Wir betrachten eine Reihe von Trivialitäts- resultaten und No-Go-Theoremen aus der Axiomatischen Quantenfeldtheorie. Von besonderem Interesse ist Haags Theorem. Im Wesentlichen sagt es aus, dass der unitäre Intertwiner des Wechselwirkungsbildes nicht existiert oder trivial ist. Als wichtigste Voraussetzung von Haags Theorem arbeiten wir die unitäre Äquivalenz heraus und unterziehen die kanonische Störungstheorie skalarer Felder einer Kritik um zu argumentieren, dass die kanonisch renormierte Quantenfeldtheorie Haags Theorem umgeht, da sie genau diese Bedingung nicht erfüllt. Der Hopfalgebraische Zugang zur perturbativen Quantenfeldtheorie bietet die Möglichkeit, Dyson-Schwinger-und Renormierungsgruppengleichungen mathematisch sauber herzuleiten, wenn auch mit rein kombinatorischem Ausgangspunkt. Wir präsentieren eine Beschreibung dieser Methode und diskutieren eine gewöhnliche Differentialgleichung für die anomale Dimension des Photons. Eine Spielzeugmodellversion dieser Gleichung lässt sich exakt lösen; ihre Lösung weist eine interessante nichtstörunsgtheoretische Eigenschaft auf, deren Auswirkungen auf die laufende Kopplung und die Selbstenergie des Photons wir untersuchen. Solche nichtperturbativen Beiträge mögen die Existenz eines Landau-Pols ausschliessen, ein Sachverhalt, den wir ebenfalls diskutieren. Unter der Arbeitshypothese, dass die anomale Dimension eines Quantenfeldes in die Klasse der resurgenten Funktionen fällt, studieren wir, welche Bedingungen die Dyson-Schwinger-und Renormierungsgruppengleichungen an ihre Transreihe stellen. Wir stellen fest, dass diese unter bestimmten Bedingungen kodieren, wie der perturbative Sektor den nichtperturbativen vollständig determiniert. / We review a package of triviality results and no-go theorems in axiomatic quantum field theory. Of particular interest is Haag''s theorem. It essentially says that the unitary intertwiner of the interaction picture does not exist unless it is trivial. We single out unitary equivalence as the most salient provision of Haag''s theorem and critique canonical perturbation theory for scalar fields to argue that canonically renormalised quantum field theory bypasses Haag''s theorem by violating this very assumption. The Hopf-algebraic approach to perturbative quantum field theory allows us to derive Dyson-Schwinger equations and the Callan-Symanzik equation in a mathematically sound way, albeit starting with a purely combinatorial setting. We present a pedagogical account of this method and discuss an ordinary differential equation for the anomalous dimension of the photon. A toy model version of this equation can be solved exactly; its solution exhibits an interesting nonperturbative feature whose effect on the running coupling and the self-energy of the photon we investigate. Such nonperturbative contributions may exclude the existence of a Landau pole, an issue that we also discuss. On the working hypothesis that the anomalous dimension of a quantum field falls into the class of resurgent functions, we study what conditions Dyson-Schwinger and renormalisation group equations impose on its resurgent transseries. We find that under certain conditions, they encode how the perturbative sector determines the nonperturbative one completely.

Começamos definindo a cohomologia clássica de De Rham e provamos alguns resultados que nos permitem calcular tal cohomologia de algumas variedades diferenciáveis. Com o intuito de provar o Teorema de De Rham, escolhemos fazer a demonstração utilizando a noção de feixes, que se mostra como uma generalização da ideia de cohomologia. Como a cohomologia de De Rham não é a única que se pode definir numa variedade, a questão da unicidade dá origem a teoria axiomática de feixes, que nos dará uma cohomologia para cada feixe dado. Mostraremos que a partir da teoria axiomática de feixes obtemos cohomologias, além das cohomologias clássicas de De Rham, a cohomologia clássica singular e a cohomologia clássica de Cech e mostraremos que essas cohomologias obtidas a partir da noção axiomática são isomorfas as definições clássicas. Concluiremos que se nos restringirmos a apenas variedades diferenciáveis, essas cohomologias são unicamente isomorfas e este será o teorema de De Rham. / We begin by defining De Rhams classical cohomology and we prove some results that allow us a calculation of the cohomology of some differentiable manifolds. In order to prove De Rhams Theorem, we chose to make a demonstration using a notion of sheaves, which is a generalization of the idea of cohomology. Since De Rhams cohomology is not a only one that can be made into a variety, the question of unicity gives rise to axiomatic theory of sheaves, which give us a cohomology for each sheaf given. We will show that from the axiomatic theory of sheaves we obtain cohomologies, besides the classical cohomologies of De Rham, a singular classical cohomology and a classical cohomology of Cech and we will show that cohomologies are obtained from the axiomatic notion are classic definitions. We will conclude that if we restrict ourselves to only differentiable manifolds, these cohomologies are uniquely isomorphic and this will be De Rhams theorem.

