6 research outputs found
Constructive aspects of Kochen\u27s theorem on p-adic closures
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
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
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Considerations in designing a cybernetic simple 'learning' model; and an overview of the problem of modelling learning
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Learning is viewed as a central feature of living systems and must be manifested in any artifact that claims to exhibit general intelligence. The central aims of the thesis are twofold: (1) - To review and critically assess the empirical and theoretical aspects of learning as have been addressed in a multitude of disciplines, with the aim of extracting fundamental features and elements. (2) - To develop a more systematic approach to the cybernetic modelling of learning than has been achieved hitherto. In pursuit of aim (1) above the following discussions are included: Historical and Philosophical backgrounds; Natural learning, both physiological and psychological aspects; Hierarchies of learning identified in the evolutionary, functional and developmental senses; An extensive section on the general problem of modelling of learning and the formal tools, is included as a link between aims (1) and (2). Following this a systematic and historically oriented study of cybernetic and other related approaches to the problem of modelling of learning is presented. This then leads to the development of a state-of-the-art general purpose experimental cybernetic learning model. The programming and use of this model is also fully described, including an elaborate scheme for the manifestation of simple learning
On Comparative Algorithmic Pathfinding in Complex Networks for Resource-Constrained Software Agents
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