Dissertations / Theses on the topic 'Linear fractals'

Texture is one of the primary visual attributes used to describe patterns found in nature. Several texture analysis methods have been used as powerful tools for real applications involving analysis and computer vision. However, existing methods do not successfully discriminate the complexity of texture patterns. Such methods disregard the possibility of describing image structures by means of measures such as the fractal dimension. Fractality-based measures allow a non-integer geometric interpretation with applications in areas such as mathematics, physics, and biology. With this gap in mind, the central hypothesis of this thesis is that textures can be described as irregular fractal surfaces due to their complex geometry; such geometry can be exploited for image analysis and computer vision. By exploring such possibilities, pushing the limits of the state-of-the-art, this thesis starts with an analysis of texture features achieved by means of agents on image surfaces. To do so, we used the Bouligand-Minkowski fractal dimension, swarm-system Artificial Crawlers, and non-linear diffusion of Perona-Malik, techniques that led to methodologies with efficacy and efficiency comparable to the state-of-the-art. Our first method combines fractal dimension with random walks on the surface of images. In a second approach, non-linear diffusion is used to represent texture images at different scales, which are described via their fractal dimension for image classification purposes. In a third proposal, we employ fractal dimension concepts over multiple scales derived from the same image for a richer texture description. One of the purposes is the automatic detection of diseases in soybean leaves. Finally, texture characteristics were exploited in a method based on complex networks used to analyze the agglomeration of particles in nanotechnology images. The results achieved in the four methodologies described in this thesis demonstrated the potential of using texture features in tasks of classification and pattern recognition. The contributions of this work shall support significant advances in materials engineering, computer vision, and agriculture.<br>A textura é um dos principais atributos visuais para a descrição de padrões encontrados na natureza. Diversos métodos de análise de textura têm sido usados como uma poderosa ferramenta para aplicações reais que envolvem análise de imagens e visão computacional. Entretanto, os métodos existentes não conseguem discriminar com sucesso a complexidade dos padrões de textura. Tais métodos desconsideram a possibilidade de se descrever estruturas de imagens por meio de medidas como a dimensão fractal. Medidas baseadas em fractalidade permitem uma interpretação geométrica não-inteira que possui aplicações encontradas em áreas como matemática, física, e biologia. Sobre esta lacuna metodológica, a hipótese central desta tese é que texturas presentes na natureza podem ser medidas como superfícies fractais irregulares devido à sua geometria complexa, o que pode ser explorado para fins de análise de imagens e visão computacional. Para superar tais limitações, avançando o estado da arte, esta tese se inicia com uma análise das características de texturas baseada em caminhadas aleatórias de agentes sobre superfícies de imagens. Esta primeira análise leva a um método que combina dimensão fractal com caminhadas de agentes sobre a superfície de imagens. Em uma segunda abordagem, usa-se a difusão não-linear para representar imagens de texturas em diferentes escalas, as quais são descritas via dimensão fractal para fins de classificação de imagens. Em uma terceira proposta, emprega-se a dimensão fractal sobre múltiplas escalas derivadas de uma mesma imagem com o propósito de se realizar a descrição multi-escala de texturas. Um dos propósitos específicos foi a detecção automática de doenças em folhas de soja. Por último, as características de textura foram exploradas segundo uma metodologia baseada em redes complexas para análise de aglomeração de partículas em imagens de nanotecnologia. Os resultados alcançados nesta tese demonstraram o potencial do uso de características de textura. Para tanto foram usadas técnicas de dimensão fractal de Bouligand-Minkowski, multiagentes Artificial Crawlerse difusão não-linear de Perona-Malik, os quais alcançaram eficácia e eficiência comparáveis ao do estado da arte. As contribuições obtidas devem suportar avanços significativos nas áreas de engenharia de materiais, visão computacional, e agricultura.

