Substitution Delone sets with pure point spectrum are inter-model sets

The paper establishes an equivalence between pure point diffraction and certain types of model sets, called inter-model sets, in the context of substitution point sets and substitution tilings. The key ingredients are a new type of coincidence condition in substitution point sets, which we call algebraic coincidence, and the use of a recent characterization of model sets through dynamical systems associated with the point sets or tilings.     

Nikolai Borisovich Delone in my life

Nikolai Delone has literally dragged me to science. I owe him my scientific carrier. In addition he taught me diligence, responsibility, purposefulness, and theoretical and experimental skills that were helpful in all my life.     

Algebras of Random Operators Associated to Delone Dynamical Systems

We carry out a careful study of operator algebras associated with Delone dynamical systems. A von Neumann algebra is defined using noncommutative integration theory. Features of these algebras and the operators they contain are discussed. We restrict our attention to a certain C ^*-subalgebra to discuss a Shubin trace formula.     

“What a Life!”—Nikolay Borisovich Delone

This essay is dedicated to my personal memories of Nikolay Borisovich Delone, the god-father to Russian strong-field science and to me, the Teacher with capital T, not only in physics but also in life.     

Discontinuities of the Integrated Density of States for Random Operators on Delone Sets

Despite all the analogies with usual random models, tight binding operators for quasicrystals exhibit a feature that clearly distinguishes them from the former: the integrated density of states may be discontinuous. This phenomenon is identified as a local effect, due to the occurrence of eigenfunctions with bounded support.Research partly supported by the DFG in the priority program Quasicrystals     

Pure Point Diffraction Implies Zero Entropy for Delone Sets with Uniform Cluster Frequencies

Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy zero if it has uniform cluster frequencies and is pure point diffractive. We also note that the patch counting entropy vanishes whenever the repetitivity function satisfies a certain growth restriction.      

Diffusion on surfaces

A commentary is presented on current progress towards understanding diffusion on surfaces. Recent work by field ion microscopy shows that interactions between diffusing adatoms lead to correlated motions and add unexpected complexity to the elementary diffusion mechanisms. Work on high temperature surface self-diffusion also suggests the importance of interactions between diffusing atoms. Mass transport, catalysed surface diffusion, giant diffusivities, impurity, tracer and adsorbed layer diffusion, diffusion in sintering and other applications, are briefly discussed.     

Diffusion on random lattices

We study the motion of a point particle along the bonds of a two-dimensional random lattice, whose sites are randomly occupied with right and left rotators, which scatter the particle according to deterministic scattering rules. We consider both a Poisson (PRL) and a vectorized random lattice (VRL) and fixed as well as flipping scatterers. On both lattices, for fixed scatterers and equal concentrations of right and left rotators the same anomalous diffusion of the particle is obtained as before for the triangular lattice, where the mean square displacement is t, the diffusion process non-Gaussian, and the particle trajectories exhibit scaling behavior as at a percolation threshold. For unequal concentrations the particle is trapped exponentially rapidly. This system can be considered as an extreme case of the Lorentz lattice gases on regular lattices discussed before or as an example of the motion of a particle along cracks or (grain or cellular) boundaries on a two-dimensional surface.     

Frames built on fractal sets

Frames of finite dimensional Hilbert spaces have recently been of great interest in applications to modern communication networks transport packets. In this note, continuous and discrete frames, living on fractal sets, of both finite and infinite dimensional separable abstract Hilbert spaces are found. In particular, we find discrete frames, robust to erasures, of finite dimensional Hilbert spaces using iterated function systems.     

Some Results on Sinai Diffusion

The diffusion of a particle in a one-dimensional random force field (Sinai diffusion) is studied using the replica method. This method, which maps the problem onto a quantum problem, is shown to be a simple and direct way to calculate the long-time diffusive behavior. Results for the distribution of the local Green's function, the particle distribution, and persistence are obtained.     

Note on Chaos and Diffusion

Using standard definitions of chaos (as positive Kolmogorov–Sinai entropy) and diffusion (that multiple time distribution functions are Gaussian), we show numerically that both chaotic and nonchaotic systems exhibit diffusion, and hence that there is no direct logical connection between the two properties. This extends a previous result for two time distribution functions.     

Epidemic Diffusion on Complex Networks

Both diffusion and epidemic are well studied in the stochastic systems and complex networks, respectively. Here we combine these two fields and study epidemic diffusion in complex networks. Instead of studying the threshold of infection, which was focused on in previous works, we focus on the diffusion behayiour. We find that the epidemic diffusion in a complex network is an anomalous superdiffusion with varying diffusion exponent and that γ is influenced seriously by the network structure, such as the clustering coefficient and the degree distribution. Numerical simulations have confirmed the theoretical predictions.     

Empirical sets

The pillar concept of Foulis and Randall's school is surely that of a manual of operations. They chose to regard an operation as the set of possible outcomes, thereby taking a manual of operations to be a family of partially overlapping operations. Our previous work is a development of their ideas in two points. First, each operation is represented not by the set of possible outcomes, but by the complete Boolean algebra of observable events. Second, since each complete Boolean algebraB possesses the Scott-Solovay modelV ^(B) of classical set theory as its higher-order companion, the Scott-Solovay universes of all the operations in the manual lump together into a family of Boolean set theories interconnected by geometric morphisms, which we suggestively designated an empirical set theory. The principal concern of this paper is to show how to get a cross-operational set concept by choosing an internal set withinV ^(B) for each operationB in the manual and bundling them up. The resulting structure is denominated an empirical set. We show that the category of empirical sets is complete, is cocomplete, has a subobject classifier for well-rounded subobjects, and has exponentials only for degraded objects.