Causes of the apparent scale independence of fractal indices associated with forest fragmentation in Bolivia

In a recent investigation, Landsat TM and ETM+ data were used to simulate different resolutions of remotely-sensed images (from 30 to 1100 m) and to analyze the effect of resolution on a range of landscape metrics associated with spatial patterns of forest fragmentation in Chapare, Bolivia since the mid-1980s. Whereas most metrics were found to be highly dependent on pixel size, several fractal metrics (DLFD, MPFD, and AWMPFD) were apparently independent of image resolution, in contradiction with a sizeable body of literature indicating that fractal dimensions of natural objects depend strongly on image characteristics. The present re-analysis of the Chapare images, using two alternative algorithms routinely used for the evaluation of fractal dimensions, shows that the values of the box-counting and information fractal dimensions are systematically larger, sometimes by as much as 85%, than the "fractal" indices DLFD, MPFD, and AWMFD for the same images. In addition, the geometrical fractal features of the forest and non-forest patches in the Chapare region strongly depend on the resolution of images used in the analysis. The largest dependency on resolution occurs for the box-counting fractal dimension in the case of the non-forest patches in 1993, where the difference between the 30 and I 100 m-resolution images corresponds to 24% of the full theoretical range (1.0 to 2.0) of the mass fractal dimension. The observation that the indices DLFD, MPFD, and AWMPFD, unlike the classical fractal dimensions, appear relatively unaffected by resolution in the case of the Chapare images seems due essentially to the fact that these indices are based on a heuristic, "non-geometric" approach to fractals. Because of their lack of a foundation in fractal geometry, nothing guarantees that these indices will be resolution-independent in general. (C) 2006 International Society for Photogrammetry and Remote Sensing, Inc. (ISPRS). Published by Elsevier B.V. All rights reserved.

Antenas de microfita com patch quase-fractal para aplicações em Redes WPAN/WLAN

The microstrip antennas are in constant evidence in current researches due to several advantages that it presents. Fractal geometry coupled with good performance and convenience of the planar structures are an excellent combination for design and analysis of structures with ever smaller features and multi-resonant and broadband. This geometry has been applied in such patch microstrip antennas to reduce its size and highlight its multi-band behavior. Compared with the conventional microstrip antennas, the quasifractal patch antennas have lower frequencies of resonance, enabling the manufacture of more compact antennas. The aim of this work is the design of quasi-fractal patch antennas through the use of Koch and Minkowski fractal curves applied to radiating and nonradiating antenna s edges of conventional rectangular patch fed by microstrip inset-fed line, initially designed for the frequency of 2.45 GHz. The inset-fed technique is investigated for the impedance matching of fractal antennas, which are fed through lines of microstrip. The efficiency of this technique is investigated experimentally and compared with simulations carried out by commercial software Ansoft Designer used for precise analysis of the electromagnetic behavior of antennas by the method of moments and the neural model proposed. In this dissertation a study of literature on theory of microstrip antennas is done, the same study is performed on the fractal geometry, giving more emphasis to its various forms, techniques for generation of fractals and its applicability. This work also presents a study on artificial neural networks, showing the types/architecture of networks used and their characteristics as well as the training algorithms that were used for their implementation. The equations of settings of the parameters for networks used in this study were derived from the gradient method. It will also be carried out research with emphasis on miniaturization of the proposed new structures, showing how an antenna designed with contours fractals is capable of a miniaturized antenna conventional rectangular patch. The study also consists of a modeling through artificial neural networks of the various parameters of the electromagnetic near-fractal antennas. The presented results demonstrate the excellent capacity of modeling techniques for neural microstrip antennas and all algorithms used in this work in achieving the proposed models were implemented in commercial software simulation of Matlab 7. In order to validate the results, several prototypes of antennas were built, measured on a vector network analyzer and simulated in software for comparison

Fractal concepts in relation to soil water diffusivity

Fractal geometry would appear to offer promise for new insight on water transport in unsaturated soils, This study was conducted to evaluate possible fractal influence on soil water diffusivity, and/or the relationships from which it arises, for several different soils, Fractal manifestations, consisting of a time-dependent diffusion coefficient and anomalous diffusion arising out of fractional Brownian motion, along with the notion of space-filling curves were gleaned from the literature, It was found necessary to replace the classical Boltzmann variable and its time t(1/2) factor with the basic fractal power function and its t(n) factor, For distinctly unsaturated soil water content theta, exponent n was found to be less than 1/2, but it approached 1/2 as theta approached its sated value, This function n = n(theta), in giving rise to a time-dependent, anomalous soil water diffusivity D, was identified with the Hurst exponent H of fractal geometry, Also, n approaching 1/2 at high water content is a behavior that makes it possible to associate factal space filling with soil that approaches water saturation, Finally, based on the fractally interpreted n = n(theta), the coalescence of both D and 8 data is greatly improved when compared with the coalescence provided by the classical Boltzmann variable.

