published_or_final_version / Mathematics / Master / Master of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy

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This thesis investigates several theoretical and practical problems in index numbers. In Chapter 2 a hedonic elementary price index for accessing the Internet in Canada is constructed. We find that the quality-adjusted price index declines at about 15% per year. Detailed data are readily available on-line. We discuss the use of different functional forms in the regression, their ease of use and performance, and compare the result with the matched model approach. Problems in using the Box-Cox transformation and in handling packages with unlimited access are also discussed. Chapter 3 studies the problems associated with the treatment of seasonal commodities in a consumer price index. Economic assumptions behind various commonly used methods are examined from the cost-of-living perspective. A new theoretical justification based on the theory of preference change is provided for the maximum overlap method. Empirical studies using a particular data set show that indices based on various approaches give substantially different results. Direct measurement techniques have recently been employed by some statistical agencies for government output components in the SNA. These methods use proxies and indicators for outputs due to the inherent lack of market valuations. Chapter 4 investigates the pros and cons of these new approaches and compares them with the traditional cost method. This leads us to take a deeper look at the purposes, objectives, and uses of the SNA. The current method can be justified from a collective household point of view, but the lack of direct output data frustrates students of productivity analysis. By taking the economic approach in index number theory, some direct measurement methods can be compatible with the cost-of-living approach in the CPI. Using implicit expected utility theory, a money metric for utility derived from playing a lottery game is developed in Chapter 5. Using a kinked parametric functional form, outputs of the Canadian Lotto 6/49 are estimated. Results show that this direct economic approach yields an average output three times that of the official GDP. The estimated price elasticity of demand -0.67 closely resembles results for the U.K. in previous studies.

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Typescript. / Thesis (Ph. D.)--University of Hawaii at Manoa, 1977. / Bibliography: leaves 266-273. / ix, 273 leaves ill. (some col.)

A long standing unanswered question in number theory concerns the existence (or not) of odd perfect numbers. Over time many properties of an odd perfect number have been established and refined. The initial goal of this research was to improve the lower bound on the number of distinct prime factors of an odd perfect number, if one exists, to at least 9. Previous approaches to this problem involved the analysis of a carefully chosen set of special cases with each then being eliminated by a contradiction. This thesis applies an algorithmic, factor chain approach to the problem. The implementation of such an approach as a computer program allows the speed, accuracy and flexibility of modern computer technology to not only assist but even direct the discovery process. In addition to considering odd perfect numbers, several related problems were investigated, concerned with (i) harmonic, (ii) even multiperfect and (iii) odd triperfect numbers. The aim in these cases was to demonstrate the correctness and versatility of the computer code and to fine tune its efficiency while seeking improved properties of these types of numbers. As a result of this work, significant improvements have been made to the understanding of harmonic numbers. The introduction of harmonic seeds, coupled with a straightforward procedure for generating most harmonic numbers below a chosen bound, expands the opportunities for further investigations of harmonic numbers and in particular allowed the determination of all harmonic numbers below 10 to the power 12 and a proof that there are no odd harmonic numbers below 10 to the power 15. When considering even multiperfect numbers, a search procedure was implemented to find the first 10-perfect number as well as several other new ones. As a fresh alternative to the factor chain search, a 0-1 linear programming model was constructed and used to show that all multiperfect numbers divisible by 2 to the power of a, for a being less than or equal to 65, subject to a modest constraint, are known in the literature. Odd triperfect numbers (if they exist) have properties which are similar to, but simpler than, those for odd perfect numbers. An extended test on the possible prime factors of such a number was developed that, with minor differences, applies to both odd triperfect and odd perfect numbers. When applicable, this test allows an earlier determination of a contradiction within a factor chain and so reduces the effort required. It was also shown that an odd triperfect number must be greater than 10 to the power 128. While the goal of proving that an odd perfect number must have at least 9 distinct prime factors was not achieved, due to mainly practical limitations, the algorithmic approach was able to show that for an odd perfect number with 8 distinct prime factors, (i) if it is exactly divisible by 3 to the power of 2a then a = 1, 2, 3, 5, 6 or a is greater than or equal to 31 (ii) if the special component is pi to the power of alpha, pi less than 10 to the 6 and pi to the (alpha+1) less than 10 to the 40, then alpha = 1.

Over the last century, the problem of a viscous flow past a sphere has received on-going attention due to its many engineering applications. These include combustion processes, sediment transport processes and atmospheric flow problems, where the sphere serves as a good model for more general bluff body particles. In these environments, particles may be subjected to both translational and rotational velocities. The objective of this study was to investigate the effect that sphere rotation, about an axis transverse to the freestream flow, has on the characteristics or the vertical wake structure and the forces exerted on the sphere. That was achieved by solving the time-dependant, incompressible Navier Stokes equations, using a highly accurate Fourier Chebyshev spectral collation method.