The speed and complexity of globalisation and reduction of natural resources on the one hand, and interests of large multinational corporations on the other, necessitates proper management of mineral resources and consumption. The need for scientific research and application of new methodologies and approaches to maximise Net Present Value (NPV) within mining operations is essential. In some cases, drill core logging in the field may result in an inadequate level of information and subsequent poor diagnosis of geological phenomenon which may undermine the delineation or separation of mineralised zones. This is because the interpretation of individual loggers is subjective. However, modelling based on logging data is absolutely essential to determine the architecture of an orebody including ore distribution and geomechanical features. For instance, ore grades, density and RQD values are not included in conventional geological models whilst variations in a mineral deposit are an obvious and salient feature. Given the problems mentioned above, a series of new mathematical methods have been developed, based on fractal modelling, which provide a more objective approach. These have been established and tested in a case study of the Kahang Cu-Mo porphyry deposit, central Iran. Recognition of different types of mineralised zone in an ore deposit is important for mine planning. As a result, it is felt that the most important outcome of this thesis is the development of an innovative approach to the delineation of major mineralised (supergene and hypogene) zones from ‘barren’ host rock. This is based on subsurface data and the utilisation of the Concentration-Volume (C-V) fractal model, proposed by Afzal et al. (2011), to optimise a Cu-Mo block model for better determination of an ultimate pit limit. Drawing on this, new approaches, referred to Density–Volume (D–V) and RQD-Volume (RQD-V) fractal modelling, have been developed and used to delineate rock characteristics in terms of density and RQD within the Kahang deposit (Yasrebi et al., 2013b; Yasrebi et al., 2014). From the results of this modelling, the density and RQD populations of rock types from the studied deposit showed a relationship between density and rock quality based on RQD values, which can be used to predict final pit slope. Finally, the study introduces a Present Value-Volume (PV-V) fractal model in order to identify an accurate excavation orientation with respect to economic principals and ore grades of all determined voxels within the obtained ultimate pit limit in order to achieve an earlier pay-back period.

Neste trabalho, estudamos uma generalização dos semigrupos gradientes, os semigrupos gradiente-like, algumas de suas propriedades e a sua invariância por pequenas perturbações; isto é, pequenas perturbações de sistemas gradiente-like continuam sendo gradiente-like. Como consequência da caracterização dos atratores para este tipo de sistema, estudamos a atração exponencial de atratores. Por fim, estudamos o concetio de dimensão de Hausdorff e dimensão fractal de atratores e apresentamos alguns resultados sobre este assunto, e estudamos a construção de uma nova classe de atratores, os atratores exponenciais fractais / In this work, we study a generalization of gradient discrete semigroups, the gradientlike semigroups, some of its properties and its invariance under small perturbations; that is, small perturbations of gradient-like semigroups are still gradient-like semigroups. As a consequence of the characterization of the attractors for this sort of semigroups, we study the exponential attraction of attractors. Finally, we study some concepts of Hausdorff dimension and fractal dimension and present some results about this subject, and we studied the construction of a new class of attractors, the exponential fractal attractors

Esta dissertação faz um estudo da aplicação da técnica estatística denominada Análise de Dados Funcionais (ADF) à geração de descritores usados em reconhecimento de padrões, mais especificamente, no reconhecimento de objetos de interesse em imagens. Estes objetos podem ser representados por vetores de características, também chamados de assinaturas, obtidos por uma técnica chamada de Dimensão Fractal Multiescala (DFM). Ocorre que estes vetores apresentam alta dimensionalidade (número de elementos), fazendo-se assim necessário o uso de uma abordagem que reduza este número de valores, sem que haja uma grande perda da informação transmitida pela assinatura. Neste contexto, diversas técnicas de extração de um reduzido conjunto de descritores da assinatura são apresentadas pela literatura. Entre estas, as mais populares são Fourier e \\emph, ambas relativamente simples de se apresentar e com resultados satisfatórios. A proposta aqui apresentada é de se utilizar ADF em combinação com DFM na geração de descritores de padrões. Os resultados obtidos com o uso desta abordagem na geração de descritores demostraram que a técnica possibilita bons resultados, mesmo em situações em que não é possível o uso de muitos descritores. Os experimentos demostraram que ADF apresenta um bom potencial para aplicação neste tipo de problema, permitindo que o método de classificação alcance bons resultados mesmo com poucos descritores. São sugeridos trabalhos futuros em que ADF possa ser usada, pesquisando-se por métodos ainda mais eficazes. / This work studies the application of a statistical technique named Functional Data Analysis (FDA) for the generation of descriptors. These descriptors can be used for pattern recognition, more specifically, for the recognition of relevant objects in an image. These objects can be represented by features vectors, also known as signatures, obtained by a technique named Multi-scale Fractal Dimension (MFD). These vectors present a high dimensionality (number of elements), causing to be necessary the use of an approach for the reduction of this number of values, but without a large loss of information carried by the signature. In this context, several techniques for the extraction of a reduced set of signature descriptors are studied in the literature. Among these techniques, the most classic are Fourier and wavelets, both with simple presentation and providing satisfactory results. The proposal presented here is the use of FDA combined with MFD for the generation of pattern descriptors. The results obtained by the use of this approach for the generation of descriptors showed that this technique allows the obtention of good results, even in situations in wich is not possible the use of many descriptors. FDA was also applied to the extraction of descriptors of MFD texture signatures. Also in this case, the results were interesting. The experiments showed the FDA presents a good potential for the application to this type of problem, allowing the obtention of good results even by using a few descriptors. It is suggested future works in which FDA can be used, researching for still more efficient methods.