Comportamento fractal de tracos de fraturas na borda sul do macico granitico de itu, estado de Sao Paulo

Fractal geometry is relevant to understand and explain many natural complex geometries. Using the fractal set concept (fig. 1) many authors have shown that shorelines, landscapes and fractures follow a fractal behaviour. These authors have developed many methods, including the Cantor's Dust Method (CDM) (VELDE et al., 1992), a linear method of analysis adapted for the determination of two-dimensional phenomena. The Itu Granitic Complex (IGC) is a wide granitic body that that crops out at northwest of Cabreuva City, Sao Paulo State (fig. 2) and was affected in its south border by dextral Itu-Jundiuvira Shear Zone (IJSZ) that produced fractures and alignment of feldspars crystals. The different types of fractures (compression, distension and shear) was discriminated from the relationship between them and medium stress ellipsoid of IJSZ (fig. 3). A modified version of CDM was used to study a possible fractal behaviour of the fracture traces in the south border of IGC. The main modification was the use only one direction of analysis (NE/SW). Four parallel profiles were traced with lengths between 9.75km and 12.75km, each one them was divided into six classes of segments (x) with 375m, 500m, 750m, 1.000m, 1.250m and 1.500m. The parameter (N) is provided by he rate between profile length and choiced segment. For each x the number of intervals is counted with at least one event (fracture intersection) which supplied the parameter(n). The n/N rate provide the parameter (p) that represents the relationship between frequency of events and x. And finally the parameters p and x were plotted in a logarithmic graphics (fig. 4) that provide a line with such a declivity (1) which is related to effective dimension (De). In theory, granitics bodies are isotropics and they would have a same fractal dimension in all segments, but the logarithmic graphics (fig. 4) show that fracture traces of IGC has a fractal behaviour in a restrict interval. This fact probably occurs from the passage of a ductil-brittle deformation condition to a more brittle deformation condition of IGC.

Análise espaço temporal da dimensão fractal de matas ciliares na alta bacia do rio Passa Cinco - centro Leste do Estado de São Paulo

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Geometria fractal: perspectivas e possibilidades no ensino de matemática

A Geometria Fractal é um ramo novo da Matemática que vem sendo estudado desde sua descoberta nos anos sessenta por Benoit Mandelbrot. Por se tratar de uma geometria essencialmente intuitiva, muito se tem comentado a respeito da possibilidade de sua introdução ainda no Ensino Fundamental e Médio de nossas escolas. Assim, um grande número de atividades envolvendo Geometria Fractal foram e ainda estão sendo desenvolvidas com o intuito de tornar o conteúdo da Matemática curricular mais significativo ao aluno. Entretanto, muitas carecem de um estudo mais aprofundado no que tange ao seu verdadeiro grau de eficácia. Para tentar vislumbrar até que ponto estas atividades podem se caracterizar como um recurso didático válido, elaboramos e ministramos um curso sobre Geometria Fractal para onze alunos do 3 ano do Ensino Médio de uma escola pública estadual na cidade de Santarém-Pa. O curso consistia de uma parte teórica sobre o assunto e algumas atividades selecionadas de tal forma que estas pudessem abranger alguns tópicos da Matemática curricular já visto por eles em suas trajetórias escolares. Aplicamos antes do curso um pré-teste e no final um pós-teste para avaliar a compreensão dos assuntos abordados. Os resultados obtidos mostram uma evolução tanto quantitativa, quanto qualitativa na (re)apropriação dos conceitos matemáticos trabalhados durante o curso. O estudo ainda sugere que a Geometria Fractal pôde proporcionar aos alunos uma relação mais forte entre os saberes do cotidiano e o escolar, além de ter proporcionado uma visão dinâmica da Matemática como uma ciência que avança, e não como um corpo de conhecimentos prontos e acabados.

