Dissertations / Theses on the topic 'Fractual'

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The Xar?u Oil Field, located in the center-southern portion of the Munda? Sub-Basin (eastern portion of the Cear? Basin), is characterized by a main Iramework of NW-trending and NE-dipping faults. The faults in the Xar?u Oil Field, among which the Xar?u Fautt stands out, are arranged according to an extensional-listriclan, rooted on a detachment surface corresponding to the Munda? Fault, the border fautt of Munda? Sub-Basin. During the tectonic-structural evolution of the Xar?u Oil Field and the Munda? Sub-Basin, the Munda? Fault played a crucial role on the control of the geometry of both compartments. The main carbonatic unit in the Xar?u Oil Field, named the Trair? Member(Paracuru Formation of Late Aptian to Early Albian age), contains the largest oil volume in the field, concentrated in structurally-controlled accumulations. The Trair? Member is composed by a variety of carbonatic rocks (massive, bedded or laminated calcilutites, ostracodites, calcarenites and carbonatic rudites, all of them presenting variable degrees of dolomitization). The carbonatic rocks are interbedded into thick packages of black shales and marls, besides local beds of siliciclastic conglomerates, sandstones, siltnes and argillites. From the spatial association and the genetic relationships between the carbonatic and siliciclastic units, it is possible to group them in three lithofacies associations (Marginal Plain, Ramp and Lacustrine Interior) that, together, were developed in a lacustrine system associated to a marginal sabkha. Structural studies based on drill coresthat sample the Trair? Member in the Xar?u Oil Field allowed to characterize two generations of meso- to microscale structures: the D1 group presents a typical hydroplastic character, being characterized by intra/interstratal to oblique-bedding shear zones. The hydroplastic character related to these structures allowed to infer their development at an early-lithilication stage of the Trair? Member, leading to infer an Early Cretaceous age to them. The second group of structures identified in the drill cores, nominated D2 and ascribed to a Neogene age, presents a strictly brttle character, being typilied by normal faults and slickenfibers of re-crystallized clayminerals, ali olthem displaying variable orientations. Although the present faults in the Xar?u Oil Field (and, consequently, in the Munda? Sub-Basin) were classically relerred as struetures of essentially normal displacement, the kinematics analysis of the meso-to microscaie D1 struetures in the drill cores led to deline oblique displacements (normal with a clockwise strike-slip component) to these faults, indicating a main tectonic transport to ENE. These oblique movements would be responsible for the installation of a transtensive context in the Munda? Sub-Basin, as part of the transcurrent to translormant opening of the Atlantic Equatorial Margin. The balancing of four struetural cross-sections ofthe Xar?u Oil Field indicates that the Munda? Fault was responsible for more than 50% of the total stretching (? factor) registered during the Early Aptian. At the initial stages of the "rifting", during Early Aptianuntil the Holocene, the Munda? Sub-Basin (and consequently the Xar?u Oil Fleld) accumulated a total stretching between 1.21 and 1.23; in other words, the crust in this segment of the Atlantic Equatorial Margin was subjeeted to an elongation of about 20%. From estimates of oblique displacements related to the faults, it ws possible to construct diagrams that allow the determination of stretching factors related to these displacements. Using these diagrams and assuming the sense 01 dominant teetonictransport towards ENE, it was possible to calculate the real stretching lactors related to the oblique movement 0 of the faults in the Munda? Sub-Basin. which reached actual values between 1.28 and 1.42. ln addnion to the tectonic-structural studies in the Xar?u Oil Field, the interpretation of remote sensing products, coupled wnh characterization of terrain analogues in seleeted areas along the northern Cear? State (continental margins of the Cear? and Potiguar basins), provided addnional data and constraints about the teetonic-structural evolution of the oil lield. The work at the analogue sites was particularly effective in the recognition and mapping, in semidetail scale, several generations of struetures originated under a brittle regime. Ali the obtained information (from the Xar?u Oil Field, the remote sensor data and the terrain analogues) were jointly interpreted, culminating with the proposnion of an evolutionary model lor this segment of the Atlantic Equatorial Margin; this model that can be applied to the whole Margin, as well. This segmentof the Atlantic Equatorial Margin was delormedin an early E-W (when considered lhe present-day position of the South American Plate) transcurrent to transform regime with dextral kinematics, started Irom, at least, the Early Aptian, which left its record in several outcrops along the continental margin of the Cear? State and specilically in the Xar?u off Field. The continuous operation of the regime, through the Albian and later periods, led to the definitive separation between the South American and African plates, with the formation of oceanic lithosphere between the two continental blocks, due to the emplacement off spreading centers. This process involved the subsequent transition of the transcurrent to a translorm dextral regime, creating lhe Equatorial Atlantic Oceano With the separation between the South American and African plates already completed and the increasing separation between lhe continental masses, other tecton ic mechanisms began to act during the Cenozoic (even though the Cretaceous tectonic regime lasted until the Neogene), like an E-W compressive stress l?eld (related to the spreading olthe oceanic floor along lhe M id-Atlantic Ridge and to the compression of the Andean Chain) effective Irom the Late Cretaceous, and a state of general extension olthe horizontal surface (due to the thermal uplift ofthe central portion of Borborema Province), effective during the Neogene. The overlap of these mechanisms during the Cenozoic led to the imprint of a complex tectonic framework, which apparently influenced the migration and entrapment 01 hydrocarbon in the Cear? Basin

Fracture propagation in rock often produces complex patterns, such as the branching patterns of fracture networks, or the irregular radiating patterns on fracture surfaces. These patterns often appear complex because they are unpredictable in detail, yet predictable in the sense that smaller pieces of the pattern, when suitably enlarged, are statistically similar to larger pieces of the pattern. This property of statistical self-similarity can be quantified using fractal geometry. The fractal nature of rock fracture patterns is related to lithological properties and to the dynamics of the fracturing processes. Joint surfaces in homogeneous rocks display rough radiating ridges. A "plumose joint" surface was analyzed using the slit island method and found to have a fractal dimension (D$\sb{\rm f}$) of 2.24 $\pm$.14 (95%). Surfaces with similar fractal dimensions (2.2-2.5) are produced by a three-dimensional computer simulation of jointing. In the simulation, randomly-distributed defects cause local mis-orientations of the stress field and local deflection of the propagating fracture front. After passing through the defect the joint surface is re-oriented relative to remote stresses, and a planar radial fracture segment (i.e. inclusion hackle) is formed. Collectively, the numerous inclusion hackle form the plumose surface pattern. For the simulation results, D$\sb{\rm f}$ increases (i.e. the surface gets rougher) in proportion to the log of the defect density. The simulation also demonstrates a complex relationship between D$\sb{\rm f}$ of the propagating fracture front and D$\sb{\rm f}$ of the fracture surface. Shatter cones are conical fracture surfaces produced during high energy events such as meteorite impact and nuclear explosion. These fractures also display radiating surface features. Using a modified version of the slit island method, the fractal dimension of a shatter cone surface in limestone is estimated to be 2.24 $\pm$ 0.09. The observation of shingled convex fracture surfaces within the conical envelope surrounding the shatter cone surface is demonstrated to support the genetic model of Johnson and Talbot (1964). Striations on these fracture surfaces are reinterpreted as micro-fracture intersections. The measured fractal dimensions of the joint and shatter cone surfaces (i.e 2.24) are within the range reported for most fracture surfaces in metals (i.e. 2.1 $\sim$ 2.3).

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior
In the last years, the fractal geometry has become widely studied. Its application in several areas increased substantially, particularly in materials engineering and science, aiming the analysis of failures and the study of the mechanical properties of materials. Some studies have shown the relationship between the fracture surfaces and their mechanical properties using the fractal geometry and its properties of fractal dimension and selfsimilarity. In this research, the fracture surface of duplex stainless steel, which was obtained by the Charpy impact test, was studied applying the fractal geometry. Considering the 475ÂC embrittlement, the steel was submitted to thermal aging to obtain the fracture surfaces, whose images were captured by the scanning electron microscope (SEM). In the fractal analysis, a study was made applying the island method and profile analysis through the digitalization of the images and the application of image analyzing software. Emphasis was given on the calculation of the fractal dimension (FD) of the surface, on the energy absorbed during the impact test, on the involved fracture mechanisms and as well on the relationship between FD and thermal aging. In order to better understanding the subject, it was done the review about fracture mechanics, of duplex stainless steel and of fractal geometry. Finishing the research, the obtained fracture surface, the absorbed energy and the obtained values of FD were analyzed. The obtained results demonstrated a relationship between the fractal dimension, the size of the dimples in fracture surfaces and the impact energy to obtain them.
A geometria dos fractais nos Ãltimos anos tem se tornado bastante difundida no meio cientÃfico. O seu emprego em diversas Ãreas aumentou substancialmente, em particular na engenharia e ciÃncia dos materiais, com o objetivo de analisar falhas e estudar as propriedades mecÃnicas dos materiais. Alguns estudos tÃm mostrado a relaÃÃo entre as propriedades mecÃnicas de superfÃcies de fratura com a geometria dos fractais e suas propriedades de dimensÃo fractal e auto-similaridade. Nesta pesquisa, foi estudada, com base na geometria dos fractais, a superfÃcie de fratura do aÃo inoxidÃvel duplex obtida atravÃs do ensaio de impacto Charpy. Considerando a fragilizaÃÃo a 475C, o aÃo foi submetido ao tratamento tÃrmico de envelhecimento para a obtenÃÃo das superfÃcies de fraturas cujas imagens foram captadas no microscÃpio eletrÃnico de varredura (MEV). Na anÃlise fractal foi feito um estudo aplicando os mÃtodos das ilhas e anÃlise de perfil atravÃs da digitalizaÃÃo das imagens e aplicaÃÃo de softwares de anÃlise de imagem. Foi dada Ãnfase na anÃlise do cÃlculo da dimensÃo fractal (Df) da superfÃcie, na energia absorvida no ensaio de impacto, nos mecanismos de fratura envolvidos, bem como na relaÃÃo entre Df e o tratamento tÃrmico de envelhecimento. Para o melhor entendimento do trabalho foi feita uma revisÃo bibliogrÃfica sobre a mecÃnica da fratura, o aÃo inoxidÃvel duplex e a geometria dos fractais. Para finalizar a pesquisa, foi feita a anÃlise da superfÃcie da fratura obtida, da energia absorvia e de valores de Df alcanÃados. Os resultados obtidos demonstraram uma relaÃÃo entre a dimensÃo fractal, o tamanho dos dimples em superfÃcies de fratura e a energia de impacto para a obtenÃÃo das mesmas.

