arXiv subjects

Rolf Sören Kraußhar

Publications and source records attributed to Rolf Sören Kraußhar.

12 recordsLinked to original sources

arXiv2025

Discrete octonionic analysis: a unified approach to the split-octonionic and classical settings

Various problems of mathematical physics consider octonions and split-octonions as a mathematical structure, which underpins the eight-dimensional nature of these problems. Therefore, it is not surprising that octonionic analysis has become an area of active research in recent years. One of the main goals of octonionic analysis is to develop tools of an octonionic operator calculus for solving boundary value problems of mathematical physics that benefit from the use of the octonionic structure. However, when we want to apply the operator calculus in practice, it becomes evident that adequate discrete counterparts of continuous constructions need to be defined. In previous works, we have proposed several approaches to discretise the classical continuous octonionic analysis. However, the split-octonionic case, which is particularly important for practical applications concretely investigated in the last years, has not been considered until now. Therefore, one of the goals of this paper is to explain how to particularly address the discrete split-octonionic setting. Additionally, we propose a general umbrella to cover all different discrete octonionic settings in one unified approach that also encompasses the different eight-dimensional algebraic structures.

math.CV↗

arXiv2024

Weyl calculus perspective on the discrete Stokes' formula in octonions

Octonions are 8-dimensional hypercomplex numbers which form the biggest normed division algebras over the real numbers. Motivated by applications in theoretical physics, continuous octonionic analysis has become an area of active research in recent year. Looking at possible practical applications, it is beneficial to work directly with discrete structures, rather than approximate continuous objects. Therefore, in previous papers, we have proposed some ideas towards the discrete octonionic analysis. It is well known, that there are several possibilities to discretise the continuous setting, and the Weyl calculus approach, which is typically used in the discrete Clifford analysis, to octonions has not been studied yet. Therefore, in this paper, we close this gap by presenting the discretisation of octonionic analysis based on the Weyl calculus.

math.CV↗

arXiv2024

The discrete octonionic Stokes' formula revisited

In a previous work [arXiv:2211.02945] we made an attempt to set up a discrete octonionic Stokes' formula. Due to an algebraic property that we have not considered in that attempt, the formula however turned out to involve an associator term in addition to a change of sign that we already observed earlier. This associator term has an impact on the final result. In this paper we carefully revise this discrete Stokes formula taking into account this additional term. In fact the result that we now obtain is much more in line with the results that one has in the continuous setting.

math.AP↗

arXiv2024

Application of the Weyl calculus perspective on discrete octonionic analysis in bounded domains

In this paper, we finish the basic development of the discrete octonionic analysis by presenting a Weyl calculus-based approach to bounded domains in $\mathbb{R}^{8}$. In particular, we explicitly prove the discrete Stokes formula for a bounded cuboid, and then we generalise this result to arbitrary bounded domains in interior and exterior settings by the help of characteristic functions. After that, discrete interior and exterior Borel-Pompeiu and Cauchy formulae are introduced. Finally, we recall the construction of discrete octonionic Hardy spaces for bounded domains. Moreover, we explicitly explain where the non-associativity of octonionic multiplication is essential and where it is not. Thus, this paper completes the basic framework of the discrete octonionic analysis introduced in previous papers, and, hence, provides a solid foundation for further studies in this field.

math.AP↗

arXiv2023

Cauchy formulae and Hardy spaces in discrete octonionic analysis

In this paper, we continue the development of a fundament of discrete octonionic analysis that is associated to the discrete first order Cauchy-Riemann operator acting on octonions. In particular, we establish a discrete octonionic version of the Borel-Pompeiu formula and of Cauchy's integral formula. The latter then is exploited to introduce a discrete monogenic octonionic Cauchy transform. This tool in hand allows us to introduce discrete octonionic Hardy spaces for upper and lower half-space together with Plemelj projection formulae.

math.CV↗

arXiv2022

Towards discrete octonionic analysis

In recent years, there is a growing interest in the studying octonions, which are 8-dimensional hypercomplex numbers forming the biggest normed division algebras over the real numbers. In particular, various tools of the classical complex function theory have been extended to the octonionic setting in recent years. However not so many results related to a discrete octonionic analysis, which is relevant for various applications in quantum mechanics, have been presented so far. Therefore, in this paper, we present first ideas towards discrete octonionic analysis. In particular, we discuss several approaches to a discretisation of octonionic analysis and present several discrete octonionic Stokes' formulae.