Modelo fractal para resistividade complexa de rochas: interpretação petrofísica e aplicação à exploração geoelétrica

Rochas contendo metálicos disseminados ou partículas de argila em ambiente natural onde soluções eletrolíticas normalmente preenchem os poros das rochas, exibem um tipo de polarização em baixas freqüências conhecido como polarização induzida. Nesta tese foi desenvolvido um novo modelo para descrever o fenômeno de polarização das rochas, não apenas em baixas freqüências, mas compreendendo todo o espectro eletromagnético, possível de utilização na prospecção geoelétrica. Este novo modelo engloba a maioria dos modelos utilizados até o momento como casos especiais, além de superar as limitações dos mesmos. Seu circuito analógico inclui uma impedância não linear do tipo r (iwtf)^-n que simula o efeito das superfícies rugosas das interfaces entre os grãos bloqueadores (partículas metálicas e/ou de argilas) e o eletrólito. A impedância de Warburg generalizada está em série com a resistência dos grãos bloqueadores da passagem de corrente e em paralelo com a impedância da dupla camada associada a essas interfaces. Esta combinação está em série com a resistência do eletrólito nas passagens dos poros bloqueados. Os canais não bloqueados são representados por uma resistência que corresponde à resistividade normal CC da rocha. A combinação desta resistência com a capacitância "global" da rocha é finalmente conectada em paralelo ao resto do circuito mencionado acima. Os parâmetros deste modelo incluem a resistividade CC (p₀), a cargueabilidade (m), três tempos de relaxação (t, Tf and T2), um fator de resistividade de grãos (δ_r), e o expoente de freqüência (η). O tempo de relaxação fractal (Tf), e o expoente de frequencia (η) estão relacionados à geometria fractal das interfaces rugosas entre os minerais condutivos (grãos metálicos e/ou partículas de argila bloqueando os canais dos poros) e o eletrólito. O tempo de relaxação (T) é um resultado da relaxação em baixa freqüência das duplas camadas elétricas formadas nas interfaces eletrólito-cristais, enquanto (T₀) é o tempo de relaxação macroscópico da amostra como um todo. O fator de resistividade dos grãos (δ_r) relaciona a resistividade dos grãos condutivos com o valor de resistividade CC da rocha. A resistividade CC da rocha (p₀), e δ_r estão relacionados à porosidade, à condutividade do eletrólito e às relações mineralógicas entre a matriz e os grãos condutivos. O modelo foi testado sobre um intervalo largo de freqüências contra dados experimentais de amplitude e fase da resistividade bem como para dados de constante dielétrica complexa. Os dados utilizados neste trabalho foram obtidos a partir da digitalização de dados experimentais publicados, obtidos por diversos autores e englobando amostras de rochas sedimentares, ígneas e metam6rficas. É mostrado neste trabalho que os parâmetros deste modelo permitem identificar diferenças texturais e mineralógicas nas rochas. Bote modelo foi introduzido, primeiramente, como propriedade intrínseca de um semiespaço homogêneo sendo demonstrado, neste trabalho, que a resposta observada em superfície reflete as propriedades intrínsecas do meio polarizável, sendo o acoplamento eletromagnético desprezível em freqüências menores que 104 Hz. Em seguida, o meio polarizável foi embebido em um pacote de N camadas sendo demonstrado que os parâmetros fractais do meio polarizável podem ser obtidos do levantamento em superfície para diferentes espessuras dessa camada. Isto justifica a utilização pura e simples de modelos de polarização desenvolvidos para amostras em laboratório para ajustar dados de campo, o que vem sendo feito sem uma justificativa bem fundamentada. Estes resultados demonstram a importância para a prospecção geolétrica do modelo proposto nesta tese.

Previsão espacial da densidade de carga nos sistemas de distribuição de energia elétrica considerando a geometria fractal da zona urbana

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Aplicações médicas das abordagens complexas não lineares: a geometria fractal do EEG

Pós-graduação em Saúde Coletiva - FMB

An introduction to flow and transport in fractal models of porous media: Part I

This special issue gathers together a number of recent papers on fractal geometry and its applications to the modeling of flow and transport in porous media. The aim is to provide a systematic approach for analyzing the statics and dynamics of fluids in fractal porous media by means of theory, modeling and experimentation. The topics covered include lacunarity analyses of multifractal and natural grayscale patterns, random packing's of self-similar pore/particle size distributions, Darcian and non-Darcian hydraulic flows, diffusion within fractals, models for the permeability and thermal conductivity of fractal porous media and hydrophobicity and surface erosion properties of fractal structures.

