Transient from crystallization to fractal growth observed in both boar bile and SnI sub 2 vapour
Zhang Ji Zhong; Xie An Jian
2003-01-01
A visual transient of the growth mechanism from crystallization to fractal growth was observed clearly in a drop of boar bile. The growing crystals were replaced by treelike fractal structures during solidification of the sample. It is fascinating to compare the transient with the result observed in SnI sub 2 vapour. They were completely identical, and revealed that under certain conditions a linear growth could be transferred spontaneously into nonlinear growth. It may be possible to consider the transient as a 'bridge' between linear and nonlinear growth, and to develop a quantitative expression of transient dynamics.
Modeling Fractal Dimension Curve of Urban Growth in Developing Countries
Chen, Yanguang
2016-01-01
The growth curve of fractal dimension of cities can be described with sigmoid function such as Boltzmann's equation and logistic function. The logistic models of fractal dimension curves have been presented for the cities in developed countries. However, these models cannot be well fitted to the observational data of fractal dimension of urban form in developing countries (e.g. China). By statistic experiments of fractal parameters, we find that the quadratic Boltzmann's equation can be used to describe fractal dimension change of Chinese cities. For the normalized fractal dimension values, the Boltzmann's equation can be reduced to a quadratic logistic function. In practice, a fractal dimension dataset of urban growth can be approximately fitted with the quadratic logistic function. Thus, a series of models of fractal dimension curve can be proposed for the cities in developing countries. The models are applied to the city of Beijing, Chinese capital, and yield satisfying trend lines of the observational dat...
Modelling Fractal Growth of Bacillus subtilis on Agar Plates
Fogedby, Hans C.
1991-02-01
The observed fractal growth of a bacterial colony of Bacillus subtilis on agar plates is simulated by a simple computer model in two dimensions. Growth morphologies are shown and the fractal dimension is computed. The concentration of nutrients and the time scale ratio of bacterial multiplication and nutrient diffusion are the variable parameters in the model. Fractal growth is observed in the simulations for moderate concentrations and time scale ratios. The simulated morphologies are similar to the ones grown in the biological experiment. The phenomenon is analogous to the fractal morphologies of lipid layers grown on a water surface.
FRACTAL PATTERN GROWTH OF METAL ATOM CLUSTERS IN ION IMPLANTED POLYMERS
Institute of Scientific and Technical Information of China (English)
ZHANG TONG-HE; WU YU-GUANG; SANG HAI-BO; ZHOU GU
2001-01-01
The fractal and multi-fractal patterns of metal atoms are observed in the surface layer and cross section of a metal ion implanted polymer using TEM and SEM for the first time. The surface structure in the metal ion implanted polyethylene terephthalane (PET) is the random fractal. Certain average quantities of the random geometric patterns contain self-similarity. Some growth origins appeared in the fractal pattern which has a dimension of 1.67. The network structure of the fractal patterns is formed in cross section, having a fractal dimension of 1.87. So it can be seen that the fractal pattern is three-dimensional space fractal. We also find the collision cascade fractal in the cross section of implanted nylon, which is similar to the collision cascade pattern in transverse view calculated by the TRIM computer program. Finally, the mechanism for the formation and growth of the fractal patterns during ion implantation is discussed.
Generalized fragmentation functions for fractal jet observables
Elder, Benjamin T.; Procura, Massimiliano; Thaler, Jesse; Waalewijn, Wouter J.; Zhou, Kevin
2017-06-01
We introduce a broad class of fractal jet observables that recursively probe the collective properties of hadrons produced in jet fragmentation. To describe these collinear-unsafe observables, we generalize the formalism of fragmentation functions, which are important objects in QCD for calculating cross sections involving identified final-state hadrons. Fragmentation functions are fundamentally nonperturbative, but have a calculable renormalization group evolution. Unlike ordinary fragmentation functions, generalized fragmentation functions exhibit nonlinear evolution, since fractal observables involve correlated subsets of hadrons within a jet. Some special cases of generalized fragmentation functions are reviewed, including jet charge and track functions. We then consider fractal jet observables that are based on hierarchical clustering trees, where the nonlinear evolution equations also exhibit tree-like structure at leading order. We develop a numeric code for performing this evolution and study its phenomenological implications. As an application, we present examples of fractal jet observables that are useful in discriminating quark jets from gluon jets.
Fractal growth in impurity-controlled solidification in lipid monolayers
DEFF Research Database (Denmark)
Fogedby, Hans C.; Sørensen, Erik Schwartz; Mouritsen, Ole G.
1987-01-01
A simple two-dimensional microscopic model is proposed to describe solidifcation processes in systems with impurities which are miscible only in the fluid phase. Computer simulation of the model shows that the resulting solids are fractal over a wide range of impurity concentrations and impurity...... diffusional constants. A fractal-forming mechanism is suggested for impurity-controlled solidification which is consistent with recent experimental observations of fractal growth of solid phospholipid domains in monolayers. The Journal of Chemical Physics is copyrighted by The American Institute of Physics....
Fractal model for simulation of frost formation and growth
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
A planar fractal model for simulation of frost formation and growth was proposed based on diffusion limited aggregation(DLA)model and the computational simulation was carried out in this paper.By changing the times of program running circulation and the ratio of random particles generated,the simulation figures were gained under different conditions.A microscope is used to observe the shape and structure of frost layer and a digital camera with high resolution is used to record the pattern of frost layer at different time.Through comparing the simulation figures with the experimental images,we find that the simulation results agree well with the experimental images in shape and the fractal dimension of simulation figures is nearly equal to that of experimental images.The results indicate that it is reasonable to represent frost layer growth time with the program circulation times and to simulate the frost layer density variation during its growth process by reducing the random particle generation probability.The feasibility of using the suggested model to simulate the process of frost formation and growth was justified.The insufficiencies and its causes of this fractal model are also discussed.
The Gompertzian curve reveals fractal properties of tumor growth
Energy Technology Data Exchange (ETDEWEB)
Waliszewski, Przemyslaw; Konarski, Jerzy
2003-06-01
The normalized Gompertzian curve reflecting growth of experimental malignant tumors in time can be fitted by the power function y(t)=at{sup b} with the coefficient of nonlinear regression r{>=}0.95, in which the exponent b is a temporal fractal dimension, (i.e., a real number), and time t is a scalar. This curve is a fractal, (i.e., fractal dimension b exists, it changes along the time scale, the Gompertzian function is a contractable mapping of the Banach space R of the real numbers, holds the Banach theorem about the fix point, and its derivative is {<=}1). This denotes that not only space occupied by the interacting cancer cells, but also local, intrasystemic time, in which tumor growth occurs, possesses fractal structure. The value of the mean temporal fractal dimension decreases along the curve approaching eventually integer values; a fact consistent with our hypothesis that the fractal structure is lost during tumor progression.
Fractal Dimension and Universality in Avascular Tumor Growth
Ribeiro, Fabiano L; Mata, Angélica S
2016-01-01
The comprehension of tumor growth is a intriguing subject for scientists. New researches has been constantly required to better understand the complexity of this phenomenon. In this paper, we pursue a physical description that account for some experimental facts involving avascular tumor growth. We have proposed an explanation of some phenomenological (macroscopic) aspects of tumor, as the spatial form and the way it growths, from a individual-level (microscopic) formulation. The model proposed here is based on a simple principle: competitive interaction between the cells dependent on their mutual distances. As a result, we reproduce many empirical evidences observed in real tumors, as exponential growth in their early stages followed by a power law growth. The model also reproduces the fractal space distribution of tumor cells and the universal behavior presented in animals and tumor growth, conform reported by West, Guiot {\\it et. al.}\\cite{West2001,Guiot2003}. The results suggest that the universal similar...
Crossover from Nonequilibrium Fractal Growth to Equilibrium Compact Growth
DEFF Research Database (Denmark)
Sørensen, Erik Schwartz; Fogedby, Hans C.; Mouritsen, Ole G.
1988-01-01
Solidification controlled by vacancy diffusion is studied by Monte Carlo simulations of a two-dimensional Ising model defined by a Hamiltonian which models a thermally driven fluid-solid phase transition. The nonequilibrium morphology of the growing solid is studied as a function of time as the s...... as the system relaxes into equilibrium described by a temperature. At low temperatures the model exhibits fractal growth at early times and crossover to compact solidification as equilibrium is approached....
An extended fractal growth regime in the diffusion limited aggregation including edge diffusion
Directory of Open Access Journals (Sweden)
Aritra Ghosh
2016-01-01
Full Text Available We have investigated on-lattice diffusion limited aggregation (DLA involving edge diffusion and compared the results with the standard DLA model. For both cases, we observe the existence of a crossover from the fractal to the compact regime as a function of sticking coefficient. However, our modified DLA model including edge diffusion shows an extended fractal growth regime like an earlier theoretical result using realistic growth models and physical parameters [Zhang et al., Phys. Rev. Lett. 73 (1994 1829]. While the results of Zhang et al. showed the existence of the extended fractal growth regime only on triangular but not on square lattices, we find its existence on the square lattice. There is experimental evidence of this growth regime on a square lattice. The standard DLA model cannot characterize fractal morphology as the fractal dimension (Hausdorff dimension, DH is insensitive to morphology. It also predicts DH = DP (the perimeter dimension. For the usual fractal structures, observed in growth experiments on surfaces, the perimeter dimension can differ significantly (DH ≠ DP depending on the morphology. Our modified DLA model shows minor sensitivity to this difference.
Origins of fractality in the growth of complex networks
Song, Chaoming; Havlin, Shlomo; Makse, Hernán A.
2006-04-01
Complex networks from such different fields as biology, technology or sociology share similar organization principles. The possibility of a unique growth mechanism promises to uncover universal origins of collective behaviour. In particular, the emergence of self-similarity in complex networks raises the fundamental question of the growth process according to which these structures evolve. Here we investigate the concept of renormalization as a mechanism for the growth of fractal and non-fractal modular networks. We show that the key principle that gives rise to the fractal architecture of networks is a strong effective `repulsion' (or, disassortativity) between the most connected nodes (that is, the hubs) on all length scales, rendering them very dispersed. More importantly, we show that a robust network comprising functional modules, such as a cellular network, necessitates a fractal topology, suggestive of an evolutionary drive for their existence.
Compact or fractal patterns in diffusion limited growth
Ben Amar, Martine
1993-02-01
Fractal viscous fingering patterns are observed in an infinite Hele-Shaw cell at long times when the capillary forces become negligible. On the contrary, growth of monocrystals from a punctual seed shows dendrites growing independently in 4 or 6 directions, according to the crystal symmetry. A close comparison of numerical and experimental data explains first the origin of the tip-splitting instability in radial growth and shows that it can be inhibited by the anisotropy of surface tension. Des structures fractales ont été observées en digitations visqueuses en cellule de Hele-Shaw et aux temps longs lorsque les forces capillaires deviennent négligeables. Par opposition, la croissance de monocristaux à partir d'un germe ponctuel fait apparaître des dendrites croissant dans 4 ou 6 directions selon la symétrie cristalline. Une comparaison entre des résultats numériques et des données expérimentales explique l'origine de l'instabilité de dédoublement des pointes en croissance radiale et montre qu'elle peut être supprimée par l'anisotropie cristalline.
Fractal dimension and universality in avascular tumor growth
Ribeiro, Fabiano L.; dos Santos, Renato Vieira; Mata, Angélica S.
2017-04-01
For years, the comprehension of the tumor growth process has been intriguing scientists. New research has been constantly required to better understand the complexity of this phenomenon. In this paper, we propose a mathematical model that describes the properties, already known empirically, of avascular tumor growth. We present, from an individual-level (microscopic) framework, an explanation of some phenomenological (macroscopic) aspects of tumors, such as their spatial form and the way they develop. Our approach is based on competitive interaction between the cells. This simple rule makes the model able to reproduce evidence observed in real tumors, such as exponential growth in their early stage followed by power-law growth. The model also reproduces (i) the fractal-space distribution of tumor cells and (ii) the universal growth behavior observed in both animals and tumors. Our analyses suggest that the universal similarity between tumor and animal growth comes from the fact that both can be described by the same dynamic equation—the Bertalanffy-Richards model—even if they do not necessarily share the same biological properties.
arXiv Generalized Fragmentation Functions for Fractal Jet Observables
Elder, Benjamin T.; Thaler, Jesse; Waalewijn, Wouter J.; Zhou, Kevin
2017-06-15
We introduce a broad class of fractal jet observables that recursively probe the collective properties of hadrons produced in jet fragmentation. To describe these collinear-unsafe observables, we generalize the formalism of fragmentation functions, which are important objects in QCD for calculating cross sections involving identified final-state hadrons. Fragmentation functions are fundamentally nonperturbative, but have a calculable renormalization group evolution. Unlike ordinary fragmentation functions, generalized fragmentation functions exhibit nonlinear evolution, since fractal observables involve correlated subsets of hadrons within a jet. Some special cases of generalized fragmentation functions are reviewed, including jet charge and track functions. We then consider fractal jet observables that are based on hierarchical clustering trees, where the nonlinear evolution equations also exhibit tree-like structure at leading order. We develop a numeric code for performing this evolution and study its phen...
Theoretical study of fractal growth and stability on surface
DEFF Research Database (Denmark)
Dick, Veronika V.; Solov'yov, Ilia; Solov'yov, Andrey V.
2009-01-01
We perform a theoretical study of the fractal growing process on surface by using the deposition, diffusion, aggregation method. We present a detailed analysis of the post-growth processes occurring in a nanofractal on surface. For this study we developed a method which describes the internal...... dynamics of particles in a fractal and accounts for their diffusion and detachment. We demonstrate that these kinetic processes are responsible for the formation of the final shape of the islands on surface after the post-growth relaxation....
Theoretical study of fractal growth and stability on surface
DEFF Research Database (Denmark)
Dick, Veronika V.; Solov'yov, Ilia; Solov'yov, Andrey V.
2009-01-01
We perform a theoretical study of the fractal growing process on surface by using the deposition, diffusion, aggregation method. We present a detailed analysis of the post-growth processes occurring in a nanofractal on surface. For this study we developed a method which describes the internal...... dynamics of particles in a fractal and accounts for their diffusion and detachment. We demonstrate that these kinetic processes are responsible for the formation of the final shape of the islands on surface after the post-growth relaxation....
Charging and Growth of Fractal Dust Grains
Matthews, Lorin S
2007-01-01
The structure and evolution of aggregate grains formed within a plasma environment are dependent upon the charge acquired by the micron-sized dust grains during the coagulation process. The manner in which the charge is arranged on developing irregular structures can affect the fractal dimension of aggregates formed during collisions, which in turn influences the coagulation rate and size evolution of the dust within the plasma cloud. This paper presents preliminary models for the charge and size evolution of fractal aggregates immersed in a plasma environment calculated using a modification to the orbital-motion-limited (OML) theory. Primary electron and ion currents incident on points on the aggregate surface are determined using a line-of-sight (LOS) approximation: only those electron or ion trajectories which are not blocked by another grain within the aggregate contribute to the charging current. Using a self-consistent iterative approach, the equilibrium charge and dipole moment are calculated for the d...
Conformal dynamics of fractal growth patterns without randomness
Davidovitch; Feigenbaum; Hentschel; Procaccia
2000-08-01
Many models of fractal growth patterns (such as diffusion limited aggregation and dielectric breakdown models) combine complex geometry with randomness; this double difficulty is a stumbling block to their elucidation. In this paper we introduce a wide class of fractal growth models with highly complex geometry but without any randomness in their growth rules. The models are defined in terms of deterministic itineraries of iterated conformal maps, generating the function Phi((n))(omega) which maps the exterior of the unit circle to the exterior of an n-particle growing aggregate. The complexity of the evolving interfaces is fully contained in the deterministic dynamics of the conformal map Phi((n))(omega). We focus attention on a class of growth models in which the itinerary is quasiperiodic. Such itineraries can be approached via a series of rational approximants. The analytic power gained is used to introduce a scaling theory of the fractal growth patterns and to identify the exponent that determines the fractal dimension.
Fractal Pattern Growth in Ti-Implanted Steel with High Ion Flux
Institute of Scientific and Technical Information of China (English)
张通和; 吴瑜光; 刘安东
2002-01-01
We report on the formation of metal nanometre phase and fractal patterns in steel using metal vapour vacuum arc source ion implantation with high ion flux. The dense nanometre phases are cylindrical and well dispersed in the Ti-implanted layer with an ion flux up to 50μA/cm2. The collision fractal pattern is formed in Ti-implanted steel with an ion flux of 25μA/cm2 and the disconnected fractal pattern is observed with an ion flux of 50μA/cm2.The average density ofnanometre phases decreases from 1.2 × 1011/cm2 to 6.5 × 1010/cm2 as the ion flux increases from 25 μA/cm2 to 50 μA/cm2. Fractal pattern growth is in remarkable agreement with Sander's diffusion-limited aggregation model. The alloy clusters have diffused and aggregated in chains forming branches to grow a beautiful tree during Ti implantation with an ion flux ranging from 75μA/cm2 to 85μA/cm2. We discuss the model of fractal pattern growth during ion implantation with high ion flux.
Hydrophobic fractal surface from glycerol tripalmitate and the effects on C6 glioma cell growth.
Zhang, Shanshan; Chen, Xuerui; Yu, Jing; Hong, Biyuan; Lei, Qunfang; Fang, Wenjun
2016-06-01
To provide a biomimic environment for glial cell culture, glycerol tripalmitate (PPP) has been used as a raw material to prepare fractal surfaces with different degrees of hydrophobicity. The spontaneous formation of the hydrophobic fractal surfaces was monitored by differential scanning calorimetry (DSC) and X-ray diffraction (XRD). The surface morphologies were observed by a scanning electron microscope (SEM), and then the fractal dimension (FD) values of the surfaces were determined with the box-counting method. C6 glioma cells were cultured and compared on different hydrophobic PPP surfaces and poly-L-lysine (PLL)-coated surface. The cell numbers as a function of incubation time on different surfaces during the cell proliferation process were measured, and the cell morphologies were observed under a fluorescence microscope. Influences of hydrophobic fractal surfaces on the cell number and morphology were analyzed. The experimental results show that the cell proliferation rates decrease while the cell morphology complexities increase with the growth of the fractal dimensions of the PPP surfaces.
Fractal Aggregation Under Rotation
Institute of Scientific and Technical Information of China (English)
WU Feng-Min; WU Li-Li; LU Hang-Jun; LI Qiao-Wen; YE Gao-Xiang
2004-01-01
By means of the Monte Carlo simulation, a fractal growth model is introduced to describe diffusion-limited aggregation (DLA) under rotation. Patterns which are different from the classical DLA model are observed and the fractal dimension of such clusters is calculated. It is found that the pattern of the clusters and their fractal dimension depend strongly on the rotation velocity of the diffusing particle. Our results indicate the transition from fractal to non-fractal behavior of growing cluster with increasing rotation velocity, i.e. for small enough angular velocity ω the fractal dimension decreases with increasing ω, but then, with increasing rotation velocity, the fractal dimension increases and the cluster becomes compact and tends to non-fractal.
Institute of Scientific and Technical Information of China (English)
吴瑜光; 张通和
1997-01-01
The fractal patterns in implanted samples are observed. Possible correlation of fractal patterns with the annealing temperature and the electrical activation ratio are given. The formation and growth process of fractal patterns are compared for implanted layers both in silicon and in SiO2/GaAsP during thermal annealing. The mechanism of formation and growth process of fractal pattern is discussed.
Fractal pattern growth simulation in electrodeposition and study of the shifting of center of mass
Energy Technology Data Exchange (ETDEWEB)
Shaikh, Yusuf H. [Shivaji Arts, Commerce and Science College, Kannad 431103 (India)], E-mail: yusufshaikh123@yahoo.com; Khan, A.R. [Dr. Rafiq Zakaria Centre for Higher Learning, Dr. Rafiq Zakaria marg, Rauza Bagh, Aurangabad 431001 (India); Pathan, J.M. [Dr. Rafiq Zakaria Campus, Dr. Rafiq Zakaria marg, Rauza Bagh, Aurangabad 431001 (India); Patil, Aruna [Viveakanand College, Aurangabad 431001 (India); Behere, S.H. [Departments of Physics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431004 (India)
2009-12-15
We presented simulation of fractal pattern in electrodeposition (Diffusion limited aggregation) using concept of off lattice walk. It is seen that the growth patterns are based on a parameter called 'bias'. This parameter 'bias' controls the growth of patterns similar to that of electric field in electrodeposition technique. In present study the fractal patterns are grown for different values of 'bias'. Dendritic patterns grown at lower value of 'bias' comprises open structure and show limited branching. As the bias is increased the growth tends to be dense and show more crowded branching. Box counting was implemented to calculate fractal dimension. The structural and textural complexities and are compared with the experimental observations. It was also noted that in the evolution of DLA patterns, the center of mass of the growth is shifted slightly. We tracked the position of the center of mass of simulated electro deposits under different electric field conditions. The center of mass exhibit random walk like patterns and it wanders around the origin or the starting point of the growth.
Fractal Aggregation Under Rotation
Institute of Scientific and Technical Information of China (English)
WUFeng-Min; WULi-Li; LUHang-Jun; LIQiao-Wen; YEGao-Xiang
2004-01-01
By means of the Monte Carlo simulation, a fractal growth model is introduced to describe diffusion-limited aggregation (DLA) under rotation. Patterns which are different from the classical DLA model are observed and the fractal dimension of such clusters is calculated. It is found that the pattern of the clusters and their fractal dimension depend strongly on the rotation velocity of the diffusing particle. Our results indicate the transition from fractal to non-fractal behavior of growing cluster with increasing rotation velocity, i.e. for small enough angular velocity ω; thefractal dimension decreases with increasing ω;, but then, with increasing rotation velocity, the fractal dimension increases and the cluster becomes compact and tends to non-fractal.
3D dendritic gold nanostructures: seeded growth of a multi-generation fractal architecture.
Pan, Ming; Xing, Shuangxi; Sun, Ting; Zhou, Wenwen; Sindoro, Melinda; Teo, Hui Hian; Yan, Qingyu; Chen, Hongyu
2010-10-14
In this report, we focus on the synthetic challenges for nanoscale 3D fractal architectures, namely the multi-generation growth with control in both size uniformity and colloidal stability; by directing the simultaneous growth of Au and polyaniline on Au seeds, fractal nanoparticles can be achieved with a topology distinctively different from those of spheres, cubes or rods.
A fractal growth model: Exploring the connection pattern of hubs in complex networks
Li, Dongyan; Wang, Xingyuan; Huang, Penghe
2017-04-01
Fractal is ubiquitous in many real-world networks. Previous researches showed that the strong disassortativity between the hub-nodes on all length scales was the key principle that gave rise to the fractal architecture of networks. Although fractal property emerged in some models, there were few researches about the fractal growth model and quantitative analyses about the strength of the disassortativity for fractal model. In this paper, we proposed a novel inverse renormalization method, named Box-based Preferential Attachment (BPA), to build the fractal growth models in which the Preferential Attachment was performed at box level. The proposed models provided a new framework that demonstrated small-world-fractal transition. Also, we firstly demonstrated the statistical characteristic of connection patterns of the hubs in fractal networks. The experimental results showed that, given proper growing scale and added edges, the proposed models could clearly show pure small-world or pure fractal or both of them. It also showed that the hub connection ratio showed normal distribution in many real-world networks. At last, the comparisons of connection pattern between the proposed models and the biological and technical networks were performed. The results gave useful reference for exploring the growth principle and for modeling the connection patterns for real-world networks.
Spatial Dynamics of Urban Growth Based on Entropy and Fractal Dimension
Chen, Yanguang
2016-01-01
The fractal dimension growth of urban form can be described with sigmoid functions such as logistic function due to squashing effect. The sigmoid curves of fractal dimension suggest a type of spatial replacement dynamics of urban evolution. How to understand the underlying rationale of the fractal dimension curves is a pending problem. This study is based on two previous findings. First, normalized fractal dimension proved to equal normalized spatial entropy; second, a sigmoid function proceeds from an urban-rural interaction model. Defining urban space-filling measurement by spatial entropy, and defining rural space-filling measurement by information gain, we can construct a new urban-rural interaction and coupling model. From this model, we can derive the logistic equation of fractal dimension growth strictly. This indicates that urban growth results from the unity of opposites between spatial entropy increase and information increase. In a city, an increase in spatial entropy is accompanied by a decrease i...
Directory of Open Access Journals (Sweden)
Bogdanov Ana
2007-01-01
Full Text Available Urban forms and processes can be observed as fractal structures since in their seemingly chaotic development and complexity it can be noticed an internal order and regularity, which could be quantified and described by the methods of fractal analysis. With determination of fractal dimension it is possible to quantify the level of irregularity, the complexity and hierarchy of the urban structures, as well as the level of urban transformations in various time intersections. The fractal geometry method has been used in analyses of spatial distribution of population, networks and utilities because it corresponds more than deterministic methods to the nature of urban settlements as open, non-linear and dynamic systems. In that sense, fractal geometry becomes the means to grasp a complex morphological urban structure of urban settlements in general, the interrelationships between the inner spatial elements, and to predict future development possibilities. Moreover on the basis of urban pattern analysis by means of fractal geometry, it is possible to evaluate the growth and development process and to perform a comparative analysis of development in spatially and temporarily different settlement settings. Having in view that complex urban fabric presumes tight connections and diversity, which is in contrast to sprawl and monotony which increasingly characterize urban growth and development, this paper is a contribution to research of potential for modern urban settlements to regain the spirit of spontaneity and human dimension through application of development models that are fractal geometry based.
Invited Article: Plasmonic growth of patterned metamaterials with fractal geometry
Directory of Open Access Journals (Sweden)
Nobuyuki Takeyasu
2016-08-01
Full Text Available Large-scale metallic three-dimensional (3D structures composed of sub-wavelength fine details, called metamaterials, have attracted optical scientists and materials scientists because of their unconventional and extraordinary optical properties that are not seen in nature. However, existing nano-fabrication technologies including two-photon fabrication, e-beam, focused ion-beam, and probe microscopy are not necessarily suitable for fabricating such large-scale 3D metallic nanostructures. In this article, we propose a different method of fabricating metamaterials, which is based on a bottom-up approach. We mimicked the generation of wood forest under the sunlight and rain in nature. In our method, a silver nano-forest is grown from the silver seeds (nanoparticles placed on the glass substrate in silver-ion solution. The metallic nano-forest is formed only in the area where ultraviolet light is illuminated. The local temperature increases at nano-seeds and tips of nano-trees and their branches due to the plasmonic heating as a result of UV light excitation of localized mode of surface plasmon polaritons. We have made experiments of growth of metallic nano-forest patterned by the light distribution. The experimental results show a beautiful nano-forest made of silver with self-similarity. Fractal dimension and spectral response of the grown structure are discussed. The structures exhibit a broad spectral response from ultraviolet to infrared, which was used for surface-enhanced Raman detection of molecules.
Invited Article: Plasmonic growth of patterned metamaterials with fractal geometry
Takeyasu, Nobuyuki; Taguchi, Natsuo; Nishimura, Naoki; Cheng, Bo Han; Kawata, Satoshi
2016-08-01
Large-scale metallic three-dimensional (3D) structures composed of sub-wavelength fine details, called metamaterials, have attracted optical scientists and materials scientists because of their unconventional and extraordinary optical properties that are not seen in nature. However, existing nano-fabrication technologies including two-photon fabrication, e-beam, focused ion-beam, and probe microscopy are not necessarily suitable for fabricating such large-scale 3D metallic nanostructures. In this article, we propose a different method of fabricating metamaterials, which is based on a bottom-up approach. We mimicked the generation of wood forest under the sunlight and rain in nature. In our method, a silver nano-forest is grown from the silver seeds (nanoparticles) placed on the glass substrate in silver-ion solution. The metallic nano-forest is formed only in the area where ultraviolet light is illuminated. The local temperature increases at nano-seeds and tips of nano-trees and their branches due to the plasmonic heating as a result of UV light excitation of localized mode of surface plasmon polaritons. We have made experiments of growth of metallic nano-forest patterned by the light distribution. The experimental results show a beautiful nano-forest made of silver with self-similarity. Fractal dimension and spectral response of the grown structure are discussed. The structures exhibit a broad spectral response from ultraviolet to infrared, which was used for surface-enhanced Raman detection of molecules.
Banerjee, Paromita; Soni, Jalpa; Ghosh, Nirmalya; Sengupta, Tapas K.
2013-02-01
It is of considerable current interest to develop various methods which help to understand and quantify the cellular association in growing bacterial colonies and is also important in terms of detection and identification of a bacterial species. A novel approach is used here to probe the morphological structural changes occurring during the growth of the bacterial colony of Bacillus thuringiensis under different environmental conditions (in normal nutrient agar, in presence of glucose - acting as additional nutrient and additional 3mM arsenate as additional toxic material). This approach combines the quantitative Mueller matrix polarimetry to extract intrinsic polarization properties and inverse analysis of the polarization preserving part of the light scattering spectra to determine the fractal parameter H (Hurst exponent) using Born approximation. Interesting differences are observed in the intrinsic polarization parameters and also in the Hurst exponent, which is a measurement of the fractality of a pattern formed by bacteria while growing as a colony. These findings are further confirmed with optical microscopic studies of the same sample and the results indicate a very strong and distinct dependence on the environmental conditions during growth, which can be exploited to quantify different bacterial species and their growth patterns.
Logistic Models of Fractal Dimension Growth for Spatio-Temporal Dynamics of Urban Morphology
Chen, Yanguang
2016-01-01
Urban form and growth can be described with fractal dimension, which is a measurement of space filling of urban evolution. Based on empirical analyses, a discovery is made that the time series of fractal dimension of urban form can be treated as a sigmoid function of time. Among various sigmoid functions, the logistic function is the most probable selection. However, how to use the model of fractal dimension growth to explain and predict urban growth is a pending problem remaining to be solved. This paper is devoted to modeling fractal dimension evolution of different types of cities. A interesting discovery is as follows: for the cities in developed countries such as UK, USA and Israel, the comparable fractal dimension values of a city's morphology in different years can be fitted to the logistic function; while for the cities in developing countries such as China, the fractal dimension data of urban form can be fitted to a quadratic logistic function. A generalized logistic function is thus proposed to mode...
Tumor growth in the space-time with temporal fractal dimension
Energy Technology Data Exchange (ETDEWEB)
Molski, Marcin [Department of Theoretical Chemistry, Faculty of Chemistry, Adam Mickiewicz University of Poznan, ul. Grunwaldzka 6, PL 60-780 Poznan (Poland)], E-mail: marcin@rovib.amu.edu.pl; Konarski, Jerzy [Department of Theoretical Chemistry, Faculty of Chemistry, Adam Mickiewicz University of Poznan, ul. Grunwaldzka 6, PL 60-780 Poznan (Poland)
2008-05-15
An improvement of the Waliszewski and Konarski approach [Waliszewski P, Konarski J. The Gompertzian curve reveals fractal properties of tumor growth. Chaos, Solitons and Fractals 2003;16:665-74] to determination of the time-dependent temporal fractal dimension b{sub t}(t) and the scaling factor a{sub t}(t) for the tumor formation in the fractal space-time is presented. The analytical formulae describing the time-dependence of b{sub t}(t) and a{sub t}(t), which take into account appropriate boundary conditions for t {yields} 0 and t {yields} {infinity}, are derived. Their validity is tested on the experimental growth curve obtained by Laird for the Flexner-Jobling rat's tumor. A hypothesis is formulated that tumorigenesis has a lot in common with the neuronal differentiation and synapse formation. These processes are qualitatively described by the same Gompertz function of growth and take place in the fractal space-time whose mean temporal fractal dimension is lost during progression.
Bacterial Fractal Growth in the Concentration Field of Nutrient
Fujikawa, Hiroshi; Matsushita, Mitsugu
1991-01-01
A Bacillus subtilis strain is found to grow through the diffusion-limited aggregation (DLA) process on agar plates. The organism is spotted on the agar plate containing a low concentration of peptone as a single nutrient and incubated at 35°C. The colony pattern grown on the plate surface is self-similar with the fractal dimension of 1.73± 0.02. Bacterial DLA branches are shown to grow in a concentration field of nutrient from the fact that they grow predominantly in the direction of higher nutrient concentration. The colonial morphology varies with the nutrient and agar concentrations of an agar plate, including DLA, a round type, dense branching morphology (DBM), and a spreading without openings. Two neighboring colonies repulse each other in DLA and DBM types only. On an agar plate containing glycerol the colony becomes remarkably round, quite different from the DLA morphology.
Lung cancer-a fractal viewpoint.
Lennon, Frances E; Cianci, Gianguido C; Cipriani, Nicole A; Hensing, Thomas A; Zhang, Hannah J; Chen, Chin-Tu; Murgu, Septimiu D; Vokes, Everett E; Vannier, Michael W; Salgia, Ravi
2015-11-01
Fractals are mathematical constructs that show self-similarity over a range of scales and non-integer (fractal) dimensions. Owing to these properties, fractal geometry can be used to efficiently estimate the geometrical complexity, and the irregularity of shapes and patterns observed in lung tumour growth (over space or time), whereas the use of traditional Euclidean geometry in such calculations is more challenging. The application of fractal analysis in biomedical imaging and time series has shown considerable promise for measuring processes as varied as heart and respiratory rates, neuronal cell characterization, and vascular development. Despite the advantages of fractal mathematics and numerous studies demonstrating its applicability to lung cancer research, many researchers and clinicians remain unaware of its potential. Therefore, this Review aims to introduce the fundamental basis of fractals and to illustrate how analysis of fractal dimension (FD) and associated measurements, such as lacunarity (texture) can be performed. We describe the fractal nature of the lung and explain why this organ is particularly suited to fractal analysis. Studies that have used fractal analyses to quantify changes in nuclear and chromatin FD in primary and metastatic tumour cells, and clinical imaging studies that correlated changes in the FD of tumours on CT and/or PET images with tumour growth and treatment responses are reviewed. Moreover, the potential use of these techniques in the diagnosis and therapeutic management of lung cancer are discussed.
Tahavvor, Ali Reza
2016-06-01
In the present study artificial neural network and fractal geometry are used to predict frost thickness and density on a cold flat plate having constant surface temperature under forced convection for different ambient conditions. These methods are very applicable in this area because phase changes such as melting and solidification are simulated by conventional methods but frost formation is a most complicated phase change phenomenon consists of coupled heat and mass transfer. Therefore conventional mathematical techniques cannot capture the effects of all parameters on its growth and development because this process influenced by many factors and it is a time dependent process. Therefore, in this work soft computing method such as artificial neural network and fractal geometry are used to do this manner. The databases for modeling are generated from the experimental measurements. First, multilayer perceptron network is used and it is found that the back-propagation algorithm with Levenberg-Marquardt learning rule is the best choice to estimate frost growth properties due to accurate and faster training procedure. Second, fractal geometry based on the Von-Koch curve is used to model frost growth procedure especially in frost thickness and density. Comparison is performed between experimental measurements and soft computing methods. Results show that soft computing methods can be used more efficiently to determine frost properties over a flat plate. Based on the developed models, wide range of frost formation over flat plates can be determined for various conditions.
Tahavvor, Ali Reza
2017-03-01
In the present study artificial neural network and fractal geometry are used to predict frost thickness and density on a cold flat plate having constant surface temperature under forced convection for different ambient conditions. These methods are very applicable in this area because phase changes such as melting and solidification are simulated by conventional methods but frost formation is a most complicated phase change phenomenon consists of coupled heat and mass transfer. Therefore conventional mathematical techniques cannot capture the effects of all parameters on its growth and development because this process influenced by many factors and it is a time dependent process. Therefore, in this work soft computing method such as artificial neural network and fractal geometry are used to do this manner. The databases for modeling are generated from the experimental measurements. First, multilayer perceptron network is used and it is found that the back-propagation algorithm with Levenberg-Marquardt learning rule is the best choice to estimate frost growth properties due to accurate and faster training procedure. Second, fractal geometry based on the Von-Koch curve is used to model frost growth procedure especially in frost thickness and density. Comparison is performed between experimental measurements and soft computing methods. Results show that soft computing methods can be used more efficiently to determine frost properties over a flat plate. Based on the developed models, wide range of frost formation over flat plates can be determined for various conditions.
Fractal Dimension and Universality in Avascular Tumor Growth
Ribeiro, Fabiano L.; Santos, Renato Vieira dos; Mata, Angélica S.
2016-01-01
The comprehension of tumor growth is a intriguing subject for scientists. New researches has been constantly required to better understand the complexity of this phenomenon. In this paper, we pursue a physical description that account for some experimental facts involving avascular tumor growth. We have proposed an explanation of some phenomenological (macroscopic) aspects of tumor, as the spatial form and the way it growths, from a individual-level (microscopic) formulation. The model propos...
Observation of a surface lattice resonance in a fractal arrangement of gold nanoparticles
Chen, Ting Lee; Segerink, Frans B; Dikken, Dirk Jan; Herek, Jennifer L
2015-01-01
The collective response of closely spaced metal particles in non-periodic arrangements has the potential to provide a beneficial angular and frequency dependence in sensing applications. In this paper, we investigate the optical response of a Sierpinski fractal arrangement of gold nanoparticles and show that it supports a collective resonance similar to the surface lattice resonances that exist in periodic arrangements of plasmonic resonators. Using back focal plane microscopy, we observe the leakage of radiation out of a surface lattice resonance that is efficiently excited when the wavenumber of the incident light matches a strong Fourier component of the fractal structure. The efficient coupling between localized surface plasmons leads to a collective resonance and a Fano-like feature in the scattering spectrum. Our experimental observations are supported by numerical simulations based on the coupled-dipole approximation and finite-difference time-domain methods. This work presents a first step towards the...
Directory of Open Access Journals (Sweden)
Jordan P. Sinclair
2015-04-01
Full Text Available Fractal symmetry is symmetry across scale. If one looks at a branch of a tree its branching pattern is reminiscent of the tree as a whole. Plants exhibit a number of different symmetries, including bilateral, rotational, translational, and fractal; deviations from each of these types has been associated with organisms developing in stressful environments. Here, we explore the utilization and meaning of fractal analysis on annual growth ring production in woody plants. Early detection of stress in plants is difficult and the compounding effects of multiple or severe stressors can lead to irreversible damage or death. Annual wood production was used to produce a time series for individuals from stands classified as either high vigor or low vigor (a general measure of health. As a measure of symmetry over time, the fractal dimension of each time series was determined and compared among vigor classes. We found that individuals obtained from low vigor sites had a significantly lower fractal dimension than those from high vigor sites. These results agree with patterns found in a variety of other organisms, and we argue that the reduced fractal dimension is related to a loss in system complexity of stressed individuals.
A principle of fractal-stochastic dualism and Gompertzian dynamics of growth and self-organization.
Waliszewski, Przemyslaw
2005-10-01
The emergence of Gompertzian dynamics at the macroscopic, tissue level during growth and self-organization is determined by the existence of fractal-stochastic dualism at the microscopic level of supramolecular, cellular system. On one hand, Gompertzian dynamics results from the complex coupling of at least two antagonistic, stochastic processes at the molecular cellular level. It is shown that the Gompertz function is a probability function, its derivative is a probability density function, and the Gompertzian distribution of probability is of non-Gaussian type. On the other hand, the Gompertz function is a contraction mapping and defines fractal dynamics in time-space; a prerequisite condition for the coupling of processes. Furthermore, the Gompertz function is a solution of the operator differential equation with the Morse-like anharmonic potential. This relationship indicates that distribution of intrasystemic forces is both non-linear and asymmetric. The anharmonic potential is a measure of the intrasystemic interactions. It attains a point of the minimum (U(0), t(0)) along with a change of both complexity and connectivity during growth and self-organization. It can also be modified by certain factors, such as retinoids.
Can fractal-like spectra be experimentally observed in aperiodic superlattices?
Maciá, Enrique; Domínguez-Adame, Francisco
1996-07-01
We numerically investigate the effects of inhomogeneities in the energy spectrum of aperiodic semiconductor superlattices, focusing our attention on Thue-Morse and Fibonacci sequences. In the absence of disorder, the corresponding electronic spectra are self-similar. The presence of a certain degree of randomness, due to imperfections occurring during the growth processes, gives rise to a progressive loss of quantum coherence, smearing out the finer details of the energy spectra predicted for perfect aperiodic superlattices and spurring the onset of electron localization. However, depending on the degree of disorder introduced, a critical size for the system exists, below which peculiar transport properties, related to the pre-fractal nature of the energy spectrum, may be measured.
Fractal Structure and Entropy Production within the Central Nervous System
Directory of Open Access Journals (Sweden)
Andrew J. E. Seely
2014-08-01
Full Text Available Our goal is to explore the relationship between two traditionally unrelated concepts, fractal structure and entropy production, evaluating both within the central nervous system (CNS. Fractals are temporal or spatial structures with self-similarity across scales of measurement; whereas entropy production represents the necessary exportation of entropy to our environment that comes with metabolism and life. Fractals may be measured by their fractal dimension; and human entropy production may be estimated by oxygen and glucose metabolism. In this paper, we observe fractal structures ubiquitously present in the CNS, and explore a hypothetical and unexplored link between fractal structure and entropy production, as measured by oxygen and glucose metabolism. Rapid increase in both fractal structures and metabolism occur with childhood and adolescent growth, followed by slow decrease during aging. Concomitant increases and decreases in fractal structure and metabolism occur with cancer vs. Alzheimer’s and multiple sclerosis, respectively. In addition to fractals being related to entropy production, we hypothesize that the emergence of fractal structures spontaneously occurs because a fractal is more efficient at dissipating energy gradients, thus maximizing entropy production. Experimental evaluation and further understanding of limitations and necessary conditions are indicated to address broad scientific and clinical implications of this work.
Spatial log-periodic oscillations of first-passage observables in fractals
Akkermans, Eric; Benichou, Olivier; Dunne, Gerald V.; Teplyaev, Alexander; Voituriez, Raphael
2012-12-01
For transport processes in geometrically restricted domains, the mean first-passage time (MFPT) admits a general scaling dependence on space parameters for diffusion, anomalous diffusion, and diffusion in disordered or fractal media. For transport in self-similar fractal structures, we obtain an expression for the source-target distance dependence of the MFPT that exhibits both the leading power-law behavior, depending on the Hausdorff and spectral dimension of the fractal, as well as small log-periodic oscillations that are a clear and definitive signal of the underlying fractal structure. We also present refined numerical results for the Sierpinski gasket that confirm this oscillatory behavior.
Nagatani, Takashi
1989-12-01
A Laplacian growth model with the third boundary condition, (1-P)∂Φ/∂n-PΦ=0, is considered in order to study the effect of the sticking probability of the diffusion-limited aggregation (DLA), where Φ is the harmonic function satisfying the Laplace equation and ∂Φ/∂n the derivative normal to the interface. The crossover from the dense structure to the DLA fractal is investigated by using a two-parameter position-space renormalization-group method. A global flow diagram in two-parameter space is obtained. It is found that there are two nontrivial fixed points, the Eden point and the DLA point. The DLA point corresponding to the DLA fractal is stable in all directions, while the Eden point is a saddle point. When the sticking probability P is not 1, the aggregate must eventually cross over to the DLA fractal. The crossover exponent φ and crossover radius r× are calculated.
Esbenshade, Donald H., Jr.
1991-01-01
Develops the idea of fractals through a laboratory activity that calculates the fractal dimension of ordinary white bread. Extends use of the fractal dimension to compare other complex structures as other breads and sponges. (MDH)
Esbenshade, Donald H., Jr.
1991-01-01
Develops the idea of fractals through a laboratory activity that calculates the fractal dimension of ordinary white bread. Extends use of the fractal dimension to compare other complex structures as other breads and sponges. (MDH)
Crossover and thermodynamic representation in the extended η model for fractal growth
Nagatani, Takashi; Stanley, H. Eugene
1990-10-01
The η model for the dielectric breakdown is extended to the case where double power laws apply. It is shown that a crossover phenomenon between the diffusion-limited aggregation (DLA) fractal and the η fractal occurs in the extended η model. Through the use of the dimensional analysis, a dimensionless parameter is found to govern the crossover. It is shown that when η1 the inverse crossover from the η fractal to the DLA fractal appears. It is also shown that the crossover radius is controlled by changing the applied field. The global flow diagram in the two-parameter space is obtained by using a two-parameter position-space renormalization-group approach. The crossover exponent and the crossover radius are calculated. The crossover phenomenon is described in terms of a thermodynamic representation of the two-phase equilibrium.
Shimokawa, Michiko; Takami, Toshiya
2014-04-01
When a droplet of a higher-density solution (HDS) is placed on top of a lower-density solution (LDS), the HDS draws a fractal pattern on the surface of the LDS. Before the fractal pattern is formed, a stick-like pattern with a periodic structure emerges in a region surrounding the surface pattern due to interfacial instability. We experimentally measure the wavelength of this stick-like pattern. The wavelength increases with the volume of the HDS and is independent of the viscosities of the two solutions. To understand the stick generation, we propose a model of miscible viscous fingering whose boundary conditions are similar to those of the experiments. The wavelength obtained from the model agrees with the experimentally obtained wavelength. The formation of the fractal pattern is discussed by comparing it with the viscous fingering.
Betancourt-Mar, J. A.; Llanos-Pérez, J. A.; Cocho, G.; Mansilla, R.; Martin, R. R.; Montero, S.; Nieto-Villar, J. M.
2017-05-01
By the use of thermodynamics formalism of irreversible processes, complex systems theory and systems biology, it is derived a relationship between the production of entropy per unit time, the fractal dimension and the tumor growth rate for human tumors cells. The thermodynamics framework developed demonstrates that, the dissipation function is a Landau potential and also the Lyapunov function of the dynamical behavior of tumor growth, which indicate the directional character, stability and robustness of the phenomenon. The entropy production rate may be used as a quantitative index of the metastatic potential of tumors. The current theoretical framework will hopefully provide a better understanding of cancer and contribute to improvements in cancer treatment.
金属锌电沉积的分形研究%Fractal growth of zinc electrodeposition
Institute of Scientific and Technical Information of China (English)
陈书荣; 张郑; 谢刚; 李金山
2006-01-01
Using zinc ring as anode and graphite as cathode, fractal growth of electrodeposited zinc in ZnSO4 solution has been studied. The fractal dimension and macroscopic morphology under various experimental conditions were analyzed. The results show that the fractal dimensions increase with the applied voltage, while the morphologies change from resembles those of DLA, thin stringy structures to open but regular grossness structures. Also, the concentration of ZnSO4 plays important role on the fractal dimension.%以环形金属锌板作为阳极,石墨棒作为阴极,对电沉积过程中金属锌的二维枝晶生长进行了研究,并对不同电沉积条件下所得到的沉积产物的分形维数和宏观形貌进行了观察、分析.研究表明,随外加电压的升高,金属锌二维沉积产物的形貌由开放的枝状晶向致密的粗大枝晶转变,分形维数也呈增大趋势;随着电解质溶液中硫酸锌浓度的增加,阴极沉积产物先后出现了类似于DLA模型模拟结果的枝状形貌、具有分叉结构的致密纤维状枝晶簇和较为粗壮的开放型规则分叉状枝晶等不同形貌,分形维数亦随之发生相应的改变.
Experimental Evidence of Dynamical Scaling in a Two-Dimensional Fractal Growth
Miyashita, Satoru; Saito, Yukio; Uwaha, Makio
1997-04-01
A dynamical scaling law of fractal aggregation is testedusing electrochemical deposition without an external electric field.Silver metal leaves grow on the edge of a Cu plate placed in a thin cell containing an AgNO3-water solution due to the difference in ionization tendency between Ag and Cu. We find that the tip height h(t) satisfies the dynamical scaling relationh(t)= c-1/(d-D_f) \\tilde{g}(tc2/(d-D_f)) with respect to the solute concentration cin the space dimension d=2 with the fractal dimension Df=1.71 of the diffusion-limited aggregation.
Hlavka, Christine A.; Strong, Laurence L.
1992-01-01
The MSS, SPOT, and AVHRR imagery of Ugandan forests were analyzed to assess the information content related to deforestation and tropical habitat fragmentation, focusing primarily on the Kibale and Mabira Forests. Analysis of actual and simulated AVHRR imagery showed that it might be possible to monitor major changes in forest extent with the relatively coarse spatial resolution of AVHRR imagery (about 1 km) provided ancillary data were available. The fractal dimension of the forest edges, measured with the Landsat and SPOT imagery, was consistently about 1.7 or 1.8. This high fractal dimension was due to the coplex pattern of clearings, remnant forest stands, and jagged forest edges caused by repeated human encroachment over centuries.
DEFF Research Database (Denmark)
Bruun Jensen, Casper
2007-01-01
. Instead, I outline a fractal approach to the study of space, society, and infrastructure. A fractal orientation requires a number of related conceptual reorientations. It has implications for thinking about scale and perspective, and (sociotechnical) relations, and for considering the role of the social...... and a fractal social theory....
Fractal Growth on the Surface of a Planet and in Orbit around it
Haranas, Ioannis; Alexiou, Athanasios
2015-01-01
Fractals are defined as geometric shapes that exhibit symmetry of scale. This simply implies that fractal is a shape that it would still look the same even if somebody could zoom in on one of its parts an infinite number of times. This property is also called self-similarity with several applications including nano pharmacology and drug nano carriers. We are interested in the study of the properties of fractal aggregates in a microgravity environment above an orbiting spacecraft. To model the effect we use a complete expression for the gravitational acceleration. In particular on the surface of the Earth the acceleration is corrected for the effect of oblateness and rotation. In the gravitational acceleration the effect of oblateness can be modeled with the inclusion of a term that contains the J2 harmonic coefficient, as well as a term that depends on the square of angular velocity of the Earth. In orbit the acceleration of gravity at the point of the spacecraft is a function of the orbital elements and incl...
Relativistic Fractal Cosmologies
Ribeiro, Marcelo B
2009-01-01
This article reviews an approach for constructing a simple relativistic fractal cosmology whose main aim is to model the observed inhomogeneities of the distribution of galaxies by means of the Lemaitre-Tolman solution of Einstein's field equations for spherically symmetric dust in comoving coordinates. This model is based on earlier works developed by L. Pietronero and J.R. Wertz on Newtonian cosmology, whose main points are discussed. Observational relations in this spacetime are presented, together with a strategy for finding numerical solutions which approximate an averaged and smoothed out single fractal structure in the past light cone. Such fractal solutions are shown, with one of them being in agreement with some basic observational constraints, including the decay of the average density with the distance as a power law (the de Vaucouleurs' density power law) and the fractal dimension in the range 1 <= D <= 2. The spatially homogeneous Friedmann model is discussed as a special case of the Lemait...
Energy Technology Data Exchange (ETDEWEB)
Xu, Mengjia; Xu, Jijin, E-mail: xujijin_1979@sjtu.edu.cn; Lu, Hao; Chen, Jieshi; Chen, Junmei; Wei, Xiao
2015-12-30
Graphical abstract: - Highlights: • Statistical and fractal analysis is applied to study the creep fracture surface. • The tensile residual stresses promote the initiation of creep crack. • The fractal dimension of a mixed mode fracture surface shows a wavy variation. • The fractal dimension increases with increasing intergranular fracture percentage. • Height coordinates of intergranular fracture surface fit Gaussian distribution. - Abstract: In order to clarify creep crack growth behavior in 2.25Cr–1.6W steel incorporating residual stresses, creep crack tests were carried out on the tension creep specimens, in which the residual stresses were generated by local remelting and cooling. Residual stresses in the specimens were measured using Synchrotron X-ray diffraction techniques. The fracture surface of the creep specimen was analyzed using statistical methods and fractal analysis. The relation between fractal dimension of the fracture surface and fracture mode of the creep specimen was discussed. Due to different fracture mechanisms, the probability density functions of the height coordinates vary with the intergranular crack percentage. Good fitting was found between Gaussian distribution and the probability function of height coordinates of the high percentage intergranular crack surface.
Dendritic design as an archetype for growth patterns in Nature: fractal and constructal views
Directory of Open Access Journals (Sweden)
Antonio F. Miguel
2014-02-01
Full Text Available The occurrence of configuration (design, shape, structure, rhythm is a universal phenomenon that occurs in every flow system. Dendritic configuration (or tree-shaped configurations is ubiquitous in nature and likely to arise in both animate and inanimate flow systems. Why is it so important? Is there a principle from which this configuration can be deduced? In this review paper we show that these systems own two of the most important properties of fractals that are self-similarity and scaling. Their configuration do not develop by chance. It´s occurrence is a universal phenomenon of physics covered by a principle. Here we also show that the emergence of dendritic configuration in flow systems constitutes a basic supportive flow path along which order need to persist is propagated.
Dewdney, A. K.
1991-01-01
Explores the subject of fractal geometry focusing on the occurrence of fractal-like shapes in the natural world. Topics include iterated functions, chaos theory, the Lorenz attractor, logistic maps, the Mandelbrot set, and mini-Mandelbrot sets. Provides appropriate computer algorithms, as well as further sources of information. (JJK)
Osler, Thomas J.
1999-01-01
Because fractal images are by nature very complex, it can be inspiring and instructive to create the code in the classroom and watch the fractal image evolve as the user slowly changes some important parameter or zooms in and out of the image. Uses programming language that permits the user to store and retrieve a graphics image as a disk file.…
Directory of Open Access Journals (Sweden)
Tatjana eStadnitski
2012-05-01
Full Text Available When investigating fractal phenomena, the following questions are fundamental for the applied researcher: (1 What are essential statistical properties of 1/f noise? (2 Which estimators are available for measuring fractality? (3 Which measurement instruments are appropriate and how are they applied? The purpose of this article is to give clear and comprehensible answers to these questions. First, theoretical characteristics of a fractal pattern (self-similarity, long memory, power law and the related fractal parameters (the Hurst coefficient, the scaling exponent, the fractional differencing parameter d of the ARFIMA methodology, the power exponent of the spectral analysis are discussed. Then, estimators of fractal parameters from different software packages commonly used by applied researchers (R, SAS, SPSS are introduced and evaluated. Advantages, disadvantages, and constrains of the popular estimators are illustrated by elaborate examples. Finally, crucial steps of fractal analysis (plotting time series data, autocorrelation and spectral functions; performing stationarity tests; choosing an adequate estimator; estimating fractal parameters; distinguishing fractal processes from short memory patterns are demonstrated with empirical time series.