math.AP↗

arXiv2021

Octonionic Kerzman-Stein operators

In this paper we consider generalized Hardy spaces in the octonionic setting associated to arbitrary Lipschitz domains where the unit normal field exists almost everywhere. First we discuss some basic properties and explain structural differences to the associative Clifford analysis setting. The non-associativity requires special attention in the definition of an appropriate inner product and hence in the definition of a generalized Szegö projection. Whenever we want to apply classical theorems from reproducing kernel Hilbert spaces we first need to switch to the consideration of real-valued inner products where the Riesz representation theorem holds. Then we introduce a generalization of the dual Cauchy transform for octonionic monogenic functions which represents the adjoint transform with respect to the real-valued inner product $\langle \cdot, \cdot \rangle_0$ together with an associated octonionic Kerzman-Stein operator and related kernel functions. Also in the octonionic setting, the Kerzman-Stein operator that we introduce turns out to be a compact operator. A motivation behind this approach is to find an approximative method to compute the Szegö projection of octonionic monogenic functions offering a possibility to tackle BVP in the octonions without the explicit knowledge of the octonionic Szegö kernel which is extremely difficult to determine in general. We also discuss the particular cases of the octonionic unit ball and the half-space. Finally, we relate our octonionic Kerzman-Stein operator to the Hilbert transform and particularly to the Hilbert-Riesz transform in the half-space case.

math.CV↗

arXiv2020

Recent and new results on octonionic Bergman and Szegö kernels

Very recently one has started to study Bergman and Szegö kernels in the setting of octonionic monogenic functions. In particular, explicit formulas for the Bergman kernel for the octonionic unit ball and for the octonionic right half-space as well as a formula for the Szegö kernel for the octonionic unit ball have been established. In this paper we extend this line of investigation by developing explicit formulas for the Szegö kernel of strip domains of the form ${\cal{S}} := \{z \in \mathbb{O} \mid 0 < \Re(z) < d\}$ from which we derive by a limit argument considering $d \to \infty$ the Szegö kernel of the octonionic right half-space. Additionally, we set up formulas for the Bergman kernel of such strip domains and relate both kernels with each other. In fact, these kernel functions can be expressed in terms of one-fold periodic octonionic monogenic generalizations of the cosecant function and the cotangent function, respectively.

math.CV↗

arXiv2020

Differential topological aspects in octonionic monogenic function theory

In this paper we apply a homologous version of the Cauchy integral formula for octonionic monogenic functions to introduce for this class of functions the notion of multiplicity of zeroes and $a$-points in the sense of the topological mapping degree. As a big novelty we also address the case of zeroes lying on certain classes of compact zero varieties. This case has not even been studied in the associative Clifford analysis setting so far. We also prove an argument principle for octonionic monogenic functions for isolated zeroes and for non-isolated compact zero sets. In the isolated case we can use this tool to prove a generalized octonionic Rouché's theorem by a homotopic argument. As an application we set up a generalized version of Hurwitz theorem which is also a novelty even for the Clifford analysis case.

math.CV↗

arXiv2018

Applications of parabolic Dirac operators to the instationary viscous MHD equations on conformally flat manifolds

In this paper we apply classical and recent techniques from quaternionic analysis using parabolic Dirac type operators and related Teodorescu and Cauchy-Bitzadse type operators to set up some analytic representation formulas for the solutions to the time depedendent incompressible viscous magnetohydrodynamic equations on some conformally flat manifolds, such as cylinders and tori associated with different spinor bundles. Also in this context a special variant of hypercomplex Eisenstein series related to the parabolic Dirac operator serve as kernel functions.

math.AP↗

arXiv2018

Automorphic Forms and Dirac Operators on Conformally Flat Manifolds

In this paper we present a summarizing description of the connection between Dirac operators on conformally flat manifolds and automorphic forms based on a series of joint work with John Ryan over the last fifteen years. We also outline applications to boundary value problems.

math.CV↗

arXiv2018

Some applications of parabolic Dirac operators to the instationary Navier-Stokes problem on conformally flat cylinders and tori in $\mathbb{R}^3$

In this paper we give a survey on how to apply recent techniques of Clifford analysis over conformally flat manifolds to deal with instationary flow problems on cylinders and tori. Solutions are represented in terms of integral operators involving explicit expressions for the Cauchy kernel that are associated to the parabolic Dirac operators acting on spinor sections of these manifolds.

math.AP↗