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Journal articles on the topic 'Mathematics-Topology - Fractals'

Author: Grafiati

Published: 4 June 2021

Last updated: 28 January 2023

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1

LAPIDUS, MICHEL L. "FRACTALS AND VIBRATIONS: CAN YOU HEAR THE SHAPE OF A FRACTAL DRUM?" Fractals 03, no. 04 (December 1995): 725–36. http://dx.doi.org/10.1142/s0218348x95000643.

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We study various aspects of the question “Can one hear the shape of a fractal drum?”, both for “drums with fractal boundary” (or “surface fractals”) and for “drums with fractal membrane” (or “mass fractals”).

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2

Semkow, Thomas M. "Neighborhood Volume for Bounded, Locally Self-Similar Fractals." Fractals 05, no. 01 (March 1997): 23–33. http://dx.doi.org/10.1142/s0218348x97000048.

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We derive the formulas for neighborhood volume (Minkowski volume in d-dimensions) for fractals which have a curvature bias and are thus bounded. Both local surface fractal dimension and local mass fractal dimension are included as well as a radius of the neighborhood volume comparable with the size of the fractal. We consider two types of the neighborhood volumes: simplified and generalized, as well as the volumes below and above the fractal boundary. The formulas derived are generalizations of the equations for isotropic unbounded fractals. Based on the simplified-volume concept, we establish the procedure for calculating a distribution of physical quantities on bounded fractals and apply it to the distribution of trace elements in soil particles. Using the concept of the generalized volume, we show how an expectation value of a physical process can be calculated on bounded fractals, and apply it to the radon emanation from solid particles.

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3

BAK, PER, and MAYA PACZUSKI. "THE DYNAMICS OF FRACTALS." Fractals 03, no. 03 (September 1995): 415–29. http://dx.doi.org/10.1142/s0218348x95000345.

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Fractals are formed by avalanches, driving the system toward a critical state. This critical state is a fractal in d spatial plus one temporal dimension. Long range spatial and temporal properties are described by different cuts in this fractal attractor. We unify the origin of fractals, 1/f noise, Hurst exponents, Levy flights, and punctuated equilibria in terms of avalanche dynamics, and elucidate their relationships through analytical and numerical studies of simple models.

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ZHANG, XIN-MIN, L. RICHARD HITT, BIN WANG, and JIU DING. "SIERPIŃSKI PEDAL TRIANGLES." Fractals 16, no. 02 (June 2008): 141–50. http://dx.doi.org/10.1142/s0218348x08003934.

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We generalize the construction of the ordinary Sierpiński triangle to obtain a two-parameter family of fractals we call Sierpiński pedal triangles. These fractals are obtained from a given triangle by recursively deleting the associated pedal triangles in a manner analogous to the construction of the ordinary Sierpiński triangle, but their fractal dimensions depend on the choice of the initial triangles. In this paper, we discuss the fractal dimensions of the Sierpiński pedal triangles and the related area ratio problem, and provide some computer-generated graphs of the fractals.

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5

LI, WEN XIA. "THE DIMENSION OF SETS DETERMINED BY THEIR CODE BEHAVIOR." Fractals 11, no. 04 (December 2003): 345–52. http://dx.doi.org/10.1142/s0218348x0300218x.

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By prescribing their code run behavior, we consider some subsets of Moran fractals. Fractal dimensions of these subsets are exactly obtained. Meanwhile, an interesting decomposition of Moran fractals is given.

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6

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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DEMÍR, BÜNYAMIN, ALI DENÍZ, ŞAHIN KOÇAK, and A. ERSIN ÜREYEN. "TUBE FORMULAS FOR GRAPH-DIRECTED FRACTALS." Fractals 18, no. 03 (September 2010): 349–61. http://dx.doi.org/10.1142/s0218348x10004919.

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Lapidus and Pearse proved recently an interesting formula about the volume of tubular neighborhoods of fractal sprays, including the self-similar fractals. We consider the graph-directed fractals in the sense of graph self-similarity of Mauldin-Williams within this framework of Lapidus-Pearse. Extending the notion of complex dimensions to the graph-directed fractals we compute the volumes of tubular neighborhoods of their associated tilings and give a simplified and pointwise proof of a version of Lapidus-Pearse formula, which can be applied to both self-similar and graph-directed fractals.

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8

XIANG, ZHIYANG, KAI-QING ZHOU, and YIBO GUO. "GAUSSIAN MIXTURE NOISED RANDOM FRACTALS WITH ADVERSARIAL LEARNING FOR AUTOMATED CREATION OF VISUAL OBJECTS." Fractals 28, no. 04 (June 2020): 2050068. http://dx.doi.org/10.1142/s0218348x20500681.

