Variance measures for symmetric positive (semi) definite tensors in two dimensions [Elektronisk resurs]

<p>Funding agencies: Swedish Foundation for Strategic Research AM13-0090, the Swedish Research Council 2016-04482, Linköping University Center for Industrial Information Technology (CENIIT), VINNOVA/ITEA3 17021 IMPACT, and National Institutes of Health P41EB015902 and R01MH074794.</p>
gratis
Calculating the variance of a family of tensors, each represented by a symmetric positive semi-definite second order tensor/matrix, involves the formation of a fourth order tensor $$R_{abcd}$$ R abcd . To form this tensor, the tensor product of each second order tensor with itself is formed, and these products are then summed, giving the tensor $$R_{abcd}$$ R abcd the same symmetry properties as the elasticity tensor in continuum mechanics. This tensor has been studied with respect to many properties: representations, invariants, decomposition, the equivalence problem et cetera. In this paper we focus on the two-dimensional case where we give a set of invariants which ensures equivalence of two such fourth order tensors $$R_{abcd}$$ R abcd and $$\widetilde{R}_{abcd}$$ R ~ abcd . In terms of components, such an equivalence means that components $$R_{ijkl}$$ R ijkl of the first tensor will transform into the components $$\widetilde{R}_{ijkl}$$ R ~ ijkl of the second tensor for some change of the coordinate system.

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