Orientador: Archimedes Perez Filho
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Geociências
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Resumo: O Cerrado se constitui em um mosaico de fitofisionomias que se distinguem entre si pelos padrões espaciais que apresentam. Apesar das características e dinâmica do Cerrado apontarem que ele deve ser compreendido como um sistema complexo, o paradigma da complexidade e os métodos dele derivados ainda são pouco aproveitados no estudo do Cerrado. O objetivo geral da presente pesquisa foi avaliar a complexidade dos padrões espaciais (texturais) e espectrais de fitofisionomias de Cerrado a fim de verificar quais propriedades relativas à sua organização e dinâmica tais padrões podem revelar. Para tanto, foram usadas imagens do sensor multi-espectral Aster relativas a unidades de conservação do estado de São Paulo situadas nos municípios de Assis, Itirapina e Luiz Antônio. Medidas de complexidade baseadas na entropia informacional e de dimensão fractal foram aplicadas às imagens e respectivas curvas de respostas espectrais de fisionomias de Cerrado presentes nas localidades estudadas. Uma área-teste foi selecionada para se avaliar as correlações entre fisionomias, seus padrões texturais e espectrais e fatores pedológicos e geomorfológicos. Os resultados obtidos para as análises textural e espectral de imagens do sensor mostram que cada fisionomia apresenta valores estatisticamente iguais nas diferentes extensões avaliadas, revelando a auto-similaridade dos padrões em várias escalas. Houve também uma forte tendência de cada fisionomia obter os mesmos valores em diferentes localidades, o que permite estabelecer intervalos de valores típicos para cada uma, independentemente da área estudada. Por outro lado, nenhuma medida foi totalmente eficiente em distinguir as diferentes fisionomias de Cerrado de uma mesma localidade, principalmente aquelas com padrões mais semelhantes. Quanto às correlações, foram encontradas associações significativas entre fisionomias e fatores pedogeomorfológicos, porém não houve nenhum fator que respondesse exclusivamente pelas características vegetacionais de determinada fisionomia e nem pela configuração de seus padrões, apontando que elas dependem das inter-relações de vários fatores. Pelos resultados alcançados na presente pesquisa, confirma-se que o Cerrado é um sistema dinâmico complexo e que, portanto, o entendimento de sua organização e dinâmica deve-se pautar nos conceitos, modelos e métodos próprios do paradigma da complexidade. Uma característica marcante aqui revelada é a invariância escalar dos padrões, a qual é indicativa de que o Cerrado apresentaria criticalidade autoorganizada, sendo algumas de suas fisionomias representativas de estados próximos a pontos críticos. Conforme apontam os resultados, fisionomias intermediárias, como cerrado denso, cerrado ss e campo cerrado, apresentariam esse tipo de organização, enquanto fisionomias situadas próximas aos extremos do gradiente vegetacional do Cerrado (como campo sujo e cerradão) representariam estados mais estáveis do sistema
Abstract: Brazilian Cerrado is characterized as a mosaic of phytophysiognomies with different spatial patterns. Despite of its characteristics and dynamics suggest that the Cerrado should be understood as a complex system, the complexity paradigm and methodologies are not widely used in Cerrado studies yet. The general objective of this research has been to evaluate the complexity of spatial (textural) and spectral patterns of Cerrado's phytophysiognomies with the purpose of verifying which properties related to organization and dynamic those patterns could show. For this, images from Aster multispectral sensor were used to study Cerrado areas in conservation reserves at São Paulo State (Brazil). Complexity measures based on information entropy and fractal dimension were applied to physiognomies images and to the correspondent spectral response curves. A test-area was selected to evaluate correlations between physiognomies, their textural and spectral patterns, and pedological-geomorphological factors. Textural and spectral image analysis pointed out that each physiognomy presents statistically equal values for different extents considered, showing self-similarity patterns in several scales. There was also a strong tendency that each physiognomy presents the same values at different localities, attributing a typical range of values for each one, independent of its localization. However, no measure was totally efficient to distinguish different Cerrado's physiognomies, especially those with similar patterns. For correlations, significant associations between physiognomies and pedological-geomorphological factors were founded, but here there was no factor responding exclusively for vegetation characteristics of a given physiognomy and for pattern configurations as well, suggesting that they depend on interrelations of many factors. Results obtained in this work confirm that Cerrado is a complex dynamical system and, therefore, comprehension of its organization and dynamics demands concepts, models, and methods related to complexity paradigm. A remarkable characteristic that was revealed here is about scale-invariance of patterns, which indicates that Cerrado presents self-organization criticality. As results show, this type of organization occurs in intermediary physiognomies, while grassland and forest physiognomies are more stable
Doutorado
Análise Ambiental e Dinâmica Territorial
Doutor em Ciências

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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.

Estudis internacionals han demostrat la utilitat de l'eina FRAX de l’Organització Mundial de la Salut (OMS) per a l'estimació del risc absolut de fractura osteoporòtica a 10 anys. S’ha evidenciat que a població femenina espanyola l'eina té una bona capacitat de discriminació per detectar dones que patiran una fractura per fragilitat de les que no, i que millora el model tradicional basat en la densitometria òssia (T-score ≤ -2.5 desviacions estandars (DE)). No obstant això, l’algoritme mostra una manca de concordança global entre la probabilitat estimada de fractura i la incidència de fractura observada individual a 10 anys, amb una tendència a infraestimar el risc. Per millorar la capacitat predictiva de fractura osteoporòtica principal s’ha desenvolupat el model FRIDEX, amb 3 categories de risc FRAX (calculat sense T-score de coll femoral) per població femenina espanyola (<5%; ≥5 a <7,5%; ≥7,5%) que iclouen la incidència de fractures observades en la cohort, les quals estaven, al seu torn, respectivament dins els nivells tradicionalment considerats com de risc baix (<10%), intermig (10-20%) i alt (> 20%) en altres estudis epidemiològics. L’objectiu d’aquesta tesi doctoral va ser validar aquest model FRIDEX de calibració del risc de fractura per fragilitat en població femenina espanyola en base a l’algoritme FRAX en la cohort ampliada FRIDEX i en la població general (cohort FROCAT). En primer lloc els resultats del treball validen les interpretacions de la capacitat discriminativa i predictiva de l’eina FRAX obtingudes en les primeres anàlisi de la cohort FRIDEX i, és important perquè d’aquests casos es va desenvolupar la proposta per categoritzar el risc de fractura en dones espanyoles. En segon lloc, validen externament el model FRIDEX de discriminació del risc de fractura principal a 10 anys en una cohort de dones de base poblacional i diferent de la cohort que ha servit per establir-lo.
The usefulness of the FRAX tool from the World Health Organization to evaluate the 10-year fracture risk of patients has been shown in many international studies. In the Spanish female population it has been demonstrated that the tool has a a good discriminative ability to detect women who will suffer from a fragility fracture and it is at least equivalent to, if not slightly higher than the bone mineral density (BMD) model (T-score ≤ -2.5 standard deviations). However, it displays weaknesses in the overall concordance between predicted fracture probability and observed fracture incidence, with a tendency to underestimate the risk. To improve the main osteoporotic fracture risk predictive capability, some FRAX thresholds have been developped to identify people with risk of osteoporotic fracture in Spain, based on the stratification from the female FRIDEX cohort into 3 levels of FRAX risk ( without femoral neck (FN) T-score) (low risk <5%, intermediate ≥5 to <7.5% and high ≥7.5%) according to the fracture incidence observed in the cohort, which in turn are related with risk ranges traditionally considered to be low (<10%), intermediate (10-20%) and high risk (> 20%) in other epidemiological studies. The aim of this thesis was to validate the FRIDEX model in the Spanish female population using the FRAX main osteoporotic fracture risk thresholds to 10-year in the updated results of the FRIDEX cohort and in the general population (FROCAT cohort). Firstly, the work validates the discriminatory and predictive ability of the FRAX tool obtained in the preliminary analysis of the FRIDEX cohort, and this is important because from these cases the categorization in the three risk groups was developed. Secondly, it externally validates the FRIDEX model to discriminate fracture risk in women from the general population.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
This work deals with the study of fractal geometry, emphasizing its main features included on natural systems that motivate them. Here some names that contributed to the emergence and development of mathematical fractals, emphasizing examples of natural fractals and the pioneer of Benoit B. Mandelbrot contribution .
Este trabalho trata do estudo da geometria fractal, enfatizando suas principais caracter sticas compreendidas com base nos sistemas naturais que as motivam. Apresentamos alguns nomes que contribuiram para o surgimento e desenvolvimento dos fractais matem aticos, enfatizando os exemplos de fractais naturais e a contribui c~ao do pioneiro Benoit B. Mandelbrot.

