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.

An analogue of the Riemannian structure of a manifold is created for an ultrametric Cantor set using the techniques of Noncommutative Geometry. In particular, a spectral triple is created that can recover much of the fractal geometry of the original Cantor set. It is shown that this spectral triple can recover the metric, the upper box dimension, and in certain cases the Hausdorff measure. The analogy with Riemannian geometry is then taken further and an analogue of the Laplace-Beltrami operator is created for an ultrametric Cantor set. The Laplacian then allows to create an analogue of Brownian motion generated by this Laplacian. All these tools are then applied to the triadic Cantor set. Other examples of ultrametric Cantor sets are then presented: attractors of self-similar iterated function systems, attractors of cookie cutter systems, and the transversal of an aperiodic, repetitive Delone set of finite type. In particular, the example of the transversal of the Fibonacci tiling is studied.

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

Made available in DSpace on 2014-06-11T19:24:53Z (GMT). No. of bitstreams: 0 Previous issue date: 2005-08-31Bitstream added on 2014-06-13T19:11:45Z : No. of bitstreams: 1 gouvea_fr_me_rcla.pdf: 3114009 bytes, checksum: 7cfd768795cfd2d4315b640578fa631f (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Neste trabalho abordamos um tema pouco explorado nos cursos de graduação em Matemática, que é a Geometria Fractal, resgatando conceitos básicos da Geometria Euclidiana, utilizando caleidoscópios e softwares educacionais. Assim, foram tecidas algumas considerações a respeito da utilização de computadores na sala de aula, através de um estudo que investigou: Que contribuições pode trazer, para o ensinoaprendizagem de Geometria, um estudo de Fractais Geométricos através de caleidoscópios e softwares de Geometria Dinâmica ?. Foram elaboradas atividades e aplicadas a alunos da Licenciatura em Matemática (do 1º e 2º semestres) da Unesp de Rio Claro, que participaram de um Curso de Extensão. A utilização de materiais diferentes do tradicional, como o caleidoscópio e o computador (este último como elemento inserido no contexto educacional), e a contextualização da Geometria contribuíram para o estabelecimento de um ambiente de aprendizagem agradável e participativo. Nosso estudo mostrou uma maneira inovadora de obterem-se fractais geométricos: através de bases caleidoscópicas, o que enseja um grande estudo sobre espelhos e caleidoscópios, e traz em si a oportunidade de estudarem-se muitos conceitos geométricos (reflexão, simetrias, transformações geométricas, bissetriz, mediatriz, seqüências, etc.). Apresentamos, ainda, alguns aspectos pedagógicos e matemáticos relacionados à aplicabilidade dos Fractais Geométricos no processo de construção de conceitos geométricos, por meio da interação aluno-aluno, aluno-computador e alunoprofessor, tendo como pano de fundo a resolução de problemas. Dessa forma, nosso estudo proporcionou para os alunos uma maior relação com os conceitos fundamentais de Geometria Euclidiana e Geometria Fractal, além de uma alternativa metodológica inerente ao ensino da Geometria. / In this work we approached a theme little explored in the degree courses in Mathematics, that it is the Fractal Geometry ransoms basic concepts of the Euclidian Geometry, using kaleidoscopic and educational softwares. At his, are some woven considerations respect the use computers in the classroom, through a study that enquired: What contributions can bring, for teaching-learning of Geometry, a study of the geometrical fractals that include kaleidoscopic and softwares of Dynamic Geometry? Activities were elaborated and applied to students of the degree in mathematics (of the 1st and 2nd semesters) of Unesp de Rio Claro, who participated in a Course of Extension. The use of different materials from the traditional as the kaleidoscopic and computer (this last one as element inserted in the education context), and the contextualization of the Geometry contributed to the establishment of an environment of the pleasing learning and interest. Our study showed an innovator way of they be obtained fractal geometrics: through of kaleidoscopic bases, that wish a great study with mirrors and kaleidoscopic, and bring in itself the opportunity of they be studied many geometric concepts (reflection, symmetric, geometric transformations, bisector, mediate, etc). We presented, still, some pedagogic and mathematic aspects related to the applicability of Fractal Geometrics in the process of construction of geometrical concepts, through the interaction student-student, student-computer and student-teacher using as backdrop the problem solve. Of this form, our study it provided for the students a bigger relation with the basic concepts of Euclidean Geometry and Fractal Geometry, beyond inherent a metodology alternative to the teaching of Geometry.

