The density of states of elastic waves in a porous amorphous “mesoglass” has been measured in the strong-scattering regime. Samples were constructed by sintering glass beads percolated on a random lattice. This structure was investigated via x-ray tomography, and fractal behaviour was observed with fractal dimension D = 2.6. Using sufficiently small samples, the individual modes of vibration could be resolved and counted in the Fourier transform of each transmitted ultrasonic pulse. A statistical treatment of the data, designed to account for the possibility of missing modes, was developed, yielding a robust method for measuring the density of states. In the strong-scattering regime, the data are in good agreement with a simple model based on mode conservation, though the density of states significantly exceeds the predictions of the Debye approximation at low frequencies. At intermediate frequencies, an average density of states of 47.1 ± 0.3 MHz⁻¹ mm⁻³ was found, with a frequency dependence of f^(0.01 ± 0.04).

The density of states of elastic waves in a porous amorphous “mesoglass” has been measured in the strong-scattering regime. Samples were constructed by sintering glass beads percolated on a random lattice. This structure was investigated via x-ray tomography, and fractal behaviour was observed with fractal dimension D = 2.6. Using sufficiently small samples, the individual modes of vibration could be resolved and counted in the Fourier transform of each transmitted ultrasonic pulse. A statistical treatment of the data, designed to account for the possibility of missing modes, was developed, yielding a robust method for measuring the density of states. In the strong-scattering regime, the data are in good agreement with a simple model based on mode conservation, though the density of states significantly exceeds the predictions of the Debye approximation at low frequencies. At intermediate frequencies, an average density of states of 47.1 ± 0.3 MHz⁻¹ mm⁻³ was found, with a frequency dependence of f^(0.01 ± 0.04).

Ο όρος μορφοκλασματικό σύνολο παρουσιάστηκε για πρώτη φορά από τον Barnsley, το 1975. Ουσιαστικά ονόμασε έτσι το όριο μιας επαναληπτικής διαδικασίας, όταν ο αριθμός των επαναλήψεων τείνει στο άπειρο. Μέσα από τη μηχανή πολλαπλών αντιγράφων ορίζουμε το μορφοκλασματικό σύνολο και τις ιδιότητές του. Τα μορφοκλασματικά σύνολα έχουν εφαρμογές σε πολλά επιστημονικά πεδία, αλλά η κυριότερη είναι ως μέσο στη συμπίεση εικόνων. Ξεκινώντας από τη συμπίεση εικόνων οδηγούμαστε στον ορισμό του επαναληπτικού συστήματος συναρτήσεων και την ανάλυση της θεωρίας του, μέσω των οποίων επιτυγχάνουμε συμπίεση εικόνων. Η ανάγκη για εξοικονόμηση μνήμης, λόγω του μεγάλου όγκου δεδομένων, έκανε τον Barnsley να σκεφτεί αντί να αποθηκεύουμε ολόκληρη την εικόνα να αποθηκεύουμε μόνο το κατάλληλο επαναληπτικό σύστημα συναρτήσεων. Η διαδικασία της συμπίεσης εικόνων με μορφοκλασματικά σύνολα συντίθεται από δύο φάσεις. Την κωδικοποίηση και την αποκωδικοποίηση. Στην πρώτη φάση, προσπαθούμε να λύσουμε ένα αντίστροφο πρόβλημα και συγκεκριμένα να βρούμε το κατάλληλο επαναληπτικό σύστημα συναρτήσεων που αν το εφαρμόσουμε σε μια αρχική εικόνα θα συγκλίνουμε στον ελκυστή. Στη δεύετρη φάση, τη φάση της αποκωδικοποίησης, με χρήση του επαναληπτικού συστήματος συναρτήσεων, προσεγγίζουμε τον ελκυστή μετά από έναν πεπερασμένο αριθμό επαναλήψεων.Η υλοποίηση αυτών των φάσεων καθώς και οι δυνατοί τρόποι μείωσης της πολυπλοκότητας που απαιτείται κατά τη φάση της κωδικοποίησης είναι και το αντικείμενο της παρούσας εργασίας, στα πλαίσια της οποίας παρουσιάζονται και υλοποιούνται παραλλαγές γνωστών τεχνικών οι οποίες παρουσιάζουν βελτίωση στο PSNR, το λόγο συμπίεσης και το χρόνο κωδικοποίησης. Τέλος, προτείνεται μια νέα τεχνική για τον προσδιορισμό του κατάλληλου επαναληπτικού συστήματος συναρτήσεων κατά τη φάση της κωδικοποίησης, η οποία βασίζεται στην ελαχιστοποίηση μιας μη γραμμικής συνάρτησης κόστους. Αποτιμάται η απόδοσή της και συγκρίνεται με αυτή άλλων γνωστών τεχνικών κωδικοποίησης. / The term ‘fractal’ was first introduced by B. Mandelbrot, in 1975. It denotes