1,165 research outputs found
Local symmetry dynamics in one-dimensional aperiodic lattices
A unifying description of lattice potentials generated by aperiodic
one-dimensional sequences is proposed in terms of their local reflection or
parity symmetry properties. We demonstrate that the ranges and axes of local
reflection symmetry possess characteristic distributional and dynamical
properties which can be determined for every aperiodic binary lattice. A
striking aspect of such a property is given by the return maps of sequential
spacings of local symmetry axes, which typically traverse few-point symmetry
orbits. This local symmetry dynamics allows for a classification of inherently
different aperiodic lattices according to fundamental symmetry principles.
Illustrating the local symmetry distributional and dynamical properties for
several representative binary lattices, we further show that the renormalized
axis spacing sequences follow precisely the particular type of underlying
aperiodic order. Our analysis thus reveals that the long-range order of
aperiodic lattices is characterized in a compellingly simple way by its local
symmetry dynamics.Comment: 15 pages, 12 figure
Alternating subgroups of Coxeter groups
We study combinatorial properties of the alternating subgroup of a Coxeter
group, using a presentation of it due to Bourbaki.Comment: 39 pages, 3 figure
A note on palindromicity
Two results on palindromicity of bi-infinite words in a finite alphabet are
presented. The first is a simple, but efficient criterion to exclude
palindromicity of minimal sequences and applies, in particular, to the
Rudin-Shapiro sequence. The second provides a constructive method to build
palindromic minimal sequences based upon regular, generic model sets with
centro-symmetric window. These give rise to diagonal tight-binding models in
one dimension with purely singular continuous spectrum.Comment: 12 page
Palindromic complexity of infinite words associated with simple Parry numbers
A simple Parry number is a real number \beta>1 such that the R\'enyi
expansion of 1 is finite, of the form d_\beta(1)=t_1...t_m. We study the
palindromic structure of infinite aperiodic words u_\beta that are the fixed
point of a substitution associated with a simple Parry number \beta. It is
shown that the word u_\beta contains infinitely many palindromes if and only if
t_1=t_2= ... =t_{m-1} \geq t_m. Numbers \beta satisfying this condition are the
so-called confluent Pisot numbers. If t_m=1 then u_\beta is an Arnoux-Rauzy
word. We show that if \beta is a confluent Pisot number then P(n+1)+ P(n) =
C(n+1) - C(n)+ 2, where P(n) is the number of palindromes and C(n) is the
number of factors of length n in u_\beta. We then give a complete description
of the set of palindromes, its structure and properties.Comment: 28 pages, to appear in Annales de l'Institut Fourie
Factor versus palindromic complexity of uniformly recurrent infinite words
We study the relation between the palindromic and factor complexity of
infinite words. We show that for uniformly recurrent words one has P(n)+P(n+1)
\leq \Delta C(n) + 2, for all n \in N. For a large class of words it is a
better estimate of the palindromic complexity in terms of the factor complexity
then the one presented by Allouche et al. We provide several examples of
infinite words for which our estimate reaches its upper bound. In particular,
we derive an explicit prescription for the palindromic complexity of infinite
words coding r-interval exchange transformations. If the permutation \pi
connected with the transformation is given by \pi(k)=r+1-k for all k, then
there is exactly one palindrome of every even length, and exactly r palindromes
of every odd length.Comment: 16 pages, submitted to Theoretical Computer Scienc
Intermittency as a universal characteristic of the complete chromosome DNA sequences of eukaryotes: From protozoa to human genomes
Large-scale dynamical properties of complete chromosome DNA sequences of
eukaryotes are considered. By the proposed deterministic models with
intermittency and symbolic dynamics we describe a wide spectrum of large-scale
patterns inherent in these sequences, such as segmental duplications, tandem
repeats, and other complex sequence structures. It is shown that the recently
discovered gene number balance on the strands is not of random nature, and a
complete chromosome DNA sequence exhibits the properties of deterministic
chaos.Comment: 4 pages, 5 figure
Palindromic complexity of trees
We consider finite trees with edges labeled by letters on a finite alphabet
. Each pair of nodes defines a unique labeled path whose trace is a
word of the free monoid . 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
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