Essa dissertação analisará uma abordagem investigativa de ensino de funções trigonométricas que prioriza a compreensão da relação entre movimentos circulares em diferentes velocidades com a formação gráfica gerada por esses movimentos. Com o auxílio do software Geogebra, diferentes movimentos foram criados, o que proporcionou a investigação gráfica por parte dos alunos. A atividade foi realizada no laboratório de informática onde, constantemente, houve investigação por parte dos alunos e intervenções significativas por parte do professor. Escolheu-se para essa pesquisa uma análise qualitativa embasada no processo descritivo das ações ocorridas em sala de aula. Para conhecer as características dessa abordagem, foi utilizado um estudo de casos. Após a atividade, os alunos conseguiram interpretar os principais movimentos gerados na circunferência e traduzi-los na sua forma gráfica. A análise mostra que os alunos não somente conseguiram desenvolver significados aos movimentos circulares, como também interpretaram corretamente situações cotidianas estabelecidas pelo professor ao fim do trabalho / This dissertation will analyze an investigative approach to the teaching of trigonometric functions that prioritizes the understanding of the relation between circular movements at different speeds with the graphical formation generated by these movements. With the help of the software Geogebra, different movements were created, which provided the graphic investigation by the students. The activity was carried out in the computer lab where, constantly, there was investigation by the students and significant interventions by the teacher. For this research, a qualitative analysis based on the descriptive process of the actions taken in the classroom was chosen. To know the characteristics of this approach, a case study was used. After the activity, the students were able to interpret the main movements generated on the circumference and translate them into their graphic form. The analysis shows that the students not only managed to develop meanings to the circular movements, but also correctly interpreted daily situations established by the teacher at the end of the work

Spherical means are a widespread model in modern imaging modalities like photoacoustic tomography. We develop Fourier based algorithms for an efficient computation of mean values. Furthermore we consider iterative reconstruction schemes, where we employ different regularization techniques.

Made available in DSpace on 2017-07-21T20:56:30Z (GMT). No. of bitstreams: 1 SUELLEN IOCHUCKI.pdf: 6944983 bytes, checksum: ba58e93c2b6c033ab9341effdee604c5 (MD5) Previous issue date: 2016-06-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This work presents an approach to trigonometry content using Geogebra and Maxima programs. It provides a road map of how trigonometry can be worked in the classroom divided into stages . It proposes applications of trigonometry in different areas of knowledge and suggests the introduction of the approach of periodic functions by trigonometric polynomials . It is a qualitative and exploratory research . Its relevance is justified because it allows the teacher to look at the process of teaching and learning significantly to the student. During the development of research, the approach of trigonometry content was the history of mathematics and the use of software. The use of computing resources was an important ally to scientific knowledge, as well as the use of applications for the contextualization. / Este trabalho apresenta uma abordagem sobre o conteúdo de Trigonometria utilizando os programas Geogebra e Maxima. Traz um roteiro de como a Trigonometria pode ser trabalhada em sala de aula dividido em etapas. Propõe aplicações da Trigonometria em diferentes áreas do conhecimento e sugere a introdução da aproximação de funções periódicas por polinômios trigonométricos. Trata-se de uma pesquisa qualitativa e exploratória. A sua relevância justificase por possibilitar ao professor olhar o processo de ensino-aprendizagem de forma significativa ao aluno. Durante o desenvolvimento da pesquisa, a abordagem do conteúdo de trigonometria ocorreu pela história da matemática e com a utilização de softwares. A utilização dos recursos computacionais mostrou-se importante aliada ao saber científico, assim como a utilização de aplicações para a contextualização do tema.

The purpose of this paper is to define and develop a set of functions of an area in such a manner as to be analogous to the trigonometric and the hyperbolic functions.

Este trabalho apresenta uma abordagem sobre as novas tecnologias no contexto escolar, com vistas para aplicação do GeoGebra em trigonometria. O objetivo é nortear professores da educação básica na preparação de aulas usando o GeoGebra, visando ao enriquecimento do tema trigonometria em sala de aula. As atividades propostas estão divididas em três grupos: trigonometria básica, funções trigonométricas e equações trigonométricas. Cada uma possui um alto nível de detalhamento, com o objetivo de incentivar o uso por professores com pouco ou nenhum conhecimento do software, bem como incentivar atividades que promovam a criação por parte dos alunos. A ideia é que os alunos construam as atividades, aprendendo a utilizar o software, interagindo por meio da movimentação dos objetos, e tirando suas conclusões pertinentes às atividades. De maneira geral, pretende-se contribuir para o desenvolvimento do raciocínio lógico do aluno por meio do ensino de Matemática agregando a utilização de tecnologia, de forma que o aluno não seja somente um expectador, mas sim, participante da construção da própria atividade. / This work presents na approach to new Technologies in the educational context, with a view to applications of the GeoGebra in trigonometry. The goal is to guide teachers of the basic education in preparing lessons using GeoGebra, aiming to enrich trigonometry the in the classroom. The proposed activities are divided into three groups : basic trigonometry, trigonometry functions and trigonometry equations. Each one has a high level of details, in order to encourage the use by teachers with little or no knowledge of the software, and also encourage activities that promote the creation by the students. The idea is that students build the activities, learning how to use the software, interacting by moving objects, and taking their conclusions about the activities. In general, one intends to contribute to the development of logical thinking of students through the teaching of Mathematics adding the use of technology, so that the student is not only a spectator, but, participant of the construction of their own activity.

