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Journal articles on the topic 'Fractal scaling'
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Anitas, Eugen Mircea, Giorgia Marcelli, Zsolt Szakacs, Radu Todoran, and Daniela Todoran. "Structural Properties of Vicsek-like Deterministic Multifractals." Symmetry 11, no. 6 (2019): 806. http://dx.doi.org/10.3390/sym11060806.
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Deterministic nano-fractal structures have recently emerged, displaying huge potential for the fabrication of complex materials with predefined physical properties and functionalities. Exploiting the structural properties of fractals, such as symmetry and self-similarity, could greatly extend the applicability of such materials. Analyses of small-angle scattering (SAS) curves from deterministic fractal models with a single scaling factor have allowed the obtaining of valuable fractal properties but they are insufficient to describe non-uniform structures with rich scaling properties such as fractals with multiple scaling factors. To extract additional information about this class of fractal structures we performed an analysis of multifractal spectra and SAS intensity of a representative fractal model with two scaling factors—termed Vicsek-like fractal. We observed that the box-counting fractal dimension in multifractal spectra coincide with the scattering exponent of SAS curves in mass-fractal regions. Our analyses further revealed transitions from heterogeneous to homogeneous structures accompanied by changes from short to long-range mass-fractal regions. These transitions are explained in terms of the relative values of the scaling factors.
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Chen, Yanguang. "Fractal Modeling and Fractal Dimension Description of Urban Morphology." Entropy 22, no. 9 (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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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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IGNATOWICH, MICHAEL J., DANIEL J. KELLEHER, CATHERINE E. MALONEY, DAVID J. MILLER, and KHRYSTYNA SERHIYENKO. "RESISTANCE SCALING FACTOR OF THE PILLOW AND FRACTALINA FRACTALS." Fractals 23, no. 02 (2015): 1550018. http://dx.doi.org/10.1142/s0218348x15500188.
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Much is known in the analysis of a finitely ramified self-similar fractal when the fractal has a harmonic structure: a Dirichlet form which respects the self-similarity of a fractal. What is still an open question is when such a structure exists in general. In this paper, we introduce two fractals, the fractalina and the pillow, and compute their resistance scaling factor. This is the factor which dictates how the Dirichlet form scales with the self-similarity of the fractal. By knowing this factor one can compute the harmonic structure on the fractal. The fractalina has scaling factor [Formula: see text], and the pillow fractal has scaling factor [Formula: see text].
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PERFECT, E., and B. DONNELLY. "BI-PHASE BOX COUNTING: AN IMPROVED METHOD FOR FRACTAL ANALYSIS OF BINARY IMAGES." Fractals 23, no. 01 (2015): 1540010. http://dx.doi.org/10.1142/s0218348x15400101.
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Many natural systems are irregular and/or fragmented, and have been interpreted to be fractal. An important parameter needed for modeling such systems is the fractal dimension, D. This parameter is often estimated from binary images using the box-counting method. However, it is not always apparent which fractal model is the most appropriate. This has led some researchers to report different D values for different phases of an analyzed image, which is mathematically untenable. This paper introduces a new method for discriminating between mass fractal, pore fractal, and Euclidean scaling in images that display apparent two-phase fractal behavior when analyzed using the traditional method. The new method, coined "bi-phase box counting", involves box-counting the selected phase and its complement, fitting both datasets conjointly to fractal and/or Euclidean scaling relations, and examining the errors from the resulting regression analyses. Use of the proposed technique was demonstrated on binary images of deterministic and stochastic fractals with known D values. Traditional box counting was unable to differentiate between the fractal and Euclidean phases in these images. In contrast, bi-phase box counting unmistakably identified the fractal phase and correctly estimated its D value. The new method was also applied to three binary images of soil thin sections. The results indicated that two of the soils were pore-fractals, while the other was a mass fractal. This outcome contrasted with the traditional box counting method which suggested that all three soils were mass fractals. Reclassification has important implications for modeling soil structure since different fractal models have different scaling relations. Overall, bi-phase box counting represents an improvement over the traditional method. It can identify the fractal phase and it provides statistical justification for this choice.
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Chauhan, Mahak Singh, Abhey Ram Bansal, and V. P. Dimri. "Scaling Laws and Fractal Geometry: Insights into Geophysical Data Interpretations." Journal Of The Geological Society Of India 101, no. 6 (2025): 983–89. https://doi.org/10.17491/jgsi/2025/174196.
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ABSTRACT Fractals, characterised by self-similarity and scale invariance, have emerged as powerful tools for understanding complex systems in geophysics. This paper highlights the applications of fractal geometry in geophysical data interpretation. For instance, fractal analysis is used in seismology to understand the fault systems, earthquake distribution, and the scaling laws governing seismic events. In potential fields, fractals are used to find the source depth, to design the optimum grid size of the survey, to detect the source and to separate signal from noise. In this paper, we first highlight the basics of fractal theory and then show how fractals are useful in various geophysical studies by showing examples from potential fields and seismology and reservoir characterisation.
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Yu, Bo, Yifei Pu, Qiuyan He, and Xiao Yuan. "Circuit Implementation of Variable-Order Scaling Fractal-Ladder Fractor with High Resolution." Fractal and Fractional 6, no. 7 (2022): 388. http://dx.doi.org/10.3390/fractalfract6070388.
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Extensive research has been conducted on the scaling fractal fractor using various structures. The development of high-resolution emulator circuits to achieve a variable-order scaling fractal fractor with high resolution is a major area of interest. We present a scaling fractal-ladder circuit for achieving high-resolution variable-order fractor based on scaling expansion theory using a high-resolution multiplying digital-to-analog converter (HMDAC). Firstly, the circuit configuration of variable-order scaling fractal-ladder fractor (VSFF) is designed. A theoretical demonstration proves that VSFF exhibits the operational characteristics of variable-order fractional calculus. Secondly, a programmable resistor–capacitor series circuit and universal electronic component emulators are developed based on the HMDAC to adjust the resistance and capacitance in the circuit configuration. Lastly, the model, component parameters, approximation performance, and variable-order characteristics are analyzed, and the circuit is physically implemented. The experimental results demonstrate that the circuit exhibits variable-order characteristics, with an operational order ranging from −0.7 to −0.3 and an operational frequency ranging from 7.72Hz to 4.82kHz. The peak value of the input signal is 10V. This study also proposes a novel method for variable-order fractional calculus based on circuit theory. This study was the first attempt to implement feasible high-resolution continuous variable-order fractional calculus hardware based on VSFF.
