5,745 research outputs found
Bounds for the discrete correlation of infinite sequences on k symbols and generalized Rudin-Shapiro sequences
Motivated by the known autocorrelation properties of the Rudin-Shapiro
sequence, we study the discrete correlation among infinite sequences over a
finite alphabet, where we just take into account whether two symbols are
identical. We show by combinatorial means that sequences cannot be "too"
different, and by an explicit construction generalizing the Rudin-Shapiro
sequence, we show that we can achieve the maximum possible difference.Comment: Improved Introduction and new Section 6 (Lovasz local lemma
Kinematic Diffraction from a Mathematical Viewpoint
Mathematical diffraction theory is concerned with the analysis of the
diffraction image of a given structure and the corresponding inverse problem of
structure determination. In recent years, the understanding of systems with
continuous and mixed spectra has improved considerably. Simultaneously, their
relevance has grown in practice as well. In this context, the phenomenon of
homometry shows various unexpected new facets. This is particularly so for
systems with stochastic components. After the introduction to the mathematical
tools, we briefly discuss pure point spectra, based on the Poisson summation
formula for lattice Dirac combs. This provides an elegant approach to the
diffraction formulas of infinite crystals and quasicrystals. We continue by
considering classic deterministic examples with singular or absolutely
continuous diffraction spectra. In particular, we recall an isospectral family
of structures with continuously varying entropy. We close with a summary of
more recent results on the diffraction of dynamical systems of algebraic or
stochastic origin.Comment: 30 pages, invited revie
Surprises in aperiodic diffraction
Mathematical diffraction theory is concerned with the diffraction image of a
given structure and the corresponding inverse problem of structure
determination. In recent years, the understanding of systems with continuous
and mixed spectra has improved considerably. Moreover, the phenomenon of
homometry shows various unexpected new facets. Here, we report on some of the
recent results in an exemplary and informal fashion.Comment: 9 pages, 1 figure; paper presented at Aperiodic 2009 (Liverpool
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