Showing papers in "Fractals in 1996"
TL;DR: In this paper, the authors studied the relationship between numerous firm characteristics and the entire distribution of growth rates, and analyzed data over many time scales, instead of just a single time interval.
Abstract: In recent years, a breakthrough in statistical physics has occurred. Simply put, statistical physicists have determined that physical systems which consist of a large number of interacting particles obey universal laws that are independent of the microscopic details. This progress was mainly due to the development of scaling theory. Since economic systems also consist of a large number of interacting units, it is plausible that scaling theory can be applied to economics. To test this possibility we study the dynamics of firm size. This may help to build a more complete characterization of the nature and processes behind firm growth. To date, the study of firm dynamics has primarily focused on whether small firms on average have higher growth rates than large firms. To a lesser extent, attention has been placed on the relationship between firm size and variation in growth rate. Our research goes beyond these questions by looking at the relationship between numerous firm characteristics and the entire distribution of growth rates. Thus, it may provide a better understanding of the mechanisms behind firm dynamics. In contrast to previous studies, this research analyzes data over many time scales, instead of just a single time interval. From a scientific standpoint, this work could be useful because it will affect the formulation of firm modeling—one of the basic building blocks of all economic analysis. In addition, this work will have practical applications. For example, there are Federal policies that are designed to encourage small businesses. While such policies might be justified on grounds other than their contribution to growth, any systematic difference in the growth rates of small and large firms might be relevant for evaluating such policies. Also, there has traditionally been a concern that an excessive amount of economic activity might become concentrated in a small number of firms. A more detailed understanding of the firm growth process will provide evidence for whether such concerns have any scientific foundation.
53 citations
TL;DR: The map images clearly show the difference between clouds and rocks, as well as between cancer parts and normal tissue in the colon, in a fractal dimension distribution from natural scenes and medical images constructed by applying the box-counting method locally.
Abstract: We construct color map images of fractal dimension distribution from natural scenes and medical images by applying the box-counting method locally. The map images clearly show the difference between clouds and rocks, as well as between cancer parts and normal tissue in the colon. The method is simple and may be expected to be applicable to a real-time video-data processing.
23 citations
TL;DR: In this article, the volcanic seismicity occurring in Campi Flegrei during the 1983-1984 volcanic crisis is quantitatively analyzed by means of fractal geometry in all its five dimensional set (magnitude, time and space).
Abstract: The volcanic seismicity occurring in Campi Flegrei during the 1983–1984 volcanic crisis is quantitatively analyzed by means of fractal (scale invariant) geometry in all its five dimensional set (magnitude, time and space). The significant power laws, relating the number of earthquakes to magnitude, time and space, within a limited range of scales, show that the Campi Flegrei region is in a critical state during the whole crisis.
21 citations
TL;DR: This paper shows the equivalence between an IFSP and a linear dynamical system driven by a white noise, and uses a multifractal analysis to obtain scaling properties of the resulting invariant measures, working within the framework of dynamical systems.
Abstract: In this paper, we focus on invariant measures arising from Iterated Function System with Probabilities (IFSP). We show the equivalence between an IFSP and a linear dynamical system driven by a white noise. Then, we use a multifractal analysis to obtain scaling properties of the resulting invariant measures, working within the framework of dynamical systems. Finally, as an application to fractal image generation, we show how this analysis can be used to obtain the most efficient choice for the probabilities to render the attractor of an IFS by applying the probabilistic algorithm known as “chaos game”.
20 citations
15 citations
TL;DR: In this article, the authors investigated correlations among pitches in several songs and pieces of piano music and found that the fundamental principle of music is the balance between repetition and contrast, which supports the belief that music is a balance between contrast and repetition.
Abstract: We investigate correlations among pitches in several songs and pieces of piano music. Real values of tones are mapped to positions within a one-dimensional walk. The structure of music, such as beat, measure and stanza, are reflected in the change of scaling exponents of the mean square fluctuation. Usually the pitches within one beat are nearly random, while nontrivial correlations are found within duration around a measure; for longer duration the mean square fluctuation is nearly flat, indicating exact 1/f power spectrum. Some interesting features are observed. Correlations are also studied by treating different tones as different symbols. This kind of correlation cannot reflect the structure of music, though long-range power-law is also discovered. Our results support the viewpoint that the fundamental principle of music is the balance between repetition and contrast.
14 citations
TL;DR: In this paper, the pore surface area fractal parameters from mercury porosimetry data on gray forest soil were estimated before and during crop development, in samples both containing and not containing soil carbohydrates known to be important structure-forming agents.
