scholarly journals On Arithmetic Progressions in Model Sets

Author(s):  
Anna Klick ◽  
Nicolae Strungaru ◽  
Adi Tcaciuc
Author(s):  
ANNA KLICK ◽  
NICOLAE STRUNGARU

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.


2019 ◽  
Vol 52 (5) ◽  
pp. 1073-1106 ◽  
Author(s):  
Tobias JÄGER ◽  
Daniel LENZ ◽  
Christian OERTEL
Keyword(s):  

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ryo Horikoshi ◽  
Hiroyuki Higashino ◽  
Yoji Kobayashi ◽  
Hiroshi Kageyama

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.


2021 ◽  
Vol 6 (2) ◽  
pp. 2373-2380
Author(s):  
Ellis Ratner ◽  
Andrea Bajcsy ◽  
Terrence Fong ◽  
Claire J. Tomlin ◽  
Anca D. Dragan
Keyword(s):  

2020 ◽  
Vol 161 (2) ◽  
pp. 507-515
Author(s):  
J. Pach ◽  
I. Tomon

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