The representation theory of the group U (1, q) is discussed in detail because of its possible application in a quaternion version of the Salam-Weinberg theory. As a consequence, from purely group theoretical arguments we demonstrate that the eigenvalues must be right-eigenvalues and that the only consistent scalar products are the complex ones. We also define an explicit quaternion tensor prod...

This paper focuses on a new type of control laws for oriented mechanical systems. The starting point is dual quaternion and its properties. Logarithm of dual quaternion is defined, based on which control laws are developed, tackling both regulation and tracking problems using logarithmic feedback. The control laws are shown to have several merits, including global asymptotically convergence, co...

In this paper a necessary and sufficient condition is established for the existence of the reflexive re-nonnegative definite solution to the quaternion matrix equation AXA∗ = B, where ∗ stands for conjugate transpose. The expression of such solution to the matrix equation is also given. Furthermore, a necessary and sufficient condition is derived for the existence of the general re-nonnegative ...

We show that the Klein-Gordon equation on the quaternion field is equivalent to a pair of DKP equations. We shall also prove that this pair of DKP equations can be taken back to a pair of new KG equations. We shall emphasize the important difference between the standard and the new KG equations. We also present some qualitative arguments, concerning the possibility of interpreting anomalous sol...

This paper introduces some construction methods for amicable orthogonal designs over the real and quaternion domain which have not been explored for code design before. The proposed construction methods generate a large number of amicable orthogonal designs of quaternions. It is also shown that amicable orthogonal designs of quaternions can be used to construct restricted quaternion orthogonal ...

1. The statement This brief note is devoted to a simple (and well-known) result in noncommutative algebra, which is not deep but nevertheless subtler than it appears. It concerns the so-called quaternion algebras: Definition 1.1. Let k be a commutative ring1. Let a ∈ k and b ∈ k. The quaternion algebra Ha,b is defined to be the k-algebra with generators i and j and relations i2 = a, j2 = b, ij ...

In this study, new highly parallel algorithm of two-sided Jacobi 8-D transformation is suggested. It is oriented on VLSI-implementation of special processor array. This array is built using 8-D CORDIC algorithm for quaternion valued matrix singular value decomposition. Accuracy analysis and simulation results are added. Such array can be utilized to speed up the Jacobi method realization to com...

We extend our previous results on the relation between quaternion-Kähler manifolds and hyperkähler cones and we describe how isometries, moment maps and scalar potentials descend from the cone to the quaternion-Kähler space. As an example of the general construction, we discuss the gauging and the corresponding scalar potential of hypermultiplets with the unitary Wolf spaces as target spaces. T...

The arithmetic of quaternions is recalled from a constructive point of view. A Hecke module is introduced, defined as a free abelian group on right ideal classes of a quaternion order, together with a natural action of Hecke operators. An equivalent construction in terms of Shimura curves is then introduced, and the quaternion construction is applied to the analysis of specific modular and Shim...