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Let U denote a finite dimensional vector space over an algebraically closed field F . In this thesis, we are concerned with rank one preservers on the r(th) symmetric product spaces r/VU and rank k preservers on the 2nd Grassmann product spaces 2/AU. The main results are as follows: (i) Let T : [formula omitted] be a rank one preserver. (a) If dim U ≥ r + 1 , then T is induced by a non-singular linear transformation on U . (This was proved by L.J. Cummings in his Ph.D. Thesis under the assumption that dim U > r + 1 and the characteristic of F is zero or greater than r .) (b) If 2 < dim U < r + 1 and the characteristic of F is zero or greater than r, then either T is induced by a non-singular linear transformation on U or [formula omitted] for some two dimensional sub-space W of U. (ii) Let [formula omitted] be a rank one preserver where r < s. If dim U ≥ s + 1 and the characteristic of F is zero or greater than s/r, then T is induced by s - r non-zero vectors of U and a non-singular linear transformation on U. (iii) Let T : [formula omitted] be a rank k preserver and char F ≠ 2. If T is non-singular or dim U = 2k or k = 2 , then T is a compound, except when dim U = 4 , in which case T may be the composite of a compound and a linear transformation induced by a correlation of the two dimensional subspaces of U. / Science, Faculty of / Mathematics, Department of / Graduate

In this thesis the solution to the variational problem of Signorini is studied, namely: (i) Δv = 0 in Ω; (ii) v ≥ ѱ on əΩ; (iii) əv/əѵ ≤ g on əΩ; (iv) (v- ѱ) (əv/əѵ – g) = 0 on əΩ where Ω is a domain in R[sup n], and v is the unit inner normal vector to əΩ. In the case n = 2 a regularity theorem is proved. It is shown that if ѱ Є C[sup 1,α] (əΩ), g Є Lip α(əΩ) then v Є C[sup 1,α] (əΩ) if α < 1/2 . An example is given to shown that this result is optimal. The method of proof relies on techniques of complex analysis and therefore does not extend to higher dimensions. For n > 2 the case where Ω, is unbounded, or equivalently, where ѱ is unbounded in a neighbourhood of some point of əΩ is considered. This is a situation where known existence theorems do not apply. Some sufficient conditions for the pair (ѱ,g) are derived that will ensure the existence of a solution in this case, thereby extending some results obtained by A. Beurling and P. Malliavin in the two dimensional case. The proof involves a variational problem in a Hilbert space analogous to the one considered by Beurling and Malliavin, and some pointwise estimates of Riesz transforms. / Science, Faculty of / Mathematics, Department of / Graduate

Research has indicated the importance of calculus knowledge for undergraduate programs in science and technology fields. Unfortunately, one of the main challenges faced by students who join science and technology fields is their knowledge of calculus concepts. The main purpose of the study is to overcome students‟ difficulties in learning calculus concepts by developing a literature informed intervention model. A design-based research approach of three phases was conducted. Grade 12 natural science stream students in one administrative zone in Ethiopia were used as the study population. Triangulated themes of students‟ difficulties and common conceptual issues that are causes of these synthesized difficulties in calculus were used as a foundation to propose an intervention model. Based on the proposed model, an intervention was prepared and administered. A pre post-test aimed to asses students‟ conceptual knowledge in calculus was used to examine the effect of the model. Quantitative analysis of the test revealed that the intervention has a positive effect. The experimental group score is better than the controlled group score with independent t-statistics, t = 4.195 with alpha =.05. In addition, qualitative analysis of the test revealed that students in the experimental group are able to overcome many of the difficulties. In particular, many students demonstrated process level conception, conceptual reasoning, qualitative justification, a consistency in reasoning, less algebraic error, and a proficiency in symbolic manipulation. The study concludes with Implications for practice that includes the use of students‟ errors and misconceptions as an opportunity for progression. Besides, students should be assisted to make sense of concepts through real-life problems, including training teachers in problem-solving approaches and mathematical thinking practice. / Mathematics Education / D. Phil (Mathematics, Science and Technology Education in the subject Mathematics Education)

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