Submitted by Maria Suzana Diniz (msuzanad@hotmail.com) on 2015-11-27T11:58:14Z No. of bitstreams: 1 arquivototal.pdf: 8972192 bytes, checksum: e0a82ad433e62b83d048d78778d60dd2 (MD5)<br>Approved for entry into archive by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2015-11-30T10:51:37Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 8972192 bytes, checksum: e0a82ad433e62b83d048d78778d60dd2 (MD5)<br>Made available in DSpace on 2015-11-30T10:51:37Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 8972192 bytes, checksum: e0a82ad433e62b83d048d78778d60dd2 (MD5) Previous issue date: 2013-04-12<br>Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES<br>Here we present a research about fractals, its history, mathematical concepts and applications. Some constructions were developed using some basic concepts of the Theory of Numbers, Trigonometry and Linear Algebra that can be applied in High School. We presented some activities that can be applied in High School’s classes in order to develop mathematical concepts such as geometric transformations, Geometry, Trigonometry, Logarithm, notions of limit among other topics. Keywords: Fractal Geometry, Geometric Transformation, Logarithm, Pascal Triangle, Arrays, Linear Algebra, High School.<br>Este trabalho apresenta uma pesquisa sobre os fractais, sua história, conceitos matemáticos utilizados e aplicações. Foram desenvolvidas algumas construções de fractais utilizando-se alguns conceitos básicos de Teoria dos Números, Trigonometria e Álgebra Linear que podem ser explorados no Ensino Médio. Foram apresentadas algumas atividades que podem ser aplicadas em sala de aula do Ensino Médio no desenvolvimento de conceitos matemáticos como Progressão Geométrica, Geometria, Trigonometria, Logaritmo e noções de limite entre outros temas.

Made available in DSpace on 2015-12-10T14:22:23Z (GMT). No. of bitstreams: 0 Previous issue date: 2015-01-03. Added 1 bitstream(s) on 2015-12-10T14:28:29Z : No. of bitstreams: 1 000854378.pdf: 6674616 bytes, checksum: d5106f3df91d25255a99661658612b89 (MD5)<br>O uso de técnicas não lineares (sistemas complexos, geometria fractal e física quântica), com enfoques mais amparados pelos princípios do caos, é uma tendência crescente no estudo das ciências naturais, e estão sendo progressivamente apresentadas como opções válidas às suas correspondentes medidas em sistemas lineares (estruturas simples, geometria euclidiana e física clássica newtoniana), com seus enfoques mais cartesianos. Neste sentido, a neurobiologia e a acupuntura podem ajudar a lançar alguma luz nesta busca. Para se avaliar o alcance do uso de técnicas não lineares - mais especificamente a dimensão fractal - foi realizado estudo piloto experimental em humanos para avaliar a sensibilidade do eletroencefalograma na identificação de alterações provocadas por pontos auriculares de acupuntura, conforme critérios clínicos, neurofisiológicos e de geometria fractal. Pacientes de ambos os sexos, tendo cefaléia como queixa comum, mas não exclusiva, foram submetidos a exames eletroencefalográficos, antes e após a estimulação de pontos de acupuntura auricular, ou após ciclos de tratamento ambulatorial por acupuntura auricular e sistêmica. As dimensões fractais pelo 'método da contagem de caixas', foram obtidas com o auxílio do sofware ImageJ, a partir de imagens tratadas dos traçados digitalizados do eletroencefalograma, e os resultados apresentados como estudos de caso. Mediante análise das variações das dimensões fractais efetuada por estudo descritivo e quantitativo, foram observados os comportamentos evolutivos da condição clínica ou sintomática do paciente, e sua relação com os conceitos de desenvolvimento caótico dos sistemas complexos dinâmicos e adaptativos<br>The use of nonlinear techniques (complex systems, fractal geometry and quantum physics), with approaches more related with the principles of chaos theory, is a growing tendency in the study of the natural sciences, and are being progressively presented as valid options to their correspondent measures in linear systems (simple structures, Euclidean geometry and newtonian classical physics), with their more Cartesian approaches. In this sense, neurobiology and acupuncture may help to shed some light upon this search. Aiming to evaluate the