This thesis considers all of Ernest McClain's published writings, from March, 1970, to September, 1976, from the standpoint of their present-day acoustical significance. Although much of the material comes from McClain's writings, some is drawn from other related musical, mathematical, and philosophical works. The four chapters begin with a biographical sketch of McClain, presenting his background which aided him in becoming a theoretical musicologist. The second chapter contains a chronological itemization of his writings and provides a synopsis of them in layman's terms. The following chapter offers an examination of some salient points of McClain's work. The final chapter briefly summarizes the findings and contains conclusions as to their germaneness to current music theory, thereby giving needed exposure to McClain's ideas.

Made available in DSpace on 2016-04-27T14:32:26Z (GMT). No. of bitstreams: 1 Alvaro A Schmidt Neto.pdf: 702535 bytes, checksum: a3ecc081039a8c1127cfaef656307928 (MD5) Previous issue date: 2009-04-22 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This research intends to link the Pythagorean cosmogony of the tetraktys with the transdisciplinar education conception. Getting from the analysis of the symbolism of Pythagorean tetraktys s numbers and from the complex thought, the research investigates the possibility of a greater integration and strengthening of the existing relationships between the subject, the nature and the other (society). These relationships and interdependences between the subject, the nature and the other (society), also known as life triangle, are related with the Pythagorean cosmology, which set out the development of the beings in a cyclical and recursive process. This movement involves the identity of the subject, the environment and the relationships the subject establishes with this environment in search of his integration and his development. The research also presents the possibility of application of this theoretical construction in the educational field, offering didactic suggestions for a learning process connected with the idea of the creation itself and the development of the beings / Esta pesquisa procura relacionar a cosmogonia pitagórica da tetraktys a uma concepção de educação transdisciplinar. A partir da análise da simbologia dos números da tetraktys pitagórica e do pensamento complexo, a pesquisa investiga a possibilidade de uma maior integração e fortalecimento das relações existentes entre o sujeito, a natureza e o outro (sociedade). Essas relações e interdependências entre o sujeito, a natureza e o outro (sociedade), também chamadas de triângulo da vida, são associadas à cosmogonia pitagórica, que explicita o desenvolvimento dos seres num processo cíclico e recursivo. Este movimento envolve a identidade do sujeito, o meio e as relações que ele estabelece com esse meio na busca de sua integração e de seu desenvolvimento. A pesquisa apresenta ainda a possibilidade da aplicação dessa construção teórica no campo educacional, oferecendo sugestões didáticas para um processo de aprendizagem que esteja associado à idéia da própria criação e desenvolvimento dos seres

Inaug.-diss.-Université de Liége. / "Table des auteurs cités": p. [269]-282.

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Les problèmes d'interpolation ont été largement étudiés dans la Conception Géométrique Assistée par Ordinateur. Ces problèmes consistent en la construction de courbes et de surfaces qui passent exactement par un ensemble de données. Dans ce cadre, l'objectif principal de cette thèse est de présenter des méthodes d'interpolation de données 2D et 3D au moyen de courbes Algébriques Trigonométriques à Hodographe Pythagorien (ATPH). Celles-ci sont utilisables pour la conception de modèles géométriques dans de nombreuses applications. En particulier, nous nous intéressons à la modélisation géométrique d'objets odontologiques. À cette fin, nous utilisons les courbes spatiales ATPH pour la construction de surfaces développables dans des volumes odontologiques. Initialement, nous considérons la construction de courbes planes ATPH avec continuité C² qui interpolent une séquence ordonnée de points. Nous employons deux méthodes pour résoudre ce problème et trouver la « bonne » solution. Nous étendons les courbes ATPH planes à l'espace tridimensionnel. Cette caractérisation 3D est utilisée pour résoudre le problème d'interpolation Hermite de premier ordre. Nous utilisons ces splines ATPH spatiales C¹ continues pour guider des facettes développables, qui sont déployées à l'intérieur de volumes tomodensitométriques odontologiques, afin de visualiser des informations d'intérêt pour le professionnel de santé. Cette information peut être utile dans l'évaluation clinique, diagnostic et/ou plan de traitement. / Interpolation problems have been widely studied in Computer Aided Geometric Design (CAGD). They consist in the construction of curves and surfaces that pass exactly through a given data set, such as point clouds, tangents, curvatures, lines/planes, etc. In general, these curves and surfaces are represented in a parametrized form. This representation is independent of the coordinate system, it adapts itself well to geometric transformations and the differential geometric properties of curves and surfaces are invariant under reparametrization. In this context, the main goal of this thesis is to present 2D and 3D data interpolation schemes by means of Algebraic-Trigonometric Pythagorean-Hodograph (ATPH) curves. The latter are parametric curves defined in a mixed algebraic-trigonometric space, whose hodograph satisfies a Pythagorean condition. This representation allows to analytically calculate the curve's arc-length as well as the rational-trigonometric parametrization of the offsets curves. These properties are usable for the design of geometric models in many applications including manufacturing, architectural design, shipbuilding, computer graphics, and many more. In particular, we are interested in the geometric modeling of odontological objects. To this end, we use the spatial ATPH curves for the construction of developable patches within 3D odontological volumes. This may be a useful tool for extracting information of interest along dental structures. We give an overview of how some similar interpolating problems have been addressed by the scientific community. Then in chapter 2, we consider the construction of planar C2 ATPH spline curves that interpolate an ordered sequence of points. This problem has many solutions, its number depends on the number of interpolating points. Therefore, we employ two methods to find them. Firstly, we calculate all solutions by a homotopy method. However, it is empirically observed that only one solution does not have any self-intersections. Hence, the Newton-Raphson iteration method is used to directly compute this \good" solution. Note that C2 ATPH spline curves depend on several free parameters, which allow to obtain a diversity of interpolants. Thanks to these shape parameters, the ATPH curves prove to be more exible and versatile than their polynomial counterpart, the well known Pythagorean-Hodograph (PH) quintic curves and polynomial curves in general. These parameters are optimally chosen through a minimization process of fairness measures. We design ATPH curves that closely agree with well-known trigonometric curves by adjusting the shape parameters. We extend the planar ATPH curves to the case of spatial ATPH curves in chapter 3. This characterization is given in terms of quaternions, because this allows to properly analyze their properties and simplify the calculations. We employ the spatial ATPH curves to solve the first-order Hermite interpolation problem. The obtained ATPH interpolants depend on three free angular values. As in the planar case, we optimally choose these parameters by the minimization of integral shape measures. This process is also used to calculate the C1 interpolating ATPH curves that closely approximate well-known 3D parametric curves. To illustrate this performance, we present the process for some kind of helices. In chapter 4 we then use these C1 ATPH splines for guiding developable surface patches, which are deployed within odontological computed tomography (CT) volumes, in order to visualize information of interest for the medical professional. Particularly, we construct piecewise conical surfaces along smooth ATPH curves to display information related to the anatomical structure of human jawbones. This information may be useful in clinical assessment, diagnosis and/or treatment plan. Finally, the obtained results are analyzed and conclusions are drawn in chapter 5.

