scholarly journals ON HIGHER DIMENSIONAL ARITHMETIC PROGRESSIONS IN MEYER SETS

2021 ◽  
pp. 1-25
Author(s):  
ANNA KLICK ◽  
NICOLAE STRUNGARU
Keyword(s):  
Arithmetic Progressions ◽  
Arbitrary Length ◽  
Model Sets ◽  
Linearly Independent ◽  
Higher Dimensional ◽  
Model Set

Abstract In this paper we study the existence of higher dimensional arithmetic progressions in Meyer sets. We show that the case when the ratios are linearly dependent over ${\mathbb Z}$ is trivial and focus on arithmetic progressions for which the ratios are linearly independent. Given a Meyer set $\Lambda $ and a fully Euclidean model set with the property that finitely many translates of cover $\Lambda $ , we prove that we can find higher dimensional arithmetic progressions of arbitrary length with k linearly independent ratios in $\Lambda $ if and only if k is at most the rank of the ${\mathbb Z}$ -module generated by . We use this result to characterize the Meyer sets that are subsets of fully Euclidean model sets.

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.

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 Boxes, extended boxes and sets of positive upper density in the Euclidean space

2021 ◽  
pp. 1-21
Author(s):  
POLONA DURCIK ◽  
VJEKOSLAV KOVAČ
Keyword(s):  
Euclidean Space ◽  
Arithmetic Progressions ◽  
Density Theorems ◽  
Higher Dimensional ◽  
Banach Density ◽  
Upper Density

Abstract We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: 2 n vertices of a fixed n-dimensional rectangular box, the same vertices extended with n points completing three-term arithmetic progressions, and the same vertices extended with n points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.

scholarly journals NONZERO VALUES OF DIRICHLET L-FUNCTIONS IN VERTICAL ARITHMETIC PROGRESSIONS

2013 ◽  
Vol 09 (04) ◽  
pp. 813-843 ◽  
Author(s):  
GREG MARTIN ◽  
NATHAN NG
Keyword(s):  
Arithmetic Progression ◽  
Positive Constant ◽  
Upper Bound ◽  
Arithmetic Progressions ◽  
Linearly Independent ◽  
Theoretical Evidence

Let L(s, χ) be a fixed Dirichlet L-function. Given a vertical arithmetic progression of T points on the line ℜs = ½, we show that at least cT/ log T of them are not zeros of L(s, χ) (for some positive constant c). This result provides some theoretical evidence towards the conjecture that all nonnegative ordinates of zeros of Dirichlet L-functions are linearly independent over the rationals. We also establish an upper bound (depending upon the progression) for the first member of the arithmetic progression that is not a zero of L(s, χ).

scholarly journals Dimensions of Sets Which Uniformly Avoid Arithmetic Progressions

2017 ◽  
Vol 2019 (14) ◽  
pp. 4419-4430 ◽  
Author(s):  
Jonathan M Fraser ◽  
Kota Saito ◽  
Han Yu
Keyword(s):  
Hausdorff Dimension ◽  
Upper Bound ◽  
Arithmetic Progressions ◽  
The Other ◽  
Kakeya Problem ◽  
Higher Dimensional ◽  
Finite Arithmetic

AbstractWe provide estimates for the dimensions of sets in $\mathbb{R}$ which uniformly avoid finite arithmetic progressions (APs). More precisely, we say $F$ uniformly avoids APs of length $k \geq 3$ if there is an $\epsilon>0$ such that one cannot find an AP of length $k$ and gap length $\Delta>0$ inside the $\epsilon \Delta$ neighbourhood of $F$. Our main result is an explicit upper bound for the Assouad (and thus Hausdorff) dimension of such sets in terms of $k$ and $\epsilon$. In the other direction, we provide examples of sets which uniformly avoid APs of a given length but still have relatively large Hausdorff dimension. We also consider higher dimensional analogues of these problems, where APs are replaced with arithmetic patches lying in a hyperplane. As a consequence, we obtain a discretized version of a “reverse Kakeya problem:” we show that if the dimension of a set in $\mathbb{R}^d$ is sufficiently large, then it closely approximates APs in every direction.

scholarly journals On Arithmetic Progressions in Model Sets

2021 ◽  
Author(s):  
Anna Klick ◽  
Nicolae Strungaru ◽  
Adi Tcaciuc
Keyword(s):  
Arithmetic Progressions ◽  
Model Sets

scholarly journals A Gas Path Fault Contribution Matrix for Marine Gas Turbine Diagnosis Based on a Multiple Model Fault Detection and Isolation Approach

Energies ◽  
2018 ◽  
Vol 11 (12) ◽  
pp. 3316 ◽  
Author(s):  
Qingcai Yang ◽  
Shuying Li ◽  
Yunpeng Cao ◽  
Fengshou Gu ◽  
Ann Smith
Keyword(s):  
Fault Detection ◽  
Gas Turbine ◽  
Gas Turbines ◽  
Operating Conditions ◽  
Fault Detection And Isolation ◽  
Multiple Model ◽  
Efficient Operation ◽  
Model Sets ◽  
Wide Range ◽  
Model Set

To ensure reliable and efficient operation of gas turbines, multiple model (MM) approaches have been extensively studied for online fault detection and isolation (FDI). However, current MM-FDI approaches are difficult to directly apply to gas path FDI, which is one of the common faults in gas turbines and is understood to mainly be due to the high complexity and computation in updating hypothetical gas path faults for online applications. In this paper, a fault contribution matrix (FCM) based MM-FDI approach is proposed to implement gas path FDI over a wide operating range. As the FCM is realized via an additive term of the healthy model set, the hypothetical models for various gas path faults can be easily established and updated online. In addition, a gap metric analysis method for operating points selection is also proposed, which yields the healthy model set from the equal intervals linearized models to approximate the nonlinearity of the gas turbine over a wide range of operating conditions with specified accuracy and computational efficiency. Simulation case studies conducted on a two-shaft marine gas turbine demonstrated the proposed approach is capable of adaptively updating hypothetical model sets to accurately differentiate both single and multiple faults of various gas path faults.

Deformed model sets and distorted Penrose tilings

2006 ◽  
Vol 221 (9) ◽  
Author(s):  
Bernd Sing ◽  
T. Richard Welberry
Keyword(s):  
Diffraction Pattern ◽  
Mathematical Concept ◽  
Model Sets ◽  
Model Set ◽  
Characteristic Features ◽  
Penrose Tilings ◽  
Insight Into

AbstractIn this article, we show how the mathematical concept of a deformed model set can be used to gain insight into the diffraction pattern of quasicrystalline structures. We explain what a deformed model set is, what its characteristic features are and how it relates to certain disorder phenomena in solids. We then apply this concept to distorted Penrose tilings,

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.

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