Relevant bibliographies by topics / Fractal dimensions

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Academic literature on the topic 'Fractal dimensions'

Author: Grafiati
Published: 4 June 2021
Last updated: 3 February 2022

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Journal articles on the topic "Fractal dimensions"

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Chen, Yanguang. "Characterizing Growth and Form of Fractal Cities with Allometric Scaling Exponents." Discrete Dynamics in Nature and Society 2010 (2010): 1–22. http://dx.doi.org/10.1155/2010/194715.

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Fractal growth is a kind of allometric growth, and the allometric scaling exponents can be employed to describe growing fractal phenomena such as cities. The spatial features of the regular fractals can be characterized by fractal dimension. However, for the real systems with statistical fractality, it is incomplete to measure the structure of scaling invariance only by fractal dimension. Sometimes, we need to know the ratio of different dimensions rather than the fractal dimensions themselves. A fractal-dimension ratio can make an allometric scaling exponent (ASE). As compared with fractal dimension, ASEs have three advantages. First, the values of ASEs are easy to be estimated in practice; second, ASEs can reflect the dynamical characters of system's evolution; third, the analysis of ASEs can be made through prefractal structure with limited scale. Therefore, the ASEs based on fractal dimensions are more functional than fractal dimensions for real fractal systems. In this paper, the definition and calculation method of ASEs are illustrated by starting from mathematical fractals, and, then, China's cities are taken as examples to show how to apply ASEs to depiction of growth and form of fractal cities.
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CHEN, YAN-GUANG. "FRACTAL TEXTURE AND STRUCTURE OF CENTRAL PLACE SYSTEMS." Fractals 28, no. 01 (February 2020): 2050008. http://dx.doi.org/10.1142/s0218348x20500085.

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The boundaries of central place models proved to be fractal lines, which compose fractal texture of central place networks. However, the fractal texture cannot be verified by empirical analyses based on observed data. On the other hand, fractal structure of central place systems in the real world can be empirically confirmed by positive studies, but there are no corresponding models. The spatial structure of classic central place models bears Euclidean dimension [Formula: see text] rather than fractal dimensions [Formula: see text]. This paper is devoted to deriving structural fractals of central place models from the textural fractals. The method is theoretical deduction based on the dimension rules of fractal sets. The main results and findings are as follows. First, the central place fractals were formulated by the [Formula: see text] numbers and [Formula: see text] numbers. Second, three structural fractal models were constructed for central place systems according to the corresponding fractal dimensions. Third, the classic central place models proved to comprise Koch snowflake curve, Sierpinski space filling curve, and Gosper snowflake curve. Moreover, the traffic principle plays a leading role in urban and rural settlements evolution. A conclusion was reached that the textural fractal dimensions of central place models can be converted into the structural fractal dimensions and vice versa, and the structural dimensions can be directly used to appraise human settlement distributions in reality. Thus, the textural fractals can be indirectly employed to characterize the systems of human settlements.
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LIAW, SY-SANG, and FENG-YUAN CHIU. "CONSTRUCTING CROSSOVER-FRACTALS USING INTRINSIC MODE FUNCTIONS." Advances in Adaptive Data Analysis 02, no. 04 (October 2010): 509–20. http://dx.doi.org/10.1142/s1793536910000598.

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Real nonstationary time sequences are in general not monofractals. That is, they cannot be characterized by a single value of fractal dimension. It has been shown that many real-time sequences are crossover-fractals: sequences with two fractal dimensions — one for the short and the other for long ranges. Here, we use the empirical mode decomposition (EMD) to decompose monofractals into several intrinsic mode functions (IMFs) and then use partial sums of the IMFs decomposed from two monofractals to construct crossover-fractals. The scale-dependent fractal dimensions of these crossover-fractals are checked by the inverse random midpoint displacement method (IRMD).
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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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Iversen, P. O., and G. Nicolaysen. "High correlation of fractals for regional blood flows among resting and exercising skeletal muscles." American Journal of Physiology-Heart and Circulatory Physiology 269, no. 1 (July 1, 1995): H7—H13. http://dx.doi.org/10.1152/ajpheart.1995.269.1.h7.

