4th Annual Workshop on Applications and Generalizations of Complex Analysis

23-24 March 2001

Room 5.5

With support from CMUC (Centro de Matemática da Universidade de Coimbra) and UI&D "Matemática e Aplicações" da Universidade de Aveiro.

Dear colleagues,

We invite you to participate in the 4th Workshop on "Applications and Generalizations of Complex Analysis" on Friday, 23rd and Saturday, 24th of March 2001, at the Department of Mathematics, University of Coimbra. As around this date the 7th European Intensive Course on "Complex Analysis and its Generalizations" takes place in Coimbra, this workshop is intended to give an opportunity for discussions between junior and senior researchers from several european countries in various fields of mathematics related to Complex, Quaternionic and Clifford Analysis (like Algebra, Geometry, Numerical Analysis, Differential Equations, etc.)

We will realize main communications of 45 minutes and short communications of 30 minutes and we would be glad if you would decide to participate and even more to provide a contribution. There will also be the possibility of publishing your contribution in a special issue of the "Cadernos de Matemática" of the Department of Mathematics of Aveiro.

In case of interest please fill the registration form  in due time (deadline March 12th) so that we are able to prepare a tentative programm.

Besides a formal inscription by replying to this announcement there will be no fee.

Looking forward to meet you at the workshop.

The Organizers:

Helmuth R. Malonek  J. Carvalho e Silva

Paula Cerejeiras  Amilcar Branquinho

WorkProgramme

 March 23 March 24 10h-10h45m Vladimir Soucek 10h50m-11h35m U. Kähler 11h55m-12h25m P. Cerejeiras 12h30m-13h Lynn Moran Lunch Time 15h30m-16h15m Ilpo Laine K. Gürlebeck 16h25m-16h55m Jacob Stordal Christiansen 17h20m-17h50m Maria das Neves Rebocho Meeting

Conferences

Author: P. Cerejeiras, Departamento de Matemática, Universidade de Aveiro, Portugal.

Title: Geometric aspects of a problem in texture goniometry

Abstract: The classical Radon transform has been, since the  fifties, an important tool in all sort of problems involving reconstruction of a function by means of its known integrals over hyperplanes, its best known application being in computer tomography. In this talk we shall consider the similar problem of reconstructing a symmetric function by means of its integrals over spheres centred at the origin - the so called spherical Radon transform - applied to a problem of texture goniometry. The spherical Radon transform, originally stated by Funk in 1916, has received a new impulse in the last decade. We  shall present here some results concerning the geometric aspect of the tri-dimensional case via treatement of quaternions.

Author: Jacob Stordal Christiansen, Department of Mathematics, Universitetsparken Copenhagen, Denmark.

Title: The q-Laguerre and Stieltjes-Wigert moment problems

Abstract: The moment problems associated with the q-Laguerre and Stieltjes-Wigert polynomials are examples of indeterminate moment problems. In the talk I will present the classical solutions to these two moment problems and take a closer  look at a transformation of the set of solutions,  which has all the classical solutions as fixed  points.  Based on generating functions, expressions for the four entire functions from the Nevanlinna parametrization will be obtained. Finally, I will look at the connection between the  two moment problems.

Author: K. Gürlebeck, Weimar, Germany.

Title: On strict inclusions of  ${\bf Q_p}$-spaces of quaternion-valued functions

Abstract: We consider a scale of weighted spaces of quaternion-valued  functions depending on real variables. This scale generalizes the idea of ${\bf Q_p}$-spaces in the complex function theory. The goal of the lecture is to show that the inclusions of spaces from the scale are strict inclusions. As a tool we prove some properties of special monogenic polynomials which do have an own importance independently on the scale of  ${\bf Q_p}$-spaces.

Author: U. Kähler, Departamento de Matemática, Universidade de Aveiro, Portugal.

Title: Beltrami equations over n-dimensional  Euclidean and hyperbolic domains

Abstract: On of the most important partial differential equation in Complex Analysis is the Beltrami equation due to its linking with quasiconformal mappings. In this talk we will consider Beltrami-type equations in the framework of Clifford analysis and discuss the existence of quasiconformal solutions over domains in the Euclidean space $R^n$ and over domains over the hyperbolae which are sitting in the Minkowski-Krain space $R^{1,n-1}$.

Author: Ilpo Laine, University of Joensuu, Finland

Title: Complex difference equations

Abstract: In a recent paper, Ablowitz, Halburd and Herbst considered some difference equations in the
complex plane related to Painlevé differential equations. An example of such difference equations
is $y (z+1) + y (z-1) = R (z,y(z))$, where $R$ is rational in $z,y$. Actually, in the frame of my
research seminar, results due to Ablowitz, Halburd and Herbst have been extended to more general difference equations of type $\displaystyle \sum_{j=1}^{n}y (z+c_j) = R (z,y(z))$, $c_j \in {\Bbb{C}}$. Considering
\begin{eqnarray*}
\sum_{j=1}^{n} a_j (z) y (z+c_j) =
\sum_{j=1}^{n} b_j (z) y^j (z) \, ,
\end{eqnarray*}
where $m \geq 2$ and $c_j \neq 0$, we also observe that, for some $K 0$, $\log M(r,y) K m^{r/C}$,
$C = \max \{ |c_1| , \dots , |c_n| \}$, for all $r$ sufficiently large, whenever $y$ has finitely many poles. If $y$ has infinitely many poles, then $n (r,y) K m^{r/C}$.

Author: Lynn Moran, University College, Dublin, Irland.

Title: Emission from the external shock in Gamma-Ray Bursts

Abstract: Gamma-ray bursts are the most energetic explosions in the universe. Afterglows from gamma-ray bursts have been detected at X-ray, optical and radio wavelengths. This suggests that the relativistic shell that produced the gamma-ray bursts itself, decelerates on encountering an external medium, producing the afterglow.  Treating the population of energetic particles as injection at the shock, the synchrotron emission is calculated. The observed spectrum is then obtained taking into consideration the relativistic nature of the source.

Author: Maria das Neves Vieiro Rebocho, Universidade da Beira Interior, Portugal.

Title: Moment Problem and Quasi-Analitycity.

Abstract

Author: Dermot Ryan, University College, Dublin, Ireland.

Title: Wild Kernels and Galois Co-Descent.

Abstract: For any number field, $F$, there is an associated group $K_2 (F)$. The wild kernel of $F$, $W(F)$, is a subgroup of $K_2 (F)$ which is the kernel of the Hilbert symbols, which are maps from $K_2 (F)$ defined at each prime ideal of the ring of integers of $F$. Suppose that $E / F$ is a Galois extension of number fields with Galois group $\Gamma$. We define $W(E)_\Gamma$ to be the largest quotient group of $W(E)$ which is invariant under the action of $\Gamma$. We find conditions under which $W(E)_\Gamma$ is isomorphic to $W(F)$.

Author: Vladimir Soucek, Charles University, Prague, Check Republic.

Title: Monogenic functions of two Clifford variables and higher spin equations.

Abstract: Conformally invariant first order systems of PDE's (generalizing the Dirac equation) for maps with values in higher spin representations of the orthogonal group have been studied recently in Clifford analysis
and basic properties of their polynomial solutions were desribed. There is an interesting  connection between these higher spin equations and the theory of monogenic functions of two Clifford variables. Certain exact complexes (studied by D.Struppa in Clifford analysis) bring important information on properties of such monogenic functions, needed for a study of properties of solutions of higher spin equations.

The Organizers:

Helmuth R. Malonek  J. Carvalho e Silva

Paula Cerejeiras  Amilcar Branquinho