Fundamental theory: Holomorphic functions and domains of holomorphy Implicit functions and analytic sets The Poincare, Cousin, and Runge problems Pseudoconvex domains and pseudoconcave sets Holomorphic mappings Theory of analytic spaces: Ramified domains Analytic sets and holomorphic functions Analytic spaces Normal pseudoconvex spaces Bibliography Index.

Function theory in several complex variables

Clifford Algebras and the Classical Groups: The classical groups

Geometric integrators for the nonlinear Schrödinger equation

The Hamiltonian and the multi-symplectic formulations of the nonlinear Schrodinger equation are considered. For the multi-symplectic formulation, a new six-point difference scheme which is equivalent to the multi-symplectic Preissman integrator is derived. Numerical experiments are also reported. (C) 2002 Elsevier Science Ltd. All rights reserved.

Symplectic and multi-symplectic methods for the nonlinear Schrodinger equation

In this chapter we study various properties of the spacesH 1 H 2 andH ∞in preparation for our study of Toeplitz operators in the following chapter. Due to the availability of several excellent accounts of this subject (see Notes), we do not attempt a comprehensive treatment and proceed in the main using the techniques which we have already introduced.

The Hardy Spaces

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 c...

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

Euclidean Clifford analysis is a higher dimensional function theory, refining harmonic analysis, centred around the concept of monogenic functions, i.e. null solutions of a first order vector valued rotation invariant differential operator, called the Dirac operator. More recently, Hermitean Clifford analysis has emerged as a new and successful branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focusses on the simultaneous null solutions of two Hermitean Dirac operators, invariant under the action of the unitary group. In this paper, a Cauchy integral formula is established by means of a matrix approach, allowing the recovering of the traditional Martinelli-Bochner formula for holomorphic functions of several complex variables as a special case.

On Cauchy and Martinelli-Bochner integral formulae in Hermitean Clifford analysis

A matrix Hilbert transform in Hermitean Clifford analysis

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

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.

H. De Schepper

Papers

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

On Cauchy and Martinelli-Bochner integral formulae in Hermitean Clifford analysis

A matrix Hilbert transform in Hermitean Clifford analysis

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

Discrete Clifford analysis : a germ of function theory