Uniform Distribution in Model Sets

2002 ◽  
Vol 45 (1) ◽  
pp. 123-130 ◽  
Author(s):  
Robert V. Moody
Keyword(s):  
Uniform Distribution ◽  
Physical Space ◽  
Pure Point ◽  
Internal Space ◽  
Compact Closure ◽  
Related Model ◽  
Model Sets ◽  
Distribution Property ◽  
Distribution Of Points ◽  
Model Set

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.

scholarly journals Dynamics on the graph of the torus parametrization

2016 ◽  
Vol 38 (3) ◽  
pp. 1048-1085 ◽  
Author(s):  
GERHARD KELLER ◽  
CHRISTOPH RICHARD
Keyword(s):  
Uniform Distribution ◽  
Haar Measure ◽  
Pure Point ◽  
Dynamical Spectrum ◽  
Maximal Density ◽  
Weak Model ◽  
Model Sets ◽  
Push Forward ◽  
Model Set ◽  
Dynamical Properties

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.

scholarly journals Design of a structure model set for inorganic compounds based on ping-pong balls linked with snap buttons

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ryo Horikoshi ◽  
Hiroyuki Higashino ◽  
Yoji Kobayashi ◽  
Hiroshi Kageyama
Keyword(s):  
High School Students ◽  
Coordination Compounds ◽  
Periodic Structures ◽  
Inorganic Compounds ◽  
Structure Model ◽  
School Students ◽  
Model Sets ◽  
Ping Pong ◽  
Hands On ◽  
Model Set

Abstract Structure model sets for inorganic compounds are generally expensive; their distribution to all students in a class is therefore usually impractical. We have therefore developed a structure model set to illustrate inorganic compounds. The set is constructed with inexpensive materials: ping-pong balls, and snap buttons. The structure model set can be used to illustrate isomerism in coordination compounds and periodic structures of ceramic perovskites. A hands-on activity using the structure model set was developed for high school students and was well-received by them. Despite the concepts being slightly advanced for them, the students’ retention of the knowledge gained through the activity was tested a week after they completed the activity and was found to be relatively high, demonstrating the usefulness of the activity based on the structure model set.

scholarly journals Three variations on a theme by Fibonacci

2020 ◽  
pp. 2140001
Author(s):  
Michael Baake ◽  
Natalie Priebe Frank ◽  
Uwe Grimm
Keyword(s):  
Direct Product ◽  
Two Dimensions ◽  
Pure Point ◽  
One Dimension ◽  
Stochastic Nature ◽  
Regular Model ◽  
Model Sets ◽  
Planar Extension ◽  
Dynamical Spectra ◽  
Variations On A Theme

Several variants of the classic Fibonacci inflation tiling are considered in an illustrative fashion, in one and in two dimensions, with an eye on changes or robustness of diffraction and dynamical spectra. In one dimension, we consider extension mechanisms of deterministic and of stochastic nature, while we look at direct product variations in a planar extension. For the pure point part, we systematically employ a cocycle approach that is based on the underlying renormalization structure. It allows explicit calculations, particularly in cases where one meets regular model sets with Rauzy fractals as windows.

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

2004 ◽  
Vol 47 (2) ◽  
pp. 168-190 ◽  
Author(s):  
Michael Baake ◽  
Bernd Sing
Keyword(s):  
Fixed Point ◽  
Pure Point ◽  
Generic Model ◽  
Diffraction Spectrum ◽  
Substitution Rule ◽  
Primitive Substitution ◽  
Diffraction Property ◽  
Model Set

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.

