Thesis (Ph. D.)--University of Wisconsin, 1966. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

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This thesis is mainly concerned with the study of hypersymplectic structures in gauge theory. These structures arise via applications of the hypersymplectic quotient construction to the action of the gauge group on certain spaces of connections and Higgs fields. Motivated by Kobayashi-Hitchin correspondences in the case of hyperkähler moduli spaces, we first study the relationship between hypersymplectic, complex and paracomplex quotients in the spirit of Kirwan's work relating Kähler quotients to GIT quotients. We then study dimensional reductions of the ASD equations on $mathbb R^{2,2}$. We discuss a version of twistor theory for hypersymplectic manifolds, which we use to put the ASD equations into Lax form. Next, we study Schmid's equations from the viewpoint of hypersymplectic quotients and examine the local product structure of the moduli space. Then we turn towards the integrability aspects of this system. We deduce various properties of the spectral curve associated to a solution and provide explicit solutions with cyclic symmetry. Hitchin's harmonic map equations are the split signature analogue of the self-duality equations on a Riemann surface, in which case it is known that there is a smooth hyperkähler moduli space. In the case at hand, we cannot expect to obtain a globally well-behaved moduli space. However, we are able to construct a smooth open set of solutions with small Higgs field, on which we then analyse the hypersymplectic geometry. In particular, we exhibit the local product structures and the family of complex structures. This is done by interpreting the equations as describing certain geodesics on the moduli space of unitary connections. Using this picture we relate the degeneracy locus to geodesics with conjugate endpoints. Finally, we present a split signature version of the ADHM construction for so-called split signature instantons on $S^2 imes S^2$, which can be given an interpretation as a hypersymplectic quotient.

Mémoire (M.A.)--Université du Québec à Chicoutimi, 1999. / Document électronique également accessible en format PDF. CaQCU

This thesis investigates the interplay between explanatory issues in special relativity and the theory's metaphysical foundations. Special attention is given to the 'dynamical approach' to relativity, promoted primarily by Harvey Brown and collaborators, according to which the symmetries of dynamical laws are explanatory of relativistic effects, inertial motion, and even the Minkowskian geometrical structure of a specially relativistic world. The thesis begins with a review of Einstein's 1905 introduction to special relativity, after which brief historical introductions are given for the standard 'geometrical' approach to relativity and the unorthodox 'dynamical' approach. After a critical review of recent literature on the topic, the dynamical approach is shown to be in need of a metaphysical package that would undergird the explanatory claims mentioned above. It is argued that the dynamical approach is best understood as a form of relationalism - in particular, as a relativistic form of 'regularity relationalism', promoted recently by Nick Huggett. According to this view, some portion of a world's geometrical structure actually supervenes upon the symmetries of the best-system dynamical laws for a material ontology endowed with a primitive sub-metrical structure. To explore the plausibility of this construal of the dynamical approach, a case study is carried out on solutions to the Klein-Gordon equation. Examples are found for which the field values, when purged of all spatiotemporal structure but their induced topology, are still arguably best-systematized by the Klein-Gordon equation itself. This bolsters the plausibility of the claim that some system of field values, endowed with mere sub-metrical structure, might have as its best-systems dynamical laws a (set of) Lorentz-covariant equation(s), on which Minkowski geometrical structure would supervene. The upshot is that the dynamical approach to special relativity can be defended as what might be called an ontologically and ideologically relationalist approach to Minkowski spacetime structure. The chapters refer regularly to three appendices, which include a brief introduction to topological and differentiable spaces.

Made available in DSpace on 2016-04-27T16:57:36Z (GMT). No. of bitstreams: 1 Acylena Coelho Costa.pdf: 4495391 bytes, checksum: 036e84e7258e8b0ddf6bc886cce6222d (MD5) Previous issue date: 2015-04-24 / Universidade do Estado do Pará / This research aims to analyze how the authors of textbooks organized the activities proposed in relation to the study of Line and Planning for the teaching of Analytic Geometry in Space. Analyses of textbooks substantiate mainly on Anthropological Theory of the Didactic (TAD), with regard to praxeologias proposed by Chevallard (1999) and in didactic variables for the teaching of Analytic Geometry in Space established by Lebeau (2009). Based on the theoretical framework adopted we conducted a qualitative investigation of documentary type, based on a literature review on four textbooks Analytic Geometry devoted to higher education. The methodology used in our research was supported in manual analysis methodology developed by Chaachoua (2014a) analyzing the textbooks the following: time of writing, representation, structure, ecological analysis and praxeological analysis. For regards the praxeological analysis identified six types present in the analyzed manual tasks, namely to: determine the equation of the line in space, determining the straight condition of parallelism in space, to determine the alignment condition of points in space, present plane equation in space as a property of orthogonality, determine a plan characterized by two intersecting lines and characterize algebraically the parallelogram. Among the findings it can be inferred that the authors favor an algebraic modeling of mathematical objects, as well as the techniques adopted by them are situated in the field of Linear Algebra and Analytic Geometry / A presente pesquisa tem como objetivo analisar como os autores de livros didáticos organizaram as atividades propostas no que se refere ao estudo da Reta e do Plano para o ensino da Geometria Analítica no Espaço. As análises dos livros didáticos fundamentaram-se essencialmente na Teoria Antropológica do Didático (TAD), quanto às praxeologias, propostas por Chevallard (1999) e nas variáveis didáticas para o ensino da Geometria Analítica no Espaço estabelecidas por Lebeau (2009). Com base no referencial teórico adotado realizamos uma investigação de caráter qualitativo do tipo documental, partindo de um levantamento bibliográfico em quatro livros didáticos de Geometria Analítica destinados ao ensino superior. A metodologia adotada em nossa pesquisa foi subsidiada na metodologia de análise de manuais desenvolvida por Chaachoua (2014a) analisando nos livros didáticos os seguintes aspectos: momento da edição, representatividade, estrutura, análise ecológica e análise praxeológica. Em relação a análise praxeológica identificamos seis tipos de tarefas presentes nos manuais analisados, a saber: determinar a equação da reta no espaço, determinar a condição de paralelismo de retas no espaço, determinar a condição de alinhamento de pontos no espaço, apresentar a equação do plano no espaço como uma propriedade da ortogonalidade, determinar um plano caracterizado por duas retas secantes e caracterizar algebricamente o paralelogramo. Dentre os resultados encontrados é possível inferir que os autores privilegiam uma modelização algébrica dos objetos matemáticos, bem como as técnicas adotadas pelos mesmos encontram-se situadas no campo da Álgebra Linear e da Geometria Analítica

