Relevant bibliographies by topics / Fractals

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  2. Dissertations / Theses
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Academic literature on the topic 'Fractals'

Author: Grafiati
Published: 4 June 2021
Last updated: 1 August 2024

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Journal articles on the topic "Fractals"

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MITINA, OLGA V., and FREDERICK DAVID ABRAHAM. "THE USE OF FRACTALS FOR THE STUDY OF THE PSYCHOLOGY OF PERCEPTION: PSYCHOPHYSICS AND PERSONALITY FACTORS, A BRIEF REPORT." International Journal of Modern Physics C 14, no. 08 (October 2003): 1047–60. http://dx.doi.org/10.1142/s0129183103005182.

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The present article deals with perception of time (subjective assessment of temporal intervals), complexity and aesthetic attractiveness of visual objects. The experimental research for construction of functional relations between objective parameters of fractals' complexity (fractal dimension and Lyapunov exponent) and subjective perception of their complexity was conducted. As stimulus material we used the program based on Sprott's algorithms for the generation of fractals and the calculation of their mathematical characteristics. For the research 20 fractals were selected which had different fractal dimensions that varied from 0.52 to 2.36, and the Lyapunov exponent from 0.01 to 0.22. We conducted two experiments: (1) A total of 20 fractals were shown to 93 participants. The fractals were displayed on the screen of a computer for randomly chosen time intervals ranging from 5 to 20 s. For each fractal displayed, the participant responded with a rating of the complexity and attractiveness of the fractal using ten-point scale with an estimate of the duration of the presentation of the stimulus. Each participant also answered the questions of some personality tests (Cattell and others). The main purpose of this experiment was the analysis of the correlation between personal characteristics and subjective perception of complexity, attractiveness, and duration of fractal's presentation. (2) The same 20 fractals were shown to 47 participants as they were forming on the screen of the computer for a fixed interval. Participants also estimated subjective complexity and attractiveness of fractals. The hypothesis on the applicability of the Weber–Fechner law for the perception of time, complexity and subjective attractiveness was confirmed for measures of dynamical properties of fractal images.
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Жихарев, Л., and L. Zhikharev. "Fractals In Three-Dimensional Space. I-Fractals." Geometry & Graphics 5, no. 3 (September 28, 2017): 51–66. http://dx.doi.org/10.12737/article_59bfa55ec01b38.55497926.

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It has long been known that there are fractals, which construction resolve into cutting out of elements from lines, curves or geometric shapes according to a certain law. If the fractal is completely self-similar, its dimensionality is reduced relative to the original object and usually becomes fractional. The whole fractal is often decomposing into a set of separate elements, organized in the space of corresponding dimension. German mathematician Georg Cantor was among the first to propose such fractal set in the late 19th century. Later in the early 20th century polish mathematician Vaclav Sierpinski described the Sierpinski carpet – one of the variants for the Cantor set generalization onto a two-dimensional space. At a later date the Austrian Karl Menger created a three-dimensional analogue of the Sierpinski fractal. Similar sets differ in a number of parameters from other fractals, and therefore must be considered separately. In this paper it has been proposed to call these fractals as i-fractals (from the Latin interfican – cut). The emphasis is on the three-dimensional i-fractals, created based on the Cantor and Sierpinski principles and other fractal dependencies. Mathematics of spatial fractal sets is very difficult to understand, therefore, were used computer models developed in the three-dimensional modeling software SolidWorks and COMPASS, the obtained data were processing using mathematical programs. Using fractal principles it is possible to create a large number of i-fractals’ three dimensional models therefore important research objectives include such objects’ classification development. In addition, were analyzed i-fractals’ geometry features, and proposed general principles for their creation.
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Жихарев and L. Zhikharev. "Generalization to Three-Dimensional Space Fractals of Pythagoras and Koch. Part I." Geometry & Graphics 3, no. 3 (November 30, 2015): 24–37. http://dx.doi.org/10.12737/14417.

