Thesis--University of London. / Bibliography: p. [192]-195.

Thesis (Ph. D., Mathematics)--University of Idaho, January 2009. / Major professor: Arie Bialostocki. Includes bibliographical references (leaves 62-65). Also available online (PDF file) by subscription or by purchasing the individual file.

Thesis (D.A.)--Illinois State University, 1980. / Title from title page screen, viewed Feb. 16, 2005. Dissertation Committee: James Parr (chair), Albert Otto, Larry Eggan, Lotus Hershberger, John Dossey, Gary Ramseyer. Includes bibliographical references (leaves 197-200) and abstract. Also available in print.

Thesis--University of London. / Bibliography: p. [192]-195.

In Monte Carlo calculations performed on electronic computers it is advantageous to use an arithmetic scheme to generate sets of numbers with "approximately" the properties of a random sequence. For many applications the local characteristics of the resulting sequence are of interest. In this thesis the concept of a pseudo-random sequence is set out, and arithmetic methods for their generation are discussed. A brief survey of some standard statistical tests of randomness is offered, and the results of empirical tests for local randomness performed on the ALWAC III-E computer at the University of British Columbia are recorded. It is demonstrated that many of the standard generating schemes do not yield sequences with suitable local properties, and could therefore be responsible for misleading results in some applications. A method appropriate for the generation of short blocks of numbers with approximately the properties of a randomly selected set is proposed and tested, with satisfactory results. / Science, Faculty of / Mathematics, Department of / Graduate

This paper is devoted to a discussion of certain aspects of an ideal theory of commutative rings. The material is divided into three chapters. Chapter I discusses necessary definitions and concepts. Chapter II deals with a theory of so called prime ideals and Chapter III is concerned with the theory of primary ideals. The entire theory is a generalization of the ideal theory of the ring of integers. / Advisor: N. Heerema, Professor Directing Paper. / Typescript. / "January, 1955." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Includes bibliographical references (leaf 30).

Algebraic geometry in positive characteristic has a quite different flavour than in characteristic zero. Many of the pathologies disappear when a variety admits a lift to characteristic zero. It is known since the sixties that such a lift does not always exist. However, for applications it is sometimes enough to lift a variety dominating the given variety, and it is natural to ask when this is possible. The main result of this dissertation is the construction of a smooth projective variety over any algebraically closed field of positive characteristic that cannot be dominated by another smooth projective variety admitting a lift to characteristic zero. We also discuss some cases in which a dominating liftable variety does exist.

Piaget observed that various stages involved in the construction of different forms of knowledge are sequential and that the same sequential order is evident in history. There seem to be three main stages in the development of algebra involving the independent and general solution of equations followed by the evolution of abstract algebra. Piaget referred to these as the intra, inter and transoperational stages but they are termed the levels of percepts, concepts and abstractions here. The perceptual level involves isolated forms, the conceptual level concerns correspondences and transformations amongst forms whilst the abstract level is characterised by the evolution of structures of forms. Historically the overall perceptual level of abstract algebra lasted from antiquity to the middle of the eighteenth century. The conceptual level followed, lasting for approximately one century and the subsequent abstract level has prevailed from the middle of the nineteenth century onwards. Each of these levels involve numerous sublevels but instead of being continually broken down into more and more sublevels, in this study a spiral of learning is being considered. Each round of the spiral contains a perceptual, conceptual and abstract level. The way in which perceptual levels can arise from previous abstract levels gives an indication of how knowledge is reorganised and expanded in new unexplored directions as the spiral is climbed. The important aspects of proof and axiomatisation are also addressed here. The historical emergence of abstract algebra reveals a significant pattern concerning the development of mathematics. The levels of thinking involved are important and reveal a general trend of algebraic thought. Hence careful consideration needs to be paid to the revelations arising from historical investigations so that these may help contribute to the encouragement of learning in students of algebra. The idea of levels of learning has been substantiated by many researchers and investigations undertaken in the past. The main characteristics of the three relevant levels and sublevels as well as insights gained from the historical emergence of algebra are being united here to form a comprehensive theory of learning algebra at both the secondary and tertiary levels of study. / Mathematical Sciences / Ph. D. (Mathematics Education)

Piaget observed that various stages involved in the construction of different forms of knowledge are sequential and that the same sequential order is evident in history. There seem to be three main stages in the development of algebra involving the independent and general solution of equations followed by the evolution of abstract algebra. Piaget referred to these as the intra, inter and transoperational stages but they are termed the levels of percepts, concepts and abstractions here. The perceptual level involves isolated forms, the conceptual level concerns correspondences and transformations amongst forms whilst the abstract level is characterised by the evolution of structures of forms. Historically the overall perceptual level of abstract algebra lasted from antiquity to the middle of the eighteenth century. The conceptual level followed, lasting for approximately one century and the subsequent abstract level has prevailed from the middle of the nineteenth century onwards. Each of these levels involve numerous sublevels but instead of being continually broken down into more and more sublevels, in this study a spiral of learning is being considered. Each round of the spiral contains a perceptual, conceptual and abstract level. The way in which perceptual levels can arise from previous abstract levels gives an indication of how knowledge is reorganised and expanded in new unexplored directions as the spiral is climbed. The important aspects of proof and axiomatisation are also addressed here. The historical emergence of abstract algebra reveals a significant pattern concerning the development of mathematics. The levels of thinking involved are important and reveal a general trend of algebraic thought. Hence careful consideration needs to be paid to the revelations arising from historical investigations so that these may help contribute to the encouragement of learning in students of algebra. The idea of levels of learning has been substantiated by many researchers and investigations undertaken in the past. The main characteristics of the three relevant levels and sublevels as well as insights gained from the historical emergence of algebra are being united here to form a comprehensive theory of learning algebra at both the secondary and tertiary levels of study. / Mathematical Sciences / Ph. D. (Mathematics Education)

This thesis addresses software evolution from the perspective of standards interoperability. We address the issue of how to apply contemporary software safety assurance standards to legacy safety-critical systems, with the aim of recertifying the legacy systems to the contemporary standards. The application of RTCA DO-178B Software Considerations in Airborne Systems and Equipment Certification to modified legacy software is the primary focus of this thesis. We present a model to capture the relationships between pre- and post-modification software and standards. The proposed formal model is then applied to the requirements for RTCA DO-178B and MIL-STD-498 as representative examples of contemporary and legacy software standards. The results provide guidance on how to achieve airworthiness certification for modified legacy software, whilst maximizing the use of software products from the previous development.