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Potapov, Alexander A. "Fractal applications in radio electronics as fractal engineering." Radioelectronics. Nanosystems. Information Technologies. 14, no. 3 (2022): 215–32. http://dx.doi.org/10.17725/rensit.2022.14.215.
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The use of the fractal paradigm is presented - the main directions for introducing textures, fractals, fractional operators, dynamic chaos and methods of nonlinear dynamics for the design and creation of real technical projects in radio electronics - fractal radio systems, taking into account the hereditarity, non-Gaussianity and scaling of physical signals and fields. The substantiation of the use of fractal-scaling and texture methods for the synthesis of fundamentally new topological texture-fractal methods for detecting signals in the space-time channel of scattering waves (a new type of radar) is discussed. It is shown that the use of fractal systems, sensors and nodes is a fundamentally new solution that significantly changes the principles of constructing intelligent radio engineering systems and devices. It is shown that the use of computational dielectric metasurfaces brings to a new level all the functional characteristics of a multifunctional system of topological texture-fractal processing of signals and fields in solving classical problems of detection, measurement, recognition and classification by intelligent radio engineering systems and devices. The concept of "fractal engineering" is introduced, the methodology of its use is discussed.
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NAIR, PRADEEP R., and MUHAMMAD A. ALAM. "KINETIC RESPONSE OF SURFACES DEFINED BY FINITE FRACTALS." Fractals 18, no. 04 (2010): 461–76. http://dx.doi.org/10.1142/s0218348x10005032.
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Historically, fractal analysis has been remarkably successful in describing wide ranging kinetic processes on (idealized) scale invariant objects in terms of elegantly simple universal scaling laws. However, as nanostructured materials find increasing applications in energy storage, energy conversion, healthcare, etc., one must reexamine the premise of traditional fractal scaling laws as it only applies to physically unrealistic infinite systems, while all natural/engineered systems are necessarily finite. In this article, we address the consequences of the 'finite-size' problem in the context of time dependent diffusion towards fractal surfaces via the novel technique of Cantor-transforms to (i) illustrate how finiteness modifies its classical scaling exponents; (ii) establish that for finite systems, the diffusion-limited reaction is decelerated below a critical dimension [Formula: see text] and accelerated above it; and (iii) to identify the crossover size-limits beyond which a finite system can be considered (practically) infinite and redefine the very notion of 'finiteness' of fractals in terms of its kinetic response. Our results have broad implications regarding dynamics of systems defined by the same fractal dimension, but differentiated by degree of scaling iteration or morphogenesis, e.g. variation in lung capacity between a child and adult.
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Pacheco, Julio César Ramırez, Homero Toral Cruz, and David Ernesto Troncoso Romero. "Rényi wavelet extropy of scaling signals." Parana Journal of Science and Education 10, no. 6 (2024): 21–25. https://doi.org/10.5281/zenodo.14232536.
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Recently, wavelet based informations tools are being used for fractals to unveil their complexities and providemore elaborate analyses. In this article, the time-domain definition of Rényi extropy is extended to the waveletdomain and a closed-form expression of this extropy for scaling signals of parameter (alfa) is obtained. Based on the theoretical results, wavelet Rényi <em>q</em>-extropy planes are obtained for various <em>q </em>and a range of the fractality parameter and, based on these, a complete characterization of the complexities of fractal signals based on Rényi extropies are obtained. Moreover, potential applications for fractal signal analysis are highlighted and useful relations amongst different wavelet Rényi extropy planes are provided.
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Potapov, Alexander A. "Mathematical Foundations of the Fractal Scaling Method in Statistical Radiophysics and Applications." Radioelectronics. Nanosystems. Information Technologies. 13, no. 3 (2021): 245–96. http://dx.doi.org/10.17725/rensit.2021.13.245.
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The system of basic mathematical concepts and constructions underlying the modern global fractal-scaling method developed by the author is presented. An overview of the main results on the creation of new information technologies based on textures, fractals (multifractals), fractional operators, scaling effects and nonlinear dynamics methods obtained by the author and his students for more than 40 years (from 1979 to the present) at the V.A. Kotelnikov Institute of Radioengineering and Electronics of RAS. It is shown that, for the first time in the world, new dimensional and topological (and not energy!) Features or invariants were proposed and then effectively applied for problems in radio physics and radio electronics, which are combined under the generalized concept of "sample topology" ~ "fractal signature". The author discovered, proposed and substantiated a new type and new method of modern radar, namely, fractal-scaling or scale-invariant radar. It should be noted that fractal radars are, in fact, a necessary intermediate stage on the path of transition to cognitive radar and quantum radar.
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Bottaccio, M., M. Montuori, and L. Pietronero. "Fractals vs. halos: Asymptotic scaling without fractal properties." Europhysics Letters (EPL) 66, no. 4 (2004): 610–16. http://dx.doi.org/10.1209/epl/i2003-10244-6.
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KARLINGER, M. R., T. M. OVER, and B. M. TROUTMAN. "RELATING THIN AND FAT-FRACTAL SCALING OF RIVER-NETWORK MODELS." Fractals 02, no. 04 (1994): 557–65. http://dx.doi.org/10.1142/s0218348x94000788.
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The fractal scaling of river networks has been described in the context of both thin and fat fractals. Whereas the thin-fractal characterization is presented as a fractal dimension, D, derived from the length properties of the river channels, a fat-fractal characterization is given as a scaling exponent, β, derived from the behavior of river-channel area. Several authors have related D to the bifurcation ratio, Rb, and length ratio, Rl, of idealized Hortonian-network trees. Here, for these types of trees, we asymptotically relate β to Rb, Rl, and to a diameter exponent, Δ, which governs the downstream channel-widening process. Using this result, we present a linkage of D to β and discuss the implications of the relative values of Rb, Rl, and Δ on this linkage and on network channel behavior. Finally, we illustrate bias in the estimation of β from preasymptotic trees.