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Fractals are geometric objects, each part of which is similar to the whole object, so that if we take a part and increase its size to the size of the whole object, it would be impossible to notice a difference. In other words, fractals are sets having scale invariance. In mathematics, they are associated primarily with non-differentiable functions. The concept of "fractal" (from the Latin "Fractus" meaning «broken») had been introduced by Benoit Mandelbrot (1924–2010), French and American mathematician, physicist, and economist. Mandelbrot had found that seemingly arbitrary fluctuations in price of goods have a certain tendency to change: it turned out that daily fluctuations are symmetrical with long-term price fluctuations. In fact, Benoit Mandelbrot applied his recursive (fractal) method to solve the problem. Since the last quarter of the nineteenth century, a large number of fractal curves and flat objects have been created; and methods for their application have been developed. From geometrical point of view, the most interesting fractals are "Koch snowflake" and "Pythagoras Tree". Two classes of analogues of the volumetric fractals were created with modern three-dimensional modeling program: "Fractals of growth” – like Pythagoras Tree, “Fractals of separation” – like Koch snowflake; the primary classification was developed, their properties were studied. Empiric data was processed with basic arithmetic calculations as well as with computer software. Among other things, for fractals of separation the task was to create an object with an infinite surface area, which in the future might acquire great importance for the development of the chemical and other industries.
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Husain, Akhlaq, Manikyala Navaneeth Nanda, Movva Sitaram Chowdary, and Mohammad Sajid. "Fractals: An Eclectic Survey, Part II." Fractal and Fractional 6, no. 7 (July 2, 2022): 379. http://dx.doi.org/10.3390/fractalfract6070379.

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Fractals are geometric shapes and patterns that can describe the roughness (or irregularity) present in almost every object in nature. Many fractals may repeat their geometry at smaller or larger scales. This paper is the second (and last) part of a series of two papers dedicated to an eclectic survey of fractals describing the infinite complexity and amazing beauty of fractals from historical, theoretical, mathematical, aesthetical and technological aspects, including their diverse applications in various fields. In this article, our focus is on engineering, industrial, commercial and futuristic applications of fractals, whereas in the first part, we discussed the basics of fractals, mathematical description, fractal dimension and artistic applications. Among many different applications of fractals, fractal landscape generation (fractal landscapes that can simulate and describe natural terrains and landscapes more precisely by mathematical models of fractal geometry), fractal antennas (fractal-shaped antennas that are designed and used in devices which operate on multiple and wider frequency bands) and fractal image compression (a fractal-based lossy compression method for digital and natural images which uses inherent self-similarity present in an image) are the most creative, engineering-driven, industry-oriented, commercial and emerging applications. We consider each of these applications in detail along with some innovative and future ready applications.
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Cherny, A. Yu, E. M. Anitas, V. A. Osipov, and A. I. Kuklin. "Scattering from surface fractals in terms of composing mass fractals." Journal of Applied Crystallography 50, no. 3 (June 1, 2017): 919–31. http://dx.doi.org/10.1107/s1600576717005696.

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It is argued that a finite iteration of any surface fractal can be composed of mass-fractal iterations of the same fractal dimension. Within this assertion, the scattering amplitude of a surface fractal is shown to be a sum of the amplitudes of the composing mass fractals. Various approximations for the scattering intensity of surface fractals are considered. It is shown that small-angle scattering (SAS) from a surface fractal can be explained in terms of a power-law distribution of sizes of objects composing the fractal (internal polydispersity), provided the distance between objects is much larger than their size for each composing mass fractal. The power-law decay of the scattering intensityI(q) ∝ q^{D_{\rm s}-6}, where 2 <Ds< 3 is the surface-fractal dimension of the system, is realized as a non-coherent sum of scattering amplitudes of three-dimensional objects composing the fractal and obeying a power-law distribution dN(r) ∝r−τdr, withDs= τ − 1. The distribution is continuous for random fractals and discrete for deterministic fractals. A model of the surface deterministic fractal is suggested, the surface Cantor-like fractal, which is a sum of three-dimensional Cantor dusts at various iterations, and its scattering properties are studied. The present analysis allows one to extract additional information from SAS intensity for dilute aggregates of single-scaled surface fractals, such as the fractal iteration number and the scaling factor.
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Fraboni, Michael, and Trisha Moller. "Fractals in the Classroom." Mathematics Teacher 102, no. 3 (October 2008): 197–99. http://dx.doi.org/10.5951/mt.102.3.0197.

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What exactly is a fractal? Traditionally, students learn about the familiar forms of symmetry: reflection, translation, and rotation. Intuitively, fractals are symmetric with respect to magnification. A magnification of a small part of the fractal looks essentially the same as the entire picture. More formally, fractals have the property of self-similarity—that is, a fractal is any shape that is made up of smaller copies of itself. Self-similarity is what distinguishes fractals from most conventional Euclidean figures and makes them appealing. Do fractals hold the same characteristics as other Euclidean objects? Fractals offer much to explore for even very young students.
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Fraboni, Michael, and Trisha Moller. "Fractals in the Classroom." Mathematics Teacher 102, no. 3 (October 2008): 197–99. http://dx.doi.org/10.5951/mt.102.3.0197.

