The dissertation uses a number of mathematical formula including de Rham cohomology with complex coefficients to state and prove extension of Brieskorn's Lemma theorem.

We study families of small sets which appear in Harmonic analysis. We focus on the systems H(N) , N ∈ N, U and U0. In particular we compare their sizes via comparing the polars of these classes, i.e. the families of measures annihilating all sets from given class. Lyons showed that in this sense, the family N∈N H(N) is smaller than U0. The main goal of this thesis is the study of the question whether this also holds when the system U0 is replaced by the much smaller system U. To this end we define a new system H(∞) and systems of sets of type N where N ∈ N∪{∞}. We then prove some of their properties, which might be useful in solving the studied question. 1

Master of Science / Department of Industrial & Manufacturing Systems Engineering / David H. Ben-Arieh / Quest for continuous improvement among the organizations and issue of missing data for data analysis are never ending. This thesis brings these two topics under one roof, i.e., to evaluate the productivity of organizations with sparse data. This study focuses on Data Envelopment Analysis (DEA) to determine the efficiency of 41 member clinics of Kansas Association of Medically Underserved (KAMU) with missing data. The primary focus of this thesis is to develop new reliable methods to determine the missing values and to execute DEA. DEA is a linear programming methodology to evaluate relative technical efficiency of homogenous Decision Making Units, using multiple inputs and outputs. Effectiveness of DEA depends on the quality and quantity of data being used. DEA outcomes are susceptible to missing data, thus, creating a need to supplement sparse data in a reliable manner. Determining missing values more precisely improves the robustness of DEA methodology. Three methods to determine the missing values are proposed in this thesis based on three different platforms. First method named as Average Ratio Method (ARM) uses average value, of all the ratios between two variables. Second method is based on a modified Fuzzy C-Means Clustering algorithm, which can handle missing data. The issues associated with this clustering algorithm are resolved to improve its effectiveness. Third method is based on interval approach. Missing values are replaced by interval ranges estimated by experts. Crisp efficiency scores are identified in similar lines to how DEA determines efficiency scores using the best set of weights. There exists no unique way to evaluate the effectiveness of these methods. Effectiveness of these methods is tested by choosing a complete dataset and assuming varying levels of data as missing. Best set of recovered missing values, based on the above methods, serves as a source to execute DEA. Results show that the DEA efficiency scores generated with recovered values are close within close proximity to the actual efficiency scores that would be generated with the complete data. As a summary, this thesis provides an effective and practical approach for replacing missing values needed for DEA.

This dissertation is a study of artificial intelligence for diagnosing the condition of high voltage bushings. The techniques include neural networks, genetic algorithms, fuzzy set theory, particle swarm optimisation, multi-classifier systems, factor analysis, principal component analysis, multidimensional scaling, data-fusion techniques, automatic relevance determination and autoencoders. The classification is done using Dissolved Gas Analysis (DGA) data based on field experience together with criteria from IEEEc57.104 and IEC60599. A review of current literature showed that common methods for the diagnosis of bushings are: partial discharge, DGA, tan- (dielectric dissipation factor), water content in oil, dielectric strength of oil, acidity level (neutralisation value), visual analysis of sludge in suspension, colour of the oil, furanic content, degree of polymerisation (DP), strength of the insulating paper, interfacial tension or oxygen content tests. All the methods have limitations in terms of time and accuracy in decision making. The fact that making decisions using each of these methods individually is highly subjective, also the huge size of the data base of historical data, as well as the loss of skills due to retirement of experienced technical staff, highlights the need for an automated diagnosis tool that integrates information from the many sensors and recalls the historical decisions and learns from new information. Three classifiers that are compared in this analysis are radial basis functions (RBF), multiple layer perceptrons (MLP) and support vector machines (SVM). In this work 60699 bushings were classified based on ten criteria. Classification was done based on a majority vote. The work proposes the application of neural networks with particle swarm optimisation (PSO) and genetic algorithms (GA) to compensate for missing data in classifying high voltage bushings. The work also proposes the application of fuzzy set theory (FST) to diagnose the condition of high voltage bushings. The relevance and redundancy detection methods were able to prune the redundant measured variables and accurately diagnose the condition of the bushing with fewer variables. Experimental results from bushings that were evaluated in the field verified the simulations. The results of this work can help to develop real-time monitoring and decision making tools that combine information from chemical, electrical and mechanical measurements taken from bushings.

