Thesis (Ph. D.)--University of Michigan, 1944. / Bibliography: p. 115-119.

Thesis (Ph. D.)--University of Michigan, 1944. / Bibliography: p. 115-119.

The subject of the study is a theory of positive and negative inquiry with emphasis in mathematics. The purposes of this study are to examine the historical development of systematic inquiry in mathematics, to identify the nature of positive and negative inquiry, to propose and develop an interrelated set of propositions regarding positive and negative inquiry, and to relate the proposition of the theory to certain basic concepts of trigonometry.

Orientador: José Carlos Rodrigues / Banca: Cristiane Nespoli de Oliveira / Banca: Larissa Ferreira Marques / Resumo: O estudo da Matemática, em especial da trigonometria é de suma importância para o desenvolvimento das ciências desde os primórdios dos tempos. Nesse trabalho abordamos o estudo das razões trigonométricas do triângulo retângulo até o círculo trigonométrico. Em análise feita na proposta curricular do Estado de São Paulo, detectamos a necessidade de novas abordagens de ensino que venham melhorar a qualidade deste nas salas de aula das escolas públicas, tornando o estudo do tema mais prazeroso e atrativo. O cenário presente na maioria das escolas públicas, destaca a falta de professor e o desinteresse dos alunos pela matemática, fatores que agravam a defasagem de conteúdo em relação ao ano/série que estão cursando. Diante desse cenário, apresentamos neste trabalho uma Sequência Didática que prevê a construção de materiais pedagógicos confeccionados pelos próprios alunos que permitam dinamizar as aulas, melhorando o estudo da trigonometria e oportunizando aos alunos à construção do próprio conhecimento, despertando mais interesse pelas aulas. A Sequência didática foi testada em uma turma da 2ª série de Ensino Médio de uma escola pública de área rural na cidade de Caiuá, São Paulo, utilizando os materiais em sala de aula e analisando os resultados obtidos. A proposta deste estudo é encontrar opção que viabilize o ensino do tema, otimizando o tempo de ensino, trabalhando estratégias como o uso de materiais concretos e ao final das atividades concluir que a aplicação dessas... / Abstract: The study of mathematics, especially trigonometry, is of paramount importance for the development of science since the dawn of time. In this paper we will study the trigonometric ratios of the rectangle triangle to the trigonometric circle. In an analysis made in the curricular proposal of the State of São Paulo, we detected the need for new teaching approaches that will improve its quality in public school classrooms, making the study of the subject more pleasant and attractive. The scenario that exists in most public schools, is the lack of teacher and students' lack of interest in mathematics, factors that aggravate the mismatch of content in relation to the year / grade they are attending. Given this scenario, we present in this paper some proposals to improve the study of the theme and two materials made, with the help of students, to streamline the classes, giving students the opportunity to seek their own knowledge, arousing more interest in the classes. The applicability was tested in a high school 2nd grade class of a rural public school in the city of Caiuá, São Paulo, using the materials in the classroom and analyzing the results obtained. The purpose of this study is to find an option that enables the teaching of the subject, optimizing the teaching time, as well as working new approaches with the use of concrete subjects and at the end of the activities conclude that the use of these approaches may contribute to the improvement of the teaching process. learning, ... . / Mestre

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This thesis is concerned with the spectral theory of the Laplacian on non-Euclidean spaces and its intimate links with harmonic analysis and the theory of special functions. More specifically, it studies the spectral theory of the Laplacian on the quotients M = Γ\G/K and X = G/K, where G is a connected semisimple Lie group, K is a maximal compact subgroup of G and Γ is a discrete subgroup of G.

This thesis uses algebraic and combinatorial methods to study subsets of the Desarguesian plane IIq = PG(2, q). Emphasis, in particular, is given to complete (k, n)-arcs and plane projective curves. Known Diophantine equations for subsets of PG(2, q), no more than n of which are collinear, have been applied to k-arcs of arbitrary degree. This yields a new lower bound for complete (k, n)-arcs in PG(2, q) and is a generalization of a classical result of Barlotti. The bound is one of few known results for complete arcs of arbitrary degree and establishes new restrictions upon the parameters of associated projective codes. New results governing the relationship between (k, 3)-arcs and blocking sets are also provided. Here, a sufficient condition ensuring that a blocking set is induced by a complete (k, 3)-arc in the dual plane q is established and shown to complement existing knowledge of relationships between k-arcs and blocking sets. Combinatorial techniques analyzing (k, 3)-arcs in suitable planes are then introduced. Utilizing the numeric properties of non-singular cubic curves, plane (k, 3)-arcs satisfying prescribed incidence conditions are shown not to attain existing upper bounds. The relative sizes of (k, 3)-arcs and non-singular cubic curves are also considered. It is conjectured that m3(2, q), the size of the largest complete (k, 3)-arc in PG(2, q), exceeds the number of rational points on an elliptic curve. Here, a sufficient condition for its positive resolution is given using combinatorial analysis. Exploiting its structure as a (k, 3)-arc, the elliptic curve is then considered as a method of constructing cubic arcs and results governing completeness are established. Finally, classical theorems relating the order of the plane q to the existence of an elliptic curve with a specified number of rational points are used to extend theoretical results providing upper bounds to t3(2, q), the size of the smallest possible complete (k, 3)-arc in PG(2, q).

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