Abstract: Fractal parameters of soils has become increasingly important in understanding and quantifying transport and adsorption phenomena in soils. It is not known how soil plant development may affect fractal characteristics of soil pores. We estimated pore surface area fractal parameters from mercury porosimetry data on gray forest soil before and during crop development, in samples both containing and not containing soil carbohydrates known to be important structure-forming agents. Two distinct intervals with different fractal dimensions were found in the range of pore radii from 4 nm to 1 μm. This could be attributed to differences in mineral composition of soil particles of different sizes. The interval of the smallest radii had the highest average fractal dimension close to 3. Smaller surface area fractal dimensions corresponding to low surface irregularity were found in the next interval of radii. The plant development affected neither fractal dimensions nor the cutoff values of soil samples. The carbohydrate oxidation caused a significant increase in the fractal dimension in the interval of larger radii, but did not affect fractal dimension in the interval of small radii. The cutoff values decreased after carbohydrate oxidation.
12 citations
TL;DR: In this paper, the effect of the density of the initial defects and the lattice size on percolation characteristics is studied by Monte Carlo simulation on a square lattice, where each initial defect is assumed to have two crack tips like the Griffith crack and percolated crack patterns in homogeneous medium.
Abstract: Hydrogen-induced fracture of steel is characterized by the formation of internal voids caused by hydrogen precipitation at an inclusion-matrix interface, followed by the formation of microcrack array under the superposed action of internal hydrogen pressure and external forces. The propagation of the hydrogen-induced fracture is considerably random and the fracture develops by stepwise linking of the microcracks. Crack growth in a solid containing many initial defects is studied by Monte Carlo simulation on a square lattice. Each initial defect is assumed to have two crack tips like the Griffith crack and percolated crack patterns in homogeneous medium are investigated. The effect of the density of the initial defects and the lattice size on percolation characteristics is also studied.
9 citations
TL;DR: In this article, the authors investigated the self-organized criticality of traffic jams in a one-dimensional traffic flow on a highway, where a car moves ahead with transition probability pt. Near pt=1, the system is driven asymptotically into a steady state exhibiting a self-organised criticality.
Abstract: Annihilation process of traffic jams is investigated in a one-dimensional traffic flow on a highway. The one-dimensional fully asymmetric exclusion model with open boundaries for parallel update is extended to take into account stochastic transition of cars, where a car moves ahead with transition probability pt. Near pt=1, the system is driven asymptotically into a steady state exhibiting a self-organized criticality. Traffic jams with various lifetimes (or sizes) appear and disappear by colliding with an empty wave. The typical lifetime of traffic jams scales as , where ∆pt=1−pt. It is shown that the cumulative lifetime distribution Nm(∆pt) satisfies the scaling form .
8 citations
TL;DR: In this article, an Abelian sandpile model is considered on the Husimi lattice of triangles with an arbitrary coordination number q. Exact expressions for the distribution of height probabilities in the Self-Organized Critical state are derived.
Abstract: An Abelian sandpile model is considered on the Husimi lattice of triangles with an arbitrary coordination number q. Exact expressions for the distribution of height probabilities in the Self-Organized Critical state are derived.
7 citations
TL;DR: In this paper, wavelet analysis is used to measure the scale exponents of atmospheric turbulence at inertial range under different stratifications, and it is found that the average values of the scale exponent under different layers are clearly different and are all lower than the Kolmogorov 1941 value.
Abstract: Wavelet analysis is used to measure directly the scale exponents of atmospheric turbulence at inertial range under different stratifications. It is found that the average values of the scale exponents under different stratifications are clearly different and are all lower than the Kolmogorov 1941 value of 1/3. The difference shows that, according to the intermittent turbulence model, turbulence under stable stratification behaves more intermittently than under unstable stratification. The reason may be that the gravity wave, which could exist only under stable stratification, modulates the state of nocturnal surface layer and causes this kind of intermittence.
TL;DR: In this paper, a general class of fractional random fields, ℬα, 0≤α<2, is defined, which can be used to model natural scenes and textures.
Abstract: In this paper, a general class of fractional random fields, ℬα, 0≤α<2, is defined. The members of ℬα can be used to model natural scenes and textures. It is shown that the fractal dimension of random fields in ℬα is a linear nonincreasing function of a for 0≤α<α0 and a linear nondecreasing function of α for α0<α<2. The number α0 corresponds to the Hausdorff-Besicovitch dimension of the random field. These linear relationships are significant for texture comparison and classification.
