12,000

We have over 12,000 students, from over 100 countries, within one of the safest campuses in the UK

93%

93% of Lancaster students go into work or further study within six months of graduating

Home > Research > Publications & Outputs > Crystal frameworks, matrix-valued functions and...
View graph of relations

« Back

Crystal frameworks, matrix-valued functions and rigidity operators

Research output: Contribution in Book/Report/ProceedingsPaper

Published

Publication date2014
Host publicationConcrete operators, spectral theory, operators in harmonic analysis and approximation: 22nd International Workshop in Operator Theory and its Applications, Sevilla, July 2011
EditorsManuel Cepedello Boiso, Håkan Hedenmalm, Marinus A. Kaashoek, Alfonso Montes Rodríguez, Sergei Treil
Place of publicationBasel
PublisherSpringer
Pages405-420
Number of pages16
ISBN (Electronic)9783034806480
ISBN (Print)9783034806473
Original languageEnglish

Publication series

NameOperator Theory: Advances and Applications
PublisherSpringer
Volume236
ISSN (Print)0255-0156
ISSN (Electronic)2296-4878

Abstract

An introduction and survey is given of some recent work on the infinitesimal dynamics of crystal frameworks, that is, of translationally periodic discrete bond-node structures in ℝd, for d = 2,3,... We discuss the rigidity matrix, a fundamental object from finite bar-joint framework theory, rigidity operators, matrix-function representations and low energy phonons. These phonons in material crystals, such as quartz and zeolites, are known as rigid unit modes, or RUMs, and are associated with the relative motions of rigid units, such as SiO4 tetrahedra in the tetrahedral polyhedral bondnode model for quartz. We also introduce semi-infinite crystal frameworks, bi-crystal frameworks and associated multi-variable Toeplitz operators.