### 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 language | English (US) |
---|---|

Title of host publication | Stochastic Processes and Applications - SPAS2017 |

Editors | Sergei Silvestrov, Anatoliy Malyarenko, Milica Rančić |

Publisher | Springer New York LLC |

Pages | 173-185 |

Number of pages | 13 |

ISBN (Print) | 9783030028244 |

DOIs | |

State | Published - Jan 1 2018 |

Event | International Conference on “Stochastic Processes and Algebraic Structures – From Theory Towards Applications”, SPAS 2017 - Västerås and Stockholm, Sweden Duration: Oct 4 2017 → Oct 6 2017 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 271 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Other

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

Country | Sweden |

City | Västerås and Stockholm |

Period | 10/4/17 → 10/6/17 |

### Fingerprint

### Keywords

- Random field
- Spectral expansion
- Symmetry class

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*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

**Random fields related to the symmetry classes of second-order symmetric tensors.** / Malyarenko, Anatoliy; Starzewski, Martin Ostoja.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Stochastic Processes and Applications - SPAS2017.*Springer Proceedings in Mathematics and Statistics, vol. 271, Springer New York LLC, pp. 173-185, International Conference on “Stochastic Processes and Algebraic Structures – From Theory Towards Applications”, SPAS 2017, Västerås and Stockholm, Sweden, 10/4/17. https://doi.org/10.1007/978-3-030-02825-1_10

}

TY - GEN

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

AU - Malyarenko, Anatoliy

AU - Starzewski, Martin Ostoja

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

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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

ER -