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Because of the self-similarity properties of nature, fractals are widely adopted as generators of natural object multimedia contents. Unfortunately, fractals are difficult to control due to their iterated function systems, and traditional researches on fractal generating visual objects focus on mathematical manipulations. In Generative Adversarial Nets (GANs), visual object generators can be automatically guided by a single image. In this work, we explore the problem of guiding fractal generators with GAN. We assume that the same category of fractal patterns is produced by a group of parameters of initial patterns, affine transformations and random noises. Connections between these fractal parameters and visual objects are modeled by a Gaussian mixture model (GMM). Generator trainings are performed as gradients on GMM instead of fractals, so that evaluation numbers of iterated function systems are minimized. The proposed model requires no mathematical expertise from the user because parameters are trained by automatic procedures of GMM and GAN. Experiments include one 2D demonstration and three 3D real-world applications, where high-resolution visual objects are generated, and a user study shows the effectiveness of artificial intelligence guidances on fractals.

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9

CRISTEA, LIGIA L., and PAUL SURER. "TRIANGULAR LABYRINTH FRACTALS." Fractals 27, no. 08 (December 2019): 1950131. http://dx.doi.org/10.1142/s0218348x19501317.

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We define and study a class of fractal dendrites called triangular labyrinth fractals. For the construction, we use triangular labyrinth pattern systems, consisting of two triangular patterns: a white and a yellow one. Correspondingly, we have two fractals: a white and a yellow one. The fractals studied here are self-similar, and fit into the framework of graph directed constructions. The main results consist in showing how special families of triangular labyrinth patterns systems, which are defined based on some shape features, can generate exactly three types of dendrites: labyrinth fractals where all nontrivial arcs have infinite length, fractals where all nontrivial arcs have finite length, or fractals where the only arcs of finite lengths are line segments parallel to a certain direction. We also study the existence of tangents to arcs. The paper is inspired by research done on labyrinth fractals in the unit square that have been studied during the last decade. In the triangular case, due to the geometry of triangular shapes, some new techniques and ideas are necessary in order to obtain the results.

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LAI, PENG-JEN. "HOW TO MAKE FRACTAL TILINGS AND FRACTAL REPTILES." Fractals 17, no. 04 (December 2009): 493–504. http://dx.doi.org/10.1142/s0218348x09004533.

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Intensive research on fractals began around 1980 and many new discoveries have been made. However, the connection between fractals, tilings and reptiles has not been thoroughly explored. This paper shows that a method, similar to that used to construct irregular tilings in ℜ2 can be employed to construct fractal tilings. Five main methods, including methods in Escher style paintings and the Conway criterion are used to create the fractal tilings. Also an algorithm is presented to generate fractal reptiles. These methods provide a more geometric way to understand fractal tilings and fractal reptiles and complements iteration methods.

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HUGHES, JAMES R. "FRACTALS IN A FIRST YEAR UNDERGRADUATE SEMINAR." Fractals 11, no. 01 (March 2003): 109–23. http://dx.doi.org/10.1142/s0218348x03001410.

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The study of basic fractal geometry can help build students' enthusiasm for learning early in their undergraduate careers. To most undergraduate students, fractals are new, visually appealing, useful, and mathematically accessible. As a result, fractals can be an effective vehicle for introducing and reinforcing multiple modes of learning, which at many institutions is one of the main goals of general first-year undergraduate education. This article describes how fractals are used in one institution's "Freshman Seminar" program to help accomplish these goals.

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Coppens, Marc-Olivier, and Gilbert F. Froment. "The Effectiveness of Mass Fractal Catalysts." Fractals 05, no. 03 (September 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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13

HUYNH, HOAI NGUYEN, and LOCK YUE CHEW. "ARC-FRACTAL AND THE DYNAMICS OF COASTAL MORPHOLOGY." Fractals 19, no. 02 (June 2011): 141–62. http://dx.doi.org/10.1142/s0218348x11005178.

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In this paper, we present an idea of creating fractals by using the geometric arc as the basic element. This approach of generating fractals, through the tuning of just three parameters, gives a universal way to obtain many novel fractals including the classic ones. Although this arc-fractal system shares similar features with the well-known Lindenmayer system, such as the same set of invariant points and the ability to tile the space, they do have different properties. One of which is the generation of pseudo-random number, which is not available in the Lindenmayer system. Furthermore, by assuming that coastline formation is based purely on the processes of erosion and deposition, the arc-fractal system can also serve as a dynamical model of coastal morphology, with each level of its construction corresponds to the time evolution of the shape of the coastal features. Remarkably, our results indicate that the arc-fractal system can provide an explanation on the origin of fractality in real coastline.

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14

JÁNOSI, IMRE M., and ANDRÁS CZIRÓK. "FRACTAL CLUSTERS AND SELF-ORGANIZED CRITICALITY." Fractals 02, no. 01 (March 1994): 153–68. http://dx.doi.org/10.1142/s0218348x94000156.

