5th Annual Workshop on Applications and Generalizations of Complex Analysis

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

 Annoucement Program Abstract Coming events

### Dear colleagues,

We invite you to participate in the 5th Workshop on "Applications and Generalizations of Complex Analysis" on Saturday, 16th of March 2002, at the Department of Mathematics, University of Coimbra. As around this date the 8th 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 1st) 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

9:30

### Opening session

9:45-10:25

– I.S.T., Technical University of Lisbon

###### Spectrum of the Cauchy singular integral operator on weighted Orlicz spaces

10:30-10:55

– U.S.T.C.

Harmonic measure characterization of BMO and VMO in the real unit sphere

11:15-11:40

– University of Beira Interior

###### Linear Quaternionic Systems

11:45-12:10

– University of Beira Interior

On Left Eigenvalues of Quaternionic Matrices

12:15-12:40

– University of Coimbra

Polynomials which are Monogenic from Both Sides

#### Lunch

14:00

###### Short visit to old Coimbra University

15:00-15:25

– University of Joensuu

On Meromorphic Solutions of the Equation f(p(z))=R(z,f(z))

15:30-15:55

– University of Algarve

A Generalization of a Result of Faber about Growth and an Application

16:00-16:25

Electrostatic Interpretation of Zeros of Orthogonal Polynomials

16:45-17:10

- University College Dublin

##### The Axiom of Choice

17:15-17:40

– University of Aveiro

Natural Dualities

18:15

### Abstract

Author: Alexei Karlovich - I.S.T., Universidade Técnica de Lisboa

Title: Spectrum of the Cauchy singular integral operator on weighted Orlicz spaces

Abstract: The Cauchy singular integral operator $S$ is one of the main actors in the theory of Toeplitz operators, Riemann-Hilbert problems, Wiener-Hopf and singular integral equations and other fields of Harmonic and Complex Analysis. We are going to describe the spectrum of the operator $S$ on weighted Orlicz spaces. These spaces are wide generalizations of classical Lebesgue spaces. During the last few years it was discovered that, in depence on the curves the operators acts on and on the weights involved Orlicz spaces, there is a surprising metamorphosis of the (local) spectra of $S$ from circular arcs via horns and logarithmic double spirals to so-called logarithmic leaves with a halo. Technical details will be omitted, but many beautiful pictures of local spectra will be shown.

Author: Guangbin Ren - University of Science and Technology of China

Title: Harmonic measure characterization of BMO and VMO in the real unit sphere

Abstract: The Garsia norm characterizations are obtained for both BMO and VMO on the real unit sphere in terms of both the harmonic measure and the hyperbolic harmonic measure in the real unit ball. The results are proved in a unified approach.

Author: José Carlos Aleixo - University of Beira Interior

Title: Linear Quaternionic Systems

Abstract: We expand the usual notions known in linear real and complex systems to the quaternionic case. The benefit of this approach is well explained by Michiel Hazewinkel in some papers which he is co-author. This approach implies the adaptation of almost theorems and definitions which carries to new demonstrations with adaptive technics.

Author: Rogério Serôdio - University of Beira Interior

Title: On Left Eigenvalues of Quaternionic Matrices

Abstract: Due to the non-commutativity of the quaternions it is necessary to distinguish between left and right eigenvalues of quaternionic
matrices.
Associated to the right eigenvalues of a quaternionic matrix there exists a real polynomial, the right characteristic polynomial,
whose zeros are the right eigenvalues. This characteristic polynomial can be determined for any quaternionic matrix, so the right eigenvalues can be computed.
The equation arising when dealing with the left eigenvalue problem is not a left (unilateral) polynomial. This fact turns the problem much harder to solve. Despite this difficulty, the 2 by 2 case has been solved using a change of variable that transforms the non-unilateral polynomial to a left polynomial which solution is known. Was in So pointed out, in an unpublished work, that the same could be done for the 3 by 3 case. For matrices of higher order, the left eigenvalue problem stays unsolved.
For the special case of quaternionic circulante matrix of any order, a closed form is given for the left eigenvalues.
All this work leads to conjecture that the left eigenvalues of quaternionic matrices are related to some left quaternionic polynomial after some change of variable.
Work supported by {\it Centro de Matem\'atica da Universidade da Beira Interior.

Author: Gil Bernardes - University of Coimbra

Title: Polynomials Monogenics from Both Sides

Abstract: We will obtain some examples of solutions of the system $$\sum_{i=1}^n e_i \partial_{x_i}f(\underline{x})=\sum_{i=1}^n e_{i+n} \partial_{x_i}f(\underline{x})=0,$$ where $f$ is a function of $n$ real variables with values in the Clifford algebra
$\R_{0,2n}$; we will pay special attention to polynomials, as they provide simple examples for this kind of functions. In particular
we will obtain explicit formulas for the generators of the space of polynomial solutions of the above mentioned system.

Author: Heli Silvennoinen - University of Joensuu

Title: On meromorphic solutions of the equation $f(p(z))=R(z,f(z))$

Abstract: I will consider meromorphic solutions of the equation \begin{displaymath} f \left(p(z)\right) = R\left(z,f(z)\right) = \frac{\sum_{i=0}^m(a_i(z)f(z)^i)}{\sum_{j=0}^n(b_j(z)f(z)^j)}, \end{displaymath} where $p(z)$ is a polynomial, $\deg(p(z))\geq2$, $R\left(z,f(z)\right)$ is an irreducible rational function in $f$, and $a_i(z)$ and $b_j(z)$ are given meromorphic functions.

Author: Maria Teresa Alzugaray - University of Algarve

Title: A generalization of a result of Faber about growth and an application

Abstract: In Classic Function Theory there are plenty of results related to the order and type of growth of an entire (or analytic)
function. The scale of growth of entire functions given by their order and type was significantly refined by Valiron when he
introduced the concept of proximate order. We generalize some classical results to the case of proximate orders and carry out an
application.

Title: Electrostatic Interpretation of Zeros of Orthogonal Polynomials

Abstract: We survey recent results on the electrostatic interpretation of the zeros of orthogonal polynomials. First, we prove that the zeros of classical orthogonal polynomials determine the equilibrium position of movable n unit charges. We show that the zeros of general orthogonal
polynomials, subject to certain integrability conditions on their weight functions, determine also the equilibrium position of a system of $n$ particles.

Author: Lisa McCarthy - University College Dublin

Title: The Axiom of Choice

Abstract: Ernst Zermelo placed a spotlight on the Axiom of Choice in 1904 by using it to prove that every set can be well ordered. This axiom tells us that we can always choose an element from each set in a collection of sets. In this talk we will see some equivalent forms and uses of the
Axiom of Choice. We will also discuss the reaction of certain French analysts to Zermelo's proof of the Well Oredering Principle.

Author: Dirk Hofmann - Universidade de Aveiro

Title: Natural Dualities

Abstract: In this talk we give a brief overview of the categorical study of duality theorems, an area of category theory which takes a great
part of its inspiration from the classical duality theorems of Pontryagin, Gelfand and Stone. We start by recalling the basic
notations of category theory and show how {\em schizophrenic objects} induce a dual adjunction between given concrete categories. This dual adjunction can always be restricted to the "fixed" subcategories and yields a dual equivalence between them. In general determining these subcategories can be quiet difficult, we give some useful results in order to describe them.

Coming events