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What exactly is a fractal? Traditionally, students learn about the familiar forms of symmetry: reflection, translation, and rotation. Intuitively, fractals are symmetric with respect to magnification. A magnification of a small part of the fractal looks essentially the same as the entire picture. More formally, fractals have the property of self-similarity—that is, a fractal is any shape that is made up of smaller copies of itself. Self-similarity is what distinguishes fractals from most conventional Euclidean figures and makes them appealing. Do fractals hold the same characteristics as other Euclidean objects? Fractals offer much to explore for even very young students.
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Joy, Elizabeth K., and Dr Vikas Garg. "FRACTALS AND THEIR APPLICATIONS: A REVIEW." Journal of University of Shanghai for Science and Technology 23, no. 07 (August 1, 2021): 1509–17. http://dx.doi.org/10.51201/jusst/21/07277.

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In this paper, I have discussed about fractals. The two key properties of fractals have been stated. A brief history about fractals is also mentioned. I have discussed about Mandelbrot fractal and have plotted it using python. A computer-generated fern is compared to a real fern to show how much fractals resemble the real-world objects. Various applications of fractal geometry have also been included.
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Chen, Yanguang. "Fractal Modeling and Fractal Dimension Description of Urban Morphology." Entropy 22, no. 9 (August 30, 2020): 961. http://dx.doi.org/10.3390/e22090961.

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The conventional mathematical methods are based on characteristic length, while urban form has no characteristic length in many aspects. Urban area is a scale-dependence measure, which indicates the scale-free distribution of urban patterns. Thus, the urban description based on characteristic lengths should be replaced by urban characterization based on scaling. Fractal geometry is one powerful tool for the scaling analysis of cities. Fractal parameters can be defined by entropy and correlation functions. However, the question of how to understand city fractals is still pending. By means of logic deduction and ideas from fractal theory, this paper is devoted to discussing fractals and fractal dimensions of urban landscape. The main points of this work are as follows. Firstly, urban form can be treated as pre-fractals rather than real fractals, and fractal properties of cities are only valid within certain scaling ranges. Secondly, the topological dimension of city fractals based on the urban area is 0; thus, the minimum fractal dimension value of fractal cities is equal to or greater than 0. Thirdly, the fractal dimension of urban form is used to substitute the urban area, and it is better to define city fractals in a two-dimensional embedding space; thus, the maximum fractal dimension value of urban form is 2. A conclusion can be reached that urban form can be explored as fractals within certain ranges of scales and fractal geometry can be applied to the spatial analysis of the scale-free aspects of urban morphology.
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BANAKH, T., and N. NOVOSAD. "MICRO AND MACRO FRACTALS GENERATED BY MULTI-VALUED DYNAMICAL SYSTEMS." Fractals 22, no. 04 (November 12, 2014): 1450012. http://dx.doi.org/10.1142/s0218348x14500121.

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Given a multi-valued function Φ : X ⊸ X on a topological space X we study the properties of its fixed fractal[Formula: see text], which is defined as the closure of the orbit Φω(*Φ) = ⋃n∈ωΦn(*Φ) of the set *Φ = {x ∈ X : x ∈ Φ(x)} of fixed points of Φ. A special attention is paid to the duality between micro-fractals and macro-fractals, which are fixed fractals [Formula: see text] and [Formula: see text] for a contracting compact-valued function Φ : X ⊸ X on a complete metric space X. With help of algorithms (described in this paper) we generate various images of macro-fractals which are dual to some well-known micro-fractals like the fractal cross, the Sierpiński triangle, Sierpiński carpet, the Koch curve, or the fractal snowflakes. The obtained images show that macro-fractals have a large-scale fractal structure, which becomes clearly visible after a suitable zooming.
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Dissertations / Theses on the topic "Fractals"

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Moraes, Leonardo Bastos. "Antenas impressas compactas para sistemas WIMAX." Universidade de São Paulo, 2012. http://www.teses.usp.br/teses/disponiveis/3/3142/tde-26122013-161125/.

