Inverse problem for the wave equation : partial data and novel boundary sources

An inverse problem for the wave equation is a mathematical formulation of the problem to convert measurements of sound waves to information about the wave speed governing the propagation of the waves. This doctoral thesis extends the theory on the inverse problems for the wave equation in cases with partial measurement data and also considers detection of discontinuous interfaces in the wave speed. A possible application of the theory is obstetric sonography in which ultrasound measurements are transformed into an image of the fetus in its mother's uterus. The wave speed inside the body can not be directly observed but sound waves can be produced outside the body and their echoes from the body can be recorded. The present work contains five research articles. In the first and the fifth articles we show that it is possible to determine the wave speed uniquely by using far apart sound sources and receivers. This extends a previously known result which requires the sound waves to be produced and recorded in the same place. Our result is motivated by a possible application to reflection seismology which seeks to create an image of the Earth s crust from recording of echoes stimulated for example by explosions. For this purpose, the receivers can not typically lie near the powerful sound sources. In the second article we present a sound source that allows us to recover many essential features of the wave speed from the echo produced by the source. Moreover, these features are known to determine the wave speed under certain geometric assumptions. Previously known results permitted the same features to be recovered only by sequential measurement of echoes produced by multiple different sources. The reduced number of measurements could increase the number possible applications of acoustic probing. In the third and fourth articles we develop an acoustic probing method to locate discontinuous interfaces in the wave speed. These interfaces typically correspond to interfaces between different materials and their locations are of interest in many applications. There are many previous approaches to this problem but none of them exploits sound sources varying freely in time. Our use of more variable sources could allow more robust implementation of the probing.

Paradoxical reflection in quantum mechanics

This article concerns a phenomenon of elementary quantum mechanics that is quite counter-intuitive, very non-classical, and apparently not widely known: a quantum particle can get reflected at a downward potential step. In contrast, classical particles get reflected only at upward steps. The conditions for this effect are that the wave length is much greater than the width of the potential step and the kinetic energy of the particle is much smaller than the depth of the potential step. This phenomenon is suggested by non-normalizable solutions to the time-independent Schroedinger equation, and we present evidence, numerical and mathematical, that it is also indeed predicted by the time-dependent Schroedinger equation. Furthermore, this paradoxical reflection effect suggests, and we confirm mathematically, that a quantum particle can be trapped for a long time (though not forever) in a region surrounded by downward potential steps, that is, on a plateau.

Widening Perspectives of Teacher Education : Studies on theory-practice relationship, reflection, research and professional development

This dissertation examined the research-based teacher education at the University of Helsinki from different theoretical and practical perspectives. Five studies focused on these perspectives separately as well as overlappingly. Study I focused on the reflection process of graduating teacher students. The data consisted of essays the students wrote as their last assignment before graduating, where their assignment was to examine their development as researchers during their MA thesis research process. The results indicated that the teacher students had analysed their own development thoroughly during the process and that they had reflected on theoretical as well as practical educational matters. The results also pointed out that, in the students’ opinion, personally conducted research is a significant learning process. -- Study II investigated teacher students’ workplace learning and the integration of theory and practice in teacher education. The students’ interviews focused on their learning of teacher’s work prior to education. The interviewees’ responses concerning their ‘surviving’ in teaching prior to teacher education were categorized into three categories: learning through experiences, school as a teacher learning environment, and case-specific learning. The survey part of the study focused on integration of theory and practice within the education process. The results showed that the students who worked while they studied took advantage of the studies and applied them to work. They set more demanding teaching goals and reflected on their work more theoretically. -- Study III examined practical aspects of the teacher students’ MA thesis research as well as the integration of theory and practice in teacher education. The participants were surveyed using a web-based survey which dealt with the participants’ teacher education experiences. According to the results, most of the students had chosen a practical topic for their MA thesis, one arising from their work environment, and most had chosen a research topic that would develop their own teaching. The results showed that the integration of theory and practice had taken place in much of the course work, but most obviously in the practicum periods, and also in the courses concerning the school subjects. The majority felt that the education had in some way been successful with regards to integration. -- Study IV explored the idea of considering teacher students’ MA thesis research as professional development. Twenty-three teachers were interviewed on the subject of their experiences of conducting research about their own work as teachers. The results of the interviews showed that the reasons for choosing the MA thesis research topic were multiple: practical, theoretical, personal, professional reasons, as well as outside effect. The objectives of the MA thesis research, besides graduating, were actual projects, developing the ability to work as teachers, conducting significant research, and sharing knowledge of the topic. The results indicated that an MA thesis can function as a tool for professional development, for example in finding ways for adjusting teaching, increasing interaction skills, gaining knowledge or improving reflection on theory and/or practice, strengthening self-confidence as a teacher, increasing researching skills or academic writing skills, as well as becoming critical and being able to read scientific and academic literature. -- Study V analysed teachers’ views of the impact of practitioner research. According to the results, the interviewees considered the benefits of practitioner research to be many, affecting teachers, pupils, parents, the working community, and the wider society. Most of the teachers indicated that they intended to continue to conduct research in the future. The results also showed that teachers often reflected personally and collectively, and viewed this as important. -- These five studies point out that MA thesis research is and can be a useful tool for increasing reflection doing with personal and professional development, as well as integrating theory and practice. The studies suggest that more advantage could be taken of the MA thesis research project. More integration of working and studying could and should be made possible for teacher students. This could be done in various ways within teacher education, but the MA thesis should be seen as a pedagogical possibility.

