arXiv subjects

Derek Courchesne

Publications and source records attributed to Derek Courchesne.

2 recordsLinked to original sources

arXiv2025

Binary Representation in Multicomplex and Clifford Algebras

Using a binary representation for basis elements of an algebra combined with a framework of multiplier and index functions, a connection has been established between the structure of a large class of algebras and the XOR componentwise operation. This result covers both multicomplex and Clifford algebras of any dimension. Necessary conditions for the existence of a diagonal basis are derived directly from this result, with an algorithm-like method to obtain it through a change-of-coordinates matrix in the commutative case. The same framework is then used to define an algebra of conjugate operations and study its properties.

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arXiv2024

Multicomplex Ideals, Modules and Hilbert Spaces

In this article we study some algebraic aspects of multicomplex numbers $\mathbb M_n$. For $n\geq 2$ a canonical representation is defined in terms of the multiplication of $n-1$ idempotent elements. This representation facilitates computations in this algebra and makes it possible to introduce a generalized conjugacy $\Lambda_n$, i.e. a composition of the $n$ multicomplex conjugates $\Lambda_n:=\dagger_1\cdots \dagger_n$, as well as a multicomplex norm. The ideals of the ring of multicomplex numbers are then studied in details, free $\mathbb M_n$-modules and their linear operators are considered and, finally, we develop Hilbert spaces on the multicomplex algebra.

math-ph↗