the limit of an iterative process, when the number of iterations is infinite. Through the multiple photocopy machine, we define fractal and its properties. Fractals have many applications, one of which is in image compression. This is the object of the present research. Starting with data compression, we aim at the definition of an iterative function system and the IFS theory, through which we can achieve image compression. The need of massive storage in memory made Barnsley think that instead of storing the whole image in memory we can store only the suitable IFS. The process consists of two phases. The encoding and decoding. In encoding we try to solve the inverse problem, i.e. finding the best IFS that when we apply it to a starting image we will get the attractor. In the second phase, after a number of iterations we approximate the attractor. Several fractal image compression methods that already exist, together with their techniques and variations are presented and implemented in this research, in order to achieve improvements in PSNR, compression ratio and encoding time. A new method for fractal image compression on grayscale images is proposed at the end of this research, based on the minimization of a cost function. The experimental as well as the comparative results are shown.

Viscous fingering occurs when one fluid displaces another fluid of a greater viscosity in a porous medium or a Hele-Shaw cell. Linear stability analysis is used to predict methods of suppressing instability. Then, experiments in which nonlinear growth dominates pattern formation are analysed to explore the nonlinear impact of strategies of suppressing finger growth. Often, chemical treatment fluid is injected into oil reservoirs in order to prevent sand production. This treatment fluid is usually followed by water injection to clean up the well. We explore the potential for viscous instability of the interface between the treatment fluid and the water, and also the treatment fluid and the oil, as a function of the volume of treatment fluid and the injection rate and viscosity ratios of the different fluids. For a given volume of treatment fluid and a given injection rate, we find the optimal viscosity of the treatment fluid to minimise the viscous instability. In the case of axisymmetric injection, the stabilisation associated with the azimuthal stretching of modes leads to a further constraint on the optimisation of the viscosity. In the case of oil production, polymers may be added to the displacing water in order to reduce adverse viscosity gradients. We also explore the case in which these polymers have a time-dependent viscosity, for example through the slow release from encapsulant. We calculate the injection flow rate profile that minimises the final amplitude of instability in both rectilinear and axisymmetric geometries. In a development of the model, we repeat the calculation for a shear-thinning rheology. Finally, experiments are analysed in which the nonlinear growth of viscous fingers develops to test the influence of different injection profiles on the development of instability. Diffusion Limited Aggregation (DLA) simulations are performed for comparison. In all cases, the evolving pattern has a saturation distribution, with an inner zone in which the fingers are static and an outer zone in which the fingers advance and grow. In the very centre of the viscous fingering patterns, there is a small fully-saturated region. In the experiments, the mass distribution in the inner zone varies with radius as a power law which relates to the fractal dimension for the analogue DLA simulations. In the outer region the saturation decreases linearly with radius. The radius of the inner frozen zone is approximately 2/3 of the outer radius in the cases of DLA and -- after a period of evolution -- the viscous fingering experiments. This allows the radial extents of the inner and outer zones to be predicted. The ratio of each radius to the extent of the fully-saturated region is independent of the injection profile and corresponds to values for DLA.