reach of the use of nonlinear techniques - more specifically the fractal dimension - a pilot experimental essay was taken in human beings in order to evaluate the sensibility of eletroencefalography in the identification of alterations resulting from the stimulation of auricular acupuncture points, according to clinical, neurophysiologic and fractal geometry criteria. Both male and female patients, having headache as a common but not exclusive complaint, were subject to eletroencefalographic examinations, before and after the stimulation of auricular acupuncture points, or after cycles of both auricular and systemic acupuncture follow up in an outpatient basis. The fractal dimensions calculated by the 'box counting method' were obtained with the aid of ImageJ software, using digitalized images from the eletroencefalographic examinations, and the results are presented as case reports. By analysing the variations in the fractal dimensions both by descriptive and quantitative approaches, observations were made on the evolutive behaviours either from the clinical or from the symptomatic condition of the patient, and their relationship with the concepts of chaotic development of the complex dynamic adaptative systems

Orientador: Karina Pavão Patrício<br>Banca: Moacir Fernandes de Godoy<br>Banca: Niura Aparecida Padula<br>Resumo: O uso de técnicas não lineares (sistemas complexos, geometria fractal e física quântica), com enfoques mais amparados pelos princípios do caos, é uma tendência crescente no estudo das ciências naturais, e estão sendo progressivamente apresentadas como opções válidas às suas correspondentes medidas em sistemas lineares (estruturas simples, geometria euclidiana e física clássica newtoniana), com seus enfoques mais cartesianos. Neste sentido, a neurobiologia e a acupuntura podem ajudar a lançar alguma luz nesta busca. Para se avaliar o alcance do uso de técnicas não lineares - mais especificamente a dimensão fractal - foi realizado estudo piloto experimental em humanos para avaliar a sensibilidade do eletroencefalograma na identificação de alterações provocadas por pontos auriculares de acupuntura, conforme critérios clínicos, neurofisiológicos e de geometria fractal. Pacientes de ambos os sexos, tendo cefaléia como queixa comum, mas não exclusiva, foram submetidos a exames eletroencefalográficos, antes e após a estimulação de pontos de acupuntura auricular, ou após ciclos de tratamento ambulatorial por acupuntura auricular e sistêmica. As dimensões fractais pelo 'método da contagem de caixas', foram obtidas com o auxílio do sofware ImageJ, a partir de imagens tratadas dos traçados digitalizados do eletroencefalograma, e os resultados apresentados como estudos de caso. Mediante análise das variações das dimensões fractais efetuada por estudo descritivo e quantitativo, foram observados os comportamentos evolutivos da condição clínica ou sintomática do paciente, e sua relação com os conceitos de desenvolvimento caótico dos sistemas complexos dinâmicos e adaptativos<br>Abstract: The use of nonlinear techniques (complex systems, fractal geometry and quantum physics), with approaches more related with the principles of chaos theory, is a growing tendency in the study of the natural sciences, and are being progressively presented as valid options to their correspondent measures in linear systems (simple structures, Euclidean geometry and newtonian classical physics), with their more Cartesian approaches. In this sense, neurobiology and acupuncture may help to shed some light upon this search. Aiming to evaluate the reach of the use of nonlinear techniques - more specifically the fractal dimension - a pilot experimental essay was taken in human beings in order to evaluate the sensibility of eletroencefalography in the identification of alterations resulting from the stimulation of auricular acupuncture points, according to clinical, neurophysiologic and fractal geometry criteria. Both male and female patients, having headache as a common but not exclusive complaint, were subject to eletroencefalographic examinations, before and after the stimulation of auricular acupuncture points, or after cycles of both auricular and systemic acupuncture follow up in an outpatient basis. The fractal dimensions calculated by the 'box counting method' were obtained with the aid of ImageJ software, using digitalized images from the eletroencefalographic examinations, and the results are presented as case reports. By analysing the variations in the fractal dimensions both by descriptive and quantitative approaches, observations were made on the evolutive behaviours either from the clinical or from the symptomatic condition of the patient, and their relationship with the concepts of chaotic development of the complex dynamic adaptative systems<br>Mestre

