2,812 research outputs found
On prefixal factorizations of words
We consider the class of all infinite words over
a finite alphabet admitting a prefixal factorization, i.e., a factorization
where each is a non-empty prefix of With
each one naturally associates a "derived" infinite word
which may or may not admit a prefixal factorization. We are
interested in the class of all words of
such that for all . Our primary
motivation for studying the class stems from its connection
to a coloring problem on infinite words independently posed by T. Brown in
\cite{BTC} and by the second author in \cite{LQZ}. More precisely, let be the class of all words such that for every finite
coloring there exist and a factorization
with for each In \cite{DPZ}
we conjectured that a word if and only if is purely
periodic. In this paper we show that so
in other words, potential candidates to a counter-example to our conjecture are
amongst the non-periodic elements of We establish several
results on the class . In particular, we show that a
Sturmian word belongs to if and only if is
nonsingular, i.e., no proper suffix of is a standard Sturmian word
On Christoffel and standard words and their derivatives
We introduce and study natural derivatives for Christoffel and finite
standard words, as well as for characteristic Sturmian words. These
derivatives, which are realized as inverse images under suitable morphisms,
preserve the aforementioned classes of words. In the case of Christoffel words,
the morphisms involved map to (resp.,~) and to
(resp.,~) for a suitable . As long as derivatives are
longer than one letter, higher-order derivatives are naturally obtained. We
define the depth of a Christoffel or standard word as the smallest order for
which the derivative is a single letter. We give several combinatorial and
arithmetic descriptions of the depth, and (tight) lower and upper bounds for
it.Comment: 28 pages. Final version, to appear in TC
A Coloring Problem for Infinite Words
In this paper we consider the following question in the spirit of Ramsey
theory: Given where is a finite non-empty set, does there
exist a finite coloring of the non-empty factors of with the property that
no factorization of is monochromatic? We prove that this question has a
positive answer using two colors for almost all words relative to the standard
Bernoulli measure on We also show that it has a positive answer for
various classes of uniformly recurrent words, including all aperiodic balanced
words, and all words satisfying
for all sufficiently large, where denotes the number of
distinct factors of of length Comment: arXiv admin note: incorporates 1301.526
Rich, Sturmian, and trapezoidal words
In this paper we explore various interconnections between rich words,
Sturmian words, and trapezoidal words. Rich words, first introduced in
arXiv:0801.1656 by the second and third authors together with J. Justin and S.
Widmer, constitute a new class of finite and infinite words characterized by
having the maximal number of palindromic factors. Every finite Sturmian word is
rich, but not conversely. Trapezoidal words were first introduced by the first
author in studying the behavior of the subword complexity of finite Sturmian
words. Unfortunately this property does not characterize finite Sturmian words.
In this note we show that the only trapezoidal palindromes are Sturmian. More
generally we show that Sturmian palindromes can be characterized either in
terms of their subword complexity (the trapezoidal property) or in terms of
their palindromic complexity. We also obtain a similar characterization of rich
palindromes in terms of a relation between palindromic complexity and subword
complexity.Comment: 7 page
On the combinatorics of finite words
AbstractIn this paper we consider a combinatorial method for the analysis of finite words recently introduced in Colosimo and de Luca (Special factors in biological strings, preprint 97/42, Dipt. Matematica, Univ. di Roma) for the study of biological macromolecules. The method is based on the analysis of (right) special factors of a given word. A factor u of a word w is special if there exist at least two occurrences of the factor u in w followed on the right by two distinct letters. We show that in the combinatorics of finite words two parameters play an essential role. The first, denoted by R, represents the minimal integer such that there do not exist special factors of w of length R. The second, that we denote by K, is the minimal length of a factor of w which cannot be extended on the right in a factor of w. Some new results are proved. In particular, a new characterization in terms of special factors and of R and K is given for the set PER of all words w having two periods p and q which are coprimes and such that ¦w¦ = p + q − 2
Characteristic morphisms of generalized episturmian words
In a recent paper with L. Q. Zamboni, the authors introduced the class of ϑ-episturmian words. An infinite word over A is standard ϑ-episturmian, where ϑ is an involutory antimorphism of A*, if its set of factors is closed under ϑ and its left special factors are prefixes. When ϑ is the reversal operator, one obtains the usual standard episturmian words. In this paper, we introduce and study ϑ-characteristic morphisms, that is, morphisms which map standard episturmian words into standard ϑ-episturmian words. They are a natural extension of standard episturmian morphisms. The main result of the paper is a characterization of these morphisms when they are injective. In order to prove this result, we also introduce and study a class of biprefix codes which are overlap-free, i.e., any two code words do not overlap properly, and normal, i.e., no proper suffix (prefix) of any code-word is left (right) special in the code. A further result is that any standard ϑ-episturmian word is a morphic image, by an injective ϑ-characteristic morphism, of a standard episturmian word
Entropy of L-fuzzy sets
The notion of “entropy” of a fuzzy set, introduced in a previous paper in the case of generalized characteristic functions whose range is the interval [0, 1] of the real line, is extended to the case of maps whose range is a poset L (or, in particular, a lattice).Some of the reasons giving rise to the non-comparability of the truth values and then the necessity of considering poset structures as range of the maps are discussed.The interpretative problems of the given mathematical definitions regarding the connections with decision theory are briefly analyzed
Harmonic and gold Sturmian words
AbstractIn the combinatorics of Sturmian words an essential role is played by the set PER of all finite words w on the alphabet A={a,b} having two periods p and q which are coprime and such that |w|=p+q−2. As is well known, the set St of all finite factors of all Sturmian words equals the set of factors of PER. Moreover, the elements of PER have many remarkable structural properties. In particular, the relation Stand=A∪PER{ab,ba} holds, where Stand is the set of all finite standard Sturmian words. In this paper we introduce two proper subclasses of PER that we denote by Harm and Gold. We call an element of Harm a harmonic word and an element of Gold a gold word. A harmonic word w beginning with the letter x is such that the ratio of two periods p/q, with p<q, is equal to its slope, i.e., (|w|y+1)/(|w|x+1), where {x,y}={a,b}. A gold word is an element of PER such that p and q are primes. Some characterizations of harmonic words are given and the number of harmonic words of each length is computed. Moreover, we prove that St is equal to the set of factors of Harm and to the set of factors of Gold. We introduce also the classes Harm and Gold of all infinite standard Sturmian words having infinitely many prefixes in Harm and Gold, respectively. We prove that Gold∩Harm contain continuously many elements. Finally, some conjectures are formulated
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