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Über dieses Buch

This book, intended to commemorate the work of Paul Dirac, highlights new developments in the main directions of Clifford analysis. Just as complex analysis is based on the algebra of the complex numbers, Clifford analysis is based on the geometric Clifford algebras. Many methods and theorems from complex analysis generalize to higher dimensions in various ways. However, many new features emerge in the process, and much of this work is still in its infancy.

Some of the leading mathematicians working in this field have contributed to this book in conjunction with “Clifford Analysis and Related Topics: a conference in honor of Paul A.M. Dirac,” which was held at Florida State University, Tallahassee, on December 15-17, 2014. The content reflects talks given at the conference, as well as contributions from mathematicians who were invited but were unable to attend. Hence much of the mathematics presented here is not only highly topical, but also cannot be found elsewhere in print. Given its scope, the book will be of interest to mathematicians and physicists working in these areas, as well as students seeking to catch up on the latest developments.

Inhaltsverzeichnis

Frontmatter

Lambda-Harmonic Functions: An Expository Account

Abstract
In this paper, we compile a variety of results on the \(\lambda -\)Laplacian operator, denoted by \(\varDelta _{\lambda }\), a generalization of the well-known Laplacian in \({\mathbb {R}^n}\). We have compiled a list of known properties for \(\varDelta _{\lambda }\) when \(\lambda = \frac{n-2}{2}\) and present analogous properties for \(\varDelta _{\lambda }\). We close by discussing the \(\lambda -\)Poisson kernel, the function that solves the Dirichlet problem on the closed ball in \({\mathbb {R}^n}\).
K. Ballenger-Fazzone, C. A. Nolder

Some Applications of Parabolic Dirac Operators to the Instationary Navier-Stokes Problem on Conformally Flat Cylinders and Tori in

Abstract
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.
P. Cerejeiras, U. Kähler, R. S. Kraußhar

From Hermitean Clifford Analysis to Subelliptic Dirac Operators on Odd Dimensional Spheres and Other CR Manifolds

Abstract
We show that the two Dirac operators arising in Hermitian Clifford analysis are identical to standard differential operators arising in several complex variables. We also show that the maximal subgroup that preserves these operators are generated by translations, dilations and actions of the unitary n-group. So the operators are not invariant under Kelvin inversion. We also show that the Dirac operators constructed via two by two matrices in Hermitian Clifford analysis correspond to standard Dirac operators in euclidean space. In order to develop Hermitian Clifford analysis in a different direction we introduce a sub elliptic Dirac operator acting on sections of a bundle over odd dimensional spheres. The particular case of the three sphere is examined in detail. We conclude by indicating how this construction could extend to other CR manifolds.
P. Cerejeiras, U. Kähler, J. Ryan

On Some Conformally Invariant Operators in Euclidean Space

Abstract
The aim of this paper is to correct a mistake in earlier work on the conformal invariance of Rarita-Schwinger operators and use the method of correction to develop properties of some conformally invariant operators in the Rarita-Schwinger setting. We also study properties of some other Rarita-Schwinger type operators, for instance, twistor operators and dual twistor operators. This work is also intended as an attempt to motivate the study of Rarita-Schwinger operators via some representation theory. This calls for a review of earlier work by Stein and Weiss.
C. Ding, J. Ryan

Notions of Regularity for Functions of a Split-Quaternionic Variable

Abstract
The utility and beauty of the theory holomorphic functions of a complex variable leads one to wonder whether analogous function theories exist for other (presumably higher dimensional) algebras. Over the last several decades it has been shown that much of Complex analysis extends to a similar theory for the family of Clifford Algebras \(C\ell _{0,n}\). However, there has yet to be a complete description for the general theory over the family of Clifford algebras \(C\ell _{p,q}\) (for \(p\ne 0\)). In this work, we describe two different approaches from the literature for finding a theory of “holomorphic” functions of a split-quaternionic variable (which is the Clifford algebra \(C\ell _{1,1}\)). We show that one approach yields a relatively trivial theory, while the other gives a richer one. In the second instance, we describe a simple subclass of “holomorphic” functions and give two examples of an analogue of the Cauchy-Kowalewski extension in this context.
J. A. Emanuello, C. A. Nolder

Decomposition of the Twisted Dirac Operator

Abstract
The classical Dirac operator is a conformally invariant first order differential operator mapping spinor-valued functions to the same space, where the spinor space is to be interpreted as an irreducible representation of the spin group. In this article we twist the Dirac operator by replacing the spinor space with an arbitrary irreducible representation of the spin group. In this way, the operator becomes highly reducible, whence we determine its full decomposition.
T. Raeymaekers

Norms and Moduli on Multicomplex Spaces

Abstract
Multicomplex analysis describes the theory of holomorphic functions on spaces generated by n commuting complex units. In this context we extend some notions of norms and moduli from the space of bicomplex numbers to the space of multicomplex numbers. This approach is meant to be used towards a meaningful theory of Riemannian and semi-Riemannian geometries built on such spaces.
M. B. Vajiac

Associated Operators to the Space of Meta-q-Monogenic Functions

Abstract
We are giving a characterization of all linear first order partial differential operators with Clifford-algebra-valued coefficients that are associated to the meta-q-monogenic operator. As an application, the solvability of initial value problems involving these operators is shown.
C. J. Vanegas, F. A. Vargas

Correction to: Clifford Analysis and Related Topics

Correction to: P. Cerejeiras et al. (eds.), Clifford Analysis and Related Topics, Springer Proceedings in Mathematics & Statistics 260, https://​doi.​org/​10.​1007/​978-3-030-00049-3
Paula Cerejeiras, Craig A. Nolder, John Ryan, Carmen Judith Vanegas Espinoza
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