43,473 research outputs found
On the Bragg Diffraction Spectra of a Meyer Set
Meyer sets have a relatively dense set of Bragg peaks and for this reason
they may be considered as basic mathematical examples of (aperiodic) crystals.
In this paper we investigate the pure point part of the diffraction of Meyer
sets in more detail. The results are of two kinds. First we show that given a
Meyer set and any intensity a less than the maximum intensity of its Bragg
peaks, the set of Bragg peaks whose intensity exceeds a is itself a Meyer set
(in the Fourier space). Second we show that if a Meyer set is modified by
addition and removal of points in such a way that its density is not altered
too much (the allowable amount being given explicitly as a proportion of the
original density) then the newly obtained set still has a relatively dense set
of Bragg peaks.Comment: 32 page
A comment on the relation between diffraction and entropy
Diffraction methods are used to detect atomic order in solids. While uniquely
ergodic systems with pure point diffraction have zero entropy, the relation
between diffraction and entropy is not as straightforward in general. In
particular, there exist families of homometric systems, which are systems
sharing the same diffraction, with varying entropy. We summarise the present
state of understanding by several characteristic examples.Comment: 7 page
Equicontinuous factors, proximality and Ellis semigroup for Delone sets
We discuss the application of various concepts from the theory of topological
dynamical systems to Delone sets and tilings. We consider in particular, the
maximal equicontinuous factor of a Delone dynamical system, the proximality
relation and the enveloping semigroup of such systems.Comment: 65 page
Diffractive point sets with entropy
After a brief historical survey, the paper introduces the notion of entropic
model sets (cut and project sets), and, more generally, the notion of
diffractive point sets with entropy. Such sets may be thought of as
generalizations of lattice gases. We show that taking the site occupation of a
model set stochastically results, with probabilistic certainty, in well-defined
diffractive properties augmented by a constant diffuse background. We discuss
both the case of independent, but identically distributed (i.i.d.) random
variables and that of independent, but different (i.e., site dependent) random
variables. Several examples are shown.Comment: 25 pages; dedicated to Hans-Ude Nissen on the occasion of his 65th
birthday; final version, some minor addition
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
- …