Velleman beweist die Konsistenz der Existenz vereinfachte Gap 2 Moraste (ein Begriff gleichwertig zu den ursprünglichen Morasten, geschafft von Jensen). Wir haben einen noch einfachen Begriff des vereinfachten Morastes in der Dissertation vorgeschlagen, Details aufgefüllt und wesentlich auch einen verschiedenen Beweis des Satzes erfunden und zwar in beide Stufe des Forcingverfahrens. Wir benötigen auch keine Squarefunktionereihenfolge (die ganz Kohärenzvoraussetzung fehlt aber ist linear und konfinal) sondern ein ``erratendes'' Verfahren für Sequenze, das nicht fest ist und nicht die ganze Kohärenzbedigung erfüllt wie bei Velleman. Wir hoffen, wir haben so eingelegt die Basis für einen zukunftigen Beweis des allgemeines Falls n in ZFC. / Velleman proved the consistency of the existence of simplified gap 2 morasses (equivalent to the concrete morasses defined by Jensen) using a two stage forcing. We give an essentially different proof of the same result and fill up some details from Velleman's paper which were not clear or imcomplete. In fact the proof uses a simpler definition of simplified gap 2 morasses. We have also eliminated the use of square-like sequences in the second stage, employing a ``guessing'' procedure for sequences which are not fixed and do not satisfy full coherence requirement. With these steps we hope to have laid the foundation for a future proof of gap n morasses in ZFC.

The notion of surreal number was introduced by J.H. Conway in the mid 1970\'s: the surreal numbers constitute a linearly ordered (proper) class No containing the class of all ordinal numbers (On) that, working within the background set theory NBG, can be defined by a recursion on the class On. Since then, have appeared many constructions of this class and was isolated a full axiomatization of this notion that been subject of interest due to large number of interesting properties they have, including model-theoretic ones. Such constructions suggests strong connections between the class No of surreal numbers and the classes of all sets and all ordinal numbers. In an attempt to codify the universe of sets directly within the surreal number class, we have founded some clues that suggest that this class is not suitable for this purpose. The present work is an attempt to obtain an \"algebraic (set) theory for surreal numbers\" along the lines of the Algebraic Set Theory - a categorial set theory introduced in the 1990\'s: to establish abstract and general links between the class of all surreal numbers and a universe of \"surreal sets\" similar to the relations between the class of all ordinals (On) and the class of all sets (V), that also respects and expands the links between the linearly ordered class of all ordinals and of all surreal numbers. We have introduced the notion of (partial) surreal algebra (SUR-algebra) and we explore some of its category theoretic properties, including (relatively) free SUR-algebras (SA, ST). We have established links, in both directions, between SUR-algebras and ZF-algebras (the keystone of Algebraic Set Theory). We develop the first steps of a certain kind of set theory based (or ranked) on surreal numbers, that expands the relation between V and On. / A noção de número surreal foi introduzida por J.H. Conway em meados da década de 1970: os números surreais constituem uma classe (própria) linearmente ordenada No contendo a classe de todos os números ordinais (On) e que, trabalhando dentro da base conjuntista NBG, pode ser definida por uma recursão na classe On. Desde então, apareceram muitas construções desta classe e foi isolada uma axiomatização completa desta noção que tem sido objeto de estudo devido ao grande número de propriedades interessantes, incluindo entre elas resultados modelos-teóricos. Tais construções sugerem fortes conexões entre a classe No de números surreais e as classes de todos os conjuntos e todos os números ordinais. Na tentativa de codificar o universo dos conjuntos diretamente na classe de números surreais, encontramos algumas pistas que sugerem que esta classe não é adequada para esse fim. O presente trabalho é uma tentativa de se obter uma \"teoria algébrica (de conjuntos) para números surreais\" na linha da Teoria dos Algébrica dos Conjuntos - uma teoria categorial de conjuntos introduzida nos anos 1990: estabelecer links abstratos e gerais entre a classe de todos números surreais e um universo de \"conjuntos surreais\" emelhantes às relações entre a classe de todos os ordinais (On) e a classe de todos os conjuntos (V), que também respeite e expanda os links entre as classes linearmente ordenadas de todos ordinais e de todos os números surreais. Introduzimos a noção de álgebra surreal (parcial) (SUR-álgebra) e exploramos algumas das suas propriedades categoriais, incluindo SUR-álgebras (relativamente) livres (SA, ST). Nós estabelecemos links, em ambos os sentidos, entre SUR-álgebras e álgebras ZF (a pedra angular da Teoria Algébrica dos Conjuntos). Desenvolvemos os primeiros passos de um determinado tipo de teoria de conjuntos baseada (ou ranqueada) em números surreais, que expande a relação entre V e On.

