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- Robert V. Moody
- 1999

This article surveys the mathematics of the cut and project method as applied to point sets, called here model sets. It covers the geometric, arithmetic, and analytical sides of this theory as well as diffraction and the connection with dynamical systems. Dedicated to the memory of Richard (Dick) Slansky The spirit of the universe is subtle and informs all… (More)

A class of translation bounded complex measures, which have the form of weighted Dirac combs, on locally compact Abelian groups is investigated. Given such a Dirac comb, we are interested in its diffraction spectrum which emerges as the Fourier transform of the autocorrelation measure. We present a sufficient set of conditions to ensure that the diffraction… (More)

- Jeong-Yup Lee, Robert V. Moody, Boris Solomyak
- Discrete & Computational Geometry
- 2003

Lattices and Z-modules in Euclidean space possess an infinitude of subsets that are images of the original set under similarity transformation. We classify such self-similar images according to their indices for certain 4D examples that are related to 4D root systems, both crystallographic and non-crystallographic. We encapsulate their statistics in terms… (More)

The similarity submodules for various lattices and modules of interest in the crystal and quasicrystal theory are determined and the corresponding semigroups, along with their zeta functions, are investigated. One application is to the colouring of crystals and quasicrystals.

- V. F. SIRVENT, Boris Solomyak, Robert V. Moody
- 2003

We consider two dynamical systems associated with a substitution of Pisot type: the usual Z-action on a sequence space, and the R-action, which can be defined as a tiling dynamical system or as a suspension flow. We describe procedures for checking when these systems have pure discrete spectrum (the “balanced pairs algorithm” and the “overlap algorithm”)… (More)

- Robert V. Moody
- 2007

Similar sublattices of the root lattice A4 are possible [9] for each index that is the square of a non-zero integer of the form m + mn − n. Here, we add a constructive approach, based on the arithmetic of the quaternion algebra H(Q( √ 5)) and the existence of a particular involution of the second kind, which also provides the actual sublattices and the… (More)

The paper is about methods of discrete Fourier analysis in the context of Weyl group symmetry. Three families of class functions are defined on the maximal torus of each compact simply connected semisimple Lie group G. Such functions can always be restricted without loss of information to a fundamental region F̌ of the affine Weyl group. The members of each… (More)

It is shown how regular model sets can be characterized in terms of regularity properties of their associated dynamical systems. The proof proceeds in two steps. First, we characterize regular model sets in terms of a certain map β and then relate the properties of β to ones of the underlying dynamical system. As a by-product, we can show that regular model… (More)

Model sets (also called cut and project sets) are generalizations of lattices, and multi-component model sets are generalizations of lattices with colourings. In this paper, we study self-similarities of multi-component model sets. The main point may be simply summarized: whenever there is a self-similarity, there are also naturally related density… (More)