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The regional blood flow distributions within single skeletal muscles are markedly uneven both at rest and during exercise hyperemia. Fractals adequately describe this perfusion heterogeneity in the resting lateral head of the gastrocnemius muscle as well as in the myocardium. Recently, we provided evidence that the fractal dimension for the blood flow distributions in this resting muscle was strongly correlated with that of the myocardium in the same rabbit. Prompted by this hitherto unknown observation, we have now examined 1) whether fractals also describe perfusion distributions within muscles with a varying metabolic activity, and 2) whether the fractal dimensions for blood flow distributions to these muscles were correlated. We used pentobarbital-anesthetized rabbits and cats. The regional distributions of blood flow within various skeletal muscles were estimated by microsphere trapping. The data unequivocally showed that the perfusion distributions could be described with fractals both in resting and in exercising muscle in both species, the corresponding fractal dimensions ranging from 1.36 to 1.41. The fractal dimensions were markedly correlated (r2 ranged from 0.82 to 0.88) when both various resting and resting plus exercising muscles were compared in the same animal. This surprising finding of high correlations for the fractal dimensions among various muscles within one animal provides a novel characteristic of blood flow heterogeneity.
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Sreenivasan, K. R., and C. Meneveau. "The fractal facets of turbulence." Journal of Fluid Mechanics 173 (December 1986): 357–86. http://dx.doi.org/10.1017/s0022112086001209.

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Speculations abound that several facets of fully developed turbulent flows are fractals. Although the earlier leading work of Mandelbrot (1974, 1975) suggests that these speculations, initiated largely by himself, are plausible, no effort has yet been made to put them on firmer ground by, resorting to actual measurements in turbulent shear flows. This work is an attempt at filling this gap. In particular, we examine the following questions: (a) Is the turbulent/non-turbulent interface a self-similar fractal, and (if so) what is its fractal dimension ? Does this quantity differ from one class of flows to another? (b) Are constant-property surfaces (such as the iso-velocity and iso-concentration surfaces) in fully developed flows fractals? What are their fractal dimensions? (c) Do dissipative structures in fully developed turbulence form a fractal set? What is the fractal dimension of this set? Answers to these questions (and others to be less fully discussed here) are interesting because they bring the theory of fractals closer to application to turbulence and shed new light on some classical problems in turbulence - for example, the growth of material lines in a turbulent environment. The other feature of this work is that it tries to quantify the seemingly complicated geometric aspects of turbulent flows, a feature that has not received its proper share of attention. The overwhelming conclusion of this work is that several aspects of turbulence can be described roughly by fractals, and that their fractal dimensions can be measured. However, it is not clear how (or whether), given the dimensions for several of its facets, one can solve (up to a useful accuracy) the inverse problem of reconstructing the original set (that is, the turbulent flow itself).
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Lutz, Neil. "Fractal Intersections and Products via Algorithmic Dimension." ACM Transactions on Computation Theory 13, no. 3 (September 30, 2021): 1–15. http://dx.doi.org/10.1145/3460948.

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Algorithmic fractal dimensions quantify the algorithmic information density of individual points and may be defined in terms of Kolmogorov complexity. This work uses these dimensions to bound the classical Hausdorff and packing dimensions of intersections and Cartesian products of fractals in Euclidean spaces. This approach shows that two prominent, fundamental results about the dimension of Borel or analytic sets also hold for arbitrary sets.
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MCGINLEY, PATTON, ROBIN G. SMITH, and JEROME C. LANDRY. "FRACTAL DIMENSIONS OF MYCOSIS FUNGOIDES." Fractals 02, no. 04 (December 1994): 493–501. http://dx.doi.org/10.1142/s0218348x94000715.