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Self-organized criticality (SOC) and fractals have been shown to be related in various ways. On the one hand, the original idea of SOC suggests that the common explanation of the origin of fractal shapes in nature may be based on self-organized processes. Thus different models exhibiting SOC result in relaxation clusters or avalanches whose geometrical characteristics could be described by fractals. On the other hand, there exist several models for fractal growth phenomena, such as viscous fingering, invasion percolation, dielectric breakdown, etc., and it is possible that the concept of SOC may help in finding the common feature of these models. In this paper we review the recent results on self-organized critical behaviour in various fractal growth models. Next we discuss the relation of fractals and self-organized criticality by concentrating on the geometrical properties of SOC clusters in 2–4 dimensions. A short analysis of the cluster growth processes is given as well.

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15

SUZUKI, MASUO. "FRACTAL FORM ANALYSIS." Fractals 04, no. 03 (September 1996): 237–39. http://dx.doi.org/10.1142/s0218348x96000327.

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16

PE, JOSEPH L. "ANA'S GOLDEN FRACTAL." Fractals 11, no. 04 (December 2003): 309–13. http://dx.doi.org/10.1142/s0218348x03002269.

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In his fascinating book Wonders of Numbers, Clifford Pickover introduces the Ana sequence and fractal, two self-referential constructions arising from the use of language. This paper answers Pickover's questions on the relative composition of sequence terms and the dimension of the fractal. In the process, it introduces a novel way of obtaining fractals from iterative set operations. Also, it presents a beautiful variant of the Ana constructions involving the golden ratio. In conclusion, it suggests ways of constructing similar fractals for the Morse-Thue and "Look and Say" sequences.

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17

TATOM, FRANK B. "THE RELATIONSHIP BETWEEN FRACTIONAL CALCULUS AND FRACTALS." Fractals 03, no. 01 (March 1995): 217–29. http://dx.doi.org/10.1142/s0218348x95000175.

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The general relationship between fractional calculus and fractals is explored. Based on prior investigations dealing with random fractal processes, the fractal dimension of the function is shown to be a linear function of the order of fractional integro-differentiation. Emphasis is placed on the proper application of fractional calculus to the function of the random fractal, as opposed to the trail. For fractional Brownian motion, the basic relations between the spectral decay exponent, Hurst exponent, fractal dimension of the function and the trail, and the order of the fractional integro-differentiation are developed. Based on an understanding of fractional calculus applied to random fractal functions, consideration is given to an analogous application to deterministic or nonrandom fractals. The concept of expressing each coordinate of a deterministic fractal curve as a “pseudo-time” series is investigated. Fractional integro-differentiation of such series is numerically carried out for the case of quadric Koch curves. The resulting time series produces self-similar patterns with fractal dimensions which are linear functions of the order of the fractional integro-differentiation. These curves are assigned the name, fractional Koch curves. The general conclusion is reached that fractional calculus can be used to precisely change or control the fractal dimension of any random or deterministic fractal with coordinates which can be expressed as functions of one independent variable, which is typically time (or pseudo-time).

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18

FANG, LINCONG, DOMINIQUE MICHELUCCI, and SEBTI FOUFOU. "EQUATIONS AND INTERVAL COMPUTATIONS FOR SOME FRACTALS." Fractals 26, no. 04 (August 2018): 1850059. http://dx.doi.org/10.1142/s0218348x18500597.

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Very few characteristic functions, or equations, are reported so far for fractals. Such functions, called Rvachev functions in function-based modeling, are zero on the boundary, negative for inside points and positive for outside points. This paper proposes Rvachev functions for some classical fractals. These functions are convergent series, which are bounded with interval arithmetic and interval analysis in finite time. This permits to extend the Recursive Space Subdivision (RSS) method, which is classical in Computer Graphics (CG) and Interval Analysis, to fractal geometric sets. The newly proposed fractal functions can also be composed with classical Rvachev functions today routinely used in Constructive Solid Geometry (CSG) trees of CG or function-based modeling.

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ZENG, QIUHUA, and HOUQIANG LI. "MULTISCALING TRANSPORT EQUATION ON FRACTALS." Fractals 07, no. 02 (June 1999): 105–11. http://dx.doi.org/10.1142/s0218348x99000128.

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The general transport equation of multiscaling disordered fractal media in three-dimensional case is derived from conservation of mass. Multiscaling fractional transport equation is obtained on the basis of discussing Brownian motion, fractional Brownian motion and standard diffusion equation of fractals, which is consistent with the result of literature.

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LI, TINGTING, KAN JIANG, and LIFENG XI. "AVERAGE DISTANCE OF SELF-SIMILAR FRACTAL TREES." Fractals 26, no. 01 (February 2018): 1850016. http://dx.doi.org/10.1142/s0218348x18500160.

