Horizontal (-δ)-Laplacian on complex Finsler manifolds

A horizontal (-δ)-Laplacian is defined on strongly pseudoconvex complex Finsler manifolds, first for functions and then for horizontal differential forms of type (p,q). The principal part of the (-δ)-Laplacian is computed in local coordinates. As an application, the (-δ)-Laplacian on strongly Kahler Finsler manifold is obtained explicitly in terms of the horizontal covariant derivatives of the Chern-Finsler conncetion.     

Geometric Origin of a Linearization of the <Emphasis Type="Italic">SU</Emphasis>(∞) Toda Equation

We explain the geometrical origin of a linearization of the continuum limit of the Toda equation with axial symmetry. A correspondence between self-dual spaces with one isometry in D=4 and Einstein-Weyl structures is the most important ingredient.     

Boundary value problems for Dirac operators and Maxwell's equations in fractal domains

Some boundary value problems are solved for time‐harmonic electromagnetic fields on fractal domains, in the framework of exploiting hypercomplex function theory. Copyright © 2014 John Wiley & Sons, Ltd.     

Diffeomorphisms taking lines to circles, and quaternionic Hopf fibrations

We list all diffeomorphisms between an open subset of the four-dimensional projective space and an open subset of the four-dimensional sphere that take all line segments to arcs of round circles. These diffeomorphisms are restrictions of quaternionic Hopf fibrations and radial projections from hyperplanes to spheres.     

Equidistribution for meromorphic transforms and the dd~c-method

In this paper we give an introduction to the notion of meromorphic transform. We describe some equidistribution problems and their solution, using the ddc-method. In particular, we give some statistical properties of the equilibrium measure for meromorphic maps on compact Kahler manifolds: K-mixing, exponential decay of correlations and central limit theorem.     

Orthogonal Appell systems of monogenic functions in the cylinder

The main goal of this paper is to construct Orthogonal Appell systems of polynomial solutions of the Riesz and Moisil‐Théodoresco systems in finite cylinders of ?³. This will be done in the spaces of square integrable functions over ? and ?. Some important properties of the systems are discussed. Copyright © 2011 John Wiley & Sons, Ltd.     

On quaternionic analysis for the Schrödinger operator with a particular potential and its relation with the Mathieu functions

It has been found recently that there exists a theory of functions with quaternionic values and in two real variables, which is determined by a Cauchy–Riemann‐type operator with quaternionic variable coefficients, and that is intimately related to the so‐called Mathieu equations. In this work, it is all explained as well as some basic facts of the arising quaternionic function theory. We establish analogues of the basic integral formulas of complex analysis such as Borel–Pompeiu's, Cauchy's, and so on, for this version of quaternionic function theory. This theory turns out to be in the same relation with the Schrödinger operator with special potential as the usual holomorphic functions in one complex variable, or quaternionic hyperholomorphic functions, or functions of Clifford analysis, are with the corresponding Laplace operator. Moreover, it is similar to that of α‐hyperholomorphic functions and the Helmholtz operator. Copyright © 2012 John Wiley & Sons, Ltd.     

Weierstrass method for quaternionic polynomial root‐finding

Quaternions, introduced by Hamilton in 1843 as a generalization of complex numbers, have found, in more recent years, a wealth of applications in a number of different areas that motivated the design of efficient methods for numerically approximating the zeros of quaternionic polynomials. In fact, one can find in the literature recent contributions to this subject based on the use of complex techniques, but numerical methods relying on quaternion arithmetic remain scarce. In this paper, we propose a Weierstrass‐like method for finding simultaneously all the zeros of unilateral quaternionic polynomials. The convergence analysis and several numerical examples illustrating the performance of the method are also presented.     

1 -convex manifolds are p- k?hler

We study here K?hler-type properties of 1-convex manifolds, using the duality between forms and compactly supported currents, and some properties of the Aeppli groups of (q-convex manifolds. We prove that, when the exceptional setS of the l-convex manifoldX has dimensionk, X is p-K?hler for everyp > k, and isk-K?hler if and only if “the fundamental class” ofS does not vanish. There are classical examples whereX is notk-K?hler even with a smoothS, but we prove that this cannot happen if2k ≥n = dimX, nor for suitable neighborhoods of S; in particular,X is always balanced (i.e.,(n - 1)-Kahler). Partially supported by MIUR research funds.