There has been considerable recent interest in geometric function theory, nonlinear partial differential equations, harmonic mappings, and the connection of these to minimal energy phenomena. This work explores Nitsche's 1962 conjecture concerning the nonexistence of harmonic mappings between planar annuli, cast in terms of distortion functionals. The connection between the Nitsche problem and the famous Grötzsch problem is established by means of a weight function. Traditionally, these kinds of problems are investigated in the class of quasiconformal mappings, and the assumption is usually made a priori that solutions preserve various symmetries. Here the conjecture is solved in the much wider class of mappings of finite distortion, symmetry-preservation is proved, and ellipticity of the variational equations concerning these sorts of general problems is established. Furthermore, various alternative interpretations of the weight function introduced herein lead to an interesting analysis of a much wider variety of critical phenomena -- when the weight function is interpreted as a thickness, density or metric, the results lead to a possible model for tearing or breaking phenomena in material science. These physically relevant critical phenomena arise, surprisingly, out of purely theoretical considerations.

This study is centered on childhood music education in Nigeria. Five research questions and five null hypotheses were formulated and tested in the study. A total of 313 subjects were used. The instrument used for data collection was a questionnaire on the childhood music education in Nigeria. Statistically weighted mean was used to answer the research questions and t - test was employed in testing the null hypotheses. At the end of the analysis, the following findings were made: <ul> <li>At least, 13 learning materials including both African and western musical instruments are available for teaching music in Nigerian primary schools. Up to 10 learning strategies were identified by the music teachers and music educators to be among the ones that could be utilized to ensure fruitful and effective acquisition of musical knowledge by the pupils in Nigeria.</li> <li>The perception of music educators and music teachers on the methodology that can best be utilized to ensure fruitful and effective acquisition of musical knowledge by the pupils at primary school level of education in Nigeria do not differ significantly.</li> <li>There is no significant difference between the music teachers and music educators’ perception on the adequacy of delivery of the music staff and learning situations in Nigerian primary schools.</li> </ul> Based on these findings a number of recommendations for improvement of the music education in Nigerian primary schools have been made. In addition to the sample teaching and learning programme, the original contribution of this study is explored in chapter six. This chapter offers lesson plans designed for grade II music teachers in primary schools in Nigeria. This programme was conducted and performed in one of the primary schools in the study area and ended with twelve contacts with pupils. (a digital video disk is attached to the thesis). / Thesis (DMus)--University of Pretoria, 2007. / Music / unrestricted

Les prairies semi-naturelles constituent des espaces multifonctionnels susceptibles de rendre de nombreux services à la société. Ce sont en particulier des agro-écosystèmes favorables à l’expression d’une grande diversité floristique et faunistique permettant de concilier enjeux écologiques et agronomiques. Ce travail de thèse vise à analyser (i) l'influence des facteurs environnementaux sur l'assemblage des communautés in situ et (ii) les relations entre la structure fonctionnelle des assemblages (i.e. valeurs de traits agrégées et diversité fonctionnelle) et leurs performances (i.e. production de biomasse et digestibilité de cette biomasse). Cette étude s'appuie sur des approches expérimentales et de terrain dans les prairies humides pâturées du Marais Poitevin. Nos résultats montrent des réponses très différentes des traits aériens et racinaires étudiés aux gradients de stress et de perturbation rencontrés dans ces prairies. Ils ont également permis de mettre en évidence l'influence prépondérante des valeurs de traits agrégées sur la fourniture de services écosystémiques (i.e. productivité et digestibilité du fourrage), en accord avec la "Biomass ratio hypothesis" énoncée par Grime. L'influence du niveau de diversité fonctionnelle sur les services rendus par les assemblages varie selon le service considéré. Contrairement aux résultats attendus selon la "Diversity hypothesis" énoncée par Tilman, le niveau de diversité fonctionnelle est lié négativement à la productivité des assemblages. En revanche, un effet positif de la diversité fonctionnelle sur la digestibilité a été mis en évidence. Cette étude a par ailleurs permis de montrer un compromis entre la productivité des assemblages et la digestibilité de la biomasse produite dans ces prairies humides. Ce travail suggère que des outils de diagnostics de la valeur fourragère du couvert peuvent être construits sur la base de la structure fonctionnelle des assemblages, et ce avec un haut niveau de fiabilité. Il met également en avant les bénéfices agronomiques associés à l'hétérogénéité des couverts végétaux rencontrés dans ces prairies. / Semi-natural grasslands form multifunctional areas susceptible to serve the society from many aspects. Indeed, they are agro-ecosystems that promote expression of a large floristic and faunistic diversity, which allows conciliating ecologic and agronomic issues. This PhD work aims at analysing (i) influence of environmental factors on natural communities' assembly rules and (ii) the relationships between the functional structure of assemblages (i.e. community weighted mean trait values and functional diversity) and their performances (i.e. biomass production and its digestibility). This study is based on both experimental and field approaches in the Marais Poitevin wet grasslands. Our results show that the studied aerial and root traits display a wide diversity of answers to stress gradients and disturbances encountered in these grasslands. Our results also highlighted the crucial influence of community weighted mean trait values on provision of ecosystem services (i.e. forage productivity and digestibility), in agreement with the Biomass ratio hypothesis set out by Grime. The influence of the functional diversity level on services provided by the assemblages differs depending on the service considered. Contrary to the results expected according to the Diversity hypothesis of Tilman, the functional diversity level is negatively correlated with the productivity of the assemblages. However, our study points to a positive feedback between functional diversity and digestibility. Furthermore, this study demonstrated the existence of trade-off between the productivity of assemblages and the digestibility of biomass produced in these wet grasslands. This work suggests that highly reliable tools can be established, based on the functional structure of assemblages, to evaluate the feeding value of a vegetal cover. This work finally illustrates the agronomic benefits associated with heterogeneous vegetal covers encountered in these grasslands.

The field of statistics is becoming increasingly more important as the amount of data in the world grows. This thesis studies the Growth Curve model in multivariate statistics which is a model that is not widely used. One difference compared with the linear model is that the Maximum Likelihood Estimators are more complicated. That makes it more difficult to use and to interpret which may be a reason for its not so widespread use. From this perspective this thesis will compare the traditional mean estimator for the Growth Curve model with the unweighted mean estimator. The unweighted mean estimator is simpler than the regular MLE. It will be proven that the unweighted estimator is in fact the MLE under certain conditions and examples when this occurs will be discussed. In a more general setting this thesis will present conditions when the un-weighted estimator has a smaller covariance matrix than the MLEs and also present confidence intervals and hypothesis testing based on these inequalities.