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In this paper, we introduce a method which can generate a family of growing symmetrical tree networks. The networks are constructed by replacing each edge with a reduced-scale of the initial graph. Repeating this procedure, we obtain the fractal networks. In this paper, we define the average geodesic distance of fractal tree in terms of some integral, and calculate its accurate value. We find that the limit of the average geodesic distance of the finite networks tends to the average geodesic distance of the fractal tree. This result generalizes the paper [Z. Zhang, S. Zhou, L. Chen, M. Yin and J. Guan, Exact solution of mean geodesic distance for Vicsek fractals, J. Phys. A: Math. Gen. 41(48) (2008) 7199–7200] for which the mean geodesic distance of Vicsek fractals was considered.

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21

BARNSLEY, MICHAEL, JOHN HUTCHINSON, and ÖRJAN STENFLO. "A FRACTAL VALUED RANDOM ITERATION ALGORITHM AND FRACTAL HIERARCHY." Fractals 13, no. 02 (June 2005): 111–46. http://dx.doi.org/10.1142/s0218348x05002799.

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We describe new families of random fractals, referred to as "V-variable", which are intermediate between the notions of deterministic and of standard random fractals. The parameter V describes the degree of "variability": at each magnification level any V-variable fractals has at most V key "forms" or "shapes". V-variable random fractals have the surprising property that they can be computed using a forward process. More precisely, a version of the usual Random Iteration Algorithm, operating on sets (or measures) rather than points, can be used to sample each family. To present this theory, we review relevant results on fractals (and fractal measures), both deterministic and random. Then our new results are obtained by constructing an iterated function system (a super IFS) from a collection of standard IFSs together with a corresponding set of probabilities. The attractor of the super IFS is called a superfractal; it is a collection of V-variable random fractals (sets or measures) together with an associated probability distribution on this collection. When the underlying space is for example ℝ2, and the transformations are computationally straightforward (such as affine transformations), the superfractal can be sampled by means of the algorithm, which is highly efficient in terms of memory usage. The algorithm is illustrated by some computed examples. Some variants, special cases, generalizations of the framework, and potential applications are mentioned.

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22

Mintz, R. I., and D. B. Berg. "Fractals in Transition Processes." Fractals 05, supp01 (April 1997): 145–52. http://dx.doi.org/10.1142/s0218348x97000711.

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The present work is devoted to the design of model physical and computer systems with competitive interactions and analysis of structures, generated by their various metastable states. As it was marked at the NATO Advanced Study Institute "Fractal image encoding and analysis" (Norway, 1995), that description and quantitative analysis of fractals forming in such systems at the self-organizing, transition processes is an unsolved problem, so called "terra incognita", which is requiring future investigations.

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23

Zair, C. E., and E. Tosan. "Fractal Geometric Modeling in Computer Graphics." Fractals 05, supp02 (October 1997): 45–61. http://dx.doi.org/10.1142/s0218348x97000826.

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Free form techniques and fractals are complementary tools for modeling respectively man-made objects and complex irregular shapes. Fractal techniques, having the advantage of describing self-similar objects, suffer from the drawback of a lack of control of the fractal figures. In contrast, free form techniques provide a high flexibility with smooth figures. Our work focuses on the definition of an IFS-based model designed to inherit the advantages of fractals and free form techniques (control by a set of control points, convex hull) in order to manipulate fractal figures in the way as classical free form shapes (BÉZIER, spline). The work is essentially based on the study of the functional equation Φ(τ * t) = T Φ (t), where Φ is a continuous function, τ and T are both contractive affine operators. We prove that there is a strong relationship between this functional equation and IFS attractors. This relationship will be used for the construction of parametric fractal attractors.

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WANG, LEI, and SHENGWEN TANG. "EDITORIAL: AN INTRODUCTION TO FRACTALS IN CONSTRUCTION MATERIALS." Fractals 29, no. 02 (February 16, 2021): 2102001. http://dx.doi.org/10.1142/s0218348x21020011.

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In recent years, the application of fractal theory in construction materials has drawn tremendous attention worldwide. This special issue section containing seven papers publishes the recent advances in the investigation and application of fractal-based approaches implemented in construction materials. The topics covered in this introduction mainly include: (1) the fractal characterization of construction materials from nano- to micro-scales; (2) combining fractals methods with other theoretical, numerical and/or experimental methods to evaluate or predict the macroscopic behavior of construction materials; (3) the relationship of fractal dimension with the macro-properties (i.e. mechanical property, shrinkage behavior, permeability, frost resistance, abrasion resistance, etc.) of construction materials.

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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 (May 28, 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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ELIA, MATTEO, and ROBERTO PEIRONE. "EIGENFORMS ON FRACTALS WITH CONNECTED INTERIOR AND THREE VERTICES." Fractals 26, no. 04 (August 2018): 1850082. http://dx.doi.org/10.1142/s0218348x18500822.

