Up until now we have learned that a rotation in R3 about an axis through the origin can be represented by a 3×3 orthogonal matrix with determinant 1. However, the matrix representation seems redundant because only four of its nine elements are independent. Also the geometric interpretation of such a matrix is not clear until we carry out several steps of calculation to extract the rotation axis...

This note provides an insight to the diophantine properties of abelian surfaces with quaternionic multiplication over number fields. We study the fields of definition of the endomorphisms on these abelian varieties and the images of the Galois representations on their Tate modules. We illustrate our results with an explicit example. 1. Abelian surfaces with quaternionic multiplication Fix Q an ...

In this paper, we present a novel concept of the quaternion discrete Fourier transform on the two-dimensional hexagonal lattice, which we call the twodimensional hexagonal quaternion discrete Fourier transform (2-D HQDFT). The concept of the right-side 2D HQDFT is described and the left-side 2-D HQDFT is similarly con sidered. We analyze and present a new approach in processing the color images...

The success of applying generalized complex orthogonal designs as space–time block codes recently motivated the definition of quaternion orthogonal designs as potential building blocks for space–timepolarization block codes. This paper offers techniques for constructing quaternion orthogonal designs via combinations of specially chosen complex orthogonal designs. One technique is used to build ...

Abstract Let X be an Enriques surface over the field of complex numbers. We prove that there exists a nontrivial quaternion algebra ???? on . Then we study moduli scheme torsion free ????-modules rank one. Finally this is étale double cover Lagrangian subscheme in corresponding associated covering K3 surface.