This thesis describes an Action Research project within the researcher's practice as a teacher educator in Rwanda. A teaching style informed by the Theory of Didactical Situations in Mathematics (Artigue, 1994; Brousseau, 1997; 2004; Douady, 1991) and by the Zone of Proximal Development (Gallimore & Tharp, 1990; Meira & Lerman, 2001; Rowlands, 2003; Vygotsky, 1978) was conducted with first-year mathematics student teachers in Rwanda. The aim of the teaching model was to develop the student teachers' understanding of the concepts of the definite and the indefinite integrals and their link through the fundamental theorem of calculus. The findings of the analysis answer the research questions, on the one hand, of what concept images (Tall & Vinner, 1981; Vinner & Dreyfus, 1989) of the underlying concepts of integrals student teachers exhibit, and how the student teachers‟ concept images evolved during the teaching. On the other hand, the findings answer the research questions of what didactical situations are likely to further student teachers' understanding of the definite and the indefinite integrals and their link through the fundamental theorem of calculus; and finally they answer the question of what learning activities student teachers engage in when dealing with integrals and under what circumstances understanding is furthered. An analysis of student teachers' responses expressed during semi-structured interviews organised at three different points of time - before, during, and after the teaching - shows that the student teachers' evoked concept images evolved significantly from pseudo-objects of the definite and the indefinite integrals to include almost all the underlying concept layers of the definite integral, namely, the partition, the product, the sum, and the limit of a sum, especially in the symbolical representation. However, only a limited evolution of the student teachers' understanding of the fundamental theorem of calculus was demonstrated after completion of the teaching. With regard to the teaching methods, after analysis of the video recordings of the lessons, I identified nine main didactical episodes which occurred during the teaching. Interactions during these episodes contributed to the development of the student teachers' understanding of the concepts of the definite and the indefinite integrals and their link through the fundamental theorem of calculus. During these interactions, the student teachers were engaged in various cognitive processes which were purposefully framed by functions of communication, mainly the referential function, the expressive function, and the cognative function. In these forms of communication, the cognative function in which I asked questions and instructed the students to participate in interaction was predominant. The student teachers also reacted by using mainly the expressive and the referential functions to indicate what knowledge they were producing. In these exchanges between the teacher and the student teachers and among the student teachers themselves, two didactical episodes in which two student teachers overtly expressed their understanding have been observed. The analysis of these didactical episodes shows that the first student teacher's understanding has been triggered by a question that I addressed to the student after a long trial and error of searching for a mistake, whereas the second student's understanding was activated by an indicative answer given by another student to the question of the student who expressed the understanding. In the former case, the student exhibited what he had understood while in the latter case the student did not. This suggests that during interactions between a teacher and a student, asking questions further the student's understanding more than providing him or her with the information to be learnt. Finally, during this study, I gained the awareness that the teacher in a mathematics classroom has to have various decisional, organisational and managerial skills and adapt them to the circumstances that emerge during classroom activities and according to the evolution of the knowledge being learned. Also, the study showed me that in most of the time the student teachers were at the center of the activities which I organised in the classroom. Therefore, the teaching methods that I used during my teaching can assist in the process of changing from a teacher-centred style of teaching towards a student-centred style. This study contributed to the field of mathematics education by providing a mathematical framework which can be used by other researchers to analyse students' understanding of integrals. This study also contributed in providing a model of teaching integrals and of researching a mathematics (integrals) classroom which indicates episodes in which understanding may occur. This study finally contributed to my professional development as a teacher educator and a researcher. I practiced the theory of didactical situation in mathematics. I experienced the implementation of some of its concepts such as the devolution, the a didactical situation, the institutionalization, and the didactical contract and how this can be broken by students (the case of Edmond). In this case of Edmond, I realised that my listening to students needs to be improved. As a researcher, I learnt a lot about theoretical frameworks, paradigms of study and analysis and interpretation of data. The theory of didactical situations in mathematics, the action research cyclical spiral, and the revised Bloom‟s Taxonomy will remain at my hand reach during my mathematics teacher educator career. However, there is still a need to improve in the analysis of data especially from the students' standpoint; that is, the analysis of the learning aspect needs to be more practiced and improved. / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2010.

