Related papers: Quaternion Rhapsody
A variety of universal similarity factorization equalities over real Clifford algebras ${\cal R}_{p,q}$ are established. On the basis of these equalities, real, complex and quaternion matrix representations of elements in ${\cal R}_{p,q}$…
Motivated by the general problem of extending the classical theory of holomorphic functions of a complex variable to the case of quater- nion functions, we give a notion of an H-derivative for functions of one quaternion variable. We show…
We present some classes of functions that are defined on the quaternions as solutions for a linear operator that resembles the Cauchy-Riemann conditions. Unlike the Fueter regular functions; in this case the identity function is a solution…
We study the structure of an algebraically closed field with extra function resembling the classical exponentiation on complex numbers.
An inifinite-dimensional representation of the double affine Hecke algebra of rank 1 and type $(C_1^{\vee},C_1)$ in which all generators are tridiagonal is presented. This representation naturally leads to two systems of polynomials that…
The study of complex functions is based around the study of holomorphic functions, satisfying the Cauchy-Riemann equations. The relatively recent field of Clifford Analysis lets us extend many results from Complex Analysis to higher…
It is known that quaternions represent rotations in 3D Euclidean and Minkowski spaces. However, product by a quaternion gives rotation in two independent planes at once and to obtain single-plane rotations one has to apply by half-angle…
This is part two of a series of four methodological papers on (bi)quaternions and their use in theoretical and mathematical physics: 1- Alphabetical bibliography, 2- Analytical bibliography, 3- Notations and terminology, and 4- Formulas and…
This treatise investigates holomorphic functions defined on the space of bicomplex numbers introduced by Segre. The theory of these functions is associated with Fueter's theory of regular, quaternionic functions. The algebras of quaternions…
We investigate the complexity of isomorphism relations for classes of finitely generated and n-generated computably enumerable (c.e.) algebras, presented via c.e. presentations -- that is, as quotients of term algebras over decidable sets…
This paper is divided in three parts. In the first part, I study the Clifford algebra associated to the hessian of a functional $f$ defined on an open subset of $\mathbb{R}^n$ \ and the Clifford algebra associated to the hessian of the…
For vectors in $\mathbb{E}_3$ we introduce an associative, commutative and distributive multiplication. We describe the related algebraic and geometrical properties, and hint some applications. Based on properties of hyperbolic (Clifford)…
The class of non-commutative hypercomplex number systems (HNS) of 4-dimension, constructed by using of non-commutative Grassmann-Clifford procedure of doubling of 2-dimensional systems is investigated in the article and established here are…
The main properties of hypercomplex generalization of quaternion system as antiquaternion are presented in this article. Definitions and studied of antiquaternions conjugation are introduced, their norm and zero divisor, and how to perform…
We describe the solutions to a family of rotationally symmetric second order partial differential equations in the complex plane that arises from a four-dimensional complex Lie algebra whose spanning set generates the algebra from which…
We study systems of quadratic forms over fields and their isotropy over 2-extensions. We apply this to obtain particular splitting fields for quaternion algebras defined over a finite field extension. As a consequence, we obtain that every…
Heisenberg groups over algebras with central involution and their automorphism groups are constructed. The complex quaternion group algebra over a prime field is used as an example. Its subspaces provide finite models for each of the real…
Two arrays form a periodic complementary pair if the sum of their periodic autocorrelations is a delta function. Finding such pairs, however, is challenging for large arrays whose entries are constrained to a small alphabet. One such…
This is an addition to a series of papers [FL1, FL2, FL3, FL4], where we develop quaternionic analysis from the point of view of representation theory of the conformal Lie group and its Lie algebra. In this paper we develop split…
The biorthogonal rational functions of the ${_3}F_2$ type on the uniform grid provide the simplest example of rational functions with bispectrality properties that are similar to those of classical orthogonal polynomials. These properties…