1,366 research outputs found
Similarity and Coincidence Isometries for Modules
The groups of (linear) similarity and coincidence isometries of certain
modules 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.Comment: 14 page
Dense Dirac combs in Euclidean space with pure point diffraction
Regular model sets, describing the point positions of ideal
quasicrystallographic tilings, are mathematical models of quasicrystals. An
important result in mathematical diffraction theory of regular model sets,
which are defined on locally compact Abelian groups, is the pure pointedness of
the diffraction spectrum. We derive an extension of this result, valid for
dense point sets in Euclidean space, which is motivated by the study of
quasicrystallographic random tilings.Comment: 18 pages. v2: final version as publishe
Colourings of planar quasicrystals
The investigation of colour symmetries for periodic and aperiodic systems
consists of two steps. The first concerns the computation of the possible
numbers of colours and is mainly combinatorial in nature. The second is
algebraic and determines the actual colour symmetry groups. Continuing previous
work, we present the results of the combinatorial part for planar patterns with
n-fold symmetry, where n=7,9,15,16,20,24. This completes the cases with values
of n such that Euler's totient function of n is less than or equal to eight.Comment: Talk presented by Max Scheffer at Quasicrystals 2001, Sendai
(September 2001). 6 pages, including two colour figure
Haldane linearisation done right: Solving the nonlinear recombination equation the easy way
The nonlinear recombination equation from population genetics has a long
history and is notoriously difficult to solve, both in continuous and in
discrete time. This is particularly so if one aims at full generality, thus
also including degenerate parameter cases. Due to recent progress for the
continuous time case via the identification of an underlying stochastic
fragmentation process, it became clear that a direct general solution at the
level of the corresponding ODE itself should also be possible. This paper shows
how to do it, and how to extend the approach to the discrete-time case as well.Comment: 12 pages, 1 figure; some minor update
Coincidence rotations of the root lattice
The coincidence site lattices of the root lattice are considered, and
the statistics of the corresponding coincidence rotations according to their
indices is expressed in terms of a Dirichlet series generating function. This
is possible via an embedding of into the icosian ring with its rich
arithmetic structure, which recently (arXiv:math.MG/0702448) led to the
classification of the similar sublattices of .Comment: 13 pages, 1 figur
Averaged shelling for quasicrystals
The shelling of crystals is concerned with counting the number of atoms on
spherical shells of a given radius and a fixed centre. Its straight-forward
generalization to quasicrystals, the so-called central shelling, leads to
non-universal answers. As one way to cope with this situation, we consider
shelling averages over all quasicrystal points. We express the averaged
shelling numbers in terms of the autocorrelation coefficients and give explicit
results for the usual suspects, both perfect and random.Comment: 4 pages, several figures, 2 tables; updated version with minor
corrections and improvements; to appear in the proceedings of ICQ
Single--crossover recombination in discrete time
Modelling the process of recombination leads to a large coupled nonlinear
dynamical system. Here, we consider a particular case of recombination in {\em
discrete} time, allowing only for {\em single crossovers}. While the analogous
dynamics in {\em continuous} time admits a closed solution, this no longer
works for discrete time. A more general model (i.e. without the restriction to
single crossovers) has been studied before and was solved algorithmically by
means of Haldane linearisation. Using the special formalism introduced by Baake
and Baake (2003), we obtain further insight into the single-crossover dynamics
and the particular difficulties that arise in discrete time. We then transform
the equations to a solvable system in a two-step procedure: linearisation
followed by diagonalisation. Still, the coefficients of the second step must be
determined in a recursive manner, but once this is done for a given system,
they allow for an explicit solution valid for all times.Comment: J. Math. Biol., in pres
A radial analogue of Poisson's summation formula with applications to powder diffraction and pinwheel patterns
Diffraction images with continuous rotation symmetry arise from amorphous
systems, but also from regular crystals when investigated by powder
diffraction. On the theoretical side, pinwheel patterns and their higher
dimensional generalisations display such symmetries as well, in spite of being
perfectly ordered. We present first steps and results towards a general frame
to investigate such systems, with emphasis on statistical properties that are
helpful to understand and compare the diffraction images. An alternative
substitution rule for the pinwheel tiling, with two different prototiles,
permits the derivation of several combinatorial and spectral properties of this
still somewhat enigmatic example. These results are compared with properties of
the square lattice and its powder diffraction.Comment: 16 pages, 8 figure
A note on palindromicity
Two results on palindromicity of bi-infinite words in a finite alphabet are
presented. The first is a simple, but efficient criterion to exclude
palindromicity of minimal sequences and applies, in particular, to the
Rudin-Shapiro sequence. The second provides a constructive method to build
palindromic minimal sequences based upon regular, generic model sets with
centro-symmetric window. These give rise to diagonal tight-binding models in
one dimension with purely singular continuous spectrum.Comment: 12 page
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