Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references (leaves 276-286). Available also in a digital version from Dissertation Abstracts.

We study quasi superharmonic functions in Brelot spaces and the relationship between a reduced function, and harmonic and Jensen measures. We introduce the concept of quasi multiply superharmonic functions on a product of two Brelot spaces and study their properties. A main result obtained is characterizing the quasi superharmonic functions in terms of harmonic, finely harmonic and Jensen measures. Then we prove that a quasi multiply superharmonic function on a product of Brelot spaces equals its lower semicontinuous regularization out side of a 2-negligible set. Further we give a sufficient condition on a Brelot space O under which O becomes an extension space for superharmonic functions. As a result we characterize the extreme Jensen measures in such spaces. Finally we study extreme Jensen measures relative to several classes of multiply superharmonic functions.

Thesis (Ph. D.)--West Virginia University, 2005. / Title from document title page. Document formatted into pages; contains vi, 72 p. Includes abstract. Includes bibliographical references (p. 69-72).

Boston University. University Professors Program Senior theses. / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / 2031-01-02

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Thesis (doctoral)--Universität Bonn, 1988. / Includes bibliographical references.

In this paper several equivalences of the axiom of choice are examined. In particular, the axiom of choice, Zorn's lemma, Tukey's lemma, the Hausdorff maximal principle, and the well-ordering theorem are shown to be equivalent. Cardinal and ordinal number theory is also studied. The Schroder-Bernstein theorem is proven and used in establishing order results for cardinal numbers. It is also demonstrated that the first uncountable ordinal space is unique up to order isomorphism. We conclude by encountering several applications of the axiom of choice. In particular, we show that every vector space must have a Hamel basis and that any two Hamel bases for the same space must have the same cardinality. We establish that the Tychonoff product theorem implies the axiom of choice and see the use of the axiom of choice in the proof of the Hahn- Banach theorem.

Esta pesquisa propõe um método que aplica a Teoria de Projeto Axiomático (PA) ao processo de desenvolvimento de software que se orientam por regras. Nesse âmbito, salienta-se que não foi encontrada na literatura, durante os esforços de pesquisa deste trabalho, a aplicação da Teoria de Projeto Axiomático a sistemas orientados a Regras. Entretanto, a Teoria de Projeto Axiomático já sim foi foco de pesquisa e aplicação no processo de desenvolvimento de software orientado a objeto, servindo de inspiração ao presente trabalho. Dito isso, este trabalho propõe o método Projeto Axiomático aplicado ao Paradigma Orientado a Notificações e ao Paradigma Orientado a Regras (PA-PON-POR) desde que as regras sigam o modelo de estruturação dado pelo PON. O método PA-PON-POR propõe a decomposição funcional de requisitos do sistema em quatro níveis que são: Casos de Uso, Subcasos de Uso Independentes de Características Técnicas, Subcasos de Uso Dependentes de Características Técnicas e Serviços Técnicos. Além disso, o método PA-PON-POR aplica o Axioma da Independência do PA em cada um dos quatro níveis de decomposição por meio das matrizes de projeto e métricas de cálculo da reangularidade e semangularidade do próprio PA. As matrizes de projeto ainda auxiliam na identificação das Premissas exclusivas, elementos esses importantes quando um sistema PON-POR possui Regras que possuem Ações que instigam a geração de fatos conflitantes. O Axioma da Informação do Projeto Axiomático também é aplicado em cada nível de decomposição avaliando as soluções de projeto quanto a sua quantidade de informação. Ainda, o método PA-PON-POR apresenta um conjunto de métricas especificas para avaliação da qualidade estrutural da composição de Regras do sistema, fornecendo critérios para tomada de decisão sobre a qualidade do projeto especificado. Além disso, o método PA-PON-POR é passível de aplicação simultânea com o método existente de projeto de software baseado em desenvolvimento de aplicações PON-POR chamado de Desenvolvimento Orientado a Notificações e Orientado a Regras (DON-DOR), auxiliando na obtenção e validação de artefatos do mesmo. O método PA-PON-POR foi aplicado no desenvolvimento de dois softwares, o primeiro software refere-se um simulador de portão eletrônico e o segundo software refere-se a um sistema de vendas. Em ambas as aplicações, o método PA-PON-POR demonstrou ser eficiente no que se propõe, auxiliando no processo de criação de Regras e de sistemas PON-POR com alguma garantia de qualidade. / This research proposes a method to apply the Axiomatic Design Theory (ADT) in the Rule-oriented software development process. In this context, it was not found in the literature, by the efforts of this work research, the application of ADT in Rule-oriented software development. However, the ADT was focus on research in Object-Oriented software development in a previous work, which was used as inspiration in this current research work. This current research proposes the method Axiomatic Design for Notification-Oriented Paradigm and Rule-Oriented Paradigm (AD-NOP-ROP) since the rules follow the NOP structural model. This method proposes a functional decomposition of system requirements in four levels which are: Use Cases, Use Subcases that are Technical Feature Independent, Use Subcases that are Technical Feature Dependent, and Technical Service . Furthermore, the method AD-NOP-ROP applies the ADT Independence Axiom in each one of the decomposition levels by means of design matrixes and metrics which calculates reangularity and semangularity from ADT. The design matrixes still aids in the identification of Exclusive Premises, which are important elements of NOP-ROP systems with Rules whose Actions instigate the creation of conflicting facts. The Information Axiom from ADT is also applied in each decomposition level in order to evaluate design solutions in terms of its amount of information. Still, the method AD-NOP-ROP presents a set of metrics which are specific for evaluation of structural quality of Rule composition, thereby providing criteria for decision making with respect to design quality. Besides, the method AD-NOP-ROP can be used in a simultaneous way with the existent method used for software design based on NOP-ROP application development, so called Notification-Oriented and Rule-Oriented Application Development (NO-RO-AD), in order to assist in the achievement and validation of artifacts. The method AD-NOP-ROP was applied during the development of two software systems, the first one refers to an Electronic Gate and the second one refers to a Sales System. In both applications the method displayed efficiency in its purposes, assisting in the Rule creation process and also in the creation of NOP-ROP software with some quality assurance.

