To see the other types of publications on this topic, follow the link: Fractal dimensions.
Dissertations / Theses on the topic 'Fractal dimensions'
Create a spot-on reference in APA, MLA, Chicago, Harvard, and other styles
Consult the top 50 dissertations / theses for your research on the topic 'Fractal dimensions.'
Next to every source in the list of references, there is an 'Add to bibliography' button. Press on it, and we will generate automatically the bibliographic reference to the chosen work in the citation style you need: APA, MLA, Harvard, Chicago, Vancouver, etc.
You can also download the full text of the academic publication as pdf and read online its abstract whenever available in the metadata.
Browse dissertations / theses on a wide variety of disciplines and organise your bibliography correctly.
1
Leifsson, Patrik. "Fractal sets and dimensions." Thesis, Linköping University, Department of Mathematics, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-7320.
Full textAbstract:
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.
2
Barros, Marcelo Miranda. "Identification of Fractal Dimensions from a Dynamical Analogy." Laboratório Nacional de Computação Científica, 2007. http://www.lncc.br/tdmc/tde_busca/arquivo.php?codArquivo=145.
Full textAbstract:
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.
3
Alrud, Beng Oscar. "Fractal spectral measures in two dimensions." Doctoral diss., University of Central Florida, 2011. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/4834.
Full textAbstract:
We study spectral properties for invariant measures associated to affine iterated function systems. We present various conditions under which the existence of a Hadamard pair implies the existence of a spectrum for the fractal measure. This solves a conjecture proposed by Dorin Dutkay and Palle Jorgensen, in several special cases in dimension 2.ID: 030422913; System requirements: World Wide Web browser and PDF reader.; Mode of access: World Wide Web.; Thesis (Ph.D.)--University of Central Florida, 2011.; Includes bibliographical references (p. 75-76).
Ph.D.
Doctorate
Mathematics
Sciences
4
Schönwetter, Moritz. "Fractal Dimensions in Classical and Quantum Mechanical Open Chaotic Systems." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2017. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-215747.
Full textAbstract:
Fractals have long been recognized to be a characteristic feature arising from chaotic dynamics; be it in the form of strange attractors, of fractal boundaries around basins of attraction, or of fractal and multifractal distributions of asymptotic measures in open systems. In this thesis we study fractal and multifractal measure distributions in leaky Hamiltonian systems. Leaky systems are created by introducing a fully or partially transparent hole in an otherwise closed system, allowing trajectories to escape or lose some of their intensity. This dynamics results in intricate (multi)fractal distributions of the surviving trajectories. These systems are suitable models for experimental setups such as optical microcavities or microwave resonators. In this thesis we perform an improved investigation of the fractality in these systems using the concept of effective dimensions. They are defined as the dimensions far from the usually considered asymptotics of infinite evolution time $t$, infinite sample size $S$, and infinite resolution (infinitesimal box-size $varepsilon$). Yet, as we show, effective dimensions can be considered as intrinsic to the dynamics of the system. We present a detailed discussion of the behaviour of the numerically observed dimension $D_mathrm{obs}(S,t,varepsilon)$. We show that the three parameters can be expressed in terms of limiting length scales that define the parameter ranges in which $D_mathrm{obs}(S,t,varepsilon)$ is an effective dimension of the system. We provide dynamical and statistical arguments for the dependence of these scales on $S$, $t$, and $varepsilon$ in strongly chaotic systems and show that the knowledge of the scales allows us to define meaningful effective dimensions. We apply our results to three main fields. In the context of numerical algorithms to calculate dimensions, we show that our findings help to numerically find the range of box sizes leading to accurate results. We further show that they allow us to minimize the computational cost by providing estimates of the required sample-size and iteration time needed. A second application field of our results is systems exhibiting non-trivial dependencies of the effective dimension $D_mathrm{eff}$ on $t$ and $varepsilon$. We numerically explore this in weakly chaotic leaky systems. There, our findings provide insight into the dynamics of the systems, since deviations from our predictions based on strongly chaotic systems at a given parameter range are a sign that the stickiness inherent to such systems needs to be taken into account in that range. Lastly, we show that in quantum analogues of chaotic maps with a partial leak, a related effective dimension can be used to explain the numerically observed deviation from the predictions provided by the fractal Weyl law for systems with fully absorbing leaks. Here, we provide an analytical description of the expected scaling based on the classical dynamics of the system and compare it with numerical results obtained in the studied quantum mapsEs ist seit langem bekannt, dass Fraktale eine charakteristische Begleiterscheinung chaotischer Dynamik sind. Sie treten in Form von seltsamen Attraktoren, von fraktalen Begrenzungen der Einzugsbereiche von Attraktoren oder von fraktalen und multifraktalen Verteilungen asymptotischer Maße in offenen Systemen auf. In dieser Arbeit betrachten wir fraktal und multifraktal verteilte Maße in geöffneten hamiltonschen Systemen. Geöffnete Systeme werden dadurch erzeugt, dass man ein völlig oder teilweise transparentes Loch im Phasenraum definiert, durch das Trajektorien entkommen können oder in dem sie einen Teil ihrer Intensität verlieren. Die Dynamik in solchen Systemen erzeugt komplexe (multi)fraktale Verteilungen der verbleibenden Trajektorien, beziehungsweise ihrer Intensitäten. Diese Systeme sind zur Modellierung experimenteller Aufbauten, wie zum Beispiel optischer Mikrokavitäten oder Mikrowellenresonatoren, geeignet. In dieser Arbeit führen wir eine verbesserte Untersuchung der Fraktalität in derartigen Systemen durch, die auf dem Konzept der effektiven Dimensionen beruht. Diese sind als die Dimensionen definiert, die weit weg von den üblicherweise betrachteten Limites unendlicher Iterationszeit $t$, unendlicher Stichprobengröße $S$ und unendlicher Auflösung, also infinitesimaler Boxgröße $varepsilon$ auftreten. Dennoch können effektive Dimensionen, wie wir zeigen, als der Dynamik des Systems inhärent angesehen werden. Wir führen eine detaillierte Diskussion der numerisch beobachteten Dimension $D_mathrm{obs}(S,t,varepsilon)$ durch und zeigen, dass die drei Parameter $S$, $t$ und $varepsilon$ in Form grenzwertiger Längenskalen ausgedrückt werden können, die die Parameterbereiche definieren, in denen $D_mathrm{obs}(S,t,varepsilon)$ den Wert einer effektiven Dimension des Systems annimmt. Wir beschreiben das Verhalten dieser Längenskalen in stark chaotischen Systemen als Funktionen von $S$, $t$ und $varepsilon$ anhand statistischer Überlegungen und anhand von auf der Dynamik basierenden Aussagen. Weiterhin zeigen wir, dass das Wissen um diese Längenskalen die Definition aussagekräftiger effektiver Dimensionen ermöglicht. Wir wenden unsere Ergebnisse hauptsächlich in drei Bereichen an: Im Kontext numerischer Algorithmen zur Dimensionsberechnung zeigen wir, dass unsere Ergebnisse es erlauben, diejenigen $varepsilon$-Bereiche zu finden, die zu korrekten Ergebnissen führen. Weiterhin zeigen wir, dass sie es uns erlauben, den Rechenaufwand zu minimieren, indem sie uns eine Abschätzung der benötigten Stichprobengröße und Iterationszeit ermöglichen. Ein zweiter Anwendungsbereich sind Systeme, die sich durch eine nichttriviale Abhängigkeit von $D_mathrm{eff}$ von $t$ und $varepsilon$ auszeichnen. Hier ermöglichen unsere Ergebnisse ein besseres Verständnis der Systeme, da Abweichungen von den Vorhersagen basierend auf der Annahme von starker Chaotizität ein Anzeichen dafür sind, dass im entsprechenden Parameterbereich die Eigenschaft dieser Systeme, dass Bereiche in ihrem Phasenraum Trajektorien für eine begrenzte Zeit einfangen können, relevant ist. Zuletzt zeigen wir, dass in quantenmechanischen Analoga chaotischer Abbildungen mit partiellen Öffnungen eine verwandte effektive Dimension genutzt werden kann, um die numerisch beobachteten Abweichungen vom fraktalen weyl'schen Gesetz für völlig transparente Öffnungen zu erklären. In diesem Zusammenhang zeigen wir eine analytische Beschreibung des erwarteten Skalierungsverhaltens auf, die auf der klassischen Dynamik des Systems basiert, und vergleichen sie mit numerischen Erkenntnissen, die wir über die Quantenabbildungen gewonnen haben
5
Wu, Shi-Ching. "Fractal analyses of some natural systems." Thesis, University of Hull, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.322455.
