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Orientador: Miriam Godoy Penteado / Banca: Claudemir Murari / Banca: Aline Maria de Medeiros Rodrigues Reali / Resumo: Esta pesquisa tem a finalidade de compreender as possibilidades para o ensino de Geometria Fractal perspectivadas por professores de matemática e alunos do curso de licenciatura em Matemática. Os dados foram provenientes da realização de duas oficinas com cinco professores de matemática os quais atuam no ensino fundamental ou médio e de vinte estudantes do curso de licenciatura em Matemática e de um questionário preenchido por eles. Essas oficinas foram organizadas de forma a introduzir a ideia de Geometria Fractal a partir do emprego de recursos tecnológicos e materiais manipuláveis. Os programas computacionais utilizados foram o SuperLogo e o Geometricks. Usou-se também materiais manipuláveis como o compasso, a régua, a tesoura e papel cartão. A pesquisa empregou os pressupostos teóricos de Shulman para o estudo da produção de saber na prática docente; os pensamentos de Mizukami e Reali sobre os aspectos da formação de professores; o uso e o emprego de maneira significativa da Tecnologia na Educação por Papert e Valente; e as concepções de Penteado sobre a formação de professores para o uso de tecnologia informática. Os resultados tratam dos seguintes aspectos: a) como os participantes das oficinas percebem os fractais como tema gerador de outros tópicos de matemática; b) a relação dos participantes da oficina com a tecnologia informática utilizada; c) as dificuldades existentes ou não com os temas matemáticos relacionados ao estudo dos fractais; d) as possíveis dificuldades para ensinar esse tópico que os participantes da oficina conseguem antecipar / Abstract: This research aims at understanding the possibilities mathematics teachers and prospective teachers consider for teaching fractal geometry in the basic school. The analysis drawn on data from workshops and questionnaire with five mathematics teachers of elementary or high school, and twenty prospective mathematics teachers. The workshops were organized to introduce the idea of fractal geometry using software as SuperLogo and Geometricks, and manipulative material as compass, ruler, scissors and cardboard. The research based on Shulman ideas of pedagogical content knowledge; on Mizukami and Reali ideas of learning for teaching; and on Papert, Valente and Penteado idea of teacher education for the use of computer for teaching The results cover the following aspects: a) how the participants perceive fractal mobilize other mathematical content; b) the relationship of participants with computers c) the difficulties the participants had with the mathematical knowledge to work with fractals; d) the possible difficulties in teaching this topic that participants could anticipate / Mestre

In this dissertation, we study fractal sets and their properties, especially the open set condition, Hausdorff dimensions and Hausdorff measures for certain fractal constructions.

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This thesis is structured as follows. Chapter 1 introduces fractal sets before recalling basic mathematical concepts from dynamical systems, measure theory, dimension theory and probability theory. In Chapter 2 we give an overview of both deterministic and stochastic sets obtained from iterated function systems. We summarise classical results and set most of the basic notation. This is followed by the introduction of random graph directed systems in Chapter 3, based on the single authored paper [T1] to be published in Journal of Fractal Geometry. We prove that these attractors have equal Hausdorff and upper box-counting dimension irrespective of overlaps. It follows that the same holds for the classical models introduced in Chapter 2. This chapter also contains results about the Assouad dimensions for these random sets. Chapter 4 is based on the single authored paper [T2] and establishes the box-counting dimension for random box-like self-affine sets using some of the results and the notation developed in Chapter 3. We give some examples to illustrate the results. In Chapter 5 we consider the Hausdorff and packing measure of random attractors and show that for reasonable random systems the Hausdorff measure is zero almost surely. We further establish bounds on the gauge functions necessary to obtain positive or finite Hausdorff measure for random homogeneous systems. Chapter 6 is based on a joint article with J. M. Fraser and J.-J. Miao [FMT] to appear in Ergodic Theory and Dynamical Systems. It is chronologically the first and contains results that were extended in the paper on which Chapter 3 is based. However, we will give some simpler, alternative proofs in this section and crucially also find the Assouad dimension of some random self-affine carpets and show that the Assouad dimension is always `maximal' in both measure theoretic and topological meanings.

The aim of this research study is to investigate different practical ways in which fractal geometry and self-similarity can be applied to art music, with reference to music composition and analysis. This specific topic was chosen because there are many misconceptions in the field of fractal and self-similar music. Analyses of previous research as well as the music analysis of several compositions from different composers in different genres were the main methods for conducting the research. Although the dissertation restates much of the existing research on the topic, it is (to the researcher‟s knowledge) one of the first academic works that summarises the many different facets of fractal geometry and music. Fractal and self-similar shapes are evident in nature and art dating back to the 16th century, despite the fact that the mathematics behind fractals was only defined in 1975 by the French mathematician, Benoit B. Mandelbrot. Mathematics has been a source of inspiration to composers and musicologists for many centuries and fractal geometry has also infiltrated the works of composers in the past 30 years. The search for fractal and self-similar structures in music composed prior to 1975 may lead to a different perspective on the way in which music is analysed. Basic concepts and prerequisites of fractals were deliberately simplified in this research in order to collect useful information that musicians can use in composition and analysis. These include subjects such as self-similarity, fractal dimensionality and scaling. Fractal shapes with their defining properties were also illustrated because their structures have been likened to those in some music compositions. This research may enable musicians to incorporate mathematical properties of fractal geometry and self-similarity into original compositions. It may also provide new ways to view the use of motifs and themes in the structural analysis of music. / Dissertation (MMus)--University of Pretoria, 2014. / lk2014 / Music / MMus / Unrestricted

The master’s thesis deals with computational fluid dynamics (CFD) of two orifices, that have different shapes of holes but similar cross-sectional flow areas. The first of them is orifice with circular-shaped hole, which is used for maintenance free measurement of flow. The second one is orifice with fractal-shaped hole, inspired by von Koch snow-flake. This thesis follows bachelor thesis, in which was experimentally examined, that fractal-shaped orifices have better hydraulic properties (hydraulic losses and lower pressure pulsations) than circle-shaped one. The main target is to confirm this conclusion based on experiment, this time using CFD with various types of turbulence modelling ap-proaches. Both single phase (cavitation free) and multiphase numerical simulations were realized. Each model was compared from perspective of hydraulic and dynamic charac-teristics.

