Expand all Collapse all | Results 1 - 8 of 8 |
1. CMB 2011 (vol 55 pp. 487)
Weighted Model Sets and their Higher Point-Correlations Examples of distinct weighted model sets with equal $2,3,4, 5$-point
correlations are given.
Keywords:model sets, correlations, diffraction Categories:52C23, 51P05, 74E15, 60G55 |
2. CMB 2011 (vol 55 pp. 98)
Similarity and Coincidence Isometries for Modules The groups of (linear) similarity and coincidence isometries of
certain modules $\varGamma$ in $d$-dimensional Euclidean space, which
naturally occur in quasicrystallography, are considered. It is shown
that the structure of the factor group of similarity modulo
coincidence isometries is the direct sum of cyclic groups of prime
power orders that divide $d$. In particular, if the dimension $d$ is a
prime number $p$, the factor group is an elementary abelian
$p$-group. This generalizes previous results obtained for lattices to
situations relevant in quasicrystallography.
Categories:20H15, 82D25, 52C23 |
3. CMB 2005 (vol 48 pp. 302)
Discrete Sets and Associated Dynamical\\ Systems in a Non-Commutative Setting We define a uniform structure on the set of discrete sets of a locally
compact topological space on which a locally compact topological group
acts continuously. Then we investigate the completeness of these
uniform spaces and study these spaces by means of topological
dynamical systems.
Categories:52C23, 37B50 |
4. CMB 2004 (vol 47 pp. 168)
Kolakoski-$(3,1)$ Is a (Deformed) Model Set Unlike the (classical) Kolakoski sequence on the alphabet $\{1,2\}$, its analogue
on $\{1,3\}$ can be related to a primitive substitution rule. Using this connection,
we prove that the corresponding bi-infinite fixed point is a regular generic model set
and thus has a pure point diffraction spectrum. The Kolakoski-$(3,1)$ sequence is
then obtained as a deformation, without losing the pure point diffraction property.
Categories:52C23, 37B10, 28A80, 43A25 |
5. CMB 2002 (vol 45 pp. 697)
Pure Discrete Spectrum for One-dimensional Substitution Systems of Pisot Type We consider two dynamical systems associated with a substitution of
Pisot type: the usual $\mathbb{Z}$-action on a sequence space, and
the $\mathbb{R}$-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'') and study the
relation between them. In particular, we show that pure discrete
spectrum for the $\mathbb{R}$-action implies pure discrete spectrum
for the $\mathbb{Z}$-action, and obtain a partial result in the
other direction. As a corollary, we prove pure discrete spectrum
for every $\mathbb{R}$-action associated with a two-symbol
substitution of Pisot type (this is conjectured for an arbitrary
number of symbols).
Categories:37A30, 52C23, 37B10 |
6. CMB 2002 (vol 45 pp. 634)
Local Complexity of Delone Sets and Crystallinity This paper characterizes when a Delone set $X$ in $\mathbb{R}^n$ is an
ideal crystal in terms of restrictions on the number of its local
patches of a given size or on the heterogeneity of their distribution.
For a Delone set $X$, let $N_X (T)$ count the number of
translation-inequivalent patches of radius $T$ in $X$ and let
$M_X(T)$ be the minimum radius such that every closed ball of radius
$M_X(T)$ contains the center of a patch of every one of these kinds.
We show that for each of these functions there is a
``gap in the spectrum'' of possible growth rates between being
bounded and having linear growth, and that having sufficiently
slow linear growth is equivalent to $X$ being an ideal crystal.
Explicitly, for $N_X(T)$, if $R$ is the covering radius of $X$
then either $N_X(T)$ is bounded or $N_X (T) \ge T/2R$ for all $T>0$.
The constant $1/2R$ in this bound is best possible in all dimensions.
For $M_X(T)$, either $M_X(T)$ is bounded or $M_X(T)\ge T/3$ for all $T>0$.
Examples show that the constant $1/3$ in this bound cannot be replaced by
any number exceeding $1/2$. We also show that every aperiodic Delone
set $X$ has $M_X(T)\ge c(n) T$ for all $T>0$, for a certain constant $c(n)$
which depends on the dimension $n$ of $X$ and is $>1/3$ when $n>1$.
Keywords:aperiodic set, Delone set, packing-covering constant, sphere packing Categories:52C23, 52C17 |
7. CMB 2002 (vol 45 pp. 483)
Diffraction of Weighted Lattice Subsets A Dirac comb of point measures in Euclidean space with bounded
complex weights that is supported on a lattice $\varGamma$ inherits
certain general properties from the lattice structure. In
particular, its autocorrelation admits a factorization into a
continuous function and the uniform lattice Dirac comb, and its
diffraction measure is periodic, with the dual lattice
$\varGamma^*$ as lattice of periods. This statement remains true
in the setting of a locally compact Abelian group whose topology
has a countable base.
Keywords:diffraction, Dirac combs, lattice subsets, homometric sets Categories:52C07, 43A25, 52C23, 43A05 |
8. CMB 2002 (vol 45 pp. 123)
Uniform Distribution in Model Sets We give a new measure-theoretical proof of the uniform distribution
property of points in model sets (cut and project sets). Each model
set comes as a member of a family of related model sets, obtained by
joint translation in its ambient (the `physical') space and its
internal space. We prove, assuming only that the window defining the
model set is measurable with compact closure, that almost surely the
distribution of points in any model set from such a family is uniform
in the sense of Weyl, and almost surely the model set is pure point
diffractive.
Categories:52C23, 11K70, 28D05, 37A30 |