Full text
6
Brock, S. T. H. "Fractal dimensions and their relationship to filtration characteristics." Thesis, Loughborough University, 2000. https://dspace.lboro.ac.uk/2134/13486.
Full textAbstract:
Work has been performed to characterise filtration systems according to their fractal properties and to construct agglomerates to mimic the filtration systems under scrutiny. The first objective was achieved by carrying out experiments examining the dead-end filtration of two separate mineral suspensions, namely calcite and talc. These minerals were chosen to represent typical incompressible (calcite) and compressible (talc) filtration systems, undergoing filtration using a range of pressures. The experimental apparatus produced filter cakes that could be sampled, sectioned and examined under high magnification. The second objective was met by developing a computer application that could construct simulated particle agglomerates in both two and three dimensions, using a seed agglomeration model as well as simulating filtration by means of a virtua1 filter cell. A large number of simulations were completed to mimic both the dead-end filtration and other agglomerate models. The computer application was also capable of characterising the fractal and Euclidean spatial nature of both the simulated and experimental particulate systems, using a variety of techniques. Euclidean spatial attributes such as porosity as well as fractal properties including surface roughness and a number of density fractal dimensions have been measured for both types of system and demonstrate that the conditions under which the trials were performed have a bearing on the final configuration of the structures. Results from both experimental and simulation work show that fractal dimensions offer a valid method of measuring the properties of filtration systems. Experimental results showed that as the filtering pressure was increased, the density fractal dimension for the system appeared to increase. This change in fractal dimension was also accompanied by a decrease in the porosity of the system (more so for talc than the calcite), confirming the compressibility of the materials under scrutiny. The characterisation of the sampled cakes also showed that the spatial characteristics vary within the individual slices of the sample,in agreement with modem filtration theory. Results from the simulations show that both the physical and fractal properties of the resulting structures varied with the parameters used to construct them. As a rule, as the particles in the simulations were able to move in a more diffusive manner (akin to Brownian motion), the agglomerates they formed had a more open, rugged structure. The simulation of filtration systems also showed a variation within the individual cake structures. In the case of the filtration simulations, the probability assigned to the particles' sticking to the growing agglomerate was the controlling factor. In addition, it was found that the simulated cakes had similar spatial properties to the experimental systems they were designed to replicate.
7
Croft, Jonathan. "Some generalisations of nested fractal constructions and associated diffusions." Thesis, University of Warwick, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.340483.
Full text
8
Patuano, Agnès. "Fractal dimensions of landscape images as predictors of landscape preference." Thesis, University of Edinburgh, 2018. http://hdl.handle.net/1842/31380.
Full textAbstract:
Many studies of natural landscape preference have demonstrated that qualities such as 'complexity' and 'naturalness' are associated with preference, but have struggled to define the key characteristics of these qualities. Recently, the development of software programs and digital techniques has offered researchers new ways of quantifying the landscape qualities associated with preference. Among them fractal geometry offers the most promising approach. Fractals have been defined as mathematical models of organic objects and patterns as opposed to the straight lines and perfect circles of Euclidean geometry found in man-made environments. Fractal patterns are mainly characterized by their dimension, which could be described as a statistical quantification of complexity. By applying this mathematical concept to digital images, several studies claim to have found a correlation between the fractal dimensions of a set of images and the images' preference ratings. Such studies have particularly focussed on demonstrating support for the hypothesis that patterns with a fractal dimension of around 1.3 induce better responses than others. However, much of this research so far has been carried out on abstract or computer-generated images. Furthermore, the most commonly used method of fractal analysis, the box-counting method, has many limitations in its application to digital images which are rarely addressed. The aim of this thesis is to explore empirically the suggestion that landscape preference could be influenced by the fractal characteristics of landscape photographs. The first part of this study was dedicated to establishing the robustness and validity of the box-counting method, and apply it to landscape images. One of the main limitations of the box-counting method is its need for image pre-processing as it can only be applied to binary (black and white) images. Therefore, to develop a more reliable method for fractal analysis of landscapes, it was necessary to compare different methods of image segmentation, i.e the reduction of greyscale photographs into binary images. Each method extracted a different structure from the original photograph: the silhouette outline, the extracted edges, and three different thresholds of greyscale. The results revealed that each structure characterized a different aspect of the landscape: the fractal dimension of the silhouette outline could quantify the height of the vegetation, while the fractal dimension of the extracted edges characterized complexity. The second part of the study focused on collecting preference ratings for the landscape images previously analysed, using an online survey disseminated in France and the UK. It was found that different groups of participants reacted differently to the fractal dimensions, and that some of those groups were significantly influenced by those characteristics while others were not. Unexpectedly, the variable most correlated with preference was the fractal dimension of the image's extracted edges, although this variable's predictive power was relatively low. The study concludes by summarising the issues involved in estimating the fractal dimensions of landscapes in relation to human response. The research offers a set of reliable and tested methods for extracting fractal dimensions for any given image. Using such methods, it produces results which challenge previous hypotheses and findings in relation to fractal dimensions that predict human preference, identifying gaps in understanding and promising future areas of research.
9
Ašeriškytė, Dovilė. "Fraktalinių dimensijų skaičiavimas kai kurioms žmogaus organizmo fiziologinių procesų realizacijoms." Master's thesis, Lithuanian Academic Libraries Network (LABT), 2005. http://vddb.library.lt/obj/LT-eLABa-0001:E.02~2005~D_20050608_171232-14237.
Full textAbstract:
For correct specification human’s physiological state, it is very important to evaluate the changes of main human organism systems. Fractal dimensions of the parameters of the human organism, according to proposed model which includes three functional elements – periphery, regulation and supplying systems were analyzed. The parameters that characterize the function of those systems, that is heart rate, JT interval, systolic and diastolic blood pressure have been studied. Interpolation of discrete data from the physical load obtained by provocative incremental bicycle ergometry stress test was made by cubic spline. For those approximated parameters fractal dimensions (capacity, information, correlation) were counted.
The differences for various groups of persons (sportsmen, healthy persons, patients with ischemic heart disease) were investigated. Fractal dimensions integrates all features of reaction to load and recovery.
The study revealed that distributions of fractal dimensions significantly differs between... [to full text]
10
Kilps, John Russel 1965. "Fractal dimensions of aggregates formed under natural and engineered fluid environments." Thesis, The University of Arizona, 1993. http://hdl.handle.net/10150/278282.
Full textAbstract:
Fractal dimensions of aggregates formed under natural and engineered fluid environments were investigated. Latex microsphere aggregates were generated under two separate hydrodynamic environments. Fractal dimensions were determined using power law relationships and relationships with slopes of aggregate size distributions. Aggregate properties were measured with a particle counter and an image analysis system. Aggregates generated in a paddle mixer and a rolling cylinder had D3 fractal dimensions of 1.92 ± 0.04 and 1.59 ± 0.16, respectively, indicating rolling cylinder aggregates are more fractal than paddle mixer aggregates. Fractal dimensions of marine snow aggregates were determined from image analysis of in-situ aggregate photographs at two different research facilities. Fractal dimensions from the two facilities were equal, indicating this analysis technique is independent of equipment and analyst. Fractal dimensions were determined for sloughed biofilm aggregates in trickling filter effluent aged under four different fluid environments. D1 and D2 fractal dimensions were 1.29 ± 0.03 and 1.71 ± 0.04, respectively, and remained unchanged.
