The main aim of this thesis has been to investigate the distinction between a nonresonant and resonant Hopf bifurcation from relative equilibria. Resonant and nonresonant Hopf bifurcations from relative equilibria posed in two spatial dimensions, in systems with Euclidean SE(2) symmetry, have been extensively studied in the context of spiral waves in a plane and are now well understood. We initially investigate Hopf bifurcations from relative equilibria posed in systems with compact S0(3) symmetry where S0(3) is the group of rotations in three dimensions/on a sphere. Unlike the SE(2) case the skew product equations cannot be solved directly and we use the normal form theory due to Fiedler and Turaev to simplify these systems. We show that the normal form theory resolves the nonresonant case, but not the resonant case. New methods developed in this thesis combined with the normal form theory resolves the resonant case. We find that the resonant Hopf bifurcation produces a motion which is strikingly different from the nonresonant case. By considering the geometric properties of the underlying relative equilibrium we give a definition of resonance directly related to the Hopf bifurcation phenomena. This yields conditions for the occurrence of a resonant Hopf bifurcation from a relative equilibrium in a system with general compact symmetry. By extending the approach used to resolve the S0(3) case we then solve the skew product equations for a general compact symmetry group. The specific case of a Hopf bifurcation from a relative equilibria with symmetry of an irregular dodecahedron is also considered. Furthermore we look at nonresonant and resonant Hopf Bifurcations from relative equilibria posed in a system with noncompact Euclidean symmetry in three spatial dimensions.

Le but dans cette thèse est d'expliciter les liens entre les propriétés analytiques, géométriques et topologiques des variétés compactes de dimension n supérieure ou égale à 3 et le comportement des valeurs propres de l'opérateur de Yamabe. On commence par étudier les propriétés de ces valeurs propres : l'un des remarques principales est que leur signe est invariant par un changement conforme de métriques. On s'intéresse plus particulièrement à la deuxième valeur propre de l'opérateur de Yamabe et on fait le lien entre son signe et l'existence des solutions nodales de l'équation de Yamabe. Pour finir, nous donnons une formule de chirurgie pour le second invariant de Yamabe, qui nous permet d'en obtenir une borne inférieure sous certaines hypothèses topologiques / The goal of this thesis is to study the relationships between the analytical, geometrical and topological properties of compact manifolds of dimension n greater or equal to 3 and the behavior of the eigenvalues of the Yamabe operator. We start by studying the properties of these eigenvalues : One of the most important observation is that their sign is a conformal invariant. We are interested particulary by the study of the seconf eigenvalue of the Yamabe operator and we enlight the relations between its sign and the existence of nodal solutions of the Yamabe equation. At last, we etablish a surgery formula for the second Yamabe invariant which allows to obtain a lower bond for the second Yamabe invariant under some topological hypothesis

Στην παρούσα διδακτορική διατριβή μελετάμε τρία Προβλήματα που αναφέρονται στην Ψευδο-Ευκλείδεια Γεωμετρία. Στα δύο πρώτα Κεφάλαια, Κεφάλαιο 1 και Κεφάλαιο 2 αναφέρουμε γνωστά αποτελέσματα και περιγράφουμε βασικές έννοιες της Ρημάννιας και Ψευδό - Ρημάννιας Γεωμετρίας. Στο Κεφάλαιο 3 μελετάμε επιφάνειες εκ περιστροφής στον τρισδιάστατο Lorentz - Minkowski χώρο ικανοποιώντας δοσμένη γεωμετρική συνθήκη. Στο Κεφάλαιο 4 βρίσκουμε όλες τις κανονικές μορφές του τελεστή σχήματος των τρισδιάστατων υπερεπιφανειών τύπου (-, +, -) του τετρασδιάστατου Ψευδο - Ευκλείδειου χώρου τύπου (-, +, -, +). Τέλος, στο Κεφάλαιο 5 μελετάμε τη σχέση που υπάρχει μεταξύ των διαρμονικών και ελαχιστικών υπερεπιφανειών που αναφέρθηκαν στο Κεφάλαιο 4, χρησιμοποιώντας τον τελεστή σχήματός τους. Ειδικότερα, αποδεικνύουμε ότι κάθε τέτοια διαρμονική υπερεπιφάνεια είναι ελαχιστική. / In the present PH.D. thesis we study three problems referred in the pseudo-Euclidean geometry. In the first two chapters, Chapter 1 and Chapter 2, we review known results and describe the basic notions of the Riemannian and Pseudo-Riemannian geometry. In Chapter 3, we study surfaces of revolution of the three dimensional Lorentz-Minkowski space satisfying given geometric condition. In Chapter 4, we find all the canonical forms of the shape operator of the three dimensional hypersurfaces of signature (-, +, -) of the four dimensional pseudo-Euclidean space of signature (-, +, -, +). Finally, in Chapter 5, we study the relation which exists between the biharmonic and minimal hypersurfaces referred in Chapter 4, by using their shape operator. Precisely, we prove that every such biharmonic hypersurface is minimal.

