On the solution of spatial generalizations of Beltrami equations

With the help of functional analytical methods complex analysis is a powerful tool in treating non-linear first-order partial differential equations in the plane. Some of the most important of these equations are the Beltrami equations. This is due to the fact that the theory of Beltrami systems is related to many problems of geometry and analysis, like non-linear subsonic two-dimensional hydrodynamics, problems of conformal and quasiconformal mappings of two-dimensional Riemannian manifolds, or complex analytic dynamics. The theory of Beltrami equations is strongly connected with the -operator. This singular integral operator is immediately recognized as two-dimensional Hilbert-transform, known also under the name of integral operator with Beurling kernel, acting as an isometry of L2(C) onto L2(C). In hypercomplex function theory the Beltrami equations have not yet this importance, but nevertheless, they are a basic condition for the transfer of complex methods and efforts for solving partial differential equations, especially of non-linear type, to the spatial case. Here we deal with hypercomplex Beltrami systems. For this we restrict ourselves to the quaternionic case, but without any loss of generality. We will show how a spatial generalization of the complex -operator can be used to solve systems of non-linear partial differential equations, in particular different types of spatial Beltrami systems. Also, the for practical purposes so important norm estimates will be derived. Some of our results are stronger as known results in the complex case, but they are applicable in the complex situation, too

Script geometry

In this paper we describe the foundation of a new kind of discrete geometry and calculus called Script Geometry. It allows to work with more general meshes than classic simplicial complexes. We provide the basic denitions as well as several examples, like the Klein bottle and the projective plane. Furthermore, we also introduce the corresponding Dirac and Laplace operators which should lay the groundwork for the development of the corresponding discrete function theory

Modern trends in hypercomplex analysis

This book contains a selection of papers presented at the session "Quaternionic and Clifford Analysis" at the 10th ISAAC Congress held in Macau in August 2015. The covered topics represent the state-of-the-art as well as new trends in hypercomplex analysis and its applications

Multiresolution analysis (MRA) on Qp-spaces

In this paper we want to outline the possibility of using methods of Wavelet analysis to study Qp-space

On the solution of spatial generalizations of Beltrami equations

With the help of functional analytical methods complex analysis is a powerful tool in treating non-linear first-order partial differential equations in the plane. Some of the most important of these equations are the Beltrami equations. This is due to the fact that the theory of Beltrami systems is related to many problems of geometry and analysis, like non-linear subsonic two-dimensional hydrodynamics, problems of conformal and quasiconformal mappings of two-dimensional Riemannian manifolds, or complex analytic dynamics. The theory of Beltrami equations is strongly connected with the -operator. This singular integral operator is immediately recognized as two-dimensional Hilbert-transform, known also under the name of integral operator with Beurling kernel, acting as an isometry of L2(C) onto L2(C). In hypercomplex function theory the Beltrami equations have not yet this importance, but nevertheless, they are a basic condition for the transfer of complex methods and efforts for solving partial differential equations, especially of non-linear type, to the spatial case. Here we deal with hypercomplex Beltrami systems. For this we restrict ourselves to the quaternionic case, but without any loss of generality. We will show how a spatial generalization of the complex -operator can be used to solve systems of non-linear partial differential equations, in particular different types of spatial Beltrami systems. Also, the for practical purposes so important norm estimates will be derived. Some of our results are stronger as known results in the complex case, but they are applicable in the complex situation, too

Die Anwendung der hyperkomplexen Funktionentheorie auf die Lösung partieller Differentialgleichungen

In der vorliegenden Arbeit wird die Methode der Anwendung der hyperkomplexen Funktionentheorie zur Behandlung partieller Differentialgleichungen über beschränkten Gebieten unter Benutzung einer orthogonalen Zerlegung des Raumes L_2(U) verallgemeinert. Zum einen kann diese Zerlegung als direkte Zerlegung über dem Raum L_p(G),p>1, verallgemeinert werden, was die Untersuchung partieller Differentialgleichungen über allgemeinen Sobolev-Räumen W_p^k(G),p>1,k natürliche Zahl, ermöglicht. Dies wird am Beispiel des Stokes-Problems demonstriert. Zum anderen wird ein modifizierter Cauchy-Kern über unbeschränkten Gebieten eingeführt, deren Komplement eine nichtleere offene Menge enthält. Grundlegende Resultate der Cliffordanalysis über beschränkten Gebieten werden auf diese Situation verallgemeinert und eine orthogonale Zerlegung des Raumes L_2(G) bewiesen. Diese Resultate werden im weiteren dazu benutzt, das stationäre Stokes- bzw. Navier-Stokes-Problem in dem allgemeinen Fall eines unbeschränkten Gebietes zu untersuchen. Im weiteren wird gezeigt, dass sich die entwickelten Methoden auch auf partielle Differentialgleichungen höherer Ordnung anwenden lassen. Dies wird am Beispiel der biharmonischen Gleichung mit Randbedingungen, die Komponenten in Normalenrichtung und tangentieller Richtung besitzen, demonstriert. Am Ende beschäftigen wir uns mit der Verallgemeinerung der komplexen Methoden von Vekua. Dazu werden hyperkomplexe Verallgemeinerungen des komplexen Pi-Operators untersucht und auf die Lösung von hyperkomplexen Beltramigleichungen angewandt.A modified Cauchy kernel is introduced over unbounded domains whose complement contain non-empty open sets. Basic results on Clifford analysis over bounded domains are now carried over to this more general context. In the end boundary value problems, e.g. for the Stokes-system or the Navier-Stokes-system, will be studied in the case of an unbounded domain without using weighted Sobolev spaces. In the latter part of this paper we deal with hypercomplex generalizations of the complex Pi-operator which turn out to have most of the useful properties of their complex origin. Afterwards the application of this operator to the solution of hypercomplex Beltrami equations will be studied

Discrete hypercomplex function theory and its applications

Recently, one can observe an increased interest in discrete function theories and their applications. Although we will give a broader overview in our talk we would like to give a closer idea on the topic and its applications. To this end we present the question of boundary values of discrete monogenic functions in this short text. We also show their applicability in the theory of discrete Riemann boundary value problems (Riemann BVP’s). The grid itself was chosen in view of applications to image processing, such as discrete monogenic functions