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Prior to an investigation of early diagnosis of mycosis fungoides (MF) using fractal geometry, we set out to see if MF lesions are fractal in nature. We analyzed three aspects of MF lesions: the dermoepidermal profile of photomicrographs of patch stage lesions and normal skin, the perimeter of patch and plaque stage lesions, and the size distribution of patch and plaque lesions on the skin surface. The perimeter of plaque lesions was measured on close-up photographs by the divider walk method using various step sizes. Based on the perimeter values, the fractal dimension was determined. The dermoepidermal profile of MF patch lesions was analyzed by the divider walk method for self-affine fractals. The size distribution of MF patch and plaque lesions was determined by counting the number of patch and plaque lesions with an area greater than or equal to a specific size A on scaled photographs of a 19.6 cm × 19.6 cm affected region. A plot of number of lesions with area greater than or equal to A vs. lesion area on log-log paper allows the detection of a power-law distribution, indicative of one type of self-similar fractals. The dermoepidermal profile of patch stage lesions and normal skin was found to be self-affine fractals. Global measurements of normal thin skin and of patch stage lesions were distinct. All observed patch and plaque lesion area distributions were a fractal set. The perimeter of non-confluent plaque lesions was not fractal. This work revealed fractal dimensions in two aspects of MF lesions. Further investigation of application of fractal geometry to the diagnosis and staging of MF is planned.
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Zhang, Pei-Lin, Bing Li, Shuang-Shan Mi, Ying-Tang Zhang, and Dong-Sheng Liu. "Bearing Fault Detection Using Multi-Scale Fractal Dimensions Based on Morphological Covers." Shock and Vibration 19, no. 6 (2012): 1373–83. http://dx.doi.org/10.1155/2012/438789.

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Vibration signals acquired from bearing have been found to demonstrate complicated nonlinear characteristics in literature. Fractal geometry theory has provided effective tools such as fractal dimension for characterizing the vibration signals in bearing faults detection. However, most of the natural signals are not critical self-similar fractals; the assumption of a constant fractal dimension at all scales may not be true. Motivated by this fact, this work explores the application of the multi-scale fractal dimensions (MFDs) based on morphological cover (MC) technique for bearing fault diagnosis. Vibration signals from bearing with seven different states under four operations conditions are collected to validate the presented MFDs based on MC technique. Experimental results reveal that the vibration signals acquired from bearing are not critical self-similar fractals. The MFDs can provide more discriminative information about the signals than the single global fractal dimension. Furthermore, three classifiers are employed to evaluate and compare the classification performance of the MFDs with other feature extraction methods. Experimental results demonstrate the MFDs to be a desirable approach to improve the performance of bearing fault diagnosis.
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XIA, YUXUAN, JIANCHAO CAI, WEI WEI, XIANGYUN HU, XIN WANG, and XINMIN GE. "A NEW METHOD FOR CALCULATING FRACTAL DIMENSIONS OF POROUS MEDIA BASED ON PORE SIZE DISTRIBUTION." Fractals 26, no. 01 (February 2018): 1850006. http://dx.doi.org/10.1142/s0218348x18500068.

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Fractal theory has been widely used in petrophysical properties of porous rocks over several decades and determination of fractal dimensions is always the focus of researches and applications by means of fractal-based methods. In this work, a new method for calculating pore space fractal dimension and tortuosity fractal dimension of porous media is derived based on fractal capillary model assumption. The presented work establishes relationship between fractal dimensions and pore size distribution, which can be directly used to calculate the fractal dimensions. The published pore size distribution data for eight sandstone samples are used to calculate the fractal dimensions and simultaneously compared with prediction results from analytical expression. In addition, the proposed fractal dimension method is also tested through Micro-CT images of three sandstone cores, and are compared with fractal dimensions by box-counting algorithm. The test results also prove a self-similar fractal range in sandstone when excluding smaller pores.
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Dissertations / Theses on the topic "Fractal dimensions"

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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.

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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.

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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.

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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.
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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.

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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
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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.

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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 maps
Es 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
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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.

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Brock, S. T. H. "Fractal dimensions and their relationship to filtration characteristics." Thesis, Loughborough University, 2000. https://dspace.lboro.ac.uk/2134/13486.

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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.
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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.

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Patuano, Agnès. "Fractal dimensions of landscape images as predictors of landscape preference." Thesis, University of Edinburgh, 2018. http://hdl.handle.net/1842/31380.

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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.
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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.

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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]
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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.

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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.
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Books on the topic "Fractal dimensions"

1

Rosenberg, Eric. Fractal Dimensions of Networks. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-43169-3.

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E, Ugalde, and Urías J, eds. Fractal dimensions for Poincaré recurrences. Amsterdam: Elsevier, 2006.

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Kaye, Brian H. A random walk through fractal dimensions. 2nd ed. Weinheim: VCH, 1994.

