Geometric Algebra (GA) has proven to be an advanced language for mathematics, physics, computer science, and engineering. This review presents a comprehensive study of works on Quaternion GA applications in science engineering from 1995 2020. After brief introduction GA, the are reviewed across many fields. We discuss characteristics various problems In addition, challenges prospects proposed b...

Let A/Q be a modular abelian variety. We establish criteria to prevent a given quaternion algebra over a totally real number field to be the endomorphism algebra of A over Q̄. We accomplish this by analyzing the representation of Gal (Q̄/Q) on the points of N-torsion of A at primes N which ramify in B and by applying descent techniques to certain covers of Shimura varieties. Our result also appli...

As is well-known, the real quaternion division algebra H is algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra O can not be algebraically isomorphic to any matrix algebras over the real number field R, because O is a non-associative algebra over R. However since O is an extension of H by the Cayley-Dickson process and is also finite-dimensional, som...

In this paper, we use four-dimensional quaternionic algebra to describing space-time field equations in curvature form. The transformation relations of a variable are established with the help basis transformations quaternion algebra. We deduced covariant derivative that explains how components vary scalar and vector fields.The metric tensor geodesic equation also discussed line element curved ...

A multi-dimensional QTRU public-key cryptosystem improves NTRU by replacing the original ring in quaternion algebra. In this paper, we introduce QMNTR an improvement of using a new mathematical structure two public keys and five private keys. This modification has enhanced key system to be more robust secure.

In this paper we compute the second Stiefel-Whitney class of the quadratic form trA/k(ax ), where A is a central simple algebra over a perfect field k of characteristic different from two, a ∈ A is a fixed element, and trA/k is the reduced trace. We show that this class is related on the one hand to the class of A in the Brauer group, and on the other hand to corestrictions of quaternion algebr...

We present a general framework whereby analysis of interior-point algorithms for semidefinite programming can be extended verbatim to optimization problems over all classes of symmetric cones derivable from associative algebras. In particular, such analyses are extendible to the cone of positive semidefinite Hermitian matrices with complex and quaternion entries, and to the Lorentz cone. We pro...

We consider Bézier-like formulas with weights in quaternion and geometric (Clifford) algebra for parametrizing rational curves and surfaces. The simplest non-trivial quaternionic case of bilinear formulas for surface patches is studied in detail. Such formulas reproduce well known biquadratic parametrizations of principal Dupin cyclide patches, and are characterized in general as special Darbou...

Geometric algebra is a consistent computational framework in which to define geometric primitives and their relationships. This algebraic approach contains all geometric operators and permits specification of constructions in a totally coordinate-free manner. Since it contains primitives of any dimensionality (rather than just vectors) it has no special cases: all intersections of primitives ar...

The algebra isomorphism between M 4 (R) and HHH, where H is the algebra of quaternions, has unexpected computational payoo: it helps construct an orthogonal similarity that 22 block-diagonalizesa 44 symmetricmatrix. Replacing plane rotations with these more powerful 4 4 rotations leads to a quaternion-Jacobi method in which thèweight' of 4 elements (in a 2 2 block) is transferred all at once on...