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.

Competence in intensive and critical care nursing - development of a basic assessment scale for graduating nursing students

Intensive and critical care nursing is a speciality in its own right and with its own nature within the nursing profession. This speciality poses its own demands for nursing competencies. Intensive and critical care nursing is focused on severely ill patients and their significant others. The patients are comprehensively cared for, constantly monitored and their vital functions are sustained artificially. The main goal is to win time to cure the cause of the patient’s situation or illness. The purpose of this empirical study was i) to describe and define competence and competence requirements in intensive and critical care nursing, ii) to develop a basic measurement scale for competence assessment in intensive and critical care nursing for graduating nursing students, and iii) to describe and evaluate graduating nursing students’ basic competence in intensive and critical care nursing by seeking the reference basis of self-evaluated basic competence in intensive and critical care nursing from ICU nurses. However, the main focus of this study was on the outcomes of nursing education in this nursing speciality. The study was carried out in different phases: basic exploration of competence (phase 1 and 2), instrumentation of competence (phase 3) and evaluation of competence (phase 4). Phase 1 (n=130) evaluated graduating nursing students’ basic biological and physiological knowledge and skills for working in intensive and critical care with Basic Knowledge Assessment Tool version 5 (BKAT-5, Toth 2012). Phase 2 focused on defining competence in intensive and critical care nursing with the help of literature review (n=45 empirical studies) as well as competence requirements in intensive and critical care nursing with the help of experts (n=45 experts) in a Delphi study. In phase 3 the scale Intensive and Critical Care Nursing Competence Scale (ICCN-CS) was developed and tested twice (pilot test 1: n=18 students and n=12 nurses; pilot test 2: n=56 students and n=54 nurses). Finally, in phase 4, graduating nursing students’ competence was evaluated with ICCN-CS and BKAT version 7 (Toth 2012). In order to develop a valid assessment scale of competence for graduating nursing students and to evaluate and establish the competence of graduating nursing students, empirical data were retrieved at the same time from both graduating nursing students (n=139) and ICU nurses (n=431). Competence can be divided into clinical and general professional competence. It can be defined as a specific knowledge base, skill base, attitude and value base and experience base of nursing and the personal base of an intensive and critical care nurse. Personal base was excluded in this self-evaluation based scale. The ICCN-CS-1 consists of 144 items (6 sum variables). Finally, it became evident that the experience base of competence is not a suitable sum variable in holistic intensive and critical care competence scale for graduating nursing students because of their minor experience in this special nursing area. ICCN-CS-1 is a reliable and tolerably valid scale for use among graduating nursing students and ICU nurses Among students, basic competence of intensive and critical care nursing was self-rated as good by 69%, as excellent by 25% and as moderate by 6%. However, graduating nursing students’ basic biological and physiological knowledge and skills for working in intensive and critical care were poor. The students rated their clinical and professional competence as good, and their knowledge base and skill base as moderate. They gave slightly higher ratings for their knowledge base than skill base. Differences in basic competence emerged between graduating nursing students and ICU nurses. The students’ self-ratings of both their basic competence and clinical and professional competence were significantly lower than the nurses’ ratings. The students’ self-ratings of their knowledge and skill base were also statistically significantly lower than nurses’ ratings. However, both groups reported the same attitude and value base, which was excellent. The strongest factor explaining students’ conception of their competence was their experience of autonomy in nursing. Conclusions: Competence in intensive and critical care nursing is a multidimensional concept. Basic competence in intensive and critical care nursing can be measured with self-evaluation based scale but alongside should be used an objective evaluation method. Graduating nursing students’ basic competence in intensive and critical care nursing is good but their knowledge and skill base are moderate. Especially the biological and physiological knowledge base is poor. Therefore in future in intensive and critical care nursing education should be focused on both strengthening students’ biological and physiological knowledge base and on strengthening their overall skill base. Practical implications are presented for nursing education, practice and administration. In future, research should focus on education methods and contents, mentoring of clinical practice and orientation programmes as well as further development of the scale.