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Alcançar altas taxas de dados em comunicações sem fio é difícil. Altas taxas de dados para redes locais sem fio tornou-se comercialmente um sucesso por volta do ano de 2000. Redes de longa distância sem fio ainda são projetados e utilizados principalmente para serviços de voz em baixas taxas. Apesar de muitas tecnologias promissoras, a realidade de uma rede de área ampla que atenda muitos usuários com altas taxas de dados e largura de banda e consumo de energia razoáveis, além de uma boa cobertura e qualidade no serviço ainda é um desafio. O objetivo do IEEE 802.16 é projetar um sistema de comunicação sem fio para obter uma internet de banda larga para usuários móveis em uma área metropolitana. É importante perceber que o sistema WIMAX tem que enfrentar desafios semelhantes aos existentes sistemas celulares e seu desempenho eventual será delimitado pelas mesmas leis da física e da teoria da informação. Em muitas áreas da engenharia elétrica, tem-se direcionado atenção à miniaturização de componentes e equipamentos. Em particular, antenas não são exceções. Desde que Wheeler iniciou estudos sobre os limites fundamentais de miniaturização de antenas, o assunto tem sido discutido por muitos estudiosos e várias contribuições nesse sentido foram feitas desde então. Os avanços das últimas décadas na área de microeletrônica permitiram a miniaturização dos demais componentes empregados no desenvolvimento de equipamentos eletrônicos e disponibilizaram o uso de aparelhos compactos, leves e com diversas funcionalidades e aplicações comerciais. No entanto, ainda que a integração de circuitos seja uma realidade, a integração completa de um sistema de comunicação sem fio, incluindo a antena, é ainda um dos grandes desafios tecnológicos. No caso de antenas impressas procura-se continuamente desenvolver antenas que, além de compactas, apresentem maior largura de banda, ou operação em múltiplas bandas dada sua inerente característica de banda estreita em projetos convencionais. Neste trabalho, o foco está na miniaturização de antenas impressas através da aplicação de fractais. São apresentadas comparações entre antenas fractais quadradas de Minkowski e fractais triangulares de Koch. Inicialmente, antenas 6 impressas com geometrias convencionais quadradas e triangulares foram projetadas para ter a mesma frequência de ressonância. Depois disso, as estruturas fractais de Minkowski Island e Koch Loop foram implementadas nas antenas quadrada e triangular, respectivamente, até a terceira iteração. As frequências escolhidas foram as de 2,4 GHz, 3,5 GHz, 5,0 GHz e 5,8 GHz. Diversos protótipos foram construídos em dois substratos de permissividade diferentes, o FR-4 e o DUROID 5870. Para validar os resultados foram construídas antenas na frequência de 3,5 GHz para as geometrias quadrada e triangular e suas iterações fractais. A contribuição deste trabalho está na análise sobre as vantagens e desvantagens de cada uma das estruturas propostas. Dependendo dos requisitos de um projeto, a opção pode ser por antenas miniaturizadas com maior largura de banda, como normalmente acontece em alguns projetos comerciais. Entretanto, o interesse por bandas estreitas muitas vezes pode ser um requisito, principalmente para emprego militar, onde por vezes a máxima discrição na transmissão é uma exigência. Além disso, também foi feita uma análise sobre as geometrias que atingiram maior miniaturização.
Achieving high data rates in wireless communication is difficult. High data rates for wireless local area networks became commercially successful only around 2000. Wide area wireless networks are still designed and used primarily for low rate voice services. Despite many promising technologies, the reality of a wide area network that services many users at high data rates with reasonable bandwidth and power consumption, while maintaining high coverage and quality of service has not been achieved. The goal of the IEEE 802.16 was to design a wireless communication system processing to achieve a broadband internet for mobile users over a wide or metropolitan area. It is important to realize that WIMAX system have to confront similar challenges as existing cellular systems and their eventual performance will be bounded by the same laws of physics and information theory. In many areas of electrical engineering, miniaturization has been an important issue. Antennas are not an exception. After Wheeler initiated studies on the fundamental limits for miniaturization of antennas, this subject has been extensively discussed by several scholars and many contributions have been made. The advances of recent decades in the field of microelectronics enabled the miniaturization of components and provided the use of compact, lightweight, equipments with many features in commercial applications. Although circuit integration is a reality, the integration of a complete system, including its antenna, is still one of the major technological challenges. In the case of patch antennas, the search is for compact structures with increased bandwidth, due to the inherent narrowband characteristic of this type of antenna. In this work the focus is on a comparison between the Minkowski and the Koch Fractal Patch Antennas. Initially, patch antennas with conventional square and triangular geometries were simulated to present the same resonance frequency. After that, fractal Minkowski and Koch Island Loop antennas were implemented in the square and triangular geometries, respectively, to the third iteration. A comparison was made for two substrates of different permittivities FR-4 and DUROID 5870 at the frequencies of 2,4 GHz; 3,5 GHz; 5,0 GHz and 5,8 GHz. 8 Prototype antennas were built using FR-4 and DUROID 5870 to resonate at a frequency of 3,5 GHz to validate simulation results. The contribution of this work is the analysis of the advantages and disadvantages of each proposed fractal structure. According to the project requirements, the best option can be use a miniaturized antenna with a wider band, as in commercial projects. Particularly in military applications, a narrow band antenna can be a requirement, as sometimes maximum discretion in transmission is a paramount. An additional analysis was performed to verify which of the geometries fulfilled the miniaturization criteria of Hansen.
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Дядечко, Алла Миколаївна, Алла Николаевна Дядечко, Alla Mykolaivna Diadechko, D. Tokar, and V. R. Tarasenko. "Fractals." Thesis, Видавництво СумДУ, 2011. http://essuir.sumdu.edu.ua/handle/123456789/13436.

