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.

<p> The notion of nondeterminism is one of the most fundamental concepts in many areas of computer science. Unambiguity, requiring that there be at most one correct sequence of nondeterministic choices, has proved to be one of the most meaningful restrictions of nondeterminism. In the context of space-bounded Turing Machines, several variants of unambiguity have been proposed and studied, and some interesting results have been established, narrowing slightly the gap between deterministic and nondeterministic logarithmic-space computation.</p> <p> We study the different variants of unambiguity in the context of computing multi-valued functions (as opposed to the usual yes/no decision problems). We propose a modification to the standard computation models of Turing Machines and configuration graphs, which allows for unambiguity-preserving composition. We introduce a unified notation, capturing the different flavors of ambiguity. Furthermore, we define a notion of reductions (based on function composition), which allows non-determinism but controls its level of ambiguity. In the light of this framework we establish some reductions between different variants of path counting problems. We also investigate more carefully the technique of inductive counting, and obtain improvement of some existing results.</p> / Thesis / Doctor of Philosophy (PhD)

Let X = (X1,.....,Xm) be an infinitely degenerate system of vector fields, we study the existence and regularity of multiple solutions of Dirichelt problem for a class of semi-linear infinitely degenerate elliptic operators associated with the sum of square operator Δx = ∑m(j=1) Xj* Xj.

This thesis focuses on efficient design of a vertex shader for per-vertex operations such as Transformation and Lighting in the OpenGL ES 2.0 graphics pipeline. The vertex shader performs these complex operations using logarithmic number system, and makes partial optimization for the hardware area based on the accuracy requirement of half-precision floating-point. The vertex shader design emphasizes low cost, and is well suited to low-accuracy embedded applications. The vertex shader is an SIMD (Single-Instruction-Multiple-Data) design with customized instruction set that allows users to write efficient vertex shader programs.

Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 44-46). Available also in a digital version from Dissertation Abstracts.

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Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2000. / Date from cover. Includes bibliographical references (p. 77).

Thesis (Ph. D.)--University of Texas at Austin, 2005. / Supervisor: Fernando Rodriguez-Villegas. Vita. Includes bibliographical references.

In this thesis, first we propose a new approach to solve Semi Infinite Linear Optimization Problems (SILP). The new algorithm uses the idea of adding violated cut or cuts at each iteration. Our proposed algorithm distinguishes itself from Luo, Roos, and Terlaky's logarithmic barrier decomposition method, in three aspects: First, the violated cuts are added at their original locations. Second, we extend the analysis to the case where multiple violated cuts are added simultaneously, instead of adding only one constraint at a time. Finally, at each iteration we update the barrier parameter and the feasible set in the same step. In terms of complexity, we also show that a good approximation of an optimal solution will be guaranteed after finite number of iterations. Our focus in this thesis is mainly on the implementation of our algorithm to approximate an optimal solution of the SILP. Our numerical experiences show that unlike other SILP solvers which are suffering from the lack of accuracy, our algorithm can reach high accuracy in a competitive time. We discuss the linear algebra involved in efficient implementation and describe the software that was developed. Our test problem set includes large scale second order conic optimization problems. / Thesis / Master of Applied Science (MASc)

This thesis discusses a development of the exponential and logarithmic functions.