Some results on traceless decomposition of tensors

Mathematics, 2023

In the present paper, we study two different approaches of tensor decomposition. The first part aims to study some properties of tensors that result from the fact that some components are vanishing in certain coordinates. It is proven that these conditions allow tensor decomposition, especially (1, σ), σ = 1, 2, 3 tensors. We apply the results for special tensors such as the Riemann, Ricci, Einstein, and Weyl tensors and the deformation tensors of affine connections. Thereby, we find new criteria for the Einstein spaces, spaces of constant curvature, and projective and conformal flat spaces. Further, the proof of the theorem of Mikeš and Moldobayev is repaired. It has been used in many works and it is a generalization of the criteria formulated by Schouten and Struik. The second part deals with the properties of a special differential operator with respect to the general decomposition of tensor fields on manifolds with affine connection. It is shown that the properties of special differential operators are transferred to the components of a given decomposition.

Computational Mathematics and Computer Modeling with Applications (CMCMA)

A tensor is called semi-symmetric if all modes but one, are symmetric. In this paper, we study the CP decomposition of semi-symmetric tensors or higher-order individual difference scaling (INDSCAL). Comon's conjecture states that for any symmetric tensor, the CP rank and symmetric CP rank are equal, while it is known that Comon's conjecture is not true in the general case but it is proved under several assumptions in the literature. In the paper, Comon's conjecture is extended for semi-symmetric CP decomposition and CP decomposition of semi-symmetric tensors under suitable assumptions. Specially, we show that if a semi-symmetric tensor has a CP rank smaller or equal to its order, or when the semi-symmetric CP rank is less than/or equal to the dimension, then the semi-symmetric CP rank is equal to the CP rank.

—We consider various methods for constructing linearly independent isotropic, gyrotropic, orthotropic, and transversally isotropic tensors. We state assertions and theorem that permit one to construct these tensors. We find linearly independent above-mentioned tensors up to and including rank six. The components of the tensor may have no symmetry or have symmetries of various types. It is known [1–5] that E ˜ E = r i r i = g ij r i r j is the only isotropic tensor of rank 2 that can be used to represent any other isotropic tensor a ˜ a of rank 2 in the form a ˜ a = aE˜E, where a is a scalar; i.e., an arbitrary isotropic tensor of rank 2 is a spherical tensor. The tensors C ˜ C ˜ C (1) = E ˜ EE˜E = r i r i r j r j , C ˜ C ˜ C (2) = r i r j r i r j , C ˜ C ˜ C (3) = r i E ˜ Er i = r i r j r j r i (1.1) are three linearly independent (irreducible to each other) tensors of rank 4. The general expression for an arbitrary isotropic tensor of rank 4 is their linear combination C ˜ C ˜ C = 3 k=1 a k C ˜ C ˜ C (k). If we pay attention to the structure of isotropic tensors of rank 2 and rank 4 in (1.1), then we easily see that they can be obtained from the corresponding multiplicative bases by pairwise convolution (contraction) of indices of the basis vectors and by exhausting all possible cases of such contraction. By way of example, let us also construct all linearly independent isotropic tensors of rank 6. The multiplicative basis of a tensor of rank 6 is r i r j r k r l r m r n. By contracting the indices pairwise arbitrarily, we obtain some isotropic tensor of rank 6. For example, r i r i r k r k r m r m = E ˜ EE˜EE˜E. (1.2) All other isotropic tensors of rank 6 can be obtained from (1.2) by permutations of basis vectors. Obviously, by rearranging the basis vectors in (1.2), we obtain 6! = 720 permutations (isotropic tensors of rank 6) in the general case. Of these tensors, only fifteen are linearly independent (irreducible to each other) [3, 5]. To obtain these linearly independent tensors of rank 6, it suffices, for example, to consider the following tensors: r i r i r k r k r m r m , r i r k r i r k r m r m , r i r k r k r i r m r m , r i r k r k r m r i r m , r i r k r k r m r m r i. (1.3) It is clear that the basis vector r i occupies all possible positions in (1.3). Now by keeping the vectors r i and r i at their positions in (1.3) and by permuting the other vectors, we obtain two additional tensors

Journal of Mathematical Sciences, 2009

Elementary information on polynomials with tensor coefficients and operations with them is given. A generalized Bezout theorem is stated and proved, and on this basis, the Hamilton-Cayley theorem is proved. Another proof of the latter theorem is also considered. Several important theorems are proved, which apply in deducing of the formula expressing the adjunct tensor B (λ) for the tensor binomial λ (2p) E − A in terms of the tensor A ∈ C2p(Ω) (elements of this module are complex tensors of rank 2p) and its invariants. Furthermore, the definitions of minimal polynomial of the tensor of module C2p(Ω), of the tensor of module Cp(Ω) (whose elements are complex tensors of rank p), and of the tensor of module Cp(Ω) with respect to the given tensor of module C2p(Ω) are given. Here, Ω is some domain of the n-dimensional Euclidean (Riemannian) space. Some theorems concerning minimal polynomials are stated and proved. Moreover, the first, second, and third theorems on the splitting of the module Cp(Ω) into invariant submodules are given. Special attention is paid to theorems on adjoint, normal, Hermitian, and unitary tensors of modules C2p(Ω) and R2p(Ω) (elements of this module are real tensors of rank 2p). The theorem on polar decomposition [4, 6, 9, 13, 14], the Schur theorem [6], and the existence theorems for a general complete orthonormal system of eigentensors for a finite or infinite set of pairwise commuting normal tensors of modules C2p(Ω) and R2p(Ω) are generalized to tensors of a complex module of an arbitrary even order. Canonical representations of normal, conjugate, Hermitian, and unitary tensors of the module C2p(Ω) are given (the definition of this module can be found in [3, 17]). Moreover, the Cayley formulas for linear operators [6] are generalized to tensors of the module C2p(Ω). CONTENTS Definition 1.1. The tensor B(λ) ∈ C 2p (Ω), whose components are polynomials with respect to λ, is called a polynomial with tensor coefficients, or tensor polynomial , or λ-tensor. By virtue of the definition, the components of the tensor polynomial B(λ) ∈ C 2p (Ω) are represented in the form B i 1 .

Indian Journal of Science and …, 2012

In this paper, we determine the nonabelian tensor square G ⊗G for special orthogonal groups SO n (F q ) and spin groups Spin n (F q ), where F q is a field with q elements.