TL;DR: In this article, the fractal dimension of the line interface is found to be 1.7-1.8, which is greater than that of turbulent/nonturbulent interface in a turbulent flow.
Abstract: Turbulent interface caused by the 2-dimensional Rayleigh-Taylor instability is investigated by direct numerical simulation. It is shown that the interface becomes fractal spontaneously in the case where there are initially multimode perturbations on the interface. The generalized dimensions and the singularity spectrum are obtained by applying the multifractal theory to the turbulent interface. The fractal dimension of the line interface is found to be 1.7–1.8, which is greater than that of turbulent/nonturbulent interface in a turbulent flow. Time evolution of the fractal dimensions of the interface is also investigated.
TL;DR: The arterial blood vessel system of kidneys is known to be fractal with a global mass dimension of 2.2 to 2.3, and the Minkowski cover was applied to the vessel surfaces and determined their scaling behavior, which yielded the surface global dimension.
Abstract: The arterial blood vessel system of kidneys is known to be fractal with a global mass dimension of 2.2 to 2.3. The global mass dimensions were determined by the mass-radius method.1 We support these results of the mass correlation analysis with an examination of the surface, which is in good approximation with the "true mass" of the border between the tissue (parenchyma) and the blood. We applied the Minkowski cover to the vessel surfaces and determined their scaling behavior, which yielded the surface global dimension. The examinations on kidney arteries1 are supplemented with new data of placentoma arteries2 and a comparison is made between their surface and mass dimensions.
TL;DR: The interface between the dermis and the epidermis in human skin is considered as a fractal-like structure and the fractal dimension quantifies this space-filling; it was found to increase from 2 to 3 as the surface folds more extensively.
Abstract: Objects in a biological body consist of structures which themselves have finer structures, and such a relation repeats several orders. We show that the interface between the dermis and the epidermis in human skin is one of them and is considered as a fractal-like structure. To determine the degree of complexity of the structure, we estimated the fractal dimension. Its fractal dimension was 2.4. Folding of the interface increases the area that occupies a three-dimensional volume, enhancing functionality and promoting strong adhesion between the dermis and the epidermis. The fractal dimension quantifies this space-filling; it was found to increase from 2 to 3 as the surface folds more extensively. We discuss the functional significance of the fractal-like structure of the human skin and the other structures in a biological body.
TL;DR: In this article, the fractal joints with different roughness (different fractal dimensions) are investigated under uniaxial and shear compression, and the results indicate that the joint roughness significantly influences the peak shear strength, the position of the maximum shear stress and the number of contact points.
Abstract: In this paper, the fractal rock joints are manufactured on the photoelastic material plate. Based on the visualized photoelastic experiments, the mechanical properties of the fractal joints with different roughness (different fractal dimensions) are investigated under uniaxial and shear compression. The research results indicate that the joint roughness (the fractal dimension of the joints) significantly influences the peak shear strength, the position of the maximum shear stress, and the number of contact points.
TL;DR: In this paper, the origin of the rugged border of thin-layer ramified copper electrodeposits influenced by chemical perturbations was studied using X-ray Bragg diffraction.
Abstract: We study the origin of the rugged border of thin-layer ramified copper electrodeposits influenced by chemical perturbations. Our main interest is to look for the smallest growth building blocks toward the understanding of the transition from a compact to a rugged structure. Using electron microscopy, we found two building blocks, well differentiated in shape and size, consisting of: (a) Spikes or rods of a diameter between 100 and 500 nm and a maximum observed length of up to 10 μm; and (b) crystalline grains with a diameter between 200 nm and 2 μm. Through X-rays Bragg diffraction, we tentatively assessed a Cu2O and metallic Cu chemical composition to the crystals and rods, respectively. The relative concentration of metallic Cu to Cu2O changes as we move from the base to the front of the aggregation and this result could be associated with the Hecker effect.
TL;DR: In this paper, the authors review a model that describes the diffusion-controlled aggregation exhibited by particles as they are deposited on a surface and analyze the effects of small cluster mobility and show that the introduction of cluster diffusion dramatically affects the dynamics of film growth.
Abstract: In this paper, we briefly review a model that describes the diffusion-controlled aggregation exhibited by particles as they are deposited on a surface. This model allows us to understand many experiments of thin film deposition. In the Sec. 1, we describe the model, which incorporates deposition, particle and cluster diffusion, and aggregation. In Sec. 2, we study the dynamical evolution of the model. Finally, we analyze the effects of small cluster mobility and show that the introduction of cluster diffusion dramatically affects the dynamics of film growth. Some of these effects can be tested experimentally.
