6 research outputs found

    Constructive aspects of Kochen\u27s theorem on p-adic closures

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    In this work we begin with a brief survey of set theory and arithmetic to provide background for a logical procedure to `cleanse\u27 the Axiom of Choice from a proof of a theorem of Kochen\u27s. We accomplish this in the following chapters. We then discuss certain theorems involving definable Skolem functions. These theorems are used in Chapter 5 to give a construction of a p-adic closure of a p-valued field. Certain further considerations and open questions are addressed in the _x000C_final chapter

    Boolean ultrapowers

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    Bibliography: leaves 121-122.The Boolean ultrapower construction is a generalisation of the ordinary ultrapower construction in that an arbitrary complete Boolean algebra replaces the customary powerset Boolean algebra. B. Koppelberg and S. Koppelberg [1976] show that the class of ordinary ultrapowers is properly contained in the class of Boolean ultrapowers thereby justifying the development of a theory for Boolean ultrapowers. This thesis is an exploration into the strategies whereby and the conditions under which aspects of the theory of ordinary ultrapowers can be extended to the theory of Boolean ultrapowers. Mansfield [1971] shows that a finitely iterated Boolean ultrapower is isomorphic to a single Boolean ultrapower under certain conditions. Using a different approach and under somewhat different conditions, Ouwehand and Rose [1998] show that the result also holds for K-bounded Boolean ultrapowers. Mansfield [1971] also proves a Boolean version of the Keisler-Shelah theorem. By redefining the notion of a K-good ultrafilter on a Boolean algebra, Benda [1974] obtains a complete generalisation of a theorem of Keisler which states that an ultrapower is K-saturated iff the ultrafilter is K-good. Potthoff [1974] defines the notion of a limit Boolean ultrapower and shows that, as is the case for ordinary ultrapowers, the complete extensions of a model are characterised by its limit Boolean ultrapowers. Upon the discovery by Frayne, Morel and Scott [1962] of an ultrapower of a simple group which is not simple, Burris and Jeffers [1978] investigate necessary and sufficient conditions for a Boolean ultrapower to be simple, or subdirectly irreducible, provided that the language is countable. Finally, Jipsen, Pinus and Rose [2000] extend the notion of the Rudin-Keisler ordering to ultrafilters on complete Boolean algebras, and prove that by using this definition, Blass' Characterisation Theorem can be generalised for Boolean ultrapowers

    On Comparative Algorithmic Pathfinding in Complex Networks for Resource-Constrained Software Agents

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    Software engineering projects that utilize inappropriate pathfinding algorithms carry a significant risk of poor runtime performance for customers. Using social network theory, this experimental study examined the impact of algorithms, frameworks, and map complexity on elapsed time and computer memory consumption. The 1,800 2D map samples utilized were computer random generated and data were collected and processed using Python language scripts. Memory consumption and elapsed time results for each of the 12 experimental treatment groups were compared using factorial MANOVA to determine the impact of the 3 independent variables on elapsed time and computer memory consumption. The MANOVA indicated a significant factor interaction between algorithms, frameworks, and map complexity upon elapsed time and memory consumption, F(4, 3576) = 94.09, p \u3c .001, h2 = .095. The main effects of algorithms, F(4, 3576) = 885.68, p \u3c .001, h2 = .498; and frameworks, F(2, 1787) = 720,360.01, p .001, h2 = .999; and map complexity, F(2, 1787) = 112,736.40, p \u3c .001, h2 = .992, were also all significant. This study may contribute to positive social change by providing software engineers writing software for complex networks, such as analyzing terrorist social networks, with empirical pathfinding algorithm results. This is crucial to enabling selection of appropriately fast, memory-efficient algorithms that help analysts identify and apprehend criminal and terrorist suspects in complex networks before the next attack

    An application of Kochen's theorem

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