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

MEDEIROS NETO. F. G. Extração automática de características de asas de mosca da espécie Drosophila melanogaster. 2017. 59 f. Dissertação (Mestrado em Engenharia Elétrica e da Computação) - Campus de Sobral, Universidade Federal do Ceará, Sobral, 2017. / Submitted by Programa de Pós-Graduação Engenharia Elétrica e de Computação (secretaria_ppgeec@sobral.ufc.br) on 2017-08-24T13:13:22Z No. of bitstreams: 1 2017_dis_fgmneto.pdf: 2887253 bytes, checksum: 71db95333b633d65ae536330c8799d28 (MD5) / Approved for entry into archive by Djeanne Costa (djeannecosta@gmail.com) on 2017-09-01T11:43:30Z (GMT) No. of bitstreams: 1 2017_dis_fgmneto.pdf: 2887253 bytes, checksum: 71db95333b633d65ae536330c8799d28 (MD5) / Made available in DSpace on 2017-09-01T11:43:31Z (GMT). No. of bitstreams: 1 2017_dis_fgmneto.pdf: 2887253 bytes, checksum: 71db95333b633d65ae536330c8799d28 (MD5) Previous issue date: 2017-08 / Biometrics techniques are used in studies of several animal species, such as the Drosophila melanogaster, popularly known as fruit fly. This species has become a model organism for the study of the impacts that other insects produce to the environment. In addition, these flies have proteins and genes similar to those of humans. A challenge for specialists in the study of these individuals is the similarity of the wings between males and females, which may be affected by mutations or variations in the genotype. This work proposes a method of discrimination for gender and genotype of flies of the species Drosophila melanogaster from features extracted from images of the wings. This approach is based on the fractal dimension extracted from the Canny filter segmentation of the components of the Stationary Wavelet Transform. The methodology is validated by dividing the data into groups with variable training and test rates and using six different classifiers: Random Forest, Support Vector Machines, Multi-layer Perceptron, Linear Discriminant Analysis, Quadratic Discriminant Analysis and K Nearest Neighbors. Then, the classification is repeated with data reduction, only females for genotype or only individuals without mutation for gender. The results were satisfactory, surpassing works of the literature with automatic methodologies. For genotype, the hit rates were lower due to the physical similarity between the wings. / Técnicas de biometria são utilizadas nos estudos de diversas espécies de animais, como exemplo a Drosophila Melanogaster, popularmente conhecida como mosca da fruta. Essa espécie tornou-se um organismo modelo para o estudo dos impactos que outros insetos produzem ao meio ambiente. Ademais, essas moscas possuem proteínas e genes similares aos dos seres humanos. Uma dificuldade para especialistas no estudo desses indivíduos é a semelhança das asas entre machos e fêmeas, que podem ser afetadas por mutações ou variações no genótipo. Este trabalho propõe um método de discriminação de gênero e genótipo de moscas da espécie Drosophila melanogaster a partir de características extraídas de imagens das asas. Essa abordagem se baseia na dimensão fractal extraída da segmentação por filtro de Canny das componentes da Transformada Wavelet Estacionária. A metodologia é validada com a divisão dos dados em grupos com taxas de treinamento e teste variáveis e na utilização de seis classificadores de abordagens diferentes: Floresta Aleatória, Máquinas de Vetor de Suporte, Perceptron Multicamadas, Análise por Discriminante Linear, Análise por Discriminante Quadrático e K Vizinhos Mais Próximos. Em seguida, a classificação é repetida com redução dos dados, apenas fêmeas para genótipo ou apenas indivíduos sem mutação para gênero. Os resultados obtidos foram satisfatórios, superando trabalhos da literatura com metodologias que não utilizam interação humana. Para genótipo, as taxas de acerto foram mais baixas devido à semelhança física entre as asas.