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Zanotto, Ricardo Anselmo. "Estudo da geometria fractal clássica." Universidade Federal de Goiás, 2015. http://repositorio.bc.ufg.br/tede/handle/tede/6058.

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Outro
This is a research about a part of the non-Euclidean geometry that has recently been very studied. It was addressed initial themes of the non-Euclidean geometry and it was exposed the studies abut fractals, its history, buildings and main fractals (known as classic fractals). It was also addressed the relation among the school years contents and how to use fractals; as well as some of its applications that have helped a lot of researches to spread and show better results.
Este trabalho é uma pesquisa sobre parte da geometria não euclidiana que há pouco vem sendo muito estudada, os fractais. Abordamos temas iniciais da geometria nãoeuclidiana e no decorrer do trabalho expomos nosso estudo sobre fractais, seu histórico, construções, principais fractais (conhecidos como fractais clássicos). Também abordamos relações entre conteúdos dos anos escolares e como usar fractais nos mesmos; como também algumas de suas aplicações que vem ajudando muitas pesquisas a se difundirem e apresentarem melhores resultados.
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LONG, LUN-HAI. "Fractals arithmetiques." Université Louis Pasteur (Strasbourg) (1971-2008), 1993. http://www.theses.fr/1993STR13249.

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Dans cette these, nous etablions d'abord un modele mathematique a partir d'un ensemble de points entiers, et d'un nombre fini de couples composes par une application strictement contractante et un ensemble borne. On applique ensuite ce modele a l'etude de la construction de la foret d'arbres. On obtient ainsi une grande classe de fractals qui contient presque tous les fractals classiques, par exemples, les ensembles a similitudes internes, les ensembles auto-affines, les ensembles recurrents de dikking, les ensembles de julia, les attracteurs, etc. Nous etudions ensuite deux familles particulieres de fractals. L'une s'inspire de l'application quasi-affine introduite par reveilles et l'autre d'une representation, analogue a celle des nombres reels dans le systeme decimal, des points du plan reel
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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.

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Fractal geometry is a new branch of mathematics. This report presents the tools, methods and theory required to describe this geometry. The power of Iterated Function Systems (IFS) is introduced and applied to produce fractal images or approximate complex estructures found in nature. The focus of this thesis is on how fractal geometry can be used in applications to computer graphics or to model natural objects.
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Mucheroni, Laís Fernandes [UNESP]. "Dimensão de Hausdorff e algumas aplicações." Universidade Estadual Paulista (UNESP), 2017. http://hdl.handle.net/11449/151653.

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Intuitivamente, um ponto tem dimensão 0, uma reta tem dimensão 1, um plano tem dimensão 2 e um cubo tem dimensão 3. Porém, na geometria fractal encontramos objetos matemáticos que possuem dimensão fracionária. Esses objetos são denominados fractais cujo nome vem do verbo "frangere", em latim, que significa quebrar, fragmentar. Neste trabalho faremos um estudo sobre o conceito de dimensão, definindo dimensão topológica e dimensão de Hausdorff. O objetivo deste trabalho é, além de apresentar as definições de dimensão, também apresentar algumas aplicações da dimensão de Hausdorff na geometria fractal.
We know, intuitively, that the dimension of a dot is 0, the dimension of a line is 1, the dimension of a square is 2 and the dimension of a cube is 3. However, in the fractal geometry we have objects with a fractional dimension. This objects are called fractals whose name comes from the verb frangere, in Latin, that means breaking, fragmenting. In this work we will study about the concept of dimension, defining topological dimension and Hausdorff dimension. The purpose of this work, besides presenting the definitions of dimension, is to show an application of the Hausdorff dimension on the fractal geometry.
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Berbiche, Amine. "Propagation d'ondes acoustiques dans les milieux poreux fractals." Thesis, Aix-Marseille, 2016. http://www.theses.fr/2016AIXM4758.