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An important problem in analysis on fractals is the existence and the determination of an eigenform on a given finitely ramified fractal. It is known that on every fractal either with three vertices or with connected interior, an eigenform exists for suitable weights on the cells. In this paper, we prove that if the fractal has three vertices and connected interior, the form having all coefficients equal to [Formula: see text] is an eigenform for suitable weights on the cells.

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Montiel, M. Eugenia, Alberto S. Aguado, and Ed Zaluska. "Fourier Series Expansion of Irregular Curves." Fractals 05, no. 01 (March 1997): 105–19. http://dx.doi.org/10.1142/s0218348x97000115.

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Fourier theory provides an important approach to shape analyses; many methods for the analysis and synthesis of shapes use a description based on the expansion of a curve in Fourier series. Most of these methods have centered on modeling regular shapes, although irregular shapes defined by fractal functions have also been considered by using spectral synthesis. In this paper we propose a novel representation of irregular shapes based on Fourier analysis. We formulate a parametric description of irregular curves by using a geometric composition defined via Fourier expansion. This description allows us to model a wide variety of fractals which include not only fractal functions, but also fractals belonging to other families. The coefficients of the Fourier expansion can be parametrized in time in order to produce sequences of fractals useful for modeling chaotic dynamics. The aim of the novel characterization is to extend the potential of shape analyses based on Fourier theory by including a definition of irregular curves. The major advantage of this new approach is that it provides a way of studying geometric aspects useful for shape identification and extraction, such as symmetry and similarity as well as invariant features.

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Shalaev, V. M., E. Y. Poliakov, V. A. Markel, V. P. Safonov, and A. K. Sarychev. "Surface-Enhanced Optical Nonlinearities of Nanostructured Fractal Materials." Fractals 05, supp02 (October 1997): 63–82. http://dx.doi.org/10.1142/s0218348x97000838.

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Localization of optical excitations in fractal metal nanocomposites, such as colloidal aggregates, self-affine and semicontinuous films, results in strong enhancements of optical nonlinearities. The localized modes of fractals cover a broad spectral range, from the visible to the far-infrared. Results of both theoretical and experimental studies of enhanced nonlinear optical responses of fractal nanostructures are considered.

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KALETA, KAMIL, MARIUSZ OLSZEWSKI, and KATARZYNA PIETRUSKA-PAŁUBA. "REFLECTED BROWNIAN MOTION ON SIMPLE NESTED FRACTALS." Fractals 27, no. 06 (September 2019): 1950104. http://dx.doi.org/10.1142/s0218348x19501044.

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For a large class of planar simple nested fractals, we prove the existence of the reflected diffusion on a complex of an arbitrary size. Such a process is obtained as a folding projection of the free Brownian motion from the unbounded fractal. We give sharp necessary geometric conditions for the fractal under which this projection can be well defined, and illustrate them by numerous examples. We then construct a proper version of the transition probability densities for the reflected process and we prove that it is a continuous, bounded and symmetric function which satisfies the Chapman–Kolmogorov equations. These provide us with further regularity properties of the reflected process such us Markov, Feller and strong Feller property.

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ZENG, QIUHUA, and HOUQIANG LI. "DIFFUSION EQUATION FOR DISORDERED FRACTAL MEDIA." Fractals 08, no. 01 (March 2000): 117–21. http://dx.doi.org/10.1142/s0218348x00000123.

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The movement of the fractal Brownian particle in isotropic and homogeneous two-dimensional assembling fractal spaces is studied by the standard diffusion equation on fractals, and we find that particle movement belongs to the anomalous diffusion. At the same time, by discussing the defectiveness of earlier proposed equations, a general form of analytic fractional diffusion equation is proposed for description of probability density of particles diffusing on fractal geometry at fractal time, and the solution connects with the ordinary solutions in the normal space time limit.

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STRUZIK, ZBIGNIEW R. "SOLVING THE TWO-DIMENSIONAL INVERSE FRACTAL PROBLEM WITH THE WAVELET TRANSFORM." Fractals 04, no. 04 (December 1996): 469–75. http://dx.doi.org/10.1142/s0218348x96000583.

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The methodology of the solution to the inverse fractal problem with the wavelet transform1,2 is extended to two-dimensional self-affine functions. Similar to the one-dimensional case, the two-dimensional wavelet maxima bifurcation representation used is derived from the continuous wavelet decomposition. It possesses translational and scale invariance necessary to reveal the invariance of the self-affine fractal. As many fractals are naturally defined on two-dimensions, this extension constitutes an important step towards solving the related inverse fractal problem for a variety of fractal types.

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MUKHERJEE, SONALI, and HISAO NAKANISHI. "EFFECT OF BOUNDARY TETHERING ON VIBRATIONAL MODES OF FRACTALS." Fractals 04, no. 03 (September 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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PAREDES, R. "CAPILLARY DISPLACEMENT ON DETERMINISTIC AND RANDOM FRACTALS." Fractals 01, no. 04 (December 1993): 887–93. http://dx.doi.org/10.1142/s0218348x93000927.

