arXiv subjects
Publications and source records attributed to John Ryan.
Monogenic functions are basic to Clifford analysis. On Euclidean space they are defined as smooth functions with values in the corresponding Clifford algebra satisfying a certain system of first order differential equations, usually referred to as the Dirac equation. There are two equally natural extensions of these equations to a Riemannian spin manifold only one of which is conformally invariant. We present a straightforward exposition.
Fundamental solutions of Dirac type operators are introduced for a class of conformally flat manifolds. This class consists of manifolds obtained by factoring out the upper half-space of $\mathbb{R}^n$ by arithmetic subgroups of generalized modular groups. Basic properties of these fundamental solutions are presented together with associated Eisenstein and Poincaré type series.
In this paper we deal with monogenic and $k$-hypermonogenic automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. Monogenic automorphic forms are exactly the 0-hypermonogenic automorphic forms. In the first part we establish an explicit relation between $k$-hypermonogenic automorphic forms and Maaß wave forms. In particular, we show how one can construct from any arbitrary non-vanishing monogenic automorphic form a Clifford algebra valued Maaß wave form. In the second part of the paper we compute the Fourier expansion of the $k$-hypermonogenic Eisenstein series which provide us with the simplest non-vanishing examples of $k$-hypermonogenic automorphic forms.
We develop basic properties of solutions to the Dirac-Hodge and Laplace equations in upper half space endowed with the hyperbolic metric. Solutions to the Dirac-Hodge equation are called hypermonogenic functions while solutions to this version of Laplace's equation are called hyperbolic harmonic functions. We introduce a Borel-Pompeiu formula and a Green's formula for hyperbolic harmonic functions. Using a Cauchy Integral formula we are able to introduce Hardy spaces of solutions to the Dirac-Hodge equation. We also provide new arguments describing the conformal covariance of hypermonogenic functions and invariance of hyperbolic harmonic functions. We introduce intertwining operators for the Dirac-Hodge operator and hyperbolic Laplacian.
This paper focuses on the development of harmonic and Clifford analysis techniques in the context of some conformally flat manifolds that arise from factoring out a simply-connected domain from $R^n$ by special arithmetic subgroups of the conformal group. Our discussion encompasses in particular the Hopf manifold $S^1 \times S^{n-1}$, conformally flat cylinders and tori and some conformally flat manifolds of genus $g \ge 2$, such as $k$-handled tori and polycylinders. This paper provides a continuation as well as an extension of our previous two papers \cite{KraRyan1,KraRyan2}. In particular, we introduce a Cauchy integral formula for hypermonogenic functions on cylinders, tori and on half of the Hopf manifold. These are solutions to the Dirac-Hodge equation with respect to the hyperbolic metric. We further develop generalizations of the Mittag-Leffler theorem and the Laurent expansion theorem for cylindrical and toroidal monogenic functions. The study of Hardy space decompositions on the Hopf manifold is also continued. Kerzman-Stein operators are introduced. Explicit formulas for the Szegö kernel, the Bergman kernel and the Poisson kernel of half the Hopf manifold are given.
We describe R-GMA (Relational Grid Monitoring Architecture) which has been developed within the European DataGrid Project as a Grid Information and Monitoring System. Is is based on the GMA from GGF, which is a simple Consumer-Producer model. The special strength of this implementation comes from the power of the relational model. We offer a global view of the information as if each Virtual Organisation had one large relational database. We provide a number of different Producer types with different characteristics; for example some support streaming of information. We also provide combined Consumer/Producers, which are able to combine information and republish it. At the heart of the system is the mediator, which for any query is able to find and connect to the best Producers for the job. We have developed components to allow a measure of inter-working between MDS and R-GMA. We have used it both for information about the grid (primarily to find out about what services are available at any one time) and for application monitoring. R-GMA has been deployed in various testbeds; we describe some preliminary results and experiences of this deployment.
We describe R-GMA (Relational Grid Monitoring Architecture) which is being developed within the European DataGrid Project as an Grid Information and Monitoring System. Is is based on the GMA from GGF, which is a simple Consumer-Producer model. The special strength of this implementation comes from the power of the relational model. We offer a global view of the information as if each VO had one large relational database. We provide a number of different Producer types with different characteristics; for example some support streaming of information. We also provide combined Consumer/Producers, which are able to combine information and republish it. At the heart of the system is the mediator, which for any query is able to find and connect to the best Producers to do the job. We are able to invoke MDS info-provider scripts and publish the resulting information via R-GMA in addition to having some of our own sensors. APIs are available which allow the user to deploy monitoring and information services for any application that may be needed in the future. We have used it both for information about the grid (primarily to find what services are available at any one time) and for application monitoring. R-GMA has been deployed in Grid testbeds, we describe the results and experiences of this deployment.
This paper is an elaboration of an introductory talk given by the author at a workshop on Clifford algebras at Tennessee Technical University, in May 2002. We give an introduction to the basic concepts of Clifford analysis, including links to harmonic analysis, Fourier analysis and conformally flat manifolds.
Here we will consider examples of conformally flat manifolds that are conformally equivalent to open subsets of the n-dimensional sphere. For such manifolds we shall introduce a Cauchy kernel and Cauchy integral formula for sections tasking values in a spinor bundle and annihilated by a Dirac operator, or generalized Cauchy-Riemann operator. Basic properties of this kernel are examined, in particular we examine links to Hardy spaces.
In this paper we study Clifford and harmonic analysis on some conformal flat spin manifolds. In particular we treat manifolds that can be parametrized by $U / Γ$ where $U$ is a simply connected subdomain of either $S^{n}$ or $R^{n}$ and $Γ$ is a Kleinian group acting discontinuously on $U$. Examples of such manifolds treated here include for example $RP^{n}$ and $S^{1}\times S^{n-1}$. Special kinds of Clifford-analytic automorphic forms associated to the different choices of $Γ$ are used to construct Cauchy kernels, Cauchy Integral formulas, Green's kernels and formulas together with Hardy spaces, Plemelj projection operators and Szegö kernels for $L^{p}$ spaces of hypersurfaces lying in these manifolds.
Plemelj projection operators are introduced for spaces of square integrable functions defined over the boundaries of a class of compact real n-dimensional manifolds lying in C^n. These manifolds posses many properties similar to domains in R^n, and are consequently called domain manifolds. The key ingredients used here are techniques from both real and complex Clifford analysis. Analogues of the Kerzman-Stein kernel and Szego projection operators ar introduced, and their conformal covariance is described.