H. De Schepper

Fundaments of Hermitean Clifford analysis part II: splitting of h -monogenic equations

Hermitean Clifford analysis focuses on h-monogenic functions taking values in a complex Clifford algebra or in a complex spinor space, where h-monogenicity is expressed by means of two complex and mutually adjoint Dirac operators, which are invariant under the action of a Clifford realization of the unitary group. In part 1 of the article the fundamental elements of the Hermitean setting have been introduced in a natural way, i.e., by introducing a complex structure on the underlying vector space, eventually extended to the whole complex Clifford algebra . The two Hermitean Dirac operators are then shown to originate as generalized gradients when projecting the gradient on invariant subspaces. In this part of the article, the aim is to further unravel the conceptual meaning of h-monogenicity, by studying possible splittings of the corresponding first-order system into independent parts without changing the properties of the solutions. In this way further connections with holomorphic functions of several complex variables are established. As an illustration, we give a full characterization of h-monogenic functions for the case n = 2. During the final redaction of this article, we received the sad news that our friend, colleague and co-author Jarolím Bureš died on 1 October 2006.

Finite element approximation for 2nd order elliptic eigenvalue problems with nonlocal boundary or transition conditions

In this paper, we use two symplectic schemes to simulate the Ablowitz-Ladik model associated to the cubic nonlinear Schrodinger equation and we compare them with nonsymplectic methods.

A new Hilbert transform on the unit sphere in

Let Ω be a bounded, simply connected domain in the complex plane, with C ∞ smooth boundary and take u ∈ C∞(∂Ω) real-valued, then there exists a unique real-valued harmonic function for which the restriction to the boundary ∂Ω is precisely u. Let be the conjugate harmonic to U in the sense that F = U+iV is holomorphic in Ω, for which moreover V(a)=0, a ∈ Ω, then the restriction v of V to the boundary ∂Ω is called the Hilbert transform of u. In this article, we extend the principles of this construction to m-dimensional space , more specifically in a Clifford analysis setting, in order to define a Hilbert-like integral transform on the unit sphere S m−1, based upon a specific notion of conjugate harmonic functions using spherical co-ordinates. ‡Dedicated to Richard Delanghe on the occasion of his 65th birthday.

On a special type of solutions of arbitrary higher spin Dirac operators

In this paper an explicit expression is determined for the elliptic higher spin Dirac operator, acting on functions f(x) taking values in an arbitrary irreducible finite-dimensional module for the group Spin(m) characterized by a half-integer highest weight. Also a special class of solutions of these operators is constructed, and the connection between these solutions and transvector algebras is explained.

Gel'fand–Tsetlin bases of orthogonal polynomials in Hermitean Clifford analysis

An explicit algorithmic construction is given for orthogonal bases for spaces of homogeneous polynomials, in the context of Hermitean Clifford analysis, which is a higher dimensional function theory centered around the simultaneous null solutions of two Hermitean conjugate complex Dirac operators. Copyright

On an inverse problem of pressure recovery arising from soil venting facilities

The technique of soil venting is commonly used for remediation of the unsaturated zone of the soil, which is contaminated by gaseous organic pollutants in the pores. We study a steady-state model with a finite number of extraction wells, where the air flow field in the subsurface is described by a linear elliptic boundary value problem, with a nonlocal Neumann boundary condition, accompanied of a Dirichlet side condition, containing unknown parameters. We develop a constructive method for the parameter identification, i.e., for the determination of the unknown pressure values at the active wells from their measured total discharges.

Orthogonal Bases of Hermitean Monogenic Polynomials: An Explicit Construction in Complex Dimension 2

In this contribution we construct an orthogonal basis of Hermitean monogenic polynomials for the specific case of two complex variables. The approach combines group representation theory, see [5], with a Fischer decomposition for the kernels of each of the considered Dirac operators, see [4], and a Cauchy‐Kovalevskaya extension principle, see [3].

Fischer Decompositions of Kernels of Hermitean Dirac Operators

In this note we describe explicitly irreducible decompositions of kernels of the Hermitean Dirac Operators. In [6], it is shown that these decompositions are essential for a construction of orthogonal (or even Gelfand‐Tsetlin) bases of homogeneous Hermitean monogenic polynomials.

Discrete Clifford Analysis: A Germ of Function Theory

We develop a discrete version of Clifford analysis, i.e., a higher-dimensional discrete function theory in a Clifford algebra context. On the simplest of all graphs, the rectangular ℤm grid, the concept of a discrete monogenic function is introduced. To this end new Clifford bases are considered, involving so-called forward and backward basis vectors, controlling the support of the involved operators. Following a proper definition of a discrete Dirac operator and of some topological concepts, function theoretic results amongst which Stokes’ theorem, Cauchy’s theorem and a Cauchy integral formula are established.

Differential forms versus multi-vector functions in Hermitean Clifford analysis

Se presentan las similitudes entre las algebras de formas diferenciales complejas y de las funciones de algebras de Clifford complejas con valores de multiples vectores aplicados en una region abierta del espacio euclidiano de dimension par.