Orientador: Prof. Dr. Marcelo Zanchetta do Nascimento<br>Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Engenharia da Informação, 2015.<br>O câncer de próstata é o segundo que provoca o maior número de vítimas fatais entre os homens, atingindo principalmente a população mundial com idades superiores a 60 anos. Entre os métodos empregados para o diagnóstico médico estão os exames clínicos, laboratoriais e o diagnóstico por imagem, o que pode indicar a necessidade da biópsia da próstata. As biópsias são avaliadas por especialistas para auxiliar na conduta mais adequada de tratamento, desta forma o estudo por imagem histológica é realizado e se destaca como um dos métodos utilizados devido a facilidade de diagnosticar a doença. Porém, ainda existem problemas que precisam ser solucionados para reduzir o número de falsos positivos. Este trabalho apresenta um conjunto de técnicas para identificar e quantificar as regiões de interesse em imagens histológicas da próstata. As análises foram realizadas com dimensão fractal de imagens coloridas e classificadas com SVM com os kernels linear, polinomial e RBF. As regiões de interesses foram segmentadas em núcleos da célula cuboide, lúmens glandulares e tecido estromal e aplicado o cálculo da dimensão fractal. A avaliação de desempenho foi baseada na área sob a curva ROC (AUC) e pela acurácia. Os resultados obtidos com essas ferramentas mostram que o grupo de imagens segmentadas por estroma com magnificação de 100x obtiveram melhores resultados de classificação, obtendo valores de AUC de 92,21% e 86,77% de acurácia para os grupos de tecido normal versus tecido tumoral, obteve 73,53% de acurácia para o grupo tecido normal versus tecido hiperplásico e de 80,00% para o grupo de tecido hiperplásico versus tecido tumoral. O método proposto quantificou tecidos histológicos da próstata com descritores baseados em técnicas não lineares multi-escala. O uso de informações dos canais de cores em conjunto com a segmentação das estruturas foi mais relevante para um sistema de apoio ao diagnóstico.<br>Prostate cancer is the second type of cancer that causes more deaths between men. It affects mainly the population over the age of 60. Laboratory exams and diagnostic imaging are among the methods used for medical diagnosis, which may indicate the need for a prostrate biopsy. Biopsies are evaluated by experts in order to indicate the most appropriate treatment strategy. Hence, the study of histological images stands out as one of the most used methods as it allows an easier diagnosis. However, there are still problems that need to be addressed to reduce the number of false positives. This work presents a set of techniques to identify and quantify regions of interest in histological images of the prostate. Color and greyscale images were analysed using fractal dimension then classified in SVM with linear, polynomial and RBF kernels. Regions of interest were segmented in basal cell cuboid, glandular lumens and stromal tissue and then a fractal dimension was applied. Performance evaluation was based on the area under the ROC curve(AUC) and accuracy. The results obtained by applying these tools show that images segmented by stroma with a magnification of 100x had better classification results, achieving AUC values of 92.21% and 86.77% accuracy for the normal tissue groups versus tumor tissue. Also, in this group of images a level of accuracy of 73.53% for hyperplastic tissue versus normal tissue and 80.00% for hyperplastic tissue versus tumor tissue. The method quantified histological prostate tissue with multi-scale techniques based on nonlinear descriptors. Therefore, the use of information from color channels together with the segmented structures are most relevant to a diagnostic support system.