11
Barros, Marcelo Miranda. "Identificação de dimensões fractais a partir de uma analogia dinâmica." Laboratório Nacional de Computação Científica, 2007. https://tede.lncc.br/handle/tede/74.
Full textAbstract:
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: 2007-03-23Several 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.
12
Inui, Kanji. "Study of the fractals generated by contractive mappings and their dimensions." Kyoto University, 2020. http://hdl.handle.net/2433/253370.
Full textAbstract:
Kyoto University (京都大学)0048
新制・課程博士
博士(人間・環境学)
甲第22534号
人博第937号
新制||人||223(附属図書館)
2019||人博||937(吉田南総合図書館)
京都大学大学院人間・環境学研究科共生人間学専攻
(主査)教授 角 大輝, 教授 上木 直昌, 准教授 木坂 正史
学位規則第4条第1項該当
13
Mougin, Pascal. "Diffusion et réaction catalytique à l'interface d'un objet fractal en deux dimensions : le peigne du diable." Vandoeuvre-les-Nancy, INPL, 1996. http://docnum.univ-lorraine.fr/public/INPL_T_1996_MOUGIN_P.pdf.
Full textAbstract:
Ce mémoire s'intéresse aux phénomènes de diffusion et de réaction catalytique dans un réseau bidimensionnel de pores à caractère fractal: le peigne du diable. La géométrie de cette structure est introduite et des grandeurs permettant la comparaison avec des supports catalytiques usuels sont définies. Ce peigne est fractal par son contour. Il présente ainsi une interface spécifique importante, justifiant son emploi pour des réactions de surface. Des simulations de marche au hasard montrent que la diffusion est modifiée en régime de Knuden. Nous avons simulé le comportement du peigne du diable dans le cas de réactions simples sur son contour. Un nouveau régime de fonctionnement apparait, le régime fractal caractérisé par un flux de production indépendant de la cinétique chimique: l'objet possède des propriétés d'auto-adaptation. Les systèmes composites ont des comportements similaires à ceux obtenus avec des supports classiques. En revanche, dans le cas de l'empoisonnement du catalyseur, la structure fractale du peigne stabilise le flux de consommation pendant une durée d'empoisonnement plus longue que celle d'un support classique. La validation des simulations nécessite la réalisation de cette structure. La réaction test est la décomposition d'ozone sur oxydes métalliques. La fabrication du peigne du diable utilise des microtechniques: électroérosion, liga, stéréophotolithographie. Des collaborations avec d'autres laboratoires de recherche développant ces techniques ont été initiées. Les résultats obtenus avec ces peignes ont permis de valider une partie des simulations numériques
14
Mucheroni, Laís Fernandes [UNESP]. "Dimensão de Hausdorff e algumas aplicações." Universidade Estadual Paulista (UNESP), 2017. http://hdl.handle.net/11449/151653.
Full textAbstract:
Submitted by LAÍS FERNANDES MUCHERONI (lais.mucheroni@gmail.com) on 2017-09-18T17:23:23Z
No. of bitstreams: 1
dissertacao_mestrado_lais.pdf: 1067574 bytes, checksum: 952e3477ef0efeafd01d052547e8f2e5 (MD5)Approved for entry into archive by Monique Sasaki (sayumi_sasaki@hotmail.com) on 2017-09-19T20:08:28Z (GMT) No. of bitstreams: 1 mucheroni_lf_me_rcla.pdf: 1067574 bytes, checksum: 952e3477ef0efeafd01d052547e8f2e5 (MD5)
Made available in DSpace on 2017-09-19T20:08:28Z (GMT). No. of bitstreams: 1 mucheroni_lf_me_rcla.pdf: 1067574 bytes, checksum: 952e3477ef0efeafd01d052547e8f2e5 (MD5) Previous issue date: 2017-08-18
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.
15
Gioveli, Izabel. "Análise e simulação de padrões de fraturas geológicas." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2010. http://hdl.handle.net/10183/29053.
Full textAbstract:
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.
16
Caby, Théophile. "Extreme value theory for dynamical systems, with applications in climate and neuroscience." Thesis, Toulon, 2019. http://www.theses.fr/2019TOUL0017.
Full textAbstract:
Tout au long de la thèse, nous discuterons, améliorerons et fournirons un cadre conceptuel dans lequel des méthodes basées sur les propriétés de récurrence de dynamiques chaotiques peuvent être comprises. Nous fournirons également de nouvelles méthodes basées sur l'EVT pour calculer les quantités d'intérêt et présenteronsr de nouveaux indicateurs utiles associés à la dynamique. Nos résultats auront une rigueur mathématique totale, même si l'accent sera mis sur les applications physiques et les calculs numériques, car l'utilisation de telles méthodes se développe rapidement. Nous commencerons par un chapitre introductif à la théorie dynamique des événements extrêmes, dans lequel nous décrirons les principaux résultats de la théorie qui seront utilisés tout au long de la thèse. Après un petit chapitre dans lequel nous introduisons certains objets caractéristiques de la mesure invariante du système, à savoir les dimensions locales et les dimensions généralisées, nous consacrons les chapitres suivants à l'utilisation de EVT pour calculer de telles quantités dimensionnelles. L'une de ces méthodes définit naturellement un nouvel indicateur global sur les propriétés hyperboliques du système. Dans ces chapitres, nous présenterons plusieurs applications numériques des méthodes, à la fois dans des systèmes réels et idéalisés, et étudierons l'influence de différents types de bruit sur ces indicateurs. Nous examinerons ensuite une question d'importance physique liée à l'EVT : les statistiques de visites dans certains sous-ensembles cibles spécifiques de l'espace de phase, en particulier pour les systèmes partiellement aléatoires et bruyants. Les résultats présentés dans cette section sont principalement numériques et hypothétiques, mais révèlent un comportement universel des statistiques de visites. Le huitième chapitre établit la connexion entre plusieurs quantités locales associées à la dynamique et calculées à l'aide d'une quantité finie de données (dimensions locales, temps de frappe, temps de retour) et les dimensions généralisées du système, calculables par les méthodes EVT. Ces relations, énoncées dans le langage de la théorie des grandes déviations (que nous exposerons brièvement), ont de profondes implications physiques et constituent un cadre conceptuel dans lequel la distribution de ces quantités locales calculées peut être comprise. Nous tirons ensuite parti de ces connexions pour concevoir d'autres méthodes permettant de calculer les dimensions généralisées d'un système. Enfin, dans la dernière partie de la thèse, qui est plus expérimentale, nous étendons la théorie dynamique des événements extrêmes à des observablesThroughout the thesis, we will discuss, improve and provide a conceptual framework in which methods based on recurrence properties of chaotic dynamics can be understood. We will also provide new EVT-based methods to compute quantities of interest and introduce new useful indicators associated to the dynamics. Our results will have full mathematical rigor, although emphasis will be placed on physical applications and numerical computations, as the use of such methods is developing rapidly. We will start by an introductory chapter to the dynamical theory of extreme events, in which we will describe the principal results of the theory that will be used throughout the thesis. After a small chapter where we introduce some abjects that are characteristic of the invariant measure of the system, namely local dimensions and generalized dimensions, w1 devote the following chapters to the use of EVT to compute such dimensional quantities. One of these method defines naturally a navel global indicator on the hyperbolic properties of the system. ln these chapters, we will present several numerical applications of the methods, bath in real world and idealized systems, and study the influence of different kinds of noise on these indicators. We will then investigate a matter of physical importanc related to EVT: the statistics of visits in some particular small target subsets of the phase-space, in particular for partly random, noisy systems. The results presented in this section are mostly numerical and conjectural, but reveal some universal behavior of the statistics of visits. The eighth chapter makes the connection betweer several local quantities associated to the dynamics and computed using a finite amount of data (local dimensions, hitting times, return times) and the generalized dimensions of the system, that are computable by EVT methods. These relations, stated in the language of large deviation theory (that we will briefly present), have profound physical implications, and constitute a conceptual framework in which the distribution of such computed local quantities can be understood. We then take advantage of these connections to design further methods to compute the generalized dimensions of a system. Finally, in the last part of the thesis, which is more experimental, we extend the dynamical theory of extreme events to more complex observables, which will allow us to study phenomena evolving over long temporal scales. We will consider the example of firing cascades in a model of neural network. Through this example, we will introduce a navel approach to study such complex systems
17
Lottin, Delphine. "Dimensions fractales, morphologie et caractéristiques dimensionnelles 2D et 3D d'agrégats de nanoparticules de suie aéronautique : Etude par microscopie électronique en transmission et tomographie électronique." Thesis, Aix-Marseille, 2013. http://www.theses.fr/2013AIXM4012/document.
