The Cartan product is a generalization of the symmetric product of two vectors. We review its definition, present some examples, and exhibit its rôle in the construction of various natural algebras.

This paper concentrates on the homogeneous (conformal) model of Euclidean space (Horosphere) with subspaces that intuitively correspond to Euclidean geometric objects in three dimensions. Mathematical details of the construction and (useful) parametrizations of the 3D Euclidean object models are explicitly demonstrated in order to show how 3D Euclidean information on positions, orientations and radii can be extracted.

This paper describes a relationship between fast real matrix representations of real universal Clifford algebras and the generalized Fast Fourier Transform for supersolvable groups. Detailed constructions of algorithms for the forward and inverse representations for Clifford algebras are given, with proof that these need at most $O(d \log d)$ operations. The algorithms have been implemented and tested in the GluCat C++ library, and some timing results are included.

A discrete version of complex Vekua type equations is considered. We construct a repre\-sentation formula for the solution of the homogeneous complex equation and investigate the inhomogeneous equation. Similar to the continuous case this representation formula is a product of two functions. We also factorize the solution of a homogeneous Vekua type equation in the quaternionic case. In the complex plane we analyse both factors in detail and study the relation between the non-holomorphic factors in the discrete and continuous case.

Let $C\ell_{n}$ be the (universal) Clifford algebra generated by $e_{1},...,e_{n}$ satisfying $e_{i}e_{j}+e_{j}e_{i}=-2\delta_{ij}$, $i,j=1,...,n$. The Dirac operator in $C\ell_{n}$ is defined by $D=\sum_{i=0}^{n}e_{i}\frac{\partial}{\partial x_{i}}$, where $e_{0}=1$. The modified Dirac operator is introduced for $k\in\mathbb{R}$ by $M_{k}f=Df+k\frac{Q^{\prime}f}{x_{n}}$, where $^{\prime}$ is the main involution and $Qf$ is given by the decomposition $f\left( x\right) =Pf\left( x\right) +Qf\left( x\right) e_{n}$ with $Pf\left( x\right) ,Qf\left( x\right) \in C\ell_{n-1}$. A continuously differentiable function $f:\Omega\rightarrow C\ell_{n}$ is called $k$-hypermonogenic in an open subset $\Omega$ of $\mathbb{R}^{n+1}$, if $M_{k}f\left( x\right) =0$ outside the hyperplane $x_{n}=0$. Note that $0$-hypermonogenic functions are monogenic and $n-1$-hypermonogenic functions are hypermonogenic defined by the author and H. Leutwiler in [10]. The power function $x^{m}$ is hypermonogenic. We prove integral formulas of hypermogenic functions.

Hyperbolic monogenics are introduced as eigenfunctions of the hyperbolic angular Dirac operator $\Gamma$. Explicit formulae for the Taylor series of these functions are given, as well as integral formulae related to the hyperbolic angular Dirac operator.

We revise a monogenic calculus for several non-commuting operators, which is defined through group representations. Instead of an algebraic homomorphism we use group covariance. The related notion of joint spectrum and spectral mapping theorem are discussed. The construction is illustrated by a simple example of calculus and joint spectrum of two non-commuting selfadjoint (n\times n) matrices.

This paper deals with a new analytic type of vector- and Clifford algebra valued automorphic forms in one and two vector variables. For hypercomplex generalizations of the classical modular group and their arithmetic congruence subgroups Eisenstein- and Poincaré type series that are annihilated by Dirac operators, and more generally, by iterated Dirac operators on the upper half-space of $\mathbb{R}^n$ are discussed. In particular we introduce (poly-)monogenic modular forms on hypercomplex generalizations of the classical theta group.

Based on observations by one or several cameras of the trajectory of a bubble it is interpolated. We show that this can be done in a very efficient way using dual quaternions. It is shown how quaternionic splines are a solution of this interpolation problem