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Kaye, Brian H. A random walk through fractal dimensions. Weinheim, Germany: VCH Verlagsgesellschaft, 1989.

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Kaye, Brian H. A random walk through fractal dimensions. Weinheim: VCH, 1989.

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Banerjee, Santo, D. Easwaramoorthy, and A. Gowrisankar. Fractal Functions, Dimensions and Signal Analysis. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-62672-3.

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A random walk through fractal dimensions. Weinheim: VCH, 1989.

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Lapidus, Michel L., and Machiel van Frankenhuijsen. Fractal Geometry, Complex Dimensions and Zeta Functions. New York, NY: Springer New York, 2006. http://dx.doi.org/10.1007/978-0-387-35208-4.

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Lapidus, Michel L., and Machiel van Frankenhuijsen. Fractal Geometry, Complex Dimensions and Zeta Functions. New York, NY: Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4614-2176-4.

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Rosenberg, Eric. A Survey of Fractal Dimensions of Networks. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-90047-6.

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Book chapters on the topic "Fractal dimensions"

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Rosenberg, Eric. "Other Dimensions." In Fractal Dimensions of Networks, 425–36. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-43169-3_20.

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Fernández-Martínez, Manuel, Juan Luis García Guirao, Miguel Ángel Sánchez-Granero, and Juan Evangelista Trinidad Segovia. "A Middle Definition Between Hausdorff and Box Dimensions." In Fractal Dimension for Fractal Structures, 85–147. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-16645-8_3.

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Rosenberg, Eric. "Other Network Dimensions." In Fractal Dimensions of Networks, 455–69. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-43169-3_22.

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Rosenberg, Eric. "Dimensions of Infinite Networks." In Fractal Dimensions of Networks, 247–66. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-43169-3_12.

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Rosenberg, Eric. "Generalized Dimensions and Multifractals." In Fractal Dimensions of Networks, 325–64. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-43169-3_16.

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Rosenberg, Eric. "Introduction." In Fractal Dimensions of Networks, 1–15. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-43169-3_1.

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Rosenberg, Eric. "Computing the Correlation Dimension." In Fractal Dimensions of Networks, 195–219. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-43169-3_10.

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Rosenberg, Eric. "Network Correlation Dimension." In Fractal Dimensions of Networks, 221–46. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-43169-3_11.

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Rosenberg, Eric. "Similarity Dimension of Infinite Networks." In Fractal Dimensions of Networks, 267–78. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-43169-3_13.

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Rosenberg, Eric. "Information Dimension." In Fractal Dimensions of Networks, 279–303. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-43169-3_14.

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Conference papers on the topic "Fractal dimensions"

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Hornbogen, E. "Fractal Dimensions of Martensitic Microstructures." In ESOMAT 1989 - Ist European Symposium on Martensitic Transformations in Science and Technology. Les Ulis, France: EDP Sciences, 1989. http://dx.doi.org/10.1051/esomat/198902008.

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Tang, Y. Y., and Yu Tao. "Feature extraction by fractal dimensions." In Proceedings of the Fifth International Conference on Document Analysis and Recognition. ICDAR '99 (Cat. No.PR00318). IEEE, 1999. http://dx.doi.org/10.1109/icdar.1999.791763.

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Wada, Yukari, and Kazunori Kuwana. "Flame Fractal Dimension Induced by Hydrodynamic Instability." In ASME/JSME 2011 8th Thermal Engineering Joint Conference. ASMEDC, 2011. http://dx.doi.org/10.1115/ajtec2011-44222.

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Premixed flames self-turbulized due to hydrodynamic instability have self-similar, fractal-like structures as evidenced by the acceleration of spherically-propagating flames. The fractal dimension of a self-turbulized premixed flame needs to be known if its apparent flame speed is to be estimated. CFD simulations of outwardly-propagating flames have been conducted to predict their fractal dimensions. There are, however, difficulties in accurately determining fractal dimension based on the flame-propagation behavior of such an outwardly-propagating flame. This paper demonstrates a newly proposed method to determine the fractal dimension based on the CFD simulation of a planar flame. The fractal dimension is computed from the dependency of apparent flame speed on the computational domain size. The computed fractal dimension well agrees with the experimental value. The box-counting method is also applied to calculate the flame’s fractal dimension. The fractal dimensions obtained by these two methods agree well, confirming the fractal nature of the self-turbulized flame.
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Pink, David, Arun S. Moorthy, and Fernanda Peyronel. "Computing the Fractal Dimensions of Aggregates." In Virtual 2020 AOCS Annual Meeting & Expo. American Oil Chemists’ Society (AOCS), 2020. http://dx.doi.org/10.21748/am20.43.