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La méthode de minimisation de l'intégrale d'action (principe variationnel) permet d’obtenir les équations de propagation des ondes. Cette méthode a été généralisée aux milieux poreux de dimensions fractales, pour étudier la propagation acoustique dans le domaine temporel, en se basant sur le modèle du fluide équivalent. L'équation obtenue réécrite dans le domaine fréquentiel représente une généralisation de l'équation d'Helmholtz. Dans le cadre du modèle d'Allard-Johnson, l'équation de propagation a été résolue de manière analytique dans le domaine temporel, dans les régimes des hautes et des basses fréquences. La résolution a été faite par la méthode de la transformée de Laplace, et a porté sur un milieu poreux semi-infini. Il a été trouvé que la vitesse de propagation dépend de la dimension fractale. Pour un matériau poreux fractal d'épaisseur finie qui reçoit une onde acoustique en incidence normale, les conditions d’Euler ont été utilisées pour déterminer les champs réfléchi et transmis. La résolution du problème direct a été faite dans le domaine temporel, par la méthode de la transformée de Laplace, et par l’usage des fonctions de Mittag-Leffler. Le problème inverse a été résolu par la méthode de minimisation aux sens des moindres carrés. Des tests ont été effectués avec succès sur des données expérimentales, en utilisant des programmes numériques développés à partir du formalisme établi dans cette thèse. La résolution du problème inverse a permis de retrouver les paramètres acoustiques de mousses poreuses, dans les régimes des hautes et des basses fréquences
The action integral minimization method (variational principle) provides the wave propagation equations. This method has been generalized to fractal dimensional porous media to study the acoustic propagation in the time domain, based on the equivalent fluid model. The resulting equation rewritten in the frequency domain represents a generalization for the Helmholtz equation. As part of the Allard-Johnson model, the propagation equation was solved analytically in the time domain, for both high and low frequencies fields. The resolution was made by the method of the Laplace transform, and focused on a semi-infinite porous medium. It was found that the wave velocity depends on the fractal dimension.For a fractal porous material of finite thickness which receives an acoustic wave at normal incidence, the Euler conditions were used to determine the reflected and transmitted fields. The resolution of the direct problem was made in the time domain by the method of the Laplace transform, and through the use of the Mittag-Leffler functions. The inverse problem was solved by the method of minimizing the least squares sense. Tests have been performed successfully on experimental data; programs written from the formalism developed in this work have allowed finding the acoustic parameters of porous foams, in the fields of high and low frequencies
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Prehl, geb Balg Janett. "Diffusion on Fractals." Master's thesis, Universitätsbibliothek Chemnitz, 2007. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200701033.

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We study anomalous diffusion on fractals with a static external field applied. We utilise the master equation to calculate particle distributions and from that important quantities as for example the mean square displacement . Applying different bias amplitudes on several regular Sierpinski carpets we obtain maximal drift velocities for weak field strengths. According to ~t^(2/d_w), we determine random walk dimensions of d_w<2 for applied external fields. These d_w corresponds to superdiffusion, although diffusion is hindered by the structure of the carpet, containing dangling ends. This seems to result from two competing effects arising within an external field. Though the particles prefer to move along the biased direction, some particles get trapped by dangling ends. To escape from there they have to move against the field direction. Due to the by the bias accelerated particles and the trapped ones the probability distribution gets wider and thus d_w<2
In dieser Arbeit untersuchen wir anomale Diffusion auf Fraktalen unter Einwirkung eines statisches äußeres Feldes. Wir benutzen die Mastergleichung, um die Wahrscheinlichkeitsverteilung der Teilchen zu berechnen, um daraus wichtige Größen wie das mittlere Abstandsquadrat zu bestimmen. Wir wenden unterschiedliche Feldstärken bei verschiedenen regelmäßigen Sierpinski-Teppichen an und erhalten maximale Driftgeschwindigkeiten für schwache Feldstärken. Über ~t^{2/d_w} bestimmen wir die Random-Walk-Dimension d_w als d_w<2. Dieser Wert für d_w entspricht der Superdiffusion, obwohl der Diffusionsprozess durch Strukturen des Teppichs, wie Sackgassen, behindert wird. Es schient, dass dies das Ergebnis zweier konkurrierender Effekte ist, die durch das Anlegen eines äußeren Feldes entstehen. Einerseits bewegen sich die Teilchen bevorzugt entlang der Feldrichtung. Andererseits gelangen einige Teilchen in Sackgassen. Um die Sackgassen, die in Feldrichtung liegen, zu verlassen, müssen sich die Teilchen entgegen der Feldrichtung bewegen. Somit sind die Teilchen eine gewisse Zeit in der Sackgasse gefangen. Infolge der durch das äußere Feld beschleunigten und der gefangenen Teilchen, verbreitert sich die Wahrscheinlichkeitsverteilung der Teilchen und somit ist d_w<2
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Yin, Qinghe. "Fractals and sumsets." Title page, contents and abstract only, 1993. http://web4.library.adelaide.edu.au/theses/09PH/09phy51.pdf.

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Beaver, Philip Frederick. "Fractals and chaos." Thesis, Monterey, California. Naval Postgraduate School, 1991. http://hdl.handle.net/10945/28232.

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Books on the topic "Fractals"

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A, Pickover Clifford, ed. Fractal horizons: The future use of fractals. New York: St. Martin's Press, 1996.

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Feder, Jens. Fractals. Boston, MA: Springer US, 1988. http://dx.doi.org/10.1007/978-1-4899-2124-6.

