Luca in 2006. In comparison to the palindromic and pseudopalindromic closure, only little is known about the generalized pseudopalindromic closure and the associated generalized pseudostandard words. In this paper we provide a necessary and sufficient condition for their periodicity over binary and ternary alphabet. More precisely, we describe how the directive bi-sequence of a generalized pseudostandard word has to look like in order to correspond to a periodic word. We state moreover a conjecture concerning a necessary and sufficient condition for periodicity over any alphabet.

math.CO↗