Weighted Model Sets and their Higher Point-Correlations

Deng, Xinghua; Moody, Robert V.

doi:10.4153/cmb-2011-106-5

Cited by 2 publications

(3 citation statements)

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“…With G set equal to F we then get the required almost periodicity properties of t → N (F ) | T t N (F ) from the corresponding almost periodicity properties of γ, by our definition of these concepts. In the reverse direction, upon linearizing the expressions N (F ) | T t N (F ) we obtain (12). From the almost periodicity properties of t → N (G) | T t N (F ) we then obtain the desired almost periodicity properties for all convolutions of the form G * F * γ(−t).…”

Section: Almost Periodicitymentioning

confidence: 99%

“…Example 13.4. Here we give an example based on an example given in [18] (see also [12]). We begin with G := Z/6Z and the two weighted Dirac combs 11δ 0 + 25δ 1 + 42δ 2 + 45δ 3 + 31δ 4 + 14δ 5 (39) 10δ 0 + 17δ 1 + 35δ 2 + 46δ 3 + 39δ 4 + 21δ 5 on G, which represent two density distributions for which we would like to find all possible homometric density distributions.…”

Section: 3mentioning

confidence: 99%

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Stationary processes and pure point diffraction

Lenz¹,

Moody²

2016

Ergod. Th. Dynam. Sys.

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We consider the construction and classification of some new mathematical objects, called ergodic spatial stationary processes, on locally compact Abelian groups, which provide a natural and very general setting for studying diffraction and the famous inverse problems associated with it. In particular we can construct complete families of solutions to the inverse problem from any given pure point measure that is chosen to be the diffraction. In this case these processes can be classified by the dual of the group of relators based on the set of Bragg peaks, and this gives a solution to the homometry problem for pure point diffraction.An ergodic spatial stationary process consists of a measure theoretical dynamical system and a mapping linking it with the ambient space in which diffracting density is supposed to exist. After introducing these processes we study their general properties and link pure point diffraction to almost periodicity.Given a pure point measure we show how to construct from it and a given set of phases a corresponding ergodic spatial stationary process. In fact we do this in two separate ways, each of which sheds its own light on the nature of the problem. The first construction can be seen as an elaboration of the Halmos-von Neumann theorem, lifted from the domain of dynamical systems to that of stationary processes. The second is a Gelfand construction obtained by defining a suitable Banach algebra out of the putative eigenfunctions of the desired dynamics. OutlineThis paper is concerned with mathematics of diffraction. More specifically we are interested in the famous inverse problem for diffraction: given something that is putatively the diffraction of something, what are all the somethings that could have produced this diffraction.Diffraction has been a mainstay in crystallography for almost a hundred years. 1 With the proliferation of extraordinary new materials with varying degrees of order and disorder, the importance of diffraction in revealing internal structure continues to be central. The precision, complexity, and variety of modern diffraction images is striking, see for instance the recent review article [53]. Nonetheless, in spite of many advances, the fundamental question of diffraction, the inverse problem of deducing physical structure from diffraction images, remains as challenging as ever.The discovery of aperiodic tilings and quasicrystals revived interest in the mathematics of diffraction, particularly since the types of diffraction images generated by such structuresessentially or exactly pure point diffraction together with symmetries not occurring in ordinary crystals -had not been foreseen by mathematicians, crystallographers, or materials research Date: October 3, 2018. RVM thanks the Natural Sciences and Engineering Council of Canada for its support of this work. 1 The Knipping-Laue experiment establishing x-ray diffraction was carried out in 1912. 1 arXiv:1111.3617v1 [math-ph] 15 Nov 2011 Definition 2.2. A stationary process N = (N, H, T ) is said to have pure point ...

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Section: Almost Periodicitymentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Stationary processes and pure point diffraction

Lenz¹,

Moody²

2016

Ergod. Th. Dynam. Sys.

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“…This phenomenon results in identical autocorrelations and hence in homometric model sets. A simple planar example was constructed in [4]; see also [16,14,15]. We skip further details here and shift our attention to continuous spectra now.…”

Section: Mathematical Setting and Pure Point Spectramentioning

confidence: 99%

Surprises in aperiodic diffraction

Baake

Grimm

2010

J. Phys.: Conf. Ser.

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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.

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Weighted Model Sets and their Higher Point-Correlations

Abstract: Abstract. Examples of distinct weighted model sets with equal 2, 3, 4, 5-point correlations are given.

Cited by 2 publications

References 9 publications

Stationary processes and pure point diffraction

Stationary processes and pure point diffraction

Surprises in aperiodic diffraction

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