527 research outputs found
Transduction of Automatic Sequences and Applications
We consider the implementation of the transduction of automatic sequences,
and their generalizations, in the Walnut software for solving decision problems
in combinatorics on words. We provide a number of applications, including (a)
representations of n! as a sum of three squares (b) overlap-free Dyck words and
(c) sums of Fibonacci representations. We also prove results about iterated
running sums of the Thue-Morse sequence
Remarks on the Spectral Properties of Tight Binding and Kronig-Penney Models with Substitution Sequences
We comment on some recent investigations on the electronic properties of
models associated to the Thue-Morse chain and point out that their conclusions
are in contradiction with rigorously proven theorems and indicate some of the
sources of these misinterpretations. We briefly review and explain the current
status of mathematical results in this field and discuss some conjectures and
open problems.Comment: 15,CPT-94/P.3003,tex,
Surface Magnetization of Aperiodic Ising Quantum Chains
We study the surface magnetization of aperiodic Ising quantum chains. Using
fermion techniques, exact results are obtained in the critical region for
quasiperiodic sequences generated through an irrational number as well as for
the automatic binary Thue-Morse sequence and its generalizations modulo p. The
surface magnetization exponent keeps its Ising value, beta_s=1/2, for all the
sequences studied. The critical amplitude of the surface magnetization depends
on the strength of the modulation and also on the starting point of the chain
along the aperiodic sequence.Comment: 11 pages, 6 eps-figures, Plain TeX, eps
Canonical Representatives of Morphic Permutations
An infinite permutation can be defined as a linear ordering of the set of
natural numbers. In particular, an infinite permutation can be constructed with
an aperiodic infinite word over as the lexicographic order
of the shifts of the word. In this paper, we discuss the question if an
infinite permutation defined this way admits a canonical representative, that
is, can be defined by a sequence of numbers from [0, 1], such that the
frequency of its elements in any interval is equal to the length of that
interval. We show that a canonical representative exists if and only if the
word is uniquely ergodic, and that is why we use the term ergodic permutations.
We also discuss ways to construct the canonical representative of a permutation
defined by a morphic word and generalize the construction of Makarov, 2009, for
the Thue-Morse permutation to a wider class of infinite words.Comment: Springer. WORDS 2015, Sep 2015, Kiel, Germany. Combinatorics on
Words: 10th International Conference. arXiv admin note: text overlap with
arXiv:1503.0618
Spectral Approximation for Quasiperiodic Jacobi Operators
Quasiperiodic Jacobi operators arise as mathematical models of quasicrystals
and in more general studies of structures exhibiting aperiodic order. The
spectra of these self-adjoint operators can be quite exotic, such as Cantor
sets, and their fine properties yield insight into associated dynamical
systems. Quasiperiodic operators can be approximated by periodic ones, the
spectra of which can be computed via two finite dimensional eigenvalue
problems. Since long periods are necessary to get detailed approximations, both
computational efficiency and numerical accuracy become a concern. We describe a
simple method for numerically computing the spectrum of a period- Jacobi
operator in operations, and use it to investigate the spectra of
Schr\"odinger operators with Fibonacci, period doubling, and Thue-Morse
potentials
Suffix conjugates for a class of morphic subshifts
Let A be a finite alphabet and f: A^* --> A^* be a morphism with an iterative
fixed point f^\omega(\alpha), where \alpha{} is in A. Consider the subshift (X,
T), where X is the shift orbit closure of f^\omega(\alpha) and T: X --> X is
the shift map. Let S be a finite alphabet that is in bijective correspondence
via a mapping c with the set of nonempty suffixes of the images f(a) for a in
A. Let calS be a subset S^N be the set of infinite words s = (s_n)_{n\geq 0}
such that \pi(s):= c(s_0)f(c(s_1)) f^2(c(s_2))... is in X. We show that if f is
primitive and f(A) is a suffix code, then there exists a mapping H: calS -->
calS such that (calS, H) is a topological dynamical system and \pi: (calS, H)
--> (X, T) is a conjugacy; we call (calS, H) the suffix conjugate of (X, T). In
the special case when f is the Fibonacci or the Thue-Morse morphism, we show
that the subshift (calS, T) is sofic, that is, the language of calS is regular
Abelian-Square-Rich Words
An abelian square is the concatenation of two words that are anagrams of one
another. A word of length can contain at most distinct
factors, and there exist words of length containing distinct
abelian-square factors, that is, distinct factors that are abelian squares.
This motivates us to study infinite words such that the number of distinct
abelian-square factors of length grows quadratically with . More
precisely, we say that an infinite word is {\it abelian-square-rich} if,
for every , every factor of of length contains, on average, a number
of distinct abelian-square factors that is quadratic in ; and {\it uniformly
abelian-square-rich} if every factor of contains a number of distinct
abelian-square factors that is proportional to the square of its length. Of
course, if a word is uniformly abelian-square-rich, then it is
abelian-square-rich, but we show that the converse is not true in general. We
prove that the Thue-Morse word is uniformly abelian-square-rich and that the
function counting the number of distinct abelian-square factors of length
of the Thue-Morse word is -regular. As for Sturmian words, we prove that a
Sturmian word of angle is uniformly abelian-square-rich
if and only if the irrational has bounded partial quotients, that is,
if and only if has bounded exponent.Comment: To appear in Theoretical Computer Science. Corrected a flaw in the
proof of Proposition
Morphisms, Symbolic sequences, and their Standard Forms
Morphisms are homomorphisms under the concatenation operation of the set of
words over a finite set. Changing the elements of the finite set does not
essentially change the morphism. We propose a way to select a unique
representing member out of all these morphisms. This has applications to the
classification of the shift dynamical systems generated by morphisms. In a
similar way, we propose the selection of a representing sequence out of the
class of symbolic sequences over an alphabet of fixed cardinality. Both methods
are useful for the storing of symbolic sequences in databases, like The On-Line
Encyclopedia of Integer Sequences. We illustrate our proposals with the
-symbol Fibonacci sequences
Quasicrystals, model sets, and automatic sequences
We survey mathematical properties of quasicrystals, first from the point of
view of harmonic analysis, then from the point of view of morphic and automatic
sequences.
Nous proposons un tour d'horizon de propri\'et\'es math\'ematiques des
quasicristaux, d'abord du point de vue de l'analyse harmonique, ensuite du
point de vue des suites morphiques et automatiques
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