Stadnitski, Tatjana
2012-01-01
WHEN INVESTIGATING FRACTAL PHENOMENA, THE FOLLOWING QUESTIONS ARE FUNDAMENTAL FOR THE APPLIED RESEARCHER: (1) What are essential statistical properties of 1/f noise? (2) Which estimators are available for measuring fractality? (3) Which measurement instruments are appropriate and how are they applied? The purpose of this article is to give clear and comprehensible answers to these questions. First, theoretical characteristics of a fractal pattern (self-similarity, long memory, power law) and the related fractal parameters (the Hurst coefficient, the scaling exponent α, the fractional differencing parameter d of the autoregressive fractionally integrated moving average methodology, the power exponent β of the spectral analysis) are discussed. Then, estimators of fractal parameters from different software packages commonly used by applied researchers (R, SAS, SPSS) are introduced and evaluated. Advantages, disadvantages, and constrains of the popular estimators ([Formula: see text] power spectral density, detrended fluctuation analysis, signal summation conversion) are illustrated by elaborate examples. Finally, crucial steps of fractal analysis (plotting time series data, autocorrelation, and spectral functions; performing stationarity tests; choosing an adequate estimator; estimating fractal parameters; distinguishing fractal processes from short-memory patterns) are demonstrated with empirical time series.
Multilayer adsorption on fractal surfaces.
Vajda, Péter; Felinger, Attila
2014-01-10
Multilayer adsorption is often observed in liquid chromatography. The most frequently employed model for multilayer adsorption is the BET isotherm equation. In this study we introduce an interpretation of multilayer adsorption measured on liquid chromatographic stationary phases based on the fractal theory. The fractal BET isotherm model was successfully used to determine the apparent fractal dimension of the adsorbent surface. The nonlinear fitting of the fractal BET equation gives us the estimation of the adsorption equilibrium constants and the monolayer saturation capacity of the adsorbent as well. In our experiments, aniline and proline were used as test molecules on reversed phase and normal phase columns, respectively. Our results suggest an apparent fractal dimension 2.88-2.99 in the case of reversed phase adsorbents, in the contrast with a bare silica column with a fractal dimension of 2.54.
Fractal Electronic Circuits Assembled From Nanoclusters
Fairbanks, M. S.; McCarthy, D.; Taylor, R. P.; Brown, S. A.
2009-07-01
Many patterns in nature can be described using fractal geometry. The effect of this fractal character is an array of properties that can include high internal connectivity, high dispersivity, and enhanced surface area to volume ratios. These properties are often desirable in applications and, consequently, fractal geometry is increasingly employed in technologies ranging from antenna to storm barriers. In this paper, we explore the application of fractal geometry to electrical circuits, inspired by the pervasive fractal structure of neurons in the brain. We show that, under appropriate growth conditions, nanoclusters of Sb form into islands on atomically flat substrates via a process close to diffusion-limited aggregation (DLA), establishing fractal islands that will form the basis of our fractal circuits. We perform fractal analysis of the islands to determine the spatial scaling properties (characterized by the fractal dimension, D) of the proposed circuits and demonstrate how varying growth conditions can affect D. We discuss fabrication approaches for establishing electrical contact to the fractal islands. Finally, we present fractal circuit simulations, which show that the fractal character of the circuit translates into novel, non-linear conduction properties determined by the circuit's D value.
Barnsley, Michael F
2012-01-01
""Difficult concepts are introduced in a clear fashion with excellent diagrams and graphs."" - Alan E. Wessel, Santa Clara University""The style of writing is technically excellent, informative, and entertaining."" - Robert McCartyThis new edition of a highly successful text constitutes one of the most influential books on fractal geometry. An exploration of the tools, methods, and theory of deterministic geometry, the treatment focuses on how fractal geometry can be used to model real objects in the physical world. Two sixteen-page full-color inserts contain fractal images, and a bonus CD of
Institute of Scientific and Technical Information of China (English)
ZhinhongLi; DongWu; Yuhansun; JunWang; YiLiu; BaozhongDong; Zhinhong
2001-01-01
Silica aggregates were prepared by base-catalyzed hydrolysis and condensation of alkoxides in alcohol.Polyethylene glycol(PEG) was used as organic modifier.The sols were characterized using Small Angle X-ray Scattering (SAXS) with synchrotron radiation as X-ray source.The structure evolution during the sol-gel process was determined and described in terms of the fractal geometry.As-produced silica aggregates were found to be mass fractals.The fractl dimensions spanned the regime 2.1-2.6 corresponding to more branched and compact structures.Both RLCA and Eden models dominated the kinetic growth under base-catalyzed condition.
Astaneh, Amin Faraji
2015-01-01
We use the Heat Kernel method to calculate the Entanglement Entropy for a given entangling region on a fractal. The leading divergent term of the entropy is obtained as a function of the fractal dimension as well as the walk dimension. The power of the UV cut-off parameter is (generally) a fractional number which indeed is a certain combination of these two indices. This exponent is known as the spectral dimension. We show that there is a novel log periodic oscillatory behavior in the entropy which has root in the complex dimension of a fractal. We finally indicate that the Holographic calculation in a certain Hyper-scaling violating bulk geometry yields the same leading term for the entanglement entropy, if one identifies the effective dimension of the hyper-scaling violating theory with the spectral dimension of the fractal. We provide more supports with comparing the behavior of the thermal entropy in terms of the temperature in these two cases.
Energy Technology Data Exchange (ETDEWEB)
Maltz, Alberto L [Departamento de Matematica, Universidad Nacional de La Plata, Casilla de Correo 172 (1900) La Plata (Argentina); Fabricius, Gabriel; Bab, Marisa A; Albano, Ezequiel V [Instituto de Investigaciones FisicoquImicas Teoricas y Aplicadas (INIFTA), CCT La Plata, UNLP, CONICET. Casilla de Correo 16, Sucursal 4, (1900) La Plata (Argentina)
2008-12-12
In this work we address the time evolution of random walks on a special type of Sierpinski carpets, which we call walk similar (WS). By considering highly symmetric fractals (symmetrically self-similar graphs (SSG)), very recently Kroen and Teufl (2003 Trans. Am. Math. Soc. 356 393) have developed a technique based on the fact that the random walk gives rise to an equivalent process in a similar subset. The method is used in order to obtain the time scaling factor ({tau}) as the average passing time (APT) of the walker from a site in the subset to any different site in the subset. For SSG, the APT is independent of the starting point. In the present work we generalize this technique under the less stringent symmetry conditions of the WS carpets, such that the APT depends on the starting point. Therefore, we calculate exactly the weighted APT ({tau}*). By performing Monte Carlo simulations on several WS carpets we verify that {tau}* plays the role of {tau} by setting the logarithmic period of the oscillatory asymptotic behaviour of dynamic observables.
Direct observation of nanowire growth and decomposition
DEFF Research Database (Denmark)
Rackauskas, Simas; Shandakov, Sergey D; Jiang, Hua
2017-01-01
knowledge, so far this has been only postulated, but never observed at the atomic level. By means of in situ environmental transmission electron microscopy we monitored and examined the atomic layer transformation at the conditions of the crystal growth and its decomposition using CuO nanowires selected...... as a model object. The atomic layer growth/decomposition was studied by varying an O2 partial pressure. Three distinct regimes of the atomic layer evolution were experimentally observed: growth, transition and decomposition. The transition regime, at which atomic layer growth/decomposition switch takes place...
Thermodynamics of Fractal Universe
Sheykhi, Ahmad; Wang, Bin
2012-01-01
We investigate the thermodynamical properties of the apparent horizon in a fractal universe. We find that one can always rewrite the Friedmann equation of the fractal universe in the form of the entropy balance relation $ \\delta Q=TdS+Td\\tilde{S}$, where $ \\delta Q $ and $ T $ are the energy flux and Unruh temperature seen by an accelerated observer just inside the apparent horizon, and $d\\tilde{S}$ is the entropy production term due to nonequilibrium thermodynamics of fractal universe. This shows that in a fractal universe, a treatment with nonequilibrium thermodynamics of spacetime may be needed. We also study the generalized second law of thermodynamics in the framework of fractal universe. When the temperature of the apparent horizon and the matter fields inside the horizon are equal, i.e. $T=T_h$, the generalized second law of thermodynamics can be fulfilled provided the deceleration and the equation of state parameters ranges either as $-1 \\leq q < 0 $, $- 1 \\leq w < - 1/3$ or as $q<-1$, $w<...
Amir, S.; Hashim Ali, S. A.; Mohamed, N. S.
2011-10-01
We initially prepared films of poly(vinylidene fluoride-co-hexafluoropropylene)/poly(ethyl methacrylate)-ammonium trifluorome-thanesulfonate dispersed with various wt.% of chromium oxide to study their properties and potential application in electrochemical devices. However, a few months later the nanocomposite polymer electrolyte membranes were found to become a sort of medium for fractal growth. This discovery led to a simulation of the fractals observed in these polymer electrolyte films using a diffusion-limited aggregation model that is based on Brownian motion theory (random walk). The fractal dimensions, D, of the fractal patterns obtained from experimental and simulation work were calculated using the box-counting method. The fractal patterns and dimensions of the simulated fractal patterns were comparable with those obtained from the original fractals observed in the polymer electrolyte films.
Combinatorial fractal Brownian motion model
Institute of Scientific and Technical Information of China (English)
朱炬波; 梁甸农
2000-01-01
To solve the problem of how to determine the non-scaled interval when processing radar clutter using fractal Brownian motion (FBM) model, a concept of combinatorial FBM model is presented. Since the earth (or sea) surface varies diversely with space, a radar clutter contains several fractal structures, which coexist on all scales. Taking the combination of two FBMs into account, via theoretical derivation we establish a combinatorial FBM model and present a method to estimate its fractal parameters. The correctness of the model and the method is proved by simulation experiments and computation of practial data. Furthermore, we obtain the relationship between fractal parameters when processing combinatorial model with a single FBM model. Meanwhile, by theoretical analysis it is concluded that when combinatorial model is observed on different scales, one of the fractal structures is more obvious.
Contour fractal analysis of grains
Guida, Giulia; Casini, Francesca; Viggiani, Giulia MB
2017-06-01
Fractal analysis has been shown to be useful in image processing to characterise the shape and the grey-scale complexity in different applications spanning from electronic to medical engineering (e.g. [1]). Fractal analysis consists of several methods to assign a dimension and other fractal characteristics to a dataset describing geometric objects. Limited studies have been conducted on the application of fractal analysis to the classification of the shape characteristics of soil grains. The main objective of the work described in this paper is to obtain, from the results of systematic fractal analysis of artificial simple shapes, the characterization of the particle morphology at different scales. The long term objective of the research is to link the microscopic features of granular media with the mechanical behaviour observed in the laboratory and in situ.
Oleshko, Klaudia; de Jesús Correa López, María; Romero, Alejandro; Ramírez, Victor; Pérez, Olga
2016-04-01
The effectiveness of fractal toolbox to capture the scaling or fractal probability distribution, and simply fractal statistics of main hydrocarbon reservoir attributes, was highlighted by Mandelbrot (1995) and confirmed by several researchers (Zhao et al., 2015). Notwithstanding, after more than twenty years, it's still common the opinion that fractals are not useful for the petroleum engineers and especially for Geoengineering (Corbett, 2012). In spite of this negative background, we have successfully applied the fractal and multifractal techniques to our project entitled "Petroleum Reservoir as a Fractal Reactor" (2013 up to now). The distinguishable feature of Fractal Reservoir is the irregular shapes and rough pore/solid distributions (Siler, 2007), observed across a broad range of scales (from SEM to seismic). At the beginning, we have accomplished the detailed analysis of Nelson and Kibler (2003) Catalog of Porosity and Permeability, created for the core plugs of siliciclastic rocks (around ten thousand data were compared). We enriched this Catalog by more than two thousand data extracted from the last ten years publications on PoroPerm (Corbett, 2012) in carbonates deposits, as well as by our own data from one of the PEMEX, Mexico, oil fields. The strong power law scaling behavior was documented for the major part of these data from the geological deposits of contrasting genesis. Based on these results and taking into account the basic principles and models of the Physics of Fractals, introduced by Per Back and Kan Chen (1989), we have developed new software (Muukíl Kaab), useful to process the multiscale geological and geophysical information and to integrate the static geological and petrophysical reservoir models to dynamic ones. The new type of fractal numerical model with dynamical power law relations among the shapes and sizes of mesh' cells was designed and calibrated in the studied area. The statistically sound power law relations were established
DEFF Research Database (Denmark)
Sørensen, Erik Schwartz; Fogedby, Hans C.; Mouritsen, Ole G.
1989-01-01
A version of the two-dimensional site-diluted spin-(1/2 Ising model is proposed as a microscopic interaction model which governs solidification and growth processes controlled by vacancy diffusion. The Ising Hamiltonian describes a solid-fluid phase transition and it permits a thermodynamic......-water interfaces....
García-Farieta, Jorge E
2016-01-01
Cosmological observations reveal that the Universe contains a hierarchy of galaxy clustering with a transition to homogeneity on large scales according to the $\\Lambda$CDM model. Some observational estimates suggest that the Universe behaves as a multifractal object, where galactic clustering is based on generalisation of the dimension in metric spaces. From this point of view, we study the spatial distribution of points by simulating galaxies on large scales in the Universe with samples from the Sloan Digital Sky Survey (SDSS), including observational holes in the masks. We build homogeneous samples following the radial selection function using the "shuffle" method for a main sample of $3,273,548$ points limited to the redshift range $0.0020$ and percentages of holes near $40\\%$, $r_H$ is displaced on scales on the order of $120~Mpc/h$. Hole percentages between $10\\%$ and $30\\%$ show an $r_H$ of $70-90~Mpc/h$, and for percentages below $10\\%$, $r_H$ decreases to become equal to the $r_H$ value of the SDSS-BO...
Mishra, Jibitesh
2007-01-01
The book covers all the fundamental aspects of generating fractals through L-system. Also it provides insight to various researches in this area for generating fractals through L-system approach & estimating dimensions. Also it discusses various applications of L-system fractals. Key Features: - Fractals generated from L-System including hybrid fractals - Dimension calculation for L-system fractals - Images & codes for L-system fractals - Research directions in the area of L-system fractals - Usage of various freely downloadable tools in this area - Fractals generated from L-System including hybrid fractals- Dimension calculation for L-system fractals- Images & codes for L-system fractals- Research directions in the area of L-system fractals- Usage of various freely downloadable tools in this area
Pelletier, J D
1997-01-01
The power spectrum S of linear transects of the earth's topography is often observed to be a power-law function of wave number k with exponent close to -2: S(k) is proportional to k^-2. In addition, river networks are fractal trees that satisfy many power-law or fractal relationships between their morphologic components. A model equation for the evolution of the earth's topography by erosional processes which produces fractal topography and fractal river networks is presented and its solutions compared in detail to real topography. The model is the diffusion equation for sediment transport on hillslopes and channels with the local diffusivity proportional to the square of the discharge. The dependence of diffusivity on discharge follows from fundamental equations of sediment transport. We study the model in two ways. In the first analysis the diffusivity is parameterized as a function of relief and a Taylor expansion procedure is carried out to obtain a differential equation for the landform elevation which i...
Khokhlov, D L
1999-01-01
The model of the universe is considered in which background of the universe is not defined by the matter but is a priori specified as a homogenous and isotropic flat space. The scale factor of the universe follows the linear law. The scale of mass changes proportional to the scale factor. This leads to that the universe has the fractal structure with a power index of 2.
Fractal analysis of sulphidic mineral
Directory of Open Access Journals (Sweden)
Miklúová Viera
2002-03-01
fractal dimensions of fine particles distribution were not observed.
Fractal patterns of fractures in granites
Velde, B.; Dubois, J.; Moore, D.; Touchard, G.
1991-01-01
Fractal measurements using the Cantor's dust method in a linear one-dimensional analysis mode were made on the fracture patterns revealed on two-dimensional, planar surfaces in four granites. This method allows one to conclude that: 1. (1)|The fracture systems seen on two-dimensional surfaces in granites are consistent with the part of fractal theory that predicts a repetition of patterns on different scales of observation, self similarity. Fractal analysis gives essentially the same values of D on the scale of kilometres, metres and centimetres (five orders of magnitude) using mapped, surface fracture patterns in a Sierra Nevada granite batholith (Mt. Abbot quadrangle, Calif.). 2. (2)|Fractures show the same fractal values at different depths in a given batholith. Mapped fractures (main stage ore veins) at three mining levels (over a 700 m depth interval) of the Boulder batholith, Butte, Mont. show the same fractal values although the fracture disposition appears to be different at different levels. 3. (3)|Different sets of fracture planes in a granite batholith, Central France, and in experimental deformation can have different fractal values. In these examples shear and tension modes have the same fractal values while compressional fractures follow a different fractal mode of failure. The composite fracture patterns are also fractal but with a different, median, fractal value compared to the individual values for the fracture plane sets. These observations indicate that the fractal method can possibly be used to distinguish fractures of different origins in a complex system. It is concluded that granites fracture in a fractal manner which can be followed at many scales. It appears that fracture planes of different origins can be characterized using linear fractal analysis. ?? 1991.
Energy Technology Data Exchange (ETDEWEB)
Amir, S; Mohamed, N S [Center for Foundation Studies in Science, University of Malaya, 50603 Kuala Lumpur (Malaysia); Hashim Ali, S A, E-mail: shahizat@um.edu.my [Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur (Malaysia)
2011-10-15
We initially prepared films of poly(vinylidene fluoride-co-hexafluoropropylene)/poly(ethyl methacrylate)-ammonium trifluorome-thanesulfonate dispersed with various wt.% of chromium oxide to study their properties and potential application in electrochemical devices. However, a few months later the nanocomposite polymer electrolyte membranes were found to become a sort of medium for fractal growth. This discovery led to a simulation of the fractals observed in these polymer electrolyte films using a diffusion-limited aggregation model that is based on Brownian motion theory (random walk). The fractal dimensions, D, of the fractal patterns obtained from experimental and simulation work were calculated using the box-counting method. The fractal patterns and dimensions of the simulated fractal patterns were comparable with those obtained from the original fractals observed in the polymer electrolyte films.
Order-fractal transitions in abstract paintings
Energy Technology Data Exchange (ETDEWEB)
Calleja, E.M. de la, E-mail: elsama79@gmail.com [Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, 91501-970, Porto Alegre, RS (Brazil); Cervantes, F. [Department of Applied Physics, CINVESTAV-IPN, Carr. Antigua a Progreso km.6, Cordemex, C.P.97310, Mérida, Yucatán (Mexico); Calleja, J. de la [Department of Informatics, Universidad Politécnica de Puebla, 72640 (Mexico)
2016-08-15
In this study, we determined the degree of order for 22 Jackson Pollock paintings using the Hausdorff–Besicovitch fractal dimension. Based on the maximum value of each multi-fractal spectrum, the artworks were classified according to the year in which they were painted. It has been reported that Pollock’s paintings are fractal and that this feature was more evident in his later works. However, our results show that the fractal dimension of these paintings ranges among values close to two. We characterize this behavior as a fractal-order transition. Based on the study of disorder-order transition in physical systems, we interpreted the fractal-order transition via the dark paint strokes in Pollock’s paintings as structured lines that follow a power law measured by the fractal dimension. We determined self-similarity in specific paintings, thereby demonstrating an important dependence on the scale of observations. We also characterized the fractal spectrum for the painting entitled Teri’s Find. We obtained similar spectra for Teri’s Find and Number 5, thereby suggesting that the fractal dimension cannot be rejected completely as a quantitative parameter for authenticating these artworks. -- Highlights: •We determined the degree of order in Jackson Pollock paintings using the Hausdorff–Besicovitch dimension. •We detected a fractal-order transition from Pollock’s paintings between 1947 and 1951. •We suggest that Jackson Pollock could have painted Teri’s Find.
Comparison of two fractal interpolation methods
Fu, Yang; Zheng, Zeyu; Xiao, Rui; Shi, Haibo
2017-03-01
As a tool for studying complex shapes and structures in nature, fractal theory plays a critical role in revealing the organizational structure of the complex phenomenon. Numerous fractal interpolation methods have been proposed over the past few decades, but they differ substantially in the form features and statistical properties. In this study, we simulated one- and two-dimensional fractal surfaces by using the midpoint displacement method and the Weierstrass-Mandelbrot fractal function method, and observed great differences between the two methods in the statistical characteristics and autocorrelation features. From the aspect of form features, the simulations of the midpoint displacement method showed a relatively flat surface which appears to have peaks with different height as the fractal dimension increases. While the simulations of the Weierstrass-Mandelbrot fractal function method showed a rough surface which appears to have dense and highly similar peaks as the fractal dimension increases. From the aspect of statistical properties, the peak heights from the Weierstrass-Mandelbrot simulations are greater than those of the middle point displacement method with the same fractal dimension, and the variances are approximately two times larger. When the fractal dimension equals to 1.2, 1.4, 1.6, and 1.8, the skewness is positive with the midpoint displacement method and the peaks are all convex, but for the Weierstrass-Mandelbrot fractal function method the skewness is both positive and negative with values fluctuating in the vicinity of zero. The kurtosis is less than one with the midpoint displacement method, and generally less than that of the Weierstrass-Mandelbrot fractal function method. The autocorrelation analysis indicated that the simulation of the midpoint displacement method is not periodic with prominent randomness, which is suitable for simulating aperiodic surface. While the simulation of the Weierstrass-Mandelbrot fractal function method has
Process simulation of fractal growth of microfiltration membrane fouling%微滤膜垢分形生长的过程模拟
Institute of Scientific and Technical Information of China (English)
张维; 许丹宇; 郑先强; 孙凯; 侯霙
2012-01-01
基于微滤膜系统污垢形成机制和分形理论,建立微滤过程膜表面混合垢生长DLA模型,并通过实验验证了模型模拟的可行性和准确性。选取不同运行周期条件下微滤膜系统中的受污染膜丝,进行膜垢污染生长的实时测试,并与不同运行条件下模型的动态模拟结果进行实际比较分析,结果表明两者分形维数相近,且分形维数与膜污染程度呈正相关,说明该模型能够动态表征膜污染水平,可揭示出微滤过程中膜垢生长的动态变化规律,预测出膜材料的受污染水平。%Based on the formation mechanism of microfiltration membrane fouling and the fractal theory,a diffusion-limited aggregation(DLA) model of microfiltration membrane fouling was established.Its accuracy and feasibility were validated with experiments.Membrane fibers of microfiltration systems contaminated under different operation cycles were taken to test the real-time growth of membrane fouling.These test results were compared with their simulation counterparts from the dynamic model for different operating conditions.The comparisons show that the fractal dimensions of the experiment results and the dynamic model are similar.Furthermore,the fractal dimensions of the dynamic model are positively correlated with the degree of membrane fouling.Therefore,it is shown that this model can simulate the dynamic of membrane fouling degrees,reveal changing dynamics of the growth of microfiltration membrane fouling during filtration processes,as well as predict the degree of fouling of membrane materials.
Segmentation of histological structures for fractal analysis
Dixon, Vanessa; Kouznetsov, Alexei; Tambasco, Mauro
2009-02-01
Pathologists examine histology sections to make diagnostic and prognostic assessments regarding cancer based on deviations in cellular and/or glandular structures. However, these assessments are subjective and exhibit some degree of observer variability. Recent studies have shown that fractal dimension (a quantitative measure of structural complexity) has proven useful for characterizing structural deviations and exhibits great potential for automated cancer diagnosis and prognosis. Computing fractal dimension relies on accurate image segmentation to capture the architectural complexity of the histology specimen. For this purpose, previous studies have used techniques such as intensity histogram analysis and edge detection algorithms. However, care must be taken when segmenting pathologically relevant structures since improper edge detection can result in an inaccurate estimation of fractal dimension. In this study, we established a reliable method for segmenting edges from grayscale images. We used a Koch snowflake, an object of known fractal dimension, to investigate the accuracy of various edge detection algorithms and selected the most appropriate algorithm to extract the outline structures. Next, we created validation objects ranging in fractal dimension from 1.3 to 1.9 imitating the size, structural complexity, and spatial pixel intensity distribution of stained histology section images. We applied increasing intensity thresholds to the validation objects to extract the outline structures and observe the effects on the corresponding segmentation and fractal dimension. The intensity threshold yielding the maximum fractal dimension provided the most accurate fractal dimension and segmentation, indicating that this quantitative method could be used in an automated classification system for histology specimens.
Hashemi, S. M.; Jagodič, U.; Mozaffari, M. R.; Ejtehadi, M. R.; Muševič, I.; Ravnik, M.
2017-01-01
Fractals are remarkable examples of self-similarity where a structure or dynamic pattern is repeated over multiple spatial or time scales. However, little is known about how fractal stimuli such as fractal surfaces interact with their local environment if it exhibits order. Here we show geometry-induced formation of fractal defect states in Koch nematic colloids, exhibiting fractal self-similarity better than 90% over three orders of magnitude in the length scales, from micrometers to nanometres. We produce polymer Koch-shaped hollow colloidal prisms of three successive fractal iterations by direct laser writing, and characterize their coupling with the nematic by polarization microscopy and numerical modelling. Explicit generation of topological defect pairs is found, with the number of defects following exponential-law dependence and reaching few 100 already at fractal iteration four. This work demonstrates a route for generation of fractal topological defect states in responsive soft matter. PMID:28117325
Neutron scattering from fractals
DEFF Research Database (Denmark)
Kjems, Jørgen; Freltoft, T.; Richter, D.
1986-01-01
The scattering formalism for fractal structures is presented. Volume fractals are exemplified by silica particle clusters formed either from colloidal suspensions or by flame hydrolysis. The determination of the fractional dimensionality through scattering experiments is reviewed, and recent small...
Thamrin, Cindy; Stern, Georgette; Frey, Urs
2010-06-01
There is increasing interest in the study of fractals in medicine. In this review, we provide an overview of fractals, of techniques available to describe fractals in physiological data, and we propose some reasons why a physician might benefit from an understanding of fractals and fractal analysis, with an emphasis on paediatric respiratory medicine where possible. Among these reasons are the ubiquity of fractal organisation in nature and in the body, and how changes in this organisation over the lifespan provide insight into development and senescence. Fractal properties have also been shown to be altered in disease and even to predict the risk of worsening of disease. Finally, implications of a fractal organisation include robustness to errors during development, ability to adapt to surroundings, and the restoration of such organisation as targets for intervention and treatment.
Hashemi, S. M.; Jagodič, U.; Mozaffari, M. R.; Ejtehadi, M. R.; Muševič, I.; Ravnik, M.
2017-01-01
Fractals are remarkable examples of self-similarity where a structure or dynamic pattern is repeated over multiple spatial or time scales. However, little is known about how fractal stimuli such as fractal surfaces interact with their local environment if it exhibits order. Here we show geometry-induced formation of fractal defect states in Koch nematic colloids, exhibiting fractal self-similarity better than 90% over three orders of magnitude in the length scales, from micrometers to nanometres. We produce polymer Koch-shaped hollow colloidal prisms of three successive fractal iterations by direct laser writing, and characterize their coupling with the nematic by polarization microscopy and numerical modelling. Explicit generation of topological defect pairs is found, with the number of defects following exponential-law dependence and reaching few 100 already at fractal iteration four. This work demonstrates a route for generation of fractal topological defect states in responsive soft matter.
Ji-Huan He
2011-01-01
A new fractal derive is defined, which is very easy for engineering applications to discontinuous problems, two simple examples are given to elucidate to establish governing equations with fractal derive and how to solve such equations, respectively.
Mukherjee, Anika; Chan, Adrian D C; Keating, Sarah; Redline, Raymond W; Fritsch, Michael K; Machin, Geoffrey A; Cornejo-Palma, Daniel; de Nanassy, Joseph; El-Demellawy, Dina; von Dadelszen, Peter; Benton, Samantha J; Grynspan, David
2016-01-01
The distal villous hypoplasia (DVH) pattern is a placental correlate of fetal growth restriction. Because the pattern seems to involve less complexity than do appropriately developed placental villi, we postulated that it may be associated with lower fractal dimension-a mathematical measure of complexity. Our study objectives were to evaluate interobserver agreement related to the DVH pattern among expert pathologists and to determine whether pathologist classification of DVH correlates with fractal dimension. A study set of 30 images of placental parenchyma at ×4 magnification was created by a single pathologist from a digital slide archive. The images were graded for the DVH pattern according to pre-specified definitions and included 10 images graded as "no DVH" (grade = 0), 10 with mild to moderate DVH (grade = 1), and 10 with severe DVH (grade = 2). The images were randomly sorted and shown to a panel of 4 international experts who similarly graded the images for DVH. Weighted kappas were calculated. For each image, fractal dimension was calculated by the Box Counting method. The correlation coefficient between (1) the averaged DVH scores obtained by the 5 pathologists and (2) fractal dimension was calculated. The mean weighted kappa score among the observers was 0.59 (range: 0.42-0.70). The correlation coefficient between fractal dimension and the averaged DVH score was -0.915 (P fractal dimension and represents an objective measure for DVH.
Fraboni, Michael; Moller, Trisha
2008-01-01
Fractal geometry offers teachers great flexibility: It can be adapted to the level of the audience or to time constraints. Although easily explained, fractal geometry leads to rich and interesting mathematical complexities. In this article, the authors describe fractal geometry, explain the process of iteration, and provide a sample exercise.…
Fractal properties of financial markets
Budinski-Petković, Lj.; Lončarević, I.; Jakšić, Z. M.; Vrhovac, S. B.
2014-09-01
We present an analysis of the USA stock market using a simple fractal function. Financial bubbles preceding the 1987, 2000 and 2007 crashes are investigated using the Besicovitch-Ursell fractal function. Fits show a good agreement with the S&P 500 data when a complete financial growth is considered, starting at the threshold of the abrupt growth and ending at the peak. Moving the final time of the fitting interval towards earlier dates causes growing discrepancy between two curves. On the basis of a detailed analysis of the financial index behavior we propose a method for identifying the stage of the current financial growth and estimating the time in which the index value is going to reach the maximum.
Fractal structures in nonlinear plasma physics.
Viana, R L; da Silva, E C; Kroetz, T; Caldas, I L; Roberto, M; Sanjuán, M A F
2011-01-28
Fractal structures appear in many situations related to the dynamics of conservative as well as dissipative dynamical systems, being a manifestation of chaotic behaviour. In open area-preserving discrete dynamical systems we can find fractal structures in the form of fractal boundaries, associated to escape basins, and even possessing the more general property of Wada. Such systems appear in certain applications in plasma physics, like the magnetic field line behaviour in tokamaks with ergodic limiters. The main purpose of this paper is to show how such fractal structures have observable consequences in terms of the transport properties in the plasma edge of tokamaks, some of which have been experimentally verified. We emphasize the role of the fractal structures in the understanding of mesoscale phenomena in plasmas, such as electromagnetic turbulence.
Directory of Open Access Journals (Sweden)
FELICIA RAMONA BIRAU
2012-05-01
Full Text Available In this article, the concept of capital market is analysed using Fractal Market Hypothesis which is a modern, complex and unconventional alternative to classical finance methods. Fractal Market Hypothesis is in sharp opposition to Efficient Market Hypothesis and it explores the application of chaos theory and fractal geometry to finance. Fractal Market Hypothesis is based on certain assumption. Thus, it is emphasized that investors did not react immediately to the information they receive and of course, the manner in which they interpret that information may be different. Also, Fractal Market Hypothesis refers to the way that liquidity and investment horizons influence the behaviour of financial investors.
Nanocrystals formation and fractal microstructural assessment in Au/Ge bilayer films upon annealing
Energy Technology Data Exchange (ETDEWEB)
Chen, Z.W. [Department of Physics and Materials Science, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, Hong Kong (China)]. E-mail: cnzwchen@yahoo.com.cn; Lai, J.K.L. [Department of Physics and Materials Science, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, Hong Kong (China); Shek, C.H. [Department of Physics and Materials Science, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, Hong Kong (China); Chen, H.D. [Department of Physics and Materials Science, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, Hong Kong (China)
2005-08-31
Nanocrystals formation and fractal microstructural assessment in Au/Ge bilayer films upon annealing have been investigated by transmission electron microscopy and high-resolution transmission electron microscopy observations. Experimental results indicated that the microstructure of the metal Au film plays an important role in metal-induced crystallization for Au/Ge bilayer films upon annealing. Synchronously, the crystallization processes of amorphous Ge accompanied by the formation of Ge fractal clusters, which were composed of Ge nanocrystals. We found that the grain boundaries of polycrystalline Au film were the initial nucleation sites of Ge nanocrystals. High-resolution transmission electron microscopy observations showed successive nucleation of amorphous Ge at Au grain boundaries near fractal tips. The crystallization process was suggested to be diffusion controlled and a random successive nucleation and growth mechanism.
Fractal Reconnection in Solar and Stellar Environments
Shibata, Kazunari
2016-01-01
Recent space based observations of the Sun revealed that magnetic reconnection is ubiquitous in the solar atmosphere, ranging from small scale reconnection (observed as nanoflares) to large scale one (observed as long duration flares or giant arcades). Often the magnetic reconnection events are associated with mass ejections or jets, which seem to be closely related to multiple plasmoid ejections from fractal current sheet. The bursty radio and hard X-ray emissions from flares also suggest the fractal reconnection and associated particle acceleration. We shall discuss recent observations and theories related to the plasmoid-induced-reconnection and the fractal reconnection in solar flares, and their implication to reconnection physics and particle acceleration. Recent findings of many superflares on solar type stars that has extended the applicability of the fractal reconnection model of solar flares to much a wider parameter space suitable for stellar flares are also discussed.
Applications of Fractal Signals
Directory of Open Access Journals (Sweden)
Ion TUTĂNESCU
2008-05-01
Full Text Available "Fractal" term - which in Latin languagedefines something fragmented anomalous - wasintroduced in mathematics by B. B. Mandelbrot in1975. He avoided to define it rigorously and used it todesignate some "rugged" and "self-similar"geometrical forms. Fractals were involved in the theoryof chaotic dynamic systems and used to designatecertain specific sets; finally, they were “captured” bygeometry and remarked in tackling of the boundaryproblems. It proved that the fractals can be of interesteven in the signal’s theory.
一种基于分形结构的树生长微带天线设计%Design of a tree-growth microstrip antenna based on fractal structure
Institute of Scientific and Technical Information of China (English)
樊磊; 骆延; 黄卡玛; 杨阳
2014-01-01
基于分形理论和自然树竞争(TGCA)思想，提出了采用树枝结构在不同生长因子下进行迭代生长的方法，对分形天线多谐振频率点进行优化控制，克服常见分形天线难于调整多个谐振频率点位置关系的缺点。树生长分形天线通过生长因子和天线尺寸的线性调整，可以方便地实现高低谐振频率的优化设计，方法简单易行。基于该方法，采用时域有限差分(FDTD)算法优化设计了一种具有 GSM900/DCS1800双频谐振点的树生长微带分形天线。从实验和仿真结果可以看出，该微带分形天线在0.91 GHz,1.81 GHz谐振频率处带宽均大于100 MHz，水平方向为全向辐射，测量所得结果和仿真数据吻合较好，验证了采用生长因子调整分形天线谐振频率的方法。%Based on the fractal theory and the idea of the Tree Growth Competition Algorithm(TGCA), a method of optimizing the resonant frequencies by iterating the branch structure at different growth factors is proposed, which overcomes the inconvenience in adjusting the position relationships of multi-band frequencies for common fractal antennas. The lower and upper resonant frequencies of the tree-growth fractal antenna can be optimized conveniently by linearly changing its size and growth factors. Based on this method, a tree-growth fractal antenna is optimized in the GSM900/DCS1800 frequency band by using Finite-Difference Time-Domain(FDTD) algorithm. The measured and simulated results indicate that the bandwidths of microwave fractal antenna are both above 100 MHz at the resonant frequencies of 0.91 GHz and 1.81 GHz with good omni-directional radiation patterns in the horizontal direction. These good agreements of the measured and simulated results validate the feasibility of the proposed method.
Fractal Geometry of Architecture
Lorenz, Wolfgang E.
In Fractals smaller parts and the whole are linked together. Fractals are self-similar, as those parts are, at least approximately, scaled-down copies of the rough whole. In architecture, such a concept has also been known for a long time. Not only architects of the twentieth century called for an overall idea that is mirrored in every single detail, but also Gothic cathedrals and Indian temples offer self-similarity. This study mainly focuses upon the question whether this concept of self-similarity makes architecture with fractal properties more diverse and interesting than Euclidean Modern architecture. The first part gives an introduction and explains Fractal properties in various natural and architectural objects, presenting the underlying structure by computer programmed renderings. In this connection, differences between the fractal, architectural concept and true, mathematical Fractals are worked out to become aware of limits. This is the basis for dealing with the problem whether fractal-like architecture, particularly facades, can be measured so that different designs can be compared with each other under the aspect of fractal properties. Finally the usability of the Box-Counting Method, an easy-to-use measurement method of Fractal Dimension is analyzed with regard to architecture.
Baryshev, Yuri
2002-01-01
This is the first book to present the fascinating new results on the largest fractal structures in the universe. It guides the reader, in a simple way, to the frontiers of astronomy, explaining how fractals appear in cosmic physics, from our solar system to the megafractals in deep space. It also offers a personal view of the history of the idea of self-similarity and of cosmological principles, from Plato's ideal architecture of the heavens to Mandelbrot's fractals in the modern physical cosmos. In addition, this invaluable book presents the great fractal debate in astronomy (after Luciano Pi
Fractal and Small-World Networks Formed by Self-Organized Critical Dynamics
Watanabe, Akitomo; Mizutaka, Shogo; Yakubo, Kousuke
2015-11-01
We propose a dynamical model in which a network structure evolves in a self-organized critical (SOC) manner and explain a possible origin of the emergence of fractal and small-world networks. Our model combines a network growth and its decay by failures of nodes. The decay mechanism reflects the instability of large functional networks against cascading overload failures. It is demonstrated that the dynamical system surely exhibits SOC characteristics, such as power-law forms of the avalanche size distribution, the cluster size distribution, and the distribution of the time interval between intermittent avalanches. During the network evolution, fractal networks are spontaneously generated when networks experience critical cascades of failures that lead to a percolation transition. In contrast, networks far from criticality have small-world structures. We also observe the crossover behavior from fractal to small-world structure in the network evolution.
Fractal and Small-World Networks Formed by Self-Organized Critical Dynamics
Watanabe, Akitomo; Yakubo, Kousuke
2015-01-01
We propose a dynamical model in which a network structure evolves in a self-organized critical (SOC) manner and explain a possible origin of the emergence of fractal and small-world networks. Our model combines a network growth and its decay by failures of nodes. The decay mechanism reflects the instability of large functional networks against cascading overload failures. It is demonstrated that the dynamical system surely exhibits SOC characteristics, such as power-law forms of the avalanche size distribution, the cluster size distribution, and the distribution of the time interval between intermittent avalanches. During the network evolution, fractal networks are spontaneously generated when networks experience critical cascades of failures that lead to a percolation transition. In contrast, networks far from criticality have small-world structures. We also observe the crossover behavior from fractal to small-world structure in the network evolution.
Paradigms of Complexity: Fractals and Structures in the Sciences
Novak, Miroslav M.
The Table of Contents for the book is as follows: * Preface * The Origin of Complexity (invited talk) * On the Existence of Spatially Uniform Scaling Laws in the Climate System * Multispectral Backscattering: A Fractal-Structure Probe * Small-Angle Multiple Scattering on a Fractal System of Point Scatterers * Symmetric Fractals Generated by Cellular Automata * Bispectra and Phase Correlations for Chaotic Dynamical Systems * Self-Organized Criticality Models of Neural Development * Altered Fractal and Irregular Heart Rate Behavior in Sick Fetuses * Extract Multiple Scaling in Long-Term Heart Rate Variability * A Semi-Continous Box Counting Method for Fractal Dimension Measurement of Short Single Dimension Temporal Signals - Preliminary Study * A Fractional Brownian Motion Model of Cracking * Self-Affine Scaling Studies on Fractography * Coarsening of Fractal Interfaces * A Fractal Model of Ocean Surface Superdiffusion * Stochastic Subsurface Flow and Transport in Fractal Fractal Conductivity Fields * Rendering Through Iterated Function Systems * The σ-Hull - The Hull Where Fractals Live - Calculating a Hull Bounded by Log Spirals to Solve the Inverse IFS-Problem by the Detected Orbits * On the Multifractal Properties of Passively Convected Scalar Fields * New Statistical Textural Transforms for Non-Stationary Signals: Application to Generalized Mutlifractal Analysis * Laplacian Growth of Parallel Needles: Their Mullins-Sekerka Instability * Entropy Dynamics Associated with Self-Organization * Fractal Properties in Economics (invited talk) * Fractal Approach to the Regional Seismic Event Discrimination Problem * Fractal and Topological Complexity of Radioactive Contamination * Pattern Selection: Nonsingular Saffman-Taylor Finger and Its Dynamic Evolution with Zero Surface Tension * A Family of Complex Wavelets for the Characterization of Singularities * Stabilization of Chaotic Amplitude Fluctuations in Multimode, Intracavity-Doubled Solid-State Lasers * Chaotic
Fractal aspects of calcium binding protein structures
Energy Technology Data Exchange (ETDEWEB)
Isvoran, Adriana [West University of Timisoara, Department of Chemistry, Pestalozzi 16, 300115 Timisoara (Romania)], E-mail: aisvoran@cbg.uvt.ro; Pitulice, Laura [West University of Timisoara, Department of Chemistry, Pestalozzi 16, 300115 Timisoara (Romania); Craescu, Constantin T. [INSERM U759/Institute Curie-Recherche, Centre Universitaire Paris-Sud, Batiment 112, 91405 Orsay (France); Chiriac, Adrian [West University of Timisoara, Department of Chemistry, Pestalozzi 16, 300115 Timisoara (Romania)
2008-03-15
The structures of EF-hand calcium binding proteins may be classified into two distinct groups: extended and compact structures. In this paper we studied 20 different structures of calcium binding proteins using the fractal analysis. Nine structures show extended shapes, one is semi-compact and the other 10 have compact shapes. Our study reveals different fractal characteristics for protein backbones belonging to different structural classes and these observations may be correlated to the physicochemical forces governing the protein folding.
Pulse regime in formation of fractal fibers
Smirnov, B. M.
2016-11-01
The pulse regime of vaporization of a bulk metal located in a buffer gas is analyzed as a method of generation of metal atoms under the action of a plasma torch or a laser beam. Subsequently these atoms are transformed into solid nanoclusters, fractal aggregates and then into fractal fibers if the growth process proceeds in an external electric field. We are guided by metals in which transitions between s and d-electrons of their atoms are possible, since these metals are used as catalysts and filters in interaction with gas flows. The resistance of metal fractal structures to a gas flow is evaluated that allows one to find optimal parameters of a fractal structure for gas flow propagation through it. The thermal regime of interaction between a plasma pulse or a laser beam and a metal surface is analyzed. It is shown that the basic energy from an external source is consumed on a bulk metal heating, and the efficiency of atom evaporation from the metal surface, that is the ratio of energy fluxes for vaporization and heating, is 10-3-10-4 for transient metals under consideration. A typical energy flux ( 106 W/cm2), a typical surface temperature ( 3000 K), and a typical pulse duration ( 1 μs) provide a sufficient amount of evaporated atoms to generate fractal fibers such that each molecule of a gas flow collides with the skeleton of fractal fibers many times.
Ultra wide band electromagnetic scattering of a fractal profile
Rouvier, S.; Borderies, P.; Chênerie, I.
1997-03-01
The relationship between the fractal dimension of a perfectly conducting bidimensionnal profile and the fractal dimension of the time domain scattered field is investigated. The first part of the paper is dedicated to the profile itself; implementation of the counting box method for fractal dimension estimation is described and improved by the adjunction of an iterative process involving a correlation criterion. The second part is about the field scattered by a fractal profile which is calculated by the method of moments; polarizations, directions of incidence and observation effects are studied. Influence of spectral window and of noise is also investigated. Results show that fractal dimensions of the field and of the profile are linked by a monotonous increasing function which weakly depends on the polarizations and on the directions of incidence and observation. Moreover, the fractal dimension shows robustness to noise.
Warchalowski, Wiktor; Krawczyk, Malgorzata J.
2017-03-01
We found the Lindenmayer systems for line graphs built on selected fractals. We show that the fractal dimension of such obtained graphs in all analysed cases is the same as for their original graphs. Both for the original graphs and for their line graphs we identified classes of nodes which reflect symmetry of the graph.
Perepelitsa, VA; Sergienko, [No Value; Kochkarov, AM
1999-01-01
Definitions of prefractal and fractal graphs are introduced, and they are used to formulate mathematical models in different fields of knowledge. The topicality of fractal-graph recognition from the point of view, of fundamental improvement in the efficiency of the solution of algorithmic problems i
Enhancement of critical temperature in fractal metamaterial superconductors
Smolyaninov, Igor I
2016-01-01
Fractal metamaterial superconductor geometry has been suggested and analyzed based on the recently developed theoretical description of critical temperature increase in epsilon near zero (ENZ) metamaterial superconductors. Considerable enhancement of critical temperature has been predicted in such materials due to appearance of large number of additional poles in the inverse dielectric response function of the fractal. Our results agree with the recent observation (Fratini et al. Nature 466, 841 (2010)) that fractal defect structure promotes superconductivity.
Enhancement of critical temperature in fractal metamaterial superconductors
Energy Technology Data Exchange (ETDEWEB)
Smolyaninov, Igor I., E-mail: smoly@umd.edu [Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742 (United States); Smolyaninova, Vera N. [Department of Physics Astronomy and Geosciences, Towson University, 8000 York Road, Towson, MD 21252 (United States)
2017-04-15
Fractal metamaterial superconductor geometry has been suggested and analyzed based on the recently developed theoretical description of critical temperature increase in epsilon near zero (ENZ) metamaterial superconductors. Considerable enhancement of critical temperature has been predicted in such materials due to appearance of large number of additional poles in the inverse dielectric response function of the fractal. Our results agree with the recent observation (Fratini et al. Nature 466, 841 (2010)) that fractal defect structure promotes superconductivity.
Multiband Terahertz Photonic Band Gaps of Subwavelength Planar Fractals
Institute of Scientific and Technical Information of China (English)
ZHAO Guo-Zhong; TIAN Yan; SUN Hong-Qi; ZHANG Cun-Lin; YANG Guo-Zhen
2006-01-01
Optical transmission properties of subwavelength planar fractals in terahertz (THz) frequency regime are studied by means of time-domain spectroscopy. The transmission spectra with multiple pass bands and stop bands are observed. The tunable photonic band gaps are realized by changing the angle between the principle axis of planar fractal and the polarization of THz wave. The possible application of the subwavelength optical component is discussed. We attribute the detected transmittance from subwavelength fractals to localized resonances.
Fractal images induce fractal pupil dilations and constrictions.
Moon, P; Muday, J; Raynor, S; Schirillo, J; Boydston, C; Fairbanks, M S; Taylor, R P
2014-09-01
Fractals are self-similar structures or patterns that repeat at increasingly fine magnifications. Research has revealed fractal patterns in many natural and physiological processes. This article investigates pupillary size over time to determine if their oscillations demonstrate a fractal pattern. We predict that pupil size over time will fluctuate in a fractal manner and this may be due to either the fractal neuronal structure or fractal properties of the image viewed. We present evidence that low complexity fractal patterns underlie pupillary oscillations as subjects view spatial fractal patterns. We also present evidence implicating the autonomic nervous system's importance in these patterns. Using the variational method of the box-counting procedure we demonstrate that low complexity fractal patterns are found in changes within pupil size over time in millimeters (mm) and our data suggest that these pupillary oscillation patterns do not depend on the fractal properties of the image viewed.
Fractals in art and nature: why do we like them?
Spehar, Branka; Taylor, Richard P.
2013-03-01
Fractals have experienced considerable success in quantifying the visual complexity exhibited by many natural patterns, and continue to capture the imagination of scientists and artists alike. Fractal patterns have also been noted for their aesthetic appeal, a suggestion further reinforced by the discovery that the poured patterns of the American abstract painter Jackson Pollock are also fractal, together with the findings that many forms of art resemble natural scenes in showing scale-invariant, fractal-like properties. While some have suggested that fractal-like patterns are inherently pleasing because they resemble natural patterns and scenes, the relation between the visual characteristics of fractals and their aesthetic appeal remains unclear. Motivated by our previous findings that humans display a consistent preference for a certain range of fractal dimension across fractal images of various types we turn to scale-specific processing of visual information to understand this relationship. Whereas our previous preference studies focused on fractal images consisting of black shapes on white backgrounds, here we extend our investigations to include grayscale images in which the intensity variations exhibit scale invariance. This scale-invariance is generated using a 1/f frequency distribution and can be tuned by varying the slope of the rotationally averaged Fourier amplitude spectrum. Thresholding the intensity of these images generates black and white fractals with equivalent scaling properties to the original grayscale images, allowing a direct comparison of preferences for grayscale and black and white fractals. We found no significant differences in preferences between the two groups of fractals. For both set of images, the visual preference peaked for images with the amplitude spectrum slopes from 1.25 to 1.5, thus confirming and extending the previously observed relationship between fractal characteristics of images and visual preference.
Fractal characterization of surface electrical discharges
Energy Technology Data Exchange (ETDEWEB)
Egiziano, L.; Femia, N.; Lupo' , G.; Tucci, V. (Salerno Univ. (Italy). Ist. di Ingegneria Elettronica Naples Univ. (Italy). Dip. di Ingegneria Elettrica)
1991-01-01
The concepts of fractal geometry have been usefully applied to describe several physical processes whose growth mechanisms are characterized by complex topological structures. The fractal characterization of electrical discharges taking place at the air/solid dielectric interface is considered in this paper. A numerical procedure allowing the reproduction the typical discharge patterns, known as Lichtenberg figures, is presented: the growth process of the discharge is simulated by solving iteratively the Laplace equation with moving boundary conditions and by considering two power probability laws whose exponents determine the ramification level of the structure. The discharge patterns are then considered as fractal sets and their characteristic parameters are determined. The dependence of the typical structures on the two exponents of the probability laws are also discussed.
Construction of Fractal Surfaces by Recurrent Fractal Interpolation Curves
Yun, Chol-Hui; O., Hyong-chol; Choi, Hui-chol
2013-01-01
A method to construct fractal surfaces by recurrent fractal curves is provided. First we construct fractal interpolation curves using a recurrent iterated functions system(RIFS) with function scaling factors and estimate their box-counting dimension. Then we present a method of construction of wider class of fractal surfaces by fractal curves and Lipschitz functions and calculate the box-counting dimension of the constructed surfaces. Finally, we combine both methods to have more flexible con...
Fractal analysis of scatter imaging signatures to distinguish breast pathologies
Eguizabal, Alma; Laughney, Ashley M.; Krishnaswamy, Venkataramanan; Wells, Wendy A.; Paulsen, Keith D.; Pogue, Brian W.; López-Higuera, José M.; Conde, Olga M.
2013-02-01
Fractal analysis combined with a label-free scattering technique is proposed for describing the pathological architecture of tumors. Clinicians and pathologists are conventionally trained to classify abnormal features such as structural irregularities or high indices of mitosis. The potential of fractal analysis lies in the fact of being a morphometric measure of the irregular structures providing a measure of the object's complexity and self-similarity. As cancer is characterized by disorder and irregularity in tissues, this measure could be related to tumor growth. Fractal analysis has been probed in the understanding of the tumor vasculature network. This work addresses the feasibility of applying fractal analysis to the scattering power map (as a physical modeling) and principal components (as a statistical modeling) provided by a localized reflectance spectroscopic system. Disorder, irregularity and cell size variation in tissue samples is translated into the scattering power and principal components magnitude and its fractal dimension is correlated with the pathologist assessment of the samples. The fractal dimension is computed applying the box-counting technique. Results show that fractal analysis of ex-vivo fresh tissue samples exhibits separated ranges of fractal dimension that could help classifier combining the fractal results with other morphological features. This contrast trend would help in the discrimination of tissues in the intraoperative context and may serve as a useful adjunct to surgeons.
Equivalent Relation between Normalized Spatial Entropy and Fractal Dimension
Chen, Yanguang
2016-01-01
Fractal dimension is defined on the base of entropy, including macro state entropy and information entropy. The generalized dimension of multifractals is based on Renyi entropy. However, the mathematical transform from entropy to fractal dimension is not yet clear in both theory and practice. This paper is devoted to revealing the equivalence relation between spatial entropy and fractal dimension using box-counting method. Based on varied regular fractals, the numerical relationship between spatial entropy and fractal dimension is examined. The results show that the ratio of actual entropy (Mq) to the maximum entropy (Mmax) equals the ratio of actual dimension (Dq) to the maximum dimension (Dmax), that is, Mq/Mmax=Dq/Dmax. For real systems, the spatial entropy and fractal dimension of complex spatial systems such as cities can be converted into one another by means of functional box-counting method. The theoretical inference is verified by observational data of urban form. A conclusion is that normalized spat...
Observational tests of Galileon gravity with growth rate
Hirano, Koichi
2016-10-01
We compare observational data of growth rate with the prediction by Galileon theory. For the same value of the energy density parameter Ω_{m,0}, the growth rate in Galileon models is enhanced compared with the Λ CDM case, due to the enhancement of Newton's constant. The smaller Ω_{m,0} is, the more suppressed growth rate is. Hence the best fit value of Ω_{m,0} in the Galileon model is 0.16 from only the growth rate data, which is considerably smaller than such value obtained from observations of supernovae Ia, the cosmic microwave background and baryon acoustic oscillations. We also find the upper limit of the Brans-Dicke parameter to be ω < -1000 (1σ ), from the growth rate data. In this paper, specific galileon models are considered, not the entire class. More and better growth rate data are required to distinguish between dark energy and modified gravity.
McAteer, R. T. J.