Made available in DSpace on 2016-07-27T13:52:25Z (GMT). No. of bitstreams: 1 THALITTA FERNANDES DE CARVALHO PERES.pdf: 6549969 bytes, checksum: 4d5738599189441a4843816f3efc080b (MD5) Previous issue date: 2010-09-29 / Mathematics is a discipline characterized by a slow learning curve, the students considering it difficult to learn its concepts. For example the teaching of spatial geometry has been distinguished by its abandonment in the classroom. This paper aims at identifying contributions and challenges of teaching spatial geometry, with an arrangement based on the theory of developmental education. What is questioned is: given the concrete conditions, how should one develop education for students in order to form the concept of spatial geometry? Moreover: What sociocultural factors are present affecting the learning of spatial geometry? How do students perceive this type of organization providing education? Does the teacher perceives it as its activity, to teach content based on this type of organization of teaching? And what is the teacher opinion of this form of education? The research was mainly based on the theories of Vygotsky and Davydov, the specific aims being: Identify and analyze the relationships between students and mathematics and with the spatial geometry a day to day basis - identify and analyze the socio-cultural factors in the specific context of school and classroom, that affect learning spatial geometry - consider the opinion of students and teacher, concerning this way of organizing the teaching of spatial geometry. For both, a qualitative study, consisting of a teaching experiment based on the assumptions of Davydov, has been carried out. Data were collected through observations, using semi-structured instruments for diagnostic assessments. The research subjects were math teacher and 28 students in a class of 2nd year of a public high school. Data analysis revealed the following results: motivation of students during the teaching experiment, enhanced knowledge of the contents after the historical logic analysis, a new alternative for providing education to the research subjects, the concept formation of the majority of students, improvement in participation of some students, not even reaching the theoretical thinking, due to various socio-cultural factors, the teaching experiment showed evidence of qualitative changes in teacher performance. It is believed that the main contribution of this research was to show an alternative way of organizing the teaching of mathematics, particularly the teaching of the concept of volume of geometric solids. It is believed that even with the difficulties and contradictions in public school and the children's school life, it is possible to organize an education grounded on the theory of developmental education and contribute to the formation of theoretical thinking in most students. / A matemática é uma disciplina marcada pelo baixo desempenho na aprendizagem, cujos conceitos são considerados difíceis de aprender. E o ensino de geometria espacial tem sido abalizado por seu abandono nas salas de aula. O presente trabalho tem como objetivo geral identificar as contribuições e os desafios de se ensinar geometria espacial organizado, com base na teoria do ensino desenvolvimental. O que se questiona é: dadas as condições concretas, como se desenvolve o ensino para que os alunos formem o conceito de geometria espacial? E ainda: Que fatores socioculturais se apresentam afetando a aprendizagem da geometria espacial? De que modo os alunos percebem este tipo de organização de ensino? E o professor, como percebe sua atividade de ensinar um conteúdo com base nesse tipo de organização do ensino, e qual seria sua visão sobre essa forma de ensino? A pesquisa fundamentando-se principalmente nas teorias de Vygotsky e Davydov. Os objetivos específicos da pesquisa foram: - identificar e analisar as relações dos alunos com a matemática e com a geometria espacial, em sua vida cotidiana; - identificar e analisar os fatores socioculturais no contexto concreto da escola e da sala de aula que interferem na aprendizagem de geometria espacial; - analisar a visão dos alunos e do professor acerca desse modo de organização do ensino de geometria espacial. Para tanto, realizou-se uma pesquisa qualitativa que consistiu num experimento de ensino baseado nos pressupostos de Davydov. Os dados foram coletados por meio de observações, entrevistas semi-estruturadas, instrumentos de avaliações diagnósticas. Os sujeitos da pesquisa foram o professor de matemática e os 28 alunos de uma turma de 2º ano do ensino médio, de uma escola pública. A análise dos dados revelou os seguintes resultados: Motivação dos alunos durante o ensino experimental; conhecimento intensificado do conteúdo após a análise lógica histórica; uma nova alternativa de organização de ensino aos sujeitos da pesquisa; a formação de conceitos da maioria dos alunos; melhora na participação de alguns alunos, mesmo não atingindo o pensamento teórico, devido a diversos fatores socioculturais; o experimento de ensino mostrou indícios de mudanças qualitativas na atuação do professor. Acredita-se que a principal contribuição desta pesquisa consistiu em mostrar um caminho alternativo de organização do ensino de matemática, particularmente o ensino do conceito de volume de sólidos geométricos. Acredita-se que mesmo com as dificuldades e contradições presentes na escola pública e na vida escolar dos alunos, é possível realizar o ensino embasado na teoria do ensino desenvolvimental e contribuir para a formação do pensamento teórico da maioria dos alunos.