One of the basic questions in harmonic analysis is to study the decay properties of the Fourier transform of measures or distributions supported on thin sets in Rn. When the support is a smooth enough manifold, an almost complete picture is available. One of the early results in this direction is the following: Let f in Cc∞(dσ), where dσ is the surface measure on the sphere Sn-1 Rn.Then the modulus of the Fourier transform of fdσ is bounded above by (1+|x|)(n-1)/2. Also fdσ in Lp(Rn) for all p > 2n/(n-1) . This result can be extended to compactly supported measure on (n-1)-dimensional manifolds with appropriate assumptions on the curvature. Similar results are known for measures supported in lower dimensional manifolds in Rn under appropriate curvature conditions. However, the picture for fractal measures is far from complete. This thesis is a contribution to the study of asymptotic properties of the Fourier transform of measures supported in sets of fractal dimension 0 < α < n for p ≤ 2n/α. In 2004, Agranovsky and Narayanan proved that if μ is a measure supported in a C1-manifold of dimension d < n, then the Fourier transform of μ is not in Lp(Rn) for 1 ≤ p ≤ 2n/d. We prove that the Fourier transform of a measure μ supported in a set E of fractal dimension α does not belong to Lp(Rn) for p≤ 2n/α. As an application we obtain Wiener-Tauberian type theorems on Rn and M(2). We also study Lp-asymptotics of the Fourier transform of fractal measures μ under appropriate conditions and give quantitative versions of the above statement by obtaining lower and upper bounds for the following limsup L∞ L-k∫|x|≤L|(fdµ)^(x)|pdx

Submitted by LAÍS FERNANDES MUCHERONI (lais.mucheroni@gmail.com) on 2017-09-18T17:23:23Z No. of bitstreams: 1 dissertacao_mestrado_lais.pdf: 1067574 bytes, checksum: 952e3477ef0efeafd01d052547e8f2e5 (MD5) / Approved for entry into archive by Monique Sasaki (sayumi_sasaki@hotmail.com) on 2017-09-19T20:08:28Z (GMT) No. of bitstreams: 1 mucheroni_lf_me_rcla.pdf: 1067574 bytes, checksum: 952e3477ef0efeafd01d052547e8f2e5 (MD5) / Made available in DSpace on 2017-09-19T20:08:28Z (GMT). No. of bitstreams: 1 mucheroni_lf_me_rcla.pdf: 1067574 bytes, checksum: 952e3477ef0efeafd01d052547e8f2e5 (MD5) Previous issue date: 2017-08-18 / Intuitivamente, um ponto tem dimensão 0, uma reta tem dimensão 1, um plano tem dimensão 2 e um cubo tem dimensão 3. Porém, na geometria fractal encontramos objetos matemáticos que possuem dimensão fracionária. Esses objetos são denominados fractais cujo nome vem do verbo "frangere", em latim, que significa quebrar, fragmentar. Neste trabalho faremos um estudo sobre o conceito de dimensão, definindo dimensão topológica e dimensão de Hausdorff. O objetivo deste trabalho é, além de apresentar as definições de dimensão, também apresentar algumas aplicações da dimensão de Hausdorff na geometria fractal. / We know, intuitively, that the dimension of a dot is 0, the dimension of a line is 1, the dimension of a square is 2 and the dimension of a cube is 3. However, in the fractal geometry we have objects with a fractional dimension. This objects are called fractals whose name comes from the verb frangere, in Latin, that means breaking, fragmenting. In this work we will study about the concept of dimension, defining topological dimension and Hausdorff dimension. The purpose of this work, besides presenting the definitions of dimension, is to show an application of the Hausdorff dimension on the fractal geometry.

