1,055 research outputs found
Doppler Tolerance, Complementary Code Sets and the Generalized Thue-Morse Sequence
We generalize the construction of Doppler-tolerant Golay complementary
waveforms by Pezeshki-Calderbank-Moran-Howard to complementary code sets having
more than two codes. This is accomplished by exploiting number-theoretic
results involving the sum-of-digits function, equal sums of like powers, and a
generalization to more than two symbols of the classical two-symbol
Prouhet-Thue-Morse sequence.Comment: 12 page
Hankel Determinant Calculus for the Thue-Morse and related sequences
The Hankel determinants of certain automatic sequences are evaluated,
based on a calculation modulo a prime number. In most cases, the Hankel
determinants of automatic sequences do not have any closed-form expressions;
the traditional methods, such as -decompo\-si\-tion and Jacobi continued
fraction, cannot be applied directly. Our method is based on a simple idea: the
Hankel determinants of each sequence equal to modulo are equal to
the Hankel determinants of modulo . The clue then consists of finding a
nice sequence , whose Hankel determinants have closed-form expressions.
Several examples are presented, including a result saying that the Hankel
determinants of the Thue-Morse sequence are nonzero, first proved by Allouche,
Peyri\`ere, Wen and Wen using determinant manipulation. The present approach
shortens the proof of the latter result significantly. We also prove that the
corresponding Hankel determinants do not vanish when the powers in the
infinite product defining the Thue--Morse sequence are replaced by
String attractors and combinatorics on words
The notion of string attractor has recently been introduced in [Prezza, 2017] and studied in [Kempa and Prezza, 2018] to provide a unifying framework for known dictionary-based compressors. A string attractor for a word w = w[1]w[2] · · · w[n] is a subset Γ of the positions 1, . . ., n, such that all distinct factors of w have an occurrence crossing at least one of the elements of Γ. While finding the smallest string attractor for a word is a NP-complete problem, it has been proved in [Kempa and Prezza, 2018] that dictionary compressors can be interpreted as algorithms approximating the smallest string attractor for a given word. In this paper we explore the notion of string attractor from a combinatorial point of view, by focusing on several families of finite words. The results presented in the paper suggest that the notion of string attractor can be used to define new tools to investigate combinatorial properties of the words
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
Surface Magnetization of Aperiodic Ising Systems: a Comparative Study of the Bond and Site Problems
We investigate the influence of aperiodic perturbations on the critical
behaviour at a second order phase transition. The bond and site problems are
compared for layered systems and aperiodic sequences generated through
substitution. In the bond problem, the interactions between the layers are
distributed according to an aperiodic sequence whereas in the site problem, the
layers themselves follow the sequence. A relevance-irrelevance criterion
introduced by Luck for the bond problem is extended to discuss the site
problem. It involves a wandering exponent for pairs, which can be larger than
the one considered before in the bond problem. The surface magnetization of the
layered two-dimensional Ising model is obtained, in the extreme anisotropic
limit, for the period-doubling and Thue-Morse sequences.Comment: 19 pages, Plain TeX, IOP macros + epsf, 6 postscript figures, minor
correction
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