Let P be a polyhedron in H$ of finite volume such that the group Γ(P) generated by reflections in the faces of P is a discrete subgroup of IsomH$. Let Γ+(P) denote the subgroup of index 2 consisting entirely of orientation-preserving isometries so that Γ+(P) is a Kleinian group of finite covolume. Γ+(P) is called a polyhedral group. As discussed in [12] and [13] for example (see §2 below), asso...

We show that over a field of characteristic 2 a central simple algebra with orthogonal involution that decomposes into a product of quaternion algebras with involution is either anisotropic or metabolic. We use this to define an invariant of such orthogonal involutions in characteristic 2 that completely determines the isotropy behaviour of the involution. We also give an example of a non-total...

This paper presents two approaches to obtain the dynamical equations of mobile manipulators using dual quaternion algebra. The first one is based on a general recursive Newton-Euler formulation and uses twists wrenches, which are propagated through high-level algebraic operations works for any type joints arbitrary parameterizations. second approach Gauss's Principle Least Constraint (GPLC) inc...

We study 2-incompressible Grassmannians of isotropic subspaces of a quadratic form, of a hermitian form over a quadratic extension of the base field, and of a hermitian form over a quaternion algebra.

A general method is presented for establishing universal factorization equalities for 2×2 and 4×4 block matrices. As applications, some universal factorization equalities for matrices over four-dimensional algebras are established, in particular, over the Hamiltonian quaternion algebra.

A general method is presented for establishing universal factorization equalities for 2×2 and 4×4 block matrices. As applications, some universal factorization equalities for matrices over four-dimensional algebras are established, in particular, over the Hamiltonian quaternion algebra.

Quaternion derivatives exist only for a very restricted class of analytic (regular) functions; however, in many applications, functions of interest are real-valued and hence not analytic, a typical case being the standard real mean square error objective function. The recent HR calculus is a step forward and provides a way to calculate derivatives and gradients of both analytic and non-analytic...

This paper discusses quaternion L geometric weighting averaging working on the multiplicative Lie group of nonzero quaternionsH∗, endowed with its natural bi-invariant Riemannian metric. Algorithms for computing the Riemannian L center of mass of a set of points, with 1 ≤ p ≤ ∞ (i.e., median, mean, L barycenter and minimax center), are particularized to the case of H∗. Two different approaches ...

We determine the norm Euclidean orders in a positive definite quaternion algebra over Q. Lagrange (1770) proved the four square theorem via Euler’s four square identity and a descent argument. Hurwitz [4] gave a quaternionic proof using the order Λ(2) with Z-basis: 1, i, j, 1 2 (1 + i + j + k). Here i = j = −1 and ij = −ji = k, the standard basis of the quaternions. The key property of Λ(2) is ...