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Problems of understanding fundamental calculus concepts by students in tertiary education colleges and universities are evidenced by a body of research studies conducted in different parts of the world. The researchers have identified, classified and analysed these problems from historical, epistemological, and learning theory perspectives. History is important because mathematical concepts are a result of the developments of the past. The way knowledge is acquired is an epistemological issue and the major purpose of learning is to acquire knowledge. Hence, these three perspectives qualify to be used as lenses in understanding problems that students encounter in a learning situation. The purpose of this study was to investigate the epistemological obstacles that mathematics students at undergraduate level encounter in coming to understand the limit concept. The role played by language and symbolism in understanding the limit concept was also investigated, because communication in the mathematics classroom takes place by using language and symbols. / Philosophiae Doctor - PhD

Three languages with polymorphic type disciplines are discussed, namely the λ-calculus with Milner's polymorphic type discipline; a language with imperative features (polymorphic references); and a skeletal module language with structures, signatures and functors. In each of the two first cases we show that the type inference system is consistent with an operational dynamic semantics. On the module level, polymorphic types correspond to signatures. There is a notion of principal signature. So-called signature checking is the module level equivalent of type checking. In particular, there exists an algorithm which either fails or produces a principal signature.

Previous work on modal analysis of mooring lines has been performed from different theoretical formulations. Most studies have focused on mooring lines of a single homogeneous material, and the effect of added mass and damping produced by the water has not been examined deeply. The variational formulation approach, employed in this research to perform a modal analysis, has been useful to study the behavior of several realistic mooring lines. The cases presented are composed from segments of materials with different mechanical characteristics, more similar to those in current offshore projects. In the newly proposed formulation, damping produced by transverse motion of the mooring line through the surrounding water has been added to the modal analysis. The modal analysis formulation applied in this work has been verified with calculations from commercial software and the results are sufficiently accurate to understand the global behavior of the dynamics of mooring lines with the damping produced by the sea water. Inclusion of linearized drag damping in the modal analysis showed that the modal periods of the mooring systems studied depend on the amplitude of the transverse motion of the mooring line. When more amplitude in the motion is expected more damping is obtained. Two realistic designs of mooring lines were compared: one made up with a main insert of steel rope, called “Steel System”, and one composed by a main insert of polyester, named “Polyester System”. Comparing the natural periods of both systems, the Steel System appears to be safer because its fundamental natural period is more distant from the wave excitation periods produced by storms. The same happens considering the wave excitation periods produced by prevailing seas. In this case the natural periods of the Polyester System are nearer to the wave excitation periods causing fatigue loads. The transverse mode shapes for lateral motions of the mooring lines are observed to be continuous and smooth across material transitions, such as transitions between chain and wire rope and transitions between chain and polyester rope. This behavior is not always observed in the tangential mode shapes for the Polyester System where significant differences in dynamic tension seem to be present in the specific cases studied.

Although modern society is critically reliant on power grids, even modern power grids are subject to unavoidable outages due to storms, lightning strikes, and equipment failures. The situation in developing countries is even worse, with frequent load shedding lasting several hours a day due to unreliable generation. We study the use of battery storage to allow a set of homes in a single residential neighbour- hood to avoid power outages. Due to the high cost of storage, our goal is to choose the smallest battery size such that, with high target probability, there is no loss of power despite a grid out- age. Recognizing that the most common approach today for mitigating outages is to use a diesel generator (genset), we study the related problem of minimizing the carbon footprint of genset operation. Drawing on recent results, we model both problems as buffer sizing problems that can be ad- dressed using stochastic network calculus. We show that this approach greatly improves battery sizing in contrast to prior approaches. Specifically, a numerical study shows that, for a neigh- bourhood of 100 homes, our approach computes a battery size, which is less than 10% more than the minimum possible size necessary to satisfy a one day in ten years loss probability (2.7 ∗ 10^4 ). Moreover, we are able to estimate the carbon footprint reduction, compared to an exact numerical analysis, within a factor of 1.7. We also study the genset scheduling problem when the rate of genset fuel consumption is given by an affine function instead of a linear function of the current power. We give alternate scheduling, an online scheduling strategy that has a competitive ratio of (k1 G/C +k2)/(k1+k2) , where G is the genset capacity, C is the battery charging rate, and k1, k2 are the affine function constants. Numerically, we show that for a real industrial load alternate scheduling is very close to the offline optimal strategy.

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The Galois group of a Schubert problem is a subtle invariant that encodes intrinsic structure of its set of solutions. These geometric invariants are difficult to determine in general. However, based on a special position argument due to Schubert and a combinatorial criterion due to Vakil, we show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. The result follows from a particular inequality of Schubert intersection numbers which are Kostka numbers of two-rowed tableaux. In most cases, the inequality follows from a combinatorial injection. For the remaining cases, we use that these Kostka numbers appear in the tensor product decomposition of sl2C-modules. Interpreting the tensor product as the action of certain Toeplitz matrices and using spectral analysis, the inequality can be rewritten as an integral. We establish the inequality by estimating this integral using only elementary Calculus.

Thesis (Ph.D.)--University of Melbourne, Dept. of Science and Mathematics Education, 2002. / Typescript (photocopy). Includes bibliographical references (leaves 193-206).

Thesis (Ph. D.)--Oregon State University, 1997. / Typescript (photocopy). Includes bibliographical references (leaves 100-103). Also available on the World Wide Web.