Made available in DSpace on 2016-04-27T16:57:47Z (GMT). No. of bitstreams: 1 EDM - Marcia Yolanda Rosalves.pdf: 3135012 bytes, checksum: 57c3dae3f651e69ce218a2790d1f5f7b (MD5) Previous issue date: 2006-10-27 / This research involves the teaching and learning of the geometry of space (three-dimensional geometry) in school mathematics. It considers, in particular, the relationships between geometrical objects and their representations in the plane. The works of Parzysz (1988; 1993), which served as the theoretical base for this study, point to the difficulties students that experience in interpreting representations of three-dimensional objects, in terms of construction (codification) and interpretation (decodification), as well as the conflict between the poles of seeing and knowing. The results presented by Parzysz concern experimentations carried out in the conventional paper-and-pencil environment. Considering the limitations of this environment and the difficulties associated with the identification of spatial relations given its static nature, the dynamic geometry environment of Cabri 3D was chosen as the context for this study. The aim was to investigate the role of the dynamic representation of this software in the study of space geometry. More precisely, the study seeks to analyse the possibilities related to the poles of seeing and knowing in the interactions of subjects with the tools and representations of Cabri 3D. The empirical part of the research involved the development of an experimental study strongly inspired by the methodology of Design Experiments, using the perspective of Steffe and Thompson (2000) and Cobb et al. (2003). High school students from a public-sector school in the city of São Paulo participated in this experiment. The results show that, in certain situation, the loss in information associated with Cabri 3D representation of spatial objects are less than in the paper-and-pencil environment. The evidence also suggested that both the dynamic aspect with the potential to manipulate and change the point of view onto the object represented and the treatment the enriching of representations made possible by the use of the construction tools aided in the process of decodification, amplifying the interpretation of the drawing on the part of the student and enabling a better use of perceptive inferences / A presente pesquisa está inserida no contexto do ensino-aprendizagem da Geometria Espacial na Educação Básica, referindo-se, em particular, às relações entre os objetos geométricos e suas representações planas. Os trabalhos de Parzysz (1988; 1993) destacam as dificuldades dos alunos com a representação de objetos tridimensionais, no que se refere à sua elaboração (codificação) e interpretação (decodificação), bem como o conflito gerado pelos pólos do visto e sabido que são as bases do presente estudo. Os resultados apresentados nas pesquisas desse autor referem-se a experimentações no ambiente convencional de papel&lápis. Considerando as limitações desse ambiente e as dificuldades de identificação de relações espaciais dado seu caráter estático, optou-se por utilizar o ambiente de geometria dinâmica Cabri 3D. Assim, o estudo teve por objetivo investigar o papel das representações dinâmicas nesse software. Mais precisamente, pretendeu-se analisar as possibilidades de gestão dos pólos do visto e do sabido nas interações dos sujeitos com as ferramentas e representações do Cabri 3D. O desenvolvimento da pesquisa deu-se por meio de um estudo experimental fortemente inspirado na metodologia do Design Experiment na perspectiva de Steffe e Thompson (2000) e Cobb et al (2003), sendo realizado com alunos do Ensino Médio de uma escola pública da cidade de São Paulo. Os resultados mostraram que, em determinadas situações, as perdas de informações no Cabri 3D são menores que no ambiente papel&lápis. Existem também evidências de que tanto o aspecto dinâmico com possibilidades de manipular e mudar o ponto de vista do objeto representado como o de tratamento , enriquecimento da representação no uso das ferramentas de construção, auxiliam no processo de decodificação, ampliando a interpretação do desenho por parte dos alunos e levando-os a um melhor aproveitamento das interferências perceptivas

The diploma thesis deals with metric tasks in space, using synthetic and analytical geometry. In addition to explaining the different approaches, there is a set of examples to practice. The solution of the examples is part of the Portal of High School Mathematics (Portál středoškolské matematiky), where we can and analytical solutions, synthetic numerical solutions and synthetic constructional solutions.

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