Os padrões de fratura são objeto de intensas investigações em várias áreas do conhecimento. Além disso, a busca por métodos matemáticos para simulação de mapas de fratura em meios geológicos tem sido alvo de muitas investigações. Este trabalho apresenta um estudo sobre a dimensão fractal de padrões de fraturas geológicas e a simulação de um mapa de fratura geológica através da análise fractal de áreas estruturalmente homogêneas na região Central do Brasil. As dimensões fractais são determinadas pelos métodos Box-counting e Cantor’s Dust num sistema anisotrópico de fraturas, a fim de caracterizar áreas geológicas selecionadas. A dimensão fractal obtida pelo método Box-counting possui valores entre um e dois, mas não permite analisar anisotropias de comprimento das fraturas, ou mesmo na sua distribuição espacial. A dimensão fractal obtida com o método de Cantor’s Dust, por outro lado, possui valores entre zero e um e fornece mecanismos adequados à avaliação da anisotropia das redes de fraturas geológicas. Deve-se observar, a partir dos resultados obtidos, que a dimensão fractal (Cantor’s Dust) depende das direções de fratura e que o número de interseções de fraturas aumenta com a diminuição da dimensão fractal. A dimensão fractal (Cantor’s Dust) também varia conforme a direção da rede de linhas ortogonais, indicando o caráter levemente anisotrópico dos padrões de fratura geológica sob análise. A análise estrutural dos padrões de fratura foi efetuada sobre as direções, o comprimento e a freqüência das fraturas de cada área homogênea. Assim, foi possível determinar uma relação entre a freqüência de fratura e a dimensão fractal. A simulação de um padrão de fraturas por método analítico levou em conta a direção e a freqüência de fratura e a dimensão fractal (Cantor’s Dust). Os resultados alcançados para o mapa de fraturas simulado são muito bons, principalmente pelo fato de não ter sido computado, nessa primeira versão, o comprimento das fraturas.
The fracture patterns are subject of large investigations in a number of knowledge areas. The mathematical methods research to simulate fractures patterns in geological media is also intense. This work aims to contribute for such investigations and presents the results of geological fracture pattern analysis (structural and fractal analysis) and an analytical procedure to simulate such fracture pattern. The fracture analysis was conducted in structurally homogeneous areas in Central Brazil. The fractal dimensions were computed by both Box-counting and Cantor’s Dust methods, in order to characterize each geological area. Box-counting fractal dimensions showed values in the range 1 and 2, but did not enable to evaluate fracture anisotropies, such as fracture length, frequency and orientation. On the other hand, Cantor’s Dust fractal dimensions showed values between 0 and 1, so it is able to determine fracture anisotropies, since orthogonal grid rotation defined different fractal dimensions. The structural analysis of the fracture patterns was conducted taking into account fracture directions, length and frequency. In this way, it is possible to correlate fracture frequency and Cantor’s Dust fractal dimensions. The achieved results for the fracture lineament map simulation are reasonable good, because it did not consider the fracture length according each direction.

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.

Dissertação de Mestrado Integrado em Medicina Veterinária
A ortopedia representa, actualmente, cerca de 10% dos casos avaliados em clínica de animais de companhia, sendo os canídeos a espécie mais representativa na ocorrência de processos traumáticos. O esqueleto, composto pelos ossos que são um importante reservatório de minerais em particular de cálcio e de fósforo, é uma estrutura altamente dinâmica cuja actividade resultante da acção de factores intrínsecos e extrínsecos é regulada por vários mecanismos endócrinos ao longo de toda a vida do indivíduo. A grande maioria do ião cálcio (99%) encontra-se no esqueleto sob a forma de cristais de hidroxiapatite estáveis e de compostos lábeis rapidamente mobilizáveis, de modo a garantirem a manutenção dos seus valores séricos. O presente estudo realizado numa população de 20 canídeos (canis familiaris) (n=20), avaliados com fracturas simples ou cominutiva do esqueleto apendicular, teve como objectivos determinar: 1) a existência ou não de diferenças nos valores de cálcio sérico ionizado (CSI) entre indivíduos com e sem fracturas; e 2) se o tipo de fractura apresentado influencia ou não os valores de CSI. Os resultados obtidos permitiram concluir que todos os doentes com fractura apresentam hipercalcemia, resultante do processo de reparação óssea; e que, os indivíduos com fractura cominutiva possuem valores de CSI tendencialmente superiores aos indivíduos com fractura simples, devido à maior área dos topos ósseos da fractura, apesar da diferença neste caso não ser estatisticamente significativa. Com base nos resultados, deverão, os clínicos das áreas médica e cirúrgica, considerar que os doentes com fracturas apresentamse sempre com uma condição de hipercalcemia de base, requerendo cuidados acrescidos na selecção dos fármacos utilizados no seu maneio e que se associam a um aumento dos níveis de CSI, assim como do quadro clinico a este associado.
ABSTRACT - INFLUENCE OF THE TYPE OF FRACTURE IN SERUM IONIZED CALCIUM IN DOGS - The orthopedics currently represents about 10% of the small animal clinical cases, being the dog the most representative species in the occurrence of traumatic processes. The skeleton, composed by bones that are an essential mineral reservoir in particular of calcium and phosphorus, is a highly dynamic structure, whose activity arising from intrinsic and extrinsic factors, is regulated by multiple endocrine mechanisms, throughout life. The majority of calcium ion (99%) is present in the skeleton by the hydroxyapatite stable crystals form and by labile compounds rapidly deployable, to promote the maintenance of it serum levels. This study was conducted in a population of 20 dogs (Canis familiaris) (n=20) assessed with simple and comminuted fractures of the appendicular skeleton, and aimed to evaluate: 1) whether there are or not differences in ionized seric calcium (ISC) levels between individuals with or without fractures; and 2) to assessed whether the type of fracture may or may not influence the ISC levels. The results showed that all patients with fracture present hypercalcemia resulting from bone healing; and patients with comminuted fractures had higher ISC levels than those with simple fractures, due to the larger area of the bone edges, despite in this case no significant statistical difference was registered. Based on the results, veterinary surgeons should consider that patients with fracture always show a hypercalcemia clinical condition, requiring increased care in the drugs selection to be used in this patient management and their potential to increase the ISC levels, as well as the clinical condition associated.

Fractal geometry is a new branch of mathematics. This report presents the tools, methods and theory required to describe this geometry. The power of Iterated Function Systems (IFS) is introduced and applied to produce fractal images or approximate complex estructures found in nature. The focus of this thesis is on how fractal geometry can be used in applications to computer graphics or to model natural objects.

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Outro
This is a research about a part of the non-Euclidean geometry that has recently been very studied. It was addressed initial themes of the non-Euclidean geometry and it was exposed the studies abut fractals, its history, buildings and main fractals (known as classic fractals). It was also addressed the relation among the school years contents and how to use fractals; as well as some of its applications that have helped a lot of researches to spread and show better results.
Este trabalho é uma pesquisa sobre parte da geometria não euclidiana que há pouco vem sendo muito estudada, os fractais. Abordamos temas iniciais da geometria nãoeuclidiana e no decorrer do trabalho expomos nosso estudo sobre fractais, seu histórico, construções, principais fractais (conhecidos como fractais clássicos). Também abordamos relações entre conteúdos dos anos escolares e como usar fractais nos mesmos; como também algumas de suas aplicações que vem ajudando muitas pesquisas a se difundirem e apresentarem melhores resultados.

