We compute the partial isomorphism rank, in the sense Scott and Karp, of a pair of ordinal structures using an Ehrenfeucht-Fraisse game. A complete formula is proven by induction given any two arbitrary ordinals written in Cantor normal form.

In this thesis, I survey 11 approximation algorithms for unweighted set cover problem. I have also implemented the three algorithms and created a software library that stores the code I have written. The algorithms I survey are: 1. Johnson's standard greedy; 2. f-frequency greedy; 3. Goldsmidt, Hochbaum and Yu's modified greedy; 4. Halldorsson's local optimization; 5. Dur and Furer semi local optimization; 6. Asaf Levin's improvement to Dur and Furer; 7. Simple rounding; 8. Randomized rounding; 9. LP duality; 10. Primal-dual schema; and 11. Network flow technique. Most of the algorithms surveyed are refinements of standard greedy algorithm.

Rough set theory (RST) is concerned with the formal approximation of crisp sets and is a mathematical tool which deals with vagueness and uncertainty. RST can be integrated into machine learning and can be used to forecast predictions as well as to determine the causal interpretations for a particular data set. The work performed in this research is concerned with using various optimisation techniques to granulise the rough set input partitions in order to achieve the highest forecasting accuracy produced by the rough set. The forecasting accuracy is measured by using the area under the curve (AUC) of the receiver operating characteristic (ROC) curve. The four optimisation techniques used are genetic algorithm, particle swarm optimisation, hill climbing and simulated annealing. This newly proposed method is tested on two data sets, namely, the human immunodeficiency virus (HIV) data set and the militarised interstate dispute (MID) data set. The results obtained from this granulisation method are compared to two previous static granulisation methods, namely, equal-width-bin and equal-frequency-bin partitioning. The results conclude that all of the proposed optimised methods produce higher forecasting accuracies than that of the two static methods. In the case of the HIV data set, the hill climbing approach produced the highest accuracy, an accuracy of 69.02% is achieved in a time of 12624 minutes. For the MID data, the genetic algorithm approach produced the highest accuracy. The accuracy achieved is 95.82% in a time of 420 minutes. The rules generated from the rough set are linguistic and easy-to-interpret, but this does come at the expense of the accuracy lost in the discretisation process where the granularity of the variables are decreased.

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This thesis presents new theorems in homotopy theory, in particular it generalizes a theorem of Saharon Shelah. We employ a technique used by Janusz Pawlikowski to show that certain Peano continua have a least nontrivial homotopy group that is finitely presented or of cardinality continuum. We also use this technique to give some relative consistency results.

The utilization of fuzzy mathematical modeling for quantification of the quality of training and educational delivery is an innovative application that can result in measurable and repeatable results. This research was designed to apply proven quantification techniques and Industrial Engineering methodologies to a nontraditional environment. The outcomes of this research provide the foundation, initial steps and preliminary validation for the development of a systematic fuzzy theoretical model to be applied for the quantification of various areas within training and education delivery. The test bed for this methodology is Orange County Public School system, the twelfth largest school district in the nation. The organizational and operational factors of a large school district are highly compatible with Systems Engineering concepts. The debate over education reform has drawn from referenced areas within the Industrial Engineering community including quality, continuous improvement, benchmarking and metrics development, data analysis, and scientific/systemic justification requirements. In spite of these applications, the literature does not reflect a consistent and broad application of these techniques in addressing the evaluation and quantification of educational delivery systems. This research draws on the previously listed areas within Industrial Engineering to apply these techniques to enhance the understanding and promote quantification of the multiple factors acting on the educational delivery system. The importance of addressing these issues is a national concern given the significant changes in the United States educational delivery system. For example, over the past 40 years there has been a more than three-hundred percent increase in per-pupil appropriations yet the academic performance gains have been limited and the quantification and measurement of those gains is even more limited. This body of work willidentify the systems, sub-systems, system factors, and factor degrees of existence necessary to quantify and measure these performance changes. Finally, the research will quantify the inputs and produce a model that provides a numeric value that represents the condition of the system and various subsystems of an educational system.