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Zhang, Xiyin. "Analysis of the Principle and Applications of Fractal." Highlights in Science, Engineering and Technology 88 (March 29, 2024): 337–42. http://dx.doi.org/10.54097/5fxx1704.
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As a matter of fact, fractal is one of the mystery in math applications which attracts a large number of scholars even in recent years. With this in mind, a brief introductions for definition of fractals and a presentation of a brief history of fractal theory will be presented in this study. To be specific, this study will provide explanations on basic principles of fractal geometry like self-similarity and scaling. At the same time, this study will also give the idea of fractal dimensions and examples of fractal shapes. In detail, the essay will include two specific fractal applications in mechanism and biomedicine. According to the analysis, challenges and limitations of fractal theory will be mentioned potential applications in various fields will be discussed. Eventually, this study will conclude summary of key points and importance of fractals in all fields. Overall, these results shed light on guiding further exploration of fractal investigation.
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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 (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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MILLÁN, H. "FRAGMENTATION OF SOIL INITIATORS: APPLICATION OF THE PORE-SOLID FRACTAL MODEL." Fractals 12, no. 04 (2004): 357–63. http://dx.doi.org/10.1142/s0218348x04002628.
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The quantification of fragmentation of natural, polidisperse, porous media using fractal models is well documented in the literature. However, in many cases, fractal exponents (fractal fragmentation dimension) and coefficients (fractal lacunarity) arising from a power law behavior do not make clear differences between different media. In the present work, the pore-solid fractal (PSF) model was used as a new fractal approach for deriving four scaling parameters (fractal dimension of the particle-size distribution, fractal fragmentation dimension of the fragment-size distribution, probability of fragmentation and scaling factor) from soil initiators subjected to different energy density input. The fractal fragmentation dimension for all soil samples was Df=2.42±0.16 without correlating with the energy expended in the fragmentation process. By contrast, probability of fragmentation and scaling factor correlated significantly with the energetic term. The PSF model is useful for estimating a group of scaling parameters more appropriate for the quantification of complex patterns associated to fragment-size distributions.
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Johnson, Bayard K., Robert F. Sekerka, and Michael P. Foley. "Scaling of fractal aggregates." Physical Review E 52, no. 1 (1995): 796–800. http://dx.doi.org/10.1103/physreve.52.796.
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Ferer, M., W. N. Sams, R. A. Geisbrecht, and Duane H. Smith. "Scaling of fractal flow." Physica A: Statistical Mechanics and its Applications 177, no. 1-3 (1991): 273–80. http://dx.doi.org/10.1016/0378-4371(91)90164-8.
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Pelous, J., R. Vacher, T. Woignier, J. L. Sauvajol, and E. Courtens. "Scaling phonon-fracton dispersion laws in fractal aerogels." Philosophical Magazine B 59, no. 1 (1989): 65–74. http://dx.doi.org/10.1080/13642818908208446.
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VIJENDER, N. "BERNSTEIN FRACTAL RATIONAL APPROXIMANTS WITH NO CONDITION ON SCALING VECTORS." Fractals 26, no. 04 (2018): 1850045. http://dx.doi.org/10.1142/s0218348x18500457.
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Fractal functions defined through iterated function system have been successfully used to approximate any continuous real-valued function defined on a compact interval. The fractal dimension is a quantifier (or index) of irregularity (non-differentiability) of fractal approximant and it depends on the scaling factors of the fractal approximant. Viswanathan and Chand [Approx. Theory 185 (2014) 31–50] studied fractal rational approximation under the hypothesis “magnitude of the scaling factors goes to zero”. In this paper, first, we introduce a new class of fractal approximants, namely, Bernstein [Formula: see text]-fractal functions which converge to the original function for every scaling vector. Using the proposed class of fractal approximants and imposing no condition on the corresponding scaling factors, we establish self-referential Bernstein [Formula: see text]-fractal rational functions and their approximation properties. In particular, (i) we study the fractal analogue of the Weierstrass theorem and Müntz theorem of rational functions, (ii) we study the one-sided approximation by Bernstein [Formula: see text]-fractal rational functions, (iii) we develop copositive Bernstein fractal rational approximation, (iv) we investigate the existence of a minimizing sequence of fractal rational approximation to a continuous function defined as a real compact interval. Finally, we introduce the non-self-referential Bernstein [Formula: see text]-fractal approximants.
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DAI, MEIFENG, YUE ZONG, XIAOQIAN WANG, WEIYI SU, YU SUN, and JIE HOU. "EFFECTS OF FRACTAL INTERPOLATION FILTER ON MULTIFRACTAL ANALYSIS." Fractals 25, no. 02 (2017): 1750024. http://dx.doi.org/10.1142/s0218348x17500244.
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Fractal interpolation filter is proposed for the first time in the literatures to transform original signals. Using the multifractal detrended fluctuation analysis (MFDFA), the authors investigate how the filter affects the multifractal scaling properties for both artificial and traffic signals. Specifically, the authors compare the multifractal scaling properties of signals before and after the transforms. It is shown that the fractal interpolation filter changes slightly the maximum value of the multifractal spectrum, while the values of spectrum width and maximum point of spectrum are much more affected by vertical scaling factor. The multifractal spectrum shrinks dramatically after the fractal interpolation filter. The fractal exponents in the signal change dramatically for the negative values of vertical scaling factor while remain stable otherwise. Thus, an appropriate vertical scaling factor can be found in order to minimize the effects of filter when one uses the fractal interpolation filter.
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Coppens, Marc-Olivier, and Gilbert F. Froment. "The Effectiveness of Mass Fractal Catalysts." Fractals 05, no. 03 (1997): 493–505. http://dx.doi.org/10.1142/s0218348x97000395.