TL;DR: In this paper, the authors show that the perimeter length of a two-dimensional diffusion-limited aggregation shrinks in a power law with time t as A(t) ~ t(d−1−D)/2 in a d-dimensional space, where D is the fractal dimension of the aggregate.
Abstract: For a realistic aggregate grown under the diffusion control, the fractal scaling holds between two cutoff lengths. These cutoff lengths often control the dynamics of aggregation and relaxation. During thermal annealing, coarsening of the aggregate structure takes place, and the lower cutoff length increases. When the relaxation is limited by kinetics, we show by a simple dimensional argument that the perimeter length (or area) A of the aggregate shrinks in a power law with time t as A(t) ~ t(d–1–D)/2 in a d-dimensional space, where D is the fractal dimension of the aggregate. This prediction is tested by Monte Carlo simulation of the thermal relaxation of a two-dimensional diffusion-limited aggregation.
TL;DR: In this article, the methodology of the solution to the inverse fractal problem with the wavelet transform was extended to two-dimensional self-affine functions, and the two dimensional wavelet maxima bifurcation representation was derived from the continuous wavelet decomposition.
Abstract: 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.
TL;DR: In this article, the steady state properties of the mean density population of infected cells in a viral spread is simulated by a general forest-like cellular automaton model with two distinct populations of cells (permissive and resistant ones) and studied in the framework of a mean field approximation.
Abstract: The steady state properties of the mean density population of infected cells in a viral spread is simulated by a general forest like cellular automaton model with two distinct populations of cells (permissive and resistant ones) and studied in the framework of the mean field approximation. Stochastic dynamical ingredients are introduced into this model to mimic cells regeneration (with probability p) and to consider infection processes by other means than contiguity (with probability f). Simulations are carried out on a L×L square lattice taking into consideration the eighth first neighbors. The mean density population of infected cells (Di) is measured as a function of the regeneration probability p, and analyzed for small values of the ratio f/p and for distinct degrees of cell resistance. The results obtained by a mean field like approach recovers the simulations results. The role of the resistant parameter R (R≥2) on the steady state properties, is investigated and discussed in comparison with the R=1 monocell case which corresponds to the self organized critical forest model. The fractal dimension of the dead cells ulcers contours was also estimated and analyzed as a function of the model parameters.
TL;DR: In this article, the authors apply a fractal description of pore surface irregularity to study the nuclear relaxation of a liquid confined in an irregular pore, and show that the long time relaxation behavior is dominated by an exponential mode characterizing a free diffusive volume.
Abstract: We apply a fractal description of pore surface irregularity to study the nuclear relaxation of a liquid confined in an irregular pore. Our calculation on a pre-fractal volume with infinite surface relaxation rate shows that the long time relaxation behavior is dominated by an exponential mode characterizing a free diffusive volume. At short time, the calculated magnetization follows closely the power-law behavior previously proposed by de Gennes. For finite surface relaxation rate, we extend our studies of the efficiency of exchange on fractal membranes to nuclear relaxation in the same geometry. We predict a decay involving two characteristic times. For very slow surface relaxation, there will exist exponential relaxation governed by the surface relaxation. These laws show that the pore surface irregularities may play an important role in nuclear relaxation.
TL;DR: In this paper, the results of extensive numerical simulations of the Dielectric Breakdown Model (DBM) with noise reduction on the hexagonal lattice are presented, and the growth probabilities are stored at the five stages of the clusters growth: at the masses of 1000, 2000, 4000, 8000 and 16 000 particles.
Abstract: The results of the extensive numerical simulations of the Dielectric Breakdown Model (DBM) with noise reduction on the hexagonal lattice are presented. Seventy-five clusters grown under different boundary conditions consisting of 16 000 particles on the lattice 1001×1001 were generated. The simulations were done for the noise reduction parameter s equal to 200 and two values of the parameter η, namely for η=0.5 and 1. For the latter case, two boundary conditions were considered: the DBM and DLA b.c. Such a growth model leads to the formation of the fractal objects resembling real snowflakes. The growth probabilities were stored at the five stages of the clusters growth: at the masses of 1000, 2000, 4000, 8000 and 16 000 particles. Multifractal analysis was performed and obtained results are presented. The comparison of two methods: the histogram and moments methods, is provided. We discover that for the η=0.5 parameter, there is a phase transition, while for η=1, there appears to be no phase transition. Besides the usual growth probability measure, the measure given by the noise reduction counters is considered and multifractality of it is presented.
