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A Cauchy-Pompeiu formula in super Dunkl-Clifford analysis

Hongfen Yuan (2017)

Czechoslovak Mathematical Journal

Using a distributional approach to integration in superspace, we investigate a Cauchy-Pompeiu integral formula in super Dunkl-Clifford analysis and several related results, such as Stokes formula, Morera's theorem and Painlevé theorem for super Dunkl-monogenic functions. These results are nice generalizations of well-known facts in complex analysis.

An integral formula of hyperbolic type for solutions of the Dirac equation on Minkowski space with initial conditions on a hyperboloid

Martin Sikora (2010)

Archivum Mathematicum

The Dirac equation for spinor-valued fields f on the Minkowski space of even dimension form a hyperbolic system of partial differential equations. In the paper, we are showing how to reconstruct the solution from initial data given on the upper sheet H + of the hyperboloid. In particular, we derive an integral formula expressing the value of f in a chosen point p as an integral over a compact cycle given by the intersection of the null cone with H + in the Minkowski space 𝕄 .

Applications of Quaternionic Holomorphic Geometry to minimal surfaces

K. Leschke, K. Moriya (2016)

Complex Manifolds

In this paper we give a survey of methods of Quaternionic Holomorphic Geometry and of applications of the theory to minimal surfaces. We discuss recent developments in minimal surface theory using integrable systems. In particular, we give the Lopez–Ros deformation and the simple factor dressing in terms of the Gauss map and the Hopf differential of the minimal surface. We illustrate the results for well–known examples of minimal surfaces, namely the Riemann minimal surfaces and the Costa surface....

Canonical bases for 𝔰𝔩 ( 2 , ) -modules of spherical monogenics in dimension 3

Roman Lávička (2010)

Archivum Mathematicum

Spaces of homogeneous spherical monogenics in dimension 3 can be considered naturally as 𝔰𝔩 ( 2 , ) -modules. As finite-dimensional irreducible 𝔰𝔩 ( 2 , ) -modules, they have canonical bases which are, by construction, orthogonal. In this note, we show that these orthogonal bases form the Appell system and coincide with those constructed recently by S. Bock and K. Gürlebeck in [3]. Moreover, we obtain simple expressions of elements of these bases in terms of the Legendre polynomials.

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