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Many porous catalysts have a fractal surface, but only rarely do they have a fractal volume, the main exceptions being extremely porous aerogels. It has been suggested that a fractal shape of their volume would be ideal, because it has an infinite area per unit mass that is easily accessible by the reactants. This paper investigates the efficiency of mass fractals by comparing them with nonfractal catalysts. It is found that the specific surface areas of comparable nonfractal catalysts are of the same order of magnitude, if not higher than those of mass fractals. Despite the high effectiveness factor of mass fractals due to the exceptionally easy accessibility of their active sites, production in a nonfractal catalyst is often higher than in a mass fractal, because of the high porosity of the latter. For some strongly diffusion limited reactions, especially in mesoporous catalysts, an added mass fractal macroporosity, with a finite scaling regime, would increase the yields beyond what is possible with a nonfractal catalyst. Nonetheless, when transport through viscous flow in macropores is very rapid the effective reaction rates in classical bimodal catalysts are higher than in fractal catalysts with their high macroporosity.
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Buriboev, Abror Shavkatovich, Djamshid Sultanov, Zulaykho Ibrohimova, and Heung Seok Jeon. "Mathematical Modeling and Recursive Algorithms for Constructing Complex Fractal Patterns." Mathematics 13, no. 4 (2025): 646. https://doi.org/10.3390/math13040646.
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In this paper, we present mathematical geometric models and recursive algorithms to generate and design complex patterns using fractal structures. By applying analytical, iterative methods, iterative function systems (IFS), and L-systems to create geometric models of complicated fractals, we developed fractal construction models, visualization tools, and fractal measurement approaches. We introduced a novel recursive fractal modeling (RFM) method designed to generate intricate fractal patterns with enhanced control over symmetry, scaling, and self-similarity. The RFM method builds upon traditional fractal generation techniques but introduces adaptive recursion and symmetry-preserving transformations to produce fractals with applications in domains such as medical imaging, textile design, and digital art. Our approach differs from existing methods like Barnsley’s IFS and Jacquin’s fractal coding by offering faster convergence, higher precision, and increased flexibility in pattern customization. We used the RFM method to create a mathematical model of fractal objects that allowed for the viewing of polygonal, Koch curves, Cayley trees, Serpin curves, Cantor set, star shapes, circulars, intersecting circles, and tree-shaped fractals. Using the proposed models, the fractal dimensions of these shapes were found, which made it possible to create complex fractal patterns using a wide variety of complicated geometric shapes. Moreover, we created a software tool that automates the visualization of fractal structures. This tool may be used for a variety of applications, including the ornamentation of building items, interior and exterior design, and pattern construction in the textile industry.
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Creffield, Charles. "Fractals on a Benchtop: Observing Fractal Dimension in a Resistor Network." Physics Teacher 60, no. 6 (2022): 410–13. http://dx.doi.org/10.1119/5.0054306.
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Our first experience of dimension typically comes in the intuitive Euclidean sense: a line is one dimensional, a plane is two dimensional, and a volume is three dimensional. However, following the work of Mandelbrot, systems with a fractional dimension, “fractals,” now play an important role in science. The novelty of encountering fractional dimension, and the intrinsic beauty of many fractals, has a strong appeal to students and provides a powerful teaching tool. I describe here a low-cost and convenient experimental method for observing fractal dimension, by measuring the power-law scaling of the resistance of a fractal network of resistors. The experiments are quick to perform, and the students enjoy both the construction of the network and the collaboration required to create the largest networks. Learning outcomes include analysis of resistor networks beyond the elementary series and parallel combinations, scaling laws, and an introduction to fractional dimension.
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Chen. "The Solutions to the Uncertainty Problem of Urban Fractal Dimension Calculation." Entropy 21, no. 5 (2019): 453. http://dx.doi.org/10.3390/e21050453.
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Fractal geometry provides a powerful tool for scale-free spatial analysis of cities, but the fractal dimension calculation results always depend on methods and scopes of the study area. This phenomenon has been puzzling many researchers. This paper is devoted to discussing the problem of uncertainty of fractal dimension estimation and the potential solutions to it. Using regular fractals as archetypes, we can reveal the causes and effects of the diversity of fractal dimension estimation results by analogy. The main factors influencing fractal dimension values of cities include prefractal structure, multi-scaling fractal patterns, and self-affine fractal growth. The solution to the problem is to substitute the real fractal dimension values with comparable fractal dimensions. The main measures are as follows. First, select a proper method for a special fractal study. Second, define a proper study area for a city according to a study aim, or define comparable study areas for different cities. These suggestions may be helpful for the students who take interest in or have already participated in the studies of fractal cities.
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Vijay and Arya Kumar Bedabrata Chand. "Convexity-Preserving Rational Cubic Zipper Fractal Interpolation Curves and Surfaces." Mathematical and Computational Applications 28, no. 3 (2023): 74. http://dx.doi.org/10.3390/mca28030074.
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A class of zipper fractal functions is more versatile than corresponding classes of traditional and fractal interpolants due to a binary vector called a signature. A zipper fractal function constructed through a zipper iterated function system (IFS) allows one to use negative and positive horizontal scalings. In contrast, a fractal function constructed with an IFS uses positive horizontal scalings only. This article introduces some novel classes of continuously differentiable convexity-preserving zipper fractal interpolation curves and surfaces. First, we construct zipper fractal interpolation curves for the given univariate Hermite interpolation data. Then, we generate zipper fractal interpolation surfaces over a rectangular grid without using any additional knots. These surface interpolants converge uniformly to a continuously differentiable bivariate data-generating function. For a given Hermite bivariate dataset and a fixed choice of scaling and shape parameters, one can obtain a wide variety of zipper fractal surfaces by varying signature vectors in both the x direction and y direction. Some numerical illustrations are given to verify the theoretical convexity results.
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Zeng, Ting-Ting, Gang Chen, Zi-Jian Dong, Zhi-Lei She, and N. A. Ragab. "Study on fractal characteristics in the e+e− collisions at s = 250GeV." International Journal of Modern Physics A 32, no. 22 (2017): 1750124. http://dx.doi.org/10.1142/s0217751x1750124x.