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Capillary displacement in deterministic and random fractals is explored numerically and analytically using Invasion Percolation (IP) and Eden growth (EG) models. Both compressible and incompressible native fluid cases are studied. It is found that capillary displacements are markedly different for supports having or lacking singly connected links at all scales. It is shown that IP and EG models fill the support if there are links at all scales, for the compressible case. For the incompressible case, on the other hand, a formula is derived which relates the fractal dimension of EG cluster and the dimension of the minimal path and other non-universal quantities. Numerical estimates of the fractal dimension of IP clusters in several supports with links are obtained, and are shown to be different from the corresponding EG values for the incompressible case.

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MILLÁN, H., M. AGUILAR, J. DOMÌNGUEZ, L. CÈSPEDES, E. VELASCO, and M. GÒNZALEZ. "A NOTE ON THE PHYSICS OF SOIL WATER RETENTION THROUGH FRACTAL PARAMETERS." Fractals 14, no. 02 (June 2006): 143–48. http://dx.doi.org/10.1142/s0218348x06003131.

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Fractals are important for studying the physics of water transport in soils. Many authors have assumed a mass fractal structure while others consider a fractal surface approach. Each model needs comparisons on the same data set in terms of goodness-of-fit and physical interpretation of parameters. In this note, it is shown, with some representative data sets, that a pore-solid interface fractal model could fit soil water retention data better than a mass fractal model. In addition to the interfacial fractal dimension, this model predicts the tension at dryness. This value is very close to 106 kPa as theoretically predicted.

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Zhao, Shanrong, Jin Tan, Jiyang Wang, Xiaohong Xu, and Hong Liu. "A Dendrite with "Sierpinski Gasket" Fractal Morphology in Matt Glaze of LiAlSiO4-SiO2 System." Fractals 11, no. 03 (September 2003): 271–76. http://dx.doi.org/10.1142/s0218348x03001525.

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In this paper, we introduce a dendritic crystal, formed in matt glaze of LiAlSiO 4- SiO 2, having "Sierpinski gasket" fractal morphology. The crystal structure of this "Sierpinski gasket" dendrite is β-quartz. β-quartz can grow two kinds of fractal patterns: snow-shaped dendrite and "Sierpinski gasket" dendrite, depending on different supercooling conditions. These two kinds of fractals can develop together in one dendritic crystal. The evolution of the boundary morphologies between these two kinds of fractal dendrites can be described by another fractal — Koch curve. The "Sierpinski gasket" dendrite is a rather new fractal growth pattern which can introduce new opportunities to fractal growth research of nonlinear sciences.

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NAIR, PRADEEP R., and MUHAMMAD A. ALAM. "KINETIC RESPONSE OF SURFACES DEFINED BY FINITE FRACTALS." Fractals 18, no. 04 (December 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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CAI, JIANCHAO, FERNANDO SAN JOSÉ MARTÍNEZ, MIGUEL ANGEL MARTÍN, and EDMUND PERFECT. "AN INTRODUCTION TO FLOW AND TRANSPORT IN FRACTAL MODELS OF POROUS MEDIA: PART I." Fractals 22, no. 03 (September 2014): 1402001. http://dx.doi.org/10.1142/s0218348x14020010.

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This special issue gathers together a number of recent papers on fractal geometry and its applications to the modeling of flow and transport in porous media. The aim is to provide a systematic approach for analyzing the statics and dynamics of fluids in fractal porous media by means of theory, modeling and experimentation. The topics covered include lacunarity analyses of multifractal and natural grayscale patterns, random packing's of self-similar pore/particle size distributions, Darcian and non-Darcian hydraulic flows, diffusion within fractals, models for the permeability and thermal conductivity of fractal porous media and hydrophobicity and surface erosion properties of fractal structures.

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RAMÍREZ, JOSÉ L., GUSTAVO N. RUBIANO, and BORUT JURČIČ ZLOBEC. "GENERATING FRACTAL PATTERNS BY USING p-CIRCLE INVERSION." Fractals 23, no. 04 (December 2015): 1550047. http://dx.doi.org/10.1142/s0218348x15500474.

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In this paper, we introduce the [Formula: see text]-circle inversion which generalizes the classical inversion with respect to a circle ([Formula: see text]) and the taxicab inversion [Formula: see text]. We study some basic properties and we also show the inversive images of some basic curves. We apply this new transformation to well-known fractals such as Sierpinski triangle, Koch curve, dragon curve, Fibonacci fractal, among others. Then we obtain new fractal patterns. Moreover, we generalize the method called circle inversion fractal be means of the [Formula: see text]-circle inversion.

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MEI, MAOFEI, BOMING YU, JIANCHAO CAI, and LIANG LUO. "A HIERARCHICAL MODEL FOR MULTI-PHASE FRACTAL MEDIA." Fractals 18, no. 01 (March 2010): 53–64. http://dx.doi.org/10.1142/s0218348x1000466x.