Full textAbstract:
Les agrégats de suie émis par les processus de combustion dans les turbines aéronautiques contribuent à modifier le bilan radiatif de l'atmosphère et la qualité de l'air. La connaissance de leurs caractéristiques physiques et chimiques est indispensable pour évaluer leur rôle dans les processus physico-chimiques atmosphériques et leur impact sur l'environnement et la santé publique. Dans ce contexte, notre étude vise à déterminer la taille et la morphologie d'agrégats de suie aéronautique à partir de mesures expérimentales menées en microscopie électronique en transmission (MET) et en tomographie électronique.Nous avons réalisé des clichés MET d'agrégats de suie émis par des turboréacteurs aéronautiques. Nous avons établi une méthode pour caractériser la morphologie des agrégats en déterminant leur allongement, leur compacité et la tortuosité de leur contour en analysant leur projection. Nous avons également développé un logiciel de traitement et d'analyse des images MET qui permet de reconstruire en 3D un agrégat à partir de ses projections et l'analyse de ses caractéristiques dimensionnelles et morphologiques à partir de sa reconstruction. Les résultats obtenus nous ont permis d'étudier la validité des relations liant les caractéristiques microphysiques 2D et 3D proposées dans la littérature et d'en proposer de nouvelles pour les agrégats étudiés.Ces résultats constituent la première caractérisation morphologique 3D d'agrégats de suie aéronautique à partir d'analyses par MET et tomographie électronique. Ils montrent que les propriétés morphologiques de ces agrégats ne permettent pas d'utiliser la méthode d'ensemble pour déterminer la dimension fractale massiqueSoot aggregates emitted by aircraft engines' combustion processes are involved in the modification of the global radiative budget and the air quality. The knowledge of their physical and chemical characteristics is a prerequisite to any evaluation of the way they may act in the atmospheric physical and chemical processes and their impact on the environment and public health. In this context, our study aims at determining the size and morphological characteristics of aircraft soot aggregates on the basis of experimental measurements by transmission electron microscopy (TEM) and electron tomography.We have acquired TEM pictures of soot aggregates emitted by aircraft engines. We have established a method to characterize the morphology of these aggregates by determining their elongation, their compacity and the tortuosity of their edge. This method is based on the analysis of their TEM projection. Besides, we have developed a software to process and analyse TEM pictures. It allows to reconstruct aggregates from their projections and to determine their size and morphological characteristics. Our results have lead us to study the validity of the relationships linking the 2D and 3D microphysical characteristics presented in the literature and to suggest new ones for the studied aggregates.These results constitute the first 3D morphological and size characterizations of aircraft soot aggregates using TEM and electron tomography. They highlight the fact that the morphological properties of these aggregates do not fulfil the hypotheses required for the use of the collective method to determine the mass fractal dimension
18
Ferreira, Filho José Roberto. "Geometria fractal : da natureza para a sala de aula." Universidade Federal de Sergipe, 2015. https://ri.ufs.br/handle/riufs/6515.
Full textAbstract:
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPESThis 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.
19
Ben, Ohoud Mohsine. "Etude comparative de l'organisation des materiaux argileux en termes de dimensions fractales." Orléans, 1988. http://www.theses.fr/1988ORLE2002.
Full textAbstract:
Caracterisation structurale et texturale des materiaux argileux selon une nouvelle approche reposant sur l'idee que l'organisation des elements de la texture a une echelle donnee se retrouve adaptes a chaque echelle. Formalisation de cette approche dans le cadre de la theorie des fractals. On montre aue kaolinite, argiles fibreuses et illite relevent d'une description triviale, basee sur un assemblage tridimensionnel desordonne, alors que les montmorillonites sont des materiaux a texture fractale. On analyse les correlations entre dimensions fractales de surface et de porosite des montmorillonitesm deshydratees et les proprietes des suspensions (permeabilite, retention en eaum rheologie)
20
Schönwetter, Moritz [Verfasser], Roland [Akademischer Betreuer] [Gutachter] Ketzmerick, Eduardo [Akademischer Betreuer] [Gutachter] Altmann, and Jan [Gutachter] Wiersig. "Fractal Dimensions in Classical and Quantum Mechanical Open Chaotic Systems / Moritz Schönwetter ; Gutachter: Jan Wiersig, Roland Ketzmerick, Eduardo Altmann ; Roland Ketzmerick, Eduardo Altmann." Dresden : Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2017. http://d-nb.info/1123931259/34.
Full text
21
Schönwetter, Moritz [Verfasser], Roland [Akademischer Betreuer] Ketzmerick, Eduardo [Akademischer Betreuer] [Gutachter] Altmann, and Jan [Gutachter] Wiersig. "Fractal Dimensions in Classical and Quantum Mechanical Open Chaotic Systems / Moritz Schönwetter ; Gutachter: Jan Wiersig, Roland Ketzmerick, Eduardo Altmann ; Roland Ketzmerick, Eduardo Altmann." Dresden : Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2017. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-215747.
Full text
22
Almeida, Filho Valdez Arag?o de. "Arranjos Log-Peri?dicos Compactos em Microfita com Elementos Fractais de Koch." Universidade Federal do Rio Grande do Norte, 2010. http://repositorio.ufrn.br:8080/jspui/handle/123456789/15315.
Full textAbstract:
Made available in DSpace on 2014-12-17T14:55:43Z (GMT). No. of bitstreams: 1
ValdezAAFi_DISSERT.pdf: 1620850 bytes, checksum: c1208b8ca13742d0d9cd3ac88c864f60 (MD5)
Previous issue date: 2010-06-14Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior
This work aims to present how the application of fractal geometry to the elements of a log-periodic array can become a good alternative when one wants to reduce the size of the array. Two types of log-periodic arrays were proposed: one with fed by microstrip line and other fed by electromagnetic coupling. To the elements of these arrays were applied fractal Koch contours, at two levels. In order to validate the results obtained some prototypes were built, which were measured on a vector network analyzer and simulated in a software, for comparison. The results presented reductions of 60% in the total area of the arrays, for both types. By analyzing the graphs of return loss, it was observed that the application of fractal contours made different resonant frequencies appear in the arrays. Furthermore, a good agreement was observed between simulated and measured results. The array with feeding by electromagnetic coupling presented, after application of fractal contours, radiation pattern with more smooth forms than the array with feeding by microstrip line
Este trabalho tem como objetivo apresentar como a aplica??o de contornos fractais aos elementos de um arranjo log-peri?dico se torna uma alternativa bastante interessante quando se deseja reduzir as dimens?es do arranjo. Foram propostos dois tipos de arranjos log-peri?dicos: um com alimenta??o por linha de microfita e outro com alimenta??o por acoplamento eletromagn?tico. Aos elementos desses arranjos foram aplicados contornos fractais de Koch, em dois n?veis. Com a finalidade de validar os resultados obtidos foram constru?dos prot?tipos, que foram caracterizados experimentalmente em um analisador de rede vetorial e simulados em software, para compara??o. Os resultados mostraram redu??es de 60% nas dimens?es dos arranjos, para ambos os tipos. Atrav?s da an?lise dos gr?ficos da perda de retorno, p?de-se observar que a aplica??o dos contornos fractais fez com que aparecessem diferentes frequ?ncias de resson?ncia nos arranjos. Al?m disso, observa-se uma boa concord?ncia entre os resultados medidos e simulados. O arranjo com alimenta??o por acoplamento eletromagn?tico apresentou, ap?s aplica??o dos contornos fractais, menores valores de diretividade do que o arranjo com alimenta??o por linha de microfita
23
Caputo, Jean-Guy. "Dimension et entropie des attracteurs associés à des écoulements réels : estimation et analyse de la méthode." Grenoble 1, 1986. http://www.theses.fr/1986GRE10057.