Submitted by (lucia.rodrigues@ufrpe.br) on 2016-05-31T12:48:57Z No. of bitstreams: 1 Daniella Tavares Pessoa.pdf: 1486789 bytes, checksum: a6f7a6497263d8419ed731a88dac28b8 (MD5)<br>Made available in DSpace on 2016-05-31T12:48:57Z (GMT). No. of bitstreams: 1 Daniella Tavares Pessoa.pdf: 1486789 bytes, checksum: a6f7a6497263d8419ed731a88dac28b8 (MD5) Previous issue date: 2012-02-28<br>Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES<br>The increased consumption of hyperlipidic diet has been an increase in obesity rates and levels of serum cholesterol and triglycerides in a large part of the population, as well as, has been linked with the development of neurodegenerative diseases, such as Alzheimer's disease. On the other hand, several studies demonstrated the importance of lipids in brain structure and activity. Epilepsy is a pathology related to the brain activity disorder, with high rate of refractoriness to conventional therapeutics, in these cases hyperlipidic diet has been used such an alternative treatment. Therefore, the investigation of possible interference from hyperlipidemic diets in TLE can add new perspectives in understanding the behavior and treatment of this pathology. In the present study we used mathematical computational methods to analyze electrographic patterns of rats in status epilepticus induced by pilocarpine fed with hyperlipidic diet. These rats were analyzed through electrographic parameters using ECoG records and determining: energies of power spectrum in the frequency of delta, theta, alpha and beta waves; Lempel-Ziv complexity; and fractal dimension of phase space. Status epilepticus induced changes in the encephalographic pattern measured by distribution of main brain waves using power spectrum, Lempel-Ziv complexity and fractal dimension of phase space. Hyperlipidic diet in normal rats also changed the values of brain waves energy in power spectrum and Lempel-Ziv complexity; however, fractal dimension of phase space showed no significant differences due to hyperlipidic diet treatment. Despite the hyperlipidic diet reduced brain activity before pilocarpine administration, the nutritional status did not change the encephalographic pattern during status epilepticus. In conclusion, hyperlipidic diet induced slower brain waves and decreased the complexity of brain activity, opposite effects of status epilepticus. Therefore, the mathematical methods were effective to detect brain hyperactivity caused by status epilepticus and reduced brain activity induced by hyperlipidic diet.<br>O aumento do consumo de dietas hiperlipídicas vem elevando os índices de obesidade e os níveis de colesterol e triglicerídeos de grande parte da população, além de estar relacionado ao desenvolvimento de doenças neurodegenerativas, como a doença de Alzheimer. Por outro lado muitas pesquisas têm comprovado a importância dos lipídeos na estrutura e atividade do cérebro. A epilepsia é uma patologia relacionada à desordem da atividade cerebral, com alto índice de refratariedade a medicamentos convencionais, nesses casos, o consumo de dietas hiperlipídica vem sendo utilizado como uma terapia alternativa. A investigação de possíveis interferências de dietas hiperlipídicas na ELT pode acrescentar novas perspectivas na compreensão do comportamento e tratamento desta condição patológica. Nesse trabalho foram analisados ratos em status epilepticus induzido pela pilocarpina submetidos à dieta hiperlipídica. Esses ratos foram analisados através de parâmetros eletrográficos utilizando os registros de ECoG e determinando as energias do seu espectro de potência nas freqüências das ondas delta, teta, alfa e beta; a complexidade de Lempel-Ziv e a dimensão fractal do espaço de fase. O status epilepticus induziu alterações no padrão encefalográfico mensuradas pela distribuição de energia das principais ondas cerebrais utilizando o espectro de potência, a complexidade de Lempel-Ziv e a dimensão fractal do espaço de fase. A dieta hiperlipídica, em ratos normais, também alterou os valores da energia das ondas cerebrais no espectro de potência e na complexidade de Lempel-Ziv; entretanto, a dimensão fractal do espaço de fase não revelou diferenças significativas devido ao tratamento com a dieta hiperlipídica. Apesar da dieta hiperlipídica ter reduzido a atividade cerebral antes da administração da pilocarpina, a condição nutricional não influenciou o padrão encefalográfico durante o status epilepticus. Em conclusão, a dieta hiperlipídica causou uma desaceleração das ondas cerebrais e diminuição da complexidade da atividade cerebral, efeitos contrários aos do status epilepticus. Portanto, os métodos matemáticos utilizados foram eficientes na detecção da hiperatividade cerebral causada pelo status epilepticus e redução da atividade cerebral induzida pela dieta hiperlipídica.

