Holor

Abstract: The fourth-rank tensors that embody the elastic or other properties of crystalline anisotropic substances can be partitioned into a number of sets in order that each shall acknowledge the symmetry of one or more of the crystal classes and moreover shall make up a closed linear associative algebra of hypercomplex numbers for the purpose of calculating the sums, products and inverses of its constituent tensors, to which end coordinate invariant expressions of the tensors are adopted. The calculations are simplified immensely, and ensuing physical analyses are well prepared for, once the structure of every algebra is unravelled completely in terms of a number of separate subalgebras isomorphic to familiar algebras such as the binary one of the complex numbers, the quaternary one of the 2x2 matrices and the octonary one of the complex quaternions. The fourth-rank tensors do not seem to have been submitted previously to the present algebraic point of view, and nor do those of any other rank: a parallel, but less intricate, development can be provided for the second-rank ones.

Abstract: A survey of hypercomplex algebras suitable for DSP is presented. Generally applicable properties are obtained, including a paraunitarity condition for hypercomplex lossless systems. Algebras of dimension n = 2N, N ∈ ℤ, are classified by generation methods, constituting families. Two algebra families, which hold commutative and associative properties for arbitrary N, are examined in more detail: The 2N-dimensional hyperbolic numbers and tessarines. Since these non-division algebras possess zero divisors, orthogonal decomposition of hypercomplex numbers is investigated in general.