The Riemann-Hilbert method applied to the theory of orthogonal polynomials

Doutoramento em Matemática

Numerical and combinatorial applications of generalized Appell polynomials

This thesis studies properties and applications of different generalized Appell polynomials in the framework of Clifford analysis. As an example of 3D-quasi-conformal mappings realized by generalized Appell polynomials, an analogue of the complex Joukowski transformation of order two is introduced. The consideration of a Pascal n-simplex with hypercomplex entries allows stressing the combinatorial relevance of hypercomplex Appell polynomials. The concept of totally regular variables and its relation to generalized Appell polynomials leads to the construction of new bases for the space of homogeneous holomorphic polynomials whose elements are all isomorphic to the integer powers of the complex variable. For this reason, such polynomials are called pseudo-complex powers (PCP). Different variants of them are subject of a detailed investigation. Special attention is paid to the numerical aspects of PCP. An efficient algorithm based on complex arithmetic is proposed for their implementation. In this context a brief survey on numerical methods for inverting Vandermonde matrices is presented and a modified algorithm is proposed which illustrates advantages of a special type of PCP. Finally, combinatorial applications of generalized Appell polynomials are emphasized. The explicit expression of the coefficients of a particular type of Appell polynomials and their relation to a Pascal simplex with hypercomplex entries are derived. The comparison of two types of 3D Appell polynomials leads to the detection of new trigonometric summation formulas and combinatorial identities of Riordan-Sofo type characterized by their expression in terms of central binomial coefficients.

Quantity-setting competition with differentiated goods under uncertain demand

We consider a quantity-setting duopoly model, and we study the decision to move first or second, by assuming that the firms produce differentiated goods and that there is some demand uncertainty. The competitive phase consists of two periods, and in either period, the firms can make a production decision that is irreversible. As far as the firms are allowed to choose (non-cooperatively) the period they make the decision, we study the circumstances that favour sequential rather than simultaneous decisions.

Flexibility and leadership advantages in a model with uncertain demand

We consider a differentiated Stackelberg model with demand uncertainty only for the first mover. We study the advantages of flexibility over leadership as the degree of the differentiation of the goods changes.

Uncertain Spaces: Virtual Configurations in Contemporary Art and Museums

Contém artigos apresentados na International Conference “Uncertain Spaces: Virtual Configurations in Contemporary Art and Museums”, na Fundação Calouste Gulbenkian (Lisboa), 31 Outubro - 1 de Novembro de 2014) de: Helena Barranha e Susana S. Martins - Introduction: Art, Museums and Uncertainty (pp.1-12); Alexandra Bounia e Eleni Myrivili - Beyond the ‘Virtual’: Intangible Museographies and Collaborative Museum Experiences (pp.15-32); Annet Dekker - Curating in Progress. Moving Between Objects and Processes (pp.33-54); Giselle Beiguelman - Corrupted Memories. The aesthetics of Digital Ruins and the Museum of the Unfinished (pp.55-82); Andrew Vaas Brooks - The Planetary Datalinks (pp.85-110); Sören Meschede - Curators’ Network: Creating a Promotional Database for Contemporary Visual Arts (pp.11-130); Stefanie Kogler - Divergent Histories and Digital Archives of Latin American and Latino Art in the United States – Old Problems in New Digital Formats (pp.131-156); Luise Reitstätter e Florian Bettel - Right to the City! Right to the Museum!(pp.159-182); Roberto Terracciano - On Geo-poetic systems: virtual interventions inside and outside the museum space (pp.183-210); e, Catarina Carneiro de Sousa e Luís Eustáquio - Art Practice in Collaborative Virtual Environments (pp.211-240).

Prime Rational Functions and Integral Polynomials

Let f(x) be a complex rational function. In this work, we study conditions under which f(x) cannot be written as the composition of two rational functions which are not units under the operation of function composition. In this case, we say that f(x) is prime. We give sufficient conditions for complex rational functions to be prime in terms of their degrees and their critical values, and we derive some conditions for the case of complex polynomials. We consider also the divisibility of integral polynomials, and we present a generalization of a theorem of Nieto. We show that if f(x) and g(x) are integral polynomials such that the content of g divides the content of f and g(n) divides f(n) for an integer n whose absolute value is larger than a certain bound, then g(x) divides f(x) in Z[x]. In addition, given an integral polynomial f(x), we provide a method to determine if f is irreducible over Z, and if not, find one of its divisors in Z[x].