ON THE STRUCTURE OF PHYSICAL SPACE

2001 ◽  
Vol 16 (24) ◽  
pp. 4045-4055 ◽  
Author(s):  
DANIEL WISNIVESKY
Keyword(s):  
Electromagnetic Coupling ◽  
Lorentz Invariance ◽  
Electroweak Interaction ◽  
Physical Space ◽  
Building Blocks ◽  
Space Time ◽  
Coupling Constants ◽  
Internal Space ◽  
External Space ◽  
Physical Phenomena

In this paper we develop a theory based on the postulate that the environment where physical phenomena take place is the space of four complex parameters of the linear group of transformations. Using these parameters as fundamental building blocks we construct ordinary space–time and the internal space. Lorentz invariance is built in the definition of external space, while the symmetry of the internal space, S(1)×SU(2) results as a consequence of the identification of the external coordinates. Thus, special relativity and the electroweak interaction symmetry ensue from the properties of the basic building blocks of physical space. Since internal and external space are derived from a common structure, there is no need to bring into the theory any additional hypothesis to account for the microscopic nature of the internal space, nor to introduce symmetry breaking mechanisms that would normally be required to force a splitting of the internal and external symmetries. As an outcome of the existence of a basic structure underlying the external space–time, the weak and electromagnetic coupling constants are not independent and the Weinberg weak mixing angle is derived from the theory. In this new theory, there is an interrelationship between external and internal transformations, which leads to the quantization of electric charge. Finally we conclude that the electroweak gauge theory can be regarded as a consequence of Einstein's general theory of relativity. The proposed theory represents an extension of the normally accepted theory and includes it as a particular case. This paper is an attempt to formulate a new framework in which the physical phenomena take place, and to explore some of its consequences.

scholarly journals Cut-and-Project Schemes for Pisot Family Substitution Tilings

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 511 ◽  
Author(s):  
Jeong-Yup Lee ◽  
Shigeki Akiyama ◽  
Yasushi Nagai
Keyword(s):  
Point Spectrum ◽  
The Other ◽  
Pure Point ◽  
Pure Point Spectrum ◽  
Internal Space ◽  
Sufficient Condition ◽  
Abstract Space ◽  
Dynamical Spectrum ◽  
Substitution Tiling ◽  
Substitution Tilings

We consider Pisot family substitution tilings in R d whose dynamical spectrum is pure point. There are two cut-and-project schemes (CPSs) which arise naturally: one from the Pisot family property and the other from the pure point spectrum. The first CPS has an internal space R m for some integer m ∈ N defined from the Pisot family property, and the second CPS has an internal space H that is an abstract space defined from the condition of the pure point spectrum. However, it is not known how these two CPSs are related. Here we provide a sufficient condition to make a connection between the two CPSs. For Pisot unimodular substitution tiling in R , the two CPSs turn out to be same due to the remark by Barge-Kwapisz.

INFLATION FROM THE SUPERSTRING VACUUM

2009 ◽  
Vol 24 (20n21) ◽  
pp. 4021-4037
Author(s):  
M. D. POLLOCK
Keyword(s):  
Background Radiation ◽  
Physical Space ◽  
Reduction Process ◽  
Tree Level ◽  
Internal Space ◽  
De Sitter ◽  
Superstring Theory ◽  
Higher Derivative ◽  
Effective Lagrangian ◽  
Microwave Background Radiation

Quartic higher-derivative gravitational terms in the effective Lagrangian of the heterotic superstring theory renormalize the bare, four-dimensional gravitational coupling [Formula: see text], due to the reduction process [Formula: see text], according to the formula [Formula: see text], where A r and B r are the moduli for the physical space gij(xk) and internal space [Formula: see text], respectively. The Euler characteristic [Formula: see text] is negative for a three-generation Calabi–Yau manifold, and therefore both the additional terms, of tree-level and one-loop origin, produce a decrease in κ-2, which changes sign when κ-2 = 0. The corresponding tree-level critical point is [Formula: see text], if we set [Formula: see text] and λ = 15π2, for compactification onto a torus. Values [Formula: see text] yield the anti-gravity region κ-2 < 0, which is analytically accessible from the normal gravity region κ-2 > 0. The only non-singular, vacuum minimum of the potential [Formula: see text] is located at the point [Formula: see text], where [Formula: see text], the quadratic trace anomaly [Formula: see text] dominates over [Formula: see text], and a phase of de Sitter expansion may occur, as first envisaged by Starobinsky, in approximate agreement with the constraint due to the effect of gravitational waves upon the anisotropy of the cosmic microwave background radiation. There is no non-singular minimum of the potential [Formula: see text].

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