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Michael Baake

Publications and source records attributed to Michael Baake.

At least 127 records · Page 7Linked to original sources
arXiv2006

Pinwheel patterns and powder diffraction

Pinwheel patterns and their higher dimensional generalisations display continuous circular or spherical symmetries in spite of being perfectly ordered. The same symmetries show up in the corresponding diffraction images. Interestingly, they also arise from amorphous systems, and also from regular crystals when investigated by powder diffraction. 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. We concentrate on properties that are accessible via an alternative substitution rule for the pinwheel tiling, based on two different prototiles. Due to striking similarities, we compare our results with the toy model for the powder diffraction of the square lattice.

math-ph↗
arXiv2006

Freely forming groups: Trying to be rare

A simple weakly frequency dependent model for the dynamics of a population with a finite number of types is proposed, based upon an advantage of being rare. In the infinite population limit, this model gives rise to a non-smooth dynamical system that reaches its globally stable equilibrium in finite time. This dynamical system is sufficiently simple to permit an explicit solution, built piecewise from solutions of the logistic equation in continuous time. It displays an interesting tree-like structure of coalescing components.

q-bio.PE↗
arXiv2006

Coincidence site modules in 3-space

The coincidence site lattice (CSL) problem and its generalization to Z-modules in Euclidean 3-space is revisited, and various results and conjectures are proved in a unified way, by using maximal orders in quaternion algebras of class number 1 over real algebraic number fields.

math.MG↗
arXiv2006

Surprises in approximating Levenshtein distances

The Levenshtein distance is an important tool for the comparison of symbolic sequences, with many appearances in genome research, linguistics and other areas. For efficient applications, an approximation by a distance of smaller computational complexity is highly desirable. However, our comparison of the Levenshtein with a generic dictionary-based distance indicates their statistical independence. This suggests that a simplification along this line might not be possible without restricting the class of sequences. Several other probabilistic properties are briefly discussed, emphasizing various questions that deserve further investigation.

q-bio.QM↗
arXiv2006

Bravais colourings of planar modules with N-fold symmetry

The first step in investigating colour symmetries for periodic and aperiodic systems is the determination of all colouring schemes that are compatible with the symmetry group of the underlying structure, or with a subgroup of it. For an important class of colourings of planar structures, this mainly combinatorial question can be addressed with methods of algebraic number theory. We present the corresponding results for all planar modules with N-fold symmetry that emerge as the rings of integers in cyclotomic fields with class number one. The counting functions are multiplicative and can be encapsulated in Dirichlet series generating functions, which turn out to be the Dedekind zeta functions of the corresponding cyclotomic fields.

math.CO↗
arXiv2006

Multiple planar coincidences with N-fold symmetry

Planar coincidence site lattices and modules with N-fold symmetry are well understood in a formulation based on cyclotomic fields, in particular for the class number one case, where they appear as certain principal ideals in the corresponding ring of integers. We extend this approach to multiple coincidences, which apply to triple or multiple junctions. In particular, we give explicit results for spectral, combinatorial and asymptotic properties in terms of Dirichlet series generating functions.

math.MG↗
arXiv2006

The structure of reversing symmetry groups

We present some of the group theoretic properties of reversing symmetry groups, and classify their structure in simple cases that occur frequently in several well-known groups of dynamical systems.

math.DS↗
arXiv2006

Solution of the coincidence problem in dimensions $d\le 4$

Discrete point sets $\mathcal{S}$ such as lattices or quasiperiodic Delone sets may permit, beyond their symmetries, certain isometries $R$ such that $\mathcal{S}\cap R\mathcal{S}$ is a subset of $\mathcal{S}$ of finite density. These are the so-called coincidence isometrie. They are important in understanding and classifying grain boundaries and twins in crystals and quasicrystals. It is the purpose of this contribution to introduce the corresponding coincidence problem in a mathematical setting and to demonstrate how it can be solved algebraically in dimensions 2, 3 and 4. Various examples both from crystals and quasicrystals are treated explicitly, in particular (hyper-)cubic lattices and quasicrystals with non-crystallographic point groups of type $H_2$, $H_3$ and $H_4$. We derive parametrizations of all linear coincidence isometries, determine the corresponding coincidence index (the reciprocal of the density of coinciding points, also called $\varSigma$-factor), and finally encapsulate their statistics in suitable Dirichlet series generating functions.