This study investigates the use of telecentres for educational purposes in telecommunications under-serviced regions of South Africa. The problem addressed by this research has both equity and efficiency aspects. In particular, the thesis examines the impact of telecentres on the formal and non-formal education outcomes of recipient communities - four telecentres, two in townships and two in rural areas. The urban telecentres are Siyabonga in Orange Farm near Johannesburg and Mamelodi Communication and Information Services (MACIS), in Mamelodi township, near Pretoria in Gauteng Province. The rural telecentres are Tombo near Port St Johns in the Eastern Cape Province and Hoxani near Bushbuckridge on the Mpumalanga-Limpopo Province border. The main aim of the research is to conduct a quantitative survey of the telecentre users’ use of telecentres in the above mentioned locations. On the basis of this, the thesis seeks to come to some conclusions about the use and effectiveness of the South African telecentre program. The quantitative analysis of users is supported by a qualitative report and analysis of data gathered through personal interviews of telecentre operators and employees. The thesis reports on the outputs of the centres, limitations in their function, inhibitors to their economic performance, and recommendations for improving their operations. Some of the findings are that telecentres established through public-private partnership (PPP) funding are more effective and successful, computer literacy is a major resource offered, Hoxani telecentre offers outcomes-based education management skills training for local teachers, while Tombo, MACIS and Siyabonga telecentres offer additional business and electronic courses. The thesis also reviews the South African telecentre program against the background of similar programs in selected Asian and Latin American countries, Australia, Canada, and Europe and against the background of an analysis of South African post-apartheid telecommunications sector reforms (1996 to 2007). Insights into the unique nature of challenges facing geographically located telecentres as well as a new model for understanding telecentre operations in South Africa are offered. / Communication Science / D.Litt. et Phil.(Communication Science)

This study investigates the use of telecentres for educational purposes in telecommunications under-serviced regions of South Africa. The problem addressed by this research has both equity and efficiency aspects. In particular, the thesis examines the impact of telecentres on the formal and non-formal education outcomes of recipient communities - four telecentres, two in townships and two in rural areas. The urban telecentres are Siyabonga in Orange Farm near Johannesburg and Mamelodi Communication and Information Services (MACIS), in Mamelodi township, near Pretoria in Gauteng Province. The rural telecentres are Tombo near Port St Johns in the Eastern Cape Province and Hoxani near Bushbuckridge on the Mpumalanga-Limpopo Province border. The main aim of the research is to conduct a quantitative survey of the telecentre users’ use of telecentres in the above mentioned locations. On the basis of this, the thesis seeks to come to some conclusions about the use and effectiveness of the South African telecentre program. The quantitative analysis of users is supported by a qualitative report and analysis of data gathered through personal interviews of telecentre operators and employees. The thesis reports on the outputs of the centres, limitations in their function, inhibitors to their economic performance, and recommendations for improving their operations. Some of the findings are that telecentres established through public-private partnership (PPP) funding are more effective and successful, computer literacy is a major resource offered, Hoxani telecentre offers outcomes-based education management skills training for local teachers, while Tombo, MACIS and Siyabonga telecentres offer additional business and electronic courses. The thesis also reviews the South African telecentre program against the background of similar programs in selected Asian and Latin American countries, Australia, Canada, and Europe and against the background of an analysis of South African post-apartheid telecommunications sector reforms (1996 to 2007). Insights into the unique nature of challenges facing geographically located telecentres as well as a new model for understanding telecentre operations in South Africa are offered. / Communication Science / D.Litt. et Phil.(Communication Science)