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The size distributions of solid particles and pores in porous media are approximately hierarchical and statistically fractals. In this paper, a model for single-phase fractal media is constructed, and the analytical expressions for area, fractal dimension and distribution function for solid particles are derived. The distribution function of solid particles obtained from the proposed model is in good agreement with available experimental data. Then, a model for approximate two-phase fractal media is developed. Good agreement is found between the predicted fractal dimensions for pore space from the two-phase fractal medium model and the existing measured data. A model for approximate three-phase fractal media is also presented by extending the obtained two-phase model.

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FORD, DANIEL J., and BENJAMIN STEINHURST. "VIBRATION SPECTRA OF THE m-TREE FRACTAL." Fractals 18, no. 02 (June 2010): 157–69. http://dx.doi.org/10.1142/s0218348x1000483x.

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We introduce a family of post-critically finite fractal trees indexed by the number of branches they possess. Then we produce a Laplacian operator on graph approximations to these fractals and use spectral decimation to describe the spectrum of the Laplacian on these trees. Lastly we consider the behavior of the spectrum as the number of branches increases.

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HUBER, GREG, MOGENS H. JENSEN, and KIM SNEPPEN. "A DIMENSION FORMULA FOR SELF-SIMILAR AND SELF-AFFINE FRACTALS." Fractals 03, no. 03 (September 1995): 525–31. http://dx.doi.org/10.1142/s0218348x9500045x.

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A geometric and very general relation between the size distribution and the fractal dimensions of a set of objects is presented. The applications are numerous, ranging from fragmentation experiments to time series. For example, it may be used to understand the fragment-size distribution of fragmenting gypsum. The formalism also generalizes to self-affine fractals, and here it is applied to the scaling properties of self-interactions in (1+1)-d directed percolation.

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BARBÉ, ANDRÉ M. "FRACTALS BY NUMBERS." Fractals 03, no. 04 (December 1995): 651–61. http://dx.doi.org/10.1142/s0218348x95000588.

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We introduce an extension of an earlier defined simple, number-based matrix substitution system for obtaining fractal matrices, by considering cyclic substitutions. The elements of the resulting matrices are related to representations of their addresses in a mixed number base. The Hutchinson operator for the limit form of a geometrical representation of the fractal matrix is derived. It is shown that the class of fractal limit sets obtainable from cyclic substitutions does not extend the class obtainable from the simple substitutions.

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BAJORIN, N., T. CHEN, A. DAGAN, C. EMMONS, M. HUSSEIN, M. KHALIL, P. MODY, B. STEINHURST, and A. TEPLYAEV. "VIBRATION SPECTRA OF FINITELY RAMIFIED, SYMMETRIC FRACTALS." Fractals 16, no. 03 (September 2008): 243–58. http://dx.doi.org/10.1142/s0218348x08004010.

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We show how to calculate the spectrum of the Laplacian operator on fully symmetric, finitely ramified fractals. We consider well-known examples, such as the unit interval and the Sierpiński gasket, and much more complicated ones, such as the hexagasket and a non-post critically finite self-similar fractal. We emphasize the low computational demands of our method. As a conclusion, we give exact formulas for the limiting distribution of eigenvalues (the integrated density of states), which is a purely atomic measure (except in the classical case of the interval), with atoms accumulating to the Julia set of a rational function. This paper is the continuation of the work published by the same authors in Ref. 1.

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DE MOL, LIESBETH. "STUDY OF FRACTALS DERIVED FROM IFS-FRACTALS BY METRIC PROCEDURES." Fractals 13, no. 03 (September 2005): 237–44. http://dx.doi.org/10.1142/s0218348x05002878.

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It is a well-known fact that when visualizing an IFS-attractor through the chaos game, it is possible that the first points mapped will come closer to but stay visibly different from the attractor. This simple fact will be analyzed in more detail, through visualizations of different aspects of this convergence process. It will be shown that, in applying on every point in a 2D-plane the same sequence of mappings and coloring each point according to convergence distance, neighboring points form structures which resemble the attractor itself. Further, it is in this way possible to generate boundaries of the attractor that vary between small and coarse-grained. Using these results, it will be shown that it is possible to, starting with an IFS-attractor, construct fractals of which this IFS-attractor is a subset.

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JAMPOUR, MAHDI, MAHDI YAGHOOBI, MARYAM ASHOURZADEH, and ADEL SOLEIMANI. "A NEW FAST TECHNIQUE FOR FINGERPRINT IDENTIFICATION WITH FRACTAL AND CHAOS GAME THEORY." Fractals 18, no. 03 (September 2010): 293–300. http://dx.doi.org/10.1142/s0218348x10005020.