Esta dissertação faz um estudo da aplicação da técnica estatística denominada Análise de Dados Funcionais (ADF) à geração de descritores usados em reconhecimento de padrões, mais especificamente, no reconhecimento de objetos de interesse em imagens. Estes objetos podem ser representados por vetores de características, também chamados de assinaturas, obtidos por uma técnica chamada de Dimensão Fractal Multiescala (DFM). Ocorre que estes vetores apresentam alta dimensionalidade (número de elementos), fazendo-se assim necessário o uso de uma abordagem que reduza este número de valores, sem que haja uma grande perda da informação transmitida pela assinatura. Neste contexto, diversas técnicas de extração de um reduzido conjunto de descritores da assinatura são apresentadas pela literatura. Entre estas, as mais populares são Fourier e \\emph, ambas relativamente simples de se apresentar e com resultados satisfatórios. A proposta aqui apresentada é de se utilizar ADF em combinação com DFM na geração de descritores de padrões. Os resultados obtidos com o uso desta abordagem na geração de descritores demostraram que a técnica possibilita bons resultados, mesmo em situações em que não é possível o uso de muitos descritores. Os experimentos demostraram que ADF apresenta um bom potencial para aplicação neste tipo de problema, permitindo que o método de classificação alcance bons resultados mesmo com poucos descritores. São sugeridos trabalhos futuros em que ADF possa ser usada, pesquisando-se por métodos ainda mais eficazes. / This work studies the application of a statistical technique named Functional Data Analysis (FDA) for the generation of descriptors. These descriptors can be used for pattern recognition, more specifically, for the recognition of relevant objects in an image. These objects can be represented by features vectors, also known as signatures, obtained by a technique named Multi-scale Fractal Dimension (MFD). These vectors present a high dimensionality (number of elements), causing to be necessary the use of an approach for the reduction of this number of values, but without a large loss of information carried by the signature. In this context, several techniques for the extraction of a reduced set of signature descriptors are studied in the literature. Among these techniques, the most classic are Fourier and wavelets, both with simple presentation and providing satisfactory results. The proposal presented here is the use of FDA combined with MFD for the generation of pattern descriptors. The results obtained by the use of this approach for the generation of descriptors showed that this technique allows the obtention of good results, even in situations in wich is not possible the use of many descriptors. FDA was also applied to the extraction of descriptors of MFD texture signatures. Also in this case, the results were interesting. The experiments showed the FDA presents a good potential for the application to this type of problem, allowing the obtention of good results even by using a few descriptors. It is suggested future works in which FDA can be used, researching for still more efficient methods.

Submitted by Jaqueline Silva (jtas29@gmail.com) on 2016-08-31T19:46:48Z No. of bitstreams: 2 Dissertação - Ricardo Anselmo Zanotto - 2015.pdf: 7706833 bytes, checksum: 26c6e884d0e3a03a3daebaa4ab5764a4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Jaqueline Silva (jtas29@gmail.com) on 2016-08-31T19:47:01Z (GMT) No. of bitstreams: 2 Dissertação - Ricardo Anselmo Zanotto - 2015.pdf: 7706833 bytes, checksum: 26c6e884d0e3a03a3daebaa4ab5764a4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2016-08-31T19:47:01Z (GMT). No. of bitstreams: 2 Dissertação - Ricardo Anselmo Zanotto - 2015.pdf: 7706833 bytes, checksum: 26c6e884d0e3a03a3daebaa4ab5764a4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2015-12-12 / 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.

Neste trabalho, estudamos uma generalização dos semigrupos gradientes, os semigrupos gradiente-like, algumas de suas propriedades e a sua invariância por pequenas perturbações; isto é, pequenas perturbações de sistemas gradiente-like continuam sendo gradiente-like. Como consequência da caracterização dos atratores para este tipo de sistema, estudamos a atração exponencial de atratores. Por fim, estudamos o concetio de dimensão de Hausdorff e dimensão fractal de atratores e apresentamos alguns resultados sobre este assunto, e estudamos a construção de uma nova classe de atratores, os atratores exponenciais fractais / In this work, we study a generalization of gradient discrete semigroups, the gradientlike semigroups, some of its properties and its invariance under small perturbations; that is, small perturbations of gradient-like semigroups are still gradient-like semigroups. As a consequence of the characterization of the attractors for this sort of semigroups, we study the exponential attraction of attractors. Finally, we study some concepts of Hausdorff dimension and fractal dimension and present some results about this subject, and we studied the construction of a new class of attractors, the exponential fractal attractors