Excessive functions, Appell polynomials and optimal stopping

The main topic of the thesis is optimal stopping. This is treated in two research articles. In the first article we introduce a new approach to optimal stopping of general strong Markov processes. The approach is based on the representation of excessive functions as expected suprema. We present a variety of examples, in particular, the Novikov-Shiryaev problem for Lévy processes. In the second article on optimal stopping we focus on differentiability of excessive functions of diffusions and apply these results to study the validity of the principle of smooth fit. As an example we discuss optimal stopping of sticky Brownian motion. The third research article offers a survey like discussion on Appell polynomials. The crucial role of Appell polynomials in optimal stopping of Lévy processes was noticed by Novikov and Shiryaev. They described the optimal rule in a large class of problems via these polynomials. We exploit the probabilistic approach to Appell polynomials and show that many classical results are obtained with ease in this framework. In the fourth article we derive a new relationship between the generalized Bernoulli polynomials and the generalized Euler polynomials.

Implementation of educational change in Finland: The case of the undivided basic education reform

For the past decades, educational large-scale reforms have been elaborated and implemented in many countries and often resulted in partial or complete failure. These results brought researchers to study policy processes in order to address this particular challenge. Studies on implementation processes brought to light an existing causal relationship between the implementation process and the effectiveness of a reform. This study aims to describe the implementation process of educational change in Finland, who produced efficient educational reforms over the last 50 years. The case study used for the purpose of this study is the national reform of undivided basic education (yhtenäinen peruskoulu) implemented in the end of the 1990s. Therefore, this research aims to describe how the Finnish undivided basic education reform was implemented. This research was carried out using a pluralist and structuralist approach of policy process and was analyzed according to the hybrid model of implementation process. The data were collected using a triangulation of methods, i.e. documentary research, interviews and questionnaires. The data were qualitative and were analyzed using content analysis methods. This study concludes that the undivided basic education reform was applied in a very decentralized manner, which is a reflection of the decentralized system present in Finland. Central authorities provided a clear vision of the purpose of the reform, but did not control the implementation process. They rather provided extensive support in the form of transmission of information and development of collaborative networks. Local authorities had complete autonomy in terms of decision-making and implementation process. Discussions, debates and decisions regarding implementation processes took place at the local level and included the participation of all actors present on the field. Implementation methods differ from a region to another, with is the consequence of the variation of the level of commitment of local actors but also the diversity of local realities. The reform was implemented according to existing structures and values, which means that it was in cohesion with the context in which it was implemented. These results cannot be generalized to all implementation processes of educational change in Finland but give a great insight of what could be the model used in Finland. Future studies could intent to confirm the model described here by studying other reforms that took place in Finland.

Implementation of educational change in Finland: The case of the undivided basic education reform

For the past decades, educational large-scale reforms have been elaborated and implemented in many countries and often resulted in partial or complete failure. These results brought researchers to study policy processes in order to address this particular challenge. Studies on implementation processes brought to light an existing causal relationship between the implementation process and the effectiveness of a reform. This study aims to describe the implementation process of educational change in Finland, who produced efficient educational reforms over the last 50 years. The case study used for the purpose of this study is the national reform of undivided basic education (yhtenäinen peruskoulu) implemented in the end of the 1990s. Therefore, this research aims to describe how the Finnish undivided basic education reform was implemented. This research was carried out using a pluralist and structuralist approach of policy process and was analyzed according to the hybrid model of implementation process. The data were collected using a triangulation of methods, i.e. documentary research, interviews and questionnaires. The data were qualitative and were analyzed using content analysis methods. This study concludes that the undivided basic education reform was applied in a very decentralized manner, which is a reflection of the decentralized system present in Finland. Central authorities provided a clear vision of the purpose of the reform, but did not control the implementation process. They rather provided extensive support in the form of transmission of information and development of collaborative networks. Local authorities had complete autonomy in terms of decision-making and implementation process. Discussions, debates and decisions regarding implementation processes took place at the local level and included the participation of all actors present on the field. Implementation methods differ from a region to another, with is the consequence of the variation of the level of commitment of local actors but also the diversity of local realities. The reform was implemented according to existing structures and values, which means that it was in cohesion with the context in which it was implemented. These results cannot be generalized to all implementation processes of educational change in Finland but give a great insight of what could be the model used in Finland. Future studies could intent to confirm the model described here by studying other reforms that took place in Finland.