On self-similarities of cut-and-project sets

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F17%3A00316073" target="_blank" >RIV/68407700:21340/17:00316073 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.14311/AP.2017.57.0430" target="_blank" >http://dx.doi.org/10.14311/AP.2017.57.0430</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.14311/AP.2017.57.0430" target="_blank" >10.14311/AP.2017.57.0430</a>

Jazyk výsledku

angličtina

Název v původním jazyce

On self-similarities of cut-and-project sets

Popis výsledku v původním jazyce

Among the commonly used mathematical models of quasicrystals are Delone sets constructed using a cut-and-project scheme, the so-called cut-and-project sets. A cut-and-project scheme (L; pi_1; pi_2) is given by a lattice L in R^s and projections pi_1, pi_2 to suitable subspaces V1, V2. In this paper we derive several statements describing the connection between self-similarity transformations of the lattice L and transformations of its projections pi_1(L), pi_2(L). For a self-similarity of a set Sigma we take any linear mapping A such that A(Sigma)subsetSigma, which generalizes the notion of self-similarity usually restricted to scaled rotations. We describe a method of construction of cut-and-project schemes with required self-similarities and apply it to produce a cut-and-project scheme such that pi_1(L) subset R2 is invariant under an isometry of order 5. We describe all linear self-similarities of this scheme and show that they form an 8-dimensional associative algebra over the ring Z. We perform an example of a cut-and-project set with linear self-similarity which is not a scaled rotation.

Název v anglickém jazyce

On self-similarities of cut-and-project sets

Popis výsledku anglicky

Among the commonly used mathematical models of quasicrystals are Delone sets constructed using a cut-and-project scheme, the so-called cut-and-project sets. A cut-and-project scheme (L; pi_1; pi_2) is given by a lattice L in R^s and projections pi_1, pi_2 to suitable subspaces V1, V2. In this paper we derive several statements describing the connection between self-similarity transformations of the lattice L and transformations of its projections pi_1(L), pi_2(L). For a self-similarity of a set Sigma we take any linear mapping A such that A(Sigma)subsetSigma, which generalizes the notion of self-similarity usually restricted to scaled rotations. We describe a method of construction of cut-and-project schemes with required self-similarities and apply it to produce a cut-and-project scheme such that pi_1(L) subset R2 is invariant under an isometry of order 5. We describe all linear self-similarities of this scheme and show that they form an 8-dimensional associative algebra over the ring Z. We perform an example of a cut-and-project set with linear self-similarity which is not a scaled rotation.

Druh

J_SC - Článek v periodiku v databázi SCOPUS

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA13-03538S" target="_blank" >GA13-03538S: Algoritmy, dynamika a geometrie numeračních systémů</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2017

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Acta Polytechnica

ISSN

1210-2709

e-ISSN

—

Svazek periodika

57

Číslo periodika v rámci svazku

6

Stát vydavatele periodika

CZ - Česká republika

Počet stran výsledku

16

Strana od-do

430-445

Kód UT WoS článku

000424516600010

EID výsledku v databázi Scopus

2-s2.0-85040042981