The generalized inverses of tensors and an application to linear models

[EN] In this paper, we recall and extend some tensor operations. Then, the generalized inverse of tensors is established by using tensor equations. Moreover, we investigate the leastsquares solutions of tensor equations. An algorithm to compute the Moore Penrose inverse of an arbitrary tensor is constructed. Finally, we apply the obtained results to higher order Gauss Markov theoremThe fourth author was supported by the National Science Foundation of China [grant number 11361009] and Grant from the High level innovation teams and distinguished scholars in Guangxi Universities [grant number 20160001].Jin, H.; Bai, M.; Benítez López, J.; Liu, X. (2017). The generalized inverses of tensors and an application to linear models. Computers & Mathematics with Applications. 74(3):385-397. doi:10.1016/j.camwa.2017.04.017S38539774

Tensor and Matrix Inversions with Applications

Higher order tensor inversion is possible for even order. We have shown that a tensor group endowed with the Einstein (contracted) product is isomorphic to the general linear group of degree $n$. With the isomorphic group structures, we derived new tensor decompositions which we have shown to be related to the well-known canonical polyadic decomposition and multilinear SVD. Moreover, within this group structure framework, multilinear systems are derived, specifically, for solving high dimensional PDEs and large discrete quantum models. We also address multilinear systems which do not fit the framework in the least-squares sense, that is, when the tensor has an odd number of modes or when the tensor has distinct dimensions in each modes. With the notion of tensor inversion, multilinear systems are solvable. Numerically we solve multilinear systems using iterative techniques, namely biconjugate gradient and Jacobi methods in tensor format

Mechanics of Systems of Affine Bodies. Geometric Foundations and Applications in Dynamics of Structured Media

In the present paper we investigate the mechanics of systems of affinely-rigid bodies, i.e., bodies rigid in the sense of affine geometry. Certain physical applications are possible in modelling of molecular crystals, granular media, and other physical objects. Particularly interesting are dynamical models invariant under the group underlying geometry of degrees of freedom. In contrary to the single body case there exist nontrivial potentials invariant under this group (left and right acting). The concept of relative (mutual) deformation tensors of pairs of affine bodies is discussed. Scalar invariants built of such tensors are constructed. There is an essential novelty in comparison to deformation scalars of single affine bodies, i.e., there exist affinely-invariant scalars of mutual deformations. Hence, the hierarchy of interaction models according to their invariance group, from Euclidean to affine ones, can be considered.Comment: 50 pages, 4 figure