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Presenta los resultados del tratamiento de las fracturas mediodiafisiarias de fémur en adultos, quirúrgicamente tratadas con clavo endomedular bloqueado, clavo de Küntscher y placa DCP en el servicio de ortopedia y traumatología del HNHU. El estudio realizado fue de tipo descriptivo, retrospectivo, de corte transversal; se consideraron a todos los pacientes hospitalizados en el servicio de Traumatología del HNHU con diagnóstico de fractura mediodiafisiarias y que cumplieron con los criterios de inclusión seleccionados para el presente estudio, los gráficos fueron elaborados en el programa SPSS versión Nº 21. Encuentra que seleccionaron 39 pacientes para el estudio, el 41.1% fueron intervenidos con clavos endomedulares bloqueados, de los cuales el 66.7% fueron varones, el 25.6% fueron menores de 20 años y entre 41 y 50 años, el 56.4% fueron afectados en la extremidad derecha, el 61.5% fueron causados por accidente de tránsito, el 48.6% presentó un trazo oblicuo, al 51.5% se les aplico tracción esquelética, el 66.6% tuvieron más de 20 días de espera antes de la cirugía, el 69.2% tuvieron un tiempo de hospitalización entre 16 y 30 días, el 53.5% tuvieron controles post operatorios de entre 2 y 5 meses, el 48.2% no presentaron secuelas. El éxito del tratamiento empleado en las facturas mediodiafisiarias ha conllevado a una reducción significativa de las secuelas, así como la adecuada decisión y aplicación de los procedimientos en los pacientes con este tipo de fracturas. Concluye que las fracturas mediodiafisiarias son más comunes en hombres que han sufrido accidente de tránsito siendo el miembro inferior derecho más afectado, el uso de clavo endomedular bloqueado es el más usado sin presentar mayores secuelas.
Trabajo académico

When a fibre-reinforced composite laminate is subjected to traction or compression, it develops a relatively large Fracture Process Zone where material toughening mechanisms such as matrix cracking, fibre-bridging and fibre pull-outs take place. The damage onset and damage propagation are well defined from a cohesive model point of view, although no standard procedure has been yet developed to characterize the translaminar Cohesive Law. The present Ph.D. thesis proposes an objective inverse method for obtaining the Cohesive Law from a single Compact Tension test. The proposed methodology has been correlated with numerical results, and has been satisfactorily used to characterize this property in various laminates, both in traction and compression
Quan un laminat compost de polímer reforçat amb fibres és sotmès a tracció o compressió, aquest desenvolupa una Zona de Procés de Fractura relativament gran, on mecanismes d'enduriment com ara l'esquerdament de la matriu, bridging i pull-out de fibres tenen lloc. La iniciació i propagació del dany estan ben definits des del punt de vista del model cohesiu, tot i que actualment encara no hi ha cap procediment estàndard per caracteritzar la Llei Cohesiva translaminar. La present tesi doctoral proposa un mètode invers per a l'obtenció de la Llei Cohesiva utilitzant únicament un assaig de la proveta Compact Tension. La metodologia proposada ha estat correlada amb resultats numèrics, i ha estat empleada satisfactòriament per a caracteritzar aquesta propietat en diversos laminats, tant a tracció com a compressió

Las antenas son dispositivos esencialmente de banda estrecha; es decir, su comportamiento es altamente dependiente de la frecuencia de operación. La relación entre la longitud de onda de trabajo y el tamaño de la antena condiciona enormemente sus prestaciones. Ello implica que es complicado realizar antenas que funcionen de la misma forma en varias frecuencias, o que tengan un tamaño mucho menor que la longitud de onda. En esta tesis se introduce el diseño fractal de antenas como una posible alternativa para abordar este tipo de problemas. En particular, se muestra como las propiedades geométricas de autosimilitud y dimensión no entera, que son características de los objetos fractales, pueden ser útiles en el diseño de antenas multifrecuencia (multibanda) y de antenas pequeñas. A lo largo de la Tesis se muestran varios resultados experimentales, numéricos y teóricos que confirman este hecho. Tras revisar las restricciones más comunes de las antenas clásicas, la tesis empieza enunciando teóricamente el comportamiento multibanda de las antenas con geometría fractal. El estudio se particulariza entonces al caso de agrupaciones fractales de antenas y a factores de agrupación fractales. Se demuestra que las agrupaciones fractales ideales tienen un comportamiento multibanda y que las versiones implementables de dichas estructuras Ideales conservan algunas de las propiedades multibanda. La antena de Sierpinski fue la primera verificación experimental de que se podía construir una antena fractal multibanda. A lo largo de la tesis se introduce la nueva geometría de la antena y sus propiedades fundamentales. Se ensayan también varias modificaciones de la estructura inicial, tales como variaciones en el ángulo del vértice de alimentación, variaciones en la inclinación de la antena, del espaciado entre bandas, etc. También se describe como otras formas fractales aleatorias poseen también propiedades multibanda.

Humans are experts at understanding what they see. Similarity and analogy play a significant role in making sense of the visual world by forming analogies to similar images encountered previously. Yet, while these acts of visual reasoning may be commonplace, the processes of visual analogy are not yet well understood. In this dissertation, I investigate the utility of representing visual information in a fractal manner for computing visual similarity and analogy. In particular, I develop a computational technique of fractal reasoning for addressing problems of visual similarity and novelty. I illustrate the effectiveness of fractal reasoning on problems of visual similarity and analogy on the Raven’s Progressive Matrices and Miller’s Analogies tests of intelligence, problems of visual novelty and oddity on the Odd One Out test of intelligence, and problems of visual similarity and oddity on the Dehaene test of core geometric reasoning. I show that the performance of my computational model on these various tests is comparable to human performance. Fractal reasoning provides a new method for computing answers to such problems. Specifically, I show that the choice of the level of abstraction of problem representation determines the degree to which an answer may be regarded as confident, and that that choice of abstraction may be controlled automatically by the algorithm as a means of seeking that confident answer. This emergence of ambiguity and its remedy via problem re-representation is afforded by the fractal representation. I also show how reasoning over sparse data (at coarse levels of abstraction) or homogeneous data (at finest levels of abstraction) could both drive the automatic exclusion of certain levels of abstraction, as well as provide a signal to shift the analogical reasoning from consideration of simple analogies (such as analogies between pairs of objects) to more complex analogies (such as analogies among triplets, or larger groups of objects). My dissertation also explores fractal reasoning in perception, including both biologically-inspired imprinting and bistable perception. In particular, it provides a computational explanation of bistable perception in the famous Necker cube problem that is directly tied to the process of determining a confident interpretation via re-representation. Thus, my research makes two primary contributions to AI theories of visual similarity and analogy. The first contribution is the Extended Analogy By Recall (ABR*) algorithm, the computational technique for visual reasoning that automatically adjusts fractal representations to an appropriate level of abstraction. The second contribution is the fractal representation itself, a knowledge representation that add the notion of self-similarity and re-representation to analogy making.

This thesis is devoted to a study about Fractals and Fractal Polynomial Interpolation. Fractal Interpolation is a great topic with many interesting applications, some of which are used in everyday lives such as television, camera, and radio. The thesis is comprised of eight chapters. Chapter one contains a brief introduction and a historical account of fractals. Chapter two is about polynomial interpolation processes such as Newton s, Hermite, and Lagrange. Chapter three focuses on iterated function systems. In this chapter I report results contained in Barnsley s paper, Fractal Functions and Interpolation. I also mention results on iterated function system for fractal polynomial interpolation. Chapters four and five cover fractal polynomial interpolation and fractal interpolation of functions studied by Navascués. Chapter five and six are the generalization of Hermite and Lagrange functions using fractal interpolation. As a concluding chapter we look at the current applications of fractals in various walks of life such as physics and finance and its prospects for the future.
M.S.
Department of Mathematics
Sciences
Mathematical Science MS

This thesis explores new approaches to the analysis of functions by combining tools from the fields of complex bases, number systems, iterated function systems (IFS) and wavelet multiresolution analyses (MRA). The foundation of this work is grounded in the identification of a link between two-dimensional non-separable Haar wavelets and complex bases. The theory of complex bases and this link are generalized to higher dimensional number systems. Tilings generated by number systems are typically fractal in nature. This often yields asymmetry in the wavelet trees of functions during wavelet decomposition. To acknowledge this situation, a class of extensions of functions is developed. These are shown to be consistent with the Mallat algorithm. A formal definition of local IFS on wavelet trees (LIFSW) is constructed for MRA associated with number systems, along with an application to the inverse problem. From these investigations, a series of algorithms emerge, namely the Mallat algorithm using addressing in number systems, an algorithm for extending functions and a method for constructing LIFSW operators in higher dimensions. Applications to image coding are given and ideas for further study are also proposed. Background material is included to assist readers less familiar with the varied topics considered. In addition, an appendix provides a more detailed exposition of the fundamentals of IFS theory.

This thesis explores new approaches to the analysis of functions by combining tools from the fields of complex bases, number systems, iterated function systems (IFS) and wavelet multiresolution analyses (MRA). The foundation of this work is grounded in the identification of a link between two-dimensional non-separable Haar wavelets and complex bases. The theory of complex bases and this link are generalized to higher dimensional number systems. Tilings generated by number systems are typically fractal in nature. This often yields asymmetry in the wavelet trees of functions during wavelet decomposition. To acknowledge this situation, a class of extensions of functions is developed. These are shown to be consistent with the Mallat algorithm. A formal definition of local IFS on wavelet trees (LIFSW) is constructed for MRA associated with number systems, along with an application to the inverse problem. From these investigations, a series of algorithms emerge, namely the Mallat algorithm using addressing in number systems, an algorithm for extending functions and a method for constructing LIFSW operators in higher dimensions. Applications to image coding are given and ideas for further study are also proposed. Background material is included to assist readers less familiar with the varied topics considered. In addition, an appendix provides a more detailed exposition of the fundamentals of IFS theory.