Full textAbstract:
On s'interesse a la caracterisation des regimes chaotiques par lesquels un ecoulement atteint la turbulence. On montre qu'un regime chaotique de convection de rayleigh-benard est decrit par un attracteur dont on determine la dimension et l'entropie. En vue de caracteriser des attracteurs de dimension plus elevee on determine les conditions d'obtention de resultats corrects sur des exemples precis
24
Azizi, Mohammed. "Simulations numériques du modèle d'Ising de 1,5 à 4 dimensions détermination des quantités critiques sur fractal, D=1,5 et en théorie des champs, D=2,3, 4." Grenoble 2 : ANRT, 1986. http://catalogue.bnf.fr/ark:/12148/cb37595627z.
Full text
25
Phillips, Jason D. "Intersections of Deleted Digits Cantor Sets With Their Translates." Wright State University / OhioLINK, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=wright1308100597.
Full text
26
Camesasca, Marco. "MULTISCALING ANALYSIS OF FLUIDIC SYSTEMS: MIXING AND MICROSTRUCTURE CHARACTERIZATION." online version, 2006. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=case1144350255.
Full text
27
Berbiche, Amine. "Propagation d'ondes acoustiques dans les milieux poreux fractals." Thesis, Aix-Marseille, 2016. http://www.theses.fr/2016AIXM4758.
Full textAbstract:
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 fréquencesThe 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
28
Dubuc, Benoit. "On estimating fractal dimension." Thesis, McGill University, 1988. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=63968.
Full text
29
Maurer, Karine. "Étude rhéologique et texturale de dispersions alimentaires : essai de quantification de leur complexité structurale au moyen du concept de géométrie fractale." Vandoeuvre-les-Nancy, INPL, 1996. http://docnum.univ-lorraine.fr/public/INPL_T_1996_MAURER_K.pdf.
Full textAbstract:
Sur le plan sensoriel, les dispersions alimentaires sont caractérisées par des propriétés telles que l'onctuosité et le caractère lisse. Ce sont des matériaux complexes, souvent hétérogènes. Leur procédé de fabrication est difficile à maitriser et la mesure de leurs propriétés rhéologiques est délicate. Il apparait donc nécessaire de mettre au point des méthodes d'observation de la complexité de cette catégorie de produits alimentaires. Dans un premier temps, on a caractérisé des dispersions modèles de type fromages frais et des dispersions commerciales par des mesures de viscoélasticité, d'écoulement et de quantification sensorielle. Dans un deuxième temps, des méthodes faisant appel au concept de géométrie fractale et d'analyse de Fourier ont été mises au point pour quantifier leur complexité structurale. Enfin, des corrélations statistiques ont été établies entre les propriétés structurales, rhéologiques et sensorielles de ces dispersions. Il est apparu, notamment, que les caractères mous, adhérent et gras étaient respectivement corrélés à la contrainte d'extrusion capillaire, au seuil d'écoulement statique et à la teneur en matière grasse. La méthode d'extrusion capillaire associée au concept de géométrie fractale et d'analyse de Fourier s'est révélée particulièrement intéressante pour quantifier les relations entre la fermeté, la rugosité et le degré d'agrégation de dispersions alimentaires grâce à l'exploitation mathématique des profils en extrusion. Le caractère lisse est apparu comme très complexe. Il est relié non seulement à d'autres perceptions de la texture comme l'homogénéité, la fermeté, ou l'onctuosité mais aussi à la dimension fractale apparente des profils et exclusion
30
Carneiro, Josà Carlos de Souza. "Study of metal transfer process in MIG / MAG through the fractal dimension of the signal voltage." Universidade Federal do CearÃ, 2005. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=15742.
Full textAbstract:
The techniques for estimating the fractal dimension of signals have been widely applied in the description of many physical systems, from studies of atmospheric turbulence, EEG signals, water systems to studies on the behavior of fractal surfaces fractured by impact. The analysis of the fractal dimension of complex phenomena has become an important tool to quantify the degree of irregularity of artificial or natural phenomena.
In this paper we investigate the fractal dimension of the signal voltage of the arc processode MIG / MAG welding. The estimation technique used for calculating the fractal dimension of the signals in the study was the method box couting. It was verified that rapid changes in the values of fractal dimension were associated with the phenomenon of metal transfer. Shows the signal analyzed in the time domain, the photographic pictures of the metal transfer, obtained by the technique of shadowgrafia, and programs used to obtain the graphs, calculating the fractal dimension as well as statistics of the signals welding.
In the literature review is made a presentation of the electric arc welding considering only its physical aspects. It appears the method used in calculating the fractal dimension of signals. Are also treated the electron emission mechanisms, the process of welding MIG / MAG, the phenomenon of metal transfer in MIG / MAG and the forces involved during metal transfer.As tÃcnicas de estimativa da dimensÃo fractal de sinais tÃm sido amplamente aplicadas na descriÃÃo de inÃmeros sistemas fÃsicos, desde estudos sobre turbulÃncia atmosfÃrica, sinais de eletroencefalograma, sistemas hidrolÃgicos atà estudos sobre o comportamento fractal de superfÃcies fraturadas por impacto. A anÃlise da dimensÃo fractal de fenÃmenos complexos passou a ser uma ferramenta importante para quantificar o grau de irregularidade de fenÃmenos artificias ou naturais. Neste trabalho investigou-se a dimensÃo fractal dos sinais de tensÃo do arco voltaico do processode soldagem MIG/MAG. A tÃcnica de estimativa empregada para o cÃlculo da dimensÃo fractal no estudo dos sinais foi o mÃtodo box couting. PÃde-se verificar que variaÃÃes bruscas nos valores da dimensÃo fractal estavam associadas com o fenÃmeno da transferÃncia metÃlica. SÃo apresentados os sinais analisados, no domÃnio do tempo, os quadros fotogrÃficos da transferÃncia metÃlica, obtidos pela tÃcnica de shadowgrafia, e os programas utilizados para obtenÃÃo dos grÃficos, cÃlculo da dimensÃo fractal e tambÃm de dados estatÃsticos dos sinais de soldagem. Na revisÃo bibliogrÃfica à feita uma apresentaÃÃo do arco elÃtrico de soldagem considerando-se apenas seus aspectos fÃsicos. à apresentado o mÃtodo empregado no cÃlculo da dimensÃo fractal dos sinais. SÃo tratados tambÃm os mecanismos de emissÃo eletrÃnica, o processo de soldagem MIG/MAG, o fenÃmeno da transferÃncia metÃlica no processo MIG/MAG e as forÃas envolvidas durante a transferÃncia metÃlica.
31
Joanpere, Salvadó Meritxell. "Fractals and Computer Graphics." Thesis, Linköpings universitet, Matematiska institutionen, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-68876.
Full textAbstract:
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.