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Fingerprints are one of the simplest and most reliable human biometric features for identification. Geometry of the fingerprint is fractal and we can classify a fingerprint database with fractal dimension, but one can't identify a fingerprint with fractal dimension uniquely. In this paper we present a new approach for identifying fingerprint uniquely; for this purpose a new fractal is initially made from a fingerprint by using Fractal theory and Chaos Game theory. While making the new fractal, five parameters that can be used in identification process can be achieved. Finally a fractal is made for each fingerprint, and then by analyzing the new fractal and parameters obtained by Chaos Game, fingerprint identification can be performed. We called this method Fingerprint Fractal Identification System (FFIS). The presented method besides having features of fractals such as stability against turning, magnifying, deleting a part of image, etc. also has a desirable speed.

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BRAMBILLA, GABRIELE, and D. BRYNN HIBBERT. "MODEL OF RAMIFICATION IN ELECTRODEPOSITED FRACTALS." Fractals 18, no. 04 (December 2010): 477–82. http://dx.doi.org/10.1142/s0218348x10005056.

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The macroscopic branching and shape of a metal, electrodeposited in quasi-two dimensions, can be represented by a Lindenmayer System of multiple iterated function systems. The motif of a central branch that continues and two side branches is shown to allow modeling of the observed shapes. The dilation (or contraction) in length, and the angle of a branch with respect to the central stem are the only parameters needed to create realistic simulations of such ramified deposits. Values for these quantities together with standard deviations measured from a number of electrodeposited copper fractals have been used to generate simulations. Measurement of the parameters of the motif, the average linear dilation factor from one generation to the next, and the angles between branches, show a 5% relative standard deviation of these factors across one growth. The values of these parameters can also indicate the transition from an open fractal form to the more directed dendritic form. The results are compared with other approaches to describing these systems.

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KAHNG, BYUNGIK, and JEREMY DAVIS. "MAXIMAL DIMENSIONS OF UNIFORM SIERPINSKI FRACTALS." Fractals 18, no. 04 (December 2010): 451–60. http://dx.doi.org/10.1142/s0218348x10005135.

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We studied the invariant fractals in convex N-gons given by N identical pure contractions at its vertices with non-overlapping images, which we called uniform Sierpinski fractals. We provided the explicit formulae for the maximal contraction ratio RN, and the maximal Hausdorff dimension hN, for the uniform Sierpinski fractals. We used maximal N-grams and principal crossing points as the main tools.

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TAYLOR, T. D., and S. ROWLEY. "CONVEX HULLS OF SIERPIŃSKI RELATIVES." Fractals 26, no. 06 (December 2018): 1850098. http://dx.doi.org/10.1142/s0218348x18500986.

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This paper presents an investigation of the convex hulls of the Sierpiński relatives. These fractals all have the same fractal dimension but different topologies. We prove that the relatives have convex hulls with polygonal boundaries with at most 12 vertices. We provide a method for finding the convex hull of a relative using its scaling and symmetry properties and present examples. We also investigate the connectivity properties of certain classes of relatives with the same convex hulls.

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FATHALLAH-SHAYKH, HASSAN M. "FRACTAL DIMENSION OF THE DROSOPHILA CIRCADIAN CLOCK." Fractals 19, no. 04 (December 2011): 423–30. http://dx.doi.org/10.1142/s0218348x11005476.

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Fractal geometry can adequately represent many complex and irregular objects in nature. The fractal dimension is typically computed by the box-counting procedure. Here I compute the box-counting and the Kaplan-Yorke dimensions of the 14-dimensional models of the Drosophila circadian clock. Clockwork Orange (CWO) is transcriptional repressor of direct target genes that appears to play a key role in controlling the dynamics of the clock. The findings identify these models as strange attractors and highlight the complexity of the time-keeping actions of CWO in light-day cycles. These fractals are high-dimensional counterexamples of the Kaplan-Yorke conjecture that uses the spectrum of the Lyapunov exponents.

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VINOY, K. J., JOSE K. ABRAHAM, and V. K. VARADAN. "IMPACT OF FRACTAL DIMENSION IN THE DESIGN OF MULTI-RESONANT FRACTAL ANTENNAS." Fractals 12, no. 01 (March 2004): 55–66. http://dx.doi.org/10.1142/s0218348x04002288.

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During the last few decades, fractal geometries have found numerous applications in several fields of science and engineering such as geology, atmospheric sciences, forest sciences, physiology and electromagnetics. Although the very fractal nature of these geometries have been the impetus for their application in many of these areas, a direct quantifiable link between a fractal property such as dimension and antenna characteristics has been elusive thus far. In this paper, the variations in the input characteristics of multi-resonant antennas based on generalizations of Koch curves and fractal trees are examined by numerical simulations. Schemes for such generalizations of these geometries to vary their fractal dimensions are presented. These variations are found to have a direct influence on the primary resonant frequency, the input resistance at this resonance, and ratios resonant frequencies of these antennas. It is expected that these findings would further enhance the popularity of the study of fractals.

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