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Graduation date: 1995

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Thesis (Ph.D.). / Written for the Dept. of Mathematics and Statistics. Title from title page of PDF (viewed 2009/06/08). Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin-Madison, 1950. / Typescript with manuscript equations. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

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A dissertation submitted in fulfilment of the requirements of the Degree of Master of Education in the Department of Mathematics, Science and Technology Education, Faculty of Education at the University of Zululand, 2019. / This study sought to determine the NSC learners’ level of understandings of exponential and logarithmic functions; grade twelve teachers’ self- assessment of their readiness to teach exponential and logarithmic functions; the relationship between the educators’ self-concept about their ability to teach exponential and logarithmic functions and the actual performance of their learners; and whether or not the educators’ MCK and PCK impacted learner achievement in exponential and logarithmic functions. The study developed a conceptual framework from literature which consisted of two major components depicting learner and educator readiness. These models illustrated factors that could possibly affect the ability of the learner to succeed in understanding instruction related to exponential equations and logarithmic functions, as well as those that would prevent educators from delivering optimum instruction to learners. This study used a mixed-methods research paradigm, as there was need to collect both quantitative and qualitative data in order to adequately answer the four research questions. The survey research design was used, and data were collected through a researcher-designed test (for learners) and a researcher-designed questionnaire for educators, focusing on their MCK and PCK. The research sample, consisting of nine school principals, nine mathematics educators, and 242 mathematics learners based in nine randomly selected schools, was drawn from a target population of high schools in the uMkhanyakude education district, KwaZulu-Natal Province, South Africa. Analysis was done using the SPSS version 23 software programme. The results revealed that learners had basic understanding of exponential and logarithmic functions in most aspects of the topic, although their performance was border line. For the educators, although all they were suitably qualified in terms of their minimum requirements for registration with the South African Council for Educators (SACE), their performance on the same test taken by their learners was only marginally above the performance of their learners. The educators’ responses to the question about their readiness to teach exponential equations and logarithmic functions were v mixed shedding some light on why many of them were unable to solve the same problems given to their learners. On the relationship between educators’ self-concept about their ability to teach exponential and logarithmic functions and their learners’ performance, the results showed that learners whose teachers considered themselves to be suitably qualified, knowledgeable and able to teach exponential and logarithmic functions performed significantly lower than learners whose teachers considered themselves not to be suitably qualified, knowledgeable and able to teach exponential and logarithmic functions. The results of the questions which sought to establish the impact of educators’ MCK and PCK on learner performance in exponential and logarithmic functions drew a blank, suggesting that there was no relationship between teachers’ MCK and PCK, on one hand, and learner performance, on another.

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Since Mclntosh's introduction of the H^∞-calculus for sectorial operators, the topic has been studied by many authors. Haase has constructed a similar functional calculus for strip-type operators, and has also developed an abstract framework which unifies both of these examples and more. In this thesis we use this abstract functional calculus setting to study two particular problems in operator theory. The first of these is concerned with operator sums. We ask the question of when the sum log A+log B is closed, where A and B are a pair of injective sectorial operators whose resolvents commute. We show that the sum is always closable and, when A and B are invertible, we determine sufficient conditions for the sum to be closed. These conditions are of Kalton-Weis type, and in fact ensure that AB is sectorial and that the identity log A + log B = log(AB) holds. We then identify an interpolation space on which these conditions are automatically satisfied. Our second problem is connected to the exponential of a strip-type operator B</e>, specifically the question of whether e^B is sectorial. When -1 ∈ p(e^B), the spectrum of e^B lies in a sector, and we obtain an estimate on the resolvent outside this sector. This estimate becomes closer to sectoriality as more restrictions are placed on the resolvents of B itself. This leads us to introduce the ideas of F-sectorial and F-strong strip-type operators, whose spectra are contained in a sector or strip, but which satisfy a different resolvent estimate from that of a sectorial or strong striptype operator. In some cases it is possible to define the logarithm of an F-sectorial operator or the exponential of an F-strong strip-type operator. We prove resolvent estimates for the resulting logarithms and exponentials, and explore the relationships between the various classes of operators considered.

The S-Boxes used in the AES algorithm are generated by field extensions of the Galois field over two elements, called GF(2). Therefore, understanding the field extensions provides a method of analysis, potentially efficient implementation, and efficient attacks. Different polynomials can be used to generate the fields, and we explore the set of polynomials x^ 2 + x + a^J over GF(2^n) where a is a primitive element of GF(2^n). The results of this work are the first steps towards a full understanding of the field that AES computation occurs in-GF(2^8). The charts created with the data we gathered detail which power of the current primitive root is equal to previous primitive roots for fields up through GF(2^16) created by polynomials of the form x^2 + x + a^i for a primitive element a. Currently, a C++ program will also provide all the primitive polynomials of the form x^2 + x+ a^i for a primitive element a over the fields through GF(2^32). This work also led to a deeper understanding of certain elements of a field and their equivalent shift register state. In addition, given an irreducible polynomial 2 f(x) = x^2 + a^i x + a^j over GF(2^n), the period (and therefore the primitivity) can be determined by a new theorem without running the shift register generated by f(x).