170 research outputs found
Ten Conferences WORDS: Open Problems and Conjectures
In connection to the development of the field of Combinatorics on Words, we
present a list of open problems and conjectures that were stated during the ten
last meetings WORDS. We wish to continually update the present document by
adding informations concerning advances in problems solving
A CMV--based eigensolver for companion matrices
In this paper we present a novel matrix method for polynomial rootfinding. By
exploiting the properties of the QR eigenvalue algorithm applied to a suitable
CMV-like form of a companion matrix we design a fast and computationally simple
structured QR iteration.Comment: 14 pages, 4 figure
Conferences WORDS, years 1997-2017: Open Problems and Conjectures
International audienceIn connection with the development of the field of Combinatorics on Words, we present a list of open problems and conjectures which were stated in the context of the eleven international meetings WORDS, which held from 1997 to 2017
Sturmian numeration systems and decompositions to palindromes
We extend the classical Ostrowski numeration systems, closely related to
Sturmian words, by allowing a wider range of coefficients, so that possible
representations of a number better reflect the structure of the associated
Sturmian word. In particular, this extended numeration system helps to catch
occurrences of palindromes in a characteristic Sturmian word and thus to prove
for Sturmian words the following conjecture stated in 2013 by Puzynina, Zamboni
and the author: If a word is not periodic, then for every it has a prefix
which cannot be decomposed to a concatenation of at most palindromes.Comment: Submitted to European Journal of Combinatoric
The sequence of open and closed prefixes of a Sturmian word
A finite word is closed if it contains a factor that occurs both as a prefix
and as a suffix but does not have internal occurrences, otherwise it is open.
We are interested in the {\it oc-sequence} of a word, which is the binary
sequence whose -th element is if the prefix of length of the word is
open, or if it is closed. We exhibit results showing that this sequence is
deeply related to the combinatorial and periodic structure of a word. In the
case of Sturmian words, we show that these are uniquely determined (up to
renaming letters) by their oc-sequence. Moreover, we prove that the class of
finite Sturmian words is a maximal element with this property in the class of
binary factorial languages. We then discuss several aspects of Sturmian words
that can be expressed through this sequence. Finally, we provide a linear-time
algorithm that computes the oc-sequence of a finite word, and a linear-time
algorithm that reconstructs a finite Sturmian word from its oc-sequence.Comment: Published in Advances in Applied Mathematics. Journal version of
arXiv:1306.225
Sturmian morphisms, the braid group B_4, Christoffel words and bases of F_2
We give a presentation by generators and relations of a certain monoid
generating a subgroup of index two in the group Aut(F_2) of automorphisms of
the rank two free group F_2 and show that it can be realized as a monoid in the
group B_4 of braids on four strings. In the second part we use Christoffel
words to construct an explicit basis of F_2 lifting any given basis of the free
abelian group Z^2. We further give an algorithm allowing to decide whether two
elements of F_2 form a basis or not. We also show that, under suitable
conditions, a basis has a unique conjugate consisting of two palindromes.Comment: 25 pages, 4 figure
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