TL;DR: This article showed that the predictability of letters in written English texts depends strongly on their position in the word, and that the first letters are usually the least easy to predict, which is consistent with the intuitive notion that words are well defined subunits in written languages, with much weaker correlations across these units than within them.
Abstract: We show that the predictability of letters in written English texts depends strongly on their position in the word. The first letters are usually the least easy to predict. This agrees with the intuitive notion that words are well defined subunits in written languages, with much weaker correlations across these units than within them. It implies that the average entropy of a letter deep inside a word is roughly 4–5 times smaller than the entropy of the first letter.
TL;DR: In this paper, the average of the masses and their qth moments within boxes of increasing size was examined for the blood vessel system as measured on kidney1 and placenta arteries.
Abstract: The blood vessel system as measured on kidney1 and placenta arteries2,3 is known to be a non-homogeneous fractal with a distribution of local dimensions We interpret this distribution as a mass multifractal property and we have therefore examined the average of the masses Mi(r) and their qth moments within boxes of increasing size r The centers i of the boxes are randomly distributed on the vessels The generalized dimensions Dq are introduced by taking the average of (Mi(r)/M0)q-1 over the centers i, according to the probability distribution Mi(r)/M0 (M0: total mass of the cluster) Thus, we have determined Dq by calculating ∝ (r/L)(q-1) Dq (L: diameter of the cluster)
TL;DR: A general method of obtaining connected replicating fractiles based on generalized digit systems in the ring of Gaussian integers is introduced.
Abstract: We introduce a general method of obtaining connected replicating fractiles based on generalized digit systems in the ring of Gaussian integers.
TL;DR: In this paper, a simple dynamical model for two-dimensional dry foam rheology is constructed for which surface tension effects and viscous dissipation at Plateau borders (intersections of three cell boundaries) are taken into account and is studied by computer simulation.
Abstract: A simple dynamical model for two-dimensional dry foam rheology is constructed for which surface tension effects and viscous dissipation at Plateau borders (intersections of three cell boundaries) are taken into account and is studied by computer simulation. Under externally applied shear strain increasing at small rates, the system exhibits avalanche-like release of stress that has been accumulating under increasing strain. There is a close similarity with earthquake models that show self-organized criticality (SOC). We discuss related simulation of two-dimensional wet foams under statically applied strain by Hutzler et al. [Phil. Mag. B71, 277 (1995)] showing critical behavior.
TL;DR: In this paper, the authors study numerically the water erosion process under a variety of conditions and propose a water erosion model that leads to a universal exponent describing the fractal basin area distribution in the steady state.
Abstract: We study numerically the water erosion process under a variety of conditions. The water erosion model that we use leads to a universal exponent describing the fractal basin area distribution in the steady state.
TL;DR: In this paper, the authors study the dynamics of growth processes of rough surfaces based on mathematical models such as the Kardar-Parisi-Zhang equation and develop the scaling theory to give an insight into the problem.
Abstract: We study the dynamics of growth processes of rough surfaces based on mathematical models such as the Kardar-Parisi-Zhang equation. The white-noise assumption in the KPZ equation is, however, noted to fail for higher dimensional cases. A careful continuum limit leads to a smooth surface solution for the cases. We develop the scaling theory to give an insight into the problem, by means of the intermediate asymptotics of the second kind, which is a very useful notion for the purpose. We find the roughness exponent and the dynamic exponent as functions of the substrate dimensionality for a model equation applicable to quenched disorder systems.
TL;DR: In this paper, the surface of superground Mn-Zn ferrite single crystal was identified as a self-affine fractal in the stochastic sense, and the roughness increased as a power of the scale from 102 nm to 106 nm.
Abstract: The surface of superground Mn-Zn ferrite single crystal may be identified as a self-affine fractal in the stochastic sense. The rms roughness increased as a power of the scale from 102 nm to 106 nm with the roughness exponent α=0.17±0.04, and 0.11±0.06, for grinding feed rate of 15 and 10 μm/rev, respectively. The scaling behavior coincided with the theory prediction well used for growing self-affine surfaces in the interested region for magnetic heads performance. The rms roughnesses increased with increase in the feed rate, implying that the feed rate is a crucial grinding parameter affecting the supersmooth surface roughness in the machining process.