Minimal Walks are a method of demonstrations in set theory and general topology. Although the main work of this document will be the construction of the L space, we intend to explain walk\'s fundamentals in a bit more detail. / Passeios mínimos são um método de demonstrações em teoria dos conjuntos e topologia geral. Apesar do trabalho principal dessa dissertação ser a existência do L espaço, a intenção é explicar os fundamentos dos passeios mínimos um tanto detalhadamente.

Neste trabalho apresentaremos o princípio seletivo S1(O;H ) que caracteriza a propriedade da diagonal Gd . Iremos também apresentar um jogo topológico G1(O;H ) induzido por esse princípio seletivo e estudaremos as relações entre esse jogo e a propriedade da diagonal Gd . Além disso, apresentaremos outros jogos topológicos e mostraremos quais são as relações destes com o jogo G1(O;H ). Finalmente, daremos algumas aplicações desses jogos e exemplos / In this work we present a selection principle S1(O;H) that characterizes the Gd -diagonal property. We also present a topological game G1(O;H ) induced by this selection principle and we study the relations between this game and the Gd -diagonal property. Besides that, we present other topological games and we show which are the relations between those topological games and the game G1(O;H ). Finally, we give some applications and examples.

Essa pesquisa tem como foco principal explorar como obras musicais de Villa-Lobos são passíveis de serem lidas ou analisadas por meio de uma racionalidade matemática. O intuito é buscar um enfoque didático alternativa didática para a abordagem de conceitos oriundos da Teoria dos Conjuntos, baseados nos estudos do matemático Georg Cantor (Teoria Ingênua dos Conjuntos) e nos estudos de Allen Forte (Teoria dos Conjuntos aplicada à Música). Busca-se trazer para o universo da Música e da Matemática ambas as teorias, por meio de um enfoque transdisciplinar, e situar o saber em regiões em que o aspecto afetivo já adquiriu níveis capazes de dar sentido ao conhecimento e propiciar a assimilação de significados relacionados à outra área. Em busca desses objetivos, e ainda estudar possíveis indicações das relações entre Matemática e Música em um cenário didático/pedagógico, essa obra lança mão da afetividade, transdisciplinaridade e pensamento analógico como forma de articular áreas aparentemente distantes, mas com forte semelhança em suas estruturas. Esse estudo pretende explorar (1) trabalhos que usaram a Teoria dos Conjuntos em análises de obras de Villa-Lobos, (2) processos criativos e composicionais presentes em obras musicais de Villa-Lobos, (3) técnicas matemáticas de análise musical, (4) tipos e estruturas matemáticas que possam auxiliar em análises musicais e verificar de que maneira a racionalidade matemática está presente na composição musical. Este estudo ao pesquisar trabalhos que usaram a Teoria dos Conjuntos em análise musical de obras de Villa-Lobos preenche uma lacuna na teoria musical; evidencia estruturas matemáticas que auxiliam na análise musical, mostrando a presença da racionalidade matemática. Uma das grandes contribuições desse trabalho é estabelecer relações de analogia entre conteúdos do currículo da matemática, frequentemente traduzidos por códigos numéricos, e aspectos da área musical, reconhecidos por sons. / This research is mainly focused on exploring how musical works by Villa-Lobos are likely to be read or analyzed by a mathematical rationality. The aim is to seek a didactic approach a teaching alternative in order to deal with concepts from the Set Theory, based on studies by mathematician Georg Cantor (Naive Set Theory), and from studies of Allen Forte (Set Theory applied to Music). It intentsthe following: to bring both theories into the world of Music and Mathematics through a transdisciplinary approach; to situate knowledge in areas where the affective aspect has already acquired levels able to make sense of such knowledge; to encourage the assimilation of related meanings from area to the other. In the pursuit of such goals, and still researching possible indications of the relationship between Mathematics and Music in a didactic/pedagogical scenario, this work makes use of affection and transdisciplinarity analogical thinking as a way of articulating seemingly distant areas with yet strong similarities in their structures. This research therefore explores (1) studies that used the Set Theory in analysis of works by Villa-Lobos, (2) creative and compositional processes present in musical works by Villa-Lobos, (3) mathematical techniques of musical analysis, (4) types and mathematical structures that can assist in musical analysis, and it verifies how the mathematical reasoning is present in the composite musical work. The present study, by researching papers that used the Set Theory in musical analysis of works by Villa-Lobos, fills a gap in music theory; it shows evidence of mathematical structures that can assist in musical analysis, showing the presence of mathematical reasoning. A major contribution of this work is to establish relations of analogy between the mathematical content of the curriculum, often translated by numerical codes, and aspects of Music recognized by sounds.