In this thesis a method for direct evaluation of Generalized B-splines (GB-splines) via the representation of these curves as piecewise functions is presented. A local structure is introduced that makes the GB-spline curves more amenable to the integration used in constructing bases of higher degree. This basis is used to perform direct computation of piecewise representation of GB-spline bases and curves. Algorithms for refinement using these local structures are also developed.

In this thesis, we make some contributions at the interface between algebra and graph theory. In Chapter 1, we give an overview of the topics and also the definitions and preliminaries. In Chapter 2, we estimate the number of possible types degree patterns of k-lacunary polynomials of degree t < p which split completely modulo p. The result is based on a rather unusual combination of two techniques: a bound on the number of zeros of lacunary polynomials and a bound on the so-called domination number of a graph. In Chapter 3, we deal with the determinant of bipartite graphs. The nullity of a graph G is the multiplicity of 0 in the spectrum of G. Nullity of a (molecular) graph (e.g., a bipartite graph corresponding to an alternant hydrocarbon) has important applications in quantum chemistry and Huckel molecular orbital (HMO) theory. A famous problem, posed by Collatz and Sinogowitz in 1957, asks to characterize all graphs with positive nullity. Clearly, examining the determinant of a graph is a way to attack this problem. In this Chapter, we show that the determinant of a bipartite graph with at least two perfect matchings and with all cycle lengths divisible by four, is zero. In Chapter 4, we first introduce an application of spectral graph theory in proving trigonometric identities. This is a very simple double counting argument that gives very short proofs for some of these identities (and perhaps the only existed proof in some cases!). In the rest of Chapter 4, using some properties of the well-known Chebyshev polynomials, we prove some theorems that allow us to evaluate the number of spanning trees in join of graphs, Cartesian product of graphs, and nearly regular graphs. In the last section of Chapter 4, we obtain the number of spanning trees in an (r,s)-semiregular graph and its line graph. Note that the same results, as in the last section, were proved by I. Sato using zeta functions. But our proofs are much shorter based on some well-known facts from spectral graph theory. Besides, we do not use zeta functions in our arguments. In Chapter 5, we present the conclusion and also some possible projects.

Recently, various approximate design problems for low-degree trigonometric regression models on a partial circle have been solved. In this paper we consider approximate and exact optimal design problems for first-order trigonometric regression models without intercept on a partial circle. We investigate the intricate geometry of the non-convex exact trigonometric moment set and provide characterizations of its boundary. Building on these results we obtain a complete solution of the exact D-optimal design problem. It is shown that the structure of the optimal designs depends on both the length of the design interval and the number of observations.

Consider the D-optimal designs for a combined polynomial of degree d and trigonometric of order m regression on a partial circle [see Graybill (1976), p. 324]. It is shown that the structure of the optimal design depends only on the length of the design interval and that the support points are analytic functions of this parameter. Moreover, the Taylor expansion of the optimal support points can be determined efficiently by a recursive procedure.

In this thesis, we make some contributions at the interface between algebra and graph theory. In Chapter 1, we give an overview of the topics and also the definitions and preliminaries. In Chapter 2, we estimate the number of possible types degree patterns of k-lacunary polynomials of degree t < p which split completely modulo p. The result is based on a rather unusual combination of two techniques: a bound on the number of zeros of lacunary polynomials and a bound on the so-called domination number of a graph. In Chapter 3, we deal with the determinant of bipartite graphs. The nullity of a graph G is the multiplicity of 0 in the spectrum of G. Nullity of a (molecular) graph (e.g., a bipartite graph corresponding to an alternant hydrocarbon) has important applications in quantum chemistry and Huckel molecular orbital (HMO) theory. A famous problem, posed by Collatz and Sinogowitz in 1957, asks to characterize all graphs with positive nullity. Clearly, examining the determinant of a graph is a way to attack this problem. In this Chapter, we show that the determinant of a bipartite graph with at least two perfect matchings and with all cycle lengths divisible by four, is zero. In Chapter 4, we first introduce an application of spectral graph theory in proving trigonometric identities. This is a very simple double counting argument that gives very short proofs for some of these identities (and perhaps the only existed proof in some cases!). In the rest of Chapter 4, using some properties of the well-known Chebyshev polynomials, we prove some theorems that allow us to evaluate the number of spanning trees in join of graphs, Cartesian product of graphs, and nearly regular graphs. In the last section of Chapter 4, we obtain the number of spanning trees in an (r,s)-semiregular graph and its line graph. Note that the same results, as in the last section, were proved by I. Sato using zeta functions. But our proofs are much shorter based on some well-known facts from spectral graph theory. Besides, we do not use zeta functions in our arguments. In Chapter 5, we present the conclusion and also some possible projects.