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The fractal characteristics of multiparticle final state are first studied in the [Formula: see text] collision of [Formula: see text], with the MC simulation generator Jetset7.4 and Herwig5.9. The results show that this multiparticle final state has a double-Hurst-exponent fractals characteristics, which means that the resulting NFM obeys the scaling law well using both the two sets of Hurst exponents to partition the three-dimensional phase space isotropically and anisotropically. Their fractal indices and the effective fluctuation strength are also predicted.
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Azizi, Tahmineh, and Sepideh Azizi. "The Fractal Nature of Drought: Power Laws and Fractal Complexity of Arizona Drought." European Journal of Mathematical Analysis 2 (May 27, 2022): 17. http://dx.doi.org/10.28924/ada/ma.2.17.
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In this study, we explore the possibility that the Drought Monitor database belongs to class of fractal process which can be characterized using a single scaling exponent. The Drought Monitor map identifies areas of drought and labels them by intensity: D0 abnormally dry, D1 moderate drought, D2 severe drought, D3 extreme drought, and D4 exceptional drought. The vibration analysis using power spectral densities (PSD) method has been carried out to discover whether some type of power-law scaling exists for various statistical moments at different scales of this database. We perform multi-fractal analysis to estimate the multi-fractal spectrum of each group. We apply Higuchi algorithm to find the fractal complexity of each group and then compare them for different time intervals. Our findings reveal that we have a wide range of exponents for D0-D4. Therefore, D0-D4 belong to class of multi-fractal process for which a large number of scaling exponents are required to characterize the scaling structure.
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Mitic, Vojislav V., Goran Lazovic, Dragoljub Mirjanic, Hans Fecht, Branislav Vlahovic, and Walter Arnold. "The fractal nature as new frontier in microstructural characterization and relativization of scale sizes within space." Modern Physics Letters B 34, no. 22 (2020): 2050421. http://dx.doi.org/10.1142/s0217984920504217.
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Today in the age of advanced ceramic civilization, there are a variety of applications for modern ceramics materials with specific properties. Our up-to date research recognizes that ceramics have a fractal configuration nature on the basis of different phenomena. The key property of fractals is their scale-independence. The practical value is that the fractal objects’ interaction and energy is possible at any reasonable scale of magnitude, including the nanoscale and may be even below. This is a consequence of fractal scale independence. This brings us to the conclusion that properties of fractals are valid on any scale (macro, micro, or nano). We also analyzed these questions with experimental results obtained from a comet, here 67P, and also from ceramic grain and pore morphologies on the microstructure level. Fractality, as a scale-independent morphology, provides significant variety of opportunities, for example for energy storage. From the viewpoint of scaling, the relation between large and small in fractal analysis is very important. An ideal fractal can be magnified endlessly but natural morphologies cannot, what is the new light in materials sciences and space.
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SERNETZ, M., M. JUSTEN, and F. JESTCZEMSKI. "DISPERSIVE FRACTAL CHARACTERIZATION OF KIDNEY ARTERIES BY THREE-DIMENSIONAL MASS-RADIUS-ANALYSIS." Fractals 03, no. 04 (1995): 879–91. http://dx.doi.org/10.1142/s0218348x95000771.
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Three-dimensional data sets of kidney arterial vessels were obtained from resin casts by serial sectioning and by micro-NMR-tomography, and were analyzed by the mass-radius-relation both for global and local scaling properties. We present for the first time the spatial resolution of local scaling and thus the dispersion of the fractal dimension within the organs. The arterial system is characterized as a non-homogeneous fractal. We discuss and relate the fractal structure to the scaling and allometry of metabolic rates in living organisms.
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Socoteanu, Radu, Mihai Anastasescu, Anabela Oliveira, Gianina Dobrescu, Rica Boscencu, and Carolina Constantin. "Aggregation Behavior of Some Asymmetric Porphyrins versus Basic Biological Tests Response." International Journal of Photoenergy 2015 (2015): 1–11. http://dx.doi.org/10.1155/2015/302587.
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Fractal analysis of free bases porphyrins was computed on atomic force microscopy (AFM) micrographs using two different methods: the correlation function method and the variable length scale method. The correlation function method provides fractal dimension only for short scale range; results indicate that only few images have fractal properties for short ranges; for the rest of them, no fractal dimension was found using the correlation function method. The variable length scale method occur information for long range scaling. All samples have fractal properties at higher scaling range. For three samples the correlation function method leads to the same fractal dimension as the variable length scale method and scaling ranges for both methods overlap. Results show the necessity to use both methods to describe the fractal properties of AB3 meso-porphyrins that may be used to predict their relative cell localization. In order to emphasize the influence of fractal and textural properties the results regarding their self-similarity and texture/morphology were further compared with their behavior in biological assessment, that is, functionality of some Jurkat cell lines.
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Bertozzi, Andrea L., and Ashvin B. Chhabra. "Cancellation exponents and fractal scaling." Physical Review E 49, no. 5 (1994): 4716–19. http://dx.doi.org/10.1103/physreve.49.4716.
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CHEN, ZHIYING, YONG LIU, and PING ZHOU. "A NOVEL METHOD TO IDENTIFY THE SCALING REGION OF ROUGH SURFACE PROFILE." Fractals 27, no. 02 (2019): 1950011. http://dx.doi.org/10.1142/s0218348x19500117.
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Scaling region identification is of great importance in calculating the fractal dimension of a rough surface profile. A new method used to identify the scaling region is presented to improve the calculation accuracy of fractal dimension. In this method, the second derivative of the double logarithmic curve is first calculated and the [Formula: see text]-means algorithm method is adopted to identify the scaling region for the first time. Then the margin of error is reasonably set to get a possible scaling region. Finally, the [Formula: see text]-means method is used again to obtain a more accurate scaling region. The effectiveness of the proposed method is compared with the existing methods. Both the simulation and experimental results show that the proposed method provides more precise results for extracting the scaling regions and leads to a higher calculation precision of fractal dimensions.