32
Collins, Patricia Jacqueline. "Three-dimensional fractal mountains." Thesis, Monterey, California. Naval Postgraduate School, 1988. http://hdl.handle.net/10945/23427.
Full textAbstract:
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.
33
Fraser, Jonathan M. "Dimension theory and fractal constructions based on self-affine carpets." Thesis, University of St Andrews, 2013. http://hdl.handle.net/10023/3869.
Full textAbstract:
The aim of this thesis is to develop the dimension theory of self-affine carpets in several directions. Self-affine carpets are an important class of planar self-affine sets which have received a great deal of attention in the literature on fractal geometry over the last 30 years. These constructions are important for several reasons. In particular, they provide a bridge between the relatively well-understood world of self-similar sets and the far from understood world of general self-affine sets. These carpets are designed in such a way as to facilitate the computation of their dimensions, and they display many interesting and surprising features which the simpler self-similar constructions do not have. For example, they can have distinct Hausdorff and packing dimensions and the Hausdorff and packing measures are typically infinite in the critical dimensions. Furthermore, they often provide exceptions to the seminal result of Falconer from 1988 which gives the `generic' dimensions of self-affine sets in a natural setting. The work in this thesis will be based on five research papers I wrote during my time as a PhD student. The first contribution of this thesis will be to introduce a new class of self-affine carpets, which we call box-like self-affine sets, and compute their box and packing dimensions via a modified singular value function. This not only generalises current results on self-affine carpets, but also helps to reconcile the `exceptional constructions' with Falconer's singular value function approach in the generic case. This will appear in Chapter 2 and is based on a paper which appeared in 'Nonlinearity' in 2012. In Chapter 3 we continue studying the dimension theory of self-affine sets by computing the Assouad and lower dimensions of certain classes. The Assouad and lower dimensions have not received much attention in the literature on fractals to date and their importance has been more related to quasi-conformal maps and embeddability problems. This appears to be changing, however, and so our results constitute a timely and important contribution to a growing body of literature on the subject. The material in this Chapter will be based on a paper which has been accepted for publication in 'Transactions of the American Mathematical Society'. In Chapters 4-6 we move away from the classical setting of iterated function systems to consider two more exotic constructions, namely, inhomogeneous attractors and random 1-variable attractors, with the aim of developing the dimension theory of self-affine carpets in these directions. In order to put our work into context, in Chapter 4 we consider inhomogeneous self-similar sets and significantly generalise the results on box dimensions obtained by Olsen and Snigireva, answering several questions posed in the literature in the process. We then move to the self-affine setting and, in Chapter 5, investigate the dimensions of inhomogeneous self-affine carpets and prove that new phenomena can occur in this setting which do not occur in the setting of self-similar sets. The material in Chapter 4 will be based on a paper which appeared in 'Studia Mathematica' in 2012, and the material in Chapter 5 is based on a paper, which is in preparation. Finally, in Chapter 6 we consider random self-affine sets. The traditional approach to random iterated function systems is probabilistic, but here we allow the randomness in the construction to be provided by the topological structure of the sample space, employing ideas from Baire category. We are able to obtain very general results in this setting, relaxing the conditions on the maps from `affine' to `bi-Lipschitz'. In order to get precise results on the Hausdorff and packing measures of typical attractors, we need to specialise to the setting of random self-similar sets and we show again that several interesting and new phenomena can occur when we relax to the setting of random self-affine carpets. The material in this Chapter will be based on a paper which has been accepted for publication by 'Ergodic Theory and Dynamical Systems'.
34
Khalil, André. "Exploration of the fractal dimension." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/tape15/PQDD_0005/MQ39464.pdf.
Full text
35
Tate, Nicholas J. "The fractal dimension of topography." Thesis, University of East Anglia, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.318094.
Full text
36
Cohen, Dolav. "An exploration of fractal dimension." Kansas State University, 2013. http://hdl.handle.net/2097/16194.
Full textAbstract:
Master of ScienceDepartment of Mathematics
Hrant Hakobyan
When studying geometrical objects less regular than ordinary ones, fractal analysis becomes a valuable tool. Over the last 30 years, this small branch of mathematics has developed extensively. Fractals can be de fined as those sets which have non-integral Hausdor ff dimension. In this thesis, we take a look at some basic measure theory needed to introduce certain de finitions of fractal dimensions, which can be used to measure a set's fractal degree. We introduce Minkowski dimension and Hausdor ff dimension as well as explore some examples where they coincide. Then we look at the dimension of a measure and some very useful applications. We conclude with a well known result of Bedford and McMullen about the Hausdor ff dimension of self-a ffine sets.
37
Florindo, João Batista. "Descritores fractais aplicados à análise de texturas." Universidade de São Paulo, 2013. http://www.teses.usp.br/teses/disponiveis/76/76132/tde-02052013-161100/.
Full textAbstract:
Este projeto descreve o desenvolvimento, estudo e aplicação de descritores fractais em análise de texturas. Nos últimos anos, a literatura vem apresentando a geometria fractal como uma ferramenta poderosa para a análise de imagens, com aplicações em variados campos da ciência. A maior parte destes trabalhos faz uso direto da dimensão fractal como um descritor do objeto representado na imagem. Entretanto, em função da complexidade de muitos problemas nesta área, algumas soluções foram propostas para melhorar essa análise, usando não apenas o valor da dimensão fractal, mas um conjunto de medidas que pudessem ser extraídas pela geometria fractal e que descrevessem as texturas com maior riqueza e precisão. Entre essas técnicas, destacam-se a metodologia de multifractais, de dimensão fractal multiescala e, mais recentemente, os descritores fractais. Esta última técnica tem se mostrado eficiente na solução de problemas relacionados à discriminação de imagens de texturas e formas, uma vez que os descritores gerados fornecem uma representação direta do padrão de complexidade (distribuição dos detalhes ao longo das escalas de observação) da imagem. Assim, essa solução permite que se tenha uma descrição rica da imagem estudada pela análise da distribuição espacial e/ou espectral dos pixels e intensidade de cores/tons de cinza, com uma modelagem que pode se aproximar da percepção visual humana para a geração de um método automático e preciso. Ocorre, entretanto, que os trabalhos apresentados até o momento sobre descritores fractais focam em métodos de estimativa de dimensão fractal mais conhecidos como Bouligand-Minkowski e Box-counting. Este projeto visa estudar mais a fundo o conceito, generalizando para outras abordagens de dimensão fractal, bem como explorando diferentes formas de se extraírem os descritores a partir da curva logarítmica associada à dimensão. Os métodos desenvolvidos são aplicados à análise de texturas, em problemas de classificação de bases públicas, cujos resultados podem ser comparados com métodos da literatura, bem como a segmentação de imagens de satélite e à identificação automática de amostras obtidas em estudos de nanotecnologia. Os resultados alcançados demonstram o potencial da metodologia desenvolvida para a solução destes problemas, mostrando tratar-se de uma nova fronteira a ser usada e explorada em análise de imagens e visão computacional como um todo.This project describes the development, study and application of fractal descriptors to texture analysis. Recently, the literature has shown fractal geometry as a powerful tool for image analysis, with applications to several areas of science. Most of these works use fractal dimension as a descriptor of the object depicted in the image. However, due to the complexity of many problems in this context, some solutions have been proposed to improve this analysis. These proposed methods use not only the value of fractal dimension, but a set of measures which could be extracted by fractal geometry to describe the textures with greater richness and accuracy. Among such techniques, we emphasize the multifractal methodology, multiscale fractal dimension and, more recently, fractal descriptors. This latter technique has demonstrated to be efficient in solving problems related to the discrimination of texture and shape images. This is possible as the extracted descriptors provide a direct representation of the complexity (the details distribution along the scales of observation) in the image. Thus, this solution allows for a rich description of the image studied by analyzing the spatial/spectral distribution of pixels and intensity of colors/gray-levels, with a model which can approximate the human visual perception, generating an automatic and precise method. However, the works about fractal descriptors presented in the literature focus on classical methods to estimate fractal dimension, such as Bouligand-Minkowski and Box-counting. This project aims at studying more deeply the concept, generalizing to other approaches in fractal dimension, as well as exploring different ways of extracting the key features from the logarithmic curve associated with the dimension. The developed methods are applied to texture analysis, in classification problems over public databases, whose results can be compared with literature methods, as well as to the segmentation of satellite images and automatically identifying samples obtained from studies on nanotechnology. The results demonstrate the potential of the methodology developed to solve such problems, showing that this is a new frontier to be explored and used in image analysis and computer vision at all.