Alcançar altas taxas de dados em comunicações sem fio é difícil. Altas taxas de dados para redes locais sem fio tornou-se comercialmente um sucesso por volta do ano de 2000. Redes de longa distância sem fio ainda são projetados e utilizados principalmente para serviços de voz em baixas taxas. Apesar de muitas tecnologias promissoras, a realidade de uma rede de área ampla que atenda muitos usuários com altas taxas de dados e largura de banda e consumo de energia razoáveis, além de uma boa cobertura e qualidade no serviço ainda é um desafio. O objetivo do IEEE 802.16 é projetar um sistema de comunicação sem fio para obter uma internet de banda larga para usuários móveis em uma área metropolitana. É importante perceber que o sistema WIMAX tem que enfrentar desafios semelhantes aos existentes sistemas celulares e seu desempenho eventual será delimitado pelas mesmas leis da física e da teoria da informação. Em muitas áreas da engenharia elétrica, tem-se direcionado atenção à miniaturização de componentes e equipamentos. Em particular, antenas não são exceções. Desde que Wheeler iniciou estudos sobre os limites fundamentais de miniaturização de antenas, o assunto tem sido discutido por muitos estudiosos e várias contribuições nesse sentido foram feitas desde então. Os avanços das últimas décadas na área de microeletrônica permitiram a miniaturização dos demais componentes empregados no desenvolvimento de equipamentos eletrônicos e disponibilizaram o uso de aparelhos compactos, leves e com diversas funcionalidades e aplicações comerciais. No entanto, ainda que a integração de circuitos seja uma realidade, a integração completa de um sistema de comunicação sem fio, incluindo a antena, é ainda um dos grandes desafios tecnológicos. No caso de antenas impressas procura-se continuamente desenvolver antenas que, além de compactas, apresentem maior largura de banda, ou operação em múltiplas bandas dada sua inerente característica de banda estreita em projetos convencionais. Neste trabalho, o foco está na miniaturização de antenas impressas através da aplicação de fractais. São apresentadas comparações entre antenas fractais quadradas de Minkowski e fractais triangulares de Koch. Inicialmente, antenas 6 impressas com geometrias convencionais quadradas e triangulares foram projetadas para ter a mesma frequência de ressonância. Depois disso, as estruturas fractais de Minkowski Island e Koch Loop foram implementadas nas antenas quadrada e triangular, respectivamente, até a terceira iteração. As frequências escolhidas foram as de 2,4 GHz, 3,5 GHz, 5,0 GHz e 5,8 GHz. Diversos protótipos foram construídos em dois substratos de permissividade diferentes, o FR-4 e o DUROID 5870. Para validar os resultados foram construídas antenas na frequência de 3,5 GHz para as geometrias quadrada e triangular e suas iterações fractais. A contribuição deste trabalho está na análise sobre as vantagens e desvantagens de cada uma das estruturas propostas. Dependendo dos requisitos de um projeto, a opção pode ser por antenas miniaturizadas com maior largura de banda, como normalmente acontece em alguns projetos comerciais. Entretanto, o interesse por bandas estreitas muitas vezes pode ser um requisito, principalmente para emprego militar, onde por vezes a máxima discrição na transmissão é uma exigência. Além disso, também foi feita uma análise sobre as geometrias que atingiram maior miniaturização.
Achieving high data rates in wireless communication is difficult. High data rates for wireless local area networks became commercially successful only around 2000. Wide area wireless networks are still designed and used primarily for low rate voice services. Despite many promising technologies, the reality of a wide area network that services many users at high data rates with reasonable bandwidth and power consumption, while maintaining high coverage and quality of service has not been achieved. The goal of the IEEE 802.16 was to design a wireless communication system processing to achieve a broadband internet for mobile users over a wide or metropolitan area. It is important to realize that WIMAX system have to confront similar challenges as existing cellular systems and their eventual performance will be bounded by the same laws of physics and information theory. In many areas of electrical engineering, miniaturization has been an important issue. Antennas are not an exception. After Wheeler initiated studies on the fundamental limits for miniaturization of antennas, this subject has been extensively discussed by several scholars and many contributions have been made. The advances of recent decades in the field of microelectronics enabled the miniaturization of components and provided the use of compact, lightweight, equipments with many features in commercial applications. Although circuit integration is a reality, the integration of a complete system, including its antenna, is still one of the major technological challenges. In the case of patch antennas, the search is for compact structures with increased bandwidth, due to the inherent narrowband characteristic of this type of antenna. In this work the focus is on a comparison between the Minkowski and the Koch Fractal Patch Antennas. Initially, patch antennas with conventional square and triangular geometries were simulated to present the same resonance frequency. After that, fractal Minkowski and Koch Island Loop antennas were implemented in the square and triangular geometries, respectively, to the third iteration. A comparison was made for two substrates of different permittivities FR-4 and DUROID 5870 at the frequencies of 2,4 GHz; 3,5 GHz; 5,0 GHz and 5,8 GHz. 8 Prototype antennas were built using FR-4 and DUROID 5870 to resonate at a frequency of 3,5 GHz to validate simulation results. The contribution of this work is the analysis of the advantages and disadvantages of each proposed fractal structure. According to the project requirements, the best option can be use a miniaturized antenna with a wider band, as in commercial projects. Particularly in military applications, a narrow band antenna can be a requirement, as sometimes maximum discretion in transmission is a paramount. An additional analysis was performed to verify which of the geometries fulfilled the miniaturization criteria of Hansen.

La méthode de minimisation de l'intégrale d'action (principe variationnel) permet d’obtenir les équations de propagation des ondes. Cette méthode a été généralisée aux milieux poreux de dimensions fractales, pour étudier la propagation acoustique dans le domaine temporel, en se basant sur le modèle du fluide équivalent. L'équation obtenue réécrite dans le domaine fréquentiel représente une généralisation de l'équation d'Helmholtz. Dans le cadre du modèle d'Allard-Johnson, l'équation de propagation a été résolue de manière analytique dans le domaine temporel, dans les régimes des hautes et des basses fréquences. La résolution a été faite par la méthode de la transformée de Laplace, et a porté sur un milieu poreux semi-infini. Il a été trouvé que la vitesse de propagation dépend de la dimension fractale. Pour un matériau poreux fractal d'épaisseur finie qui reçoit une onde acoustique en incidence normale, les conditions d’Euler ont été utilisées pour déterminer les champs réfléchi et transmis. La résolution du problème direct a été faite dans le domaine temporel, par la méthode de la transformée de Laplace, et par l’usage des fonctions de Mittag-Leffler. Le problème inverse a été résolu par la méthode de minimisation aux sens des moindres carrés. Des tests ont été effectués avec succès sur des données expérimentales, en utilisant des programmes numériques développés à partir du formalisme établi dans cette thèse. La résolution du problème inverse a permis de retrouver les paramètres acoustiques de mousses poreuses, dans les régimes des hautes et des basses fréquences
The action integral minimization method (variational principle) provides the wave propagation equations. This method has been generalized to fractal dimensional porous media to study the acoustic propagation in the time domain, based on the equivalent fluid model. The resulting equation rewritten in the frequency domain represents a generalization for the Helmholtz equation. As part of the Allard-Johnson model, the propagation equation was solved analytically in the time domain, for both high and low frequencies fields. The resolution was made by the method of the Laplace transform, and focused on a semi-infinite porous medium. It was found that the wave velocity depends on the fractal dimension.For a fractal porous material of finite thickness which receives an acoustic wave at normal incidence, the Euler conditions were used to determine the reflected and transmitted fields. The resolution of the direct problem was made in the time domain by the method of the Laplace transform, and through the use of the Mittag-Leffler functions. The inverse problem was solved by the method of minimizing the least squares sense. Tests have been performed successfully on experimental data; programs written from the formalism developed in this work have allowed finding the acoustic parameters of porous foams, in the fields of high and low frequencies

This thesis aims at getting a better understanding of the properties, scalings and similarities of turbulent axisymmetric wakes, as well as possible applications that arise from the information learnt. Over the last 60 years, axisymmetric wakes have been generated using axisymmetric bodies, such as disks, spheres and bodies of revolution, and key parameters such as the drag coefficient, shedding frequency and similarity and scaling of the wake width and velocity deficit have been documented and verified by numerous experimental and numerical studies. However, in this thesis the aim is to use asymmetric wake generators to generate the axisymmetric wakes and see if this has any effect on the results. These asymmetric wake generators are made up of a square plate and a number of fractal plates, where the perimeter of the plates can be increased by as much as 16 times that of the square. As well as increasing the perimeter, the irregularity, or fractal dimension, is also increased. It is found that the drag coefficient of the fractal plates is increased to beyond the values observed for regular polygons and a theory is presented that could explain this possible change in the drag coefficient, whereby the drag coefficient is the product of the volume of the wake and the dissipation of the turbulent kinetic energy within the wake. Wake profiles were taken over a moderate downstream distance of up to 50l, where l is the characteristic length of the plates, defined as the square root of the frontal area. Using the measured integral width of the wake directly, it was found that the volume of the wake decreased with increasing fractal dimension and iteration. Using these values, the similarity and scaling of the wake was carried out and a new high local Reynolds number scaling for turbulent axisymmetric wakes was discovered and for which the data from the fractal plates fit very well. The intensity of the vortex shedding is also shown to decrease with increasing perimeter and fractal dimension and it is found that the rate at which these vortices are shed is the same for all plates if the characteristic length is used to normalise the frequency. It is also discussed how the decrease in the energy of the vortex shedding is linked to the volume of the wake. Finally, the use of fractal geometries to manipulate the wake to reduce noise is also investigated, with emphasis placed on various aspects of an aircrafts wing.