Scale is an essential notion in geography and geographic information science (GIScience). However, the complex concepts of scale and traditional Euclidean geometric thinking have created tremendous confusion and uncertainty. Traditional Euclidean geometry uses absolute size, regular shape and direction to describe our surrounding geographic features. In this context, different measuring scales will affect the results of geospatial analysis. For example, if we want to measure the length of a coastline, its length will be different using different measuring scales. Fractal geometry indicates that most geographic features are not measurable because of their fractal nature. In order to deal with such scale issues, the topological and scaling analyses are introduced. They focus on the relationships between geographic features instead of geometric measurements such as length, area and slope. The scale change will affect the geometric measurements such as length and area but will not affect the topological measurements such as connectivity.   This study uses three case studies to demonstrate the scale issues of geographic features though fractal analyses. The first case illustrates that the length of the British coastline is fractal and scale-dependent. The length of the British coastline increases with the decreased measuring scale. The yardstick fractal dimension of the British coastline was also calculated. The second case demonstrates that the areal geographic features such as British island are also scale-dependent in terms of area. The box-counting fractal dimension, as an important parameter in fractal analysis, was also calculated. The third case focuses on the scale effects on elevation and the slope of the terrain surface. The relationship between slope value and resolution in this case is not as simple as in the other two cases. The flat and fluctuated areas generate different results. These three cases all show the fractal nature of the geographic features and indicate the fallacies of scale existing in geography. Accordingly, the fourth case tries to exemplify how topological and scaling analyses can be used to deal with such unsolvable scale issues. The fourth case analyzes the London OpenStreetMap (OSM) streets in a topological approach to reveal the scaling or fractal property of street networks. The fourth case further investigates the ability of the topological metric to predict Twitter user’s presence. The correlation between number of tweets and connectivity of London named natural streets is relatively high and the coefficient of determination r2 is 0.5083.   Regarding scale issues in geography, the specific technology or method to handle the scale issues arising from the fractal essence of the geographic features does not matter. Instead, the mindset of shifting from traditional Euclidean thinking to novel fractal thinking in the field of GIScience is more important. The first three cases revealed the scale issues of geographic features under the Euclidean thinking. The fourth case proved that topological analysis can deal with such scale issues under fractal way of thinking. With development of data acquisition technologies, the data itself becomes more complex than ever before. Fractal thinking effectively describes the characteristics of geographic big data across all scales. It also overcomes the drawbacks of traditional Euclidean thinking and provides deeper insights for GIScience research in the big data era.

Made available in DSpace on 2014-06-11T19:24:52Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-04-27Bitstream added on 2014-06-13T20:32:24Z : No. of bitstreams: 1 baldovinotti_nj_me_rcla.pdf: 3213302 bytes, checksum: 13742e4d477d10dc20ffc3a7daaa50cb (MD5) / Fundesp / Esta pesquisa tem a finalidade de compreender as possibilidades para o ensino de Geometria Fractal perspectivadas por professores de matemática e alunos do curso de licenciatura em Matemática. Os dados foram provenientes da realização de duas oficinas com cinco professores de matemática os quais atuam no ensino fundamental ou médio e de vinte estudantes do curso de licenciatura em Matemática e de um questionário preenchido por eles. Essas oficinas foram organizadas de forma a introduzir a ideia de Geometria Fractal a partir do emprego de recursos tecnológicos e materiais manipuláveis. Os programas computacionais utilizados foram o SuperLogo e o Geometricks. Usou-se também materiais manipuláveis como o compasso, a régua, a tesoura e papel cartão. A pesquisa empregou os pressupostos teóricos de Shulman para o estudo da produção de saber na prática docente; os pensamentos de Mizukami e Reali sobre os aspectos da formação de professores; o uso e o emprego de maneira significativa da Tecnologia na Educação por Papert e Valente; e as concepções de Penteado sobre a formação de professores para o uso de tecnologia informática. Os resultados tratam dos seguintes aspectos: a) como os participantes das oficinas percebem os fractais como tema gerador de outros tópicos de matemática; b) a relação dos participantes da oficina com a tecnologia informática utilizada; c) as dificuldades existentes ou não com os temas matemáticos relacionados ao estudo dos fractais; d) as possíveis dificuldades para ensinar esse tópico que os participantes da oficina conseguem antecipar / This research aims at understanding the possibilities mathematics teachers and prospective teachers consider for teaching fractal geometry in the basic school. The analysis drawn on data from workshops and questionnaire with five mathematics teachers of elementary or high school, and twenty prospective mathematics teachers. The workshops were organized to introduce the idea of fractal geometry using software as SuperLogo and Geometricks, and manipulative material as compass, ruler, scissors and cardboard. The research based on Shulman ideas of pedagogical content knowledge; on Mizukami and Reali ideas of learning for teaching; and on Papert, Valente and Penteado idea of teacher education for the use of computer for teaching The results cover the following aspects: a) how the participants perceive fractal mobilize other mathematical content; b) the relationship of participants with computers c) the difficulties the participants had with the mathematical knowledge to work with fractals; d) the possible difficulties in teaching this topic that participants could anticipate