Theorizing the Nature of Reflection

For the past decades reflection has been the buzzword of adult and higher education. Reflection is facilitated in many practices and there is abundant research on the issue. Despite the popularity of the concept, the reasons why bringing about reflection in educational practices is difficult remain unclear. The prevailing theories inform of the process in its ideal form. However, to a great extent, they fail to offer conceptual tools for understanding and working with the actualities of reflection. The aim of the doctoral thesis was to explore the challenges and prerequisites of reflection in order to theorize the nature of reflection. By the term reflection it is here referred to becoming aware of and questioning the assumptions that orient our thinking, feelings and actions. The doctoral thesis consists of five studies that approach these questions from different viewpoints and within different contexts. The methods involve both a philosophical and an empirical approach. This multifaceted approach embodies the aim of both gaining a more thorough grasp of the phenomenon and to develop the methodology of researching reflection. The theory building is based on conceptual analysis and rational reconstruction (see Davia 1998; Habermas 1979; Rorty1984) of Mezirow s (1981; 1991; 2000; 2009) theory of transformative learning. In order to explore the aspects which, based on the analysis, appeared insufficiently considered within Mezirow s theory, Damasio s (1994; 1999; 2003; 2010) theory on emotions and consciousness as well as Clausewitz s (1985) view on friction are used as complementary theories. Empirical analyses are used in dialogue with the theoretical, in order to challenge and refine the emerging theorization. Reflection is examined in three different contexts; regarding university teachers pedagogical growth, involuntarily childless women recovering from a life-event crisis, and soldiers preparing to act in chaotic situations of the battlefield as well as recovering from it. The choice of these contexts is based on Mezirow s notion of disorienting dilemma as a trigger for reflection. This notion indicates that reflection may more naturally emerge in association to life-event crises or other cumulative sets of instances, which bring our worldview and beliefs under question. Nevertheless, reflection is often being promoted in educational contexts in which the trigger conditions may not readily prevail. These contextual issues as well as the differences between the facilitated and non-facilitated contexts have not, however, been considered in detail within the research on reflection (or transformative learning). The doctoral thesis offers a new perspective into reflection which, as a further development on Mezirow s transformative learning theory, theorizes the nature of reflection. The developed theory explicates the prerequisites and challenges to reflection. The theory suggests that the challenges of reflection are fundamentally connected to the way the biological life-support system affects our thinking through emotions. While depicting the mechanisms that function as a counterforce to reflection, the developed theory also opens a perspective for considering possibilities for carrying out reflection, and suggests ways to locate and deal with the assumptions to be reflected on. The basic dynamic of the challenges to reflection was explicated by conceptually bridging the gap between Mezirow s and Damasio s theories, through exploring the connections between the meaning perspective and the biological functions of emotions. The concepts of comfort zone and edge-emotions were formed so as to depict the emotional orientation of our thinking, as part of the explanation of the nature of reflection.