Made available in DSpace on 2014-06-11T19:25:31Z (GMT). No. of bitstreams: 0 Previous issue date: 2008-02-11Bitstream added on 2014-06-13T19:47:47Z : No. of bitstreams: 1 tavares_df_me_rcla.pdf: 652000 bytes, checksum: 980f827df5cfeb7ee90e034a218f06ea (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Estudamos, neste trabalho, algumas propriedades dinâmicas do Modelo do Acelerador de Fermi, em três versões distintas: uma versão conservativa e duas versões dissipativas. A primeira versão dissipativa com amortecimento proporcional à velocidade da partícula e a segunda versão dissipativa com amortecimento proporcional ao quadrado da velocidade. Nas versões dissipativas, a força de dissipação é introduzida via arrasto viscoso e age ao longo de toda a trajetória da partícula. O modelo de Fermi é um modelo unidimensional que consiste de uma partícula clássica de massa m que move-se com velocidade constante e sofre colisões elásticas com duas paredes rígidas. Uma delas é fixa ao passo que a posição da outra é dependente do tempo. A descrição da dinâmica dos modelos é feita todas as vezes que a partícula colide com a parede móvel, de modo que o conhecimento dos valores da velocidade da partícula e do tempo no instante da colisões descrevem toda a dinâmica. O mapeamento que descreve a dinâmica é bidimensional, não linear e, para os casos dissipativos, obtidos via soluçãoo de equações diferenciais. Analisamos os modelos numericamente e apresentamos e discutimos nossos resultados. / Some dynamic properties of the one-dimensional Fermi Accelerator Model under the presence and absence of frictional force are studied. We have considered three different versions, namely: a conservative and two dissipative versions. The first dissipative version consists in considering the damping to be proportional to the particle’s velocity while the second one assumes the damping to be proportional to the square particle’s velocity. For the dissipative versions, we have introduced dissipation via a drag force, like a gas, that act during all trajectory of the particle. The Fermi accelerator model consists of a classical particle of mass m that is confined in and suffers elastic collisions with two walls. One of them is fixed while the other is time dependent. The description of the dynamics of either models is made all the times the particle bounces the moving wall. It is made via a two dimensional non linear mapping. For the dissipative cases the mapping is obtained via differential equations. We investigate both models analytically and numerically, present and discuss our results.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2015-01-14T13:36:25Z No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Dissertação - Marília Gomes Negri - 2014.pdf: 4966490 bytes, checksum: 2bf5278c0fe3b2a2ede7195a75c3584e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-01-14T14:12:43Z (GMT) No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Dissertação - Marília Gomes Negri - 2014.pdf: 4966490 bytes, checksum: 2bf5278c0fe3b2a2ede7195a75c3584e (MD5) / Made available in DSpace on 2015-01-14T14:12:43Z (GMT). No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Dissertação - Marília Gomes Negri - 2014.pdf: 4966490 bytes, checksum: 2bf5278c0fe3b2a2ede7195a75c3584e (MD5) Previous issue date: 2014-08-01 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work was developed with the aim of studying fractals with their characteristics - self - similarity, in nite complexity and fractional dimension . In this sense , we study some geometric fractals whose approaches can be built on paper , and other fractals which the complexity of formation can only be represented and generated by means of computational resource. Thus, for the realization fo this work was done a literature research of this subject complemented with the calculation data only indicated in his references . We also cited some examples of applications of fractal geometry giving up deepening the theme would require the scope of scienti c research , but on the other hand have the merit of fractal geometry as a tool to analyse the world in which we live. We can see in this work the importance of fractal geometry , that is a geometry of complex structures with unique properties and linked to forms of nature that di ers in several aspects from traditional geometry. / Este trabalho foi desenvolvido com o objetivo de estudar os fractais com suas caracter ísticas - auto-semelhança, complexidade in nita e dimensão fracionada. Nesse sentido, estudamos alguns fractais geométricos cujas aproximações podem ser construídas em papel, e outros fractais nos quais sua complexidade de formação só pode ser representada e gerada por meio de recurso computacionais. Dessa forma, para realização deste trabalho foi feito um levantamento bibliográ co deste assunto complementado com o cálculo de dados apenas indicados em suas referências. Procuramos também, citar alguns exemplos de aplicações da geometria fractal abrindo mão do aprofundamento que o tema exigiria no escopo de uma pesquisa cientí ca, mas que por outro lado apresentam o mérito de colocar a geometria fractal como uma ferramenta para analisar o mundo onde vivemos. Podemos constatar no decorrer deste trabalho a importância da Geometria Fractal, ou seja, uma geometria de estruturas complexas, com propriedades exclusivas e ligada às formas da natureza e que diferencia em vários aspectos da geometria tradicional.

