A Symbolic Algorithm for Computation of Non-degenerate Clifford Algebra Matrix Representations

Clifford algebras are an active area of mathematical research. The main objective of the paper is to exhibit a construction of a matrix algebra isomorphic to a Clifford algebra of signature (p,q), which can be automatically implemented using general purpose linear algebra software. While this is not the most economical way of implementation for lower-dimensional algebras it offers a transparent mechanism of translation between a Clifford algebra and its isomorphic faithful real matrix representation. Examples of lower dimensional Clifford algebras are presented.Comment: 220 page

Fractional velocity as a tool for the study of non-linear problems

Singular functions and, in general, H\"older functions represent conceptual models of nonlinear physical phenomena. The purpose of this survey is to demonstrate the applicability of fractional velocity as a tool to characterize Holder and in particular singular functions. Fractional velocities are defined as limit of the difference quotient of a fractional power and they generalize the local notion of a derivative. On the other hand, their properties contrast some of the usual properties of derivatives. One of the most peculiar properties of these operators is that the set of their non trivial values is disconnected. This can be used for example to model instantaneous interactions, for example Langevin dynamics. Examples are given by the De Rham and Neidinger's functions, represented by iterative function systems. Finally the conditions for equivalence with the Kolwankar-Gangal local fractional derivative are investigated.Comment: 21 pages; 2 figure

Algorithmic computation of multivector inverses and characteristic polynomials in non-degenerate Clifford algebras

The power of Clifford or, geometric, algebra lies in its ability to represent geometric operations in a concise and elegant manner. Clifford algebras provide the natural generalizations of complex, dual numbers and quaternions into non-commutative multivectors. The paper demonstrates an algorithm for the computation of inverses of such numbers in a non-degenerate Clifford algebra of an arbitrary dimension. The algorithm is a variation of the Faddeev-LeVerrier-Souriau algorithm and is implemented in the open-source Computer Algebra System Maxima. Symbolic and numerical examples in different Clifford algebras are presented.Comment: 11 pages. arXiv admin note: text overlap with arXiv:1904.0008