202 research outputs found
Open and Closed Prefixes of Sturmian Words
A word is closed if it contains a proper factor that occurs both as a prefix
and as a suffix but does not have internal occurrences, otherwise it is open.
We deal with the sequence of open and closed prefixes of Sturmian words and
prove that this sequence characterizes every finite or infinite Sturmian word
up to isomorphisms of the alphabet. We then characterize the combinatorial
structure of the sequence of open and closed prefixes of standard Sturmian
words. We prove that every standard Sturmian word, after swapping its first
letter, can be written as an infinite product of squares of reversed standard
words.Comment: To appear in WORDS 2013 proceeding
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
Coding rotations on intervals
We show that the coding of rotation by on intervals with
rationally independent lengths can be recoded over Sturmian words of angle
More precisely, for a given an universal automaton is constructed such
that the edge indexed by the vector of values of the th letter on each
Sturmian word gives the value of the th letter of the coding of rotation.Comment: LIAFA repor
Local Rules for Computable Planar Tilings
Aperiodic tilings are non-periodic tilings characterized by local
constraints. They play a key role in the proof of the undecidability of the
domino problem (1964) and naturally model quasicrystals (discovered in 1982). A
central question is to characterize, among a class of non-periodic tilings, the
aperiodic ones. In this paper, we answer this question for the well-studied
class of non-periodic tilings obtained by digitizing irrational vector spaces.
Namely, we prove that such tilings are aperiodic if and only if the digitized
vector spaces are computable.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
Distributing Labels on Infinite Trees
Sturmian words are infinite binary words with many equivalent definitions:
They have a minimal factor complexity among all aperiodic sequences; they are
balanced sequences (the labels 0 and 1 are as evenly distributed as possible)
and they can be constructed using a mechanical definition. All this properties
make them good candidates for being extremal points in scheduling problems over
two processors. In this paper, we consider the problem of generalizing Sturmian
words to trees. The problem is to evenly distribute labels 0 and 1 over
infinite trees. We show that (strongly) balanced trees exist and can also be
constructed using a mechanical process as long as the tree is irrational. Such
trees also have a minimal factor complexity. Therefore they bring the hope that
extremal scheduling properties of Sturmian words can be extended to such trees,
as least partially. Such possible extensions are illustrated by one such
example.Comment: 30 pages, use pgf/tik
A measure of transcendence for singular points on conics
A singular point on a plane conic defined over is a
transcendental point of the curve which admits very good rational
approximations, uniformly in terms of the height. Extremal numbers and Sturmian
continued fractions are abscissa of such points on the parabola . In
this paper we provide a measure of transcendence for singular points on conics
defined over which, in these two cases, improves on the measure
obtained by Adamczewski et Bugeaud. The main tool is a quantitative version of
Schmidt subspace theorem due to Evertse.Comment: 8 page
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
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