New Insight into Quaternions and Their Matrices

The aim of this paper is to bring together quaternions and generalized complex numbers. Generalized quaternions with generalized complex number components are expressed and their algebraic structures are examined. Several matrix representations and computational results are introduced. As a crucial part, alternative approach for generalized quaternion matrix with elliptic number entries are developed.

And Where are Fluctuations In Quantum-Mechanical Wave Function?

The generalized complex numbers containing weight coefficients for a paraxial case are described. An evident analogy between the wavy systems of trajectories, the module and an argument of complex numbers and the module and an argument of wave functions is drawn.Their geometrical interpretation is given on the plane and on the sphere. The symmetric and asymmetrical systems of wavy trajectories are considered. The examination of different types of wavy and zigzag trajectories of our systems of rays is presented. An evident analogy of these systems and wave functions describing finding of an electron in a potential well and near an atomic nucleus is drawn. The possible arrangement of a wave of de Broglie of an electron on the sphere of Bohr radius is shown. In the offered work great mathematicians whose works are connected with researches of complex numbers are marked out in {braces}: {Cardano, Hamilton, Gauss, Cauchy and Euler}.

On duality theory and pseudodifferential techniques for Colombeau algebras: generalized delta functionals, kernels and wave front sets

Summarizing basic facts from abstract topological modules over Colombeau generalized complex numbers we discuss duality of Colombeau algebras. In particular, we focus on generalized delta functional and operator kernels as elements of dual spaces. A large class of examples is provided by pseudodifferential operators acting on Colombeau algebras. By a refinement of symbol calculus we review a new characterization of the wave front set for generalized functions with applications to microlocal analysis. AMS Mathematics Subject Classification (2000): 46F30, 46A20, 47G30.