Related papers: Aperiodic order and spectral properties
This article explains the relationship between analytic and algebraic order in case of abstract pseudo-differential operators for a regular spectral triple.
Kinematic diffraction is well suited for a mathematical approach via measures, which has substantially been developed since the discovery of quasicrystals. The need for further insight emerged from the question of which distributions of…
In this paper, we establish a condition on the coefficients of differential operators generated in the space of square-integrable functions on the entire real line by an ordinary differential expression with periodic, complex-valued…
Aperiodic order refers to the mathematical formalisation of quasicrystals. Substitutions and cut and project sets are among their main actors; they also play a key role in the study of dynamical systems, whether they are symbolic, generated…
A brief review of the manifestations of classical chaos observed in atomic systems is presented. Particular attention is paid to the analysis of atomic spectra by periodic orbit-type theories. For diamagnetic non-hydrogenic Rydberg atoms,…
Given an $\alpha > 1$ and a $\theta$ with unbounded continued fraction entries, we characterise new relations between Sturmian subshifts with slope $\theta$ with respect to (i) an $\alpha$-H\"oder regularity condition of a spectral metric,…
Optical methods are most convenient to analyze spatially periodic patterns with wavevector $\bm q$ in a thin layer of a nematic liquid crystal. In the standard experimental setup a beam of parallel light with a 'short' wavelength $\lambda…
We introduce and study the class of spherically ordered groups. The notions of spherically ordered groups and their spectra of spherical orderability are introduced. Values of these spectra are found for a series of natural groups.
Mathematical diffraction theory is concerned with the analysis of the diffraction image of a given structure and the corresponding inverse problem of structure determination. In recent years, the understanding of systems with continuous and…
This paper continues our study of quasicrystals initiated in Part I. We propose a general mechanism for constructing quasicrystals, existing globally in time, in spatially-extended systems (partial differential equations with Euclidean…
This article gives a short description of pattern formation and coarsening phenomena and focuses on recent experimental and theoretical advances in these fields. It serves as an introduction to phase ordering kinetics and it will appear in…
This paper is devoted to the mathematical analysis of a time-domain electromagnetic scattering by periodic structures which are known as diffraction gratings. The scattering problem is reduced equivalently into an initial-boundary value…
Periodic optical structures, such as diffraction grating and numerous photonic crystals, are one of the staples of modern nanophotonics for the manipulation of electromagnetic radiation. The array of subwavelength dielectric rods is one of…
This text is a slightly edited version of lecture notes for a course I gave at ETH, during the Winter term 2000-2001, to undergraduate Mathematics and Physics students. Contents: Chapter 1 - Examples of Dynamical Systems Chapter 2 -…
The main goal of the present paper is to convince that it is feasible to construct a `periodic orbit theory' of localization by extending the idea of classical action correlations. This possibility had been questioned by many researchers in…
We introduce a construction to embed a quasiperiodic lattice of obstacles into a single unit cell of a higher-dimensional space, with periodic boundary conditions. This construction transparently shows the existence of channels in these…
This is a chapter in an incoming book on aperiodic order. We review results about the topology, the dynamics, and the combinatorics of aperiodically ordered tilings obtained with the tools of noncommutative geometry.
Stochastic point processes relevant to the theory of long-range aperiodic order are considered that display diffraction spectra of mixed type, with special emphasis on explicitly computable cases together with a unified approach of…
In Optics it is common to split up the formal analysis of diffraction according to two convenient approximations, in the near and far fields (also known as the Fresnel and Fraunhofer regimes, respectively). Within this scenario, geometrical…
This paper is about the tiling dynamical systems approach to the study of aperiodic order. We compare and contrast four related types of systems: ordinary (one-dimensional) symbolic systems, one-dimensional tiling systems, multidimensional…