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Dekking, Michel, Jacques Lévy Véhel, Evelyne Lutton, and Claude Tricot, eds. Fractals. London: Springer London, 1999. http://dx.doi.org/10.1007/978-1-4471-0873-3.

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O'Connell, Richard. Fractals. Newport: Atlantis Editions, 2002.

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Stephen, Pollock, and British Broadcasting Corporation, eds. Fractals. [London]: [British Broadcasting Corporation, 1990.

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Feder, Jens. Fractals. New York: Plenum Press, 1988.

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Feder, Jens. Fractals. New York, NY: Plenum Press, 1988.

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Mac Cormac, Earl, and Maxim I. Stamenov, eds. Fractals of Brain, Fractals of Mind. Amsterdam: John Benjamins Publishing Company, 1996. http://dx.doi.org/10.1075/aicr.7.

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Barnsley, Michael. Fractals everywhere. 2nd ed. Boston: Academic Press, 1993.

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Barnsley, Michael. Fractals everywhere. Mineola, N.Y: Dover Publications, 2012.

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Book chapters on the topic "Fractals"

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Hergarten, Stefan. "Fractals and Fractal Distributions." In Self-Organized Criticality in Earth Systems, 1–24. Berlin, Heidelberg: Springer Berlin Heidelberg, 2002. http://dx.doi.org/10.1007/978-3-662-04390-5_1.

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Courtens, Eric, and René Vacher. "Fractons in Real Fractals." In Random Fluctuations and Pattern Growth: Experiments and Models, 20–26. Dordrecht: Springer Netherlands, 1988. http://dx.doi.org/10.1007/978-94-009-2653-0_4.

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Feder, Jens. "Introduction." In Fractals, 1–5. Boston, MA: Springer US, 1988. http://dx.doi.org/10.1007/978-1-4899-2124-6_1.

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Feder, Jens. "Self-Similarity and Self-Affinity." In Fractals, 184–92. Boston, MA: Springer US, 1988. http://dx.doi.org/10.1007/978-1-4899-2124-6_10.

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Feder, Jens. "Wave-Height Statistics." In Fractals, 193–99. Boston, MA: Springer US, 1988. http://dx.doi.org/10.1007/978-1-4899-2124-6_11.

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Feder, Jens. "The Perimeter-Area Relation." In Fractals, 200–211. Boston, MA: Springer US, 1988. http://dx.doi.org/10.1007/978-1-4899-2124-6_12.

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Feder, Jens. "Fractal Surfaces." In Fractals, 212–28. Boston, MA: Springer US, 1988. http://dx.doi.org/10.1007/978-1-4899-2124-6_13.

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Feder, Jens. "Observations of Fractal Surfaces." In Fractals, 229–43. Boston, MA: Springer US, 1988. http://dx.doi.org/10.1007/978-1-4899-2124-6_14.

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Feder, Jens. "The Fractal Dimension." In Fractals, 6–30. Boston, MA: Springer US, 1988. http://dx.doi.org/10.1007/978-1-4899-2124-6_2.

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Feder, Jens. "The Cluster Fractal Dimension." In Fractals, 31–40. Boston, MA: Springer US, 1988. http://dx.doi.org/10.1007/978-1-4899-2124-6_3.

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Conference papers on the topic "Fractals"

1

Wang, Yan. "3D Fractals From Periodic Surfaces." In ASME 2010 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2010. http://dx.doi.org/10.1115/detc2010-29081.

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Fractals are ubiquitous as in natural objects and have been applied in designing porous structures such as micro antenna and porous silicon. The seemingly complex and irregular structures can be generated based on simple principles. In this paper, we present three approaches to construct 3D fractal geometries using a recently proposed periodic surface model. By applying iterated function systems to the implicit surface model in the Euclidean or parameter space, 3D fractals can be constructed efficiently. Porosity is also proposed as a metric in fractal design.
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"BACK MATTER." In Fractals 2006. WORLD SCIENTIFIC, 2006. http://dx.doi.org/10.1142/9789812774217_bmatter.

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WEST, BRUCE J. "MODELING FRACTAL DYNAMICS." In Fractals 2006. WORLD SCIENTIFIC, 2006. http://dx.doi.org/10.1142/9789812774217_0002.

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MAINZER, KLAUS. "COMPLEXITY IN NATURE AND SOCIETY: Complexity Management in the Age of Globalization." In Fractals 2006. WORLD SCIENTIFIC, 2006. http://dx.doi.org/10.1142/9789812774217_0010.

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PEARSON, MICHAEL. "FRACTALS, COMPLEXITY AND CHAOS IN SUPPLY CHAIN NETWORKS." In Fractals 2006. WORLD SCIENTIFIC, 2006. http://dx.doi.org/10.1142/9789812774217_0011.