Made available in DSpace on 2016-03-15T19:37:49Z (GMT). No. of bitstreams: 1 Paulo Cesar da Costa.pdf: 4253113 bytes, checksum: 8fe9dda35243a879336e5fa6037d55cb (MD5) Previous issue date: 2014-02-18 / The concepts of fractal geometry, which were developed in 1975 by Mandelbrot, complement Euclidean geometry as they provide theoretical dimensional fundamentals for shapes whose irregularities cannot be properly interpreted by classical topological definitions. The standards of this irregular morphology, found in abundance in nature, can be recognized in the construction processes of various fractals anthropic figures, mainly when combined with computational technology. Professionals and researchers from several knowledge fields are exploring these concepts to simulate or create models of irregular shapes, with detail levels hitherto inconceivable. Earth sciences supported by geotechnologies have great potential for applying Mandelbrot s theories as a conceptual reference for analyzing phenomena presenting fractal behaviors, belonging to different application areas as geomorphology, climatology and urban and landscapes studies. In this approach, the fractal dimension of the territorial space occupied by cities is considered an indicator for understanding occupation patterns as one of the factors to be considered in urban planning policies, e.g., to propose projects for more appropriate investments distribution and development of infrastructure. In this work, these concepts were applied by using the box-counting method to calculate the fractal dimension of urban forms from eight Brazilian state capitals by using the Municipal Human Development Index (MHDI) as the selection criteria: four cities ranked among the top of the list and four cities classified among the last positions of the state capitals. The results were analyzed in order to identify possible relationships or trends among these indicators, namely, fractal dimension and MHDI, that may support future studies and urban planning. These relationships were not confirmed: the eight studied cities presented heterogeneous fractal dimension values and no trends were verified. However, it was concluded that the fractal dimension of urban form is related to its scattering pattern and occupation over the geographical territorial space and may be an indicator of the occupation density from the urban area. / Os conceitos da geometria fractal, desenvolvida em 1975 por Mandelbrot, complementam os da geometria euclidiana à medida que fornecem um arcabouço teórico de cálculo dimensional para diversas formas cujas irregularidades apresentam características que não podem ser interpretadas adequadamente pelas definições topológicas clássicas. Os padrões dessa morfologia irregular, encontrada em abundância na natureza, podem ser reproduzidos em processos de construção de figuras fractais desenvolvidas pelo homem, especialmente quando combinadas com a tecnologia computacional. Profissionais e pesquisadores de diversas áreas de conhecimento passaram a explorar esses conceitos a fim de simular ou criar modelos de formas irregulares, com níveis de detalhe até então inconcebíveis. As ciências da Terra, com o apoio de ferramentas de geotecnologia, apresentam grande potencial de se apropriar das teorias da geometria de Mandelbrot, como referência conceitual aos estudos de fenômenos que apresentam comportamento fractal, pertencentes a várias áreas de aplicação como a geomorfologia, climatologia e os estudos urbanos e de paisagens. Nessa abordagem, a dimensão fractal do espaço territorial ocupado pelas cidades vem sendo considerada indicador importante para entendimento desse padrão de ocupação, contribuindo como um dos fatores a ser considerados nas políticas de planejamento urbano e, por exemplo, propor projetos de distribuição de investimentos e desenvolvimento de infraestrutura mais adequados. Neste trabalho, esses conceitos foram aplicados por meio da utilização do método da contagem de quadrados para calcular a dimensão fractal das formas urbanas de oito capitais brasileiras, utilizando o Índice de Desenvolvimento Humano Municipal (IDHM) de 2010, publicado em julho de 2013, como critério de seleção. Foram escolhidas quatro cidades classificadas entre as primeiras da lista e outras quatro que ocupam as últimas posições entre as capitais estaduais, com o objetivo de encontrar possíveis relações ou tendências entre esses indicadores dimensão fractal e IDHM que pudessem subsidiar futuros estudos e planejamentos urbanos. Os resultados obtidos não revelaram a existência de tais relações: as oito cidades estudadas apresentaram valores dimensionais heterogêneos, sem registro de tendências. No entanto, foi possível concluir que a dimensão fractal de uma forma urbana está relacionada ao seu padrão de espalhamento e ocupação do espaço territorial geográfico e pode ser considerada um indicador da densidade de ocupação da área urbana.

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\'