The necessary appearance of Clifford algebras in the quantum description of fermions has prompted us to re-examine the fundamental role played by the quaternion Clifford algebra, C0,2. This algebra is essentially the geometric algebra describing the rotational properties of space. Hidden within this algebra are symplectic structures with Heisenberg algebras at their core. This algebra also enab...

Let Fq be a finite field of q elements, F a p-adic field and D is a quaternion division algebra over F . This paper proves uniqueness of Shalika models for GL2n(Fq) and GL2n(D), and re-obtain uniqueness of Shalika models for GL2n(F), 1 for any n ∈ N.

We present explicit models for non-elliptic genus one Shimura curves X0(D, N) with Γ0(N)-level structure arising from an indefinite quaternion algebra of reduced discriminant D, and Atkin-Lehner quotients of them. In addition, we discuss and extend Jordan’s work [10, Ch. III] on points with complex multiplication on Shimura curves.

In this paper we prove an analogue of Mordell’s inequality for lattices in finite-dimensional complex or quaternionic Hermitian space that are modules over a maximal order in an imaginary quadratic number field or a totally definite rational quaternion algebra. This inequality implies that 16dimensional Barnes-Wall lattice has optimal density among all 16-dimensional lattices with Hurwitz struc...

In this paper, we present quaternion matrix algebra techniques that can be used to process the eigen analysis of a color image. Applications of Principal Component Analysis (PCA) in image processing are numerous, and the proposed tools aim to give material for color image processing, that take into account their particular nature. For this purpose, we use the quaternion model for color images a...

In this paper, we give an algorithm for presenting S-unit groups of an order O in a definite rational quaternion algebra B such that, for every p ∈ S at which B splits, the localization of O at p is maximal and all left ideals of O of norm p are principal. We then apply this to give presentations for projective S-unit groups of the Hurwitz order in Hamilton’s quaternions over the rational field...

In the paper a new structure of Multi-Layer Perceptron, able to deal with quaternion-valued signal, is proposed. A learning algorithm for the proposed Quaternion MLP (QMLP) is also derived. Such a neural network allows to interpolate functions of quaternion variable with a smaller number of connections with respect to the corresponding real valued MLP. INTRODUCTION In the last few years, neural...

Quaternion is a four-dimensional and an extension of the complex number system. It often viewed from various fields, such as analysis, algebra, geometry. Several applications quaternions are related to object’s rotation motion in three-dimensional space form differential equation. In this paper, we do systematic literature review on development quaternion equations. We utilize PRISMA (preferred...

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...