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SING, BERND. "ITERATED FUNCTION SYSTEMS IN MIXED EUCLIDEAN AND 𝔭-ADIC SPACES." In Fractals 2006. WORLD SCIENTIFIC, 2006. http://dx.doi.org/10.1142/9789812774217_0024.

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LIEBOVITCH, L. S., V. K. JIRSA, and L. A. SHEHADEH. "STRUCTURE OF GENETIC REGULATORY NETWORKS: EVIDENCE FOR SCALE FREE NETWORKS." In Fractals 2006. WORLD SCIENTIFIC, 2006. http://dx.doi.org/10.1142/9789812774217_0001.

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GORENFLO, RUDOLF, and FRANCESCO MAINARDI. "FRACTIONAL RELAXATION OF DISTRIBUTED ORDER." In Fractals 2006. WORLD SCIENTIFIC, 2006. http://dx.doi.org/10.1142/9789812774217_0003.

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ALLEGRINI, P., F. BARBI, P. GRIGOLINI, and P. PARADISI. "FRACTIONAL TIME: DISHOMOGENOUS POISSON PROCESSES VS. HOMOGENEOUS NON-POISSON PROCESSES." In Fractals 2006. WORLD SCIENTIFIC, 2006. http://dx.doi.org/10.1142/9789812774217_0004.

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PAPASIMAKIS, NIKITAS, and FOTINI PALLIKARI. "MARKOV MEMORY IN MULTIFRACTAL NATURAL PROCESSES." In Fractals 2006. WORLD SCIENTIFIC, 2006. http://dx.doi.org/10.1142/9789812774217_0005.

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Reports on the topic "Fractals"

1

Haussermann, John W. An Introduction to Fractals and Chaos. Fort Belvoir, VA: Defense Technical Information Center, June 1989. http://dx.doi.org/10.21236/ada210257.

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Driscoll, John. Fractals as Basis for Design and Critique. Portland State University Library, January 2000. http://dx.doi.org/10.15760/etd.7059.

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Moore, Charles. A Quantitative Description of Soil Microstructure Using Fractals. Fort Belvoir, VA: Defense Technical Information Center, July 1992. http://dx.doi.org/10.21236/ada337825.

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Kostoff, Ronald N., Dustin Johnson, J. A. Del Rio, Louis A. Bloomfield, Michael F. Shlesinger, and Guido Malpohl. Duplicate Publication and 'Paper Inflation' in the Fractals Literature. Fort Belvoir, VA: Defense Technical Information Center, January 2005. http://dx.doi.org/10.21236/ada440622.

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Oppenheim, Alan V., and Gregory W. Wornell. Signal Analysis, Synthesis and Processing Using Fractals and Wavelets. Fort Belvoir, VA: Defense Technical Information Center, November 1995. http://dx.doi.org/10.21236/ada305490.

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Rao, C. R., and S. R. Kumara. Shape and Image Analysis using Neural Networks Fractals and Wavelets. Fort Belvoir, VA: Defense Technical Information Center, May 2000. http://dx.doi.org/10.21236/ada392772.

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Yortsos, Y. C., and J. A. Acuna. Numerical construction and flow simulation in networks of fractures using fractals. Office of Scientific and Technical Information (OSTI), November 1991. http://dx.doi.org/10.2172/6283188.

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Pardo Igúzquiza, Eulogio. Karst y fractales. Ilustre Colegio Oficial de Geólogos, December 2022. http://dx.doi.org/10.21028/eog.2022.12.05.

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¿Qué tienen en común la distribución de galaxias en el universo, la red del metro de Madrid y la estrategia de caza de la tribu de los Hadza en el norte de Tanzania? La respuesta es que las tres están conectadas con el karst, tal y como se describe en este trabajo, a través del carácter fractal del mismo. En efecto, el karst presenta un comportamiento fractal tanto en superficie como en el subsuelo. En superficie, tanto la topografía kárstica como las depresiones cerradas (dolinas) que caracterizan el paisaje kárstico son fractales. En el karst subterráneo, la red de conductos kársticos (cuevas) presenta asimismo un comportamiento fractal.
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Aminzadeh, Fred, Charles Sammis, Mohammad Sahimi, and David Okaya. Characterizing Fractures in Geysers Geothermal Field by Micro-seismic Data, Using Soft Computing, Fractals, and Shear Wave Anisotropy. Office of Scientific and Technical Information (OSTI), April 2015. http://dx.doi.org/10.2172/1185274.

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Fisher, Yuval, and Albert Lawrence. Fractal Image Encoding. Fort Belvoir, VA: Defense Technical Information Center, March 1992. http://dx.doi.org/10.21236/ada248003.

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