OPERTI, Felipe Gioachino. Interpolation strategy based on Dynamic Time Warping. 2015. 53 f. Dissertação (Mestrado em Física) - Programa de Pós-Graduação em Física, Departamento de Física, Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2015.<br>Submitted by Edvander Pires (edvanderpires@gmail.com) on 2015-04-14T22:11:14Z No. of bitstreams: 1 2015_dis_fgoperti.pdf: 5361657 bytes, checksum: b47dae9c4d72accf5fe2c50b89abaae4 (MD5)<br>Approved for entry into archive by Edvander Pires(edvanderpires@gmail.com) on 2015-04-16T18:35:49Z (GMT) No. of bitstreams: 1 2015_dis_fgoperti.pdf: 5361657 bytes, checksum: b47dae9c4d72accf5fe2c50b89abaae4 (MD5)<br>Made available in DSpace on 2015-04-16T18:35:49Z (GMT). No. of bitstreams: 1 2015_dis_fgoperti.pdf: 5361657 bytes, checksum: b47dae9c4d72accf5fe2c50b89abaae4 (MD5) Previous issue date: 2015<br>In oil industry, it is essential to have the knowledge of the stratified rocks’ lithology and, as consequence, where are placed the oil and the natural gases reserves, in order to efficiently drill the soil, without a major expense. In this context, the analysis of seismological data is highly relevant for the extraction of such hydrocarbons, producing predictions of profiles through reflection of mechanical waves in the soil. The image of the seismic mapping produced by wave refraction and reflection into the soil can be analysed to find geological formations of interest. In 1978, H. Sakoe et al. defined a model called Dynamic Time Warping (DTW)[23] for the local detection of similarity between two time series. We apply the Dynamic Time Warping Interpolation (DTWI) strategy to interpolate and simulate a seismic landscape formed by 129 depth-dependent sequences of length 201 using different values of known sequences m, where m = 2, 3, 5, 9, 17, 33, 65. For comparison, we done the same operation of interpolation using a Standard Linear Interpolation (SLI). Results show that the DTWI strategy works better than the SLI when m = 3, 5, 9, 17, or rather when distance between the known series has the same order size of the soil layers.

Nous avons construit des lasers qui ont servi a des experiences de melange a quatres ondes (reseau transitoire). Nous avons d'abord etudie la reponse non lineaire du troisieme ordre lorsque le milieu presente un continuum large d'etats (colorants). Le signal obtenu sur le cresyl-violet en film mince de colle cellulosique a ete analyse. La technique du reseau transitoire a ete ensuite utilisee pour caracteriser la dynamique de molecules dans un milieu poreux (vycor) et etudier la geometrie des connexions entre pores. Nous avons etudie la rotation de l'azobenzene en solution soit libre soit contenue a l'interieur de l'espace poreux. Ces experiences peuvent etre realisees sur des echelles allant de la femtoseconde a la seconde, permettant ainsi l'etude tant du temps de coherence que des proprietes de diffusion des materiaux

The existing world is becoming more disrupted and is falling apart. For its resurrection and restoration, a new way of thinking is necessary. This new type of thinking is needed to be able to open up its mind and to think about the process of thinking itself; it must understand what is happening in other systems, where processes seem to be taking place by themselves without any other visible interference. First Chaos is the title for an intermedia installation which contains 90 black and white photographs, both digital and analogue, all of which were taken between the years 2008 and 2012. Together, the photographs create one coherent piece – a kind of sculpture. They can evoke a "still film" with a non-linear, cyclical storyline, whilst the images can simultaneously function individually, without any connection to other photographs.

碩士<br>中原大學<br>國際貿易研究所<br>94<br>Based on the conclusion, derived by Teräsvirta, Dick and Medeiros(2003), that LSTAR model appears to have better probability coverage than the neural network model, this paper adopts LSTARX and ESTARX models to find out the technical indicator variables influencing little wavelet transfers from stock price fractal dimension and use them to forecast little wavelet in reverse point of stock porice. Under no strong theorical model, we first use stepwise regression model to obtain statistically significant exogenous technical indicator variables affecting returns of stock price index, from TAIEX, electronics-index and finance-index. Further using these significant variables to form their lag terms and LSTARX and ESTARX models, then we can analyze wavelet in reverse point of stock price. Empirical evidences show that LSTARX and ESTARX models outperform linear model. For LSTARX model, the location parameters of little wavelet reversing are identified significant. And for ESTARX model, transition function variables and variables’ lag terms are significant. That is to say, LSTARX model supports there are little wavelet transition reversed in three major type stock price index returns and ESTARX model can find out the important independent variables to explain the little wavelet transfer of the stock price return.