The aesthetic aspect of a functional product is growing to be an important reason for the consumers choice to buy the product. Despite this importance, aesthetics has not generally been incorporated into engineering design which makes much sense of functional and ergonomic designs. The study presented in this thesis aims to remedy this observed gap. The study focuses on the integration of aesthetic attributes with functional attributes of a product and on the quantification of the aesthetic principle from fine arts into design variables of the product. In particular, two hypotheses underlie this study: (1) design variables can be classified in terms of their relevance to functional, ergonomic, and aesthetic attributes, and (2) a particular aesthetic principle, namely visual balance, helps to achieve an improved aesthetic product.<p>The cell phone is used to ground this study. A statistic experiment using the cell phone product positively tests the first hypothesis, resulting in two design variable which are only related to the aesthetic attribute of the cell phone product. The study of the visual balance principle results in a more general formula which relates design variables to visual balance with consideration of both geometry and color of the cell phone product. Finally, another statistic experiment is designed, which positively tests the second hypothesis.<p>This study concludes: (1) the effective integration of aesthetics with function and ergonomics requires an analysis and classification of design variables, and (2) there is a potential to quantify all aesthetic principles from fine arts into design variables.

The aesthetic aspect of a functional product is growing to be an important reason for the consumers choice to buy the product. Despite this importance, aesthetics has not generally been incorporated into engineering design which makes much sense of functional and ergonomic designs. The study presented in this thesis aims to remedy this observed gap. The study focuses on the integration of aesthetic attributes with functional attributes of a product and on the quantification of the aesthetic principle from fine arts into design variables of the product. In particular, two hypotheses underlie this study: (1) design variables can be classified in terms of their relevance to functional, ergonomic, and aesthetic attributes, and (2) a particular aesthetic principle, namely visual balance, helps to achieve an improved aesthetic product.<p>The cell phone is used to ground this study. A statistic experiment using the cell phone product positively tests the first hypothesis, resulting in two design variable which are only related to the aesthetic attribute of the cell phone product. The study of the visual balance principle results in a more general formula which relates design variables to visual balance with consideration of both geometry and color of the cell phone product. Finally, another statistic experiment is designed, which positively tests the second hypothesis.<p>This study concludes: (1) the effective integration of aesthetics with function and ergonomics requires an analysis and classification of design variables, and (2) there is a potential to quantify all aesthetic principles from fine arts into design variables.

In the past little work has been done to characterize the models of various mereotopological systems. This thesis focuses on Asher and Vieu's first-order mereotopology which evolved from Clarke's Calculus of Individuals. Its soundness and completeness proofs with respect to a topological translation of the axioms provide only sparse insights into structural properties of the mereotopological models. To overcome this problem, we characterize these models with respect to mathematical structures with well-defined properties - topological spaces, lattices, and graphs. We prove that the models of the subtheory RT− are isomorphic to p-ortholattices (pseudocomplemented, orthocomplemented). Combining the advantages of lattices and graphs, we show how Cartesian products of finite p-ortholattices with one multiplicand being not uniquely complemented (unicomplemented) gives finite models of the full mereotopology. Our analysis enables a comparison to other mereotopologies, in particular to the RCC, of which lattice-theoretic characterizations exist.

In the past little work has been done to characterize the models of various mereotopological systems. This thesis focuses on Asher and Vieu's first-order mereotopology which evolved from Clarke's Calculus of Individuals. Its soundness and completeness proofs with respect to a topological translation of the axioms provide only sparse insights into structural properties of the mereotopological models. To overcome this problem, we characterize these models with respect to mathematical structures with well-defined properties - topological spaces, lattices, and graphs. We prove that the models of the subtheory RT− are isomorphic to p-ortholattices (pseudocomplemented, orthocomplemented). Combining the advantages of lattices and graphs, we show how Cartesian products of finite p-ortholattices with one multiplicand being not uniquely complemented (unicomplemented) gives finite models of the full mereotopology. Our analysis enables a comparison to other mereotopologies, in particular to the RCC, of which lattice-theoretic characterizations exist.

