Tensors and their applications

Abstract

In recent several decades, tensors have more and more important applications in both mathematical field and physical field. This thesis devotes to tensors and their applications in several research areas. These applications include positive (semi)-definiteness of structure tensors (hypermatrices), strong ellipticity condition for elasticity tensors and tensor representation theory in physics. In details, these three topics are: 1. finding a new class of positive (semi-)definiteness tensors and verifying their properties; 2. constructing a kind of elasticity tensor with a special structure such that the strong ellipticity condition can be verified more easily; 3. presenting an irreducible isotropic function basis of a third order three-dimensional symmetric tensor and proposing a minimal isotropic integrity basis and an irreducible isotropic function basis of a Hall tensor. For the first topic, motivated by a kind of positive definite test matrix, the Moler matrix, we introduce a new class of positive semi-definite tensor, the MO tensor, which is a generalization of the Moler matrix. We pay our attention to two special cases of the MO tensors: the essential MO tensor and the Sup-MO tensor. Both of them are proved to be positive definite. Especially, the definition of the Sup-MO tensor is based on the concepts of the MO value, the MO set and the Sup-MO value which are all defined in this work. Furthermore, an essential MO tensor is also a completely positive tensor. Furthermore, the properties of the H-eigenvalues of the Sup-MO tensor are presented. We show that the smallest H-eigenvalue of a Sup-MO tensor is positive and approaches to zero as its dimension tends to infinity. In the second topic, we focus on the verification for the strong ellipticity of a fourth order elasticity tensor. The problem of verifying the strong ellipticity is converted to an optimization problem of verifying the M-positive semi-definiteness of a partially symmetric tensor. Hence, a kind of tensors which satisfy the strong ellipticity condition is proposed. The elasticity M-tensor is constructed with respect to the M-eigenvalues of elasticity tensors. After proposing a Perron-Frobenius-type theorem for M-spectral radii of the nonnegative elasticity tensors, we are able to show that any nonsingular elasticity M-tensor satisfies the strong ellipticity condition. Furthermore, several equivalent definitions for nonsingular elasticity M-tensors are established in this topic. In the last topic, we turn our attention to tensor representation theory in the physical field. An isotropic irreducible function basis with 11 invariants of a third order three-dimensional symmetric tensor, which is a proper subset of the Olive-Auffray minimal integrity basis of that tensor, are presented. This result is essential to further investigation for the irreducible function basis of higher order tensors. What is more, the representations of the Hall tensor are also investigated. The Hall tensor, which comes from the Hall effect, an important magnetic effect observed in electric conductors and semiconductors, is a third order three-dimensional tensor whose first two indices are skew-symmetric. We build a connection between its hemitropic and isotropic invariants and invariants of a second order three-dimensional tensor via the third order permutation tensor, i.e., the Levi-Civita tensor. Then, a minimal integrity basis with 10 isotropic invariants for the Hall tensor is proposed and it is proved to be an irreducible function basis for that Hall tensor as well.