The notion of a curve played important role in the history of mathematical thought. This dissertation is focused on the conception of a curve in analysis, point set theory and topology. The rectification of curves and the notion of arc length are considered in connection with the history of analysis from antiquity to the beginning of the 20th century. "Measurement of curves" is also discussed from the measure-theoretic viewpoint and various definitions of linear measure and fractional dimension are described. Historically, there are two main approaches to understanding curves. Jordan defined a curve as a continuous image of a closed interval. However, his definition appeared to be too wide, since it was met by objects such as the Peano curve. In the point set theory, a curve is considered to be a one-dimensional continuum. The development of the dimension theory and the continuum theory, starting with the pioneering work of Bolzano, was motivated by the search for rigorous topological definition of a curve, a surface etc. Among "pathological" curves, that were often introduced as counterexamples in the development of modern analysis, we can find early examples of fractals. The fractal theory motivated further study of mathematical properties of these curves in the late 20th century, such as self-similarity and...

Η ποιότητα πολλών εκ των αποτελεσμάτων της σύγχρονης έρευνας εξαρτώνται άμεσα από την «ποιότητα» και την ποσότητα των τυχαίων αριθμών που χρησιμοποιούνται. Ειδικότερα σε τομείς όπως η στοχαστική μοντελοποίηση και προσομοίωση προτιμώνται οι ντετερμινιστικές γεννήτριες τυχαίων αριθμών, ή αλλιώς γεννήτριες ψευδοτυχαίων αριθμών λόγω της δυνατότητας αναπαραγωγής των αποτελεσμάτων και της μεταφερσιμότητας τους. Επομένως, μας είναι χρήσιμο να εντοπίσουμε ψευδοτυχαίες γεννήτριες αριθμών με αυξημένη φαινόμενη τυχαιότητα αποτελεσμάτων. Για το λόγο αυτό, στη διπλωματική εργασία προτείνεται και εξετάζεται η καταλληλότητα της θραυσματικής διάστασης (fractal dimension) για την αξιολόγηση ψευδοτυχαίων γεννητριών τυχαίων αριθμών (Pseudorandom Number Generators). Η θραυσματική διάσταση αποτελεί μία μετρική που δύναται να εκφράσει την τυχαιότητα των αποτελεσμάτων μιας γεννήτριας ψευδοτυχαίων αριθμών καθώς «ποσοτικοποιεί» την κατανομή των ψευδοτυχαίων αριθμών στον ευκλείδειο χώρο. Σε πρώτο στάδιο γίνεται μία επισκόπηση των υπαρχουσών μεθοδολογιών παραγωγής τυχαίων αριθμών καθώς και των προσεγγίσεων για την αξιολόγηση της απόδοσης των ψευδοτυχαίων γεννητριών τυχαίων αριθμών. Οι καθιερωμένες τεχνικές που εφαρμόζονται για την αξιολόγηση μιας γεννήτριας εστιάζουν σε στατιστικά χαρακτηριστικά που έχουν ως στόχο να μετρήσουν πόσο απρόβλεπτα είναι τα αποτελέσματά της, ή χαρακτηριστικά όπως η περίοδος μιας γεννήτριας. Ακολούθως, μελετάται η θραυσματική διάσταση και οι προτεινόμενες στη βιβλιογραφία μέθοδοι υπολογισμού της. Στο στάδιο αυτό επιλέγεται η κατάλληλη μέθοδος για τον υπολογισμό της θραυσματικής διάστασης. Στο τελευταίο πειραματικό στάδιο παρουσιάζονται τα αποτελέσματα της μέτρησης της μορφοκλασματικής διάστασης. Οι ψευδοτυχαίες γεννήτριες προς αξιολόγηση που μετείχαν στα υπολογιστικά πειράματα ήταν η Γραμμική Αναλογική γεννήτρια, η γεννήτρια Blum-Blum-Shub, η γεννήτρια που βασίζεται στο κρυπτοσύστημα RSA και η γεννήτρια που βασίζεται στο πρόβλημα του διακριτού λογαρίθμου. Τα υπολογιστικά πειράματα επιχειρούν να ανακαλύψουν την απόδοση των εξεταζόμενων γεννητριών αλλά και την ευαισθησία της συμπεριφοράς τους ως προς τις παραμέτρους εισόδου των γεννητριών.