Proof of Brlek-Reutenauer conjecture

Brlek and Reutenauer conjectured that any infinite word u with language closed under reversal satisfies the equality 2D(u) = \sum_{n=0}^{\infty}T_u(n) in which D(u) denotes the defect of u and T_u(n) denotes C_u(n+1)-C_u(n) +2 - P_U(n+1) - P_u(n), where C_u and P_u are the factor and palindromic complexity of u, respectively. This conjecture was verified for periodic words by Brlek and Reutenauer themselves. Using their results for periodic words, we have recently proved the conjecture for uniformly recurrent words. In the present article we prove the conjecture in its general version by a new method without exploiting the result for periodic words.Comment: 9 page

Return Words and Recurrence Function of a Class of Infinite Words

Many combinatorial and arithmetical properties have been studied for infinite words ub associated with ß-integers. Here, new results describing return words and recurrence function for a special case of ub will be presented. The methods used here can be applied to more general infinite words, but the description then becomes rather technical.

Asymptotic behavior of beta-integers

Beta-integers (``$\beta$-integers'') are those numbers which are the counterparts of integers when real numbers are expressed in irrational basis $\beta > 1$. In quasicrystalline studies $\beta$-integers supersede the ``crystallographic'' ordinary integers. When the number $\beta$ is a Parry number, the corresponding $\beta$-integers realize only a finite number of distances between consecutive elements and somewhat appear like ordinary integers, mainly in an asymptotic sense. In this letter we make precise this asymptotic behavior by proving four theorems concerning Parry $\beta$-integers.Comment: 17 page

Repetitions in beta-integers

Classical crystals are solid materials containing arbitrarily long periodic repetitions of a single motif. In this paper, we study the maximal possible repetition of the same motif occurring in beta-integers -- one dimensional models of quasicrystals. We are interested in beta-integers realizing only a finite number of distinct distances between neighboring elements. In such a case, the problem may be reformulated in terms of combinatorics on words as a study of the index of infinite words coding beta-integers. We will solve a particular case for beta being a quadratic non-simple Parry number.Comment: 11 page

Palindromic complexity of trees

We consider finite trees with edges labeled by letters on a finite alphabet $\varSigma$. Each pair of nodes defines a unique labeled path whose trace is a word of the free monoid $\varSigma^*$. The set of all such words defines the language of the tree. In this paper, we investigate the palindromic complexity of trees and provide hints for an upper bound on the number of distinct palindromes in the language of a tree.Comment: Submitted to the conference DLT201

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

Struktura osobnosti učitele, jeho prestiž, společenské postavení a sociální role v komunistické výchově mladé generace.

Katedra sociologieFilozofická fakult

Return Words and Recurrence Function of a Class of Infinite Words

Many combinatorial and arithmetical properties have been studied for infinite words ub associated with ß-integers. Here, new results describing return words and recurrence function for a special case of ub will be presented. The methods used here can be applied to more general infinite words, but the description then becomes rather technical.