Esta pesquisa propõe um método que aplica a Teoria de Projeto Axiomático (PA) ao processo de desenvolvimento de software que se orientam por regras. Nesse âmbito, salienta-se que não foi encontrada na literatura, durante os esforços de pesquisa deste trabalho, a aplicação da Teoria de Projeto Axiomático a sistemas orientados a Regras. Entretanto, a Teoria de Projeto Axiomático já sim foi foco de pesquisa e aplicação no processo de desenvolvimento de software orientado a objeto, servindo de inspiração ao presente trabalho. Dito isso, este trabalho propõe o método Projeto Axiomático aplicado ao Paradigma Orientado a Notificações e ao Paradigma Orientado a Regras (PA-PON-POR) desde que as regras sigam o modelo de estruturação dado pelo PON. O método PA-PON-POR propõe a decomposição funcional de requisitos do sistema em quatro níveis que são: Casos de Uso, Subcasos de Uso Independentes de Características Técnicas, Subcasos de Uso Dependentes de Características Técnicas e Serviços Técnicos. Além disso, o método PA-PON-POR aplica o Axioma da Independência do PA em cada um dos quatro níveis de decomposição por meio das matrizes de projeto e métricas de cálculo da reangularidade e semangularidade do próprio PA. As matrizes de projeto ainda auxiliam na identificação das Premissas exclusivas, elementos esses importantes quando um sistema PON-POR possui Regras que possuem Ações que instigam a geração de fatos conflitantes. O Axioma da Informação do Projeto Axiomático também é aplicado em cada nível de decomposição avaliando as soluções de projeto quanto a sua quantidade de informação. Ainda, o método PA-PON-POR apresenta um conjunto de métricas especificas para avaliação da qualidade estrutural da composição de Regras do sistema, fornecendo critérios para tomada de decisão sobre a qualidade do projeto especificado. Além disso, o método PA-PON-POR é passível de aplicação simultânea com o método existente de projeto de software baseado em desenvolvimento de aplicações PON-POR chamado de Desenvolvimento Orientado a Notificações e Orientado a Regras (DON-DOR), auxiliando na obtenção e validação de artefatos do mesmo. O método PA-PON-POR foi aplicado no desenvolvimento de dois softwares, o primeiro software refere-se um simulador de portão eletrônico e o segundo software refere-se a um sistema de vendas. Em ambas as aplicações, o método PA-PON-POR demonstrou ser eficiente no que se propõe, auxiliando no processo de criação de Regras e de sistemas PON-POR com alguma garantia de qualidade. / This research proposes a method to apply the Axiomatic Design Theory (ADT) in the Rule-oriented software development process. In this context, it was not found in the literature, by the efforts of this work research, the application of ADT in Rule-oriented software development. However, the ADT was focus on research in Object-Oriented software development in a previous work, which was used as inspiration in this current research work. This current research proposes the method Axiomatic Design for Notification-Oriented Paradigm and Rule-Oriented Paradigm (AD-NOP-ROP) since the rules follow the NOP structural model. This method proposes a functional decomposition of system requirements in four levels which are: Use Cases, Use Subcases that are Technical Feature Independent, Use Subcases that are Technical Feature Dependent, and Technical Service . Furthermore, the method AD-NOP-ROP applies the ADT Independence Axiom in each one of the decomposition levels by means of design matrixes and metrics which calculates reangularity and semangularity from ADT. The design matrixes still aids in the identification of Exclusive Premises, which are important elements of NOP-ROP systems with Rules whose Actions instigate the creation of conflicting facts. The Information Axiom from ADT is also applied in each decomposition level in order to evaluate design solutions in terms of its amount of information. Still, the method AD-NOP-ROP presents a set of metrics which are specific for evaluation of structural quality of Rule composition, thereby providing criteria for decision making with respect to design quality. Besides, the method AD-NOP-ROP can be used in a simultaneous way with the existent method used for software design based on NOP-ROP application development, so called Notification-Oriented and Rule-Oriented Application Development (NO-RO-AD), in order to assist in the achievement and validation of artifacts. The method AD-NOP-ROP was applied during the development of two software systems, the first one refers to an Electronic Gate and the second one refers to a Sales System. In both applications the method displayed efficiency in its purposes, assisting in the Rule creation process and also in the creation of NOP-ROP software with some quality assurance.

The axiomatic theory of ` Zelazko defines a variety of general spectra where specified axioms are satisfied. However, there arise a number of spectra, usually defined for a single element of a Banach algebra, that are not covered by the axiomatic theory of ` Zelazko. V. Kordula and V. M¨uller addressed this issue and created the theory of regularities. Their unique idea was to describe the underlying set of elements on which the spectrum is defined. The axioms of a regularity provide important consequences. We prove that the set of Koliha-Drazin invertible elements, which includes the Drazin invertible elements, forms a regularity. The properties of the spectrum corresponding to a regularity are also investigated. / Mathematical Sciences / M. Sc. (Mathematics)