2013-06-01
When Mandelbrot, the father of modern fractal geometry, made this seemingly obvious statement he was trying to show that we should move out of our comfortable Euclidean space and adopt a fractal approach to geometry. The concepts and mathematical tools of fractal geometry provides insight into natural physical systems that Euclidean tools cannot do. The benet from applying fractal geometry to studies of Self-Organized Criticality (SOC) are even greater. SOC and fractal geometry share concepts of dynamic n-body interactions, apparent non-predictability, self-similarity, and an approach to global statistics in space and time that make these two areas into naturally paired research techniques. Further, the iterative generation techniques used in both SOC models and in fractals mean they share common features and common problems. This chapter explores the strong historical connections between fractal geometry and SOC from both a mathematical and conceptual understanding, explores modern day interactions between these two topics, and discusses how this is likely to evolve into an even stronger link in the near future.
Objetos fractales y arquitectura
MARTÍNEZ REQUENA, CELIA ANA
2015-01-01
Este trabajo final de grado versa acerca de la fractalidad y su posible aplicación arquitectónica. Se parte del concepto de fractal quedándose con la idea de que “un fractal es un diseño que se repite indefinidamente hacia el infinito cada vez a escala menor” y se presentan los diferentes conjuntos haciendo especial hincapié en los fractales clásicos. La fractalidad se puede apreciar en la naturaleza (p.e: un árbol tiene un tronco, este se divide en ramas, cada una de ellas en ...
Directory of Open Access Journals (Sweden)
M. A. Navascués
2013-01-01
Full Text Available This paper tackles the construction of fractal maps on the unit sphere. The functions defined are a generalization of the classical spherical harmonics. The methodology used involves an iterated function system and a linear and bounded operator of functions on the sphere. For a suitable choice of the coefficients of the system, one obtains classical maps on the sphere. The different values of the system parameters provide Bessel sequences, frames, and Riesz fractal bases for the Lebesgue space of the square integrable functions on the sphere. The Laplace series expansion is generalized to a sum in terms of the new fractal mappings.
Objetos fractales y arquitectura
MARTÍNEZ REQUENA, CELIA ANA
2015-01-01
Este trabajo final de grado versa acerca de la fractalidad y su posible aplicación arquitectónica. Se parte del concepto de fractal quedándose con la idea de que “un fractal es un diseño que se repite indefinidamente hacia el infinito cada vez a escala menor” y se presentan los diferentes conjuntos haciendo especial hincapié en los fractales clásicos. La fractalidad se puede apreciar en la naturaleza (p.e: un árbol tiene un tronco, este se divide en ramas, cada una de ellas en ...
Fractal structures and fractal functions as disease indicators
Escos, J.M; Alados, C.L.; Emlen, J.M.
1995-01-01
Developmental instability is an early indicator of stress, and has been used to monitor the impacts of human disturbance on natural ecosystems. Here we investigate the use of different measures of developmental instability on two species, green peppers (Capsicum annuum), a plant, and Spanish ibex (Capra pyrenaica), an animal. For green peppers we compared the variance in allometric relationship between control plants, and a treatment group infected with the tomato spotted wilt virus. The results show that infected plants have a greater variance about the allometric regression line than the control plants. We also observed a reduction in complexity of branch structure in green pepper with a viral infection. Box-counting fractal dimension of branch architecture declined under stress infection. We also tested the reduction in complexity of behavioral patterns under stress situations in Spanish ibex (Capra pyrenaica). Fractal dimension of head-lift frequency distribution measures predator detection efficiency. This dimension decreased under stressful conditions, such as advanced pregnancy and parasitic infection. Feeding distribution activities reflect food searching efficiency. Power spectral analysis proves to be the most powerful tool for character- izing fractal behavior, revealing a reduction in complexity of time distribution activity under parasitic infection.
Smitha, K A; Gupta, A K; Jayasree, R S
2015-09-07
Glioma, the heterogeneous tumors originating from glial cells, generally exhibit varied grades and are difficult to differentiate using conventional MR imaging techniques. When this differentiation is crucial in the disease prognosis and treatment, even the advanced MR imaging techniques fail to provide a higher discriminative power for the differentiation of malignant tumor from benign ones. A powerful image processing technique applied to the imaging techniques is expected to provide a better differentiation. The present study focuses on the fractal analysis of fluid attenuation inversion recovery MR images, for the differentiation of glioma. For this, we have considered the most important parameters of fractal analysis, fractal dimension and lacunarity. While fractal analysis assesses the malignancy and complexity of a fractal object, lacunarity gives an indication on the empty space and the degree of inhomogeneity in the fractal objects. Box counting method with the preprocessing steps namely binarization, dilation and outlining was used to obtain the fractal dimension and lacunarity in glioma. Statistical analysis such as one-way analysis of variance and receiver operating characteristic (ROC) curve analysis helped to compare the mean and to find discriminative sensitivity of the results. It was found that the lacunarity of low and high grade gliomas vary significantly. ROC curve analysis between low and high grade glioma for fractal dimension and lacunarity yielded 70.3% sensitivity and 66.7% specificity and 70.3% sensitivity and 88.9% specificity, respectively. The study observes that fractal dimension and lacunarity increases with an increase in the grade of glioma and lacunarity is helpful in identifying most malignant grades.
FRACTAL IMAGE FEATURE VECTORS WITH APPLICATIONS IN FRACTOGRAPHY
Directory of Open Access Journals (Sweden)
Hynek Lauschmann
2011-05-01
Full Text Available The morphology of fatigue fracture surface (caused by constant cycle loading is strictly related to crack growth rate. This relation may be expressed, among other methods, by means of fractal analysis. Fractal dimension as a single numerical value is not sufficient. Two types of fractal feature vectors are discussed: multifractal and multiparametric. For analysis of images, the box-counting method for 3D is applied with respect to the non-homogeneity of dimensions (two in space, one in brightness. Examples of application are shown: images of several fracture surfaces are analyzed and related to crack growth rate.
Trabajando fractales con Winlogo
Sabogal, Sonia; Arenas, Gilberto
2007-01-01
Después de una breve introducción en la cual se establecerán algunos conceptos teóricos básicos de la geometría fractal, se realizarán talleres en los cuales, con ayuda de las herramientas que trabaja el software WinLogo, se construirán diversos fractales, analizando sus principales características (autosimilitud, dimensión, etc.)
Turcotte, Donald L.
Tectonic processes build landforms that are subsequently destroyed by erosional processes. Landforms exhibit fractal statistics in a variety of ways; examples include (1) lengths of coast lines; (2) number-size statistics of lakes and islands; (3) spectral behavior of topography and bathymetry both globally and locally; and (4) branching statistics of drainage networks. Erosional processes are dominant in the development of many landforms on this planet, but similar fractal statistics are also applicable to the surface of Venus where minimal erosion has occurred. A number of dynamical systems models for landforms have been proposed, including (1) cellular automata; (2) diffusion limited aggregation; (3) self-avoiding percolation; and (4) advective-diffusion equations. The fractal statistics and validity of these models will be discussed. Earthquakes also exhibit fractal statistics. The frequency-magnitude statistics of earthquakes satisfy the fractal Gutenberg-Richter relation both globally and locally. Earthquakes are believed to be a classic example of self-organized criticality. One model for earthquakes utilizes interacting slider-blocks. These slider block models have been shown to behave chaotically and to exhibit self-organized criticality. The applicability of these models will be discussed and alternative approaches will be presented. Fragmentation has been demonstrated to produce fractal statistics in many cases. Comminution is one model for fragmentation that yields fractal statistics. It has been proposed that comminution is also responsible for much of the deformation in the earth's crust. The brittle disruption of the crust and the resulting earthquakes present an integrated problem with many fractal aspects.
Nanoparticles dynamics on a surface: fractal pattern formation and fragmentation
DEFF Research Database (Denmark)
Dick, Veronika V.; Solov'yov, Ilia; Solov'yov, Andrey V.
2010-01-01
In this paper we review our recent results on the formation and the post-growth relaxation processes of nanofractals on surface. For this study we developed a method which describes the internal dynamics of particles in a fractal and accounts for their diffusion and detachment. We demonstrate...... that these kinetic processes determine the final shape of the islands on surface after post-growth relaxation. We consider different scenarios of fractal relaxation and analyze the time evolution of the island's morphology....
Nanoparticles dynamics on a surface: fractal pattern formation and fragmentation
DEFF Research Database (Denmark)
Dick, Veronika V.; Solov'yov, Ilia; Solov'yov, Andrey V.
2010-01-01
In this paper we review our recent results on the formation and the post-growth relaxation processes of nanofractals on surface. For this study we developed a method which describes the internal dynamics of particles in a fractal and accounts for their diffusion and detachment. We demonstrate...... that these kinetic processes determine the final shape of the islands on surface after post-growth relaxation. We consider different scenarios of fractal relaxation and analyze the time evolution of the island's morphology....
Fractal 1/f Dynamics Suggest Entanglement of Measurement and Human Performance
Holden, John G.; Choi, Inhyun; Amazeen, Polemnia G.; Van Orden, Guy
2011-01-01
Variability of repeated measurements in human performances exhibits fractal 1/f noise. Yet the relative strength of this fractal pattern varies widely across conditions, tasks, and individuals. Four experiments illustrate how subtle details of the conditions of measurement change the fractal patterns observed across task conditions. The results…
Interdiffusion assessment of nanoparticles in fat fractal patterns
Energy Technology Data Exchange (ETDEWEB)
Chen, Z W; Lai, J K L; Shek, C H; Chen, H D [Department of Physics and Materials Science, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, Hong Kong (China)
2004-10-07
Nanoparticles of polycrystalline Ge have been grown in a freshly cleaved single crystal NaCl (100) substrate, starting from Au/Ge bilayer films prepared using the evaporation method during annealing. The experimental results indicate that fat fractal Ge patterns can be formed in Au/Ge bilayer films by annealing at 100 deg. C for 60 and 70 min. Here, we report in detail interdiffusion assessment of nanoparticles in fat fractal patterns. The scaling exponent (or fractal dimension) of polycrystalline Ge clusters in fat fractal patterns is larger than that of the conventional diffusion-limited aggregation. The formation of fractal patterns and the perplexing scaling behaviour may result from the random successive nucleation and growth mechanism.
Hierarchical socioeconomic fractality: The rich, the poor, and the middle-class
Eliazar, Iddo; Cohen, Morrel H.
2014-05-01
Since the seminal work of the Italian economist Vilfredo Pareto, the study of wealth and income has been a topic of active scientific exploration engaging researches ranging from economics and political science to econophysics and complex systems. This paper investigates the intrinsic fractality of wealth and income. To that end we introduce and characterize three forms of socioeconomic scale-invariance-poor fractality, rich fractality, and middle-class fractality-and construct hierarchical fractal approximations of general wealth and income distributions, based on the stitching of these three forms of fractality. Intertwining the theoretical results with real-world empirical data we then establish that the three forms of socioeconomic fractality-amalgamated into a composite hierarchical structure-underlie the distributions of wealth and income in human societies. We further establish that the hierarchical socioeconomic fractality of wealth and income is also displayed by empirical rank distributions observed across the sciences.
Fractal Weyl law for quantum fractal eigenstates.
Shepelyansky, D L
2008-01-01
The properties of the resonant Gamow states are studied numerically in the semiclassical limit for the quantum Chirikov standard map with absorption. It is shown that the number of such states is described by the fractal Weyl law, and their Husimi distributions closely follow the strange repeller set formed by classical orbits nonescaping in future times. For large matrices the distribution of escape rates converges to a fixed shape profile characterized by a spectral gap related to the classical escape rate.
Observation of Single Colloidal Platinum Nanocrystal Growth Trajectories
Energy Technology Data Exchange (ETDEWEB)
Zheng, Haimei; Smith, Rachel; Jun, Young-wook; Kisielowski, Christian; Dahmen, Ulrich; Alivisatos, A. Paul
2009-02-09
It is conventionally assumed that the growth of monodisperse colloidal nanocrystals requires a temporally discrete nucleation followed by monomer attachment onto the existing nuclei. However, recent studies have reported violations of this classical growth model, and have suggested that inter-particle interactions are also involved during the growth. Mechanisms of nanocrystal growth still remain controversial. Using in situ transmission electron microscopy, we show that platinum nanocrystals can grow either by monomer attachment from solution onto the existing particles or by coalescence between the particles. Surprisingly, an initially broad size distribution of the nanocrystals can spontaneously narrow. We suggest that nanocrystals take different pathways of growth based on their size- and morphology-dependent internal energies. These observations are expected to be highly relevant for other nanocrystal systems.
The fractal forest: fractal geometry and applications in forest science.
Nancy D. Lorimer; Robert G. Haight; Rolfe A. Leary
1994-01-01
Fractal geometry is a tool for describing and analyzing irregularity. Because most of what we measure in the forest is discontinuous, jagged, and fragmented, fractal geometry has potential for improving the precision of measurement and description. This study reviews the literature on fractal geometry and its applications to forest measurements.
Realization of Fractal Affine Transformation
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
This paper gives the definition of fractal affine transformation and presents a specific method for its realization and its cor responding mathematical equations which are essential in fractal image construction.
Fractal Patterns and Chaos Games
Devaney, Robert L.
2004-01-01
Teachers incorporate the chaos game and the concept of a fractal into various areas of the algebra and geometry curriculum. The chaos game approach to fractals provides teachers with an opportunity to help students comprehend the geometry of affine transformations.
Marks-Tarlow, Terry
Linear concepts of time plus the modern capacity to track history emerged out of circular conceptions characteristic of ancient and traditional cultures. A fractal concept of time lies implicitly within the analog clock, where each moment is treated as unique. With fractal geometry the best descriptor of nature, qualities of self-similarity and scale invariance easily model her endless variety and recursive patterning, both in time and across space. To better manage temporal aspects of our lives, a fractal concept of time is non-reductive, based more on the fullness of being than on fragments of doing. By using a fractal concept of time, each activity or dimension of life is multiply and vertically nested. Each nested cycle remains simultaneously present, operating according to intrinsic dynamics and time scales. By adding the vertical axis of simultaneity to the horizontal axis of length, time is already full and never needs to be filled. To attend to time's vertical dimension is to tap into the imaginary potential for infinite depth. To switch from linear to fractal time allows us to relax into each moment while keeping in mind the whole.
Fractal and multifractal analyses of bipartite networks.
Liu, Jin-Long; Wang, Jian; Yu, Zu-Guo; Xie, Xian-Hua
2017-03-31
Bipartite networks have attracted considerable interest in various fields. Fractality and multifractality of unipartite (classical) networks have been studied in recent years, but there is no work to study these properties of bipartite networks. In this paper, we try to unfold the self-similarity structure of bipartite networks by performing the fractal and multifractal analyses for a variety of real-world bipartite network data sets and models. First, we find the fractality in some bipartite networks, including the CiteULike, Netflix, MovieLens (ml-20m), Delicious data sets and (u, v)-flower model. Meanwhile, we observe the shifted power-law or exponential behavior in other several networks. We then focus on the multifractal properties of bipartite networks. Our results indicate that the multifractality exists in those bipartite networks possessing fractality. To capture the inherent attribute of bipartite network with two types different nodes, we give the different weights for the nodes of different classes, and show the existence of multifractality in these node-weighted bipartite networks. In addition, for the data sets with ratings, we modify the two existing algorithms for fractal and multifractal analyses of edge-weighted unipartite networks to study the self-similarity of the corresponding edge-weighted bipartite networks. The results show that our modified algorithms are feasible and can effectively uncover the self-similarity structure of these edge-weighted bipartite networks and their corresponding node-weighted versions.
Fractal and multifractal analyses of bipartite networks
Liu, Jin-Long; Wang, Jian; Yu, Zu-Guo; Xie, Xian-Hua
2017-01-01
Bipartite networks have attracted considerable interest in various fields. Fractality and multifractality of unipartite (classical) networks have been studied in recent years, but there is no work to study these properties of bipartite networks. In this paper, we try to unfold the self-similarity structure of bipartite networks by performing the fractal and multifractal analyses for a variety of real-world bipartite network data sets and models. First, we find the fractality in some bipartite networks, including the CiteULike, Netflix, MovieLens (ml-20m), Delicious data sets and (u, v)-flower model. Meanwhile, we observe the shifted power-law or exponential behavior in other several networks. We then focus on the multifractal properties of bipartite networks. Our results indicate that the multifractality exists in those bipartite networks possessing fractality. To capture the inherent attribute of bipartite network with two types different nodes, we give the different weights for the nodes of different classes, and show the existence of multifractality in these node-weighted bipartite networks. In addition, for the data sets with ratings, we modify the two existing algorithms for fractal and multifractal analyses of edge-weighted unipartite networks to study the self-similarity of the corresponding edge-weighted bipartite networks. The results show that our modified algorithms are feasible and can effectively uncover the self-similarity structure of these edge-weighted bipartite networks and their corresponding node-weighted versions. PMID:28361962
Fractal and multifractal analyses of bipartite networks
Liu, Jin-Long; Wang, Jian; Yu, Zu-Guo; Xie, Xian-Hua
2017-03-01
Bipartite networks have attracted considerable interest in various fields. Fractality and multifractality of unipartite (classical) networks have been studied in recent years, but there is no work to study these properties of bipartite networks. In this paper, we try to unfold the self-similarity structure of bipartite networks by performing the fractal and multifractal analyses for a variety of real-world bipartite network data sets and models. First, we find the fractality in some bipartite networks, including the CiteULike, Netflix, MovieLens (ml-20m), Delicious data sets and (u, v)-flower model. Meanwhile, we observe the shifted power-law or exponential behavior in other several networks. We then focus on the multifractal properties of bipartite networks. Our results indicate that the multifractality exists in those bipartite networks possessing fractality. To capture the inherent attribute of bipartite network with two types different nodes, we give the different weights for the nodes of different classes, and show the existence of multifractality in these node-weighted bipartite networks. In addition, for the data sets with ratings, we modify the two existing algorithms for fractal and multifractal analyses of edge-weighted unipartite networks to study the self-similarity of the corresponding edge-weighted bipartite networks. The results show that our modified algorithms are feasible and can effectively uncover the self-similarity structure of these edge-weighted bipartite networks and their corresponding node-weighted versions.
Ghost quintessence in fractal gravity
Indian Academy of Sciences (India)
Habib Abedi; Mustafa Salti
2015-04-01
In this study, using the time-like fractal theory of gravity, we mainly focus on the ghost dark energy model which was recently suggested to explain the present acceleration of the cosmic expansion. Next, we establish a connection between the quintessence scalar field and fractal ghost dark energy density. This correspondence allows us to reconstruct the potential and the dynamics of a fractal canonical scalar field (the fractal quintessence) according to the evolution of ghost dark energy density.
Structural five-fold symmetry in the fractal morphology of diffusion-limited aggregates
Arneodo, A.; Argoul, F.; Muzy, J. F.; Tabard, M.
1992-09-01
The statistical self-similarity of the geometry of diffusion-limited aggregates and the multifractal nature of the growth probability distribution on the surface of the growing clusters are investigated using the wavelet transform. This study reveals the existence of a predominant structural five-fold symmetry in the internal frozen region as well as in the active outer region of the interface. This observation is corroborated by a statistical analysis of the screening effects that govern diffusion-limited aggregation (DLA) growth in linear and sector-shaped cells. The existence of this symmetry is likely to be a clue to a hierarchichal fractal ordering. We report on the discovery of Fibonacci sequences in the inner extinct region of large mass off-lattice DLA clusters, with a branching ratio which converges asymptotically to the golden mean. We suggest an interpretation of the DLA morphology as a “quasifractal” counterpart of the well-ordered snowflake fractal architecture.
Fractal geometry and stochastics IV
Bandt, Christoph
2010-01-01
Over the years fractal geometry has established itself as a substantial mathematical theory in its own right. This book collects survey articles covering many of the developments, like Schramm-Loewner evolution, fractal scaling limits, exceptional sets for percolation, and heat kernels on fractals.
Fractal Models of Earthquake Dynamics
Bhattacharya, Pathikrit; Kamal,; Samanta, Debashis
2009-01-01
Our understanding of earthquakes is based on the theory of plate tectonics. Earthquake dynamics is the study of the interactions of plates (solid disjoint parts of the lithosphere) which produce seismic activity. Over the last about fifty years many models have come up which try to simulate seismic activity by mimicking plate plate interactions. The validity of a given model is subject to the compliance of the synthetic seismic activity it produces to the well known empirical laws which describe the statistical features of observed seismic activity. Here we present a review of two such models of earthquake dynamics with main focus on a relatively new model namely The Two Fractal Overlap Model.
Fractal Cosmology in an Open Universe
Joyce, M; Montuori, M; Pietronero, L; Sylos-Labini, F
2000-01-01
The clustering of galaxies is well characterized by fractal properties, withthe presence of an eventual cross-over to homogeneity still a matter ofconsiderable debate. In this letter we discuss the cosmological implications ofa fractal distribution of matter, with a possible cross-over to homogeneity atan undetermined scale R_{homo}. Contrary to what is generally assumed, we showthat, even when R_{homo} -> \\infty, this possibility can be treatedconsistently within the framework of the expanding universe solutions ofFriedmann. The fractal is a perturbation to an open cosmology in which theleading homogeneous component is the cosmic background radiation (CBR). Thiscosmology, inspired by the observed galaxy distributions, provides a simpleexplanation for the recent data which indicate the absence of deceleration inthe expansion (q_o \\approx 0). Correspondingly the `age problem' is alsoresolved. Further we show that the model can be extended back from thecurvature dominated arbitrarily deep into the radiation dom...
Relaxation Dynamics of Semiflexible Fractal Macromolecules
Directory of Open Access Journals (Sweden)
Jonas Mielke
2016-07-01
Full Text Available We study the dynamics of semiflexible hyperbranched macromolecules having only dendritic units and no linear spacers, while the structure of these macromolecules is modeled through T-fractals. We construct a full set of eigenmodes of the dynamical matrix, which couples the set of Langevin equations. Based on the ensuing relaxation spectra, we analyze the mechanical relaxation moduli. The fractal character of the macromolecules reveals itself in the storage and loss moduli in the intermediate region of frequencies through scaling, whereas at higher frequencies, we observe the locally-dendritic structure that is more pronounced for higher stiffness.
Interstellar extinction by fractal polycrystalline graphite clusters?
Andersen, A C; Pustovit, V N; Niklasson, G A
2001-01-01
Certain dust particles in space are expected to appear as clusters of individual grains. The morphology of these clusters could be fractal or compact. To determine how these structural features would affect the interpretation of the observed interstellar extinction peak at $\\sim 4.6~\\mu$m, we have calculated the extinction by compact and fractal polycrystalline graphite clusters consisting of touching identical spheres. We compare three general methods for computing the extinction of the clusters, namely, a rigorous solution and two different discrete-dipole approximation methods.
The fractal nature of vacuum arc cathode spots
Energy Technology Data Exchange (ETDEWEB)
Anders, Andre
2005-05-27
Cathode spot phenomena show many features of fractals, for example self-similar patterns in the emitted light and arc erosion traces. Although there have been hints on the fractal nature of cathode spots in the literature, the fractal approach to spot interpretation is underutilized. In this work, a brief review of spot properties is given, touching the differences between spot type 1 (on cathodes surfaces with dielectric layers) and spot type 2 (on metallic, clean surfaces) as well as the known spot fragment or cell structure. The basic properties of self-similarity, power laws, random colored noise, and fractals are introduced. Several points of evidence for the fractal nature of spots are provided. Specifically power laws are identified as signature of fractal properties, such as spectral power of noisy arc parameters (ion current, arc voltage, etc) obtained by fast Fourier transform. It is shown that fractal properties can be observed down to the cutoff by measurement resolution or occurrence of elementary steps in physical processes. Random walk models of cathode spot motion are well established: they go asymptotically to Brownian motion for infinitesimal step width. The power spectrum of the arc voltage noise falls as 1/f {sup 2}, where f is frequency, supporting a fractal spot model associated with Brownian motion.
Energy Technology Data Exchange (ETDEWEB)
Benenti, Giuliano; Casati, Giulio; Guarneri, Italo; Terraneo, Marcello
2001-07-02
We numerically analyze quantum survival probability fluctuations in an open, classically chaotic system. In a quasiclassical regime and in the presence of classical mixed phase space, such fluctuations are believed to exhibit a fractal pattern, on the grounds of semiclassical arguments. In contrast, we work in a classical regime of complete chaoticity and in a deep quantum regime of strong localization. We provide evidence that fluctuations are still fractal, due to the slow, purely quantum algebraic decay in time produced by dynamical localization. Such findings considerably enlarge the scope of the existing theory.
Fractal actors and infrastructures
DEFF Research Database (Denmark)
Bøge, Ask Risom
2011-01-01
-network-theory (ANT) into surveillance studies (Ball 2002, Adey 2004, Gad & Lauritsen 2009). In this paper, I further explore the potential of this connection by experimenting with Marilyn Strathern’s concept of the fractal (1991), which has been discussed in newer ANT literature (Law 2002; Law 2004; Jensen 2007). I...... under surveillance. Based on fieldwork conducted in 2008 and 2011 in relation to my Master’s thesis and PhD respectively, I illustrate fractal concepts by describing the acts, actors and infrastructure that make up the ‘DNA surveillance’ conducted by the Danish police....
Deppman, Airton
2016-01-01
The non extensive aspects of $p_T$ distributions obtained in high energy collisions are discussed in relation to possible fractal structure in hadrons, in the sense of the thermofractal structure recently introduced. The evidences of self-similarity in both theoretical and experimental works in High Energy and in Hadron Physics are discussed, to show that the idea of fractal structure of hadrons and fireballs have being under discussion for decades. The non extensive self-consistent thermodynamics and the thermofractal structure allow one to connect non extensivity to intermittence and possibly to parton distribution functions in a single theoretical framework.
Direct observation of episodic growth in an abyssal xenophyophore (Protista)
Gooday, A. J.; Bett, B. J.; Pratt, D. N.
1993-11-01
Three specimens of the xenophyophore Reticulammina labyrinthica were photographed on the Madeira Abyssal Plain (31°6.1'N, 21°10.9'W; 4944 m) using the Bathysnap time-lapse camera system. During the 8 month observation period, the specimens underwent an estimated 3-10 fold increase in volume. Growth occurred episodically in several distinct phases, each lasting 2-3 days, during which sediment was collected and incorporated into the test. These phases were separated by fairly regular periods of about 2 months when the organisms showed little obvious activity. The growth phases were approximately synchronous between specimens. However, it is not clear whether the periodicity and apparent synchronization of these events resulted from an external (environmental) cue or whether growth is internally controlled and the synchronization arose by chance. These unique observations, which represent the first direct measurement of growth in any abyssal organism living outside a hydrothermal vent field, suggest that xenophyophores combine test growth with deposit feeding. The tests appear to grow more quickly, and to be more active, dynamic structures, than previously believed.
Namazi, Hamidreza; Kulish, Vladimir V.; Akrami, Amin
2016-01-01
One of the major challenges in vision research is to analyze the effect of visual stimuli on human vision. However, no relationship has been yet discovered between the structure of the visual stimulus, and the structure of fixational eye movements. This study reveals the plasticity of human fixational eye movements in relation to the ‘complex’ visual stimulus. We demonstrated that the fractal temporal structure of visual dynamics shifts towards the fractal dynamics of the visual stimulus (image). The results showed that images with higher complexity (higher fractality) cause fixational eye movements with lower fractality. Considering the brain, as the main part of nervous system that is engaged in eye movements, we analyzed the governed Electroencephalogram (EEG) signal during fixation. We have found out that there is a coupling between fractality of image, EEG and fixational eye movements. The capability observed in this research can be further investigated and applied for treatment of different vision disorders. PMID:27217194
Namazi, Hamidreza; Kulish, Vladimir V.; Akrami, Amin
2016-05-01
One of the major challenges in vision research is to analyze the effect of visual stimuli on human vision. However, no relationship has been yet discovered between the structure of the visual stimulus, and the structure of fixational eye movements. This study reveals the plasticity of human fixational eye movements in relation to the ‘complex’ visual stimulus. We demonstrated that the fractal temporal structure of visual dynamics shifts towards the fractal dynamics of the visual stimulus (image). The results showed that images with higher complexity (higher fractality) cause fixational eye movements with lower fractality. Considering the brain, as the main part of nervous system that is engaged in eye movements, we analyzed the governed Electroencephalogram (EEG) signal during fixation. We have found out that there is a coupling between fractality of image, EEG and fixational eye movements. The capability observed in this research can be further investigated and applied for treatment of different vision disorders.
Fractal elements and their applications
Gil’mutdinov, Anis Kharisovich; El-Khazali, Reyad
2017-01-01
This book describes a new type of passive electronic components, called fractal elements, from a theoretical and practical point of view. The authors discuss in detail the physical implementation and design of fractal devices for application in fractional-order signal processing and systems. The concepts of fractals and fractal signals are explained, as well as the fundamentals of fractional calculus. Several implementations of fractional impedances are discussed, along with comparison of their performance characteristics. Details of design, schematics, fundamental techniques and implementation of RC-based fractal elements are provided. .
Fractal Representation of Exergy
Directory of Open Access Journals (Sweden)
Yvain Canivet
2016-02-01
Full Text Available We developed a geometrical model to represent the thermodynamic concepts of exergy and anergy. The model leads to multi-scale energy lines (correlons that we characterised by fractal dimension and entropy analyses. A specific attention will be paid to overlapping points, rising interesting remarks about trans-scale dynamics of heat flows.
Earnshow, R; Jones, H
1991-01-01
This volume is based upon the presentations made at an international conference in London on the subject of 'Fractals and Chaos'. The objective of the conference was to bring together some of the leading practitioners and exponents in the overlapping fields of fractal geometry and chaos theory, with a view to exploring some of the relationships between the two domains. Based on this initial conference and subsequent exchanges between the editors and the authors, revised and updated papers were produced. These papers are contained in the present volume. We thank all those who contributed to this effort by way of planning and organisation, and also all those who helped in the production of this volume. In particular, we wish to express our appreciation to Gerhard Rossbach, Computer Science Editor, Craig Van Dyck, Production Director, and Nancy A. Rogers, who did the typesetting. A. J. Crilly R. A. Earnshaw H. Jones 1 March 1990 Introduction Fractals and Chaos The word 'fractal' was coined by Benoit Mandelbrot i...
New particle growth and shrinkage observed in subtropical environments
Directory of Open Access Journals (Sweden)
L.-H. Young
2012-07-01
Full Text Available We present the first systematic analysis for new particle formation (NPF, growth and shrinkage of new particles observed at four different sites in subtropical Central Taiwan. A total of 14 NPF events were identified during 137 days of ambient measurements during a cold and warm season. The derived nucleation rates of 1 nm particles (J_{1} and growth rates were in the range of 39.6–252.9 cm^{−3} s^{−1} and 6.5–14.5 nm h^{−1}, respectively. The NPF events occurred on days either with low condensation sink (CS, increased morning traffic emissions and the breakup of nocturnal inversion layer (type A, or with high CS, minimum levels of primary traffic emissions and enhanced atmospheric dilution (type B. On non-event days, the particle number concentrations were mostly driven by traffic emissions. We have also observed shrinkage of new particles (type A-S and B-S, reversal of growth, during five out of the 14 NPF events. In intense shrinkage cases, the grown particles shrank back to the smallest measurable size of ~10 nm, thereby creating a unique "arch-like" shape in the size distribution contour plot. The particle shrinkage rates ranged from 5.1 to 7.6 nm h^{−1}. The ratios of shrinkage-to-growth rates were mostly in the range of 0.40–0.65, suggesting that a large fraction of the condensable species that contributed to growth were likely semi-volatile. The particle shrinkage was related to air masses with low CS due to atmospheric dilution, high ambient temperature and low relative humidity and such atmospheric conditions may have facilitated the evaporation of semi-volatile species from the particles to the gas phase. Our observations show that the new particle growth may be a~reversible process and the evaporating semi-volatile species are important for the growth of new particles to cloud condensation nuclei sizes.
Entrainment to a real time fractal visual stimulus modulates fractal gait dynamics.
Rhea, Christopher K; Kiefer, Adam W; D'Andrea, Susan E; Warren, William H; Aaron, Roy K
2014-08-01
Fractal patterns characterize healthy biological systems and are considered to reflect the ability of the system to adapt to varying environmental conditions. Previous research has shown that fractal patterns in gait are altered following natural aging or disease, and this has potential negative consequences for gait adaptability that can lead to increased risk of injury. However, the flexibility of a healthy neurological system to exhibit different fractal patterns in gait has yet to be explored, and this is a necessary step toward understanding human locomotor control. Fifteen participants walked for 15min on a treadmill, either in the absence of a visual stimulus or while they attempted to couple the timing of their gait with a visual metronome that exhibited a persistent fractal pattern (contained long-range correlations) or a random pattern (contained no long-range correlations). The stride-to-stride intervals of the participants were recorded via analog foot pressure switches and submitted to detrended fluctuation analysis (DFA) to determine if the fractal patterns during the visual metronome conditions differed from the baseline (no metronome) condition. DFA α in the baseline condition was 0.77±0.09. The fractal patterns in the stride-to-stride intervals were significantly altered when walking to the fractal metronome (DFA α=0.87±0.06) and to the random metronome (DFA α=0.61±0.10) (both p<.05 when compared to the baseline condition), indicating that a global change in gait dynamics was observed. A variety of strategies were identified at the local level with a cross-correlation analysis, indicating that local behavior did not account for the consistent global changes. Collectively, the results show that a gait dynamics can be shifted in a prescribed manner using a visual stimulus and the shift appears to be a global phenomenon.
Transmission and reflection properties of terahertz fractal metamaterials
DEFF Research Database (Denmark)
Malureanu, Radu; Lavrinenko, Andrei; Cooke, David
2010-01-01
We use THz time-domain spectroscopy to investigate transmission and reflection properties of metallic fractal metamaterial structures. We observe loss of free-space energy at certain resonance frequencies, indicating excitation of surface modes of the metamaterial.......We use THz time-domain spectroscopy to investigate transmission and reflection properties of metallic fractal metamaterial structures. We observe loss of free-space energy at certain resonance frequencies, indicating excitation of surface modes of the metamaterial....
Spatial behavior analysis at the global level using fractal geometry.
Sambrook, Roger C
2008-01-01
Previous work has suggested that an estimate of fractal dimension can provide a useful metric for quantifying settlement patterns. This study uses fractal methods to investigate settlement patterns at a global scale showing that the scaling behavior of the pattern of the world's largest cities corresponds to that typically observed for coastlines and rivers. This serves to validate the use of fractal dimension as a scale-independent measure of settlement patterns which can be correlated with other physical features. Such a measure may be a useful validation criterion for models of human settlement and spatial behavior.
Signatures of fractal clustering of aerosols advected under gravity.
Vilela, Rafael D; Tél, Tamás; de Moura, Alessandro P S; Grebogi, Celso
2007-06-01
Aerosols under chaotic advection often approach a strange attractor. They move chaotically on this fractal set but, in the presence of gravity, they have a net vertical motion downwards. In practical situations, observational data may be available only at a given level, for example, at the ground level. We uncover two fractal signatures of chaotic advection of aerosols under the action of gravity. Each one enables the computation of the fractal dimension D(0) of the strange attractor governing the advection dynamics from data obtained solely at a given level. We illustrate our theoretical findings with a numerical experiment and discuss their possible relevance to meteorology.
Plant development in space: Observations on root formation and growth
Levine, H. G.; Kann, R. P.; Krikorian, Abraham D.
1990-01-01
Root growth in space is discussed and observations on root production from plants flown as part of the Chromex project that were defined as to their origin, stage of development and physiological status, are presented. Roots were generated from fully differentiated, aseptically maintained individuals of Haplopappus gracilis (Compositae) under spaceflight conditions. Results are compared for tissue culture generated plantlets and comparably sized seedling clone individuals, both of which had their roots trimmed on Earth before they were loaded into NASA's plant growth unit and subjected to a 5 day shuttle flight (STS-29). Asepsis was maintained throughout the experiment. Overall root production was 40 to 50 percent greater under spaceflight conditions than during ground control tests. However, root formation slowed down towards the end of the flight. This decrease in new roots did not occur in the ground controls that sought to simulate flight except for microgravity.
Kinetic properties of fractal media
Chumak, Oleg V
2016-01-01
Kinetic processes in fractal stellar media are analyzed in terms of the approach developed in our earlier paper (Chumak, Rastorguev, 2016) involving a generalization of the nearest neighbor and random force distributions to fractal media. Diffusion is investigated in the approximation of scale-dependent conditional density based on an analysis of the solutions of the corresponding Langevin equations. It is shown that kinetic parameters (time scales, coefficients of dynamic friction, diffusion, etc.) for fractal stellar media can differ significantly both qualitatively and quantitatively from the corresponding parameters for a quasi-uniform random media with limited fluctuations. The most important difference is that in the fractal case kinetic parameters depend on spatial scale length and fractal dimension of the medium studied. A generalized kinetic equation for stellar media (fundamental equation of stellar dynamics) is derived in the Fokker-Planck approximation with the allowance for the fractal properties...
Fractals in geology and geophysics
Turcotte, Donald L.
1989-01-01
The definition of a fractal distribution is that the number of objects N with a characteristic size greater than r scales with the relation N of about r exp -D. The frequency-size distributions for islands, earthquakes, fragments, ore deposits, and oil fields often satisfy this relation. This application illustrates a fundamental aspect of fractal distributions, scale invariance. The requirement of an object to define a scale in photograhs of many geological features is one indication of the wide applicability of scale invariance to geological problems; scale invariance can lead to fractal clustering. Geophysical spectra can also be related to fractals; these are self-affine fractals rather than self-similar fractals. Examples include the earth's topography and geoid.
Fractals in several electrode materials
Energy Technology Data Exchange (ETDEWEB)
Zhang, Chunyong, E-mail: zhangchy@njau.edu.cn [Department of Chemistry, College of Science, Nanjing Agricultural University, Nanjing 210095 (China); Suzhou Key Laboratory of Environment and Biosafety, Suzhou Academy of Southeast University, Dushuhu lake higher education town, Suzhou 215123 (China); Wu, Jingyu [Department of Chemistry, College of Science, Nanjing Agricultural University, Nanjing 210095 (China); Fu, Degang [Suzhou Key Laboratory of Environment and Biosafety, Suzhou Academy of Southeast University, Dushuhu lake higher education town, Suzhou 215123 (China); State Key Laboratory of Bioelectronics, Southeast University, Nanjing 210096 (China)
2014-09-15
Highlights: • Fractal geometry was employed to characterize three important electrode materials. • The surfaces of all studied electrodes were proved to be very rough. • The fractal dimensions of BDD and ACF were scale dependent. • MMO film was more uniform than BDD and ACF in terms of fractal structures. - Abstract: In the present paper, the fractal properties of boron-doped diamond (BDD), mixed metal oxide (MMO) and activated carbon fiber (ACF) electrode have been studied by SEM imaging at different scales. Three materials are self-similar with mean fractal dimension in the range of 2.6–2.8, confirming that they all exhibit very rough surfaces. Specifically, it is found that MMO film is more uniform in terms of fractal structure than BDD and ACF. As a result, the intriguing characteristics make these electrodes as ideal candidates for high-performance decontamination processes.
ABC of multi-fractal spacetimes and fractional sea turtles
Energy Technology Data Exchange (ETDEWEB)
Calcagni, Gianluca [Instituto de Estructura de la Materia, CSIC, Madrid (Spain)
2016-04-15
We clarify what it means to have a spacetime fractal geometry in quantum gravity and show that its properties differ from those of usual fractals. A weak and a strong definition of multi-scale and multi-fractal spacetimes are given together with a sketch of the landscape of multi-scale theories of gravitation. Then, in the context of the fractional theory with q-derivatives, we explore the consequences of living in a multi-fractal spacetime. To illustrate the behavior of a non-relativistic body, we take the entertaining example of a sea turtle. We show that, when only the time direction is fractal, sea turtles swim at a faster speed than in an ordinary world, while they swim at a slower speed if only the spatial directions are fractal. The latter type of geometry is the one most commonly found in quantum gravity. For time-like fractals, relativistic objects can exceed the speed of light, but strongly so only if their size is smaller than the range of particle-physics interactions. We also find new results about log-oscillating measures, the measure presentation and their role in physical observations and in future extensions to nowhere-differentiable stochastic spacetimes. (orig.)
ABC of multi-fractal spacetimes and fractional sea turtles
Calcagni, Gianluca
2016-04-01
We clarify what it means to have a spacetime fractal geometry in quantum gravity and show that its properties differ from those of usual fractals. A weak and a strong definition of multi-scale and multi-fractal spacetimes are given together with a sketch of the landscape of multi-scale theories of gravitation. Then, in the context of the fractional theory with q-derivatives, we explore the consequences of living in a multi-fractal spacetime. To illustrate the behavior of a non-relativistic body, we take the entertaining example of a sea turtle. We show that, when only the time direction is fractal, sea turtles swim at a faster speed than in an ordinary world, while they swim at a slower speed if only the spatial directions are fractal. The latter type of geometry is the one most commonly found in quantum gravity. For time-like fractals, relativistic objects can exceed the speed of light, but strongly so only if their size is smaller than the range of particle-physics interactions. We also find new results about log-oscillating measures, the measure presentation and their role in physical observations and in future extensions to nowhere-differentiable stochastic spacetimes.
Eliazar, Iddo; Klafter, Joseph
2008-06-01
We explore six classes of fractal probability laws defined on the positive half-line: Weibull, Frechét, Lévy, hyper Pareto, hyper beta, and hyper shot noise. Each of these classes admits a unique statistical power-law structure, and is uniquely associated with a certain operation of renormalization. All six classes turn out to be one-dimensional projections of underlying Poisson processes which, in turn, are the unique fixed points of Poissonian renormalizations. The first three classes correspond to linear Poissonian renormalizations and are intimately related to extreme value theory (Weibull, Frechét) and to the central limit theorem (Lévy). The other three classes correspond to nonlinear Poissonian renormalizations. Pareto's law--commonly perceived as the "universal fractal probability distribution"--is merely a special case of the hyper Pareto class.
Turbulent wakes of fractal objects.
Staicu, Adrian; Mazzi, Biagio; Vassilicos, J C; van de Water, Willem
2003-06-01
Turbulence of a windtunnel flow is stirred using objects that have a fractal structure. The strong turbulent wakes resulting from three such objects which have different fractal dimensions are probed using multiprobe hot-wire anemometry in various configurations. Statistical turbulent quantities are studied within inertial and dissipative range scales in an attempt to relate changes in their self-similar behavior to the scaling of the fractal objects.
Statistical mechanics and fractals
Dobrushin, Roland Lvovich
1993-01-01
This book is composed of two texts, by R.L. Dobrushin and S. Kusuoka, each representing the content of a course of lectures given by the authors. They are pitched at graduate student level and are thus very accessible introductions to their respective subjects for students and non specialists. CONTENTS: R.L. Dobrushin: On the Way to the Mathematical Foundations of Statistical Mechanics.- S. Kusuoka: Diffusion Processes on Nested Fractals.
Martin, Demetri
2015-03-01
Demetri Maritn prepared this palindromic poem as his project for Michael Frame's fractal geometry class at Yale. Notice the first, fourth, and seventh words in the second and next-to-second lines are palindromes, the first two and last two lines are palindromes, the middle line, "Be still if I fill its ebb" minus its last letter is a palindrome, and the entire poem is a palindrome...
Fractal multifiber microchannel plates
Cook, Lee M.; Feller, W. B.; Kenter, Almus T.; Chappell, Jon H.
1992-01-01
The construction and performance of microchannel plates (MCPs) made using fractal tiling mehtods are reviewed. MCPs with 40 mm active areas having near-perfect channel ordering were produced. These plates demonstrated electrical performance characteristics equivalent to conventionally constructed MCPs. These apparently are the first MCPs which have a sufficiently high degree of order to permit single channel addressability. Potential applications for these devices and the prospects for further development are discussed.
Darwinian Evolution and Fractals
Carr, Paul H.
2009-05-01
Did nature's beauty emerge by chance or was it intelligently designed? Richard Dawkins asserts that evolution is blind aimless chance. Michael Behe believes, on the contrary, that the first cell was intelligently designed. The scientific evidence is that nature's creativity arises from the interplay between chance AND design (laws). Darwin's ``Origin of the Species,'' published 150 years ago in 1859, characterized evolution as the interplay between variations (symbolized by dice) and the natural selection law (design). This is evident in recent discoveries in DNA, Madelbrot's Fractal Geometry of Nature, and the success of the genetic design algorithm. Algorithms for generating fractals have the same interplay between randomness and law as evolution. Fractal statistics, which are not completely random, characterize such phenomena such as fluctuations in the stock market, the Nile River, rainfall, and tree rings. As chaos theorist Joseph Ford put it: God plays dice, but the dice are loaded. Thus Darwin, in discovering the evolutionary interplay between variations and natural selection, was throwing God's dice!
Spina, Maria E; Saraceno, Marcos
2010-01-01
It has been conjectured that for a class of piecewise linear maps the closure of the set of images of the discontinuity has the structure of a fat fractal, that is, a fractal with positive measure. An example of such maps is the sawtooth map in the elliptic regime. In this work we analyze this problem quantum mechanically in the semiclassical regime. We find that the fraction of states localized on the unstable set satisfies a modified fractal Weyl law, where the exponent is given by the exterior dimension of the fat fractal.
Fractals a very short introduction
Falconer, Kenneth
2013-01-01
Many are familiar with the beauty and ubiquity of fractal forms within nature. Unlike the study of smooth forms such as spheres, fractal geometry describes more familiar shapes and patterns, such as the complex contours of coastlines, the outlines of clouds, and the branching of trees. In this Very Short Introduction, Kenneth Falconer looks at the roots of the 'fractal revolution' that occurred in mathematics in the 20th century, presents the 'new geometry' of fractals, explains the basic concepts, and explores the wide range of applications in science, and in aspects of economics.This is esse
Fractal characterization and wettability of ion treated silicon surfaces
Yadav, R. P.; Kumar, Tanuj; Baranwal, V.; Vandana, Kumar, Manvendra; Priya, P. K.; Pandey, S. N.; Mittal, A. K.
2017-02-01
Fractal characterization of surface morphology can be useful as a tool for tailoring the wetting properties of solid surfaces. In this work, rippled surfaces of Si (100) are grown using 200 keV Ar+ ion beam irradiation at different ion doses. Relationship between fractal and wetting properties of these surfaces are explored. The height-height correlation function extracted from atomic force microscopic images, demonstrates an increase in roughness exponent with an increase in ion doses. A steep variation in contact angle values is found for low fractal dimensions. Roughness exponent and fractal dimensions are found correlated with the static water contact angle measurement. It is observed that after a crossover of the roughness exponent, the surface morphology has a rippled structure. Larger values of interface width indicate the larger ripples on the surface. The contact angle of water drops on such surfaces is observed to be lowest. Autocorrelation function is used for the measurement of ripple wavelength.
Besselink, R.; Stawski, T. M.; Van Driessche, A. E. S.; Benning, L. G.
2016-12-01
Densely packed surface fractal aggregates form in systems with high local volume fractions of particles with very short diffusion lengths, which effectively means that particles have little space to move. However, there are no prior mathematical models, which would describe scattering from such surface fractal aggregates and which would allow the subdivision between inter- and intraparticle interferences of such aggregates. Here, we show that by including a form factor function of the primary particles building the aggregate, a finite size of the surface fractal interfacial sub-surfaces can be derived from a structure factor term. This formalism allows us to define both a finite specific surface area for fractal aggregates and the fraction of particle interfacial sub-surfaces at the perimeter of an aggregate. The derived surface fractal model is validated by comparing it with an ab initio approach that involves the generation of a "brick-in-a-wall" von Koch type contour fractals. Moreover, we show that this approach explains observed scattering intensities from in situ experiments that followed gypsum (CaSO4 ṡ 2H2O) precipitation from highly supersaturated solutions. Our model of densely packed "brick-in-a-wall" surface fractal aggregates may well be the key precursor step in the formation of several types of mosaic- and meso-crystals.
Thermodynamic fractals and formalism. Fractales y formalismo termodinamico
Energy Technology Data Exchange (ETDEWEB)
Chacon, R.; Morales, J.J.
1994-01-01
We give a brief introduction to the so called ''thermodynamical description of fractals'' restricting our attention to Cantor sets generated by chaotic motion of a dynamical system. In particular, an entropy function and a free energy are introduced for multi fractals. (Author) 14 refs.
Experimental control of scaling behavior: what is not fractal?
Likens, Aaron D; Fine, Justin M; Amazeen, Eric L; Amazeen, Polemnia G
2015-10-01
The list of psychological processes thought to exhibit fractal behavior is growing. Although some might argue that the seeming ubiquity of fractal patterns illustrates their significance, unchecked growth of that list jeopardizes their relevance. It is important to identify when a single behavior is and is not fractal in order to make meaningful conclusions about the processes underlying those patterns. The hypothesis tested in the present experiment is that fractal patterns reflect the enactment of control. Participants performed two steering tasks: steering on a straight track and steering on a circular track. Although each task could be accomplished by holding the steering wheel at a constant angle, steering around a curve may require more constant control, at least from a psychological standpoint. Results showed that evidence for fractal behavior was strongest for the circular track; straight tracks showed evidence of two scaling regions. We argue from those results that, going forward, the goal of the fractal literature should be to bring scaling behavior under experimental control.
Energy Technology Data Exchange (ETDEWEB)
Guzman-Castaneda, J.I.; Garcia-Borquez, A. [Instituto Politecnico Nacional, ESFM, 07738 Mexico D.F. (Mexico); Arizabalo-Salas, R.D. [Instituto Mexicano del Petroleo, Direccion de Investigacion y Posgrado, 07730 Mexico D.F. (Mexico)
2012-06-15
Optical and scanning electron microscopy (OM and SEM) are techniques that are normally used for 2D-analysis of surface features. By fractal dimension analysis of the gray-scale OM and SEM images, it is possible to get quantitative topographical measurements. In this work, three different surface topographies (polished, eroded, and oxidized) were analyzed on FeCrAl alloy by OM and SEM. Clear surface topographical changes can be qualitatively observed. In order to quantify such changes, two steps were followed: (i) a gray-scale digitalization from each image was used to reproduce topographical features on the analyzed surface, and (ii) from this information, the fractal dimension (D) was determined using fractal3e software. The fractal dimension determined in this form follows coherently the topographical changes produced on the FeCrAl alloy after polishing, erosion, and oxidizing processes. The variations of fractal dimension values against the temperature of the oxidizing processes reflect well the oxide growth changes. Moreover, a minimum D-value is registered at 750 C, which corresponds to the {delta}-{theta} alumina phase transition temperature as determined by differential thermal analysis (DTA) on the same alloy. (Copyright copyright 2012 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
Fractal analysis: methodologies for biomedical researchers.
Ristanović, Dusan; Milosević, Nebojsa T
2012-01-01
Fractal analysis has become a popular method in all branches of scientific investigations including biology and medicine. Although there is a growing interest in the application of fractal analysis in biological sciences, questions about the methodology of fractal analysis have partly restricted its wider and comprehensible application. It is a notable fact that fractal analysis is derived from fractal geometry, but there are some unresolved issues that need to be addressed. In this respect, we discuss several related underlying principles for fractal analysis and establish the meaningful relationship between fractal analysis and fractal geometry. Since some concepts in fractal analysis are determined descriptively and/or qualitatively, this paper provides their exact mathematical definitions or explanations. Another aim of this study is to show that nowadays fractal analysis is an independent mathematical and experimental method based on Mandelbrot's fractal geometry, Euclidean traditiontal geometry and Richardson's coastline method.
Fractal Particles: Titan's Thermal Structure and IR Opacity
McKay, C. P.; Rannou, P.; Guez, L.; Young, E. F.; DeVincenzi, Donald (Technical Monitor)
1998-01-01
Titan's haze particles are the principle opacity at solar wavelengths. Most past work in modeling these particles has assumed spherical particles. However, observational evidence strongly favors fractal shapes for the haze particles. We consider the implications of fractal particles for the thermal structure and near infrared opacity of Titan's atmosphere. We find that assuming fractal particles with the optical properties based on laboratory tholin material and with a production rate that allows for a match to the geometric albedo results in warmer troposphere and surface temperatures compared to spherical particles. In the near infrared (1-3 microns) the predicted opacity of the fractal particles is up to a factor of two less than for spherical particles. This has implications for the ability of Cassini to image Titan's surface at 1 micron.
Fractal analysis of circulating platelets in type 2 diabetic patients.
Bianciardi, G; Tanganelli, I
2015-01-01
This paper investigates the use of computerized fractal analysis for objective characterization by means of transmission electron microscopy of the complexity of circulating platelets collected from healthy individuals and from type 2 diabetic patients, a pathologic condition in which platelet hyperreactivity has been described. Platelet boundaries were extracted by means of automatically image analysis. Local fractal dimension by box counting (measure of geometric complexity) was automatically calculated. The results showed that the platelet boundary observed by electron microscopy is fractal and that the shape of the circulating platelets is significantly more complex in the diabetic patients in comparison to healthy subjects (p fractal analysis of platelet shape by transmission electron microscopy can provide accurate, quantitative, data to study platelet activation in diabetes mellitus.
Simoson, Andrew J.
2009-01-01
This article presents a fun activity of generating a double-minded fractal image for a linear algebra class once the idea of rotation and scaling matrices are introduced. In particular the fractal flip-flops between two words, depending on the level at which the image is viewed. (Contains 5 figures.)
Brothers, Harlan J.
2015-03-01
Benoit Mandelbrot always had a strong feeling that music could be viewed from a fractal perspective. However, without our eyes to guide us, how do we gain this perspective? Here we discuss precisely what it means to say that a piece of music is fractal.
Vialidad, conectividad y fractales
Pineda Paz, Eduardo; Guerrero Torrenegra, Alejandro
2014-01-01
La morfología urbana es posible analizarla mediante ecuaciones no lineales que aparentemente reflejan el comportamiento del hombre. La teoría del caos, la incertidumbre y los fractales, aportan nuevas posibilidades al planificador urbano. El estudio es descriptivo y analítico, siguiendo pautas fenomenológicas, combinando teoría y práctica urbanística, con matemática sencilla. La parroquia Olegario Villalobos de Maracaibo es el caso de estudio. La investigación abordó la dimensión ...
