Random fields related to the symmetry classes of second-order symmetric tensors

Anatoliy Malyarenko, Martin Ostoja-Starzewski

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Under the change of basis in the three-dimensional space by means of an orthogonal matrix g, a matrix A of a linear operator is transformed as A → gAg -1 Mathematically, the stationary subgroup of a symmetric matrix under the above action can be either (Formula Presented), when all three eigenvalues of A are different, or (Formula Presented), when two of them are equal, or O(3), when all three eigenvalues are equal. Physically, one typical application relates to dependent quantities like a second-order symmetric stress (or strain) tensor. Another physical setting is that of dependent fields, such as conductivity with such three cases is the conductivity (or, similarly, permittivity, or anti-plane elasticity) second-rank tensor, which can be either orthotropic, transversely isotropic, or isotropic. For each of the above symmetry classes, we consider a homogeneous random field taking values in the fixed point set of the class that is invariant with respect to the natural representation of a certain closed subgroup of the orthogonal group. Such fields may model stochastic heat conduction, electric permittivity, etc. We find the spectral expansions of the introduced random fields.

Original languageEnglish (US)
Title of host publicationStochastic Processes and Applications - SPAS2017
EditorsSergei Silvestrov, Anatoliy Malyarenko, Milica Rančić
PublisherSpringer New York LLC
Pages173-185
Number of pages13
ISBN (Print)9783030028244
DOIs
StatePublished - Jan 1 2018
EventInternational Conference on “Stochastic Processes and Algebraic Structures – From Theory Towards Applications”, SPAS 2017 - Västerås and Stockholm, Sweden
Duration: Oct 4 2017Oct 6 2017

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume271
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Other

OtherInternational Conference on “Stochastic Processes and Algebraic Structures – From Theory Towards Applications”, SPAS 2017
CountrySweden
CityVästerås and Stockholm
Period10/4/1710/6/17

Keywords

  • Random field
  • Spectral expansion
  • Symmetry class

ASJC Scopus subject areas

  • Mathematics(all)

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  • Cite this

    Malyarenko, A., & Ostoja-Starzewski, M. (2018). Random fields related to the symmetry classes of second-order symmetric tensors. In S. Silvestrov, A. Malyarenko, & M. Rančić (Eds.), Stochastic Processes and Applications - SPAS2017 (pp. 173-185). (Springer Proceedings in Mathematics and Statistics; Vol. 271). Springer New York LLC. https://doi.org/10.1007/978-3-030-02825-1_10