2022-10-01 | A new n-dimensional associative and commutative, but non-distributive, algebra
A. L. Vrba
This paper introduces a new n-dimensional associative and commutative, but non-distributive, algebra. We define the spatial operator $_sj$ that manipulates numbers in a multidimensional number-space (hyper-complex) according to the spatial angle $_s\theta$, a tuple of angles $_s(\theta_1, \theta_2, \ldots)$. The spatial number, which is expressed symbolically as $\textrm{e}^{_s\theta}$, belongs to both the additive and multiplicative Abelian groups. They are non-distributive in multiplication with respect to addition, thus forming a non-distributive ring. Spatial numbers could have applications in vector algebra allowing the algebraic product of two vector quantities. Furthermore, they could be of interest in physics, and towards that purpose, I present a novel multi-dimensional solution of the wave equation that describes a spherical wave object whose centre propagates at a velocity $c$ in a vector space. Introducing n-dimensional numbers that form an associative and commutative,
but non-distributive, algebra. They could find application in defining multidensional wave structures. Full abstract ... BibLaTeX@Online{Vrba-2022-987, |