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 ...

The classical uncertainty principle of harmonic analysis states that a nontrivial function and its Fourier transform cannot both be sharply localized. It plays an important role in signal processing and physics. This paper generalizes the uncertainty principle for measurable sets from complex domain to hypercomplex domain using quaternion algebras, associated with the Quaternion Fourier transfo...

We construct explicit examples of quaternion-Kähler and hypercomplex structures on bundles over hyperKähler manifolds. study the infinitesimal symmetries these associated Galicki–Lawson moment map. By performing QK reduction we produce several metrics. Moreover are led to a new proof hyperKähler/quaternion-Kähler type correspondence. also give other Einstein metrics balanced Hermitian bundles.

This paper presents a new moment-preserving thresholding technique, called the binary quaternion-moment-preserving (BQMP) thresholding, for color image data. Based on representing color data by the quaternions, the statistical parameters of color data can be expressed through the definition of quaternion moments. Analytical formulas of the BQMP thresholding can thus be determined by using the a...

A 4D rotation can be decomposed into a left-isoclinic and a right-isoclinic rotation. This decomposition, known as Cayley’s factorization of 4R rotations, can be performed using Elfrinkhof-Rosen method. In this paper, we present a more straightforward alternative approach based on the fact that there is an orthogonal basis, in the sense of Hilbert-Schmidt, for the space of 4×4 real orthonormal ...

Three-dimensional coordinate transformation problem is the most frequent problem in photogrammetry, geodesy, mapping, geographical information science (GIS), and computer vision. To overcome the drawback that traditional solution of the problem based on rotation angles depends strongly on initial value of parameter, which makes the method ineffective in the case of super-large rotation angle, t...

This paper gives a necessary criterion in terms of Hodge theory for representability by quaternion algebras of certain 2-torsion classes in the unramified Brauer group of a complex function field. This criterion is used to give examples of threefolds with unramified Brauer group elements which are the classes of biquaternion division algebras. DOI: https://doi.org/10.1112/S0024609302001546 Post...