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TAKEHARA, TAKUMA, FUMIO OCHIAI, and NAOTO SUZUKI. "FRACTALS IN EMOTIONAL FACIAL EXPRESSION RECOGNITION." Fractals 10, no. 01 (2002): 47–52. http://dx.doi.org/10.1142/s0218348x02001087.
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Following the Mandelbrot's theory of fractals, many shapes and phenomena in nature have been suggested to be fractal. Even animal behavior and human physiological responses can also be represented as fractal. Here, we show the evidence that it is possible to apply the concept of fractals even to the facial expression recognition, which is one of the most important parts of human recognition. Rating data derived from judging morphed facial images were represented in the two-dimensional psychological space by multidimensional scaling of four different scales. The resultant perimeter of the structure of the emotion circumplex was fluctuated and was judged to have a fractal dimension of 1.18. The smaller the unit of measurement, the longer the length of the perimeter of the circumplex. In this study, we provide interdisciplinarily important evidence of fractality through its application to facial expression recognition.
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JOHANSSON, BENNY, and AGNES LUKACS. "FIBONACCI OPTICAL LATTICE: A FRACTAL BIOMODULATION DEVICE." Fractals 27, no. 03 (2019): 1950039. http://dx.doi.org/10.1142/s0218348x19500397.
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Natural scenes or structures share fractal-like geometries with scale-invariant statistical patterns that exhibit useful functional physiological properties in humans. This paper explores the self-assembling of alumina microelements on a nanostructured silica-embedded PET substrate and the application of Fibonacci fractal geometry as a photonic wave-guiding device. Two concentric Fibonacci circles impinged on the top of embedded silica clusters were investigated using fractal analysis. Calculating the fractal dimension of spatial scaling properties demonstrated the potential of a fractal photonic element. Simulating the propagation of spectral visible or low NIR incident light through the fractal trajectory shows that the fractal scaling properties generate novel nonlinear and double-twisted electromagnetic waves with biophilic interconnecting potential.
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Zborovský, I. "A conservation law, entropy principle and quantization of fractal dimensions in hadron interactions." International Journal of Modern Physics A 33, no. 10 (2018): 1850057. http://dx.doi.org/10.1142/s0217751x18500574.
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Fractal self-similarity of hadron interactions demonstrated by the [Formula: see text]-scaling of inclusive spectra is studied. The scaling regularity reflects fractal structure of the colliding hadrons (or nuclei) and takes into account general features of fragmentation processes expressed by fractal dimensions. The self-similarity variable [Formula: see text] is a function of the momentum fractions [Formula: see text] and [Formula: see text] of the colliding objects carried by the interacting hadron constituents and depends on the momentum fractions [Formula: see text] and [Formula: see text] of the scattered and recoil constituents carried by the inclusive particle and its recoil counterpart, respectively. Based on entropy principle, new properties of the [Formula: see text]-scaling concept are found. They are conservation of fractal cumulativity in hadron interactions and quantization of fractal dimensions characterizing hadron structure and fragmentation processes at a constituent level.
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Kim, Anthony Y., and John C. Berg. "Fractal Aggregation: Scaling of Fractal Dimension with Stability Ratio." Langmuir 16, no. 5 (2000): 2101–4. http://dx.doi.org/10.1021/la990841n.
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Feng, Feng, Kexin Zhang, Xinghui Li, Yousheng Xia, Meng Yuan, and Pingfa Feng. "Scaling Region of Weierstrass-Mandelbrot Function: Improvement Strategies for Fractal Ideality and Signal Simulation." Fractal and Fractional 6, no. 10 (2022): 542. http://dx.doi.org/10.3390/fractalfract6100542.
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Fractal dimension (D) is widely utilized in various fields to quantify the complexity of signals and other features. However, the fractal nature is limited to a certain scope of concerned scales, i.e., scaling region, even for a theoretically fractal profile generated through the Weierstrass-Mandelbrot (W-M) function. In this study, the scaling characteristics curves of profiles were calculated by using the roughness scaling extraction (RSE) algorithm, and an interception method was proposed to locate the two ends of the scaling region, which were named corner and drop phenomena, respectively. The results indicated that two factors, sampling length and flattening order, in the RSE algorithm could influence the scaling region length significantly. Based on the scaling region interception method and the above findings, the RSE algorithm was optimized to improve the accuracy of the D calculation, and the influence of sampling length was discussed by comparing the lower critical condition of the W-M function. To improve the ideality of fractal curves generated through the W-M function, the strategy of reducing the fundamental frequency was proposed to enlarge the scaling region. Moreover, the strategy of opposite operation was also proposed to improve the consistency of generated curves with actual signals, which could be conducive to practical simulations.
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Somogyi, Ildiko, and Anna Soos. "Graph-directed random fractal interpolation function." Studia Universitatis Babes-Bolyai Matematica 66, no. 2 (2021): 247–55. http://dx.doi.org/10.24193/subbmath.2021.2.01.
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"Barnsley introduced in [1] the notion of fractal interpolation function (FIF). He said that a fractal function is a (FIF) if it possess some interpolation properties. It has the advantage that it can be also combined with the classical methods or real data interpolation. Hutchinson and Ruschendorf [7] gave the stochastic version of fractal interpolation function. In order to obtain fractal interpolation functions with more exibility, Wang and Yu [9] used instead of a constant scaling parameter a variable vertical scaling factor. Also the notion of fractal interpolation can be generalized to the graph-directed case introduced by Deniz and Ozdemir in [5]. In this paper we study the case of a stochastic fractal interpolation function with graph-directed fractal function."
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Kavitha, C., and A. Gowrisankar. "Fractional integral approach on nonlinear fractal function and its application." Mathematical Modelling and Control 4, no. 3 (2024): 230–45. http://dx.doi.org/10.3934/mmc.2024019.
