Differential equations over octonions

Differential equations with constant and variable coefficients over octonions are investigated. It is found that different types of differential equations over octonions can be resolved. For this purpose non-commutative line integration is used. Such technique is applied to linear and non-linear partial differential equations in real variables. Possible areas of applications of these results are outlined.Comment: 50 page

A note on totally regular variables and Appell sequences in hypercomplex function theory

Series title : Lecture notes in computer science, vol. 7971, ISSN 0302-9743The aim of our contribution is to call attention to the relationship between totally regular variables, introduced by R. Delanghe in 1970, and Appell sequences with respect to the hypercomplex derivative. Under some natural normalization condition the set of all paravector valued totally regular variables defined in the three dimensional Euclidean space will be completely characterized. Together with their integer powers they constitute automatically Appell sequences, since they are isomorphic to the complex variables.Fundação para a Ciência e a Tecnologia (FCT

Bloch's Theorem in the Context of Quaternion Analysis

The classical theorem of Bloch (1924) asserts that if $f$ is a holomorphic function on a region that contains the closed unit disk $|z|\leq 1$ such that $f(0) = 0$ and $|f'(0)| = 1$, then the image domain contains discs of radius $3/2-\sqrt{2} > 1/12$. The optimal value is known as Bloch's constant and 1/12 is not the best possible. In this paper we give a direct generalization of Bloch's theorem to the three-dimensional Euclidean space in the framework of quaternion analysis. We compute explicitly a lower bound for the Bloch constant.Comment: The present article is a preliminary version, submitted to Computational Methods and Function Theor

3D mappings by generalized joukowski transformations

The classical Joukowski transformation plays an important role in di erent applications of conformal mappings, in particular in the study of ows around the so-called Joukowski airfoils. In the 1980s H. Haruki and M. Barran studied generalized Joukowski transformations of higher order in the complex plane from the view point of functional equations. The aim of our contribution is to study the analogue of those generalized Joukowski transformations in Euclidean spaces of arbitrary higher dimension by methods of hypercomplex analysis. They reveal new insights in the use of generalized holomorphic functions as tools for quasi-conformal mappings. The computational experiences focus on 3D-mappings of order 2 and their properties and visualizations for di erent geometric con gurations, but our approach is not restricted neither with respect to the dimension nor to the order.Financial support from "Center for Research and Development in Mathematics and Applications" of the University of Aveiro, through the Portuguese Foundation for Science and Technology (FCT), is gratefully acknowledged. The research of the first author was also supported by the FCT under the fellowship SFRH/BD/44999/2008. Moreover, the authors would like to thank the anonymous referees for their helpful comments and suggestions which improved greatly the final manuscript

Applications of Bergman kernel functions

In this paper we revisit the so-called Bergman kernel method (BKM) for solving conformal mapping problems. This method is based on the reproducing property of the Bergman kernel function. The construction of reproducing kernel functions is not restricted to real dimension 2. Results concerning the construction of Bergman kernel functions in closed form for special domains in the framework of hypercomplex function theory suggest that BKM can also be extended to mapping problems in higher dimensions. We describe a 3-dimensional BKM-approach and present two numerical examples.Fundação para a Ciência e a Tecnologia (FCT) - Programa Operacional "Ciência, Tecnologia, Inovação" (POCTI)

Source integrals of asymptotic multipole moments

We derive source integrals for multipole moments that describe the behaviour of static and axially symmetric spacetimes close to spatial infinity. We assume isolated non-singular sources but will not restrict the matter content otherwise. Some future applications of these source integrals of the asymptotic multipole moments are outlined as well.Comment: 9 pages, 1 figure, contribution to the proceedings of the conference "Relativity and Gravitation - 100 Years after Einstein in Prague", June 25-29, 2012, Pragu

A 3-dimensional Bergman Kernel method with applications to rectangular domains

In this paper we revisit the so-called Bergman kernel method - BKM - for solving conformal mapping problems and propose a generalized BKM-approach to extend the theory to 3-dimensional mapping problems. A special software package for quaternions was developed for the numerical experiments.Fundação para a Ciência e a Tecnologia (FCT