In this paper, we obtain analogues of Jørgensen’s inequality for non-elementary groups of isometries of quaternionic hyperbolic n-space generated by two elements, one of which is loxodromic. Our result gives some improvement over earlier results of Kim [10] and Markham [15]. These results also apply to complex hyperbolic space and give improvements on results of Jiang, Kamiya and Parker [7]. As...

The aim of this paper is to present the basics of a mathematical theory of a new Fourier transform of colour images from the point of view of Clifford analysis. Quaternionic-valued signals have long been used as a model for colour images and Fourier transforms for such signals have been studied in [3], [?], [6], [?], [7]. In this paper we use a version of the Fourier kernel markedly from those ...

The classical matrix groups are of fundamental importance in many parts of geometry and algebra. Some of them, like Sp.n/, are most conceptually defined as groups of quaternionic matrices. But, the quaternions not being commutative, we must reconsider some aspects of linear algebra. In particular, it is not clear how to define the determinant of a quaternionic matrix. Over the years, many peopl...

We introduce a model of computation based on quaternions, which is inspired on the quantum computing model. Pure states are vectors of a suitable linear space over the quaternions. Every other aspect of the theory is the same as in quantum computing: superposition and linearity of the state space, unitarity of the transformations, and projective measurements. We then show that this model is no ...

Differential calculus on the quantum quaternionic group GL(1, Hq) is introduced. E-mail: [email protected]

1. Introduction. The richness of the theory of functions over the complex field makes it natural to look for a similar theory for the only other non-trivial real asso-ciative division algebra, namely the quaternions. Such a theory exists and is quite far-reaching, yet it seems to be little known. It was not developed until nearly a century after Hamilton's discovery of quaternions. Hamilton him...

Index theorems are discussed and applied to coupled Dirac operators in a quaternionic setting.