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The shape and dimension of the fractal function have been significantly influenced by the scaling factor. This paper investigated the fractional integral of the nonlinear fractal interpolation function corresponding to the iterated function systems employed by Rakotch contraction. We demonstrated how the scaling factors affect the flexibility of fractal functions and their different fractional orders of the Riemann fractional integral using certain numerical examples. The potentiality application of Rakotch contraction of fractal function theory was elucidated based on a comparative analysis of the irregularity relaxation process. Moreover, a reconstitution of epidemic curves from the perspective of a nonlinear fractal interpolation function was presented, and a comparison between the graphs of linear and nonlinear fractal functions was discussed.
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Chen, Yanguang, and Yuqing Long. "Spatial Signal Analysis Based on Wave-Spectral Fractal Scaling: A Case of Urban Street Networks." Applied Sciences 11, no. 1 (2020): 87. http://dx.doi.org/10.3390/app11010087.
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A number of mathematical methods have been developed to make temporal signal analyses based on time series. However, no effective method for spatial signal analysis, which are as important as temporal signal analyses for geographical systems, has been devised. Nonstationary spatial and temporal processes are associated with nonlinearity, and cannot be effectively analyzed by conventional analytical approaches. Fractal theory provides a powerful tool for exploring spatial complexity and is helpful for spatio-temporal signal analysis. This paper is devoted to developing an approach for analyzing spatial signals of geographical systems by means of wave-spectrum scaling. The traffic networks of 10 Chinese cities are taken as cases for positive studies. Fast Fourier transform (FFT) and ordinary least squares (OLS) regression methods are employed to calculate spectral exponents. The results show that the wave-spectrum density distribution of all these urban traffic networks follows scaling law, and that the spectral scaling exponents can be converted into fractal dimension values. Using the fractal parameters, we can make spatial analyses for the geographical signals. The wave-spectrum scaling methods can be applied to both self-similar fractal signals and self-affine fractal signals in the geographical world. This study has implications for the further development of fractal-based spatiotemporal signal analysis in the future.
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Chen, Yanguang, and Jiejing Wang. "Describing Urban Evolution with the Fractal Parameters Based on Area-Perimeter Allometry." Discrete Dynamics in Nature and Society 2016 (2016): 1–14. http://dx.doi.org/10.1155/2016/4863907.
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The area-perimeter allometric scaling is an important approach for researching fractal cities, and the basic ideas and models have been researched for a long time. However, the fractal parameters based on this scaling relation have not been efficiently utilized in urban studies. This paper is devoted to developing a description method of urban evolution using the fractal parameter sets based on the area-perimeter measure relation. The novelty of this methodology is as follows: first, the form dimension and boundary dimension are integrated to characterize the urban structure and texture; second, the global and local parameters are combined to characterize an urban system and individual cities; third, an entire analytical process based on the area-perimeter scaling is illustrated. Two discoveries are made in this work: first, a dynamic proportionality factor can be employed to estimate the local boundary dimension; second, the average values of the local fractal parameters are approximately equal to the corresponding global fractal parameters of cities. By illustrating how to carry out the area-perimeter scaling analysis of Chinese cities in Yangtze River Delta in the case of remote sensing images with low resolution, we propose a possible new approach to exploring fractal systems of cities.
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ZAMMOURI, IKBAL, ERIC TOSAN, and BÉCHIR AYEB. "A THEORETICAL FRAMEWORK MAPPING GRAMMAR BASED SYSTEMS AND FRACTAL DESCRIPTION." Fractals 16, no. 04 (2008): 389–401. http://dx.doi.org/10.1142/s0218348x08004058.
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In this work, we propose a new method to describe fractal shapes using parametric L-systems. This method consists of introducing scaling factors in the production rules of the parametric L-system grammars. We present a turtle monoid on which we base our calculations to show the exact mathematical relation between L-systems and iterated function systems (IFS); we then establish the conditions for the scaling factors to produce plants' and curves' fractal shapes from parametric L-systems. We demonstrate that with specific values of the scaling factors, we find the exact relationship established by Prusinkiewicz and Hammel between L-systems and IFS. Finally, we present some examples of fractal plant forms and curves created using parametric L-systems with scaling factors.
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Priyanka, T. M. C., R. Valarmathi, Kishore Bingi, and A. Gowrisankar. "On Approximation Properties of Fractional Integral for A-Fractal Function." Mathematical Problems in Engineering 2022 (August 24, 2022): 1–17. http://dx.doi.org/10.1155/2022/6409656.
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In this paper, the Riemann–Liouville fractional integral of an A-fractal function is explored by taking its vertical scaling factors in the block matrix as continuous functions from 0,1 to ℝ . As the scaling factors play a significant role in the generation of fractal functions, the necessary condition for the scaling factors in the block matrix is outlined for the newly obtained function. The resultant function of the fractional integral is demonstrated as an A-fractal function if the scaling factors obey the necessary conditions. Furthermore, this article proposes a fractional operator which defines the Riemann–Liouville fractional integral of an A-fractal function for each continuous function on C I , ℝ 2 , where C I , ℝ 2 is the space of all continuous functions from closed interval I ⊂ ℝ to ℝ 2 . In addition, the approximation properties such as linearity, boundedness, and semigroup property of the proposed fractional operator are investigated.
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Susanti, Eka, Fitri Maya Puspita, Siti Suzlin Supadi, Evi Yuliza, and Ahmad Farhan Ramadhan. "Improve Fuzzy Inventory Model of Fractal Interpolation with Vertical Scaling Factor." Science and Technology Indonesia 8, no. 4 (2023): 654–59. http://dx.doi.org/10.26554/sti.2023.8.4.654-659.
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The inventory model is used to determine the optimal inventory of a product. In certain cases, parameters in the inventory model are uncertain. Fractal interpolation techniques can be used to overcome parameter with uncertainty. Fractal interpolation results are affected by the fractal interpolation function and the vertical scaling factor. The vertical scaling factor is positive and less than 1. In this study, fractal interpolation techniques are introduced with variations in vertical scaling factor to overcome the uncertainty of demand data in inventory models. Furthermore, the interpolation results are used in fuzzy inventory models and expressed by Trapezoidal Fuzzy Number. This paper considers an inventory model with varying demand to optimize rice inventory. Based on the data obtained, the accuracy level will increase for the vertical scaling factor values close to 1. Optimal rice inventory of each successive fuzzy parameter is 1447963, 1013914, 504950, 215312. If the cost parameter is increased, then the amount of inventory is decreases.
