In this thesis, the geometry of curved surfaces is studied using the methods of differential geometry. The introduction of manifolds assists in the study of classical two-dimensional surfaces. To study the geometry of a surface a metric, or way to measure, is needed. By changing the metric on a surface, a new geometric surface can be obtained. On any surface, curves called geodesics play the role of "straight lines" in Euclidean space. These curves minimize distance locally but not necessarily globally. The curvature of a surface at each point p affects the behavior of geodesics and the construction of geometric objects such as circles and triangles. These fundamental ideas of manifolds, geodesics, and curvature are developed and applied to classical surfaces in Euclidean space as well as models of non-Euclidean geometry, specifically, two-dimensional hyperbolic space.

The Fibonacci sequence and Cantor's ternary set are two objects of study in mathematics. There is much theory published about these two objects, individually. This paper provides a fascinating relationship between the Fibonacci sequence and Cantor's ternary set. It is a fact that every natural number can be expressed as the sum of distinct Fibonacci numbers. This expression is unique if and only if no two consecutive Fibonacci numbers are used in the expression--this is known as Zekendorf's representation of natural numbers. By Zekendorf's representation, a function F from the natural numbers into [0,0.603] will be defined which has the property that the closure of F(N) is homeomorphic to Cantor's ternary set. To accomplish this, it is shown that the closure of F(N) is a perfect, compact, totally disconnected metric space. This then shows that the closure of F(N) is homeomorphic to Cantor's ternary set and thereby establishing a relationship between the Fibonacci sequence and Cantor's ternary set.

The terminology and ideas involved in the statistical analysis of survival data are explained including the survival function, the probability density function, the hazard function, censored observations, parametric and nonparametric estimations of these functions, the product limit estimation of the survival function, and the proportional hazards estimation of the hazard function with explanatory variables. In Appendix A these ideas are applied to the actual analysis of the survival data for 54 cervical cancer patients.

Thesis (MBA)--Stellenbosch University, 2007. / ENGLISH ABSTRACT: is a common misconception that marketing is all about advertising and other promotional activities. These activities are only part of the marketing process which is in fact about identifying the wants and needs of certain customers and satisfying them better than the competition. However, it must also be borne in mind that these needs and wants change over time as customers change positions in the customer lifecycle. The customer lifecycle of a Woolworths Financial Services personal loan customer is not unlike the generic customer lifecycle of acquisition, development and retention and as such aligns with the principles associated with this theory – acquire, develop and retain as many customers as possible at the lowest cost. The current marketing practices of Woolworths Financial Services are not targeted, focusing on the entire base of prospective customers without any differentiation between those who would most likely respond and those who would not. These practices, however, have provided the opportunity to gain valuable information about the characteristics of those who do respond and those who do not. The use of data mining and regression analysis models (developed on this data) can provide the ability to effectively predict the response curve – ranking and assigning probabilities based on customer behaviour. That being said, these probabilities are of little use if they are not used to optimise the structure of marketing campaigns by determining where to differentiate an offer or not. It is here that mathematical science provides a solution through the optimisation process – the process of computing the mix of variables that provides the best result – in this case the highest number of customers acquired, developed and retained within the constraints of a defined marketing budget. Finally, the result of this process culminates in an increase in shareholder value which would not have been achieved through existing marketing products. The risk associated with implementing this process is negligible, based on empirical evidence regarding the impact of differentiated offers on response rates and as such must be implemented wherever possible. / AFRIKAANSE OPSOMMING: Dit is ‘n algemene wanopvatting dat bemarking net oor advertensie en ander reklame aktiwitiete gaan. Hierdie aktiwiteite is slegs ‘n deel van die bemarkingsproses wat eintlik gaan oor die identifisering van die behoeftes van sekere kliënte en om hulle beter as die opposisie te kan bedien. Dit moet egter in gedagte gehou word dat die behoeftes van kliente oor tyd verander as hulle posisies skuif in die kliëntelewenssiklus. Die kliëntelewenssiklus van ‘n Woolworths Finansiële Dienste persoonlikeleningskliënt is nie anders as die gemiddelde kliëntelewensiklus van verkryging, ontwikkeling en retensie nie, en vergelyk dus met die beginsels wat met hierdie teorie geassosieer word – verkry, ontwikkel en behou so veel as moontlik kliënte teen die laagste koste. Die huidige bemarkingsprosesse van Woolworths Finansiële Dienste is nie beperk nie en fokus op die algehele moontlike kliëntebasis sonder om te onderskei tussen dié wat heel moontlik sal reageer en dié wat nie sal nie. Hierdie praktyk het egter die geleentheid geskep om waardevolle inligting te bekom oor dié wat reageer en dié wat nie reageer nie. Die gebruik van dataontleding en regressie analise modelle (ontwikkel op hierdie data) kan die vermoë skep om effektief die reaksiekurwe te voorspel – rangskikking en aanwysing van waarskynlikhede gebaseer op kliëntegedrag. Gegewe dié afleiding, is hierdie waarskynlikhede van geringe belang as dit nie aangewend word om die struktuur van die bemarkingsveldtog te optimeer deur te bepaal of ‘n aanbieding gewysig moet word of nie. Dit is hier waar wiskundige metodes ‘n oplossing bied deur die optimeringsproses – in hierdie geval die meeste kliënte wat verkry, ontwikkel en behou kan word binne die beperkinge van ‘n geïdentifiseerde bemarkingsbegroting. Ten slotte, die resultaat van hierdie proses lei tot ‘n toename in aandeelhouerwaarde wat nie moontlik sou wees deur bestaande bemarkingsprodukte nie. Die risiko met die implementering van hierdie proses is weglaatbaar klein, gebaseer op die empiriese bewyse aangaande die impak van gedifferensieerde aanbiedings op reaksiekoerse en moet dus geïmplementeer word waar moontlik.