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Kinsner, W. "A unified approach to fractal dimensions." In Fourth IEEE Conference on Cognitive Informatics, 2005. (ICCI 2005). IEEE, 2005. http://dx.doi.org/10.1109/coginf.2005.1532616.

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Liu, Jing, Weicai Zhong, Fang Liu, and Licheng Jiao. "Image retrieval algorithm using fractal dimensions." In Second International Conference on Image and Graphics, edited by Wei Sui. SPIE, 2002. http://dx.doi.org/10.1117/12.477114.

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Symonds, P. S., and Jae-Yeong Lee. "Fractal Dimensions in Elastic-Plastic Beam Dynamics." In ASME 1993 Design Technical Conferences. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/detc1993-0285.

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Abstract The final midpoint displacement of a two-degree-of-freedom beam model subjected to a short pulse of transverse loading may be either in the direction of the initial impulse or in the opposite (“negative”) direction, when moderately small plastic deformations occur. In the range where chaotic vibrations occur, the result depends with great sensitivity on the impulse magnitude. Considering a pulse of duration 0.5 × 10−3 sec, 100 calculations have been made for pulse forces P starting at 2500 N and increasing by increments of 2.0, 10−2, 10−4, and 10−6 N. It is found that the proportion and distribution of negative final displacements remain, on average, the same, independent of the size of the force increment. A fractal dimension representing a self-similarity property is calculated for the four choices of the force increment, and is found to be approximately 0.78 in each case. A correlation fractal dimension is also computed for undamped responses.
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Angulo, R. F., V. Alvarado, and H. Gonzalez. "Fractal Dimensions from Mercury Intrusion Capillary Tests." In SPE Latin America Petroleum Engineering Conference. Society of Petroleum Engineers, 1992. http://dx.doi.org/10.2118/23695-ms.

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Liu, Weiqiang, Jinzhe Xie, Jiulong Xiong, and Ying Li. "Analysis of Fractal Dimensions against Noise Interference." In 2018 IEEE 3rd Advanced Information Technology, Electronic and Automation Control Conference (IAEAC). IEEE, 2018. http://dx.doi.org/10.1109/iaeac.2018.8577812.

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Tumer, Irem Y., R. S. Srinivasan, Kristin L. Wood, and Ilene Busch-Vishniac. "Fractal Precision Models of Lathe-Type Turning Machines." In ASME 1993 Design Technical Conferences. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/detc1993-0425.

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Abstract The primary objective of this research is to develop surface models of machining processes to simulate machined surface profiles and analyze their structure. In this paper, fractal analysis is used to discover and characterize the variational pattern of turned surfaces. This analysis provides a means of explicitly stating the precision of a machining process. The results are used to further test the validity of the concept of fractal dimensions at tolerance scales and to establish a relation between surface models, experimental surfaces, and fractals. Based on these results, a more complete model of turning is available to designers for choosing process and design parameters and for comparing the precision between competing machining processes.
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Reports on the topic "Fractal dimensions"

1

Friesen, W. I., and R. J. Mikula. Fractal dimensions of coal particles. Natural Resources Canada/ESS/Scientific and Technical Publishing Services, 1986. http://dx.doi.org/10.4095/304962.

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Loehle, C. Estimation of fractal dimensions from transect data. Office of Scientific and Technical Information (OSTI), April 1994. http://dx.doi.org/10.2172/10141758.

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England, A. W. The Fractal Dimension of Diverse Topographies and the Effect of Spatial Windowing. Natural Resources Canada/ESS/Scientific and Technical Publishing Services, 1992. http://dx.doi.org/10.4095/133648.

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Gureghian, A. B. FRACVAL: Validation (nonlinear least squares method) of the solution of one-dimensional transport of decaying species in a discrete planar fracture with rock matrix diffusion. Office of Scientific and Technical Information (OSTI), August 1990. http://dx.doi.org/10.2172/6468991.

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