The notion of surreal number was introduced by J.H. Conway in the mid 1970\'s: the surreal numbers constitute a linearly ordered (proper) class No containing the class of all ordinal numbers (On) that, working within the background set theory NBG, can be defined by a recursion on the class On. Since then, have appeared many constructions of this class and was isolated a full axiomatization of this notion that been subject of interest due to large number of interesting properties they have, including model-theoretic ones. Such constructions suggests strong connections between the class No of surreal numbers and the classes of all sets and all ordinal numbers. In an attempt to codify the universe of sets directly within the surreal number class, we have founded some clues that suggest that this class is not suitable for this purpose. The present work is an attempt to obtain an \"algebraic (set) theory for surreal numbers\" along the lines of the Algebraic Set Theory - a categorial set theory introduced in the 1990\'s: to establish abstract and general links between the class of all surreal numbers and a universe of \"surreal sets\" similar to the relations between the class of all ordinals (On) and the class of all sets (V), that also respects and expands the links between the linearly ordered class of all ordinals and of all surreal numbers. We have introduced the notion of (partial) surreal algebra (SUR-algebra) and we explore some of its category theoretic properties, including (relatively) free SUR-algebras (SA, ST). We have established links, in both directions, between SUR-algebras and ZF-algebras (the keystone of Algebraic Set Theory). We develop the first steps of a certain kind of set theory based (or ranked) on surreal numbers, that expands the relation between V and On. / A noção de número surreal foi introduzida por J.H. Conway em meados da década de 1970: os números surreais constituem uma classe (própria) linearmente ordenada No contendo a classe de todos os números ordinais (On) e que, trabalhando dentro da base conjuntista NBG, pode ser definida por uma recursão na classe On. Desde então, apareceram muitas construções desta classe e foi isolada uma axiomatização completa desta noção que tem sido objeto de estudo devido ao grande número de propriedades interessantes, incluindo entre elas resultados modelos-teóricos. Tais construções sugerem fortes conexões entre a classe No de números surreais e as classes de todos os conjuntos e todos os números ordinais. Na tentativa de codificar o universo dos conjuntos diretamente na classe de números surreais, encontramos algumas pistas que sugerem que esta classe não é adequada para esse fim. O presente trabalho é uma tentativa de se obter uma \"teoria algébrica (de conjuntos) para números surreais\" na linha da Teoria dos Algébrica dos Conjuntos - uma teoria categorial de conjuntos introduzida nos anos 1990: estabelecer links abstratos e gerais entre a classe de todos números surreais e um universo de \"conjuntos surreais\" emelhantes às relações entre a classe de todos os ordinais (On) e a classe de todos os conjuntos (V), que também respeite e expanda os links entre as classes linearmente ordenadas de todos ordinais e de todos os números surreais. Introduzimos a noção de álgebra surreal (parcial) (SUR-álgebra) e exploramos algumas das suas propriedades categoriais, incluindo SUR-álgebras (relativamente) livres (SA, ST). Nós estabelecemos links, em ambos os sentidos, entre SUR-álgebras e álgebras ZF (a pedra angular da Teoria Algébrica dos Conjuntos). Desenvolvemos os primeiros passos de um determinado tipo de teoria de conjuntos baseada (ou ranqueada) em números surreais, que expande a relação entre V e On.