The principle of parsimonious modelling of Internet traffic states that a minimal number of descriptors should be used for its characterization. Until early 1990s, the conventional Markovian models for voice traffic had been considered suitable and parsimonious for data traffic as well. Later with the discovery of strong correlations and increased burstiness in Internet traffic, various self-similar count models have been proposed. But, in fact, such models are strictly mono-fractal and applicable at coarse time scales, whereas Internet traffic modelling is about modelling traffic at fine and coarse time scales; modelling traffic which can be mono and multi-fractal; modelling traffic at interarrival time and count levels; modelling traffic at access and core tiers; and modelling all the three structural components of Internet traffic, that is, packets, flows and sessions. The philosophy of this thesis can be described as: “the renewal of renewal theory in Internet traffic modelling”. Renewal theory has a great potential in modelling statistical characteristics of Internet traffic belonging to individual users, access and core networks. In this thesis, we develop an Internet traffic modelling framework based on fractal renewal processes, that is, renewal processes with underlying distribution of interarrival times being heavy-tailed. The proposed renewal framework covers packets, flows and sessions as structural components of Internet traffic and is applicable for modelling the traffic at fine and coarse time scales. The properties of superposition of renewal processes can be used to model traffic in higher tiers of the Internet hierarchy. As the framework is based on renewal processes, therefore, Internet traffic can be modelled at both interarrival times and count levels.

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\'

Made available in DSpace on 2015-03-04T18:50:54Z (GMT). No. of bitstreams: 1 Dissertacao Marcelo Barros.pdf: 906132 bytes, checksum: 67f089fdd05da5a2f2ab6d807fbbf51b (MD5) Previous issue date: 2007-03-23 / Several areas of knowledge use fractal geometry to help to understand natural objects and phenomena. Irregular self-similar - in which parts resemble the whole - objects may be better understood through fractal dimensions which provide how a property varies with resolution or scale. We present a new approach to calculate fractal dimensions that, instead of the frequently used methods based on covering, seeks geometry information from physical characteristics. Here, we treat the element of a fractal sequence as structures. Imposing constraints on the structures, we build simple harmonic oscillators. The variation of the period of these oscillators with respect to a determined measure of length provides a fractal dimension. This techinique was tested for a family of continuous self-similar plane curves, including the classical Koch triadic. We show that this dynamical dimension may be related to Hausdorff-Besicovitch dimension. With random geometry, the techinique besides providing a fractal dimension, identifies randomness. A new kind of fractal is also presented. The ideia is to use more than one generator in the generation process of a fractal to obtain mixed fractals. / Diversas áreas do conhecimento têm utilizado a geometria fractal para melhor entender muitos objetos e fenômenos naturais. Objetos irregulares com padrão auto-similar onde as partes se assemelham ao todo podem ser melhor compreendidos através de dimensões fractais que fornecem como o valor de uma propriedade varia dependendo da resolução, ou escala, em que o objeto é observado ou medido. Apresentamos uma nova abordagem para calcular dimensões fractais através de características físicas. Neste trabalho busca-se uma caracterização da dinâmica de estruturas lineares com geometria fractal. Trata-se os elementos de uma sequência geradora de um fractal como estruturas. Osciladores harmônicos simples são construídos com tais estruturas. A variação do período de vibração desses osciladores com uma determinada medida de comprimento nos fornece uma dimensão fractal. A técnica foi testada para a família de curvas contínuas e auto-similares no plano, onde está incluída a clássica triádica de Koch. Mostramos que essa dimensão dinâmica pode ser relacionada à dimensão de Hausdorff-Besicovitch. Com geometria aleatória, a técnica além de fornecer a dimensão fractal, identifica a aleatoriedade. Um novo tipo de fractal é apresentado. A idéia é usar mais de um gerador no processo de geração de um fractal para obter os fractais mistos.

The dissertation is organized into two main parts. The first part considers fractal extension operators. Although extension operators are available for general subsets of Euclidean domains or metric spaces, our extension operator is unique in that it utilizes both the iterative nature of the fractal and finite element approximations to construct the operator. The resulting operator is especially well suited for future numerical work on domains with prefractal boundaries. In the dissertation we prove the existence of a linear extension operator, Π from the space of Hölder continuous functions on a fractal set S to the space of Hölder continuous functions on a larger domain Ω. Moreover this same extension operator maps functions of finite energy on the fractal to H1 functions on the larger domain Ω. In the second part, we consider boundary value problems in domains with fractal boundaries. First we consider the Sierpinski prefractal and how we might apply the technique of singular homogenization to thin layers constructed on the prefractal. We will also discuss numerical approximation in domains with fractal boundaries and introduce a finite element mesh developed for studying problems in domains with prefractal Koch boundaries. This mesh exploits the self-similarity of the Koch curve for arbitrary rational values of α and its construction is crucial for future numerical study of problems in domains with prefractal Koch curve boundaries. We also show a technique for mesh refinement so that singularities in the domain can be handled and present sample numerical results for the transmission problem.