When the variance of a single population needs to be assessed, the well-known chi-squared test of variance is often used but relies heavily on its normality assumption. For non-normal populations, few alternative tests have been developed to conduct left tailed hypothesis tests of variance. This thesis outlines a method for generating new test statistics using a saddlepoint approximation. Several novel test statistics are proposed. The type-I error rates and power of each test are evaluated using a Monte Carlo simulation study. One of the proposed test statistics, R_gamma2, controls type-I error rates better than existing tests, while having comparable power. The only observed limitation is for populations that are highly skewed with heavy-tails, for which all tests under consideration performed poorly.

This thesis presents the Weil pairing on elliptic curves as a tool to implement a tripartite Diffie-Helman key exchange. Elliptic curves are introduced, as well as the addition operation that creates a group structure on its points. In leading to the definition of the Weil pairing, divisors of rational functions are studied, as well as the Weierstrass }-function, which shows the complex lattice as isomorphic to elliptic curves. Several important qualities of the Weil pairing are proved, and Miller's algorithm for quick calculation is shown. Next, the bipartite Diffie-Helman key exchange is discussed over finite fields and elliptic curves. Finally an example of a modifed Weil pairing is defined, which leads to the tripartite Diffie-Helman key exchange.

In this paper, we present and implement a method to assess the mathematical synergy of two-drug combinations based on a stochastic model. The drugs in question are two isomers that are applied to the human eye via a liquid eye drop. Techniques applied to the data in this paper can be applied to other two-drug combination studies. We derive the mean and the variance terms of the drug combination "effects" in closed form using Ito's method of stochastic differential equations. The model fit of the data to the individual subject is examined by both statistical and graphical methods. Two estimation methods in SAS, PROC NUN and PROC NLMIXED, are used to estimate model parameters. We perform simulation and power studies using R software to show the strengths of the proposed approach in estimating the model parameters. From this research, we find that the combination of drugs under study is synergistic in nature. We also confirm that the proposed stochastic model is appropriate.

Singular values have been found to be useful in the theory of unitarily invariant norms, as well as many modern computational algorithms. In examining singular value inequalities, it can be seen how these can be related to eigenvalues and how several algebraic inequalities can be preserved and written in an analogous singular value form. We examine the fundamental building blocks to the modern theory of singular value inequalities, such as positive matrices, matrix norms, block matrices, and singular value decomposition, then use these to examine new techniques being used to prove singular value inequalities, and also look at existing conjectures.

Missing data bias results if adjustments are not made accordingly. This thesis addresses this issue by exploring a scenario where data is missing at random depending on a covariate x. Four methods for comparing groups while adjusting for missingness are explored by conducting simulations: independent samples t-test with predicted mean stratification, independent samples t-test with response propensity stratification, independent samples t-test with response propensity weighting, and an analysis of covariance. Results show that independent samples t-test with response propensity weighting and analysis of covariance can appropriately adjust for bias. ANCOVA is the stronger method when the ANCOVA assumptions are met. When the ANCOVA assumptions are not met, a t-test with inverse response propensity score weighting is the superior method.

This thesis starts with the fundamentals of matrix theory and ends with applications of the matrix singular value decomposition (SVD). The background matrix theory coverage includes unitary and Hermitian matrices, and matrix norms and how they relate to matrix SVD. The matrix condition number is discussed in relationship to the solution of linear equations. Some inequalities based on the trace of a matrix, polar matrix decomposition, unitaries and partial isometies are discussed. Among the SVD applications discussed are the method of least squares and image compression. Expansion of a matrix as a linear combination of rank one partial isometries is applied to image compression by using reduced rank matrix approximations to represent greyscale images. MATLAB results for approximations of JPEG and .bmp images are presented. The results indicate that images can be represented with reasonable resolution using low rank matrix SVD approximations.