Numerical Modeling of Dendrite Growth in Al Alloys
Institute of Scientific and Technical Information of China (English)
许庆彦; 柳百成
2004-01-01
Dendritic grains are the most often observed microstructure in metals and alloys. In the past decade, more and more attention has been paid to the modeling and simulation of dendritic microstructures. This paper describes a modified diffusion-limited aggregation model to simulate the complex shape of the dendrite grains during metal solidification. The fractal model was used to simulate equiaxed dendrite growth. The fractal dimensions of simulated Al alloy structures range from 1.63-1.88 which compares well with the experimentally-measured fractal dimension of 1.85; therefore, the model accurately predicts not only the dendritic structure morphology, but also the fractal dimension of the dendrite structure formed during solidification.
Fractals properties of EEG during event-related desynchronization of motor imagery.
Nguyen, Ngoc Quang; Truong, Quang Dang Khoa; Kondo, Toshiyuki
2015-01-01
Chaos and fractal dimension are emerging modalities for the research of electroencephalogram (EEG) signal processing. The capability of measuring non-linear characteristics of the fractal dimension enables new methodologies to identify distinct brain activities. Recent studies on the topic focus on utilizing various types of fractals as features in order to design better brain state classification system. However, we have little insight about the EEG signals projected in fractal dimension. In this paper, we investigate the relationship between the non-linear characteristics of ongoing EEG signals and event-related desynchronization (ERD) during motor imagery. We observed a considerable synchronization between ERD and fractal dimension. This finding suggests further usage of chaos and fractal theory in investigating brain activities.
Fractal Dimension of Fracture Surface in Rock Material after High Temperature
Directory of Open Access Journals (Sweden)
Z. Z. Zhang
2015-01-01
Full Text Available Experiments on granite specimens after different high temperature under uniaxial compression were conducted and the fracture surfaces were observed by scanning electron microscope (SEM. The fractal dimensions of the fracture surfaces with increasing temperature were calculated, respectively. The fractal dimension of fracture surface is between 1.44 and 1.63. Its value approximately goes up exponentially with the increase of temperature. There is a quadratic polynomial relationship between the rockburst tendency and fractal dimension of fracture surface; namely, a fractal dimension threshold can be obtained. Below the threshold value, a positive correlativity shows between rockburst tendency and fractal dimension; when the fractal dimension is greater than the threshold value, it shows an inverse correlativity.
Patricio, Pedro; Duarte, Jorge; Januario, Cristina
2015-01-01
We investigate the rheology of a fractal network, in the framework of the linear theory of viscoelasticity. We identify each segment of the network with a simple Kelvin-Voigt element, with a well defined equilibrium length. The final structure retains the elastic characteristics of a solid or a gel. By considering a very simple regular self-similar structure of segments in series and in parallel, in 1, 2 or 3 dimensions, we are able to express the viscoelasticity of the network as an effective generalised Kelvin-Voigt model with a power law spectrum of retardation times, $\\phi\\sim\\tau^{\\alpha-1}$. We relate the parameter $\\alpha$ with the fractal dimension of the gel. In some regimes ($0<\\alpha<1$), we recover the weak power law behaviours of the elastic and viscous moduli with the angular frequencies, $G'\\sim G''\\sim w^\\alpha$, that occur in a variety of soft materials, including living cells. In other regimes, we find different and interesting power laws for $G'$ and $G''$.
Wicks, Keith R
1991-01-01
Addressed to all readers with an interest in fractals, hyperspaces, fixed-point theory, tilings and nonstandard analysis, this book presents its subject in an original and accessible way complete with many figures. The first part of the book develops certain hyperspace theory concerning the Hausdorff metric and the Vietoris topology, as a foundation for what follows on self-similarity and fractality. A major feature is that nonstandard analysis is used to obtain new proofs of some known results much more slickly than before. The theory of J.E. Hutchinson's invariant sets (sets composed of smaller images of themselves) is developed, with a study of when such a set is tiled by its images and a classification of many invariant sets as either regular or residual. The last and most original part of the book introduces the notion of a "view" as part of a framework for studying the structure of sets within a given space. This leads to new, elegant concepts (defined purely topologically) of self-similarity and fracta...
Prediction of osteoporosis using fractal analysis on periapical radiographs
Energy Technology Data Exchange (ETDEWEB)
Park, Gum Mi; Jung, Yun Hoa; Nah, Kyung Soo [Pusan National University College of Medicine, Busan (Korea, Republic of)
2005-03-15
To purpose of this study was to investigate whether the fractal dimension and radiographic image brightness of periapical radiograph were useful in predicting osteoporosis. Ninety-two postmenopausal women were classified as normal, osteopenia and osteoporosis group according to the bone mineral density of lumbar vertebrae and periapical radiographs of both mandibular molar areas were taken. The ROIs of 358 areas were selected at periapical and interdental areas and fractal dimension and radiographic image brightness were measured. The fractal dimension in normal group was significantly higher than that in osteoporosis group at periapical ROI (p<0.05). The radiographic image brightness in normal group was higher than that in osteopenia and osteoporosis group. There was significant difference not only between normal and osteopenia group (p<0.05) but also within osteopenia and osteoporosis group (p<0.01) at periapical ROI. Significant difference was observed not only between normal and osteopenia group but also between normal and osteoporosis group at interdental ROI (p<0.01). Positive linear relationship was weakly shown at Pearson correlation analysis between fractal dimension and radiographic image brightness. BMD significantly correlated with fractal dimension at periapical ROI (p<0.01), and BMD and radiographic image brightness significantly correlated at both periapical and interdental ROIs (p<0.01). This study suggests that the fractal dimension and radiographic image brightness of periapical ROI may predict BMD.
Enhanced Graphene Photodetector with Fractal Metasurface
DEFF Research Database (Denmark)
Fang, Jieran; Wang, Di; DeVault, Clayton T
2017-01-01
Graphene has been demonstrated to be a promising photodetection material because of its ultrabroadband optical absorption, compatibility with CMOS technology, and dynamic tunability in optical and electrical properties. However, being a single atomic layer thick, graphene has intrinsically small...... optical absorption, which hinders its incorporation with modern photodetecting systems. In this work, we propose a gold snowflake-like fractal metasurface design to realize broadband and polarization-insensitive plasmonic enhancement in graphene photodetector. We experimentally obtain an enhanced...... photovoltage from the fractal metasurface that is an order of magnitude greater than that generated at a plain gold-graphene edge and such an enhancement in the photovoltage sustains over the entire visible spectrum. We also observed a relatively constant photoresponse with respect to polarization angles...
Static friction between rigid fractal surfaces.
Alonso-Marroquin, Fernando; Huang, Pengyu; Hanaor, Dorian A H; Flores-Johnson, E A; Proust, Gwénaëlle; Gan, Yixiang; Shen, Luming
2015-09-01
Using spheropolygon-based simulations and contact slope analysis, we investigate the effects of surface topography and atomic scale friction on the macroscopically observed friction between rigid blocks with fractal surface structures. From our mathematical derivation, the angle of macroscopic friction is the result of the sum of the angle of atomic friction and the slope angle between the contact surfaces. The latter is obtained from the determination of all possible contact slopes between the two surface profiles through an alternative signature function. Our theory is validated through numerical simulations of spheropolygons with fractal Koch surfaces and is applied to the description of frictional properties of Weierstrass-Mandelbrot surfaces. The agreement between simulations and theory suggests that for interpreting macroscopic frictional behavior, the descriptors of surface morphology should be defined from the signature function rather than from the slopes of the contacting surfaces.
Fractal Structures Driven by Self-Gravity Molecular clouds and the Universe
Combes, F
1998-01-01
In the interstellar medium, as well as in the Universe, large density fluctuations are observed, that obey power-law density distributions and correlation functions. These structures are hierarchical, chaotic, turbulent, but are also self-organizing. The apparent disorder is not random noise, but can be described by a fractal, with a deterministic fractal dimension. We discuss the theories advanced to describe these fractal structures, and in particular a new theory of the self-gravity thermodynamics, that could explain their existence, and predict their fractal dimension. The media obeying scaling laws can be considered critical, as in second order phase transitions for instance.
Fractal and Multifractal Time Series
Kantelhardt, Jan W
2008-01-01
Data series generated by complex systems exhibit fluctuations on many time scales and/or broad distributions of the values. In both equilibrium and non-equilibrium situations, the natural fluctuations are often found to follow a scaling relation over several orders of magnitude, allowing for a characterisation of the data and the generating complex system by fractal (or multifractal) scaling exponents. In addition, fractal and multifractal approaches can be used for modelling time series and deriving predictions regarding extreme events. This review article describes and exemplifies several methods originating from Statistical Physics and Applied Mathematics, which have been used for fractal and multifractal time series analysis.
Exterior dimension of fat fractals
Grebogi, C.; Mcdonald, S. W.; Ott, E.; Yorke, J. A.
1985-01-01
Geometric scaling properties of fat fractal sets (fractals with finite volume) are discussed and characterized via the introduction of a new dimension-like quantity which is called the exterior dimension. In addition, it is shown that the exterior dimension is related to the 'uncertainty exponent' previously used in studies of fractal basin boundaries, and it is shown how this connection can be exploited to determine the exterior dimension. Three illustrative applications are described, two in nonlinear dynamics and one dealing with blood flow in the body. Possible relevance to porous materials and ballistic driven aggregation is also noted.
Thermal collapse of snowflake fractals
Gallo, T.; Jurjiu, A.; Biscarini, F.; Volta, A.; Zerbetto, F.
2012-08-01
Snowflakes are thermodynamically unstable structures that would ultimately become ice balls. To investigate their dynamics, we mapped atomistic molecular dynamics simulations of small ice crystals - built as filled von Koch fractals - onto a discrete-time random walk model. Then the walkers explored the thermal evolution of high fractal generations. The in silico experiments showed that the evolution is not entirely random. The flakes step down one fractal generation before forfeiting their architecture. The effect may be used to trace the thermal history of snow.
Moghilevsky, Débora Estela
2011-01-01
A lo largo de los últimos años del siglo veinte se ha desarrollado la teoría de la complejidad. Este modelo relaciona las ciencias duras tales como la matemática, la teoría del caos, la física cuántica y la geometría fractal con las llamadas seudo ciencias. Dentro de este contexto podemos definir la Psicología Fractal como la ciencia que estudia los aspectos psíquicos como dinámicamente fractales.
On the fractal properties microaccelerations
Sedelnikov, A V
2012-01-01
In this paper the fractal property of the internal environment of space laboratory microaccelerations that occur. Changing the size of the space lab leads to the fact that the dependence of microaccelerations from time to time has the property similar to the self-affinity of fractal functions. With the help of microaccelerations, based on the model of the real part of the fractal Weierstrass-Mandelbrot function is proposed to form the inertial-mass characteristics of laboratory space with a given level of microaccelerations.
Fractal structure and fractal dimension determination at nanometer scale
Institute of Scientific and Technical Information of China (English)
张跃; 李启楷; 褚武扬; 王琛; 白春礼
1999-01-01
Three-dimensional fractures of different fractal dimensions have been constructed with successive random addition algorithm, the applicability of various dimension determination methods at nanometer scale has been studied. As to the metallic fractures, owing to the limited number of slit islands in a slit plane or limited datum number at nanometer scale, it is difficult to use the area-perimeter method or power spectrum method to determine the fractal dimension. Simulation indicates that box-counting method can be used to determine the fractal dimension at nanometer scale. The dimensions of fractures of valve steel 5Cr21Mn9Ni4N have been determined with STM. Results confirmed that fractal dimension varies with direction at nanometer scale. Our study revealed that, as to theoretical profiles, the dependence of fractal dimension with direction is simply owing to the limited data set number, i.e. the effect of boundaries. However, the dependence of fractal dimension with direction at nanometer scale in rea
DEFF Research Database (Denmark)
Malureanu, Radu; Jepsen, Peter Uhd; Xiao, S.
2010-01-01
The concept of metamaterials (MTMs) is acknowledged for providing new horizons for controlling electromagnetic radiations thus their use in frequency ranges otherwise difficult to manage (e.g. THz radiation) broadens our possibility to better understand our world as well as opens the path for new...... frequency range as well as a clear differentiation between one polarisation and another. Based on theoretical predictions we fabricated and measured a fractal based THz metamaterial that shows more than 60% field transmission at around 1THz for TE polarized light while the TM waves have almost 80% field...... wavelength of THz radiation, the resolution requirements for fabrication of metamaterials are within the optical lithography range. However, the high aspect ratio of such structures as well as the substrate thickness pose challenges in the fabrication process. The measurements were made using terahertz time...
Eliazar, Iddo; Klafter, Joseph
2008-09-01
The Central Limit Theorem (CLT) and Extreme Value Theory (EVT) study, respectively, the stochastic limit-laws of sums and maxima of sequences of independent and identically distributed (i.i.d.) random variables via an affine scaling scheme. In this research we study the stochastic limit-laws of populations of i.i.d. random variables via nonlinear scaling schemes. The stochastic population-limits obtained are fractal Poisson processes which are statistically self-similar with respect to the scaling scheme applied, and which are characterized by two elemental structures: (i) a universal power-law structure common to all limits, and independent of the scaling scheme applied; (ii) a specific structure contingent on the scaling scheme applied. The sum-projection and the maximum-projection of the population-limits obtained are generalizations of the classic CLT and EVT results - extending them from affine to general nonlinear scaling schemes.
Diophantine approximations on fractals
Einsiedler, Manfred; Shapira, Uri
2009-01-01
We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the natural measure) contains all finite patterns (hence is well approximable). Similarly, we show that for a variety of fractals in [0,1]^2, possessing some symmetry, almost any point is not Dirichlet improvable (hence is well approximable) and has property C (after Cassels). We then settle by similar methods a conjecture of M. Boshernitzan saying that there are no irrational numbers x in the unit interval such that the continued fraction expansions of {nx mod1 : n is a natural number} are uniformly eventually bounded.
Ofstatistical and Fractal Properties of Semiconductor Surface Roughness
Directory of Open Access Journals (Sweden)
Stanislav Jurecka
2008-01-01
Full Text Available Surface morphology evolution is of primary significance for the thin-film growth and modification of surface andinterface states. Surface and interface states substantially influence the electrical and optical properties of the semiconductorstructure. Statistical and fractal properties of semiconductor rough surfaces were determined by analysis of the AFM images.In this paper statistical characteristics of the AFM height function distribution, fractal dimension, lacunarity and granulometric density values are used for the surface morphology of the SiC samples description. The results can be used for solution ofthe microstructural and optical properties of given semiconductor structure.
Quantification of fungal growth: models, experiments, and observations
Lamour, A.
2002-01-01
This thesis is concerned with the growth of microscopic mycelial fungi (Section I), and that of macroscopic fungi, which form specialised hyphal structures such as rhizomorphs (Section II). A growth model is developed in Section I in relation to soil organic matter decompos
Fractal geometry and computer graphics
Sakas, Georgios; Peitgen, Heinz-Otto; Englert, Gabriele
1992-01-01
Fractal geometry has become popular in the last 15 years, its applications can be found in technology, science, or even arts. Fractal methods and formalism are seen today as a general, abstract, but nevertheless practical instrument for the description of nature in a wide sense. But it was Computer Graphics which made possible the increasing popularity of fractals several years ago, and long after their mathematical formulation. The two disciplines are tightly linked. The book contains the scientificcontributions presented in an international workshop in the "Computer Graphics Center" in Darmstadt, Germany. The target of the workshop was to present the wide spectrum of interrelationships and interactions between Fractal Geometry and Computer Graphics. The topics vary from fundamentals and new theoretical results to various applications and systems development. All contributions are original, unpublished papers.The presentations have been discussed in two working groups; the discussion results, together with a...
Configuration entropy of fractal landscapes
National Research Council Canada - National Science Library
Rodríguez‐Iturbe, Ignacio; D'Odorico, Paolo; Rinaldo, Andrea
1998-01-01
.... The spatial arrangement of two‐dimensional images is found to be an effective way to characterize fractal landscapes and the configurational entropy of these arrangements imposes demanding conditions for models attempting to represent these fields.
Anomalous diffusion in fractal globules.
Tamm, M V; Nazarov, L I; Gavrilov, A A; Chertovich, A V
2015-05-01
The fractal globule state is a popular model for describing chromatin packing in eukaryotic nuclei. Here we provide a scaling theory and dissipative particle dynamics computer simulation for the thermal motion of monomers in the fractal globule state. Simulations starting from different entanglement-free initial states show good convergence which provides evidence supporting the existence of a unique metastable fractal globule state. We show monomer motion in this state to be subdiffusive described by ⟨X(2)(t)⟩∼t(αF) with αF close to 0.4. This result is in good agreement with existing experimental data on the chromatin dynamics, which makes an additional argument in support of the fractal globule model of chromatin packing.
Fractals endlessy repeated geometrical figures
Lauwerier, Hans
1991-01-01
Provides a basic mathematical introduction to fractal geometry, the mathematics that lie behind chaos theory. This book attempts to communicate the relatively simple understanding of the subject to an audience with a basic mathematical education.
Turbulence on a Fractal Fourier set
Lanotte, Alessandra Sabina; Biferale, Luca; Malapaka, Shiva Kumar; Toschi, Federico
2015-01-01
The dynamical effects of mode reduction in Fourier space for three dimensional turbulent flows is studied. We present fully resolved numerical simulations of the Navier-Stokes equations with Fourier modes constrained to live on a fractal set of dimension D. The robustness of the energy cascade and vortex stretching mechanisms are tested at changing D, from the standard three dimensional case to a strongly decimated case for D = 2.5, where only about $3\\%$ of the Fourier modes interact. While the direct energy cascade persist, deviations from the Kolmogorov scaling are observed in the kinetic energy spectra. A model in terms of a correction with a linear dependency on the co-dimension of the fractal set, $E(k)\\sim k^{- 5/3 + 3 -D }$, explains the results. At small scales, the intermittent behaviour due to the vorticity production is strongly modified by the fractal decimation, leading to an almost Gaussian statistics already at D ~ 2.98. These effects are connected to a genuine modification in the triad-to-tri...
The contact mechanics of fractal surfaces.
Buzio, Renato; Boragno, Corrado; Biscarini, Fabio; Buatier de Mongeot, Francesco; Valbusa, Ugo
2003-04-01
The role of surface roughness in contact mechanics is relevant to processes ranging from adhesion to friction, wear and lubrication. It also promises to have a deep impact on applied science, including coatings technology and design of microelectromechanical systems. Despite the considerable results achieved by indentation experiments, particularly in the measurement of bulk hardness on nanometre scales, the contact behaviour of realistic surfaces, showing random multiscale roughness, remains largely unknown. Here we report experimental results concerning the mechanical response of self-affine thin films indented by a micrometric flat probe. The specimens, made of cluster-assembled carbon or of sexithienyl, an organic molecular material, were chosen as prototype systems for the broad class of self-affine fractal interfaces, today including surfaces grown under non-equilibrium conditions, fractures, manufactured metal surfaces and solidified liquid fronts. We observe that a regime exists in which roughness drives the contact mechanics: in this range surface stiffness varies by a few orders of magnitude on small but significant changes of fractal parameters. As a consequence, we demonstrate that soft solid interfaces can be appreciably strengthened by reducing both fractal dimension and surface roughness. This indicates a general route for tailoring the mechanical properties of solid bodies.
Dynamic contact interactions of fractal surfaces
Jana, Tamonash; Mitra, Anirban; Sahoo, Prasanta
2017-01-01
Roughness parameters and material properties have significant influence on the static and dynamic properties of a rough surface. In the present paper, fractal surface is generated using the modified two-variable Weierstrass-Mandelbrot function in MATLAB and the same is imported to ANSYS to construct the finite element model of the rough surface. The force-deflection relationship between the deformable rough fractal surface and a contacting rigid flat is studied by finite element analysis. For the dynamic analysis, the contacting system is represented by a single degree of freedom spring mass-damper-system. The static force-normal displacement relationship obtained from FE analysis is used to determine the dynamic characteristics of the rough surface for free, as well as for forced damped vibration using numerical methods. The influence of fractal surface parameters and the material properties on the dynamics of the rough surface is also analyzed. The system exhibits softening property for linear elastic surface and the softening nature increases with rougher topography. The softening nature of the system increases with increase in tangent modulus value. Above a certain value of yield strength the nature of the frequency response curve is observed to change its nature from softening to hardening.
Steady laminar flow of fractal fluids
Balankin, Alexander S.; Mena, Baltasar; Susarrey, Orlando; Samayoa, Didier
2017-02-01
We study laminar flow of a fractal fluid in a cylindrical tube. A flow of the fractal fluid is mapped into a homogeneous flow in a fractional dimensional space with metric induced by the fractal topology. The equations of motion for an incompressible Stokes flow of the Newtonian fractal fluid are derived. It is found that the radial distribution for the velocity in a steady Poiseuille flow of a fractal fluid is governed by the fractal metric of the flow, whereas the pressure distribution along the flow direction depends on the fractal topology of flow, as well as on the fractal metric. The radial distribution of the fractal fluid velocity in a steady Couette flow between two concentric cylinders is also derived.
Visible parts of fractal percolation
Arhosalo, I; Järvenpää, M; Rams, M; Shmerkin, P
2009-01-01
We study dimensional properties of visible parts of fractal percolation in the plane. Provided that the dimension of the fractal percolation is at least 1, we show that, conditioned on non-extinction, almost surely all visible parts from lines are 1-dimensional. Furthermore, almost all of them have positive and finite Hausdorff measure. We also verify analogous results for visible parts from points. These results are motivated by an open problem on the dimensions of visible parts.
The fractal nature materials microstructure influence on electrochemical energy sources
Directory of Open Access Journals (Sweden)
Mitić V.V.
2015-01-01
Full Text Available With increasing of the world energy crisis, research for new, renewable and alternative energy sources are in growth. The focus is on research areas, sometimes of minor importance and applications, where the different synthesis methods and microstructure properties optimization, performed significant improvement of output materials’ and components’ electro-physical properties, which is important for higher energy efficiency and in the electricity production (batteries and battery systems, fuel cells and hydrogen energy contribution. Also, the storage tanks capacity improvement, for the energy produced on such way, which is one of the most important development issues in the energy sphere, represents a very promising research and application area. Having in mind, the results achieved in the electrochemical energy sources field, especially electrolyte development, these energy sources, materials fractal nature optimization analysis contribution, have been investigated. Based on materials fractal structure research field, particularly electronic materials, we have performed microstructure influence parameters research in electrochemistry area. We have investigated the Ho2O3 concentration influence (from 0.01wt% to 1wt% and sintering temperature (from 1320°C to 1380°C, as consolidation parameters, and thus, also open the electrochemical function fractalization door and in the basic thermodynamic parameters the fractal correction introduced. The fractal dimension dependence on additive concentration is also investigated. [Projekat Ministarstva nauke Republike Srbije, br. 172057: Directed synthesis, structure and properties of multifunctional materials
Fractal Analyses of Steady Infiltration and Terrain on an Undulating Agricultural Field
Fractal scaling behaviors have been observed in systems where interacting factors cause nested spatial structures. Surface water infiltration affects spatial patterns of soil water, nutrients, and plant development and crop yield. Here, we explored simple fractal scaling of quasi-steady infiltrati...
Cantorian Fractal Patterns, Quantum-Like Chaos and Prime Numbers in Atmospheric Flows
Selvam, A M; Fadnavis, Suvarna
1998-01-01
Atmospheric flows exhibit cantorian fractal space-time fluctuations signifying long-range spatiotemporal correlations. A recently developed cell dynamical system model shows that such non-local connections are intrinsic to quantum-like chaos governing flow dynamics. The dynamical evolution of fractal structures can be quantified in terms of ordered energy flow described by mathematical functions which occur in the field of number theory. The quantum-like chaos in atmospheric flows can be quantified in terms of the following mathematical functions / concepts: (1) The fractal structure of the flow pattern is resolved into an overall logarithmic spiral trajectory with the quasiperiodic Penrose tiling pattern for the internal structure and is equivalent to a hierarchy of vortices. The incorporation of Fibonacci mathematical series, representative of ramified bifurcations, indicates ordered growth of fractal patterns. (2) The steady state emergence of progressively larger fractal structures incorporates unique pri...
Iannaccone, Stephen; Zhou, Yue; Walterhouse, David; Taborn, Greg; Landini, Gabriel; Iannaccone, Philip
2012-01-01
The production of organ parenchyma in a rapid and reproducible manner is critical to normal development. In chimeras produced by the combination of genetically distinguishable tissues, mosaic patterns of cells derived from the combined genotypes can be visualized. These patterns comprise patches of contiguously similar genotypes and are different in different organs but similar in a given organ from individual to individual. Thus, the processes that produce the patterns are regulated and conserved. We have previously established that mosaic patches in multiple tissues are fractal, consistent with an iterative, recursive growth model with simple stereotypical division rules. Fractal dimensions of various tissues are consistent with algorithmic models in which changing a single variable (e.g. daughter cell placement after division) switches the mosaic pattern from islands to stripes of cells. Here we show that the spiral pattern previously observed in mouse cornea can also be visualized in rat chimeras. While it is generally held that the pattern is induced by stem cell division dynamics, there is an unexplained discrepancy in the speed of cellular migration and the emergence of the pattern. We demonstrate in chimeric rat corneas both island and striped patterns exist depending on the age of the animal. The patches that comprise the pattern are fractal, and the fractal dimension changes with the age of the animal and indicates the constraint in patch complexity as the spiral pattern emerges. The spiral patterns are consistent with a loxodrome. Such data are likely to be relevant to growth and cell division in organ systems and will help in understanding how organ parenchyma are generated and maintained from multipotent stem cell populations located in specific topographical locations within the organ. Ultimately, understanding algorithmic growth is likely to be essential in achieving organ regeneration in vivo or in vitro from stem cell populations.
Directory of Open Access Journals (Sweden)
Stephen Iannaccone
Full Text Available The production of organ parenchyma in a rapid and reproducible manner is critical to normal development. In chimeras produced by the combination of genetically distinguishable tissues, mosaic patterns of cells derived from the combined genotypes can be visualized. These patterns comprise patches of contiguously similar genotypes and are different in different organs but similar in a given organ from individual to individual. Thus, the processes that produce the patterns are regulated and conserved. We have previously established that mosaic patches in multiple tissues are fractal, consistent with an iterative, recursive growth model with simple stereotypical division rules. Fractal dimensions of various tissues are consistent with algorithmic models in which changing a single variable (e.g. daughter cell placement after division switches the mosaic pattern from islands to stripes of cells. Here we show that the spiral pattern previously observed in mouse cornea can also be visualized in rat chimeras. While it is generally held that the pattern is induced by stem cell division dynamics, there is an unexplained discrepancy in the speed of cellular migration and the emergence of the pattern. We demonstrate in chimeric rat corneas both island and striped patterns exist depending on the age of the animal. The patches that comprise the pattern are fractal, and the fractal dimension changes with the age of the animal and indicates the constraint in patch complexity as the spiral pattern emerges. The spiral patterns are consistent with a loxodrome. Such data are likely to be relevant to growth and cell division in organ systems and will help in understanding how organ parenchyma are generated and maintained from multipotent stem cell populations located in specific topographical locations within the organ. Ultimately, understanding algorithmic growth is likely to be essential in achieving organ regeneration in vivo or in vitro from stem cell populations.
Herrmann, Richard
2014-01-01
The dark silicon problem, which limits the power-growth of future computer generations, is interpreted as a heat energy transport problem when increasing the energy emitting surface area within a given volume. A comparison of two 3D-configuration models, namely a standard slicing and a fractal surface generation within the Menger sponge geometry is presented. It is shown, that for iteration orders $n>3$ the fractal model shows increasingly better thermal behavior. As a consequence cooling pro...
On the fractal morphology of combustion-generated soot aggregates
Energy Technology Data Exchange (ETDEWEB)
Koylu, U.O. [Yale Univ., New Haven, CT (United States)
1995-12-31
The fractal properties of soot aggregates were investigated using ex-situ and in-situ experimental methods as well as computer simulations. Ex-situ experiments involved thermophoretic sampling and analysis by transmission electron microscopy (TEM), while in-situ measurements employed angular static light scattering and data inversion based on Rayleigh-Debye-Gans (RDG) approximation. Computer simulations used a sequential algorithm which mimics mass fractal-like structures. So from a variety of hydrocarbon-fueled laminar and turbulent nonpremixed flame environments were considered in the present study. The TEM analysis of projected soot images sampled from fuel-rich conditions of buoyant and weakly-buoyant laminar flames indicated that the fractal dimension of soot was relatively independent of position in flames, fuel type and flame condition. These measurements yielded an average fractal dimension of 1.8, although other structure parameters such as the primary particle diameters and number of primary particles in aggregates had wide range of values. Fractal prefactor (lacunarity) was also measured for soot sampled from the fuel-lean conditions of turbulent flames, considering the actual morphology by tilting the samples during TEM analysis. These measurements yielded a fractal dimension of 1.65 and a lacunarity of 8.5, with experimental uncertainties (95% confidence) of 0.08 and 0.5, respectively. Relationships between the actual and projected structure properties of soot were also developed by combining TEM observations with numerical simulations. Practical approximate formulae were suggested to find radius of gyration of an aggregate from its maximum dimension, and number of primary particles in an aggregate from projected area. Finally, the fractal dimension and lacunarity of soot were obtained using light scattering for the same conditions of the above TEM measurements.
Self-organized network of fractal-shaped components coupled through statistical interaction.
Ugajin, R
2001-09-01
A dissipative dynamics is introduced to generate self-organized networks of interacting objects, which we call coupled-fractal networks. The growth model is constructed based on a growth hypothesis in which the growth rate of each object is a product of the probability of receiving source materials from faraway and the probability of receiving adhesives from other grown objects, where each object grows to be a random fractal if isolated, but connects with others if glued. The network is governed by the statistical interaction between fractal-shaped components, which can only be identified in a statistical manner over ensembles. This interaction is investigated using the degree of correlation between fractal-shaped components, enabling us to determine whether it is attractive or repulsive.
Zipf's law, 1/f noise, and fractal hierarchy
Chen, Yanguang
2011-01-01
Fractals, 1/f noise, Zipf's law, and the occurrence of large catastrophic events are typical ubiquitous general empirical observations across the individual sciences which cannot be understood within the set of references developed within the specific scientific domains. All these observations are associated with scaling laws and have caused a broad research interest in the scientific circle. However, the inherent relationships between these scaling phenomena are still pending questions remaining to be researched. In this paper, theoretical derivation and mathematical experiments are employed to reveal the analogy between fractal patterns, 1/f noise, and the Zipf distribution. First, the multifractal process follows the generalized Zipf's law empirically. Second, a 1/f spectrum is identical in mathematical form to Zipf's law. Third, both 1/f spectra and Zipf's law can be converted into a self-similar hierarchy. Fourth, fractals, 1/f spectra, Zipf's law, and the occurrence of large catastrophic events can be d...
Observation of dendritic growth under the influence of forced convection
Roshchupkina, O.; Shevchenko, N.; Eckert, S.
2015-06-01
The directional solidification of Ga-25wt%In alloys within a Hele-Shaw cell was visualized by X-ray radioscopy. The investigations are focused on the impact of melt convection on the dendritic growth. Natural convection occurs during a bottom up solidification because lighter solute is rejected during crystallization. Forced convection was produced by a specific electromagnetic pump. The direction of forced melt flow is almost horizontal at the solidification front. Melt flow induces various effects on grain morphology primarily caused by convective transport of solute, such as a facilitation of the growth of primary trunks or lateral branches, dendrite remelting, fragmentation or freckle formation depending on the dendrite orientation, the flow direction and intensity. Forced flow eliminates solutal plumes and damps local fluctuations of solute. A preferential growth of the secondary arms occurs at the upstream side of the dendrites, whereas high solute concentration at the downstream side inhibits the formation of secondary branches.
Tremberger, George, Jr.; Flamholz, A.; Cheung, E.; Sullivan, R.; Subramaniam, R.; Schneider, P.; Brathwaite, G.; Boteju, J.; Marchese, P.; Lieberman, D.; Cheung, T.; Holden, Todd
2007-09-01
The absorption effect of the back surface boundary of a diffuse layer was studied via laser generated reflection speckle pattern. The spatial speckle intensity provided by a laser beam was measured. The speckle data were analyzed in terms of fractal dimension (computed by NIH ImageJ software via the box counting fractal method) and weak localization theory based on Mie scattering. Bar code imaging was modeled as binary absorption contrast and scanning resolution in millimeter range was achieved for diffusive layers up to thirty transport mean free path thick. Samples included alumina, porous glass and chicken tissue. Computer simulation was used to study the effect of speckle spatial distribution and observed fractal dimension differences were ascribed to variance controlled speckle sizes. Fractal dimension suppressions were observed in samples that had thickness dimensions around ten transport mean free path. Computer simulation suggested a maximum fractal dimension of about 2 and that subtracting information could lower fractal dimension. The fractal dimension was shown to be sensitive to sample thickness up to about fifteen transport mean free paths, and embedded objects which modified 20% or more of the effective thickness was shown to be detectable. The box counting fractal method was supplemented with the Higuchi data series fractal method and application to architectural distortion mammograms was demonstrated. The use of fractals in diffusive analysis would provide a simple language for a dialog between optics experts and mammography radiologists, facilitating the applications of laser diagnostics in tissues.
Analysis of fractals with combined partition
Dedovich, T. G.; Tokarev, M. V.
2016-03-01
The space—time properties in the general theory of relativity, as well as the discreteness and non-Archimedean property of space in the quantum theory of gravitation, are discussed. It is emphasized that the properties of bodies in non-Archimedean spaces coincide with the properties of the field of P-adic numbers and fractals. It is suggested that parton showers, used for describing interactions between particles and nuclei at high energies, have a fractal structure. A mechanism of fractal formation with combined partition is considered. The modified SePaC method is offered for the analysis of such fractals. The BC, PaC, and SePaC methods for determining a fractal dimension and other fractal characteristics (numbers of levels and values of a base of forming a fractal) are considered. It is found that the SePaC method has advantages for the analysis of fractals with combined partition.
Institute of Scientific and Technical Information of China (English)
吴则焰; 刘金福; 洪伟; 郑世群; 何中声
2012-01-01
The average increment of diameter at breast height ( DBH) and tree height of Glyptostrobus pensilis from different provenances were studied by combining geostatistical methods with fractal theory. The fractal dimensions of DBH and tree height growth of G. pensilis were calculated in order to reveal the rule of spatial distribution variation. Results showed that the fractal dimensions of DBH and tree height were 1.635 and 1. 824, respectively. DBH can be used as an index for evaluating different provenances of G. pensilis to reflect the spatial variability.%以珍稀濒危植物水松(Glyptostrobus pensilis)不同种源树高和胸径平均生长量为研究对象,将分形理论与地统计学原理相结合,计算水松种源树高和胸径生长的分形维数,揭示其空间分布变异规律和分形特征.结果表明:水松种源胸径、树高生长特性的分维值分别为1.635和1.824,胸径的分维值小于树高的分维值.为反映水松种源的空间差异性,在评价水松种源时应选取胸径生长指标.
Fractals and finite scales; Fractales et echelles finies
Energy Technology Data Exchange (ETDEWEB)
Aubry, J.M
1997-08-01
Fractal description is used in various scientific domains and in particular in the modeling of particle aggregates and in the modeling of the Rayleigh-Taylor instabilities in turbulent two-phase flows. In particular, the interface geometry between fluids in a turbulent mixing is a crucial parameter for the modeling of mixtures in inertial confinement fusion devices. In this paper, a review of the various fractal dimensions is given first. Then, for a more rigorous use, a probabilistic description of the dimension of an ensemble which is known only up to a finite scale is proposed. This description is based on a probabilistic measurement of the overall fractals. (J.S.) 22 refs.
The fractal aggregation of asphaltenes.
Hoepfner, Michael P; Fávero, Cláudio Vilas Bôas; Haji-Akbari, Nasim; Fogler, H Scott
2013-07-16
This paper discusses time-resolved small-angle neutron scattering results that were used to investigate asphaltene structure and stability with and without a precipitant added in both crude oil and model oil. A novel approach was used to isolate the scattering from asphaltenes that are insoluble and in the process of aggregating from those that are soluble. It was found that both soluble and insoluble asphaltenes form fractal clusters in crude oil and the fractal dimension of the insoluble asphaltene clusters is higher than that of the soluble clusters. Adding heptane also increases the size of soluble asphaltene clusters without modifying the fractal dimension. Understanding the process of insoluble asphaltenes forming fractals with higher fractal dimensions will potentially reveal the microscopic asphaltene destabilization mechanism (i.e., how a precipitant modifies asphaltene-asphaltene interactions). It was concluded that because of the polydisperse nature of asphaltenes, no well-defined asphaltene phase stability envelope exists and small amounts of asphaltenes precipitated even at dilute precipitant concentrations. Asphaltenes that are stable in a crude oil-precipitant mixture are dispersed on the nanometer length scale. An asphaltene precipitation mechanism is proposed that is consistent with the experimental findings. Additionally, it was found that the heptane-insoluble asphaltene fraction is the dominant source of small-angle scattering in crude oil and the previously unobtainable asphaltene solubility at low heptane concentrations was measured.
Fractal properties of the lattice Lotka-Volterra model.
Tsekouras, G A; Provata, A
2002-01-01
The lattice Lotka-Volterra (LLV) model is studied using mean-field analysis and Monte Carlo simulations. While the mean-field phase portrait consists of a center surrounded by an infinity of closed trajectories, when the process is restricted to a two-dimensional (2D) square lattice, local inhomogeneities/fluctuations appear. Spontaneous local clustering is observed on lattice and homogeneous initial distributions turn into clustered structures. Reactions take place only at the interfaces between different species and the borders adopt locally fractal structure. Intercluster surface reactions are responsible for the formation of local fluctuations of the species concentrations. The box-counting fractal dimension of the LLV dynamics on a 2D support is found to depend on the reaction constants while the upper bound of fractality determines the size of the local oscillators. Lacunarity analysis is used to determine the degree of clustering of homologous species. Besides the spontaneous clustering that takes place on a regular 2D lattice, the effects of fractal supports on the dynamics of the LLV are studied. For supports of dimensionality D(s)<2 the lattice can, for certain domains of the reaction constants, adopt a poisoned state where only one of the species survives. By appropriately selecting the fractal dimension of the substrate, it is possible to direct the system into a poisoned or oscillatory steady state at will.
Fractal differential equations and fractal-time dynamical systems
Indian Academy of Sciences (India)
Abhay Parvate; A D Gangal
2005-03-01
Differential equations and maps are the most frequently studied examples of dynamical systems and may be considered as continuous and discrete time-evolution processes respectively. The processes in which time evolution takes place on Cantor- like fractal subsets of the real line may be termed as fractal-time dynamical systems. Formulation of these systems requires an appropriate framework. A new calculus called -calculus, is a natural calculus on subsets ⊂ R of dimension , 0 < ≤ 1. It involves integral and derivative of order , called -integral and -derivative respectively. The -integral is suitable for integrating functions with fractal support of dimension , while the -derivative enables us to differentiate functions like the Cantor staircase. The functions like the Cantor staircase function occur naturally as solutions of -differential equations. Hence the latter can be used to model fractal-time processes or sublinear dynamical systems. We discuss construction and solutions of some fractal differential equations of the form $$D^{}_{F,t} x = h(x, t),$$ where ℎ is a vector field and $D^{}_{F,t}$ is a fractal differential operator of order in time . We also consider some equations of the form $$D^{}_{F,t} W(x, t) = L[W(x, t)],$$ where is an ordinary differential operator in the real variable , and $(t, x) F × \\mathbf{R}^{n}$ where is a Cantor-like set of dimension . Further, we discuss a method of finding solutions to -differential equations: They can be mapped to ordinary differential equations, and the solutions of the latter can be transformed back to get those of the former. This is illustrated with a couple of examples.
Enhanced Graphene Photodetector with Fractal Metasurface
DEFF Research Database (Denmark)
Fan, Jieran; Wang, Di; DeVault, Clayton
2016-01-01
We designed and fabricated a broadband, polarization-independent photodetector by integrating graphene with a fractal Cayley tree metasurface. Our measurements show an almost uniform, tenfold enhancement in photocurrent generation due to the fractal metasurface structure.......We designed and fabricated a broadband, polarization-independent photodetector by integrating graphene with a fractal Cayley tree metasurface. Our measurements show an almost uniform, tenfold enhancement in photocurrent generation due to the fractal metasurface structure....
Fractal Structures For Fixed Mems Capacitors
Elshurafa, Amro M.
2014-08-28
An embodiment of a fractal fixed capacitor comprises a capacitor body in a microelectromechanical system (MEMS) structure. The capacitor body has a first plate with a fractal shape separated by a horizontal distance from a second plate with a fractal shape. The first plate and the second plate are within the same plane. Such a fractal fixed capacitor further comprises a substrate above which the capacitor body is positioned.
Fractal harmonic law and waterproof/dustproof
Directory of Open Access Journals (Sweden)
Kong Hai-Yan
2014-01-01
Full Text Available The fractal harmonic law admits that the friction between the pure water and the moving surface is the minimum when fractal dimensions of water in Angstrom scale are equal to fractal dimensions of the moving surface in micro scale. In the paper, the fractal harmonic law is applied to demonstrate the mechanism of waterproof/ dustproof. The waterproof phenomenon of goose feathers and lotus leaves is illustrated to verify our results and experimental results agree well with our theoretical analysis.
The observed peripheral growth of disc galaxies from z ~ 1
Gadotti, Dimitri A.; Sachdeva, Sonali; Saha, Kanak; Singh, Harinder P.
2017-03-01
Using images from the Hubble Space Telescope and Sloan Digital Sky Survey, we have computed both parametric and non-parametric measures, and examined the evolution in size, concentration, stellar mass, effective stellar mass density and asymmetry for a sample of 600 disc galaxies from z ~ 1 till z ~ 0. We find that disc galaxies have gained more than 50 per cent of their present stellar mass over the last 8 Gyr. Also, the increase in disc size is found to be peripheral. While the average total (Petrosian) radius almost doubles from z ~ 1 to z ~ 0, the average effective (half-light) radius undergoes a marginal increase in comparison. This indicates that galaxies grow more substantially in their outskirts, and is consistent with the inside-out growth picture. The substantial increase in mass and size indicates that accretion of external material has been a dominant mode of galaxy growth, where the circumgalactic environment plays a significant role.
Special phase transformation and crystal growth pathways observed in nanoparticles†
Directory of Open Access Journals (Sweden)
Finnegan Michael P
2003-11-01
Full Text Available Phase transformation and crystal growth in nanoparticles may happen via mechanisms distinct from those in bulk materials. We combine experimental studies of as-synthesized and hydrothermally coarsened titania (TiO2 and zinc sulfide (ZnS with thermodynamic analysis, kinetic modeling and molecular dynamics (MD simulations. The samples were characterized by transmission electron microscopy, X-ray diffraction, synchrotron X-ray absorption and scattering, and UV-vis spectroscopy. At low temperatures, phase transformation in titania nanoparticles occurs predominantly via interface nucleation at particle–particle contacts. Coarsening and crystal growth of titania nanoparticles can be described using the Smoluchowski equation. Oriented attachment-based crystal growth was common in both hydrothermal solutions and under dry conditions. MD simulations predict large structural perturbations within very fine particles, and are consistent with experimental results showing that ligand binding and change in aggregation state can cause phase transformation without particle coarsening. Such phenomena affect surface reactivity, thus may have important roles in geochemical cycling.
Fractal characterization of fracture surfaces in concrete
Saouma, V.E.; Barton, C.C.; Gamaleldin, N.A.
1990-01-01
Fractal geometry is used to characterize the roughness of cracked concrete surfaces through a specially built profilometer, and the fractal dimension is subsequently correlated to the fracture toughness and direction of crack propagation. Preliminary results indicate that the fracture surface is indeed fractal over two orders of magnitudes with a dimension of approximately 1.20. ?? 1990.
Fractal analysis of time varying data
Vo-Dinh, Tuan; Sadana, Ajit
2002-01-01
Characteristics of time varying data, such as an electrical signal, are analyzed by converting the data from a temporal domain into a spatial domain pattern. Fractal analysis is performed on the spatial domain pattern, thereby producing a fractal dimension D.sub.F. The fractal dimension indicates the regularity of the time varying data.
Fractal Structures For Mems Variable Capacitors
Elshurafa, Amro M.
2014-08-28
In accordance with the present disclosure, one embodiment of a fractal variable capacitor comprises a capacitor body in a microelectromechanical system (MEMS) structure, wherein the capacitor body has an upper first metal plate with a fractal shape separated by a vertical distance from a lower first metal plate with a complementary fractal shape; and a substrate above which the capacitor body is suspended.
Speaker Identification Based on Fractal Dimensions
Institute of Scientific and Technical Information of China (English)
侯丽敏; 王朔中
2003-01-01
This paper discusses application of fractal dimensions to speech processing. Generalized dimensions of arbitrary orders and associated fractal parameters are used in speaker identification. A characteristic vactor based on these parameters is formed, and a recognition criterion definded in order to identify individual speakers. Experimental results show the usefulness of fractal dimensions in characterizing speaker identity.
Steady laminar flow of fractal fluids
Energy Technology Data Exchange (ETDEWEB)
Balankin, Alexander S., E-mail: abalankin@ipn.mx [Grupo Mecánica Fractal, ESIME, Instituto Politécnico Nacional, México D.F., 07738 (Mexico); Mena, Baltasar [Laboratorio de Ingeniería y Procesos Costeros, Instituto de Ingeniería, Universidad Nacional Autónoma de México, Sisal, Yucatán, 97355 (Mexico); Susarrey, Orlando; Samayoa, Didier [Grupo Mecánica Fractal, ESIME, Instituto Politécnico Nacional, México D.F., 07738 (Mexico)
2017-02-12
We study laminar flow of a fractal fluid in a cylindrical tube. A flow of the fractal fluid is mapped into a homogeneous flow in a fractional dimensional space with metric induced by the fractal topology. The equations of motion for an incompressible Stokes flow of the Newtonian fractal fluid are derived. It is found that the radial distribution for the velocity in a steady Poiseuille flow of a fractal fluid is governed by the fractal metric of the flow, whereas the pressure distribution along the flow direction depends on the fractal topology of flow, as well as on the fractal metric. The radial distribution of the fractal fluid velocity in a steady Couette flow between two concentric cylinders is also derived. - Highlights: • Equations of Stokes flow of Newtonian fractal fluid are derived. • Pressure distribution in the Newtonian fractal fluid is derived. • Velocity distribution in Poiseuille flow of fractal fluid is found. • Velocity distribution in a steady Couette flow is established.
Magnetic reconnection rate in space plasmas: a fractal approach.
Materassi, Massimo; Consolini, Giuseppe
2007-10-26
Magnetic reconnection is generally discussed via a fluid description. Here, we evaluate the reconnection rate assuming a fractal topology of the reconnection region. The central idea is that the fluid hypothesis may be violated at the scales where reconnection takes place. The reconnection rate, expressed as the Alfvén Mach number of the plasma moving toward the diffusion region, is shown to depend on the fractal dimension and on the sizes of the reconnection or diffusion region. This mechanism is more efficient than prediction of the Sweet-Parker model and even Petschek's model for finite magnetic Reynolds number. A good agreement also with rates given by Hall MHD models is found. A discussion of the fractal assumption on the diffusion region in terms of current microstructures is proposed. The comparison with in-situ satellite observations suggests the reconnection region to be a filamentary domain.
Multiband Fractal Antenna : Application to Wi-Max
Directory of Open Access Journals (Sweden)
Miss. Awalekar Madhavi J.
2016-01-01
Full Text Available In this paper configuration of multiband fractal antenna for Wi-Max application is presented and analyzed. Three circle triangle iterations are configured and observed. The feeding method used for antenna is CPW (co-planar waveguidefeed. To provide the Wireless technologies like Wi-MAX and other advanced applications through the antennas by using Fractal technology to the microstrip antennas. By using the fractal technology on the microstrip antennas we can get several advantages like wide band operation, less power consumption, less return loss and many more. The antenna characteristics were simulated using full-wave electromagnetic simulator (IE3D. According to simulations, the proposed antenna can provide proper response at 2.4 GHz for third iteration. Return loss values according to simulated results obtained at 2.4GHz Simulated and practically are -15.8db,-11.5db respectively and VSWR values are practical 1.5 and 1.95 respectively.
Sandwich type plasmonic platform for MEF using silver fractals
DEFF Research Database (Denmark)
Raut, Sangram L.; Rich, Ryan; Shtoyko, Tanya
2015-01-01
In this report, we describe a plasmonic platform with silver fractals for metal enhanced fluorescence (MEF) measurements. When a dye containing surface was brought into contact with silver fractals, a significantly enhanced fluorescence signal from the dye was observed. Fluorescence enhancement...... was studied with the N-methyl-azadioxatriangulenium chloride salt (Me-ADOTA·Cl) in PVA films made from 0.2% PVA (w/v) solution spin-coated on a clean glass coverslip. The Plasmonic Platforms (PP) were assembled by pressing together silver fractals on one glass slide and a separate glass coverslip spin......-coated with a uniform Me-ADOTA·Cl in PVA film. In addition, we also tested ADOTA labeled human serum albumin (HSA) deposited on a glass slide for potential PP bioassay applications. Using the new PP, we could achieve more than a 20-fold fluorescence enhancement (bright spots) accompanied by a decrease...
Random curds as mathematical models of fractal rhythm in architecture
Directory of Open Access Journals (Sweden)
Ćirović Ivana
2014-01-01
Full Text Available The author Carl Bovill has suggested and described a method for generating rhythm in architecture with the help of random curds, as they are the mathematical models of unpredictable and uneven groupings which he recognizes in natural shapes and in natural processes. He specified the rhythm generated in this way as the fractal rhythm. Random curds can be generated by a simple process of curdling, as suggested by B. Mandelbrot. This paper examines the way in which the choice of probability for every stage or level of the curdling process, and the number of stages in the procedure of curdling, affect the characteristics of the obtained fractal object as a potential mathematical model of rhythm in the design process. At the same time, this paper examines the characteristics of rhythm in architecture which determine whether the obtained fractal object will be accepted as an appropriate mathematical model of the observed rhythm.
Herrmann, Richard
2015-01-01
The dark silicon problem, which limits the power-growth of future computer generations, is interpreted as a heat energy transport problem when increasing the energy emitting surface area within a given volume. A comparison of two 3D-configuration models, namely a standard slicing and a fractal surface generation within the Menger sponge geometry is presented. It is shown, that for iteration orders $n>3$ the fractal model shows increasingly better thermal behavior. As a consequence cooling problems may be minimized by using a fractal architecture. Therefore the Menger sponge geometry is a good example for fractal architectures applicable not only in computer science, but also e.g. in chemistry when building chemical reactors, optimizing catalytic processes or in sensor construction technology building highly effective sensors for toxic gases or water analysis.
Fractal Structure of Debris Flow
Institute of Scientific and Technical Information of China (English)
LI Yong; LIU Jingjing; HU Kaiheng; CHEN Xiaoqing
2007-01-01
One of the most remarkable characteristics of debris flow is the competence for supporting boulders on the surface of flow, which strongly suggests that there should be some structure in the fluid body. This paper analyzed the grain compositions from various samples of debris flows and then revealed the fractal structure. Specifically, the fractality holds in three domains that can be respectively identified as the slurry, matrix, and the coarse content. Furthermore, the matrix fractal, which distinguishes debris flow from other kinds of flows, involves a hierarchical structure in the sense that it might contain ever increasing grains while the total range of grain size increases. It provides a possible mechanism for the boulder suspension.
Pulmonary vasculature in dogs assessed by three-dimensional fractal analysis and chemometrics
DEFF Research Database (Denmark)
Müller, Anna V; Marschner, Clara B; Kristensen, Annemarie T
2017-01-01
angiogram, applying fractal analyses of these vascular trees in dogs with and without diseases that are known to predispose to thromboembolism, and testing the hypothesis that diseased dogs would have a different fractal dimension than healthy dogs. A total of 34 dogs were sampled. Based on computed...... tomographic pulmonary angiograms findings, dogs were divided in three groups: diseased with pulmonary thromboembolism (n = 7), diseased but without pulmonary thromboembolism (n = 21), and healthy (n = 6). An observer who was aware of group status created three-dimensional pulmonary artery vascular trees...... for each dog using a semiautomated segmentation technique. Vascular three-dimensional reconstructions were then evaluated using fractal analysis. Fractal dimensions were analyzed, by group, using analysis of variance and principal component analysis. Fractal dimensions were significantly different among...
Energy Technology Data Exchange (ETDEWEB)
Das, Bipul; Bag, Swarup; Pal, Sukhomay [Indian Institute of Technology Guwahati, Assam (India)
2017-05-15
Providing solutions towards the improvisation of welding technologies is the recent trend in the Friction stir welding (FSW) process. We present a monitoring approach for ultimate tensile strength of the friction stir welded joints based on information extracted from process signals through implementing fractal theory. Higuchi and Katz algorithms were executed on current and tool rotational speed signals acquired during friction stir welding to estimate fractal dimensions. Estimated fractal dimensions when correlated with the ultimate tensile strength of the joints deliver an increasing trend with the increase in joint strength. It is observed that dynamicity of the system strengthens the weld joint, i.e., the greater the fractal dimension, the better will be the quality of the weld. Characterization of signals by fractal theory indicates that the single-valued indicator can be an alternative for effective monitoring of the friction stir welding process.
Moisture Content Determination Using Microstrip Fractal Resonator Sensor
Directory of Open Access Journals (Sweden)
Beulah Jackson
2014-04-01
Full Text Available This paper presents a novel method for determining the moisture content in spices by means of compact microstrip fractal resonator. The proposed resonator is designed at 1 GHz using FR4 substrate with dielectric constant of 4.6. A resonator sensor in the shape of fractal designed using the tool ADS and working at 0.9 GHz to 3 GHz band is demonstrated. The relative shift in frequency and attenuation was observed to vary with respect to the moisture content/permittivity of the test material independent of its density.