Dendrites are tree-like topological spaces, and in this thesis, the physical characteristics of various random fractal versions of this type of set are investigated. This work will contribute to the development of analysis on fractals, an area which has grown considerably over the last twenty years. First, a collection of random self-similar dendrites is constructed, and their Hausdorff dimension is calculated. Previous results determining this quantity for random self-similar structures have often relied on the scaling factors being bounded uniformly away from zero. However, using a percolative argument, and taking advantage of the tree-like structure of the sets considered here, it is shown that this condition is not necessary; a simple condition on the tail of the distribution of the scaling factors at zero is all that is assumed. The scaling factors of these recursively defined structures form what is known as a multiplicative cascade, and results about the height of this random object are also obtained. With important physical and probabilistic applications, the heat equation has justifiably received a substantial amount of attention in a variety of settings. For certain types of fractals, it has become clear that a key factor in estimating the heat kernel is the volume growth with respect to the resistance metric on the space. In particular, uniform polynomial volume growth, which occurs for many deterministic self-similar fractals, immediately implies uniform (on-diagonal) heat kernel behaviour. However, in the random fractal setting, this is frequently not the case, and volume fluctuations are often observed. Motivated by this, an analysis of how volume fluctuations lead to corresponding heat kernel fluctuations for measure-metric spaces equipped with a resistance form is conducted here. These results apply to the aforementioned random self-similar dendrites, amongst other examples. The continuum random tree (CRT) of Aldous is an important random example of a measure-metric space, and fits naturally into the framework of the previous paragraph. In this thesis, quenched (almost-sure) volume growth asymptotics for the CRT are deduced, which show that the behaviour in almost-every realisation is not uniform. Applying the results introduced above, these yield heat kernel bounds for the CRT, demonstrating that heat kernel fluctuations occur almost-surely. Finally, a new representation of the CRT as a random self-similar dendrite is presented.

Nous nous intéressons à la modélisation de formes complexes de type structures arborescences, formes lacunaires ou surfaces rugueuses. Ces formes sont intéressantes de par leurs propriétés physiques particulières :objets légers, économie de matière, résistance mécanique, absorption acoustique importante. Les modèles basés sur le concept de la géométrie fractale permettent de générer de telles formes et notamment les formes auto-similaires. A partir des travaux de Barnsley sur les systèmes itérés de fonctions, Tosan et al, ont proposé une extension, Boundary Controled Iterated Funcions Systems (BCIFS) pour contrôler plus facilement les formes et faciliter leur description. Nous nous intéressons aux propriétés différentielles des formes décrites par BCIFS. Nous proposons une définition plus générale d'espace tangent qui permet de caractériser le comportement de cas non-classiquement différentiables.Nous montrons que l'étude du comportement différentiel peut alors se faire simplement par analyse des valeurs propres et vecteurs propres généralisés des opérateurs de subdivision. Il devient alors possible de contrôler ces propriétés différentielles. Nous présentons une application de nos résultats, en proposant une méthode pour construire des raccords entre deux structures définies par des processus de subdivision différents. Cette méthode est appliquée pour la construction d'un raccord entre une surface de subdivision de Doo-Sabin(schéma dual) et une surface de subdivision de Catmull-Clark (schéma primal)

Au cours des 20 dernières années, le développement des méthodes de télédétection et le succès des missions spatiales ont considérablement enrichi nos connaissances sur les surfaces planétaires révélant une immense diversité de morphologies. Etant le reflet de l'interaction et de la compétition entre des processus géologiques dont les modalités sont variables d'un corps à l'autre, elles sont largement étudiées pour retracer l'histoire géologique des planètes telluriques. En particulier, des informations précieuses sur la nature des processus et sur les lois générales qui contrôlent la formation et l'évolution des paysages sont enregistrées dans le champ topographique qui peut être analysé en tant que champ statistique. Nous rapportons dans cette thèse les résultats d'une étude comparative des propriétés statistiques de la topographie des principaux corps du système solaire en nous appuyant sur le volume croissant de données altimétriques et photogrammétriques. Notre approche est centrée sur la notion de loi d'échelle qui vise à caractériser les symétries du champ en traduisant le caractère auto-similaire des surfaces naturelles : les détails d'une surface ressemblent en général à des versions réduites de l'ensemble. Nous mettons en oeuvre plusieurs méthodes d'analyse de données dites «multifractales» pour dégager le meilleur modèle statistique capable de décrire la topographie dans différents contexte et proposons de nouveaux indicateurs de rugosité à l'échelle globale, régionale et locale. Nous montrons qu'en dépit de leur diversité, les surfaces du système solaire respectent des lois statistiques similaires que nous explicitons. En particulier nous montrons que la distribution globale des pentes d'un corps respecte en général des lois multifractales pour les échelles supérieures à 10-20km et présente une structure statistique différente pour les échelles inférieures. Enfin nous proposons une méthode pour générer des topographies synthétique sphériques dont le propriétés statistiques sont similaires aux topographie planétaire du système solaire
Over the past 20 years, the development of remote sensing methods and the success of space missions have considerably enriched our knowledge of planetary surfaces revealing an immense diversity of morphologies. Being the reflection of the interaction and the competition between geological processes whose modalities are variable from one body to the other, they are widely studied to trace the geological history of the telluric planets. In particular, precise information on the nature of processes and general laws controlling the formation and evolution of landscapes is recorded in the topographic field which can be analyzed as a statistical field. We report in this thesis the results of a comparative study of the statistical properties of the topography of the main bodies of the solar system based on the increasing volume of altimetric and photogrammetric data. Our approach focuses on the notion of scaling law which aims to characterize the symmetries of the field by translating the self-similar nature of natural surfaces: the details of a surface generally look like reduced versions of the whole. We use several methods of analyzing so-called "multifractal" to derive the best statistical model capable of describing the topography in different contexts and propose new indicators of roughness at the global, regional and local scale. We show that in spite of their diversity, the surface of the solar system respects similar statistical laws. In particular, we show that the overall distribution of the slopes of a body generally respects multifractal laws for scales greater than 10-20 km and presents a different statistical structure for the lower scales. Finally, we propose a method for generating spherical synthetic topographies whose statistical properties are similar to the topographies in the solar system

La caractéristique la plus évidente de l’interface terrestre est son hétérogénéité. Phénoménologiquement, celle-ci, anthropique ou naturelle, transparaît de limites qui définissent les formes déployées dans l’espace géographique. Ces limites sont certes dues à une dynamique temporelle, tout autant que à une dynamique scalaire. Celle-ci se manifeste dans les rapports possibles et changeant existant entre échelles dans la mesure où tout échelle ne peut se concevoir que comme relative à une autre servant de référence. Ceci conduit à la relativité d’échelle (R.E.) qui devrait permettre de définir intrinsèquement l’espace géographique. Le premier objectif est de montrer la possibilité d’utiliser la R.E. en géographie. Au coeur de la R.E., on trouve la géométrie fractale qui reste indispensable pour essayer de comprendre l’organisation scalaire du monde. Jusqu’à présent les fractales n’étaient utilisées que comme un outil de description plus ou moins pertinent. En R.E., les formes fractales deviennent une conséquence d’un espace formel intrinsèquement irrégulier. La fractalité peut donc être une voie de compréhension du monde utilisant l’espace de ses échelles, c’est-à-dire de ses résolutions. L’objectif central de cette étude est donc de construire une méthodologie fractale générale nécessaire à l’étude d’une morphologie quelconque à travers divers exemples issus de la géographie physique, de la géographie urbaine, de la géohistoire et de la géographie du peuplement. L’objectif final est d’aboutir à des solutions formelles accessibles à une large communauté de géographes, ce qui n’est pas le cas de la théorie de la R.E. dans son formalisme actuel. D’un point de vue épistémologique, le développement en géographie de la R.E. pose la question de la renaturalisation de cette discipline des Sciences humaines et sociales et de sa constitution en science analytique, donc plus largement de proposer une nouvelle définition de la géographie
The most obvious characteristic of the terrestrial interface is its heterogeneity. Phenomenologically, this one, human or natural, show limits that define the forms deployed in geographic space. These limits are certainly due to temporal dynamic, all as much as a scale dynamic. This one is manifested in the possible relationships and variables that exist between scales inasmuch as every scale can no conceive than relative to another that is used as reference. This leads to the scale relativity (SR) which should allow to define intrinsically the geographical space. The first objective is to show the possibility of using SR in geography. At the heart of the SR, we found that fractal geometry is indispensable to try to understand the organization in the scales of the world. So far fractals no were used than as a tool of describing more or less relevant. In SR, the fractal forms become a consequence of a formal space intrinsically irregular. The fractality can be thus a way of understanding of the world using the space of scales, that is to say its resolutions. The central objective of this study is thus to build a general fractal methodology necessary under investigation of an unspecified morphology through various examples resulting from the physical geography, the urban geography, the geohistory and the geography of the settlement. The final objective is to lead to solutions formal accessible at a broad community from geographers, which is not the case of the theory of the R.E in its current formalism. From an epistemological point of view, the development in geography of the R.E raises the question of the Re-naturalization of this discipline of the human and social sciences and of the constitution in analytical science, therefore more largely to propose a new definition of the geography