Fractal parameters of pore space from CT images of soils under contrasting management practices

Soil structure plays an important role in flow and transport phenomena, and a quantitative characterization of the spatial heterogeneity of the pore space geometry is beneficial for prediction of soil physical properties. Morphological features such as pore-size distribution, pore space volume or pore?solid surface can be altered by different soil management practices. Irregularity of these features and their changes can be described using fractal geometry. In this study, we focus primarily on the characterization of soil pore space as a 3D geometrical shape by fractal analysis and on the ability of fractal dimensions to differentiate between two a priori different soil structures. We analyze X-ray computed tomography (CT) images of soils samples from two nearby areas with contrasting management practices. Within these two different soil systems, samples were collected from three depths. Fractal dimensions of the pore-size distributions were different depending on soil use and averaged values also differed at each depth. Fractal dimensions of the volume and surface of the pore space were lower in the tilled soil than in the natural soil but their standard deviations were higher in the former as compared to the latter. Also, it was observed that soil use was a factor that had a statistically significant effect on fractal parameters. Fractal parameters provide useful complementary information about changes in soil structure due to changes in soil management. Read More: http://www.worldscientific.com/doi/abs/10.1142/S0218348X14400118?queryID=%24%7BresultBean.queryID%7D&

Teaching as a fractal: from experience to model

The aim of this work is to improve students’ learning by designing a teaching model that seeks to increase student motivation to acquire new knowledge. To design the model, the methodology is based on the study of the students’ opinion on several aspects we think importantly affect the quality of teaching (such as the overcrowded classrooms, time intended for the subject or type of classroom where classes are taught), and on our experience when performing several experimental activities in the classroom (for instance, peer reviews and oral presentations). Besides the feedback from the students, it is essential to rely on the experience and reflections of lecturers who have been teaching the subject several years. This way we could detect several key aspects that, in our opinion, must be considered when designing a teaching proposal: motivation, assessment, progressiveness and autonomy. As a result we have obtained a teaching model based on instructional design as well as on the principles of fractal geometry, in the sense that different levels of abstraction for the various training activities are presented and the activities are self-similar, that is, they are decomposed again and again. At each level, an activity decomposes into a lower level tasks and their corresponding evaluation. With this model the immediate feedback and the student motivation are encouraged. We are convinced that a greater motivation will suppose an increase in the student’s working time and in their performance. Although the study has been done on a subject, the results are fully generalizable to other subjects.

Fractal Analysis for Cancer Research: Case Study and Simulation of Fractals

2010 Mathematics Subject Classification: 65D18.

Estudo de arranjos de antenas de microfita com Patch quase-fractal para comunicações sem fio

In this dissertation, are presented two microstrip antennas and two arrays for applications in wireless communication systems multiband. Initially, we studied an antenna and a linear array consisting of two elements identical to the patch antenna isolated. The shape of the patch used in both structures is based on fractal geometry and has multiband behavior. Next a new antenna is analyzed and a new array such as initial structure, but with the truncated ground plane, in order to obtain better bandwidths and return loss. For feeding the structures, we used microstrip transmission line. In the design of planar structures, was used HFSS software for the simulation. Next were built and measures electromagnetic parameters such as input impedance and return loss, using vector network analyzer in the telecommunications laboratory of Federal University of Rio Grande do Norte. The experimental results were compared with the simulated and showed improved return loss for the first array and also appeared a fourth band and increased directivity compared with the isolated antenna. The first two benefits are not commonly found in the literature. For structures with a truncated ground planes, the technique improved impedance matching, bandwidth and return loss when compared to the initial structure with filled ground planes. Moreover, these structures exhibited a better distribution of frequency, facilitating the adjustment of frequencies. Thus, it is expected that the planar structures presented in this study, particularly arrays may be suitable for specific applications in wireless communication systems when frequency multiband and wideband transmission signals are required.

An old language for a new landscape

Because aesthetics can have a profound effect upon the human relationship to the non-human environment the importance of aesthetics to ecologically sustainable designed landscapes has been acknowledged. However, in recognition that the physical forms of designed landscapes are an expression of the social values of the time, some design professionals have called for a new aesthetic ― one that reflects these current ecological concerns. To address this, some authors have suggested various theoretical design frameworks upon which such an aesthetic could be based. Within these frameworks there is an underlying theme that the patterns and processes of natural systems have the potential to form a new aesthetic for landscape design —an aesthetic based on fractal rather than Euclidean geometry. Perry, Reeves and Sim (2008) have shown that it is possible to differentiate between different landscape forms by fractal analysis. However, this research also shows that individual scenes from within very different landscape forms can possess the same fractal properties. Early data, revealed by transforming landscape images from the spatial to the frequency domain, using the fast Fourier transform, suggest that fractal patterning can have a significant effect within the landscape. In fact, it may be argued that any landscape design that includes living processes will include some design element whose ultimate form can only be expressed through the mathematics of fractal geometry. This paper will present ongoing research into the potential role of fractal geometry as a basis for a new form language – a language that may articulate an aesthetic for landscape design that echoes our ecological awakening.