A presente tese explora, como solução para o controvertido quadro geral da moderna história da crítica sobre Pitágoras e seu movimento, a definição do pitagorismo como categoria historiográfica. Superando tanto o dilema entre ceticismo e confiança nas fontes, como a pretensão de alcançar uma única chave hermenêutica que permita resolver a questão pitagórica, procura percorrer a história da tradição em busca de uma imagem suficientemente plural a ponto de possibilitar a compreensão do pitagorismo em sua irredutível articulação de bíos e theoría e não apesar dela. A configuração da comunidade e de seu bíos é percebida como elemento central de identificação do pitagorismo. A análise das duas teorias que mais decididamente contribuíram para a definição do pitagorismo ao longo da história, a transmigração da alma imortal e a doutrina dos números, procura definir as condições de possibilidade de atribuí-las ao pitagorismo mais antigo e as formas pelas quais ambas teriam contribuído, ao longo da história, para a definição do pitagorismo como categoria historiográfica. As fontes pré-socráticas, a platonização do pitagorismo, o testemunho aristotélico sobre os assim chamados pitagóricos, a literatura pseudoepigráfica helenística e o pitagorismo de época imperial são entendidos como momentos de um percurso histórico que resulta em uma imagem poliédrica de um dos maiores fenômenos intelectuais da história ocidental. / This thesis explores the definition of Pythagoreanism as historiographical category, seen as solution for the controversial general framework of modern history of criticism about Pythagoras and his movement. Overcoming both the dilemma between skepticism and faith in the sources and the claim to achieve a single hermeneutical key to solve the Pythagorean question, in it searches are made throughout the history of tradition looking for an image sufficiently plural in order to understand Pythagoreanism in accordance with its irreducible articulation of bíos and theoría, not despite it. The setting of the community and its bíos is understood as central element for Pythagorean identification. An analysis of the two theories that more decisively contributed to the definition of Pythagoreanism throughout history, the transmigration of the immortal soul and the doctrine of numbers, attempts to define the conditions of possibility to assign them to the earlier Pythagoreanism and the ways in which these have contributed throughout history to the definition of Pythagoreanism as historiographical category. Presocratic sources, the platonization of Pythagoreanism, Aristotle\'s testimony about the \"so-called\" Pythagoreans, the hellenistic Pseudoepigrapha and Pythagoreanism in imperial age are understood as moments of an historical route resulting in a polyhedral image of one of the greatest intellectual phenomena in Western history.