<br>Scientific experimental results are highly dependent on the "quality" and quantity of random numbers used for these experiments. Especially in areas such as stochastic modeling and simulation, deterministic random number generators, known as pseudorandom number generators are preferred because of reproducibility of the results and their portability. Trying to identify pseudorandom number generators sequences which appear to be random, we examine the suitability of Fractal Dimension measurement for assessing Pseudorandom Number Generators. The established techniques that are used to evaluate a generator are focused on statistical features that are designed to detect correlations into generated pseudorandom number sequences. On the other hand, Fractal Dimension is a metric that can express the randomness of the results of a pseudorandom number generator as it "quantifies" the distribution of pseudorandom numbers in Euclidean space. We attempt to evaluate some Pseudorandom Number Generators, like classical Linear Congruential generator, Blum-Blum-Shub generator, the generator based on RSA cryptosystem and the generator based on the Discrete Logarithm problem. The computational experiments presented in our work attempt to assess the performance and the sensitivity of the examined generators.

Στη συγκεκριμένη μεταπτυχιακή εργασία γίνεται ανάλυση των ηλεκτροεγκεφαλογραφικών καταγραφών επιληπτικών ασθενών με βάση την κλασματική διάσταση για τον εντοπισμό της έναρξης των επιληπτικών κρίσεων καθώς και για την ανίχνευση των επερχόμενων επιληπτικών κρίσεων. Πιο συγκεκριμένα, χρησιμοποιούνται τέσσερις μέθοδοι υπολογισμού της κλασματικής διάστασης στο πεδίο του χρόνου. Πραγματοποιείται μία συγκριτική μελέτη των συγκεκριμένων μεθόδων χρησιμοποιώντας δύο συνθετικά σήματα γνωστής κλασματικής διάστασης και στη συνέχεια οι μέθοδοι εφαρμόζονται σε ηλεκτροεγκεφαλογραφικές καταγραφές ασθενών που πάσχουν από επιληψία. Η συγκεκριμένη εργασία αποδεικνύει ότι όλες οι μέθοδοι μπορούν και εντοπίζουν την έναρξη των επιληπτικών κρίσεων ενώ η στατιστική ανάλυση δείχνει ότι κάποιες από αυτές τις μεθόδους μπορούν να χρησιμοποιηθούν και ως δείκτες πρόβλεψης των επερχόμενων επιληπτικών κρίσεων. Επίσης, διαπιστώνεται ότι ανάλογα με το είδος των ηλεκτροεγκεφαλογραφικών καταγραφών, το μέγεθος του παραθύρου και τη ζώνη διέλευσης του φίλτρου τα αποτελέσματα της κλασματικής διάστασης που προκύπτουν επηρεάζονται.<br>Τhe fractal dimension is a powerful tool in the analysis of electroencephalograms. In this work, four methods of estimating the fractal dimension of electroencephaolographic recordings of epileptic patients directly in the time domain are analyzed and compared. The analysis is performed over both synthetic data and real recordings of epileptic patients. The effects of the type of recordings, the window size and the frequencies in the passband filter are examined. This study shows that the four methods detect the beginning of the epileptic seizures, while the statistic analysis proves that some of these methods can be used for the prediction of the next seizures.