Multi-fractal analysis of highway traffic data
Institute of Scientific and Technical Information of China (English)
Shang Peng-Jian; Shen Jin-Sheng
2007-01-01
The purpose of the present study is to investigate the presence of multi-fractal behaviours in the traffic time series not only by statistical approaches but also by geometrical approaches. The pointwise H(o)lder exponent of a function is calculated by developing an algorithm for the numerical evaluation of H(o)lder exponent of time series. The traffic time series observed on the Beijing Yuquanying highway are analysed. The results from all these methods indicate that the traffic data exhibit the multi-fractal behaviour.
Energy Technology Data Exchange (ETDEWEB)
Wilson, T.H.; Dominic, J.; Halverson, J.
1996-12-31
The primary goal of this study is to evaluate the possibility that the fractal characteristics of reservoir fracture systems might be inferred from the fractal characteristics of the reservoir reflector. Results discussed in the summary below provide support for such a view. The matter will, however, remain unresolved until fracture data acquired from core or FMS logs can be compared to reflection seismic data from the core areas. A series of cross sections along the Middle Mountain syncline and Elkhorn Mountain anticline were evaluated. Near-surface deformation in the Middle Mountain and Elkhorn mountain area of the Valley and Ridge province is significant. In this area the fractal dimension of topography is linearly related to the fractal dimension of underlying structure. Comparison of the fractal variability of Valley and Ridge structures with those observed in seismic data from the Plateau indicate that the increased fractal dimension of reflection events implies greater relative abundance of higher order or smaller wavelength structures. Results from the seismic evaluation suggest that fractal analysis might provide a useful exploration tool in cases where one is interested in locating subtle detached structures or identifying fractured reservoirs. Results from the Valley and a Ridge area suggest that, in active tectonic areas, fractal analysis may provide a means to assess the relative frequency of earthquake activity over time periods that extend beyond the historical record.
The Fractal Patterns of Words in a Text: A Method for Automatic Keyword Extraction.
Najafi, Elham; Darooneh, Amir H
2015-01-01
A text can be considered as a one dimensional array of words. The locations of each word type in this array form a fractal pattern with certain fractal dimension. We observe that important words responsible for conveying the meaning of a text have dimensions considerably different from one, while the fractal dimensions of unimportant words are close to one. We introduce an index quantifying the importance of the words in a given text using their fractal dimensions and then ranking them according to their importance. This index measures the difference between the fractal pattern of a word in the original text relative to a shuffled version. Because the shuffled text is meaningless (i.e., words have no importance), the difference between the original and shuffled text can be used to ascertain degree of fractality. The degree of fractality may be used for automatic keyword detection. Words with the degree of fractality higher than a threshold value are assumed to be the retrieved keywords of the text. We measure the efficiency of our method for keywords extraction, making a comparison between our proposed method and two other well-known methods of automatic keyword extraction.
Hu, Kun; Meijer, Johanna H; Shea, Steven A; vanderLeest, Henk Tjebbe; Pittman-Polletta, Benjamin; Houben, Thijs; van Oosterhout, Floor; Deboer, Tom; Scheer, Frank A J L
2012-01-01
The mammalian central circadian pacemaker (the suprachiasmatic nucleus, SCN) contains thousands of neurons that are coupled through a complex network of interactions. In addition to the established role of the SCN in generating rhythms of ~24 hours in many physiological functions, the SCN was recently shown to be necessary for normal self-similar/fractal organization of motor activity and heart rate over a wide range of time scales--from minutes to 24 hours. To test whether the neural network within the SCN is sufficient to generate such fractal patterns, we studied multi-unit neural activity of in vivo and in vitro SCNs in rodents. In vivo SCN-neural activity exhibited fractal patterns that are virtually identical in mice and rats and are similar to those in motor activity at time scales from minutes up to 10 hours. In addition, these patterns remained unchanged when the main afferent signal to the SCN, namely light, was removed. However, the fractal patterns of SCN-neural activity are not autonomous within the SCN as these patterns completely broke down in the isolated in vitro SCN despite persistence of circadian rhythmicity. Thus, SCN-neural activity is fractal in the intact organism and these fractal patterns require network interactions between the SCN and extra-SCN nodes. Such a fractal control network could underlie the fractal regulation observed in many physiological functions that involve the SCN, including motor control and heart rate regulation.
Fractal Structure of Molecular Clouds
Datta, Srabani
2001-01-01
Compelling evidence exists to show that the structure of molecular clouds is fractal in nature. In this paper, the author reiterates this view and, in addition, asserts that not only is cloud geometry fractal, but that they also have a common characteristic - they are similar in shape to the Horsehead nebula in Orion. This shape can be described by the Julia function f(x)= z^2 + c,where both z and c are complex quantities and c = -0.745429 + 0.113008i. The dynamical processes responsible for ...
Time evolution of quantum fractals
Wojcik; Bialynicki-Birula; Zyczkowski
2000-12-11
We propose a general construction of wave functions of arbitrary prescribed fractal dimension, for a wide class of quantum problems, including the infinite potential well, harmonic oscillator, linear potential, and free particle. The box-counting dimension of the probability density P(t)(x) = |Psi(x,t)|(2) is shown not to change during the time evolution. We prove a universal relation D(t) = 1+Dx/2 linking the dimensions of space cross sections Dx and time cross sections D(t) of the fractal quantum carpets.
Time Evolution of Quantum Fractals
Wójcik, D; Zyczkowski, K; Wojcik, Daniel; Bialynicki-Birula, Iwo; Zyczkowski, Karol
2000-01-01
We propose a general construction of wave functions of arbitrary prescribed fractal dimension, for a wide class of quantum problems, including the infinite potential well, harmonic oscillator, linear potential and free particle. The box-counting dimension of the probability density $P_t(x)=|\\Psi(x,t)|^2$ is shown not to change during the time evolution. We prove a universal relation $D_t=1+D_x/2$ linking the dimensions of space cross-sections $D_x$ and time cross-sections $D_t$ of the fractal quantum carpets.
Synergetics and fractals in tribology
Janahmadov, Ahad Kh
2016-01-01
This book examines the theoretical and practical aspects of tribological process using synergy, fractal and multifractal methods, and the fractal and multifractal models of self-similar tribosystems developed on their basis. It provides a comprehensive analysis of their effectiveness, and also considers the method of flicker noise spectroscopy with detailed parameterization of surface roughness friction. All models, problems and solutions are taken and tested on the set of real-life examples of oil-gas industry. The book is intended for researchers, graduate students and engineers specialising in the field of tribology, and also for senior students of technical colleges.
Fat fractal percolation and k-fractal percolation
Broman, Erik; Camia, Federico; Joosten, Matthijs; Meester, Ronald
2011-01-01
We consider two variations on the Mandelbrot fractal percolation model. In the k-fractal percolation model, the d-dimensional unit cube is divided in N^d equal subcubes, k of which are retained while the others are discarded. The procedure is then iterated inside the retained cubes at all smaller scales. We show that the (properly rescaled) percolation critical value of this model converges to the critical value of site percolation in L^d as N tends to infinity. This is analogous to the result of Falconer and Grimmett that the critical value for Mandelbrot fractal percolation converges to the critical value of site percolation in L^d. In the fat fractal percolation model, subcubes are retained with probability p_n at step n of the construction, where (p_n) is a non-decreasing sequence with \\prod p_n > 0. The Lebesgue measure of the limit set is positive a.s. given non-extinction. We show that with probability 1 either the set of "dust" points or the set of connected components larger than one point has positi...
Designing fractal nanostructured biointerfaces for biomedical applications.
Zhang, Pengchao; Wang, Shutao
2014-06-06
Fractal structures in nature offer a unique "fractal contact mode" that guarantees the efficient working of an organism with an optimized style. Fractal nanostructured biointerfaces have shown great potential for the ultrasensitive detection of disease-relevant biomarkers from small biomolecules on the nanoscale to cancer cells on the microscale. This review will present the advantages of fractal nanostructures, the basic concept of designing fractal nanostructured biointerfaces, and their biomedical applications for the ultrasensitive detection of various disease-relevant biomarkers, such microRNA, cancer antigen 125, and breast cancer cells, from unpurified cell lysates and the blood of patients.
Order-Fractal transition in abstract paintings
De la Calleja, E. M.; Cervantes, F.; De la Calleja, J.
2015-01-01
We report the degree of order of twenty-two Jackson Pollock's paintings using \\emph{Hausdorff-Besicovitch fractal dimension}. Through the maximum value of each multi-fractal spectrum, the artworks are classify by the year in which they were painted. It has been reported that Pollock's paintings are fractal and it increased on his latest works. However our results show that fractal dimension of the paintings are on a range of fractal dimension with values close to two. We identify this behavio...
Fractality in a Perturbed Einstein-de Sitter Cosmology
Abdalla, Elcio; Ribeiro, M B; Abdalla, Elcio; Mohayaee, Roya; Ribeiro, Marcelo B.
1999-01-01
This work presents the first step of an attempt to check the validity of a hypothesis known as the ``apparent fractal conjecture'' (Ribeiro 1999), according to which the observed fractal structure of large-scale distribution of galaxies arises when observational quantities are calculated along the past light cone. Inasmuch as general relativity states that astronomical observations are carried out in this spacetime hypersurface, observational quantities relevant for direct comparison with astronomical data must be calculated along it. Implementing this condition profoundly changes the behaviour of many observables in the standard cosmological models. In particular, the observed average density, becomes inhomogeneous, even in the spatially homogeneous spacetime of standard cosmology. Such a change in the observational quantities has already been fully analysed in previous works by Ribeiro (1992b, 1993, 1994, 1995) for a non-perturbed cosmological model. Here we derive observational relations in a perturbed Ein...
Institute of Scientific and Technical Information of China (English)
何池全; 赵魁义
2003-01-01
By using the principles and methods of fractal geometry theory,the relationship between aboveground biomass and plant length or sheath height of Carex lasiocarpa population was studied,The results showed that there was a good staic fractal relationship between them.and the resulted fractal dimension was an efficient description of the accumulation of aboveground biomass in each organ.The dynamic fractal relationship showed that during the whole growing season,the increase of aboveground biomass had a self-similarity,being a fractal growth process and the pattern of its increase was the fractal dimension D.Based on these results,a fractal growth model of Carex lasiocarpa population was established ,which regarded the bigger grass as the result of the amplification of seedling growth.
Fractal and its application to sedimentology
Institute of Scientific and Technical Information of China (English)
余继峰; 李增学; 韩美莲
2002-01-01
In the paper,the foundation,development,basic conception and general characteristics of fractal and the calculating method of the fractional dimension are expounded briefly, and the current situation and prospect of the fractal application in sedimentology are discussed stressly. Both sedimentary process and sedimentary record behave the fractal feature of the self-similarity structure. External form and internal texture of the sediments and the distribution of the grain-size of the sediments are of fractal feature very well, and the size of the fractional dimension is the quantitative index of the complexity of the background when they are formed. The further analysis on the multi-fractal feature of the sedimentary body is the base of the fractal simulation and forecast, and it is the key of the application of the fractal theory to sedimentology.
Emergence of fractal scaling in complex networks
Wei, Zong-Wen; Wang, Bing-Hong
2016-09-01
Some real-world networks are shown to be fractal or self-similar. It is widespread that such a phenomenon originates from the repulsion between hubs or disassortativity. Here we show that this common belief fails to capture the causality. Our key insight to address it is to pinpoint links critical to fractality. Those links with small edge betweenness centrality (BC) constitute a special architecture called fractal reference system, which gives birth to the fractal structure of those reported networks. In contrast, a small amount of links with high BC enable small-world effects, hiding the intrinsic fractality. With enough of such links removed, fractal scaling spontaneously arises from nonfractal networks. Our results provide a multiple-scale view on the structure and dynamics and place fractality as a generic organizing principle of complex networks on a firmer ground.
Morphology and functions of astrocytes cultured on water-repellent fractal tripalmitin surfaces.
Hu, Wei-wei; Wang, Zhe; Zhang, Shan-shan; Jiang, Lei; Zhang, Jing; Zhang, Xiangnan; Lei, Qun-fang; Park, Hyun-Joo; Fang, Wen-jun; Chen, Zhong
2014-08-01
In the brain, astrocytes play an essential role with their multiple functions and sophisticated structure, as surrounded by a fractal environment which has not been available in our traditional cell culture. Water-repellent fractal tripalmitin (PPP) surfaces can imitate the fractal environment in vivo, so the morphology and biochemical characterization of astrocytes on these surfaces are examined. Water-repellent fractal PPP surface can induce astrocytes to display sophisticated morphology with smaller size of cell area, longer and finer filopodium-like processes, and higher morphological complexity. The super water-repellent fractal PPP surface with water contact angle of 150°∼160° produces the maximal effects compared with other surfaces at lower water contact angles. The trends of characteristic protein expression, including that of nestin, vimentin, GFAP and glutamine synthetase, for astrocytes cultured on super water-repellent fractal PPP surfaces approximate more to in vivo pattern. The super water-repellent PPP surface also render astrocytes to perform more pronounced promotion of neurogenesis by increasing the release of nerve growth factor in a co-culture system. Altogether, our results suggest that the super water-repellent fractal PPP surface facilitates the astrocytes to mimic their in vivo performance, thus provides a closer-to-natural culture environment for experimental assessment of glial structure and functions.
Laboratory observation of magnetic field growth driven by shear flow
Energy Technology Data Exchange (ETDEWEB)
Intrator, T. P., E-mail: intrator@lanl.gov; Feng, Y.; Sears, J.; Weber, T. [Los Alamos National Laboratory, M.S. E526, Los Alamos, New Mexico 87545 (United States); Dorf, L. [Applied Materials, Inc., Santa Clara, CA 95054 (United States); Sun, X. [University of Science and Technology, Hefei (China)
2014-04-15
Two magnetic flux ropes that collide and bounce have been characterized in the laboratory. We find screw pinch profiles that include ion flow v{sub i}, magnetic field B, current density J, and plasma pressure. The electron flow v{sub e} can be inferred, allowing the evaluation of the Hall J×B term in a two fluid magnetohydrodynamic Ohm's Law. Flux ropes that are initially cylindrical are mutually attracted and compress each other, which distorts the cylindrical symmetry. Magnetic field is created via the ∇×v{sub e}×B induction term in Ohm's Law where in-plane (perpendicular) shear of parallel flow (along the flux rope) is the dominant feature, along with some dissipation and magnetic reconnection. We predict and measure the growth of a quadrupole out-of-plane magnetic field δB{sub z}. This is a simple and coherent example of a shear flow driven dynamo. There is some similarity with two dimensional reconnection scenarios, which induce a current sheet and thus out-of-plane flow in the third dimension, despite the customary picture that considers flows only in the reconnection plane. These data illustrate a general and deterministic mechanism for large scale sheared flows to acquire smaller scale magnetic features, disordered structure, and possibly turbulence.
Laboratory observation of magnetic field growth driven by shear flow
Intrator, T. P.; Dorf, L.; Sun, X.; Feng, Y.; Sears, J.; Weber, T.
2014-04-01
Two magnetic flux ropes that collide and bounce have been characterized in the laboratory. We find screw pinch profiles that include ion flow vi, magnetic field B, current density J, and plasma pressure. The electron flow ve can be inferred, allowing the evaluation of the Hall J ×B term in a two fluid magnetohydrodynamic Ohm's Law. Flux ropes that are initially cylindrical are mutually attracted and compress each other, which distorts the cylindrical symmetry. Magnetic field is created via the ∇×ve×B induction term in Ohm's Law where in-plane (perpendicular) shear of parallel flow (along the flux rope) is the dominant feature, along with some dissipation and magnetic reconnection. We predict and measure the growth of a quadrupole out-of-plane magnetic field δBz. This is a simple and coherent example of a shear flow driven dynamo. There is some similarity with two dimensional reconnection scenarios, which induce a current sheet and thus out-of-plane flow in the third dimension, despite the customary picture that considers flows only in the reconnection plane. These data illustrate a general and deterministic mechanism for large scale sheared flows to acquire smaller scale magnetic features, disordered structure, and possibly turbulence.
Fractal black holes and information
Energy Technology Data Exchange (ETDEWEB)
El Naschie, M.S. [Department of Physics, University of Alexandria, Alexandria (Egypt); Department of Astrophysics, Cairo University (Egypt); Department of Physics, Mansura University (Egypt)
2006-07-15
If nature is fractal as it evidently is, at classical resolution and if it is suspected to also be fractal at the quantum resolution as it is now a days generally believed to be, then we must have over looked at least two points or so in our physical model building of mini black holes. To start with at such ultra high resolution, the mini black hole geometry must be a fractal. Consequently we have zero volume and only a fractal surface area. Second because we cannot take the differential limit for the -bar {sub p}{sup 2} covering the transfinite surface area, there will be many gaps between the (-bar {sub p}){sup 2} tilings. In other words we must introduce transfinite corrections to the final result. Proceeding this way the information entropy unit of a black hole should be a=I=(7+{phi}{sup 3})(10){sup -66}cm{sup 2}=7.23606799(10){sup -66}cm{sup 2}The nearest classical result to the above is that obtained by Gerard 't Hoofta=I=(0.724)(10){sup -65}cm{sup 2}The paper ends with a general discussion of E-infinity theory and its possible relation with 't Hooft's holographic principle and his gluons-quark strings.
Fractal Characterization of Hyperspectral Imagery
Qiu, Hon-Iie; Lam, Nina Siu-Ngan; Quattrochi, Dale A.; Gamon, John A.
1999-01-01
Two Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) hyperspectral images selected from the Los Angeles area, one representing urban and the other, rural, were used to examine their spatial complexity across their entire spectrum of the remote sensing data. Using the ICAMS (Image Characterization And Modeling System) software, we computed the fractal dimension values via the isarithm and triangular prism methods for all 224 bands in the two AVIRIS scenes. The resultant fractal dimensions reflect changes in image complexity across the spectral range of the hyperspectral images. Both the isarithm and triangular prism methods detect unusually high D values on the spectral bands that fall within the atmospheric absorption and scattering zones where signature to noise ratios are low. Fractal dimensions for the urban area resulted in higher values than for the rural landscape, and the differences between the resulting D values are more distinct in the visible bands. The triangular prism method is sensitive to a few random speckles in the images, leading to a lower dimensionality. On the contrary, the isarithm method will ignore the speckles and focus on the major variation dominating the surface, thus resulting in a higher dimension. It is seen where the fractal curves plotted for the entire bandwidth range of the hyperspectral images could be used to distinguish landscape types as well as for screening noisy bands.
Marks-Tarlow, Terry
2010-01-01
In this article, the author draws on contemporary science to illuminate the relationship between early play experiences, processes of self-development, and the later emergence of the fractal self. She argues that orientation within social space is a primary function of early play and developmentally a two-step process. With other people and with…
Fractals in DNA sequence analysis
Institute of Scientific and Technical Information of China (English)
Yu Zu-Guo(喻祖国); Vo Anh; Gong Zhi-Min(龚志民); Long Shun-Chao(龙顺潮)
2002-01-01
Fractal methods have been successfully used to study many problems in physics, mathematics, engineering, finance,and even in biology. There has been an increasing interest in unravelling the mysteries of DNA; for example, how can we distinguish coding and noncoding sequences, and the problems of classification and evolution relationship of organisms are key problems in bioinformatics. Although much research has been carried out by taking into consideration the long-range correlations in DNA sequences, and the global fractal dimension has been used in these works by other people, the models and methods are somewhat rough and the results are not satisfactory. In recent years, our group has introduced a time series model (statistical point of view) and a visual representation (geometrical point of view)to DNA sequence analysis. We have also used fractal dimension, correlation dimension, the Hurst exponent and the dimension spectrum (multifractal analysis) to discuss problems in this field. In this paper, we introduce these fractal models and methods and the results of DNA sequence analysis.
Heritability of Retinal Vascular Fractals
DEFF Research Database (Denmark)
Vergmann, Anna Stage; Broe, Rebecca; Kessel, Line
2017-01-01
Purpose: To determine the genetic contribution to the pattern of retinal vascular branching expressed by its fractal dimension. Methods: This was a cross-sectional study of 50 monozygotic and 49 dizygotic, same-sex twin pairs aged 20 to 46 years. In 50°, disc-centered fundus photographs, the reti......Purpose: To determine the genetic contribution to the pattern of retinal vascular branching expressed by its fractal dimension. Methods: This was a cross-sectional study of 50 monozygotic and 49 dizygotic, same-sex twin pairs aged 20 to 46 years. In 50°, disc-centered fundus photographs......, the retinal vascular fractal dimension was measured using the box-counting method and compared within monozygotic and dizygotic twin pairs using Pearson correlation coefficients. Falconer's formula and quantitative genetic models were used to determine the genetic component of variation. Results: The mean......, the branching pattern of the retinal vessels demonstrated a higher structural similarity in monozygotic than in dizygotic twin pairs. The retinal vascular fractal dimension was mainly determined by genetic factors, which accounted for 54% of the variation. The genetically predetermination of the retinal...
Thermal transport in fractal systems
DEFF Research Database (Denmark)
Kjems, Jørgen
1992-01-01
Recent experiments on the thermal transport in systems with partial fractal geometry, silica aerogels, are reviewed. The individual contributions from phonons, fractons and particle modes, respectively, have been identified and can be described by quantitative models consistent with heat capacity...... data. The interpretation in the particle mode regime sheds light on the mechanisms for thermal conductivity in normal vitreous silica....
Fractal Aggregation of Copper Particles using Electroless Cell
Directory of Open Access Journals (Sweden)
S.Q.Chishty
2013-12-01
Full Text Available A phenomenon In which the particles performing Brownian motion when hit the aggregates become the part of it is known as Diffusion limited aggregation (DLA, which produces a fractal shape. Experimental efforts are discussed through which some DLA shapes arc obtained. For this purpose different electrolytic solutions are used. Electro less cells are also designed and constructed using standard methods. The cells have to be flexible in the sense that changing of plates and solutions should be easier for photography. We used compounds of copper,. for growth of fractals. Within a very short time metallic dendrites appeared in the cell at different operating conditions. These images were photographed, while desired branching structures in copper sulphate solution were seen. Results thus obtained are compared with the growth of DLA.
Dendritic gold nanowire growth observed in liquid with transmission electron microscopy.
Kraus, Tobias; de Jonge, Niels
2013-07-02
The growth of nanoscale gold dendrites was studied in situ in a thin liquid film with transmission electron microscopy (TEM) using a liquid cell with silicon nitride (SiN) windows. Gold nanoparticle seeds were covered by a thin liquid layer containing precursor solution. Dendrite nucleation was induced by the electron beam leading to an initial burst of growth. The growth then settled at tip velocities between 0.1 and 2.0 nm/s for different dendrites. Tip velocities fluctuated as different dendrite geometries grew from the tips. Those dendrites showing granularities in their structure experienced the largest growth speed. Comparison of the observed velocities with diffusion-limited growth rates suggests that dendrite growth in thin films at this scale is limited by diffusion. The described method may find application in research on the mechanisms behind dendrite growth and also to study other types of anisotropic growth of nanomaterials driven by crystal and twin geometries.
Sandwich type plasmonic platform for MEF using silver fractals.
Raut, Sangram L; Rich, Ryan; Shtoyko, Tanya; Bora, Ilkay; Laursen, Bo W; Sørensen, Thomas Just; Borejdo, Julian; Gryczynski, Zygmunt; Gryczynski, Ignacy
2015-11-14
In this report, we describe a plasmonic platform with silver fractals for metal enhanced fluorescence (MEF) measurements. When a dye containing surface was brought into contact with silver fractals, a significantly enhanced fluorescence signal from the dye was observed. Fluorescence enhancement was studied with the N-methyl-azadioxatriangulenium chloride salt (Me-ADOTA·Cl) in PVA films made from 0.2% PVA (w/v) solution spin-coated on a clean glass coverslip. The Plasmonic Platforms (PP) were assembled by pressing together silver fractals on one glass slide and a separate glass coverslip spin-coated with a uniform Me-ADOTA·Cl in PVA film. In addition, we also tested ADOTA labeled human serum albumin (HSA) deposited on a glass slide for potential PP bioassay applications. Using the new PP, we could achieve more than a 20-fold fluorescence enhancement (bright spots) accompanied by a decrease in the fluorescence lifetime. The experimental results were used to calculate the extinction (excitation) enhancement factor (GA) and fluorescence radiative rate enhancements factor (GF). No change in emission spectrum was observed for a dye with or without contact with fractals. Our studies indicate that this type of PP can be a convenient approach for constructing assays utilizing metal enhanced fluorescence (MEF) without the need for depositing the material directly on metal structures platforms.
Observation of growth modes during metal-organic chemical vapor deposition of GaN
Energy Technology Data Exchange (ETDEWEB)
Stephenson, G.B.; Eastman, J.A.; Thompson, C.; Auciello, O.; Thompson, L.J. [Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439 (United States); Munkholm, A. [Chemistry Division, Argonne National Laboratory, Argonne, Illinois 60439 (United States); Fini, P.; DenBaars, S.P.; Speck, J.S. [Materials Department, University of California, Santa Barbara, California 93106 (United States)
1999-05-01
We present real-time surface x-ray scattering measurements during homoepitaxial growth of GaN by metal-organic chemical vapor deposition. We observed intensity oscillations corresponding to the completion of each monolayer during layer-by-layer growth. The growth rate was found to be temperature independent and Ga-transport limited. Transitions between step-flow, layer-by-layer, and three-dimensional growth modes were determined as a function of temperature and growth rate. {copyright} {ital 1999 American Institute of Physics.}
Region-Based Fractal Image Coding with Freely-Shaped Partition
Institute of Scientific and Technical Information of China (English)
SUNYunda; ZHAOYao; YUANBaozong
2004-01-01
In Fractal image coding (FIC), a partitioning of the original image into ranges and domains is required, which greatly affects the coding performance. Usually, the more adaptive to the image content the partition is, the higher performance it can achieve. Nowadays, some alleged Region-based fractal coders (RBFC) using split-and-merge strategy can achieve better adaptivity andperformance compared with traditional rectangular block partitions. However, the regions are still with linear contour. In this paper, we present a Freely-shaped Regionbased fractal coder (FS-RBFC) using a two-step partitioning, i.e. coarse partitioning based on fractal dimension and fine partitioning based on region growth, which brings freely-shaped regions. Our highly image-adaptive scheme can achieve better rate-distortion curve than conventional scheme, even more visually pleasing results at the same performance.
Interface fractal construction in Ni⧸KBr⧸Ni system
Shang, Chang He; Li, Heng De
1994-04-01
Non-equilibrium aggregation behavior on a surface has attracted increasing attention among researchers. It was found that at the early stage of film formation particles could coagulate into the form of diffusion-limited aggregation (DLA) on a free substrate surface. By using scanning tunnelling microscopy, Hwang, Schroder, Gunther and Behm [Phys. Rev. Lett. 67 (1991) 3279] recently showed that on the clean Ru surface, Au atoms grew into irregular islands of a fractal character as well. Similar monolayer growth was also found on a constrained surface, i.e., the interface between two lattice planes. In this paper, we report on the cluster aggregation behavior between two metallic layers. Samples were prepared by alternatively depositing pure constituent materials onto freshly cleaved NaCl single crystals in a high vacuum. Transmission electron microscopy was used to characterize the surface construction. It was found that KBr dendritic islands on the constrained surface also had a fractal geometry. The growth dynamics could be modified by adding interface impurities. Our experiment showed that Al addition onto the constrained surface could accelerate the growth process and degraded the fractal dimension. Detailed results will be presented, and some possible mechanisms will also be discussed.
Brief communication: age and fractal dimensions of human sagittal and coronal sutures
DEFF Research Database (Denmark)
Lynnerup, Niels; Jacobsen, Jens Christian Brings
2003-01-01
The fractal dimensions of human sagittal and coronal sutures were calculated on 31 complete skulls from the Terry Collection. The aim was to investigate whether the fractal dimension, relying on the whole sutural length, might yield a better description of age-related changes in sutural morphology......, as opposed to other methods of quantification, which generally rely on more arbitrary scoring systems. However, the fractal dimension did not yield better age correlations than other previously described methods. At best, the results reflected the general observation that young adults below age 40 years...
Hierarchical fractal Weyl laws for chaotic resonance states in open mixed systems.
Körber, M J; Michler, M; Bäcker, A; Ketzmerick, R
2013-09-13
In open chaotic systems the number of long-lived resonance states obeys a fractal Weyl law, which depends on the fractal dimension of the chaotic saddle. We study the generic case of a mixed phase space with regular and chaotic dynamics. We find a hierarchy of fractal Weyl laws, one for each region of the hierarchical decomposition of the chaotic phase-space component. This is based on our observation of hierarchical resonance states localizing on these regions. Numerically this is verified for the standard map and a hierarchical model system.
Z-Scaling, Fractality and Principle of Relativity in Relativistic Collisions of Hadrons and Nuclei
Zborovský, I; Panebratsev, Yu A; Skoro, G P
2001-01-01
The formation length of particles produced in the relativistic collisions of hadrons and nuclei has relevance to the fundamental principles of physics at small interaction distances. The relation is phenomenologically expressed by a z-scaling observed in the differential cross sections for the inclusive reactions at high energies. The scaling variable reflects the length of the elementary particle trajectories in terms of a fractal measure. Characterizing the fractal approach, we demonstrate the relativity principle in space-time with broken isotropy. We derive relativistic transformations accounting for the asymmetry of space-time induced in the interactions by various parton fractal structures of hadrons and nuclei.
Quantum Fractals: From Heisenberg's Uncertainty to Barnsley's Fractality
Jadczyk, Arkadiusz
2014-07-01
This book brings together two concepts. The first is over a hundred years old -- the "quantum", while the second, "fractals", is newer, achieving popularity after the pioneering work of Benoit Mandelbrot. Both areas of research are expanding dramatically day by day. It is somewhat amazing that quantum theory, in spite of its age, is still a boiling mystery as we see in some quotes from recent publications addressed to non-expert readers:...
Dellino, P.; Liotino, G.
2002-03-01
Image processing analysis is used to check the ability of the fractal dimension for quantitatively describing the shape of volcanic ash particles. Digitized scanning electron microscopy images of fine pyroclasts from the eruptions of Monte Pilato-Rocche Rosse (Lipari, Italy) are investigated to test the efficiency of the fractal dimension to discriminate between particles of different eruptive processes. Multivariate analysis of multiple fractal components allows distinction between magmatic particles and phreatomagmatic particles, which however is less significant than the discrimination obtained in previous studies by the use of simple 'adimensional' shape parameters. Approximation of the actual particle boundary and the not rotation invariant nature of the fractal data frequently result in a significant scatter of data points in the Mandelbrot-Richardson plot. Such behavior obscures in some cases the actual information of particle shape and renders the discriminating power of fractal analysis less effective than classical shape descriptors. Data less affected by scatter reveal that phreatomagmatic particles of the Monte Pilato-Rocche Rosse eruptions are true (mono) fractals, whereas magmatic particles are multifractals. The textural (small-scale) fractal of magmatic particles is similar to the fractal dimension value of phreatomagmatic particles, and is attributed to the rheological behavior of melt upon brittle fragmentation. The structural (large-scale) fractal of magmatic particles refers to the walls of ruptured vesicles that lay on the particle outline. The high difference between the values of the textural and structural fractals of magmatic particles of the Monte Pilato-Rocche Rosse eruptions suggests two distinct and independent processes in the formation of such pyroclasts. At the scales corresponding to the textural fractal, the fragmentation process is independent of vesicles. Magmatic fragmentation is not simply related to growth, expansion
Fractal characteristics of surface crack evolution in the process of gas-containing coal extrusion
Institute of Scientific and Technical Information of China (English)
Chen Peng; Wang Enyuan; Ou Jianchun; Li Zhonghui; Wei Mingyao; Li Xuelong
2013-01-01
In this paper,simulated experiment device of coal and gas outburst was employed to perform the experiment on gas-containing coal extrusion.In the experiment,coal surface cracks were observed with a highspeed camera and then the images were processed by sketch.Based on the above description,the paper studied the fractal dimension values from different positions of coal surface as well as their changing laws with time.The results show that there is a growing parabola treen of crack dimension value in the process of coal extrusion.Accordingly,we drew the conclusion that extruded coal crack evolution is a process of fractal dimension value increase.On the basis of factal dimension values taken from different parts of coal masses,a fractal dimension of the contour map was drawn.Thus,it is clear that the contour map involves different crack fractal dimension values from different positions.To be specific,where there are complicated force and violent movement In coal mass,there are higher fractal dimension values,i.e.,the further the middle of observation surface is from the exit of coal mass,and the lower the fractal dimension value is.In line with fractal geometry and energy theory of coal and gas outburst,this study presents the relation between fractal dimension and energy in the process of extruding.In conclusion,the evolution of crack fractal dimension value can signify that of energy,which has laid a solid foundation for the quantification research on the mechanism of gas-containing coal extrusion.
Fractal Properties in Economics
2000-01-01
Leschhorn, P. Maass, M. A. Salinger and H. E. Stanley, Scaling behavior in the growth of companies, Nature 379 (1996) 804. 16. H. Takayasu and K. Okuyama...Amaral, S. V. Buldyrev, S. Havlin, M. A. Salinger and H. E. Stanley, Power law scaling in a system of interacting units with complex internal
Flocculation control study based on fractal theory
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
A study on flocculation control based on fractal theory was carried out. Optimization test of chemical coagulant dosage confirmed that the fractal dimension could reflect the flocculation degree and settling characteristics of aggregates and the good correlation with the turbidity of settled effluent. So that the fractal dimension can be used as the major parameter for flocculation system control and achieve self-acting adjustment of chemical coagulant dosage. The fractal dimension flocculation control system was used for further study carried out on the effects of various flocculation parameters, among which are the dependency relationship among aggregates fractal dimension, chemical coagulant dosage, and turbidity of settled effluent under the conditions of variable water quality and quantity. And basic experimental data were obtained for establishing the chemical coagulant dosage control model mainly based on aggregates fractal dimension.
FRACTAL KINEMATICS OF CRACK PROPAGATION IN GEOMATERIALS
Institute of Scientific and Technical Information of China (English)
谢和平
1995-01-01
Experimental results indicate that propagation paths of cracks in geomaterials are often irregular, producing rough fracture surfaces which are fractal. A formula is derived for the fractal kinematics of crack propagation in geomaterials. The formula correlates the dynamic and static fracture toughnesses with crack velocity, crack length and a microstructural parameter, and allows the fractal dimension to be obtained. From the equations for estimating crack velocity and fractal dimension it can be shown that the measured crack velocity, Vo , should be much smaller than the fractal crack velocity, V. It can also be shown that the fractal dimension of the crack propagation path can be calculated directly from Vo and from the fracture toughness.
Fractal Weyl law for Linux Kernel architecture
Ermann, L.; Chepelianskii, A. D.; Shepelyansky, D. L.
2011-01-01
We study the properties of spectrum and eigenstates of the Google matrix of a directed network formed by the procedure calls in the Linux Kernel. Our results obtained for various versions of the Linux Kernel show that the spectrum is characterized by the fractal Weyl law established recently for systems of quantum chaotic scattering and the Perron-Frobenius operators of dynamical maps. The fractal Weyl exponent is found to be ν ≈ 0.65 that corresponds to the fractal dimension of the network d ≈ 1.3. An independent computation of the fractal dimension by the cluster growing method, generalized for directed networks, gives a close value d ≈ 1.4. The eigenmodes of the Google matrix of Linux Kernel are localized on certain principal nodes. We argue that the fractal Weyl law should be generic for directed networks with the fractal dimension d < 2.
Conference on Fractals and Related Fields III
Seuret, Stéphane
2017-01-01
This contributed volume provides readers with an overview of the most recent developments in the mathematical fields related to fractals, including both original research contributions, as well as surveys from many of the leading experts on modern fractal theory and applications. It is an outgrowth of the Conference of Fractals and Related Fields III, that was held on September 19-25, 2015 in île de Porquerolles, France. Chapters cover fields related to fractals such as harmonic analysis, multifractal analysis, geometric measure theory, ergodic theory and dynamical systems, probability theory, number theory, wavelets, potential theory, partial differential equations, fractal tilings, combinatorics, and signal and image processing. The book is aimed at pure and applied mathematicians in these areas, as well as other researchers interested in discovering the fractal domain.
Patell, Hilla
2016-01-01
In order to achieve the goal of observation, preparation of the adult, the observer, is necessary. This preparation, says Hilla Patell, requires us to "have an appreciation of the significance of the child's spontaneous activities and a more thorough understanding of the child's needs." She discusses the growth of both the desire to…
A Fractal Model for the Capacitance of Lunar Dust and Lunar Dust Aggregates
Collier, Michael R.; Stubbs, Timothy J.; Keller, John W.; Farrell, William M.; Marshall, John; Richard, Denis Thomas
2011-01-01
Lunar dust grains and dust aggregates exhibit clumping, with an uneven mass distribution, as well as features that span many spatial scales. It has been observed that these aggregates display an almost fractal repetition of geometry with scale. Furthermore, lunar dust grains typically have sharp protrusions and jagged features that result from the lack of aeolian weathering (as opposed to space weathering) on the Moon. A perfectly spherical geometry, frequently used as a model for lunar dust grains, has none of these characteristics (although a sphere may be a reasonable proxy for the very smallest grains and some glasses). We present a fractal model for a lunar dust grain or aggregate of grains that reproduces (1) the irregular clumpy nature of lunar dust, (2) the presence of sharp points, and (3) dust features that span multiple scale lengths. We calculate the capacitance of the fractal lunar dust analytically assuming fixed dust mass (i.e. volume) for an arbitrary number of fractal levels and compare the capacitance to that of a non-fractal object with the same volume, surface area, and characteristic width. The fractal capacitance is larger than that of the equivalent non-fractal object suggesting that for a given potential, electrostatic forces on lunar dust grains and aggregates are greater than one might infer from assuming dust grains are sphericaL Consequently, electrostatic transport of lunar dust grains, for example lofting, appears more plausible than might be inferred by calculations based on less realistic assumptions about dust shape and associated capacitance.
The fractal globule as a model of chromatin architecture in the cell.
Mirny, Leonid A
2011-01-01
The fractal globule is a compact polymer state that emerges during polymer condensation as a result of topological constraints which prevent one region of the chain from passing across another one. This long-lived intermediate state was introduced in 1988 (Grosberg et al. 1988) and has not been observed in experiments or simulations until recently (Lieberman-Aiden et al. 2009). Recent characterization of human chromatin using a novel chromosome conformational capture technique brought the fractal globule into the spotlight as a structural model of human chromosome on the scale of up to 10 Mb (Lieberman-Aiden et al. 2009). Here, we present the concept of the fractal globule, comparing it to other states of a polymer and focusing on its properties relevant for the biophysics of chromatin. We then discuss properties of the fractal globule that make it an attractive model for chromatin organization inside a cell. Next, we connect the fractal globule to recent studies that emphasize topological constraints as a primary factor driving formation of chromosomal territories. We discuss how theoretical predictions, made on the basis of the fractal globule model, can be tested experimentally. Finally, we discuss whether fractal globule architecture can be relevant for chromatin packing in other organisms such as yeast and bacteria.
The fractal dimension of architecture
Ostwald, Michael J
2016-01-01
Fractal analysis is a method for measuring, analysing and comparing the formal or geometric properties of complex objects. In this book it is used to investigate eighty-five buildings that have been designed by some of the twentieth-century’s most respected and celebrated architects. Including designs by Le Corbusier, Eileen Gray, Frank Lloyd Wright, Robert Venturi, Frank Gehry, Peter Eisenman, Richard Meier and Kazuyo Sejima amongst others, this book uses mathematics to analyse arguments and theories about some of the world’s most famous designs. Starting with 625 reconstructed architectural plans and elevations, and including more than 200 specially prepared views of famous buildings, this book presents the results of the largest mathematical study ever undertaken into architectural design and the largest single application of fractal analysis presented in any field. The data derived from this study is used to test three overarching hypotheses about social, stylistic and personal trends in design, along...
Chaos, Fractals and Their Applications
Thompson, J. Michael T.
2016-12-01
This paper gives an up-to-date account of chaos and fractals, in a popular pictorial style for the general scientific reader. A brief historical account covers the development of the subject from Newton’s laws of motion to the astronomy of Poincaré and the weather forecasting of Lorenz. Emphasis is given to the important underlying concepts, embracing the fractal properties of coastlines and the logistics of population dynamics. A wide variety of applications include: NASA’s discovery and use of zero-fuel chaotic “superhighways” between the planets; erratic chaotic solutions generated by Euler’s method in mathematics; atomic force microscopy; spontaneous pattern formation in chemical and biological systems; impact mechanics in offshore engineering and the chatter of cutting tools; controlling chaotic heartbeats. Reference is made to a number of interactive simulations and movies accessible on the web.
Fractal model of anomalous diffusion.
Gmachowski, Lech
2015-12-01
An equation of motion is derived from fractal analysis of the Brownian particle trajectory in which the asymptotic fractal dimension of the trajectory has a required value. The formula makes it possible to calculate the time dependence of the mean square displacement for both short and long periods when the molecule diffuses anomalously. The anomalous diffusion which occurs after long periods is characterized by two variables, the transport coefficient and the anomalous diffusion exponent. An explicit formula is derived for the transport coefficient, which is related to the diffusion constant, as dependent on the Brownian step time, and the anomalous diffusion exponent. The model makes it possible to deduce anomalous diffusion properties from experimental data obtained even for short time periods and to estimate the transport coefficient in systems for which the diffusion behavior has been investigated. The results were confirmed for both sub and super-diffusion.
The Fractal Dimension of the ρ Ophiucus Molecular Cloud Complex
Lee, Yongung; Yi, Di; Kim, Y. S.; Jung, J. H.; Kang, H. W.; Lee, C. H.; Yim, I. S.; Kim, H. G.
2016-12-01
We estimate the fractal dimension of the ρ Ophiuchus Molecular Cloud Complex, associated with star forming regions. We selected a cube (v, l, b) database, obtained with J=1-0 transition lines of \\coand tco at a resolution of 22'' using a multibeam receiver system on the 14-m telescope of the Five College Radio Astronomy Observatory. Using a code developed within IRAF, we identified slice-clouds with two threshold temperatures to estimate the fractal dimension. With threshold temperatures of 2.25 K (3σ) and 3.75 K (5σ), the fractal dimension of the target cloud is estimated to be D = 1.52-1.54, where P ∝ A^{D/2} , which is larger than previous results. We suggest that the sampling rate (spatial resolution) of observed data must be an important parameter when estimating the fractal dimension, and that narrower or wider dispersion around an arbitrary fit line and the intercepts at NP = 100 should be checked whether they relate to rms noise level or characteristic structure of the target cloud. This issue could be investigated by analysing several high resolution databases with different quality (low or moderate sensitivity).
Fractals of graphene quantum dots in photoluminescence of shungite
Razbirin, B. S.; Rozhkova, N. N.; Sheka, E. F.; Nelson, D. K.; Starukhin, A. N.
2014-05-01
Viewing shungite as loosely packed fractal nets of graphene-based (reduced graphene oxide, rGO) quantum dots (GQDs), we consider photoluminescence of the latter as a convincing proof of the structural concept as well as of the GQD attribution to individual rGO fragments. We study emission from shungite GQDs for colloidal dispersions in water, carbon tetrachloride, and toluene at both room and low temperatures. As expected, the photoluminescence of the GQD aqueous dispersions is quite similar to that of synthetic GQDs of the rGO origin. The morphological study of shungite dispersions shows a steady trend of GQDs to form fractals and to drastically change the colloid fractal structure caused by the solvent exchange. Spectral study reveals a dual character of the emitting centers: individual GQDs are responsible for the spectra position while the fractal structure of GQD colloids ensures high broadening of the spectra due to structural inhomogeneity, thus causing a peculiar dependence of the photoluminescence spectra on the excitation wavelength. For the first time, photoluminescence spectra of individual GQDs were observed in frozen toluene dispersions, which paves the way for a theoretical treatment of the GQD photonics.
Fractal texture analysis of the healing process after bone loss.
Borowska, Marta; Szarmach, Janusz; Oczeretko, Edward
2015-12-01
Radiological assessment of treatment effectiveness of guided bone regeneration (GBR) method in postresectal and postcystal bone loss cases, observed for one year. Group of 25 patients (17 females and 8 males) who underwent root resection with cystectomy were evaluated. The following combination therapy of intraosseous deficits was used, consisting of bone augmentation with xenogenic material together with covering regenerative membranes and tight wound closure. The bone regeneration process was estimated, comparing the images taken on the day of the surgery and 12 months later, by means of Kodak RVG 6100 digital radiography set. The interpretation of the radiovisiographic image depends on the evaluation ability of the eye looking at it, which leaves a large margin of uncertainty. So, several texture analysis techniques were developed and used sequentially on the radiographic image. For each method, the results were the mean from the 25 images. These methods compute the fractal dimension (D), each one having its own theoretic basis. We used five techniques for calculating fractal dimension: power spectral density method, triangular prism surface area method, blanket method, intensity difference scaling method and variogram analysis. Our study showed a decrease of fractal dimension during the healing process after bone loss. We also found evidence that various methods of calculating fractal dimension give different results. During the healing process after bone loss, the surfaces of radiographic images became smooth. The result obtained show that our findings may be of great importance for diagnostic purpose.
Fractal Reference Signals in Pulse-Width Modulation
Lurie, Boris; Lurie, Helen
2005-01-01
A report proposes the use of waveforms having fractal shapes reminiscent of sawteeth (in contradistinction to conventional regular sawtooth waveforms) as reference signals for pulse-width modulation in control systems for thrusters of spacecraft flying in formation. Fractal reference signals may also be attractive in some terrestrial control systems - especially those in which pulse-width modulation is used for precise control of electric motors. The report asserts that the use of fractal reference signals would enable the synchronous control of several variables of a spacecraft formation, such that consumption of propellant would be minimized, intervals between thruster firings would be long (as preferred for performing scientific observations), and delays in controlling large-thrust maneuvers for retargeting would be minimized. The report further asserts that whereas different controllers would be needed for different modes of operation if conventional pulsewidth modulation were used, the use of fractal reference signals would enable the same controller to function nearly optimally in all regimes of operation, so that only this one controller would be needed.
SIERPIENSKI & CROWN SQUARE FRACTAL SHAPES SLOTTED MICROSTRIP PATCH ANTENNA
Directory of Open Access Journals (Sweden)
Dr. Yogesh Bhomia
2014-01-01
Full Text Available A new Sierpienski & Crown Square Fractal Shapes Slotted Microstrip Patch Antenna is proposed. A patch antenna is a narrowband, wide-beam antenna. These antennas are low profile, conformal to planar and non-planar surface, simple and inexpensive to manufacture using modern printed circuit technology, mechanically robust when mounted on rigid surface, compatible with MMIC designs and when the particular shape and mode are selected they are very versatile in terms of resonant frequency, polarization, field pattern and impedance. Microstrip patch antenna consist of a very thin metallic strip (patch placed a small fraction of a wavelength above a ground plane. The patch is generally made of conducting material such as copper or gold and can take any possible shape. This paper presents a design of Sierpienski & Crown Square Fractal Shapes Slotted Microstrip Patch Antenna and experimentally studied on IE3D software. This design is achieved by cutting Sierpienski & Crown Square Fractal Shapes Slottes in a patch. With Sierpienski & Crown Square Fractal Shapes patch antenna is designed on a FR4 substrate of thickness 1.524 mm and relative permittivity of 4.4 and mounted above the ground plane at a height of 6 mm. Bandwidth as high as 36.6% are achieved with stable pattern characteristics, such as gain and cross polarization, within its bandwidth. Impedance bandwidth, antenna gain and return loss are observed for the proposed antenna. Details of the measured and simulated results are presented and discussed.
Fractal methods in image analysis and coding
Neary, David
2001-01-01
In this thesis we present an overview of image processing techniques which use fractal methods in some way. We show how these fields relate to each other, and examine various aspects of fractal methods in each area. The three principal fields of image processing and analysis th a t we examine are texture classification, image segmentation and image coding. In the area of texture classification, we examine fractal dimension estimators, comparing these methods to other methods in use, a...
Fractal characteristics of electric properties of coal
Institute of Scientific and Technical Information of China (English)
LIU Cheng-lun; XU Long-jun; XIAN Xue-fu
2006-01-01
In the light of fractal geometry theory, the characteristics of coal's electric parameters (including dielectric constant, alternating conductivity, dielectric loss angle tangent and electric polarization constant) were studied by using literature data. The results are shown that the electrical properties of coal have fractal characteristic. The fractal dimensions of dielectric, alternating conductivity, dielectric loss angle tangent were obtained, and are relative to the content of pyrite sulfur, heat and ash content of coal.
Wideband irregular-shaped fractal antennas
Kolesov, V. V.; Krupenin, S. V.
2007-01-01
This paper proposes an algorithm of generating fully reproducible irregular fractal structures for antenna design. Three types of pseudorandom fractal clusters are introduced. The multi-frequency behavior of the irregular-shaped fractal antennas is studied by means of numerical analysis. The antenna behavior is studied under feeder displacement. As shown by numerical results feeder displacements allow one to control the spatial-frequency antenna characteristics.
Fractals and Scars on a Compact Octagon
Levin, J; Levin, Janna; Barrow, John D.
2000-01-01
A finite universe naturally supports chaotic classical motion. An ordered fractal emerges from the chaotic dynamics which we characterize in full for a compact 2-dimensional octagon. In the classical to quantum transition, the underlying fractal can persist in the form of scars, ridges of enhanced amplitude in the semiclassical wave function. Although the scarring is weak on the octagon, we suggest possible subtle implications of fractals and scars in a finite universe.
Fractal properties of nanostructured semiconductors
Energy Technology Data Exchange (ETDEWEB)
Zhanabaev, Z.Zh. [Al-Farabi Khazakh National University, Tole bi Street, 96, Almaty 050012 (Kazakhstan); Grevtseva, T.Yu. [Al-Farabi Khazakh National University, Tole bi Street, 96, Almaty 050012 (Kazakhstan)]. E-mail: kenwp@mail.ru
2007-03-15
A theory for the temperature and time dependence of current carrier concentration in semiconductors with different non-equilibrium nanocluster structure has been developed. It was shown that the scale-invariant fractal self-similar and self-affine laws can exist near by the transition point to the equilibrium state. Results of the theory have been compared to the experimental data from electrical properties of semiconductor films with nanoclusters.
THE DISTRIBUTIONAL DIMENSION OF FRACTALS
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
In the book [1] H.Triebel introduces the distributional dimension of fractals in and distributional dimension, respectively. Thus we might say that the distributional dimension is an analytical definition for Hausdorff dimension. Therefore we can study Hausdorff dimension through the distributional dimension analytically.By discussing the distributional dimension, this paper intends to set up a criterion for estimating the upper and lower bounds of Hausdorff dimension analytically. Examples illustrating the criterion are included in the end.
Fractal Solutions of the Nizhnik-Novikov-Veselov Equation
Institute of Scientific and Technical Information of China (English)
楼森岳; 唐晓艳; 陈春丽
2002-01-01
Considering that some types of fractal solutions may appear in many (2+ l )-dimensional soliton equations because some arbitrary functions can be included in the exact solutions, we use some special types of lower dimensional fractal functions to construct higher dimensional fractal solutions of the Nizhnik Novikov-Veselov equation. The static eagle-shape fractal solutions, fractal dromion solutions and the fractal lump solutions are given in detail.
Fractal Metrology for biogeosystems analysis
Directory of Open Access Journals (Sweden)
V. Torres-Argüelles
2010-11-01
Full Text Available The solid-pore distribution pattern plays an important role in soil functioning being related with the main physical, chemical and biological multiscale and multitemporal processes of this complex system. In the present research, we studied the aggregation process as self-organizing and operating near a critical point. The structural pattern is extracted from the digital images of three soils (Chernozem, Solonetz and "Chocolate" Clay and compared in terms of roughness of the gray-intensity distribution quantified by several measurement techniques. Special attention was paid to the uncertainty of each of them measured in terms of standard deviation. Some of the applied methods are known as classical in the fractal context (box-counting, rescaling-range and wavelets analyses, etc. while the others have been recently developed by our Group. The combination of these techniques, coming from Fractal Geometry, Metrology, Informatics, Probability Theory and Statistics is termed in this paper Fractal Metrology (FM. We show the usefulness of FM for complex systems analysis through a case study of the soil's physical and chemical degradation applying the selected toolbox to describe and compare the structural attributes of three porous media with contrasting structure but similar clay mineralogy dominated by montmorillonites.
Fractal Metrology for biogeosystems analysis
Torres-Argüelles, V.; Oleschko, K.; Tarquis, A. M.; Korvin, G.; Gaona, C.; Parrot, J.-F.; Ventura-Ramos, E.
2010-11-01
The solid-pore distribution pattern plays an important role in soil functioning being related with the main physical, chemical and biological multiscale and multitemporal processes of this complex system. In the present research, we studied the aggregation process as self-organizing and operating near a critical point. The structural pattern is extracted from the digital images of three soils (Chernozem, Solonetz and "Chocolate" Clay) and compared in terms of roughness of the gray-intensity distribution quantified by several measurement techniques. Special attention was paid to the uncertainty of each of them measured in terms of standard deviation. Some of the applied methods are known as classical in the fractal context (box-counting, rescaling-range and wavelets analyses, etc.) while the others have been recently developed by our Group. The combination of these techniques, coming from Fractal Geometry, Metrology, Informatics, Probability Theory and Statistics is termed in this paper Fractal Metrology (FM). We show the usefulness of FM for complex systems analysis through a case study of the soil's physical and chemical degradation applying the selected toolbox to describe and compare the structural attributes of three porous media with contrasting structure but similar clay mineralogy dominated by montmorillonites.
Triangular Constellations in Fractal Measures
Wilkinson, Michael
2014-01-01
The local structure of a fractal set is described by its dimension $D$, which is the exponent of a power-law relating the mass ${\\cal N}$ in a ball to its radius $\\epsilon$: ${\\cal N}\\sim \\epsilon^D$. It is desirable to characterise the {\\em shapes} of constellations of points sampling a fractal measure, as well as their masses. The simplest example is the distribution of shapes of triangles formed by triplets of points, which we investigate for fractals generated by chaotic dynamical systems. The most significant parameter describing the triangle shape is the ratio $z$ of its area to the radius of gyration squared. We show that the probability density of $z$ has a phase transition: $P(z)$ is independent of $\\epsilon$ and approximately uniform below a critical flow compressibility $\\beta_{\\rm c}$, but for $\\beta>\\beta_{\\rm c}$ it is described by two power laws: $P(z)\\sim z^{\\alpha_1}$ when $1\\gg z\\gg z_{\\rm c}(\\epsilon)$, and $P(z)\\sim z^{\\alpha_2}$ when $z\\ll z_{\\rm c}(\\epsilon)$.
