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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)

- 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 2 + mn − n 2. 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)

- Jeong-Yup Lee, Robert V. Moody
- Discrete & Computational Geometry
- 2001

Learning is but an adjunct to ourself And where we are our learning likewise is. Abstract The paper studies ways in which the sets of a partition of a lattice in R n become regular model sets. The main theorem gives equivalent conditions which assure that a matrix substitution system on a lattice in R n gives rise to regular model sets (based on p-adic-like… (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.

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)

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 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 regioň F of the affine Weyl group. The members of each… (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)

- Michael Baake, Robert V. Moody, Martin Schlottmann, Peter Kramer
- 1998

Model sets (or cut and project sets) provide a familiar and commonly used method of constructing and studying nonperiodic point sets. Here we extend this method to situations where the internal spaces are no longer Euclidean, but instead spaces with p-adic topologies or even with mixed Euclidean/p-adic topologies. We show that a number of well known tilings… (More)

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