Related papers: The vector algebra war: a historical perspective
In this paper we find a Clifford algebra associated to generalized Fibonacci quaternions. In this way, we provide a nice algorithm to obtain a division quaternion algebra starting from a quaternion non-division algebra and vice-versa.
We introduce the concept of paravectors to describe the geometry of points in a three dimensional space. After defining a suitable product of paravectors, we introduce the concepts of biparavectors and triparavectors to describe line…
The expectation values of the first and second moments of the quantum mechanical spin operator can be used to define a spin vector and spin fluctuation tensor, respectively. The former is a vector inside the unit ball in three space, while…
This paper is the third of a series of three, and it is the continuation of math-ph/0412074 and math-ph/0412075. After reviewing the conformal spacetime structure, conformal maps are described in Minkowski spacetime as the twisted adjoint…
Fefferman and Graham showed some time ago that four dimensional conformal geometries could be analyzed in terms of six dimensional, ambient, Riemannian geometries admitting a closed homothety. Recently it was shown how conformal geometry…
The tools, ideas, and insights from linear algebra, abstract algebra, and functional analysis can be extremely useful to signal processing and system theory in various areas of engineering, science, and social science including…
Classical vector analysis is the predominant formalism used by engineers of computational electromagnetism, despite the fact that manifold as a theoretical concept has existed for a century. This paper discusses the benefits of manifolds…
Quantum mechanics is among the most important and successful mathematical model for describing our physical reality. The traditional formulation of quantum mechanics is linear and algebraic. In contrast classical mechanics is a geometrical…
Standard formulation is unable to distinguish between the (+++-) and (---+) spacetime metric signatures. However, the Clifford algebras associated with each are inequivalent, R(4) in the first case (real 4 by 4 matrices), H(2) in the latter…
Based on a family of indefinite unitary representations of the diffeomorphism group of an oriented smooth $4$-manifold, a manifestly covariant $4$ dimensional and non-perturbative algebraic quantum field theory formulation of gravity is…
Large amount of multidimensional data represented by multiway arrays or tensors are prevalent in modern applications across various fields such as chemometrics, genomics, physics, psychology, and signal processing. The structural complexity…
Many basis sets for electronic structure calculations evolve with varying external parameters, such as moving atoms in dynamic simulations, giving rise to extra derivative terms in the dynamical equations. Here we revisit these derivatives…
In order to ask for future concepts of relativity, one has to build upon the original concepts instead of the nowadays common formalism only, and as such recall and reconsider some of its roots in geometry. So in order to discuss 3-space…
We begin from the generalised eight-dimensional Minkowski spacetime structure, previously developed in Clifford geometric algebra $ C\ell(\Re^3) $. We propose that this is the correct algebraic representation for physical three-dimensional…
Let $V$ be a $n$-dimensional real vector space. In this paper we introduce the concept of \emph{euclidean} Clifford algebra $\mathcal{C\ell}(V,G_{E})$ for a given euclidean structure on $V,$ i.e., a pair $(V,G_{E})$ where $G_{E}$ is a…
We investigate the properties of the Extended Fock Basis (EFB) of Clifford algebras introduced in [1]. We show that a Clifford algebra can be seen as a direct sum of multiple spinor subspaces that are characterized as being left…
The paper explains how a unit generalized quaternion is used to represent a rotation of a vector in 3-dimensional space. We review of some algebraic properties of generalized quaternions and operations between them and then show their…
The space-time geometry is considered to be a physical geometry, i.e. a geometry described completely by the world function. All geometrical concepts and geometric objects are taken from the proper Euclidean geometry. They are expressed via…
Clifford algebras have broad applications in science and engineering. The use of Clifford algebras can be further promoted in these fields by availability of computational tools that automate tedious routine calculations. We offer an…
In this paper I shall consider various possible scalar-vector-tensor field theories which might be used to describe the Universe. After imposing numerous constraints of a physical and mathematical nature on the theories under consideration,…