Fractal metrology for biogeosystems analysis
Directory of Open Access Journals (Sweden)
V. Torres-Argüelles
2010-06-01
Full Text Available The solid-pore distribution pattern plays an important role in soil functioning being related with the main physical, chemical and biological multiscale and multitemporal processes. In the present research, this pattern is extracted from the digital images of three soils (Chernozem, Solonetz and "Chocolate'' Clay and compared in terms of roughness of the gray-intensity distribution (the measurand quantified by several measurement techniques. Special attention was paid to the uncertainty of each of them and to the measurement function which best fits to the experimental results. Some of the applied techniques are known as classical in the fractal context (box-counting, rescaling-range and wavelets analyses, etc. while the others have been recently developed by our Group. The combination of all these techniques, coming from Fractal Geometry, Metrology, Informatics, Probability Theory and Statistics is termed in this paper Fractal Metrology (FM. We show the usefulness of FM through a case study of soil physical and chemical degradation applying the selected toolbox to describe and compare the main structural attributes of three porous media with contrasting structure but similar clay mineralogy dominated by montmorillonites.
Fractal Geometry and Stochastics V
Falconer, Kenneth; Zähle, Martina
2015-01-01
This book brings together leading contributions from the fifth conference on Fractal Geometry and Stochastics held in Tabarz, Germany, in March 2014. The book is divided into five sections covering different facets of this fast developing area: geometric measure theory, self-similar fractals and recurrent structures, analysis and algebra on fractals, multifractal theory, and random constructions. There are state-of-the-art surveys as well as papers highlighting more specific recent advances. The authors are world-experts who present their topics comprehensibly and attractively. The book provides an accessible gateway to the subject for newcomers as well as a reference for recent developments for specialists. Authors include: Krzysztof Barański, Julien Barral, Kenneth Falconer, De-Jun Feng, Peter J. Grabner, Rostislav Grigorchuk, Michael Hinz, Stéphane Jaffard, Maarit Järvenpää, Antti Käenmäki, Marc Kesseböhmer, Michel Lapidus, Klaus Mecke, Mark Pollicott, Michał Rams, Pablo Shmerkin, and András Te...
Fractal zone plates with variable lacunarity.
Monsoriu, Juan; Saavedra, Genaro; Furlan, Walter
2004-09-06
Fractal zone plates (FZPs), i.e., zone plates with fractal structure, have been recently introduced in optics. These zone plates are distinguished by the fractal focusing structure they provide along the optical axis. In this paper we study the effects on this axial response of an important descriptor of fractals: the lacunarity. It is shown that this parameter drastically affects the profile of the irradiance response along the optical axis. In spite of this fact, the axial behavior always has the self-similarity characteristics of the FZP itself.
Experimental Study of Fractal Image Compression Algorithm
Directory of Open Access Journals (Sweden)
Chetan R. Dudhagara
2012-08-01
Full Text Available Image compression applications have been increasing in recent years. Fractal compression is a lossy compression method for digital images, based on fractals. The method is best suited for textures and natural images, relying on the fact that parts of an image often resemble other parts of the same image. In this paper, a study on fractal-based image compression and fixed-size partitioning will be made, analyzed for performance and compared with a standard frequency domain based image compression standard, JPEG. Sample images will be used to perform compression and decompression. Performance metrics such as compression ratio, compression time and decompression time will be measured in JPEG cases. Also the phenomenon of resolution/scale independence will be studied and described with examples. Fractal algorithms convert these parts into mathematical data called "fractal codes" which are used to recreate the encoded image. Fractal encoding is a mathematical process used to encode bitmaps containing a real-world image as a set of mathematical data that describes the fractal properties of the image. Fractal encoding relies on the fact that all natural, and most artificial, objects contain redundant information in the form of similar, repeating patterns called fractals.
A Fast Fractal Image Compression Coding Method
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
Fast algorithms for reducing encoding complexity of fractal image coding have recently been an important research topic. Search of the best matched domain block is the most computation intensive part of the fractal encoding process. In this paper, a fast fractal approximation coding scheme implemented on a personal computer based on matching in range block's neighbours is presented. Experimental results show that the proposed algorithm is very simple in implementation, fast in encoding time and high in compression ratio while PSNR is almost the same as compared with Barnsley's fractal block coding .
Fractal signatures in the aperiodic Fibonacci grating.
Verma, Rupesh; Banerjee, Varsha; Senthilkumaran, Paramasivam
2014-05-01
The Fibonacci grating (FbG) is an archetypal example of aperiodicity and self-similarity. While aperiodicity distinguishes it from a fractal, self-similarity identifies it with a fractal. Our paper investigates the outcome of these complementary features on the FbG diffraction profile (FbGDP). We find that the FbGDP has unique characteristics (e.g., no reduction in intensity with increasing generations), in addition to fractal signatures (e.g., a non-integer fractal dimension). These make the Fibonacci architecture potentially useful in image forming devices and other emerging technologies.
Fractal Dimension of Voice-Signal Waveforms
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
The fractal dimension is one important parameter that characterizes waveforms. In this paper, we derive a new method to calculate fractal dimension of digital voice-signal waveforms. We show that fractal dimension is an efficient tool for speaker recognition or speech recognition. It can be used to identify different speakers or distinguish speech. We apply our results to Chinese speaker recognition and numerical experiment shows that fractal dimension is an efficient parameter to characterize individual Chinese speakers. We have developed a semiautomatic voiceprint analysis system based on the theory of this paper and former researches.
Mineral resource analysis by parabolic fractals
Institute of Scientific and Technical Information of China (English)
XIE Shu-yun; YANG Yong-guo; BAO Zheng-yu; KE Xian-zhong; LIU Xiao-long
2009-01-01
Elemental concentration distributions in space have been analyzed using different approaches. These analyses are of great significance for the quantitative characterization of various kinds of distribution patterns. Fractal and multi-fiactal methods have been extensively applied to this topic. Traditionally, approximately linear-fractal laws have been regarded as useful tools for characterizing the self-similarities of element concentrations. But, in nature, it is not always easy to fred perfect linear fractal laws. In this paper the parabolic fractal model is used. First a two dimensional multiplicative multi-fractal cascade model is used to study the concentration patterns. The results show the parabolic fractal (PF) properties of the concentrations and the validity of non-linear fractal analysis. By dividing the studied area into four sub-areas it was possible to show that each part follows a non-linear para-bolic fractal law and that the dispersion within each part varies. The ratio of the polynomial coefficients of the fitted parabolic curves can reflect, to some degree, the relative concentration and dispersal distribution patterns. This can provide new insight into the ore-forming potential in space. The parabolic fractal evaluations of ore-forming potential for the four subareas are in good agreement with field investigation work and geochemical mapping results based on analysis of the original data.
Fractal geometry mathematical foundations and applications
Falconer, Kenneth
2013-01-01
The seminal text on fractal geometry for students and researchers: extensively revised and updated with new material, notes and references that reflect recent directions. Interest in fractal geometry continues to grow rapidly, both as a subject that is fascinating in its own right and as a concept that is central to many areas of mathematics, science and scientific research. Since its initial publication in 1990 Fractal Geometry: Mathematical Foundations and Applications has become a seminal text on the mathematics of fractals. The book introduces and develops the general theory and applica
Hecht, C. A.
Fractal packing and highly irregular shaped particles increase the mechanical properties of rocks and building materials. This suggests that fractal methods are good tools for modeling particle mixes with efficient properties like maximum strength and maximum surface area or minimum porosity and minimum permeability. However gradings and packings are calculated by ``Euclidean'' disk models and sphere models. Surprisingly even the simplest models are far more complex than they appear. The fractal ``Appolonian packing model'' is proposed as the most universal two-dimensional packing model. However the inhomogeneity of gradings and the irregularity of natural grain shapes and surfaces are not reflected by these models. Consequently calculations are often far from empirical observations and experimental results. A thorough quantification of packings and gradings is important for many reasons and still a matter of intense investigation and controversial discussion. This study concentrates on fractal models for densely packed non-cohesive rocks, crushed mineral assemblages, concrete and asphalt mixtures. A summary of fractal grain size distributions with linear cumulative curves on log-log plots is presented for these mixtures. It is shown that fractal two-dimensional and three-dimensional models for dense packings reflect different physical processes of material mixing or geological deposition. The results from shear-box experiments on materials with distinct grain size distributions show a remarkable increase of the mechanical strength from non-fractal to fractal mixtures. It is suggested that fractal techniques need more systematical application and correlation with results from material testing experiments in engineering geology. The purpose of future work should lead towards the computability of dense packings of angular particles in three dimensions.
Inkjet-Printed Ultra Wide Band Fractal Antennas
Maza, Armando Rodriguez
2012-05-01
In this work, Paper-based inkjet-printed Ultra-wide band (UWB) fractal antennas are presented. Three new designs, a combined UWB fractal monopole based on the fourth order Koch Snowflake fractal which utilizes a Sierpinski Gasket fractal for ink reduction, a Cantor-based fractal antenna which performs a larger bandwidth compared to previously published UWB Cantor fractal monopole antenna, and a 3D loop fractal antenna which attains miniaturization, impedance matching and multiband characteristics. It is shown that fractals prove to be a successful method of reducing fabrication cost in inkjet printed antennas while retaining or enhancing printed antenna performance.
A variational principle for the Hausdorff dimension of fractal sets
DEFF Research Database (Denmark)
Olsen, Lars; Cutler, Colleen D.
1994-01-01
Matematik, fraktal (fractal), Hausdorff dimension, Renyi dimension, pakke dimension (packing dimension)......Matematik, fraktal (fractal), Hausdorff dimension, Renyi dimension, pakke dimension (packing dimension)...
Broadband enhanced graphene photodetector with fractal metasurface (Conference Presentation)
Wang, Di; Fang, Jieran; DeVault, Clayton T.; Chung, Ting-Fung; Chen, Yong P.; Boltasseva, Alexandra; Shalaev, Vladimir M.; Kildishev, Alexander V.
2016-09-01
Graphene has been demonstrated to be a promising photo-detection material because of its ultra-broadband absorption, compatibility with CMOS technology, and dynamic tunability. There are multiple known photo-detection mechanisms in graphene, among which the photovoltaic effect has the fastest response time thus is the prioritized candidate for ultrafast photodetector. There have been numerous efforts to enhance the intrinsically low sensitivity in graphene photovoltaic detectors using metallic plasmonic structures, but such plasmonic enhancements are mostly narrowband and polarization dependent. In this work, we propose a gold Cayley-tree fractal metasurface design that has a multi-band resonance, to realize broadband and polarization-insensitive plasmonic enhancement in graphene photovoltaic detectors. When illuminated with visible light, the fractal metasurface exhibits multiple hotspots at the metal-graphene interface, where the electric field of the incident electromagnetic wave is enhanced and contributes to generating excess electron-hole pairs in graphene. The large metal-graphene interface length in the fractal metasurface also helps to harvest at a higher efficiency the electron-hole pairs by built-in electric field due to metal-graphene potential gradient. To demonstrate the concept, we carried out experiment using Ar-Kr CW laser, an optical chopper, and lock-in amplifier. We obtained experimentally an almost constant ten-fold enhancement of photocurrent generated on the fractal metasurface compared to that generated on the plain gold-graphene edge, at all tested wavelengths (488 nm, 514 nm, 568 nm, and 647 nm). We also observed an unchanged photoresponse with respect to incident light polarization angles, which is a result of the highly symmetric geometry of the fractal metasurface.
Fractal dimension in percolating Heisenberg antiferromagnets
Energy Technology Data Exchange (ETDEWEB)
Itoh, S. [Neutron Science Laboratory, High Energy Accelerator Research Organization, Tsukuba 305-0810 (Japan)]. E-mail: shinichi.itoh@kek.jp; Kajimoto, R. [Quantum Beam Science Directorate, Japan Atomic Energy Agency, Tokai 319-1195 (Japan); Adams, M.A. [ISIS Facility, Rutherford Appleton Laboratory, Didcot, Oxon OX11 0QX (United Kingdom); Bull, M.J. [ISIS Facility, Rutherford Appleton Laboratory, Didcot, Oxon OX11 0QX (United Kingdom); Iwasa, K. [Department of Physics, Tohoku University, Sendai 980-8578 (Japan); Aso, N. [Neutron Science Laboratory, Institute for Solid State Physics, University of Tokyo, Tokai 319-1106 (Japan); Yoshizawa, H. [Neutron Science Laboratory, Institute for Solid State Physics, University of Tokyo, Tokai 319-1106 (Japan); Takeuchi, T. [Low Temperature Center, Osaka University, Toyonaka 560-0043 (Japan)
2007-03-15
We investigated static and dynamical properties in the three-dimensional percolating Heisenberg antiferromagnets, RbMn{sub c}Mg{sub 1-c}F{sub 3}, with the magnetic concentration close to the percolation threshold, c{sub P}=0.312, around the superlattice point well below T{sub N}. In neutron diffraction experiment, the wave number dependence of the elastic scattering component was well fitted to q{sup -x}. Magnetic fractons were also studied using inelastic neutron scattering, and the observed fractons showed the dispersion relation of q{sup z}. The determined exponents, x=2.43+/-0.05 and z=2.5+/-0.1, were in good agreement with the fractal dimension (D{sub f}=2.48)
Fractal Analysis in Agrophysics
The geometric irregularity is an intrinsic property of soils and plants. This geometric irregularity is easy to perceive and observe, but quantifying it has long presented a daunting challenge. Such quantifying is imperative because the geometric irregularity is the cause and the reflection of spati...
Fractal EEG analysis with Higuchi's algorithm of low-frequency noise exposition on humans
Panuszka, Ryszard; Damijan, Zbigniew; Kasprzak, Cezary
2004-05-01
Authors used methods based on fractal analysis of EEG signal to assess the influence of low-frequency sound field on the human brain electro-potentials. The relations between LFN (low-frequency noise) and change in fractal dimension EEG signal were measured with stimulations tones. Three types of LFN stimuli were presented; each specified dominant frequency and sound-pressure levels (7 Hz at 120 dB, 18 Hz at 120 dB, and 40 Hz at 110 dB). Standard EEG signal was recorded before, during, and after subject's exposure for 35 min. LFN. Applied to the analysis fractal dimension of EEG-signal Higuchis algorithm. Experiments show LFN influence on complexity of EEG-signal with calculated Higuchi's algorithm. Observed increase of mean value of Higuchi's fractal dimension during exposition to LFN.
Fractal space-time fluctuations: A signature of quantumlike chaos in dynamical systems
Selvam, A M
2004-01-01
Dynamical systems in nature such as fluid flows, heart beat patterns, rainfall variability, stock market price fluctuations, etc. exhibit selfsimilar fractal fluctuations on all scales in space and time. Power spectral analyses of fractal fluctuations exhibit inverse power law form indicating long-range space-time correlations, identified as self-organized criticality. The author has proposed a general systems theory, which predicts the observed self-organized criticality as signatures of quantumlike chaos. The model shows that (1) the fractal fluctuations result from an overall logarithmic spiral trajectory with the quasiperiodic Penrose tiling pattern for the internal structure. Conventional power spectral analysis of such a logarithmic spiral trajectory will show a continuum of eddies with progressive increase in phase. (2) Power spectral analyses of fractal fluctuations of dynamical systems exhibit the universal inverse power law form of the statistical normal distribution. Such a result indicates that th...
Hütt, M.-Th.; Rascher, U.; Lüttge, U.
Crassulacean acid metabolism (CAM) serves as a plant model system for the investigation of circadian rhythmicity. Recently, it has been discovered that propagating waves and, as a result, synchronization and desynchronization of adjacent leaf areas, contribute to an observed temporal variation of the net CO2 uptake of a CAM plant. The underlying biological clock has thus to be considered as a spatiotemporal product of many weakly coupled nonlinear oscillators. Here we study the structure of these spatiotemporal patterns with methods from fractal geometry. The fractal dimension of the spatial pattern is used to characterize the dynamical behavior of the plant. It is seen that the value of the fractal dimension depends significantly on the dynamical regime of the rhythm. In addition, the time variation of the fractal dimension is studied. The implications of these findings for our understanding of circadian rhythmicity are discussed.
Fractal characteristics of cracks and fragments generated in unloading rockburst tests
Institute of Scientific and Technical Information of China (English)
Li Dejian; Zhao Fei; Zheng Maojiong
2014-01-01
True triaxial rockburst experiments with four different unloading rates were performed on four prism specimens of granite sampled from Beishan, China. The damage evolution in the rockburst test was investigated from two aspects including fracture surface crack and fragment characteristics. The scanning electron microscopy was used to observe the micro crack information on fragment surface. Combing binarization and box counting dimensions, the fractal dimensions of cracks were obtained. Meanwhile, the fragments were collected and a sieving experiment was conducted. We weighed the fragments qualities, counted the amount of fragments and measured the fragments length, width and thickness. Utilizing four methods to calculate damage fractal dimensions of fragments, the trend of fractal value changing with unloading rates can be roughly described. It can be concluded from these experiments that the fractal dimension either for crack or for fragment holds a decreasing trend with the decreasing unloading rate, indicating a reduction of damage level.
Directory of Open Access Journals (Sweden)
Javier Rodríguez
Full Text Available La geometría fractal caracteriza objetivamente los grados de irregularidad de los objetos naturales. De otro lado, las dimensiones fractales permiten definir matemáticamente la irregularidad de las formas naturales, como por ejemplo las estructuras cardiacas. El ventrículo izquierdo se estudia a través del ventriculograma, y es a partir de este examen con la aplicación de la geometría fractal, que se puede calcular el grado de irregularidad, de forma objetiva y reproducible para cualquier paciente. A partir de 17 ventriculogramas de 6 de pacientes con fracción de eyección normal y 11 con fracción de eyección fracción menor a 40%, con diagnóstico de compromiso ventricular severo, se desarrolló una medida cuantitativa de los ventriculogramas en la que se evaluaron los grados de similitud entre las dimensiones fractales de los contornos ventriculares izquierdos durante la dinámica cardíaca, en sístole, diástole y totalidad. Se observó que el grado de similitud entre las dimensiones fractales de las comparaciones hechas en los contornos de un ventrículo sano, varía entre 20,9 y 210, mientras que las de un ventrículo con fracción de eyección menor a 40% se encuentra entre 210 y 2500 al menos en uno de los cotejos realizados.Fractal geometry is the geometry that objectively characterizes the degrees of irregularity of natural objects. On the other hand, fractal dimensions allow defining mathematically the irregularity of natural forms such as those of the heart structures. The left ventricle is studied through ventriculography, and by the application of fractal geometry to this exam, it is possible to calculate the degree of irregularity in an objective and reproducible way in any patient. From 17 ventriculographies, 6 from patients with normal ejection fraction and 11 with ejection fraction <40%, with diagnosis of severe ventricular involvement, a quantitative measurement from the ventriculographies was developed in which the
Design of LTCC Based Fractal Antenna
AdbulGhaffar, Farhan
2010-09-01
The thesis presents a Sierpinski Carpet fractal antenna array designed at 24 GHz for automotive radar applications. Miniaturized, high performance and low cost antennas are required for this application. To meet these specifications a fractal array has been designed for the first time on Low Temperature Co-fired Ceramic (LTCC) based substrate. LTCC provides a suitable platform for the development of these antennas due to its properties of vertical stack up and embedded passives. The complete antenna concept involves integration of this fractal antenna array with a Fresnel lens antenna providing a total gain of 15dB which is appropriate for medium range radar applications. The thesis also presents a comparison between the designed fractal antenna and a conventional patch antenna outlining the advantages of fractal antenna over the later one. The fractal antenna has a bandwidth of 1.8 GHz which is 7.5% of the centre frequency (24GHz) as compared to 1.9% of the conventional patch antenna. Furthermore the fractal design exhibits a size reduction of 53% as compared to the patch antenna. In the end a sensitivity analysis is carried out for the fractal antenna design depicting the robustness of the proposed design against the typical LTCC fabrication tolerances.
Interface after explosion welding: Fractal analysis
Greenberg, B. A.; Ivanov, M. A.; Pushkin, M. S.; Patselov, A. M.; Volkova, A. Yu.; Inozemtsev, A. V.
2015-10-01
The interfaces (plain, wavy) in the welding joints formed by explosion welding are investigated. Various types of fractals, namely, islands, multifractals, and a coastline, are found. The fractal dimensions of islands in the case of a plain interface and a coastline in the case of a wavy interface are calculated.
Chaos and fractals an elementary introduction
Feldman, David P
2012-01-01
For students with a background in elementary algebra, this text provides a vivid introduction to the key phenomena and ideas of chaos and fractals, including the butterfly effect, strange attractors, fractal dimensions, Julia sets and the Mandelbrot set, power laws, and cellular automata.
Fractal Trigonometric Polynomials for Restricted Range Approximation
Chand, A. K. B.; Navascués, M. A.; Viswanathan, P.; Katiyar, S. K.
2016-05-01
One-sided approximation tackles the problem of approximation of a prescribed function by simple traditional functions such as polynomials or trigonometric functions that lie completely above or below it. In this paper, we use the concept of fractal interpolation function (FIF), precisely of fractal trigonometric polynomials, to construct one-sided uniform approximants for some classes of continuous functions.
THE FRACTAL GEOMETRY AND TRIGONOMETRIC SERIES
Institute of Scientific and Technical Information of China (English)
YuJiarong
1994-01-01
Mathemstics is used to study the nature. Straight lines, circles, ellipses,continuous and differentiable curves and surfaces etc. are the first approximations of forms of concrete objects. But in reality, these forms are very irregular. Consequentily B. Mandebrot introduces since 1975 fractals and the fractal geometry to study the second approximaions of such forms. Si
Fractal Image Coding with Digital Watermarks
Directory of Open Access Journals (Sweden)
Z. Klenovicova
2000-12-01
Full Text Available In this paper are presented some results of implementation of digitalwatermarking methods into image coding based on fractal principles. Thepaper focuses on two possible approaches of embedding digitalwatermarks into fractal code of images - embedding digital watermarksinto parameters for position of similar blocks and coefficients ofblock similarity. Both algorithms were analyzed and verified on grayscale static images.
Fractal Music: The Mathematics Behind "Techno" Music
Padula, Janice
2005-01-01
This article describes sound waves, their basis in the sine curve, Fourier's theorem of infinite series, the fractal equation and its application to the composition of music, together with algorithms (such as those employed by meteorologist Edward Lorenz in his discovery of chaos theory) that are now being used to compose fractal music on…
BIVARIATE FRACTAL INTERPOLATION FUNCTIONS ON RECTANGULAR DOMAINS
Institute of Scientific and Technical Information of China (English)
Xiao-yuan Qian
2002-01-01
Non-tensor product bivariate fractal interpolation functions defined on gridded rectangular domains are constructed. Linear spaces consisting of these functions are introduced.The relevant Lagrange interpolation problem is discussed. A negative result about the existence of affine fractal interpolation functions defined on such domains is obtained.
Fractal Model of the Spheroidal Graphite
Institute of Scientific and Technical Information of China (English)
Z.Y.HE; K.Z.HWANG
1996-01-01
In this paper,a fractal model about the microstructure of spheroidal-graphite is presented through the research on the surface form and the analysis to microregion.The fractal dimension is calculated and the forming mechanism is also discussed.
Fractal basins in an ecological model
Directory of Open Access Journals (Sweden)
I. Djellit
2013-09-01
Full Text Available Complex dynamics is detected in an ecological model of host-parasitoid interaction. It illustrates fractalization of basins with self-similarity and chaotic attractors. This paper describes these dynamic behaviors, bifurcations, and chaos. Fractals basins are displayed by numerical simulations.
Fractal analysis of polar bear hairs
Directory of Open Access Journals (Sweden)
Wang Qing-Li
2015-01-01
Full Text Available Hairs of a polar bear (Ursus maritimus are of superior properties such as the excellent thermal protection. Why do polar bears can resist such cold environment? The paper concludes that its fractal porosity plays an important role, and its fractal dimensions are very close to the golden mean, 1.618, revealing the possible optimal structure of polar bear hair.
Dirichlet Form of Product of Variational Fractals
Institute of Scientific and Technical Information of China (English)
刘源
2003-01-01
Much effort has gone into constructing Dirichlet forms to define Laplacians on self-similar sets. However, the results have only been successful on p.c.f. (post critical finite) fractals. We prove the existence of a Dirichlet form on a class of non-p.c.f. sets that are the product of variational fractals.
Undergraduate Experiment with Fractal Diffraction Gratings
Monsoriu, Juan A.; Furlan, Walter D.; Pons, Amparo; Barreiro, Juan C.; Gimenez, Marcos H.
2011-01-01
We present a simple diffraction experiment with fractal gratings based on the triadic Cantor set. Diffraction by fractals is proposed as a motivating strategy for students of optics in the potential applications of optical processing. Fraunhofer diffraction patterns are obtained using standard equipment present in most undergraduate physics…
Riemann zeta function is a fractal
Woon, S C
1994-01-01
Voronin's theorem on the "Universality" of Riemann zeta function is shown to imply that Riemann zeta function is a fractal (in the sense that Mandelbrot set is a fractal) and a concrete "representation" of the "giant book of theorems'' that Paul Halmos referred to.
Fractal fluctuations in gaze speed visual search.
Stephen, Damian G; Anastas, Jason
2011-04-01
Visual search involves a subtle coordination of visual memory and lower-order perceptual mechanisms. Specifically, the fluctuations in gaze may provide support for visual search above and beyond what may be attributed to memory. Prior research indicates that gaze during search exhibits fractal fluctuations, which allow for a wide sampling of the field of view. Fractal fluctuations constitute a case of fast diffusion that may provide an advantage in exploration. We present reanalyses of eye-tracking data collected by Stephen and Mirman (Cognition, 115, 154-165, 2010) for single-feature and conjunction search tasks. Fluctuations in gaze during these search tasks were indeed fractal. Furthermore, the degree of fractality predicted decreases in reaction time on a trial-by-trial basis. We propose that fractality may play a key role in explaining the efficacy of perceptual exploration.
A random walk through fractal dimensions
Kaye, Brian H
2008-01-01
Fractal geometry is revolutionizing the descriptive mathematics of applied materials systems. Rather than presenting a mathematical treatise, Brian Kaye demonstrates the power of fractal geometry in describing materials ranging from Swiss cheese to pyrolytic graphite. Written from a practical point of view, the author assiduously avoids the use of equations while introducing the reader to numerous interesting and challenging problems in subject areas ranging from geography to fine particle science. The second edition of this successful book provides up-to-date literature coverage of the use of fractal geometry in all areas of science.From reviews of the first edition:''...no stone is left unturned in the quest for applications of fractal geometry to fine particle problems....This book should provide hours of enjoyable reading to those wishing to become acquainted with the ideas of fractal geometry as applied to practical materials problems.'' MRS Bulletin
Fractal organization of feline oocyte cytoplasm.
De Vico, G; Peretti, V; Losa, G A
2005-01-01
The present study aimed at verifying whether immature cat oocytes with morphologic irregular cytoplasm display self-similar features which can be analytically described by fractal analysis. Original images of oocytes collected by ovariectomy were acquired at a final magnification of 400x with a CCD video camera connected to an optic microscope. After greyscale thresholding segmentation of cytoplasm, image profiles were submitted to fractal analysis using FANAL++, a program which provided an analytical standard procedure for determining the fractal dimension (FD). The presentation of the oocyte influenced the magnitude of the fractal dimension with the highest FD of 1.91 measured on grey-dark cytoplasm characterized by a highly connected network of lipid droplets and intracellular membranes. Fractal analysis provides an effective quantitative descriptor of the real cytoplasm morphology, which can influence the acquirement of in vitro developmental competence, without introducing any bias or shape approximation and thus contributes to an objective and reliable classification of feline oocytes.
Fractal organization of feline oocyte cytoplasm
Directory of Open Access Journals (Sweden)
G De Vico
2009-06-01
Full Text Available The present study aimed at verifying whether immature cat oocytes with morphologic irregular cytoplasm display selfsimilar features which can be analytically described by fractal analysis. Original images of oocytes collected by ovariectomy were acquired at a final magnification of 400 X with a CCD video camera connected to an optic microscope. After greyscale thresholding segmentation of cytoplasm, image profiles were submitted to fractal analysis using FANAL++, a program which provided an analytical standard procedure for determining the fractal dimension (FD. The presentation of the oocyte influenced the magnitude of the fractal dimension with the highest FD of 1.91 measured on grey-dark cytoplasm characterized by a highly connected network of lipid droplets and intracellular membranes. Fractal analysis provides an effective quantitative descriptor of the real cytoplasm morphology, which can influence the acquirement of in vitro developmental competence, without introducing any bias or shape approximation and thus contributes to an objective and reliable classification of feline oocytes.
Fractals Generated by Statistical Contraction Operators
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
In the theory of random fractal, there are two important classes of random sets, one is the class of fractals generated by the paths of stochastic processes and another one is the class of factals generated by statistical contraction operators. Now we will introduce some things about the probability basis and fractal properties of fractals in the last class. The probability basis contains (1) the convergence and measurability of a random recursive set K(ω) as a random element, (2) martingals property. The fractal properties include (3) the character of various similarity, (4) the separability property, (5) the support and zero-one law of distribution Pk=PK-1, (6) the Hausdorff dimension and Hausdorff exact measure function.
FRACTAL PROPERTIES OF ROCK FRACTURE SURFACES
Institute of Scientific and Technical Information of China (English)
王金安; 谢和平; MarekA．KWASNIEWSKI
1996-01-01
To give a better understanding of the morphological features of rock fracture surfaces within the framework of fractal geometry, the fractal characters of the rough surfaces in" rock are analyzed according to the variogram method. The study elaborates the significance of the geometric parameters-fractal dimension D and the intercept A on a log-log plot to the surface structure. Investigation extends to the anisotropy and heterogeneity of rock fracture surfaces, and the scale effect on the fractal estimation. The present study indicates that fractal dimension alone may not be sufficient to characterize the surface roughness of rock joints. A reliable estimation should take into account the combination of D and A.
Fractal lattice of gelatin nanoglobules
Novikov, D. V.; Krasovskii, A. N.
2012-11-01
The globular structure of polymer coatings on a glass, which were obtained from micellar solutions of gelatin in the isooctane-water-sodium (bis-2-ethylhexyl) sulfosuccinate system, has been studied using electron microscopy. It has been shown that an increase in the average globule size is accompanied by the formation of a fractal lattice of nanoglobules and a periodic physical network of macromolecules in the coating. The stability of such system of the "liquid-in-a-solid" type is limited by the destruction of globules and the formation of a homogeneous network structure of the coating.
On fractal space-time and fractional calculus
Directory of Open Access Journals (Sweden)
Hu Yue
2016-01-01
Full Text Available This paper gives an explanation of fractional calculus in fractal space-time. On observable scales, continuum models can be used, however, when the scale tends to a smaller threshold, a fractional model has to be adopted to describe phenomena in micro/nano structure. A time-fractional Fornberg-Whitham equation is used as an example to elucidate the physical meaning of the fractional order, and its solution process is given by the fractional complex transform.
Cancer Detection via Determination of Fractal Cell Dimension
Bauer, W; Bauer, Wolfgang; Mackenzie, Charles D.
1995-01-01
We utilize the fractal dimension of the perimeter surface of cell sections as a new observable to characterize cells of different types. We propose that it is possible to distinguish cancerous from healthy cells with the aid of this new approach. As a first application we show that it is possible to perform this distinction between patients with hairy-cell lymphocytic leukemia and those with normal blood lymphocytes.
Cosmological observables, IR growth of fluctuations, and scale-dependent anisotropies
Giddings, Steven B
2011-01-01
We extend semiclassical methods in inflationary cosmology that capture leading IR corrections to correlators. Such large IR effects can be absorbed into a coordinate change when examining sufficiently local observables, but not when comparing observations at large separation in scales, such as seen by a late-time observer. The analysis is facilitated by definition of a scale-dependent metric and physical momentum. These assist definition of "IR-safe" observables seen by a post-inflationary observer, which are contrasted to those based on the local geometry of the reheating surface. For such observables, the observer's horizon provides an effective IR cutoff. IR growth contributes to enhanced statistical inhomogeneities/anisotropies at short scales, observation of which by a present day observer might be sought in 21 cm measurements. Such IR corrections are argued to grow large for a very late-time observer.
Cheng, C.-Y.; Perevedentseva, E.; Tu, J.-S.; Chung, P.-H.; Cheng, C.-L.; Liu, K.-K.; Chao, J.-I.; Chen, P.-H.; Chang, C.-C.
2007-04-01
This letter presents direct observation of growth hormone receptor in one single cancer cell using nanodiamond-growth hormone complex as a specific probe. The interaction of surface growth hormone receptor of A549 human lung epithelial cells with growth hormone was observed using nanodiamond's unique spectroscopic signal via confocal Raman mapping. The growth hormone molecules were covalent conjugated to 100nm diameter carboxylated nanodiamonds, which can be recognized specifically by the growth hormone receptors of A549 cell. The Raman spectroscopic signal of diamond provides direct and in vitro observation of growth hormone receptors in physiology condition in a single cell level.
Fractal THz slow light metamaterial devices
Ito, Shoichi
Scope and Method of Study: The goal of this study is to investigate the time delay of the fractal H metamaterials in the terahertz regime. This metamaterial contains resonators with two different sizes of H structures which mimic Electromagnetically Induced Transparency and create a transmission window and the corresponding phase dispersion, thus producing slow light. The Al structures were fabricated on silicon wafer and Mylar by using microelectronic lithography and thermal evaporation technique. By using terahertz time-domain spectroscopy, the phase change caused by the slow light system and the actual time delay were obtained. Numerical simulations were carried out to systematize the effect of permittivity and structure dimensions on the optical properties. Findings and Conclusions: We experimentally demonstrated the numerical time delay of the fractal H metamaterial as a slow light device. When permittivity of the substrates increases, the peak position of the transmission window shifts to lower frequency and the bandwidth becomes broader. As a result, silicon performed larger time delay than that of Mylar. By changing the length of the resonator, the bandwidth and the peak position of the transmission window is controllable. At the edges of the transmission window, the negative time delays (fast light) were also observed. Mylar acts as a quaci-free standing structure and allows higher spectral measurement. Moreover, metamaterials fabricated on multiple Mylar films can potentially act as a more effective slow light device. As applications, slow light metamaterials are expected to be used for high-capacity terahertz communication networks, all-optical information processing and sensing devices.
In-Situ Observation of SiC Bulk Single Crystal Growth by XRD System
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
In-situ analysis for SiC bulk single crystal growth was reported using vertical X-ray diffractometer system. A furnace for SiC sublimation growth combined with the XRD system which possessed three kinds of functions including topography, rocking curve measurement and crystal growth rate monitoring was developed. These functions could contribute as a powerful tool finding the optimum growth condition by dynamic observation in the crucible. In this study, the in-situ X-ray topographs succeeded to capture dynamic elongation of defects and dislocation generated in the SiC growing crystals. The in-situ rocking curve measurement reviled appearance of mosaic structure in the SiC crystal grown with high growth rate. The in-situ growth rate monitoring also succeeded very precisely using the direct X-ray beam absorption. On the base of findings and facts obtained by the in-situ observations, the importance for the SiC growth was discussed.
Local fractality: The case of forest fires in Portugal
Kanevski, Mikhail; Pereira, Mário G.
2017-08-01
The research deals with a study of local fractality in spatial distribution of forest fires in Portugal using the sandbox method. The general procedure is the following: (a) define a circle centred in each and all events with increasing radius R; (b) count the number of other events located within the circle of radius R, N(R) ; (c) plot the growth curve which is the functional dependence of N(R) versus R; and (d) estimate the local fractal dimension as the slope on log[ N(R) ] versus log[ R]. The computation is carried out by using the location of every fire event as a centre but without the final averaging over all the fires for a given R, which is usually performed to get a global fractal dimension and to estimate global clustering. Sandbox method is widely used in many applications in physics and other subjects. The local procedure has the ability to provide the most complete information regarding the spatial distribution of clustering and avoiding non-homogeneity and non-stationarity problems. Most of the analysis was performed using the National Mapping Burnt Area (NMBA) database which accounts for 32 156 fires during the 1975-2013 period. The results of local analysis are compared with a randomly generated pattern in forest zones (validity domain). The results demonstrate interesting local spatial patterns of clustering. Some results on global measures are reported as well.
Diagnosis of Lung Cancer by Fractal Analysis of Damaged DNA
Directory of Open Access Journals (Sweden)
Hamidreza Namazi
2015-01-01
Full Text Available Cancer starts when cells in a part of the body start to grow out of control. In fact cells become cancer cells because of DNA damage. A DNA walk of a genome represents how the frequency of each nucleotide of a pairing nucleotide couple changes locally. In this research in order to study the cancer genes, DNA walk plots of genomes of patients with lung cancer were generated using a program written in MATLAB language. The data so obtained was checked for fractal property by computing the fractal dimension using a program written in MATLAB. Also, the correlation of damaged DNA was studied using the Hurst exponent measure. We have found that the damaged DNA sequences are exhibiting higher degree of fractality and less correlation compared with normal DNA sequences. So we confirmed this method can be used for early detection of lung cancer. The method introduced in this research not only is useful for diagnosis of lung cancer but also can be applied for detection and growth analysis of different types of cancers.
Local fractality: the case of forest fires in Portugal
Kanevski, Mikhail
2016-01-01
The research deals with a study of local fractality in spatial distribution of forest fires in Portugal using the sandbox method. The general procedure is the following: (a) define a circle centred in each and all events with increasing radius R; (b) count the number of other events located within the circle of radius R, N(R); (c) plot the growth curve which is the functional dependence of N(R) versus R; and (d) estimate the local fractal dimension as the slope on log[N(R)] versus log[R]. The computation is carried out by using the location of every fire event as a centre but without the final averaging over all the fires for a given R, which is usually performed to get a global fractal dimension and to estimate global clustering. Sandbox method is widely used in many applications in physics and other subjects. The local procedure has the ability to provide the most complete information regarding the spatial distribution of clustering and avoiding non-homogeneity and non-stationarity problems. Most of the ana...
Pre-Service Teachers' Concept Images on Fractal Dimension
Karakus, Fatih
2016-01-01
The analysis of pre-service teachers' concept images can provide information about their mental schema of fractal dimension. There is limited research on students' understanding of fractal and fractal dimension. Therefore, this study aimed to investigate the pre-service teachers' understandings of fractal dimension based on concept image. The…
Fractal gait patterns are retained after entrainment to a fractal stimulus.
Rhea, Christopher K; Kiefer, Adam W; Wittstein, Matthew W; Leonard, Kelsey B; MacPherson, Ryan P; Wright, W Geoffrey; Haran, F Jay
2014-01-01
Previous work has shown that fractal patterns in gait can be altered by entraining to a fractal stimulus. However, little is understood about how long those patterns are retained or which factors may influence stronger entrainment or retention. In experiment one, participants walked on a treadmill for 45 continuous minutes, which was separated into three phases. The first 15 minutes (pre-synchronization phase) consisted of walking without a fractal stimulus, the second 15 minutes consisted of walking while entraining to a fractal visual stimulus (synchronization phase), and the last 15 minutes (post-synchronization phase) consisted of walking without the stimulus to determine if the patterns adopted from the stimulus were retained. Fractal gait patterns were strengthened during the synchronization phase and were retained in the post-synchronization phase. In experiment two, similar methods were used to compare a continuous fractal stimulus to a discrete fractal stimulus to determine which stimulus type led to more persistent fractal gait patterns in the synchronization and post-synchronization (i.e., retention) phases. Both stimulus types led to equally persistent patterns in the synchronization phase, but only the discrete fractal stimulus led to retention of the patterns. The results add to the growing body of literature showing that fractal gait patterns can be manipulated in a predictable manner. Further, our results add to the literature by showing that the newly adopted gait patterns are retained for up to 15 minutes after entrainment and showed that a discrete visual stimulus is a better method to influence retention.
Fractal dimensions of flocs between clay particles and HAB organisms
Institute of Scientific and Technical Information of China (English)
WANG Hongliang; YU Zhiming; CAO Xihua; SONG Xiuxian
2011-01-01
The impact of harmful algal blooms (HABs) on public health and related economics have been increasing in many coastal regions of the world. Sedimentation of algal cells through flocculation with clay particles is a promising strategy for controlling HABs. Previous studies found that removal efficiency (RE) was influenced by many factors, including clay type and concentration, algal growth stage,and physiological aspects of HAB cells. To estimate the effect of morphological characteristics of the aggregates on HAB cell removal, fractal dimensions were measured and the RE of three species of HAB organism, Heterosigma akashiwo, Alexandrium tamarense, and Skeletonema costatum, by original clay and modified clay, was determined. For all HAB species, the modified clay had a higher RE than original clay.For the original clay, the two-dimensional fractal dimension (D2) was 1.92 and three-dimensional fractal dimension (D3) 2.81, while for the modified clay, D2 was 1.84 and D3 was 2.50. The addition of polyaluminum chloride (PACI) lead to a decrease of the repulsive barrier between clay particles, and resulted in lower D2 and D3. Due to the decrease of D3, and the increase of the effective sticking coefficient,the flocculation rate between modified clay particles and HAB organisms increased, and thus resulted in a high RE. The fractal dimensions of flocs differed in HAB species with different cell morphologies. For example, Alexandrium tamarense cells are ellipsoidal, and the D3 and D2 of flocs were the highest, while for Skeletonema costatum, which has filamentous cells, the D3 and D2 of flocs were the lowest.
Cosmological observables, IR growth of fluctuations, and scale-dependent anisotropies
DEFF Research Database (Denmark)
B. Giddings, Steven; Sloth, Martin Snoager
2011-01-01
We extend semiclassical methods in inflationary cosmology that capture leading IR corrections to correlators. Such large IR effects can be absorbed into a coordinate change when examining sufficiently local observables, but not when comparing observations at large separation in scales, such as seen...... by a late-time observer. The analysis is facilitated by definition of a scale-dependent metric and physical momentum. These assist definition of "IR-safe" observables seen by a post-inflationary observer, which are contrasted to those based on the local geometry of the reheating surface....... For such observables, the observer's horizon provides an effective IR cutoff. IR growth contributes to enhanced statistical inhomogeneities/anisotropies at short scales, observation of which by a present day observer might be sought in 21 cm measurements. Such IR corrections are argued to grow large for a very late...
Measures and dimensions of fractal sets in local fields
Institute of Scientific and Technical Information of China (English)
QIU Hua; SU Weiyi
2006-01-01
The study of fractal analysis over the local fields as underline spaces is very important since it can motivate new approaches and new ideas, and discover new techniques in the study of fractals. To study fractal sets in a local field K, in this paper, we define several kinds of fractal measures and dimensions of subsets in K. Some typical fractal sets in K are constructed. We also give out the Hausdorff dimensions and measures, Box-counting dimensions and Packing dimensions, and stress that there exist differences between fractal analysis on local fields and Euclidean spaces. Consequently, the theoretical foundation of fractal analysis on local fields is established.
Complex Patterns in Financial Time Series Through HIGUCHI’S Fractal Dimension
Grace Elizabeth Rani, T. G.; Jayalalitha, G.
2016-11-01
This paper analyzes the complexity of stock exchanges through fractal theory. Closing price indices of four stock exchanges with different industry sectors are selected. Degree of complexity is assessed through Higuchi’s fractal dimension. Various window sizes are considered in evaluating the fractal dimension. It is inferred that the data considered as a whole represents random walk for all the four indices. Analysis of financial data through windowing procedure exhibits multi-fractality. Attempts to apply moving averages to reduce noise in the data revealed lower estimates of fractal dimension, which was verified using fractional Brownian motion. A change in the normalization factor in Higuchi’s algorithm did improve the results. It is quintessential to focus on rural development to realize a standard and steady growth of economy. Tools must be devised to settle the issues in this regard. Micro level institutions are necessary for the economic growth of a country like India, which would induce a sporadic development in the present global economical scenario.
MODELING OF THE EMULSION STABILITY USING FRACTAL DIMENSIONS
Directory of Open Access Journals (Sweden)
PREDRAG JOVANIĆ
2008-09-01
Full Text Available There are many developed strategies in the emulsion stability evaluation, for purpose of determining the life circle of emulsions. Most of them are based on the reological properties of the emulsions. There are very few which relay on the direct emulsion observations. In this paper we present the developed method for the emulsion stability evaluation by the direct observation of optical properties. As the stability quantification measure we propose the fractal dimension approach. The method is based on the measure of the emulsion transmittance properties, which are directly dependent on the emulsion stability at the moment of measurement. As the test emulsion the oil in the water emulsion was used. The system is classified as the stable emulsion and our intention was to find the moment when the emulsion starts to break. The emulsion transmittance properties were measured using an acquisition system, consisting of a CCD camera and a fast PC configuration equipped with the capturing software. The fractal dimensions were determined by the so called box counting method. The experimental emulsions were measured continuously within the period of 1200 h, from the moment of the emulsion creation. The changes of fractal dimensions were observed which indicates that the emulsion changed its state and therefore the stability during the time. Three regions of the emulsion life circle were divided according to the fractal dimensions measurement, which can be connected with the stable, unstable, and meta-stable states of the emulsion life circle. In the end, the model of the emulsion behavior was developed for the purpose of quantifying the changes in the experimental emulsion.
Scale relativity and fractal space-time: theory and applications
Nottale, Laurent
2008-01-01
In the first part of this contribution, we review the development of the theory of scale relativity and its geometric framework constructed in terms of a fractal and nondifferentiable continuous space-time. This theory leads (i) to a generalization of possible physically relevant fractal laws, written as partial differential equation acting in the space of scales, and (ii) to a new geometric foundation of quantum mechanics and gauge field theories and their possible generalisations. In the second part, we discuss some examples of application of the theory to various sciences, in particular in cases when the theoretical predictions have been validated by new or updated observational and experimental data. This includes predictions in physics and cosmology (value of the QCD coupling and of the cosmological constant), to astrophysics and gravitational structure formation (distances of extrasolar planets to their stars, of Kuiper belt objects, value of solar and solar-like star cycles), to sciences of life (log-p...
Searching for fractal phenomena in multidimensional phase-spaces
Blažek, Mikuláš
2000-07-01
A unified point of view on the fractal analysis in d-dimensional phase-spaces is presented. It is applicable to the data coming from the counting experiments. Explicit expressions are formulated for the fundamental types of factorial moments characterizing the presence of the fractal phenomena, their number being given by (2 d+1 - 1), as well as for a variety of associated statistical moments; special attention is paid to two and three dimensions. In particular, it is found that scaling properties of the modified dispersion moments are directly related with the presence of empty bins in the corresponding distributions. As to the high-energy experiments, those expressions can be applied to the data presently available, e.g. from LEP, as well as to the data arising in the near future from heavy-ion collisions performed at the CERN collider and from the pp collisions observed at the Tevatron, Fermilab.
Fractal pattern formation in metallic ink sessile droplets
Hadj-Achour, Miloud; Brutin, David
2014-11-01
We report a fingering instability that occurs during the spreading and evaporation of a nanosuspension droplet. The patterns has a fractal structure similar to those reported by N. Shahidzadeh-Bonn et al. (2008) for salt crystallisation, during evaporation of saturated Na2SO4 on a hydrophilic surface. The fingering instability has been widely studied for both Newtonian and non-Newtonian fluids. However, we describe for the first time that a fingering instability is observed for the spreading of a nanosuspension sessile droplet. We demonstrate that in certain cases, the contact line evolves through different spreading regimes according to J. De Coninck et al. (2001) with an enhancement in the evaporation rate due the formation of the fractal patterns.
Fractal characterization and optimization of electroless Ni-P coatings
Energy Technology Data Exchange (ETDEWEB)
Sahoo, Prasanta [Department of Mechanical Engineering, Jadavpur University, Kolkata 700032 (India)
2008-01-21
This paper presents an experimental study of fractal characteristics of electroless Ni-P (EN) coatings and optimization of coating process parameters based on the Taguchi method. Experiments are carried out by utilizing the combination of process parameters based on the L{sub 27} Taguchi orthogonal design with three process parameters, namely, bath temperature, concentration of nickel source solution and concentration of reducing agent. It has been observed that the concentration of nickel source solution and the interaction of the bath temperature with the concentration of nickel source solution and reducing agent have a significant influence on controlling fractal dimension characteristics of EN coatings. The surface morphology and composition of coatings are also studied with the help of scanning electron microscopy, energy dispersed x-ray analysis and x-ray diffraction analysis.
Predicting beauty: fractal dimension and visual complexity in art.
Forsythe, A; Nadal, M; Sheehy, N; Cela-Conde, C J; Sawey, M
2011-02-01
Visual complexity has been known to be a significant predictor of preference for artistic works for some time. The first study reported here examines the extent to which perceived visual complexity in art can be successfully predicted using automated measures of complexity. Contrary to previous findings the most successful predictor of visual complexity was Gif compression. The second study examined the extent to which fractal dimension could account for judgments of perceived beauty. The fractal dimension measure accounts for more of the variance in judgments of perceived beauty in visual art than measures of visual complexity alone, particularly for abstract and natural images. Results also suggest that when colour is removed from an artistic image observers are unable to make meaningful judgments as to its beauty.
Fractal Character of China Bedrock Coastline
Institute of Scientific and Technical Information of China (English)
朱晓华
2004-01-01
Fractal theory was applied to a preliminary discussion of the fractal character and formation mechanism of the coastline of the bedrock coast of China on the basis of GIS (Geographical Information System). Some significant conclusions were drawn:(1) The fractal dimensions of the coastline and linear structures of Liaodong Peninsula are 1.0093 and 1.0246 respectively, those of Shandong Peninsula are 1.019 and 1.021 respectively, etc.(2) The fractal dimensions of coastlines of Liaodong Peninsula, Shandong Peninsula, Zhejiang and Fujian-Guangdong tend to increase with the spatial change from north to south.(3)The regional linear structures(including faults)control the basic trends and fractal dimensions of coastlines as a whole in the regions of the bedrock coast of China:the more the controlling effect of linear structures, the smaller the fractal dimensions of coastlines.(4)The substantial constituents of coast and biologic function both play an important role in affecting the fractal dimensions of coastlines of Liaodong Peninsula, Shandong Peninsula, Zhejiang, Fujian-Guangdong and Taiwan Island.
Kinetic properties of fractal stellar media
Chumak, O. V.; Rastorguev, A. S.
2017-01-01
Kinetic processes in fractal stellar media are analysed in terms of the approach developed in our earlier paper involving a generalization of the nearest neighbour and random force distributions to fractal media. Diffusion is investigated in the approximation of scale-dependent conditional density based on an analysis of the solutions of the corresponding Langevin equations. It is shown that kinetic parameters (time-scales, coefficients of dynamic friction, diffusion, etc.) for fractal stellar media can differ significantly both qualitatively and quantitatively from the corresponding parameters for a quasi-uniform random media with limited fluctuations. The most important difference is that in the fractal case, kinetic parameters depend on spatial scalelength and fractal dimension of the medium studied. A generalized kinetic equation for stellar media (fundamental equation of stellar dynamics) is derived in the Fokker-Planck approximation with the allowance for the fractal properties of the spatial stellar density distribution. Also derived are its limit forms that can be used to describe small departures of fractal gravitating medium from equilibrium.
Band structure characteristics of T-square fractal phononic crystals
Institute of Scientific and Technical Information of China (English)
Liu Xiao-Jian; Fan You-Hua
2013-01-01
The T-square fractal two-dimensional phononic crystal model is presented in this article.A comprehensive study is performed for the Bragg scattering and locally resonant fractal phononic crystal.We find that the band structures of the fractal and non-fractal phononic crystals at the same filling ratio are quite different through using the finite element method.The fractal design has an important impact on the band structures of the two-dimensional phononic crystals.
Fractal Dimension Invariant Filtering and Its CNN-based Implementation
Xu, Hongteng; Yan, Junchi; Persson, Nils; Lin, Weiyao; Zha, Hongyuan
2016-01-01
Fractal analysis has been widely used in computer vision, especially in texture image processing and texture analysis. The key concept of fractal-based image model is the fractal dimension, which is invariant to bi-Lipschitz transformation of image, and thus capable of representing intrinsic structural information of image robustly. However, the invariance of fractal dimension generally does not hold after filtering, which limits the application of fractal-based image model. In this paper, we...
Microtopographic Inspection and Fractal Analysis of Skin Neoplasia
Costa, Manuel F. M.; Hipolito, Alberto Valencia; Gutierrez, Gustavo Fidel; Chanona, Jorge; Gallegos, Eva Ramón
2008-04-01
Early detection of skin cancer is fundamental to a successful treatment. Changes in the shape, including the relief, of skin lesions are an indicator of a possible malignity. Optical microtopographic inspection of skin lesions can be used to identify diagnostic patterns of benign and malign skin' lesions. Statistical parameters like the mean roughness (Ra) may allow the discrimination between different types of lesions and degree of malignity. Fractal analysis of bi-dimensional and 3D images of skin lesions can validate or complement that assessment by calculation of its fractal dimensions (FD). On the study herein reported the microtopographic inspection of the skin lesions were performed using the optical triangulation based microtopographer developed at the Physics Department of the University of Minho, MICROTOP.03.MFC. The patients that participated in this research work were men and women older than 15 years with the clinical and histopathology diagnoses of: melanoma, basocellular carcinoma, epidermoide carcinoma, actinic keratosis, keratoacantosis and benign nevus. Latex impressions of the lesions were taken and microtopographically analyzed. Characteristic information for each type of studied lesion was obtained. For melanoma it was observed that on the average these tumors present an increased roughness of around 67 percent compared to the roughness of the healthy skin. This feature allows the distinction from other tumors as basocellular carcinoma (were the roughness increase was in the average of 49 percent) and benign lesions as the epidermoide cyst (37 percent) or the seborrhea keratosis (4 percent). Tumor size and roughness are directly proportional to the grade of malignality. The characterization of the fractal geometry of 2D (histological slides) and 3D images of skin lesions was performed by obtaining its FD evaluated by means of the Box counting method. Results obtained showed that the average fractal dimension of histological slide images (FDh
On the ubiquitous presence of fractals and fractal concepts in pharmaceutical sciences: a review.
