scholarly journals Kolakoski-(3, 1) Is a (Deformed) Model Set

2004 ◽  
Vol 47 (2) ◽  
pp. 168-190 ◽  
Author(s):  
Michael Baake ◽  
Bernd Sing

AbstractUnlike 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 biin finite 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.

2002 ◽  
Vol 45 (1) ◽  
pp. 123-130 ◽  
Author(s):  
Robert V. Moody

AbstractWe 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 themodel set ismeasurable 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.


2016 ◽  
Vol 38 (3) ◽  
pp. 1048-1085 ◽  
Author(s):  
GERHARD KELLER ◽  
CHRISTOPH RICHARD

Model sets are projections of certain lattice subsets. It was realized by Moody [Uniform distribution in model sets. Canad. Math. Bull. 45(1) (2002), 123–130] that dynamical properties of such a set are induced from the torus associated with the lattice. We follow and extend this approach by studying dynamics on the graph of the map that associates lattice subsets to points of the torus and then we transfer the results to their projections. This not only leads to transparent proofs of known results on model sets, but we also obtain new results on so-called weak model sets. In particular, we prove pure point dynamical spectrum for the hull of a weak model set of maximal density together with the push forward of the torus Haar measure under the torus parametrization map, and we derive a formula for its pattern frequencies.


1996 ◽  
Vol 61 (1) ◽  
pp. 147-176 ◽  
Author(s):  
Martin Otto

AbstractWe study the expressive power in the finite of the logic Fixed-Point+Counting, the extension of first-order logic which is obtained through adding both the fixed-point constructor and the ability to count.To this end an isomorphism preserving (‘generic’) model of computation is introduced whose PTime restriction exactly corresponds to this level of expressive power, while its PSpace restriction corresponds to While+Counting. From this model we obtain a normal form which shows a rather clear separation of the relational vs. the arithmetical side of the algorithms involved.In parallel, we study the relations of Fixed-Point+Counting with the infinitary logics and the corresponding pebble games.The main result, however, involves the concept of an arithmetical invariant. By this we mean a functor taking every finite relational structure to an expansion of (an initial segment of) the standard arithmetical structure. In particular its values are linearly ordered structures. We establish the existence of a family of arithmetical invariants with the following properties:• The invariants themselves can be evaluated in polynomial time.• A class of finite relational structures is definable in Fixed-Point+Counting if and only if membership can be decided in polynomial time on the basis of the values of one of the invariants.• The invariant r classifies all finite relational structures exactly up to equivalence with respect to the logic We also give a characterization of Fixed-Point+Counting in terms of sequences of formulae in the : It corresponds exactly to the polynomial time computable families (φn)n ∈ ω in these logics.Towards a positive assessment of the expressive power of Fixed-Point+Counting, it is shown that the natural extension of fixed-point logic by Lindström quantifiers, which capture all the PTime computable properties of cardinalities of definable predicates, is strictly weaker than what we get here. This implies in particular that every extension of fixed-point logic by means of monadic Lindström quantifiers, which stays within PTime, must be strictly contained in Fixed-Point+Counting.


2008 ◽  
Vol 18 (06) ◽  
pp. 1759-1765 ◽  
Author(s):  
J. HIZANIDIS ◽  
R. AUST ◽  
E. SCHÖLL

We study the effect of a time-delayed feedback within a generic model for a saddle-node bifurcation on a limit cycle. Without delay the only attractor below this global bifurcation is a stable node. Delay renders the phase space infinite-dimensional and creates multistability of periodic orbits and the fixed point. Homoclinic bifurcations, period-doubling and saddle-node bifurcations of limit cycles are found in accordance with Shilnikov's theorems.


2005 ◽  
Vol 7 (1) ◽  
pp. 17-29 ◽  
Author(s):  
Hervé P. Morvan

The focus of the paper is on demonstrating how it is possible to automate complex CFD simulations using scripting language around and within the structure of the CFD command files. To illustrate this, the concept of an atmospheric pollution case is used and, more specifically, that of a water treatment plant. The code that is used is CFX-5 with PERL as a scripting ‘language’. The simulation of the factory atmospheric environment and its fluctuating conditions are fully automated. The simulation is based on a pre-defined generic CFD model, for which initial conditions, boundary conditions and source terms of atmospheric pollutant release are written automatically by the scripts using data recorded by measuring devices and stored on computers every half an hour as the simulation runs. When the correct amount of time has elapsed, the simulation pauses and the script updates the set-up using the newly recorded data. It then proceeds further, restarting from the appropriate result files. At each pause, a HTML report is also produced, which contains pictures of the area and summary tables. If a suitable criterion is defined in the post-treatment algorithm, such as a critical concentration for example, an alarm bell can be started, so that the technician knows the simulation has found a potential problem within the large domain that is thus monitored. The implications of this work are numerous. Firstly, non-CFD experts can run and use results from a CFD simulation without having to implement the models, run the simulation or fully understand the intricacy of the physics and mathematics that it contains. Going further, it is even possible to parametrize the generic model set-up, e.g. the domain dimensions or the location of emission sources, to make the case more flexible. Running the application remotely is also possible, using a web browser to submit the necessary input to the CFD code. Secondly, a very wide area can be monitored numerically, which would not be commercially viable with physical devices and field monitoring campaigns. Thirdly, such a simulation can be used to learn the general behaviour of, and the potential problems associated with, the region of interest and eventually set up a response plan to any given situation known to cause discomfort or form a health hazard to the neighbourhood. This feedback can be used to improve the operation of the plant and its safety, but also to enhance the model set-up for future simulations.


2014 ◽  
Vol 35 (7) ◽  
pp. 2017-2043 ◽  
Author(s):  
MICHAEL BAAKE ◽  
DANIEL LENZ ◽  
AERNOUT VAN ENTER

It is well known that the dynamical spectrum of an ergodic measure dynamical system is related to the diffraction measure of a typical element of the system. This situation includes ergodic subshifts from symbolic dynamics as well as ergodic Delone dynamical systems, both via suitable embeddings. The connection is rather well understood when the spectrum is pure point, where the two spectral notions are essentially equivalent. In general, however, the dynamical spectrum is richer. Here, we consider (uniquely) ergodic systems of finite local complexity and establish the equivalence of the dynamical spectrum with a collection of diffraction spectra of the system and certain factors. This equivalence gives access to the dynamical spectrum via these diffraction spectra. It is particularly useful as the diffraction spectra are often simpler to determine and, in many cases, only very few of them need to be calculated.


2003 ◽  
Author(s):  
Robin R. Vallacher ◽  
Andrzej Nowak ◽  
Matthew Rockloff
Keyword(s):  

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