TY - GEN

T1 - Random fields related to the symmetry classes of second-order symmetric tensors

AU - Malyarenko, Anatoliy

AU - Ostoja-Starzewski, Martin

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

KW - Random field

KW - Spectral expansion

KW - Symmetry class

UR - http://www.scopus.com/inward/record.url?scp=85058569471&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85058569471&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-02825-1_10

DO - 10.1007/978-3-030-02825-1_10

M3 - Conference contribution

AN - SCOPUS:85058569471

SN - 9783030028244

T3 - Springer Proceedings in Mathematics and Statistics

SP - 173

EP - 185

BT - Stochastic Processes and Applications - SPAS2017

A2 - Silvestrov, Sergei

A2 - Malyarenko, Anatoliy

A2 - Rančić, Milica

PB - Springer New York LLC

T2 - International Conference on “Stochastic Processes and Algebraic Structures – From Theory Towards Applications”, SPAS 2017

Y2 - 4 October 2017 through 6 October 2017

ER -