Abstract not found

A (fractional) repetition in a word w is a subword with the period of at most half of the subword length. We study maximal repetitions occurring in w, that is those for which any extended subword of w has a bigger period. The set of such repetitions represents in a compact way all repetitions in w. We first study maximal repetitions in Fibonacci words -- we count their exact number, and estimate the sum of their exponents. These quantities turn out to be linearly-bounded in the length of the word. We then prove that the maximal number of maximal repetitions in general words (on arbitrary alphabet) of length n is linearly-bounded in n, and we mention some applications and consequences of this result.

The study of combinatorics on words, or finite sequences of symbols from a finite alphabet, finds applications in several areas of biology, computer science, mathematics, and physics. Molecular biology, in particular, has stimulated considerable interest in the study of combinatorics on partial words that are sequences that may have a number of “do not know ” symbols also called “holes”. This paper is devoted to a fundamental result on periods of words, the Critical Factorization Theorem, which states that the period of a word is always locally detectable in at least one position of the word resulting in a corresponding critical factorization. Here, we describe precisely the class of partial words w with one hole for which the weak period is locally detectable in at least one position of w. Our proof provides an algorithm which computes a critical factorization when one exists. A World Wide Web server interface at

this paper, it is shown that the subword complexity of a D0L language is bounded by cn (resp. cn log n, cn) if the morphism that generates the languages is arbitrary (resp. growing, uniform). This result was extended in [Pan84a]: Theorem 6.7. The subword complexity of an in nite word generated by iterating a morphism is of one of the following types: (n), (n log n), (n log n log n), (n ), or (1)

We study the minimal proportion (density) of one letter in n-th power-free binary words. First, we introduce and analyse a general notion of minimal letter density for any innite set of words which don't contain a specied set of \prohibited" subwords. We then prove that for n-th power-free binary words the density function is 1 n + 1 n 3 + 1 n 4 + O( 1 n 5 ). We also consider a generalization of n-th power-free words for fractional powers (exponents): a word is x-th power-free for a real x, if it does not contain subwords of exponent x or more. We study the minimal proportion of one letter in x-th power-free binary words as a function of x and prove, in particular, that this function is discontinuous at 7 3 as well as at all integer points n 3. Finally, we give an estimate of the size of the jumps. Keywords: Unavoidable patterns, power-free words, exponent, minimal density. 1 Introduction One of classical topics of formal language theory and word combinatorics is th...

Some shuffle-like operations on infinite (!-words) are investigated. The operations are introduced using a uniform method based on the notion of an

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pg. 8 Elenco di alcune pubblicazioni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pg. 9 2 Dati personali Data e luogo di nascita Residenza Recapito di lavoro Telefono / fax email 17 dicembre 1963, Palermo. Viale Regione Siciliana 10750, 90147 Palermo, telefono 091-6911068. Dipartimento di Matematica e Applicazioni, Via Archira 34, Universita di Palermo 90123 Palermo. 091-6040414 / 091-6165425 mignosi@altair.math.unipa.it 3 Titoli di studio e professionali Aprile 2000 Iscritto dalla Sezione 27 (Informatica) del C.N.U. (Consiglio Nazionale delle Universita`) nella lista nazionale francese dei qualicati alle funzioni di Professore delle Universita. Dal 20/12/99 ai ni giuridici e dal 14/06/2000 agli eetti economici, professore associato per il settore scientico disciplinare K05B (Informatica) presso la Facolta di Scien

We study the Shallit's conjecture which states that an infinite word ! is ultimately periodic if and only if lim n!1 inf ju n (!)j n ? 3 \Gamma p 5 2 ; where u n (!) is the longest suffix of length n prefix of ! which occurs also inside length n \Gamma 1 prefix of !. We prove that a weaker condition holds, namely that the conjecture is true if the constant 3\Gamma p 5 2 is replaced by 13\Gamma p 69 10 . TUCS Research Group Mathematical Structures of Computer Science 1 Introduction One of the most fundamental notions of words is that of the periodicity. In addition to classical results, such as Periodicity Lemma of Fine and Wilf and Critical Factorization Theorem, cf. [5, 8], a very interesting periodicity property was introduced in [11, 10]. Namely that an infinite word ! is ultimately periodic if and only if each long enough prefix of ! contains as a suffix a repetition of order 1 + OE, where OE is the golden ratio 1+ p 5 2 . A related but different characteriza...

We introduce and investigate some sets of !-trajectories that have the following properties: each of them defines an associative and commutative operation of parallel composition (shuffle) of !-words and, moreover, each of them satisfies a certain condition of fairness. The fairness conditions considered involve the operation of parallel composition of !-words.

Abstract not found