On sait que la matrice d'incidence associée à une substitution ne suffit pas pour déterminer complètement le système dynamique associé, même dans des cas très simples, il existe plusieurs substitutions associées à une même matrice car il existe de nombreux mots ayant le même abélianisé. Dans cette thèse, on étudie les points communs de deux lignes brisées associées à deux substitutions $sigma_1$ et $sigma_2$ irréductibles unimodulaires de type Pisot qui ont la même matrice d'incidence. On identifie les points communs de ces deux lignes brisées à partir d'un algorithme. On montre ainsi que l'intersection de ces deux lignes brisées est aussi une ligne brisée associée au point fixe d'une nouvelle substitution. On montre plus précisément que si $sigma_1$ vérifie la conjecture Pisot et $0$ est un point intérieur à son fractal de Rauzy alors ces points communs peuvent être engendrés par une substitution définie sur un alphabet appelé alphabet des paires équilibrées. Cette substitution est obtenue à partir d'un algorithme, l'algorithme des paires équilibrées. On obtient ainsi l'intersection des intérieurs des deux fractals de Rauzy. En prenant la clôture de cet ensemble on obtient un ensemble substitutif. La condition que $0$ est un point intérieur au fractal de Rauzy associé à la substitution $sigma_1$ nous permet de montrer que l'intersection des deux fractals de Rauzy est de mesure positive. Dans une deuxième partie du travail on s'intéresse à l'étude de la frontière du fractal de Rauzy. Le fractal de Rauzy est dit fractal mais c'est en fait sa frontière qui est fractale
The matrix of a substitution is not sufficient to completely determine the dynamics associated with it, even in the simplest cases since there are many words with the same abelianization. In this paper we study the common points of the canonical broken lines associated with two different irreducible Pisot unimodular substitutions σ1 and σ2 having the same incidence matrix. We prove that if σ1 verifies the Pisot conjecture and 0 is an inner point to the Rauzy fractal associated with the substitution σ1 then these common points can be generated with a substitution on an alphabet of so-called balanced pairs, and we obtain in this way the intersection of the interior of two Rauzy fractals

Estudamos as propriedades aritméticas, geométricas e topológicas de uma classe dos chamados Fractais de Rauzy. Estudamos partucularmente o azulejamento periódico do plano complexo C induzido por eles, assim como a dimensão de Hausdorff de suas fronteiras. Tal trabalho exige um estudo detalhado da fronteira destes conjuntos, que está associada às propriedades aritméticas da \'alpha\' -representação dos números complexos com respeito a um certo número algébrico \'alfa\'
We study the arithmetic, geometric and topological properties of a class of the so-called Rauzy\'s fractals. In particular we study the periodic tiling of the complex plane C induced by them and the Hausdorff dimension of its boundary. Such work is connected to a detailed study of the boundary of such sets and the arithmetic properties of the \'alpha\' representation of complex numbers with respect to a certain algebraic number \'alpha\'

L’oxydation des échantillons de tungstène chauffés par effet Joule, est réalisée dans un spectrophotomètre infrarouge sous atmosphère contrôlée (10^-2 torr d'oxygène) et à des températures T ≥ 1100 K pour créer des surfaces rugueuses. La morphologie de la couche d'oxyde est caractérisée par analyseur d'images à partir du cliché d'une coupe réalisée par microscopie électronique à balayage. On assimile l'oxyde à une superposition de films minces, dont l'un hétérogène, contenant la rugosité, est étudié par la théorie de BRUGGMAN en vue de déterminer sa constante diélectrique. Les propriétés optiques d'un corps hétérogène sont interprétées généralement à partir de la théorie des milieux dispersés qui ne tient compte que très partiellement de la rugosité des surfaces. Ce travail expose la conception et l'élaboration d'outils destinés à caractériser l'influence de la complexité et de la forme de la rugosité sur l'émissivité d'une surface, selon une approche originale qui tient compte de la morphologie réelle des surfaces (profils fractals). Le calcul de l'émissivité d'une surface rugueuse est mené de la manière suivante: numérisation des profils expérimentaux calcul de la constante diélectrique d'une couche homogène optiquement équivalente contenant la rugosité du profil considéré calcul du facteur moyen de dépolarisation gm par une méthode originale de remplissage du profil par des ellipsoïdes utilisation de la théorie des films minces, pour calculer l'émissivité monochromatique et directionnelle de cette surface. Une étude analogue a été réalisée sur des profils théoriques crées par interpolation fractale, pour la validation théorique du modèle. Nous mettons en évidence une relation entre la dimension fractale, l'émissivité et le facteur de dépolarisation