Orientador: Ermínia de Lourdes Campello Fanti / Banca: Evelin Meneguesso Barbaresco / Banca: Tomas Edson Barros / Resumo: O objetivo principal deste trabalho é estudar algumas extensões de um dos teoremas mais importantes e divulgados da matemática elementar: o Teorema de Pitágoras, que tem suas aplicações em diversas áreas do conhecimento. A princípio é realizado um breve resgate histórico da vida de Pitágoras, o surgimento do teorema e suas aplicações. Por possuir mais de 400 demonstrações, elencamos algumas delas e as reproduzimos. Algumas demonstrações podem ser feitas de maneira lúdica, em forma de quebra-cabeça e outras que se tornaram famosas ao longo da história. São feitas várias extensões do teorema, para polígonos regulares, polígonos semelhantes e figuras não retilíneas. A generalização de Polya também é enunciada e demonstrada, situação em que o padrão pitagórico (relação entre as áreas) é válido para quaisquer tipos de figuras semelhantes construídas sobre os lados de um triângulo retângulo, sendo o Teorema de Pitágoras um caso particular, bem como a generalização de Pappus. Com o uso do software GeoGebra, foram propostas e desenvolvidas atividades em sala de informática, explorando o Teorema de Pitágoras e algumas de suas extensões. Por fim, é analisado como o Teorema de Pitágoras e o seu ensino são abordados em certos documentos oficiais de ensino no Brasil (PCNs, Currículo do Estado de São Paulo, matrizes de referências do SARESP, SAEB e ENEM) / Abstract: The main objective of this work is to study some extensions of one of the most important and published elementary mathematics theorem: the Pythagorean Theorem, which has applications in many areas of knowledge. We begin with a brief historical review of the life of Pythagoras, the emergence of the theorem and its applications. From over 400 existing proofs, we list some of them and reproduce. Some proofs can be made in a playful manner, as shaped puzzle and others have become famous throughout the history. Several extensions of the theorem are presented for regular polygons, similar polygons and non-rectilinear figures. The generalization of Polya is also stated and demonstrated, in which the Pythagorean pattern (area ratio) is valid for any kind of similar figures constructed on the sides of a right triangle, being the Pythagorean theorem a particular case, as well as the generalization of Pappus. By using the GeoGebra software, we proposed and developed activities in a computer lab, exploring the Pythagorean Theorem and some of its extensions. Finally, it is analyzed how the Pythagorean Theorem and its teaching are cited in some official documents concerning education in Brazil (PCNs, Curriculum of São Paulo, matrices of references SARESP, SAEB and ENEM) / Mestre

Practice of mathematical proof increase the understanding of mathematics anddevelop creativity skills, problem solving, communication, logical thinking andreasoning which are all important tools not only within the subject of mathematicsbut also important tools for the society in which we are living. The aim of this projectwas to investigate whether it is accurate that proof and proving has a subordinate rolein mathematic education in the upper secondary school in Sweden. This was done byconstructing of a digital survey that was sent to approximately 100 practicingmathematics teachers in a normal size city located in the middle of Sweden. Theresults of the survey show that the teachers consider themselves comfortable withtheir own skills in teaching proof. Paradoxically, the results also show that there is alack of teaching of proof and proving in the upper secondary school, although the newcurriculum puts more focus on proof and proving. / Matematiska bevis ger eleven en ökad förståelse för matematiken och utvecklarförmågor som kreativitet, problemlösning, kommunikation, logiskt tänkande ochresonemang, vilka alla är viktiga även utanför matematiken och för det samhälle vilever i. Syftet med detta arbete var att undersöka om det stämmer att bevis ochbevisföring har en underordnad roll i matematikundervisningen i den svenskagymnasieskolan vilket gjordes med hjälp av en enkät som skickades digitalt till cirka100 gymnasiematematiklärare i en medelstor stad i Mellansverige. Det visade sig attlärarna själva anser sig tillräckligt kunniga för att undervisa i bevis men det förefallerinte som att undervisningen är tillräcklig trots att dagens läroplaner sätter bevis istörre fokus än tidigare.

Sports are getting more and more benefit from use of information technologies. Information technologies can provide not only collected sport event statistics but also can give more complex analysis results that can be used for planning sport strategies and tactics. Also information technologies are used in sports monitoring, data gathering and result data analysis. Great results are achieved in basketball monitoring and data analysis. Most popular and widely used basketball data analysis methods are rating systems. Other analysis methods can provide predictions on forthcoming basketball games. With aim to provide new possibilities in data analysis area some popular basketball data analysis methods are introduced with discussed possibilities of methods improvements. As a research result possibilities for basketball rating systems improvements are proposed. Also framework for basketball games prediction methods precision research is designed and developed. Using developed framework prediction methods are researched and main results provided.

We give some background information about the Cauchy-Schwarz inequality including its history. We then continue by providing a number of proofs for the inequality in its classical form using various proof techniques, including proofs without words. Next we build up the theory of inner product spaces from metric and normed spaces and show applications of the Cauchy-Schwarz inequality in each content, including the triangle inequality, Minkowski's inequality and Hölder's inequality. In the final part we present a few problems with solutions, some proved by the author and some by others.