Começamos definindo a cohomologia clássica de De Rham e provamos alguns resultados que nos permitem calcular tal cohomologia de algumas variedades diferenciáveis. Com o intuito de provar o Teorema de De Rham, escolhemos fazer a demonstração utilizando a noção de feixes, que se mostra como uma generalização da ideia de cohomologia. Como a cohomologia de De Rham não é a única que se pode definir numa variedade, a questão da unicidade dá origem a teoria axiomática de feixes, que nos dará uma cohomologia para cada feixe dado. Mostraremos que a partir da teoria axiomática de feixes obtemos cohomologias, além das cohomologias clássicas de De Rham, a cohomologia clássica singular e a cohomologia clássica de Cech e mostraremos que essas cohomologias obtidas a partir da noção axiomática são isomorfas as definições clássicas. Concluiremos que se nos restringirmos a apenas variedades diferenciáveis, essas cohomologias são unicamente isomorfas e este será o teorema de De Rham. / We begin by defining De Rhams classical cohomology and we prove some results that allow us a calculation of the cohomology of some differentiable manifolds. In order to prove De Rhams Theorem, we chose to make a demonstration using a notion of sheaves, which is a generalization of the idea of cohomology. Since De Rhams cohomology is not a only one that can be made into a variety, the question of unicity gives rise to axiomatic theory of sheaves, which give us a cohomology for each sheaf given. We will show that from the axiomatic theory of sheaves we obtain cohomologies, besides the classical cohomologies of De Rham, a singular classical cohomology and a classical cohomology of Cech and we will show that cohomologies are obtained from the axiomatic notion are classic definitions. We will conclude that if we restrict ourselves to only differentiable manifolds, these cohomologies are uniquely isomorphic and this will be De Rhams theorem.

Manufacturing facilities need control for sustainability and longevity. If no control is provided for the manufacturing facility, then chaos can be unleashed causing much alarm. Therefore, it is essential to understand how control can be utilized to support the manufacturing facility and the corresponding manufacturing processes. This thesis will walk through a tool to help provide control and that tool is a Manufacturing Execution System (MES). Thisthesis will start with research to defineMESand its implications, then will work into the development of MES from the ground up. The design process willbe systematic and utilize the Collective System Design (CSD) approach with the aiding tool of the axiomatic decomposition map. Then examples will be given for the implementation and execution of the decomposition map as it relates to inventory and traceability. Finalwork will show the 7 FRs ofmanufacturing and how they are applicable to MES with given examples. Throughout the entire design and implementation, the initial hypothesis will be evaluated to determine if MES can provide the control requiredfor a robust manufacturing facility.

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)

Made available in DSpace on 2016-04-27T16:57:50Z (GMT). No. of bitstreams: 1 EDM - Michaela C Schon.pdf: 1931458 bytes, checksum: 5cde0886ff87d5dafb588e52ab96ed50 (MD5) Previous issue date: 2006-11-06 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This paper aimed the realization of a study concerning the philosophical epistemology of the concept of number, in which it still makes sense to ask: What is number? In this perspective, we have assumed as problematic the philosophical duality of the conceptualizations of numbers, according to Axiomatic (proposed by Peano) e by the Set Theory and Logics (proposed by Russell), being the Conceptualization of Number the problem of this research, concerning the possibility of introducing an ultimate definition to this concept. The focus of this research is in the polemics that exists about the number introduced by Russell (1872-1970) contrary to Piano s (1858-1932), taking as a basis Otte s criticism, introduced in the article: B. Russell Introduction to Mathematical Philosophy , 2001. The research was developed using, as a reference, the sense of Complementarity, as well as using proper qualitative methodological research procedures. As a conclusion, we are able to claim that numbers are: on one hand, characteristics of certain classes and, on the other hand, operative concepts. This way, the existence of polemics between philosophers like Frege and Russell, who have favored predicative aspects, that is, they define number in terms of cardinality and, others like Grassmann, Dedekind and Peano who have highlighted the ordinal numbers, justify Otto s proposition of complementarity between the approaches. The possibility of having cognitive and didactical consequences on the teaching in the use of one or another approach of conceptualization of the number or both, as Otte intends, makes this study a contribution to Mathematical Education / Este trabalho objetivou realizar um estudo sobre a epistemologia filosófica do conceito de número, na qual ainda faz sentido o questionamento: O que é número? Nesta perspectiva, assumiu-se como problemática a dualidade filosófica das conceituações de número, sustentadas pela Axiomática (proposta por Peano) e pela Teoria dos Conjuntos e Lógica (proposta por Russell), sendo o problema de pesquisa a Conceituação de Número frente a essa dualidade e à possibilidade de ser apresentada uma definição em definitivo ao conceito de número. O foco da presente pesquisa está na polêmica existente entre a concepção de número apresentada por Russell (1872-1970) contraposta à de Peano (1858-1932), tomando-se por base as críticas de Otte, apresentadas no artigo: B. Russell Introduction to Mathematical Philosophy , de 2001. A pesquisa desenvolveu-se tendo por referência a noção de Complementaridade, tendo sido utilizados procedimentos metodológicos adequados às pesquisas qualitativas. Como conclusão pode-se afirmar que os números são: por um lado, características de certas classes e, por outro, conceitos operativos. Deste modo, a existência da polêmica entre filósofos como Frege e Russell, que favoreceram os aspectos predicativos, isto é, definem os números em termos de cardinalidade e, outros como Grassmann, Dedekind e Peano que destacam os números ordinais, justifica a proposição de Otte da complementaridade entre as abordagens. A possibilidade de existirem conseqüências cognitivas e didáticas na utilização no ensino de uma ou outra abordagem da conceituação de número ou de ambas como pretende Otte torna, este estudo, uma contribuição para a Educação Matemática