Map of the Lower Geyser Basin of the Upper Madison River

Kartta kuuluu A. E. Nordenskiöldin kokoelmaan

Quasiconformal mappings and inequalities involving special functions

This PhD thesis in Mathematics belongs to the field of Geometric Function Theory. The thesis consists of four original papers. The topic studied deals with quasiconformal mappings and their distortion theory in Euclidean n-dimensional spaces. This theory has its roots in the pioneering papers of F. W. Gehring and J. Väisälä published in the early 1960’s and it has been studied by many mathematicians thereafter. In the first paper we refine the known bounds for the so-called Mori constant and also estimate the distortion in the hyperbolic metric. The second paper deals with radial functions which are simple examples of quasiconformal mappings. These radial functions lead us to the study of the so-called p-angular distance which has been studied recently e.g. by L. Maligranda and S. Dragomir. In the third paper we study a class of functions of a real variable studied by P. Lindqvist in an influential paper. This leads one to study parametrized analogues of classical trigonometric and hyperbolic functions which for the parameter value p = 2 coincide with the classical functions. Gaussian hypergeometric functions have an important role in the study of these special functions. Several new inequalities and identities involving p-analogues of these functions are also given. In the fourth paper we study the generalized complete elliptic integrals, modular functions and some related functions. We find the upper and lower bounds of these functions, and those bounds are given in a simple form. This theory has a long history which goes back two centuries and includes names such as A. M. Legendre, C. Jacobi, C. F. Gauss. Modular functions also occur in the study of quasiconformal mappings. Conformal invariants, such as the modulus of a curve family, are often applied in quasiconformal mapping theory. The invariants can be sometimes expressed in terms of special conformal mappings. This fact explains why special functions often occur in this theory.

Test Anxiety, Coping, Gender and Academic Performance in English Exams: A Study of Finnish Upper Secondary School Students

In this MA thesis, test anxiety related to English exams among Finnish upper secondary school students was studied. In addition, the ways students try to cope with test anxiety were investigated. The purpose of the study was to investigate gender differences in test anxiety, the effects of test anxiety on academic performance and relationships between test anxiety, academic performance and coping strategies. Test anxiety and coping strategies were analysed as scores of questionnaire responses. Coping strategies comprised of three categories – task-orientation and preparation, seeking social support and avoidance. Academic performance was analysed as teacher ratings of general performance in English exams. In total 67 subjects were studied. The subjects were Finnish general upper secondary school students. The data were collected by using online questionnaires. This data were mainly quantitative, but also qualitative elements were included. The quantitative data were analysed by using statistical methods. The results showed that females experienced statistically significantly more test anxiety than males. In addition, a statistically significant correlation was found between test anxiety levels and academic performance ratings of the subjects: the higher the test anxiety score, the lower the academic performance rating. A meaningful correlation was found between test anxiety and seeking social support as a coping strategy: a higher test anxiety score was related to using social support as a coping strategy. However, no relationships were found between academic performance and the three coping strategies when quantitative and qualitative data were analysed. Therefore, different coping strategies per se did not seem to be related to academic performance, but instead it was assumed that the effectiveness of coping strategies is dependent on individual differences. In order to obtain more generalisable results and to gain more understanding of test anxiety and coping with it, a larger number of subjects form different areas of Finland and of different ages could be examined in future studies. Moreover, cross-national and cross-cultural studies could provide valuable information. As a practical recommendation for educational purposes, the results of this study indicated that a more individualised approach is needed.