38
Morse, D. R. "Fractals in ecology : The effect of the fractal dimension of trees on the body length distribution of arboreal arthropods." Thesis, University of York, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.234971.
Full text
39
Martins, Bruna de Castro Lobo Sousa. "Os fractais no urbanismo." Master's thesis, Universidade de Lisboa. Faculdade de Arquitetura, 2014. http://hdl.handle.net/10400.5/12226.
Full textAbstract:
Dissertação para obtenção do grau de Mestre em Arquitectura com Especialização em Urbanismo, apresentada na Universidade de Lisboa - Faculdade de Arquitectura.O objetivo desta dissertação é o de, recorrendo à geometria fractal e às suas ferramentas de cálculo, comparar malhas urbanas que definem a cidade de Lisboa, relativamente, aos espaços que a compõem – construídos, privados e públicos – com o propósito de compreender como as dimensões fractais variam e se essas variações refletem o tipo de vivência característico de cada zona. Para cumprir este objetivo apresentam-se cinco capítulos. No primeiro, faz-se uma breve apresentação dos fundamentos da geometria euclidiana e das geometrias não euclidianas, bem como da geometria fractal, explicando em que consiste, como se caracteriza, como se medem os objetos fractais e como estes se aplicam na natureza, arquitetura e urbanismo. O segundo, corresponde à análise de três trabalhos práticos que servem de exemplo de como a geometria fractal é aplicada ao urbanismo. O terceiro, que consiste na apresentação do conceito de Dimensão Fractal, da Dimensão de Hausdorff e do método de Contagem de Quadrículas (Box-Counting), dos programas de cálculo testados e, de alguns conceitos estatísticos relevantes para o trabalho. O quarto, é composto pelos casos de estudo, bem como as análises comparativas feitas entre eles. O quinto, no qual se apresentam as conclusões finais e perspetivas de desenvolvimento futuro.
ABSTRACT: The objective of this dissertation is to, using fractal geometry and its calculation tools, compare urban meshes which define the city of Lisbon, according to the spaces that compose them – built, private and public – with the purpose of understanding how the fractal dimensions vary and how those variations reflect the type of life which characterizes each zone. To achieve this goal five chapters are presented. The first, consisting of a brief presentation of the fundamentals of Euclidian geometry and non-euclidian geometries, as well as fractal geometry, explaining what it consists on, how it is characterized, how to measure fractal objects and how it is applied to nature, architecture and urban planning; the second, corresponding to the study of three practical papers which serve as examples on how fractal geometry is applied to urban planning; the third, consisting on the presentation of the concept of Fractal Dimension, the Hausdorff Dimension and the Box- Counting method, the calculation programs studied and, a few statistical concepts relevant for this work; the fourth, composed by the case studies, as well as the comparative analyses made between them; and, the fifth, in which the conclusions are presented along with some future development prospects for the work.
40
Pesquet-Popescu, Béatrice. "Modélisation bidimensionnelle de processus non stationnaires et application à l'étude du fond sous-marin." Cachan, Ecole normale supérieure, 1998. http://www.theses.fr/1998DENS0021.
Full textAbstract:
Dans cette thèse, nous proposons des généralisations anisotropes des champs 2D de type fractal. Tout d'abord, nous introduisons les champs 2D à accroissements stationnaires fractionnaires et nous montrons que le mouvement brownien fractionnaire appartient à cette classe de processus. L'intérêt d'une analyse multi résolution de ces champs est démontré théoriquement et sur un exemple d'application à la localisation sous-marine. Pour la modélisation de données, un moyen efficace pour caractériser les textures à accroissements stationnaires est fourni par la fonction de structure. Nous soulignons la possibilité de contrôler l'anisotropie de ces champs par le biais de cette fonction, dont nous proposons également plusieurs modèles. La fonction de structure est aussi employée pour l'interpolation des champs non stationnaires à accroissements stationnaires. Un autre aspect de ce travail concerne les extensions bidimensionnelles des processus ARIMA fractionnaires et leurs liens avec les champs continus présentés. Finalement, nous considérons des processus auto-similaires non-gaussiens et étudions les statistiques de leurs coefficients d'ondelettes.
41
Brandão, Daniela Teresa Quaresma Santos. "Dimensões fractais e dimensão de correlação." Master's thesis, Universidade de Évora, 2008. http://hdl.handle.net/10174/17740.
Full textAbstract:
O objetivo deste trabalho é o estudo da dimensão fractal, nomeadamente a dimensão de Hausdorff, dimensão de capacidade e dimensão de correlação, relacionando-as e efetuando o cálculo em alguns exemplos. Sempre que se considera indispensável, são apresentadas noções introdutórias para uma melhor compreensão dos conceitos analisados. O Capítulo 2 é dedicado ao estudo da dimensão de Hausdorff, introduzindo, previamente, uma noção de medida, de Hausdorff. No Capítulo 3 analisamos a dimensão de capacidade, suas propriedades e inconvenientes, relacionando, no final, esta dimensão com a dimensão de Hausdorff. O Capítulo 4 estuda técnicas para calcular dimensões. São estudados subconjuntos de medida. Finita, sistemas de funções iteradas, conjuntos auto-semelhantes e auto-afins e dimensões de gráficos. O Capítulo 5 é dedicado à dimensão de correlação. Estuda o expoente de correlação Introduzido por Grassberger e Procaccia. São analisadas funções de dimensão 1 e no plano. Terminamos com o estudo de séries temporais de variável única. ABSTRACT: The aim of this work is the study of the fractal dimension, namely the Hausdorff dimension, the box-counting dimension and the correlation dimension, relating and computing them in some examples. Everytime it is necessary we introduce the basic concepts to a better understanding of the concepts analysed in this work. Chapter 2 is dedicated to the study of the Hausdorff dimension, introducing first the notion of Hausdorff measure. Chapter 3 is concerned with the box-counting dimension, its properties and problems. Then we relate this dimension With Hausdorff dimension studied in Chapter 2. Chapter 4 is dedicated to the techniques for calculating dimensions. We study subsets of finite measure, iterated function schemes, self-similar and self-affine sets and dimensions of graphs. Finally, in Chapter 5 we present the correlation dimension. We study the correlation exponent, introduced by Grassberger and Procaccia. We finish this Chapter with a study of single-variable time series.
42
McClure, Mark. "Fractal measures on infinite-dimensional sets /." The Ohio State University, 1994. http://rave.ohiolink.edu/etdc/view?acc_num=osu148785391310164.
Full text
43
CAMPOS, CLAUDENIZE FRANCISCA JAPIASSU. "AN ALGORITHM FOR COMPUTING IMAGE FRACTAL DIMENSION." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 1996. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=19751@1.
Full textAbstract:
PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIROCONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO
Neste trabalho, é apresentado um algoritmo eficiente de cálculo da Dimensão Fractal (DF) de imagens digitais. Este algoritmo fornece valores em toda a região teoricamente admissível (DF E [2,3]). É investigada a possibilidade de utilização deste método como uma ferramenta para identificação de falhas em tecidos. A DF caracteriza o grau de complexidade de um objeto. Esta característica têm sido usada recentemente na segmentação e classificação de texturas, na análise de formas e outros problemas. Este trabalho apresenta uma nova possibilidade de uso deste parâmetro, ainda não observado em outro trabalho. Foram realizados experimentos para verificar a eficiência do algoritmo desenvolvido: em imagens reais e sintéticas; na identificação de parâmetros de variação do cálculo; e verificação da influência da posição e da rotação do padrão da imagem na estimativa da imperfeição.
