Weighted Dirac combs with pure point diffraction

Baake, Michael; Moody, Robert V.

doi:10.1515/crll.2004.064

Cited by 140 publications

(380 citation statements)

References 33 publications

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“…(3) Both paperfolding and period-doubling sequences form model sets, obtained by a cut and project scheme, with an internal space of 2-adic numbers. [6][7][8]34]. This agrees with the existence of a connection between Parisi's replica-symmetry-breaking ideas and p-adic numbers as was suggested in [28].…”

Section: Resultssupporting

confidence: 77%

An Ultrametric State Space with a Dense Discrete Overlap Distribution: Paperfolding Sequences

Enter

Groote

2010

J Stat Phys

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We compute the Parisi overlap distribution for paperfolding sequences. It turns out to be discrete, and to live on the dyadic rationals. Hence it is a pure point measure whose support (as a closed set) is the full interval [−1, +1]. The space of paperfolding sequences has an ultrametric structure. Our example provides an illustration of some properties which were suggested to occur for pure states in spin glass models.

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Section: Resultssupporting

confidence: 77%

An Ultrametric State Space with a Dense Discrete Overlap Distribution: Paperfolding Sequences

Enter

Groote

2010

J Stat Phys

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“…The Dirac comb ω = δ Λ is pure point diffractive, by an application of the model set diffraction theorem 23,67,115 mentioned before. The diffraction measure γ is explicitly given by Eq.…”

Section: One-dimensional Examplesmentioning

confidence: 95%

Mathematical diffraction of aperiodic structures

Baake

Grimm

2012

Chem. Soc. Rev.

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Kinematic diffraction is well suited for a mathematical approach via measures, which has substantially been developed since the discovery of quasicrystals. The need for further insight emerged from the question of which distributions of matter, beyond perfect crystals, lead to pure point diffraction, hence to sharp Bragg peaks only. More recently, it has become apparent that one also has to study continuous diffraction in more detail, with a careful analysis of the different types of diffuse scattering involved. In this review, we summarise some key results, with particular emphasis on non-periodic structures. We choose an exposition on the basis of characteristic examples, while we refer to the existing literature for proofs and further details. IntroductionDiffraction techniques have dominated the structure analysis of solids for the last century, ever since von Laue and Bragg employed X-ray diffraction to determine the atomic structure of crystalline materials. Despite the availability of direct imaging techniques such as electron and atomic force microscopy, diffraction by X-rays, electrons and neutrons continues to be the method of choice to detect order in the atomic arrangements of a substance; see Cowley's book 35 and references therein for background.In its full generality, the diffraction of a beam of X-rays, electrons or neutrons from a macroscopic piece of solid is a complicated physical process. It is the presence of inelastic and multiple scattering, prevalent particularly in electron diffraction, which makes it essentially impossible to arrive at a complete mathematical description of the process. Here, we restrict to kinematic diffraction in the far-field or Fraunhofer limit. In this case, powerful tools of harmonic analysis are available to attack the direct problem of calculating the (kinematic) diffraction pattern of a given structure.In contrast, the inverse problem of determining a structure from its diffraction intensities is extremely involved. A diffraction pattern rarely determines a structure uniquely, as there can be homometric structures sharing the same autocorrelation (and hence the same diffraction). 9,54,57,103 We are far away from a complete understanding of the homometry classes of structures, in particular if the diffraction spectrum contains continuous components. At present, a picture is emerging, based on the analysis of explicit examples, which highlight how large the homometry classes may be.Originally, much of the effort concentrated on the pure point part of diffraction, also called the Bragg diffraction, for † Part of a themed issue on Quasicrystals in honour of the 2011 Nobel Prize in Chemistry winner, Professor Dan Shechtman.

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“…It is a subset of D r for some r > 0 and it is pure point diffractive [13,28,3]. The orbit closure X = R d + Λ(W ) is uniquely ergodic.…”

Section: Regular Model Setsmentioning

confidence: 99%

Dworkin’s argument revisited: Point processes, dynamics, diffraction, and correlations

Deng

Moody

2008

Journal of Geometry and Physics

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Abstract. The setting is an ergodic dynamical system (X, µ) whose points are themselves uniformly discrete point sets Λ in some space R d and whose group action is that of translation of these point sets by the vectors of R d . Steven Dworkin's argument relates the diffraction of the typical point sets comprising X to the dynamical spectrum of X. In this paper we look more deeply at this relationship, particularly in the context of point processes.We show that there is an R d -equivariant, isometric embedding, depending on the scattering strengths (weights) that are assigned to the points of Λ ∈ X, that takes theWe examine the image of this embedding and give a number of examples that show how it fails to be surjective. We show that full information on the measure µ is available from the weights and set of all the correlations (that is, the 2-point, 3-point, . . . , correlations) of the typical point set Λ ∈ X.We develop a formalism in the setting of random point measures that includes multi-colour point sets, and arbitrary real-valued weightings for the scattering from the different colour types of points, in the context of Palm measures and weighted versions of them. As an application we give a simple proof of a square-mean version of the Bombieri-Taylor conjecture, and from that we obtain an inequality that gives a quantitative relationship between the autocorrelation, the diffraction, and the ǫ-dual characters of typical element of X. The paper ends with a discussion of the Palm measure in the context of defining pattern frequencies.

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Weighted Dirac combs with pure point diffraction

Cited by 140 publications

References 33 publications

An Ultrametric State Space with a Dense Discrete Overlap Distribution: Paperfolding Sequences

An Ultrametric State Space with a Dense Discrete Overlap Distribution: Paperfolding Sequences

Mathematical diffraction of aperiodic structures

Dworkin’s argument revisited: Point processes, dynamics, diffraction, and correlations

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