Introducció L'any 1990 es van produir 1,6 milions de fractures de maluc a tot el món i s'estima que aquesta xifra augmentarà a 6 milions l'any 2050. A la Unió Europea es produeixen 600.000 fractures de maluc a l'any, aproximadament, amb un cost global anual de 13.000 milions d'Euros. La incidència a Espanya és de 517 casos per cada 100.000 habitants i per any. L'edat mitjana de 82 anys i un 78% dels pacients són de sexe femení. Els objectius del tractament de la fractura de maluc són preservar la vida i aconseguir una recuperació funcional que permeti el pacient integrar-se de nou en el seu medi habitual. Però, en aquests pacients, la taxa de mortalitat s'eleva durant el primer any d'un 8,4% a un 36%. A l'any de la fractura el 50% presenten dificultats per caminar, el 38-39% presenten dificultat per fer transferències i el 17-19% presenten dificultats per a l’higiene. Fins a un 90% dels pacients presenten múltiples comorbiditats, entre les quals la malaltia pulmonar obstructiva crònica, la demència, la hipertensió arterial, la patologia cardíaca isquèmica i la diabetis són les més comunes. Les característiques d'aquests pacients, ancians i amb múltiples comorbiditats, va fer sorgir la idea de proporcionar-los una atenció compartida entre cirurgians ortopèdics i geriatres. Aquesta idea inicial ha evolucionat a la tendència actual d'implementar unitats de ortogeriatria que integrin un tractament multidisciplinari. En aquest model a geriatres i traumatòlegs se sumen també anestesistes, rehabilitadors, fisioterapeutes, infermeres i nutricionistes, entre d'altres. Objectius L'objectiu principal d'aquesta tesi doctoral és determinar la demora per a intervenció quirúrgica dels pacients amb fractura de fèmur proximal després de la implementació d'una unitat de ortogeriatria. Els objectius secundaris són determinar el temps d'estada hospitalària, la mortalitat intrahospitalària i als 30 dies, i els reingressos que es produeixen per complicacions mèdiques i traumatològiques. Material i mètodes Durant l'any 2013 (juny-desembre), 2014 i 2015, van ingressar al nostre servei 534 fractures de maluc, de manera consecutiva. Mentre que en els anys 2011, 2012 i 2013 (gener-maig), quan encara no existia la Unitat de Traumatologia Geriàtrica (UTG) van ingressar 501. Les dades recollides prospectivament en el segon període, després de la implementació de la UTG, han estat comparats amb les dades dels pacients que van ingressar en el primer període. Resultats La demora mitjana per a ser intervingut quirúrgicament prèviament a la implementació de la UTG va ser de 2,27 dies (DE = 2,35), mentre que posteriorment va ser de 1,84 (DE = 1,73). (P = 0,0004). L'estada mitjana prèviament a la implementació de la UTG va ser de 11,39 dies (DE = 9,05), mentre que posteriorment va ser de 10,08 (DE = 5,43). (P = 0,0024). La mortalitat en els primers 30 dies després de la fractura de maluc, prèviament a la implementació de la UTG va ser del 7,7%, mentre que posteriorment va ser de l'4,8%. (P = 0,027). Conclusió La implementació d'una unitat de ortogeriatria, per al tractament dels pacients amb fractura de fèmur proximal, que inclou un conjunt de mesures entre les que destaquen la introducció de circuits ràpids de tractament, tractament multidisciplinari integrat i protocols de rehabilitació primerenca postoperatòria, ha permès disminuir de 2,27-1,84 dies el temps de demora mitjà per ser intervingut quirúrgicament després de l'ingrés. El temps d'estada hospitalària s'ha reduït en un temps mitjà d'un dia. La mortalitat dels pacients als 30 dies s'ha reduït en un 2,9%. Els reingressos per complicacions mèdiques o quirúrgiques no s'han incrementat.
Introducción En el año 1990 se produjeron 1,6 millones de fracturas de cadera en todo el mundo y se estima que esa cifra aumentará a 6 millones en el año 2050. En la Unión Europea se producen 600.000 fracturas de cadera al año, aproximadamente, con un coste global anual de 13.000 millones de Euros. La incidencia en España es de 517 casos por cada 100.000 habitantes y por año. La edad media de 82 años y un 78% de los pacientes son de sexo femenino. Los objetivos del tratamiento de la fractura de cadera son preservar la vida y conseguir una recuperación funcional que permita al paciente integrarse de nuevo en su medio habitual. Pero, en estos pacientes, la tasa de mortalidad se eleva durante el primer año de un 8,4% a un 36%. Al año de la fractura el 50% presentan dificultades para caminar, el 38-39% presentan dificultades para realizar transferencias y el 17-19% presentan dificultades para el aseo. Hasta un 90% de los pacientes presentan múltiples comorbilidades, entre las que la enfermedad pulmonar obstructiva crónica, la demencia, la hipertensión arterial, la patología cardiaca isquémica y la diabetes son las más comunes. Las características de estos pacientes, ancianos y con múltiples comorbilidades, hizo surgir la idea de proporcionarles una atención compartida entre cirujanos ortopédicos y geriatras. Esa idea inicial ha evolucionado a la tendencia actual de implementar unidades de ortogeriatría que integren un tratamiento multidisciplinar. En este modelo a geriatras y traumatólogos se suman también anestesistas, rehabilitadores, fisioterapeutas, enfermeras y nutricionistas, entre otros. Objetivos El objetivo principal de esta tesis doctoral es determinar la demora para intervención quirúrgica de los pacientes con fractura de fémur proximal tras la implementación de una unidad de ortogeriatría. Los objetivos secundarios son determinar el tiempo de estancia hospitalaria, la mortalidad intrahospitalaria y a los 30 días, y los reingresos que se producen por complicaciones médicas y traumatológicas. Material y métodos Durante el año 2013 (Junio-Diciembre), 2014 y 2015, ingresaron en nuestro servicio 534 fracturas de cadera, de forma consecutiva. Mientras que en los años 2011, 2012 y 2013 (Enero-Mayo), cuando todavía no existía la Unidad de Traumatología Geriátrica (UTG) ingresaron 501. Los datos recogidos prospectivamente en el segundo periodo, tras la implementación de la UTG, han sido comparados con los datos de los pacientes que ingresaron en el primer periodo. Resultados La demora media para ser intervenido quirúrgicamente previamente a la implementación de la UTG fue de 2,27 días (DE=2,35), mientras que posteriormente fue de 1,84 (DE=1,73). (p=0,0004). La estancia media previamente a la implementación de la UTG fue de 11,39 días (DE=9,05), mientras que posteriormente fue de 10,08 (DE=5,43). (p=0,0024). La mortalidad en los primeros 30 días tras la fractura de cadera, previamente a la implementación de la UTG fue del 7,7%, mientras que posteriormente fue del 4,8%. (p = 0,027). Conclusión La implementación de una unidad de ortogeriatría, para el tratamiento de los pacientes con fractura de fémur proximal, que incluye un conjunto de medidas entre las que destacan la introducción de circuitos rápidos de tratamiento, tratamiento multidisciplinar integrado y protocolos de rehabilitación temprana postoperatoria, ha permitido disminuir de 2,27 a 1,84 días el tiempo de demora medio para ser intervenido quirúrgicamente tras el ingreso. El tiempo de estancia hospitalaria se ha reducido en un tiempo medio de un día. La mortalidad de los pacientes a los 30 días se ha reducido en un 2,9%. Los reingresos por complicaciones médicas o quirúrgicas no se han incrementado.
Introduction In 1990, there were 1.6 million hip fractures worldwide. This number is expected to reach 6 million by 2050. In the European Union, osteoporosis causes approximately 600.000 hip fractures per year. The annual estimated economic burden for healthcare systems is 13.000 million Euros. The incidence of hip fractures in Spain is 517 cases per 100.000 inhabitants and year. The average age is 82 years and 78% are women. The goal of hip fracture treatment is to return the patient to preoperative levels of function, facilitating return to pre-fracture residence and supporting long-term wellbeing. Mortality rates in hip fracture patients rise from 8.4 to 36% in the first year after surgery. One year after the fracture, 50% have difficulties in walking, 38-39% are not able to transfer from a bed to a chair and 17-19% require aids for bathing and grooming. Up to 90% of patients have several comorbidities. Commonly, these include chronic obstructive pulmonary disease, dementia, high blood pressure, ischemic heart disease, and diabetes. Elderly patients with several comorbidities could benefit from shared care approaches provided by orthopedic surgeons and geriatricians. This cooperation has triggered the current trend of implementing orthogeriatric units that integrate multidisciplinary teams. In this model, several disciplines, besides surgeons and geriatricians, are involved in the care of the patients including anesthesiologists, physical therapists, nurses, and nutritionists. Objectives The main objective of this study is to determine the delay for surgical intervention of patients with proximal femur fracture after the implementation of an orthogeriatric unit. Secondary objectives are to determine the length of hospital stay, in-hospital and 30-day mortality, and readmissions resulting from medical and trauma complications. Material and methods During 2013 (June-December), 2014, and 2015, 534 consecutive hip fractures were treated in our hospital. While in 2011, 2012, and 2013 (January-May), before the orthogeriatric unit (OGU) was created, 501 hip fractures were treated. Data collected prospectively in the second period, after the implementation of the OGU, have been compared with the first period data. Results The mean delay to undergo surgery before the implementation of the OGU was 2.27 days (SD = 2.35), compared to 1.84 (SD = 1.73). (p = 0.0004) for the second period. The average in-hospital stay before the implementation of the OGU was 11.39 days (SD = 9.05), compared to 10.08 (SD = 5.43). (p = 0.0024) after the orthogeriatric model of care was established. 30-day mortality rate after hip fracture, before OGU implementation, was 7.7%, and 4.8% afterward. (p = 0.027). Conclusion The implementation of an orthogeriatric unit for the treatment of patients with a hip fracture which requires a series of measures including the introduction of fast treatment circuits, integrated multidisciplinary treatment, and early postoperative rehabilitation protocols, has allowed a decrease from 2.27 to 1.84 days in the average time to surgery after admission. The length of hospital stay was reduced by an average time of one day. 30-day mortality was reduced by 2.9%. Readmissions for medical or surgical complications did not increase.

Fractal analysis is an important tool when we need to study geometrical objects less regular than ordinary ones, e.g. a set with a non-integer dimension value. It has developed intensively over the last 30 years which gives a hint to its young age as a branch within mathematics.

In this thesis we take a look at some basic measure theory needed to introduce certain definitions of fractal dimensions, which can be used to measure a set's fractal degree. Comparisons of these definitions are done and we investigate when they coincide. With these tools different fractals are studied and compared.

A key idea in this thesis has been to sum up different names and definitions referring to similar concepts.

Engineering conceptual design is a knowledge-intensive process that generates solutions to a product specification. It is a process that can benefit from past experience of similar designs. In reality however, designers often have limited time to build up the necessary experience and are, in any event, unlikely to become experts in all relevant fields. Hence there is a need to capture, store and reuse valuable knowledge. Currently available conventional CAD systems offer limited possibilities for the re-use of existing designs. Techniques from the field of Artificial Intelligence (Al) may be applied to aid the conceptual design phase, which is known as the area of intelligent computer-aided design. The aim of this work is to identify and externalise design knowledge using a fractal-like model, to understand the role of design knowledge in conceptual design and to use design knowledge as a guide for every stage of concept development. This research provides a framework for supporting conceptual design, which uses the techniques of Case-Based Reasoning (CBR) and fractal theory, for reasoning about the design and development of computer-based design aids. The framework is comprised of three parts. The first is case representation. This research proposes a new representation technique, Fractal-like Design Modelling (FDM), which integrates design knowledge in a graph-based form and has fractal-specific characteristics. The second is case retrieval. Based on FDM, the similarity between a new design and the existing designs is assessed by concurrently applying a feature-based similarity measure and a structure-based similarity measure. The third is case adaptation. With the help of fractal characteristics, an approach of adaptive design is developed by performance revision and by goal-oriented substitution. These three parts work together to achieve an automated, case-based, conceptual design method: Fractal-Based Re-design.

Fractal antennas have the potential to provide multi-band solution through the property of self similarity that the fractal shape possess. This thesis deals with some modifications of the original Sierpinski fractal patch antenna, providing additional parameters to optimize in order to get the required radiation characteristics. A systematic study of the original Sierpinski monopole was done to gain an insight on the operation and multi-band behavior of the antenna and a similar study was done to find out why the patch version of Sierpinski lacks multi-band behavior. A Sierpinski antenna is designed by merging a grid and the fractal restoring the multi-band behavior. A stacked Sierpinski antenna is then designed by introducing an intermediate layer containing the triangular patch whose size is equal to that of the triangular hole in the fractal structure. The introduction of a gap in this structure improves the overall radiation pattern. The antennas are analyzed using the commercial IE3D package and the simulated radiation patterns are presented.
M.S.E.E.
Department of Electrical and Computer Engineering
Engineering and Computer Science
Electrical Engineering

This study provides a guide to a series of systematic techniques used to create fractal mountains. The fractal mountains are created through an Interactive System for Fractal Mountains (ISFM) . To create the fractal mountains in ISFM a modified midpoint displacement technique in three dimensions is used. Augmenting the midpoint displacement algorithm is a random number generator that provides randomness in the displacement so as to simulate nature. These two algorithms plus an algorithm for lighting and for shading allow the user to develop different types of fractal mountains. When creating a fractal mountain with ISFM, the user has the options of placing the location of the light source for the time of day, of determining the ruggedness or texture of the mountain and of positioning the outline for a mountain range. ISFM generates a fractal mountain or a fractal mountain range on an IRIS workstation. ISFM provides a systematic and tutorial approach to creating fractal mountains that can be easily repeated by others.

Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2013.
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Com o objetivo de obter atratores de semigrupos em espaço de Banach de dimensão infinita como objetos em espaços de dimensão finita estudamos condições sobre o semigrupo que asseguram que o atrator global possui dimensão de Hausdorff ou fractal ("upper"box-couting dimension) finita. ______________________________________________________________________________ ABSTRACT
In order to obtain the attractors of semigroups in in nite dimensional Banach spaces as objects in nite dimensional spaces, we study conditions on the semigroups which guarantee nite Hausdorff, or fractal ("upper"box-counting dimension), dimension for the attractors.