Efficiency in Uncertain Cooperative Games

A contingent contract in a transferable utility game under uncertainty specifies an outcome for each possible state. It is assumed that coalitions evaluate these contracts by considering the minimal possible excesses. A main question of the paper concerns the existence and characterization of efficient contracts. It is shown that they exist if and only if the set of possible coalitions contains a balanced subset. Moreover, a characterization of values that result in efficient contracts in the case of minimally balanced collections is provided.

Certain problems concerning polynomials and transcendental entire functions of exponential type

Soit $\displaystyle P(z):=\sum_{\nu=0}^na_\nu z^{\nu}$ un polynôme de degré $n$ et $\displaystyle M:=\sup_{|z|=1}|P(z)|.$ Sans aucne restriction suplémentaire, on sait que $|P'(z)|\leq Mn$ pour $|z|\leq 1$ (inégalité de Bernstein). Si nous supposons maintenant que les zéros du polynôme $P$ sont à l'extérieur du cercle $|z|=k,$ quelle amélioration peut-on apporter à l'inégalité de Bernstein? Il est déjà connu [{\bf \ref{Mal1}}] que dans le cas où $k\geq 1$ on a $$(*) \qquad |P'(z)|\leq \frac{n}{1+k}M \qquad (|z|\leq 1),$$ qu'en est-il pour le cas où $k < 1$? Quelle est l'inégalité analogue à $(*)$ pour une fonction entière de type exponentiel $\tau ?$ D'autre part, si on suppose que $P$ a tous ses zéros dans $|z|\geq k \, \, (k\geq 1),$ quelle est l'estimation de $|P'(z)|$ sur le cercle unité, en terme des quatre premiers termes de son développement en série entière autour de l'origine. Cette thèse constitue une contribution à la théorie analytique des polynômes à la lumière de ces questions.

From cognition to action : a psychological investigation of action regulation in complex and uncertain social systems

Cette thèse présente une revue des réflexions récentes et plus traditionnelles provenant de la théorie des systèmes, de la créativité en emploi, des théories d’organisation du travail et de la motivation afin de proposer une perspective psychologique de la régulation des actions des individus au sein d’environnements de travail complexes et incertains. Des composantes de la Théorie de la Régulation de l’Action (Frese & Zapf, 1994) ainsi que de la Théorie de l’Auto-Détermination (Deci & Ryan, 2000) sont mises en relation afin d’évaluer un modèle définissant certains schémas cognitifs clés associés aux tâches individuelles et collectives en emploi. Nous proposons que ces schémas cognitifs, organisés de manière hiérarchique, jouent un rôle central dans la régulation d’une action efficace au sein d’un système social adaptatif. Nos mesures de ces schémas cognitifs sont basées sur des échelles de mesure proposées dans le cadre des recherches sur l’ambiguïté de rôle (eg. Sawyer, 1992; Breaugh & Colihan, 1994) et sont mis en relation avec des mesures de satisfaction des besoins psychologiques (Van den Broeck, Vansteenkiste, De Witte, Soenens & Lens, 2009) et du bien-être psychologique (Goldberg, 1972). Des données provenant de 153 employés à temps plein d’une compagnie de jeu vidéo ont été récoltées à travers deux temps de mesure. Les résultats révèlent que différents types de schémas cognitifs associés aux tâches individuelles et collectives sont liés à la satisfaction de différents types de besoin psychologiques et que ces derniers sont eux-mêmes liés au bien-être psychologique. Les résultats supportent également l’hypothèse d’une organisation hiérarchique des schémas cognitifs sur la base de leur niveau d’abstraction et de leur proximité avec l’exécution concrète de l’action. Ces résultats permettent de fournir une explication initiale au processus par lequel les différents types de schémas cognitifs développés en emplois et influencé par l’environnement de travail sont associés à l’attitude des employés et à leur bien-être psychologique. Les implications pratiques et théoriques pour la motivation, l’apprentissage, l’habilitation, le bien-être psychologique et l’organisation du travail dans les environnements de travail complexes et incertains sont discutés.

Development of Shape Descriptors Based on Legendre Polynomials and Content-Based Retrieval of Scoliosis Images

The wealth of information available freely on the web and medical image databases poses a major problem for the end users: how to find the information needed? Content –Based Image Retrieval is the obvious solution.A standard called MPEG-7 was evolved to address the interoperability issues of content-based search.The work presented in this thesis mainly concentrates on developing new shape descriptors and a framework for content – based retrieval of scoliosis images.New region-based and contour based shape descriptor is developed based on orthogonal Legendre polymomials.A novel system for indexing and retrieval of digital spine radiographs with scoliosis is presented.