Pippa, Natassa; Dokoumetzidis, Aristides; Demetzos, Costas; Macheras, Panos
2013-11-18
Fractals have been very successful in quantifying nature's geometrical complexity, and have captured the imagination of scientific community. The development of fractal dimension and its applications have produced significant results across a wide variety of biomedical applications. This review deals with the application of fractals in pharmaceutical sciences and attempts to account the most important developments in the fields of pharmaceutical technology, especially of advanced Drug Delivery nano Systems and of biopharmaceutics and pharmacokinetics. Additionally, fractal kinetics, which has been applied to enzyme kinetics, drug metabolism and absorption, pharmacokinetics and pharmacodynamics are presented. This review also considers the potential benefits of using fractal analysis along with considerations of nonlinearity, scaling, and chaos as calibration tools to obtain information and more realistic description on different parts of pharmaceutical sciences. As a conclusion, the purpose of the present work is to highlight the presence of fractal geometry in almost all fields of pharmaceutical research.
Institute of Scientific and Technical Information of China (English)
YANG Zhiyuan; ZHOU Anning
2005-01-01
The characteristics of broken surfaces were researched by a scanning electron microscope (SEM) and a reflection microscope, and the fractal dimensions of broken surfaces were measured by the Slit Island method. The experimental results indicate that the broken surface of aluminum electric porcelain is a fractal body in statistics, and the fractal dimensions of broken surfaces are different with the different amplification multiple value.In all of measured fractal dimensions,both of values measured in 100× under reflection microscope and in 500× under SEM are maximum, whereas the values measured in 63× under reflection microscope and in 2000× under SEM are obviously minimum. The fractal dimensions of broken surfaces are also affected by the degrees of gray comparison and the kinds of measuring methods. The relationships between the fractal dimensions of broken surfaces and porcelain bend strengths are that they are in positive correlation on the low multiples and in negative correlation on the high multiples.
Fractal and complexity measures of heart rate variability.
Perkiömäki, Juha S; Mäkikallio, Timo H; Huikuri, Heikki V
2005-01-01
Heart rate variability has been analyzed conventionally with time and frequency domain methods, which measure the overall magnitude of RR interval fluctuations around its mean value or the magnitude of fluctuations in some predetermined frequencies. Analysis of heart rate dynamics by methods based on chaos theory and nonlinear system theory has gained recent interest. This interest is based on observations suggesting that the mechanisms involved in cardiovascular regulation likely interact with each other in a nonlinear way. Furthermore, recent observational studies suggest that some indexes describing nonlinear heart rate dynamics, such as fractal scaling exponents, may provide more powerful prognostic information than the traditional heart rate variability indexes. In particular, the short-term fractal scaling exponent measured by the detrended fluctuation analysis method has predicted fatal cardiovascular events in various populations. Approximate entropy, a nonlinear index of heart rate dynamics, that describes the complexity of RR interval behavior, has provided information on the vulnerability to atrial fibrillation. Many other nonlinear indexes, e.g., Lyapunov exponent and correlation dimensions, also give information on the characteristics of heart rate dynamics, but their clinical utility is not well established. Although concepts of chaos theory, fractal mathematics, and complexity measures of heart rate behavior in relation to cardiovascular physiology or various cardiovascular events are still far away from clinical medicine, they are a fruitful area for future research to expand our knowledge concerning the behavior of cardiovascular oscillations in normal healthy conditions as well as in disease states.
Global Observation of Substorm Growth Phase Processes in the Polar Caps
Brittnacher, M.; OFillingim, M. O.; Chua, D.; Wilber, M.; Parks, G. K.; Germany, G. A.; Spann, J. F.
1998-01-01
Global images of the polar cap region during the substorm growth phase by the Polar Ultraviolet Imager reveals evidence of the processes which are not completely explained by current models. In particular, it was found that size of the polar cap region increases during the growth phase even if the interplanetary magnetic field has no southward component. Three phenomena were observed to produce an increase in the size of the polar cap: (1) motion of the auroral oval to lower latitude, (2) thinning of the auroral oval, and (3) reduction of intense aurora[ precipitation in the polar region. Correlation of image intensities with in situ particle measurements from the FAST satellite are being conducted to study the three growth phase phenomena; and to help identify the source regions of the particles, the mechanisms involved in producing the auroral structures and what may be reducing the polar cap precipitation during the substorm growth phase.
Measurement Based Quantum Computation on Fractal Lattices
Directory of Open Access Journals (Sweden)
Michal Hajdušek
2010-06-01
Full Text Available In this article we extend on work which establishes an analology between one-way quantum computation and thermodynamics to see how the former can be performed on fractal lattices. We find fractals lattices of arbitrary dimension greater than one which do all act as good resources for one-way quantum computation, and sets of fractal lattices with dimension greater than one all of which do not. The difference is put down to other topological factors such as ramification and connectivity. This work adds confidence to the analogy and highlights new features to what we require for universal resources for one-way quantum computation.
An Optical Demonstration of Fractal Geometry
Scannel, Billy; Taylor, Richard
2012-01-01
We have built a Sinai cube to illustrate and investigate the scaling properties that result by iterating chaotic trajectories into a well ordered system. We allow red, green and blue light to reflect off a mirrored sphere, which is contained in an otherwise, closed mirrored cube. The resulting images are modeled by ray tracing procedures and both sets of images undergo fractal analysis. We offer this as a novel demonstration of fractal geometry, utilizing the aesthetic appeal of these images to motivate an intuitive understanding of the resulting scaling plots and associated fractal dimensions.
Fractal image encoding based on adaptive search
Institute of Scientific and Technical Information of China (English)
Kya Berthe; Yang Yang; Huifang Bi
2003-01-01
Finding the optimal algorithm between an efficient encoding process and the rate distortion is the main research in fractal image compression theory. A new method has been proposed based on the optimization of the Least-Square Error and the orthogonal projection. A large number of domain blocks can be eliminated in order to speed-up fractal image compression. Moreover, since the rate-distortion performance of most fractal image coders is not satisfactory, an efficient bit allocation algorithm to improve the rate distortion is also proposed. The implementation and comparison have been done with the feature extraction method to prove the efficiency of the proposed method.
The Dimension of Projections of Fractal Percolations
Rams, Michał; Simon, Károly
2014-02-01
Fractal percolation or Mandelbrot percolation is one of the most well studied families of random fractals. In this paper we study some of the geometric measure theoretical properties (dimension of projections and structure of slices) of these random sets. Although random, the geometry of those sets is quite regular. Our results imply that, denoting by a typical realization of the fractal percolation on the plane, If then for all lines ℓ the orthogonal projection E ℓ of E to ℓ has the same Hausdorff dimension as E,
Robinson, P. A.; Cairns, I. H.; Gurnett, D. A.
1993-01-01
Detailed comparisons are made between the Langmuir-wave properties predicted by the recently developed stochastic-growth theory of type III sources and those observed by the plasma wave experiment on ISEE 3, after correcting for the main instrumental and selection effects. Analysis of the observed field-strength distribution confirms the theoretically predicted form and implies that wave growth fluctuates both spatially and temporally in sign and magnitude, leading to an extremely clumpy distribution of fields. A cutoff in the field-strength distribution is seen at a few mV/m, corresponding to saturation via nonlinear effects. Analysis of the size distribution of Langmuir clumps yields results in accord with those obtained in earlier work and with the size distribution of ambient density fluctuations in the solar wind. This confirms that the inhomogeneities in the Langmuir growth rate are determined by the density fluctuations and that these fluctuations persist during type III events.
An Ensemble Kalman Filter-based data assimilation framework that links a crop growth model with active and passive (AP) microwave models was developed to improve estimates of soil moisture (SM) and vegetation biomass over a growing season of soybean. Complementarities in AP observations were incorpo...
Prediction of Future Observations in Polynomial Growth Curve Models. Part 1.
1983-03-01
UNIT NUMBERS University of Pittsburgh, Ninth Floor, PE6llO2F; 2304/A5 Schenley Hall, Pittsburgh PA 15260 It CONTROLLING OFFICE NAME AND ADDRESS 12...8217. DSIM Enitvd, ’ SR-TR. 8 3 0491 PREDICTION OF FUTURE OBSERVATIONS IN POLYNOMIAL GROWTH CURVE MODELS PART - 1 C. Radhakrishna Rao University of Pittsburgh
Directory of Open Access Journals (Sweden)
D. M. Westervelt
2012-05-01
Full Text Available Aerosol nucleation occurs frequently in the atmosphere and is an important source of particle number. Observations suggest that nucleated particles are capable of growing to sufficiently large sizes that they act as cloud condensation nuclei (CCN, but some global models have reported that CCN concentrations are only modestly sensitive to large changes in nucleation rates. Here we present a novel approach for using long-term size distribution observations to evaluate the contribution of nucleation and growth to the tropospheric CCN budget. We derive from observations at five locations nucleation-relevant metrics such as nucleation rate of particles at diameter of 3 nm (J_{3}, diameter growth rate (GR, particle survival probability (SP, condensation and coagulation sinks, and CCN formation rate. These quantities are also derived for a global microphysical model and compared to the observations on a daily basis to evaluate the model's CCN budget. Using the GEOS-Chem-TOMAS global aerosol model we simulate nucleation events predicted by ternary (with a 10^{−5} tuning factor or activation nucleation over one year and find that the model does not understate the contribution of boundary layer nucleation to CCN concentrations. Model-predicted annual-average formation rates of 50 nm and 100 nm particles due to nucleation are always within 50% and show a slight tendency to over-estimate the observations. Because it is rare for observations to track the growth of a nucleation mode across several days, it is difficult to assess CCN formation when growth requires multiple days. To address multi-day growth, we present three cases of survival of particles beyond one day: single-day growth, partial multi-day survival, and total multi-day survival. For the single-day growth case, only particles that reach a CCN size (50 or 100 nm on the same day are counted as contributing to the CCN budget, which represents a low estimate of CCN
LFN, QPO and fractal dimension of X-ray light curves from black hole binaries
Prosvetov, Art; Grebenev, Sergey
The origin of the low frequency noise (LFN) and quasi-periodic oscillations (QPO) observed in X-ray flux of Galactic black hole binaries is still not recognized in spite of multiple studies and attempts to model this phenomenon. There are known correlations between the QPO frequency, X-ray power density, X-ray flux and spectral state of the system, but there is no model that can do these dependences understandable. For the low frequency (~1 Hz) QPO we still have no even an idea capable to explain their production and don't know even what part of an accretion disc is responsible for them. Here we attempted to measure the fractal dimension of X-ray light curves of several black hole X-ray binaries and to study its correlation with the frequency of quasi periodic oscillations observed in their X-ray light-curves. The fractal dimension is a measure of the space-filling capacity of the light curves' profile. To measure the fractal dimension we used R/S method, which is fast enough and has good reputation in financial analytic and materials sciences. We found that if no QPO were observed in X-ray flux from the particular source, the fractal dimension is equal to the unique value which is independent on the source, its luminosity or its spectral state. On the other hand if QPO were detected in the flux, the fractal dimension deviated from its usual value. Also, we found a clear correlation between the QPO frequency and the fractal dimension of the emission. The relationship between these two parameters is solid but nonlinear. We believe that the analysis of X-ray light curves of black hole binaries using the fractal dimension has a good scientific potential and may provide an addition information on the geometry of accretion flow and fundamental physical parameters of the system.
Directory of Open Access Journals (Sweden)
María Eugenia Torres
2007-01-01
Full Text Available En este trabajo comparamos tres métodos diferentes utilizados para estimar el exponente de Hurst, y analizamos su eficiencia cuando son aplicados a series de datos de diferentes longitudes. Se analizan series temporales de fBm sintetizada pura y con tendencias sinusoidales superpuestas. Mostraremos que los tres métodos aquí discutidos, DFA, basado en wavelets y de variaciones discretas, no sólo son altamente dependientes de la longitud de la señal, sino también del orden o número de los momentos (polinómico, regularidad wavelet o variaciones discretas. Para longitudes de datos suficientemente grandes (superiores a 212, los métodos basados en wavelets y de variaciones discretas mostraron ser menos sesgados y más estables para señales fBm simuladas. Mostraremos que el método de DFA, más utilizado en el ambiente biomédico, es el que proporciona peores estimaciones, arrojando resultados ambiguos cuando son aplicados a señales biológicas de diferentes longitudes o con diferentes parámetros de estimación, sin que pueda considerarse a ninguno de los otros dos como métodos confiables en el momento de desear obtener resultados de relevancia física o fisiológica. Los resultados obtenidos indican que debería procederse con más cautela cuando se trata de obtener conclusiones fisiológicas a partir de estimaciones realizadas a partir de señales reales.
USDA 846-1 fractal melon and derived recombinant inbred lines
The Agricultural Research Service, United States Department of Agriculture announces the release of a melon (Cucumis melo L.) breeding line with highly branched, fractal-type architectural growth habit and 81 derived recombinant inbred lines (RIL). The indeterminate, monoecious USDA 846-1 produces 2...
Structural and Fractal Dimensions are Reliable Determinants of Grain Yield in Soybean
Reliable models are needed to describe plants with complex geometric structures, quantify the impact of management strategies on the plant’s geometric distribution in space and time, and predict yield as a function of fractal dimension. We measured growth and development variables on single soybean ...
Fractal cartography of urban areas.
Encarnação, Sara; Gaudiano, Marcos; Santos, Francisco C; Tenedório, José A; Pacheco, Jorge M
2012-01-01
In a world in which the pace of cities is increasing, prompt access to relevant information is crucial to the understanding and regulation of land use and its evolution in time. In spite of this, characterization and regulation of urban areas remains a complex process, requiring expert human intervention, analysis and judgment. Here we carry out a spatio-temporal fractal analysis of a metropolitan area, based on which we develop a model which generates a cartographic representation and classification of built-up areas, identifying (and even predicting) those areas requiring the most proximate planning and regulation. Furthermore, we show how different types of urban areas identified by the model co-evolve with the city, requiring policy regulation to be flexible and adaptive, acting just in time. The algorithmic implementation of the model is applicable to any built-up area and simple enough to pave the way for the automatic classification of urban areas worldwide.
Energy Technology Data Exchange (ETDEWEB)
Liu Haoliang [State Key Laboratory of Magnetism and Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190 (China); He Wei, E-mail: hewei@aphy.iphy.ac.cn [State Key Laboratory of Magnetism and Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190 (China); Du Haifeng; Wu Qiong; Fang Yapeng [State Key Laboratory of Magnetism and Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190 (China); Zhu Yun [College of Physics and Electronic Information Science, Tianjin Normal University, Tianjin 300387 (China); Cai Jianwang [State Key Laboratory of Magnetism and Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190 (China); Cheng Zhaohua, E-mail: zhcheng@aphy.iphy.ac.cn [State Key Laboratory of Magnetism and Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190 (China)
2011-09-15
We present the experimental results on thermally activated magnetization reversal for [Co{sub 0.9}Fe{sub 0.1}(5.0 A)/Pt(20 A)]{sub 4} multilayer. Direct domain observations show that magnetization reversal is initiated with rare nucleation and followed by dendritic growth of domain walls. Based on macroscopic magnetic parameters from experimental data, the dendritic domain growth mode is qualitatively interpreted by Monte Carlo simulations in terms of a simple uniaxial magnetic anisotropy model. Moreover, both time evolution of domain growth observation and magnetic relaxation measurements reveal that CoFe/Pt multilayer has a relatively large activation volume compared with Co/Pt multilayers. - Highlights: > We investigate magnetization reversal of [Co{sub 0.9}Fe{sub 0.1}(5.0 A)/Pt(20 A)]{sub 4} multilayer. > Magnetization reversal is governed by thermally activated mechanism. > Magnetic domains evolve in dendritic domain growth mode. > Relatively large activation volume is obtained for the multilayer. > Monte Carlo simulation reproduces the domain growth mode well.
Charney, Noah D; Babst, Flurin; Poulter, Benjamin; Record, Sydne; Trouet, Valerie M; Frank, David; Enquist, Brian J; Evans, Margaret E K
2016-09-01
Predicting long-term trends in forest growth requires accurate characterisation of how the relationship between forest productivity and climatic stress varies across climatic regimes. Using a network of over two million tree-ring observations spanning North America and a space-for-time substitution methodology, we forecast climate impacts on future forest growth. We explored differing scenarios of increased water-use efficiency (WUE) due to CO2 -fertilisation, which we simulated as increased effective precipitation. In our forecasts: (1) climate change negatively impacted forest growth rates in the interior west and positively impacted forest growth along the western, southeastern and northeastern coasts; (2) shifting climate sensitivities offset positive effects of warming on high-latitude forests, leaving no evidence for continued 'boreal greening'; and (3) it took a 72% WUE enhancement to compensate for continentally averaged growth declines under RCP 8.5. Our results highlight the importance of locally adapted forest management strategies to handle regional differences in growth responses to climate change. © 2016 John Wiley & Sons Ltd/CNRS.
Directory of Open Access Journals (Sweden)
Lesley A. Judd
2015-07-01
Full Text Available The study, characterization, observation, and quantification of plant root growth and root systems (Rhizometrics has been and remains an important area of research in all disciplines of plant science. In the horticultural industry, a large portion of the crops grown annually are grown in pot culture. Root growth is a critical component in overall plant performance during production in containers, and therefore it is important to understand the factors that influence and/or possible enhance it. Quantifying root growth has varied over the last several decades with each method of quantification changing in its reliability of measurement and variation among the results. Methods such as root drawings, pin boards, rhizotrons, and minirhizotrons initiated the aptitude to measure roots with field crops, and have been expanded to container-grown plants. However, many of the published research methods are monotonous and time-consuming. More recently, computer programs have increased in use as technology advances and measuring characteristics of root growth becomes easier. These programs are instrumental in analyzing various root growth characteristics, from root diameter and length of individual roots to branching angle and topological depth of the root architecture. This review delves into the expanding technologies involved with expertly measuring root growth of plants in containers, and the advantages and disadvantages that remain.
Riemann zeros, prime numbers, and fractal potentials.
van Zyl, Brandon P; Hutchinson, David A W
2003-06-01
Using two distinct inversion techniques, the local one-dimensional potentials for the Riemann zeros and prime number sequence are reconstructed. We establish that both inversion techniques, when applied to the same set of levels, lead to the same fractal potential. This provides numerical evidence that the potential obtained by inversion of a set of energy levels is unique in one dimension. We also investigate the fractal properties of the reconstructed potentials and estimate the fractal dimensions to be D=1.5 for the Riemann zeros and D=1.8 for the prime numbers. This result is somewhat surprising since the nearest-neighbor spacings of the Riemann zeros are known to be chaotically distributed, whereas the primes obey almost Poissonlike statistics. Our findings show that the fractal dimension is dependent on both level statistics and spectral rigidity, Delta(3), of the energy levels.
Characterizations of PSD Fractal of Porous Medium
Institute of Scientific and Technical Information of China (English)
黄国强; 徐世民; 李鑫钢
2003-01-01
A volume-based method for measuring particle-size distribution (PSD) fractal dimensions of porous mediums was developed by employing laser size-analyzing technology. Compared with conventional approaches of using hydrometer or screen to determine PSD, this method can avoid calculation errors and measure smaller size-scale porous medium. In this paper the experimental porous mediums were brown soil, kaolin and sand soil. A micro-order of magnitude (10-5 m) in particle-size interval could be shown in PSD results of brown soil and kaolin. The experiments indicated that brown soil had a nearly mono-fractal PSD character, while kaolin and sand soil showed multi-fractal PSD characters. By the adsorption isotherm experiments, the PSD fractal dimensions of the sand soil were also found to keep a linearly increasing relation with the linear adsorptive parameters of the soils in different intervals to adsorb benzene from aqueous solution.
Fractal measures of female caribou movements
Directory of Open Access Journals (Sweden)
Steven H. Ferguson
2011-03-01
Full Text Available Understanding caribou movement during short-term searches for specific habitats, potential mates, and refugia against predators can help resolve ecological questions on how individual caribou perceive their environment. We used measures of fractal dimension and standardized pathlength to compare the movement pathways of female caribou. Satellite telemetry locations were collected over a 2-year study, March 1994 to mid-May 1996, for a caribou population in central Saskatchewan living in the southern boreal forest. Female caribou displayed more random searching behaviour during winter and more regular dispersal movements during early winter/spring and autumn periods. Females with a calf showed no difference in movement pattern (fractal dimension relative to females without a calf but their standardized path length was shorter. We discuss the advantages of using fractal dimension as a measure of the tortuosity of movement pathways and how changes in fractal dimension over a range of scales can define domains of consistent ecological processes.
Fractal Fluctuations and Statistical Normal Distribution
Selvam, A M
2008-01-01
Dynamical systems in nature exhibit selfsimilar fractal fluctuations and the corresponding power spectra follow inverse power law form signifying long-range space-time correlations identified as self-organized criticality. The physics of self-organized criticality is not yet identified. The Gaussian probability distribution used widely for analysis and description of large data sets underestimates the probabilities of occurrence of extreme events such as stock market crashes, earthquakes, heavy rainfall, etc. The assumptions underlying the normal distribution such as fixed mean and standard deviation, independence of data, are not valid for real world fractal data sets exhibiting a scale-free power law distribution with fat tails. A general systems theory for fractals visualizes the emergence of successively larger scale fluctuations to result from the space-time integration of enclosed smaller scale fluctuations. The model predicts a universal inverse power law incorporating the golden mean for fractal fluct...
Using texture synthesis in fractal pattern design
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
Traditional fractal pattern design has some disadvantages such as inability to effectively reflect the characteristics of real scenery and texture. We propose a novel pattern design technique combining fractal geometry and image texture synthesis to solve these problems. We have improved Wei and Levoy (2000)'s texture synthesis algorithm by first using two-dimensional autocorrelation function to analyze the structure and distribution of textures, and then determining the size of L neighborhood.Several special fractal sets were adopted and HSL (Hue, Saturation, and Light) color space was chosen. The fractal structure was used to manipulate the texture synthesis in HSL color space where the pattern's color can be adjusted conveniently. Experiments showed that patterns with different styles and different color characteristics can be more efficiently generated using the new technique.
Finite element contact analysis of fractal surfaces
Energy Technology Data Exchange (ETDEWEB)
Sahoo, Prasanta; Ghosh, Niloy [Department of Mechanical Engineering, Jadavpur University, Kolkata 700032 (India)
2007-07-21
The present study considers finite element analysis of non-adhesive, frictionless elastic/elastic-plastic contact between a rigid flat plane and a self-affine fractal rough surface using the commercial finite element package ANSYS. Three-dimensional rough surfaces are generated using a modified two-variable Weierstrass-Mandelbrot function with given fractal parameters. Parametric studies are done to consider the general relations between contact properties and key material and surface parameters. The present analysis is validated with available experimental results in the literature. Non-dimensional contact area and displacement are obtained as functions of non-dimensional load for varying fractal surface parameters in the case of elastic contact and for varying rates of strain hardening in the case of elastic-plastic contact of fractal surfaces.
Accelerated expansion in a stochastic self-similar fractal universe
Energy Technology Data Exchange (ETDEWEB)
Santini, Eduardo Sergio [Centro Brasileiro de Pesquisas Fisicas-MCT, Coordenacao de Cosmologia, Relatividade e Astrofisica: ICRA-BR, Rua Dr. Xavier Sigaud 150, Urca 22290-180, Rio de Janeiro, RJ (Brazil) and Comissao Nacional de Energia Nuclear-MCT, Rua General Severiano 90, Botafogo 22290-901, Rio de Janeiro, RJ (Brazil)]. E-mail: santini@cbpf.br; Lemarchand, Guillermo Andres [Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, C.C. 8-Sucursal 25, C1425FFJ Buenos Aires (Argentina)]. E-mail: lemar@correo.uba.ar
2006-05-15
In a recent paper, a cosmological model based on El Naschie E infinity Cantorian space-time was presented [Iovane G. Varying G, accelerating universe, and other relevant consequences of a stochastic self-similar and fractal universe. Chaos, Solitons and Fractals 2004;20:657-67]. In that work, it was claimed that the present accelerated expansion of the universe can be obtained as the effect of a scaling law on Newtonian cosmology with a certain time-dependent gravitational constant (G). In the present work we show that such a cosmological model actually describes a decelerated universe. Then starting from the scenario presented in that paper, we realize a complementary approach based on an extended Friedmann model. In fact, we apply the same scaling law and a time-dependent gravitational constant, that follows from the observational constraints, to relativistic cosmology, i.e. a (extended) Friedmann's model. We are able to show that for a matter-dominated flat universe, with the scaling law and a varying G, an accelerated expansion emerges in such a way that the function luminosity distance vs redshift can be made close to the corresponding function that comes from the usual Friedmann's model supplemented with a cosmological constant, of value {omega} {sub {lambda}} {approx_equal} 0.7. Then the measurements of high redshift supernovae, could be interpreted as a consequence of the fractal self-similarity of the G varying relativistic universe.
Fractal Characterization of Chromatin Decompaction in Live Cells.
Yi, Ji; Stypula-Cyrus, Yolanda; Blaha, Catherine S; Roy, Hemant K; Backman, Vadim
2015-12-01
Chromatin organization has a fundamental impact on the whole spectrum of genomic functions. Quantitative characterization of the chromatin structure, particularly at submicron length scales where chromatin fractal globules are formed, is critical to understanding this structure-function relationship. Such analysis is currently challenging due to the diffraction-limited resolution of conventional light microscopy. We herein present an optical approach termed inverse spectroscopic optical coherence tomography to characterize the mass density fractality of chromatin, and we apply the technique to observe chromatin decompaction in live cells. The technique makes it possible for the first time, to our knowledge, to sense intracellular morphology with length-scale sensitivity from ∼30 to 450 nm, thus primarily probing the higher-order chromatin structure, without resolving the actual structures. We used chromatin decompaction due to inhibition of histone deacytelases and measured the subsequent changes in the fractal dimension of the intracellular structure. The results were confirmed by transmission electron microscopy and confocal fluorescence microscopy.
Lévy processes on a generalized fractal comb
Sandev, Trifce; Iomin, Alexander; Méndez, Vicenç
2016-09-01
Comb geometry, constituted of a backbone and fingers, is one of the most simple paradigm of a two-dimensional structure, where anomalous diffusion can be realized in the framework of Markov processes. However, the intrinsic properties of the structure can destroy this Markovian transport. These effects can be described by the memory and spatial kernels. In particular, the fractal structure of the fingers, which is controlled by the spatial kernel in both the real and the Fourier spaces, leads to the Lévy processes (Lévy flights) and superdiffusion. This generalization of the fractional diffusion is described by the Riesz space fractional derivative. In the framework of this generalized fractal comb model, Lévy processes are considered, and exact solutions for the probability distribution functions are obtained in terms of the Fox H-function for a variety of the memory kernels, and the rate of the superdiffusive spreading is studied by calculating the fractional moments. For a special form of the memory kernels, we also observed a competition between long rests and long jumps. Finally, we considered the fractal structure of the fingers controlled by a Weierstrass function, which leads to the power-law kernel in the Fourier space. This is a special case, when the second moment exists for superdiffusion in this competition between long rests and long jumps.
Directory of Open Access Journals (Sweden)
Tushar Tyagi
2016-11-01
Full Text Available This paper presents solution of multi-objective optimal dispatch (MOOD problem of solar-wind-thermal system by improved stochastic fractal search (ISFSA algorithm. Stochastic fractal search (SFSA is inspired by the phenomenon of natural growth called fractal. It utilizes the concept of creating fractals for conducting a search through the problem domain with the help of two main operations diffusion and updating. To improve the exploration and exploitation capability of SFSA, scale factor is used in place of random operator. The SFSA and proposed ISFSA is implemented and tested on six different multi objective complex test systems of power system. TOPSIS is used here as a decision making tool to find the best compromise solution between the two conflicting objectives. The outcomes of simulation results are also compared with recent reported methods to confirm the superiority and validation of proposed approach.
Institute of Scientific and Technical Information of China (English)
张玮; 梁成浩
2004-01-01
Experiments were performed to study the pitting corrosion morphology of 304 stainless steel exposed to FeCl3 environments and SEM micrographs of the pitting corrosion morphology were obtained. The image processing technique combining with the fractal method was employed to analyze these pitting corrosion images and the self-similarity of pits morphology was observed. It indicates that fractal characteristics exist in pitting corrosion of 304 stainless steel. The self-similarity and complexity of the pitting morphology phenomenon were described in terms of fractal dimension which can also be an important parameter related to characterize pitting morphology qualitatively and quantitatively.
MULTI SEGMENT CIRCULAR FRACTAL REFLECT ARRAY ANTENNA
Directory of Open Access Journals (Sweden)
Bahareh Baghani BAJGIRAN
2014-01-01
Full Text Available in this paper with using novel fractal structure which is composed of multi segment circular fractal. A unit cell and then reflectarray antenna have been designed. The unit cell of reflect array has been designed in 4.4 GHz with 24*24*1 mm3 dimension. The reflectarray is consist of 400 (20* 20 elements that even element is placed in the locus has been calculated. Maximum gain of antenna is 12.9 dBi.
Fractal dimension and architecture of trabecular bone.
Fazzalari, N L; Parkinson, I H
1996-01-01
The fractal dimension of trabecular bone was determined for biopsies from the proximal femur of 25 subjects undergoing hip arthroplasty. The average age was 67.7 years. A binary profile of the trabecular bone in the biopsy was obtained from a digitized image. A program written for the Quantimet 520 performed the fractal analysis. The fractal dimension was calculated for each specimen, using boxes whose sides ranged from 65 to 1000 microns in length. The mean fractal dimension for the 25 subjects was 1.195 +/- 0.064 and shows that in Euclidean terms the surface extent of trabecular bone is indeterminate. The Quantimet 520 was also used to perform bone histomorphometric measurements. These were bone volume/total volume (BV/TV) (per cent) = 11.05 +/- 4.38, bone surface/total volume (BS/TV) (mm2/mm3) = 1.90 +/- 0.51, trabecular thickness (Tb.Th) (mm) = 0.12 +/- 0.03, trabecular spacing (Tb.Sp) (mm) = 1.03 +/- 0.36, and trabecular number (Tb.N) (number/mm) = 0.95 +/- 0.25. Pearsons' correlation coefficients showed a statistically significant relationship between the fractal dimension and all the histomorphometric parameters, with BV/TV (r = 0.85, P fractal dimension shows that trabecular bone exhibits fractal properties over a defined box size, which is within the dimensions of a structural unit for trabecular bone. Therefore, the fractal dimension of trabecular bone provides a measure which does not rely on Euclidean descriptors in order to describe a complex geometry.
Fractal Weyl law for Linux Kernel Architecture
Ermann, L; Shepelyansky, D L
2010-01-01
We study the properties of spectrum and eigenstates of the Google matrix of a directed network formed by the procedure calls in the Linux Kernel. Our results obtained for various versions of the Linux Kernel show that the spectrum is characterized by the fractal Weyl law established recently for systems of quantum chaotic scattering and the Perron-Frobenius operators of dynamical maps. The fractal Weyl exponent is found to be $\
Fractal Basins in the Lorenz Model
Institute of Scientific and Technical Information of China (English)
I.Djellit; J.C.Sprott; M. R. Ferchichi
2011-01-01
@@ The Lorenz mapping is a discretization of a pair of differential equations.It illustrates the pertinence of compu- tational chaos.We describe complex dynamics, bifurcations, and chaos in the map.Fractal basins are displayed by numerical simulation.%The Lorenz mapping is a discretization of a pair of differential equations. It illustrates the pertinence of computational chaos. We describe complex dynamics, bifurcations, and chaos in the map. Fractal basins are displayed by numerical simulation.
Institute of Scientific and Technical Information of China (English)
Liang Shichu; Wang Bosun
2003-01-01
The fractal characteristics of the canopy structure of B. gvmnorrhiza population are investigated by fractal dimension analysis in the National Shankou Mangrove Nature Reserve. The 3-year-old branches have box dimensions between 1.22 and 1.55, showing the complexity degree of branching structure and the ability of occupying and utilizing ecological space. It may be considered that fractal dimension provides a useful index for the study of light utilization efficiencies and growth processes of B. gymnorrhiza. Calculated by using the two-surface method, the fractal dimensions for the crown pattern of individuals with ages of 20 to 50 years range from 2.21 to 2.54, indicating that the filling degree of foliage to a tree crown is relatively low and B. gymnorrhiza has the property of a sun plant.Along with the increase of ages of individuals, the filling degree of foliage to a tree crown changes from high to low, and so does the fractal dimension. The box dimensions obtained from the grayscale curves of population canopy are between 1.47 and 1.61. The greater the box dimension, the more loosely organized the canopy spatial structure, and the more the light spots. The canopy structural information and complexity of a population can be effectively captured by box dimensions obtained from canopy grayscale curves.
Sui, Jize; Zhao, Peng; Bin-Mohsin, Bandar; Zheng, Liancun; Zhang, Xinxin; Cheng, Zhengdong; Chen, Ying; Chen, Goong
2016-12-01
Nano-suspensions (NS) exhibit unusual thermophysical behaviors once interparticle aggregations and the shear flows are imposed, which occur ubiquitously in applications but remain poorly understood, because existing theories have not paid these attentions but focused mainly on stationary NS. Here we report the critical role of time-dependent fractal aggregation in the unsteady thermal convection of NS systematically. Interestingly, a time ratio λ = tp/tm (tp is the aggregate characteristic time, tm the mean convection time) is introduced to characterize the slow and fast aggregations, which affect distinctly the thermal convection process over time. The increase of fractal dimension reduces both momentum and thermal boundary layers, meanwhile extends the time duration for the full development of thermal convection. We find a nonlinear growth relation of the momentum layer, but a linear one of the thermal layer, with the increase of primary volume fraction of nanoparticles for different fractal dimensions. We present two global fractal scaling formulas to describe these two distinct relations properly, respectively. Our theories and methods in this study provide new evidence for understanding shear-flow and anomalous heat transfer of NS associated non-equilibrium aggregation processes by fractal laws, moreover, applications in modern micro-flow technology in nanodevices.
Modelo fractal de substâncias húmicas Fractal model of humic substances
Directory of Open Access Journals (Sweden)
Alessandro Costa da Silva
2001-10-01
Full Text Available A teoria fractal, por meio da determinação da dimensão fractal (D, tem sido considerada como uma alternativa para explicar a conforma��ão de agregados moleculares. Sua utilização no estudo de substâncias húmicas (SH reside na tentativa de descrever (representar a estrutura ramificada ou a superfície rugosa e distorcida destas substâncias. A presença de um modelo fractal por sistemas naturais implica que este pode ser decomposto em partes, em que cada uma, subseqüentemente, é cópia do todo. Do ponto de vista experimental, a dimensão fractal de sistemas húmicos pode ser determinada a partir de técnicas como turbidimetria, raios x, espalhamento de neutrons, dentre outras. Este trabalho pretende facilitar o entendimento sobre a aplicação de fractais ao estudo conformacional de SH, introduzindo conceitos e informações sobre o fundamento dos modelos fractais, bem como sobre o uso da técnica turbidimétrica na determinação do valor D.Fractal theoria application by determination of fractal dimension has been considered an alternative tool to explain the conformation of molecular aggregates. Its utilization in the study of humic substances (HS aims the attempt to describe the limbed structure or the rugous and distorced surface of these substances. The presence of fractal models indicates that the system may be decomposed in parts, each part being a copy of the whole. In the experimental point of view the fractals models of natural systems may be measured through techniques as turbidimetry, x- ray and neutrons scattering. This paper seeks to facilitate the understanding on the application of the fractals in the conformational study of HS, supply information about fractal models foundation and use of the turbidimetry in the determination of fractal dimension.
Fractals and Spatial Statistics of Point Patterns
Institute of Scientific and Technical Information of China (English)
Frederik P Agterberg
2013-01-01
The relationship between fractal point pattern modeling and statistical methods of parameter estimation in point-process modeling is reviewed.Statistical estimation of the cluster fractal dimension by using Ripley's K-function has advantages in comparison with the more commonly used methods of box-counting and cluster fractal dimension estimation because it corrects for edge effects,not only for rectangular study areas but also for study areas with curved boundaries determined by regional geology.Application of box-counting to estimate the fractal dimension of point patterns has the disadvantage that,in general,it is subject to relatively strong "roll-off" effects for smaller boxes.Point patterns used for example in this paper are mainly for gold deposits in the Abitibi volcanic belt on the Canadian Shield.Additionally,it is proposed that,worldwide,the local point patterns of podiform Cr,volcanogenic massive sulphide and porphyry copper deposits,which are spatially distributed within irregularly shaped favorable tracts,satisfy the fractal clustering model with similar fractal dimensions.The problem of deposit size (metal tonnage) is also considered.Several examples are provided of cases in which the Pareto distribution provides good results for the largest deposits in metal size-frequency distribution modeling.
Fractal parameters and vascular networks: facts & artifacts
Directory of Open Access Journals (Sweden)
Maniero Fabrizio
2008-07-01
Full Text Available Abstract Background Several fractal and non-fractal parameters have been considered for the quantitative assessment of the vascular architecture, using a variety of test specimens and of computational tools. The fractal parameters have the advantage of being scale invariant, i.e. to be independent of the magnification and resolution of the images to be investigated, making easier the comparison among different setups and experiments. Results The success of several commercial and/or free codes in computing the fractal parameters has been tested on well known exact models. Based on such a preliminary study, we selected the code Frac-lac in order to analyze images obtained by visualizing the angiogenetic process occurring in chick Chorio Allontoic Membranes (CAM, assumed to be paradigmatic of a realistic 2D vascular network. Among the parameters investigated, the fractal dimension Df proved to be the most robust estimator for CAM vascular networks. Moreover, only Df was able to discriminate between effective and elusive increases in vascularization after drug-induced angiogenic stimulations on CAMs. Conclusion The fractal dimension Df is likely to be the most promising tool for monitoring the effectiveness of anti-angiogenic therapies in various clinical contexts.
Fractal phenomena in powder injection molding process
Institute of Scientific and Technical Information of China (English)
郑洲顺; 曲选辉; 李云平; 雷长明; 段柏华
2003-01-01
The complicated characteristics of the powder were studied by fractal theory. It is illustrated that powder shape, binder structure, feedstock and mold-filling flow in powder injection molding process possess obvious fractal characteristics. Based on the result of SEM, the fractal dimensions of the projected boundary of carbonylic iron and carbonylic nickel particles were determined to be 1.074±0.006 and 1.230±0.005 respectively by box counting measurement. The results show that the fractal dimension of the projected boundary of carbonylic iron particles is close to smooth curve of one-dimension, while the fractal dimension of the projected boundary of carbonylic nickel particle is close to that of trisection Koch curve, indicating that the shape characteristics of carbonylic nickel particles can be described and analyzed by the characteristics of trisection Koch curve. It is also proposed that the fractal theory can be applied in the research of powder injection molding in four aspects.
Fractal Structure in Galactic Star Fields
Elmegreen, B G; Elmegreen, Bruce G.; Elmegreen, Debra Meloy
2001-01-01
The fractal structure of star formation on large scales in disk galaxies is studied using the size distribution function of stellar aggregates in kpc-scale star fields. Achival HST images of 10 galaxies are Gaussian smoothed to define the aggregates, and a count of these aggregates versus smoothing scale gives the fractal dimension. Fractal and Poisson models confirm the procedure. The fractal dimension of star formation in all of the galaxies is ~2.3. This is the same as the fractal dimension of interstellar gas in the Milky Way and nearby galaxies, suggesting that star formation is a passive tracer of gas structure defined by self-gravity and turbulence. Dense clusters like the Pleiades form at the bottom of the hierarchy of structures, where the protostellar gas is densest. If most stars form in such clusters, then the fractal arises from the spatial distribution of their positions, giving dispersed star fields from continuous cluster disruption. Dense clusters should have an upper mass limit that increase...
An Active Region Model for Capturing Fractal Flow Patterns inUnsaturated Soils: Model Development
Energy Technology Data Exchange (ETDEWEB)
Liu, Hui-Hai; Zhang, R.; Bodvarsson, Gudmundur S.
2005-06-11
Preferential flow commonly observed in unsaturated soils allows rapid movement of solute from the soil surface or vadose zone to the groundwater, bypassing a significant volume of unsaturated soil and increasing the risk of groundwater contamination. A variety of evidence indicates that complex preferential patterns observed from fields are fractals. In this study, we developed a relatively simple active region model to incorporate the fractal flow pattern into the continuum approach. In the model, the flow domain is divided into active and inactive regions. Flow occurs preferentially in the active region (characterized by fractals), and inactive region is simply bypassed. A new constitutive relationship (the portion of the active region as a function of saturation) was derived. The validity of the proposed model is demonstrated by the consistency between field observations and the new constitutive relationship.
In situ observations of austenite grain growth in Fe-C-Mn-Si super bainitic steel
Institute of Scientific and Technical Information of China (English)
Feng Liu; Guang Xu; Yu-long Zhang; Hai-jiang Hu; Lin-xin Zhou; Zheng-liang Xue
2013-01-01
In situ observations of austenite grain growth in Fe-C-Mn-Si super bainitic steel were conducted on a high-temperature laser scanning confocal microscope during continuous heating and subsequent isothermal holding at 850, 1000, and 1100◦C for 30 min. A grain growth model was proposed based on experimental results. It is indicated that the austenite grain size increases with austenitizing temperature and holding time. When the austenitizing temperature is above 1100◦C, the austenite grains grow rapidly, and abnormal austenite grains occur. In addition, the eff ect of heating rate on austenite grain growth was investigated, and the relation between austenite grains and bainite morphology after bainitic transformations was also discussed.
Indian Academy of Sciences (India)
Kandhasamy Durai Murugan; Arlin Jose Amali; Paramasivam Natarajan
2012-03-01
Interpolymer adducts of poly(methacrylic acid), (PMAA), with poly(vinylpyrrolidone) in presence of sodium chloride or potassium chloride form highly ordered fractal patterns in films on glass surface on drying at ambient temperature. The structure, morphology and the conditions under which the formation of fractal patterns occurs were investigated by SEM, EDX and confocal microscopic techniques. Self-organization of PMAA with complementary polymers such as poly(vinylpyrrolidone) is well-known and in the presence of sodium chloride formation of the fractals in films of the adducts is a novel observation. Fractal formation occurs due to the aggregation of interpolymer adducts. The composition of the fractals in the film is studied by EDX and confocal microscopic images of the fluorophores covalently bound to PMAA. In presence of salts, sodium chloride or potassium chloride, micellar like entities of 80 nm size were formed which further aggregate to form fractal patterns. It is suggested that the fractals result from the interpolymer adduct by Diffusion Limited Aggregation mechanism.
Image compression with a hybrid wavelet-fractal coder.
Li, J; Kuo, C J
1999-01-01
A hybrid wavelet-fractal coder (WFC) for image compression is proposed. The WFC uses the fractal contractive mapping to predict the wavelet coefficients of the higher resolution from those of the lower resolution and then encode the prediction residue with a bitplane wavelet coder. The fractal prediction is adaptively applied only to regions where the rate saving offered by fractal prediction justifies its overhead. A rate-distortion criterion is derived to evaluate the fractal rate saving and used to select the optimal fractal parameter set for WFC. The superior performance of the WFC is demonstrated with extensive experimental results.
The utility of fractal analysis in clinical neuroscience.
John, Ann M; Elfanagely, Omar; Ayala, Carlos A; Cohen, Michael; Prestigiacomo, Charles J
2015-01-01
Physicians and scientists can use fractal analysis as a tool to objectively quantify complex patterns found in neuroscience and neurology. Fractal analysis has the potential to allow physicians to make predictions about clinical outcomes, categorize pathological states, and eventually generate diagnoses. In this review, we categorize and analyze the applications of fractal theory in neuroscience found in the literature. We discuss how fractals are applied and what evidence exists for fractal analysis in neurodegeneration, neoplasm, neurodevelopment, neurophysiology, epilepsy, neuropharmacology, and cell morphology. The goal of this review is to introduce the medical community to the utility of applying fractal theory in clinical neuroscience.
Unveiling the Multi-fractal Structure of Complex Networks
Jalan, Sarika; Sarkar, Camellia; Boccaletti, Stefano
2016-01-01
The fractal nature of graphs has traditionally been investigated by using the nodes of networks as the basic units. Here, instead, we propose to concentrate on the graph edges, and introduce a practical and computationally not demanding method for revealing changes in the fractal behavior of networks, and particularly for allowing distinction between mono-fractal, quasi mono-fractal, and multi-fractal structures. We show that degree homogeneity plays a crucial role in determining the fractal nature of the underlying network, and report on six different protein-protein interaction networks along with their corresponding random networks. Our analysis allows to identify varying levels of complexity in the species.
Shrestha, Bhushan; Lee, Won-Ho; Han, Sang-Kuk
2006-01-01
Characteristic growth patterns of Cordyceps militaris isolates on various media, under varying light conditions and at varying incubation periods were examined. Light was found to be the most critical single factor in determining the density, texture, and pigmentation of the mycelial culture of the fungus. However, under the light condition, the degree of pigmentation and mycelial density were found to be affected by the incubation period and type of medium. Irrespective of the variations in medium type or incubation period, there was no pigmentation of the mycelium under dark condition. Radial growth of the mycelium was faster under dark incubation rather than under light incubation. Abundant mycelial density and darkest pigmentation of C. militaris isolates were produced in nutritionally rich media like SDAY, SMAY and CZYA, suggesting that these media may fulfill all the requirements for vegetative growth of the fungus. Growth characteristics of C. militaris isolates could be easily observed by the simple agar culture method, which would be useful to characterize the phenotypic characteristics of large number of pure cultures of the fungus under given conditions of growth factors such as medium, light and temperature. PMID:24039476
Fractal Dimension in Epileptic EEG Signal Analysis
Uthayakumar, R.
Fractal Analysis is the well developed theory in the data analysis of non-linear time series. Especially Fractal Dimension is a powerful mathematical tool for modeling many physical and biological time signals with high complexity and irregularity. Fractal dimension is a suitable tool for analyzing the nonlinear behaviour and state of the many chaotic systems. Particularly in analysis of chaotic time series such as electroencephalograms (EEG), this feature has been used to identify and distinguish specific states of physiological function.Epilepsy is the main fatal neurological disorder in our brain, which is analyzed by the biomedical signal called Electroencephalogram (EEG). The detection of Epileptic seizures in the EEG Signals is an important tool in the diagnosis of epilepsy. So we made an attempt to analyze the EEG in depth for knowing the mystery of human consciousness. EEG has more fluctuations recorded from the human brain due to the spontaneous electrical activity. Hence EEG Signals are represented as Fractal Time Series.The algorithms of fractal dimension methods have weak ability to the estimation of complexity in the irregular graphs. Divider method is widely used to obtain the fractal dimension of curves embedded into a 2-dimensional space. The major problem is choosing initial and final step length of dividers. We propose a new algorithm based on the size measure relationship (SMR) method, quantifying the dimensional behaviour of irregular rectifiable graphs with minimum time complexity. The evidence for the suitability (equality with the nature of dimension) of the algorithm is illustrated graphically.We would like to demonstrate the criterion for the selection of dividers (minimum and maximum value) in the calculation of fractal dimension of the irregular curves with minimum time complexity. For that we design a new method of computing fractal dimension (FD) of biomedical waveforms. Compared to Higuchi's algorithm, advantages of this method include
Directory of Open Access Journals (Sweden)
Renkuan Liao
2014-01-01
Full Text Available The water absorption capacity of superabsorbent polymers (SAPs is important for agricultural drought resistance. However, herbicides may leach into the soil and affect water absorption by damaging the SAP three-dimensional membrane structures. We used 100-mesh sieves, electron microscopy, and fractal theory to study swelling and water absorption in SAPs in the presence of three common herbicides (atrazine, alachlor, and tribenuron-methyl at concentrations of 0.5, 1.0, and 2.0 mg/L. In the sieve experiments it was found that 2.0 mg/L atrazine reduces the capacity by 9.64–23.3% at different swelling points; no significant diminution was observed for the other herbicides or for lower atrazine concentrations. We found that the hydrogel membrane pore distributions have fractal characteristics in both deionized water and atrazine solution. The 2.0 mg/L atrazine destroyed the water-retaining polymer membrane pores and reduced the water-absorbing mass by modifying its three-dimensional membrane structure. A linear correlation was observed between the fractal analysis and the water-absorbing mass. Multifractal analysis characterized the membrane pore distribution by using the range of singularity indexes Δα (relative distinguishing range of 16.54–23.44%, which is superior to single-fractal analysis that uses the fractal dimension D (relative distinguishing range of 2.5–4.0%.
Directory of Open Access Journals (Sweden)
APPELBAUM S.
1998-04-01
(x = 0.167 ± 0.006 g. After 45 days of rearing, the weight differences between the species became insignificant. Except for the different transportation mortalities on arrival, the European and the American eels demonstrated similar mortality rates during the experiment ( 1 2 . 5 % , food conversion rate (2.3 and specific growth rate (0.90 and 0.99 respectively. No parasites or diseases were detected. No differences were observed in general behaviour and feeding between the two species.
Di Ieva, Antonio; Grizzi, Fabio; Tschabitscher, Manfred; Colombo, Piergiuseppe; Casali, Massimiliano; Simonelli, Matteo; Widhalm, Georg; Muzzio, Pier Carlo; Matula, Christian; Chiti, Arturo; Rodriguez y Baena, Riccardo
2010-09-01
Neuroradiological and metabolic imaging is a fundamental diagnostic procedure in the assessment of patients with primary and metastatic brain tumors. The correlation between objective parameters capable of quantifying the neoplastic angioarchitecture and imaging data may improve our understanding of the underlying physiopathology and make it possible to evaluate treatment efficacy in brain tumors. Only a few studies have so far correlated the quantitative parameters measuring the neovascularity of brain tumors with the metabolic profiles measured by means of amino acid uptake in positron emission tomography (PET) scans. Fractal geometry offers new mathematical tools for the description and quantification of complex anatomical systems, including microvascularity. In this study, we evaluated the microvascular network complexity of six cases of human glioblastoma multiforme quantifying the surface fractal dimension on CD34 immunostained specimens. The microvascular fractal dimension was estimated by applying the box-counting algorithm. As the fractal dimension depends on the density, size and shape of the vessels, and their distribution pattern, we defined it as an index of the whole complexity of microvascular architecture and compared it with the uptake of (11)C-methionine (MET) assessed by PET. The different fractal dimension values observed showed that the same histological category of brain tumor had different microvascular network architectures. Fractal dimension ranged between 1.19 and 1.77 (mean: 1.415+/-0.225), and the uptake of (11)C-methionine ranged between 1.30 and 5.30. A statistically significant direct correlation between the microvascular fractal dimension and the uptake of (11)C-methionine (p=0.02) was found. Our preliminary findings indicate that that vascularity (estimated on the histologic specimens by means of the fractal dimension) and (11)C-methionine uptake (assessed by PET) closely correlate in glioblastoma multiforme and that microvascular
Fractal gait patterns are retained after entrainment to a fractal stimulus.
Directory of Open Access Journals (Sweden)
Christopher K Rhea
Full Text Available Previous work has shown that fractal patterns in gait can be altered by entraining to a fractal stimulus. However, little is understood about how long those patterns are retained or which factors may influence stronger entrainment or retention. In experiment one, participants walked on a treadmill for 45 continuous minutes, which was separated into three phases. The first 15 minutes (pre-synchronization phase consisted of walking without a fractal stimulus, the second 15 minutes consisted of walking while entraining to a fractal visual stimulus (synchronization phase, and the last 15 minutes (post-synchronization phase consisted of walking without the stimulus to determine if the patterns adopted from the stimulus were retained. Fractal gait patterns were strengthened during the synchronization phase and were retained in the post-synchronization phase. In experiment two, similar methods were used to compare a continuous fractal stimulus to a discrete fractal stimulus to determine which stimulus type led to more persistent fractal gait patterns in the synchronization and post-synchronization (i.e., retention phases. Both stimulus types led to equally persistent patterns in the synchronization phase, but only the discrete fractal stimulus led to retention of the patterns. The results add to the growing body of literature showing that fractal gait patterns can be manipulated in a predictable manner. Further, our results add to the literature by showing that the newly adopted gait patterns are retained for up to 15 minutes after entrainment and showed that a discrete visual stimulus is a better method to influence retention.
The fractal method of the lunar surface parameters analysis
Nefedev, Yuri; Demina, Natalia; Petrova, Natalia; Demin, Sergey; Andreev, Alexey
2016-10-01
Analysis of complex selenographic systems is a complicated issue. This fully applies to the lunar topography. In this report a new method of the comparative reliable estimation of the lunar maps data is represented. The estimation was made by the comparison of high-altitude lines using the fractal analysis. The influence of the lunar macrofigure variances were determined by the method of fractal dimensions comparison.By now the highly accurate theories of the lunar movement have been obtained and stars coordinates have been determined on the basis of space measurements with the several mas accuracy but there are factors highly influencingon the accuracy of the results of these observations. They are: exactitude of the occultation moment recording, errors of the stars coordinates, accuracy of lunar ephemeris positions and unreliability of lunar marginal zone maps. Existing charts of the lunar marginal zone have some defects. To resolve this task thecomparison method in which the structure of the high-altitude lines of data appropriated with identical lunar coordinates can use. However, such comparison requires a lot of calculations.In order to find the variations of irregularities for the limb points above the mean level of lunar surface were computed the position angles of this points P and D by Hayn' coordinates. Thus the data of our studies was obtained by identical types.Then the first, segments of a lunar marginal zone for every 45" on P were considered. For each segment profile of the surface for a constant D were constructed with a step of 2". Thus 80 profiles were obtained. Secondly the fractal dimensions d for each considered structure was defined. Third the obtained values d were compared with the others maps considered in this work.The obtained results show some well agreement between the mean fractal dimensions for maps. Thus it can be concluded that the using of fractal method for lunar maps analysis to determine the accuracy of the presented to
2013-01-01
Changes in the concentration profiles of β-carotene caused by diffusion through parenchymatic dried apple tissue were characterized by image and fractal analysis. Apple slices were dried by convection, and then impregnated with an aqueous β-carotene solution. Scanning electron microscopy images of dried apple slices were captured and the fractal dimension (FD) values of the textures of the images were obtained (FDSEM). It was observed that the microstructure of the foodstuff being impregnated...
