On the frequency module of the hull of a primitive substitution tiling

2022 ◽  
Vol 78 (1) ◽  
Author(s):  
April Lynne D. Say-awen ◽  
Dirk Frettlöh ◽  
Ma. Louise Antonette N. De Las Peñas
Keyword(s):  
Statistical Properties ◽  
Absolute Frequency ◽  
Aperiodic Tilings ◽  
Primitive Substitution ◽  
Substitution Tiling ◽  
Tiling Spaces ◽  
The Family ◽  
Frequency Module ◽  
Substitution Tilings ◽  
The Absolute

Understanding the properties of tilings is of increasing relevance to the study of aperiodic tilings and tiling spaces. This work considers the statistical properties of the hull of a primitive substitution tiling, where the hull is the family of all substitution tilings with respect to the substitution. A method is presented on how to arrive at the frequency module of the hull of a primitive substitution tiling (the minimal {\bb Z}-module, where {\bb Z} is the set of integers) containing the absolute frequency of each of its patches. The method involves deriving the tiling's edge types and vertex stars; in the process, a new substitution is introduced on a reconstructed set of prototiles.

scholarly journals Pure discrete spectrum and regular model sets in d-dimensional unimodular substitution tilings

2020 ◽  
Vol 76 (5) ◽  
pp. 600-610
Author(s):  
Dong-il Lee ◽  
Shigeki Akiyama ◽  
Jeong-Yup Lee
Keyword(s):  
Discrete Spectrum ◽  
Additional Condition ◽  
Internal Space ◽  
Representative Point ◽  
Primitive Substitution ◽  
Substitution Tiling ◽  
Regular Model ◽  
Model Sets ◽  
Project Scheme ◽  
Substitution Tilings

Primitive substitution tilings on {\bb R}^d whose expansion maps are unimodular are considered. It is assumed that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, a cut-and-project scheme can be constructed with a Euclidean internal space. Under some additional condition, it is shown that if the substitution tiling has pure discrete spectrum, then the corresponding representative point sets are regular model sets in that cut-and-project scheme.

Measurement of the absolute frequency of the methane E-line at 88 THz

1998 ◽  
Vol 151 (4-6) ◽  
pp. 229-234 ◽  
Author(s):  
P.S. Ering ◽  
D.A. Tyurikov ◽  
G. Kramer ◽  
B. Lipphardt
Keyword(s):  
Absolute Frequency ◽  
The Absolute

scholarly journals Introspective cosmopolitanism: The family in the Greek Weird Wave

2020 ◽  
Vol 6 (1) ◽  
pp. 3-27 ◽  
Author(s):  
Dimitris Eleftheriotis
Keyword(s):  
Conceptual Model ◽  
Theoretical Work ◽  
Critical Discourse ◽  
Greek Crisis ◽  
The Family ◽  
The Absolute ◽  
Critical Interpretation ◽  
Self Examination

This article reframes the critical discourse around the ‘Greek Weird Wave’ using an approach informed by theoretical work on cosmopolitanism. Focussing on Yorgos Lanthimos’s Dogtooth (2009) and Athena-Rachel Tsangari’s Attenberg (2010), the critical interpretation of the role of the family is radically rethought. I argue that the privileging of allegorical readings of the family in the Weird Wave films constitutes a form of critical denial of the deeply problematic and specifically Greek ways in which the family (dys)functions. I challenge the absolute and exclusive power that the Greek ‘crisis’ holds over interpretations and evaluations of Weird Wave films, which discursively displaces the problems of the family to broader sociopolitical frameworks. In reclaiming the importance of literal readings of the films, I reposition them as manifestations of a specific cosmopolitan disposition, that of introspection, a process of self-examination that overcomes denial. In turn, the critical reframing of the films outlines the contours of a complex agonistics of introspective cosmopolitanism, an inward investigative disposition that is dialectically linked to cosmopolitan positioning. Jean François Lyotard’s 1989 theorization of the oikos (home/house) provides a conceptual model for understanding the family (oikogeneia), which, in its Greek specificities, is central to the films under discussion.

scholarly journals Characterization of the absolute frequency stability of an individual reference cavity

2009 ◽  
Vol 34 (2) ◽  
pp. 190 ◽  
Author(s):  
T. Liu ◽  
Y. N. Zhao ◽  
V. Elman ◽  
A. Stejskal ◽  
L. J. Wang
Keyword(s):  
Frequency Stability ◽  
Absolute Frequency ◽  
Reference Cavity ◽  
The Absolute

FRACTAL TILINGS FROM SUBSTITUTION TILINGS

Fractals ◽  
2019 ◽  
Vol 27 (02) ◽  
pp. 1950009
Author(s):  
XINCHANG WANG ◽  
PEICHANG OUYANG ◽  
KWOKWAI CHUNG ◽  
XIAOGEN ZHAN ◽  
HUA YI ◽  
...  
Keyword(s):  
Short Paper ◽  
Self Similarity ◽  
Substitution Rule ◽  
Substitution Tiling ◽  
Simple Strategy ◽  
Substitution Tilings

A fractal tiling or [Formula: see text]-tiling is a tiling which possesses self-similarity and the boundary of which is a fractal. By substitution rule of tilings, this short paper presents a very simple strategy to create a great number of [Formula: see text]-tilings. The substitution tiling Equithirds is demonstrated to show how to achieve it in detail. The method can be generalized to every tiling that can be constructed by substitution rule.

scholarly journals On a Generalization of Hofstadter’s Q-Sequence: A Family of Chaotic Generational Structures

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Altug Alkan
Keyword(s):  
Statistical Technique ◽  
Growth Characteristics ◽  
Statistical Properties ◽  
Global Properties ◽  
The Family ◽  
Successive Generations

Hofstadter Q-recurrence is defined by the nested recurrence Qn=Qn−Qn−1+Qn−Qn−2, and there are still many unanswered questions about certain solutions of it. In this paper, a generalization of Hofstadter’s Q-sequence is proposed and selected members of this generalization are investigated based on their chaotic generational structures and Pinn’s statistical technique. Solutions studied have also curious approximate patterns and considerably similar statistical properties with Hofstadter’s famous Q-sequence in terms of growth characteristics of their successive generations. In fact, the family of sequences that this paper introduces suggests the existence of conjectural global properties in order to classify unpredictable solutions to Q-recurrence and a generalization of it.

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