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Dobson, F. Stephen, Bertram Zinner, and Marina Silva. "Testing models of biological scaling with mammalian population densities." Canadian Journal of Zoology 81, no. 5 (2003): 844–51. http://dx.doi.org/10.1139/z03-060.
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Two hypotheses have been suggested to explain the form of interspecific scaling of organismal characteristics to body size, such as the well-known increase in total metabolism with body mass. A hypothesis based on simple Euclidean geometry suggests that the scaling of many biological variables to body size should have a scaling exponent of 2/3, or [Formula: see text]0.667. On the other hand, according to a hypothesis based on fractal dimensions, the relationship between biological variables and body mass should have a scaling exponent of 0.750. We conducted a power analysis of the predicted exponents of scaling under the Euclidean and fractal hypotheses, using average adult body masses and population densities collected from the published literature on mammalian species. The collected data reflect 987 mammal populations from a broad variety of terrestrial habitats. Using statistical methods we determined the sample sizes required to decide between the values of the scaling exponent of the density-to-mass relationship based on the Euclidean (0.667) and fractal (0.750) hypotheses. Non-linearities in the dataset and insufficient power plagued our tests of the predictions. We found that mammalian species weighing less than 100 kg had a linear scaling pattern, sufficient power to reveal a difference between the scaling coefficients 0.667 and 0.750, and an actual scaling coefficient of 0.719 (barely significantly different from 0.667 but not from 0.750). Thus, our results support the fractal hypothesis, though the support was not particularly strong, which suggests that the relationship between body mass and population density should have a scaling exponent of 0.750.
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MUKHERJEE, SONALI, and HISAO NAKANISHI. "EFFECT OF BOUNDARY TETHERING ON VIBRATIONAL MODES OF FRACTALS." Fractals 04, no. 03 (1996): 273–78. http://dx.doi.org/10.1142/s0218348x96000376.
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We introduce a mapping between a tethered scalar elastic network and a diffusion problem with permanent traps. Various vibrational properties of progressively tethered fractals are discussed using this analogy both in terms of scaling ansatz and numerically by approximately diagonalizing the corresponding large random matrices, with the critical percolation cluster as an example of a fractal.
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Li, Q., and G. M. Xu. "Local scaling property of seismicity: an example of getting valuable information from complex hierarchical system." Nonlinear Processes in Geophysics 17, no. 5 (2010): 423–29. http://dx.doi.org/10.5194/npg-17-423-2010.
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Abstract. In order to get valuable information about the scale invariance in the process of seismogeny of Lijiang M 7.0 earthquake, the scaling property of the interevent time series of the seismic sequences for Lijiang area in China were studied by using the method of the local scaling property, the generalised dimension spectrum, and the correlation dimension. It is found that there is a clear characteristic variation of local scaling property prior to Lijiang M 7.0 earthquake while there is no characteristic variation of the generalised dimension spectrum and the correlation dimension. The reason for producing this phenomenon is that the fractal seismic system is a complex hierarchical system. For such a system, searching for a relevant choice in application of the three methodologies is needed. Compared with the generalised dimension spectrum and mono-fractal dimension which focus on the global description of the scaling properties of fractal objects, the local scaling property emphasizes the local features, and can give the local information of the singularity of the fractal system, therefore, it is easier for us to get valuable information from complex hierarchical structure with this method.
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CHEN, YANGUANG. "THE SPATIAL MEANING OF PARETO'S SCALING EXPONENT OF CITY-SIZE DISTRIBUTIONS." Fractals 22, no. 01n02 (2014): 1450001. http://dx.doi.org/10.1142/s0218348x14500017.
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The scaling exponent of a hierarchy of cities used to be regarded as a fractional dimension. The Pareto exponent was treated as the fractal dimension of size distribution of cities, while the Zipf exponent was considered to be the reciprocal of the fractal dimension. However, this viewpoint is not exact. In this paper, I will present a new interpretation of the scaling exponent of rank-size distributions. The ideas from fractal measure relation and the principle of dimension consistency are employed to explore the essence of Pareto's and Zipf's scaling exponents. The Pareto exponent proved to be a ratio of the fractal dimension of a network of cities to the average dimension of city population. Accordingly, the Zipf exponent is the reciprocal of this dimension ratio. On a digital map, the Pareto exponent can be defined by the scaling relation between a map scale and the corresponding number of cities based on this scale. The cities of the United States of America in 1900, 1940, 1960, and 1980 and Indian cities in 1981, 1991, and 2001 are utilized to illustrate the geographical spatial meaning of Pareto's exponent. The results suggest that the Pareto exponent of city-size distributions is a dimension ratio rather than a fractal dimension itself. This conclusion is revealing for scientists to understand Zipf's law on the rank-size pattern and the fractal structure of hierarchies of cities.
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Viswanathan, P., A. K. B. Chand, and K. R. Tyada. "Lacunary Interpolation by Fractal Splines with Variable Scaling Parameters." Numerical Mathematics: Theory, Methods and Applications 10, no. 1 (2017): 65–83. http://dx.doi.org/10.4208/nmtma.2017.m1514.
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AbstractFor a prescribed set of lacunary data with equally spaced knot sequence in the unit interval, we show the existence of a family of fractal splines satisfying for v = 0, 1, … ,N and suitable boundary conditions. To this end, the unique quintic spline introduced by A. Meir and A. Sharma [SIAM J. Numer. Anal. 10(3) 1973, pp. 433-442] is generalized by using fractal functions with variable scaling parameters. The presence of scaling parameters that add extra “degrees of freedom”, self-referentiality of the interpolant, and “fractality” of the third derivative of the interpolant are additional features in the fractal version, which may be advantageous in applications. If the lacunary data is generated from a function Φ satisfying certain smoothness condition, then for suitable choices of scaling factors, the corresponding fractal spline satisfies , as the number of partition points increases.