Birnstiel, T; Trotta, F; Dullemond, C P; Natta, A; Testi, L; Dominik, C; Henning, T; Ormel, C W; Zsom, A
2010-01-01
Context. Observations at sub-millimeter and mm wavelengths will in the near future be able to resolve the radial dependence of the mm spectral slope in circumstellar disks with a resolution of around a few AU at the distance of the closest star-forming regions. Aims. We aim to constrain physical models of grain growth and fragmentation by a large sample of (sub-)mm observations of disks around pre-main sequence stars in the Taurus-Auriga and Ophiuchus star-forming regions. Methods. State-of-the-art coagulation/fragmentation and disk-structure codes are coupled to produce steady-state grain size distributions and to predict the spectral slopes at (sub-)mm wavelengths. Results. This work presents the first calculations predicting the mm spectral slope based on a physical model of grain growth. Our models can quite naturally reproduce the observed mm-slopes, but a simultaneous match to the observed range of flux levels can only be reached by a reduction of the dust mass by a factor of a few up to about 30 while ...
Fractal modeling of natural fracture networks
Energy Technology Data Exchange (ETDEWEB)
Ferer, M.; Dean, B.; Mick, C.
1995-06-01
West Virginia University will implement procedures for a fractal analysis of fractures in reservoirs. This procedure will be applied to fracture networks in outcrops and to fractures intersecting horizontal boreholes. The parameters resulting from this analysis will be used to generate synthetic fracture networks with the same fractal characteristics as the real networks. Recovery from naturally fractured, tight-gas reservoirs is controlled by the fracture network. Reliable characterization of the actual fracture network in the reservoir is severely limited. The location and orientation of fractures intersecting the borehole can be determined, but the length of these fractures cannot be unambiguously determined. Because of the lack of detailed information about the actual fracture network, modeling methods must represent the porosity and permeability associated with the fracture network, as accurately as possible with very little a priori information. In the sections following, the authors will (1) present fractal analysis of the MWX site, using the box-counting procedure; (2) review evidence testing the fractal nature of fracture distributions and discuss the advantages of using the fractal analysis over a stochastic analysis; and (3) present an efficient algorithm for producing a self-similar fracture networks which mimic the real MWX outcrop fracture network.
Fractal analysis of cervical intraepithelial neoplasia.
Directory of Open Access Journals (Sweden)
Markus Fabrizii
Full Text Available INTRODUCTION: Cervical intraepithelial neoplasias (CIN represent precursor lesions of cervical cancer. These neoplastic lesions are traditionally subdivided into three categories CIN 1, CIN 2, and CIN 3, using microscopical criteria. The relation between grades of cervical intraepithelial neoplasia (CIN and its fractal dimension was investigated to establish a basis for an objective diagnosis using the method proposed. METHODS: Classical evaluation of the tissue samples was performed by an experienced gynecologic pathologist. Tissue samples were scanned and saved as digital images using Aperio scanner and software. After image segmentation the box counting method as well as multifractal methods were applied to determine the relation between fractal dimension and grades of CIN. A total of 46 images were used to compare the pathologist's neoplasia grades with the predicted groups obtained by fractal methods. RESULTS: Significant or highly significant differences between all grades of CIN could be found. The confusion matrix, comparing between pathologist's grading and predicted group by fractal methods showed a match of 87.1%. Multifractal spectra were able to differentiate between normal epithelium and low grade as well as high grade neoplasia. CONCLUSION: Fractal dimension can be considered to be an objective parameter to grade cervical intraepithelial neoplasia.
Multirate diversity strategy of fractal modulation
Institute of Scientific and Technical Information of China (English)
Yuan Yong; Shi Si-Hong; Luo Mao-Kang
2011-01-01
Previous analyses of fractal modulation were carried out mostly from a signle perspective or a subband, but the analyses from the perspective of multiscale synthesis have not been found yet;while multiscale synthesis is just the essence of the mutlirate diversity which is the most important characteristic of fractal modulation. As for the mutlirate diversity of fractal modulation, previous studies only dealt with the general outspread of its concept, lacked the thorough and intensive quantitative comparison and analysis.In light of the above fact, from the perspective of multiscale synthesis, in this paper we provide a comprehensive analysis of the multirate diversity of fractal modulation and corresponding quantitative analysis. The results show that mutlirate diversity, which is a fusion of frequency diversity and time diversity, pays an acceptable price in spectral efficiency in exchange for a significant improvement in bit error rate. It makes fractal modulation particularly suitable for the channels whose bandwidth and duration parameters are unknown or cannot be predicted to the transmitter. Surely it is clearly of great significance for reliable communications.Moreover, we also attain the ability to flexibly make various rate-bandwidth tradeoffs between the transmitter and the receiver, to freely select the reception time and to expediently control the total bandwidth. Furthermore, the acquisitions or improvements of these fine features could provide support of the technical feasibility for the electromagnetic spectrum control technology in a complex electromagnetic environment.
Rheological and fractal hydrodynamics of aerobic granules.
Tijani, H I; Abdullah, N; Yuzir, A; Ujang, Zaini
2015-06-01
The structural and hydrodynamic features for granules were characterized using settling experiments, predefined mathematical simulations and ImageJ-particle analyses. This study describes the rheological characterization of these biologically immobilized aggregates under non-Newtonian flows. The second order dimensional analysis defined as D2=1.795 for native clusters and D2=1.099 for dewatered clusters and a characteristic three-dimensional fractal dimension of 2.46 depicts that these relatively porous and differentially permeable fractals had a structural configuration in close proximity with that described for a compact sphere formed via cluster-cluster aggregation. The three-dimensional fractal dimension calculated via settling-fractal correlation, U∝l(D) to characterize immobilized granules validates the quantitative measurements used for describing its structural integrity and aggregate complexity. These results suggest that scaling relationships based on fractal geometry are vital for quantifying the effects of different laminar conditions on the aggregates' morphology and characteristics such as density, porosity, and projected surface area.
From dendrimers to fractal polymers and beyond
Directory of Open Access Journals (Sweden)
Charles N. Moorefield
2013-01-01
Full Text Available The advent of dendritic chemistry has facilitated materials research by allowing precise control of functional component placement in macromolecular architecture. The iterative synthetic protocols used for dendrimer construction were developed based on the desire to craft highly branched, high molecular weight, molecules with exact mass and tailored functionality. Arborols, inspired by trees and precursors of the utilitarian macromolecules known as dendrimers today, were the first examples to employ predesigned, 1 → 3 C-branched, building blocks; physical characteristics of the arborols, including their globular shapes, excellent solubilities, and demonstrated aggregation, combined to reveal the inherent supramolecular potential (e.g., the unimolecular micelle of these unique species. The architecture that is a characteristic of dendritic materials also exhibits fractal qualities based on self-similar, repetitive, branched frameworks. Thus, the fractal design and supramolecular aspects of these constructs are suggestive of a larger field of fractal materials that incorporates repeating geometries and are derived by complementary building block recognition and assembly. Use of terpyridine-M2+-terpyridine (where, M = Ru, Zn, Fe, etc connectivity in concert with mathematical algorithms, such as forms the basis for the Seirpinski gasket, has allowed the beginning exploration of fractal materials construction. The propensity of the fractal molecules to self-assemble into higher order architectures adds another dimension to this new arena of materials and composite construction.
Multirate diversity strategy of fractal modulation
Yuan, Yong; Shi, Si-Hong; Luo, Mao-Kang
2011-04-01
Previous analyses of fractal modulation were carried out mostly from a signle perspective or a subband, but the analyses from the perspective of multiscale synthesis have not been found yet; while multiscale synthesis is just the essence of the mutlirate diversity which is the most important characteristic of fractal modulation. As for the mutlirate diversity of fractal modulation, previous studies only dealt with the general outspread of its concept, lacked the thorough and intensive quantitative comparison and analysis. In light of the above fact, from the perspective of multiscale synthesis, in this paper we provide a comprehensive analysis of the multirate diversity of fractal modulation and corresponding quantitative analysis. The results show that mutlirate diversity, which is a fusion of frequency diversity and time diversity, pays an acceptable price in spectral efficiency in exchange for a significant improvement in bit error rate. It makes fractal modulation particularly suitable for the channels whose bandwidth and duration parameters are unknown or cannot be predicted to the transmitter. Surely it is clearly of great significance for reliable communications. Moreover, we also attain the ability to flexibly make various rate-bandwidth tradeoffs between the transmitter and the receiver, to freely select the reception time and to expediently control the total bandwidth. Furthermore, the acquisitions or improvements of these fine features could provide support of the technical feasibility for the electromagnetic spectrum control technology in a complex electromagnetic environment.
Changes in fractal dimension and lacunarity as early markers of UV-induced apoptosis.
Pantic, Igor; Harhaji-Trajkovic, Ljubica; Pantovic, Aleksandar; Milosevic, Nebojsa T; Trajkovic, Vladimir
2012-06-21
The aim of our study was to employ fractal analysis for evaluation of ultrastructural changes during early stages of apoptosis. Apoptosis was induced in U251 human glioma cell line by exposure to UVB light. The cells were visualized by optical phase-contrast microscopy and photographed before the UV treatment, immediately after the treatment, as well as at 30 min intervals during 5h observation period. For each of the 32 cells analyzed, cellular and nuclear fractal dimension, as well as nuclear lacunarity, were determined at each time point. Our data demonstrate that cellular ultrastructural complexity determined by fractal dimension and lacunarity significantly decreases after the UV irradiation, with the nuclear lacunarity being a particularly sensitive parameter in detecting early apoptosis. Importantly, fractal analysis was able to detect cellular apoptotic changes earlier than conventional flow cytometric analysis of phosphatidylserine exposure, DNA fragmentation and cell membrane permeabilization. These results indicate that fractal analysis might be a powerful and affordable method for non-invasive early identification of apoptosis in cell cultures. Copyright © 2012 Elsevier Ltd. All rights reserved.
Fractals in the Neurosciences, Part I: General Principles and Basic Neurosciences.
Di Ieva, Antonio; Grizzi, Fabio; Jelinek, Herbert; Pellionisz, Andras J; Losa, Gabriele Angelo
2014-08-01
The natural complexity of the brain, its hierarchical structure, and the sophisticated topological architecture of the neurons organized in micronetworks and macronetworks are all factors contributing to the limits of the application of Euclidean geometry and linear dynamics to the neurosciences. The introduction of fractal geometry for the quantitative analysis and description of the geometric complexity of natural systems has been a major paradigm shift in the last decades. Nowadays, modern neurosciences admit the prevalence of fractal properties such as self-similarity in the brain at various levels of observation, from the microscale to the macroscale, in molecular, anatomic, functional, and pathological perspectives. Fractal geometry is a mathematical model that offers a universal language for the quantitative description of neurons and glial cells as well as the brain as a whole, with its complex three-dimensional structure, in all its physiopathological spectrums. For a holistic view of fractal geometry of the brain, we review here the basic concepts of fractal analysis and its main applications to the basic neurosciences.
Inversion problem for the dimension of fractal rough surface
Institute of Scientific and Technical Information of China (English)
ZHAO Donghua; CAI Zhijie; RUAN Jiong
2005-01-01
In the present paper, the fractal rough surface is described by a band-limited Weierstrass-Mandelbrot function. By using the Monte Carlo method and optimal method,a minimal target function method is applied to inverting the fractal dimension of the fractal rough surface. Numerical simulations show that the method can avoid the influence of the fractal characteristic scale, and that the method is of high precision.
Some Properties of Fractals Generated by Linear Cellular Automata
Institute of Scientific and Technical Information of China (English)
倪天佳
2003-01-01
Fractals and cellular automata are both significant areas of research in nonlinear analysis. This paper studies a class of fractals generated by cellular automata. The patterns produced by cellular automata give a special sequence of sets in Euclidean space. The corresponding limit set is shown to be a fractal and the dimension is independent of the choice of the finite initial seed. As opposed to previous works, the fractals here do not depend on the time parameter.
Fractal Characteristics of Round Jets in Steady Crossflow
Institute of Scientific and Technical Information of China (English)
YuliangLI; ChaoquanCHEN; 等
1998-01-01
The fractal dimensions of turbulent round jets in steady crossflow has been analyzed by using planar laser-induced fluorescence(PLIF) method.The relation between the fractal dimension and the momentum ratio,the variation of the fractal dimension with the elevation of jet and the dilution have been investigated.The comparison of the fractal characteristics between the multiple jets and the signal jet has been carried out.
Fractal Description of the Shearing-Surface of Tools
Institute of Scientific and Technical Information of China (English)
WANG Bing-cheng; JING Chang; REN Zhao-hui; REN Li-yi
2004-01-01
In this paper, the basic methods are introduced to calculate the fractal dimensions of the shearing surface of some tools. The fractal dimension of the shearing surface of experimental sampling is obtained and the fractal characteristics are also discussed. We can apply the fractal method to identify types of tools used by burglars and to do the job of individual recognition. New theories and methods are provided to measure and process the shearing surface profile of tools.
Direct observations of sigma phase growth and dissolution in 2205 duplex stainless steel
Energy Technology Data Exchange (ETDEWEB)
Palmer, T.A.; Elmer, J.W.; Babu, S.S.; Specht, E.D. (LLNL); (ORNL)
2007-10-10
The formation and growth of sigma ({sigma}) phase in a 2205 duplex stainless steel is monitored during an 850 C isothermal heat treatment using an in situ synchrotron x-ray diffraction technique. At this temperature, {sigma} phase is first observed within approximately 40 seconds of the start of the isothermal heat treatment and grows rapidly over the course of the 3600 second heat treatment to a volume fraction of approximately 13%. A simultaneous increase in the austenite ({gamma}) volume fraction and a decrease in the ferrite ({delta}) volume fraction are observed. The {sigma} phase formed at this temperature is rapidly dissolved within approximately 200 seconds when the temperature is increased to 1000 C. Accompanying this rapid dissolution of the {sigma} phase, the {delta} and {gamma} volume fractions both approach the balanced (50/50) level observed in the as-received material.
Experimental observations of root growth in a controlled photoelastic granular material
Barés, Jonathan; Mora, Serge; Delenne, Jean-Yves; Fourcaud, Thierry
2017-06-01
We present a novel root observation apparatus capable of measuring the mechanical evolution of both the root network and the surrounding granular medium. The apparatus consists of 11 parallel growth frames, two of them being shearable, where the roots grow inside a photo-elastic or glass granular medium sandwiched between two pieces of glass. An automated system waters the plant and image each frame periodically in white light and between crossed polarisers. This makes it possible to follow (i) the root tips and (ii) the grain displacements as well as (iii) their inner pressure. We show how a root networks evolve in a granular medium and how it can mechanically stabilize it. This constitutes a model experiment to move forward in the understanding of the complex interaction between root growth and surrounding soil mechanical evolution.
Modeling Soil Water Retention Curve with a Fractal Method
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
Many empirical models have been developed to describe the soil water retention curve (SWRC). In this study, a fractal model for SWRC was derived with a specially constructed Menger sponge to describe the fractal scaling behavior of soil; relationships were established among the fractal dimension of SWRC, the fractal dimension of soil mass, and soil texture; and the model was used to estimate SWRC with the estimated results being compared to experimental data for verification. The derived fractal model was in a power-law form, similar to the Brooks-Corey and Campbell empirical functions. Experimental data of particle size distribution (PSD), texture, and soil water retention for 10 soils collected at different places in China were used to estimate the fractal dimension of SWRC and the mass fractal dimension. The fractal dimension of SWRC and the mass fractal dimension were linearly related. Also, both of the fractal dimensions were dependent on soil texture, i.e., clay and sand contents. Expressions were proposed to quantify the relationships. Based on the relationships, four methods were used to determine the fractal dimension of SWRC and the model was applied to estimate soil water content at a wide range of tension values. The estimated results compared well with the measured data having relative errors less than 10% for over 60% of the measurements. Thus, this model, estimating the fractal dimension using soil textural data, offered an alternative for predicting SWRC.
Determination of permeability using fractal method for porous media
Institute of Scientific and Technical Information of China (English)
施明恒; 陈永平
2001-01-01
A theoretical formulation was developed to express permeability as a function of different fractal dimensions and other scales for porous media . The effective fractal void ratio, the spectral dimension and the fractal dimension of particle mass distribution were introduced. The permeabilities for different soils in China are calculated. The predicted permeability for rice soil was compared with the measured data available in literature.
Fractals and the irreducibility of consciousness in plants and animals.
Gardiner, John
2013-08-01
In both plants and animals consciousness is fractal. Since fractals can only pass information in one direction it is impossible to extrapolate backward to find the rule that governs the fractal. Thus, similarly, it will be impossible to completely determine the rule or rules that govern consciousness.
ON FRACTAL MECHANISM OF COASTLINE -A Case Study of China
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
MANDELBROT enunciated the uncertainty of the length of a coastline in his paper" How long is the coastline of Britain?" published in " Science" in 1967. The fractal concept was presented for the first time in that paper and has been applied to many fields ever since. According to the fractal theory and conditions of fractal research of coastline, the controls of faults and biologic function on the fractal character of coastline are preliminarily discussed on the basis of GIS in this paper . Finally, some significant conclusions are drawn: 1) the faults control the basic trends of coastlines of two study areas;2) the fractal dimension of coastline of Taiwan is smaller than that of Changle- Lufeng, because the faults of Taiwan more intensely control the trend and fractal dimension of the coastline;3) the larger the fractal dimension of the faults or the major faults, the more the controlling effect of them on the trend and fractal dimension of coastline; 4) the larger fractal dimension of the coastline of Changle- Lufeng indicates that the biologic function intensely shapes the coastline. In a word, the controls of faults and biologic function on the fractal character of coastline are discussed with a case study of China in this paper, it can be seen that faults and biologic function both have influence over the trend and fractal dimension of coastline, the fractal mechanism of coastline of two study areas may be so.
Improved Fractal Method for Singularity Detection in Fingerprint Images
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
A new technique that uses Discrete Fractal Brownian Motion todescribe a fingerprint is presented. By computing certain fractal parameters, a fingerprints core and delta fields can be roughly detected. Experimental results demonstrate this method to be not only more efficient than the single fractal dimension method, but also more noise-resistant than the traditional schemes.
New Approach to Fractal Approximation of Vector-Functions
Directory of Open Access Journals (Sweden)
Konstantin Igudesman
2015-01-01
Full Text Available This paper introduces new approach to approximation of continuous vector-functions and vector sequences by fractal interpolation vector-functions which are multidimensional generalization of fractal interpolation functions. Best values of fractal interpolation vector-functions parameters are found. We give schemes of approximation of some sets of data and consider examples of approximation of smooth curves with different conditions.
Fractal Image Filters for Specialized Image Recognition Tasks
2010-02-11
The Fractal Geometry of Nature, [24], Mandelbrot argues that random frac- tals provide geometrical models for naturally occurring shapes and forms...Fractal Properties of Number Systems, Period. Math. Hungar 42 (2001) 51-68. [24] Benoit Mandelbrot , The Fractal Geometry of Nature, W. H. Freeman, San
Directory of Open Access Journals (Sweden)
Kendra A Batchelder
Full Text Available The 2D Wavelet-Transform Modulus Maxima (WTMM method was used to detect microcalcifications (MC in human breast tissue seen in mammograms and to characterize the fractal geometry of benign and malignant MC clusters. This was done in the context of a preliminary analysis of a small dataset, via a novel way to partition the wavelet-transform space-scale skeleton. For the first time, the estimated 3D fractal structure of a breast lesion was inferred by pairing the information from two separate 2D projected mammographic views of the same breast, i.e. the cranial-caudal (CC and mediolateral-oblique (MLO views. As a novelty, we define the "CC-MLO fractal dimension plot", where a "fractal zone" and "Euclidean zones" (non-fractal are defined. 118 images (59 cases, 25 malignant and 34 benign obtained from a digital databank of mammograms with known radiologist diagnostics were analyzed to determine which cases would be plotted in the fractal zone and which cases would fall in the Euclidean zones. 92% of malignant breast lesions studied (23 out of 25 cases were in the fractal zone while 88% of the benign lesions were in the Euclidean zones (30 out of 34 cases. Furthermore, a Bayesian statistical analysis shows that, with 95% credibility, the probability that fractal breast lesions are malignant is between 74% and 98%. Alternatively, with 95% credibility, the probability that Euclidean breast lesions are benign is between 76% and 96%. These results support the notion that the fractal structure of malignant tumors is more likely to be associated with an invasive behavior into the surrounding tissue compared to the less invasive, Euclidean structure of benign tumors. Finally, based on indirect 3D reconstructions from the 2D views, we conjecture that all breast tumors considered in this study, benign and malignant, fractal or Euclidean, restrict their growth to 2-dimensional manifolds within the breast tissue.
Batchelder, Kendra A; Tanenbaum, Aaron B; Albert, Seth; Guimond, Lyne; Kestener, Pierre; Arneodo, Alain; Khalil, Andre
2014-01-01
The 2D Wavelet-Transform Modulus Maxima (WTMM) method was used to detect microcalcifications (MC) in human breast tissue seen in mammograms and to characterize the fractal geometry of benign and malignant MC clusters. This was done in the context of a preliminary analysis of a small dataset, via a novel way to partition the wavelet-transform space-scale skeleton. For the first time, the estimated 3D fractal structure of a breast lesion was inferred by pairing the information from two separate 2D projected mammographic views of the same breast, i.e. the cranial-caudal (CC) and mediolateral-oblique (MLO) views. As a novelty, we define the "CC-MLO fractal dimension plot", where a "fractal zone" and "Euclidean zones" (non-fractal) are defined. 118 images (59 cases, 25 malignant and 34 benign) obtained from a digital databank of mammograms with known radiologist diagnostics were analyzed to determine which cases would be plotted in the fractal zone and which cases would fall in the Euclidean zones. 92% of malignant breast lesions studied (23 out of 25 cases) were in the fractal zone while 88% of the benign lesions were in the Euclidean zones (30 out of 34 cases). Furthermore, a Bayesian statistical analysis shows that, with 95% credibility, the probability that fractal breast lesions are malignant is between 74% and 98%. Alternatively, with 95% credibility, the probability that Euclidean breast lesions are benign is between 76% and 96%. These results support the notion that the fractal structure of malignant tumors is more likely to be associated with an invasive behavior into the surrounding tissue compared to the less invasive, Euclidean structure of benign tumors. Finally, based on indirect 3D reconstructions from the 2D views, we conjecture that all breast tumors considered in this study, benign and malignant, fractal or Euclidean, restrict their growth to 2-dimensional manifolds within the breast tissue.
Sawant, S. A.; Chakraborty, M.; Suradhaniwar, S.; Adinarayana, J.; Durbha, S. S.
2016-06-01
Satellite based earth observation (EO) platforms have proved capability to spatio-temporally monitor changes on the earth's surface. Long term satellite missions have provided huge repository of optical remote sensing datasets, and United States Geological Survey (USGS) Landsat program is one of the oldest sources of optical EO datasets. This historical and near real time EO archive is a rich source of information to understand the seasonal changes in the horticultural crops. Citrus (Mandarin / Nagpur Orange) is one of the major horticultural crops cultivated in central India. Erratic behaviour of rainfall and dependency on groundwater for irrigation has wide impact on the citrus crop yield. Also, wide variations are reported in temperature and relative humidity causing early fruit onset and increase in crop water requirement. Therefore, there is need to study the crop growth stages and crop evapotranspiration at spatio-temporal scale for managing the scarce resources. In this study, an attempt has been made to understand the citrus crop growth stages using Normalized Difference Time Series (NDVI) time series data obtained from Landsat archives (http://earthexplorer.usgs.gov/). Total 388 Landsat 4, 5, 7 and 8 scenes (from year 1990 to Aug. 2015) for Worldwide Reference System (WRS) 2, path 145 and row 45 were selected to understand seasonal variations in citrus crop growth. Considering Landsat 30 meter spatial resolution to obtain homogeneous pixels with crop cover orchards larger than 2 hectare area was selected. To consider change in wavelength bandwidth (radiometric resolution) with Landsat sensors (i.e. 4, 5, 7 and 8) NDVI has been selected to obtain continuous sensor independent time series. The obtained crop growth stage information has been used to estimate citrus basal crop coefficient information (Kcb). Satellite based Kcb estimates were used with proximal agrometeorological sensing system observed relevant weather parameters for crop ET estimation. The
Cassini CAPS-ELS observations of carbon-based anions and aerosol growth in Titan's ionosphere
Desai, Ravindra; Coates, Andrew; Wellbrock, Anne; Kataria, Dhiren; Jones, Geraint; Lewis, Gethyn; Waite, J.
2016-06-01
Cassini observations of Titans ionosphere revealed an atmosphere rich in positively charged ions with masses up to > 350 amu and negatively charged ions and aerosols with mass over charge ratios as high as 13,800 amu/q. The detection of negatively charged molecules by the Cassini CAPS Electron Spectrometer (CAPS-ELS) was particularly surprising and showed how the synthesis of large aerosol-size particles takes place at altitudes much greater than previously thought. Here, we present further analysis into this CAPS-ELS dataset, through an enhanced understanding of the instrument's response function. In previous studies the intrinsic E/E energy resolution of the instrument did not allow specific species to be identified and the detections were classified into broad mass ranges. In this study we use an updated fitting procedure to show how the ELS mass spectrum can be resolved into specific peaks at multiples of carbon-based anions up to > 100 amu/q. The negatively charged ions and aerosols in Titans ionosphere increase in mass with decreasing altitude, the lightest species being observed close to Titan's exobase of ˜1,450km and heaviest species observed at altitudes < 950km. We identify key stages in this apparent growth process and report on key intermediaries which appear to trigger the rapid growth of the larger aerosol-size particles.
Fractal Systems of Central Places Based on Intermittency of Space-filling
Chen, Yanguang
2011-01-01
The central place models are fundamentally important in theoretical geography and city planning theory. The texture and structure of central place networks have been demonstrated to be self-similar in both theoretical and empirical studies. However, the underlying rationale of central place fractals in the real world has not yet been revealed so far. This paper is devoted to illustrating the mechanisms by which the fractal patterns can be generated from central place systems. The structural dimension of the traditional central place models is d=2 indicating no intermittency in the spatial distribution of human settlements. This dimension value is inconsistent with empirical observations. Substituting the complete space filling with the incomplete space filling, we can obtain central place models with fractional dimension D
Fractal features of dark, maintained, and driven neural discharges in the cat visual system
Lowen, S B; Kaplan, E; Saleh, B E A; Teich, M C; Lowen, Steven B.; Ozaki, Tsuyoshi; Kaplan, Ehud; Saleh, Bahaa E. A.; Teich, Malvin C.
1999-01-01
We employ a number of statistical measures to characterize neural discharge activity in cat retinal ganglion cells (RGCs) and in their target lateral geniculate nucleus (LGN) neurons under various stimulus conditions, and we develop a new measure to examine correlations in fractal activity between spike-train pairs. In the absence of stimulation (i.e., in the dark), RGC and LGN discharges exhibit similar properties. The presentation of a constant, uniform luminance to the eye reduces the fractal fluctuations in the RGC maintained discharge but enhances them in the target LGN discharge, so that neural activity in the pair no longer mirror each other. A drifting-grating stimulus yields RGC and LGN driven spike trains similar in character to those observed in the maintained discharge, with two notable distinctions: action potentials are reorganized along the time axis so that they occur only during certain phases of the stimulus waveform, and fractal activity is suppressed. Under both uniform-luminance and drift...
Fractal design concepts for stretchable electronics.
Fan, Jonathan A; Yeo, Woon-Hong; Su, Yewang; Hattori, Yoshiaki; Lee, Woosik; Jung, Sung-Young; Zhang, Yihui; Liu, Zhuangjian; Cheng, Huanyu; Falgout, Leo; Bajema, Mike; Coleman, Todd; Gregoire, Dan; Larsen, Ryan J; Huang, Yonggang; Rogers, John A
2014-01-01
Stretchable electronics provide a foundation for applications that exceed the scope of conventional wafer and circuit board technologies due to their unique capacity to integrate with soft materials and curvilinear surfaces. The range of possibilities is predicated on the development of device architectures that simultaneously offer advanced electronic function and compliant mechanics. Here we report that thin films of hard electronic materials patterned in deterministic fractal motifs and bonded to elastomers enable unusual mechanics with important implications in stretchable device design. In particular, we demonstrate the utility of Peano, Greek cross, Vicsek and other fractal constructs to yield space-filling structures of electronic materials, including monocrystalline silicon, for electrophysiological sensors, precision monitors and actuators, and radio frequency antennas. These devices support conformal mounting on the skin and have unique properties such as invisibility under magnetic resonance imaging. The results suggest that fractal-based layouts represent important strategies for hard-soft materials integration.
Higuchi fractal properties of onset epilepsy electroencephalogram.
Khoa, Truong Quang Dang; Ha, Vo Quang; Toi, Vo Van
2012-01-01
Epilepsy is a medical term which indicates a common neurological disorder characterized by seizures, because of abnormal neuronal activity. This leads to unconsciousness or even a convulsion. The possible etiologies should be evaluated and treated. Therefore, it is necessary to concentrate not only on finding out efficient treatment methods, but also on developing algorithm to support diagnosis. Currently, there are a number of algorithms, especially nonlinear algorithms. However, those algorithms have some difficulties one of which is the impact of noise on the results. In this paper, in addition to the use of fractal dimension as a principal tool to diagnose epilepsy, the combination between ICA algorithm and averaging filter at the preprocessing step leads to some positive results. The combination which improved the fractal algorithm become robust with noise on EEG signals. As a result, we can see clearly fractal properties in preictal and ictal period so as to epileptic diagnosis.
Higuchi Fractal Properties of Onset Epilepsy Electroencephalogram
Directory of Open Access Journals (Sweden)
Truong Quang Dang Khoa
2012-01-01
Full Text Available Epilepsy is a medical term which indicates a common neurological disorder characterized by seizures, because of abnormal neuronal activity. This leads to unconsciousness or even a convulsion. The possible etiologies should be evaluated and treated. Therefore, it is necessary to concentrate not only on finding out efficient treatment methods, but also on developing algorithm to support diagnosis. Currently, there are a number of algorithms, especially nonlinear algorithms. However, those algorithms have some difficulties one of which is the impact of noise on the results. In this paper, in addition to the use of fractal dimension as a principal tool to diagnose epilepsy, the combination between ICA algorithm and averaging filter at the preprocessing step leads to some positive results. The combination which improved the fractal algorithm become robust with noise on EEG signals. As a result, we can see clearly fractal properties in preictal and ictal period so as to epileptic diagnosis.
Fractals in Spatial Patterns of Vegetation Formations
Institute of Scientific and Technical Information of China (English)
SONG Zhiyuan; HUANG Daming; Masae Shiyomi; WANG Yusheng; Shigeo Takahashi; Hori Yoshimichi; Yasuo Yamamuru; CHEN Jun
2006-01-01
The spatial distribution patterns of species are always scale-dependent and spatially self-similar in ecological systems. In this work, vegetation distribution data collected from the vegetation map of the Xigazê region was analyzed using a box-counting method. The power law of the box-counting dimension (DB) across a range of scales (5-160 km) confirms the fractal patterns for most vegetation formations, while the fluctuations of the scale-specific DB among the different abundance groups indicate limitations of fractal coherence. The fractal method is shown to be a useful tool for measuring the distribution patterns of vegetation formations across scales, which provides important information for both species and habitat conservation, especially in landscape management.
Computer Security: The dilemma of fractal defence
Stefan Lueders, Computer Security Team
2015-01-01
Aren’t mathematical fractals just beautiful? The Mandelbrot set and the Julia set, the Sierpinski gasket, the Menger sponge, the Koch curve (see here)… Based on very simple mathematical rules, they quickly develop into a mosaic of facets slightly different from each other. More and more features appear the closer you zoom into a fractal and expose similar but not identical features of the overall picture. Computer security is like these fractals, only much less pretty: simple at first glance, but increasingly complex and complicated when you look more closely at the details. The deeper you dig, the more and more possibilities open up for malicious people as the attack surface grows, just like that of “Koch’s snowflakes”, where the border length grows exponentially. Consequently, the defensive perimeter also increases when we follow the bits and bytes layer by layer from their processing in the CPU, trickling up the software stack thro...
Exotic topological order from quantum fractal code
Yoshida, Beni
2014-03-01
We present a large class of three-dimensional spin models that possess topological order with stability against local perturbations, but are beyond description of topological quantum field theory. Conventional topological spin liquids, on a formal level, may be viewed as condensation of string-like extended objects with discrete gauge symmetries, being at fixed points with continuous scale symmetries. In contrast, ground states of fractal spin liquids are condensation of highly-fluctuating fractal objects with certain algebraic symmetries, corresponding to limit cycles under real-space renormalization group transformations which naturally arise from discrete scale symmetries of underlying fractal geometries. A particular class of three-dimensional models proposed in this paper may potentially saturate quantum information storage capacity for local spin systems.
Exotic topological order in fractal spin liquids
Yoshida, Beni
2013-09-01
We present a large class of three-dimensional spin models that possess topological order with stability against local perturbations, but are beyond description of topological quantum field theory. Conventional topological spin liquids, on a formal level, may be viewed as condensation of stringlike extended objects with discrete gauge symmetries, being at fixed points with continuous scale symmetries. In contrast, ground states of fractal spin liquids are condensation of highly fluctuating fractal objects with certain algebraic symmetries, corresponding to limit cycles under real-space renormalization group transformations which naturally arise from discrete scale symmetries of underlying fractal geometries. A particular class of three-dimensional models proposed in this paper may potentially saturate quantum information storage capacity for local spin systems.
Fractal design concepts for stretchable electronics
Fan, Jonathan A.; Yeo, Woon-Hong; Su, Yewang; Hattori, Yoshiaki; Lee, Woosik; Jung, Sung-Young; Zhang, Yihui; Liu, Zhuangjian; Cheng, Huanyu; Falgout, Leo; Bajema, Mike; Coleman, Todd; Gregoire, Dan; Larsen, Ryan J.; Huang, Yonggang; Rogers, John A.
2014-02-01
Stretchable electronics provide a foundation for applications that exceed the scope of conventional wafer and circuit board technologies due to their unique capacity to integrate with soft materials and curvilinear surfaces. The range of possibilities is predicated on the development of device architectures that simultaneously offer advanced electronic function and compliant mechanics. Here we report that thin films of hard electronic materials patterned in deterministic fractal motifs and bonded to elastomers enable unusual mechanics with important implications in stretchable device design. In particular, we demonstrate the utility of Peano, Greek cross, Vicsek and other fractal constructs to yield space-filling structures of electronic materials, including monocrystalline silicon, for electrophysiological sensors, precision monitors and actuators, and radio frequency antennas. These devices support conformal mounting on the skin and have unique properties such as invisibility under magnetic resonance imaging. The results suggest that fractal-based layouts represent important strategies for hard-soft materials integration.
A fractal-based image encryption system
Abd-El-Hafiz, S. K.
2014-12-01
This study introduces a novel image encryption system based on diffusion and confusion processes in which the image information is hidden inside the complex details of fractal images. A simplified encryption technique is, first, presented using a single-fractal image and statistical analysis is performed. A general encryption system utilising multiple fractal images is, then, introduced to improve the performance and increase the encryption key up to hundreds of bits. This improvement is achieved through several parameters: feedback delay, multiplexing and independent horizontal or vertical shifts. The effect of each parameter is studied separately and, then, they are combined to illustrate their influence on the encryption quality. The encryption quality is evaluated using different analysis techniques such as correlation coefficients, differential attack measures, histogram distributions, key sensitivity analysis and the National Institute of Standards and Technology (NIST) statistical test suite. The obtained results show great potential compared to other techniques.
Vibrations of strongly irregular or fractal resonators
Sapoval, B.; Gobron, Th.
1993-05-01
It is shown on a specific example that fractal boundary conditions drastically alter the properties of wave excitations in space. The low-frequency part of the vibration spectrum of a finite-range fractal drum is computed using an analogy between the Helmoltz equation and the diffusion equation. The irregularity of the frontier is found to influence strongly the density of states at low frequency. The fractal perimeter generates a specific screening effect. Very near the frontier, the decrease of the wave form is related directly to the behavior of the harmonic measure. The possibility of localization of the vibrations is qualitatively discussed and we show that localized modes may exist at low frequencies if the geometrical structures possess narrow paths. Possible application of these results to the interpretation of thermal properties of binary glasses is briefly discussed.
Fractal dynamics in chaotic quantum transport.
Kotimäki, V; Räsänen, E; Hennig, H; Heller, E J
2013-08-01
Despite several experiments on chaotic quantum transport in two-dimensional systems such as semiconductor quantum dots, corresponding quantum simulations within a real-space model have been out of reach so far. Here we carry out quantum transport calculations in real space and real time for a two-dimensional stadium cavity that shows chaotic dynamics. By applying a large set of magnetic fields we obtain a complete picture of magnetoconductance that indicates fractal scaling. In the calculations of the fractality we use detrended fluctuation analysis-a widely used method in time-series analysis-and show its usefulness in the interpretation of the conductance curves. Comparison with a standard method to extract the fractal dimension leads to consistent results that in turn qualitatively agree with the previous experimental data.
Fractal Analysis on Human Behaviors Dynamics
Fan, Chao; Zha, Yi-Long
2010-01-01
The study of human dynamics has attracted much interest from many fields recently. In this paper, the fractal characteristic of human behaviors is investigated from the perspective of time series constructed with the amount of library loans. The Hurst exponents and length of non-periodic cycles calculated through Rescaled Range Analysis indicate that the time series of human behaviors is fractal with long-range correlation. Then the time series are converted to complex networks by visibility graph algorithm. The topological properties of the networks, such as scale-free property, small-world effect and hierarchical structure imply that close relationships exist between the amounts of repetitious actions performed by people during certain periods of time, especially for some important days. Finally, the networks obtained are verified to be not fractal and self-similar using box-counting method. Our work implies the intrinsic regularity shown in human collective repetitious behaviors.
Fractals on IPv6 Network Topology
Directory of Open Access Journals (Sweden)
Bo Yang
2013-02-01
Full Text Available The coarse-grained renormalization and the fractal analysis of the Internet macroscopic topology can help people better understand the relationship between the part and whole of the Internet, and it is significant for people to understand the essence of the research object through a small amount of information. Aiming at the complexity of Internet IPv6 IP-level topology, we put forward a method of core-threshold coarse-grained to renormalize its topology. By analyzing the degree distribution and degree correlation characteristics in each k-core network topology, the scale invariance of the networks of coarse-grained renormalization was illustrated. The fractal dimension of Internet IPv6 IP-level topology was further computed which shows that the Internet IPv6 IP-level topology has got fractals.
Fractal Beauty in Xinjiang Folk Art Patterns
Institute of Scientific and Technical Information of China (English)
PENG Hong; ZHAO Hai-ying
2014-01-01
Xinjiang folk art patterns and designs are the art treasures of Chinese cultural treasure-house as well as the precious humanistic resources of Western China. In the process of collecting, sorting out and studying Xinjiang folk art patterns, the elegant simplicity as well as the good taste stands out impressively, and the pattern shape as well as the layout composition shows a distinctive national trait and a strong local color. As “The Geometry of Nature”, fractal geometry brings about a new performing method. Various fractal graphs are created by different generators. Their dynamic pictures contain visual information of great magnitude and their artistic effect is similar to Xinjiang folk art patterns, which fully proves the fractal beauty in Xinjiang folk art patterns.
A Fractal and Scale-free Model of Complex Networks with Hub Attraction Behaviors
Kuang, Li; Li, Deyi; Li, Yuanxiang; Sun, Yu
2013-01-01
It is widely believed that fractality of complex networks origins from hub repulsion behaviors (anticorrelation or disassortativity), which means large degree nodes tend to connect with small degree nodes. This hypothesis was demonstrated by a dynamical growth model, which evolves as the inverse renormalization procedure proposed by Song et al. Now we find that the dynamical growth model is based on the assumption that all the cross-boxes links has the same probability e to link to the most connected nodes inside each box. Therefore, we modify the growth model by adopting the flexible probability e, which makes hubs have higher probability to connect with hubs than non-hubs. With this model, we find some fractal and scale-free networks have hub attraction behaviors (correlation or assortativity). The results are the counter-examples of former beliefs.
Probing fractal magnetic domains on multiple length scales in Nd2Fe14B.
Kreyssig, A; Prozorov, R; Dewhurst, C D; Canfield, P C; McCallum, R W; Goldman, A I
2009-01-30
Using small-angle neutron scattering, we demonstrate that the complex magnetic domain patterns at the surface of Nd2Fe14B, revealed by quantitative Kerr and Faraday microscopy, propagate into the bulk and exhibit structural features with dimensions down to 6 nm, the domain wall thickness. The observed fractal nature of the domain structures provides an explanation for the anomalous increase in the bulk magnetization of Nd2Fe14B below the spin-reorientation transition. These measurements open up a rich playground for studies of fractal structures in highly anisotropic magnetic systems.
Radial distribution function for hard spheres in fractal dimensions. A heuristic approximation
Santos, Andrés
2016-01-01
Analytic approximations for the radial distribution function, the structure factor, and the equation of state of hard-core fluids in fractal dimension $d$ ($1 \\leq d \\leq 3$) are developed as heuristic interpolations from the knowledge of the exact and Percus-Yevick results for the hard-rod and hard-sphere fluids, respectively. In order to assess their value, such approximate results are compared with those of recent Monte Carlo simulations and numerical solutions of the Percus-Yevick equation for fractal dimension [M. Heinen et al., Phys. Rev. Lett. \\textbf{115}, 097801 (2015)], a good agreement being observed.
Is fractal 1/f scaling in stream chemistry universal?
Hrachowitz, Markus
2016-04-01
Stream water chemistry data from catchments worldwide suggest that catchments act as filters that transform white noise, i.e. random, input signals such as in precipitation, into 1/f^α noise whose slope in a power spectrum typically ranges between -0.5>α>-1.5. This previously lead to the hypothesis that catchments act as fractal filters. In other words, it was posed that considering uncertainty, a slope of α=-1 may be a universal and intrinsic property of catchments. Such fractal scaling characteristics would have considerable implications on the predictability of stream water chemistry, as both, temporal short- and long-range interdependence and memory control the system response. While short memories and thus flatter slopes with α closer to 0 indicate poor short term but good long-term predictability, steeper slopes with values of α short and long-term response patterns. The hypothesis of catchments acting as fractal filters (α=-1), however, remains to be tested more profoundly. It is, for example, not yet clear, if the observed inter-catchment variations in α indeed need to be interpreted as uncertainty and noise in the signal or if the variations underlie a systematic pattern and can be explained by some characteristic of catchment function, as was recently suggested in a modelling study based two experimental catchments (Hrachowitz et al., 2015). Here we will therefore further test the hypothesis that the spectral slope of stream water chemistry is not necessarily α=-1 and that catchments therefore do not inherently act as fractal filters. Further, it will be tested if closer links between the variations in spectral slope and hydrological function of catchments can be identified. The combined data-analysis and modelling study uses hydrochemical data (i.e. Cl- and O-18) from a wide range of catchments worldwide to allow a robust inter-comparison of response characteristics. The high number of study catchments is chosen to represent physically
Communities and classes in symmetric fractals
Krawczyk, M J
2014-01-01
Two aspects of fractal networks are considered: the community structure and the class structure, where classes of nodes appear as a consequence of a local symmetry of nodes. The analysed systems are the networks constructed for two selected symmetric fractals: the Sierpinski triangle and the Koch curve. Communities are searched for by means of a set of differential equations. Overlapping nodes which belong to two different communities are identified by adding some noise to the initial connectivity matrix. Then, a node can be characterized by a spectrum of probabilities of belonging to different communities. Our main goal is that the overlapping nodes with the same spectra belong to the same class.
Fractal characterization of neural correlates of consciousness
Ibañez-Molina, A. J.; Iglesias-Parro, S.
2013-01-01
In this work we present a novel experimental paradigm, based on binocular rivalry, to address the study of internally and externally generated conscious percepts. Assuming the nonlinear nature of the EEG signals, we propose the use of fractal dimension to characterize the complexity of the EEG associated with each percept. Data analysis showed significant differences in complexity between the internally and externally generated percepts. Moreover, EEG complexity of auditory and visual percepts was unequal. These results support fractal dimension analyses as a new tool to characterize conscious perception.
The virtual education fractality: nature and organization
Directory of Open Access Journals (Sweden)
Osbaldo Turpo Gebera
2013-04-01
Full Text Available The potential generated by ICT in education raises reflect on the underlying frameworks. In this sense, the fractal is an opportunity to explain how it organizes and manages virtual education.This approach recognizes that educational dynamics are recursive and iterative processes instituted as progressive sequences, by way of fractals. This understanding enables becoming as mediated and articulated successive levels. In each dimension are embodied own activities and in turn, involves the recurrence of subsequent levels as possible solving of problem situations. Thus, the knowledge built in response to a collaborative action, participation in networks, ranging from autonomous to the cultural level or conversely.
Preparation of Nickel Materials with Fractal Structure
Institute of Scientific and Technical Information of China (English)
无
2003-01-01
A way of manufacturing nickel material with fractal structure has been studied. Some algae with natural fractalstructure were used as the basic substrates. The nickel was coated on the substrates by both electroless depositionand electrodeposition. After elimination of the foundational algae by erosion, dissolution etc, the pure nickel materialswith fractal structure were obtained. At last, the specific surface area was analyzed by BET analyses and the fractaldimension of the nickel material was calculated by means of box-counting technique. The comparison of fractaldimension between Ni structure and natural algae was also given.
Fractal Symmetries: Ungauging the Cubic Code
Williamson, Dominic J
2016-01-01
Gauging is a ubiquitous tool in many-body physics. It allows one to construct highly entangled topological phases of matter from relatively simple phases and to relate certain characteristics of the two. Here we develop a gauging procedure for general submanifold symmetries of Pauli Hamiltonians, including symmetries of fractal type. We show a relation between the pre- and post- gauging models and use this to construct short range entangled phases with fractal like symmetries, one of which is mapped to the cubic code by the gauging.
The Fractal Simulation Of Biological Shapes
Pickover, Clifford A.
1989-04-01
This paper provides a light introduction to simple graphics techniques for visualizing a large class of biological shapes generated from recursive algorithms. In order to capture some of the structural richness inherent in organisms, the algorithms produce not only extreme variability but also a high level of organization. The material primarily comes from previous published works of the author. For a general background on fractal methods in mathematics and science, see Mandelbrot's famous book. For research on the fractal characterization of other biological structures, such as the lung's bronchial tree and the surfaces of protein molecules.
Analysis of Fractal Parameters of the Lunar Surface
Nefedyev, Yuri; Petrova, Natalia; Andreev, Alexey; Demina, Natalya; Demin, Sergey
2016-07-01
Analysis of complex selenographic systems is a complicatedissue. This fully applies to the lunar topography. In this report a new method of the comparative reliable estimation of thelunar mapsdata is represented. The estimation was made by the comparison of high-altitude lines using the fractal analysis. The influence of the lunar macrofigure variances were determined by the method of fractal dimensions comparison. It should be noted the investigations of the lunar figure and rotation implystudy itsmarginal zone charts constructionwith various methods and this is traditionally carried out at the Engelhardt Astronomical Observatory (EAO). In particular this research is important for lunar occultations reductions and on the basis of that it is possible to solve a number of astrometric and astrophysical problems. By now the highly accurate theories of the lunar movement have been obtained and stars coordinates have been determined on the basis of space measurements with the several multiarcseconds accuracy but there are factors highly influencingon the accuracy of the results of these observations. They are: exactitude of the occultation moment recording, errors of the stars coordinates, accuracy of lunar ephemeris positions and unreliability of lunar marginal zone charts. Therefore difficulties arise during the reduction process of lunar occultations by the reason of irregularities of lunar limb. Existing charts of the lunar marginal zone have some defects. The researching of lunar marginal zone maps is very difficult. First of all, it concernsthe reliability of maps data. To resolve this task thecomparison method in which the structure of the high-altitude lines of data appropriated with identical lunar coordinates can used. However, such comparison requires a lot of calculations. In addition there is a large number of the marginal zone maps constructed by different methods and the accuracy of their data causes many questions. In other words, the lunar relief has a
Observational study of aerosol hygroscopic growth factors over rural area near Beijing mega-city
Directory of Open Access Journals (Sweden)
X. L. Pan
2009-02-01
Full Text Available We investigated aerosol hygroscopic growth property and its influence on scattering coefficient using M9003 nephelometers in coupling with a relative humidity controlled inlet system at a rural site near Beijing mega-city (Jingjintang from 24th April to15th May 2006. Inlet relative humidity was controlled in an increasing range of 40%–90% while the aerosol hygroscopic growth factor, f(RH=80%, varied in a range of 1.07–2.35 during the measurement. Estimated periodic mean values of aerosol hygroscopic growth factors are 1.27–1.34, 1.17–1.23, 1.55–1.59 and 2.33–2.48 for clean, dust, urban pollution and mixed pollution periods respectively. An examination of chemical composition of daily filter samples highlighted that aerosol hygroscopicity was generally enhanced with the increasing ratio of ammonium sulfate (AS to organic matter (OMC. Furthermore, strong hygroscopic organic aerosols were observed on 11th (f(RH=80%=2.23 and 15th (f(RH=80%=2.21 of May with organic carbon proportions of PM_{2.1} reaching 42.3% and 43.0% respectively. Back-trajectory analysis indicated that solar radiation and vertical convective movement along the air mass pathway might strongly influence the hygroscopic properties of organic matter.
In-Situ Observation of Surface Phenomena During Sr(NO3)2 Crystal Growth
Institute of Scientific and Technical Information of China (English)
陈万春; 李宝霞; 李超荣
2003-01-01
The reflected differential interference phase contrast microscope is used to study a growing crystal surface. The surface phenomena on the {111} and {100} faces of Sr(NO3)2, such as the propagation of steps, the bunches of surface steps, the impurity stopper and the growth hillocks, have been observed during the crystal growth. It was found that: (1) The macrosteps velocity is from 0.86μm/s to 9.8 × 10-2 μm/s on the {111} face at σ = 5.33 × 10-3 to 2.13 × 10-3. (2) If the propagating directions of the steps are in opposition, the velocity of the macrosteps will be increased after they bunched. These phenomena first provide the evidence for the existence of the mutual acceleration effect of macroscopic steps. (3) The growth hillocks include a concentric step which evidently results from successive acts of a two-dimension nucleation on surface.
Quantitative observations of hydrogen-induced, slow crack growth in a low alloy steel
Nelson, H. G.; Williams, D. P.
1973-01-01
Hydrogen-induced slow crack growth, da/dt, was studied in AISI-SAE 4130 low alloy steel in gaseous hydrogen and distilled water environments as a function of applied stress intensity, K, at various temperatures, hydrogen pressures, and alloy strength levels. At low values of K, da/dt was found to exhibit a strong exponential K dependence (Stage 1 growth) in both hydrogen and water. At intermediate values of K, da/dt exhibited a small but finite K dependence (Stage 2), with the Stage 2 slope being greater in hydrogen than in water. In hydrogen, at a constant K, (da/dt) sub 2 varied inversely with alloy strength level and varied essentially in the same complex manner with temperature and hydrogen pressure as noted previously. The results of this study provide support for most of the qualitative predictions of the lattice decohesion theory as recently modified by Oriani. The lack of quantitative agreement between data and theory and the inability of theory to explain the observed pressure dependence of slow crack growth are mentioned and possible rationalizations to account for these differences are presented.
Fractal Self-Organization of Bacteria-Inspired Agents
Huang, Yufeng; Krumanocker, Ian; Coppens, Marc-Olivier
2012-06-01
We develop an agent-based model as a preliminary theoretical basis to guide the synthesis of a new class of materials with dynamic properties similar to bacterial colonies. Each agent in the model is representative of an individual bacterium capable of: the uptake of chemicals (nutrients), which are metabolized; active movement (part viscous, part diffusive), consuming metabolic energy; and cellular division, when agents have doubled in size. The agents grow in number and self-organize into fractal structures, depending on the rules that define the actions of the agents and the parameter values. The environment of the agents includes chemicals responsible for their growth and is described by a diffusion-reaction equation with Michaelis-Menten kinetics. These rules are modeled mathematically by a set of equations with five dimensionless groups that are functions of physical parameters. Simulations are performed for different parameter values. The resulting structures are characterized by their fractal scaling regime, box-counting and mass-radius dimensions, and lacunarity. Each parameter influences the overall structure in a unique way, generating a wide spectrum of structures. For certain combinations of parameter values, the model converges to a steady state, with a finite population of agents that no longer divide.
Modelling Applicability of Fractal Analysis to Efficiency of Soil Exploration by Roots
Walk, Thomas C.; van Erp, Erik; Lynch, Jonathan P.
2004-01-01
• Background and Aims Fractal analysis allows calculation of fractal dimension, fractal abundance and lacunarity. Fractal analysis of plant roots has revealed correlations of fractal dimension with age, topology or genotypic variation, while fractal abundance has been associated with root length. Lacunarity is associated with heterogeneity of distribution, and has yet to be utilized in analysis of roots. In this study, fractal analysis was applied to the study of root architecture and acquisi...
Fractal dimensions the digital art of Eric Hammel
Hammel, Eric
2014-01-01
The concept behind fractal geometry is extremely difficult to explain . . . but easy to see and enjoy. Eric Hammel, a professional author of military history books, is unable to explain fractals in a way that will be clear to anyone else, but most mathematicians can't explain fractals in language most people can understand. The simplest explanation is that fractals are graphic representations of high-order mathematical formulas that repeat patterns to infinity.Don't get hung up on the math. It's really all in the seeing. Like Volume 1 of Eric Hammel's Fractal Dimensions, Volume 2 is filled wit