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A General Framework for Tilings, Delone Sets, Functions, and Measures and Their Interrelation

We define a general framework that includes objects such as tilings, Delone sets, functions, and measures. We define local derivability and mutual local derivability (MLD) between any two of these objects in order to describe their interrelation. This is a generalization of the local derivability an...

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Detalles Bibliográficos
Autor principal: Nagai, Yasushi
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer US 2019
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6951494/
https://www.ncbi.nlm.nih.gov/pubmed/31983797
http://dx.doi.org/10.1007/s00454-019-00081-2
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author Nagai, Yasushi
author_facet Nagai, Yasushi
author_sort Nagai, Yasushi
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description We define a general framework that includes objects such as tilings, Delone sets, functions, and measures. We define local derivability and mutual local derivability (MLD) between any two of these objects in order to describe their interrelation. This is a generalization of the local derivability and MLD (or S-MLD) for tilings and Delone sets which are used in literature, under a mild assumption. We show that several canonical maps in aperiodic order send an object [Formula: see text] to one that is MLD with [Formula: see text] . Moreover, we show that, for an object [Formula: see text] and a class [Formula: see text] of objects, a mild condition on them ensures that there exists some [Formula: see text] that is MLD with [Formula: see text] . As an application, we study pattern-equivariant functions. In particular, we show that the space of all pattern-equivariant functions contains all the information on the original object up to MLD, in a quite general setting.
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spelling pubmed-69514942020-01-23 A General Framework for Tilings, Delone Sets, Functions, and Measures and Their Interrelation Nagai, Yasushi Discrete Comput Geom Article We define a general framework that includes objects such as tilings, Delone sets, functions, and measures. We define local derivability and mutual local derivability (MLD) between any two of these objects in order to describe their interrelation. This is a generalization of the local derivability and MLD (or S-MLD) for tilings and Delone sets which are used in literature, under a mild assumption. We show that several canonical maps in aperiodic order send an object [Formula: see text] to one that is MLD with [Formula: see text] . Moreover, we show that, for an object [Formula: see text] and a class [Formula: see text] of objects, a mild condition on them ensures that there exists some [Formula: see text] that is MLD with [Formula: see text] . As an application, we study pattern-equivariant functions. In particular, we show that the space of all pattern-equivariant functions contains all the information on the original object up to MLD, in a quite general setting. Springer US 2019-07-17 2019 /pmc/articles/PMC6951494/ /pubmed/31983797 http://dx.doi.org/10.1007/s00454-019-00081-2 Text en © The Author(s) 2019 Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
spellingShingle Article
Nagai, Yasushi
A General Framework for Tilings, Delone Sets, Functions, and Measures and Their Interrelation
title A General Framework for Tilings, Delone Sets, Functions, and Measures and Their Interrelation
title_full A General Framework for Tilings, Delone Sets, Functions, and Measures and Their Interrelation
title_fullStr A General Framework for Tilings, Delone Sets, Functions, and Measures and Their Interrelation
title_full_unstemmed A General Framework for Tilings, Delone Sets, Functions, and Measures and Their Interrelation
title_short A General Framework for Tilings, Delone Sets, Functions, and Measures and Their Interrelation
title_sort general framework for tilings, delone sets, functions, and measures and their interrelation
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6951494/
https://www.ncbi.nlm.nih.gov/pubmed/31983797
http://dx.doi.org/10.1007/s00454-019-00081-2
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