Related papers: Angles between subspaces computed in Clifford Alge…

Given the real Clifford algebra of a quadratic space with a given signature, we define a new product in this structure such that it simulates the Clifford product of a quadratic space with another signature different from the original one.…

This paper is to serve as a key to the projective (homogeneous) model developed by Charles Gunn (arXiv:1101.4542 [math.MG]). The goal is to explain the underlying concepts in a simple language and give plenty of examples. It is targeted to…

I apply the algebraic framework developed in [1] to study geometry of hyperbolic spaces in 1, 2, and 3 dimensions. The background material on projectivised Clifford algebras and their application to Cayley-Klein geometries is described in…

In this article, we discuss the equality of two inner products on a vector space. Particularly, we look at some geometric properties that are given to a vector space by an inner product namely, length and angle, and we ask under what…

We consider relationships between cubic algebras and implication algebras. We first exhibit a functorial construction of a cubic algebra from an implication algebra. Then we consider an collapse of a cubic algebra to an implication algebra…

We try to create a wise definition of 'angle spaces'. Based on an idea of Ivan Singer, we introduce a new concept of an angle in real Banach spaces, which generalizes the euclidean angle in Hilbert spaces. With this angle it is shown that…

Let K \subset L be a field extension. Given K-subspaces A,B of L, we study the subspace spanned by the product set AB = {ab | a \in A, b \in B}. We obtain some lower bounds on the dimension of this subspace and on dim B^n in terms of dim A,…

Of the various formalisms developed to treat relativistic phenomena, those based on Clifford's geometric algebra are especially well adapted for clear geometric interpretations and computational efficiency. Here we study relationships…

This article provides a pedagogically oriented introduction to geometric (Clifford) calculus on pseudo-Riemannian manifolds. Unlike usual approaches to the topic, which rely on embedding the geometric algebra either within a tensor algebra…

We consider the angle in mathematics and arrive at a conclusion that there are two concepts on the issue. One is a descriptive geometrical one, while the other is from functional analysis. They are somewhat different, allow for different…

Let $V$ be a real inner product space and $C[V]$ its ${\rm C}^*$ Clifford algebra. We prove that if $Z$ is a subspace of $V$ then $C[Z^{\perp}]$ coincides with the supercommutant of $C[Z]$ in $C[V]$.

Involutions of the Clifford algebra of a quadratic space induced by orthogonal symmetries are investigated.

We give a definition of the Clifford algebra of an antiautomorphism of a central simple algebra, and compute it for the algebras of degree 2.

We study the relationship between the areas of the consecutive quadrilaterals cut from a convex quadrilateral in the plane by means of a finite or infinite number of straight lines intersecting two of its opposite sides. Moreover, we obtain…

In a recent paper, algebraic descriptions for all non-relativistic spins were derived by elementary means directly from the Lie algebra $\specialorthogonalliealgebra{3}$, and a connection between spin and the geometry of Euclidean…

There are two basic angles associated with a pair of linear subspaces: the Diximier angle and the Friedrichs angle. The Dixmier angle of the pair of orthogonal complements is the same as the Dixmier angle of the original pair provided that…

In "A note on generalized Clifford algebras and representations" (Caenepeel, S.; Van Oystaeyen, F., Comm. Algebra 17 (1989) no. 1, 93--102.) generalized Clifford algebras were introduced via Clifford representations; these correspond to…

Krylov subspace methods are a powerful tool for efficiently solving high-dimensional linear algebra problems. In this work, we study the approximation quality that a Krylov subspace provides for estimating the numerical range of a matrix.…

We investigate the construction and properties of Clifford algebras by a similar manner as our previous construction of the octonions, namely as a twisting of group algebras of Z_2^n by a cocycle. Our approach is more general than the usual…

The Subspace Theorem is a powerful tool in number theory. It has appeared in various forms and been adapted and improved over time. It's applications include diophantine approximation, results about integral points on algebraic curves and…