math.MG↗
arXiv2005

Planar coincidences for N-fold symmetry

The coincidence problem for planar patterns with $N$-fold symmetry is considered. For the N-fold symmetric module with $N<46$, all isometries of the plane are classified that result in coincidences of finite index. This is done by reformulating the problem in terms of algebraic number fields and using prime factorization. The more complicated case $N \ge 46$ is briefly discussed and N=46 is described explicitly. The results of the coincidence problem also solve the problem of colour lattices in two dimensions and its natural generalization to colour modules.

math.MG↗
arXiv2005

Averaged coordination numbers of planar aperiodic tilings

We consider averaged shelling and coordination numbers of aperiodic tilings. Shelling numbers count the vertices on radial shells around a vertex. Coordination numbers, in turn, count the vertices on coordination shells of a vertex, defined via the graph distance given by the tiling. For the Ammann-Beenker tiling, we find that coordination shells consist of complete shelling orbits, which enables us to calculate averaged coordination numbers for rather large distances explicitly. The relation to topological invariants of tilings is briefly discussed.

math-ph↗
arXiv2005

Signal analysis of impulse response functions in MR- and CT-measurements of cerebral blood flow

The impulse response function (IRF) of a localized bolus in cerebral blood flow codes important information on the tissue type. It is indirectly accessible both from MR- and CT-imaging methods, at least in principle. In practice, however, noise and limited signal resolution render standard deconvolution techniques almost useless. Parametric signal descriptions look more promising, and it is the aim of this contribution to develop some improvements along this line.

q-bio.TO↗
arXiv2005

Recombination semigroups on measure spaces

The dynamics of recombination in genetics leads to an interesting nonlinear differential equation, which has a natural generalization to a measure valued version. The latter can be solved explicitly under rather general circumstances. It admits a closed formula for the semigroup of nonlinear positive operators that emerges from the forward flow and is, in general, embedded in a multi-parameter semigroup.

math.CA↗
arXiv2005

Deformation of Delone dynamical systems and pure point diffraction

This paper deals with certain dynamical systems built from point sets and, more generally, measures on locally compact Abelian groups. These systems arise in the study of quasicrystals and aperiodic order, and important subclasses of them exhibit pure point diffraction spectra. We discuss the relevant framework and recall fundamental results and examples. In particular, we show that pure point diffraction is stable under ``equivariant'' local perturbations and discuss various examples,including deformed model sets. A key step in the proof of stability consists in transforming the problem into a question on factors of dynamical systems.

math.DS↗
arXiv2004

SCD Patterns Have Singular Diffraction

Among the many families of nonperiodic tilings known so far, SCD tilings are still a bit mysterious. Here, we determine the diffraction spectra of point sets derived from SCD tilings and show that they have no absolutely continuous part, that they have a uniformly discrete pure point part on the z-axis, and that they are otherwise supported on a set of concentric cylinder surfaces around this axis. For SCD tilings with additional properties, more detailed results are given.

math-ph↗
arXiv2004

Diffraction spectrum of lattice gas models above T_c

The diffraction spectra of lattice gas models on Z^d with finite-range ferromagnetic two-body interaction above T_c or with certain rates of decay of the potential are considered. We show that these diffraction spectra almost surely exist, are Z^d-periodic and consist of a pure point part and an absolutely continuous part with continuous density.

math-ph↗
arXiv2004

Weighted Dirac combs with pure point diffraction

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 measure is a pure point measure. Simultaneously, we establish a natural link to the theory of the cut and project formalism and to the theory of almost periodic measures. Our conditions are general enough to cover the known theory of model sets, but also to include examples such as the visible lattice points.

math.MG↗