Orientador: Claudemir Murari / Banca: Geraldo Perez / Banca: Ruy Madsen Barbosa / Resumo: Neste trabalho abordamos um tema pouco explorado nos cursos de graduação em Matemática, que é a Geometria Fractal, resgatando conceitos básicos da Geometria Euclidiana, utilizando caleidoscópios e softwares educacionais. Assim, foram tecidas algumas considerações a respeito da utilização de computadores na sala de aula, através de um estudo que investigou: "Que contribuições pode trazer, para o ensinoaprendizagem de Geometria, um estudo de Fractais Geométricos através de caleidoscópios e softwares de Geometria Dinâmica ?". Foram elaboradas atividades e aplicadas a alunos da Licenciatura em Matemática (do 1º e 2º semestres) da Unesp de Rio Claro, que participaram de um Curso de Extensão. A utilização de materiais diferentes do tradicional, como o caleidoscópio e o computador (este último como elemento inserido no contexto educacional), e a contextualização da Geometria contribuíram para o estabelecimento de um ambiente de aprendizagem agradável e participativo. Nosso estudo mostrou uma maneira inovadora de obterem-se fractais geométricos: através de bases caleidoscópicas, o que enseja um grande estudo sobre espelhos e caleidoscópios, e traz em si a oportunidade de estudarem-se muitos conceitos geométricos (reflexão, simetrias, transformações geométricas, bissetriz, mediatriz, seqüências, etc.). Apresentamos, ainda, alguns aspectos pedagógicos e matemáticos relacionados à aplicabilidade dos Fractais Geométricos no processo de construção de conceitos geométricos, por meio da interação aluno-aluno, aluno-computador e alunoprofessor, tendo como pano de fundo a resolução de problemas. Dessa forma, nosso estudo proporcionou para os alunos uma maior relação com os conceitos fundamentais de Geometria Euclidiana e Geometria Fractal, além de uma alternativa metodológica inerente ao ensino da Geometria. / Abstract: In this work we approached a theme little explored in the degree courses in Mathematics, that it is the Fractal Geometry ransoms basic concepts of the Euclidian Geometry, using kaleidoscopic and educational softwares. At his, are some woven considerations respect the use computers in the classroom, through a study that enquired: "What contributions can bring, for teaching-learning of Geometry, a study of the geometrical fractals that include kaleidoscopic and softwares of Dynamic Geometry?" Activities were elaborated and applied to students of the degree in mathematics (of the 1st and 2nd semesters) of Unesp de Rio Claro, who participated in a Course of Extension. The use of different materials from the traditional as the kaleidoscopic and computer (this last one as element inserted in the education context), and the contextualization of the Geometry contributed to the establishment of an environment of the pleasing learning and interest. Our study showed an innovator way of they be obtained fractal geometrics: through of kaleidoscopic bases, that wish a great study with mirrors and kaleidoscopic, and bring in itself the opportunity of they be studied many geometric concepts (reflection, symmetric, geometric transformations, bisector, mediate, etc). We presented, still, some pedagogic and mathematic aspects related to the applicability of Fractal Geometrics in the process of construction of geometrical concepts, through the interaction student-student, student-computer and student-teacher using as backdrop the problem solve. Of this form, our study it provided for the students a bigger relation with the basic concepts of Euclidean Geometry and Fractal Geometry, beyond inherent a metodology alternative to the teaching of Geometry. / Mestre

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O presente trabalho consiste em um relato sobre a origem da Geometria Fractal, tendo em destaque a figura de Benoît Mandelbrot, identificado como pioneiro nesta área, cujo fractal leva seu nome. Mostra os fractais pioneiros, assim como a construção destes através da ferramenta computacional "Asymptote". É necessário dizer que, a partir da construção destes, percebe-se, com facilidade um intenso uso de conteúdos presentes no currículo escolar do ensino básico, como por exemplo o cálculo de perímetro e de áreas de figuras planas, potenciação, problemas de contagens, entre outros, os quais podem ser abordados com o intuito de introduzir tal conteúdo ou mesmo aprofundá-lo. Por fim, faremos uso de Indução Matemática para demonstrar algumas destas fórmulas encontradas. / This work consists the historic report of the origin of Fractal Geometry, and highlighted the figure of Benoît Mandelbrot, identified as pioneer in this area, whose fractal bears his name. Shows the pioneers fractals, as well as the construction of these using the computational tool "Asymptote". It must be said that, from the construction of these, it is noted, easily a intense use of contents present in the curriculum of basic education, such as the calculation of perimeter and area of plane figures, potentiation, in counts problems, among others, they can be addressed in order to start the study of such content or to same deepen it. Finally, we will make use of Mathematical Induction to demonstrate some of the formulas found.

The theoretical focus of this study is Meaning in Architecture. The study does not draw from history or popular culture for meaning, but will aim to transcend Post-modern concepts by focusing on the contemporary condition and archetypal forms as a result of history and popular culture. This is done through the use of fractal geometry and deals with the questions that arise regarding signification. Fractals can be considered a subdivision of the language of mathematics and will be utilised as mediation between the reality of our world and the generation of form for the purpose of design. This is explored in two analogies, Architecture and Mathematics; and Architecture and Language. Through this process, subjectivity relating to form is removed as the design was developed in conjunction with the area in which it manifests, through the transformation of quantifiable entities into form. The programme and the process have become a unified whole in that mathematical concepts were utilised to design a building to house people involved with mathematics. The aim of the design proposal is to contribute to the urban landscape of Tshwane by allowing access to facilities in which training in mathematics and computer science can be achieved and to allow individuals to come to a place of self-actualization. The design is defined as a snapshot of the contemporary condition and therefore makes use of passive and active technologies to create a habitable environment. It is imperative to realise that society is in a process of transition and this can be embraced by combining ‘green’ and ‘non-green’ design approaches, while working towards more energy-efficient design solutions. Copyright / Dissertation (MArch(Prof))--University of Pretoria, 2010. / Architecture / unrestricted

京都大学 / 0048 / 新制・課程博士 / 博士(人間・環境学) / 甲第22534号 / 人博第937号 / 新制||人||223(附属図書館) / 2019||人博||937(吉田南総合図書館) / 京都大学大学院人間・環境学研究科共生人間学専攻 / (主査)教授 角 大輝, 教授 上木 直昌, 准教授 木坂 正史 / 学位規則第4条第1項該当 / Doctor of Human and Environmental Studies / Kyoto University / DFAM