A generic polynomial solution for the differential equation of hypergeometric type and six sequences of orthogonal polynomials related to it

In this work, we present a generic formula for the polynomial solution families of the well-known differential equation of hypergeometric type s(x)y"n(x) + t(x)y'n(x) - lnyn(x) = 0 and show that all the three classical orthogonal polynomial families as well as three finite orthogonal polynomial families, extracted from this equation, can be identified as special cases of this derived polynomial sequence. Some general properties of this sequence are also given.

Orthogonal polynomials and recurrence equations, operator equations and factorization

This article surveys the classical orthogonal polynomial systems of the Hahn class, which are solutions of second-order differential, difference or q-difference equations. Orthogonal families satisfy three-term recurrence equations. Example applications of an algorithm to determine whether a three-term recurrence equation has solutions in the Hahn class - implemented in the computer algebra system Maple - are given. Modifications of these families, in particular associated orthogonal systems, satisfy fourth-order operator equations. A factorization of these equations leads to a solution basis.

A generic formula for the values at the boundary points of monic classical orthogonal polynomials

In a previous paper we have determined a generic formula for the polynomial solution families of the well-known differential equation of hypergeometric type σ(x)y"n(x)+τ(x)y'n(x)-λnyn(x)=0. In this paper, we give another such formula which enables us to present a generic formula for the values of monic classical orthogonal polynomials at their boundary points of definition.

Some New Classes of Orthogonal Polynomials and Special Functions

In dieser Dissertation präsentieren wir zunächst eine Verallgemeinerung der üblichen Sturm-Liouville-Probleme mit symmetrischen Lösungen und erklären eine umfassendere Klasse. Dann führen wir einige neue Klassen orthogonaler Polynome und spezieller Funktionen ein, welche sich aus dieser symmetrischen Verallgemeinerung ableiten lassen. Als eine spezielle Konsequenz dieser Verallgemeinerung führen wir ein Polynomsystem mit vier freien Parametern ein und zeigen, dass in diesem System fast alle klassischen symmetrischen orthogonalen Polynome wie die Legendrepolynome, die Chebyshevpolynome erster und zweiter Art, die Gegenbauerpolynome, die verallgemeinerten Gegenbauerpolynome, die Hermitepolynome, die verallgemeinerten Hermitepolynome und zwei weitere neue endliche Systeme orthogonaler Polynome enthalten sind. All diese Polynome können direkt durch das neu eingeführte System ausgedrückt werden. Ferner bestimmen wir alle Standardeigenschaften des neuen Systems, insbesondere eine explizite Darstellung, eine Differentialgleichung zweiter Ordnung, eine generische Orthogonalitätsbeziehung sowie eine generische Dreitermrekursion. Außerdem benutzen wir diese Erweiterung, um die assoziierten Legendrefunktionen, welche viele Anwendungen in Physik und Ingenieurwissenschaften haben, zu verallgemeinern, und wir zeigen, dass diese Verallgemeinerung Orthogonalitätseigenschaft und -intervall erhält. In einem weiteren Kapitel der Dissertation studieren wir detailliert die Standardeigenschaften endlicher orthogonaler Polynomsysteme, welche sich aus der üblichen Sturm-Liouville-Theorie ergeben und wir zeigen, dass sie orthogonal bezüglich der Fisherschen F-Verteilung, der inversen Gammaverteilung und der verallgemeinerten t-Verteilung sind. Im nächsten Abschnitt der Dissertation betrachten wir eine vierparametrige Verallgemeinerung der Studentschen t-Verteilung. Wir zeigen, dass diese Verteilung gegen die Normalverteilung konvergiert, wenn die Anzahl der Stichprobe gegen Unendlich strebt. Eine ähnliche Verallgemeinerung der Fisherschen F-Verteilung konvergiert gegen die chi-Quadrat-Verteilung. Ferner führen wir im letzten Abschnitt der Dissertation einige neue Folgen spezieller Funktionen ein, welche Anwendungen bei der Lösung in Kugelkoordinaten der klassischen Potentialgleichung, der Wärmeleitungsgleichung und der Wellengleichung haben. Schließlich erklären wir zwei neue Klassen rationaler orthogonaler hypergeometrischer Funktionen, und wir zeigen unter Benutzung der Fouriertransformation und der Parsevalschen Gleichung, dass es sich um endliche Orthogonalsysteme mit Gewichtsfunktionen vom Gammatyp handelt.