In this work an efficient algorithm for estimation of the Fractal Dimension (FD) of images is presented. At first, the approach is tested on the synthetic images. It is expected that the PD range is 2.0 – 3.0. A good method, as this approach, should reflect this desirable feature. The utilization of such algorithm on textile imperfection identification is investigated. The FD is a feature proposed recently to characterize roughness, self-similiarity and the complexity degree in a picture. This characteristic has been used in textures segmentation and classification, shape analysis and other problems. However, its utilization on image change characterization is a new feacture. Experiments has been done, not only on synthetic images, but also on real textile. The relation of a picture scanned at various different orientation and relative rotation of digital images are also discussed.
44
Araújo, Anderson Tadeu Gonçalves de. "Noções de geometria fractal elementar." Mestrado Profissional em Matemática, 2014. http://ri.ufs.br/jspui/handle/riufs/7383.
Full textAbstract:
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPESIn this work we present some of the main elemental fractals, highlighting some Math patterns and their autosimilarities. We make suggestions of activities that can be applied in the classroom of Elementary and / or High School in order to awaken the interest of students and teachers for Math, showing its applicability in the day to day, in addition to providing students with the creation and elaboration of concepts from a di erent view of the traditional one. In addition to this presentation, we analyzed basic mathematical tools studied on the Cartesian plane and used linear algebra in order to understand the initial concepts necessary for Elemental Fractal Geometry. Finally, we have developed a brief study about one of the fundamental characteristics of a fractal, the dimension of elemental fractals.
Neste trabalho apresentamos alguns dos principais fractais elementares, ressaltando alguns padrões matemáticos e suas autossimilaridades. Fazemos sugestões de atividades que podem ser aplicadas em sala de aula do Ensino Fundamental e/ou Ensino Médio com nalidade de despertar o interesse de alunos e professores pela matemática, evidenciando sua aplicabilidade no dia-a-dia, além de proporcionar aos alunos a criação e elaboração de conceitos a partir de uma visão diferente da tradicional. Além dessa apresentação, analisamos ferramentas matemáticas básicas estudadas no plano cartesiano e recorremos à álgebra linear a m de compreender conceitos iniciais necessários à Geometria Fractal elementar. Por m, desenvolvemos um breve estudo sobre uma das características fundamentais que um fractal possui, a dimensão de fractais elementares.
São Cristóvão, SE
45
Iwai, Marceli Megumi Hamazi. "Geometria fractal." reponame:Repositório Institucional da UFABC, 2015.
Find full text
46
Kaiser, Tashniba. "Node Localization using Fractal Signal Preprocessing and Artificial Neural Network." WorldComp, International Conference on Security and Management, 2012, 2012. http://hdl.handle.net/1993/22730.
Full textAbstract:
This thesis proposes an integrated artificial neural network based approach to classify the position of a wireless device in an indoor protected area. Our experiments are conducted in two different types of interference affected indoor locations. We found that the environment greatly influences the received signal strength. We realized the need of incorporating a complexity measure of the Wi-Fi signal as additional information in our localization algorithm.
The inputs to the integrated artificial neural network were comprised of an integer dimension representation and a fractional dimension representation of the Wi-Fi signal. The integer dimension representation consisted of the raw signal strength, whereas the fractional dimension consisted of a variance fractal dimension of the Wi-Fi signal.
The results show that the proposed approach performed 8.7% better classification than the “one dimensional input” ANN approach, achieving an 86% correct classification rate. The conventional Trilateration method achieved only a 47.97% correct classification rate.
47
Paxitzis, James T. Jr. "Entropy and Fractal Dimension of Swallow Acceleration Signals." University of Akron / OhioLINK, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=akron1313410479.
Full text
48
Wang, Nancy. "Fractal Sets: Dynamical, Dimensional and Topological Properties." Thesis, KTH, Skolan för teknikvetenskap (SCI), 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-233147.
Full textAbstract:
Fractals is a relatively new mathematical topic which received thorough treatment only starting with 1960's. Fractals can be observed everywhere in nature and in day-to-day life. To give a few examples, common fractals are the spiral cactus, the romanesco broccoli, human brain and the outline of the Swedish map. Fractal dimension is a dimension which need not take integer values. In fractal geometry, a fractal dimension is a ratio providing an index of the complexity of fractal pattern with regard to how the local geometry changes with the scale at which it is measured. In recent years, fractal analysis is used increasingly in many areas of engineering and technology. Among others, fractal analysis is used in signal and image compression, computer and video design, neuroscience and fractal based cancer modelling and diagnosing. This study consists of two main parts. The first part of the study aims to understand the appearance of an irregular Cantor set generated by the chaotic dynamical system generated by the logistic function on the unit interval [0,1]. In order to understand this irregular Cantor set, we studied the topological properties of the Cantor Middle-thirds set and the generalised Cantor sets, all of which have zero length. The necessity to compare these sets with regard to their size led us to the second part of this paper, namely the dimension studies of fractals. More complex fractals were presented in the second part, three definitions of dimension were introduced. The fractal dimension of the irregular Cantor set generated by the logistic mapping was estimated and we found that the Hausdorff dimension has the widest scope and greatest flexibility in the fractal studies.Fraktaler är ett relativt nytt ämne inom matematik som fick sitt stora genomslag först efter 60-talet. En fraktal är ett självliknande mönster med struktur i alla skalor. Några vardagliga exempel på fraktaler är spiralkaktus, romanescobroccoli, mänskliga hjärnan, blodkärlen och Sveriges fastlandskust. Bråktalsdimension är en typ av dimension där dimensionsindexet tillåts att anta alla icke-negativa reella tal. Inom fraktalgeometri kan dimensionsindexet betraktas som ett komplexitetsindex av mönstret med avseende på hur den lokala geometrin förändras beroende på vilken skala mönstret betraktas i. Under det senaste decenniet har fraktalanalysen använts alltmer flitigt inom tekniska och vetenskapliga tillämpningar. Bland annat har fraktalanalysen använts i signal- och bildkompression, dator- och videoformgivning, neurovetenskap och fraktalbaserad cancerdiagnos. Denna studie består av två huvuddelar. Den första delen fokuserar på att förstår hur en fraktal kan uppstå i ett kaotiskt dynamiskt system. För att vara mer specifik studerades den logistiska funktionen och hur denna ickelinjära avbildning genererar en oregelbunden Cantormängd på intervalet [0,1]. Vidare, för att förstå den oregelbundna Cantormängden studerades Cantormängden (eng. the Cantor Middle-Thirds set) och de generaliserade Cantormängderna, vilka alla har noll längd. För att kunna jämföra de olika Cantormängderna med avseende på storlek, leds denna studie vidare till dimensionsanalys av fraktaler som är huvudämnet i den andra delen av denna studie. Olika topologiska fraktaler presenterades, tre olika definitioner av dimension introducerades, bland annat lådräkningsdimensionen och Hausdorffdimensionen. Slutligen approximerades dimensionen av den oregelbundna Cantormängden med hjälp av Hausdorffdimensionen. Denna studie demonstrerar att Hausdorffdimensionen har större omfattning och mer flexibilitet för fraktalstudier.
49
Cooper, Jonathan Craig. "The potential of chaos and fractal analysis in urban design." Thesis, Oxford Brookes University, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.325285.
Full text
50
Chagroune, Lakhdar. "Modélisation de l'émissivité d'une surface en utilisant une approche fractale." Vandoeuvre-les-Nancy, INPL, 1995. http://www.theses.fr/1995INPL115N.
Full textAbstract:
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